text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
{"url":"https:\/\/proofwiki.org\/wiki\/Inverse_of_Hilbert_Matrix","text":"# Inverse of Hilbert Matrix\n\n## Theorem\n\nLet $H_n$ be the Hilbert matrix of order $n$:\n\n$\\begin{bmatrix} a_{i j} \\end{bmatrix} = \\begin{bmatrix} \\dfrac 1 {i + j - 1} \\end{bmatrix}$\n\nThen its inverse $H_n^{-1} = \\sqbrk b_n$ can be specified as:\n\n$\\begin{bmatrix} b_{i j} \\end{bmatrix} = \\begin{bmatrix} \\dfrac {\\paren {-1}^{i + j} \\paren {i + n - 1}! \\paren {j + n - 1}!} {\\paren {\\paren {i - 1}!}^2 \\paren {\\paren {j - 1}!}^2 \\paren {n - j}! \\paren {n - i}! \\paren {i + j - 1} } \\end{bmatrix}$\n\n## Proof\n\nFrom Hilbert Matrix is Cauchy Matrix, $H_n$ is a special case of a Cauchy matrix:\n\n$\\begin{bmatrix} c_{i j} \\end{bmatrix} = \\begin{bmatrix} \\dfrac 1 {x_i + y_j} \\end{bmatrix}$\n\nwhere:\n\n$x_i = i$\n$y_j = j - 1$\n\nFrom Inverse of Cauchy Matrix, the inverse of the square Cauchy matrix of order $n$ is:\n\n$\\begin{bmatrix} b_{i j} \\end{bmatrix} = \\begin{bmatrix} \\dfrac {\\ds \\prod_{k \\mathop = 1}^n \\paren {x_j + y_k} \\paren {x_k + y_i} } {\\ds \\paren {x_j + y_i} \\paren {\\prod_{\\substack {1 \\mathop \\le k \\mathop \\le n \\\\ k \\mathop \\ne j} } \\paren {x_j - x_k} } \\paren {\\prod_{\\substack {1 \\mathop \\le k \\mathop \\le n \\\\ k \\mathop \\ne i} } \\paren {y_i - x_k} } } \\end{bmatrix}$\n\nThus $H_n^{-1}$ can be specified as:\n\n$\\begin{bmatrix} b_{i j} \\end{bmatrix} = \\begin{bmatrix} \\dfrac {\\ds \\prod_{k \\mathop = 1}^n \\paren {i + k - 1} \\paren {j + k - 1} } {\\ds \\paren {i + j - 1} \\paren {\\prod_{\\substack {1 \\mathop \\le k \\mathop \\le n \\\\ k \\mathop \\ne i} } \\paren {i - k} } \\paren {\\prod_{\\substack {1 \\mathop \\le k \\mathop \\le n \\\\ k \\mathop \\ne j} } \\paren {j - k} } } \\end{bmatrix}$\n\nFirst, from Product of Products:\n\n$\\ds \\prod_{k \\mathop = 1}^n \\paren {i + k - 1} \\paren {j + k - 1} = \\prod_{k \\mathop = 1}^n \\paren {i + k - 1} \\prod_{k \\mathop = 1}^n \\paren {j + k - 1}$\n\nWe address in turn the various factors of this expression for $b_{i j}$.\n\n $\\ds \\prod_{k \\mathop = 1}^n \\paren {i + k - 1}$ $=$ $\\ds \\prod_{k \\mathop = 0}^{n - 1} \\paren {i + k}$ Translation of Index Variable of Product $\\ds$ $=$ $\\ds i^{\\overline n}$ Definition of Rising Factorial $\\ds$ $=$ $\\ds \\frac {\\paren {i + n - 1}!} {\\paren {i - 1}!}$ Rising Factorial as Quotient of Factorials\n\nand similarly:\n\n$\\ds \\prod_{k \\mathop = 1}^n \\paren {j + k - 1} = \\frac {\\paren {j + n - 1}!} {\\paren {j - 1}!}$\n\nThen:\n\n $\\ds \\prod_{\\substack {1 \\mathop \\le k \\mathop \\le n \\\\ k \\mathop \\ne i} } \\paren {i - k}$ $=$ $\\ds \\paren {\\prod_{1 \\mathop \\le k \\mathop < i} \\paren {i - k} } \\paren {\\prod_{i \\mathop < k \\mathop \\le n} \\paren {i - k} }$ $\\ds$ $=$ $\\ds \\paren {i - 1}! \\paren {\\prod_{i \\mathop < k \\mathop \\le n} \\paren {i - k} }$ Definition of Factorial $\\ds$ $=$ $\\ds \\paren {i - 1}! \\paren {\\prod_{0 \\mathop < k \\mathop \\le n - i} \\paren {-k} }$ Translation of Index Variable of Product $\\ds$ $=$ $\\ds \\paren {i - 1}! \\paren {-1}^{n - i} \\paren {\\prod_{0 \\mathop < k \\mathop \\le n - i} k}$ $\\ds$ $=$ $\\ds \\paren {i - 1}! \\paren {-1}^{n - i} \\paren {n - i}!$ Definition of Factorial\n\nand similarly:\n\n$\\ds \\prod_{\\substack {1 \\mathop \\le k \\mathop \\le n \\\\ k \\mathop \\ne j} } \\paren {j - k} = \\paren {j - 1}! \\paren {-1}^{n - j} \\paren {n - j}!$\n\nThus we can write:\n\n $\\ds \\begin{bmatrix} b_{i j} \\end{bmatrix}$ $=$ $\\ds \\frac {\\paren {\\dfrac {\\paren {i + n - 1}!} {\\paren {i - 1}!} } \\paren {\\dfrac {\\paren {j + n - 1}!} {\\paren {j - 1}!} } } {\\paren {i + j - 1} \\paren {i - 1}! \\paren {-1}^{n - i} \\paren {n - i}! \\paren {j - 1}! \\paren {-1}^{n - j} \\paren {n - j}!}$ $\\ds$ $=$ $\\ds \\frac {\\paren {-1}^{i + j} \\paren {i + n - 1}! \\paren {j + n - 1}!} {\\paren {\\paren {i - 1}!}^2 \\paren {\\paren {j - 1}!}^2 \\paren {n - i}! \\paren {n - j}! \\paren {i + j - 1} }$\n\n$\\blacksquare$","date":"2022-05-19 20:52:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 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\": 0.8773742914199829, \"perplexity\": 429.75873738317813}, \"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-2022-21\/segments\/1652662530066.45\/warc\/CC-MAIN-20220519204127-20220519234127-00447.warc.gz\"}"} | null | null |
Wicked: The Life and Times of the Wicked Witch of the West traducido al español como Wicked: Memorias de una bruja mala es una novela publicada en 1995, escrita por Gregory Maguire. Ha vendido millones de ejemplares, inspirando también el musical de Brodway con el mismo título Wicked. Está basado en la popular historia de 1900 El Maravilloso Mago de Oz escrita por L. Frank Baum y luego del popular film de 1939: El Mago de Oz. Es el primero de una saga de cuatro libros que Maguire ha nombrado como "The Wicked Years". La segunda parte se llama Hijo de Bruja (Son of a Witch), el tercer volumen Un León entre los Hombres (A Lion Among Men) y el cuarto y último libro se llama Fuera de Oz (Out of Oz).
Wicked está ambientada antes de la llegada de Dorothy Gale a la tierra de Oz. Narra la historia de Elphaba, una niña nacida de color verde y cómo se convierte en la Malvada Bruja del Oeste, pasando por los sucesos que experimenta desde su nacimiento, infancia y juventud en la Universidad de Shiz hasta llegar a la edad adulta, momento de la llegada de Dorothy a la tierra de Oz.
Sinopsis
El libro comienza con el inicio nacimiento de Elphaba Thropp, una niña con la epidermis verde. Es la primera hija de Melena Thropp, y su padre es un clérigo unionista (religión descrita en la historia, muy parecida al catolicismo). Se insinúa que la causa de su piel verde es porque Melena bebió una poción que un viajero le ofreció antes de su concepción.
Melena se prepara para dar a luz, y la noche en la cual nace Elphaba, la futura Bruja, coincide con la llegada de un espectáculo blasfemo de títeres llamado "El Reloj del Dragón del Tiempo" al pueblo donde viven. Su padre, como buen clérigo, abandona a su esposa a punto de parir para evitar que sus seguidores caigan en la tentación que muestra semejante espectáculo.
Elphaba nace y aparentemente es una niña normal salvo por su piel verde y los abominables dientes de tiburón con los que nace y con los cuales corta un dedo de la comadrona.
No es fácil ser verde en el mundo de Oz, así que a Elphaba le cuesta mucho adaptarse a la sociedad, la cual no es muy diferente de nuestro mundo (descrito en el libro como "La Otra Tierra", un lugar misterioso donde se cree que van las almas al morir y lugar de donde el famoso Mago de Oz asegura venir). En Oz, la discriminación y los prejuicios están a la orden del día, desde los ricos hasta los pobres, hasta las diferencias entre nacionalidades, especies, animales y Animales. Animales con mayúscula porque pueden caminar en dos patas, tienen el don del habla, poseen alma y conciencia humana, pero que son constantemente atacados por la sociedad, especialmente por el Mago, quien promulga leyes para su arresto y exterminio.
A lo largo de su vida Elphaba hace amigos, como Boq, un munchkin con quien jugaba de niña, dos muchachos bromistas llamados Crope y Tibbett y Fiyero, un príncipe del Vinkus.
En su vida, Elphaba Thropp se ve inmersa en medio de su escepticismo, al considerarse atea y aespiritualista, dudando sobre la existencia del alma, la magia y la religión que profesa su padre, y siempre pensando en lo que realmente es bueno y malo, sin poder encontrar una respuesta. A los diecisiete años entra a la Universidad de Shiz, cerca de Ciudad Esmeralda, en el país del norte, Gillikin. Ahí conocerá a Galinda, en un futuro llamándose a sí misma "Glinda", quien se convertirá en La Bruja Buena del Norte (del Sur en la historia de Baum).
En esta historia, el Mago de Oz es un dictador que atenta contra los derechos de los animales parlantes, por lo tanto Elphaba se presenta como una revolucionaria que lucha en contra de su régimen. El nombre de Elphaba es una invención del autor, y el color verde de su piel corresponde a la imagen que de ella se hizo en la película de 1939, que por supuesto nada tiene que ver con la imagen de la novela de L. Frank Baum, donde su color de piel es blanco, es tuerta y viste de amarillo (representando así el color de su país, el de los Winkies). El carácter del personaje original nada tiene que ver tampoco con el del personaje descrito por Maguire.
Elphaba descubre un complot político entre la Señora Morrible, la directora de su escuela, y el Mago de Oz, en contra de los Animales. Durante toda su vida aboga siempre por los derechos de los Animales, se va en contra de las injusticias y con deseos de derrocar al Mago, y termina dejando la escuela para pasarse a la lucha clandestina, abandonando a sus amigos en la universidad e incluso a su inválida hermana menor, Nessarose Thropp, la dueña de los famosos zapatos rojos (plateados en la historia original), quien se considera superior en rectitud moral, es la favorita de papá, la santa local y que nace sin brazos, y quien se convertiría en la Malvada Bruja del Este, para después ser aplastada por la casa de Dorothy.
Enlaces externos
Novelas de Estados Unidos
Novelas de 1995 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,209 |
Q: Download file using REST I want to download a file using REST API.
I have a file in a document library /folder. I want download that file, but with a different name.
Edit 1 : Below is my code. i am able to download file on Chrome browser but data gets corrupted and also not working on IE 10+.
function SaveToDisk_blob(blobURL, fileName)
{
var reader = new FileReader();
console.log(blobURL.body)
console.log(fileName.split(".")[1])
var blob = new Blob([blobURL.body], { type: "application/"+fileName.split(".")[1]+"" });
reader.readAsDataURL(blob);
reader.onload = function (event) {
var save = document.createElement('a');
save.href = event.target.result;
save.target = '_blank';
save.download = fileName || 'unknown file';
var event = document.createEvent('Event');
event.initEvent('click', true, true);
save.dispatchEvent(event);
(window.URL || window.webkitURL).revokeObjectURL(save.href);
};
reader.onerror = function (e) {
console.log(e)
}
}
function downloadFile(url, fileName)
{
// executes cross domain request
var appweburl = _spPageContextInfo.webAbsoluteUrl
$.getScript(appweburl + "/_layouts/15/SP.RequestExecutor.js", function () {
var executor = new SP.RequestExecutor(appweburl);
executor.executeAsync(
{
url: url,
type: "GET",
binaryStringResponseBody: true,
success: function (blobURL)
{
SaveToDisk_blob(blobURL, fileName)
},
error: function (xhr)
{
console.log(xhr);
console.log("downloadFile" + xhr.status + ": " + xhr.statusText);
}
});
});
}
Edit 2: i have made some more change in code.
function call_to_rest_binarystring(rest_url)
{
return $.ajax({
url: rest_url,
method: "GET",
});
}
function downloadFile(rest_url, filepath, fileName)
{
var dfd = $.Deferred();
if (!window.ActiveXObject)
{
var save = document.createElement('a');
save.href = filepath;
save.target = '_blank';
save.download = fileName || 'unknown';
var event = document.createEvent('Event');
event.initEvent('click', true, true);
save.dispatchEvent(event);
(window.URL || window.webkitURL).revokeObjectURL(save.href);
dfd.resolve(true);
}
else if (!!window.ActiveXObject && document.execCommand)
{
var rest_return_call = call_to_rest_binarystring(rest_url)
rest_return_call.done(function (response, status, xhr)
{
var blob = new Blob([response], { type: "application/" + fileName.split(".")[1] + "" });
window.navigator.msSaveBlob(blob, fileName);
});
}
return dfd.promise()
}
var rest_url = "<site name>/_api/web/GetFileByServerRelativeUrl('/ShareDocument/111.pdf')/openbinarystream"
downloadFile(rest_url, filepath, fileName)
still while we download file it is coming blank. i think i am missing something. can anyone help me with this.
A: I usually use AttachmentFiles endpoint to do this, it is provided when you create or select Item with API.
<link rel="http://schemas.microsoft.com/ado/2007/08/dataservices/related/AttachmentFiles" type="application/atom+xml;type=feed" title="AttachmentFiles" href="Web/Lists(guid'12f9fd2c-0eea-4440-b9d3-a6e445839ba3')/Items(66)/AttachmentFiles" />
Main part is
href="Web/Lists(guid'12f9fd2c-0eea-4440-b9d3-a6e445839ba3')/Items(66)/AttachmentFiles"
You can use both
To download /Items(66)/AttachmentFiles('file.docx')
To upload /Items(66)/AttachmentFiles/add(FileName='file.docx')
Then u can use a GET method to download file content :
executor.executeAsync(
{
url: UsePreviousEndPoint,
type: "GET",
contentType: "application/atom+xml;type=entry",
headers: {
"Accept": "application/atom+xml",
"Authorization": "Bearer " + UserOrAppToken
},
success: function (blobURL)
{
SaveToDisk_blob(blobURL, fileName)
},
error: function (xhr)
{
console.log(xhr);
console.log("downloadFile" + xhr.status + ": " + xhr.statusText);
}
});
I guess Digest is only mandatory for POST request.
A: below are my final working code for downloading file from REST.
var dfd = $.Deferred();
var xhr = new XMLHttpRequest();
xhr.open("GET", filepath);
xhr.responseType = "blob";
//setting response-type header to be blob so that we get the file as blob
xhr.onload = function ()
{
//async call
var blobobj = xhr.response;
window.navigator.msSaveBlob(blobobj, fileName);
//save using msSaveBlob.
dfd.resolve(true);
}
xhr.send();
return dfd.promise()
A: If you have direct link to the file, then you can use below approach. But the filename will be the same
<script type="text/javascript">
function download(url) {
var a = document.createElement('a');
a.setAttribute('href', url);
document.body.appendChild(a);
a.click();
document.body.removeChild(a);
}
</script>
A: I am not sure about your method. I am using REST and needed to add $value to the URL get the file content, instead of the file metadata.
rc.spActionURL = "https://{mysite}.sharepoint.com/sites/{mysites}/#this.sp365.orisApp#/_api/Web/GetFileByServerRelativeUrl('/sites/{mysites}/#this.sp365.orisApp#/#arguments.libraryPath#/#arguments.folderName#/#arguments.fileName#')/$value";
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,664 |
namespace naive {
namespace base {
using HandlerFunc = std::function<void()>;
class Looper {
public:
virtual void AddFd(int fd, HandlerFunc func) = 0;
virtual void AddHandler(HandlerFunc func) = 0;
};
} // namespace base
} // namespace naive
#endif // BASE_LOOPER_H_
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,520 |
{"url":"http:\/\/connectedpicture.blogspot.com\/","text":"## Wednesday, January 6, 2016\n\n### Dissemination of Math in the internet age\n\nI'm at the Joint Mathematics Meetings in Seattle. One of the first talks (8:00 AM!) of the first day was given by Prof. Tim Gowers on How might Mathematics be better disseminated (slight change in the title from what was originally published). Prof. Gowers highlighted several themes on better ways of doing mathematical research as well as better ways of disseminating the same, which he has been working on, and popularizing on the web, in recent years. Here are some of my own interpretations of what he talked about.\n\nIn the current setting of mathematical research, most emphasis is on being the \"first to prove the theorem\", and the basic unit of discourse is the peer-reviewed journal article. There are more than a few things wrong with this set up, the main one being that the wheel is reinvented repeatedly! Here is a type of result which could be very useful. If Lemma A is true, then BIG RESULT B is true, which would be fantastic news!. But, I have a counterexample for Lemma A :-(. Unfortunately, such a negative result cannot be \"published\" in a peer-reviewed journal article. Hence, others fumble around and reinvent the same result!\n\nAnother major drawback of the current system is that mathematical conversation happens at an inordinately slow pace. Someone proves a theorem, which appears in the journal in two years time. Then someone else reads it, comes up with a modification or a simpler proof, which appears in another journal two more years later! But in the current internet age, it's only natural to expect conversations occurring at a much faster pace (live tweets, any one?).\n\nWhile electronic publication has made all research easily searchable, the search capacity is strong only in one direction, so to speak. If you know what you're looking for in terms of keywords, then it's easy to find it by search, e.g., you want to know what Szemer\u00e9di's theorem states, a quick search pulls up multiple relevant web pages. What would be very useful is a way to search \"in reverse\" using some limited keywords or partial statements (and not the name itself!) to see if such a result is already known. In other words, the community needs a mathematics research database that allows semantic search. Today, forums such as the mathoverflow often gets us accurate answers to questions of the form \"has this been done before?\".\n\nIn current times, mathematics research should use the internet - both for conducting it as well as to disseminate it. But someone having a high rating on the Math StackExchange for posting numerous answers, or who writes popular blog articles on otherwise difficult to understand math papers, is not rewarded for these activities in the current system. To get tenure, you better have the required number of papers in the top(-enough!) journals! One could have a potentially huge impact by writing easy-to-understand expository blog posts on otherwise hard-to-read set of mathematical papers written by other authors. These blog posts could in turn spur new contributions from others, who would not have had the inclination otherwise to digest the original papers. As such, this effort could potentially be worth much more than publishing a paper with a new theorem! But then again, the current system has no means of rewarding such expository efforts.\n\nIn follow-up conversation, Prof. Gowers agreed that senior\/reputed mathematicians such as himself could afford to spend more time and effort on such endeavors on the internet without worrying about the rewards or evaluations. For tenure-track faculty or other junior researchers, the best course might well be to do both - publish via the conventional means, but also spend some effort on internet-based activities. Further down the line, we as a community would want to be able to judge how \"impactful\" a series of blog posts have been, so as to reward the same. But we must be careful not to go down the same path as overusing journal impact factors (so, don't judge the impact of a blog post by the number of times it has been re-tweeted, or +1-ed!).\n\nIn the latter part of the talk, Prof. Gowers highlighted four of his personal efforts in this regard - a reform of the journal system (Discrete Analysis), informal mathematical writing (via his blog), polymath projects, and automating proof discovery. He also presented his ideas of what mathematics research would be like in 20-30 years from now, and then in 50-60 years from now. Not surprisingly, computers would be expected to do most of the heavy (and light:-) lifting in the future.\n\nA question from the audience inquired about the place of real, i.e., face-to-face, conversations on mathematics that happen at conferences. Would there be less of a place for them in mathematics research in the future? While agreeing that such conversations have their place, Prof. Gowers observed that may be the back-and-forth postings on a polymath project (with time-stamps!) allows the posters the time to understand the subject better, and think through before posting what they want to post. An instant face-to-face conversation would not give that luxury!\n\nIn summary, much need to be changed for mathematics research to be done right and be impactful in the internet age. Individual researchers need to be a bit bold, and not worry too much about rewards and ratings when spending time and effort on the internet. The more people who do so, the sooner the inevitable transition would happen!\n\n## Wednesday, November 4, 2015\n\n### Discrete optimization @ Oaxaca - II\n\nSome snippets from days 2 and 3 at the BIRS-CMO Workshop on Discrete Optimization in Oaxaca.\n\nThomas Rothvoss presented his work on constructive discrepancy minimization for convex sets (slides are available). The basic problem is that of assigning one of two colors $$\\chi(i)$$ to each $$i \\in [n]:=\\{1,\\dots,n\\}$$ of $$n$$ items represented by $$\\{-1,+1\\}$$ such that for a system of sets $$\\cal{S} = \\{S_1,\\dots,S_m\\}$$ with $$S_i \\in [n]$$ we minimize the maximum \"mismatch\", defined as the discrepancy:\n\n$$\\rm{disc}(\\cal{S}) = \\min\\limits_{\\chi(i) = \\pm 1} \\max\\limits_{S \\in \\cal{S}} \\left| \\sum_{i \\in S} \\chi(i) \\right|$$.\n\nI found the techniques developed fairly deep, and would find use in lots of applications (i.e., for proving results on other optimization problems; Thomas talked about an application to bin packing). We had previously looked at the somewhat related problem of number partitioning. There, we assign a set of integers $$\\{a_1, \\dots, a_n\\}$$ to two sets such that the sums of the numbers over the two sets are as close to each other as possible (the difference between these two sums is the discrepancy here). At the same time, the corresponding alternative definition of discrepancy given as\n$$\\rm{disc'}(\\cal{S}) = \\min\\limits_{\\chi(i) = \\pm 1} \\sum\\limits_{S \\in \\cal{S}} \\left| \\sum_{i \\in S} \\chi(i) \\right|$$\n\nwould make the problem sort of \"easy\" here. With that objective, one could prove that a random assignment of $$\\pm 1$$ would perhaps do as well as we can. Nonetheless, an appropriately defined notion of \"weighted\" discrepancy, where each element now has, say, nonnegative weights, would be interesting to consider. It appears a generalization to more than two colors would be interesting, but perhaps tricky to establish the building blocks of results.\n\nTamon Stephen talked about a variant of the Hirsch conjecture using circuit diameter of polyhedra, instead of the default graph diameter. See the preprint for an illustration of circuit distance between vertices of a polyhedron - unlike the graph diameter, it's not symmetric. The idea is that one is allowed to take \"shortcuts\" along the interior of the polyhedron along with the walks along the edges. In that sense, it's mixing the ideas of the default simplex method and the interior point method for solving linear programs (LPs). The authors show that the most basic counterexample to the Hirsch conjecture, the Klee-Walkup polyhedron, in fact satisfies the Hirsch bound. It would be interesting for software programs that solve LPs to be able to seamlessly and intelligently switch back and forth between interior point and simplex methods (a basic ability to do so is already provided by some of the state-of-the-art solvers)\n\nJuan Pablo Vielma talked about when are Minkowski sums good\/bad in the context of formulations for unions of polyhedra, and for unions of convex sets in general (the slides should be up soon here; but other versions are already available). For unions of polyhedra, aggregated formulations are often \"short\", but are not as tight as disaggregated ones (also termed extended formulations). The latter formulations are sharp\/ideal, but are often too large in size (see the excellent review on MIP formulation techniques by JP Vielma, or lectures 5-7 from my IP class for a shorter overview). This interesting line of work tries to find a better middle ground by finding the sharpest formulations that are not extended, i.e., without having to add extra variables. Things get very interesting when one considers unions of convex sets (in place of polyhedra)!\n\n## Monday, November 2, 2015\n\n### Discrete optimization @ Oaxaca - I\n\nI'm at the BIRS-CMO Workshop on Discrete Optimization in beautiful Oaxaca (in Mexico). Unlike other typical workshops, the organizers have tried hard to encourage lots of discussion and interactions among the participants - big props to Jesus De Loera and Jon Lee! I'm hoping to write snippets on talks\/discussion that I found particularly interesting (yes, it'll be a biased view :-).\n\nThe meeting started with an apt talk by Dan Bienstock on LP formulations for polynomial optimization problems (based on this paper; Dan has also posted the slides). He started with a motivating problem - the optimal power flow (OPF) problem, which motivates the use of the treewidth of the intersection graph of the constraints as the parameter which controls the complexity of the proposed reformulation operator. And real-life power grids often have small treewidths. The reformulation operator produces linear programming approximations that attain provable bounds for mixed integer polynomial problems where the variables are either binary, or require $$0 \\leq x_j \\leq 1$$.\n\nInformally, the intersection graph of a system of constraints has one vertex for each variable $$x_j$$, and edge $$(x_i,x_j)$$ is present when both $$x_i$$ and $$x_j$$ appear in some constraint. The simple example of a subset sum (or knapsack) problem was insightful. With the single constraint being $$x_1+\\dots+x_n \\leq \\beta$$, a constraint graph could have $$n+1$$ vertices $${0,1,\\dots,n}$$, with the $$n$$ edges $$(0,j)$$ for each $$j$$ corresponding to $$x_j$$ (node $$0$$ is a \"dummy\" node here). The treewidth of this \"star\" graph is $$1$$. The other key trick employed is the use of binary variables to approximate a continuous variable $$0 \\leq x \\leq 1$$ (attributed originally to Glover). For a given error term $$0 < \\gamma < 1$$, we can approximate $$x \\approx \\sum_{j=1}^L \\left(1\/2^h\\right) y_h$$, where $$y_h$$ are binary variables. With $$L = \\lceil \\log_2 (1\/\\gamma) \\rceil$$, we can get $$x \\leq \\sum_{j=1}^L \\left(1\/2^h\\right) y_h \\leq x + \\gamma$$. This step helps to get pure binary problems in place of the mixed integer problems. The treewidth gets blown up by $$L$$, but things still work out nicely. This paper seems to have lots of nice and deep \"tricks\" (I hope to study it in detail).\n\nThere were several other interesting talks, and a very interactive problem session to conclude the day. Oh, and I learned a new terminology used in the power industry from Shabbir Ahmed's talk: a prosumer is someone who both produces and consumes power. I'm wondering why that is more apt than a conducer...\n\nThere was some lively discussion over dinner about how the optimization and operations research communities have failed to sell itself as well as the CS community (as a whole, or even the CS theory community by itself). Large membership sizes of ACM vs INFORMS and similarly large NSF budgets for CS vs OR were cited as indicators. There was also an anecdote mentioned about how back in the 1980s when NSF funding for algorithms\/CS theory was on the decline, a group of several top big names from that field submitted a memo\/petition to the lawmakers in DC, and also convinced them in person that \"algorithms\/theory is as fundamental as cosmology\" (needless to say, I'm paraphrasing to a huge extent here!). And yes, they managed to restore the funding flow. The optimization community tried a bit of the same trick with Karmarkar's interior point algorithm for LP. May be we optimizers should try harder - not just from the point of view of securing funding, but also from the point of view of our students getting better industry jobs!\n\n## Saturday, September 12, 2015\n\n### Math of Data Science @ ICERM\n\nNote: This is a re-post of the blog post I wrote originally on WordPress last month. It appears that Blogger is a much better platform for the kind of posts I'd like to make. Yes, I'm still learning ...\n\nI recently attended the topical workshop on Mathematics in Data Sciences at ICERM. The attendance was good mix of students, postdocs, and researchers\/faculty from academic institutions and national labs along with a sizable number of industry folks. The line of talks involved a similar mixture as well (abstracts\/slides from many of the talks are available from the workshop page linked above). In particular from the industry side, there were talks from data scientists (or \"engineers\" with similar roles) from Ayasdi, LinkedIn (formerly), Netflix, New York Times, and Schlumberger-Doll, to name a few. Indeed, I found this diversity a direct indicator of the young age of the discipline in question, i.e., data science. And yes, the usual jokes about data science\/big data were not spared, including the one about how big data is like teenage sex, how big data is very much a man's game since you usually hear men boasting \"my data is bigger than yours\", and how data scientists are mostly data janitors!\n\nComing from the academic side, what I found most interesting at this workshop were the panel (and open) discussion sessions. In the first such discussion, the group tried to come up with a (not so short!) list of topics that a program in data science should train the (undergraduate\/masters) student in. After starting with the usual suspects such as calculus and linear algebra, probability and statistics, algorithms, machine learning, and databases, the group expanded the scope. Next came high dimensional geometry, information theory, data visualization\/exploration, experimental design, and communication\/business \"skills\". But many in the audience appeared to be surprised by the suggestion of a class on inverse problems, and electromagnetism (yes!). The topics and then associated skills to be taught soon filled up two large panels of white board (recall the \"teenage sex\" joke, any one?). To wrap up the session, it was suggested that the student be trained (at least) in Python, GitHub, Sql (or something similar), all from the point of view of industry readiness. As far as the mathematicians are concerned, it was suggested that they could start by making a wish list of all results (related to data science) one wants to see as theorems, and such a list will keep them busy for more than a life time. But to see any such effort make huge impact, one should ideally work with a domain expert. One particular subtopic of much importance in this context (no pun intended!) is that of textual data - very important for data science, and as yet not well explored by mathematicians.\n\nThe panel discussion about careers in data science was quite popular as well. A majority of the panelists were junior (read \"young\") data scientists from the industry, and were able to shed a lot of light on what a typical work day looks like for a data scientist. One aspect of their work that particularly appealed to me (coming from traditional academia) is how quickly and directly they are able to see the impact of their ideas and work. For instance, a data scientist in a social media company could brainstorm for 2 hours, write the code in 2 more hours, and see thousands of users enjoying the benefits before the end of the day! On the other hand, academicians often wait years, if not months, to just count citations of their papers.\n\nIf there was one take home message from the data scientists to (young and old) aspirants, it was to just play with data - of many types and from many sources, and not to worry so much about all the different classes\/training (or proving theorems). Be ever-ready to dive into any data that you come across, manipulate\/analyze it quickly, and get the first insights.\n\nI'm not attempting to list any summary\/thoughts on the Mathematics involved (as meant in the title of the workshop). The list of relevant Math\/Stat\/CS topics has been huge already, and is not getting any shorter in this era of data science. I doubt if we're going to precisely define what data science is any time soon!","date":"2017-03-28 15:52:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.4852578938007355, \"perplexity\": 934.532563854878}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-13\/segments\/1490218189802.18\/warc\/CC-MAIN-20170322212949-00015-ip-10-233-31-227.ec2.internal.warc.gz\"}"} | null | null |
{"url":"http:\/\/linear.ups.edu\/jsmath\/0222\/fcla-jsmath-2.22li56.html","text":"### Section VR\u00a0\u00a0Vector Representations\n\nFrom A First Course in Linear Algebra\nVersion 2.22\nhttp:\/\/linear.ups.edu\/\n\nWe begin by establishing an invertible linear transformation between any vector space V of dimension m and {\u2102}^{m}. This will allow us to \u201cgo back and forth\u201d between the two vector spaces, no matter how abstract the definition of V might be.\n\nDefinition\u00a0VR\nVector Representation\nSuppose that V is a vector space with a basis B = \\left \\{{v}_{1},\\kern 1.95872pt {v}_{2},\\kern 1.95872pt {v}_{3},\\kern 1.95872pt \\mathop{\\mathop{\u2026}},\\kern 1.95872pt {v}_{n}\\right \\}. Define a function {\u03c1}_{B}: V \u2192 {\u2102}^{n} as follows. For w \u2208 V define the column vector {\u03c1}_{B}\\left (w\\right ) \u2208 {\u2102}^{n} by\n\n\\eqalignno{ w & ={ \\left [{\u03c1}_{B}\\left (w\\right )\\right ]}_{1}{v}_{1} +{ \\left [{\u03c1}_{B}\\left (w\\right )\\right ]}_{2}{v}_{2} +{ \\left [{\u03c1}_{B}\\left (w\\right )\\right ]}_{3}{v}_{3} + \\mathrel{\u22ef} +{ \\left [{\u03c1}_{B}\\left (w\\right )\\right ]}_{n}{v}_{n} & & }\n\n(This definition contains Notation VR.)\n\nThis definition looks more complicated that it really is, though the form above will be useful in proofs. Simply stated, given w \u2208 V , we write w as a linear combination of the basis elements of B. It is key to realize that Theorem\u00a0VRRB guarantees that we can do this for every w, and furthermore this expression as a linear combination is unique. The resulting scalars are just the entries of the vector {\u03c1}_{B}\\left (w\\right ). This discussion should convince you that {\u03c1}_{B} is \u201cwell-defined\u201d as a function. We can determine a precise output for any input. Now we want to establish that {\u03c1}_{B} is a function with additional properties - it is a linear transformation.\n\nTheorem\u00a0VRLT\nVector Representation is a Linear Transformation\nThe function {\u03c1}_{B} (Definition\u00a0VR) is a linear transformation.\n\nProof\u00a0\u00a0 We will take a novel approach in this proof. We will construct another function, which we will easily determine is a linear transformation, and then show that this second function is really {\u03c1}_{B} in disguise. Here we go.\n\nSince B is a basis, we can define T : V \u2192 {\u2102}^{n} to be the unique linear transformation such that T\\left ({v}_{i}\\right ) = {e}_{i}, 1 \u2264 i \u2264 n, as guaranteed by Theorem\u00a0LTDB, and where the {e}_{i} are the standard unit vectors (Definition\u00a0SUV). Then suppose for an arbitrary w \u2208 V we have,\n\n\\eqalignno{ {\\left [T\\left (w\\right )\\right ]}_{i} & ={ \\left [T\\left ({\\mathop{\u2211 }}_{j=1}^{n}{\\left [{\u03c1}_{ B}\\left (w\\right )\\right ]}_{j}{v}_{j}\\right )\\right ]}_{i} & &\\text{@(a href=\"#definition.VR\")Definition VR@(\/a)} & & & & \\cr & ={ \\left [{\\mathop{\u2211 }}_{j=1}^{n}{\\left [{\u03c1}_{ B}\\left (w\\right )\\right ]}_{j}T\\left ({v}_{j}\\right )\\right ]}_{i} & &\\text{@(a href=\"fcla-jsmath-2.22li51.html#theorem.LTLC\")Theorem LTLC@(\/a)} & & & & \\cr & ={ \\left [{\\mathop{\u2211 }}_{j=1}^{n}{\\left [{\u03c1}_{ B}\\left (w\\right )\\right ]}_{j}{e}_{j}\\right ]}_{i} & & & & \\cr & ={ \\mathop{\u2211 }}_{j=1}^{n}{\\left [{\\left [{\u03c1}_{ B}\\left (w\\right )\\right ]}_{j}{e}_{j}\\right ]}_{i} & &\\text{@(a href=\"fcla-jsmath-2.22li23.html#definition.CVA\")Definition CVA@(\/a)} & & & & \\cr & ={ \\mathop{\u2211 }}_{j=1}^{n}{\\left [{\u03c1}_{ B}\\left (w\\right )\\right ]}_{j}{\\left [{e}_{j}\\right ]}_{i} & &\\text{@(a href=\"fcla-jsmath-2.22li23.html#definition.CVSM\")Definition CVSM@(\/a)} & & & & \\cr & ={ \\left [{\u03c1}_{B}\\left (w\\right )\\right ]}_{i}{\\left [{e}_{i}\\right ]}_{i} +{ \\mathop{\u2211 }}_{\\begin{array}{c}j=1 \\\\ j\\mathrel{\u2260}i \\end{array}}^{n}{\\left [{\u03c1}_{ B}\\left (w\\right )\\right ]}_{j}{\\left [{e}_{j}\\right ]}_{i} & &\\text{@(a href=\"fcla-jsmath-2.22li23.html#property.CC\")Property CC@(\/a)} & & & & \\cr & ={ \\left [{\u03c1}_{B}\\left (w\\right )\\right ]}_{i}\\left (1\\right ) +{ \\mathop{\u2211 }}_{\\begin{array}{c}j=1 \\\\ j\\mathrel{\u2260}i \\end{array}}^{n}{\\left [{\u03c1}_{ B}\\left (w\\right )\\right ]}_{j}\\left (0\\right ) & &\\text{@(a href=\"fcla-jsmath-2.22li28.html#definition.SUV\")Definition SUV@(\/a)} & & & & \\cr & ={ \\left [{\u03c1}_{B}\\left (w\\right )\\right ]}_{i} & & & & }\n\nAs column vectors, Definition\u00a0CVE implies that T\\left (w\\right ) = {\u03c1}_{B}\\left (w\\right ). Since w was an arbitrary element of V , as functions T = {\u03c1}_{B}. Now, since T is known to be a linear transformation, it must follow that {\u03c1}_{B} is also a linear transformation.\n\nThe proof of Theorem\u00a0VRLT provides an alternate definition of vector representation relative to a basis B that we could state as a corollary (Technique\u00a0LC): {\u03c1}_{B} is the unique linear transformation that takes B to the standard unit basis.\n\nExample\u00a0VRC4\nVector representation in {\u2102}^{4}\nConsider the vector y \u2208 {\u2102}^{4}\n\n y = \\left [\\array{ 6\\cr 14 \\cr 6\\cr 7 } \\right ]\n\nWe will find several vector representations of y in this example. Notice that y never changes, but the representations of y do change.\n\nOne basis for {\u2102}^{4} is\n\n B = \\left \\{{u}_{1},\\kern 1.95872pt {u}_{2},\\kern 1.95872pt {u}_{3},\\kern 1.95872pt {u}_{4}\\right \\} = \\left \\{\\left [\\array{ \u22122\\cr 1 \\cr 2\\cr \u22123 } \\right ],\\kern 1.95872pt \\left [\\array{ 3\\cr \u22126 \\cr 2\\cr \u22124 } \\right ],\\kern 1.95872pt \\left [\\array{ 1\\cr 2 \\cr 0\\cr 5 } \\right ],\\kern 1.95872pt \\left [\\array{ 4\\cr 3 \\cr 1\\cr 6 } \\right ]\\right \\}\n\nas can be seen by making these vectors the columns of a matrix, checking that the matrix is nonsingular and applying Theorem\u00a0CNMB. To find {\u03c1}_{B}\\left (y\\right ), we need to find scalars, {a}_{1},\\kern 1.95872pt {a}_{2},\\kern 1.95872pt {a}_{3},\\kern 1.95872pt {a}_{4} such that\n\n y = {a}_{1}{u}_{1} + {a}_{2}{u}_{2} + {a}_{3}{u}_{3} + {a}_{4}{u}_{4}\n\nBy Theorem\u00a0SLSLC the desired scalars are a solution to the linear system of equations with a coefficient matrix whose columns are the vectors in B and with a vector of constants y. With a nonsingular coefficient matrix, the solution is unique, but this is no surprise as this is the content of Theorem\u00a0VRRB. This unique solution is\n\n\\eqalignno{ {a}_{1} & = 2 &{a}_{2} & = \u22121 &{a}_{3} & = \u22123 &{a}_{4} & = 4 & & & & & & & & }\n\nThen by Definition\u00a0VR, we have\n\n { \u03c1}_{B}\\left (y\\right ) = \\left [\\array{ 2\\cr \u22121 \\cr \u22123\\cr 4 } \\right ]\n\nSuppose now that we construct a representation of y relative to another basis of {\u2102}^{4},\n\n C = \\left \\{\\left [\\array{ \u221215\\cr 9 \\cr \u22124\\cr \u22122 } \\right ],\\kern 1.95872pt \\left [\\array{ 16\\cr \u221214 \\cr 5\\cr 2 } \\right ],\\kern 1.95872pt \\left [\\array{ \u221226\\cr 14 \\cr \u22126\\cr \u22123 } \\right ],\\kern 1.95872pt \\left [\\array{ 14\\cr \u221213 \\cr 4\\cr 6 } \\right ]\\right \\}\n\nAs with B, it is easy to check that C is a basis. Writing y as a linear combination of the vectors in C leads to solving a system of four equations in the four unknown scalars with a nonsingular coefficient matrix. The unique solution can be expressed as\n\n y = \\left [\\array{ 6\\cr 14 \\cr 6\\cr 7 } \\right ] = (\u221228)\\left [\\array{ \u221215\\cr 9 \\cr \u22124\\cr \u22122 } \\right ]+(\u22128)\\left [\\array{ 16\\cr \u221214 \\cr 5\\cr 2 } \\right ]+11\\left [\\array{ \u221226\\cr 14 \\cr \u22126\\cr \u22123 } \\right ]+0\\left [\\array{ 14\\cr \u221213 \\cr 4\\cr 6 } \\right ]\n\nso that Definition\u00a0VR gives\n\n { \u03c1}_{C}\\left (y\\right ) = \\left [\\array{ \u221228\\cr \u22128 \\cr 11\\cr 0} \\right ]\n\nWe often perform representations relative to standard bases, but for vectors in {\u2102}^{m} its a little silly. Let\u2019s find the vector representation of y relative to the standard basis (Theorem\u00a0SUVB),\n\n D = \\left \\{{e}_{1},\\kern 1.95872pt {e}_{2},\\kern 1.95872pt {e}_{3},\\kern 1.95872pt {e}_{4}\\right \\}\n\nThen, without any computation, we can check that\n\n y = \\left [\\array{ 6\\cr 14 \\cr 6\\cr 7 } \\right ] = 6{e}_{1}+14{e}_{2}+6{e}_{3}+7{e}_{4}\n\nso by Definition\u00a0VR,\n\n { \u03c1}_{D}\\left (y\\right ) = \\left [\\array{ 6\\cr 14 \\cr 6\\cr 7 } \\right ]\n\nwhich is not very exciting. Notice however that the order in which we place the vectors in the basis is critical to the representation. Let\u2019s keep the standard unit vectors as our basis, but rearrange the order we place them in the basis. So a fourth basis is\n\n E = \\left \\{{e}_{3},\\kern 1.95872pt {e}_{4},\\kern 1.95872pt {e}_{2},\\kern 1.95872pt {e}_{1}\\right \\}\n\nThen,\n\n y = \\left [\\array{ 6\\cr 14 \\cr 6\\cr 7 } \\right ] = 6{e}_{3}+7{e}_{4}+14{e}_{2}+6{e}_{1}\n\nso by Definition\u00a0VR,\n\n { \u03c1}_{E}\\left (y\\right ) = \\left [\\array{ 6\\cr 7 \\cr 14\\cr 6 } \\right ]\n\nSo for every possible basis of {\u2102}^{4} we could construct a different representation of y.\n\nVector representations are most interesting for vector spaces that are not {\u2102}^{m}.\n\nExample\u00a0VRP2\nVector representations in {P}_{2}\nConsider the vector u = 15 + 10x \u2212 6{x}^{2} \u2208 {P}_{ 2} from the vector space of polynomials with degree at most 2 (Example\u00a0VSP). A nice basis for {P}_{2} is\n\n B = \\left \\{1,\\kern 1.95872pt x,\\kern 1.95872pt {x}^{2}\\right \\}\n\nso that\n\n u = 15 + 10x \u2212 6{x}^{2} = 15(1) + 10(x) + (\u22126)({x}^{2})\n\nso by Definition\u00a0VR\n\n { \u03c1}_{B}\\left (u\\right ) = \\left [\\array{ 15\\cr 10 \\cr \u22126 } \\right ]\n\nAnother nice basis for {P}_{2} is\n\n B = \\left \\{1,\\kern 1.95872pt 1 + x,\\kern 1.95872pt 1 + x + {x}^{2}\\right \\}\n\nso that now it takes a bit of computation to determine the scalars for the representation. We want {a}_{1},\\kern 1.95872pt {a}_{2},\\kern 1.95872pt {a}_{3} so that\n\n 15 + 10x \u2212 6{x}^{2} = {a}_{ 1}(1) + {a}_{2}(1 + x) + {a}_{3}(1 + x + {x}^{2})\n\nPerforming the operations in {P}_{2} on the right-hand side, and equating coefficients, gives the three equations in the three unknown scalars,\n\n\\eqalignno{ 15 & = {a}_{1} + {a}_{2} + {a}_{3} & & \\cr 10 & = {a}_{2} + {a}_{3} & & \\cr \u2212 6 & = {a}_{3} & & }\n\nThe coefficient matrix of this sytem is nonsingular, leading to a unique solution (no surprise there, see Theorem\u00a0VRRB),\n\n\\eqalignno{ {a}_{1} & = 5 &{a}_{2} & = 16 &{a}_{3} & = \u22126 & & & & & & }\n\nso by Definition\u00a0VR\n\n { \u03c1}_{C}\\left (u\\right ) = \\left [\\array{ 5\\cr 16 \\cr \u22126 } \\right ]\n\nWhile we often form vector representations relative to \u201cnice\u201d bases, nothing prevents us from forming representations relative to \u201cnasty\u201d bases. For example, the set\n\n D = \\left \\{\u22122 \u2212 x + 3{x}^{2},\\kern 1.95872pt 1 \u2212 2{x}^{2},\\kern 1.95872pt 5 + 4x + {x}^{2}\\right \\}\n\ncan be verified as a basis of {P}_{2} by checking linear independence with Definition\u00a0LI and then arguing that 3 vectors from {P}_{2}, a vector space of dimension 3 (Theorem\u00a0DP), must also be a spanning set (Theorem\u00a0G). Now we desire scalars {a}_{1},\\kern 1.95872pt {a}_{2},\\kern 1.95872pt {a}_{3} so that\n\n 15 + 10x \u2212 6{x}^{2} = {a}_{ 1}(\u22122 \u2212 x + 3{x}^{2}) + {a}_{ 2}(1 \u2212 2{x}^{2}) + {a}_{ 3}(5 + 4x + {x}^{2})\n\nPerforming the operations in {P}_{2} on the right-hand side, and equating coefficients, gives the three equations in the three unknown scalars,\n\n\\eqalignno{ 15 & = \u22122{a}_{1} + {a}_{2} + 5{a}_{3} & & \\cr 10 & = \u2212{a}_{1} + 4{a}_{3} & & \\cr \u2212 6 & = 3{a}_{1} \u2212 2{a}_{2} + {a}_{3} & & }\n\nThe coefficient matrix of this sytem is nonsingular, leading to a unique solution (no surprise there, see Theorem\u00a0VRRB),\n\n\\eqalignno{ {a}_{1} & = \u22122 &{a}_{2} & = 1 &{a}_{3} & = 2 & & & & & & }\n\nso by Definition\u00a0VR\n\n { \u03c1}_{D}\\left (u\\right ) = \\left [\\array{ \u22122\\cr 1 \\cr 2 } \\right ]\n\nTheorem\u00a0VRI\nVector Representation is Injective\nThe function {\u03c1}_{B} (Definition\u00a0VR) is an injective linear transformation.\n\nProof\u00a0\u00a0 We will appeal to Theorem\u00a0KILT. Suppose U is a vector space of dimension n, so vector representation is of the form {\u03c1}_{B}: U \u2192 {\u2102}^{n}. Let B = \\left \\{{u}_{1},\\kern 1.95872pt {u}_{2},\\kern 1.95872pt {u}_{3},\\kern 1.95872pt \\mathop{\\mathop{\u2026}},\\kern 1.95872pt {u}_{n}\\right \\} be the basis of U used in the definition of {\u03c1}_{B}. Suppose u \u2208K\\kern -1.95872pt \\left ({\u03c1}_{B}\\right ). We write u as a linear combination of the vectors in the basis B where the scalars are the components of the vector representation, {\u03c1}_{B}\\left (u\\right ).\n\n\\eqalignno{ u& ={ \\left [{\u03c1}_{B}\\left (u\\right )\\right ]}_{1}{u}_{1} +{ \\left [{\u03c1}_{B}\\left (u\\right )\\right ]}_{2}{u}_{2} +{ \\left [{\u03c1}_{B}\\left (u\\right )\\right ]}_{3}{u}_{3} + \\mathrel{\u22ef} +{ \\left [{\u03c1}_{B}\\left (u\\right )\\right ]}_{n}{u}_{n} & &\\text{@(a href=\"#definition.VR\")Definition VR@(\/a)} & & & & \\cr & ={ \\left [0\\right ]}_{1}{u}_{1} +{ \\left [0\\right ]}_{2}{u}_{2} +{ \\left [0\\right ]}_{3}{u}_{3} + \\mathrel{\u22ef} +{ \\left [0\\right ]}_{n}{u}_{n} & &\\text{@(a href=\"fcla-jsmath-2.22li52.html#definition.KLT\")Definition KLT@(\/a)} & & & & \\cr & = 0{u}_{1} + 0{u}_{2} + 0{u}_{3} + \\mathrel{\u22ef} + 0{u}_{n} & &\\text{@(a href=\"fcla-jsmath-2.22li18.html#definition.ZCV\")Definition ZCV@(\/a)} & & & & \\cr & = 0 + 0 + 0 + \\mathrel{\u22ef} + 0 & &\\text{@(a href=\"fcla-jsmath-2.22li37.html#theorem.ZSSM\")Theorem ZSSM@(\/a)} & & & & \\cr & = 0 & &\\text{@(a href=\"fcla-jsmath-2.22li37.html#property.Z\")Property Z@(\/a)} & & & & }\n\nThus an arbitrary vector, u, from the kernel ,K\\kern -1.95872pt \\left ({\u03c1}_{B}\\right ), must equal the zero vector of U. So K\\kern -1.95872pt \\left ({\u03c1}_{B}\\right ) = \\left \\{0\\right \\} and by Theorem\u00a0KILT, {\u03c1}_{B} is injective.\n\nTheorem\u00a0VRS\nVector Representation is Surjective\nThe function {\u03c1}_{B} (Definition\u00a0VR) is a surjective linear transformation.\n\nProof\u00a0\u00a0 We will appeal to Theorem\u00a0RSLT. Suppose U is a vector space of dimension n, so vector representation is of the form {\u03c1}_{B}: U \u2192 {\u2102}^{n}. Let B = \\left \\{{u}_{1},\\kern 1.95872pt {u}_{2},\\kern 1.95872pt {u}_{3},\\kern 1.95872pt \\mathop{\\mathop{\u2026}},\\kern 1.95872pt {u}_{n}\\right \\} be the basis of U used in the definition of {\u03c1}_{B}. Suppose v \u2208 {\u2102}^{n}. Define the vector u by\n\n u ={ \\left [v\\right ]}_{1}{u}_{1} +{ \\left [v\\right ]}_{2}{u}_{2} +{ \\left [v\\right ]}_{3}{u}_{3} + \\mathrel{\u22ef} +{ \\left [v\\right ]}_{n}{u}_{n}\n\nThen for 1 \u2264 i \u2264 n\n\n\\eqalignno{ {\\left [{\u03c1}_{B}\\left (u\\right )\\right ]}_{i} & ={ \\left [{\u03c1}_{B}\\left ({\\left [v\\right ]}_{1}{u}_{1} +{ \\left [v\\right ]}_{2}{u}_{2} +{ \\left [v\\right ]}_{3}{u}_{3} + \\mathrel{\u22ef} +{ \\left [v\\right ]}_{n}{u}_{n}\\right )\\right ]}_{i} & & & & \\cr & ={ \\left [v\\right ]}_{i} & &\\text{@(a href=\"#definition.VR\")Definition VR@(\/a)} & & & & }\n\nso the entries of vectors {\u03c1}_{B}\\left (u\\right ) and v are equal and Definition\u00a0CVE yields the vector equality {\u03c1}_{B}\\left (u\\right ) = v. This demonstrates that v \u2208\u211b\\kern -1.95872pt \\left ({\u03c1}_{B}\\right ), so {\u2102}^{n} \u2286\u211b\\kern -1.95872pt \\left ({\u03c1}_{ B}\\right ). Since \u211b\\kern -1.95872pt \\left ({\u03c1}_{B}\\right ) \u2286 {\u2102}^{n} by Definition\u00a0RLT, we have \u211b\\kern -1.95872pt \\left ({\u03c1}_{B}\\right ) = {\u2102}^{n} and Theorem\u00a0RSLT says {\u03c1}_{B} is surjective.\n\nWe will have many occasions later to employ the inverse of vector representation, so we will record the fact that vector representation is an invertible linear transformation.\n\nTheorem\u00a0VRILT\nVector Representation is an Invertible Linear Transformation\nThe function {\u03c1}_{B} (Definition\u00a0VR) is an invertible linear transformation.\n\nProof\u00a0\u00a0 The function {\u03c1}_{B} (Definition\u00a0VR) is a linear transformation (Theorem\u00a0VRLT) that is injective (Theorem\u00a0VRI) and surjective (Theorem\u00a0VRS) with domain V and codomain {\u2102}^{n}. By Theorem\u00a0ILTIS we then know that {\u03c1}_{B} is an invertible linear transformation.\n\nInformally, we will refer to the application of {\u03c1}_{B} as coordinatizing a vector, while the application of {\u03c1}_{B}^{\u22121} will be referred to as un-coordinatizing a vector.\n\n#### Subsection CVS: Characterization of Vector Spaces\n\nLimiting our attention to vector spaces with finite dimension, we now describe every possible vector space. All of them. Really.\n\nTheorem\u00a0CFDVS\nCharacterization of Finite Dimensional Vector Spaces\nSuppose that V is a vector space with dimension n. Then V is isomorphic to {\u2102}^{n}.\n\nProof\u00a0\u00a0 Since V has dimension n we can find a basis of V of size n (Definition\u00a0D) which we will call B. The linear transformation {\u03c1}_{B} is an invertible linear transformation from V to {\u2102}^{n}, so by Definition\u00a0IVS, we have that V and {\u2102}^{n} are isomorphic.\n\nTheorem\u00a0CFDVS is the first of several surprises in this chapter, though it might be a bit demoralizing too. It says that there really are not all that many different (finite dimensional) vector spaces, and none are really any more complicated than {\u2102}^{n}. Hmmm. The following examples should make this point.\n\nExample\u00a0TIVS\nTwo isomorphic vector spaces\nThe vector space of polynomials with degree 8 or less, {P}_{8}, has dimension 9 (Theorem\u00a0DP). By Theorem\u00a0CFDVS, {P}_{8} is isomorphic to {\u2102}^{9}.\n\nExample\u00a0CVSR\nCrazy vector space revealed\nThe crazy vector space, C of Example\u00a0CVS, has dimension 2 by Example\u00a0DC. By Theorem\u00a0CFDVS, C is isomorphic to {\u2102}^{2}. Hmmmm. Not really so crazy after all?\n\nExample\u00a0ASC\nA subspace characterized\nIn Example\u00a0DSP4 we determined that a certain subspace W of {P}_{4} has dimension 4. By Theorem\u00a0CFDVS, W is isomorphic to {\u2102}^{4}.\n\nTheorem\u00a0IFDVS\nIsomorphism of Finite Dimensional Vector Spaces\nSuppose U and V are both finite-dimensional vector spaces. Then U and V are isomorphic if and only if \\mathop{ dim}\\nolimits \\left (U\\right ) =\\mathop{ dim}\\nolimits \\left (V \\right ).\n\nProof\u00a0\u00a0 () This is just the statement proved in Theorem\u00a0IVSED.\n\n() This is the advertised converse of Theorem\u00a0IVSED. We will assume U and V have equal dimension and discover that they are isomorphic vector spaces. Let n be the common dimension of U and V . Then by Theorem\u00a0CFDVS there are isomorphisms T : U \u2192 {\u2102}^{n} and S : V \u2192 {\u2102}^{n}.\n\nT is therefore an invertible linear transformation by Definition\u00a0IVS. Similarly, S is an invertible linear transformation, and so {S}^{\u22121} is an invertible linear transformation (Theorem\u00a0IILT). The composition of invertible linear transformations is again invertible (Theorem\u00a0CIVLT) so the composition of {S}^{\u22121} with T is invertible. Then \\left ({S}^{\u22121} \u2218 T\\right ): U \u2192 V is an invertible linear transformation from U to V and Definition\u00a0IVS says U and V are isomorphic.\n\nExample\u00a0MIVS\nMultiple isomorphic vector spaces\n{\u2102}^{10}, {P}_{9}, {M}_{2,5} and {M}_{5,2} are all vector spaces and each has dimension 10. By Theorem\u00a0IFDVS each is isomorphic to any other.\n\nThe subspace of {M}_{4,4} that contains all the symmetric matrices (Definition\u00a0SYM) has dimension 10, so this subspace is also isomorphic to each of the four vector spaces above.\n\n#### Subsection CP: Coordinatization Principle\n\nWith {\u03c1}_{B} available as an invertible linear transformation, we can translate between vectors in a vector space U of dimension m and {\u2102}^{m}. Furthermore, as a linear transformation, {\u03c1}_{B} respects the addition and scalar multiplication in U, while {\u03c1}_{B}^{\u22121} respects the addition and scalar multiplication in {\u2102}^{m}. Since our definitions of linear independence, spans, bases and dimension are all built up from linear combinations, we will finally be able to translate fundamental properties between abstract vector spaces (U) and concrete vector spaces ({\u2102}^{m}).\n\nTheorem\u00a0CLI\nCoordinatization and Linear Independence\nSuppose that U is a vector space with a basis B of size n. Then S = \\left \\{{u}_{1},\\kern 1.95872pt {u}_{2},\\kern 1.95872pt {u}_{3},\\kern 1.95872pt \\mathop{\\mathop{\u2026}},\\kern 1.95872pt {u}_{k}\\right \\} is a linearly independent subset of U if and only if R = \\left \\{{\u03c1}_{B}\\left ({u}_{1}\\right ),\\kern 1.95872pt {\u03c1}_{B}\\left ({u}_{2}\\right ),\\kern 1.95872pt {\u03c1}_{B}\\left ({u}_{3}\\right ),\\kern 1.95872pt \\mathop{\\mathop{\u2026}},\\kern 1.95872pt {\u03c1}_{B}\\left ({u}_{k}\\right )\\right \\} is a linearly independent subset of {\u2102}^{n}.\n\nProof\u00a0\u00a0 The linear transformation {\u03c1}_{B} is an isomorphism between U and {\u2102}^{n} (Theorem\u00a0VRILT). As an invertible linear transformation, {\u03c1}_{B} is an injective linear transformation (Theorem\u00a0ILTIS), and {\u03c1}_{B}^{\u22121} is also an injective linear transformation (Theorem\u00a0IILT, Theorem\u00a0ILTIS).\n\n() Since {\u03c1}_{B} is an injective linear transformation and S is linearly independent, Theorem\u00a0ILTLI says that R is linearly independent.\n\n() If we apply {\u03c1}_{B}^{\u22121} to each element of R, we will create the set S. Since we are assuming R is linearly independent and {\u03c1}_{B}^{\u22121} is injective, Theorem\u00a0ILTLI says that S is linearly independent.\n\nTheorem\u00a0CSS\nCoordinatization and Spanning Sets\nSuppose that U is a vector space with a basis B of size n. Then u \u2208\\left \\langle \\left \\{{u}_{1},\\kern 1.95872pt {u}_{2},\\kern 1.95872pt {u}_{3},\\kern 1.95872pt \\mathop{\\mathop{\u2026}},\\kern 1.95872pt {u}_{k}\\right \\}\\right \\rangle if and only if {\u03c1}_{B}\\left (u\\right ) \u2208\\left \\langle \\left \\{{\u03c1}_{B}\\left ({u}_{1}\\right ),\\kern 1.95872pt {\u03c1}_{B}\\left ({u}_{2}\\right ),\\kern 1.95872pt {\u03c1}_{B}\\left ({u}_{3}\\right ),\\kern 1.95872pt \\mathop{\\mathop{\u2026}},\\kern 1.95872pt {\u03c1}_{B}\\left ({u}_{k}\\right )\\right \\}\\right \\rangle .\n\nProof\u00a0\u00a0 () Suppose u \u2208\\left \\langle \\left \\{{u}_{1},\\kern 1.95872pt {u}_{2},\\kern 1.95872pt {u}_{3},\\kern 1.95872pt \\mathop{\\mathop{\u2026}},\\kern 1.95872pt {u}_{k}\\right \\}\\right \\rangle . Then there are scalars, {a}_{1},\\kern 1.95872pt {a}_{2},\\kern 1.95872pt {a}_{3},\\kern 1.95872pt \\mathop{\\mathop{\u2026}},\\kern 1.95872pt {a}_{k}, such that\n\n u = {a}_{1}{u}_{1} + {a}_{2}{u}_{2} + {a}_{3}{u}_{3} + \\mathrel{\u22ef} + {a}_{k}{u}_{k}\n\nThen,\n\n\\eqalignno{ {\u03c1}_{B}\\left (u\\right ) & = {\u03c1}_{B}\\left ({a}_{1}{u}_{1} + {a}_{2}{u}_{2} + {a}_{3}{u}_{3} + \\mathrel{\u22ef} + {a}_{k}{u}_{k}\\right ) & & & & \\cr & = {a}_{1}{\u03c1}_{B}\\left ({u}_{1}\\right ) + {a}_{2}{\u03c1}_{B}\\left ({u}_{2}\\right ) + {a}_{3}{\u03c1}_{B}\\left ({u}_{3}\\right ) + \\mathrel{\u22ef} + {a}_{k}{\u03c1}_{B}\\left ({u}_{k}\\right ) & &\\text{@(a href=\"fcla-jsmath-2.22li51.html#theorem.LTLC\")Theorem LTLC@(\/a)} & & & & }\n\nwhich says that {\u03c1}_{B}\\left (u\\right ) \u2208\\left \\langle \\left \\{{\u03c1}_{B}\\left ({u}_{1}\\right ),\\kern 1.95872pt {\u03c1}_{B}\\left ({u}_{2}\\right ),\\kern 1.95872pt {\u03c1}_{B}\\left ({u}_{3}\\right ),\\kern 1.95872pt \\mathop{\\mathop{\u2026}},\\kern 1.95872pt {\u03c1}_{B}\\left ({u}_{k}\\right )\\right \\}\\right \\rangle .\n\n() Suppose that {\u03c1}_{B}\\left (u\\right ) \u2208\\left \\langle \\left \\{{\u03c1}_{B}\\left ({u}_{1}\\right ),\\kern 1.95872pt {\u03c1}_{B}\\left ({u}_{2}\\right ),\\kern 1.95872pt {\u03c1}_{B}\\left ({u}_{3}\\right ),\\kern 1.95872pt \\mathop{\\mathop{\u2026}},\\kern 1.95872pt {\u03c1}_{B}\\left ({u}_{k}\\right )\\right \\}\\right \\rangle . Then there are scalars {b}_{1},\\kern 1.95872pt {b}_{2},\\kern 1.95872pt {b}_{3},\\kern 1.95872pt \\mathop{\\mathop{\u2026}},\\kern 1.95872pt {b}_{k} such that\n\n {\u03c1}_{B}\\left (u\\right ) = {b}_{1}{\u03c1}_{B}\\left ({u}_{1}\\right ) + {b}_{2}{\u03c1}_{B}\\left ({u}_{2}\\right ) + {b}_{3}{\u03c1}_{B}\\left ({u}_{3}\\right ) + \\mathrel{\u22ef} + {b}_{k}{\u03c1}_{B}\\left ({u}_{k}\\right )\n\nRecall that {\u03c1}_{B} is invertible (Theorem\u00a0VRILT), so\n\n\\eqalignno{ u & = {I}_{U}\\left (u\\right ) & &\\text{@(a href=\"fcla-jsmath-2.22li54.html#definition.IDLT\")Definition IDLT@(\/a)} & & & & \\cr & = \\left ({\u03c1}_{B}^{\u22121} \u2218 {\u03c1}_{ B}\\right )\\left (u\\right ) & &\\text{@(a href=\"fcla-jsmath-2.22li54.html#definition.IVLT\")Definition IVLT@(\/a)} & & & & \\cr & = {\u03c1}_{B}^{\u22121}\\left ({\u03c1}_{ B}\\left (u\\right )\\right ) & &\\text{@(a href=\"fcla-jsmath-2.22li51.html#definition.LTC\")Definition LTC@(\/a)} & & & & \\cr & = {\u03c1}_{B}^{\u22121}\\left ({b}_{ 1}{\u03c1}_{B}\\left ({u}_{1}\\right ) + {b}_{2}{\u03c1}_{B}\\left ({u}_{2}\\right ) + {b}_{3}{\u03c1}_{B}\\left ({u}_{3}\\right ) + \\mathrel{\u22ef} + {b}_{k}{\u03c1}_{B}\\left ({u}_{k}\\right )\\right ) & & & & \\cr & = {b}_{1}{\u03c1}_{B}^{\u22121}\\left ({\u03c1}_{ B}\\left ({u}_{1}\\right )\\right ) + {b}_{2}{\u03c1}_{B}^{\u22121}\\left ({\u03c1}_{ B}\\left ({u}_{2}\\right )\\right ) + {b}_{3}{\u03c1}_{B}^{\u22121}\\left ({\u03c1}_{ B}\\left ({u}_{3}\\right )\\right ) & & & & \\cr &\\quad \\quad + \\mathrel{\u22ef} + {b}_{k}{\u03c1}_{B}^{\u22121}\\left ({\u03c1}_{ B}\\left ({u}_{k}\\right )\\right ) & &\\text{@(a href=\"fcla-jsmath-2.22li51.html#theorem.LTLC\")Theorem LTLC@(\/a)} & & & & \\cr & = {b}_{1}{I}_{U}\\left ({u}_{1}\\right ) + {b}_{2}{I}_{U}\\left ({u}_{2}\\right ) + {b}_{3}{I}_{U}\\left ({u}_{3}\\right ) + \\mathrel{\u22ef} + {b}_{k}{I}_{U}\\left ({u}_{k}\\right ) & &\\text{@(a href=\"fcla-jsmath-2.22li54.html#definition.IVLT\")Definition IVLT@(\/a)} & & & & \\cr & = {b}_{1}{u}_{1} + {b}_{2}{u}_{2} + {b}_{3}{u}_{3} + \\mathrel{\u22ef} + {b}_{k}{u}_{k} & &\\text{@(a href=\"fcla-jsmath-2.22li54.html#definition.IDLT\")Definition IDLT@(\/a)} & & & & }\n\nwhich says that u \u2208\\left \\langle \\left \\{{u}_{1},\\kern 1.95872pt {u}_{2},\\kern 1.95872pt {u}_{3},\\kern 1.95872pt \\mathop{\\mathop{\u2026}},\\kern 1.95872pt {u}_{k}\\right \\}\\right \\rangle .\n\nHere\u2019s a fairly simple example that illustrates a very, very important idea.\n\nExample\u00a0CP2\nCoordinatizing in {P}_{2}\nIn Example\u00a0VRP2 we needed to know that\n\n D = \\left \\{\u22122 \u2212 x + 3{x}^{2},\\kern 1.95872pt 1 \u2212 2{x}^{2},\\kern 1.95872pt 5 + 4x + {x}^{2}\\right \\}\n\nis a basis for {P}_{2}. With Theorem\u00a0CLI and Theorem\u00a0CSS this task is much easier. First, choose a known basis for {P}_{2}, a basis that forms vector representations easily. We will choose\n\n B = \\left \\{1,\\kern 1.95872pt x,\\kern 1.95872pt {x}^{2}\\right \\}\n\nNow, form the subset of {\u2102}^{3} that is the result of applying {\u03c1}_{B} to each element of D,\n\n F = \\left \\{{\u03c1}_{B}\\left (\u22122 \u2212 x + 3{x}^{2}\\right ),\\kern 1.95872pt {\u03c1}_{ B}\\left (1 \u2212 2{x}^{2}\\right ),\\kern 1.95872pt {\u03c1}_{ B}\\left (5 + 4x + {x}^{2}\\right )\\right \\} = \\left \\{\\left [\\array{ \u22122\\cr \u22121 \\cr 3 } \\right ],\\kern 1.95872pt \\left [\\array{ 1\\cr 0 \\cr \u22122 } \\right ],\\kern 1.95872pt \\left [\\array{ 5\\cr 4 \\cr 1 } \\right ]\\right \\}\n\nand ask if F is a linearly independent spanning set for {\u2102}^{3}. This is easily seen to be the case by forming a matrix A whose columns are the vectors of F, row-reducing A to the identity matrix {I}_{3}, and then using the nonsingularity of A to assert that F is a basis for {\u2102}^{3} (Theorem\u00a0CNMB). Now, since F is a basis for {\u2102}^{3}, Theorem\u00a0CLI and Theorem\u00a0CSS tell us that D is also a basis for {P}_{2}.\n\nExample\u00a0CP2 illustrates the broad notion that computations in abstract vector spaces can be reduced to computations in {\u2102}^{m}. You may have noticed this phenomenon as you worked through examples in Chapter\u00a0VS or Chapter\u00a0LT employing vector spaces of matrices or polynomials. These computations seemed to invariably result in systems of equations or the like from Chapter\u00a0SLE, Chapter\u00a0V and Chapter\u00a0M. It is vector representation, {\u03c1}_{B}, that allows us to make this connection formal and precise.\n\nKnowing that vector representation allows us to translate questions about linear combinations, linear independence and spans from general vector spaces to {\u2102}^{m} allows us to prove a great many theorems about how to translate other properties. Rather than prove these theorems, each of the same style as the other, we will offer some general guidance about how to best employ Theorem\u00a0VRLT, Theorem\u00a0CLI and Theorem\u00a0CSS. This comes in the form of a \u201cprinciple\u201d: a basic truth, but most definitely not a theorem (hence, no proof).\n\nThe Coordinatization Principle Suppose that U is a vector space with a basis B of size n. Then any question about U, or its elements, which ultimately depends on the vector addition or scalar multiplication in U, or depends on linear independence or spanning, may be translated into the same question in {\u2102}^{n} by application of the linear transformation {\u03c1}_{B} to the relevant vectors. Once the question is answered in {\u2102}^{n}, the answer may be translated back to U (if necessary) through application of the inverse linear transformation {\u03c1}_{B}^{\u22121}.\n\nExample\u00a0CM32\nCoordinatization in {M}_{32}\nThis is a simple example of the Coordinatization Principle, depending only on the fact that coordinatizing is an invertible linear transformation (Theorem\u00a0VRILT). Suppose we have a linear combination to perform in {M}_{32}, the vector space of 3 \u00d7 2 matrices, but we are adverse to doing the operations of {M}_{32} (Definition\u00a0MA, Definition\u00a0MSM). More specifically, suppose we are faced with the computation\n\n 6\\left [\\array{ 3 & 7\\cr \u22122 & 4 \\cr 0 &\u22123 } \\right ]+2\\left [\\array{ \u22121&3\\cr 4 &8 \\cr \u22122&5 } \\right ]\n\nWe choose a nice basis for {M}_{32} (or a nasty basis if we are so inclined),\n\n B = \\left \\{\\left [\\array{ 1&0\\cr 0&0 \\cr 0&0 } \\right ],\\kern 1.95872pt \\left [\\array{ 0&0\\cr 1&0 \\cr 0&0 } \\right ],\\kern 1.95872pt \\left [\\array{ 0&0\\cr 0&0 \\cr 1&0 } \\right ],\\kern 1.95872pt \\left [\\array{ 0&1\\cr 0&0 \\cr 0&0 } \\right ],\\kern 1.95872pt \\left [\\array{ 0&0\\cr 0&1 \\cr 0&0 } \\right ],\\kern 1.95872pt \\left [\\array{ 0&0\\cr 0&0 \\cr 0&1 } \\right ]\\right \\}\n\nand apply {\u03c1}_{B} to each vector in the linear combination. This gives us a new computation, now in the vector space {\u2102}^{6},\n\n 6\\left [\\array{ 3\\cr \u22122 \\cr 0\\cr 7 \\cr 4\\cr \u22123 } \\right ]+2\\left [\\array{ \u22121\\cr 4 \\cr \u22122\\cr 3 \\cr 8\\cr 5 } \\right ]\n\nwhich we can compute with the operations of {\u2102}^{6} (Definition\u00a0CVA, Definition\u00a0CVSM), to arrive at\n\n \\left [\\array{ 16\\cr \u22124 \\cr \u22124\\cr 48 \\cr 40\\cr \u22128 } \\right ]\n\nWe are after the result of a computation in {M}_{32}, so we now can apply {\u03c1}_{B}^{\u22121} to obtain a 3 \u00d7 2 matrix,\n\n 16\\left [\\array{ 1&0\\cr 0&0 \\cr 0&0 } \\right ]+(\u22124)\\left [\\array{ 0&0\\cr 1&0 \\cr 0&0 } \\right ]+(\u22124)\\left [\\array{ 0&0\\cr 0&0 \\cr 1&0 } \\right ]+48\\left [\\array{ 0&1\\cr 0&0 \\cr 0&0 } \\right ]+40\\left [\\array{ 0&0\\cr 0&1 \\cr 0&0 } \\right ]+(\u22128)\\left [\\array{ 0&0\\cr 0&0 \\cr 0&1 } \\right ] = \\left [\\array{ 16&48\\cr \u22124 & 40 \\cr \u22124&\u22128 } \\right ]\n\nwhich is exactly the matrix we would have computed had we just performed the matrix operations in the first place. So this was not meant to be an easier way to compute a linear combination of two matrices, just a different way.\n\n1. The vector space of 3 \u00d7 5 matrices, {M}_{3,5} is isomorphic to what fundamental vector space?\n2. A basis for {\u2102}^{3} is\n B = \\left \\{\\left [\\array{ 1\\cr 2 \\cr \u22121} \\right ],\\kern 1.95872pt \\left [\\array{ 3\\cr \u22121 \\cr 2} \\right ],\\kern 1.95872pt \\left [\\array{ 1\\cr 1 \\cr 1} \\right ]\\right \\}\n\nCompute {\u03c1}_{B}\\left (\\left [\\array{ 5\\cr 8 \\cr \u22121} \\right ]\\right ).\n\n3. What is the first \u201csurprise,\u201d and why is it surprising?\n\n#### Subsection EXC: Exercises\n\nC10 In the vector space {\u2102}^{3}, compute the vector representation {\u03c1}_{B}\\left (v\\right ) for the basis B and vector v below.\n\n\\eqalignno{ B & = \\left \\{\\left [\\array{ 2\\cr \u22122 \\cr 2 } \\right ],\\kern 1.95872pt \\left [\\array{ 1\\cr 3 \\cr 1 } \\right ],\\kern 1.95872pt \\left [\\array{ 3\\cr 5 \\cr 2 } \\right ]\\right \\} &v & = \\left [\\array{ 11\\cr 5 \\cr 8 } \\right ] & & & & }\n\nContributed\u00a0by\u00a0Robert\u00a0Beezer Solution\u00a0[1664]\n\nC20 Rework Example\u00a0CM32 replacing the basis B by the basis\n\n C = \\left \\{\\left [\\array{ \u221214&\u22129\\cr 10 & 10 \\cr \u22126 &\u22122 } \\right ],\\kern 1.95872pt \\left [\\array{ \u22127&\u22124\\cr 5 & 5 \\cr \u22123&\u22121 } \\right ],\\kern 1.95872pt \\left [\\array{ \u22123&\u22121\\cr 0 &\u22122 \\cr 1 & 1 } \\right ],\\kern 1.95872pt \\left [\\array{ \u22127&\u22124\\cr 3 & 2 \\cr \u22121& 0 } \\right ],\\kern 1.95872pt \\left [\\array{ 4 & 2\\cr \u22123 &\u22123 \\cr 2 & 1 } \\right ],\\kern 1.95872pt \\left [\\array{ 0 & 0\\cr \u22121 &\u22122 \\cr 1 & 1 } \\right ]\\right \\}\n\nContributed\u00a0by\u00a0Robert\u00a0Beezer Solution\u00a0[1665]\n\nM10 Prove that the set S below is a basis for the vector space of 2 \u00d7 2 matrices, {M}_{22}. Do this choosing a natural basis for {M}_{22} and coordinatizing the elements of S with respect to this basis. Examine the resulting set of column vectors from {\u2102}^{4} and apply the Coordinatization Principle.\n\n S = \\left \\{\\left [\\array{ 33&99\\cr 78 &\u22129 } \\right ],\\kern 1.95872pt \\left [\\array{ \u221216&\u221247\\cr \u221236 & 2 } \\right ],\\kern 1.95872pt \\left [\\array{ 10&27\\cr 17 & 3 } \\right ],\\kern 1.95872pt \\left [\\array{ \u22122&\u22127\\cr \u22126 & 4 } \\right ]\\right \\}\n\nContributed\u00a0by\u00a0Andy\u00a0Zimmer\n\n#### Subsection SOL: Solutions\n\nC10 Contributed\u00a0by\u00a0Robert\u00a0Beezer Statement\u00a0[1662]\nWe need to express the vector v as a linear combination of the vectors in B. Theorem\u00a0VRRB tells us we will be able to do this, and do it uniquely. The vector equation\n\n { a}_{1}\\left [\\array{ 2\\cr \u22122 \\cr 2 } \\right ]+{a}_{2}\\left [\\array{ 1\\cr 3 \\cr 1 } \\right ]+{a}_{3}\\left [\\array{ 3\\cr 5 \\cr 2 } \\right ] = \\left [\\array{ 11\\cr 5 \\cr 8 } \\right ]\n\nbecomes (via Theorem\u00a0SLSLC) a system of linear equations with augmented matrix,\n\n \\left [\\array{ 2 &1&3&11\\cr \u22122 &3 &5 & 5 \\cr 2 &1&2& 8 } \\right ]\n\nThis system has the unique solution {a}_{1} = 2, {a}_{2} = \u22122, {a}_{3} = 3. So by Definition\u00a0VR,\n\n { \u03c1}_{B}\\left (v\\right ) = {\u03c1}_{B}\\left (\\left [\\array{ 11\\cr 5 \\cr 8 } \\right ]\\right ) = {\u03c1}_{B}\\left (2\\left [\\array{ 2\\cr \u22122 \\cr 2 } \\right ] + (\u22122)\\left [\\array{ 1\\cr 3 \\cr 1 } \\right ] + 3\\left [\\array{ 3\\cr 5 \\cr 2 } \\right ]\\right ) = \\left [\\array{ 2\\cr \u22122 \\cr 3 } \\right ]\n\nC20 Contributed\u00a0by\u00a0Robert\u00a0Beezer Statement\u00a0[1662]\nThe following computations replicate the computations given in Example\u00a0CM32, only using the basis C.\n\n\\eqalignno{ {\u03c1}_{C}\\left (\\left [\\array{ 3 & 7\\cr \u22122 & 4 \\cr 0 &\u22123 } \\right ]\\right ) & = \\left [\\array{ \u22129\\cr 12 \\cr \u22126\\cr 7 \\cr \u22122\\cr \u22121 } \\right ] &{\u03c1}_{C}\\left (\\left [\\array{ \u22121&3\\cr 4 &8 \\cr \u22122&5 } \\right ]\\right ) & = \\left [\\array{ \u221211\\cr 34 \\cr \u22124\\cr \u22121 \\cr 16\\cr 5} \\right ] & & & & \\cr 6\\left [\\array{ \u22129\\cr 12 \\cr \u22126\\cr 7 \\cr \u22122\\cr \u22121 } \\right ] + 2\\left [\\array{ \u221211\\cr 34 \\cr \u22124\\cr \u22121 \\cr 16\\cr 5} \\right ] & = \\left [\\array{ \u221276\\cr 140 \\cr \u221244\\cr 40 \\cr 20\\cr 4} \\right ] &{\u03c1}_{C}^{\u22121}\\left (\\left [\\array{ \u221276\\cr 140 \\cr \u221244\\cr 40 \\cr 20\\cr 4} \\right ]\\right ) & = \\left [\\array{ 16&48\\cr \u22124 & 30 \\cr \u22124&\u22128 } \\right ] & & & & }","date":"2017-11-25 00:15:53","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\": 0.990393340587616, \"perplexity\": 5314.340873987327}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-47\/segments\/1510934809160.77\/warc\/CC-MAIN-20171124234011-20171125014011-00040.warc.gz\"}"} | null | null |
Prostitute () also known as Slain by Life () is a 1927 Soviet silent drama film directed by Oleg Frelikh.
Plot
The film is set in Moscow during the mid 1920s, heyday of the New Economic Policy, or NEP. Some live the high life while others barely survive. A young girl, Lyuba lives with her elderly Aunt Barbara. The aunt abuses the girl, and later, "sells" her to a neighbor and kicks her out of the house. But Lyuba does not stay in the street for long, she is sheltered by a woman she meets, who turns out to be a brothel madam. The madam also imposes a contract of adhesion upon the girl.
The Tyrkin family lives next to Aunt Barbara. Pyotr Tyrkin works for the businessman-butcher Kondratiev. Tyrkin's everyday life is well-adjusted. His wife Vera keeps house and raises their two young children. Working for Kondratiev brings a regular income. Tyrkin is killed when drunk. Left without a livelihood, Vera is forced by the situation to give herself to the butcher (the boss of her deceased husband), and then to sell her body.
On the street she meets with the veteran prostitute Manka. Manka tells Vera that when she worked as a maid, she was seduced by the son of the mistress. After getting kicked out of the house by the mistress for having relations with her son, she became homeless. On the street she came to work at a whorehouse and contracted a venereal disease from which she is still recovering.
Vera is unable to earn money by prostitution. Both of her children fall seriously ill. In desperation she tries to commit suicide by throwing herself into an ice-hole. But she does not succeed and is rescued. Among the saviors is Lyuba who managed to escape from the brothel and now works in a sewing workshop at a venereal dispensary. She has a new boyfriend is Shura who is a member of the Komsomol. The brothel keeper does not want to just let Lyuba go. She threatens to tell Shura all about her past. So, the teary-eyed girl tells Shura everything herself. Shura sympathizes with her and helps to write a letter to the prosecutor. The police break up the den.
Lyuba and Shura are happy. Life is getting better for Vera too because Shura helps her get a job as a railway points operator and her children begin to go the kindergarten. Manka is housed in a venereal hospital.
Cast
Olga Bonus as Aunt Varvara (as O. Bonus)
Mark Donskoy
L. Krasina
Ivan Lagutin as Vasiliy Dmitrich (as I. Lagutin)
Aleksandr Ledashchev
Vera Orlova
E. Sheremetyeva
Pavel Tamm as Kondratiev - the butcher (as P. Tamm)
E. Toeplitz
Elisaveta Yarosh Elisaveta Yarosh ... Lyuba
Vasili Yaroslavtsev as Peter Stupin (as V. Yaroslavtsev)
Interesting facts
"Prostitute" shares the glory of being the first Belarusian feature film together with "Forest Story". Work on the film "Prostitute" started earlier and it was also released first in the all-union movie theaters, but "Forest Story" was shown in Minsk first.
Despite the film's success with the audience, the critics of that time did not receive the film too positively, and some saw in it the influence of the "bourgeois pseudo-scientific German films".
A circulaire dated 1937 from "Belgoskino" reported to Boris Shumyatsky in Moscow, the head of the main department of the film industry of the USSR about neutralizing "enemy" films. The ban for the film "Prostitute" was motivated by the fact that the film was seen as being "politically incorrect".
A significant episode (by the length of the film) is dedicated to the lecture "about the dangers of prostitution and social ways of getting rid of it." The lecture uses the statistics available at the time and simple animation.
The intertitles are in three languages, Russian, English and Chinese.
References
Literature
Y. S. Kalashnikov Essays on the history of Soviet Cinema: 1917—1934, — Moscow: Iskusstvo, 1956.
Igor Avdeev, Larisa Zaitseva All Belarus Films: Catalog-Handbook. Feature Films (1926—1970). — Minsk: Belaruskaya navuka, 2001. — Volume 1. — 240 pages. —
External links
mubi.com/films/prostitutka
Soviet black-and-white films
1927 drama films
1927 films
Soviet drama films
Belarusfilm films
Belarusian drama films
Soviet-era Belarusian films
Soviet silent feature films
Films about prostitution in Russia
Silent drama films | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,045 |
Marlon Gaillard (Chinon, 9 mei 1996) is een Frans voormalig wielrenner.
Carrière
Gaillard liep stage bij TotalEnergie in 2016 en 2018. In 2019 won hij de eerste etappe in de Ronde de l'Oise. In zowel 2020 als 2021 reed hij als prof voor de ploeg. Nadat hij geen nieuw contract kreeg in 2021 besloot hij te stoppen als prof.
Palmares
2018
Jongerenklassement Ronde van Marokko
2019
1e etappe Ronde de l'Oise
Resultaten in voornaamste wedstrijden
Ploegen
2016 – ↑Direct Énergie (stagiair per 1-8)
2018 – ↑Direct Énergie (stagiair per 1-8)
2020 – Total Direct Energie
2021 – Total Direct Energie
Frans wielrenner | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,521 |
{"url":"http:\/\/mathhelpforum.com\/pre-calculus\/56350-right-triangle-print.html","text":"# Right Triangle\n\nPrintable View\n\n\u2022 Oct 29th 2008, 09:00 AM\nmagentarita\nRight Triangle\nA right triangle has one vertex on the graph of\ny = 9 - x^2, x > 0, at (x, y), another at the origin, and the thiurd on the positive x-axis at (x, 0). Express the area A of the triangle as a function of x.\n\u2022 Oct 29th 2008, 09:17 AM\nHenderson\nThe base of your triangle is your x-coordinate, and the height of the triangle is your y-coordinate. For a triangle, the area is given by\n\n$A = \\frac{1}{2}(base)(height)$, so\n\n$A = \\frac{1}{2}(x)(y)$\n\n$A = \\frac{1}{2}(x)(9-x^2)$\n\n$\nA = \\frac{9x - x^3}{2}\n$\n\u2022 Oct 29th 2008, 09:54 AM\nSoroban\nHello, magentarita!\n\nHenderson is absolutely correct!\nDid you make a sketch? . . . It's very simple.\n\nQuote:\n\nA right triangle has one vertex on the graph of $y \\:=\\: 9 - x^2,\\;\\;x > 0,\\text{ at }(x, y),$\nanother at the origin, and the third on the positive x-axis at (x, 0).\nExpress the area $A$ of the triangle as a function of $x.$\n\nCode:\n\n\u00a0 \u00a0 \u00a0 \u00a0 | \u00a0 \u00a0 \u00a0 9 ** \u00a0 \u00a0 \u00a0 \u00a0 |\u00a0 \u00a0 *\u00a0 P \u00a0 \u00a0 \u00a0 \u00a0 |\u00a0 \u00a0 \u00a0 *(x,y) \u00a0 \u00a0 \u00a0 \u00a0 |\u00a0 \u00a0 *:| * \u00a0 \u00a0 \u00a0 \u00a0 |\u00a0 *:::|\u00a0 * \u00a0 \u00a0 \u00a0 \u00a0 | *:::::| \u00a0 \u00a0 \u00a0 - + - - - + - * - - \u00a0 \u00a0 \u00a0 \u00a0 O\u00a0 \u00a0 \u00a0 Q\u00a0 3\n\nThe area of a triangle is: . $A \\;=\\;\\tfrac{1}{2}\\text{(base)(height)}$\n\nThe base is $x.$\nThe height is $y$, where $y \\:=\\:9-x^2$\n\nTherefore: . $A \\;=\\;\\tfrac{1}{2}x(9-x^2)$\n\n\u2022 Oct 29th 2008, 07:13 PM\nmagentarita\ngreat work....\nQuote:\n\nOriginally Posted by Henderson\nThe base of your triangle is your x-coordinate, and the height of the triangle is your y-coordinate. For a triangle, the area is given by\n\n$A = \\frac{1}{2}(base)(height)$, so\n\n$A = \\frac{1}{2}(x)(y)$\n\n$A = \\frac{1}{2}(x)(9-x^2)$\n\n$\nA = \\frac{9x - x^3}{2}\n$\n\nI thank you very much.\n\u2022 Oct 29th 2008, 07:14 PM\nmagentarita\nSoroban....\nQuote:\n\nOriginally Posted by Soroban\nHello, magentarita!\n\nHenderson is absolutely correct!\nDid you make a sketch? . . . It's very simple.\n\nCode:\n\n\u00a0 \u00a0 \u00a0 \u00a0 | \u00a0 \u00a0 \u00a0 9 ** \u00a0 \u00a0 \u00a0 \u00a0 |\u00a0 \u00a0 *\u00a0 P \u00a0 \u00a0 \u00a0 \u00a0 |\u00a0 \u00a0 \u00a0 *(x,y) \u00a0 \u00a0 \u00a0 \u00a0 |\u00a0 \u00a0 *:| * \u00a0 \u00a0 \u00a0 \u00a0 |\u00a0 *:::|\u00a0 * \u00a0 \u00a0 \u00a0 \u00a0 | *:::::| \u00a0 \u00a0 \u00a0 - + - - - + - * - - \u00a0 \u00a0 \u00a0 \u00a0 O\u00a0 \u00a0 \u00a0 Q\u00a0 3\nThe area of a triangle is: . $A \\;=\\;\\tfrac{1}{2}\\text{(base)(height)}$\n\nThe base is $x.$\nThe height is $y$, where $y \\:=\\:9-x^2$\n\nTherefore: . $A \\;=\\;\\tfrac{1}{2}x(9-x^2)$\n\nIt's always great to receive an answer from you.","date":"2016-12-10 21:50:35","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\": 21, \"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\": 0.8633193969726562, \"perplexity\": 6787.4271763555125}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-50\/segments\/1480698543567.64\/warc\/CC-MAIN-20161202170903-00218-ip-10-31-129-80.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
Magnetic systems with geometric frustration have been studied
experimentally and theoretically for decades \cite{Lacroix11}.
In particular, systems on networks of triangles or tetrahedra,
such as triangular \cite{Wannier50}, kagom\'{e} \cite{Shyozi51,Qi2008},
and pyrochlore \cite{Gardner10} lattices, show interesting behavior
due to the frustration.
Among them classical spin ice on the pyrochlore lattice \cite{Bramwell01}
has been investigated in depth
from viewpoints of the finite zero-point entropy of water ice \cite{Ramirez99},
field-induced two-dimensional (2D) kagom\'{e} ice \cite{MatsuhiraJPCM2002,Tabata06,Fennell2007},
emergent magnetic monopoles \cite{Castelnovo08,Kadowaki09},
topological sectors \cite{Jaubert2013}, etc.
In recent years quantum spin liquid (QSL) states \cite{Lee08,Balents10},
where conventional long-range orders (LRO) are suppressed
by quantum fluctuations, are being intensively studied \cite{Savary2017}.
A QSL state is theoretically predicted
for spin-ice like systems \cite{Hermele04,Benton12,Lee12,Gingras14},
where transverse spin interactions transform
the classical spin ice into QSL.
Among frustrated magnetic pyrochlore oxides \cite{Gardner10}
Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ (TTO) has attracted much attention
as a QSL candidate, because
no conventional magnetic orders have been found \cite{Gardner99,Kadowaki2018},
and a quantum version of spin ice was theoretically proposed \cite{Molavian07,Gingras14}.
Recently we showed that the putative QSL state of TTO
is limited in a range of the small off-stoichiometry parameter
$x < x_{\text{c}} \simeq -0.0025$ \cite{Taniguchi13,Wakita16,Kadowaki2018}.
While in the other range $x_{\text{c}} < x$ TTO undergoes a phase transition
most likely to an electric multipolar (or quadrupolar) state
($T<T_{\text{c}}$) \cite{Takatsu2016prl}
which is described by an effective pseudospin-$\frac{1}{2}$
Hamiltonian for non-Kramers ions \cite{Onoda11}.
The estimated parameter set of this Hamiltonian \cite{Takatsu2016prl}
is close to the theoretical phase boundary
between the electric quadrupolar state and a U(1) QSL state \cite{Onoda11,Lee12},
which is hence a theoretical QSL candidate for TTO.
In our previous investigations
using a TTO crystal sample with $T_{\text c}=0.53$ K \cite{Takatsu2016prl,Takatsu16,Takatsu2017},
specific heat and magnetization
under [111] and [100] magnetic fields
were measured and
finite-temperature phase-transitions
were semi-quantitatively analyzed using
classical Monte-Carlo (CMC) simulation techniques.
Despite the quantum nature of the pseudospin-$\frac{1}{2}$ Hamiltonian \cite{Onoda11,Lee12},
the classical treatment provided us good arguments
that TTO can be described by the Hamiltonian \cite{Takatsu2016prl}.
Although quantum (e.g. \cite{Bojesen2017}) and classical (e.g. \cite{Zhitomirsky14})
properties of these types of pseudospin-$\frac{1}{2}$ Hamiltonians
for non-Kramers and Kramers pyrochlore magnets are of interest,
they have not been fully investigated \cite{Rau_Gingras2018}.
In this paper we present detailed studies of CMC simulations
to complement our previous study of the quadrupole orders in TTO \cite{Takatsu2016prl}.
In particular, order parameters and finite-temperature phase transitions
of the quadrupolar states were remained to be elucidated
from a theoretical standpoint \cite{Takatsu2016prl}.
We have shown that under zero and low [111] fields
the quadrupole ordered states have three dimensional (3D) and 2D
characters, respectively.
Nature of these phase transitions in zero and low fields
is shown to be first order and second order
with the 2D Ising universality class, respectively.
Implication of the CMC simulation results is discussed
with respect to experimental data of TTO.
\section{Effective pseudospin-$\frac{1}{2}$ Hamiltonian and CMC simulation}
The minimal pseudospin-$\frac{1}{2}$ Hamiltonian
for TTO \cite{Takatsu2016prl,Kadowaki15} is described by
\begin{align}
\mathcal{H} =
J_{\text{nn,eff}} \sum_{\langle {\bm r} , {\bm r}^{\prime} \rangle}
& \sigma_{\bm{r}}^{z} \sigma_{\bm{r}^{\prime}}^{z}
- J_{\text{nn,eff}} \bm{H} \cdot \sum_{\bm r} \bm{z}_{\bm{r}} \sigma_{\bm{r}}^{z} \nonumber \\
+ J_{\text{nn,eff}} \sum_{\langle {\bm r} , {\bm r}^{\prime} \rangle}
&[2 \delta ( \sigma_{\bm{r}}^+ \sigma_{\bm{r}^{\prime}}^-
+ \sigma_{\bm{r}}^- \sigma_{\bm{r}^{\prime}}^+ ) \nonumber \\
&+ 2 q ( e^{2 i \phi_{\bm{r},\bm{r}^{\prime}} }
\sigma_{\bm{r}}^+ \sigma_{\bm{r}^{\prime}}^+ + \text{H.c.} ) ] \; ,
\label{H_effective}
\end{align}
where the first and second terms are magnetic interactions:
nearest-neighbor (NN) superexchange interaction of magnetic moment operators
$\sigma_{\bm{r}}^{z}$ (the Pauli matrix) acting on the crystal field (CF)
ground state doublet at a site $\bm{r}$,
and Zeeman energy under dimensionless external magnetic field $\bm{H}$.
These magnetic terms has been used as the model of spin ice
with the effective coupling constant $J_{\text{nn,eff}}$ ($>0$) \cite{Hertog00}.
The third term of Eq.~(\ref{H_effective})
represents NN superexchange interaction of quadrupole moment operators
$\sigma_{\bm{r}}^{\pm}= ( \sigma_{\bm{r}}^x \pm i \sigma_{\bm{r}}^y )/2$ \cite{Onoda11}.
This term induces quantum fluctuations to the classical spin ice for
the non-zero dimensionless parameters $\delta$ and $q$.
Other detailed definitions of Eq.~(\ref{H_effective}),
the lattice site, its local axes etc. \cite{Kadowaki15,Takatsu2016prl},
are described in the appendix.
In Eq.~(\ref{H_effective}) we omit the dipolar interaction included in Eq.~(1) of
Ref.~\cite{Takatsu2016prl} in order to perform CMC simulations with larger system sizes.
In this simplification, the typical parameters of the Hamiltonian for TTO are
$J_{\text{nn,eff}}=1.48$ K, $\delta=0$, and $q = 0.57$ \cite{ParametersJdeltaq}.
In zero field, the classical ground state of Eq.~(\ref{H_effective}) with these parameters
is LRO of $xy$-components of the pseudospins (quadrupole order),
which is denoted by the planar antiferropseudospin (PAF) phase
(Fig.~7 in Ref.~\cite{Onoda11}).
By treating the pseudospin $\bm{\sigma}_{\bm{r}}$ as a classical unit vector \cite{LandauBinder15},
we carried out CMC simulations of the classical spin model described by Eq.~(\ref{H_effective}).
Since critical behaviors of finite-temperature phase-transitions
are expected to be the same for classical and quantum models \cite{Sachdev11,Zhitomirsky14},
CMC simulations can be used to shed light on experimental data.
For present CMC simulations we used parameter sets
in a range relevant to TTO: $-0.1 \le \delta \le 0.1$
and $0.2 \le q \le 0.7$ \cite{Takatsu2016prl},
which encompasses the PAF and classical spin ice states \cite{Onoda11}.
These simulations were performed typically
with $\sim 4 \times 10^6$ MC steps per spin
and for periodic clusters with $N= 12 L \times L \times L^{\prime} \leq 629856$ spins,
where $L$ and $L^{\prime}$ stand for linear dimensions
perpendicular and parallel to a [111] direction, respectively.
The magnetic field was applied parallel to this [111] direction,
along which there are $3 L^{\prime}$ triangular layers and $3 L^{\prime}$ kagom\'{e}
layers within the periodic boundary (Fig.~\ref{3DPAF_2DPAF}).
We used the Metropolis single spin-flip updates \cite{LandauBinder15}
and the exchange Monte-Carlo method \cite{Hukushima96}.
The CMC simulation software \cite{kadowaki_CMC_OH_2018} is based on
an example of a Heisenberg model
distributed by the ALPS project \cite{Bauer11,Albuquerque2007}.
We note that the parameter set ($\delta, q$) had
the substantial experimental uncertainty in Ref.~\cite{Takatsu2016prl},
which is shown by the elongated region enclosed by the dotted line
in Fig.~1(a) of Ref.~\cite{Takatsu2016prl}.
This uncertainty was concluded,
because CMC simulations with small $\delta \ne 0$ show very similar results
to those with $\delta = 0$ by adjusting the parameter $q$ \cite{Takatsu2016prl}.
\section{Order parameters}
Long range orders of magnetic dipole and electric quadrupole moments
expressed by pseudospin LRO
$(\langle \sigma_{ \bm{r} }^x \rangle , \langle \sigma_{ \bm{r} }^y \rangle , \langle \sigma_{ \bm{r} }^z \rangle )$
were discussed using a classical mean-field analysis
in zero field \cite{Onoda11}.
It was shown that the PAF ordering has
the highest mean-field critical temperature $T_{\text{c}}$ with
degeneracy lines along [111] directions \cite{Onoda11},
more specifically, pseudospin LRO of non-zero $\langle \sigma_{ \bm{r} }^x \rangle$
and $\langle \sigma_{ \bm{r} }^y \rangle$
with modulation wavevectors $\bm{k}= (h, h, h)$ ($|h| \le \tfrac{1}{2}$).
We summarize details of these classical mean-field LROs in the appendix.
In addition, it was suggested \cite{Onoda11} that
orders with the wavevector $\bm{k}=0$ can be selected
from the infinitely degenerate mean-field PAF orders
by an energetic \cite{PalmerChalker2000} or an order-by-disorder mechanism.
The mean-field PAF order \cite{Onoda11} with a wavevector $\bm{k}= (h, h, h)$
is expressed by a pseudospin LRO
\begin{equation}
\langle \bm{\sigma}_{ \bm{t}_n + \bm{d}_i } \rangle \propto \bm{v}_{i}^{\text{2D}} e^{i \bm{k} \cdot (\bm{t}_n + \bm{d}_i) }
\label{PAF}
\end{equation}
with
\begin{equation}
\bm{v}_{i}^{\text{2D}} =
\begin{cases}
\bm{0} & (i=0) \\
\tfrac{\sqrt{3}}{2} \bm{x}_i + \tfrac{1}{2} \bm{y}_i & (i=1) \\
-\tfrac{\sqrt{3}}{2} \bm{x}_i + \tfrac{1}{2} \bm{y}_i & (i=2) \\
- \bm{y}_i & (i=3) \; ,
\end{cases}
\label{2DPAFk}
\end{equation}
where $\bm{x}_i$ and $\bm{y}_i$ stand for local axes
at a crystallographic site $\bm{d}_i$
in the unit cell (Table~\ref{local_axis}),
and $\bm{t}_n$ is an FCC translation vector.
We note that these mean-field PAF orders have the zero amplitude on triangular
lattice layers ($i=0$ sites in Fig.~\ref{3DPAF_2DPAF}),
which implies that the PAF order is essentially 2D LRO
on each kagom\'{e} lattice layer (appendix).
\subsection{Order parameter under zero magnetic field}
\begin{figure}
\begin{center}
\includegraphics[width=7.0cm]{Fig1.pdf}
\end{center}
\caption{
(a) 3D PAF [Eqs.~(\ref{3DPAF}) and (\ref{3DPAF_0})] and
(b) 2D PAF [Eqs.~(\ref{2DPAF}) and (\ref{2DPAFk})]
electric quadrupole orders are
schematically illustrated by deformation of the $f$-electron
change density from that of the paramagnetic phase \cite{Takatsu2016prl,Kadowaki15}.
}
\label{3DPAF_2DPAF}
\end{figure}
The one-fold degeneracy of the mean-field PAF order with a wavevector
$\bm{k}= (h, h, h)$ ($h>0$) is increased to three-fold
in the limit of $h \rightarrow 0$.
These three pseudospin LRO structures
with $\bm{k}=0$ are expressed by (appendix)
\begin{equation}
\langle \bm{\sigma}_{ \bm{t}_n + \bm{d}_i } \rangle \propto \bm{v}_{i}^{(j)} \; ,
\label{3DPAF}
\end{equation}
where $j=0,1,2$ with
\begin{equation}
\bm{v}_{i}^{(0)}=
\begin{cases}
\bm{y}_i & (i=1,2) \\
-\bm{y}_i & (i=0,3) \; ,
\end{cases}
\label{3DPAF_0}
\end{equation}
\begin{equation}
\bm{v}_{i}^{(1)}=
\begin{cases}
\tfrac{\sqrt{3}}{2} \bm{x}_i - \tfrac{1}{2} \bm{y}_i & (i=1,3) \\
-\tfrac{\sqrt{3}}{2} \bm{x}_i + \tfrac{1}{2} \bm{y}_i & (i=0,2) \; ,
\end{cases}
\label{3DPAF_1}
\end{equation}
and
\begin{equation}
\bm{v}_{i}^{(2)}=
\begin{cases}
\tfrac{\sqrt{3}}{2} \bm{x}_i + \tfrac{1}{2} \bm{y}_i & (i=0,1) \\
-\tfrac{\sqrt{3}}{2} \bm{x}_i - \tfrac{1}{2} \bm{y}_i & (i=2,3) \; .
\end{cases}
\label{3DPAF_2}
\end{equation}
Under zero field, these 3D PAF orders can
be stabilized energetically or by an order-by-disorder mechanism \cite{Onoda11},
which will be shown by CMC simulations.
Their order parameters may be decomposed into
\begin{equation}
m^{(j)}=\frac{\sum_{n,i} \bm{\sigma}_{ \bm{t}_n + \bm{d}_i } \cdot \bm{v}_{i}^{(j)} } {\sum_{n,i} 1} \; ,
\label{3DPAF_3op}
\end{equation}
where the summation runs over all sites $\bm{t}_n + \bm{d}_i$.
In the limit of $T \rightarrow 0$,
$(\langle m^{(0)} \rangle,\langle m^{(1)} \rangle,\langle m^{(2)} \rangle)$ becomes $(\pm 1,0,0)$, $(0,\pm 1,0)$, or $(0,0,\pm 1)$.
In CMC simulations we measure the average of
\begin{equation}
m_{\text{3DPAF}} = \sqrt{ [m^{(0)}]^2 + [m^{(1)}]^2 + [m^{(2)}]^2 } \; ,
\label{3DPAF_op}
\end{equation}
which represents the amplitude of the 3D PAF ordering.
In Fig.~\ref{3DPAF_2DPAF}(a) we schematically illustrate
the electric quadrupole order expressed by the pseudospin structure
[Eqs.~(\ref{3DPAF}) and (\ref{3DPAF_0})].
We note that this ``3D PAF'' state is
expressed by the ``$T_{2g}$'' state in Fig.~2(a) of Ref.~\cite{Rau_Gingras2018},
where a different notation is used:
$J_{zz} = 4 J_{\text{nn,eff}}$,
$J_{\pm}/J_{zz} = - \delta /2 $,
and $J_{\pm \pm}/J_{zz} = q/2$;
the local $\bm{x}_i$ and $\bm{y}_i$
are rotated by 120 degrees from our definition.
\subsection{Order parameter under [111] magnetic field}
By taking a linear combination of Eq.~(\ref{PAF}) with various wavevectors $\bm{k}= (h, h, h)$
one can construct a 2D PAF pseudospin LRO
which is non-zero only on an $\ell$-th kagom\'{e} lattice layer ($\ell=1,2, \cdots$)
\begin{equation}
\langle \bm{\sigma}_{ \bm{t}_n + \bm{d}_i } \rangle \propto
\bm{v}_{i}^{\text{2D}} \delta_{\ell,\hat{\bm{k}} \cdot (\bm{t}_n + \bm{d}_i) } \; ,
\label{2DPAF}
\end{equation}
where $\hat{\bm{k}}$ is a vector parallel to the [111] direction
such that $\hat{\bm{k}} \cdot (\bm{t}_n + \bm{d}_i) = 1,2, \cdots$ on
the kagom\'{e} layers.
In Fig.~\ref{3DPAF_2DPAF}(b) we schematically illustrate
the electric quadrupole order
expressed by the pseudospin structure Eq.~(\ref{2DPAF}).
Since mean fields on the triangular layers ($i=0$ sites)
vanish for the 2D PAF order,
magnetic dipole moments on the triangular layers,
$\langle \sigma_{\bm{t}_n + \bm{d}_0}^z \rangle \bm{z}_0$,
can be easily induced by applying [111] magnetic field.
When this magnetized state is stabilized against
the 3D PAF state by low [111] magnetic fields,
one can expect that the system behaves as
a 2D PAF state on each kagom\'{e} layer,
which is decoupled by field-induced ferromagnetic triangular layers.
Since $\bm{v}_{i}^{\text{2D}}$
[Eq.~(\ref{2DPAFk})] in
Eq.~(\ref{2DPAF}) is expressed by
$ \bm{v}_{i}^{\text{2D}} = \tfrac{1}{2} \left[ \bm{v}_{i}^{(0)} + \bm{v}_{i}^{(1)} + \bm{v}_{i}^{(2)} \right] $,
we can define an order parameter of the 2D PAF order on a kagom\'{e} layer as
\begin{equation}
m_{\text{2DPAF}} = \tfrac{2}{3} \left( m^{(0) \prime} + m^{(1) \prime} + m^{(2) \prime} \right)
\label{2DPAF_op}
\end{equation}
with
\begin{equation}
m^{(j) \prime}=\frac{\sum_{n,i} \bm{\sigma}_{ \bm{t}_n + \bm{d}_i } \cdot \bm{v}_{i}^{(j)}}{\sum_{n,i} 1} \; ,
\label{2DPAF_3op}
\end{equation}
where the summation runs over sites
on a single kagom\'{e} layer and an adjacent triangular layer.
Under low [111] fields they become
$\langle m^{(0) \prime} \rangle=\langle m^{(1) \prime} \rangle=\langle m^{(2) \prime} \rangle \simeq \pm \tfrac{1}{2}$
and $\langle m_{\text{2DPAF}} \rangle \simeq \pm 1$ at low temperatures.
We will show that $m_{\text{2DPAF}}$
is the order parameter under low [111] fields by CMC simulations.
\section{Results of CMC simulations}
\subsection{Zero magnetic field}
\begin{figure}
\begin{center}
\includegraphics[width=8.7cm]{Fig2.pdf}
\end{center}
\caption{
Temperature dependence of
3D PAF order parameter $\langle m_{\text{3DPAF}} \rangle$
and specific heat $C$ under zero field
calculated by CMC simulations for various $q$ values.
Shown in (a,b) are results with $\delta = 0$; (c,d) and (e,f) are results
with $\delta = 0.1$, and $-0.1$, respectively.
}
\label{m3DPAF_C}
\end{figure}
Under zero magnetic field, it was shown that
the classical ground state of the model
for
small $\delta$
changes from the classical spin ice state
($q < q_{\text{c}}=(1-\delta)/2 $)
to the PAF state ($q > q_{\text{c}}$) \cite{Onoda11}.
We performed CMC simulations using several
parameter sets of the effective Hamiltonian to
clarify whether the energetic or the order-by-disorder selection
mechanism stabilizes the 3D PAF order.
The simulations were performed
with a lattice size of $L=12$ and $L^{\prime}=4$
($12 \times 12 \times 4$).
In Fig.~\ref{m3DPAF_C} we plot
the 3D-PAF order parameter $\langle m_{\text{3DPAF}} \rangle$
and
the specific heat $C=(<E^2>-<E>^2)/(N T^2)$, where $E$ is the internal energy,
as a function of temperature for
$\delta=0, \pm 0.1$
and various $q$ values under zero field.
One can see from Fig.~\ref{m3DPAF_C}(a) that
$\langle m_{\text{3DPAF}} \rangle$ discontinuously increases
below a critical temperature $T_{\text{c}}$ for
$q \ge q_{\text{c}}$.
This implies that the phase transition is first order and
that the $\bm{k}=0$ order (3D PAF) occurs as expected.
At the transition temperatures the specific heat [Fig.~\ref{m3DPAF_C}(b)] shows
very sharp peaks.
The CMC simulations with non-zero
$\delta=0.1$ [Figs.~\ref{m3DPAF_C}(c) and (d)]
and $\delta=-0.1$ [Figs.~\ref{m3DPAF_C}(e) and (f)]
show parallel results with those of $\delta=0$.
This confirms previous CMC simulations \cite{Takatsu2016prl}
and is consistent with a mean-field result (appendix)
that small $\delta$ only changes
$T_{\text{c}}$ [the largest eigenvalue Eq.~(\ref{largest_eigenvalue})]
as $T_{\text{c}}(q,\delta) = T_{\text{c}}(q,\delta=0) [1 + \delta /(2q)]$,
without affecting eigenvectors
Eqs.~(\ref{eigenvector2D}) and (\ref{eigenvector3D}).
\begin{figure}
\begin{center}
\includegraphics[width=7.5cm]{Fig3.pdf}
\end{center}
\caption{
Size dependence of 3D PAF order parameter $\langle m_{\text{3DPAF}} \rangle$
as a function of temperature.}
\label{uu3D_size}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7.5cm]{Fig4.pdf}
\end{center}
\caption{
$T$-$q$ phase diagram determined by CMC simulations
shown in Fig.~\ref{m3DPAF_C}.
Red and blue thick lines are $T_{\text{c}}$ and
broad peak of specific heat, respectively, obtained by simulations with $\delta = 0$.
Dashed and dotted thin lines are those with $\delta = \pm 0.1$.
}
\label{T-q_PhaseDiagram}
\end{figure}
Further CMC simulations with $(\delta,q) = (0,0.7)$
were performed to study size dependence of the 3D PAF order parameter.
These results are shown in Fig.~\ref{uu3D_size},
which obviously demonstrates that the phase transition is first order.
In Fig.~\ref{T-q_PhaseDiagram} three curves of $T_{\text{c}}$
are plotted as a function of $q$ for
$\delta = -0.1$, $0.0$, and $0.1$.
It discontinuously decreases to $T_{\text{c}}=0$ at the critical value
$q_{\text{c}} = -0.45$, $0.5$, and $0.55$ for $\delta = -0.1$, $0.0$, and $0.1$, respectively.
This agrees with the first-order nature of the quantum phase
transition, which was investigated by a quantum treatment \cite{Lee12}.
In the range $q < q_{\text{c}}$ the specific heat
shows only a broad peak at about $T/J_{\text{nn,eff}} \sim 0.2$,
which can be interpreted as the behavior of the classical spin ice
model \cite{Onoda11}.
We note that this peak temperature is significantly lower
(about $1/4$) than that of the quantum MC simulation
of the same model with parameters
$q=0$ and $\delta \ne 0$ \cite{Kato15}.
This implies that the temperature scale
of the present CMC simulations is considerably reduced.
Thereby one has to take account of this fact
when comparing the CMC simulations
with experimental data.
\subsection{Under [111] magnetic field}
\begin{figure}
\begin{center}
\includegraphics[width=7.5cm]{Fig5.pdf}
\end{center}
\caption{
$H$-$T$ phase diagram under [111] magnetic field.
There are the paramagnetic state and two LRO states
of electric quadrupole moments denoted by 3D PAF and 2D PAF.
Inset shows $H$ dependence of specific heat $C$ and
2D PAF order parameter $\langle | m_{\text{2DPAF}} | \rangle$
at $T/J_{\text{nn,eff}} = 0.38$,
which are calculated by simulations with lattice size $12 \times 12 \times 4$.
}
\label{HT_phase}
\end{figure}
To study finite-temperature phase-transitions under [111] magnetic fields,
we performed CMC simulations
with a parameter set $(\delta,q)=(0,0.7)$
under various fields $H$.
Figure~\ref{HT_phase} shows an approximate $H$-$T$ phase diagram
obtained from peaks of the specific heat and
jumps of the order parameter $\langle m_{\text{3DPAF}} \rangle$,
which are calculated by simulations with lattice sizes
$12 \times 12 \times 4$ and/or $6 \times 6 \times 2$.
From the high-temperature paramagnetic phase
the system undergoes a phase transition to one of the
two quadrupole ordered phases denoted by 3D PAF and 2D PAF,
which will be discussed later.
These 3D and 2D PAF phases are separated by a phase transition line,
a crossover line, or multiple phase transitions
(the dashed curve in Fig.~\ref{HT_phase}).
These three possibilities could not be clarified by the present simulation
techniques, because the single-spin-flip simulations suffer
from a freezing problem at low temperatures.
We note that the boundary line between 3D PAF and 2D PAF states
depicted by the dashed curve in Fig.~\ref{HT_phase}
corresponds to the low-field kink of the $M$-$H$ curve
shown in Fig.~5(b) of Ref.~\cite{Takatsu2016prl}.
Simulated $M(H,T)$ data suggest that there may
be intermediate magnetization plateau states
between zero field and the low-field kink.
\begin{figure}
\begin{center}
\includegraphics[width=7.5cm]{Fig6.pdf}
\end{center}
\caption{
Temperature and size dependence of
(a) specific heat $C$,
(b) 3D PAF order parameter $\langle m_{\text{3DPAF}} \rangle$, and
(c) 2D PAF order parameter $\langle | m_{\text{2DPAF}} | \rangle$
calculated by CMC simulations
under three typical [111] fields $H=0.1$, 0.4, and 3. }
\label{C_uu3D_uu2D_h111}
\end{figure}
Figure~\ref{C_uu3D_uu2D_h111} shows
temperature dependence of the specific heat $C$,
the 3D-PAF order parameter $\langle m_{\text{3DPAF}} \rangle$,
and the 2D-PAF order parameter $\langle | m_{\text{2DPAF}} | \rangle$
under three typical magnetic fields:
$H=0.1$, 0.4, and 3.
At the low field $H=0.1$
it is evident that the system shows
the same first-order phase transition as zero field,
and that LRO is the 3D PAF order.
On the other hand, at the high field $H=3$,
the size dependence of $C(T)$ and $\langle | m_{\text{2DPAF}} | \rangle(T)$
[Figs.~\ref{C_uu3D_uu2D_h111}(a) and \ref{C_uu3D_uu2D_h111}(c)]
show typical behaviors of a second-order phase-transition.
These indicate that $\langle | m_{\text{2DPAF}} | \rangle$ is the order parameter
of the second-order phase-transition,
in agreement with the initial expectation.
At the intermediate field $H = 0.4$
the temperature dependence of
the specific heat [Fig.~\ref{C_uu3D_uu2D_h111}(a)] implies that
two successive phase transitions occur.
At the higher $T_{\text{c}1}/J_{\text{nn,eff}} \simeq 0.35$,
$C(T)$ and $\langle | m_{\text{2DPAF}} | \rangle(T)$
[Figs.~\ref{C_uu3D_uu2D_h111}(a) and \ref{C_uu3D_uu2D_h111}(c)]
show that
the phase transition is the same kind as that for $H=3$.
On the other hand, characteristics of the lower
$T_{\text{c}2}/J_{\text{nn,eff}} \simeq 0.28$
are less clear owing to the freezing problem.
The simulated $C(T)$, $\langle m_{\text{3DPAF}} \rangle(T)$,
and $\langle | m_{\text{2DPAF}} | \rangle(T)$
(Fig.~\ref{C_uu3D_uu2D_h111})
suggest that $T_{\text{c}2}$ is
a continuous phase transition between 2D PAF and 3D PAF states,
which could not be further investigated using the present techniques.
In addition to the constant $H$ plots (Fig.~\ref{C_uu3D_uu2D_h111}),
magnetic field dependence of $C$ and $\langle | m_{\text{2DPAF}} | \rangle$
with constant $T=0.38 J_{\text{nn,eff}}$
are shown in the inset of Fig.~\ref{HT_phase}.
At this temperature reentrant phase transitions occur
at lower and upper critical fields,
$H_{\text{c1}} \simeq 0.7$ and $H_{\text{c2}} \simeq 5.2$.
\begin{figure}
\begin{center}
\includegraphics[width=7.5cm]{Fig7.pdf}
\end{center}
\caption{
Temperature dependence of the Binder cumulant
$\left< m_{\text{2DPAF}}^4 \right>/ \left< m_{\text{2DPAF}}^2 \right>^2$
close to $T_{\text{c}}$
for lattice sizes $L=18,24,36$, and $54$ under [111] field $H = 1.5$.
}
\label{BC_cross}
\end{figure}
Since the 2D PAF order breaks a $Z_2$ symmetry of $m_{\text{2DPAF}}$,
one can naturally expect that its second-order phase-transition at $T_{\text{c}}$
belongs to the universality class of the 2D Ising model.
To confirm this universality
we performed standard finite-size scaling
analyses \cite{LandauBinder15}
on CMC simulation data taken
under a typical [111] field $H=1.5$.
These simulations were carried out
on clusters with lattice sizes $L \times L \times (L/3)$
with $L=18,24,36$, and $54$.
Figure~\ref{BC_cross} shows the Binder cumulant
$U_4 = \left< m_{\text{2DPAF}}^4 \right>/ \left< m_{\text{2DPAF}}^2 \right>^2$
as a function of temperature.
These curves with different lattice sizes cross at
a single point, which enables us to
determine the critical temperature $T_{\text{c}}/J_{\text{nn,eff}}=0.4088(2)$.
\begin{figure}
\begin{center}
\includegraphics[width=7.5cm]{Fig8.pdf}
\end{center}
\caption{
Finite size scaling of
(a) the Binder cumulant $\left< m_{\text{2DPAF}}^4 \right>/ \left< m_{\text{2DPAF}}^2 \right>^2$,
(b) 2D PAF order parameter $\langle | m_{\text{2DPAF}} | \rangle$, and
(c) 2D PAF susceptibility $\chi_{\text{2DPAF}}$.}
\label{FS_scaling_plots}
\end{figure}
The theory of the finite-size scaling indicates that
the Binder cumulant,
the order parameter $\left< |m_{\text{2DPAF}}| \right>$,
and the susceptibility
$\chi_{\text{2DPAF}} = N_{\text{2D}} \left( \left< m_{\text{2DPAF}}^2 \right> - \left< |m_{\text{2DPAF}}| \right>^2 \right)/T$
show the scaling forms
\begin{align}
U_4 &= f( L^{1/\nu} (T-T_{\text{c}})/T_{\text{c}} ) \; , \nonumber \\
\left< |m_{\text{2DPAF}}| \right> &= L^{-\beta/\nu} g( L^{1/\nu} (T-T_{\text{c}})/T_{\text{c}} ) \; , \\
\chi_{\text{2DPAF}} &= L^{2-\eta} h( L^{1/\nu} (T-T_{\text{c}})/T_{\text{c}} ) \; , \nonumber
\label{FSscaling}
\end{align}
where $f$, $g$, and $h$ are universal functions \cite{LandauBinder15}.
In Fig.~\ref{FS_scaling_plots} we show these finite-size scaling plots
using the exact critical exponents $\nu=1$, $\beta=1/8$, and $\eta=1/4$
for the 2D Ising model.
These figures show excellent data collapse,
which proves the finite-size scaling relations of the 2D Ising model.
Therefore we conclude that the second-order phase-transition
of the 2D PAF state
belongs to the 2D Ising universality class.
To complement the argument of the 2D Ising universality class
we calculated squares of the Fourier transform of
$m_{\text{2DPAF}}$ [Eq.~(\ref{2DPAF_op})],
which is defined on each $\ell$-th kagom\'{e} lattice layer,
with wavevectors $\bm{k}=(h,h,h)$ ($0 \le h \le 1$)
\begin{equation}
|m_{\text{2DPAF}}(\bm{k})|^2 = \left| \sum_{\ell} \left[ m_{\text{2DPAF}} \right]_{\ell} \, e^{i \bm{k} \cdot \bm{r}} \right|^2 ,
\label{Bragg_int}
\end{equation}
where $\bm{r}$ is a lattice position on the $\ell$-th kagom\'{e} lattice layer.
If $m_{\text{2DPAF}}$ has really 2D character,
simulated averages of $|m_{\text{2DPAF}}(\bm{k})|^2$ do not depend on $h$.
In terms of a scattering experiment (assuming that the quadrupole moment would be visible),
$\langle |m_{\text{2DPAF}}(\bm{k})|^2 \rangle$ is constant
between two $\Gamma$ points $\bm{k}= (0,0,0)$ and $(1,1,1)$.
In Fig.~\ref{2D_Bragg} we show CMC averages $\langle |m_{\text{2DPAF}}(\bm{k})|^2 \rangle$
close to $T_{\text{c}}$, which were computed with a lattice size $12 \times 12 \times 4$.
These curves show independence of $h$ and thereby the two dimensionality
of the order parameter.
We note that the freezing problem of the present CMC techniques
prohibited us from performing simulations with larger system sizes
and from obtaining the averages at low temperatures ($T \ll T_{\text{c}}$).
This difficulty is seen as the large error estimation
of the low-temperature data ($T \le T_{\text{c}}$) shown in Fig.~\ref{2D_Bragg}.
Despite this large error,
we also note that one may see slight wavevector dependence
for the curve at $T=0.40 J_{\text{nn,eff}} < T_{\text{c}}$.
This may suggest that the 2D PAF order is weakly modulated
along the [111] direction at low temperatures.
\begin{figure}
\begin{center}
\includegraphics[width=7.5cm]{Fig9.pdf}
\end{center}
\caption{
Wavevector dependence of $\langle |m_{\text{2DPAF}}(\bm{k})| \rangle^2$
along [111] direction above and below $T_{\text{c}}$ computed by CMC simulations
with lattice size $12 \times 12 \times 4$.
Size of symbol represents estimated error of data.
}
\label{2D_Bragg}
\end{figure}
\section{Discussion}
In previous investigations \cite{Takatsu2016prl,Takatsu16}
we showed that the simple pseudospin-$\frac{1}{2}$ Hamiltonian
described by Eq.~(\ref{H_effective})
qualitatively and semi-quantitatively
accounts for most of the experimental observations of the TTO sample
with $T_{\text{c}}>0$ by selecting the appropriate model parameters.
The agreement between experiments and theories
was surprisingly better than our initial expectation.
This means that the model Hamiltonian essentially explains
the experimentally observed properties of TTO.
Although there remain problems of oversimplifications caused by
the classical approximations for the quantum model
and by neglecting effects of higher-energy CF states \cite{Rau_Gingras2018}
and Jahn-Teller effects due to
the phonon mechanism \cite{Bonville11}.
We would like to make a few comments
on the the present CMC simulation results
in relation to experimental observations.
A first comment is on the natural question:
how does the off-stoichiometry parameter
of Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$, $x$ (and/or $y$),
function as the tuning parameter between
QSL and quadrupolar states?
Our experiments using both poly- and single-crystalline samples
showed that $x_{\text{c}} \simeq -0.0025$ is
the quantum critical point \cite{Taniguchi13,Wakita16}.
They also showed that by approaching to $x_{\text{c}}$
from the quadrupolar side $x > x_{\text{c}}$,
the large specific-heat peak observed in
$C(T)$ data (e.g. Fig.~4(a) in Ref.~\cite{Takatsu2016prl})
abruptly becomes smaller peaks
as shown in Fig.~2 of Ref.~\cite{Taniguchi13}
and Fig.~4(a) of Ref.~\cite{Wakita16}.
By assuming that the change of $x$ is equivalent to that of $q$,
the experimental behavior of $C(T)$ is approximately reproduced
by the simulated $C(T)$ shown in Fig.~\ref{m3DPAF_C}(b).
Therefore an answer to the question may be that
$x$ tunes the ratio of the magnitude of
the quadrupole interaction to that of the magnetic interaction.
A second comment is on susceptibilities under zero field.
We calculated the magnetic susceptibility
$\chi_{\parallel [111]} = N \left( \langle m_{\parallel [111]}^2 \rangle - \langle |m_{\parallel [111]}|\rangle^2 \right) /T$
using the same parameter sets as those of Fig.~\ref{m3DPAF_C}(a).
These results are shown in Fig.~\ref{chim3DPAF_chi111}(a).
The curve with $q=0.55$
bears resemblance to the experimental data
of the TTO sample with $T_{\text{c}} = 0.53$ K
(Fig.~2(a) of Ref.~\cite{Takatsu2016prl}).
If we take account of the reduction of the temperature scale
for the CMC simulation
the resemblance becomes more striking.
This also can justify the interpretation of TTO
using the model Hamiltonian and the CMC simulation.
We also calculated the electric quadrupole susceptibility
corresponding to the 3D PAF order
$\chi_{m_{\text{3DPAF}}} = N \left( \langle m_{\text{3DPAF}}^2 \rangle - \langle |m_{\text{3DPAF}}| \rangle^2 \right) /T$.
Temperature dependence of this quadrupole susceptibility is
shown in Fig.~\ref{chim3DPAF_chi111}(b).
The large increase of $\chi_{m_{\text{3DPAF}}}$ close to $T_{\text{c}}$
can be measured by ultrasonic experiments of TTO,
for example, extending measurements of Ref.~\cite{Nakanishi2011} down to 0.3 K.
\begin{figure}
\begin{center}
\includegraphics[width=7.5cm]{Fig10.pdf}
\end{center}
\caption{
Temperature dependence of
(a) magnetic susceptibility parallel to [111] direction $\chi_{\parallel [111]}$
and (b) susceptibility of $m_{\text{3DPAF}}$ under zero field
calculated by CMC simulations.
}
\label{chim3DPAF_chi111}
\end{figure}
A third comment is on
the first-order nature of the zero-field phase-transition
of the CMC simulations.
This does not agree with experimental $C(T)$,
which shows a second-order behavior \cite{Takatsu2016prl}.
In addition, the second-order phase-transition under [111] field
seems to be somewhat smeared out
for the the experimental data (Fig.~4(a,b) of Ref.~\cite{Takatsu2016prl})
compared to the CMC simulations.
These disagreements remain to be explained, e.g.,
by adding a higher-order term in the Hamiltonian \cite{Zhitomirsky2012},
by a disorder effect \cite{Imry75},
or possibly by a quantum effect.
\section{Conclusions}
We have studied phase transitions of pyrochlore magnets
with non-Kramers ions under [111] magnetic field
represented by the effective pseudospin-$\frac{1}{2}$ Hamiltonian \cite{Onoda11}
from a viewpoint of relevance to electric quadrupolar states
of Tb$_{2}$Ti$_{2}$O$_{7}$ \cite{Takatsu2016prl}.
Order parameters and finite-temperature
phase-transitions of this frustrated model system
are investigated using classical Monte-Carlo simulations.
In zero field, the model undergoes a first-order phase-transition
from the paramagnetic state to a 3D quadrupolar state
with an antiparallel arrangement of pseudospins.
This 3D order is selected energetically or by an order-by-disorder mechanism
from degenerate $\bm{k}=(h,h,h)$ mean-field orders.
Under [111] magnetic field this 3D state
is transformed to a 2D quadrupolar state on each kagom\'{e} lattice,
which is separated by field-induced ferromagnetic triangular lattices.
This 2D system undergoes a second-order phase-transition
belonging to the 2D Ising universality class.
\\
\begin{acknowledgments}
We wish to thank S. Onoda and Y. Kato for useful discussions.
This work was supported by JSPS KAKENHI grant numbers 25400345 and 26400336.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 349 |
Welcome to Dragons (DR Congo - Super League) statistics. Below you find a lot of statistics for this team. Last and next matches, top scores, best players, under/over stats, handicap etc.
Dragons is currently on the 15 place in the Super League table. Last game played with Sanga Balende, which ended with result: Win Dragons 1:0.
Streaks for for all matches played in Super League is undefeated from 3 matches in a row.
Streaks for for matches played only away in Super Leaguefrom recent 19 matches in a row can't win a match.
Statistics of matches that the team Dragons won or lost with a particular goal difference. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,645 |
Mining Equipment Accessories - Suppliers & Manufacturers in IndiaGold Concentartor : STAR TRACE Gold concentrator is a kind of centrifugal concentration equipment. It can be used not only for placer gold mining, but also for hard rock mining to recover the natural gold, replacing amalgamation. It is also more.. Rs 1.2 Lakh/ Unit. Star Trace Private Limited. Redhills, ChennaiNo.suppliers of gold mining equipments india,suppliers of gold mining equipments india,Gold Mining Equipment - Silver Mining Machinery Exporter from .We are the reputed manufacturer, supplier of the largest range of Silver Mining Machinery for precious metals and are built to withstand years of the most rugged field conditions and heavy use, designed for highest efficiency and recovery yield of gold andsilver and other precious metals. We take into consideration the.
Jun 24, 2014 . The total number of small, medium and large suppliers of mining equipment and accessories in India exceeds 200. Many of these companies also export mining equipment and accessories to other developing countries. Many foreign companies including ones from the U.S. have established manufacturing.
Krishnapatnam Port sets India record for coal discharging. infomine - mining equipment and supplier news | Jun. 20, 2011, 4:02 PM | | 0. Krishnapatnam Port has set an all India record for discharging 95,528 tons of steam coal in just 24 hrs using the conventional unloading system in form of advanced Mobile Harbor cranes.
Jan 24, 2017 . India's been the number one source of global gold demand for decades importing close to a 1,000 tonnes in good years. But, following the closure of the iconic Kolar Gold Field in 2001 after more than 120 years and 800 tonnes or 26 million troy ounces of production, India is home to a single gold mine.
Amazon : Gold Buddy Mini Highbanker - Gold Mining Equipment : Other Products : Everything Else.
Infra Business - Earth Moving Equipments, Construction Equipments .
Earth Moving Equipments, Rock Breakers, Backhoe Loaders, Road Construction Equipments, Pocklain, JCB On Hire, Rent, Maintenance, Repair Services, Manufacturer, Exporter, Supplier, Dealer, Used Equipments, India.
Results 1 - 48 of 1403 . Blue Bowl Concentrator Kit with Pump and Leg Levelers Gold Mining Equipment . This unit is loaded with features to make your panning experience not only fun, but potentially profitable because the Gold Miner will get the smallest speck of gold out of even the most .. Gold Panning Supplies.
Fuel, precious metals and gems, and mineral resources are among the most commonly mined goods across Canada and North America. Whether mining equipment manufacturers produce placer mining equipment, industrial mining equipment, heavy mining equipment, industrial gold mining equipment, safety and precise.
From Australia to Zambia and underground gold to open-pit coal, Minestories is our visual storytelling portal that serves to inform and inspire the global mining industry . As the leading supplier of rock drill technology in the world, we design, produce and deliver these core components and their spare parts for underground. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,660 |
{"url":"https:\/\/www.socialscienceregistry.org\/trials\/4651\/history\/99415\/changes","text":"### Fields Changed\n\n##### Trial\nField Before After\nStatus In Development Completed\nAbstract A choice is most tempting when it is both immediate and certain. Prevailing experimental measures of static present-bias rely on the random incentive scheme (RIS) to gather rich data from subjects. Recent studies show that RIS adds background risk to decisions, thereby increasing risk-taking behavior. Evidence has shown that the immediacy effect is significantly moderated (or even eliminated) by uncertainty. Thus estimates of the present-bias factor $\\beta$ are possibly biased upward toward unity, underestimating present-bias intensity. I conduct a framed field experiment that determines whether uncertainty indeed diminishes the immediacy effect by using real-effort tasks to approximate immediate and certain utility flows. Subjects allocate tasks using convex time budgets under both RIS and certain implementation. I compare within-subject estimates of the quasi-hyperbolic present-bias factor under these two mechanisms, assuming convex effort costs. I type individuals as present- or future-biased under each mechanism, which provides data on the interaction between the immediacy and certainty effects within subjects. I also report how the introduction of uncertainty affects the present-bias parameter on average, which would be some of the first incentivized evidence of the effect of uncertainty on immediacy. Is an option is especially tempting when it is both immediate and certain? To study the effect of risk on present bias, I conduct an online experiment in which workers allocate about thirty minutes of real-effort tasks between two weeks. I compare choices made two days before the first workday against choices made when work is imminent. In baseline treatments, one choice is randomly implemented; meanwhile, one treatment implements a particular allocation with certainty. By assuming that effort costs are not affected by the mechanism (and thus independent of risk preferences), my novel design permits estimation of present bias using a decision with a consequence both immediate and certain. I find the average intensity of present bias is far greater under certainty than under risk. I find no evidence that present bias is more pervasive among individuals, suggesting instead that present-biased individuals become more myopic.\nTrial Start Date September 15, 2019 October 28, 2019\nTrial End Date November 08, 2019 November 06, 2019\nLast Published October 28, 2019 02:20 PM September 10, 2021 01:38 AM\nIntervention Completion Date November 06, 2019\nData Collection Complete Yes\nData Collection Completion Date November 06, 2019\nIntervention Start Date September 15, 2019 October 28, 2019\nIntervention End Date November 08, 2019 November 06, 2019\nAdditional Keyword(s) Immediacy effect, certainty effect, Allais paradox, common-ratio effect, non-expected utility, independence axiom, intertemporal choice, dynamic inconsistency, nonstationarity, myopia, present-bias, future-bias intertemporal choice, time preferences, risk preferences, dynamic inconsistency, present bias, nonstationarity, immediacy effect, certainty effect, common difference effect, common ratio effect, non-expected utility, Allais paradox\nKeyword(s) Labor Behavior, Labor","date":"2021-12-08 07:06:29","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\": 0.3139902651309967, \"perplexity\": 7888.750801054558}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964363445.41\/warc\/CC-MAIN-20211208053135-20211208083135-00070.warc.gz\"}"} | null | null |
Jacobus Rolandus (* 1562 in Delft; † 1632 in Leiden) war ein niederländischer evangelisch-reformierter Geistlicher während des Achtzigjährigen Krieges.
Biografie
Jacobus Rolandus studierte Theologie in Genf und Heidelberg. Seine erste Stelle als reformierter Pfarrer trat er 1587 in Wiesloch und Germersheim in der Kurpfalz an. 1594 wurde er Pfarrer in Delft. Da er eine schwache Stimme hatte und überhaupt bei schlechter Gesundheit war, konnte er den Dienst in der dortigen Grote Kerk nur schwer versehen und wechselte 1598 nach Frankenthal, 1603 dann nach Amsterdam. Dort trat er an der Oude Kerk die Nachfolge von Jacobus Arminius an. Gleich die erste Predigt befasste sich mit der Prädestinationslehre und rief den Widerspruch prominenter Remonstranten hervor. Rolandus, der als eher furchtsamer Gelehrter beschrieben wird, war stark in der Auseinandersetzung um den Arminianismus engagiert, auch emotional. Er äußerte, wenn der bereits verstorbene Philipp Melanchthon seine (den Remonstranten ähnliche) Prädestinationslehre öffentlich vertreten wollte, würde er ihn an der Predigt hindern. Gemeinsam mit Jacobus Trigland war er in der überregionalen Organisation der Kontraremonstranten aktiv, und beide Theologen wurden von der Provinzialsynode von Nord-Holland 1618 als Delegierte zur Dordrechter Synode entsandt. Dort wurde er als erster Assessor ins Moderamen gewählt und nahm während der Synode Leitungsaufgaben wahr. Die Synode wählte ihn in die Kommission für die geplante Neuübersetzung der Bibel ins Niederländische (Staatenübersetzung); er war bei diesem Projekt für das Neue Testament zuständig, nachträglich kam die Korrektur der Übersetzung des Alten Testaments hinzu. Seine Übersetzeraufgaben führten dazu, dass er nach Leiden umzog, wo er seine letzten Lebensjahre verbrachte. Die Fertigstellung der Staatenübersetzung erlebte er nicht mehr.
Jacobus Rollandus war dreimal verheiratet. Seine erste Ehefrau war Maria Doucy, eine hugenottische Adlige. Das Paar hatte neun Kinder und begründete damit eine große Pfarrerdynastie. Die weiteren Ehen mit Hendrika Pietersz. de Boer und Barbara de Weerdt blieben kinderlos.
Literatur
R.B. Evenhuis: Rolandus, Jacobus. In: D. Nauta (Hrsg.): Biografisch lexicon voor de geschiedenis van het Nederlands protestantisme. Kok, 1. Band Kampen 1978, S. 292–293.
Donald Sinnema, Christian Moser, Herman J. Selderhuis (Hrsg.): Acta et Documenta Synodi Nationalis Dordrechtanae (1618–1619). Vandenhoeck & Ruprecht, Göttingen 2015.
(abgerufen über De Gruyter Online)
Weblinks
Biografisch portaal van Nederland: Jacobus Rolandus
Reformierter Theologe (17. Jahrhundert)
Bibelübersetzer
Übersetzer ins Niederländische
Niederländer
Geboren 1562
Gestorben 1632
Mann | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,262 |
"use strict";
var logger = {
debug: function() {}
};
logger.info = logger.warn = logger.error = logger.debug;
module.exports = logger;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,068 |
Hoppers Crossing doing cartwheels over new gymnastics facility
Share this page: Twitter Facebook Linkedin Email
Hoppers Crossing has been making the most of a new gymnastics' facility at the Grange P-12 College, made possible by a $5.35 million loan to Belgravia Group through the Victorian Government's innovative Community Sports Infrastructure Loan Scheme.
The new Grange P-12 College facility features the latest gymnastics equipment and an innovative layout, designed to stimulate and encourage young people to be active.
Grange P-12 Primary School Principal, Meredith Clencie said this came about through us being able to access a community infrastructure loan. We're just so pleased to have this facility.
"Parents love it and we push it as an alternative to screen time."
It's already inspiring the local students with over 965 tumbling and somersaulting their way through a 6-week gymnastics course earlier this year to improve physical literacy, whilst having some rolling fun.
Emma Coogan, PE and gymnastics teacher at the school said the space is awesome and the kids absolutely love it.
Grange Primary school student, Aarna said it has everything we need to do gymnastics and that's nice.
"With the gymnasium we can do a lot more things and many people at our school are now gymnasts and other people who are not so good can have a try."
Whilst outside of school hours, over 230 junior home-grown talents are also hitting the bars and mat weekly to participate in the Belgravia Kids gymnastics programs in Hoppers Crossing.
And there's more to it than just advancing gymnastics skills, the facility has provided some additional employment opportunities for students at the Grange with operator Belgravia Leisure offering complimentary accredited gymnastics coaching courses to senior students.
Thanks to this initiative, Belgravia have gone on to employ 15 senior students who help coordinate the gymnastics program at the new Grange P-12 College gymnastics facility with opportunities for more in the future.
Belgravia CEO, Mark Rendell said thanks to the Community Sports Loan Scheme through Sport and Recreation Victoria have put in place we were able to fund the fit out of this gym through that loan scheme.
"That's fast tracked the process, but it also allowed us to create 15 jobs for the high school students here.
"Eight of those have gone to intermediate qualification. Four of them have become supervisors. So it's provided some great leadership opportunities and first employment for a lot of young community members
Jamie Parsons, CEO of Gymnastics Victoria said gymnastics provides a life-long love of physical fitness and activity and that's what we want to instil in children from a very young age.
"Venues like this provide a great pathway, it starts with kindergarten kids, gymnastics helps with transition to Primary school and transition in to club activity after school here using the same venue. So it's a really great way to keep children involved in sport as much as possible."
The Community Sports Infrastructure Loans Scheme provides low-interest loans for councils, clubs and organisations, to deliver and improve community sport and recreation infrastructure.
It's part of our commitment to promoting sports and active recreation and encouraging all Victorians to get out there and get active.
Across Victoria we've invested more than $850 million in community sport and recreation facilities since 2014 to support sport and active recreation at all levels.
For further information on the Community Sports Infrastructure Loans Scheme
Australian Open 2020 - The Grand Slam for Everyone
Score a sporting club program grant
Moe celebrates new cricket centre of excellence
Bumping in and out at the Disability Sport and Recreation Festival
Supporting Victoria's sport and recreation sector through Together More Active
Victorians urged to fill the MCG for Women's T20 final | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,181 |
Divine Mercy Adoration Chapel
Sacred Heart & St. Stanislaus Catholic Churches
Catholic Churches in Minto & Warsaw
The community churches of Sacred Heart and St. Stanislaus enjoy a rich mission and long-standing history.
The Churches' Mission
Sacred Heart and St. Stanislaus Catholic churches serve as a living witness to the Gospel of Christ, providing Christians with the fullness of truth for the salvation of souls.
The churches form a community committed to ministering to the people of God so that they may learn to generously know, love, and serve God our Father by the inspiration of the Holy Spirit.
Both parishes offer the Sacraments, spiritual guidance and nourishment, community building, and a Christian way of life. As a parish family, they seek to know God our Father and enter more deeply into the eternal marriage feast made present in the Holy Eucharist by growing in union with the Sacred Heart of Jesus through the Immaculate Heart of Mary.
A people without the knowledge of their past history, origin and culture is like a tree without roots.
-Marcus Garvey
Yesterday is history, tomorrow is a mystery, today is God's gift, that's why we call it the present.
-Joan Rivers
St. Stanislaus & Sacred Heart
Over 250 combined years of Catholic history
St. Stanislaus, established 1878
St. Stanislaus, located in Warsaw, ND, is listed on the National Historic Register and has merited the nickname "The Cathedral on the Prairie" due to its extraordinary grandeur and breathtaking ability to point the Christian heavenward.
In the 1870s, Polish immigrants flooded the grassy wilderness of southeast Walsh County, staking claims, breaking and plowing land, and seeding wheat, oats, potatoes, and rutabagas.
Warsaw, first named Pulaski after the great Polish patriot and hero of the American Revolution, became the new home to these deeply religious Roman Catholics. Naturally concerned about Sunday worship, a priest named Father Considine was summoned to pastor the people of St. Stanislaus Bishop & Martyr Church, the original name of St. Stanislaus.
Sacred Heart, established 1905
As early as the 1870s, the faith and spirit of settlers stemmed from the religious life of the Old World, and the devout worshipers pursued Catholic community life. After several years of traveling to Warsaw for Mass, the Polish Catholics wanted to establish their own church in Minto.
On April 28, 1903, the Polish Catholics acquired an old Baptist church in Minto. Sacred Heart Catholic Church was founded in 1905. By 1912, the structure was far too small for the congregation, so a noble brick building was erected in the Byzantine style.
A few years after construction, the church was gutted by a fire that started in the tower. A few items were removed undamaged including the pews that remain almost one hundred years later. Rebuilding commenced immediately, and the same church has stood since 1916.
Priests of St. Stanislaus
1878-1882 - Rev. Klement Grynolc & Rev. John Considine
1883-1886 - Rev. Alexander Michanowski
1886-1888 - Rev. Damin Kolasinski
1889-1890 - Rev. Stanislaus Tokarski
1891-1893 - Rev. Mateusz Grochowski
1893-1895 - Rev. Roman Wawrzykowski
1896-1905 - Rev. Francis Gawlowicz
1905-1910 - Rev. Boleslaus Waldowski
1910-1911 - Rev. W. St. Majer
1911-1926 - Rev. Theodore Kupka
1926-1959 - Monsignor John Maluski
1959-1971 - Monsignor Petr Lekavy
1971-1975 - Rev. Michael McNamee
1975-1976 - Rev. Francis Kuttner, S.A.C.
1976-1994 - Rev. Stanislaus Duda
1994-1997 - Rev. Joseph Gregory
1997-2005 - Rev. Damian Hils
2005-2016 - Rev. John Kleinschmidt
2016-present - Rev. Brian Moen
Priests of Sacred Heart
1905-1906 - Rev. Joseph Karpinski
1906-1908 - Rev. Wenceslaus Krzywonos
1908-1909 - Rev. Vincent S. Mayer
1910-1917 - Rev. Stanislaus W. Majer
1917-1922 - Rev. Francis Olzewski
1922-1923 - Rev. Stephen Bryalski
1923-1940 - Rev. Francis J. Slominski
1940-1957 - Rev. John J. Stempel
1957-1966 - Rev. Hilarion J. Mikalofsky
1967-1988 - Rev. Antonio J. Richard
1988-2002 - Rev. Feliks Lubas
2002-2004 - Rev. Timothy Schroeder
The Fargo Diocese
Our two dynamic parishes, both over 100 years old, have striven to teach and uphold the Catholic faith for several generations. St. Stanislaus and Sacred Heart serve under the service of Most Rev. John Folda, the bishop of the Fargo Diocese. He has been entrusted with the authority of the diocesan bishop to teach, sanctify, and lead Catholics residing within Eastern North Dakota. The diocese, established in 1889, comprises 30 counties over more than 35,000 square miles in the eastern half of North Dakota.
Franciscans of Mary Immaculate
On May 31, 2011, the Franciscans of Mary Immaculate was founded and is based in Warsaw, ND, under the auspices of Most Reverend John T. Folda, Bishop of the Diocese of Fargo, ND. The men profess the Vows of Poverty, Chastity, and Obedience and strive to live a life of simplicity, humility, and penance for God's glory, for their own conversion and holiness and for the sake of the personal holiness of the whole world.
Fr. Joseph Christensen established FMI and serves as chaplain for St. Gianna's Maternity Home in Warsaw.
The community is dedicated to Jesus through the Blessed Virgin Mary, according to the teachings and writings of St. Maximilian Kolbe, a Polish priest, Franciscan Friar, and Martyr for love of neighbor in Auschwitz Concentration Camp.
Website | www.fmifriars.com
Facebook | Franciscans of Mary Immaculate-Third Order
YouTube | Franciscans of Mary Immaculate
The Sisters of the Resurrection
A religious order founded in Rome, the Sisters of the Resurrection from Chicago, Illinois, were welcomed to the Diocese of Fargo in 1918. Bishop James O'Reilly selected Warsaw as a suitable location for the sisters to reside.
The sisters immediately began the construction of St. Anthony School and Convent which opened in 1921. In 1961 the school was renamed the St. Stanislaus Parochial School. However, in 1971 the school was closed due to low attendance. The building currently serves the community as the pro-life ministry Saint Gianna & Pietro Molla Maternity Home in honor of Saint Gianna Beretta Molla and her husband Pietro.
Saint Gianna & Pietro Molla Maternity Home
Since 2003, Saint Gianna & Pietro Molla Maternity Home (formerly known as Saint Gianna's Maternity Home) has served the Body of Christ as a pro-life home of formation for pregnant women and their children. The home offers a place for peaceful reflection and redirection as a woman determines a plan for herself and her baby.
Saint Gianna & Pietro Molla provides women with access to medical, educational, and professional services. The home embodies a spiritual environment. Women of all faiths are welcome at SGPM Maternity Home.
For more information, visit www.sgpmollahome.com.
Facebook | Saint Gianna & Pietro Molla Maternity Home
Instagram | @sgpmollahome
Jesus is adored in Eucharistic Adoration in the beautiful chapel of Saint Gianna & Pietro Molla Maternity Home in Warsaw, ND.
St. Stanislaus' & Sacred Heart's treasures from the past
A picture is worth a thousand words.
God's grace is free!
Learn more about the Catholic faith and how you can become a part of it. Contact us!
2023 © Sacred Heart & St. Stanislaus Catholic Churches
629 3rd Street, Minto, ND 58261, USA | 7012483589 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,436 |
\section{Introduction}
Two disjoint partial Latin squares $T$ and $T^{*}$ of the same order, with the same set of filled cells and satisfying the property that
corresponding rows (corresponding columns) contain the same entry values, form a {\sf Latin trade} and its disjoint mate. The pair $(T,T^{*})$ is called a
{\sf Latin bitrade}. In earlier papers the word ``Latin trade'' is used for ``Latin bitrade'', but we keep the word ``trade'' for each partial Latin square of a Latin bitrade.
The
study of Latin trades and combinatorial trades in general, has generated much interest in recent years. For a
survey on the topic see \cite{MR2041871},
\cite{MR2048415}, and \cite{CavenaghMathSlovac}.
A Latin bitrade which is obtained from another one by deleting
its empty rows and empty columns, is called a {\sf
$k$--homogeneous Latin bitrade}, if in each row and each column it
contains exactly $k$ elements, and each element appears exactly
$k$ times. The number of filled cells in a Latin trade is referred to
as its {\sf volume}.
The following question is of interest.
\begin{question}
\label{existence}
For given $m$ and $k$, $m \ge k$, does there exist a $k$--homogeneous Latin bitrade of volume $km$?
\end{question}
In the sequel we need some more notations and definitions.
Concepts not defined here may be found in~\cite{anderson90a}. We
can represent each Latin square as a set of $3$--tuples $L =
\{(i,j;k)\,|\,\mbox{element $k$ is located in position} \
(i,j)\}.$ A Latin bitrade $(T,T^{*})$ is said to be {\sf primary}
if whenever $(U,U^{*})$ is a Latin bitrade such that $U\subseteq
T$ and $U^{*}\subseteq T^{*}$, then $(T,T^{*})=(U,U^{*})$. A
Latin trade $T$ is said to be $\sf minimal$ if whenever
$(U,U^{*})$ is a Latin bitrade such that $U\subseteq T$, then
$T=U$. So if $T$ is a minimal Latin trade in a Latin bitrade
$(T,T^{*})$, then $(T,T^{*})$ is a primary Latin bitrade. A
Latin bitrade of volume $4$ is called an {\sf intercalate}. In
Figure~\ref{2x2} an intercalate $(T,T^{*})$ is shown. The
elements of $T^{*}$ is written as subscripts in the same array as
$T$.
\def1.3{1.1}
\begin{center}
\begin{tabular}
{|@{\hspace{1pt}}c@{\hspace{1pt}} |@{\hspace{1pt}}c@{\hspace{1pt}}
|@{\hspace{1pt}}c@{\hspace{1pt}} |@{\hspace{1pt}}c@{\hspace{1pt}}
|@{\hspace{1pt}}c@{\hspace{1pt}} |@{\hspace{1pt}}c@{\hspace{1pt}}
|} \hline
\m{1}{2}&\m{2}{1}\\\hline
\m{2}{1}&\m{1}{2}\\\hline
\end{tabular}
\end{center}
\begin{center}
\begin{figure}[ht]
\label{2x2} \vspace*{-7mm} \caption{An intercalate}
\end{figure}
\end{center}
\vspace*{-7mm}
We call a Latin bitrade {\sf circulant} if it can be obtained from the elements of its first row, called
{\sf base row}, by permuting them diagonally.
See Figure~2.
\def1.3{1.0}
\begin{center}
\begin{tabular}
{|@{\hspace{1pt}}c@{\hspace{1pt}} |@{\hspace{1pt}}c@{\hspace{1pt}}
|@{\hspace{1pt}}c@{\hspace{1pt}}|@{\hspace{1pt}}c@{\hspace{1pt}}
|@{\hspace{1pt}}c@{\hspace{1pt}}|} \hline
\m{2}{1}&\m{1}{3}&\m{3}{2}&\x.&\x.\\\hline
\x.&\m{3}{2}&\m{2}{4}&\m{4}{3}&\x.\\\hline
\x.&\x.&\m{4}{3}&\m{3}{5}&\m{5}{4}\\\hline
\m{1}{5}&\x.&\x.&\m{5}{4}&\m{4}{1}\\\hline
\m{5}{2}&\m{2}{1}&\x.&\x.&\m{1}{5}\\\hline
\end{tabular}
\hspace*{10mm}
\begin{tabular}
{|@{\hspace{1pt}}c@{\hspace{1pt}} |@{\hspace{1pt}}c@{\hspace{1pt}}
|@{\hspace{1pt}}c@{\hspace{1pt}}|@{\hspace{1pt}}c@{\hspace{1pt}}
|@{\hspace{5pt}}c@{\hspace{5pt}}|} \hline
\m{2}{1}&\m{1}{3}&\m{3}{2}&\x.&\x.\\\hline
\x.&$\searrow$&$\searrow$&$\searrow$&\x.\\\hline
\x.&\x.&\ &\ &\ \\\hline
\ &\x.&\x.&\ &\ \\\hline
\ &\ &\x.&\x.&\ \\\hline
\end{tabular}
\\\vspace*{-.01mm}
\end{center}
\begin{center}
\begin{figure}[ht]
\label{circulant} \vspace*{-6mm} \caption{A circulant
$3$--homogeneous Latin bitrade of volume $15$ and its base row}
\end{figure}
\end{center}
\begin{example}\label{k=4}
The following is a base row of a circulant $4$--homogeneous Latin
bitrade of volume $4m$ for $m > 4$:
$$D_m^{4}=\{(3,2)_1,(1,4)_2,(4,1)_3,(2,3)_4\}.$$
Note that since in a base row of a circulant Latin
bitrade $T= (T_1, T_2 )$, all the elements are in the {\em first} row, we
use the notation $(i,j)_c$ for $(1,c;i) \in T_1$ and
$(1,c;j)\in T_2$.
\end{example}
It is proved in \cite{MR2170114} that
$3$--homogeneous Latin bitrades of volume $3m$ exist for all $m\geq
3$, and in~\cite{MR2139816} they have discussed minimal $4$--homogeneous Latin bitrades. In~\cite{MR2220235},
among other results it is shown that
the answer for Question~\ref{existence} is positive for all $m \ge k$, where $3 \le k \le 8$. While there is
an error in Theorem~6 of~\cite{MR2220235}, but the results are valid and we will explain this in the
last section (Section 4.1). The following results from~\cite{MR2220235} will be used in this paper.
\begin{oldtheorem} {\rm (\cite{MR2220235})}.\label{sum}
If $\ell \neq 2, 6$ and for each $k\in\{k_1,\ldots,k_{\ell}\}$
there exists a $k$--homogeneous Latin bitrade of volume $kp$,
then a $(k_1+\cdots+k_{\ell})$--homogeneous Latin bitrade of
volume $(k_1+\cdots+k_{\ell}){\ell}p$ exists. {\rm(}Some $k_i$s
can possibly be zero{\rm)}.
\end{oldtheorem}
\begin{oldtheorem} {\rm (\cite{MR2220235})}.\label{k(k+1)}
For each $k>2$, a $k$--homogeneous Latin bitrade of volume $k(k+1)$
exists.
\end{oldtheorem}
For the case of $k=2$ the following holds.
\begin{oldtheorem} {\rm (\cite{MR2220235})}.
\label{2hom} For any $m\geq 1$, there exists a $2$--homogeneous
Latin bitrade of volume $2m$ if and only if $m$ is an even
integer.
\end{oldtheorem}
\begin{oldtheorem} {\rm (\cite{MR2220235})}.
\label{5m} For any $m=5{\ell}$ and $3\leq k \leq m$, there exists
a $k$--homogeneous Latin bitrade of volume $km$.
\end{oldtheorem}
\begin{oldtheorem}\label{2k-1} {\rm (\cite{MR2220235})}.
Consider an arbitrary integer $k$. If for any ${k+1}\leq m\leq
2k-1$ there exists a $k$--homogeneous Latin bitrade of volume
$km$, then for any $m \geq k$ there exists a $k$--homogeneous
Latin bitrade of volume $km$.
\end{oldtheorem}
Here we prove that for each given odd integer $k\ge 3$ and for $m
\ge k$, all $k$--homogeneous Latin bitrades of volume $km$ exist
and for all even integers $k>2$ and $m \ge \min\{k+u,
\frac{3k}{2}\}$, where $u$ is any odd integer which divides $k$,
all $k$--homogeneous Latin bitrades of volume $km$ exist. We also
show that for $3\le k\leq 37$ and $m\geq k$, $k$--homogeneous
Latin bitrades of volume $km$ exist.
\section{Constructions and general results}
We discuss our constructions depending on the parity of $k$.
\subsection{$k$ is odd}
\begin{theorem}\label{odd}
A $k$--homogeneous Latin bitrade of volume $km$ exists for all
odd integers $k$ and $m \ge k\ge 3$.
\end{theorem}
\begin{proof}{
Assume $k=2{\ell}+1$ and $m\geq k$. The following is a base row
of a circulant $k$--homogeneous Latin bitrade of volume $km$:
$$B_m^{2{\ell}+1}=
(\bigcup_{i=1}^{{\ell}+1}({\ell}+i,i)_{2i-1})\bigcup(\bigcup_{i=1}^{{\ell}}(i,{\ell}+1+i)_{2i}).$$
\vspace*{-1.1cm}}\end{proof}
\vspace*{0.1mm}
\begin{theorem}\label{ primary odd}
All constructed circulant $k$--homogeneous Latin bitrades in
Theorem~\ref{odd}, are primary.
\end{theorem}
\begin{proof}{Suppose $(T,T^{*})$ is the Latin bitrade constructed in the proof of
Theorem~\ref{odd}. Let $(U,U^{*})$ be a Latin bitrade such that
$U\subseteq T$ and $U^{*}\subseteq T^{*}$, we show that
$(U,U^{*})=(T,T^{*})$. Without loss of generality assume that
$(1,1;{\ell}+1) \in U$ and therefore $(1,1;1) \in U^{*}$. Since
$1$ must appear in the first row of $U$ and since $U \subseteq
T$, the only possibility is $(1,2;1) \in U$. Then we must have
$(1,2;{\ell}+2) \in U^{*}$. Similarly $(1,3;{\ell}+2) \in U$,
thus $(1,3;2) \in U^{*}$. Following this process results that
$(1,2{\ell}+1;2{\ell}+1) \in U$, and then
$(1,2{\ell}+1;{\ell}+1) \in U^{*}$. Therefore all the elements in
the first row of $T$ ($T^{*}$) are the same as all the elements
in the first row of $U$ ($U^{*}$). With the similar argument the
first column of $T$ ($T^{*}$) is the same as the first column of
$U$ ($U^{*}$). Finally this reasoning ends up showing that $U=T$
and $U^{*}=T^{*}$. }\end{proof}
\subsection{$k$ is even}
\begin{theorem}\label{even 3k/2}
A $k$--homogeneous Latin bitrade of volume $km$ exists for all
even integers $k>4$ and $m \ge \frac{3k}{2}.$
\end{theorem}
\begin{proof}{
Let $k=2a$ ($k>4$) and $m \ge \frac{3k}{2}$. The following is a
base row of a circulant $k$--homogeneous Latin bitrade of volume
$km$, when ${\ell}=a-2$:
$$B_m^{2{\ell}+1}\bigcup\{(3a-1,3a-2)_{2a-2},(3a-2,3a)_{2a-1},(3a,3a-1)_{2a}\}.$$
\vspace*{-1cm}}\end{proof}
\noindent {\bf Notation.} Note that a base row $B_m^{2{\ell}+1}$
was defined in Theorem~\ref{odd}. We use a more general
notation, $B_m^{(r)(2{\ell}+1)}$, for a base row obtained from
$B_m^{2{\ell}+1}$ by adding $2(r-1)(2{\ell}+1)$ for both elements
in each cell of $B_m^{2{\ell}+1}$ and moving entry of each cell
$x$ to the cell $x+(r-1)(2{\ell}+1)$. Also for even $k>2$ we
denote by $C_m^{k(r)(2{\ell}+1)}$ a base row obtained from
$B_m^{2{\ell}+1}$, by adding $(2r-1)(2{\ell}+1)$ for both
elements in each cell of $B_m^{2{\ell}+1}$ and moving entry of
each cell $y$ to the cell $y+k/2+r(2{\ell}+1)$.
\begin{theorem}\label{even k+u}
A $k$--homogeneous Latin bitrade of volume $km$ exists for all
even integers $k>2$ and $m \ge (k+u)$, where $u$ is any odd
integer greater than $1$ that divides $k$.
\end{theorem}
\begin{proof}{
If $u=2{\ell}+1$ then let $s=k/2u$. The following is a base row
of a circulant $k$--homogeneous Latin bitrade of volume $km$:
$$(\bigcup_{r=1}^{s+1} B_m^{(r)(2{\ell}+1)})\bigcup(\bigcup_{r=1}^{s-1}
C_m^{k(r)(2{\ell}+1)}).$$
\vspace*{-1cm}}\end{proof}
\section{More constructions}
The following theorem is very useful recursive construction.
\begin{theorem}\label{kmxln}
Let $m\geq k$ and $n\ge {\ell}$. If there exist a
$k$--homogeneous Latin bitrade of volume $km$, and an
${\ell}$--homogeneous Latin bitrade of volume ${\ell}n$, then
there exists a $k{\ell}$--homogeneous Latin bitrade of volume
$(k{\ell})(mn).$
\end{theorem}
\begin{proof}{
We construct a $k{\ell}$--homogeneous Latin bitrade of volume $(k{\ell})(mn)$
in the following way. Suppose $(T_1,T_2)$ is a $k$--homogeneous
Latin bitrade of volume $km$. We replace each $i$ in $T_1$ and
$T_2$ with an ${\ell}$--homogeneous Latin trade of volume
${\ell}n$ whose elements are from
$\{(i-1)n+1,(i-1)n+2,\ldots,in\}$; and the empty cells in $T_1$
and $T_2$ with an empty $n\times n$ array. As a result we
obtain a $k{\ell}$--homogeneous Latin bitrade of volume
$(k{\ell})(mn).$
}\end{proof}
\begin{example}\label{example3x5}
The existence of a
$2$--homogeneous Latin bitrade of volume $4$ (an intercalate), and a
$3$--homogeneous Latin bitrade of volume $15$ imply the existence of a
$6$--homogeneous Latin bitrade of volume $60$. Indeed we take a Latin trade
of an intercalate of the following form:
\begin{center}
\begin{tabular}
{|@{\hspace{5pt}}c@{\hspace{5pt}} |@{\hspace{5pt}}c@{\hspace{5pt}}
|} \hline a&b\\\hline
b&a\\\hline
\end{tabular}
\end{center}
then for $i=1,2,3,4,5$ let $a=2(i-1)$ and $b=2(i-1)+1$, we replace them by the filled cells of the $3$--homogeneous
Latin bitrade of volume $15$ (of Figure~$2$) and obtain the following.
\def1.3{1.3}
\begin{center}
\begin{tabular}
{||@{\hspace{2.5pt}}c@{\hspace{2.5pt}} |@{\hspace{2.5pt}}c@{\hspace{2.5pt}}
||@{\hspace{2.5pt}}c@{\hspace{2.5pt}} |@{\hspace{2.5pt}}c@{\hspace{2.5pt}}
||@{\hspace{2.5pt}}c@{\hspace{2.5pt}} |@{\hspace{2.5pt}}c@{\hspace{2.5pt}}
||@{\hspace{2.5pt}}c@{\hspace{2.5pt}} |@{\hspace{2.5pt}}c@{\hspace{2.5pt}}
||@{\hspace{2.5pt}}c@{\hspace{2.5pt}}
|@{\hspace{1pt}}c@{\hspace{1pt}} ||} \hline\hline
\m{2}{0}&\m{3}{1}&\m{0}{4}&\m{1}{5}&\m{4}{2}&\m{5}{3}&\x.&\x.&\x.&\x.\\\hline
\m{3}{1}&\m{2}{0}&\m{1}{5}&\m{0}{4}&\m{5}{3}&\m{4}{2}&\x.&\x.&\x.&\x.\\\hline\hline
\x.&\x.&\m{4}{2}&\m{5}{3}&\m{2}{6}&\m{3}{7}&\m{6}{4}&\m{7}{5}&\x.&\x.\\\hline
\x.&\x.&\m{5}{3}&\m{4}{2}&\m{3}{7}&\m{2}{6}&\m{7}{5}&\m{6}{4}&\x.&\x.\\\hline\hline
\x.&\x.&\x.&\x.&\m{6}{4}&\m{7}{5}&\m{4}{8}&\m{5}{9}&\m{8}{6}&\m{9}{7}\\\hline
\x.&\x.&\x.&\x.&\m{7}{5}&\m{6}{4}&\m{5}{9}&\m{4}{8}&\m{9}{7}&\m{8}{6}\\\hline\hline
\m{0}{8}&\m{1}{9}&\x.&\x.&\x.&\x.&\m{8}{6}&\m{9}{7}&\m{6}{0}&\m{7}{1}\\\hline
\m{1}{9}&\m{0}{8}&\x.&\x.&\x.&\x.&\m{9}{7}&\m{8}{6}&\m{7}{1}&\m{6}{0}\\\hline\hline
\m{8}{2}&\m{9}{3}&\m{2}{0}&\m{3}{1}&\x.&\x.&\x.&\x.&\m{0}{8}&\m{1}{9}\\\hline
\m{9}{3}&\m{8}{2}&\m{3}{1}&\m{2}{0}&\x.&\x.&\x.&\x.&\m{1}{9}&\m{0}{8}\\\hline\hline
\end{tabular}
\end{center}
\begin{center}
\begin{figure}[ht]
\label{3x5} \vspace*{-6mm} \caption{A $6$--homogeneous Latin bitrade of volume $60$}
\end{figure}
\end{center}
\end{example}
In the following we will improve the interval given in Theorem~\ref{2k-1}. First we need a lemma and a corollary.
\begin{lemma}\label{k+3}
A $k$--homogeneous Latin bitrade of volume $km$ exists for all
integers $k$ and $m = k+3.$
\end{lemma}
\begin{proof}{If $k$ is odd, the statement follows from Theorem
\ref{odd}. For $k=2\ell$, in each case in the following, we
introduce a base row of a circulant $k$--homogeneous Latin
bitrade of volume $km$, depending on the modulo classes of $k$.
First we define two types for the \underline{first row} in $T$:
{\bf Type I}. For $1 \le i \le \ell -1$, in the $(2i-1)$-th cell
($2i$-th cell, respectively) we put $i$ ($\ell +2+i$,
respectively). In the $(k-1)$-th and $m$-th cells we put $\ell$
and $\ell +2$, respectively.
{\bf Type II}. For $1 \le i \le \ell -1$, in the $(2i-1)$-th cell
($2i$-th cell, respectively) we put $i$ ($\ell +2+i$,
respectively). In the $k$-th and $m$-th cells we put $\ell +1$
and $\ell +2$, respectively.
Now we introduce the base rows.
\begin{enumerate}
\item \qquad
$k \equiv 1 \ ({\rm mod \ 7})$
Let the first row of $T$ be as in Type I. For $T^*$, in the first
row and in the $(7i+3)$-th cell ($i\ge 0$ and $7i+3 < k$) we let
$a+4$ ({\rm mod $m$}), where $a$ is the element of $T$ in the
same cell. Now in the $(k-1)$-th and $m$-th cells of $T^*$ we put
$1$ and $\ell +4$, respectively. Finally in each cell $c$ of the
first row in $T^*$ which is filled in $T$ but is so far empty in
$T^*$, we let the entry of $(c+1)$-th cell of $T$.
\item \qquad
$k \equiv 2 \ ({\rm mod \ 7})$
Let the first row of $T$ be as in Type I. For $T^*$, in the first
row and in the $(7i+4)$-th cell ($i\ge 0$ and $7i+4 < k$) we let
$a+4$ ({\rm mod $m$}), where $a$ is the element of $T$ in the
same cell. Now in the $(k-1)$-th and $m$-th cells of $T^*$ we put
$1$ and $3$, respectively. Finally in each cell $c$ of the first
row in $T^*$ which is filled in $T$ but is so far empty in $T^*$,
we let the entry of $(c+1)$-th cell of $T$.
\item \qquad
$k \equiv 3 \ ({\rm mod \ 7})$
Let the first row of $T$ be as in Type II. For $T^*$, in the first
row and in the $(7i+3)$-th cell ($i\ge 0$ and $7i+3 < k$) we let
$a+4$ ({\rm mod $m$}), where $a$ is the element of $T$ in the
same cell. Now in the $(k-2)$-th, $k$-th and $m$-th cells of
$T^*$ we put $1$, $\ell +2$ and $\ell +4$, respectively. Finally
in each cell $c$ of the first row in $T^*$ which is filled in $T$
but is so far empty in $T^*$, we let the entry of $(c+1)$-th cell
of $T$.
\item \qquad
$k \equiv 4 \ ({\rm mod \ 7})$
Let the first row of $T$ be as in Type II. For $T^*$, in the first
row and in the $(7i+4)$-th cell ($i\ge 0$ and $7i+4 < k$) we let
$a+4$ ({\rm mod $m$}), where $a$ is the element of $T$ in the
same cell. Now in the $(k-2)$-th, $k$-th and $m$-th cells of
$T^*$ we put $1$, $\ell +2$ and $3$, respectively. Finally in
each cell $c$ of the first row in $T^*$ which is filled in $T$
but is so far empty in $T^*$, we let the entry of $(c+1)$-th cell
of $T$.
\item \qquad
$k \equiv 5 \ ({\rm mod \ 7})$
Let the first row of $T$ be as in Type I. For $T^*$, in the first
row and in the $(7i+r)$-th cell ($i\ge 0$, $r=1,2,3$ and $7i+r <
k$) we let $a+4$ ({\rm mod $m$}), where $a$ is the element of
$T$ in the same cell. Now in the $(k-1)$-th and $m$-th cells of
$T^*$ we put $\ell +2$ and $\ell +4$, respectively. Finally in
each cell $c$ of the first row in $T^*$ which is filled in $T$
but is so far empty in $T^*$, we let the entry of $(c+1)$-th
cell of $T$.
\item \qquad
$k \equiv 6 \ ({\rm mod \ 7})$
Let the first row of $T$ be as in Type I. For $T^*$, in the first
row and in the $(7i+r)$-th cell ($i\ge 0$, $r=2,4$ and $7i+r < k$)
we let $a+4$ ({\rm mod $m$}), where $a$ is the element of $T$ in
the same cell. Now in the $(k-1)$-th and $m$-th cells of $T^*$
we put $\ell +2$ and $3$, respectively. Finally in each cell $c$
of the first row in $T^*$ which is filled in $T$ but is so far
empty in $T^*$, we let the entry of $(c+1)$-th cell of $T$.
\item \qquad
$k \equiv 0 \ ({\rm mod \ 7})$
Let the first row of $T$ be as in Type I. For $T^*$, in the first
row and in the $(7i+3)$-th cell ($i\ge 0$ and $7i+3 < k$) we let
$a+4$ ({\rm mod $m$}), where $a$ is the element of $T$ in the
same cell. Now in the $(k-1)$-th and $m$-th cells of $T^*$ we put
$\ell +2$ and $\ell +4$, respectively. Finally in each cell $c$
of the first row in $T^*$ which is filled in $T$ but is so far
empty in $T^*$, we let the entry of $(c+1)$-th cell of $T$.
\end{enumerate}
\vspace*{-1cm}}\end{proof}
\begin{corollary}\label{k+6}
A $k$--homogeneous Latin bitrade of volume $km$ exists for all
integers $k$ and $m = k+6$.
\end{corollary}
\begin{proof}{
If $k$ is an odd integer, then the statement follows from
Theorem~\ref{odd}. In case $k=2{\ell}$ we know that by
Lemma~\ref{k+3} there exist an ${\ell}$--homogeneous Latin
bitrade of volume ${\ell}({\ell}+3)$ and a $2$--homogeneous Latin
bitrade of volume $4$, therefore by Theorem~\ref{kmxln} there
exists a $2{\ell}$--homogeneous Latin bitrade of volume
$2{\ell}(2{\ell}+6)$.
\vspace*{-0.7cm}}\end{proof}
\begin{lemma}\label{k+2,4}
A $k$--homogeneous Latin bitrade of volume $km$ exists for all
integers $k$ and $m = k+2,k+4.$
\end{lemma}
\begin{proof}{
If $k$ is an odd integer then the statement follows from
Theorem~\ref{odd}.
Let $k=2{\ell}$.
\begin{itemize}
\item
$m=k+2$
By Theorem~\ref{k(k+1)} and Theorem~\ref{kmxln} there exists a
$2{\ell}$--homogeneous Latin bitrade of volume
$2{\ell}(2{\ell}+2)$.
\item $m=k+4$
By previous case and by Theorem~\ref{kmxln} there exists a
$2{\ell}$--homogeneous Latin bitrade of volume
$2{\ell}(2{\ell}+4)$.
\end{itemize}
\vspace*{-1cm}}\end{proof}
The following theorem follows from Theorem~\ref{even 3k/2}, Lemmas~\ref{k+3}~and~\ref{k+2,4}.
%
\begin{theorem}\label{k+5}
Let $k$ be an integer. If for all $m$, ${k+5}\leq m<
3k/2$, there exists a $k$--homogeneous Latin bitrade of volume
$km$, then for any $m \geq k$ there exists a $k$--homogeneous
Latin bitrade of volume $km$.
\end{theorem}
\section{The intervals}
From Theorems~\ref{even k+u} and ~\ref{k+5} a result follows which
is very useful in the constructions of the needed bitrades:
\begin{corollary}\label{3k',5k'}
If $k$ is a multiple of $3$ or $5$, then there exists a $k$--homogeneous
Latin bitrade of volume $km$ for all $m \geq k$.
\end{corollary}
\subsection{$2\leq k\leq8$}
The `proof' of Theorem~6 in \cite{MR2220235} is false, but we may
apply Theorem~\ref{kmxln} above, and Theorem~\ref{sum}
(Theorem~1 in \cite{MR2220235}) to correct all results in that paper where ever its Theorem~6 is used.
For example for the Case~1 in the proof of Theorem~9
(in~\cite{MR2220235}), we take the following parameters in
Theorem~\ref{sum} (Theorem~1 in \cite{MR2220235}): $k_i=5$ for $1
\leq i \leq {\ell}^{'} $, \ $k_i=0$ for ${\ell}^{'}+1 \leq i \leq
{\ell}$ and $p=5$. Or for the Case~4 in the proof of Main
Theorem~2 (in \cite{MR2220235}), since there exist a
$4$--homogeneous Latin bitrade of volume $24$ and a
$2$--homogeneous Latin bitrade of volume $4$, so by
Theorem~\ref{kmxln} above, there exists an $8$--homogeneous Latin
bitrade of volume $96$.
So for the interval $2\leq k\leq8$, Example~\ref{k=4}, Theorem~\ref{2hom} and the following theorem answer Question~\ref{existence}.
\begin{oldtheorem}
{\rm\bf (Main Theorem~2 of ~\cite{MR2220235}).}
\label{kolli} For any $k$, $5\leq k\leq 8$ and $m\geq k $, there
exists a $k$--homogeneous Latin bitrade of volume $km$.
\end{oldtheorem}
\subsection{$9\leq k\leq37$}
\begin{theorem}\label{37}
If $9\leq k\leq 37 $ then there exists a $k$--homogeneous Latin
bitrade of volume $km$ for any $m \geq k$.
\end{theorem}
\begin{proof}{
Note that the case $k$ odd follows by Theorem~\ref{odd}. The
cases$\linebreak$ $k=10,12,18,20,24,30,36$ follow by
Corollary~\ref{3k',5k'}. For $k=14$, by$\linebreak$
Theorem~\ref{k+5} we only need to show for $m=19$ and $m=20$.
For $m=20$ we apply Theorem~\ref{5m}. The following base row is for $m=19$:
$D_{19}^{14}=\{(1,11)_1,(11,2)_2,(2,12)_3,(12,3)_4,(3,13)_5,(13,4)_6,
(4,14)_7,$ \\
\hspace*{17.7mm}$(14,5)_8,(5,1)_9,(6,7)_{11},
(7,8)_{13},(8,9)_{15},(9,10)_{17},(10,6)_{19}\}.$
For $k=16$, again by Theorem~\ref{k+5}, it suffices to show the
existence of $16$--homogeneous Latin bitrades of volume $16m$,
where $21 \le m \le 23$. The case $m=21$ follows from
Theorem~\ref{sum} by letting $k_1=4$, $k_2=k_3=6$ and $p=7$. The
case $m=22$ follows from Theorem~\ref{kmxln}.
And the following base row is for $m=23$:
$D_{23}^{16}=\{(1,13)_1,(13,2)_2,(2,14)_3,(14,3)_4,(3,15)_5,(15,4)_6,
(4,16)_7,$ \\
\hspace*{17.7mm}$(16,5)_8,(5,17)_9,(17,6)_{10},(6,1)_{11},(7,8)_{13},(8,10)_{16},
(10,11)_{19},$ \\
\hspace*{17.7mm}$(11,12)_{21},(12,7)_{23}\}$.
Similarly for $k=22,26,28,32,34$ we include the base rows in
the $\linebreak$ Appendix for odd integers ${k+5}\leq m< 3k/2$ such
that $m\ne 5{\ell}$. By Theorems~\ref{kmxln} and~\ref{5m} the
proof is complete. }\end{proof}
The results above motivates us to conjecture that:
\begin{conjecture}
\label{existenceconjecture} For all $m$ and $k$, $m \ge k\ge
3$, there exists a $k$--homogeneous Latin bitrade of volume $km$.
\end{conjecture}
\section*{Appendix}
The followings are base rows of bitrades needed in the proof of Theorem~\ref{37}:
\begin{itemize}
\item{$\bf k=22$\\
$D_{27}^{22}=\{(1,19)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},\\
\hspace*{12.6mm}(14,17)_{9},(16,27)_{10},(7,18)_{11},(19,6)_{12},(21,10)_{13},(9,24)_{14},\\
\hspace*{12.6mm}(24,21)_{15},(10,11)_{16},(27,13)_{17},(12,15)_{22},(15,1)_{23},(13,16)_{24},\\
\hspace*{12.6mm}(18,14)_{25},(17,8)_{26}\}$
$D_{29}^{22}=\{(1,21)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,16)_{9},(16,15)_{10},(7,17)_{11},(19,8)_{12},(21,6)_{13},(9,24)_{14},$\\
\hspace*{12.6mm}$(24,10)_{15},(10,13)_{16},(27,11)_{17},(29,27)_{18},(12,1)_{19},(13,29)_{21},$\\
\hspace*{12.6mm}$(15,14)_{24},(17,19)_{27}\}$
$D_{31}^{22}=\{(1,24)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,16)_{9},(16,15)_{10},(7,1)_{11},(19,21)_{12},(21,19)_{13},(9,11)_{14},$\\
\hspace*{12.6mm}$(24,10)_{15},(10,27)_{16},(27,8)_{17},(29,14)_{18},(12,13)_{19},(13,29)_{21},$\\
\hspace*{12.6mm}$(15,17)_{24},(17,6)_{27}\}$ }
\item{$\bf k=26$\\
$D_{31}^{26}=\{(1,19)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,1)_{12},(21,20)_{13},(9,22)_{14},$\\
\hspace*{12.6mm}$(24,10)_{15},(10,27)_{16},(27,13)_{17},(29,11)_{18},(31,29)_{19},(12,31)_{22}$\\
\hspace*{12.6mm}$,(13,16)_{25},(15,14)_{26},(18,8)_{27},(20,18)_{28},(22,21)_{29},(17,24)_{30}\}$
$D_{33}^{26}=\{(1,22)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,1)_{10},(7,17)_{11},(19,20)_{12},(21,18)_{13},(9,24)_{14},$\\
\hspace*{12.6mm}$(24,10)_{15},(10,27)_{16},(27,29)_{17},(29,14)_{18},(12,11)_{19},(32,13)_{20}$\\
\hspace*{12.6mm}$,(13,15)_{21},(15,32)_{25},(18,16)_{29},(17,19)_{30},(22,21)_{31},(20,8)_{32}\}$
$D_{37}^{26}=\{(1,27)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,19)_{11},(19,18)_{12},(21,24)_{13},(9,21)_{14},$\\
\hspace*{12.6mm}$(24,10)_{15},(10,29)_{16},(27,8)_{17},(29,32)_{18},(12,13)_{19},(32,16)_{20},$\\
\hspace*{12.6mm}$(13,11)_{21},(35,14)_{22},(37,35)_{23},(15,17)_{24},(17,37)_{27},(18,1)_{29}\}$
}
\item{$\bf k=28$\\
$D_{33}^{28}=\{(1,20)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,8)_{12},(21,1)_{13},(9,24)_{14},$\\
\hspace*{12.6mm}$(24,23)_{15},(10,11)_{16},(27,29)_{17},(29,27)_{18},(31,13)_{19},(33,31)_{20},$\\
\hspace*{12.6mm}$(12,14)_{21},(13,33)_{26},(15,19)_{27},(17,18)_{28},(22,16)_{29},(20,10)_{30},$\\
\hspace*{12.6mm}$(23,22)_{31},(18,21)_{32}\}$
$D_{37}^{28}=\{(1,27)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,37)_{12},(21,22)_{13},(9,21)_{14},$\\
\hspace*{12.6mm}$(24,10)_{15},(10,24)_{16},(27,29)_{17},(29,11)_{18},(12,32)_{19},(32,14)_{20}$\\
\hspace*{12.6mm}$,(13,35)_{21},(35,18)_{22},(37,13)_{23},(15,16)_{24},(17,1)_{27},(18,20)_{29}$\\
\hspace*{12.6mm}$,(20,19)_{32},(22,8)_{35}\}$
$D_{39}^{28}=\{(1,27)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,20)_{12},(21,1)_{13},(9,24)_{14},$\\
\hspace*{12.6mm}$(24,22)_{15},(10,8)_{16},(27,13)_{17},(29,11)_{18},(12,14)_{19},(32,29)_{20},$\\
\hspace*{12.6mm}$(13,32)_{21},(35,16)_{22},(37,35)_{23},(15,37)_{24},(17,18)_{27},(18,19)_{29}$\\
\hspace*{12.6mm}$,(20,21)_{32},(22,10)_{35}\}$
$D_{41}^{28}=\{(1,32)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,21)_{12},(21,20)_{13},(9,29)_{14},$\\
\hspace*{12.6mm}$(24,27)_{15},(10,24)_{16},(27,11)_{17},(29,13)_{18},(12,8)_{19},(32,16)_{20},$\\
\hspace*{12.6mm}$(13,35)_{21},(35,10)_{22},(37,14)_{23},(15,37)_{24},(40,18)_{25},(17,19)_{27}$\\
\hspace*{12.6mm}$,(18,40)_{29},(20,1)_{32}\}$ }
\item{$\bf k=32$\\
$D_{37}^{32}=\{(1,22)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,8)_{12},(21,20)_{13},(9,1)_{14},$\\
\hspace*{12.6mm}$(24,10)_{15},(10,25)_{16},(27,29)_{17},(29,26)_{18},(12,13)_{19},(32,35)_{20},$\\
\hspace*{12.6mm}$(35,32)_{21},(37,36)_{22},(36,16)_{23},(13,37)_{27},(17,14)_{29},(15,18)_{30},$\\
\hspace*{12.6mm}$(18,23)_{31},(22,21)_{32},(25,19)_{33},(23,24)_{34},(26,11)_{35},(20,27)_{36}\}$\\
$D_{39}^{32}=\{(1,24)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,8)_{12},(21,20)_{13},(9,22)_{14},$\\
\hspace*{12.6mm}$(24,39)_{15},(10,25)_{16},(27,11)_{17},(29,32)_{18},(12,29)_{19},(32,13)_{20},$\\
\hspace*{12.6mm}$(13,16)_{21},(35,14)_{22},(37,35)_{23},(39,37)_{24},(15,1)_{25},(17,18)_{28},$\\
\hspace*{12.6mm}$(18,19)_{33},(20,21)_{34},(26,23)_{35},(23,27)_{36},(25,26)_{37},(22,10)_{38}\}$\\
$D_{41}^{32}=\{(1,26)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,8)_{12},(21,20)_{13},(9,22)_{14},$\\
\hspace*{12.6mm}$(24,1)_{15},(10,25)_{16},(27,11)_{17},(29,32)_{18},(12,29)_{19},(32,10)_{20},$\\
\hspace*{12.6mm}$(13,16)_{21},(35,13)_{22},(37,35)_{23},(15,37)_{24},(40,18)_{25},(17,19)_{27},$\\
\hspace*{12.6mm}$(18,40)_{29},(20,21)_{32},(22,24)_{37},(25,23)_{38},(23,27)_{39},(26,14)_{40}\}$\\
$D_{43}^{32}=\{(1,29)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,8)_{12},(21,1)_{13},(9,23)_{14},$\\
\hspace*{12.6mm}$(24,22)_{15},(10,27)_{16},(27,25)_{17},(29,13)_{18},(12,32)_{19},(32,14)_{20},$\\
\hspace*{12.6mm}$(13,10)_{21},(35,37)_{22},(37,35)_{23},(15,40)_{24},(40,16)_{25},(42,18)_{26},$\\
\hspace*{12.6mm}$(17,20)_{27},(18,19)_{29},(20,42)_{32},(22,21)_{35},(23,24)_{37},(25,11)_{40}\}$\\
$D_{47}^{32}=\{(1,35)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,8)_{12},(21,20)_{13},(9,24)_{14},$\\
\hspace*{12.6mm}$(24,23)_{15},(10,11)_{16},(27,29)_{17},(29,27)_{18},(12,32)_{19},(32,13)_{20},$\\
\hspace*{12.6mm}$(13,10)_{21},(35,37)_{22},(37,40)_{23},(15,16)_{24},(40,19)_{25},(42,14)_{26},$\\
\hspace*{12.6mm}$(17,18)_{27},(45,42)_{28},(18,45)_{29},(20,22)_{32},(22,21)_{35},(23,1)_{37}\}$
}
\item{$\bf k=34$\\
$D_{39}^{34}=\{(1,24)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,8)_{12},(21,20)_{13},(9,22)_{14},$\\
\hspace*{12.6mm}$(24,10)_{15},(10,25)_{16},(27,39)_{17},(29,28)_{18},(12,13)_{19},(32,11)_{20},$\\
\hspace*{12.6mm}$(34,32)_{21},(37,34)_{22},(39,38)_{23},(38,37)_{24},(13,1)_{26},(15,16)_{30},$\\
\hspace*{12.6mm}$(17,21)_{31},(20,19)_{32},(23,18)_{33},(25,23)_{34},(18,27)_{35},(28,29)_{36},$\\
\hspace*{12.6mm}$(26,14)_{37},(22,26)_{38}\}$ \\
$D_{41}^{34}=\{(1,25)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,8)_{12},(21,20)_{13},(9,22)_{14},$\\
\hspace*{12.6mm}$(24,10)_{15},(10,41)_{16},(27,26)_{17},(29,28)_{18},(12,13)_{19},(32,35)_{20},$\\
\hspace*{12.6mm}$(13,32)_{21},(35,14)_{22},(37,16)_{23},(39,37)_{24},(41,39)_{25},(15,1)_{26},$\\
\hspace*{12.6mm}$(17,18)_{27},(18,19)_{33},(22,23)_{35},(20,21)_{36},(28,24)_{37},(26,27)_{38},$\\
\hspace*{12.6mm}$(25,29)_{39},(23,11)_{40}\}$ \\
$D_{43}^{34}=\{(1,27)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,8)_{12},(21,20)_{13},(9,22)_{14},$\\
\hspace*{12.6mm}$(24,42)_{15},(10,1)_{16},(27,26)_{17},(29,28)_{18},(12,14)_{19},(32,35)_{20},$\\
\hspace*{12.6mm}$(13,32)_{21},(35,10)_{22},(37,16)_{23},(15,40)_{24},(40,37)_{25},(42,18)_{26},$\\
\hspace*{12.6mm}$(17,21)_{27},(18,19)_{29},(20,23)_{32},(22,24)_{37},(23,25)_{39},(26,29)_{40},$\\
\hspace*{12.6mm}$(28,11)_{41},(25,13)_{42}\}$ \\
$D_{47}^{34}=\{(1,32)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,8)_{12},(21,20)_{13},(9,22)_{14},$\\
\hspace*{12.6mm}$(24,25)_{15},(10,1)_{16},(27,29)_{17},(29,27)_{18},(12,13)_{19},(32,37)_{20},$\\
\hspace*{12.6mm}$(13,10)_{21},(35,14)_{22},(37,18)_{23},(15,35)_{24},(40,16)_{25},(42,40)_{26},$\\
\hspace*{12.6mm}$(17,42)_{27},(45,21)_{28},(18,19)_{29},(20,45)_{32},(22,23)_{35},(23,24)_{37},$\\
\hspace*{12.6mm}$(25,26)_{40},(26,11)_{42}\}$ \\
$D_{49}^{34}=\{(1,35)_{1},(3,2)_{2},(2,4)_{3},(6,3)_{4},(8,7)_{5},(4,9)_{6},(11,5)_{7},(5,12)_{8},$\\
\hspace*{12.6mm}$(14,6)_{9},(16,15)_{10},(7,17)_{11},(19,8)_{12},(21,20)_{13},(9,27)_{14},$\\
\hspace*{12.6mm}$(24,23)_{15},(10,25)_{16},(27,29)_{17},(29,13)_{18},(12,37)_{19},(32,10)_{20},$\\
\hspace*{12.6mm}$(13,32)_{21},(35,14)_{22},(37,11)_{23},(15,18)_{24},(40,42)_{25},(42,40)_{26},$\\
\hspace*{12.6mm}$(17,16)_{27},(45,19)_{28},(18,22)_{29},(48,45)_{30},(20,48)_{32},(22,21)_{35},$\\
\hspace*{12.6mm}$(23,24)_{37},(25,1)_{40}\}$ }
\end{itemize}
\newpage
\noindent {\bf Acknowledgement.} This research was in part
supported by a grant (\#86050213) from the
Department of Mathematics of
Institute for Studies in Theoretical Physics and Mathematics (IPM)
P. O. Box 19395-5746, Tehran, I. R. Iran. We also appreciate
the help of Amir Hooshang Hosseinpoor for his computer
programming.
\def$'${$'$}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,065 |
\subsection{Event reconstruction}
Analyzer consists of several \verb+Task+s for each step of analysis;
each \verb+Task+ can be switched on/off.\\
In the first step, raw data are read and calibrations are applied to waveforms.
In the second step, waveform analysis specialized for each sub-detector are
performed to extract time and charge of pulses. Waveforms are also used to
identify pileup events and for particle identifications.\\
In the third and last step, events are reconstructed using algorithms implemented by
experts of each sub-detector. Several different algorithms are implemented
to reconstruct each kinematic parameter for crosschecks. Each \verb+Task+ may have
a dedicated \verb+Folder+ to write its result. \verb+Task+s share a \verb+Folder+ to hold results of
a standard choice among those algorithms; this choice is specified by a configuration file.
\verb+Task+s are executed in the same process and results are written in an
output file together.
Figure \ref{eventdisplay} shows a reconstructed experimental event.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=.95\linewidth]{3dev2.png}
\caption{A $\mu^+\to e^+\gamma$ reconstructed event and closer views.
Reconstructed hits in drift chambers and timing
counters, a positron track and a $\gamma$-ray are shown. Color-code of Calorimeter
PMTs represents output of each PMT.}
\label{eventdisplay}
\end{center}
\end{figure}
\subsection{Visualization}
Data quality is monitored for various time-spans: event-by-event, run-by-run or in days.
For event-by-event monitoring, several displays are implemented. Figure \ref{display}
shows one of them. The displays show waveforms, status of trigger, reconstructed hits and tracks
and any other information useful for monitoring. Those displays are used for both online and offline.
When it is used for online monitoring, Analyzer and DAQ run in parallel and data are
transferred over a socket connection. Hard copies of the displays are saved periodically
for remotely monitoring using web-browsers.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=.95\linewidth]{tic2d.png}
\caption{A graphical display of timing counter hits, waveforms. A reconstructed positron track is also shown.}
\label{display}
\end{center}
\end{figure}
Two types of portable document format (PDF) files are automatically prepared by macros,
which read histogram files made by Analyzer.
The first type shows histograms to describe
the run and is made automatically for each run soon after the run is finished.
The second type shows strip charts to monitor time variations of the status
of the detector and of the electronics in a day or a week.
\subsection{Calibration}
Analyzer is used also to compute calibration constants (photomultiplier gains, time-offsets, etc.).
Each calibration constant is associated to a \verb+Task+. The calibration \verb+Task+s are usually run on
events already processed with a preliminary set of calibration constants. The updated
calibration constants can be made available in a variety of format: histograms, text file
or SQL macro.
They can be stored in the database, and used in the next round of reconstruction.
\subsection{Physics analysis}
Event preselection and blinding for physics analysis, described in
section \ref{sec_offline}, are implemented in Analyzer.
On events in the analysis region, likelihood analysis is performed to
calculate the best estimate of the number of $\mu^+\rightarrow e^+\gamma$ signal candidates, its
confidence interval and the significance.
The 90\% confidence interval of the number of signal events is calculated using the unified
approach \cite{feldman_1998}.
We made independent likelihood analysis tools with different statistical methods
or parametrization of probability density functions for cross checks.
\section{Introduction}
\input{introduction}
\section{The MEG software structure}
\input{structure}
\section{REM: a FORTRAN 77 framework }
\input{rem}
\section{GEM: the Monte Carlo simulation }
\input{gem}
\section{The database}
\input{database}
\section{ROME: a framework generator }
\input{rome}
\section{Readout simulation and event mixing}
\input{bartender}
\section{The reconstruction and analysis program}
\input{analyzer}
\section{Offline processing}\label{sec_offline}
\input{offline}
\section{Conclusion}
\input{conclusion}
\section*{Acknowledgment}
We acknowledge the role of Dr.~Stefan~Ritt from PSI,
who is the main author of the online software MIDAS.\\
Integration of each sub-detector part was done by many collaborators; forty
of them have contributed to the MEG software.
\bibliographystyle{unsrt}
\subsection{Implementation of a FORTRAN 77 framework}
The detector simulation section GEM of the MEG software is written in FORTRAN 77, that
was designed for procedure oriented structured programming, not for OO programming.\\
Nevertheless a programming paradigm can be implemented in a variety of programming
languages, even not designed for it. A limited but satisfactory support to the OO
paradigm is at reach also in FORTRAN 77 on the basis of the following list of
approximate equivalences between procedure oriented and OO concepts
\begin{itemize}
\item Class $\leftrightarrow $ Library
\item Class data $\leftrightarrow $ Data structure (FORTRAN 77 Common block)
\item Class interface $\leftrightarrow $ Set of library routines
\item Base Class $\leftrightarrow $ Module standardization
\item Virtual Class $\leftrightarrow$ Alternate choice of libraries
\end{itemize}
\subsection{Modules}
The Module is the basic unit manipulated by the framework that corresponds
to an OO class. Each Module is implemented concretely in a library.
There are different types of Modules, that can be classified as
\begin{itemize}
\item Basic Module : empty Module
\item Steerable Module : Module steerable by configuration files (cards)
\item Data Module : contains only data
\item Algorithm Module : implements an algorithm using other Modules
\item Service Module : provides interface to external libraries
\end{itemize}
These types share a common set of routines and differ by additional functionalities
depending on the Module type implementing the OO paradigm of class hierarchy.
\subsection{The framework: REM}
The framework is a Module with an event loop.
The Modules associated to the framework are accessed in sequence by calling
their routines in the corresponding framework routines.\\
Three module are provided by default in REM
\begin{itemize}
\item Steering cards: FFREAD package
\item I/O : ZEBRA I/O
\item Histogramming : HBOOK package
\end{itemize}
The others Modules are project dependent and their routines are called in
the corresponding framework user routines.
These user routines, provided empty by default, are called by the framework routines.
They can be overwritten implementing the OO inheritance mechanism.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,813 |
Ви́ктор Ю́рьевич Кала́шников (род. 9 марта 1968 года) — российский диабетолог, , член-корреспондент РАН (2016).
Биография
Родился 9 марта 1968 года.
В 1993 году — окончил Московскую медицинскую академию (ММА) имени И. М. Сеченова, затем там же проходил обучение в клинической ординатуре по специальности «кардиология» на кафедре факультетской терапии.
С 1995 по 2008 годы — врач анестезиолог-реаниматолог отделения реанимации и интенсивной терапии клиники кардиологии ММА имени И. М. Сеченова, там же работал в должности научного, затем старшего научного и ведущего научного сотрудника отдела кардиологии научно-исследовательского центра.
В 1996 году — защитил кандидатскую диссертацию, тема: «Отдалённые результаты хирургического лечения злокачественных желудочковых тахикардий».
В 2008 году — защитил докторскую диссертацию, тема: «Использование клинико-экономического анализа в выборе тактики обследования и лечения больных с сердечно-сосудистыми заболеваниями».
С 2008 года по настоящее время — заведующий отделом кардиологии и сосудистой хирургии Национального медицинского исследовательского центра кардиологии.
В январе 2016 года — присвоено звание профессора РАН.
В октябре 2016 году — избран членом-корреспондентом РАН.
Научная деятельность
Разработал концепцию и предложил алгоритм обследования и лечения ишемической болезни сердца у больных сахарным диабетом и критической ишемией нижних конечностей.
Автор более 50 печатных работ, в том числе монографии «Сахарный диабет: острые и хронические осложнения», 4 учебных пособий и 1 патента, соавтор алгоритмов специализированной помощи больным сахарным диабетом, методических рекомендаций.
Под его руководством защищено 7 кандидатских диссертаций.
Награды
Примечания
Ссылки
Выпускники 1-го МГМУ
Преподаватели 1-го МГМУ
Национальный медицинский исследовательский центр кардиологии
Профессора РАН | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,128 |
Дубасова — деревня в составе Горноуральского городского округа и Пригородного района Свердловской области России.
Географическое положение
Деревня расположена на левом берегу реки Бродовки, вблизи её впадения в Петрокаменский пруд (р. Нейва). Деревня находится к северу от Екатеринбурга, в 45 километрах на юго-восток от Нижнего Тагила (по дорогам — в 59 километрах), вблизи большого села Петрокаменского.
Вблизи деревни проходит шоссе местного значения Николо-Павловское — Петрокаменское — Алапаевск.
Население
Примечания
Ссылки
http://semantic.uraic.ru/object/objectedit.aspx?object_id=5583&project=1
Населённые пункты Горноуральского городского округа
Населённые пункты на Бродовке (притоке Нейвы) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,914 |
{"url":"https:\/\/socratic.org\/questions\/how-to-use-the-discriminant-to-find-out-what-type-of-solutions-the-equation-has--16","text":"# How to use the discriminant to find out what type of solutions the equation has for x^2 + 25 = 0?\n\nMar 22, 2018\n\nThe discriminant equals -100. Therefore the equation has 0 solutions.\n\n#### Explanation:\n\nThe discriminant is ${b}^{2} - 4 \\times a \\times c \\text{ }$, and the form of that equation is $a {x}^{2} + b x + c$. Therefore the discriminant is\n\n${0}^{2} - 4 \\times 1 \\times 25 = - 100$\n.\nTherefore the discriminant is -100. This means that the equation has 0 solutions.\n\nDiscriminant $> 0 \\to 2$ Solutions\nDiscriminant$= 0 \\textcolor{w h i t e}{.} \\to 1$ Soltion\nDiscriminant $< 0 \\to 0$ Solutions\n\nMar 22, 2018\n\nThe solution type for this question is such that it belongs to the 'Complex' number set of values.\n\nThe graph does NOT cross the x-axis\n\n#### Explanation:\n\nConsider the standardised form of $y = a {x}^{2} + b x + c = 0$\n\nThe formula is $x = \\frac{- b \\pm \\sqrt{{b}^{2} - 4 a c}}{2 a}$\n\nThe determinate is the part ${b}^{2} - 4 a c$\n\nWrite the given equation as: $y = {x}^{2} + 0 x + 25 = 0$\n\nIn this case: a=1; b=0 and c=25\n\nSo the determinate $\\to {0}^{2} - 4 \\left(1\\right) \\left(25\\right) = - 100$\n\nso we end up with $\\sqrt{- 100}$\n\nAs this is negative we have a complex number solution.\n\nThat is $x \\in \\mathbb{C}$","date":"2021-06-13 21:53:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 14, \"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\": 0.8173929452896118, \"perplexity\": 785.7006569034126}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487610841.7\/warc\/CC-MAIN-20210613192529-20210613222529-00432.warc.gz\"}"} | null | null |
Q: Find Error Line Number in VBA I'm trying to find the line number where my code crashes but many explanation on this site seems to complicated for my level.
My code is basically as below and I have no idea where it's breaking.
Sub1
Call function1
Call function2
End Sub
Other answers on this website seems to be just a short function. But I don't know where to call the function in my code or how to get a popup message. If I'm meant to put my sub1 code into their function, I don't know where either. Beginner here.
A: If your code doesn't have line numbers, then VBA has no way of giving you line numbers.
You can write VBA and make it look 1980-like to do this:
Sub1
On Error GoTo 100
10 Call Function1
20 Call Function2
90 Exit Sub
100 Debug.Print Err.Message & " on line " & Erl
End Sub
But you don't want to do that. Really, you don't need a line number.
You need smaller functions that handle runtime errors.
On Error GoTo ErrHandler
When a runtime error occurs, execution jumps to the line label called ErrHandler.
...
Exit Sub
ErrHandler: '<< the line label is denoted with a colon
What goes in that handler? If you're debugging, you might want to just Stop execution there and inspect your locals:
Stop
Then add Resume on the next line, and press F8 to step into it. Resume will return to the call that caused the error. If that's a function call, then you need to handle runtime errors in that function.
Make sure you never leave Stop and Resume instructions in production code:
Sub WhenWillThisEnd()
On Error GoTo ErrHandler
Debug.Print 42/0
Exit Sub
ErrHandler:
Resume 'jumps back to the line that caused the error
Resume Next 'resumes execution on the line right after the one that went boom
End Sub
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,234 |
An Indian settlement is a census subdivision outlined by the Canadian government Department of Aboriginal Affairs and Northern Development Canada for census purposes. These areas have at least 10 status Indian or non-status Indian people who live, more or less, permanently in the given area. They are usually located on Crown land owned by the federal or provincial government and have not been set apart for the use and the benefit of an Indian band, as is the case with Indian reserves.
See also
Indian Land Claims Settlements
List of Indian settlements in Alberta
List of Indian settlements in Quebec
References
Populated places in Canada
First Nations
Census divisions of Canada | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,963 |
Николай Васильевич Гноевой (24 августа 1922 год, село Святославка — январь 1998 год) — советский казахский партийный и государственный деятель, первый секретарь Урицкого райкома Компартии Казахстана, Кустанайская область, Казахская ССР. Герой Социалистического Труда (1972).
Биография
Родился в 1922 году в крестьянской семье в селе Святославка (сегодня — Карабалыкский район Костанайской области). После окончания семилетки поступил в Казахстанский техникум механизации сельского хозяйства. С 1939 года — заведующий зернохранилища Актюбинского семеноводческого хозяйства в селе Белоглиновка Карабалыкского района. В 1940 году призван на срочную службу в Красную Армию. Участвовал в Великой Отечественной войне. Командовал артиллерийской батареей. Получил тяжёлое ранение во время Курской битвы. После штурма Берлина был назначен комендантом города Шверин. После демобилизации в 1947 году возвратился в Казахстан.
В 1947 году назначен заведующим отделом Пешковского райисполкома. С 1952 года служил в Туркестанском военном округе.
С 1956 года на различных партийных и государственных должностях:
С 1956 по 1961 года председатель исполкома Пешковского, Комсомольского, Карабалыкского, Убаганского и Урицкого райсоветов.
С 1961 по 1970 — начальник Урицкого районного производственного управления сельского хозяйства.
С 1970 года — первый секретарь Урицкого райкома Компартии Казахстана.
Будучи первым секретарём Урицкого райкома партии занимался организацией сельского хозяйства в Урицком районе. Указом Президиума Верховного Совета СССР N 3630-VIII «О присвоении звания Героя Социалистического Труда передовикам сельского хозяйства Казахской ССР» от 13 декабря 1972 года за успехи, достигнутые в увеличении производства и продаже государству сельскохозяйственной продукции удостоен звания Героя Социалистического Труда с вручением ордена Ленина и золотой медали «Серп и Молот».
В 1983 году вышел на пенсию. Скончался в январе 1998 года.
Награды
Орден Ленина — дважды (1972)
Орден Красной Звезды (1943)
Орден Отечественной войны 1 (1985) и 2 (1944) степеней
Орден Трудового Красного Знамени — трижды (1966,1971,1976)
Орден «Знак Почёта» (1957)
Медаль «За освобождение Варшавы»
Медаль «За взятие Берлина»
Почётный гражданин посёлка Урицкий (1988)
Источники
Гноевой Николай Васильевич// Сарыкольская ЦБС
Гноевой Николай Васильевич
Человек — легенда// , Архивист, № 1, 2016 (январь — июнь), Костанай, 2016, стр. 57 — 58
Председатели районных исполкомов Казахской ССР
Первые секретари районных комитетов КП Казахстана
Депутаты Верховного Совета Казахской ССР 9-го созыва | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,942 |
Star of David necklace available in 15 mosaic colors. The pendant is about 3/4" in size and includes a silver chain in your choice of length. Shown in color "S", turquoise. Ships for free via 1st class mail ($36). | {
"redpajama_set_name": "RedPajamaC4"
} | 304 |
{"url":"https:\/\/forums.ankiweb.net\/t\/issue-with-latex\/4960","text":"# Issue with latex\n\nI am starting in this of anki, my motivation to use anki is to learn mathematics, but for almost 1 month I have not been able to solve the problem of using latex in anki, I followed the anki manual as it was, installing miktex and configuring it as it says in the manual, I downloaded an algebra deck, but at the time of studying it appeared that I was missing some components, I decided to download all the miktex packages, but it appeared that not found \u201clibertian-tipe1\u201d, I tried texlive but the Same mistake, do I need to do something else? Is there any other way to use latex in anki?, sorry for my english\n\nYou may have more luck with MathJax.\n\n2 Likes\n\nthank you!","date":"2022-07-06 07:37:11","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\": 0.8211394548416138, \"perplexity\": 958.0991824832574}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104668059.88\/warc\/CC-MAIN-20220706060502-20220706090502-00209.warc.gz\"}"} | null | null |
namespace WarMachines.Common
{
using System;
public static class Validator
{
public static void CheckIfNull(object obj, string msg = null)
{
if (obj == null)
{
throw new NullReferenceException(msg);
}
}
public static void CheckIfStringIsNullOrEmpty(string text, string msg = null)
{
if (string.IsNullOrEmpty(text))
{
throw new NullReferenceException(msg);
}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,981 |
As cats age, they seem to slow down or not do activities as they used to. This could be because of pain in their joints. This video shows some cats that are impaired due to degenerative joint disease or arthritis. It can affect cats as early as 1 year old but more typically cats over 10-especially if there is a weight problem. Our lab works to find ways to assess and treat pain with the help of grants, donations and University funding. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,760 |
Fear Itself: Unboxing!
In: Fear Itself, Horror
Friends of Maria Anderson, a popular vlogger who buys antiques from online auction sites and showcases them, have been invited to her house. Maria has found a wonderful artifact for her video channel! She has invited all of you for her unboxing video. It should be a fun time for everyone.
This game was run for patrons of the RPPR Patreon. You can play in these games if you become a member.
http://media.blubrry.com/rpg_actual_play/p/tabletoptales.roleplayingpublicradio.com/podcasts/RPPR-Fear-Itself-Unboxing-Tabletop-Tales.mp3
Alien: Red Skies
In: alien, Sci-Fi
You are all members of the USS Thesus, a cargo spaceship on its way back from picking up a shipment of tellurium from an asteroid mining operation. You have been given a new mission on your way back. A space station in orbit over planet GX190 in the HD110113 system has requested a VIP priority transport. The Thesus is the closest available ship and you are obligated to fulfill the request. Your crew chief, a superstitious woman named Barringer claims that this is a "red sky" omen, although she won't reveal if it's good or bad luck…
hostile as dylain steiger, company intern
Ken as Liam Flyn, medic. Check out PRRP podcast!
mr hellspawn as Rupert, company agent
This was run for members of the RPPR Patreon via our Youtube channel.
I and other members of the RPPR podcast run games for patrons at the RPPR Patreon. . You can play in one of these games by joining at the $10 level! All of our most recent games are on our Youtube channel.
http://media.blubrry.com/rpg_actual_play/p/tabletoptales.roleplayingpublicradio.com/podcasts/RPPR-Alien-Red-Skies.mp3
Monster of the Week: Jollification Zone
In: Horror, Monster of the Week
I and other members of the RPPR podcast run games for patrons at the RPPR Patreon. These games are recorded to Youtube but I realize I might as well put some of them on here. You can play in one of these games by joining at the $10 level! All of our most recent games are on our Youtube channel.
You are all freelance monster hunters/paranormal experts. Ashley Bradshaw, the manager of Steve Warwick, a popular Youtuber, needs your help because Steve has gone missing. He went into Jollification Zone, an abandoned theme park in New Jersey. It's supposed to be haunted but you aren't sure what could be lurking in there.
This adventure used monsters from the Deck of Monsters, available now!
Squirrel as Brock the wronged
Chris as Packing Dan mundane
Hostile as Kafka Valentine professional
Adaptive as Zoya the monstrous
http://media.blubrry.com/rpg_actual_play/p/tabletoptales.roleplayingpublicradio.com/podcasts/RPPR-MOTW-Jollification-Zone.mp3
Red Markets: The Mavericks – Episode 12
In: Horror, Red Markets
StopGap. The Dam Job. The Score to End All Scores. Bartlett Reservoir is deep enough within the YA reservation to be lost in harsh terrain, near enough to Phoenix to arouse fear of massed herds of casualties, and whispered about enough to create its own graboid-like Loch Ness monster myth! It's also one of the few plentiful sources of water within reasonable travelling distance, and control of it could set up the Waterhole to supply fresh water to the area for years to come. But how could anyone pass up that kind of water source in this hellish desert without good reason. . .
http://media.blubrry.com/rpg_actual_play/p/tabletoptales.roleplayingpublicradio.com//podcasts/RPPR-Red-Markets-Mavericks-Episode12.mp3
Mr. JOLS is approaching, but one last job is required to get all of the Mavericks ready for their big score. The enclave Zeta Station is being plagued by alien cattle rustlers! The SETI researchers are hard pressed for resources and can ill afford to lose what little they have. Despite their best efforts, they've failed to make first contact. Perhaps they need a cruder mouth piece to get through to our extraterrestrial guests. . .
The Mavericks finally approach the Yavapai-Apache Tribal Council. The Council are riding the high of peace between the factions of the enclave following the successful defense of the Waterhole and are looking to build upon it! Their tribal casino was a failed attempt at an early enclave and remains a dark mark upon their past. Inside lies a treasure trove of their cultural heritage, links to their past that they would rather have in their possession for preservation. Also, there is an extensive collection of fine alcohol stored within the depths of the casino, a worthy contribution to the celebrations to show the council's renewed investment in the enclave. Both caches are equally valuable to the council, for different reasons obviously, and they are willing to pay a fine price to retrieve them all. . .
Red Markets: The Mavericks – Episode 9
(Due to technical errors, Session 8 was unfortunately lost).The sins of Stagecoach's idealism came back to bite the group. Agent Lynch, of the DHQS, was not thrilled with the Maverick's selling government servers to the Moths. But he's a generous man, if they do a job for him, he'll forget all about this little indiscretion. And so, the Mavericks are sent across the unknown wilds of the YA reservation with a Moth wetwork specialist whose trial, and summary execution, awaits him back in the Recession. Stagecoach was forced to face the darker realities of resistance while the rest of the crew learned the finer points of detainee handling! HORDE! A stampede of casualties thousands strong is sweeping towards the Waterhole! While the Mavericks dealt with the DHQS, the enclave and all of its resident taker groups has been preparing for a tidal wave of Blight to crash upon their walls. Now with little time left before the arrival of the horde, the Maverick's must slot themselves into the defensive works to save their home! This was a playtest of early-build War Campaign rules, it offers a sneak peak of the upcoming supplement The Racket though specifics in design and execution may have changed dramatically between the two versions.
The Mavericks are going independent! When a lone stranger arrives at the Waterhole with information to share, the opportunity to set up their own score and the profit margin that goes with it is just too tempting. But what seemed an innocuous USGS research facility is in fact something far more foreboding. Have the Mavericks been goaded into a trap?
Red Markets: The Mavericks – Episode 6 part 2
Secrets and Lies! The Mavericks have uncovered the perpetrators of the murders at Earthship Serenity, but an inconclusive gunfight with the Cowboys has left them with more questions than answers. With their foes still loose and out for blood in the wilds, and the inner workings of Earthship Serenity obstructing them at every turn, will there be any justice to find here?
http://media.blubrry.com/rpg_actual_play/p/tabletoptales.roleplayingpublicradio.com//podcasts/RPPR-Red-Markets-Mavericks-Episode06b.mp3
Due to technical errors, Session 5 was unfortunately lost. The Takers took on a second job from The Brothers, utilizing their electric semi-truck to transport clean water to a newly contacted enclave in hopes of building trade and diplomatic relations. The Mavericks instead found a massive convoy; the new enclave was entirely mobile and had survived via migration tactics. Shenanigans ensued. * The Maverick's are trying their hand at frontier justice! Their nearest rival enclave, Earthship Serenity, has suffered a series of grizzly murders, and the leadership of Serenity have charged the Waterhole with proving their innocence by solving the murders and bringing the guilty to justice. Sheriff Graham Lightfoot is looking for competent takers to go down and sort the mess out quickly and professionally. Getting the job done quickly, no problem for the Mavericks. Professionally . . . | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,516 |
Q: Issue with using python exec() inside discord bot command I have basically identical code, one written in idle;
The command is a simple eval command so I can run python code from my bot. When I was testing this I learned how to do it in the idle so I just copied it over to the bot. For some reason the code in IDLE works yet in the bot it doesn't.
My Bot code:
try:
cmd = "\n".join(f" {i}" for i in command.splitlines())
func = "def eval_cmd():\n"+cmd
print(func)
exec(func)
embed = discord.Embed(title=" ", description="Code Evaluation: ", color=discord.Color.green())
embed.add_field(name=":inbox_tray: Input", value="```{}```".format(command), inline=False)
embed.add_field(name=":outbox_tray: Output", value="```{}```".format(eval_cmd()), inline=False)
embed.set_author(name=message.author.name, icon_url=message.author.avatar_url)
await client.send_message(message.channel, embed=embed)
except Exception as ex:
template = "An exception of type {0} occurred. Arguments:\n{1!r}"
error_message = template.format(type(ex).__name__, ex.args)
await client.send_message(message.channel, embed=discord.Embed(color=discord.Color.red(), description="You encountered an error: \n```{}```".format(error_message)))
My IDLE code:
command = '''
pi = 3.14
rad = 5
area = pi * rad**2
return area'''
cmd = "\n".join(f" {i}" for i in command.splitlines())
func = "def eval_cmd(): "+cmd
print(func)
exec(func)
print(eval_cmd())
The code in IDLE works perfectly and the value of eval_cmd() is 78.5 - What it should be. Inside the bot when exec(func) is run it just returns None.
Is this an issue with discord? or am I missing something.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,375 |
Q: dsym more info displays -- I am attempting to symbolicate a crash file but it appears xcode can not match the dsym file to the crash report.
Having done a bit of research I found the dsym file for my archive and when I go into getinfo and look at "more info:" it displays "--", I believe this should be showing the uuid and that this is the reason xcode cannot find it.
I'm guessing I need to change something in my build settings but don't know what.
A: You can find out the UUID of the dSYM by using the following command in the terminal:
dwarfdump --uuid YourDSYMPackage.app.dSYM
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,076 |
Q: Regular Expression for line starting with a number followed by tab I have a really simple question. I figured out that the regular expression for a line that starts with a number is ^(\s+)?\d+
I want a regular expression for lines that starts with a number, followed by a tab. Naturally I assumed that ^(\s+)?\d+\t would do the job. Unfortunately it is not working. I would appreciate if somebody could point out the error in my expression.
Regards,
SS
A: (^|(?<=\n))\d+\t
breakin' it down:
^ is the start of the string.
(?<=\n) is a positive lookbehind on \n.
so either you match the start of the string, or you match something following a new line.
and what you're matching after either of those conditions is \d+\t, which means one or more digits followed by a tab.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,455 |
So here is an idea: Instead of goose birth control or gassing the birds, how about people STOP FEEDING THEM.
Geese are migratory animals that should move from one place to another. One of the major reasons they stay and reproduce like rabbits is because our society cannot get the idea into their heads that they are WILD animals. They don't need the leftover bread from your pantry.
Stop making the geese provide some hallmark moment with you and your children/grandchildren. The kids only get bit, and the geese continue their takeover.
Maybe we should consider a fine if you are caught feeding the geese. Do we really need the extra patrol and extra fines? No, but we don't need a million geese taking over our town either. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,426 |
Number 357211821 has 9 digits. Number 357211821 can be formatted as 357,211,821 or 357.211.821 or 357 211 821 or in case this was a phone number 357-211-821 or 35-721-1821 to be easier to read. Number 357211821 in English words is "three hundred and fifty-seven million, two hundred and eleven thousand, eight hundred and twenty-one". Number 357211821 can be read by triplets (groups of 3 digits) as "three hundred and fifty-seven, two hundred and eleven, eight hundred and twenty-one". Number 357211821 can be read digit by digit as "three five seven two one one eight two one". Number 357211821 is odd. Number 357211821 is divisible by: three. Number 357211821 is a composite number (non-prime number).
Number 357211821 in binary code is 10101010010101001111010101101. Number 357211821 in octal code is: 2522517255. Number 357211821 in hexadecimal (hexa): 154a9ead.
The sum of all digits of this number is 30. The digital root (repeated digital sum until you get single-digit number) is 3. Number 357211821 divided by two (halved) equals 178605910.5. Number 357211821 multiplied by two (doubled) equals 714423642. Number 357211821 multiplied by ten equals 3572118210. Number 357211821 raised to the power of 2 equals 1.2760028506214E+17. Number 357211821 raised to the power of 3 equals 4.5580330187165E+25. The square root (sqrt) of 357211821 is 18900.048174542. The sine (sin) of 357211821 degree is 0.35836794953709. The cosine (cos) of 357211821 degree is 0.93358042650035. The base-10 logarithm of 357211821 equals 8.5529258223053. The natural logarithm of 357211821 equals 19.693839499924. The number 357211821 can be encoded to characters as CEGBAAHBA. The number 357211821 can be encrypted to chemical element names as lithium, boron, nitrogen, helium, hydrogen, hydrogen, oxygen, helium, hydrogen.
295830287, 531419693, 522180376, 307169767, 816280243, 708296785, 9296586713, 115795407, 59870148, 503846458, 490088699, 187131315, 558131619, 639785696, 4380235159, 693128034, 894122678, 7100861081, 146151715, 2551819630, 787743157, 207276785, 387473623, 299113993, 271467078, 824934002, 252573708, 826845709, 552457021, 8707281, 625912157, 724334341, 926383044, 936337891, 4516763254, 349667188, 1549986037, 9568213121, 565315698, 989815999, 58895758, 598338555. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,444 |
import unittest
import sys
import lit
import lit.discovery
input_path = sys.argv[1]
unittest_suite = lit.discovery.load_test_suite([input_path])
runner = unittest.TextTestRunner(verbosity=2)
runner.run(unittest_suite)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,856 |
I have purchased your theme and manipulated it on the domain of rushcampusministries.org. The main page looks great on the desktop, however the ipad/tablet versions of it do not show the main image that so clearly is seen on the desktop rendering (i.e. the Image that says "RUSH FAITH + ACTION = CHANGE"). Is there something I can do in the wordpress dashboard to change this?
Try to make the header image with two layers (for the first layer in back you will need to insert black and white image, in front of that image you will need to insert image with red text in .png format with transparent background). All the procedure are being done from Frontend Builder.
Also you can send us your WP login info via e-mail to support@shindiristudio.com so we could see and made that for you? Also, you must include your Ticket ID so we can identify your purchasing. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,738 |
Proba de 100 de metri feminin de la Jocurile Olimpice de vară din 2016 a avut loc în perioada 12-13 august pe Stadionul Olimpic.
Recorduri
Înaintea acestei competiții, recordurile mondiale și olimpice erau următoarele:
Program
<small>Orele sunt ora Braziliei (UTC-3)</small>
Rezultate
Etapa preliminară
În runda preliminară au intrat în competiție sportivele invitate să concureze și care nu au realizat timpul de calificare necesar. Sportivele care au realizat timpul de calificare au intrat direct în primul tur. Reguli de calificare: primele două din fiecare serie (C) și următoarele două cu cel mai rapid timp (c) au avansat în Runda 1.
Seria 1
Seria a 2-a
Seria a 3-a
Runda 1
Reguli de calificare: primele două din fiecare serie (C) și următoarele 8 atlete cu cel mai bun timp (c) se califică în semifinale.
Seria 1
Seria a 2-a
Seria a 3-a
Seria a 4-a
Seria a 5-a
Seria a 6-a
Seria a 7-a
Seria a 8-a
Semifinale
Semifinala 1
Semifinala a 2-a
Semifinala a 3-a
Finala
Legendă
RA Record african | AM Record american | AS Record asiatic | RE Record european | OCRecord oceanic | RO Record olimpic | RM Record mondial | RN Record național | SA Record sud-american | RC Record al competiției | DNF Nu a terminat | DNS Nu a luat startul | DS Descalificare | EL Cea mai bună performanță europeană a anului | PB Record personal | SB Cea mai bună performanță a sezonului | WL Cea mai bună performanță mondială a anului
Referințe
Legături externe
Rezultate runda preliminară
Rezultate runda 1
Rezultate semifinale
Rezultate finală
Atletism la Jocurile Olimpice de vară din 2016 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,578 |
Home Books Heroes Season 5 Prologue: Brave New World / Fan Fiction
HEROES -- Pictured: "Heroes" Logo -- NBC Photo
Heroes Season 5: Brave New World
Change is inevitable. We Grow. We Learn. We Evolve. The Most Ordinary People, Discovering They Can Do The Most Extraordinary Things. The World Has Become An Amazing And More Dangerous Place. All That We Know Is At An End. Yet, We Find Hope. It's A Brave New World.
After Claire's insane reveal to the local news of her abilities, many of our "heroes" go underground. HRG does what he does best. Protect Claire by any means necessary. With the help of Angelia Petrelli, HRG and Claire are brought into government protective custody. Once again working for the government they are tasked with finding others to join the cause, of creating "stability" between humans and specials. HRG agrees but Claire is now his new partner. Her "Company Training" begins. Tracy Strauss and The Haitian also work with HRG, as shadow agents. Meaning the government doesn't know of her involvement.
Samuel is taken in to custody. Of course he will be the prime test subject. Most of the carnival specials find a way to make a new life. Many are brought in by HRG and his New Company. Hiro and Ando head back to Japan. They hope to use their powers in secret, shutting down their little heroes for hire building, to work from the shadows. Although things are going well for them, Hiro worries about the others. Especially Peter and Sylar. Hiro's sister disappears and Ando takes it upon himself to find her……without Hiro's help. With no one else to turn to Hiro teleports back to ancient Japan to visit Yaiko.
Meanwhile Peter Petrelli visits his mother to discuss what will happen now that Claire has outed them all. Angela is finished explaining things to Peter. He must grow up and move beyond Nathans shadow. "Nathan is dead. And you are the only man left in the Petrelli family now. Grow up, Peter. And fast. Or we'll all soon be dead." Angela presses him to take control of himself and his abilities. "You are the only one who can stop what is to come." Furious Peter departs. Teleporting from the home he once grew up in. And Angela cries silently not knowing if she has sent her last son to his destiny or his death.
Matt Parkman is the first to be recruited by HRG. The New York detective will make a great asset to his new team. With mind reading and manipulation Parkman will keep the government officials in place while the New Company works. Having worked with HRG on a few occasions, Parkman is worried he will be forced to do something he will soon regret. But it's better to have more control by working for the Company. That will keep his family safe. He hopes.
Gabriel Grey…..that is what he has been calling himself these past few months. It's hard to think that he can go back to his old life again. He was given another chance. HGR had told him, "Go back to your old life, Sylar. You can still fix watches, can't you?! It would be unfortunate to have another incident, don't you agree?" He was given a bank card and all the help needed to open a new clock repair shop. Very upscale. As Gabriel looks out onto the streets of Manhattan, fondling the glasses he no longer really needed, he could still hear the low ticking that has always been in sync with the beat of his heart. He knows how things work. The Power. The Hunger. But he was Gabriel Grey again. Despite that, the ticking of his mind was not off-set by the ticking of the many clocks and watches of his shop. He stared out at the city and knew that eventually, someone would come. To seek power. To control it. Or to try and destroy it. It was only a matter of TIME.
Micah Sanders stands in his new apartment looking over the various monitor screens that make up more than half the space of the over-sized loft. The quiet hum of computers is the only sound in the room. He watches. His teammates of Rebel. He watches. All of the specials he has been able to track down. He watches. The New Company as they begin to round up other specials. Recruiting some. Incarcerating much more. Few actually go free. But they are set free. Micah knows not to get involved. Team Rebel has its own goals. Noah Bennett (HRG) contacted him recently. Rebel would be a way of keeping a small number of specials out of government hands. Save them and ask them to join to help others like themselves. Unlike the New Company, Rebel only asked. If they declined they were given a new identity, and asked to make themselves scarce. Micah thought, "I'm the Special Harriet Tubman. She probably was a special also. Who knew????" But he had other problems. Ando had gone on a rescue mission, alone. And Hiro had disappeared, again. Team Rebel had their own hands full. And his cousin Monica pretty much took care of New Orleans by herself. If only he could find Peter Petrelli. He was like Sylar, but a good guy and the greatest Hero of us all. So many powers collected into one person. He could send him to help Ando. Again he thought, "If only I could find him." Micah calls out into the air," Angela Petrelli". Somewhere in the room a phone dialed. He thought to himself, "Who needs a Bluetooth".
Nothing can stop Claire Bennett. With a body that will repair from virtually any wound, she becomes a machine of fear and balance in the hands of HRG. Even though he hates having her fight by his side, she's a great weapon. Noah continues to train her; with guns and small arms. Close quarters combat and knife training and of course her favorite, demolition. She pretty much becomes a Kamikaze. She has become hardened by her training and the dozen or so missions she's been on. But Claire does fear for her father. He's not powerless, but…….. After an intense sparring session Claire meets HRG in the New Company ready room. Nothing more than a room filled with a huge 10 foot monitor with a ever moving digital map. The tables are over flowing with boxes of files. A few desks with computers sit unused. And a bulletin board with hundreds of faces on it. Some crossed out. HRG waits for her, alone. "Claire, I hope you had a good work because you're not going to like this", Noah says. "I never will like it, dad. Who's the victim this time?" The monitor switches from map display to a file read out with various pictures of one person. Peter Petrelli. Claire knows what Peter is capable of. She knows that in reality she has no choice. Why does it always come back to him? All he ever wanted to do was help people. But it was her job to help the world. Not just one person at a time. Keep the balance between the specials and the regular population settled. Looking at the monitor and her father she says, "What about Sylar? Isn't he the real threat?" Noah says nothing. They've gone through this before. His answer has and always will be the same, even though this time it is different. Finally Claire says is, "How many powers does he have"? Noah turns the monitor off and walks toward the exit. "He can only have a single ability at a time. He shouldn't be any problem at all. You know he would never hurt his niece." She glares at him, but remains silent. "Sometimes", she thought. "I really do hate him".
Peter Petrelli had no idea what to do. "As usual", he thought to himself. Back in his apartment, still fuming over his mother's words Peter sat down and just thought. He thought OF EVERYTHING. "These powers, what good are they? Every time I try do something courageous or good it always turns out wrong. Powers I have no knowledge of. No control of. Exploding in New York. Being tricked by Adam Monroe. Jumping through time and losing Caitlin. Caitlin!??!" After all this time. Peter remembers the one person that stood with him and never wanted to do anything but love him. "How could I have forgotten her?!? Adam. The Shante Virus. The return of my thought dead father. Being stripped of my powers. Only having to give myself the shot to save Nathan. Now I can only use a single ability at a time. Nathan's dead. Sylar gets a new lease on life. Claire has let the world know about us. And my mother, trying to manipulate me again. Maybe this time she was right." He didn't believe her. He was always weak. Always looking for someone to lead him. Tell him what to do. "Maybe Claude could help me get the answers." Lost in thought he did not hear the footsteps near his door. Or see the three red dots trained on him. But he felt the sharp pain of the darts that pierced his arm, leg and neck. The last thing Peter remembers is the men in black masks with rifles and stun batons. And the exploding pain as electricity raced through him, until he fell into the blackness of unconsciousness.
Claire Bennett
Heroes Season 5
Matt Parkman
Peter Petrelli
The Haitian
Tracy Strauss
Previous articleTGS 2016: TEKKEN 7 Trailer
Next articleBIOHAZARD: VENDETTA Trailer and Posters
Terminator: Dark Fate – Official Teaser Trailer
VENOM – Official Trailer & Poster
E3 2017: The Last Night on Xbox One Trailer
Middle-earth: Shadow of War | Dominate the Open World Trailer
Resident Evil 2 – TGS 2018 Trailer
Netflix Castlevania Trailer: Vengeance
Blade Runner 2049 Teaser Trailer
E3 2017: Evil Within 2 Announcement Trailer
Titanfall 2 – Become One Official Launch Trailer | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,875 |
How entrepreneurs can learn from the Wheeltons
by ImageMaker | Sep 4, 2018 | Career, Entrepreneurship, Giving, Leadership, Philanthropy, Success
Angela and Paul Wheelton are the kind of people you don't meet every day. Both highly successful in business in their own rights, the two have now joined forces in philanthropy, and it is this that has seen them both recognised under the Order of Australia.
Angela came from a background as a business leader in science and healthcare. As CEO (ANZ) for Straumann (the world's leading manufacturer of dental implant systems), she was the first and only female executive in the company, leading her division to become the top performer.
Earlier this year, Angela was awarded the Medal of the Order of Australia as part of the 2018 Queen's Birthday Honours for her significant contributions to the community.
"I tend to look at philanthropists as being the next stage in the entrepreneurial life cycle and it is leadership that has led to their success in the business world. Bringing that leadership skill to philanthropic activities enables the success, that is being achieved, to be possible." – Paul Wheelton. Tweet
Whilst both Angela and Paul are distinguished leaders in business, Paul Wheelton has never been to a job interview in his life. After graduating from Camberwell Grammar School, and whilst working at a Box Hill service station, he met Bob Ansett who had just started up Budget car and truck rentals. Bob hired Paul to work with him at Budget Rent a Car on the condition that he went to university part time and studied business and accounting. Paul passed his degree and progressed on in Budget to become Chief Accountant.
At this point Bob funded him to buy his first Budget car and truck rental franchise, and Paul was to gain his fortune from his ownership in 12 Budget Car Rental franchises.
Now, Paul and Angela are running Wheelton Philanthropy, looking at key trends around the globe and seeking to spread the message of positive and engaged philanthropy for families and the broader community.
So what prompted the Wheeltons' shift from business to philanthropy? Here is how Paul sees it.
"I tend to look at philanthropists as being the next stage in the entrepreneurial life cycle and it is leadership that has led to their success in the business world. Bringing that leadership skill to philanthropic activities enables the success, that is being achieved, to be possible."
"Philanthropy is often stated as being the big picture cousin of charity. It is less of the emotional, immediate action and relief focus and more focused on the long term, strategic and problem solving initiatives. By definition then, leadership is an important and essential part of success in philanthropy."
When you get, give
Paul says that his lightbulb moment came when after years of business success, he paused to ask himself a very personal question.
"It was the fundamental question of 'When is enough, enough?'" he says.
"I looked around and saw people who'd been on the same entrepreneurial journey as me who just kept building more businesses and I thought, 'I've got enough. Do I spend the rest of my life doing the same thing or should I go down a different path and try to make a difference with what I've been fortunate enough to receive?'"
"I began to talk with my business mentor Jon Michail, about ways that I could work on the business, rather than spending all my time working in the business, at the coalface, seven days a week."
This shift in focus is what allowed Paul to spend the majority of his time in philanthropic work.
"As a leader, I strive to instil the passion of education and learning to give others the confidence and courage to embrace their dreams and turn them into reality." – Angela Wheelton. Tweet
"Our plan for the future is that one hundred per cent of what we make in our businesses is going to be donated. We don't need more money personally for ourselves – we know when enough is enough."
Though philanthropy came as a natural extension of his entrepreneurial career, Paul says he was brought up with a giving mindset.
"Being a practising Anglican, I was also exposed at an early age to the spirit of giving and would listen with intrigue to where a portion of the weekly collection would be going and who it would help. This also introduced me to the concept of tithing," he says.
Giving generously out of what he had received was always a part of Paul's life, so that it was only natural that when his earnings increased, so too would his philanthropic endeavours.
When you learn, teach
After many years as a senior business leader, Angela believes that education and learning form the lifelong process of acquiring vital skills and tools to build a sustainable, successful and prosperous life.
"As a leader, I strive to instil the passion of education and learning to give others the confidence and courage to embrace their dreams and turn them into reality," she says.
"Ultimately, the cornerstones of life experience, work experience and education empower us with the capability to actively contribute to the growth and development of ourselves and others. All aspects of my past and current career choices, professional associations and more recently, involvement in charitable organisations, have been driven by this philosophy."
Balinese students. Image: Bali Children Foundation Australia.
Paul is also passionate about lifelong learning and education, and so the Wheeltons place particular emphasis on supporting organisations such as Life Education Australia, and the Bali Children Foundation – an organisation that values education, and the positive impact that this can have, not only on the children who participate, but on their wider community and future generations.
Empowering women and girls
"I learned early on in our journey that the difference between underdeveloped nations and those moving to developing and developed status was the opportunity given to women," Paul says.
As a father to three daughters, he was well aware that even in Australia his son got a fairer go than his girls. But it was in Bali that he really saw the vast gap in opportunity between the children of poorer families.
Working to build community centres, orphanages and bring educational opportunities to children in remote Balinese villages, Paul was struck by the gender divide. Families sent their boys to the programs Paul and Angela had established, while their sisters stayed to work with the rest of the family.
"The great thing is I have not met one person who does not want their child to get an education, it's just that they can't afford it. The girls are working the fields at the age of ten and that's the end of the story."
"When we've got that research, I'm confident we'll be able to convince the government that supporting women and girls is not just about gender equity, but that there are also economic benefits." – Paul Wheelton. Tweet
By addressing economic powerlessness, the playing field levels.
"Our experience has taught us that the girls are yearning to learn, and more so than the boys. The girls put in an incredible effort."
But it doesn't end there, as these girls move on to greater things thanks to the education they have received, they pass on knowledge and economic benefit to their communities as well.
"The statistics from the UN are that when women work they reinvest 90% in their families compared to 30 or 40% for a man," says Julie Reilly, CEO of the Women Donors Network.
Julie Reilly, CEO of the Women Donors Network (far left), fellow Women Moving Millions member, Jo Kirk, and Angela and Paul Wheelton at the 2017 Women Moving Millions Annual Summit. Image: Generosity Magazine.
Wheelton Philanthropy is currently involved in a study following ten Balinese girls who are supported to go to university for the next decade in order to evaluate outcomes and impacts upon the girls' local villages, families, and themselves.
"When we've got that research, I'm confident we'll be able to convince the government that supporting women and girls is not just about gender equity, but that there are also economic benefits," Paul says. "I think that lesson will be applicable here too, not just in Indonesia."
Plan carefully, give thoughtfully, partner wisely
"The biggest thing I've learned is that there are a lot of passionate people out there starting not-for-profits but more than passion is needed to make change," Paul says. "There are too many not-for-profits, so many of them have the concept right, but end up duplicating the work of others.
"Sometimes I've funded things where, if I'd done a bit more due diligence, I would've picked up on the fact that there was already another organisation doing it better and I would've encouraged them to join up with them instead of creating another administration to do the same thing."
The Wheeltons have found that it is important, not only to give generously, but also to give wisely. There will always be worthy causes to which they can give funds, but in order to make their philanthropic work achieve the greatest impact they now consider carefully how they can best partner with people and organisations who have the runs on the board, in addition to having their hearts in the right place.
After all, philanthropy is not about throwing money at something to feel better about yourself, or rushing in headlong with a saviour complex. Just as entrepreneurs carefully evaluate what the market wants, and who their product will be serving, and what they must achieve for and with partners and stakeholders, so any philanthropic venture must take the same considered approach.
As Aboriginal elder, activist and educator, Lila Watson once said, "If you have come here to help me you are wasting your time, but if you have come because your liberation is bound up with mine, then let us work together."
So, what do I think entrepreneurs can learn from the Wheeltons? I believe we would all do well to learn from their leadership style. To see that unconventional roads can become the path to success, to value lifelong learning and wise leadership, to be good stewards of our finances and to use the skills and resources that we acquire to improve the lives of others.
What do you think about your role and responsibility as a leader or an entrepreneur? I would love to hear from you!
Jon Michail, Image Consultant | Personal Branding Coach | Business and Personal Branding Strategist | Author | Group CEO and Founder, Image Group International |
Image Group International are Entrepreneurial Activists… Rebels With a Cause!
Jon Michail, Founder and award-winning Chief Imagemaker together with his team at Image Group International helps executives and entrepreneurs to build, grow, and monetise their personal brands, (online and offline) by positioning them to stand out so they attract their ideal clients.
They are committed to maximising an individual's personal impact, influence, and value in the ever-changing and disruptive business environment.
Image Group International is recognised as Australasia's leading personal brand image advisory, with over 29 years of proven results.
www.imagegroup.com.au
Did you enjoy this article? Keep reading!
Paul Wheelton – The Philanthropist
Cool in a crisis: How all entrepreneurs can learn from ex-Fire-Chief Alan Quinton
How entrepreneurs can learn from creatives
Why social entrepreneurship matters and empowering the poor is good for society
Share this article with your friends: | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,044 |
{"url":"http:\/\/en.wikipedia.org\/wiki\/User_talk:Billlion","text":"# User talk:Billlion\n\nA slug\n\narchive\n\n## photos of alan turing building\n\nHi. Now that it's summer (you wanted to wait for a sunny day...) maybe you would be able to take some new photos? both inside and outside are more finished now. regards, Marmelad (talk) 20:08, 6 July 2008 (UTC)\n\n## Ole Nydahl article\n\nHello, I see that you have been active in editing the Ole Nydahl article. Recently it seems that unbalanced edits are being made to this article, mainly that well-sourced quotes are being removed from the controversies section of the article. I have recently reverted some of these edits before creating this account, but I can only do so much without breaking the 3 revert rule. I ask that you help keep the article balanced, and in accord with Wikipedia guidelines.\n\nBest regards, 87.56.199.8 (talk) 01:12, 22 July 2008 (UTC)\n\n## Orphaned non-free media (Image:Oxford Brookes University Logo.jpg)\n\nThanks for uploading Image:Oxford Brookes University Logo.jpg. The media description page currently specifies that it is non-free and may only be used on Wikipedia under a claim of fair use. However, it is currently orphaned, meaning that it is not used in any articles on Wikipedia. If the media was previously in an article, please go to the article and see why it was removed. You may add it back if you think that that will be useful. However, please note that media for which a replacement could be created are not acceptable for use on Wikipedia (see our policy for non-free media).\n\nIf you have uploaded other unlicensed media, please check whether they're used in any articles or not. You can find a list of 'image' pages you have edited by clicking on the \"my contributions\" link (it is located at the very top of any Wikipedia page when you are logged in), and then selecting \"Image\" from the dropdown box. Note that all non-free media not used in any articles will be deleted after seven days, as described on criteria for speedy deletion. Thank you. BJBot (talk) 05:44, 6 August 2008 (UTC)\n\n## JCM telescope\n\nthis article, on which you have edited, contains a statement re. the completion of scuba-2 in early 2008. this needs updating. is the detector online? thx.Toyokuni3 (talk) 14:33, 5 October 2008 (UTC)\n\nSorry I don't know.Billlion (talk) 14:52, 5 October 2008 (UTC)\n\n## Image:Palace_westminster_pano_corrected.jpg listed for deletion\n\nAn image or media file that you uploaded or altered, Image:Palace_westminster_pano_corrected.jpg, has been listed at Wikipedia:Images and media for deletion. Please see the discussion to see why this is (you may have to search for the title of the image to find its entry), if you are interested in it not being deleted. Thank you. Calliopejen1 (talk) 16:23, 19 November 2008 (UTC)\n\n## Nice work with electrical impedance tomography\n\nGood work with the EIT article. I found it to be surprisingly informative and well-written given the relatively small number of editors who have given it attention. I did make a few fixups mostly to bring it into conformity with MoS. Robert\u00a0K\u00a0S (talk) 18:56, 12 December 2008 (UTC)\n\n## Ancoats\n\nI think the URL in question is www.mm2manchester.com. Whether it's any good as a source, I have no idea. Regards, Mr Stephen (talk) 18:11, 28 March 2009 (UTC)\n\n## higher-dimensional M\u00f6bius transformations\n\nHi Billion -- I've added a comment at Talk:M\u00f6bius_transformation#Higher_dimensions regarding higher-dimensional M\u00f6bius transformations. Joriki (talk) 08:57, 10 April 2009 (UTC)\n\n## 4th floor Alan Turing\n\nHi Bill, the 4th floor is not just plant space, there is a kitchen, 3 large labs, clean room, shielded room, network hub room, server room, cleaners cupboard and a loo\u00a0;-) - All 'user space' so I think is worth including. Cheers, Ant Holloway.\n\nThanks, you are right my mistake. Is it documented somewhere?Billlion (talk) 18:19, 30 April 2009 (UTC)\n\n## File:Fish at Lakes Aquarium Cumbria.jpg\n\nMy pleasure to help you illustrate your article. Thanks for the Flickr courtesy note. Computerjoe's talk 17:11, 31 May 2009 (UTC)\n\n## Thanks\n\nThanks for your input at the SSEM article on the thorny issue of what arithmetic operations it implemented. I'm conscious that I may be too close to it to always be entirely objective, and perhaps sometimes too protective, but I was deeply unhappy about the recent addition. --Malleus Fatuorum 23:24, 16 June 2009 (UTC)\n\nI hope I added to the clarity. Its not clear to everyone that subtraction and conditional branching is enough so it could be explained a bit better.Billlion (talk) 23:28, 16 June 2009 (UTC)\nI agree, and thanks once again for clarifying what I must admit I thought was obvious. --Malleus Fatuorum 23:30, 16 June 2009 (UTC)\n\n## Help\n\nHi, I'm posting this on your (and other members of the Maths Wikiproject) talk as we need editors who are knowledgeable about Mathematics to evaluate the following discussion and check out the editors and articles affected. Please follow the link below and comment if you can help.\n\nThankyou. Exxolon (talk) 18:01, 1 July 2009 (UTC)\n\nHi. Could you please have a look over at Radon transform. There is an editor insistent upon adding another reference to the article, to a recent book by Gabor Herman. (This same editor, incidentally, is spamming the same reference over many related articles.) Initially I removed it because it already has enough references\u2014many of which were presumably used in writing it, some were added awhile back by you I think. The editor then re-inserted the reference, claiming on my talk page that it was superior to the others (a dubious claim that I have replied to). At any rate, while I will certainly defend my decision to remove the reference, I'm ultimately not that invested in it one way or the other. An outside opinion would be helpful here. Thanks, S\u0142awomir Bia\u0142y (talk) 16:40, 2 January 2010 (UTC)\n\nSee Talk:Radon transformBilllion (talk) 19:28, 2 January 2010 (UTC)\n\n## Unreferenced BLPs\n\nHello Billlion! Thank you for your contributions. I am a bot alerting you that 1 of the articles that you created is tagged as an Unreferenced Biography of a Living Person. The biographies of living persons policy requires that all personal or potentially controversial information be sourced. In addition, to ensure verifiability, all biographies should be based on reliable sources. If you were to bring this article up to standards, it would greatly help us with the current 1,004 article backlog. Once the article is adequately referenced, please remove the {{unreferencedBLP}} tag. Here is the article:\n\n1. Doug Scott - Find sources:\u00a0\"Doug Scott\"\u00a0\u2013\u00a0books\u00a0\u00b7 scholar\u00a0\u00b7 JSTOR\u00a0\u00b7 free images\n\nThanks!--DASHBot (talk) 01:23, 18 January 2010 (UTC)\n\n## math notation\n\nHello. This edit prompts some comments.\n\nTeX is sophisticated. If you write 4\u00a0\\mathrm{sup}_f\u00a0A, it looks like this:\n\n$4 \\mathrm{sup}_f A\\,$\n\nBut if you write 4\u00a0\\sup_f\u00a0A, it looks like this:\n\n$4 \\sup_f A\\,$\n\nThe difference is not only that the \u0192 appears directly under \"sup\", but also that spacing before and after \"sup\" is automatically provided according to standard style conventions (note the space between 4 and sup in the second case and the lack of such space in the first case), and in some contexts, the size gets adjusted according to standard usages. Michael Hardy (talk) 03:54, 13 March 2010 (UTC)\n\nThanks Michael, that was a subtlety I missed.Billlion (talk) 06:17, 21 June 2010 (UTC)\n\n## You are now a Reviewer\n\nHello. Your account has been granted the \"reviewer\" userright, allowing you to review other users' edits on certain flagged pages. Pending changes, also known as flagged protection, is currently undergoing a two-month trial scheduled to end 15 August 2010.\n\nReviewers can review edits made by users who are not autoconfirmed to articles placed under pending changes. Pending changes is applied to only a small number of articles, similarly to how semi-protection is applied but in a more controlled way for the trial. The list of articles with pending changes awaiting review is located at Special:OldReviewedPages.\n\nWhen reviewing, edits should be accepted if they are not obvious vandalism or BLP violations, and not clearly problematic in light of the reason given for protection (see Wikipedia:Reviewing process). More detailed documentation and guidelines can be found here.\n\nIf you do not want this userright, you may ask any administrator to remove it for you at any time. Courcelles (talk) 05:14, 20 June 2010 (UTC)\n\nThanks! Billlion (talk) 06:18, 21 June 2010 (UTC)\n\n## File copyright problem with File:CT of human thorax showing current paths for EIT.jpg\n\nThank you for uploading File:CT of human thorax showing current paths for EIT.jpg. However, it currently is missing information on its copyright status. Wikipedia takes copyright very seriously. It may be deleted soon, unless we can determine the license and the source of the file. If you know this information, then you can add a copyright tag to the image description page.\n\nIf you have uploaded other files, consider checking that you have specified their license and tagged them, too. You can find a list of files you have created in your upload log.\n\nIf you have any questions, please feel free to ask them at the media copyright questions page. Thanks again for your cooperation. ww2censor (talk) 00:29, 3 November 2010 (UTC)\n\n## Proposed Tibetan naming conventions\n\nA few months ago, I posted a new proposal for Tibetan naming conventions, i.e. conventions that can be used to determine the most appropriate titles for articles related to the Tibetan region. This came out of discussions about article titles on Talk:Qamdo and Talk:Lhoka (Shannan) Prefecture. I hope that discussions on the proposal's talk page will lead to consensus in favour of making these conventions official, but so far only a few editors have left comments. If you would be interested in taking a look at the proposed naming conventions and giving your opinion, I would definitely appreciate it. Thanks \u2014 Nat Krause(Talk!\u00b7What have I done?) 22:08, 11 November 2010 (UTC)\n\n## Fran\u00e7oise Tisseur\n\nThanks for the changes you've made to the article. If you add a stub template to an article then it's a good idea to check the talk page to see what class the article was carrying. It was listed as a start-class. I've changed that to stub-class to avoid inconsistencies. It's worth making sure you check the talk page before making changes to articles. Thanks again for the edits, and keep up the good work. 03:22, 16 January 2011 (UTC)\n\n## Autopatrolled\n\nHello, this is just to let you know that I have granted you the \"autopatrolled\" permission. This won't affect your editing, it just automatically marks any page you create as patrolled, benefiting new page patrollers. Please remember:\n\n\u2022 This permission does not give you any special status or authority\n\u2022 Submission of inappropriate material may lead to its removal\n\u2022 You may wish to display the {{Autopatrolled}} top icon and\/or the {{User wikipedia\/autopatrolled}} userbox on your user page\n\u2022 If, for any reason, you decide you do not want the permission, let me know and I can remove it\nIf you have any questions about the permission, don't hesitate to ask. Otherwise, happy editing! Acalamari 11:29, 27 January 2011 (UTC)\nThanksBilllion (talk) 13:08, 28 January 2011 (UTC)\n\n## Category:Mathematical physicists\n\nHi,\n\nfor some strange reason, Jean Ginibre appears in the category under 'J' (and not under 'G'). Do you accidentally know how to fix this? (I am asking you since you created the category, I apologise if this is the completely wrong address).\n\nBest, Sasha (talk) 03:32, 24 June 2011 (UTC)\n\nI fixed it with the Defaultsort template, have a look and see how I did that. Billlion (talk) 05:41, 24 June 2011 (UTC)\nthanks!Sasha (talk) 05:50, 24 June 2011 (UTC)\n\n## Isn't that other article a POV fork?\n\nRe 'Mongol invasions'. There are 2 POVs. The Tibetan religious historical tradition see negotiations with the Mongols from 1206 to 1251 as precluding invasion by an agreement between the Mongols as the Sa-skya religious authority. The essentially Sinocentric view cites a cavalry incursion that burnt two monasteries to the north of Lhasa and killed several hundred monks in 1240 as an 'invasion'. 34 years ago, Wylie showed these are just POVs simplying a period which, for the obscurity of testimonies, we know little about, and the misinformation is repeated in many secondary sources. But it is certain that there was no 'Mongol invasion' of Tibet (the whole ununified plateau) in the sense that there was a Mongol invasion of 19 other countries in that century. The 'main article' is no such thing. It cites a few sources, but is thinner in substantive detail than the History of Tibet section I was editing. A lot of these pages seem intent on establishing retroactively the modern Chinese case for sovereignty over Tibet (which it indeed now exercises) on the basis of (a)Mongols unified China (b)Tibet acceded to Mongol claims for submission (c)therefore China's claim to Tibet dates from these invents. On this principle, China's soverenty extends to Hunbgary and Baghdad. I'm not interested in POV battles. I'm interested in what the best academic intelligence says about the specific facts of any period, and the word 'invasion' here is inappropriate.Nishidani (talk) 13:27, 17 July 2011 (UTC)\n\nI see your argument and it certainly has a lot of merit, but I think you should edit the main article first Mongol invasions of Tibet(especially change the title to Mongol incursions in to Tibet) even if it is a fairly poor atrticle, or at least have the debate on the talk page and then change History of Tibet to agree. As you say you will be up against editors who seem to advocate a view that supports the Chinese calim to soverenty over Tibet on that atricle as well. Perhaps citing academic articles that confirm at least that there are two points of view might be the way forward?Billlion (talk) 14:50, 17 July 2011 (UTC)\n\n## Metal detector split tag\n\nI am trying to clear up the backlog of split tags. Taking a quick look, I think the proposal to split the article was opposed and would therefore have removed the split tag. If you have a clear idea on what you want then I suggest that you do it, otherwise the split tag ought to go. It has been on rather a long time. regards Op47 (talk) 18:31, 30 December 2011 (UTC)\n\nI suggest we drop the split tag. I only know about certain non-controversial aspects of metal detectors and it needs some diplomacy and understanding to do the split well, so as to reduce rather than fuel edit wars. Billlion (talk) 23:10, 30 December 2011 (UTC)\n\n## Possibly unfree File:The Stork stalactite at Treak Cavern.JPG\n\nA file that you uploaded or altered, File:The Stork stalactite at Treak Cavern.JPG, has been listed at Wikipedia:Possibly unfree files because its copyright status is unclear or disputed. If the file's copyright status cannot be verified, it may be deleted. You may find more information on the file description page. You are welcome to add comments to its entry at the discussion if you are interested in it not being deleted. Thank you. Stefan2 (talk) 11:22, 8 January 2012 (UTC)\n\n## Nomination of Royal Society Wolfson Research Merit Award for deletion\n\nA discussion is taking place as to whether the article Royal Society Wolfson Research Merit Award is suitable for inclusion in Wikipedia according to Wikipedia's policies and guidelines or whether it should be deleted.\n\nThe article will be discussed at Wikipedia:Articles for deletion\/Royal Society Wolfson Research Merit Award until a consensus is reached, and anyone is welcome to contribute to the discussion. The nomination will explain the policies and guidelines which are of concern. The discussion focuses on high-quality evidence and our policies and guidelines.\n\nUsers may edit the article during the discussion, including to improve the article to address concerns raised in the discussion. However, do not remove the article-for-deletion template from the top of the article. ConcernedVancouverite (talk) 21:30, 3 June 2012 (UTC)\n\n## File source problem with File:Akong Tulku Rinpoche throne.jpg\n\nIf the necessary information is not added within the next days, the image will be deleted. If the file is already gone, you can still make a request for undeletion and ask for a chance to fix the problem.\n\nPlease refer to the image use policy to learn what images you can or cannot upload on Wikipedia. Please also check any other files you have uploaded to make sure they are correctly tagged. Here is a list of your uploads. If you have any questions or are in need of assistance please ask them at the Media copyright questions page. Thank you. Stefan2 (talk) 13:59, 5 September 2012 (UTC)\n\nThanks I have changed the tag to GFDL-self to make it unambiguous. I think the message is a bit strange as it seems to ask fro a source, as though copying images from other websites rather than making them oneself was the norm. Billlion (talk) 16:45, 5 September 2012 (UTC)\n\n## Bach, Cotton, Lanczos, Schouten tensors\n\nHi Billlion. I noticed you are one of few active editors to have meaningfully edited the Cotton tensor article. I've recently made some major changes to the Lanczos tensor article. Perhaps you'd be interested in improving it or know someone else who is. Similar pages that could use some attention are Bach tensor and Schouten tensor. Teply (talk) 23:41, 13 October 2012 (UTC)\n\n## File:George Henry Livens cropped.jpg missing description details\n\nDear uploader: The media file you uploaded as:\n\nis missing a description and\/or other details on its image description page. If possible, please add this information. This will help other editors make better use of the image, and it will be more informative to readers.\n\nIf the information is not provided, the image may eventually be proposed for deletion, a situation which is not desirable, and which can easily be avoided.\n\nIf you have any questions, please see Help:Image page. Thank you. Theo's Little Bot (error?) 22:56, 12 April 2013 (UTC)\n\n## CFD\n\nHi Billlion -- I posted the Adams Prize recipients category on CFD for discussion; per WP:OCAT#Award, award winners are usually better listed on the award pages rather than categorized. Please feel free to join the discussion; notice + link is posted below. --Lquilter (talk) 02:27, 5 May 2013 (UTC)\n\n## Disambiguation link notification for June 24\n\nHi. Thank you for your recent edits. Wikipedia appreciates your help. We noticed though that when you edited Metal detector, you added a link pointing to the disambiguation page Little Big Horn (check to confirm\u00a0|\u00a0fix with Dab solver). Such links are almost always unintended, since a disambiguation page is merely a list of \"Did you mean...\" article titles. Read the FAQ\u00a0\u2022 Join us at the DPL WikiProject.\n\nIt's OK to remove this message. Also, to stop receiving these messages, follow these opt-out instructions. Thanks, DPL bot (talk) 11:02, 24 June 2013 (UTC)\n\n## commons:File:Dr Akong Tulku Rinpoche.jpg\n\nIf you created this file, please mark it with {{own}} or something similar so we know who the author is. Thanks. Magog the Ogre (t c) 09:06, 9 October 2013 (UTC)","date":"2014-03-09 17:25:25","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\": 2, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4135439395904541, \"perplexity\": 2672.754991277428}, \"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-2014-10\/segments\/1394009903935\/warc\/CC-MAIN-20140305085823-00034-ip-10-183-142-35.ec2.internal.warc.gz\"}"} | null | null |
## My Weird School #11
Mrs. Kormel Is Not Normal!
## Dan Gutman
Pictures by
Jim Paillot
To Emma
## Contents
Never Kiss Your Mom in Public
2 Mrs. Kormel's Secret Language
3 My Head Almost Exploded
4 Are We There Yet?
5 The Middle of Nowhere
6 The Nude Kid's Dad
7 Fighting Evil Under the Bus
8 Striker Smith's Final Battle
9 We Are Survivors
10 Mrs. Kormel Is Driving Us Crazy
11 We Finally Meet the Nude Kid
About the Author and the Illustrator
Credits
Copyright
About the Publisher
##
## Never Kiss Your Mom in Public
My name is A.J. and I hate school.
Do you know which is the worst day of the week? If you ask me, it's Monday. Because Monday is the start of five days of school in a row. That's horrible!
Tuesday and Wednesday aren't so great either.
Thursday is a pretty good day, because then we only have one day of school left before the weekend.
Friday is really good, because that's when the school week is over.
But the best day of the week is Saturday. I play peewee football on Saturday, and we don't have school again for two whole days.
Too bad it was Monday morning. I was waiting in front of my house for the school bus with my mom.
"You be a good boy, A.J.," my mom told me.
"I will."
"Don't get into any trouble, A.J.," my mom told me.
"I won't."
"Remember to raise your hand when you want to talk, A.J.," my mom told me.
"I will."
"Don't shoot straw wrappers at the girls, A.J.," my mom told me.
"I won't."
My mom told me about a million hundred other things I wasn't allowed to do until I saw the yellow school bus coming around the corner.
"Mom, I promise not to have any fun at all," I said. "Bye!"
The bus pulled up. Mrs. Kormel, the bus driver, pushed a button and made the little STOP sign pop out the side of the bus so the cars on the street will stop. We call it the magic STOP sign. That thing is cool.
"Give Mommy a kiss, A.J."
No way I was going to kiss my mother in front of all the kids staring out the bus window. That's the first rule of being a kid. Don't ever kiss your mother when other kids are watching!
"Uh, I don't want to be late for school, Mom."
"Give Mommy a kiss, A.J."
"That's not gonna happen, Mom."
"Give Mommy a kiss, A.J."
"Over my dead body, Mom."
"Give Mommy a kiss, A.J."
"I will if you give me a hundred dollars, Mom," I said.
My mother tried to wrap her arms around me, but I know how to get away from tacklers. When Mom went to grab me, I threw her a head fake, spun away, and gave her a few of my best fancy foot-work moves that I learned playing peewee football. She didn't have a chance! I sidestepped her and ran on the bus before she could hug or kiss me.
Ha-ha-ha! My mom can't play football for beans. Nah-nah-nah boo-boo on her!
##
## Mrs. Kormel's Secret Language
I dashed on the bus and there was Mrs. Kormel, the school bus driver. She was wearing a crash helmet on her head and a silver whistle around her neck.
"Bingle boo, A.J.!" she said.
"Bingle boo, Mrs. Kormel."
"Bingle boo" is Mrs. Kormel's way of saying "hello." One time I asked her why she doesn't just say "hello" like normal people.
"I'm inventing my own secret language," she told me. "Everybody says 'hello.' But I think 'hello' is boring. I'm trying to get people to switch from saying 'hello' to saying 'bingle boo.' Secret languages are fun!"
Mrs. Kormel is not normal.
"Limpus kidoodle," said Mrs. Kormel. That means "sit down" in Mrs. Kormel's secret language.
I looked around the bus. There was a snot-covered kindergartner in the front row behind Mrs. Kormel, and a few angry fifth graders in the back row.
Fifth graders are really mean because they get a lot of homework. The more homework you get, the meaner you are. That's why fifth graders are meaner than fourth graders, and fourth graders are meaner than third graders, and third graders are meaner than second graders.
You don't want to go near seventh or eighth graders. They get lots of homework, and they just hate the world. I hope I never get to high school.
I sat down in the middle by myself. Mrs. Kormel stopped the bus at the next corner, and a few other kids got on. At the stop after that, my friends Ryan and Michael got on.
"Bingle boo!" Mrs. Kormel said to Ryan and Michael. "Limpus kidoodle."
Ryan and Michael sat down next to me.
"What did you bring in for Show and Share?" Ryan asked. "I brought in an old light switch."
"I brought in a ball of string," said Michael.
Show and Share is when we bring something from home that starts with a certain letter of the alphabet and talk about it in class. Today's letter was s.
I took my Show and Share thing out of my backpack. It was an action figure called Striker Smith. He's a superhero from the future who travels through time and fights bad guys with a sharp sword that's attached to his hand. He can turn into a jet plane, too, and fly when you push a button. I saw a commercial for Striker Smith on TV and bugged my parents until they finally got it for me.
"Striker Smith belongs to a secret organization of crime fighters," I told Ryan and Michael, in case they didn't see the commercial.
"You should get extra credit," Ryan said, "because Striker Smith has two S 's."
"He's cool," said Michael. "Sometimes I take my old action figures down to the basement and my dad lets me saw them in half or torture them with his power drill."
"I take mine out in the sun and melt their faces with a magnifying glass," said Ryan.
Michael and Ryan are weird.
At the next stop, this really annoying girl in my class named Andrea who thinks she knows everything got on the bus with curly brown hair. Well, the bus didn't have curly brown hair. Andrea did.
"Bingle boo, Andrea!" said Mrs. Kormel.
"Bingle boo," Andrea said. "I'll go limpus kidoodle now."
What a brownnoser! Andrea plopped her dumb self down in the seat right in front of me, like always.
"Good morning, Arlo," she said.
I hate her.
Andrea's mother found out that A.J. stands for Arlo Jervis, so Andrea went and told everybody. It was the worst day of my life. I thought I was gonna die. I wanted to switch schools or move to Antarctica and go live with the penguins, but my mom wouldn't let me.
Penguins are cool.
"Are you boys ready for the big spelling test this afternoon?" Andrea asked.
Oh no. I forgot all about the big dumb spelling test! How can I be expected to remember stuff over the weekend? Weekends are for having fun, not for studying for tests. I hate spelling.
"Do you know how to spell 'spelling,' A.J.?" asked Andrea.
"Sure," I said. "I-H-A-T-E-Y-O-U."
Michael and Ryan laughed.
"I made my own spelling flash cards," Andrea told us, "and I'm going to use them for Show and Share, too. Because spelling begins with an s."
I was going to tell Andrea that "stupid" also begins with an s and that's what she is, but I decided I would save that and use it the next time she really got me mad.
Andrea turned around so she wasn't facing us anymore. I picked up Striker Smith and pretended that he was going to attack the back of her head with his sword. It was hilarious. Michael and Ryan laughed. But Andrea turned around suddenly, before I could take Striker Smith away from her head.
"That's a nice doll, Arlo," she said.
"It's not a doll!" I told her. "It's an action figure!"
"My mother told me that action figures are dolls for boys," said Andrea.
"They are not!" I said.
"Are too!" said Andrea.
We went back and forth like that for a while.
"Striker Smith is a one-man wrecking machine," I told Andrea. "He belongs to a secret organization of crime fighters. If Striker got into a fight with one of your dumb dolls, he would rip its head off."
"Dolls don't fight," Andrea said.
"Striker Smith does," I said.
"I thought you said he wasn't a doll, Arlo."
Why can't a bus filled with spelling flash cards fall on Andrea's head?
##
## My Head Almost Exploded
"Bingle boo! Limpus kidoodle," said Mrs. Kormel.
Andrea's equally annoying crybaby friend Emily got on the bus in front of her house. She sat down next to Andrea, and they studied Andrea's dumb flash cards together.
"Is everybody here?" asked Mrs. Kormel after she picked up a few more kids.
"Yes," we all said.
"If you're not here, raise your hand."
I knew that was a trick question, because if somebody wasn't there we wouldn't be able to see if their hand was up. But just to be on the safe side, I got up in my seat to see if anybody who wasn't there had their hand up.
"Limpus kidoodle, A.J.," said Mrs. Kormel.
In case you don't remember, that means "sit down." Mrs. Kormel doesn't like it when we get out of our seats.
"No standing on the seats," said Mrs. Kormel.
"Can I kneel on my seat?" I asked.
"You can only kneel on your seat if your name is Neil. Anyone named Neil may kneel."
"Can I stand if my name is Stan?" asked Ryan.
"Okay," said Mrs. Kormel. "If your name is Stan, you can stand."
"I can't stand sitting down," Michael said.
"Nobody can stand sitting down," Ryan said. "If you're sitting down, you're not standing."
"Can you crouch if your name is Crouch?" I asked.
"There's nobody named Crouch!" Andrea told me. She thinks she knows everything.
"Oh yeah?" I said. "What about that guy on Sesame Street named Oscar the Crouch?"
"That's Oscar the Grouch, dumbhead!" Andrea said.
I knew that.
"Now that we're all here, how about singing a song to make the ride go quicker?" suggested Mrs. Kormel.
"Let's sing 'The Wheels on the Bus'!" said Andrea. "I love that song."
"I hate that song," I said. "Can we sing 'Ninety-nine Bottles of Beer on the Wall'?"
Mrs. Kormel said we couldn't sing about beer because kids aren't allowed to drink beer. I wouldn't want to drink beer even if I was allowed to. My dad gave me a sip of his beer once. I thought I was gonna throw up. Mrs. Kormel said we could sing "Ninety-nine Bottles of Pop on the Wall" if we wanted to.
The girls started singing "The Wheels on the Bus." The boys started singing "Ninety-nine Bottles of Pop on the Wall." Me and Michael and Ryan tried to sing louder than all the girls. Andrea and Emily tried to sing louder than all the boys. It was really loud in there.
Soon everybody on the bus was screaming, and kids were bouncing around like Mexican jumping beans. I covered my ears so my head wouldn't explode.
Something about being on a school bus makes you want to go crazy. Maybe it's all that yellow.
I'll bet Mrs. Kormel was sorry she told us to sing. Suddenly she blew her whistle really loud.
"Zingy zip!" she yelled.
That's her way of saying "quiet down" in her secret language. Everybody stopped singing.
"Shhhhh, my cell phone is ringing," said Mrs. Kormel. "It's Mr. Klutz."
Mr. Klutz is our principal. He is like the king of the school. He's bald, too. One time he kissed a pig. This other time he got stuck on the top of the flagpole. Another time he was climbing the school, and the custodian had to rescue him by sticking one of those toilet plungers on his head. We saw it live and in person. Mr. Klutz is nuts!
"What does Mr. Klutz want?" somebody yelled.
Mrs. Kormel finished talking to Mr. Klutz and turned around to whisper something to the boy in the row behind her. He turned around and whispered something to the girl in the row behind him. She turned and whispered something to the girl in the row across from her, who then turned and whispered something to Michael.
"We have to go pick up a nude kid," Michael whispered.
"A nude kid?" I said. "That's disgusting!"
"That nude kid better not sit next to me," said Ryan.
"Tell the nude kid to put some clothes on!" said Michael.
##
## Are We There Yet?
We hadn't even met him yet, but suddenly everybody on the bus was buzzing about the nude kid.
"Some people don't believe in wearing clothes," said Andrea, who thinks she knows everything. "They're called nudists."
"I call 'em freaks," said Ryan.
"My parents say we should respect and celebrate people's differences," Emily said.
"Your parents are weird," I told her.
"What's the big deal?" Andrea asked. "After all, we were born without clothing."
"You were born without a brain," I couldn't resist adding. Anytime anyone says anything about being born, always say they were born without a brain. That's the first rule of being a kid.
"I'm nude under my clothes," Ryan told us.
"Thanks for sharing that with us," I told him. "Now I'm totally grossed out."
Andrea said that nudists save a lot of money because they don't have to buy clothes. But Michael said that nudists have to spend a lot of money because they always have to buy sunscreen. But Emily said that nudists save a lot of money because they don't have to do laundry. But Ryan said that nudists have to spend a lot of money to heat their houses because they're so cold.
"The nude kid probably doesn't even have a closet at home," I said, "because he doesn't have any clothes to put in it."
"Do you think nude kids are allowed to wear hats?" asked Michael.
"Zingy zip!" yelled Mrs. Kormel.
Mrs. Kormel had pulled out a big map and she had it opened up on top of the steering wheel. She told us we had to be quiet while she figured out the directions. I bet it's hard to drive and look at a map at the same time.
To make things even worse, it started raining. Mrs. Kormel put the windshield wipers on. All the girls started to sing, "'The wipers on the bus go swish swish swish, swish swish swish, swish swish swish!'"
"Zingy zip!" Mrs. Kormel yelled again. "I can't concentrate."
We drove for a long time in the rain. It was hard to go so long without talking.
"Are we there yet?" somebody asked.
"No," said Mrs. Kormel.
"Are we lost?" Emily asked.
"Of course not," said Mrs. Kormel. "I just don't know where we are."
"Oh no, we're going to miss Show and Share!" Andrea said to Emily, like she was all worried. "And I spent all weekend making my spelling flash cards."
Ha-ha-ha! I spent all weekend playing football. Nah-nah-nah boo-boo on her! We were going to be late for school. It was great! I hoped Mrs. Kormel wouldn't find the right way to the nude kid's house for a while. Because right after Show and Share, we have math.
And I hate math.
##
## The Middle of Nowhere
Mrs. Kormel drove for a million hundred hours, and we still hadn't reached the nude kid's house. Where was it? Now Show and Share was over for sure. We probably missed math, too. If we didn't get to school soon, we would miss our DEAR time. That stands for Drop Everything and Read.
I hate reading.
Finally we drove by some big trucks and there was a sign that said DETOUR.
"Hey, we're going to take a tour," said Ryan. "Tours are cool!"
"Detours aren't tours, dumbhead!" said Andrea. "Workers are fixing the road, so Mrs. Kormel has to get off this road and go another way to the nude kid's house."
"Bix blattinger!" said Mrs. Kormel as she slammed her fist against the steering wheel.
I don't know what "bix blattinger" means, but that's what Mrs. Kormel always says when she gets really angry or frustrated.
"What does 'bix blattinger' mean, Mrs. Kormel?" I asked.
"Never you mind!" said Mrs. Kormel.
"I think Mrs. Kormel must have said a bad word," said Ryan.
"But she said it in her secret language," Michael said, "so we won't know what it means."
You shouldn't say bad words. We all tried to figure out which bad word Mrs. Kormel said. I had heard most of the bad words, but Michael and Ryan knew a few that I didn't know and they taught them to me. So even though we were really late for school, it was still a learning experience.
After we got off the regular road, the road we were on wasn't even a real road. It was more like a jungle or a swamp or something. There were these big plants slapping against the bus windows. The ride was bumpy, and it was still raining. Mrs. Kormel could barely see in front of her.
"This is like the rainforest," Andrea said.
"I hope we don't get eaten by alligators," said Michael.
"Look!" somebody yelled. A family of ducks was crossing the road in front of us.
"They're cute!" shouted all the girls.
"Let's run them over!" shouted all the boys.
I didn't really want Mrs. Kormel to run over the ducks, but it was hilarious anyway.
Mrs. Kormel kept on driving, but I didn't see the nude kid's house anywhere. In fact, I didn't see any houses. Or human beings.
"Where are we?" somebody asked.
"The middle of nowhere," said Mrs. Kormel.
I said it was too bad we weren't at the edge of nowhere. Because if we were at the edge of nowhere, we'd be right next to the edge of somewhere. And the edge of somewhere is near the middle of somewhere, which is where we wanted to be.
Ryan said we were probably close to the nude kid's house. Because if you were nude, you'd probably live in the middle of nowhere. That way, nobody would see you. And that's where we were. It made sense to me.
Andrea looked at her watch and got all upset.
"We missed DEAR time!" she complained. "We're very late for school now."
Ha-ha-ha! This was the greatest day of my life!
"Don't worry, Andrea," said Mrs. Kormel. "I'll get you to school if it's the last thing I do."
Just then I had a genius thought. Right after DEAR time is spelling. I hate spelling. And if we'd missed DEAR time and we weren't even at the nude kid's house yet, there was a good chance that we were going to miss spelling, too. And that meant we would miss the big spelling test, which I didn't study for!
I decided that the nude kid was the coolest kid in the history of the world. Thanks to him, we were going to miss the big spelling test.
"Bix blattinger!" said Mrs. Kormel.
"Ooooh!" we all said. "Mrs. Kormel said that bad word again!"
##
## The Nude Kid's Dad
Finally Mrs. Kormel stopped the bus in front of a house and pushed the button to make the magic STOP sign pop out.
"This must be the nude kid's house," Ryan said.
It looked like a pretty normal house. There was a swing set on the lawn and a car in the driveway. You would never know that nudists lived there.
Nobody came out of the house, so Mrs. Kormel honked the horn. We all craned our necks to get a look at the nude kid, but he didn't come out.
"Where is he?" Ryan asked.
Suddenly the front door of the house opened. A guy came out with an umbrella. And it was the most amazing thing in the history of the world. You know why?
Because the guy had clothes on!
The guy with clothes on came over to the bus. He climbed up the steps and said something to Mrs. Kormel.
"Bix blattinger!" muttered Mrs. Kormel after the guy got off the bus. She was probably mad at Mr. Klutz because he made her drive all the way out to the nude kid's house and the nude kid wasn't even there.
"What did he say?" we all asked. "What did he say?"
"He said his wife drove their son to school today on her way to work," Mrs. Kormel told us. "She didn't want him to be late on his first day of school."
"Maybe she took him to buy some clothes," said Michael.
Mrs. Kormel closed the bus door and made the magic STOP sign go back. Before the nude kid's dad went inside his house, me and Ryan leaned out the window and yelled to him.
"Hey, mister! Do you sleep with clothes on and then take them off after you wake up?"
"Where does your son keep his lunch money if he doesn't have pockets?"
It was hilarious. The nude kid's dad just looked at us with this confused expression on his face. Then he went inside.
Nudists are weird.
##
## Fighting Evil Under the Bus
Finally we were back on the road. Mrs. Kormel was mad at Mr. Klutz for making her drive to the nude kid's house for nothing. I was mad because we would be at school soon. Andrea was mad because spelling was over and she didn't get the chance to take the big spelling test and show everybody how smart she was. Well, nah-nah-nah boo-boo on Andrea! She would go to school on the weekend if it was open.
The rain stopped. I was getting hungry. We had missed snack time. It was probably close to lunchtime. We were bored, too, and sick of sitting on the bus.
Ryan flipped his light switch on and off. Michael played with his ball of string. I played with Striker Smith. That's when I got the most genius idea in the history of the world! We could tie Michael's string to Striker Smith's leg and fly him out the window!
Michael and Ryan realized what a genius I was. We tied the string to Striker Smith and opened the window.
"You're going to get in trouble, A.J.," said Andrea. "We're not supposed to hold things out the window."
"Can you possibly be any more boring?" I asked Andrea. "We're not holding anything out the window. The string will hold him."
Ryan tossed Striker Smith out the window.
"Look!" Ryan said. "He's flying!"
It was cool. Striker Smith was doing loops in the air. You should have been there.
"He's fighting evil outside the bus!" said Michael.
"Mrs. Kormel!" Andrea whined. "A.J. threw his doll out the window!"
"It's not a doll!" I told Andrea. "It's an action figure. And it's none of your beeswax."
What is her problem?
It didn't matter what Andrea said, because Mrs. Kormel didn't hear her anyway. Striker Smith was flying outside the bus, dipping and diving in the wind. He is so cool.
The only problem was that suddenly Striker Smith dove down so far that we couldn't see him anymore.
"Where is he?" asked Ryan.
"He's fighting evil under the bus," I said.
But I don't think Striker Smith was fighting evil under the bus. Because that's when I heard a pop, and then a hissing sound.
Hisssssssss!
##
## Striker Smith's Final Battle
Ryan pulled the string up. There was nothing on the other end! Striker Smith was gone!
Suddenly the ride got all bumpy. Mrs. Kormel pulled off the side of the road. She opened the door and got out to see what was the matter.
"You're in big trouble, A.J.," Andrea said.
"So is your face," I replied.
Mrs. Kormel came back on the bus. She was holding Striker Smith in her hand. Or what was left of him, anyway. His head was gone. So was one of his legs and the arm that used to hold his sword.
I felt bad. My parents probably paid a lot of money to get me this cool action figure, and now it was totally crushed. On the other hand, it is also a well-known fact that crushing stuff and pulling the limbs off action figures is cool.
All in all, I was just happy that Show and Share was over. It wouldn't be very cool to show the class an action figure that was missing an arm, a leg, and his head.
"Who does this belong to?" asked Mrs. Kormel.
Andrea looked at me. I looked at Ryan. Ryan looked at me. Mrs. Kormel looked at me. I didn't know what to say. I didn't know what to do. I had to think fast.
"I said, who does this belong to?"
"Striker Smith belongs to a secret organization of crime fighters," I said.
I thought Mrs. Kormel was going to be really mad. But she just told us all to get off the bus. She said we had a flat tire and she was going to call Mr. Klutz to send somebody out to fix it. In the meantime, we'd have to wait outside.
"Now we're going to miss lunch!" one of the mean fifth graders complained.
"Who cares about lunch?" said somebody else. "We're going to miss recess!"
"It's all Arlo's fault," said Andrea.
"It is not," I said.
"It is too."
"Oh yeah?" I said. "Well, stupid begins with an s and that's what you are."
Ha-ha-ha! In her face!
We all got off the bus. It was a quiet road, and there were no other cars or houses or people around. Me and Ryan and Michael went to look at the flat tire. Striker Smith's sword was stuck right in the tire with his arm still attached to it. It was cool. It was like that story "The Sword in the Stone," except with a tire.
Mrs. Kormel tried to call Mr. Klutz on her cell phone, but something was wrong, and she started stamping her feet and yelling.
"Bix blattinger!" she yelled. "My cell phone battery is dead!"
Mrs. Kormel said she would have to fix the flat tire herself. She told us to get our lunches and have a little picnic on the sidewalk while she got out her tools and the spare tire.
Not everybody had brought a lunch bag, because some kids buy the school lunch. They must be nuts. The school lunch is usually rubber hot dogs, chicken nuggets that bounce, and nachos that glow in the dark. I wouldn't eat the school lunch if I was starving and there was no other food left in the world.
Mrs. Kormel asked us to share some of our food with kids who didn't bring a lunch. I gave my tuna sandwich to one of the first graders, but I kept my pudding treat.
I always eat my treat first anyway. You should always eat your treat first because if an asteroid hits the earth in the middle of lunch and destroys the planet, well, at least you got to eat your dessert. That's the first rule of being a kid. It would be a major bummer if the earth was destroyed by an asteroid and you didn't have the chance to eat dessert.
"Hey," Ryan said, "look what I found!"
It was Striker Smith's head! Ryan found it at the side of the road.
We decided right away to hold a funeral for the head. Michael dug a little hole in the dirt, and we dropped the head into it. Some of the other boys on the bus gathered around.
"Farewell, Striker," Ryan said solemnly. "You defeated the mighty tire. You sacrificed your life, so that others might not have to go to school. You paid the ultimate price, made the ultimate sacrifice so that we can live in freedom from reading, writing, and arithmetic. Long live Striker Smith. We will always remember you."
It was really sad. I almost cried when Michael said a little prayer:
Ashes to ashes,
Dust to dusted.
We buried Striker Smith,
Because he was busted.
He was really cool,
But now he's dead.
It's hard to live
When you don't have a head.
We covered up Striker's grave, and Ryan said we should have a moment of silence in honor of our fallen superhero.
It was really quiet. Then, in the middle of our moment of silence, Andrea said, "Boys are dumbheads."
##
## We Are Survivors
Finally Mrs. Kormel fixed the flat tire and said we could get back on the bus. She was all sweaty, and her hair was messed up, and her hands were covered with grease. She looked too tired to be mad at me or Mr. Klutz or anybody else. She just got into her bus driver's seat and hit the gas. The bus lurched forward, and we all fell back in our seats.
It was really late. Andrea complained that we might have missed social studies. Ha-ha-ha! That was fine with me. I hate social studies. Why is it called social studies anyway?
Mrs. Kormel was driving fast! We were far from school. It looked to me like we were still in the middle of nowhere. The road was really bumpy, and it was wet from the rain. Mrs. Kormel was having trouble keeping the bus in the middle of the road. I was afraid she might drive right off the side of the road.
And what happened next was the most amazing thing in the history of the world.
Do you want to know what happened?
I'm not going to tell you.
Well, okay, I'll tell you.
Mrs. Kormel drove right off the side of the road!
"Bix blattinger!" shouted Mrs. Kormel.
The bus skidded to a stop. Some kids even fell out of their seats! It was hilarious. You should have been there.
"Is everybody okay?" Mrs. Kormel asked.
"Yeah!" me and Michael and Ryan said. "That was fun. Can we do it again?"
"We're stuck in a ditch," said Mrs. Kormel. "We're not going anywhere."
"What are we going to do now?" asked Emily. She looked like she was going to cry. I was amazed that Emily hadn't cried yet. She usually can't go five minutes without crying about something.
"I don't know what to do," Mrs. Kormel said sadly. "My cell phone is dead. I guess we'll just have to wait for help to arrive."
"Too bad Striker Smith isn't here," I said. "He would know what to do."
"If you hadn't thrown that dumb doll out the window, none of this would have happened!" yelled Andrea.
"He's not a doll!" I yelled right back at her.
"Zingy zip!" yelled Mrs. Kormel.
Everybody was really depressed. We just sat there on the bus. There was nobody around. No houses. No stores. No nothing. Nobody was going to rescue us. It felt like we had been on the bus a million hundred hours.
It occurred to me that we might not only miss the rest of the school day, we might miss the rest of our lives! We could sit there forever. We could die out there!
Suddenly I felt hungry. I wished I hadn't given my sandwich to that first grader. I was starving. I was afraid my stomach might eat itself.
My friend Billy who lives around the corner from me and was in second grade last year told me he once heard about some guy who was stranded on an airplane, and he ate a seat cushion to survive.
"We might have to eat the seat cushions," I told Michael and Ryan.
Ryan looked at the seat cushion.
There's something you need to know about Ryan. He will eat anything, even stuff that is not food. One time we gave him a dollar to eat dirt.
Ryan got down on the floor and took a little bite from the corner of the cushion.
"Ugh," he said. "It's horrible."
"Put some ketchup on it," suggested Michael. "Ketchup makes anything taste good."
Michael gave Ryan a little ketchup packet from his lunch bag. Ryan put it on the seat cushion and took a tiny bite.
"It's not bad, actually," Ryan said.
Ryan is weird.
It was so boring sitting there waiting for somebody to rescue us. I almost wished we were at school. Almost.
"I saw this reality TV show where some people were stranded on a dessert island," Michael said.
"It's not a dessert island, dumbhead," Andrea turned around to say. "It's a desert island. One s. 'Desserts' is one of our spelling words this week." Then she held up her dumb flash card with the word "desserts" on it.
"Who asked you?" Ryan asked.
"A dessert island would be cool to be stranded on," I said. "There would be ice cream and candy and treats everywhere."
"Hey, look," said Emily, "I just noticed that 'desserts' is 'stressed' spelled backward."
"So what?" asked Ryan. "'Backward' is 'drawkcab' backward."
"Who cares what 'backward' is backward?" asked Andrea.
"'Bus' is 'sub' backward," I mentioned.
"Bix blattinger!" shouted Mrs. Kormel. "Will you please zingy zip?"
I couldn't blame Mrs. Kormel for being mad. It was a rough day for her. We had to whisper after that.
"Hey," whispered Michael, "I just thought of something. Maybe we're on a reality TV show right now and we don't even know it."
"That's impossible," Ryan whispered. "If we didn't know we were on a reality TV show, we wouldn't be talking about us being on a reality TV show."
"I think Michael is right," I said, looking around to see if there were any hidden cameras. "Maybe this was all planned in advance for the reality TV show we're on right now."
"Oh yeah?" said Andrea. "Who planned for you to throw your doll out the window and cause a flat tire?"
"It's not a doll!" I said. "Don't you have any respect for the dead?"
"If this is a reality TV show," Ryan said, "we're going to have to vote somebody off the bus."
"Why?" asked Emily.
"Because that's what they always do on reality TV shows, dumbhead!" I said. "You vote somebody off and they have to leave."
"But why?" asked Emily.
"Because that's the rule!" Michael told Emily. She doesn't know anything about reality TV shows.
"I vote for Andrea," I said.
"I vote for A.J.," Andrea said.
"Ooooh!" Ryan said. "A.J. and Andrea voted for each other. They must be in love!"
"When are you gonna get married?" asked Michael.
If those guys weren't my best friends, I would hate them.
##
## Mrs. Kormel Is Driving Us Crazy
We couldn't just sit around on the bus forever. Soon we would die of starvation or kill each other, like they do in the movies all the time.
"We've got to do something!" Emily said.
For once she was right. That's when I got the most genius idea in the history of the world. There must have been at least twenty kids on the bus. If we all got out and pushed, maybe we could push the bus out of the ditch!
I got up and and told Mrs. Kormel about my genius idea. At first she thought I was crazy and told me to go limpus kidoodle. But I guess she thought it over and decided to give it a shot.
"Okay, everybody off the bus!" she yelled.
We all got off the bus and went to the back.
"When I say push, everybody push," Mrs. Kormel yelled out the window.
"One...two...three...PUSH!"
I pushed with all my might. Everybody was grunting and groaning and moaning. The bus didn't move.
"Harder!" yelled Ryan.
And then the most amazing thing in the history of the world happened.
The bus started moving!
"Hooray!" everybody yelled.
Mrs. Kormel steered the bus back onto the road, and we all piled in. She told me my idea was great, and promised to drive carefully the rest of the way to school.
The only one who wasn't happy was Andrea. She was looking at her watch.
"Now we've missed music class," she complained.
That was fine with me. I hate music.
"Y'know," Michael said, "maybe Mrs. Kormel isn't a bus driver at all. Did you ever think of that?"
"Yeah," Ryan said. "Maybe she captured our real bus driver and has her tied up in a cave. Stuff like that happens all the time, you know."
"Stop trying to scare Emily," Andrea said.
"Maybe we're being kidnapped," I added. "Maybe Mrs. Kormel is driving us to her secret underground hideout at the North Pole, where she's going to do unspeakable things to us."
"Like what things?" Emily asked, all worried.
"I can't tell you," I told her. "They're unspeakable!"
"We've got to do something!" Emily said. "I don't want to go to the North Pole!"
That girl will fall for everything. Emily probably wanted to run away. But there was no place to run. She was stuck on the bus. So she started crying. What a baby!
Once Emily started crying, it set off a chain reaction and other kids started crying, too. Some of the first graders said they wanted their mommies. Some kid peed in his pants. Everyone was freaking out.
The fifth graders made a sign and put it in the back window—HELP! OUR BUS DRIVER IS DRIVING US CRAZY!
I didn't cry. I figured that it would be pretty horrible to be kidnapped and driven to the North Pole, but at least we wouldn't have to go to school anymore. And they have penguins at the North Pole too. Or maybe that's the South Pole. Either way, penguins are cool.
"Are we there yet?" somebody asked.
"KNOCK IT OFF!" yelled Mrs. Kormel.
##
## We Finally Meet the Nude Kid
The bus turned a corner, and we saw the big sign—ELLA MENTRY SCHOOL.
"We're there yet!" announced Mrs. Kormel.
"Yippee!" yelled all the girls.
"Boo!" yelled all the boys.
Mrs. Kormel pulled the bus up to the curb, and Mr. Klutz came running over.
"Bingle boo!" he said. "What—"
But he never got the chance to finish his sentence because, at that moment, the weirdest thing in the history of the world happened.
Mrs. Kormel must have leaned against the magic STOP sign button by accident. Because the magic STOP sign on the side of the bus swung out at the exact same time as Mr. Klutz arrived. The STOP sign smacked Mr. Klutz on the side of his bald head! He fell down! It was a real Kodak moment.
Those STOP signs are dangerous!
We all rushed off the bus to see if Mr. Klutz was okay. He stood up slowly. He looked like he'd been in a fight, or he'd drunk too much beer.
"W-what happened?" he asked.
"The STOP sign hit you in the head," said Mrs. Kormel. "I'm so sorry."
"No, I mean why were you so late?" asked Mr. Klutz.
Everybody started telling Mr. Klutz what happened.
"We had a flat tire!"
"We got kidnapped and drove to the North Pole!"
"We went to the nude kid's house!"
"A.J. threw a doll out the window!"
"Ryan ate his seat cushion!"
"We got lost in the rainforest!"
"We pushed the bus out of a ditch!"
"We had a funeral for Striker Smith's head!"
"Well, I'm just glad you're all safe!" said Mr. Klutz.
"Did we miss the big spelling test?" asked Andrea.
"Oh, your teacher Miss Daisy was out sick today," said Mr. Klutz. "So your class had a substitute teacher named Ms. Todd. You'll have your spelling test tomorrow."
"Yippee!" yelled all the girls.
"Boo!" yelled all the boys.
While we were yelling, the school bell rang. The front door opened, and kids started pouring out.
"It's three o'clock!" said Mr. Klutz. "Everybody back on the bus. It's time to go home."
"Home?" said Andrea. "But we just got here!"
"Bix blattinger!" yelled Mrs. Kormel.
"Hey, wait a minute," I said to Mr. Klutz. "Where's the nude kid?"
"Nude kid?" said Mr. Klutz. "What are you talking about, A.J.?"
"You know," I said, "the kid we were going to pick up before we got lost."
"Ohhhhh!" said Mr. Klutz. "You mean the new kid. He's not nude. He's new. Here he comes now."
This kid came over to the bus. He looked pretty normal. He even had clothes on.
"What's your name?" Ryan asked him.
"Neil."
"Really?" Michael asked.
"Yeah, my name is Neil Crouch."
"Is that your real name?" I asked. "Neil Crouch?"
"Sure it is," he said. "Why?"
"Well," I explained, "if your name is Neil Crouch, that means you can kneel and crouch on the bus."
"Why would anybody want to kneel or crouch on a bus?" asked Neil Crouch.
"Because we're not allowed to!" we all yelled.
Sheesh! The nude kid has a lot to learn about being a kid. He's weird. Me and Michael and Ryan decided to keep calling Neil Crouch "the nude kid" even if he did wear clothes.
We all got back in the bus, and Mrs. Kormel pulled out of the driveway. Maybe she'll get us back home before we die of starvation. Maybe we'll find out what "bix blattinger" means. Maybe I'll talk my parents into getting me another Striker Smith action figure. Maybe Neil Crouch will learn how to be a kid. Maybe I'll pass the big spelling test tomorrow.
But it won't be easy!
## About the Author and the Illustrator
DAN GUTMAN has written many weird books for kids. Dan lives in New Jersey (a very weird place) with his weird wife and two weird children. You can visit him on his weird website at www.dangutman.com
JIM PAILLOT lives in Arizona (another weird place) with his weird wife and two weird children. Isn't that weird? You can visit him on his weird website at www.jimpaillot.com
Visit www.AuthorTracker.com for exclusive information on your favorite HarperCollins author.
## Credits
Cover art © 2006 by Jim Paillot
## Copyright
MY WEIRD SCHOOL #11: MRS. KORMEL IS NOT NORMAL!. Text copyright © 2006 by Dan Gutman. Illustrations copyright © 2006 by Jim Paillot. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.
EPub © Edition SEPTEMBER 2008 ISBN: 9780061973314
10 9 8 7 6 5 4 3 2 1
## About the Publisher
Australia
HarperCollins Publishers (Australia) Pty. Ltd.
25 Ryde Road (PO Box 321)
Pymble, NSW 2073, Australia
http://www.harpercollinsebooks.com.au
Canada
HarperCollins Publishers Ltd.
2 Bloor Street East - 20th Floor
Toronto, ON, M4W 1A8, Canada
http://www.harpercollinsebooks.ca
New Zealand
HarperCollinsPublishers (New Zealand) Limited
P.O. Box 1
Auckland, New Zealand
http://www.harpercollinsebooks.co.nz
United Kingdom
HarperCollins Publishers Ltd.
77-85 Fulham Palace Road
London, W6 8JB, UK
http://www.harpercollinsebooks.co.uk
United States
HarperCollins Publishers Inc.
10 East 53rd Street
New York, NY 10022
http://www.harpercollinsebooks.com
| {
"redpajama_set_name": "RedPajamaBook"
} | 4,472 |
The J.M. Johnson House in Boise, Idaho, is a -story Queen Anne house designed by John E. Tourtellotte and constructed in 1898. The house includes a sandstone foundation and features a Tuscan column porch with a prominent, corner entry at 10th and Franklin Streets. A side gable with a shingled dimple window above a prominent beveled window bay are central to the Franklin Street exposure. The house was added to the National Register of Historic Places in 1982.
J.M. Johnson was a wool buyer from Mountain Home. He moved to Boise in 1898, the year of construction of the J.M. Johnson House. Among his business activities, Johnson traded in real estate and engaged in ranching. In 1919 he sold the J.M. Johnson House after purchasing a 7000-acre cattle ranch in Alturas.
See also
Fort Street Historic District
References
External links
1898 Queen Anne Cottage in Boise includes interior photographs of the J.M. Johnson House at Captivating Houses, a real estate website.
National Register of Historic Places in Boise, Idaho
Queen Anne architecture in Idaho
Houses in Boise, Idaho
Houses completed in 1898 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 335 |
Lynmar Capital to close
Barry B. Burr
Equity manager Lynmar Capital Group will close as of April 30, according to a fax the firm sent March 19 to the Illinois State Board of Investment, Chicago.
After receiving the announcement, ISBI terminated Lynmar, which was managing $115 million in U.S. active large-cap growth equities for the board, said William R. Atwood, executive director of the $10.1 billion board.
Marilyn J. Dicks-Riley, founder, president and CEO of Marlton, N.J.-based Lynmar, said in the fax that she "will retire April 30 from the investment management business" and "will be permanently closing Lynmar."
She couldn't be reached for comment.
Lynmar has $619.9 million in assets under management in 46 accounts, mostly from pension funds, according to its investment advisory registration with the Securities and Exchange Commission.
The SEC filing lists Ms. Dick-Riley as owning at least 75% of the firm and Keith Augustus Graham, executive vice president, and Sherri E. Jones, chief compliance officer, as owning less than 5% each.
ISBI had planned to place Lynmar on a watchlist during its March 19 meeting because of performance reasons, Mr. Atwood said.
ISBI will place the $115 million from Lynmar into an existing Russell 1000 Growth equity index fund managed by RhumbLine Advisers, raising it to $352 million, Mr. Atwood said.
"Long term we haven't figured out yet what we will do with the funds," he said. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,411 |
"""
Copyright (C) 2016, MuChu Hsu
Contributed by Muchu Hsu (muchu1983@gmail.com)
This file is part of BSD license
<https://opensource.org/licenses/BSD-3-Clause>
"""
import logging
import smtplib
from email.mime.text import MIMEText
from email.header import Header
class EmailUtility:
#建構子
def __init__(self):
#default email setting
self.DEFAULT_SMTP = "smtp.gmail.com:587"
self.DEFAULT_ACCOUNT = "public.muchu1983@gmail.com"
self.DEFAULT_PASSWORD = "bee520520bee"
#寄送 email
def sendEmail(self, strSubject=None, strFrom=None, strTo=None, strMsg=None, lstStrTarget=None, strSmtp=None, strAccount=None, strPassword=None):
if not (strSmtp and strAccount and strPassword):
strSmtp = self.DEFAULT_SMTP
strAccount = self.DEFAULT_ACCOUNT
strPassword = self.DEFAULT_PASSWORD
#郵件內容
msg = MIMEText(strMsg, "html", "utf-8")
msg["Subject"] = Header(strSubject, "utf-8")
msg["From"] = Header(strFrom, "utf-8")
msg["To"] = Header(strTo, "utf-8")
#傳送
server = smtplib.SMTP(strSmtp)
server.ehlo()
server.starttls()
logging.info("smtp login: %s %s"%(strAccount, strPassword))
server.login(strAccount, strPassword)
server.sendmail(strAccount, lstStrTarget, msg.as_string())
server.quit() | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,635 |
Q: AngularJS directive: put a call function in an attribute, without including another attribute What I'm after
I would like to create a ngLoad directive for images on my webpage. This is my preferred markup:
<img ng-src="{{ src }}" ng-load="onLoad()">
What I have
JSFiddle
Right now, I have a imgLoad directive with ngLoad specified in the scope, like so:
var app = angular.module('app', []);
app.directive('imgLoad', [function() {
return {
restrict: 'A',
scope: {
loadHandler: '&ngLoad'
},
link: function (scope, element, attr) {
element.on('load', scope.loadHandler);
}
};
}]);
The resulting markup is:
<img ng-src="{{ src }}" img-load ng-load="onLoad()">
Edit: I previously assumed that the name of the directive (i.e. imgLoad) needed to be different from the name of my attribute (i.e. ngLoad). This is not the case. The solution is to name my directive ngLoad.
What needs to change
I want to get rid of the imgLoad attribute. I want ngLoad to work regardless of any other attributes.
What I've already seen
My implementation is based on:
*
*angularjs directive call function specified in attribute and pass an argument to it
*AngularJS - Image "onload" event
*AngularJS: introduction to directives and $compile documentation
Any help is much appreciated!
A: Simple Solution Using Isolate Scope
Thanks to @smk for this answer
Give the directive and the scope property the same name.
app.directive('imgLoad', function() { // 'imgLoad'
return {
restrict: 'A',
scope: {
loadHandler: '&imgLoad' // 'imgLoad'
},
link: function (scope, element, attr) {
element.on('load', scope.loadHandler);
}
};
});
HTML:
<img img-load="onLoad()">
JSFiddle • AngularJS Guide to Isolate Scopes
While this solution is practical for most situations, it prevents you from using another directive with an isolate scope on the same element. Which brings us to…
More Control Using $parse
Use $parse to process the attributes yourself. The effect will be the same, and there won't be any conflicts with isolate scopes.
app.directive('imgLoad', ['$parse', function($parse) { // Inject $parse
return {
restrict: 'A',
link: function(scope, element, attr) {
var loadHandler = $parse(attr.imgLoad); /* Parse value of
'imgLoad' attribute */
element.on('load', function() {
loadHandler(scope); /* Run the function returned by $parse.
It needs the scope object
to operate properly. */
});
}
};
}]);
HTML (looks the same as before):
<img img-load="onLoad()">
JSFiddle • AngularJS $parse Documentation
Side Note: I didn't use ngLoad because Angular advises against it
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,944 |
Die Amagi-Klasse () war eine geplante Klasse von vier Schlachtkreuzern der Kaiserlich Japanischen Marine, die in Folge des Washingtoner Flottenvertrags von 1922 nicht gebaut wurden. Das einzige Schiff der Klasse, das je in den aktiven Dienst übernommen wurde, war die Akagi, allerdings erst nach dem Umbau zum Flugzeugträger. Von den übrigen drei Schiffen wurden zwei kurz nach Baubeginn abgebrochen und eines während der Bauphase durch ein Erdbeben zerstört.
Geschichte
Entwurf und Bau
Die Amagi-Klasse wurde im Zuge des japanischen 8-8-Plans entwickelt. Dieser Plan sah den Aufbau einer Flotte aus acht Schlachtschiffen und acht Schlachtkreuzern vor. Weil zwei Schiffe der Kongō-Klasse, Hiei und Kongō, 1923 ihre Altersgrenze erreichten, hätten die vier geplanten Einheiten der Amagi-Klasse Japan zu einer Flotte aus sechs Schlachtkreuzern verholfen.
Die Klasse war entwickelt worden, um die neuesten Schlachtkreuzertypen – die britische Hood und die amerikanische Lexington-Klasse – zu übertreffen.
Das Konzept orientierte sich an den Entwürfen der Schlachtschiffe der Tosa-Klasse. Wie diese sollte die Amagi-Klasse fünf Geschütztürme mit je zwei 41-cm-Geschützen als Hauptbewaffnung tragen. Die Türme sollten entlang der Mittelschifflinie aufgestellt werden, zwei auf dem Vorschiff und drei auf dem Achterschiff.
Schiffe der Amagi-Klasse
Amagi
Die Amagi wurde am 16. Dezember 1920 in der Marinewerft Yokosuka auf Kiel gelegt. Nach den Vereinbarungen im Flottenvertrag von Washington 1922 wurde der Bau des Schlachtkreuzers gestoppt und der Rumpf zum Umbau zu einem Flugzeugträger vorbereitet. Während dieser Umbauarbeiten wurden die Werftanlagen am 1. September 1923 um 11:58 Uhr vom Großen Kantō-Erdbeben getroffen. Da sich der Rumpf nicht im Wasser befand, sondern auf dem Boden des Docks aufsaß, übertrugen sich die Erschütterungen auf die Schiffsstruktur. Die Schäden waren so groß, dass die Amagi als Totalverlust eingestuft und abgewrackt werden musste. Ihre vertraglich festgelegte Rolle als Flugzeugträger wurde später von der Kaga, einem Schlachtschiff der Tosa-Klasse, übernommen.
Akagi
Die Akagi wurde am 6. Dezember 1920 von der Marinewerft in Kure auf Kiel gelegt. Nach den Beschlüssen des Flottenvertrages von Washington vom Februar 1922 wurde sie vom November 1923 an zum Flugzeugträger umgebaut. Sie wurde im März 1927 in Dienst gestellt. In dieser ursprünglichen Version besaß sie ein durchgängiges Flugdeck und zwei kleinere Abflugdecks. Als sich dieses Konzept als unzureichend erwiesen hatte, wurde sie 1935 umgebaut. Mit nur einem Flugdeck wurde sie letztlich zu einem der wichtigsten japanischen Flugzeugträger und nahm im Pazifikkrieg an zahlreichen Einsätzen teil. Am Morgen des 4. Juni 1942 wurde sie während der Schlacht um Midway von amerikanischen Trägerflugzeugen angegriffen und erhielt dabei einen einzigen Bombentreffer. Die Bombe detonierte zwischen aufgetankten und bewaffneten Flugzeugen auf dem Flugdeck der Akagi und löste Brände und Sekundärexplosionen aus, die nicht eingedämmt werden konnten, so dass das ausgebrannte Wrack am Morgen des 5. Juni von japanischen Zerstörern versenkt werden musste.
Atago
Atago war der Name, den das dritte Schiff der Amagi-Klasse erhalten sollte. Sie wurde am 22. November 1921 auf der Kawasaki-Werft in Kobe auf Kiel gelegt. Als Folge der Flottenverträge von Washington wurde ihr Bau im Juli 1922 abgebrochen. Auf den vorgesehenen Namen wurde später der 1932 in Dienst gestellte Schwere Kreuzer Atago getauft.
Takao
Takao war der Name, den das vierte Schiff der Amagi-Klasse erhalten sollte. Sie wurde am 19. Dezember 1921 auf der Mitsubishi-Werft in Nagasaki auf Kiel gelegt. Als Folge der Flottenverträge von Washington wurde ihr Bau im Juli 1922 abgebrochen. Auf den vorgesehenen Namen wurde später der 1932 in Dienst gestellte Schwere Kreuzer Takao getauft.
Technische Beschreibung
Rumpf
Der Rumpf eines Schlachtkreuzers der Amagi-Klasse, unterteilt in wasserdichte Abteilungen und genietet, sollte über alles 251,8 Meter lang, 30,8 Meter breit und hätte bei einer geplanten Einsatzverdrängung von 47.754 Tonnen einen Tiefgang von 9,5 Metern gehabt.
Panzerung
Struktureller Schutz
Die Amagi-Klasse erhielt ein integriertes, strukturelles Schutzsystem ohne zusätzliche Torpedowülste. Das System entsprach dem bei den Vorgängerklassen verwendeten Konzept: Eine äußere Hülle, aus vergleichsweise dünnem Stahl, ein Expansionsraum mit Tanks, die Luft oder Treibstoff enthielten, ein 75 Millimeter starkes Torpedoschott und dahinter eine weitere Lage mit Tanks, die durch ein abschließendes Längsschott von den Maschinenräumen und Magazinen getrennt waren.
Um die Wirkung von Splittern und Druckwellen, die bei Explosionen am Unterwasserrumpf entstehen konnten, zu minimieren, wurde mittschiffs, auf Höhe der Maschinenräume, der strukturelle Schutz durch Knautschrohre verstärkt, mit denen man eine der Abteilung vor dem Torpedoschott füllte. Knautschrohre erlaubten es, die Dicke der nachfolgenden Panzerung, ohne eine Verminderung der Schutzwirkung, in diesen Bereichen um bis zu 30 Prozent zu reduzieren.
Panzerschutz
Der Gürtelpanzer war an seiner stärksten Stelle 250 Millimeter dick, er neigte vom Oberdeck um rund 15 Grad nach innen und reichte etwa bis zur Oberkante des Torpedoschotts. Der horizontale Schutz bestand aus einem 95 Millimeter starken Panzerdeck. Die Barbetten, also die zylindrischen Strukturen unterhalb der Türme, durch die die Munition transportiert wurde, waren durchgehend bis zum Panzerdeck mit bis zu 230 Millimeter Panzerstahl geschützt, der an einigen Stellen bis auf 280 Millimeter Dicke aufwuchs. Der Gefechtsstand, also die kleine Befehlszentrale im Brückenaufbau, unmittelbar hinter Turm "B", von der im Notfall die wichtigsten Schiffssysteme gesteuert werden konnten, sollte einen Schutz von 356 Millimeter Stahl erhalten.
Der verwendete Panzerstahl für den größten Teil der Panzerung wäre vom Typ "NVNC" (New Vickers, Non Cemented), der nicht durch Einsatzhärtung nachbearbeitet worden wäre. Der Gürtelpanzer hätte dagegen eine Einsatzhärtung erhalten, wäre aber ebenfalls nach dem Herstellungsverfahren des britischen Vickers-Armstrog-Konzerns produziert worden und deshalb die Bezeichnung "VC" (Vickers, Cemented) getragen. Die NVNC-Panzerungselemente waren flexibler als die VC-Panzerung und, wegen des fehlenden Arbeitsschrittes, auch preiswerter herzustellen und zu verarbeiten. Die VC-Panzerung vermochte dagegen eher, Granatsplitter und direkte Treffer ohne Beschädigungen am Schiff abzuweisen.
Antrieb
Der Antrieb sollte durch acht kohle- und elf ölbefeuerte Dampferzeuger – Kampon-Kesseln des Yarrow-Typs – und vier Gijutsu-Hombu-Getriebeturbinensätze erfolgen, mit denen eine Leistung von 131.200 PS (96.497 kW) erreicht werden sollte. Diese hätten ihre Leistung an vier Wellen mit je einer dreiflügligen Schraube abgegeben. Die Höchstgeschwindigkeit hätte 30 Knoten (56 km/h) betragen und die maximale Fahrstrecke 8.000 Seemeilen (14.816 km) bei 14 Knoten, wofür 2.500 Tonnen Kohle und 3.900 Tonnen Schweröl gebunkert werden können sollten.
Bewaffnung
Schwere Artillerie
Als schwere Artillerie sollten zehn 41-cm-Seezielgeschütze Typ 3 in Kaliberlänge 45 verbaut werden, die in fünf Zwillingsgeschütztürmen entlang der Schiffsmittellinie aufgestellt werden sollten. Dabei wäre Turm "B" und Turm "D" überhöht positioniert worden, während der zusätzliche Turm "C" auf dem Wetterdeck aufsaß und dementsprechend, nach vorn durch die Aufbauten und nach achtern durch die Barbette von Turm "D", ein eingeschränktes Schussfeld besaß.
Das verwendete Geschütz hatte eine Feuerrate von 1,5 bis 2,5 Schuss die Minute und eine Lebensdauer von rund 250 Schuss. Es konnte eine 1.000 kg schwere Granate bis zu 38 Kilometer weit schießen. Das zum Einbau geplante Turmmodell entsprach dem, das auf der Tosa-Klasse verwendet werden sollte. Es hatte eine Seitenrichtgeschwindigkeit von 3° pro Sekunde, eine Höhenrichtgeschwindigkeit von 5° pro Sekunde und einen Höhenrichtbereich von −3° bis +35°. Die Panzerung hätte an der Front 460 mm, an den Seite 280 mm, am Rücken 190 mm und auf dem Dach 230 bis 250 mm betragen.
Mittelartillerie
Als Mittelartillerie sollten sechzehn 14-cm-Seezielgeschütze Typ 3 mit Kaliberlänge 50 in Kasematten verbaut werden. Dieses 1916 eingeführte Geschütz hatte eine Feuerrate von 6 bis 10 Schuss die Minute und eine Lebensdauer von 800 Schuss.
Flugabwehrbewaffnung
Zur Flugabwehr waren vier 12-cm-Geschütze Typ 10 in Einzellafetten geplant. Dieses Flugabwehrgeschütz erreichte eine effektive Kadenz von 6 bis 8 Schuss pro Minute und die maximale Reichweite betrug etwa 10 Kilometer bei 75° Rohrerhöhung. Die 7,8 Tonnen schwere Mittelpivotlafette war um 360° drehbar und hatte einen Höhenrichtbereich von −10° bis +75°.
Torpedobewaffnung
Es war vorgesehen acht Unterwassertorpedorohre des Kalibers 61-cm für Torpedos des Typ 8 zu verbauen. Je zwei Rohre beidseitig im Vorschiff – unmittelbar vor Turm "A" – und im Achterschiff – unmittelbar hinter Turm "E". Die Torpedoräume hätten dabei ein Deck oberhalb der Wasserlinie, das Torpedolager ein Deck tiefer befunden. Die Grundidee dieser Räume war, dass die Schlachtschiffe in lange andauernde Gefechte mit anderen Großkampfschiffen verwickelt werden konnten, bei denen beide Kontrahenten längere Zeit auf parallelen Kursen liefen, so dass sich die Möglichkeit ergeben hätte, den Gegner auch mit Torpedos zu beschießen.
Besatzung
Die Besatzung hatte eine geplante Stärke von 1.600 Offizieren, Unteroffizieren und Mannschaften.
Literatur
Weblinks
World Battleships List: Japanese Dreadnoughts auf hazegray.org (englisch)
Einzelnachweise
Militärschiffsklasse (Japanisches Kaiserreich)
Schlachtkreuzer-Klasse
Nicht realisiertes Projekt (Schiffbau) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,013 |
Cloiselia is a genus of flowering plants belonging to the family Asteraceae.
Its native range is Madagascar.
Species:
Cloiselia carbonaria
Cloiselia humbertii
Cloiselia madagascariensis
Cloiselia oleifolia
References
Asteraceae
Asteraceae genera | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,353 |
{"url":"https:\/\/ftp.aimsciences.org\/article\/doi\/10.3934\/eect.2022002","text":"# American Institute of Mathematical Sciences\n\nOctober\u00a0 2022,\u00a011(5):\u00a01829-1871. doi:\u00a010.3934\/eect.2022002\n\n## Local well-posedness of the coupled KdV-KdV systems on $\\mathbb{R}$\n\n 1 Department of Mathematics, University of California, Riverside, Riverside, CA 92521, USA 2 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA\n\n*Corresponding author: Xin Yang\n\nReceived\u00a0 August 2021 Revised\u00a0 November 2021 Published\u00a0 October\u00a02022 Early access\u00a0 January 2022\n\nInspired by the recent successful completion of the study of the well-posedness theory for the Cauchy problem of the Korteweg-de Vries (KdV) equation\n $u_t +uu_x +u_{xxx} = 0, \\quad \\left. u \\right |_{t = 0} = u_{0}$\nin the space\n $H^{s} (\\mathbb{R})$\n(or\n $H^{s} (\\mathbb{T})$\n), we study the well-posedness of the Cauchy problem for a class of coupled KdV-KdV (cKdV) systems\n $\\left\\{\\begin{array}{rcl} u_t+a_{1}u_{xxx} & = & c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \\\\ v_t+a_{2}v_{xxx}& = & c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, \\\\ \\left. (u, v)\\right |_{t = 0} & = & (u_{0}, v_{0}) \\end{array}\\right.$\nin the space\n $\\mathcal{H}^s (\\mathbb{R}) : = H^s (\\mathbb{R})\\times H^s (\\mathbb{R})$\n. Typical examples include the Gear-Grimshaw system, the Hirota-Satsuma system and the Majda-Biello system, to name a few.\nIn this paper we look for those values of\n $s\\in \\mathbb{R}$\nfor which the cKdV systems are well-posed in\n $\\mathcal{H}^s ( \\mathbb {R})$\n. The key ingredients in the proofs are the bilinear estimates in both divergence and non-divergence forms under the Fourier restriction space norms. Sharp results are established for all four types of the bilinear estimates that are associated to the cKdV systems. In contrast to the lone critical index\n $-\\frac{3}{4}$\nfor the single KdV equation, the critical indexes for the cKdV systems are\n $-\\frac{13}{12}$\n,\n $-\\frac{3}{4}$\n,\n $0$\nand\n $\\frac{3}{4}$\n.\nAs a result, the cKdV systems are classified into four classes, each of which corresponds to a unique index\n $s^{*}\\in\\{-\\frac{13}{12}, \\, -\\frac{3}{4}, \\, 0, \\, \\frac{3}{4}\\}$\nsuch that any system in this class is locally analytically well-posed if\n $s>s^{*}$\nwhile the bilinear estimate fails if\n $s . Citation: Xin Yang, Bing-Yu Zhang. Local well-posedness of the coupled KdV-KdV systems on$ \\mathbb{R} $. Evolution Equations and Control Theory, 2022, 11 (5) : 1829-1871. doi: 10.3934\/eect.2022002 ##### References: [1] B. Alvarez and X. Carvajal, On the local well-posedness for some systems of coupled KdV equations, Nonlinear Anal., 69 (2008), 692-715. doi: 10.1016\/j.na.2007.06.009. [2] J. M. Ash, J. Cohen and G. Wang, On strongly interacting internal solitary waves, J. Fourier Anal. Appl., 2 (1996), 507-517. doi: 10.1007\/s00041-001-4041-4. [3] D. Bekiranov, T. Ogawa and G. Ponce, Weak solvability and well-posedness of a coupled Schr\u00f6dinger-Korteweg de Vries equation for capillary-gravity wave interactions, Proc. Amer. Math. Soc., 125 (1997), 2907-2919. doi: 10.1090\/S0002-9939-97-03941-5. [4] J. L. Bona, G. Ponce, J.-C. Saut and M. M. Tom, A model system for strong interaction between internal solitary waves, Comm. Math. Phys., 143 (1992), 287-313. [5] J. L. Bona and R. Scott, Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces, Duke Math. J., 43 (1976), 87-99. [6] J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601. doi: 10.1098\/rsta.1975.0035. [7] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schr\u00f6dinger equations, Geom. Funct. Anal., 3 (1993), 107-156. doi: 10.1007\/BF01896020. [8] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007\/BF01895688. [9] E. Cerpa and E. Cr\u00e9peau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain, Ann. Inst. H. Poincar\u00e9 Anal. Non Lin\u00e9aire, 26 (2009), 457-475. doi: 10.1016\/j.anihpc.2007.11.003. [10] M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293. doi: 10.1353\/ajm.2003.0040. [11] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on$\\Bbb R$and$\\Bbb T$, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090\/S0894-0347-03-00421-1. [12] P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439. doi: 10.1090\/S0894-0347-1988-0928265-0. [13] J.-M. Coron and E. Cr\u00e9peau, Exact boundary controllability of a nonlinear KdV equation with a critical length, J. Eur. Math. Soc., 6 (2004), 367-398. [14] X. Feng, Global well-posedness of the initial value problem for the Hirota-Satsuma system, Manuscripta Math., 84 (1994), 361-378. doi: 10.1007\/BF02567462. [15] J. A. Gear and R. Grimshaw, Weak and strong interactions between internal solitary waves, Stud. Appl. Math., 70 (1984), 235-258. doi: 10.1002\/sapm1984703235. [16] Z. Guo, Global well-posedness of Korteweg-de Vries equation in$H^{-3\/4}(\\Bbb R)$, J. Math. Pures Appl., 91 (2009), 583-597. doi: 10.1016\/j.matpur.2009.01.012. [17] R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A, 85 (1981), 407-408. doi: 10.1016\/0375-9601(81)90423-0. [18] Y. Kametaka, Korteweg -de vries equation, i, ii, iii, iv, Proc. Japan Acad., 45 (1969), 661-665. [19] T. Kappeler and P. Topalov, Global wellposedness of KdV in$H^{-1}(\\Bbb T, \\Bbb R)$, Duke Math. J., 135 (2006), 327-360. doi: 10.1215\/S0012-7094-06-13524-X. [20] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Lecture Notes in Math., 448 (1975), 25-70. [21] T. Kato, On the Korteweg-de\\thinspace Vries equation, Manuscripta Math., 28 (1979), 89-99. doi: 10.1007\/BF01647967. [22] T. Kato, The Cauchy problem for the Korteweg-de Vries equation, In Nonlinear Partial Differential Equations and Their Applications, Res. Notes in Math., 53 (1981), 293\u2013307. [23] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, 8 (1983), 93-128. [24] C. E. Kenig, G. Ponce and L. Vega, On the (generalized) Korteweg-de Vries equation, Duke Math. J., 59 (1989), 585-610. doi: 10.1215\/S0012-7094-89-05927-9. [25] C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512\/iumj.1991.40.40003. [26] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347. doi: 10.1090\/S0894-0347-1991-1086966-0. [27] C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21. doi: 10.1215\/S0012-7094-93-07101-3. [28] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002\/cpa.3160460405. [29] C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090\/S0894-0347-96-00200-7. [30] R. Killip and M. Vi\u015fan, KdV is well-posed in$H^{-1}$, Ann. of Math., 190 (2019), 249-305. doi: 10.4007\/annals.2019.190.1.4. [31] N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464. [32] C. Laurent, L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic Domain, Comm. Partial Differential Equations, 35 (2010), 707-744. doi: 10.1080\/03605300903585336. [33] F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations, Commun. Pure Appl. Anal., 3 (2004), 417-431. doi: 10.3934\/cpaa.2004.3.417. [34] A. J. Majda and J. A. Biello, The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves, J. Atmospheric Sci., 60 (2003), 1809-1821. doi: 10.1175\/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2. [35] L. Molinet, A note on ill posedness for the KdV equation, Differential Integral Equations, 24 (2011), 759-765. [36] L. Molinet, Sharp ill-posedness results for the KdV and mKdV equations on the torus, Adv. Math., 230 (2012), 1895-1930. doi: 10.1016\/j.aim.2012.03.026. [37] T. Oh, Diophantine conditions in global well-posedness for coupled KdV-type systems, Electron. J. Differential Equations, (2009), 48 pp. [38] T. Oh, Diophantine conditions in well-posedness theory of coupled KdV-type systems: Local theory, Int. Math. Res. Not. IMRN, 18 (2009), 3516-3556. doi: 10.1093\/imrn\/rnp063. [39] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Cal. Var., 2 (1997), 33-55. doi: 10.1051\/cocv:1997102. [40] L. Rosier and B.-Y. Zhang, Control and stabilization of the korteweg-de Vries equation: Recent progress, J. Syst. Sci. Complex, 22 (2009), 647-682. doi: 10.1007\/s11424-009-9194-2. [41] D.-L. Russell and B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans. Amer. Math. Soc., 348 (1996), 3643-3672. doi: 10.1090\/S0002-9947-96-01672-8. [42] J. C. Saut and R. Temam, Remarks on the Korteweg-de Vries equation, Israel J. Math., 24 (1976), 78-87. doi: 10.1007\/BF02761431. [43] J.-C. Saut and N. Tzvetkov, On a model system for the oblique interaction of internal gravity waves, M2AN Math. Model. Numer. Anal., Special issue for R. Temam's 60th birthday 34 (2000), 501\u2013523. doi: 10.1051\/m2an:2000153. [44] A. Sj\u00f6berg, On the Korteweg-de Vries equation: Existence and uniqueness, Department of Computer Sciences, Uppsala University, Uppsala, Sweden, 1967. [45] A. Sj\u00f6berg, On the Korteweg-de Vries equation: Existence and uniqueness, J. Math. Anal. Appl., 29 (1970), 569-579. doi: 10.1016\/0022-247X(70)90068-5. [46] P. Sj\u00f6lin, Regularity of solutions to the Schr\u00f6dinger equation, Duke Math. J., 55 (1987), 699-715. doi: 10.1215\/S0012-7094-87-05535-9. [47] T. Tao, Multilinear weighted convolution of$L^2$-functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908. doi: 10.1353\/ajm.2001.0035. [48] L. Tartar, Interpolation non lin\u00e9aire et r\u00e9gularit\u00e9, J. Functional Analysis, 9 (1972), 469-489. doi: 10.1016\/0022-1236(72)90022-5. [49] R. Temam, Sur un probl\u00e8me non lin\u00e9aire, J. Math. Pures Appl., 48 (1969), 159-172. [50] M. Tsutsumi and T. Mukasa, Parabolic regularizations for the generalized Korteweg-de Vries equation, Funkcial. Ekvac., 14 (1971), 89-110. [51] M. Tsutsumi, T. Mukasa and R. Iino, On the generalized Korteweg-de Vries equation, Proc. Japan Acad., 46 (1970), 921-925. [52] B.-Y. Zhang, Analyticity of solutions of the generalized Kortweg-de Vries equation with respect to their initial values, SIAM J. Math. Anal., 26 (1995), 1488-1513. doi: 10.1137\/S0036141093242600. [53] B.-Y. Zhang, A remark on the Cauchy problem for the Korteweg-de Vries equation on a periodic domain, Differential Integral Equations, 8 (1995), 1191-1204. [54] B.-Y. Zhang, Taylor series expansion for solutions of the Korteweg-de Vries equation with respect to their initial values, J. Funct. Anal., 129 (1995), 293-324. doi: 10.1006\/jfan.1995.1052. [55] B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation, SIAM J. Cont. Optim., 37 (1999), 543-565. doi: 10.1137\/S0363012997327501. [56] B.-Y. Zhang, Well-posedness and control of the Korteweg-de Vries equation on a bounded domain, Fifth International Congress of Chinese Mathematicians, AMS\/IP Stud. Adv. Math. AMS, Providence, RI, 51 (2012), 931\u2013956. show all references ##### References: [1] B. Alvarez and X. Carvajal, On the local well-posedness for some systems of coupled KdV equations, Nonlinear Anal., 69 (2008), 692-715. doi: 10.1016\/j.na.2007.06.009. [2] J. M. Ash, J. Cohen and G. Wang, On strongly interacting internal solitary waves, J. Fourier Anal. Appl., 2 (1996), 507-517. doi: 10.1007\/s00041-001-4041-4. [3] D. Bekiranov, T. Ogawa and G. Ponce, Weak solvability and well-posedness of a coupled Schr\u00f6dinger-Korteweg de Vries equation for capillary-gravity wave interactions, Proc. Amer. Math. Soc., 125 (1997), 2907-2919. doi: 10.1090\/S0002-9939-97-03941-5. [4] J. L. Bona, G. Ponce, J.-C. Saut and M. M. Tom, A model system for strong interaction between internal solitary waves, Comm. Math. Phys., 143 (1992), 287-313. [5] J. L. Bona and R. Scott, Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces, Duke Math. J., 43 (1976), 87-99. [6] J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601. doi: 10.1098\/rsta.1975.0035. [7] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schr\u00f6dinger equations, Geom. Funct. Anal., 3 (1993), 107-156. doi: 10.1007\/BF01896020. [8] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007\/BF01895688. [9] E. Cerpa and E. Cr\u00e9peau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain, Ann. Inst. H. Poincar\u00e9 Anal. Non Lin\u00e9aire, 26 (2009), 457-475. doi: 10.1016\/j.anihpc.2007.11.003. [10] M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293. doi: 10.1353\/ajm.2003.0040. [11] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on$\\Bbb R$and$\\Bbb T$, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090\/S0894-0347-03-00421-1. [12] P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439. doi: 10.1090\/S0894-0347-1988-0928265-0. [13] J.-M. Coron and E. Cr\u00e9peau, Exact boundary controllability of a nonlinear KdV equation with a critical length, J. Eur. Math. Soc., 6 (2004), 367-398. [14] X. Feng, Global well-posedness of the initial value problem for the Hirota-Satsuma system, Manuscripta Math., 84 (1994), 361-378. doi: 10.1007\/BF02567462. [15] J. A. Gear and R. Grimshaw, Weak and strong interactions between internal solitary waves, Stud. Appl. Math., 70 (1984), 235-258. doi: 10.1002\/sapm1984703235. [16] Z. Guo, Global well-posedness of Korteweg-de Vries equation in$H^{-3\/4}(\\Bbb R)$, J. Math. Pures Appl., 91 (2009), 583-597. doi: 10.1016\/j.matpur.2009.01.012. [17] R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A, 85 (1981), 407-408. doi: 10.1016\/0375-9601(81)90423-0. [18] Y. Kametaka, Korteweg -de vries equation, i, ii, iii, iv, Proc. Japan Acad., 45 (1969), 661-665. [19] T. Kappeler and P. Topalov, Global wellposedness of KdV in$H^{-1}(\\Bbb T, \\Bbb R)$, Duke Math. J., 135 (2006), 327-360. doi: 10.1215\/S0012-7094-06-13524-X. [20] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Lecture Notes in Math., 448 (1975), 25-70. [21] T. Kato, On the Korteweg-de\\thinspace Vries equation, Manuscripta Math., 28 (1979), 89-99. doi: 10.1007\/BF01647967. [22] T. Kato, The Cauchy problem for the Korteweg-de Vries equation, In Nonlinear Partial Differential Equations and Their Applications, Res. Notes in Math., 53 (1981), 293\u2013307. [23] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, 8 (1983), 93-128. [24] C. E. Kenig, G. Ponce and L. Vega, On the (generalized) Korteweg-de Vries equation, Duke Math. J., 59 (1989), 585-610. doi: 10.1215\/S0012-7094-89-05927-9. [25] C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512\/iumj.1991.40.40003. [26] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347. doi: 10.1090\/S0894-0347-1991-1086966-0. [27] C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21. doi: 10.1215\/S0012-7094-93-07101-3. [28] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002\/cpa.3160460405. [29] C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090\/S0894-0347-96-00200-7. [30] R. Killip and M. Vi\u015fan, KdV is well-posed in$H^{-1}$, Ann. of Math., 190 (2019), 249-305. doi: 10.4007\/annals.2019.190.1.4. [31] N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464. [32] C. Laurent, L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic Domain, Comm. Partial Differential Equations, 35 (2010), 707-744. doi: 10.1080\/03605300903585336. [33] F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations, Commun. Pure Appl. Anal., 3 (2004), 417-431. doi: 10.3934\/cpaa.2004.3.417. [34] A. J. Majda and J. A. Biello, The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves, J. Atmospheric Sci., 60 (2003), 1809-1821. doi: 10.1175\/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2. [35] L. Molinet, A note on ill posedness for the KdV equation, Differential Integral Equations, 24 (2011), 759-765. [36] L. Molinet, Sharp ill-posedness results for the KdV and mKdV equations on the torus, Adv. Math., 230 (2012), 1895-1930. doi: 10.1016\/j.aim.2012.03.026. [37] T. Oh, Diophantine conditions in global well-posedness for coupled KdV-type systems, Electron. J. Differential Equations, (2009), 48 pp. [38] T. Oh, Diophantine conditions in well-posedness theory of coupled KdV-type systems: Local theory, Int. Math. Res. Not. IMRN, 18 (2009), 3516-3556. doi: 10.1093\/imrn\/rnp063. [39] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Cal. Var., 2 (1997), 33-55. doi: 10.1051\/cocv:1997102. [40] L. Rosier and B.-Y. Zhang, Control and stabilization of the korteweg-de Vries equation: Recent progress, J. Syst. Sci. Complex, 22 (2009), 647-682. doi: 10.1007\/s11424-009-9194-2. [41] D.-L. Russell and B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans. Amer. Math. Soc., 348 (1996), 3643-3672. doi: 10.1090\/S0002-9947-96-01672-8. [42] J. C. Saut and R. Temam, Remarks on the Korteweg-de Vries equation, Israel J. Math., 24 (1976), 78-87. doi: 10.1007\/BF02761431. [43] J.-C. Saut and N. Tzvetkov, On a model system for the oblique interaction of internal gravity waves, M2AN Math. Model. Numer. Anal., Special issue for R. Temam's 60th birthday 34 (2000), 501\u2013523. doi: 10.1051\/m2an:2000153. [44] A. Sj\u00f6berg, On the Korteweg-de Vries equation: Existence and uniqueness, Department of Computer Sciences, Uppsala University, Uppsala, Sweden, 1967. [45] A. Sj\u00f6berg, On the Korteweg-de Vries equation: Existence and uniqueness, J. Math. Anal. Appl., 29 (1970), 569-579. doi: 10.1016\/0022-247X(70)90068-5. [46] P. Sj\u00f6lin, Regularity of solutions to the Schr\u00f6dinger equation, Duke Math. J., 55 (1987), 699-715. doi: 10.1215\/S0012-7094-87-05535-9. [47] T. Tao, Multilinear weighted convolution of$L^2$-functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908. doi: 10.1353\/ajm.2001.0035. [48] L. Tartar, Interpolation non lin\u00e9aire et r\u00e9gularit\u00e9, J. Functional Analysis, 9 (1972), 469-489. doi: 10.1016\/0022-1236(72)90022-5. [49] R. Temam, Sur un probl\u00e8me non lin\u00e9aire, J. Math. Pures Appl., 48 (1969), 159-172. [50] M. Tsutsumi and T. Mukasa, Parabolic regularizations for the generalized Korteweg-de Vries equation, Funkcial. Ekvac., 14 (1971), 89-110. [51] M. Tsutsumi, T. Mukasa and R. Iino, On the generalized Korteweg-de Vries equation, Proc. Japan Acad., 46 (1970), 921-925. [52] B.-Y. Zhang, Analyticity of solutions of the generalized Kortweg-de Vries equation with respect to their initial values, SIAM J. Math. Anal., 26 (1995), 1488-1513. doi: 10.1137\/S0036141093242600. [53] B.-Y. Zhang, A remark on the Cauchy problem for the Korteweg-de Vries equation on a periodic domain, Differential Integral Equations, 8 (1995), 1191-1204. [54] B.-Y. Zhang, Taylor series expansion for solutions of the Korteweg-de Vries equation with respect to their initial values, J. Funct. Anal., 129 (1995), 293-324. doi: 10.1006\/jfan.1995.1052. [55] B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation, SIAM J. Cont. Optim., 37 (1999), 543-565. doi: 10.1137\/S0363012997327501. [56] B.-Y. Zhang, Well-posedness and control of the Korteweg-de Vries equation on a bounded domain, Fifth International Congress of Chinese Mathematicians, AMS\/IP Stud. Adv. Math. AMS, Providence, RI, 51 (2012), 931\u2013956. Range of$ s $and$ b $when$ s<-\\frac{3}{4} $Main Results Case$ r=\\frac{a_{2}}{a_{1}} $Coefficients$ b_{ij} $,$ c_{ij} $and$ d_{ij} s $(1)$r <0(c_{ij})=0$,$d_{11}=d_{12}$and$d_{21}=d_{22}$Otherwise$s\\geq -\\frac{13}{12}s>-\\frac{3}{4}$(2)$0 -\\frac{3}{4}s\\geq 0$(3)$r=\\frac{1}{4}c_{21}=d_{11}=d_{12}=0$Otherwise$s\\geq 0s\\geq\\frac{3}{4}$(4)$\\frac{1}{4} -\\frac{3}{4}s>0$(6)$1 4c_{21}=d_{11}=d_{12}=0$Otherwise$s>-\\frac{3}{4}s\\geq 0$ Case$ r=\\frac{a_{2}}{a_{1}} $Coefficients$ b_{ij} $,$ c_{ij} $and$ d_{ij} s $(1)$r <0(c_{ij})=0$,$d_{11}=d_{12}$and$d_{21}=d_{22}$Otherwise$s\\geq -\\frac{13}{12}s>-\\frac{3}{4}$(2)$0 -\\frac{3}{4}s\\geq 0$(3)$r=\\frac{1}{4}c_{21}=d_{11}=d_{12}=0$Otherwise$s\\geq 0s\\geq\\frac{3}{4}$(4)$\\frac{1}{4} -\\frac{3}{4}s>0$(6)$1 4c_{21}=d_{11}=d_{12}=0$Otherwise$s>-\\frac{3}{4}s\\geq 0$LWP Results Case Coefficient$ a_2 s $(1)$ a_2\\in(-\\infty, 0)\\cup \\{1\\} \\cup \\{4, \\infty\\} s>-\\frac34 $(2)$ a_2\\in(0, 1)\\cup(1, 4) s\\geq 0 $(3)$ a_2=4 s\\geq\\frac{3}{4} $ Case Coefficient$ a_2 s $(1)$ a_2\\in(-\\infty, 0)\\cup \\{1\\} \\cup \\{4, \\infty\\} s>-\\frac34 $(2)$ a_2\\in(0, 1)\\cup(1, 4) s\\geq 0 $(3)$ a_2=4 s\\geq\\frac{3}{4} $GWP Results Case Coefficient$ a_2 s $(1)$ a_2=1 s>-\\frac34 $(2)$ a_2\\not\\in\\{1, 4\\} s\\geq 0 $(3)$ a_2=4 s\\geq1 $ Case Coefficient$ a_2 s $(1)$ a_2=1 s>-\\frac34 $(2)$ a_2\\not\\in\\{1, 4\\} s\\geq 0 $(3)$ a_2=4 s\\geq1 $LWP Results Case Coefficients$ a_1 $and$ c_{12} s $(1)$ a_1\\in(-\\infty, 0)\\cup(0, \\frac14) s>-\\frac34 $(2)$ a_1\\in(\\frac14, 1)\\cup(1, \\infty) s\\geq 0 $(3)$ a_1=1 s>0 $(4)$ a_1=\\frac14 s\\geq\\frac{3}{4} $ Case Coefficients$ a_1 $and$ c_{12} s $(1)$ a_1\\in(-\\infty, 0)\\cup(0, \\frac14) s>-\\frac34 $(2)$ a_1\\in(\\frac14, 1)\\cup(1, \\infty) s\\geq 0 $(3)$ a_1=1 s>0 $(4)$ a_1=\\frac14 s\\geq\\frac{3}{4} $GWP Results Case Coefficients$ a_1 $and$ c_{12} s $(1)$ a_1\\not\\in\\{\\frac14, 1\\} $,$ c_{12}>0 s\\geq 0 $(2)$ a_1=\\frac14 $,$ c_{12}>0 s\\geq 1 $ Case Coefficients$ a_1 $and$ c_{12} s $(1)$ a_1\\not\\in\\{\\frac14, 1\\} $,$ c_{12}>0 s\\geq 0 $(2)$ a_1=\\frac14 $,$ c_{12}>0 s\\geq 1 $LWP Results Case$ \\rho_1 $,$ \\rho_2 $and$ \\sigma_{i} (1\\leq i\\leq 4) s $(1)$ \\sigma_3=0 $,$ \\rho_1=1 s>-\\frac34 $(2)$ \\rho_2\\sigma_3^2>1 s>-\\frac34 $(3)$ \\rho_2\\sigma_3^2<1 $, (1.13) fails$ s\\geq 0 $(4)$ \\rho_2\\sigma_3^2<1 $, (1.13) holds$ s\\geq\\frac{3}{4} $ Case$ \\rho_1 $,$ \\rho_2 $and$ \\sigma_{i} (1\\leq i\\leq 4) s $(1)$ \\sigma_3=0 $,$ \\rho_1=1 s>-\\frac34 $(2)$ \\rho_2\\sigma_3^2>1 s>-\\frac34 $(3)$ \\rho_2\\sigma_3^2<1 $, (1.13) fails$ s\\geq 0 $(4)$ \\rho_2\\sigma_3^2<1 $, (1.13) holds$ s\\geq\\frac{3}{4} $GWP Results Case$ \\rho_1 $,$ \\rho_2 $and$ \\sigma_{i} (1\\leq i\\leq 4) s $(1)$ \\rho_2\\sigma_3^2\\neq 1 $, (1.13) fails$ s\\geq 0 $(2)$ \\rho_2\\sigma_3^2\\neq 1 $, (1.13) holds$ s\\geq 1 $ Case$ \\rho_1 $,$ \\rho_2 $and$ \\sigma_{i} (1\\leq i\\leq 4) s $(1)$ \\rho_2\\sigma_3^2\\neq 1 $, (1.13) fails$ s\\geq 0 $(2)$ \\rho_2\\sigma_3^2\\neq 1 $, (1.13) holds$ s\\geq 1 $Bilinear Estimates Type$ r<0 0\\frac{1}{4} $,$ r\\neq 1 r=1 $(D1): (3.4)$ s>-\\frac{3}{4} s>-\\frac{3}{4} s\\geq \\frac{3}{4} s\\geq 0 s>-\\frac{3}{4} $(D2): (3.5)$ s>-\\frac{3}{4} s\\geq \\frac{3}{4} s\\geq 0 s>-\\frac{3}{4} $(ND1): (3.6)$ s>-\\frac{3}{4} s>-\\frac{3}{4} s\\geq \\frac{3}{4} s\\geq 0 s>0 $(ND2): (3.7)$ s>-\\frac{3}{4} s>-\\frac{3}{4} s\\geq \\frac{3}{4} s\\geq 0 s>0 $ Type$ r<0 0\\frac{1}{4} $,$ r\\neq 1 r=1 $(D1): (3.4)$ s>-\\frac{3}{4} s>-\\frac{3}{4} s\\geq \\frac{3}{4} s\\geq 0 s>-\\frac{3}{4} $(D2): (3.5)$ s>-\\frac{3}{4} s\\geq \\frac{3}{4} s\\geq 0 s>-\\frac{3}{4} $(ND1): (3.6)$ s>-\\frac{3}{4} s>-\\frac{3}{4} s\\geq \\frac{3}{4} s\\geq 0 s>0 $(ND2): (3.7)$ s>-\\frac{3}{4} s>-\\frac{3}{4} s\\geq \\frac{3}{4} s\\geq 0 s>0 $Sharpness of Bilinear Estimates Type$ r<0 0\\frac{1}{4} $,$ r\\neq 1 r=1 $(D1): (3.4)$ s<-\\frac{3}{4} s<-\\frac{3}{4} s< \\frac{3}{4} s< 0 s<-\\frac{3}{4} $(D2): (3.5)$ s<-\\frac{3}{4} s< \\frac{3}{4} s< 0 s<-\\frac{3}{4} $(ND1): (3.6)$ s<-\\frac{3}{4} s<-\\frac{3}{4} s< \\frac{3}{4} s< 0 s<0 $(ND2): (3.7)$ s<-\\frac{3}{4} s<-\\frac{3}{4} s<\\frac{3}{4} s< 0 s<0 $ Type$ r<0 0\\frac{1}{4} $,$ r\\neq 1 r=1 $(D1): (3.4)$ s<-\\frac{3}{4} s<-\\frac{3}{4} s< \\frac{3}{4} s< 0 s<-\\frac{3}{4} $(D2): (3.5)$ s<-\\frac{3}{4} s< \\frac{3}{4} s< 0 s<-\\frac{3}{4} $(ND1): (3.6)$ s<-\\frac{3}{4} s<-\\frac{3}{4} s< \\frac{3}{4} s< 0 s<0 $(ND2): (3.7)$ s<-\\frac{3}{4} s<-\\frac{3}{4} s<\\frac{3}{4} s< 0 s<0 $Troubles and Critical Indexes ($ r = \\frac{{\\alpha}_2}{{\\alpha}_1} $) $ r<0 0\\frac{1}{4} $,$ r\\neq 1 r=1 $(D1): (3.4) (T2)$-\\frac{3}{4}$(T2)$-\\frac{3}{4}$(T1)+(T2)$\\frac{3}{4}$(T1)$ 0$(T2)$-\\frac{3}{4}$(D2): (3.5) None$ -\\frac{13}{12}$(T2)$-\\frac{3}{4}$(T1)+(T2)$ \\frac{3}{4}$(T1)$ 0$(T2)$-\\frac{3}{4}$(ND1): (3.6) (T3)$-\\frac{3}{4}$(T2) or (T3)$-\\frac{3}{4}$(T1)+(T2)$ \\frac{3}{4}$(T1)$ 0$(T2)+(T3)$0$(ND2): (3.7) (T3)$-\\frac{3}{4}$(T2) or (T3)$-\\frac{3}{4}$(T1)+(T2)$ \\frac{3}{4}$(T1)$ 0$(T2)+(T3)$0$$ r<0 0\\frac{1}{4} $,$ r\\neq 1 r=1 $(D1): (3.4) (T2)$-\\frac{3}{4}$(T2)$-\\frac{3}{4}$(T1)+(T2)$\\frac{3}{4}$(T1)$ 0$(T2)$-\\frac{3}{4}$(D2): (3.5) None$ -\\frac{13}{12}$(T2)$-\\frac{3}{4}$(T1)+(T2)$ \\frac{3}{4}$(T1)$ 0$(T2)$-\\frac{3}{4}$(ND1): (3.6) (T3)$-\\frac{3}{4}$(T2) or (T3)$-\\frac{3}{4}$(T1)+(T2)$ \\frac{3}{4}$(T1)$ 0$(T2)+(T3)$0$(ND2): (3.7) (T3)$-\\frac{3}{4}$(T2) or (T3)$-\\frac{3}{4}$(T1)+(T2)$ \\frac{3}{4}$(T1)$ 0$(T2)+(T3)$0$ [1] Roberto de A. Capistrano-Filho, Vilmos Komornik, Ademir F. Pazoto. Pointwise control of the linearized Gear-Grimshaw system. Evolution Equations and Control Theory, 2020, 9 (3) : 693-719. doi: 10.3934\/eect.2020029 [2] Xin Yang, Bing-Yu Zhang. Well-posedness and critical index set of the Cauchy problem for the coupled KdV-KdV systems on$ \\mathbb{T} $. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934\/dcds.2022090 [3] Timur Akhunov. Local well-posedness of quasi-linear systems generalizing KdV. Communications on Pure and Applied Analysis, 2013, 12 (2) : 899-921. doi: 10.3934\/cpaa.2013.12.899 [4] Cezar Kondo, Ronaldo Pes. Well-posedness for a coupled system of Kawahara\/KdV type equations with polynomials nonlinearities. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2615-2641. doi: 10.3934\/cpaa.2022063 [5] Shengfu Deng. Generalized multi-hump wave solutions of Kdv-Kdv system of Boussinesq equations. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3671-3716. doi: 10.3934\/dcds.2019150 [6] Hartmut Pecher. Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3303-3321. doi: 10.3934\/cpaa.2020146 [7] Andreia Chapouto. A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3915-3950. doi: 10.3934\/dcds.2021022 [8] Hung Luong. Local well-posedness for the Zakharov system on the background of a line soliton. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2657-2682. doi: 10.3934\/cpaa.2018126 [9] Akansha Sanwal. Local well-posedness for the Zakharov system in dimension d \u2264 3. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1067-1103. doi: 10.3934\/dcds.2021147 [10] Hartmut Pecher. Local well-posedness for the Maxwell-Dirac system in temporal gauge. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 3065-3076. doi: 10.3934\/dcds.2022008 [11] M\u00e1rcio Cavalcante, Chulkwang Kwak. Local well-posedness of the fifth-order KdV-type equations on the half-line. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2607-2661. doi: 10.3934\/cpaa.2019117 [12] Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5047-5066. doi: 10.3934\/dcds.2016019 [13] Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934\/dcds.2020393 [14] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934\/dcds.2020382 [15] Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934\/cpaa.2015.14.1259 [16] Jerry L. Bona, Didier Pilod. Stability of solitary-wave solutions to the Hirota-Satsuma equation. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1391-1413. doi: 10.3934\/dcds.2010.27.1391 [17] Hartmut Pecher. Local well-posedness for the Klein-Gordon-Zakharov system in 3D. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1707-1736. doi: 10.3934\/dcds.2020338 [18] Sirui Li, Wei Wang, Pingwen Zhang. Local well-posedness and small Deborah limit of a molecule-based$Q\\$-tensor system. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2611-2655. doi: 10.3934\/dcdsb.2015.20.2611 [19] Jishan Fan, Yueling Jia. Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum. Kinetic and Related Models, 2018, 11 (1) : 97-106. doi: 10.3934\/krm.2018005 [20] Jingjing Zhang, Ting Zhang. Local well-posedness of perturbed Navier-Stokes system around Landau solutions. Electronic Research Archive, 2021, 29 (4) : 2719-2739. doi: 10.3934\/era.2021010\n\n2021\u00a0Impact Factor:\u00a01.169","date":"2022-08-19 14:24:46","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\": 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\": 0.860099196434021, \"perplexity\": 10995.793563843059}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882573699.52\/warc\/CC-MAIN-20220819131019-20220819161019-00059.warc.gz\"}"} | null | null |
"""Unit tests for image utils."""
import contextlib
import mox
import textwrap
from cinder.image import image_utils
from cinder import test
from cinder import utils
class TestUtils(test.TestCase):
def setUp(self):
super(TestUtils, self).setUp()
self._mox = mox.Mox()
self.addCleanup(self._mox.UnsetStubs)
def test_resize_image(self):
mox = self._mox
mox.StubOutWithMock(utils, 'execute')
TEST_IMG_SOURCE = 'boobar.img'
TEST_IMG_SIZE_IN_GB = 1
utils.execute('qemu-img', 'resize', TEST_IMG_SOURCE,
'%sG' % TEST_IMG_SIZE_IN_GB, run_as_root=False)
mox.ReplayAll()
image_utils.resize_image(TEST_IMG_SOURCE, TEST_IMG_SIZE_IN_GB)
mox.VerifyAll()
class TestExtractTo(test.TestCase):
def test_extract_to_calls_tar(self):
mox = self.mox
mox.StubOutWithMock(utils, 'execute')
utils.execute(
'tar', '-xzf', 'archive.tgz', '-C', 'targetpath').AndReturn(
('ignored', 'ignored')
)
mox.ReplayAll()
image_utils.extract_targz('archive.tgz', 'targetpath')
mox.VerifyAll()
class TestSetVhdParent(test.TestCase):
def test_vhd_util_call(self):
mox = self.mox
mox.StubOutWithMock(utils, 'execute')
utils.execute(
'vhd-util', 'modify', '-n', 'child', '-p', 'parent').AndReturn(
('ignored', 'ignored')
)
mox.ReplayAll()
image_utils.set_vhd_parent('child', 'parent')
mox.VerifyAll()
class TestFixVhdChain(test.TestCase):
def test_empty_chain(self):
mox = self.mox
mox.StubOutWithMock(image_utils, 'set_vhd_parent')
mox.ReplayAll()
image_utils.fix_vhd_chain([])
def test_single_vhd_file_chain(self):
mox = self.mox
mox.StubOutWithMock(image_utils, 'set_vhd_parent')
mox.ReplayAll()
image_utils.fix_vhd_chain(['0.vhd'])
def test_chain_with_two_elements(self):
mox = self.mox
mox.StubOutWithMock(image_utils, 'set_vhd_parent')
image_utils.set_vhd_parent('0.vhd', '1.vhd')
mox.ReplayAll()
image_utils.fix_vhd_chain(['0.vhd', '1.vhd'])
class TestGetSize(test.TestCase):
def test_vhd_util_call(self):
mox = self.mox
mox.StubOutWithMock(utils, 'execute')
utils.execute(
'vhd-util', 'query', '-n', 'vhdfile', '-v').AndReturn(
('1024', 'ignored')
)
mox.ReplayAll()
result = image_utils.get_vhd_size('vhdfile')
mox.VerifyAll()
self.assertEquals(1024, result)
class TestResize(test.TestCase):
def test_vhd_util_call(self):
mox = self.mox
mox.StubOutWithMock(utils, 'execute')
utils.execute(
'vhd-util', 'resize', '-n', 'vhdfile', '-s', '1024',
'-j', 'journal').AndReturn(('ignored', 'ignored'))
mox.ReplayAll()
image_utils.resize_vhd('vhdfile', 1024, 'journal')
mox.VerifyAll()
class TestCoalesce(test.TestCase):
def test_vhd_util_call(self):
mox = self.mox
mox.StubOutWithMock(utils, 'execute')
utils.execute(
'vhd-util', 'coalesce', '-n', 'vhdfile'
).AndReturn(('ignored', 'ignored'))
mox.ReplayAll()
image_utils.coalesce_vhd('vhdfile')
mox.VerifyAll()
@contextlib.contextmanager
def fake_context(return_value):
yield return_value
class TestTemporaryFile(test.TestCase):
def test_file_unlinked(self):
mox = self.mox
mox.StubOutWithMock(image_utils, 'create_temporary_file')
mox.StubOutWithMock(image_utils.os, 'unlink')
image_utils.create_temporary_file().AndReturn('somefile')
image_utils.os.unlink('somefile')
mox.ReplayAll()
with image_utils.temporary_file():
pass
def test_file_unlinked_on_error(self):
mox = self.mox
mox.StubOutWithMock(image_utils, 'create_temporary_file')
mox.StubOutWithMock(image_utils.os, 'unlink')
image_utils.create_temporary_file().AndReturn('somefile')
image_utils.os.unlink('somefile')
mox.ReplayAll()
def sut():
with image_utils.temporary_file():
raise Exception()
self.assertRaises(Exception, sut)
class TestCoalesceChain(test.TestCase):
def test_single_vhd(self):
mox = self.mox
mox.StubOutWithMock(image_utils, 'get_vhd_size')
mox.StubOutWithMock(image_utils, 'resize_vhd')
mox.StubOutWithMock(image_utils, 'coalesce_vhd')
mox.ReplayAll()
result = image_utils.coalesce_chain(['0.vhd'])
mox.VerifyAll()
self.assertEquals('0.vhd', result)
def test_chain_of_two_vhds(self):
self.mox.StubOutWithMock(image_utils, 'get_vhd_size')
self.mox.StubOutWithMock(image_utils, 'temporary_dir')
self.mox.StubOutWithMock(image_utils, 'resize_vhd')
self.mox.StubOutWithMock(image_utils, 'coalesce_vhd')
self.mox.StubOutWithMock(image_utils, 'temporary_file')
image_utils.get_vhd_size('0.vhd').AndReturn(1024)
image_utils.temporary_dir().AndReturn(fake_context('tdir'))
image_utils.resize_vhd('1.vhd', 1024, 'tdir/vhd-util-resize-journal')
image_utils.coalesce_vhd('0.vhd')
self.mox.ReplayAll()
result = image_utils.coalesce_chain(['0.vhd', '1.vhd'])
self.mox.VerifyAll()
self.assertEquals('1.vhd', result)
class TestDiscoverChain(test.TestCase):
def test_discovery_calls(self):
mox = self.mox
mox.StubOutWithMock(image_utils, 'file_exist')
image_utils.file_exist('some/path/0.vhd').AndReturn(True)
image_utils.file_exist('some/path/1.vhd').AndReturn(True)
image_utils.file_exist('some/path/2.vhd').AndReturn(False)
mox.ReplayAll()
result = image_utils.discover_vhd_chain('some/path')
mox.VerifyAll()
self.assertEquals(
['some/path/0.vhd', 'some/path/1.vhd'], result)
class TestXenServerImageToCoalescedVhd(test.TestCase):
def test_calls(self):
mox = self.mox
mox.StubOutWithMock(image_utils, 'temporary_dir')
mox.StubOutWithMock(image_utils, 'extract_targz')
mox.StubOutWithMock(image_utils, 'discover_vhd_chain')
mox.StubOutWithMock(image_utils, 'fix_vhd_chain')
mox.StubOutWithMock(image_utils, 'coalesce_chain')
mox.StubOutWithMock(image_utils.os, 'unlink')
mox.StubOutWithMock(image_utils, 'rename_file')
image_utils.temporary_dir().AndReturn(fake_context('somedir'))
image_utils.extract_targz('image', 'somedir')
image_utils.discover_vhd_chain('somedir').AndReturn(
['somedir/0.vhd', 'somedir/1.vhd'])
image_utils.fix_vhd_chain(['somedir/0.vhd', 'somedir/1.vhd'])
image_utils.coalesce_chain(
['somedir/0.vhd', 'somedir/1.vhd']).AndReturn('somedir/1.vhd')
image_utils.os.unlink('image')
image_utils.rename_file('somedir/1.vhd', 'image')
mox.ReplayAll()
image_utils.replace_xenserver_image_with_coalesced_vhd('image')
mox.VerifyAll()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,491 |
\section{Introduction}
Over the last century, general relativity (GR) has been exceedingly successful as a theory in describing gravitational phenomena. The discovery of gravitational waves resulting from a binary black hole merger \cite{wave}, and the first released image of a black hole shadow by the Event Horizon Telescope \cite{shadow}, are some of the recent success stories. However, there are some theoretical and observational issues which suggest modification of GR. For example, it is found that the one-loop renormalization of GR needs to supplement Einstein-Hilbert action with second-order curvature invariants \cite{stell}. Also, some cosmological problems such as the inflationary era in the early universe and recent accelerated expansion of the universe, motivate us to seek alternative ideas beyond GR.
Among generalizations of GR are the scalar-tensor theories where a scalar field is non-minimally coupled to curvature and interfere in the generation of gravitational interaction \cite{scalar}. It is worth mentioning that theories such as Kaluza-Klein and string which attempt to unify fundamental forces naturally result in scalar-tensor generalization of GR.
The study of compact objects such as black holes and neutron stars in modified theories of gravity is an important subject and the question arises as to whether a scalar field has an imprint on black holes to distinguish them from those in GR. An interesting mechanism which has attracted renewed attention in recent years and leads to different black hole solutions from GR is the spontaneous scalarization. This phenomena was first proposed for neutron stars in the context of standard scalar-tensor theory in 1990 \cite{Damour}, where the coupling between matter and scalar field leads to an effective mass for the scalar field. If such an effective mass becomes tachyonic, the GR solution will be unstable against scalar perturbations. Therefore, in such a scenario perturbations grow spontaneously and finally settle to a stable configuration with a non-trivial profile for the scalar field. The generalization of this study to the case of massive scalar fields or vector fields has been considered in \cite{Fethi1}--\cite{Cardoso}. Black holes cannot be scalarized in this way, because the Ricci scalar vanishes in vacuum. However as shown in \cite{Pani}, when a black hole is surrounded by matter, a similar mechanism can also occur. In a similar fashion, Einstein-Maxwell-scalar model was studied in \cite{Font}--\cite{Astefanesei1} where the scalar field is non-minimally coupled to electromagnetic field for which the spontaneous scalarization of charged black holes may occur. In addition to the previous cases, a recent new type of scalarization, the so-called curvature-induced scalarization has been introduced \cite{Doneva1}--\cite{Kanti1}. In Einstein-scalar-Gauss-Bonnet (EsGB) theory containing a non-minimal coupling between the scalar field and Gauss-Bonnet (GB) invariant and under specific conditions on the coupling function, hairy black hole solutions as well as GR solutions may exist. In these models the tachyonic instability of GR solutions can be triggered due to the scalar-GB coupling and new black hole solutions with non-trivial scalar fields may form. The stability of these solutions depends on the functional form of the scalar coupling and has been studied in \cite{Doneva2}--\cite{Ikeda}. The curvature-induced scalarization of charged black holes, rotating black holes, black holes with a massive scalar field and black holes with cosmological constant has been studied in \cite{charge}--\cite{lambda2}, respectively. The study of scalarization of neutron stars and charged wormholes in EsGB gravity has been carried out in \cite{star} and \cite{wormhole}. Also, a complete analysis of higher dimensional generalizations of spontaneous scalarization models was recently studied in \cite{Astefanesei2}.
It is well known that in a binary system the accretion process can take place around compact objects (a black hole or a neutron star) where strong gravitational effects are important \cite{x1}--\cite{x2}. In this process the hot gas falls into the gravitational potential of the compact object and releases the energy in the form of heat and radiation. The properties of the emission spectra from the disk depend on the geodesic motion of particles and may also be associated with the structure of the central object. Therefore the study of disk properties can be used to test gravity in these extreme regions and explore possible deviations from GR and generalized theories of gravity. In this regard thin accretion disks have been investigated in $f(R)$ modified gravity, Horava-Lifshitz gravity, scalar-vector-tensor gravity, brane-world scenarios and Einstein-Maxwell-dilaton gravity \cite{fR}--\cite{dilaton2}. Thin accretion disks around gravastars, boson and fermion stars, naked singularities and exotic matter such as wormholes have been studied in \cite{grava}--\cite{wormhole2}. It has also been shown that the continuum-fitting method and analysis of relativistic iron line profiles are available techniques which can be used to distinguish different astrophysical objects through their accretion disks \cite{m1}--\cite{m3}. Moreover, the effects on relevant accretion properties for black hole solutions with non-trivial scalar fields have also been studied. For instance, In Einstein-dilaton-Gauss-Bonnet gravity it is pointed out that depending on the values of the dilatonic charge and mass of the solution, the ISCO location can be different relative to that of GR \cite{pani1}. Also, for scalarized neutron stars in scalar-tensor theories the effects of scalarization on the epicyclic frequencies, ISCO location and shape of the iron line have been investigated in \cite{s1} and \cite{s2}, respectively. In the present paper we study the accretion process in thin disks around scalarized black holes in EsGB gravity and investigate the effects of scalarization on their properties.
The paper is structured as follows. In Section 2, we present the EsGB theory of gravity and introduce the spontaneous scalarization mechanism in this theory. A brief review of geodesic motion and thin accretion disk model in a general static and spherically symmetric space-time is given in section 3. Our results are presented in section 4, where the disk properties around a scalarized black hole in EsGB gravity are obtained and the effects of the scalar hair are discussed. The paper ends with drawing conclusions.
\section{Einstein-scalar-Gauss-Bonnet gravity}
The action for EsGB gravity reads
\begin{equation}
{\cal S}=\frac{1}{16\pi}\int d^4x \sqrt{-g}\left[R-\frac{1}{2}g^{\mu \nu}\partial_{\mu}\varphi\partial_{\nu}\varphi+\lambda ^2f(\varphi){\cal G}\right],
\label{action}
\end{equation}
where $R$ is the Ricci scalar and ${\cal G}=R^2-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ is the GB invariant. The parameter $\lambda$ is the GB coupling constant and has the dimension of length. A well-known model of the theory with coupling function $f(\varphi)\sim e^{-\alpha\varphi}$ is that of the Einstein-dilaton-Gauss-Bonnet gravity which arises in the low energy limit of string theory. The first black hole solutions for this model were obtained in \cite{Mignemi}--\cite{Torii} and later on extended to slowly rotating and rapidly rotating black holes in \cite{pani1} and \cite{slow2}--\cite{rapid2}.
Now, varying action (\ref{action}) with respect to the metric $g_{\mu\nu}$ gives the gravitational field equations
\begin{equation}
G_{\mu\nu}=\frac{1}{2}\nabla_{\mu}\varphi\nabla_{\nu}\varphi-\frac{1}{4}g_{\mu\nu}\nabla ^{\rho}\varphi\nabla _{\rho}\varphi-4\lambda^2(\nabla^{\rho}\nabla^{\sigma}f(\varphi))P_{\mu\rho\nu\sigma},
\label{grav}
\end{equation}
where
\begin{equation}
P_{\mu\nu\rho\sigma}=R_{\mu\nu\rho\sigma}+g_{\mu\sigma}R_{\rho\nu}-g_{\mu\rho}R_{\sigma\nu}+g_{\nu\rho}R_{\sigma\mu}-g_{\nu\sigma}R_{\rho\mu}+
\frac{R}{2}(g_{\mu\rho}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\rho}).
\label{P}
\end{equation}
It is important to note that the field equations are of second-order as in GR. Therefore the Ostrogradski instability is avoided and the theory is free from ghosts. Variation of the action with respect to the scalar field gives the generalized Klein-Gordon equation
\begin{equation}
\Box\varphi+\lambda ^2f'(\varphi){\cal G}=0,
\label{scalar}
\end{equation}
where a prime denotes the derivative with respect to the scalar field $\varphi$. Numerical black hole solutions of the field equations have been found in \cite{Doneva1}--\cite{Kanti1}. It was shown that if the coupling function satisfies the condition $f'(\varphi_0)=0$ at some $\varphi_0=const$, the scalar equation (\ref{scalar}) is trivially satisfied and equation (\ref{grav}) reduces to that of the GR field equations. Thus, the field equations resulting from (\ref{action}) admit a GR solution with a constant scalar field $\varphi_0$. Now by considering small scalar perturbations $\delta\varphi$ around the GR solution, the linearized form of equation (\ref{scalar}) becomes
\begin{equation}
(\Box_{(0)}-m_{\rm eff}^2)\delta\varphi=0,
\label{}
\end{equation}
where $m_{\rm eff}^2=-\lambda ^2f''(\varphi){\cal G}_{(0)}$ is the effective mass squared for the scalar perturbations and subscript '$0$' refer to the background geometry. When $m_{\rm eff}^2<0$, the tachyonic instability leads to a GR solution being unstable. Therefore the perturbed scalar field grows spontaneously and new black hole solutions with a non-trivial scalar field will be produced. These scalarized solutions are characterized by the number of nodes, $n$, of the scalar field. In \cite{Doneva2}, it was shown that the solutions with $n>0$ are unstable under radial perturbations, while the first non-trivial solution with $n=0$ has a different behavior. The stability of this solution strongly depends on the choice of the scalar-GB coupling and although the solution is unstable for a quadratic coupling function it can be stable for an exponential one. Indeed, adding higher powers of the scalar field to the coupling function can stabilize the solution, \cite{Silva2}--\cite{Ikeda}.
In this work, following \cite{Doneva1}, we will use the coupling function
\begin{equation}
f(\varphi)=\frac{1}{12}[1-\exp(-6\varphi^2)],
\label{b4}
\end{equation}
which leads to non-negligible deviations from GR solutions in the strong field regime as well as satisfies the conditions for the existence of solutions with non-trivial profile of the scalar field. We will consider static and spherically symmetric solutions of the field equations and investigate the properties of the accretion disks around them.
\section{Thin accretion disk model}
In this section we first briefly review the equatorial circular orbits in a general static, spherically symmetric space-time and then derive the basic equations that describe the electromagnetic properties of a thin accretion disk \cite{Horava}.
The geodesic motion of test particles moving around a compact object is governed by the Lagrangian
\begin{equation}
{\cal L}= \frac{1}{2}g_{\mu\nu}\dot{x^{\mu}}\dot{x^{\nu}},
\label{lagrangi}
\end{equation}
where $g_{\mu\nu}$ is the metric of the space-time and a dot denotes differentiation with respect to the affine parameter. Let us consider a static, spherically symmetric space-time with a metric in the general form
\begin{equation}
ds^2=g_{tt}dt^2+g_{rr}dr^2+g_{\theta\theta}d\theta ^2+g_{\phi\phi}d\phi ^2,
\label{100}
\end{equation}
for which we assume the components $g_{tt}, g_{rr}, g_{\theta\theta}$ and $g_{\phi\phi}$ only depend on the radial coordinate $r$. Using the Euler-Lagrange equations, in the equatorial plane $\theta=\frac{\pi}{2}$, one obtains
\begin{equation}
\dot{t}=-\frac{\tilde{E}}{g_{tt}},
\label{2}
\end{equation}
\begin{equation}
\dot{\phi}=\frac{\tilde{L}}{g_{\phi\phi}},
\label{3}
\end{equation}
where $\tilde{E}$ and $\tilde{L}$ are the specific energy and the specific angular momentum, respectively. Now, taking $2{\cal L}=-1$ for test particles and using equations (\ref{2}) and (\ref{3}) we find
\begin{equation}
-g_{tt}g_{rr}\dot{r}^2+V_{\rm eff}(r)=\tilde{E}^2,
\label{4}
\end{equation}
where the effective potential is defined as
\begin{equation}
V_{\rm eff}(r)=-g_{tt}\left(1+\frac{\tilde{L}^2}{g_{\phi\phi}}\right).
\label{5}
\end{equation}
For stable circular orbits using conditions $V_{\rm eff}(r)=0$ and $V_{\rm eff,r}(r)=0$, one obtains the specific energy, specific angular momentum and angular velocity $\Omega$ of particles moving in the gravitational potential of the central object as follows
\begin{equation}
{\tilde{E}}=-\frac{g_{tt}}{\sqrt{-g_{tt}-g_{\phi\phi}\Omega^2}},
\label{6}
\end{equation}
\begin{equation}
{\tilde{L}}=\frac{g_{\phi\phi}\Omega}{\sqrt{-g_{tt}-g_{\phi\phi}\Omega^2}},
\label{7}
\end{equation}
\begin{equation}
\Omega=\frac{d\phi}{dt}=\sqrt{\frac{-g_{tt,r}}{g_{\phi\phi,r}}}.
\label{8}
\end{equation}
Also, the innermost stable circular orbit, $r_{\rm isco}$, of the central object can be determined from the condition $V_{\rm eff,rr}(r)=0$ which leads to equation
\begin{equation}
{\tilde{E}^2g_{\phi\phi,rr}}+{\tilde{L}^2g_{tt,rr}}+(g_{tt}g_{\phi\phi})_{,rr}=0. \label{9}
\end{equation}
The general relativistic model of thin accretion disks has been developed by Novikov, Thorne and Page \cite {Novikov}--\cite{Thorne} which is an extension of Newtonian approach of Shakura-Sunyaev \cite{Shakura}. The model presents a geometric description of thin accretion disks for which its vertical size, $h$, is negligible compared to its horizontal size, $h\ll r$. The accretion disk is in the equatorial plan of the central object and the accreting matter moves in Keplerian orbits. Also, the disk is considered to be in a steady state, that is the accretion mass rate, $\dot{M}_0$, is assumed to be constant in time.
In the steady-state model the accreting matter is assumed to be in thermodynamical equilibrium. Therefore, the disk temperature $T(r)$ is related to energy flux, $F(r)$ through the Stefan-Boltzmann law
\begin{equation}
F(r)=\sigma_{\rm SB} T^4(r),
\label{10}
\end{equation}
where $\sigma_{\rm SB}=5.67\times10^{-5}\rm erg$ $\rm s^{-1} cm^{-2} K^{-4}$ is the Stefan-Boltzmann constant. From the conservation equations for the mass, energy and angular momentum, we obtain an expression for the radiant energy flux in terms of the specific energy, angular momentum and angular velocity of the orbiting particles. Let us first consider the energy-momentum tensor of the accreting matter in the form \cite{Novikov}--\cite{Page}
\begin{equation}
T^{\mu\nu}=\rho_0u^{\mu}u^{\nu}+2u^{(\mu}q^{\nu)}+t^{\mu\nu},\label{n1}
\end{equation}
where $u_{\mu}q^{\mu}=0$ and $u_{\mu}t^{\mu\nu}=0$. The four-velocity of the orbiting particles is denoted by $u^{\mu}$ whereas $\rho_0$, $q^{\mu}$ and $t^{\mu\nu}$ are respectively the rest mass density, energy flow vector and stress tensor of the accreting matter. From the rest-mass conservation, $\nabla_{\mu}(\rho_0u^{\mu})=0$, we find that the time averaged accretion rate, $\dot{M}_{0}$, is independent of the disk radius
\begin{equation}
\dot{M}_{0}=-2\pi\sqrt{-g}\Sigma u^r=\rm const.,\label{n2}
\end{equation}
where the time averaged surface density is defined as
\begin{equation}
\Sigma(r)=\int_{-h}^{h}\langle\rho_0\rangle dz,\label{n3}
\end{equation}
with $z$ being the cylindrical coordinate. Using the conservation laws for energy, $\nabla_{\mu}E^{\mu}=0$, and angular momentum, $\nabla_{\mu}J^{\mu}=0$, we find
\begin{equation}
[\dot{M}_{0}\tilde{E}-2\pi\sqrt{-g}\Omega W^{r}_{\phi}]_{,r}=4\pi rF(r)\tilde{E},\label{n4}
\end{equation}
and
\begin{equation}
[\dot{M}_{0}\tilde{L}-2\pi\sqrt{-g}W^{r}_{\phi}]_{,r}=4\pi rF(r)\tilde{L},\label{n5}
\end{equation}
where the averaged torque $W^{r}_{\phi}$ is given by
\begin{equation}
W^{r}_{\phi}=\int_{-h}^{h}\langle t^{r}_{\phi}\rangle dz,\label{n6}
\end{equation}
and $\langle t^{r}_{\phi}\rangle$ is the $(\phi,r)$ component of the stress tensor, averaged over the time scale $\Delta t$ and over angle $\Delta\phi=2\pi$. Now, by employing the
energy-angular momentum relation $\tilde{E}_{,r}=\Omega\tilde{L}_{,r}$ and eliminating $W^{r}_{\phi}$ from equations (\ref{n4}) and (\ref{n5}), the time averaged energy flux emitted from the surface of an accretion disk is given by
\begin{equation}
F(r)=-\frac{\dot{M}_{0}\Omega_{,r}}{4\pi\sqrt{-g}\left(\tilde{E}-\Omega \tilde{L}\right)^2}\int^r_{r_{\rm isco}}\left(\tilde{E}-\Omega \tilde{L}\right) \tilde{L}_{,r}dr.
\label{11}
\end{equation}
The observed luminosity $L(\nu)$ has a red-shifted black body spectrum
\begin{equation}
L(\upsilon)=4\pi d^2 I(\nu)=\frac{8\pi h \cos\gamma}{c^2}\int_{r_{\rm in}}^{r_{\rm out}}\int_0^{2\pi}\frac{ \nu_e^3 r dr d\phi}{\exp{[\frac{h\nu_e}{k_{\rm B} T}]}-1},\label{12}
\end{equation}
where $h$ and $k_{\rm B}$ are the Planck and Boltzmann constants, respectively. Also, $\gamma$ is the disk inclination angle and $r_{\rm in}$ and $r_{\rm out}$ are inner and outer radii of the edge of the disk. The emitted frequency is given by $\nu_{e}=\nu (1+z)$, where the redshift factor $z$ can be written as
\begin{equation}
1+z=\frac{1+\Omega r\sin\phi\sin\gamma}{\sqrt{-g_{tt}-\Omega ^2g_{\phi\phi}}}.\label{120}
\end{equation}
Now, we define the accretion efficiency, $\epsilon$, which demonstrates the capability of the central object to convert rest mess into radiation. This quantity is defined as the ratio of the rate of the energy of photons escaping from the disk surface to infinity, and the rate at which mass-energy is transported to the black hole \cite{Novikov}--\cite{Page}. Assuming all emitted photons can escape to infinity, the radiative efficiency in terms of the specific energy measured at the ISCO radius is given by
\begin{equation}
\epsilon=1-\tilde{E}_{\rm isco}.\label{13}
\end{equation}
\section{Numerical results}
In the following, we will consider static, spherically symmetric black hole solutions in EsGB gravity with an ansatz for the metric as follows
\begin{equation}
ds^2=-e^{2\Phi(r)}dt^2+e^{2\Lambda(r)}dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2).
\label{b1}
\end{equation}
Then, Einstein's equations (\ref{grav}) read
\begin{equation}
\frac{2}{r}\left[1+\frac{2}{r}(1-3e^{-2\Lambda})\Psi_r\right]\frac{d\Lambda}{dr}+\frac{(e^{2\Lambda}-1)}{r^2}
-\frac{4}{r^2}(1-e^{-2\Lambda})\frac{d\Psi_r}{dr}-\left(\frac{d\varphi}{dr}\right)^2=0,
\label{c1}
\end{equation}
\begin{equation}
\frac{2}{r}\left[1+\frac{2}{r}(1-3e^{-2\Lambda})\Psi_r\right]\frac{d\Phi}{dr}-\frac{(e^{2\Lambda}-1)}{r^2}-\left(\frac{d\varphi}{dr}\right)^2=0,
\label{c2}
\end{equation}
\begin{equation}
\frac{d^2\Phi}{dr^2}+\left(\frac{d\Phi}{dr}+\frac{1}{r}\right)\left(\frac{d\Phi}{dr}-\frac{d\Lambda}{dr}\right)+\frac{4e^{-2\Lambda}}{r}
\left[3\frac{d\Phi}{dr}\frac{d\Lambda}{dr}-\frac{d^2\Phi}{dr^2}-\left(\frac{d\Phi}{dr}\right)^2\right]\Psi_r
-\frac{4e^{-2\Lambda}}{r}\frac{d\Phi}{dr}\frac{d\Psi_r}{dr}+\left(\frac{d\varphi}{dr}\right)^2=0,
\label{c3}
\end{equation}
where $\Psi_r=\lambda ^ 2f'(\varphi)\frac{d\varphi}{dr}$ and the scalar field equation (\ref{scalar}) is given by
\begin{equation}
\frac{d^2\varphi}{dr^2}+\left(\frac{d\Phi}{dr}-\frac{d\Lambda}{dr}+\frac{2}{r}\right)\frac{d\varphi}{dr}-\frac{2\lambda ^2}{r^2}f'(\varphi)
\left[(1-e^{-2\Lambda})\left[\frac{d^2\Phi}{dr^2}+\frac{d\Phi}{dr}\left(\frac{d\Phi}{dr}-\frac{d\Lambda}{dr}\right)\right]+2e^{-2\Lambda}\frac{d\Phi}{dr}\frac{d\Lambda}{dr}\right]=0
.\label{c5}
\end{equation}
To obtain the black hole solutions with non-trivial scalar hair we have to solve the above field equations numerically by employing the shooting method. In our numerical analysis, for simplicity, we set $r_H=1$ and integrate the field equations (\ref{c1})-(\ref{c5}) from $r=r_{H}+\epsilon$ with $\epsilon=10^{-5}$, to $r\rightarrow\infty$. Here, we only state the results and refer the interested reader to \cite{Doneva1} for the details of calculations. The requirement for the regularity of the scalar field and its first and second derivatives on the horizon is given by
\begin{equation}
r_H^4>24\lambda^4f'^2(\varphi_H),
\label{b2}
\end{equation}
where $\varphi_H$ is the value of the scalar field on the horizon, $r_H$. The mass of the black hole $M$ and the dilaton charge $D$ can be obtained from asymptotic behavior of functions $\Lambda$, $\Phi$ and $\varphi$ which are given by
\begin{equation}
\Lambda\approx\frac{M}{r}+{\cal{O}}(1/r^2),\quad \Phi\approx-\frac{M}{r}+{\cal{O}}(1/r^2), \quad \varphi\approx\frac{D}{r}+{\cal{O}}(1/r^2).
\label{b3}
\end{equation}
Also as pointed out in \cite{Doneva1}, the dilaton charge is a secondary hair; it is not an independent quantity but instead it depends on the black hole mass.
Now, one is ready to study properties of thin accretion disks around scalarized black holes in EsGB gravity and compare the results with that of the Schwarzschils in GR. As was mentioned before, these new GB black hole solutions with curvature induced scalarization were first obtained numerically in \cite{Doneva1}. Since the non-trivial solutions with a scalar field that has one or more nodes are unstable \cite{Doneva2}, in our study we will consider only the first non-trivial solutions with the scalar field with zero nodes.
The profile of the scalar field as a function of radial coordinate is shown for some values of the coupling constant $\lambda$ in the left panel of figure 1. Also, the behavior of the effective potential for these scalarized black holes is shown in the right panel of figure 1 and for comparison we have plotted the corresponding results for the Schwarzschild black hole. As is clear, the effective potential is larger in EsGB gravity than in GR and with increasing $\lambda$, the deviation from GR also increases.
\begin{figure}[H]
\centering
\includegraphics[width=3.0in]{fig1-1.pdf}
\includegraphics[width=3.0in]{fig1-2.pdf}
\caption{The scalar field, left panel and the effective potential multiplied by $r_{H}^{2}$, right panel, as functions of the normalized radial coordinate $r/r_{H}$ for some black hole solutions with different values of $\lambda$. In the right panel the solid curve corresponds to the Schwarzschild black hole.}
\label{potential}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=3.0in]{fig2-1.pdf}
\includegraphics[width=3.0in]{fig2-2.pdf}
\caption{The energy flux $F(r)$ of a disk around a static scalarized black hole with mass accretion rate $\dot{M}=2\times10^{-6}M_{\odot}yr^{-1}$ for different values of $\lambda$, left panel, and the disk temperature $T(r)$, right panel, as functions of the normalized radial coordinate $r/r_{H}$. In each panel the solid curve corresponds to the Schwarzschild black hole.}
\label{flux}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=3.0in]{fig3.pdf}
\caption{The emission spectrum $\nu L(\nu)$ of the accretion disk around a static scalarized black hole with mass accretion rate $\dot{M}=2\times10^{-6}M_{\odot}yr^{-1}$ and inclination $\gamma=0^{\circ}$ for different values of $\lambda$, as a function of frequency $\nu$. The solid curve represents the disk spectrum for a Schwarzschild black hole.}
\label{spectra}
\end{figure}
The effects of scalarization on the energy flux over the surface of the disk for some values of $\lambda$ is shown in the left panel of figure 2. It is easy to see that for scalarized black holes the energy flux is more pronounced than in the Schwarzschild black hole (the scalar-free solution) and as the values of the GB coupling become larger the maximum of the energy flux increases. This is because with increasing GB coupling the ISCO radii of scalarized black holes become smaller and shift closer to the horizon. Therefore, for smaller ISCO radii, disk particles experience stronger gravitational fields and thus more energy is released from the gravitational potential energy. The disk temperature is plotted in the right panel of the figure and the same behavior is also observed there.
In figure 3, we display the disk spectra for scalarized black holes and for comparison we also present the emission spectra for the Schwarzschild black hole. Similar to the case of the energy flux, deviation of $L(\nu)$ for scalarized black holes from that of the Schwarzschild becomes more pronounced with increasing values of $\lambda$. Also, it is clear that with increasing GB coupling, the cut-off frequencies for scalarized black holes shift to higher frequencies.
In Table 1, we present numerical results for the dilaton charge, ISCO radius, $r_{\rm isco}$, orbital frequency, $\Omega_{\rm isco}$ and radiative efficiency, $\epsilon$ of scalarized black holes for some values of the GB coupling. We have found that the first branch solution with non-trivial scalar field bifurcates from the Schwarzschild solution at the point $\lambda=0.42$. Also the stability analysis shows that black hole solutions with $\lambda\geq5.15$ are unstable against radial perturbations. That is why, in Table 1, we have chosen values of $\lambda$ to be in the range $0.42\leq\lambda\leq5.15$. As is clear, the ISCO radius in scalarized black holes becomes smaller and $\Omega$ becomes larger as the value of the GB coupling increases. This is due to the fact that the GB term effectively counteracts gravity, thus the instability area around scalarized black holes decreases and the ISCO radius takes smaller values. Also it is worth stressing that we have presented $\frac{r_{\rm isco}}{M}$ in Table 1 where $M$ is the physical mass defined in equation (\ref{b3}). Such a mass is measured at infinity and is interpreted as due to the contribution from the scalar field. Only in the case of the Schwarzschild black hole the physical mass is equal to the bare mass in GR and we have $\frac{r_{\rm isco}}{M}=6$. But the scalarized black holes have a larger physical mass and thus the ratio $\frac{r_{\rm isco}}{M}$ for these black holes is smaller than that for the Schwarzschild black hole. Moreover we note that $\lambda$ can be constrained by electromagnetic and gravitational wave observations. For instance, it is found that the least massive black hole observed in x-ray binary A0620-00 and the remnant of GW170817 lead to constrains, $\lambda\leq27$ $\rm km$ and $\lambda\leq24$ $\rm km$, respectively \cite{Doneva2}. Therefore, $\lambda$ is in units of km in Table 1.
In Table 2, The maximum values of the energy flux, temperature distribution and emission spectra for scalarized black holes are presented and compared to that of the Schwarzschild's. It is easy to see that the peak values of $F_{\rm max}(r)$, $T_{\rm max}(r)$ and $\nu L(\nu)_{\rm max}$ grow with increasing $\lambda$. Also, the cut-off frequencies for which the corresponding maxima occur, shift to higher values.
\begin{table}[H]
\centering
\caption{The $r_{\rm isco}$, orbital frequency, $\Omega_{\rm isco}$ and the efficiency for thin accretion disk around static scalarized black hole.}
\begin{tabular}{l l l l l l}
\hline
$\lambda$&$D/M$& $r_{\rm isco}/M$&$M\Omega_{\rm isco}$&$\epsilon$\\ [0.5ex]
\hline
-&0& 6.0&0.0680&0.0572\\
\\
0.5& 0.0088& 5.9761&0.0681&0.0572\\
\\
0.75& 0.2798& 5.8467&0.0719&0.0578\\
\\
1& 0.7964& 5.5872&0.0787&0.0583\\
\\
1.25& 0.9558& 5.1757&0.0855&0.0605\\
\\
1.5& 1.0935& 4.6545&0.0958&0.0632\\
\\
1.75& 1.2045& 4.2958&0.1029&0.0697\\
\\
2& 1.3029& 4.1065&0.1055&0.0885\\
\hline
\end{tabular}
\end{table}
It should be noted that this study can be extended to the case of other scalar-GB coupling functions such as even and odd polynomials and inverse polynomial functions, considered in \cite{Kanti1}, which satisfy the conditions for the existence of scalarized black holes. Moreover, we note that geodesic equations and relevant accretion properties depend only on the metric components. Therefore, one can easily apply this model to charged or spinning scalarized black holes and study thin accretion disks around them. Here we would like to mention the effects of the spin parameter on disk properties. As is well known, the accretion process around rotating black holes in GR was first studied in \cite{Novikov}. It was found that in the presence of a rotation parameter, the ISCO radius of the disk decreases in comparison to Schwarzschild black holes and thus the radiative efficiency of an accretion disk increases from 6\% for the Schwarzschild black hole to 42\% for a co-rotating disk around an extremal Kerr black hole \cite{Thorne}. Similarly it is expected that in the case of scalarized rotating black holes the rotation parameter also causes the ISCO radius of the disk to decrease. Indeed, as is shown in \cite{rotating1}, the relative frequencies at the ISCO for spinning scalarized black holes and Kerr black holes (namely $\frac{\Omega^{(\rm s)}}{\Omega^{(\rm GR)}}$) is greater than one. Therefore, the spinning scalarized black holes have smaller ISCO radius and thus larger energy flux in comparison with Kerr black holes. However, it is pointed out that there is a spin selection effect, so that black holes with a large spin are indistinguishable from Kerr black holes and only low spin scalarized black holes have significant deviation from GR.
Finally, we mention that the emission spectra from accretion disks have imprints of the background space-time and can be used as astrophysical probes to constrain modified theories of gravity \cite{new1}--\cite{new3}. This is called the continuum fitting method and was first proposed in \cite{new4}. In addition to rotating black holes, the continuum fitting method can also be used for non-rotating space-times to test the metric around a black hole \cite{new5}--\cite{new6}. In order to investigate whether the continuum spectrum observations can constrain the EsGB gravity, one has to theoretically estimates the luminosity and compare it with observation, using a minimum chi-squared, $\chi^2$, approach. The reduced $\chi^2$ is given by
\begin{equation*}
\chi_{\rm red}^2=\Sigma_{i}\frac{L_{\rm obs}(\nu_{i})-L_{\rm theo}(\nu_{i},\lambda)}{\sigma(\nu_{i})}
\end{equation*}
where $L_{\rm obs}(\nu_{i})$ denotes the observed data with possible errors $\sigma$ and $L_{\rm theo}(\nu_{i},\lambda)$ is the model estimates of the luminosity. The value of $\lambda$ that minimizes the reduced $\chi_{\rm red}^2$ is the most favored value of the GB coupling. This study is out of the scope of the present paper and will be considered elsewhere.
\begin{table}[H]
\centering
\caption{ The maximum values of the radiant energy flux $F(r)$, temperature distribution $T(r)$ and the emission spectra. The cut-off frequency is also shown in column 5. }
\begin{tabular}{l l l l l l}
\hline
$\lambda$ &$F_{\rm max}$ [\rm erg $\rm s^{-1}$ $\rm cm^{-2}$]$\times 10^{14}$& $T_{\rm max}$ [\rm K] $\times 10^{4}$ &$\nu L(\nu)_{\rm max}$ [\rm erg $\rm s^{-1}$]$\times 10^{35}$ &$\nu_{crit}$[\rm Hz]$\times 10^{15}$\\ [0.5ex]
\hline
- & 1.6271 & 2.0084 & 3.9048 & 6.1776\\
\\
0.5 & 1.2370$\times 10$ & 3.3327 & 2.0867$\times 10$ & 9.9594\\
\\
0.75 & 1.3375$\times 10$ & 3.3780 & 2.1339$\times 10$ & 1.0002$\times 10$\\
\\
1 & 1.5553$\times 10$ & 3.5141 & 2.3150$\times 10$ & 1.0041$\times 10$\\
\\
1.25 & 1.9781$\times 10$ & 3.7409 & 2.5729$\times 10$ & 1.0079$\times 10$\\
\\
1.5 & 2.7111$\times 10$ & 4.0359 & 2.9017$\times 10$ & 1.0266$\times 10$\\
\\
1.75 & 3.5553$\times 10$ & 4.3308 & 3.1982$\times 10$ & 1.0821$\times 10$\\
\\
2 & 4.3775$\times 10$ & 4.5576 & 3.4539$\times 10$ & 1.1364$\times 10$\\
\hline
\end{tabular}
\end{table}
\section{Conclusions}
In this paper we have studied the properties of thin accretion disks around scalarized black holes in EsGB theory of gravity. We considered the coupling function $f(\varphi)=\frac{1}{12}[1-\exp(-6\varphi^2)]$ which can lead to large deviations from GR in the strong field regime and numerically obtained physical properties of accretion disks such as the energy flux, temperature distribution and the emission spectra for some values of the GB coupling constant, using the steady-state Novikov-Thorne model, and displayed the relevant results in figures 2 and 3 for accreting scalarized black holes. We also studied the effects of non-trivial scalar fields on the ISCO radius and angular frequency $\Omega$ and showed that for scalarized black holes the ISCO radius is smaller than that of the Schwarzschild, in contrast to $\Omega$ which becomes larger, with the results summarized in Table 1. We presented characteristics of the emissivity profile of thin disks including the maximum values of the radiation energy flux, temperature distribution and emission spectra in Table 2 and showed that with increasing $\lambda$, the peaks of $F_{\rm max}(r)$, $T_{\rm max}(r)$ and $\nu L(\nu)_{\rm max}$ grow as well. Comparing the results to that in GR we found that thin accretion disks around scalarized black holes in EsGB gravity are hotter and more luminous than in GR. In this work, we focused attention on the structure and electromagnetic properties of relativistic thin accretion disks. It would be interesting to investigate whether one can practically test the EsGB gravity and constrain the model parameter with continuum spectrum observations. We will address such problems in a future work.
\section*{Acknowledgements}
We would like to thank the anonymous referee for valuable comments.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,358 |
{"url":"https:\/\/artofproblemsolving.com\/wiki\/index.php?title=1987_AHSME_Problems\/Problem_2&diff=85411&oldid=65669","text":"# Difference between revisions of \"1987 AHSME Problems\/Problem 2\"\n\n## Problem\n\nA triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle ABC of side length $3$. The perimeter of the remaining quadrilateral is\n\n$[asy] draw((0,0)--(2,0)--(2.5,.87)--(1.5,2.6)--cycle, linewidth(1)); draw((2,0)--(3,0)--(2.5,.87)); label(\"3\", (0.75,1.3), NW); label(\"1\", (2.5, 0), S); label(\"1\", (2.75,.44), NE); label(\"A\", (1.5,2.6), N); label(\"B\", (3,0), S); label(\"C\", (0,0), W); label(\"D\", (2.5,.87), NE); label(\"E\", (2,0), S); [\/asy]$\n\n$\\text{(A)} \\ 6 \\qquad \\text{(B)} \\ 6\\frac{1}{2} \\qquad \\text{(C)} \\ 7 \\qquad \\text{(D)} \\ 7\\frac{1}{2} \\qquad \\text{(E)} \\ 8$\n\n## Solution\n\n$\\triangle DBE$ is similar to $\\triangle ABC$ by AA, so $\\overline{DE}$ = 1 by similarity, and $\\overline{CE} = \\overline{AD} = 2$, by subtraction. Thus the perimeter is $3+2+2+1 = 8$, or $\\boxed{E}$. -slackroadia","date":"2022-01-28 12:03:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 10, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6619279384613037, \"perplexity\": 1549.9365907475947}, \"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\/1642320305494.6\/warc\/CC-MAIN-20220128104113-20220128134113-00278.warc.gz\"}"} | null | null |
A new Thames waterfront neighbourhood with 800 homes surrounded by water on three sides where the Lea flows into the Thames has been started on the Leamouth Peninsula.
The Goodluck Hope scheme ready by 2020 is now going on the market, with the first batch of suites, warehouse and tower apartments ready by 2019, from £335,000.
It extends Ballymore developers' massive London City Island complex on the loop of the Lea tributary, dubbed 'Little Manhatten', where English National Ballet and the London Film School are moving to next year.
"Goodluck Hope completes the Leamouth Peninsula transformation," Ballymore managing director John Mulryan said.
The scheme includes a business centre, leisure centre, swimming pool, gym, arts club, private cinema, restaurant and 20th-floor viewing lounge.
It even has a cottage brewery on the site of the old Orchard House pub along with its own brewers' school teaching the traditional beer-making craft. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,226 |
Q: Where can I view tools or packages availble in Homebrew I am trying homebrew for the first time. I used to use macports.
In macports, it is quick clear which ports are available, and their versions (http://www.macports.org/ports.php).
Is there any such website / tools in hombrew?
Thanks!
A: You can browse the Formula directory on GitHub. Or type brew search.
There is even a website braumeister.org.
A: Following Jerome Mahuet's answer, I could not find how to locate various versions of a particular formula.
here is what my problem was:
*
*I installed homebrew as instructed on the website
*This installed the very basics of homebrew, but did not install the .git repository folder
*As a result, brew versions ruby did not work
*to fix this, simply type brew update
You can then view the versions via
brew versions ruby
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,094 |
vaastudoshremedies.com is created only for spreading information about the Vaastu Shastra, Vaastu dosh, and defects and with simple remedies. This is best of our knowledge, we're not assuring or guarantying you, the remedies given in this website may or may not resolve your Vaastu issues. Please consult a Vaastu expert before making any alteration or changes. We do not take any responsibility thereafter.
vaastudoshremedies.com Knowledge base on vaastu shastra-doshas-remedies, it's free and always will be. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,026 |
#include "core/ChTrasform.h"
#include "physics/ChMarker.h"
#include "physics/ChGlobal.h"
#include "physics/ChBody.h"
#include "physics/ChExternalObject.h"
namespace chrono
{
#define MARKER_BDF_STEP 0.0001
//////////////////////////////////////
//////////////////////////////////////
// CLASS FOR MARKERS
// Register into the object factory, to enable run-time
// dynamic creation and persistence
ChClassRegister<ChMarker> a_registration_ChMarker;
ChMarker::ChMarker()
{
Body = NULL;
motion_X = new ChFunction_Const (0); // default: no motion
motion_Y = new ChFunction_Const (0);
motion_Z = new ChFunction_Const (0);
motion_ang = new ChFunction_Const (0);
motion_axis = VECT_Z;
rest_coord= CSYSNORM;
motion_type = M_MOTION_FUNCTIONS;
last_rel_coord = CSYSNORM;
last_rel_coord_dt = CSYSNULL;
last_time = 0;
SetIdentifier(GetUniqueIntID()); // mark with unique ID
UpdateState();
}
ChMarker::ChMarker (char myname[], ChBody* myBody, Coordsys myrel_pos, Coordsys myrel_pos_dt, Coordsys myrel_pos_dtdt)
{
SetName (myname);
Body = myBody;
motion_X = new ChFunction_Const (0); // default: no motion
motion_Y = new ChFunction_Const (0);
motion_Z = new ChFunction_Const (0);
motion_ang = new ChFunction_Const (0);
motion_axis = VECT_Z;
rest_coord= CSYSNORM;
motion_type = M_MOTION_FUNCTIONS;
SetCoord (myrel_pos);
SetCoord_dt (myrel_pos_dt);
SetCoord_dtdt (myrel_pos_dtdt);
last_rel_coord = CSYSNORM;
last_rel_coord_dt = CSYSNULL;
last_time = 0;
SetIdentifier(GetUniqueIntID()); // mark with unique ID
UpdateState();
};
ChMarker::~ChMarker ()
{
if (motion_X) delete motion_X;
if (motion_Y) delete motion_Y;
if (motion_Z) delete motion_Z;
if (motion_ang) delete motion_ang;
}
void ChMarker::Copy(ChMarker* source)
{
// first copy the parent class data...
ChObj::Copy(source);
// first copy the parent class data...
ChFrameMoving<double>::operator=(*source);
Body = NULL;
// Replace the default functions.
if (motion_X) delete motion_X;
if (motion_Y) delete motion_Y;
if (motion_Z) delete motion_Z;
if (motion_ang) delete motion_ang;
motion_X = source->motion_X->new_Duplicate();
motion_Y = source->motion_Y->new_Duplicate();
motion_Z = source->motion_Z->new_Duplicate();
motion_ang = source->motion_ang->new_Duplicate();
motion_axis= source->motion_axis;
rest_coord = source->rest_coord;
motion_type = source->motion_type;
abs_frame = source->abs_frame;
last_rel_coord = source->last_rel_coord;
last_rel_coord_dt = source->last_rel_coord_dt;
last_time = source->last_time;
}
// setup the functions when user changes them.
void ChMarker::SetMotion_X (ChFunction* m_funct)
{
if (motion_X) delete motion_X;
motion_X = m_funct;
}
void ChMarker::SetMotion_Y (ChFunction* m_funct)
{
if (motion_Y) delete motion_Y;
motion_Y = m_funct;
}
void ChMarker::SetMotion_Z (ChFunction* m_funct)
{
if (motion_Z) delete motion_Z;
motion_Z = m_funct;
}
void ChMarker::SetMotion_ang (ChFunction* m_funct)
{
if (motion_ang) delete motion_ang;
motion_ang = m_funct;
}
void ChMarker::SetMotion_axis (Vector m_axis)
{
motion_axis = m_axis;
}
// Coordinate setting, for user access
void ChMarker::Impose_Rel_Coord (const Coordsys& m_coord)
{
Quaternion qtemp;
// set the actual coordinates
SetCoord(m_coord);
// set the resting position coordinates
rest_coord.pos.x = m_coord.pos.x - motion_X->Get_y(ChTime);
rest_coord.pos.y = m_coord.pos.y - motion_Y->Get_y(ChTime);
rest_coord.pos.z = m_coord.pos.z - motion_Z->Get_y(ChTime);
qtemp = Q_from_AngAxis (-(motion_ang->Get_y(ChTime)), motion_axis);
rest_coord.rot = Qcross (m_coord.rot, qtemp); // ***%%% check
// set also the absolute positions, and other.
UpdateState () ;
}
void ChMarker::Impose_Abs_Coord (const Coordsys& m_coord)
{
ChBody* my_body;
my_body = GetBody();
Coordsys csys;
// coordsys: trasform the representation from the parent reference frame
// to the local reference frame.
csys.pos= ChTrasform<>::TrasformParentToLocal (m_coord.pos, my_body->GetCoord().pos, *my_body->GetA());
csys.rot= Qcross (Qconjugate (my_body->GetCoord().rot), m_coord.rot);
// apply the imposition on local coordinate and resting coordinate:
Impose_Rel_Coord (csys);
}
//// Utilities for coordinate transformations
///
Vector ChMarker::Point_World2Ref (Vector* mpoint)
{
return abs_frame / *mpoint;
}
Vector ChMarker::Point_Ref2World (Vector* mpoint)
{
return *(ChFrame<double>*)&abs_frame * *mpoint;
}
Vector ChMarker::Dir_World2Ref (Vector* mpoint)
{
return abs_frame.GetA()->MatrT_x_Vect (*mpoint);
}
Vector ChMarker::Dir_Ref2World (Vector* mpoint)
{
return abs_frame.GetA()->Matr_x_Vect (*mpoint);
}
// This handles the time-varying functions for the relative
// coordinates
void ChMarker::UpdateTime (double mytime)
{
Coordsys csys, csys_dt, csys_dtdt;
Quaternion qtemp;
double ang, ang_dt, ang_dtdt;
ChTime = mytime;
// if a imposed motion (keyframed movement) affects the marker postion (example,from R3D animation system),
// compute the speed and acceleration values by BDF (example,see the UpdatedExternalTime() function, later)
// so the updating via motion laws can be skipped!
if (motion_type == M_MOTION_KEYFRAMED) return;
// skip realtive-position-functions evaluation also if
// someone is already handling this from outside..
if (motion_type == M_MOTION_EXTERNAL) return; // >>>>
// positions:
// update positions: rel_pos
csys.pos.x= motion_X->Get_y(mytime);
csys.pos.y= motion_Y->Get_y(mytime);
csys.pos.z= motion_Z->Get_y(mytime);
if (motion_X->Get_Type() != FUNCT_MOCAP)
csys.pos += rest_coord.pos;
// update speeds: rel_pos_dt
csys_dt.pos.x= motion_X->Get_y_dx(mytime);
csys_dt.pos.y= motion_Y->Get_y_dx(mytime);
csys_dt.pos.z= motion_Z->Get_y_dx(mytime);
// update accelerations
csys_dtdt.pos.x= motion_X->Get_y_dxdx(mytime);
csys_dtdt.pos.y= motion_Y->Get_y_dxdx(mytime);
csys_dtdt.pos.z= motion_Z->Get_y_dxdx(mytime);
// rotations:
ang = motion_ang->Get_y(mytime);
ang_dt = motion_ang->Get_y_dx(mytime);
ang_dtdt= motion_ang->Get_y_dxdx(mytime);
if ((ang !=0)||(ang_dt !=0)||(ang_dtdt !=0))
{
// update q
Vector motion_axis_versor = Vnorm(motion_axis);
qtemp = Q_from_AngAxis (ang, motion_axis_versor);
csys.rot = Qcross (qtemp, rest_coord.rot);
// update q_dt
csys_dt.rot = chrono::Qdt_from_AngAxis (csys.rot, ang_dt, motion_axis_versor);
// update q_dtdt
csys_dtdt.rot = chrono::Qdtdt_from_AngAxis (ang_dtdt, motion_axis_versor, csys.rot, csys_dt.rot);
}
else
{
csys.rot = coord.rot; //rel_pos.rot;
csys_dt.rot = QNULL;
csys_dtdt.rot = QNULL;
}
// Set the position, speed and acceleration in relative space,
// automatically getting also the absolute values,
if (!(csys == this->coord))
SetCoord(csys);
if (!(csys_dt == this->coord_dt) || !(csys_dt.rot == QNULL))
SetCoord_dt(csys_dt);
if (!(csys_dtdt == this->coord_dtdt) || !(csys_dtdt.rot == QNULL))
SetCoord_dtdt(csys_dtdt);
};
void ChMarker::UpdateState ()
{
if (!GetBody()) return;
GetBody()->TrasformLocalToParent(*this, abs_frame);
};
void ChMarker::Update (double mytime)
{
UpdateTime (mytime);
UpdateState ();
}
void ChMarker::UpdateExternalGeometry ()
{
// tell the R3 object to move itself where needed
if (GetExternalObject())
GetExternalObject()->onChronoChanged();
}
void ChMarker::UpdatedExternalTime (double prevtime, double mtime)
{
double mstep = mtime-prevtime;
Coordsys m_rel_pos_dt;
Coordsys m_rel_pos_dtdt;
// do not try to switch on the M_MOTION_KEYFRAMED mode if
// we are already in the M_MOTION_EXTERNAL mode, maybe because
// a link point-surface is already moving the marker and
// it will handle the accelerations by itself
if (this->motion_type == M_MOTION_EXTERNAL) return; // >>>>
// otherwise see if a BDF is needed, cause an external 3rd party is moving the marker
this->motion_type = M_MOTION_FUNCTIONS;
if ( (!(Vequal(coord.pos, last_rel_coord.pos))||
!(Qequal(coord.rot, last_rel_coord.rot)) )
&& (fabs(mstep) < 0.1)
&& (mstep != 0)
) // if POSITION or ROTATION ("rel_pos") has been changed in acceptable time step,.
{
if ((motion_X->Get_y(mtime)==0)
&& (motion_Y->Get_y(mtime)==0)
&& (motion_Z->Get_y(mtime)==0)
&& (motion_ang->Get_y(mtime)==0)
&& (motion_X->Get_Type() == FUNCT_CONST)
&& (motion_Y->Get_Type() == FUNCT_CONST)
&& (motion_Z->Get_Type() == FUNCT_CONST)
&& (motion_ang->Get_Type() == FUNCT_CONST)
) // .. and if motion wasn't caused by motion laws, then it was a keyframed movement!
{
// compute the relative speed by BDF !
m_rel_pos_dt.pos = Vmul ( Vsub (coord.pos, last_rel_coord.pos), 1/mstep);
m_rel_pos_dt.rot = Qscale ( Qsub (coord.rot, last_rel_coord.rot), 1/mstep);
// compute the relative acceleration by BDF !
m_rel_pos_dtdt.pos = Vmul ( Vsub (m_rel_pos_dt.pos, last_rel_coord_dt.pos), 1/mstep);
m_rel_pos_dtdt.rot = Qscale ( Qsub (m_rel_pos_dt.rot, last_rel_coord_dt.rot), 1/mstep);
// Set the position, speed and acceleration in relative space,
// automatically getting also the absolute values,
SetCoord_dt (m_rel_pos_dt);
SetCoord_dtdt (m_rel_pos_dtdt);
// update the remaining state variables
this->UpdateState();
// remember that the movement of this guy won't need further update
// of speed and acc. via motion laws!
this->motion_type = M_MOTION_KEYFRAMED;
}
}
// restore state buffers and that's all.
last_time = ChTime;
last_rel_coord = coord;
last_rel_coord_dt = coord_dt;
}
//////////////// FILE I/O
void ChMarker::StreamOUT(ChStreamOutBinary& mstream)
{
// class version number
mstream.VersionWrite(2);
// serialize parent class too
ChObj::StreamOUT(mstream);
// deserialize parent class too
ChFrameMoving<double>::StreamOUT(mstream);
// stream out all member data
mstream << abs_frame;
mstream << rest_coord;
mstream << (int)motion_type;
mstream.AbstractWrite(GetMotion_X());
mstream.AbstractWrite(GetMotion_Y());
mstream.AbstractWrite(GetMotion_Z());
mstream.AbstractWrite(GetMotion_ang());
mstream << motion_axis;
}
void ChMarker::StreamIN(ChStreamInBinary& mstream)
{
Coordsys cfoo;
ChFunction* ffoo;
Vector vfoo;
int menum;
// class version number
int version = mstream.VersionRead();
// deserialize parent class too
ChObj::StreamIN(mstream);
// stream in all member data
if(version==1)
{
mstream >> cfoo; SetCoord(cfoo);
mstream >> cfoo; SetCoord_dt(cfoo);
mstream >> cfoo; SetCoord_dtdt(cfoo);
mstream >> cfoo; SetAbsCoord(cfoo);
mstream >> cfoo; SetAbsCoord_dt(cfoo);
mstream >> cfoo; SetAbsCoord_dtdt(cfoo);
mstream >> cfoo; //SetRestCoord(cfoo); not needed?
}
if(version >1)
{
// deserialize parent class too
ChFrameMoving<double>::StreamIN(mstream);
mstream >> abs_frame;
mstream >> rest_coord;
mstream >> menum; SetMotionType((eChMarkerMotion)menum);
}
mstream.AbstractReadCreate(&ffoo); SetMotion_X(ffoo);
mstream.AbstractReadCreate(&ffoo); SetMotion_Y(ffoo);
mstream.AbstractReadCreate(&ffoo); SetMotion_Z(ffoo);
mstream.AbstractReadCreate(&ffoo); SetMotion_ang(ffoo);
mstream >> motion_axis;
}
void ChMarker::StreamOUT(ChStreamOutAscii& mstream)
{
//***TO DO***
}
} // END_OF_NAMESPACE____
////////// end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,803 |
Q: Schema Conversion in JAXB Let's assume, I have a sample class foo with an attribute bar. I have set annotations for JAXB, so I can export it to XML and re-import it again.
@XmlRootElement
public class foo {
private int bar;
@XmlElement()
public int getBar() {
return bar;
}
public void setBar(int bar) {
this.bar = bar;
}
}
The XML file should look somewhat like this:
<?xml version="1.0" encoding="UTF-8" standalone="yes"?>
<foo>
<bar>5</bar>
</foo>
Now I have learned, I can also make bar an attribute. Using @XmlAttribute instead would make it look like <foo bar="5"/>, which I would prefer?
I already have a lot of files using the old schema. Changing them by hand would be very cumbersome, so I'd like to do it automatically.
Does JAXB offer a convenient way to convert the files? Maybe different annotations for unmarshalling and marshalling? What else could I do to achieve this without too much effort?
A: Not an expert with JAXB and there propably is more elegant solution. But here is anyway one - maybe not so elegant - solution.
In class foo you will anyway change the
@XmlElement()
public int getBar() {
to
@XmlAttribute()
public int getBar() {
You could then inherit foo for unmarshalling the old files
@XmlRootElement
public class foo2 extends foo {
@Override
@XmlElement()
public int getBar() {
return super.getBar();
}
@Override
public void setBar(int bar) {
super.setBar(bar);
}
}
Then it might be possible to marshall foo2 as its super foo by
JAXBContext context = JAXBContext.newInstance(foo.class);
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,122 |
<?php
class Mage_Core_Model_Message_Success extends Mage_Core_Model_Message_Abstract
{
public function __construct($code)
{
parent::__construct(Mage_Core_Model_Message::SUCCESS, $code);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 0 |
**ALSO BY**
**SCOTT DOUGLAS**
**_Advanced Marathoning_**
(with Pete Pfitzinger)
**_Bill Rodgers' Lifetime Running Plan_**
(with Bill Rodgers)
**_The Complete Idiot's Guide to Running_**
(with Bill Rodgers)
**_The Little Red Book of Running_**
**_Road Racing for Serious Runners_**
(with Pete Pfitzinger)
TO ALL
AMBITIOUS RUNNERS
WITH
OPEN MINDS
# CONTENTS
**CHAPTER 1**
**AN INTRODUCTION TO MINIMALISM**
The ideas behind running in less shoe, and how this book will help you safely implement them
**CHAPTER 2**
**WHY BOTHER?**
Reasons for running in less shoe
**CHAPTER 3**
**A BRIEF HISTORY OF MINIMALISM**
There was a time when all shoes were minimalist
**CHAPTER 4**
**FACTS ON FORM, FOOTSTRIKE, AND FOOTWEAR**
What research shows—and doesn't show—about running shoes, injury rates, barefoot running, and more
**CHAPTER 5**
**THE MANY MODES OF MINIMALISM**
The characteristics and categories of minimalist shoes
**CHAPTER 6**
**STEPS TO MINIMALISM**
How to transition safely to running in less shoe
**CHAPTER 7**
**REASONABLE BAREFOOTING**
The theory and practice of running without shoes
**CHAPTER 8**
**MINIMALISM FOR LIFE**
How to stay healthy long-term while running in less shoe
**CHAPTER 9**
**MINIMALISM IN THE LONG RUN**
Where are minimalist shoes headed?
**A MINIMALISM GLOSSARY**
**ONLINE MINIMALISM RESOURCES**
**ACKNOWLEDGMENTS**
**ABOUT THE AUTHOR**
**INDEX**
# **CHAPTER**
# 1
# AN INTRODUCTION TO MINIMALISM
## The ideas behind running in less shoe, and how this book will help you safely implement them
**MARK TOMKINSON'S STORY** is one you hear a lot these days.
A resident of Huntington Beach, California, Tomkinson started running in 2006. He ran half-marathons and marathons but battled injuries on and off. While training for a marathon in 2011, he had shin pain and tendinitis in his knee. Intrigued by stories of runners who'd become injury-free after switching from conventional running shoes to lower, lighter, flatter models, he ditched his old running shoes. He switched to doing all his running in a few popular minimalist shoes and barefoot. In 2012, he returned to the marathon he'd run in conventional shoes the year before and improved his time from 3:57 to 3:29. "I've gotten much faster, more efficient and 95 percent of the knee and shin problems have gone away," Tomkinson says. "I am a huge advocate of minimalist running and preach it to anyone who will listen."
Mimi Englander's story is also one you hear a lot these days.
In 2010, the Littleton, Massachusetts, resident started running in Vibram FiveFingers. At first, things went well. She was able to run 7 miles at a time, 3 miles farther than she ever had in conventional running shoes. Longtime knee, back, and bunion issues improved. She stepped up her training even more and ran her first half-marathon in FiveFingers. Then the foot pain started. She kept running. About a year into her switch to FiveFingers, she stopped running because of pain in both Achilles tendons. She got an x-ray on her painful foot and learned that she had a metatarsal stress fracture. A year later, Englander was still getting back to running. "It's slow going, as it should be," she says. "My experience hasn't dissuaded me; I know what I did wrong and am listening to my feet much more carefully, working to refine my form at the same time."
Tomkinson and Englander are two of the millions of runners who've become intrigued by minimalist and barefoot running. In the past few years, no aspect of running has been discussed more than what type and how much shoe is best. Barefoot running has gone mainstream. Minimalism, or running in something other than conventional training shoes, has spread from a small segment of competitive runners to account for 11 percent of the US running shoe market in the first quarter of 2012.
## WHAT'S IT ALL ABOUT?
In the early 2000s, running message boards began to fill with complaints about the state of running shoes. Longtime runners were frustrated by most of what was commercially available. The shoes, they said, were too heavy, too high off the ground, too heeled, too full of gadgetry purported to correct flaws in people's running form. The shoes, they said, hurt rather than helped performance and increased rather than decreased injury rates. Runners shared experiences of switching to other types of shoes for all their mileage. Some opted for retro running shoes from the 1970s marketed as urban fashion wear. Some started doing even their slowest training runs in racing shoes. Others took a longtime practice of occasional barefoot running and did more and more of their running unshod.
The differences between a conventional running shoe (Saucony Ride) and a minimalist shoe (Vivobarefoot Evo) are obvious when you put the shoes next to each other.
The critique of modern running shoes went like this: By placing a large amount of soft foam and an elevated heel between the runner's foot and the ground, modern running shoes change people's running form for the worse. Modern shoes encourage a hard heel landing rather than a softer midfoot landing; the latter is how people run when barefoot, the critique continued. Hard heel landings are like braking with every step, which slows you down and sends the impact forces of landing up your body, leading to injury.
In addition, the critique continued, the large amounts of cushioning rob the body of its natural means of stability, because the feet can no longer get good sensory feedback as they roll through the gait cycle. Being so big, the shoes add weight that makes a quick, light cadence more difficult. The critique ended with the point that all the add-ons, like motion-control devices and midsoles of different densities, add more weight and rob the foot that much more of its ability to run naturally.
This isn't as radical a critique as it might seem at first. The type of running shoe described above—heavily cushioned, sloping significantly from heel to toe—became widely available only at the end of the 1970s. Because pronation—how much your foot rolls in as you move from landing to toe-off—could be easily measured in labs, shoe companies made controlling pronation one of the guiding paradigms for designing shoes.
But the primary elements of conventional running shoe design were never proven to be meaningful before they were foisted on millions of runners. "Not every injury is related to pronation and overpronation," says Brian Fullem, a sport podiatrist with 3 decades of running experience and a 14:25 5-K PR. "You need a certain amount of pronation to help absorb shock. And the heel height, I don't know where that came from. The running shoe industry had a 12-millimeter heel-to-toe drop as standard, and there was never any basis for that."
By the end of last decade, this rejection of conventional running shoes was no longer confined to a small group of longtime competitive runners. Nike had introduced the Free, which it marketed as a "training tool" to help strengthen the feet and lower legs. Christopher McDougall's _Born to Run_ became an unlikely bestseller and introduced minimalism to the masses. Research suggesting that modern running shoes contributed to poor form and increased risk of injury got mainstream press. Some runners, like Mark Tomkinson, found their running reborn and spoke of the switch to minimalism in evangelical terms. But many more runners, like Mimi Englander, discovered that simply switching shoes wasn't a cure-all and could lead to problems of its own.
As more runners started running in less shoe, blogs and message boards filled with vigorous, often rancorous debate on the topic. As often happens with suddenly hot topics, people at the extremes were the loudest and most involved in the debate, regardless of their level of knowledge. People often universalized from their experience and shouted past each other. The terms of the discussion got oversimplified: Is barefoot running good? Should you throw away your old running shoes? Aren't those people prancing around in those toe shoes going to wreck their feet?
## ABOUT THIS BOOK
Meanwhile, most runners read and heard about barefoot running and minimalism, and kept right on doing whatever they'd always done. After all, minimalist shoes accounting for 11 percent of running shoe sales means that conventional running shoes account for 89 percent of running shoe sales.
This book is for those runners, the vast majority of whom are curious about minimalism and barefoot running but still have some basic questions: Why is changing from conventional shoes worth considering? What is the most relevant information on the topic? How can I experiment with minimalism in a safe, sustainable way that will improve my running?
I've organized this book to answer these questions in a logical way. Before we get started on the specifics, I want to give you an idea where I'm coming from.
The basic premise of this book is that minimalism and barefoot running are means to an end. That end is running with better form and less injury, both of which should make you faster and help you enjoy your running more.
It's important to keep this means-to-an-end framework in mind. Minimalism and barefoot running are tools, not magic bullets. As Jay Johnson, who has coached national champions, collegiate runners, and recreational runners, says, "I think most people want the easy fix. There's no easy answer in running. Ever!"
That's another way of saying that, in running, there are no secrets—either of modern elites or of supposedly lost tribes. There are, however, best practices worked out through experimentation by ambitious, experienced, open-minded runners. The distinction matters because secrets imply, "Do this one thing and everything will be fixed." Best practices imply, "Here's a process that you can implement to improve as a runner."
"There are no secrets" also means keeping the importance of this or any aspect of running in perspective. There's no one element of running that deserves obsessive focus while you underemphasize other contributors to successful running. What you have on your feet when you run matters a lot. So do a lot of other things: how much and how far you run, how strong and flexible you are, your diet, your running form, and how you spend your nonrunning time. Zealotry never works out over the long term in running.
Throughout this book, our conceptual framework will be a model that former Nike Oregon Project coach Steve Magness, a real student of the sport, uses to judge the merits of ideas. He calls it the three legs of the stool, with the legs being theory, research, and practical. "If you have all three legs of the stool aligned, then you can be pretty sure an idea is legit," Magness says. "If you have two legs of the stool, you can sometimes make it work, depending on how strong those two legs are. If you have just one solid leg, then it's obviously not going to stand."
In running terms, the practical leg of the stool most often means looking at what the most successful runners have been doing over the longest period of time. Elite runners aren't a different species from the rest of us. Yes, they have more genetic running aptitude than most people. But they do the same sport we all do; like everyone, they're subject to gravity and apathy. Through unfathomable hours of practice and thought, their trial-and-error process reveals what works best most of the time for most runners. The same is true of runners who've been consistently able to meet their goals over long stretches of time. They have things to teach other runners.
That's why the experts we'll hear from in this book are for the most part not the ones usually cited when you read about minimalism. Our guides will be running experts with deep knowledge and experience throughout the sport, not just in minimalism. They're best able to place minimalism in perspective and show how it fits in with other aspects of long-term healthy and happy running.
In most chapters, we'll also meet runners who describe themselves as minimalists. My goal in presenting these "Meet a Minimalist" profiles is to highlight the range of discoveries—and mistakes—runners often make when they start experimenting with running in less shoe. In some cases, the experiment seems to have gone well; in other cases, not so well. Read the profiles to learn ordinary runners' experiences with minimalism, while bearing in mind that there are many variables that affect success and failure in running. That is, read the profiles more to gather general ideas than to find models to mimic.
## A NOTE FROM THE AUTHOR
I've run more than 100,000 miles since I started running as a teenager in 1979. I've run almost all those miles in light, low-to-the-ground shoes. At first that was because such shoes were pretty much all that was available. But even after more cushioned shoes became the norm, I instinctively gravitated toward the light, low models. They just made running more enjoyable. When I ran in bulkier models—most often while testing them for magazine shoe reviews—I spent most of the run noticing the shoes—how they felt too heavy, how I felt suspended off the ground and tilted forward by the large heel.
I began barefoot running not long after I started the sport. In 10th grade my running geek friends and I read advice from coaches in _Runner's World_ , _Running Times_ , and _The Runner_ (now defunct). They recommended small amounts of barefoot running to strengthen the feet and improve form. And we knew our history—how Abebe Bikila won the 1960 Olympic Marathon barefoot and how world-class Europeans sometimes raced on the track barefoot. On most easy days, our coach had us do 10 100-meter strides on grass. My friends and I started doing these sessions barefoot. We loved how the grass felt under our feet, and we had that eager-teen belief that we were training smarter than our rivals.
As it turned out, my school district didn't allow athletes to leave school grounds for practice. So instead of hitting the roads on days we weren't working out on the track, we were confined to a 1-mile perimeter of the school. It was almost entirely grass. Around and around and around we'd go, 5, 8, 10 times. This was also where we warmed up and cooled down on track days. My friends and I enjoyed our barefoot striders so much that we started doing our cooldowns barefoot. That felt good. We started doing some regular runs on easy days barefoot. That felt good, too. One Saturday I drove to school and did a barefoot 14-miler on that perimeter loop. It remains one of the most enjoyably memorable solo runs of my life.
The point of all this is that I've been what would now be called a minimalist my entire running life. Long before I started thinking about writing a book on minimalism, I watched the movement's evolution with great interest. In the early part of this century, I was the kook. Friends would see me doing regular runs in racing shoes and ask, "How can you run in those?" I'd point at the huge midsoles and heels of their shoes and ask, "How can you run in those?" When they'd say something like, "I like the cushioning," I'd respond, "Why that amount? Why not twice as much, or three times as much?" They would just shrug and we'd get on with our run. Like I said, I was the kook.
By the end of the decade, I was the boring old fart. Acquaintances would tell stories of reading _Born to Run_ , switching immediately from their conventional running shoes to a barely there model, or even starting to run in Vibram FiveFingers after years of inactivity, and finding themselves injured. "Shocking," I'd say in my unhelpful way. Casual runners—even nonrunners like my in-laws—would ask, "What do you think about barefoot running?" and I'd either give an answer far more long and involved than they wanted, full of caveats and codicils, or just sigh and try to change the topic.
Of course, all that time, my opinion and practices were more or less unchanged: Running in as little shoe as you can while staying a healthy, happy runner is a good goal. This isn't a new development in running, a secret, or a magic bullet. It's simply a goal that may help you enjoy your running more. And really, what more do any of us want from our running than to enjoy it on our terms for the rest of our lives?
To work toward that goal, use the same slow-burn ambition that the late Grete Waitz, nine-time winner of the New York City Marathon, advised for success in any area of running. "Hurry slowly," Waitz said. "Be dedicated and disciplined and work hard, but take your time. Move ahead, but be patient."
The first step toward meeting any goal is understanding why it's worthwhile. Let's start by looking at the most basic question about minimalism: Why bother?
# **CHAPTER**
# 2
# WHY BOTHER?
## Reasons for running in less shoe
**THE FOCUS ON MINIMALIST FOOTWEAR** sometimes obscures an important point: Running in lighter, lower, flatter shoes (or no shoes) isn't an end goal, but a means to an end. That end is a more efficient, more effective running gait. Better running form should translate to increased performance, decreased risk of injury, and, harder to quantify but still important, greater enjoyment of your running. (Who wouldn't like to feel more fluid and flowing on their runs?) Many runners who've successfully transitioned to running in minimalist shoes—by which I mean they're running the mileage they want largely injury-free—report achieving these benefits.
These are compelling reasons to consider running in minimalist shoes. Anything you can do to lower your risk of injury should translate to increased performance, in that you'll be able to accumulate long stretches of consistent training. If there's one thing that running experts agree on, it's that consistency is key to running your best.
As always in running, there are caveats. Yes, running in minimalist shoes can help change your form for the better. And yes, running with good form is better than running with poorer form. But there's more to running than your form. There's mileage and hard workouts, strength and flexibility, diet and attitude, and many other contributors to success.
For too long, the majority of non-elite runners barely thought about their form. Now the pendulum has quickly swung to the other extreme, and form has become an obsession for some runners. They ask: Should I switch to a midfoot strike? Should I increase my cadence? Should I run upright or with a forward lean? How should I hold my thumbs? It's easy to find runners who spend more time online discussing running form than they do running.
Later in this chapter we'll get experts' thoughts on the benefits of minimalist shoes. But first, let's take a step back and get a better understanding of what you should know about running form, no matter what's on your feet.
## DON'T JUST RUN, BABY
When Bill Rodgers was the best marathoner in the world in the late 1970s, a biomechanist named Peter Cavanagh tested him in his lab at Penn State. As part of the test, Cavanagh had Rodgers "fix" his trademark across-the-body right arm swing. The result? Running with more textbook form, Rodgers consumed more oxygen at the same pace. That is, changing Rodgers's form to something thought to be better made it more "costly" for him to run a given pace; what's known as his running economy worsened.
In the 3½ decades since that lab experiment, a take-home message from it has been endlessly repeated: Don't mess with your running form. Over time, your body will find its best way of running. The more you run, the more your body will find its natural form. Just run, baby.
Why, then, do almost all top coaches have their runners spend time working on their form? Why do most elites, already blessed with enviable technique, think that working on their form will make them faster, either directly or by allowing them to train more by avoiding injury? And why should you?
For starters, let's go back to the Rodgers experiment. No reputable source claims that, at any one instant, significantly altering your form from what your body is used to will make you faster. Coaching legend and longtime exercise-science lab rat Jack Daniels, PhD, has tested thousands of runners over the last 40 years. "I have tested runners' economy of running with their hands in their pockets, on their hips, folded on top of their heads, etc., and it always costs more than when using a normal arm swing," he says.
But that doesn't mean the logical conclusion is that the form your body naturally gravitates toward is what will make you fastest. "We all run as children and assume that we are doing it correctly," says two-time Olympic marathoner Pete Pfitzinger, now general manager of capacity and expertise at High Performance Sport NZ in New Zealand. "That is usually not a bad assumption, but there is a difference between doing something reasonably well and maximizing performance." Pfitzinger says that many runners can improve their running economy—again, their oxygen cost at a given pace—by 2 to 4 percent through improved form. "If you have been training hard for several years, it can be an easier way to improve than doing more repeat miles."
Nor does it mean that your "natural" form is in your best long-term interest. "When we go out and run, we have a pattern of form that follows our skeleton and is dictated by our muscles and range of motion," says veteran coach Roy Benson, who has worked with high schoolers, Olympians, beginners, and everyone in between. "Over the course of lots of running, it's like an electrical current—your body follows the path of least resistance."
Running with "least resistance" sounds great, right? Doesn't this mean you're running as efficiently as possible?
Not necessarily. Pete Magill, who holds several American age-group records and has coached runners for more than 2 decades, says, "This belief system that just doing it over and over is somehow going to make us better is really crazy. Longtime runners actually suffer from the body's ability to become efficient. You become so efficient that you start recruiting fewer muscle fibers to do the same exercise, and as you begin using fewer muscle fibers, you start to get a little bit weaker. Over time, that can become significant. Once you've stopped recruiting as many fibers, you start exerting too much pressure on the fibers you are recruiting to perform the same action. And then you start getting muscle imbalance injuries—calf strains, little hamstring pulls, things like that."
Magill adds, with more than a little frustration in his voice, "Running is the one sport where people think, 'I don't have to worry about my technique. I'm not carrying a ball, I'm not swinging a bat, I'm not on skates, so my form doesn't matter.' We also have a sport where people don't always listen to what the top people are doing. They're far more interested in what the local Pose guru (see "Form Schools") might be telling them than in what [two-time Olympic marathoner] Ryan Hall is doing. I would say all top runners work to improve their form."
Certainly that's been my observation. In the past 2 decades, professional responsibilities have given me the privilege of observing the training of scores of runners. I've seen every top runner I've spent more than a little time around work on form, either directly through technique drills, indirectly through strengthening work, or simply by being mindful of form while running.
## WHAT'S GOOD RUNNING FORM?
It's important when discussing running form to remember that there's no "perfect" form that we should all aspire to. And, adds Pfitzinger, "No one can look at you and say whether your running economy is good or bad. We would all try to 'fix' Paula Radcliffe if classic running technique was synonymous with good running economy." That's Pfitzinger's nice way of saying that the women's marathon world record holder can be tiring just to watch running, given how her head bobs and weaves like she's in a sparring match. And yet looks often deceive when it comes to running form. In one experiment, Daniels tested a group's running economy, then showed footage of the runners to coaches and had them rank who, based on running form, had the best running economy. The coaches' answers were no more accurate than if they had guessed randomly.
So how to know if you should bother working on your form? And again, why do elites spend time doing so?
University of Illinois coach Jeremy Rasmussen puts it this way: "I bet that if I went out and said we're going to do functional testing on a sample of people, you're going to find weaknesses in every single one of them. The body has adapted to who you are, but has the body adapted to the best possible thing you can offer it? No, because you have inefficiencies somewhere, so if you can change those inefficiencies and make them strengths, then your body will start to change naturally for the better." Rasmussen works on form with all his runners, including three-time NCAA champion Angela Bizzarri, who won those championships and overcame a history of injury only after she and Rasmussen worked to improve how she covers the ground.
Magill agrees, especially for the many over-40 runners he works with. "I assume that any runner who's been away from youthful activities like basketball, Frisbee, football, tennis—been away from a wide variety of activities that actually work on muscle balance—I assume that they haven't been trained for a full range of motion and that they've developed muscle imbalances."
### FORM SCHOOLS
**Various prescriptions for running form** have become popular in recent years. ChiRunning and the Pose Method are the best known.
These methods have much good to say about running form, including working toward a lighter, quicker cadence and avoiding heavy heel-striking. Where the vast majority of successful coaches and runners find fault with the schools is the idea that there's one precise, perfect way to run that's universally applicable. Most also question the near-myopic emphasis on form, rather than considering it one of several elements of peak performance. And, it's fair to ask, where are all the world-class runners who follow these methods to a T?
Many non-elite runners have found their running invigorated and improved by ChiRunning and the Pose Method. This isn't surprising, in the same way that people who go on a diet often lose weight, regardless of whether it's high-carb or low-carb, gluten-free or grain-heavy. Simply paying attention to what they're eating is enough of a change from their normal habits for many people to lose weight. Similarly, working on their form is different enough from many runners' past practices to cause significant improvements.
While there's no perfect form, there are common elements of good form, based on basic principles of physics and biomechanics. These include:
**Footstrike.** A true forefoot landing is rare and can be as inefficient as a heavy heel landing. Somewhere between a slight heel landing and midfoot is best for most people. A midfoot landing allows release at toe-off of energy stored in your calf muscles and Achilles tendons. Over time, your footstrike should move toward what's natural for you if you're running in shoes that don't negatively change your gait and if you have proper strength.
**Landing position.** Make ground contact near your body's center of mass, with your landing knee slightly bent and your lower leg close to perpendicular to the ground.
**Midflight.** When both feet are off the ground, you still want to be traveling forward more than you are vertically. Too much vertical motion (for example, if you see the horizon bobbing up and down) means you're wasting energy to go in a different direction than you want to be.
**Knee lift.** Your amount of knee lift should be in sync with how fast you're going, from almost none when you're jogging easily, on up to your thigh being parallel to the ground when you're close to sprinting, and with an appropriate increase as you move from your slowest pace to your fastest pace.
### BORN TO HEEL-STRIKE?
**Coach Steve Magness says** that runners are heel-strikers for different reasons. Here's how he distinguishes them.
"I have someone who's a heavy heel-striker do strides with their shoes on, then have them take their shoes off and do strides," Magness says. "If they still heel-strike barefoot, then you know it's a motor programming thing, because it's not the weight of the shoe or heel-toe drop or feedback blocked by the shoe. So then it's, 'All right, this is how this person tends to run.' Then you have to take an approach of 'We've got to do some actual mechanical work and give some cues and try to consciously change things.'
"If that's not the case, then I think you go the other route and adjust the surroundings via strengthening work, shoe choice, and dynamic flexibility."
**Turnover.** There's no magic number of strides per minutes that's universally desirable, but most people will run most efficiently at a cadence of 170 to 180 steps per minute (that is, each foot striking the ground 85 to 90 times per minute). Your turnover should increase somewhat the faster you're going, but that 170-to-180 range is appropriate for most of your runs. When I did some easy runs in Kenya with guys who'd run under 13:00 for 5-K, their turnover was the same as when they got rolling.
**Upper body.** Run straight with a slight forward lean (which is different from bent over at the waist). Your arms should be at 90 degrees, your shoulders low and level, your hands held loosely cupped, neither in tight fists nor flopping around. Swing your arms up from the hip, not out, and try to keep them from crossing your body laterally. The insides of your wrists should come near your waist as your arms come through.
What deviations from this basic model do experts most often see?
Daniels says that in young and old runners alike he's worked with, "The most common form problem was stride rate—bounding over the ground too slowly, with long strides. Runners are often told to work on a long stride, but that is more a function of getting fitter rather than just doing it. I never had a runner perform worse when I felt they needed a faster rhythm and they learned to use a faster cadence." Minimalist shoes help many runners develop this quicker cadence.
Benson and Pfitzinger also see more overstriding than they would like. Says Pfitzinger, "My observation of runners in road races is that hardly any of the elite runners overstride, but up to 20 percent of the runners slower than 40:00 for 10-K overstride. Increasing stride rate by a few percent and decreasing stride length by a few percent can improve running economy in most overstriders." Here, too, minimalist shoes are often helpful.
### ARE YOU OVERSTRIDING?
**There's a difference between overstriding** and having a long stride. Overstriding means that your feet land significantly in front of your center of mass. When this happens, you're unable to make full use of your fitness, because you're braking with every step. And you might soon be breaking with every step, in that overstriding amplifies the already-strong impact forces of running and therefore can contribute to more strain on your bones, muscles, and ligaments.
Veteran coach Roy Benson suggests these two methods of determining if you're overstriding: First, have a friend with a video camera stand 20 yards back from the side of a level surface. Run past your friend for 30 to 40 yards at an easy pace. Then run past the camera at around 10-K race pace. Finally, run at a near sprint.
Says Benson, "When you watch yourself, even though you might not be able to stop action and analyze it at that level, just by seeing your form you can recognize whether you're overstriding. As long as your knee is bent and your foot is coming down back underneath you or close to you, there's probably not much inefficiency and not much risk."
Second, have a friend stand in front of you while you run toward her at the three effort levels in the above exercise. Says Benson, "The friend looks to see how much of the sole of your shoe is showing on impact. If there's 4, 5, 6 inches of daylight between your toe and the ground when your heel hits, you're overstriding."
The three effort levels are important, Benson explains, because many runners, especially those who didn't compete in school, become overstriders only when they try to go faster.
To improve a tendency to overstride, practice running fast while landing over your center of mass. This is often best done by going to a field or other safe, soft surface and shedding your shoes.
Says Benson, "At first, jog in place. You'll be landing on the ball of your foot. That's what it feels like to be a midfoot-striker. Now stay up there and jog in place and lean over and slowly accelerate over the next 50 yards or so. Don't go so fast that you forget to stay up there and land on the ball of your foot. When you do them right, strides like these are fast enough to be a good way to teach midfoot-striking." Then stay conscious of what that footstrike feels like, especially when you do track workouts and other faster sessions.
Pfitzinger says other form problems he often sees include:
* Leaning forward at the waist, which causes the quads to work harder to keep you from falling forward.
* Obviously not using the glute muscles. "When the glutes aren't working, the leg typically does not straighten behind the body, so the stride is more in front and under the body than behind the body," says Pfitzinger. "It looks like the runner is running just with the quads and hamstrings. Often the calves also don't do much, because they are the last push at the back of the stride. There is very little push behind the body, and the stride is relatively short."
* Holding the head forward of the center of gravity, which makes the neck and upper-back muscles fire to keep the head from falling forward.
Magill says, for longtime runners, "I assume you're not getting the same knee lift you used to get. Even for people who do tempo runs and reps, they rarely run faster than the race pace they're expecting to go. Let's say your shortest distance is 5-K and you almost never regularly run faster than 5-K race pace. Well, if that's 100 percent of what you're training your body to do, then it's a 100 percent effort for your body to lift your knees to the level you have to at 5-K race pace. Your body's going to find it's easier to hit 90 percent of that max effort, and you're not going to get the knee lift you need to run as fast as you want, and that's just going to compound over time."
### DON'T FORCE FORM CHANGES
**If your car is out of alignment,** you can keep driving it by continually jerking it in a direction it doesn't want to go, or you can get the alignment fixed.
If you want to significantly change an aspect of your running form, such as switching from a heel to midfoot landing, don't take the jerking-the-steering-wheel approach. "I'm not a real fan of contrived changes," says physiotherapist and 2:23 marathoner Phil Wharton. "They're not sustainable—the body will just keep reverting if you don't have the necessary flexibility and strength. Or, neurologically you're not yet ready to change that movement pattern. I feel like so many people are jumping ahead to the foot placement or some other isolated part of their form before they have the functioning body to sustain it. Get the functioning, and then the form will just start to naturally come."
There's a difference between being mindful of the elements of good form and forcing your body to run in a way it's not yet ready to. All good training is a gradual accumulation of small gains. Be diligent about monitoring your form to correct less-than-ideal habits. But also be diligent about strengthening and other work that will improve your running body and enable you to run naturally with better form.
## FORM FIXES
There's more to improving your running form than switching shoes. Some people can run with poor form barefoot; others can have exquisite mechanics in moon boots. That being the case, if you want to improve your form, there are things to address beyond footwear.
First, become an overall better athlete through regular short bouts of running at faster than race pace, strengthening your core and other key body parts, and by performing form drills. Says Magill, "If you can strengthen your muscles so that you can move strongly through a fuller range of motion, you can take the fitness you already have and run faster." Benson agrees, saying, "As you get general strength, you get better form."
If you're thinking, "That's all well and good for college runners and pros, who can train all day, but I have only an hour a day total for my running, so I'm better off spending that time just getting in the miles," Magill has an answer for you.
"That would be a great argument," he says, "if it were true. But if you have only an hour a day to devote to your running, the first thing you've got to do is learn to run. If you bring bad form into your running, all you're going to be doing for that hour a day is reinforcing bad form. If you spent even 1 of those days per week, or just a bit of time in those sessions, now you would be spending time actually training with good form. You'd then be using that hour every day to train with the form that's going to apply to your race speed and to your efficiency when running.
"A lot of people waste far more time being injured from running with muscle imbalances and poorly developed form than they do spending time doing drills or exercises or short hills or setting aside a short period each week to work on form itself."
We'll look at drills and targeted strengthening in Chapter 8.
A second way to have better running form is to minimize the deleterious effects of your nonrunning life. Spending hours a day slumped over in front of a computer wreaks havoc on everyone's body, but especially on a runner's body. Excessive sitting, even with the best posture, shuts off muscle activity along the back of your body; physiotherapist Phil Wharton calls this "glutes in hibernation." These are the very muscles you need to use to run with good form. In addition, the posture most of us adopt when working, texting, driving, and watching TV throws off our body's alignment. In Chapter 8 we'll see how to deal with these potential vectors of inefficiency and injury.
Third, you can work on improving specific parts of your form while running. Rasmussen does much of his form-improvement work by giving runners cues ("fast feet," "shoulders low," etc.) while they do "striders." Throughout this book you'll hear experts recommend striders. These are runs of 60 to 100 meters, done at about the pace you could hold in a mile race. Striders (or "strides" or "strideouts") are done on flat, level ground. The key is to run fast but relaxed—mile race pace, not sprinting. Doing striders once or twice a week after an easy run is a fun, easy way to improve many aspects of your running, including your form. If there's one type of fast running that all runners should do, even runners who never intend to race, it's striders.
Rasmussen also advises short bits of form work on regular runs. "When you go out for your run, for part of your run, pick a light pole that's about 100 meters out," he says. "Pick an aspect of form you want to improve. Focus on that one particular thing for that period of time, and then go back to just running, and then a few minutes later find another light pole and do it again, and bring it into your normal runs that way. Over time you can feel the difference."
Finally, when this all starts to seem too much to worry about for what's a basic human motion, relax. Literally. Says Daniels, "While running, go over your body from head to toe and ask yourself: 'Am I relaxed in the eyes? Am I relaxed in the jaw? Am I relaxed in the neck and shoulders? Am I relaxed in the arms and hands? Am I relaxed in the hips, in the knees, in the ankles, in the feet?' You may find some tight areas that may lead to better economy if fixed."
## THE BEEF WITH BEEFY SHOES
Now we can circle back to shoes. If you had to give the 30-second pro-minimalism pitch, it would be some version of this:
Because of their high heels and plush cushioning, conventional running shoes alter many elements of good running form. They encourage runners to run differently than if they were running barefoot—in big shoes, runners tend to land hard on their heels, to overstride, and to lose valuable feedback that comes from "feeling" the ground. These changes can lead to less efficient running, eventual injury, and gradual weakening of the feet and lower limbs. The onus is on conventional shoes, not minimalist shoes, to prove their value, because minimalist shoes encourage and allow more natural running form while providing necessary protection from surface hazards.
Experts from throughout running agree with this argument.
"I definitely think that some people benefit from the minimalist shoes," says sport podiatrist Brian Fullem. "Especially if they're able to change their stride and get off their heels, that may help eliminate some running pains people have in their knees, or shin splints or other injuries that may be increased by landing on your heels."
"Midfoot is the best place to land, and obviously if the shoe has too much in the rear, your propensity is to land more toward the back of the foot," says Wharton. "My clinical experience over the last 24 years is that people's feet get weaker by wearing high heels, by wearing shoes that don't fit properly, by having too narrow a toebox. There are all kinds of conditions that come from the wrong shoes—you get hammertoe, you get hallux valgus, where the big toe comes in. And that's one of the most important joints to have working, to get up on your metatarsals to propel yourself properly when walking or running."
Wharton's comments here touch on another common minimalist-versus-conventional-shoe design difference, in that minimalist shoes tend to be constructed to allow the foot to work naturally. That includes a wide toebox to allow the toes to splay as you roll through the gait cycle. Most traditional running shoes restrict this movement because they taper from the ball of the foot to the front of the shoe.
Referring to the three-legged-stool means of weighing scientific evidence discussed in Chapter 1, coach Steve Magness says, "If you look at the theory leg of the stool, you have some good theory—that conventional training shoes change mechanics and might block feedback, and that minimalist shoes can help strengthen you and let you better utilize elastic energy."
In Chapter 4, we'll look in depth at another leg of that stool—research—and see exactly what the data show on how people run differently when shod or barefoot, as well as what the research says about injury rates and running economy in different types of footwear. For now, it's enough to acknowledge that most runners run differently when they switch from conventional running shoes to barely there models. Just ask the calves of someone who immediately runs for an hour in Vibram FiveFingers after years in conventional shoes.
"One of the main effects of going to the more minimalist footwear is that it pretty much forces you to shorten your stride a bit," says minimalist blogger and college biology professor Peter Larson. "That's the quick fix that changing shoes can bring, and I think that's why some people may see an almost immediate result by changing shoes, because it forces you to change your stride in certain ways."
It's worth reemphasizing that this is a change for the better. It's a change toward a more natural running form, the form you would use barefoot and if you didn't have years of bad habits ingrained in your muscle memory.
Minimalist shoes can help bring about another positive change—reviving your feet's ability to work to their full mechanical potential. That can occur in two ways. First, you can improve your proprioceptive ability, which is sensing what's underfoot and making minute, instant changes to adapt optimally to changes in terrain. Second, your feet can relearn how to change from a flexible platform when you land to a rigid propulsive lever as you toe off. These mechanical gains can help increase efficiency, lower your risk of injury, and just make running feel better.
Embracing minimalism is also a healthy change in mind-set—to emphasizing that running is a natural, healthy activity, rather than something we dare to do only in "protective" and "supportive" shoes.
### **MEET A MINIMALIST**
### **CAMILLE HERRON**
### **WARR ACRES, OKLAHOMA**
Camille Herron is the rare national-class marathoner who's a full-blown minimalist. She credits ditching orthotics and conventional running shoes with allowing her to overcome a history of stress fractures, run prodigious mileage, lower her marathon PR to 2:37, and, to top it off, frequently win marathons.
After suffering her third stress fracture in her left foot as a high school runner, Herron was given orthotics and told to wear them with traditional trainers. Four more stress fractures prevented her from seriously competing in college. She became, in her words, "a hobby jogger."
At the same time, Herron's physics, biomechanics, and kinesiology classes got her thinking about the basics of the running gait. She also read stories of Kenyans running long distances barefoot. After yet another injury in the fall of 2003, she'd had enough—she committed to starting from scratch and rebuilding herself as a minimalist runner years before most runners had heard of the concept.
"I decided to go cold turkey and ditch my orthotics and the shoes I'd been training in," she says. "I think the first run I tried to do was in house slippers. I thought, 'Yeah, that might be a little too minimal,' so I ended up getting a pair of retro Asics flats."
Frustrated by so many years of setbacks, Herron had the patience to progress slowly. She started by running 2 or 3 miles every other day. In her first month, she hit 10 miles a week. In her second month, 20 miles a week. She added 10 miles per week each month, and by June 2004, at age 22, she was running 70 miles a week injury-free.
But not necessarily ache-free, at least at first. "During the first few months, I had very sore ankles and calves," Herron says. "My arches were pretty sore, too, but it was the kind of soreness or stiffness that felt okay, like the kind you might get with any new training program. I'd hobble out of bed in the morning wondering when my feet were going to stop bothering me. Then one day in March 2004, I woke up and got out of bed and I felt fine."
Herron also started experimenting with barefoot running at this time. Here, too, she took a gradual approach—5 minutes at a time, then 10 minutes, then up to 20 to 30 minutes a few times a week, on grass, within a few months. She ran barefoot as therapy when her legs felt tired or, her word, quirky. "If my Achilles would feel tweaked or my plantar felt off, I'd do some barefoot running, and it felt like it healed my body," she says.
These days, she still does some barefoot running, weather permitting, but mostly in the form of strides and drills. Herron is sponsored by inov-8 and says, "The inov-8s have more of a minimal build than the shoes I wore when I was sponsored by Brooks, so I feel like I haven't had to do any barefoot running to reset my body like I used to."
Herron consistently runs 120 to 140 miles a week in the inov-8 230s or 233s after initially trying slighter inov-8 models, the 155s and 195s. "It's interesting, I've gone from less shoe to a little more shoe," she says. "I think it's a matter of over time for the marathon you need a little bit more cushioning because you're spending more time on your feet."
In 2011, Herron did indeed spend a lot of time on her feet—she wound up with just under 6,000 miles for the year. That included running six marathons, and winning three of them. Does she therefore think everyone should take her approach?
"I don't know if other people can do what I did and be successful with it, but it's something to try if all else fails," she says. "If someone feels like they're okay, I would stick with what works. The key is to be healthy and to be able to train consistently, regardless of what's on your feet. Definitely if you keep getting hurt, it could be your shoes, especially if you have a lot of foot injuries, Achilles problems, plantar problems."
For herself, though, she's sold. "To have gone from seven stress fractures to no stress fractures and being able to handle high mileage," Herron says, "it definitely says something about how it's changed my gait, made me stop overstriding, and changed the stresses on my feet and lower legs."
## WHERE ARE ALL THE ELITE MINIMALISTS?
Runners who are skeptical about minimalism often ask a simple question: If minimalism leads to better form and performance and fewer injuries, why do the best runners in the world do a lot of their mileage in conventional shoes? Even if full-on minimalism caused only a fraction of a percentage of improvement, shouldn't the people whose races are decided by hundredths of a second be among its greatest adherents? Given everything else that elites do to get even the slightest edge, why not this?
These are fair questions that merit an in-depth answer. To start, let's make clear that we're not talking here about trail ultramarathoners or other excellent niche runners, but the sort who contend for Olympic teams and national titles. That is, the best at the type of running that most competitive runners do.
The observation that there are few world-class full-time minimalists is accurate. Elite runners run a lot in conventional running shoes. This is true not just in the United States, where today's stars grew up as conventionally shod young runners. I once spent a month in Iten, Kenya, a small town on the edge of the Rift Valley that's the epicenter of Kenyan running. Nary a minimalist was in sight. Kenyans may grow up barefoot, but once they have access to conventional running shoes, they wear them. A few years ago, one of the organizations that provide donated running shoes to needy Kenyan runners was given 300 pairs of Vibram FiveFingers to distribute. None of the Kenyans wanted them.
There are a few key points to consider in unraveling why most elites aren't minimalists:
**If minimalist shoes are a means to an end, most elites already have attained that end.** That is, almost all elites have the form that non-elites running in minimalist shoes aspire to. One reason they've been above-average runners since their first step is because their bodies are inherently better at running efficiently.
"For the most part they're already very biomechanically sound," says Wharton. "They're already landing up front, in the mid- to forefoot, so they don't really need to do everything in minimalist shoes to help them get there. You look at [world champion and Olympic medalist] Bernard Lagat and it's hard to see anything you'd want to change." Also, remember from earlier in this chapter that almost all elites do regular form work, another means toward this end.
**For elites, the potential payoff in slight form improvement isn't worth the risk.** That risk comes in two forms—injury from a too-rapid transition and reduced mileage and/or intensity while transitioning. "Out of 52 weeks a year, the average elite is healthy maybe 48 weeks a year," says Jay Johnson, who's coached runners to national titles in cross-country and indoor track. "They're going to say, 'I could potentially lose a lot of training getting used to doing all my running in these flimsy little shoes, and I don't know that it's going to make me any better.'"
Magness, who used to work with double Olympic gold medalist Mo Farah and Olympic silver medalist Galen Rupp, says, "The bang for the buck isn't there. If you can train in a mixture of shoes and be injury-free, then why make any sort of transition? That's why the few elites and sub-elites who you see do all their training in minimalist shoes are the ones who don't have anything to lose. They were always injured, and they were like, 'This is a last-ditch effort. It's worth a shot.'"
**Cushioning has its merits.** Running easily, for as much or as little as you feel like doing on any given day, is a natural activity. But training to be the best in the world is extraordinarily hard. It places varied demands on the body. World-class runners require different solutions—and different shoes—depending on their training.
"Sometimes the smartest thing to do with shoes is to match the shoe to fatigue level," says Magness. "If Galen Rupp has just done a long track workout in spikes and his calves are completely beat up, then they're not going to do any of the cushioning job they normally do. So you throw on some heavy, padded trainers and let the legs recover."
**Elites have always done a lot of running in minimalist shoes.** They just don't call them "minimalist shoes." They call them spikes and racing flats and lightweight trainers, and they wear them for almost all their hard workouts. Someone like Rupp might run 30 or 40 miles a week in light, low-to-the-ground shoes, the same amount as many minimalists' weekly mileage. ("And with a heck of a lot more force!" Johnson points out.) It's just that Rupp runs another 70 to 80 miles on top of that.
"That's the crux of this," says Magness. "Elites spend a lot of time in flats and spikes and lightweight trainers. And they'll do some easy barefoot mileage. I think most coaches agree that's going to get us all the bang for our buck in terms of performance enhancement or injury prevention or mechanics changes. Are we really going to gain from switching to minimalist shoes for the regular easy runs?"
Gold medalist Frank Shorter once told his fellow Olympic marathoner Kenny Moore, "You say you don't believe in high mileage, but you sure as hell run high mileage." Look at what successful runners do, not what arbitrarily defined camp they or others put them in. In the case of minimalism, that means recognizing that elites run a lot of miles in what most people would call minimalist shoes, while acknowledging that they also run a lot of miles in what most people would call conventional running shoes.
As it turns out, there was a time when all elites did all their running in minimalist shoes. In fact, all runners did. A brief survey of running shoe history is in order.
# **CHAPTER**
# 3
# A BRIEF HISTORY OF MINIMALISM
## There was a time when all shoes were minimalist
**HERE ARE SOME USER COMMENTS** I recently read about a popular New Balance running shoe:
"Flexibility gives feeling of next-to-nothingness," one runner wrote. Another praised the shoe's "ground-hugging feel." Then again, another runner pointed out, the shoe's "soles wear out too fast," "every grain of sand feels like an egg," and they offer "insufficient support."
Hmm, maybe a popular Adidas shoe would be better. It "conforms to the natural shape of the foot," one runner wrote. It "makes you want to run fast," a second commented. But, a third warned, it's "too thin for races as long as a marathon."
I read on, hoping an upstart company's innovative model, designed by a leading coach, might deliver nirvana. It's "made on a last conforming to the natural shape of the foot," one runner wrote, while others said it provides "excellent inside support and balance, good cushioning, plenty of toe room, relief from recurrent foot and leg problems." And yet, despite being one of the most expensive running shoes on the market, its "soles and heels wear too quickly," another runner cautioned.
I read these insights not in the latest _Running Times_ shoe review or on a message board, but in a special _Runner's World_ "booklet of the month," _All About Distance Running Shoes_.
If the title of the booklet—not to mention the idea of a booklet of the month—sounds a little off, that's probably because said booklet was published in July 1971. The New Balance shoe reviewed was the Jogster ("formerly Trackster II," the booklet informs, but you probably already knew that). The Adidas model was the Marathon. The new kid on the block was the Lydiard Road Runner, developed with input from the legendary New Zealand coach Arthur Lydiard. Perhaps his consulting fee was why the shoe cost $19.95, a dollar more than the Jogster. The most expensive shoe in the booklet was the Puma Marathon, which, at $26, cost more than twice as much as the Tiger Marathon, a steal at $11.95. (Imagine three companies today giving a shoe the same name.)
_All About Distance Running Shoes_ shows that much of minimalism isn't new. For that matter, in the 4 decades since its publication, not much has changed in runners' attitudes toward shoes. One runner quoted in the booklet noted, "Each time a shoe comes along, I eagerly anticipate the wearing of them and hope they solve my problems. I am always disappointed." Another put it this way: "Light shoes are nice, but my feet get awful sore in a long race, and thus reduces my morale and 'fight' badly sometimes. Heavy shoes reduce the amount of foot pain but increase leg fatigue. You can't win."
It's worth learning a bit about running shoes of old, for fun and context. Whether the topic is training or diet, gadgets or gear, being a student of the sport helps to provide perspective when weighing others' statements about running.
## MINIMALISM AS THE DEFAULT
In _All About Distance Running Shoes_ , the Tiger Cortez was praised for its "high heel" and being "effective for absorbing shock on hard pavement." The shoe "feels like running on pillows," one runner gushed. Not so fast, others countered; the Cortez was "too mushy" and "feels like a logging boot."
That reads like what today's runners might say, good and bad, about current conventional shoes. The Cortez is the anomaly in the booklet. The rest of the road running shoes look like what today would be classified as minimalist models—low to the ground, a simple upper and outsole construction, little or no difference in heel height and forefoot height. One big difference: Because of manufacturing processes and materials of 4 decades ago, many of the models in _All About Distance Running Shoes_ are heavy by today's standards. Whereas the current New Balance Minimus Road weighs 6.1 ounces in a men's size 9, the New Balance Jogster of old weighed 11 ounces in a size 8. (One clue about running demographics of that time is that weights for women's sizes aren't listed in the booklet, as there were almost no women runners, and certainly not mass-produced shoes made specifically for women to run in.)
The Tiger Marathon, from the early 1970s, was one of the most popular models before highly cushioned running shoes with large heels became widely available.
But there were some light shoes. The Adidas Marathon weighed 8.5 ounces in a size 8, in line with many of today's minimalist models. The Tiger Marathon, the booklet informs in an oddly precise detail, weighed 6.24 ounces in a size 8. I measured its heel height at 10 millimeters and forefoot height at 8 millimeters. Compare that with a current shoe like the Vivobarefoot Evo II, which is one of the lowest shoes on the market, with heel and forefoot heights of 7 millimeters. If available today, the Tiger Marathon would appeal as much to minimalist runners as to urban hipsters.
I borrowed _All About Distance Running Shoes_ from Dave Kayser, a retired museum curator for the National Park Service who lives in Danvers, Massachusetts. In addition to marbles, advertising pins, and telephone pole insulators (!), Kayser collects old running shoes. He estimates his collection at about 100. For 12 years he had what he calls "a running tree" in his yard—a fence post with thin strips tacked to the ends of branches that he attached shoes to. (These days Kayser keeps the shoes in his basement.) The earliest mass-production model he owns is the New Balance Trackster, from the 1960s. The oldest models in his collection are spikes from the 1930s and a pair of road shoes from the 1940s, "with kind of a gum sole," he says.
### WHO WERE THE RUNNERS OF OLD?
**The 1971 booklet** _All About Distance Running Shoes_ included input from _Runner's World_ subscribers. About 800 of them, 15 percent of the magazine's circulation at the time, filled out a questionnaire about their shoes, mileage, and injuries. (Another constant in the sport—runners' love for talking about their running to anyone who'll listen.)
The average respondent was 29 years old, stood 5′9″, weighed 145 pounds, had been running for more than 5 years, and averaged nearly 50 miles a week. The respondents were overwhelmingly, if not entirely, male. The differences in demographics compared with today's runners are worth tucking away somewhere in your brain.
Injuries were the bane of runners' existence then as now. Using as the definition of injury a condition that required time off from running and that was caused by running, the _Runner's World_ readers reported that the most common injuries were "knee damage" (17.9 percent of respondents), Achilles tendinitis (14 percent), and shin splints (10.6 percent).
Kayser has been a runner for more than 45 years, with PRs of 53:19 for 10 miles and 2:30 for the marathon. "When I started running, you didn't have much of a selection," he says. "You took what was foisted on you and hoped for the best. Almost all the shoes were low-profile, what today I guess would be called minimalist shoes. That's what almost everyone ran in."
Kayser is bemused by the trends he's seen in shoe preference, as runners rejected the shoes of his youth for heavily cushioned models that were marketed as runners' saviors. "What happened to all the science that was supposedly put into running shoes?" he asks. "Over the years they'd say, 'Oh, this is the greatest shoe because our studies show blah, blah, blah,' and 'It's got all this extra cushioning and protection.' But then all of a sudden minimalism is the way to go. When you've been around awhile, you just kind of hold your head and go, 'Oh my gosh.'"
The Tiger Obori, from the early 1970s, had an undercut heel.
As Kayser's collection shows, many minimalist shoe features marketed today as innovations were present decades ago. The outsole of the Tiger Obori, which came out in the early 1970s, wrapped around the upper in the outside ball of the foot area. The outsole also, bizarrely, extended a third of the way up the heel counter. More significantly, the heel was slightly undercut—the heel counter extended slightly over the back of the outsole. Today some minimalist shoes have this design to move runners away from a heavy heel landing.
The Lydiard Road Runner that we met at the beginning of this chapter had an asymmetrical, wide forefoot. Then, as now, this design was touted as allowing more natural foot motion by letting the toes spread and push off optimally. Etonic and Nike models from the 1970s had side lacing, to take pressure off the top of the foot. In the late 1960s, a British shoe was advertised with the tagline "run barefoot on the road." And almost 60 years before _Born to Run_ and Vibram FiveFingers became bestsellers, a Japanese runner named Shigeki Tanaka won the 1951 Boston Marathon in shoes that had a separate compartment for his big toe. "I don't want to sound cynical," Kayser says, "but it's all been done."
I asked Kayser if he's ever tempted to run in shoes from his collection. "No," he said, "it's hard to put most of them on. Over time they get stiff. Some of them might fall apart. Some of the ones on my shoe tree started delaminating. The soles and uppers were coming off."
He has another reason not to run in the shoes of his salad days. "My knees hurt so much," Kayser says. "Back then you didn't realize how much pounding your body took in those shoes because you were young and springy. I thought I was indestructible." Kayser and his friend Phil Stewart, race director of the Cherry Blossom 10-miler in Washington, DC, and a 2:19 marathoner in the 1970s, often discuss minimalism. Stewart, in his 60s and still a daily runner, tells Kayser, "I need cushioning!"
## WHEN BIGGER WAS BETTER
That desire for cushioning existed in the '70s. The Nike LD 1000, for example, came out in 1976 and had a high, flared heel and plush midsole. "I remember the first time I tried them on," Kayser says. I thought, 'Oh my god, this is wonderful.' When running started getting more popular and the more cushioned shoes started coming out, that was a godsend."
The first running boom, underscored by Jim Fixx's _Complete Book of Running_ hitting number one on bestseller lists, broadened the sport's demographics. The average _Runner's World_ subscriber who submitted comments in 1971 for _All About Distance Running Shoes_ was a 29-year-old male who weighed 145 pounds. Within the decade, the running population started looking much more like the American population.
Longtime runners like Kayser and Stewart appreciated shoes like the LD 1000 because they were such a contrast to the thin models they'd been pounding the pavement with for years. New runners—and increasingly, nonrunners—responded to the more cushioned shoes' immediate running-on-pillows sensation when they tried them on. An arms race in cushioning developed among companies hoping to capture that market.
The Nike Air Rift was released in 1995 with an "articulated toe" and came with a similarly designed pair of socks.
Bob Roncker has been a runner since 1958 and owns four running stores in the Cincinnati area. His running shoe collection trumps even Kayser's; most of the 300 models in it are on display at one of his stores. "When you look at the evolution of shoes in our museum," he says, "you see no flare, then a little flare, then all of a sudden a lot of flare after the Nike LD 1000."
Shoe companies were motivated by more than customer try-on satisfaction. "The thought was that higher heel heights would decrease Achilles tendinitis," says Joe Rubio, a partner in the online running store RunningWarehouse.com. Despite the lack of scientific evidence in support, cushioned soles were said to reduce common running injuries stemming from repetitive impact forces.
Thickly cushioned shoes were also touted as preventing footstrike hemolysis, or the breakdown of red blood cells in the feet. "That was a large impetus for more cushioned shoes and greater amounts of padding—preventing elite athletes from getting anemic," says Rubio. That benefit hasn't been proven. Knowledgeable minimalists such as college biology professor Peter Larson say that the shift in footstrike most minimalists achieve should eliminate concern about footstrike hemolysis. "Impact would seem to be the major driver of hemolysis, and this is vastly reduced among forefoot-strikers relative to heel-strikers."
The Taras were made in the 1990s and featured a hard Goodyear outsole, no midsole, and an asymmetrical forefoot.
Even though shoes kept getting bigger in the 1980s and '90s, companies occasionally released models that today's minimalists might smile upon. In 1995, Nike put out the Air Rift. Its distinguishing feature was what Nike called an "articulated toe," by which they meant a separate compartment for the big toe, like the shoes Shigeki Tanaka wore to win the 1951 Boston Marathon. In this case, Nike said the design was inspired by Kenyan runners who grew up barefoot; the shoe, Nike said, helped you run with more of a barefoot gait. The shoes were secured across the top of the foot and behind the heel with Velcro straps, and much of the upper was open. They came with a pair of socks with a separate compartment for the big toe.
The Air Rift was, I thought, a delight to run in and, as so often happens with such shoes, was soon taken off the market. Nike never got around to explaining why, if the articulated-toe design improved performance, it wasn't made standard throughout its running line, or even maintained in one shoe.
A year before the Air Rift came out, the owner of a California running store called Movin' Shoes sent me a pair of oddities called the Taras. (Yes, they were named after the Tarahumara. See "The Tarahumara through the Years.") The shoes have a leather upper, a thin, ridged outsole of Goodyear rubber, and a wide, asymmetrical forefoot to match the natural shape of the foot. There's no real heel or forefoot height to measure. One of my size-9.5 models weighs a bit more than 5 ounces. At the time, they retailed for $85.
After I did some running in the Taras, I called the owner to let him know how much I liked them, especially the generous forefoot, given how wide my feet are. He said the shoes were selling well. Satisfied customers would often report, "The shoes work great!" to which he would respond, "It's not the shoes, it's your feet!"
### THE TARAHUMARA THROUGH THE YEARS
**For a lost tribe,** the Tarahumara sure do get written about a lot.
The Mexican tribe of runners central to Christopher McDougall's 2009 book _Born to Run_ have been part of running lore for decades. Running historian Roger Robinson notes that former British elite runner Bruce Tulloh visited the Tarahumara in 1971 and wrote articles about them for a British newspaper. Another elite Brit, Tim Johnston, lived in Mexico in 1967 to train for the 1968 Olympic Marathon, held in Mexico City. He told Robinson about his interactions with the Tarahumara: "They were with the Mexican pre-Olympic squad, happy to have nothing to do but run all day, but they were very slow, just shuffling along in traditional huaraches."
A 1976 book, _The Joy of Running_ , by Thaddeus Kostrubala, MD, includes sections on the Tarahumara, as well as speculation about running's role in human evolution. The following year, the March 1977 issue of _Running Times_ included an interview with Jonathon Cassel, who, the magazine noted, "spent several months living with the Tarahumara." Cassel related stories of—see if this sounds familiar—persistence hunting and running as transportation. "Wherever they go, whether it is across the canyon or 100 yards or 200 miles, they run," Cassel said. "They chase down the game for food, they run wherever they go, their only sport or entertainment is a race in which, of course, they run—and this could be upward of 175 miles."
When I started reading running magazines in the early 1980s, the Tarahumara were presented more as common knowledge than as a paradigm-changing discovery. In 1993 and 1994, many runners now know, members of the Tarahumara won the Leadville Trail 100 Run, a 100-miler in Colorado. Many runners also know that aptitude in trail ultras at altitude is one of several forms of running competency, not the _ne plus ultra_ of athletic accomplishment.
As Robinson puts it, "We have long been familiar with the Tarahumara Indians . . . their prowess and their limitations as runners are well known in the running world."
## BAREFOOT RUNNING: LONG COMMON
Ah yes, the feet.
It shouldn't be surprising to find college runners doing barefoot strides even if their team is sponsored by Nike. Rather than a mild form of protest against the shoes, that's simply what competitive runners have done for decades.
The most famous barefoot running accomplishment, of course, is Ethiopian Abebe Bikila's victory at the 1960 Olympic Marathon in Rome. People trying to temper the enthusiasm of barefoot runners often point out that when Bikila repeated as Olympic Marathon champion 4 years later, he wore shoes, and ran faster, as if that one data point has a canceling effect on his barefoot win in 1960. A better view is that, like any intelligent competitive runner, Bikila did what he thought gave him the greatest chance of success on the day. He probably would have won shod in 1960 and barefoot in 1964.
Zola Budd of South Africa is the other most famous barefoot elite runner. She won the 1985 and 1986 World Cross-Country Championships barefoot, and wasn't wearing shoes when she and Mary (Decker) Slaney famously tangled in the 1984 Olympic 3000-meter final. (Budd got blamed for many things in this incident, but at least no one could accuse her of spiking her rivals.) Now in her mid-40s and a resident of South Carolina with the last name Pieterse, she's been a spokesperson for Newton running shoes.
_All About Distance Running Shoes_ notes that many British elites of that era often raced barefoot on the track. (Bear in mind that tracks then were almost always made of cinders, not today's more forgiving surfaces.) Among the occasionally barefoot Brits were Ron Hill, who won the Commonwealth, European, and Boston Marathons and set world records for 10 miles, 15 miles, and 25 kilometers; Bruce Tulloh, who set European records at 3 miles and 6 miles; and Jim Hogan, another winner of the European Marathon Championships and a world record holder at 30 kilometers.
It wasn't just elites who regularly ran barefoot, nor was racing the only time runners went unshod. In _All About Distance Running Shoes_ , former _Runner's World_ editor Joe Henderson reflected on his high school running in the late 1950s in Iowa, where he was a state champion.
"In that era, running shoes were poorly constructed, cumbersome, and expensive," Henderson wrote a decade after graduating from high school. "I naturally chose not to wear any shoes when running if I could help it. No one on our cross-country team wore them. We trained on the dirt and grass. Our feet got as tough as the soles of some of the shoes I now pay $20 for."
In the 1971 booklet, Henderson lamented that high schools had recently made it mandatory to wear shoes while competing. "International rules have no such requirement," he pointed out wistfully.
As the shoes of the time improved and most adult runners participated in road racing—not track and cross-country—regular barefoot running declined. This didn't make Henderson happy. "We've become so civilized in the 1970s that the sight of a barefoot runner is considered foolhardy, odd, abnormal," he wrote. "This is particularly true if he's seen racing on the roads."
The _All About Distance Running Shoes_ survey asked respondents if they ever ran barefoot. "Probably 99.9 percent said no," Henderson wrote. But as one respondent, Dick Cordone, put it, "If the prices of shoes keep going up, I might be forced to."
# **CHAPTER**
# 4
# FACTS ON FORM, FOOT STRIKE, AND FOOTWEAR
## What research shows—and doesn't show—about running shoes, injury rates, barefoot running, and more
**"NOBODY KNOWS ANYTHING,"** screenwriter William Goldman is said to have quipped about Hollywood. If Goldman had been commenting on running-shoe research, he might instead have said, "Nobody can prove anything."
People trying to convince others of minimalism's benefits or risks often cite research to back their claims. This study proves barefoot running is best, that study proves running shoes with heels cause injuries, and so on. Unfortunately, the presentation of the research is often lacking—findings are misrepresented, taken out of context, or given conclusions that aren't in the research. Sometimes all three happen.
This occurs more because people are passionate about the topic and misunderstand the nature of research than because they're being intentionally misleading. But it still doesn't help runners make good decisions.
Here are a few reasons why research on running shoes should be taken in stride:
* The studies are almost always short-term, whereas running is a lifetime sport. Even if you show that something changes over a 10-week study period, that doesn't tell you what continuing that practice for years will do.
* The pool of subjects is often small and of different demographics (age, gender, running history, weekly mileage, etc.) than a lot of runners.
* Study results usually compare mean values between groups. As an individual runner, you may be in line with these mean values, or far afield, and you probably have no way of knowing which is true. As sport podiatrist Kevin Kirby says, "There are many ways to run comfortably and without injury."
* The study design is seldom blind, much less the scientific gold standard of being double-blind (where neither the researchers nor subjects know who's in the experimental group and who's in the control group). If you're running barefoot, you know it. Even acknowledging that the research will have to rely on prospective studies—which follow groups of similar people with one important difference between the groups—it's hard to see how a valid long-term study could be designed. How would you isolate one running-related variable in just one person, not to mention a statistically significant group, over months or years?
* The studies are seldom replicated by other researchers. A key element of scientific research is that others can and do try to replicate your results. This seldom happens with running research, mostly because there's no incentive. As much as we love running, we must admit that whether heeled shoes cause iliotibial band syndrome isn't a top public health priority compared with looking into causes of heart disease.
* Related to the above, the studies are sometimes done or funded by people or companies with a stake in the outcomes. For example, Casey Kerrigan, MD, whose work we'll read about below, has helped a company called OESH design zero-drop shoes. Some of "barefoot professor" Daniel Lieberman's research at Harvard has been funded by Vibram. This isn't as nefarious as it might sound. Potential conflicts of interest are disclosed in peer-reviewed research, and scientists such as Kerrigan and Lieberman don't fudge data. Industry funding isn't inherently bad and can lead to discoveries that might otherwise not be made. But it's undeniable that, for example, Kerrigan has an interest in presenting data that might make you question the merit of conventional running shoes. Researchers who are completely removed from the matter don't have much of an incentive to try to replicate this research.
And from the everyday runner's perspective, here's the most compelling reason to keep all this in perspective: research that shows something doesn't necessarily prove anything.
On that last point, consider one of any number of studies I could cite. A few years ago Kerrigan conducted a study in which she measured external joint torques at the ankle, knee, and hip of people running barefoot and in conventional stability running shoes. When they ran in shoes, the subjects had increased joint torques at the three sites compared with when they ran barefoot.
Greater external joint torques sound bad. It seems reasonable to think that they should lead to greater frequency of injury. But it also sounds reasonable to think that running on soft surfaces is always "safer" than running on hard surfaces, when that's not necessarily so. (More on that below.) Researchers are almost always reluctant—sometimes maddeningly so—to draw big-picture conclusions from their work. Kerrigan hasn't said her finding of greater torque in running shoes proves that conventionally shod runners get injured more often. But the rest of the world isn't that careful, and the researchers aren't in the business of sending out perspective-providing press releases when their work is simplified or taken out of context. So in the case of Kerrigan's torque study, headlines such as "Running Shoes Worse than High Heels" are what people remember. To take another example, a study that Lieberman helped conduct found that heel-strikers on Harvard's cross-country team got injured more than their forefoot-striking teammates. It gets presented by the minimalist shoe company Vivobarefoot's Web site as "It's official—barefoot is best."
None of this helps the average runner make sense of these matters. We hear about these studies most often from nonscientists who cherry-pick research to support a conclusion they've already reached. The more ardent online barefoot advocates don't mention studies like the one published in the _British Journal of Sports Medicine_ in 2009 that showed that runners prescribed custom orthotics had less plantar pain over an 8-week period than a control group not running in orthotics. "Ah, but that was a short-term study," the response might be, and that's a valid point. But intellectual honesty compels reading peer-reviewed research consistently. You don't get to tout the results of studies you like but then switch to a design critique of ones you don't like.
Elite coach Steve Magness, who has a master's degree in exercise science and spends his noncoaching hours immersed in this sort of stuff, says, "The research, honestly, hasn't been done to the degree it needs to be done. There's some decent comparative data but there's nothing long-term related to performance or injury prevention that says, 'Here's what minimalism does compared to regular shoes.'"
Any honest assessment of research done to date has to include these three statements:
* Nobody has proven running shoes cause or prevent injuries.
* Nobody has proven running barefoot causes or prevents injuries.
* Nobody has proven that runners who wear conventional running shoes get injured more than barefoot runners or that barefoot runners get injured more than conventionally shod runners.
That doesn't mean the research is worthless (or uninteresting). It's better to be well informed than poorly informed, both in terms of facts and in how to put those facts in context. Below I've summarized some key findings in five topic areas. For a collection of links to the original research, go to www.runnersworld.com/minimalismlinks.
## RUNNING SURFACES
Research on running surfaces might not seem the most obvious place to start, but I'm leading with it because the research results lead to another matter of perspective to keep in mind. (As if everything I've laid out so far isn't enough.)
Research consistently shows that runners run differently on different surfaces. Using sensory feedback, we try to regulate impact forces by adjusting footstrike, joint stiffness, and muscle activation based on the firmness of the surface. (Muscle activation refers to the degree of tenseness in muscles before landing.) On harder surfaces such as asphalt, we land more softly than on softer surfaces such as grass.
The most interesting study in this regard involved jumping, not running. People jumped off a bench onto mats of different colors; the colors, they were told, corresponded to different cushioning levels in the mats. When they thought they were jumping on a well-cushioned mat, impact forces were higher than when they thought they were jumping on a less-cushioned mat. This is consistent with the research on adjustments to varying running surfaces. Well, except for one detail: The athletes were lied to—the mats were all equally cushioned. But when they thought they'd be landing on a softer mat, they presumably made subtle adjustments in muscle activation and joint stiffness to let the "softer" mat do some of the work.
This same sort of errant adjustment is thought to occur from wearing cushioned running shoes. Perhaps because of blocked sensory feedback, runners in cushioned shoes stiffen their knees when landing compared to when running barefoot. Impact forces wind up being higher than would be expected; it's as if running in the shoe causes your body to make adjustments that cancel out the supposedly beneficial cushioning.
So the research shows that we make complex, integrated adjustments to account for running surface and what's between us and that surface. Barefoot running on asphalt probably isn't as harsh as it might seem, and running in heavily cushioned shoes on soft surfaces probably isn't as protective as it might seem. Now for the additional bigger-picture thought on research stemming from this.
Nearly all running research of this sort looks at one element in isolation, such as impact forces, joint torque, or running economy. Yet we just saw via one such isolated research focus how the human body while running is an organic whole, with mind and muscles working in sync. This increases the chances of misinterpreting and overemphasizing any one bit of research.
The onus for intellectual honesty in this regard is especially on the most ardent pro-barefoot adherents. They point out, rightfully, that the running body is an amazingly adaptive thing. So why can't those adaptations be made in a healthy way when running in cushioned shoes? Sport podiatrist Stephen Pribut says, "Why do we assume the body suddenly becomes 'dumb' when you put on shoes? Shouldn't we assume it stays smart and makes the correct adjustments?"
## MECHANICS OF BAREFOOT AND SHOD RUNNING
The findings on how biomechanics change when running barefoot versus shod are consistent. The best research, such as Lieberman's, includes people who are experienced at running barefoot, rather than just having subjects who always wear shoes suddenly run barefoot. The findings are usually presented as how runners adjust their form when switching from shoes to barefoot, although if you wanted to be persnickety, you could argue it should be the other way around, given that barefoot is the natural state.
When runners ditch their shoes, stride length decreases (by about 6 percent), stride rate increases, and ground-contact time decreases. In Lieberman's famous study published in 2010, runners who were accustomed to conventional running shoes ("the habitually shod," Lieberman called them) were overwhelmingly heel-strikers in their running shoes. When they ran barefoot, they still mostly landed on their heels, but did so with a flatter foot than in their shoes—the angle of dorsiflexion (toes pointed toward the shin) of their feet went from 28 degrees to 16 degrees, and at the ankle went from 9 degrees to a bit over 0 degrees. That last bit means that their ankles were ever so slightly plantarflexed (pointing away from the shin) when they ran barefoot. Meanwhile, the angle at their knee upon landing went from 9 to 12 degrees.
That study also looked at these differences in runners Lieberman called "habitually barefoot"—that is, who regularly ran barefoot or in barefoot-style shoes like the Vibram FiveFingers. (Vibram helped to fund this study.) When running in shoes, half of them landed on their heels, the rest midfoot or forefoot. When barefoot, three-quarters of them used what Lieberman classified as a forefoot strike, and the remaining 25 percent landed on their heels. The joint angles were also quite different from those of the habitually shod. The only situation resulting in dorsiflexion was foot angle when running in shoes, and that was an angle of only 2.2 degrees. Compare that with the 28 degrees of dorsiflexion the habitually shod had in their feet when landing wearing running shoes. The angle of their knee at landing was much more similar barefoot versus shod than was the case with the runners used to conventional running shoes.
This is perhaps the most interesting finding of the study, suggesting that regular barefoot or minimalist runners have what are thought to be better foot and ankle mechanics regardless of what they wear on any one run. We'll return to this notion in Chapter 8.
It's worth remembering that this study, published in _Nature_ and given huge amounts of mainstream press, had a small subject pool. There were eight runners in each of the habitually shod and habitually barefoot groups. So when you say half of the habitually barefoot were heel-strikers in shoes but only 25 percent were when barefoot, bear in mind you're talking about changes in two runners.
## RUNNING ECONOMY
Things seem pretty straightforward concerning how shoes affect running economy, or the oxygen "cost" of running a given pace.
One commonly cited finding is that every 100 grams (just more than 3.5 ounces) of weight added to a bare foot increases the oxygen cost of running 7:00 mile pace by 1.2 percent. Another is that shoes constituting 1 percent of a runner's weight increase oxygen cost by 3.1 percent. (That would mean 12-ounce trainers for a 150-pound runner.) Another fun one to throw around is that models weighing just over 12 ounces per shoe increase the oxygen cost of running 8:00 mile pace by 4.7 percent. A recent study from Lieberman's lab looked not at bare feet versus shoes, but at FiveFingers versus the Asics Cumulus (a typical conventional running shoe). Regardless of whether they were forefoot- or rearfoot-striking, the runners were 2 to 3 percent more economical in the FiveFingers.
Competitive runners have known this intuitively for years. It's why someone sponsored by Asics might train in the Cumulus (11.2 ounces in a men's size 9), wear the Hyperspeed (7 ounces) for a marathon, and switch to the Piranha (4.7 ounces) for a 5-K.
As for why more runners without shoe contracts don't race barefoot, note that research on running economy hasn't been conducted with large numbers of runners moving at race pace. It's possible the shorter stride most runners adopt barefoot limits performance once you're going faster than a certain race pace. We'll consider this idea more in Chapter 7.
Also, recent research has suggested there's not necessarily a predictable decrease in economy as you move from barefoot to light shoe to heavy shoe. A study from the University of Colorado published in 2012 measured running economy in a small (12) group of men who were midfoot-strikers and used to running barefoot. The researchers measured running economy when the runners ran 8:00 mile pace "barefoot" (actually, they wore thin yoga socks) and in the Nike Mayfly, a racing shoe that weighs a bit more than 5 ounces. Eight of the runners were more economical in the Mayfly, four weren't, and when the results were pooled (as they always are), the difference in running economy when they were "barefoot" compared to in the Mayflys wasn't statistically significant.
By now you're probably tired of my offering five caveats for every research result. So I'll just say that while the Colorado study is interesting and intriguing, it's another example of a small study with precise parameters that gets spun into a supposed game-changer. For example, I'm chagrined to report that the _Running Times_ Web site linked to the study with the headline "Here's Proof Barefoot Isn't Better."
## INJURY CAUSES AND RATES
This is where things start to get really hazy.
To date, there have been no studies with satisfyingly clear conclusions on why runners get hurt. One research review—meaning that the study authors surveyed existing research to find commonalities—resulted in "strong evidence" for two culprits in lower-leg injuries: mileage and a history of injury. So, one of the leading contributors to injury is having been injured; that clarifies that!
The impact forces of running would seem to be a likely explanation for injury. Except that they're not. Some studies have suggested that runners with what the researchers call "higher vertical loading rate" have more of some injuries, including stress fractures. But other studies have found fewer injuries in runners with a higher vertical loading rate than in those with lower impact forces. As Kirby points out, research has also found seemingly counterintuitive results—such as that running on hard surfaces doesn't lead to more injuries than running on soft surfaces (see the earlier section "Running Surfaces") and that cushioned insoles don't appear to reduce the incidence of stress fractures. Further muddying things is that isolating a primary cause of one type of injury wouldn't necessarily say anything about other injuries. Stress fractures likely have different causes than the rusty-coil sensation many longtime runners feel at their hamstring insertions.
The study on Harvard cross-country runners—the one that Vivobarefoot says makes it official that "barefoot is best"—looked at form, not forces, for insight on injuries. Of the 52 runners in the study, 36 were said to be primarily heel-strikers, the others forefoot-strikers. Nearly three-quarters of the team members were said to get injured every year, and the rearfoot-strikers had roughly twice as many repetitive stress injuries as the forefoot-strikers.
This sounds pretty clear-cut. (Vivobarefoot certainly thinks so.) But as usual, this really just leads to more questions. Maybe the heel-strikers are injured more often because they have some structural weakness that leads them to heel-strike and that would be present no matter how they run. Maybe heel-striking isn't as good as forefoot-striking when you're training for and racing in collegiate cross-country races, but is if your focus in running is elsewhere. Who knows, maybe if the Harvard cross-country team switched to road ultras, the forefoot-strikers would be the ones getting injured more often. And maybe we shouldn't draw grand conclusions from a study on 52 thin runners in their early 20s.
Even if we could prove what causes injury, there's another reason to be leery of leaning too heavily on research on the topic.
A common element of the most rigorous scientific paper and the most superficial newspaper article on minimalism is some version of the statement "Every year, between _X_ and _Y_ percent of runners get injured." The numbers used for _X_ and _Y_ vary significantly, from less than 20 percent at the low end to more than 80 percent at the high end. The numbers are unsatisfactorily vague for a good reason: Nobody can say with confidence how many runners get injured every year.
There are several reasons why. First, there's no commonly agreed-upon definition of an injury. Does it mean an overuse injury bad enough to merit time off from running? That's a reasonable definition, but is it explained as such when runners are asked how often they're injured?
Second, even if that were the universal definition, it's flawed as a means of gathering meaningful data. What might lead you to take a week off from running could be the sort of condition I try to run through. Pushed to its extreme, the definition would mean that people who have running streaks never get injured, because they never miss a day. So, if you don't want to get injured, start a running streak!
Third, this is by necessity self-reported data. Even if there were a universal definition of injury, and even if all runners had the same standard for taking time off because of injury, not all runners have accurate records of all their runs. Without a log, can you say how many days you ran in 2010 and how many days you missed to injury?
All of this drifts even farther away from accuracy when people try to compare injury rates over time. As we saw in Chapter 3, the average _Runner's World_ subscriber in the early 1970s was a 145-pound 29-year-old man who ran 50 miles per week. The demographics of running have fundamentally changed since then. Today's _Runner's World_ subscribers are evenly split between men and women, with an average age of 42 and average weekly mileage of 20. Today there are more older runners, more new runners, more slow runners, and, let's face it, more runners who weigh more than 145 pounds. Even if accurate injury rates for a given year were possible, comparing them from year to year, not to mention decade to decade, would be meaningless.
Do runners get injured? Of course. Do we know why? Not really. Do we know how many runners got injured last year, and how that compares with how many got injured 5, 10, or 30 years ago? No. Take with a healthy dose of salt any advice on how to proceed in running that's based on injury research.
### **MEET A MINIMALIST**
### **PAUL MANGO**
### **SEATTLE, WASHINGTON**
Paul Mango's story is a good example of how minimalism can help with, but not necessarily solve, long-simmering injury issues.
Mango began running in ninth grade in the early 1990s, competed throughout high school and part of college, and has been a runner ever since. While training 25 to 35 miles a week and recording PRs such as 18:49 for 5-K and 30:55 for 8-K, he encountered typical bodily complaints; iliotibial band issues have been his greatest challenge.
While running in conventional shoes, Mango always had knee pain when he ran more than 12 miles. Figuring more support and cushioning would help, he moved from conventional but not massive Asics 2100-series shoes to the Asics Nimbus, then on up to the even heftier Asics Kayano. His knee pain didn't go away, and his iliotibial band started to bother him. He began to see a physical therapist.
While waiting in the therapist's office one day in the fall of 2010, he read an article about Vibram FiveFingers. Soon after, he read an Internet debate on barefoot running. "That's when I decided to give minimalist running a shot," Mango says. "I figured that wearing minimalist shoes would help strengthen my feet and calves, reduce the pain in my knees on long runs, and let me run with better form. I figured that if my lower limbs were stronger and I ran with better form, my IT band issues would go away."
Mango transitioned gradually, starting with one or two runs a week and switching to full-time after a couple of months. His IT band issue subsided.
In the spring of 2011, however, Mango twisted his ankle during a 20-mile trail race. "It was pretty sore after the race, but by the time I got home, I could barely walk back from my car to my apartment. I saw a doctor, and he said that I'd suffered a stress fracture in my foot. One thing I learned from that is that I should've worn a more protective shoe to keep the rocks in the trail at bay and to be more aware of the pain in my body. Muscle soreness is one thing, but any pain in the joints is a big red flag."
Mango suffered a couple of minor setbacks while returning from the trail-race injury, because of trying to increase mileage and intensity too soon. And his IT band still bothers him some.
"One thing that I've learned from switching to minimalist running is that it exposes a lot of your weak points," he says. "I've learned that I need to regularly do cross-training to strengthen my whole body. Having a good core and strengthening the muscles around your legs, especially the hips, can really help a lot in staying injury-free."
## FORCES AT FOOTSTRIKE
And now we enter the black hole of running research.
What impact forces the body incurs, and how those forces are incurred, is where online discussions of minimalism often deteriorate into people shouting past each other. For those who enjoy spending time arguing such things, there's good news: Both sides can cite research to support their claims. Some studies show that vertical loading rates from impact forces are greater when you run in shoes. Other studies show that vertical loading rates from impact forces are greater when you run barefoot.
One part of this topic that's often discussed is differences in what peak impact forces are like in a heel landing versus a midfoot landing. (This often gets interchanged with what happens when running in conventional shoes versus barefoot.) Lieberman's famous study published in _Nature_ included data that peak vertical impact forces were about three times less in forefoot-striking runners who were used to running barefoot, compared with heel-striking runners used to running in conventional shoes. Overall, Lieberman wrote, "in the majority of [forefoot-striking] runners, rates of loading were approximately half those of shod [rearfoot-striking] runners."
But, referring to "Injury Causes and Rates", these different impact-force measurements might not mean what we tend to think they do. Leading running mechanist Benno Nigg has produced research showing that people with low peak impact forces get injured as often as people with high peak impact forces. (And, of course, we have to consider all the notions about accuracy of reported injury rates. See what I mean about the black hole?) After surveying research on the topic, Nigg wrote, "One cannot conclude that impact forces are important factors in the development of chronic and/or acute running-related injuries." Instead, Nigg proposed, impact forces are "input signals that produce muscle tuning shortly before the next contact with the ground to minimize soft-tissue vibration and/or reduce joint and tendon loading."
One graphic you'll see often if you delve into this topic compares the peak loading rate of a heel strike versus a midfoot strike. The heel-strike visual has a distinct spike upon initial contact, while the midfoot-strike visual shows a much more even distribution of force from landing to toe-off. It's natural to look at the heel-strike visual and envision the heel crashing into the ground and imagine the bad things this causes compared with the visually pleasing midfoot-strike curve.
By now you won't be surprised to hear it's not that simple—some think more impact forces can lead to increased bone density. "What is the proven danger of the initial 'spike' in the rearfoot strike?" Pribut asks. "I've thought this is a 'signal' to the osteocytes [a type of cell found in bone] and bone in general to produce more bone."
In the "Running Surfaces" section, we saw how the body makes complex, near-instant changes when it's running. Research can offer valuable and interesting insights into isolated aspects of what happens when we run; it's unclear that it'll do more than that any time soon. Certainly runners shouldn't feel compelled to change things because this or that study "proved" something. I'm not big on clichés, and I wish our society as a whole had more respect for science. But what shoes to run in is definitely an area where the "we're all an experiment of one" bromide trumps the latest from the labs.
# **CHAPTER**
# 5
# THE MANY MODES OF MINIMALISM
## The characteristics and categories of minimalist shoes
**WHAT DO WE MEAN** when we talk about minimalist shoes?
As minimalist and barefoot running have exploded in popularity the last few years, most manufacturers have introduced models said to be in that category. But one manufacturer's minimalist model is another manufacturer's idea of a conventional running shoe. And, of course, different runners have different ideas about what minimalism means. Barefoot devotees apply the term only to barely there models such as the Vibram FiveFingers or Merrell Trail Glove, while runners who are used to 14-ounce trainers with high heels call the Saucony Kinvara a minimalist shoe.
Who's right? They all are. Remember, minimalism and barefooting are a means to the end of better running form. If you find that a given shoe within the broad brush of minimalism helps you to achieve that goal, then that's a minimalist shoe for you. Rigid classifications and debates about what is and isn't a minimalist shoe miss the point of why runners should pay attention to the matter in the first place.
That said, there are certain characteristics that most minimalist shoes share. In this chapter, we'll look at those characteristics and how they can be combined to produce a few broad categories under the minimalism umbrella. In doing so, we'll see how minimalism applies to trail shoes. We'll also examine a question many longtime runners have: If you want to run in a light, low-to-the-ground shoe that encourages getting off your heels, why not just do all your training in racing flats?
## THE MARKS OF A MINIMALIST SHOE
Ask 100 people to define the phrase "serious runner," and you'll get at least 30 different answers. Some will emphasize speed, others distance. Some will focus on competitive record, others daily dedication. But when considering any one runner, most people would get to their answer not through some theoretical definition, but by the way Supreme Court Justice Potter Stewart defined pornography—you know it when you see it.
That thinking is applicable to minimalist shoes—you know one when you see it. So in the following description of shared characteristics among minimalist shoes, I'm working more from the standpoint of "What do shoes most people consider minimalist have in common?" than "For a shoe to be considered minimalist, it must meet these criteria or we're not going to talk about it." The former approach not only broadens the types of shoes to consider, but also gets us past arguments about shoe specifics so that we can concentrate on the more important matter of which shoe is right for you.
Here are the key characteristics that most minimalist running shoes have to some degree. Some characteristics will be more important to some runners than others. And some shoes will have more of some characteristics and less of others. That's good, because it increases the chances of finding one that's right for where you are in your minimalist adventure.
**Low to the ground.** Minimalist shoes have less foam or other material between you and the ground than do conventional running shoes. What's known as a lower "stack height" results in better road feel and encourages your feet to work more naturally. A higher stack height potentially introduces instability, because there's that much more between your proprioceptive muscles and the ground. Look for heels that are less than around 25 millimeters high and a forefoot height not greater than around 17 millimeters, while bearing in mind these are general guidelines rather than deal-breakers for a given shoe.
**Not much difference between heel and forefoot height.** The difference between a shoe's heel height and forefoot height is known as the ramp angle. Research has shown that too great a ramp angle—as found in most conventional running shoes—can lead to overstriding, heel landing in runners who might otherwise be midfoot-strikers, and other undesirable aspects of running form.
Most minimalist models retain a slightly higher heel than forefoot. Even more moderate "transitional" shoes—conventional-looking models that many runners use to experiment with minimalism—can have just a slight ramp angle. The Saucony Kinvara, for example, has a reported 4-millimeter drop from heel to forefoot, while the Brooks PureConnect has a reported 5-millimeter drop. (Note that these are the measurements reported by manufacturers. Measurements in the _Runner's World_ Shoe Lab often yield different values, such as a 7-millimeter drop for the Kinvara and a 4-millimeter drop for the PureConnect.)
A few shoes, such as most Altras and Merrells, are what are known as "zero-drop," meaning no difference between heel and forefoot height.
**Light weight.** As noted in Chapter 4, too much weight on your feet reduces your running economy. Almost by definition, minimalist shoes should be among the lightest models on the market. As a general guideline, think less than 9 ounces for a men's size 9 and less than 8 ounces for a women's size 8.
**A simple upper.** The top part of your shoe should do little more than secure your foot to the bottom of the shoe. Thick overlays and heavily padded tongues add weight without increasing performance.
**A wider-than-average toebox.** Many conventional running shoes (as well as many racing flats) taper at the front of the shoe. This horribly conceived design inhibits your foot's natural flexion in the forward arch and constrains your toes' ability to splay. When arch flexion and toe splaying are allowed, these parts of the foot can better help in propelling you forward. The widest part of most people's feet is across the heads of the metatarsals (toe bones). A shoe designed to encourage a natural running gait should reflect that.
**A lack of gadgetry.** Running shoe industry mainstays like dual-density midsoles and plastic devices intended to control pronation are exactly what minimalist runners are trying to run away from. These add-ons not only increase a shoe's weight but can also inhibit the foot's natural motion.
**Flexibility.** A minimalist shoe ideally should be flexible in two ways—from front to back and from side to side. This allows your foot to move naturally through the gait cycle and to adapt instantly to the different ground conditions you encounter on a typical run.
Information on some of these characteristics is relatively easy to find on your own through manufacturers' Web sites, magazine shoe reviews, and online retailers. (The site RunningWarehouse.com sets the standard for giving stack height and weight for every model it sells.) Other characteristics are harder to judge without seeing and touching the shoe. This can be tricky for many models, as most retailers still lean heavily toward carrying conventional running shoes over a full range of minimalist models.
Also consider that some of the most innovative minimalist shoes, such as Altra, are small companies that have yet to make their way into most stores. In these cases, your best bet is to find a retailer who will work with you on a fair return policy if you decide a shoe won't work for you once you've seen it and tried it on.
### **MEET A MINIMALIST**
### **BRANDON WOOD**
### **ANCHORAGE, ALASKA**
Brandon Wood became a minimalist not long after he became a runner. Initial overeagerness toward running in general and switching to less shoe left him licking his wounds, but now he's found a middle ground sustainable for the long run.
Wood ran his first marathon in December 2010 at age 28, less than a year after becoming what he considers a regular runner. After that race, he was sidelined for a month and a half with an iliotibial band injury. "During those 6 weeks, I was going crazy and began reading everything I could about injuries, running form, minimalist shoes, etc.," he says. "One thing that kept coming up again and again was the shoes I was wearing, and it got me thinking that switching to a more minimal shoe may be the key."
Unfortunately, Wood's body wasn't prepared to handle his understandable enthusiasm that he'd found a solution. "I bought into the hype, and I dove headfirst into minimal running," he says. "I bought a pair of Vibram FiveFingers and began running in them regularly. I broke the cardinal rule, doing too much too soon, and almost injured myself again with a metatarsal fracture. After that, I regrouped and reassessed my thinking.
"The second time around, I took a much more sane and gradual approach to transitioning to minimal shoes. Today, I run most of my miles in shoes like the Saucony Kinvara and Altra Instinct—both relatively minimal shoes, but they still offer some cushioning." Wood does about 75 percent of his 30 to 40 weekly miles in those models, and uses a heavier Montrail shoe for trail running.
Wood looks at what he calls "this happy medium" solely as a means to the end of injury-free running. Although he says he likes the feel of minimal shoes, especially the wide toebox of the Altra Instinct, that's still more a functional appreciation than anything else. "My initial motivation was injury prevention, and that's pretty much what continues to drive this decision for me," he says. "If running in minimal shoes hadn't helped me stay injury-free through four marathons, I would be looking at other solutions."
## MINIMALIST CATEGORIES
Within the broad rubric of minimalism, it's helpful to think of three main categories: barely there/barefoot-style, moderate minimalists, and transitional/gateway shoes. Here's more on each of those categories, again with the caveat that these are general groupings more than distinct silos.
### Barely There/Barefoot-Style Shoes
Examples of barefoot-style shoes (clockwise from bottom center): Vibram FiveFingers SeeYa; inov-8 Bare-X Lite 135; Vivobarefoot Evo; Adidas AdiPure Adapt; Merrell Road Glove.
These are the minimalist's minimalists, so to speak. They have the no real midsole to speak of, just an outsole for abrasion resistance, putting your foot no more than 10 millimeters off the ground. They have almost no cushioning. They also have a tiny or nonexistent ramp angle and a wide toebox. The whole idea is to let your foot work as naturally as possible while providing a bit of protection from underfoot items. Examples include the Vibram FiveFingers, Vivobarefoot Evo, and Merrell Road Glove.
### Moderate Minimalist Shoes
Examples of moderate minimalist shoes (left to right): Altra Instinct 1.5; Skechers GoRun; New Balance Minimus Road.
These shoes share many characteristics of the barely there models, including a slight ramp angle, wide forefoot, and, usually, firm midsole. Their stack height is greater than those of the barefoot-style models but noticeably lower than those of conventional running shoes. Most runners new to minimalism will feel like their mechanics are different in these shoes, but because the shoes retain some elements of conventional running shoes, won't find them to be as abrupt a change as they do the barely there shoes. Examples include the New Balance Minimus Road, Altra Instinct, Nike Free, and Skechers GoRun.
### Transitional/Gateway Shoes
Examples of transitional minimalist shoes (left to right): Saucony Kinvara; Newton Distance; Brooks PureConnect.
These shoes retain many of the features of conventional running shoes, including a relatively high stack height and soft midsole. Their defining feature is a low ramp angle. Many runners use these shoes as their entrée into minimalism—hence the category name—by first adapting to the low ramp angle, then moving closer to the ground in more minimalist models if so inspired. Examples include the Saucony Kinvara, Brooks PureConnect, and Newton Gravitas.
## MINIMALISM ON THE TRAIL
In the 1990s, Ann Trason was one of the top ultramarathoners in the world. She won the storied Western States 100-miler 14 times; the course record she set there in 1994 lasted until 2012.
Trason was sponsored by Nike and appeared in an ad for Nike trail shoes in the mid-'90s. So when I spent a few days with her in 1996, I was surprised that for a long run on the Western States trail we started together, she had on the Nike Air Skylon T/C, an excellent lightweight road shoe of the time. When I asked her why someone in a Nike trail shoe ad was going to do a 44-mile (!) trail run in road shoes, she said, "A good shoe is a good shoe."
So take it from Trason, not me: A good shoe for running on the roads should also work well on trails. Trason's comment was especially pertinent in the mid-'90s. Companies were just starting to introduce trail running shoes, most of which were more hiking boot than performance-oriented running shoe. The heavy, sturdy shoes seemed counterintuitive—on trails, wouldn't you want something that much more flexible, nimble, and low to the ground to help quickly adjust to varying terrain?
Since then, trail running shoes have markedly improved, but it really took the minimalist movement to get trail shoes toward what they should have been from the start. Now you can find the same range of minimalist trail shoes—from barely there to borderline conventional—as you can road shoes. Of course, Trason's tenet still holds. But if you are interested in minimalist trail shoes, what should you look for?
### ORTHOTICS AND MINIMALISM
**Orthotics—customized insoles** designed to address an injury or structural weakness—seem antithetical to the let-the-body-do-its-thing ethos of minimalism. Can the two coexist?
Sometimes, says Brian Fullem, DPM, a sport podiatrist in Tampa, Florida, who ran 14:25 for 5-K while competing for Bucknell. If you've been prescribed orthotics, he says, you should consider whether you still need them, regardless of what kind of shoe you're thinking about running in.
"I have a lot of patients come in to see me who say, 'I've been wearing orthotics for 10 years,' and I'll ask them why they got them, and they don't remember," Fullem says. "When I make someone orthotics, it's almost always either to treat a specific injury or because they've had a history of injuries, and I'm trying to correct what I perceive to possibly be the cause of the injury."
If you received orthotics for a given injury but no longer have that injury, Fullem would generally advise weaning yourself off them. If, however, you have an injury in the acute phase, he advises continuing to wear orthotics and holding off on experimenting with minimalism until the injury is resolved.
"I'll give you a specific example—the posterior tibial tendon," he says. "The posterior tibial tendon helps to support the arch. If it's injured, a little tendon like the posterior tibial tendon can't resist the hundreds of pounds of force that happen every time you land running. I'm sorry, but doing drills and strengthening your foot and running in a Vibram FiveFingers isn't going to help you overcome that injury while it's acute, I believe.
"I think the same is true with something like plantar fasciitis. Let's get rid of the inflammation first. Once the injury is better, then start thinking about switching to a minimalist shoe."
If you have a pattern of injury but aren't currently in an acute phase, Fullem still advises sticking with the orthotics you've been prescribed to deal with that problem. Over time, through strengthening work and reaping some of the benefits of minimalism, you should be able to transition out of the orthotics.
## TRAIL SHOES VS. ROAD SHOES
The first thing to consider is the difference between a road shoe and a trail shoe. The largest difference is usually in the outsole. A trail shoe is going to have a more rugged, aggressive design to better handle the constantly changing surface underfoot.
Before minimalist trail shoes were widely available, I tried some trail runs in road racing flats, on the theory that their light, flexible construction would give me greater agility over roots and rocks. And they did. But the flat, uncontoured outsole of the racing shoes also meant that I felt every root and rock I stepped on. These weren't the most enjoyable trail runs of my life—if I wasn't landing on something hard and sharp, I was cherry-picking my way down the trail in more of a prancing than running motion. Some trail shoes have a rock plate, a thin layer of reinforced material directly above the outsole, to allow more cruising and less bruising over tricky terrain.
For similar reasons, trail shoes tend to have more of a protective upper. The reasoning here is that a sturdier wrap across the top of the foot will give greater protection from various rocks and twiggery protruding from the trail and will shelter the side and top of your foot if you roll an ankle while navigating uncertain ground. Reinforced sidewalls are common to help keep trail debris from entering the shoe.
Throw in other features, like a firm midsole to increase stability, and conventional trail running shoes start to outweigh their road-running counterparts by 2 to 3 ounces.
Examples of minimalist trail shoes (clockwise from bottom center): New Balance Minimus Amp; Brooks PureGrit; Merrell Trail Glove; Adidas Adizero XT-10; inov-8 f-lite 230.
## MINIMALIST TRAIL SHOES VS. CONVENTIONAL TRAIL SHOES
These days, one of the worst-performing categories in the running-shoe business is traditional trail shoes, while one of the hottest is minimalist trail shoes. One retailer I spoke with lamented that he couldn't unload conventional trail shoes even at liquidation prices.
It makes sense that some of the greatest enthusiasm for minimalism has come from trail runners. What better place to get back to more natural running than in the woods or around a lake or atop a mountain ridge? Also, on a soft surface like a bridle path or the floor of a pine forest, a shoe's lack of cushioning isn't as noticeable as on asphalt. And as I mentioned above, for many runners, traditional trail shoes harmed rather than helped the more agile, no-two-strides-the-same gait that's appropriate on many trails.
That said, I can vouch that less isn't always better on trails. This breakthrough insight stems from some short runs I did in the Maine woods in an early iteration of the Vibram FiveFingers. The experience was much the same as when I ran on the rutted, rooty ground in road flats—at times more of a walk-and-jump motion than something resembling running. Fortunately, there are now plenty of light, flexible, low-to-the-ground shoes with a trail-appropriate outsole.
How minimal to go with a trail shoe depends more on what the trails you run on are like than on the cushioning question that might mostly guide what minimalist road shoe you seek. No matter how adept a trail runner you are, some venues—such as a single-track trail over lots of rocks and roots—are going to get tiresome after a while in some of the more barely there models. There's a freedom in trail running that can be diminished by having to be aware of every footplant; if you're running amid great scenery, don't you want to be able to look around at times? There are enough minimalist trail models available that you should be able to find a happy medium between a close-to-barefoot feel and enough underfoot protection so that you can lose yourself in the run.
### SOCKS AND MINIMALISM
**Whether to wear socks** as you experiment with running in less shoe is an individual matter. Going sockless seems to be more in the spirit of letting your feet do their thing free of accretions; some minimalists will tell you that socks, especially thick ones, add another barrier to really feeling and reacting to the ground. A University of Toledo study published in 2011 found that people's single-leg static balance was better barefoot than while wearing thin conventional socks or five-toe socks. Sport podiatrist Brian Fullem, however, says he doesn't think that socks interfere with a runner's proprioception while in motion.
With their elemental design, many minimalist shoes have uppers that shouldn't irritate an otherwise bare foot. Still, even the slightest irritant, like a low toebox rubbing against a nail, can have major consequences by the end of a 2-hour run. Blisters can form when you go sockless, especially when it's hot and your feet are sweating a lot. At other times of the year, going sockless, especially in a highly ventilated shoe, can lead to uncomfortably cold feet, at least at the beginning of your run. (If you doubt me, come on up to Maine for an early-morning run next January.)
Coolmax, merino wool, and other wicking materials in socks help draw sweat away from your feet. "If there is more sweat being absorbed by the shoe, then that might set a person up for a greater chance of getting athlete's foot infections," says Fullem. Odor absorption is another reason to consider socks.
If you're so inspired, experiment with running without socks, but don't feel like you're less of a minimalist if you revert to wearing them.
## WHAT ABOUT RACING SHOES?
In the early years of this century, I found myself gravitating toward doing almost all my running in racing shoes. This wasn't because I was working from a creedal statement of what shoes are acceptable to run in, and it certainly wasn't because I was cranking out daily tempo runs. It was simply that, as training shoes kept getting bigger and bigger, the only widely available models that still felt like shoes I wanted to run in were racing shoes.
I wasn't alone in this solution. Message boards of the time were full of runners sharing advice (and frustrations) on which racing flats worked best as daily trainers. Given where shoes were heading then—toward ever-thicker stack heights and greater ramp angles—saying, "I do all my running in racing shoes" wasn't as dramatic a pronouncement as it might appear. A lot of racing shoes were also trending up in height and weight. Shoes like the original Saucony Fastwitch and the Fila Racer were more or less what lightweight trainers had been a dozen years before.
Now that there's a wide variety of minimalist shoes designed for daily running, does training in racing shoes still make sense?
Given what most people are looking for in a minimalist shoe, yes. The average racing flat is responsive and low to the ground, and doesn't have much of a ramp angle. Consider something like the Asics Hyperspeed, the shoe that Ryan Hall races marathons in. In a men's size 9, it weighs 7 ounces and has a stack height of 21 millimeters in the heel and 15 millimeters in the forefoot. Now consider the first generation of the New Balance Minimus Road, one of the poster children of mainstream minimalism. In a men's size 9, it weighs 8 ounces and has a stack height of 19 millimeters in the heel and 14 millimeters in the forefoot. The Hyperspeed is marketed as a racing shoe, the Minimus Road as a daily trainer, but in terms of two of the most important factors minimalists look for in a shoe, there's not much to distinguish the two models.
Examples of racing flats (clockwise from bottom center): Newton MV2; Mizuno Universe; Asics Hyperspeed; Adidas Adizero Hagio; New Balance 1400.
At the really minimal end of the spectrum, to most people a racing flat like the Mizuno Wave Universe (4 ounces in a men's size 9, stack height of 19 millimeters in the heel, 14 millimeters in the forefoot) isn't going to feel that different from something like the Merrell Road Glove (6.9 ounces in a men's size 9, stack height of 11 millimeters in the heel and forefoot).
A few caveats: One way that manufacturers keep weight down in racing flats is with a blown rubber outsole, which wears out more quickly than a standard outsole. Then again, the Hyperspeed, which has a blown rubber outsole, costs $75, compared with more than $100 for most shoes marketed as minimalist models. If you tend to switch shoes for reasons other than outsole wear, then racing flats can make good financial sense as an everyday option.
Perhaps more important for many minimalists' purposes, many racing flats are built on a traditional spike last (basically, the mold or skeleton around which the shoe is built). Models made this way tend to taper toward the front of the foot and have a low toebox. This is in contrast to the wide forefoot that's a key part of the design of many minimalist shoes. (Why racing flats, which are supposedly created to encourage best-possible mechanics, retain a design that inhibits natural foot motion remains a mystery.)
Also, many racing flats have a higher ramp angle than even the gateway minimalist shoes. For example, the Adidas Adios 2, which Patrick Makau wore to set a marathon world record at Berlin in 2011, has a heel-to-toe differential of 9 millimeters, compared with a 4-millimeter drop for the Saucony Kinvara or 5-millimeter drop for the Brooks PureFlow.
Put simply, consider racing shoes as an option when deciding on a minimalist shoe, even if you never intend to compete. As we'll see in the next section, the best approach is to focus on criteria, not category, when deciding what shoes are best for you.
## WHAT SHOE IS RIGHT FOR YOU?
We've just spent several pages looking at various categories of minimalist shoes. That's good and necessary information, because it's important to know your options and the thinking that goes into them. But as we saw with racing shoes, sometimes shoes are in a given category more because of a company's way of looking at the world than because of where they fit in with everything else on the market. When you go to buy a minimalist shoe, your best bet is to look past rigid categories and focus on which characteristics matter most to you.
A personal example: If I had to pick one running shoe from the last 3½ decades as my favorite, it would be the first generation of the Brooks Cheetah. Marketed as a lightweight trainer, it came out in the early 1990s. It was low to the ground and had a minimal ramp angle, a wide forefoot, and, for me, just the right combination of responsiveness and cushioning. So of course in 1996 Brooks nearly doubled the height of the midsole in the next iteration, the Cheetah 2. It was the same shoe in name only. I bought 10 pairs of the originals and mourned the day I retired the last pair.
Remembering what I loved about the Cheetah helps me pick shoes now. If a shoe's midsole is toward the firm end of things, I know I'll wish for a little more softness. If the ramp angle and stack height are much greater than in the Cheetah, I know I'll feel tilted forward and suspended above the ground. If the weight is much more than that of the Cheetah, I know I won't have that free-floating feeling, especially at the end of long runs or when I'm tired.
It's not that I have Brooks Cheetah stats lying around that I compare all shoes with. Rather, knowing that the Cheetah was, to date, the best shoe for me ever designed, I can quickly assess how closely other shoes come to its combination of attributes. Whether a particular model is marketed as a lightweight trainer or racing flat or minimalist shoe is irrelevant. Rather, based on knowing what's most important to me, I try to find a shoe from what's available that feels like the closest match to my memory of the Cheetah. Sometimes that's what's marketed as a minimalist shoe, sometimes it's what's marketed as a lightweight trainer, and sometimes it's what's marketed as a racing flat.
You should take the same approach when trying to choose from among the broad array of minimalist shoes. Of course, if you've always worn conventional running shoes, there will be some differences as you move toward minimalism. Most likely this will have to do with stack height and ramp angle. Still, try to be able to articulate the key features that the running shoes you've most liked have shared. Think about things like:
* Do I like a firm midsole or a soft midsole?
* Do I like a wide toebox or a narrow toebox?
* Do I like a high toebox or a low toebox?
* Do I like a firm heel counter or a soft heel counter?
In an ideal world, you'd then be able to go to a running store, tell them your general preferences, and ask them to bring out the five minimalist models that best match these criteria. Then you could try on each one, see which feels best, and go from there. If you take it home and can run in it without problems, proceed. If you get hurt or the shoe seems to get in the way of enjoyable running, cut your losses and start over.
In the likely case that that's not possible, make use of fellow runners' experiences via message boards and see what magazine shoe reviews and online running shops say about similar models.
But wait, you might be thinking: How do I know how much or how little of a minimalist shoe to look for? How do I know what minimalist shoe I can safely start running in? And, come to think of it, how should I go about starting to run in whatever minimalist shoe I buy?
Determining how ready your body is to start running in minimalist shoes and how to integrate minimalist shoes into your training program is the subject of the next chapter.
# **CHAPTER**
# 6
# STEPS TO MINIMALISM
## How to transition safely to running in less shoe
**IN MARCH 2012,** a woman runner came to sport podiatrist Brian Fullem's office in Tampa, Florida, with a sore foot. While talking with the woman, Fullem suspected she had a stress fracture. He asked her about her running. She said she had recently started training for a marathon and had switched shoes.
"I asked why she'd switched shoes," Fullem says, "and she said, 'I read _Born to Run_ and saw we're not supposed to be running in regular shoes, and the Kenyans run barefoot, and so I figured I'd start running in minimalist shoes.'" She had discarded her conventional training shoes to do all her running in a barefoot-style model. Fullem knew what to tell this patient because she wasn't the first such case he'd seen in recent years.
"I told her that you can't just go from wearing running shoes with a 12-millimeter heel-to-toe drop to a shoe that doesn't have any cushioning, any support," he says. "I told her _Born to Run_ isn't coming from any sort of science perspective, but from the perspective of telling a story about the Tarahumara, who run all day in these handmade sandals. I told her that's not her, and that there has to be a transition into minimalist shoes."
Up in Washington, DC, and New York City, physiotherapist Phil Wharton was seeing patients with similar pains—and recent histories. "Because of all the awareness, we've seen people jump in too fast or without a proper progression plan," he says about injured would-be minimalists. "Think about it like this: You get a device like this phone I'm using now. I certainly don't read the manual or the fine print. I don't even look at the quick-start guide. I just get it and start using it, and I don't know what I'm doing.
"And that's kind of what's happening with people in minimalism. They remember parts of _Born to Run_. They don't remember the part where Christopher McDougall says he had a personal trainer and did strengthening. What they remember is 'getting into a minimalist shoe changed my life and stopped all these injuries.' And so they just jump right in.
"The biggest pitfall is that we're a culture that's not designed to look at process," Wharton continues. "When we want to do something, we want to do it 100 percent, starting today."
In this chapter, we'll look at a better way of becoming a minimalist than the all-at-once mode that felled Fullem's and Wharton's patients and thousands of other overeager runners. We'll see how to gradually integrate minimalism into your running to minimize your chance of injury and maximize your chance of long-term success.
## ARE YOU READY?
As we've seen in earlier chapters, running in shoes that are lower to the ground, more level, and less cushioned than conventional running shoes places different demands on your body. To quickly recap: Your feet need to be stronger, your plantar fasciae and Achilles tendons need to be longer, your postural muscles need to be functioning well, and so on. Even your neuromuscular system needs to be ready to work differently, as you use more proprioceptive feedback than when running in thickly cushioned shoes that blunt those messages among muscles, nerves, and brain.
So why not just suck it up, start running in minimalist shoes, and allow your body to adapt? Wharton puts it this way: "What I see a lot is people wearing minimalist shoes and running with terrible form. So they go to a form clinic and they fix their form. That's going to last about 2 weeks before it wears off because they can't hold their form if their bodies aren't ready. People have to take a step back and make sure their body is working correctly first.
"We know minimalism is good, but here's the rider to the contract," Wharton continues. "The precursor is you gotta make sure your body is working correctly. Otherwise, you're going to get the injuries we see with minimalism. That's where the real gung-ho folks are going to come up against a brick wall if they're not addressing this."
There's enormous variability among runners in how prepared their bodies are for minimalism. This is true even among runners of the same age, gender, build, and lifetime mileage, among other factors. There are, however, a few common limiting factors in runners successfully transitioning to minimalism.
Below are five tests. Some were developed by Wharton, some by physical therapist Jay Dicharry, formerly the director of the SPEED Performance Clinic at the University of Virginia. Each targets one of the prime bodily needs for long-term, injury-free minimalism. If you can pass all these tests, you should have little to no difficulty in transitioning to minimalism if you follow the guidelines laid out later in this chapter.
If you fail a test, that means you're lacking that key aspect of functional strength or flexibility—that is, strength or flexibility directly related to the demands of what you're trying to do. Your chances of getting injured while running barefoot or in minimalist shoes are greater than if you could pass the test and followed the same transition plan. If you currently fail most of the tests, then your road to healthy minimalism will be even longer.
Failing one or more of the tests, however, doesn't mean you're not made for minimalism. After each test you'll see a simple exercise to improve that area of functioning. In some cases you can make dramatic improvements in as little as a week. Almost everyone can eventually get to where they can pass all the tests.
So, as of today, how ready are you to run in minimalist shoes? Let's find out.
## **TESTS**
### **TEST #1:**
#### **A NKLE DORSIFLEXION/POSTERIOR CHAIN RANGE OF MOTION**
Testing ankle dorsiflexion.
**Why test:** If your Achilles tendon lacks sufficient flexibility, you not only limit your ability to push off effectively, but also increase your chance of injuring the tendon or surrounding soft tissue. This test also indicates how well your gastrocnemius (outer calf muscle) and hamstrings work together when your ankle is dorsiflexed (pointed toward your shin) and your upper back is flexed; restriction in the calves and hamstrings when the ankle is dorsiflexed will place additional strain on your Achilles tendon.
**How to test:** Sit with both legs straight on the floor. Lock the knee of the leg you're going to test; keeping your leg straight fixes the hamstring at the knee and pelvis to isolate the gastrocnemius where it inserts at the heel. Loop a strap or towel around the ball of the foot to be tested. Let your thoracic spine (middle to upper back) roll forward naturally and comfortably. Flex your foot toward your shin, using the strap only to gently guide the movement.
To pass this test, your ankle should be able to flex 20 degrees toward your shin, and your upper body should be able to flex 45 degrees forward.
Stretching the Achilles and calf muscles.
**How to improve:** Even though the test might seem focused on the Achilles tendon, "what we need here is for the entire gastrocnemius/Achilles lever to relax, reset, and lengthen," says Wharton. Therefore, do the following stretch, which targets both parts of the lever. It's essentially the test exercise, but done gently and repeatedly to gradually lengthen the tissue.
Sit with both legs straight. Loop a strap or towel around the foot of the leg to be stretched and grasp both ends with your hands. Use the muscles in the front of your lower leg to flex your foot toward your knee. Use the strap only to assist at the end of the movement to get an additional slight stretch. Hold the stretch for 2 seconds and return to the starting position. Exhale as you stretch, and inhale as you return the foot to the starting position. Do 10 reps on each side daily.
### **TEST #2:**
#### **B IG-TOE (HALLUX) DORSIFLEXION**
Testing big-toe dorsiflexion.
**Why test:** An inability to move your big toe toward your shin can be an indicator of a tight plantar fascia, says Wharton. Poor big-toe dorsiflexion limits your ability to roll through smoothly to toeing off and can make your foot rotate enough to cause lower-leg injuries.
**How to test:** Sit with your knees bent at 90 degrees and your feet flat on the floor. Slide your hips forward so that your knees are slightly ahead of your toes. Reach down and grab your big toe while keeping the ball of your foot on the floor.
To pass this test, you should be able to raise your big toe 30 degrees off the floor without the ball of your foot coming off the floor.
Loosening the plantar fascia.
**How to improve:** Use massage rather than stretching to loosen the plantar fascia. Sit with one leg crossed over the other, with the outside ankle of the foot to be massaged on the other knee. Press into the bottom of your foot with your thumbs. Wherever you feel a sore spot, press down for a few seconds while flexing your toes up and down. "The massage needs to be focused pressure where you feel the microbundles of fascia release," says Wharton. Spend a few minutes per foot daily.
### **TEST #3:**
#### **B IG-TOE (HALLUX) ISOLATION**
Testing big-toe isolation.
**Why test:** About 85 percent of foot control comes from the big toe, Dicharry says. If your big toe can't operate independently, your foot can't properly adjust itself during the stance phase (between landing and toeing off). An unstable arch will transmit that instability up your leg. "Like the glute is the big push muscle in your hip apparatus, the big toe is the big extensor flexor for your lower leg," Wharton says.
**How to test:** Stand tall but relaxed. Keep all the toes of one foot on the floor. On the other foot, press the big toe into the floor and raise your other toes while keeping your ankle stable.
To pass this test, you should be able to keep the big toe flat on the ground (instead of bending it) while you raise the other toes, and your ankle shouldn't roll in or out.
Increasing toe extensor and flexor flexibility.
**How to improve:** Increase the flexibility of your toe extensors and flexors. Sit with one leg straight and the other bent at 90 degrees. Grab the toes of the bent leg while keeping the heel on the floor. Curl your toes away from your body, using your hand only to assist gently at the end of the motion. Bring your toes back to the starting position and then curl them toward your body, again using your hand only at the end of the motion. That's 1 repetition. Do 10 repetitions on each foot daily.
### **TEST #4:**
#### **A NKLE INVERSION AND EVERSION**
Testing ankle inversion.
**Why test:** If your ankle can't move adequately toward the midline of your body (inversion) and away from the midline of your body (eversion), "you won't utilize your arch's shock absorber or spring to withstand the impact of footstrike," says Wharton.
**How to test:** Sit with one leg bent at 90 degrees and the outside ankle of the foot to test resting just above the opposite knee. While holding that foot on the outside forefoot, rotate it inward and point the sole of the foot up.
Now bring that foot up so that it's just in front of your butt. From the ankle, rotate the foot outward and away from your body's midline.
To pass the first test, you should be able to rotate your foot inward 15 degrees. To pass the second test, you should be able to rotate your foot outward 5 degrees.
**How to improve:** Do inversion/eversion walking. First, walk like a pigeon: Turn your feet in toward each other at about 45 degrees while not bending your knees. Taking long strides, walk for 20 yards, staying on the outside of your arches as much as possible. Stop and turn around. Now walk like a duck: Turn your feet away from each other at about 45 degrees while not bending your knees. Taking small strides, walk back the 20 yards to your starting point, staying on the inside of your arches as much as possible. Do two or three of these circuits daily.
Testing ankle eversion.
Walking like a pigeon to build ankle inversion.
Walking like a duck to build ankle eversion.
### **TEST #5:**
#### **S INGLE-LEG BALANCE**
**Why test:** Imbalances and weaknesses in your hip and trunk areas introduce instability into your gait and require your feet and lower legs to absorb more impact forces than they should.
**How to test:** Stand tall but relaxed with your hands on your hips. Lift one leg so that its foot is slightly below the other knee. Keep the heel and the inside and outside of the ball of the foot standing on the floor. Repeat with the other leg.
To pass this test, you should be able to hold the raised-foot position for 30 seconds while keeping your weight evenly distributed over your foot. Your leg and upper body should remain still.
**How to improve:** Practice the test a few times a day until you can pass it. Once you can hold the position for 30 seconds, try doing it with your eyes closed. You'll be in for a treat!
Let's say you're one of the biomechanically blessed and have passed all the tests. Now you need to decide what shoe to start your transition in, and how to go about safely incorporating it into your running.
### BODY IN BALANCE
**While doing these tests,** pay attention to whether one side of your body performs better than the other. It's likely that if you can pass a test on one side of your body, you'll be able to on the other side. But you still might be surprised by the differences, which are evidence of imbalances you would do well to address independently of these tests. For example, I do much better on all these tests on my left side. It's no coincidence that almost every lower-leg injury I've had in the last 3½ decades has been in my right leg.
## HOW MINIMAL A SHOE TO START?
As we saw in Chapter 5, the phrase "minimalist shoe" covers a wide range of options among features like heel-to-toe drop, height of midsole, and amount of cushioning. Deciding which minimal shoe to use when starting your transition begins with the general questions noted in Chapter 5, such as whether you like a wide or narrow toebox. Once you've found some options within the broad characteristics you want, then you can decide how minimal a shoe to go to compared with your current one.
There are no hard-and-fast rules here. If you passed the five tests above, you could probably go right to as barefoot-style a shoe as you want, assuming you listen to your body during the transition. If your performance on the tests was shakier, you should probably consider a more gradual step down the shoe spectrum.
Another factor is your running history. If you've been running for only a few years and did well on the tests, you can probably handle a bigger jump down than someone who's been running for 15 years in nothing but conventional training shoes. Your body won't have adapted as much as the veteran's to the altered mechanics caused by conventional running shoes. The rationale here is similar to most experts saying young runners can get away with running in less shoe because they're starting with a cleaner biomechanical slate, as we'll see in Chapter 8.
Also consider your typical training. If you do almost no hard workouts or races, where your feet and lower legs work through a fuller range of motion and generate more force, you're probably not ready for as minimal a shoe as someone who regularly does faster running. Similarly, if you're used to wearing racing flats or lightweight trainers for races and hard workouts, your Achilles, plantar fasciae, and calf muscles will be better prepared for a more minimal model than those of someone who wears more built-up training shoes for everything.
In terms of shoe features, focus on the ramp angle, or the difference in height from heel to forefoot. A good rule of thumb is to try a minimalist shoe with a ramp angle that's about half that of your current model. To take just one example, say you know you like Asics, and have been in a traditional model like their Gel DS Trainer. It has a reported 10-millimeter heel-to-toe drop. You could try their Asics Gel Hyperspeed, which has a reported 5-millimeter heel-to-toe drop.
Of course, it's possible to pull off a greater jump in ramp angle. In those cases, walk before you run to see how conservative a transition you should attempt. "If you're gung-ho on going right to something like the FiveFingers, spend a day walking around in your new shoes," says coach Jay Johnson. "For a lot of people, your feet are going to get sore afterward. That'll give you an idea how much strength they need for you to run well in them."
### WEIGHT FOR IT
**In addition to your performance** on these tests, consider your weight when deciding how aggressively to transition to minimalism. Experts like Jay Johnson, who offers online coaching advice for Nike, and Phil Wharton agree that if you're not at a good running weight, you should take a more conservative approach while working to get your weight down. As Wharton says, "If you're not at your optimal weight, there's so much extra stress on the joints, the 26 bones in the foot, all those little muscles down there."
## A CONSERVATIVE TRANSITION PLAN
For almost everyone transitioning to minimalism, slower will get you there faster. Gradually integrating minimal shoes into your running will allow your body to adapt to the new stresses better than plunging right in. A month into the transition, you might not be running as much as you'd like in your new shoes, but 3 months in, you're more likely to still be progressing, and running healthfully and happily, than if you switch over too quickly. What's the rush? You have the rest of your running life to make the transition.
What constitutes a gradual transition is going to vary by runner. There are so many factors that will affect how you respond to running in minimalist shoes, including your strength and flexibility, running history, mileage, injury history, running surface, and more. This is definitely an area where the adage "we're all an experiment of one" is true. Consider the following strategies while conducting your experiment:
* **Start by walking.** This is something you can do from day 1. Wear your new minimalist shoes as daily footwear, as your professional responsibilities allow. The impact forces of walking are about one-third those of running, so your risk of injury is next to nothing. Spending a week walking in minimalist shoes will prepare your feet and lower legs for when you start running in them.
* **Go barefoot around the house.** Why wear shoes inside? Strengthen your feet and keep the floor clean at the same time! This is a good habit to stay in for the rest of your running life, regardless of what shoe you run in, because of the foot-strengthening benefits it provides.
* **When you start running in minimalist shoes, wear them at most every other day for the first 2 to 3 weeks.** "Fascia takes 48 hours to recover," says Wharton. Fascia is connective tissue that envelops muscles, bones, nerves, blood vessels, and organs. Among other roles, healthy fascia enables smooth movement of muscles by reducing friction. Tight fascia inhibits good running form and can lead to compensatory imbalances. You don't want overly tight fascia at any time in your running career, but especially when you're adding the new stress of running in minimalist shoes.
* **Start with just a mile or two at a time.** Logistically, this can be accomplished in a couple of ways. You can do the bulk of your run in your old shoes and wind up back at your starting point with a mile or two to go. Then change into your minimalist shoes for the remainder of the day's mileage. If you do workouts involving a warmup and cooldown, you could start by wearing them for one or both of those bookends, says Fullem.
* **Consider striders.** Striders (fast but relaxed runs of about 100 meters at the pace you can hold for a mile) are an essential part of most competitive runners' training. They help to build and maintain basic speed and are a great opportunity once or twice a week to concentrate on good running form. Wearing minimalist shoes for striders after an easy run helps you get used to the shoes in small doses while neurologically associating the shoes with good running form from the start.
* **Increase on a weekly, not per-run with basis.** Whatever amount of running you start, in your new shoes, hold at that level for at least a week. Then increase the next week only by whatever your original amount was. For example, if in the first week you ran in minimalist shoes for a mile at a time, go up to no more than 2 miles at a time the second week, 3 miles at a time the third week, and so on. Eventually you'll reach a time or mileage amount in your minimalist shoes that's at the low end of your range of regular runs, such as an easy 4-miler the day before your weekly long run.
* **Gradually integrate minimalist shoes into harder running.** If you regularly do track sessions, tempo runs, and other types of faster running, and don't already wear racing flats for these workouts, take the same conservative approach here as with mileage. If you're doing a track workout, you could quickly change shoes before the last one or two repeats. If it's a tempo day, you could break the tempo run into two or three segments with just enough rest between to change shoes. If you're doing a hill workout, you could put on your minimalist shoes for the last few charges uphill. Then, as with mileage, gradually increase the number of repeats or total amount of time spent running hard in minimalist shoes.
* **Always be ready to take a step back.** "If at any point you feel some discomfort that's beyond your muscles being sore, back off," says Peter Larson, author of the popular minimalist blog Runblogger. If you're an experienced runner, you probably know how to distinguish between acceptable niggles and pain that merits attention. Most muscular soreness should get better with gentle running. If you have a transition-related ache that doesn't improve once you're warmed up, then stop running in your minimalist shoes. Return to your regular shoes until the pain goes away, and start again from scratch.
## A RADICAL TRANSITION PLAN
The above conservative transition plan is for the vast majority of runners. A more radical mode is that advocated by Mark Cucuzzella, MD, owner of a minimalism-focused store in Shepherdstown, West Virginia. This plan entails immediately doing all your running in minimalist shoes but starting with, well, minimal mileage, such as a mile at a time. From there you gradually build to 2 miles per run, then 3 miles, and so on, until you're back to your normal mileage in a few months. In common with the conservative transition plan, here you also spend as much of your nonrunning time as possible barefoot or in zero-drop casual shoes.
This more extreme program makes the most sense for more desperate runners. It's what 2:37 marathoner Camille Herron, whom we met in Chapter 2, did after suffering seven stress fractures in a few years. The thinking here is that if you're chronically injured in regular running shoes, you need a more dramatic change in your routine. Once you're over your latest injury and are ready to resume running, make minimalist shoes an integral part of your new running life from the get-go, Cucuzzella advises. Of course, you don't have to be returning from injury to take this approach, but few healthy runners are willing to go cold turkey on mileage simply for the sake of adapting to new shoes.
Counterintuitive though it might seem, the best shoe for this approach is often a more minimal model than when transitioning gradually. You're essentially starting your running program from scratch. So your mileage will be quite low for some time, and your body should be able to handle the small amount of running you're doing even in a zero-drop, minimally cushioned model. With that safety factor built in, it makes sense to attempt to rewire your running mechanics in a shoe that allows for the most natural gait.
## COMMON INJURIES WHILE TRANSITIONING TO MINIMALISM
Most runners are going to feel muscular aches in the early days of their transition to minimalist shoes. Unfortunately, some will experience more acute injuries, despite carrying out what seemed like a conservative plan. As Wharton explains, those minimalism-specific injuries are predictably from the knee down.
"As the heel has to drop, you're going to utilize a lot more of the calf unit, the gastrocnemius and soleus," he says, "and you're stretching the Achilles a little bit more, which is great long-term, but at the beginning it's really tough because you're not getting help from other muscles in close proximity on the kinetic chain. There's a lot more responsibility on these lower-leg muscles. So we get a lot of posterior tibial problems, tendinitis and ligament strains, peroneal tendon pain, even fractures."
Here are the three most common injuries runners encounter while transitioning to minimalist shoes, and what to do about them. In all these situations, once you have the acute phase under control, reboot your transition to minimalism by starting from scratch instead of diving back into where you were when you got injured.
### Plantar Fasciitis
The plantar fascia is a thick, fibrous band of connective tissue that supports the arch. Running in a shoe with a lower heel and running with more of a midfoot strike require the plantar fascia to absorb more impact forces. "The arch was designed as a spring, and if that spring isn't strong and flexible, then you're not going to be able to translate the shock of running to a specific mechanical change," says Wharton. Many runners have tight, weak plantar fasciae from decades of wearing shoes with heels, both while running and in daily life. A dramatic increase in the plantar fascia's workload can lead to almost immediate injury.
You'll feel a sharp, tearing pain along the inside bottom of your foot anywhere from the heel through the arch. Many times, the pain is the worst when you step out of bed in the morning or when you've been sitting for a long time; it tends to lessen with mild activity but then be present after you run.
You can usually run through plantar fasciitis, but you need to protect the fascia while it's inflamed. That means returning to conventional shoes until the pain subsides. Rolling your foot along a glass bottle you keep in the freezer provides a simultaneous icing and massage. Anti-inflammatories such as ibuprofen can also help to calm the fascia.
### BUILD YOUR RUNNING BODY
**During your transition—** and for the rest of your running life, for that matter—be dedicated about maintaining and increasing functional strength and flexibility. Regardless of how you performed on the five tests earlier in this chapter, do the "fix" exercise from each test a couple times a week. In addition, see "The Minimalist's Maintenance Tool Kit" in Chapter 8 for a series of simple but highly effective exercises you can do anywhere to stay a healthy minimalist runner.
### Achilles Tendinitis
The Achilles tendon is key to running with a midfoot landing and push-off. Like the plantar fascia, in most modern runners the Achillies has been shortened and weakened by elevated shoes for running, work, and leisure. So, like the plantar fascia, it's easily overloaded if you suddenly start running in shoes with a minimal ramp angle. Bloodflow to the tendon is relatively poor; the tendon is slower to warm up than some other key running body parts.
Like plantar fasciitis, Achilles tendinitis produces a sharp, tugging sensation, in this case from the back of your heel up to the bottom of your calf muscles. In severe cases, you'll be able to see swelling. Unlike with plantar fasciitis, the pain tends to get worse, not better, with running.
Icing and anti-inflammatories can help to reduce the inflammation. Mild stretching—never to the point of producing pain—can help increase bloodflow to the tendon. You can try to run through Achilles tendinitis in your conventional shoes, but unless you want it to drag on forever, you'll need to lower your mileage dramatically, do nothing but run slowly, and avoid hills as much as possible. If the tendon is so aggravated that it's noticeably larger than your healthy one, you're usually better off not running until you get the most severe inflammation under control.
### Metatarsal Stress Fractures and Stress Reactions
Running in minimalist shoes often reveals underlying weaknesses throughout the body. In many runners, says Wharton, "a lot of the muscles that are supposed to be doing their job, like the glutes, hamstrings, hip rotators, and iliopsoas, aren't doing their jobs. This tends to lead to a harder footplant." Conventional running shoes tend to accommodate this form flaw better than minimalist shoes, thanks to plush cushioning. With less material between you and the ground, running in minimalist shoes puts more of the impact force from bad form on the bones of your feet, especially the metatarsals, the five long bones running from midfoot to the base of the toes.
Stress reactions are precursors to stress fractures. Initially you'll feel a dull ache over a small area in the front or middle of your foot. As with most stress reactions and fractures, pressing on the sore spot will likely cause pinpoint pain. Although you may have heard that you can't run on a stress fracture, we runners are a tough, dedicated bunch, and it's possible to run on a fractured foot, at your normal mileage, no matter how ill-advised doing so is. If the pain is worse after you run, then you're well along the path of a stress reaction becoming a full-blown fracture.
Although you can run on a stress fracture, you shouldn't. There's no finessing your way through the process of getting a weight-bearing bone to heal. Metatarsal stress fractures usually require 1 to 2 months of no running for proper healing. If you've caught the problem early (you can produce pinpoint pain by pressing, you feel it when you run, but it's not worse hours after a run), then you can get away with less time off. But as with transitioning to minimalist shoes, a little more conservatism in the short run can lead to fewer setbacks a couple of months down the road.
### How Low Should You Go?
Let's say you've successfully transitioned from a conventional training shoe into one of the gateway minimalist models like the Saucony Kinvara, or even something more minimal. Should you keep moving through the minimalist spectrum to a lighter, lower shoe? Should everyone's goal be running solely in barely there shoes or barefoot?
As with so much in running, there are no universal answers here. Start by remembering that minimalist shoes are a means to the goal of better running form. They're part of a tool kit that, ideally, also includes core strength and functional flexibility, training at a variety of paces and being at a good running weight, and other factors that contribute to being a healthy, efficient runner. We can all run with better form in whatever shoes we have. Focusing only on the ramp angle and stack height of your shoes is, frankly, being irresponsible. "You can't just get to a certain point and say, 'Okay, I've got these shoes that are a 6-millimeter drop, I'm running a little better, but I just don't feel like doing drills anymore,'" says Brian Metzler, coauthor of _Natural Running_. "You've got to keep it up and do everything across the board."
Minimalist blogger Larson says, "I wouldn't say, 'You're in the Kinvara now, so you have to go down to something like the New Balance 1600 [a light racing flat].' I think that's what some people do. They feel like, 'I have to continue this progression until I'm running in nothing.'
"That doesn't have to be the end point. When the Kinvara came out, I was talking with one of the guys from Saucony, and he was telling me they have this internal debate about that—what do you tell people? Is this an end-point shoe or a gateway shoe? My response is it could be either. You need to decide that for yourself.
"Initially my own goal was 'Maybe I should be doing all my running in something like the Vibram FiveFingers,'" Larson continues. "I've come back to the realization that wearing a shoe with a little bit of cushioning is not an evil thing if it allows you to run the way you want to run."
That last point is key. Just as there are other aspects of running with good form than your shoes, there are other aspects of running than your form. "Of course running form is important," says Joe Rubio, who coaches several national-class runners in California. "But to think about it all the time . . . sometimes you just want to go for a run, you know?"
"That's especially true on the trails," says Metzler, who was the founding editor of _Trail Runner_ magazine. "Depending on the trails and shoes, you can run as nimbly as possible but still feel every pebble, stalk, or notch on the trail. To go through a whole run like that isn't always the most enjoyable thing to do.
"But even on the roads, there are a lot of obstacles out there, whether it's stepping off a curb or on a pebble," Metzler continues. "Do you want to have every step of the way be this aware, eyes-on-the-road thing, or do you just want to run and zone out? I think for most runners there's a happy medium where you're in a shoe that promotes good form but also offers enough protection so that most days you're just running and not thinking about your shoes and form the whole way."
If, for most runners, the transition to minimalist shoes doesn't necessarily mean running in as little as possible, where does that leave barefoot running? Is there a role for running without shoes in most runners' programs? The answer is yes, and that's the subject of the next chapter.
### **MEET A MINIMALIST**
### **ADRIENNE LEIGH MENDENHALL**
### **SINGAPORE**
An American living in the city-state of Singapore, Adrienne Leigh Mendenhall notes, "There is definitely a strong following for Vibram [FiveFingers] over here, both on roads and trails." She hopes to eventually be one of the Singapore runners wearing them, but is working toward that goal gradually after her first go-round with the shoes was too sudden and led to setbacks.
Mendenhall began running as a high school student in the mid-1990s. She wore conventional shoes, usually beefy models like the Asics Kayano, for the next 15 years. After reading _Born to Run_ , in 2010 she bought a pair of FiveFingers in the hope that they would help with hip and knee pain that had been plaguing her for years. Although she's a marathon veteran, including in the soupy climate of Singapore, Mendenhall's main motivation is to be able to run for years. She thought a switch to Vibrams would help.
"I started out slow, with 1 mile at a time," she says, "built up to 7-mile runs with a tiny bit of fifth metatarsal pain, but figured if the pain didn't get worse, then it would eventually get better. Smart? No. Typical impatience of a runner? Yes."
At the time, she was living in Minnesota, and one icy night she opted for the treadmill. Figuring that the need for the slight surficial protection that Vibram provides on roads wasn't needed on the treadmill, she ran barefoot. As Mendenhall puts it, "Inaugural barefoot running plus treadmill boredom—playing with speed and incline—equals ouch." She had strained a tendon or ligament in her already-hurting foot. "So, we can chalk up the next 2 months of sedentary wistfulness to stupidity," she says.
When Mendenhall's foot got better, she returned to conventional Asics, the 2150, and, she says, "decided to approach minimalist running with a logical plan. I've been stepping down the cushioning in each pair of running shoes." From the Asics 2150 she went to the Brooks PureFlow while using the Asics Sky Speed for long runs. About the latter, she says, "Frankly, I detest their weight and bulk, but I'm still hesitant about lightweight shoes and the potential for injury if my feet and ankles aren't ready."
Then Mendenhall bought the Merrell Pace Glove and began to work them into her training conservatively, starting with just a few 3-milers. "I'll slowly add distance to these short runs until they equal the distance of a normal run," she says. If as she adapts she feels like the difference between the fairly cushioned Brooks and barely there Merrell is too great, she's open to a shoe that lies between the two to aid the transition.
And what about those FiveFingers so many others in Singapore run in? "I haven't put them on for over a year but am getting antsy to give them another shot," Mendenhall says. "I liked running in them before, and the goal is to eventually run regularly in them again. But first I need to be pain-free."
# **CHAPTER**
# 7
# REASONABLE BAREFOOTING
## The theory and practice of running without shoes
**ONE DAY IN THE EARLY 1980s,** I found myself at my sister's house without running gear. For reasons I can't recall, there was only this window of time that day when I'd be able to run. So I borrowed a T-shirt and pair of tennis shorts from my sister and ran barefoot on the roads for an hour. I remember getting stares. Whether that was because of the lack of shoes on asphalt or the women's tennis shorts, I can't say. Certainly I was an anomaly.
The thing is, I would be an anomaly today as well, even in the most staid running attire. For all you hear about barefoot running, few runners regularly do it.
That's a shame. Running barefoot has many benefits, including simply feeling good when you can find a suitable surface. In Chapter 4, we saw how most runners who are used to going unshod adopt a midfoot strike, compared with the heel strike most runners in shoes use, and that barefoot runners don't overstride as often as shod runners do. We also saw that most runners have better running economy when barefoot.
This chapter is about the intersection of barefoot running theory and modern running. We'll start by looking at an elegant hypothesis by Harvard anthropologist Daniel Lieberman on the role of (barefoot) running in human evolution, then move on to practical recommendations for regularly adding barefoot running to your training.
## THE RUNNING MAN THEORY
If you've read _Born to Run_ , or even read about _Born to Run_ , you know that Lieberman thinks that running was central to our species' development. Call it the Running Man theory. Briefly stated, it goes like this.
Humans are better than other mammals at dissipating heat. Running in the heat of the day, our African ancestors were able to chase game until the prey collapsed. Because the most successful subsistence hunters were the most likely to survive and pass on their genes, natural selection favored proficiency in long-distance running. At the same time, the meat from the animals provided enough high-quality protein to stimulate brain growth, which further separated humans from other primates. The next thing you know, humans had spread throughout the world and invented the Internet, where people can spend all day debating what shoes to run in. Running helped to make us human.
The news-you-can-use corollary to the main theory is that most of this running occurred barefoot on hard, uneven surfaces. Even once humans started running in footwear, it was minimal footwear, from the Paleolithic Era of 45,000 years ago until the invention of the modern running shoe in the 1970s. So humans must be well adapted to running long distances barefoot. Modern running shoes are the aberration; running barefoot is the default. The form that humans use when running barefoot is the natural way to run and will lead to less injury, proponents say. The best way to run with that form, obviously, is to run barefoot.
## SOME "YEAH, BUT . . ." THOUGHTS
The evolutionary part of the Running Man theory is wonderful. We've all had runs when everything's clicking and we feel the elemental rightness of what we're doing. And of course it's flattering to hear our beloved sport posited as integral to human progress.
Lieberman, who does a lot of his running barefoot, hasn't said the Running Man theory means everyone should run barefoot. Others have, and hold up Running Man as the ideal to which all runners should aspire. Here are a few points to consider about how Running Man might differ from his 21st-century counterparts.
**Evolution continues.** Modern humans are thought to have started branching out from Africa 125,000 years ago. UCLA professor Jared Diamond, author of the 1997 bestseller _Gun, Germs, and Steel_ , notes that food production began to replace the hunter-gatherer mode of existence 11,000 years ago. These are, of course, blips on the timeline of human evolution over a couple of million years. But whatever evolutionary selection may have once occurred in favor of barefoot running, it hasn't been nearly as significant in most of the world for hundreds of generations now. Human evolution didn't stop at some random date in the past; natural selection has favored traits other than running aptitude, in environments other than the African savanna, for quite a while now.
**Experiment of one.** When Running Man adherents wonder about injury rates, that seems partly based on the notion that we should all be able to run without getting hurt. But we have no way of knowing how often Running Man was injured, nor do we know if everyone was a regular hunter. Perhaps only the Mo Farahs of the day were sent out to hunt, and as the meat providers, they had the highest status and most often passed along their genes. (The best distance runners having alpha status—talk about differences with modern runners!)
There's a difference between what the species has evolved to do and what any one member of the species is capable of. It's pretty easy to think of ways in which aptitude in spatial relations would have an evolutionary advantage, yet you're reading a book written by someone who didn't learn to tie his shoes until third grade. Running with good mechanics is another evolved trait that some individuals lack. That should be increasingly true over time when that trait is no longer being worked on by natural selection.
**Gender of the runner.** Although we don't have demographic info on the Great Antelope Hunt 10-Miler of 20,000 BC, it's logical to think that all or nearly all the participants were men. (Young men, for that matter. More on that in a bit.) In 2011, 53 percent of finishers in US half-marathons were women. This is as it should be. But women have wider pelvises and less muscle than men and can have different running mechanics.
**Age of the runner.** How to spend his senior years wasn't a pressing issue for Running Man. Skeletal evidence from Neanderthal times through the beginning of the Common Era (i.e., what we call year 0) indicates that the mean human life span was about 30 years. The median age of today's _Runner's World_ subscriber is 41. People in their 80s regularly finish marathons. With age come physiological changes, such as loss of muscle mass, that can affect one's running mechanics.
**History of the runner.** In Running Man days, running would have been a regular part of one's life soon after becoming bipedal. Many modern runners do the first half-hour run of their life in their 30s, 40s, or older. This is quite different from the state that the Running Man theory suggests is the default human condition.
**Size of the runner.** Even as recently as the late 19th century, the average American looked different than today. The average Union soldier in the Civil War is estimated to have stood 5′8″ and weighed 143 pounds. Running Man was almost certainly smaller than that. Sport podiatrist Kevin Kirby says, "When the average adult male runner now weighs 185 pounds versus probably the average male weight of 125 pounds thousands of years ago, the injury rates during running will naturally increase, regardless if they were barefoot or in shoes. This seems so obvious to me both from a biomechanical modeling aspect and from my experience as a clinician, but none of the barefoot advocates tend to mention body weight as being a very significant factor in injury production."
**Road versus savanna.** An obvious and almost always immediate objection to the Running Man theory is some form of "Our ancestors didn't evolve running on asphalt." True, but that doesn't necessarily scuttle the theory. Lieberman likes to point out that the savanna on which Running Man did his running was hard, not soft like grass or sand. We should also keep in mind the fascinating finding from Chapter 4 that we make adjustments to account for the firmness of the surface we're running on.
At the same time, it should be noted that Lieberman's assessment of the ancestral running surface is based on his time running in Africa, not durometer readings of ancient soil. Like most of the longtime Western runners who've run in Kenya, I think modern American roads are harder than the African earth. (I'm not counting Kenya's clay roads, which obviously didn't exist in Running Man's time.) I did barefoot striders on random Kenyan ground with what felt like normal mechanics. If I were to do them on my street, I would run them with slightly different form. Running comfortably and injury-free on asphalt is eventually within most people's capacity, but it will probably look a little different than running barefoot on a natural surface. And this leads us to perhaps the major caveat concerning the Running Man theory.
**Ancient versus modern running.** Running as would have occurred in the evolutionary setting is significantly different from the running that most of us do. Running Man didn't do track workouts early in the morning before heading off to work. He no doubt ran long and fast at times, but he ran only when he had to, and only as long and hard as was necessary. He didn't follow 16-week schedules leading up to a marathon. Put another way, running is a natural activity. Training to run 26.2 miles as fast as you can on the roads, not so much. The Running Man theory asks why so many people get injured despite what variables are present in their running. Training for excellence is a variable that seems an obvious means of injury.
As for Running Man and racing, he didn't do it like we do. In terms of mechanics, remember from Chapter 4 that most runners shorten their stride and increase their turnover when they go from shod to barefoot. For well-trained runners, this could be a performance limiter at faster speeds. When you run markedly faster than your normal training pace, your turnover increases. At some point you'll bump up against the upper limit of cadence you can sustain for the duration of a race. If, at the same time, you're doing something (running barefoot) that shortens your stride length, then you won't be running as fast as you could. When asked about barefoot running, two-time Olympic marathoner Ryan Hall has said, "The best guys in the world are wearing shoes and we're running fast. [When] some barefoot dude comes by me at 26 miles, then maybe I'll look into it."
If the Running Man theory inspires you to explore daily barefoot running, go for it. But don't feel subhuman if you decide to wear something on your feet for most of your runs.
### **MEET A MINIMALIST**
### **GREG DIAMOND**
### **CORTLANDT MANOR, NEW YORK**
Greg Diamond is a great example of a longtime runner who's worked principles of minimalism into an established, successful running program. Now in his mid-50s, he's running faster on an age-graded basis than he was 15 years ago. A few years ago, he was the New York Road Runners Runner of the Year in the 50-to-54 age group. He credits part of that accomplishment to running more in less shoe, supplemented by small doses of barefoot running.
Diamond started doing short barefoot stints simply because "it just sounded right to me," he says. "I believe that we have weakened our feet with footwear." He began extremely conservatively, running just a 10th of a mile on a treadmill the first time, and increasing by another 10th of a mile every other run. "I knew that doing it very gradually had to be the only way," he says. "I knew better than to run a mile first time out."
These days he does about 4 miles a week on the treadmill—some barefoot, the rest in the ZEMgear Ninja Split Toe High, a 2.3-ounce, slipperlike, ankle-high shoe that provides him abrasion protection on the moving belt of the treadmill. ("I tried the Vibram FiveFingers, an early version, and found them too clunky," he says.) He also runs in one of the lightest racing flats on the market, the Mizuno Wave Universe, and has worn the Saucony Kinvara for lots of long runs and a marathon. In all, Diamond estimates that he does 20 to 35 percent of his mileage, which can get as high as 100 a week, barefoot or in minimalist shoes.
Although Diamond started his barefoot experiment for its own sake, "I realized how strong my feet were becoming, how I was switching to forefoot/midfoot strike even in regular shoes, how this was translating into less lower-leg soreness in races," he says. "Now my major motivation is to keep my feet strong and my stride healthy as I get older. I ran a half-marathon in the Mizuno Wave Universe with no soreness at all. I remember running a half-marathon in a shoe twice as heavy at 41 and barely walking away from it. It is so amazing to me that I can run all these hard miles in 3.5-ounce shoes with no cushioning."
Diamond's transition was eased by years of good habits in his nonrunning hours. "I have always walked around barefoot or in socks," he says. (His wife calls his practice "socks as shoes.") Also, as a longtime competitive runner—he ran a 2:41 marathon in 1995—he was used to running in racing flats. So his feet and lower legs were stronger than average and readier than most people's when he shifted to more minimalist and barefoot running.
## BEST USES OF BAREFOOT RUNNING
I could go out today and, more than 30 years after doing so from my sister's house, run for an hour barefoot on the roads with no issues. I just don't want to. For me, watching every step for pebbles or glass or other common features of American roads detracts from enjoying the run.
You, of course, should do what you want in this regard. You'll quickly learn your preferences for where—or whether—to do everyday runs barefoot, and in what range of conditions. Here in Maine, I've run barefoot on grass as early in the year as March and as late as November. There's a regular barefoot-on-roads runner in my area whom I've seen running in December. There's great variability in runners' tolerance for cold, and that includes when running barefoot.
But let's step back and assume that, like most people, you haven't run barefoot in a long time. What's the best way to get started?
Begin by keeping in mind the overall message of my nitpicky points from above about the Running Man theory. Modern runners are fairly far removed from the ancestral setting. As sport scientist Ross Tucker puts it, injury-free barefoot running should be considered a skill. As with all skills, some will acquire it almost immediately, many will acquire it with more practice, and some might never acquire it. (The last group might be those who heel-strike when running barefoot and overload the plantar area.) Whatever caution you may exercise when moving from conventional running shoes to minimalist models, be even more conservative when you start experimenting with barefoot running.
Your first barefoot runs should be quite short, in the neighborhood of 5 minutes of easy running. Logistically, you could do this by starting your run from a playing field, heading out on the roads for the bulk of your run, then returning to your starting point and shedding your shoes for the last few minutes. Because there's such great variance in acquiring the skill of barefoot running, universal rules aren't possible. But again, err on the side of caution. Give yourself a few days to a week until you try again. Add in small increments—think minutes, not miles. Respect the difference between a little tenderness the day after and pain during or after.
Easy running on the infield of a track is a safe, convenient way to introduce barefoot running to your body.
### Cooldowns and More
If you do regular track workouts, experiment with doing some of your cooldown barefoot on the interior of the track. Elite coach Steve Magness says, "I like the barefoot cooldown as a way to introduce barefoot running. It's going to be a little lower impact because you're going pretty slow after a workout. It's an easy conditioning and strengthening tool for the feet."
Jay Johnson, who used to coach a group of national-class runners in Boulder, Colorado, also favors barefoot cooldowns. "They all loved the sensation of running barefoot after track workouts," he says about his former runners, who included national champions in indoor track and cross-country. "We would do 10 laps of the interior of the field, which is basically 2 miles of running. They really felt like it strengthened their feet and lower legs, and I think, most importantly, they liked how it feels."
Another common use of barefoot running is strides. If you enjoy easy barefoot running on natural surfaces and want to try to further strengthen your feet, consider barefoot strides on an unrutted field or the interior of the track. Here the logistics would be similar to ending a run with easy barefoot running. Finish a regular easy-to-moderate run at a place where you can safely and comfortably run barefoot, and do 8 to 10 100-meter strides at mile race-pace effort.
Barefoot striders on grass strengthen your feet, teach good running form, and feel great.
"I think the best use for barefoot running is strengthening the arch with strides," says physiotherapist Phil Wharton. "Easy barefoot running, like in a cooldown, is the starting point and segue. But there's nothing like barefoot strides if you can find a great grass surface. It really does help rebuild the feet. Afterward it can be almost the same feeling you get when you do a whole session of foot/ankle towel curls and ankle inversion/eversion exercises with a weighted sock. You can just feel that lever strengthening." (We'll learn all about towel curls and weighted-sock exercises in the next chapter.)
Magness has another use for barefoot strides. "I like them as a feedback mechanism if you're working on form," he says. "Take the shoes off, do some strides, throw the shoes back on and do some strides and you say, 'Oh, okay, there is a difference.' It's a feedback cue. You get the sensation of what it feels like barefoot and then try to recall that on other runs."
As Magness points out, you can't accumulate as much volume of barefoot running doing strides as you can in a cooldown. As you continue to adapt to barefoot running, you can get good barefoot volume and strengthening benefits by doing a Kenyan specialty called diagonals.
A session of diagonals consists of mixing strides and jogging. Envision a rectangular grass surface such as a football field. When doing diagonals, you jog from one end of the goal line to the other, then run fast from that corner to the opposite corner of the other goal line. Then you jog to the end of that goal line, and then run fast from that corner to where you started jogging on the other goal line. Your path will have made an X across the interior of the field, plus solid lines at the top and bottom of the X. For most runners, it will take roughly as long to run fast diagonally across the field as it will to jog from one end of the goal line to the other.
Everyone in Kenya, from milers to marathoners, does diagonals. They're fun, short workouts that help build basic speed and smooth out your running form. Doing them barefoot adds a foot-strengthening aspect. Do diagonals in units of time, not repeats, so that you can concentrate on good running form rather than how many times you've crisscrossed the field. Start with 5 or 10 minutes. That's plenty of time to get the benefits. If you come to enjoy diagonals, consider making the sessions longer and treating them like fartlek sessions that replace a harder workout in your schedule.
A few years ago, Halloween was freakishly warm in Maine. I knew how best to savor the day—I headed to the waterside field where I do diagonals. For 40 minutes I strode and jogged barefoot while the sun set over the Casco Bay. For that brief time, I felt like Running Man. And if you ever want to feel the difference between running barefoot and in shoes, do 40 minutes of barefoot diagonals and then put on your trainers for a 15-minute run home.
Small bits of regular barefoot running can safely and enjoyably become part of most runners' lifelong programs. What else to do to be a healthy, happy minimalist for the rest of your running life is the subject of the next chapter.
# **CHAPTER**
# 8
# MINIMALISM FOR LIFE
## How to stay healthy long-term while running in less shoe
**ALMOST ALL RUNNERS** hope to run for the rest of their lives and to never be injured while doing so. Working toward that ideal is probably your main motivation for running in minimalist shoes. This chapter is about what you can do to marry minimalism with a lifetime of running.
First we'll see how to do some key exercises that help you run with good form in any shoes, but especially in minimalist shoes. Then we'll learn a few drills that will help hardwire improved form into your daily runs. We'll see what to do when not running to best preserve your running body, and consider whether minimalism is right for runners of all ages. We'll end by addressing this question: Should you commit to doing every run for the rest of your life in minimalist shoes?
## THE MINIMALIST'S MAINTENANCE TOOL KIT
"If you want to become a better runner, begin by running better," says Pete Magill, a California coach and the oldest American to break 15:00 for 5-K, which he did a few months before his 50th birthday.
By that Magill means doing things that improve your ability to run more efficiently and with good form. Learning to run well in minimalist shoes is one means toward that goal. But for almost all runners, switching shoes isn't sufficient. Typical Western lifestyles, with lots of time spent sitting and little time spent doing a wide variety of activities that work the whole body through a variety of motions, make many of the muscles needed to run with good form weak and tight.
Put another way, all modern runners should consider regular bodywork integral to their running, not something only elites who have all day to train or people returning from injury do. Consider strengthening key running-specific muscles as both a performance enhancer and a form of insurance. "Any runner, no matter what shoe they wear, if they want to do one thing to try to prevent injury, I would recommend targeted strengthening," says sport podiatrist Brian Fullem.
Minimalist runners are especially good candidates for this work. As we saw in Chapter 6, weakness and/or tightness in key lower-leg muscles and your core can increase your risk of injury when you start running in minimalist shoes. What you need in these areas is functional strength and mobility—that is, strength and mobility to meet the specific demands of maintaining good running form and being resistant to injury. "Sorry, but these aren't your beach muscles," says physiotherapist Phil Wharton. "A lot of them are tiny muscles no one's ever going to see, but that absolutely need to be functioning to run with good form in minimalist shoes."
Although we tend to focus on the lower legs when we talk about the requirements of running in minimalist shoes, core strength—in your hips, glutes, lower back, and abdomen—is vital. Good core strength will help you maintain a stable pelvic position while running, saving your lower legs from getting overloaded by the impact forces of running. If you notice that you start runs with good form when wearing minimalist shoes but then get sloppy after a few miles, that's an indication that you need to improve your core strength.
The following exercises strengthen and improve flexibility in the key running muscles in your feet, legs, and core that contribute to good running form. Do this group of exercises at least twice and preferably three times a week. The whole set won't take you longer than 15 minutes and makes for a nice postrun routine. If you're motivated to go beyond the exercises here, Wharton (whartonperformance.com) and coach Jay Johnson (runningdvds.com) have excellent, extensive resources in the form of books, DVDs, and short instructional videos.
### **W EIGHTED-SOCK SWING**
#### Weighted-sock swings build lower-leg and foot strength.
**Why:** This exercise strengthens your lower-leg muscles and the muscles that support and form your arch.
**How:** Stuff a 1- or 2-pound weight into the toe of a long sock. (Finally, a nonembarrassing use for compression socks!) Place the sock between your big toe and second toe with the weight dangling under the ball of your foot. Make a stirrup by wrapping the end of the sock around the outside of your foot, under your arch, and back up over the top of your foot. Fasten it securely by tying under the top wrap.
Sit on a high enough surface so that the weighted sock doesn't touch the floor, with your back straight, your knees slightly apart, and your lower legs dangling. Point your foot down so that your big toe is pointing at the floor. Sweep your foot up and toward the midline of your body, pointing your big toe to the ceiling. Pause, then sweep your foot back down to the starting position, with your big toe pointing straight down at the floor.
Without pausing, sweep your foot up and away from your body as far as possible. Slowly return to the starting position. Imagine the full movement forming a U in the air. The whole pendulum sweep should take you about 8 to 10 seconds. Do 10 full sweeps (10 in each direction). Repeat on the other foot. Do two sets of 10 for each foot. (For convenience's sake, consider getting enough weights to have a weighted sock wrapped around each foot.)
Over time, you can add more weight to the socks. Don't add so much weight that the sweeping movement becomes strained. It should feel like a good assisted stretch, not a taxing weight-lifting movement.
### **S EATED CALF RAISE**
#### Seated calf raises strengthen the soleus.
**Why:** This exercise strengthens your soleus, the calf muscle that's the main supplier of blood to your lower legs, ankles, and feet.
**How:** Sit with your legs bent at 90 degrees and your feet flat on the floor. Place a weight on top of one knee. The weight should be heavy enough so that you feel the soleus working when you do this exercise, but not so heavy that your calf is sore after. When in doubt, start with a lighter weight.
Keeping your toes on the floor and your knee bent at 90 degrees, raise your heel as far as you can by pushing up with your toes. Take 2 to 3 seconds to go up and another 2 to 3 seconds to come down. Do 10 reps, switch the weight to the other leg and do 10 reps, and then do another set of 10 reps with each leg.
Over time, add more weight.
### **T OWEL PULL**
#### Towel pulls strengthen the arch.
**Why:** This exercise strengthens your medial and lateral arch, especially the medial (inside) arch.
**How:** On a slick surface such as a wood or tile floor, place a towel so that the short end is in front of a chair you're sitting in. Sit straight with bare feet and with your legs bent at 90 degrees. Place your foot on the edge of the towel; throughout the exercise, keep your heel flat on the towel. Using only your foot muscles, reach out with your toes and contract them to grab a bit of the towel and pull it toward you. Concentrate on spreading your toes as you pull the towel toward you. Do 10 pulls, then straighten the towel to the starting position. Do another set of 10. Repeat with your other foot.
Once you've mastered regular towel pulls, make them more challenging by placing a small weight on the middle of the towel.
### **C LAM**
#### Clams strengthen the glutes and hip rotators.
**Why:** This exercise strengthens your glutes and hip rotators. Better strength in these crucial core muscles will lessen common form flaws like splayed feet, an uneven pelvis, excessive lateral rotation, and leaning forward at the waist.
**How:** Lie on your left side with your legs together and bent at a 90-degree angle. Hold your arms together, extended in front of you. While keeping your feet together and your left leg on the floor, lift your right knee as if your legs were an open clamshell. Do the movement slowly—2 to 3 seconds going up, 2 to 3 seconds coming down—rather than rushing through as quickly as you can. Do 10 reps on that side, then switch sides and do 10 with the other leg.
Over time you can make the exercise more challenging by attaching a Theraband around your legs near the knees.
### **K NEE CIRCLE**
#### Knee circles improve hip mobility.
**Why:** This exercise will dramatically increase range of motion in your hip joints.
**How:** Start on your hands and knees on the floor, hands under your shoulders, knees under your hips. Keeping your right knee bent at 90 degrees, lift your right leg and lead with your knee moving forward to sweep the leg in a circular motion. Ideally your knee will come near your hip at the top of the circle. Do 10 forward circles, return to the starting position, and do 10 backward circles. Backward circles will probably be more challenging. Repeat with the other leg.
### **S IDE LEG RAISE**
#### Side leg raises strengthen stabilizing muscles in the hips and butt.
**Why:** In its three variations, this exercise strengthens the small stabilizing hip and butt muscles, especially the gluteus medius.
**How:** Lie on your left side with your legs extended out straight. Your left arm can rest under your head; your top arm can rest on your hip. With your right foot parallel to your left foot, lift your right leg. Keep your hips steady and facing forward. Lower the leg. Take 2 to 3 seconds to go up and 2 to 3 seconds to come down. Do 5 reps with your right foot in this neutral position. Then maintain the original position except turn your right foot out so that the toes point toward the ceiling. Do 5 reps in this position. Finally, maintain the original position except turn your right foot in so that your toes point toward the floor. Do 5 reps in this position. (This last position will be the most challenging for most runners.) Do the same sequence of five reps in each of the three foot positions with the other leg.
When you can do all three positions without your working leg shaking, make the exercise more challenging by wearing ankle weights.
### **D ONKEY KICK**
#### Donkey kicks strengthen the glutes.
**Why:** This exercise strengthens your glutes, thereby enhancing your hip extension.
**How:** Start on your hands and knees on the floor, with your back straight but relaxed. Your hands should be under your shoulders, and your knees should be under your hips. Keeping your right knee bent, kick your right foot back and up so that the bottom of your foot faces the ceiling and then, without pausing, moves toward your back like a hook. Keeping your knee bent, return to the starting position and then move past it so that your knee nears your chest. Do 10 times with each leg.
### **P RONE PEDESTAL**
#### The prone pedestal strengthens deep abdominal muscles.
**Why:** This exercise strengthens your deep abdominal muscles. (You may have heard it and the other pedestal positions that follow called planks.)
**How:** Balance on your forearms and toes on the floor. Your elbows should be bent at 90 degrees and your toes should be slightly dorsiflexed (pointing up). Keep your shoulders, neck, and head stable but relaxed; don't scrunch up your shoulders or place all your weight on your elbows. Keep a straight line from your shoulders to your ankles. Concentrate on keeping your hips level, neither sagging toward the floor nor elevated so that your butt points up. Hold for 30 seconds.
Over time increase to holding for 60 seconds.
### **S UPINE PEDESTAL**
#### The supine pedestal strengthens core muscles along the backside.
**Why:** This exercise strengthens your hamstrings, glutes, and lower-back muscles.
**How:** This is basically the flip side of the prone pedestal position. Balance on your elbows and heels, belly button pointing up. As with the prone pedestal, keep a straight line from your shoulders to your feet. Tuck your chin slightly toward your chest. Hold for 30 seconds.
Over time increase to holding for 60 seconds.
### **S IDE PEDESTAL**
#### The side pedestal strengthens lower-back and abdominal muscles.
**Why:** This exercise strengthens your lower-back and side abdominal muscles.
**How:** Balance yourself on your left elbow and your feet, with the right foot stacked on top of the left foot. Place your right hand on your waist. Your left elbow should be under your left shoulder, with plenty of space between your left shoulder and head. Keep a straight line from your right shoulder to your feet. Don't allow your hips to drop. Hold for 30 seconds, then repeat on the other side.
There are several ways to make this exercise more challenging. First, increase to holding for 60 seconds. Then switch to balancing on your hand instead of your elbow. In that position, the closer you keep your top arm to your body, the more difficult the side pedestal will be. Finally, you can progress to doing side pedestals with your eyes closed.
### Running Form Drills
Most elite runners do drills at least a couple of times a week. Most recreational runners don't. That's a shame, because devoting just a little time each week to these exercises has so many benefits—you'll recruit muscle fibers needed to run with your best form, correct muscle imbalances, become accustomed to moving through a fuller range of motion, and retrain your nervous system. These gains will help you run with better form at all speeds.
Runners moving to minimalist shoes are especially good candidates for drills. Remember, minimalist shoes are a means to the end of running more efficiently and effectively. Drills are another means toward that goal of improved form. Refining your running body with drills will make it easier to hold the improved form you're hoping to achieve by running in minimalist shoes.
Do drills at least once and preferably twice a week. When you're new to drills, do them after one of your easier runs of the week. As you get used to doing drills, you can do them pretty much any day. Many competitive runners include drills as part of their warmup before races and hard workouts, to help prime their bodies to operate at peak capacity.
You can do drills anywhere you have a flat, level surface for 30 to 50 yards—road, track, grass. Drills on a grass field feel good, but make sure there aren't divots or other hazards waiting to trip you up. In terms of convenience, your street is a great venue if it doesn't have a lot of traffic—you finish your run, do your drills, then head in for the day. And the neighbors will enjoy watching you.
There are an infinite number of drills you can do. The six that follow complement each other by working key aspects of having and holding the form you're trying to attain by running in minimalist shoes. If you'd like to learn more drills, I recommend DVDs put out (separately) by Johnson and another coach, Greg McMillan.
Do these drills in your minimalist shoes or barefoot. Complete each drill twice. Perform the drill as described, turn around and jog back to the starting point, then turn around again and run fast but relaxed over the same stretch. Adding that quick strideout (not a sprint!) after doing the drill will help to hardwire the movement pattern into your form. After doing the strideout, walk back to the starting point or walk around the area where you finished long enough so that you're sufficiently recovered to do the next drill with good form. Your heart rate can get quite high in the short term while you're doing the drill and postdrill stride; allow it to lower before starting the next drill. After you've done the drill and postdrill strideout the second time, recover, then move on to the next exercise.
The order of the drills moves from gentlest to most challenging in terms of dynamic range of motion.
### **B ACKWARD WALK**
#### Backward walking activates the glutes and hamstrings.
**Why:** This exercise activates the hamstrings and glute muscles so that you become more accustomed to using them to power the swing phase of your stride; the result is a stronger, more flowing stride than when you rely on the flexor muscles, such as the quadriceps, along the front of your body.
**How:** While staying on your forefoot, bend the leg you're going to lead with at the knee. Use your hamstrings and deep buttock muscles to extend that leg behind you. Walk backward as you continue to contract your glutes. Take long walking strides while keeping your upper body upright but relaxed. Cover 30 yards.
### **F AST-FEET SHUFFLE**
#### Fast-feet shuffles improve stride frequency.
**Why:** This exercise will help improve your stride frequency by getting your nervous system used to firing more quickly. The shuffle can also help reduce a tendency to overstride. Finally, it works the peroneal muscles, on the outside of your lower legs; better activation of those muscles will help you get off your toes more quickly and with more power.
**How:** Staying on the balls of your feet, shuffle forward as quickly as you can while skimming over the ground just inches at a time. Keep your knee lift at a minimum. Keep your arms bent at a 90-degree angle and loose, and your torso upright and relaxed. Cover 10 yards. Proper recovery between drills is especially important with this exercise, so that your nervous system can fire as quickly as possible.
### **G D RILL**
#### G drills lessen contact time.
**Why:** This exercise will teach you how to get off the ground more quickly, thereby improving your stride rate and lessening the amount of time you spend in the stance phase of the running gait. It's especially helpful for runners whose feet splay out as they toe off.
**How:** Stand with one foot flat and the other leg bent at a 90-degree angle, with your arms in the appropriate corresponding positions. As quickly as possible, do an "exchange," getting into the same position on the other foot. While learning to do the drill, focus on doing 10 exchanges with the best balance possible. As you become better at the drill, see how many you can do with good balance in 15 seconds.
### **B UTT KICK**
#### Butt kicks improve the swing phase.
**Why:** This exercise helps to lessen the time it takes to bring your foot up and behind you after you toe off. It also teaches you to use your calf muscles more at toe-off and to keep your trail leg close to your backside rather than extended far behind you. Finally, it improves driving your arms quickly in sync with your legs.
**How:** While staying on your toes with your ankles dorsiflexed (feet pointed toward your shins), quickly snap your feet back to kick your butt. The first few times you do this drill, you might not be able to reach your butt; you should be able to with practice. Keep your head up and over your shoulders, and keep your upper body erect but not overly stiff. Although you want the foot-to-butt motion to be quick, don't worry about how swiftly you're moving across the ground. Cover 30 yards.
### **H IGH KNEE**
#### High knees improve knee lift.
**Why:** This exercise works your quadriceps and hip flexors so that your knee lift is greater and more efficient.
**How:** Use a pistonlike motion to rapidly lift your knees and drive your toes back toward the ground. Stay on your toes throughout the drill. Hold your arms straight out in front of you as if you're carrying a pizza box; imagine trying to keep the box flat. It's okay to lean back a little if that helps with your knee lift. Cover 30 yards.
### **C ARIOCA**
#### Cariocas improve hip range of motion.
**Why:** This exercise works full range of motion in your hips.
**How:** If you're new to this drill, do it walking your first few times. Walk sideways, bringing your right foot in front of your left foot, then your right foot behind your left foot, repeating this pattern for 20 yards. Turn around and continue to walk sideways in the same direction, but now leading with your left leg—left foot in front of right foot, then left foot behind right foot, again repeating the pattern for 20 yards.
As the movement becomes familiar, convert the walk into a skip. Move laterally while swiveling your hips and swinging your arms across your body, alternately moving one leg behind your body, then bringing it across the front and lifting your knee high in the front swivel. Do 20 yards focusing on one leg, then turn 180 degrees and focus on the opposite leg for another 20 yards.
Over time the movement will feel more natural and you can concentrate on increasing the speed of the lateral skip and the distance you travel with each stride.
### **MEET A MINIMALIST**
### **SHOSHANNA COHEN**
### **PORTLAND, OREGON**
Shoshanna Cohen's story is a good reminder that we're not just runners when we're running, but all day. Put another, perhaps less obsessive-sounding way, Cohen has learned the hard way that what we do in our nonrunning hours can have significant effects on our running.
Cohen began running in 2004, when she was in her mid-20s. In the fall of 2011, encouraged by the injury-prevention promises of minimalism, she gradually switched from conventional stability shoes to barely there Merrells, after a transition period of walking and standing in the Merrells while working retail. "Once I started, it just felt really good and was more fun than running in regular shoes, and that became my motivation, in addition to injury prevention," she says. "So much lighter and nimbler than the stability shoes, like going from a station wagon to a sports car."
But then she apparently threw too much change at her body at once. "I got a pair of Vivobarefoot casual shoes to wear walking around town and starting wearing them everywhere," Cohen says. "I thought I'd be fine since I'd transitioned to Merrells already, but they were apparently different. At the same time, I had begun increasing my minimalist running gradually and started a new job where I wore high heels all day for a couple of days. I should have realized that switching between barefoot shoes and high heels was a really big difference and been more careful with that. It's tough because I wanted to look professional for my new job, and women's dress slacks are designed to be worn with heels only—they are too long to wear with flats. I didn't have time to try to come up with an alternative or get my pants tailored.
"I developed a weird top-of-foot injury. I'm not sure which variable caused it."
Another variable was that Cohen was returning to running after injury at this time. As she notes, "It's tricky because when you're beginning to run again after taking a break to heal, there are inevitable aches and pains that you have to run through as your body gets used to it again. It's tough to know what is a normal pain and what is the beginning of an injury."
Cohen's foot took months to feel better. During that time she developed a few guidelines for herself that most modern runners would do well to heed.
"The lesson definitely seems to be that for some people, going supergradual with all transitions is needed, and don't underestimate the importance of the casual time you spend on your feet when you're not running," she says. "I still want to get back to running in minimalist-ish shoes, although I am not committed to going all the way and only running in Merrells.
"I had thought that pure barefoot-style, zero-drop shoes were the only way to go and everything else was dumb and bad for you. But now I am realizing that, for some people, transitioning can take a really, really long time, and you might not be ready for that little of a shoe for a while, or ever, and that the in-between types of shoes can be really helpful. I also see the value in switching among several different types of shoes to make your feet stronger in a wider variety of situations and work them at different angles.
"I'm more committed now to being an all-around athlete and doing as many different activities as I can, because I hope it will help me be stronger, more flexible, and more resistant to injury, and it's fun to do new things!"
## MODERN LIFE AND MINIMALISM
With age, many of us become increasingly removed from regularly moving through all planes of motion, as children tend to do while playing and participating in a wide variety of games and sports. At the same time, we tend to spend more and more time sitting, either in a car or at a desk. In the latter case, we're often slumped in front of a computer, perhaps with our head bent down and/or thrust forward.
None of this is good for our running form. Former Olympic marathoner and current office worker Pete Pfitzinger says that sitting all day at a desk means "the hamstrings become short and weak and the core muscles do not have to work as you lean back in your chair." Magill says, "It plays murder on our hips, and can also cause iliotibial band syndrome. Anything we do for a long time strains certain muscles, and they're going to go into spasm." Veteran coach Roy Benson adds, "As we spend less time being active and more time being passive, like sitting at a computer, even though we run, the less control we have of our skeleton by our muscular system, and that is a big problem." Wharton describes the effect of the sitting most of us do as "glutes in hibernation"—what should be the powerful muscles in your butt and hamstrings are rarely activated. Over time you lose the ability to use them effectively when running.
There are two modes of attack here—address the problems and prevent the problems. Addressing them includes regularly doing the strengthening exercises and drills outlined above. In addition, if you run after work, undo some of the day's damage with a dynamic warmup (leg swings, plus the milder drills, like backward walking) that will activate your running muscles and help you start the run with better form.
In addition to regular strengthening work, much of prevention comes down to day-to-day habits. Set up your monitor or other workstation so that it's at eye level. Move your monitor close enough so that you're not straining to see it (and therefore thrusting your head forward). Position your keyboard so that your elbows are bent at 90 degrees to minimize strain on your shoulders. When your shoulders and neck are tight and out of alignment, they'll throw off your hips, and your running form will suffer.
Sit with your center of gravity over your hips and your feet flat on the floor. Angle your chair so that your knees are slightly lower than your hips. (As much as possible, try to achieve the same posture while driving.) And no matter how good your sitting posture is, get up and move around at least once an hour to undo some of the chronic low-level strain on your shoulders, neck, and head.
Then there's the matter of what's on your feet during the vast majority of your life when you're not running.
Kevin Kirby, a sport podiatrist and marathoner in northern California, believes minimalists would do well to think about more than just their running shoes. "They are so worried about everyone's extra 8 millimeters of heel height during their 30- to 60- minute runs, but are saying nothing about the health effects that wearing shoes with 75 millimeters of heel height and overly tight toe boxes for 8 hours per day has on a woman's feet, knees, and lower back."
A British study published in 2010 documented the structural changes that high heels cause. It measured gastrocnemius length and Achilles tendon agility in women who regularly wear high heels versus women who don't. The heel wearers' calf muscles were about 12 percent shorter, and their Achilles tendons were more than 10 percent more rigid than those of the women who wore flat shoes. As a result, the heel wearers had significantly less range of motion in their ankles.
A Finnish study published in 2012 showed how these sorts of structural changes have functional implications. Researchers gathered two groups: one made up of women who had worn heels of at least 5 centimeters for 40 hours a week for at least 2 years, and the other made up of women who wore heels for less than 10 hours a week. While the women walked over level ground, the researchers measured ankle and knee motion and lower-leg muscle activity. The heel wearers had measurable increases in fascial strain and muscle activity compared with the other group. Their history of wearing heels for much of the workweek had altered their basic walking mechanics. While the study looked at the effect of heels on walking, not running, it's reasonable to think the mechanical deficiencies would be that much more amplified when running.
Wearing flat shoes and going barefoot in sedentary hours is a key part of the transition-to-minimalism plan we looked at in Chapter 6. Continuing that practice is a key part of maintaining your running body. If heels are unavoidable in your profession, do the best you can to minimize the time you spend in them, such as wearing other shoes while commuting and taking the heels off when you know you'll be at your desk for a while.
## MINIMALISM FOR YOUNG AND OLD
Parents used to be told their children should wear "supportive" shoes to help young feet adapt properly to increased activity. Fortunately, that thinking has been set aside by knowledgeable people. In running terms, most experts agree that young runners are ideal candidates for minimalist shoes, both because of less initial injury risk and because of the long-term gains to be had.
"I would encourage it for any adolescent," says Fullem. "My 10-year-old son just ran his first 5-K, and I try to get him in as minimalist a shoe as possible. I think if you start with the younger kids, their muscles and their tendons and everything get stronger."
"I think the high school kid should maybe push the envelope a little bit more in terms of minimalism," says Johnson, who conducts annual high school running camps in Boulder, Colorado. "I think sometimes kids get sold shoes that are made for heavier, middle-aged people training for a marathon when they're a light, waify kid whose longest race is 5-K." The "protection" that "supportive" shoes traditionally were said to provide should come via the strengthening work laid out earlier in this chapter, says Johnson.
Young runners have less injury risk running in minimal shoes because they haven't done years of mileage in bigger shoes that alter their mechanics and weaken and tighten key running muscles. For the same reason, they're also ideal candidates for extensive form work, says Steve Magness, a former coach with the Nike Oregon Project who's also worked with leading high school runners.
"They don't have years of 'This is how my running form is,'" Magness says. "It's not ingrained as motor programming like with someone who's been running a long time. They're much more of a blank slate and easy to change with low risk." As Magness points out, young runners usually aren't doing much mileage, making their situation analogous to the start-from-scratch transition plan we looked at in Chapter 6.
"I'd generally go with a lightweight trainer for most high school kids," says Magness. "Not something crazy minimal, but more like what older runners might wear for long tempo runs and marathon-pace runs."
Johnson thinks that young runners will benefit from minimalism not only now, but also when they're adult runners. "We're at a really cool time in our country where kids are into the sport and the shoe companies are putting out 'less' shoes," he says. "I think we can have kids have fewer injuries in the next 10, 20 years as long as they're doing the strengthening work as well."
At the other end of the age-group spectrum, don't be dissuaded from running in minimalist shoes simply because you're middle-aged or older.
"All other things being equal, if you're an older runner, you should have better bone density than a sedentary person of your age," says Fullem. "It's known as Wolff's Law—your body responds to the stress of the pounding by strengthening the bone." You shouldn't be at a higher risk of stress fractures than younger runners from running in shoes with less cushioning.
If you're a new runner in your 40s or older and have been sedentary most of your life, then it's a good idea to have your bone density checked regardless of what shoes you run in.
As we saw in Chapter 6, younger new runners can sometimes transition to minimalist shoes quicker than their contemporaries who have run for many years. If you're an older new runner and your bone density is good, you should be able to use minimalist shoes as one of your footwear options without undue concern for getting a stress fracture.
## MINIMALISM IN YOUR ARSENAL
You're probably familiar with the fact that it's easier to maintain fitness than to obtain it. That principle also applies to improvements in your running form—once it's gotten better because of minimalist shoes, strengthening exercises, and drills, it's easier to run with good form in whatever shoes you wear. Certainly that's the experience of elite Kenyans, who grow up covering lots of miles in little or no shoes, and then maintain beautiful form once they start running in conventional shoes.
Of course, you need to be diligent about the strengthening work and form drills, especially if you have a typical Western lifestyle. But running in minimalist shoes isn't an all-or-nothing matter any more than most other aspects of running are. If you enjoy doing all your running in minimalist shoes, great. If you enjoy running in minimalist shoes and more conventional models, feel free to do so. You won't be kicked out of any club (at least any worth being part of). And you'll be putting into practice what successful runners have long done—rotating shoes.
Joe Rubio, coach of the Asics Aggies and a 2:18 marathoner at his prime, says, "I've been coaching a long time, and 80 to 85 percent of the people are going to have the most success alternating different shoes for different purposes. When I was really racing, you had spikes for really fast stuff, a really light pair of racing flats for 5-K race-pace workouts, a little more substantial shoe for tempo runs, a lightweight trainer for most of your regular runs, and a padded pair of bricks for your true recovery days, because you wanted to run slow and not take the pounding. And you did your barefoot strides.
"It's like interval training back in the '60s—you did it every frickin' day!" Rubio continues. "It worked for some people, but it didn't work for a huge number of people. In my mind, it's the same thing with real minimalist shoes. Some people can train in them for everything and be fine, but most people are going to do better by picking certain days to train in that type of footwear. They'll still get the advantages."
Many runners find that once they've become accustomed to running in less shoe, they no longer enjoy running in traditional trainers. (I'm one of them.) If that's you, consider rotating shoes within the minimalist spectrum. Popular minimalist blogger Peter Larson says, "If I had to pick three or four shoes, I'd probably want something with maybe a 4-millimeter heel rise and some decent cushioning for longer runs; a racing flat; something that has no cushioning as a training tool and for form work and shorter runs; and a trail shoe."
Larson, who teaches college courses in anatomy, points out a key value of shoe rotation beyond the performance aspects Rubio outlined above.
"If you ask me why injuries happen, I think it's because we run lots of miles on terrain that doesn't vary at all," Larson says. "And that's not necessarily because it's hard, but we go out and run on the road and there's no variation and we pound our legs in the same shoes on the same runs every single day and there's no variability. I haven't seen any studies comparing injury rates in trail runners versus road runners, but running on trails varies things. If you want to try to get that variation on the roads, running in a few different pairs of shoes might be one way to do that."
If you find you're a happy, healthy runner doing all your mileage in minimalist shoes, great. If you find you're a happy, healthy runner doing two runs a week in minimalists, or somewhere between that and all your running, that's great, too. There are no rules you have to follow about how often to wear minimalist shoes. Five years from now, you're more likely to be a healthy runner enjoying your sport if you avoid extremes and rigid but arbitrary standards. That applies to mileage, diet, competitive goals, stretching and strengthening, and pretty much everything else, including what shoes you choose to run in.
Five years from now, if you've made minimalism part of a sustainable approach to your running, what will your minimalist shoe choices be? That's the subject of our final chapter.
# **CHAPTER**
# 9
# MINIMALISM IN THE LONG RUN
## Where are minimalist shoes headed?
**AS THE PHYSICIST NIELS BOHR** once said, perhaps while channeling Yogi Berra, prediction is very difficult, especially about the future. Who in 2005 would have predicted that 5 years later the Vibram FiveFingers would be the top seller in outdoor specialty retail sales? Or that shoes with massive heel heights like the Nike Shox, which runners were then buying by the boatload, would soon be viewed as evil incarnate?
Still, it's fun and often educational to forecast how today's reality might morph into tomorrow's trends. In this chapter we'll get expert opinion on where minimalist shoes are heading. Will barely there shoes like the Vibram FiveFingers still be popular? Will minimalist shoes constitute more or less of the overall running-shoe market? Where are running shoes in general headed, and how might manufacturing innovations affect them and their relationship with minimalism? And finally, will the minimalist movement cause shoe manufacturers to throw out their traditional paradigm for designing shoes?
## THINNING THE MINIMALIST HERD
As we saw in Chapter 5, minimalism encompasses a broad range of shoes, from barely there models like the FiveFingers and Merrell Trail Glove to moderate minimalists like the New Balance Minimus Road and Nike Free to gateway or transitional shoes like the Saucony Kinvara and Brooks PureConnect.
At the beginning of the decade, the barely there shoes were the hottest. Vibram had nudged its way into constituting 2 percent of running-shoe sales; that's a remarkable achievement for a brand that basically had 0 percent 5 years earlier. Solely because of Vibram, the big seven running-shoe companies—Nike, Asics, Adidas, New Balance, Brooks, Saucony, and Mizuno—saw their traditional 96 percent share of the market slip to 94 percent. Vibram's market share might have gone even a little higher if the firm had been able to keep up with demand. In late 2009 and early 2010, the company was frantically adding staff and production facilities, but some stores still couldn't keep FiveFingers in stock. Sales were roughly doubling every year.
That growth trend has stopped. In the first quarter of 2012, according to industry analyst SportsOneSource.com, Vibram sales declined by more than half. That happened for two key reasons.
First, there are more choices now within the barefoot-style neighborhood of minimalism. Merrell has created a well-liked line of barely there road and trail shoes, niche brands like Vivobarefoot are more widely available, and the big players now have this ground covered if they so choose (for example, the Adidas Adipure Trainer, which features Vibramesque toe pockets).
Second, the pendulum that swung so quickly from traditional running shoes to the other extreme is now moving back. "I don't know how many people are buying a second or third pair of Five Fingers," says Joe Rubio, a partner in the online running store RunningWarehouse.com and one of the most astute industry watchers around.
"I think we may have reached the end of people's infatuation with them," says Peter Larson, author of the popular minimalist shoe blog Runblogger, about the FiveFingers. "The fit is so hard to get right. The toe pockets, you either love those or hate those when you run in them. I think a lot of them are sitting in people's closets." Regardless of its resolution, a spring 2012 proposed class-action lawsuit against Vibram for false advertising claims certainly symbolized a potential end of the love affair with the less-is-always-better approach.
"What we're going to see is a growth in the quasi-minimalist area," Larson says. "I think the no-cushion-at-all Merrell Trail Glove area will be a small part of the market. The majority of people are going to want some amount of cushion under their feet. I think the growth will be from the Kinvara type down to flat cushioned shoes like what Altra is producing."
Rubio says market realities will hasten the pendulum's swing away from the extreme end of minimalism.
"There was an argument after Vibrams got big that the big brands didn't want to do these minimal shoes," he says with a scoff. "Okay, if you're Brooks and you're looking at this whole thing and you've got an Adrenaline that has a three-piece medial post with all the plastic pieces in there . . . you know how expensive it is to make that shoe? Why would you want to make that shoe versus a one-piece midsole/outsole with no added technology? Your cost has got to be 60 percent of a traditional shoe, maybe less."
The result of big brands moving in, Rubio says, will be the death of some smaller ones. "Any big idea that the small brands come up with will migrate to the big brands, who have lots of resources and access to the end consumer," he says. "That makes it very hard for any of the small brands to gain significant traction in the marketplace. You've got a bazillion small companies knocking it out for that final 4, 5 percent of the marketplace. Some of them are going to have to go away."
It's not so much that all the big companies will make shoes to compete toe-to-toe with the niche minimalist brands, but that their minimalist offerings will seem minimal enough to most runners. That is, people who want only the FiveFingers will go looking for the FiveFingers, but people looking for "less shoe" will be much more likely to find the big brands' minimalist offerings and deny the smaller brands a chance at a sale. That's especially true if you consider Larson's point about growth in minimalism coming in the slightly cushioned segment instead of the barely there models, which tend to be made by the smaller companies. And then there's this: About 70 percent of sales within the minimalist category go to one shoe, the Nike Free.
In any industry, Rubio points out, economies of scale usually favor larger companies. In the running-shoe industry, he says, "The basics of the business are that a small company doesn't deliver necessarily on time, whereas a big company generally does. And if you're running a business, you need to have stuff show up on time. And what if something goes wrong with the product? A smaller company can't deal with a big blunder if they come up with a product that doesn't hit, whereas a big company can absorb it." Translation: If the Adidas Adipure Trainer doesn't take off, Adidas will survive. Niche brand Somnio invested heavily in a minimalist shoe, the Nada, that never made it to stores; the project helped drive the company from the US running market.
Rubio adds, "The big companies, just the sheer number of shoes they have to choose from is incredible. They have their traditional shoes, their minimal shoes, their racing flats, their trail shoes. Some of these little brands make only one or two models, and what if you don't like them?"
So a few of the many small companies that were making barefoot-style running shoes while I was writing this book might not exist by the time you're reading it. Will other companies follow the Skechers plan—use their deep pockets to develop a minimalist running shoe and try to make inroads into an industry they've not historically been part of? Skechers' first running shoe, the lightweight GoRun, has gotten mostly positive reviews from initially incredulous minimalists. The brand scored a coup by signing as their one sponsored runner Meb Keflezighi, who won the 2012 US Olympic Marathon Trial in Skechers.
Rubio is dubious. "Every 6 months you have another company that tries," he says. "We're not carrying Skechers at [RunningWarehouse.com]. The guy keeps calling us asking why not. I say, 'I know you have Meb, and Meb's a friend of mine, but you got a crappy brand name. You got Kim Kardashian as a spokesperson. There's no credibility with that brand. You have a decent shoe, but until the brand name gets better, there's no point in carrying it.'
"If someone like Puma, with a tradition and decent product, can't sell shoes, no way can I see Skechers making it," Rubio continues. "If you're a brick-and-mortar store, why would you give up shelf space for something that might move versus something that is moving? And besides, how much room do I have in my back room to store this thing?"
The break-in brands then have to look for sales in what the industry calls "other channels," in this case meaning outside of specialty running stores and large sporting goods chains, or largely through online sales. Rubio says, "There aren't enough people looking for it in those other channels. The only thing in recent memory that made it that way is the FiveFingers. Long-term, you gotta get in brick-and-mortar."
## MINIMALISM IN THE MARKET
If you paid attention only to news stories, you'd be excused for thinking that minimalist shoes are pretty much the only running shoes people have bought the last few years. And yes, in the spring of 2012, minimalist shoes were still hot, with sales increasing by 70 percent or more most months. But here's a reality check: In the first quarter of 2012, minimalist shoes accounted for about 11 percent of the US running-shoe market. Remove the Nike Free from consideration, and sales of the remaining minimalist models constituted 4 percent of the US market.
"It's similar to hybrid and electric cars," Rubio says. "There's a huge amount of press, and every ad you see touts these technologies, but if you look at the sales figures, it's 2 to 3 percent of the industry."
Rubio's livelihood depends on intimate knowledge of the US running-shoe scene. Ignoring hard data and what most people want simply isn't an option in his line of work. So I asked him what he thought would be the biggest sellers on RunningWarehouse. com in 2015.
"It'll still be traditional trainers," he says. "They make up 80 percent of our sales. In the industry as a whole, they're more like 90 percent of sales.
"There are two customers in the processes: the people who sell the shoes and the public. The people selling running shoes to the customer, they need to make business decisions more on what's definitely going to work versus what might work. And the guys who are making the shoes for the dealers are only going to respond by making what's popular. So I don't see minimalism getting any bigger than it is now."
But what about all the press for minimalism? I asked. Aren't customers influenced by that, and wouldn't that change what they ask for when shopping for running shoes?
"The people going to brick-and-mortar tend to be the people at the last third of the race, the beginning athlete who needs a lot of guidance," Rubio responds. "They might have heard about this FiveFingers thing, and they might buy a pair. And maybe they'll do as recommended and start off walking around or hiking in them, and go, 'Wow, this is a lot of work and kind of painful, versus my [Asics] Nimbus, which are really comfortable and plush and they look good and I can go out afterward in them.' There's always going to be a place for traditional shoes that are soft and comfortable and look good."
When considering minimalism's share of the overall market, it's helpful to keep in mind where most running shoes are sold. Sixty percent of sales occur in department stores like Nordstrom. Thirty percent are through what the industry calls "big box"—large chains like Sports Authority or Dick's Sporting Goods. Only 10 percent are through running specialty stores or online outfits like Rubio's.
"The final 10 percent is where a lot of the smaller brands would hope to get shelf space," Rubio says. "No way Nordstrom brings in a small, unknown but trending brand."
Most running-shoe sales occur elsewhere than what most readers of this book would consider a good place to buy running shoes. Because of this simple market reality, it's hard for knowledgeable industry watchers like Rubio to see minimalist shoes making inroads beyond their current share of about 10 percent of sales. And yet . . .
## BEYOND CATEGORY
"They're hugely influential," Rubio says of the minimalist shoes it might seem he's just been deriding. "If you look at things that are happening in the auto industry, you're seeing Porsches that get 72 miles per gallon for the fastest production car they've ever made. The same thing's happening in the running industry. You've got these really lightweight minimal shoes that are having a huge influence on how all shoes are made."
That influence is showing in two key aspects of running-shoe construction: weight and ramp angle, or the difference between heel and forefoot height. For example, after the tremendous success of the Kinvara, with its reported 4-millimeter ramp angle, Saucony is lowering the heel-to-toe differential throughout its line from 12 millimeters to 8 millimeters. Combine that with less weight, and you might find yourself asking, Is this a conventional shoe? A minimalist model? What?
"I think the movement toward lighter and faster and more responsive isn't going to end," says Rubio. "It's just going to be a natural part of shoe development. It's similar to the bike industry, where you gotta keep making things lighter every year. Any one bike from year to year might not be significantly different, but over the course of 5 years, you notice a pretty big change in the weight and responsiveness. Every industry goes lighter over time. I'm just surprised it took running this long to get there."
The gains in lightness are coming not only from minimalist-driven consumer demand. They're also driven by improvements in manufacturing processes that would be lowering shoe weight regardless of whether models like the FiveFingers had ever become popular.
At just over 3 ounces, the New Balance RC 5000 weighs the same as this small screwdriver.
"There are so many new materials being created over in Asia that are superlightweight, superstrong, superresilient," says Brian Metzler, a former _Running Times_ senior editor who's wear-tested more than 1,000 shoes. "So whatever you need a piece of a shoe to do, whether it's add stretchiness for comfort or add firmness, there are these new materials playing a huge role in the shoe revolution."
New manufacturing processes, like welding instead of stitching, will lead to lighter shoes, Metzler says. Welding ultrathin synethic overlays instead of sewing heavier ones is one reason why New Balance was able to make a racing flat, the RC 5000, weigh in at just 3.3 ounces when it was released in the summer of 2012. Lighter shoes will also happen as companies move from "heavier overlays and various plastics and vinyls that weren't really that conducive to a running shoe, resulting in these 14-, 15-ounce bricks," Metzler says. For example, as Nike developed its Fly Knit technology, it found lessons from that process it could carry into its entire running-shoe line. By trying to create a snug, seamless upper—a development unrelated to minimalism—Nike found a way for all its shoes to lose 10 percent of their weight.
As such developments happen, differences among shoes will become more a matter of degree than category. "You'll start to see some of these categories overlapping," says Metzler. "Take racing flats—now you see a lot of people wearing the modern minimalist shoes, like the New Balance Minimus Road, instead of traditional racing flats."
Rubio sees the lines blurring most in the already-ambiguous distinction between basic training shoe and lightweight trainer. "Now you get a [Nike] Pegasus under 10 ounces, so an everyday training shoe starts moving toward what we would have called a lightweight trainer a few years ago," he says. "Something like the New Balance 890, an everyday trainer, not a minimalist shoe, that's significantly lighter—that's where I think you'll see things going. Brands that aren't doing that are losing sales significantly."
## MOVING PAST THE PRONATION PARADIGM
For the last couple of decades, the model for conceiving and designing running shoes has been based largely on pronation control. The thinking has been that what happens to your foot as it hits the ground is what's most important. If your foot rolls in a little between landing and pushing off, you're said to have normal pronation, and have been told to go with a "neutral" shoe, sort of the Goldilocks version of running shoes—not too soft, not too rigid, providing just the right amount of cushioning and stability. If your foot rolls in too much, we've been told, you need a motion-control shoe that will arrest some of the overpronation. And if your foot barely rolls in but instead remains rigid upon landing, you're said to be an underpronator, or supinator, and therefore need a flexible, cushioned shoe to absorb some of the shock that would be dissipated if you had a normal amount of pronation.
The minimalist movement has led an increasing number of people throughout running to question this model. Many runners who were told to wear stiff, heavy motion-control shoes have felt liberated (and remained injury-free) by moving to barely there shoes that allow their feet to work naturally. Sales in the motion-control category of running shoes have been in decline the last few years, and good riddance to all that, say many running experts.
"One of the things with the traditional line that you'll probably see—and are already starting to see—is the dissolution of the pronation-control category," says minimalist blogger and college biology professor Peter Larson. "There's never really been evidence supporting those, and in fact there's now research that came out recently showing that those don't work."
Here Larson is referring to studies such as one with Marines that followed the traditional practice of assigning shoe type based on arch height; the study found little effect on incidence of injury even after other injury factors were considered. Even more interesting, a Canadian study randomly assigned three types of shoes—neutral, stability or motion-control—to a group of women runners. The researchers found that there was no relationship between the type of shoe the women wore and their incidence of running-induced pain. That is, women who were "supposed" to be in motion-control shoes reported as much or more pain when wearing those shoes as did women who were "supposed" to be in motion-control shoes but were in one of the other, supposedly less suitable types. "The findings of this study suggest that our current approach of prescribing in-shoe pronation control systems on the basis of foot type is overly simplistic and potentially injurious," the researchers concluded.
Steve Magness, formerly an assistant coach with the Nike Oregon Project and holder of a master's degree in exercise sciences, agrees, noting, "I think in the lab you're seeing more of the thought that this model probably doesn't work, so let's come up with another theory." Magness likes to point out that the focus on pronation came about at least as much because it could be measured in the lab as because those measurements yielded meaningful data.
"The question is, Do you keep using a model that most people think is broken just because it's what we've always done?" asks Larson. "What do we do instead? There you kinda get stuck. That's a huge question right now, and I wish I had the answer."
Indeed, it's difficult to see how the great running insight "we're all an experiment of one" can be matched with modern industry's penchant for flow charts and categorization. Which company is more likely to get the average consumer's purchase: one whose promotional materials provide quick, easy guidance on navigating their product line, or one that more or less admits "your guess is as good as ours"? As Magness says, "We know different shoes work for different people based on a couple of different things. But how do we translate that into designing shoes and classifying shoes?"
Consider flat feet, says Larson. Under the pronation-control model, nearly everyone with flat feet has been put in motion-control shoes, because the assumption has been that flat feet are prone to overpronation. As it turns out, "there are different reasons why people can have flat feet," Larson says, "so just putting everyone with a flat foot in one type of shoe doesn't make any sense. We're learning that flat feet when you're standing might not mean anything about what the foot does when you're running."
Magness, a voracious consumer of research, offers another example: "With muscle activity, what they find is some people, if they have a lot of activity in their calves and their calves are really tight before footstrike, it makes them more economical," he says. "Some people it's the exact opposite. The question is why and what makes people different, and does that mean they need a different shoe? Would someone who relies entirely on reactive elastic response in their calves and Achilles need a really hard shoe that lets that work, where someone who doesn't rely on it, maybe they need a soft shoe to take some of the load?"
The muscle-activity thinking stems from the work of the Canadian biomechanist Benno Nigg, a professor at the University of Calgary. If the conventional paradigm considers running shoes the primary actor, a piece of equipment that gets the body to do what it's "supposed" to do, Nigg represents the opposite end: He believes footwear shouldn't affect footstrike, or even muscle activity before the foot hits the ground.
Good luck systematizing that in a way that can lead to mass production in Asian factories and helpful guidance in American stores, all at prices runners are accustomed to. And, notes Magness, even if this _X_ factor could be isolated, it will be hard to keep its value in perspective. "We can measure pronation easily, so that's what everything is based on," he says. "There's going to be this shift of, well, maybe we can measure this other new thing really well, so let's make everything entirely based on that.
"The reality is, what people need in a running shoe is probably from some crazy combination of foot mechanics and pronation and muscle activity and structure," Magness says. "It's hard to tease out all these things and say, 'All right, here's the perfect combination.'"
## THE ROAD AHEAD
"I fear that I may have been too negative in this attack, but there are times when a pendulum has swung far enough and needs a strong push in the other direction to restore equilibrium."
That's the famous evolutionary biologist Richard Dawkins in his book _Unweaving the Rainbow_. Although he was writing about how science gets taught, Dawkins's desire for restoring balance happens in most matters where people have passionate beliefs. A movement bursts on the scene, spearheaded by the most ardent and committed members. People are impressed by the purity of the message and the clear direction it provides on how to act. The newest recruits become some of the strongest proponents. Seemingly overnight, what had been obscure becomes common knowledge, and what the new movement is reacting against is seen as the folly of a less enlightened time.
Over time, however, most people find that the new movement pushed the pendulum too far. Through experimentation and going about their lives, people find what's most useful from the movement's message. What's helpful, they keep; what's not, they begin to ignore. The pendulum starts to move back to the center.
Have the movement's efforts been a waste? Most people would say no—through the process of the pendulum swing, they changed what's considered normal. Vegans get people to reconsider their use of animal products. Environmentalists make recycling mainstream. Back-to-the-landers show suburban homeowners the pleasure of a small garden plot. And, yes, hardcore minimalists get regular runners to reconsider what they run in.
The more extreme end of the minimalist movement was a necessary corrective. Although some runners were, on their own, finding a way around the ever-bigger shoes of the 1990s and early 2000s, most accepted the conventional line that if some cushioning is good, more is better. It took that big push of the pendulum to bring the issue of footwear fundamentals into the mainstream.
But Dawkins's quote applies to both ends of the pendulum's arc. As we've seen, that first pendulum swing is now over, and things are moving back to homeostasis. It's a new homeostasis, however, one that incorporates the minimalist message of running shoes serving the runner, rather than the other way around. That's been my belief throughout my 3-plus decades as a runner. I hope this book has helped you see the wisdom of that view and shown you how to implement it in your running in a way that works for you in the real world.
# A MINIMALISM GLOSSARY
Here are some key words and phrases you'll encounter in this book.
### BARELY THERE/BAREFOOT-STYLE SHOE
The most minimal of minimalist running shoes. These shoes generally have little to no cushioning between your foot and the outsole, and a slight stack height. As such, running in them replicates barefoot running better than running in other shoes, while providing protection from surface hazards. Most runners who have done the bulk of their lifetime mileage in conventional running shoes will need a slow, gradual transition to be able to run in barely there/barefoot-style shoes without risking injury.
### BIOMECHANICS
How the parts of your body work together to create movement. In running, the term is sometimes used synonymously with running form, but it's more accurate to say that your biomechanics determine what your running form looks like. When you develop the needed strength and flexibility, running in minimalist shoes or barefoot can improve your biomechanics and thereby improve your running form.
### DUAL-DENSITY MIDSOLE
A shoe-construction method in which the midsole is firmer in one section than elsewhere; usually, the firmer section is on the inside, near the heel. The idea is that the harder part of the midsole helps to control overpronation better. Dual-density midsoles are anathema to most minimalists, who say such midsoles make conventional running shoes that much more likely to interfere with the foot's natural motion.
### FOREFOOT STRIKE
Landing with the front of your foot first when running. A true forefoot strike at normal running speed (as opposed to racing) is rare, but switching to minimalist shoes and working on running form can move many runners away from heavy heel-striking and encourage them to use their forefoot more effectively.
### GROUND CONTACT TIME
How long your foot stays on the ground when you're running. A longer ground contact time is generally associated with slower turnover, and it increases deceleration as you move through the gait cycle. Running barefoot or in minimalist shoes tends to lessen runners' ground contact time and can retrain your nervous system to run this way even when you're in more conventional running shoes.
### HEEL COUNTER
The part of a running shoe that wraps around the back of your foot. Minimalist shoes tend to have a more flexible heel counter than conventional running shoes to encourage more natural foot motion. Some models' heel counters are collapsible.
### HEEL STRIKE
Landing with your heel first when running. Although heel-striking isn't inherently bad, minimalists contend that modern running shoes can cause people who wouldn't heel-strike when running barefoot to do so when running shod. Most experts agree that too much of a heel strike can be a source of injury and poorer performance.
### HEEL-TO-TOE DROP
The difference between a shoe's stack height in the heel and its stack height at the lowest point of the forefoot. Minimalists contend that shoes with too great a heel-to-toe drop encourage severe heel-striking and inhibit the calf muscles and Achilles tendons from working through their full range of motion.
### MEDIAL POST
An addition of firmer material along the inside of a shoe's midsole, usually near the heel. Medial posts are supposed to help control overpronation. Minimalists contend that, rather than helping runners avoid injury, medial posts usually keep your feet and lower legs from working naturally and can increase your risk of injury.
### MIDFOOT STRIKE
Move accurately called "flat-footed striking," or landing with the heel and forefoot at the same time. Midfoot-striking is less common than heel-striking but more common than forefoot-striking. Many runners find that, by switching to minimalist shoes and working on their running form, over time they adopt more of a midfoot strike. This, in turn, can mean less impact force and, many runners say, simply makes their stride feel more flowing.
### MIDSOLE
The part of a running shoe between the outsole and upper. Conventional running shoes tend to have thick, cushioned midsoles. Minimalists contend that the midsoles in modern running shoes interfere with natural running mechanics because they're too soft, too high off the ground, and too tilted forward. Minimalist shoes tend to have firmer, lower, and flatter midsoles.
### MINIMALISM
The belief that lighter, flatter, more flexible shoes allow people to run more like they're running barefoot than do conventional running shoes and thereby improve their running form and reduce their injury risk. For more details, read the preceding nine chapters!
### MODERATE MINIMALIST SHOE
A shoe with construction features between a transitional minimalist shoe and a barefoot-style/barely there shoe. Moderate minimalist shoes have some midsole cushioning and a greater stack height than barely there models but are still a marked change for runners accustomed to conventional running shoes.
### ORTHOTICS
Devices, often customized, put inside running shoes in the hope of preventing or overcoming injury. Most minimalists say that runners should wean themselves off orthotics and let the feet and lower legs learn how to run naturally. Sport podiatrists tend to agree that orthotics should be considered part of the fix for an acute situation rather than a lifelong presence, but they also think that runners can increase their injury risk if they abandon prescription orthotics too hastily.
### OUTSOLE
The part of the shoe that contacts the running surface and provides traction and stability. Most minimalist shoes have a relatively level outsole to maximize the foot's ability to feel and adapt to the running surface. Minimalist trail shoes usually have a heavier, more lugged outsole than minimalist road shoes so that you don't feel every rock and twig on the trail.
### OVERSTRIDING
Landing with the heel far out in front of the body. Note that overstriding has to do with foot position on landing, not stride length. Runners can have long stride lengths but not overstride. Most runners find it almost impossible to overstride while running barefoot. Wearing minimalist shoes, increasing functional strength and flexibility, and being mindful of running form can help overcome a tendency to overstride.
### PRONATION
Rolling in of the foot after landing. Almost all runners pronate. One of the guiding ideas behind conventional running shoes is that too much pronation leads to injury and needs to be controlled by shoes; models built to control overpronation are heavier and stiffer than most other running shoes. Most experts now agree that overpronation as a driving factor in shoe design is a bad idea, and minimalists believe that motion-control shoes do much more harm than good.
### RACING FLAT
A shoe designed to be worn in road races. Most racing flats have little heel-to-toe drop, a relatively small midsole, little cushioning, and an outsole built more for traction than for durability. Competitive runners have long worn racing flats during their faster workouts, and many minimalists do much of their running in racing flats. One drawback to wearing racing flats for daily running is that some flats wear out more quickly than shoes designed specifically as minimalist shoes.
### RAMP ANGLE
The angle formed by the difference between a shoe's stack height in the heel and its stack height at the lowest point of the forefoot. Minimalists contend that shoes with too great a ramp angle encourage severe heel-striking and inhibit the calf muscles and Achilles tendons from working through their full range of motion. A given model's ramp angle usually differs among various shoe sizes because most models have the same heel-to-toe drop regardless of how long the shoe is; hence, smaller shoe sizes usually have a steeper ramp angle than larger sizes.
### STACK HEIGHT
A measurement of everything between the bottom of your foot and the top of the road, including the midsole and outsole. Heel-to-toe drop (or ramp angle) is calculated from the stack height. For example, a shoe with a stack height of 19 millimeters in the heel and 15 millimeters in the forefoot has a heel-to-toe drop of 4 millimeters. Generally speaking, minimalists contend that the greater a shoe's stack height, the greater the chance the midsole cushioning will interfere with how you run barefoot; the reasons given include that a large amount of cushioning does some of the work your feet and lower legs are designed to do, and too much material between your feet and your running surface introduces instability.
### STRIDE RATE
The number of steps you take per minute while running. Stride rate—also called turnover—is usually expressed as the total number of steps both feet take in a minute. Stride rate varies to some degree with your pace (higher stride rate at faster paces). Although there's no ideal stride rate, most experts advise working toward a stride rate of 170 or more for moderate-paced and faster running; stride rates of 160 or less usually indicate overstriding. Most runners find that their stride rate increases at least a few steps a minute when they run barefoot or in minimalist shoes.
### TOE BOX
The area at the front of a running shoe that houses your toes. Most conventional running shoes taper toward the front. This construction can cramp toes and prevent them from spreading as you land and push off, as occurs when you run barefoot. Many minimalist shoes have a wider-than-average toe box that allows more natural foot motion.
### TOE SHOES
Running shoes with a separate "pocket" or container for each toe: Vibram FiveFingers are the most popular. Proponents of toe shoes contend that this construction best allows the toes to work as they do when you run barefoot. Some minimalist runners say a satisfying fit can be difficult if one's toes aren't the same length as the separate pockets.
### TOE SPRING
The upward curvature of a running shoe at the front. Toe spring is usually more pronounced in conventional running shoes than in minimalist models. Many minimalists look for shoes with not much toe spring on the theory that too much toe spring inhibits the toes' ability to flatten and spread naturally and encourages more heel-striking. But because most minimalist shoes have a more flexible midsole than conventional running shoes, what might look like a large toe spring becomes functionally not as much of a factor as it would be in shoes with a stiffer construction.
### TRANSITIONAL/GATEWAY SHOE
A shoe designed to ease a runner's transition into minimalism. These models maintain many features of conventional running shoes, including a relatively high midsole and relatively soft cushioning, while having a lower heel-to-toe drop than most running shoes. Although minimalist purists might scoff at transitional shoes, many runners find them helpful in learning how to better use their feet and lower legs while avoiding the soreness that can accompany a too-sudden switch to lighter, lower minimalist models. Many runners happily stop their minimalism journey at transitional shoes.
### UPPER
The part of a running shoe that covers the top of the foot. Most minimalist shoes have a light, spare upper designed only to secure the foot to the rest of the shoe. The overlays and thick padded tongues found in the uppers of many conventional running shoes add weight and interfere with the foot's natural motion, minimalists contend.
### ZERO-DROP SHOE
A shoe that has a heel-to-toe drop (or ramp angle) of 0; the heel and forefoot stack heights are the same. Proponents of zero-drop shoes say this construction allows the feet and lower legs to work like they do when running barefoot. Most runners need to work gradually toward running in zero-drop shoes to minimize their risk of calf and Achilles soreness or injury. A zero-drop shoe can still provide a lot of cushioning, depending on its stack height.
# ONLINE MINIMALISM RESOURCES
The Web is awash with materials on minimalism. The short list below focuses on resources that value the tenets of minimalism while avoiding stridency and maintaining perspective on how the modern minimalist movement fits in with running history and the larger running world.
### _RUNNING TIMES'_ MINIMALIST CHANNEL
_Running Times_ has been a leader in presenting real-world information on the merits of minimalism. The magazine's minimalist channel (runningtimes.com/minimalism) contains several articles with use-at-home information on topics like how to know if you're ready for minimalism, how to make the transition safely, and how to make sure shoes fit properly to allow natural foot movement. There are also a lot of shoe reviews collected there.
### COLLECTED MINIMALISM RESEARCH LINKS
As mentioned in Chapter 4, I've collected the most important research on minimalism-related topics such as foot strike, injury rates, and shoe type at runnersworld.com/minimalismlinks. There you'll find a brief description of the main findings of each study listed and a link to its abstract or full text, depending on the policy of the journal in which the study appeared.
### PETER LARSON'S BLOG: RUNBLOGGER
Peter Larson, whose thoughts you'll find throughout this book, is a professor of biology at St. Anselm College in New Hampshire and is the coauthor of the book _Tread Lightly_. He maintains a deservedly popular blog that's mostly about minimalism at runblogger.com. Larson's blog frequently highlights new minimalism-related research; his posts do an excellent job of explaining a certain study and putting it in context given what other studies on similar matters have found. Larson is also an enthusiastic shoe collector who provides extensive wear-test reports on most minimalist shoes soon after—or sometimes before—they hit the market.
### STEVE MAGNESS'S BLOG: THE SCIENCE OF RUNNING
Steve Magness, another source for this book, is the cross-country coach at the University of Houston and a former assistant coach under Alberto Salazar at the Nike Oregon Project. A 4:01 miler in high school, Magness has a master's degree in exercise science and is an inveterate consumer of physiology research. At his blog, scienceofrunning.com, he combines his coaching and athletic experience with his education to provide real-world takes on running, including, frequently, shoe choice and running form. Given his background and interests, Magness's analysis of research often centers on whether a given approach will answer what are most runners' two most important questions: Will this help me run faster and will this keep me from getting injured?
### ROSS TUCKER'S AND JONATHAN DUGAS'S BLOG: THE SCIENCE OF SPORT
Ross Tucker and Jonathan Dugas each have a PhD in exercise science; they cowrite the blog sportsscientists.com. Their interests are wide ranging, but they write frequently about running. Their posts on footwear and barefoot running are models of balancing experience and research and placing studies that others trumpet as game changers in proper context. Authors of the book _The Runner's Body_ , Tucker, and Dugas are also adept at showing how topics like running form relate to the bigger picture of what happens to our bodies when we run ambitiously.
### ALEX HUTCHINSON'S BLOG: SWEAT SCIENCE
Alex Hutchinson has represented Canada in international competition, and he has a PhD in physics. He knows how to read scientific literature, assess its positives and shortcomings, and describe its relevance to runners. At his blog, sweatscience.runnersworld.com, Hutchinson often writes about studies done on injury rates, running form, and footwear.
### _RUNNER'S WORLD_ FORUMS
On the _RUNNER'S WORLD_ site, there are two popular forums where runners of all backgrounds and abilities discuss all things minimalism: the Barefoot Running forum (runnersworld.com/barefoot-forums) and the Shoes forum (runnersworld.com/shoes-forums).
# ACKNOWLEDGMENTS
**PHIL LATTER PROVIDED THOUGHTFUL** reading of the chapters at draft stage and reporting for a section of Chapter 6.
Phil Wharton, Steve Magness, Brian Fullem, Joe Rubio, Peter Larson, Jay Johnson, and Pete Magill freely shared their knowledge and time.
Jeff Dengate provided a thorough and insightful read at the layout stage.
Dave Kayser trusted me with his babies, otherwise known as the old shoes seen in Chapter 3.
Amby Burfoot and Frank Brooks gave helpful advice during difficult writing patches.
Robert Gomez (a 2:23 marathoner) and Julia Kirtland (the 1997 national marathon champion) were patient and willing models.
Eric Alexander and Steve Davis make music that revives flagging spirits.
Stacey Cramp worked her usual photographic magic and, when not holding a camera, gave above-and-beyond spousal support.
# ABOUT THE AUTHOR
**SCOTT DOUGLAS IS NEWS EDITOR** for _Runner's World_ and a former editor of _Running Times_. He's the author or coauthor of five other running books, including _Advanced Marathoning_. Douglas has run more than 100,000 miles since taking up the sport as a teen in 1979. He lives in South Portland, Maine.
# INDEX
**Boldface** page references indicate illustrations.
Underscored references indicate boxed text.
**A**
Achilles tendinitis, 114–16
Adidas shoes
Adipure Adapt, ****
Adizero Hagio, ****
Adizero XT-10, ****
Marathon, 38,
Aging
cushioning need and,
minimalism and, 162–63
Running Man theory and,
_All About Distance Running Shoes_ , , 44, 50–51
Altra shoes
Instinct, , ****,
zero-drop,
Ankles
dorsiflexion test, 97–99, ****, ****
inversion and eversion test, 102–3, ****, ****
plantarflexion,
Arm swing, 15,
Asics shoes
Cumulus,
Gel DS,
Gel Hyperspeed,
Hyperspeed, 62, 87, ****,
Piranha,
**B**
Backward Walk, 152, ****
Balance, single-leg, 85, 104–5
Barefoot running, 121–34
author's experience with, 121–22, 129–30, 133–34
best uses of, 129–34
biomechanics changed by, 60–62
cooldowns for, 131,
decades-long history of, 49–51
diagonals for, 133–34
gradual approach to,
on grass, ****
joint torques reduced in,
premise of this book, 6–7
Running Man theory, 122–29
starting out, 130–31
strides for, 131–33
track infield for, ****
Barefoot Running forum,
Barely there/barefoot-style shoes, 43, ****, 78,
Benson, Roy, 16, 22–23, 25,
Big-toe dorsiflexion test, 99–100, ****, ****
Big-toe isolation test, 101–2, ****, ****
Bikila, Abebe,
Biomechanics. _See also specific aspects_
basic principles of, 19–22
changed when barefoot, 60–62
defined,
of elite runners,
Bizzarri, Angela,
Blogs, 5–6, 196–97
Bohr, Niels,
Bone density, impact and,
_Born to Run_ , 5, , 94,
Braking when running, 4,
_British Journal of Sports Medicine_ ,
Brooks shoes
Cheetah, 89–90
Cheetah 2,
PureConnect, 74, 79, ****
PureGrit, ****
Budd, Zola, 49–50
Butt Kicks, 155, ****
**C**
Carioca, 157, ****
Cassel, Jonathon,
Cavanagh, Peter,
ChiRunning school,
Clams, 142, ****
Cohen, Shoshanna, 150–51
_Complete Book of Running_ ,
Computer work, 26, 158,
Cooldowns, barefoot, 131,
Cordone, Dick,
Core strength,
Cucuzzella, Mark, 110–11
Cushioning
aging and need for,
barefoot-style shoes,
critique of,
determining the need for,
footstrike hemolysis and, 46–47
impact forces higher with,
introduction of, 44–45
merits of,
moderate minimalist shoes,
**D**
Daniels, Jack, 15, 18, 21,
Dawkins, Richard, 182,
Diagonals, barefoot, 133–34
Diamond, Greg, 126–27
Diamond, Jared,
Dicharry, Jay,
Donkey Kicks, 145, ****
Dorsiflexion, 61, 97–100
Douglas, Scott (author of this book)
barefoot experience of, 121–22, 129–30, 133–34
early shoes tried by, 47,
favorite shoe of, 89–90
minimalist experience of, 8–10
racing flats use by, 82–83,
Drills. _See_ Form drills
Dual-density midsole, 76, 183–84
Dugas, Jonathan,
**E**
Elites
barefoot running by, 49–50
defined,
minimalists lacking among, 32–35
Tarahumara tribe visits by,
Englander, Mimi, 2,
**F**
Fast-Feet Shuffle, 153, ****
Feeling the ground, 28,
Fixx, Jim,
Flexibility of shoes,
Footstrike. _See also_ Heel strike; Impact forces; Midfoot strike
adjustments made for surfaces, 58–59
best for most people,
changed when barefoot, 60–61
correcting,
cushioning and injury from,
efficiency of,
forcing changes, avoiding,
forefoot vs. heel, 57,
impact forces and,
injury likelihood and,
natural,
with overstriding,
research on forces at, 68–70
shoes and, 4, 20, 27–28, 60–61
Footstrike hemolysis, 46–47
Forefoot strike
defined,
efficiency of,
heel strike vs., 57,
impact forces and,
increased by barefoot running,
injury likelihood and,
Form. _See also specific elements_
aspects other than, 14, 118–19
benefits of working on, 16,
common elements of, 19–21
common problems with, 21–25
correcting overstriding, 21, 22,
drills, 149–57
forcing changes, avoiding,
general strength aiding,
importance of, 16–17, 18,
improving, 25–27
individual nature of, 17–18
minimalist shoes improving, 13–14, 28–29
natural, 15–16
nonrunning life's affect on,
obsession with,
office work damaging, 26,
readiness for minimalism and, 95–96
risks of changing shoes for,
running economy and,
schools of,
Form drills, 149–57
about, 149–51
Backward Walk, 152, ****
Butt Kicks, 155, ****
Carioca, 157, ****
Fast-Feet Shuffle, 153, ****
G Drill, 154, ****
High Knees, 156, ****
Fullem, Brian
abrupt switch not advised by, 93–94
critique of shoes,
on minimalism for older runners,
minimalism for youths supported by,
minimalist shoe advice,
orthotics advice, 80–81
socks not a problem for,
strengthening advised by,
**G**
Gateway shoes, 79, ****,
G Drill, 154, ****
Glutes, 23,
Goldman, William,
Ground contact time, 60,
_Guns, Germs, and Steel_ ,
**H**
Hall, Ryan,
Hands, cupped when running,
Head during running, 18,
Heel counter,
Heel of shoe
everyday shoes, 159–60
high, said to reduce injuries, 45–46
today vs. 1970s, 40–41,
too high, feet weakened by,
undercut, ****, 43,
Heel strike
bodily adjustments for,
braking due to,
changed when barefoot, 60–61
conventional shoes causing, 4, 27–28
decrease with barefoot running,
defined,
footstrike hemolysis and, 46–47
forefoot strike vs., 57,
impact forces and,
injury likelihood and,
Lieberman's research on, 57, 60–61,
loading with midfoot strike vs., 69–70
with overstriding,
reasons for,
Heel-to-toe drop. _See also_ Ramp angle
calculation of, 190–91
defined,
of minimalist shoes, 73–74
racing flats,
ramp angle and,
recommended amounts,
zero-drop shoes,
Henderson, Joe, 50,
Herron, Camille, 30–31,
High Knees, 156, ****
Hill, Ron,
Hogan, Jim,
Hurrying slowly, 10–11
Hutchinson, Alex,
**I**
Iliotibial (IT) band issues, 64–65,
Impact forces. _See also_ Footstrike
adjustments made for surfaces, 58–59
bone density increased by,
cushioning and injury from,
cushioning's affect on,
footstrike hemolysis and, 46–47
for heel-striking vs. forefoot-striking,
higher vertical loading rate, 65–66,
as input signals, 69,
research on injury and, 65–66, 68–70
Individuality
of form, 17–18
research results and, 54,
Injury
abrupt switch to minimalism leading to, 10, 93–94,
Achilles tendinitis, 114–16
conclusions possible from research, 57–58
contributors to,
counterintuitive research results on,
defined for this book,
definitional problems with,
heel strike leading to,
high heels and, 45–46
impact forces and, 46, 65–66, 68–70
impact-related, 46–47
joint torque research and,
Mango's story, 64–65
minimalist shoes helping eliminate,
most common in 1971,
orthotics for, 80–81
plantar fasciitis, 112–13
prevalence of,
pronation and,
research on causes and rates of, 63–68
Running Man theory and, 124–25
shoe-related, , 31
stress fractures, , , 93–94, 116–17
while transitioning to minimalism, 112–17
inov-8 shoes
Bare-X Lite 135, ****
f-lite 230, ****
Internet resources. _See_ Online resources
IT band issues, 64–65,
**J**
Johnson, Jay
barefoot cooldowns advised by,
easy answers negated by,
minimalism for youths supported by, 161,
running weight advice,
shoe change risks noted by,
on walking in new shoes,
Johnston, Tim,
Joints
barefoot running and, 56,
running weight and stress on,
stiffness adjustments made for surfaces, 58–59
_The Joy of Running_ ,
Jumping, adjustments for surfaces, 58–59
**K**
Kayser, Dave, 41–44
Keflezighi, Meb,
Kenyan runners, 33, 47,
Kerrigan, Casey, 55, 56–57
Kirby, Kevin, 54, 66,
Knee angle upon landing,
Knee Circles, 143, ****
Knee lift, 20–21, 24–25
Kostrubala, Thaddeus,
**L**
Landing position,
Larson, Peter
on footstrike hemolysis, 46–47
on how minimal to go,
minimalism used to fix stride length by,
pronation control questioned by, 179,
rotating shoes advised by, 164–65
runblogger blog, 194
trends noted by, 169, 170,
Lieberman, Daniel
heel strike research, 57, 60–61,
Running Man theory, 122–23
Vibram funding for, 55,
Lower-leg injury contributors,
Lydiard, Arthur,
Lydiard Road Runner, 38,
**M**
Magill, Pete
form improvement advised by, 16–17, 18,
on ills of sitting too much,
knee lift advice, 24–25
on running better,
strengthening advised by,
Magness, Steve
barefoot running tips, 132–33
minimalism for youths supported by, 161–62
research critiqued by,
The Science of Running blog, 194
shoe change risks noted by,
shoe design queries of, 179, 180–81
three-legged stool model, 7–8,
Maintenance. _See_ Minimalist's Maintenance Toolkit
Mango, Paul, 64–65
McDougall, Christopher, 5, , 94,
Mechanics. _See_ Biomechanics
Medial post,
Mendenhall, Adrienne Leigh, 114–15
Merrell shoes
road and trail,
Road Glove, ****, 78,
Trail Glove, ****
zero-drop,
Metzler, Brian, 117, 118–19,
Midflight, horizontal motion in,
Midfoot strike
as best for most people,
conventional shoes and,
defined, 185–86
getting the feeling of,
high heels discouraging,
loading with heel strike vs., 69–70
Midsole
critique of add-ons, 4,
defined,
dual-density, 76, 185–87
medial post inside,
in moderate minimalist shoes,
toe spring and flexibility of,
in trail shoes,
Mileage, 64, 109,
Minimalism. _See also_ Transitioning to minimalism
characteristics of shoes, 72–74, 76–77
defined,
determining how minimal to go, 117–19
future of, 182–83
gradual approach to, 10–11, ,
industry trends in, 168–72
influence of, 175–77
lack of elite minimalists, 32–35
market share of, 173–74
modern life and, 158–60
for older runners, 162–63
orthotics and, 80–81
premise of this book, 6–7
rapid spread of, 2–3
variations in shoes, 71–72
weak points exposed by,
for young runners, 161–62
Minimalist channel,
Minimalist's Maintenance Toolkit, 136–57
about, 136–38
Backward Walk, 152, ****
Butt Kicks, 155, ****
Carioca, 157, ****
Clams, 142, ****
Donkey Kicks, 145, ****
Fast-Feet Shuffle, 153, ****
G Drill, 154, ****
High Knees, 156, ****
Knee Circles, 143, ****
Prone Pedestal, 146, ****
running form drills, 149–57
Seated Calf Raises, 140, ****
Side Leg Raises, 144, ****
Side Pedestal, 148, ****
Supine Pedestal, 147, ****
Towel Pulls, 141, ****
Weighted-Sock Swings, 139, ****
Mizuno shoes
Universe, ****
Wave Universe, 87–88
Moderate minimalist shoes, 78–79, ****,
Moore, Kenny,
Muscle activation, 58–59
**N**
New Balance shoes
890 model,
Jogster, 38,
Minimus Amp, ****
Minimus Road, 40, 74, ****, 79, 87,
RC1400, ****,
RC5000, ****
Newton shoes
Distance, 79, ****
MV racing flats, ****
with negative ramp angle,
Nigg, Benno, 69,
Nike shoes
Air Rift, ****,
Air Skylon T/C,
Free, 5, 79,
LD 1000, 44,
Mayfly,
Pegasus,
**O**
Office work, 158,
Older runners. _See_ Aging
Online resources
collected minimalist links,
links to original research,
runblogger blog,
_Running Times'_ minimalist channel,
The Science of Running blog,
The Science of Sport blog,
shoe specifications,
Sweat Science blog,
Orthotics, 57, 80–81, 188–89
Outsole
defined,
racing flats,
today vs. 1970s,
trail shoes, 82–83, 84,
Overstriding
braking due to,
as common problem, 21–22
conventional shoes causing,
correcting, 21, 22,
defined, ,
determining, 22–23
long stride vs., 21,
**P**
Pace. _See also_ Stride rate
for improving knee lift, 24–25
matching knee lift to, 20–21
stride rate affected by,
for striders, 26–27
Pfitzinger, Pete
common form problems noticed by, 22–24
form improvement advised by,
on ills of sitting too much,
on individuality of form, 17–18
overstriding noticed by, 21–22
Pieterse, Zola (Zola Budd), 49–50
Plantar fasciitis, 112–13
Pose Method,
Pribut, Stephen, 60,
Pronation
defined,
injury and,
lack of evidence for controlling, 178–80
shoe design based on, 4–5, 76, 177–81, 189–90
Prone Pedestal, 146, ****
Proprioceptive ability, 28,
Puma Marathon,
**R**
Racing flats
blown rubber outsole of,
defined,
durability issues for,
elites' use of,
examples of, ****
minimalist shoes replacing,
road shoes vs., 87–88
tapered toebox of,
use as minimalist shoes, 86–88
Radcliffe, Paula,
Ramp angle. _See also_ Heel-to-toe drop
defined,
finding information on,
for first minimalist shoe, 106–7
heel-to-toe drop and,
of minimalist shoes, 73–74
recommended amounts,
Rassmussen, Jeremy, 18, 26,
Rearfoot strike. _See_ Heel strike
Relaxation during runs,
Research
biomechanics, 60–62
challenges for designing,
cherry-picking,
conclusions possible from, 57–58
counterintuitive results of,
footstrike forces, 68–70
funding of, 55–56
individual variations and, 54,
injury causes and rates, 63–68
limitations of, 54–56,
links to original research,
misrepresentation of, 54, 59–60,
pool of subjects for,
replication of results,
running economy, 62–63
running surfaces, 58–60
showing vs. proving in, 56–57
time period of,
Road shoes
racing flats vs., 87–88
trail shoes vs., 80–81, 82–83
Robinson, Roger,
Rodgers, Bill,
Roncker, Bob,
Rotating shoes, 163–65
Rubio, Joe
on big vs. small companies, 171,
cushioning critiqued by,
on not obsessing about form,
rotating shoes advised by, 163–64
trends noted by, 170, 173–74, 175,
Runblogger blog,
Runner profiles
Adrienne Leigh Mendenhall, 114–15
Brandon Wood,
Camille Herron, 30–31
Greg Diamond, 126–27
Paul Mango, 64–65
Shoshanna Cohen, 150–51
Runners. _See also_ Runner profiles
Running Man theory, 122–29
today vs. 1971, , 44,
weight of,
_The Runner's Body_ ,
_Runner's World_
_All About Distance Running Shoes_ , 38–41, , 44, 50–51
Barefoot Running forum,
minimalist links online,
Shoes forum,
Running economy, 15, 16, 18, 62–63
Running form. _See_ Form
Running history
Running Man theory and,
transitioning to minimalism and, 105–6
Running Man theory
critique of, 123–29
overview of, 122–23
Running surfaces, 58–60
_Running Times_ , , 63,
_RunningWarehouse. com_ ,
Rupp, Galen,
**S**
Saucony Kinvara
going more minimal, 117,
heel-to-toe drop,
as transitional shoe, 79, ****
Wood's choice of,
Saucony Ride, ****
The Science of Running blog,
The Science of Sport blog,
Seated Calf Raises, 140, ****
Secrets, nonexistence of,
Shoes. _See also specific characteristics and brands_
articulated toe, 43, ****,
author's favorite, 89–90
barely there/barefoot-style, 43, ****, 78,
bodily adjustments for, 59,
characteristics of minimalist, 72–74, 76–77
choosing a pair to start, 105–7
collections of, 41–42,
early minimalist shoes, 47,
everyday, 159–60
finding a favorite shoe, 89–91
heel strike and,
how minimal to go, 117–19
industry trends in, 168–72
market share of minimalist, 173–74
matching to fatigue level,
as means to an end, 13–14,
minimalist vs. conventional, ****, 27–29, 37–38,
mixed reviews common for, 37–39
moderate minimalist, 78–79, ****,
overlapping categories of,
pronation-based design of, 4–5, 76, 177–81, 187–88
racing flats, 35, 86–88, ****, 177,
road vs. racing flats, 87–88
road vs. trail, 80–81, 82–83
rotating kinds of, 163–65
running economy and weight of, 62–63
spikes, elites' use of,
stride rate affected by,
today vs. 1970s or earlier, 39–41, 43–44
toe, 2, 33, 43, 189–90
trail, 80–86, ****,
transitional/gateway, 79, ****,
zero-drop, 74,
Shoes forum,
Shorter, Frank,
Side Leg Raises, 144, ****
Side Pedestal, 148, ****
Single-leg balance, , 104–5
Sitting, ill effects of, 26, 158,
Skechers GoRun, 74, ****, 79,
Slaney, Mary (Decker), 49–50
Socks
articulated toe in, ****,
wearing vs. going sockless,
Spikes, elites' use of,
Stack height
benefits of lower,
defined,
finding information on,
heel-to-toe drop calculation from, 190–91
of minimalist shoes,
moderate minimalist shoes,
racing flats vs. minimalist shoes, 87–88
recommended amounts,
Stewart, Phil,
Stool, three-legged, 7–8,
Strength training. _See_ Minimalist's Maintenance Toolkit
Stress fractures, , , 93–94, 116–17
Stress reactions, 116–17
Stride length. _See also_ Overstriding
decreased when barefoot,
as function of fitness,
shortened by minimalist shoes,
Stride rate. _See also_ Pace
defined,
increased when barefoot,
overstriding fixed by, 21,
recommended rate, 21,
Striders or strides, 26–27, 109, 131–33
Strike. _See_ Footstrike
Supine Pedestal, 147, ****
Surfaces, 58–60
Sweat Science blog,
**T**
Tanaka, Shigeki, 43,
Tarahumara tribe,
Taras (early minimalist shoes), 47,
Tests. _See_ Transitioning to minimalism
Tiger shoes
Cortez,
Marathon, 38, 40, ****
Obori, ****,
Toebox
articulated toe, 43, ****,
benefits of wide, 29,
defined,
of minimalist shoes,
in older shoes,
of racing flats,
Toe shoes
articulated toe, 43, ****,
defined, 191–92
Englander's experience with,
1951 example,
refused by Kenyan runners,
Toe spring,
Tomkinson, Mark, 1–2,
Toolkit. _See_ Minimalist's Maintenance Toolkit
Towel Pulls, 141, ****
Trail shoes, 80–86, ****,
Trail ultras,
Transitional/gateway shoes, 79, ****,
Transitioning to minimalism, 93–119
abrupt, injury due to, 10, 93–94,
allowing recovery time,
choosing a first shoe, 105–7,
Englander's experience, 2,
going barefoot at home,
gradual approach, 10–11, , , 95–96, 107–10
harder running, 109–10
how minimal to go, 117–19
injuries to beware of, 112–17
Mendenhall's experience, 114–15
mileage considerations, 109,
need for,
radical plan for, 110–11
ramp angle to start with, 106–7
running history and, 105–6
stepping back when needed,
striders for,
Test #1: ankle dorsiflexion/posterior chain range of motion, 97–99, ****, ****
Test #2: big-toe dorsiflexion, 99–100, ****, ****
Test #3: big-toe isolation, 101–2, ****, ****
Test #4: ankle inversion and eversion, 102–3, ****, ****
Test #5: single-leg balance, 104–5, ****
tests, body imbalances during,
tests, failing,
Tomkinson's experience, 1–2,
training regimen and,
walking in new shoes, 107,
Trason, Ann, 80–81
Tucker, Ross, 130,
Tulloh, Bruce, ,
Turnover. _See_ Stride rate
**U**
Undercut heel, ****, 43,
_Unweaving the Rainbow_ ,
Upper body position, 21,
Upper of shoe, 76, 83,
**V**
Vertical motion, avoiding in midflight,
Vibram company
lawsuit against, 169–70
Lieberman funded by, 55,
market share of, 168–69
Vibram FiveFingers
Asics Cumulus compared to,
as barefoot-style shoe, ****,
early popularity of,
Englander's experience with,
market competition for, 170–71
refused by Kenyan runners,
Seeya model, ****
as trail shoes,
wane of, 169–70
VivoBarefoot, study touted by,
VivoBarefoot Evo, ****, ****,
**W**
Waitz, Grete, 10–11
Web sites. _See_ Online resources
Weighted-Sock Swings, 139, ****
Weight of runners, optimal,
Weight of shoes
critique of,
finding information on,
minimalist,
racing flats vs. minimalist shoes, 87–88
recommended amounts,
running economy and,
today vs. 1971,
trail vs. road,
trends in, 175–77
Wharton, Phil
barefoot running tips,
elites' biomechanics praised by,
gradual approach advised by, 94–96,
on ills of sitting too much,
minimalism readiness tests developed by,
on minimalism-specific injuries,
poor shoes critique,
running weight advice,
Wolff's Law,
Wood, Brandon,
**Y**
Young runners, minimalism for, 161–62
**Z**
Zero-drop shoes, 74,
The information in this book is meant to supplement, not replace, proper exercise training. All forms of exercise pose some inherent risks. The editors and publisher advise readers to take full responsibility for their safety and know their limits. Before practicing the exercises in this book, be sure that your equipment is well maintained, and do not take risks beyond your level of experience, aptitude, training, and fitness. The exercise and dietary programs in this book are not intended as a substitute for any exercise routine or dietary regimen that may have been prescribed by your doctor. As with all exercise programs, you should get your doctor's approval before beginning.
Mention of specific companies, organizations, or authorities in this book does not imply endorsement by the author or publisher, nor does mention of specific companies, organizations, or authorities imply that they endorse this book, its author, or the publisher.
Internet addresses and telephone numbers given in this book were accurate at the time it went to press.
© 2013 by Scott Douglas
All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any other information storage and retrieval system, without the written permission of the publisher.
_Runner's World_ is a registered trademark of Rodale Inc.
Book design by Christopher Rhoads
Photographs by Stacey Cramp
Library of Congress Cataloging-in-Publication Data is on file with the publisher.
ISBN–13: 978–1–60961–222–1
eISBN–13: 978–1–60961–227–6
**We inspire and enable people to improve their lives and the world around them.**
rodalebooks.com
| {
"redpajama_set_name": "RedPajamaBook"
} | 7,429 |
// @flow
/* eslint no-underscore-dangle: 0 */
/* **********************************************************
* File: Footer.spec.js
*
* Brief: Test for the react footer component
*
* Author: George Whitfield
* Date: 2017.07.27
*
********************************************************* */
import * as React from 'react';
import { shallow } from 'enzyme';
import { __RewireAPI__ as Rewire } from '../../app/components/Footer';
import Footer from '../../app/components/Footer';
// The default export from Fotoer.js needs to be imported seperately or else Jest throws an error
function setup() {
const component = shallow(<Footer />);
return {
component
};
}
const mitLogoStyle = Rewire.__get__('mitLogoStyle');
const bilabLogoStyle = Rewire.__get__('bilabLogoStyle');
const footerStyle = Rewire.__get__('footerStyle');
// Begin test
describe('Footer test', () => {
const { component } = setup();
const grid = component.find('Grid');
describe('Grid JSX Element', () => {
it('Has correct values for props', () => {
expect(grid.at(0).prop('className')).toEqual('Footer');
expect(grid.at(0).prop('style')).toBe(footerStyle);
expect(grid.at(0).prop('fluid')).toBe(true);
});
it('Colomn elements have correct props', () => {
const col = component.find('Col');
expect(col.at(0).prop('xs')).toBe(4);
expect(col.at(1).prop('xs')).toBe(4);
expect(col.at(2).prop('xs')).toBe(4);
});
it('Image elements match snapshot', () => {
const img = component.find('img');
expect(img.at(0)).toMatchSnapshot();
expect(img.at(1)).toMatchSnapshot();
});
it('variables test', () => {
expect(footerStyle).toMatchSnapshot();
expect(mitLogoStyle).toMatchSnapshot();
expect(bilabLogoStyle).toMatchSnapshot();
});
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,209 |
{"url":"https:\/\/stats.libretexts.org\/Courses\/Taft_College\/PSYC_2200%3A_Elementary_Statistics_for_Behavioral_and_Social_Sciences_(Oja)\/Unit_1%3A_Description\/5%3A_Using_z\/5.06%3A_The_Write-Up","text":"# 5.6: The Write-Up\n\n\nThrough the practice examples, I hope that you have realized that when conducting statistics for the social sciences, the answer is never just the number. We do the statistics to answer questions, to the final answer needs enough information to answer that question, and to let other statisticians know a little bit about the sample and the calculations. Based on what we\u2019ve learned so far, here\u2019s what you might include in a concluding sentence, as well as what should be included in a paper describing a distribution.\n\n## Concluding Sentence\n\nFor any conclusion, you should include the results of your calculations, what was measured, and the answer to the original research question. Sometimes, this might be as simple as:\n\n\u2022 Research Question: What is the average final exam score?\n\u2022 Conclusion: The average final exam score was 77.7 points.\n\nThe Research Question from Exercise 5.4.1 was:\n\n\u2022 Research Question: How many students earned 90 points or higher on the Final Exam?\n\nSo the Conclusion should be:\n\n\u2022 Conclusion: Based on the mean, standard deviation, and size of this sample, 1 of the 20 students should earn 90 points or higher on the Final Exam.\n\nThe important pieces of information to include in these concluding sentences are the research question (rephrase as an answer), the calculation results, and what was measured. For Exercise 5.4.1, those are:\n\n1. Research Question: \u201cHow many students earned 90 points or higher on the Final Exam?\u201d was turned in to \u201c1 student should earn 90 points or higher on the Final Exam.\u201d\n2. Calculation: This is the \u201c1 of the 20 students\u201d part of the conclusion.\n3. What was measured: This is your DV, your outcome variable. In this Exercise, it was points earned on the Final Exam.\n\nFor something a little more advanced you will need to include more information. We'll cover that a little later!\n\n## Paper Describing a Distribution\n\nFor a full paper to describe a distribution, you will combine the conclusion for everything that we\u2019ve covered so far. This should include:\n\n1. Describing who and what was measured. This should include:\n1. Naming the sample. Who provided the data? How many participants were there?\n2. Naming who you think the population could be. In other words, name who is the biggest group that the sample can represent?\n3. Naming what was measured (quantitative DV).\n2. Interpreting what the measures of central tendency mean.\n1. Make sure that you calculate the mean, median, and mode correctly!\n2. To interpret the measures of central tendency, describe what does knowing the mean, median, and mode collectively tell you? Maybe answering these questions will help: Are the mean, median, and mode similar? What could that tell us about the shape of the distribution? Is one smaller or bigger? What could that tell us about the shape of the distribution? Are they all very different? What could that tell us about the shape of the distribution?\n3. Interpreting what the standard deviation can tell us.\n1. Make sure that you calculated the standard deviation correctly!\n2. An interpretation of standard deviation should include:\n1. An evaluation of whether one standard deviation above and below the mean really includes about 68% of the scores, like it should if the data was normally distributed.\n2. Use the standard deviation to predict the shape of the distribution (tall\/narrow, medium\/normal, or wide\/flat), then compare the predicted shape to the actual shape.\n3. If the standard deviation seems large, you would expect a platykurtic distribution (wide and flat). For example, if you look at your frequency chart, does the shape seem wide and flat? Small samples are often not normally distributed, so what we might expect based on the standard deviation is not the actual shape of the distribution. Plus, outliers can cause skewed shapes and large standard deviations. The standard deviation gives us a general idea of how different each score is from the mean, but there\u2019s nothing better than looking at the actual distribution.\n4. Providing and describing\/interpreting appropriate frequency charts.\n1. Don\u2019t forget to format the chart number and title in appropriate APA Style!\n2. You should mention the chart (by Figure by number) in the paper body.\n3. Just like we did for each different types of chart, you should describe\/interpret the frequency chart by saying something about what you see or what the chart makes you wonder.\n\nAnd don\u2019t forget that a paper in your statistics class is still a paper! You should have an introduction with some sort of hook (Why is this topic interesting?), the body (which should include everything above), and a concluding paragraph. Concluding paragraphs often include why this topic is important (which may refer back to the hook from the introduction), or who would want to know this information. A one- or two-sentence summary of what was found could also be included, but don\u2019t get too hung up on that.\n\nThe next chapter will discuss how to format this paper describing a data set in APA Style.\n\n5.6: The Write-Up is shared under a CC BY-SA 4.0 license and was authored, remixed, and\/or curated by Michelle Oja.","date":"2022-06-29 10:20:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5391746163368225, \"perplexity\": 670.2431796721531}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103626162.35\/warc\/CC-MAIN-20220629084939-20220629114939-00694.warc.gz\"}"} | null | null |
{"url":"http:\/\/openstudy.com\/updates\/526f0316e4b0e209601e3259","text":"osanseviero 2 years ago In a geometric series, a2=6 a5=48 Which is the first term, which is the constant and write the progression\n\n1. osanseviero\n\nI know that $an=a1*r ^{n-1}$ What goes next?\n\n2. osanseviero\n\n6=a1*r^(n-1) ?\n\n3. osanseviero\n\n6=a1*r^(2-1), 6=a1*r\n\n4. anonymous\n\nAs you said : $a_n=a_1\\times q^{n-1}$ where q is the constant of the serie, so we have : $a_2=a_1\\times q\\\\a_5=a_1\\times q^4$ Is that true ?\n\n5. osanseviero\n\nYep\n\n6. anonymous\n\nso we get : $6=a_1\\times q~~~~~~(1)\\\\48=a_1\\times q^4~~~(2)$ Now, we can divide the equation (2) over (1) , what should we get ?\n\n7. osanseviero\n\n8=q^3, q=2!\n\n8. anonymous\n\nGood. Now the 1st term can be found easily, can't it ?\n\n9. osanseviero\n\nyepp :D\n\n10. osanseviero\n\n3 :) thanks\n\n11. osanseviero\n\nSo an=3*2^n-1 ?\n\n12. anonymous\n\nYes, it is ! And you are welcome !","date":"2016-05-24 19:35:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9497207403182983, \"perplexity\": 3232.13546013916}, \"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-2016-22\/segments\/1464049273643.15\/warc\/CC-MAIN-20160524002113-00085-ip-10-185-217-139.ec2.internal.warc.gz\"}"} | null | null |
Q: Using 'where' in Sequelize Many-to-Many query Just started using sequelize, how and 'where' do you use where to specify condition on a table? I have the belongstomany relationship defined like this:
db.users.belongsToMany(db.groups, {through: "users_groups", foreignKey:"user_id"});
db.groups.belongsToMany(db.users, {through: "users_groups",foreignKey:"group_id"});
1) I want to be able to be able to see if a user is in a group given a user id
2) I want to retrieve all groups and the groups info attended by a user given a user id
where do I put the "where" in these cases?
I tried
Group.findOne({
through: {where:{id:req.body.group_id}},
include: [{
model: User,
through:{
where:{id:req.body.user_id}
}
}]
}).then(data=>{
res.send(data);
})
A: through property expects a junction model or an alias, you are giving it a where caluse.
const { group_id: groupId, user_id: userId } = req.body;
Group.findOne({
where: { id: groupId },
include: [{
model: User,
through: 'users_groups',
where: { id: userId },
}]
}).then(data=>{
res.send(data);
})
For more information read here
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,336 |
//BSD, 2014-present, WinterDev
//MatterHackers
#define USE_CLIPPING_ALPHA_MASK
using System;
using PixelFarm.CpuBlit.VertexProcessing;
using PixelFarm.CpuBlit.PixelProcessing;
using PixelFarm.Drawing;
using Mini;
namespace PixelFarm.CpuBlit.Sample_LionAlphaMask
{
[Info(OrderCode = "05")]
[Info(DemoCategory.Bitmap, "Clipping to multiple rectangle regions")]
public class LionAlphaMask3 : DemoBase
{
int _maskAlphaSliderValue = 100;
double _angle = 0;
double _lionScale = 1.0;
double _skewX = 0;
double err_skewY = 0;
bool _isMaskSliderValueChanged = true;
MemBitmap lionImg;
MemBitmap _alphaBitmap;
public LionAlphaMask3()
{
string imgFileName = "Data/lion1.png";
if (System.IO.File.Exists(imgFileName))
{
lionImg = MemBitmapExt.LoadBitmap(imgFileName);
}
this.Width = 800;
this.Height = 600;
#if USE_CLIPPING_ALPHA_MASK
//alphaMask = new AlphaMaskByteClipped(alphaMaskImageBuffer, 1, 0);
#else
//alphaMask = new AlphaMaskByteUnclipped(alphaMaskImageBuffer, 1, 0);
#endif
}
public override void Init()
{
}
void SetupMaskPixelBlender(int width, int height)
{
//----------
//same size
_alphaBitmap = new MemBitmap(width, height);
var alphaPainter = AggPainter.Create(_alphaBitmap, new PixelBlenderBGRA());
alphaPainter.Clear(Color.Black);
//------------
System.Random randGenerator = new Random(1432);
int i;
int num = (int)_maskAlphaSliderValue;
num = 50;
int elliseFlattenStep = 64;
using (Tools.BorrowVxs(out var v1))
using (Tools.BorrowEllipse(out var ellipseForMask))
{
for (i = 0; i < num; i++)
{
if (i == num - 1)
{
////for the last one
ellipseForMask.Set(Width / 2, (Height / 2) - 90, 110, 110, elliseFlattenStep);
ellipseForMask.MakeVxs(v1);
alphaPainter.FillColor = new Color(255, 255, 255, 0);
alphaPainter.Fill(v1);
v1.Clear();
//
ellipseForMask.Set(ellipseForMask.originX, ellipseForMask.originY, ellipseForMask.radiusX - 10, ellipseForMask.radiusY - 10, elliseFlattenStep);
ellipseForMask.MakeVxs(v1);
alphaPainter.FillColor = new Color(255, 255, 0, 0);
alphaPainter.Fill(v1);
v1.Clear();
//
}
else
{
ellipseForMask.Set(randGenerator.Next() % width,
randGenerator.Next() % height,
randGenerator.Next() % 100 + 20,
randGenerator.Next() % 100 + 20,
elliseFlattenStep);
ellipseForMask.MakeVxs(v1);
alphaPainter.FillColor = new Color(100, 255, 0, 0);
alphaPainter.Fill(v1);
v1.Clear();
}
}
}
maskPixelBlender.SetMaskBitmap(_alphaBitmap);
maskPixelBlenderPerCompo.SetMaskBitmap(_alphaBitmap);
}
[DemoConfig(MinValue = 0, MaxValue = 255)]
public int MaskAlphaSliderValue
{
get
{
return _maskAlphaSliderValue;
}
set
{
_maskAlphaSliderValue = value;
_isMaskSliderValueChanged = true;
}
}
PixelBlenderWithMask maskPixelBlender = new PixelBlenderWithMask();
PixelBlenderPerColorComponentWithMask maskPixelBlenderPerCompo = new PixelBlenderPerColorComponentWithMask();
public override void Draw(Painter p)
{
AggPainter painter = p as AggPainter;
if (painter == null) return;
//
painter.Clear(Color.White);
int width = painter.Width;
int height = painter.Height;
//change value ***
if (_isMaskSliderValueChanged)
{
SetupMaskPixelBlender(width, height);
_isMaskSliderValueChanged = false;
//
//painter.DestBitmapBlender.OutputPixelBlender = maskPixelBlender; //change to new blender
painter.DestBitmapBlender.OutputPixelBlender = maskPixelBlenderPerCompo; //change to new blender
}
//1. alpha mask...
//p2.DrawImage(alphaBitmap, 0, 0);
//2.
painter.FillColor = Color.Black;
painter.FillRect(0, 0, 200, 100);
//3.
painter.FillColor = Color.Blue;
painter.FillCircle(300, 300, 100);
painter.DrawImage(lionImg, 20, 20);
////4.
//painter.FillColor = Color.Black;
////this test lcd-effect => we need to draw it 3 times with different color component, on the same position
////(same as we do with OpenGLES rendering surface)
//maskPixelBlenderPerCompo.SelectedMaskComponent = PixelBlenderColorComponent.B;
//maskPixelBlenderPerCompo.EnableOutputColorComponent = EnableOutputColorComponent.B;
//painter.FillRect(0, 0, 200, 100);
//maskPixelBlenderPerCompo.SelectedMaskComponent = PixelBlenderColorComponent.G;
//maskPixelBlenderPerCompo.EnableOutputColorComponent = EnableOutputColorComponent.G;
//painter.FillRect(0, 0, 200, 100);
//maskPixelBlenderPerCompo.SelectedMaskComponent = PixelBlenderColorComponent.R;
//maskPixelBlenderPerCompo.EnableOutputColorComponent = EnableOutputColorComponent.R;
//painter.FillRect(0, 0, 200, 100);
}
public override void MouseDown(int x, int y, bool isRightButton)
{
doTransform(this.Width, this.Height, x, y);
}
public override void MouseDrag(int x, int y)
{
doTransform(this.Width, this.Height, x, y);
base.MouseDrag(x, y);
}
void doTransform(double width, double height, double x, double y)
{
x -= width / 2;
y -= height / 2;
_angle = Math.Atan2(y, x);
_lionScale = Math.Sqrt(y * y + x * x) / 100.0;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,807 |
\section{Introduction}
Turbulent convection occurs frequently in stellar interiors and other
astrophysical fluid flows. While convective motion naturally
transports heat and chemical elements, the transport of angular
momentum by convection in rotating bodies is a more subtle issue. It
is of particular interest in the case of the Sun, where the internal
pattern of rotation has been measured but remains incompletely
understood. It may also play an significant role in accretion flows.
Numerical simulations of astrophysical convection are becoming
increasingly powerful and capable of resolving a widening range of
length and time scales. Nevertheless, a simpler, statistical
description of turbulent transport is desirable in order to treat the
effects of convection on the structure and evolution of stars. It
almost goes without saying that such a description cannot be derived
strictly from the equations of fluid dynamics but must involve some
modelling or parametrization.
The mixing-length theory of turbulent transport was developed by
Prandtl (1925) and applied to stellar convection by Biermann (1932).
It is still the basic model used in most calculations of stellar
structure and evolution, usually in the form devised by B\"ohm-Vitense
(1958). The main purpose of mixing-length theory is to relate the
convective heat flux to the superadiabatic gradient; in this context
it does not usually deal with the transport of (angular) momentum that
arises in the presence of shear or rotation.
A standard theoretical approach to convection in differentially
rotating stars is set out in the monograph by R\"udiger (1989).
Angular momentum transport is described by a Reynolds stress tensor
whose components can be related to the large-scale mean flows and
thermodynamical gradients. A first contribution to the Reynolds stress
is typically proportional to the angular velocity gradient through
a turbulent viscosity coefficient. An important additional contribution
comes from the $\Lambda$-effect (named after Lebedinsky),
whereby even uniformly rotating convection transports angular momentum
by virtue of its anisotropy. Attempts to constrain or parameterize
these quantities have been made through local
numerical simulations (e.g.\ K\"apyl\"a, Korpi \& Tuominen
2004) or theoretical models (e.g.\ Kitchatinov \& R\"udiger
1993). Mean-field models of stellar rotation
(e.g. Kitchatinov \& R\"udiger 1999, Rempel 2005) have
been developed which use such parameterized expressions for the Reynolds
stress and heat flux.
Reynolds-stress models of turbulent flows have been developed in the
engineering community over several decades (e.g.\ Pope 2000).
The exact equation governing the Reynolds stress in a turbulent fluid
cannot be solved because of the well known closure problem whereby an
infinite hierarchy of correlations is involved. Nevertheless, by
parametrizing the difficult terms in this equation, models can be
constructed that bear some fidelity to the turbulent dynamics. From a
more physical point of view, what is obtained is a time-dependent
constitutive equation for the turbulent fluid, which relates the
turbulent stress to the local history of deformation. There is a
close similarity with models of non-Newtonian fluids (Ogilvie \& Proctor 2003).
The advection and deformation of the turbulent stress are accurately represented
since they derive from linear terms in the Reynolds-stress equation,
while the nonlinear `relaxation' effects are only modelled (as is also
true for non-Newtonian fluids).
A similar approach can be applied to turbulent convection in which
buoyancy forces play an essential role. The additional correlations
that must be considered are the flux and the variance of entropy (or
temperature, in the Boussinesq approximation). This approach offers
some benefits over the conventional description in terms of a
turbulent viscosity and a $\Lambda$-effect. It can be formulated in a
covariant manner and is not tied to the spherical geometry of a slowly
rotating star. It starts from a more fundamental description and
allows phenomena such as the $\Lambda-$effect to emerge in a natural way
from more elementary considerations. It may also allow a more unified
approach to be taken towards problems involving astrophysical
turbulence.
In this paper we explore some of the consequences of a simple
dynamical model of astrophysical convection of this type. The model
derives from one originally conceived for magnetohydrodynamic
turbulence in accretion discs (Ogilvie 2003) and later applied to
rotating shear flows without magnetic fields (Garaud \& Ogilvie 2005,
GO05 herafter).
Our motivation is to develop and test a model that can be applied to
the convective zone of the Sun, to other stars or to accretion discs.
We emphasize, however, that our model is chosen to be as simple as
possible for the purposes of this investigation. In contrast with
some of the engineering literature, we restrict the algebraic
complexity in order to retain a physical understanding of the terms in
the equations. Further refinements are likely to be required in order
to provide an accurate match to a wide range of data.
In comparing a closure model of astrophysical convection with
experimental and numerical results, we face certain difficulties.
Astrophysical convection usually takes place at very high Rayleigh
number, in a highly turbulent regime. Experiments have been conducted
at very high Rayleigh number but mainly for the Rayleigh--B\'enard
problem in which the flow is dominated by boundary layers, which may
not be relevant in an astrophysical context, or by mean flows not
represented in the closure model. An alternative system is provided
by the homogeneous Rayleigh--B\'enard problem, which has periodic
boundary conditions in all directions. This model, however, has
certain peculiarities of its own. These issues will be addressed in
the sections that follow.
In the remainder of the paper, we develop the closure model first in
the Boussinesq approximation (Section~2) and apply it to the standard
Rayleigh--B\'enard problem (Section~3). We then consider the
homogeneous Rayleigh--B\'enard system with triply periodic boundary
conditions (Section~4); in this section we also introduce rotation and
discuss the $\Lambda-$effect. We then adapt the model to the anelastic
approximation for use in stars and other astrophysical flows
(Section~5) and finally draw conclusions (Section~6). A number of
technical details are covered in the appendices.
\section{Closure model in the Boussinesq system}
\subsection{Basic equations}
In the Boussinesq approximation (e.g. Chandrasekhar 1961) the
equations governing the motion of the fluid are
\begin{equation}
\partial_iu_i=0,
\end{equation}
\begin{equation}
\rho_0(\partial_t+u_j\partial_j)u_i=\rho g_i-\partial_ip+
\rho_0\nu\partial_{jj}u_i,
\end{equation}
\begin{equation}
\rho=\rho_0\left[1-\alpha(T-T_0)\right],
\end{equation}
\begin{equation}
(\partial_t+u_i\partial_i)T=\kappa\partial_{ii}T,
\end{equation}
where we have adopted a Cartesian tensor notation. The dynamical
variables are the velocity $\mbox{\boldmath$u$}$, the density $\rho$, the pressure $p$
and the temperature $T$. Quantities regarded as constant in the
Boussinesq approximation are the reference density $\rho_0$, the
reference temperature $T_0$, the coefficient of expansion $\alpha$,
the gravitational acceleration $\mbox{\boldmath$g$}$, the kinematic viscosity $\nu$,
and the thermal diffusivity $\kappa$.
A simple, static basic state is possible when the temperature is
uniform and the pressure gradient balances gravity, i.e.
\begin{equation}
T=T_0,
\end{equation}
\begin{equation}
p=p_0+\rho_0g_ix_i,
\end{equation}
where $p_0$ is a reference pressure. To examine departures from this
state we define
\begin{equation}
\Theta=T-T_0,
\end{equation}
\begin{equation}
\psi={{p-(p_0+\rho_0g_ix_i)}\over{\rho_0}},
\end{equation}
obtaining the governing equations
\begin{equation}
\partial_iu_i=0,
\label{boussinesq1}
\end{equation}
\begin{equation}
(\partial_t+u_j\partial_j)u_i=-\alpha\Theta g_i-\partial_i\psi+
\nu\partial_{jj}u_i,
\label{boussinesq2}
\end{equation}
\begin{equation}
(\partial_t+u_i\partial_i)\Theta=\kappa\partial_{ii}\Theta.
\label{boussinesq3}
\end{equation}
\subsection{Fluctuations}
We now adopt a standard procedure and separate the dynamical variables
into mean and fluctuating parts, e.g.
\begin{equation}
u_i=\bar u_i+u_i',\qquad
\langle u_i'\rangle=0,
\end{equation}
where the angle brackets or the overbar are interchangeably used to
denote a suitable averaging operation such as
a temporal, spatial or ensemble average. The mean parts of the
governing equations are
\begin{equation}
\partial_i\bar u_i=0,
\label{eq:meancont}
\end{equation}
\begin{equation}
(\partial_t+\bar u_j\partial_j)\bar u_i=-\alpha\bar\Theta g_i-
\partial_i\bar\psi+\nu\partial_{jj}\bar u_i-\partial_j\bar R_{ij},
\end{equation}
\begin{equation}
(\partial_t+\bar u_i\partial_i)\bar\Theta=\kappa\partial_{ii}\bar\Theta-
\partial_i\bar F_i,
\label{eq:meanenergy}
\end{equation}
where
\begin{equation}
R_{ij}=u_i'u_j'
\end{equation}
is the Reynolds tensor, representing (minus) the turbulent stress, and
\begin{equation}
F_i=\Theta'u_i'
\end{equation}
represents the turbulent heat flux density. The problem at hand is to
determine $\bar R_{ij}$ and $\bar F_i$ and thereby close the system
of mean equations. We also introduce the quantity
\begin{equation}
Q=\Theta^{\prime2},
\end{equation}
representing the temperature variance.
It should be noted that all three quadratic correlations $R_{ij}$, $F_i$ and
$Q$ will be redefined when we
move on to the (more relevant) anelastic system in which the reference density is non-uniform, but these definitions
are convenient for the Boussinesq system.
The fluctuating parts of the governing equations are
\begin{equation}
\partial_iu_i'=0,
\label{divu'}
\end{equation}
\begin{eqnarray}
\lefteqn{(\partial_t+\bar u_j\partial_j)u_i'+u_j'\partial_j\bar u_i=
-\alpha\Theta'g_i-\partial_i\psi'+\nu\partial_{jj}u_i'}&\nonumber\\
&&-\partial_j(R_{ij}-\bar R_{ij}),
\label{u'}
\end{eqnarray}
\begin{equation}
(\partial_t+\bar u_i\partial_i)\Theta'+
u_i'\partial_i\bar\Theta=\kappa\partial_{ii}\Theta'-
\partial_i(F_i-\bar F_i).
\label{t'}
\end{equation}
From these we can obtain exact equations for $\bar R_{ij}$, $\bar F_i$
and $\bar Q$ in the form
\begin{eqnarray}
&&(\partial_t+\bar u_k\partial_k)\bar R_{ij}+
\bar R_{ik}\partial_k\bar u_j+\bar R_{jk}\partial_k\bar u_i\nonumber \\
&& \qquad +
\alpha(\bar F_ig_j+\bar F_jg_i) - \nu \partial_{kk} \bar R_{ij}
=-\langle u_i'\partial_j\psi'+u_j'\partial_i\psi'\rangle\nonumber\\
&&\qquad-
\langle u_i'\partial_kR_{jk}+u_j'\partial_kR_{ik}\rangle - 2\nu\langle \partial_{k} u_i'\partial_{k }u_j'\rangle,
\end{eqnarray}
\begin{eqnarray}
\lefteqn{(\partial_t+\bar u_j\partial_j)\bar F_i+
\bar R_{ij}\partial_j\bar\Theta+\bar F_j\partial_j\bar u_i+
\alpha\bar Qg_i - {\textstyle\frac{1}{2}}(\nu + \kappa) \partial_{jj} \bar F_i} &\nonumber\\
&&\qquad=-\langle\Theta'\partial_i\psi'\rangle-
\langle\Theta'\partial_jR_{ij}+u_i'\partial_jF_i\nonumber\rangle\\
&&\qquad + {\textstyle\frac{1}{2}}(\nu-\kappa) \langle \partial_j (\Theta'\partial_{j}u_i' - u_i' \partial_j \Theta') \rangle\nonumber\\
&&\qquad -
(\nu+ \kappa) \langle \partial_j u_i' \partial_{j}\Theta'\rangle,
\end{eqnarray}
\begin{eqnarray}
&& (\partial_t+\bar u_i\partial_i)\bar Q+2\bar F_i\partial_i\bar\Theta - \kappa \partial_{ii} \bar Q \nonumber \\
&&\qquad =
-2\langle\Theta'\partial_iF_i\rangle-
2\kappa\langle (\partial_{i}\Theta')^2 \rangle.
\end{eqnarray}
The left-hand sides of these equations represent the linear
interaction of $\bar R_{ij}$, $\bar F_i$ and $\bar Q$ with the mean
velocity gradient, the mean temperature gradient and the gravitational
field, as well as their diffusion by the microscopic transport coefficients. There is no difficulty in treating such terms exactly as they
appear. The right-hand sides of these equations contain difficult
terms of three sorts: those involving correlations with the pressure
fluctuation $\psi'$, those involving triple correlations of
fluctuating quantities, and dissipative terms involving the microscopic
diffusivities $\nu$ and $\kappa$. These effects can all be regarded
as `non-linear'; although viscous diffusion, for example, is a linear
process, when the Reynolds number is large the viscous terms can be
significant only when a turbulent cascade has forced structure to
appear on the dissipative scales. None of the terms on the right-hand
sides of these equations can be written in terms of $\bar R_{ij}$,
$\bar F_i$ and $\bar Q$ without further knowledge of the statistical
properties of the fluctuating quantities, such as the spectrum of the
turbulence, which are determined by the non-linear physics of the
turbulent cascade.
\subsection{Proposed closure model}
We therefore attempt to model the system by retaining the exact forms
of the left-hand sides and proposing simple closures for the
right-hand sides, i.e.
\begin{eqnarray}
&& (\partial_t+\bar u_k\partial_k)\bar R_{ij}+
\bar R_{ik}\partial_k\bar u_j+\bar R_{jk}\partial_k\bar u_i\nonumber\\
&&\qquad+\alpha(\bar F_ig_j+\bar F_jg_i)-\nu \partial_{kk} \bar R_{ij}\nonumber\\
&&\qquad =\mathcal{F}_{ij}(\bar R_{ij},\bar F_i,\bar Q,\dots),
\label{dtrij}
\end{eqnarray}
\begin{eqnarray}
\lefteqn{(\partial_t+\bar u_j\partial_j)\bar F_i+
\bar R_{ij}\partial_j\bar\Theta+
\bar F_j\partial_j\bar u_i+\alpha\bar Qg_i - {\textstyle\frac{1}{2}}(\nu + \kappa)\partial_{jj} \bar F_i }&\nonumber\\
&&=\mathcal{F}_i(\bar R_{ij},\bar F_i,\bar Q,\dots),
\end{eqnarray}
\begin{equation}
(\partial_t+\bar u_i\partial_i)\bar Q+2\bar F_i\partial_i\bar \Theta - \kappa \partial_{ii} \bar Q =
\mathcal{F}(\bar R_{ij},\bar F_i,\bar Q,\dots),
\label{dtq}
\end{equation}
where the quantities $\mathcal{F}$ are non-linear tensorial functions of their
arguments. The dots represent the parameters of the problem, on which
the functions $\mathcal{F}$ may depend.
A simple example of such a model is
\begin{eqnarray}
&& (\partial_t+\bar u_k\partial_k)\bar R_{ij}+
\bar R_{ik}\partial_k\bar u_j+\bar R_{jk}\partial_k\bar u_i\nonumber\\
&&\qquad+
\alpha(\bar F_ig_j+\bar F_jg_i)
- \nu \partial_{kk} \bar R_{ij}\nonumber\\
&&\qquad=- \frac{C_1}{L}\bar R^{1/2}\bar R_{ij} -
\frac{C_2}{L}\bar R^{1/2}
(\bar R_{ij}-{\textstyle{{1}\over{3}}}\bar R\delta_{ij}) - \nu \frac{C_{\nu}}{ L^2} \bar R_{ij},\nonumber\\
\label{eq:Rprop}
\end{eqnarray}
\begin{eqnarray}
\lefteqn{(\partial_t+\bar u_j\partial_j)\bar F_i+
\bar R_{ij}\partial_j\bar \Theta+\bar F_j\partial_j\bar u_i+
\alpha\bar Qg_i - {\textstyle\frac{1}{2}}(\nu+\kappa) \partial_{jj} \bar F_{i}}&\nonumber\\
&&=-\frac{C_6}{L}\bar R^{1/2}\bar F_i - {\textstyle\frac{1}{2}}(\nu + \kappa) \frac{C_{\nu\kappa}}{ L^2} \bar F_{i} ,
\label{eq:Fprop}
\end{eqnarray}
\begin{equation}
(\partial_t+\bar u_i\partial_i)\bar Q+2\bar F_i\partial_i\bar \Theta - \kappa \partial_{ii} \bar Q =
-\frac{C_7}{L}\bar R^{1/2}\bar Q - \kappa \frac{C_{\kappa}}{ L^{2}} \bar Q,
\label{eq:Qprop}
\end{equation}
where $R=R_{ii}$ is the trace of the Reynolds tensor, which is twice
the turbulent kinetic energy per unit mass, and $C_1$, $C_2$, $C_6$
and $C_7$ are positive dimensionless coefficients of order unity, of a
universal nature. (Coefficients $C_3$, $C_4$ and $C_5$ are reserved
for a magnetohydrodynamic extension of the model, see Ogilvie 2003)
The justification for introducing non-linear terms of the above form
is similar to that used in the model of magnetorotational turbulent
stresses originally introduced by Ogilvie (2003).
The term involving $C_1$ causes a dissipation of
turbulent kinetic energy, and allows for the free decay of
hydrodynamic turbulence. The term involving $C_2$ redistributes energy among
the components of $\bar R_{ij}$, and corresponds to the tendency of
hydrodynamic turbulence to return to isotropy through the effect of
the pressure--strain correlation. Both are constructed assuming that these
effects occur on a timescale related to the eddy turnover time, $ L / \bar R^{1/2}$, where $L$ is defined
as the typical scale of the largest turbulent eddies. Terms
$C_6$ and $C_7$, related to the transport of heat,
are advanced by simple analogy. The coefficients must satisfy certain conditions
to ensure the realizability of the model, as discussed in Appendix~A.
The terms proportional to the microscopic diffusion coefficients are introduced
to allow a modelling of the correlation terms $2\nu\langle \partial_{k} u_i'\partial_{k }u_j'\rangle$, $(\nu+ \kappa) \langle \partial_j u_i' \partial_{j}\Theta'\rangle$ and $2\kappa\langle (\partial_{i}\Theta')^2 \rangle$ at moderate Reynolds number, i.e.
close to the onset of convection. In such a situation a turbulent cascade does not form and the dissipative terms are proportional to, rather than independent of, the diffusion coefficients. In a similar way, for turbulent shear flows, GO05
proposed to model the momentum diffusion term as
\begin{equation}
2\nu\langle \partial_{k} u_i'\partial_{k }u_j'\rangle \rightarrow \nu \frac{C_\nu}{L^2} \bar R_{ij}
\label{eq:cnumodel}
\end{equation}
on dimensional grounds. Indeed, it is expected that near the onset of convection, most fluid motions will be on the largest scales of the system ($L$). By analogy, we model the other two terms here as
\begin{eqnarray}
&& (\nu+ \kappa) \langle \partial_j u_i' \partial_{j}\Theta'\rangle \rightarrow {\textstyle\frac{1}{2}}(\nu+ \kappa) \frac{C_{\nu\kappa}}{L^2} \bar F_i \mbox{ , } \\
&& 2\kappa\langle (\partial_{i}\Theta')^2 \rangle \rightarrow \kappa \frac{C_\kappa}{L^2} \bar Q \mbox{ . }
\label{eq:ckappamodel}
\end{eqnarray}
Therefore the dissipative term in each of equations (\ref{dtrij})--(\ref{dtq}) is modelled by a sum of two terms, one that is independent of the diffusivity and dominates at high Reynolds numbers, and another that is proportional to the diffusivity and dominates at moderate Reynolds numbers.
This completes the justification for the form of the closure model proposed in
equations (\ref{eq:Rprop}), (\ref{eq:Fprop}) and (\ref{eq:Qprop}).
\section{Rayleigh--B\'enard convection}
\subsection{Model setup}
We now apply the closure model to the problem of Rayleigh--B\'enard convection.
We consider a horizontally infinite, plane-parallel system, where
the bottom plate is located at height $z=0$ and the top plate at
height $z=h$. The relative temperature of the bottom plate
is $\bar \Theta = \Delta T$ while that of the top
plate is $\bar \Theta= 0$.
In this setup, we look for statistically steady
and horizontally homogeneous solutions assuming
that mean quantities and correlations between fluctuating
quantities vary only with $z$. We also assume that there are no mean
flows in the system. Equations (\ref{eq:meancont})-(\ref{eq:meanenergy})
and (\ref{eq:Rprop})-(\ref{eq:Qprop})
reduce to a set of ordinary differential equations (ODEs)
which can be solved to obtain the temperature profile
$\bar \Theta(z)$ between the two plates, the profiles of the turbulent kinetic
energy, $\bar R(z) / 2$, and the temperature variance, $\bar Q(z)$ (for example).
By analogy with Prandtl's mixing-length formulation (Prandtl, 1932)
we set $L$, the size of the largest eddies, to be equal
to the distance to the nearest wall, i.e.\ $L(z)=\min(z,h-z)$ (see GO05 for
applications of the same principle to pipe flows and to Couette--Taylor flows).
It can be shown with little effort that $\bar R_{xy} = \bar R_{xz}
= \bar R_{yz} = 0$, as well as $\bar F_x = \bar F_y = 0$. The remaining
set of five second-order ODEs fully characterizes the system:
\begin{eqnarray}
\label{eq:RBmodeleq}
\nu \frac{{\rm d}^2 \bar{R}}{{\rm d} z^2}
&=& \nu \frac{C_\nu}{L^{2}} \bar{R}
+ \frac{C_1}{ L} \bar{R}^{3/2}
- 2 \alpha \bar{F}_z g, \nonumber \\
\nu \frac{{\rm d}^2 \bar{R}_{zz}}{{\rm d} z^2} &=& \nu \frac{C_\nu}{L^2} \bar{R}_{zz}
+ \frac{C_1+C_2}{ L} \bar{R}^{1/2} \bar{R}_{zz}
- \frac{ C_2}{3 L} \bar{R}^{3/2}\nonumber \\
&& - 2 \alpha \bar{F}_z g, \nonumber \\
{\textstyle\frac{1}{2}}(\nu + \kappa) \frac{{\rm d}^2 \bar{F}_{z}}{{\rm d} z^2} &=& {\textstyle\frac{1}{2}}(\nu + \kappa) \frac{C_{\nu\kappa}}{ L^2} \bar{F}_{z}
+ \frac{C_6}{ L} \bar R^{1/2}\bar{F}_z \nonumber \nonumber \\
&& - \alpha \bar{Q} g
+ \bar{R}_{zz} \frac{{\rm d} \Theta}{{\rm d} z}, \nonumber \\
\kappa \frac{{\rm d}^2 \bar{Q}}{{\rm d} z^2} &=& \kappa \frac{C_\kappa}{ L^2} \bar{Q}
+ \frac{C_7}{ L} \bar R^{1/2} \bar{Q} +
2 \bar{F}_z \frac{{\rm d} \bar{\Theta}}{{\rm d} z}, \nonumber \\
\kappa \frac{{\rm d}^2 \bar{\Theta}}{{\rm d} z^2} &=& \frac{{\rm d} \bar{F}_z}{{\rm d} z},
\label{eq:RBODE}
\end{eqnarray}
where $g = -g_z$. In the case of no-slip boundaries with fixed temperature on each plate as listed above, $\bar R$, $\bar R_{zz}$, $\bar F_z$ and $\bar Q$ are zero on both boundaries.
This system of ODEs with associated boundary conditions can be solved with a two-point boundary-value solver. Typical solutions are shown in Fig.~\ref{fig:raplots} for various Rayleigh numbers, defined here as
\begin{equation}
{\rm Ra} = \frac{\alpha g h^3 \Delta T}{\nu \kappa}.
\end{equation}
We set the Prandtl number
\begin{equation}
\mathrm{Pr}=\frac{\nu}{\kappa}
\end{equation}
to~$1$ for the purposes of illustration.
Note the appearance of the characteristically flat temperature profile between
the two plates as Ra $\rightarrow \infty$ and of the thin
thermal boundary layers. We now study in more detail the structure of the
solution.
\begin{figure}
\epsfig{file=f1.eps,width=8cm,height=17cm}
\caption{Vertical profiles $\bar \Theta(z)$ (in units of $\Delta T$), $\bar R(z)$
(in units of $\kappa^2/h^2$), $\bar F_z(z)$ (in units of $\kappa \Delta T/h$) and
$\bar Q(z)$ (in units of $(\Delta T)^2$) for $\mathrm{Ra}=10^6$ (dotted line),
$\mathrm{Ra}=10^8$ (dashed line) and $\mathrm{Ra}=10^{10}$ (solid line).
In all cases, $\mathrm{Pr} = 1$. }
\label{fig:raplots}
\end{figure}
\subsection{Universal profile of convection from a wall}
\label{subsec:blbehavior}
As in the case of shear flows past a wall (see GO05), we can derive
a universal profile for convection away from a wall. Let us consider a
semi-infinite domain $z>0$, in which case $L = z$, and
let $F_0$ be the convective heat flux through the system.
We define dimensionless variables via
\begin{eqnarray}
&& z = \left[ \frac{\kappa^2 \nu}{\alpha g F_0} \right]^{1/4} \eta\mbox{ , } \nonumber \\
&& \bar F_z = F_0 f(\eta), \qquad \bar Q = \left[ \frac{F_0^3 \nu}{\alpha g \kappa^2} \right]^{1/2} q(\eta),\nonumber\\
&& \bar \Theta - \Delta T = \left[ \frac{F_0^3 \nu}{\alpha g \kappa^2} \right]^{1/4} \theta(\eta) ,\nonumber \\
&& \bar R_{ij} = \left[ \frac{\alpha g F_0 \kappa^2}{\nu} \right]^{1/2} r_{ij}(\eta) ,
\label{eq:nondim}
\end{eqnarray}
so that the system of equations (\ref{eq:RBODE}) becomes
\begin{eqnarray}
r'' &=& \frac{C_{\nu}}{\eta^2} r + \frac{1}{{\rm Pr}} \frac{C_1}{\eta} r^{3/2} - 2f, \nonumber \\
r''_{zz} &=& \frac{C_{\nu}}{\eta^2} r_{zz} + \frac{1}{{\rm Pr}} \frac{C_1+C_2}{\eta} r^{1/2}r_{zz} -\frac{1}{{\rm Pr}} \frac{C_2}{3\eta} r^{3/2}- 2f, \nonumber \\
\frac{{\rm Pr} + 1}{2} f'' &=& \frac{{\rm Pr} + 1}{2} \frac{C_{\nu\kappa}}{\eta^2} f + \frac{C_6}{\eta} r^{1/2} f - {\rm Pr}\, q + r_{zz} \theta', \nonumber \\
q'' &=& \frac{C_{\kappa}}{\eta^2} q + \frac{C_7}{\eta} r^{1/2} q + 2 f \theta' \nonumber, \\
\theta' &=& f-1.
\end{eqnarray}
The boundary conditions at $\eta = 0$
are $r=r_{zz}=f=q=\theta = 0$.
Solutions very close to the wall $(\eta \ll 1)$ satisfy:
\begin{eqnarray}
&& r \mbox{ and } r_{zz} \propto \eta^{\alpha_{\nu}} \mbox{ with } \alpha_\nu(\alpha_\nu - 1) = C_\nu \nonumber, \\
&& f \propto \eta^{\alpha_{\nu\kappa}} \mbox{ with } \alpha_{\nu\kappa}(\alpha_{\nu\kappa} - 1) = C_{\nu\kappa}, \nonumber\\
&& q \propto \eta^{\alpha_{\kappa}} \mbox{ with } \alpha_{\kappa}(\alpha_{\kappa} - 1) = C_{\kappa}.
\label{sublayerprofile}
\end{eqnarray}
These simple relationships provide an ideal way of calibrating each of
the three constants $C_\nu$, $C_{\nu\kappa}$ and $C_\kappa$ individually
(see Section~\ref{s:calibration}), by analysing the power-law behaviour of the
near-wall profiles of experimental or numerical data.
Solutions far away from the boundary layer can be expanded as
\begin{eqnarray}
&& r = r_0 \eta^{2/3} + O(\eta^{-2/3}), \nonumber \\
&& r_{zz} = r_{zz0} \eta^{2/3} + O(\eta^{-2/3}), \nonumber \\
&& f = 1 - f_1 \eta^{-4/3} + O(\eta^{-8/3}), \nonumber \\
&& q = q_0 \eta^{-2/3} + O(\eta^{-4/3}), \nonumber \\
&& \theta = \theta_0 + 3f_1 \eta^{-1/3} + O(\eta^{-5/3}),
\label{eq:powerlaweta}
\end{eqnarray}
where
\begin{eqnarray}
r_0 &=& \left( \frac{2{\rm Pr}}{C_1} \right)^{2/3},\qquad
r_{zz0} = \frac{3C_1+C_2}{3(C_1 + C_2)} r_0, \nonumber \\
f_1 &=& \frac{C_6}{\frac{C_1}{C_7} + \frac{3C_1 + C_2}{3(C_1+C_2)} } r_0^{-1/2},\qquad
q_0 = \frac{2 f_1}{C_7 r_0^{1/2} }.
\end{eqnarray}
However, unlike $r_0$, $f_1$ and $q_0$
the constant $\theta_0$ cannot be determined without a numerical calculation of the
boundary-layer solution for $\eta = O(1)$.
The scaling laws obtained for $r$, $f$, $\theta$ and $q$ far from the wall
are expected on dimensional grounds, and recover the well-known solution
of Priestley (1954). They are analogous to the universal ``log-law''
solutions for turbulent shear flows past a wall (e.g. Schlichting, 1979).
By comparing profiles of $r$, $f$ and $q$ with laboratory or numerical
experiments, one can constrain some of the unknown
coefficients $\{C_i\}$ (see Section~\ref{s:calibration}).
\subsection{Nusselt--Rayleigh number relationship}
\label{subsec:RaNu}
The heat flux through the system in Rayleigh--B\'enard convection is commonly
measured by the dimensionless Nusselt number
\begin{equation}
{\rm Nu} = 1 + \frac{h F_0}{\kappa \Delta T } ,
\end{equation}
which compares the total heat flux with the conductive one in the absence of convection.
The universal convection-from-a-wall solution calculated
in the previous section can be used to derive
the relationship between the Nusselt number and the Rayleigh number.
Indeed, by selecting a Rayleigh number we set the relative
temperature at the midpoint $z=h/2$ to be
$\bar \Theta=\Delta T /2$ which implies through (\ref{eq:nondim}) that
\begin{equation}
\frac{\Delta T}{2} - \Delta T = \left[ \frac{F_0^3 \nu}{\alpha g \kappa^2} \right]^{1/4} \theta\left( \left[ \frac{\alpha g F_0}{\kappa^2 \nu} \right]^{1/4} \frac{h}{2} \right) \mbox{ , }
\end{equation}
yielding an equation for the (unknown) constant heat flux $F_0$.
In dimensionless terms, we have the implicit equation for $\mathrm{Nu}$:
\begin{equation}
\frac{1}{2}[\mathrm{Ra}(\mathrm{Nu}-1)^{-3}]^{1/4}=\theta\left({\textstyle\frac{1}{2}}[\mathrm{Ra}(\mathrm{Nu}-1)]^{1/4}\right),
\end{equation}
which can be solved to find $\mathrm{Nu}(\mathrm{Ra})$.
In the limit of very large Rayleigh number
the mid-point of the system is very far from
the boundary layer, so $\theta\approx\theta_0$
which then recovers the standard scaling law (Malkus 1954)
\begin{equation}
{\rm Nu} = 1 + \left( \frac{ {\rm Ra}}{16 \theta_0^4} \right)^{1/3} \mbox{ . }
\end{equation}
The constant $\theta_0$ depends only on Pr and on the closure
parameters $\{ C_i\}$, but cannot easily be expressed analytically
in terms of these parameters.
\subsection{Comparison with data and estimation of the model parameters}
\label{s:calibration}
The aim of this section is to estimate, in a rough sense,
the parameters $\{C_i\}$ by comparing the model predictions
with numerical simulations and laboratory experiments.
This approach was successfully used in GO05 on pipe
flow data and Couette--Taylor data, yielding:
\begin{equation}
C_1 \simeq 0.4 \mbox{ , } C_2 \simeq 0.6 \mbox{ , } C_\nu \simeq 12 .
\label{eq:C1C2}
\end{equation}
Under the assumption that the closure parameters are
universal properties of the turbulent cascade, these estimated values
should also apply to the case of turbulent convection
without need for re-calibration. The remaining parameters $C_6$, $C_7$,
$C_{\nu\kappa}$ and $C_{\kappa}$ may then be independently estimated.
In the following sections, we first discuss this assumption
in the light of known model limitations. We then select
appropriate experimental datasets and
use them to constrain the remaining parameters.
\subsubsection{Discussion of the model limitations}
As discussed by Ogilvie (2003) and GO05
the closure model proposed has two intrinsic limitations:
it ignores some (but not all) of the effects of
pressure-strain correlations $<u'_i \partial_j \psi' >$,
and assumes that the effect of all modelled terms
(such as the triple-correlations
in (\ref{eq:Rprop})-(\ref{eq:Qprop})) is local both in
time and space. As a result, it
may poorly represent strongly sheared systems
or systems where the turbulent eddies exhibit a strong degree of
spatial or temporal coherence.
The neglected effects of the pressure-strain correlations are not thought to be important in
turbulent convection, except in the presence of strong rotation or of
an externally driven strong mean shear (where the timescale of rotation and shear is comparable to
that of the convection). The closure should be
well-suited to model convection in stellar interiors, but maybe less so for
convectively unstable accretion discs. We defer this particular case to subsequent work.
However, for similar reasons these effects are also likely to be important in pipe flows or
Couette--Taylor flow, which were used as a basis for calibrating the constants
$C_1$ and $C_2$ (see GO05). Consequently, the estimates given in (\ref{eq:C1C2})
could be somewhat biased, in particular $C_2$ which contains information on the rate of
return to isotropy. Comparing the model with turbulent convection experiments (see below) can
therefore help refine the estimates for $C_1$ and $C_2$ using more appropriate data.
As mentioned above, the closure is also less reliable when applied to systems where the turbulence
exhibits coherence over large scales or long timescales. This might pose some problems
when applied to convection in a finite domain, since large-scale coherent plumes which span
the whole system are commonly observed in most cases ranging from Boussinesq
to fully compressible systems. Comparisons with experiments can help reveal which aspects
of convective transport are adequately described by the model, and which are not.
\subsubsection{Available experimental data}
Our application of the closure model to Rayleigh-B\'enard convection in Sections
\ref{subsec:blbehavior} and \ref{subsec:RaNu} assumes for simplicity
that the system is horizontally invariant, while all laboratory and numerical
experiments have a limited horizontal extent.
The presence, nature and geometry of the side-walls are known to affect
various properties of the turbulent convection, in particular
through the generation of large-scale circulations (often called ``wind'').
This wind influences the overall heat transport properties
by changing the nature of
the boundary layers (Castaing et al. 1989; Cioni et al. 1997;
Grossmann \& Lohse 2000, 2001, 2002, 2004).
It also induces large-scale
horizontal inhomogeneities, so that the measured vertical profiles
of mean quantities and higher-order moments may vary
with position (Maystrenko, Resagk \& Thess 2007).
While our formalism can in principle be applied to finite
geometries and self-consistently model the effect of large-scale
flows, such an extension is beyond the scope
of the present paper.
In order to minimize the effect of side-walls
we restrict the model comparison to experimental setups
with very large aspect ratios (defined as the ratio of the horizontal
to vertical extent of the domain, and denoted as $\Gamma$).
There are a few large aspect ratio, high Rayleigh number
experimental studies which provide measurements of the Nusselt number.
Of particular interest are results of
F\"unfschilling et al. (2005) for convection
in water (Pr = 4.38) in a cylindrical
enclosure of aspect ratio up to $\Gamma= 6$,
for Ra up to a few times $10^{10}$.
Niemela \& Sreenivasan (2006) provide similar information
for convection in Helium (0.7 $<$ Pr $<$ 8)
in a cylindrical container with $\Gamma = 4$,
for Rayleigh numbers between $10^8$ and $10^{13}$.
Finally, the Ilmenau barrel experiments of
DuPuits, Resagk \& Thess (2007) provide Nu(Ra) for convection in
air (Pr = 0.7) in a cylindrical enclosure with variable aspect ratio up
to 11.3, for Rayleigh numbers up to a few times $10^8$ (in the case of the
largest aspect ratio).
By contrast, only very few large aspect ratio experimental measurements of
the boundary-layer profiles of velocity and temperature correlations
(such as $\bar R_{ij}$, $\bar F_i$ or $\bar Q$) have been reported.
The largest aspect ratio experiments available ($\Gamma =11.3$)
with fully resolved boundary layer profiles are presented by
DuPuits, Resagk \& Thess (2007) although the
data provided is limited to the mean and rms temperature profiles.
Taking a different approach, direct numerical experiments are a powerful tool
for ``idealized'' experiments. Horizontally periodic simulations minimize
the effect of side-walls (although retain a finite aspect ratio) and permit
resolved and precise measurements of all desired mean and fluctuating
quantities within the flow. The main drawback is the limited range
of parameter space for which resolved simulations can be run
(typically, Ra $< 10^8$ for large aspect ratio simulations at Pr $= O(1)$).
For these reasons, we use a combination of
experimental data (DuPuits, Resagk \& Thess 2007) and numerical
simulations to calibrate the remaining model parameters.
Our numerical simulations are all run for Pr = 1,
in a horizontally periodic domain with aspect ratio $L_x/L_z = L_y/L_z = 4$,
using a spectral method briefly described in Appendix~B. The largest
Rayleigh number achieved in this case is Ra $= 2.1 \times 10^7$.
Figure \ref{fig:RBeyecandy} shows a typical snapshot of the results, in this
parameter regime, for the temperature field for example. The
results of the simulations are globally consistent with those
of Hartlep (PhD thesis, 2005, G\"ottingen).
\begin{figure}
\epsfig{file=f2.epsf,width=8cm}
\caption{Volume-rendered visualization of the temperature field in
our numerical simulation of Rayleigh-B\'enard convection for
${\rm Ra} = 2.1 \times 10^7$, and $\mathrm{Pr} = 1$. The system is doubly-periodic
in the horizontal direction, with aspect ratio 4,
and has no-slip boundary conditions at the top and bottom boundary. The colour and
opacity scheme has been selected to emphasize structures near the lower boundary layer.}
\label{fig:RBeyecandy}
\end{figure}
\subsubsection{Near-wall profiles and estimation of $C_\nu$, $C_\kappa$ and $C_{\nu\kappa}$}
Very close to the wall ($\eta \ll 1$), the closure model solutions for
the normalized
correlations $r$, $r_{zz}$, $f$, $q$ and $\theta$ are well approximated
by power laws,
as described in equation~(\ref{sublayerprofile}). These relationships can
be compared with data and provide a simple
way of individually estimating each of the model constants
$C_\nu$, $C_\kappa$ and $C_{\nu\kappa}$
from laboratory or numerical experiments.
Comparisons of (\ref{sublayerprofile}) with the experimental
near-wall profile for $r_{zz}(\eta)$, $f(\eta)$ and $q(\eta)$
yield slopes $\alpha_\nu$ close to 4
(see Fig.~\ref{fig:rzzcal}), $\alpha_{\nu\kappa}$ close to 3 (see Fig.~\ref{fig:fzcal}),
and $\alpha_\kappa$ close to 2 (see Fig.~\ref{fig:qcal}).
Note that while the amplitude of the power-law observed in
the near-wall profile for $q(\eta)$ is seen to depend on the experiment considered,
the slope $\alpha_\kappa$ appears to be universal.
We then adopt the following values for the constants $C_\nu$, $C_{\nu\kappa}$ and $C_\kappa$:
\begin{eqnarray}
&& C_\nu = 12\pm 1 \mbox{ , } \nonumber \\
&& C_{\nu\kappa} = 6 \pm 0.5 \mbox{ , } \nonumber \\
&& C_{\kappa} = 2 \pm 0.2 \mbox{ . }
\label{eq:Cd}
\end{eqnarray}
Given the experimental and model uncertainties, these values
and their errorbars should be thought of as rough estimates rather
than precise calibrations.
It is comforting to note that this independent comparison recovers the
value of $C_\nu$ found by GO05. Moreover, we find that
within fitting errors $C_{\nu\kappa}
\simeq (C_\nu C_\kappa)^{1/2}$. Given the quantities
modelled by the associated diffusive terms (see equations
(\ref{eq:cnumodel})--(\ref{eq:ckappamodel})), this result is not
entirely surprising.
On the other hand, Fig. \ref{fig:rzzcal} reveals an important caveat
of the closure model when applied to Rayleigh-B\'enard convection.
The universal solution for the two horizontal stress components
$r_{xx}(\eta)$ and
$r_{yy}(\eta)$ can easily be deduced from $r_{xx} = r_{yy} = 0.5 (r -
r_{zz})$. These horizontal stresses should therefore be identical to
one another and have the same power-law dependence on $\eta$ as $r$
and $r_{zz}$, close to the wall and far from the wall. However,
Fig. \ref{fig:rzzcal} clearly shows that the numerical data is at odds
with the model. We attribute the discrepancy to the presence of
large-scale coherent convective plumes in the system, which span the
entire domain and create strong horizontally correlated fluctuations
as they crash against each boundaries. As a result, the fluid in the
viscous sublayer is much more strongly anisotropic than predicted.
\begin{figure}
\epsfig{file=f3.eps,width=8cm}
\caption{Comparison of the universal ``convection from a wall''
solution with numerical data for the dimensionless Reynolds stress
components $r_{zz}$, $r_{xx}$ and $r_{yy}$. The large symbols
represent $r_{zz}(\eta)$ for Ra = $2.1 \times 10^6$ (triangles) and
$2.1 \times 10^7$ (diamonds). The two sets of smaller diamonds show
$r_{xx}(\eta)$ and $r_{yy}(\eta)$ for the case where Ra= $2.1 \times
10^7$. Note that theoretically these should be lying on the same curve
-- the difference can be attributed to limited statistics. In all
cases $\mathrm{Pr}=1$. The dotted line shows the asymptotic solution
$r_{zz} = r_{0zz} \eta^{2/3}$ using the value of $C_1$ estimated by
GO05, while the solid line shows a numerical integration of the full
universal profile, for our estimated parameter values as listed in
(\ref{eq:C1C2}), (\ref{eq:Cd}), and (\ref{eq:C6C7}). }
\label{fig:rzzcal}
\end{figure}
\begin{figure}
\epsfig{file=f4.eps,width=8cm}
\caption{Comparison of the predicted dimensionless convective heat
flux profile $f(\eta)$ with our numerical data.
The symbols have the same meaning as in Fig.~\ref{fig:rzzcal}. Note
that the scatter for $\eta < 0.1$ comes from imperfect statistics very
close to the wall. This plot was used to fit $C_{\nu\kappa}$ to
capture the near-wall solution correctly. The solid line shows a
numerical integration of the full universal profile, for our estimated
parameter values as listed in (\ref{eq:C1C2}), (\ref{eq:Cd}), and
(\ref{eq:C6C7}).}
\label{fig:fzcal}
\end{figure}
\subsubsection{Far-field solution and estimation of $C_6$ and $C_7$.}
Fig.~\ref{fig:rzzcal} compares the predicted profile for
$r_{zz}(\eta)$ with data from our numerical simulations. The dotted
line shows the model prediction for the solution far from the wall
$r_{zz} = r_{zz0} \eta^{2/3}$. Note that $r_{zz0}$ depends only on two
numbers, the Prandtl number (which is known) and the model parameter
$C_1$. It is reassuring to see that the value of $C_1$ estimated by
GO05 from wall-bounded shear flow data adequately
fits the far-from wall solution for $r_{zz}$ in this convection
problem.
\begin{figure}
\epsfig{file=f5.eps,width=8cm}
\caption{Comparison of the predicted dimensionless temperature
variance $q$ with experimental and numerical data. The open symbols
represent the results of our numerical simulations (Pr = 1) for Ra =
$2.1 \times 10^6$ (triangles) and $2.1 \times 10^7$ (diamonds). The
plus symbols are experimental data from DuPuits, Resagk \& Thess
(2007) for Ra = 8.14$\times 10^8$ for air (Pr = 0.7) in a cylindrical
box at aspect ratio 11.3. The discrepancy between the numerical
solutions and the experimental data is attributed to the difference
between periodic side-walls and impermeable side-walls.
The near-wall solution was used to fit $C_\kappa$ while the
far-from-the-wall data was used to provide a constraint between $C_6$
and $C_7$. The solid line show a numerical integration of the full
universal profile as in Figs. \ref{fig:rzzcal} and \ref{fig:qcal} for
Pr = 1. }
\label{fig:qcal}
\end{figure}
The universal profiles away from the wall listed in equation
(\ref{eq:powerlaweta}) can also be used in conjunction with numerical and laboratory
experiments to constrain $C_6$ and $C_7$.
These constants are unfortunately difficult to extract
directly from our numerical simulations. The highest Rayleigh number
available (Ra = 2.1 $\times 10^7$) only has a short asymptotic ($\eta
\gg 1$) range, so that estimates of $C_6$ and $C_7$ from these
datasets are unreliable\footnote{This statement can be verified using
a simple test problem in which artificial data are created {\it using}
the closure model, and then used blindly to reconstruct $C_6$ and
$C_7$.}. The rms temperature data measured in various laboratory experiments at
higher Rayleigh number provides a more adequate point of comparison.
We use the rms temperature data of the highest aspect ratio
experiments of DuPuits, Resagk \& Thess (2007), for Ra = $8.14 \times
10^8$. This dataset exhibits a significant asymptotic range, with a
power law close to the one predicted by the closure model ($q \sim
q_0 \eta^{-2/3}$). Fitting the data yields $q_0 = 0.95 \pm 0.05$,
which provides a first constraint between $C_6$ and $C_7$ (see
Fig.~\ref{fig:C6C7}). Note that other datasets (from Maystrenko,
Resagk \& Thess, 2007, for example) are generally consistent with this
estimate for $q_0$.
A second constraint between $C_6$ and $C_7$ is obtained by comparing
the model predictions with experimental measurements of $\mathrm{Nu}(\mathrm{Ra})$.
The closure model implies that ${\rm Nu} = 1 + K {\rm Ra}^{1/3}$ where
the constant $K$ is a function of the model parameters (and the
Prandtl number). The data from F\"unfschilling et al. (2005),
Niemela \& Sreenivisan (2006) and DuPuits, Resagk \& Thess (2007) are
reasonably well approximated by taking $K=0.06 \pm 0.003$. Variations
of $K$ with Prandtl number, for the range of experiments discussed,
are within the errorbars. Given that $C_1$, $C_2$, $C_\nu$,
$C_\kappa$ and $C_{\nu\kappa}$ are now known, for fixed Prandtl
number, fitting $K$ provides a unique relationship between $C_6$ and
$C_7$, as seen in Fig.~\ref{fig:C6C7}.
\begin{figure}
\epsfig{file=f6.eps,width=8cm}
\caption{Calibration of the constants $C_6$ and $C_7$. The straight
lines show the relationship between $C_6$ and $C_7$ when the constant
$q_0$ is equal to 0.95 (solid line), 0.9 or 1.0 (dashed lines, top
and bottom respectively). The curves show the value of $K$ in the
relationship ${\rm Nu} \sim 1 + K {\rm Ra}^{1/3}$, as predicted by
numerical integrations of the closure model equations
(\ref{eq:RBmodeleq}) for no-slip boundary conditions. The area marked
by the intersection of the 4 dashed lines, and centred on the point
where the two solid lines cross, provides estimates for $C_6$ and
$C_7$.}
\label{fig:C6C7}
\end{figure}
By combining these two constraints, we conclude that a good fit to the
data can be obtained with
\begin{equation}
C_6 = 1.4 \pm 0.1 \mbox{ , } C_7 = 1.4 \pm 0.1.
\label{eq:C6C7}
\end{equation}
The values for $\{C_i\}$ quoted in equations (\ref{eq:C1C2}),
(\ref{eq:Cd}), and (\ref{eq:C6C7}) form from here on our selected set
of parameters. These values are to be taken as indicative estimates,
rather than precise calibrations. We note that the parameters derived
do satisfy realizability (see Appendix~A). The solid lines shown in
Figs. \ref{fig:rzzcal}, \ref{fig:fzcal} and \ref{fig:qcal} are the
universal boundary layer profiles calculated using these parameters,
and are seen to fit all datasets (except for $r_{xx}$ and $r_{yy}$, as
discussed above) satisfactorily.
Fig.~\ref{fig:RaNu} compares our closure model prediction for the
Nu(Ra) relationship, using the estimated parameters, with various
available datasets for large aspect ratio experiments ($\Gamma \ge
4$). It also shows (as dashed lines), for comparison, strict upper
bounds obtained by Plasting \& Kerswell (2003) and by Ierley, Kerswell
\& Plasting (2006) for transport by convection at finite and infinite
Prandtl numbers respectively. It is reassuring to see that the
$\mathrm{Pr}\rightarrow\infty$ prediction from our own closure model
remains below the strict upper bound for the same limit.
\begin{figure}
\epsfig{file=f7.eps,width=8cm}
\caption{Comparison of the model predictions with data for the Nusselt
number as a function of the Rayleigh number. The square symbols are
experimental data from Niemela \& Sreenivisan (2006) with Pr $\simeq
1$ (Helium), and aspect ratio $\Gamma= 4$. The diamond symbols are
the data from F\"unfschilling et al. (2005) with $\Gamma= 6$,
Pr $= 4.38$ (water). The triangles are data from DuPuits et al.
(2007), for $4\le \Gamma \le 11.3$, for Pr = 0.7 (air). The plus
symbols are numerical data from Hartlep et al. (2007), with
$\Gamma = 10$ and for Pr = 0.7. Finally, the star symbols are our own
numerical simulations. The various thin lines shows the closure model
predictions for fiducial values of the parameters $C_i$, for Pr = 1
(solid line), Pr = 4.38 (dashed line) and Pr $\rightarrow \infty$
(dotted line). In addition, the two thick solid lines correspond to
strict upper bound limits: the Nu$=1+0.133\,\mathrm{Ra}^{1/3}$ line
is a strict upper bound obtained by Ierley, Kerswell \& Plasting
(2006) for Rayleigh--B\'enard convection at infinite Prandtl number,
while the Nu $= 1+0.0264\,\mathrm{Ra}^{1/2}$ line is a strict upper
bound obtained by Plasting \& Kerswell (2003) for Rayleigh--B\'enard
convection at arbitrary (finite) Prandtl number. }
\label{fig:RaNu}
\end{figure}
In conclusion, our model successfully reproduces
most measurable features pertaining to laboratory and numerical
experiments of Rayleigh-B\'enard convection, for reasonable values of
the model parameters $\{C_i\}$. Furthermore, comparison of the
estimated parameter values across a range of experiments in
other systems (pipe flows, Couette--Taylor flows) shows that they are
indeed of a universal nature, a results which can only increase
confidence in our approach.
\section{Homogeneous Rayleigh--B\'enard convection}
\label{A minimal model system}
\subsection{Introduction}
Another system that is of interest, and possibly more relevant to
astrophysical applications, consists of an unbounded layer in which
there is no mean flow, while the mean temperature gradient
$\nabla\bar\Theta$ is uniform and parallel to the gravitational
acceleration (taken to be in the $z$-direction). The
evolution of perturbations to this mean state can be described by the
following set of Boussinesq equations:
\begin{eqnarray}
\frac{\partial \mbox{\boldmath$u$}'}{\partial t} + \mbox{\boldmath$u$}' \cdot \nabla \mbox{\boldmath$u$}' = - \alpha
\Theta' g_z\,\mbox{\boldmath$e$}_z -\nabla\psi' + \nu \nabla^2 \mbox{\boldmath$u$}' \mbox{ , }
\nonumber \\ \frac{\partial \Theta'}{\partial t} + \mbox{\boldmath$u$}' \cdot \nabla
\Theta' + u_z' \frac{{\rm d} \bar \Theta}{{\rm d} z} = \kappa \nabla^2 \Theta'
\mbox{ , } \nonumber \\ \nabla \cdot \mbox{\boldmath$u$}'= 0 \mbox{ , }
\label{eq:HRBorig}
\end{eqnarray}
where all perturbations are triply periodic, as for example
\begin{eqnarray}
\mbox{\boldmath$u$}'(x,y,z,t) &=& \mbox{\boldmath$u$}'(x + L_x,y,z,t) \nonumber\\ &=&
\mbox{\boldmath$u$}'(x,y+L_y,z,t) \nonumber\\ &=& \mbox{\boldmath$u$}'(x,y,z+L_z,t).
\end{eqnarray}
This model setup is now commonly referred to as Homogeneous
Rayleigh--B\'enard (HRB) convection (Borue \& Orszag 1997; Lohse \&
Toschi 2003; Calzavarini et al.\ 2005; Calzavarini et al.\ 2006).
While this system cannot be studied using laboratory experiments, it
lends itself relatively easily to numerical experimentation using
spectral methods in particular. The relevant dimensionless parameters
are the Prandtl number ${\rm Pr}=\nu/\kappa$, the Rayleigh number, now defined as
\begin{equation}
{\rm Ra}=\frac{\alpha g_z L_z^4\frac{{\rm d} \bar\Theta}{{\rm d}
z}}{\nu\kappa},
\end{equation}
and the aspect ratio(s) $\Gamma =L_{x,y}/L_z$.
The microscopic diffusivities are included in the original equations
(\ref{eq:HRBorig}) to regularize the system by allowing for
dissipation and irreversibility. However, note that the periodic
boundary conditions forbid the formation of boundary layers, so it
may be conjectured that the macroscopic statistical properties of the
turbulent convection should be well defined and independent of $\nu$
and $\kappa$ in the limits ${\rm Ra}\to\infty$ (Spiegel 1971).
Furthermore, we may expect the turbulence to be statistically steady
and homogeneous, although anisotropic. These properties have been
argued to be more relevant to convection in astrophysical systems
than standard Rayleigh--B\'enard convection. The HRB model may
therefore provide a suitable local model of convection deep inside a
star or planet.
On dimensional grounds, the rms turbulent velocity, for example, must
be expressible in the form
\begin{equation}
\langle u^{\prime2}\rangle^{1/2}= \left(\alpha g_z{{{\rm
d}\bar\Theta}\over{{\rm d}z}}\right)^{1/2}L_z\, f({\rm Ra},{\rm
Pr},\Gamma),
\end{equation}
where $f$ is a dimensionless function. According to the discussion
above, $f$ should tend to a non-zero function of $\Gamma$ alone in
the limit $\mathrm{Ra}\to\infty$. It is tempting to conjecture that
$f$ also becomes independent of $\Gamma$ in the limit of large aspect
ratio, $\Gamma\to\infty$. This would imply that the vertical
length-scale $L_z$ plays a fundamental role in determining the
saturation level of the turbulent convection, presumably by limiting
the size of coherent structures (`eddies'). For convection deep
inside a star or planet, it is the pressure scale-height that imposes
a characteristic vertical scale on the turbulence (see
Section~\ref{sec:anelastic}); in the local model, the vertical extent
of the box plays an equivalent role. In practice, owing to some peculiarities
of the HRB system discussed below, the role of the aspect ratio in the
behaviour of the solutions is not so straightforward.
\subsection{Closure model for HRB}
\subsubsection{Governing equations}
Applying our closure model to HRB, and noting that all statistical
averages are now independent of position, we obtain the system of ODEs
for the temporal evolution of the second-order correlations $\bar
R_{ij}$, $\bar F_i$ and $\bar Q$:
\begin{eqnarray}
&&\partial_t\bar R_{xx} = -\frac{C_1+C_2}{L}\bar R^{1/2}\bar R_{xx}+
\frac{C_2}{3L}\bar R^{3/2} ,\nonumber\\ &&\partial_t\bar R_{xy} =
-\frac{C_1+C_2}{L}\bar R^{1/2}\bar R_{xy},\nonumber\\ &&\partial_t\bar
R_{xz}+\alpha\bar F_xg_z = -\frac{C_1+C_2}{L}\bar R^{1/2}\bar R_{xz}
,\nonumber\\ &&\partial_t\bar R_{yy} =-\frac{C_1+C_2}{L}\bar
R^{1/2}\bar R_{yy}+ \frac{C_2}{3L}\bar R^{3/2} ,\nonumber\\
&&\partial_t\bar R_{yz}+\alpha\bar F_yg_z = -\frac{C_1+C_2}{L}\bar
R^{1/2}\bar R_{yz} ,\nonumber\\ &&\partial_t\bar R_{zz}+2\alpha\bar
F_zg_z = -\frac{C_1+C_2}{L}\bar R^{1/2}\bar R_{zz}+
\frac{C_2}{3L}\bar R^{3/2} ,\nonumber\\ &&\partial_t\bar F_x+\bar
R_{xz}{{{\rm d}\bar\Theta}\over{{\rm d}z}} = -\frac{C_6}{L}\bar
R^{1/2}\bar F_x ,\nonumber\\ &&\partial_t\bar F_y+\bar R_{yz}{{{\rm
d}\bar\Theta}\over{{\rm d}z}} = -\frac{C_6}{L}\bar R^{1/2}\bar F_y
,\nonumber\\ &&\partial_t\bar F_z+\bar R_{zz}{{{\rm d}\bar
\Theta}\over{{\rm d}z}}+ \alpha\bar Q g_z = -\frac{C_6}{L}\bar
R^{1/2}\bar F_z ,\nonumber\\ &&\partial_t\bar Q+2\bar F_z{{{\rm
d}\bar \Theta}\over{{\rm d}z}} = -\frac{C_7}{L}\bar R^{1/2}\bar Q.
\label{hrb}
\end{eqnarray}
where we have ignored for simplicity contributions from terms
including $C_\nu$, $C_\kappa$ and $C_{\nu\kappa}$ which do not
contribute to the high-Rayleigh number dynamics of HRB convection.
Note that the resulting equation for $\bar R$ is
\begin{equation}
\partial_t\bar R + 2\alpha\bar F_zg_z =- \frac{C_1}{L}\bar R^{3/2}
\mbox{ ,}
\end{equation}
so that these equations consist of a main system for $(\bar R,\bar
R_{zz},\bar F_z,\bar Q)$, decoupled systems for $(\bar R_{xz},\bar
F_x)$ and $(\bar R_{yz},\bar F_y)$, and prognostic equations for $\bar
R_{xx}, \bar R_{yy}$ and $\bar R_{xy}$.
\subsubsection{Choice of $L$ and consequences for the coefficients $\{C_i\}$}
While selecting $L$ as the distance to the wall is a natural choice
for wall-bounded convection or shear flows, a different approach must
be used for triply periodic flows. The largest eddy size in this case
is limited by the horizontal and vertical scales in the box, so that
$L$ can be assumed to be proportional to $\min(L_x,L_y,L_z)$.
It is important to note that the selection of a different $L$ implies
a potential rescaling of the $\{C_i\}$ coefficients. For example, had we
selected $L = z/2$ in the wall-bounded case instead of $L=z$, then
the estimated $C_1$, $C_2$, $C_6$ and $C_7$ would all be half the
values quoted in Section~\ref{s:calibration}
since these parameters enter the model in the combinations
$C_1/L$, etc.
Nevertheless, the ratios of any pairs of constants within
the group $\{C_1,C_2,C_6,C_7\}$
should (presumably) be preserved.
Following these considerations, we elect to keep the estimated values
of the $\{C_i\}$ given in equations (\ref{eq:C1C2}) and
(\ref{eq:C6C7}), and calibrate instead the value of the
proportionality constant $\delta$ in the expression $L= \delta
\min(L_x,L_y,L_z)$.
\subsubsection{High Rayleigh number HRB convection}
A search for non-trivial fixed points of the dynamical system
(\ref{hrb}) (with $\bar R > 0$) reveals they are the (positive)
solutions of a quartic equation. In the limit of large Ra
it can be shown that there is only one positive fixed point with
\begin{eqnarray}
&&\bar R_{xx}=\bar R_{yy}=\left({{C_2}\over{C_1+C_2}}\right)
{{\bar R}\over{3}},\nonumber\\
&&\bar R_{zz}=\left({{3C_1+C_2}\over{C_1+C_2}}\right)
{{\bar R}\over{3}},\nonumber\\
&&\bar R_{xy}=\bar R_{xz}=\bar R_{yz}=0,\nonumber\\
&&\bar F_z=-{{C_1\bar R^{3/2}}\over{2L(-N^2)}}
{{{\rm d}\bar\Theta}\over{{\rm d}z}},
\nonumber\\
&&\bar F_x=\bar F_y=0,\nonumber\\
&&\bar Q={{C_1\bar R}\over{C_7(-N^2)}}
\left({{{\rm d}\bar \Theta}\over{{\rm d}z}}\right)^2,
\label{eq:dimsols1}
\end{eqnarray}
with
\begin{equation}
\bar R={{2}\over{C_1C_6}}\left[{{C_1}\over{C_7}}+
{{3C_1+C_2}\over{3(C_1+C_2)}}\right]L^2(-N^2).
\label{eq:dimsols2}
\end{equation}
Note that, in this case,
\begin{equation}
\bar Q = \frac{2}{C_6 C_7} \left[{{C_1}\over{C_7}}+
{{3C_1+C_2}\over{3(C_1+C_2)}} \right] L^2 \left({{{\rm d}\bar \Theta}\over{{\rm d}z}}\right)^2 \propto |\nabla \bar T|^2 .
\end{equation}
This solution represents a state of fully developed turbulent convection, which is statistically steady and homogeneous. The solution exists in the statistically
axisymmetric subspace in which $\bar R_{xx}=\bar R_{yy}$ and $\bar
R_{xy}=\bar R_{xz}=\bar R_{yz}=\bar F_x=\bar F_y=0$, and is stable
with respect to perturbations transverse to this subspace. It has the
desired properties that the vertical motion is dominant $(\bar
R_{zz}>\bar R_{xx}=\bar R_{yy})$, while the heat flux is purely
vertical and directed down the temperature gradient. Moreover,
numerical integrations suggest that, where it exists, this state is
stable and universally attracting.
Defining the Nusselt number Nu as the ratio of the total to the conducted
heat flux,
\begin{equation}
{\rm Nu} = \frac{\bar F_z - \kappa {{{\rm d}\bar\Theta}\over{{\rm d}z}} }{-\kappa {{{\rm d}\bar\Theta}\over{{\rm d}z}} } ,
\end{equation}
we have, in the limit Ra $\gg$ Pr,
\begin{eqnarray}
{\rm Nu} &=& \sqrt{2} C_1 \left[ \frac{1}{C_1 C_6} \left( \frac{C_1}{C_7} + \frac{3C_1 + C_2}{3(C_1+C_2)} \right) \right]^{3/2}\nonumber\\
&&\quad\times ({\rm Pr Ra})^{1/2} \left( \frac{L}{L_z} \right)^2.
\label{eq:RaNuHRB}
\end{eqnarray}
This scaling recovers the ``ultimate turbulence'' regime,
where the turbulent transport properties are independent of microscopic
diffusivities (Spiegel 1971). Defining the turbulent Reynolds number
Re as Re = $L \bar R^{1/2}/\nu$, we have
\begin{equation}
{\rm Re} = \left[ {{2}\over{C_1C_6}}\left({{C_1}\over{C_7}}+
{{3C_1+C_2}\over{3(C_1+C_2)}}\right)\right]^{1/2} \left( \frac{\rm Ra}{\rm Pr} \right)^{1/2} \left( \frac{L}{L_z} \right)^2,
\label{eq:RaReHRB}
\end{equation}
again reproducing the standard scaling for the ultimate regime of convection.
\subsection{Comparison with numerical experiments}
Numerical simulations of HRB convection were first performed by Borue
\& Orzag (1997). More recently, Toschi \& Lohse (2003)
and Calzavarini et al. (2005) performed a range of
Lattice--Boltzmann simulations in a cubic geometry,
for various values of the Rayleigh and Prandtl numbers, and report
on the first evidence for scalings consistent with the ``ultimate regime''
of convection, namely ${\rm Nu} \propto ({\rm Ra Pr})^{1/2}$ and
${\rm Re} \propto \left({\rm Ra }/{\rm Pr} \right)^{1/2} $.
However, it is now recognized that the dynamics of HRB convection are
more subtle than previously thought. As discussed by Calzavarini
et al. (2006), simulations at unit aspect ratio
show huge fluctuations in the
instantaneous Nusselt and Reynolds numbers arising from the intermittent
or quasi-periodic (depending on Ra) exponential
growth of so-called ``elevator modes''. These modes are thus named because
they are independent of $z$, and have the peculiar property of being
exact nonlinear and exponentially growing solutions of the governing
equations (\ref{eq:HRBorig}). The most unstable mode has a horizontal
wavelength equal to the larger horizontal dimension
of the box. Hence, the aspect ratio of the system directly influences
the macroscopic solution.
This phenomenon has a close parallel
in shearing-box studies of the magnetorotational instability. In that case,
forcing by a constant velocity gradient plays the role of the
constant temperature gradient, while perturbations to the background fields
are also assumed to be triply periodic. This system is unstable to equivalent
``channel modes'', exact nonlinear and exponentially growing solutions
of the equations and
associated periodic boundary conditions (Goodman \& Xu, 1994). In this case,
it is known that the channel modes are themselves subject to secondary
shearing
instabilities which limit their growths.
However, the existence and growth rates
of shearing instabilities depend sensitively on aspect ratio:
they are strongly inhibited in systems where the streamwise
direction is smaller than the cross-stream directions. As a result,
systems with roughly cubic geometry are dominated by the channel
modes and are found to have
very strongly fluctuating large-scale transport properties, but for
larger aspect ratio the fluctuations are much smaller and the channel modes
are inhibited (Bodo et al. 2008).
For these reasons, we performed a series of HRB simulations of
various aspect ratios, in order to determine whether the same phenomenon
occurs, and to provide a better point of comparison for the closure model.
Appendix~C provides a brief description of the numerical algorithm
used, and the results are summarized in Fig.~\ref{fig:HRBRaNu}. We studied
5 cases, with $L_x = L_y$ and $L_x/L_z=$1/2, 2/3, 9/10, 1/1 and 4/3.
In the last case, the elevator modes
continue growing unaffected by perturbations until the code fails, which
seems to corroborate the premise that the secondary instabilities
are inhibited
in wider-than-tall boxes. For $\Gamma < 1$, the measured Nusselt number
eventually converges to a meaningful average and
is found to scale as predicted by the closure model, namely proportional
to (Pr Ra)$^{1/2} \Gamma^2$.
A good fit with the model predictions is found by selecting
$L = \delta L_x = L_x/\sqrt{\pi}$. For the purpose of illustration,
a snapshot of the temperature field for our largest Rayleigh number,
${\rm Ra} = 5 \times 10^6$ (with Pr = 1) and aspect ratio 1/2 is shown
in Figure \ref{fig:HRBeyecandy}.
\begin{figure}
\epsfig{file=f8.eps,width=8cm}
\caption{Variation of the Nusselt number with rescaled Rayleigh number for
$\mathrm{Pr}=1$
for homogeneous convection. The diamond symbols show
the data from our 3D HRB numerical simulations for Ra = $5\times 10^6$ and
the stars for Ra = $2.16 \times 10^5$. In all cases
$\mathrm{Pr}=1$. The error bars show the measurement uncertainty due to
the finite integration time of the simulation. The aspect ratio $\Gamma = L_x/L_z$ of each
simulation is indicated near the corresponding symbol. The solid line shows
the asymptotic analytical solution (\ref{eq:RaNuHRB}),
using the values of the parameters $\{ C_i\}$ as listed
in equations (\ref{eq:C1C2}), (\ref{eq:Cd}), and (\ref{eq:C6C7}). A good fit
to the data is found by choosing
$L = L_x/\sqrt{\pi}$.}
\label{fig:HRBRaNu}
\end{figure}
\begin{figure}
\centerline{\epsfig{file=f9.epsf,width=7cm}}
\caption{Volume-rendered visualization of the temperature field
for ${\rm Ra} = 5\times 10^6$ and $\mathrm{Pr}=1$
for homogeneous convection in a box of aspect ratio 1/2. Note how,
even at this high Rayleigh number, the size of the dominant structures
is equal to the box size.}
\label{fig:HRBeyecandy}
\end{figure}
\subsection{The effect of rotation on homogeneous turbulent convection}
\label{RHRB}
We now consider the effect of rotation on HRB convection,
where the rotation axis
lies at an angle $\gamma$ from the vertical direction:
${\bf \Omega} = (0,\Omega \sin\gamma, \Omega \cos\gamma)$.
In this section it is more convenient to work with dimensionless
variables so we select the following scalings:
\begin{eqnarray}
&& \bar R_{ij} = L^2 \tilde{N}^2 \, \hat R_{ij}, \nonumber \\
&& \bar F_{i} = - \frac{{\rm d} \bar \Theta}{{\rm d} z} L^2 \tilde{N} \, \hat F_{i}, \nonumber \\
&& \bar Q = \left( \frac{{\rm d} \bar \Theta}{{\rm d} z}\right)^2 L^2 \, \hat Q, \nonumber \\
&& \Omega_k = \Omega \, \hat \Omega_k,\qquad g_k = g \, \hat g_k,
\label{eq:nondimrot}
\end{eqnarray}
where for convenience
$\tilde{N}$ is defined as $\tilde{N}^2 = - N^2$, and is {\it positive}
when the fluid is convectively unstable. The convective Rossby number
is then defined as
\begin{equation}
{\rm Ro} = \tilde{N}/\Omega.
\end{equation}
Stationary solutions of the closure model far from onset of convection satisfy the following equations:
\begin{eqnarray}
&& 2\, {\rm Ro}^{-1} (\epsilon_{ikl} \hat R_{lj} + \epsilon_{jkl} \hat R_{li} ) \hat \Omega_k + \hat g_i \hat F_j + \hat g_j \hat F_i \nonumber \\
&& \qquad = - C_1 \hat R^{1/2} \hat R_{ij} - C_2 \hat R^{1/2} \left(\hat R_{ij} - \frac{\hat R}{3} \delta_{ij} \right),
\end{eqnarray}
\begin{equation}
- \hat R_{iz} + 2\, {\rm Ro}^{-1} \epsilon_{ijk} \hat\Omega_j \hat F_k + \hat Q \hat g_i = - C_6 \hat R^{1/2} \hat F_i,
\end{equation}
\begin{equation}
2 \hat F_{z} = C_7 \hat R^{1/2} \hat Q.
\end{equation}
In the infinite Rossby number limit (equivalently in the non-rotating limit),
the solution of these equations reduces to the non-dimensional form of (\ref{eq:dimsols1})
and (\ref{eq:dimsols2}).
Should all of the quantities be expanded in terms of the inverse Rossby number as (for example)
\begin{equation}
\hat{R} = \hat{R}^{(0)} + {\rm Ro}^{-1} \hat{R}^{(1)} + {\rm Ro}^{-2} \hat{R}^{(2)} + \cdots,
\end{equation}
then we find that
\begin{equation}
\hat{R} = {{2}\over{C_1C_6}}\left[{{C_1}\over{C_7}}+
{{3C_1+C_2}\over{3(C_1+C_2)}}\right] + O( {\rm Ro}^{-2}) \mbox{ ,}
\label{eq:norotlimit}
\end{equation}
(and similarly for all diagonal components of $\hat R$).
Our expressions for the non-diagonal
terms, to first order, recover the equivalent of the well-known $\Lambda$-effect (see R\"udiger, 1989)
in the coefficient $\hat R_{xz}$:
\begin{eqnarray}
&&\hat R_{xz} = 2 \frac{\hat F_z^{(0)} + C_6 \sqrt{\hat R^(0)} (\hat R_{zz}^{(0)} - \hat R_{xx}^{(0)})}{1-C_6 \hat R^{(0)} (C_1+C_2)} \sin \gamma\, {\rm Ro}^{-1}\nonumber\\
&&\qquad\qquad + O({\rm Ro}^{-3} ).
\label{eq:leffect}
\end{eqnarray}
The $\Lambda$-effect, as seen in the above equation, describes how rotationally
constrained turbulent motions can drive differential rotation, through
the non-diagonal component of the stress-tensor
$\hat R_{xz}$. As expected from dimensional analysis and geometrical
arguments, its amplitude scales linearly with $\sin \gamma \,\Omega$.
The other two components $\hat R_{xy} $ and $\hat R_{yz} $ only become
important for more
rapidly rotating systems as they are both $O( {\rm Ro}^{-3})$. Finally,
a non-negligible horizontal heat flux is generated in the direction of
${\bf\Omega} \times \mbox{\boldmath$g$}$, namely
\begin{eqnarray}
&& \hat F_{x} = 2 \frac{ (\hat R_{zz}^{(0)} - \hat R_{xx}^{(0)}) + (C_1+C_2) \sqrt{\hat R^(0)} \hat F_z^{(0)} }{1-C_6 \hat R^{(0)} (C_1+C_2)} \sin \gamma\, {\rm Ro}^{-1} \nonumber \\
&&\qquad\qquad + O({\rm Ro}^{-3} )\mbox{ ,}
\end{eqnarray}
although note that when applied to stellar convection zones, this effect is relevant only for non-axisymmetric
heat transport. The ``latitudinal'' heat flux $\hat F_y$ on the other hand is of higher order in Ro$^{-1}$.
In the opposite limit of very low Rossby number
(the rapidly rotating limit) an expansion in powers of Ro reveals that
\begin{equation}
\hat{R} = {{2 \cos^2 \gamma }\over{C_1C_6}}\left[{{C_1}\over{C_7}}+
{{3C_1+C_2}\over{3(C_1+C_2)}}\right] + O({\rm Ro}) \mbox{ , }
\label{eq:largerotlimit}
\end{equation}
so that the rms velocity is reduced by a factor $\cos \gamma$
compared with the non-rotating case. Note, however, how the expected
reduction (and potential suppression) of the convective heat flux
in rapidly rotating systems where gravity is {\it aligned}
with the rotation axis (Chandrasekhar, 1961) so that $\gamma = 0$
is not captured by this closure model. This problem, which was identified
by Miller \& Garaud (2007), can presumably be attributed to the
incomplete modeling of the effects of the
pressure-strain correlations which are known to play
an important role in the limit of rapid rotation. It is therefore likely
that these effects also
cause our model to yield inaccurate predictions for $\gamma \neq 0$ in the same limit.
A full resolution of the issue must eventually involve the
derivation of a better closure for the pressure-strain correlation terms.
For completeness note that in this limit
the model predicts that a significant heat flux is
carried horizontally along $\mbox{\boldmath$e$}_y$, with amplitude
$\hat{F}_y = \tan \gamma \, \hat{F}_z$, and that
\begin{equation}
\hat{R}_{yz} = \frac{C_1}{C_1+C_2} \sin\gamma \cos\gamma \, \hat{R} + O({\rm Ro}) \mbox{ ,}
\end{equation}
while $\hat{R}_{xy}$ and $\hat{R}_{xz}$ are both $O($Ro$)$.
Fig.~\ref{fig:rvsro} shows the variation of the normalized $\hat R$
as a function of both $\gamma$ and Ro$^{-1}$, while Fig.~\ref{fig:rxzoverrvsro}
shows the variation of the normalized $-\hat R_{xz}/\hat R$
as a function of both $\gamma$ and Ro$^{-1}$, illustrating the dependence of the $\Lambda$-effect on
both parameters as predicted by our model.
\begin{figure}
\epsfig{file=f10.eps,width=8cm}
\caption{Variation of $\hat R$ with Ro$^{-1}$, for various values of $\gamma$, for values of the $\{C_i\}$ parameters given in (\ref{eq:C1C2}) and (\ref{eq:C6C7}). The Ro$^{-1}\rightarrow 0$ and Ro$^{-1}\rightarrow \infty$ asymptotes satisfy equations (\ref{eq:norotlimit}) and (\ref{eq:largerotlimit}) respectively.}
\label{fig:rvsro}
\end{figure}
\begin{figure}
\epsfig{file=f11.eps,width=8cm}
\caption{Variation of $-\hat R_{xz}/\hat R$ with Ro$^{-1}$, for various values of $\gamma$, for values of the $\{C_i\}$ parameters given in (\ref{eq:C1C2}) and (\ref{eq:C6C7}). Note that $\hat R_{xz}/\hat R \propto \Omega$ for low rotation rates, and to $\Omega^{-1}$ for large rotation rates.}
\label{fig:rxzoverrvsro}
\end{figure}
\subsection{Comparison with previous second-order models}
We now compare our findings with the commonly used model for
convective stresses originally proposed by
R\"udiger \& Kitchatinov (1993) and later extended by R\"udiger et al.
(2005, Ral05 hereafter). Note that the related theory
of Kitchatinov \& R\"udiger (2005) relies on the presence
of a background density stratification to explain the $\Lambda$-effect. As such it
is not an appropriate point of comparison for our Boussinesq calculation.
R\"udiger \& Kitchatinov (1993) and Ral05
assume the presence of a ``background'' turbulence caused
by a given (unspecified) forcing mechanism, which, in the absence of
rotation, is described by an eddy
turnover time $\tau$, a mixing length $l$ and a turbulent diffusivity $\nu_{\rm t}=l^2/\tau$.
This background turbulence also may also have some degree of
anisotropy, controlled by the parameter $a$ defined in our notation as
\begin{equation}
a = \frac{\bar R_{xx}^{(0)} + \bar R_{yy}^{(0)} - 2\bar R_{zz}^{(0)} }{\bar R_{zz}^{(0)} }\mbox{ , }
\end{equation}
where the superscript $(0)$ denotes turbulent
quantities of the non-rotating system. Note how $a=0$ for isotropic turbulence.
Ral05 show how the presence of rotation (where the rotation
axis lies at an angle $\gamma$ from the vertical) modifies the background turbulence, an effect which
gives rise to non-diagonal components in the stress tensor. They
argue that this phenomenon is controlled
by the Coriolis number $\Omega^*$ defined as
\begin{equation}
\Omega^* = 2\tau \Omega \mbox{ . }
\end{equation}
Their eddy turnover time $\tau$ is naturally
related to $L/\sqrt{R^{(0)}}$ in our notation, so that, for the purpose
of comparison we have
\begin{equation}
\Omega^* \propto \frac{\Omega L}{\sqrt{R^{(0)}}} \mbox{ , }
\label{eq:omegastar}
\end{equation}
where the proportionality constant is of order unity.
In the limit of slow rotation, Ral05
predict a $\Lambda-$effect through the following term:
\begin{equation}
\bar R_{xz} \propto \frac{2 a }{5} \sin\gamma \frac{\Omega L}{\sqrt{R^{(0)}}}
\bar R_{zz}^{(0)} \mbox{ , }
\end{equation}
where the proportionality constant is the same as
in equation (\ref{eq:omegastar}). Meanwhile, our model when written in
dimensional form and recast in terms of the anisotropy factor $a$ yields
\begin{equation}
\bar R_{xz} = \frac{\left(C_1 - C_6 \frac{a}{a+3}\right) \hat R^{(0)} }{1-C_6 \hat R^{(0)} (C_1+C_2)} \frac{3(C_1 + C_2)}{3C_1 + C_2} \sin \gamma\, \frac{\Omega L}{\sqrt{R^{(0)}}} \bar R_{zz}^{(0)} \mbox{ .}
\end{equation}
where $ \hat R^{(0)} $ is a dimensionless constant which depends only on the
model parameters, and is given by equation~(\ref{eq:norotlimit}) with Ro$^{-1}=0$.
In the same slow-rotation limit, the other off-diagonal components of the stress
tensor are $O({\rm Ro}^{-2})$ or higher order in both our and their models.
Overall, the two formalisms agree on the dependence of the stresses on the rotation rate and
on latitude, as expected on dimensional and geometrical grounds. In addition, both
models explicitly demonstrate the importance of the anisotropy of the background non-rotating
turbulence
in controlling the amplitude of the $\Lambda$-effect. However, the dependence of $\bar R_{xz}$
on the anisotropy factor $a$ superficially appears to be different in the two theories.
We interpret this in two ways. First, note that the anisotropy factor
$a$ is a ``free'' parameter in the works of Ral05.
In our model by contrast, there is no freedom in independently
specifying $a$ since it is a solution of the model once
the system is specified (e.g. shearing flow, convective flow) and
depends on the $\{C_i\}$ parameters. In the HRB system for example
$a = -6C_1 /(3C_1 + C_2) $.
Secondly, $\bar R_{xz}$ is directly proportional to $a$ in the model of Ral05 while our model
reveals an additional contribution to the $\Lambda$-effect arising from the background turbulent
heat flux (see equation (\ref{eq:leffect}) for a more explicit expression).
This contribution is missing from the model of Ral05 which
does not take into account the heat equation. As a result, one may
superficially conclude that the $\Lambda$-effect could exist even
for isotropic background turbulent convection.
In practice, it is difficult to
conceive of a naturally occurring isotropic turbulent
system which has a non-zero vertical heat flux, so the
term $\hat F_z^{(0)}$ is in fact also indirectly related
to the anisotropy of the system, although perhaps not exactly in the same way.
Finally, we emphasize that in the limit of rapid rotation, neither theory
is expected to be accurate because of the extreme induced anisotropy of the
rotating turbulent motions. Nevertheless it is interesting to note that the
predicted dependence of the stresses on the rotation rates now no
longer agree with one another. We find that $\bar R_{yz}$ tends
to a constant independent of rotation rate while Ral05
find that $\bar R_{yz} \propto$ Ro. For the other off-diagonal
components $\bar R_{xz}$ and $\bar R_{xy}$ we find a dependence on Ro,
while they predict a dependence on Ro$^{2}$.
We conclude this section by emphasizing the success of our closure model in reproducing
numerical experiments of HRB convection at various aspect ratios and Rayleigh numbers. Furthermore
our model predictions are exactly proportional to those of Ral05 (with a proportionality
constant of order unity) for convection in a slowly rotating system. Hence we
expect to recover many of the results and successes of these authors in modeling
differential rotation in stars, albeit with an extended model which self-consistently includes
heat transport in addition to angular momentum transport. In preparation of this future
modeling endeavour, we finally turn to the next natural step of this work, namely the extension of the
model to the anelastic and fully compressible equations.
\section{The anelastic system and compressible flows}
\label{sec:anelastic}
So far we have worked within the Boussinesq approximation, which is
applicable only to a shallow layer of fluid whose depth is much less
than the density scaleheight. In order to apply our model to stars
we must first adapt it to the anelastic approximation
(Ogura \& Phillips 1962; Gough 1969), which is relevant
to subsonic convection in a deep layer.
Here we follow the more standard derivation of the anelastic approximation where
the reference state is taken to be an
adiabatically stratified fluid in hydrostatic equilibrium. The reference density
$\rho_0(\mbox{\boldmath$r$})$ and temperature $T_0(\mbox{\boldmath$r$})$ may vary
substantially, while the specific entropy $s_0$ is uniform. In place
of equations (\ref{boussinesq1})--(\ref{boussinesq3}) we have
\begin{equation}
\partial_i(\rho_0u_i)=0,
\label{anelastic1}
\end{equation}
\begin{equation}
(\partial_t+u_j\partial_j)u_i=-(s-s_0)\partial_iT_0-\partial_i\psi+\cdots,
\label{anelastic2}
\end{equation}
\begin{equation}
(\partial_t+u_i\partial_i)(s-s_0)=\cdots,
\label{anelastic3}
\end{equation}
where the dots represent terms due to viscosity (in the equation of
motion) and thermal conduction (in the thermal energy equation), while
$\psi$ is, again, a modified pressure. Viscous dissipation can also be included in the thermal energy equation, although it is usually omitted in the Boussinesq approximation. A derivation of these
equations, omitting diffusive effects, is given in Appendix~B.
The anelastic system is formally very similar to the Boussinesq system
except for the variable density of the reference state. However, the
entropy perturbation and background temperature gradient respectively play the
roles taken by the temperature perturbation and $\alpha g_i$ in the
Boussinesq approximation. A very similar analysis to that carried out for
the Boussinesq system leads to equations for $\bar R_{ij}$, $\bar F_i$
and $\bar Q$ of the form
\begin{eqnarray}
\lefteqn{(\partial_t+\bar u_k\partial_k)\bar R_{ij}+
\bar R_{ik}\partial_k\bar u_j+\bar R_{jk}\partial_k\bar u_i+
\bar R_{ij}\partial_k\bar u_k}&\nonumber\\
&&+\bar F_i\partial_jT_0+\bar F_j\partial_iT_0=\cdots,
\end{eqnarray}
\begin{eqnarray}
\lefteqn{(\partial_t+\bar u_j\partial_j)\bar F_i+
\bar R_{ij}\partial_j\bar s+\bar F_j\partial_j\bar u_i+
\bar F_i\partial_j\bar u_j+\bar Q\partial_iT_0}&\nonumber\\
&&=\cdots,
\end{eqnarray}
\begin{equation}
(\partial_t+\bar u_i\partial_i)\bar Q+
2\bar F_i\partial_i\bar s+\bar Q\partial_i\bar u_i=\cdots,
\end{equation}
where the dots represent terms that require a closure model. In the
anelastic system the relevant definitions of the Reynolds stress $\bar R_{ij}$,
flux $\bar F_i$ and variance $\bar Q$ are
\begin{equation}
\bar R_{ij}=\langle\rho_0u_i'u_j'\rangle,\qquad
\bar F_i=\langle\rho_0u_i's'\rangle,\qquad
\bar Q=\langle\rho_0s^{\prime2}\rangle.
\end{equation}
Note that $\bar R_{ij}$ now has the correct dimensions for a stress
tensor, and that $\bar F_i$ is really an entropy flux density. Some
additional linear terms arise in the anelastic system because
$\partial_i\bar u_i\ne0$.
We apply the same closure model as for the Boussinesq system, except
that the relaxation timescale which was proportional to $L/\bar R^{1/2}$ is now
proportional to $L/(\bar R/\rho_0)^{1/2}$ because of
the redefinition of $\bar R_{ij}$:
\begin{eqnarray}
\lefteqn{(\partial_t+\bar u_k\partial_k)\bar R_{ij}+
\bar R_{ik}\partial_k\bar u_j+\bar R_{jk}\partial_k\bar u_i+
\bar R_{ij}\partial_k\bar u_k}&\nonumber\\
&&+\bar F_i\partial_jT_0+\bar F_j\partial_iT_0=
-\frac{C_1}{L}\left({{\bar R}\over{\rho_0}}\right)^{1/2}\bar R_{ij}\nonumber\\
&&\qquad-\frac{C_2}{L}\left({{\bar R}\over{\rho_0}}\right)^{1/2}
(\bar R_{ij}-{\textstyle{{1}\over{3}}}\bar R\delta_{ij}),
\label{rij_anelastic}
\end{eqnarray}
\begin{eqnarray}
\lefteqn{(\partial_t+\bar u_j\partial_j)\bar F_i+
\bar R_{ij}\partial_j\bar s+\bar F_j\partial_j\bar u_i+
\bar F_i\partial_j\bar u_j+\bar Q\partial_iT_0}&\nonumber\\
&&=-\frac{C_6}{L}\left({{\bar R}\over{\rho_0}}\right)^{1/2}\bar F_i,
\label{fi_anelastic}
\end{eqnarray}
\begin{equation}
(\partial_t+\bar u_i\partial_i)\bar Q+
2\bar F_i\partial_i\bar s+\bar Q\partial_i\bar u_i=
-\frac{C_7}{L}\left({{\bar R}\over{\rho_0}}\right)^{1/2}\bar Q.
\label{q_anelastic}
\end{equation}
We do not include any of the terms proportional to $\nu$ or $\kappa$ here
because we consider the high-Rayleigh number limit in the absence of rigid boundaries only.
The question arises as to how the length-scale $L$ should be
identified for anelastic convection in a deep layer. It should probably
related to the pressure scaleheight or density scaleheight, as in the stellar mixing-length theory.
Indeed, numerical simulations of convection in spherical shells with a substantial density variation indicate that the convective cells are much smaller near the outer surface where the scaleheight is small; nevertheless, there may be situations in which convective plumes can span several scaleheights.
Equations~(\ref{rij_anelastic})--(\ref{q_anelastic}) can then be combined with equations for the mean variables in the form
\begin{equation}
\partial_i(\rho_0\bar u_i)=0,
\end{equation}
\begin{equation}
\rho_0(\partial_t+\bar u_j\partial_j)\bar u_i=-(\bar s-s_0)\partial_iT_0-
\rho_0\partial_i\bar\psi-\partial_j\bar R_{ij},
\end{equation}
\begin{equation}
\rho_0T_0(\partial_t+\bar u_i\partial_i)(\bar s-s_0)=
\frac{C_1}{L}\left({{\bar R}\over{\rho_0}}\right)^{1/2}{{\bar R}\over{2}}
-T_0\partial_i\bar F_i.
\end{equation}
In the last equation we have included the turbulent viscous heating.
These equations could be applied to studying convection and meridional circulation in rotating stars. The solution can be assumed to be axisymmetric and independent of time, although for practical purposes it may be easier to evolve the equations forwards in time until a steady state is reached (if it is) rather than directly seeking such a solution.
In the absence of rotation the problem becomes spherically symmetric, the mean flow disappears, the stress becomes diagonal (although anisotropic) and we obtain the local algebraic system
\begin{equation}
2\bar F_r\partial_rT_0=-\frac{C_1+C_2}{L}\left({{\bar R}\over{\rho_0}}\right)^{1/2}\bar R_{rr}+\frac{C_2}{3L}\left({{\bar R}\over{\rho_0}}\right)^{1/2}\bar R,
\end{equation}
\begin{equation}
2\bar F_r\partial_rT_0=-\frac{C_1}{L}\left({{\bar R}\over{\rho_0}}\right)^{1/2}\bar R,
\end{equation}
\begin{equation}
\bar R_{rr}\partial_r\bar s+\bar Q\partial_rT_0=
-\frac{C_6}{L}\left({{\bar R}\over{\rho_0}}\right)^{1/2}\bar F_r,
\end{equation}
\begin{equation}
2\bar F_r\partial_r\bar s=
-\frac{C_7}{L}\left({{\bar R}\over{\rho_0}}\right)^{1/2}\bar Q.
\end{equation}
The solution is, by direct analogy with equations~(\ref{eq:dimsols1})--(\ref{eq:dimsols2}),
\begin{eqnarray}
&&\bar R_{rr}=\left({{3C_1+C_2}\over{C_1+C_2}}\right)
{{\bar R}\over{3}},\nonumber\\
&&\bar R_{\theta\theta}=\bar R_{\phi\phi}=\left({{C_2}\over{C_1+C_2}}\right)
{{\bar R}\over{3}},\nonumber\\
&&\bar F_r=-{{C_1(\bar R/\rho_0)^{3/2}}\over{2L(-N^2)}}\rho_0\partial_r\bar s,\nonumber\\
&&\bar Q={{C_1\bar R}\over{C_7(-N^2)}}(\partial_r\bar s)^2,
\end{eqnarray}
with
\begin{equation}
\bar R={{2}\over{C_1C_6}}\left[{{C_1}\over{C_7}}+
{{3C_1+C_2}\over{3(C_1+C_2)}}\right]\rho_0L^2(-N^2),
\end{equation}
where now
\begin{equation}
-N^2=(\partial_rT_0)\partial_r\bar s.
\end{equation}
In this situation the entropy gradient $\bar s$ is not known in
advance. However, to balance the thermal energy equation,
$\partial_i\bar F_i=0$, which implies that $r^2\bar F_r$ is a
constant, determined by the luminosity generated by the core of the
star. (This conclusion is modified if the radiative flux or any
sources of energy such as turbulent viscous dissipation make an
important contribution to the thermal energy equation.) Then the
above equations can be solved algebraically to find $\partial_r \bar s$,
$\bar R$, etc., at each
radius, assuming that a prescription for $L$ is given. The result is
equivalent to a version of mixing-length theory.
Rotation couples radial and latitudinal transport of heat and momentum
and induces large-scale entropy gradients and mean flows.
However, if we assume that their effects can be ignored in the overall
turbulent dynamics controlling the properties of the stresses, then
the local $\Lambda-$effect is easily recovered as an anelastic version of
equation (\ref{eq:leffect}). As before, the
only differences with the Boussinesq case is that (i) the two terms containing
$\hat R^{(0)}$, which have their origin in the eddy turnover time, should be replaced by
$\hat R^{(0)} / \rho_0$ and (ii) in expressing (\ref{eq:leffect}) in dimensional form
(see equation (\ref{eq:nondimrot})), one must
also replace $\tilde N^2$ by $(\partial_rT_0)\partial_r\bar s$ and
${\rm d} \bar \Theta/{\rm d}z$ by $\rho_0 \partial_r \bar s$,
as seen above. The resulting expression then directly links the turbulent transport of
angular momentum and of heat to one another. Since heat transport in this model
is very similar to mixing-length theory, our formalism now provides a simple
framework in which to combine models of stellar structure with models of
internal stellar dynamics. Note that in practice mean flows and especially
latitudinal entropy gradients could play a role
in the global dynamics of the system. The whole model should therefore be solved
self-consistently and globally instead of using (\ref{eq:leffect}). This can only be done
numerically and is deferred to a subsequent paper.
It is also possible to `import' the model of anelastic convection into
the full set of equations governing the motion of a compressible
fluid. The idea here is that, while the convection might be assumed
to be subsonic and to obey the anelastic approximation, the mean flow
need not obey these constraints. An example is convection in an
accretion disc, where the accretion flow, although slow, cannot be
treated in the anelastic approximation with a reference density
profile. Omitting now the bars on all quantities, and neglecting
self-gravitation (although it can easily be restored), we propose a
system of equations consisting of the equation of mass conservation,
\begin{equation}
\partial_t\rho+\partial_i(\rho u_i)=0,
\end{equation}
the equation of motion,
\begin{equation}
\rho(\partial_t+u_j\partial_j)u_i=-
\rho\partial_i\Phi-\partial_ip-\partial_jR_{ij},
\end{equation}
and the thermal energy equation,
\begin{equation}
\rho T(\partial_t+u_i\partial_i)s=
\frac{C_1}{L}\left({{R}\over{\rho}}\right)^{1/2}{{R}\over{2}}
-T\partial_iF_i,
\end{equation}
together with the equations of the closure model,
\begin{eqnarray}
\lefteqn{(\partial_t+u_k\partial_k)R_{ij}+
R_{ik}\partial_ku_j+
R_{jk}\partial_ku_i+R_{ij}\partial_ku_k}&\nonumber\\
&&+F_i\partial_jT+F_j\partial_iT=
-\frac{C_1}{L}\left({{R}\over{\rho}}\right)^{1/2}R_{ij}\nonumber\\
&&\qquad-\frac{C_2}{L}\left({{R}\over{\rho}}\right)^{1/2}
(R_{ij}-{\textstyle{{1}\over{3}}}R\delta_{ij}),
\end{eqnarray}
\begin{eqnarray}
\lefteqn{(\partial_t+u_j\partial_j)F_i+R_{ij}\partial_js+
F_j\partial_ju_i+F_i\partial_ju_j+
Q\partial_iT}&\nonumber\\
&&=-\frac{C_6}{L}\left({{R}\over{\rho}}\right)^{1/2}F_i,
\end{eqnarray}
\begin{equation}
(\partial_t+u_i\partial_i)Q+2F_i\partial_is+
Q\partial_iu_i=
-\frac{C_7}{L}\left({{R}\over{\rho}}\right)^{1/2}Q.
\end{equation}
The total energy is then exactly conserved in the form
\begin{eqnarray}
\lefteqn{\partial_t\left[\rho({\textstyle{{1}\over{2}}}u^2+\Phi+
e)+{\textstyle{{1}\over{2}}}R\right]}&\nonumber\\
&&+\partial_i\left[\rho({\textstyle{{1}\over{2}}}u^2+\Phi+h)
u_i+
{\textstyle{{1}\over{2}}}Ru_i+R_{ij}u_j+
TF_i\right]=0,\nonumber\\
\end{eqnarray}
where $e$ and $h$ are the specific internal energy and the specific
enthalpy, respectively, and the gravitational potential $\Phi$ is
assumed to be independent of time. The existence of this conservation
law implies a certain self-consistency in the equations of the model.
The terms that were added in passing to the compressible model are
required to have the form that they do in order that energy be
conserved. We note again that $F_i$ is really an entropy
flux density, and that $TF_i$ is the corresponding energy flux density.
The physical content of this model is that the turbulent convecting
fluid behaves similarly to a complex, non-Newtonian material in which
there is a dynamical constitutive equation that relates the stress
tensor to the deformation history of the fluid. The above equation
for $\partial_tR_{ij}$ (along with those for $\partial_tF_i$ and
$\partial_tQ$) plays this role.
\section{Conclusions and future prospects}
We have laid out the foundations of a new second-order closure model
for the dynamics of turbulent convection, with future applications
to stellar convective regions in mind. This model is a direct extension
of the work of Ogilvie (2003) and GO05, and has similar
properties. The proposed closure has a straightforward physical interpretation,
and well-understood limitations.
Comparison with laboratory and numerical experiments reveals
good overall agreement of the model predictions with known properties
of rotating shear flows (GO05) and high Rayleigh-number
rotating convection (this work). In particular, our model naturally
reproduces the standard scaling relationships between the Rayleigh and Nusselt
numbers for Rayleigh-B\'enard convection and for Homogeneous Rayleigh-B\'enard
convection, and contains the well-known $\Lambda$-effect describing
angular momentum transport in a rotating turbulent fluid.
When extended to the anelastic (or fully compressible) case, our formalism
can be applied to study convection in stellar interiors. Note that the effects of
Maxwell stresses can also straightforwardly be included following Ogilvie (2003)
if needed. We show that the model naturally reduces to a version of mixing-length theory when applied
in a one-dimensional framework. In the presence of rotation
it becomes a powerful tool to study within a single framework the multi-dimensional
balance involving large-scale mean quantities such as the entropy
profile, the meridional circulation and the differential rotation.
Future work applying our closure model in a spherical shell geometry
will help understand some of the trends seen in the increasingly
large number of available observations of stellar differential rotation.
\section*{Acknowledgements}
The authors thank N. Brummell and C. Doering for stimulating discussions,
and DuPuits et al. for providing
electronic tables of their data. This work was supported by
NSF-AST-0607495 and NSF-CAREER.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,011 |
Thank you to the poet Caitlin Doyle for tagging me for "The Next Big Thing" interview series! You can read her self-interview here. For "The Next Big Thing," each participating poet or fiction writer engages the same set of questions pertaining to a recently published book, a soon-to-be-published book, or a book-in-progress. Here are my responses regarding the development of my first poetry collection.
1. What is your title of your book?
2. Where did the idea come from for the book?
I wrote the title poem of the collection a long time ago, probably in 1998, and ever since then, this family of poems has gathered around that flag. Along the way, I considered Sandymount Strand (since that beach in Dublin features in the three poem-sequence of that name, and is referenced in one other poem). The Irish College at Salamanca was another possible title. But, luckily, the Brendan title stuck, and it's the right one.
There was no "idea", in that the book didn't start as a deliberate project. It evolved naturally out of my concerns and obsessions. Each poem came about because at the time, I needed to write it. But, what's been amazing has been how the poems have emerged. After the title poem, and other poems about emigration, I wrote a poem about a visit to Norway, and a found poem ("The Paradise of Birds") taken from the Irish medieval geographer, Dicuil's book De Mensura Orbis, an extract which supposedly describes the Faroe Islands. The grouping, or placing, of the poems evokes the stepping stone route that Saint Brendan may have taken to North America, via the Faroes and Iceland. The Norway setting evokes that, however indirectly. (An inspiring background source is Tim Severin's The Brendan Voyage.)
Similarly, the remora sub-theme in the book is something that evolved organically. This kind of development gives immense satisfaction, because it comes, most probably, subconsciously. Here's an example. I noticed that I had, at times, longer poems that were followed by shorter, "baby poems" that glossed or commented on the same experience, but from another angle. I remembered the remora, or "sucking fish" that swims with, and sticks to sharks, and decided to subtitle these short poems "remora". In the end, I decided to remove this subtitling, as it seemed somewhat obsessive. I'd also come up with the idea to adapt a Wikipedia entry about the fish into a found poem. One day, during the MFA at NYU, I brought it to my workshop teacher, Kimiko Hahn, during office hours. She thought it was interesting, but said that she felt that the idea needed to break through into another level of story, or psychodrama, in order to "live" – rather than simply be the reworking of an encylopaedia entry. Just as I left her office, as I was walking down the stairs, the line came to me. In a moment of almost lucid day dreaming, I actually said out loud: "I was your remora", and I had found the first line of "To a Predator". In the end, I cut that line, but it was the start of that poem, and of a theme in the book relating to predation, and gave me a whole metaphorical structure: a way of using prey and predator, in the animal kingdom, to speak about psychological and sexual predation in the human world.
How we found the cover is another story – not quite of synchronicity, but of something falling into place that is satisfying on a number of levels. Siobhán Hutson of Salmon, who designs the books (beautifully), asked for my ideas. I started looking at British Admiralty maps of the coast of County Kerry, where a lot of the poems are set, and where Saint Brendan has a presence in placenames (he was from nearby Fenit); I then moved on to look at portraits or paintings, but nothing really satisfied. Then the Harry Clarke stained glass "Saint Brendan Meets the Sorrowful Judas" jumped out at me, and that was the one.
3. What genre does your book fall under?
4. Which actors would you choose to play your characters in a movie rendition?
Liam Neeson(!) or Viggo Mortensen, would play Saint Brendan; perhaps a modified version of the CGI Gollum would play the Judas of the cover (who doesn't appear in the book, by the way). The speaker of a lot of these would be played by my own kind of Matrix residual self image. The villiain of the Digesting a Scorpion sequence in the middle of the book (the "he" / "you" addressed throughout in these sequence of what I have heard Sharon Olds call, not confessional but "accusatory" poetry) could be played by any number of actors. The important thing would be that the actor would need to convey that sentiment in The Lord of the Rings, where the "goodies" often note that evil often comes masked as too "fair".
Frodo: "I would think that a servant of the Enemy would look fair and feel foul."
Aragorn: "Ah," laughed Strider, "and I look foul and feel fair?"
This is interesting, given that it echoes Macbeth's witches, with their "Fair is foul and foul is fair. Hover through the fog and filthy air." Interestingly, in order to suggest the utter deviousness of the grooming process, the poem "Unrhymed Sonnet" is epigraphed with the quote "To beguile the time, / Look like the time" (from Macbeth, Act 1, Scene 5)
5. What is the one-sentence synopsis of your book?
An embedded report back from the Irish Diaspora of the 1980s, with several poems spoken through the voices of Lazarus, St. Christopher, St. James; poems of travel that are at the same time exile, displacement and spiritual journey; poems that explore betrayal, parasitism, sexual violence and evil (told through a number of zoological metaphors reflecting domination and submission (shark and remora; cleaner wrasse and "client fish")); poems of emergence from that hypnosis into healing and greater selfhood.
6. Will your book be self-published or represented by an agency?
The book was published by Salmon Poetry in July of 2012.
7. How long did it take you to write the first draft of your manuscript?
At the beginning, I didn't exactly think I was writing a book. I felt / hoped that the poems would be a book, but I didn't know when, especially because the first publishers I sent it to weren't interested. I think it's important to write the poems for themselves, and after that, to think about a book. Otherwise, one starts to put "product before process", and that is fatal, because it can bleed the pleasure out of the writing. (Ultimately, why else do we write?) The earliest poem in terms of chronology is "In the underground carpark", a poem about a friend's grandmother's funeral, and it was written probably in 1996. The first section of the book was written largely between then and 2008; the middle section ("Digesting a Scorpion") was written while I was studying an MFA at NYU from 2010 to 2012. (Having Sharon Olds as a teacher, and thesis advisor, was hugely helpful, as was being able to attend office hours with Marie Howe and Yusef Komunyakaa.) I also reworked a lot of the poems from sections 1 and 3 during my MFA. I think a lot of the poems had already been finished, but I think what I learnt at NYU about craft helped me to drill the decay out of the poems, and I was hugely lucky to be able to do that. Friends and fellow poets like Thomas Dooley, and my fiancée, the short story writer Adrienne Brock, were a big help. The workshop process also helped me to lose "custody" or overprotectiveness. Jessie Lendennie had accepted the book for publication in January 2010, but it was an entirely different kettle of fish then, pardon the pun, in that it didn't have the middle section. So, even though the book was on the back burner from 1996 until 2012 (I wrote the last poem in April 2012), in many ways it was largely "written" as a book, between 2010 and 2012.
One last thing: I wrote my second collection at NYU also, so I had to work out where to draw the line in the sand between what would go into Brendan, and what would belong to some future collection. It was difficult at times, enjoyably difficult, especially because the "Digesting a Scorpion" vein was still spawning poems. At times it came down to: "is this poem finished yet? No? Then leave it." 8 months after publication, I'm happy that what seemed arbitrary in terms of cutting the umbilical cord between me and the book has a satisfying aesthetic symmetry in how the poems in the book speak to each other. It's a kind of deep satisfaction that gives creative peace of mind, because although no work of art is ever finished, you've done your best.
8. What other books would you compare this story to within your genre?
The middle section would have a lot in common with an aspect of Sharon Old's Satan Says, in that it has unity of theme and purpose, and is built around an evolution within the same family of imagery. The first and third (last) sections of the book would have something in common with aspects of Irish poets like Seamus Heaney's work: the importance of landscape or, loosely Dinnseanchas, "place lore" in the Irish tradition. This is not to say that I go deeply into mythology or history, although I do reference it, but simply that landscape, for me, is both a symbol, and reality, of alienation, and belonging. Place is hugely important to the Irish.
9. Who or what inspired you to write this book?
What most inspired me was the Yeatsian argument with the self that poetry constitutes. Each poem was, at the time I wrote it, a healing of that argument, or an equation that gave me equilibrium. So, a poem like "The Irish College at Salamanca" was a search for belonging. Since I felt unmoored by having left Ireland as a child, when I lived in Spain as an adult, I was drawn to the Irish College in Salamanca. The physical place became hugely important as an emblem for Irish exile, or presence abroad, and gave me a metaphor for my own experience of the complexity of travel, and having left home and trying to return. Another example: the poem "Saint James as a Young Man" came out of a visit to the Prado Museum in Madrid. I noticed that out of Ribera's paintings of the apostles, only one was a young man. It was Saint James, patron saint of Spain, who is supposed to be buried in Santiago de Compostela cathedral, where I lived for a year at the age of 21. So, the poem reflects the paintings, indirectly, but it is also my way of merging my "studenty" experiences with something outside myself—in this case, Spanish apocryphal history, which is a place I think poetry thrives, in the interstices between fact and myth. Even though the short stories I have tried to write haven't seemed to have worked as fictions, I think fiction works best in my poetry. That's not to say that I am not a confessional poet: in almost every poem in the collection the speaker is me, let's not be coy. Then again, the confessional in the most personal of the poems is mediated via image, metaphor, story, and the tension and hold of craft, the hawsers and ropes that craft is, that allows the ship to sail. Without it, the sails would puddle on the poop deck, as it were!
10. What else about your book might pique the reader's interest?
I would hope that the reader would be piqued, or interested, to know that I strive to write for the kind of person who often says "oh, I don't read poetry." I don't go for the lowest common denominator in terms of my word choices; and, sometimes, indeed, people who don't read much poetry say that they don't understand a particular image or idea. I am happy, though, when a non-poet friend-of-a-friend comes up to me at a reading and says that they enjoyed hearing my work, and that that was their first time at a reading. Although I write for myself, when that moment happens is another reason why I write.
As a pique, here's a link to some of the poems in the book. You can even watch a short movie in which the first poem of the book, "Dún Chaoin", features: click here.
FOR THE NEXT ROUND, I've "tagged" some talented poets. You can click on their names (as the links become available) to read their interviews: Cat Richardson, Ken L. Walker (whose interview is hosted on this blog), & Thomas Dooley.
Posted on March 14, 2013 April 17, 2013 Author David McBloglinCategories WritingTags David McLoghlin, Interview, process, The Next Big Thing, Waiting for Saint Brendan, writing
One thought on "The Next Big Thing"
Pingback: The Next Big Thing Interview, with Ken L. Walker | New York Peristalsis
Previous Previous post: Rediscovering Luis Cernuda (1902 – 1963)
Next Next post: Who is Saint Patty? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,730 |
Brede Hangeland: Ex-Fulham And Crystal Palace Hero Doesn't Disappoint After Being Asked To Name Laziest XI Teammates
27th, January 2017
Waffling away on a Norwegian podcast, Brede Hangeland spent a little time reminiscing about his career – specifically the laziest players he's ever played alongside.
Speaking in a light-hearted interview with Heia Fotball, Hangeland was asked to name XI of the most work-shy fops he'd ever played with.
The former Fulham and Crystal Palace defender didn't disappoint…
Wayne Hennessey
Used to just lie on one of those thick blue mats in the gym while we were working out.
Chris Baird
Totally uninterested in gym work. Whenever we did cardio, he asked the coach: 'When can we go and play football?'
Zdenek Grygera
Great guy, good friend. When he arrived at Fulham, he told the coach: 'I don't do weights.' He didn't.
As a centre-back, 'Panzer' was big, strong but it must have all come from his genes because he never went near the gym.
Amazing physique, athletic, huge potential. Some Mondays, he'd come over to me and say: 'I'm starting my programme now!'
He'd do five push-ups, sigh, then leave. He would have been incredible if he was serious.
Jimmy Bullard
Great player but incredibly weak. Never interested in the gym. Clear-cut pick for my laziest team!
Mousa Dembele
Maybe the best I played with, but struggles with his physique. Never lifted weights. Has incredible balance. Earned the right to not work out.
Bryan Ruiz
Don't think he even knew where the gym was. He was from Costa Rica and didn't like feeling the least bit uncomfortable.
Always wore long sleeves and gloves. If it was cold or away to Stoke, he just wouldn't come along.
Bobby Zamora
Strong but hated the gym. Whenever it was time for dead lifts, he'd start feeling his hamstring – every single time!
Never seen a man get so many massages in my life. Whenever we were in the gym, Berbatov was getting a massage.
I knew the guy who gave him the massages. Usually at the end of the season, the players would give all the physios a gift but he'd massage Berbatov for hundreds of hours during the season and he would get nothing.
I was marking Adebayor in midfield. Suddenly he said: 'Ah, I'm hungry.' I replied: 'What?' He said: 'I can't wait for the game to finish. I'm so hungry. Do you know a good restaurant in London?'
At Palace, when we had strength workouts, he would sit in the gym with a cup of coffee and a muffin. He was being paid by City, Tottenham and Palace at the same time, and he was sitting in the gym drinking coffee.
Thank god Adebayor was blessed with those golden genes, eh?
Funnily enough, Zaha has since thrown a tantrum over Hangeland's comments, accusing his former Palace teammate of "lying to stay relevant".
Dare we suggest that he might have touched a raw nerve?
Posted in Crystal Palace, Fulham, Newsnow
"… If it was cold or away to Stoke, he just wouldn't come along…" — looool i literally fell out of my chair reading this at work lol #dead
FA Cup: Quarter Final Draw Sees Man City Travel To Swansea And Wolves Host Man Utd | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,078 |
Danuser acknowledges that, so far, the narrative has been focused on the Horde. "All wars have ebbs and flows," he said. "One side takes the upper hand for a while, and the other side gets their turn." The Horde fired the first volley, but the upcoming raid and new warfront are both centered around telling an Alliance story. Right now, the story's focus is on the Night Elves. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,050 |
Q: How to print out an element which has been processed by an function. Good day everyone, I'm finishing some studycode and I'm stuck at the last step!
That is, to print out an element from an array which has been randomly generated by calling upon a function which does so.
I'm scratching my head here, since the statement must pick out a certain element from the array and then print it out, and I've run it through an visualizer and it does everything correct right up to the end.
So I'm hoping if you guys(and gals) can help me understand a bit better what's going on.
public class randomfylki
{
public static int veljaEitt(int [] a)
{
int x = (int)(Math.random()*10);
return a[x];
}
public static void main(String[] args)
{
int a[] = new int[10];
int i;
for( i = 0; i < 10; i++)
{
a[i] = (int)(Math.random()*100);
}
veljaEitt(a);
System.out.print(a);
}
}
I do hope you'll forgive me for having a bad title, I'm not quite sure on what it should be ._.
A: instead of
veljaEitt(a);
System.out.print(a);
do this
System.out.print(veljaEitt(a));
A: Your method returns an int, you can store that into a variable and print the variable:
int result = veljaEitt(a);
System.out.print(result);
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,596 |
Q: How to debug a node module part of a build? In many of my applications I use the NPM package.json to manage my build tools. I've found that one of the modules probably has a bug. I'd like to debug it but I don't know how to debug the application in the context of the build task.
Specifically, in this case, I'm using Ember-cli. Ember-cli has a build command: ember build that builds the app using a bunch of modules defined in package.json such as ember-cli-compass-compiler.
I want to be able to add breakpoints or some sort of logging at certain points of the ember-cli-compass-compiler module that are triggered when the build runs so that I can inspect values.
A: Looks like according to https://github.com/ember-cli/ember-cli/blob/c8934ab0f2eb3aab03ce4557a36c317887245b95/lib/models/project.js as part of the build step is to look at the project's package.json and check for ember-cli-compass. After which, it would presumably use your project's local version of ember-cli-compass-compiler to execute some tasks. The simplest way to debug it is to use console.log() and log various points inside ember-cli-compass-compiler to see what code paths are being triggered.
The codebase for the compiler is very small, and you'd probably want to start with the index.js https://github.com/quaertym/ember-cli-compass-compiler/blob/master/index.js
A: node --inspect-brk ./node_modules/.bin/ember build
will launch the program in the debugger. After you attach, it will stop at the entrypoint - if you haven't already set your breakpoints that is a convenient time to set them.
You may have to skip through some "false" errors that are normal and are being handled correctly - they seem random and can be confusing if you are not expecting them. You can uncheck "caught exceptions" to avoid that, but then you could miss important caught exceptions in ember which occur before your breakpoint(s).
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,864 |
namespace Castle.SvnHooks
{
using System;
using System.IO;
/// <summary>
/// Summary description for IRepositoryFile.
/// </summary>
public class RepositoryFile : IDisposable
{
public RepositoryFile(IRepository repository, String path, RepositoryStatus contentsStatus, RepositoryStatus propertiesStatus)
{
if (path == null)
throw new ArgumentNullException("path");
if (path.Trim().Length == 0)
throw new ArgumentException("Path must be set to a valid path", "path");
if (path[path.Length-1] == '/')
throw new ArgumentException("Path must be set to a file, not a directory", "path");
if (propertiesStatus == RepositoryStatus.Added ||
propertiesStatus == RepositoryStatus.Deleted)
{
throw new ArgumentException("Properties status cannot be set to Added or Deleted, use Updated", "propertiesStatus");
}
this.contentsStatus = contentsStatus;
this.propertiesStatus = propertiesStatus;
this.repository = repository;
SetPathRelatedFields(path);
if (fileName.EndsWith(" "))
throw new ArgumentException("Filename cannot end with trailing spaces", "path");
if (fileName.StartsWith(" "))
throw new ArgumentException("Filename cannot begin with leading spaces", "path");
}
private void SetPathRelatedFields(String path)
{
// Set the path
this.path = path;
// Extract the file name from the path
if (path.IndexOf('/') == -1)
{
fileName = path;
}
else
{
fileName = path.Substring(path.LastIndexOf('/')+1);
}
// Extract the extension from the file name
if (fileName.IndexOf('.') == -1)
{
extension = String.Empty;
}
else
{
extension = fileName.Substring(fileName.LastIndexOf('.') + 1);
}
}
#region IDisposable Members
public void Dispose()
{
if(contents != null)
{
contents.Close();
contents = null;
}
}
#endregion
public override string ToString()
{
return Path;
}
public Stream GetContents()
{
if (!IsText)
throw new InvalidOperationException("Cannot get the contents of a binary file");
if (contents == null)
{
this.contents = repository.GetFileContents(Path);
}
contents.Seek(0, SeekOrigin.Begin);
return new ReadOnlyStreamWrapper(contents);
}
public String[] GetProperty(string name)
{
if (name == null)
throw new ArgumentNullException("name");
return repository.GetProperty(Path, name);
}
public RepositoryStatus ContentsStatus
{
get { return contentsStatus; }
}
public RepositoryStatus PropertiesStatus
{
get { return propertiesStatus; }
}
public String Extension
{
get { return extension; }
}
public String FileName
{
get { return fileName; }
}
public String Path
{
get { return path; }
}
/// <value>
/// The MIME type of the file it must always
/// return a valid string, if the MIME type
/// is not known by Subversion it should
/// default to "text/plain"
/// </value>
public String MimeType
{
get
{
if (mimeType == null)
{
String[] props = GetProperty("svn:mime-type");
if (props == null)
{
mimeType = "text/plain";
}
else
{
mimeType = props[0];
}
}
return mimeType;
}
}
public bool IsText
{
get
{
return MimeType.StartsWith("text/");
}
}
private IRepository repository;
private RepositoryStatus contentsStatus;
private RepositoryStatus propertiesStatus;
private String path;
private String fileName;
private String extension;
private String mimeType = null;
private Stream contents = null;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,373 |
Q: Setting Queue Arguments For All Consumers In Mass Transit I have a simple Mass Transit setup using RabbitMQ and am taking advantage of IRabbitMqBusFactoryConfigurator.ConfigureEndpoints to automatically create endpoints for my consumers.
The problem is I also want to set some queue arguments i.e. "x-max-length", "x-overflow" on all these queues. Using ConfigureEndpoints it doesn't seem to copy those arguments across.
I'm using MassTransit 5.5.6. I see that in 7.1.6 I could use IConfigureReceiveEndpoint however I cannot upgrade.
Is this possible or do I need to manually specify each endpoint?
A: You can upgrade, or you'll need to specify it manually for each endpoint.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,496 |
{"url":"https:\/\/stats.stackexchange.com\/questions\/491505\/two-procedures-for-hypothesis-testing","text":"# Two procedures for hypothesis testing\n\nI would like to know if these two procedures for hypothesis testing are equivalent.\n\nFirst, build a test statistic. Then with the data calculate it. And then compare it with the value of the distribution for a given level of significance. And so reject or accept the null hypothesis.\n\nSecond, calculate the p-value. Then reject or accept the null hypothesis if the p-value is greater or less than the given significance level.\n\nThe two procedures are the same but expressed in different ways? Can in any case the null hypothesis be rejected with one procedure and accepted with the other?\n\n[Your p value method (i) is exactly backward (implying you reject if p is large by listing both first), and (ii) doesn't identify what happens if p is exactly $$\\alpha$$ - which is reject. Some books have this border case wrong]","date":"2021-03-06 10:24:27","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\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 1, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6480051279067993, \"perplexity\": 381.89105174771265}, \"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-2021-10\/segments\/1614178374686.69\/warc\/CC-MAIN-20210306100836-20210306130836-00288.warc.gz\"}"} | null | null |
Kendrick Lamar collaborator and producer Terrace Martin has announced a new album and released a new single that features fellow-Kendrick collaborator (and accomplished artist in his own right) Kamasi Washington. The album is called Velvet Portraits, and it hits on April 1 via Sounds of Crenshaw/Ropeadope. "Think of You" is indeed velvety, a catchy, smooth jazz song with a great melody and some distinctive Kamasi sax-playing. You can stream it below.
Meanwhile, another Kendrick collaborator Anna Wise released a new song and video. It's called "BitchSlut" — a feminist anthem that tackles slutshaming and the double standards women have to deal with. She directed the video herself (it was shot on an iPhone). You can watch it below.
Anna is also opening for Petite Noir when he plays Mercury Lounge on March 26. Tickets for that are still on sale. She's also opening for the producer Photay with her group Sonnymoon at a midnight show at Baby's All Right on April 9. Tickets for that are on sale now.
Speaking of Kendrick, his new album untitled unmastered debuted at #1 on the Billboard charts, moving 178,000 units and marking his 2nd chart-topper in less than a year with To Pimp a Butterfly. He also has been in the news as his song "Alright," which has acted as a protest anthem since its release, was chanted by protestors at the cancelled Donald Trump rally in Chicago.
There's also a fun new Kendrick video, which was put out by Reebok, that finds Kendrick surprising fans at a school in England and engaging in a freestyle battle with some of them. You can watch that, along with a clip of the Chicago protests, and stream the new Terrace Martin and Anna Wise songs below. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,872 |
The 2011 ITF Men's Circuit is the 2011 edition of the third tier tour for men's professional tennis. It is organised by the International Tennis Federation and is a tier below the ATP Challenger Tour. During the months of April 2011 and June 2011 over 150 tournaments were played with the majority being played in the month of May.
Key
April
May
June
References
04-06 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,480 |
FRONTIER DRILLING
OUR RIG FLEET
THE FD ADVANTAGE
© 2020 | Frontier Drilling, LLC
THE FRONTIER TEAM
With over 200 years of combined experience Frontier has exactly the right people to make your drilling operation a success.
Mac McAlister
Glen McAlister has more than 40 years of business ownership in the oil field industry. After establishing Rig Air, in 1976 at age 23, Mr. McAlister transformed Rig Air into the largest independent air and gas compression company by territory in the United States. He then formed Advanced Air Systems, later led to the creation of Advanced Rig & Equipment in 1985. Advanced Rig & Equipment completed and deployed rigs ranging from 900hp portable operating rigs in Africa and Europe to 1500hp state-of-the-art diesel electric rigs stateside. Mr. McAlister Formed partnerships to begin Advanced Drilling Technologies, and later Strata Drilling. Following the sale of his 44% interest in Strata Drilling to Bronco Drilling (BRNC), Mr. McAlister formed Frontier Drilling. He has taken the company from 1 rig to 32 rigs. Mr. McAlister's 30 years of experience have enabled him to accumulate a unique insight into every phase of development and deployment of drilling rigs and equipment.
Robert Hammons
Drilling Equipment Manager
Jerry Brown has over 40 years of experience in the Drilling Industry. Starting out as the Rig Superintendent for Big Chief Drilling Company in 1966 in Los Angeles, CA. Mr. Brown was responsible for supervising drilling operations ensuring safety standards on government land were met. Mr. Brown then worked from 1980 to 1985 as a V.P. of Operations and Drilling Manager for Cromling Drilling Company. From 1985-1990, Mr. Brown was a Rig Manager for Welch & Howell Drilling Company where he supervised the drilling of geothermal wells in the deserts of southern California. In 1990, Mr. Brown joined Mac McAlister at Advanced Rig & Equipment. With his extensive experience in the Drilling Industry, Mr. Brown now oversees the building of new Rigs for Frontier Drilling's fleet. He has proven to be a truly valuable asset to the company.
Billy Postma
Drilling Superintendent
Robert Hammons has built his career in Accounting and Finance and has more than 25 years in the oilfield. Robert oversees financial decisions, contract negotiations and the general day-to-day operations of Frontier Drilling. Robert is an Oklahoma native and has worked with Mac McAlister since the 1980's. Robert plays a key role in the trajectory of Frontier Drilling and ensures the seamless integration of both management and field offices.
Chris McMannis
Texas Superintendent
Chris O'Driscoll
Western Superintendent | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,664 |
{"url":"https:\/\/cran.rapporter.net\/web\/packages\/tab\/vignettes\/tab.html","text":"# Summary Tables with \u2018tab\u2019\n\n## Installation\n\nYou can install and load tab from GitHub via the following code:\n\ndevtools::install_github(\"vandomed\/tab\")\nlibrary(\"tab\")\n\n## Functions\n\nThe main purpose of tab is to create neatly formatted summary tables for papers and presentations. The following functions are included:\n\n\u2022 glm_v prints a GLM summary table to the RStudio Viewer\n\u2022 tabglm summarizes generalized linear models (GLM\u2019s) fit via glm or survey::svyglm\n\u2022 tabgee summarizes generalized estimating equation models (GEE\u2019s) fit via gee::gee\n\u2022 tabcoxph summarizes Cox Proportional Hazards models fit via survival::coxph or survey::svycoxph\n\u2022 tabmulti compares variables across two or more groups, e.g.\u00a0to create a \u201cTable 1\u201d\n\u2022 tabmulti.svy does the same thing as tabmulti but for complex survey data\n\n## Regression summaries with just 2 extra keystrokes\n\nTo summarize a fitted generalized linear model, simply call glm_v as you would glm. The result will be a formatted summary table printed to the RStudio Viewer. Here\u2019s an example for logistic regression:\n\nglm_v(\ndeath_1yr ~ poly(Age, 2, raw = TRUE) + Sex * BMI,\ndata = tabdata,\nfamily = binomial\n)\n\nFrom here, you can \u201csnip\u201d the summary table and save it as a figure (as I did for this README) or copy directly from the Viewer and paste outside of R.\n\nFor more flexibility, see tabglm. That function lets you control things like what columns to present, how categorical predictors are presented, and so on.\n\n## Summary tables for continuous and categorical variables\n\nYou can use tabmulti to summarize variables across two or more groups, using a formula interface. Here\u2019s an example:\n\ntabmulti(Age + Sex + Race + BMI ~ Group, data = tabdata)\n\n## Compatibility with Markdown\/Knitr\n\nThe functions all return kable objects, so they should work perfectly well in R Markdown and knitr documents.\n\n## References\n\nXie, Yihui. 2014. \u201cKnitr: A Comprehensive Tool for Reproducible Research in R.\u201d In Implementing Reproducible Computational Research, edited by Victoria Stodden, Friedrich Leisch, and Roger D. Peng. Chapman; Hall\/CRC. http:\/\/www.crcpress.com\/product\/isbn\/9781466561595.\n\n\u2014\u2014\u2014. 2015. Dynamic Documents with R and Knitr. 2nd ed. Chapman; Hall\/CRC.\n\n\u2014\u2014\u2014. 2021. Knitr: A General-Purpose Package for Dynamic Report Generation in R.","date":"2021-09-18 23:29:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4427594244480133, \"perplexity\": 12089.093576476931}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780056578.5\/warc\/CC-MAIN-20210918214805-20210919004805-00328.warc.gz\"}"} | null | null |
Membership/
The Affiliate Circle
Join the adventure of research and discovery as we explore through science and engineering
Who are Sigma Xi affiliates?
Individuals who value scientific research and do not qualify for Sigma Xi membership* such as citizen scientists and science journalists.
Students who aspire to become researchers and do not qualify to become a Sigma Xi explorer or associate member
Teachers, school administrators, and parents who educate and encourage future scientists and engineers
Health practitioners and pharmacists who would like to stay connected with the research community
*Check if you qualify for Sigma Xi membership on the Becoming a Member page.
Join the Affiliate Circle today to:
Engage with the research community
Promote science and engineering
Support the mission of Sigma Xi, The Scientific Research Honor Society to enhance the health of the research enterprise, foster integrity in science and engineering, and promote the public's understanding of science for the purpose of improving the human condition
Subscription to American Scientist magazine
This bimonthly, illustrated magazine is written and edited for scientists, engineers, and science enthusiasts who are curious about developments across the spectrum of science and technology.
Access to Sigma Xi chapters
Connect with researchers and research supporters in your area through Sigma Xi chapters, groups of Sigma Xi members, explorers, and affiliates in a community. Chapters are located at colleges and universities as well as government and private laboratories.
Additional perks:
A 50% discount on an institutional gift subscription to American Scientist for your schools or organizations
A discounted rate of $4.99 plus $2.50 for shipping of the book The Rightful Place of Science: Citizen Science (2016, Consortium for Science, Policy & Outcomes at Arizona State University), edited by Darlene Cavalier and Eric B. Kennedy. A purchase of this book comes with membership in the citizen science project, SciStarter
Registration discounts to events organized by Sigma Xi headquarters, such as annual meetings, symposia, the Student Research Conference, and the Student Research Showcase.
Sigma Xi SmartBrief
An email, Monday through Friday, with a summary of science and engineering news
Sigma Xi Newsletter
A biweekly electronic newsletter with updates on the Society's news and activities
Online community: coming soon!
Affiliates will have access to the online communication platform, Sigma Xi Communities
Sigma Xi Affiliate categories and annual dues
Professional – US $50
Individuals who work or have earned a degree in science or engineering, or non-scientists or non-engineers who support the role of science and technology in society
College or University Students – US $35
K – 12 Students – US $20
Click here to join the Affiliate Circle or download the Affiliate Circle Form, complete it and email it to affiliates@sigmaxi.org.
Questions? Email affiliates@sigmaxi.org or call us at 1-800-243-6534
Welcome to the Sigma Xi Affiliate Circle! | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,562 |
Career Development Right Arrow
EDUCAUSE Awards Right Arrow
Rising Star Award Right Arrow
Rising Star Award Recipients Right Arrow
2018 Rising Star Award Recipient
Leadership Award
Community Leadership Award
DEI Leadership Award
Damian M. Doyle
Assistant Vice President of Enterprise Infrastructure Services at the University of Maryland, Baltimore County
For demonstrating exemplary leadership in the implementation and deployment of information technology in support of the educational mission; for strong mentorship and promotion of diversity for those aspiring to the IT profession; for modeling collaborative partnerships within his institution and in the greater higher education community
The EDUCAUSE Rising Star Award for 2018 is presented to Damian M. Doyle, Assistant Vice President of Enterprise Infrastructure Services at the University of Maryland, Baltimore County (UMBC), to recognize his steady progression of achievement within UMBC and his expanding influence in the profession as a role model for collaborative partnerships to achieve success.
Damian's connection with UMBC began when he was a student and scholar in the university's Honors College. He was a student employee in the Division of Information Technology (DoIT) and joined the department full-time upon his graduation. Over time, he distinguished himself by taking on some of the university's most foundational and difficult technical assignments, beginning with supporting the NSF grant to implement the vBNS (very high speed Backbone Network Service, which later formed part of the backbone of Internet2) and continuing with his leadership on numerous other projects to establish or upgrade the technology to support the educational process and maintain network security on campus.
In recent years, Damian was promoted to director, senior director, and recently Assistant Vice President of Enterprise Infrastructure Services (EIS) and also assumed responsibility for support for the university's high-performance computing facility. In this role, he has led campus efforts to move to cloud services and has spearheaded efforts to prepare technical staff in their support of this move. More recently, Damian closely collaborated with staff involved in business services, enrollment management, and institutional research to help manage upgrades to Amazon Web Services involving the university's financial and analytical platforms.
On a campus with a strong tradition of staff participation in community-shared governance, Damian stands out as an especially insightful and effective leader. Serving as the first UMBC Professional Staff Senate president to come from DoIT, Damian has exhibited an inclusive and positive leadership style that has inspired trust and collaboration as the staff and campus community face complex challenges. He has made key contributions on several high-level committees and groups on campus, always demonstrating an outlook that is thoughtful, substantive, and collegial.
Damian's commitment to UMBC's ethos of "inclusive excellence" resonates through his work to expand opportunities for under-represented groups in all disciplines. Damian serves on the internal board of UMBC's Center for Women in Technology (CWIT) and has collaborated with CWIT to develop internship programs for women and minority student employees with DoIT. As part of this same effort, CWIT students gain valuable hands-on experience working in the university's cybersecurity, Unix infrastructure, and advanced networking teams—all to help prepare them for their future careers. Damian also volunteers extensively with Maryland's First Lego League, serving as the chair of the state planning committee and one of the head referees at competitions throughout the state.
Damian has participated in the EDUCAUSE Institute Management, Leadership, and Leading Change Programs, as well as contributed to the Cloud Computing Constituency Group and the ECAR Cloud Working Group. He has also presented in or moderated several sessions at EDUCAUSE conferences and has served as a reviewer for both the Connect and the Annual Conference Program Committees. He will be among the first faculty cohort in next year's EDUCAUSE Institute program for senior directors.
Damian's demonstrated technical and managerial aptitudes in an academic computing setting, as well as his commitment to the promotion of diversity and inclusion and his consummate focus on building relationships as a bridge to working collaboratively to leverage technology, make him worthy of recognition as an EDUCAUSE Rising Star—and as a leader who has much more to contribute to the higher education IT community.
This EDUCAUSE Award is sponsored by Moran Technology Consulting, Gold Partner. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,375 |
Three Army Soldiers Killed in NY Helicopter Crash
TRAGIC
Flags in New York will be lowered to half staff in their honor Thursday.
Megan Fox's Private Parts
Oversharers
Updated Apr. 25, 2017 1:55PM ET / Published Sep. 16, 2009 10:14AM ET
Krista Kennell, Sipa Press / AP Photo
Someone please make Megan Fox go away: "Men are scared of vaginas," she says in the latest issue of Rolling Stone, lauding her own "powerful, confident vagina" as the secret to her power. She goes on to say that "I don't really want to share myself with the public. I want to deflect attention from my reality." Away from reality, toward her vagina, apparently.
Read it at Rolling Stone | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,020 |
\section{Introduction}
In Seyfert Active Galactic Nuclei (AGNs), the observation of ubiquitous
rapid variability of the X-ray continuum emission
has established that these X-rays must originate very close to the central supermassive black hole,
in the central tens of $R_{\rm Sch}$ ($\equiv 2GM_{\rm BH}/c^2$), and
in the region of extreme gravity. The X-rays are generally though to originate in some type of optically-thin,
hot ($10^{\sim9}$ K) corona; plausible geometrical configurations include, but are not limited to,
the hot, ionized outer layers of the optically-thick, geometrically-thin accretion disk which feeds the black hole
(Nayakshin et al.\ 2000),
part of a central, advection-dominated, radiatively-inefficient accretion flow (e.g., Narayan \& Yi 1994)
or the base of an outflowing jet (Markoff et al.\ 2005),
A common tool to quantify the aperiodic variability as a function of temporal frequency
is the power spectral density (PSD) function. The {\it Rossi X-ray Timing Explorer} (\textit{RXTE}), launched in 1995,
has allowed multi-time scale monitoring campaigns to be executed, yielding
high-quality, broadband X-ray PSDs probing X-ray variability on temporal frequencies from
$\sim 10^{-4}$ to $\sim 10^{-8}$ Hz (time scales from
hours to years). These PSDs commonly show "breaks" at temporal frequencies $f_{\rm b}$,
with PSD power-law slopes breaking from $\sim$--2 to $\sim$--1 above and below $f_{\rm b}$, respectively.
Values of $f_{\rm b}$ are usually in the range $10^{-6}$ to $10^{-3}$ Hz, i.e.,
break time scales $T_{\rm b}$ ($\equiv 1/f_{\rm b}$) are usually in the range $\sim$0.01 to a few days
(e.g., Edelson \& Nandra 1999; Uttley, M$^{\rm c}$Hardy \& Papadakis 2002, hereafter U02;
Markowitz et al.\ 2003, hereafter M03;
M$^{\rm c}$Hardy et al.\ 2004, 2005, 2007;
Markowitz et al.\ 2007; Porquet et al.\ 2007; Chatterjee et al.\ 2009).
These papers also demonstrated that break frequencies were consistent with
a correlation between $T_{\rm b}$ and black hole masses $M_{\rm BH}$,
estimated by optical measurements such as reverberation mapping (e.g., Peterson et al.\ 2004) or via
host galaxy stellar velocity dispersion $\sigma_*$ (e.g.,
Merritt \& Ferrarese 2001, Tremaine et al.\ 2002). The correlation extrapolates down to
stellar-mass, Galactic black hole X-ray binary (BH XRB) systems, and, combined with
the remarkable similarity in PSD shapes between Seyferts and BH XRBs, strongly
supports the notion of identical variability mechanisms operating
in both classes of objects.
However, as first suggested by M$^{\rm c}$Hardy et al.\ (2004), there is an additional dependence on the accretion
relative to Eddington $\dot{m} \equiv L_{\rm Bol}/L_{\rm Edd}$, such that for a given
$M_{\rm BH}$, $T_{\rm b}$ decreases as $\dot{m}$ increases.
This "fundamental plane" between $T_{\rm b}$, $M_{\rm BH}$ and $\dot{m}$
signifies a sort of "unification" across supermassive and stellar-mass
accreting black hole systems, and has been
quantified in an empirically-derived relation by M$^{\rm c}$Hardy et al.\ (2006) from a sample of
Seyferts and BH XRB X-ray PSDs.
In this paper, we present the broadband X-ray PSD of the Seyfert AGN
NGC 7469 and present evidence for a PSD break at time scale of $\lesssim$ 10 days.
As this source's black hole mass is well known both from reverberation mapping
the optical broad line region (from an International AGN Watch monitoring program, e.g.,
Peterson et al.\ 2004) and from the $M_{\rm BH}$--$\sigma_*$ relation, we can
test if the break is consistent with the M$^{\rm c}$Hardy et al.\ (2006) empirical relation.
Nandra \& Papadakis (2001; hereafter NP01) used an {\it RXTE}\ intensive monitoring campaign
on NGC 7469 to measure preliminary high temporal frequency PSDs,
over the temporal frequency range $10^{-6}$ to $10^{-3}$ Hz. They did not
find evidence for a break in the PSD over those temporal frequencies,
but they claimed that the PSD slope flattened
as photon energy increased. The PSDs we present here
cover a wider temporal frequency range, and so we can revisit this claim.
The rest of this paper is organized as follows: the light curve sampling and data reduction
are described in Section 2.
The PSD measurement and fit results are presented in Section 3. The results are discussed in
Section 4, and a summary of our main conclusions is given in Section 5.
Appendix A contains the results of spectral fits to summed spectra derived from the
\textit{RXTE} campaigns.
Appendix B describes the ``surrogate'' Monte Carlo method of Press et al.\ (1992)
used to derive estimates of
PSD model parameter uncertainties.
\section{Observing Strategy, Observations, and Data Reduction}
Most of the high dynamic range Seyfert PSDs have been based on similar observing strategies:
regularly sampling the source flux over multiple time scales such that the resulting set of
light curves yields individual PSD segments covering complementary ranges in temporal frequency
(e.g., Edelson \& Nandra 1999)\footnote{Exceptions can occur for objects whose PSD break frequencies
lie at relatively high temporal frequencies, $10^{\sim -4}$ Hz;
for those objects, a PSD derived from an uninterrupted \textit{XMM-Newton} long-look of $\sim$ a hundred ks in duration
can reveal the break, e.g., Vaughan \& Fabian (2003) and Porquet et al.\ (2007).}. We follow the same strategy for NGC 7469.
We monitored the source with {\it RXTE} once every 4.27 days
(64 satellite orbits) for a duration of 6.3 years, from 2003 April 8 to 2009 July 15
(Modified Julian Day [MJD] 52737--55027;
observation identifiers 80152-05-*, 90154-02-*, 91138-02-*, 92108-04-*, 93144-06-*, and 94144-06-*).
This sampling, which probes variability on
time scales from $\sim$ a week to a few years, is henceforth called
``long-term'' sampling. Each observation lasted approximately 1 ks.
There were six gaps due to satellite sun-angle viewing constraints
in mid-February to early April of each year; each gap was $\sim50$ days long.
For this light curve, the average deviation from ephemeris was 0.23 days (5$\%$).
More intensive monitoring with {\it RXTE} was done to probe
variability on time scales from a few hours to a month
(``medium-term'' sampling).
{\it RXTE} observed NGC 7469 from 1996 June 10 at 00:55 UT until 1996 July 12 at 00:21 UT
(MJD 50244.0--50276.0; observation identifiers 10315-01-*), obtaining 1-ks snapshots
every orbit (6 ks) for approximately the first half of the campaign, followed by
2-ks snapshots every other orbit (13 ks).
Finally, we used the light curves from two continuous \textit{XMM-Newton} observations
to quantify variability on time scales from $\sim$ 1 hour to $\sim$ 1 day.
For the \textit{RXTE} sampling,
we used data from the Proportional Counter Array (PCA; Swank 1998, Jahoda et al.\ 2006);
extraction of background-subtracted light curves followed standard extraction procedures
(we refer the reader to, e.g., M03 for additional details) and HEASOFT version 6.7 software.
PCA STANDARD-2 data were collected from proportional counter units (PCUs)
0, 1, and 2 for the medium-term data (as PCUs 3 and 4 frequently exhibit breakdown during on-source time)
and PCU 2 only for the long-term data (as by 1998, PCU 1 also began to exhibit repeated breakdown during on-source time, and PCU 0 lost its propane veto layer in 2000 May). PCU 2 is thus the best-calibrated PCU.
We used standard screening criteria, including rejecting
data gathered within 20 minutes of satellite passage through the South Atlantic Anomaly (SAA).
The PCA background was estimated using the ``L7-240'' background models, appropriate for faint sources.
We extracted flux light curves;
spectral fitting for each observation was done using \textsc{xspec} version 12.5.1n,
assuming a Galactic column of $4.5 \times 10^{20}$ cm$^{-2}$ (Kalberla et al.\ 2005),
the abundances of Wilms et al.\ (2000), and the cross sections of Verner et al.\ (1996).
Response files were generated for each separate observation using \textsc{pcarsp} version 7.10 to
account for the gradual hardening of the PCA response due to the gradual leak of xenon gas into the propane layer
in each PCU.
NGC 7469 contains a complex warm absorber system lying along the line of sight
to the continuum source, but these absorbers do not significantly impact the spectrum above 2--3 keV
($\lesssim 1\%$ of continuum flux absorbed, e.g., Blustin et al.\ 2007) and are ignored in our modeling.
Errors on each flux point were derived from the standard error of
16 s count rate light curve bins within each observation.
For the long-term light curve, the total number of data points after screening was
538, with 85 (15.8$\%$) missing due to, e.g., sun-angle constraints
or screening. The medium-term light curve, binned to a time scale of 6 ks,
contained 479 pts, with 73 pts (15.2$\%$) missing due to screening or
time-critical observations of other sources; most of the missing points were in the second half of the
campaign as the sampling time was decreased (see NP01).
We extracted light curves in the 2--10, 2--5, 5--10, and 10--20 keV bands for both the
long- and medium-term time scales.
For the short-term sampling, we used public archive data
from two continuous \textit{XMM-Newton} observations
(hereafter referred to as ``short1'' and ``short2'').
\textit{XMM-Newton} observed NGC 7469 starting at
2004 November 30 at 21:12 UT (revolution 912) for a duration of 85 ks,
and again starting at 2004 December 3 at 01:28 (revolution 913) for a duration of 79 ks.
We downloaded the pipeline processed data (version 6.6.0) from the HEASARC archive, and used
data from the European Photon Imaging Camera (EPIC) pn, which observed in Small Window mode
with the medium filter. No MOS data were used as the MOS cameras were in Timing Uncompressed mode;
further details of the observation can be found in Blustin et al.\ (2007).
Using \textsc{xselect} version 2.4a, we extracted
source photons from a circular region of radius 40$\arcsec$ for each observation; backgrounds were
extracted from circular regions of identical size, centered $\sim$3$\arcmin$ away.
We searched for background flares by inspecting the 10--13 keV pn light curves, finding none.
We extracted light curves in the 2--10, 2--5, and 5--10 keV bands
binned to 2000 s; variability at shorter time scales was
dominated by Poisson noise (see below).
Light curve sampling parameters, including
mean net source and background count rates and average observed fluxes
for all light curves are listed in Table 1.
Also listed in Table 1 are fractional variability amplitudes $F_{\rm var}$ (see Vaughan et al.\ 2003 for
definition of $F_{\rm var}$ and its error) for each light curve.
\textit{XMM-Newton} does not have coverage $>$ 10 keV, so we do not have
short-term sampling in the 10--20 keV range. We investigated if the longest uninterrupted
10--20 keV PCA light curves during the longest duration observations from
the 1996 intensive monitoring could be used. However, we found the resulting
PSDs to be dominated by Poisson noise at temporal frequencies above $\sim10^{-3.4}$ Hz, typically,
and not highly suitable for PSD analysis.
We also examined 20--40 keV light curves from the High Energy X-ray Timing Experiment (HEXTE) detectors
aboard \textit{RXTE} on all three time scales, but found the resulting PSDs to
generally be dominated by Poisson noise over most temporal frequencies of interest,
so we do not investigate PSDs in that energy band.
2--10 keV light curves are displayed in Figure 1 for all time scales.
\section{PSD Measurement and Fit Results}
The PSD measurement procedure is summarized briefly here; the reader is referred to U02 or M03 for details.
Light curves were linearly interpolated across gaps.
Periodograms were constructed using a discrete Fourier transform (e.g., Oppenheim \& Shafer 1975)
and using the normalization of Miyamoto et al.\ (1991) and van der Klis (1997).
Following Papadakis \& Lawrence (1993) and Vaughan (2005),
the periodogram was logarithmically binned
every factor of 1.4 in $f$ (0.15 in the logarithm)
to produce the observed PSD $P(f)$; the two lowest
temporal frequency bins were widened to accommodate three periodogram points, yielding 13, 12, 6, and 6 binned PSD
points for the long, medium, short1, and short2 light curves, respectively.
The observed PSDs are plotted in Figure 2 for the 2--10 keV band and in Figure 3 for the
sub-bands.
The $\sim$5 lowest temporal frequency
bins in each individual PSD typically contained less than 15 periodogram points, precluding us from
assigning normal errors.
To estimate proper errors on each binned PSD point, to account for the PSD measurement distortion effects of
red noise leak and aliasing, and to account for the effect of Poisson noise, we use the
Monte Carlo procedure outlined by U02 (based on
Done et al.\ 1992). The vast majority of AGN X-ray PSD analyses conducted since 2002
have used this procedure. For each PSD model shape tested, an average model $\overline{P_{{\rm sim}}(f)}$ is calculated
based on simulated PSDs; $\overline{P_{{\rm sim}}(f)}$ accounts for the distortion effects and has errors assigned based on the
rms spread of the individual simulated PSDs within each temporal frequency bin.
For each model, the value of the test statistic
$\chi^2_\textrm{dist}$ between $P_{{\rm sim}}(f)$ and the observed PSD is
compared to an empirical distribution of simulated $\chi^2_\textrm{dist}$ values.
The ``rejection probability'' $R$ is a goodness of fit measure defined as
the percentile of simulated $\chi ^2 _{{\rm dist}}$ values
exceeded by the value of the observed $\chi ^2 _{{\rm dist}}$;
below we list the likelihood of acceptance $L \equiv 1 - R$.
The constant level of power due to Poisson noise is not subtracted from the data,
but instead is added to each model discussed below.
For the long- and medium-term PSDs, the power due to Poisson noise is estimated using $P_{\rm Psn} = 2(\mu+B)/\mu^2$,
where $\mu$ and $B$ are the total net and background count rates,
respectively; as these light curves are non-continuous light curves we multiply our estimate of
$P_{\rm Psn}$ by the average value of the ratio $\Delta$$T_{\rm samp}$, the average sampling time,
to the average exposure time per snapshot.
For each short-term PSD, we used light curves binned to 300 s to
measure the PSD out to $10^{-2.8}$ Hz, well into the temporal frequency range
dominated by Poisson noise (above 10$^{-3.6}$ Hz, typically).
A best fit to all binned PSD points in the range 10$^{-3.6}$ to $10^{-2.8}$ Hz yielded
the $P_{\rm Psn}$ values listed in Table 1.
Given that there is no overlap in time between the long-, medium-, and short-term
light curves, we implicitly assume that the intrinsic variability process
is only weakly non-stationary
over 13 years. That is, we assume that the intrinsic PSD has remained constant in both shape and normalization for a given energy band,
and the expectation value of $F_\textrm{var}$ for a given energy band also remains constant.
This is a reasonable assumption:
PSDs in BH XRBs tend to display significant changes in the shape or normalization of the
components comprising the PSD (e.g., Lorentzians and/or power-laws) on time scales of $\sim$ a day or longer (e.g., Pottschmidt et al.\ 2003; Remillard \& McClintock 2006);
scaling with black hole mass and luminosity, AGN light curves may be expected to display strong
non-stationarity, with significant changes in the observed
PSD, on time scales of at least centuries to millennia.
As the long-term monitoring spans a six-year duration, we can test the assumption of
weak non-stationarity over this duration by splitting the long-term light curve
in half (before and after MJD 53882), calculating the periodogram for each half, and
using the procedure outlined by Papadakis \& Lawrence (1995)
to determine if the periodograms are consistent.
In this method, a statistic $S$ is calculated
based on the sum of the differences in power at each temporal frequency
(see Appendix A of Papadakis \& Lawrence 1995 for the definition of $S$); for
consistent periodograms, $S$ has zero mean and a variance of 1.
For the periodograms derived from each half of the 2--10, 2--5, 5--10, and 10--20 keV light curves, we
calculate $S$ values of +0.08, --1.08, +0.20, and --1.10, respectively,
consistent with the notion that at each energy range studied, the PSD
has remained constant in shape and normalization between 2003 and 2009.
The procedure of Papadakis \& Lawrence (1995) cannot be used to directly compare
the long- and medium-term PSDs. However, in BH XRBs, a change in the PSD is commonly
accompanied by a significant change in the energy spectrum.
We thus performed fits to summed energy spectra
derived from the long- and medium-term \textit{RXTE} monitoring.
These results are presented in Appendix A and
demonstrate that the energy spectra are highly similar in form on all time scales
and are thus consistent with the notion of only weak non-stationarity
between 1996 and 2009.
\subsection{PSD Model Fits}
The PSDs of BH XRBs are usually of sufficient quality to model multiple components, including
Lorentzians and quasi-periodic oscillations (e.g., Remillard \& McClintock 2006).
However, the temporal frequency resolution of AGN PSDs usually means that simple
unbroken or singly-broken PSD model shapes provide an adequate fit (the double-peaked
profile of Ark 564 measured by M$^{\rm c}$Hardy et al.\ 2007 is a notable exception).
For NGC 7469, we tested three PSD model shapes: an unbroken power-law,
a broken power-law consisting of a sharp break, and a
broken power-law with a gradual bend connecting the high- and low-frequency portions.
The unbroken power-law model was of the form $P(f) = A_0 (f/f_0)^{-\alpha}$,
where $\alpha$ is the power-law slope
and the normalization $A_0$ is the PSD amplitude at $f_0$, arbitrary chosen to be $10^{-6}$ Hz.
We stepped through $\alpha$ in increments of 0.02, testing the
range of slopes 0.5 to 2.5, and using $N_{\rm trial}$ = 200 simulations each time
to calculate $\overline{P_{{\rm sim}}(f)}$.
Best-fit model parameters, likelihoods of acceptance $L_{\rm unbr}$, and values of observed
$\chi^2_{\rm dist}$/dof are listed in Table 2.
The data--model residuals are plotted in Figure 2(c) for the 2--10 keV band and Figure 3(c) for the
sub-bands.
In Appendix B, we discuss various methods for estimating the (one-dimensional)
confidence ranges for each fitted model parameter for this and all subsequent PSD models.
There, we discuss implementation of
the ``surrogate'' Monte Carlo method of Press et al.\ (1992);
the resulting confidence ranges are reported in parentheses in Table 2 for the unbroken
power-law model and in Table 3 for the broken-power models discussed below.
The values of $\chi^2_{\rm dist}$/dof are generally poor for the 2--10, 2--5, and 5--10 keV PSDs,
and the likelihoods of acceptance are low ($<$0.3$\%$).
The 10--20 keV PSD, lacking short-term sampling, covers less dynamic range
and the likelihood of acceptance is much higher.
For the lower-energy bands, the residuals plotted in Figures 2(c) and 3(c) suggest that a more
complex PSD model shape is appropriate.
To test for the presence of a PSD break, we employed a power-law model with a sharp break
of the form
\[P(f)= \left\{ \begin{array}{ll}
A_1(f/f_{\rm b})^{- \alpha_{\rm lo}}, & f \le f_{\rm b} \\
A_1(f/f_{\rm b})^{- \alpha_{\rm hi}}, & f > f_{\rm b} \end{array}
\right. \]
where the normalization $A_1$ is the PSD
amplitude at the break frequency $f_{\rm b}$, and
--$\alpha_{\rm lo}$ and --$\alpha_{\rm hi}$ are the low- and high-frequency power law slopes, respectively,
with the constraint $\alpha_{\rm lo} < \alpha_{\rm hi}$.
Break frequencies were tested in the log from --7.4 to --4.9
in increments of 0.1. $\alpha_{\rm hi}$ and $\alpha_{\rm lo}$
were both tested in increments of 0.1, over the ranges 1.0--3.2 and 0.0--2.0, respectively.
One hundred simulated PSDs were used to determine $\overline{P_{{\rm sim}}(f)}$.
The best-fit model parameters, along with likelihoods of acceptance $L_{\rm brkn}$, are listed in Table 3.
Errors listed are for one interesting parameter and
were determined assuming that other parameters (except for $A_1$) were fixed.
Data--model residuals are plotted in Figure 2(d) for the 2--10 keV PSD and
in Figure 3(d) for the sub-band PSDs.
Figure 4 shows contour plots of $\alpha_{\rm hi}$ versus $f_{\rm b}$ for these three PSDs
at the respective best-fit values of $\alpha_{\rm lo}$.
Best-fit values of $f_{\rm b}$ lie near $2 \times 10^{-6}$ Hz for all PSDs, with best-fit values of
$\alpha_{\rm lo}$ close to 1.0. Excluding the 10--20 keV PSD, best-fit values of
$\alpha_{\rm hi}$ are found to be 1.8--1.9.
We can use the ratio of the likelihoods of acceptance $L_{\rm brkn}/L_{\rm unbr}$ between the broken
and unbroken power law model fits to establish that incorporating a break into the model
yields a significant improvement. The likelihood ratio test
$D \equiv -2$ln($L_{\rm unbr}/L_{\rm brkn}$)
is distributed as a $\chi^2$ distribution with $n$ degrees of freedom, where $n$ is the difference in degrees of freedom between
the two models, here equal to 2.
For the 2--10, 2--5, and 5--10 keV PSDs,
values of $L_{\rm brkn}/L_{\rm unbr}$ are in the range 17--67 and values of $D$ span 5.7--8.4,
indicating an improvement in fit with respect to
the null hypothesis model (no break required) at confidence levels spanning 94.0--98.5$\%$.
However, for the 10--20 keV PSD, the improvement in fit
when adding a break to the model is not significant, with $L_{\rm brkn}/L_{\rm unbr} = 2.4$ and $D=1.72$,
an improvement in fit over the null hypothesis model at only 58$\%$ confidence.
Combined with the fact that the best-fit values of
$\alpha_{\rm lo}$ and $\alpha_{\rm hi}$ are consistent with other, this signifies that a break
has not been robustly detected in the 10--20 keV PSD.
We also tested a more slowly-bending PSD
model of the form $P(f) = (A_1 f^{-\alpha_{\rm lo}})/( ( 1 + f/f_{\rm b})^{(\alpha_{\rm hi} - \alpha_{\rm lo})}$,
testing the same range and increments of $f_{\rm b}$, $\alpha_{\rm hi}$ and $\alpha_{\rm lo}$
as for the sharply-broken power-law model.
The best-fit model parameters and likelihoods of acceptance $L_{\rm slow}$ are listed in Table 3,
with data--model residuals plotted in Figure 2(e) for the 2--10 keV PSD and Figure 3(e) for the sub-band PSDs.
Contour plots of $\alpha_{\rm hi}$ versus $f_{\rm b}$ at the respective best-fit values of $\alpha_{\rm lo}$
are shown in Figure 4.
As with the sharply-broken PSD model fits, the improvement in fit when adding a break
is significant: for the 2--10, 2--5 and 5--10 keV PSDs,
$L_{\rm brkn}/L_{\rm unbr}$ spans 29--75, $D$ spans 6.8--8.6, and the improvement with
respect to the unbroken power-law is at confidence levels spanning 96.5--98.6$\%$. The evidence for a break in the
10--20 keV PSD is again unconvincing, with consistent best-fit values of
$\alpha_{\rm lo}$ and $\alpha_{\rm hi}$ and with $L_{\rm brkn}/L_{\rm unbr} = 2.4$
and $D=1.77$, an improvement over the null hypothesis model at only 59$\%$ confidence.
As both classes of broken power-law models fit roughly similarly and yield significant evidence
for a PSD break in the 2--10, 2--5 and 5--10 keV bands, we henceforth treat both models equally in the paper.
\subsection{The PSD as a function of photon energy}
We now examine the behavior in PSD parameters with photon energy; specifically,
we focus on the apparent lack of a significant detection of a break in the 10--20 keV PSD.
To quantify differences between the observed
behavior of the 10--20 keV PSD and that of the PSDs at lower energies,
we define $\Delta\alpha$ as the difference between the power-law slopes below and above the best-fit PSD break,
using the parameter errors as listed in Table 3.
For the singly-broken power-law model,
the average of $\Delta\alpha$ for the 2--5 and 5--10 keV PSDs is $1.0 \pm 0.3$,
while for the 10--20 keV PSD, $\Delta\alpha$ is $0.1 \pm 0.2$.
The difference in $\Delta\alpha$ from below 10 keV to above 10 keV
suggests that the absence of a break in the 10--20 keV PSD
and the appearance of a break in the $<$ 10 keV PSDs
are significant at approximately the 4$\sigma$ confidence level.
Similarly, for the slowly-bending model,
the average of $\Delta\alpha$ for the 2--5 and 5--10 keV PSDs is $ 1.4 \pm 0.3$,
while for the 10--20 keV PSD, $\Delta\alpha$ is $0.2\pm0.3$;
the difference between these $\Delta\alpha$ values also suggests
that the observed difference in PSD behavior is significant at the $\sim$4$\sigma$ confidence level.
Figure 5 shows an overplot of the observed PSDs in each band in data-space
and in ``model-space'', along with the best-fitting sharply-broken and slowly-bending power-law models.
Above $\sim10^{-6}$ Hz, the 10--20 keV PSD is much flatter than the 2--10, 2--5, and 5--10 keV PSDs.
Similar behavior at high temporal frequencies was reported by NP01.
They constructed PSDs from the medium-term sampling, covering temporal
frequencies from $10^{-5.8}$ to $10^{-4.1}$ Hz
and used uninterrupted PCA light curves with a time resolution of 16 s
to probe $10^{-3.5}$ Hz to $10^{-1.5}$ Hz in the 2--4, 4--10, and 10--15 keV PSDs.
NP01 fixed $P_{\rm Psn}$ to the expected noise power level instead of leaving it as a
free parameter, but this yielded a good fit at high temporal frequencies
(as stated earlier, we do not use PCA to measure the 10--20 keV PSD because $P_{\rm Psn}$
dominates). However, NP01 did not take into account PSD distortion measurement effects
(the effects of which will have an energy dependence if the
PSD shape itself is energy-dependent). Nonetheless, the current work and NP01 both
find a similar result: the 10--20 keV PSD is much flatter at higher temporal frequencies.
While the 10--20 keV band lacks short-term sampling,
this behavior cannot be due to an effect of sampling on the medium or long time scales, as \textit{RXTE}
sampled all four bands equally.
We also compared the 2--5 and 5--10 keV bands to search for any PSD evolution in
energy between those two bands. We first tested for any dependence of $f_{\rm b}$ with energy
assuming a universal PSD shape whose power-law slopes
are independent of energy.
Assuming the best-fit sharply-broken model to the 2--10 keV PSD,
we find consistent values for $f_{\rm b}$ for the 2--5 and 5--10 keV bands, with log($f_{\rm b}$,Hz) = $-5.6 \pm 0.3$
for each band. We find log($f_{\rm b}$,Hz) = --($6.0 \pm 0.4$) and --($6.0\pm0.3$) for the 2--5 and 5--10 keV bands, respectively,
using the slowly-bending model.
We then tested for an energy dependence of $\alpha_{\rm hi}$ assuming that $f_{\rm b}$ is energy-independent.
For a sharply-broken model with log($f_{\rm b}$,Hz) fixed at --5.7, the values of $\alpha_{\rm hi}$ are consistent for
the 2--5 and 5--10 keV bands: $1.8 \pm 0.2$ and $1.8^{+0.1}_{-0.2}$, respectively
($1.8 \pm 0.4$ for each band for the slowly-bending model assuming log($f_{\rm b}$,Hz) is fixed at --6.0).
In conclusion, there is no significant evidence for PSD evolution in energy between the 2--5 and 5--10 keV bands.
\section{Discussion}
\subsection{NGC 7469's Place in the $T_{\rm b}$--$M_{\rm BH}$--$\dot{m}$ Plane}
The primary result of this paper is that we detect a turnover in the 2--10 keV PSD of NGC 7469 at a
temporal frequency of $f_{\rm b}= 2.0^{+3.0}_{-0.8} \times 10^{-6}$ Hz or $1.0^{+3.0}_{-0.6} \times 10^{-6}$ Hz
for the sharply- or slowly-bending power-law model, respectively (using the 68$\%$ confidence limits from the P92 method);
these frequencies correspond to turnover time scales $5.8 \pm 3.5$ days or $11.6^{+17.5}_{-8.7}$ days, respectively.
We first determine if the measured PSD turnovers are consistent with the empirical relation
between $T_{\rm b}$, $M_{\rm BH}$, and $\dot{m}$ quantified by M$^{\rm c}$Hardy et al.\ (2006).
As noted above, NGC 7469's black hole mass is well studied.
We use the reverberation-mapped mass from Vestergaard \& Peterson (2006), who provide
a "calibrated" mass estimate based on H$\beta$ width and optical luminosity measurements of
$M_{\rm BH} = 3.34^{+0.69}_{-0.68} \times 10^7 \hbox{$\rm\thinspace M_{\odot}$}$.
Our conclusions below do not change if we use the mass estimate from
the $M_{\rm BH}$--$\sigma_*$ relation: $\sigma_* = 131 \pm 5$ km s$^{-1}$ (Nelson et al.\ 2004), and
using the $M_{\rm BH}$--$\sigma_*$ relation of Tremaine et al.\ (2002)
yields $M_{\rm BH} = 2.46^{+0.44}_{-0.34} \times 10^7 \hbox{$\rm\thinspace M_{\odot}$}$.
For an estimate of the bolometric luminosity, we use the value from
Vasudevan \& Fabian (2009) based on \textit{XMM-Newton} EPIC/Optical Monitor spectral energy distribution
fitting, $L_{\rm bol} = 6 \times 10^{44}$ erg s$^{-1}$.
An independent estimate of $L_{\rm Bol}$ can be based on the
\textit{RXTE} monitoring data: a model fit jointly to the summed spectrum
\textit{RXTE} PCA and HEXTE spectra constructed from all available \textit{RXTE} observations of NGC 7469
(Rivers et al., ApJS, submitted) indicates that
the modeled average unabsorbed 2--10 keV flux is $3.1 \times 10^{-11}$ erg cm$^{-2}$ s$^{-1}$.
Using a luminosity distance of 62.7 Mpc (from the NED database, using the reference frame defined by the 3K cosmic microwave
background radiation),
the 2--10 keV luminosity $L_{2-10}$ is $1.7 \times 10^{43}$ erg s$^{-1}$.
Using Marconi et al.\ (2004), $L_{\rm bol} \sim 22 L_{2-10} = 3.7 \times 10^{44}$ erg s$^{-1}$,
very close to the value from Vasudevan \& Fabian (2009), which we use below.
Using the best-fit values of the coefficients in the
M$^{\rm c}$Hardy et al.\ (2006) relation,
log($T_{\rm b,pred}$) = 2.10log($M_{\rm BH}$/($10^6 \hbox{$\rm\thinspace M_{\odot}$}$)) -- 0.98log($L_{\rm bol}/(10^{44} {\rm erg~s^{-1}}$) -- 2.32,
with $T_{\rm b,pred}$ in units of days, and taking into the account the uncertainty in $M_{\rm BH}$ listed above,
we obtain $T_{\rm b,pred} = 1.25^{+0.60}_{-0.48}$ days.
Taking into account the uncertainties on
the coefficients in the M$^{\rm c}$Hardy et al.\ (2006) relation,
$2.10\pm0.15$, $0.98\pm0.15$, and $2.32\pm0.2$, respectively, we can obtain
$T_{\rm b,pred}$ values from 0.2 to 9.3 days, meaning the predicted
value is consistent with both the observed sharply-broken and slowly-bending PSD break time scales.
\subsection{Corresponding Physical Time Scales}
Detailed discussions on the likely physical mechanisms responsible for the
PSD turnovers have appeared in many of the PSD papers cited above; here,
we provide only a brief review.
We assume that the bulk of the X-ray emission originates 10 $R_{\rm Sch}$ from the black hole.
The Keplerian orbital time scale $t_{\rm orb}$ at this radius for a $3 \times 10^7 \hbox{$\rm\thinspace M_{\odot}$}$ black hole is
1 day. Following, e.g., Treves et al.\ (1988), the thermal time scale $t_{\rm th}$ is roughly $t_{\rm orb} / \alpha$, where
$\alpha$ is the accretion disk viscosity parameter. For values of $\alpha$ of $\sim 0.1$--$0.2$,
$t_{\rm th}$ will be $\sim 5-10$ days, consistent with the observed PSD break time scales in NGC 7469.
One variability model that has had a measure of success in explaining
the observed variability properties of Seyferts and BH XRBs,
including PSD shapes, temporal frequency-dependent lags, and the rms--flux relation, involves
inwardly-propagating fluctuations in the local mass accretion rate (Lyubarskii 1997;
Kotov, Churazov \& Gilfanov 2001; Ar\'{e}valo \& Uttley 2006).
In this model, the fluctuations originate across a range of annuli in the disk and travel
inward toward the central X-ray source, eventually modifying the X-ray continuum emission.
The PSD breaks could correspond to the local viscosity time scale $t_{\rm visc}$ at the outer radius of X-ray emission.
$t_{\rm visc} = t_{\rm th} / (H/R)^2$, where $H/R$ is the ratio of the disk scale height
to the radius (e.g, Treves et al.\ 1988).
For a geometrically thin disk surrounding
a $3 \times 10^7 \hbox{$\rm\thinspace M_{\odot}$}$ black hole, with $H/R = 1/100$, and
with $\alpha \sim 0.1-0.2$, $t_{\rm visc} \sim 5-10 \times 10^4$ days,
far too long to be associated with the observed PSD breaks. However, for a geometrically thick disk where
$H/R$ approaches 1, $t_{\rm visc}$ approaches the thermal time scale and can match the
observed PSD time scale for NGC 7469.
However, Ar\'{e}valo \& Uttley (2006) caution that if the X-ray emission region is radially extended,
one can still get a bend in the PSD due to the radial distribution in variability fluctuations,
and the PSD bend does not have to correspond to a singular characteristic time scale.
\subsection{Energy Dependence of the PSD}
Rapid variability in the Compton hump is not a likely cause for the observed energy-dependence of the PSD.
The Compton hump contributes only $\sim$20--30$\%$ of the total 10--20 keV emission (see Table 4 in Appendix A), and so
such extreme variability in the total 10--20 keV emission
would require the bulk of the Compton-thick reflecting material to originate within
light-hours of the X-ray continuum source and to be responding to continuum variations much larger than those we observe.
Nandra et al.\ (2000) explored variability in the absolute normalization $A_{\rm ref}$ of the Compton hump down to time scales of 1 day,
but found the observed variability in $A_{\rm ref}$ consistent with being due to model degeneracy between $\Gamma$ and $A_{\rm ref}$.
Finally, Papadakis, Nandra \& Kazanas (2001) demonstrated that the emission in 2--10 and 10--15 keV bands show high coherence over $10^{-5.5}$ to $10^{-4}$ Hz, suggesting a common variability mechanism for both bands.
Flattening of the power-law slope at temporal frequencies above the break
with increasing energy has been claimed for some PSDs
published previously, such as Mkn 766 (Vaughan \& Fabian 2003, Markowitz et al.\ 2007)
and MCG--6-30-15 (Vaughan, Fabian \& Nandra 2003); $\alpha_{\rm hi}$ was
typically 2.5 for bandpasses centered near 0.5 keV and 2.1 near 5 keV.
Moreover, modeling the broadband PSD of Ark 564 using a double-Lorentzian profile,
M$^{\rm c}$Hardy et al.\ (2007) found the normalization of the
higher-temporal frequency Lorentzian to increase by a factor of 1.5
from the 0.6--2.0 to the 2--10 keV bands.
Interestingly, evolution in PSD shape as a function of photon energy is not
uncommonly observed in BH XRBs, and is
commonly attributed to changes in the normalizations and/or peak temporal frequencies
of the Lorentzians components used to fit the PSD. Energy-dependent
PSD changes do not tend to be confined to any particular energy spectral state
(Done \& Gierli\'{n}ski 2005). For example, Kalemci et al.\ (2003, their Figure 5)
studied the PSD of XTE J1650--500 during an outburst decay, modeling the PSD
using a single Lorentzian profile. They noted that the
6--15 keV Lorentzian peaks at a temporal frequency
a factor of 3 higher than that in the 2--6 keV band,
and the normalization also increased by a factor of 3, yielding a much flatter
PSD above a temporal frequency of $\sim$3 Hz.
B\"{o}ck et al.\ (2009), modeling the 4.5--5.8 keV and 9.5--15 keV PSDs
of an intermediate state of Cyg X-1 using a double-Lorentzian profile,
noted that the lower-temporal frequency Lorentzian decreased in normalization
toward the higher-energy band, while the normalization of the other Lorentzian remained
the same, yielding a flatter overall PSD above $\sim$3 Hz in the higher-energy band.
Similar changes between PSDs measured over the 2--4 and 15--71 keV bands were seen
across several spectral states of Cyg X-1 by Pottschmidt et al.\ (2006).
In addition, many quasi-periodic oscillations are
detected only above a threshold photon energy (e.g.,
Strohmayer 2001a, 2001b; Montanari et al.\ 2009).
Though we have modeled the PSD of NGC 7469 using a simple singly-broken power-law,
we cannot rule out the possibility that multiple broadband components may exist in this PSD, and, speculatively,
the observed PSD flattening with energy in NGC 7469 and other Seyferts
could be due to changes in the relative normalizations of these components, as in Ark 564 (M$^{\rm c}$Hardy et al.\ 2007).
Another possibility is that the break time scale could increase dramatically from
approximately $10^{-6}$ Hz in the 2--10 keV band to $10^{-4}$ Hz or greater in the 10--20 keV band.
In the context of a model incorporating inwardly-propagating disk fluctuations,
if the bulk of the 10--20 keV emission originates at a smaller radius compared to the 2--10 keV emission
(e.g., Kotov, Churazov \& Gilfanov 2001), then
a jump in the physical parameters of the disk with radius (e.g., if $H/R$ increases by 10)
could cause vastly different values for $t_{\rm visc}$ at different radii of the disk.
A final possibility is that the break frequency in the PSD of NGC 7469 occurs at the same temporal frequency
in all bands, with the power-law slope flattening with photon energy.
This behavior is (at least qualitatively) consistent with
the jet model of Giannios et al.\ (2004; see also Kylafis et al.\ 2008), wherein
the Comptonizing corona is identified with the base of a jet,
and a PSD break corresponds to the Keplerian frequency at the radius of the jet base.
PSD flattening above the break is attributed to having the temperature of jet decrease with increasing radius
and having the variability of the soft input photons governed by the radius at which they are emitted from the disk.
\section{Conclusions}
We have presented the broadband X-ray PSD of the X-ray-typical Seyfert 1.2 NGC 7469.
Preliminary PSDs for this object were published by NP01 and based on
a 1996 month-long \textit{RXTE} campaign which quantified variability over temporal frequencies greater
than approximately 10$^{-6}$ Hz. We combined these data with
sampling obtained from long-term monitoring with \textit{RXTE} spanning 2003--2009
and two \textit{XMM-Newton} long-looks obtained in 2004.
The resulting high dynamic range of the PSD, $9 \times 10^{-9}$ to $2 \times 10^{-4}$ Hz,
allowed us to test simple unbroken and singly-broken power-law model shapes.
In the 2--10 keV PSD, we find significant evidence for a break at a best-fit temporal frequency
$ f_{\rm b} = 1-2 \times 10^{-6}$ Hz, depending on the form of the power-law break.
This corresponds to a time scale of $T_{\rm b} = 6-12$ days.
Given NGC 7469's well-constrained black hole mass of $3 \times 10^{7} \hbox{$\rm\thinspace M_{\odot}$}$, the PSD break is consistent with the
empirical relation between $M_{\rm BH}$, $L_{\rm bol}$ and $T_{\rm b}$ of M$^{\rm c}$Hardy et al.\ (2006) for Seyfert PSDs.
Our results include reliably derived confidence regions on the best-fit parameters
on models fit to the 2--10 keV PSD. We applied the ``surrogate'' method of Press et al.\ (1992), wherein
one uses Monte Carlo simulations to create a large number of synthetic data sets based on the best-fit model,
and fits each data set to map out the probability distributions of model parameters.
The parameter confidence ranges listed in Tables 2 and 3 thus lack the
ambiguity inherent in parameter confidence regions derived from previously derived methods
(Markowitz et al.\ 2003; Uttley et al.\ 2002),
though we emphasize that work is still ongoing in this area (e.g., Mueller \& Madejski 2009).
A break is not confirmed in the 10--20 keV band PSD, which is significantly flatter than the
2--10 keV PSD (and the 2--5 and 5--10 keV sub-bands), consistent with claims by NP01.
Including the current result for NGC 7469, evidence for energy-dependent PSD evolution in Seyferts has been accumulating,
and includes an energy-dependent change in the normalization of a Lorentzian component
(as in Ark 564; M$^{\rm c}$Hardy et al.\ 2007),
and detections of evolution in power-law slope at temporal frequencies above the break (e.g., Mkn 766, Vaughan \& Fabian 2003, Markowitz et al.\ 2007;
MCG--6-30-15, Vaughan, Fabian \& Nandra 2003).
The accumulated evidence suggests that energy-dependent evolution may not be uncommon in Seyfert PSDs and
may be similar to energy-dependent changes observed in the (higher-quality) PSDs of several BH XRBs, thereby
corroborating other timing-based observational links between Seyferts and BH XRBs
(broadband PSD shapes and the scaling of PSD break frequencies with black hole mass and accretion rate
relative to Eddington; coherence; phase lags)
which support the notion of similar variability mechanisms at work
in both classes of objects.
For further progress, the community needs a large sample of AGN PSDs observed with the same quality as XRB PSDs in order to perform
the same level of detailed model fitting of Lorentzian components, power-law components, etc.,
and determine if and how those components vary in energy.
The required PSDs would need adequately high temporal frequency resolution covering both a broad temporal frequency range
and a broad photon energy range, including obtaining low Poisson noise measurements above at least 10 keV
and covering temporal frequencies up to at least $\sim$10$^{-3}$ Hz.
\acknowledgements
A.M.\ thanks the {\it RXTE} Science Operations staff, particularly the {\it RXTE}
schedulers for ensuring that the long-term monitoring observations were scheduled
so evenly all these years, thereby ensuring straightforward PSD measurement.
A.M.\ also thanks Martin Mueller, Katja Pottschmidt, and Moritz B\"{o}ck for useful discussions
which helped guide the manuscript. This work has made use of HEASARC online services, supported by
NASA/GSFC, and the NASA/IPAC Extragalactic Database,
operated by JPL/California Institute of Technology under
contract with NASA.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,840 |
Q: Extrapolate "middle" file contents using Python So, here's the facts:
I'm iterating over a bunch of files inside a directory.
The first thing I need doing on these files is extracting the first block of N bytes (let's call this "heading" block and let's say its 16 bytes long, just to say a number, that's not the point here) and compare it with a "reference" block of bytes of the same size to see if they're equal. If that's not the case ('cause the blocks contents differ or the file is even smaller than the reference-block size), I'll skip the file and go on with next one; in case the blocks equals then I found a "candidate" file.
In my case, a "good candidate" file is one that also has a "tail" block, let's have it the same size as the heading one to simplify things, and also let's say I can be sure that if a "candidate" file in that directory is "big enough" (= bigger than head+tail blocks size) the file is "good" to be processed (= no need to compare its "tail" block) so now I need to extrapolate its "middle" content (= the data found after the heading block and before the tail block) to do some further stuffs with it ... and here comes the question: suppose I already did the head/reference block comparison and I just found a "candidate", how can I read/get the contents in the "middle" of the file now? (when the file is "big enough", of course)
(Real things are a little more complicated than that, in the way that the "bad candidates" in the first directory loop should be moved to a different directory to be iterated over again with a smaller reference block... but that's definitely not the point here.)
I need using only python (>= 3) to do all this.
I searched here on StackOverflow but seems something like this hasn't been asked/replied already before.
P.S.
I'm posting an answer to this myself as I found a way to do it and hope others might find it useful as a starting point idea maybe. Yet I'd really appreciate to see more if possible, I'm always happy to see different/better approach to achieve the needs or any comments you'd like to share of course, thanks
A: So far this is a very simple python3 script you could use to do that:
block = 16 # this is the block size (using same size for "head" and "tail" in this example)
f = open("path/to/file", "rb")
# ...
# ... (code for head/reference block comparison will be here)
# ...
try:
# if I get this far I found a "candidate" file
f.seek(-block,2) # put offset right before "tail" block
p = f.tell() - block # p is storing "middle" contents size
except OSError:
p = -1 # something went wrong while calculating "middle" contents size
if p <= 0: # (= the candidate file was a "bad" one)
print("either file size is too small or block size is too big")
else: # (= the candidate file is a "good" one, let's process it now)
f.seek(block) # put offset right after "head" block
middle = f.read(p) # finally retrieved "middle" file contents
print(middle) # (process data as needed instead here)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,845 |
The other three involved in the crash were a 24-year-old man from Worden, a 26-year-old man from Edwardsville and a 29-year-old woman from Maryville, according to Dye.
Illinois State Police continue an investigation into the crash. The Madison County Coroner's Office will conduct a toxicology test of Wieda, according to Chief Deputy Coroner Roger D. Smith.
Irwin Chapel of Glen Carbon is handling funeral arrangements for Wieda. | {
"redpajama_set_name": "RedPajamaC4"
} | 872 |
Q: Is there a Python function to calculate the diff based on the first element? I have a pandas dataframe, like this example:
df = pd.DataFrame({
'steps': ['step1','step2', 'step3', 'step4','step5'],
'qty': [100, 95, 92, 87, 78]},
index=[0,1,2,3,4])
I would like to calculate the percentage of abandonments in each step, based on the first value.
Output:
Steps qty Tx %
Step1 100 0,00%
Step2 95 5,00%
Step3 92 3,00%
Step4 87 5,00%
Step5 78 9,00%
I thought about using pd.pct_change(), but it doesn't work as expected.
The manual calculation would be something like:
values = [(1 - df['qty'][0]/df['qty'][0]) - (1-df['qty'][0]/df['qty'][0]),
(1 - df['qty'][1]/df['qty'][0]) - (1-df['qty'][0]/df['qty'][0]),
(1 - df['qty'][2]/df['qty'][0]) - (1-df['qty'][1]/df['qty'][0]),
(1 - df['qty'][3]/df['qty'][0]) - (1-df['qty'][2]/df['qty'][0]),
(1 - df['qty'][4]/df['qty'][0]) - (1-df['qty'][3]/df['qty'][0])]
However, I believe that this is not scalable, especially considering increasing the number of steps or time periods.
Could someone help me think of some function or show a rationale that can make this calculation simpler?
A: You can use pd.Series.shift to shift the qty column one element down. Then simply calculate the difference between the shifted column and itself:
import pandas as pd
df = pd.DataFrame({
'steps': ['step1','step2', 'step3', 'step4','step5'],
'qty': [100, 95, 92, 87, 78],
})
df['Tx %'] = df.qty.shift() - df.qty
print(df)
# output:
# steps qty Tx %
# 0 step1 100 NaN
# 1 step2 95 5.0
# 2 step3 92 3.0
# 3 step4 87 5.0
# 4 step5 78 9.0
A: This works:
# you have
values = [(1 - df['qty'][0]/df['qty'][0]) - (1-df['qty'][0]/df['qty'][0]),
(1 - df['qty'][1]/df['qty'][0]) - (1-df['qty'][0]/df['qty'][0]),
(1 - df['qty'][2]/df['qty'][0]) - (1-df['qty'][1]/df['qty'][0]),
(1 - df['qty'][3]/df['qty'][0]) - (1-df['qty'][2]/df['qty'][0]),
(1 - df['qty'][4]/df['qty'][0]) - (1-df['qty'][3]/df['qty'][0])]
# 1s cancel out, so the above is equivalent to
values = [( - df['qty'][0]/df['qty'][0]) + (df['qty'][0]/df['qty'][0]),
( - df['qty'][1]/df['qty'][0]) + (df['qty'][0]/df['qty'][0]),
( - df['qty'][2]/df['qty'][0]) + (df['qty'][1]/df['qty'][0]),
( - df['qty'][3]/df['qty'][0]) + (df['qty'][2]/df['qty'][0]),
( - df['qty'][4]/df['qty'][0]) + (df['qty'][3]/df['qty'][0])]
# since every element is divided by df['qty'][0], you can take it out, so the above is equivalent to
values = [( - df['qty'][0] + df['qty'][0]) / df['qty'][0],
( - df['qty'][1] + df['qty'][0]) / df['qty'][0],
( - df['qty'][2] + df['qty'][1]) / df['qty'][0],
( - df['qty'][3] + df['qty'][2]) / df['qty'][0],
( - df['qty'][4] + df['qty'][3]) / df['qty'][0]]
#the RHS is the LHS shifted one level down, so it's equivalent to:
values = (- df['qty'] + df['qty'].shift()).fillna(0) / df['qty'][0]
# in summary
df['Tx %'] = (df['qty'].shift() - df['qty']).fillna(0) / df['qty'][0] * 100
steps qty Tx %
0 step1 100 0.0
1 step2 95 5.0
2 step3 92 3.0
3 step4 87 5.0
4 step5 78 9.0
A: This response my doubt:
df['Tx %'] = round(((1 - df.qty / df.qty[0]) - (1- df.qty.shift() / df.qty[0]))*100, 2)
Thanks to @jfaccioni
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,529 |
Q: Any way to get unigrams AND bigrams in my TDM with RTextTools? RTextTools is very handy if you want to compare algorithms for classifying text documents. there is an option: ngramLength with which one can specify, if one want to use 1-grams, 2-grams, 3-grams and so on. Now i'd like to have 1-grams AND 2-grams in my Term Document Matrix! is there a way to do that?
myTDM <- create_matrix(labeled$CLEAN, language="english",
removeNumbers=FALSE, stemWords=FALSE, ngramLength=1,
removePunctuation=FALSE, removeSparseTerms=0.98,
removeStopwords=FALSE, stripWhitespace=FALSE, toLower=FALSE)
in another forum, i've seen "Tim J" (probably Tim Jurka, one of the creators of RTextTools) write that this functionality will be in a new version of RTextTools. He wrote it will probably be released in mid-January 2013. Now more than a year later, i wonder if there really is this new functionality and how it's used!
Thank You!!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,689 |
{"url":"https:\/\/physics.stackexchange.com\/questions\/229720\/how-do-we-get-a-2deg-in-a-remote-doped-heterostructure?noredirect=1","text":"How do we get a 2DEG in a remote doped heterostructure?\n\nI have a question regarding the way in which one often constructs a two-dimensional electron gas in heterostructures. I have a specific example in mind, although I believe this is quite a common way to build the system if I can trust the literature. As shown in the diagram below (borrowed from Semiconductor Nanostructures by Thomas Ihn) we have a type 1 heterostructure with GaAs and AlGaAs. On top of that there is a $\\delta$ doped donor layer (so there is one layer with heavy doping), above which there is another GaAs layer, an AlAs layer and another GaAs layer.\n\nNow, I have a few questions about this. First of all I'm not entirely sure why we don't just make the 2DEG by sandwiching the GaAs layer between two AlGaAs layers (ABA) so that you have a quantum well in your conduction band, which then gives you your 2DEG, like in the diagram below. In my mind this is the most simple way of going about such a problem, but perhaps it is just too simplistic. You're not taking the surface into account and such.\n\nBut okay, let us continue. On top of the AlGaAs layer we have the $\\delta$ donor layer. Why do we use this? I know that the idea of sheet doping is that you create an electrostatic potential and thus that (in the absence of other factors) you have donor electrons that are bound to the plane. But now in this system, because the layer is on top of the AlGaAs which has a larger band gap than GaAs, it is energetically favorable for the donor electrons to move towards the GaAs. On the other hand the positively charged donors also pull on these electrons. Is this then how we get our 2DEG? The donor electrons somehow get into a bound state at the interface?\n\nI am not sure if this is the case, but perhaps it is. My final question would then be, if this is indeed so, why do we have all these additional layers on top? The GaAs, the AlAs, more AlGaAs, and more GaAs. I don't get the purpose of this. Perhaps it is related to surface states and Fermi level pinning or something?\n\nRegarding your question on sandwiching GaAs between two AlGaAs barriers:\n\nIf you do this for a narrow quantum well (like you sketched above), the electron wavefunction protrudes into the barrier quite a bit. As the barrier material is a ternary alloy, the electrons are exposed to alloy scattering. This is simply due to the fact that Ga and Al atoms are assumed to be randomly distributed. A second important contribution is interface roughness scattering, caused by an imperfect interface. Even in highest quality structures, you will still have some roughness at the order of one monolayer (RMS ~ 0.3 nm). So these are the arguments against a more complex heterostructure.\n\nNow, in fact state-of-the-art high-mobility 2DEGs are realized with quantum wells, thus barriers on both sides. This is due to the necessity of getting as many electrons as possible into the channel. It sounds contradictory, but within certain limits, more electrons in the channel lead to higher mobility. Due to technical limits, you can only put a certain amount of doping in one sheet. Therefore, people used two so called delta-dopings on either side of the GaAs channel. Why does this work then? They simply use a rather wide quantum well (at the order of 50 nm), which only yields small electronic confinement. The electronic wavefunction is then mostly localized in the GaAs, which is why this still can be a high mobility structure.\n\nIn any case, you need to take the surface into account. GaAs doped ~ 1e16\/cm\u00b3 leads to a depleted region, which is roughly 1 \u00b5m thick from the surface. Now if you talk about high mobility structures, your background doping is far below 1e16, therefore your depleted layer grows in thickness. If you want realistic structures, where you can fabricate contacts and gate electrodes, you can not bury the 2DEG infinitely deep. Therefore, you can't neglect surface effects. Usually, you compensate surface states by an additional doping sheet closer to the surface.\n\nThe band bending issue was discussed already in the answer by @ignacio. Basically, you have to solve the Schr\u00f6dinger-Poisson equation in a self-consistent way to obtain the correct band profile. Basically, you solve Schr\u00f6dinger's equation, which gives you bound states. Based on these, you distribute your electrons, which then allows you to solve Poissons equation, which gives you band bending and therefore a new conduction band profile on which you can solve Schr\u00f6dinger's equation again. You do this in an iterative way, until your solution converges to the hopefully correct one. This is somewhat tedious on paper, but there are free programs available. E.g. the famous solver by Greg Snider. But it's also not too difficult to write such a simulation by yourself. Actually its a good occasion to practice and see, if you understood the problem.\n\nThe structure, you sketched here, is a bit more complex. Obviously, they use an AlAs layer to prevent electrons from leaking to the surface and to push them towards the 2D channel.\n\nThe first case you mention where you just sandwich GaAs inside AlGaAs is worked out in page 66 of Semiconductor Nanostructures. By adding a doping layer you control the carrier density and with that the resistivity. Like you say, it's necessary to draw the electrons away from the doping layer using attraction to the smaller-gapped GaAs. This creates bound states in the spacer layer. For more detail check out page 75:\n\n\u2022 I indeed had a look at that page, but I don't completely follow the diagram. From left to right, we start with the conduction band of AlGaAs, that I get. How is it that by going further right, we first increase in energy? Is this due to the electrostatic attraction? The energy the drops at the interface, because GaAs has a smaller band gap, that I get. But why does it then bend up again as we go further to the right, to the original level of AlGaAs? \u2013\u00a0user129412 Jan 15 '16 at 19:31\n\u2022 Yes, the attraction to the donors gives you the positive slope next to the doping layer. Electrons flow into the 2DEG until their charge bends the bands in the GaAs enough to match Fermi levels. \u2013\u00a0ignacio Jan 15 '16 at 20:28\n\u2022 Hmm, that is a concept that I most definitly still struggle with, the band bending that is seen here. It really isn't explained at that point in the book, is it? I really don't seem to understand why it bends upwards here. I do understand it in the case of Schottkey contacts, because of the donor ionization in the viscinity of the metal\/semiconductor surface, but I don't really see how it works here.. I understand that this wasn't my original question, but do you perhaps have a tip on how to understand that further? \u2013\u00a0user129412 Jan 15 '16 at 20:57\n\u2022 As for the main question, your answer does solve my first two issues, but I'm still wondering why we have all these additional layers on top. The GaAs, the AlAs, more AlGaAs, and more GaAs. \u2013\u00a0user129412 Jan 15 '16 at 20:58\n\u2022 The charge of the electrons in the 2DEG is bending the bands. No idea about the AlAs though. \u2013\u00a0ignacio Jan 15 '16 at 21:05","date":"2020-01-26 17:04:04","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\": 0.784692108631134, \"perplexity\": 588.9239367850591}, \"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-2020-05\/segments\/1579251690095.81\/warc\/CC-MAIN-20200126165718-20200126195718-00119.warc.gz\"}"} | null | null |
Q: Make sense of numbers in perf stat output I've been trying to use perf to profile my running process, but I cannot make sense of some numbers output by perf, here is the command I used and output I got:
$ sudo perf stat -x, -v -e branch-misses,cpu-cycles,cache-misses sleep 1
Using CPUID GenuineIntel-6-55-4
branch-misses: 7751 444665 444665
cpu-cycles: 1212296 444665 444665
cache-misses: 4902 444665 444665
7751,,branch-misses,444665,100.00,,
1212296,,cpu-cycles,444665,100.00,,
4902,,cache-misses,444665,100.00,,
May I know what event does the number "444665" represent?
A: -x format of perf stat is described in man page of perf-stat, section CSV FORMAT. There is fragment of this man page without optional columns:
CSV FORMAT top
With -x, perf stat is able to output a not-quite-CSV format output
Commas in the output are not put into "". To make it easy to parse it
is recommended to use a different character like -x \;
The fields are in this order:
· counter value
· unit of the counter value or empty
· event name
· run time of counter
· percentage of measurement time the counter was running
Additional metrics may be printed with all earlier fields being
empty.
So, you have value of counter, empty unit of counter, event name, run time, percentage of counter being active (compared to program running time).
By comparing output of these two commands (recommended by Peter Cordes in comment)
perf stat awk 'BEGIN{for(i=0;i<10000000;i++){}}'
perf stat -x \; awk 'BEGIN{for(i=0;i<10000000;i++){}}'
I think than run time is nanoseconds for all time this counter was active. When you run perf stat with non-conflicting set of events, and there are enough hardware counters to count all required events, run time will be almost total time of profiled program being run on CPU. (Example of too large event set: perf stat -x , -e cycles,instructions,branches,branch-misses,cache-misses,cache-references,mem-loads,mem-stores awk 'BEGIN{for(i=0;i<10000000;i++){}}' - run time will be different for these events, because they were dynamically multiplexed during program execution; and sleep 1 will be too short to have multiplexing to activate.)
For sleep 1 there is very small amount of code to be active on CPU, it is just libc startup code and calling syscall nanosleep for 1 second (check strace sleep 1). So in your output 444665 is in ns or is just 444 microseconds or 0.444 milliseconds or 0.000444 seconds of libc startup for sleep 1 process.
If you want to measure whole system activity for one second, try adding -a option of perf stat (profile all processes), optionally with -A to separate events for cpu cores (or with -I 100 to have periodic printing):
perf stat -a sleep 1
perf stat -Aa sleep 1
perf stat -a -x , sleep 1
perf stat -Aa -x , sleep 1
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,640 |
With the construction of the H-5 Wet Weather Pump Station nearing completion, this newsletter will be the final volume that NHSA will publish. Since the beginning of the project in June 2015, thirty-two (32) newsletters have been published in a concerted effort to keep the community informed about the construction process. The final construction activities, which will occur over the next two weeks, are detailed below. The project will be completed on time, in conformity with the original construction schedule.
North Hudson would like to thank the Maxell HOA and residents for their patience and cooperation over the last 17 months.
All construction activities along 11th Street between Sinatra Drive North and Hudson Street will be finished by the end of October. The contractor is currently completing the surface restoration and paving. This will be followed by the striping of the repaved roadway and landscaping next week, thus fully restoring the Maxwell islands. The fencing and fencing cover for the NHSA island between Hudson and Washington Streets, which houses the emergency generator for the H-5 WWPS, will not be installed until the second week of November.
The HOA has approved work on Saturday, October 22nd, weather permitting, to install the roadway striping. The work will begin at 9:00 a.m. and end at 3:30 p.m.. The site will be locked up by 4:00 p.m.
Pump station—The contractor partially completed the wet tests of the new pumps. We are awaiting more wet weather to complete the final performance testing protocol.
Transition vault—The contractor completed the installation of the Tideflex Valve and restored the surface of the below grade chamber.
Control vault—The functions of the completed control vault have been tested and are functional.
Emergency generator—Installation completed. The contractor will test the generator now that the new pumps are energized.
Restoration/landscape planning—The landscaping plan has been approved by the HOA and NHSA. The contractor is procuring the planting materials and will install them once they are received. This is planned to occur the week of October 24th.
The project's three (3) phases are now essentially completed, with restoration of the site planned for the next two weeks. The three phases are: 1) the construction of the wet weather pump station; 2) the construction of the control vault; and 3) the installation of the emergency generator.
The following construction submittals were reviewed the week of October 21, 2016.
The photograph below depicts the latest construction activity. The beginning of the restoration can be seen.
In the next three (3) weeks, the contractor will complete the remaining work. As noted above, the fencing at the 11th Street Pump Station, between Hudson and Washington Streets, will be completed in November. Once the fencing is completed, all project work will be finished and the contractor will be off the site.
Pump station—Completed and operational. Final performance test the main pumps.
Site restoration—Underway with scheduled completion by end of October.
Construction activities—Completed as scheduled, with site restoration underway and expected to be completed on the Maxwell site by the end of October.
Performance testing of the H-5 WWPS continues.
Sidewalks are now open to pedestrians. Full auto access will be available by October 31st, if not sooner.
Air monitoring of the site began on Thursday, August 27th, 2015 and continued in accordance with the LSRP-approved plan.
The program was discontinued as of September 16th, 2016.
Sidewalks are now open to pedestrians.
Full auto access will be available by October 31st, if not sooner. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,412 |
{"url":"https:\/\/collegephysicsanswers.com\/openstax-solutions\/how-wide-single-slit-produces-its-first-minimum-633-nm-light-angle-280circ-b","text":"Change the chapter\nQuestion\n(a) How wide is a single slit that produces its first minimum for 633-nm light at an angle of $28.0^\\circ$? (b) At what angle will the second minimum be?\n1. $1.35 \\textrm{ }\\mu\\textrm{m}$\n2. $69.9^\\circ$\nSolution Video\n\n# OpenStax College Physics Solution, Chapter 27, Problem 45 (Problems & Exercises) (1:38)\n\nRating\n\nNo votes have been submitted yet.\n\n## Calculator Screenshots\n\nVideo Transcript\nThis is College Physics Answers with Shaun Dychko. We're dealing with light have a wavelength 633 nanometers and the first minimum is at an angle of 28 degrees we're told and we have this formula for finding the minima in the diffraction pattern. And we have capital D is the slit width times sine of the angle equals the order of the minimum times the wavelengths of light. And so we're going to solve for capital D because we want to know how wide is the slit and so will divide both sides by sine theta and we get D is m lambda over sine theta. So the first order minimum which is one occurs at an angle of 28 degrees and so we have one times 633 times ten to the minus nine meters divided by sine of 28 degrees giving us 1.35 micrometers is the width of the slit. And then part b asks us to figure out what would the angle be to the second order minimum so we can rearrange this formula now and solve it for theta because we know what capital D is now because we've just found it in part a. And we'll divide both sides by D and take the inverse sine of both sides and that gives us this formula that the angle to a minimum is inverse sine of the order times a wavelength divided by the slit width and so the angle to the second order minimum is the inverse sine of two times the wavelength divided by the slit width, and I'm using extra digits here to avoid intermediate rounding error, 1.3483 times ten to the minus six meters slit width and this gives us an angle of 69.9 degrees.\n\nSubmitted by eethie5 on Wed, 12\/02\/2020 - 08:20\n\nSo I thought that when it said first minimum then m=0 and when it says first-order minimum that means m=1. Do I have this confused?\n\nSubmitted by ShaunDychko on Fri, 12\/11\/2020 - 16:57\n\nThanks for the question @eethie5.\nIt's important to distinguish the situation between double slit interference vs single slit diffraction. With single slit diffraction, as in this question, the formula for minima is $D\\sin{\\theta}=m\\lambda \\textrm{ for } m = 1,-1,2,-2,3, ...$, where $D$ is the width of the single slit. There is no $m = 0$ in this case. The \"first minimum\" is also referred to as the \"first order minimum\" in this case. The \"order\" is the value of $m$.\nFor double slit interference (destructive), the formula is $d \\sin{\\theta} = \\left ( m + \\dfrac{1}{2} \\right ) \\lambda \\textrm{ for } m = 0,1,-1,2,-2,...$, where $d$ is the distance between the two slits. The first minimum in this scenario would be the \"zeroth order minimum\".\nHope this helps,\nShaun","date":"2021-03-03 04:34:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.8045487999916077, \"perplexity\": 451.0961020083731}, \"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-2021-10\/segments\/1614178365454.63\/warc\/CC-MAIN-20210303042832-20210303072832-00001.warc.gz\"}"} | null | null |
\section{Introduction}
Since 1989 it has been known that the magnitude of direct CP
violation in the $K_L \to 2 \pi$ decays crucially depends
on the relative strengths
of the imaginary parts of
the QCD-penguin (QCDP) and the electroweak-penguin
(EWP) contributions to the amplitude \cite{1}.
The reason for this sensitivity
is that the contributions to
$\varepsilon'$ from the two diagrams
have opposite signs and partially cancel one another.
As the dynamical structures of the amplitudes for
$K^{\pm} \to (3\pi)^{\pm}$ differ
from those for $K_L \to 2\pi$, there is no immediate relation
between the
strengths of direct CP violation in the $K_L \to 2\pi$ and the
$K^{\pm}\to (3\pi)^{\pm}$
decays. In particular, it was observed in refs.~\cite{2,3}
that in contrast to
the situation in $K_L \to 2\pi$,
the CP violating effect
in $ K^{\pm} \to \pi^{\pm} \pi^{\pm} \pi^{\mp}$
produced by the QCDP contribution is enhanced by the EWP contribution.
However, in the present note, we shall demonstrate
that the $K^{\pm} \to \pi^0 \pi^0 \pi^{\pm}$
decays are similar to the $K_L \to 2\pi$ decay
in that the EWP contribution cancels part
of the QCDP contribution. Due to this circumstance,
we suggest that a simultaneous study of the decays
$K^{\pm} \to \pi^0 \pi^0 \pi^{\pm} $ and
$ K^{\pm} \to \pi^{\pm} \pi^{\pm}\pi^{\mp} $
could throw new light on the relative strengths
of the QCDP and
the EWP mechanisms in direct CP violation.
We shall estimate, in the framework of
the Standard Model,
the CP violating contributions to the slope
parameters $g^+$ and $g^-$ characterizing the charged pion energy
distributions in the $K^{\pm} \to \pi^0 \pi^0 \pi^{\pm}$ decays
(formerly $\tau'$ decay). The slope parameters are defined by
the expansion
\begin{equation}
|M(K^{\pm}(k) \to \pi^0(p_1) \pi^0 (p_2) \pi^{\pm}
(p_3))|^2 \propto 1+g^{\pm}Y+..., \label{slope-def}
\end{equation}
where
\begin{equation}
Y=(s_3-s_0)/m^2_{\pi},\quad s_i=(k-p_i)^2, \quad s_0=m^2_K/3 +m^2_\pi.
\label{kinvar}
\end{equation}
Our tenet is corroborated by a calculation of
the amplitudes to leading non-vanishing order in a momentum expansion.
As was previously found for the $K^{\pm} \to \pi^{\pm} \pi^{\pm}
\pi^{\mp}$ decays \cite{2,3}, higher-order corrections do not considerably
change the conclusion concerning the
relative magnitudes of the QCDP and EWP contributions to
the difference $(g^+ -g^-)_{\tau}$. The role of higher-order corrections
in the ${\tau}'$ decays will be considered elsewhere.
\section{The $K^{\pm} \to \pi^0 \pi^0 \pi^{\pm} $ amplitude}
Our starting point is the
$\Delta S=1$ effective non-leptonic Lagrangian proposed in ref.~\cite{4}
\begin{equation}
L(\Delta S=1) =\sqrt{2}G_{\rm F}\sin\theta_{\rm C} \cos \theta_{\rm C}
\sum_i c_iO_i\,, \label{1}
\end{equation}
where $ O_{1-6}$ are effective four-quark operators represented by the operator
products
\begin{align}
O_1=&\bar s_L\gamma_{\mu}d_L \cdot \bar u_L \gamma_{\mu}u_L-\bar s_L
\gamma_{\mu} u_L \cdot \bar u_L \gamma_{\mu}d_L \ ; \qquad(\{8_f\},
\Delta I=1/2), \\
O_2 =& \bar s_L \gamma_{\mu}d_L \cdot \bar u_L \gamma_{\mu} u_L +
\bar s_L \gamma_{\mu} u_L \cdot \bar u_L \gamma_{\mu} d_L +2 \bar s_L
\gamma_{\mu} d_L \cdot \bar d_L \gamma_{\mu} d_L \nonumber\\
&+2 \bar s_L \gamma_{\mu} d_L \cdot \bar s_L \gamma_{\mu} s_L \ ; \qquad
(\{8_d\}, \Delta I=1/2),\\
O_3=&\bar s_L \gamma_{\mu} d_L \cdot \bar u_L \gamma_{\mu}u_L +\bar s_L
\gamma_{\mu}u_L \cdot \bar u_L \gamma_{\mu} d_L +
2 \bar s_L \gamma_{\mu}
d_L \cdot \bar d_L \gamma_{\mu} d_L \nonumber\\
& -3 \bar s_L \gamma_{\mu} d_L \cdot
\bar s_L \gamma_{\mu} s_L \ ; \qquad(\{27\}, \Delta I=1/2), \\
O_4=&\bar s_L
\gamma_{\mu} d_L \cdot \bar u_L \gamma_{\mu} u_L + \bar s_L \gamma_{\mu}
u_L \cdot \bar u_L \gamma_{\mu} d_L \nonumber\\
& -\bar s_L \gamma_{\mu} d_L \cdot
\bar d_L \gamma_{\mu} d_L\ ; \qquad (\{27\}, \Delta I =3/2),\\
O_5=& \bar s_L \gamma_{\mu} \lambda^a
d_L(\sum_{q=u,d,s} \bar q_R \gamma_{\mu} \lambda^a q_R)\ ; \qquad (\{8\},
\Delta I=1/2),\\
O_6=&\bar s_L \gamma_{\mu} d_L(\sum_{q=u,d,s} \bar q_R \gamma_{\mu} q_R)
\ ; \qquad (\{8\}, \Delta I =1/2) \ .
\end{align}
Among these operators, only $O_4$ generates $\Delta I=3/2$ transitions.
The operators $O_{5,6}$ originate from the QCDP diagrams.
To calculate CP-violating effects, also the operators $ O_{7,8}$
generated by the EWP diagrams must be added,
\begin{align}
O_7=&\frac{3}{2} \bar s\gamma_{\mu}(1+\gamma_5)d (\sum_{q=u,d,s}e_q
\bar q \gamma_{\mu}(1-\gamma_5)q) ; \qquad (\Delta I=1/2, 3/2),\\
O_8=&-12\sum_{q=u,d,s} e_q (\bar s_L
q_R)(\bar q_R d_L) ; \quad e_q=(\frac{2}{3}, -\frac{1}{3}, -\frac{1}{3});
\quad (\Delta I=1/2, 3/2).
\end{align}
The corresponding Wilson coefficients $c_{7,8}$ are small, being
proportional to $\alpha_{em}$. The coefficients $c_{5-8}$ contain the
imaginary parts necessary for CP violation.
Bosonization of the operators $O_i$ is achieved through the substitutions
\cite{5}
\begin{align}
\bar q_j(1+\gamma_5)q_k =& -\frac{1}{\sqrt{2}}
F_{\pi}r(U-\frac{1}{\Lambda^2} \partial^2 U)_{kj}\ ,\\
\bar q_j \gamma_{\mu}(1+\gamma_5)q_k=& i[(\partial_{\mu} U)U^{\dag}
-U(\partial_{\mu}U^{\dag}) -\frac{rF_{\pi}}{\sqrt{2}
\Lambda^2}(m(\partial_{\mu}U^{\dag})-(\partial_{\mu}U)m)]_{kj}\ .
\end{align}
Here, $m$ is the diagonal quark-mass matrix,
$$
m={\rm Diag}\{m_u,m_d,m_s\} \ ,
$$
and the remaining parameters are defined as
$$
r=2m^2_{\pi}/(m_u+m_d),\quad \Lambda \approx 1\; \mbox{GeV},\quad
F_{\pi}=93\; \mbox{MeV}. $$
The $3\times3$ $U$-matrix is written as an expansion
\begin{equation}
U=\frac{F_{\pi}}{\sqrt{2}}\left(1+\frac{i\sqrt{2}\hat
\pi}{F_{\pi}}-\frac{\hat \pi^2}{F_{\pi}^2}+a_3\left(\frac{i\hat
\pi}{\sqrt{2} F_{\pi}}\right)^3 +2(a_3-1)\left(\frac{i \hat \pi}{\sqrt{2}
F_{\pi}}\right)^4 +....\right)\
\end{equation}
in the pseudoscalar nonet-meson-field matrix $\hat \pi$
\begin{equation}
\hat \pi=
\left (
\begin{array}{ccc}
\displaystyle{
\frac{\pi_0}{\sqrt{3}}+\frac{\pi_8}{\sqrt{6}}+\frac{\pi_3}{\sqrt{2}} }&
\pi^+ & K^+ \\ \\
\pi^- &
\displaystyle{
\frac{\pi_0}{\sqrt{3}}+\frac{\pi_8}{\sqrt{6}}-\frac{\pi_3}{\sqrt{2}} } &
K^0 \\ \\
K^- & \bar K^0 &
\displaystyle{ \frac{\pi_0}{\sqrt{3}}-\frac{2\pi_8}{\sqrt{6}} }
\end{array}
\right)\ .
\end{equation}
The PCAC condition demands $a_3 =0$ \cite{6} and we adopt this condition
as well,
bearing in mind, that on mass shell, the values of the mesonic amplitudes
are independent of the parameter $a_3$.
In the calculation of the $K\to 3 \pi$ amplitudes
we make use of the Fierz identities for the colour matrices
\begin{align}
\delta^{\alpha}_{\beta}
\delta^{\gamma}_{\delta}=&\third \delta^{\alpha}_{\delta}
\delta^{\gamma}_{\beta} +\half \lambda^{\alpha}_{\delta}
\lambda^{\gamma}_{\beta}\nonumber \\
\lambda^{\alpha}_{\beta} \lambda^{\gamma}_{\delta}=&\extrabrak
\delta^{\alpha}_{\delta} \delta^{\gamma}_{\beta} -\third
\lambda^{\alpha}_{\delta} \lambda^{\gamma}_{\beta}\nonumber
\end{align}
as well as the Fierz identities for the Dirac matrices
$$
\bar s \gamma_{\mu} (1+\gamma_5)d \cdot \bar q \gamma_{\mu} (1-\gamma_5)q=
-2\bar s (1-\gamma_5)q \cdot \bar q(1+\gamma_5)d \ .
$$
Thus, in leading order non-vanishing approximation our result for the
matrix element can be expressed as
\begin{align}
M\left(K^{\pm}\right.&\left. \to \pi^0(p_1) \pi^0 (p_2) \pi^{\pm}(p_3)\right)
= \nonumber \\
& = \kappa \left[1 \pm ia_{KM}
+\frac{3m^2_{\pi}}{m^2_K}\left(1-\frac{9c_4}{2c_0}\right) Y (1 \pm
ib_{KM})+...\right] \ .\label{K3pi}
\end{align}
The kinematic variable $Y$ is defined in eq.(\ref{kinvar}). The overall
strength is regulated by the parameter
\begin{equation}
\kappa=\frac{G_Fm^2_K}{6\sqrt{2}} c_0\sin \theta_C \cos \theta_C \ ,
\end{equation}
and the remaining parameters are functions of the following combinations
\begin{equation}
c_0=c_1-c_2-c_3-c_4+\frac{32}{9} \beta \mbox{Re} \tilde c_5 \ ,
\end{equation}
\begin{equation} \tilde
c_5=c_5+\frac{3}{16}c_6, \quad \tilde c_7=c_7+3c_8 \ ,
\end{equation}
\begin{equation}
\beta=\frac{2m^4_{\pi}}{\Lambda^2 (m_u+m_d)^2} \ .
\end{equation}
The terms $a_{KM}$ and $b_{KM}$ in the amplitude
(\ref{K3pi}) are the imaginary parts of the amplitude
generated by the Kobayashi-Maskawa phase. Explicitely, we find
\begin{eqnarray}
a_{KM}&=& \beta\left(\frac{32}{9} \mbox{Im} \tilde c_5 +\frac{6
\Lambda^2 \mbox{Im}\tilde c_7}{m^2_K} \right)/c_0 \label{aKM-def}\\
b_{KM}&=& \beta \left( \frac{32}{9} \mbox{Im} \tilde c_5 +\frac{3
\Lambda^2 \mbox{Im} \tilde c_7}{m^2_K-m^2_{\pi}}
\right)/(c_0-\frac{9c_4}{2}) \ , \label{bKM-def}
\end{eqnarray}
with coefficients as above.
Our approach can also be used to caculate the $K\to 2\pi$ amplitudes.
For their real parts we get
\begin{eqnarray}
M(K^0_1 \to \pi^+ \pi^-)& =&\frac{G_F F_{\pi}}{\sqrt{2}}\sin \theta_C \cos
\theta_C (m^2_K-m^2_{\pi}) c_0 \ , \label{K2pi1} \\
M(K^+ \to \pi^+ \pi^0)&=&\frac{G_F F_{\pi}}{\sqrt{2}} \sin \theta_C \cos
\theta_C (m^2_K-m^2_{\pi}) (\threehalf c_4 ) \ .
\end{eqnarray}
A comparison between the real parts of the amplitudes
of eqs (\ref{K3pi}) and (\ref{K2pi1}) shows that their ratio
is nothing more than a reflection of the well-known relation
\begin{equation}
\begin{array}{rrr}
M\left(K^+(k)\to \pi^0(p_1) \pi^0(p_2) \pi^+(p_3)\right)
=\displaystyle{ \frac{1}{6F_{\pi}} } M(K^0_1 \to
\pi^+\pi^-)\\ \\
\times \left[1+ \displaystyle{
\frac{3m^2_{\pi}}{m^2_K} \left(1-\frac{3M(K^+ \to
\pi^+\pi^0)}{M(K^0_1 \to \pi^+\pi^-)} \right) } Y
\right]
\end{array}
\end{equation}
obtained earlier
\cite{7,8,9}\footnote{In \cite{9} there is a
misprint: a factor $y$ is missing after the factor $(1+
\frac{3\delta}{1+ \theta})$ in the expression for the $K^+ \to \pi^0
\pi^0 \pi^+$ amplitude, eq.(6.9).} using soft-pion techniques and current
algebra.
From the data on the $K\to 2\pi$ decay rates \cite{10},
it follows that
\begin{eqnarray}
c_1-c_2-c_3 +\frac{32}{9}\beta\, \mbox{Re} \tilde c_5& =& -10.13 \ ,\\
c_4&=&0.328 \ .
\end{eqnarray}
Furthermore, the combination $\beta\, \mbox{Re} \tilde c_5$
can be determined separately, provided we are willing to accept the
estimate of Shifman {\it et al.}~\cite{4,11},
\begin{equation}
c_1-c_2-c_3 =-2.89 \ ,
\end{equation}
which leads to the value
\begin{equation}
\frac{32}{9}\beta\, \mbox{Re} \tilde c_5 = -7.24\ .
\end{equation}
To estimate CP-odd effects in $K^{\pm} \to (3\pi)^{\pm}$ decays,
described
by the coefficients $a_{KM}$ and by $b_{KM}$,
we need a certain combination of the parameters $\mbox{Im} \tilde c_5$
and $\mbox{Im} \tilde c_7$. The theoretical preditions for these
parameters are very uncertain and different authors
(see \cite{3}) give different results. Fortunately, the combination
entering the $K^{\pm} \to \pi^0 \pi^0 \pi^{\pm}$
amplitude turns out to be similar to the combination determining the
parameter $\varepsilon'$ in $K_L \to 2\pi$ decay \cite{2}.
This circumstance allows us to
obtain a reliable estimate of $(g^+ -g^-)_{\tau'}$.
\section{Estimate of the CP-odd difference $(g^+ -g^-)_{\tau'}$}
Although the amplitude (\ref{K3pi}) incorporates the imaginary
terms necessary for
CP violation, this is not sufficient for producing observable
CP-violating effects. In fact, the observable effects
arise from the interference between these CP-odd terms and the
CP-even imaginary terms created by the strong-interaction final-state
rescattering between the pions. The strong-interaction effects are
introduced into the
$K\to 3\pi$ amplitudes of eq.(\ref{K3pi}) by adding two terms,
$a_{\tau'}$ and $b_{\tau'}$, so that
\begin{equation}
\begin{array}{rrr}
M\left(K^{\pm} (k) \to \pi^0(p_1) \pi^0(p_2) \pi^{\pm}(p_3) \right)=\kappa
\displaystyle{ \frac{1\pm ia_{KM}}{(1+a^2_{KM})^{1/2}} }
[1+ia_{\tau'} + \\ \\
+\displaystyle{ \frac{3m^2_{\pi}}{m^2_K} \left(1-\frac{9c_4}{2c_0}
\right) } Y(1+ib_{\tau'} \pm
i(b_{KM}-a_{KM}))+...]\ .\label{Kto3pi-full}
\end{array}
\end{equation}
This assumption is valid as long as the rescattering contribution
can be treated in the linear approximation.
The slope parameters $g^+$ and $g^-$ were defined in
eq.(\ref{slope-def}).
From this definition and eq.(\ref{Kto3pi-full}) we get for the
relative difference in slope parameter for the $K\to 3\pi$ decays
\begin{equation}
\Delta g_{\tau'}=\left(\frac{g^+ -g^-}{g^++ g^-} \right)_{\tau'}=
\frac{a_{\tau'}(b_{KM}-a_{KM})_{\tau'}}{1+a_{\tau'}b_{\tau'}} \label{Deltag}
\end{equation}
The strong-interaction-rescattering parameters,
$a_{\tau'}$ and $b_{\tau'}$, are
determined by calculating the imaginary parts of the loop diagrams of
Fig.~1. Putting the intermediate pions on shell (see Appendix)
yields, in leading approximation,
\begin{equation}
a_{\tau'}=0.12, \qquad b_{\tau'}=0.49 .
\end{equation}
The CP-odd numerator of eq.(\ref{Deltag})
can be calculated from the expressions
in eqs (\ref{aKM-def}) and (\ref{bKM-def}), and is found
being equal to
\begin{eqnarray}
(b_{KM}-a_{KM})_{\tau'}&=&\frac{16c_4\beta\;
\mbox{Im} \tilde c_5 }{c_0(c_0-\ninehalf c_4)}
-\frac{6\beta \Lambda^2\; \mbox{Im} \tilde c_7}{m^2_K
c_0} \left(1-\frac{c_0m^2_K}{2(m^2_K-m^2_{\pi})(c_0-\ninehalf c_4)}
\right) \nonumber \\ && \nonumber \\
&=& 0.042 \beta\, \mbox{Im} \tilde c_5\, (1+27.8\; \mbox{Im}
\tilde c_7/\mbox{Im} \tilde c_5) \ .\label{abCP-odd}
\end{eqnarray}
The combination of Wilson coefficients in this formula,
\begin{equation}
\beta\; \mbox{Im}\tilde c_5\,(1+27.8\; \mbox{Im} \tilde c_7/\mbox{Im} \tilde
c_5) \ ,\label{Comb3pi}
\end{equation}
is very similar to another combination
\begin{equation}
\beta\, \mbox{Im} \tilde c_5(1+\frac{24.36}{1-\Omega} \cdot
\frac{\mbox{Im}\tilde c_7}{\mbox{Im} \tilde c_5}) =-\frac{(1.63 \pm
0.25)\cdot10^{-4}}{1-\Omega} \beta\, \mbox{Re} \tilde c_5
\label{Comb2pi}
\end{equation}
defining the direct CP-violating parameter $\varepsilon'$ in
$K_L \to 2\pi$ decay \cite{2,3}.
The parameter $\Omega$ takes into account isospin-breaking
contributions generated by the two-step transition $K^0
\to \pi^0 \eta(\eta') \to \pi^0 \pi^0$.
At $\Omega=0.124$ expressions (\ref{Comb3pi}) and (\ref{Comb2pi})
coincide, giving
\begin{equation}
\Delta g_{\tau'}=(1.8 \pm 0.28)\cdot 10^{-6} \ ,
\end{equation}
and at $\Omega$=0.25
\begin{equation}
\Delta g_{\tau'}=2.1\cdot 10^{-6}(1-\frac{4.7\, \mbox{Im} \tilde
c_7/\mbox{Im} \tilde c_5}{1+32.48\, \mbox{Im} \tilde c_7/\mbox{Im}
\tilde c_5})\ .
\end{equation}
Both values of
$\Omega$ are in line with estimates figurating in the literature
(see \cite{12} and references therein).
\section{The CP-odd difference $(g^+ -g^-)_{\tau}$}
As we shall now show, our result for the slope-parameter
difference in $\tau'$ decay, as embodied in
eq.(\ref{abCP-odd}), enables us to draw quite precise conclusions
concerning the magnitude of
the slope-parameter difference in another decay, namely the
$K^{\pm} \to \pi^{\pm}\pi^{\pm}\pi^{\mp}$ decay, or
$\tau$ decay. From eqs (33) and (34) in ref.~\cite{3},
we derive the following relation
\begin{equation}
(b_{KM} -a_{KM})_{\tau}=-2\left[ \frac{16c_4\beta\,
\mbox{Im} \tilde c_5 }{c_0(c_0+9c_4)}
+ \frac{3\beta \Lambda^2\, \mbox{Im}
{\tilde c}_7 }{m^2_K c_0}
\left(1+\frac{12c_4 m^2_K}{\Lambda^2 (c_0+9c_4)} \right) \right] \ .
\label{abCP-odd-tau}
\end{equation}
The slope-parameter difference $\Delta g_{\tau}$ is again given by
expression (\ref{Deltag}), provided index $\tau'$ is everywhere
replaced by $\tau$. The value of the rescattering parameter
$a$ does not change, but that of $b$ does,
\begin{equation}
a_{\tau}=0.12, \qquad b_{\tau}=0.714 .
\end{equation}
Combining eqs (\ref{Deltag}), (\ref{abCP-odd}) and (\ref{abCP-odd-tau})
we can form the ratio of the slope parameters differences,
\begin{equation}
\frac{-\Delta g_{\tau}}{\Delta
g_{\tau'}}=2\frac{c_0-9c_4/2}{c_0+9c_4} \cdot \frac{1-14.34\,
\mbox{Im} \tilde c_7/\mbox{Im} \tilde c_5}{1+ 27.8\, \mbox{Im}\tilde
c_7/\mbox{Im} \tilde c_5}\cdot
\frac{1+a_{\tau'}b_{\tau'}}{1+a_{\tau}b_{\tau}}\ .\label{Dgprim/Dg}
\end{equation}
Now, only negative values of the ratio $\mbox{Im} \tilde
c_7/\mbox{Im} \tilde c_5$ appear in the literature \cite{3}.
If furthermore, we assume that the numerical value of
this ratio is so small that the sign of the right hand side
of eq.(\ref{Dgprim/Dg}) is positive,
we may conclude that
\begin{equation} -\Delta g_{\tau} \ge 3.1 \Delta g_{\tau'} > 0.56
\cdot 10^{-5}.
\end{equation}
This result differs from other
estimates, as exemplified by refs.~\cite{13}
and \cite{14}
\begin{equation} -\Delta g_{\tau}=1.8 \Delta
g_{\tau'} \quad \cite{13},\qquad - \Delta g _{\tau}=2.2 \Delta
g_{\tau'} \quad \cite{14}.
\end{equation}
Moreover, as follows from our discussion in Sect.~3,
we strongly believe $\Delta g_{\tau'}$ to be
of order $ 10^{-6}$. In contrast, $\Delta g_{\tau}$ can
reach values of order $10^{-5}$ , providing the EWP
contribution cancels out a considerable part of the
QCDP contribution (see eq.(\ref{Dgprim/Dg})).
For example, if the EWP cancels half of the
QCDP contribution, then
\begin{equation} -\Delta g_{\tau} =7.8
\Delta g_{\tau'} \ge 1.4 \cdot 10^{-5}
\end{equation}
and if the EWP
cancels three-quarters of the QCDP contribution, then
\begin{equation}
-\Delta g_{\tau}=17.2 \Delta g_{\tau'}\ge 3.1 \cdot 10^{-5} .
\end{equation}
\begin{center}
\begin{table}[h]
\begin{tabular}{|c|c|c|}
\hline \quad $\Delta g_{\tau}$ (in units $10^{-5}$) & $\Delta g_{\tau'}$
(in units $10^{-5}$) & Refs. \\ \hline \hline $-700 \pm 500$ & $-15 \pm
275$ & [15] \\ \hline $|\Delta g_{\tau}|_{LO} \le 0.7$ & - & [16]
\\ \hline -0.16 & - & [17] \\ \hline $|\Delta g_{\tau}|=38.2$ &
$|\Delta g_{\tau'}|=31.5$ & [18] \\ \hline $-0.23 \pm 0.06$ &
$0.13 \pm 0.04$ & [13] \\ \hline $(-4.9 \pm 0.9)\sin \delta$ & - &
[2] \\ \hline $-2.4 \pm 1.2$ & $1.1 \pm 0.7$ & [14]
\\ \hline $-(3.0 \pm 0.5)x; \quad 0.5< x < 5.0$ & - & [3] \\ \hline
\hline
$(- \Delta g_{\tau})_{LO}>(0.56\pm0.09) f(x)$ & $0.18 \pm 0.03$ &
present
\\ At $x=1, \quad (-\Delta g_{\tau}) =2.9 \pm 0.6$ & & work
\\ \hline
\end{tabular}
\caption{ Values for the slope-parameter ratios
$\Delta g_{\tau}$ and $\Delta g_{\tau'}$ in $\tau$ and $\tau'$ decays,
in units of $10^{-5}$.}
\end{table}
\end{center}
These examples show that a simultaneous measurement of $\Delta
g_{\tau'}$ and $\Delta g_{\tau}$ can clear up the question
about the true relative
strength of EWP and QCDP mechanisms in direct CP violation. The
estimates of the values $\Delta g_{\tau}$ and $\Delta g_{\tau'}$
as obtained in other investigations are summarised in Table 1.
\section{Concluding remarks}
We have calculated the CP-odd difference of slope parameters,
$\Delta g_{\tau'}$ of eq.(\ref{Deltag}), in the ${\tau'}$ decays
$ K^{\pm} \to \pi^0 \pi^0 \pi^{\pm}$
in leading non-vanishing approximation in a momentum expansion
of the decay amplitude.
We observe that the difference of slope parameters $\Delta
g_{\tau'}$ in $ K^{\pm} \to \pi^0 \pi^0 \pi^{\pm}$ decay
and the parameter $\varepsilon'$ in $K_L \to2\pi$ decay both depend
practically on one and the same combination of the Wilson
coefficients $\mbox{Im} \tilde c_5$ and
$\mbox{Im} \tilde c_7$. This observation permits a reliable estimate
of $\Delta g_{\tau'}$
using the known magnitude of $\varepsilon'$.
A comparison with the value of the
corresponding parameter $\Delta g_{\tau}$ in the ${\tau}$ decays
$K^{\pm} \to \pi^{\pm} \pi^{\pm} \pi^{\mp}$ shows that
$\Delta g_{\tau}$ is expected to be at least 3 times larger than
$\Delta g_{\tau}'$.
In fact, it may be even one order of magnitude
larger, provided there is a sizeable cancellation between
the electroweak-penguin and the QCD-penguin contributions
to the parameter $\varepsilon'$. Such a cancellation
is not excluded \cite{3,19}.
We have not considered the possibility of a sequential decay
$K^{\pm}\to \pi^0 \eta \pi^{\pm} \to \pi^0 \pi^0 \pi^{\pm}$
through an intermediate $ \eta \to \pi^0$ transition, a correction
which is of order $p^4$.
We shall study this possibility elsewhere.
In the case of $K^{\pm} \to \pi^{\pm} \pi^{\pm}
\pi^{\mp}$ decay, higher-order corrections increase $\Delta
g_{\tau}$ by 20\%, but change very little the relation between
electroweak-penguin and QCD-penguin
contributions \cite{3}. We expect a similar increase of
$\Delta g_{\tau'}$ in the
$K^{\pm} \to \pi^0 \pi^0 \pi^{\pm} $ decay,
since $a_{\tau'}\approx a_{\tau}$, and according to ref.\cite{3}
$p^4$ corrections increase the value of $a_{\tau}$ by 30\%.
\vspace{1cm}
{\large \bf Acknowledgments.}
We would like to thank the Swedish Research Council for financial support.
One of us (E.Sh) would also like to acknowledge a partial financial support
from the Grant RFBR-02-02 16957.
\section{Appendix}
Here, we shall calculate the CP-even imaginary part coming from the
pion-rescattering diagrams displayed in fig.~1. The imaginary
part of a diagram is obtained by cutting the internal lines as shown.
\begin{figure}[h]
\begin{tabular}{c@{}c@{}c@{}}
\scalebox{.70}{\includegraphics{cp1.eps}}&
\scalebox{.70}{\includegraphics{cp2.eps}}&
\scalebox{.70}{\includegraphics{cp3.eps}} \\
a) &
b) &
c)+d)
\end{tabular}
\caption{Rescattering diagrams for the imaginary part.
Diagrams are cut along the dashed line. Diagrams c) and d)
are related through $ \pi^0(p_1)\leftrightarrow
\pi^0(p_2)$.}
\end{figure}
We start from the amplitudes
\begin{eqnarray}
M\left(K^+(k) \to \pi^0(p_1) \pi^0 (p_2) \pi^+(p_3)\right)
&=&A+B(s_0-s_3) \label{tau-prime}\\
M\left(K^+(k) \to \pi^+(p_1) \pi^+ (p_2) \pi^-(p_3)\right)
&=&A'+B'(s_0-s_3) \label{tau-no-prime}
\end{eqnarray}
with kinematic variables as defined in eq.(\ref{kinvar}). The $\tau'$
decay amplitude, eq.(\ref{tau-prime}), is given in eq.(\ref{K3pi}),
of which we only need the leading real term.
In the $\tau$ decay amplitude, eq.(\ref{tau-no-prime}),
the parameter $A'$ is twice as large as $A$.
For the $\pi \pi$ scattering amplitudes we
insert the leading-order approximations,
\begin{eqnarray}
M\left( \pi^+(q_1)\pi^-(q_2) \to \pi^0(q_3)\pi^0(q_4)\right)
&=&\frac{i}{F_{\pi}^2}(s-m_{\pi}^2) \\
M\left( \pi^0(q_1)\pi^0(q_2) \to \pi^0(q_3)\pi^0(q_4)\right)
&=&\frac{i}{F_{\pi}^2}(s+t+u-3m_{\pi}^2) \\
M\left( \pi^0(q_1)\pi^+(q_2) \to \pi^0(q_3)\pi^+(q_4)\right)
&=&\frac{i}{F_{\pi}^2}(t-m_{\pi}^2)
\end{eqnarray}
with $s=(q_1+q_2)^2,$ $t=(q_1-q_3)^2$ and $u=(q_1-q_4)^2$
as usual.
First, we calculate diagram a) with a $\pi^+\pi^-$ pair in
the loop. The result is an imaginary contribution to the
$\tau'$ decay
\begin{equation}
\delta M_a=\frac{i}{16\pi F_{\pi}^2}(s_3-\mu^2)
\sqrt{1-\frac{4\mu^2}{s_3}}
\left[ A'+\half B'(s_3-s_0) \right] \ . \label{diag-a-exact}
\end{equation}
However, we are not interested in the exact value of
$\delta M_a$. The slope parameters, eq.(\ref{slope-def}),
are defined through an expansion in $Y=(s_3-s_0)/m_{\pi}^2$.
Moreover, we normalise the $K^+\to\pi^+\pi^+\pi^-$ decay
parameters as
\begin{eqnarray}
A'&=&\twothird m_K^2 \\
B'&=&1+9c_4/c_0=0.718\ ,
\end{eqnarray}
so that a short algebraic calculation gives as result
\begin{equation}
\delta M_a=i\frac{m_K^4}{72\pi F_{\pi}^2}
\sqrt{\frac{s_0-4\mu^2}{s_0}}
\left[ 1+ \frac{3(s_3-s_0)}{m_K^2}
\left\{1+\frac{2m_K^2m_{\pi}^2}{3s_0(s_0-4m_{\pi}^2)}
+ \fourth B'\right\} \right] \ .
\end{equation}
Diagram b) with two neutral pions in the loop give a contribution to
the imaginary part
\begin{equation}
\delta M_b=\frac{i}{32\pi F_{\pi}^2}\mu^2
\sqrt{1-\frac{4\mu^2}{s_3}}
\left[ A+ B(s_0-s_3) \right] \ .\label{diag-b-exact}
\end{equation}
The parameters for the decay $K^+\to\pi^+\pi^0\pi^0$ are
\begin{eqnarray}
A&=&\third m_K^2 \\
B&=&-(1-9c_4/2c_0)= -1.14\ .
\end{eqnarray}
The expansion of this contribution yields the result
\begin{equation}
\delta M_b=i\frac{m_K^2 m_{\pi}^2}{96\pi F_{\pi}^2}
\sqrt{\frac{s_0-4\mu^2}{s_0}}
\left[ 1+ \frac{s_3-s_0}{m_K^2}
\left\{\frac{2m_K^2m_{\pi}^2}{3s_0(s_0-4m_{\pi}^2)}
- 3 B\right\} \right] \ .
\end{equation}
There are two contributions, diagrams c) and d),
with $\pi^+\pi^0$ in the loop, since the final
state is symmetric in the two neutral pions, $\pi^0(p_1)$ and
$\pi^0(p_2)$. We shall not give the exact expressions,
corresponding to eqs (\ref{diag-a-exact}) and
(\ref{diag-b-exact}), since they are
somewhat complicated. The expansion of the sum of the two
amplitudes results in an imaginary contribution
\begin{align}
\delta M_c+\delta M_d = &
i\frac{m_K^4}{144\pi F_{\pi}^2}
\sqrt{\frac{s_0-4\mu^2}{s_0}}
\left[ -(1-\frac{3m_{\pi}^2}{m_K^2}) \right. \nonumber \\
& + \left.\frac{3(s_3-s_0)}{2m_K^2}
\left\{1+\frac{2m_{\pi}^2(s_0-2m_{\pi}^2)}{s_0(s_0-4m_{\pi}^2)}
+ \frac{3m_{\pi}^2}{m_K^2} B\right\} \right] \ .
\end{align}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,921 |
\section{Background} \label{sec:background}
\begin{figure*}[!htb]
\centering
\includegraphics[width=\textwidth]{Figures/CAVA_Pipeline.png}
\caption{Architecture of the 5-stage camera ISP pipeline.}
\label{fig:cava}
\end{figure*}
\subsection{The 5-Stage Camera ISP Pipeline} \label{ssoc:cava}
To perform our case study, we use the 5-stage Camera ISP pipeline that is shown in Figure \ref{fig:cava}~\cite{yaoyuannnn}. It takes in a raw image that is produced by camera sensors, and generates a useful image that can be displayed. The main stages of the pipeline are described below:
\textit{Demosaic:} This stage applies a Bayer Filter color filter array on each raw pixel to interpolate it's true R-G-B values. It produces a mosaic of RGB pixel intensities. \cite{wiki:demosaic}
\textit{Denoise:} This stage applies a local nonlinear interpolation denoising algorithm to reduce the level of noise in the image. \cite{wiki:denoise}
\textit{Color Space Transform / White Balancing:} This stage performs color balancing by multiplying the RGB color value at each point with a 3x3 diagonal matrix whose values are configurable. It preserves the neutrality of neutral colors. \cite{wiki:transform}
\textit{Gamut Mapping:} This stage maps the colors of the original image to a set of restricted available colors of an output device without compromising the original image. To do so, it first computes the L2-norm from each pixel to the set of control points that represent the target gamut. Then, the L2 distances are weighted and summed. Finally, a bias is added to implement a radial basis function. This is the most computationally-intensive kernel and serves as the bottleneck for the pipeline. \cite{wiki:gamut}
\textit{Tone Mapping:} This stage approximates images with a higher dynamic range than the output device. This is done by using a Tone Map Operator to squeeze the original dynamic range of the image into the lower range of the output device. \cite{wiki:tone}
\textit{Profile of the pipeline:} The execution time breakdown of the pipeline is as follows: 99\% in Gamut Map, 0.8\% in Denoise, 0.04\% in Transform, 0.03\% in Demosaic, and 0.02\% in Tone Map. Therefore, we can see that Gamut Map significantly outweighs all the other kernels, and we focus our efforts on achieving the best speedup for this kernel, while still studying the tradeoffs in applying different optimizations on the other kernels.
\subsection{Intel FPGA SDK for OpenCL} \label{ssec:aocl}
The Intel FPGA SDK for OpenCL is an HLS tool that allows hardware designers to use OpenCL, instead of HDLs, for programming Intel FPGAs. The main tool in the SDK is the Altera OpenCL Compiler (AOC), an HLS compiler that compiles the OpenCL kernels into RTL, then runs them through Intel Quartus to synthesize them and generate an FPGA bitstream. AOC's strong suit is in automatically pipelining loops in a kernel and trying to achieve perfect pipelining with an initiation interval (II) equal to 1. As such, it is recommended that designers implement their kernels as Single Work Item Kernels (SWIK), as opposed to the traditional multithreaded kernels that OpenCL is known for, in order to maximize AOC's ability to pipeline the kernel's execution, and extract as much parallelism as possible. To facilitate the process of optimizing kernels using Intel's OpenCL SDK, AOC provides three key resources: 1) An Optimization Report, which can be generated through a quick intermediate compilation step, that contains information about estimated resource utilization and the status of all the loops (unrolled, pipelined -- along with the estimated II). 2) A Profile Report, which requires a full synthesis and place-and-route to be completed, and provides profiling information related to the kernel performance (operating frequency, execution time, memory bandwidth, etc...). 3) A Best Practices Guide, which is a document that provides different optimization techniques to assist the designer in selecting optimization decisions based on the results of the optimization and profiling reports. Next, we will describe the main information that is reported by the Optimization Report and Profiling Report.
\textit{Understanding the Optimization Report:}
The Optimization Report provides an analysis of the loops in the kernel, identifying which loops were unrolled, and which were pipelined along with their corresponding II. If the II > 1, the report provides an explanation of what might be the bottlenecks that prevented perfect pipelining of the loops, along with pointers to the corresponding sections in the Intel guides that might provide techniques to improve the II. In addition to loop information, the report provides a ``System Viewer''. This viewer shows the basic blocks of the kernel and provides information about the start cycle, end cycle, and latency of each basic block, along with the structure of local memories.
\textit{Understanding the Profile Report:}
The Profile Report is an essential tool for understanding the performance of the generated kernel. It provides the total execution time of the kernel, the operating frequency, and the global bandwidth to DRAM. It also provides details about each load and store in the kernel.
\section{Conclusion} \label{sec:conclusion}
In this paper, we presented a study of the automatic optimization potential of the AOC compiler, and the tradeoffs in using differnt optimization techniques. We show that there is limited potential in automatic optimziations that AOC can perform, even driven with programmer directives (2.8$\times$ speedup vs. CPU in the best case), and that great performance benefits can be achieved by combining them with other optimizations that require manual hand-tuning of code (36.5$\times$ speedup vs. CPU). This motivates the need for more extensive compiler optimizations in commercial HLS tools, an area of research that we are interested in. We also show that different optimizations have different effects on different kernels, and that while some have clear-cut effects, others depend on the behavior of the kernel.
\section{Introduction} \label{sec:intro}
In recent years, High-Level Synthesis (HLS) has gained a lot of traction in the accelerator design community as a faster means of designing high-performance accelerators on Field Programmable Gate Arrays (FPGAs) and Application Specific Integrated Circuits (ASICs) \cite{VivadoHLS,SDAccel,zhang2008autopilot,IntelFPGAOpenCL,czajkowski2012opencl,maxcompiler,koeplinger2018spatial,pu2017programming, canis2011legup, canis2013legup, hegarty2014darkroom, wei2013improving}. With the right optimizations and tuning, HLS allows designers to reach the same end goal as Hardware Descriptive Languages (HDLs) like Verilog and VHDL, in a fraction of the design time with minimal loss of performance. In fact, recent literature has been showing a wide adoption of HLS in designing accelerators for new applications in a multitude of domains \cite{nakahara2018lightweight,Cabal:2018:CFP:3174243.3174250,zohouri2018combined}.
A successful example of an HLS system is AOC, the Intel FPGA SDK for OpenCL \cite{IntelFPGAOpenCL}. AOC allows designers to use OpenCL for designing their accelerator while targeting Intel's family of FPGAs. However, getting good performance with AOC is non-trivial, requiring extensive tuning. Intel has released two supporting documents, a Programming Guide~\cite{AOCLProgrammingGuide} and a Best Practices Guide~\cite{AOCLBestPractices} to provide designers with optimization techniques for achieving the best possible performance with AOC. In addition, AOC provides two reporting mechanisms: 1) a pre-synthesis Optimization Report that gives the designer a quick estimate of performance, allowing for rapid code modification without having to go through synthesis, and 2) a Profile Report that gives profiling information of the kernel physically running on the FPGA (post synthesis and place-and-route).
However, even with the guidance that Intel provides in its manuals, getting the best performance possible still presents at least three challenges. First, the programmer must often use compiler directives (in the form of pragmas) to provide information that may not be tractable for the compiler to prove automatically, such as for interprocedural pointer aliasing or loop-carried dependences. Second, it is often the case that different optimizations come with different tradeoffs. Navigating the design space of these optimizations requires the programmer to go through many iterations of their designs, many of which need to go through the complete synthesis flow, because the Optimization Report is not always indicative of the actual performance in hardware. Third, and most problematic, some important optimizations may be lacking in the compiler entirely (for a variety of reasons), and programmers may need to make significant manual changes to the code in order to accomplish those optimizations.
In this work, we study the tradeoffs of different optimization techniques, and the potential of automatic directive-driven optimizatons in AOC, on a 5-stage Image Signal Processing (ISP) pipeline that takes a raw image produced by camera sensors, and converts it into a viewable image format. Our target device is an Intel Arria 10 GX FPGA Development Kit which carries a GX 1150 FPGA. We show through our study that while some optimizations have clear-cut benefits and always yield improvements, others behave differently depending on the kernel being optimized. We also show that the amount of performance improvement that can be achieved automatically by the compiler, with only simple directives and minimal code modifications, is useful but limited, and that, in most cases, achieving the best performance requires significant modifications to the code.
In order to perform our study, we implemented the stages of the ISP pipeline as separate kernels in OpenCL using the Intel-recommended single work item model. Next, we used the pre-synthesis Optimization Report to choose different optimizations from Intel's Best Practices Guide that would help us achieve perfect pipelining of the outer loops in the kernels. We profiled the resulting design and found that significant performance improvements were possible but not achievable using automated optimizations implemented in the compiler (even with programmer directives), and therefore required further hand-tuned optimzation of the kernels. While some of these optimizations are described in the Best Practices Guide (like buffering of inputs), a very important transformation (similar to unroll-and-jam~\cite{callahan1988estimating}) is not. We performed both kinds of optimizations through manual hand-tuning of the code. Our final results show that for the full pipeline, automatic optimizations give a 3.3$\times$ slowdown, while hand tuning the kernels gives us up to 4.6$\times$ speedup compared to CPU execution. Looking at individual kernel performance, we find that automatic optimizations can achieve up to 2.72$\times$ speedup, while hand tuning the kernels allows us to achieve close to 36.5$\times$ speedup compared to CPU.
In our study, we treated the 5 stages of the ISP pipeline as distinct kernels, each exhibiting its own characteristics, to try and identify patterns in the effect of the optimization techniques that we apply. Through comparing the behaviors of the 5 kernels, we were able to learn the following lessons:
\begin{enumerate}
\item Our most important finding is that automatic optimizations that are performed by AOC, even those guided by directives, are not sufficient to achieve the best performance possible without extensive manual modifications by the designer.
\item In spite of the above limitation, using `\texttt{restrict}' and `\texttt{ivdep}' keywords almost always provide a decent boost in performance by allowing AOC to avoid memory dependence assumptions.
\item Rewriting the kernels to increase the amount of independent operations within the body of a pipelined outerloop provides significant performance benefits by increasing spatial parallelism.
\item If read-only data fits on the FPGA, manually buffering the data gives better performance than depending on AOC's constant memory, in all but one case, since it removes the overhead of the cache structure.
\item Finally, if inner loops are present in the body of the pipelined loop, even partially unrolling them gives a boost in performance since it increases the amount of parallel operations.
\end{enumerate}
By performing this study, we identified the limitations of automatic optimization in a modern commercial HLS tool like AOC, which motivates the need for more extensive compiler optimizations in these tools. This is an area of research that we are interested in and are pursuing as future work.
The rest of the paper is organized as follows: Section \ref{sec:background} provides a background about the ISP pipeline and the Intel FPGA SDK for OpenCL, followed by the details of how the optimizations we used, and how we applied them to each kernel in Section \ref{sec:opt}. Next, Section \ref{sec:eval} provides our experimental evaluation, followed by some of the related work in Section \ref{sec:related}, after which we conclude in Section \ref{sec:conclusion}.
\section{Optimizing the ISP pipeline} \label{sec:opt}
In this section, we will first describe our baseline implementation of the pipeline, followed by a description of the different optimizations that we applied along with our optimization strategy.
\subsection{Baseline} \label{ssec:baseline}
For our baseline, we use a simple `single work item' OpenCL implementation that is a direct mapping of the original C implementation of the pipeline into OpenCL. In this implementation, all arrays are stored in OpenCL global memory (FPGA on-board memory), and kernels communicate through global memory as well -- each kernel writes its output to global memory, which is then read by the next kernel. Our pipeline implimentation operates on images with three channels (`R', `G', and `B'). Images are stored in memory in row-major order such that all the rows of the `R' channel come first, followed by the rows of the `G' channel, which are then followed by the rows of the `B' channel. In the baseline implementation, all the kernels, except Demosaic, operate on the 3 channels sequentially, and produce all the pixels of `R' followed by `G', then `B'.
\subsection{Automatic Optimizations} \label{ssec:autoopt}
Some of the optimization techniques that are provided by Intel's guides, and that we use in this study, require minimal code modifications in the form of directives, and rely on the compiler to automatically optimize the kernel based on the information that these directives provide. We will now describe the three optimizations that we use:
\begin{itemize}
\item \textit{The ``\texttt{restrict}'' keyword:} Is an attribute that can be used to mark the different pointer operands of a kernel as non-aliasing. This allows AOC to avoid making conservative assumptions about whether or not the operands may alias.
\item \textit{The ``\texttt{ivdep}'' pragma:} Is a directive that can be used to instruct AOC to ignore any assumed loop-carried dependecies that enforce serialization of loop iterations, thus limiting the ability of AOC to pipeline the loops.
\item \textit{The ``\texttt{constant}'' attribute:} This attribute can be added to pointer operands of the kernel that are read-only. It instructs AOC to cache accesses to these operands in a global Constant Memory Cache that it creates using the FPGA Block RAM. A single constant cache of configurable size -- the default of which is 16KB -- is shared among all the ``constant'' operands of all the kernels that would be running on the FPGA at the same time.
\item \textit{The ``\texttt{unroll}'' pragma:} Is a directive that can be used to instruct AOC to attempt unrolling a loop. A specific unroll factor can be specified, otherwise AOC attempts to fully unroll the loop. If done on an inner loop of a pipelined outerloop, it gives AOC the potential to increase spatial parallelism by having more parallel computations to perform per iteration.
\end{itemize}
\subsection{Manual Optimizations} \label{ssec:manopt}
In addition to the automatic optimization techniques, Intel's guides recommend other optimizations that require significant code modifications, thus relying more heavily on the designer. We also combine in this category other hand-tuned optimizations that are not specifically recommended by Intel, but rather stem from the natural design considerations of spatial parallelism on the FPGA. The following describe the manual optimizations that we use in our study:
\begin{itemize}
\item \textit{Using local memory when possible:} This represents a set of optimizations recommended by Intel that involve moving computation to local memory when possible. These techniques include copying read-only kernel inputs that possess temporal locality from global to local memory, and privatizing any kernel operands that are only generated and accessed in the kernel.
\item \textit{Manual loop modifications:} There are scenarios where loop nests could be rewritten with a combination of unrolling, interchange, and fusion in order to maximize the amount of spatial parallelism by exposing more independent computations per iteration of a pipelined loop. We perform a variety of these optimizations as described Section \ref{ssec:strategy} below.
\item \textit{Using Intel Channels:} Intel Channels are FIFO buffers that can be used to transfer data between kernels. This can be useful when multiple kernels are running on the FPGA at the same time, and they exhibit a producer-consumer behavior. By using channels, all the data can be transferred between the kernels locally on the FPGA without having to write to/read from global memory. In many cases, using channels require modifications to the kernel code to make sure that the memory access pattern of the producer and consumer match.
\end{itemize}
\subsection{Optimization Strategy} \label{ssec:strategy}
Starting with the baseline from Section \ref{ssec:baseline}, we performed different combinations of the optimizations that we discussed in Sections \ref{ssec:autoopt} and \ref{ssec:manopt} on each kernel. We carefully chose optimizations to apply based on the results of the AOC Optimization Report and the Profile Report. We break down our study of the optimizations into four steps:
\begin{enumerate}
\item First, we apply the `\texttt{restrict}' and `\texttt{ivdep}' keywords, to achieve perfect pipelining of the outer loop, and study the extent of which AOC can automatically optimize the kernels with these keywords in place.
\item Next, we apply manual rewrites to the code, such that each kernel generates the corresponding pixel of the `R', `G', and `B' channels in the same iteration as opposed to generating them sequentially. This step involves a combination of loop unrolling, loop interchange, and loop fusion. By performing this step, we are increasing the amount of parallel computations in a single iteration of the outerloop, which in turn should allow AOC to exploit the spatial parallelism that is available.
\item After that, we try to further optimize the kernels by optimizing the memory accesses. In this step, we study the effects of using `constant' memory versus manually buffering for all the read-only inputs of each kernel that demonstrate temporal locality.
\item Finally, we use the `\texttt{unroll}' pragma on any inner loops to increase the spatial parallelism that can be exploited in each iteration, and as such further improve the performance.
\end{enumerate}
Once we have the fully optimized version of each kernel, we make an additional modification for the full pipeline, where we use Intel Channels to pass data between the kernels on-chip rather than using global memory. We use this to study the effect of channels on our pipeline. We also study two other versions of the full pipeline, one having the best performing version of each kernel, and one having the best performing version of each kernel using only the autmatic optimizations that we described in Section \ref{ssec:autoopt} above.
We will now describe how each kernel in the pipeline was optimized.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Figures/Full.pdf}
\caption{Full pipeline results. Best: best version of each kernel, Auto: best version of each kernel with only automatic optimizations, C\_CH: Intel channels with constant memory, B\_CH: Intel channels with manual buffering.}
\label{fig:full}
\end{figure}
\subsubsection{Demosaic:}
This kernel has a doubly-nested loop that goes over the rows and columns of the image and calculates the `R', `G', and `B' values of each pixel, while reading the input from global memory, and writing the output to global memory. As such, we only apply step 1 from our optimziation strategy to it, since it does not have read-only operands, and it cannot benefit from rewrites.
\subsubsection{Denoise:}
This kernel has a triply-nested loop that iterates over the image channels, rows, and columns. For each input pixel (which is coming from the output of the previous kernel), it reads a 3x3 tile centered around that pixel, sorts the values, and sets the corresponding output pixel to the median of the sorted values. Optimizing the sorting algorithm is beyond the scope of this paper; as such, we can only apply steps 1 and 2 from our strategy to this kernel.
\subsubsection{Color Space Transform:}
This kernel loops over the channels, rows, and columns of the image, and for each output pixel it reads the `R', `G', and `B' values of input pixel, multiplies them by a column from a 3x3 matrix (one column for each channel), and sums up the products. The 3x3 matrix is a read-only input, as such we apply steps 1, 2, and 3 from our strategy to this kernel. We will refer to this kernel as Transform in the rest of the paper.
\subsubsection{Gamut Map:}
For each input pixel, the Gamut Map kernel calculates the L2 distance between the `R', `G', and `B' values of that pixel and each control point. Then, the L2 distances are weighted and summed. Finally, a bias is added to that sum by multiplying each `R', `G', and `B' value with a coefficient and adding them up. The final result is the corresponding output pixel. As such, the Gamut Map kernel is the most computationally intensive kernel in the pipeline. Given that the L2 distance calculation is performed in an innermost loop, we apply all 4 steps of our strategy to this kernel.
\subsubsection{Tone Map:}
The Tone Map kernel reads each input pixel, uses its value to access a \texttt{tone\_map} matrix, and sets the corresponding value from the matrix as the output pixel value. Given that this kernel has a read-only input and potential for rewrite, we apply steps 1, 2, and 3 from our strategy to it.
\section{Related Work} \label{sec:related}
One recent work that also analyzes AOC is presented in \cite{wang2016performance}. In this work the authors build an analysis framework for modeling the effects of different AOC optimization techniques. Their goal is to provide a tool for designers to be able to analyze the performance effects of difference AOC optimzations on their code. However, their work does not focus on determining the potential of AOC's automatic optimizations. Their work is that they focus on the multi-threaded execution model of AOC, rather than the Single Work Item model that we focus our study on because it is recommended by Intel to allow AOC to perform more automatic optimizaiton. Given that the multi-threaded model of AOC depends on the programmer to manually express the parallelism, most of the optimizations there require manual hand-tuning.
There are recent surveys that focus on comparing HLS tools \cite{campbell2017new, nane2016survey}. These surveys provide a description of the abilities of different HLS tools, and compare ability of these tools in generating high-performing FPGA designs. However, they do not highlight what is possible with automatic optimizations in the different tools, versus what requires extensive hand-tuning.
Multiple academic papers present new HLS tool flows that can perform better automatic optimizations than AOC ~\cite{canis2013legup,canis2011legup,lee2016openacc,Zuo:2013:IPC:2555692.2555707,gupta2004coordinated,wei2013improving}. However, these papers do not specifically study the potential of automatic optimizaiton in AOC (or other commercial tools) compared to non-automatic optimizations. Instead, they show an overall improvement in generated code quality, and design efficiency. In addition, some of them are domain-specific targeting specific \cite{hegarty2014darkroom,pu2017programming,koeplinger2018spatial}. We focus on evaluating automating optimizations, and use AOC as a leading example of a commercial HLS tool that provides such optimizations.
\section{Experimental Evaluation} \label{sec:eval}
In this section, we start by presenting our evaluation methodology. Next we present the results of the full pipeline, which as an overview of the optimizations. Finally, we show the results of optimizing each kernel seperately, to highlight the effect of each optimization.
\subsection{Methodology}
Our software infrastructure consists of the Intel FPGA SDK for OpenCL version 18.1 (along with Quartus Pro 18.1) running on Ubuntu 16.04 LTS. Our target device is an Intel Arria 10 GX FPGA Developtment Kit which houses an Arria 10 GX1150 FPGA. Our CPU implementation is single-threaded running on an Intel Xeon E3-1240 V2 @ 3.40GHz. The operating frequency of the kernels on the FPGA ranges between 210MHz and 360MHz.
We synthesized different versions of each kernel, and of the entire pipeline, according to our optimization strategy described in Section \ref{sec:opt}. We tested the execution with multiple different input images of the same size, and confirmed that the results are similar. As such, we present the results for a single image, averaged over 10 runs.
\subsection{Full Pipeline Result}
\begin{figure*}[!h]
\centering
\includegraphics[width=\textwidth,height=4cm]{Figures/RI.pdf}
\caption{Effect of `\texttt{restrict}' (R) and `\texttt{ivdep}' (I) keywords on each kernel. (RI=R+I)}
\label{fig:RI}
\end{figure*}
\begin{figure} [!h]
\includegraphics[width=\columnwidth]{Figures/W.pdf}
\caption{Effect of rewrites (W) on each kernel (see Sections \ref{ssec:manopt} and \ref{ssec:strategy}). Does not apply to Demosaic. }
\label{fig:W}
\end{figure}
\begin{figure} [!h]
\includegraphics[scale=0.6]{Figures/CB.pdf}
\caption{Effect of using constant memory (C) vs manual buffering (B) for read-only data. Does not apply to Denoise.}
\label{fig:BC}
\end{figure}
\begin{figure}[h]
\includegraphics[scale=0.6]{Figures/GamutMap.pdf}
\caption{Different versions of Gamut Map kernel.}
\label{fig:gamut}
\end{figure}
In this section, we provide an overview of the performance of the full pipeline, before diving into the different steps of our optimization strategy and analyzing each kernel separately. Figure \ref{fig:full} (a) presents the execution time for the different versions of the full pipeline, normalized to CPU. Two versions were tested: \textbf{Best:} has the best performing version of each kernel constituting the pipeline; \textbf{Auto:} has the best performing version of each kernel, with automatic optimizations only (as described in Section \ref{ssec:autoopt}), constituting the pipeline. Note that since the Baseline described in Section \ref{sec:opt} is very unoptimal, we do not use it as a comparison point in the overall performance study of the pipeline. The results in the graph clearly show that there is a big gap in the overall performance benefits of automatic optimizations that can be performed by a tool like AOC (3.3$\times$ slowdown over CPU), and the best performance possible using manual modifications of the code (21$\times$ speedup over CPU).
We also study the effects of channels on the pipeline (Figure \ref{fig:full} (b)). We compare two versions against the best case from Figure \ref{fig:full} (a), one that uses constant memory with channels (C\_CH) and another that uses manual buffering with channels (B\_CH). We see that in our application Intel channels do not provide us with much more speedup compared to the best version of each kernel that uses global memory. The reason behind that is the imbalance between the kernels of the pipeline, which in this case causes increased stalling where the consumers are waiting on the producers to generate the data. We also show that manual buffering of the data gives better performance than using constant memory. This is due to the overhead in accessing the cache structure that is generated for constant memory compared to accessing the local memory that would be used for buffering.
\subsection{Single Kernel Results}
As discussed in Section \ref{sec:opt}, our optimization strategy involves four steps, applied incrementally to the baseline. We discuss the analysis of our strategy, one step at a time, in this section.
\subsubsection{Step 1:} This step studies the effect of the `\texttt{restrict}' (R) and `\texttt{ivdep}' (I) keywords on each kernel. Figure \ref{fig:RI} shows the three versions (R, I, and RI) compared to baseline and CPU. Although we also applied this step to Gamut Map, we do not show any results because these three versions of Gamut Map did not fit on our FPGA, and therefore we were not able to gather results for them. Our results show that there is always benefit in adding the two keywords, which is expected since this allows AOC to avoid making any assumptions related to memory aliasing and inter-loop memory dependence, thus allowing it to parallelize memory accesses to the fullest. Although RI isn't always the best-performing version, we chose it as the version to build on in our next steps in order to maintain uniformity accross all kernels. We also believe that R and I together will be beneficial for AOC to exploit more memory parallelism when we apply further optimizations.
\subsubsection{Step 2:} This step studies the effect of performing manual code modifications on top of RI, these include a combination of loop unrolling, interchange, and fusion, in order to increase the amount of parallel operations that can be scheduled in one iteration of the pipelined outerloop (as described in Section \ref{ssec:manopt} above). Our results are presented in Figure \ref{fig:W}, where W corresponds to manual rewrites. The results show that manual code modifications are crucial, as they can provide huge performance benefits (2.16$\times$-3.34$\times$ speedup vs RI) because of their ability to extract more independent operations per outerloop iteration, which can be spatially parallelized. Note that this optimization cannot be applied to Demosaic due to the nature of the kernel. Also, Gamut Map's version of RIW does not fit on our FPGA.
\subsubsection{Step 3:} This step studies the effect of further optimizing accesses to read-only memory after performing manual code modifications. This can either be done using constant memory (C) or manual buffering of the data in local memory (B). Figure \ref{fig:BC} and Figure \ref{fig:gamut} present the results for applying C and B on top of RIW. For Tone Map, our read only-data is larger and less frequently accessed, thus we believe that the overhead of the constant memory cache accesses slightly hurt performance. We also see that RIW performs slightly better than using either C or B, which we found is due to the fact that AOC was able to schedule it at a faster frequency compared to the other two. Another interesting observation is that with C and B added, the Gamut Map kernel now fits in our FPGA. We added an extra configuration to Gamut Map (\_128), which corresponds to manually configuring the size of the constant cache to 128KB, instead of the default 16KB, since the amount of read-only data accessed by this kernel is closer to 128KB (cache size can only be set to a power of 2). We see that by moving from a cache size that is smaller than the data set, to one that allows the data set to fit, greatly improves performance (\textasciitilde6$\times$ speedup vs \textasciitilde2$\times$ slowdown, when compared to CPU.). We also see that, although they are close, using buffering is slightly better than using constant memory. Finally, for Transform, the amount of read-only memory is small enough (9 floats), and the variation in the execution time is small enough, making the effect of C vs B negligible to the bigger picture. Therefore, the results show that it is almost always better to use buffering instead of constant memory when the data can fit on the FPGA, especially when the amount of data being accessed is significant.
\subsubsection{Step 4:} In addition to all the above optimizations, we perform an extra step for Gamut Map, which is unrolling its inner loop by a factor of 6. This optimization is not possible for the other kernels since they do not posses such an inner loop. We see in Figure \ref{fig:gamut} that by unrolling, we further improve the performance compared to buffering and using constant memory (\textasciitilde 36$\times$ vs CPU with unrolling, \textasciitilde 6$\times$ without). This is expected, since by unrolling the inner loop we are increasing the amount of independent operations that can benefit from the spatial parallelism on the FPGA.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,742 |
December 14, 2014 February 6, 2019 Averica Discovery Services, Inc.
Averica's method development services help ensure that your team makes the best decisions and hits drug development milestones.
If you want methods that are robust, transferable and designed to help achieve drug development timelines.
As an independent analytical development and testing company, our focus is on methods that support your drug development program.
We recognize that our clients use many resources to bring a drug to market. Those resources: your internal team, CROs and CMOs need methods that produce consistent, reliable data for their work. We develop analytical methods that are robust, transferable, reproducible, and designed to generate quality results.
Our clients rely on our methods, and our business is to make sure that they can.
Averica evaluates and carefully considers each of the performance parameters and critical quality attributes (CQA) when developing a method.
Averica's scientists understand both the requirements of the method and the requirements of the compound. We define and evaluate key method parameters and identify critical quality attributes. Methods we develop are fit for purpose, ranged properly, and can be quickly validated.
Robust methods for fast method transfer.
Emphasis on definition: making sure that a chromatographic method meets your needs and serves your development effort.
Depth of experience with method development suited to challenging compounds and projects.
Custom assay design and physical properties testing.
Small scale assay development for when material is limited.
Click for more information about how we use our technology systems.
If you want methods that help you achieve your drug development timelines.
Development, validation, verification, recovery and inter-laboratory transfer of analytical methods.
Averica helps late stage lead development teams understand the physiochemical properties of their lead compound.
Our highly responsive team is constantly developing and optimizing methods so they are the best fit for your goals. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,250 |
{"url":"http:\/\/jvmwriter.org\/not-found\/latex-error-not-found.html","text":"## Contents\n\nIn the first case, c is a superscript to the complete expression a^b. This is likely due to some earlier error. not loaded; Not enough room left ! If stumped, try the general tricks. ! http:\/\/jvmwriter.org\/not-found\/latex-error-fullpage-sty-not-found.html\n\nToo late for ... LaTeX Error: \\caption outside float A \\caption must inside a \"float\" like a figure or a table. Like \\load{\\textsize}{\\sc}. If you really need the \\\\ try putting in a \\strut before it. http:\/\/tex.stackexchange.com\/questions\/45900\/texmaker-file-not-found-when-viewing-as-pdf\n\nLater you can view your file in the same folder. :)1.3k Views \u00b7 View Upvotes \u00b7 Answer requested by Krishna Kanth YenumulaRelated QuestionsMore Answers BelowHow do you fix a file not Do I have to do everything here myself? But it listed in the terminal as having no file-extension (for some strange reason).\n\nCheck everything before \\begin{document} for typos or other errors. Not somewhere else. Check for still-open opening brackets. Texmaker Pdflatex Not Working was complete !\n\nBut you could actually forget the \\begin{document} line itself, I guess. Texmaker Log File Not Found If you really need it there, try putting \\protect right in front of it. To fix this, put the \\label{...} in equations at the end, followed by a %. http:\/\/tex.stackexchange.com\/questions\/58043\/texmaker-error-file-not-found Huge page cannot be shipped out Never seen this one in my life.\n\nPatterns can be loaded only by INITEX ! Texmaker Pdf Not Updating However, [ and ] are normal, while $and$ surround math. If stumped, try the general tricks. ! Dimension too large I think this means a length is specified that is larger than possible.\n\nBut when you load several different packages, and you will, they might conflict. see here If stumped, try the general tricks. ! Latex Error File Not Found Includegraphics Try to reinstall the font? Latex Pdf File Not Found Missing { inserted LaTeX decided it really needed a { and inserted one.","date":"2018-02-18 17:58:35","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\": 0.9444198608398438, \"perplexity\": 5103.5349558729995}, \"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-2018-09\/segments\/1518891812247.50\/warc\/CC-MAIN-20180218173208-20180218193208-00686.warc.gz\"}"} | null | null |
{"url":"http:\/\/math.stackexchange.com\/questions\/172465\/coordinate-free-method-to-determine-local-maxima-minima","text":"# Coordinate-free method to determine local maxima\/minima?\n\nIf there is a function $f : M \\to \\mathbb R$ then the critical point is given as a point where\n\n$$d f = 0$$\n\n$df$ being 1-form (btw am I right here?). Is there a coordinate independent formulation of a criteria to determine if this point is a local maximum or minimum (or a saddle point)?\n\n-\nThere are coordinate free definitions of second derivatives. Positive\/negative definiteness of this object is a coordinate free criterion. \u2013\u00a0 user20266 Jul 18 '12 at 16:11\n\nLet $p$ be a critical point for a smooth function $f:M\\to \\mathbb{R}.$\nLet $(x_\\,\\ldots,x_n)$ be an arbitrary smooth coordinate chart around $p$ on $M.$\nFrom multivariate calculus we know that a sufficient condition for $p$ to be a local maximum (resp. minimum) of $f$ is the positiveness (resp. negativeness) of the Hessian $H(f,p)$ of $f$ at $p$ which is the bilinear map on $T_pM$ defined locally by $$H(f,p)=\\left.\\frac{\\partial^2f}{\\partial x_i\\partial x_j}dx^i\\otimes dx^j\\right|_p,$$ here the Einstein convention on summation is working.\n\nHowever, as Thomas commented, the Hessian of a function at a critical point has a coordinate-free espression.\nInfact, $H(f,p): T_pM\\times T_pM\\to\\mathbb{R}$ is characterized by $$H(f,p)(X(p),Y(p))=(\\left.\\mathcal{L}_X(\\mathcal{L}_Y f))\\right|_p$$ for any smooth vector fields $X$ and $Y$ on $M$ around $p.$\n\nNote that without a Riemannian metric on $M$ you cannot invariantly define the Hessian of a function at a non-critical point.\n\n-\nDear Giuseppe, I have upvoted you but I certainly don't believe that the hessian is an alternating $2$-form as in the displayed equality you wrote: it is a symmetric form (or, equivalently, a quadratic form). \u2013\u00a0 Georges Elencwajg Jul 18 '12 at 21:17\nand we know the signature of a quadratic form is co-ordinate independent by Sylvester's law of inertia. \u2013\u00a0 Kris Jul 18 '12 at 21:39\n\nThe generalization of the Hessian matrix to functions on smooth manifolds is\n\n$$H(X,Y) = X (Yf) - df (\\nabla_X Y)$$\n\nwhere $X$ and $Y$ are vector fields, i.e. the Hessian is a bilinear form. The definition for positive\/negative definiteness for bilinear forms is the usual one. $H$ is positive definite if\n\n$$H(X,X)>0$$\n\nfor all vector fields $X$, and similarly for negative definite. As usual, a critical point is a local maximum if $H$ is negative definite, a local minimum if $H$ is positive definite, and a saddle otherwise.\n\n-","date":"2015-08-02 23:24:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9667884707450867, \"perplexity\": 134.62426430393938}, \"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-2015-32\/segments\/1438042989301.17\/warc\/CC-MAIN-20150728002309-00262-ip-10-236-191-2.ec2.internal.warc.gz\"}"} | null | null |
The Danish pride themselves on art, architecture and design, with architects such as Jørn Utzon, Arne Jacobsen, and Henning Larsen counting among the elite. This preoccupation is clearly reflected in the culture of Copenhagen, where during the summer months, the Danish Architectural Center offers tours around the city to showcase its developing community. Today, Copenhagen is considered one of the hottest spots for interesting architecture in Europe. Having managed to maintain a fine balance between historic buildings and the continued growth of the metropolis, Copenhagen has preserved its historical identity whilst looking to the future.
The architectural journey of Copenhagen starts from the moment you touch down at the airport. Kastrup Lufthavn Terminal 3 is an intriguing structure, inspired by the wing of an airplane. The structure has been completed using glass, steel, and aluminum; giving the building a light, elegant impression as if it might take off in flight. It follows traditional Scandinavian design principles of open space and simple lines. The best thing about this building, however, is that you can view it from all angles, provided you get a window seat.
Not too far from the airport is one of the most extraordinary community developments know as, Ørestad. The development is nearly 20 years in the making, and stretches 600 meters across. The development of this community was based on the ideas proposed by a Finnish architectural company, which won a competition for the project in 1994. The proposed ideas include a world class, holistic community that incorporates high-end architecture and a city plan. The project is split into four parts with master-planning and architecture designed by world-renowned architects including, Daniel Liebeskind, Jean Nouvel, Zaha Hadid, Norman Foster, and MVRDV. Each of the four districts is linked by the surrounding natural resources. It offers residents great opportunity in terms of employment, schooling, housing, and great outdoor activities. With only a ten-minute metro ride to the center of Copenhagen, it has all the facilities required of a developing area. Part of the reasoning behind Ørestad was to offer a contrast to the traditional sections of Copenhagen: a development that could showcase potential architectural genius to the world. One development in Ørestad garnering attention is Steven Holl's T-houses. This American architect has designed a series of T-shaped apartment complexes, which maximize green space beneath the structure by elevating the majority of the living structure to the horizontal bar of the T. This brilliant design idea enhances the ability to utilize the Scandinavian light and gives more residents a stunning view compared to traditional buildings.
Another building gaining publicity in Ørestad is the beehive or Bikuben-literally translated to "bee cube"-presumably named so because of its cube-like structure. Non-traditional gaps in the building structure were designed to make room for social areas, which are shared by all the residents. The building houses communal student accommodation for students from surrounding universities. It distances itself from traditional student living by placing the communal facilities on different stories toward the center courtyard of the building, making it possible to see from one common area to another. The student rooms spiral around the communal areas enhancing the social experience. The architects at A.A.R.T., the masterminds behind Bikuben, intended the structure to be an eye-catching building, which they have unquestionably achieved, giving Danish youth culture a quirky building fit for challenging conventional belief.
In 1999, an astonishing building was built in Copenhagen as an addition to the Royal Library. The building has been coined Den Sorte Diamant (The Black Diamond) as a result of its angular exterior walls made from highly polished black granite imported from Zimbabwe. Situated on the harbor front, entrance is facilitated by "harbor-busses" that sail straight to the front door of the structure; this is by far the best way to see it. The majestic building is reflected in the water of the harbor producing impressive geometric shapes, which contrast traditional surrounding edifices faced in oxidized copper that rise from behind the black sculpture. The Diamond's apparent ethos is of cultural education and this is reflected throughout. The impressive décor includes a 200-square-meter painting by Per Kirkeby. Aside from interior art, the building itself offers a great homage to abstract art. It has been finished using steel, glass, and blue oyster sandstone from Spain, which have been brought together in a simple, nearly understated approach, so as not to distract from the building as a whole. The light interior exposes the angles from the black exterior with the help of slowly rising escalators and walking bridges. The interior, however, unlike the exterior, is very organic with smooth, curving lines. Architects, Schmidt, Hammer and Lassen, designed the interior to mimic the current of the adjacent harbor, while granting harbor visibility via a massive wall of glass in the center of the building.
The Black Diamond shares its place on the harbor front with the Copenhagen Operahuset or Opera House, which was designed by Henning Larsen and erected in 2004. During the initial stages of the project, it experienced a number of public debates over its location and the aesthetic of the exterior. The Opera House is one of the largest buildings in Denmark, containing over 1100 rooms, consisting of nine stories above grade and a five additional floors below grade. The hardscape surrounding the building is comprised of Jura Gelb, a chalkstone from southern Germany, and Chinese granite cobblestones. This gives the building a very grandiose appearance that is fitting of the building's intended purpose, yet still complements the surrounding community. The roof has a 32-meter long extension, on which projected light reflects at night accenting the building's beautiful glass façade. It casts a warm glow, making the roof line appear as though it is floating. From the front of the Opera House the roof overlay and column-like structures hint at influence from Asian temples with a modern twist. The entire structure can be seen from across of the harbor in the gardens of the royal castle, Amaliehaven.
The Tycho Brahe Planetariet, a planetarium designed by Knud Munk, is another extraordinary building located in the midst of Copenhagen. Upon approaching the building, one is immediately aware the building was built for a unique purpose. The eye is initially drawn to its cylindrical shaped tower and angled roofline, finished with a symmetrical pattern made from glazed tiles. It is only when you come near that you find a daring contrasting building, which houses the entrance to the planetarium. For most children and young-minded adults, the excitement lies within this building as it houses the planetarium's cinema; a 360-degree screen with numerous projectors hidden behind screens, where illusions like that of a desert sky can be digitally recreated.
The key to many of these buildings and their success in Copenhagen comes down to traditional Danish mentality. Art, household items, and buildings all must follow the same criteria: simple, purposeful, and made with quality. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,927 |
{"url":"https:\/\/in.mathworks.com\/help\/audio\/ref\/audiofeatureextractor.html","text":"audioFeatureExtractor\n\nStreamline audio feature extraction\n\nDescription\n\n`audioFeatureExtractor` encapsulates multiple audio feature extractors into a streamlined and modular implementation.\n\nCreation\n\nSyntax\n\n``aFE = audioFeatureExtractor()``\n``aFE = audioFeatureExtractor(Name=Value)``\n\nDescription\n\n````aFE = audioFeatureExtractor()` creates an audio feature extractor with default property values.```\n\nexample\n\n````aFE = audioFeatureExtractor(Name=Value)` specifies nondefault properties for `aFE` using one or more name-value arguments.```\n\nProperties\n\nexpand all\n\nMain Properties\n\nAnalysis window, specified as a real vector.\n\nData Types: `single` | `double`\n\nOverlap length of adjacent analysis windows, specified as an integer in the range [0, `numel(Window)`).\n\nData Types: `single` | `double`\n\nFFT length, specified as an integer. The default value of `[]` means that the FFT length is equal to the window length `numel(Window)`.\n\nData Types: `single` | `double`\n\nInput sample rate in Hz, specified as a positive scalar.\n\nData Types: `single` | `double`\n\nInput to spectral descriptors, specified as `\"linearSpectrum\"`, `\"melSpectrum\"`, `\"barkSpectrum\"`, or `\"erbSpectrum\"`.\n\nSpectral descriptors affected by this property are:\n\nThe spectrum input to the spectral descriptors is the same as output from the corresponding feature:\n\nFor example, if you set `SpectralDescriptorInput` to `\"barkSpectrum\"`, and `spectralCentroid` to `true`, then `aFE` returns the centroid of the default Bark spectrum.\n\n```[audioIn,fs] = audioread(\"Counting-16-44p1-mono-15secs.wav\"); aFE = audioFeatureExtractor(SampleRate=fs, ... SpectralDescriptorInput=\"barkSpectrum\", ... spectralCentroid=true); barkSpectralCentroid = extract(aFE,audioIn);```\nIf you specify a nondefault `barkSpectrum` using `setExtractorParameters`, then the nondefault Bark spectrum is the input to the spectral descriptors. For example, if you call `setExtractorParameters(aFE,\"barkSpectrum\",NumBands=40)`, then `aFE` returns the centroid of a 40-band Bark spectrum.\n\n```setExtractorParameters(aFE,\"barkSpectrum\",NumBands=40) bark40SpectralCentroid = extract(aFE,audioIn);```\n\nData Types: `char` | `string`\n\nTotal number of features output from `extract` for the current object configuration, specified as a positive integer. `FeatureVectorLength` is equal to the second dimension of the output from the `extract` function.\n\nData Types: `single` | `double`\n\nFeatures to Extract\n\nExtract the one-sided linear spectrum, specified as `true` or `false`.\n\nTo set parameters of the linear spectrum extraction, use `setExtractorParameters`:\n\n`setExtractorParameters(aFE,\"linearSpectrum\",Name=Value)`\nSettable parameters for the linear spectrum extraction are:\n\n\u2022 `FrequencyRange` \u2013\u2013 Frequency range of the extracted spectrum in Hz, specified as a two-element vector of increasing numbers in the range [0, SampleRate\/2]. If unspecified, `FrequencyRange` defaults to `[0, SampleRate\/2]`.\n\n\u2022 `SpectrumType` \u2013\u2013 Spectrum type, specified as `\"power\"` or `\"magnitude\"`. If unspecified, `SpectrumType` defaults to `\"power\"`.\n\n\u2022 `WindowNormalization` \u2013\u2013 Apply window normalization, specified as `true` or `false`. If unspecified, `WindowNormalization` defaults to `true`.\n\nData Types: `logical`\n\nExtract the one-sided mel spectrum, specified as `true` or `false`.\n\nTo set parameters of the mel spectrum extraction, use `setExtractorParameters`:\n\n`setExtractorParameters(aFE,\"melSpectrum\",Name=Value)`\nSettable parameters for the mel spectrum extraction are:\n\n\u2022 `FrequencyRange` \u2013\u2013 Frequency range of the extracted spectrum in Hz, specified as a two-element vector of increasing numbers in the range [0, SampleRate\/2]. If unspecified, `FrequencyRange` defaults to `[0, SampleRate\/2]`.\n\n\u2022 `SpectrumType` \u2013\u2013 Spectrum type, specified as `\"power\"` or `\"magnitude\"`. If unspecified, `SpectrumType` defaults to `\"power\"`.\n\n\u2022 `NumBands` \u2013\u2013 Number of mel bands, specified as an integer. If unspecified, `NumBands` defaults to `32`.\n\n\u2022 `FilterBankNormalization` \u2013\u2013 Normalization applied to bandpass filters, specified as `\"bandwidth\"`, `\"area\"`, or `\"none\"`. If unspecified, `FilterBankNormalization` defaults to `\"bandwidth\"`.\n\n\u2022 `WindowNormalization` \u2013\u2013 Apply window normalization, specified as `true` or `false`. If unspecified, `WindowNormalization` defaults to `true`.\n\n\u2022 `FilterBankDesignDomain` \u2013\u2013 Domain in which the filter bank is designed, specified as either `\"linear\"` or `\"warped\"`. If unspecified, `FilterBankDesignDomain` defaults to `\"linear\"`.\n\nData Types: `logical`\n\nExtract the one-sided Bark spectrum, specified as `true` or `false`.\n\nTo set parameters of the Bark spectrum extraction, use `setExtractorParameters`:\n\n`setExtractorParameters(aFE,\"barkSpectrum\",Name=Value)`\nSettable parameters for the Bark spectrum extraction are:\n\n\u2022 `FrequencyRange` \u2013\u2013 Frequency range of the extracted spectrum in Hz, specified as a two-element vector of increasing numbers in the range [0, SampleRate\/2]. If unspecified, `FrequencyRange` defaults to `[0, SampleRate\/2]`.\n\n\u2022 `SpectrumType` \u2013\u2013 Spectrum type, specified as `\"power\"` or `\"magnitude\"`. If unspecified, `SpectrumType` defaults to `\"power\"`.\n\n\u2022 `NumBands` \u2013\u2013 Number of Bark bands, specified as an integer. If unspecified, `NumBands` defaults to `32`.\n\n\u2022 `FilterBankNormalization` \u2013\u2013 Normalization applied to bandpass filters, specified as `\"bandwidth\"`, `\"area\"`, or `\"none\"`. If unspecified, `FilterBankNormalization` defaults to `\"bandwidth\"`.\n\n\u2022 `WindowNormalization` \u2013\u2013 Apply window normalization, specified as `true` or `false`. If unspecified, `WindowNormalization` defaults to `true`.\n\n\u2022 `FilterBankDesignDomain` \u2013\u2013 Domain in which the filter bank is designed, specified as either `\"linear\"` or `\"warped\"`. If unspecified, `FilterBankDesignDomain` defaults to `\"linear\"`.\n\nData Types: `logical`\n\nExtract the one-sided ERB spectrum, specified as `true` or `false`.\n\nTo set parameters of the ERB spectrum extraction, use `setExtractorParameters`:\n\n`setExtractorParameters(aFE,\"erbSpectrum\",Name=Value)`\nSettable parameters for the ERB spectrum extraction are:\n\n\u2022 `FrequencyRange` \u2013\u2013 Frequency range of the extracted spectrum in Hz, specified as a two-element vector of increasing numbers in the range [0, SampleRate\/2]. If unspecified, `FrequencyRange` defaults to `[0, SampleRate\/2]`.\n\n\u2022 `SpectrumType` \u2013\u2013 Spectrum type, specified as `\"power\"` or `\"magnitude\"`. If unspecified, `SpectrumType` defaults to `\"power\"`.\n\n\u2022 `NumBands` \u2013\u2013 Number of ERB bands, specified as an integer. If unspecified, `NumBands` defaults to `ceil(hz2erb(FrequencyRange(2))-hz2erb(FrequencyRange(1)))`.\n\n\u2022 `FilterBankNormalization` \u2013\u2013 Normalization applied to bandpass filters, specified as `\"bandwidth\"`, `\"area\"`, or `\"none\"`. If unspecified, `FilterBankNormalization` defaults to `\"bandwidth\"`.\n\n\u2022 `WindowNormalization` \u2013\u2013 Apply window normalization, specified as `true` or `false`. If unspecified, `WindowNormalization` defaults to `true`.\n\nData Types: `logical`\n\nExtract mel-frequency cepstral coefficients (MFCC), specified as `true` or `false`.\n\nTo set parameters of the MFCC extraction, use `setExtractorParameters`:\n\n`setExtractorParameters(aFE,\"mfcc\",Name=Value)`\nSettable parameters for the MFCC extraction are:\n\n\u2022 `NumCoeffs` \u2013\u2013 Number of coefficients returned for each window, specified as a positive integer. If unspecified, `NumCoeffs` defaults to `13`.\n\n\u2022 `DeltaWindowLength` \u2013\u2013 Delta window length, specified as an odd integer greater than 2. If unspecified, `DeltaWindowLength` defaults to `9`. This parameter affects the `mfccDelta` and `mfccDeltaDelta` features.\n\n\u2022 `Rectification` \u2013\u2013 Type of nonlinear rectification, specified as `\"log\"` or `\"cubic-root\"`.\n\nThe mel-frequency cepstral coefficients are calculated using the melSpectrum.\n\nData Types: `logical`\n\nExtract delta of MFCC, specified as `true` or `false`.\n\nThe delta MFCC is calculated based on the extracted MFCC. Parameters set on `mfcc` affect `mfccDelta`.\n\nData Types: `logical`\n\nExtract delta-delta of MFCC, specified as `true` or `false`.\n\nThe delta-delta MFCC is calculated based on the extracted MFCC. Parameters set on `mfcc` affect `mfccDeltaDelta`.\n\nData Types: `logical`\n\nExtract gammatone cepstral coefficients (GTCC), specified as `true` or `false`.\n\nTo set parameters of the GTCC extraction, use `setExtractorParameters`:\n\n`setExtractorParameters(aFE,\"gtcc\",Name=Value)`\nSettable parameters for the GTCC extraction are:\n\n\u2022 `NumCoeffs` \u2013\u2013 Number of coefficients returned for each window, specified as a positive integer. If unspecified, `NumCoeffs` defaults to `13`.\n\n\u2022 `DeltaWindowLength` \u2013\u2013 Delta window length, specified as an odd integer greater than 2. If unspecified, `DeltaWindowLength` defaults to `9`. This parameter affects the `gtccDelta` and `gtccDeltaDelta` features.\n\n\u2022 `Rectification` \u2013\u2013 Type of nonlinear rectification, specified as `\"log\"` or `\"cubic-root\"`.\n\nThe gammatone cepstral coefficients are calculated using the erbSpectrum.\n\nData Types: `logical`\n\nExtract delta of GTCC, specified as `true` or `false`.\n\nThe delta GTCC is calculated based on the extracted GTCC. Parameters set on `gtcc` affect `gtccDelta`.\n\nData Types: `logical`\n\nExtract delta-delta of GTCC, specified as `true` or `false`.\n\nThe delta-delta GTCC is calculated based on the extracted GTCC. Parameters set on `gtcc` affect `gtccDeltaDelta`.\n\nData Types: `logical`\n\nExtract spectral centroid, specified as `true` or `false`.\n\nThe spectral centroid is calculated on one of the following spectral representations, as specified by the SpectralDescriptorInput property:\n\nData Types: `logical`\n\nExtract spectral crest, specified as `true` or `false`.\n\nThe spectral crest is calculated on one of the following spectral representations, as specified by the SpectralDescriptorInput property:\n\nData Types: `logical`\n\nExtract spectral decrease, specified as `true` or `false`.\n\nThe spectral decrease is calculated on one of the following spectral representations, as specified by the SpectralDescriptorInput property:\n\nData Types: `logical`\n\nExtract spectral entropy, specified as `true` or `false`.\n\nThe spectral entropy is calculated on one of the following spectral representations, as specified by the SpectralDescriptorInput property:\n\nData Types: `logical`\n\nExtract spectral flatness, specified as `true` or `false`.\n\nThe spectral flatness is calculated on one of the following spectral representations, as specified by the SpectralDescriptorInput property:\n\nData Types: `logical`\n\nExtract spectral flux, specified as `true` or `false`.\n\nThe spectral flux is calculated on one of the following spectral representations, as specified by the SpectralDescriptorInput property:\n\nTo set parameters of the spectral flux extraction, use `setExtractorParameters`:\n\n`setExtractorParameters(aFE,\"spectralFlux\",Name=Value)`\nSettable parameters for the spectral flux extraction are:\n\n\u2022 `NormType` \u2013\u2013 Norm type used to calculate the spectral flux, specified as `1` or `2`. If unspecified, `NormType` defaults to `2`.\n\nData Types: `logical`\n\nExtract spectral kurtosis, specified as `true` or `false`.\n\nThe spectral kurtosis is calculated on one of the following spectral representations, as specified by the SpectralDescriptorInput property:\n\nData Types: `logical`\n\nExtract spectral rolloff point, specified as `true` or `false`.\n\nThe spectral rolloff point is calculated on one of the following spectral representations, as specified by the SpectralDescriptorInput property:\n\nTo set parameters of the spectral rolloff point extraction, use `setExtractorParameters`:\n\n`setExtractorParameters(aFE,\"spectralRolloffPoint\",Name=Value)`\nSettable parameters for the spectral flux extraction are:\n\n\u2022 `Threshold` \u2013\u2013 Threshold of the rolloff point, specified as a scalar in the range (0, 1). If unspecified, `Threshold` defaults to `0.95`.\n\nData Types: `logical`\n\nExtract spectral skewness, specified as `true` or `false`.\n\nThe spectral skewness is calculated on one of the following spectral representations, as specified by the SpectralDescriptorInput property:\n\nData Types: `logical`\n\nExtract spectral slope, specified as `true` or `false`.\n\nThe spectral slope is calculated on one of the following spectral representations, as specified by the SpectralDescriptorInput property:\n\nData Types: `logical`\n\nExtract spectral spread, specified as `true` or `false`.\n\nThe spectral spread is calculated on one of the following spectral representations, as specified by the SpectralDescriptorInput property:\n\nData Types: `logical`\n\nExtract pitch, specified as `true` or `false`.\n\nTo set parameters of the pitch extraction, use `setExtractorParameters`:\n\n`setExtractorParameters(aFE,\"pitch\",Name=Value)`\nSettable parameters for the pitch extraction are:\n\n\u2022 `Method` \u2013\u2013 Method used to calculate the pitch, specified as `\"PEF\"`, `\"NCF\"`, `\"CEP\"`, `\"LHS\"`, or `\"SRH\"`. If unspecified, `Method` defaults to `\"NCF\"`. For a description of available pitch extraction methods, see `pitch`.\n\n\u2022 `Range` \u2013\u2013 Range within to search for the pitch in Hz, specified as a two-element row vector of increasing values. If unspecified, `Range` defaults to `[50,400]`.\n\n\u2022 `MedianFilterLength` \u2013\u2013 Median filter length used to smooth pitch estimates over time, specified as a positive integer. If unspecified, `MedianFilterLength` defaults to `1` (no median filtering).\n\nData Types: `logical`\n\nExtract harmonic ratio, specified as `true` or `false`.\n\nData Types: `logical`\n\nExtract zero-crossing rate, specified as `true` or `false`.\n\nTo set parameters of the zero-crossing rate extraction, use `setExtractorParameters`:\n\n`setExtractorParameters(aFE,\"zerocrossrate\",Name=Value)`\nSettable parameters for the zero-crossing rate extraction are:\n\n\u2022 `Method` \u2013\u2013 Method for computing the zero-crossing rate, specified as `\"difference\"` or `\"comparison\"`. If unspecified, `Method`, defaults to `\"difference\"`. For more information, see `zerocrossrate`.\n\n\u2022 `Level` \u2013\u2013 Signal level for which the crossing rate is computed, specified as a real scalar. `audioFeatureExtractor` subtracts the `Level` value from the signal and then finds the zero crossings. If unspecified, `Level` defaults to `0`.\n\n\u2022 `Threshold` \u2013\u2013 Threshold above and below the `Level` value over which the crossing rate is computed, specified as a real scalar. `audioFeatureExtractor` sets all the values of the input in the range ```[\u2013Threshold, Threshold]``` to `0` and then finds the zero crossings. If unspecified, `Threshold` defaults to `0`.\n\n\u2022 `TransitionEdge` \u2014 Transitions to include when counting zero crossings, specified as `\"falling\"`, `\"rising\"`, or `\"both\"`. If you specify `\"falling\"`, only negative-going transitions are counted. If you specify `\"rising\"`, only positive-going transitions are counted. If unspecified, `TransitionEdge` defaults to `\"both\"`.\n\n\u2022 `ZeroPositive` \u2014 Sign convention, specified as a logical scalar. If you specify `ZeroPositive` as `true`, then `0` is considered positive. If you specify `ZeroPositive` as `false`, then `audioFeatureExtractor` considers `0`, `\u20131`, and `+1` to have distinct signs following the convention of the `sign` function. If unspecified, `ZeroPositive` defaults to `false`.\n\nData Types: `logical`\n\nExtract short-time energy, specified as `true` or `false`. The short-time energy is computed using\n\n`sTE = sum(xbw.^2,1)`,\n\nwhere `xbw` is the buffered and windowed signal.\n\nExample: Chirp Function\n\nGenerate a chirp sampled at 1 kHz for 3 seconds. The instantaneous frequency is 100 Hz at $\\mathit{t}=0$ and crosses 200 Hz at $\\mathit{t}=1$ second. Divide the signal into 103-sample segments with 43 samples of overlap between adjoining segments. Window each segment with a periodic Hamming window.\n\n```fs = 1e3; x = chirp(0:1\/fs:3,100,1,200)'; win = hamming(103,\"periodic\"); nover = 43; [xb,~] = buffer(x,length(win),nover,\"nodelay\"); xbw = xb.*win;```\n\nCompute the short-time energy using the definition.\n\n`Edef = sum(xbw.^2,1)';`\n\nUse `audioFeatureExtractor` to compute the short-time energy.\n\n```EaFE = extract(audioFeatureExtractor(shortTimeEnergy=true, ... SampleRate=fs,Window=win,OverlapLength=nover),x);```\n\nVerify that both procedures give the same short-time energy.\n\n`dff = max(abs(EaFE-Edef))`\n```dff = 0 ```\n\nData Types: `logical`\n\nObject Functions\n\n `extract` Extract audio features `setExtractorParameters` Set nondefault parameter values for individual feature extractors `info` Output mapping and individual feature extractor parameters `generateMATLABFunction` Create MATLAB function compatible with C\/C++ code generation\n\nExamples\n\ncollapse all\n\n`[audioIn,fs] = audioread(\"Counting-16-44p1-mono-15secs.wav\");`\n\nCreate an `audioFeatureExtractor` object that extracts the MFCC, delta MFCC, delta-delta MFCC, pitch, spectral centroid, zero-crossing rate, and short-time energy of the signal. Use a 30 ms analysis window with 20 ms overlap.\n\n```aFE = audioFeatureExtractor( ... SampleRate=fs, ... Window=hamming(round(0.03*fs),\"periodic\"), ... OverlapLength=round(0.02*fs), ... mfcc=true, ... mfccDelta=true, ... mfccDeltaDelta=true, ... pitch=true, ... spectralCentroid=true, ... zerocrossrate=true, ... shortTimeEnergy=true);```\n\nCall `extract` to extract the audio features from the audio signal.\n\n`features = extract(aFE,audioIn);`\n\nUse `info` to determine which column of the feature extraction matrix corresponds to the requested pitch extraction.\n\n`idx = info(aFE)`\n```idx = struct with fields: mfcc: [1 2 3 4 5 6 7 8 9 10 11 12 13] mfccDelta: [14 15 16 17 18 19 20 21 22 23 24 25 26] mfccDeltaDelta: [27 28 29 30 31 32 33 34 35 36 37 38 39] spectralCentroid: 40 pitch: 41 zerocrossrate: 42 shortTimeEnergy: 43 ```\n\nPlot the detected pitch over time.\n\n```t = linspace(0,size(audioIn,1)\/fs,size(features,1)); plot(t,features(:,idx.pitch)) title(\"Pitch\") xlabel(\"Time (s)\") ylabel(\"Frequency (Hz)\")```\n\nPlot the zero-crossing rate over time.\n\n```plot(t,features(:,idx.zerocrossrate)) title(\"Zero-Crossing Rate\") xlabel(\"Time (s)\")```\n\nPlot the short-time energy over time.\n\n```plot(t,features(:,idx.shortTimeEnergy)) title(\"Short-Time Energy\") xlabel(\"Time (s)\")```\n\nCreate an audio datastore that points to audio samples included with Audio Toolbox\u00ae.\n\n```folder = fullfile(matlabroot,\"toolbox\",\"audio\",\"samples\"); ads = audioDatastore(folder);```\n\nFind all files that correspond to a sample rate of 44.1 kHz and then `subset` the datastore.\n\n```keepFile = cellfun(@(x)contains(x,\"44p1\"),ads.Files); ads = subset(ads,keepFile);```\n\nConvert the data to a `tall` array. `tall` arrays are evaluated only when you request them explicitly using `gather`. MATLAB\u00ae automatically optimizes the queued calculations by minimizing the number of passes through the data. If you have Parallel Computing Toolbox\u2122, you can spread the calculations across multiple workers. The audio data is represented as an M-by-1 tall cell array, where M is the number of files in the audio datastore.\n\n`adsTall = tall(ads)`\n```Starting parallel pool (parpool) using the 'local' profile ... Connected to the parallel pool (number of workers: 6). adsTall = M\u00d71 tall cell array { 539648\u00d71 double} { 227497\u00d71 double} { 8000\u00d71 double} { 685056\u00d71 double} { 882688\u00d72 double} {1115760\u00d72 double} { 505200\u00d72 double} {3195904\u00d72 double} : : : : ```\n\nCreate an `audioFeatureExtractor` object to extract the mel spectrum, Bark spectrum, ERB spectrum, and linear spectrum from each audio file. Use the default analysis window and overlap length for the spectrum extraction.\n\n```aFE = audioFeatureExtractor(SampleRate=44.1e3, ... melSpectrum=true, ... barkSpectrum=true, ... erbSpectrum=true, ... linearSpectrum=true);```\n\nDefine a `cellfun` function so that audio features are extracted from each cell of the tall array. Call `gather` to evaluate the tall array.\n\n```specsTall = cellfun(@(x)extract(aFE,x),adsTall,UniformOutput=false); specs = gather(specsTall);```\n```Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 14 sec Evaluation completed in 14 sec ```\n\nThe `specs` variable returned from gather is a numFiles-by-1 cell array, where numFiles is the number of files in the datastore. Each element of the cell array is a numHops-by-numFeatures-by-numChannels array, where the number of hops and number of channels depends on the length and number of channels of the audio file, and the number of features is the requested number of features from the audio data.\n\n`numFiles = numel(specs)`\n```numFiles = 12 ```\n`[numHops1,numFeaturesFile1,numChanelsFile1] = size(specs{1})`\n```numHops1 = 1053 ```\n```numFeaturesFile1 = 620 ```\n```numChanelsFile1 = 1 ```\n`[numHops2,numFeaturesFile2,numChanelsFile2] = size(specs{2})`\n```numHops2 = 443 ```\n```numFeaturesFile2 = 620 ```\n```numChanelsFile2 = 1 ```\n\nAlgorithms\n\nThe `audioFeatureExtractor` creates a feature extraction pipeline based on your selected features. To reduce computations, `audioFeatureExtractor` reuses intermediary representations and outputs some intermediate representations as features.\n\nFor example, to create an object that extracts the centroid of the Bark spectrum, the flux of the Bark spectrum, the pitch, the harmonic ratio, and the delta-delta of the MFCC, specify the `audioFeatureExtractor` as follows.\n\n```aFE = audioFeatureExtractor( ... SpectralDescriptorInput=\"barkSpectrum\", ... spectralCentroid=true, ... spectralFlux=true, ... pitch=true, ... harmonicRatio=true, ... mfccDeltaDelta=true)```\n```aFE = audioFeatureExtractor with properties: Properties Window: [1024\u00d71 double] OverlapLength: 512 SampleRate: 44100 FFTLength: [] SpectralDescriptorInput: 'barkSpectrum' Enabled Features mfccDeltaDelta, spectralCentroid, spectralFlux, pitch, harmonicRatio Disabled Features linearSpectrum, melSpectrum, barkSpectrum, erbSpectrum, mfcc, mfccDelta gtcc, gtccDelta, gtccDeltaDelta, spectralCrest, spectralDecrease, spectralEntropy spectralFlatness, spectralKurtosis, spectralRolloffPoint, spectralSkewness, spectralSlope, spectralSpread To extract a feature, set the corresponding property to true. For example, obj.mfcc = true, adds mfcc to the list of enabled features.```\nThis configuration corresponds to the highlighted feature extraction pipeline.\n\nNote\n\nBecause `audioFeatureExtractor` reuses intermediary representations, the features output from `audioFeatureExtractor` might not correspond with the default configuration of features output by corresponding individual feature extractors.\n\nVersion History\n\nIntroduced in R2019b\n\nexpand all\n\nBehavior changed in R2020b","date":"2022-05-20 11:28:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 2, \"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\": 0.8522218465805054, \"perplexity\": 5137.9071852995}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662531779.10\/warc\/CC-MAIN-20220520093441-20220520123441-00285.warc.gz\"}"} | null | null |
Формали́н — водный раствор формальдегида (метаналь), стабилизированный метанолом.
Основные данные
Наиболее распространённой стала форма, содержащая 40 % формальдегида, 8 % метилового спирта и 52 % воды. Источник формальдегида, дезинфицирующее и дезодорирующее средство, жидкость для сохранения анатомических препаратов и дубления кож. Ирритант, токсичен.
Формалин технический марка ФМ ГОСТ 1625-2016 «Формалин технический. Технические условия» — водометанольный раствор формальдегида — бесцветная прозрачная жидкость. При хранении допускается образование мути или белого осадка, растворимого при температуре не выше 40° С. Такой раствор содержит 37 % формальдегида по массе и объему имеет плотность около 1,1г/см³, зависящую от количества стабилизирующего метанола.
Используется в производстве: синтетических смол, синтетического каучука, поверхностно-активных веществ, многоатомных спиртов, формалей и других метиленовых производных.
Широкое применение находит: в бумажной промышленности для улучшения прочности и качества бумаги; в кожевенной — для дубления кожи; в текстильной — для повышения сопротивляемости изделий к смятию и усадке; в сельском хозяйстве — для обработки семян и корнеплодов, дезинфекции почвы и животноводческих помещений; в медицине — в качестве дезинфицирующего средства.
Упаковка — железнодорожные и автоцистерны с алюминиевыми или нержавеющими котлами, полиэтиленовые бутыли, бидоны; алюминиевые, нержавеющие или стальные с антикоррозионным покрытием бочки вместимостью до 200 дм³.
Транспортируют железнодорожным или автомобильным транспортом в крытых транспортных средствах в соответствии с правилами перевозки грузов, действующими на данном виде транспорта.
Технический формалин хранят в обогреваемых емкостях, изготовленных из материалов, обеспечивающих сохранение качества продукта при температуре 10-25° С. В упаковке изготовителя — в отапливаемых складских помещениях при температуре 10-25° С.
Гарантийный срок хранения — 3 месяца со дня изготовления.
Применение
Формалин свёртывает белки и предотвращает их разложение. Поэтому он применяется для дубления желатина при производстве кинофотоплёнки, для сохранения анатомических и зоологических влажных препаратов, используется при бальзамировании, как фиксатор в микроскопии, а также как антисептик. Не способствует разрушению латекса, потому зачастую применяется в качестве лубриканта (смазки) в секс-индустрии. Широко применяется для инактивации бактерий и вирусов при производстве инактивированных вакцин.
Используется в патологоанатомических отделениях для дезинфекции трупов, также для обработки одежды для захоронения и гробов, позволяет защитить тело умершего и препараты для патогистологических исследований от гниения.
Используется для производства фенолформальдегидных олигомеров.
См. также
Формальдегид
Метандиол
Карбамидоформальдегидный концентрат
Примечания
Химические смеси
Альдегиды | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,344 |
WELCOME TO THE BIBB COUNTY FAMILY - GO CHOCTAWS!!!
BIBB COUNTY YOUTH FOOTBALL
Helmets for Christ
"Helmets for Christ is a non-profit organization founded by Coach Steve Bowman who played on the 1964 & 1965 National Championship teams at the University of Alabama. He led the SEC in rushing in 1964 & 1965. The organization seeks to provide resources for helmets for youth and provide Bibles for them as well."
-Emliy Bowman
Volunteer Profile
Store Uniform
Background Cons...
Bibb County Youth Football League
Centreville, Alabama 35042
Copyright © 2023 Bibb County Youth Football | Privacy Statement | Terms Of Use | License Agreement | Children's Privacy Policy Login | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,769 |
\section*{Abstract}
Decisions in the cell that lead to its ultimate fate are important for cellular
functions such as proliferation, growth, differentiation, development and death.
Understanding this decision process is imperative for advancements in the
treatment of diseases such as cancer. It is clear that underlying gene regulatory
networks and surrounding environments of the cells are crucial for function. The
self-repressor is a very abundant gene regulatory motif, and is often believed to
have only one cell fate. In this study, we elucidate the effects of
microenvironments mimicking the epigenetic effects on cell fates through the
introduction of inducers capable of binding to a self-repressing gene product
(protein), thus regulating the associated gene. This alters the effective
regulatory binding speed of the self-repressor regulatory protein to its
destination DNA without changing the gene itself. The steady state observations
and real time monitoring of the self-repressor expression dynamics reveal the
emergence of the two cell fates, The simulations are consistent with the
experimental findings. We provide physical and quantitative explanations for the
origin of the two phenotypic cell fates. We find that two cell fates, rather than
a single fate, and their associated switching dynamics emerge from a change in
effective gene regulation strengths. The switching time scale is quantified. Our
results reveal a new mechanism for the emergence of multiple cell fates. This
provides an origin for the heterogeneity often observed among cell states, while
illustrating the influence of microenvironments on cell fates and their
decision-making processes without genetic changes.
Keywords: gene expression $\vert{}$ self-repressor $\vert{}$ biomodality
$\vert{}$ cell fate decision making
\section*{Significance}
It is often believed that genotypes determine phenotypes. Many studies have
focused on genetic mutations rather than environmental changes or epigenetics.
Here, we design a simple self-repressing gene circuit in \textit{Escherichia
coli}. We elucidate the effects of microenvironments or epigenetics on gene
expressions through the introduction of inducers capable of binding to the
self-repressor regulatory protein. This slows down the effective binding
(regulation strength) of the regulatory protein to DNA. Despite the long-held
belief that only one cell fate is present for the self-repressor, we observe that
at some induction conditions, cells show two expression states, indicating two
cell fates. Real-time monitoring of self-repressor expression during cell growth
reveals the switching dynamics for cell fate decision-making between these two
populations.
\section*{Introduction}
Uncovering the origin of the phenotypes or fates of the cell and their
associated switching is important for the full understanding of cell functions
such as proliferation, growth, differentiation, development, and death. This
remains a challenging issue in biology. It is clear that the underlying gene
regulatory networks are crucial in determining the function of the cell (1-6),
and it is often believed that the genotype determines the phenotype (7-11).
Recently, some studies have indicated that microenvironments or epigenetics can
also alter the fates of the cell or its phenotypes even with the same genotypes
(12-18). In other words, there is a possibility that apart from mutating the
genes or the nodes themselves in the gene circuit, changing the underlying gene
regulatory wirings among the genes or nodes in the regulatory network can alter
the cell phenotypes or fates. In this study, we aim to study how altering gene
regulation determines cell fates.
Negative auto-regulation is abundant: it is found in nearly 50\% of
the feedback loops in gene regulatory networks. It is widely believed that
negative auto-regulation leads to a reduction of the gene expression noise, an
increase of gene response times, an induction of possible oscillatory gene
expression, and an improvement of the stability~of proteins produced by the
underlying gene networks (19-26). Despite these novel findings, most experimental
studies have been focused on the influences of the genetic structures themselves,
rather than the environmental or the epigenetic effects on the self-repressor.
For a self-repressing system, the expression distribution is commonly more
concentrated and well-distributed (27). Many previous investigations have reached
similar conclusions, observing only one cell fate (25, 28-30). However, these
experiments were performed mostly in simple organisms such as bacteria, for which
it is often assumed that the speed of regulatory protein binding to the
corresponding DNA for switching is significantly faster than the synthesis and
degradation of the corresponding regulatory proteins. In fact, in most organisms,
cell complexes such as the nuclei inside mammalian cells may give rise to
effectively slower processes of the underlying gene regulatory binding, due to
environmental complexities such as epigenetic effects through histone
modification or DNA methylation. That is, the effective rates of
binding/unbinding of the regulatory proteins to the DNA can be comparable to, or
even slower than, the production and degradation rate of the regulatory proteins
(31){\footnotesize}. Modeling studies (32, 33) indicate that, in this case, the
protein expressions of a negative feedback loop may not always show a simple
single steady state, but instead can show two steady states, resulting in two
different cell fates. Since the auto-regulation circuit involves only a single
gene, it is the simplest gene regulation in \textit{vivo}. We will show
experimentally that this simple gene auto-regulation circuit can lead to
different cell fates or phenotypes under specific conditions, rather than that of
only one cell fate as is commonly expected.
\section*{ Results}
\subsection*{Self-Repressing Gene Circuit and Non-Regulatory Gene Circuit}
In this study, we have designed and constructed a purely negative
auto-regulation feedback loop circuit (self-repressing gene circuit) in
\textit{Escherichia coli }(\textit{E}. \textit{coli}). The Ptet promoter
including two\textit{ tetO} operons controls the production of its repressor,
\textit{TetR}. Meanwhile, the \textit{TetR} was fused with a fluorescence protein
(Venus) for experimental measurements of the \textit{TetR} expressions. The
inducer, aTc (anhydrotetracycline), was introduced to mimic environmental
influences on expressions of the self-repression system. In the presence of an
inducer, the repressor \textit{TetR} can change its conformation and dissociate
from specific binding sequences of the DNA (T\textit{etO}). This allows for the
transcription of \textit{TetR}-Venus (Figure 1A). In order to avoid fluctuations
in copy numbers of the plasmids, the constructed circuit in the plasmid was
integrated into the chromosome of E. coli. We also constructed a series of
self-repressing circuits with different affinities to the \textit{TetR} protein
(MG::PR-WT, MG::PR-1G) (Figure S2). We chose MG::PR-8T as the main circuit of
this study for its stability and bimodal behavior. To compare this with our
self-repressing circuit construction MG::PR-8T, we designed a non-regulatory
circuit as a control group: the MG::PR-8T-P39K circuit (Fig. 1B).
\subsection*{The Expression Distributions of the Self-Repressor Gene Circuit under
Microscopy}
To obtain the expressions of \textit{TetR} under different induction conditions,
we measured the average fluorescence signals of the reporter protein Venus for
the strain of MG::PR-8T at different inducer concentrations (300 ng/mL-1500
ng/mL) across cell populations using a wide-field fluorescence microscope. Cells
were collected and measured after being cultured in M9 medium and induced by aTc
for 4\textasciitilde{}6 hours to a logarithmic phase. To ensure accuracy of the
expression distribution, we collected no less than 10$^{3}$ cells to measure for
each sample. All expression distributions under different induction
concentrations are shown in Figure 2. The results indicate that \textit{TetR}
expression distributions vary with inducer (aTc) concentrations. Under low
inducer concentrations, the expression levels of the negative regulated gene
circuit were quite low, and this gene can be considered to be in the ``off''
state for a long time. With increased inducer concentrations, the expression
levels were significantly enhanced (Fig. 2A). From the results shown in the
microscope, we can clearly see that when inducers are added to the system, the
repressor\textit{ TetR} can no longer prevent the transcription of \textit{TetR}.
When the inducer concentrations are high enough (such as 1400ng/mL and
1500ng/mL), the steady state expression distribution can become bimodal, with two
states of low and high expression levels. Meanwhile, the percentage of the cells
in the low expression state gradually increases with the increase of the inducer
concentrations (Fig. 2A). Under high inducer concentrations, the coexistence of
both phenotypes characterized by the bimodal steady state distributions of the
fluorescence intensities can be clearly seen (Fig. 2E). When we further compare
the images in Fig. 2D and 2E, it is clear to see that one section of the cells in
Fig. 2E is brighter, while other sections were dimmer, compared to most of the
cells in Fig. 2D. As can be seen from the microscopy images, the morphologies of
the bacteria cells are not influenced by the aTc inducers at a concentration
level of 1500 ng/mL (Fig. 2E). The corresponding distributions of those images
are given in Fig.2A. In our control experiments, similar behaviors are not found
in the MG::PR-8T-P39K non-self-repressing gene circuit under the same conditions
(Figure S7). This indicates that the two expression states of \textit{TetR} were
due to the self-repressing circuit, rather than other factors such as the
influences of the inducers on the cells.
\subsection*{Fano Factor and Inhibition Curve }
To further understand our experimental observations, we need to quantify the
degrees of fluctuations. This can be measured by the Fano factor quantified as
the variance of the observable divided by the mean value (34). The Fano factor is
equal to one (\textit{F} = 1) if the distribution of the observable is exactly
Poisson. A large Fano factor implies significant statistical fluctuations
deviating from Poisson (Figure 3A). Qualitatively, the Poisson distribution
should be a good approximation for the individual ``on'' and ``off'' states when
the observed distribution of fluorescence intensity is bimodal, because each gene
state can produce proteins almost independently of gene switching. However, the
overall Fano factor for the combined probability distribution of ``on'' and
``off'' states is much larger than 1. This is because the system is close to a
two peak (Non-Poisson) distribution with different means summed together,
producing large statistical fluctuations deviating from the single Poisson
distribution. This indicates that two Poisson processes added together will not
lead to a Poisson distribution. The analysis of the coefficient of variation (CV)
in Figure S6 also illustrates this same conclusion.
Furthermore, we investigated the inhibition curve, which describes the
proportion of the bacteria with a fluorescence intensity lower than a certain
value (Figure 3B). We can see that the proportion of the gene in its inhibited
state first decreases at low concentrations of inducer (up to aTc concentration
at 1200 ng/mL) and then increases as the inducer concentration becomes higher.
More inducers introduce more interactions with the \textit{TetR} molecules. This
slows down the effective binding of the \textit{TetR} to the DNA. Therefore, the
gene has more times to be in its ``on'' state and less of a chance of being at
the inhibition state (less inhibition capability from aTc 300 ng/mL to 1200
ng/mL). More \textit{TetR} proteins will be synthesized as a result. At certain
concentrations of inducer aTc (1200 ng/mL), the number of free \textit{TetR}
molecules synthesized from the gene's ``on'' state increases, resulting in a
comparable number of \textit{TetR} molecules to aTc molecules. This will lead to
more effective regulatory binding of \textit{TetR} to DNA. Finally, there are
more chances of the gene being in its ``off'' state. We suggest that the
increasing number of the proteins produced as a result of the presence of more
inducers at this concentration range of aTc (1200-1500 ng/mL) will eventually
promote the probability of inhibition for gene switches, since more regulatory
proteins are synthesized and available for inhibition when the inducer
concentration becomes higher. At this condition, although the total protein
expression is higher with the increase of inducer concentration, it is not high
enough that the proportion of the bacteria in the inhibited state increases due
to self-repressing regulation, leading to stronger effective inhibition. At
extremely high aTc concentrations (beyond 1900 ng/mL), one expects that the
number of the available regulatory molecules becomes far beyond the one needed
for inhibition and high expression peak should dominate. However, the aTc the
toxicity from aTc as antibacterial agent to the cells becomes effective. It is
therefore not feasible to observe the healthy cell expression distribution at
this extremely high concentration of aTc.
\subsection*{The Dynamics of \textit{TetR} Expression in Real Time}
We have seen that the self-repressing circuit can give a bimodal distribution.
In order to further explore the underlying mechanism of this behavior, we
monitored the dynamics of \textit{TetR} expression in real time. We tracked cells
during their growth and division on a microscope with a FCS2 (Focht Chamber
System 2, Bioptechs) system which provides aTc continuously to guarantee the
cells growing in the right environments (continuous flow of adequate nutrients
from fresh medium (M9) through the cells on agarose pad) and avoids potential
issue of heterogeneity of the environments. As shown in Fig. 4B, upon aTc
induction, two types of cell responses were observed: the fluorescence intensity
either changed significantly or almost remained the same. When we track cells in
real time, we can see that, some cells switch between bright and dim, while other
cells stay with similar brightness (Fig. 4B). The resulting fluorescence
distribution is thus bimodal and a fluorescence threshold can be defined for each
cell in its most probable induction state. The use of a microfluidic device,
coupled with cell tracking and fluorescence measurements, allows us to generate
fluorescence trajectories for a single cell on reasonable time scales
(\textasciitilde{}300 minutes) for a single trajectory. Based on this, we
collected 28 micro-colony movies and chose 163 fluorescence trajectories. We
observed that the trajectories of a single cell fluorescence fluctuated
significantly. We collected about 8200 fluorescence intensity data points
corresponding to the selected trajectories. Several representative trajectories
with significant fluctuations were shown to demonstrate the existence of two
states (From Figure 4A-B, Figure S8, Movie S1 and Movie S2).
\subsection*{Two Cell State Identifications by Hidden Markov Chain Modeling}
In order to explore the underlying mechanism of the bimodality, we collected the
statistics of the fluorescence intensity obtained from the trajectories. The
distribution of these intensities exhibits two peaks, suggesting that most of the
initial cells are either in a high expression state or in a low expression state
in their progeny. We then used a Hidden Markov Chain Model (HMM) (35) to fit the
real time trajectories and identify the cell states, and then simulate the
distribution of the fluorescence intensity (Figure 4C). To assign protein
expression states and the rates of inter-conversion between them, we performed
data fitting using the HMM. From the HMM analysis, we obtained a correlation
coefficient of 0.975 between the measured and simulated trajectories after
identifying the cell states and quantifying their switching rates. The simulated
distribution fits with the measured distribution well. From the HMM analysis, we
further determined the center positions of the peaks to be at 2.690 and at 2.933
in logarithm of fluorescence intensity. The variances of the individual peak
distributions are at 0.085 and at 0.080, respectively. For our system, the
probability in the high expression state is around 0.401, and we can also see
that the probability in the low expression state is around 0.599.
In the high expression state, the system will continue its behavior
with a probability of 0.963 (the switching or residence time will be discussed in
the next section). There is a small chance, with the probability of 0.037, to
switch to the low expression state from the high expression state. Meanwhile,
there is additionally a probability of 0.023 that the system will switch to the
high expression state from the low expression state, instead of remaining in the
low expression state.
\subsection*{The Average Residence Times of the Protein Expression States }
To estimate the average residence times of the protein expression state, we
distinguished the states from the trajectories using HMM analysis and calculated
the residence times of each state (Figure S11). For each trajectory, we counted
the total residence times and the number of the state changes. The average
residence times were calculated as the quotient of the total residence times and
the number of states changed.
The length of the test fluorescence trajectory is finite and limited. This may
lead to some errors in estimating the transition times. We take this into account
in determining the time scale of the transitions. The average residence time of
the high expression state is estimated to be about 92\textasciitilde{}103
minutes, and that of the low expression state is estimated to be about
151\textasciitilde{}182 minutes. The average residence time can be used to
quantify the switching time between two cell fates. Therefore, the switching time
from high (low) expression to low (high) expression can be estimated to be about
92\textasciitilde{}103 (151\textasciitilde{}182) minutes. Through fluctuations,
the bimodal distribution can be maintained in a dynamic balance between the high
expression ``on'' state and the low expression ``off'' state. When the inducer
concentration is fixed, the increasing number of proteins will promote the
inhibition probability of gene switching. Therefore, the cells in the high
expression state will have a tendency to migrate to the low expression state.
Conversely, the cells with low expressions will be more likely to move towards
the ``on'' state. Therefore, the cells in the low expression state will also have
a tendency to migrate to the high expression state.
\subsection*{Physical Origin of the Two Cell Fates }
Intuitively, from a molecular perspective, we know that the transcription
process is suppressed when the promoter site of the DNA is occupied by a
repressor (the gene is ``off''), and enhanced when the repressor is dissociated
from DNA (the gene is ``on''). When the inducer concentration is low, increasing
the inducer concentration will increase the binding of aTc to \textit{TetR} and
slow down the effective binding of \textit{TetR} to the promoter. This lessens
the chance of the genes being in an ``off state'' and conversely increases the
possibilities of the gene being at the ``on state'', resulting in higher
expressions. This explains the shift of the expression peak from low to high as
inducer concentration increases. When the inducer concentration further increases
to sufficiently high values, the chance of having free \textit{TetR} molecules
will be higher (comparable number of \textit{TetR} molecules to that of aTc
molecules) as a result of synthesis. More \textit{TetR} molecules will have
increased chances of binding to the promoter site and will therefore display more
repressive activity. This will lead to the emergence of the low expression peak
and therefore bimodal distribution of the copy number in mRNA and proteins.
Further increases of the inducer concentrations will lead to more free
\textit{TetR} molecules, with a resulting greater weighting of low, rather than
high, expression peaks. This explains the trend of expression peaks as seen in
Fig. 2A.
When the effective binding/unbinding is much faster compared to the
synthesis/degradation, the gene state changes rapidly. The interactions and the
mixings become stronger between the two gene states, and therefore also between
the two corresponding protein concentration peaks. For the self-repressor,
decreasing the effective binding (increasing the inducer concentrations in our
study) promotes the generation of more proteins which in turn shows greater
repressive activity. This leads to the high concentration peak moving towards a
lower concentration. On the other hand, increasing the binding (decreasing the
inducer concentrations in our study) represses the generation of the proteins,
and so fewer proteins produced bind effectively to DNA. This in turn promotes
production of \textit{TetR} molecules. It leads to the low concentration peak
moving outward towards a higher concentration. As a result, the two peaks from
the non-adiabatic limit (e.g. high aTc concentrations at 1400 ng/mL, slower
binding) meet in the adiabatic limit (e.g. lower aTc concentration at 1300 ng/mL,
faster binding) of the fast binding and merge into a single peak.
Gene switching is often rapid in bacterial cells. However, slow gene switching
controlled by regulatory proteins binding/unbinding to the promoters can also be
significant for gene expression dynamics. In eukaryotic cells and some
prokaryotic cells, binding/unbinding may be comparable to or even slower than the
corresponding synthesis and degradation due to epigenetic effects or complex
microenvironments. By studying how the introduction of the inducers effectively
weakens gene regulation in bacteria, we mimicked gene regulation dynamics in more
complex eukaryotic cells. Through increasing inducer concentrations, we achieved
effectively slower regulatory binding relative to synthesis and degradation. In
other words, the introduction of the inducers in the bacteria leads to an
additional time scale for regulatory binding. This mimicked the additional time
scales for regulatory binding from including the histone modifications and DNA
methylations in eukaryotic cells. This slower regulatory binding to inducers will
lead to prolonged times of genes being in the ``on state'' in addition to the
time spent in the ``off state'', originated from the fast binding without
inducers. As a result, both ``on'' and ``off'' states of genes may emerge. This
is the physical mechanism of bimodality. In other words, the fast regulatory
binding mimicked stronger interactions while the slow regulatory binding mimicked
the weaker interactions among genes. While stronger interactions give more
constraints to the system and therefore fewer degrees of freedom for the
expressions (single peak expression), the weaker interactions will constrain the
system less and therefore result in more degrees of freedom for the expressions
(e.g., double peak expressions). Through the steady state and the real time
observations of the dynamics of the self-repressor in the experiments, we
observed the robust emergence of the bimodal gene expression distribution for the
self-repressor.
\subsection*{Stochastic Simulations of Bimodality }
We further explored the stochastic dynamics of self-regulative feedback genes
through a mathematical model, which can be used to explain and simulate the
experimental observations (Figure 3C). The mathematical model clarifies the
underlying mechanism of how bimodality emerges. Under faster regulation binding,
the self-repressor is forced to stay in the repressed state. This is because once
produced, the regulatory protein immediately binds to the gene and therefore
represses protein production. In our study, slower binding of the regulatory
protein to the gene is realized through the inducer binding to the regulatory
protein, which effectively blocks the ability of the protein to bind to the
promoter. Under slower regulatory binding, the self-repressor may function in two
different ways: it may bind to the DNA for some time and repress protein
production, or unbind from the DNA for some time, leading to increased protein
production. This generates two cell phenotypes. Furthermore, due to the intrinsic
statistical fluctuations of the number of proteins, there is a possibility of
switching between the high expression and low expression state. We have observed
such phenotypic switching in real time experiments. The simulation results are
consistent with the experimental observations.
On the other hand, the trajectories in Figure 4A and Fig. S8 showed comparable
growth rates in high expression state and in low expression state. It is possible
that high expression cells in our study have not reached the threshold for
significant metabolic burden to slow down the growth. The inhibition curves of
the different inducer concentrations in Fig. 3B and the dynamic balance by
intrinsic fluctuations also imply that the bimodality of the protein expression
distribution is not due to cell growth.
\section*{ Discussion}
For self-repressor gene network, even when the gene is fixed, there can still be
new cell phenotypes. Our study shows explicitly in this concrete gene circuit
that different cell fates can emerge not only from the changes in the genes (such
as mutations) but also from the changes in regulatory wirings or links through
microenvironments without altering the gene itself. In fact, even when the
topology of the wiring for the underlying gene regulatory network is fixed, there
is still a possibility of cell phenotypic changes due to the changes in the
regulation strengths induced by the environment. Furthermore, we observed both in
real time experiments and simulations that the cell phenotypes or fates can be
switched from one to the other. We also obtained the average time of this
switching which quantifies how difficult it is to communicate globally from one
cell fate to the other. Therefore, using real time trajectories, we determine
both the speed and the underlying processes of the cell fate
decision-making/phenotypic state-switching.
Epigenetic effects are often challenging to study in eukaryotic
cells. Our study in bacteria illustrates how the environments can influence the
cell fates and cell fate decision-making in a controllable way. The experiments
in bacteria are relatively easy and straightforward to perform and control. The
epigenetic and micro-environmental effects can be mimicked through the modulation
of inducers in our study. This is an advantage of our approach. We plan to apply
our method to a variety of core regulatory motifs and modules in the gene
networks to investigate how the environments or epigenetics influence the cell
fates and the cell fate decision-making processes.
\section*{Acknowledgments}
We thank H. Bujard and B.L.Wanner for providing pZE11 and CRIM vector systems
respectively, as well as for providing detailed information on their origins and
growth conditions. We would also like to thank X. S. Xie for providing \textit{E.
coli} SX4. This work was supported by the National Science Foundation (NSF) with
grant number 0947767, the National Science Foundation of China (NSFC) with grant
number 91430217 and the Ministry of Science and Technology (MOST) of China with
grant number 2016YFA0203200.
\section*{Author contributions}
Z.L. Jiang and L. Tian, X.N. Fang, Q. Z. Dong, and J. Wang contributed to the
experimental design. Z.L. Jiang, L. Tian, X. N. Fang and Q. Z. Dong conducted the
experiments. K. Zhang, Q. Liu, Z.L. Jiang and L. Tian, X. Fang, and J. Wang
contributed to data interpretation. Z.L. Jiang, L. Tian, X.N. Fang, E. K. Wang
and J. Wang contributed to writing and revising the manuscript.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,656 |
Morpheus is a 1987 shoot 'em up developed by Graftgold for the Commodore 64 and published by Rainbird. The game's designer, Andrew Braybrook, wrote a series of articles on the game's creation for the magazine Zzap!64 over eight months.
Gameplay
Morpheus features 50 subuniverses called Aithers, each of which consists of a central Nucleus surrounded by obstacles called Orbitals. The aim of the game is for players to break through the obstacles to destroy the Nucleus, and by the end of level 50, Morpheus as a whole. Players will also encounter enemies such as Morphii and aliens which can be killed for points. Morpheus allows players to use points earned to purchase weapons, and customize various features of their ships, such as auxiliary weapons and expansion ports to allow various gameplay possibilities.
Reception
Zzap!64 magazines positively reviewed the game and awarded it a Sizzler rating with an overall score of 90%. They described it as "without a doubt one of the most finely constructed games ever written for the 64". The graphics were praised, but reviewers noted that the gameplay style significantly departed from Braybrook's previous games. Reviewers recommended that readers try the game before purchasing. Although technically accomplished, it might not appeal to everyone. Many Commodore Users were also enthusiastic about the game, saying it "reeks of quality". Their only criticism was the lack of a game save and load facility. Zzap!64 gave it a 9 out of 10 rating.
Computer and Video Games' reviews were less enthusiastic about the game, which they considered a disappointment, noting "the originality of the game is great, but the gameplay is extremely laborious." The reviewer believed the music and sound effects were the best thing about the game, saying it was "visually very good." They advised potential purchasers to try the game first. It was rated 7/10.
References
External links
Morpheus box and manual at C64Sets.com
1987 video games
Commodore 64 games
Commodore 64-only games
Graftgold games
Multiplayer and single-player video games
Shoot 'em ups
Telecomsoft games
Video games developed in the United Kingdom
sv:Morpheus (datorspel) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,686 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.