text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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<p>Contains classes for accessing and publishing data on a device. It includes three main
categories of APIs:</p>
<ul>
<dt>Content sharing ({@link android.content})</dt>
<dd>For sharing content between application components. The most important classes are:
<ul>
<li>{@link android.content.ContentProvider} and {@link android.content.ContentResolver}
for managing and publishing persistent data associated with an application.</li>
<li>{@link android.content.Intent} and {@link android.content.IntentFilter}, for delivering
structured messages between different application components—allowing components to initiate
other components and return results.</li>
</ul>
</dd>
<dt>Package management ({@link android.content.pm})</dt>
<dd>For accessing information about an Android package (an {@code .apk}), including information
about its activities, permissions, services, signatures, and providers. The most important class for
accessing this information is {@link android.content.pm.PackageManager}.</dd>
<dt>Resource management ({@link android.content.res})</dt>
<dd>For retrieving resource data associated with an application, such as strings, drawables,
media, and device configuration details. The most important class for accessing this data is {@link
android.content.res.Resources}.</dd>
</ul>
</body>
</html>
| {
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When you wake up in the morning you pray for flow of income no matter the kind of work you do.
Today there was an argument at Ikeja between an engineer and those ikeja boys who woo you to patronize them. This happened as one of those boys picked a customer and took him to the engineers shop.
The boy and the engineer both had an agreement to split the money equally since he brought him customer.
They agreed #2500 with the customer.when the engineer finished the job. The boy then came to him to tell him that the customer has changed his mind that he his not with enough cash that he his giving them #2000 the engineer was totally pissed off.
He had bought new cable before the phone could work, after much pleading from the boy.
The engineer agreed and collected the #2000.so he gave the customer his phone.
some minutes after the customer had left the boy who brough the customer came to the engineer to have his share..
The engineer told him he bought cable with the money #1000 so they are both going to share the other half equally which will be #500 each.
According to the engineer,he his giving him the #500,since he was the one who did the job and if anything should go wrong with the phone it would be on him.and it will bad for him to take loss after his effort on the phone.
While the boy who brought the customer said he knows the customer would now be patronizing him all thanks to him.
and since he knows the engineer would not call him or tell him if the customer ever comes to patronize him. So he his to receive his full share and precisely it is the engineer that would be using the cable as well.. Which means the engineer will make more money from the cable and also from the new customer he has gotten him as far as he his concerned he knows the customer would surely bring him more work another time. So he should be paid his own share now.
He also got into a hot argument with anyone who tries to interfere in the matter.
Is the engineer wrong or the boy? | {
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{"url":"https:\/\/www.gradesaver.com\/textbooks\/science\/physics\/physics-10th-edition\/chapter-2-kinematics-in-one-dimension-problems-page-50\/49","text":"## Physics (10th Edition)\n\nLet up be the positive direction: $x=ut+\\frac{1}{2}at^{2}$ $(-3.00)=(2.50)t+\\frac{1}{2}(-9.8)t^{2}$ $4.9t^{2}-2.50t-3.00=0$ Using quadratic formula: $t=\\frac{-(-2.50)\u00b1\\sqrt {((-2.50)^{2}-4(4.9)(-3.00)}}{2(4.9)}$ $t\\approx1.08$ or $t\\approx-0.57$ $t\\approx1.08\\; s$ since $t\\gt0$","date":"2018-10-20 06:06:40","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\": 0.9691134691238403, \"perplexity\": 736.0943357417411}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-43\/segments\/1539583512592.60\/warc\/CC-MAIN-20181020055317-20181020080817-00489.warc.gz\"}"} | null | null |
ERIC Number: ED334876
Record Type: RIE
Publication Date: 1989-Oct-31
Abstractor: N/A
ISSN: N/A
EISSN: N/A
Mandate for a New Century: Reshaping the Research University's Role in Social Policy. Eleventh David Dodds Henry Lecture.
Shalala, Donna E.; And Others
A speech by Chancellor Donna E. Shalala of the University of Wisconsin, Madison, is presented; it calls for a broadening university commitment to societal issues and for the university to engage in basic research on social policy issues in a fashion similar to the fundamental science and technology research currently undertaken. The lecture articulates a vision of the land-grant mission for public research universities of the 21st century and also calls for the research university, receiving funding from the general public, to work with a greater sense of public service in helping society to grow healthy and prosperous. In addition to the speech, three responses by administrators and scholars from the University of Illinois, Urbana-Champaign are provided, as well as questions from the audience, and Chancellor Shalala's responses to these questions. The formal responses are from the following personnel at the University of Illinois at Urbana-Champaign campus: Morton W. Weir, Chancellor of the university; P. David Pearson, Dean, College of Education; and Dianne M. Pinderhughes, Associate Professor of Political Science and Acting Director, Afro-American Studies and Research Program. (GLR)
Descriptors: Higher Education, Institutional Role, Policy Formation, Problem Solving, Research Needs, Research Universities, Research Utilization, Scientific Methodology, Social Problems, Social Studies, Speeches
Publication Type: Speeches/Meeting Papers; Opinion Papers
Education Level: N/A
Audience: N/A
Sponsor: N/A
Authoring Institution: Illinois Univ., Urbana.
Grant or Contract Numbers: N/A
Note: Speech given at the University of Urbana-Champaign (Urbana, IL, October 31, 1989). | {
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Skorpion war eine US-amerikanische Automobilmarke, die von 1952 bis 1954 von der Viking-Craft Manufacturing Company in Anaheim (Kalifornien) gebaut wurde.
Beschreibung
Gebaut wurde ein zweisitziger Roadster, bei dem nur die GFK-Karosserie von Viking-Craft kam. Die Kunden hatten die Wahl zwischen verschiedenen Fahrgestell-Motor-Kombinationen von Crosley, Ford und Studebaker.
Die kompletten Fahrzeuge kosteten US$ 2495,–; für komplette GFK-Karosserien waren US$ 1200,– zu zahlen und als Kits waren sie für US$ 645,– erhältlich.
Literatur
John Gunnell: Standard Catalog of American Cars 1946–1975. Krause Publications, Iola 2002, ISBN 0-87349-461-X. (englisch)
Ehemaliger Pkw-Hersteller (Kalifornien)
Unternehmen (Anaheim)
Gegründet 1952
Aufgelöst 1954 | {
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{"url":"https:\/\/tex.stackexchange.com\/questions\/355404\/creating-a-custom-beamer-theme","text":"# Creating a custom Beamer theme\n\nI'm trying to write a custom Beamer theme and I'm having trouble figuring out how to manage the title page to allow an arbitrary assignment of authors or titles.\n\nI've managed to figure out how to place the title, date, and author information in relative position, but the text must be of a certain number of lines to make everything line up properly with the display boxes (i.e. the Title in the center of the blue box and the author and date information in the center of the green box.) If the title is more or less than 2 lines, it breaks the formatting. If the authors span more than 2 lines it breaks the formatting. I ultimately want the Title to resize within a certain bounding box within the blue area and the author and date to resize within a bounding box in the green box.\n\nWhat is the best way to accomplish this?\n\nMy beamerinnerthemeUniversity.sty file contains the following:\n\n\\definecolor{UniversityGreen}{RGB}{0, 133, 66}\n\\definecolor{UniversityOrange}{RGB}{199, 91, 18}\n\\definecolor{UniversityBlue}{RGB}{0, 161, 222}\n\n\\setbeamertemplate{background}{\n\\begin{tikzpicture}\n\\useasboundingbox (0,0) rectangle(\\the\\paperwidth,\\the\\paperheight);\n\\fill[color=UniversityBlue] (0,2) rectangle (\\the\\paperwidth,\\the\\paperheight);\n\\fill[color=UniversityOrange] (0,0) rectangle(2.95,1.9);\n\\node[inner sep=0] at (1.475,0.95) {\\includegraphics[width=.25\\textwidth]{University_logo.pdf}};\n\\fill[color=UniversityGreen] (3.05,0) rectangle(\\the\\paperwidth,1.9);\n\\ifnum\\thepage>1\\relax%\n\\fill[white,opacity=1] (0,0) rectangle(\\the\\paperwidth,\\the\\paperheight);\n\\fi\n\\end{tikzpicture}\n}\n\n\\defbeamertemplate*{title page}{university}[1][]\n{\n\\vskip4cm%\n\\usebeamerfont{title}\\inserttitle\\par%\n\\end{beamercolorbox}%\n\\vskip1.75cm%\n\\begin{beamercolorbox}[wd=12cm,leftskip=3cm,#1]{author}\n\\usebeamerfont{author}\\insertauthor%\n\\end{beamercolorbox}\n\\vskip0.2cm%\n\\begin{beamercolorbox}[wd=12cm,leftskip=3cm,#1]{date}\n\\usebeamerfont{author}\\insertdate%\n\\end{beamercolorbox}\n\\vfill\n}\n\n\\setbeamertemplate{items}[square]\n\\setbeamertemplate{sections\/subsections in toc}[square]\n\n\\mode\n<all>\n\n\nHere is an example of the output when everything is lined up with a 2-line title and 2 authors with short names:\n\n\\documentclass[xcolor={table}]{beamer}\n\\usepackage[T1]{fontenc}\n\n\\usetheme{default}\n\\RequirePackage{tikz}\n\\useinnertheme{University}\n\n\\begin{document}\n\\title[A Presentation of Great Import]{A Presentation of Great Import:\\\\Important discoveries in science}\n\\author{Bob Smith \\and Sally Johnson}\n\\institute{State University}\n\n\\date{February 24, 2016}\n\n\\frame{\\titlepage}\n\\end{document}\n\n\nHere is an example of the overflow with a longer title and more authors\n\n\\documentclass[xcolor={table}]{beamer}\n\\usepackage[T1]{fontenc}\n\n\\usetheme{default}\n\\RequirePackage{tikz}\n\\useinnertheme{University}\n\n\\begin{document}\n\\title[A Presentation of Great Import]{A Presentation of Great Import:\\\\Important discoveries in science\\\\and a further digression on important things}\n\\author{Bob Smith \\and Sally Johnson \\and Uma Thirugnanasampanthan}\n\\institute{State University}\n\n\\date{February 24, 2016}\n\n\\frame{\\titlepage}\n\\end{document}\n\n\n\u2022 Maybe \\parbox[b][3ex][c]{.7\\paperwidth}{\\centering \\insertframetitle} \u2013\u00a0Bobyandbob Feb 23 '17 at 20:47\n\u2022 Can you make a compilable minimal working example (MWE)? \u2013\u00a0user36296 Feb 23 '17 at 20:52\n\u2022 I tried to make the MWE as small as possible while still compiling and illustrating the basic question. \u2013\u00a0WildGunman Feb 23 '17 at 21:24\n\nTo keep the text inside it's respective boxes, I placed the text inside minipages, which means you can simply control the alignment via \\begin{minipage}[c][.8\\textheight][c]{\\textwidth}, in this example the text is vertically centred.\n\n\\documentclass{beamer}\n\n\\usepackage{tikz}\n\n\\definecolor{UniversityGreen}{RGB}{0, 133, 66}\n\\definecolor{UniversityOrange}{RGB}{199, 91, 18}\n\\definecolor{UniversityBlue}{RGB}{0, 161, 222}\n\n\\setbeamertemplate{background}{\n\\begin{tikzpicture}\n\\useasboundingbox (0,0) rectangle(\\the\\paperwidth,\\the\\paperheight);\n\\fill[color=UniversityBlue] (0,2) rectangle (\\the\\paperwidth,\\the\\paperheight);\n\\fill[color=UniversityOrange] (0,0) rectangle(2.95,1.9);\n\\node[inner sep=0] at (1.475,0.95) {\\includegraphics[width=.2\\textwidth]{example-image}};\n\\fill[color=UniversityGreen] (3.05,0) rectangle(\\the\\paperwidth,1.9);\n\\end{tikzpicture}\n}\n\n\\defbeamertemplate*{title page}{university}[1][]{%\n\\begin{minipage}[c][.8\\textheight][c]{\\textwidth}\n\\usebeamerfont{title}\n\\inserttitle\\par\n\\end{minipage}\n\\vskip1pt\n\\hskip.2\\paperwidth\n\\begin{minipage}[c][.2\\textheight][c]{.75\\textwidth}\n\\usebeamerfont{author}\n\\raggedright\n\\insertauthor\\par\n\\insertdate\\par\n\\end{minipage}\n}\n\n\\setbeamertemplate{items}[square]\n\\setbeamertemplate{sections\/subsections in toc}[square]\n\n\\mode\n<all>\n\n\\title[A Presentation of Great Import]{A Presentation of Great Import:\\\\Important discoveries in science}\n\\author{Bob Smith \\and Sally Johnson}\n\\institute{State University}\n\n\\date{February 24, 2016}\n\n\\begin{document}\n\n\\begin{frame}[plain]\n\\titlepage\n\\end{frame}\n\n\\title[A Presentation of Great Import]{A Presentation of Great Import:\\\\Important discoveries in science\\\\and a further digression on important things}\n\\author{Bob~Smith \\and Sally~Johnson \\and Uma~Thirugnanasampanthan}\n\\institute{State University}\n\n\\begin{frame}[plain]\n\\titlepage\n\\end{frame}\n\n\\end{document}\n\n\n\u2022 This works great for properly bounding the location of the text. Thanks! I was wondering if there was any way to auto shrink the text if the text was going to be larger than the bounding box. So imagine that there was another line of authors. Could LaTeX auto-size the text to be small enough to fit into the green box? \u2013\u00a0WildGunman Feb 23 '17 at 22:30\n\u2022 @WildGunman I would not do automatic resizing (although possible, for example \\resizebox), but either replace the author list by one with initials for the first names or adjust the size with \\setbeamerfont{author}{...} \u2013\u00a0user36296 Feb 23 '17 at 22:36","date":"2019-10-15 09:28:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.5068024396896362, \"perplexity\": 2823.1234655698877}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986657949.34\/warc\/CC-MAIN-20191015082202-20191015105702-00036.warc.gz\"}"} | null | null |
Arm lift surgery is a procedure used to reduce excess skin and/or fat in the upper arms. Also known as a brachioplasty, arm lift surgery aims to improve the appearance of the upper arms through surgically contouring better proportions and making the skin smoother and tighter.
Arm lift surgery addresses the area from the upper arm under the armpit down to the elbow. During arm lift surgery, excess skin and fat are removed while the underlying supportive tissue is smoothed and tightened re-defining the shape of the arms and reducing localized fat pockets in the process.
Where is it done? Your arms lift surgery will be performed at Honeysuckle Day Hospital in Newcastle, a fully licensed and accredited state of the art day hospital which specializes in cosmetic surgery.
Length of surgery: Usually between 2-3 hrs. Maybe longer depending on type and complexity of breast enhancement surgery that is performed.
Type of anaesthesia: General Anaesthesia or Twilight Sedation coupled with Local Anaesthesia.
Costs will vary according to the complexity and amount of time required for the surgery. Dr. Verma will discuss this with you at the time of your consultation and his team will provide you with a detailed quote after your visit.
Do I qualify for arm lift surgery?
If you have excess sagging skin on your upper arms, arm lift surgery might be a suitable treatment option.
The ideal candidate for arm lift surgery is someone who has excess skin, but minimal fat. This is often the result of weight loss as skin elasticity is reduced with fat loss. The ageing process also causes a loss of skin elasticity and sagging skin on the upper arms.
As weight changes will influence the shape of your body, arm lift surgery is best undertaken when you have reached your goal weight and it has been stabilised for 3 months.
Dr. Verma will take a full medical history and examine your arms. He will discuss your concerns with you and then advise of the best treatment option or series of treatments to achieve results. During your consultation, Dr. Verma will take photographs to document your appearance pre and post surgery and record the healing process and cosmetic results.
You will have a scar on each arm running from the armpit to the elbow. Dr. Verma will limit the scar length and will work to make the scar in an inconspicuous position. Scars will fade over time.
Depending upon the amount of skin and fat to be removed, the type of anaesthesia used will be either general anaesthesia or twilight sedation. Dr. Verma will be able to advise which anaesthesia will be best suited to your needs at the time of your consultation.
What are side effects of arm lift surgery?
With any surgical procedure, there are risks involved – however, most problems that occur are usually easily treated and settle with time.
General risks of any surgery include excessive bleeding, reaction to any of the medications (including anesthesia), infection, poor healing or excessive scarring.
What are the risks and complications of arm lift surgery?
How is arm lift surgery performed? Are there different techniques?
Full arm lift. This is a traditional method where an incision is made from the upper armpit to the elbow. The fatty tissue is then contoured and the excess skin removed. The remaining skin is pulled taut and then sutured in place. Due to the larger length of the incision, this technique leaves a bigger scar but it is most suitable for patients who have larger amounts of tissue to be removed.
Limited incision arm lift. In this method, a smaller incision is made in the armpit and from there excess tissue is removed. This is better suited to patients who have a smaller amount of excess tissue and minimal sagging of the arm.
You will be provided with detailed instructions by Finesse Cosmetic Surgery with regards to care of your sutures, wounds etc. however, please note the following.
After your surgery, you will have dressings and bandages applied to your wounds. You will also be wrapped in a compression garment designed to minimize swelling and provide support to your surgical site during healing. It may sometimes be necessary to provide a thin tube under the skin to drain away excess blood or fluid that can collect at the site.
After your bandages are removed your results will be visible immediately but will become more noticeable when the swelling and bruising from your operation subsides which may take up to 6 weeks.
Similar to your consultation visit (see above), Dr. Verma will be taking photographs of your arms in order to document its healing during follow up visits.
How long do the effects of arm lift surgery last for?
Arm lift surgery is a permanent procedure but is dependent upon stabilized weight. Any significant weight gain or weight loss after your surgery will affect the results. | {
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Kajdas – osada w Polsce umiejscowiona na lewym brzegu Warty, w województwie łódzkim, w powiecie wieluńskim, w gminie Osjaków.
W tej niewielkiej osadzie w XIX stuleciu znajdowało się tylko jedno gospodarstwo zamieszkałe przez siedmiu mieszkańców. Utrzymywali się oni zapewne z młyna rzecznego, którego obecność potwierdzona została w źródłach historycznych już w XIX w. Obecnie oprócz młyna można podziwiać rzekę Wartę z kilkunastometrowych skarp tworzących naturalne punkty widokowe.
W latach 1975–1998 miejscowość administracyjnie należała do województwa sieradzkiego.
Przypisy
Linki zewnętrzne
Osjaków (gmina) | {
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Q: UTF-8 encoding validator online Hello I would like to test an URL encoding in UTF-8 Is there some online tools to do that?
The closest I have found is in this question which finally suggest to test on the client side using GNU.
note: I need to do that by code. Like sending the chars to enconde on a GET petition.
| {
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I have a circuit with atmega8 + basic . I want to use "sleep mode" and need your help.
my cricuit have a keyboard 3*4 . my microcontroler wait for recive a nomber from keyboard whole time, I want my microcontroler go to sleep mod when not switched any keyboard figure . how can i use "sleep mode" in progeram (basic) ?
Have you read up on the sleep command? You will get more help if you post the work you have already done, otherwise no one will do your project/homework for you.
Power used by microcontroller .
Also I use a internal crystal in program .
Not up on that compiler. I use RVKBASIC from bastoc.com for AVR projects.
Are you trying to wake-up on a key-press? Really need to see a schematic. Maybe someone that knows bascom will pop in here. Or hit AVRfreaks.com.
If I get time I will see if I can get the manual for it online later. Always open to a new compiler.
it uses atmega162, but it discusses how to wake-up the controller from sleep when a key is pressed.
thanks my friends : mramos & CCDCCD / this link not useful to learn how to use a sleep command in progeram.
here is another document from atmel which perfectly describes what you are looking for (only that the code is in assembly).
for your application, you can load MCUCR with 0xa0 or 0xb0 for power-down or power-save mode respectively and enable external interrupt.
After setting MCUCR, just put the SLEEP instruction in your code where ever you want the controller to go to sleep.
the controller will wake-up whenever the external interrupt signal is applied (i.e. a key is pressed).
The micro project you are making are impressive.
That document is quite old. If you have 4x4 keypad project for Atmel AVR, try Googling "avr 4x4 keypad".. you will find lots of good tutorials and C-code. | {
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Everything You Need to Know About the Latest Version of the Google Algorithm
In what Google has billed as a "regular update," a new change to its search ranking algorithm was rolled out in full on August 1, 2018. The update took roughly one week to reach full impact across the web and it seems as though this latest change has had little impact on a broad basis. Taking a more focused look at its impact though, analysts have found that certain industries or niches have seen more side effects from the new Google algorithm roll out. Here is every important detail that companies should know about the latest version of the web giant's search algorithm.
What's it Called?
Life at Google wouldn't be fun without a pet name for every project. This time though, Google lacks a name for it, but that doesn't mean others haven't given the new Google algorithm update a pet name for the sake of clarity when discussing the topic. Yoast.com refers to it as the Medic Update, without citing any real reason. However, SearchEngineLand refers to it as the Medic Update because sites in the health and medical fields have seen the most significant impact so far.
What's the Point?
This is where things get a little murkier. In truth, the new Google algorithm update is about as clear as mud. Google has stated on countless occasions following the August 1st release that it routinely puts out updates to its core algorithms to ensure that the web remains a fair and balanced place. Some analysts and users have found that this particular update, beyond seemingly targeting a handful of particular niches, is also impacting organic search results over other types of search results.
Google has emphasized time and again that the new algorithm update requires no fix by webmasters. The purpose, again according to Google, is not to punish any type of site design or industry, but rather to enhance those sites which have been previously undervalued.
Focused or Not, Medic Update has had an Impact
As alluded to above, Google has consistently pointed out that there was no targeted niche or website type with this latest algorithm change. However, it has so far proven to require adjustments for those companies and brands in the medical field, health field, and so-called "Your Money, Your Life" niches.
Your Money, Your Life, or "YMYL," sites contain content focused on money and life events. Other niches that have felt the pinch so far as a result of the new Google algorithm update include gaming, business-to-business, and e-commerce sites. With that said, many of the e-commerce sites are tied to the medical and health fields in one way or another.
What can Brands do About it?
While Google is sticking to its usual talking points, insisting that there is no "fix" for the update, there is always something that brands can do react and adjust following a change in the Google algorithm. In this case, the best bet is to focus on creating stronger content that is more relevant to users. Moreover, Google seems to be focusing on the intent of website content. If brands want to avoid suffering a drop in visibility as a result of the new algorithm, it would be wise to ensure that existing content and newly developed content going forward matches the intent of searches within a given niche. Again, relevance to the viewer is important.
How to improve your PPC campaign? | {
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{"url":"http:\/\/lilypond.1069038.n5.nabble.com\/Polyphony-confuses-change-td229963.html","text":"# Polyphony confuses \\change\n\n8 messages\nOpen this post in threaded view\n|\n\n## Polyphony confuses \\change\n\n Hi again, \u00a0 \u00a0In this example, LilyPond complains and \\change has no effect. However, \u00a0 \u00a0if you move the \\change between the second and the third measure (i.e. \u00a0 \u00a0one line lower), then all is fine. So, it seems that \\change does not \u00a0 \u00a0work right after the end of simultaneous voices. \u00a0 \u00a0\\version \"2.21.0\" \u00a0 \u00a0<< \u00a0 \u00a0 \u00a0\\new Staff = \"up\" { \u00a0 \u00a0 \u00a0 \u00a0<< { c'' } \\\\ { c' } >> \u00a0 \u00a0 \u00a0 \u00a0\\change Staff = \"down\" \u00a0 \u00a0 \u00a0 \u00a0c'4 \u00a0 \u00a0 \u00a0 \u00a0c'4 \u00a0 \u00a0 \u00a0} \u00a0 \u00a0 \u00a0\\new Staff = \"down\" { \u00a0 \u00a0 \u00a0 \u00a0s4*3 \u00a0 \u00a0 \u00a0} \u00a0 \u00a0>> \u00a0 \u00a0The log contains: \u00a0 \u00a0warning: Change_iterator::process (): Staff = `up': \u00a0 \u00a0-:8:5: warning: cannot change `Staff' to `down': not changing to same \u00a0 \u00a0context type: Staff \u00a0 \u00a0\\change Staff = \"down\" \u00a0 \u00a0A workaround is to add a \\grace s8 \u00a0in all voices, between the end of \u00a0 \u00a0simultaneous voices and the \\change invocation. \u00a0 \u00a0Regards, \u00a0 \u00a0Jean Abou Samra _______________________________________________ bug-lilypond mailing list [hidden email] https:\/\/lists.gnu.org\/mailman\/listinfo\/bug-lilypond\nOpen this post in threaded view\n|\n\n## Re: Polyphony confuses \\change\n\n Jean Abou Samra <[hidden email]> writes: > \u00a0 \u00a0Hi again, > > \u00a0 \u00a0In this example, LilyPond complains and \\change has no effect. However, > \u00a0 \u00a0if you move the \\change between the second and the third measure (i.e. > \u00a0 \u00a0one line lower), then all is fine. So, it seems that \\change does not > \u00a0 \u00a0work right after the end of simultaneous voices. Wrong. \u00a0\\change does not work to change a Staff from Staff to Staff, and at the point of your first \\change command, you do not yet have a Voice (the temporary voices started in << \\\\ >> have just ended). So the solution is to actually start a voice. > \u00a0 \u00a0\\version \"2.21.0\" > > \u00a0 \u00a0<< > \u00a0 \u00a0 \u00a0\\new Staff = \"up\" { try \u00a0 \u00a0\\new Staff = \"up\" \\new Voice { and you'll actually _have_ a Voice that can change to the other Staff. > \u00a0 \u00a0 \u00a0 \u00a0<< { c'' } \\\\ { c' } >> > \u00a0 \u00a0 \u00a0 \u00a0\\change Staff = \"down\" > \u00a0 \u00a0 \u00a0 \u00a0c'4 > \u00a0 \u00a0 \u00a0 \u00a0c'4 > \u00a0 \u00a0 \u00a0} > \u00a0 \u00a0 \u00a0\\new Staff = \"down\" { > \u00a0 \u00a0 \u00a0 \u00a0s4*3 > \u00a0 \u00a0 \u00a0} > \u00a0 \u00a0>> > > \u00a0 \u00a0The log contains: > > \u00a0 \u00a0warning: Change_iterator::process (): Staff = `up': > \u00a0 \u00a0-:8:5: warning: cannot change `Staff' to `down': not changing to same > \u00a0 \u00a0context type: Staff > \u00a0 \u00a0\\change Staff = \"down\" > > \u00a0 \u00a0A workaround is to add a \\grace s8 \u00a0in all voices, between the end of > \u00a0 \u00a0simultaneous voices and the \\change invocation. > > \u00a0 \u00a0Regards, > > \u00a0 \u00a0Jean Abou Samra > _______________________________________________ > bug-lilypond mailing list > [hidden email] > https:\/\/lists.gnu.org\/mailman\/listinfo\/bug-lilypond-- David Kastrup _______________________________________________ bug-lilypond mailing list [hidden email] https:\/\/lists.gnu.org\/mailman\/listinfo\/bug-lilypond\nOpen this post in threaded view\n|\n\n## Re: Polyphony confuses \\change\n\n In reply to this post by Jean ABOU SAMRA > Wrong. \u00a0\\change does not work to change a Staff from Staff to Staff, and > at the point of your first \\change command, you do not yet have a Voice > (the temporary voices started in << \\\\ >> have just ended). > So the solution is to actually start a voice. > and you'll actually _have_ a Voice that can change to the other Staff. OK, thanks, I do understand now. It's logical after all. The warning is not very helpful though because in this case, the problem was not the type of the context we were changing to but that of the context that had to be switched. But that's quibbling. Regards, Jean Abou Samra _______________________________________________ bug-lilypond mailing list [hidden email] https:\/\/lists.gnu.org\/mailman\/listinfo\/bug-lilypond\nOpen this post in threaded view\n|\n\n## Re: Polyphony confuses \\change\n\n Jean Abou Samra <[hidden email]> writes: >> Wrong. \u00a0\\change does not work to change a Staff from Staff to Staff, and >> at the point of your first \\change command, you do not yet have a Voice >> (the temporary voices started in << \\\\ >> have just ended). > >> So the solution is to actually start a voice. > >> and you'll actually _have_ a Voice that can change to the other Staff. > > OK, thanks, I do understand now. It's logical after all. > > The warning is not very helpful though because in > this case, the problem was not the type of the context we > were changing to but that of the context that had to be > switched. But that's quibbling. Well, the less reason for quibbling (which I myself tend to indulge in more than appropriate) ends up on our list and the less confusion ends up with our users, the friendlier LilyPond will appear to everyone. \u00a0So if you have a good suggestion of how to improve that warning in a manner that would have increased the likelihood of you understanding just what the problem was, feel free to suggest a different wording, possibly depending on the situation (we don't need the same warning text for everything: we can special-case some cases). And rather than wasting unnecessary time with LilyPond (like you probably feel the current text made you do), you'll invest time for an actual improvement benefitting users that may see the same problem in future. All the best -- David Kastrup My replies have a tendency to cause friction. \u00a0To help mitigating damage, feel free to forward problematic posts to me adding a subject like \"timeout 1d\" (for a suggested timeout of 1 day) or \"offensive\". _______________________________________________ bug-lilypond mailing list [hidden email] https:\/\/lists.gnu.org\/mailman\/listinfo\/bug-lilypond\nOpen this post in threaded view\n|\n\n## Re: Polyphony confuses \\change\n\n Hi, \u00a0 \u00a0Le 3 mars 2020 \u00e0 20:58, David Kastrup < [1][hidden email]> a \u00e9crit : \u00a0 \u00a0Well, the less reason for quibbling (which I myself tend to indulge in \u00a0 \u00a0more than appropriate) ends up on our list and the less confusion ends \u00a0 \u00a0up with our users, the friendlier LilyPond will appear to everyone. So \u00a0 \u00a0if you have a good suggestion of how to improve that warning in a \u00a0 \u00a0manner \u00a0 \u00a0that would have increased the likelihood of you understanding just what \u00a0 \u00a0the problem was, feel free to suggest a different wording, possibly \u00a0 \u00a0depending on the situation (we don't need the same warning text for \u00a0 \u00a0everything: we can special-case some cases). \u00a0 \u00a0And rather than wasting unnecessary time with LilyPond (like you \u00a0 \u00a0probably feel the current text made you do), you'll invest time for an \u00a0 \u00a0actual improvement benefitting users that may see the same problem in \u00a0 \u00a0future. \u00a0 \u00a0All the best \u00a0 \u00a0Yes, I know. I would have suggested a different wording if I \u00a0 \u00a0had had a brilliant idea, but this was not the case. \u00a0 \u00a0Anyway, I hope the attached patch will help. \u00a0 \u00a0Best regards, \u00a0 \u00a0Jean Abou Samra References \u00a0 \u00a01. mailto:[hidden email] _______________________________________________ bug-lilypond mailing list [hidden email] https:\/\/lists.gnu.org\/mailman\/listinfo\/bug-lilypond\nOpen this post in threaded view\n|\n\n## Re: Polyphony confuses \\change\n\n Jean Abou Samra <[hidden email]> writes: > Hi, > > \u00a0Le 3 mars 2020 \u00e0 20:58, David Kastrup < [hidden email]> a \u00e9crit : > > \u00a0Well, the less reason for quibbling (which I myself tend to indulge in > \u00a0more than appropriate) ends up on our list and the less confusion ends > \u00a0up with our users, the friendlier LilyPond will appear to everyone. So > \u00a0if you have a good suggestion of how to improve that warning in a manner > \u00a0that would have increased the likelihood of you understanding just what > \u00a0the problem was, feel free to suggest a different wording, possibly > \u00a0depending on the situation (we don't need the same warning text for > \u00a0everything: we can special-case some cases). > > \u00a0And rather than wasting unnecessary time with LilyPond (like you > \u00a0probably feel the current text made you do), you'll invest time for an > \u00a0actual improvement benefitting users that may see the same problem in > \u00a0future. > > \u00a0All the best > > Yes, I know. I would have suggested a different wording if I > had had a brilliant idea, but this was not the case. > > Anyway, I hope the attached patch will help. Since you did not just send to the list but also sent a private copy to my mail account, I am pretty sure that the patch never attached itself. We've had various occurences of patches not making it through the list server, but in this case, my personal copy arrived without going through the list server, so you likely just forgot. \u00a0Happens. -- David Kastrup My replies have a tendency to cause friction. \u00a0To help mitigating damage, feel free to forward problematic posts to me adding a subject like \"timeout 1d\" (for a suggested timeout of 1 day) or \"offensive\". _______________________________________________ bug-lilypond mailing list [hidden email] https:\/\/lists.gnu.org\/mailman\/listinfo\/bug-lilypond","date":"2020-04-06 15:42:22","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\": 0.8662816286087036, \"perplexity\": 9861.186924908081}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585371637684.76\/warc\/CC-MAIN-20200406133533-20200406164033-00147.warc.gz\"}"} | null | null |
Mediaserv était un fournisseur d'accès à Internet français ultramarin. Les offres était disponibles pour les départements et collectivités d'outre-mer suivants : Guadeloupe, Martinique, Guyane française, Saint-Barthélemy, Saint-Martin et La Réunion.
Historique
À l'origine, Mediaserv a repris le réseau de Cegetel en 2001.
La société appartenait au holding Loret jusqu'en où elle tombe dans l'influence du groupe Canal+.
Annoncé en après validation de l'Autorité de la concurrence, le Canal+ Overseas (Groupe Canal+) acquiert 51 % de Mediaserv au Groupe Loret.
Les offres Mediaserv ne sont plus commercialisés depuis le 13 avril 2014, remplacés par Canalbox.
Caractéristiques techniques de la Box Mediaserv
Les Box Mediaserv sont des Sagem FAST 3304 ou 3504.
2 ports téléphone : Phone 1 / Phone 2 (ivoire)
1 port ADSL 2+ (gris)
2 ports Ethernet RJ45 PC : ETH1 et ETH2 (jaune)
2 ports Ethernet TV : ETH3 et ETH4 (orange)
1 port USB 1
Connexion filaire ou sans fil (WiFi 802.11 b/g) à l'ordinateur
Possibilité de connexion en CPL (vendu séparément)
Fonction routeur
Fonction pare-feu
Services
La Box de Mediaserv est une offre triple-play proposant :
Accès à Internet :
Internet illimité jusqu'à 20 méga
WiFi
Contrôle parental
Une messagerie de 100mo
Une adresse email Mediaserv + 4 adresses liées
Téléphone :
Sans abonnement téléphonique
Téléphonie illimitée vers 53 destinations et les appels vers les mobiles locaux et métropole en fonction du forfait choisi (soir et WE ou 24H/24).
Messagerie vocale, affichage du numéro, double appel, restriction d'appel
Télévision :
Un bouquet basique de 41 chaînes de télévision (location du décodeur TV à par mois)
Image et son en qualité numérique,
Quelques chaînes métropolitaines
Liste des chaînes de Mediaserv
Annexes
Articles connexes
Mediaserv
Liste des chaînes de Mediaserv
Liens externes
Site officiel
Références
Fournisseur d'accès à Internet en Martinique
Modem triple play | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 577 |
{"url":"https:\/\/www.nature.com\/articles\/s41396-019-0463-3?error=cookies_not_supported&code=27a7b978-8f09-4e35-80b2-b06f4aa375f1","text":"Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.\n\n# How sample heterogeneity can obscure the signal of microbial interactions\n\n## Abstract\n\nMicrobial community data are commonly subjected to computational tools such as correlation networks, null models, and dynamic models, with the goal of identifying the ecological processes structuring microbial communities. A major assumption of these methods is that the signs and magnitudes of species interactions and vital rates can be reliably parsed from observational data on species\u2019 (relative) abundances. However, we contend that this assumption is violated when sample units contain any underlying spatial structure. Here, we show how three phenomena\u2014Simpson\u2019s paradox, context-dependence, and nonlinear averaging\u2014can lead to erroneous conclusions about population parameters and species interactions when samples contain heterogeneous mixtures of populations or communities. At the root of this issue is the fundamental mismatch between the spatial scales of species interactions (micrometers) and those of typical microbial community samples (millimeters to centimetres). These issues can be overcome by measuring and accounting for spatial heterogeneity at very small scales, which will lead to more reliable inference of the ecological mechanisms structuring natural microbial communities.\n\n## Common \u201cpattern-to-process\u201d inferential methods yield erroneous results\n\nAdvances in sequencing technology offer microbiologists unprecedented access to the composition and dynamics of microbial communities [1]. Marker gene and metagenomic surveys regularly chronicle hundreds to thousands of taxa, many previously unknown, all seemingly co-occurring within their respective habitats. In possession of these large observational datasets, we have adapted theory and methods developed from plant and animal ecology to investigate how species interactions\u2014such as competition, predation, and facilitation\u2014structure microbial communities [2, 3].\n\nWithout experimental systems in which competition (or any other interaction) may be directly manipulated and detected, researchers often employ randomization-based null models, correlation networks, and population dynamic models to identify and quantify putative interspecific interactions from observational sequence data [4,5,6,7]. Here, negative covariation between the (relative) abundances of taxa are assumed to result from negative interspecific interactions such as competition, though practitioners generally agree that precisely parsing ecological mechanisms from observational data remains unreliable. In fact, the utility of these methods for reliably isolating and quantifying signals of competition from alternative community assembly processes such as habitat filtering and trophic interactions has been disputed in the community ecology literature for decades [8].\n\nRecently, a number of studies have challenged null model and correlation-based methods to recapitulate known interactions in well-studied marine intertidal habitats [9,10,11]. In all cases, these tests revealed troubling inaccuracies and discrepancies among the various methods, calling into question their ability to reliably identify true ecological interactions. For microbial communities, the only successful validations of these methods have occurred in simple, well-mixed liquid cultures [7] or in situations where direct, inter-kingdom chemical inhibition occurs [12]. Taken in concert, these studies highlight potential pitfalls in our ability to correctly identify species interactions when communities are sampled over underlying spatial heterogeneity. Most natural microbial communities are spatially structured and exhibit marked heterogeneity at multiple spatial scales. We argue that failure to account for this underlying spatial heterogeneity in environmental samples can undermine our conclusions about the ecological processes structuring microbial assemblages [13].\n\n## Causes and consequences of heterogeneity in microbial samples\n\nTypical sample volumes used for environmental marker gene and metagenomics studies are rarely smaller than 0.1\u2009ml, but can be as large as 100\u2009l of seawater and 100\u2009g of soil in low-DNA habitats. Unless these samples come from a perfectly mixed, completely homogeneous medium, they will contain at least some amount of spatial structure. For example, a typical 0.25\u2009g sample of soil containing particles 1\u2009mm in diameter (i.e., a very coarse sand) will inevitably contain hundreds to thousands of discrete granules on which microbial communities can assemble. These discrete habitats can represent a heterogeneous array of environments or resources, each selecting for their own unique local microbial communities [13]. However, even a physicochemically homogeneous collection of particles can contain a mosaic of distinct microbial communities owing to the effects of limited or asymmetric dispersal, priority effects, and successional turnover.\n\nFine-scale heterogeneity in microbial communities appears to be a general property of environmental samples, having been repeatedly documented in aquatic, soil, fecal, leaf surface, and wastewater habitats [14,15,16,17,18,19]. Owing to this, marker gene samples commonly represent sums or averages of sequence reads made over underlying environmental heterogeneity, leaving us with a bulk inventory of operational taxonomic units (OTUs) and their (often relative) abundances without information on spatial context. Because microbial interactions such as resource competition, phage predation, DNA transfer, and syntrophy are hypothesized to take place at spatial scales much smaller than that of the typical bulk sample [20], it can be argued that many marker gene samples actually measure the metacommunity\u2014a collection of semi-autonomous communities linked through dispersal [21]. In the following sections, we illustrate how collecting samples at the metacommunity scale can introduce errors into computational estimates of interspecific interactions by virtue of three phenomena: Simpson\u2019s paradox, context-dependence, and nonlinear averaging. Note that although we present total abundance data throughout our scenarios, these phenomena also apply to compositional (i.e., relative abundance) data, which are more commonly collected in environmental marker gene surveys.\n\nSimpson\u2019s paradox refers to the reversal or negation of a statistical association between two variables, X and Y, when conditioned on a third variable, Z [22,23,24,25]. In ecology, this Z variable might include information on spatial variation among local patches, which, if accounted for, changes the direction of a trend at larger spatial scales [26]. Computational approaches to inferring microbial interactions can be sensitive to the effects of Simpson\u2019s paradox. For instance, the inferred signs of interspecific correlation coefficients might change when comparing analytic results obtained from bulk community samples with results that have statistically accounted for underlying variation in microhabitats or resource availability within bulk samples.\n\nTo illustrate this point, consider a hypothetical study that uses data obtained from bulk soil samples to infer the sign of interspecific interaction between two microbial OTUs. If the true nature of this interaction is competitive, then our results are anticipated to reveal a negative correlation between the abundances of the OTUs. To add some realism to this scenario, let us assume that each of our samples represent collections of discrete microhabitats on which our focal taxa grow (Fig. S1). Finally, we might also make the realistic assumption that both OTUs respond similarly to these discrete microhabitats such that sub-optimal habitats support fewer individuals of both species. If we create bulk soil samples by subsampling a particulate, heterogeneous soil habitat containing populations of our two competitors (Fig.\u00a01a), we find that these samples\u2019 OTU abundances reflect both local habitat composition, as well as negative interspecific interactions (Fig.\u00a01b). As a result, our interspecific correlation estimates from bulk samples predict positive correlations between our two OTUs, whereas their true, competitive interactions result in negative correlations when the local environment is accounted for (Fig.\u00a01c). Furthermore, by repeating this experiment many times using the same parameters but different habitat configurations (Fig. S2), each time re-assembling our bulk samples, we encounter an overwhelming majority of cases where the inferred sign of interaction between our two OTUs (positive) is the opposite of its true sign (negative), leading us to erroneously conclude that these species are not strong competitors when, in truth, they are (Fig.\u00a01d; details in\u00a0supplemental information).\n\nCrucially, detection methods relying on false-discovery rate correction will fail to identify this error as such, as statistical significance is decoupled from the effects of Simpson\u2019s Paradox [27]. Because of these issues, we contend that unless the assumption of homogeneity within and among microbial community samples is justified, interspecific interaction coefficients derived from correlation or model-based approaches should be interpreted with extreme caution, and should always include a statement concerning the spatial context of the sample including potential sources of underlying spatial heterogeneity.\n\n### Context dependence\n\nA common assumption of computational models for identifying species interactions is that the sign and strength of interactions are immutable across time and space, yet this simplifying assumption is widely acknowledged not to hold. Often invoked for statistical convenience, this assumption reduces the sample sizes required for estimating correlation coefficients or population parameters, and permits the use of graph theoretic descriptors of network structure (connectance, nestedness, etc.). However, numerous laboratory experiments have documented context-dependent interactions arising from variation in population densities, community composition, or environmental context, such that interactions measured at one place and time cannot reliably be extrapolated across habitats [28,29,30,31]. For instance, a recent study documented predictable shifts in the sign of species interactions with changing resource concentrations in experimental yeast communities as cross-feeding gave way to competition [32] (Fig.\u00a02a). The presence of predators can also mediate the sign of interspecific interactions through a variety of mechanisms [33] (e.g., Fig.\u00a02b). Likewise, a meta-analysis of hundreds of experiments uncovered a strong effect of spatial heterogeneity on context-dependent species interactions [34]. Consequently, it is not unreasonable to expect the signs of microbial interactions to change across gradients of resource density, predation pressure, or other indicators of habitat quality (Fig.\u00a02c). While temporal correlation network approaches might be used to circumvent the static interactions assumption at larger spatial scales or in well-mixed samples, they cannot account for variable interactions arising from underlying, unmeasured spatial heterogeneity within individual samples.\n\nFrom a theoretical perspective, context-dependence is hypothesized to be a critical factor for maintaining diversity in spatially structured communities. For instance, the abilities of two competing microbial strains to coexist will be enhanced if the negative impacts of competition experienced by each strain are stronger in more favorable habitat patches [35]. Given that microbial species richness appears to peak in particulate, heterogeneous habitats (soil, sediments) [1], context-dependent interactions within these habitats may be quite common and important in promoting high levels of diversity. Currently, the extent of context-dependent interactions in spatially structured microbial communities remains largely unknown. We note, though, that correlation network approaches have been successfully used to identify context-dependent interactions robust to experimental ground-truthing [12, 36]. However, until the prevalence and magnitude of context-dependent microbial interactions are better understood, we encourage researchers to exercise discretion when making general statements concerning any local estimates of interspecific interactions, ideally contextualizing results to the specific environment and scale at which measurements were taken.\n\n### Nonlinear averaging\n\nThe previous two sections concerned issues that arise when quantifying local microbial interactions from heterogeneous samples. However, we also face difficulties when using microbial community data collected at very small scales to quantify the aggregate behavior of aggregate microbial communities. Imagine that we are now able to obtain measurements of microbial populations at the scale of the individual microhabitat patches. Importantly, these data are collected at the spatial scale over which intraspecific interactions play out, which, in a heterogeneous sample experiencing dispersal among particles, is at the scale of individual microhabitat patches or particles. Called the characteristic scale, it is the scale which maximizes the ratio of deterministic signal to the influences of stochasticity and spatial heterogeneity [37], making it the optimal scale for measuring and characterizing the effects of deterministic species interactions.\n\nLet us now envision a scenario where we wish to quantify whether a microbial OTU\u2019s competitive ability is a function of its local soil habitat. Since accurately estimating the strength of competition in our samples is of paramount importance, suppose we have conducted our sequencing surveys at appropriately small characteristic scales and have generated time series data from this assortment of individual particles. We then fit a population dynamic model to these data in order to estimate our OTU\u2019s growth rate and competitive interactions among different soil types, adequately replicated within each type. The generalized Lotka-Volterra (gLV) population dynamic model is increasingly being utilized for this purpose [38]. Fitting such a differential equation model requires estimating parameters describing a focal species\u2019 growth rates and interspecific interactions. The gLV model commonly takes the form\n\n$$\\frac{{dN_i}}{{dt}} = N_i\\left( {\\mu _i + \\mathop {\\sum}\\nolimits_{j = 1}^M {\\alpha _{ij}N_j} } \\right),\\quad i = 1, \\ldots ,M,$$\n(1)\n\nwhere Ni is the abundance of OTU i, \u03bci is its maximum per capita growth rate, and \u03b1ij is a parameter describing the proportional change in its growth rate with conspecific or heterospecific densities. Values of \u03b1ij greater than zero imply that OTU j has a positive effect on OTU i, which might stem from interactions such as syntrophy, whereas values less than zero can signify interactions such as competition or chemical inhibition.\n\nFor illustrative purposes, let us simplify our problem of estimating competition among soil types by assuming that only our single focal OTU occupies our habitats, and so is only capable of experiencing intraspecific competition. This permits us to simplify our model to the case where (i\u2009=\u2009j), and define \u03b1ij\u2009=\u2009\u2212\u03bciKi\u22121, where Ki represents the local carrying capacity of our OTU i. This results in the familiar logistic population growth model describing decelerating microbial population growth with increasing population density. Expanding this model across a spatially structured array of n individual particles, we obtain the equation\n\n$$\\frac{{dN_x}}{{dt}} = \\mu N_x\\left( {1 - \\frac{{N_x}}{K}} \\right),\\quad \\quad \\left( {x = 1, \\ldots ,n} \\right),$$\n(2)\n\nwhere Nx are the local sub-populations of our focal OTU on habitat particle x.\n\nWith a collection of population dynamic equations describing our individual particles, we can now aggregate these local dynamics to obtain general growth parameters to compare growth across soil types. This scaling-up process requires a spatial averaging of local population dynamics (Fig.\u00a03a). Crucially, because the average of a nonlinear function is not equal to the function of its averaged covariates (i.e., $$\\overline {f\\left( N \\right)} \\ne f\\left( {\\overline N } \\right)$$ when f\u2033(N)\u2009\u2260\u20090), to scale up microbial population dynamics\u2014which are almost unanimously nonlinear\u2014by averaging across spatially variable local populations will result in biases proportional to the spatial population variation and model\u2019s nonlinearity. This principle, called Jensen\u2019s inequality, has important consequences for our abilities to accurately estimate scaled-up model parameters and make predictions from any gLV model when the local populations contain spatial heterogeneity, as is often the case.\n\nThe consequences of this spatial averaging process are illustrated in Fig.\u00a03b. For notational simplicity, we replace the growth function in Eq.\u00a02, \u03bcNx(1\u2212Nx\/K), with G(Nx). The spatially averaged dynamical equation that we wish to obtain is $$d\\overline N {\\mathrm{\/}}dt = \\overline {G\\left( N \\right)}$$. Calculating our population dynamic model using the spatial averages of the populations we have measured, $$G\\left( {\\overline N } \\right)$$, overestimates the correctly scaled-up population growth function, $$\\overline {G\\left( N \\right)}$$. In Fig.\u00a03c, we generated four collections of particles in which spatially explicit populations have been randomly drawn from lognormal distributions having equal means but different variances (\u03c32). We then used these simulated data to fit four spatially averaged population growth functions, $$\\overline {G\\left( N \\right)}$$. Our results demonstrate how increasing the spatial variation among local populations has the effect of changing our scaled-up estimates of carrying capacity. The challenge for microbiologists is to correctly estimate $$\\overline {G\\left( N \\right)}$$ using their measured population densities, Nx. Fortunately, if we have already collected these values, and if they can be reasonably fit to a population dynamic model, we can use the tools of scale transition theory [39, 40] to correctly obtain scaled-up population parameters. We introduce these methods in the following section and provide a more thorough overview in the\u00a0supplemental information.\n\n## Recommendations moving forward\n\nDespite the various ways in which spatial heterogeneity can subvert our interpretation or complicate our assessment of microbial community interactions and dynamics, we are optimistic that these issues can be surmounted with prudent data collection, analysis, and interpretation. The lurking effects of habitat heterogeneity are most effectively mitigated by quantifying microbial populations or communities at the spatial scales over which cell-cell interactions occur, which is on the scale of micrometers to millimeters. Micron-scale sampling has successfully been accomplished using individual grains of sand [14], aquatic organic particles [41], and sludge granules [42, 43]\u2014 all of which encountered marked heterogeneity among particles. Sampling at this scale is facilitated by technologies such as fluorescence-activated cell sorting and laser-assisted microdissection, which offer opportunities to precisely and efficiently capture individual microscopic particles for sequencing, as well as fluorescence in situ hybridization microscopy to determine the spatial associates of individual populations. A noteworthy example of such an approach is the in situ microscopic validation of a network-predicted mutualistic interaction [44]. However, as we have seen, even measurements made at the appropriate characteristic scales can be challenging to generalize due to the issues of nonlinear averaging and context-dependence.\n\nThe restrictive assumptions of most correlation network and null models hinder our reliable assessment of microbial interactions in all but the most homogeneous samples. However, the influence of Simpson\u2019s paradox and context-dependence may be surmounted by measuring and statistically accounting for the confounding effects of environmental and\/or community variation among samples. Though methods to control for environmental factors using partial correlations or joint species distribution models are available [4, 45, 46], unless environmental characterization is made at the appropriate scales, they cannot account for the spatial variation contained within a single heterogeneous sample. These methods have also delivered underwhelming results when challenged to predict an empirically measured interaction web, after accounting for the confounding effects of environmental variation [10]. Though it will be challenging to collect environmental data at such fine spatial grains, such data could be used to test the alternative hypotheses of habitat filtering, competition, and dispersal limitation\u2014all of which can feasibly manifest as identical metacommunity patterns in the presence of spatial variation.\n\nWhile creative new statistical and mechanistic modeling approaches for identifying nonlinear and context-dependent species interactions are becoming available [46,47,48,49,50], we suggest these methods be ground-truthed with more complex and realistic data than are currently in use. For example, rather than using time series simulated from equilibrial, non-spatial Lotka-Volterra equations to benchmark a new method, a more powerful validation routine could use data simulated from spatially explicit agent-based models, which can test methods\u2019 robustness to spatial heterogeneity, scale-dependence, and demographic stochasticity. We also encourage the inclusion of dynamic parameters in generalized Lotka-Volterra models. While it is challenging to estimate these parameters from observational data, experiments consistently show that microbial growth rate, carrying capacity, and interaction parameters are functions of their underlying environments. A benefit of including environmentally dependent growth parameters in gLV models is that these models can then be used to quantify the effects of various coexistence-promoting mechanisms [51]. Context-dependent parameters also allow us to investigate the effects of environmental change on microbial populations and communities.\n\nThe increasing use of gLV models in microbial ecology also prompts us to account for the effects of nonlinear spatial averaging on scaled up population dynamics (see section Nonlinear averaging). Chesson\u2019s scale transition theory [34, 35] provides a mathematical framework for tackling the issues of spatial heterogeneity and nonlinearities in gLV models. We introduce the scale transition using two simple models, but refer interested readers to the original papers [39, 40] and the supplementary information section for general scale transition approaches. Continuing from section Nonlinear averaging, we can calculate the scaled-up population dynamics, $$\\overline {G\\left( N \\right)}$$, by accounting for the nonlinearity in G(Nx) using its second derivative, G\u2033(Nx), as well as the spatial variation in Nx, measured by the spatial variance, Var(N). The full, spatially averaged population model can be approximated as\n\n$$\\frac{{d\\overline N }}{{dt}} = \\overline {G\\left( N \\right)} \\approx G\\left( {\\overline N } \\right) + \\frac{1}{2}G\\prime\\prime \\left( {\\overline N } \\right){\\mathrm{Var}}\\left( N \\right)\\\\ \\approx \\left[ {g\\left( {\\overline N } \\right) + \\frac{1}{2}g\\prime\\prime \\left( {\\overline N } \\right){\\mathrm{Var}}\\left( N \\right)} \\right]\\overline N + g\\prime \\left( {\\overline N } \\right){\\mathrm{Var}}\\left( N \\right),$$\n(3)\n\nWhere g(N)\u2009=\u2009\u03bc[1\u2212N\/K] and $$\\frac{1}{2}G\\prime\\prime \\left( {\\overline N } \\right) = g\\prime \\left( {\\overline N } \\right) = - \\mu {\\mathrm{\/}}K$$. This approximation is exact when the growth function is quadratic (as is the case for logistic growth).\n\nA similar, albeit more complicated scale transition can be calculated for a multispecies gLV model (Eq.\u00a01) [34]. This model is commonly used to identify interspecific interactions, denoted by the \u03b1ij parameters. By defining $$W_i = \\mathop {\\sum}\\nolimits_{j = 1}^M {\\alpha _{ij}N_j}$$ and g(Wi)\u2009=\u2009\u03bci+Wi, the scaled up version of Eq.\u00a01 can be written as a function of mean field terms, a nonlinearity term, and spatial variances and covariances:\n\n$$\\frac{{d\\overline N _i}}{{dt}} \\approx g\\left( {\\overline W _i} \\right)\\overline N _i + g{\\prime} \\left( {\\overline W _i} \\right){\\mathrm{Cov}}\\left( {W_i,\\upsilon _i} \\right)\\overline N _i\\\\ \\approx \\left( {\\mu _i + \\mathop {\\sum}\\nolimits_{j = 1}^M {\\alpha _{ij}\\overline N _j} } \\right)\\overline N _i + \\mathop {\\sum}\\nolimits_{j = 1}^M {\\alpha _{ij}{\\mathrm{Cov}}\\left( {N_i,N_j} \\right)},$$\n(4)\n\nwhere $$\\upsilon _i = N_{ix}{\\mathrm{\/}}\\overline N _i$$. Once again, we see that the spatially averaged population dynamics are not simply a function of average populations across space. However, the only extra information needed to calculate the scale transition are the spatial variances and covariances of the populations, which we can approximate by measuring local population densities across a sufficient number of particles within a sample. Thus, the calculation of scale transition terms is straightforward once they are defined for a particular dynamic model.\n\nGiven the potential for biases and errors stemming from the joint effects of underlying spatiotemporal heterogeneity and other methodological choices (e.g., relative abundance transformations, normalization techniques) [52], it may seem like the inference of species interactions from observational microbial data represents an underdetermination problem. That is, there may be multiple, or even infinite potential mechanisms capable of generating an observed community pattern. However, this problem, like many in ecology and evolution, can more precisely be described as an example of contrast failure [53]. Instead of a solution-free, underdetermined system, we instead have one where our failure to parse competing hypotheses is a transient consequence of data insufficiency. Access to better, more contrastive data, derived either experimentally or observationally at the appropriate spatiotemporal scales, will refine our ability to discriminate among alternative hypotheses. To clarify, we do not advocate for the abandonment of \u2018pattern-to-process\u2019 approaches for parsing microbial communities, which have already proven useful for identifying the \u2018keystone taxa\u2019 associated with alternative community configurations and ecosystem functioning [12, 36]. On the contrary, we are optimistic about continued methodological development in this area\u2014particularly with regard to the spatial scale of sample characterization. In the meantime, we implore researchers to consider and confront the lurking effects of spatial structure on their inferred microbial interaction networks and growth parameters. At minimum, this could simply comprise a comment on the spatiotemporal scale over which the results are anticipated to hold and a description of the spatial structure contained within a sample unit.\n\n## References\n\n1. 1.\n\nThompson LR, Sanders JG, McDonald D, Amir A, Ladau J, Locey KJ, et al. A communal catalogue reveals Earth\u2019s multiscale microbial diversity. Nature. 2017;551:457\u201363.\n\n2. 2.\n\nProsser JI, Bohannan BJM, Curtis TP, Ellis RJ, Firestone MK, Freckleton RP, et al. The role of ecological theory in microbial ecology. Nat Rev Microbiol. 2007;5:384\u201392.\n\n3. 3.\n\nNemergut DR, Schmidt SK, Fukami T, O\u2019Neill SP, Bilinski TM, Stanish LF, et al. Patterns and processes of microbial community assembly. Microbiol Mol Biol Rev. 2013;77:342\u201356.\n\n4. 4.\n\nFaust K, Raes J. Microbial interactions: from networks to models. Nat Rev Micro. 2012;10:538\u201350.\n\n5. 5.\n\nB\u00e1lint M, Bahram M, Eren AM, Faust K, Fuhrman JA, Lindahl B, et al. Millions of reads, thousands of taxa: microbial community structure and associations analyzed via marker genes. FEMS Microbiol Rev. 2016;40:686\u2013700.\n\n6. 6.\n\nLayeghifard M, Hwang DM, Guttman DS. Disentangling interactions in the microbiome: a network perspective. Trends Microbiol. 2017;25:217\u201328.\n\n7. 7.\n\nXiao Y, Angulo MT, Friedman J, Waldor MK, Weiss ST, Liu Y-Y. Mapping the ecological networks of microbial communities. Nat Commun. 2017;8:2042.\n\n8. 8.\n\nRoughgarden J. Competition and theory in community ecology. Am Natural. 1983;122:583\u2013601.\n\n9. 9.\n\nSander EL, Wootton JT, Allesina S. Ecological network inference from long-term presence-absence data. Sci Rep. 2017;7:7154.\n\n10. 10.\n\nBarner AK, Coblentz KE, Hacker SD, Menge BA. Fundamental contradictions among observational and experimental estimates of non-trophic species interactions. Ecology. 2018;99:557\u201366.\n\n11. 11.\n\nFreilich MA, Wieters E, Broitman BR, Marquet PA, Navarrete SA. Species co-occurrence networks: Can they reveal trophic and non-trophic interactions in ecological communities? Ecology. 2018;99:690\u20139.\n\n12. 12.\n\nDur\u00e1n P, Thiergart T, Garrido-Oter R, Agler M, Kemen E, Schulze-Lefert P, et al. Microbial interkingdom interactions in roots promote Arabidopsis survival. Cell. 2018;175:973\u201383.e14.\n\n13. 13.\n\nBerry D, Widder S. Deciphering microbial interactions and detecting keystone species with co-occurrence networks. Front Microbiol. 2014;5:219.\n\n14. 14.\n\nProbandt D, Eickhorst T, Ellrott A, Amann R, Knittel K. Microbial life on a sand grain: from bulk sediment to single grains. ISME J. 2018;12:623\u201333.\n\n15. 15.\n\nHunt DE, David LA, Gevers D, Preheim SP, Alm EJ, Polz MF. Resource partitioning and sympatric differentiation among closely related bacterioplankton. Science. 2008;320:1081\u20135.\n\n16. 16.\n\nSessitsch A, Weilharter A, Gerzabek MH, Kirchmann H, Kandeler E. Microbial population structures in soil particle size fractions of a long-term fertilizer field experiment. Appl Environ Microbiol. 2001;67:4215\u201324.\n\n17. 17.\n\nSwidsinski A, Loening\u2013Baucke V, Verstraelen H, Osowska S, Doerffel Y. Biostructure of fecal microbiota in healthy subjects and patients with chronic idiopathic diarrhea. Gastroenterology. 2008;135:568\u201379.e2.\n\n18. 18.\n\nRemus-Emsermann MNP, Tecon R, Kowalchuk GA, Leveau JHJ. Variation in local carrying capacity and the individual fate of bacterial colonizers in the phyllosphere. ISME J. 2012;6:756\u201365.\n\n19. 19.\n\nGonzalez-Gil G, Holliger C. Aerobic granules: microbial landscape and architecture, stages, and practical implications. Appl Environ Microbiol. 2014;80:3433\u201341.\n\n20. 20.\n\nCordero OX, Datta MS. Microbial interactions and community assembly at microscales. Curr Opin Microbiol. 2016;31:227\u201334.\n\n21. 21.\n\nLeibold MA, Holyoak M, Mouquet N, Amarasekare P, Chase JM, Hoopes MF, et al. The metacommunity concept: a framework for multi-scale community ecology. Ecol Lett. 2004;7:601\u201313.\n\n22. 22.\n\nThorndike EL. On the fallacy of imputing the correlations found for groups to the individuals or smaller groups composing them. Am J Psychol. 1939;52:122\u20134.\n\n23. 23.\n\nSimpson EH. The Interpretation of interaction in contingency tables. J R Stat Soc Ser B. 1951;13:238\u201341.\n\n24. 24.\n\nBlyth CR. On Simpson\u2019s paradox and the sure-thing principle. J Am Stat Associat. 1972;67:364\u20136.\n\n25. 25.\n\nAppleton DR, French JM, Vanderpump MPJ. Ignoring a covariate: an example of Simpson\u2019s paradox. Am Stat. 1996;50:340\u20131.\n\n26. 26.\n\nScheiner SM, Cox SB, Willig MR, Mittelbach GG, Osenberg CW, Kaspari M. Species richness, species\u2013area curves and Simpson\u2019s paradox. Evol Ecol Res. 2000;2:791\u2013802.\n\n27. 27.\n\nHeydtmann M. The nature of truth: Simpson\u2019s paradox and the limits of statistical data. QJM. 2002;95:247\u20139.\n\n28. 28.\n\nde Muinck EJ, Stenseth NC, Sachse D, Roost J, vander, R\u00f8nningen KS, Rudi K, et al. Context-dependent competition in a model gut bacterial community. PLoS ONE. 2013;8:e67210.\n\n29. 29.\n\nLiu A, Archer AM, Biggs MB, Papin JA. Growth-altering microbial interactions are responsive to chemical context. PLoS ONE. 2017;12:e0164919.\n\n30. 30.\n\nTecon R, Ebrahimi A, Kleyer H, Levi SE, Or D. Cell-to-cell bacterial interactions promoted by drier conditions on soil surfaces. PNAS. 2018;115:9791\u20136.\n\n31. 31.\n\nGould AL, Zhang V, Lamberti L, Jones EW, Obadia B, Gavryushkin A, et al. High-dimensional microbiome interactions shape host fitness. Proc Nat Acad Sci USA. 2018;115 E11951\u2013E11960.\n\n32. 32.\n\nHoek TA, Axelrod K, Biancalani T, Yurtsev EA, Liu J, Gore J. Resource availability modulates the cooperative and competitive nature of a microbial cross-feeding mutualism. PLoS Biol. 2016;14:e1002540.\n\n33. 33.\n\nChesson P, Kuang JJ. The interaction between predation and competition. Nature. 2008;456:235\u20138.\n\n34. 34.\n\nChamberlain SA, Bronstein JL, Rudgers JA. How context dependent are species interactions? Ecol Lett. 2014;17:881\u201390.\n\n35. 35.\n\nChesson P. General theory of competitive coexistence in spatially-varying environments. Theor Population Biol. 2000;58:211\u201337.\n\n36. 36.\n\nAgler MT, Ruhe J, Kroll S, Morhenn C, Kim S-T, Weigel D, et al. Microbial hub taxa link host and abiotic factors to plant microbiome variation. PLoS Biol. 2016;14:e1002352.\n\n37. 37.\n\nPascual M, Levin SA. From individuals to population densities: searching for the intermediate scale of nontrivial determinism. Ecology. 1999;80:2225\u201336.\n\n38. 38.\n\nStein RR, Bucci V, Toussaint NC, Buffie CG, R\u00e4tsch G, Pamer EG, et al. Ecological modeling from time-series inference: Insight into dynamics and stability of intestinal microbiota. PLoS Comput Biol. 2013;9:e1003388.\n\n39. 39.\n\nChesson P, Donahue MJ, Melbourne BA, Sears ALW. Scale transition theory for understanding mechanisms in metacommunities. In: Holyoak M, Leibold MA, Holt RD, editors. Metacommunities: spatial dynamics and ecological communities. Chicago: University Of Chicago Press; 2005. p. 279\u2013306.\n\n40. 40.\n\nChesson P. Scale transition theory: Its aims, motivations and predictions. Ecol Complex. 2012;10:52\u201368.\n\n41. 41.\n\nBi\u017ei\u0107-Ionescu M, Ionescu D, Grossart H-P. Organic particles: heterogeneous hubs for microbial interactions in aquatic ecosystems. Front Microbiol. 2018;9:2569.\n\n42. 42.\n\nKuroda K, Nobu MK, Mei R, Narihiro T, Bocher BTW, Yamaguchi T, et al. A single-granule-level approach reveals ecological heterogeneity in an upflow anaerobic sludge blanket reactor. PLoS ONE. 2016;11:e0167788.\n\n43. 43.\n\nLeventhal GE, Boix C, Kuechler U, Enke TN, Sliwerska E, Holliger C, et al. Strain-level diversity drives alternative community types in millimetre-scale granular biofilms. Nat Microbiol. 2018;3:1295.\n\n44. 44.\n\nLima-Mendez G, Faust K, Henry N, Decelle J, Colin S, Carcillo F, et al. Determinants of community structure in the global plankton interactome. Science. 2015;348:1262073.\n\n45. 45.\n\nOvaskainen O, Hottola J, Siitonen J. Modeling species co-occurrence by multivariate logistic regression generates new hypotheses on fungal interactions. Ecology. 2010;91:2514\u201321.\n\n46. 46.\n\nHarris DJ. Inferring species interactions from co-occurrence data with Markov networks. Ecology. 2016;97:3308\u201314.\n\n47. 47.\n\nBiswas S, McDonald M, Lundberg DS, Dangl JL, Jojic V. Learning microbial interaction networks from metagenomic count data. In: Przytycka TM editor. Research in computational molecular biology. Springer International Publishing; 2015. pp. 32\u201343.\n\n48. 48.\n\nMomeni B, Xie L, Shou W. Lotka-Volterra pairwise modeling fails to capture diverse pairwise microbial interactions. Elife. 2017;6:e25051.\n\n49. 49.\n\nTackmann J, Rodrigues JFM, von Mering C. Rapid inference of direct interactions in large-scale ecological networks from heterogeneous microbial sequencing data. bioRxiv 2018;390195. https:\/\/doi.org\/10.1101\/390195.\n\n50. 50.\n\nMa B, Wang H, Dsouza M, Lou J, He Y, Dai Z, et al. Geographic patterns of co-occurrence network topological features for soil microbiota at continental scale in eastern China. ISME J. 2016;10:1891\u2013901.\n\n51. 51.\n\nChesson P. Mechanisms of maintenance of species diversity. Ann Rev Ecol Systemat. 2000;31:343\u201366.\n\n52. 52.\n\nWeiss S, Treuren WV, Lozupone C, Faust K, Friedman J, Deng Y, et al. Correlation detection strategies in microbial data sets vary widely in sensitivity and precision. ISME J. 2016;10:1669\u201381.\n\n53. 53.\n\nForber P. Spandrels and a pervasive problem of evidence. Biol Philos. 2008;24:247.\n\n## Acknowledgements\n\nThe authors thank members of the Jones Lab, J. Prosser, and three anonymous referees for helpful discussion and feedback. Financial support was provided by the US National Science Foundation (DEB-1442246).\n\n## Author information\n\nAuthors\n\n### Corresponding author\n\nCorrespondence to David W. Armitage.\n\n## Ethics declarations\n\n### Conflict of interest\n\nThe authors declare that they have no conflict of interest.\n\nPublisher\u2019s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.\n\n## Rights and permissions\n\nReprints and Permissions\n\nArmitage, D.W., Jones, S.E. How sample heterogeneity can obscure the signal of microbial interactions. ISME J 13, 2639\u20132646 (2019). https:\/\/doi.org\/10.1038\/s41396-019-0463-3\n\n\u2022 Revised:\n\n\u2022 Accepted:\n\n\u2022 Published:\n\n\u2022 Issue Date:\n\n\u2022 ### Microbiome of vineyard soils is shaped by geography and management\n\n\u2022 Emanuela Coller\n\u2022 , Alessandro Cestaro\n\u2022 , Roberto Zanzotti\n\u2022 , Daniela Bertoldi\n\u2022 , Massimo Pindo\n\u2022 , Simone Larger\n\u2022 , Davide Albanese\n\u2022 , Enzo Mescalchin\n\u2022 \u00a0&\u00a0Claudio Donati\n\nMicrobiome (2019)","date":"2021-09-18 08:51:37","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\": 2, \"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\": 0.6756271123886108, \"perplexity\": 6474.356119556164}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780056348.59\/warc\/CC-MAIN-20210918062845-20210918092845-00586.warc.gz\"}"} | null | null |
/* Based on reflectored code coming from Microsoft.IdentityModel.Protocols.WSTrust.Bindings.WSTrustBindingBase class */
using System;
using System.ServiceModel;
using System.ServiceModel.Channels;
using System.ServiceModel.Security;
using System.ServiceModel.Security.Tokens;
namespace OfficeDevPnP.Core.IdentityModel.WSTrustBindings
{
public abstract class WSTrustBinding : Binding
{
private bool _enableRsaProofKeys;
private SecurityMode _securityMode;
private TrustVersion _trustVersion;
protected abstract void ApplyTransportSecurity(HttpTransportBindingElement transport);
protected abstract SecurityBindingElement CreateSecurityBindingElement();
protected WSTrustBinding(SecurityMode securityMode) : this(securityMode, TrustVersion.WSTrust13)
{
}
protected WSTrustBinding(SecurityMode securityMode, TrustVersion trustVersion)
{
this._securityMode = SecurityMode.Message;
this._trustVersion = TrustVersion.WSTrust13;
if (trustVersion == null)
{
throw new ArgumentNullException("trustVersion");
}
this.ValidateTrustVersion(trustVersion);
ValidateSecurityMode(securityMode);
this._securityMode = securityMode;
this._trustVersion = trustVersion;
}
protected virtual SecurityBindingElement ApplyMessageSecurity(SecurityBindingElement securityBindingElement)
{
if (securityBindingElement == null)
{
throw new ArgumentNullException("securityBindingElement");
}
if (TrustVersion.WSTrustFeb2005 == this._trustVersion)
{
securityBindingElement.MessageSecurityVersion = MessageSecurityVersion.WSSecurity11WSTrustFebruary2005WSSecureConversationFebruary2005WSSecurityPolicy11BasicSecurityProfile10;
}
else
{
securityBindingElement.MessageSecurityVersion = MessageSecurityVersion.WSSecurity11WSTrust13WSSecureConversation13WSSecurityPolicy12BasicSecurityProfile10;
}
if (this._enableRsaProofKeys)
{
RsaSecurityTokenParameters item = new RsaSecurityTokenParameters
{
InclusionMode = SecurityTokenInclusionMode.Never,
RequireDerivedKeys = false
};
securityBindingElement.OptionalEndpointSupportingTokenParameters.Endorsing.Add(item);
}
return securityBindingElement;
}
public override BindingElementCollection CreateBindingElements()
{
BindingElementCollection elements = new BindingElementCollection();
elements.Clear();
if ((SecurityMode.Message == this._securityMode) || (SecurityMode.TransportWithMessageCredential == this._securityMode))
{
elements.Add(this.ApplyMessageSecurity(this.CreateSecurityBindingElement()));
}
elements.Add(this.CreateEncodingBindingElement());
elements.Add(this.CreateTransportBindingElement());
return elements.Clone();
}
protected virtual MessageEncodingBindingElement CreateEncodingBindingElement()
{
return new TextMessageEncodingBindingElement { ReaderQuotas = { MaxArrayLength = 0x200000, MaxStringContentLength = 0x200000 } };
}
protected virtual HttpTransportBindingElement CreateTransportBindingElement()
{
HttpTransportBindingElement element;
if (SecurityMode.Message == this._securityMode)
{
element = new HttpTransportBindingElement();
}
else
{
element = new HttpsTransportBindingElement();
}
element.MaxReceivedMessageSize = 0x200000L;
if (SecurityMode.Transport == this._securityMode)
{
this.ApplyTransportSecurity(element);
}
return element;
}
protected static void ValidateSecurityMode(SecurityMode securityMode)
{
if (((securityMode != SecurityMode.None) && (securityMode != SecurityMode.Message)) && ((securityMode != SecurityMode.Transport) && (securityMode != SecurityMode.TransportWithMessageCredential)))
{
throw new ArgumentOutOfRangeException("securityMode");
}
if (securityMode == SecurityMode.None)
{
throw new InvalidOperationException("ID3224");
}
}
protected void ValidateTrustVersion(TrustVersion trustVersion)
{
if ((trustVersion != TrustVersion.WSTrust13) && (trustVersion != TrustVersion.WSTrustFeb2005))
{
throw new ArgumentOutOfRangeException("trustVersion");
}
}
public bool EnableRsaProofKeys
{
get
{
return this._enableRsaProofKeys;
}
set
{
this._enableRsaProofKeys = value;
}
}
public override string Scheme
{
get
{
TransportBindingElement element = this.CreateBindingElements().Find<TransportBindingElement>();
if (element == null)
{
return string.Empty;
}
return element.Scheme;
}
}
public SecurityMode SecurityMode
{
get
{
return this._securityMode;
}
set
{
ValidateSecurityMode(value);
this._securityMode = value;
}
}
public TrustVersion TrustVersion
{
get
{
return this._trustVersion;
}
set
{
if (value == null)
{
throw new ArgumentNullException("value");
}
this.ValidateTrustVersion(value);
this._trustVersion = value;
}
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,090 |
El Sultanato de Joló o Sultanato de Sulú (árabe: سلطنة سولو; joloano: Kasultanan Sūg; malayo: Kesultanan Suluk) era un estado musulmán que abarcaba gran parte de los territorios en el área del Mar de Joló, en el Suroeste de Filipinas y nordeste de Malasia.
Se cree que el Sultanato fue fundado en 1450, aunque algunos historiadores musulmanes afirman que ya existía siglos atrás, desde los tiempos del Rajá Baguinda Alí.
En su apogeo, se extendía sobre las islas que bordean la península occidental de Mindanao en el este, hasta el actual estado de Sabah en Malasia en el oeste y el sur, y la isla de Palawan en el norte.
Las fuerzas coloniales españolas consiguieron establecer una fortaleza en Zamboanga que significó un apoyo crucial para las sucesivas expediciones a diferentes territorios del Sultanato.
Hasta el día de hoy la cuestión sobre quién es el legítimo sultán de Joló es disputada por los gobiernos de Filipinas y Malasia y varias ramas de la familia real, aunque la línea de sucesión recayó en la rama de la familia real de los Kiram desde 1823 hasta la muerte del último sultán soberano en 1936.
Historia
Antes del Islam
El área actual del Sultanato de Sulu estuvo una vez bajo la influencia del Imperio de Brunéi antes de que obtuviera su propia independencia en 1578. Más tarde, el primer asentamiento conocido en esta área que pronto sería ocupada por el sultanato estaba en Maimbung, Sulu, Maimbung, Jolo. Durante este tiempo, Sulu fue llamado Lupah Sug. El principado de Maimbung, poblado por la gente de Buranun (o Budanon, literalmente significa "habitantes de la montaña"), fue gobernado primero por cierto rajah que asumió el título de Rajah Sipad el Viejo. Según Majul, los orígenes del título rajah sipad provienen del hindú sri pada, que simboliza autoridad. El principado fue instituido y gobernado mediante el sistema de rajás. Sipad el Viejo fue sucedido por Sipad el Joven.
Algunos Chams que emigraron a Sulu se llamaban Orang Dampuan. La Civilización Champa y el puerto-reino de Sulu se involucraron en el comercio entre sí, lo que resultó en que los comerciantes Chams se establecieran en Sulu, donde eran conocidos como Orang Dampuan desde los siglos X-XIII. Los Orang Dampuan fueron asesinados por los envidiosos Sulu Buranuns nativos debido a la riqueza de los Orang Dampuan. Los Buranun fueron luego sometidos a una matanza de represalia por parte de los Orang Dampuan. Más tarde se restableció el comercio armonioso entre Sulu y Orang Dampuan. Los Yakans eran descendientes de los Orang Dampuan de Taguima que llegaron a Sulu desde Champa. Sulu received civilization in its Indic form from the Orang Dampuan.
Durante el reinado de Sipad el Joven, un místico llamado Tuan Mashā′ikha llegó a Jolo en 1280 d.C. Poco se sabe sobre los orígenes y la biografía temprana de Tuan Mashā′ikha, excepto que es un musulmán "que vino de tierras extranjeras" al frente de una flota de comerciantes musulmanes. o salió de un tallo de bambú y fue considerado un profeta, por lo tanto muy respetado por la gente. Otros informes, sin embargo, insistieron en que Tuan Mashā′ikha junto con sus padres, Jamiyun Kulisa e Indra Suga, fueron enviados a Sulu por Alejandro Magno (conocido como Iskandar Zulkarnain en Anales malayos). Sin embargo, Najeeb Mitry Saleeby, un médico estadounidense libanés que escribió "Una historia de Sulu" en 1908 y otros estudios de los Moros, descarta esta afirmación al concluir que Jamiyun Kulisa e Indra Suga eran nombres míticos. Según tarsila, durante la llegada de Tuan Mashā′ikha, la gente de Maimbung adoraba tumbas y piedras de cualquier tipo. Después de predicar el Islam en la zona, se casó con la hija de Sipad el Joven, Idda Indira Suga y tuvo tres hijos: Tuan Hakim, Tuan Pam y 'Aisha. Tuan Hakim, a su vez, engendró cinco hijos. De la genealogía de Tuan Mashā′ikha, comenzó en Sulu otro sistema titular de aristocracia llamado "tuanship". Además de Idda Indira Suga, Tuan Mashā′ikha también se casó con otra "mujer no identificada" y engendró a Moumin. Tuan Mashā′ikha murió en 710 año Hijri (equivalente a 1310 dC), y fue enterrado en Bud Dato cerca de Jolo, con una inscripción de Tuan Maqbālū.
A Durante la llegada de Tuan Mashā′ikha, el pueblo Tagimaha (literalmente significa "el partido del pueblo") proveniente de Basilan y varios lugares en Mindanao, también llegó y se estableció en Buansa. Después de Tagimaha llegó el pueblo Baklaya (que significa "habitantes de la costa"), que se cree que se originó en Sulawesi y se asentó en Patikul. Después de estos vinieron los pueblo Bajau (o Samal) de Johor. Los Bajau fueron conducidos accidentalmente hacia Sulu por un fuerte monzón, algunos de ellos a las costas de Brunéi y otros a Mindanao. La población de Buranun, Tagimaha y Baklaya en Sulu creó tres partidos con distintos sistemas de gobierno y súbditos. En los años 1300, los anales chinos, Nanhai zhi, informaron que Brunéi invadió o administró los reinos filipinos de Butuan, Sulu y Ma-i (Mindoro) que recuperarían su independencia en una fecha posterior. A
Según el Nagarakretagama, el Imperio Majapahit bajo el emperador Hayam Wuruk, invadió Sulu en el año 1365. Sin embargo, en 1369, los Sulus se rebelaron y recuperaron la independencia y en venganza, asaltaron el Imperio Majapahit y su provincia. 'Po-ni (Brunei), y había invadido la costa noreste de Borneo y luego fue a la capital, saqueándola de tesoros y oro. En el saqueo de Brunei, los Sulus habían robado 2 perlas sagradas del rey de Brunei. Una flota de la capital de Majapahit logró ahuyentar a los Sulus, pero "Po-ni" quedó más débil después del ataque. Hacia 1390 d. C., Rajah Baguinda Ali, un príncipe del Reino de Pagaruyung llegó a Sulu y se casó con miembros de la nobleza local. Al menos en 1417, según los anales chinos, tres reyes (o monarcas) gobernaban tres reinos civilizados en la isla. Patuka Pahala (Paduka Batara) gobernaba el reino del este, era el más poderoso; el reino del oeste fue gobernado por Mahalachi (Maharajah Kamal ud-Din); y el reino cerca de la cueva (o Rey de la Cueva) era Paduka Patulapok. Los colonos de Bajau se distribuyeron entre los tres reinos.
Los descendientes de Moumin, el hijo de Tuan Mashā′ikha poblaron Sulu. Después de algún tiempo, un tal Timway Orangkaya Su'il fue mencionado por la segunda página de tarsila, que recibió cuatro esclavos Bisaya (Personas de los Kedatuan de Madja-as) de Manila (presumiblemente Reino de Maynila) como una señal de amistad entre los dos países Los descendientes de Timway Orangkaya Su'il heredaron el título timway, que significa "jefe". En la tercera página de tarsila, se cuenta el hecho de que los esclavos fueron los antepasados de los habitantes de la isla de Parang, Lati, Gi'tung y Lu'uk respectivamente.
Luego, la cuarta página narra la llegada de los Buranun (a los que se hace referencia en la "tarsila" como "el pueblo Maimbung") Tagimaha, Baklaya y luego los inmigrantes Bajau de Johor. La condición de Sulu antes de la llegada del Islam se puede resumir como tal: La isla estaba habitada por varias culturas y era gobernada por tres reinos independientes gobernados por los pueblos Buranun, Tagimaha y Baklaya. Asimismo, los sistemas sociopolíticos de estos reinos se caracterizaron por varias instituciones distintas: rajahship, datusship, tuanship y timwayship. La llegada de Tuan Mashā′ikha luego estableció una comunidad islámica central en la isla.
Establecimiento
Durante el siglo XIII, colonizadores de Minangkabau comenzaron a establecerse a lo largo de la costa oeste de la isla de Sumatra, de Meulaboh hasta Bengkulu, mientras que comerciaban en especias con en el Aceh. En Aceh, se les conoce como Aneuk Jamee Según algunas teorías, el Rajá Baguinda emigró al sur de Filipinas y fundó el Sultanato de Joló en 1390.
Otra fuente afirma que durante la década de 1450, Shariful Hashem Syed Abu Bakar, un árabe nacido en Johor, llegó a Joló desde Malaca. En 1457, fundó el Sultanato de Joló, y luego se rebautizó a sí mismo como "Paduka Maulana Mahasari Sharif Sultan'' Hashem Abu Bakar", adornando su nombre con no menos de cinco títulos consecutivos deferentes: "Paduka" es un término local para "Amo", "Maulana" es una palabra de origen árabe, que significa "lo mismo", "Mahasari" significa "Su Majestad", "Sharif (Jerife)" es una palabra árabe que significa "gobernante local", mientras que "Sultán" es un título utilizado en algunos países islámicos equivalente al de rey o monarca.
En 1658 (aunque otras fuentes afirman que fue en 1703) el Sultanato de Joló recibió la región de Borneo del Norte del Sultán de Brunéi, en agradecimiento después de que el Sultanato de Joló hubiera enviado ayuda para sofocar una rebelión en Brunéi. En ese mismo año, el Sultanato de Joló entregó las Islas de la Paragua a Qudarat, Sultán de Maguindanao, quien se casó con una princesa de Joló, formando una alianza. El Sultán Qudarat finalmente cedió Paragua al Imperio Español en 1705 y Basilán en 1762.
Sabah
En 1865, el cónsul de Estados Unidos en Brunéi, Claude Lee Moses, obtuvo un contrato de arrendamiento por diez años para el territorio de Borneo septentrional. Sin embargo, Moses vendió sus derechos a la American Trading Company en Hong Kong. Asediado con dificultades financieras, la compañía cedió sus derechos en el Norte de Borneo al cónsul del Imperio austrohúngaro en Hong Kong, el barón Von Overbeck. Von Overbeck consiguió una renovación de diez años del contrato de arrendamiento por parte del "Temenggong" de Brunéi, y un tratado similar del Sultán de Joló el 22 de enero de 1878.
Para financiar sus planes para el norte de Borneo, Overbeck consiguió el apoyo financiero de los hermanos Dent - Alfred y Edward Dent. Sin embargo, fue incapaz de involucrar a su gobierno en el desarrollo del territorio. Von Overbeck se retiró de la operación en 1880, dejando a Alfred Dent en el control. Dent fue entonces apoyado por Rutherford Alcock y el almirante Harry Keppel. En julio de 1881, Alfred Dent y su hermano formaron la British North Borneo Company y en mayo de 1882 la empresa procedió a organizar los asentamientos y la administración del territorio, a pesar de las protestas diplomáticas del gobierno holandés, español y de Sarawak.
Aunque en 1888, el norte de Borneo, junto con Sarawak y Brunéi adquirieron el estatus de protectorados de Gran Bretaña, su administración sin embargo, permaneció en manos de la British North Borneo Company. En retrospectiva, el Ministerio de Asuntos Exteriores británico admitió que la cuestión de la soberanía seguía dependiendo de la voluntad del Sultán de Joló y no podía ser delegada en cualquiera de las partes, porque el tratado de 1878 lo prohibía expresamente. En concreto, el tratado de 1878 se específica claramente que "los derechos y facultades de arrendamiento no se transferirán a otra nación o otra empresa de otra nacionalidad, sin el consentimiento del Sultanato de Brunéi y el Sultanato de Joló".
A pesar de que mediante un referéndum supervisado por las Naciones Unidas, la parte de Borneo del Norte conocida como Sabah, se adhirió a Malasia en 1963, su estado es disputado tanto por los herederos de la rama Kiram, así como por el gobierno filipino. Sin embargo, todos los intentos para resolver el problema en la Corte Internacional de Justicia están siendo bloqueados por la falta de voluntad del gobierno de Malasia.
Véase también
Sabah
Historia de Filipinas
Historia de Brunéi
Historia de Malasia
Referencias
Enlaces externos
Lista oficial de sultanes de Joló
El antiguo sitio web oficial del Sultanato - Sultán Esmail Dalus Kiram II
El nuevo sitio web oficial del Sultanato - Sultán Esmail Dalus Kiram II
Estados y territorios fundados en 1450
Estados y territorios desaparecidos en 1917
Antiguos sultanatos de Asia
Nación Mora
Estados desaparecidos de Asia en el siglo XX
Estados desaparecidos del Sudeste Asiático
Historia de Filipinas
Provincia de Joló | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,853 |
<div class="modal-header">
<h3>Edit Device</h3>
</div>
<div class="modal-body">
<section>
<form name="createDeviceForm" novalidate="novalidate" class="form-horizontal">
<div class="form-group">
<label for="modal-name" class="col-sm-2 control-label modal-label"><span>Device Name:</span></label>
<div class="col-sm-10">
<input id="modal-name" name="deviceName" type="text" data-ng-model="deviceCopy.name" placeholder="Enter device name" data-ng-required="true" class="form-control"/>
</div>
</div>
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<div class="modal-footer">
<button data-ng-click="$dismiss()" class="btn btn-default"><i class="glyphicon glyphicon-remove"></i><span> Cancel</span></button>
<button data-ng-click="$close(deviceCopy)" class="btn btn-primary"><i class="glyphicon glyphicon-plus"></i><span> Update</span></button>
</div> | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,388 |
{"url":"https:\/\/protopixel.net\/doc\/create\/protopixel_labs.html","text":"# ProtoPixel Labs\u00b6\n\nThis section has documentation on optional features that are in protoPixel Create, but not yet stable to be a in a regular part of the documentation. You can use them, if something breaks please report it to us!\n\nWarning\n\nThe functionality documented in this section is still experimental.\n\n## OSC Interface\u00b6\n\nProtoPixel has a native Open Sound Control interface exposed in the port 2345.\n\nYou can affect contents in your project by sending OSC messages to this port. The OSC address schema is like follows:\n\n\/<type>\/<name>\/<param> <value>\n\/<type>\/<name>\/<param>\n\n\nWith those parameters:\n\n\u2022 <type> is the entity type, and at the moment it can only be Content.\n\u2022 <name> is the name of the entity.\n\u2022 <param> is the parameter of the entity to be modified. You can see those parameters names by accessing to the Content section in the WebApp. If the parameter is inside a parameter group, you can use \/ to separate group from parameter name. See the examples.\n\u2022 <value> is the new value of the parameter. It can be omitted if the parameter is a button.\n\n### OSC Messages Examples\u00b6\n\n# enable content\n\/Content\/mycontent\/enabled 1\n\n# disable content\n\/Content\/mycontent\/enabled 0\n\n# play video content\n\/Content\/rainbow.mp4\/params\/play\n\n# stop video content\n\/Content\/rainbow.mp4\/params\/stop\n\n# change color for a color content (R, G, B, A)\n\/Content\/color\/params\/color 255 100 100 255\n\n\n### Custom OSC bindings\u00b6\n\nCustom OSC bindings are also available in scripts. See custom_osc.py example in the examples\/scripting folder for more details.\n\n## UDP Interface\u00b6\n\nThere is also a UDP interface in the port 2344. The mechanics are the same as the OSC Interface. The messages are composed like the following:\n\n\/<type>\/<name>\/<param> <JSON-encoded value>\n\n\nFor instance:\n\n# enable content\n\/Content\/mycontent\/enabled 1\n\n# change color for a color content (R, G, B, A)\n\/Content\/color\/params\/color [255 100 100 255]","date":"2021-04-13 07:58:15","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\": 0.2083682119846344, \"perplexity\": 5508.802498480546}, \"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-2021-17\/segments\/1618038072175.30\/warc\/CC-MAIN-20210413062409-20210413092409-00156.warc.gz\"}"} | null | null |
<?php
namespace Engine;
class Module
{
public function getDependencyModules()
{
return array(
'Blog',
'Oauth',
);
}
public function getAutoloaderConfig()
{
return array(
'Zend\Loader\StandardAutoloader' => array(
'namespaces' => array(
__NAMESPACE__ => __DIR__ . '/src/' . __NAMESPACE__,
),
),
);
}
public function getConfig()
{
return include __DIR__ . '/config/module.config.php';
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,670 |
// ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ
// TNT Basic
// CWrappingViewport.cpp
// © Mark Tully 2002
// 6/2/02
// ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ
/*
An extension of the viewport idea that allows a viewport to represent a larger scrollable area than the true canvas area
by wrapping around when scrolling off the edge. If you scroll off the right edge by 10 pixels, you get a band of 10 pixels
from the left edge instead.
This is useful for performing functions such as tile scrolling.
A wrapped viewport has a real area and an affective area (which is usually bigger). The affective area is created by
wrapping the coordinates that are off the real area back onto the other side.
This is useful for tile scrolling because it means that if continuously scrolling you just
keep drawing 1 row of tiles on the edge and moving your rect, as if you were tile scrolling with an infinite canvas.
I decided not to add wrapping viewports to TNT Basic, except for use for tile scrolling. The reason being that it upsets
the user's perception of sprites because they're aware the canvas they opened is only so large and could be confused by
sprites being wrapped (as they would be).
With tile scrolling this is less of a problem because the user thinks of the canvas as being as large as the map and so
sprites wrapping goes unnoticed.
If it could all be explained with less difficulty than the usefulness it would grant then we could add generic use of
wrapped viewports, but I think that tmap only is sufficient; being both easier to explain and to visualise.
*/
#include "CWrappingViewport.h"
#include "CGraphicsContext16.h"
#include "CGraphicsContextGL.h"
#include "CGLCanvas.h"
#include "CTBSpriteGL.h"
// ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ
// ¥ Constructor
// ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ
// The effective width/height are simulated width of the canvas being scrolled. The max offset is effectiveWidth-width of viewport
CWrappingViewport::CWrappingViewport(
CProgram &inProgram,
TTBInteger inCanvasId,
const Rect &inDestRect,
TTBInteger inEffectiveWidth,
TTBInteger inEffectiveHeight) :
mEffectiveWidth(inEffectiveWidth),
mEffectiveHeight(inEffectiveHeight),
CViewport(inProgram,inCanvasId,inDestRect)
{
}
// ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ
// ¥ Render
// ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ
// Renders the viewport. This can involve up to four individual renders depending on how the viewport is offset over the edges
// of the canvas
void CWrappingViewport::Render()
{
CGraphicsContext *graphics=mProgram->CheckGraphicsMode();
if (graphics->GetType()==CGraphicsContext::kBlastGraphicsContext)
RenderBlastViewport();
else
RenderOpenGLViewport();
}
// ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ
// ¥ RenderBlastViewport
// ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ
void CWrappingViewport::RenderBlastViewport()
{
CGraphicsContext *graphics=mProgram->CheckGraphicsMode();
CCanvas *fromCanvas=graphics->mCanvasManager.GetCanvas(mCanvas);
if ((mUpdateAll) || (((CBLCanvas*)fromCanvas)->GetDrawBuffer()->GetUsedBlitRects()))
{
CCanvas *toCanvas=graphics->mCanvasManager.GetCanvas(0);
Rect copyRect=GetViewportRect();
TTBInteger srcCanvasWidth=fromCanvas->GetWidth(),srcCanvasHeight=fromCanvas->GetHeight();
// if scrolled right off the edge of the canvas, wrap the source rect back in bounds
if (copyRect.left>=srcCanvasWidth)
{
copyRect.left%=srcCanvasWidth;
copyRect.right=copyRect.left+GetWidth();
}
if (copyRect.top>=srcCanvasHeight)
{
copyRect.top%=srcCanvasHeight;
copyRect.bottom=copyRect.top+GetHeight();
}
// Left side split copy, when the viewport has scrolled of the right edge
if (copyRect.right>srcCanvasWidth)
{
if (copyRect.bottom>srcCanvasHeight) // scrolled off the bottom too
{
// we're in a state of 4 quarters now
// '''''''''''''''''''''''
// ' (1) | (2) '
// ' | '
// 'ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ'
// ' (3) |(0,0) (4) '
// ' | '
// '''''''''''''''''''''''
// copy on screen (2) and (4)
TTBInteger widthOfOnScreenRightHalf=copyRect.right%srcCanvasWidth; // no pixels it's overlapping on the right border
TTBInteger widthOfOnScreenLeftHalf=FRectWidth(copyRect)-widthOfOnScreenRightHalf;
TTBInteger heightOfOnScreenBottomHalf=copyRect.bottom%srcCanvasHeight; // no pixels it's overlapping on the bottom border
TTBInteger heightOfOnScreenTopHalf=FRectHeight(copyRect)-heightOfOnScreenBottomHalf;
Rect tempCopyRect;
tempCopyRect.left=0;
tempCopyRect.right=widthOfOnScreenRightHalf;
tempCopyRect.top=0;
tempCopyRect.bottom=heightOfOnScreenBottomHalf;
toCanvas->Copy(fromCanvas,tempCopyRect,mBounds.left+widthOfOnScreenLeftHalf,mBounds.top+heightOfOnScreenTopHalf);
tempCopyRect.bottom=srcCanvasHeight;
tempCopyRect.top=srcCanvasHeight-heightOfOnScreenTopHalf;
toCanvas->Copy(fromCanvas,tempCopyRect,mBounds.left+widthOfOnScreenLeftHalf,mBounds.top);
// copy on screen (1) and (3)
tempCopyRect.right=srcCanvasWidth;
tempCopyRect.left=srcCanvasWidth-widthOfOnScreenLeftHalf;
tempCopyRect.top=0;
tempCopyRect.bottom=heightOfOnScreenBottomHalf;
toCanvas->Copy(fromCanvas,tempCopyRect,mBounds.left,mBounds.top+heightOfOnScreenTopHalf);
tempCopyRect.bottom=srcCanvasHeight;
tempCopyRect.top=srcCanvasHeight-heightOfOnScreenTopHalf;
toCanvas->Copy(fromCanvas,tempCopyRect,mBounds.left,mBounds.top);
}
else
{
// two half split
// '''''''''''''''''''''''
// ' |(0,0) '
// ' | '
// ' | '
// ' | '
// ' | '
// '''''''''''''''''''''''
// on screen left rect is sourced from the right side of the source canvas
// on screen right rect is sourced from the left side of the source canvas
// copy on screen right rect
TTBInteger widthOfOnScreenRightHalf=copyRect.right%srcCanvasWidth; // no pixels it's overlapping on the right border
TTBInteger widthOfOnScreenLeftHalf=FRectWidth(copyRect)-widthOfOnScreenRightHalf;
Rect tempCopyRect=copyRect;
tempCopyRect.left=0;
tempCopyRect.right=widthOfOnScreenRightHalf;
toCanvas->Copy(fromCanvas,tempCopyRect,mBounds.left+widthOfOnScreenLeftHalf,mBounds.top);
// copy on screen left rect
tempCopyRect.right=srcCanvasWidth;
tempCopyRect.left=srcCanvasWidth-widthOfOnScreenLeftHalf;
toCanvas->Copy(fromCanvas,tempCopyRect,mBounds.left,mBounds.top);
}
}
else if (copyRect.bottom>srcCanvasHeight)
{
// two half vertical split
// '''''''''''''''''''''''
// ' '
// ' '
// 'ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ'
// '(0,0) '
// ' '
// '''''''''''''''''''''''
// on screen top rect is sourced from the bottom side of the source canvas
// on screen bottom rect is sourced from the top side of the source canvas
// copy on screen bottom rect
TTBInteger heightOfOnScreenBottomHalf=copyRect.bottom%srcCanvasHeight; // no pixels it's overlapping on the bottom border
TTBInteger heightOfOnScreenTopHalf=FRectHeight(copyRect)-heightOfOnScreenBottomHalf;
Rect tempCopyRect=copyRect;
tempCopyRect.top=0;
tempCopyRect.bottom=heightOfOnScreenBottomHalf;
toCanvas->Copy(fromCanvas,tempCopyRect,mBounds.left,mBounds.top+heightOfOnScreenTopHalf);
// copy on screen top rect
tempCopyRect.bottom=srcCanvasHeight;
tempCopyRect.top=srcCanvasHeight-heightOfOnScreenTopHalf;
toCanvas->Copy(fromCanvas,tempCopyRect,mBounds.left,mBounds.top);
}
else
{
// completely on canvas - cool
toCanvas->Copy(fromCanvas,copyRect,mBounds.left,mBounds.top);
}
((CBLCanvas*)fromCanvas)->GetDrawBuffer()->ClearBlitList();
mUpdateAll=false;
}
}
// ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ
// ¥ RenderOpenGLViewport
// ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ
void CWrappingViewport::RenderOpenGLViewport()
{
CGraphicsContextGL *graphics=(CGraphicsContextGL*)mProgram->CheckGraphicsMode();
CGLCanvas *fromCanvas=(CGLCanvas*)graphics->mCanvasManager.GetCanvas(mCanvas);
float screenWidth=graphics->GetWidth()/2.0f;
float screenHeight=graphics->GetHeight()/2.0f;
float across,down;
Rect copyRect=GetViewportRect();
TTBInteger srcCanvasWidth=fromCanvas->GetWidth(),srcCanvasHeight=fromCanvas->GetHeight();
// if scrolled right off the edge of the canvas, wrap the source rect back in bounds
if (copyRect.left>=srcCanvasWidth)
{
copyRect.left%=srcCanvasWidth;
copyRect.right=copyRect.left+GetWidth();
}
if (copyRect.top>=srcCanvasHeight)
{
copyRect.top%=srcCanvasHeight;
copyRect.bottom=copyRect.top+GetHeight();
}
glPixelStorei(GL_UNPACK_ROW_LENGTH,fromCanvas->GetWidth());
// Left side split copy, when the viewport has scrolled of the right edge
if (copyRect.right>srcCanvasWidth)
{
if (copyRect.bottom>srcCanvasHeight) // scrolled off the bottom too
{
// we're in a state of 4 quarters now
// '''''''''''''''''''''''
// ' (1) | (2) '
// ' | '
// 'ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ'
// ' (3) |(0,0) (4) '
// ' | '
// '''''''''''''''''''''''
TTBInteger widthOfOnScreenRightHalf=copyRect.right%srcCanvasWidth; // no pixels it's overlapping on the right border
TTBInteger widthOfOnScreenLeftHalf=FRectWidth(copyRect)-widthOfOnScreenRightHalf;
TTBInteger heightOfOnScreenBottomHalf=copyRect.bottom%srcCanvasHeight; // no pixels it's overlapping on the bottom border
TTBInteger heightOfOnScreenTopHalf=FRectHeight(copyRect)-heightOfOnScreenBottomHalf;
// Copy section 1
across=(mBounds.left/screenWidth)-1.0f;
down=-(((mBounds.bottom-heightOfOnScreenBottomHalf)/screenHeight)-1.0f);
glPixelStorei(GL_UNPACK_SKIP_ROWS,0);
glPixelStorei(GL_UNPACK_SKIP_PIXELS,fromCanvas->GetWidth()-widthOfOnScreenLeftHalf);
glRasterPos2f(across,down);
glDrawPixels(widthOfOnScreenLeftHalf,heightOfOnScreenTopHalf,GL_RGB,GL_UNSIGNED_BYTE,fromCanvas->GetPixels());
// Copy section 2
across=((mBounds.left+widthOfOnScreenLeftHalf)/screenWidth)-1.0f;
glPixelStorei(GL_UNPACK_SKIP_PIXELS,0);
glRasterPos2f(across,down);
glDrawPixels(widthOfOnScreenRightHalf,heightOfOnScreenTopHalf,GL_RGB,GL_UNSIGNED_BYTE,fromCanvas->GetPixels());
// Copy section 3
across=(mBounds.left/screenWidth)-1.0f;
down=-((mBounds.bottom/screenHeight)-1.0f);
glPixelStorei(GL_UNPACK_SKIP_ROWS,fromCanvas->GetHeight()-heightOfOnScreenBottomHalf);
glPixelStorei(GL_UNPACK_SKIP_PIXELS,fromCanvas->GetWidth()-widthOfOnScreenLeftHalf);
glRasterPos2f(across,down);
glDrawPixels(widthOfOnScreenLeftHalf,heightOfOnScreenBottomHalf,GL_RGB,GL_UNSIGNED_BYTE,fromCanvas->GetPixels());
// Copy section 4
across=((mBounds.left+widthOfOnScreenLeftHalf)/screenWidth)-1.0f;
glPixelStorei(GL_UNPACK_SKIP_PIXELS,0);
glRasterPos2f(across,down);
glDrawPixels(widthOfOnScreenRightHalf,heightOfOnScreenBottomHalf,GL_RGB,GL_UNSIGNED_BYTE,fromCanvas->GetPixels());
}
else
{
// two half split
// '''''''''''''''''''''''
// ' |(0,0) '
// ' | '
// ' | '
// ' | '
// ' | '
// '''''''''''''''''''''''
// on screen left rect is sourced from the right side of the source canvas
// on screen right rect is sourced from the left side of the source canvas
TTBInteger widthOfOnScreenRightHalf=copyRect.right%srcCanvasWidth; // no pixels it's overlapping on the right border
TTBInteger widthOfOnScreenLeftHalf=FRectWidth(copyRect)-widthOfOnScreenRightHalf;
// copy on screen right rect
across=((mBounds.left+widthOfOnScreenLeftHalf)/screenWidth)-1.0f;
down=-((mBounds.bottom/screenHeight)-1.0f);
glPixelStorei(GL_UNPACK_SKIP_ROWS,fromCanvas->GetHeight()-GetHeight()-(GetYOffset()%srcCanvasHeight));
glPixelStorei(GL_UNPACK_SKIP_PIXELS,0);
glRasterPos2f(across,down);
glDrawPixels(widthOfOnScreenRightHalf,GetHeight(),GL_RGB,GL_UNSIGNED_BYTE,fromCanvas->GetPixels());
// copy on screen left rect
across=(mBounds.left/screenWidth)-1.0f;
glPixelStorei(GL_UNPACK_SKIP_ROWS,fromCanvas->GetHeight()-GetHeight()-(GetYOffset()%srcCanvasHeight));
glPixelStorei(GL_UNPACK_SKIP_PIXELS,fromCanvas->GetWidth()-widthOfOnScreenLeftHalf);
glRasterPos2f(across,down);
glDrawPixels(widthOfOnScreenLeftHalf,GetHeight(),GL_RGB,GL_UNSIGNED_BYTE,fromCanvas->GetPixels());
}
}
else if (copyRect.bottom>srcCanvasHeight)
{
// two half vertical split
// '''''''''''''''''''''''
// ' '
// ' '
// 'ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ'
// '(0,0) '
// ' '
// '''''''''''''''''''''''
// on screen top rect is sourced from the bottom side of the source canvas
// on screen bottom rect is sourced from the top side of the source canvas
TTBInteger heightOfOnScreenBottomHalf=copyRect.bottom%srcCanvasHeight; // no pixels it's overlapping on the bottom border
TTBInteger heightOfOnScreenTopHalf=FRectHeight(copyRect)-heightOfOnScreenBottomHalf;
// copy on screen bottom rect
across=(mBounds.left/screenWidth)-1.0f;
down=-((mBounds.bottom/screenHeight)-1.0f);
glPixelStorei(GL_UNPACK_SKIP_ROWS,fromCanvas->GetHeight()-heightOfOnScreenBottomHalf);
glPixelStorei(GL_UNPACK_SKIP_PIXELS,GetXOffset()%srcCanvasWidth);
glRasterPos2f(across,down);
glDrawPixels(GetWidth(),heightOfOnScreenBottomHalf,GL_RGB,GL_UNSIGNED_BYTE,fromCanvas->GetPixels());
// copy on screen top rect
down=-(((mBounds.bottom-heightOfOnScreenBottomHalf)/screenHeight)-1.0f);
glPixelStorei(GL_UNPACK_SKIP_ROWS,0);
glPixelStorei(GL_UNPACK_SKIP_PIXELS,GetXOffset()%srcCanvasWidth);
glRasterPos2f(across,down);
glDrawPixels(GetWidth(),heightOfOnScreenTopHalf,GL_RGB,GL_UNSIGNED_BYTE,fromCanvas->GetPixels());
}
else
{
// completely on canvas - cool
across=(mBounds.left/screenWidth)-1.0f;
down=-((mBounds.bottom/screenHeight)-1.0f);
glPixelStorei(GL_UNPACK_SKIP_ROWS,fromCanvas->GetHeight()-GetHeight()-(GetYOffset()%srcCanvasHeight));
glPixelStorei(GL_UNPACK_SKIP_PIXELS,GetXOffset()%srcCanvasWidth);
glRasterPos2f(across,down);
glDrawPixels(GetWidth(),GetHeight(),GL_RGB,GL_UNSIGNED_BYTE,fromCanvas->GetPixels());
}
glRasterPos2f(0,0);
glPixelStorei(GL_UNPACK_SKIP_PIXELS,0);
glPixelStorei(GL_UNPACK_SKIP_ROWS,0);
glPixelStorei(GL_UNPACK_ROW_LENGTH,0);
RenderSprites();
}
// ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ
// ¥ VerifyNewOffsets /*e*/
// ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ
// Check that the current offsets are valid for the source canvas
// Don't store or change any viewport vars, just check the x,y and throw if it's bad
void CWrappingViewport::VerifyNewOffsets(
TTBInteger inNewX,
TTBInteger inNewY)
{
// Check the viewport is still inside the canvas
if ((inNewY<0) || (inNewX<0) ||
((inNewY+GetHeight())>mEffectiveHeight) || ((inNewX+GetWidth())>mEffectiveWidth))
{
UTBException::ThrowViewportOutOfCanvas();
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,771 |
if [ -z "$1" ]; then
echo "This script must be called with an argument" >&2
exit 1
fi
# Use Tools -> Create Command-Line Launcher
/usr/local/bin/idea $(script/find_spec_or_impl.rb $1) | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,496 |
Q: pycharm - how to import directory as a module, keeping directory name as part of the namespace I have on my local machine following structure
all_code_root\
- utils\
-- some_specific_util.py
- project1\
-- app1.py
- project2\
-- app2.py
(utils and each project hosted in own git repository if that matters)
I'd like to set pycharm to work on a project1, while being able to recognize utils as a module, i.e. to be able to write in app1.py
from utils import some_specific_util
I've tried several options (marking the utils dir a source root, opening and attaching it as my project dependency etc), but I always end up only able to do this in app1
import some_specific_util # referring directly to the file works
#from utils import some_specific_util # this does NOT work, 'utils' is not recognized as a module name to import from
Is using utils.some_specific_util actually possible in pycharm? I see only workaround/hack to create artificial pycharm 'project' on all_code_root level, or create unneeded subdir for utils) (Also tried adding __ init __.py or using File - Invalidate caches from menu; my pycharm version is 2022.2.2)
A: If both python file and directory in same folder then this should be sufficient.
import utils
It will import all the Python files in the utils directory.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,373 |
Henry George Thomas Perry (March 18, 1889 – December 26, 1959) was an English-born real estate and insurance broker, journalist and political figure in British Columbia, Canada. He represented Fort George in the Legislative Assembly of British Columbia from 1920 to 1928 and from 1933 to 1945 as a Liberal.
He was born in Whitwick, Leicester, the son of Samuel Perry and Annie Ward, was educated in Coalville and Loughborough, and came to Canada in 1910, settling in Prince George in 1912. In 1911, Perry married Florence Smith. He was mayor of Prince George from 1917 to 1918, in 1920 and in 1924. He was owner and editor of the Fort George Tribune, The Prince George Citizen, The Nechako Chronicle and the Prince Rupert Daily News. Perry was defeated by Frederick Parker Burden when he ran for reelection in 1928. He ran unsuccessfully for a seat in the House of Commons in 1930. Perry was speaker for the assembly from 1934 to 1937 and served in the British Columbia cabinet as Minister of Education from 1941 to 1945. He was defeated by John McInnis when he ran for reelection to the provincial assembly in 1945.
Perry died in Victoria of a heart attack at the age of 70.
Election results (partial)
References
1889 births
1959 deaths
Speakers of the Legislative Assembly of British Columbia
British Columbia Liberal Party MLAs
Mayors of Prince George, British Columbia
People from Whitwick
British emigrants to Canada | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,060 |
Involuntary admission order; transportation, transfer to local law enforcement.
Provides that in cases in which an alternative transportation provider providing transportation of a minor or a person who is subject to an involuntary admission order becomes unable to continue providing transportation, local law enforcement shall take custody of the minor or person and provide transportation to the proper facility.
02/05/2020 Senate: Printed as engrossed 20105321D-E Open
01/07/2020 Senate: Prefiled and ordered printed; offered 01/08/20 20105321D Open
Senate: Printed as engrossed 20105321D-E
20105321D
SENATE BILL NO. 603
Senate Amendments in [ ] � February 5, 2020
A BILL to amend and reenact �� 16.1-340.2, 16.1-345, 37.2-810,
and 37.2-829 of the Code of Virginia, relating to involuntary admission;
transportation; transfer to local law enforcement.
Patron Prior to Engrossment--Senator Hanger
Referred to Committee on the Judiciary
Be it enacted by the General Assembly of Virginia:
1. That �� 16.1-340.2, 16.1-345, 37.2-810, and 37.2-829 of
the Code of Virginia are amended and reenacted as follows:
� 16.1-340.2. Transportation of minor in the temporary
detention process.
A. In specifying the primary law-enforcement agency and
jurisdiction for purposes of this section, the magistrate shall specify in the
temporary detention order the law-enforcement agency of the jurisdiction in
which the minor resides to execute the order and, in cases in which
transportation is ordered to be provided by the primary law-enforcement agency,
provide transportation. However, if the nearest boundary of the jurisdiction in
which the minor resides is more than 50 miles from the nearest boundary of the
jurisdiction in which the minor is located, the law-enforcement agency of the
jurisdiction in which the minor is located shall execute the order and provide
transportation.
B. The magistrate issuing the temporary detention order shall
specify the law-enforcement agency to execute the order and provide
transportation. However, the magistrate may authorize transportation by an
alternative transportation provider, including a parent, family member, or
friend of the minor who is the subject of the temporary detention order, a
representative of the community services board, or other transportation provider
with personnel trained to provide transportation in a safe manner upon
determining, following consideration of information provided by the petitioner;
the community services board or its designee; the local law-enforcement agency,
if any; the minors treating physician, if any; or other persons who are
available and have knowledge of the minor, and, when the magistrate deems
appropriate, the proposed alternative transportation provider, either in person
or via two-way electronic video and audio or telephone communication system,
that the proposed alternative transportation provider is available to provide
transportation, willing to provide transportation, and able to provide
transportation in a safe manner. When transportation is ordered to be provided
by an alternative transportation provider, the magistrate shall order the
specified primary law-enforcement agency to execute the order, to take the
minor into custody, and to transfer custody of the minor to the alternative
transportation provider identified in the order.
In such cases any case in which a magistrate authorizes
transportation of a minor subject to a temporary detention order by an
alternative transportation provider, a copy of the temporary detention order
shall accompany the minor being transported pursuant to this section at all
times and shall be delivered by the alternative transportation provider to the
temporary detention facility. The temporary detention facility shall return a
copy of the temporary detention order to the court designated by the magistrate
as soon as is practicable. Delivery of an order to a law-enforcement officer or
alternative transportation provider and return of an order to the court may be
accomplished electronically or by facsimile.
The order may include transportation of the minor to such
other medical facility as may be necessary to obtain further medical evaluation
or treatment prior to placement as required by a physician at the admitting
temporary detention facility. Nothing herein shall preclude a law-enforcement
officer or alternative transportation provider from obtaining emergency medical
treatment or further medical evaluation at any time for a minor in his custody
as provided in this section. Such medical evaluation or treatment shall be
conducted immediately in accordance with state and federal law.
C. If an alternative transportation provider providing
transportation of a minor who is the subject of �[ an emergency custody �a
temporary detention ] order becomes unable to
continue providing transportation of the minor at any time after taking custody
of the minor, �[ local law enforcement �the
primary law-enforcement agency ]
for the jurisdiction in which the alternative transportation provider is
located at the time he becomes unable to continue providing transportation
shall take custody of the minor and shall transport the minor to the facility
of temporary detention. In such cases, [ (i) ]
a copy of the temporary detention order shall accompany the minor being
transported and shall be delivered to and returned by the temporary detention facility
in accordance with the provisions of subsection B [ and (ii) if the
alternative transportation provider originally authorized to provide
transportation is a person other than the minors parent, the alternative
transportation provider shall notify the minors parent (a) that primary law
enforcement has taken custody of the minor and is transporting the minor to the
facility of temporary detention and (b) of the name of the law-enforcement
officer providing transportation of the minor and the jurisdiction that such
local law-enforcement officer represents ] .
D. In cases in which an alternative facility of
temporary detention is identified and the law-enforcement agency or alternative
transportation provider identified to provide transportation in accordance with
subsection B continues to have custody of the minor, the local law-enforcement
agency or alternative transportation provider shall transport the minor to the
alternative facility of temporary detention identified by the employee or
designee of the local community services board. In cases in which an
alternative facility of temporary detention is identified and custody of the
minor has been transferred from the law-enforcement agency or alternative
transportation provider that provided transportation in accordance with
subsection B to the initial facility of temporary detention, the employee or
designee of the local community services board shall request, and a magistrate
may enter an order specifying, an alternative transportation provider or, if no
alternative transportation provider is available, willing, and able to provide
transportation in a safe manner, the local law-enforcement agency for the
jurisdiction in which the minor resides or, if the nearest boundary of the
jurisdiction in which the minor resides is more than 50 miles from the nearest
boundary of the jurisdiction in which the minor is located, the law-enforcement
agency of the jurisdiction in which the minor is located, to provide
D. E. A law-enforcement officer may lawfully go
or be sent beyond the territorial limits of the county, city, or town in which
he serves to any point in the Commonwealth for the purpose of executing any
temporary detention order pursuant to this section. Law-enforcement agencies
may enter into agreements to facilitate the execution of temporary detention
orders and provide transportation.
E. F. No person who provides alternative
transportation pursuant to this section shall be liable to the person being
transported for any civil damages for ordinary negligence in acts or omissions
that result from providing such alternative transportation.
� 16.1-345. Involuntary commitment; criteria.
After observing the minor and considering (i) the
recommendations of any treating or examining physician or psychologist licensed
in Virginia, if available, (ii) any past actions of the minor, (iii) any past
mental health treatment of the minor, (iv) any qualified evaluators report,
(v) any medical records available, (vi) the preadmission screening report, and (vii)
any other evidence that may have been admitted, the court shall order the
involuntary commitment of the minor to a mental health facility for treatment
for a period not to exceed 90 days if it finds, by clear and convincing
evidence, that:
1. Because of mental illness, the minor (i) presents a serious
danger to himself or others to the extent that severe or irremediable injury is
likely to result, as evidenced by recent acts or threats or (ii) is
experiencing a serious deterioration of his ability to care for himself in a
developmentally age-appropriate manner, as evidenced by delusionary thinking or
by a significant impairment of functioning in hydration, nutrition,
self-protection, or self-control;
2. The minor is in need of compulsory treatment for a mental
illness and is reasonably likely to benefit from the proposed treatment; and
3. If the court finds that inpatient treatment is not the
least restrictive treatment, the court shall consider entering an order for
mandatory outpatient treatment pursuant to � 16.1-345.2.
Upon the expiration of an order for involuntary commitment,
the minor shall be released unless he is involuntarily admitted by further
petition and order of a court, which shall be for a period not to exceed 90
days from the date of the subsequent court order, or the minor or his parent
rescinds the objection to inpatient treatment and consents to admission
pursuant to � 16.1-338 or subsection D of � 16.1-339 or the minor is ordered to
A minor who has been hospitalized while properly detained by a
juvenile and domestic relations district court shall be returned to the
detention home, shelter care, or other facility approved by the Department of
Juvenile Justice by the sheriff serving the jurisdiction where the minor was
detained within 24 hours following completion of a period of inpatient
treatment, unless the court having jurisdiction over the case orders that the
minor be released from custody. However, such a minor shall not be eligible for
mandatory outpatient treatment.
In conducting an evaluation of a minor who has been properly
detained, if the evaluator finds, irrespective of the fact that the minor has been
detained, that the minor meets the criteria for involuntary commitment in this
section, the evaluator shall recommend that the minor meets the criteria for
involuntary commitment.
If the parent or parents with whom the minor resides are not
willing to approve the proposed commitment, the court shall order inpatient
treatment only if it finds, in addition to the criteria specified in this
section, that such treatment is necessary to protect the minors life, health,
safety, or normal development. If a special justice believes that issuance of a
removal order or protective order may be in the childs best interest, the
special justice shall report the matter to the local department of social
services for the county or city where the minor resides.
Upon finding that the best interests of the minor so require,
the court may enter an order directing either or both of the minors parents to
comply with reasonable conditions relating to the minors treatment.
If the minor is committed to inpatient treatment, such
placement shall be in a mental health facility for inpatient treatment
designated by the community services board which serves the political
subdivision in which the minor was evaluated pursuant to � 16.1-342. If the
community services board does not provide a placement recommendation at the
hearing, the minor shall be placed in a mental health facility designated by
the Commissioner of Behavioral Health and Developmental Services.
When a minor has been involuntarily committed pursuant to this
section, the judge shall determine, after consideration of information provided
by the minors treating mental health professional and any involved community
services board staff regarding the minors dangerousness, whether
transportation shall be provided by the sheriff or may be provided by an
friend of the minor, a representative of the community services board, a
representative of the facility at which the minor was detained pursuant to a temporary
detention order, or other alternative transportation provider with personnel
trained to provide transportation in a safe manner. If the judge determines
that transportation may be provided by an alternative transportation provider,
the judge may consult with the proposed alternative transportation provider
either in person or via two-way electronic video and audio or telephone
communication system to determine whether the proposed alternative
transportation provider is available to provide transportation, willing to
provide transportation, and able to provide transportation in a safe manner. If
the judge finds that the proposed alternative transportation provider is
available to provide transportation, willing to provide transportation, and
able to provide transportation in a safe manner, the judge may order
transportation by the proposed alternative transportation provider. In all
other cases, the judge shall order transportation by the sheriff of the
jurisdiction where the minor is a resident unless the sheriffs office of that
jurisdiction is located more than 100 road miles from the nearest boundary of
the jurisdiction in which the proceedings took place. In cases where the
sheriff of the jurisdiction in which the minor is a resident is more than 100
road miles from the nearest boundary of the jurisdiction in which the
proceedings took place, it shall be the responsibility of the sheriff of the
latter jurisdiction to transport the minor.
If the judge determines that the minor requires transportation
by the sheriff, the sheriff, as specified in this section shall transport the
minor to the proper facility. In no event shall transport commence later than
six hours after notification to the sheriff or alternative transportation
provider of the judges order.
If an alternative transportation provider providing
transportation of a minor becomes unable to continue providing transportation
of the minor at any time after taking custody of the minor, local law
enforcement for the jurisdiction in which the alternative transportation
provider is located at the time he becomes unable to continue providing
transportation shall take custody of the minor and shall transport the minor to
the proper facility. [ In such cases, if the alternative transportation
provider originally authorized to provide transportation is a person other than
the minors parent, the alternative transportation provider shall notify the
minors parent (a) that primary law enforcement has taken custody of the minor
and is transporting the minor to the facility of temporary detention and (b) of
the name of the law-enforcement officer providing transportation of the minor
and the jurisdiction that such local law-enforcement officer represents.
No person who provides alternative transportation pursuant to
this section shall be liable to the person being transported for any civil
damages for ordinary negligence in acts or omissions that result from providing
such alternative transportation.
� 37.2-810. Transportation of person in the temporary
which the person resides, or any other willing law-enforcement agency that has
agreed to provide transportation, to execute the order and, in cases in which
which the person resides is more than 50 miles from the nearest boundary of the
jurisdiction in which the person is located, the law-enforcement agency of the
jurisdiction in which the person is located shall execute the order and provide
transportation. However, the magistrate shall consider any request to authorize
transportation by an alternative transportation provider in accordance with
this section, whenever an alternative transportation provider is identified to
the magistrate, which may be a person, facility, or agency, including a family
member or friend of the person who is the subject of the temporary detention
order, a representative of the community services board, or other
transportation provider with personnel trained to provide transportation in a
safe manner upon determining, following consideration of information provided by
the petitioner; the community services board or its designee; the local
law-enforcement agency, if any; the persons treating physician, if any; or
other persons who are available and have knowledge of the person, and, when the
magistrate deems appropriate, the proposed alternative transportation provider,
communication system, that the proposed alternative transportation provider is
able to provide transportation in a safe manner. When transportation is ordered
to be provided by an alternative transportation provider, the magistrate shall
order the specified law-enforcement agency to execute the order, to take the
person into custody, and to transfer custody of the person to the alternative
In such cases, a copy of the temporary detention order shall
accompany the person being transported pursuant to this section at all times
and shall be delivered by the alternative transportation provider to the
The order may include transportation of the person to such
treatment or further medical evaluation at any time for a person in his custody
transportation of a person who is the subject of �[ an emergency custody �a
continue providing transportation of the person at any time after taking
custody of the person, local law enforcement for the jurisdiction in which the
alternative transportation provider is located at the time he becomes unable to
continue providing transportation shall take custody of the person and shall
transport the person to the facility of temporary detention. In such cases, a
copy of the temporary detention order shall accompany the person being
transported and shall be delivered to and returned by the temporary detention
facility in accordance with the provisions of subsection B.
subsection B continues to have custody of the person, the local law-enforcement
agency or alternative transportation provider shall transport the person to the
designee of the community services board. In cases in which an alternative facility
of temporary detention is identified and custody of the individual has been
transferred from the law-enforcement agency or alternative transportation
provider that provided transportation in accordance with subsection B to the
initial facility of temporary detention, the employee or designee of the
community services board shall request, and a magistrate may enter an order
specifying, an alternative transportation provider or, if no alternative
transportation provider is available, willing, and able to provide
jurisdiction in which the person resides or, if the nearest boundary of the
jurisdiction in which the person resides is more than 50 miles from the nearest
boundary of the jurisdiction in which the person is located, the
law-enforcement agency of the jurisdiction in which the person is located, to
provide transportation.
to or be sent beyond the territorial limits of the county, city, or town in
which he serves to any point in the Commonwealth for the purpose of executing
any temporary detention order pursuant to this section. Law-enforcement
agencies may enter into agreements to facilitate the execution of temporary
detention orders and provide transportation.
� 37.2-829. Transportation of person in civil admission
When a person has volunteered for admission pursuant to �
37.2-814 or been ordered to be admitted to a facility under �� 37.2-815 through
37.2-821, the judge or special justice shall determine after consideration of
information provided by the persons treating mental health professional and
any involved community services board or behavioral health authority staff
regarding the persons dangerousness, whether transportation shall be provided
by the sheriff or may be provided by an alternative transportation provider,
including a family member or friend of the person, a representative of the
community services board, a representative of the facility at which the person
was detained pursuant to a temporary detention order, or other alternative
safe manner. If the judge or special justice determines that transportation may
be provided by an alternative transportation provider, the judge or special
justice may consult with the proposed alternative transportation provider
the judge or special justice finds that the proposed alternative transportation
provider is available to provide transportation, willing to provide
transportation, and able to provide transportation in a safe manner, the judge
or special justice may order transportation by the proposed alternative
transportation provider. In all other cases, the judge or special justice shall
order transportation by the sheriff of the jurisdiction where the person is a
resident unless the sheriffs office of that jurisdiction is located more than
100 road miles from the nearest boundary of the jurisdiction in which the
proceedings took place. In cases where the sheriff of the jurisdiction of which
the person is a resident is more than 100 road miles from the nearest boundary
of the jurisdiction in which the proceedings took place, it shall be the responsibility
of the sheriff of the latter jurisdiction to transport the person.
If the judge or special justice determines that the person
requires transportation by the sheriff, the person may be delivered to the care
of the sheriff, as specified in this section, who shall transport the person to
the proper facility. In no event shall transport commence later than six hours
after notification to the sheriff or alternative transportation provider of the
judges or special justices order.
If any state hospital has become too crowded to admit any such
person, the Commissioner shall give notice of the fact to all community
services boards and shall designate the facility to which sheriffs or
alternative transportation providers shall transport such persons.
transportation of a person becomes unable to continue providing transportation
of the person at any time after taking custody of the person, local law
transportation shall take custody of the person and shall transport the person
to the proper facility.
Emmett W. Hanger R-Richmond Sponsor
01/07/2020 Senate Senate: Prefiled and ordered printed; offered 01/08/20 20105321D
01/07/2020 Senate Senate: Referred to Committee on the Judiciary | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,453 |
\section{\label{sec:intro}Introduction}
The increase of available computational power, together with
the development of more accurate and efficient simulation algorithms,
have made it possible to reliably predict the properties of materials
and molecules of increasing levels of complexity.
Furthermore, high-throughput computational screening of existing
and hypothetical compounds promises to dramatically accelerate
the development of materials with the better performances or
custom-tailored properties \cite{PhysRevLett.114.105503,
PhysRevB.92.014106,PhysRevB.92.094306,kusne15screport,
ramkrsinan_2014sd,PhysRevB.90.155136,PhysRevB.90.155136}.
These developments have made even more urgent the need
for automated tools to analyze, classify~\cite{rodr-laio14science,
clustering-rui,gang_clustering,cartography,prasanna-sd}
and represent~\cite{ferg+10pnas,ceri+11pnas,trib+12pnas,ceri+13jctc,rohr+arpc13}
large amounts of structural data, as well as techniques to
leverage this wealth of information to estimate inexpensively
the properties of materials using machine-learning techniques,
circumventing the need for computationally demanding
quantum mechanical calculations \cite{PhysRevLett.108.058301,
PhysRevB.90.104108,PhysRevB.89.235411,
pilania13screport,PhysRevB.88.054104,rupp+07jcim,
hirn+15arxiv,qm7b,PhysRevLett.108.253002,PhysRevB.92.045131,anatoleIJQC,bag.of.bonds}.
At the most fundamental level, the crucial ingredient for
all these techniques is a mathematical formulation of the
concept of (dis)similarity between atomic configurations,
that can take the form of a distance - that can be used
for dimensionality reduction or clustering -
or of a kernel function, that could be used for
ridge regression or automated classification.\cite{krr-plant,krr-face,rasm05book,hast09book}
The most obvious choice for a metric to compare atomic structures
would involve the Euclidean distance between the Cartesian
coordinates of the atoms, commonly known as root mean square
displacement (RMSD) distance, that can be easily made
invariant to relative translations and rotations. It
is however highly non-trivial to extend the RMSD to
deal with situations in which atoms in the two structures
cannot be mapped unequivocally with each other. The
deterministic evaluation of a ``permutationally invariant''
RMSD scales combinatorially with the size of the molecules
to be compared~\cite{sade+13jcp}, and introduces cusps at locations where the mapping of atom identities changes. Furthermore, as we will
discuss later on, the RMSD is perhaps the most straightforward,
but not necessarily the most flexible or effective strategy
to compare molecular and condensed-phase configurations.
In the last few years, a large number of ``fingerprint''
functions have been developed to represent the state of
structures, or of groups of atoms within a structure.
Structural descriptors have been developed based on graph-theoretic
procedures (e.g. SPRINTs~\cite{sprint}), as well as on
analogies with electronic structrure methods (e.g.
Hamiltonian matrix, Hessian matrix, Overlap matrix of Gaussian
type Orbitals (GTO) or even Kohn-Sham eigenvalues fingerprints~\cite{sade+13jcp}).
Most of these approaches have been introduced to provide a fast and
reliable estimate of the dissimilarity between structures.
Several other descriptors have been also used in machine learning,
to predict properties of materials and molecules circumventing
the need for an expensive electronic structure calculation.
A non-comprehensive list of such methods include Coulomb
matrices~\cite{rupp+07jcim}, bags of bonds~\cite{bag.of.bonds},
``symmetry functions''~\cite{behlerNNP}, scattering transformation
applied on a linear superposition of atomic densities~\cite{hirn+15arxiv}.
A particularly promising approach to compare structures
in a way that is invariant to rotations, translations,
and permutations of equivalent atoms, is to start from
descriptors designed to represent \emph{local} atomic
environments and that fulfill these requirements, and
combine them to yield a \emph{global} measure of
similarity between structures. This idea typically relies
on finding the best match between pairs of environments
in the two configurations~\cite{rupp+07jcim,sade+13jcp,newstefan}, and
can also be traced back to methods developed to compare
images based on the matching of local features~\cite{grau05ieee}.
In the present work we start from a recently-developed
strategy to define a similarity kernel between local
environments -- the smooth overlap of atomic positions
(SOAP)\cite{bart+13prb} -- and discuss the different ways one
can process the set of all possible matchings between atomic environments
to generate a global kernel to compare two structures.
In particular, we introduce a regularized entropy match
(REMatch) strategy that is based on
techniques in optimal-transport
theory~\cite{cutu13nips}, and that is both more efficient
and tunable than previously-applied methods.
We discuss the relative merits of different approaches,
and generalize this strategy to the comparison between
structures with different numbers and kinds of atoms.
We demonstrate the behavior of the different global
kernels when applied to completely different classes of problems,
ranging from elemental clusters, to bulk structures, to
the conformers of oligopeptides and to a heterogeneous
database of small organic molecules. We visualize
the behavior of the distance associated with these kernels
using sketch-map~\cite{ceri+11pnas}, a non-linear dimensionality reduction
technique, and demonstrate the great promise shown
by the straightforward application of the REMatch-SOAP
kernel to the machine-learning of molecular properties.
Finally, we present our conclusions.
\section{\label{sec:theory}Theory}
Let us start by introducing the notation we will employ in the
rest of the paper. We will label structures to be compared by
capital letters, use a lowercase Latin letter to indicate
the index of an atom, and when necessary use a
Greek lowercase letter to mark its chemical identity.
For instance, the position of the $i$-th atom
within the structure $A$ will be labeled as $\mathbf{x}^A_{i}$.
The \emph{environment} of that atom, i.e. the abstract descriptor of the
arrangement of atoms in its vicinity will be labelled with
a calligraphic upper case letter, e.g. $\mathcal{X}^A_{i}$,
and the sub-set of such environment that singles out atoms of
species $\alpha$ will be indicated as $\mathcal{X}^{A,\alpha}_{i}$.
Among the many descriptors of local environments that have been
developed in the recent years\cite{PhysRevLett.114.105503,
PhysRevB.92.014106,PhysRevB.92.094306,ramkrsinan_2014sd,
PhysRevB.90.155136,PhysRevB.90.155136,rupp+07jcim,sade+13jcp,
newstefan,PhysRevLett.108.058301,PhysRevB.90.104108,
PhysRevB.89.235411,pilania13screport,PhysRevB.88.054104,qm7b,
PhysRevLett.108.253002,PhysRevB.92.045131,anatoleIJQC,bag.of.bonds},
we will refer in particular to the SOAP fingerprints \cite{bart+13prb}, that have
been proven to be a very elegant and robust strategy to describe
coordination environments in a way that is naturally invariant
with respect to translations, rotations and permutations of atoms.
We will use the notation $k(\mathcal{X},\mathcal{X}')$ to
indicate the similarity kernel between two environments
-- which one would use in a kernel ridge regression method~\cite{hast09book,scho+98nc,rasm05book} --
and $d(\mathcal{X},\mathcal{X}')^2=2-2k(\mathcal{X},\mathcal{X}')$
to indicate the (squared) kernel distance between the environments
-- which one would use in a dimensionality reduction
method~\cite{ceri+11pnas,rohr+arpc13}.
In what follows we will discuss different ways by which
environment kernels can be combined to yield a
a \emph{global} similarity kernel between two structures $K(A,B)$,
and the associated squared distance $D(A,B)^2=2-2K(A,B)$.
\subsection{SOAP similarity kernels and local environment distance}
We will first focus on the comparison between the environment
of two atoms in a pure compound made up of single atomic species $\alpha$.
The crucial ingredient in making the comparison is a kernel
function based on the distribution of atoms in the two
environments. In the context of SOAP kernels
one represents the local density of atoms within the
environment $\mathcal{X}$ as a sum of Gaussian functions
with variance $\sigma^2$, centered on each of the
neighbors of the central atom,
as well as on the central atom itself:
\begin{equation}\label{eq:atomic-density}
\rho_\mathcal{X}(\mathbf{r}) = \sum_{i\in\mathcal{X}}
\exp\left(-\frac{(\mathbf{x}_i-\mathbf{r})^2}{2\sigma^2}\right).
\end{equation}
The SOAP kernel is then defined as the overlap of the two local atomic
neighbour densities, integrated over all three-dimensional rotations $\hat R$,
\begin{equation}
\tilde{k}(\mathcal{X},\mathcal{X}')=
\int \ensuremath{\mathrm{d}}\hat{R} \left| \int \rho_\mathcal{X}(\mathbf{r}) \rho_\mathcal{X'}(\hat{R}\mathbf{r}) \ensuremath{\mathrm{d}} \mathbf{r}\right|^n.
\label{eq:soap-kernel-nonorm}
\end{equation}
Note that in the $n=1$ case the two integrals can be switched, and therefore the kernel
looses all angular information, so we focus on the $n=2$ case exclusively.
For most applications it is helpful to normalise the kernel
so that the self-similarity of any environment is unity, giving the final kernel
\begin{equation}
k(\mathcal{X},\mathcal{X}')= \tilde{k}(\mathcal{X},\mathcal{X}')/
\sqrt{\tilde{k}(\mathcal{X},\mathcal{X})\tilde{k}(\mathcal{X}',\mathcal{X}')}
\label{eq:soap-kernel-int}
\end{equation}
It is a remarkable property of the SOAP kernel that the integration over all rotations can be carried out analytically. First the atomic neighbour density is expanded in a basis composed of spherical harmonics and a set of orthogonal radial basis functions $\{g_b(r)\}$,
\begin{equation}
\rho_\mathcal{X}(\mathbf{r}) = \sum_{blm} c_{blm} g_b(|{\mathbf r}|) {\mathrm{Y}}_{lm}(\hat {\mathbf r}),
\end{equation}
then the rotationally invariant {\em power spectrum} is given by
\begin{equation}
p(\mathcal{X})_{b_1 b_2l} = \sum_{m} (c_{b_1l})^\dag c_{b_2l}.
\end{equation}
Collecting the elements of the power spectrum into a unit-length vector $\hat{\mathbf p}(\mathcal{X})$, the SOAP kernel is shown\cite{bart+13prb} to be given by
\begin{equation}
k(\mathcal{X},\mathcal{X}')=\hat{\mathbf p}(\mathcal{X})\cdot \hat{\mathbf p}(\mathcal{X'})
\end{equation}
eventually leaving a definition of the distance as
\begin{equation}
d\left(\mathcal{X},\mathcal{X}'\right)=\sqrt{2-2\hat{\mathbf{p}}(\mathcal{X})\cdot \hat{\mathbf{p}}(\mathcal{X}')}
\label{eq:soap-distance}
\end{equation}
The SOAP kernel can be written in the form of a dot product,
therefore it is manifestly positive definite, which implies that
the distance function~\eqref{eq:soap-distance} is a proper metric.
\subsection{From local descriptors to structure matching \label{sec:global}}
The vectors that enter the definition of the environments
are defined in such a way that their dot product is the
overlap of (smoothed) atomic distributions. Given two structures
with the same number $N$ of atoms, we can compute an
\emph{environment covariance matrix}
that contains all the possible pairing of environments
\begin{equation}
C_{ij}(A,B)=k\left(\mathcal{X}^A_{i},\mathcal{X}^B_{j}\right),
\end{equation}
This matrix contains the complete information on the pair-wise
similarity of all the environments between the two systems.
Based on it, one can introduce a global kernel
to compare two structures or molecules. We will discuss
and compare four different approaches. All of them are
meant to be normalized, i.e. the given expressions
for $K(A,B)$ are to be divided by $\sqrt{K(A,A) K(B,B)}$
whenever such normalization is not automatically one.
\paragraph{Average structural kernel}
A first possibility to compare two structures involves computing
an \emph{average kernel}
\begin{equation}
\begin{split}
\bar{K}(A,B) =& \frac{1}{N^2} \sum_{ij} C_{ij}(A,B) =\\
=&
\left[\frac{1}{N}\sum_{i}\mathbf{p}(\mathcal{X}^A_{i})\right]
\cdot
\left[\frac{1}{N}\sum_{j}\mathbf{p}(\mathcal{X}^B_{j})\right] .
\end{split}
\label{eq:k-avg}
\end{equation}
One sees that $\bar{K}$ can be computed inexpensively by just storing
the average SOAP fingerprint between all environments of the two structures.
This kernel is also positive-definite, being based on a scalar product~\cite{berg84book},
and therefore induces a metric $\bar{D}(A,B)=\sqrt{2-2\bar{K}(A,B)}$.
On the other hand, it is not a very sensitive metric:
two very different structures can appear to be the same
if they are composed of environments
that give the same fingerprint upon averaging.
\paragraph{Best-match structural kernel}
Another possibility, that has been used previously with different kinds of structural
fingerprints~\cite{De2011,sadeghi2013a,De2014,rupp+07jcim} is to identify the best match between the environments
of the two structures,
\begin{equation}
\hat{K}(A,B) = \frac{1}{N} \max_{\pi} \sum_{i} C_{i\pi_i}(A,B)
\label{eq:k-match}.
\end{equation}
which can be accomplished with an $\mathcal{O}(N^3)$
effort using the Munkres algorithm~\cite{kuhn55nrlq}.
The corresponding distance has the properties of a metric,
which means it can still be safely used to assess similarity
between structures and molecules.
Unfortunately, this ``best-match'' kernel is not
guaranteed to be positive-definite, which makes it
less than ideal for use in machine-learning
applications. Furthermore, the distance obtained
by a best-match strategy is continuous, but has
discontinuous derivatives whenever the matching
of environments changes.
These problems can be solved or alleviated
by matching the environments based on a different strategy,
that combines features of the average and the best-match kernels.
\paragraph{Regularized entropy match kernel}
The best match problem can be also stated in an alternative
form, namely
\begin{equation}
\hat{K}(A,B) = \max_{\mathbf{P}\in \mathcal{U}(N,N)} \sum_{ij} C_{ij}(A,B) P_{ij}
\label{eq:k-match-b}.
\end{equation}
where $\mathcal{U}(N,N)$ is the set of $N\times N$
(scaled) doubly stochastic matrices,
whose rows and columns sum to $1/N$,
i.e. $\sum_i P_{ij} = \sum_j P_{ij}=1/N$.
We can then borrow an idea that was recently
introduced in the field of optimal
transport\cite{cutu13nips} to \emph{regularize} this
problem, adding a penalty that instead aims
at maximizing the information entropy for the matrix
$\mathbf{P}$ subject to the aforementioned
constraints on its marginals.
Such ``regularized-entropy match'' (REMatch)
kernel is defined as
\begin{equation}
\begin{split}
&\hat{K}^{\gamma}(A,B) = \operatorname{Tr}\mathbf{P}^\gamma\mathbf{C}(A,B), \\
&\mathbf{P}^\gamma =\operatorname*{argmin}_{\mathbf{P}\in \mathcal{U}(N,N)} \sum_{ij} P_{ij} \left(1-C_{ij}-\gamma \ln P_{ij}\right)
\end{split}
\label{eq:k-regmatch},
\end{equation}
where the regularization is given by an entropy term $E(\mathbf{P})=\sum_{ij}P_{ij}\ln P_{ij}$.
$\mathbf{P}^\gamma$ can be computed very efficiently,
with $\mathcal{O}(N^2)$ effort,
by the Sinkhorn algorithm~\cite{cutu13nips}
(see Appendix~\ref{app:sinkhorn}).
For $\gamma\rightarrow 0$, the entropic
penalty becomes negligible, and
$\hat{K}^{\gamma}(A,B)\rightarrow \hat{K}(A,B)$.
For $\gamma\rightarrow \infty$,
one selects the $\mathbf{P}$ with
the least information content, that is one with constant
$P_{ij}=1/N^2$. Hence, in this limit
$\hat{K}^{\gamma}(A,B)\rightarrow \bar{K}(A,B)$.
\paragraph{Permutation structural kernel}
For the sake of completeness, we also discuss a
fourth option: rather than summing over all possible
pairs of environments, one can consider each pairing
of environments separately, and sum over all the
$N!$ possible permutations that define the pairings.
In order to kill off more rapidly the combinations
of environments that contain bad matches, one can
\emph{multiply} the kernels that appear in each pairing,
and define a \emph{permutation kernel}
\begin{equation}
\breve{K}(A,B) = \frac{1}{N!} \sum_{\pi}\prod_{i} C_{i\pi_i}(A,B) = \operatorname{perm}{\mathbf{C}(A,B)}.
\label{eq:k-perm}
\end{equation}
This choice corresponds to the evaluation of the permanent
of the environment kernel matrix, and has some appeal
as it is guaranteed to yield a positive-definite kernel~\cite{cutu07proc}.
The evaluation of the permanent of a matrix, however, has
combinatorial computational complexity\footnote{Although stochastic algorithms do exist
to compute it to a desired precision in polynomial
time~\cite{jerr+04jacm}}.
Its application is limited to small molecules, and we will
not discuss it further in the present work.
\subsection{Matching structures containing multiple species}
When comparing structures that contain different atomic
species, the first problem that has to be addressed is that
of extending the local environment metric so that the presence
of multiple species is properly accounted for.
SOAP descriptors provide a very natural way to do this:
a separate density can be built for each atomic species
\begin{equation}
\rho_\mathcal{X}^\alpha(\mathbf{r}) = \sum_{i\in\mathcal{X}^{}\alpha}
\exp\left(-\frac{(\mathbf{x}_{i}-\mathbf{r})^2}{2\sigma^2}\right),
\end{equation}
and a (non-normalized) kernel be defined by matching
separately the different species:
\begin{equation}
\begin{split}
\tilde{k}(\mathcal{X},\mathcal{X}')=&
\int \ensuremath{\mathrm{d}}\hat{R} \left| \int \sum_\alpha \rho_\mathcal{X}^\alpha(\mathbf{r}) \rho_{\mathcal{X}'}^{\alpha}(\hat{R}\mathbf{r}) \ensuremath{\mathrm{d}} \mathbf{r}\right|^2\\
=&\sum_{\alpha\beta} \mathbf{p}_{\alpha\beta}(\mathcal{X})\cdot
\mathbf{p}_{\alpha\beta}(\mathcal{X}'). \label{eq:k-multispecies}
\end{split}
\end{equation}
Here we have introduced ``partial'' power spectra that encode
information on the relative arrangement of pairs of species,
\begin{equation}
\mathbf{p}_{\alpha\beta}(\mathcal{X})\cdot
\mathbf{p}_{\alpha\beta}(\mathcal{X}')
=
\int \ensuremath{\mathrm{d}}\hat{R} \left| \int \rho_\mathcal{X}^\alpha(\mathbf{r}) \rho_{\mathcal{X}'}^{\beta}(\hat{R}\mathbf{r}) \ensuremath{\mathrm{d}} \mathbf{r}\right|^2.
\end{equation}
The species-resolved power spectrum $\mathbf{p}_{\alpha\beta}$ can be
written as
\begin{equation}
p(\mathcal{X})^{\alpha\beta}_{b_1 b_2l} = \sum_{m} (c^{\alpha}_{b_1lm})^\dag c^\beta_{b_2lm},
\end{equation}
and combines the expansion coefficients
\begin{equation}
\rho_\mathcal{X}^\alpha(\mathbf{r}) = \sum_{blm} c^\alpha_{blm} g_b(|{\mathbf r}|) {\mathrm{Y}}_{lm}(\hat {\mathbf r})
\end{equation}
of the two densities. The kernel in Eq.~\eqref{eq:k-multispecies} can then be
normalized as in Eq.~\eqref{eq:soap-kernel-int}.
Note that the SO(3) power spectrum vectors contain mixed-species
components due to the squaring of the density
overlap within the rotational average. These
mixed terms guarantee that the kernel is sensitive
to the relative correlations of different species,
although the overlap between the environments of
the different species is considered to be zero.
One could however introduce a notion of ``alchemical
similarity'' between different species. For instance, when
comparing structures of III-V semiconductors one could
disregard the chemical information on the identity of
an atom as long as it belongs to the same column of the
periodic table.
Such a notion can be readily implemented,
defining an alchemical similarity kernel $\kappa_{\alpha\beta}$
which is one for pairs that should be considered
interchangeable, and tend to zero for pairs that
one wants to consider as completely unrelated.
The expression then becomes
\begin{equation}
\begin{split}
\tilde{k}(\mathcal{X},\mathcal{X}')=&
\int \ensuremath{\mathrm{d}}\hat{R} \left| \int \sum_{\alpha\beta} \kappa_{\alpha\beta} \rho_\mathcal{X}^\alpha(\mathbf{r}) \rho_{\mathcal{X}'}^{\beta}(\hat{R}\mathbf{r}) \ensuremath{\mathrm{d}} \mathbf{r}\right|^2\\
=&
\sum_{\alpha\beta\alpha'\beta'} \mathbf{p}_{\alpha\beta}(\mathcal{X})
\mathbf{p}_{\alpha'\beta'}(\mathcal{X}')\kappa_{\alpha\alpha'} \kappa_{\beta\beta'}.
\end{split}
\end{equation}
The original expression~\eqref{eq:k-multispecies}
can be recovered by setting $\kappa_{\alpha\beta}=\delta_{\alpha\beta}$.
Global similarity kernels can then be transparently introduced to
compare structures composed of different atomic species, with structure
and alchemical composition treated on the same footings and the possibility
of adapting the definition of similarity to the system and application.
\subsection{Matching structures with different numbers of atoms}
The definitions above can be readily extended to compare structures
containing different numbers of atoms $N_A$ and $N_B$. We discuss two possible
strategies. When comparing crystalline, periodic structures, it may be the
case that one of the structures corresponds to a slight distortion of the
other, that needs a larger unit cell for a proper representation.
Comparing the structures using the average kernel~\eqref{eq:k-avg}
does automatically the ``right thing'', that is performing the
comparison in a way that is independent of the number of times
the two structures have to be replicated to match atom counts.
In the case of the permutation kernel and of the best-match
kernel, the most effective way to perform the
comparison is to evaluate the least common multiple $N$ of $N_A$ and $N_B$, and
replicate the environment similarity matrix to form a square
matrix. One can then proceed to compute the permanent,
or the linear assignment problem, based on such replicated
matrix. The advantage of this procedure is that one does not
need to explicitly
find the relation between the shape of the two unit cells and replicate
them to perform the comparison: the environment similarities can be
evaluated including periodic replicas, and the minimum number of
comparisons will be naturally performed among any pairs of structures.
However, the least common multiple can become very large,
making even the best-match kernel~\eqref{eq:k-match}
impractically demanding, although the cost can be reduced by exploiting the
redundancy in the extended environment covariance matrix.
As shown in the Appendix, the REMatch kernel~\eqref{eq:k-regmatch}
can be computed easily also for a rectangular matrix,
which constitutes an additional advantage of formulating
the environment matching problem in terms of a regularized
transport optimization.
When comparing molecules or molecular fragments, it may
be advisable to proceed differently. One could consider an idealized
pool (``kit'') of isolated atoms from which a number of molecules will be built,
and use them to ``top up'' each
molecule so as to obtain a set of structures, all having
the same number of atomic environments.
The ``reference kit'' could be chosen dynamically for
each pair of molecules, or -- when working with a well
defined database -- fixed globally as the smallest set of
atoms needed to generate all the relevant structures.
One can then define the kernel between an actual atomic
environment and those within this ``virtual atom reservoir'',
$\tilde{k}(\mathcal{X},\emptyset)=e^{-\mu}$, in terms
of a ``chemical potential'' parameter $\mu$.
Since the SO(3) fingerprints that underlie the definition of
the SOAP kernel can also be evaluated for isolated atoms\footnote{The density of atoms defined in equation \eqref{eq:atomic-density} contains the central atom.},
it is also possible to introduce a natural definition
of the covariance between an environment and an isolated
atom, which has the advantage that the global kernels
will then vary smoothly during an actual atomization process.
\subsection{Representing (al)chemical landscapes}
In this work we will demonstrate the flexibility, transferability
and effectiveness of the framework we have just introduced to compare
molecular and condensed-phase structures. To this aim, we will build
two dimensional maps that represent proximity relations between the
structures -- as assessed by the kernel-induced metric -- using
sketch-map~\cite{ceri+11pnas}, a non-linear
dimensionality reduction (NLDR) scheme specifically designed
to deal with atomistic simulation data.
As we will demonstrate, the combination
of SOAP-based structural metrics and NLDR representation provides a
broadly applicable protocol to generate an insightful representation
of the structural and alchemical landscape of complex molecular and
condensed-phase systems. Of course, one could use
the SOAP-based global kernels, or the corresponding distances,
as the basis of other non-linear dimensionality reduction
techniques, such as multi-dimensional scaling~\cite{cox-cox10boox}
or diffusion maps~\cite{coif+05pnas,ferg+10pnas,rohr+arpc13}.
We refer the reader to the relevant literature for a
detailed explanation of the sketch-map
algorithm\cite{ceri+11pnas,trib+12pnas,ceri+13jctc}. The main idea
derives from multi-dimensional scaling, and is based on optimizing
a non-linear objective function
\begin{equation}
S^2 = \sum_{ij} \left[F\left[D(X_i,X_j)\right]-
f\left[d(x_i,x_j)\right]\right]^2 \label{eq:smap}
\end{equation}
where $\left\{X_i\right\}$ and $\left\{x_i\right\}$ correspond
respectively to high-dimensional reference structures
and to vectors in a low-dimensional space. The metric $d$ in
low dimension is typically taken to be the Euclidean distance,
whereas the metric in high dimension could be more complex. In
this case, $X_i$ can be regarded as an abstract descriptor of
a structure or molecule, and $D$ is one of the kernel-based distance
metrics discussed above.
$F$ and $f$ are non-linear sigmoid functions of the form
\begin{equation}
F(r) = 1 - ( 1 + (2^{a/b} - 1)(r/\sigma)^a )^{-b/a},
\end{equation}
which serve to focus the optimization of~\eqref{eq:smap} on
the most significant, intermediate distances, disregarding local
distortion (e.g. induced by thermal fluctuations) and the relation
between completely unrelated portions of configuration landscape.
The choice of the parameters in the sigmoid functions
is discussed in Ref.~\cite{ceri+13jctc}. Here we will label
synthetically each sketch-map representation using the
notation {\texttt{$\sigma$-A\_B-a\_b}} where $A$ and $B$ denote
the exponents used for the high-dimensional function $F$,
$a$ and $b$ denote the exponents for the low-dimensional function $f$,
and $\sigma$ the threshold for the switching function.
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{fig1_small.pdf}
\caption{
The figure compares the value of global structural similarities
for different pairs of structures taken from the
80 local energy minima of C$_{60}$ discussed in Ref.~\cite{De2011}
The structural similarities considered include
the absolute difference in energy per atom, the
(permutation invariant\cite{sade+13jcp}) RMSD per atom,
and the best-match combination of SOAP kernels
computed with different cutoff distances
(2\AA{}, 3.5\AA{}, 7\AA{}).
The correlation between RMSD and $\hat{D}$ based
on 2\AA{}-cutoff SOAP is enlarged, color-coded based on
energy differences and annotated with selected pairs of
structures corresponding to different distances.
}
\label{fig:C60-CORR}
\end{figure*}
\begin{figure*}[tbhp]
\centering
\includegraphics[width=1.0\textwidth]{fig2_small.pdf}
\caption{
The figure compares the value of global structural distances
induced by the average, best-match, and REMatch kernels discussed in Section~\ref{sec:global}, for
80 local-minimum structures of C$_{60}$. On the diagonal
we report the sketch-map projections of the structural landscape
based on the three metrics, colored according to the energy
of each structure, as obtained by Sandip et all ~\cite{De2011}.
Eight representative structures and their positions on the Sketch-maps have been indicated with alphabets on color coded disks. The numeric value on the top of each structure represents their energy in eV, relative to the global minimum. SOAP descriptors were computed using a cutoff of 3.5\AA{} and the Sketch-map parameters are indicated on the map according to the syntax described in the text.
}
\label{fig:C60}
\end{figure*}
\section{Examples and applications}
After having described the theoretical and algorithmic
background of our strategy to define a structural similarity
kernel, let us present a series of applications.
In order to demonstrate that our approach can be seamlessly
applied to the most diverse atomistic simulation problems, we have chosen
examples of increasing complexity, from clusters, to
crystalline and amorphous solids, to biological molecules
and a database of small organic compounds, containing varying number of both atoms and atomic species.
\subsection{The energy landscape of C$_{60}$ clusters}
Let us start with a relatively simple test case.
We consider the same set of 80 local minima for C$_{60}$
discussed in Ref.~\cite{De2011},
which were obtained by exploring the Density Functional Theory energy
landscape of C$_{60}$ using the Minima Hopping ~\cite{minhop} global structure
search algorithm. Figure:~\ref{fig:C60-CORR} contrasts different
similarity matrices: the permutation-invariant RMSD\cite{sadeghi2013a},
the absolute difference between the potential energy, and
the best-match distances obtained from SOAP descriptors
computed with different environment cutoff.
RMSD distance does not correlate very well with
SOAP-based metrics, particularly for the smaller cutoff
value. The $\bar{D}(2\text{\AA})$-RMSD correlation plot is
enlarged, and allows us to discuss the source of this discrepancy.
Hollow fullerene-like structures
(A, with reference to the labeling
in the figure) and compact structures
containing internal connections
(A,G) are extremely different from the point
of view of the short-range
connectivity, but differ comparatively less in terms of RMSD,
since they are both fairly compact. On the other hand, flake-like
structures based on a honeycomb motif (F,E) have the same basic
first-neighbor connectivity as the defective fullerene structures (C,D)
but have much different spatial extent.
Then, one sees that the discrepancy between RMSD and small-cutoff
$\hat{D}$ indicates just the focus on different structural features:
the global arrangement of atoms in the first case, and the local
connectivity in the latter. In the case of SOAP-based metrics, however
it is easy to extend the sensitivity of the metric to longer distances
just by increasing the cutoff: by going from 2\AA{} to 3.5 and 7,
one sees that $\hat{D}$ and RMSD become progressively more
correlated, as the focus shifts from the nearest-neighbor
coordination to the overall geometry of the cluster.
It is worth stressing that the RMSD, albeit a very natural
measure of structural similarity, is not necessarily the best
metric to compare configurations.
To see why, consider the absolute energy difference as
a measure of similarity: even though one can obviously
have configurations with very different geometry and similar
energies, in general one would expect that on the contrary large
energy differences should be
associated with highly dissimilar structures in a given system
-- which is not the case for RMSD.
One sees that the intermediate-cutoff $\bar{D}(3.5\text{\AA})$
shows a nice correlation between energetic and structural differences.
These considerations underline a theme that will recur
in other examples: SOAP-based structural metrics offer a
mathematically sound framework
that can be transparently adapted to focus on the aspects that are
most relevant to a given application. For instance, power-spectrum
based environment kernels are invariant to mirror symmetry, and
therefore the derived metrics cannot distinguish enantiomers.
If one needed to do so, however, it would be sufficient to use
a bispectrum-based SOAP kernel~\cite{bart+13prb} --
which corresponds to $n=3$ in
eq. (\ref{eq:soap-kernel-nonorm}) and is
invariant to rotations but not to mirror symmetry operations --
as the basis for obtaining a global comparison
that is sensitive to chirality.
Having established a connection between traditional structural
similarity metrics and the best-match SOAP kernel, let us use
the example of C$_{60}$ to compare the three main strategies
we propose to build a global kernel: the average kernel $\bar{K}$,
the best-match kernel $\hat{K}$, and the regularized entropy match kernel
$\hat{K}^\gamma$ with an intermediate regularization parameter
$\gamma=0.1$.
The distance-distance correlation plot for each pair of structures,
that compares the distances induced by the three kernels,
is reported in Fig.\ref{fig:C60}.
The $\bar{D}-\hat{D}$ plot shows overall linear correlation
except for very small values of $\bar{D}$.
This is expected as the average kernel is under-determined, and
could in principle label two structures as identical even though
they might composed of different environments. The best-match
kernel, therefore, provides better resolving power.
As we will discuss in more detail later on, the regularized
best-match kernel $\hat{D}^\gamma$ can be
tuned to interpolate between these
two extremes. As an example, we chose here an intermediate
value $\gamma=0.1$: as shown in Fig. \ref{fig:C60}, the
resulting distance correlates strongly with both $\bar{D}$
and the conventional best-match distance $\hat{D}$.
\begin{figure*}[tbhp]
\centering
\includegraphics[width=\textwidth]{fig3_small.pdf}
\caption{Sketch-map of 1274 crystalline and amorphous silicon structures
obtained by sampling different phases from the
phase diagram (disks), polymorphs obtained by ab initio random
structure search\cite{airss} ($+$ signs) and by minima hopping\cite{sipoly} ($\times$ signs).
The color and size of the points varies according to their
atomic energy and atomic volumes respectively. Regions
of the plot which represents different phases have been outlined
with dotted contours.
\label{fig:silicon}
}
\end{figure*}
Fig.\ref{fig:C60} also shows annotated sketch-maps obtained
based on the three metrics.
Once the sketch-map parameters have
been adjusted following the guidelines in Ref.~\cite{ceri+13jctc},
the three maps are effectively equivalent -- indicating
that the three kernels give similar qualitative information on the similarity
between different structures. Given the much lower
computational cost associated with the evaluation of the average
kernel, this observation suggests it might conveniently be used to
preliminarily screen a dataset before proceeding to a more accurate
comparison of similar structures based on the best-match, or REMatch distance.
\subsection{Natural and hypothetical polymorphs
of silicon}
As a second example, let us move on to a condensed-phase
application. Here we start from a database of 1274 bulk silicon
structures containing ideal and distorted structures
from the phase diagram (e.g. diamond, simple hexagonal,
$\beta$-tin, liquid and quenched amorphous structures).
SOAP environment kernels with a 5 \AA{} cutoff distance
were used, and combined with a best-match strategy
to obtain the (dis)similarity matrix
\footnote{Some of the structures in the overall data set have
numbers of atoms in the unit cell that would lead to a
large least common multiple when repeating the environment
similarity matrix to form a square matrix. One can keep
the cost of computing the similarity matrix low by
exploiting the redundancy in the similarity matrix,
or by approximating $\hat{K}$ using a REMatch kernel
with a small entropy regularization.}
We selected 100 landmark configurations out of this
data set (using farthest point sampling based on
kernel distance) and built a sketch-map, on which
the rest of configurations were projected.
The outcome of such mapping procedure is shown in Fig.~\ref{fig:silicon},
where points are colored according to the DFT atomic energy,
and point sizes are scaled to a size proportional to volume per atom.
As seen in the Fig.~\ref{fig:silicon} the map is extremely well
correlated with both atomic energy and density.
Furthermore, structures that were obtained by distorting and
heating up structures coming from different portions of the
phase diagram are clustered together: rough outlines have
been drawn on the map to indicate different phases.
Although the map has been built using only reference
configurations from a few of the conventional Si phases, we have also projected
on it (using out-of-sample embedding) two sets of
hypothetical configurations obtained by minima hopping~\cite{sipoly} and by ab initio random structure search (AIRSS)~\cite{airss,rapp-ncom-2015}. These structures were not included
in the landmarks selection phase. Still, the out-of-sample
embedding procedure correctly identifies not only that in
most cases AIRSS structures differ significantly from
stable phases of silicon, but also clusters together
hypothetical polymorphs that share common features.
For instance, the AIRSS structures outlined in the lower
portion of the map are all taken from Ref.~\cite{rapp-ncom-2015}.
The structures were proposed as possible metastable polymorphs
arising as a result of microexplosion (powerful ultrashort
and tightly focused laser pulse) in crystalline cubic diamond
silicon phase, hence their structural motif naturally
carries resemblance with silicon diamond phase. It is
interesting to see that they indeed are projected
close to the diamond phase on the map.
All of the minima hopping low-density Si polymorphs
are also clustered together, which is consistent with
the fact that they are all based on combinations of a few
base motifs.
Thus, Figure~\ref{fig:silicon} shows not only that
SOAP-based structural similarity distances can be very
effective in the study of bulk crystalline structures,
but also testify the extrapolative power of a sketch-map
representation based on such a metric.
\begin{figure*}[btph]
\centering
\includegraphics[width=\textwidth]{fig4_small.pdf}
\caption{Sketch-map representation of locally stable
arginine dipeptide conformers, without (top) and with (bottom) a Ca$^{2+}$ ion.
Left-hand panels are colored according to the energy relative to the minimum energy form, while the smaller maps on the right are colored according to the values of different dihedral
angles, as indicated in the legend.
}
\label{fig:arg-cfg}
\end{figure*}
\begin{figure}[tbph]
\centering
\includegraphics[width=\columnwidth]{fig5_small.pdf}
\caption{Sketch-map representation of stable configurations of Arginine dipeptide complexed with a Ca$^{2+}$ ion. The
structures that have undergone a proton transfer reaction
relative to the neutral molecule have been highlighted, and
a few representative snapshots of the molecular structure
are also reported.}
\label{fig:arg-chem}
\end{figure}
\subsection{Arginine Dipeptide}
Having shown that SOAP-based structural similarity kernels
are equally effective for clusters and for bulk configurations
of elemental materials, let us consider a case of a multi-species
chemical compound. We selected a library of 5062 locally stable
conformers of arginine dipeptide (845 with and 4217 without a Ca$^{2+}$
counterion) from a public database of
oligopeptides structures developed
by Matti Ropo et al \cite{aminodb}.
We used a cut-off of 3.5\AA{} in the definition of environment
SOAP kernels, and combined them using a best-match strategy.
Since H atoms stay at almost fixed positions relative to their
neighboring atoms, we decided to include them in the
environment descriptors of other atoms, but did not include them
explicitly as centers of atomic environments. This is another
example of how SOAP-based structural metrics are effective in
a broad variety of contexts, but at the same time can be
easily and transparently refined based on intuition,
prior experience, or a clear understanding of the objectives
of the structural comparison.
In Fig.~\ref{fig:arg-cfg} we show the sketch-map representation
for these two sets of structures, highlighting the correlation
between the location on the map and structural and energetic
properties of the conformers. In the absence of a complexing
cation, the dipeptide can exist in a very large number of
local minima, spanning a relatively narrow range of energies.
The map shows very clearly partitioning of configuration space
in four disconnected regions.
Conventional wisdom~\cite{ramachandran-plot} assumes
that the C$_\alpha$ dihedral angles $\phi$ and $\psi$
are the most important descriptors of oligopeptide
structure. One quickly realizes, however, that
the order parameters corresponding to the
four lobes are connected to the cis-trans isomerization
of the two peptide bonds. Within each of the lobes, configurations
with different $\phi$-$\psi$ dihedral angles are clearly clustered
together, but in this case they constitute features of
secondary importance. This observation demonstrates
the advantages of using a general-purpose descriptor,
that does not rely on pre-conceived assumptions on
the behavior of the molecule being studied,
but instead captures automatically
the intrinsic structural hierarchy of minima in the configuration landscape.
The presence of a Ca$^{2+}$ cation has a dramatic
impact on the landscape for the dipeptide. The distribution of
configurations becomes considerably more sparse and span a broader
range of energies. The strong electrostatic interaction with the
cation means that there is not a clear separation anymore between
the energy scale for $\phi$-$\psi$ flexibility of the backbone and
the isomerization of the peptide bonds.
A remarkable observation in this analysis is the
realization that the presence of the cation catalyzed
unexpected proton transfer reactions, that change the
chemical structure of the molecule. Configurations that
underwent a chemical reaction are clustered on one side of the map
(Fig.~\ref{fig:arg-chem}),
with further internal structure reflecting the fact that
SOAP-based structural metrics treat on the same footing
information on the chemical bonding and on the conformational
variability of the molecule. It is again worth noting that
by changing the cut-off value for the SOAP descriptors, one
can ``focus'' the structural metric on different molecular
features. A short cutoff of 2\AA{} makes the chemically
different structures stand out more as outliers -- which would
for instance be useful to detect automatically this kind
of unwanted transitions in an automatically-generated data set --
while on the contrary a longer cutoff would give more
importance to the difference between collapsed and extended
molecular conformers.
\begin{figure}[tbph]
\centering
\includegraphics[width=0.9\columnwidth]{fig6_small.pdf}
\caption{Correlations between structural similarity
distances induced by the average kernel $\bar{K}$,
the best-match kernel $\hat{K}$, and \emph{regularized}
best-match kernels $\hat{K}^\gamma$ with different
regularization parameters $\gamma$. Distances are
computed between pairs of 200 structures, randomly
selected from the QM7b database\cite{qm7b,qm7b-father}.}
\label{fig:qm7b-corr}
\end{figure}
\begin{figure*}[tbph]
\centering
\includegraphics[width=0.9\textwidth]{fig7_small.pdf}
\caption{Sketch-map representation of minimum-energy
structures from a database of molecules containing
up to 7 heavy atoms (C O N S Cl), and saturated with
hydrogen to a different degree\cite{qm7b}.
Left-hand panels
show the map colored according to the atomization
energy as computed by DFT. In the right-hand images,
the points are colored according to the number
of constituent C, O, N, S atoms.
The top row corresponds to an alchemical kernel
that treats all species as different, the
middle row treats all the heavy atoms as the
same species, whereas the bottom row introduces
an alchemical kernel that depends on the
difference in electronegativity between
species.}
\label{fig:qm7b}
\end{figure*}
\subsection{Mapping (al)chemical space}
As a final example of the evaluation of a
structural and alchemical similarity metric, and its
use to represent complex ensembles of compounds,
let us consider the QM7b database~\cite{qm7b}.
This set of compounds contains 7211 minimum-energy
structures for small organic compounds containing
up to seven heavy atoms (C, N, O, S, Cl),
saturated with H to different degrees. This
database constitutes a small fraction of
a larger chemical library that contains millions
of hypothetical structures screened for accessible
synthetic pathways~\cite{qm7b-father}.
This is an extremely challenging data set to
benchmark a structural similarity metric: molecules
differ by number of atoms, chemical composition,
bonding and conformation. To simplify the description,
we decided to use
SOAP descriptors with a cutoff of 3\AA{},
and to include H atoms in the environments but
not as environment centers, to simplify the
description -- considering also that in the case of arginine
dipeptide this choice did not prevent clear
identification of isomers that only differed by
a proton transfer reaction. We used a best-match
strategy to compare configurations, and topped
them up with isolated atoms up to the maximum number
of each species that is present in the database.
This effectively corresponds to choosing a
``kit'' (in other terms, a fully atomized reference state)
starting from which all of the compounds can be assembled.
This is a fairly extreme case for the application of
our idea of compounding local structure matching to
obtain a global structural metric, so it is worth
returning on a comparison of the different strategies
we proposed. Fig.~\ref{fig:qm7b-corr} compares
average and best-match distances, together with
the REMatch using different regularization
parameters. Despite the very different context, the
outcome is similar to what we observed in
Figure~\ref{fig:C60} for C$_{60}$ clusters. The
average kernel is reasonably well correlated
with the more demanding best-match kernel, although
in most cases it has poorer resolution. By
varying $\gamma$, the regularized match distance
$\hat{D}^\gamma$ varies between these two extremes,
and for $\gamma<1$ provides a smooth, inexpensive
approximation to the best-match distance.
For the sake of simplicity (and given we reduced the
size of the environment covariance matrix $\mathbf{C}$ by considering only heavy atoms
as environment centers) we used the
conventional best-match distance for the rest of
our analyses.
As shown in Fig.~\ref{fig:qm7b}, the SOAP-based
metric nicely separates out ``islands'' with homogeneous
composition in terms of the number of heavy atoms.
Within each group of atoms, one can recognize some
sub-structure, with configuration roughly arranged in terms
of the atomization energy -- which in turns strongly
correlates with the degree of H saturation.
As it can be seen from inspection of the database
(see the SI) in many cases one can notice that structures
with similar chemical skeleton (presence of cycles,
chemical groups, etc.) are clustered close to
each other in the map. However, it is of course very
difficult to quantitatively assess how well the map
corresponds to chemical intuition, and how much
departures from it are to be considered a failure of
the metric, of the sketch-map procedure or of
the notion of ``chemical intuition''.
Our objective here is more to demonstrate how the
fingerprint-based structural metric we introduced
can cope with widely different classes of problems,
and how it can treat on the same footings alchemical and
structural variability. As an example
we have also computed the similarity matrix
and mapped the QM7b landscape using a modified
alchemical similarity metric between the heavy
atoms (we always take $\kappa_{\alpha\text{H}}=\delta_{\alpha\text{H}}$).
First, we set
$\kappa_{\alpha\beta}=1$ (which means we are treating
species $\alpha$ and $\beta$ as the same species) for all of heavy
atoms.
The clear separation of the map into islands with the
same stoichiometry is lost. However, there is now
near-perfect correlation between position on the map and
atomization energy, and at the same time one can see some
residual clustering of molecules with similar composition.
This can be explained because information on the
alchemical identity of the atoms is encoded in their
atomic coordination and bond lengths.
This is for instance evident for sulfur,
that has considerably larger bond lengths, leading to
a clustering of sulfur-containing compounds that is
considerably better than for instance in the case of
oxygen or nitrogen.
Obviously, assuming that all atom kinds
are interchangeable is an extreme choice, and it is
hard to imagine circumstances in which this ``element
agnostic'' metric would be advantageous over one that
exploited knowledge of the chemical identity of atoms.
On the other hand, one could foresee to encode
information on the ``alchemical similarity'' using
one of the many quantities chemists have used historically
to rationalize trends in reactivity across the periodic
table. As an example, we used the electronegativity $E_\alpha$ to define
\begin{equation}
\kappa_{\alpha\beta}=e^{-(E_\alpha -E_\beta)^2/2\Delta^2}
\end{equation}
where $\Delta$ is a parameter that determines how sensitive
is the alchemical kernel to differences in
electronegativities. We used $\Delta=1$ to
generate the last set of maps in Fig.~\ref{fig:qm7b}.
The map now separates out quite accurately regions
with homogeneous stoichiometry. Whereas in the
$\kappa_{\alpha\beta}=\delta_{\alpha\beta}$ the
different ``islands'' were roughly arranged according
to a square grid pattern corresponding to $n_\text{O}$
and $n_\text{N}$ along two orthogonal directions, now
stripe-shaped islands are arranged in 1D, following
numbers of $n_\text{O}$ and $n_\text{C}$, with
the number of
nitrogen atoms coming out clustered in adjacent
``stripes'', but less clear-cut partitioning than
for the other two elements. This is perhaps unsurprising
given that nitrogen has an intermediate electronegativity
between that of oxygen and carbon, and the metric tries
to separate most efficiently the elements that
differ most based on the alchemical similarity kernel.
This last example gives perhaps the most compelling
demonstration of how a structural similarity metric
based on a combination of SOAP kernels gives an effective,
broadly applicable and easily customizable strategy
to assess the similarity of materials and molecules,
and how a sketch-map construction based on such metric
provides an insightful representation of structural
and alchemical landscapes.
\subsection{Learning molecular properties}
In this paper we focused mainly on the definition of
a compound structural similarity kernel, and on characterizing
its behavior by means of sketch-map representations.
It is however important to keep in mind that an
effective tool to compare atomic structures can find
application to a broad range of problems - one of the
most intriguing being the inexpensive prediction of
physical-chemical properties of materials and molecules.
To demonstrate the great promise of REMatch-SOAP kernels
for machine-learning of molecular properties, we used
a standard kernel-ridge regression (KRR) method~\cite{hast09book} to reproduce
the 14 properties that had been reported in Ref.~\cite{qm7b}
for the 7211 molecules we described in the previous paragraph.
We randomly selected 5000 training structures,
and used the remainder as an out-of-sample validation set.
After having computed the REMatch-SOAP kernel matrix
$\mathbf{K}$ between all the structures, using a cutoff
of 3\AA{} and a regularization parameter $\gamma=0.5$
-- in this case including also H atoms in the list of environments --
we computed the KRR weights vector
\begin{equation}
\mathbf{w}=\left(\mathbf{K}_\text{train}^\xi+\sigma \mathbf{1}\right)^{-1}
\mathbf{y}_\text{train}.
\end{equation}
Here $\mathbf{K}_\text{train}$ and $\mathbf{y}_\text{train}$ are
the kernel matrix and property values restricted to the training set,
$\xi$ indicates entry-wise exponentiation to tune the spatial range of the kernel, and
$\sigma$ is a regularization hyperparameter.
The prediction of the properties for the test set can then
be obtained as $\mathbf{y}_\text{test}=\mathbf{K}_\text{test}^\xi \mathbf{w}$,
where $\mathbf{K}_\text{test}$ is the matrix containing
the REMatch-SOAP kernels between the test points and the training points.
The procedure was repeated 10 times, and the average mean absolute
error (MAE) and root mean square error (RMSE) on the test set
were computed.
We optimized the $\xi$ and $\sigma$ hyperparameters
by minimising the MAE on the atomization energy,
and then used the same values to perform a KRR for all
the other molecular properties. Since we did not further adjust the
choice of kernel and the $\xi$ exponent, all the properties
could be estimated at the same time, as discussed e.g. in
\cite{rama-vonl15cijc}.
The results of this procedure are reported in
Table~\ref{tab:krr-data}, and demonstrate the extraordinary
performance of REMatch-SOAP for machine-learning applications.
For the atomization energy we can obtain
a MAE of less than 1kcal/mol --
a four-fold improvement relative to
previous results that were based on a Coulomb matrix
representation of structures and a deep-neural-network
learning strategy. What is more, even without separately tuning the
KRR hyperparameters, we can improve or match the
performance of prior methods for almost all of the
properties, the only exceptions being some of the
properties computed with semi-empirical methods.
The fact we can obtain such a dramatic
improvement using a standard regression technique is a
testament to the effectiveness of our kernel.
The crucial importance of the choice of descriptors
is also apparent by noting that a MAE of about 1.5 kcal/mol was recently obtained by regression based
on a ``bag of bonds'' description of molecules, coupled with a Laplacian kernel~\cite{bag.of.bonds}.
Reaching chemical accuracy in the automated prediction
of atomization energies is an important milestone, and the
fact that we could achieve that without fully exploring
the flexibility of the REMatch-SOAP framework (e.g. by optimizing
the entropy regularization parameter, the environment cutoff,
eliminating the outliers, combining multiple layers
of description or using a non-diagonal alchemical
similarity matrix) is a testament to the potential of
our approach. Future work will be devoted to
analyzing the performance, convergence and limits for the
machine-learning of molecular and materials' properties using
our SOAP-based structural similarity kernel.
\begin{table}[]
\centering
\begin{tabular}{lccccc}
\hline\hline
Property & SD & MAE & RMSE & MAE\cite{qm7b} & RMSE\cite{qm7b} \\ \hline
$E$ (PBE0) & 9.70 & 0.04 & 0.07 & 0.16 & 0.36 \\
$\alpha$ (PBE0) & 1.34 & 0.05 & 0.07 & 0.11 & 0.18 \\
$\alpha$ (SCS) & 1.47 & 0.02 & 0.04 & 0.08 & 0.12 \\
HOMO (GW) & 0.70 & 0.12 & 0.17 & 0.16 & 0.22 \\
HOMO (PBE0) & 0.63 & 0.11 & 0.15 & 0.15 & 0.21 \\
HOMO (ZINDO) & 0.96 & 0.13 & 0.18 & 0.15 & 0.22 \\
LUMO (GW) & 0.48 & 0.12 & 0.17 & 0.13 & 0.21 \\
LUMO (PBE0) & 0.68 & 0.08 & 0.12 & 0.12 & 0.20 \\
LUMO (ZINDO) & 1.31 & 0.10 & 0.15 & 0.11 & 0.18 \\
IP (ZINDO) & 0.96 & 0.19 & 0.28 & 0.17 & 0.26 \\
EA (ZINDO) & 1.41 & 0.13 & 0.18 & 0.11 & 0.18 \\
$E^{*}_{1^{st}}$ (ZINDO) & 1.87 & 0.18 & 0.41 & 0.13 & 0.31 \\
$E_{max}^{*}$ (ZINDO) & 2.82 & 1.56 & 2.16 & 1.06 & 1.76 \\
$I_{max}$ (ZINDO) & 0.22 & 0.08 & 0.12 & 0.07 & 0.12 \\
\hline\hline
\end{tabular}
\caption{Mean absolute errors (MAEs) and root mean square errors (RMSE)
for the KRR estimation of 14 molecular properties, together with previously published estimation~\cite{qm7b} for the same data set.
The standard deviation of the values of the properties across all
7211 molecules in the database is shown in the second column.
Errors in the KRR estimation refer to a test set of ~2200 randomly selected configurations,
while the remaining structures were used for training.
Property labels refer to the level of theory and molecular property, i.e. atomization energy (E), averaged molecular
polarizability ($\alpha$), HOMO and LUMO eigenvalues, ionization potential (IP),
electron affinity (EA), first excitation energy ($E^{∗}_{1^{st}}$), excitation frequency of
maximal absorption ($E^{∗}_{max}$) and the corresponding maximal absorption intensity
($I_{max}$).
Energies, polarizabilities and intensities are in eV, $\AA^3$
and arbitrary units, respectively.}
\label{tab:krr-data}
\end{table}
\section{Conclusions}
Distances between atomic structures based on combinations
of local similarity kernels provide a flexible framework
to define a metric in structural and alchemical space.
Atom-centered environment information can be combined
to provide a global measure of (dis)similarity. An average
kernel $\bar{K}$ provides an inexpensive strategy to do so, with
a cost that scales linearly with the size of the structures
to be compared, but might under-estimate the difference
between two configurations -- since in principle two
different structures might yield zero $\bar{D}$.
Alternatively, one can compute the local kernel between
every possible pair of environments (which itself entails
a cost scaling with the square of the number of environments),
and then build a compound kernel $\hat{K}$ by finding the best-match
permutation of the environments -- which gives a
metric with better resolving power, but entails solving a
cubic-scaling linear assignment problem.
Introducing an entropy regularization makes it possible
at the same time to reduce the size-scaling to quadratic,
and to obtain a better behaved, smoothly varying
metric, that interpolates - depending on the regularization
parameter - between the average and best-match limit.
This strategy to compare atomic configurations builds
on the very general notion that complex bulk and molecular
structures arise from the combination of local building
blocks, and can be applied seamlessly to systems
as diverse as clusters, bulk phases of an element,
conformation of a biomolecules and an assembly of
small chemical compounds with varying atom kinds and
number. At the same time, the structure of the
underlying SOAP kernels allows for very effective fine-tuning.
For instance, by choosing the cutoff radius over which
atomic densities are compared between environments, one
can make the metric more sensitive to the first-neighbor
chemical connectivity, or vice versa, include information
on the long-range conformation of flexible molecules.
What is more, it is possible to treat structural and
alchemical complexity on the same footing, by introducing
an alchemical similarity kernel that makes it possible to
specify whether atoms of different species should be
considered completely separate, or whether a notion of chemical
distance (based e.g. on the difference in
electronegativity) should be introduced to
give different weights to substitutions
between elements with similar reactivity.
We also demonstrate that straightforward application
of the REMatch-SOAP kernel to the ridge-regression
evaluation of molecular properties matches or outperforms
all previously-presented approaches,
reaching chemical accuracy in the prediction of
the atomization energies of a set of small organic
molecules. We believe that in this respect we are
only scratching the surface of the potential
applications of our approach to machine-learning,
since these results were obtained without using
any of the more sophisticated techniques (e.g. introducing
a hierarchy of models to capture the variance of properties
at different structural scales) that have
been shown to significantly improve this kind of
procedures when using different kinds of structural
descriptors.
The similarity metric we introduce could find application
as the workhorse of a number of simulation protocols,
machine-learning algorithms and data mining strategies.
For instance, it could be used to detect outliers in
automated high-throughput screenings of materials,
to cluster similar configurations together, to accelerate
the exploration of chemical and conformational space
of materials and molecules. Here, we show in particular
how it can be combined with a non-linear dimensionality
reduction technique such as sketch-map, to give
simple and insightful two-dimensional representation of
a given molecular or structural data set. As atomistic
modelling adventures into larger-scale structures, and
unsupervised exploration of materials space,
maps such as these can provide a valuable tool to
convey intuitive information on complex structural
and alchemical landscapes, to rationalize
structure-property relations, and to predict
physical observables of novel compounds by
training machine-learning models to libraries
of known materials.
\section{Acknowledgments}
S.D and M.C would like to acknowledge support from
the NCCR MARVEL. A.P.B. was supported by a Leverhulme
Early Career Fellowship with joint funding from
the Isaac Newton Trust. We would like to thank M. Cuturi
and C. Ortner for insightful discussion. We thank C. Pickard
for providing silicon structures found with AIRSS, and S.
Goedecker and M. Amsler for sharing with us the crystal
structures of low-density silicon polymorphs.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,847 |
{"url":"https:\/\/www.researchgate.net\/scientific-contributions\/Marco-Del-Giudice-2053144711","text":"# Marco Del Giudice's research while affiliated with Italian Institute for Genomic Medicine and other places\n\n## Publications (22)\n\nArticle\nFull-text available\nArtificial intelligence, or the discipline of developing computational algorithms able to perform tasks that requires human intelligence, offers the opportunity to improve our idea and delivery of precision medicine. Here, we provide an overview of artificial intelligence approaches for the analysis of large-scale RNA-sequencing datasets in cancer....\nArticle\nFull-text available\nIndividual cells exhibit specific proliferative responses to changes in microenvironmental conditions. Whether such potential is constrained by the cell density throughout the growth process is however unclear. 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However, an alternative is that diversity arises in pa...\nArticle\nFull-text available\nHeterogeneity is a fundamental feature of complex phenotypes. So far, genomic screenings have profiled thousands of samples providing insights into the transcriptome of the cell. However, disentangling the heterogeneity of these transcriptomic Big Data to identify defective biological processes remains challenging. Here we present GSECA, a method e...\nPreprint\nFull-text available\nCell-to-cell variation in gene expression is a common feature of developmental processes. Yet, it remains unclear whether molecular mediators can generate variation and how this process is coordinated across loci to allow the emergence of new cell states. Using embryonic stem cells (ESCs) as a model of development, we found interconverting cell sta...\nChapter\nNon-coding RNAs play a key role in the post-transcriptional regulation of mRNA translation and turnover in eukaryotes. miRNAs, in particular, interact with their target RNAs through protein-mediated, sequence-specific binding, giving rise to extended and highly heterogeneous miRNA\u2013RNA interaction networks. Within such networks, competition to bind...\nPreprint\nFull-text available\nProstate cancer (PCa) is genomically driven by dysregulation of transcriptional networks involving the transcriptional factors (TFs) FOXA1, ERG, AR, and HOXB13. However, the role of these specific TFs in the regulation of alternative pre-mRNA splicing (AS), which is a proven therapeutic vulnerability for cancers driven by the TF MYC, is not describ...\nPreprint\nFull-text available\nNon-conding RNAs play a key role in the post-transcriptional regulation of mRNA translation and turnover in eukaryotes. miRNAs, in particular, interact with their target RNAs through protein-mediated, sequence-specific binding, giving rise to extended and highly heterogeneous miRNA-RNA interaction networks. Within such networks, competition to bind...\nArticle\nFull-text available\nDirectional transport of recycling cargo from early endosomes (EE) to the endocytic recycling compartment (ERC) relies on phosphatidylinositol 3-phosphate (PtdIns(3)P) hydrolysis and activation of the small GTPase Rab11. However, how these events are coordinated is yet unclear. By using a novel genetically-encoded FRET biosensor for Rab11, we repor...\nArticle\nFull-text available\nMany biological processes are known to be based on molecular sequestration. This kind of dynamics involves two types of molecular species, namely targets and sequestrants, that bind to form a complex. In the simple framework of mass-action law, key features of these systems appear to be threshold-like profiles of the amounts of free molecules as a...\nData\nThe role of the offset. (A) Example of two average mRNA profiles, for a regulated (orange) and an unregulated (blue) mRNA. (B) Same profile as panel A but with the curves shifted upwards by an arbitrary offset of 10 mRNA molecules. (C) Fold repression (ratio of blue to orange curve from panel A) without the offset. (D) Fold repression (ratio of blu...\nData\nAnalytical prediction for the coefficient of variation. Analytical predictions for the target coefficient of variation in case of one (A) or two (B) targets. In (A) the parameters are k\u00afS=1.2\u00d710-3nMmin-1, \u03c3 = 4.8 \u00d7 10\u22124 nM min\u22121, gS = 1.2 \u00d7 10\u22122 min\u22121, gR = 2.4 \u00d7 10\u22122 min\u22121, g = 1.2 \u00d7 102 nM\u22121 min\u22121, kP = 6.0 min\u22121, gP = 1.2 \u00d7 10\u22122 min\u22121, \u03b1 = 0.5....\nData\nDetails on different approximation methods, supplementary analysis, and simulations. (PDF)\nData\nComparison between Van Kampen and Gaussian approximations. In (A-C) mRNA and protein distributions for unstable and stable proteins are shown together with the two approximations. In (D) the mean number of mRNA molecules as a function of the miRNA transcription rate is shown, the blue line corresponds to numerical simulations, while the red and bla...\nData\nBimodality phase diagram. The plot shows the bimodality phase diagram for the mRNA 1 in a system with two targets competing for the same miRNA. The parameters here used are the following: k\u00afS=1.2\u00d710-3 nM min\u22121, \u03c3S = 2.4 \u00d7 10\u22124 nM min\u22121, g1 = 1.2 \u00d7 102 nM\u22121 min\u22121, kR1 and kR2 range from 0 nM min\u22121 to 5.1 \u00d7 10\u22123 nM min\u22121, gS = 1.2 \u00d7 10\u22122 min\u22121, gR1 =...\nData\nComparison between the bimodal mRNA noisy distribution and the weighted superposition of distributions obtained without noise for different miRNA transcription rates. The parameters are the following: kR = 3.1 \u00d7 10\u22123 nM min\u22121, gS = 1.2 \u00d7 10\u22122 min\u22121, gR = 2.4 \u00d7 10\u22122 min\u22121, g = 1.2 \u00d7 102 nM\u22121 min\u22121, kP = 6.0 min\u22121, gP = 1.2 \u00d7 10\u22122 min\u22121 and \u03b1 = 0.5....\nData\nBimodality amplitude phase diagram. Phase diagram of the bimodality amplitude of the mRNA distribution as a function of the mRNA transcription rate kR and of the extrinsic noise level. The parameters here used are the following: gS = 1.2 \u00d7 10\u22122 min\u22121, gR = 2.4 \u00d7 10\u22122 min\u22121, g = 3.0 \u00d7 102 nM\u22121 min\u22121, kP = 6.0 min\u22121, gP = 1.2 \u00d7 10\u22122 min\u22121, \u03b1 = 0.5. T...\nArticle\nFull-text available\nWe document the initial-density dependence of the growth rate achieved by Jurkat cell cultures in a standard growth medium with fixed carrying capacity. As the density $N_0$ of the inoculum varies over 4 orders of magnitude, three distinct growth regimes appear. At small $N_0$, the growth rate $\\lambda$ is roughly constant and displays small sample...\nArticle\nFull-text available\nSeveral studies pointed out the relevance of extrinsic noise in molecular networks in shaping cell decision making and differentiation. Interestingly, bimodal distributions of gene expression levels, that may be a feature of phenotypic differentiation, are a common phenomenon in gene expression data. The modes of the distribution often correspond t...\nArticle\nFull-text available\nIn view of the relation between information and thermodynamics we investigate how much information about an external protocol can be stored in the memory of a stochastic measurement device given an energy budget. We consider a layered device with a memory component storing information about the external environment by monitoring the history of a se...\n\n## Citations\n\n... Auslander et al. reviewed machine learning\/deep learning approaches incorporated to establish bioinformatics and computational biology frameworks in the areas of molecular evolution, protein structure analysis, systems biology, and disease genomics [19]. Del Giudice et al. comprehensively reviewed machine learning\/deep learning solutions for computational problems in bulk and single-cell RNA-sequencing data analysis [20]. Banegas-Luna et al. discussed the interpretability of machine learning\/deep learning methods in cancer research [21]. ...\n... As a second application of TOLOMEO, we discuss the case of biological images, where the progression of microscopy and multiplexed fluorescence imaging techniques allows one to take snapshots with enough resolution to distinguish cell populations [31][32][33][34] or even cellular compounds [35,36] and their respective spatial organization [37]. ...\n... Another open question is which kinesins are required for the transport of Hrs and STAM1 under basal conditions, as KIF13A knockdown does not seem to impair this process. Several kinesins in addition to KIF13A and KIF13B, including KIF5A\/B\/ C, KIFC1, and KIF16B, have been implicated in endosomal trafficking (Bonifacino & Neefjes, 2017;Li et al, 2020;Villari et al, 2020) and may also have roles in the axonal transport of ESCRT-0 proteins. Clearly, additional work is needed to identify other cargoes of KIF13A, as well as the specific motor proteins responsible for anterograde and retrograde transport of Hrs+ vesicles. ...\n... These variations have functional consequences such as altered differentiation potentials of stem cells and drug resistance of cancer cells (1)(2)(3)6,7). For progenitor cells that exhibit multimodal expression patterns, a small subpopulation with a relatively homogenous expression profile recovers the parental population's heterogeneity of individual gene products after several days or longer (2,(8)(9)(10) (Figure 1 box). Although stochasticity in transcriptional activities can cause expression variation and associated cell state changes (3,11), this type of noise influences expression at a much faster timescale (minutes) than the fluctuations required to achieve observed cell state transitions (days) (1,2,8,12,13). ...\n... The development of AI industry enables easier and more visually appealing solutions for SCS technology. For example, AI can be widely exploited in all aspects of the SCS workflow, such as batch correction for technical heterogeneity [133,134], feature extraction [135,136], data distribution transformation [137,138], classification of cancer subtypes [139,140], and biomarker identification [141][142][143]. Most notably, SCS in combination with AI is also widely used to identify and analyze CTCs, a class of cells that can be used for searching therapeutic targets for tumor metastasis [133][134][135][136][137][138][139][140][141][142][143][144]. ...\n... (23)(24)(25) in the limit of small perturbations. [100] The above program has been carried out starting in 2013 over a series of papers by various authors, covering aspects ranging from the off-equilibrium dynamics of small miRNA-mediated circuits to the typical system-level properties of crosstalk in the human transcriptome (see Ref. [101] for a thorough review). For sakes of clarity, we focus here on two sets of results with high biological significance, concerning the roles of (a) transcriptional heterogeneities and (b) topological heterogeneities in shaping the emergent network-scale expression profiles in the human miRNA interactome mapped via the CLASH protocol (Crosslinking, Ligation And Sequencing of Hybrids), accounting for O(10 7 ) potential miRNA-mediated cross-regulatory interactions, each quantifiable by the value of \u03c7 ij . ...\n... A recent study based on a combination of theory and experiments [23] showed that selective pressure might even increase expression noise and the positively selected genes with elevated noise are also those highly regulated by transcription factors. On the same idea that cells do not necessarily buffer noise, a recent work showed that introduction of extrinsic noise in microRNA-mediated regulatory networks, i.e., increased variability in gene expression, can instead favour cell differentiation [24][25][26][27]. These recent studies arouse the possibility that cells do not only buffer noise but they rather take advantage of stochasticity to optimise specific needs, e.g., cell-to-cell variability, protein number precision, information flow [28,29], etc. ...\n... Class III PI3K, Vps34, is required for GPCRs to be recycled from the endosomal compartments to the plasma membrane during resensitization ( 93 ). PtdIns3P activates Rab11, a master regulator of endocytic recycling ( 94 ), and recruits the 3-phosphatase MTM1, which dephosphorylates PtdIns3P in the sorting of endosomes ( 95 ) ( Figure 4 ). Some GPCRs can recruit 14-3-3 proteins after endocytosis, which are scaffold proteins that direct GPCRs from recycling endosomes to the cell membrane ( 96 ) ( Figure 4 ). ...\n... Few previous studies have searched for bimodality in large-scale gene expression data (Bessarabova et al., 2010;Mason et al., 2011;Shalek et al., 2013) and causes for such bimodality have been discussed, including: i) differential action of transcription factors (Ochab-Marcinek and Tabaka;, ii) regulation by microRNAs (Bosia et al., 2017;Del Giudice et al., 2018); iii) regulation by circular RNA (Hu and Zhou;) and even iv) stochastic events (Samoilov et al., 2005). For an extensive review of the different methods developed for detection of bimodality, please see Moody et al. (2019). ...\n... For any A \u2286 N I define K x x (A; t) := i\u2208A K x x (i; t). As a concrete illustration, consider the scenario investigated in [33,42], in which receptors in the wall of a cell sense the concentration of a ligand in the intercellular medium, and those receptors are in turn observed by a \"memory\" subsystem inside the cell. 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ACCEPTED
#### According to
The Catalogue of Life, 3rd January 2011
#### Published in
null
#### Original name
Talauma minor Urb.
### Remarks
null | {
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BIG TOM is ready for adventure. This 1:10 scale remote control 4 x 4 rock crawler features oversized high grip off-road rubber tires and a high travel suspension for navigation through rough terrain. Built tough with oversized shocks and a muscular energy absorbing composite roll cage that helps protect the truck. A front mounted LED light bar illuminates surrounding terrain enabling driving even in low light conditions! | {
"redpajama_set_name": "RedPajamaC4"
} | 970 |
\section{Introduction}
Beams have been used since ancient times to reinforce structures such as bridges, buildings, and others. Through the millennia, understanding the dynamics and controllability of beams, including bending and vibration has been of great importance. Pioneering studies goes back to 1493 Leonardo da Vinci's manucript that identified properly the stresses and strains in a beam subject to bending \cite{DaVinci} and Galileo Galilei's writings that identified the principle of virtual work as a general law but made incorrect assumptions \cite{Timoshenko:1921aa}. It was not until the late 17th century with the elasticity theory evolution that Leonhard Euler and Daniel Bernoulli provided a second-order spatial derivatives mathematical model that later, in 1921, Stephen Timoshenko improved by including a shear deformation and rotational inertia effects, obtaining fourth order mathematical model (see \cite{Timoshenko:1921aa,Timoshenko:1922aa, Timoshenko:1953aa} for details).
Nowadays, adjustments of the Timoshenko beam model, in mechanical engineering and nanotechnology design \cite{Wang-Tan:2006aa,Wang-Zhang:2006aa}, yield to the impulsive semilinear beam equations of the form \eqref{eq:beam} where the memory and delay provide information of the viscoelasticity property and response of the materials.
In this paper, we are exploring the approximate controllability on a bounded domain $\Omega \subseteq \R^{N} \,(N\geq1)$ of
\begin{equation}\label{eq:beam}
w_{tt} - 2\beta\Delta w_t + \Delta^{2}w = u(t,x) + f(t,w(t-r),w_{t}(t-r),u)+\displaystyle \int_{0}^{t}M(t-s)g(w(s-r,x))ds,
\end{equation}
subjected to the initial-boundary conditions and impulses
\begin{equation}\label{eq:initial}
\left\{\begin{array}{ll}
w(t,x)=\Delta w (t,x)=0, &\mbox{in}\; (0, \tau) \times \partial \Omega,\\
\begin{split}
&w(s,x)=\phi_1(s,x),\\
&w_{t}(s,x)=\phi_2(s,x),
\end{split} & \mbox{in} \; [-r,0] \times \Omega,\\
w_{t}(t_{k}^{+},x) = w_{t}(t_{k}^{-},x)+I_{k}(t_k,w(t_{k},x),w_{t}(t_{k},x),u(t_{k},x)), &t \neq t_k, \; k=1, \dots, p,
\end{array}
\right.
\end{equation}
where $\Delta w=\sum_{j=1}^{N}\frac{\partial^2w}{\partial x_j^2}$ and $\Delta^2 w=\sum_{j=1}^{N}\frac{\partial^4w}{\partial x_j^4}$. Additionally, the damping coefficient $\beta > 1$ and the real-valued functions $w = w(t,x)$ in $(0, \tau] \times \Omega$ represents the beam deflection, $u$ in $(0, \tau] \times \Omega$ is the distributed control, $M$ acts as convolution kernel with respect to the time variable, the impulses $I_k$ are defined on $[0, \tau] \times \R^3$ and the nonlinearities $g$ on $\R$, $f$ on $[0, \tau] \times \R^3$. Under the assumptions:
\begin{description}
\item[H$_1$] $M\in L^{\infty}((0,\tau)\times \Omega)$, and $g, f, I_{k}$ are smooth enough, in order that, for all $\phi, \psi\in \C([-r,0],L^{2}(\Omega))$ and $u\in L^{2}([0,\tau]; L^{2}(\Omega))$ the equation \eqref{eq:beam} admits only one mild solution on $[-r,\tau]$.
\item [H$_2$] $t\!\in\! [0,\tau],$\; $a, b \geq \!0$ and $u, v, y \in \R$, the nonlinearity $f$ satisfies
\begin{equation}\label{eq:f1}
\begin{array}{ll}
|f(t,y,v,u)| & \leq a\sqrt{|y |^2+ |v |^2} +b.
\end{array}
\end{equation}
\end{description}
N.~Abada, M.~Benchohra, and H.~Hammouche \cite{Abada-Benchohra:2010aa} and R.~S. Jain and M.~B. Dhakne in \cite{Jain-Dhakne:2013aa} works showed the existence of solutions for impulsive evolution equations with delays. Balachandran, Kiruthika, and Trujillo \cite{Balachandran-Kiruthika:2011aa} supplied existence results for the fractional impulsive integrodifferential equations and finally for the Beam equation with variable coefficients J.~L{\'\i}maco, H.~Clark, and A.~Feitosa \cite{Limaco-Clark:2005aa} showed the existence and uniqueness of non-local strong solutions and the existence of a unique global weak solution with decay rate energy.
Inspired in a series of papers from A. Carrasco, H. Leiva, N. Merentes and J.
Sanchez on the approximate controllability of semilinear beam equation \cite{Carrasco-Leiva:2013aa,Carrasco-Leiva:2014aa,Carrasco-Leiva:2016aa} and the works on the approximate controllability for the semilinear heat and strongly damped wave equations with memory and delays by C. Guevara and H. Leiva \cite{Guevara-Leiva:2016aa, Guevara-Leiva:2017aa}. We prove the approximate controllability of the beam equation \eqref{eq:beam} under the initial-boundary condition \eqref{eq:initial} with memory, impulses and delay terms by applying A.E. Bashirov, N. Ghahramanlou, N. Mahmudov, N. Semi and H. Etikan technique \cite{Bashirov-Ghahramanlou:2014aa, Bashirov-Ghahramanlou:2015aa, Bashirov-Jneid:2013aa, Bashirov-Etikan:2010aa}, and avoiding the Rothe's fixed point theorem used in \cite{Carrasco-Leiva:2013aa,Carrasco-Leiva:2016aa} and the Schauder fixed point theorem applied in \cite{Carrasco-Leiva:2014aa}.
The structure of this paper is as follow: In section \ref{sec:formulation}, we present the abstract formulation of the beam equation \eqref{eq:beam}. Section \ref{sec:lineal}, recalls the linear controllability characterization of the problem. In section \ref{sec:semilineal}, the approximated controllability of the beam equation with memory, delay and impulses is proved.
\section{Abstract Formulation of the Problem} \label{sec:formulation}
In this section, we choose the appropiate Hilbert space where the Cauchy problem \eqref{eq:beam}-\eqref{eq:initial} can be written as an abstract differential equation.
First of all, notice that the term $-2\beta\Delta w_t$ in the equation \eqref{eq:beam} acts as a damping force, thus the energy space used to set up the wave equation is not suitable here. Even so, in \cite{Oliveira:1998aa}, Oliveira shows that the uncontrolled linear equation can be transformed into a system of parabolic equations of the form $w_{t} = D \Delta w$, obtaining that corresponding space for the abstract formulation of the problem is $\mathcal{Z}^{1}=\left[H^{2}(\Omega) \bigcap H^{1}_{0}(\Omega) \right] \times L^{2}(\Omega)$ and proving that the linear part of this system generates a strongly continuous analytic semigroup in this space.
Consider the Hilbert space $\mathcal{X} = L^{2}(\Omega)$, and denote $\mathcal{A}=-\Delta$ with eigenvalues $0 <
\lambda_{1}<\lambda_{2}<...<\lambda_{j}\to \infty,$ with multiplicity $\gamma_{j}<\infty$ equal to its corresponding eigenspace dimension. Recall, $\mathcal{A}$ satisfies the following properties:
\begin{enumerate}[(i)]
\item There exists a complete orthonormal set $\left\{
\phi_{j_k} \right\}$ of eigenvectors of $\mathcal{A}$.
\item For all $x \in D(\mathcal{A})$,
\begin{equation*} \label{prop}
\mathcal{A} x = \sum_{j = 1}^{\infty} \lambda_{j} \sum_{k =
1}^{\gamma_j} \inner{\xi, \phi_{j_k}} \phi_{j_k} =\sum_{j = 1}^{\infty}
\lambda_{j}E_{j}x,
\end{equation*}
where $\inner{\cdot, \cdot}$ denotes the inner product in $\mathcal{X}$, $\displaystyle E_{n}x = \sum_{k = 1}^{\gamma_j} \inner{z, \phi_{j_k}} \phi_{j_k},$ and $\{ E_j \}$ is a family of complete orthogonal projections in
$\mathcal{X}$.
\item $-\mathcal{A}$ generates an analytic semigroup $\{
S(t) \}_{t \geq 0}$ given by
$$
S(t)x = \sum_{j = 1}^{\infty} e^{-\lambda_j t}E_{j}x \quad \mbox{and} \quad \norm {S(t)} \leq e^{-\lambda_{1}t}.
$$
\item For $\alpha\geq0$ the fractional powered spaces $\mathcal{X}^{\alpha}$ are given by
\begin{equation*}
\mathcal{X}^{\alpha} =D(\mathcal{A}^{\alpha}) = \left\{x \in \mathcal{X} : \sum_{j = 1}^{\infty} \lambda_{j} ^{2
\alpha} \norm{ E_{j}x}^2 < \infty \right\}
\end{equation*}
equipped with the norm $\displaystyle
\norm{x}_{\alpha}^2 = \norm{\mathcal{A}^{\alpha}x}^2= \sum_{j = 1}^{\infty}
\lambda_{j}^{2 \alpha} \norm{ E_{j}x}^2
$, where $\displaystyle \mathcal{A}^{\alpha}x = \sum_{j = 1}^{\infty} \lambda_{j}^{ \alpha} E_{j}x$.
\end{enumerate}
In particular, $\alpha=2$ yields $\displaystyle \mathcal{A}^{2}x = \sum_{j = 1}^{\infty} \lambda_{j}^{ 2} E_{j}x=(-\Delta)^2x=\Delta^2x$. And for $\alpha=1,$ the Hilbert space $\mathcal{Z}^{1}=\mathcal{X}^{1}\times \mathcal{X}$ has the norm
$$
\displaystyle \norm {\left(
\begin{array}{c}
w \\
v \\
\end{array}
\right)}^{2}_{\mathcal{Z}^{1}}=\|w\|^{2}_{1}+\|v\|^{2}.
$$
Using the above notation, we rewrite the system \eqref{eq:beam}-\eqref{eq:initial} as the second-order ordinary differential equations in the Hilbert
space $\mathcal{X}$
\begin{equation} \label{eq:ODE}
\left\{%
\begin{array}{ll}
\begin{split}
w''(t) = &-\mathcal{A}^{2}w(t) - 2\beta \mathcal{A} w'(t) + u(t)+ \displaystyle \int_{0}^{t}M(t,s)g^{e}(w(s-r))ds \\
&\ \ + f^{e}(t,w(t-r),w'(t-r),u(t)),
\end{split}
& t>0,t \neq t_k,\\
\begin{split}
w(s) &= \phi_1(s), \\
w'(s)&=\phi_2(s),
\end{split}
&s \in [-r,0],\\
w' (t_{k}^{+}) = w' (t_{k}^{-})+I^{e}_{k}(t_k,w(t_{k}),w' (t_{k}),u(t_{k},)), & k=1, \dots, p,
\end{array}%
\right.
\end{equation}
where $\mathcal{U}=\mathcal{X}=L^{2}(\Omega)$, and
\begin{eqnarray*}
I_{k}^{e}:&[0, \tau]\times \mathcal{Z}^{1} \times \mathcal{U} &\longrightarrow \qquad \mathcal{X} \\
&(t,w,v,u)(\cdot)&\longmapsto \quad I_{k}(t,w(\cdot),v(\cdot),u(\cdot)),
\end{eqnarray*}
\begin{eqnarray*}
f^{e}:&[0, \tau]\times \C (-r,0; \mathcal{Z}^{1} ) \times \mathcal{U} & \longrightarrow \qquad \mathcal{X}\\
&(t,\Phi,u)(\cdot)&\longmapsto \quad f(t,\phi_1(-r, \cdot),\phi_2(-r, \cdot),u(\cdot)),
\end{eqnarray*}
and
\begin{eqnarray*}
g^{e}:&\C(-r,0; \mathcal{Z}^{1} ) &\longrightarrow \mathcal{Z}^{1} \\
&\Phi=\left(\begin{array}{c}
\phi_1\\
\phi_2
\end{array}\right)&\longmapsto g(\phi_1(\cdot-r)).
\end{eqnarray*}
\noindent Changing variables, $v=w',$ the systems \eqref{eq:ODE} can be written as an abstract first order
functional differential equations with memory, impulses and delay in $\mathcal{Z}^{1}$
\begin{equation}\label{eq:abstract}
\left\{
\begin{array}{lr}
z' = -\mathbb{A} z+ \mathbb{B} u + \displaystyle \int_{0}^{t}\mathbb{M}_g(t,s ,z_{s}(-r))ds + \mathbb{F}(t,z_{t}(-r),u(s)) ,& z\in Z^{1},\; t\geq 0, \\
z(s) = \Phi(s), & s \in [-r,0],\\
z(t_{k}^{+}) = z(t_{k}^{-})+\mathbb{I}_{k}(t_k, z(t_{k}),u(t_{k})), & k=1,2,3, \dots, p,
\end{array}
\right.
\end{equation}
where $ z =\left(\begin{array}{c}
w\\
v
\end{array}\right)$,
$\Phi=\left(\begin{array}{c}
\phi_1\\
\phi_2
\end{array}\right) \in \C\left(-r,0; \mathcal{Z}^{1} \right),$ $u\in L^{2}(0,\tau;\mathcal{U})$, $\mathbb{A} = \left(
\begin{array}{rr}
0 & I_{\mathcal{X}} \\ - \mathcal{A}^2 & -2\beta \mathcal{A}
\end{array}\right)$ is a unbounded linear operator with domain
$$
D(\mathbb{A})=\{w\in H^{4}(\Omega):\:w=\Delta w=0\}\times D(\mathcal{A}),
$$ and $I_{\mathcal{X}}$ being the identity in $\mathcal{X} $.
$\mathbb{B}: \mathcal{U} \longrightarrow \mathcal{Z}^{1}$ is the bounded linear operator defined by
$\mathbb{B} u=
\left(\begin{array}{c}
0\\
u
\end{array}\right),$ and the functions
\begin{eqnarray*}
\mathbb{I}_{k}:&[0, \tau]\times \mathcal{Z}^{1} \times \mathcal{U}& \longrightarrow \qquad \mathcal{Z}^{1} \\
&(t, z,u)&\longmapsto \quad
\left(\begin{array}{c}
0\\
I_{k}^{e}(t,w,v,u)
\end{array}\right)
\end{eqnarray*}
\begin{eqnarray}
\label{functionF}
\mathbb{F}:& [0, \tau] \times \C(-r,0; \mathcal{Z}^{1} ) \times \mathcal{U} & \longrightarrow \qquad \mathcal{Z}^{1}\\
&(t, \Phi,u)&\longmapsto \quad
\left(
\begin{array}{c}
0 \\f^{e}(t,\phi_1(-r),\phi_2(-r),u)
\end{array}
\right),\notag
\end{eqnarray}
and
\begin{eqnarray*}
\mathbb{M}_g:&[0,\tau]\times [0,\tau]\times \C(-r,0; \mathcal{Z}^{1} ) &\longrightarrow \mathcal{Z}^{1} \\
&(t,s,\Phi)&\longmapsto \left(
\begin{array}{c}
0 \\M(t,s) g^{e}(\Phi)
\end{array}
\right).
\end{eqnarray*}
Moreover, this abstract formulation together with condition \eqref{eq:f1} and the continous imbeding $\mathcal{X}^1 \subset \mathcal{X}$ yields
\begin{prop}\label{prop:cotaF}
There exist constants $\tilde{a},\tilde{b}>0$ such that, for all $(t, \Phi,u) \in [0, \tau] \times \C(-r,0; \mathcal{Z}^{1} ) \times \mathcal{U}$ the following inequality holds
\begin{equation}\label{eq:bound}
\norm{\mathbb{F}(t,\Phi,u)}_{\mathcal{Z}^{1}} \leq \tilde{a}\| \Phi(-r) \|_{\mathcal{Z}^1}+\tilde{b}.
\end{equation}
\end{prop}
A.~ Carrasco, H.~ Leiva, and J.~Sanchez \cite[Theorem 2.1]{Carrasco-Leiva:2013aa} proved that the linear unbounded operator
$\mathbb{A}$ generates a strongly continuous compact semigroup
in the space $\mathcal{Z}^1$ which decays exponentially to zero, precisely:
\begin{prop}\label{semigroup}
The operator $\mathcal{A}$ is the infinitesimal generator
of a strongly continuous compact semigroup $\{T(t)\}_{t\geq0}$
represented by
\begin{equation}\label{repre}
T(t)z=\displaystyle\sum_{j=1}^{\infty}e^{\mathbb{A}_{j}t}P_{j}z,\qquad z\in
\mathcal{Z}^{1},\;t\geq 0,
\end{equation}
where $\{P_{j}\}_{j\geq0}$ is a complete family of orthogonal
projections in the Hilbert space $\mathcal{Z}^{1}$ given by
\begin{equation}\label{proyecciones}
P_{j} = diag(E_{j},E_{j}),
\end{equation}
and
$$
\mathbb{A}_{j}=K_{j}P_{j},\qquad
K_{j}=\left(
\begin{array}{cc}
0 & 1 \\
-\lambda_{j}^{2} & -2\beta\lambda_{j} \\
\end{array}
\right),\qquad j\geq 1,
$$
and there exists $M \geq 1$ and $\mu >0$ such that
$$
\parallel T(t)\parallel\leq Me^{-\mu t},\qquad t\geq0.
$$
\end{prop}
\section{Approximate Controllability of the Linear System }\label{sec:lineal}
This section is devoted to characterize the approximate controllability of the linear system. Thus, for all $z_{0}\in \mathcal{Z}^{1}$ and $u\in
L^{2}([0,\tau];\mathcal{U})$ consider the initial value problem
\begin{equation}\label{eq:linear}
\left\{%
\begin{array}{lll}
z'(t) = \mathbb{A} z(t) + \mathbb{B} u(t),\\
z(t_{0}) = z_{0},
\end{array}%
\right.
\end{equation}
obtained from \eqref{eq:abstract}. It admits only one mild solution on $0\leq t_{0}\leq t\leq \tau$ given by
\begin{equation}\label{eq:mild-linear}
z(t)=T(t-t_{0})z_{0} + \displaystyle\int_{t_{0}}^{t}T(t-s)\mathbb{B} u(s)ds
\end{equation}
\begin{definition} \label{def2}
({\bf Approximate Controllability of (\ref{eq:linear})}) The system (\ref{eq:linear}) is said
to be approximately controllable on $[t_{0},\tau]$ if for every $z_0$,
$z_1\in \mathcal{Z}$, $\varepsilon>0$ there exists $u\in L^{2}(t_{0},\tau;\mathcal{U})$ such
that the solution $z(t)$ of (\ref{eq:mild-linear}) corresponding to $u$
verifies: $$\|z(\tau)-z_1\|<\varepsilon.$$
\end{definition}
For the system \eqref{eq:linear} and $\tau>0$, we have the following notions:
\begin{enumerate}
\item $G_{\tau\delta}$ is the controllability operator defined by
\begin{eqnarray*}
G_{\tau\delta}: L^2(\tau-\delta,\tau;\mathcal{U}) \longrightarrow& \mathcal{Z}^{1}\\
u\longmapsto&\displaystyle \int_{\tau-\delta}^{\tau}T(\tau-s)\mathbb{B} u(s)ds,
\end{eqnarray*}
with corresponding adjoint $G^*_{\tau\delta}$ given by
\begin{eqnarray*}
G^*_{\tau\delta}: \mathcal{Z}^{1} \longrightarrow& L^2(\tau-\delta,\tau;\mathcal{U})\\
z\longmapsto& \mathbb{B} ^{*}T^{*}(\tau-\cdot)z.
\end{eqnarray*}
\item The Gramian controllability operator is
\begin{equation*}
Q_{\tau \delta*} = G_{\tau\delta}G_{\tau\delta}^{*}= \int_{\tau-\delta}^{\tau}T(\tau-t)\mathbb{B} \mathbb{B} ^{*}T^{*}(\tau-t)dt.
\end{equation*}
\end{enumerate}
In general, for linear bounded operator $G$ between Hilbert spaces $\mathcal{W}$ and $\mathcal{Z}$, the following lemma holds (see \cite{Bashirov-Kerimov:1997aa,Bashirov-Mahmudov:1999aa, Leiva-Merentes:2013aa}).
\begin{lemma}
The approximate controllability of the linear system \eqref{eq:linear} on $[\tau-\delta,\tau]$ is equivalent to any of the following statements
\begin{enumerate}[(a)]
\item $\overline{Rang(G_{\tau\delta})}=\mathcal{Z}^{1}.$
\item $\ker(G_{\tau\delta}^{*})={0}.$
\item For $0\neq z \in\ Z^{1}, \ \ \inner{ Q_{\tau\delta}z,z}>0.$
\end{enumerate}
\end{lemma}
The controllability of the linear system \eqref{eq:linear} on $[0,\tau]$ was proved by A. Carrasco and H. Leiva in \cite{Carrasco-Leiva:2013aa}. Theorem \ref{A1.5} and Lemma \ref{lema} characterized the controllability of the system \eqref{eq:linear}, their proofs and details can be found in \cite{Bashirov-Kerimov:1997aa,Bashirov-Mahmudov:1999aa, Curtain-Pritchard:1978aa, Curtain-Zwart:1995aa, Leiva-Merentes:2013aa}
\begin{teo}\label{A1.5} The system \eqref{eq:linear} is approximately
controllable on $[0,\tau]$ if and only if any one of the following conditions hold:
\begin{enumerate}
\item $\displaystyle\lim_{\alpha \to 0^+} \alpha(\alpha I +Q_{\tau\delta}^{*})^{-1}z =0 $.\\
\item If $z\in Z^{1}$, $0<\alpha \leq 1$ and $u_{\alpha}=G_{\tau\delta}^{*}(\alpha I +
Q_{\tau\delta}^{*})^{-1}z$, then
$$
G_{\tau\delta}u_{\alpha}=z - \alpha(\alpha I+ Q_{\tau\delta})^{-1}z \quad \mbox{and} \quad
\displaystyle\lim_{\alpha\to 0}G_{\tau\delta}u_{\alpha}=z.
$$
Moreover, for each $v\in L^{2}([\tau-\delta,\tau];\mathcal{U})$, the sequence of controls
$$
u_{\alpha}=G_{\tau\delta}^{*}(\alpha I +
Q_{\tau\delta}^{*})^{-1}z + (v-G_{\tau\delta}^{*}(\alpha I +
Q_{\tau\delta}^{*})^{-1}G_{\tau\delta}v),
$$
satisfies
$$
G_{\tau\delta}u_{\alpha}=z-\alpha(\alpha I +
Q_{\tau\delta}^{*})^{-1}(z-G_{\tau\delta}v)
\quad
\mbox{and}
\quad
\displaystyle\lim_{\alpha\to 0}G_{\tau\delta}u_{\alpha}=z,
$$
with the error $E_{\tau\delta}z=\alpha(\alpha I +
Q_{\tau\delta})^{-1}(z+G_{\tau\delta}v),\;\alpha\in(0,1].
$
\end{enumerate}
\end{teo}
Theorem \ref{A1.5} indicates that the family of linear operators $
\Gamma_{\tau\delta}=G_{\tau\delta}^{*}(\alpha I +
Q_{\tau\delta}^{*})^{-1}
$
is an approximate right inverse for the $G_{\tau\delta}$, in
the sense that
$$
\displaystyle\lim_{\alpha\longrightarrow 0}G_{\tau\delta}\Gamma_{\tau\delta}=I,
$$
in the strong topology.
\begin{lemma}\label{lema}
$Q_{\tau\delta}> 0$, if and only if, the linear system \eqref{eq:linear} is controllable on $[\tau-\delta, \tau]$.
Moreover, for given initial state $y_0$ and final state $z_{1}$, there exists a sequence of controls $\{u_{\alpha}^{\delta}\}_{0 <\alpha \leq 1}$ in the space $L^2(\tau-\delta,\tau;\mathcal{U})$, defined by
$$
u_{\alpha}=u_{\alpha}^{\delta}= G_{\tau\delta}^{*}(\alpha I+ G_{\tau\delta}G_{\tau\delta}^{*})^{-1}(z_{1} - T(\tau)y_0),
$$
such that the solutions $y(t)=y(t,\tau-\delta, y_0, u_{\alpha}^{\delta})$ of the initial value problem
\begin{equation}\label{IVL}
\left\{
\begin{array}{l}
y'=\mathbb{A} y+\mathbb{B} u_{\alpha}(t), \ \ y \in \mathcal{Z}^{1}, \ \ t>0,\\
y(\tau-\delta) = y_0,
\end{array}
\right.
\end{equation}
satisfies
\begin{equation}\label{eq:limit}
\lim_{\alpha \to 0^{+}}y(\tau
= \lim_{\alpha \to 0^{+}}\left(T(\delta)y_0 + \int_{\tau-\delta}^{\tau}T(\tau-s)\mathbb{B} u_{\alpha}(s)ds \right)= z_{1}.
\end{equation}
\end{lemma}
\section{Controllability of the Semilinear System}\label{sec:semilineal}
This section is devoted to prove the main result of this paper, the approximate controllability of the beam equation (Theorem \ref{main}), which is it is equivalent to prove the controllability of the abstract system \eqref{eq:abstract} under the condition \eqref{eq:bound}. Recall
\begin{definition} \label{def2}{\rm (}{\sf Approximate Controllability}{\rm )} The system \eqref{eq:abstract} is said
to be approximately controllable on $[0,\tau]$
if for every $\epsilon>0$, every
$\Phi\in \C\left(-r,0; \mathcal{Z}^{1} \right)$ and a given initial state $z_{1}\in \mathcal{Z}^{1}$ there exists $u\in L^{2}(0,\tau;\mathcal{U})$, such that, the corresponding mild solution
\begin{align}\label{eq:mild}
z^{u}(t) = & \displaystyle T(t)\Phi(0)+\int_{0}^{t}T(t-s)\left[\mathbb{B} u(s)+\left(\int_{0}^{s}\mathbb{M}_g(s,l,z(l-r))dl\right)\right]ds \\
& \;+ \displaystyle \int_{0}^{t}T(t-s)\mathbb{F}(s,z(s-r),u(s))ds + \sum_{0 < t_k < t} T(t-t_k )\mathbb{I}_{k}(t_k,z(t_k), u(t_k)), \nonumber
\end{align}
satisfies $z(0)=\Phi(0)$ and
\begin{equation}\label{eq:goal}
\norm{ z^u(\tau) - z_{1}}_{\mathcal{Z}^1}<\epsilon.
\end{equation}
\end{definition}
The approach to obtain \eqref{eq:goal} consist in construct a sequence of controls conducting the system from the initial condition $\Phi$ to a small ball around $z_1.$ This is achieved taking advantage of the delay, which allows us to pullback the corresponding family of solutions to a fixed trajectory in short time interval. Now, we are ready to present the proof of our main result
\begin{teo} \label{main}
Under the condition \eqref{eq:f1} the impulsive semilinear beam equation with memory and delay \eqref{eq:beam}-\eqref{eq:initial} is approximately
controllable on $[0,\tau]$.
\end{teo}
\noindent {\bf Proof.} \mbox{} Let $\epsilon>0$, and given $\Phi\in \mathcal{C}$ and a final state $z_{1}$. By section \ref{sec:formulation}, we have that the semilinear beam equation in consideration can be represented as the abstract system \eqref{eq:abstract} under the condition \eqref{eq:bound}. Thus,
consider any $u\in L^{2}([0,\tau];\mathcal{U})$ and the corresponding mild solution \eqref{eq:mild} of the initial value problem \eqref{eq:abstract}, denoted by $z(t)=z(t,0,\Phi,u)$.
\noindent For $0\leq\alpha \leq 1,$ define the control $u_{\alpha}^{\delta}\in L^{2}([0,\tau];\mathcal{U})$ as follows
$$
u_{\alpha}^{\delta}(t)=\left\{\begin{array}{ccl}
u(t), &&0\leq t\leq \tau-\delta, \\
u_{\alpha}(t), &\quad& \tau-\delta\leq t\leq \tau,
\end{array}\right.
$$
with $
u_{\alpha}= \mathbb{B}^{*}T^{*}(\tau-t)(\alpha I+ G_{\tau\delta}G_{\tau\delta}^{*})^{-1}(z_{1} - T(\delta)z(\tau-\delta)).
$
For,
$0<\delta<\tau-t_{p}$ its corresponding mild solution at time $\tau$ can be written as follows:
\begin{eqnarray*}
\displaystyle
z^{\delta,\alpha}(\tau) &=&
\displaystyle T(\tau)\Phi(0) +\int_{0}^{\tau}T(\tau-s)
\left[ \mathbb{B} u_{\alpha}^{\delta} (s)
+ \int_0^s \mathbb{M}_g(z^{\delta,\alpha}(l-r))dl\right]ds+ \\
&&+ \int_{0}^{\tau}T(\tau-s)\mathbb{F}(s,z^{\delta,\alpha}(s-r),u_{\alpha}^{\delta}(s))ds+
\sum_{0 < t_k < \tau} T(t-t_k )\mathbb{I}_{k}(t_k,z^{\delta,\alpha}(t_k), u_{\alpha}^{\delta}(t_k))\\
&=&T(\delta)\left\{T(\tau-\delta)\Phi(0)
+\int_{0}^{\tau-\delta}T(\tau-\delta-s) \left(\mathbb{B} u_{\alpha}^{\delta} (s)+\mathbb{F}(s,z^{\delta,\alpha}(s-r),u_{\alpha}^{\delta}(s))\right)ds\right.\\
&&\qquad\quad+\int_{0}^{\tau-\delta}T(\tau-\delta-s) \int_0^s \mathbb{M}_g(s,l, z^{\delta,\alpha}(l-r))dlds\\
&&\qquad\quad\left.+ \sum_{0 < t_k < \tau-\delta} T(t-\delta-t_k )\mathbb{I}_{k}(t_k,z^{\delta,\alpha}(t_k), u_{\alpha}^{\delta}(t_k))\right\}+\\
&& + \int_{\tau-\delta}^{\tau}T(\tau-s)\left(\mathbb{B} u_{\alpha}(s)+
\mathbb{F}(s,z^{\delta,\alpha}(s-r),u_{\alpha}^{\delta}(s))+\int_0^s\mathbb{M}_g(s,l,z^{\delta,\alpha}(l-r))dl\right)ds.
\end{eqnarray*}
Therefore,
\begin{align*}
z^{\delta,\alpha}(\tau) = &T(\delta)z(\tau-\delta)+ \int_{\tau-\delta}^{\tau}T(\tau-s)\left(\mathbb{B} u_{\alpha}(s)+
\mathbb{F}(s,z^{\delta,\alpha}(s-r),u_{\alpha}^{\delta}(s))\right)ds \\
&+ \int_{\tau-\delta}^{\tau}T(\tau-s)\int_0^s\mathbb{M}_g(s,l,z^{\delta,\alpha}(l-r))dlds.
\end{align*}
Observing that the corresponding solution $y^{\delta,\alpha}(t)=y(t,\tau-\delta,z(\tau-\delta),u_{\alpha})$ of the initial value problem \eqref{IVL} at time $\tau$ is:
$$
y^{\delta,\alpha}(\tau)=T(\delta)z(\tau-\delta)+ \int_{\tau-\delta}^{\tau}T(\tau-s)\mathbb{B}_{\varpi} u_{\alpha}(s)ds,
$$
yields,
$$
z^{\delta,\alpha}(\tau)-y^{\delta,\alpha}(\tau)=
\int_{\tau-\delta}^{\tau}T(\tau-s)\left(\int_{0}^{s}\mathbb{F}(s,z^{\delta,\alpha}(s-r),u_{\alpha}^{\delta}(s))+\mathbb{M}_g(s,l,z^{\delta,\alpha}(l-r))dl)\right)ds,
$$
and together with condition \eqref{eq:bound}, we obtain
\begin{align*}
\norm{ z^{\delta,\alpha}(\tau)-y^{\delta,\alpha}(\tau)} & \leq \int_{\tau-\delta}^{\tau} \norm{ T(\tau-s)}\left( \tilde{a}\norm{\Phi(s-r)}+\tilde{b}\right)ds \\
& + \int_{\tau-\delta}^{\tau}\norm{ T(\tau-s)}\int_{0}^{s}\norm{\mathbb{M}_g(s,l,z^{\delta,\alpha}(l-r))}dlds.
\end{align*}
Observe that
$0< \delta< r$ and $\tau-\delta \leq s\leq \tau$, thus $$l-r \leq s-r \leq \tau-r< \tau-\delta.
$$
Therefore,
$
z^{\delta,\alpha}(l-r)=z(l-r) $ and $ z^{\delta,\alpha}(s-r)=z(s-r),
$ implying that for $\epsilon>0,$ there exists $\delta>0$ such that
\begin{align*}
\norm{z^{\delta,\alpha}(\tau)-y^{\delta,\alpha}(\tau)} & \leq \int_{\tau-\delta}^{\tau}\norm{ T(\tau-s)}\left( \tilde{a}\norm{z(s-r)}+\tilde{b}\right)ds \\
&\quad + \int_{\tau-\delta}^{\tau}\norm{T(\tau-s)}\int_{0}^{s}\norm{ \mathbb{M}_g(s,l,z(l-r))} dlds \\
& < \displaystyle\frac{\epsilon}{2}.
\end{align*}
Additionally, for $0<\alpha <1$, Lemma \ref{lema} \eqref{eq:limit} yields
$$
\norm{ y^{\delta,\alpha}(\tau)-z_{1}} < \frac{\epsilon}{2}.
$$
Thus,
$$
\begin{array}{lll}
\norm{ z^{\delta,\alpha}(\tau)-z_{1}} & \leq & \norm{ z^{\delta,\alpha}(\tau)-y^{\delta,\alpha}(\tau)} + \norm{ y^{\delta,\alpha}(\tau)-z_{1}} < \frac{\epsilon}{2}+ \frac{\epsilon}{2}=\epsilon,
\end{array}
$$
which completes our proof.
\section{Final Remarks}\label{final}
\noindent
We believe this technique can be applied for controlling diffusion processes systems involving compact semigroups. In particular, our result can be formulated in a more general setting for the semilinear evolution equation with impulses, delay and memory in a Hilbert space $\mathcal{Z}$
\begin{equation*}\label{eq:class}
\left\{
\begin{array}{lr}
z' = -\mathbb{A} z+ \mathbb{B} u + \displaystyle \int_{0}^{t}\mathbb{M}_g(t,s ,z(s-r))ds + \mathbb{F}(t,z_{t}(-r),u(s)) ,& z\in Z^{1},\; t\geq 0, \\
z(s) = \Phi(s), & s \in [-r,0],\\
z(t_{k}^{+}) = z(t_{k}^{-})+\mathbb{I}_{k}(t_k, z(t_{k}),u(t_{k})), & k=1,2, \dots, p,
\end{array}
\right.
\end{equation*}
where $u\in L^{2}(0,\tau;\mathcal{U})$, $\mathcal{U}$ is another Hilbert space, $\mathbb{B} :\mathcal{U} \longrightarrow \mathcal{Z}$ is a bounded linear operator, $\mathbb{I}_{k}, \mathbb{F}:[0, \tau]\times \C(-r,0; \mathcal{Z}) \times \mathcal{U} \rightarrow \mathcal{Z}$, $\mathbb{A} :D(\mathbb{A}) \subset \mathcal{Z} \rightarrow \mathcal{Z}$ is an unbounded linear operator in $\mathcal{Z}$ that generates a strongly continuous semigroup \cite[Lemma 2.1]{Leiva:2003aa}
\begin{equation*}\label{damp2}
T(t)z =\sum_{nj=1}^{\infty}e^{\mathbb{A}_{j}t}P_jz%
\mbox{, } \ \ z\in \mathcal{Z} \mbox{, } \ \ t \geq 0,
\end{equation*}
where $\left\{ P_j\right\} _{j \geq 0}$ is a complete family of orthogonal projections in the Hilbert space $\mathcal{Z}$ and
\begin{equation*}
\|\mathbb{F}(t,\Phi,u) \|_{\mathcal{Z}} \leq \tilde{a} \|\Phi(-r)\|_{\mathcal{Z}} +\tilde{b},
\end{equation*}
for all $(t, \Phi, u) \in [0, \tau]\times \C(-r,0; \mathcal{Z} ) \times \mathcal{U}.$
\begin{center} {\sc Acknowledgments}
\end{center}
The authors are thankful to the anonymous referees for valuable comments
that help improve the quality of the paper. This work has been supported by Louisiana State University, Universidad YachayTech and Universidad Centroccidental Lisandro Alvarado.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,472 |
South African Air Force » Museum Flight
This shot of Dakota 6859 of the SAAF Museum Flight was taken on 17 September 2005 during the annual Swartkop Airshow. The aircraft of the Museum Flight are kept in immaculate condition. Up to this point this aircraft is one of the few Museum aircraft that have not been registered on the ZU register. Only the last two digits, which appear in yellow on the nose of the aircraft, of the serial number identify the aircraft as Dakota 6859. During the Border War era this was the only marking identifying some aircraft. 70 years after seeing the light of day, the Dakota is still going strong, albeit in this case with a museum and not in operational service. The Dakota is one of the most popular aircraft ever manufactured and it is quite appropriate that the SAAF Museum has a flying example. 6859 has con number 12586 and was originally built as a Douglas C-47A-DK.
This image of Atlas C-4M Kudu 969 (registration ZU-CWZ) of the SAAF Museum Flight was taken on 17 September 2005 at Swartkop (ICAO code FASK) during the Swartkop Airshow. One of the great features of the Swartkop Airshow is that the sunlight is absolutely beautiful on the aircraft in the afternoon. The aircraft is finished in the scheme as used by 41 Squadron in the 1980s and it has the c/n 19.
Here North American Harvard III 7001 (registration ZU-CXV) of the SAAF Museum Flight is seen on 17 September 2005 at Swartkop (ICAO code FASK). It is actually 7001 number 2. The previous 7001 was a Harvard IIA. The previous identity of 7001(2) is 7559, which went to Gabon in 1970. Below the cockpit is the name �Inkwazi�, which means �fish eagle�, and note the eagle symbol on the rear fuselage. At one stage (from about 1993 to about 2003) the eagle was used as the national marking of the SAAF.
This Mirage IIIBZ spent many years as 817 with 2 Squadron SAAF. After the Mirage III was officially retired from the SAAF towards the end of 1990, the aircraft fulfilled a test role before being transferred to the SAAF Museum. It is still with the Museum, but now with civilian registration ZU-DMD and in this all-flag colour scheme. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,946 |
Q: How do I uninstall/purge php 7.x completely? (Kali/Debian) On my distribution of Kali (2016.2) there was a php package v7.0.11 preinstalled. I'd like to use version 5.6 instead. Could anyone guide me through on how to remove it? I've already tried, looking for some guides online, but php7 is still pretty fresh, so there are not too many threads about it, especially while using Kali. Nevertheless, I found some clues but I didn't manage to do it properly, apparently. A lot of mess with dependencies, and overall, my whole Kali install is now broken.
After purging php7, I'd like to install php 5.6, is there an easy way to do it with apt?
P.S. This was previously posted on Stack Overflow. My bad.
A: If I am understanding this correctly, you're having sort of the same problem that I had when I tried to do the same with Apache2. Here's how to purge it and remove it. First, enter.service --status-all To see the exact name of it, just find it in the list. Next, type apt-get --purge nameofprogramyouwanttopurge DO NOT CONTINUE UNTIL YOU READ THIS NEXT PART THROUGH Now, when that command initializes, it will give you the option of (y,n) and you choose yes or no. Before you choose yes or no, fully read the entire output in your terminal and make sure of every file it is preparing to remove is what you want to remove. Don't hit yes until you are sure there's no packages or dependencies being purged that you don't want purged. Don't be lazy and skip the reading. If you are satisfied with all the files it lists being purged, then select yes And move to the next step. After the purging and removal is done, you will want to enter apt-get upgrade Reply back here with the output of that line before you say yes to the upgrade.
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"redpajama_set_name": "RedPajamaStackExchange"
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//----------------------------------------------------------------------
/*
* The hashlib++ SHA384 and SHA512 implementations are derivative from the
* sourcecode published by Aaron D. Gifford
*
* Copyright (c) 2000-2001, Aaron D. Gifford
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. Neither the name of the copyright holder nor the names of contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTOR(S) ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTOR(S) BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
//----------------------------------------------------------------------
/**
* @file hl_sha2ext.h
* @brief This file contains the declaration of the SHA384 and
* SHA512 classes
* @date Mo 12 Nov 2007
*/
//----------------------------------------------------------------------
//include protection
#ifndef SHA2ext_H
#define SHA2ext_H
//----------------------------------------------------------------------
//lenght defines
#define SHA384_BLOCK_LENGTH 128
#define SHA384_DIGEST_LENGTH 48
#define SHA384_DIGEST_STRING_LENGTH (SHA384_DIGEST_LENGTH * 2 + 1)
#define SHA512_BLOCK_LENGTH 128
#define SHA512_DIGEST_LENGTH 64
#define SHA512_DIGEST_STRING_LENGTH (SHA512_DIGEST_LENGTH * 2 + 1)
#define SHA512_SHORT_BLOCK_LENGTH (SHA512_BLOCK_LENGTH - 16)
//----------------------------------------------------------------------
//hl includes
#include "hl_types.h"
//----------------------------------------------------------------------
//typedefs
/**
* Exactly 1 byte
*/
typedef hl_uint8 sha2_byte;
/**
* Exactly 4 bytes
*/
typedef hl_uint32 sha2_word32;
/**
* Exactly 8 bytes
*/
typedef hl_uint64 sha2_word64;
/**
* @brief This struct represents a SHA512-hash context
*/
typedef struct SHA512_CTX
{
hl_uint64 state[8];
hl_uint64 bitcount[2];
hl_uint8 buffer[SHA512_BLOCK_LENGTH];
} SHA512_CTX;
/**
* @brief This struct represents a SHA384-hash context
*/
typedef SHA512_CTX SHA384_CTX;
//----------------------------------------------------------------------
/**
* @brief This class represents the implementation of
* the SHA384 and SHA512 algorithm.
*
* Basically the class provides six public member-functions
* to create a hash: SHA384_Init(), SHA384_Update(), SHA384_End(),
* SHA512_Init(), SHA512_Update() and SHA512_End().
* If you want to create a hash based on a string or file quickly
* you should use the sha384wrapper or sha512wrapper classes.
*/
class SHA2ext
{
private:
/**
* @brief Finalize the sha384 operation
* @param digest The digest to finalize the operation with.
* @param context The context to finalize.
*/
void SHA384_Final(hl_uint8 digest[SHA384_DIGEST_LENGTH],
SHA384_CTX* context);
/**
* @brief Finalize the sha512 operation
* @param digest The digest to finalize the operation with.
* @param context The context to finalize.
*/
void SHA512_Final(hl_uint8 digest[SHA512_DIGEST_LENGTH],
SHA512_CTX* context);
/**
* @brief Internal method
*
* used by SHA512 and SHA384
* @author Benjamin Grüdelbach
* @param context The context of the operation
*/
void SHA512_Last(SHA512_CTX* context);
/**
* @brief Internal data transformation
* @param context The context to use
* @param data The data to transform
*/
void SHA512_Transform(SHA512_CTX* context,
const sha2_word64* data);
public:
/**
* @brief Initialize the SHA384 context
* @param context The context to init.
*/
void SHA384_Init(SHA384_CTX* context);
/**
* @brief Initialize the SHA512 context
* @param context The context to init.
*/
void SHA512_Init(SHA512_CTX* context);
/**
* @brief Updates the SHA512 context
* @param context The context to update.
* @param data The data for updating the context.
* @param len The length of the given data.
*/
void SHA384_Update(SHA384_CTX* context,
const hl_uint8* data,
unsigned int len);
/**
* @brief Updates the SHA284 context
* @param context The context to update.
* @param data The data for updating the context.
* @param len The length of the given data.
*/
void SHA512_Update(SHA512_CTX* context,
const hl_uint8* data,
unsigned int len);
/**
* @brief Ends the SHA384 operation and return the
* created hash in the given buffer.
* @param context The context to end.
* @param buffer This OUT-Parameter contains the created
* hash after ending the operation.
*/
char* SHA384_End(SHA384_CTX* context,
char buffer[SHA384_DIGEST_STRING_LENGTH]);
/**
* @brief Ends the SHA512 operation and return the
* created hash in the given buffer.
* @param context The context to end.
* @param buffer This OUT-Parameter contains the created
* hash after ending the operation.
*/
char* SHA512_End(SHA512_CTX* context,
char buffer[SHA512_DIGEST_STRING_LENGTH]);
};
//----------------------------------------------------------------------
//end of include protection
#endif
//----------------------------------------------------------------------
//EOF
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"redpajama_set_name": "RedPajamaGithub"
} | 9,475 |
void jumpToLocation(double x, double y, double z);
void acceptDoublePointer(double* _Nonnull ptr) __attribute__((swift_name("accept(_:)")));
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,997 |
{"url":"https:\/\/mathoverflow.net\/questions\/244949\/a-combinatorial-game-the-snake-and-the-hunter","text":"# A Combinatorial Game: the Snake and the Hunter\n\nThe Snake and the Hunter is a game for two players who play in two rounds interchanging the roles of snake and hunter. The game is played in a rectangular grid of points, say $6 \\times 6$. In both rounds the snake plays first and begins the game by joining with a straight line any two adjacent (either vertically or horizontally, but not diagonally) points. Thus the snake is born. From then on players take turns to lengthen the snake by one line on either of its extremes. At no stage can the snake \u201cbite\u201d itself. The round finishes when no further lengthening of the snake is possible, at which point the length of the snake, i.e., the number of its lines, is recorded.\n\nIn the second round players switch roles and the game proceeds in the same fashion. The winner of the game is whoever achieves the longest snake while playing the role of the snake. (The figure below shows a complete game won by the first player.)\n\nFirst player's snake of length 25\n\nSecond player's snake of length 17\n\nIf both players play optimally during a given round, what is the length of the snake obtained?\n\nNote: I originally posted this question in Mathematics Stack Exchange but received no answers.\n\n\u2022 I think this site is a lot more suitable for this question than puzzling.SE. \u2013\u00a0domotorp Jul 24 '16 at 18:15\n\u2022 I would formulate this problem as follows. First of all, let's forget about the second round completely. Two players play on an $n\\times n$ grid according to the above rules, and denote by $s(n)$ the length of the longest snake the first player can achieve with optimal strategies. What do we know about $s(n)$? Is it quadratic in $n$? What about small values? What about the analogous problem for other graphs than the grid? \u2013\u00a0domotorp Jul 24 '16 at 19:38\n\u2022 s(2) = 3 is trivial and s(3) = 8 is easy to show. \u2013\u00a0Bernardo Recam\u00e1n Santos Jul 24 '16 at 21:09\n\u2022 The first player can achieve a full length snake in the 4x4 grid too. The first player starts with down (wlog). If the second player follows with a horizontal edge (right), the first one can win in such a way that the second player from now on has only forced moves. Otherwise the first player wins by: down-down(otherwise we won)-down-right(forced)-up, and from here it is easy in both cases. \u2013\u00a0Daniel Solt\u00e9sz Jul 28 '16 at 2:15\n\u2022 @WhatsUp: your variant 2 is Slither, and was solved by William N. Anderson, Jr. here sciencedirect.com\/science\/article\/pii\/009589567490029X Your variant 1 seems to be the game called Trap here arxiv.org\/pdf\/1505.07485.pdf \u2013\u00a0Zack Wolske Sep 7 '16 at 19:15","date":"2019-06-17 11:40:44","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\": 0.499294251203537, \"perplexity\": 563.0331368746146}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-26\/segments\/1560627998473.44\/warc\/CC-MAIN-20190617103006-20190617125006-00458.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.physicsforums.com\/threads\/can-you-take-the-inverse-of-any-function.710336\/","text":"Can you take the inverse of any function?\n\n1. Sep 14, 2013\n\nZeroPivot\n\ni know when you are dealing with limits you can take the inverse to fit the standard limit equations.\n\nhow about integrals? can u take the inverse for instance: integral(f(x)dx)\n\nturn it into integral((1\/f(x))dx)^-1) get the answer and then reverse it back?\n\nwhen can u or cant you take the inverse???\n\n2. Sep 14, 2013\n\nHallsofIvy\n\nFirst, not every function has an inverse. But if a function has an inverse, then, theoretically, you can find it. Exactly how you would find that inverse and how hard it would be depends on the function.\n\nDo you have some specific reason for asking about the inverse of an integral? In general the inverse of the integral of a function is NOT the integral of the inverse of the function. Your last expression has an unmatched right parenthesis so it is hard to tell exactly what you intend.\n\n3. Sep 14, 2013\n\nZeroPivot\n\nlets say the fuction is continous and has an inverse.\n\nsomething easy say y=2x+1 and u want the integral integral(2x+1)dx can u switch it around and say (integral(1\/(2x+1))dx)^-1, are there any integrals u can do this to or is it a no no?\n\n4. Sep 14, 2013\n\nverty\n\nIntegrate that and see what you get. It'll have nothing to do with the inverse of y = 2x+1.\n\n5. Sep 14, 2013\n\nMandelbroth\n\nYou can take the inverse of any function. (pause for shock value :tongue:)\n\nHOWEVER, it is important to note that the inverse of a function is not necessarily a function. For example, consider the $\\sin$ function. Since $\\sin(x)=\\sin(x+2\\pi)$, $\\sin^{-1}(\\sin(x))$ will not be unique. In fact, $\\sin^{-1}(\\sin(x))=x+2\\pi n$, where $n\\in\\mathbb{Z}$.","date":"2018-07-15 23:31:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8328946828842163, \"perplexity\": 377.6137496154725}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676589022.38\/warc\/CC-MAIN-20180715222830-20180716002830-00108.warc.gz\"}"} | null | null |
\section{Introduction}
Orlicz spaces represent an important class of Banach function spaces considered in mathematical analysis. This class naturally arises as a generalization of $L^p$-spaces and contains, for example, the well-known Zygmund space $L \log^+ L$ which is a Banach space related to Hardy-Littlewood maximal functions. Orlicz spaces can also contain certain Sobolev spaces as subspaces. Linear properties of Orlicz spaces have been studied thoroughly (see \cite{rao} for example). However, until recently, little attention has been paid to their possible algebraic properties, particularly, if they are considered over translation-invariant measurable spaces. One reason might be that, on its own, an Orlicz space is rarely an algebra with respect to a natural product! For instance, it is well-known that for a locally compact group $G$, $L^p(G)$ ($1<p<\infty$) is an algebra under the convolution product exactly when $G$ is compact \cite{S}. Similar results have also been obtained for other classes of Orlicz algebras (see \cite{AM}, \cite{HKM}, \cite{S} for details).
The preceding results indicate that, in most cases, Orlicz spaces over locally compact groups are simply ``too big" to become algebras under convolution. However, it turned out that it is possible for ``weighted" Orlicz spaces to become algebras. In fact, weighted $L^p$-algebras and their properties have been studied by many authors including J. Wermer on the real line and
Yu. N. Kuznetsova on general locally compact groups (see, for example, \cite{K1}, \cite{K2}, \cite{KM}, \cite{jW} and the references therein). These spaces have various properties and numerous applications in harmonic analysis. For instance, by applying the Fourier transform, we know that Sobolev spaces $W^{k,2}(\T)$ are nothing but certain weighted $l_\om^2(\Z)$ spaces.
Recently, in \cite{OO}, A. Osan\c{c}l{\i}ol and S. \"{O}ztop considered weighted Orlicz algebras over locally compact groups and studied their properties, extending, in part, the results of \cite{K1} and \cite{K2}. In \cite{OS1} and \cite{OS2}, the first two-named authors initiated a more general approach by considering the twisted convolution coming from a 2-cocycle $\Om$ with values in $\C^*$, the multiplicative group of complex numbers. Sufficient conditions on $\Om$ were found ensuring that the twisted convolution coming from $\Om$ turns the Orlicz space to a Banach algebra or a Banach $*$-algebra
\cite[Theorems 3.3 and 4.5]{OS1}. These methods produce abundant families of Arens regular, symmetric dual Banach $*$-algebras in the form of weighted Orlicz algebras, mostly on compactly generated groups with polynomial growth (see \cite[Theorems 5.2 and 5.8]{OS1} and \cite[Theorem 4.2 and 5.3]{OS2}). Certain cohomological properties of weighted Orlicz algebras were also investigated in \cite{OS2} where it was shown that either property can not happen unless $G$ is finite \cite[Theorems 6.2 and 6.3]{OS2}.
In the present paper, we continue our investigation of the cohomology of weighted Orlicz algebras focusing on the concept of weak amenability. In~\cite{Z}, Y. Zhang provided a necessary and sufficient condition on the weight $\om$ ensuring that the weighted group algebra $(L^1_\om(G),*)$ on a locally compact abelian group $G$ is weakly amenable (see also \cite{BCD} and \cite{G} for earlier results). We show that, for a large family of Young functions, the condition presented in \cite{Z} is necessary and sufficient for weak amenability of weighted Orlicz algebras.
As an example, we apply our results to $\Z^d$, the group of $d$-dimensional integers, and characterize when $l^p_\om(\Z^d)$ ($1<p<\infty$) is weakly amenable. We also present many more classes of (non-) weakly amenable weighted Orlicz algebras.
We finish the introductory part by pointing out that throughout this paper we concern ourselves with the
theory of ``bounded multiplications" for Banach algebras and Banach modules, as opposed to ``contractive multiplications". Also weights for us are ``weakly submultiplicative" as opposed to ``submultiplicative".
\section{Preliminaries}
In this section, we give some definitions and state some technical results that will be crucial in the rest of the paper. In what follows, $G$ always denotes a locally compact group with a fixed left Haar measure $ds$.
\subsection{Orlicz Spaces}
We first recall some facts concerning Young functions and Orlicz spaces. Our main reference is \cite{rao}.
A nonzero function $\Phi:[0,\infty) \to[0,\infty]$ is called a Young
function if $\Phi$ is convex, $\Phi(0)=0$, and $\lim_{x\to \infty} \Phi(x)=\infty$. For a Young function $\Phi$, the complementary function $\Psi$ of
$\Phi$ is given by
\begin{align}\label{Eq:Young function-complementary}
\Psi(y)=\sup\{xy-\Phi(x):x\ge0\}\quad(y\geq 0).
\end{align}
It is easy to check that $\Psi$ is also a Young function. Also, if $\Psi$ is the complementary function of $\Phi$, then $\Phi$ is
the complementary function of $\Psi$ and $(\Phi,\Psi)$ is called a
complementary pair. We always have the Young inequality
\begin{align}\label{Eq:Young inequality}
xy\le\Phi(x)+\Psi(y)\quad(x,y\ge0)
\end{align}
for complementary functions $\Phi$ and $\Psi$. By
definition, Young function can have the value $\infty$ at some point,
and hence be discontinuous at this point. However, we always consider the pair of complementary Young
functions $(\Phi,\Psi)$ with both $\Phi$ and $\Psi$ being continuous and strictly increasing. In particular, they attain positive values on $(0,\infty)$. In this case, if we let $\Phi^{-1}$ and $\Psi^{-1}$ to be the inverse functions of $\Phi$ and $\Psi$, respectively, then, by \cite[Lemma~4.8.16]{BS} we have
\begin{align}\label{Eq:Inverse Young inequality}
x\leq \Phi^{-1}(x)\Psi^{-1}(x)\leq 2x \ \ \ (0\leq x<\infty).
\end{align}
Now suppose that $G$ is a locally compact group with a fixed Haar measure $ds$ and $(\Phi,\Psi)$ is a complementary pair of Young functions. We define
\begin{align}\label{Eq:Orlicz defn-0}
\mathcal{L}^\Phi(G)=\left\{f:G\to\C:f \ \text{is measurable and}\ \int_G\Phi(|f(s)|)\,ds <\infty
\right\}.
\end{align}
Since $\mathcal{L}^\Phi(G)$ is not always a linear space, we define the Orlicz space $L^\Phi(G)$ to be
\begin{align}\label{Eq:Orlicz defn}
L^\Phi(G)=\left\{f:G\to\C:\int_G\Phi(\alpha|f(s)|)\,ds <\infty
\mbox{ for some }\alpha>0\right\},
\end{align}
where $f$ indicates a member in equivalence class of measurable functions with respect to the Haar measure $ds$. When $G$ is discrete, we simply use the standard terminology and write $l^\Phi(G)$ instead of $L^\Phi(G)$. The Orlicz space is a Banach space under the (Orlicz) norm $\|\cdot\|_\Phi$
defined for $f\in L^\Phi(G)$ by
\begin{align}\label{Eq:Orlicz norm}
\|f\|_\Phi=\sup\left\{\int_G|f(s)v(s)|\,ds: \int_G\Psi(|v(s)|)\,ds \le1\right\},
\end{align}
where $\Psi$ is the complementary function of $\Phi$. One can also
define the (Luxemburg) norm $N_\Phi(\cdot)$ on $L^\Phi(G)$ by
\begin{align}\label{Eq:Orlicz Luxemburg defn}
N_\Phi(f)=\inf\left\{k>0:\int_G\Phi\left(\frac{|f(s)|}{k}\right)
\,ds \le1\right\}.
\end{align}
It is known that these two norms are equivalent, i.e.,
\begin{align}\label{Eq:Orlicz norm-Luxemburg relation}
N_\Phi(\cdot)\le \|\cdot\|_\Phi\le2 N_\Phi(\cdot)
\end{align}
and
\begin{align}\label{Eq:Orlicz norm-defn relation}
N_\Phi(f)\le1 \ \ \text{if and only if}\ \ \int_G\Phi(|f(s)|)\,ds \le1.
\end{align}
Let $\Sm^\Phi(G)$ be the closure of the linear
space of all step functions in $L^\Phi(G)$. Then $\Sm^\Phi(G)$ is a Banach space
and contains $C_c(G)$, the space of all continuous functions on $G$ with compact support, as a dense subspace \cite[Proposition 3.4.3]{rao}. Moreover, $\Sm^\Phi(G)^*$, the dual of $\Sm^\Phi(G)$, can be identified with $L^\Psi(G)$ in a natural way \cite[Theorem 4.1.6]{rao}. Another useful characterization of $\Sm^\Phi(G)$ is that
$f\in \Sm^\Phi(G)$ if and only if for every $\alpha>0$, $\alpha f\in \mathcal{L}^\Phi(G)$ \cite[Definition 3.4.2 and Proposition 3.4.3]{rao}.
A Young function $\Phi$ satisfies the $\Delta_2$-condition
if there
exists a constant $K>0$
such that
\begin{align}\label{Eq:Delta 2 condition}
\Phi(2x)\le K\Phi(x) \ \ \text{for all}\ \ x\ge 0.
\end{align}
In this case we write
$\Phi\in\Delta_2$.
If $\Phi\in\Delta_2$, then it follows that $L^\Phi(G)=\Sm^\Phi(G)$ so that $L^\Phi(G)^*=L^\Psi(G)$ \cite[Corollary 3.4.5]{rao}. If, in addition, $\Psi\in\Delta_2$, then the Orlicz space $L^\Phi(G)$ is a reflexive Banach space.
As in \cite[Page 20]{rao}, we say that two Young functions $\Phi_1$ and $\Phi_2$ are {\it strongly equivalent} and write $\Phi_1 \approx \Phi_2$ if there exists $0<a\leq b<\infty$ such that
\begin{equation}\label{Eq:equivalent young functions}
\Phi_1(ax)\leq \Phi_2(x)\leq \Phi_1(bx) \ \ \ (x\geq 0).
\end{equation}
It is clear from the definition of the Orlicz space \eqref{Eq:Orlicz defn} that the strongly equivalent Young functions generate the same Orlicz space.
We will frequently use the (generalized) H\"{o}lder's inequality for Orlicz spaces
\cite[Remark 3.3.1]{rao}.
More precisely, for any complementary pair of Young functions $(\Phi,\Psi)$
and any $f\in L^\Phi(G)$ and $g\in L^\Psi(G)$, we have
\begin{align}\label{Eq:Holder inequality}
\|fg\|_1:=\int_G |f(s)g(s)|ds \leq \min\{N_\Phi(f)\|g\|_\Psi , \|f\|_\Phi N_\Psi(g)\}.
\end{align}
This, in particular, implies that $fg\in L^1(G)$.
In general, there is a straightforward method to construct various complementary pairs of strictly increasing continuous Young functions as described in
\cite[Theorem 1.3.3]{rao}. Suppose that $\varphi: [0,\infty)\to [0,\infty)$ is a continuous strictly increasing function with $\varphi(0)=0$ and
$\lim_{x\to \infty} \varphi(x)=\infty.$
Then $$\Phi(x)=\int_0^x \varphi(y)dy$$ is a continuous strictly increasing Young function and
$$\Psi(y)=\int_0^y \varphi^{-1}(x)dx$$
is the complementary Young function of $\Phi$ which is also continuous and strictly increasing.
Here $\varphi^{-1}(x)$ is the inverse function of $\varphi$. Below are several families of examples obtained using the above construction (see \cite[Proposition 2.11]{ML} and \cite[Page 15]{rao} for more details):
$(1)$ For $1< p,q<\infty$ with $\frac{1}{p}+\frac{1}{q}=1$, if $\Phi(x)=\frac{x^p}{p}$, then $\Psi(y)=\frac{y^q}{q}$. In this case,
the space $L^{\Phi}(G)$ becomes the Lebesgue space $L^p(G)$ and the norm $\|\cdot\|_{\Phi}$ is equivalent to the
classical norm $\|\cdot\|_{p}$.
(2) If $\Phi(x)=\cosh x-1$, then $\Psi(x)\approx x\ln (1+x)$.
(3) If $\Phi(x)=(1+x)\ln(1+x)-x$, then $\Psi(x)=e^x-x-1$.
\subsection{Weighted Orlicz algebras}\label{S:Twisted Orlicz alg}
In this section, we present and summarize what we need from the theory of twisted Orlicz algebras. These are taken from \cite{OS1}.
A {\bf weight} on $G$ is a locally integrable measurable function $\om : G \to \R_+$
with $\om(e)=1$ and $1/\om\in L^\infty(G)$ such that there is $C>0$ satisfying
\begin{align}\label{Eq:defn-weight}
\om(st)\leq C\om(s)\om(t) \ \ \ (s,t\in G)
\end{align}
In this case, we define
$$\Om(s,t)=\frac{\om(st)}{\om(s)\om(t)} \ \ \ (s,t\in G).$$
For a weight $\om$ on $G$, we define the weighted spaces
$$L^\Phi_\om(G):=\{f: f\om \in L^\Phi(G) \},$$
and
$$\Sm^\Phi_\om(G):=\{f: f\om \in \Sm^\Phi(G) \}.$$
It is clear that with the norm
$$\|f\|_{\Phi,\om}:=\|f\om\|_\Phi,$$
$L^\Phi_\om(G)$ becomes a Banach space having $\Sm^\Phi_\om(G)$ as a closed subspace. We say that $L^\Phi_\om(G)$ is a {\bf weighted Orlicz algebra} if it is a Banach algebra under the convolution product. In this case, $\Sm^\Phi_\om(G)$ becomes a closed subalgebra of $L^\Phi_\om(G)$ which we call the {\bf maximal essential subalgebra of} $L^\Phi_\om(G)$.
In \cite{OS1}, sufficient conditions on $\om$ were found under which the convolution turns a weighted Orlicz space into a Banach algebra (see \cite[Lemma 3.2 and Theorem 3.3]{OS1}). We summarize them below:
\begin{thm}\label{T:twisted Orlicz alg}
Let $G$ be a locally compact group and $\om$ a weight on $G$. \\
$(i)$ $L_\om^\Phi(G)$ is a Banach $L_\om^1(G)$-bimodule with respect to the convolution having $\Sm_\om^\Phi(G)$ as an essential Banach $L_\om^1(G)$-submodule.\\
$(ii)$ Suppose that there exist non-negative measurable functions $u$ and $v$
in $L^\Psi(G)$ such that
\begin{align}\label{Eq:2-cocycle bdd sum}
|\Om(s,t)|\leq u(s)+v(t) \ \ \ (s,t\in G).
\end{align}
Then for every $f,g\in L_\om^\Phi(G)$, the convolution is well-defined on $L_\om^\Phi(G)$
so that $(L_\om^\Phi(G),\tw)$ becomes a twisted Orlicz algebra having $\Sm_\om^\Phi(G)$ as a closed subalgebra.
\end{thm}
Let $G$ be a compactly generated abelian group. Then, by the Structure Theorem,
\begin{align}\label{Eq:Compactly gen abelian group-structure thm}
G\cong \R^k \times \Z^m \times T
\end{align}
where $k,m \in \N \cup \{0\}$ and $T$ is a compact (abelian) group. We can then consider the
generating set
\begin{align}\label{Eq:Compactly gen abelian group-generating set}
U=(-1,1)^k\times \{-1,0,1\}^m\times T
\end{align}
and define a {\it length function} $|\cdot|_U : G \to \N\cup \{0\}$ by
\begin{align}\label{Eq:length function}
|s|_U=\inf \{n\in \N : s\in U^n \} \ \ \text{for} \ \ s \neq e, \ \ |e|=0.
\end{align}
When there is no fear of ambiguity, we write $|\cdot|$ instead of $|\cdot|_U$.
It is straightforward to verify that $|\cdot|$ is a symmetric subadditive function on $G$, i.e.
\begin{align}\label{Eq:lenght func-trai equality}
|s+t|\leq |s|+|t| \ \ \text{and} \ \ |s|=|-s|\ \ \ (s,t\in G).
\end{align}
Now if $\nu:\N\cup \{0\}\to \R^+$ is a continuous increasing subadditive function with $\nu(0)=0$ and $\lim_{n\to \infty}\nu(n)=\infty$, then
\begin{align}\label{Eq:weight-lenght function}
\om(t)=e^{\nu(|t|)} \ \ \ (t\in G)
\end{align}
is a weight on $G$. This gives rise to a number of different weights on $G$.
For example, for every $0< \alpha \leq 1$, $\beta \geq 0$, $\gamma >0$, and $C>0$,
we can define the {\it polynomial weight} $\om_\beta$ on $G$ of order $\beta$ by
\begin{align}\label{Eq:poly weight-defn}
\om_\beta(s)=(1+|s|)^\beta \ \ \ \ (s\in G),
\end{align}
and the {\it subexponential weights} $\sg_{\alpha, C}$ and $\rho_{\beta,C}$ on $G$ by
\begin{align}\label{Eq:Expo weight-defn}
\sg_{\alpha,C}(s)=e^{C|s|^\alpha} \ \ \ \ (s\in G)
\end{align}
and
\begin{align}\label{Eq:Expo weight II-defn}
\rho_{\gamma,C}(s)=e^\frac{C|s|}{(\ln (1+|s|))^\gamma} \ \ \ \ (s\in G).
\end{align}
We can apply Theorem~\ref{T:twisted Orlicz alg}(ii) to these weights to obtain weighted Orlicz algebras. In particular, we have the following result (\cite[Corollary 5.3 and Theorem 5.8]{OS1}).
\begin{thm}\label{T:twisted Orlicz alg-poly and exp weight-Poly growth}
Let $G$ be a compactly generated abelian group, and $\om$ be a weight on $G$.
Then $(L_\om^\Phi(G),\tw)$ is a weighted Orlicz algebra if $\om$ is either one of the following weights:\\
$(i)$ $\om=\om_\beta$, the polynomial weight \eqref{Eq:poly weight-defn} with $1/\om \in \Sm^\Psi(G)$;\\
$(ii)$ $\om=\sg_{\alpha,C}$, the subexponential weight \eqref{Eq:Expo weight-defn};\\
$(iii)$ $\om=\rho_{\gamma,C}$, the subexponential weight \eqref{Eq:Expo weight II-defn}.
\end{thm}
We finish this section by pointing out that in the proof of \cite[Corollary 5.3]{OS1}, it was shown that $1/\om_\beta \in \Sm^\Psi(G)$ if $\beta> \frac{k+m}{l}$,
where $k$ and $m$ come from the representation \eqref{Eq:Compactly gen abelian group-structure thm} and
$l\geq 1$ is such that $\lim_{x\to 0^+}\frac{\Psi(x)}{x^l}$ exists.
\section{Weak amenability}
A Banach algebra $A$ is {\it weakly amenable} if every bounded derivation $D$ from $A$ into $A^*$ is inner \cite[Definition 4.2.1]{Run}. When $A$ is commutative, one can use the equivalent formulation that $A$ is weakly amenable if every bounded derivation from $A$ into any symmetric Banach $A$-module is zero \cite[Theorem 2.8.63]{D}. The class of weakly amenable Banach algebras is much larger than the class of amenable ones. For instance, all C$^*$-algebras and group algebras are weakly amenable \cite[Theorems 4.2.3 and 4.2.4]{Run}.
In \cite[Thereom 3.1]{Z}, Y. Zhang has found a necessary and sufficient condition, formulated below in \eqref{Eq:group homo vs weak amen}, for weak amenability of weighted group algebras on abelian groups. His work extends the previous partial results in this direction (see \cite{Z} and the reference therein). Our goal is to further extend the result of Zhang to weighted Orlicz algebras. Interestingly, we will see that, in most cases, the criterion found in \cite[Theorem 3.1]{Z} also works in our settings.
We start with the following theorem which shows that the sufficient condition formulated in \cite{Z} will imply weak amenability of the maximal essential subalgebras of weighted Orlicz algebras. Note that in what follows we always consider $\mathbb{C}$ as an additive group.
\begin{thm}\label{T:weighted Orlicz algebra-weak amenable}
Let $G$ be a locally compact abelian group, $\om$ be a weight on $G$, and $(\Phi,\Psi)$ be a complementary pair of Young functions such that $(L^\Phi_\om(G),*)$ is a Banach algebra. Suppose that there exists no nonzero continuous group
homomorphism $\xi:G\to \C$ such that $\txi \in L^\infty(G)$, where
\begin{align}\label{Eq:unbounded group homo vs weak amen}
\txi(s):=\frac{\xi(s)}{\om(s)\om(s^{-1})} \ \ \ \ (s\in G).
\end{align}
Then $\Sm^\Phi_\om(G)$ is weakly amenable.
\end{thm}
\begin{proof}
We first note that, by the main result of \cite[Theorem 3.1]{Z}, the nonexistence of group homomorphisms $\xi:G\to \C$ satisfying \eqref{Eq:unbounded group homo vs weak amen} will imply (in fact, it is equivalent to) the weak amenability of $L^1_\om(G)$. Now consider the space
$$\Sm_\om^{1,\Phi}(G):=L_\om^1(G)\cap \Sm_\om^\Phi(G).$$
It is clear that $\Sm_\om^{1,\Phi}(G)$ with the norm
$$\|\cdot\|_\om^{1,\Phi}:=\|\cdot\|_{1,\om}+\|\cdot\|_{\Phi,\om}$$
is a Banach space. Moreover, it follows from \cite[Lemma 3.2]{OS1} and \cite[Lemma 3.1]{OS2} that
$\Sm_\om^{1,\Phi}(G)$ becomes an abstract Segal subalgebra of $L_\om^1(G)$ in the sense of \cite[Definition 4.1.8]{D}. Hence, by \cite[Theorem 4.1.10]{D}, $\Sm_\om^{1,\Phi}(G)$ is weakly amenable. In particular, this implies that $\Sm_\om^{\Phi}(G)$ is weakly amenable by \cite[Proposition 2.8.64]{D} and the fact that $\Sm_\om^{1,\Phi}(G)$ is dense in $\Sm_\om^\Phi(G)$.
\end{proof}
The rest of this section is devoted to showing that, in many instances, the condition in Theorem \ref{T:weighted Orlicz algebra-weak amenable} is also necessary for weak amenability of weighted Orlicz algebras and their maximal essential subalgebras. After proving a technical lemma, the main results will be presented in Theorems~\ref{T:weighted Orlicz algebra-non weak amenable}~and~\ref{T:weighted Orlicz algebra-non weak amenable-compactly generated group}.
\begin{lem}\label{L:Young functions-square root}
Let $(\Phi,\Psi)$ be a complementary pair of Young functions.
Suppose that $\tPs(x):=\Psi(\sqrt{x})$ is a Young function and $\tPh$ is the complementary Young function of $\tPs$. Then\\
$(i)$ for every $x>0$,
\begin{eqnarray}\label{Eq:Young functions-square root-comparision I}
\Phi(x)\leq \tPh\left(\frac{2x^2}{\Phi(x)}\right), \ \ \ \tPh\left(\frac{x^2}{4\Phi(x)}\right)\leq \Phi(x).
\end{eqnarray}
$(ii)$ We have the following jointly continuous inclusions:
\begin{eqnarray}\label{Eq:Orlicz spaces-convolution product}
L^{\tPs}(G)*L^{\Phi}(G) \subseteq L^{\Psi}(G),
\end{eqnarray}
\begin{eqnarray}\label{Eq:Orlicz spaces-pointwise product}
L^{\tPh}(G)L^{\Psi}(G) \subseteq L^{\Phi}(G).
\end{eqnarray}
\end{lem}
\begin{proof}
Since $\Phi$ and $\Psi$ are strictly increasing, then so are their inverse functions. Hence, it is clear
that inequalities \eqref{Eq:Young functions-square root-comparision I} are equivalent to
\begin{eqnarray}\label{Eq:Young functions-square root-comparision II}
\frac{\tPh^{-1}(x)}{2}\leq \frac{\Phi^{-1}(x)^2}{x},\ \ \ \frac{\Phi^{-1}(x)^2}{x}\leq 4\tPh^{-1}(x)\quad(x>0).
\end{eqnarray}
Since $\tPs(x):=\Psi(\sqrt{x})$, we have that
\begin{eqnarray}\label{Eq:Young functions-square root-comparision III}
\tPs^{-1}(x)=\Psi^{-1}(x)^2 \ \ \ (x\geq 0).
\end{eqnarray}
Combining \eqref{Eq:Inverse Young inequality} and \eqref{Eq:Young functions-square root-comparision III}, we obtain \eqref{Eq:Young functions-square root-comparision II}
and the following inequality:
\begin{equation}\label{Eq:5}
\tPs^{-1}(x) \Phi^{-1}(x) \leq 2 x\Psi^{-1}(x) \ \ \ (x\geq 0).
\end{equation}
According to \cite[Theorem 3.3.9]{rao}, if Young functions $\Phi_1, \,\Phi_2,\,\Phi_3$ are such that
$$\Phi_1^{-1}(x) \Phi_2^{-1}(x) \leq x\Phi_3^{-1}(x) \ \ \ (x\geq 0), $$
then $L^{\Phi_1}*L^{\Phi_2}\subseteq L^{\Phi_3}$, and the embedding is jointly continuous. Because of (\ref{Eq:5}), the above condition is satisfied for $\Phi_1=\tilde{\Psi}$, $\Phi_2=\Phi$, and $\Phi_3(x)=\Psi(x/2)$. So in order to prove (\ref{Eq:Orlicz spaces-convolution product}), we just note that $\Phi_3$ is strongly equivalent to $\Psi$ in the sense of (\ref{Eq:equivalent young functions}), and hence they produce the same Orlicz space.
Finally, using \eqref{Eq:Inverse Young inequality} and (\ref{Eq:Young functions-square root-comparision III}), we can also get
$$\tPh^{-1}(x) \Psi^{-1}(x) \leq \frac{2x\Psi^{-1}(x)}{\tPs^{-1}(x)}=\frac{2x}{\Psi^{-1}(x)}\leq 2\Phi^{-1}(x)\ \ \ (x\ge0),$$
which implies the jointly continuous inclusion
$$L^{\tPh}(G)L^\Psi(G)\subseteq L^\Phi(G)$$
by \cite[Theorem 3.3.7]{rao} and the same kind of argument as above to get rid of the constant $2$.
\end{proof}
\begin{thm}\label{T:weighted Orlicz algebra-non weak amenable}
Let $G$ be a locally compact abelian group, $\om$ be a weight on~$G$, and $(\Phi,\Psi)$ be a complementary pair of Young functions. Suppose that there exists a nonzero continuous group homomorphism $\xi:G\to \C$ such that $\txi \in L^\infty(G)$, where $\txi$ is defined in \eqref{Eq:unbounded group homo vs weak amen}. If $(L^\Phi_\om(G),*)$ is a Banach algebra, then neither $L^\Phi_\om(G)$ nor $\Sm^\Phi_\om(G)$ is weakly amenable in any of the following cases:\\
$(i)$ $\tPs(x):=\Psi(\sqrt{x})$ is a Young function;\\
$(ii)$ $\tPh(x):=\Phi(\sqrt{x})$ is a Young function and $\txi\in L^{\tPs}$, where $\tPs$ is the complementary Young function of $\tPh$.
\end{thm}
\begin{proof}
Let $\xi:G\to \C$ be a nonzero continuous group homomorphism such that $\txi \in L^\infty(G)$, where $\txi$ is defined in \eqref{Eq:unbounded group homo vs weak amen}.
Let $U$ be a compact neighborhood of the identity of $G$. For every $f\in C_c(G)$, define
$$D(f)=1_U*(\check{f}\check{\xi}),$$
where, as usually, $\check{g}(x)=g(x^{-1})$ $(x\in G)$. It is easy to see that $D$ is a linear operator that ranges in $L^\Phi_{\om}(G)^*$. Moreover, a similar argument to the one presented in \cite[Theorem 3.1]{Z} shows that $D$ is a non-zero derivation on $C_c(G)$. Hence it suffices to show that $D$ can be extended to a bounded linear operator on $L_\om^\Phi(G)$.
For every $g\in L_\om^\Phi(G)$, we have
\begin{eqnarray*}
\la D(f) \,, \,g \ra &=& \int_G\int_G 1_U(ts)f(s)\xi(s)g(t) dsdt \\
&=& \int_G\int_G \frac{1_U(ts)\om(s^{-1})}{\om(t)}(f\om)(s)\txi(s)(g\om)(t) dsdt.
\end{eqnarray*}
Applying the inequality $\om(s^{-1})\leq C \om(s^{-1}t^{-1})\om(t)$ (see \eqref{Eq:defn-weight}) and the Fubini's theorem, we get
\begin{eqnarray*}
|\la D(f) \,, \,g \ra| &\leq & C\int_G\int_G 1_U(ts)\om((ts)^{-1})|f\om|(s)|\txi(s)||g\om|(t) dtds \\
&=& C\,\la 1_U\check{\om}*|\check{g}\check{\om}| \ , \, |f\om\txi| \ra.
\end{eqnarray*}
Now, if (i) holds, then by \eqref{Eq:Orlicz spaces-convolution product} of Lemma~\ref{L:Young functions-square root} we have
$$L^{\tPs}(G)* L^\Phi(G)\subseteq L^\Psi(G).$$
Since $f\om,\, \check{g}\check{\om} \in L^\Phi(G)$, $\txi \in L^\infty(G)$, and $1_U\check{\om} \in L^{\tPs}(G)$, it follows that $1_U\check{\om}*|\check{g}\check{\om}|\in L^\Psi(G)$ and $f\om\txi \in L^\Phi(G)$, so that
$$ |\la D(f) \,, \,g \ra| \leq C_1 \|1_U\|_{\tPs,\check{\om}} \|\txi\|_\infty \|g\|_{\Phi,\om}\|f\|_{\Phi,\om}.$$
Now suppose that (ii) holds. By \eqref{Eq:Orlicz spaces-pointwise product} of Lemma~\ref{L:Young functions-square root} we have that
$$L^{\tPs}(G)L^\Phi(G)\subseteq L^\Psi(G).$$
Since $f\om,\, \check{g}\check{\om} \in L^\Phi(G)$, $1_U\check{\om}\in L^1(G)$, and $\txi \in L^{\tPs}(G)$, it follows that $1_U\check{\om}*|\check{g}\check{\om}|\in L^\Phi(G)$ and $f\om\txi \in L^\Psi(G)$, so that
$$ |\la D(f) \,, \,g \ra| \leq C_2 \|1_U\|_{1,\check{\om}} \|\txi\|_{\tPs} \|g\|_{\Phi,\om}\|f\|_{\Phi,\om}.$$
Thus, in either case, $D$ has a continuous extension to a linear operator from $L^\Phi_\om(G)$ to $L^\Phi_{\om}(G)^*$, and so neither $L^\Phi_\om(G)$ nor $\Sm_\om^\Phi(G)$ is weakly amenable.
\end{proof}
\begin{thm}\label{T:weighted Orlicz algebra-non weak amenable-compactly generated group}
Let $G$ be a non-compact compactly generated abelian group, $(\Phi,\Psi)$ be a complementary pair of Young functions, and $\om$ be the weight on $G$ defined in \eqref{Eq:weight-lenght function}. Suppose that $(L^\Phi_\om(G),*)$ is a Banach algebra and $1/\om \in L^\Psi(G)$. If $\tPh(x):=\Phi(\sqrt{x})$ is a Young function, then neither $L^\Phi_\om(G)$ nor $\Sm^\Phi_\om(G)$ is weakly amenable.
\end{thm}
\begin{proof}
Let $\xi:G\to \C$ be any nonzero continuous group homomorphism (its existence follows, for example, from the structural
description of $G$) and let $\txi$ be the function defined in \eqref{Eq:unbounded group homo vs weak amen}. By Theorem \ref{T:weighted Orlicz algebra-non weak amenable}, it suffices to prove that $\txi\in L^{\tPs}(G)$.
Our assumption that $1/\om \in L^\Psi(G)$ means that there exists $\alpha>0$ such that
\begin{align}\label{Eq:1}
\int_G \Psi\left(\frac{\alpha}{\om(s)}\right)ds < \infty.
\end{align}
Now put
\begin{align}\label{Eq:2}
f(s)=\frac{\alpha}{\om(s)}:=\frac{\alpha}{e^{\nu(|s|)}} \ \ \ (s\in G).
\end{align}
We first prove that
\begin{align}\label{Eq:3}
\xi (\Psi\circ f) \in L^\infty(G).
\end{align}
Since $\xi$ is a group homomorphism, for every $n\in \N$ and $s\in U^n \setminus U^{n-1}$, we have
\begin{align}\label{Eq:4}
\xi(s) \Psi(f(s)) \leq Cn a_n,
\end{align}
where
$$C=\sup \{\xi(s): s\in U \} \ \ \text{and} \ \ a_n=\Psi\Biggl(\frac{\alpha}{e^{\nu(n)}}\Biggr).$$
On the other hand, $\nu$ and $\Psi$ are both increasing so that $\{a_n\}_{n\in \N}$ is a decreasing sequence of positive numbers. Therefore, if the Haar measure of $G$ is denoted by $\lambda$ and we assume that $\lambda(U)=1$, then
\begin{eqnarray*}
C^{-1}\xi(s) \Psi(f(s)) &\leq& n a_n \ \ \ (\text{by}\ \eqref{Eq:4}) \\
&\leq& a_1+\cdots+ a_n \\
&\leq & \sum_{n=1}^\infty \displaystyle \lambda(U^n \setminus U^{n-1})a_n \\
&=& \sum_{n=1}^\infty \int_{U^n\setminus U^{n-1}} \Psi(f(t)) dt \\
&\leq& \int_{G} \Psi(f(t)) dt < \infty \ \ \ (\text{by}\ \eqref{Eq:1}\ \text{and}\ \eqref{Eq:2}).
\end{eqnarray*}
Hence, \eqref{Eq:3} is verified. Next, since $\tPh(x):=\Phi(\sqrt{x})$ is a Young function with the complementary function $\tPs$, it follows from \eqref{Eq:Young functions-square root-comparision I} of Lemma \ref{L:Young functions-square root}, \eqref{Eq:1}, and \eqref{Eq:2} that
$$\int_G \tPs\left(\frac{f(t)^2}{4\Psi(f(t))}\right) dt \leq \int_G \Psi(f(t)) dt<\infty.$$
Hence, by definition \eqref{Eq:Orlicz defn}, $\frac{f^2}{\Psi(f)}\in L^{\tPs}(G)$. This, together with \eqref{Eq:3}, implies that
$$\xi f^2=\xi \Psi(f)\frac{f^2}{\Psi(f)}\in L^\infty(G)L^{\tPs}(G)\subseteq L^{\tPs}(G).$$
Finally, $\txi\in L^{\tPs}(G)$ since $$\txi=\frac1{\alpha^2} \xi f^2.$$
\end{proof}
We can summarize the preceding results and also provide an equivalent and easy-to-check criterion on Young functions so that Theorems \ref{T:weighted Orlicz algebra-non weak amenable} and \ref{T:weighted Orlicz algebra-non weak amenable-compactly generated group} may be applied.
\begin{cor}\label{C:weighted lp algebra-(non) weak amenable1}
Let $G$ be a non-compact, compactly generated abelian group, $\om$ be a weight on $G$, and $(\Phi,\Psi)$ be a complementary pair of Young functions such that $\Psi'$ exists on $\R^+$ and $\Psi'(x)/x$ is increasing on $\R^+$.\\
$(i)$ If $(L^\Phi_\om(G),*)$ is a Banach algebra, then $\Sm^\Phi_\om(G)$ is weakly amenable if and only if there exists no nonzero continuous group
homomorphism $\xi:G\to \C$ such that $\txi \in L^\infty(G)$, where
\begin{equation*}\label{Eq:group homo vs weak amen}
\txi(s):=\frac{\xi(s)}{\om(s)\om(s^{-1})} \ \ \ \ (s\in G).
\end{equation*}
$(ii)$ If $(L^\Psi_\om(G),*)$ is a Banach algebra, $1/\om \in L^\Phi(G)$, and $\om$ is of the form \eqref{Eq:weight-lenght function}, then neither $L^\Psi_\om(G)$ nor $\Sm^\Psi_\om(G)$ is weakly amenable.
\end{cor}
\begin{proof}
Note that if $\tilde{\Psi}(x)=\Psi(\sqrt x)$, then $\tilde{\Psi}'(x)=\frac{\Psi'(\sqrt x)}{2\sqrt x}$ is increasing because we assumed that ${\Psi'(x)}/{x}$ is increasing. Therefore, $\tilde{\Psi}$ is convex and hence it is a Young function. We then immediately obtain (ii) from Theorem~\ref{T:weighted Orlicz algebra-non weak amenable-compactly generated group} and one direction of (i)~--- from Theorem~\ref{T:weighted Orlicz algebra-non weak amenable}(i). The other direction of (i) follows from Theorem~\ref{T:weighted Orlicz algebra-weak amenable}.
\end{proof}
\begin{rem}
It is straightforward to verify that Corollary~\ref{C:weighted lp algebra-(non) weak amenable1} can be applied to various Young functions to determine the weak amenability of their weighted Orlicz algebras. Below, we give few families of examples of such Young functions ($p\geq 1$ is arbitrary):
(1) $\Psi(x)=[x^2\ln (1+x)]^p$.
(2) $\Psi(x)=(e^x-x-1)^p$.
(3) $\Psi(x)=(e^{x^p}-1)$
(4) $\Psi(x)=(\cosh x-1)^p$.
\end{rem}
We finish this section by highlighting, in the following corollary and the example afterward, that when weighted Orlicz spaces are just weighted $L^p$ spaces, we have fairly general results on their weak amenability.
\begin{cor}\label{C:weighted lp algebra-(non) weak amenable}
Let $G$ be a non-compact compactly generated abelian group, $\om$ be a weight on $G$, and $1<p,\,q<\infty$ be such that $1/p+1/q=1$. Suppose that $(L^p_\om(G),*)$ is a Banach algebra. \\
$(i)$ For $1<p\le 2$, $L^p_\om(G)$ is weakly amenable if and only if there exists no nonzero continuous group
homomorphism $\xi:G\to \C$ such that $\txi \in L^\infty(G)$, where
\begin{equation*}\label{Eq:group homo vs weak amen}
\txi(s):=\frac{\xi(s)}{\om(s)\om(s^{-1})} \ \ \ \ (s\in G).
\end{equation*}
$(ii)$ For $p>2$, if $\om$ is of the form \eqref{Eq:weight-lenght function} and $1/\omega\in L^q(G)$, then
$L^p_\om(G)$ is not weakly amenable.
\end{cor}
\begin{proof} As was mentioned in the Preliminaries, $L^p_\om(G)=L^{\Phi}_{\omega}(G)$ for $\Phi(x)={x^p}/{p}$ and the corresponding complementary Young function is $\Psi(x)=x^q/q$. Since $\Phi\in\Delta_2$, we have that $L^p_\om(G)=L^{\Phi}_{\omega}(G)=\Sm^{\Phi}_{\omega}(G)$. For $1<p\le2$ we have $q\ge2$ implying that $\Psi'(x)/x=x^{q-2}$ is increasing, and so (i) follows directly from Corollary~\ref{C:weighted lp algebra-(non) weak amenable1}(i).
Likewise, to obtain (ii), we apply Corollary~\ref{C:weighted lp algebra-(non) weak amenable1}(ii) to $\Phi(x)={x^q}/{q}$ and $\Psi(x)=x^p/p$.
\end{proof}
\begin{exm}\label{E:twisted lp alg on integers-poly and exp weight-Operator alg}
Let $\Z^d$ be the group of $d$-dimensional integers.
The standard choice of generating set for $\Z^d$ is
$$F=\{(x_1,\ldots, x_d) \mid x_i\in \{-1,0,1\} \},$$
It is straightforward to verify that
\begin{align}\label{Eq:growth generating set of integers}
|F^n|=(2n+1)^d \ \ \ (n=0,1,2,\ldots).
\end{align}
Now suppose $1< p,q<\infty$ with $\frac{1}{p}+\frac{1}{q}=1$. Let $\om_\beta$ be the polynomial weight on $\Z^d$ defined in \eqref{Eq:poly weight-defn}. Then
\begin{align}\label{Eq:poly weight-d dim integers}
\om_\beta(\mathbf{x})=(1+\max\{|x_i|:i=1,\ldots,d\})^\beta \ \ \ (\mathbf{x}=(x_1,\ldots,x_d)\in \Z^d).
\end{align}
Hence, by Theorem \ref{T:twisted Orlicz alg-poly and exp weight-Poly growth} and \cite[Theorem 3.2]{K3}, $l^p_{\om_\beta}(\Z^d)$ is a Banach algebra if and only if $1/\om_\beta \in l^q(\Z^d)$ and, by \eqref{Eq:growth generating set of integers}, the latter happens exactly when $\beta >d/q$.
Therefore, by Corollary \ref{C:weighted lp algebra-(non) weak amenable}, $l^p_{\om_\beta}(\Z^d)$ is weakly amenable if $1<p<2$ and
$d/q <\beta <1/2$
and $l^p_{\om_\beta}(\Z^d)$ is not weakly amenable if $p> 2$ and $\beta >d/q$.
On the other hand, if we let $\om$ to be either of the subexponential weights
defined in \eqref{Eq:Expo weight-defn} or \eqref{Eq:Expo weight II-defn}, then $\txi\in l^\infty(\Z^d)$ (defined in \eqref{Eq:group homo vs weak amen}) for any nonzero group homomorphism $\xi: \Z^d \to \C$ . Thus, again by applying Corollary \ref{C:weighted lp algebra-(non) weak amenable}, it follows that $l^p_\om(\Z^d)$ is not weakly amenable. One can also obtain similar statements for $L^p_\om(\R^d)$.
\end{exm}
\section{Acknowledgements}
This work was initiated while the second named author visited Istanbul University under a fellowship from Tubitak for visiting scientists and scientists on sabbatical leave. The first named author would like to acknowledge the support of Tubitak and express his deep gratitude toward his host, Serap
\"{O}ztop. Also, the third named author would like to acknowledge the support from PIMS Institute as she is currently holding a PIMS postdoctoral Fellowship. The authors would like to thank the referee for carefully reading the paper and providing comments that have improved the exposition of the paper. This includes, in particular, finding a minor error in the earlier version of Corollary \ref{C:weighted lp algebra-(non) weak amenable}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,803 |
* * *
Proust as Philosopher
* * *
Marcel Proust's _In Search of Lost Time_ has long fascinated philosophers for its complex accounts of time, personal identity and narrative, amongst many other themes. _Proust as Philosopher: The Art of Metaphor_ is the first book to try and connect Proust's implicit ontology of experience with the question of style, and of metaphor in particular.
Miguel de Beistegui begins with an observation: throughout _In Search of Lost Time_ , the two main characters seem prone to chronic dissatisfaction in matters of love, friendship and even art. Reality always falls short of expectation. At the same time, the narrator experiences unexpected bouts of intense elation, the cause and meaning of which remain elusive. Beistegui argues we should understand these experiences as acts of artistic creation, and that this is why Proust himself wrote that true life is the life of art.
He goes on to explore the nature of these joyful and pleasurable experiences and the transformation required of art, and particularly literature, if it is to incorporate them. He concludes that Proust revolutionises the idea of metaphor, extending beyond the confines of language to understand the nature of lived, bodily experience.
**Miguel de Beistegui** is Professor of Philosophy at the University of Warwick, UK. His books include _Aesthetics After Metaphysics: From Mimesis to Metaphor_ (Routledge, 2012), _Immanence and Philosophy: Deleuze_ (2010), _Truth and Genesis: Philosophy as Differential Ontology_ (2005) and _Heidegger and the Political_ (Routledge, 1998). He is editor (with Simon Sparks) of _Philosophy and Tragedy_ (Routledge, 2000).
* * *
Proust as Philosopher
* * *
The Art of Metaphor
Miguel de Beistegui
Translated by Dorothée Bonnigal Katz, with Simon Sparks and Miguel de Beistegui
LONDON AND NEW YORK
Originally published 2007 _as Jouissance de Proust pour une esthétique
de la métaphore_
This translation first published 2013 by Routledge
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Simultaneously published in the USA and Canada by Routledge
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Copyright © LES BELLES LETTRES — collection "Encre Marine"
Translation © Routledge, 2013
All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers.
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_British Library Cataloguing in Publication Data_
A catalogue record for this book is available from the British Library
_Library of Congress Cataloging in Publication Data_
Beistegui, Miguel de, 1966-
[Jouissance de Proust. English]
Proust as philosopher : the art of metaphor / by Miguel de Beistegui;
translated by Dorothée Bonnigal Katz, with Simon Sparks and Miguel de
Beistegui.
p. cm.
Includes bibliographical references and index.
1. Proust, Marcel, 1871–1922—Criticism and interpretation. 2. Proust, Marcel,
1871–1922. À la recherche du temps perdu. 3. Metaphor in literature. I. Title.
PQ2631.R63Z525713 2012
843′.912-dc23
2012007041
ISBN: 978-0-415-58431-9 (hbk)
ISBN: 978-0-415-58432-6 (pbk)
Typeset in Sabon
by Taylor & Francis Books
* * *
Contents
* * *
1. Looking for joy
2. Proust among the psychologists
3. Finding joy (involuntary memory)
4. Giving joy (metaphor)
5. Dress or patchwork?
_Notes_
_Bibliography_
_Index_
Chapter 1
* * *
Looking for joy
* * *
What sort of existence would raise our hopes and awaken our desires simply in order to dash them? What sort of world would clip imagination's wings? Answer: this one.
Whenever we think we're sowing the seeds of happiness, life's busy planting those of disillusionment. Whenever we think we're working carefully towards contentment we are, in fact, hurtling towards our doom. A single life can hold more disillusion and disappointment than we can imagine: every second of happiness will eventually fade, every fleeting joy will quickly be replaced by ever increasing sorrow; every desire fulfilled will end up either boring us or making us insatiable, in thrall to the ever more urgent exercise of the will.
And? Is _this_ the lesson of Proust's vast novel? Is this _really_ what _In Search of Lost Time_ establishes as a universal law, setting itself up in the process as _the_ novel of suffering and melancholia? Well, from one perspective at least, it _does_ unfold as a novel of hopelessness and despair. Seen in this way, reality — or our idea of it, at least — never fails to disappoint, never lives up to its promise or to the expectations to which we feel it has entitled us. But maybe this is all just Proust's fault. His privileging of certain types of experience might account for this rather pessimistic assessment. Even experiences like the experience of art, from which Marcel expects so much, don't escape the rule: in his joy at the prospect of seeing La Berma in _Phèdre_ he expects his first night out at the theatre to reveal "truths which dwelt in a world more real than my own." Having such absurdly high expectations from art and from life in general is only ever going to open us up to the threat of disappointment. The book's moral, itself a disappointment — how could it be anything else? — would be that, by expecting less from life, by settling for a little bit less, we might avoid being disappointed; by letting go of our desires and breaking free of our will, we might be spared innumerable sufferings.
Nothing, though, could be further from the truth, nothing more at odds with the spirit of Proust than this sort of pseudo-Stoic or -Schopenhauerian lesson. Why? Because it's precisely this sort of suffering that hones our senses and sharpens our intelligence. Life itself sets us off on a quest for what such suffering hides, making that same discomfort instrumental in discovering its buried truths. And this is why it doesn't make much sense to say that, through the narrator, the novel simply presents us with a type, one psychological profile among many; instead, it tries to unveil a truth that lies dormant within the type in question. Ultimately, Proust's point is to show that the dissatisfaction — whether in the form of suffering or in the form of boredom — that defines our relation to the world actually stems from an even deeper lack, one inscribed at the heart of reality itself. Which means what, exactly? That reality itself and alone is responsible for our misfortune? That the conditions of our disappointment are structural and not circumstantial, that they are inherent to our relation to the world rather than to any given "character trait"? Well, yes, providing that we're clear on what "reality" means, and its meaning is, as I'll try to show, a paradoxical one. Implicitly universal and unshakeable, it leads directly to a feeling of separation and alienation, of an irreversible lack. Overcoming this feeling will involve seeing reality differently and, in truth, creating it.
My point of departure, then, consists in identifying an ontological deficiency, a deficiency with respect to being, which I'd define in the following way: at the heart of our relation to the world there's a lack. This lack isn't nothing, however, but is, rather, a lack or want of being, a lack or want that functions as the sign of a truth that lies beyond or, more accurately, at the heart of present reality. This lack is original and structural and so isn't something that could be remedied by a strategy of compensation, by recapturing or reproducing the "thing" that's lacking. It's precisely by lacking that what's lacking "functions" and "structures." And it's precisely this lack or this deficiency that we experience, precisely this lack or this deficiency that we cannot help but feel. Indeed, I'd go so far as to say that it actually defines the very meaning of experience, that is, the meaning of the sensible. At the same time, though, it signals, if only implicitly, what lies beyond or on the other side of this experience, its hidden face, as it were, from which Proust manages to draw the meaning of literature and art in general. Literature, then — and this is, after all, the novel's real subject — doesn't take us out of the real world, thereby leaving life behind; rather, it transfigures life, reversing it, not into its opposite but into its other or its flip side. Literature is the flip side of the side that coincides with reality, the wrong side or the inside of the real and the sign of another meaning of experience. Far from fleeing the real, then, literature actually tracks it and weaves it, spinning and following its thread. The threads that make up its text or its fabric (the Latin _textus_ refers to something woven, something entwined) are the threads of the real itself, and its mission is to trace and disentangle them. In the process, literature lets itself be carried off to where the real flees its own self-presence. Ultimately, the real just _is_ that very self-absence. And if it always disappoints, it's not because we always expect too much of it but because we expect it where it actually isn't, because it's never where we expect it to be, because it can only be grasped in its own drift or constitutive gap. We always want it to be in its rightful place but that place is precisely where it's not, precisely where it's lacking. We would like it to be here, in front of us, in the flesh. But it's in that very immediacy or fullness that it steals away and goes missing. Which doesn't mean that it has in some way disappeared; rather, this absence or this lack is the key to its mystery, the secret of its functioning. This is why we should follow it, why we should surrender to its drift and get caught up in that gap, that shift in being which is also a shift in meaning: in fact the drift of meaning and being itself, being and meaning _as_ drift. As for literature, it's a matter of understanding how it arises from this release, from this letting go; it traces and spins the real in its drift. It always seeks to find the real where it isn't, since this is the only place where it's likely to find it. This self-escaping presence constitutes its only reality. The rest is illusion. In other words, literature doesn't believe in the solidity of being, in raw being, in short in what is commonly referred to as reality or life, and which so many forms of literature claim as their subject-matter. Its "faith" isn't that of simple perception. Instead, it takes being to be that which, from the outset, is carried away and caught in a system of reference devoid of any actual origin or end. And it's from this fundamental structure that it draws its own poetic law, through it that it elevates style beyond mere technique, elevating it to the status of "vision." It's through the thread of metaphor — the only one that isn't illusory — that it relates to the real. As such, the metaphor that it weaves isn't the product of _fancy_ , as Coleridge has it, or the creation of "the part of the human being which dominates, this mistress of error and falsehood" that so unsettled Pascal. Rather, it's the figure of the real in its self-transposition or transfiguration. Metaphor believes in transubstantiation, in the conversion of matter into spirit, which it carries out, but only as an implicit dimension of matter itself, inscribed within it from the start.
***
None of which, of course, necessarily prevents reality from being a source of disappointment. And, were it not for one or two clues and for the way in which these clues come together in the final revelation of a very different state of affairs, it would be easy to see in Proust's work a true abhorrence of time, a true source of bitterness and resentment. His world would be one teetering on the verge of nihilism. Isn't Swann's whole life — just like his study of Vermeer that remains constantly imminent, constantly postponed — undermined by a point — lessness that nothing's going to be able to refute? Doesn't Jean Santeuil fail to find any pleasure in writing down the memory of his walks at Beg-Meil? And doesn't Marcel find the trees seen from the train "tedious to observe and to describe"? The psychological law that lies at the heart of Proust's entire aesthetics dictates that it's only long after Jean's disappointment at the seaside show at Beg-Meil that the sea will seem beautiful to him and only in retrospect that the row of trees will become dear to Marcel. It's as if we can't appreciate a sistuation, love someone, in short, enjoy reality without telling it to someone else. Infinitely free and vast, imagination has always anticipated the real and begun to adorn it, to produce a specific image of it, one full of details, reflections and contrasts that reality reduces to nothing. The disappointment that the narrator can't help but experience when confronted with the real would be entirely of a piece with the conflict between imagination and perception. Whether imagination is anticipating the real or, in its presence, carrying it elsewhere and transforming it, thereby giving it a meaning and a purpose, it is always a prosthesis or supplement to the real. It could be, then, that the real isn't self-sufficient and can't, in fact, proceed on its own. It's always waiting on something else, always _truly_ elsewhere. Odette de Crécy is neither truly beautiful nor particularly moving, in Swann's eyes at least. And yet, once she starts to remind him of Botticelli's Zipporah, his pleasure in seeing her is justified and her beauty established. Once the resemblance is established, Odette takes on the capacity to evoke Botticelli's pictorial universe, the Renaissance ideal of beauty itself, setting her within "a world of dreams to which she had not had access until then and where she was steeped in nobility." It is as if, remorseful at having "limited his life to worldly relationships, to conversation" and ashamed of the frivolity of his own existence, Swann's able to elevate it, to grant it some value by imagining his world as the reflection of a great artist's. Anyone who fails to see Odette in such a light or fails to see her through that other, magical and distorting lens, a lens ground by imagination, isn't going to find her all that attractive. In other people's eyes, she's common, just like Rachel after her. In fact, though, she's no more common than anyone else. It's reality itself, reality as a whole, that's mundane. It only becomes something interesting when we contribute to it; it only acquires value inasmuch as it can shore up our imagination. From which it follows that pretty women are the province of men with no imagination. The paradox of Proust's real is that it needs this supplement, this prosthesis, in order to be something. When Albertine wears a plain Fortuny dress, Marcel doesn't see the woman he loves; all he sees is its tempting evocation of Venice, a Venice desired and dreamt of, a Venice he's dying to visit but to which his jealous love for Albertine is preventing him from going. Once Albertine's gone, Marcel can finally go to Venice. And? Will he be happy there? Not really. In front of Carpaccio's paintings, which Elstir described so delightfully, he lapses into a state of deep melancholy: reminding him of the dress he once gave Albertine as a present, the paintings simply reawaken his love for the one who no longer is.
Just as disappointing is the initial impression given by the Duchesse de Guermantes when compared to the imagery built up by the young Marcel over the years, imagery that's based on the tapestry hanging on the wall of the church in Combray, on the stained-glass window featuring Gilbert the Bad, and, finally, on the profile of Geneviève de Brabant on her magic lantern — two ancestors of the Guermantes. Should we conclude, then, that art, which feeds the imagination and makes us despair of reality, can alleviate our disappointment since it's not a mediation but the very object of our experience? No. The real Berma is but the pale reflection of the imagined one, just as Bergotte's red bottle-nose can, in Marcel's eyes, eradicate "those things of beauty, his wonderful works" that he had worshipped for so long.
And how much more disappointing are the women we love and this dull feeling we call love This is where the work of the imagination is the greatest. The feeling of love has nothing to do with the emotion we can derive from the beauty, the intelligence or the kindness of the one we love. Of his mistresses, the narrator says this:
> When I saw them, when I listened to them, I found nothing in them that could resemble my love or be able to explain it. Yet my one joy was to see them, my one anxiety to wait for them. It was as if a virtue having no connection with them had been adjoined to them incidentally by nature, and that this virtue, this electricity-like power, had the effect on me of exciting my love, that is to say of directing all my actions and causing all my suffereings. But from this, the beauty, or the intelligence, or the goodness of these women were wholly distinct.
A lack of intelligence or elegance can constitute just so many obstacles to love, as evidenced by Swann's love for Odette, who isn't even his type. But these obstacles aren't insurmountable. Spotting a resemblance between a lover and the representation of a biblical character in a great Master's painting, and letting the imagination bring these two worlds together, can easily do away with the flaws (and even the qualities) of the woman in question. The object of desire then no longer has any actual reference to desire itself, which is fuelled by its own appetite and constructs its object on the basis of fragments. If anything, the object might occasion this desire, but it's certainly not its efficient cause. What we love in a woman is that kind of life, unknown to us, to which love would grant us access; it is the promise of a new world, or a lost world perhaps, which she encapsulates and might hand over to us as one hands over a treasure. In fact, the narrator himself only starts loving Albertine when he "suddenly" sees in "the real Albertine, the one [he] saw every day, [and] who [he] thought was hidebound in bourgeois prejudices," the embodiment of the imaginary Albertine, far more attractive than the real one, namely the Albertine "who, at a time when [he] did not even know her, [he] had thought was taking furtive looks at [him] on the esplanade, the one who, when she saw [him] walking off, had seemed to be wending so reluctantly her own way home." Meeting the gaze of an unknown woman is enough to make him fall in love with her, since those eyes contain everything he could ever know of a thought, a wish, a memory. "The hope of taking possession of all that," Proust writes, "is what gives her eyes their value, much more than any mere material beauty." And if Marcel indiscriminately loves all the young girls in flower, it's because the first time he caught a glimpse of them on the beach at Balbec, they were a "little gang" and, for him, such multiplicity has the power to evoke "the towering blue waves or the shapes of a parade passing in front of the sea." What he wants to recapture is the place where they are and he's not, the world that is theirs at this particular moment and not his, recapture by finding them again. Making out Albertine's gaze as she is surrounded by her friends, Marcel immediately wants to possess everything that shines in it and that's unknown to him: the things, people and places she calls on, the thoughts she harbours towards them, "her desires, her likes and dislikes, the power of her inscrutable and inexhaustible will." In other words, Marcel falls in love with Albertine as a whole, with her ability to inhabit a world that's different from his. But if that's true, we might as well fall in love with a checkout girl as a duchess, so long as we live on the Faubourg Saint-Germain and hang out with the right sort of crowd. A homely checkout girl can awaken the memory of a long gone — and so deeply prized — point in time or a world that's unknown and alien, pricking the imagination and letting it run free. And so it's not all that surprising that Swann's ready to sacrifice everything to Miss Sacripan, that Saint-Loup's willing to squander his entire fortune on another _cocotte_ , Rachel, just like Marcel when he's struggling to keep Albertine from leaving him.
This life, however, isn't one we'll ever be able to know completely; the world of the loved one will never truly be _our_ world. The reason why "we find desiring innocent, and hideous that the other should desire" actually stems from the fact that what we desire in the other is ultimately the other's own capacity for desire. To love is to desire the other's desire; it's to turn the other's desire into our own, to want to enter the world that the other holds within them and to possess thereby that which cannot be possessed. Right from the start Marcel says this about Albertine:
> My desire for her was desire for her whole life: a desire that was full of pain, because I sensed it was unattainable, but also full of heady excitement, because what had been my life up to that moment had suddenly ceased to be all of life, had turned into a small corner of a great space opening up for me, which I longed to explore and which was composed of the lives led by these young girls, because what was laid out now before my eyes was that extension and potential multiplication of self which we know as happiness.
Love might well be the promise of happiness that Stendhal claimed for beauty, but it's a promise we're not going to be able to keep. Why? Because love only lasts as long as the other remains _terra incognita_ , uncharted territory. As long as the other still seems to be hiding something from me, I love him or her out of a possessive kind of love that torments me. If love's essentially jealous, it's because it aims to seize the other as a whole, to know that part of the other that was shown to someone else, to experience the other's experiences as my own and make the other's world my own. Precisely because the other is someone who's different from me, his or her perspective on and knowledge of the world differ from mine and neither will ever be mine, regardless of how intelligent I might be and how stupid he or she might be in turn:
> How many people, towns, pathways jealousy makes us desperate to know! It is a thirst for knowledge thanks to which we come to have, on a series of isolated points, all possible information except the information we really want.
The mere fact that the loved one is an other implies that I'll never know the world as he or she does. And yet where else did my love originate if not in the promise of that new world? A part of that world, of that life will always escape me; the object of my desire will always have the power to steal away from me, thereby renewing and spurring my own desire. My desire, as the desire of the other, is fuelled by its structural dissatisfaction, by its inability to reach the presence of its object:
> Having a liking for someone is one thing; but to be afflicted with the sadness, the feeling of something irreparable having happened, the anguish which all accompany the onset of love, what is necessary is the risk [... ] of an impossibility.
Jealousy is the passion — the illness or madness — for that impossible possession, an impossibility one nonetheless cannot settle for. "Possessing" a woman implies wanting to take hold of what is ungraspable within her. And with this we've just doomed ourselves to the worst possible torment. Love only endures as long as the quest remains unfinished, as long as the thirst remains unquenched. But the fulfilment of the promise that constituted love actually brings love to an end: as soon as the loved one is no longer unknown to me, as soon as I am no longer jealous, I stop loving. When it's not jealous, love is indifferent.
Of Swann, Proust says that
> When Odette ceased to be for him a creature always absent, longed for, imaginary, [... ] when normal relationships were established between what would put an end to his madness and his gloom, then no doubt the actions of Odette's daily life would appear to him of little interest in themselves [... ] [H]e told himself that when he had recovered his health what Odette might be doing would leave indifferent.
Similarly, Albertine was as wonderful and desirable as she was unattainable. When he first kisses her, however, when she finally gives in, she's already begun to lose some of her mystery:
> I should really have liked, before kissing her, to have been able to re-create in her the mystery she had held for me on the beach before I knew her, to discover in her the place where she had lived before that; in default of such mystery, I could at least insinuate all the memories of the time we had spent in Balbec, the sound of the waves breaking beneath my window, the shouts of the children.
Don't think for a moment that kissing and touching lips that until now have been only longed for could reawaken the mystery. There's no such thing as "knowledge through the lips." The kiss, crushing our ability to see and to feel against folds of skin and flesh, only underlines the cruel absence of a true organ of knowledge. The instant Albertine is held and captured, she becomes boring. The narrator remembers that
> [i]t was because I had seen her as a mysterious bird, then as a great actress of the beach scene, desired and, who knows, enjoyed, that she had seemed wonderful to me. Once a captive in my house, the marvellous bird [... ] had lost all its colours [... ]. Little by little she had lost her beauty.
Once imagination's jealousy doesn't need to rear its ugly head, love itself dies and is replaced by boredom, that kind of expectation that's devoid of any object: "I could feel that my life with Albertine was nothing but, on the one hand, when I was not jealous, boredom, and on the other, when I was, suffering." Love, then, if not life as a whole, as Schopenhauer thought, would alternate between suffering and boredom, between the ultimate and paroxystic expression of the imagination (thanks to which one may be concerned with the world and with beings in particular) and indifference or longing where the world and beings fall away from us and that we call happiness only insofar as they put an end to our pain. If life disturbs me, I'm unhappy; if I'm happy, it's only because I'm indifferent to life. Continually lauded, love would ultimately be the most tenacious illusion of all, the most overrated feeling and the most disappointing experience. As soon as they're possessed, women, elevated and seemingly magnified by the power of our imagination and the strength of our desire, end up crushed by an invisible force that leads them to trip and fall "on the flat earth of vulgar reality."
What we have here, though, is an indication of what insists, what stands behind love and, more generally, behind the work of the imagination, namely our desire as a desire to know. The other is not _what_ I love, but only _who_ I love: there _is_ something I desire through the other but it's not the other. In its pathology, in its sometimes delusional behaviour, love needs to be distinguished from what the pathology reveals. And what it actually reveals is our deepest tendency, our _conatus_ , as the tendency to know all that there is to know about every being and every kind of being — a kind of knowledge that's never simply or immediately given, however, but that we have to wrestle from the world. The real significance of love, perhaps the most extreme and futile expression of the imagination, is epistemological: what drives us to imagine is being unable to know. Knowing — knowing _everything_ — is what we want. The ultimate horizon of our relation to beings and to others, to the natural world as to the cultural one, is the Absolute. If "the most exclusive love for any person is always love for something else," then it's in the sense that no particular instance of love will ever fulfil the desire for an absolute. The reality of love is exhausted in this demand for an absolute which it fails to meet. Its actualization lies in its transcendence. Equally, for the one who can understand it, love is indicative of an infinite desire and one step towards the contemplation of this "divinity (or Idea)" that the one we love (and, as a result, because of whom we suffer) is "only a fragmentary reflection at the lowest level." And, instead of — but also because of — the pain we once felt, this whole process of contemplation can even make us happy. This is precisely how Swann ends up at his mistress's house in order to spy on her. Desperate to know whether she's in another man's arms, he decides to knock at the shutters of the flat which he believes to be hers:
> And perhaps, what he was feeling at this moment, which was almost pleasant, was also something different from the assuaging of a doubt and a distress: it was _a pleasure in knowledge_. If, ever since he had fallen in love, things had regained for him a little of the delightful interest they had once had for him, but only insofar as they were illuminated by the memory of Odette, now it was another of the faculties of his studious youth that his jealousy revived, _a passion for truth_ , but for a truth that was likewise interposed between him and his mistress, taking its light only from her, a completely individual truth whose sole object, of an infinite value and almost disinterested in its beauty, was Odette's actions, her relationships, her plans, her past [... ] the curiosity he now felt awakening in him concerning the smallest occupations of this woman, was the same curiosity he had once had about History. And all these things that would have shamed him up to now, such as spying, tonight, outside a window, tomorrow, perhaps, for all he knew, cleverly inducing neutral people to speak, bribing servants, listening at doors, now seemed to him to be fully as much as were the deciphering of texts, the weighing of evidence and the interpretation of old monuments, merely methods of scientific investigation with a real intellectual value and appropriate to _a search for the truth_.
Loving passionately would therefore involve loving more than the one being loved; it would involve loving what demands to be loved within them but what they are not, namely truth. The only possible love, at least in the sense of a love that doesn't just lead to suffering and disappointment, would be the love of truth. This is a pretty philosophical view of love — a Platonic one, to be exact — in which _Eros_ , once properly understood and applied, leads to _aletheia_ and _sophia_ , and passion is a pathway to knowledge. But the opposite is equally true: philosophy (or hermeneutics) is itself an erotica and (jealous) love a propaedeutic. Imagination, then, would actually be a baton to be handed on, a back-up solution, a mode of knowledge — an inadequate one, according to Spinoza — that would reveal our thirst for absolutes, a tendency seeking to _go beyond itself_. But in what direction? Is it geared towards a lessening or a transformation? Trying to live better, should we stoically desire less — or perhaps just desire what depends only on us — or should we just work to understand ourselves better until our knowledge becomes in some way absolute? "The whole art of living", as Proust claims at the end of the book, "is to use the people who make us suffer simply as steps enabling us to obtain access to their divine form and thus joyfully to people our lives with divinities." This life, joyful, divine, is, at its peak, the life of art. And it's this hymnic dimension of art (against which, no doubt, there needs to be set a melancholic or elegiac view) that the book sets out to find.
If Proust's attitude towards love seems somewhat abrasive, his take on friendship is harsher still. It's hard to imagine anything more cruel than his exposure of friendship as a mirage and his denunciation of its pointlessness. While friendship can, like love, lead to suffering — we can feel betrayed by a friend, for example — it's not in the same way or with the same intensity insofar as jealousy plays a lesser role. Jealousy can at least broaden our scope and quench our thirst for knowledge, even through the very suffering it entails. Without ever bringing us to the intensity of love or without provoking the excesses of our imagination, friendship gives the narrator a feeling that's "halfway between fatigue [and not suffering, from which there is so much to learn] and boredom." It stems from the fact that friendship seeks some sense of self-confirmation in the other, as opposed to the discovery of true difference; it privileges superficial exchanges over in-depth exploration. If friendship, as it is ordinarily conceived, implies a sharing and an enjoyment of what we have in common, it does not teach us anything more than what we could have known through self-observation. It is therefore devoid of value. If friendship can enable us to enter another world, access an unknown life, it is certainly very valuable but it can also have nothing to do with the deep reasons underlying the feelings of friendship that the other harbours towards us and believes to be sharing with us. In this case, friendship is predicated on a misunderstanding and we are "inescapably alone" anyway; friendship is nothing but the lie that consists in making us believe that there are two of us. There will therefore always be something better to do than talking with a friend, however delightful this might be, and the pleasure that can be drawn from such conversation — which is actually a superficial kind of pleasure (insofar as it furthers nothing) and a selfish one (insofar as it comforts us in the idea of our own self-worth) — is nothing compared to the pleasure derived from solitary work:
> Even conversation, which is friendship's mode of expression, is a superficial digression, through which we can make no acquisition. We may converse our whole life away, without speaking anything other than the interminable repetitions that fill the vacant minute; but the steps of thought which we take during the lonely work of artistic creation all lead us downwards, deeper into ourselves, the only direction which is not closed to us, the only direction in which we advance, albeit with much greater travail, towards an outcome of truth.
Unlike jealous love, then, which is likely to trigger or spur our desire for knowledge and our thirst for truth, friendship indulges us with the illusion of a communication between souls. We believe that something is shared when in fact we're merely wasting our time. We believe that we are connecting with someone, that a real feeling is experienced, when we're mired in misunderstanding. There's no real difference between friendship and worldliness or any other form of diversion or entertainment.
Such a harsh if not altogether fierce diagnosis only makes sense once the true and genuine aim of life has been identified — the aim of the specific life that the narrator intuits throughout the book and which is finally unveiled in the last pages. Such a radical and sovereign condemnation of friendship can only be voiced, then, in the context of a revelation concerning the deep and singular meaning of human life — itself fully inherent in artistic creation and the search for truth. Its full meaning will only arise by the end of our analysis, in keeping, in fact, with the narrator's experience at the end of the narrative.
The friendship between Marcel and Saint-Loup is like the love between Swann and Odette: it consists in a misunderstanding that brings some joy to the narrator only insofar as Saint-Loup appears to him, as in a work of art, under the guise of the "nobleman," i.e. as a type or essence "for [his] thoughts to toy with in an idle moment." And if Marcel manages to experience intense joy in his company, it's not, as Saint-Loup would have hoped, as a result of his intellectual or moral qualities, but "to glimpse through him the earlier, immemorial, aristocratic self that Robert _sought to avoid being_." Marcel's interested in the young Guermantes for a reason that is the exact opposite of what Saint-Loup might have hoped for, therefore. The aristocrat effectively succeeds in arousing the narrator's curiosity and intelligence not as a result of what he actually thinks but because of who he is: the aristocrat " _par excellence_ ," an object of study whose distinctive features the narrator, like a portraitist, attempts to point out. For Marcel, then, the value of this friendship is heuristic, if not hermeneutical. It's _interested_ ; not in the sense that it might afford Marcel certain practical advantages or elevate him socially, but in the psychological and sociological sense:
> Although I felt I was better able than most to practise the virtues of friendship (in that I would always see the interests of my friends as counting for more than the sort of selfish advantage that motivates others, but which I ignored), I was aware of my own inability to find joy in a feeling which, rather than enhancing the differences between my mind and the minds of others — differences which exist among all minds — would abolish them.
It's easy to see how Saint-Loup's enthusiasm for his new friend, the numerous attentions he lavishes on him and the delicate courtesies with which he treats him, should "in a way" sadden Marcel. There's a deep and insuperable gap between this friendship that Saint-Loup calls "the greatest joy in his life" and the only joy that Marcel feels able to experience, namely "the happiness [he] was capable of deriving from being without company." It's only by the yardstick of such solitude and of the work of reflection and creation it elicits that friendship might be seen as errancy. If friendship is an "abdication of self" it's only as regards the duty of "living for oneself" (which stems from our attachment to the search for truth), a duty from which it believes it can exempt us by granting us the illusion of communication between beings and of the nobility of the heart.
Let's draw an initial conclusion, then: so far as the power of our imagination's concerned, as well as our desire for the absolute, the reality of the world turns out to be a disappointment and confirms both Marcel's and Swann's sense that the reality of the world is devoid of value. Despite this, however, reality always seems to have the last word. Whatever power the imaginary might have, there always comes a time when the real reclaims pride of place: Marcel's snobbery was the product of his medieval imaginary and of a worldly mythology he had created for himself; then came the moment when he became accepted and even envied amid the Guermantes circle, and the prestige that this world had held until this point started to crumble until it collapsed altogether. The snobbery then turned against itself, dialectically so to speak, morphing into its harsh and pungent critique. In the same way, there comes a time when Swann fails to imagine Odette, as Marcel fails to imagine Albertine. This is when the undermining effect of ordinariness, unrelenting and haunting, ends up being felt; when disappointment, indifference and boredom prevail over passion. Passion can in fact only be nurtured by jealousy, a realm into which it can retreat as if into a fortress which, although impregnable, eventually becomes intolerable, so unbearable can the suffering it entails become. Jealousy can even manage to sustain the illusion of love by supplying that factitious intensity that Swann and later Marcel deep down recognize as completely unhelpful. The torment it involves is the cost of the feeling of existence that it provides. It certainly gives us the impression that we are experiencing something genuine and true; ultimately, though, it's no more than a pretence of the imagination in the face of the _structural_ incapacity to love, i.e. to understand one another. All that is left to do then is let oneself live, wait for time to pass by, and entertain oneself. Conversation, jokes, clique mentalities, naughtiness, gambling, lovers: these are all still there at a moment's notice. But that's all there is: entertainment, expectations devoid of object, more or less conscious despair; in short boredom. Granted, imagination can always migrate and find an occasion to soar. But for how long? And how many more times after that? Imagination now knows that reality, led by its cruel indifference and its power to crush, will eventually clip its wings. Incapable of enjoying reality for what it is, of embarking on a search for its truth, Swann must settle for sublimation, just as Marcel himself does (until the necessity and the meaning of this work become evident to him at the end of the novel). This raises, once again, the issue of values and, with it, that of nihilism: up to a certain point (which coincides with the turning-point of the novel and a possible reversal of nihilism) Proust's heroes are incapable of seeing any inherent value to real existence and so they find refuge in the imaginary and in art above all. But Swann's view, and even Bergotte's view, of art is a false one: disconnected from life, the support and the value that art is supposed to bring about, these are actually negations of art. Escaping boredom through art — at least following Swann's or Madame Verdurin's perspective — provides a false escape, for which Marcel will strive to find a substitute. Restoring the true value of art and the true place of life will be the direction of "the invisible vocation which is the subject of this book."
***
At this point, I'd like to advance a hypothesis regarding the relation between reality and imagination as hitherto envisioned. Reality, as I suggested, seems disappointing, in the sense that it seems limited in time and space, compared to the imagination which deals with the merely possible and always anticipates reality. Without any curb or limit, the imagination is infinitely more rich, lush and satisfying for the mind. By anticipating the present and enriching the real through the mediation of the imagination, our encounter with reality always runs the risk of disappointment: "[R]eality," the narrator confides at the end of the book, as he looks, in retrospect, at his experience as a whole, "has disappointed me because at the moment when I perceived it, my imagination, which was my only organ for the enjoyment of beauty, could not be applied to it, by virtue of the inevitable law which means that one can imagine only what is absent." This is a paradoxical if not aporetic situation in which the imagination, seemingly the only true organ of joy, would only yield joy in the absence of its object. Joy would therefore only be possible _in absentia_ and the full presence of the object of desire or of the object of fantasy would quite simply imply the loss of the corresponding joy. Is there any solution to this aporia and, if so, what's the miracle that brings it about? The answer to this question will be the focus of the following chapters. Let me note for now that the issue, as it's raised here, has a temporal origin: if we only grasped the most immediate present, if we had no way of projecting ourselves into the future through the imagination or no way to picture what is absent, the real would yield neither suffering nor boredom. As a result of our temporality, however, and I would even argue of the singularity of our being as a temporal being, our present always runs the risk of seeming like a simple ontological residue to us, of appearing, in any case, as the mark of our structural capacity to overcome it. This makes reconciling oneself with the present extremely difficult. And ultimately, then, the problem would consist in knowing how to reconcile the present and the future, the real and the imaginary: are we irrevocably mired in this tension, doomed to this split, or is there a way to suture it? Would joy eventually be a problem of time? Would time be the very object of joy? For that to be so, time would have to be something more than nothing, all the while not being present. Moreover, I pointed out that the imagination isn't limited to the straightforward anticipation of the real; it contributes to it as well, extending it and connecting it to another reality (to the world of art, for example, as evidenced by the resemblance between Odette and Botticelli's Zipporah posited by Swann, or to the world of History). Here again, though, the point is to prove that the reality of the present — raw and material reality — always aims to surpass itself in the direction of something else, to transcend itself in the direction of another universe, a world implicitly inscribed within it and that would constitute its hidden face. It all happens as if the only value or interest of the present reality consisted in its capacity to transport us elsewhere, to translate us into another reality. The latter might well be a possible or forthcoming reality, as I suggested; but it can also be a past reality, a completed era of history or a forgotten episode of our life, as I'll show later on. There's an enormous difference between the two, perhaps even a difference in nature: memory will bring us to the edge of a joy that the imagination wished for but failed to fulfil. What's important at this point is to stress how the present is devoid of value and empty of meaning, how it's therefore wholly driven towards its own surpassing. Why? Because it's the realm where our perception and our action are exercised and our interest is played out. As such, it can be a source of satisfaction, but only in a physical and vital sense. Perception is only concerned with the visible and, through the visible, with what's immediately useful to us. It does direct us towards the world but mainly the world as that which one can — and most of the time _must_ — act upon. Perception lives in the present since we can only ever act in and on the present alone. Its link with the world (and its knowledge _of_ the world) is wholly material: what I perceive is matter and matter saturates my being in the world as an immediately present being. Perception can act on the present but it can't dream the present or imagine it. Such a dream, such an imagining of the present even, requires that perception should loosen its grip on the material world and let the mind expand, wander until it connects with a different world. The imagination — like memory, an even more determining phenomenon, as I'll try to show later on — presupposes, then, a transformation of the world of perception into that of the mind. Now, the novel's constitutive horizon is, in fact, the mind, if not truth, and that's mainly because, through the narrator, it tries to understand the world, to grasp what it takes to be the world's hidden truth while present reality, even though the locus of this attempt and the thing that it ultimately aims to enlighten, is a source of deception. Perception and the present can only ever play a minor role in the search for truth; yes, they're our initial gateway into the world, our gateway into matter, but they're unable to take us much further than that. Perception can only serve as an introduction, as a springboard, so to speak, from which the mind can then take flight. Right up until the final revelation, though, however foreshadowed it's been throughout the novel, the narrator can't reconcile his thirst for truth and his disappointment, the power of the imagination and the deficiency of perception, possible time and real time.
The second hypothesis I'd like to advance is this: in order to escape the nihilism underlying this experience of disappointment in the face of reality and so as not to end up in despair at existence, the novel has to find a solution to the miserable human condition — the condition that alternates between suffering and boredom — without actually settling for some metaphysics of the immediate present or for a merely descriptive, "realistic" form of literature. What has to be retained, in other words, is the thought of a present that's not self-sufficient and that's meaningful only to the exent that it overcomes and exceeds itself. At the same time, though, what needs to be rejected is the very condition to which this way of thinking looks to have condemned us. From now on, it's a matter of knowing whether the paucity of the present can be reversed into something rich and whether the suffering (and boredom) that the present always entails can morph into joy.
What are the conditions under which we can stop despairing about life and start rejoicing in it? What's involved in this question is nothing so much as a reversal of nihilism, Schopenhauer's version of nihilism especially, which Proust however seemed to endorse. Whatever some have wanted to say, Proust's "solution" isn't Schopenhauer's: while both are solutions mediated by art, for Proust art doesn't coincide with any sort of suspension or neutralization of the will, of desire, of sensibility, in short; instead, it coincides with the latter's actualization and with its truth. At the same time, such a reversal of nihilism is as far from Platonism as it's possible to be, despite some commentators having seen the novel as decidedly Platonic in places: because Proust talks about Ideas as those realities that are truer and richer than the reality of the immediate present, it was assumed that he posited the existence of the intelligible world above or behind the world of appearances. As I'll show, though, Proust's Ideas are embodied and sensible. They are less a negation of sensible reality, an indication of the existence of a supersensible world, and more its other side or its lining and the sign of its own overcoming. They don't oppose perception or the present that perception uncovers; rather, they excavate them, probe them, until their depth is plumbed. Such a deepening of sensible and material reality will in fact coincide with the discovery of a new meaning of time, a discovery to which the very possibility of artistic and literary creation will find itself bound. The process of overcoming despair and the very possibility of joy will be mediated by the experience of a temporal dimension hitherto unknown, one that has memory at its core. There's no radical and final opposition between memory and perception, between an alternative form of absence and the raw presence of the world. Instead, there's an entanglement and a structure of substitution.
As a transitional stage, I'd like now to show how Proust's novel works out a solution to the issue I just raised. Throughout the novel, like a parallel narrative that will eventually merge with the first one, the narrator experiences revelations, single moments that are at odds with the disappointments that I've been talking about so far and whose reflection involves such intense pleasure. Everything happens as if that experience was entirely different in nature, as if a new meaning was emerging from it, a meaning that the narrator himself will try to interpret. The reconciliation with reality that I've been talking about as the novel's problematic horizon originates from that other experience, or that other meaning of experience. The moments in question all seem grounded in sensations or in strong affects. Should we be surprised by this? Isn't sensation the gateway through which the world reaches us, our initial point of contact with it, the point at which our being and the world mingle, as if the two lightly touched? Doesn't sensation mediate my joy at the real itself (and not at its representation) and does so _immediately_? Doesn't Proust situate himself as an apologist of sensation and — why not? — of sensualism as the solution to the traps of the imagination? But how, given what I've already said about perception as wholly directed towards what's useful for me? Proust doesn't reject the temptation of the imaginary to surrender to the cult of strong feelings. True joy will be neither the joy of the senses nor that of the imagination.
In order to answer this question, I'd like to spend some time on a celebrated passage from _In Search of Lost Time_ , one that features a pivotal experience, somewhere between the type of experience I've been describing so far — a type that could be described as nihilistic — and one that will function as its dialectical reversal. My sense is that this reversal actually takes place over the course of the passage, a passage that operates on several levels and across various stages. The passage I'm talking about is the one devoted to the hawthorns in "Combray."
The first stage seems to confirm the imagination as the only way of eliciting the joy of beauty through the shift of the present reality into another one. The second and third stages of the experience deal with the feeling of plenitude that the narrator has when he comes into contact with those flowers and which seems to transcend the power of the imagination alone. This raises as a result the issue of deciding whether this feeling originates in its material reality, in sensation itself, or if it indicates something else. It's an enlightening passage, then, insofar as it deepens the sense of the perceptible world and of beauty in particular, insofar as it investigates the nature of certain sensations. It points towards a reconciliation with sensible reality, without ensnaring us yet again in the illusion that its presence is something real and immediate. Albertine, remember, was only beautiful when she was the subject of the imagination. She was desirable only insofar as she was mysterious. However precisely or realistically Proust might have described her features, her gestures, her habits, she still wouldn't have appeared beautiful since her beauty isn't objective in the slightest and depends entirely on the narrator's ability to invent it. And? Does the same go for nature in general as these hawthorns for the young Marcel would seem to suggest or the sea in Beg-Meil for Jean Santeuil? The beauty of a flower's no more objective than that of a woman and our taste for it is no more immediate either. The pleasure we derive from a flower doesn't ground us in the immediate present any more than our desire to possess someone else physically. The pleasure we might experience in the other's presence — or, indeed, in the presence of the hawthorns in Marcel's case — is obviously mediated by the senses, but can't be attributed to them. As a sense object the hawthorns are endowed, Proust claims, with a certain "charm" that's perhaps inexhaustible. Their beauty, though, isn't constituted by this charm. Something can be pleasant and can address the senses alone without ever being beautiful. Marcel intuits that something's being pleasant isn't enough to make us love it, whether we're talking about a woman, a flower or a work of art. For this to happen, they have to be beautiful.
Kant basically said everything that needs to be said about the essential difference between what's merely agreeable and what's beautiful (or what's declared to be such, since this question only makes sense in the context of judgment). The judgment of taste in relation to the beautiful and the kind of judgment relating to what's merely agreeable are similar insofar as they both involve a feeling of pleasure. However, the sense of pleasure in question differs radically depending on whether it applies to one type of judgment or the other. Following Kant, this calls for a distinction between two meanings of the word "sensation." Every satisfaction is itself a sensation, namely the sensation of pleasure. Consequently, everything that pleases is agreeable precisely because it pleases. When it comes to defining what's agreeable, there's a rich vocabulary to draw from, and Proust does precisely that: "gracious," "charming," "delightful," "lovely," etc., are the sort of epithets among others that denote nuances or differences in degree of the same type of sensation. The sensation in question — or rather _this meaning_ of the word "sensation" — is described by Kant as "feeling" ( _Gefühl_ ). It denotes what's always merely subjective and what's strictly sensible. When the pleasure principle is in the sensation of the object, the feeling is of the agreeable kind. The agreeable, Kant contends in the _Critique of Judgment_ , represents the object only in relation to the senses: "the rose — or the hawthorn, if you like — is (or smells) agreeable" is a judgment pertaining to the senses ( _ein Sinnenurteil_ ) and not a judgment of taste. In other words, the pleasure principle is altogether located in the sensation or in the materiality of the object. "Agreeable," Kant affirms, "is what the senses like in sensation." From which we could infer that there might be something in sensation that appeals to something other than the senses, that there might be a source of pleasure which, even though it's combined with a sensation, remains distinguished from it. In fact, Proust's narrative underscores precisely this sort of hypothesis. We often — all too often — rely on this language of pleasantness in order to describe the effect of a work of art, a person or some natural phenomenon, implying thereby a satisfaction pertaining to a "feeling." Now, from the point of view of the effect the difference between what's (merely) agreeable and what's (really) beautiful isn't easy to establish. The difference is even non-existent: it always consists in the enjoyment experienced in the sensation of our state. But the effect is precisely not the level at which we should dwell (and, through the hardships and the questioning experienced by the narrator in the episode of the hawthorns, Proust invites us to envision the move to another level). If it were, we'd have to assume that every single activity should aim to achieve this state and that the only way things and their value might be appreciated would relate to the pleasure that they bring. It would no longer be a matter of knowing how the goal was reached, the goal being the only value. The ends justifying the means, every possible mean would be good so long as it led to the goal everyone's trying to reach: pleasure. So far as Kant's concerned, though, this is not only unacceptable, but also fails to tally with what we see around us: not everything we do is predicated on the search for pleasure. Moreover, the experience of beauty is, for Kant, an entirely disinterested one. In sensation _per se_ , conversely, the representation of a thing relates to the form of the object, not to its content, and pleasure is experienced when the representation of the object is in agreement with its end. As opposed to the expression of the agreeable, the judgment of taste isn't merely subjective: on the one hand, it aims at the object itself (in its form); on the other hand, it requires the adherence of others. Without being a statement of knowledge, it's an objective and universal (or intersubjective) judgment:
> It would be ridiculous if someone who prided himself on his taste tried to justify [it] by saying: This object [... ] is beautiful _for me_. For he must not call it beautiful if [he means] only [that] _he_ likes it. Many things may be charming and agreeable to him; no one cares about that. But if he proclaims something to be beautiful, then he requires the same liking from others; he then judges not just for himself but for everyone, and speaks of beauty as if it were a property of things [ _Dinge_ ]. That is why he says: The _thing_ [ _die Sache_ ] is beautiful, and does not count on other people to agree with this judgment of liking on the ground that he has repeatedly found them agreeing with him; rather, he _demands_ that they agree...
The narrator claims that the hawthorns are beautiful. He even wants to explain the mystery of this beauty and understand the meaning of this distinctive experience. By finding the flowers more than just pleasing, more than just pleasant, he endows them with a metaphysical value and draws out a new meaning of our being in the world; by experiencing, in such personal terms and in such a localized way in space and time, something universal, he discovers the possibility of a communion, not just with the object of his judgment but with every rational being as well. In doing so he discovers the possibility of a true communication: not the communion of love or of friendship, which rest on the illusion of transparency, but the kind of communication postulated by the aesthetic experience and the judgment of taste. He even discovers, albeit implicitly, the only real community, namely the aesthetic community. The hawthorns, then, come alive to Marcel and acquire aesthetic value only once, through their association with the "Month of Mary" and their decorating the altar in the church of Combray, they embody devotion to the Virgin Mary and the celebration of feminine mystery:
> Higher up, their corollas opened here and there with a careless grace, still holding so casually, like a last and vaporous adornment, the bouquets of stamens, delicate as gossamer, which clouded them entirely, that in following, in trying to mime deep inside myself the motion of their flowering, I _imagined_ it as the quick and thoughtless movement of the head, with coquettish glance and contracted eyes, of a young girl in white, dreamy and alive.
Whatever aesthetic value Proust's description may have — and, for the narrator, whatever epistemological value it has — it gets from the fact that the world to which it exposes the narrator is evocative of another one and from the fact that it suggests thereby a secret correspondence between two strictly distinct worlds and the capacity of one to contain or imply the other. Nothing here seems to invalidate the relation between reality and imagination as I have envisioned it so far: reality has value only in its ability to stir our imagination. But, as we saw earlier, this joy comes at a cost, namely the distance it requires so far as reality's concerned: absence is an integral part of the workings of the imagination — we can only imagine what's absent — and what's present — that is, reality — never coincides, so to speak, with what we'd imagined. The object of joy isn't the real in itself or as such, therefore; instead, it's the world into which, on the basis of present reality, our imagination can transport us.
A few pages further on, though, as, by chance, Marcel comes across a hawthorn hedge during one of his walks, the intense pleasure he experiences in their presence doesn't seem to come from his imagination alone or from the world that these flowers evoke for him. The fact that seeing one thing leads us to represent another isn't necessarily reflected by a feeling of pleasure. Sure, in the narrator's eyes, the hedge does look like "a sort of series of chapels," and the sun above them looks like "a grid of brightness" reminiscent of the kind we might see through a greenhouse; the flowers themselves, "adorned also, each held out with a distracted air its sparkling bunch of stamens, delicate radiating ribs in the flamboyant style like those which, in the church, perforated the balustrade of the rood screen..." The hawthorns are undoubtedly once again evocative of the Virgin worship, along, this time, with the architectural materiality of the church at Combray. Yet, the narrator's pure pleasure, untarnished by any shadow (except for the shadow of not knowing its origin), cannot be attributed to the sole world connoted by the flowers, a world which, in itself, is not tantamount to happiness. Marcel therefore sees himself, to some extent, as distracted from this world and returned to the present, material and sensory reality of the flowers, as if the key to his pleasure and to the flowers' beauty dwelled hidden somewhere within them, in their shape, their colour or their fragrance, withdrawn and anxious to arise from them. The narrator intuits the presence of something else behind the raw, physical and material presence of the flowers. But what is it? The hawthorns are all the more captivating in the church. Why is that? It might be because they are simply more present and because Marcel may now come in close contact with them, inhale their fragrance, and examine their colour. It might also be that, experiencing the absolute presence of the phenomenon, Marcel is now saturated with it. This sheds light on the young protagonist's repeated attempts to delve into the fragrance of the hawthorns, over and over again, in the hope of grasping the mystery of their appeal. But this is to no avail as nothing helps, nothing more that is, or nothing new — the _enjoyment_ remains the same while its cause remains unknown:
> But though I remained there in front of the hawthorns breathing in their invisible, unchanging smell, bringing it into the presence of my thoughts, which did not know what to do with it, then losing it, and then finding it again, absorbing myself in the rhythm that tossed their flowers here and there with youthful high spirits and at unexpected intervals like certain intervals in music, they offered me the same charm endlessly and with an inexhaustible profusion, but without letting me study it more deeply, like the melodies you replay a hundred times in succession without descending farther into their secrets. [... ] [A]lthough I formed a screen for myself with my hands so that I would have only them before my eyes, the feeling they awakened in me remained obscure and vague, seeking in vain to detach itself...
The insistence on not going beyond the impression or the sensation, on getting as close as possible to the phenomenon in the hope of grasping the feeling that arises from it, necessarily results in missing the latter. At other points in the novel, the narrator makes fun of those who allege that, upon listening to some sonata for the hundredth time, they can still experience the same pleasure and exclaim "how beautiful! What a gem," thus giving themselves the feeling of being immersed in this music, of becoming one with it and merging with it completely, without ever wondering about the origin of their pleasure or providing themselves with the means to look into its meaning. There's no point in shaking all over, like Madame Verdurin, and claiming that, if the music does not stop, she will end up crying or even falling ill because she "feels" the music with such intensity. This doesn't help our understanding of the feeling of pleasure that we're experiencing. Until something useful can be extracted from this impression, we'll grow old "useless and unsatisfied, like celibates at the shrine of art." Because his pleasure is wholly contained in the flowers under his eyes, Marcel thinks that its origin can be grasped through the closest possible encounter with their materiality, by drinking in their presence and, intoxicated, embracing it completely. He wants to hold onto them at all costs. How? And what, exactly, does he want to hold onto? The flowers are here, present and available. He can come back to them, throw himself into them and plunge his senses into them all over again. Something in them, though, escapes him. His pleasure's wholly located in these flowers; it depends on them unconditionally. Something else, though, something that's seemingly bound up in this pleasure, at once preserved and embodied in it, seems to want to escape it. But to where? And for whom? His thought process itself is no help at all. But how could it be since it's the process of _thought_ and thoughts aren't sensations? He doesn't know what to think of these flowers, what to make of them. If sensation exposes us to a mystery, if it paves the way for another reality or, rather, for a truth buried in the very ribs and stamens of the hawthorns, it leaves us on the doorstep, as it were, without any way of getting any further. And whatever thought thinks about taking over, pushing the door open and breaking through the secret, it lacks purchase in the sensible, the sting of the senses. Sensation yields only charm; thought, on the other hand, remains abstract and glides over the experience.
Neither perception, nor imagination, nor even intelligence seem to exhaust this intense pleasure or explain its origin. Marcel now suffers from another kind of frustration: not the kind of frustration arising from a reality that's deficient in relation to the image that we form of it, but from the kind that stems from a reality that seems to pass over the imagination and the intelligence alike and call upon some unknown faculty within us, a faculty whose nature and origin are wholly unclear. Right from the start of the book, then, there's an attempt to articulate a relationship with the real that would be the exact counterpoint of the dissatisfaction that presides over the novel as a whole. And while this alternative relationship is itself the object of some frustration up to the very last pages of the book, this takes place in a completely different way: not because it emerges from some ontological deficiency as regards our desire, but because it takes the narrator a long time — a lifetime, in fact — to break through the mystery of this excess, this over-abundance, this surplus of meaning that emerges from the real itself and becomes embodied in an impression. The economy of scarcity that I've depicted so far is now juxtaposed with an economy of excess from which literature will ultimately draw its _raison d'être_. The disappointment and even the depression of the first economy will gradually give way to the true joy or joyfulness inherent in the second. It's just as though a surplus of meaning came from sensation itself, as though sensation endured beyond its capacity for imagination (for transfiguration) and grew deeper. There's a sort of persistence, of insistence, even, of sensible reality where this pleasure seems to be embodied. The intensity of impression, its engulfing aspect, is the sign of a depth, a reality that's deepening and surpassing itself in the direction of something else that, in truth, is more like what it actually is, namely its accomplishment or its reality. Reality's no longer that lack. It's become this surplus of phenomenality that pulls us up before its mystery and compels us to break through it. The difference between this and the previous experiences is that now we're dealing with an enduring kind of pleasure: persistent, it urges us to grasp its origin and wonder whether there might not always be more than the present in the present, not that absence with which we can (through our imagination) invest the present, or that raw presence that emerges from perception, but this extra reality, this residue. As for this surplus, it might well constitute the "substance" or the "matter" of literature, as well as the truth of the real. _Finding Time Again_ eventually sums it up like this:
> Impressions of the sort that I was trying to stabilize would simply evaporate if they came into contact with a direct pleasure [ _jouissance_ ] which was powerless to bring them into being. The only way to continue to appreciate them was to try to understand them more completely just as they were, that is to say within myself, to make them transparent enough to see right down into their depths.
Joy can only ever be indirect, therefore, and the _Search_ is the guide to this indirect way.
At this point, I should deal with what I referred to as the third stage of the experience insofar as it affords a distinctive hypothesis regarding the real origin of the feeling of pleasure experienced in the presence of the hawthorns. The solution to the problem that I'm interested in here was already hinted at in the initial description:
> I felt that [...] it was nature herself who, by carving those indentations in the leaves, by adding the supreme ornament of those white buds, had made the decorations worthy of what was at once a popular festivity and a mystical celebration.
In other words, it's not simply the narrator's fancy that compels him to see the flowers as he does. The pleasure he gets from seeing them in this way is specifically _not_ derived from his ability to imagine them as young girls or from the power of the imagination and, consequently, from the fascination that this power exerts upon us; instead, it stems from his ability to sense some secret agreement linking the imagination and nature, to feel that nature itself is the inspiration behind these images. It seems as if — and let's consider "as if" as pivotal here — nature itself and nature as such presented itself in this form, as if it wished to surpass itself, to extend itself into the spiritual world (symbolized here by the religious celebration). It's _as if_ nature, as the set of laws subject to strict determinism, aimed to suggest some compatibility or, rather, some convergence, with the spiritual laws, beyond the clear opposition that Kant establishes between mechanical and free causality, or between the phenomenal and noumenal world. To Kant as to many in his wake (to Schiller and Schelling among others), aesthetic pleasure signals this compatibility, indicating that nature itself is led by a dynamic that we experience as a purpose or as the natural, sensible confirmation of a spiritual reality. This is reflected by the feeling of being at home, as a spiritual being, in nature itself. It's a feeling of wonder and disinterested pleasure, i.e. one that's in no way attached to a desire to possess or to appropriate. It's the feeling of being in the world as being in the midst of an ordered and rational reality, without, however, being mechanical, of gazing at and understanding the world as if it were subjected to an end, not a natural one but a spiritual one. In other words, it's the feeling of the realization of the natural world _in_ the spiritual one (the world of the sacred). The point is not to claim, as Hegel would have, that this is actually the case, that nature is actually in and as spirit, following a dialectical dynamic. Rather, the point is to indicate, on the basis of a feeling of pleasure (and so a sensation) and of the judgment of taste in which it is expressed, how nature reveals itself to us following an otherwise inconceivable purpose (or one that is conceivable without any relation to the experience whatsoever). It does seem as if nature itself were purposive, but purposive without purpose, as Kant claimed.
The confirmation of nature's intentionality can be found in the following passage, which tries to account for the origin of the pleasure — a pleasure that's even more intense — that the narrator experiences when, during the same outing, he comes across pink hawthorns:
> And certainly, I had felt at once, as I had felt in front of the white hawthorns but with more wonder, that it was in no artificial manner, by no device of human fabrication that the festive intention of the flowers was expressed, but that nature had spontaneously expressed it...
If what's so far been seen as a metaphor had been nothing more than deliberate artifice, a figment of the narrator's over-active imagination, the intensity of the pleasure experienced in the presence of the hawthorns couldn't possibly be accounted for. Why not? Because even though the pleasure we're talking about here would, thanks to the narrator's imagination, have gone beyond the sort of feeling derived from the merely agreeable, he would ultimately have been unable to believe in the appropriateness of his metaphor. Why not? Well, for a metaphor to be appropriate and, even, true, it must, as we know from experience, be universal and, as such, detached from any form of arbitrariness. It must be natural itself, "spontaneous" and somewhat inspired by its very object. If it does translate the latter into another language — the language of poetic ideality — and transposes it into another world, it's a language that translates its essence and reveals its end. Deeming the flowers festive while reflecting on the origin of the pleasure they invoke in us implies the discovery or the extraction of the festive intention that nature somehow spontaneously locates in the flowers. It implies seeing that there's more than a flower in the flower and more in sensation than what my senses alone can recognize. The way the essence of the flower can be reached and the good metaphor can be woven doesn't lie in immersing ourselves in sensation itself but in following the thread of the flower's meaning.
But why is the pink hawthorn even more festive? Why is it the source of an even greater pleasure? Because it allows us to penetrate even further into the spiritual world, to reveal a new dimension, a new aspect of reality in a simple shade of colour. Between the pink and the white of the thorn, there's no apparent difference: we're still dealing with colours and hawthorns. And yet, the pink seems to reveal another reality altogether, one that's still compatible with the first one and that brings with it the promise of an infinite number of possible worlds living as if they were embedded in each nuance of a quality, in each variation on a theme, however minute it might be. Proust continues to weave the festive and religious metaphors ("It, too, wore finery for a holiday — for the only true holidays, which are the religious holidays"), adds another one alongside: the pink of the hawthorns are reminiscent of the most precious biscuits from _chez Camus_ , which are pink as well, and of the pink cream cheese into which strawberries often get mixed. We're taken from one (sacred) world to another (a sweet one): the flowers themselves have the appearance of something edible. As such, they evoke something stronger, something more immediately tangible, something that can be assimilated fully; they return us to the sensible sphere, but to a sensible sphere that's been modified, transformed even, by the diversion or the mediation of the sacred: the sacred is now something that can take shape in my body, as suggested by the mystery of the eucharist. In truth, it consists in the transubstantiation of the real itself, i.e. the incarnation of the sacred and, more generally, of the ideal world, as well as the spiritualization of the sensible world. Religious and even deeply Christian in its structure — a structure generalized by Proust, as we'll see — the experience of the hawthorns isn't religious in its content. Its link with the cult of the Virgin and the church at Combray signals and inscribes the general structure of experience in the strongest possible sense, namely the one that consists in extracting the ideal world from the sensible world, in drawing the spiritual world out of the world of the flesh. The hawthorn has become something more personal, something more true: the displacement of one semantic world to the other is more than a mere transverse displacement, a mere metaphor: it's a deepening, a pathway pointing in the direction of the discovery of something essential. Of truth, then? Absolutely, even though the narrator isn't yet able to see it for what it is. What the pink of the hawthorn reveals, through the unctuous, fresh, sweet appearance of the puddings and the biscuits of childhood, is the reason for our attachment to it: i.e. the childhood that's settled and that can be found there again at will, the promise of a unique world (as opposed to a divided if not torn one) where everything communicates with everything, accessible _de jure_. What comes back to the surface in the experience of the hawthorns, what emerges before these flowers, is the singularity of childhood, its world of colours, tastes, flavours, its religion, its myths, in short all the differences that constitute "Marcel" the subject. This isn't something he's able to realize yet. He only realizes it halfway, so to speak. His wonder, his joy, his emotion are markers of it, however, unquestionable signs.
***
Throughout the novel, Marcel would like to get away from disappointment, suffering and boredom and to turn the few instants of joy he experiences into lasting moments; he even suspects that writing might provide him with the means to do so. He'd like to become a writer and has the sense that writing is a privileged mode of access to the real. The real, though, can't be understood as that disappointing present which requires the support of the imagination in order to move forward. If writing is to bring us anything, it is bound to surpass that immediacy. It must elicit a deepening of the real and not a diversion from it. If it simply diverts us from the real it's no more than a form of "entertainment." No, if literature's going to exist, it must be as a probing device, a depth-exploring tool. Granted, Marcel thinks, it must have some relation with the pleasure that the sight of a roof, a reflection on a stone, the smell of a path or a flower might evoke in us. But it's not enough to note, in the immediate aftermath of his experience of the steeples at Martinville, "the shape of their spires, the shifting of their lines, the sunlight on their surfaces," as Marcel does in his first go at writing. On the contrary, as he senses he has to do in the presence of the hawthorns, he needs to exhaust the impression, to understand what is hiding behind this movement, behind this light, to grasp what seems to dwell and steal away simultaneously in its midst — something the young Marcel's unable to do for practically the whole novel. For him, if art's to exist at all, it must yield happiness, reverse the course of dissatisfaction and bitterness; in short, it must reconcile us with life. This implies that art has to enclose the promise of an access to another meaning of the real, a meaning that's deeper and more true but also more hidden than the meaning of everyday life and of its transformation by the imaginary. But how could art ever access this meaning if it feeds on the imaginary? In order to escape dissatisfaction, art would have to be able to escape the imaginary and discover a reality that would be neither the negation of its anticipated image nor the alleged reason for its flight. But what are the means by which it could reach that reality? What would be the mediation via which it could be grasped? Involuntary memory is the emergence of that other reality or, rather, of reality as unanticipated and unimagined. It implies the insistent and somewhat miraculous presence of what was thought to be dead, the rebirth of a forgotten past. The value of this experience is wholly included in the return itself and not in the content of what returns. It signals the existence of a time that isn't the time of anticipation and desire, ultimately doomed to an ever disappointing reality, or the time of the sole present and of perception, devoid of meaning in themselves, but the time of the contiguity of the present and the past. We know that, through the mediation of involuntary memories, Proust ultimately assents that we can surpass the disappointment entailed by the real. This view implies the perennialization of these fleeting moments through the creation of a work of art. Thanks to art, which feeds on the raw matter constituted by unconscious recollection, we can escape the structural disappointment imposed by the real and become reconciled with life. We can extricate ourselves from the throes of dissatisfaction and bitterness and devote ourselves to the evanescent and fleeting share of beauty and truth which life encloses.
Chapter 2
* * *
Proust among the psychologists
* * *
_In Search of Lost Time_ depends on a theory of memory that involves Proust in a conversation with the psychologists and philosophers of his time, Taine, Ribot and Bergson in particular. For example, his critique of the view that it's " _intelligence_ " that affords us access to the truth of the world and to our experience of it, as well as the positive role that he gives memory in all this, wouldn't have been possible without the ideas developed in _On Intelligence_ ( _De l'intelligence_ ), in which Taine defines intelligence as understanding or as intellect, i.e. as the faculty of knowing. Like some of his contemporaries, Taine distinguishes the phenomenon of memory from intelligence or the will (two things that are actually equivalent for Proust), and opposes the hitherto dominant tradition of intellectualist psychology epitomized by Maine de Biran. And Taine's at pains, too, to distinguish it from the other sort of memory that's discussed by the psychologists and philosophers of the time, Bergson included, namely automatic or habitual memory. It's only as _involuntary_ memory — or, in the language of Ribot and his school, as _affective_ memory, as the repetitition of past affective states produced by sensations experienced in the present — that memory's going to reveal itself as the solution to the mystery of the Ego, no longer understood in terms of its continuity and self-identity but in terms of its periodic or discontinuous character and comprising, as Proust himself points out, the very "matter" of the book.
Is this enough to make the novel a psychological one, perhaps, even, as some have suggested, a straightforward illustration of associationist psychology's prevailing theses? I don't think so. Instead, I'll want to show that while the notion of involuntary memory so central to Proust _is_ anchored in an already well-established psychological context, it exceeds any purely physiological interpretation as well as the primacy of perception that any such interpretation's going to involve, and calls instead for a specifically _ontological_ interpretation. As such, it affords a solution to the existential impasse that I discussed in the last chapter (the tension, remember, between perception and imagination as the source of suffering and boredom), a solution that's only fully realized by artistic creation. In other words, ontology's extended and completed in an aesthetics that defines what's most distinctive about the book.
So, how does Proust's view of memory differ from that of his contemporaries? Well, as far as associationist psychology's concerned, there are three decisive differences.
First of all, even though instances of involuntary or affective memory are rare, they don't concern, at least not for the most part, the sorts of pathologies that involve hypermnesia; rather, they concern the very nature of the Ego, which they unveil in its truth and distinctiveness.
Second, while memory, at least in the pyschologist's view, simply reawakens a past that's already been experienced and while that past is itself understood as a collection of lived experiences, Proust sees memory as the quite possibly infinite reservoir of what was never actually experienced, so to speak, but which returns all the same: a sort of _un_ lived experience, if you like, which Proust is going to make the very matter of literature. Before going any further, I should explain what I mean whenever I use the term "matter": the matter at stake is what we'd normally call "spirit," namely the _residue_ of a sensation which, while related to the sensation, can't be reduced to it. What Proust refers to as the Idea is, in fact, this very "residue" that exceeds lived experience, the supplement or the share of _un_ lived experience that's particular to the lived experience and that only appears as such by coming back, in other words, by being tied to a new experience.
Lastly, I'll argue that the overcoming of the psychological empiricism of a Taine or a Ribot is intimately related to this discovery and that it tends not in the direction of the sort of idealism that involves a constituting consciousness or a supersensible world but towards a kind of naturalist ontology for which the meaning of the being of the world is disclosed as an immemorial past, one wider and more vast than any subjective experience and irreducible to mere "psychological fact." In terms of his basic commitments and despite appearances to the contrary, Proust's actually closer to Bergson than to the psychologists: not in terms of his view of duration — he doesn't have one — or his view of the involuntary aspect of memory — which Bergson doesn't develop — but because of his intuition of a fundamental reality, namely Time, that's irreducible to any matter or sensation.
Memory provides the novel with its matter in more ways than one, perhaps because memory itself's plural and its various levels need to be articulated.
***
First, there's this form of memory based on sedimentation. It's a bodily and completely involuntary form of memory (although it's also completely different from what Proust calls involuntary memory). It's a kind of memory, too, that proceeds by way of accretion and that takes place in the here and now. It's aimed mainly at action and retains from the world only what's useful to it. It's selective, therefore, and its ultimate goal is survival. This form of memory is what we most normally think of in terms of that process of picking up habits by which our body can then deal with the world, those habits thanks to which the body's worldly surroundings become familiar to it, allowing it to find its bearings, its points of reference, without ever even having to think about or envision the process of doing so. Because some phenomenon, some object or some circumstance has already occurred and left its mark on a body that's essentially been readied or prepared by it, I can now respond to its recurrence and not be surprised by it. This type of memory is a sort of recording. Ribot describes it as acquired automatic memory (as opposed to the innate organic form of memory that he'll go on to refer to as genetic). To Ribot, as to Taine, memory is simply the process through which we acquire automatic functions. The basis of this process is association: gradually, in children, as they develop, "there are formed, in the nervous elements corresponding to the motor organs, secondary dynamical associations, more or less stable (that is to say a memory), which unite with the primitive and permanent anatomical associations." Today we'd call this sort of "automatic" memory cybernetic and describe it in terms of information exchange and feedback. It's the kind of memory based on adaptation and compromise that we have in common with all living organisms. Just as our muscles have their memory, therefore, so the single cell of the amoeba has its own. Life is very much a habit or, more accurately, a series and a succession of habits. Breathing's a habit. We _live_ out of habit. "Habit," as Beckett has it in his wonderful book on Proust, "is the ballast that chains the dog to his vomit." Between something living and its environment, there's a constantly renewed "pact" as Beckett goes on to say. Since the surrounding world is continually changing and evolving, this treaty is constantly being renegotiated, the pact amended. The transition periods that make up the successive processes of adaptation to the fluctuating world each represent a hazardous, precarious, painful and mysterious zone in the life of an individual. These are the moments when, again according to Beckett, "the boredom of living is replaced by the suffering of being." And when a new situation crops up? When the body's taken by surprise? Nature quickly kicks back in, life demands its due and the body continues its ongoing process of reconfiguration and reprogramming until it's finally attuned with its new environment. Habit's a full-time occupation; there are no time outs. It's oriented towards the present and towards full presence, not the night of absence and anxiety. It wants to take up all the time there is and control every square inch of space. The book's packed with illustrations of such transitional periods of adaptation and throughout the novel we get the sense that these might lead somewhere, to some revelation or to the discovery of some buried beauty or truth, since what they disclose is the very possibility of experience irrespective of its immediate cause; in other words, we have the sense that a flash of light, a sky, a room hitherto unknown might make its appearance — always assuming that the mechanism of habit, whose only real interest is in ending the time out as it were, and negotiating an alliance with a new set of circumstances, would allow such discoveries the time to develop. Momentarily disrupted, habit quickly wrests back its control of the situation and imposes a new synthesis on the body. Remember the famous description of the process of waking up and the state of being half-asleep with which the novel actually opens:
> A sleeping man holds in a circle around him the sequence of the hours, the order of the years and worlds. He consults them instinctively as he wakes and reads in them in a second the point on the earth he occupies, the time that has elapsed up to his waking; but their ranks can be mixed up, broken. If towards morning, after a bout of insomnia, sleep overcomes him as he is reading, in a position different from the one in which he usually sleeps, his raised arm alone is enough to stop the sun and make it retreat, and, in the first minute of his waking, he will no longer know what time it is, he will think he has only just gone to bed. If he dozes off in a position still more displaced and divergent, for instance after dinner sitting in an armchair, then the confusion among the disordered worlds will be complete, the magic armchair will send him travelling at top speed through time and space, and at the moment of opening his eyelids, he will believe he went to bed several months earlier in another country. But it was enough if, in my own bed, my sleep was deep and allowed my mind to relax entirely; then it would let go of the map of the place where I had fallen asleep and, when I woke up in the middle of the night, since I did not know where I was, I did not even understand in the first moment who I was; all I had, in its original simplicity, was the sense of existence as it may quiver in the depths of an animal; I was more bereft than a caveman; but then the memory — not yet of the place where I was, but of several of those where I had lived and where I might have been — would come to me like help from on high to pull me out of the void from which I could not have got out on my own; I passed over centuries of civilization in one second, and the image confusedly glimpsed of oil lamps, then of wing-collar shirts, gradually, recomposed my self's original features.
Even fast asleep, the body retains its memory of the surrounding world — of space and time. Imprinted into its musculature are the landmarks of the world it inhabits, a world that wraps around it like so many heavenly bodies revolving around a virtual sun. Even while it's sleeping, it's orienting itself. The body's sleeping but habit's wakeful and alert. And that's why, when I wake up, the world that wakes up with me is the same as it was before, the familiar world that I know is mine.
Habit can get pushed off track, of course. Most of us have experienced one of those rare moments when, waking up, we wake up to a world that's momentarily unrecognizable. Proust's surely right when he says that these sorts of moments are fleeting and that habit, momentarily pulled up short, quickly regains control by adapting, by seeking out, as he says, similar situations in the past, by urging the body to get the better of the new situation which, at least at the start, is cause for anxiety and alienation. So strong is this ability to "incorporate" the past, to put it at the service of the immediate present and of whatever task at hand there might be, the body very quickly comes around and habit regains control. Confronted by any new situation, the body has access to a supply of experiences which it can adapt and use to turn an unfamiliar world into a largely familiar one. Drawing on this memory bank that it carries with it at all times, the body selects the image that's going to be most useful to it. What I'm trying to say here is that there is such a thing as the body's memory, then, one that's tied to its practical side and from which it can draw whatever help it needs: "the memory — not yet of the place where I was, but of several of those where I had lived and where I might have been." Proust goes on to describe the work of such memory as geared towards easing existing discomfort by extracting possible solutions from the body itself:
> My body, too benumbed to move, would try to locate, according to the form of its fatigue, the position of its limbs in order to deduce from this the direction of the wall, the location of the furniture, in order to reconstruct and name the dwelling in which it found itself. _Its memory, the memory of its ribs, its knees, its shoulders_ , offered in succession several of the rooms where it had slept, while around it the invisible walls, changing place according to the shape of the imagined room, spun through the shadows. And even before my mind, which hesitated on the threshold of times and shapes, had identified the house by reassembling the circumstances, it — my body — would recall the kind of bed in each one, the location of the doors, the angle at which the light came in through the windows, the existence of a hallway, along with the thought that I had had as I fell asleep and that I had recovered upon waking. My stiffened side, trying to guess its orientation, would imagine, for instance, that it lay facing the wall in a big canopied bed and immediately I would say to myself: "Why, I went to sleep even though Mama didn't come to say goodnight to me," I was in the country at the house of my grandfather, dead for many years; _and my body, the side on which I was resting, faithful guardians of a past my mind ought never to have forgotten_ , recalled to me the flame of the nightlight of Bohemian glass, in the shape of an urn...
It would be a mistake to think, then, that the body's entirely trapped in the present. Although it's oriented towards it, of course — and towards the resolution of the problems that the present raises — it does so on the basis of its own past. The present is always riddled with the past, therefore, perception always mixed with memories. In the hollow of my ribs and my shoulders, in the crook of my arms and the bend of my knees, in the palm of my hand and on the line of my back, my present is fed, virtually at least, by all the bedrooms I've ever visited, the beds I've lain on, the objects I've handled and the caresses I've received. Which might well cause us to wonder whether the past could ever be expressed outside of whatever task it is that's currently occupying us. Is memory nothing more than the reinforcements called in to help perception's efforts? Is life itself nothing but habit? Put a bit differently, and in terms this time of the "moment" or those "few seconds" during which the body strives to adapt to what it doesn't recognize, the question here, which I'm not going to try to answer right now, would consist in knowing what this moment is actually all about and whether the time out that it signals is simply a measurable duration, something that lasts for a brief moment before it's gone, or something that opens onto another time, another reality, onto a _depth_ and truth that might be allowed to spread and to encroach on a habit that's always looking to contain them. But does it even make sense to talk in these terms? Is it even within our power to leap into another time, to overcome habit and return, perhaps, to a time that would be other than present and alive? Might it be that the ability to do so is imposed on us from elsewhere with at least as much strength as the force of habit? From the very start of the novel, there's a familial struggle set up between the "anaesthetizing influence of habit" and "the immense edifice of memory."
Here's another example. In another bedroom — the setting here's the Grand Hotel at Balbec where the narrator stays for the first time with his grandmother — the protagonist wants to rest. "Feeling exhausted" after a long journey, he lies on a bed in which he already knows he won't be able to sleep and discovers a bedroom that he can't bring himself to call his "own." Filled with objects that don't know him and seem to be watching him warily as if he were an intruder, this room testifies to a routine, a habit that has no place for him. The clock gives the impression of mocking him "in some unfamiliar tongue" and "the tall violet curtains" seem to be shrugging "their shoulders to show they are irked by the mere sight of someone." The small glass-fronted bookcases, the tall cheval-glass, the high ceiling, all contribute to this discomfort, this sense of strangeness, if not downright hostility, that overwhelms him, giving him the urge to flee, even to die. In fact, this same bedroom at Balbec's mentioned at the very start of the book:
> the small bedroom with the very high ceiling, hollowed out in the form of a pyramid two storeys high [... ], where from the first second I had been mentally poisoned by the unfamiliar odour of the vetiver, convinced of the hostility of the violet curtains and the insolent indifference of the clock chattering loudly as though I were not there; [... ] where my mind, struggling for hours to dislodge itself, to stretch upwards so as to take the exact shape of the room and succeed in filling its gigantic funnel to the very top, had suffered many hard nights, while I lay at full length in my bed, my eyes lifted, my ear anxious, my nostril restive, my heart pounding...
That unknown and hostile bedroom, which feels like a country we're visiting for the first time and whose language we don't speak, stands in sharp contrast to the bedroom in Paris, so familiar and comforting, where objects are like "accessories of my own organs, extensions of myself," or with the Louis XVI bedroom (the one at Doncières, perhaps), "so gay that even the first night I had not been too unhappy there and where the slender columns that lightly supported the ceiling stood aside with such grace to show and reserve the place where the bed was." But there's nowhere, no new situation, so to speak, that habit won't eventually conquer and where we won't end up feeling at home. Habit changes the curtains, silences the clock, covers up the smell of vetiver and brings the ceiling down:
> Habit! — that skilful but very slow housekeeper who begins by letting our mind suffer for weeks in a temporary arrangement; but whom we are nevertheless very happy to find, for without habit and reduced to no more than its own resources, our mind would be powerless to make a lodging habitable.
Habitual memory is, as Bergson said before Proust, a bodily and useful form of memory, therefore, one geared towards practical action; it's a tool developed in order to solve the problems encountered by the living world and characterized by a superior capacity for adaptation. Ultimately, it's only meaningful within the problematic of life (which is really nothing other than the problematic of survival) and of evolution in particular. This is certainly Ravaisson's view in his short but highly influential doctoral dissertation, _On Habit_ ( _De l'Habitude_ ). Underlying the issue of habit is the issue of mechanism (and of the possibility of moving beyond it in the direction of spiritualism in a vitalist sense), along with, as a result, the question of the meaning of nature itself. In a long article devoted to his predecessor at the Collège de France, Bergson states that, behind its very modest title, _De l'Habitude_ provides us with a true philosophy of nature. To my mind, the same sort of naturalistic ambition's revived and achieved by Proust, albeit in his own way. Once adopted, the motor habit is a mechanism, namely,
> [a] series of movements which determine one another: it is that part of us which is inserted into nature and which coincides with nature; it is nature itself. Now, our inner experience shows us in habit an activity which has passed, by imperceptible degrees, from consciousness to unconsciousness and from will to automatism. Should we not then imagine nature, in this form, as an obscured consciousness and a dormant will? Habit thus gives us the living demonstration of this truth, that mechanism is not sufficient to itself: it is, so to speak, only the fossilised residue of a spiritual activity.
There's no point in dwelling here on whether Ravaisson's view of habit had any influence on Bergson. On the other hand, so far as Proust's concerned it's worth pointing out Ravaisson's emphasis on the involuntary, the unconscious and the automatic or, more accurately, his emphasis on the shift from voluntary movement to instinct. In his essay, Ravaisson stresses the temporal dimension of habit, developed as a result of a change that it outlives. As such, habit's not only a "state" but also a "disposition." What really generates habits isn't so much change understood as a modification of things but change understood as something that takes place over time. It's the _continuity_ or _repetition_ of the same change that generates habit. Unlike instincts, which are also tendencies, habits are "tendencies generated by the continuity or the repetition of the act." As I'll show, this repetition is very different from the repetition of involuntary memory, as Proust understands it. Facing a repetition of movement (a swinging or rocking motion, for example), the subject is passive. He or she is acted on. This passivity, however, is _itself_ a sort of activity and the mutual interplay between these two dimensions can't be reduced to simple mechanism or pure reflective freedom. Habit, as Ravaisson tells us, is what lies between nature, understood as pure mechanism, and will, understood as pure freedom, i.e. the "infinitesimal _differential_ ," moving towards each one without ever connecting fully, "from Will to Nature." It's an acquired nature, a second nature, as Aristotle already suggested, and its goal is to turn what were originally voluntary movements into instincts. In habit, Ravaisson tells us, we're dealing with a kind of "spontaneity" that's "at once passive and active," one whose activity is super-voluntary and distinct from action in the classical sense of the term.
> If, therefore, in the process of repetition, the motion increasingly changes into an involuntary motion, it is not in will but in the passive element of the motion itself that a secret activity gradually develops. It is not the action _per se_ which the continuity or the repetition of locomotion yields or fortifies; it is an ever more obscure and unreflected tendency, which drops increasingly further into the organism and becomes concentrated there ever more. Habit only exerts an indirect influence on the simple acts of will and intelligence, lowering the obstacles before them and subjecting the means to them.
The kind of repetition specific to habit develops an "unreflected spontaneity that penetrates and increasingly settles in the passivity of the organisation, outside, beneath the region of will, of personality, and consciousness." Without exactly belonging to the order of instincts — of which memory, as Ribot will say later, is the heir — habit is a form of memory that's bodily and unconscious while still being spontaneous.
***
The transition towards the second type of memory — voluntary — is easy to see. Will and intelligence are identified and criticized from the viewpoint of their capacity to restore the true past: "[f]or me," Proust says, "voluntary memory, which is above all a memory of the intellect and of the eyes, gives us only facets of the past that have no truth." Considering the nature of our relationship to the past, we might expect more from an analysis of habit-memory than from an analysis of voluntary memory. Which Marcel confirms at the beginning of the _Search_ , when evoking Combray:
> The fact is, I would have answered anyone who asked me that Combray also included other things and existed at other hours. But since what I recalled would have been supplied to me only by my voluntary memory, the memory of the intelligence, and since the information it gives about the past preserves nothing of it, I would never have had any desire to think about the rest of Combray.
To Taine and Ribot, the kind of memory in question is the one that goes "together with consciousness" and which only differs from the previous one on this count. It basically remains a physiological and biological memory: the roots and the origin of the psychological fact are wholly biological. The scholastics call this "recognition"; Ribot prefers to call it "localisation in time" or "vision in time," the equivalent of vision in space, made up of primitive perceptions (the colourful surface) and acquired perceptions (secondary data like direction, distance, or shape). The memory of recognition allows us to "locate" images in time by appearing to us "as part of a more or less extensive series leading to the present."
This sort of memory is a source of constant frustration for the young narrator: there's a pleasant experience, the special pleasure that's found in front of a hedge of hawthorns, for example, that yields some intensity and then there's the attempt to hold onto this sensation, to make it last. All in vain. Then there's the attempt to revive it, to produce an "image" of it. And so we picture it. But this picture's never more than a crude approximation. In terms that already make him sound like Proust, Taine evokes the spectacle of the previous day of the sky darkening in the setting sunlight on the embankment along the Arsenal:
> I saw this yesterday; and now as I write I see it again — dimly, it is true, but still I see it. The colours, forms, sounds, which struck me yesterday, are now renewed, or nearly so. Yesterday, I experienced sensations excited by the immediate contact of things and immediate action of the nerves. Today, impressions analogous to those sensations, thought remotely so, arise in me, notwithstanding the want of this action and contact, notwithstanding the presence of other actions and contacts. It is a semi-revival of my experience; different terms might be used to express it, we might call it an after-taste, an echo, a representation, a phantom, an _image_ of the primitive sensation; it matters little; all these comparisons mean no more than that after a sensation excited by the outer world, which resembles the sensation, and is accompanied, though not so forcibly, with the same emotions, which is pleasurable or the reverse, but in a less degree, and is followed by some, but not all, the same mental conclusions. The sensation repeats itself, though with less distinctness and force...
As we saw when we looked at the hawthorns earlier on, the immediacy of a sensation or an impression is what memory strives to retain or actually revive through the mediation of an image or a simulacrum. During another stroll around Guermantes, filled with sorrow at the realization that he'll never be able to write anything to match the poems and novels he admires, Marcel experiences a real and deep pleasure at the sight of a roof, the reflection of the sun on a stone, from the scent of a path. He naturally tries to make his pleasure last by staying where he is, allowing these delightful impressions to penetrate fully. In the lines I'm going to cite, he hasn't even asked about the origin of such pleasure and whether it's something that's attributable to the senses alone:
> But the moral duty imposed on me by the impressions I received from form, fragrance or colour was so arduous — to try to perceive what was concealed behind them — that I would soon look for excuses that would allow me to save myself from this effort and spare myself this fatigue.
This will come later and the answer will also turn out to be the answer to the question — to the very possibility — of writing: once this puzzle's solved he'll have solved the mystery of artistic creation as well. Right now, though, Marcel has to catch up with his grandfather. "[B]y closing [his] eyes" and by trying to "concentrate on recalling precisely the line of the roof, the shade of the stone" he tries to hold onto his first impressions and the pleasure that they produced. In the absence of the stimulus, memory tries to represent it: it emerges where the sensation vanishes, as if trying to make up for the lack of it. It is, as Taine says it is, the "voluntary revival" of sensations in an image. All sensations, he adds, "whatever... the nerve by whose action they are excited, have their images." This is how we can "mentally" hear a tune. But this image is never anything more than that, never anything more than the less intense, less precise and less rich image of an original, the substitute for a physical sensation, i.e. a nervous sensation triggered by present contact with things. The image is a derivative, spontaneous sensation generated from within and is always deficient as regards the original impression. Even young Marcel's description of his return to Combray in a horse-drawn carriage during which, sitting next to the coachman, he makes out the two steeples at Martinville — a rather successful account, as he himself admits — fails to reproduce the intense pleasure he experienced when he saw them and when he witnessed their transformation as the carriage travelled on. It's worth pointing out that the young Marcel begins by transcribing what he just saw without further ado and with uncontrollable excitement in the hope that he might be able to retain and forever hold onto the rapidly receding object of his emotion: writing itself is seen precisely _as_ memory, as _the_ way of retaining what has a natural tendency to fade away. Considering that Proust himself expends inordinate effort on the resurrection of a time past through what Taine would call "an extreme concentration of attention," a concentration that allows the narrator to evoke "the plenitude of details, the distinctness, the energy" of the sensations and emotions once experienced, the novel as a whole really does seem to follow the classical view and practice of writing as recollection and of narrative as the search for lost time. The same sort of effort of recollection's also evoked in _Against Sainte-Beuve_ :
> I recall on a journey one day struggling to extract impressions of the countryside passing before me from the window of the compartment. I wrote as I watched the little country graveyard go past, I noted bars of sunlight on the trees, the flowers along the way like those in _Le Lys dans la vallée_. Subsequently I tried often to evoke that day, by thinking again of those trees striped with light, and of the village graveyard, I mean the day _itself_ and not its lifeless ghost. I never achieved it and had given up hope of succeeding...
If Proust hadn't gone beyond this, however, he would have conceived of writing as no more than the repetition of an original impression, as the increasingly less intense and rich revival of a primary sensation. In keeping with the theory of writing and memory set out in the _Phaedrus_ , he would have understood writing only in terms of absence, either as the substitute for an original sensation that would attest to our place in the world, to the fact that we were here and _did_ belong here, despite the transience of that belonging, or as the supplement that memory needs if it's going to retain what it needs to. Proust's loyalty to this tradition, on which Taine's associationist psychology also implicitly rests, would have completely failed to advance either the metaphysics of the novel or its aesthetics. If the _Search_ hadn't gone beyond this and had only been the narrative of a life from childhood to maturity, of a writer attempting a journey of self-discovery and wasting his time, if it had been simply the narrative of this writer's gradual awakening to the world, to love, society, even to art, it wouldn't have involved anything particularly new outside of a certain stylistic distinctiveness, its dizzying length and the extraordinary wealth of its descriptions. Its only achievement would have been to have carried out a variation — already a pretty significant operation in itself — on the theme of recollection or the theme of the literary transcription of a lived experience. This is what Proust himself says quite explicitly in a letter to Jacques Rivière of 6 February 1914: "... if I simply sought to remember and, thanks to these memories, duplicate the real-life days, I would not, ill as I am, bother to write." Proust in fact becomes a writer precisely because he stops seeing writing as the revival of a lived experience (what in German is termed _Erlebnis_ ) and capturing what I'd call its eventuality, i.e. the aspect of this lived experience that's still likely to surprise us. By wresting the very work of writing away from the work of recollection and orienting it to the work of reminiscence, Proust breaks new ground. In doing so, he uncovers another meaning of experience, one that Benjamin refers to as _Erfahrung_ : the experience of what remains to be lived and what's still to come in any lived experience, the experience of what's unlived.
> [... ] but should a smell or a taste, met with again in quite different circumstances, reawaken the past in us, in spite of ourselves, we sense how different that past was from what we thought we had remembered, our voluntary memory having painted it, like a bad painter, in false colours.
This remark, one of Proust's most well known, should be enough to convey the importance that's conferred in the novel on the form of memory known as involuntary, less as a simple theme of the work and more as its very "matter." What still remains to be seen is just how it's put to work there.
By "involuntary memory" Proust understands something like those "unconscious remembrances" which, randomly triggered by some sensation, some spoken word or some encounter, arise from the very depths of our being and from the forgetting to which a wilful intelligence and a wilful memory, both oriented wholly towards practical action, would have condemned them. On such memories, Proust declares, rests his "whole theory of art," a theory that the final volume of the book reveals and which I'll address in the fourth chapter of this study. Proust's claim involves less the discovery of a psychological law and more a rule for art, a foundation for modern literature. Still, doesn't this suggest that Proust's aesthetics is ultimately based on the psychology of involuntary memory and that the novel is primarily a psychological one? It certainly seems so, as Proust himself admits:
> You know that there is plane geometry and solid geometry. Well, for me, the novel is not only plane psychology, but psychology in time. I have attempted to isolate the invisible substance of time...
What needs to be asked, though, is whether the kind of psychology that's at stake in the book is actually a transposed version of associationist psychology or whether the novel itself succeeds in revealing some aspects of human experience, some quality of the soul, inaccessible to physical or neurophysiological psychology. Whatever Proust himself might have said, we need to wonder whether, when it comes to aesthetics, psychology has the last word or whether what Proust calls unconscious remembrance reflects a view of the unconscious — and so of the subject — that can't be reduced to the notion of an "I" or of some personal lived experience.
For now, and so as to clarify the context specific to the period's interest in what was then called "affective memory," I simply speculate on the extent to which Proust's taste for this form of memory might have stemmed from considerations that aren't purely literary. As we'll see, while making good use of some of the theses from the psychology and the philosophy of his time, Proust doesn't just transcribe such theories into literary terms and towards artistic goals; rather, he extends them and expands their scope.
We know that Proust read Taine and that Taine's theses, which were widely adopted by Ribot and his school, featured heavily in philosophy textbooks and classes. Even though affective memory — a term that Proust adopts — seems first to have appeared in Ribot's writings, Taine should be credited as its rightful originator, in the French context at least. Opposing the intellectualist psychology epitomized by de Maine de Biran and largely in agreement with associationist psychology, an offspring of the Scottish school, Taine acknowledges the existence of a kind of memory which is neither the memory of sensations (the bodily memory of acquired automatic behaviour) nor the memory of intelligence (the voluntary memory that locates images in time), but the memory of feelings. This threefold distinction features at various stages in the novel. Don't forget, though, that for this school of thought, affective memory always originates in a sensation. It's always possible, therefore — at least in principle — to reduce the former to the latter: psychology's rooted in esthesiology, the scientific study of sensations, and esthesiology is rooted in turn in neurology and neurophysiology. While recognizing the existence and significance of affective memory, Taine and his successors continue to rely on Hume's empiricism. Later on, I'll look at whether the novel itself abides by this kind of materialism or whether it actually opens onto a very distinctive form of spiritualism. As the hawthorn episode shows, the narrator fails to locate the origin of his pleasure and wonderment in the sensation alone and, as a result, in the materiality of the object. Which raises two questions: how can we express this kind of memory and how can we explain it?
Taine's thought in general, and his theory of involuntary memory in particular, are developed in the context of the neurophysiology and psychophysiology of his time, which tend to negate the autonomy of the ego. This tendency leads him to see mental health as the always fragile outcome of an unstable relation between forces. Whence, most notably, the idea of the discontinuous ego, an idea that we know is going to play a major role in Proust, but which others also happen to develop. By acting as if what we refer to as _I_ or _me_ were a stable, autonomous substance,
> we allow ourselves to be duped by language; we forget that its permanence is apparent, that, if it appears fixed, it is because it is incessantly repeated, that it is in itself nothing more than an extract from internal events, that it derives from them its entire being, that this detached being, detached by fiction, isolated by the oblivion of its connections, is nothing in itself and apart.
The point here is to see that this so-called permanent substance is nothing outside its "events" (among which Taine includes sensations, external perceptions, memories, volitions, voluntary motions, etc.) and that its being is identical to its becoming. In Perrin's terms, this involves a demystification of the ego, a demystification that Proust will eventually carry out in his own way.
Such demystification results in a decentring of memory itself, shifting the latter towards the involuntary sphere: images come back to life on their own accord, following some "spontaneous revival" which can take place at very great distances. For Taine, memories are endowed with their own stamina and with an autonomy that's absolute and utterly independent from intelligence. Drawing on all sorts of personal examples, which are actually reminiscent of the passages from _Les Mémoires d'outre-tombe_ and _Les Filles du feu_ that I mentioned earlier, Taine shows how the strongest memories always originate in sensations and how the memory of feelings remains a memory of the body:
> On returning, after many years' absence, to one's father's house, or to one's native village, numbers of forgotten objects and facts unexpectedly reappear. The mind, suddenly thronged by their stirring crowd, resembles a box of dried rotifera, which have lain inert some ten years, but when sprinkled with water, at once revive and twist about [an image reminiscent of the "Japanese papers" in the _Search_ ]. We mount the dark staircase, we know where to find the handle of the door, we imagine ourselves seated at table in our accustomed place, we see the water-bottle on the right, and the salt-cellar on the left, we seem to taste the flavour of some Sunday dish. We look up at the wall and are surprised not to find there some old engraving we had stared at as children. We see the gesture and stoop of some former guest, the square body and long folds of a blue dress; we almost hear the tones of voices which have long been; we come to the well, and recall the vague terror with which, as children, mounting on tiptoe, we looked on the obscure depth and the trembling reflection of the cold water at what seemed to us an infinite distance below.
Following a long and winding road that leads us from the re-emergence of habits that settle at the bottom of the body ("we know where to find the handle of the door") and become covered up by so many others, to the re-emergence of long-forgotten images ("we see the water-bottle on the right, and the salt-cellar on the left," then "the gesture and stoop of some former guest") and sensations ("the flavour of some Sunday dish") and, at last, to the re-emergence of feelings ("the vague terror"), Taine wonderfully evokes the journey taken by involuntary memory and the richness of the time that re-emerges and is recaptured despite ourselves, intact as it were. Proust himself evokes the same sort of involuntary memories as "tremors" which "throw older layers back up to the surface": a sensible object in a world of sensible objects, our life is merged with a depth of sensations which are experienced upon interaction with them. One sensation, one object, gives way to another. The experience of the object and its matching sensation never actually disappear completely, however. True, they're no longer here and no longer now, but for all that they remain stored at the bottom of our consciousness, layered according to a process of sedimentation. They're there, all together and yet totally distinct, in a space that can't be accessed by our intelligence, ready to rise back up again, to re-emerge and come to the surface of consciousness, ready to enrich our present experiences with the past with which they coincide but from which they're still distinct. They're there, like the trace of that immemorial past, always linked with objects or a series of objects, rumbling beneath the surface. Most of them remain buried, absolutely past, and the objects to which they're connected are forgotten: "That's because the objects in question are so small, so lost in the world, and the odds of stumbling upon them are so low!" But sometimes something exceptional occurs and it's in these exceptions that the very possibility of art is born.
Ribot's to be credited with exploring further the mechanism of affective memory, which he always understands in terms of the laws of association. The images arising from our past do become associated within us, either through resemblance or through contiguity. In the case of involuntary and affective memories, Ribot claims, the association's one of resemblance. Contiguity, on the other hand, is more like the body's automatic memory. The same law applies to voluntary memory, which locates some past image in its corresponding series and recreates the sequence of the series itself. When it comes to affective memory, however, things are very different. Making use of already established terms, Ribot understands affective memory in terms of the category of "displacement" or "transference," now applied to feelings. Under its most general form, Ribot claims in the _Psychology of Feelings (Psychologie des sentiments_ ), "the law of transference consists in the direct attribution of a feeling to an object which is not itself the cause of the feeling." There are two possible ways in which this can happen. The first way, referred to by Ribot as "transference via contiguity," happens when, from among the various intellectual states that coexisted and that entered, by way of contiguity, into a series of relations, any one arouses the same feeling as another. In such cases, the lover transfers the feeling first induced, say, by his mistress onto her clothes, her furniture, her house. Which leads Ribot to an example taken from Herbert Spencer: the call of the crow isn't a particularly pleasant sound in itself; quite the opposite, in fact. At the same time, though, the caw can produce pleasant impressions if it's associated with picking wild flowers in childhood, days out in the country during the summer and so on. Now such a sound, even though it's not causally linked to all these past pleasures, numerous and diverse as they are, awakens — simply because it's associated with them — some obscure awareness of these pleasures. The second type of transference which, to my mind, is much closer to the way Proust sees involuntary memory working, follows the other principle of association, namely resemblance. It's an emotion, then, a past affective situation that leads to the association of two very different realities, a colour and a feeling of well-being, for example, or a sound and a sorrow. Ribot here refers to another associationist psychologist, Théodore Flournoy:
> By affective association, I mean the kind that takes place between two representations, not because of some qualitative resemblance (they can be disparate such as a sound and a colour) or on account of their regular or frequent encounters in consciousness, but as a result of an analogy in their emotional character. Each sensation or perception indeed possesses, besides its objective quality or its intellectual content, a sort of subjective ratio, stemming from the roots it lays down in our being and from the very distinctive way in which it impresses us, appeals to us or repels us, excites us or soothes us, in short the way in which it shakes us all up... Two sensations that are radically heterogeneous and incomparable in terms of their objective content, such as the colour and the sound _i_ , can be envisioned as comparable and as resembling each other more or less through the impact they have on the body; and at the same time, one can envision that this emotional factor might become some hyphen between them, some link of association thanks to which one will awaken the other.
It's precisely this phenomenon on which the psychologists dwell at such length that Proust sees at work in Chateaubriand, for instance, or in Nerval:
> When Chateaubriand is — if I remember aright — at Montboisier, he all of a sudden hears a thrush singing. And this song, which he had so often listened to in his youth, at once takes him back to Combourg, prompting him to change, and make the reader change with him, both the time and the province.
Similarly, he adds,
> the first part of _Sylvie_ takes place in front of a stage and describes Gérard de Nerval's love for an actress. All of a sudden his gaze falls on an announcement: "Tomorrow the archers of Loisy", _etc._ These words evoke a memory, or rather two childhood loves: the scene of the novella at once shifts.
What seems to hold Proust's attention, from a literary perspective at least, is the capacity of these kind of memories to alter the time and place of the narrative quite unexpectedly, to transpose narrator and reader alike from one space-time to another, without any transition. But isn't this power of transposition the same as what we call metaphor in literature? Later on, I'll need to look at whether there's not some phenomenon, rooted in unconscious remembrance as well as in metaphor, that unites the two. I'll need to consider the particularly deep rooted connection between memory and writing.
It seems difficult _not_ to connect the examples that Proust draws from literature with the infinite despair and sorrow that overwhelms Swann when he listens to Vinteuil's septet at the Marquise de Saint-Euverte's. He heard the septet's sonata for the first time at Madame Verdurin's and its melody drifts through the early days of his love affair with Odette, like a blessing. That affair's now crushed, wrecked on the rock of Odette's indifference and cruelty and, at that very moment, as he listens to this beautiful and cruel music, Swann understands that it's gone forever. By reminding him of the joy he once knew, Vinteuil's music makes his suffering unbearable when, until then, it was something that could be explained away by Odette's mere absence:
> But suddenly it was as though she had appeared in the room, and this apparition caused him such harrowing pain that he had to put his hand on his heart. What had happened was that the violin had risen to a series of high notes on which it lingered as though waiting for something, holding on to them in a prolonged expectancy, in the exaltation of already seeing the object of its expectation approaching, and with a desperate effort to try to endure until it arrived, to welcome it before expiring, to keep the way open for it another moment with a last bit of strength so that it could come through, as one holds up a trap-door that would otherwise fall back. And before Swann had time to understand, and say to himself: "It's the little phrase from the sonata by Vinteuil; don't listen!" all his memories of the time when Odette was in love with him, which he had managed until now to keep out of sight in the deepest part of himself, deceived by this sudden beam of light from the time of love which they believed had returned, had awoken and flown swiftly back up to sing madly to him, with no pity for his present misfortune, the forgotten refrains of happiness.
With Taine, Ribot and others, memory had already begun what one might call its metamorphosis. From a conscious and voluntary form of memory, i.e. the faculty of (re)cognition of a centred subject in control of his or her own house, a gradual shift has given way to a spontaneous, involuntary and affective form of memory, one that would be "truer" than the one it's just replaced. However intermittent, it still remains the key to getting at the truth of the ego, a truth that doesn't necessarily imply its permanence. With Proust, as we'll see, this phenomenon is taken even further: from a psychological fact, it's going to become an ontological problem, if not the cornerstone of a new aesthetics.
Chapter 3
* * *
Finding joy (involuntary memory)
* * *
The hawthorns already pointed to the "presence" of a reality beyond merely physical presence, beyond their raw materiality, whilst at the same time pointing to something that unfolds right before the narrator's eyes, as it were, right there in the stems, in the petals, the fragrance. Somehow this experience announced deeper ones in which the transubstantiation of matter and its corresponding joy happen to be even more intense and even more manifest. The taste of a madeleine dipped in tea, the way the uneven paving-stones feel underfoot in the courtyard of the Hôtel de Guermantes, the tapping of a knife against a plate, a napkin's dry stiffness: what makes each one of these distinctive is the way in which it can arouse something in us (or, of course, in the narrator). And this is why they're even more significant than the encounter with the hawthorns: Proust's discovery, however much a commonplace it might be today, is that a sensation can give rise to the very real experience of our own temporality and that matter can then transcend itself and morph into some immaterial reality, bringing back to life, with unrivalled purity and clarity, a world that we thought we'd lost forever. In order for something so insignificant as a madeleine or a paving-stone to reawaken a past in us, that past must have been latent within us, intact and completely forgotten, like those amazing cities that are sometimes discovered after hundreds of years, frozen in time beneath the vegetation that both covered and protected them. But if the reawakening of this past depends on objects as contingent as a madeleine, a paving-stone or a spoon and not on us, then what we're dealing with here must be a discovery far more profound than the one that says that the most precious part of our lives is made up of a myriad of little things, little specks of matter or tiny fragments of reality that live outside us, not within. Such a part (precious because it's preserved in its entirety despite the passage of time that carries us inexorably towards death) is one about which we have no real idea and which we can only intuit through that sensible shock that comes to us from outside, in this involuntary memory. It lies in what was forgotten and discarded from the outset, in what always escapes our attention or our perception and so which never gets used by consciousness. Far from divesting me of myself like a depersonalizing memory would, involuntary memories actually bring me back to myself. What they restore isn't so much my past as such, my lived past, but a past hitherto unknown and which, in that instant, I experience as if for the first time: time in its pure state.
It's not the past that involuntary memories bring back to me, then, but that part of the past that hasn't yet passed. And it's not yet passed because it's never been present, at least in the sense that we'd ordinarily understand it. Isn't that what we mean with the word "essence"? Essence, in Proust's rather nice turn of phrase, is "the part which, after all our tears seem to have dried, can make us weep again." It seems, then, that we ought to infer the existence of another time or, rather, the insistence of time against the grain of its own unfolding. To chronological time, along which our existence unrolls and which moves only forward, carrying us inexorably towards our death, we ought at least to add — if not to oppose — a time that curls up, that records and holds onto every detail of our existence, every fragment of our experience, hoarding their presence in a virtual compartment of our memory. To the time of existence, to the time of death, the time that runs and slides through the world, we ought to juxtapose the time of insistence, which weighs on it, and alongside the time of erosion and decay we'd need to set the time of sedimentation. For while there _is_ a time that slides, there's another that dives. While there _is_ a time of surfaces, there's also a time of depths. In short, Kronos has to recognize Mnemosyne. Memory isn't the time of life, neither the time of what's alive nor lived time; it's not conscious memory, the sort of memory that retains and recollects. Rather, it's the one that simply returns. The time of life is still Kronos. The time of memory, by contrast, is that of unlived experience. Such is, to my mind, Proust's most radical discovery: the fact that time is always divided in two, into present and past, just as life's always divided into lived and unlived experience. And it's the latter, this unlived experience, that's the concern of literature.
It's in this sense, then, that Mnemosyne, Kronos' sister, is the mother of all muses, ruling over the very notion of poetry. For both Proust and Hesiod, anamnesis yields a sort of transformation of temporal experience: this is how it allows us to escape from the present and its usual round of weariness, misery and anxiety. On this basis, the drama of chronological time — which Goya depicts so clearly in the image of Saturn (Kronos) devouring his children and which haunts the narrator as much as the writing of the novel itself — becomes just a little less horrific once we know that time comprises more, infinitely more, than what's _lived_ about time and once we know as well that the time of existence never exhausts that of essence. The great Proustian lesson as regards time is the lesson of time's split or doubling.
***
"Where could it have come to me from — this powerful joy?" Marcel asks after dipping his piece of madeleine into his spoon of tea. The answer to his question comes, as we know, only at the end of the novel and it's a one-word answer: time, and especially that time that, because we misconceived it, we thought we'd lost forever, namely the past.
Let's leave this idea aside for the moment, though, and focus on what Proust says about the joyous sensation he has when he tastes the madeleine: "I sensed it was connected to the taste of the tea and the cake, but that it went infinitely far beyond it, could not be of the same nature." The experience of the madeleine is a lot like the experience of the hawthorn which, you'll remember, also exceeded the scope of a purely physiological interpretation (the one that empirical psychology might want to maintain, for example) and owed its intensity to the intuition we have of some agreement between our faculty of feeling and an end in nature posited by our faculty of knowledge. In this case, however, it's the experience of time that's the decisive factor and constitutes the real object of the feeling of pleasure. In this regard, then, Proust's analysis is both an extension of Kant's interpretation of the judgment of taste as well as an innovation woven around it. There's a feeling of pleasure that "infinitely" exceeds the simple taste of the madeleine, the mere materiality of the object, pointing thereby to the existence of a spiritual reality at the heart of the material; while going infinitely beyond the taste of a cup of tea, the feeling still needs its material support. The material object of the experience isn't really the source of pleasure. What's more, what it refers to, namely the analogous sensation experienced in the past, isn't itself especially pleasant and didn't involve much in the way of pleasure then. It's only the repetition of a sensation experienced in the past that entails this happiness. It seems fair to wonder, however, how the repetition of some sensation that initially left us indifferent could ever provide us with the opportunity for such joy. Such sensations can be useful, of course, as evidenced by the benefits of habit and the mechanical behaviour specific to it. And it can leave us indifferent and even bore us (and we've already seen how boredom's the horizon of repetition for Swann or for Marcel), especially if it's ordinary to begin with. What's repeated, then, isn't repeated _identically_ and what returns in the return of the sensation has to be more than the sensation itself (or perhaps even something else entirely). A past — _whatever it may be_ — can be the reason for our happiness, provided it returns differently. In fact, if it initially involves an intense sensation, its identical repetition will never be as intense because of its distance from the source. And if it involves sadness or some feeling of relative indifference, its identical return won't ever be miraculously reversed into its opposite. So the nature of this unique experience won't be uncovered through its repetition, by taking another sip of tea or by managing to trip again on the paving-stones, something the young Marcel eventually realizes as he's smelling the hawthorns:
> I continued, even at the risk of making myself the laughing-stock of the crowd of chauffeurs, to stagger, as I had done a moment before, one foot on the raised paving-stone, the other foot on the lower one. Each time I simply repeated the _outward form_ of this movement, nothing helpful occurred...
This wholly voluntary and material sort of repetition — let's call it identical repetition — won't help anything.
The joy that's experienced — re-experienced might be a better way of putting it — is a function of a contemplative dimension that's free from the restraints of perception; at the same time, though, the object of this contemplation is always an embodied one. As a result, even though it's always linked to a sensation, no "madeleine experience" is going to owe its delights to the delights of the sensation itself. Thinking back to the psychologists of the previous chapter, we might even want to consider the possibility of a profound and tangible joy that's not directly linked to a pleasant sensation. After all, putting our foot on a paving-stone that's a touch lower than the previous one, hearing the sound of a spoon on a plate or feeling the starched stiffness of a napkin doesn't generally involve a feeling of "bliss." In _Finding Time Again_ , however, such events really do occasion bliss and "elation." Determined this time — about time, an impatient reader might think — not to let the mystery of this escape him, the narrator does all that he can to locate its origin. And, as chance (or providence) would have it:
> I wiped my mouth with the napkin [the butler] had given me; but immediately, like the character in the _Arabian Nights_ who, without knowing it, performs precisely the ritual which makes appear, visible to himself alone, a docile genie ready to take him far away, a new vision of azure passed in front of my eyes; but it was pure and saline, and billowed into a bluish, bosomy swell; the impression was so strong that the moment I was reliving seemed actually to be the present; [... ] I thought that the servant had just opened the window on to the beach and that everything was inviting me to go down and stroll along the sea-front at high tide; the napkin which I had taken to wipe my mouth had exactly the same stiffness and the same degree of starch as the one with which I had had so much trouble drying myself in front of the window, the first day after my arrival in Balbec, and, now in this library in the Guermantes' _hôtel_ , it displayed, spread across its folds and creases, the plumage of an ocean green and blue as a peacock's tail. And _it was not just these colours which filled me with joy_ , but a whole moment of my life which aroused them, which had probably been an aspiration towards them, which some sense of fatigue or of sadness had perhaps prevented me from enjoying at Balbec, and which now, freed of _whatever was imperfect in the external perception, pure and disembodied, filled me with delight_.
The lesson here — a lesson that the hawthorns already hinted at — is this: the experience of joy — of that higher form of joy, the kind that's synonymous with delight, that makes us indifferent to death — doesn't come from colours, smells and flavours as such, from objects _qua_ material objects; rather, it comes from that part of those objects which exceeds this materiality and does so from the very heart of the material itself. And here's where Proust's account differs from that of the psychologists, who still think of experience as physiological data. As long as Marcel insists on seeing _material_ joy as the means to happiness and on the present as the condition of his joy, he's bound to be disappointed. What the experiences of involuntary memory reveal is the — non-temporal or at least not present — truth that can be drawn from any perception. Perception, though, insofar as it's immersed in the here and now, isn't ever going to be able to do this. The tendency to see anything that's not present as past is contingent on the fact that, for us and for the most part, the present is what's essential about time; as a result, we tend to mistake the fact that a given expression might have been forgotten simply because it no longer had any role to play in the present situation with the fact that its corresponding reality has now vanished.
This is how Proust's able to evoke what he calls the "general laws of remembering," laws that
> are themselves subject to the more general laws of habit. Habit weakens all things; but the things which are best at reminding us of a person are those which, because they were insignificant [i.e. useless], we have forgotten and which have therefore lost none of their power. Which is why the greater part of our memory exists outside us, in a dampish breeze, in the musty air of a bedroom or the smell of autumn's first fires, things through which we can retrieve any part of us that the reasoning mind, _having no use for it_ , disdained, the last vestige of the past, the best of it, the part which, after all our tears seem to have dried, can make us weep again.
Underneath habit, and forgotten by it since it served no purpose, impressions lie unaltered, tucked away but always ready to resurface: a rainy breeze, the smell of mould or smoke, the taste of a cake dipped in a cup of tea. Habit's selective; it leaves all sorts of impressions, sensations and perceptions aside, on the edge of consciousness which, through its intelligence, strives to find its place within the world. So it's selective, but it doesn't erase such perceptions. True, they might not be incorporated, but they're not simply eradicated either. They're kept somewhere, like a virtual presence:
> The broad daylight of habitual memory gradually fades our images of the past, wears them away until nothing is left of them and the past becomes irrecoverable. Or rather, it would be irrecoverable, were it not that a few words [... ] had been carefully put away and forgotten [ _enfermés dans l'oubli_ ], much as a copy of a book is deposited in the Bibliothèque nationale against the day when it may become unobtainable.
Surprising? Well, yes. Not the statements concerning habitual memory, of course, but those concerning the forgetting into which images of the past are "carefully put away." We might have expected Proust to appeal to the other form of memory, involuntary memory, but instead he appeals to this notion of forgetting, not in negative terms, but in the terms that are usually ascribed to memory itself: forgetting isn't just what erases and destroys, but it's also what maintains and preserves, protects and safeguards. Images from the past don't take refuge in memory and, even less so, in consciousness; rather, they end up in the unconscious. "Forgetting" something's the exact opposite of obliterating it — or, rather, it's better understood as obliterating an image from the _surface_ of our being, from what we'd normally call consciousness and to which actual (organic and habitual) memory's limited, and letting it sink down beneath the surface where its transformation can take place and from which it might, one day, re-emerge. This form of forgetting's reminiscent of the Greek _lethe_ : far from being what it'll later become, namely a destructive force that separates us from the past, forgetting here is that place into which temporal beings like ourselves can withdraw and take refuge. It's a withdrawal, however, that's never final; instead, it's the very condition for the return — the virtual return, at least — of what had withdrawn. And if time really is the meaning of being and the truth of the world, forgetting, by contrast, is the depths in which such truth is prepared. Far from being opposed to true memory, then, to unconscious recollection, forgetting is its dark side or its inner lining. With this, Proust reconnects with another dimension of the Greeks' archaic and mythical view of time. In an article I mentioned earlier, Vernant looks at the cult of Lethe that sprang up around the oracle of Trophonios at Lebadeia. At the site of this oracle, in Trophonios' cave, there was said to be an entrance to Hades itself. And remember, too, that before entering, the person consulting the oracle was taken to two springs, Lethe and Mnemosyne. Drinking from the first, Vernant explains, the person would forget everything about his or her human life and, like a dead person, would enter the domain of Night. Drink from the second, however, and you hold onto the memory of everything seen and heard in the underworld. You had to forget in order to be able to drink from the spring of that other time, then, in order to experience a temporal mode essentially different from the existential world: the world that the archaic mind associated with the beyond eventually became the world of the atemporal and of eternal divinity for Plato. Proust will view it in the same way, but again with the significant difference that going back in time is completely unrelated to the exercise of recollection: the time that constitutes the flip side of existence isn't transcendence but insistence, namely that part of the past that doesn't pass away. Besides, it's only with Plato that forgetting comes to be seen in a negative light, as the basic flaw of the soul, its disease — ignorance, precisely — that anamnesis is designed to fix. With Plato, memory becomes the very faculty of knowing.
In the waters of Lethe, at least according to the way the myth's presented in the _Republic_ (621a), souls lose the memory of eternal truths on which they gazed before falling back to earth and which anamnesis is supposed to allow them to recapture, bringing them back to their true nature. Knowing, then, is nothing other than remembering and reverting to the immutable and eternal world of ideas ( _Meno_ , 81c–d). And for the Pythagoreans as well, forgetting's closely linked to our mortal condition, hence the emphasis on "memory exercises." Time as something fleeting and ungraspable, made up of an indefinite succession of endlessly repeated cycles, time as the time of suffering and death, is finally brought to a close by the methodical recollection of the day's events, of an entire life, and from one life to the next. The exaltation of memory is aimed at leaving the time of existence — the time of forgetting — and returning to the divine, to a time that doesn't age, to the immortal and imperishable time celebrated in the Orphic verses under the name _Chronos agèraos_. In Proust, it's reminiscence, rather than recollection, that prompts this departure, this removal from destructive time that returns us to a time that's at once more impersonal and more true. It is, however, a removal that comes to us from the world itself, referring us back to the world as well as to its condition. Eternity withdraws into forgetting, therefore, and wraps itself in the world. More than a departure, it's a matter of plunging into the world and, through that, transforming it. This is how Proust avoids idealism: although he sees memory in the same way as Plato, namely as the faculty of knowing, it's still not an internal faculty and the truth that it uncovers is not one available to a method; perhaps it's not even a faculty at all, therefore. Knowing might well be remembering and remembering might well be recognizing, just as Plato thought, but in a sense entirely different to the one articulated in the _Meno_. There, you'll remember, Socrates shows how knowing involves the overcoming of forgetting, something he calls recollection. Truth's innate, therefore, and its forgetting is a consequence of purely contingent factors. I can recognize the truth (Descartes will call this "evidence") only insofar as I already know it. Between knowledge that's innate but forgotten or simply covered up and actually grasping that knowledge, actually recognizing it, there'd be no obstacle, no separation, that the right method couldn't overcome, even in the case of uneducated slaves. Ideas lie there within our soul, and all it needs is someone like Socrates to lead them to the surface. Proust's entire project, as Deleuze consistently stresses, is aimed at locating ideas not in myself but in the world; or, more accurately, I think, in myself as a being in the world, that is, myself as essentially _not_ independent of the world's being, of its contingency and its sensible exteriority. Thought, truth, even memory all come to us from outside, as it were, surprising us and transporting us in what amounts to a learning process rather than one of recognition. There's a degree of violence to memory, a shock that forces us to think. There's nothing natural about thinking, knowing or remembering, nothing presupposing an effort, the exercise of a faculty, nothing likely to take place simply because some method's been used. Using his notion of _involuntary_ memory, Proust aims to show the experience of a truth which, in returning, happens as if for the first time and which comes from the world itself. What reminiscence — as opposed to recollection — reveals is the creative power of time: ideas, even thought itself emerges from an encounter, a chance, an event. The world of essences isn't constituted from the start and then gradually discovered or recollected, as the _Meno_ suggests (86b). Rather, it's something that's always in the process of being constituted. By the same token, truth isn't there from the start, ready made, as it were, but always in the making. Besides, the time that we reach with Proust isn't the cyclical time of reincarnation, the time of the bad return that memory would restore by bringing it to a close. Rather, it's the time that doubles this time and doubles it from the start; it's a fold of the real in which we're curled and whose unfolding is something that we can think of as experience, even as truth.
What's important, though, is for us to think of forgetting not as the result of our imperfection as human beings, but as the condition for accessing the experience of a time that's other than the time that passes and erodes. Genuine memory's not what's retained (this is how we tend to understand it today, and it's even a view that phenomenology pushes on us) but what returns; it's not what unfolds and spreads out on the surface, but what periodically resurfaces until it breaks through. And any such return presupposes forgetting.
We need, then, to make a distinction between two forms of forgetting. First, there's forgetting linked to the passage of time, forgetting that happens as a result of time having passed: time gets rid of certain impressions in order to make room for others that are useful in more immediate ways. Second, there's forgetting as the very passage of time itself. Lost time, the time that the novel goes in search of, definitely relates to the former and implies an effort of memory, of reconstruction on the part of the narrator. But time regained is completely different from time lost: it's regained only by chance, through involuntary memory. And since more's regained in it than was ever experienced, forgetting is somehow its condition of possibility:
> Yes, if the memory, thanks to oblivion, has not been able to make a single connection, to throw up a single link between it and the present moment, if it stayed in its place, at its date, if it has kept its distance, its isolation in the depths of a valley or at the very peak of a summit, it suddenly makes us breathe a new air, new precisely because it is an air we have breathed before...
Time regained is a virtual state of time, a past that's not actually been experienced yet, preserved in forgetting and reawakened by involuntary memory. The time that's found again isn't exactly the time that was lost: by finding it again we find that lost time, which is lost as if for the first time; but we also find more than that time, since we experience it through a kind of shortcut, condensed and intense. Time regained isn't just past time, therefore, lived time, but it's also and above all time in its pure state which, through the juxtaposition or, more accurately, through the collision of a present moment and a lived past, is elevated above lived time. Time in its pure state is time freed from the law of Kronos, time insisting beneath the time that passes (or exists). It's the sort of time that digs tunnels in order to connect through a sort of paradoxical contiguity events that Kronos works to drive apart (and, as we'll see later on, this is what, in the realm of art, we call metaphor). And if true time really is this kind of time, namely the kind of time that we can only ever find again and that we can only ever find again involuntarily, then it presupposes a sort of forgetting that's not strictly negative but is, rather, the condition for its happening: forgetting's needed so that time in its pure state can come about; total recall would mean the impossibility of time's return in its difference, that is, in its eternity. By the same token, the memory of the time we're looking at right now isn't the memory of what was, some archival form of memory, but the memory of what could have been or, better put — insofar as it's a matter not of the conditional but of what's held back in the present itself, not as another present but as its eventuality — the memory of a virtual event or the virtual memory of the event. In other words, forgetting and lost time are the condition _sine qua non_ for the happening of time in its pure state. On this basis, there's no need for us to fret about the passage of time: every past corresponds to something that can return and return in force, not as it once was or even as it could have been, but as it is, in some sort of a-temporality or _wholly other_ temporality: what returns is the virtuality of the past and not its actuality. In other words, alongside the time that flows or perhaps wrapped around it there's a time that plunges, a time of depths that can surface at any moment. What emerges in involuntary memory is a time that, although different from lost time, still presupposes it. It's this process of forgetting time that allows it to return, but the return in question's rather peculiar in the sense that what returns isn't the same as what was lost. But doesn't hitching the possibility of an experience of time in its pure state to some kind of time that's itself impure amount to contaminating the purity of eternity itself? What's clear is that these moments of eternity only take place in the context of chronological time, banning any _direct_ access to time in its pure state establishing some kind of dialectic between the two forms of time.
So images of the past never disappear entirely. Forgotten, covered over by habitual memory, buried under the sediments of a watchful consciousness, they carry within themselves an unprecedented power to bounce back, a power of becoming, of eventfulness. Which is exactly what Marcel sees after his first experience of involuntary memory at Combray:
> But, when nothing subsists of an old past, after the death of people, after the destruction of things, alone, frailer but more enduring, more immaterial, more persistent, more faithful, smell and taste remain for a long time, like souls, remembering, waiting, hoping, on the ruin of all the rest, bearing without giving way, on their almost impalpable droplet, the immense edifice of memory.
Carefully locked and preserved in forgetting — and who knows if they'll ever get out — such images of the past can still arise at any time. And if they do, they do so with the speed and violence of a volcano on the surface of consciousness, a surface that they break through, submerging and exceeding the narrow field of organic memory. If they return, or they happen again, as it were, it's not in the same way that they happened in the first place. Maybe they originally happened in relation to a perception, were combined with a habitual memory and a potential action. Regardless, such images only return by chance, when there's nothing left to achieve. Only chance, whether through an equivalent impression or through a sensation that's in some way similar to the past impression, can awaken the past, not as it was (or as it was perceived to be), but as it might have been and as it _is_ now. This "is," though, isn't the "is" of the present, of present impressions or perceptions. Rather, it's the "is" of the past that returns and _insists_ while the present, by contrast, endlessly dissipates and passes away in the latest present of _existence_. Once a spur to action, the images of the past have become a spur to contemplation as a result of their having been forgotten. Their return signals a metamorphosis of the real itself, its transformation: from an image tied to a material situation and a body acting on the world, it's become an image of an image, immaterial and immemorial. And this is why it involves art (and specifically metaphor, as I'll try to show later on) and not science: it's a spiritual object, an essence.
If my relation to the world is principally a perceptual one — often helped by my memory which, as Bergson suggests, selects the images that are most likely to be useful to it — this doesn't mean that the two are the same and that my experience of time leads to the connection and incorporation of perceptions according to my actions. As well as perceptions and alongside them, there are contemplations that come to me from inactive time. The mistake is to believe in the passing of time, to think that there's only room for the present. Provided there's _some_ past, the past isn't a present that has stopped being a present, but a past that's past _from the outset_ and that'll return as such. In reality, time's never lost for the simple reason that it never _ended_. Put another way: there is something that passes in time (this lived present that's replaced by another one while still being retained by it); but there's also something that doesn't pass, something that remains, not as present but as a power of becoming, as what can return, surprising consciousness and exceeding the reduced scope of our perception. Time's split from the outset, then (into linear time and sedimented time, into incorporated and incorporeal time), the present fractured (into the lived and the unlived present). What returns in involuntary memories is the virtual part of our being, that part that the real has, unknown to us, set down deep inside us. The extent of this virtuality's unimaginable and unfathomable. It's bound to surprise us and bring us back to ourselves, beyond the control that we have over the world and beyond this set of experiences that we talk about as if they were our own. It's the most valuable part of ourselves, the part that's always exceeded the narrow scope of our practical and utilitarian relation to the world — in which seeing is perceiving and perceiving is acting — and that's plunged us into a reality that's at once more personal and more distinctive, one in which seeing is tantamount to dreaming. Our joy wasn't tied to perception but to memory, not the senses but essence.
This should help us understand what Proust means by the infinite surpassing of perception. In fact, it refers to perception's own overcoming, to the capacity of perception and of the thing to which it's tied to expand and to disengage from the immediate present in which it's usually immersed. But putting it like this still makes things sound too intentional: it seems to me that the originality of Proust's position comes from his seeing that while perception's usually active and entirely focused on whatever task it's involved with, it can sometimes find itself faced with a passivity that interrupts it; this passivity doesn't come from another, older perception but from another dynamic or temporal logic. The passivity in the face of which the present of perception relaxes and renounces action, switching into a force of contemplation, couldn't be that of another present. The passive "force" that submerges the present and takes its place is, in fact, the past. This past, though, doesn't draw its force from the fact that it used to be present, from the fact that it was once a perception and tied to something: the paving stones at Saint-Marc that return in the courtyard of the Hôtel de Guermantes, the row of trees that returns with the sound of the spoon, all of Combray and Balbec that return through the moist madeleine and the stiff napkin, these things had gone unnoticed so to speak. This past derived its strength solely from the fact that, doubling each perception, by producing a virtual and wholly past image, it constituted, behind our backs as it were, this extraordinary power of becoming. In other words, we never know what the past will be made of or what surprises it might have in store for us. Its return is a first since what returns is the immaterial (and eternal) part of the present and what causes our joy is the experience of a reality _in itself_ or an _essence:_ some concentrate of Combray or Balbec, childhood or adolescence of the purest kind. Marcel's joy consists precisely in _being_ this pure past, this moment that doesn't pass. Benjamin's right when he says that Proust's eternity is a "convoluted" one, one that's curled tight in the present.
Before going any further, I want to draw a distinction between what I'm calling the "present moment" and the notion of the "instant," between the time of existence and the time of essence (or of insistence). The moment disappears, replaced by another. The instant, by contrast, endures beyond this. The present is the immediate to which I'm chained, the habit to which my body's tethered like the dog to his vomit, as we saw Beckett put it. The past, however, is what emerges from behind it, what plunges into the depths of our being as the present moves on. Memory doesn't just go back in time and writing doesn't just run around in search of lost time. Proust's novel isn't oriented towards the past in the way that a nostalgic novel would be. By returning the past makes something else and something more happen: it creates. Kronos procreates, yes, but he also devours his offspring. Unlike Mnemosyne, however, who rescues something from each existence and gives birth to the muses. Time's an artist, then, even providing art with its model. The narrator's joy comes from seeing that, while subject to the Saturnian time that effaces, destroys and kills the reality it creates — a truly monstrous form of infanticide and one that, despite the best efforts of human memory, whether natural or technical (writing, printing, photography, cinema, computer science), despite the whole culture, morality, religion and politics of memory, remains irreversible — he's also subject to this other time that's created somehow behind the back of the first and at the same time as it, which it doubles and carries elsewhere, traversing it, interrupting it and turning man into a god.
Such would be, then, the paradoxes of joy:
1. Joy comes not from the senses but from essence.
2. Joy comes not from perception but from recollection.
3. Joy only ever happens the second time around.
Present paradises are impossible since a paradise is eternal and the present is ephemeral. As for paradises yet to come, they're an illusion insofar as they stem from some structural dissatisfaction and from the workings of our imagination. When the past returns, however, it delivers the past in its pure state and we can say: that was paradise! Such paradises aren't lost forever, therefore. In fact, they're only lost insofar as they're found again and they're paradises only insofar as they are found again.
The past found again by involuntary memory isn't a lived present, therefore. Most often, memory brings the past back to life and the pleasure we might get from remembering comes from the fact that, as in a film, we can linger on any particular image before resuming the film of our life. But the discovery of involuntary memories involves an even more unsettling discovery, namely that lived time does not exhaust the experience of time. Experience itself, if not in the broadest sense of the term then at least in its most decisive sense, can't be reduced to lived experience. There is, then, the possibility that I can uncover something broader and "older" than myself as a collection of lived experiences. This is what Taine or Ribot's empirical psychology — even what Husserl's transcendental psychology — failed to grasp. Deleuze puts it like this:
> Of course, there is an objective aspect of the phenomenon, for example, the flavor of the madeleine as the quality common to two moments. There is also a subjective aspect: the associative chain that links to this flavor all of Combray as it was actually experienced. But if the resonance has both objective and subjective conditions, what it produces is of an altogether different nature: the Essence, the spiritual Equivalent, the Combray that was never seen and that breaks with the subjective chain.
When it comes to this notion of essence, then, it's no longer the subject's point of view that determines the value of the phenomena; rather, we're dealing now with a unique point of view that has to be specific to the object and in which the subject is captured as well: what drifts up from the teacup is a Combray that's not been lived up until now. And as for the narrator, he seems more of an effect or an expression than anything else.
This is the substance of Benjamin's 1939 essay on Baudelaire: the distinction Proust draws between voluntary and involuntary memory helps to disclose the distinction between the two notions of experience with which Benjamin's concerned, namely _Erlebnis_ and _Erfahrung_. The first refers to experience that's expressly and consciously lived, one that can be translated and recorded, but only as the experiences of someone else, from the outside as it were. This is experience related to voluntary memory. It's devoid of what Benjamin calls aura. The second, by contrast, is auratic insofar as it consists not in remembering distinct events very precisely fixed in memory (what I'd refer to as archival memory), but in data accumulated without our knowledge and brought together in memory. Benjamin marks the distinction by using two different terms, _Erinnerung_ or what I'd call "memory" as recollection, and _Eingedenken_ , which I'd translate as "reminiscence." As he rightfully points out, "only what has not been expressed explicitly and consciously, what has not happened to the subject as an experience" can become part of involuntary memory. Benjamin's pointing to the crux of the matter here, namely the unlived dimension that always doubles and accompanies the lived one. Involuntary memories are the starting point for experience in the strongest sense of the term. But what does it mean to talk about experience "in the strong sense"? It means the communication between two temporal strata or layers, between two spaces as well, something like a secret correspondence (what, in _Against Sainte-Beuve_ , Proust calls "harmony"), which cancels out all distance, and temporal distance in particular. Experience needs to be understood as the trans-temporal communication between two events that only communicate via their difference. The time that separates them is annulled and they are carried onto another level, another temporality. They converge in a sort of instantaneousness that runs parallel to the flow of time, a sort of "time outside time" that is at the same time the very essence of time, what Proust calls "time in its pure state." Kronos is overcome and the time of eternity begins, a time in which nothing ever passes any more except for the pure gathering or the infinite contraction of time itself. What we need to bear in mind, then, is that experience in the strongest sense, as well as the possibility of transcending innerworldly time and accessing the time of eternity, are in no way contingent upon an effort of the intellect or some sort of voluntary act, but on chance alone: what's afforded us by what's outside of time — which is the same thing as time in its pure state — isn't some recollection, something like a representation of a more or less distant past, but the emergence of this past with and within the present through a shock, a sensation felt in response to a material object that enjoys analogy with a past sensation (what I'm calling analogy here is more than just a similarity between two sensations, more than one simply being fashioned in the image of the other). What we feel comes from outside, yes, but this outside affects and disrupts the sedimented form of memory that I was talking about earlier on. It's not sensation as such that's the source of the quasi-infinite joy that the narrator fails to understand at first: it's not the taste of the madeleine dipped in tea or the motion of his foot against the paving stone, the trees striped with light, the square of green fabric obstructing part of the windows; it's not any one of these but all of them insofar as they're evocative of a past that's otherwise inaccessible to him, insofar as they form a bridge above the river of oblivion and allow him access to the hitherto unknown banks of pure memory, of the other time. The pleasure that's experienced, then, is tied to the experience of this excess that releases sensation and which we never directly experience: we never apprehend time in its pure state; it's never given to us in the purity of a noetic intuition. Its purity is wholly dependent on some sensation or other or on a series of sensations and accessible through them alone. This is why the book's best seen as an exploration of the sensible world. The sensible world contains more than itself, more than the simple play of sensations. This play opens onto another reality, one that's somewhat coded, inscribed in the world but still to be deciphered: the reality of a pure memory, of a metaphysical time in addition to a physical one. From the beginning, I've been insisting on the fact that what Proust refers to as the world's truth isn't something beyond the world but the world's flip side or its doubling, and this is something that needs to be understood literally: if truth isn't some sort of transcendence, if the Idea, as the name of the true, isn't above life but coiled up within it, that's because it consists very precisely in that part of unlived experience that doubles each lived one, this virtual side of an existing reality which is only ever present through its return or its repetition. The present's always and already splitting into two, into an existing present and a virtual one, into a visible and an invisible one, into something retained (whether voluntarily or automatically) and something forgotten. Obviously enough, the doubling's the work of the flip side, the invisible side of what we see (and perceive), that follows what we see (and perceive) as closely as possible and which, on its return (a return that's also its reversal), radiates or becomes visible in a light that's no longer that of perception but is instead that of memory. The doubling is what we "see" with the eyes of memory alone in a "stereoscopic" vision. But it's also what emerges behind the living present and replaces it — not in the way one present ordinarily replaces another by taking its place, but as the self- _absence_ of the present itself, revealing its ontological lack and imposing itself as this implicitly inscribed reality, that pure past which, however real, escapes the economy of perception and the logic of the present time (chrono-logy).
This is how Swann, when he unexpectedly hears Vinteuil's sonata at the marquise de Saint-Euverte's, is able to relive his happiness with an intensity and a lucidity that he's not known until now:
> In the place of the abstract expressions "the time when I was happy", "the time when I was loved", which he had so often used before now without suffering too much, for his mind had enclosed within them only spurious extracts of the past that preserved nothing of it, he now recovered everything which had fixed for ever the specific, volatile essence of that lost happiness; he saw everything again, the snowy curled petals of the chrysanthemum that she had tossed to him in his carriage, that he had held against his lips — the embossed address of the "Maison Dorée" on the letter in which he had read: "My hand is shaking so hard as I write to you" — the way her eyebrows had come together when she said to him with a supplicating look: "It won't be too long before you send word to me?"; he smelled the fragrance of the hairdresser's iron by which he would have his "brush-cut" straightened while Lorédan went to fetch the young working-girl, the stormy rains that fell so often that spring, the icy drive home in his victoria [... ]. And Swann saw, motionless before that relived happiness, a miserable figure who filled him with pity because he did not recognize him right away, and he had to lower his eyes so that no one would see they were full of tears. It was himself.
What Swann recaptures is his _entire_ happiness, namely his happiness in its pure state, its "specific, volatile essence." We need to understand essence here not as the sort in relation to which Swann's and Odette's happiness would have been the ephemeral difference, but as absolute singularity, itself specific and irreducible to any other form of happiness, made up of a multitude of details and instances. The happiness Swann finds himself facing is more than the happiness he experienced with Odette: it's a concentration of happiness, once scattered moments of happiness now delivered in huge blocks and clusters. It's almost too much happiness insofar as it's the whole of Marcel's happiness brought together in one single instant: his love _in itself_ , as Deleuze would put it, the way Marcel's teacup comprises Combray in itself. By coming back his happiness appears to him with a doubled intensity, widening the gap with his present unhappiness, revealing it for what it is. The fact that this happiness has ceased to be simply underlines his suffering. But what this suffering reveals, the metaphysical discovery that it brings about (and which Swann, unlike the narrator at the end of the book, is unable to understand), is cause for joy. Even though it's tied to a sensation and a topical cause, this rejoicing can't be reduced to the latter. It's even further removed from the simple sensation of physical pleasure that the young narrator experienced when he saw the hawthorns, a sensation that he'd already concluded couldn't account for how he was feeling. What emerges here is the possibility of rejoicing even at our darkest moments, our greatest misfortunes. It's always possible to extract the "Idea" from a state of things or a state of mind, to draw out the "divinity" that, although distinct from it, still lives coiled within it (and nowhere else). Love itself, whose tendency to lead astray, to disappoint and to cause suffering seems so significant to Marcel and to Swann we've already seen, is rehabilitated, thereby, along with any affective reality. This is what Vinteuil's music potentially bestowed on Swann. It told him of his love, all his love, and even the law of all love, expressing it for him:
> For the little phrase, [... ] whatever its opinion of the brief duration of the conditions of the soul, did not see them as people did, as something less serious than the events of everyday life, but on the contrary, regarded them as so superior that they alone were worth expressing.
It represents or, rather, embodies his love to such an extent that it becomes even more real than the love he experienced, that "real time" love. The only truly real time isn't that of the living present, however. Rather, it's the time that settles elsewhere and returns with tremendous force, purity and lucidity: the time of essence:
> These charms of an intimate sadness — these were what it sought to imitate, to recreate, and their very essence, even though it is to be incommunicable and to seem frivolous to everyone but the one who is experiencing them, had been captured by the little phrase and made visible.
Making the invisible visible, turning essence into an object of experience and not just an object of contemplation, this is what art ultimately involves. Its world is a world of ideas, yes, but ideas that are _sensible_ , as Merleau-Ponty would have put it — the ideas "of another world, of another order [than that of the intelligence], ideas veiled in shadows, unknown, impenetrable to the intelligence, but not for all that less perfectly distinct from one another, unequal among themselves in value and significance."
It's Husserl who paved the way for Merleau-Ponty's view of perception and of the world of subjectivity as the _sensible_ world. While giving sensible perception (perception, strictly speaking) a prominent role in what's commonly called the grasping or original "givenness" of the object, the being there in the flesh ( _leibhaftig_ ) of the object which consciousness somehow intends, for Husserl there's also a true givenness and, as a result, some perception, in the wider sense of the term, of categories or essences that are actually neither sensible nor real. In what really does amount to a reversal of Platonism, phenomenology sees the Idea, the essence, in short everything that's not immediately sensible (so everything that Platonism deals with under the heading of the intelligible, then, and thinks of as prior to and more elevated than the sensible) as being derived _from_ the sensible, which is now seen as the only real origin. And this, in passing, is why Merleau-Ponty ends up not mentioning the sensible and the intelligible, preferring instead to speak of the visible and the invisible, insofar as it's not a matter of a simple opposition or of a hierarchy, but of an extension of one within the other or, put another way, of the development of a unique structure: the invisible always is the invisible of the visible itself, which only the visible can access. Seen in this way, the intelligible isn't a lesser mode of being, something that somehow lacks being and that's arrived at through a process of weakening or abstraction and that's therefore reducible to the sensible; rather, it's its extension or its flip side. It's the intelligible that's perceived, then, and it's this claim that distinguishes phenomenology from straightforward empiricism. That said, sensible experience is still held up as the pre-eminent or archetypal meaning of flesh, as _the_ form of perception and not as one modality among others, as the intellectualist or Platonic would suggest by seeing it as the adulteration of an intelligible reality that's only accessible to a purely intellectual intuition. The flesh is at one with the sensible, then, even though it exceeds it. The sensible, as Merleau-Ponty says, "is not only things but also everything sketched out there, even virtually, everything which leaves its trace there, everything which figures there, even as divergence and a certain absence." The rupture or self-absence typical of the sensible is, once again, far from accidental; rather, it's structural. Exceeding the sensible, the flesh also contains it as one of its modalities. The sensible transcends itself, therefore, and does so in a way that's no longer vertical, intelligible, supersensible, but horizontal: the sensible goes beyond itself in depth, implicitly offering itself as meaning. Meaning is the frame, the very lining of the sensible. It might seem a bit surprising, then, that Merleau-Ponty sees Vinteuil's sonata and Swann's delightful and painful experience of it at the Sainte-Euvertes' as illustrating that relation of reciprocity, of mutual encroachment and intertwining between the sensible and its meaning, between raw existence and essence, or between the real and the ideal. The misery, the joy, they definitely go beyond anything the sonata has to offer. And yet they vanish as soon as Swann tries to isolate them, to draw them out of the sonata. When Swann tries to understand it, to pick apart the music itself as if his feelings were caught up within it, the whole experience begins to slip away from him. His love's certainly there, embodied in the sonata. The sonata even comprises and preserves the very essence of that love. And yet it's there only as some "discrepancy," namely its absence. This inevitably evokes the _absente de tous bouquets_ , the "absent from all bouquets" of Mallarmé's flower that denoted the form of transformation specific to poetic language. The sonata definitely performs this kind of poetic "work" for Swann's benefit, showing him the very essence of his love, the extra-temporal time in which it exists (or insists, as Deleuze has it), and the very meaning of the experience. The difference is that, unlike the narrator who will ultimately turn this work into the starting point of his life as a writer, Swann's unable to do anything with it and soon returns to his distractions. He never actually manages to avail himself in any creative way of his intuitions and recollections. To a certain extent he's understood everything that needs to be understood and knows what's still left to do. After that evening, and having understood that the feeling Odette once had for him will never be revived, he nonetheless chooses to stay in Paris where she just happens to be rather than taking the trips that he needs to take if he's going to carry on with his study on Vermeer.
Just as Vinteuil's little phrase contained all of Odette's and Swann's love and was able to bring it back to life, so the young Marcel's boot in some way contains the presence of his grandmother, her true face, her tenderness, her very nature, none of which had been erased by her death, a year before her return in Marcel's involuntary memory. In her absence, however, the narrator had ended up if not forgetting then at least picturing her in a way that was artificial and abstract and that retained only the most inessential part of her. The situation here's like when we say that we're "thinking about" someone who's died even though we're fully aware that our thoughts don't reflect what this person might have meant for us or accurately convey the depths of the emotions that tied us to him or to her, the visceral attachment we felt. Unbuttoning his boot, Marcel's seized by a feeling of immense sadness, sobbing uncontrollably as he "sees her again" unbuttoning his boots for him many years ago at Balbec: just as Combray arose from his teacup, now his grandmother emerges from his boots, an unknown and divine presence, her _true_ face appearing to him and
> not that of the one whom [he] had been surprised and self-reproachful at having missed so little, who had nothing of her but her name, but of [his] true grandmother, the living reality of whom, for the first time since the Champs-Élysées, where she had suffered her stroke, [he] had rediscovered in a complete and involuntary memory.
What we have with involuntary memory, then, is the juxtaposition of two temporal orders, namely the past and the present, or, rather, what's left of the past, what can come back from the past and so what's in no way simply past. For example, there's Swann's present, dominated by his sadness and melancholy, and the instant of happiness that both contradicts and enhances his sadness. Alternatively, there's Marcel's present as an adult at the Grand Hôtel at Balbec and the true image (not just the abstract thought) of his departed grandmother, which provokes such an intense grief in him. There's the time of facts, actual and chronological time, with its linear, objective, abstract and thus impersonal logic. And then there's the virtual time of "events," the time, that is, of feelings and impressions that settled at the bottom of things and were forgotten there, living within things, coiled up. This eventful time exceeds the time of facts: "It seems that events extend further than the moments in which they happen, and cannot be completely contained in them." So even though it happened over a year ago, the death of Marcel's grandmother wasn't an event until Marcel's experience of the involuntary memory in his room at Balbec:
> [T]hus, in a wild desire to hurl myself into her arms, it was only at this instant — more than a year after her funeral, on account of the anachronism which so often prevents the calendar of facts from coinciding with that of our feelings — that I had just learned she was dead. I had spoken of her often since that time and thought of her also, but beneath the words and thoughts of an ungrateful, selfish and cruel young man there had never been anything that might resemble my grandmother, for, in my frivolity, my love of pleasure, and accustomed as I was to seeing her as an invalid, I contained within me the memory of what she had been only in a virtual state.
In reality, the calendar of facts never matches the calendar of events. Before the episode at Balbec, Marcel's "grandmother's death" as a fact hadn't brought about any event; it hadn't entailed any grief or triggered any change in the narrator. Why? Because whatever we might think, facts are anonymous, insignificant and abstract. Only when the grandmother's found missing from her place at Balbec does the narrator understand his misery and his solitude. What all this suggests is that his memory of her isn't a representation in his mind but wholly embodied in a place. Until "her death" as a reality overlay the reality of Balbec, it was merely abstract, if not entirely devoid of meaning. His best efforts at remembering, efforts that stem from the guilt he feels over not being more grieved, are all in vain. What we call _le devoir de mémoire_ , the duty to remember, is really just an illusion considering the extent to which the content and the truth of an attachment, the feeling of loyalty towards someone, can't ever depend on a moral imperative. So far as Marcel's concerned, his grandmother didn't die in Paris, but in Balbec, in the place that's permeated with her presence, her affection and her tenderness. Let's go even further: this event of the death of a loved one, an event that ultimately catches up with him, is at the same time the revelation of what this person was to him, the event that she represented for him. When she was alive, his grandmother unfolded her essence, which the present both concealed and revealed. Now she's dead, her essence appears in a unique intuition, more complete than when she was alive, more present than any present moment. From an actual presence inscribed in the sequence of facts, she's become a virtual presence, an event that sets itself apart or extracts itself from the facts and that, like Swann's love, is concentrated in some extra-temporal vision. In her absence, the grandmother's more present than she ever was, essentially present, if you like, like this event that she was. She reveals for the first time what she was and continues to be: love, true love, unconditional and infinite. It takes that discrepancy, that rupture at the heart of what's real, for Marcel to be able to say to himself: so, that's what my grandmother was, and take the measure of this event.
***
Everything happens as if time unfolds simply in order to repeat itself, as if it passed simply in order to pass again. And by passing again, time highlights that the past is indeed past. At the same time, though, it indicates that there's a measure of atemporality to any present, to the time that doesn't pass and reveals itself as such only in its repetition. Proust's present, which is the same present as the narrator of the book, is completely hollowed out by this twofold absence, torn between a future that alone carries the keys to the present but that still remains inaccessible, and a past that it struggles in vain to hold onto. The return of lost time, on the other hand, doesn't depend on any effort to remember or on its representation; rather, it depends on a couple of sensations in which, once upon a time and unknown to us, some unlived experience embedded itself. As Proust puts it, while we do hang onto our inner joys, our past assets, and all our pains, they mostly dwell in unknown corners where they're not of much use to us and where memories of an entirely different order repress even the most customary joys or pains and exclude their actually coinciding with consciousness. However, as soon as "the framework of sensations in which they are preserved [is] recaptured, they have in their turn the same capacity to expel all that is incompatible with them, to install in us, on its own, the self that experienced them." Once the network of sensations that harboured the memory of his grandmother returns, a network whose existence was unknown until this return, Marcel finds himself in the presence of a long-forgotten, long-vanished self, one that's infinitely more present than his "present" self — exactly the same, then, as at the end of the novel when he finds himself thrown back into his childhood at Combray when he sees Sand's _The Country Waif_ in the Prince de Guermantes' library. Everything happens, then, as if there were different and parallel series of time, as if chronological time were nothing more than a fiction, useful to be sure, but one that's always going to fail to match any aspect of the truth of lived time, the time of the eventful series and their mysterious composition and communication. What characterizes involuntary memory is the fact that what resurfaces like this hasn't ever been experienced or perceived as such. It's a paradox: something resurfaces or returns that has never actually happened. The "matter" of involuntary memory only exists in its return, therefore, in its capacity to return. Voluntary memory, on the other hand, can bring things back, but only what actually took place, some lived experience or other. That experience was recorded in a present and then "retained" in the Husserlian sense, that is, intentionally. And so it can always be recalled as a second present. In voluntary memory, then, two presents coexist, a present present and a past present. What's remarkable about involuntary memory is that the past returns _as if for_ _the first time_ — not as a duplicate since it was never experienced or perceived as such, but in an entirely different way.
Events alone, and certainly not facts, not even childhood memories, will yield (good) literature. Unlike facts, which are given and always situable in time, events need to be drawn out from the way, each time distinct, in which they affect bodies and impress minds. An event's not a date but a process, a becoming, through which the subject experiences his or her own transformation. There's quite clearly a chronology at work in Proust's novel, one that's been reconstructed often enough and which Proust doesn't try to hide or obliterate. This would be to give it too much importance. The time that the novel's in search of and to whose meaning the narrator only gradually awakens is the time of events — a time of slow gestation and evolution broken up by sudden accelerations and violent turmoils. It's the geological time that lies beneath chronological time. All the characters in the novel are subjects _and_ events. What their duration in the novel sketches out is their own eventuality or eventfulness, namely the way in which their virtual being becomes actualized, revealing in the process their "essence" or their difference: a grandmother's love for her grandson, Charlus's greatness and decline, as well as his virility and masochistic femininity, Madame Verdurin's triumphant cruelty, Saint-Loup's homosexuality, the fading glory of the Duchesse de Guermantes and her whole clan, Elstir's truth in painting. Sifted through time, the subjects exude their eventuality or their eventfulness and reveal their essence. The two "ways," Guermantes and Méséglise, are the two structural events on which the whole novel hinges and which constitute Marcel as a subject. Absolutely separate — even incompatible — at the start, they end up by coming together. And the intersection of these two events, these two eventual or eventful series, creates a new one. Extracting the eventual or eventful aspect of any chronological "fact," extracting what seems to be made up of discrete and successive moments from the process of becoming, is an act of creation or of work (work in the sense of _oeuvre_ ). In this sense, the work of time is much the same as the work of photography: originally we only knew the negatives of things and, once dipped in the developing bath of time, each being's revealed in its true colours. The negative's obviously something that exists, something that's clearly identified on a certain level; virtually, though, it's something else as well, something that's only revealed or developed in certain circumstances which are themselves totally haphazard. While existing on the level of actuality, on the virtual level, the level of unlived experience, things "insist." Time's the true developer — not chronological time, which records facts and sets up sequences, but Time the artist, which creates as it develops, that is, as it _transforms_ : only Time the artist can turn the duc de Guermantes into a character by Molière, the baron de Charlus into King Lear and monsieur d'Argencourt into a sublime dodderer. What Time reveals is nothing but the negative from which it started; in the end, though, it's something else altogether. If Time's truly artistic, then it's in its capacity to make things recognizable, whether particular beings or the past itself, by changing them into different beings altogether.
So art's completely wrong when it thinks it's dealing with the immediacy of lived experiences, and calls _that_ the real, for then it fails to see how they always stem from a concrete and practical situation which psychology (whether empirical or transcendental) or neurophysiology aim to grasp. That kind of life, which imagination aims to transcend, is always a source of disappointment and boredom. In the return of the lived experience, on the other hand, literature finds something worthwhile inasmuch as this return signals the happening not of the same lived experience but of this transformed lived experience, metamorphosed by forgetting, lived again and always unlived. The lived experience returns _differently_ and this difference is literature's only real concern. In each lived experience there's a sense of living beyond or in excess of life, and this is what art calls life, true life. It's the life of the unlived, to which it lends life. Art prolongs life, not in the sense of extending lived life, and not just in the sense of life as survival. Rather, art prolongs life in the sense that it takes something that's _less_ than life, something that is somehow life-deficient and that escapes the grip of life, something that's forgotten and sheltered in life, it takes, in short, that past in which the promise of some future is held and the promise of some event is generated. The life of literature isn't added to life as an appendage or an addendum. Rather, life exceeds itself in art (exceeds in the double sense of a going out from oneself — _excedere_ — and overcoming oneself) and art exceeds life (in the sense of exacerbating it and firing it up). The excess of life that each one of us carries within us lies in the unlived experience that "doubles" every lived experience. And this doubling's never more apparent than in involuntary memories which make clear the structure of experience as irreducibly divided into the lived and the unlived present (or, since we can't really talk about an unlived present, the unlived past to come), into what's actual and what's virtual, into what actually took place and what will only take place in its repetition, in its return, into what happens for the first time and what only happens the second time around. Reminiscences, as Deleuze rightly points out, are metaphors of life and metaphors are reminiscences of art. What they have in common is the fact that they determine a relation between two radically different objects, "in order to protect them from the contingencies of time." Reminiscence and art (understood as metaphor) have the same relation to the world of essences: reminiscence is the analogue of art and involuntary memory is the analogue of metaphor. At least that's what I'd like to try to show now.
Chapter 4
* * *
Giving joy (metaphor)
* * *
> "So that at that moment, lying in bed with my eyes shut, I would say to myself that everything can be transposed..."
Overwhelmed by memories and sensations, Marcel's forced to wonder about the deeper meaning of his lived experience and, in doing so, becomes the writer he's always felt he was destined to be. It's a seemingly paradoxical situation: this excessive wealth of sensations and feelings that comes from sensible reality itself is precisely what pushes him to cross the threshold of a reality that's merely sensed or felt and to express its truth. Up until that point, the happy moments that occurred throughout the novel had gone without explanation, failing every time to set our hero on the path to literature. The very possibility of writing — longed for, yes, but frustrated until now — is mediated by the resolution of what we might call the mystery of experience: the fact that the truth of an impression, a sensation or a memory is always revealed through something else. The possibility of creating a work of art endowed with "infinite philosophical meaning," as Marcel puts it at the beginning of the novel, depends on its ability to retain and fix in an image the true, immaterial or spiritual content of what's being played out on the level of the sensible. The point is to show how the sensible can be overcome by the creation of a work of art and its meaning extended and to show how, as the horizon of transcendence proper to the sensible world itself, it becomes immaterial without heading in the direction of a supersensible world. There's a reality of transubstantiation, a metamorphosis of matter itself, which the experience of recollection had revealed and which art only deepens.
So what Marcel actually hears in the drawing-room of the Hôtel de Guermantes is the sound of the spoon against the plate. But something emerges from this perception, as if curled up within it, ungraspable and unsuspected until now, something that's much more than a perception, much more than a sound. Marcel's effectively transported somewhere else; he's transformed. The feeling of tremendous warmth, mixed with the smell of smoke and soothed by the coolness of forest fragrances, is completely different from the sound on the plate from which it emerges. The narrator has to concentrate in order to identify the missing link in this association, the link that allows these two groups of sensations to communicate with one another. This link — the sound of the hammer against the wheel of the train echoing the sound of the spoon against the plate — was imperceptible at first, as if the association depended on one term being withdrawn. We might even go so far as to say that the first sound wasn't actually perceived during the train trip; unperceived, though, it was still there from the outset, in a virtual state, waiting to be realized, waiting to happen, as it were. Insofar as it opens the door from one group of sensations to another, the link's quite real, almost sensible, perceptible; it's not the product of a synthesis of the intelligence, the concept of a genre or a species of which the group or series of sensations would be the difference. It's nothing general, nothing that could be applied to another situation. It belongs to this very specific situation, which it designates in its singularity. And if it unites or draws the two series together, it's not in the way an external power would or like a term that would subsume them; rather, it joins from within and as a sensation. True, this sensation _is_ a distant echo and reminder of the sound of the spoon against the plate: it's through this vague resemblance that the two series begin to communicate with each other. Maybe we need this minimal degree of resemblance, this vaguely common trait, for this resonance to take place. But even if we do it's not the object of experience and we shouldn't see it as pivotal. Its whole job is like that of the spark that sets the flame or the lightning that joins the sky and the earth. And when the two series touch and are tied together through this link or this common element what they communicate to each other are their _differences_ and not their identity; they express their _divergence_ and not their resemblance. All that matters here is this: how can differences resonate without dissolving into a new identity (whether immediately or somewhere down the road)? How can they communicate and exchange their very difference? What involuntary memory reveals is the coexistence and communication, _on the same level of consciousness_ , of divergent if not entirely heterogeneous groups or series of sensations. What they reveal, in other words, is the joining together of differences. The two groups or series of sensations communicate with one another by way of a certain sound; _what_ they communicate, however, isn't just this sound, but the differences that surround it on either side. The sound brings the series together much like a bridge between two banks that underlines the abyss that separates them as much as the link that joins them. It joins without blurring and unites without identifying. Yes, it draws a bridge between one bank and another, between the past and the present. What it doesn't do, however, is negate either one; instead, it reveals the space and time through which they communicate with each other, through which they become compatible. What's important here is the fact of this connection and this sudden communication, both of which show that we can't know in advance what's compatible and what's not, what can coexist and what can't. Just like in the spoon/plate/butler series and the intense heat/smoke smell/fresh forest smell series, we're never going to be able to say that these differences are incompatible a priori or compatible only when they're referred to a subsuming identity (species or genus, say): each lived experience holds a share of unlived material, of possible surprises. In short, each lived experience is eventful. The relation between the series isn't based on resemblance or analogy, even if what motivates the communication between them is similar on either side. Moreover, if we could climb back up the slope on which the series were individuated in actual time and space, we might come to the conclusion that everything coexists: there's a level, then (a strictly virtual one), where experiences as a whole — I'd go so far as to say _events_ as a whole — coexist in a totality and a coherence that's only accessible to the divine gaze even though, every now and then, we can ourselves have an intuition and a revelation of such coexistence.
And so the mystery of experience is solved. It consists of two closely related principles: (1) in itself the present is devoid of value; (2) whatever value it does have is always located somewhere else. This is going to have a major consequence for aesthetics: there's nothing beautiful _in itself_ about a row of trees or a hawthorn bush, nothing there that's interesting for art. Equally, though, the most ordinary things can possess incredible beauty and power when they're able to evoke other things, that is, to transport us elsewhere, to a faraway land or a remote corner of our past. Which is, of course, Plato's definition of beauty, not only as the most radiant ( _to ekphanestaton_ ), but also as the most desirable and ravishing ( _to erasmiotaton_ ), as what transports us and ravishes us. Beauty's ravishing insofar as it takes us away from the ordinary and from ourselves, doing so the better to return us to the world. Yes, I know that's a truism, but it's only become a truism because it's a constant feature of Western aesthetics. Proust himself endorses it. But he doesn't just endorse it. He also extends it and offers a decidedly singular reading of it. As I said, for Plato beauty is what transports us elsewhere. But what's the significance of this transport? What does it depend on? As I'll show in a bit, this notion of transport and of dislocation, of disorientation and estrangement, lies at the heart of Aristotle's reflection on metaphor, a reflection that Proust's going to reinforce, but in a way that goes beyond its Aristotelian framework.
Saying this already implies that the kind of art that focuses on the present as the only existing reality and that's happy just to describe that present — or assumes that's what it's doing — is a total illusion: it never gets beyond the surface of things, never even suspecting that there are depths still to be plumbed:
> Thus the sort of literature which is content to "describe things," to provide nothing more of them than a miserable list of lines and surfaces, despite calling itself realist, is the furthest away from reality, the most impoverishing and depressing, because it ceremoniously cuts all communication between our present self and the past, the essence of which is retained in things, and the future, where things prompt us to enjoy it afresh. It is this that any art worthy of the name must express...
The job of literature, then, is to overcome vulgar realism. "Realist" or "purely descriptive literature" — nicely skewered by Proust in a pastiche of the Goncourts' _Journal_ — is no more than a simulacrum of reality. How, Proust wonders, can this sort of literature
> have any value at all, when reality lies hidden beneath the surface of little things of the sort it documents (grandeur in the distant sound of an aeroplane, or in the outline of the steeple of Saint-Hilaire, the past in the taste of a madeleine, etc.) so that the things have no meaning in themselves until it is disentangled from them.
For this kind of literature, reality's no more than "a residue of experience, more or less identical for everybody." True, everyone knows what we mean when we talk about "bad weather, a war, a cab-stand, a brightly lit restaurant, a garden in flower." But is this all that reality is, especially when we're talking about literature's reality? If it were, then "some sort of cinematographic film of things would be enough and 'style' and 'literature' which departed from their simple data would be an artificial irrelevance." If this _were_ reality, there would be no need for literature. And if this _were_ the only way literature could ever be "realistic," Plato was right to condemn it. Literature's necessary — and it's actually this necessity or this uselessness that's been at stake all along — only insofar as there's a reality that can be _probed_ and _translated_ , only insofar as there's a reality that, presented in the form of a strong impression, simultaneously reveals and conceals its meaning, a meaning that it's the writer's job to extract or to "express." "Purely descriptive literature" isn't literature because real literature only begins where description ends, namely at the point when, under the impulse of an impression, a sensation, a memory, in short, a shock that comes from outside, we're no longer content simply to take note of these impressions but instead pause, throw ourselves into them and look behind them until they've given up their secret and revealed their depth. Art can't be happy with impressions alone: its job is to extract a law from them. This is what I'm going to call Proust's _expressionism_. By that, I mean an elevated form of realism. Obviously our impressions bring us face to face with the things themselves and allow us to experience reality itself, but this reality isn't reducible to its present state: it can always be transposed or transported, and us along with it. The expressionist truth, for which the truth of something can only be found in something else, is different from the realist lie, which, certain of its absolute fidelity to the real, ends up only repeating or paraphrasing it and, in doing so, falls under the hammer of mimesis and Plato's prohibition on it. When we try to hold onto what's "lived" and to keep things as simple as life itself, we end up with nothing more than a "tedious and pointless duplication of what our eyes see and our intellect records," and there's nothing beautiful about that. And what's the point, after all, of simply reproducing what's already there? If art does have a meaning it lies in its ability to go beyond what's already there, not into a world or a life beyond this one, but into their pristine depths and mysterious folds. "The greatness of true art,"Proust tell us, "the sort of art that M. de Norpois would have called dilettante amusement, lies in rediscovering, grasping hold of, and making us recognize this reality, distant as it is from our daily life" — so used are we to substituting conventional knowledge for it — "and which is quite simply our life." There are two kinds of life, therefore: there's the life that we live, this life that we reproduce (by way of habit) and can even relive when we remember it, and then there's the other, true life — I'd call it _real_ life if the term wasn't so loaded — the life that's buried under things and that it's literature's job to express. True life just _is_ literature, therefore, but only insofar as we understand literature as _expression_ , expression not in the sense of a simple repetition of lived experience in a work of art, but in the sense of an extraction and translation of that part of an impression that points to what it's not, to what's beyond it and which, despite all that, is still more _of_ that experience than all the lived qualities of that experience put together. The artist develops what the real envelops; he or she is the positive of that negative that the real actually is, the "revelation" (still in the photographic sense of the term) of the "qualitative difference in the ways we perceive the world."
Although literature's opposed to vulgar realism, that doesn't mean that it's a form of idealism, either in its objective (that is, Platonic) or subjective (Kantian) guise. Literature isn't idealism in the Platonic sense because, as I argued earlier on, the fact that it surpasses the real doesn't mean that it ends up stranded in the intelligible world of ideas, ideas that are the original and true form of their sensible expression. All the same, literature wants to say that it does have access to the world of ideas and essences and can compete with philosophy. The ideas that it uncovers, however, are _in_ the world; they are _embodied_ ideas. They are the substance and thickness of the world. And if literature isn't a form of idealism in the Kantian sense of the term, it's because, through involuntary memory, it recognizes that the real has the power to awaken us to the world of essences: the discovery of ideas isn't a matter of introspection any more than the generation of ideas is a matter of pure spontaneity. Ideas come to us from the world. And it's wrong to think, then, that Proust sees literature as turning away from the world in order to focus on the ego. The ego's caught up in the world from the outset: it's in the world and of it, tied to it by the threads of perception and memory.
Whence the necessity of metaphor in literature.
Let me try to make that a little clearer. Proust's view of metaphor and the way that he uses it are unique inasmuch as they have a specifically _ontological_ source and not just a rhetorical one. Proust's poetics is an onto-poetics. It involves a conception of time and space — one that I sketched out in the last chapter and that I'll develop further in this one — as a reality that's split or differentiated from the outset. There is, in effect, a law of displacement and transposition to which we're subject, as demonstrated by, amongst other things, the experience of involuntary memory. If style, as Proust says and as I'll need to show, really is a matter of "vision" and not of "technique," it'll have to follow the law of this originary split and work through metaphors. This law's an effect of a structural lack of ontological deficit: we could say about being what Lacan said of the phallus, namely that it lacks its proper place, that it's never where (or when) we expect it to be. What Proust calls beauty is the realization and expression (which is the task of art) of the potentially positive side of such a seemingly negative inference. If metaphor turns out to be structural and not just ornamental it's because what we ordinarily call "the present" or "the real" is, in fact, an empty case or an empty set, an ontological lack that's only filled out through a process of transposition. This transposition, however, is the result not of an external intervention but of the "work" of the real itself.
***
Seeing as how the sense of metaphor that I'm trying to develop here is rooted in the very structure of experience and not simply a rhetorical trope, it's obvious why we're going to need to compare and possibly contrast what I'm saying with what's classically been said. And it's equally obvious that I'm thinking here of the sense worked out by Aristotle in the _Poetics_ and the _Rhetoric_ and then adopted — and modified and commented on — by the tradition.
Aristotle develops the analysis of metaphor because he sees it as a useful tool as regards two kinds of discourse: the discourse of persuasion or eloquence, and the discourse of catharsis or tragedy. Metaphor pertains both to poetry and to rhetoric. And while it's endowed with a unique structure — namely the process of transferring the meanings of words — it has two functions: a rhetorical one and a poetic one (I'm only really interested in the second here). As for the definition itself, it applies to two domains and has to be qualified in relation to each. Here's what Aristotle says:
> Metaphor consists in giving something an alien name [ _onomatos allotriou_ ] by transference [ _epiphora_ ] either from genus to species, or from species to genus, or from species to species, or by analogy.
From this definition, at least four things follow:
First of all, metaphor doesn't actually concern modes of elocution (orders, pleas, threats, queries, replies and so on); rather, it concerns a particular element or specific constituent of utterances, namely nouns (as opposed to letters, syllables, conjunctions, articles, verbs, cases or locution). So, and as Paul Ricoeur rightly suggests, "the destiny of metaphor is sealed for centuries to come: henceforth it is connected to poetry and rhetoric, not at the level of discourse, but at the level of a segment of discourse, the name or noun." Now, a noun, as Aristotle tells us, is "a complex sound [which distinguishes it from the 'letter,' what linguistics calls a 'phoneme'] endowed with signification [that's what distinguishes it from the syllable], 'a composite significant sound not involving the idea of time, with parts which have no significance by themselves in it.'" Now, sticking just with Proust and his desire to turn time into the very matter or substance of literature, the question I want to ask is how metaphor needs to be transformed in order to become temporal. Can nouns ever designate time or does metaphor's scope need to be extended to other elements of poetic discourse, to verbs, for example? I'm going to try to answer these sorts of questions by demonstrating that the roots of metaphor lie not just in poetics but in ontology, too.
Second, metaphor can be defined as a very particular type of movement since it consists in a sort of displacement from... to... ( _epiphora_ ). Bear in mind that, for Aristotle, the term "metaphor" doesn't really refer to a particular trope among others: metaphor, synecdoche, metonymy and so on; instead, it applies to _any_ transposition of terms. This is why we find it cropping up in all sorts of contexts, even ethics. The same goes for Proust: we're never going to be able to understand his extended use of metaphor — not strictly tropic or even semantic, but onto-poetic — through the lens of classical rhetoric or modern linguistics. Besides, Aristotle himself resorts to metaphor when he's trying to describe what metaphors are: he displaces the common meaning of a word and transposes it into another context. The term _phora's_ borrowed from the _Physics_ , in which it refers to change, specifically with respect to location. So far as Proust and the relation between metaphor and time's concerned, memory especially, the issue seems an increasingly thorny one: if metaphor's a certain type of operation involving the displacement of one term onto another and so an operation that involves a change with respect to place, it's the spatial dimension and local movement in particular that's privileged here. So metaphor isn't just incapable of accounting for temporal change; it's also rooted in change in the local spatial sense of the term. Time would be doubly excluded from metaphor, then, and from its poetic resources. All of which raises the question of how we should understand the type of displacement that's specific to Proust's concept of metaphor. Displacement with respect to location, perhaps? Or does metaphor involve another sense of space, one that would be real without being local? And is this space the same as the space of literature?
Third, metaphor is the transposition of a word into a context that Aristotle qualifies as alien or improper ( _allotrios_ ): it involves the use of an "alien name" (1457b31), a name that "belongs to something else" (1457b7). Bear in mind here that _allotrios_ doesn't mean "figurative," as it does in classical rhetoric, and it's not opposed to the "proper" in the sense of the literal, the primary, original or indigenous. If _allotrios is_ opposed to anything it's to the proper ( _kurion_ ) in the sense of the ordinary meaning of the word, its common use. The "primary" or "proper" sense of a word is the way in which it's generally used (1457b3). Metaphor implies a displacement in the direction of the foreign, then, entailing something like a semantic displacement. It presupposes a certain distance, some _deviation_ from or _break_ with the ordinary meaning of the word. And it's this break that allows the poet to "escape banality" (1458a21), this deviation that generates what's surprising about the poem. As far as Proust's concerned, we'll need to show how metaphor and art in general thrive on this deviation or this break from ordinary language and everyday meaning, how the space of literature and the world that it reveals grow out of this uncanniness, this exile, this displacement that'll prove to be the very meaning of the everyday world. In this deviation, in which metaphor's continually at work, what's disclosed is the secret and extraordinary side of the world. The true world — the world of literature — is wholly caught up in this deviation from the ordinary world. Far from signifying a departure or a flight from the world, an abstraction or a whim, metaphor articulates the world in what we might call its worldly truth, in its deviation or its irreducible self-absence. We tend to assume — and Proust himself says as much — that when we're reading we're looking to be "taken out of our surroundings [ _se dépayser_ ]." But how should we understand our need here, and how should we understand this dislocation? After all, we've already seen that reading's much more than a form of entertainment and, more importantly, altogether different from it.
What distinguishes metaphor from other linguistic deviations (rare words and neologisms, for example), though, is the fact that it gives a thing a name that belongs to something else. It is, in other words, a value that's _borrowed_. The common name's opposed to the borrowed one. The unusual or extraordinary meaning of the word that's introduced by metaphor has its own field, its usual meaning. This field's displaced in order to generate an effect. The big question here for Proust is whether this borrowed value is simply a matter of ornamentation or poetic effect or whether it's a more complex and entirely different phenomenon. In other words, whether metaphor's a linguistic and literary resource or, on a deeper level, whether it's itself the effect of metaphor's _original_ space and time. This is where it might be useful to distinguish between two ways of understanding the borrowing process that's specific to metaphor. The first one, the classic one, shared by Aristotle and his successors (including Ricoeur), looks at it in terms of the original or borrowed domain, on the basis of which the two terms of the comparison can be carefully distinguished. The second one, the one that I'm advancing here, looks instead at what allows this borrowing to take place and wonders, more specifically, whether it emerges not just from the writer's imagination but from a certain necessity, from the acknowledgement of a lack or deficiency of the real in its raw or immediate state. Does metaphor borrow for the sake of pleasure alone (is it something simply pleasant or enjoyable, as Kant would have it?) or does the pleasure we experience with metaphor correspond to an economy that governs the real as a whole? If we recognize metaphors as exact (or beautiful), as opposed to merely nice or pleasant turns of phrase, if we recognize them as revealing something about the world, isn't it to the extent that the original domain is only revealed through its own dislocation? Isn't the aptness of this dislocation and this borrowing a consequence of the fact that the original domain signalled a lack or absence more than a full and established presence, a failure of being that metaphor both reveals and transcends? If a word only becomes poetic by borrowing a domain from another word, isn't it because the way that it's commonly used actually masks its truth more than it reveals it? Doesn't the fact that we can recognize the aptness of a dislocation suggest that the original signals a deficiency rather than a plenitude and that we're more likely to find the original, and find it better, precisely where, technically speaking, it's actually not?
Finally, and still following Ricoeur, we should bear in mind that the notion of "alien" usage implies the notion of _substitution_. This idea, which has close ties to the one I've just been discussing (it's a central idea in classical rhetoric), isn't a necessary one, however. I'll try to show that Proust's view of metaphor and his use of it escape what we might call the law of substitution (or at least what used to be seen as just such a law). The law states that any metaphorical term could be replaced with an ordinary word belonging to the original semantic domain of the first term. Put differently: the metaphorical word has replaced a non-metaphorical one that could have been used instead. As Ricoeur points out, it's doubly alien, therefore, borrowing a word that's present and substituting for a word that's absent. The law's often seen as an essential one, and it's one that I want to examine in the light of my reading of Proust. It actually posits, if only indirectly, the derivative and secondary character not just of the metaphorical process but of poetic language in general: _de jure_ , it should always be possible to substitute an original ordinary term for the metaphorical term and, in the process, ban poetic speech from having any cognitive value, restricting it to the role of something purely ornamental and pleasurable. As I'll show, though, metaphor's not a source of pleasure, at least in the sense of being something merely pleasant or charming; rather, it's a source of _joy_. Metaphor's joyful because it reveals something about the world that's otherwise inaccessible and so its value ultimately _is_ cognitive.
So far as Proust's concerned, we'll need to ask about this threefold process of deviation, borrowing and substitution. We'll see that, while metaphor _does_ entail a displacement of the ordinary in the direction of the extraordinary, it doesn't so much displace an original meaning as it actually replaces its proper or ownmost meaning: only true dislocation enables a return home; only expropriation affords appropriation and the experience of the proper; only deviation as regards what's proper affords access to it. Put differently, the experience of the foreign in fact coincides with the experience of essence, and essence is only given in the distance or gap that separates the real from itself. The manifestation of essence passes through metaphor's movement of expropriation and the very composition of the work of art, as a work of metaphor, constitutes the expropriation of the artist himself.
The fourth and final feature of metaphor that I'd like to retain from Aristotle's definition relates to metaphor's distinctive form of transference either from genus to species, or from species to genus, or from species to species, whether by analogy (or proportion). It's worth pointing out that only the latter type of transference — the only one, in fact, that really refers to resemblance — is what classical rhetoric actually calls metaphor: _D_ is to C in the same way ( _omoiôs ekhei_ ) (1457b20) as _B_ is to _A_ : old age is to life as evening is to day. It's this restricted definition of metaphor that leads Derrida to say that "resemblance or analogy... is the distinctive source of metaphor, from Aristotle to Fontanier." As far as Proust's concerned, we'll have to identify a new break with the classical tradition. Far from being a case of resemblance, agreement or identity, Proust's use of metaphor stresses and reveals dissemblance, discord, and difference. It's worth wondering, moreover, whether Proust's practice of metaphor actually is a process of transposing one term into another or whether it's an indication of a space — the very space of literature — from which the terms of the metaphor can be envisaged _differently_ , namely from the perspective of their difference and so as if for the first time. What I've said about the relation between sensation and memory, between present and past, has largely paved the way for just such a way of thinking about metaphor — an original way, in my view. Far from being a matter of resemblance and the straightforward similarity between terms, metaphor would operate like a power of dissemblance capable of bringing differences together without the mediation of any sort of identity (resemblance, genus, species and so on). Proust would agree with Aristotle, then, in seeing an appropriate use of poetic forms as a good thing (1459a4) and in seeing it as even better to be "a master of metaphor [ _eu metapherein_ ]" (1459a7). But does making good metaphors simply involve having an eye for the similarity or the resemblance ( _to omoion_ ) in what is dissimilar? To what extent are the terms of metaphor alike? What does resemblance mean? If it's not mimetic, is it simply analogical?
I'll come back to the issue of resemblance later on. For now, let's simply point out its problematic dimension. Everyone agrees that, through metaphors, things that were "a long way apart" suddenly seem "closely related." Is this proximity a consequence of resemblance, though? Maybe I can compare the sea and the boats on it to a "quiet roof, where dove-sails saunter by" like Valéry does in _Le Cimetière marin_. Why? Because the sea and the roof look alike and stem from the same Form, in whose image both would be constituted? This is where there's disagreement. In _M'tonymie et Métaphore_ , Albert Henry includes the following, eminently Proustian remark from the poet Reverdy:
> The image is a pure creation of the mind. — It cannot arise from a comparison but from the bringing together of two realities that are remote from each other. — The more distant and exact the relations between the two realities brought together, the more powerful the image — the more emotional power and poetic reality it will hold.
Is this a matter of resemblance? I don't think so. When two things are associated with each other they might well end up looking similar. Resemblance, though, is absolutely not the condition for their having been brought together. And it's this dimension of metaphor that I'm most keen to explore. Aristotle, warning against "far fetched" metaphors, recommends that metaphors be derived from what's related or "kindred" ( _sungenôn_ ) and "of like form" ( _homoeidôn_ ). A far-fetched metaphor is one for which there's no visible kinship (as defined by genus or species), no formal (or _eidetic_ ) identity behind the differences, in short no resemblance. All of which shows that poetics is, right from the start, subject to a philosophical decision regarding the limits of human discourse and the architecture of being: insofar as metaphor's located between genus and species, it's still subject to the concept without ever disrupting the ontological order that runs from the identity of the genus to the difference of the species. The notion of generic kinship situates us right in the middle of a notion of family resemblance that's very different from the poetic spirit of Reverdy, Claudel or Proust for the matter, as I'll try to show. I have to disagree with Ricoeur, then, when, quoting all three, he claims that they all share Aristotle's view of metaphor. I'd like to show that, from a properly poetic perspective, one I'd qualify as onto-poetic, metaphor no longer involves comparison and involves resemblance even less. Making successful metaphors doesn't actually have anything to do with having an eye for resemblances, provided we understand resemblance in the way that Aristotle does, as involving some generic, specific or analogical identity. For Aristotle, we have an eye for resemblance when we can see the same in what's different. In metaphorical statements, then, "'the similar' is perceived _despite_ difference, _in spite of_ contradiction." Which naturally implies that resemblance is
> the logical category corresponding to the predicative operation in which "approximation" (bringing close) meets the resistance of "being distant." In other words, metaphor displays the work of resemblance because the literal contradiction preserves the difference within the metaphorical statement; "same" and "different" are not just mixed together, they also remain opposed.
I'd argue that seeing the similar isn't the same as looking beyond differences in order to see what's the same; rather, it's a matter of seeing how differences emerge as a result of the same, as I already suggested in my discussion of the notion of the series in involuntary memory. Metaphor doesn't reveal shared or common qualities and it's not concerned with comparisons (we can only compare species, after all, and doing that doesn't involve much in the way of poetry). Instead, it discloses another level of the real, another state of matter, one that moves beneath the fixed world of genres and species. Metaphor's oriented towards differences themselves insofar as they signify the limit or the threshold of a certain state of things, beyond which we're then dealing with something else altogether; it's oriented towards the transformative or transpositional power inherent in the real; it's oriented towards differences as singular points and events, and not as something that arises from generic or analogical identity. For Aristotle, metaphor's a form of comparison, and this means that it's always geared towards extracting an identity from two distinct terms, a similarity from something dissimilar. It's another story altogether with Proust, who doesn't compare anything since what allows him to bring two terms or two series of terms together isn't of the same _kind_ or _nature_ of those terms or series in their respective, individuated and ordinary state, which alone can be compared with another. If we can still speak of something "similar" it's only in the sense of another level of reality that _emerges_ from metaphor and emerges from it as its pre-individual horizon, thereby making it essentially different from a genus or a species. To my mind, metaphor will turn out to be the transposition to the poetic level of an otherwise philosophical issue, namely that of difference. The distinctiveness of metaphor lies in its ability to present or schematize difference: metaphor is the sensible figure of difference, its poetic schema.
In the previous chapter as well as earlier on in this one, I cast a brief critical eye over the claims of the concept of resemblance to shed light on the phenomenon of involuntary or affective memory. The concept of contiguity, which is the other governing principle of associationism, seemed equally unable to get to the heart of things. It's hardly surprising that the issue of resemblance should crop up again in the context of metaphor. Rhetoric, linguistics and psychology have a long — and ongoing — history of close cooperation. The detailed analysis of the reasons underlying such an alliance would take up too much space for me to attempt it here. Let's just bear in mind, then, that Fontanier sees metaphor as a trope based on "resemblance" and metonymy as a trope based on "correspondence," a term closely related to contiguity. Structural linguistics goes even further and, unlike the position that I'll adopt here, sees no real contradiction in the exchanges between semantics and psychology which Ricoeur traces back to the chapter on "Mechanism of Language" in Saussure's _Cours de linguistique générale_. However, it's only with Roman Jakobson that the distinction between metaphor and metonymy gets grafted on the distinction, established by psychology, between resemblance and contiguity, and that language was finally structured according to these two poles:
> The development of a discourse may take place along two different semantic lines: one topic may lead to another either through their similarity or through their contiguity. The metaphoric way would be the most appropriate term for the first case and the metonymic way for the second, since they find their most condensed expression in metaphor and metonymy respectively.
This bipolar structure of language would also lie at the origin of various literary genres and traditions. Hence the privileging of metaphor in Romantic and Symbolist poetry and the primarily metonymic nature of "realist" literature. I've already alluded to Proust's critique of realism in literature. Combined with his positive view of metaphor, which he takes to be a symbol of style, this critique would seem to confirm Jakobson's thesis. As we'll see, though, Proust's way of thinking about metaphor will call into question the classic distinction between metaphor and metonymy as well as the structuring value that Jakobson lends this opposition. To the extent that Proust's aesthetics is one of metaphor and not of metonymy, we'll also need to see him as complicating the relation between poetry and prose by way of a poetics of the novel. His critique of realist literature isn't unrelated to this, and what he refers to as "style," that is to say metaphor, isn't a distancing of reality but a deepening of it, the key to accessing its truth. Nothing's more superficial than realism in literature, therefore, and nothing's more true than metaphor, provided we see metaphor not just as a matter of style or technique but as the expression of the deepest meaning of experience. It's interesting that, in the same essay, Jakobson considers Freud's _Traumdeutung_ , which involves an interpretation of the mechanisms of dreams in terms of the process of "transposition" ( _Entstellung_ ) from a latent content to a manifest content, analysing them through the lens of this bipolarity. The processes of "displacement" ( _Verschiebung_ ) and "condensation" ( _Verdichtung_ ) would relate to metonymy, therefore, while "identification" and "symbolism" would relate to metaphor. The conclusion that the unconscious, at least as it's revealed in dreams, is structured like a language seems like the next logical step, and it's a step that Lacan's more than happy to take. According to him, it's not just the work of dreams that's organized around this bipartition that comes from linguistics; the unconscious as a whole is as well. The symptom's a metaphor, then, and desire a metonymy. If the symptom's a metaphor, that's a function of the law of substitution: the symptom substitutes one term for another, one thing stands for something else. But this substitution's been in place _from the start_. I'm more than happy to subscribe to the theory of substitution if we define it like this: if there _is_ a substitution, it's because the thing doesn't have a place to call its own; it's always lacking its own place and can only ever be found somewhere else. _As such_ , it's necessarily impossible to find. And I'd say the same about beauty and truth, both of which are at stake in metaphor, at least according to Proust.
Conversely, I'd like to show that this psychological approach, at least as far as its associationist residue is concerned, limits the understanding of metaphor; we need to approach it differently, then. What I wanted to do in the last chapter was to free memory of its purely associationist interpretation (all the while recognizing the impact that such interpretation might have had on Proust). What I want to do now is to tear metaphor away from an interpretation based on classical rhetoric and modern semantics, an interpretation that's itself largely indebted to associationism. It's symptomatic that Jakobson, in the article I mentioned earlier and in the context in which he's writing—namely the context of aphasia and of the light linguistics might be led to shed on this question — lays the two metonymic and the metaphorical poles of language over the two fundamental laws of associationist psychology. How could linguistics illuminate a psychological problem, to wit aphasia, if the only terms in which this operation can be illuminated are themselves psychological ones? How can a problem from psychology be explained by a linguistics that borrows its concepts from psychology? Any such approach is symptomatic of an alliance, one that's more compulsory than truly constructed, between psychology and linguistics. And this is what we need to overcome if we want to understand the singularity of Proust's aesthetics. We're only going to be able to carry this out if we're prepared to offer a strictly philosophical interpretation of that aesthetics, and an ontological one in particular. My aim here, though, isn't to go back to Aristotle's ontology, constructed around the concept of _ousia_ (substance, essence and presence at once), or its corresponding logic of identity. Rather, the ontology that corresponds to this new metaphorics (or this new aesthetics) is that of the event and its logic is the logic of difference.
That said, we need to emphasize the extent to which Proust's view of metaphor is a general one. In that respect, it's closer to Aristotle's view than to the view of modern rhetoric. So, against Fontanier, for example, who restricts the field of metaphor in order to contrast it with the field of metonymy, or against Du Marsais, who sees metonymy as the figure that encompasses all others insofar as it signifies "the transposition or change in name, a name replacing another," Proust doesn't oppose metonymy and metaphor, and considers the latter to be the encompassing poetic figure. This brings him closer to Condillac, who pointed out that "if one looks at the etymology, all the tropes are metaphors: for metaphor properly means that a word is transported from one meaning to another."
***
Maybe it's surprising that the first explicit reference to metaphor in the novel comes in the context of a reflection on painting, not poetry or even literature in general. Marcel's visiting Elstir in his studio in Balbec. From which we might conclude that the specificity of metaphor, at least as Proust understands it, lies in the fact that it defines the nature of art as such, at least from the viewpoint of style. The law of metaphor applies to aesthetics in general. Here's Marcel's discovery as he walks through the door:
> Almost all the works I could see about me in the studio were, of course, seascapes done recently here in Balbec. But I could see that their charm lay in a kind of metamorphosis of the things depicted, analogous to the poetical device known as metaphor, and that, if God the Father had created things by naming them, Elstir recreated them by removing their names, or by giving them another name.
The key-term here is obviously _metamorphosis_. It's the only one likely to lead us to a real understanding of metaphor. In metamorphosis, which is clearly similar to what's referred to here as "the poetical device known as metaphor," it's the things themselves that are changed. It's the things themselves that are given in the seascapes. And yet, those things only really become themselves by being changed, by undergoing a metamorphosis. They are more themselves, if you like, closer to what they truly are in Elstir's painting than in the images that we usually have of them — images that are diverted and perverted by the practical and utilitarian nature of our perception, by habit and by the theoretical understanding that we have of the world. It's a discovery and a paradox. It's not enough for God to have created the world. For Proust, God's act of creation is unfinished until it's resumed and completed by the artist on another level. Why would there be artists, after all, if the created world were perfect? What's the point of recreating if, by recreating, we're simply imitating and, by imitating, simply adulterating? Art's long been confined to this mimetic and creationist framework, justifying the pleasure it generates by way of the perfection (and so the beauty) of its model. But the relation between the work of art and nature is different from the relation between an image and the original or between a copy and the model. If the work of art remains an image it does so only insofar as it discloses what's otherwise invisible, namely nature in its power of transformation, evolution and creation. When He created the world God might well have created it in His own image, i.e. as something creative. The artist's the one who recreates the created world, not identically according to some sort of resemblance requirement, but differently and on another plane, with a difference that's the difference of nature itself, of nature in its own power of transformation. The work of the artist consists in capturing the world in its difference and not in its identity.
So long as they're not simply arbitrary, the transformation and transfer of meaning that's so characteristic of metaphor emerge from this potentiality of the world, from the world precisely _as_ potentiality. The world's an unfinished system, one that the world of art reveals as this power of metamorphosis. The work of art, whether poetic or pictorial, then, can't be said to complete the world. Saying this is to continue to think of the world in theological terms: at first unfinished, the world would still contain the germ of its own fulfilment and the idea of its completion that the artist would eventually realize. Though still pending, the end would already be inscribed and determined from the start. Art, though, has no time for this sort of finalism. It recreates the world on the basis of its virtualities and its overlooked differences. We don't know what the world's capable of; we don't know what differences it already harbours. We do know, however, that the life of art lies in these differences and that it's these metamorphoses that the artist most closely pursues. Between metaphor and metamorphosis, between transposition (of one word into another) and transformation (of one thing into another), there's much more than a vague resemblance: if metaphor has anything to say about the world itself, about its worldly truth, then it's only insofar as the world _is_ this very power of transformation; if, beneath the world of completed and individuated forms, there's the chaotic world of the formless, which nonetheless gives birth to forms, then metaphor is its schema. This relation only begins to make sense once we see Time as its ultimate stake: against the grain of the time that destroys, this metamorphosis signals a time that's creative and artistic.
For the narrator, the memory of seeing Elstir's seascapes evokes a moment in the morning or the evening when, from his window at the hotel in Balbec, the sunlight meant that he saw "a darker area of the sea as a distant coastline, or be filled with joy at the sight of a zone of liquid blue which it was impossible to say was either sea or sky." Such moments, which are rare because they're limited to pure impression and don't allow the intelligence to reinstate the order and separations that it sees as coinciding with objective nature, afford a means of access to poetry and to the domain of metaphor. If there _is_ something like an impressionism in Elstir's art — and in the novel's aesthetics in general, as the hawthorn episode suggests — it's only in the sense that impressions, which are the way in which we access the world in its raw materiality, are always caught up in their own displacement or transcendence, only in the sense that it demands that we linger on them and delve into them. In the same way that Elstir recreates the world by turning it back over to our impressions, before our intelligence has had a chance to select and order them according to its own needs; in the same way that, in his works, nature seems to undergo the most extraordinary metamorphoses — metamorphoses that are still recognized as truths — so the poet who works with words, words that are attributed by that same intelligence, has to recreate each of them and return them to the raw and wild experience that we have of the world. Metaphor's instrumental, then, in extricating language and the world along with it from the objective, distant, and cold gaze of the intelligence, bringing it back to the truth of our evanescent and fugitive impressions. Elstir's work is made up of just such moments, moments that can uniquely reveal to us "nature as it is, poetically." This isn't the same nature that we represent for ourselves and manipulate; it's the one that we inhabit, the one in which we dwell. Dwelling poetically means capturing the world as it's born, before it's overtaken by utilitarian perception and objective intelligence. But it also means seeing the world in terms of its transformative power, taking in all its virtualities, letting ourselves be snatched up by this universe before any demarcation and any finite individuation. It means going beyond the perspetive of outlines, lines and surfaces so as to plunge into the world's depths. And here I'm thinking, for example, of the long description of one of Elstir's seascapes, a description from which I'd like to quote the following lines:
> It was to a metaphor of this sort — in a painting showing the harbour of Carquethuit, which he had finished only a few days before, and which I looked at for a long time — that Elstir alerted the mind of the spectator, by using marine terminology to show the little town, and urban terms for the sea [... ] On the beach in the foreground, the painter had accustomed the eye to distinguish no clear frontier, no line of demarcation, between the land and the ocean. Men pushing boats out moved in the tide as on the sand, which being wet reflected the hulls as though it was water [...] Though the whole painting gave the impression of seaports where the waves advance into the land, where the land almost belongs to the sea and the population is amphibious, the power of the marine element was everywhere manifest.
What we need to hold onto from this description is the way in which certain elements, ones that would usually be perfectly distinct and that no one would ever think of confusing, come together, encroach on one another and end up merging, as if the whole thing took place somewhere upstream of the stable world that our intelligence recognizes and divides up according to its own demands. It's a bit like if the colour orange, gradually stretching beyond its limits on all sides, merged with the colour red on the one hand and with the colour yellow on the other, thereby unveiling the world to which it continues to belong but which we end up forgetting, remembered only at the end, in its completed stage so to speak. Isn't this kind of orange more rich, more concrete, more true, even, than the one we're used to seeing? Isn't Elstir's nature, in its very metaphor and metamorphosis, more open, more fluid, more "becoming" than the fixed, rigid version that meets the needs of knowledge and of practical life and that we tend to think of as the real world? The elements in the seascape communicate with and echo one another; they even blend, all the while without ever becoming blurred in an amorphous and undifferentiated mass, without reducing the world to chaos. Seen as a totality, as what I'd call a _virtual_ landscape and not a series of discrete and contiguous states, the seascape appears as a multiplicity of terms linked together by relations of reciprocity. The prevailing Law here, the only poetic law and the one that's realized in metaphor, isn't the law of contiguity or resemblance; rather, it's the law of encroachment. There's a point — a threshold, rather — beyond which land and sea aren't just placed alongside one another but, in fact, _become_ one another. This doesn't imply that they've become _the same thing_ or that they've started to look so much alike that they're at the point of becoming one and the same. No, the fact that they've been brought together doesn't stem from their resemblance or even from their proximity in _actual_ space. I'm not saying that they've become impossible to distinguish from one another or that they've melted into each other. In the episode where the magic lantern fills the bedroom in Combray with a Merovingian light, both delightful and alarming, it's essential that the white doorknob stays the same old doorknob where Golo's "pale face as noble and melancholy as ever" is reflected or even embedded. Both worlds, both spaces and both eras blend and join into a single reality that's irreducible to either one or the other, all the while remaining tangible and alive. In the same way, Elstir's seascapes disclose a plane where, before they are actually differentiated into distinct and fully individuated entities, the land and the sea, are eminently compatible or, as Leibniz would have it, "compossible." The plane in question isn't a simple point of contact between the two elements, a line of demarcation, but a horizon that's "situated at a somewhat deeper point, beyond appearances themselves, in a zone slightly further back," whence their mutual and virtual belonging together. To pass from one plane to the other is to cross the threshold of poetic space — the very space that the bedroom at Combray unveiled — and dwelling in metaphor. The land and the sea, the seascape suggests to us, have a common past, a common origin that endures through their individuation and the divisions to which we subject them. And it's on this past or this plane that the artist focuses. In Elstir's seascape, we find the sea in the land and the land in the small town; a world of tiny elements, of details and specificities that are shared by the various components and environments, gradually emerges, suggesting another reality or another plane of the real, one that only art's able to make visible. Why? Because the artist takes us beyond — it's probably more accurate to say beneath — the world already determined and divided up into substances with fixed outlines and permanent attributes (the world that we tend to think of as the _true_ world). The world the artist gives us isn't a world of static and stable objects, of individuated things. The artist does give us the real world, but it's a world that's only available to a gaze that's able to see the virtual correspondences that underlie existing resemblances. If, for example, the actual sea and seaport that Elstir paints have nothing to do with the sea and the seaport on the canvas, this isn't because he's translating something solid, wet and fragrant into something purely visual. Rather, it's because the setting he's painting is one in which land, sea and men communicate, one that's prior to any division and that allows for the most surprising, even insane correspondences. Such madness, though, is the very essence and wisdom of poetry, as Baudelaire so clearly saw.
What's peculiar about this experience — like so many others related throughout the novel and which, taken together, make up the very meaning of experience — is that a term or a series of terms turns out to have been contained in another term right from the start, comprising thereby its most intimate content and one that might well never see the light of day. One series of terms is contained in another like a myriad of secrets, echoes and correspondences that, taken together, make up an alternative world, a parallel and uncharted universe. Once intuited, it captivates the narrator and consumes him with the desire to mine its infinite wealth, to put it into new words, to depict the thousand and one ways in which this world combines impressions and sensations, individuals and situations. The metaphorical sequence set off by the view of the sea at Balbec from Marcel's hotel room affords a powerful illustration of what Merleau-Ponty calls "the occult trading of the metaphor" and a stunning echo of Elstir's work:
> I went to my room: the painting on show in the window-frame kept changing as the season advanced [ _first image:_ the window-frame as the frame of a painting, pictorial and somewhat classical]. At the beginning of my stay, it was broad daylight, its tones sometimes dulled by bad weather; the sea, through the glaucous glass [ _second image:_ the sea is at once caught in the thickness of the glass and seen through it; it's made visible by its glaucous transparency, which, in turn, owing to its opacity, leads one to make it out] bulging [ _third image:_ an extension of the previous one, perfectly blending the semantic field of glassmaking and that of the sea: glass is blown and the sea is said to swell or balloon; two different registers converge completely and become one, disclosing thereby a new reality, namely the sea-glass] or with its round waves, held between the iron uprights of the frame as though set in the lead of a latticed window [ _fourth image_ and a new register: the sea, seen as wrought by a glass-maker, is itself set like a jewel, the work of a goldsmith, which is immediately likened to the work of a stained glass artist, an image that brings us back to glass, all the while adding on a _fifth image_ of an architectural and religious nature] teased out along the deep, rocky fringe of the bay triangles plumed with spray which hung motionless, touched in with the delicacy of down or a feather [ _a new image:_ the foam as the feather on an arrow or a cluser of small arrows that, motionless, evoke the quiet of the sea] pencilled by Pisanello, and fixed by the creamy white never-fading enamel used for a fall of snow in glass-works by Gallé [the images of the painter and the glassmaker, of drawing and glass, merge with a _sixth image_ , a final semantic shift, namely the shift from sea to land, from foam to snow, thus epitomizing, at the level of writing, what Elstir accomplishes through painting].
For the narrator, the only way to remain faithful to these experiences, to the joy they provide as well as to the world they unveil, is to become a writer. Metaphor is the image that crystallizes this promised land, the tool that allows him to explore its treasures. Whether he's discussing Carpaccio's _Legend of St. Ursula_ or making his own forays into the subject, Elstir always comes back to the structural metaphoricity of nature. Of Carpaccio he says that "it was unclear where the land finished and the water began, what was still palace or possibly ship, a caravel, a galleas, Bucentaur." And of an afternoon spent at the races, he tells Marcel that he'd have liked to have painted the transformation of every individual that he saw, as well as the deep truth that emerged from that afternoon as a whole:
> Look at how all things are transformed in that vast and luminous space of the race-course [... ] with women of exceptional elegance, amid a wash of moist light, a Dutch light, and you could sense the penetrating chill from the water reaching up into the sunlight. [... ] How I wished I could capture it
Here, metaphor becomes oxymoron, Elstir's appeal to the sun's coldness just as appropriate as Jean Santeuil's evocation of the dryness of the sea at Réveillon.
None of this, however, amounts to a merely subjective view and none of it's a product of what could casually be described as a "poetic temperament." Rather, metaphor's rooted in the _being_ of the world and comes out of it. It's nature itself that shifts like that, that's swept away in its own transposition, its own excess or overflow. And this, in fact, is where its beauty lies. So far as Proust's concerned, if we want, like Elstir, to capture the beauty of the sea, we need to avoid fixating on it since, like everything else, it's essentially _not_ in its place. It's _really_ not where we'd expect it to be, even though it's _actually_ there. It _is_ and _reveals itself_ only through the process that characterizes it, to wit its movement of expansion and encroachment, of displacement and deviation, in short, of _transposition_. When I talk about this process in terms of essence, though, I'm not talking about an ideal and atemporal reality; rather, I'm talking about a line through which a substance escapes its own existence as defined by actual spatial and temporal coordinates, and enters the realm of virtual compossibility I was just evoking. Essence is on the side of life only insofar as it signifies difference. When Proust talks about "essence" he's not referring to an idea in the Platonic sense, to what's itself and nothing else, to a generality extracted from a collection of specificities. Quite the opposite, in fact: he's referring to a difference, a singularity that defines the point or the threshold where a thing communicates with and becomes something else. It is, as I'll show, a matter not of being (or existence) but of becoming. The _essence_ of the sea is in the sun. But the sun's also somewhere else, and so on and so on and so on. It's a chain (but not an associative one) that holds everything together, but in which nothing or nobody is ever in its place, once and for all as it were. If essence refers to a point of view, it is, as Deleuze claims, a privileged point of view, at once individuating and preindividual. Deleuze offers an apt summary of all this when he highlights, without actually naming it, the singularity of Proust's view of and use of metaphor in relation to associationist psychology on the one hand and Aristotelian ontology on the other:
> Style begins with two _different_ objects, distant even if they are contiguous: it may be that these two objects resemble each other objectively, are of the same kind; it may be that they are linked subjectively by a chain of association. Style will have to sweep all this on, like a river bearing the substances of its bed; but that is not what is essential. What is essential occurs when the sentence achieves a viewpoint proper to each of the two objects, but precisely a viewpoint that we must call proper to the object because the object is already dislocated by it, as if the viewpoint were divided into a thousand various non-communicating viewpoints, so that, the same operation being performed for the other object, the viewpoints can be set within each other, setting up resonance among themselves, a little as the land and the sea exchange their viewpoint in Elstir's paintings.
So, reaching the point of view of essence isn't a matter of identifying the genus or the Idea that two terms or two series of terms have in common. Genette's perfectly clear about this, even though he infers from it that there's an irreducible tension in Proust between the work of metaphor and the work of essence: he wonders whether essence isn't going to be found "more on the side that differs and resists, on the irreducible and _refractory_ side of things?" I'd agree, simply adding that it's only at the cost of just such a radical displacement of the sense of essence _and_ of metaphor that metaphor can be seen as the way of getting to essence. There's no such thing as the realism of essences, any more than there's such a thing as the impressionism of metaphor, or even such a thing as the reality of things and the reality of impressions. Rather, there's one reality, the reality of expressionism, through which essences, wrapped up or implicated in things, are developed and explicated through memories and sensations. To say that there's a relation of expression between essences and the phenomenal world is also to say, following Deleuze, that every essence corresponds to a movement of explication and complication within things: if the sensible "explicates" essences, essences "complicate" things: if things are "inherent" in essences, essences "implicate themselves" in things. Metaphor is the image or schema of this "complex" dynamic.
In its extraordinary capacity to draw together vastly different places — even vastly different eras — metaphor is "like a horse-shoe," as Proust claims in the _Cahiers_. As an image, it discloses the gathering power of nature, something that intelligence, which is essentially diachronic and practical, can't do. With metaphor, a different nature's disclosed or nature's disclosed differently: a poetic and sensible nature, populated by resonances and echoes. Before this, before the discovery of the laws of poetic nature that govern the relations between things, sensations, and memories, there's nothing and, most importantly, no style. Here's an example. Proust tells how, in the Preface to his translation of _Sesame and Lilies_ , he evokes the cakes he ate on Sunday and their "indolent, sugary aroma." As he points out, he could have just as easily have described the shop, the closed shutters, how the cakes smelled and tasted, in short all of the things that seem "real." If he had, though, that would have been the degree zero of style and, consequently, of literary reality. However, by using phrases like "indolent, sugary aroma," Proust goes on, "I establish, above this flow [of the various sensations involved], a relation that brings them together, holds them together, immobilises them." It's not much, he adds, but it _is_ the beginning of style. It's the moment when intelligence, which normally stands back from its object and hovers on the threshold of the world, actually "incorporates" matter, merges with it and, in doing so, embraces movement. It's the moment when, as Bergson would have it, intelligence becomes intuition. The words I just cited are taken from a section in which Proust lingers on an exquisite metaphor from _L'Éducation sentimentale_ , a metaphor that's all the more remarkable, he stresses, given how weak Flaubert's work is when it comes to images. He'll go so far as to say that "in the whole of Flaubert, there is perhaps not one good metaphor," leaving to one side, he quickly adds, that we're dealing with a grammatical genius "who, by the entirely new and personal use that he made of the past definite, the past indefinite, the present participle and certain pronouns and prepositions, renewed our vision of things almost as much as Kant, with his Categories, renewed our theories of Cognition and of the Reality of the external world":
> It looks as though the word _talent_ , when used with reference to the arts, consists of bringing the artist into an ever closer relationship with the object to be expressed. So long as there is any degree of separation between them, the task of the artist is incomplete [... ] It seems that, in previous centuries, there was always some degree of distance between the _object_ and the great spirits who discoursed upon it. But, in, for example, the case of Flaubert, the intelligence — which was not, perhaps, his strongest point — manages to _identify itself_ with, say, the shuddering movement of a steamboat, with the colour of churned foam, with an islet lying out in the bay. A moment comes, in reading him, when one is no longer conscious of the writer's intelligence (even when, as with Flaubert, that intelligence is somewhat mediocre), but only of the moving ship — "running into floating bales of timber which bobbed up and down in the brisk agitation of the waves." That word "bobbed" shows us intelligence transformed, intelligence that has become part and parcel of the physical scene. Similarly, it can penetrate the tangle of heath, the trunks of trees, the silence and the light of the underbrush. Is it not the first concern of any artist intent on style to achieve just this transformation of energy in which the thinker disappears, and the objects which he is busy depicting become real and actual to our eyes?
As we saw with Elstir, style doesn't involve brute impressions. Style's the work of intelligence, even thinking, but an intelligence or a thinking that's integrated into the movement and the becoming of matter. As Bergson puts it, style involves a sympathetic form of intelligence, not just an analytic one. Proust's critique of intelligence only applies to a certain kind of intelligence, namely the kind of intelligence that, by representing and analysing the world, translates sensations and perceptions into objective data, sweeping over the world instead of plunging into it. And while impression's the only "criterion of truth" for the writer, it's still not enough: it needs the assistance of the mind in order to "elucidate its truth." Metaphorical effort or "style" aren't the simple expression of a brute impression: they are its continuation, its depth and lining. Beginning with an impression, they lead us beyond it. Sympathizing with matter isn't the same as representing it. But it's also not clinging to it in some sort of immediate and pre-linguistic presence. Rather, it means transposing and translating it ("the writer's task and duty are those of a translator"), not into a radically different language or another reality altogether — thinking along these lines simply reinstates the idea of a world in itself and a world of phenomena, a supersensible world and a sensible world — but into this implicit or tacit language that's the language of the world itself. Ultimately, the impression is "for the writer what an experiment is for the scientist, except that for the scientist the work of the intelligence precedes it, and for the writer it comes afterwards." Proust's aesthetics might well be a form of impressionism, provided we understand the latter as extending the meaning of impression, as disclosing the means by which the impression transcends or exceeds itself — expresses itself — in its own sense: this is the moment when impressionism overcomes itself and is actualized as expressionism. And metaphor's the figure through which such an excess or overcoming is schematized.
The sentence from _L'Éducation sentimentale_ that Proust uses as his example explains, formally speaking, the nature of "that something which brings the artist into an ever closer relationship with the object to be expressed": a substantive referring to a mere thing is taken as the subject of a verb of action and, through that, acquires the character of something "animated." On the other hand, the metaphor bears directly on the verb through which it illuminates the whole sentence. Remember that Aristotle restricted the use of metaphor to nouns which couldn't possibly specify time or action. However, the kind of metaphor we're dealing with now talks of things as verbs and takes movement itself as its object. So have we gone beyond metaphor, gone over its horizon? No. All we've done is drawn that horizon back so that metaphor can now behave like a verb and describe the world as one that's in a state of becoming: and it's here that the fluidity of language merges with the fluidity of matter itself.
It shouldn't come as a surprise, then, to find that the novel's filled with moments like this: while Albertine's sleeping she's caught becoming a flower and Marcel, as he holds the copy of _François le Champi_ in the Prince de Guermantes' library, feels within him, as if against his will, the child he's always remained. By the same token, Albertine doesn't look like a bird (unlike Madame Verdurin who, like a grotesque parrot, sits on her perch spraying clichés and hackneyed judgments). Held captive, however, under the yoke of her jealous lover who wants to possess every inch of her soul, she grows wings, numerous wings, and an infinite number of ways to escape him. The more Marcel wants to possess her, the more unattainable she becomes. She becomes a bird that, infinitely free, turns the tables and tortures and possesses her lover. As soon as she's asleep, though, she becomes something entirely different; now she's a plant that Marcel can finally possess. Odd as it might seem, Albertine only becomes truly lovable once she gets rid of her human garb and takes on that of a plant:
> Lying at full length on my bed, in a pose so natural that it could never have been adopted deliberately, she seemed to me like a long, flowering stem that had been laid there; and that was what she was: normally I could dream only when she was not there, but at these times the power of dreaming returned as I lay next to her, as if in her sleep she had turned into a plant. In that way her sleep realized, to a certain degree, the promise of love; when I was alone, I could think about her, but she was not there, she was not mine. When she was there, I could speak to her, but was too removed from myself to be able to think. When she was asleep, I did not have to speak any more, I knew she could not see me, I did not have to live on the surface of myself. By closing her eyes, by losing consciousness, Albertine had put off, one by one, the various marks of humanity which had so disappointed me in her, from the day that we first met. She was animated only by the unconscious life of plants, of trees, a life more different from my own, stranger, and yet which I possessed more securely. Her individuality did not break through at every moment, as it did when we talked, through unconfessed thoughts and unguarded looks. [... ] Watching her, holding her in my hands, I felt that I possessed her completely, in a way I never did when she was awake. Her life was subject to me, was breathing out its light breath in my direction.
When she's asleep, Albertine has a completely different sort of energy, an alternative material configuration: she's slowed down to the point of embracing a modality that's more akin to that of a flowering stem. She doesn't just _look_ like a flower; instead, she actually _becomes_ one. Motionless, her lines and contours still suggest growth, the very kind of movement that Aristotle associated with life. This life, though, is now that of the inanimate world: her figure, her features, her posture all make up a flower. But this process of becoming goes even further: as she falls deeper and deeper asleep, Albertine-the-plant becomes something else again:
> I listened to that mysterious, murmuring emanation, gentle as a soft breeze over the sea, fairylike as the moonlight: the sound of her sleep. So long as it continued I could dream of her and look at her at the same time, and when her sleep became deeper, touch her and kiss her. What I experienced then was a love for something as pure, as immaterial, as mysterious as if I had been before those inanimate creatures that we call the beauties of nature. And indeed, once she had fallen into a deeper sleep, she was no longer just a plant; her slumber, on the edge of which I dreamed, experiencing a new, limpid pleasure of which I would never have tired and which I could have gone on enjoying indefinitely, had become for me a whole landscape. Having her asleep at my side offered something as sensually delicious as my moonlit nights on the bay at Balbec, when the water was calm as a lake amid scarcely moving branches, and one could lie on the beach forever, listening to the sound of the sea.
Sleep isn't just a state of body and mind: from the lover's point of view it transports the loved — and love itself — onto another plane: it signals the shift to a new organization of bodies and minds, to a new reality. At the very heart of sleep there are a number of stages, all of which coincide with a series of breaks and metamorphoses: starting with its organized and usual state, the body becomes disorganized and recomposes itself, reinventing itself as it changes. It becomes a plant (again) or a landscape, and who knows what other virtualities it has in it? Can the sleeping Albertine be the same person who torments Marcel when she's awake? To a degree, yes: the novel involves a unique process of becoming and its real ontological and artistic merit is that of letting us see one state transform into another. This sameness isn't what matters, though; what matters isn't the possiblity of identifying an essence or an eidetic core beyond or even through these metamorphoses. The point isn't to record the variations of an irreducible identity that's there from start to finish and that would be disclosed either at the start or at the end. The point isn't to determine what Albertine is, to draw out her essence, to turn her into a substance and then say that we can see what happens to her, her accidents. No: Albertine's complete throughout all her metamorphoses; she's a bird, a plant, a landscape, she's Odette, Andrée, Marcel, all of them at once and even simultaneously, while we can never say what she truly is, _essentially_. Like the world in general she's always in a state of becoming, a composition of matter that connects with others and, doing so, becomes something else entirely.
And the same goes for Marcel:
> if in the library I take down _François le Champi_ , a child immediately rises up within me and takes my place, the only one who has the right to read the title _François le Champi_ and who reads it then, with the same impressions of the weather outside in the garden, the same dreams as he formed then about other countries and about life, the same anxiety about the future.
"A child," he's careful to say, just _some_ child or other, and not "myself as a child," or "my own childhood." This child isn't something that _belongs_ to the narrator, but the presence in him of an anonymous or impersonal being that possesses _him_. In place of a personal pronoun, what we have is an indefinite article that suggests the emergence of a stranger or an intruder from the depths of the narrator's soul. That intruder, though, as he realizes a little later on, is actually himself, and more so than the self we usually call our own: it's a self that's at once forgotten and preserved, eternal and intact, and that sits alongside the self that's advancing inevitably towards its own death. It's a self that returns and comes to the surface from a place that the narrator cannot recognize. And if that self rises up and takes the place of the narrator's consciousness and lived experience, it's because, up until that point, he occupied a place that he thought was his, but where he didn't belong. That place belonged to the self, never announced, never expected, that returns to haunt him. In these sorts of reminiscences, as we saw earlier on, I come back to myself, but only as that part of myself that's relegated to the bottom of my soul and that surprises me, like ghosts always do: I thought that self had gone, living on only in memory — a faint and dead representation of a _tableau vivant_ — but here it is re-emerging, returning me to myself and to the past, taking my place, taking over my present. My place, though, or what I normally think of as my place, is actually the place of the dying self, and the present that gives way to the past is the place of my death: death isn't still to come; it's already here, in each present moment insofar as their inexorable sequence draws me in towards it. Life, the life that's not survival and that doesn't grow older, is the life of the pure past. True time is the time which, doubling the present, never fades away and becomes eternal from the start. In this new configuration, it's the present that fades away before the past, and not, as we tend to think, the past that fades away to make room for the present. At the end of the day, the present is the place where I am farthest from myself, and as it were alienated from myself. If, as we recall from Marcel's numerous disappointments and disillusions, the present really is a source of suffering, it's because it suffers from the fact that it _is not_ , from the fact that it _is_ a lack, or an empty shell. All the same, we need this deficiency or poverty of the present so that the past can sneak up from behind it and, on the odd occasion, take its place; the present has to be porous for eternity to penetrate. It's not the present that's real, then, or the immediate impression. What's real, rather, is "a certain relationship between these sensations and the memories which surround us simultaneously," a "unique" relationship, as Proust promptly stresses, which requires, as I showed earlier, that our vision, ordinarily so monoscopic, cinematographical and chronological, becomes _stereoscopic_.
This way of seeing things is the artist's way, a way of seeing that brings "two different terms together permanently." Such is, it seems to me, precisely the work of style and of metaphor in particular. And metaphor's at its peak when it schematizes involuntary memory as the supplement to the lack that the present itself is. When it turns time into the very object of its figure, it becomes the very symbol of literature and art in general:
> One can list indefinitely in a description all the objects that figured in the place described, but the truth will begin only when the writer takes two different objects, establishes their relationship, the analogue in the world of art of the unique relation created in the world of science by the laws of causality, and encloses them within the necessary armature of a beautiful style. Indeed, just as in life, it begins at the moment when, by bringing together a quality shared by two sensations, he draws out their common essence by uniting them with each other, in order to protect them from the contingencies of time, in a metaphor.
The work of metaphor is more "true," then, than what's achieved in a realist description because it captures in an image and fixes for all eternity a unique relationship between the present and the past, between sensation and memory. As we've already seen, it's the schema of time itself and of this poetic law of nature that's been our real concern up to now:
> Had not nature herself, from this point of view, set me on the way to art, wasn't she herself the beginning of art, she who made it possible for me, often after a long interval, to recognize the beauty of one thing only in another, noon at Combray only in the sound of its bells, mornings at Doncières only in the hiccuping of our water-heater?
Like truth, beauty happens belatedly and not where we'd expect it to. It responds to a law of deviation and absence, of differing and difference: it's always missing in its own time (the present) and place (presence). And if the writer's someone who tries to look for something where it's not and to look for the truth of something in something else, it's because he or she knows that it's only by doing so that we have a chance of finding it: this is how Combray springs out of a cup of tea, how little Marcel emerges years later from the Guermantes' library and how a row of trees, once so tedious to look at and to describe, comes out of the sound of a knife against a plate. The time-space of metaphor, of the image or the figure that both gathers and separates, that connects and disconnects, is the translation into the domain of art of an ontological spatio-temporality where certain impressions, certain different realities coexist, despite their separation in actual space and time. Art owes its power to this alternative temporality and spatiality. As a trope or a figure, metaphor's nothing but the transposition of this transformation into the artistic sphere. But the reason why this transposition is justified, why art's not only possible but necessary, lies in the fact that metaphor itself is _of_ the world and, in reality, is nature's enigma and poetic law. Proust's use of metaphor finds its impetus and its justification in an essentially dia- and metaphorical view of nature itself: Proust's poetics just _is_ a metaphorics and metaphor itself draws from an onto-diaphorical source. It's nature itself that discloses the beauty of one thing in another, that operates through this shift, a shift that's also a transformation. Nature's what puts us on the path to difference. Far from reducing or effacing the gap that creates a difference between terms, nature sees this distance as what brings them together and this gap as the condition of their relationship. What it does is probe and deepen this gap; this is where it dwells. Metaphor's a gathering force, then, a _legein_ , perhaps even the completed form of human _logos_. But the terms that it draws together are gathered around their difference, not above them. The poet doesn't bring things together by extracting a quality or a genus that would be common or identical to the terms with which it's dealing; rather, he or she does so by delving into difference, dramatizing and exposing it as this gathering space (and time). Metaphor's not concerned with comparing terms to one another or with drawing out resemblances or common traits, thereby suggesting that its unity is one of genus or species, of identity in short. This sort of unity is that of _logos_ (and even of poetic _logos_ ) in the Aristotelian sense. What it's not, though, is _Proust's logos_. Rather than reducing each term to the qualities that it shares with others, the poetic _logos_ reveals the side through which each one differs from itself and communicates with the other via this difference. The poetic _logos_ gathers by connecting two terms in their difference, by isolating the time and the space that simultaneously separates them and brings them together, extracting the singular points through which they communicate, beneath their resemblance and despite their differences. Ultimately, the poetic _logos_ is the supreme gathering force ( _legein_ ) — not by means of concepts, or synthesis, but by way of metaphors — which allows us to overcome the opposition between metaphor and metonymy, between synchrony and diachrony. The relationship that's uncovered in this way is one of virtual mutual belonging together, beyond whatever divisions actual reality might want to make. The poet or the writer does two things, then: he or she reveals a distance — a distance between two perfectly individuated moments or situations — that's also a proximity. He draws them together precisely by keeping them apart and disclosing their hidden correspondence. Poetic correspondence differs from the scientific or epistemological correspondence in the sense that it does not relate to a concept (or a function) and a thing, or a state of affairs, but to an image which reveals that two heterogeneous and seemingly incompatible impressions (or series of impressions) belong together.
Knowing or recognizing one thing in another: such is, to my mind, the most economical definition of Proust's notion of metaphor. But while this definition reflects an original conception of metaphor, it also involves a new conception of recognition and representation.
We need to distinguish between two senses of recognition. According to the first sense, recognition takes place on the basis of common identities and differences are established on the basis of what's alike. This is the normal way of thinking about recognition. It's recognition as defined by classical metaphysics. It's what allows Plato to condemn painting as the mere representation or imitation of a more truthful original, and even as twice removed from the actual origin of things (the Ideas). So far as Plato's concerned, the only way we can recognize a phenomenon is through prior knowledge — diffuse or forgotten, yes, but always innate — of its true essence or Idea. Only the Idea grasps the thing in its identity as a thing, the thing as what it is and nothing else. Identity's distinguished from differences in the same way that the substance is distinguished from its accidents, or the essence from the non-essential. Given the direct and irreducible connection between classical metaphysics and the understanding of objects in terms of their self-identity (in terms, that is, of their identity to what's essential to them), it's hardly surprising that classical metaphysics ends up using recognition as its ideal model. And it's precisely this sense of recognition that Deleuze criticizes. When we're talking about recognition, he says, we're talking about "the harmonious exercise of all the faculties upon a supposed same object." As Descartes says of the piece of wax, then, "it is of course the same wax which I see, which I touch, which I picture in my imagination, in short the same wax which I thought it to be from the start." Recognizing an object in this sense, then, means seeing it in its very identity through different faculties and beyond whatever differences it may happen to manifest. Only this identity can be the object of real knowledge. And so knowing something always involves recognizing it. Deleuze sees "recognition" as one of _the_ fundamental characteristics of the classical "image of thought" or of dogmatic metaphysics, and works accordingly to expose it and replace it with another image altogether. What this postulate of recognition reveals is the presupposition (which Deleuze calls "common sense") of a free and natural agreement of the faculties directed towards the object's form of identity and constituting the subject in his or her identity as a knowing subject. To this view of knowledge as recognition and of thought as a natural and spontaneous exercise, Deleuze opposes a view of thought as _learning_. Learning, he claims, is a result not of an agreement or a convergence between faculties, or between a faculty and its object, but of a disagreement and a divergence. Thought (as opposed to "recognition") stems from a shock that comes from outside and is, by definition, impossible to anticipate; we're _forced_ to think, to create tools (concepts) that can measure up to what's happening to us. In every true creation, there's an element of violence, an event to which we're summoned to respond. According to Deleuze, this is precisely the logic of Proust's novel. _In Search of Lost Time_ is a _Bildungsroman_ , then, and not a novel of recognition insofar as the narrator finds himself forced to _produce_ the meaning of what happens to him. As we saw earlier, the truth that's found isn't the truth that was looked for: we never find what we're looking for, only something else. Put differently: what we're looking for is never where it should be, never in its place, but always somewhere else and authentic experience is naturally metaphorical.
So, alongside this first sense of recognition, there's another sense, the very one that Proust focuses on in his use of metaphor, and the true principle of knowledge, as he himself states. It's no longer a matter of knowing what we already know or of recognizing what's familiar to us, as Gadamer rightly points out. To my mind, and contrary to what Gadamer says about Plato, it's no longer even a matter of recognizing an essence, irrespective of its accidents. What we're talking about now, rather, is a recognition in which differences prevail over substance and transform it beyond its primary and false identity in order to disclose it as this reality in the making or in becoming. As far as this second meaning goes, only differences identify; only differences designate the thing in its being (which is precisely not a quiddity). What we recognize, then, aren't identities but differences, virtual tranformations or becomings.
All of which is to say that "recognizing one thing in another" implies:
1. that the first thing was embedded or enveloped in the other thing from the start;
2. that a particular gaze — the gaze of the artist — can see that first thing more clearly in another than in itself, that is, in its self-presence. The thing's revealed for what it is in its deviation from and difference with itself, then; its essence is disclosed in its own displacement or transposition, and the revealing of this truth presupposes a creative gesture;
3. that essence can't be captured immediately, only indirectly.
In this sense the structure of recognition's rather like the structure of variation as it's understood in music, even more so than it's understood phenomenologically (which, after all, implies the identification of an invariant or a stable essence). But we need to be precise here, and the question of variation is exactly what Barthes evokes when he's talking about Proust, suggesting that we need to make a distinction here. Think of what Brahms does with Haydn, starting with a strong theme that's formed at the start and that's easily identifiable in order to develop it without ever making it unrecognizable, the theme remaining identical to itself throughtout its variations. But think, too, of what Beethoven does with Diabelli's waltz, starting from a weak, almost insignificant theme and allowing it to materialize, shaping it through a series of rich and differentiated variations which lead it towards something else altogether and through which it actually takes shape, forcing us to take notice of it, to recognize it. Maybe we need this minimal degree of identity if things are going to remain legible. As Barthes explains, a theme's given at the start, but it's "a very basic theme," "a little bit as a form of mockery." But this isn't what's important; what's important is the fact that, when we recognize the theme, we say: who could have imagined it capable of such variations and such twists? As with Elstir's seascapes, the point is to recognize "a kind of new creation of the world," a world that's brought to light as if for the first time. What matters is the eventual power of the work of art, in other words its capacity to produce a surplus or an excess — and not a lack, as Plato assumed — as regards the original. The lack isn't on the side of representation and recognition; rather, it's on the side of the real or of being itself. All of which implies a real reversal or inversion of Platonism: the work of art has now become an event that draws its ontological value _not_ from what it represents, but from what it generates, namely a new time and space.
What's at stake when we (re)cognize the beauty of something in something else is more than just an aesthetic law and more, too, than the very definition of metaphor. What's at stake is the possibility of inverting the nihilistic course that constitutes the deepest tendency of the novel. The reason why Marcel's so disappointed by reality, the reason why it seems infinitely less inspiring than the reality invoked by his imagination and the power of his desire has to do with the fact that he's too attached to life as it's immediately given. He expects reality to match his sublimated image of it and it hurts him when he sees that the present can never live up to his hopes. The present, he thinks, is indifferent; only the past and the future are truly alive with infinite possibilities. As soon as he realizes that the present is caught in its own displacement, though, as soon as he sees that it's not an indifferent vacuum and that this lack is an opportunity, he can affirm time in its totality. There's truth only insofar as truth returns and it always returns differently — in the Work of Art. The beauty of Combray will always only be found in its bells and Albertine in Venice, after her death.
Chapter 5
* * *
Dress or patchwork?
* * *
Readers of Proust will no doubt have in mind the page from _Finding Time Again_ in which Marcel compares his book with a dress, and a cathedral. Thanks in part to Luc Fraisse's _L'Œuvre cathédrale. Proust et l'architecture médiévale_ , we know that Proust's concern with cathedrals goes back to the years when he was planning and writing _Jean Santeuil_ , and so precedes his translation of Ruskin's _The Bible of Amiens_ by a few years. We may even want to agree with Fraisse's claim that "for the writer, the study of cathedrals is a way of conjuring the failure of _Jean Santeuil_ , which occurred as an excess of dispersion and fragmentation. The cathedral embodies stability, concentration, continuity also, and is opposed to the perpetual risk of incompletion." Yet aren't dispersion, fragmentation and discontinuity defining and _constitutive_ features of _À la Recherche_ itself? Is Proust's great novel not also a work that integrates the very excess that led him to abandon his first novel? Further still, does this novel not assume the risk of incompletion fully, affirming it as such? If the image of the dress, as well as that of the cathedral, characterizes indeed a certain level of unity of the _Recherche_ , and accounts for a certain plane on which the novel unfolds, it exhausts neither its nature nor its architecture. Juxtaposed to the first plane, or perhaps cutting across it, lies a second plane, less apparent, and less controlled, almost by definition. This second plane reveals a distinct coherence, as well as a unity, which challenges the classical conception of the novel, if not the classical concepts of philosophy itself — those very concepts and values which Fraisse claims Proust saw in cathedrals: continuity, stability and concentration. By contrast with the unity of the cathedral, or that of the dress, I'll compare the unity of the second plane with that of the patchwork. Let me stress from the start, however, that the singularity of Proust's great novel, whether at the level of its architecture or of its characters, lies perhaps in its ability to gather those two planes in a kind of productive tension, one that elevates each plane to a superior degree of expression. We'll see this tension reach its climax in Marcel's first kiss to Albertine, which I'll analyse in some detail. The truly philosophical point I intend to make is that this tension and intersection between the two planes of the novel has an ontological dimension, applicable to — and verifiable in — a number of other fields. If there is a philosophical conclusion to draw from the question we'll be concerned with, and if there is, for that matter, something like a philosophy of Proust's novel (one we should be careful to distinguish from Proust's so-called "philosophy," or "philosophical views"), it will have to do with the manner in which two different types of planes — a plane of organization and a plane of fragmentation — coexist and interact. It will have to do with the manner in which the structure, outline, and organic unity of the work find themselves confronted with an excess they cannot integrate, a fracture they cannot reduce. It will be a matter, therefore, of analysing the manner in which Proust's novel opens itself to an excess that once threatened to annihilate it, but which it is now in a position to integrate and recognize in its ontological dimension and its artistic potential.
In the second part of _Proust and Signs_ , written for the second edition, Deleuze raises quite explicitly the question I'm concerned with, namely, that of the unity of the _Recherche_. Specifically, he raises the question of what I would call the _other_ unity of the _Recherche_ , thus taking up the challenge that Proust's novel poses for philosophical thought, and to which I was just alluding. This unity, he suggests, is not the one that Proust had in mind initially. It is not even, I would claim, the type of unity that one discovers only retrospectively, almost despite oneself. Such a unity is precisely the one that Proust evokes in _The Prisoner_ , in a passage where he discusses Wagner and nineteenth-century literature:
>... Wagner, as he took from his desk a delicious fragment to introduce, as a _retrospectively_ necessary theme, into a work of which he had not yet dreamed, when he was composing it, and when, having written one mythological opera, then a second, then more, he realized he had composed a Ring Cycle, must have known something of the same intoxication Balzac felt when, casting over his novels the eyes of both a stranger and a father, and seeing in one the purity of a Raphael and in another the simplicity of the Gospel, he suddenly saw, _with the light of hindsight_ [ _une illumination rétrospective_ ] that they would be even more beautiful if brought together in a cycle in which the same characters would recur, and added to his work the final brushstroke, the most sublime of all. This unity was an _afterthought_ , but not artificial.
However remarkable, this unity revealed only in the end, this "retrospective illumination," which discovers a necessity all the more effective that it does not abolish chance, is not the unity I want to discuss here. For such a unity still partakes of an ideal of organicity, where beginning and end, however distant, finally meet up, where every piece fits in, every part refers to the whole, and where the work as a whole follows a thread, however hidden. This ideal governed the construction of cathedrals, as well as that of the _Recherche_ itself, up to a point. The French word _plan_ , which Deleuze turns into a concept, summarizes it rather nicely. The _plan_ can refer to an architectural plan, a battle plan, or the structure and outline of a book. In each case, it carries a sense of purpose and goal. At school, French children are told never to even begin a piece of writing before having a clear _plan_ , that is, without knowing exactly where they are going, what their final goal is, and how their writing is going to be organized. Oriented to and by its own end, the _plan_ is essentially teleological. In that respect, the _plan_ also refers to the unity or coherence of the piece in question. Inevitably, it contains a number of parts — most often an introduction, a central part, and a conclusion — which all work together to achieve a common end. It constitutes an organic totality, each part reflecting the whole and containing it virtually. It is both linear (it has a beginning, a middle and an end) and circular (the end justifies and gathers the beginning).
The unity I'm interested in highlighting, however, is of a different kind. It belongs to another logic, and another conception of the work, perhaps less rooted in the nineteenth century. It reveals another kind of _plan_ , a plane that is altogether different from the plane of organicity I've just alluded to. In what follows, I'd like to analyse it from a twofold perspective: that of the structure of the novel as a whole, and that of some of its characters and episodes.
While complex and intricate, the structure and unity of the _Recherche_ are well known. We know that, from the start, Proust knew how his novel was going to end. Initially, the novel was to comprise two volumes only, _Time Lost_ and _Time Regained_. The end was planned from the beginning, with the beginning. This means that, from the arche-teleological point of view, the novel did not evolve much from the moment it was first conceived. Its unity, outline and circular structure were firmly in place from the start. What was missing, however, was the middle. And it is from the middle, precisely, that the novel grew, almost out of proportion, from two to seven volumes, between the publication of the first volume in 1913 and that of _Time Regained_ in 1922. It grew in part thanks to a technique of pasting and folding which Céleste Albaret, then Proust's housekeeper, _confidante_ and secretary, came up with when, having already filled the margins and the space between the lines of a given page of his manuscript, Proust was distraught at the thought of not being able to introduce further corrections and additions. To be sure, there were contingent reasons for such considerable additions: soon after the publication of _The Way by Swann_ , the war broke out, Grasset (Proust's publisher) was drafted in 1916 and the publishing house had to close down temporarily, thus finally allowing Gallimard to convince Proust to change publishers and give Proust the chance to revise and expand his novel continuously. This is how the character of Albertine was born, and how the second half of _In the Shadow of Young Girls in Flower, The Prisoner_ and _The Fugitive_ were written. But, we may wonder, since when does a delay in publication translate automatically into developments of that magnitude? It is rather as if the novel were animated by a logic of growth of its own — an unexpected, unplanned and yet irrepressible need to outgrow its initial dimensions, not by extending its life beyond its planned end, not, that is, by adding segments at the end of the book, but by growing from the middle. Specifically, the novel grew by a process of swelling, thus revealing a remarkable elasticity, which must have been there from the start, virtually as it were, and which materialized by chance and with the help of the contingencies of time. As for the time of the novel itself, it was forced to slow down, at times almost infinitely, and reveal a different quality. As readers, we find ourselves trapped, bogged down almost, in an infinitely elastic and uncanny duration. The sense of duration that emerges from the novel is the result not of a time that keeps pressing ahead along a straight line, but of a time that halts, moves downward, and drags us into unsuspected depths. The _Recherche_ is not (only) what the French call a _roman fleuve_ , that is, an endless, river-like novel; it is (also) what we could call a _roman marais_ , or a _roman delta_ , that is, something resembling a swamp, or a delta. In addition, when Proust receives the proofs of his book, he keeps rewriting the novel, writing new pages, inserting relative and conjunctive propositions in between principal clauses: the Proustian sentence also swells up, grows and expands from the middle, exceeding and deforming the classical sentence, disorienting the reader, literally losing him in its meanders. Proust's sentence ceases to be a line, and becomes a rhizome. Ultimately, it resembles the swollen face of Rembrandt in the self-portrait briefly mentioned in _Finding Time Again_ : there is something deformed, almost monstrous, about it. It is subjected to an excess that makes us feel dizzy. Similarly, doubling Proust's novel as it were, Marcel's book grows anarchically, cancerously, like an organism gone mad, hovering between organization and disorganization, between order and chaos:
> Because I often had to glue one piece on to another, the papers that Françoise called my manuscribbles [ _paperoles_ ] kept getting torn. But wouldn't Françoise be able to help me mend them, just as she would put patches on the worn-out parts of her dresses or, while she was waiting for the glazier, as I was for the printer, she would stick a piece of newspaper over a broken pane in the kitchen window? Françoise would say to me, pointing to my note-books, eaten away like wood that insects have got into: "It's all moth-eaten, look, that's a pity, there's a page here that looks like lace," and examining closely like a tailor: "I don't think I can mend this, it's too far gone. It's a shame, those might have been your best ideas..."
Something decisive takes place in that space, _between_ beginning and end, and in that process that the arche-teleo-logic (the structure and unity) of the novel could neither anticipate nor master. Besides the logic of organization, besides the genesis and planned structure of the novel, or submerging them from within, an unanticipated and somewhat unmasterable logic of deformation and disarticulation unfolds. Besides the circular unity of the novel, then, the unity that was there from the start and organized the novel as a whole, there is another unity, one that is not the principle or the cause of the novel, but its _effect_.
In addition, and more significantly still, there is a level or plane internal to the novel itself, involving local characters or situations. This is the plane that draws Deleuze's attention. On the one hand, he claims, the unity of the characters of the novel is one of envelopment, implication or encasing ( _emboîtement_ ): every character, name or thing is like a box containing an infinite number of other boxes. The voice of M. de Charlus, "that motley character, pot-bellied and closed, like some box of exotic and suspect origin," contains, in the words of Deleuze, "broods of young girls and tutelary feminine souls." Similarly, "the name Guermantes is also like one of those tiny balloons in which oxygen or some other gas has been stored" or else like one of those "little tubes" from which we "squeeze" the right colour. And the taste of the madeleine, of course, envelops the essence of Combray. The Proustian image that best characterizes this type of unity is that of the Japanese papers which, when thrown in water, open up and reveal an unexpected variety of shapes and colours. It is the incessant curiosity of the narrator that, above all, allows the characters, names, and places, to reveal their hidden reality. With respect to this first figure of encasing, or envelopment, the role of the narrator is, quite literally, to explicate or develop a content that is incommensurable with its container. The explication in question must be understood quite literally: the narrator explicates what is implicated, makes explicit what is only implicit, or develops what is initially enveloped. In so doing, he reveals what was initially hidden, much like a developer in photography.
The second type of unity is that of _complication_. Neo-Platonic in origin, and particularly developed in the Renaissance, this concept takes on a distinctive meaning in Deleuze's thought. This is a concept that testifies to a new and decisive stage with respect to Deleuze's earlier thought, and to the problematic of implication and explication he develops in the first part of _Proust and Signs_ , and in the fourth chapter in particular. Between the plane of explication and that of complication, between the first and the second edition of _Proust and Signs_ , Deleuze's thought matured. With the unity of complication, we are no longer dealing with open boxes but with separate and mutually exclusive worlds, with closed vessels that do not communicate amongst themselves (or at least not according to the classical conception of communication). These vessels cannot be brought under the unity of a higher or more general instance, or be seen as the expression of a more or less manifest totality. And yet, if we're going to speak of a unity of the novel in that respect, the closed vessels must communicate in one way or another. At a certain level, they need to be compatible or, better said perhaps, compossible. Their unity, Deleuze insists, is not one of inclusion, of encasing, but of juxtaposition. Their relation is no longer one of contained and container, but of transversality. In that respect, they are more like fragments than parts of a whole. However, they are not fragments in the Greek sense, that is, fragments of an obscure, hidden totality, or pieces of a puzzle that can be assembled, or reassembled. A fragment, Deleuze claims, can be understood in two ways at least:
> When a part is valid for itself, when a fragment speaks in itself, when a sign appears, it may be in two very different fashions: either because it permits us to divine the whole from which it is taken, to reconstitute the organism or the statue to which it belongs, and to seek out the other part that belongs to it — or else, on the contrary, because there is no other part that corresponds to it, no totality into which it can enter, no unity from which it is torn and to which it can be restored.
A fragment can be a part that reveals the whole; it can be the microcosm of a macrocosm. But it can also signal a reality of its own, juxtaposed in relation to another, but not leading to a higher unity: a multiplicity of differences, or a set of relations, rather than a gathering of identities, or an organization of units. We need only think of the little phrase of the Vinteuil sonata ("Why do you need the rest? Just that is _our_ piece [ _morceau_ ]," Odette says to Swann): it is a fragment that somehow escapes and exceeds the sonata, and doesn't belong to a lost unity. It does not present or schematize the sonata as a whole, much like "the little patch of yellow wall" from Vermeer's _View of Delft_ , which Proust compares to "a precious work of Chinese art, of an entirely self-sufficient beauty," does not symbolize the painting as a whole. In this instance, Proust is not interested in the painting or the sonata as a whole, and thus not in the relation between the particular piece and the whole: he is only interested in the fragment and its ability to draw within itself people, places and affects that are essentially heterogeneous. The unity in question, then, will have to be a unity of fragmentation, a unity of the multiple itself.
As a result, we can begin to understand why Deleuze claims that those "parts," whilst _décousues_ , do not carry any negative connotations. A narrative (whether real or fictitious, literary or cinematographic) that's _décousu_ (literally, unstitched or unsewn, but let us say, disjointed) is normally understood to be lacking in unity and direction, planning and coherence. A well-structured thought, a well-assembled narrative, we're told, must avoid this condition at all cost. By not being sufficiently or tightly woven, an argument, like a narrative, threatens to fall apart. By contrast, the French speak of the _trame narrative_ (the woof of the narrative), thereby reinforcing the connection between text and weft, a connection that is there in the word _textus_ itself, which, after all, refers precisely to something woven. In Italian, for example (and unsurprisingly), the common origin between text ( _testo_ ) and fabric ( _tessuto_ ) is most obvious. And yet, in this instance, it is a matter of affirming the reality and the positive nature of this disjointed plane, and of exploiting the stylistic and narrative resources of this unity of juxtaposition. In saying that — in one respect at least — the Proustian novel is _décousu_ , Deleuze is not claiming that it lacks in unity or coherence. On the contrary: he is affirming that the unity — or at least one of the two senses of unity it reveals — is precisely one of disjointedness. In saying that, he is not denying that the Proustian text is a woven fabric of some kind, but only that the manner in which it is woven or put together does not conform to the usual methods and techniques. If the more classical conception of weaving, especially in relation to writing and thinking (one might think of Plato here), privileges an intricate and continuous fabric, such as a carpet or garment, with its recurrent patterns and coordinated colours, the Proustian text resembles (at least in part) something like a patchwork, that is, an assemblage of disparate and heterogeneous parts, roughly stitched together. Whilst voicing his taste for classical dress-making, and for Fortuny's couture in particular, Proust also innovates in that department: he invents or creates a way of stitching together not just the pages of his book — his famous _paperoles_ — but also his characters, the names they bear and their attributes. The vessels, fragments or pieces of cloth that constitute the fabric are not sewn together in a manner that would give a sense of continuity and homogeneity. They are loosely connected with one another, and do not constitute a synthetic totality. They coexist in a kind of spatial and temporal proximity, but without dialectical exchange. Their juxtaposition is not the prelude to their reunification. Their point of contact — their seam — emphasizes their difference and their separation, more than their common qualities. It is _either_ the way by Swann, _or_ the Guermantes way; it is _either_ the Verdurin clique, _or_ the Guermantes clan. Each world has its language, its code, and its rites. They are mutually incompatible, and exclude one another (in fact, Madame Verdurin is a priest of sorts who does not hesitate to excommunicate those who betray the trust of the "family," by venturing into another world, for example). As a result, one finds oneself having to choose between them, often at great cost, as Swann's affair with Odette, followed by their marriage, reveals most clearly. And yet, as we can gather from the end of _The Way by Swann_ , and that of the novel as a whole, worlds that first appeared to be mutually exclusive eventually converge, and even merge. There is always a point at which a piece or a part breaks free from the set to which it belonged and, more or less directly, drifts into another set. There are passages, bridges, or tunnels, therefore, which some of Proust's characters use, or even represent: the violinist Morel, discovered by the Verdurin, is pursued by Charlus; Gilberte Swann marries Saint-Loup; Odette, formerly "Miss Sacripan" and wife of Swann, rejected by the Guermantes clan, becomes the mistress of the duc de Guermantes; and the old Verdurin widow ends up marrying a "bore," the prince de Guermantes.
At a more local level, this phenomenon is particularly visible when, after a long and frustrating wait, Albertine finally allows Marcel to kiss her. The reality of the kiss fails to measure up to Marcel's hopes: far from conforming to the ideal of unity and totality Marcel had — quite understandably — projected into such a moment, the actual kiss amounts to its (rather traumatic) shattering. At the very moment at which Albertine is finally in reach of Marcel's lips, she escapes him, her face breaking into a series of disparate and disconnected snapshots, or profiles. The promise of a final and total possession, which that kiss represented for Marcel, vanishes. Moreover, the fragmentation of Albertine's physical appearance, and the experience of (also comical) discomfort it creates, is not followed by a moment of reunification. It's a fragmentation beyond synthesis. The passage is remarkable in that it illustrates the tension that's the result of the coexistence of the two senses of multiplicity in one and the same system (the Albertine system): the multiplicity of synthesis, and that of dispersion, or fragmentation. Albertine is indeed a box, which contains Balbec, and the beach, and her little gang, as well as the secrets of an existence that escapes the grasp of the narrator. But she's also a multiplicity of closed vessels, of fragments.
Before turning to the passage in question, let me simply recall that Marcel had previously recorded his passion for Albertine's cheeks, "as one can have for a variety of flower," a passion that evolved into an obsession: "Her cheeks often looked pale; but seen from the side, as I could see them now, they were suffused and brightened by blood which gave them the glow of those brisk winter mornings when, out for a walk, we see stone touched and ruddied by the sun, looking like pink granite and filling us with joy." At the sight of Albertine's cheeks, Marcel's desire is not for a walk, however, but for a kiss, the very kiss he hopes to give her when invited to visit her in bed that evening. At that point, "the sight of her naked throat and her excessively pink cheeks had so intoxicated [him]" that he leans over to kiss Albertine, who rings by "pulling the bell for all she was worth." There's considerable anticipation built up, then, as well as frustration, which all crystallize on the cheek. On one level, the cheek's function is metonymical: it is the part that signals the whole, the part in which the whole — not just the whole of Albertine's body, but the whole of her life, everything that she contains and envelops, and which is the real object of Marcel's desire — is gathered and reflected: "What a difference there is between possessing a woman with one's body alone, because she is no more than a piece of flesh, and possessing the girl one used to see on the beach with her friends on certain days, without even knowing why it was on those days and not on others, so that one trembled to think one might not see her again." The cheek is the very promise of an absolute that Marcel hopes to reach, the bodily part on which his love has crystallized. Yet it is this absolute that crumbles before Marcel's eyes. Beyond the desire to possess Albertine, and even to possess what she possesses, namely, a vantage point, a window onto the world, different from his own, what Marcel truly desires is the world as such and as a whole. All desire is desire of the Whole, or the Absolute. But the illusion is to believe that such a desire can be realized in love, since the object of one's love — the desire of the Other — always escapes us. This is what Swann, and subsequently Marcel, learn, at the cost of considerable suffering. From that point of view, the scene of the first kiss signals the impossibility of a total union, and the shattering of that ideal. By kissing that cheek, Marcel hopes to possess an entire life, a life that far exceeds Albertine herself. Kissing her will be like "kissing the whole Balbec seashore." Could we think of a better image of unity and fusion? By kissing the cheek of the beloved, Marcel hopes to be able to grasp her entire essence, much in the way that, later on in the novel, he will grasp the essence of Combray by biting into the madeleine (with the following, crucial difference, however, that the essence of Combray is discovered involuntarily, in a reminiscence, whereas it is denied to erotic desire). It is precisely this hope that is crushed, however, this ambition that is frustrated, some four hundred pages after having been expressed for the first time — not, as had happened previously, as a result of Albertine's refusal to let herself be kissed, but as a result of her acquiescence. The much-awaited kiss tolls the bell of the organic totality. The cheek is no longer the metonymical organ it once was, and in which the whole of Albertine was gathered; it has now become a differentiating factor that brings about a new and fragmented reality. There is no longer one Albertine, but ten Albertines, all unknown and uncanny. Marcel finds himself transported from one plane to another — from Albertine as a box to Albertine as a multiplicity of vessels, from Albertine as an organized body to Albertine as a disjointed body. The narrator's organs themselves, far from working together towards a clear and distinct perception, break down and betray him, leaving him lost and utterly disappointed:
> [A]s my mouth began to move towards the cheeks my eyes had led it to want to kiss, my eyes changed position and saw different cheeks; the neck, observed at closer range and as if through a magnifying glass, became coarse-grained and showed a sturdiness which altered the character of the face [... ] now what I saw, in the brief trajectory of my lips towards her cheek, was ten Albertines; because this one girl was like a many-headed goddess, the head I had seen last, when I tried to draw near, gave way to another. As long as I had not touched it, I could at least see this head, and a faint perfume came to me from it. But alas — for when we kiss our nostrils and eyes are as ill-placed as our lips are ill-made — suddenly, my eyes ceased to see, and my nose in turn, crushed against her cheek, no longer smelled anything, so, without my efforts bringing me any clearer notion of the taste of the rose I desired, I discovered, from these abominable signs, that I was finally in the process of kissing Albertine's cheek.
The scene of the first kiss marks the end of a certain ideal — the ideal of organicity and totality. Yet it does not signal the death of unity as such. Rather, it signals the appearance of a different kind of unity: the fragmentary unity.
And if, as far as literature is concerned, metonymy is the trope that designates the organic plane, it is, as we saw in great detail in the previous chapter, metaphor which corresponds to the fragment, and to the way in which it communicates and resonates with other fragments. Like the sea in Elstir's painting, which cannot be distinguished from earth, like the sound of the knife against the plate in the hôtel de Guermantes, like the cheek of Albertine or the patch of yellow wall, metaphor transposes us into another reality. It is the trope through which closed vessels and fragments enter into resonance, the trope of transposition and transference. Its force lies in its ability to displace and gather through differentiation. Whereas metonymy gathers by concentrating the whole into one of its parts, metaphor gathers by extracting heterogeneous parts from their totality, and allowing them to enter into a kind of resonance. Turning to Charlus once again, we could say that he is the Guermantes _par excellence_ , the finest example of their blood, wit, disdain and depravity. In that sense, he is their emblem, or, to use a Kantian concept, their schema. At the same time, however, by virtue of his infatuation with Morel, he is the bridge or the tunnel through which the Guermantes clan communicates and resonates with the Verdurin clique, the mole in the Guermantes fortress, and the cause of his own downfall. But these are not exceptions: in the end, every closed system, every vessel turns out to be perforated. There is no pure closure. The vessels can't be opened by the narrator, or synthesized, of course. That is because the mode of openness and disclosure of those systems is not one of unfolding, but of transversal and subterranean communication, linking or stringing fragments together, not in an organic totality, but a motley necklace, or a patchwork.
There is, therefore, a double unity of the _Recherche_ — that of the cathedral, or the dress, and that of the patchwork. The novel as a whole oscillates between the classical ideal of organicity, for which a work is a totality in which every part fits, or a body not lacking any organ, and the fragmentation and disorganization of such a body. On the one hand, Proust perpetuates the ideal of the work as organic; on the other hand, he introduces a plane of fragmentation and unmasterable excess. Generalizing this double structure, we could say that the quest for organicity, and the development of organs, will always have to reckon with a force of disorganization, with an anorganic and anarchic principle. Every totality, every organic unity, in the moment in which it is constituted, is swept away by a power of fragmentation and dissemination; every organic unity is traversed by a body without organs. A such, Proust's novel can be seen to oscillate between reason — or rather _logos_ , as a power of gathering and a faculty of intellectual synthesis — and anti- _logos_ , as a force of dissemination and fragmentation. Stylistically, the novel hovers between metonymy and metaphor. But the two forces are irreducibly bound together, creating a reality between order and chaos, between reason and madness, or _logos_ and _mania_. In its own way, the Proustian novel testifies to this eternal _polemos_ and this conflicting harmony, this irreducible intertwining that is chaosmos: it reveals a macro-plan and an organic unity, but one that is overcome from the middle, by a rhizomatic and truly monstrous expansion (and it is not an overstatement to say that Proust died of this exhausting swelling, that the life of the novel took over his own); it reveals macro-assemblages, highly structured and rigid sets, but which crumble in the face of mere fragments. Bodies — whether physical, social, or artistic — fall apart, and give birth to new assemblages. In the end, every major character, every name and every place turns out to be on both sides of the divide. Beneath this living metonymy, Charlus, who represents the Guermantes in their mannerisms, their wit, their history, there's this undercurrent of madness, that is, this force pulling him in the opposite direction, towards the abyss. In thus hovering between the classical ideal of organicity and the more contemporary experience of madness in literature, Proust draws a century to an end, and opens up another.
* * *
Notes
* * *
#### **1 Looking for joy**
It is, after all, a very Schopenhauerian sort of a book, as some people have tried to show; see, for example, Anne Henry, _Marcel Proust. Théories pour une esthétique_ , Paris: Klincksieck, 1981. _The World as Will and Representation_ certainly seems to afford an accurate description of the book: "[t]he never-fulfilled wishes, the frustrated efforts, the hopes mercilessly blighted by fate, the unfortunate mistakes of the whole life, with increasing suffering and death at the end translated by E.F.J. Payne, New York: Dover Publications, 1969, vol. I, p. 322.
M. Proust, _In the Shadow of Young Girls in Flower, In Search of Lost Time_ , II, translated by James Grieve, London: Allen Lane, The Penguin Press, 2002, p. 16 (translation slightly modified). Further references to the _Search_ (abbreviated as _SLT_ ) will be made to the six-volume Penguin Press edition (2002), under the general editorship of Christopher Prendergast, with indications of volume and page numbers (e.g. _SLT_ , II, p. 78).
And, despite the importance of Schopenhauer to Proust, it's here that the difference between them needs to be seen. For Schopenhauer, desire is the indication of a lack and such a lack is the kernel, the very essence of life. If desire encounters any impediment, suffering inevitably ensues. If desire reaches its aim, satisfaction, well-being and happiness follow. And insofar as desire stems from lack, from a state that dissatisfies us, it remains suffering as long as it's not satisfied. As soon as it becomes satisfied, however, our desire discovers a new aim. But if will happens to be without an object, if some speedy satisfaction happens to remove any cause for desire, then will and satisfaction lapse into a dreadful void, into boredom. Like a pendulum, life therefore swings between suffering and boredom ( _The World as Will and Representation_ , op. cit., §§ 56 and 57). Proust's genius reverses this orientation first by making lack a structure of the real itself and _not_ a function of desire or will and second by making satisfaction and even _jouissance_ not an illusory and transient form of relief, but an ability to delve into lack and draw positive aesthetics from it — not aesthetics based on a neutralizing or smothering of desire as suggested by Schopenhauer, or aesthetics based on unrestrained _jouissance_ as advocated by common hedonism or epicurianism.
Proust, _Finding Time Again, SLT_ , VI, p. 204.
Samuel Taylor Coleridge, _Biographia literaria_ , London: Dent, 1965, p. 167.
Pascal, _Pensées and Other Writings_ , translated by Honor Levi, World's Classics, Oxford UP, 1995, p. 16.
Proust, _Jean Santeuil_ , Paris: Gallimard, "Bibliothèque de la Pléiade," 1971, p. 398.
Proust, _Finding Time Again, SLT_ , VI, p. 176.
In this, the book has a very Kantian view of the imagination. Kant divides sensibility into sense ( _Sinn_ ) and imagination, sense referring to our "faculty of intuition in the presence of an object" and imagination referring to "the intuition without the presence of the object" ( _Critique of Pure Reason_ , A § 15). The absence of the object for the imagination can be understood in two ways: either the object was present but no longer is or it hasn't yet become present. From the perspective of empirical imagination, this double possibility entails the faculties of memory and anticipation whose task is to recall and predict the presence of discrete objects. In Marcel's view, this double faculty of recollection and anticipation largely exceeds the frame of perception and empirical reality which it reveals. Faced with the power of imagination, the present itself suffers from a lack in being. The object is never more present or more alive than in its absence, therefore.
Proust, _The Way by Swann's, SLT_ , I, p. 227.
Ibid., p. 226.
Proust, _The Fugitive, SLT_ , V, p. 407.
Proust, _The Fugitive, SLT_ , V, p. 611.
"I was very disappointed. My disappointment came from the fact that I had never noticed, when I thought of Mme de Guermantes, that I was picturing her to myself in the colours of a tapestry or a stained-glass window. In another century, of a material different from that of other living people. I had never realized that she might have a red face, a mauve tie like Mme Sazerat, and the oval of her cheeks reminded me so much of people I had seen at our house that the suspicion touched me, dissipating immediately, however, that this lady, in her generative principle, in all her molecules, was perhaps not essentially the Duchesse de Guermantes, that instead, her body, unaware of the name applied to it, belonged to a certain female type that also included the wives of doctors and shopkeepers. 'So that's Mme de Guermantes — that's what it is, that's all it is!' said the attentive and astonished expression with which I contemplated [such] an image..." ( _The Way by Swann's, SLT_ , I, p. 175).
"La Berma in Racine's _Andromaque_ or _Phèdre, in Les Caprices de Marianne_ by Musset, these were the stirring things that I had gloated on in imagination" ( _In the Shadow of Young Girls in Flower, SLT_ , II, p. 14). "To be sure, right up until the moment when I saw La Berma act, I enjoyed the day" (ibid., p. 19) — an enjoyment wholly made up of anticipation, which the sight of the theatre itself, the box-office and the staff, the layout of the theatre, his own seat, the curtain and the vague and strange sounds that could be heard behind it further enhance. "This first matinée was, alas, a great disappointment" (ibid., p. 18).
"The whole Bergotte I had slowly and painstakingly constructed for myself, a drop at a time, like a stalactite, out of the limpid beauty of his books, had suddenly been rendered useless by the need to include the bottle-nose and the black goatee." ( _In the Shadow of Young Girls in Flower, SLT_ , II, p. 123). The comical aspect of such emergence of the reality principle in the narrator's imaginary is to be stressed: it is indeed as if, in the face of the common, if not almost grotesque physical appearance of the famous actor, his work could no longer exert its magic. The work and the bottle-nose cannot be accommodated at once.
In _The Prisoner_ , the narrator evokes "[t]he disappointment [he] had experienced with women whom [he] had known" (op. cit., _SLT_ , V, p. 153). And _Sodom and Gomorrah_ posits love as "a sentiment which (whatever its cause) is always erroneous" (op. cit., _SLT_ , IV, p. 199).
Proust, _Sodom and Gomorrah, SLT_ , IV, pp. 518–19.
Proust, _The Way by Swann's, SLT_ , I, p. 383.
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 508. Later in Paris, when he is about to kiss Albertine for the first time, Marcel ponders on the following, which is the same as wondering whether, in love, one ever possesses the loved one, namely the imaginary being, or only her flesh: "What a difference there is between possessing a woman with one's body alone, because she is no more than a piece of flesh, and possessing the girl one used to see on the beach with her friends on certain days, without even knowing why it was on those days and not on others, so that one trembled to think one might not see her again" ( _The Guermantes Way, SLT_ , III, p. 360).
Proust, _The Prisoner, SLT_ , V, p, 154.
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 413.
Ibid., p. 375.
Proust, _The Way by Swann's, SLT_ , I, p. 157.
Proust, _The Prisoner, SLT_ , V, p. 153.
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 375.
Proust, _The Prisoner, SLT_ , V, p.75.
The phrase I'm using here is Lacan's phrase, borrowed in turn from Kojève's reading of the master-slave dialectics in Hegel's _Phenomenology of Spirit_. In Book I of his Seminar, Lacan identifies at least two possible meanings in this expression: I obviously desire to be desired by the other but, equally and most importantly, I desire what the other desires, i.e. the desire which is his or hers and not mine. This double definition of love (as imaginary or specular love) can naturally be reflected by narcissistic and aggressive tendencies, as evidenced by Swann's attitude towards Odette or by how Marcel relates to Albertine: while recognizing the other's desire (a recognition consequently inscribed in the symbolic economy of the Law, which arises from the recognition of the image of the Father as irreducibly other than my own image and than the ideal ego I was able to form thanks to its projection), I aim to seize and reduce it at once. To love is thus to want the other to recognize me in my place, but it also consists in wanting to take the other's place. Hence an irreducible and destructive tension for which there might be a solution in Lacan's theory, unlike in Proust, where it remains unresolved. Should we infer from this that Proust himself could never get beyond the imaginary stage, or the barely symbolic, and could never fully recognize the other's alterity? The relevance of and the possible answer to such a question is up to the reader to decide.
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, pp. 411–12.
The equivalence between love and chronic pain, which pathologizes love, is a constant theme in Proust: Swann's love for Odette is a "holy evil" ( _The Way by Swann s, SLT_ , I, p.233), "his morbid state" (ibid., p.302), "his madness" (ibid., p.302). The narrator's love for Albertine is described as "an incurable ailment" ( _The Prisoner, SLT_ , V, p. 74), "a kind of mutual torture" (ibid., p. 96) in which the other's existence "is no longer anything but a cause of pain" ( _The Fugitive, SLT_ , V, pp. 442–43). On this question, see Nicolas Grimaldi's compelling study, _La jalousie. Étude sur l'imaginaire proustien_ (Paris: Le Méjan, Actes Sud, 1993).
Proust, _The Way by Swann's, SLT_ , I, p. 302.
Proust, _The Guermantes Way, SLT_ , III, p. 362.
Ibid.
Proust, _The Prisoner, SLT_ , V, p. 153.
Ibid., p. 364.
Proust evokes "the tedium that we find in the midst of happiness" ( _The Fugitive, SLT_ , V, p. 400).
Proust, _The Prisoner, SLT_ , V, p. 155.
Proust's thesis would once again bring us back to Kant, for whom the work of the imagination can only be grasped in relation to the faculty of knowledge (whether theoretical or practical). According to Kant, "some of man's desires involve him in self-contradiction" (Kant, _Critique of Judgment_ , trans. Werner S. Pluhar, Cambridge, Mass.: Hackett Publishing Co., 1987, p. 17). As for an understanding of why we are doomed to live in this gap between our power to desire and our power to act, of why, in other words, "our nature was given a propensity [ _Hang_ ] towards desires of whose futility we are aware" (ibid.), Kant admits that it is an anthropological and teleological question and not a philosophical one.
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 413.
Proust, _Finding Time Again, SLT_ , VI, p. 207.
Ibid.
Proust, _The Way by Swann s, SLT_ , I, p. 276 (emphasis mine). Regarding this section, see Malcolm Bowie's insightful reading "Proust, jealousy, knowledge" in _Freud, Proust, and Lacan_ , Cambridge: Cambridge University Press, 1987, pp. 49 ff., along with Deleuze's _Proust et les signes_ , Paris: Presses Universitaires de France, 1964 ( _Proust and Signs_ , translated by Richard Howard, Minneapolis: University of Minnesota Press, 2000). For Deleuze, the world of Proust is a world made of sensible signs waiting to be deciphered. While the discovery of truth depends on intelligence, the emergence or instigation of truth conversely depends on the sensible.
Proust, _Finding Time Again, SLT_ , VI, p. 207.
This is a recurrent refrain in Proust: a lover's grief may damage the body but it can also nurture the spirit (Proust, _Finding Time Again, SLT_ , VI, pp. 214–15 and 225), virtually at least, in a way that friendship never can.
Proust, _The Guermantes Way, SLT_ , III, p. 394.
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, pp. 483–84.
Ibid., pp. 484–85.
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, pp. 483–84.
Ibid., p. 316.
Ibid. Emphasis mine.
Ibid. Also see Proust's letter to Emmanuel Berl in _Lettres_ , Paris: Plon, 2004, pp. 749–51. The thread of the fundamental problem that this study offers to unravel has now been posited: how does the work of literature, whose goal consists in the discovery of essences and of the truth, focus on the world of differences? Friendship and even more so love only find their true value in their ability to carry us to the heart of this world, the only object of true knowledge. For loving truly, as Proust states elsewhere, consists in learning how to distinguish, how to differentiate.
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 315.
Ibid.
"And, while the purely fleshly view he had had of this woman, by perpetually renewing his doubts about the quality of her face, her body, her whole beauty, had weakened his love, these doubts were vanquished, that love confirmed when he had instead, for a foundation, the principles of an unquestionable aesthetic; while the kiss and the possession that would seem natural and ordinary if they had been granted by damaged flesh, if they came to crown the adoration of a museum piece, appeared to him necessarily supernatural and delicious" (Proust, _The Way by Swann s, SLT_ , I, p. 227).
Proust, _The Guermantes Way, SLT_ , III, p. 395.
Proust, _Finding Time Again, SLT_ , VI, p. 180. The imagination is in fact the organ of _jouissance_ in general and not only of beauty. As we suggested, there is true _jouissance_ included in the torment of jealousy and that comes from having the certainty that we love and live.
Proust, _Finding Time Again, SLT_ , VI, p. 180. The imagination is in fact the organ of joy in general and not only of beauty. As I suggested, there's true joy included in the torment of jealousy and that comes from the certainty that we love and live.
Proust, _The Way by Swann's, SLT_ , I, p. 139.
Kant, _Critique of Judgment_ , op. cit., p. 47.
Ibid., pp. 55–56.
Proust, _The Way by Swann's, SLT_ , I, pp. 113–14. Emphasis mine.
In this instance, flowers are evocative of the human world. Later, Balbec's young girls and Albertine, more specifically, will be evocative of the world of flowers. When accounting for the complexity of Charlus's sexuality and even of men in general, flowers are once again to provide Proust with his model. More than a theme or a motif, flowers are a _mise en abîme_ and a distillate of the _Search_ itself.
Proust, _The Way by Swann's, SLT_ , I, p. 139.
"Then I came back to stand in front of the hawthorns as you do in front of those masterpieces which, you think, you will be able to see more clearly when you have stopped looking at them for a moment, but although I formed a screen for myself with my hands so that I would have only them before my eyes, the feeling they awakened in me remained obscure and vague, seeking in vain to detach itself, to come and adhere to the flowers" (ibid., p. 140).
Proust, _The Way by Swann's, SLT_ , I, p. 139.
Proust, _Finding Time Again, SLT_ , IV, p. 200. Here's how Proust describes those "celibates at the shrine of art" that attend every concert and every exhibit: "They get more excited by works of art than real artists do, because their excitement, not being for them the result of hard introspective investigation, bursts outwards, overheats their conversation and makes them go red in the face." Unused and sterile, their love for art "overflows even into their calmest conversations, makes them make grand gestures, and grimace and toss their heads, whenever they talk about art." It is often preceded with "anxious concern" (as if they were somehow thinking: "I can see sparks, there's a small smell of burning, something must be on fire") and accompanied by "head on one side" and "gesticulation" that mime "the whole absurd pantomime of a gosling with half-grown winglets which has not solved the problem of wings but is nonetheless tormented with a desire to soar into the air" (ibid., pp. 200–201).
Proust, _Finding Time Again, SLT_ , IV, p. 185.
Proust, _The Way by Swann's, SLT_ , I, p. 113.
As is well known, such is the programme of German idealism, from Schiller to Fichte, Schelling and Hegel. Divergences emerge regarding the way in which this reconciliation ( _Versöhnung_ ) may be achieved, not regarding the very idea of a reconciliation.
Proust, _The Way by Swann's, SLT_ , I, p. 141.
The issue of metaphor will be discussed at length in the fourth chapter.
Proust, _The Way by Swann's, SLT_ , I, p. 140.
Ibid., p. 180.
#### **2 Proust among the psychologists**
Hippolyte Taine, _On Intelligence_ , translated by Robert H. Wozniak, Bristol: Thoemmes Press, 1998, p. ix.
Proust, _Against Sainte-Beuve and Other Essays_ , translated by John Sturrock, London: Penguin Books, 1994, p. 235.
Everyone knows that Proust denied having written a "Bergsonian" novel, which didn't stop some people, sometimes rightfully so (look, in particular, at what Deleuze has to say), trying to discern in the novel some connection with Bergson. As far as I know, Proust didn't set out his views on the associationist theses about memory, theses that some have seen as very close to Proust's own. On this, see F. Léger, _Monsieur Taine_ , Paris: Criterion, 1993, J.-F. Perrin, "La scène de la réminiscence avant Proust," _Poétique_ n° 102, 1995, pp. 193–213, and "Taine et la mémoire involontaire," _Romantisme_ n° 82, 1993, pp. 73–81.
As Ribot has it, memory "is, _per se_ , a biological fact — by accident, a psychological fact" (T. Ribot, _The Diseases of Memory_ , trans. W. Huntington Smith, New York: Appleton, 1887, p. 10). For Proust, memory's not even a psychological fact; it'san ontological event.
Ribot, _The Diseases of Memory_ , op. cit., p. 14.
Ibid., p. 16.
Samuel Beckett, _Proust_ [1931], London: Calder & Boyers, 1965, p. 19.
Ibid. "Habitual" reality therefore oscillates between boredom — the state corresponding to the moments when one is completely in sync with the real — and suffering — the state corresponding to the periods of adjustment, to the intermediary stages between two habitual states.
Proust, _The Way by Swann's, SLT_ , I, p.9.
Proust, _The Way by Swann's, SLT_ , I, p. 9. Emphasis mine.
Proust, _The Way by Swann's, SLT_ , I, p. 14.
Ibid., p. 50.
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 245.
Ibid.
Proust, _The Way by Swann's, SLT_ , I, p. 12.
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 246.
Proust, _The Way by Swann's, SLT_ , I, pp. 11–12.
Ibid., p. 12.
Félix Ravaisson, _De l'Habitude_ , Paris: Payot & Rivages, 1997. Proust met Ravaisson shortly before the latter passed away, writing in a letter to Paul Marais in January 1889 of "Monsieur Ravaisson, whom I had the honour to meet." Proust was probably familiar with Ravaisson's dissertation as well, either indirectly, via the Lachelier-Darlu lineage (Alphonse Darlu, Proust's philosophy professor at the Lycée Condorcet, was a disciple of Jules Lachelier, himself one of Ravaisson's disciples), or directly and, for example, through a lecture given by Charles Secrétan and which included Ravaisson himself, along with Brochard and Séailles. Proust attended this lecture on 6 January 1893, a year before he was awarded his _licence_ degree in Philosophy. On this, see Marco Piazza, _Passione e conoscenza in Proust_ , Milan: Guerini, 1998, pp. 92–95 and Annamaria Contini, _La biblioteca di Proust_ , Bologna: Nuova Alfa Editoriale, 1988, pp. 57–65.
Henri Bergson, _The Creative Mind, An Introduction to Metaphysics_ , trans. Mabelle L. Andison, New York: The Wisdom Library, 1946, pp. 231–32.
Félix Ravaisson, _De l'Habitude_ , op. cit., pp. 81–82.
Ibid., p. 83.
Ibid., p. 76.
Ibid., pp. 73–74.
Ibid., p. 75.
Proust, _Against Sainte-Beuve and Other Essays_ , op. cit., p. 235.
Proust, _The Way by Swann's, SLT_ , I, p. 46.
T. Ribot, _The Diseases of Memory_ , op. cit., p. 42.
Ibid., p. 34. It's well known that Proust sees the idea of the vision of time as an important one, along with the odd mix of perception and intuition, of space and time to which the book aspires. His view of time presupposes a double, stereoscopic vision of the sort described in Roger Shattuck's _Proust's Binoculars. A Study of Memory, Time and Recognition in À la recherche du temps perdu_ , London: Chatto & Windus, 1964 and in S. Guindani, _Lo stereoscopio di Proust_ , Milan: Mimesis, 2005.
T. Ribot, _The Diseases of Memory_ , op. cit., p. 44.
H. Taine, _On Intelligence_ , op. cit., p. 36.
Proust, _The Way by Swann's, SLT_ , I, p. 179.
Ibid.
Taine, _On Intelligence_ , op. cit., p. 40.
Proust, _The Way by Swann's, SLT_ , I, pp. 181–82. This archival and archivistic view of writing is a pretty classical one that goes back at least to Plato's _Phaedrus_. But Proust will go further: the involuntary memory and the material that it bestows on writing are precisely the path he will take to depart from such a view, thus tearing writing away from its secondary and derivative status in relation to some primary time which it would merely repeat. Involuntary memory will ultimately constitute some primary trace or, following Derrida's conceptuality, "archi-writing."
Proust, _Against Sainte-Beuve and Other Essays_ , op. cit., p. 5.
If I link the two things together here, it's because Plato's _mimesis_ , from which western aesthetics derives, is rooted in the aforementioned metaphysics: as the image of an image (as a simulacrum), the work of art is, in this context, even far more remote from the original than any mental representation and, in particular, any recollection.
Proust, _Lettres_ , Paris: Plon, 2004, p. 667. In a letter to Gaston Gallimard written shortly after 1 March 1916, he says "my book is not a book of childhood memories" (ibid., p. 757).
Walter Benjamin, "On Some Motifs in Baudelaire," translated by Harry Zohn in _Illuminations: Essays and Reflections_ , edited by Hannah Arendt, New York: Schocken, 1968.
The fact that the Marcel of the book isn't the I of its author, that the experiences that are recounted and analysed aren't experiences that were actually lived, changes nothing: the logic of the narrative largely remains the logic of the lived experience, as lived by the narrator. As I'll try to show, this logic of lived experience is in fact doubled by a logic of the unlived experience which itself depends on an alternative form of memory and an alternative sense of time. And here's where we find Proust's real originality, which disengages his novel from the aesthetics of mimesis. Involuntary memory just _is_ the way out of the mimetic view of art; by breaking with the view of memory as representation, Proust also breaks with the view of art as simulacrum or as the image of an image.
Proust, _Against Sainte-Beuve and Other Essays_ , op. cit., p. 235.
Ibid., p. 273.
Ibid., p. 234.
From this perspective, I have to disagree with François Léger's conclusion that, "deeply instilled with the reading of Taine," Proust's novel "offers a magnificent illustration of some of the main claims made in _On Intelligence_ " (F. Léger, op. cit., p. 359). Besides, we have to take Proust seriously when, discussing Sainte-Beuve's "method" and the influence it had on Taine's work, an influence Taine himself acknowledged, he says that "in art there are no initiators or precursors (at least in the scientific sense). Everything is in the individual, each individual starts the artistic or literary endeavour over again, on his own account; the works of his predecessors do not constitute, unlike in science, an acquired truth from which he who follows after may profit. A writer of genius today has everything to do. He is not much further advanced than Homer" ( _Against Sainte-Beuve and Other Essays_ , op. cit., p. 11).
On this subject, see J.-F. Perrin, "La scène de la réminiscence avant Proust" (op. cit.) and "Taine et la mémoire involontaire" (op. cit.). Proust knew Taine'smajor book, _On Intelligence_ , before 1908 and, in _Against Sainte-Beuve_ (p. 11), refers to the book's "Preface." Also see Proust's letter to Rosny Aîné on 14 June 1921, _Corr._ , XX, p. 335. _On Intelligence_ was well known at the time, abundantly cited in Èlie Rabier's textbook _Leçons de Philosophie, vol. I: Psychologie_ , Paris: Hachette, 1884, which young Marcel studied in high school.
This school originates with David Hume and _A Treatise of Human Nature_. Hume effectively reduces all ideas to the impressions of which they're copies and which they resemble and represent. Two ideas become connected either because of their resemblance or because the impressions they are copies of were contiguous or, finally, because one represents a cause whose effect is represented by the other. However, Hume's not an "associationist" in the sense that the term comes to have in the context of the school and among its most celebrated adherents (Erasmus Darwin, James Mill, Dugal Stewart, Thomas Brown, William Hamilton). Its mental attraction isn't actually universal: this lack of universality first stems from the fact that attention has the power to stop the series on one idea, secondly from the fact that there sometimes is some true irregularity in the imagination and that the arbitrary union, without any connection, of two or several ideas can take place in fantasy. Furthermore, Hume considers these associative connections as one of the main causes of our errors.
Cf. Taine, _On Intelligence_ , op. cit.
Perrin, "Taine et la mémoire involontaire," op. cit., pp. 74 ff.
Taine, _On Intelligence_ , op. cit., p. 367.
Ibid., p. 151.
Ibid., pp. 75–76.
Proust, _The Fugitive, SLT_ , V, p. 509.
In _L'Inconscient cérébral_ (Paris: Seuil, 1992), Marcel Gauchet locates the origin of memory's stratigraphic and geological metaphor in the neurophysiology and psychology of Proust's time. It can notably be found in Taine (cf. J.-F. Perrin, "Taine et la mémoire involontaire", op. cit., pp. 73–81).
Proust, _Against Sainte-Beuve and Other Essays_ , op. cit., p. 1. Translation modified.
"Displacement" is a translation of the term _Verschiebung_ , used by Alfred Lehmann in _Die Hauptgesetze des menschlichen Gefühlslebens_ , Leipzig: O. R. Reisland, 1892, pp. 268–75 and found, this time along with the term Transference, in _Grundzüge der Psychophysiologie_ , Leipzig: O. R. Reisland, 1912, pp. 682 and 702; on the other hand, the concept of "transference" comes from James Sully, _Outlines of Psychol_ ogy, London: Longmans, Green & Co., 1884, pp. 486 ff. and _The Human Mind_ , 2 volumes, London: Longmans, Green & Co., 1892, II, pp. 78 ff.
T. Ribot, _Psychologie des sentiments_ , Paris: Félix Alcan, 1896, p. 175.
Herbert Spencer, _The Principles of Psychology_ , 2 volumes, London: Williams & Norgate, 1870–72, II, § 519.
Théodore Flournoy, _Des Phénomènes de synopsie_ , Paris: Alcan, 1893, p. 20.
Proust, _Against Sainte-Beuve and Other Essays_ , op. cit., p. 273.
Ibid.
The following admission gives us a hint of how we might think of this connection: ". I believe that it is really only to involuntary memories that the artist should go for the raw material of his work. First, precisely, because they are involuntary and take shape of their own accord, drawn by the resemblance of some identical moment, they alone bear the hallmark of authenticity. Then, they bring things back to us in exact proportions of memory and oblivion. And finally, since they give us to enjoy the same sensation in quite other circumstances, they release it from all contingency, they give us its extratemporal essence, which is the very content of good style, that general and necessary truth that the beauty of a style alone can reveal" (ibid., pp. 235–36).
Proust, _The Way by Swann's, SLT_ , I, p. 347.
#### **3 Finding joy (involuntary memory)**
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 222.
On the development of the theory of memory in ancient Greece, from the archaic period to the classical era, see Jean-Pierre Vernant, "Mythical Aspects of Memory and Time", in _Myth and Thought among the Greeks_ , translated by Janet Lloyd, New York: Zone Books, 2006, pp. 115–56. I should point out one real difference, though: while the Greek notion of anamnesis, whether in Hesiod or in Plato, involves an effort of memory and is, properly speaking, a matter of recollection, the emergence of the time of always or, alternatively, of divine time ( _aion_ ), is, for Proust, the fact of an involuntary memory, a reminiscence. Moreover, divine time doesn't refer, as it did in ancient Greece and in Hesiod in particular, to the time of a mythical and completed era, of a primary age that poetic memory might bring back to life; rather, it refers to the time of each and every one of us, at once affective and impersonal.
Proust, _The Way by Swann's, SLT_ , I, pp. 47–48.
Proust, _Finding Time Again, SLT_ , VI, p. 175. Emphasis mine.
Ibid., p. 177.
Proust, _Finding Time Again, SLT_ , VI, p. 177. Emphasis mine.
"[... ] I knew", Marcel claims in _Finding Time Again, "_ that the beauty of Balbec was something I had never experienced when I was there, and that the beauty it left me with, the beauty of memory, was something I was unable to discover when I went back there to stay for the second time" ( _SLT_ , VI, p. 185). "I had been unable to know pleasure at Balbec, any more than the pleasure of living with Albertine [... ]. And the recapitulation I was making of all the disappointments of my life, as I had lived it, and which made me believe that its reality must reside somewhere else than in action, was not bringing the different disappointments together in a purely fortuitous manner in accordance with the circumstances of my existence. I felt very strongly that the disappointments of love were not different disappointments, but the varied aspect taken on, according to circumstances which bring it into play, by our powerlessness to realize ourselves in material pleasure or real action" (ibid., pp. 185–86).
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 222. Emphasis mine.
Ibid. A book like _François le Champi_ , for example, comprises all of Marcel's childhood, however virtually.
This is a lot like what Walter Benjamin argues, although in quite different terms, in "The Image of Proust" (translated by Harry Zohn in _Illuminations: Essays and Reflections_ , edited by Hannah Arendt, New York: Schocken, 1968), and what Beckett notices two years later in _Proust_ : "The man with a good memory does not remember anything because he does not forget anything" (op. cit., p. 29).
Vernant, "Mythical Aspects of Memory and Time," op. cit., pp. 115 ff.
J.-P. Vernant, "Mythical Aspects of Memory and Time," op. cit., pp. 117 ff.
Proust, _Finding Time Again, SLT_ , VI, p. 178 (translation modified).
Proust, _The Way by Swann's, SLT_ , I, pp. 49–50.
The insistence of the sensation linked to the presence of the hawthorns, this surplus or excess of reality that can't be reduced to _Gefühl_ and to the flower that's immediately there, was already this pure past, that dimension of being that exceeds actuality.
Let's be clear about this: there's undoubtedly a nostalgic dimension to the book and that dimension's even an essential one: faced with the disappointment of the present, by a reality that frustrates the imagination, the only available paradise looks to be the one that was lost. Such is the time, the lost time, that Marcel struggles to find again. This search, though, which involves voluntary memory, is ultimately frustrated too insofar as it never manages to retain that time in all its glory. The time that's redemptive and regained, however, is different to the one that was lost and the involuntary memory thanks to which it's found again isn't a modification of its voluntary counterpart. The book reveals two distinct but related kinds of time (unless what it actually reveals is the virtual split of a single time): Saturnian, destructive time and the creative, artistic and redemptive time of eternity; the time of life and the time of what is unlived within life. To this doubling of time there corresponds a vision that's itself stereoscopic.
Gilles Deleuze, _Proust and Signs_ , translated by Richard Howard, Minneapolis: University of Minnesota Press, 2000, p. 154.
This is suggested by remarks like this, for example: "[... ] I understood only too well that what the sensation of the uneven flagstones, the stiffness of the napkin, the taste of the madeleine, had awoken within me bore no relation to what I was trying to remember about Venice, about Balbec and about Combray, with the help of a uniform memory" ( _Finding Time Again, SLT_ , VI, pp. 177–78).
Walter Benjamin, "On Some Motifs in Baudelaire," translated by Harry Zohn in _Illuminations: Essays and Reflections_ , op. cit., pp. 157–202, especially pp. 159–62.
Ibid., pp. 162–63.
Proust, _Against Sainte-Beuve and Other Essays_ , op. cit., pp. 92–93. More precisely, harmony's what finds its way "between" (between two sensations, between a sensation and a memory, between two ideas) and which the artist succeeds in perceiving. The artist, Proust tells us, lives on these harmonies and his or her preferred space is precisely this space in-between: "What is in one picture by a painter cannot sustain him, nor in one book by an author, nor in a second picture by that painter, nor a second book by that author. But if in the second picture or second book, he perceives something that is not in the second nor the first but somehow between the two, in a sort of ideal picture, which he sees taking on shape in a spiritual substance independently of the picture, he has received his sustenance and starts to exist once more and to be happy. Because for him to exist and to be happy is all one. And if he discovers a higher connection still between the ideal picture and the ideal book, either of which suffices to make him happy, his joy increases still further" (ibid., p. 93).
Proust, _Finding Time Again, SLT_ , VI, p. 180.
Proust, _The Way by Swann's, SLT_ , I, pp. 347–49.
Deleuze, _Proust and Signs_ , op. cit., pp. 55–61.
Proust, _The Way by Swann's, SLT_ , I, p. 351.
Ibid.
Cf. Maurice Merleau-Ponty, _The Visible and the Invisible_ , translated by Alphonso Lingis, Evanston, Ill.: Northwestern UP, 1968 and _L'Institution, la passivité. Notes de cours au Collège de France_ [1954–55], Paris: Belin, 2003, pp. 63–77. Also see Mauro Carbone's fine discussion of sensible ideas in Proust, _Una deformazione senza precedenti. Marcel Proust e le idee sensibili_ , Macerata: Quodlibet, 2004, chapter II.
Proust, _The Way by Swann's, SLT_ , I, p. 351.
Maurice Merleau-Ponty, "The Philosopher and his Shadow," _Signs_ , translated by Richard McCleary, Evanston, Ill.: Northwestern UP, 1964, p. 172.
"She took such pleasure in any trouble that spared me trouble, such delight in a moment of rest and peace for my weary limbs, that when I tried to prevent her helping me untie my laces and get ready for bed, making as though to undress myself, her pleading glance halted my hands, which were already on my boots and the first buttons of my jacket" ( _In the Shadow of Young Girls in Flower, SLT_ , II, p. 247).
Proust, _Sodom and Gomorrah, SLT_ , IV, p. 158.
Proust, _The Prisoner, SLT_ , V, p. 371.
Proust, _Sodom and Gomorrah, SLT_ , IV, p. 158.
This love's most movingly conveyed in the passage where Proust describes grandmother's reaction to the teasing that sometimes came her way: "[... ] and my grandmother would go off again, sad, discouraged, yet smiling, for she was so humble at heart and so gentle that her tenderness for others and the little fuss she made over her own person and her sufferings came together in her gaze in a smile in which, unlike what one sees in the faces of so many people, there was irony only for herself, and for all of us a sort of kiss from her eyes which could not see those she cherished without caressing them passionately with her gaze" ( _The Way by Swann's, SLT_ , I, p. 15).
Proust, _Sodom and Gomorrah, SLT_ , IV, p. 159.
Proust, _Finding Time Again, SLT_ , VI, p. 243.
Deleuze, _Proust and Signs_ , op. cit., p. 55.
Proust, _Finding Time Again, SLT_ , VI, p. 198.
#### **4 Giving joy (metaphor)**
Proust, _The Prisoner, SLT_ , V, p.73.
Proust, _The Way by Swann's, SLT_ , I, p. 173.
Plato, _Phaedrus_ , 250d-e. Also see John Sallis's fine reading in _Force of Imagination. The Sense of the Elemental_ , Bloomington and Indianapolis: Indiana University Press, 2000, p. 228.
Proust, _Finding Time Again, SLT_ , VI, p. 193.
Ibid., p. 203.
Ibid.
Ibid., p. 198.
Ibid.
Ibid.
Ibid., pp. 203–4.
Ibid., p. 204.
Ibid.
Aristotle, _Poetics_ , translated by Ingram Bywater, _The Basic Works of Aristotle_ (New York: Random House, 1941), 1457b6–9. Translation modified. On the relation between philosophy and rhetoric and the interpretation of Aristotle'sdefinition, see Paul Ricoeur, _La Métaphore vive_ , Paris: Èditions du Seuil, 1975 ( _The Rule of Metaphor_ , translated by Robert Czerny, Toronto: Toronto UP, 1977). What I'm going to say largely endorses Ricoeur's analysis, despite some occasional differences.
Ricoeur, _The Rule of Metaphor_ , op. cit., p. 14.
Ibid. and Aristotle, _Poetics_ , 1457a10–11 (Ricoeur's translation).
Pierre Fontanier's the first to extend the notion of metaphor to words other than nouns: "[... ] not only the noun, but also the adjective, the participle, the verb, in short all the possible kinds of words pertain to it [... ] The kinds likely to be used _metaphorically_ as _figures_ are the noun, the adjective, the participle, the verb and perhaps also the adverb, though not very often" (Pierre Fontanier, _Les Figures du discours_ [1830], Paris: Flammarion, 1977, p. 99). As Ricoeur points out, this doesn't change the fact that metaphors are limited to _words_ (even if they're not limited to nouns), as opposed to sentences or speech as such. According to Fontanier, metaphors can apply to any possible object of thought, whether physical, natural, abstract, moral or metaphysical; they can be drawn from anything in our surroundings: from nature or from the arts, from purely fictional, imaginary beings, even from purely intellectual or moral beings. For the sake of simplicity, metaphors fall into one of two broad categories: the physical metaphor and the moral metaphor. In the first kind, two physical objects, whether animate or inanimate, it doesn't matter, are compared. In the second, something abstract, metaphysical, moral is compared to something physical that affects the senses, "either from the second to the first or from the first to the second" (p. 103). Proust's use of metaphor, I'll show later on, is much more in keeping with Fontanier's extended view of metaphor than with Aristotle's. We'll see how Proust in fact systematizes it, to the point of confusing metaphor with the artistic gesture itself. In so doing, he also exceeds the context of simple comparisons or analogies and he identifies metaphor at the level of univocity: being as such will be said to be metaphorically structured. As I'll try to show, the real itself is a metaphor and it's not a metaphor to say so.
Cf. Aristotle, _Eudemian Ethics_ , 1221b12–13; _Nicomachean Ethics_ , III, 15, 1119a36–b3.
Aristotle, _Physics_ , III, 1, 201a15; V, 2, 225a32–b2.
Proust, _Finding Time Again, SLT_ , VI, p. 197.
Following Ricoeur, it's worth stressing that, on one occasion at least, Aristotle mentions a case when there's no word that can be substituted for the metaphorical one; so, the phrase "sowing divine light" can be analysed according to the rules of proportional metaphor (B is to A what D is to C): what the sun is in relation to sunlight is what sowing is in relation to seeds. But there's no name for term B (in Greek at least). In this instance, Aristotle recognizes one of the functions of metaphor, i.e. its ability to make up for semantic lack, a function added to the ornamental function in the later tradition. In fact, the essence of metaphor will eventually be defined through the idea of analogical relation. My question is whether the exception to the rule of substitution only pertains to the analogy of proportion, or whether, conversely, it's able to overcome this lack in other ways. In other words, I want to know whether, in the absence of a substitutable term, it's only analogy that can signify or if metaphor can escape the rule of substitution as well as the rule of analogy. How? By freeing up a sense of being that is neither plurivocal nor analogical, but univocal. In order to deal with this, we'll need to generalize metaphor, to turn it into a law and not just a simple rule. We'll have to show how it belongs to a logic of "primary" and systematic replacement. There's metaphor because there's deficiency, because there's a lack. This lack, though, isn't the negation of a positive, a self-present and self-identical origin: the real as a whole is lacking and shifting. The world itself is a metaphor.
Jacques Derrida, "White Mythology" in _Margins of Philosophy_ , translated by Alan Bass, Chicago: Chicago UP (1982 [1972]), p. 234, note 40. Derrida's quoting chapter III of Fontanier's _Les Figures du discours_ , a chapter called "On the tropes by resemblance, that is, metaphors" and in which we find the following definition: "Tropes by resemblance consist in presenting an idea under the sign of another idea that is more striking or better known, and which, moreover, has no other tie to the first idea than that of a certain conformity of analogy" (op. cit., p. 99).
Albert Henry, _Métonymie et Métaphore_ , Paris: Klincksieck, 1971, p. 57. In the same book, Henry also quotes Claudel's _Journal_ in which we find the following: "Metaphor, like reasoning, brings together, but from farther away" ( _Journal_ , Èd. de la Pléiade, I, p. 42).
Ricoeur, _The Rule of Metaphor_ , op. cit., p. 196.
Ibid.
On difference, albeit more in terms of science than of art, see my _Truth and Genesis: Philosophy as Differential Ontology_ , Bloomington and Indianapolis: Indiana University Press, 2004, the goal of which was to bypass the onto-ethical primacy of identity (the idea as well as the genus). What I try to do is to show how non-linear dynamics break with Aristotelian formalism and essentialism, which have always subordinated the comparison between phenomena to the form that they're supposed to represent. For instance, while there's a connection between a bubble of soap and a crystal, comparing their respective geometrical shape (the sphere and the cube) will always fail to establish any such connection. Rather, it's a matter of showing that what elicits their association is also what supports their difference, in this case an economic principle of free energy (the bubble of soap strives to minimize the surface tension and the crystal strives to minimize the energy required for crystallization). The sphericity of the soap bubble is in no way a _function_ of the equi-distance of its points from its centre, therefore, any more than the cubic aspect of the crystal is a function of the formal properties of cubes, namely the fact that it'sa solid figure with six sides. Their shapes are the results of topological and not geometical data, therefore. What conditions the geometrical shape of the bubble and the crystal and elicits their association isn't anything geometrical. We might say that the "essence" of these phenomena doesn't resemble them in any way, that it's not something we can "see" as we can "see" their image but is, rather, their invisible and yet determining side. It seems to me that the same goes for metaphor, insofar as it foregrounds something shared by two phenomena or two images but on the basis of a completely different reality: a principle of heterogeneity presides over their relation and what unites them is a singular reality (not a generality). Between this reality (or this condition) and these images there's something like a relation of _expression_ : what they have in common is that they are different expressions of the same singularity, which neither one actually resembles. What's distinctive both to this reality and to these images is their ability to display the difference that they convey and hence the expressive power of difference itself. In my view, it's here that we run up against the univocal sense of being _as_ difference and break with the plurivocal and analogical conceptions of being.
Ferdinand de Saussure, _Cours de linguistique générale_ , Paris: Èditions Payot, 1972, pp. 176–84; trans. Roy Harris, _Course in General Linguistics_ , La Salle, Ill.: Open Court, 1983, pp. 126–32; Ricoeur, _The Rule of Metaphor_ , op. cit., p. 117. In my view, such exchanges in fact relate to the previous chapter, "Syntagmatic Relations and Associative Relations." Associative relations are mental, often mnemonic associations that we perform in the presence of certain words or groups of words.
Roman Jakobson, "Two Aspects of Language and two Types of Aphasic Disturbances," _On Language_ , Harvard UP, 1995, p. 129. Among Jakobson's predecessors who see association as the mechanism presiding over semantic innovations, Léonce Roudet's work is worth noting ("Sur la classification psychologique des changements sémantiques," _Journal de psychologie_ , XVIII, 1921, pp. 676–92), Z. Gombocz ( _Jelentéstan_ , Pécs, 1926), as is Stephen Ullman's (see _The Principles of Semantics_ , Glasgow and Oxford: Jackson & Blackwell, 1951 and 1959 for the 2nd revised edition). A detailed discussion of these texts is available in Ricoeur, _The Rule of Metaphor_ , op. cit., pp. 110–25.
I tackled the issue of _Verschiebung_ in the last chapter.
Jacques Lacan, "The Instance of the Letter in the Unconscious or Reason Since Freud" (1957), in _Ècrits_ , translated by Bruce Fink, New York: Norton, 2006, pp. 438–39.
César Chesneau Du Marsais, _Traité des tropes_ (1818), Paris: Le Nouveau Commerce, 1977, Second Chapter, II, p. 61.
In this sense, I agree with Julia Kristeva's remarks in _Time and Sense: Proust and the Experience of Literature_ , translated by Ross Guberman, New York: Columbia UP, 1996, pp. 215–16. In more than one way, this study could be read as a supplement to the chapter that Kristeva devotes to metaphor ("A Tribute to the Metaphor", pp. 199–226). My only reservation would concern the nature of the ontology — analogical in Kristeva's view, univocal in my opinion — that Proust's conception of metaphor implies. This is what underlies — crucially in my mind — Proust's break with Aristotle's poetics and ontology, paving the way for an aesthetics of originary difference.
Etienne Bonnot de Condillac, _Traité de l'Art d'écrire_ (Orléans: Èditions Le Pli, 2002), II, VI.
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 415.
Ibid.
Ibid.
Ibid., pp. 415–16.
Proust, _The Way by Swann's, SLT_ , I, p. 14.
Compossible things, Leibniz writes in a piece from the pre-Paris period, are "those, one of which being given, it does not follow that the other is negated; or those of which one is possible, the other being assumed" (Gottfried Wilhelm Leibniz: _Sämtliche Schriften und Briefe_ , edited by the German Academy of Science, Darmstadt and Berlin: Berlin Academy, 1923, 6.2.498). As such, compossibility is necessary for what Leibniz calls the harmony of substances. By contrast, as he makes clear in his Paris notes, something is incompossible when it is impossible not in itself or its essence, because its concept implies a contradiction, but in relation to the existence of other things (AK 6.3.463–64). When applied to Proust, this view should stipulate that artistic vision reveals as compossible, and even as belonging to the same essence, two or more substances ordinarily thought to be mutually exclusive.
Proust, _Finding Time Again, SLT_ , VI, p. 25.
See Charles Baudelaire, "Correspondences," "Spleen and Ideal" in _Les Fleurs du Mal_ , translated by Richard Howard, David R. Godine Publisher, 1983, p. 15:
> Like long-held echoes, blending somewhere else
> into one deep and shadowy unison
> as limitless as darkness and as day,
> the sounds, the scents, the colors correspond.
M. Merleau-Ponty, _The Visible and the Invisible_ , op. cit., p. 125.
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 383. I've tried to indicate and comment on the series of sematic shifts operating here.
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 476.
Ibid., p. 475.
Proust, _Jean Santeuil_ , op. cit., p. 398: ".... and the sea there was as if dry".
Deleuze, _Proust and Signs_ , op. cit., p. 167.
Ibid., pp. 166–67.
Gérard Genette, "Proust Palimpsest," in _Figures of Literary Discourse_ , trans. Alan Sheridan, New York: Columbia UP, 1982, p. 208. In _Proust ou le réel retrouvé_ , Anne Simon also acknowledges and analyses this "tension" (op. cit., pp. 34 ff.), but she ultimately denounces it as being only apparent (p. 43). In fact, Proust's novel finds a way to overcome this dispute (the dispute opposing idealism and materialism that was raging at the time) by setting a rich and nuanced theory of sensations that stands in stark contrast to Rabier's simplistic view and whose philosophical expression will be found in Merlau-Ponty's late work. The whole point here is that "since the sensible experience itself already pertains to the spiritual, the writing thus appears as a deepening of the sensory work" (p. 44). From this perspective, Anne Simon's analysis is indebted to Carbone's work that I mentioned earlier on. I agree that the contradiction between essence and impression is only apparent, but I'm suggesting a different way of lifting it (one that involves the mediation of difference).
Gilles Deleuze, _Expressionism in Philosophy: Spinoza_ , trans. Martin Joughin, New York: Zone Books, 1990.
Quoted by Jean Milly in _Proust et le style_ , Paris: Minard, 1970, p. 89.
Proust, _Against Sainte-Beuve and Other Essays_ , op. cit., p. 204.
Ibid., p. 261.
"Preface to _Tendres Stocks_ " in _Marcel Proust: A Selection from his Miscellaneous Writings_ , translated by Gerard Hopkins, London: Allan Wingate, 1948, pp. 218–19. These pages were first published in _La Revue de Paris_ on 15 November 1920 under the title "Pour un ami (remarques sur le style)." Flaubert's image is taken from _L'Èducation sentimentale_ , Folio Gallimard, Paris, p. 20.
A summary of this critique can be found in the "Prefaces," _Against Sainte-Beuve and Other Essays_ , pp. 3–9 and in the _Search_ itself ( _Finding Time Again, SLT_ , VI, pp. 187–88).
Proust, _Finding Time Again, SLT_ , VI, p. 188.
Ibid., p. 199.
Ibid., p. 188.
This is why I don't agree with Luc Fraisse ( _L Esthétique de Marcel Proust_ , SEDES collection "Esthétique," 1995, p. 37), who sees Proust's view of art as "essentially impressionistic." If art originates in impressions, then the artist's "work" lies in its expression, in the sense indicated above. For a philosophical interpretation of impressionism which doesn't exactly match Proust's aesthetics, see John Sallis's shrewd and insightful study on Monet's "Haystacks" (J. Sallis, _Shades — of Painting at the Limit_ , Bloomington and Indianapolis: Indiana University Press, 1998, pp. 22–56).
Jean Milly, _Proust et le style_ , op. cit., p. 98.
Proust, _The Prisoner, SLT_ , V, pp. 59–60.
Ibid., p. 60.
Proust, _Finding Time Again, SLT_ , VI, p. 194.
Ibid., pp. 197–98.
Ibid.
Ibid.
Ibid.
Gilles Deleuze, _Difference and Repetition_ , translated by Paul Patton, London: Continuum, 2004, pp. 169–76.
Ibid., p. 169.
Deleuze, _Proust and Signs_ , "The Image of Thought." Finding ourselves on the path to truth doesn't happen naturally. Proust's novel teaches us that truth isn't ceded: it betrays itself; it's not communicated: it's interpreted; it's not deliberate: it's involuntary. We think under the influence of external signs. Some violence presides over the discovery of truth.
Hans Georg Gadamer, _Truth and Method_ , Second Revised Edition, Translation revised by Joel Weinsheimer and Donald G. Marshall, London: Continuum, 2004, pp. 113–14.
Barthes' remark comes from a round-table discussion published in the _Cahiers Marcel Proust_ , Nouvelle Série, no 7, Paris: Gallimard, 1975, pp. 87–116.
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 414.
A few pages before his visit to Elstir's studio, Marcel evokes a sunset that leaves him indifferent, so engaged is his mind with imagining dinner at Rivebelle on the same evening and pondering on how to please the guests: "With the bored, disdainful superficiality of the sated expert, or the elegant lady glancing in at an exhibition during her crowded day of visits to fashionable friends, I dismissed it with the thought: 'Quite an interesting sunset, rather different, but I've seen plenty of others that are just as delicately done and every bit as striking'" (Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 385).
#### **5 Dress or patchwork?**
A slightly different version of this chapter was published in Miguel de Beistegui, _Immanence: Deleuze and Philosophy_ , Edinburgh: Edinburgh University Press, 2010, pp. 160–72.
"... I should construct my book, I don't dare say, ambitiously, as if it were a cathedral, but simply as if it were a dress I was making." Proust, _Finding Time Again, SLT_ , VI, p. 343.
Luc Fraisse, _L'Œuvre cathédrale. Proust et l'architecture médiévale_ , Paris: Librairie José Corti, 1990, p. 79.
G. Deleuze, _Proust et les signes_ , Paris: Presses Universitaires de France, 1964/1972. Translated by Richard Howard as _Proust and Signs_ , London: The Athlone Press, 2000. My analysis will focus on the chapter entitled "Cells and Vessels."
Proust, _The Prisoner, SLT_ , V, pp. 666–67. My emphasis.
Those bits of paper pasted onto the manuscript were known to the printers as " _béquets_ ," and are referred to in the novel as " _paperoles_."
Proust, _Finding Time Again, SLT_ , VI, p. 485.
Proust, _Finding Time Again, SLT_ , VI, p. 344.
Cited in G. Deleuze, _Proust and Signs_ , p. 116.
Ibid., p. 116.
The chapter in question is entitled "Essences and The Signs of Art"; the fact that Deleuze mentions "complication" in that chapter shouldn't fool us, for it is used interchangeably with "implication."
G. Deleuze, _Proust and Signs_ , p. 112.
In that sense, the fragments that are here in question resemble the fragments of German Romanticism, and of Friedrich Schlegel in particular, for whom the total or absolute work, the fragment, reveals an incompletion that is universal and essential. See Friedrich Schlegel, _Philosophical Fragments_ , Minneapolis: Minnesota University Press, 1991, translated by Peter Firchow, and the Foreword by Rodolphe Gasché. See also M. Blanchot, "L'Athenaeum" in _L'entretien infini_ , Paris: Gallimard, 1969, pp. 515–27; Ph. Lacoue-Labarthe and J.L. Nancy, _L absolu littéraire_ , Paris: Èditions du Seuil, 1978, pp. 57–80, in particular "L'exigence fragmentaire". It is this very "fragmentary exigency" that Blanchot takes up again and develops in _L'écriture du désastre_ , Paris: Gallimard (pp. 96–102 in particular), which he connects with the notions of " _désastre_ " and " _désœuvrement_." One could argue that, in his own way, Proust himself responds to this exigency.
Proust, _The Prisoner, SLT_ , V, 169.
All of this, of course, takes us back to Homer's _Odyssey_ , and to the paradigmatic image of Penelope's woof. Like Penelope, Proust undoes with one hand what he has done with the other hand, or adds onto the tight fabric of the novel a looser one.
If the conjunction that characterizes the cell or the box is the coordination "and" (Albertine is youth itself _and_ the little gang _and_ Balbec _and_ the waterfront _and_ an infinity of other boxes, which Marcel's jealous curiosity seeks — in vain — to discover in their entirety), that of the closed vessels is the disjunctive "or."
Proust, _In the Shadow of Young Girls in Flower, SLT_ , II, p. 466.
Ibid., p. 507.
Ibid., p. 509.
Ibid., p. 510.
Proust, _The Guermantes Way, SLT_ , III, p. 360.
The absolute that the narrator eventually reaches is of a different kind, namely, of the kind that's been used to characterize the ambition of German Romanticism — a _literary_ absolute, which makes room and integrates the reality of the fragment. The literary absolute, therefore, is not synonymous with the dialectical absolute, or with a self-differentiated totality, inasmuch as it affirms the unsurpassable reality of the fragment.
Proust, _The Guermantes Way, SLT_ , III, p. 361.
Ibid., pp. 362–63.
* * *
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* * *
Index
* * *
absence ; and imagination , ; _see also_ self-absence
absolute, desire for , ,
acquired automatic memory
aesthetics , , , , , , ,
affective association
affective memory , , , , ,
_Against Sainte-Beuve_ (Proust) ,
agreeable , ,
Albaret, Céleste
_allotrios_
anamnesis , –
aphasia
Aristotle ; and metaphor , –, , , , ; and ontology/ontological , –; _Physics_
art , , , , , –, –, ; escaping boredom and dissatisfaction through –, , ; experience of ; and imaginary ; prolonging of life ; and reality –; theory of ; work of , , , , , , –
art of living
artistic time –
association, laws of –
associationist psychology , , , ,
Barthes, Roland
beauty/beautiful , , , , ,
Beckett, Samuel: _Proust_ ,
Beethoven, Ludwig van
Benjamin, Walter ; 'On Some Motifs in Baudelaire'
Bergson, Henri , , , , ,
Biran, Maine de ,
boredom , ; escaping of through art ; and love ; tension between imagination and perception as source of
borrowed value –
cathedrals , ,
Chateaubriand
Chesneau Du Marsais, César
chronological time , ,
_Chronos agèraos_
classical metaphysics –
Coleridge, Samuel
common sense
complication: unity of –
Condillac, Étienne Bonnot de
contiguity , , , –
Deleuze, Gilles , –, , , –, , –, –, ; _Proust and Signs_ ,
Derrida, Jacques
Descartes, René ,
desire , , , ,
difference ; and metaphor –, ; and nature
disappointment , ; alleviation of by art –; reality as source of , , ,
discontinuous ego –
dress –
ego , , ; demystification of ; discontinuous –
_Eingedenken_
Elstir's seascapes –
empiricism
_Erfahrung_ ,
_Erinnerung_
_Erlebnis_ ,
Eros
essence , , , , –, , , ; and metaphor , –
esthesiology
events
experience , , , ; mystery of –; unlived , , , , ,
expressionism –,
facts ,
Flaubert, Gustave: _Education sentimnentale, L'_ ,
Flournoy, Théodore
Fontanier, Pierre ,
forgetting –
fragments
Fraisse, Luc
Freud, Sigmund
friendship –
Gadamer, Hans Georg
Gallimard, Folio
Genette, Gérard
Goya
Greeks, ancient
habit –
habitual memory –, , –, ,
happiness , , , ,
harmony ,
hawthorn scene –, , –
Henry, Albert: _Métonymie et Métaphore_ –
Hume, David
Husserl, Edmund
Idea/Ideas , , , , ,
idealism ; and literature
imaginary , , , ,
imagination , , , ; and absence , ; conflict between perception and , ; and nature ; and reality , , ,
instant
intelligence , , , , –, –, –
invisible: and visible
involuntary memory , , , , , –, , , –, ,
Jakobson, Roman ,
jealousy , , ,
_Jean Santeuil_ (Proust)
joy: finding –; giving –; looking for –; paradoxes of
Kant, Immanuel –, , ; _Critique of Judgement_
kiss: between Marcel and Albertine –
knowing , –; memory as faculty of
Kronos , , ,
Lacan, Jacques ,
lack/lacking –, –
learning
Lethe, cult of
life: kinds of
linguistics , ,
literature ; and embodied ideas ; and idealism ; and reality –, , –,
lived experience , –, , , , , , ,
lived past ,
lived present , ,
lived time ,
_logos_ ,
lost time , ,
love –, ,
'madeleine experience' –, ,
material joy
memory , , , –; acquired automatic ; affective , , , , , ; based on sedimentation –; body's ; connection with writing ; differing of Proust's view of with contemporaries ; as faculty of knowing ; habitual –, , –, , ; involuntary , , , , , –, , , –, , ; organic , , ; and sensation , ; time of ; voluntary –, , , ,
memory exercises
_Meno_ ,
Merleau-Ponty, Maurice , ,
metamorphosis –, ,
metaphor –; and Aristotle , –, , , , ; and borrowed value –; and contiguity –; definition and features –; and difference –, ; and dissemblance ; distinction between metonymy and –, ; and Elstir's seascapes –; and essence –; functions ; onto-diaphorical source –; ontological roots ; and recognition –; and resemblance –, –; as source of joy ; and substitution , ; and time ; and transference
metonymy –, ,
Mnemosyne ,
nature –; and difference ; and imagination ; intentionality of –
Nerval, Gérard de
nihilism ; reversal of , –
onto-poetics –,
ontology/ontological , , , , , , , , ; and Aristotle , –
organic memory , ,
organicity , , ,
paradise
Pascal, Blaise
passion
past , –, –, –, , –; immemorial , ; lived , and present , –, , , , , , ,
patchwork –
perception , , ; conflict between imagination and , ; infinite surpassing of ; and the present –; sensible
Perrin, Jean-François ,
phenomenology ,
philosophy , ,
plan –
Plato , , , , –, ; _Republic_ –
Platonism , –,
pleasantness
pleasure –
pleasure principle –
poetic _logos_
poetics –
present , , , , , ; and past , –, , , , , , , ; and perception –; unlived ,
present moment
_Prisoner, The_ (Proust)
psychology/psychologists –
Pythagoreans
Ravaisson, Félix ; _On Habit_ –
reality , ; and art –; and imagination , , , ; and literature –, , –, ; reconciliation with sensible –; as source of disappointment , , ,
recognition , –
recollection ,
remembering, general laws of
reminiscence , –, ,
repetition –
resemblance , , ; and metaphor –, –
residue
Reverdy
Ribot, T. , , , , ; _Psychology of Feelings_ –
Ricoeur, Paul , , ,
Ruskin, John: _The Bible of Amiens_
Schopenhauer, Arthur ,
Scottish school
self-absence , , ,
sensation , –, –, ; and memory , ; and pleasure principle –
sensible , –,
_Sesame and Lilies_ –
Socrates
Spencer, Herbert
Spinoza
spiritualism
style , , –, –,
subjectivity
substitution , ,
suffering , ,
Taine, Hippolyte , , , , , –; _On Intelligence_ ,
taste –, ; and beautiful
_textus_
time , , –, –; artistic –; chronological , , ; lived , ; lost , , ; and metaphor ; regained –
time lost
time regained –
transference: and metaphor
transubstatiation , ,
_Traumdeutung_
Trophonios oracle
true life
truth –, ; and intelligence ; love of –
unconscious ,
unity , –, ; of complication –
unlived experience , , , , , ,
unlived present ,
Vermeer: _View of Delft_
Vernant, Jean-Pierre
virtual memory
visible: and the invisible , ,
voluntary memory –, , , ,
Wagner, Richard
weaving –
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{"url":"https:\/\/robinryder.wordpress.com\/2010\/02\/","text":"Archive for February, 2010\n\nObama\u2019s skin seems darker to his opponents and lighter to his\u00a0proponents\n\n26\/02\/2010\n\nThis is a couple of months old, but still worth reposting:\n\nIn three studies, participants rated the representativeness of photographs of a hypothetical (Study 1) or real (Barack Obama; Studies 2 and 3) biracial political candidate. Unbeknownst to participants, some of the photographs had been altered to make the candidate\u2019s skin tone either lighter or darker than it was in the original photograph. Participants whose partisanship matched that of the candidate they were evaluating consistently rated the lightened photographs as more representative of the candidate than the darkened photographs, whereas participants whose partisanship did not match that of the candidate showed the opposite pattern.\n\nCNRS: short interview!\n\n25\/02\/2010\n\nI got news yesterday that I passed the first stage of the CNRS hiring process (admissibilit\u00e9), which was definitely the easy part!\n\nI was surprised by the description of the forthcoming interview: it will only last 7 minutes, including a 2 minute presentation of my research project! I guess this is simply a consequence of the large number of candidates, but I wonder how much the jury can take out of such a short interview. In a way, this makes it less stressful for me: I suppose the interview only has a small influence on the final decision.\n\nLe Monde puzzle on sums of\u00a0products\n\n25\/02\/2010\n\nChristian Robert is disappointed by last week-end\u2019s Le Monde\u2019s mathematical problem :\n\nTake the integers 1 to 10. Group them together; the product of these numbers is 3,628,800. This is the first term of your sum. Now group these numbers in 9s. There are 10 different ways of doing this. For each group, take the product of the 9 numbers and substract the results of each of these 10 products from your sum. Repeat the process by grouping the numbers in 8s (add the product), in 7s (substract them), \u2026, in pairs (add them) and in singletons (substract them). What is the final total? What happens if, instead of 1 to 10, you apply the same (even more interminable) procedure to the first 49 squares (1, 4, 9, 16, \u2026 2401)?\n\nChristian solves this the brutal way, which gives the answer but isn\u2019t very satisfying. Here are a couple of alternative ways of getting the product\u00a0 (call it $P$).\n\n1. Consider the following 11\u00d711 matrix:\n\n$\\begin{tabular}{*{11}{c}} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 0& 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 0& 0 & 3 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 0& 0 & 0 & 4 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 0& 0 & 0 & 0 & 5 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 0& 0 & 0 & 0 & 0 & 6 & 1 & 1 & 1 & 1 & 1 \\\\ 0& 0 & 0 & 0 & 0 & 0 & 7 & 1 & 1 & 1 & 1 \\\\ 0& 0 & 0 & 0 & 0 & 0 & 0 & 8 & 1 & 1 & 1 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 1 & 1 \\\\ 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 1 \\\\1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\end{tabular}$\n\nThe determinant is obviously 0 (the first and last row are identical). On the other hand, using the Leibniz formula to calculate the determinant would yield all the terms in the product calculated above, plus one term given by the bottom-leftmost 1 and the 1s on the diagonal above the main diagonal. Hence $0=P+1$.\n\n2. This solution is more obvious, and\u00a0 I expect Le Monde will give a solution similar to this. Let $P_n$ be the product given when using the numbers 1 to $n$. Then\n\n$P_n=nP_{n-1} -P_{n-1} +(-1)^{n-1} \\times n$\n\nwhence $P_n=(-1)^{n-1}$.\n\n3. I was hoping to find a counting argument to solve this problem, because the alternating signs reminded me of the formula for the cardinal of a union of sets. Here\u2019s the best I have come up with, but I\u2019m sure there\u2019s a better way of seeing this.\n\nConsider the set of all sequences of 10 integers where the $i$th entry is an integer between 1 and $i$; this corresponds to the sequences which lie in the lexicographic order between $(1,1,1,\\ldots , 1)$ and $(1, 2, 3, \\ldots , 10)$. There are clearly $10!$ such sequences, but let\u2019s count them in a more stupid way, noting that each sequence contains at least one $1$..\n\nLet $A_i$ be the number of such sequences with the additional constraint that the $i$th entry is equal to $1$. Then, using $\\vert \\cdot \\vert$ to denote the cardinal function, we have\n\n$10! = | \\cup A_{i} | = | A_1 | + | A_2 | + \\ldots + | A_{10} | - |A_1 \\cap A_2 | - |A_1 \\cap A_3| \\ldots + |A_1 \\cap A_2 \\cap A_3| \\ldots$\n\nEach of these terms corresponds to a term in $P$, with the exception of the last term $|A_1\\cap A_2 \\cap \\ldots \\cap A_{10}|$, which is worth $1$ and comes with the sign of $(-1)^{n-1}$.\n\nA very similar reasoning holds if we take numbers other than 1, 2, \u2026 10, as long as the first number is a $1$.","date":"2017-10-23 20:59:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 23, \"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\": 0.8294453024864197, \"perplexity\": 421.8847015965193}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-43\/segments\/1508187826642.70\/warc\/CC-MAIN-20171023202120-20171023222120-00848.warc.gz\"}"} | null | null |
<?php //-->
use Cradle\CommandLine\Index as CommandLine;
use Cradle\Sink\Faucet\Schema;
return function($request, $response) {
$cwd = $request->getServer('PWD');
$schemaRoot = $cwd . '/schema';
if(!is_dir($schemaRoot)) {
return CommandLine::error('Schema folder not found. Generator Aborted.');
}
//Available schemas
$available = [];
$paths = scandir($schemaRoot, 0);
foreach($paths as $path) {
if(strpos($path, '.') === 0) {
continue;
}
if(!is_dir($schemaRoot . '/' . $path)
&& !file_exists($schemaRoot . '/' . $path)
)
{
continue;
}
$available[] = pathinfo($path, PATHINFO_FILENAME);
}
if(empty($available)) {
return CommandLine::error('No available schemas found in ' . $schemaRoot);
}
//determine the active schema
$active = $request->getStage('schema');
if(!$active) {
CommandLine::info('Available schemas:');
foreach($available as $name) {
CommandLine::info(' - ' . $name);
}
$active = CommandLine::input('Which schema to use?');
}
if(!in_array($active, $available)) {
return CommandLine::error('Invalid schema. Generator Aborted.');
}
//it is possible that the active schema has multiple schemas
$schemas = [];
if(file_exists($schemaRoot . '/' . $active . '.php')) {
$schemas[] = $active;
} else if(is_dir($schemaRoot . '/' . $active)) {
$paths = scandir($schemaRoot . '/' . $active, 0);
foreach($paths as $path) {
if($path === '.' || $path === '..' || substr($path, -4) !== '.php') {
continue;
}
$schemas[] = $active . '/' . pathinfo($path, PATHINFO_FILENAME);
}
}
CommandLine::system('Generating REST...');
$handlebars = include __DIR__ . '/helper/handlebars.php';
//get source and destination root
$sourceRoot = __DIR__ . '/template/rest';
$destinationRoot = $cwd . '/app/api/src/controller/rest';
foreach($schemas as $schema) {
//get the template data
$data = (new Schema($schemaRoot, $schema))->getData();
//get all the files
$paths = new RecursiveIteratorIterator(new RecursiveDirectoryIterator($sourceRoot));
foreach ($paths as $source) {
//is it a folder ?
if($source->isDir()) {
continue;
}
//it's a file, determine the destination
// if /template/module/src/events.php, then /path/to/file
$destination = $destinationRoot . substr($source->getPathname(), strlen($sourceRoot));
$destination = str_replace('NAME', $data['name'], $destination);
//does it not exist?
if(!is_dir(dirname($destination))) {
//then make it
mkdir(dirname($destination), 0777, true);
}
//if the destination exists
if(file_exists($destination)) {
//ask questions
$overwrite = CommandLine::input($destination .' exists. Overwrite?(n)', 'n');
if($overwrite === 'n') {
CommandLine::warning('Skipping...');
continue;
}
}
CommandLine::info('Making ' . $destination);
$contents = file_get_contents($source->getPathname());
$template = $handlebars->compile($contents);
$contents = $template($data);
$contents = str_replace('{{ ', '{{', $contents);
file_put_contents($destination, $contents);
}
//add to cradle.php
$cradleFile = $cwd . '/app/api/.cradle.php';
if(file_exists($cwd . '/app/api/.cradle')) {
$cradleFile = $cwd . '/app/api/.cradle';
}
if(file_exists($cradleFile)) {
$flag = '//START: GENERATED CONTROLLERS';
$add = 'include_once __DIR__ . \'/src/controller/rest/' . $data['name'] . '.php\';';
$contents = file_get_contents($cradleFile);
if(strpos($contents, $flag) !== false && strpos($contents, $add) === false) {
$contents = str_replace($flag, $flag . PHP_EOL . $add, $contents);
}
CommandLine::info('Updating ' . $cradleFile);
file_put_contents($cradleFile, $contents);
}
}
CommandLine::success($active . ' REST was generated.');
};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,910 |
Spectacular sea views home above the Mediterranean Sea in Altea. The property features amazing views from every window. Wood burning fireplace in the spacious living/dining room, very large terraces, kitchen, 3 bedrooms and 2 bathrooms are located in the main house. There is a fully equipped guest apartment of modern design with a large living room with fully equipped design kitchen, 1 bedroom, 1 bathroom and a private terrace. This lovely home features a heated swimming pool that allows you to enjoy the Mediterranean climate year round. Garage for 1 car and parking space for several vehicles on the plot. Are you interested in this property? Please complete the contact form, and we will get in touch with you as soon as possible. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,384 |
Trump Is Scaring Indian Americans Into Finding Their Political Voice
america, immigration, india, law, national security, politics
The Atlantic | March 17, 2017 | By Emma Green
Highly educated immigrants from South Asia have often been able to live comfortably in America. With a new wave of hate crimes, that's changing. Manik Suri is the archetypical overachiever from an Indian American family. The 34-year-old runs a start-up in Silicon Valley. He speaks four languages. He's got two Ivy League degrees. And yet, when the windows at an Indian restaurant near his house were shot out late February, along with those of an Eritrean place nearby, he felt shaken. Read more here.
Of Challenges Tempered With Optimism
development, economics, india, politics
India Abroad | August 23, 2013 | By Manik Suri
On the occasion of India's 66th anniversary, the world's largest democracy has made significant strides in its economic, social, and political development. Notwithstanding serious governance challenges in the near term, there are fundamental reasons to remain optimistic about India's long term prospects. Read the full piece here.
From Crowdsourcing Potholes to Community Policing
america, cyber, economics, innovation, law, politics, technology
Berkman Center for Internet & Society at Harvard University | Interoperability Case Studies | August 15, 2013 | By Manik Suri
Open311 is a state-of-the-art technology platform that provides a uniform base to expand existing "311" services, which provide information tracking and monitoring in cities around the world. Over the past decade, these 311 services have allowed cities to respond to millions of citizen-generated inputs, creating better and smarter governance. This paper applies "interoperability" theory to consider the promises and perils of Open311, explaining how we can unlock the full potential of this powerful civic technology platform in the future. Read the full piece here.
Reorienting the Principal-Agent Frame: Adopting the "Hartian" Assumption in Understanding and Shaping Legal Constraints on the Executive
america, constitutional law, law, politics
Harvard Law and Policy Review | Volume 7, No. 2: Summer 2013 | By Manik Suri
Debate over whether law can, and indeed should, constrain presidential power is as old as the Republic. This article claims that liberal legalists, who adopt a consequentialist "Holmesian" view of the law, ignore the possibility that law – as an internalized normative commitment or duty – can restrain the executive. This alternative "Hartian" view may help explain how laws constrain presidential power at key moments in history. Recognizing the difficulty in establishing causation, the article nonetheless concludes that much is at stake in how we frame the relationship between law and the executive. Read the full piece here.
Why the Indian American Vote Mattered
america, immigration, india, politics
India America Today | November 21, 2012 | By Manik Suri
While Indian Americans were solidly within the Obama camp, the challenge in this election lay in making sure their voice was heard – only 63 percent had voted in 2008. The three million-strong community's widespread distribution – particularly in contested states – meant that they could help move the needle where it counted. Now, looking back at the 2012 election, Indian Americans who voted should feel proud: they were joined by record numbers of voters across key liberal constituencies, including Latinos and African Americans, who delivered at the ballot box – toppling conservative pundits' electoral models, challenging long-held assumptions, and igniting a firestorm within the Republican leadership over their party's ability to connect with an electorate that is increasingly diverse. Read the full piece here.
Time to Get Out the Indian American Vote
India America Today | October 27, 2012 | By Manik Suri
Indian Americans are amongst President Obama's most committed backers, but less than two-thirds of the 3-million strong community's eligible voters showed up at the ballot box in 2008. This time around, no one can afford to stay on the sidelines. Each of us must head to the polls not only because we believe in a better future – the very reason our families came to this country – but because we are committed to shaping it ourselves. Doing so will strengthen the community's political voice. But more importantly, it could help decide an election where the stakes are high, margins are razor-thin, and every vote counts. Read the full piece here. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,807 |
Would a 51% attack be profitable for the attacker?
Is attack of 51% just a concept or a real threat?
What is an attack of 51%?
A lot of cryptocurrencies, are switching from POW mining to POS. What is the difference?
Is there any place where all hardforks and important announcements are gathered?
How can I join groups of blockchain developers? Where do they communicate? | {
"redpajama_set_name": "RedPajamaC4"
} | 311 |
\section{SECTION 1 HEADING TYPE HERE}
Rapid experimental progress in monitoring of single quantum systems
renewed the interest in the quantum mechanical limitation of
measurement accuracy\cite{rus}. This limitation is originated from
the uncertainty relation between observables of a measured
microscopic system, the so-called standard quantum limit of
measurement\cite{brag}. Yet, a single quantum system is not observed
directly, but through the interaction with a measurement device
(detector). This implies that the quantum limit of measurement is
not related to a measured system only, but rather to an entire
system, including detector\cite{brag}.
A measurement in the quantum mechanical formalism corresponds to
projection of the wave function of an entire system on eigenstates
of the {\em detector}, accessible by an observer. This is the
so-called projection postulate\cite{neu}, analogues to the Bayes
principle in any probabilistic description. Since the detector and
the measured system are interacting, the above projection on the
{\em detector} states affects the measured microscopic system. If
the latter is projected on one of its own states, then the
microscopic system can be measured with any accuracy. However, if
this system is projected on the superposition of its states, the
measurement cannot be precise. Its accuracy is given by a size of
the corresponding wave packet.
In this Letter we apply the above described procedure for a
determination of the precision limit of quantum measurements. As an
example we take an electron trapped inside a double-dot
(electrostatic qubit) and continuous monitored by a point-contact
detector\cite{qb1}. The total system can be treated entirely quantum
mechanically, although the detector represents a macroscopic
(mesoscopic) device\cite{gur}. This allows us to obtain the detector
eigenstates and then to perform projections of the total wave
function on these eigenstates without any additional assumptions.
Consider a qubit, represented by an electron in the double-dot,
placed near a point-contact that separated two reservoirs (the
point-contact detector), Fig.~1. When the first dot, which is far
away from to the point-contact, is occupies (Fig.~1a), the current
through the point-contact is $I_1$. If the second dot, close to the
point contact, is occupied (Fig.~1b), the current decreases
($I_2<I_1$) due to the electrostatic repulsion. The entire system
can be described by the following tunneling Hamiltonian\cite{gur}:
$H=H_{PC}+H_{DD}+H_{int}$, where
\begin{eqnarray}
\label{a1} {\cal H}_{PC}&=&\sum_l E_la_l^\dagger a_l+\sum_r
E_ra_r^\dagger a_r+\sum_{l,r}\Omega_{lr}(a_l^\dagger a_r+H.c.),
\label{a1a}\\
{\cal H}_{DD}&=&E_1 c_1^{\dagger}c_{1}+E_2 c_2^{\dagger}c_{2}+
\Omega (c_2^{\dagger}c_{1}+ c_1^{\dagger}c_{2})\, ,
\label{a1b}\\
{\cal H}_{int}&=&\sum_{l,r}\delta\Omega_{lr}c_2^{\dagger}
c_2(a^{\dagger}_la_r+H.c.)\, , \label{a1c}
\end{eqnarray}
and $\delta\Omega_{lr}=\Omega'_{lr}-\Omega_{lr}$. Here
$a_{l,r}^\dagger (a_{l,r})$ is the creation (annihilation) operator
for an electron at the level $l$ or $r$ in the left or right
reservoir, and $c^\dagger_{1,2}(c_{1,2})$ is the same operator for
the electron inside the double-dot. $\Omega_{lr}$ is the hopping
amplitude between the states $l$ and $r$ of the reservoirs, and
$\Omega$ is the hopping amplitude between the states $E_1$ and $E_2$
of the qubit. For simplicity we consider electrons as spin-less
fermions. The interaction term $H_{int}$ generates variation of the
hopping amplitude, $\delta\Omega_{lr}={\Omega'}_{lr}-\Omega_{lr}$,
resulting in a decrease of the detector current from $I_1$ to $I_2$,
Fig.~1.
\begin{figure}
{\centering{\psfig{figure=fig1.eps,height=5cm,width=12cm,angle=0}}}\\
{\bf Fig.~1:} The electrostatic qubit monitored by the point-contact
detector. The detector current decreases when the left dot of the
qubit is occupied. Here $n$ denotes the number of electrons which
have arrived at the right reservoir by time $t$.
\end{figure}
The wave function describing the entire system can be written as
\begin{eqnarray}
&&|\Psi (t)\rangle = \left [ b_1(t)c_1^{\dagger} + \sum_{l,r}
b_{1lr}(t)c_1^{\dagger}a_r^{\dagger}a_l +\sum_{l<l'\atop r<r'}
b_{1ll'rr'}(t)
c_1^{\dagger}a_r^{\dagger}a_{r'}^{\dagger}a_la_{l'}\right.\nonumber\\
&&\left.+b_2(t)c_2^{\dagger} +\sum_{l,r}
b_{2lr}(t)c_2^{\dagger}a_r^{\dagger}a_l+\sum_{l<l'\atop r<r'}
b_{2ll'rr'}(t)
c_2^{\dagger}a_r^{\dagger}a_{r'}^{\dagger}a_la_{l'}+\cdots \right
]|0\rangle, \label{a2}
\end{eqnarray}
where $b_\alpha (t)$ is the amplitude of finding the system in the
state $\alpha$ determined by the corresponding creation and
annihilation operators. These operators act on the initial
(``vacuum'') state, $|0\rangle$. For simplicity we assume that the
reservoirs are initially at zero temperature and filled with
electrons up to the Fermi energies $\mu_{L,R}$, respectively. All
the amplitudes $b_\alpha (t)$ can be obtained from the
time-dependent Schr\"odinger equation, $\partial_t|\Psi (t)\rangle
=H|\Psi (t)\rangle $.
Now we project the total wave function (\ref{a2}) on the detector
eigenstates. There exists an uncertainty however, about a choice of
the eigenstates, since different detector variables can be
recorded\cite{zur}. In fact, the point-contact detector is not
recorded directly, but via another readout device (``pointer''). The
latter can single out a particular set of the eigenstates through a
coupling with the corresponding detector variables. Yet, in this
work we do not extend our system by including such a pointer in the
Schr\"odinger equation. We assume instead that the detector states
are directly accessible to an ``observer''. Let us examine different
alternatives of the measurement.
Consider first the measurement of number of electrons ($n$) in the
right reservoir (the accumulated charge), Fig.~1. This implies that
the wave function, $|\Psi (t)\rangle$, Eq.~(\ref{a2}), is projected
on the eigenstates of the operator $\hat
N=\sum_ra_{r}^{\dagger}a_{r}$, which can be written as
\begin{equation}
|n\rangle ={\underbrace{a_{r_1}^\dagger a_{r_2}^\dagger\cdots
a_{r_n}^\dagger}_n} |0\rangle\, . \label{a3}
\end{equation}
(Note, that the state $|n\rangle$ is strongly degenerate:
$|n\rangle\equiv |n,\alpha \rangle$, where $\alpha$ corresponds to a
particular configuration of $n$ electrons in the collector). Thus,
$|\Psi (t)\rangle\to {\cal N}^{-1/2} \hat P_n|\Psi (t)\rangle$,
where $\hat P_n=\sum_\alpha |n,\alpha \rangle \langle n,\alpha|$ is
a projection operator and ${\cal N}$ is a normalization factor. One
finds from Eqs.~(\ref{a2}) and (\ref{a3}) that
\begin{eqnarray}
&&\hat P_n|\Psi (t)\rangle = \sum_{l,\ldots\atop r,\ldots }\left
[b_{1{\underbrace{l\ldots}_n}{\underbrace{r\ldots}_n}}(t)
c_1^\dagger a_{\underbrace{r\ldots}_n}^\dagger
a_{\underbrace{l\ldots}_n}+b_{2{\underbrace{l\ldots}_n}{\underbrace{r\ldots}_n}}(t)
c_2^\dagger a_{\underbrace{r\ldots}_n}^\dagger
a_{\underbrace{l\ldots}_n} \right ]|0\rangle \ \ \ \label{a4}
\end{eqnarray}
where the corresponding normalization factor ${\cal
N}=\sum_{l,\ldots ,r,\ldots }[|b_{1l\ldots r}(t)|^2 +|b_{2l\ldots
r}(t)|^2]$ is a probability of finding $n$ electrons in the
collector by time $t$. Eq.~(\ref{a4}) shows that the measurement
leaves the qubit in a linear superposition of its two states.
Therefore one cannot determine the state of the qubit by measuring
the number of electrons in the right reservoirs.
Consider now the measurement of electric current in the right
reservoir. The latter is given by a commutator of $\hat N$ with the
total Hamiltonian of the system, $\hat I=i[H,\hat N]$ (we choose
$e=1$). Using Eqs.~(\ref{a1a})-(\ref{a1c}) we obtain
\begin{equation}
\hat I=i\sum_{l,r}(\Omega_{lr}+\delta\Omega_{lr}c_2^\dagger c_2)
(a_l^\dagger a_r-a_r^\dagger a_l)\equiv \sum_{l,r}\hat I_{lr}
\label{a5}
\end{equation}
The eigenstates of the energy resolved-current operator, $\hat
I_{lr}|I^\pm_{lr}(q)\rangle =I^\pm_{lr}(q)|I^\pm_{lr}(q)\rangle$ are
\begin{equation}
|I^\pm_{lr}(q)\rangle ={1\over\sqrt{2}}(a^\dagger_ra_l\pm
i)c_q^\dagger |0\rangle\, , \label{a6}
\end{equation}
where $q=1,2$ denotes the qubit state, $c^\dagger_q |0\rangle$ and
$I^\pm_{lr}(q)= \pm [\Omega_{lr}+(q-1)\delta\Omega_{lr}]$.
Respectively, the eigenstates of the total current, $|I\rangle$, are
given by a product of $|I^\pm_{lr}\rangle$.
It follows from Eqs.~(\ref{a5}), (\ref{a6}) that by measuring the
energy-resolved current $\hat I_{lr}$ (or the total current, $\hat
I$), one projects the wave function (\ref{a2}) on a certain state of
the qubit, corresponding to an observed value of the current. This
implies that the qubit position can be determined with any accuracy,
in principle, by monitoring directly the detector current (via its
magnetic field\cite{lev}).
If, however, such a direct measurement of the detector current
cannot be performed, one can determine it indirectly, via a
variation of the collector charge. Let us assume that we recorded
$n_0$ electrons in the collector at time $t$. As a result, the
entire system is projected to the state $|n_0\rangle$: $|\psi
(t)\rangle\to \hat P_{n_0}|\psi (t)\rangle$, Eq.~(\ref{a4}), which
is an eigenstate of the operator $\hat N$, Eq.~(\ref{a3}). (We
omitted the index of degeneracy $\alpha$). Next we detect the
accumulated charge $n$ at the time $t+\Delta t$. The final state of
the detector (up to the normalization factor) is
\begin{equation}
\hat P_ne^{-iH\Delta t}\hat P_{n_0}|\Psi (t)\rangle
=|n\rangle\langle\varphi_n(\Delta t)|n_0\rangle\langle n_0|\Psi
(t)\rangle\, , \label{a7}
\end{equation}
where $|\varphi_n(\Delta t)\rangle =\exp (iH\Delta t)|n\rangle$ is
an eigenstate of the operator $\hat N(\Delta t)=\exp (iH\Delta
t)\hat N\exp (-iH\Delta t)$, corresponding to an eigenvalue $n$.
This operator can be expanded in powers of $\Delta t$
\begin{equation}
\hat N(\Delta t)=\hat N +\hat I\Delta t +i[H,\hat I]{(\Delta
t)^2\over 2}+\cdots \, , \label{a8}
\end{equation}
where the current $\hat I$ is given by Eq.~(\ref{a5}). Thus the
time-dependent operator $\hat N(\Delta t)$ includes the qubit
position operator, $c_q^\dagger c_q$, in contrast with $\hat N\equiv
N(0)$. As a result the projection on the eigehstates of $\hat
N(\Delta t)$, Eq.~(\ref{a7}), could determine the qubit position.
Let us take small $\Delta t$ (``measurement time'') in
Eq.~(\ref{a8}) such that $\hat N(\Delta t)\simeq \hat N +\hat
I\Delta t$. If $\hat N$ and $\hat I$ commute, then the eigenstates
of $\hat N(\Delta t)$ would be a product of eigenstates of these
operators: $|\varphi_n(\Delta t)\rangle=|n',I(q)\rangle$, where
$q=\{ 1,2\}$ denotes the qubit state and $n=n'+I(q)\Delta t$. It
follows from Eq.~(\ref{a7}) that $n'=n_0$ and $I(q)\equiv I_{\Delta
n}(q)=\Delta n/\Delta t$, where $\Delta n=n-n_0$. As a result the
qubit is projected in the state $q$ corresponding to the variation
of the collector charge, $\Delta n$.
In fact, the operators $\hat N$ and $\hat I$ do not commute. In this
case it is only the average current, $\overline{I}_{\Delta
n}=\overline{\Delta n}/\Delta t$ which can be determined from
ensemble measurements of $\Delta n$, where its dispersion, $\delta
I_{\Delta n}=[\overline{(\Delta n)^2}-(\overline{\Delta
n})^2]^{1/2}/\Delta t$, diverges as $(\Delta t)^{-1/2}$ for $\Delta
t\to 0$. The latter restricts the accuracy of the qubit
measurements, respectively.
The measurement accuracy, however, increases by increasing the
measurement time, since $\delta I_{\Delta n}(q)\to 0$ for $\Delta
t\to \infty$\cite{fn1}. Yet, this can be done only if the qubit is
not ``moving'', (i.e. the hopping amplitude $\Omega =0$, Fig.~1).
Then $[H,\hat I]=0$, so that the higher order terms in the expansion
(\ref{a8}) vanish. If it is not the case ($\Omega\not =0$), the
current it driven by the qubit, so that $[H,\hat I]$ and the higher
order commutators in Eq.~(\ref{a8}) are not zero. The average
contribution from these terms to $I_{\Delta n}$, which we denote as
$\delta_1 I_{\Delta n}(\Delta t)$, increase with $\Delta t$. This
suggests that the quantum limit of the qubit measurement is
determined by the optimal measurement time ($\Delta t$) which
minimizes the total error,
\begin{equation}
[\delta_2 I_{\Delta n}(\Delta t)]^2 =[\delta I_{\Delta n}(\Delta
t)]^2+[\delta_1 I_{\Delta n}(\Delta t)]^2\, . \label{aa8}
\end{equation}
In order to perform this procedure we introduce the density matrix
$\sigma_{qq'}^{(n)}(t)=\langle n,q'|\Psi (t)\rangle\langle\Psi
(t)|n,q\rangle$, where the wave function $|\Psi (t)\rangle$ is given
by Eq.~(\ref{a2}). It was demonstrated in\cite{gur,gur1} that for
large large bias voltage, $V=\mu_L-\mu_R$ (Fig.~1), the
time-dependent Schr\"odinger equation, $\partial_t|\Psi (t)\rangle
=H|\Psi (t)\rangle$, can be reduced to the following Bloch-type rate
equations for the density matrix $\sigma_{qq'}^{(n)}(t)$ by assuming
weak energy dependence of the transition amplitudes
($\Omega_{lr}=\bar\Omega,~ \Omega'_{lr}=\bar\Omega'$)\cite{gur},
\begin{eqnarray}\label{a10a}
\dot\sigma_{11}^{(n)} &=&
-D_1\sigma_{11}^{(n)}+D_1\sigma_{11}^{(n-1)} +i\Omega
(\sigma_{12}^{(n)}-\sigma_{21}^{(n)})\\ \label{a10b}
\dot\sigma_{22}^{(n)}
&=&-D_2\sigma_{22}^{(n)}+D_2\sigma_{22}^{(n-1)} -i\Omega
(\sigma_{12}^{(n)}-\sigma_{21}^{(n)})
\\ \label{a10c}
\dot\sigma_{12}^{(n)} &=& i(E_2-E_1)\sigma_{12}^{(n)}+
i\Omega(\sigma_{11}^{(n)}-\sigma_{22}^{(n)})
-\frac{D_1+D_2}{2}\sigma_{12}^{(n)}
+(D_1D_2)^{1/2}\sigma_{12}^{(n-1)}
\end{eqnarray}
where $D_{1,2}=T_{1,2}V$ and $T_{1,2}$ is the transmission
probability of the barrier: $T_1=(2\pi)^2\bar\Omega^2\rho_L\rho_R$
and $T_2=(2\pi)^2(\bar\Omega')^2\rho_L\rho_R$, where
$\rho_{L,R}$ is the density of states in the left (right)
reservoir, Fig.~1.
Solving Eqs.~(\ref{a10a})-(\ref{a10c}) one can find all quantities
needed for evaluation of $\delta I_{\Delta n}$ and
$\delta_1I_{\Delta n}$. For instance, in order to evaluate the
average value of $\Delta n$ and its dispersion we have to solve
these equations with the initial condition $n=n_0$. It follows from
Eqs.~(\ref{a10a})-(\ref{a10c}) that the density matrix
$\sigma_{qq'}^{(n)}(t)$ depends only on $\Delta n=n-n_0$. Thus we
can take $n_0=0$ and $\Delta n =n$. Then the average values
$\overline{\Delta n}=\overline{n}$ and $\overline{(\Delta
n)^2}=\overline{n^2}$ are given by
\begin{eqnarray}
\overline{n}(t)&=&\sum_nnP_n(t)=\overline{n}_{11}(t)+\overline{n}_{22}(t)\,
, \label{a11a}\\
\overline{n^2}(t)&=&\sum_nn^2P_n(t)=\overline{n_{11}^2}(t)+\overline{n_{22}^2}(t)\,
, \label{a11b}
\end{eqnarray}
where $P_n(t)=\sigma_{11}^{(n)}(t)+\sigma_{22}^{(n)}(t)$ is the
probability of finding $n$ electron in the collector. Multiply
Eqs.~(\ref{a10a})-(\ref{a10c}) by $n$ and sum over $n$ one finds
\begin{eqnarray}
\dot{\overline{n}}_{11} &=& D_1\sigma_{11}
+i\Omega (\overline{n}_{12}-\overline{n}_{21})\\
\label{a12a} \dot{\overline{n}}_{22} &=& D_2\sigma_{22}
-i\Omega (\overline{n}_{12}-\overline{n}_{21})\\
\label{a12b} \dot{\overline{n}}_{12} &=&
i(E_2-E_1)\overline{n}_{12}+ i\Omega(\overline{n}_{11}-
\overline{n}_{22}) -\frac{\Gamma_d}{2}
\overline{n}_{12}+(D_1D_2)^{1/2}\sigma_{12}\, , \label{a12c}
\end{eqnarray}
where $\sigma_{qq'}(t)=\sum_n\sigma_{qq'}^{(n)}(t)$ is the qubit
density matrix, and $\Gamma_d=(\sqrt{D_1}-\sqrt{D_2})^2$ is the
decoherence rate\cite{gur}.
Similarly multiplying Eqs.~(\ref{a10a})-(\ref{a10c}) by $n^2$ and
sum over $n$ one obtains
\begin{eqnarray}\label{a13a}
\dot{\overline{n_{11}^2}} &=&2 D_1\overline{n}_{11}+
D_1\sigma_{11}
+i\Omega (\overline{n_{12}^2}-\overline{n_{21}^2})\\ \label{a13b}
\dot{\overline{n_{22}^2}} &=&2 D_2\overline{n}_{22}+
D_2\sigma_{22}
-i\Omega (\overline{n_{12}^2}-\overline{n_{21}^2})\\ \label{a13c}
\dot{\overline{n_{12}^2}}&=&
i(E_2-E_1)\overline{n_{12}^2}+ i\Omega(\overline{n_{11}^2}-
\overline{n_{22}^2}) -\frac{\Gamma_d}{2}
\overline{n_{12}^2}+(D_1D_2)^{1/2}(2\overline{n}_{12}+\sigma_{12})\,
.
\end{eqnarray}
The qubit density matrix $\sigma_{qq'}(t)$ can be easily found from
Eqs.~(\ref{a10a})-(\ref{a10c}) by tracing it over $n$,
\begin{eqnarray}\label{aa10a}
\dot\sigma_{11} &=&i\Omega (\sigma_{12}-\sigma_{21})\\ \label{aa10b}
\dot\sigma_{12} &=& i\epsilon_{21}\sigma_{12}+
i\Omega(2\sigma_{11}-1) -\frac{\Gamma_d}{2}\sigma_{12}\, ,
\end{eqnarray}
and $\sigma_{22}(t)=1-\sigma_{11}(t)$. This quantity,
$\sigma_{qq'}(t)$, determines the detector average current,
$\overline{I}(t)=\langle\Psi (t)|\hat I|\Psi (t)\rangle$. Indeed, it
follows from Eqs.~(\ref{a2}), (\ref{a5}) that
\begin{equation}
\overline{I}(t)=D_1\dot{\overline{n}}_{11}(t)+D_2\dot{
\overline{n}}_{22}(t)=D_2+\Delta D\, \sigma_{11}(t)\, , \label{a14}
\end{equation}
where $\Delta D=D_1-D_2$ is an average ``signal''. Obviously,
$\overline{I}(\Delta t)=\overline{n}/\Delta t\equiv
\overline{I}_n(\Delta t)$ for small $\Delta t$. The dispersion of
$I_n(\Delta t )$ can be found from Eqs.~(\ref{a11b}),
(\ref{a13a})-(\ref{a13c}),
\begin{equation}
(\delta I_n)^2=\overline{n^2}(\Delta t)/(\Delta
t)^2-\overline{I}_n^2\simeq I_n(0)/\Delta t\, . \label{a15}
\end{equation}
As expected, $\delta I_n(\Delta t)$ diverges as $1/\sqrt{\Delta t}$
for $\Delta t\to 0$.
Let us evaluate the contribution from higher order terms in the
expansion (\ref{a8}), which generate variation of the average
current, $\delta_1 I_n$. (In our case this variation is produced by
the qubit only). Therefore, one can write $\delta_1I_n(\Delta
t)=|\overline{I}_n(\Delta t)-\overline{I}_n(0)|$. Using
Eq.~(\ref{a14}) we evaluate this quantity as
\begin{equation}
\delta_1 I_n =\Delta D\,|\sigma_{11}(\Delta t)-\sigma_{11}(0)|\, ,
\label{a16}
\end{equation}
where $\sigma_{11}(t)$ is obtained from
Eqs.~(\ref{aa10a})-(\ref{aa10b}). For instance, for aligned levels,
$E_1=E_2$, and $\sigma_{11}(0)=1$,
\begin{equation}
\delta_1 I_n={\Delta D\over2}\left |e^{-{\Gamma_d\over 4}\Delta
t}\left[\cos (\omega \Delta t)+\eta\sin (\omega \Delta
t)\right]-1\right |\, ,
\label{a17}
\end{equation}
where $\eta=\Gamma_d/4\omega$ and $\omega
=2\Omega\sqrt{1-(\Gamma_d/8\Omega )^2}$ is the Rabi frequency. As
expected, $\delta_1 I_n\to 0$ for $\Omega\to 0$.
Eqs.~(\ref{a15}) and (\ref{a17}) allow us to evaluate the optimal
$\Delta t$ by minimizing $[\delta_2 I_n(\Delta t)]^2$,
Eq.~(\ref{aa8}). We take for simplicity $\Delta D\ll D$, where
$D=(D_1+D_2)/2$. As a result $\Gamma_d\simeq (\Delta D)^2/4D$. We
first consider weak distortion of the qubit, $\Gamma_d/8\ll\Omega$.
Then expanding Eq.~(\ref{a17}) in powers of $\Delta t$ we easily
obtain for the optimal measurement time and for the corresponding
measurement limit
\begin{equation}
\Delta t= {1\over 2\Omega}\, \left({2\Omega\over\Gamma_d}\right
)^{1/5},~~~(\delta_2 I_n)^2={5D\Omega \over 2}\left({\Gamma_d\over
2\Omega}\right )^{1/5}\, .
\label{a18}
\end{equation}
In order to observe Rabi oscillations of the qubit in a single
measurement run one needs $\delta_2 I_n\ll \Delta D$, at least. It
follows from Eq.~(\ref{a18}) that this condition corresponds to
$\Omega\ll 2\Gamma_d$. This, however cannot be combined with the
condition of weak qubit distortion, used in Eq.~(\ref{a18}). Hence,
one cannot observe Rabi oscillations in a single run, but only in an
ensemble average of different runs.
Consider now large decoherence limit, $\Gamma_d/8\gg\Omega$. Then
the qubit is strongly affected by the detector. As a result, the
electron stays in the same dot for a long time,
$t\sim\Gamma_d/8\Omega^2$ (``quantum Zeno'' effect). Indeed, one
finds from Eqs.~(\ref{aa10a})-(\ref{aa10b}) that $\sigma_{11}=
[1+\exp(-8\Omega^2t/\Gamma_d)]/2$ for $t\gg 1/\Gamma_d$.
Respectively, the optimal measurement time and the measurement limit
are given by
\begin{equation}
\Delta t= {1\over 4\Omega }\, \left ({\Gamma_d\over 2\Omega}\right
)^{1/3},~~~(\delta_2 I_n)^2=6D\Omega \left ({2\Omega
\over\Gamma_d}\right )^{1/3}\, .
\label{a19}
\end{equation}
In contrast with the previous case, Eq.~(\ref{a18}), the measurement
time $\Delta t$ increases with $\Gamma_d$. This is not surprising
since large decoherence generated by the detector, localizes the
qubit for a long time. Therefore it behaves as a static one so that
the measurement time increases.
It follows from our arguments that the quantum precision limit
depends on a particular set of the detector observables which the
total wave function is projected on. In this Letter we discussed two
alternative sets related to charge and current states of the
point-contact detector. It was demonstrated that single projection
on charge states cannot measure the qubit state. However, two
consecutive projections of the entire system on the charge states
can measure the qubit state, but only with a limited accuracy. On
the other hand, direct projection on the current state of the
detector can determine the qubit state with absolute precision, in
principle. If this could be realized, one would arrive to the Zeno
paradox\cite{zeno}, i.e. to complete freezing of a system as a
result of continuous measurement.
This shows a necessity of including a ``pointer'' in the total
Hamiltonian coupled with current states of the detector (von Neumann
hierarchy\cite{neu}). In this case two consecutive projections of
the total wave function on the pointer states, used for a
determination of the detector current, would restrict the
measurement accuracy in a total analogy with the previous case. If
the pointer is coupled with charge states of the detector, it
obviously cannot decrease the quantum measurement limit found in our
calculations. We assume also that the pointer cannot essentially
increase this limit, since otherwise the von Neumann hierarchy of
measurements would not converge. This problem, however needs a
special attention.
Our final results on quantum limit of measurement involved only
average quantities, which were obtained without any explicit resort
to the projection postulate. Indeed, the latter is related only to a
single measurement. Nevertheless, as we demonstrated in this Letter,
the use of the projection postulate was very useful in a
determination of quantum limit of measurement. In particular, it
clearly displayed the detector variables which would allow us to
measure a microscopic system with maximal accuracy. Such variables
are usually represent commutators (time derivatives) of operators
describing the detector states.
An appropriate choice of this variable depends of a particular
measurement apparatus. For instance, for a single electron
transistor (SET) detector, one needs to use the second commutator
(``acceleration'') of the accumulated charge. In contrast with the
point-contact detector, the projection to current states of the SET
would not determine the qubit state. The measurement of the charge
``acceleration'' can be very useful if the corresponding operator
would commute with the charge operator. In this case one can design
an appropriate procedure of projecting on the charge states at
different times which would diminish the quantum measurement limit.
This however must be a subject of a separate investigation.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,270 |
243.6: Arena Football League Founder Jim Foster - Part Two [Archive Re-Release]
[More holiday fun with a re-release of a fan favorite episode from January 2018!]
We conclude our two-part journey into the early history of the Arena Football League with founder and inventor Jim Foster, who recounts some of the most notable events of the league's formative years – including a memorable 1987...
243.5: Arena Football League Founder Jim Foster [Archive Re-Release]
[We celebrate the holidays with a re-release of a fan favorite episode from January 2018!]
As the new year beckons, the fate of the Arena Football League – one of America's most innovative modern-day professional sports concepts – hangs in the balance. With only four teams (the mutually-owned Washington Valor and...
243: The 3rd Annual Year-End Holiday Roundtable Spectacular!
We try to make sense of a decidedly bipolar 2021 with our third-annual Holiday Roundtable Spectacular - featuring three of our favorite fellow defunct sports enthusiasts Paul Reeths (OurSportsCentral.com, StatsCrew.com & Episode 46); Andy Crossley (Fun While It Lasted & Episode 2); and Steve Holroyd (Episodes 92, 109,...
242: Pittsburgh's Civic Arena ("The Igloo") - With Dave Finoli
Our "tour" of lost pro sports venues continues with another stop in the Keystone State, this time for a loving look back at the life and times of Pittsburgh's legendary Civic Arena - aka "The Igloo" - with Steel City native Dave Finoli (editor, "Pittsburgh's Civic Arena: Stories from the Igloo").
Originally... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 76 |
Q: Can I call solr through thrift API? Solution:
solr version: 5.40
thrift : 0.9.3
How can I call solr request through thrift api?
(1)is exist api?
(2)or implement service above solr ?
A: I don't know much about thrift. Looks like it could directly use the solrj-Examples. Otherwise you could use solr as a restfull Service via http (nur REST Service, because only POST and GET are used and you could use GET to change data).
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,044 |
Anyone who likes vintage jewelry knows the allure of Bakelite. There's just something special about it, the colors, designs, feel, and that sound. That wonderful clunky sound.
Bakelite or it's cousin Catalin was used for lots more than jewelry though. It was originally created for use in the industrial world. Think electrical insulators, distributor caps on cars, radio parts, telephones, etc.
Which brings us to that eternal vintage jewelry lover's question: how do you tell if it's Bakelite, fakelite, or something else all together? There are a few simple tests that can answer your question.
One quick thing to look for, Bakelite will never have mold or form markings on it. If you feel a line running around the center of a bangle, it is not bakelite.
The easiest and safest way to test Bakelite is the Hot Water Test. As simple as it sounds, just place your item in hot water (not boiling) for about 15 seconds. Now smell it. If it's true Bakelite it should have a somewhat sweet old chemical smell to it similar to formaldehyde. Many types of plastic and resin have distinct smells. So this one is going to be a bit of a learning curve. Once you know that distinct smell however, you will never forget it.
Another way to test Bakelite is with Simichrome Polish. Simichrome polish is a pink polish made for cleaning metals. It is a bit pricey, but will last for a very long time. It can be purchased online and it works well as a way to test most Bakelite. But, there are some Bakelite items that will not test positive with Simichrome. These include newly polished or made items that do not have an aged patina, items that have been overdyed and some black or red Bakelite items.
To test for authentic Bakelite with Simichrome polish, take a cotton swab or clean rag and dab a bit of polish on it. Rub you piece for a few seconds and take a look at your swab or rag. Is the area yellow? If so, it's Bakelite. If not, it's something else.
I have also read about and heard that Original 409 works similarly to Simichrome polish. As I have not personally tried it, it's not something I can vouch for however.
There are online sellers who sell Bakelite testing pads that you can purchase and carry with you to test items before you purchase. You can do a Google search for "Bakelite testing pads" to find out where to buy.
Another note regarding Bakelite. There are a number of wonderfully talented artists using old bakelite stock to create new items. Some involve intricate carvings, overdyes, and laminates or inserts that were never seen in the past. These true artisans create both new and old designs of such high quality, they are well worth the price and will retain their value over time.
A quick word about "fakelite": There are many unscrupulous sellers out there passing off newly manufactured items that resemble Bakelite but are not. Buyer beware. If it seems too good to be true, it probably is. However, there are many fabulous new items made to look like vintage bakelite. If a seller is honest about what they are selling, I find no reason not to enjoy these new products.
Just because your item isn't Bakelite doesn't mean it isn't vintage or a worthwhile purchase.There are many forms of vintage plastics and resins that were used to create fabulous pieces of jewelry, purses, household items, and more. Part 2 of this post covers those other vintage plastics. | {
"redpajama_set_name": "RedPajamaC4"
} | 513 |
{"url":"https:\/\/www.physicsforums.com\/threads\/problem-with-dimensionality.31863\/","text":"# Problem with dimensionality\n\n1. Jun 20, 2004\n\n### alexepascual\n\nIn the following equation:\n$$\\left\\langle {\\phi } \\mathrel{\\left | {\\vphantom {\\phi \\Psi }} \\right. \\kern-\\nulldelimiterspace} {\\Psi } \\right\\rangle = \\int {\\phi ^ * } \\left( x \\right)\\psi \\left( x \\right)dx$$\nmy understanding would have been that the bracket represents a probability amplitude. But the dx on the integral gives it a dimension of length. OK, the square of the absolute value of the braket is not a probability but a probability density. Wouldn't then the units be probability over unit of length?\nI am a little confused. Any help will be appreciated.\nBy the way, this equation is from Feynman Lectures, Vol III, 16-6\n\nLast edited: Jun 20, 2004\n2. Jun 21, 2004\n\n### turin\n\nDon't forget about normalization. To interpret it as a probability, you are already assuming that it is normalized. In other words, implicit in the interpretation is that an integral over all of space has been done and divided by.\n\n3. Jun 21, 2004\n\n### alexepascual\n\nIf both state vectors are normalized, so would the bracket, as well as the integral on the right. Wouldn't it?. Well, I mean they would reflect the correct value which would have an absolute value typically less than one. On the other hand, doesn't normalization only affect the absolute value of the vector and not any units?. As far as I know, the state vector as well as a bracket are dimensionless. Now, it appears that the integral would give you a dimensionless quantity only if you drop the units of length from dx. If you do that, everything would be fine and the right side of the equation would be dimensionally consistent with the left.\nNow, in a different situation of integration, such as W= integral F*dx , or W= integral P*dv, you do consider the dimension of the diferential. As a matter of fact, in these cases you have to consider it in order for the equation to be dimensionally consistent. So, I would assume that in physics in general, when you do an integration, if you are integrating over some variable that has units, you keep the units and use them in your integration.\nIn the quantum mechanics case I posted, the only apparent way to keep it dimensionally correct would be to add a factor of (1\/unit of length) in front of the integral. I wonder if there is some convention where this factor is ommited even it would be needed to make the equation dimensionally consistent.\n\n4. Jun 22, 2004\n\n### tavi_boada\n\nYour mistake is supposing that the wave functions dont have dimensions. They do. The dimensionality is correct. Take a square well problem. The wave functions (sin(nx) and cos(nx)) must have normalization inversely proportional to the square root of a lenght constant. This way the product of two wave functions has dimensionality of L to the minus one, keeping the dimensionality of the scalar product correct. If you think about it you will see that normalization constants have to have dimensionality. In momentum representation it's 1 divided by the square root of momentum units and in space representation one divided by the square root of space units.keep it up.\n\nciao!\n\nLast edited: Jun 22, 2004\n5. Jun 22, 2004\n\n### somy\n\ndear tavi boada:\ncan you explain more.I didn't understand your explanation.\nthanks in advanced.\n\n6. Jun 22, 2004\n\n### tavi_boada\n\nhey somy\n\nWHen constructing the solutions to the infinite square well, we get from the time independent Schr\u00f6dinger equation + boundary value considerations that the wave functions are sin(n*pie*x\/L), where L is the size of the well. THe probability of finding the particle inside the well is 100%, the integral of the wave function squared must be equal to one. THus the earlier given solution is not complete, we must modify it with a factor wich is the inverse square root of the value of the above integral. THus, the scalar product of the two wave funcions has the dimensions of nothing, as expected, because both wave functions are multiplied by constants with dimensionality of length to the minus one half.\n\n7. Jun 22, 2004\n\n### alexepascual\n\nThanks Tavi,\nYour explanation is very clear and it solves my problem.\nNow I can continue reading chapters 16 and 20 from Feynman's Lectures.\n\n8. Jun 24, 2004\n\n### alexepascual\n\nTavi,\nI haven't had time to do much reading, but I was thinking about the following. The expression <x|psi> is equivalent to psi(x). Now, psi(x) has units, according to your post. This would make me think that <x|psi> also has units. But in that case where do the units come from?. I was not aware of bras or kets having units. I guess the ket psi by itself could not come with any units as it is a state vector that can be represented in different bases. Could the units be in the <x| bra? I tought this bra just had a label, which is the value of x, and a bare complex number without units.\nMaybe you can clarify this for me. I'll appreciate it.\n\n9. Jun 25, 2004\n\n### selfAdjoint\n\nStaff Emeritus\nWhy do you say $$\\psi(x)$$ has units?\nHere's what QM says about $$\\psi$$:\n\n- It's a square summable function forming a point in a complex Hilbert space\n- It satisfies the Schroedinger equation\n- $$\\psi\\bar{\\psi}$$ maps into the configuration space of an observable and gives the probability for the observable to be in the state indicated by the point on the configuration space.\n\nNone of this allows the wave function to assume the attributes of a real measurement in spacetime. It requires the action of a self-adjoint operator to create a number that can do that.\n\n10. Jun 25, 2004\n\n### alexepascual\n\nSelf Adjoint:\nThanks for your response. If you look to the previous posts you'll see what my concern is. But in short: when integrating the wave function over all space, if properly normalized, we are supposed to get just the number one. But the integral has the differential element dx which has units of volume, ( length in one-dimensional case). So Tavi suggested that the wave function has units of lenght to the minus 1\/2 (one dimensional case again) . I can accept that. But I had some doubts as to how this concept translates when using Dirac notation. I never suggested you could get a measurable out of a wave function without using an operator.\n\n11. Jun 25, 2004\n\n### tavi_boada\n\nSuppose [psi> represents the state an electron in a given potential. We know that <x[psi> times it's complex conjugate represents the probability density of the electron being at x, not the probability of finding it there. The probability of finding it there is\nP(x)dx=<x[psi>\u00b7<psi[x>dx=<x[psi>^2dx.\nTherefore, psi(x) must have units because a probability, by definition can't have any.\nYou know that a state can be expressed in terms of the eigenvectors of a hermitian operator. The position operator,x, has eigenvectors [x>, so psi(x) is the coefficient of the expansion of [psi> in terms of the proper base of x. THat's why <x[psi> squared is the probability density of measuring the position of the electron and finding it at x.\n\n12. Jun 25, 2004\n\n### alexepascual\n\nTavi,\nI am not questioning that $$\\psi(x)$$ has units. I accepted that.\n(see my post before SelfAdjoints's)\nMy concern is, in the bracket <x|psi> , where are the units hiding?\nSo, see, my problem comes when expressing things in dirac notation, not when expressed as a wave function.\nPerhaps I am looking at this the wrong way. I usually think of the bracket as an inner product. Maybe in this case I should look at it as just the components in x-space of |psi>, together with its units.\n\nLast edited: Jun 25, 2004\n13. Jun 25, 2004\n\n### Eye_in_the_Sky\n\nSolution to Problem with dimensionality\n\nAny ket |f> belonging to the Hilbert space (regardless of whether or not |f> is normalized) is dimensionless. A \"generalized\" ket, such as |x>, however, is not a member the Hilbert space ... and it has dimension. If x has dimension L, then |x> has dimension L^(-1\/2), and likewise for the corresponding \"generalized\" bra.\n\n--------------------------\n\nYou can convince yourself of this by writing\n\n<x|x'> = delta(x-x')\n\nand observing that\n\nIntegral { delta(x-x') dx }\n\nis dimensionless, implying that delta(x-x') has dimension 1\/L.\n\n--------------------------\n\nIn this way, a function f(x) = <x|f> gets the dimension it needs.\n\nSimilarly, <x|p>, where |p> is a \"generalized\" eigenket of the momentum operator, gets the dimension of reciprocal square-root of action.\n\n14. Jun 25, 2004\n\n### alexepascual\n\nThanks a lot Eye_in_the_Sky. Your explanation is easy to understand and straight-to-the-point. I think it completely answers my question.\nThanks again,\nAlex\n\n15. Jun 27, 2004\n\n### turin\n\nI guess I'm not appreciated at all around here.\n\n16. Jun 28, 2004\n\n### alexepascual\n\nTurin,\nWhy do you say that?\nYou posted the first response to this post and I answered the same day. Your post was apparently correct but it did not totally answer my question. When Tavi replied, that solved part of my puzzle, but some questions remained, which were addressed by Eye_in_the_Sky. The response by SelfAdjoint was also correct but was not answering my question.\nSome times it is hard to understand what a person asking for help on a topic is really looking for. Part of this may be because the original poster has not communicated the problem in enough detail. Part might be that the person responding is looking at the subject from a different angle, which is satisfactory for him but not for the original poster.\nYou should not take it personally. I appreciate all responses to my posts and I remember you very well from an interesting discussion we had in another thread.\nI view this forum as a collaborative effort where we are all learning from each other. In my physics classes, where instructors graded on a curve, I found that there was more competition than collaboration. Here I have enjoyed a different atmosphere, where collaboration prevails over competition.\nI also think that due to the complexity of the subject and its abstract character, it is not always easy to communicate effectively, partly also due to the fact that different people have different angles of approach which others may not find satisfactory.\nI, personally like to think in terms of pictures, and I like to always connect the abstract features of the theory to concrete examples. There are people who are perfectly comfortable exploring all the mathematical apparatus without making contact with \"reality\". That is their preference and I respect it.\nAs an added benefit of this forum, we get exposed to different views which we may not find useful today but may gradually start having more appeal in the future. All the disagreement and difficulties in communication may not be that counterproductive after all.\nWell, Turin, you are always welcome and certainly appreciated.\n-Alex Pascual-\n\n17. Jun 28, 2004\n\n### tavi_boada\n\nAlex, this has nothing to do with the thread, but I find it remarcable you can think pictorialy about QM. I've tried that aproach because it's always been easier for me that way in other subjects, but failed completely. I cannot attack QM prloblems without use of mathematical weapons, I envy you. In fact, come to think of it, I guess I wont be able to adress any problem in modern physics pictorialy. I guess I'm a bit depressed 'cause I can't for the life of me grasp the spirit of Misner\/Thorne\/Wheelers gravitation. How can they say 2-forms are like honycombs!!???Any GR expert fell up to explain?\n\n18. Jun 28, 2004\n\n### robphy\n\nI posted these links on this other thread regarding visualizing field lines\nhttp:\/\/www.ee.byu.edu\/ee\/forms\/\nhttp:\/\/www.lgep.supelec.fr\/mse\/perso\/ab\/IEEEJapan2.pdf [Broken] [see p. 10]\n\nIn addition, you may find this useful\nhttp:\/\/clifford-algebras.org\/Beijing2000\/page127-154.pdf [Broken]\nhttp:\/\/physics.syr.edu\/courses\/vrml\/electromagnetism\/references.html [Broken]\nI believe the field of 2-forms visualized as a honeycomb is due to Jan A Schouten.\n\n(This is now off-topic for this thread.)\n\nLast edited by a moderator: May 1, 2017\n19. Jun 28, 2004\n\n### alexepascual\n\nTavi,\nThe fact that I have as a preference visualizing things, does not mean that I am being successful in doing so in quantum mechanics. But I try. Memorizing rules to manipulate symbols on a piece of paper does not give me the feeling that I am really learning about nature. On the other hand, when I have a good explanation of the math in terms of \"things\" in the real world, that helps. With respect to pictures, even seing the state vector as an arrow, and a basis as a set of 3-D coordinate axes gives me a more intuitive description. When you do the math, I like to know what is actually happening to the state. Is anything happening to it or are we simply changin our way to describe it?. I am not really asking here, I am saying that every time that's the kind of question I ask.\nIn this thread, though, my question was purelly mathematical, which means that it did not concern what is happening to the state but how we use the symbols to describe it.\nIn this respect I have my suspicion that the explanation given by you and Eye-in-the-Sky, (which I am sure represents common knowledge in traditional quantum mechanics) may not be the only one, or the only possible way to do things. I think this may be a matter of convention, an I am not knowlegeable enough about the subject to challenge the conventional approach, but I could venture an opinion to see what you guys think. I'll do that my next post.\nWith respect to the Gravitation book, I haven't read it. Probably I will within the next two years, but I think it is kind of heavy reading.\n\n20. Jun 28, 2004\n\n### alexepascual\n\nRobphy,\nI read a little about Geometric Algebra a few weeks ago and it sounded interesting. It wonder how much it is used in quantum mechanics and what are it's advantages. It appears it may be a better way to conceptualize some operations such as products of vectors, but I am not sure about it yet. I have that in my to-do list of things to read. There is so much to learn and so little time!\n\nKnow someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook","date":"2018-01-23 12:52:48","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\": 0.7620497345924377, \"perplexity\": 613.412557931215}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084891926.62\/warc\/CC-MAIN-20180123111826-20180123131826-00704.warc.gz\"}"} | null | null |
\section{Introduction}\label{section_int}
Tomographic imaging techniques visualize the inner structure of probes. Particularly relevant for this work are Optical Coherence Tomography (OCT) and Photoacoustic (PAT). In OCT a sample is placed in an interferometer and is illuminated by light pulses. Then, the backscattered light is measured far from the medium, see for instance \cite{Bre06, DreFuj15, Fer96}.
PAT visualizes the capability of a medium to transform optical (infrared) waves into ultrasound waves to be measured on the surface of the medium \cite{HalSchBurPal04, Wan08, XuWan06}. PAT is called coupled physics imaging technique since it combines two kind of waves \cite{ArrSch12}.
As stand alone imaging techniques PAT and OCT are not capable of recovering all diagnostically relevant physical parameters, but only some combinations of them, see \cite{BalRen11a} for PAT and \cite{ElbMinSch15} for OCT.
Recently setups which combine different imaging modalities, have been investigated mathematically with the objective to reconstruct more diagnostically relevant physical parameters from the measurements.
Particular applications are coupled physics imaging systems
and elastography \cite{Bal12, Kuc12, WidSch12}, to name but a few.
We refer to these techniques as hybrid imaging or multi-modal imaging systems. Note that in the mathematical literature
the name hybrid imaging is also used for coupled physics imaging.
In this work we consider the multi-modal PAT/OCT system, developed for imaging biological tissues, see \cite{DreLiuKumKam14, LiuCheZab16, LiuSchmSanZab14, LiuSchmSanZab13, ZhaPovLauAleHof11}.
We show that with such a system, in contrast to the single modality setups, we obtain sufficient measurements which allow us to extract quantitative information on the electric susceptibility and the Grüneisen parameter of the sample.
In the multi-modal PAT/OCT system, two different excitation laser systems, both operating in the same wavelength range, are used.
The PAT and OCT scans are performed sequentially and vary a lot in acquisition times (around 5 minutes in PAT and less than 30 seconds in
OCT). The obtained PAT and OCT images are co-registered afterwards.
In \autoref{section_com}, we describe mathematically the multi-modal PAT/OCT setup. We use the model, from \cite{ElbMinSch17}, based on Maxwell's equations for the electric permittivity.
In \autoref{sec_inv}, we present the equivalence of the inverse problem of recovering both optical parameters with the
solution of a Fredholm integral equation of the first kind for the Gr\"uneisen parameter. Here the kernel of the
integral operator depends on the PAT measurements.
We propose a numerical reconstruction method based on a Galerkin method using a series expansion of the unknown
functions with respect to Hermite functions, see \autoref{section_numerics}. The discretization of the continuous
integral operator results in a system of linear algebraic equations.
We solve the matrix equation using Tikhonov regularization.
Numerical results which justify the feasibility of the proposed method are presented in \autoref{section_results}.
\section{The multi-modal PAT/OCT system}\label{section_com}
We consider the two modalities independently. Full field illumination is used in PAT and focused in OCT.
The medium in OCT is illuminated by a Gaussian light. However, we can assume that the plane wave illumination
is still valid \cite{Fer96}.
\subsection{Light propagation}
We consider macroscopic Maxwell's equations in order to model the interaction of the incoming light with the sample. These equations describe the time evolution of the electric and magnetic fields $E$ and $B$ for given charge density $\rho$ and electric current $J$:
\begin{subequations}\label{eqMaxwellMacro}
\begin{alignat}{2}
\div_{x} D(t,x) &= 4\pi \rho (t,x),\quad &&t\in\mathbbm{R},\;x\in\mathbbm{R}^3,\label{eqMaxwellMacro1} \\
\div_{x} B(t,x) &= 0,\quad &&t\in\mathbbm{R},\;x\in\mathbbm{R}^3,\label{eqMaxwellMacro2a} \\
\curl_{x} E(t,x) &= -\frac1c\partial_t B(t,x),\quad &&t\in\mathbbm{R},\;x\in\mathbbm{R}^3,\label{eqMaxwellMacro3} \\
\curl_{x} H (t,x) &= \frac1c\partial_t D(t,x)+\frac{4\pi}c J (t,x),\quad &&t\in\mathbbm{R},\;x\in\mathbbm{R}^3 ,\label{eqMaxwellMacro4}
\end{alignat}
\end{subequations}
where $D \equiv E + 4\pi P$ is the electric displacement and $H \equiv B - 4\pi M$ denotes the effective magnetic field, related to the electric and magnetic polarization fields $P$ and $M,$ respectively. We specify the material properties of the medium.
\begin{definition}\label{def_medium}
The medium is called non-magnetic if $M=0$, and perfect linear dielectric and isotropic if there exist a scalar function
$\chi\in C^\infty_{\mathrm c}(\mathbbm{R}\times\mathbbm{R}^3;\mathbbm{R})$ the electric susceptibility, with $\chi(t,x)=0$ for all $t<0$, $x\in\mathbbm{R}^3,$ (this property is referenced
as causality), such that
\begin{subequations}\label{eq_dielectric}
\begin{align}
P (t,x) &= \int_\mathbbm{R} \chi (\tau , x) E(t-\tau, x) \d \tau , \label{eq_polar}\\
J(t,x) &= 0 .
\end{align}
\end{subequations}
The electric susceptibility describes the optical properties of the medium and is the parameter to be determined. In addition, we assume that the medium has no free charges, meaning $\rho = 0$ in \eqref{eqMaxwellMacro1}.
\end{definition}
Under the assumptions \eqref{eq_dielectric} of \autoref{def_medium}, combining equations \eqref{eqMaxwellMacro3} and \eqref{eqMaxwellMacro4} we obtain the vector Helmholtz equation for the electric field
\begin{equation}\label{vector1}
\curl_{x}\curl_{x} E(t,x) +\frac1{c^2}\partial_{tt} E(t,x) = -\frac{4\pi}{c^2} \int_\mathbbm{R} \partial_{tt}\chi (\tau , x) E(t-\tau, x) \d \tau .
\end{equation}
Let $\Omega \subset \mathbbm{R}^3$ denote the domain where the object is located, meaning $\supp \chi (t,\cdot) \subset \Omega$ for all $t\in\mathbbm{R}.$
\begin{definition}\label{definition_initial}
We call $E^{(0)}$ an initial field if it satisfies the wave equation
\begin{equation}\label{wave_incident}
\Delta_x E^{(0)} (t,x)-\frac1{c^2}\partial_{tt} E^{(0)}(t,x) = 0,
\end{equation}
and $\div_{x} E^{(0)} = 0,$ and
does not interact with the medium until the time $t=0,$ meaning
\[
\supp E^{(0)} (t,\cdot) \cap \Omega = \emptyset \quad \text{for all} \quad t<0 .
\]
\end{definition}
The initial pulse $E^{(0)}$ is a vacuum solution of Maxwell's equations, meaning it satisfies \eqref{vector1} with $\chi \equiv 0.$ Indeed, using the vector identity $\curl_x\curl_x E= \mbox{grad}_{x}\div_x E-\Delta_x E$ in (\ref{vector1}) we obtain the wave equation (\ref{wave_incident}), since $\div_x E^{(0)} = 0.$
Then, if the medium is given by \autoref{def_medium}, we consider $E$ as the solution of \eqref{vector1} with initial condition
\begin{equation}\label{eq_initial}
E(t,x) = E^{(0)} (t,x) \quad \text{for all} \quad t<0 , \, x\in \mathbbm{R}^3 .
\end{equation}
\subsubsection{Specific illumination}
For both imaging modalities we consider the same incoming field. We use the convention
\[
\hat f (\omega ,x) = \int_\mathbbm{R} f(t,x) \mathrm e^{\i\omega t} \d t,
\]
for the Fourier transform of a integrable function $f$ with respect to time $t.$
The multiple laser pulses centered around different frequencies $\nu$, are described by the initial electric fields
\begin{equation}\label{eqLocalisedPulse1}
E^{(0)}_\nu(t,x)=f_\nu(t+\tfrac{x_3}c)\eta, \quad \nu>0,
\end{equation}
which describe linearly polarized plane waves moving in the direction $-e_3 ,$ for some $f_\nu \in C^\infty_{\mathrm c}(\mathbbm{R})$ and fixed polarization vector $\eta\in\mathbbm{R}^2\times\{0\},$ with $\vert\eta \vert =1.$ These fields satisfy \eqref{wave_incident} for every $\nu$.
We assume that the Fourier transform of $f_\nu$ satisfies
\begin{equation}\label{eqLocalisedPulse}
\supp\hat f_\nu\subset[-\nu-\varepsilon,-\nu+\varepsilon]\cup[\nu-\varepsilon,\nu+\varepsilon] ,
\end{equation}
for some sufficiently small $\varepsilon>0$. We denote by $E_\nu$ the solution of (\ref{vector1}) for the specific initial field $E^{(0)}_\nu .$
The multiple illuminations result to multi-frequency PAT measurements, but they do not provide extra information in OCT, see \cite[Lemma 3.6]{ElbMinSch17}.
\subsection{PAT measurements}
Let the medium be defined as in Definition \ref{def_medium}.
Then, we estimate the averaged change in energy density around a point $x,$ for every $\nu,$ by
\begin{equation}\label{eqAbsorbedEnergy}
\partial_t\mathcal E_\nu (t,x) \approx \left<E_\nu(t,x),\partial_t P_\nu (t,x)\right>.
\end{equation}
In order to derive the above formula we have to consider the interaction of the medium with the incoming electromagnetic wave locally. For a derivation, using microscopic Maxwell's equations, see for instance \cite[Section 4]{ElbMinSch17}.
The laser pulse is absorbed by the medium and part of it is transformed into heat. This generates a pressure wave which is then measured on the object surface. Since the laser pulse is typically very short, the propagation of the acoustic wave during thermal absorption can be neglected. Then, we consider as PAT measurements the initial pressure density $p$ which is proportional to the absorbed energy
\begin{equation}\label{pat_data}
p_\nu(x) = \Gamma (x) \int_\mathbbm{R} \partial_t\mathcal E_\nu (\tau ,x) \d \tau .
\end{equation}
The proportionality factor $\Gamma$ is the Gr\"uneisen parameter, a parameter which, together with the susceptibility $\chi,$ describes the optical properties of our medium.
\subsection{OCT measurements}
In the frequency domain, the equation \eqref{vector1} and the condition \eqref{eq_initial} result to an integral equation of Lippmann-Schwinger type \cite{ColKre98, ElbMinSch15}.
\begin{lemma}
Let the medium be defined as in \autoref{def_medium} and $E_\nu^{(0)}$ as in \autoref{definition_initial}. If $E_\nu$ is a solution of \eqref{vector1} with initial values \eqref{eq_initial}, then its Fourier transform solves the Lippmann-Schwinger integral equation
\begin{equation}\label{Lippmann}
\hat E_\nu (\omega ,x) = \hat E_\nu^{(0)}(\omega,x) + \left(\frac{\omega^2}{c^2}+\grad_x\div_x\right)\int_{\mathbbm{R}^3}\frac{\mathrm e^{\i\frac\omega c|x-y|}}{ |x-y|}\hat\chi (\omega,y)\hat E_\nu (\omega,y)\d y .
\end{equation}
\end{lemma}
Due to the limiting penetration depth of OCT (1 to 2 millimeters), the medium can be considered as weakly scattering, since only single scattering events will be measured. In addition, in OCT the measurements are performed in a distance much larger compared to the size of the medium.
The Born approximation allows us to obtain an explicit form for $\hat E_\nu$ from the Lippmann-Schwinger equation \eqref{Lippmann}. In the limiting case $\hat\chi \rightarrow 0,$ we take the first order approximation of the electric field by replacing $\hat E_\nu$ with $\hat E_\nu^{(0)}$ in the integrand of \eqref{Lippmann}.
We write $x$ in spherical coordinates $x = \rho \vartheta , \, \rho >0, \, \vartheta \in S^2 .$ Under the far-field approximation,
we consider the asymptotic behavior of the expression \eqref{Lippmann} for $\rho\rightarrow \infty,$ uniformly in $\vartheta$
\begin{equation*}
\hat E_\nu (\omega,\rho\vartheta) \simeq\hat E_\nu^{(0)}(\omega,\rho\vartheta)- \mathrm e^{\i\frac\omega c\rho} \frac{\omega^2}{\rho c^2}\int_{\mathbbm{R}^3}\vartheta\times\big(\vartheta\times(\hat\chi(\omega,y)\hat E_\nu (\omega,y))\big)\mathrm e^{-\i\frac\omega c\left<\vartheta,y\right>}\d y.
\end{equation*}
Then, we define
\begin{equation}\label{eqFarField}
\hat E_\nu^{(1)}(\omega,\rho\vartheta) := \hat E_\nu^{(0)}(\omega,\rho\vartheta)- \mathrm e^{\i\frac\omega c\rho} \frac{\omega^2}{\rho c^2}\int_{\mathbbm{R}^3}\vartheta\times\big(\vartheta\times(\hat\chi(\omega,y)\hat E_\nu^{(0)}(\omega,y))\big)\mathrm e^{-\i\frac\omega c\left<\vartheta,y\right>}\d y,
\end{equation}
as the electric field considering both approximations.
The approximated backscattered light $\hat E_\nu^{(1)}- \hat E_\nu^{(0)}$ is combined with a known back-reflected field
and its correlation is measured at each point on the detector surface. Under some assumptions on the incident
illumination we state that what we actually measure in OCT is the backscattered light at a detector placed far from the
medium \cite[Proposition 8]{ElbMinSch15}.
Then, we formulate the direct problem as:
\begin{definition}[direct problem]
Given a medium as in \autoref{def_medium} with susceptibility $\chi$ and Gr\"uneisen parameter $\Gamma,$ and incident illumination $E_\nu^{(0)}$ of the form \eqref{eqLocalisedPulse1}, the direct problem is to find the PAT measurements $p_\nu (x), \, x \in \Omega, \, \nu >0,$ given by \eqref{pat_data}, and the OCT measurements $$(\hat E_\nu^{(1)}-\hat E_\nu^{(0)}) (\omega,\rho\vartheta), \, \omega\in\mathbbm{R}\setminus\{0\}, \, \vartheta\in S^2_+ =\{\vartheta\in S^2\mid\vartheta_3>0\},\, \nu >0,$$ given by \eqref{eqFarField}.
\end{definition}
\section{The Inverse Problem}\label{sec_inv}
In the following the assumptions on the medium (\autoref{def_medium}) hold and especially the causality of $\chi.$
We denote by $\tilde\chi $ the three-dimensional Fourier transform of $\hat\chi$ with respect to space
\[
\tilde\chi(\omega,k ) = \int_{\mathbbm{R}^3} \hat \chi(\omega,x ) \mathrm e^{-\i \langle k,x\rangle} \d x.
\]
The OCT system, by replacing $E_\nu^{(0)}$ in (\ref{eqFarField}) and simple calculations, see \cite[Proposition 9]{ElbMinSch15}, provide us with the data
\begin{equation}\label{eqMeasurementDataIsotropic}
\tilde\chi (\omega,\tfrac\omega c(\vartheta+e_3)),\quad\omega\in\mathbbm{R}\setminus\{0\},\;\vartheta\in S^2_+ .
\end{equation}
However, in practice, these data are incomplete because of the band-limited source and size of the detector. Thus, we get the spatial and temporal Fourier transform of $\chi$ only in a subset of $\mathbbm{R} \times \mathbbm{R}^3 .$
Then, the inverse problem we address here reads:
\begin{definition}[inverse problem]\label{def_inverse}
Given a medium as in \autoref{def_medium} and incident fields $E^{(0)}_\nu$ of the form \eqref{eqLocalisedPulse1} for all $\nu >0,$ the inverse problem is to recover the parameters $\hat\chi$ and $\Gamma$ given the internal PAT measurements $p_\nu(x),$ for $x\in \Omega,$ and all $\nu>0,$ given by \eqref{pat_data}, and the external OCT data $\tilde\chi (\omega,\tfrac\omega c(\vartheta+e_3)), \, \omega\in\mathbbm{R}\setminus\{0\}, \, \vartheta\in S^2_+ ,$ given by \eqref{eqMeasurementDataIsotropic}.
\end{definition}
Similar inverse problems have been considered in \cite{BalRenUhlZho11, BalZho14} where the far-field measurements from OCT are replaced by boundary measurements and in \cite{BalUhl12} for the diffusion approximation of the radiative transfer equation.
To present an equivalent formulation of the inverse problem, we assume that in both imaging techniques, we illuminate with multiple laser pulses with small spectrum centered around different frequencies. This setup describes swept-source OCT and multi-frequency PAT measurements.
First we describe the PAT measurements for multiple laser pulses. We
combine (\ref{eqAbsorbedEnergy})
and (\ref{pat_data}) to get
\[
p_\nu (x) = \Gamma(x)\int_\mathbbm{R} \left<E_\nu(t,x),\partial_t P_\nu(t,x)\right>\d t ,
\]
where $E_\nu$ is the electric field generated by the laser pulse $E^{(0)}_\nu .$ Using the Fourier transform of \eqref{eq_polar} we derive
\begin{equation*
p_\nu(x)=\Gamma(x)\frac1{2\pi} \int_\mathbbm{R} -\i\omega \hat\chi(\omega, x)|\hat E_\nu(\omega,x)|^2\d\omega.
\end{equation*}
\begin{remark}
In the case of nonlinear medium, the polarization field $P$ is usually expressed as a power series of the electric field $E.$ Then, the third order term contributes to the so-called two-photon absorbed energy \cite{Fri82}. We refer to \cite{RenZha16} for reconstructions in two-photon PAT.
\end{remark}
As in OCT, we replace $E_\nu$ by the initial pulse $E_\nu^{(0)}$ and we approximate the PAT data by
\[
p_\nu(x) \approx \Gamma(x)\frac1{2\pi} \int_\mathbbm{R} -\i\omega \hat\chi(\omega, x)|\hat f_\nu (\omega)|^2\d\omega,
\]
since $|
\eta |=1.$ The support of $\hat f_\nu$ is localized around the frequency $\nu$, see \eqref{eqLocalisedPulse}. Thus, we get in the limit $\varepsilon\to0$ (for constant norm $\|\hat f_\nu\|_2$) that
\begin{equation*}
p_\nu(x) \simeq \frac1{2\pi}\|\hat f_\nu\|_2^2\Gamma(x)(-\i\nu \hat\chi(\nu,x)+\i \nu \hat\chi (-\nu,x)) = \frac1{\pi}\|\hat f_\nu\|_2^2 \Gamma(x) \nu \Im(\hat\chi(\nu,x)).
\end{equation*}
We define $p ( \nu , x) := \frac{\pi}{\nu} \|\hat f_\nu \|_2^{-2} \, p_\nu (x).$ Then, we get asymptotically
\begin{equation}\label{eqPATOCTmeasSimplified}
\boxed{p(\nu ,x) \simeq \Gamma(x) \Im(\hat\chi(\nu ,x)). }
\end{equation}
We assume measurements for all frequencies $\nu>0.$ Recall that $\chi$ is a causal real valued function. Then the real part of $\hat\chi$ can be completely determined from the imaginary part via the Kramers--Kronig relation
\begin{equation*}
\Re(\hat\chi (\omega,x)) = \mathcal{H}[\Im \hat\chi ] (\omega,x) .
\end{equation*}
Here, $\mathcal{H}$ denotes the Hilbert transform with respect to frequency
\[
\mathcal{H}[ f] (\omega ,x ) = \frac{1}{\pi} \int_{\mathbbm{R}} \frac{ f(\tilde{\omega},x)}{\tilde{\omega}-\omega} \d \tilde{\omega}.
\]
We use the above two equations in order to describe the OCT data \eqref{eqMeasurementDataIsotropic}. Then we end up with the Fredholm integral equation
\begin{equation}\label{com_fre}
\boxed{\int_{\mathbbm{R}^3} \left(\mathcal{H} [p] (\omega , y) + \i p (\omega , y) \right) \, \mathrm e^{-\i \tfrac{\omega}c \langle \vartheta + e_3, y\rangle} \frac1{\Gamma (y)} \d y = \tilde\chi (\omega,\tfrac\omega c(\vartheta+e_3)) ,}
\end{equation}
for the Gr\"uneisen parameter $\Gamma.$ Once \eqref{com_fre} is solved, we can easily recover the imaginary part of $\hat\chi$ from equation~\eqref{eqPATOCTmeasSimplified}.
\begin{remark} If the medium is a perturbation of a single material then the above equation is transformed to a Fredholm integral equation of the second kind for a new function depending on $\tfrac1\Gamma$ \cite{ElbMinSch17}. At least in this simplified setting, we find that using the multi-modal model PAT/OCT we can (uniquely) determine the Gr\"uneisen parameter and the susceptibility $\chi$ describing the absorption and scattering properties of the medium.
\end{remark}
Observing the formulas \eqref{eqPATOCTmeasSimplified} and \eqref{com_fre}, we rewrite the inverse problem (\autoref{def_inverse}) in its simplified form:
\begin{definition}[simplified inverse problem]
Find $\Gamma (x)$ and $\hat \chi (\omega,x),$ given $\tilde\chi (\omega,\tfrac\omega c(\vartheta+e_3)),$ for all $\omega\in\mathbbm{R}\setminus\{0\},\;\vartheta\in S^2_+ $ (approximated OCT data) and the product $\Gamma (x) \Im(\hat\chi(\omega,x)),$ for all $\omega\in\mathbbm{R}\setminus\{0\},\;x\in \Omega$ (approximated PAT data).
\end{definition}
In the following section we present a Galerkin type method for the numerical solution of equation \eqref{com_fre} considering two types of media. There exist also other projection methods for the numerical solution of integral equations, the collocation method and the method of moments and quadrature methods, like the Nystr\"om method \cite{Hac95, Kre99, PolMan98}.
\section{Numerical Implementation}\label{section_numerics}
Without loss of generality we set $c=1$ and we specify $\Omega =[-l,l]^3 .$
For the numerical examples we have to introduce the parameter $\tilde{\Gamma},$ related to the physical parameter $\Gamma,$ which satisfies
\[
\tilde{\Gamma} (x) = \frac{1}{\Gamma(x)}, \quad \mbox{for } x\in \Omega, \quad \mbox{and} \quad \supp \tilde{\Gamma} \subset \Omega_L ,
\]
where $\Omega_L =[-L,L]^3,$ for $L>l.$ This is possible since $\Gamma \geq \Gamma_0 >0,$ with $\Gamma_0 \sim 1$ in biological tissues.
In addition, we do not consider the restrictions on the frequency in the following analysis.
\subsection{Medium with depth-dependent coefficients}\label{section_depth}
In the first example, we assume that both parameters $\Gamma$ and $\hat\chi$ are only depth-dependent, meaning that they vary only in the incident direction.
Then, $\tilde\Gamma$ and $\hat\chi$ admit the forms
\begin{equation*}
\tilde\Gamma (x) = \mathbbm{1}_{[-L,L]^2} (x_1 ,x_2) \gamma (x_3) , \quad \text{and} \quad \hat\chi (\omega , x) = \mathbbm{1}_{[-l,l]^2} (x_1 ,x_2) \psi (\omega ,x_3 ) ,
\end{equation*}
respectively. Here $\mathbbm{1}$ denotes the characteristic function. This case represents media which have a multilayer structure with depth-dependent properties, like the human skin.
If the illumination is focused to a small region inside the object and this region is small enough such that the functions can be assumed constant in both directions $e_1$ and $e_2,$ we get the above forms.
Thus the problem reduces to the problem of recovering a one-dimensional function $\gamma$. We do not consider the two-dimensional detector array but only the measurements at the single point detector located at the position $(0,\,0,\,d),$ meaning we set $\vartheta = e_3. $ Then, the equation \eqref{com_fre} takes the simplified form
\begin{equation}\label{foc_fredholm}
\int_{\mathbbm{R}} \left(\mathcal{H} [p] (\omega , y_3) + \i p (\omega , y_3) \right) \mathrm e^{- \i 2\omega y_3 } \gamma (y_3) \d y_3 = m (\omega) ,
\end{equation}
where $m (\omega) := (\tfrac{\pi}{l})^2 \tilde\chi (\omega,2 \omega e_3 ). $
Let $p \in (L^2 (\mathbbm{R}))^2$ and $\gamma \in L^2 (\mathbbm{R}). $ Since the kernel of the integral operator and the right-hand side in \eqref{foc_fredholm} have specific structures containing Hilbert and Fourier transforms we consider as orthonormal basis of $L^2 (\mathbbm{R})$ the Hermite functions $h_k , \, k\in \mathbbm{N}_0$. In addition the multi-dimensional Hermite functions can be written as sum of products of the usual Hermite functions. Their properties are given in \autoref{section_her}.
Other choices are also possible, especially when we treat the three-dimensional problem with real data, for instant using wavelets as basis functions.
Let $x\in \mathbbm{R}.$ The Hermite polynomials are defined by the formula
\begin{equation*
H_k (x) = (-1)^k \frac{d^k}{dx^k} (\mathrm e^{-x^2}) \mathrm e^{x^2}, \quad k \in \mathbbm{N}_0 .
\end{equation*}
The normalized Hermite functions are given by
\begin{equation}\label{her_fun}
h_k (x) = \alpha_k H_k (x) \mathrm e^{-\tfrac12 x^2}, k \in \mathbbm{N}_0 ,
\end{equation}
where $\alpha_k = (2^k k! \sqrt{\pi})^{-\tfrac12}.$ The functions $h_k$ satisfy the orthonormality condition
\begin{equation*
\int_\mathbbm{R} h_k (x) h_l (x) \d x = \delta_{k,l} .
\end{equation*}
\begin{proposition}\label{propo_1d}
Let $x\in\mathbbm{R}.$ We consider the expansion
\begin{equation}\label{foc_expan_gamma}
\gamma (\tfrac{x}2) = \sum_{k=0}^\infty \gamma_k h_k (x) ,
\end{equation}
with coefficients $\gamma_k \in \mathbbm{R}, \, k \in \mathbbm{N}_0$ and
\begin{equation}\label{foc_expan_pat}
p (\omega, \tfrac{x}2) = \sum_{k,l=0}^\infty p_{k,l} h_k (\omega) h_l (x) ,
\end{equation}
with coefficients
$ p_{k,l} \in \mathbbm{R},\, k,l \in \mathbbm{N}_0 . $
Then, if $\gamma$ satisfies the integral equation \eqref{foc_fredholm}, the coefficients $\gamma_k ,\, k\in\mathbbm{N}_0$ solve the equation
\begin{equation}\label{1d_full_7}
\sum_{j=0}^\infty \gamma_j A_{j,s} = m_s , \quad s\in \mathbbm{N}_0 ,
\end{equation}
where
\begin{equation*
\begin{aligned}
m_s &= \int_\mathbbm{R} 2 \mathrm e^{
\tfrac{\omega^2}4} m (\omega) h_s (\omega) \d \omega, \\
A_{j,s} &:= \sum_{k,l=0}^\infty (\tilde p_{k,l} +\i p_{k,l} ) \sum_{n=0}^{\min (j,l)} \beta_{j,l,n} \,\zeta_{j+l-2n}\frac{(j+l-2n)!}{2^{j+l-2n}}
\sum_{r=0}^{\left[ \tfrac{j+l-2n}{2}\right] } \frac{1}{r!q! \,\alpha_q}
\\
&\phantom{=} \times \sum_{s=0}^\infty \mathbbm{1}_{[\vert k-q\vert,k+q ]}(s) \beta_{k,q,\tfrac{k+q-s}2} , \quad \mbox{and } q := j+l-2n-2r.
\end{aligned}
\end{equation*}
\end{proposition}
Before proving this Proposition, we state the following lemma. Its proof is presented in \autoref{section_her}.
\begin{lemma}\label{lemma1}
Let $k\in \mathbbm{N}_0 .$ Then
\begin{equation}\label{eq_lemma1}
\int_\mathbbm{R} \mathrm e^{-\tfrac{x^2}2} h_k (x) \mathrm e^{- \i \omega x } \d x = \zeta_k \mathrm e^{-
\tfrac{\omega^2}4} \omega^k ,
\end{equation}
where $\zeta_k = 2\sqrt{\pi^3 } (-\i)^k \alpha_k ,$ for $\alpha_k$ as in \eqref{her_fun}.
\end{lemma}
\begin{proof}[\autoref{propo_1d}]
Using the expansion \eqref{foc_expan_pat} and considering \eqref{her_hil}, we get
\begin{equation*
\mathcal{H} [p] (\omega , x) + \i p (\omega , x) = \sum_{k,l=0}^\infty (\tilde p_{k,l} +\i p_{k,l} ) h_k (\omega) h_l (2x).
\end{equation*}
The coefficients $\tilde p_{k,l},$ using \eqref{her_coe_hil}, are given by
\begin{equation*
\tilde p_{k,l} = (-\i)^{k+1} \sum_{m=0}^\infty p_{m,l} (-\i)^m \int_\mathbbm{R} \sign (\omega) h_k (\omega) h_m (\omega) \d \omega .
\end{equation*}
We substitute the above expansions and \eqref{foc_expan_gamma} in \eqref{foc_fredholm} and we obtain
\begin{equation*
\sum_{j=0}^\infty \gamma_j \sum_{k,l=0}^\infty (\tilde p_{k,l} +\i p_{k,l} ) h_k (\omega) \int_\mathbbm{R} h_j (2y_3) h_l (2y_3) \mathrm e^{- \i 2\omega y_3 } \d y_3 = m (\omega) .
\end{equation*}
We rewrite the product of the two Hermite functions in the integrand using the formula \eqref{her_pro_fun} and we change variables to get
\begin{equation}\label{1d_full_2}
\sum_{j=0}^\infty \gamma_j \sum_{k,l=0}^\infty (\tilde p_{k,l} +\i p_{k,l} ) h_k (\omega) \sum_{n=0}^{\min (j,l)} \beta_{j,l,n}\int_\mathbbm{R} \mathrm e^{-\tfrac{x^2}2} h_{j+l-2n} (x) \mathrm e^{- \i \omega x } \d x = 2 m (\omega) .
\end{equation}
Then, equation \eqref{1d_full_2} using \eqref{eq_lemma1} takes the form
\begin{equation}\label{1d_full_3}
\sum_{j=0}^\infty \gamma_j \sum_{k,l=0}^\infty (\tilde p_{k,l} +\i p_{k,l} ) h_k (\omega) \sum_{n=0}^{\min (j,l)} \beta_{j,l,n} \,\zeta_{j+l-2n} \omega^{j+l-2n} = \tilde m (\omega) ,
\end{equation}
where $\tilde m (\omega )= 2 \mathrm e^{
\tfrac{\omega^2}4} m (\omega).$
Using \eqref{her_fun} and \eqref{her_inv}, we get
\begin{equation}\label{her_inv_fun}
\omega^k = \frac{k!}{2^k} \mathrm e^{\tfrac{\omega^2}2}\sum_{q=0}^{\left[ \tfrac{k}{2}\right] } \frac{1}{q!(k-2q)! \alpha_{k-2q}} h_{k-2q} (\omega) .
\end{equation}
We substitute this expansion in \eqref{1d_full_3} to obtain
\begin{multline}\label{1d_full_5}
\sum_{j=0}^\infty \gamma_j \sum_{k,l=0}^\infty (\tilde p_{k,l} +\i p_{k,l} ) \sum_{n=0}^{\min (j,l)} \beta_{j,l,n} \,\zeta_{j+l-2n}\frac{(j+l-2n)!}{2^{j+l-2n}} \mathrm e^{\tfrac{\omega^2}2} \\
\times \sum_{r=0}^{\left[ \tfrac{j+l-2n}{2}\right] } \frac{1}{r!q! \alpha_q} h_k (\omega) h_q (\omega)= \tilde m (\omega) ,
\end{multline}
where for simplicity we set $q: = j+l-2n-2r .$ Again the last product using \eqref{her_pro_fun} admits the form
\[
h_k (\omega) h_q (\omega)= \mathrm e^{-\tfrac{\omega^2}2} \sum_{u=0}^{\min (k,q)} \beta_{k,q,u} h_{k+q-2u} (\omega) .
\]
We expand also the data using the same basis functions
\begin{equation*
\tilde m (\omega) = \sum_{k=0}^\infty m_k h_k (\omega), \quad \text{for} \quad m_k = \int_\mathbbm{R} \tilde m (\omega) h_k(\omega) \d
\omega ,
\end{equation*}
and in order to obtain a linear equation for $\gamma_j$ we have to enlarge the index of the last sum. We set $s: = k+q-2u$ and for $\tfrac{k+q-s}2 \in \mathbbm{N}_0$ we reformulate \eqref{1d_full_5} using the above formulas as
\begin{multline*
\sum_{j=0}^\infty \gamma_j \sum_{k,l=0}^\infty (\tilde p_{k,l} +\i p_{k,l} ) \sum_{n=0}^{\min (j,l)} \beta_{j,l,n} \,\zeta_{j+l-2n}\frac{(j+l-2n)!}{2^{j+l-2n}}
\sum_{r=0}^{\left[ \tfrac{j+l-2n}{2}\right] } \frac{1}{r!q! \,\alpha_q} \\ \times \sum_{s=0}^\infty \mathbbm{1}_{[\vert k-q\vert,k+q ]} (s) \beta_{k,q,\tfrac{k+q-s}2} h_s (\omega) = \sum_{s=0}^\infty m_s h_s (\omega) .
\end{multline*}
Equating the coefficients in the above equation yields \eqref{1d_full_7}.
\end{proof}
The final step, for the Galerkin method, is to consider a finite dimensional subset of $L^2 (\mathbbm{R} ),$ meaning restrict ourselves to a finite number of coefficients. Let $j,k,l = 0,...,N-1 .$ Then the definitions used in the above analysis gives $q = 0,...,2(N-1)$ and $s=0,...,3(N-1).$ Finally, the discrete linear system of \eqref{1d_full_7} reads
\begin{equation}\label{eq_final_1d}
\b A \bm \gamma = \b m ,
\end{equation}
where $\b A = (A_{s,j}) \in \mathbbm{C}^{(3N-2)\times N}, \bm \gamma =(\gamma_j)\in \mathbbm{R}^N$ and $\b m = (m_s)\in \mathbbm{C}^{3N-2}.$
\subsection{Medium with coefficients constant in one direction}\label{section_2d}
In this case, we assume that both parameters are constant only in one direction, let us say in $e_1 .$ Then, $\tilde\Gamma$ and $\hat\chi$ take the forms
\begin{equation*}
\tilde\Gamma (x) = \mathbbm{1}_{[-L,L]} (x_1 ) \gamma (x_2, x_3) , \quad \text{and} \quad \hat\chi (\omega , x) = \mathbbm{1}_{[-l,l]} (x_1 ) \psi (\omega , x_2, x_3 ) ,
\end{equation*}
respectively. This assumption results to a two-dimensional function $\gamma,$ a case more involved compared to \autoref{section_depth} that approximates better the unconditional general problem.
Here, we need two-dimensional data, thus we have to consider measurements for all frequencies in a one-dimensional array, modeling measurement points on a line.
The equation \eqref{com_fre}, for $c=1,$ now takes the form
\begin{equation}\label{foc_fredholm_2d}
\int_{\mathbbm{R}} \int_{\mathbbm{R}} \left(\mathcal{H} [p] (\omega , y_2 , y_3) + \i p (\omega ,y_2 , y_3) \right) \mathrm e^{- \i \omega (\vartheta_2 y_2 + \tilde\vartheta_3 y_3 )} \gamma (y_2 , y_3) \d y_2 \d y_3 = m ( \omega, \vartheta ) ,
\end{equation}
where $\tilde\vartheta_3 = \vartheta_3 +1,$ and $m (\omega, \vartheta ) := \frac{\pi}{l} \tilde\chi (\omega,\tfrac\omega c(\vartheta+e_3)). $
\begin{proposition}\label{propo_2d}
Let $x=(x_1 , x_2)\in\mathbbm{R}^2.$ We use (\ref{her_3d}), for $\bm k = (k,\,l)$, and we expand $\gamma$ as
\begin{equation}\label{foc_expan_gamma_2d}
\gamma (x) = \sum_{\bm k=0}^\infty \gamma_{\bm k} h_{\bm k} (x) = \sum_{k,l=0}^\infty \gamma_{ k,l} h_{k} (x_1) h_l (x_2),
\end{equation}
where the coefficients $\gamma_{k,l}$ are defined by
\begin{equation*
\gamma_{k,l} = \int_\mathbbm{R} \int_\mathbbm{R} \gamma (x_1 ,x_2) h_k (x_1 ) h_l (x_2 ) \d x_1 \d x_2 ,\quad k,l \in \mathbbm{N}_0 .
\end{equation*}
and we assume the expansion
\begin{equation}\label{foc_expan_pat_2d}
p (\omega, x) = \sum_{k,l,a=0}^\infty p_{k,l,a} h_k (\omega) h_l (x_1) h_a (x_2) ,
\end{equation}
where
\[
p_{k,l,a} = \int_{\mathbbm{R}}\int_{\mathbbm{R}}\int_{\mathbbm{R}} p (\omega, x_1 ,x_2) h_k (\omega) h_l (x_1) h_a (x_2) \d \omega \d x_1 \d x_2 , \quad k,l,a \in \mathbbm{N}_0
\]
Then, if $\gamma$ solves the integral equation (\ref{foc_fredholm_2d}), its coefficients $\gamma_{k,l}, \, k,l\in N_0$ satisfy the equation
\begin{equation}\label{2d_full_7}
\sum_{k,l=0}^\infty \gamma_{k,l} B_{k,l,\mu} (\vartheta) = m_\mu (\vartheta), \quad \mu \in \mathbbm{N}_0,
\end{equation}
for
\begin{equation*
\begin{aligned}
m_\mu (\vartheta) &= \int_\mathbbm{R} \mathrm e^{\frac{\omega^2 \tilde\vartheta_3 }2 } m (\vartheta , \omega) h_\mu (\omega) \d \omega ,\\
B_{k,l,\mu} (\vartheta) &= \sum_{a,n,u=0}^\infty (\tilde p_{a,n,u} +\i p_{a,n,u} ) \sum_{r=0}^{\min (k,n)} \beta_{k,n,r}
\, \zeta_{k+n-2r} \vartheta_2^{k+n-2r} \\
&\phantom{=}\times \sum_{q=0}^{\min (l,u)} \beta_{l,u,q} \,\zeta_{l+u-2q} \tilde\vartheta_3^{l+u-2q}
\frac{s!}{2^s} \sum_{j=0}^{\left[ \frac{s}{2}\right] } \frac{1}{j!(s-2j)! \alpha_{s-2j}} \\ &\phantom{=}\times \mathbbm{1}_{[\vert a-s+2j \vert, a+s-2j]} (\mu) \beta_{a,s-2j,\frac{a+s-2j-\mu}2} \,,
\end{aligned}
\end{equation*}
with $s:= k+n+l+u-2r-2q.$
\end{proposition}
\begin{proof}
The equation (\ref{foc_fredholm_2d}) using the expansions (\ref{foc_expan_gamma_2d}) and (\ref{foc_expan_pat_2d}) results to
\begin{multline*
\sum_{k,l=0}^\infty \gamma_{k,l} \sum_{a,n,u=0}^\infty (\tilde p_{a,n,u} +\i p_{a,n,u} ) h_a (\omega)
\int_\mathbbm{R} \int_\mathbbm{R} h_k (y_2 ) h_l (y_3) h_n (y_2 ) \\
\times h_u (y_3) \mathrm e^{- \i \omega (\vartheta_2 y_2 + \tilde\vartheta_3 y_3 )} \d y_2 \d y_3 = m ( \omega, \vartheta) .
\end{multline*}
We apply twice the formula (\ref{her_pro_fun}) for the product of two Hermite functions, to obtain
\begin{multline*
\sum_{k,l=0}^\infty \gamma_{k,l} \sum_{a,n,u=0}^\infty (\tilde p_{a,n,u} +\i p_{a,n,u} ) h_a (\omega) \sum_{r=0}^{\min (k,n)} \beta_{k,n,r}
\int_\mathbbm{R} \mathrm e^{-\frac{y_2^2}2} h_{k+n-2r} (y_2) \mathrm e^{- \i \omega \vartheta_2 y_2} \d y_2
\\
\times \sum_{q=0}^{\min (l,u)} \beta_{l,u,q}
\int_\mathbbm{R} \mathrm e^{-\frac{y_3^2}2} h_{l+u-2q} (y_3) \mathrm e^{- \i \omega \tilde\vartheta_3 y_3} \d y_3 = m ( \omega, \vartheta) .
\end{multline*}
The last two integrals can be again simplified using lemma \ref{lemma1}. We get
\begin{multline*
\sum_{k,l=0}^\infty \gamma_{k,l} \sum_{a,n,u=0}^\infty (\tilde p_{a,n,u} +\i p_{a,n,u} ) h_a (\omega) \sum_{r=0}^{\min (k,n)} \beta_{k,n,r}
\zeta_{k+n-2r} \, \mathrm e^{-
\frac{(\omega \vartheta_2 )^2}4} (\omega \vartheta_2)^{k+n-2r}
\\
\times \sum_{q=0}^{\min (l,u)} \beta_{l,u,q} \zeta_{l+u-2q} \, \mathrm e^{-
\frac{(\omega \tilde\vartheta_3 )^2}4} (\omega \tilde\vartheta_3)^{l+u-2q}
= m ( \omega, \vartheta),
\end{multline*}
which for $s:= k+n+l+u-2r-2q,$ can be rewritten as
\begin{multline*
\sum_{k,l=0}^\infty \gamma_{k,l} \sum_{a,n,u=0}^\infty (\tilde p_{a,n,u} +\i p_{a,n,u} ) h_a (\omega) \sum_{r=0}^{\min (k,n)} \beta_{k,n,r}
\zeta_{k+n-2r} \vartheta_2^{k+n-2r}
\\
\times \sum_{q=0}^{\min (l,u)} \beta_{l,u,q} \zeta_{l+u-2q} \tilde\vartheta_3^{l+u-2q} \omega^s
= \tilde m ( \omega, \vartheta),
\end{multline*}
where $\tilde{m} = \mathrm e^{\frac{\omega^2 \tilde\vartheta_3 }2 } m,$ using that $|\vartheta |= 1$ and $\vartheta_1 = 0.$ The term $\omega^s$ can be analysed using the inverse explicit expression (\ref{her_inv_fun}) resulting to
\begin{eqnarray}\label{2d_full_5}
\sum_{k,l=0}^\infty \gamma_{k,l} \sum_{a,n,u=0}^\infty (\tilde p_{a,n,u} +\i p_{a,n,u} ) h_a (\omega) \sum_{r=0}^{\min (k,n)} \beta_{k,n,r}
\zeta_{k+n-2r} \vartheta_2^{k+n-2r}
\sum_{q=0}^{\min (l,u)} \beta_{l,u,q} \zeta_{l+u-2q} \nonumber\\
\times \tilde\vartheta_3^{l+u-2q} \frac{s!}{2^s} \mathrm e^{\frac{\omega^2}2}\sum_{j=0}^{\left[ \frac{s}{2}\right] } \frac{1}{j!(s-2j)! \alpha_{s-2j}} h_{s-2j} (\omega)
= \tilde m ( \omega, \vartheta).
\end{eqnarray}
We expand again the product $h_a (\omega)h_{s-2j}(\omega)$ using (\ref{her_pro_fun}) as
\[
h_a (\omega)h_{s-2j}(\omega) = \mathrm e^{-\frac{\omega^2}2} \sum_{t=0}^{\min (a,s-2j)}
\beta_{a,s-2j,t} h_{a+s-2j-2t} (\omega).
\]
We set $\mu:= a+s-2j-2t$ and for $\frac{a+s-2j-\mu}2 \in \mathbbm{N}_0$ the above sum can be rewritten as
\[
\sum_{t=0}^{\min (a,s-2j)}
\beta_{a,s-2j,t} h_{a+s-2j-2t} (\omega) = \sum_{\mu=0}^\infty \mathbbm{1}_{[\vert a-s+2j \vert, a+s-2j]} (\mu)\beta_{a,s-2j,\frac{a+s-2j-\mu}2} h_\mu (\omega) .
\]
We expand the right-hand side of (\ref{2d_full_5}) using the same basis functions
\begin{equation*
\tilde m ( \omega, \vartheta) = \sum_{\mu=0}^\infty m_\mu (\vartheta) h_\mu (\omega), \quad \mbox{for} \quad m_\mu (\vartheta) = \int_\mathbbm{R} \tilde m ( \omega, \vartheta) h_\mu(\omega) \d
\omega .
\end{equation*}
Then, the equation (\ref{2d_full_5}) using the above formulas and equating the coefficients results in equation (\ref{2d_full_7}).
\end{proof}
Let $k,l,a = 0,...,N-1 ,$ then we get $s = 0,...,4(N-1)$ and $\mu=0,...,5(N-1).$ Thus, the discrete linear system of \eqref{2d_full_7} admits the form
\begin{equation}\label{2d_full_final}
\bm \Gamma \, \b B (\vartheta ) = \b m (\vartheta), \quad \vartheta \in S^2_+ ,
\end{equation}
for the matrix-valued unknown function $\bm \Gamma =(\gamma_{k,l})\in \mathbbm{R}^{N\times N},$
where $\b B = (B_{k,l,\mu}) \in \mathbbm{C}^{ N\times N\times (5N-4)}$ and $\b m = (m_\mu)\in \mathbbm{C}^{5N-4}.$
To bring the above equation into a form similar to \eqref{eq_final_1d}, we define the vector
\[
\bm\zeta = (\gamma_{0,0},...,\gamma_{0,N-1},\gamma_{1,0},...,\gamma_{1,N-1},...,\gamma_{N-1,0},...,\gamma_{N-1,N-1})^\top \in \mathbbm{R}^{N^2},
\]
and we rearrange $\b B$ to create the matrix $\b C \in \mathbbm{C}^{(5N-4)\times N^2}$ given by
\[
\b C = \begin{pmatrix}
B_{0,0,0} & \hdots & B_{0,N-1,0} & \hdots & \hdots & B_{N-1,0,0} & \hdots & B_{N-1,N-1,0} \\
B_{0,0,1} & \hdots & B_{0,N-1,1} & \hdots & \hdots & B_{N-1,0,1} & \hdots & B_{N-1,N-1,1} \\
\vdots & & \vdots & & & \vdots & & \vdots \\
B_{0,0,5(N-1)} & \hdots & B_{0,N-1,5(N-1)} & \hdots & \hdots & B_{N-1,0,5(N-1)} & \hdots & B_{N-1,N-1,5(N-1)} \\
\end{pmatrix} .
\]
Then we rewrite \eqref{2d_full_final} as
\begin{equation*
\b C (\vartheta ) \bm\zeta = \b m (\vartheta),
\end{equation*}
and we consider $K$ detection directions, meaning $\vartheta^{(k)} , \, k = 1,...,K ,$ such that the system
\begin{equation}\label{eq_final_2d}
\b D \bm\zeta = \b d ,
\end{equation}
for
\begin{equation*}
\b D = \begin{pmatrix}
\b C (\vartheta^{(1)} ) \\
\vdots \\
\b C (\vartheta^{(K)} )
\end{pmatrix} \in \mathbbm{C}^{K(5N-4)\times N^2}, \quad \text{and} \quad
\b d = \begin{pmatrix}
\b m (\vartheta^{(1)} ) \\
\vdots \\
\b m (\vartheta^{(K)} )
\end{pmatrix} \in \mathbbm{C}^{K(5N-4)},
\end{equation*}
is at least exactly determined.
\section{Numerical Results}\label{section_results}
Both linear systems derived in the previous section admit the general form
\begin{equation}\label{equation_final}
\b G \b x = \b g .
\end{equation}
In the case of depth-dependent coefficients, see \autoref{section_depth}, we have
\[
\b G := \b A \in \mathbbm{C}^{(3N-2)\times N}, \quad \b x := \bm \gamma \in \mathbbm{R}^N , \quad \b g := \b m \in \mathbbm{C}^{3N-2},
\]
and in the case of constant in one direction coefficients, see \autoref{section_2d}, we get
\[
\b G := \b D \in \mathbbm{C}^{K(5N-4)\times N^2}, \quad \b x := \bm \zeta \in \mathbbm{R}^{N^2}, \quad \b g := \b d \in \mathbbm{C}^{K(5N-4)}.
\]
We approximate the solution of \eqref{equation_final} by minimizing the Tikhonov functional
\[
\| \b G \b x - \b g\|_2^2 + \lambda \| \b x \|_2^2 ,
\]
where $\lambda >0$ is the regularization parameter. Since $\b x$ is in both case a real-valued function we actually solve the following regularized equation
\[
\left( \Re (\b G)^\top \Re (\b G) + \Im (\b G)^\top \Im (\b G)+ \lambda \b I \right) \b x = \Re (\b G)^\top \Re (\b g) + \Im (\b G)^\top \Im (\b g) ,
\]
where $\b I$ is the identity matrix with dimensions depending on each problem. We consider also noisy data for both measurements data, the pressure $p$ and the OCT data $m,$ with respect to the $L^2$ norm
\[
p_\delta = p + \delta_p \frac{\| p \|_2}{\| v \|_2} v , \quad
\text{and} \quad m_\delta = m + \delta_m \frac{\| m \|_2}{\| w \|_2} w ,
\]
for given noise levels $\delta_p , \delta_m$ and $v = v_1 + \i v_2, \, w = w_1 + \i w_2,$ for $v_1 , v_2 , w_1$ and $w_2$ normally identically distributed, independent random variables.
We present reconstructions for different functions $\gamma$ (related to $1/\Gamma$) and $\psi$ (related to $\hat\chi$) for both cases of media. As OCT data we consider the function $\tilde \chi$ (using the Fourier transform and the Kramers--Kronig relation) and to construct the simulated PAT data we have to assume that both functions have similar behavior such that ratio $\hat\chi / \gamma$ (see \eqref{eqPATOCTmeasSimplified}) is still integrable. In all figures we plot the spatial domain $\Omega_L.$
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.7]{1d_example1_crop.pdf}
\caption{Reconstruction of $\gamma,$ given by equation \eqref{gamma_example1}, for increasing number of Fourier coefficients. }\label{Fig1}
\end{center}
\end{figure}
\subsection{Examples with depth-dependent coefficients (see \autoref{section_depth})}\label{section_depth_numerics}
In the following figures the true curve is represented by a dashed red line and the reconstructed by a solid blue line. Let $x\in\mathbbm{R}.$ In the first example we consider
\begin{equation}\label{gamma_example1}
\gamma (x) = (2x^4 + 1) \,\mathrm e^{-x^2},
\end{equation}
and
\[
\Im (\psi (\omega,x)) = h_1 (\omega) (x^4 + x^3 + x^2 + 0.1) \, \mathrm e^{-2x^2} .
\]
We set $\Omega = [-3.5,3.5]$ and $\Omega_L = [-4,4]$ such that
$\supp \psi (\omega,\cdot ) \subset \Omega,$ and
$\supp \gamma \subset \Omega_L,$ and we restrict ourselves to $\omega \in \mathcal{W}:= [-4,4].$ We consider data with $\delta_p = \delta_m = 3\%$ noise. The results are presented in \autoref{Fig1} for regularization parameter $\lambda = 10^{-4}$ and different values of $N.$ Here, we see the improvement in the reconstructions as $N$ increases.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.7]{1d_example2_crop.pdf}
\caption{Reconstruction of $\gamma,$ given by equation \eqref{gamma_example2}, for increasing number of Fourier coefficients. }\label{Fig2}
\end{center}
\end{figure}
In the second example, we use the same function $\psi$ and we consider as $\gamma$ the function
\begin{equation}\label{gamma_example2}
\gamma (x) = h_0 (x) + h_0 (2x) + h_1 (3x) .
\end{equation}
We keep all the parameters the same as in the first example. In \autoref{Fig2}, we see the reconstructions for different number of coefficients.
In the third example, we consider
\begin{equation}\label{susce_example3}
\Im (\psi (\omega,x)) = (h_1 (\omega)+h_1 (2\omega) )(x^2 + 0.1) \, \mathrm e^{-2x^2} ,
\end{equation}
such that again $\supp \psi (\omega,\cdot ) \subset [-3.5,3.5],$ see the left picture in \autoref{Fig3}. We present the reconstructions of $\Im(\psi)$ using the form \eqref{gamma_example1} for $\gamma,$ while keeping all the other parameters the same. We set $N=15$ coefficients. We present the results for $\Im (\psi (\omega,1)), \, \omega \in \mathcal{W},$ (center picture) and $\Im (\psi (3,x)), \, x\in \Omega ,$
(right picture) in \autoref{Fig3}.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.7]{1d_example3_crop.pdf}
\caption{Reconstruction of $\Im (\psi ),$ see equation \eqref{susce_example3}, for $N=15$. The true imaginary part (left), the cross section of the reconstruction at the line $x=1$ (center) and at the line $\omega=3$ (right).}\label{Fig3}
\end{center}
\end{figure}
\subsection{Examples with coefficients constant in one direction (see \autoref{section_2d})}
Here, the measurements are given at points on a line. We consider the minimum amount of measurement points in order to have an exactly determined system (\ref{eq_final_2d}) in our examples.
In the following examples we keep the same noise levels $\delta_p = \delta_m = 3\%$ and we obtain the regularization parameter using the L-curve criterion \cite{HanOle93}.
Let $x,y \in \mathbbm{R} .$ In the fourth example, we consider
\begin{equation}\label{gamma_example4}
\gamma (x,y) = \mathrm e^{ -(x+1.5)^2- (y+1.5)^2},
\end{equation}
and
\begin{equation}\label{susce_example4}
\Im (\psi (\omega,x,y)) = 0.7 (h_1 (\omega)+h_1 (2\omega)) \, \mathrm e^{ -(x+1.6)^4- \tfrac12 (y+1.6)^4}.
\end{equation}
We set $\Omega = [-4,4]^2, \, \Omega_L = [-4.5,4.5]^2$ and $\mathcal{W}=[-3,3].$ The reconstructions of $\gamma$ for $N=5$ and $\vartheta^{(1)}=(0, \, 0,\, 1)^\top$ are presented in \autoref{Fig4}. The results for the cross-section of the imaginary part of $\psi$, given by equation \eqref{susce_example4}, at frequency $\omega=0$ are presented in \autoref{Fig5}.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.7]{ex4_new_mu_crop.pdf}
\caption{The exact function $\gamma,$ see equation \eqref{gamma_example4}, (left) and the reconstructed for $N=5$ and one measurement point (right).}\label{Fig4}
\end{center}
\end{figure}
In the last example the unknown function is given by
\begin{equation}\label{gamma_example5}
\gamma (x,y) = \mathrm e^{ -(x+0.5)^2- (y+2)^2} + 0.8\,\mathrm e^{ -(x+2)^2- (y+0.5)^2} + \mathrm e^{ -(x-2)^2- (y-2)^2} .
\end{equation}
The size of the medium is kept the same as in the previous example and we set $\mathcal{W}=[-2,2]$. Here, we want to test the performance of our numerical scheme with respect to the number of the detection directions $\vartheta^{(k)}.$ In the case of three measurement directions, see \eqref{eq_final_2d}, we consider
\[
\vartheta^{(1)} = \left(0, \, \cos (\tfrac{5\pi}{12} ),\, \sin (\tfrac{5\pi}{12}) \right)^\top , \quad \vartheta^{(2)} = \left(0, \,0,\,1\right)^\top , \quad \vartheta^{(3)} = \left(0, \, -\cos (\tfrac{5\pi}{12}),\, \sin (\tfrac{5\pi}{12}) \right)^\top .
\]
The reconstructions for $N=5$ coefficients are presented in \autoref{Fig6}, where we set to zero the negative values.
We set the imaginary part of $\psi$ to be
\begin{equation}\label{susce_example5}
\Im (\psi (\omega,x,y)) = h_1 (\omega) \left(\mathrm e^{ -(x+0.5)^4- (y+2)^4} + \mathrm e^{ -0.6(x+2)^4- (y+0.5)^4} + 0.8\, \mathrm e^{ -(x-2)^4- (y-2)^4} \right).
\end{equation}
The reconstruction for $\mathcal{W}=[-3,3]$ are given in \autoref{Fig7}, where we see the improvement of the results with respect to the Fourier coefficients. In the first case we set $N=5$ and we consider one detection direction. In the second case, we use $N=10$ coefficients and two measurement points in the directions:
\[
\vartheta^{(1)} = \left(0, \, \cos (\tfrac{7\pi}{16}),\, \sin (\tfrac{7\pi}{16}) \right)^\top , \quad \vartheta^{(2)} = \left(0, \, -\cos (\tfrac{7\pi}{16}),\, \sin (\tfrac{7\pi}{16}) \right)^\top .
\]
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.7]{ex4_new_chi_crop.pdf}
\caption{Reconstruction of $\Im (\psi ),$ given by equation \eqref{susce_example4}, for $N=5$. The true imaginary part (left), the cross section of the true value at the plane $\omega=0$ (center) and the reconstructed (right).}\label{Fig5}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.7]{fig6_crop.pdf}
\caption{The exact function $\gamma,$ see equation \eqref{gamma_example5}, (left) and the reconstructed for $N=5$ and one measurement point $K=1$ (center) and $K=3$ (right).}\label{Fig6}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.7]{fig7_new_close.pdf}
\caption{Reconstruction of $\Im (\psi ),$ see equation \eqref{susce_example5}. In the left column we see the cross-section of the true imaginary part at the plane $\omega = 0$ (first row) and at the plane $y=-2$ (second row). The reconstructions for $N=5$ and one detection direction are presented in the second column. The results in the third column are for $N=10$ and two detection directions. }\label{Fig7}
\end{center}
\end{figure}
\section{Conclusions}
In this work we considered the inverse problem to reconstruct quantitatively the electric susceptibility and the Gr\"uneisen parameter of a non-magnetic linear dielectric medium from measurements with the multi-modal tomographic system of Photoacoustic and Optical Coherence Tomography.
Our scheme is based on the numerical solution of a Fredholm integral equation of the first kind for the Gr\"uneisen parameter using a Galerkin type method. We presented numerical results for different kinds of media.
\section*{Acknowledgements}
The work of OS has been supported by the Austrian Science Fund (FWF), Project P26687-N25 (Interdisciplinary Coupled Physics Imaging).
\section*{Appendix}\label{section_her}
We recall Hermite functions and we present their properties which are used in this work. We connect the Fourier and Hilbert transforms of a function with expansions in terms of Hermite functions.
Let $x\in \mathbbm{R}.$ The normalized Hermite functions $h_k , \, k \in \mathbbm{N}_0 $ are eigenfunctions of the inverse Fourier transform
\[
\check f (x) = \frac1{2\pi}\int_\mathbbm{R} \hat f(\omega) \mathrm e^{-\i\omega t} \d \omega ,
\]
meaning they satisfy
\begin{equation*
\check{h}_k (x) = (-\i)^k h_k (x) .
\end{equation*}
The product of two Hermite polynomials admits the following series expansion
\begin{equation*
H_k (x) H_l (x) = k! l! \sum_{m=0}^{\min (k,l)}
\frac{2^m}{m! (k-m)! (l-m)!} H_{k+l-2m} (x) ,
\end{equation*}
also known as Feldheim's identity. Using \eqref{her_fun} we see that the product of two Hermite functions can be written as
\begin{equation}\label{her_pro_fun}
h_k (x) h_l (x) = \mathrm e^{-\tfrac{x^2}{2}} \sum_{m=0}^{\min (k,l)}
\beta_{k,l,m} h_{k+l-2m} (x) ,
\end{equation}
for
\[
\beta_{k,l,m} = \pi^{-\tfrac14}\frac{(k! l! (k+l-2m)!)^{\tfrac12}}{m! (k-m)! (l-m)!}
\]
We recall the addition formula \cite{Fel43, MagObeSon66}
\begin{equation}\label{her_sum}
H_k (x+y) = \sum_{m=0}^k \frac{k!}{(k-m)!m!} (2y)^{k-m} H_m(x) ,
\end{equation}
the multiplication formula
\begin{equation*
H_k (\rho x) = k!\sum_{m=0}^{\left[ \tfrac{k}{2}\right] } \frac{\rho^k}{m! (k-2m)!} \left( 1-\frac{1}{\rho^2}\right)^m H_{k-2m} (x),
\end{equation*}
and the inverse explicit expression
\begin{equation}\label{her_inv}
x^k = \frac{k!}{2^k}\sum_{m=0}^{\left[ \tfrac{k}{2}\right] } \frac{1}{m!(k-2m)!} H_{k-2m} (x) .
\end{equation}
Let $f \in L^2 (\mathbbm{R}).$ We consider the expansion
\begin{equation*
f(x) = \sum_{k=0}^\infty f_k h_k (x) ,
\end{equation*}
where the coefficients $f_k$ are defined by
\begin{equation*
f_k = \int_\mathbbm{R} f(x) h_k (x) \d x .
\end{equation*}
The Hilbert transform of $f$ admits the expansion
\begin{equation}\label{her_hil}
\mathcal{H} [f] (x) = \sum_{k=0}^\infty \tilde f_k h_k (x) ,
\end{equation}
where $\tilde f_k$ are given by \cite{PorSchuKin13}
\begin{equation}\label{her_coe_hil}
\tilde f_k = (-\i)^{k+1} \sum_{m=0}^\infty f_m (-\i)^m \int_\mathbbm{R} \sign (x) h_k (x) h_m (x) \d x .
\end{equation}
For $\b k \in \mathbbm{N}_0^d$ and $ x \in \mathbbm{R}^d,$ we define the $\b k$th Hermite polynomial as
\begin{equation}\label{her_3d}
H_{\b k} ( x) = \prod_{j=1}^{d} H_{k_j} (x_j) .
\end{equation}
Now we present the proof of \autoref{lemma1}.
\begin{proof}[\autoref{lemma1}]
We consider the convolution theorem for the inverse Fourier transform and the above properties.
\begin{equation*}
\begin{aligned}
\int_\mathbbm{R} \mathrm e^{-\tfrac{x^2}2} h_k (x) \mathrm e^{- \i \omega x } \d x &= \left( \int_\mathbbm{R} \mathrm e^{-\tfrac{x^2}2} \mathrm e^{- \i \omega x } \d x\right) \ast \check{h}_k (\omega) \\
&= 2\pi (-\i)^k \mathrm e^{-
\tfrac{\omega^2}2} \ast h_k (\omega) \\
&= 2\pi (-\i)^k \int_\mathbbm{R} \mathrm e^{-
\tfrac{(\omega-y)^2}2} h_k (y) \d y \\
&= 2\pi (-\i)^k \alpha_k \mathrm e^{-
\tfrac{\omega^2}4} \int_\mathbbm{R} \mathrm e^{-(y-
\tfrac{\omega}2 )^2} H_k (y) \d y \\
&= 2\pi (-\i)^k \alpha_k \mathrm e^{-
\tfrac{\omega^2}4} \int_\mathbbm{R} \mathrm e^{-z^2} H_k (z + \tfrac{\omega}2) \d z .
\end{aligned}
\end{equation*}
To compute the last integral we apply the formula \eqref{her_sum}
\begin{equation*}
\begin{aligned}
\int_\mathbbm{R} \mathrm e^{-z^2} H_k (z + \tfrac{\omega}2) \d z &= \sum_{m=0}^k \frac{k!}{(k-m)!m!} \omega^{k-m} \int_\mathbbm{R} \mathrm e^{-z^2} H_m(z) \d z \\
&= \sum_{m=0}^k \frac{k!}{(k-m)!m!} \omega^{k-m} \int_\mathbbm{R} \mathrm e^{-z^2} H_m(z) H_0 (z) \d z \\
&= \sum_{m=0}^k \frac{k!}{(k-m)!m!} \omega^{k-m} a_m^{-2} \delta_{m,0 } \\
&= \sqrt{\pi} \omega^k .
\end{aligned}
\end{equation*}
The last two equations result to \eqref{eq_lemma1}.
\end{proof}
\section*{References}
\printbibliography[heading=none]
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 500 |
\section{Introduction}
Axion-like particles (ALPs) are ubiquitous in a large number of high-energy completions of the Standard Model (SM).
They are for instance generic predictions in the low-energy spectrum of string compactifications \cite{Svrcek:2006yi,Arvanitaki:2009fg}.
ALPs denote any particles which are a pseudo-Nambu Goldstone Bosons enjoying a discrete symmetry. They are light and have weak self-interactions.
Interestingly, they can be used to solve open problems of the SM as they are well-motivated candidates to explain \textit{e.g.} dark matter~\cite{Preskill:1982cy, Abbott:1982af, Dine:1982ah}, inflation~\cite{Pajer:2013fsa, Adshead:2015pva, Domcke:2019qmm, Adshead:2019lbr}, and baryogenesis~\cite{Servant:2014bla,Domcke:2019mnd}. They were originally introduced to solve the Strong CP problem \cite{Wilczek:1977pj,Peccei:1977hh} and still remain the most popular explanation to this puzzle. Independently of their virtues in providing solutions to some major open problems in the SM, they are also interesting purely from their specific phenomenological properties, notably in cosmology. They can leave imprints in many different ways such as in the Cosmic Microwave Background, in Large Scale Structures, in Gravitational Waves, in stellar astrophysics and they can be searched for directly in dedicated laboratory experiments such as haloscopes (see e.g \cite{Marsh:2015xka, Irastorza:2018dyq} for recent reviews).
The role of axions in cosmology has been the subject of a large number of studies.
Specifically, zero-mode axion oscillations around the minimum of the axion potential can provide a large component of the energy density of the universe and mimic the effect of dark matter.
Surprisingly, the effect of axion quantum fluctuations in the early universe has been mostly overlooked so far in the literature where only the zero mode homogeneous mode has been considered. In this paper, we investigate in details the effect of axion particle production during the evolution of the homogeneous zero-mode.
The production of axion quanta when the zero-mode field oscillates around one minimum of the potential is generally suppressed unless the initial position of the field is extremely close to the maximum of the potential~\cite{Greene:1998pb}.%
\footnote{As this work was being completed, Ref.~\cite{Arvanitaki:2019rax} appeared, which considers the production of axion quantum fluctuations during oscillations about the minimum of the potential, as in~\cite{Greene:1998pb}.
This effect is important only if the initial position of the axion field is tuned very close to the top of the barrier of the axion potential, at the level of $\sim 10^{-7}$. Such peculiar initial position was recently motivated by some inflation dynamics in \cite{Co:2018mho} and by anthropic arguments in \cite{Freivogel:2008qc}.}
In this work, we are instead investigating the exponential production of axion quanta when the axion is rolling down its potential with a large velocity and the axion is crossing a large number of minima/maxima during its evolution. Such situation was somehow rarely investigated before, although it is quite generic. We are interested in the resulting friction force on the zero-mode. Our findings do not require any tuning of initial conditions.
Our only assumption is the existence of a small slope together with some wiggles, and an initial kinetic energy larger than the barriers potential energy.
In fact, this is precisely the type of potential introduced in the relaxion mechanism proposed in
\cite{Graham:2015cka}
to resolve the electroweak scale hierarchy problem.
This type of axion potential, with a linear term plus a cosine, was first considered in the context of string cosmology, in models of axion
monodromy~\cite{McAllister:2008hb, Flauger:2009ab}%
\footnote{In Ref.~\cite{McAllister:2008hb}, particle production was not alluded to. In the follow-up paper~\cite{Flauger:2009ab},
the axion is the inflaton
and axion particle production was mentioned,
but in the case of small wiggles,
which does not lead to exponential particle production.
Friction from backreaction on the zero mode was therefore not discussed.
Later, Ref.~\cite{Jaeckel:2016qjp} studied a quadratic potential with wiggles.
The particle production, from the zero-mode oscillations around the global minimum with large amplitude, was discussed at the linearized level.
However, the backreaction on the zero mode was not.
In a follow-up study \cite{Berges:2019dgr}, the particle production was discussed by non-perturbative numerical analysis. The aim is not to exploit this effect as a stopping condition (the field eventually stops at a global minimum) but as a dark matter production mechanism.
It is there suggested that axions quanta produced during the rolling stage could be (warm) dark matter candidates.}.
In this paper, we show that the production of axion particles generates a friction that decelerates the rolling of the field.
This occurs if the field velocity is large enough to overcome the sinusoidal term and the field goes over a large number of wiggles.
As we will see, the equation of motion for the axion fluctuation can be described by the Mathieu equation and parametric resonance gives an exponential production of the particles with specific wave numbers.
We denote this phenomenon as \textit{axion fragmentation.}
This provides a novel mechanism to stop axion rolling.
We focus our attention on the most dramatic effect of fragmentation, \textit{i.e.}, the regime in which the axion stops its motion by transferring all of its kinetic energy to the fluctuations. Other effects can arise from fragmentation. For example, in situation in which a field rolls down a steep potential, fragmentation can be a way of generating a slow-roll regime not sustained by the sole Hubble friction. At the same time, the effect discussed in this paper is mathematically very similar to the amplification of fluctuations in oscillons~\cite{ Hertzberg:2010yz, Amin:2010xe, Amin:2010dc, Antusch:2017flz, Olle:2019kbo}%
\footnote{A recent analysis \cite{Olle:2019kbo}, although not related to axions, suggests that quanta produced by oscillons could be the dark matter candidates.}.
The phenomenon of scalar field fragmentation has been studied in the context of preheating \cite{Dolgov:1989us, Traschen:1990sw, Kofman:1994rk, Shtanov:1994ce, Kofman:1997yn}.
The excitation of gauge field quanta from an axion has been studied extensively in different contexts, mainly in axion inflation models where the axion has a coupling to gauge fields, see \textit{e.g.}~\cite{Pajer:2013fsa,Domcke:2019qmm}.
We provide a model-independent detailed analytical treatment of axion fragmentation that can be applied to various setups.
We discuss the precise conditions for fragmentation to stop the field (even far away from the global minimum), which we check over a numerical analysis.
We discuss the implications of these findings for the relaxation mechanisms of the electroweak scale.
In this case, the {\it relaxion} scans the Higgs mass-squared term in the early universe,
and dynamically realizes an electroweak scale which is suppressed compared to the cutoff scale.
A key ingredient of this scenario is the friction that slows down the relaxion field rolling.
In the original paper~\cite{Graham:2015cka}, Hubble friction is responsible for the slow roll of the relaxion.
Alternatively, a coupling to SM gauge bosons $\phi F_{\mu\nu} \tilde F^{\mu\nu}$ can provide the necessary friction, through tachyonic particle production~\cite{Hook:2016mqo, Choi:2016kke, Tangarife:2017rgl, Matsedonskyi:2017rkq, Fonseca:2018xzp, Fonseca:2018kqf}. In~\cite{Ibe:2019udh} the necessary friction is provided by parametric resonance of the Higgs zero mode. In~\cite{Wang:2018ddr} the field is stopped thanks to a potential instability. Finally, in~\cite{Kadota:2019wyz} the relaxion is slowed down via the production of dark fermions.
So far, the excitation of axion particle themselves, was not considered, although they are present in the most minimal models where the relaxion has no extra couplings to gauge fields.
We analyse in details how this can be used as a stopping mechanism for the relaxion in our companion paper~\cite{Fonseca:2019lmc}, the results of which are summarised here.
The paper is organized as follows.
In section \ref{sec:intuitive derivation},
we describe our setup and derive the equation of motion of the fluctuations at the leading order.
We also give an intuitive discussion on the particle production effect.
Then, we derive semi-analytic formula in section \ref{sec:analytical discussion},
and show numerical results in section \ref{sec:numeircal analysis}.
We discuss the analysis beyond the leading order in section \ref{sec:beyond linear}.
In Sec.~\ref{sec:Consequences} we discuss how our results apply to the relaxion mechanism.
Our conclusions are drawn in Sec.~\ref{sec:conclusions}. In the appendices \ref{sec:technical details} and \ref{sec:detail on phiddot} we provide further details on the derivation of our results.
\section{Axion fragmentation in a nutshell} \label{sec:intuitive derivation}
In this section we discuss how the axion field $\phi$ evolution is affected by the axion fragmentation phenomenon.
The dynamics of axion quantum fluctuation is described by a Mathieu equation with time varying coefficients, from which we can estimate how the fluctuations back-react on the zero mode. Our goal is to study the dynamics of axion particle production when the axion field is rolling down a wiggly potential.
For concreteness, we consider the following potential
\begin{align}
V(\phi) = \Lambda_b^4 \cos \frac{\phi}{f} - \mu^3 \phi, \label{eq:potential}
\end{align}
and we assume the height of the barrier $\Lambda_b$ as a constant for simplicity (see \Fig{fig:potential} for a sketch of the potential).
For simplicity, we do not include any cosmological constant term in Eq.~(\ref{eq:potential}), and we assume that the vacuum energy is cancelled in the late universe by some other mechanism, about which we remain agnostic.
We are interested in the case in which the barriers are large, \textit{i.e.}
\begin{equation} \label{eq:local minima}
\frac{\Lambda_b^4}{f} > \mu^3
\end{equation}
which corresponds to say that the potential has local minima. Additionally, we assume that the kinetic energy of $\phi$ is large enough to overcome the barriers, $\dot\phi>\Lambda_b^2$.
\begin{figure}[t]
\centering
\includegraphics[width=0.6\hsize]{figures/potential}
\caption{
Sketch of the axion potential given in \Eq{eq:potential}. Once \emph{axion fragmentation} starts, the field $\phi$ takes a time $\Delta t_{\rm frag}$ and moves a distance $\Delta\phi_{\rm frag}$ until it gets trapped in one of the wiggles.
}\label{fig:potential}
\end{figure}
The equation of motion of the axion $\phi(x,t)$ is given by
\begin{align}
\ddot\phi + 3H\dot\phi - \frac{1}{a^2} \nabla^2\phi + V'(\phi)=0, \label{eq:EOM}
\end{align}
where $a$ is the scale factor of the Friedmann-Lema\^itre-Robertson-Walker metric and $H=\dot a/a$ is the Hubble expansion rate.
Let us decompose $\phi(x,t)$ into a classical homogeneous mode $\phi(t)$ and small fluctuations $\delta\phi$
(with no risk of confusion, we will denote the homogeneous mode as $\phi(t)$):
\begin{align}
\phi(x,t)
= \phi(t) + \delta\phi(x,t)
= \phi(t) + \left( \int\frac{d^3 k}{(2\pi)^3} a_k u_{k}(t) e^{ikx} + h.c. \right), \label{eq:phi mode expansion}
\end{align}
with $a_k$, $a_k^\dagger$ being respectively the annihilation and creation operators which satisfy
\begin{align}
[a_k, a_{k'}^\dagger] = (2\pi)^3 \delta^{(3)}(k-k'),
\end{align}
and the initial condition of the mode function $u_k$ at $t \to -\infty$ is given by
\begin{align}
u_k(t) = \frac{e^{-i(k/a) t}}{a\sqrt{2k}}. \label{eq:initial}
\end{align}
In the analysis of this work, we treat $\delta\phi$ as small perturbation.
We discuss the validity of this approximation in Sec.~\ref{sec:beyond linear}.
By using this approximation, we expand the last term of the LHS of Eq.~(\ref{eq:EOM}) as $V'(\phi) + V''(\phi) \delta\phi + (1/2) V'''(\phi)\delta\phi^2 + \cdots$.
The third term of this expansion gives the dominant source of the backreaction to the zero mode from the particle production.
The equations of motion of $\phi$ and $\delta\phi$ are given by
\begin{align}
\ddot{\phi} + 3H\dot{\phi} + V'(\phi) + \frac{1}{2} V'''(\phi) \int \frac{d^3 x}{V_{\rm vol}} \langle \delta\phi(x)^2 \rangle &= 0, \label{eq:EOM1}\\
\ddot{\delta\phi} + 3H\dot{\delta\phi} + \left[ -\frac{1}{a^2} \nabla^2 + V''(\phi) \right] \delta\phi &= 0, \label{eq:EOM2}
\end{align}
where $\langle\dots\rangle$ indicates a quantum expectation value. Equations~(\ref{eq:EOM1}) and~(\ref{eq:EOM2}) can be rewritten in terms of the mode functions $u_k$ as
\begin{align}
\ddot{\phi} + 3H\dot{\phi} + V'(\phi) + \frac{1}{2}V'''(\phi) \int\frac{d^3 k}{(2\pi)^3} |u_k|^2 &=0, \label{eq:zeromode}\\
\ddot{u_k} + 3H\dot{u}_k + \left[ \frac{k^2}{a^2} + V''(\phi)\right] u_k &= 0. \label{eq:fluctuation}
\end{align}
Let us denote with $\dot\phi_0$ the initial velocity of the field $\phi$. We assume that $\dot{\phi}_0$ is large enough to overcome the barrier, \textit{i.e.}, $$\dot\phi_0 \gtrsim \Lambda_b^2,$$ otherwise $\phi$ is trapped in the first valley.
This marks a crucial difference with the well-studied case of parametric resonance due to a scalar field which oscillates coherently at the minimum of its potential~\cite{Dolgov:1989us, Traschen:1990sw, Kofman:1994rk, Shtanov:1994ce, Kofman:1997yn, Greene:1998pb, Hertzberg:2010yz, Amin:2010xe, Amin:2010dc, Antusch:2017flz, Olle:2019kbo}.
Our case of study is sketched in \Fig{fig:potential}. The field $\phi$ rolls over many wiggles until it gets trapped.
To get insights about the axion particle production, let us estimate the effect of friction.
In the limit of constant $\dot\phi$ and $H=0$, Eq.~(\ref{eq:fluctuation}) is simplified to the Mathieu equation \cite{mclachlan}:
\begin{align}\label{eq:Mathieu uk constant phidot}
\ddot u_k + \left( k^2 - \frac{\Lambda_b^4}{f^2} \cos \frac{\dot\phi}{f}t \right) u_k = 0.
\end{align}
Solutions to the Mathieu equation have instability and grow exponentially if the parameters are in some specific regions.%
\footnote{See \textit{e.g.}, figure 8 (A) of \cite{mclachlan} and section~IV of \cite{Kofman:1997yn}}
For $\dot\phi^2_0 > \Lambda_b^4$, the instability region presents a band structure, as it is shown in Fig.~\ref{fig:Mathieu instability}.
\begin{figure}[t]\label{fig:Mathieu instability}
\centering
\includegraphics[width=.6\textwidth]{figures/instabilityband}
\caption{In grey, the instability bands of \Eq{eq:Mathieu uk constant phidot}. Inside those bands, the solution grows exponentially.
For the calculation of the boundaries of the instability bands, see \textit{e.g.}, Ref.~\cite{kovacic2018mathieu}.
}
\end{figure}
In the limit $\dot\phi^2_0 \gg \Lambda_b^4$, the solution has an instability if the momentum $k$ is close to $n \dot\phi_0/2f$ with integer $n\geq 1$.
For $n\geq 2$, the speed of the growth is slow and the size of the instability band is small.
Hence, the instability band with $n=1$ gives the most important source of the friction to decelerate the axion rolling, which is given by
\begin{align}
\frac{\dot\phi^2}{4f^2} - \frac{\Lambda_b^4}{2f^2}
< k^2 <
\frac{\dot\phi^2}{4f^2} + \frac{\Lambda_b^4}{2f^2}. \label{eq:instability band}
\end{align}
Equivalently, one can write the instability band as $|k-k_{\rm cr}| < \delta k_{\rm cr}$ for $\dot\phi^2 \gg \Lambda_b^4$, where $k_{\rm cr}$ and $\delta k_{\rm cr}$ are defined as
\begin{align}\label{eq:k crit}
k_{\rm cr} = \frac{\dot\phi}{2f}, \quad
\delta k_{\rm cr} = \frac{\Lambda_b^4}{2f\dot\phi}\,,
\end{align}
where, initially, $\dot\phi=\dot\phi_0$. Inside the instability band, the asymptotic behavior of $u_k$ at large $t$ is given by
\begin{align}
u_k \propto \exp\left( \sqrt{ (\delta k_{\rm cr} )^2 - \left( k- k_{\rm cr} \right)^2 } t \right) \sin \left(k_{\rm cr} t + \arctan\sqrt{ \frac{\delta k_{\rm cr} + (k - k_{\rm cr} ) }{ \delta k_{\rm cr} - (k - k_{\rm cr} ) } } \right). \label{eq:asymptotic uk}
\end{align}
Let us now estimate the energy of the growing modes.
The number of modes which exponentially grow per unit volume is $\sim (k_{\rm cr}^0)^2\, \delta k_{\rm cr}^0$.
The energy density of the fluctuations is
\begin{equation}
\delta\rho = \frac{1}{2} \langle(\dot{\delta\phi})^2\rangle + \frac{1}{2} \langle(\vec\nabla \,\delta\phi)^2\rangle \,.
\end{equation}
As long as $\dot\phi$ is constant, this grows as
\begin{align} \label{eq:rho fluctuations}
\rho_{\rm fluc}(t) \sim (k_{\rm cr}^0)^3 \,\delta k_{\rm cr}^0 \exp(2 \, \delta k_{\rm cr}^0 \, t ).
\end{align}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\hsize]{figures/sketch.png}
\caption{Sketch of different time frames showing how the instability band moves due to particle production. For $t=0$, $|k-k_{\rm cr}| \lesssim \delta k_{\rm cr}$ is satisfied so there is an exponential growth of the fluctuations for the mode $k_{\textrm{cr}}$. Subsequently, the zero mode looses kinetic energy and the instability band starts moving towards smaller values of momentum. The exponential growth of mode with $k_{\textrm{cr}}$ stops when it moves beyond the instability band.
}\label{fig:sketch}
\end{figure}
The homogeneous mode gradually looses its kinetic energy because of back-reaction,
and the instability band moves towards the region of small values of $k$ (see \Fig{fig:sketch}).
The exponential growth of the modes with the wave number $k_{\rm cr}^0$ stops when this mode goes out from the instability band.
At that time, the critical wave number has changed by $\delta k_{\rm cr}^0$. Using the definition of $k_{\rm cr}$, the kinetic energy of the zero mode decreases by
\begin{equation}
\delta K \approx \frac{1}{2}4 f^2 [(k_{\rm cr}^0-\delta k_{\rm cr}^0)^2 - (k_{\rm cr}^0)^2] \approx - 4 f^2 k_{\rm cr}^0 \delta k_{\rm cr}^0 = - \dot\phi_0^2 \frac{\delta k_{\rm cr}^0}{k_{\rm cr}^0}\, ,
\end{equation}
with the new $\dot\phi$ given by
\begin{equation} \label{eq:phidot variation}
\dot\phi \sim \dot\phi_0 \left(1-\frac{\delta k_{\textrm{cr}}^0}{k_{\textrm{cr}}^0} \right) \,.
\end{equation}
The energy density of the fluctuations $\rho_{\rm fluc}$ is
\begin{align}\label{eq:rhofluc estimate}
\rho_{\rm fluc} \approx - \delta K \approx \dot\phi_0^2 \times \frac{\delta k_{\rm cr}^0}{k_{\rm cr}^0}\,.
\end{align}
The timescale that the mode with wave number $k_{\rm cr}^0$ spends inside the instability band can be estimated combining Eqs.~(\ref{eq:rhofluc estimate}) and~(\ref{eq:rho fluctuations}):
\begin{align} \label{eq:deltatime ampl}
\delta t_{\rm amp} = \frac{1}{2\,\delta k_{\rm cr}^0} \log \frac{\dot\phi_0^2}{(k_{\rm cr}^0)^4}.
\end{align}
The time evolution of kinetic energy $d\rho/dt$ is roughly given as $\sim \delta K / \delta t_{\rm amp} $:
\begin{align}\label{eq:phi double dot approx}
\frac{d}{dt} \left(\frac{1}{2}\dot\phi^2\right)
\sim -\dot\phi^2 \frac{2(\delta k_{\rm cr})^2}{k_{\rm cr}} \left( \log\frac{\dot\phi^2}{k_{\rm cr}^4} \right)^{-1}
\sim -\frac{\Lambda_b^8}{f \dot\phi} \left( \log\frac{16f^4}{\dot\phi^2} \right)^{-1} \,,
\end{align}
where we have dropped the subscript $0$ because this equation of motion is now valid at any velocity. Eq.~(\ref{eq:phi double dot approx}) can be integrated exactly from $\dot\phi_0$ to $0$, giving the time $\Delta t_{\rm frag}$ and the field excursion $\Delta\phi_{\rm frag}$ from the beginning of particle production until the field stops:
\begin{align}
\Delta t_{\rm frag} &\sim \frac{f \dot\phi_0^3}{\Lambda_b^8} \log\frac{f^4}{\dot\phi_0^2}, \label{eq:Delta t frag approx} \\
\Delta \phi_{\rm frag} &\sim \frac{f \dot\phi_0^4}{\Lambda_b^8} \log\frac{f^4}{\dot\phi_0^2} . \label{eq:Delta phi frag approx}
\end{align}
Here $\dot\phi_0$ is the velocity of $\phi$ at the beginning of the particle production.
In the equations above we neglected $\mathcal{O}(1)$ factors as the calculation above was only approximate. In the next section, we will derive these expressions using a more precise treatment. The correct numerical factors are the ones of Eqs.~(\ref{eq:dtppapprox}) and~(\ref{eq:dphippapprox}) below, which reproduce the parametric dependence of Eqs.~(\ref{eq:Delta t frag approx}) and ~(\ref{eq:Delta phi frag approx}).
\section{Analytical discussion} \label{sec:analytical discussion}
In what follows we discuss in detail the axion fragmentation dynamics introduced in the previous section.
We establish the conditions to decelerate an axion field uniquely due to particle production friction from the axion field itself.
The approximate analytical formulae derived here will be compared with the numerical solutions in the next section.
In the intuitive discussion in \Sec{sec:intuitive derivation} we considered the limit in which the Hubble expansion is negligible, \textit{i.e.}, $H=0$.
Before deriving the conditions to make the field decelerate, let us consider the effect of the Hubble friction in the equation of motion for the fluctuation in Eq.~(\ref{eq:fluctuation}).
We can anticipate two additional effects once we consider the cosmic expansion. Most importantly, the growth of the modes is suppressed by the friction term $3H\dot u_k$ in Eq.~(\ref{eq:fluctuation}).
In addition, since the instability band moves towards smaller values of momentum when the zero-mode decelerates, Hubble expansion makes a given mode to spend more time inside the instability band due to the red-shift of the physical momentum $k/a(t)$.
Let us assume for the moment $H$ to be constant.
By defining $\tilde u_k \equiv e^{3Ht/2} u_k$, Eq.~(\ref{eq:fluctuation}) can be rewritten as
\begin{align} \label{eq:eomfluctuations}
{\ddot{\tilde u}}_k + \left( e^{-2Ht} k^2 + V''(\phi) + \frac{9}{4}H^2 \right) \tilde u_k = 0.
\end{align}
For $H=0$, the last equation simply reproduces~\Eq{eq:Mathieu uk constant phidot}.
According to Eq.~(\ref{eq:asymptotic uk}), the exponential growth of $ u_k$ is at most $\exp( \Lambda_b^4 t/2f\dot\phi )$.
Thus, in order for $\tilde u_k$ to grow, the Hubble expansion rate should be bounded by
\begin{align}
H < \frac{\Lambda_b^4}{3f\dot\phi}~. \label{eq:condition 0}
\end{align}
Equivalently, the last equation can be rewritten as
\begin{align}
\dot\phi < \frac{\Lambda_b^4}{\mu^3 f} \dot\phi_{\mathrm{SR}}, \label{eq:condition 0'}
\end{align}
where
\begin{equation}
\dot\phi_{\mathrm{SR}} \equiv \frac{\mu^3}{3H}
\end{equation}
is the slow-roll velocity of the field in the linear potential $-\mu^3 \phi$ for a constant Hubble rate $H$. With a slight abuse of notation, in the following we will use this definition also in the case in which $H$ is not constant or the potential is not linear, and this quantity will turn out to be useful, even without representing a proper slow-roll velocity.
In addition, to have particle production active, the field should go over the barriers, $H < \dot\phi / 2\pi f$.%
\footnote{
This condition guarantees that the time needed to go over one wiggle is shorter than one Hubble time.
If this were not the case, the effect of Hubble friction would be dominant with respect to fragmentation.
In particular, if the wiggles are large enough, instead of rolling over many of them the field would stopped as soon as $V'=0$, just due to cosmic expansion. For a more detailed discussion about this point, we refer the reader to Ref.~\cite{Fonseca:2019lmc}.}
However, this condition is trivially satisfied when both of Eq.~(\ref{eq:condition 0}) and $\dot\phi > \Lambda_b^2$ are satisfied.
Hence, we assume
\begin{equation}\label{eq:assumptions}
\dot\phi_0^2 > \Lambda_b^4 \qquad \text{and} \qquad H \ll \frac{\Lambda_b^4 }{3 f\dot\phi_0 } \,.
\end{equation}
The first of this assumption is valid until the field keeps rolling. As $\dot\phi\approx\Lambda_b^2$, the field stops.
Contrary to the former, the second becomes easier to satisfy as the velocity decreases, therefore it is enough to assume that it is valid for the initial conditions.
The assumptions above allow us to simplify the analysis due to the following three reasons.
First, we can regard $\dot\phi$ as a smooth function of the time $t$.
The numerical solution of $\dot\phi$ has a smooth component and a rapidly oscillating component with frequency $\dot\phi / 2\pi f$.
This oscillating component is caused by the wiggles and its relative size compared to the smooth component is $\sim \Lambda_b^4/\dot\phi^2$. We then neglect this oscillating term in this section.
Second, we can assume that $\ddot\phi$ is constant during the amplification time
$\delta t_{\rm amp}$, which we defined before as the time it takes for the mode $k_\mathrm{cr}$ to exit the instability band. This can be calculated, using \Eq{eq:phidot variation}, as
\begin{equation}\label{eq:delta time ampl 2}
\delta t_{\rm amp} \sim \frac{\dot\phi(\delta t_{\rm amp}) - \dot\phi}{\ddot\phi} \sim \frac{\Lambda_b^4}{\dot\phi|\ddot\phi|}.
\end{equation}
Using the result in (\ref{eq:delta time ampl 2}), we impose that $|\dddot\phi| \delta t_{\rm amp} \ll |\ddot\phi|$, which we will justify later in \Eq{eq:phi double dot 2}. This condition can be rewritten as
\begin{align}
\left| \frac{\Lambda_b^4}{\dot\phi} \frac{d \log |\ddot\phi|}{d\dot\phi} \right| \ll 1\,, \label{eq:constant ddot phi}
\end{align}
which shows that if $\ddot\phi$ depends on $\dot\phi$ polynomially, this condition is satisfied if $\Lambda_b^4 \ll \dot\phi^2$.
As a third simplification, we can drop the friction term $3H \dot{u}_k$ in Eq.~(\ref{eq:fluctuation}) without changing the physical momentum by ${\cal O}(1)$ fraction during the amplification.
\medskip
Now that we have presented our simplifying assumptions, let us go back to discuss the equation of motion.
For a given velocity $\dot\phi$, once cosmic expansion is taken into account, the critical mode $k_\mathrm{cr}$ and the width $\delta k_\mathrm{cr}$ are obtained dividing the left hand side of Eq.~(\ref{eq:k crit}) by the scale factor $a$:
\begin{align}\label{eq:k crit expansion}
\frac{k_{\rm cr}}{a} = \frac{\dot\phi}{2f}, \quad
\frac{\delta k_{\rm cr}}{a} = \frac{\Lambda_b^4}{2f\dot\phi}\,.
\end{align}
The condition $H<\dot\phi/(2\pi f)$, which was discussed above Eq.~(\ref{eq:assumptions}), ensures that $k_\mathrm{cr}> a \,H$. In other words, the amplification process takes place and ends when the modes are well inside the horizon. This marks an important difference with the case in which quantum fluctuations grow until they exit the horizon, as it happens, for example, with the gauge fields generated at the end of axion inflation~\cite{Pajer:2013fsa}.
For our discussion, we do not need to specify when the axion fragmentation dynamics takes place. The latter can be embedded in the cosmological history of the universe at different epochs. The usual dynamics of the modes crossing the horizon during inflation and then re-entering after the Big Bang does not affect our discussion.
By taking the initial condition Eq.~(\ref{eq:initial}) and assuming constant $\ddot\phi$,
the asymptotic behaviour of $u_k$ with $k/a = k_\mathrm{cr}/a = \dot\phi/2f$ after the amplification is
\begin{align}
u_{k_\mathrm{cr}}(t) \simeq \frac{1}{a} \sqrt{\frac{2}{k_\mathrm{cr}}} \exp\left( \frac{\pi \Lambda_b^8}{4f\dot\phi^2\,|\ddot\phi + H\dot\phi|} \right) \sin\left( \frac{1}{a} k_\mathrm{cr} t + \delta \right)\,. \label{eq:asymptotic uk 2}
\end{align}
Equation~(\ref{eq:asymptotic uk 2}) is derived in Appendix \ref{sec:technical details} in the following way. The equation of motion Eq.~(\ref{eq:eomfluctuations}) is solved by means of a WKB approximation in three separate time intervals, before the mode $k_\mathrm{cr}$ enters the instability band, when the mode is well inside the instability band and after it has left it. In the two transition regions, when the mode enters and exit the instability band, the solution is found in terms of Airy functions. Finally, the five intervals are matched by imposing the continuity of the solution.
From Eq.~(\ref{eq:asymptotic uk 2}) we see that for exponential particle production occurs, one needs:
\begin{align}
\frac{\pi \Lambda_b^8 }{ 2f\dot\phi^2\,|\ddot\phi + H\dot\phi|} > 1 \,, \label{eq:condition exponent}
\end{align}
where the factor of $2$ difference compared to Eq.~(\ref{eq:asymptotic uk 2}) depends on the fact that the energy density scales with $|u_k|^2$.
We discuss the validity of this assumption later, around Eq.~(\ref{eq:exponential pp condition phidot}).
Using \Eq{eq:phi mode expansion} and \Eq{eq:asymptotic uk 2},
we can estimate the energy density per volume in comoving momentum space right after the end of the amplification as
\begin{align}
\frac{d^3 \rho_{\textrm{fluc}}}{d k^3} \biggr|_{k=k_\mathrm{cr}} =
\frac{1}{(2\pi)^3} \frac{1}{2 a^2} k_\mathrm{cr}^2 \left| u_{k_\mathrm{cr}} \right|^2 \approx
\frac{1}{(2\pi)^3} \frac{k_\mathrm{cr}}{a^4} \exp\left( \frac{\pi \Lambda_b^8}{2f\dot\phi^2\,|\ddot\phi + H\dot\phi|} \right).
\label{eq:d3rhodk3}
\end{align}
Using the initial condition for $u_k$ in \Eq{eq:initial}, the energy density before the amplification is $d^3\rho_{\textrm{fluc}}/dk^3 = k/[2 a^4 (2\pi)^3]$. Then,
the energy of the fluctuation is amplified by a factor of
\begin{equation} \label{eq:amplification factor}
2\exp\left( \frac{\pi \Lambda_b^8} { 2 f\dot\phi^2 |\ddot\phi + H\dot\phi| }\right).
\end{equation}
When this factor is much larger than $1$, particle production is efficient to provide the friction for the homogeneous mode.
The increment of the energy density of the fluctuations because of the particle production is
\begin{align}
\frac{d\rho_{\textrm{fluc}}}{dt} = \left|\frac{dk_{cr}}{dt}\right| \times 4\pi k_\mathrm{cr}^2 \, \frac{d^3 \rho_{\textrm{fluc}}}{dk^3} \biggr|_{k = k_\mathrm{cr}},
\end{align}
where $dk_{\rm cr}/dt$ is the velocity of the instability band in the momentum space.
For non-zero $H$, $k_{\mathrm{cr}}$ is determined as
$$k_{\mathrm{cr}}/a = \dot\phi / (2f).$$
Therefore, the velocity of the instability band is given as $dk_{\mathrm{cr}}/dt = a(\ddot\phi + H \dot\phi)/(2f)$.
Thus,
\begin{align}
\frac{d\rho_{\textrm{fluc}}}{dt}
&= a \left| \frac{\ddot\phi + H\dot\phi}{2f} \right| \times 4\pi k_\mathrm{cr}^2 \times \frac{1}{(2\pi)^3} \frac{k_\mathrm{cr}}{a^4} \exp\left( \frac{\pi \Lambda_b^8}{2f\dot\phi^2|\ddot\phi + H\dot\phi|} \right) \nonumber\\
&= \frac{1}{32\pi^2 f^4} \dot\phi^3 |\ddot\phi + H\dot\phi| \exp\left( \frac{\pi \Lambda_b^8}{2f\dot\phi^2|\ddot\phi + H\dot\phi|} \right).
\label{eq:fluctuation production}
\end{align}
The kinetic energy of the homogeneous mode is $\dot\phi^2/2$ and its time derivative is $\dot\phi\ddot\phi$.
As a result, from conservation of energy, we obtain the following equation:
\begin{align}
\boxed{
\dot\phi\ddot\phi
=
- 3 H \dot\phi^2
+ \mu^3 \dot\phi
- \frac{1}{32\pi^2 f^4} \dot\phi^3 |\ddot\phi + H\dot\phi| \exp\left( \frac{\pi \Lambda_b^8}{2f\dot\phi^2|\ddot\phi + H\dot\phi|} \right).}
\label{eq:eq for phi double dot}
\end{align}
Compared to Eqs.~(\ref{eq:EOM1}) and~(\ref{eq:zeromode}), Eq.(\ref{eq:eq for phi double dot}) describes the effect of fragmentation after averaging over many oscillations of the sinusoidal potential, and is more suitable for our analysis.
Using this equation we will determine the general conditions to stop the field due to axion fragmentation, which are obtained in the next sections.
The first and second term in the right-handed side of \Eq{eq:eq for phi double dot} are the effect of Hubble friction and acceleration by the slope, respectively.
This equation can be regarded as a consistency condition for $\ddot\phi$ during the fragmentation phase.
By solving the above equation, $\ddot\phi$ can be calculated as a function of $\dot\phi$, $\Lambda_b$, $f$, $\mu^3$, and $H$.
\subsection{General condition to stop the axion}
Let us discuss conditions to stop the axion field.
Here we summarize the results, while the details of the derivation are given in Appendix \ref{sec:detail on phiddot}.
$\ddot\phi<0$ must hold from the initial time until the field has come to a complete stop. As detailed in Appendix~\ref{sec:stopping condition}, this is realized if (and only if) the following condition holds for the initial velocity:
\begin{empheq}[box=\widefbox]{align}
\mu^3 <
2H\dot\phi_0 + \displaystyle\frac{\pi\Lambda_b^8}{2f\dot\phi_0^2} \left( W_0\left( \displaystyle\frac{32\pi^2f^4}{e\dot\phi_0^2} \right) \right)^{-1}\,.
\label{eq:condition for no positive phiddot}
\end{empheq}
Here $W_n(x)$ is the product logarithm function (also known as Lambert $W$ function),%
\footnote{
The product logarithm function is the inverse function of $W e^W = x$.
In general, there exist infinite number of solutions for this equation, and $W_0$ and $W_{-1}$ are the two real ones.
In particular, $W_0(x)$ is real for $-e^{-1} \leq x$ and $W_{-1}$ is real for $-e^{-1} \leq x < 0$.
Also, $W_0(x) = \log x - \log\log x + \cdots$ for large $x$.
A plot of $W_0$ and $W_{-1}$ for small values of $x$ is shown in Fig.~\ref{fig:Lambert}.
This function is available \texttt{ProductLog} in \texttt{Mathematica} or \texttt{special.lambertw} in \texttt{SciPy}.
See \textit{e.g.}, \url{http://mathworld.wolfram.com/LambertW-Function.html}.
}
whose real branches are plotted in Fig.~\ref{fig:Lambert}.
\begin{figure}
\centering
\includegraphics[width=.55\textwidth]{figures/plotLambert}
\caption{\label{fig:Lambert}The two real branches of the product logarithm function.}
\end{figure}
\Eq{eq:condition for no positive phiddot} expresses an equilibrium between the slope and the Hubble expansion which allows for efficient fragmentation: if the slope increases, in order to avoid the field acceleration, the friction due to cosmic expansion should compensate this effect. Alternatively, for $\dot\phi_0 < \mu^3/(2 H) = 3/2 \dot\phi_\mathrm{SR}$, one can see \Eq{eq:condition for no positive phiddot} as a lower bound on $\Lambda_b$ that expresses, for given $\mu^3$ and $H$, the necessary amount of fragmentation needed in order to slow down the field.
If Eq.~(\ref{eq:condition for no positive phiddot}) is satisfied, Eq.~(\ref{eq:eq for phi double dot}) has only one solution, with negative $\ddot\phi$, which is given by%
\footnote{Before fragmentation is active, the field is only subject to its potential and to Hubble friction, and its equation of motion is simply $\ddot\phi = \mu^3 - 3 H \dot\phi$, where we neglected a small oscillating term.
This cannot be obtained as the $\Lambda_b\to0$ limit of Eq.~(\ref{eq:solution phiddot}). The reason is that the equation of motion Eq.~(\ref{eq:eq for phi double dot}) was derived assuming fragmentation is active. In particular, we assumed $\ddot\phi\ll\Lambda_b^4/f$. The acceleration is initially $\ddot\phi = \mu^3 - 3 H \dot\phi$, and asymptotes to Eq.~(\ref{eq:solution phiddot}) as fragmentation starts. }
\begin{align}
\ddot\phi &=
\begin{cases}
- H\dot\phi + \displaystyle\frac{\pi \Lambda_b^8}{2f\dot\phi^2}\left[ b + W_{-1}(-ab e^{-b} ) \right]^{-1} & (b>0) \\
- H\dot\phi + \displaystyle\frac{\pi \Lambda_b^8}{2f\dot\phi^2}\left[ b + W_0(-ab e^{-b} ) \right]^{-1} & (b<0) \\
\end{cases}
.
\label{eq:solution phiddot}
\end{align}
Here $a$ and $b$ are dimensionless parameters which are defined as
\begin{align}
a \equiv \frac{\dot\phi^2}{32\pi^2 f^4},\quad
b \equiv \frac{\pi\Lambda_b^8}{2f\dot\phi^2(\mu^3-2H\dot\phi)}.
\label{eq:a b definition}
\end{align}
Let us discuss the validity of the assumption Eq.~(\ref{eq:condition exponent}).
The effect of the axion fragmentaion in Eq.~(\ref{eq:eq for phi double dot}) has
a exponential factor with a exponent $N \equiv \pi \Lambda_b^8 / 2f\dot\phi^2 |\ddot\phi + H \dot\phi|$.
As we have discussed, for exponential particle production to occur, $N$ should be larger than 1.
By using Eq.~(\ref{eq:solution phiddot}), we obtain
\begin{align}
N =
\begin{cases}
|b + W_{-1}(-abe^{-b})| & (b>0) \\
|b + W_{ 0}(-abe^{-b})| & (b<0)
\end{cases}. \label{eq:exponent}
\end{align}
As we can see in Fig.~\ref{fig:exponent}, for fixed $a$, $N$ is an monotonously increasing function of $1/b$.
For $a\ll 1$, as we will always assume, $N$ becomes small at $1/b\lesssim 0$, and is well approximated as $N \simeq -b$.
Thus, by requiring $N > 1$, we obtain
\begin{align}
\frac{2f\dot\phi^2 (\mu^3 - 2H\dot\phi)}{\pi\Lambda_b^8} > -1.
\label{eq:exponential pp condition phidot}
\end{align}
If $\dot\phi < (3/2)\dot\phi_{\rm SR}$, the LHS of Eq.~(\ref{eq:exponential pp condition phidot}) is positive and the inequality is satisfied.
Thus, the above condition can be rewritten as
\begin{align}
\dot\phi < \frac{3}{2}\dot\phi_{\rm SR}
\qquad{\rm or}\qquad
\mu^3 > 2H\dot\phi - \frac{\pi \Lambda_b^8}{2f\dot\phi^2}.
\label{eq:exponential pp condition phidot 2}
\end{align}
As long as Eq.~(\ref{eq:exponential pp condition phidot}) (or equivalently Eq.~(\ref{eq:exponential pp condition phidot 2})) is satisfied,
we can safely use $\ddot\phi$ given in Eq.~(\ref{eq:solution phiddot}).
Note that $2H\dot\phi - \pi \Lambda_b^8/(2f\dot\phi^2)$ is a monotonously increasing function of $\dot\phi$.
Thus, we can see that Eq.~(\ref{eq:exponential pp condition phidot 2}) is satisfied for any $\dot\phi < \dot\phi_0$ if (and only if)
\begin{align}
\dot\phi_0 < \frac{3}{2}\dot\phi_{\rm SR}
\qquad{\rm or}\qquad
\mu^3 > 2H\dot\phi_0 - \frac{\pi \Lambda_b^8}{2f\dot\phi_0^2}.
\label{eq:exponential pp condition}
\end{align}
If this condition is satisfied, Eq.~(\ref{eq:solution phiddot}) for the acceleration $\ddot\phi$ can be used to describe the fragmentation process from its begininng until the end of that.
In phenomenologically interesting applications $\dot\phi \leq \dot\phi_{\rm SR}$, and Eq.~(\ref{eq:exponential pp condition}) is always satisfied.
\begin{figure}[ht]
\centering
\includegraphics[width=0.69\hsize]{figures/exponent2.pdf}
\caption{The exponent $N$ given in Eq.~(\ref{eq:exponent}) as a function of $1/b$.}\label{fig:exponent}
\end{figure}
\medskip
To summarize, in order for fragmentation to stop the rolling of the axion field we need to simultaneously impose the conditions of \Eq{eq:condition 0'} and \Eq{eq:condition for no positive phiddot} to stop the axion. We are interested in the case in which the potential has local minima (\textit{i.e.} $\Lambda_b^4/f > \mu^3$), and we will assume $\dot\phi_0 \leq \dot\phi_{\mathrm{SR}}$. In this case, \Eq{eq:condition 0'} is trivially satisfied, and the only condition that must hold is \Eq{eq:condition for no positive phiddot}.
Figure~\ref{fig:conditions and parameter space 1} and Fig.~\ref{fig:conditions and parameter space 2} show
examples of the parameter space in which Eq.~(\ref{eq:condition for no positive phiddot}) is satisfied.
In the case of $\dot\phi_0 \lesssim \dot\phi_{\mathrm{SR}}$, there is an upperbound on $\mu^3$ because the acceleration effect by the slope should be weaker than the particle production effect.
On the other hand, the bound on $\mu^3$ is very weak for $\dot\phi \gtrsim \dot\phi_{\mathrm{SR}}$
and it becomes trivial for $\dot\phi_0 \geq (3/2)\dot\phi_{\mathrm{SR}}$.
Indeed, for $\dot\phi_0 \geq (3/2)\dot\phi_{\mathrm{SR}}$, Eq.~(\ref{eq:condition for no positive phiddot}) is always satisfied, and the field is slowed down by Hubble friction without the need of fragmentation. In this region, the only bound comes from imposing that the exponential amplification of the fluctuations is active, as we do in Eq.~(\ref{eq:exponential pp condition}).
It is interesting to investigate whether Eq.~(\ref{eq:eq for phi double dot}) admits constant velocity solutions. If such solutions exist the field can reach a steady state, and fragmentation can not stop the evolution.
Equation~(\ref{eq:eq for phi double dot}) with $\ddot\phi = 0$ has no solution for $\dot\phi$ if (and only if):
\begin{align}
H < H_{\rm cr} \simeq \frac{\pi \Lambda_b^8}{2f\dot\phi_{\rm SR}^3} \left( W_0\left( \frac{32\pi^2f^4}{e\dot\phi_{\rm SR}^2} \right) \right)^{-1}
\label{eq:Hcr}
\end{align}
is satisfied. This means that the axion zero mode cannot roll with constant velocity in such cases. For details, see the Appendix \ref{sec:modified slow roll velocity}. Note that \Eq{eq:Hcr} represents an upper bound on $H$ for fixed $\dot\phi_\mathrm{SR}$, but for fixed $\mu^3$ it can be rewritten as a lower bound on $H$.
On the other hand, for $H > H_{\rm cr}$, Eq.~(\ref{eq:eq for phi double dot}) with $\ddot\phi=0$ admits solutions. To distinguish it from the slow roll velocity $\dot\phi_{\rm SR} = \mu^3/3H$, we denote this velocity as the modified slow roll velocity $\dot\phi_{\rm SR(frag)}$. We show this velocity in red in Fig.~\ref{fig:conditions and parameter space 1} and Fig.~\ref{fig:conditions and parameter space 2}. As long as $\dot\phi_{\rm SR} \ll f^2$, the modification to the slow roll velocity is small, \textit{i.e.}, $\dot\phi_{\rm SR(frag)} \simeq \dot\phi_{\rm SR}$.
\begin{figure}[t]
\centering
\includegraphics[width=0.69\hsize]{figures/solutions.pdf}
\caption{
The parameter space which is excluded by Eq.~(\ref{eq:condition for no positive phiddot}).
The ratio between the decay constant and the slow roll velocity is fixed to $f^2/\dot\phi_{\textrm{SR}} = 10^4$.
The coefficient of the slope is fixed to $\mu^3 = 3 H \dot\phi_{\textrm{SR}}$.
Eq.~(\ref{eq:condition for no positive phiddot}) is \textit{not} satisfied in the gray regions.
The red solid line shows the modified slow roll velocity $\dot\phi_{\rm SR(frag)}$ given in Eq.~(\ref{eq:modifield slow roll velocity}).
$H_{\rm cr}$ is defined in Eq.~(\ref{eq:Hcr}).
The magenta line shows a condition given in Eq.~(\ref{eq:exponential pp condition}).
The axion can be successfully stopped in the white region.
}\label{fig:conditions and parameter space 1}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.69\hsize]{figures/solutions2.pdf}
\caption{
Same as Fig.~\ref{fig:conditions and parameter space 1}.
The decay constant and Hubble expansion rate are fixed to $f/\Lambda_b = 10^3$, $H/\Lambda_b = 10^{-7}$.
}\label{fig:conditions and parameter space 2}
\end{figure}
\subsection{Stopping conditions in several limits}
In the previous section, we described the generic negative solution of $\ddot\phi$ and the stopping conditions.
Here we discuss several cases in which the conditions can be simplified by taking some limits of the parameters.
\subsubsection{$H=0$ and $\mu^3=0$}
First, let us discuss the limit of $H=0$ and $\mu^3=0$.
In this limit, Eqs.~(\ref{eq:condition 0}) and (\ref{eq:condition for no positive phiddot}) are satisfied automatically.
The expression for $\ddot\phi$ given in Eq.~(\ref{eq:solution phiddot}) is simplified to%
\footnote{
Eq.~(\ref{eq:phi double dot 2}) can be obtained by separately taking the limits $\mu^3-2H\dot\phi \to 0^+$ and $\mu^3-2H\dot\phi \to 0^-$ of the two branches of Eq.~(\ref{eq:solution phiddot}), respectively, and using the expansions $W_0(x)\approx \log x -\log\log x$ for $x\to+\infty$ and $W_1(x)\approx \log(-x) - \log(-\log(-x))$ for $x\to 0^-$, from which we get
\[
\lim_{b\to+\infty} b + W_{-1}(-abe^{-b}) =
\lim_{b\to-\infty} b + W_{0}(-abe^{-b}) = \log a
\]
}
\begin{align}
\ddot\phi = -\frac{\pi\Lambda_b^8}{2\dot\phi^2 f}\left( \log\frac{32\pi^2 f^4}{\dot\phi^2} \right)^{-1}. \label{eq:phi double dot 2}
\end{align}
This can be regarded as a refinement of Eq.~(\ref{eq:phi double dot approx}).
In this case, the amplification factor of the fluctuation energy in \Eq{eq:amplification factor} is given by
\begin{align}\label{eq:amplification mu=0 H=0}
2\exp\left( \frac{\pi \Lambda_b^8}{2 f\dot\phi^2 |\ddot\phi|} \right) = \frac{64\pi^2 f^4}{\dot\phi^2}.
\end{align}
In the derivation of Eq.~(\ref{eq:asymptotic uk 2}) we assumed that $\dot\phi\ll f^2$, which is necessary for the validity of the low-energy EFT of the axion field, thus the amplification factor is much larger than $1$, enhancing the efficiency of fragmentation.
The time scale and the field excursion during the axion particle production are
\begin{align}
\Delta t_{\rm frag} &\equiv \int^0_{\dot\phi_0}\frac{d\dot\phi}{\ddot\phi}
\simeq \frac{2f \dot\phi_0^3}{3\pi \Lambda_b^8} \log\frac{32\pi^2f^4}{\dot\phi_0^2},
\label{eq:dtppapprox}\\
\Delta \phi_{\rm frag} &\equiv \int^0_{\dot\phi_0}\frac{\dot\phi\, d\dot\phi}{\ddot\phi}
\simeq \frac{f \dot\phi_0^4}{2\pi \Lambda_b^8} \log\frac{32\pi^2f^4}{\dot\phi_0^2} ,
\label{eq:dphippapprox}
\end{align}
where we dropped subleading terms.
The number of wiggles which the axion travels until it stops is
\begin{align}
\frac{\Delta\phi_{\rm frag}}{2\pi f}
=
\frac{\dot\phi_0^4}{4\pi^2 \Lambda_b^8} \log\frac{32\pi^2f^4}{\dot\phi_0^2}.
\end{align}
For example, for $\dot\phi_0 / \Lambda_b^2 = 10^2$ and $\log \,(32\pi^2 f^4/\dot\phi_0^2) = 10$, this number is $\sim 3 \times 10^7$.
\subsubsection{$H\simeq 0$ and $\mu^3 \neq 0$}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\hsize]{figures/phiddot_ratio.pdf}
\caption{
The ratio between $\ddot\phi$ with non-zero slope $\mu^3$ (Eq.~(\ref{eq:phi double dot with H=0}))
and $\ddot\phi$ with $\mu^3=0$ (Eq.~(\ref{eq:phi double dot 2})).
We take $H=0$ in this figure.
}\label{eq:phiddot ratio}
\end{figure}
Next, let us discuss the case in which $\mu^3\neq0$ and $H$ is small enough to be neglected.
For $\dot\phi_0 \ll \dot\phi_\mathrm{SR}$, Eq.~(\ref{eq:condition for no positive phiddot}) can be simplified as
\begin{align}
\mu^3 < \mu_\mathrm{max}^3\equiv \frac{\pi\Lambda_b^8}{2f\dot\phi_0^2} \left( W_0\left( \frac{32\pi^2 f^4}{e\dot\phi^2_0} \right) \right)^{-1}. \label{eq:slope bound (H=0)}
\end{align}
This corresponds to the left part of Figs.~\ref{fig:conditions and parameter space 1} and \ref{fig:conditions and parameter space 2}, which show how $H$ is irrelevant if $\dot\phi_0 \ll \dot\phi_{\textrm{SR(frag)}} \approx \dot\phi_{\textrm{SR}}$.
In the limit of $H=0$, the negative $\ddot\phi$ solution Eq.~(\ref{eq:solution phiddot}) is simplified as
\begin{align}
\ddot\phi \simeq \frac{\pi \Lambda_b^8}{2f\dot\phi^2} [ b + W_{-1}(-abe^{-b}) ]^{-1}\,, \label{eq:phi double dot with H=0}
\end{align}
where $a,b$ are defined in \Eq{eq:a b definition}. In the limit $\mu^3 =0$ we recover Eq.~(\ref{eq:phi double dot 2}).
The acceleration $|\ddot\phi|$ monotonically decreases as a function of $\mu^3$.
In Fig.~\ref{eq:phiddot ratio}, we show the relative variation of $\ddot\phi$ when the slope $\mu^3$ goes from $0$ to $\mu_\mathrm{max}$ defined in Eq.~(\ref{eq:slope bound (H=0)}).
As long as $\dot\phi < f$, the decrements of $\ddot\phi$ is at most $\sim 15$ \%.
Hence, the estimates for the fragmentation time and the total field excursion of Eqs.~(\ref{eq:dtppapprox}, \ref{eq:dphippapprox}) still provide a reliable approximation.
\subsubsection{$\dot\phi = \dot\phi_{\textrm{SR}}$}
Let us discuss the case in which the initial velocity is equal to the slow roll velocity, \textit{i.e.}, $\dot\phi_0 = \dot\phi_{\textrm{SR}}$. This is the case if the dynamics takes place during inflation, since the velocity is exponentially driven to the attractor slow-roll velocity irrespectively of the initial conditions.
In this case, the condition Eq.~(\ref{eq:condition for no positive phiddot}) can be simplified to the following equation:
\begin{align}
H < \frac{\pi \Lambda_b^8}{2f\dot\phi_{\textrm{SR}}^3} \left( W_0\left( \frac{32\pi^2 f^4}{e \dot\phi_{\textrm{SR}}^2} \right) \right)^{-1}.
\label{eq:condition 3}
\end{align}
By replacing $\dot\phi_{\textrm{SR}} = \mu^3/(3 H)$, the above condition Eq.~(\ref{eq:condition 3}) can be equivalently rewritten as a condition for $\mu^3$ and $H$ for given $\Lambda_b$ and $f$:
\begin{align}
\mu^3 < \mu_{th}^3 \equiv \frac{3e \Lambda_b^8}{64\pi f^5}
\quad
{\rm or}
\quad
H > H_{min} \equiv \sqrt{ \frac{2}{27\pi} } \frac{f^{1/2} \mu^{9/2}}{\Lambda_b^4} \sqrt{ \log\frac{64\pi f^5 \mu^3}{3 e\Lambda_b^8} }.
\label{eq:Hmin mu3th}
\end{align}
This condition tells us that
the Hubble friction is required to prevent acceleration in order to work the fragmentation mechanism
if the slope is steeper than the threshold value $\mu^3_{th}$.
Alternatively, one can think of Eq.~(\ref{eq:condition 3}) as a lower bound on $\Lambda_b$ to have enough fragmentation to stop the field, for given $\mu^3$ and $H$.
The negative $\ddot\phi$ solution Eq.~(\ref{eq:solution phiddot}) for $\dot\phi = \dot\phi_\mathrm{SR}$ is given as
\begin{align}\label{eq:phi double dot SR}
\ddot\phi = \frac{\pi \Lambda_b^8}{2f\dot\phi_{\textrm{SR}}^2} \left( -\frac{1}{\tilde b} + \frac{1}{\tilde b + W_{-1}(- a\tilde b e^{-\tilde b} ) }\right),\quad
\tilde b \equiv \frac{\pi\Lambda_b^8}{2f H \dot\phi_{\textrm{SR}}^3}.
\end{align}
In Fig.~\ref{eq:phiddot ratio slowroll}, we show the ratio between
$\ddot\phi$ evaluated with Eq.~(\ref{eq:phi double dot SR}) and $\ddot\phi$ computed in Eq.~(\ref{eq:phi double dot 2}) with $H,\mu^3=0$, with the same velocity $\dot\phi = \dot\phi_{\rm SR}$.
Again, as long as Eq.~(\ref{eq:Hmin mu3th}) (or equivalently Eq.~(\ref{eq:condition 3})) is satisfied, the acceleration is well described by the $H=0$ equation~(\ref{eq:phi double dot 2}), thus the estimates for the fragmentation time and the total field excursion of Eqs.~(\ref{eq:dtppapprox}, \ref{eq:dphippapprox}) are reliable also in this case.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\hsize]{figures/phiddot_ratio_slowroll_new.pdf}
\caption{
The ratio between $\ddot\phi$ by Eq.~(\ref{eq:phi double dot SR})
and $\ddot\phi$ by Eq.~(\ref{eq:phi double dot 2}) with $\dot\phi = \mu^3/3H$.
$H_{min}$ and $\mu^3_{th}$ are defined in Eq.~(\ref{eq:Hmin mu3th}).
This figure is independent on the choice of $f$.
}
\label{eq:phiddot ratio slowroll}
\end{figure}
\section{Numerical analysis of the equations of motion} \label{sec:numeircal analysis}
In this section, we test the analytical understanding developed in Sec.~\ref{sec:analytical discussion} against a numerical solution of the equations of motion for the homogeneous mode $\phi(t)$, Eq.~(\ref{eq:zeromode}), and for the fluctuations $\delta \phi(x,t)$, Eq.~(\ref{eq:fluctuation}). We still limit ourselves to a linear level analysis, the validity of which will be further discussed in Sec.~\ref{sec:beyond linear}.
For this calculation, we discretize the integral in Eq.~(\ref{eq:zeromode})
and take 10000 modes whose momentum is between $10^{-4} k_{\rm cr}^0 $ and $k_{\rm cr}^0 + 10 \delta k_{\rm cr}^0$.
The momenta are evenly spaced in logarithmic scale.
The differential equations are solved numerically by the fourth order Runge-Kutta method.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\hsize]{figures/timeevol_1000_10_0_0_energy.pdf}
\caption{
Time evolution of energy of zero-mode and fluctuations.
We take $f = 1000 \Lambda_b$, $\dot\phi_0 = 10 \Lambda_b^2$, $\mu^3=0$, and $H=0$.
}\label{fig:time evolution of energy}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\hsize]{figures/timeevol_1000_10_0_0.pdf}
\caption{
Time evolution of $\dot\phi$ with $f = 1000 \Lambda_b$, $\dot\phi = 10 \Lambda_b^2$, $\mu^3=0$, and $H=0$.
The blue line is obtained by solving Eqs.~(\ref{eq:zeromode}) and (\ref{eq:fluctuation}),
and the orange line is the solution of Eq.~(\ref{eq:phi double dot 2}).
We take the same parameters as Fig.~\ref{fig:time evolution of energy}.
}\label{fig:time evolution of phidot}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\hsize]{figures/dphidt.pdf}
\caption{
$\ddot\phi$ as a function of $\dot\phi$ with $\mu^3=0$ and $H=0$.
The lines are obtained from Eq.~(\ref{eq:phi double dot 2}).
The dots are calculated from the numerical solution of Eqs.~(\ref{eq:zeromode}) and (\ref{eq:fluctuation}).
}\label{fig:phi doubledot as a function of phidot}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\hsize]{figures/timeevol_fluctuation.pdf}
\caption{
Time evolution of the energy of the fluctuations with $f = 1000 \Lambda_b$, $\dot\phi_0 = 10 \Lambda_b^2$, and $\mu^3 = 3H\dot\phi_0$. For small $H$, the evolution is fairly insensitive to it, while for larger values Hubble friction progressively reduces the efficiency of fragmentation. A transition is clearly visible around $H\simeq 6.4\times 10^{-8} \Lambda_b$.
}\label{fig:time evolution of fluctuations}
\end{figure}
In Fig.~\ref{fig:time evolution of energy}, we show the time evolution of the energy of zero-mode and of the fluctuations, for $\mu^3 = 0$ and $H=0$.
The total energy is conserved and the figure shows that the energy of the zero-mode is successfully transferred to the fluctuations.
In Figs.~\ref{fig:time evolution of phidot} and \ref{fig:phi doubledot as a function of phidot} we show the time evolution of $\dot\phi$ and $\ddot\phi$, again with $\mu^3=0$ and $H=0$ The numerical solution of Eqs.~(\ref{eq:zeromode}), (\ref{eq:fluctuation}) is compared to the result of Eq.~(\ref{eq:phi double dot 2}).
In Fig.~ \ref{fig:phi doubledot as a function of phidot}, we show $\ddot\phi$ as a function of $\dot\phi$ and $f$, again comparing the numerical solution with the analytical results.
Both Figs.~\ref{fig:time evolution of phidot} and \ref{fig:phi doubledot as a function of phidot} show that Eq.~(\ref{eq:phi double dot 2}) is consistent with the direct numerical calculation with Eqs.~(\ref{eq:zeromode}, \ref{eq:fluctuation}).
The effect of Hubble is shown in Fig.~\ref{fig:time evolution of fluctuations}, where we plot the time evolution of the energy of fluctuations for several value of $H$, with $\dot\phi_0 = \dot\phi_{\rm SR}$.
As long as Eq.~(\ref{eq:condition 3}) is satisfied, Eq.~(\ref{eq:eq for phi double dot}) has only one solution and $\ddot\phi$ mildly depends on $H$ in this regime.
However, when $H$ becomes larger than the critical value, the additional solutions given in Eqs.~(\ref{eq:phiddot 2}), (\ref{eq:phiddot 3}) appear.
Fig.~\ref{fig:time evolution of fluctuations} shows this transition behavior
and the fragmentation process becomes slower for large value of $H$.
\begin{figure}[ht]
\centering
\includegraphics[width=0.65\hsize]{figures/selffriction_startornot.pdf}
\caption{
Phase diagram of axion fragmentation with $H=0$.
Blue: Particle production is efficient enough to stop the axion rolling.
Green: The axion is accelerated by the slope and particle production is not efficient enough to stop its rolling.
Red: The initial kinetic energy is not large enough to overcome the first barrier.
The black line shows the condition in Eq.~(\ref{eq:slope bound (H=0)}), which reproduces the boundary between the blue and the green regions.
}\label{fig:phase diagram H=0}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.65\hsize]{figures/selffriction_startornot_hubble.pdf}
\caption{
Phase diagram of axion particle production effect with $H\neq 0$.
The initial velocity is taken to be the slow roll velocity $\mu^3 / 3H$.
Blue: Particle production is efficient enough to stop the axion rolling.
Green: The velocity of the axion is fixed to the slow roll velocity.
Red: The initial kinetic energy is not large enough to overcome the first barrier.
The black line shows the condition in Eq.~(\ref{eq:condition 3}), which reproduces the boundary between the blue and the green regions.
}\label{fig:phase diagram H!=0}
\end{figure}
The phase diagrams of the axion particle production are shown in Figs.~\ref{fig:phase diagram H=0} and \ref{fig:phase diagram H!=0}
for general values of $\mu^3$ and $H$.
In these figures, we take $\phi = -\pi f/2$ as the initial condition so that $\Lambda_b^4 \cos \phi/f = 0$ at the beginning.
In Fig.~\ref{fig:phase diagram H=0}, we take $H=0$ and show the parameter region
in which the particle production is efficient in $\dot\phi$-$\mu^3$ plane.
The figure shows that the condition Eq.~(\ref{eq:slope bound (H=0)}) successfully reproduces the numerical result for the maximal slope $\mu^3$ that allows stopping due to fragmentation.
In Fig.~\ref{fig:phase diagram H!=0}, we take nonzero $H$ and show the parameter region
in which the particle production is efficient in the $\dot\phi_\mathrm{SR}-\mu^3$ plane.
The figure shows the excellent agreement between Eq.~(\ref{eq:condition 3}) and the numerical results.
\begin{figure}
\centering
\includegraphics[width=0.7\hsize]{figures/spectrum_fluctuation.pdf}
\caption{
Energy spectrum of the fluctuations.
We took the same parameters as Fig.~\ref{fig:time evolution of phidot}.
The black line shows Eq.~(\ref{eq:drhodk}).
}\label{fig:fluctuation spectrum}
\end{figure}
In Fig.~\ref{fig:fluctuation spectrum}, we show the time evolution of the energy spectrum of the fluctuations. This quantity can be easily estimated as follows.
For $H=0$ and $\mu^3 = 0$, by using Eqs.~(\ref{eq:asymptotic uk 2}) and~(\ref{eq:phi double dot 2}), we get
\begin{align}
u_{k_\mathrm{cr}}(t) = \frac{4 \pi f}{k_\mathrm{cr}^{3/2}} \sin(k_\mathrm{cr} t+\delta).
\label{eq:uk H=0 mu3=0}
\end{align}
Then, we can calculate the energy spectrum after fragmentation as
\begin{align}
\frac{d\rho}{dk} = 4 k f^2 \,, \label{eq:drhodk}
\end{align}
where we dropped the oscillating terms.
Fig.~\ref{fig:fluctuation spectrum} shows that this estimation agrees with the result of the numerical calculation.
From Eq.~(\ref{eq:drhodk}), we see that the peak frequency coincides with the initial position of the instability band, $k_\mathrm{cr}^0 = \dot\phi_0/(2 f)$. Since $\dot\phi_0> \Lambda_b^2$, the emitted particles are typically relativistic.
We expect that non-perturbative effects will broaden this spectrum (see the discussion in the next section).
\section{Beyond the perturbative analysis}\label{sec:beyond linear}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\hsize]{figures/deltaphi_over_f.pdf}
\caption{Time evolution of $\sqrt{\langle \delta\phi^2\rangle}/f$.
The parameters are same as Fig.~\ref{fig:time evolution of phidot}.}\label{fig:deltaphi}
\end{figure}
Let us comment on the validity of the leading order expansion of $\delta\phi$.
For $H=0$ and $\mu^3=0$,
the size of fluctuation which is generated by the axion particle production
can be estimated by using
Eq.~(\ref{eq:uk H=0 mu3=0}) as
\begin{align}
\langle \delta\phi^2 \rangle
= \int^{\dot\phi_0/2f}_{\dot\phi/2f} dk \frac{4\pi k^2}{(2\pi)^3} |u_k|^2
\approx 4 f^2 \log \frac{\dot\phi_0}{\dot\phi} \,,
\end{align}
where we averaged out the oscillating term in the integral.
The time evolution of the ratio between $\sqrt{\langle\delta\phi^2\rangle}$ and $f$ is shown in Fig.~\ref{fig:deltaphi}.
We can see that $\sqrt{\langle\delta\phi^2\rangle}$ at late times becomes large and
we need to use non-perturbative methods for a concrete analysis in this regime.
However, Fig.~\ref{fig:deltaphi} shows that $\sqrt{\langle\delta\phi^2\rangle} / f \lesssim {\cal O}(1)$ is satisfied during most of fragmentation process.
Thus, we can expect that the estimation on the time scale Eq.~(\ref{eq:dtppapprox}) and field excursion Eq.~(\ref{eq:dphippapprox})
during fragmentation do not considerably vary from the ones obtained using a non-perturbative analysis, unless non-linear effects lead to a significant suppression of the growth of the field perturbations.
This suppression is not expected to happen in our scenario due to the periodic potential, as we elaborate in the next paragraph.
A non-linear analysis would require a dedicated lattice study, on which we are currently working~\cite{lattice}. Preliminary results indicate that our main quantitative result, namely the estimate of the total field excursion before the axion is stopped Eq.~(\ref{eq:dphippapprox}), is correct up to a factor of order 1. The main difference that we expect to emerge from a non-linear analysis is the spectrum of excited fluctuations, which will be broadened by interactions among different modes.
Moreover, the final field configuration will be inhomogeneous, with the scalar field populating more than one minima of the potential. This observation does not change significantly the picture we described so far, since the spread $\sqrt{\langle\delta\phi^2\rangle}$ will always be much smaller compared to the total field excursion $\Delta\phi_\mathrm{frag}$. Interestingly, formation of domain walls can occur in this scenario, depending on the non-linear evolution of the system and on other model dependent inputs such as the axion lifetime. We postpone this discussion to a future publication~\cite{lattice}.
A thorough comparison with other resonant systems is non-trivial, and it can hardly provide any insight on the non-linear behaviour of our model. Many models which were studied in the context of preheating feature a strong suppression of the growth at the non-linear level, generically due to the appearance of large effective mass terms. As an example, in Ref.~\cite{Prokopec:1996rr}, the resonant production of scalar particles $\chi$ from the oscillations of the inflaton $\varphi$ is studied, and it is shown that both the perturbations of the inflaton field $\delta\varphi$ (which are generated non-linearly) and the inclusion of the quartic coupling $\chi^4$, suppress the further growth of $\chi$ modes through a mass term $(\langle\delta\phi^2\rangle + \langle\chi^2\rangle)\chi^2$ (with appropriate coupling constants). Moreover, as fluctuations grow and drain energy from the zero-mode, the amplitude of the latter decreases, and hence the force driving the growth also progressively decreases.
The case under study here has two peculiarities with respect to the one above. First, since the field traverses many periods of the periodic potential, the oscillating term that stimulates the growth of fluctuations has effectively a constant amplitude. Secondly, because of the approximate shift symmetry, no effective mass scale is generated in our case. Instead, all corrections enter the equation of motions only through the cosine potential, and thus there is no reason to expect any suppression.
A setup similar to ours is discussed in~\cite{Berges:2019dgr}, in which a monodromy potential $m^2\phi^2 + \Lambda_b^4\cos\phi/f$ is studied. The paper shows that the evolution of the zero-mode stops shortly after the fluctuations have entered the non-linear regime, in accordance with our expectations.
\section{Consequences: Relaxation of the electroweak scale}\label{sec:Consequences}
The axion fragmentation dynamics explored in this work should be taken into account in the evolution of any axion field which rolls down a wiggly potential. This phenomenon can
fundamentally impact on a broad range of models, such as axion monodromy constructions and relaxion scenarios.
In this section, we consider the effects of axion fragmentation on the relaxation mechanisms of the electroweak scale.
The relaxion mechanism is a solution to the electroweak hierarchy problem in which the Higgs mass term is controlled by the evolution of an axion-like field, the relaxion \cite{Graham:2015cka}. This field evolves classically in the early universe until it stops close to a critical point, defined as the field value at which the Higgs VEV is zero. A key ingredient in this picture is a potential that features periodic wiggles, similar to the one discussed in this work. Relaxion fragmentation affects this construction in a substantial way~\cite{Fonseca:2019lmc}, as we detail below
in the two main implementations of the relaxion idea which have been discussed in the literature so far.
\paragraph{Higgs dependent barriers}
In the original proposal~\cite{Graham:2015cka}, the cosine term in the relaxion potential \Eq{eq:potential} has an amplitude dependent on the Higgs VEV, $\Lambda_b^4 \propto \langle h \rangle^n$, with $n = 1, 2$. For $n=1$ (QCD relaxion), $\Lambda_b^4 \sim m_q \, \Lambda_{\textrm{QCD}}^3$ where $m_q$ is the light quark masses. For the case with $n=2$, the scale $\Lambda_b$ cannot be far from the electroweak scale, satisfying $ \Lambda_b\lesssim \mathrm{TeV}$.
The potential contains an interaction between the Higgs and the relaxion:
\begin{equation}
V\supset -g\Lambda^3 \phi + \frac{1}{2}(\Lambda^2 - g'\Lambda\phi) h^2 +\frac{\lambda}{4}h^4 + \Lambda_b^4\cos\frac{\phi}{f} \,.
\end{equation}
Initially, the Higgs mass term $\mu_h^2\equiv \Lambda^2 - g'\Lambda\phi$ is positive, and the VEV is zero. As soon as $\phi>\Lambda/g'$, $\mu_h^2$ turns negative, a VEV develops, and the cosine term grows. In Ref.~\cite{Graham:2015cka}, it is assumed that the entire evolution takes place during a long period of inflation, and that Hubble friction is strong enough to stop the field as soon as the wiggles become larger than the average slope and the potential develop local minima, \textit{i.e.} for $g \Lambda^3 \approx \Lambda_b^4 / f$. In particular, this happens when the time that it takes to roll over one period of the cosine term is longer than one Hubble time, \textit{i.e.} for
\begin{equation}\label{eq:slow roll 1 period}
\Delta t_1 = \frac{2\pi f}{g\Lambda^3/3 H} > H^{-1} \,.
\end{equation}
Relaxion fragmentation offers an additional source of friction for the relaxion rolling. As discussed in Ref.~\cite{Fonseca:2019lmc}, this opens up two possibilities: on the one hand, the relaxion can be stopped by fragmentation even when \Eq{eq:slow roll 1 period} is not satisfied.
On the other hand, it is possible to stop the relaxion field with a much shorter period of inflation or even in the absence of an inflationary background, with a negligible Hubble friction.
This opens new possibilities for relaxion model building, independent from constraints on the inflationary sector. If relaxation takes place after inflation, it is possible, at least in principle, to conceive a model in which this phase has observable features, most probably in gravitational waves.This study could open the way to observable relaxion models.
\paragraph{Higgs independent barriers}
An alternative relaxion construction was proposed in~\cite{Hook:2016mqo}, in which the amplitude of the cosine term is Higgs-independent, and the friction is mainly provided by gauge boson particle production.
The relaxion couples to the Chern-Simons term of the massive SM $Z$ boson, through a term
\begin{equation}
\frac{\phi}{F} Z_{\mu\nu} \widetilde Z^{\mu\nu} \,.
\end{equation}
In the presence of this coupling, the equation of motion for the transverse polarization of the $Z$ has a tachyonic instability for small mass $m_Z$
\begin{equation}
\ddot Z_\pm + \left(k^2 + m_Z^2 \pm k \frac{\dot\phi}{F} \right) Z_\pm = 0
\end{equation}
Contrarily to the case discussed above, initially the Higgs has a large VEV, and the SM particles are heavy. As the relaxion approaches the critical point and the gauge bosons become lighter, the tachyonic instability is triggered and the relaxion kinetic energy is dissipated through the production of $Z$ bosons.
Fragmentation poses a serious threat to this model \cite{Fonseca:2019lmc}. Since the amplitude of the cosine term is constant, fragmentation is always active, and the relaxion can be slowed down and stopped when the Higgs mass is large and close to the cut-off $\Lambda$, thus spoiling the successful relaxation of the Higgs VEV to its current value.
In particular,
\begin{itemize}
\item if relaxation takes place after inflation~\cite{Fonseca:2018xzp}, the parameter space is restricted by the condition of avoiding excessive fragmentation. Moreover, once cosmological constraints are taken into account, the mechanism is excluded at least for a cutoff larger than few TeV.
\item If relaxation happens during inflation, the constraints from fragmentation reduce the available parameter space but do not exclude the model. The dark matter scenario discussed in~\cite{Fonseca:2018kqf}, in particular, is not affected.
\end{itemize}
\section{Summary and outlook}\label{sec:conclusions}
In this paper, we discussed the production of quantum fluctuations
during the evolution of an axion-like field rolling down a potential featuring
wiggles, as given in Eq.~(\ref{eq:potential}). We refer to this effect as axion fragmentation.
While the production of quanta is suppressed when an axion oscillates around the minimum of its potential, unless the initial amplitude is very large and the initial position of the field is tuned close to the maximum of the sinusoidal potential, the effect is very large in the case where the axion field crosses many of its maxima.
We studied in detail under which conditions axion fragmentation can efficiently stop the evolution of the field.
We computed the time scale needed for stopping and the corresponding field excursion.
The wavefunction of the fluctuations obeys the Mathieu equation
and the energy of the modes within the instability band Eq.~(\ref{eq:instability band}) grows exponentially.
If both the slope of the potential and the Hubble expansion rate are sufficiently small,
this particle production effect decelerates the homogeneous mode.
The condition is given as \Eq{eq:condition 0'} and \Eq{eq:condition for no positive phiddot} in terms of the initial field velocity, the linear slope of the potential, the Hubble rate, the size of the barriers and the periodicity of the sinusoidal potential.
The corresponding acceleration $\ddot\phi$ is given in \Eq{eq:solution phiddot}.
Axion fragmentation is a generic effect
which can have interesting phenomenological implications.
It is particularly relevant for the mechanism of cosmological relaxation of the electroweak scale.
We dedicate a separate paper to study in details these implications in Ref.~\cite{Fonseca:2019lmc}, where we conclude that new regions of parameter space open and novel directions for relaxion model building are offered by this effect.
In the present work we study the regime in which the potential has local minima, see \Eq{eq:local minima}. It would be interesting to investigate the effect of fragmentation in the case in which the oscillating term in potential does not generate local minima. This may have implications for some relaxion models where loop effects generate small Higgs-independent barriers, so that there are constant wiggles with small amplitude during the whole scanning of the Higgs mass parameter.
Another promising direction will be to explore the impact of axion quanta on the cosmological history of the universe. As discussed in Ref.~\cite{Fonseca:2019lmc}, depending on the equation of the state of the universe during axion rolling, the produced quanta may represent a significant fraction of the energy density of the universe. Whether they can be viable dark matter candidates, depends on the time of fragmentation. Such quanta may in turn induce gravitational waves.
They may be diluted or leave observable imprints.
These effects deserve detailed studies which we postpone for future work.
\section*{Acknowledgements}
The authors thank Yohei Ema, Hyungjin Kim, Kyohei Mukaida, and Gilad Perez, Alexander Westphal for useful discussions. We are grateful to Sven Krippendorf for important discussions in the initial stages of this work.
This work is supported by the Deutsche Forschungsgemeinschaft under Germany's Excellence Strategy - EXC 2121 ``Quantum Universe'' - 390833306. Research in Mainz is supported by the Cluster of Excellence ``Precision Physics, Fundamental Interactions, and Structure of Matter'' (PRISMA+ EXC 2118/1) funded by the German Research Foundation(DFG) within the German Excellence Strategy (Project ID 39083149).
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{"url":"https:\/\/devel.isa-afp.org\/entries\/Promela.html","text":"# Promela Formalization\n\n Title: Promela Formalization Author: Ren\u00e9 Neumann (rene \/dot\/ neumann \/at\/ in \/dot\/ tum \/dot\/ de) Submission date: 2014-05-28 Abstract: We present an executable formalization of the language Promela, the description language for models of the model checker SPIN. This formalization is part of the work for a completely verified model checker (CAVA), but also serves as a useful (and executable!) description of the semantics of the language itself, something that is currently missing. The formalization uses three steps: It takes an abstract syntax tree generated from an SML parser, removes syntactic sugar and enriches it with type information. This further gets translated into a transition system, on which the semantic engine (read: successor function) operates. BibTeX: @article{Promela-AFP, author = {Ren\u00e9 Neumann}, title = {Promela Formalization}, journal = {Archive of Formal Proofs}, month = may, year = 2014, note = {\\url{https:\/\/isa-afp.org\/entries\/Promela.html}, Formal proof development}, ISSN = {2150-914x}, } License: BSD License Depends on: CAVA_Automata, LTL Status: [ok] This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.","date":"2021-07-30 04:46:43","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\": 0.27248623967170715, \"perplexity\": 2518.930009675375}, \"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-2021-31\/segments\/1627046153931.11\/warc\/CC-MAIN-20210730025356-20210730055356-00660.warc.gz\"}"} | null | null |
\section{Introduction}
A new type of classical field theories has been intensively investigated during the several last years.
These theories, named k-field models, are usually endowed with nonstandard kinetic terms that change the dynamics
of the model under investigation. The $k$-field models have application in cosmology \cite{APDM, APDM1, APDM11, APDM2, APDM4},
strong gravitational waves \cite{MV}, dark matter \cite{APL} and ghost condensates \cite{a1, a2, a3, a4, a5}
and others. In particular, an interesting issue concern to the study of topological structures, where topologically nontrivial
configurations, named topological k-solutions, can exist \cite{SG}-\cite{SG1012}.
In the recent years, theories with nonstandard kinetic term, named $k$-field models, have received much attention.
The $k$-field models are mainly in connection with effective cosmological models \cite{APDM, APDM1, APDM11, APDM2, APDM4}
as well as the tachyon matter\cite{12}
and the ghost condensates \cite{a1, a2, a3, a4, a5}. The strong gravitational waves \cite{MV} and dark matter \cite{APL}, are
also examples of
non-canonical fields in cosmology. Also, topological structure of these models was analyzed \cite{SG}-\cite{SG1012}, showing that
the
$k$-theories
can support topological soliton solutions both in models of matter as in gauged models.
\\[3mm]
In this paper we propose to study a nonrelativistic Higgs $k$-model. Here, the nonstandard kinetic terms are
introduced by a function $\omega$, which depend on the Higgs field. In particular we show that $\omega(\rho)$,
where $\rho=\phi^\dagger \phi$, is Galilean invariant and we will
construct the conserved charges associated with this invariance. We, also, show that the model realize the algebra of the
Galilean group, if we choose a particular $\omega$, i.e.
\begin{equation}
\omega(\rho)= \rho^n
\end{equation}
Finally, we analyze a nonrelativistic gauge model with nonstandard kinetic
term. In particular, we will
concentrate on the Jackiw-Pi model \cite{JP, JP1} with nonstandard kinetic terms. We, also,
show that this model is Galilean invariant, realizing the algebra of the group.
\section{The model and its symmetries}
\label{4v}
Let us start by considering the $(2+1)$-dimensional Schr\"{o}dinger model governed by the action,
\begin{equation}
S =\int d^{3}x\Big( i\phi^\dagger \partial_0 \phi -\frac{1}{2m}|\partial_i \phi|^2 + \lambda |\phi|^2 \Big)\;,
\label{Ac1}
\end{equation}
Here, $\phi(x)$ is a complex scalar field and $\lambda$ is a strength coupling constant.
Also,the metric tensor is $g^{\mu \nu}=(1,-1,-1)$.
\\
It is well know that the model (\ref{Ac1}) presents Galilean invariance \cite{Ha}. This means that action
(\ref{Ac1}) is invariant under time and space translation, rotations, Galilean boost and the $U(1)$ symmetry. More precisely, the
Schr\"{o}dinger model remains invariant under the following symmetry transformations::
\begin{enumerate}
\item
The infinitesimal time-translation of the field
\begin{equation}
\delta \phi= a\partial_0 \phi\;,
\label{S1}
\end{equation}
where the Hamiltonian is the conserved charge associated to this symmetry
\begin{equation}
H= \int d^2x \Big( \frac{1}{2m}|\partial_i \phi|^2 + \lambda |\phi|^2 \Big)
\label{Ac2}
\end{equation}
\vspace{0.1cm}
\item
The infinitesimal translation of the field
\begin{equation}
\delta \phi= a_i \partial_i \phi\;,
\label{S2}
\end{equation}
which leads to the conservation of linear momentum
\begin{equation}
P_i=\frac{i}{2} \int d^2 x \Big(\phi^\dagger \partial_i \phi - \partial_i \phi^\dagger \phi \Big)
\end{equation}
\vspace{0.1cm}
\item
The infinitesimal field transformation due to a rotation
\begin{equation}
\delta \phi= \theta {\bf r}\times {\bf \partial}\phi \;,
\label{7}
\end{equation}
being $\theta$ the rotation angle. Here the conserved charge obtained from the Noether theorem is angular momentum,
\begin{eqnarray}
J= \int d^2x \Big( -\mathcal{P}_1 x_2 + \mathcal{P}_2 x_1 \Big)\;,
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{P}_i = \frac{i}{2}\Big(\phi^\dagger \partial_i \phi - \partial_i \phi^\dagger \phi \Big)
\end{eqnarray}
\vspace{0.1cm}
\item
The infinitesimal field transformation due to Galilean boost
\begin{equation}
\delta \phi= i m v_i r_i \phi -t v_i\partial_i \phi
\label{}
\end{equation}
which leads to the conservation of the following charge
\begin{eqnarray}
&&G_i=\int d^2x \Big(\mathcal{P}_i t -m x_i \rho \Big)
\label{gi}
\\[3mm]
&&\rho=\phi^\dagger \phi
\end{eqnarray}
\vspace{0.1cm}
\item
The Galilean invariance is completed with the inclusion of $U(1)$ symmetry
\begin{equation}
\delta \phi = i\alpha \phi
\end{equation}
Here, the mass operator $M=m\int d^2x \rho$ is the conserved charge associated to this transformation.
\vspace{0.1cm}
\end{enumerate}
The algebra of the Galilean group may be realized by using the Poisson brackets for functions of the matter fields, which are
defined from the symplectic structure of the Lagrangian at fixed time to be
\begin{eqnarray}
\lbrace F,G
\rbrace_{PB}=i\int d^2x \left( \frac{\delta F}{\delta \phi^\dagger(r)}
\frac{\delta G}{\delta \phi(r)}- \frac{\delta F}{\delta \phi(r)}
\frac{\delta G}{\delta \phi^\dagger(r)}\right)
\
\label{poisson}
\end{eqnarray}
In the particular case in which $F=\phi$ and $G=\phi^\dagger$ we have
\begin{eqnarray}
[\phi(x), \phi(x')^\dagger] = -i \delta^2(x- x')
\label{cr}
\end{eqnarray}
Using the Poisson bracket relations the above conserved charges can be shown to realize the algebra of the Galilean group
\begin{eqnarray}
&&[P_i, P_j]= [P_i, H]= [J, H]= [G_i, G_j]= 0
\nonumber \\[3mm]
&&[J, P_i] =\epsilon^{ij}P_j
\nonumber \\[3mm]
&&[J, G_i] =\epsilon^{ij}G_j
\nonumber \\[3mm]
&&[P_i, G_j] = \delta^{ij} mN
\nonumber \\[3mm]
&&[H, G_i]= P_i
\label{galalb}
\end{eqnarray}
In this section we are interested in exploring a generalization of the model (\ref{Ac1}). Following the same idea of the works
cited in Ref.\cite{SG, SG0, SG2, SG9, SG10, SG101}, we modify the
model (\ref{Ac1}) by changing both the canonical kinetic term of the scalar field and the potential term, so that the proposed
model is described by the action
\begin{equation}
S = \int d^{3}x \;\;\omega(\rho) \mathcal{L}_{NR}=\int d^{3}x \;\;\omega(\rho)\Big( i\phi^\dagger \partial_0 \phi
-\frac{1}{2m}|\partial_i \phi|^2 + \lambda |\phi|^2
\Big)=S_1 + S_2 +S_3\;,
\label{Ac3}
\end{equation}
where
\begin{eqnarray}
&&S_1 = \int d^{3}x \;\;\omega(\rho) i\phi^\dagger \partial_0 \phi
\nonumber \\[3mm]
&&S_2 = -\int d^{3}x \;\;\omega(\rho)\frac{1}{2m}|\partial_i \phi|^2
\nonumber \\[3mm]
&&S_3 = \int d^{3}x \;\;\omega(\rho) \lambda |\phi|^2
\label{three}
\end{eqnarray}
Here, the function $\omega(\rho)$ is in principle an arbitrary dielectric function of the complex scalar field $\phi$,
and $\rho$ is related $\phi$ by
\begin{equation}
\rho = \phi^\dagger \phi\;,
\label{}
\end{equation}
where $n$ is a positive integer.
\\
In the next we will
calculate the variation of the action (\ref{Ac3}) under time and space translation, angular rotation, Galilean boost and U(1)
transformation.
We begin to calculate the variation of the action (\ref{Ac3}) under time and space translation
\begin{equation}
\delta \phi= a\partial_0 \phi
\label{S11}
\end{equation}
\begin{equation}
\delta \phi= a_i \partial_i \phi
\label{S22}
\end{equation}
The variation (\ref{S11}) implies
\begin{equation}
\delta \omega(\rho)= \frac{\delta \omega}{\delta \rho}\delta \rho= a\frac{\partial \omega}{\partial \rho} \partial_0 \rho =
a\partial_0 \omega
\label{v1}
\end{equation}
So that,
\begin{eqnarray}
\delta S_1 &=& \int d^{3}x \;\;\Big[i\delta\omega(\rho) \phi^\dagger
\partial_0 \phi + i\omega(\rho)\delta (\phi^\dagger \partial_0 \phi) \Big]= \int d^{3}x \;\;\Big[ia \partial_0 \omega(\rho)
\phi^\dagger
\partial_0 \phi + i\omega(\rho)\delta (\phi^\dagger \partial_0 \phi) \Big]
\nonumber \\
&=&
\int d^{3}x \;\;\Big[ia \partial_0 \omega(\rho)
\phi^\dagger
\partial_0 \phi + i\omega(\rho)(a \partial_0\phi^\dagger \partial_0 \phi +a \phi^\dagger \partial^2̣_0\phi) \Big]
\end{eqnarray}
By integration by parts, the last term of this integral, we have,
\begin{eqnarray}
\delta S_1 =0
\end{eqnarray}
The variation of $S_2$ under (\ref{S11}) leads to
\begin{eqnarray}
\delta S_2 =-\frac{a}{2m}\int d^{3}x \;\;\Big[(\partial_i \partial_0\phi^\dagger \partial_i \phi + \partial_i
\phi^\dagger\partial_i\partial_0 \phi)\omega(\rho)+|\partial_i \phi|^2 \partial_0 \omega(\rho)\Big]
\end{eqnarray}
Then, integrating by parts the first term of this integral, we immediately arrive to
\begin{eqnarray}
\delta S_2 =0
\end{eqnarray}
Finally, it easy to check that,
\begin{eqnarray}
\delta S_3 =\lambda \int d^{3}x \;\; [\delta \omega(\rho) \rho + \omega(\rho)
\delta \rho] = a\lambda \int d^{3}x \;\;
\partial_0[ \omega(\rho) \rho]= 0\;
\end{eqnarray}
where we have supposed the boundary condition
\begin{eqnarray}
\lim_{t,x \to \infty}\phi=0
\end{eqnarray}
Thus, the model is invariant under time translation. Space translation, involves
\begin{equation}
\delta \omega(\rho)= a_i\partial_i\omega(\rho)
\label{v2}
\end{equation}
Then we have,
\begin{eqnarray}
\delta S_1 =\int d^{3}x \;\;\Big[ia_i \partial_i \omega(\rho)
\phi^\dagger\partial_0 \phi- ia_i\omega(\rho)(\partial_i \phi^\dagger \partial_0 \phi +\phi^\dagger \partial_0 \partial_i \phi)
\Big]
\end{eqnarray}
Integrating by parts the last term of this integral we get
\begin{eqnarray}
\delta S_1 =0
\end{eqnarray}
The variation with respect to $S_2$ is
\begin{eqnarray}
\delta S_2 =-\frac{1}{2m}\int d^{3}x \;\;\Big[\Big(a_i \partial_i^2
\phi^\dagger\partial_i \phi + a_i \partial_i \phi^\dagger \partial_i^2 \phi\Big)\omega(\rho) + |\partial_i \phi|^2 a_i
\partial_i \omega(\rho)\Big]
\Big]
\end{eqnarray}
It can be easily seen that integrating by parts the first term, the variation becomes zero.
\\
For $S_3$, we have
\begin{eqnarray}
\delta S_3 =\lambda \int d^{3}x \;\; [\delta \omega(\rho) \rho + \omega(\rho) \delta \rho] = a_i\lambda \int d^{3}x \;\;
\partial_i[ \omega(\rho) \rho] = 0
\end{eqnarray}
The model is also invariant under rotations. Indeed, we have from (\ref{7})
\begin{eqnarray}
\delta \phi= x_1\partial_2 \phi -x_2 \partial_1 \phi
\label{r}
\end{eqnarray}
Such that,
\begin{eqnarray}
\delta \omega(\rho)= \frac{\partial_\omega}{\partial \rho} (x_1 \partial_2 \rho -x_2\partial_1 \rho)= x_1 \partial_2
\omega(\rho) -x_2\partial_1\omega(\rho)
\label{v3}
\end{eqnarray}
\begin{eqnarray}
\delta S_1 =i \int d^{3}x&\Big[&[x_1 \partial_2 \omega(\rho) -x_2 \partial_1 \omega(\rho)]\phi^\dagger \partial_0 \phi +
\omega(\rho)\partial_0 \phi[ x_1 \partial_2\phi^\dagger -x_2\partial_1 \phi^\dagger]
\nonumber \\
&+&\omega(\rho)\phi^\dagger \partial_0[x_1 \partial_2\phi -x_2 \partial_1 \phi]\Big]
\end{eqnarray}
Integrating by parts, in $x_1$ and $x_2$, the last term of this variation, we easily arrive to
\begin{eqnarray}
\delta S_1 =0
\end{eqnarray}
Variation with respect to $S_2$, requires a bit more attention. By using (\ref{r}) and (\ref{v3}) we have
\begin{eqnarray}
\delta S_2 =-\frac{1}{2m}\int d^{3}x &\Big[&\partial_i ( x_1 \partial_2\phi^\dagger -x_2\partial_1 \phi^\dagger) \partial_i
\phi \omega(\rho)+ \omega(\rho) \partial_i \phi^\dagger \partial_i (x_1 \partial_2\phi -x_2 \partial_1 \phi)
\nonumber \\
&+&
|\partial_i\phi|^2 [x_1\partial_2 \omega(\rho) - x_2 \partial_1 \omega(\rho)]
\Big]
\label{1.2}
\end{eqnarray}
Developing the first two terms of this integral and after some algebra we can check that $\delta S_2 =0$,
\\
The invariance under rotation of the model is completed by the variation of $S_3$,
\begin{eqnarray}
\delta S_3 =\lambda \int d^{3}x \delta[\omega(\rho) \rho]&=& \lambda\int d^{3}x \Big[x_1\partial_2 \omega(\rho) -x_2 \partial_1
\omega(\rho)\Big] \rho + \omega(\rho)\Big[x_1 \partial_2 \rho - x_2 \partial_1 \rho\Big]
\nonumber \\
&=& \lambda\int d^{3}x \Big[\partial_2[\omega(\rho) \rho x_1] - \partial_1[\omega(\rho) \rho x_2]\Big] =0
\end{eqnarray}
Let us concentrate on the Galilean boost,
\begin{eqnarray}
\delta \phi = (im v_i x_i -tv_i \partial_i)\phi
\label{g1}
\end{eqnarray}
Under this transformation the variation of $\omega(\rho)$ is
\begin{eqnarray}
\delta \omega(\rho) =\frac{\partial \omega}{\partial \rho}\delta \rho =-tv_i\partial_i \rho \frac{\partial \omega}{\partial \rho}
= -tv_i\partial_i
\omega(\rho)
\label{g2}
\end{eqnarray}
Thus, we have for $S_1$ the following variation,
\begin{eqnarray}
\delta S_1 = i \int d^{3}x \Big[-t v_i\partial_i \omega(\rho) \phi^\dagger \partial_0 \phi - \omega(\rho)
tv_i\partial_i\phi^\dagger \partial_0 \phi - \omega(\rho) \phi^\dagger tv_i\partial_i\partial_0\phi\Big]
\end{eqnarray}
Integrating by parts the last term, it is easy to check $\delta S_1 = 0$
\\
The variation of $S_2$ may be evaluated by using (\ref{g1}) and (\ref{g2}), so that
\begin{eqnarray}
\delta S_2 = -\frac{1}{2m} \int d^{3}x \Big[-t v_i\partial_i\omega(\rho)|\partial_i \phi|^2 + (-tv_i\partial_i^2\phi^\dagger
\partial_i\phi - tv_i\partial_i \phi^\dagger \partial_i^2 \phi) \omega(\rho)\Big]\;,
\end{eqnarray}
which vanish after integrating by parts the last term of this integral.
\\
The $S_3$ is also invariant under Galilean boost. Indeed we have,
\begin{eqnarray}
\delta S_3 = \lambda \int d^{3}x \Big[-t v_i\rho \partial_i\omega(\rho) - tv_i \omega(\rho) \partial_i \phi^\dagger \phi + tv_i
\omega(\rho) \phi^\dagger \partial_i \phi \Big]\;,
\end{eqnarray}
where the last term may be integrated by parts, arriving to $\delta S_3 = 0$
\\
Finally, the $U(1)$ invariance of (\ref{Ac3}) is automatically satisfied, since $\omega(\rho)$ and $\mathcal{L}_{NR}$ are $U(1)$
invariant, and then
\begin{eqnarray}
\delta S = \int d^{3}x \Big(\delta \omega(\rho) \mathcal{L}_{NR} + \omega(\rho) \delta \mathcal{L}_{NR}\Big) = 0
\end{eqnarray}
Using the Noether it is not difficult to obtain the conserved charges associated to the Galilean symmetries. In particular we
arrive
to the following quantities:
\begin{eqnarray}
H= \int d^{2}x\;\; j_0 dx^2 = \int d^{2}x\;\; i\phi^\dagger \partial_0 \phi \omega(\rho) -\mathcal{L} = \int d^{2}x\;\;
\Big(\frac{1}{2m}|\partial_i \phi|^2 + \lambda |\phi|^2 \Big) \omega(\rho)\;,
\label{Homega}
\end{eqnarray}
which is the Hamiltonian of the model (\ref{Ac3}).
\\
\begin{eqnarray}
P_i = \frac{i}{2}\int d^{2}x \Big[\Big(\phi^\dagger\partial_i
\phi -\partial_i\phi^\dagger\phi\Big)\omega(\rho) -\rho \partial_i \omega(\rho)\Big]
\label{Pomega}
\end{eqnarray}
This is the conserved charge associated to space-translations, which differs from the usual nonrelativistic $P_i$ in the fact
that here we have the function $\omega(\rho)$ multiplying the term $\phi^\dagger\partial_i
\phi -\partial_i\phi^\dagger\phi$.
\\
\begin{eqnarray}
J =\int d^{2}x \Big(-\mathcal{P}_1 x_2 + \mathcal{P}_2 x_1
\Big)\;,
\label{Jomega}
\end{eqnarray}
which is the usual expression of the Angular momentum.
\\
\begin{eqnarray}
G =\int d^{2}x \Big(-m
x_i\rho + \mathcal{P}_i
t \Big)\omega(\rho)
\label{Gomega}
\end{eqnarray}
which differs from (\ref{gi}) only on the factor $\omega(\rho)$.
\\
\begin{eqnarray}
N = -\alpha\int d^{2}x\;\; \omega(\rho)\rho
\label{Nomega}
\end{eqnarray}
which is a generalization of the usual mass operator.
\section{The Galilean Algebra}
\label{2v}
In this section we shall study the algebra of the generators associate to the symmetry transformations studied in section
(\ref{4v}). We have seen in section (\ref{4v}) that the algebra of the Galilean group is realized by the Poisson bracket
(\ref{poisson}). Also, the Poisson bracket (\ref{poisson}) implies the commutation relation (\ref{cr}), which is the fundamental
relation to construct the algebra (\ref{galalb}). The commutator (\ref{cr}), is the usual commutator between the fundamental
field of the theory and its canonical conjugate, which is usually defined as
\begin{eqnarray}
\pi = \frac{\partial\mathcal{L}}{\partial(\partial_0\phi)}= i\phi^\dagger
\end{eqnarray}
So that,
\begin{eqnarray}
[\phi (x), \pi(x')] = \delta^2(x-x')
\end{eqnarray}
However, if we apply this commutation relation to construct the Galilean algebra of the model (\ref{Ac3}), it is not difficult
to see that we can not construct the Galilean algebra (\ref{galalb}). For instance we can check easily that,
\begin{eqnarray}
[ P_i, H] \not= 0
\end{eqnarray}
where $P_i$ and $H$ are given by the expressions (\ref{Homega}) and (\ref{Pomega}).
The problem lies in the fact that, here, $\pi$ is
not $i\phi^\dagger$. Indeed,
\begin{eqnarray}
\pi = \frac{\partial\mathcal{L}}{\partial(\partial_0\phi)}= i\phi^\dagger \omega(\rho)
\end{eqnarray}
So that $\pi$ is a function of $\phi^\dagger$ and $\phi$ and therefore the definition (\ref{poisson}) of the Poisson bracket does
not apply. For this reason, we must redefine the theory in terms of new fundamental fields. In general, this is difficult due to
the arbitrariness of the function $\omega(\rho)$. However, if we choose
\begin{eqnarray}
\omega(\rho)= \rho^n \;,
\end{eqnarray}
where, $n$ is an arbitrary positive real number, we can rewrite the model (\ref{Ac3}) as follows
\begin{eqnarray}
S&=&\int d^{3}x \Big(i\phi^\dagger \partial_0\phi -\frac{1}{2m}|\partial_i\phi|^2 +\lambda \rho \Big)\omega(\rho) = \int d^{2}x
\Big(i\phi^\dagger \partial_0\phi \rho^n -\frac{1}{2m}|\partial_i \phi|^2 \rho^n + \lambda\rho^{n+1}\Big)
\nonumber \\
&=&\int d^{3}x \Big(i(\phi^{n+1})^\dagger \phi^{n}\partial_0\phi - \frac{1}{2m}\phi^n \partial_i \phi
(\phi^n)^\dagger \partial_i\phi^\dagger + \lambda (\phi^{n+1})^\dagger \phi^{n+1}\Big)
\nonumber \\
&=&\int d^{3}x \Big(\frac{i}{n+1} (\phi^{n+1})^\dagger \partial_0\phi^{n+1} - \frac{1}{2m(n+1)^2}
\partial_i(\phi^{n+1})\partial_i(\phi^{n+1})^\dagger + \lambda (\phi^{n+1})^\dagger \phi^{n+1}
\label{Ac5}
\end{eqnarray}
From (\ref{Ac5}), it is natural to define new fields, such that
\begin{eqnarray}
\psi = \phi^{n+1}\;, \;\;\;
\;\;\;
\psi^\dagger = (\phi^\dagger)^{n+1}\;,
\label{55}
\end{eqnarray}
Thus, the action (\ref{Ac5}) is rewritten as
\begin{eqnarray}
S = \int d^{3}x \Big( \frac{i}{n+1} \psi^\dagger \partial_0\psi - \frac{1}{2m(n+1)^2}
\partial_i(\psi)\partial_i(\psi)^\dagger + \lambda \psi^\dagger \psi \Big)
\label{}
\end{eqnarray}
Comparing this action with the nonrelativistic action (\ref{Ac1}), we see immediately that both are very similar. Then, the
canonical conjugate field is
\begin{eqnarray}
\pi = \frac{\partial\mathcal{L}}{\partial(\partial_0\psi)}= \frac{i}{n+1}\psi^\dagger\;,
\end{eqnarray}
and we can
define the Poisson bracket, in terms of the new fields, following the definition (\ref{poisson}),
\begin{eqnarray}
\lbrace F,G
\rbrace_{PB}=i\int d^2x \left( \frac{\delta F}{\delta \psi^\dagger}
\frac{\delta G}{\delta \psi}- \frac{\delta F}{\delta \psi}
\frac{\delta G}{\delta \psi^\dagger}\right)
\
\end{eqnarray}
In particular, if $F=\psi$ and $G=\psi^\dagger$ we recover the usual commutation relation between the fundamental field and its
canonical conjugate,
\begin{eqnarray}
\lbrace \psi,\psi^\dagger
\rbrace_{PB}=[\psi, \psi^\dagger] = i\int d^2x \Big(-\delta^2(x- x')\delta^2(x- x')\Big)
= -i \delta^2(x- x')
\end{eqnarray}
We can proceed in the same way as with the action and check that the conserved charges (\ref{Homega}),
(\ref{Pomega}), (\ref{Jomega}), (\ref{Gomega}), (\ref{Nomega}), writing in terms of the fields $\psi$ and $\psi^\dagger$, are
identical to the conserved charges of the model (\ref{Ac1}).
Thus, the conserved charges written in terms of $\psi$ and $\psi^\dagger$ as well as the commutation relation between $\psi$ and
$\psi^\dagger$, lead us to similar context of the nonrelativistic case analyzed in section (\ref{4v}). Therefore, it is easy to
understand that the generalized model (\ref{Ac3}), with $\omega(\rho) = \rho^n$ satisfies the algebra of the Galilean group
expressed in (\ref{galalb}).
\section{Gauged model}
Let us consider the model, in which Higgs field is coupled to a gauge field $A_\mu(x)$,
\begin{equation}
S = S_A + \int d^{3}x \;\;\omega(\rho) \mathcal{L}_{NR}= S_A + \int d^{3}x \;\;\omega(\rho)\Big( i\phi^\dagger D_0 \phi
-\frac{1}{2m}|D_i \phi|^2 + \lambda |\phi|^4
\Big)\;,
\label{Ac6}
\end{equation}
where the covariant derivative is
\begin{eqnarray}
D_{\mu}= \partial_{\mu} + ieA_{\mu}\;\;\;\;\;\;,(\mu =0,1,2)
\end{eqnarray}
and $S_A$ denote the dynamics of the gauge field. In particular we will assume that $S_A$ is a $2+1$ dimensional Chern-Simons
action, given by,
\begin{eqnarray}
S_{cs}=
\frac{\kappa}{4} \int d^3x \epsilon^{\mu \nu \alpha}A_\mu F_{\nu
\alpha}= \kappa \int d^3x \left( A_0 F_{12} + A_2 \partial_0 A_1
\right)
\end{eqnarray}
In the same form that in the model (\ref{Ac3}), it is not difficult to see that, the model (\ref{Ac6}) is invariant under time
and space translations, angular rotation, Galilean boost and $U(1)$ transformation. In addition if we choose, $\omega(\rho) =
\rho^n$, the model (\ref{Ac6}) may be rewritten as
\begin{eqnarray}
S &=& \int d^{3}x \Big( i(\phi^{n+1})^\dagger [\frac{1}{n+1}\partial_0 \phi^{n+1} + ieA_0\phi^{n+1}]
\nonumber \\
&-&\frac{1}{2m}(\frac{1}{n+1}\partial_i(\phi^\dagger)^{n+1} -ieA_i (\phi^\dagger)^{n+1})
(\frac{1}{n+1}\partial_i\phi^{n+1} + ieA_i \phi^{n+1})
\nonumber \\
&+& \lambda \rho^{n+2}\Big) + S_{cs}
\end{eqnarray}
In terms of the fields $\psi$ and $\psi^\dagger$,
\begin{eqnarray}
S &=& \int d^{3}x \Big( i \psi^\dagger [\frac{1}{n+1}\partial_0 \psi + ieA_0\psi]
\nonumber \\
&-&\frac{1}{2m}(\frac{1}{n+1}\partial_i\psi^\dagger -ieA_i \psi^\dagger)
(\frac{1}{n+1}\partial_i\psi + ieA_i \psi)
+ \lambda (\psi^\dagger \psi)^2\Big) + S_{cs}
\end{eqnarray}
Let us, now, define the action $S^{'}$, such that $S^{'}= (n+1)S$,
\begin{eqnarray}
S^{'} = \int d^{3}x \Big( i \psi^\dagger D_0^{'}\psi
-\frac{1}{2m}|D_i^{'} \psi|^2
+ \lambda_1 (\psi^\dagger \psi)^2\Big) + S_{cs}^{'}
\label{Ac7}
\end{eqnarray}
where, here, the covariant derivative is defined as
\begin{eqnarray}
D_{\mu}^{'}\psi = \partial_{\mu}\psi + ie_1 A_{\mu}\psi\;\;\;\;\;\;,(\mu =0,1,2)\;,
\end{eqnarray}
the $S_{cs}^{'}$ is
\begin{eqnarray}
S_{cs}^{'}=
\frac{\kappa_1}{4} \int d^3x \epsilon^{\mu \nu \alpha}A_\mu F_{\nu
\alpha}= \kappa_1 \int d^3x \left( A_0 F_{12} + A_2 \partial_0 A_1
\right)\;,
\end{eqnarray}
and the coupling constants $e_1$, $\kappa_1$ and $\lambda_1$ are
\begin{eqnarray}
e_1= e(n+1),\;\;\;\;\;\; \kappa_1=\kappa (n+1) ,\;\;\;\;\;\; \lambda_1 = \lambda (n+1)
\end{eqnarray}
Thus, the model (\ref{Ac6}) may be rewritten in terms of fields $\psi$ and $\psi^\dagger$ as
\begin{eqnarray}
S = \int d^{3}x \frac{1}{n+1}\Big( i \psi^\dagger D_0^{'}\psi -\frac{1}{2m}|D_i^{'} \psi|^2
+ \lambda_1 (\psi^\dagger \psi)^2\Big) + S_{cs}
\label{Ac8}
\end{eqnarray}
This is the well know Jackiw-Pi model \cite{JP, JP1}, which is Galilean invariant and satisfies the algebra of the formula
(\ref{poisson}) inherent to the Galilean group. So, the model (\ref{Ac6}) realize the Galilean algebra.
\section{Twin models}
We can also modify the model (\ref{Ac1}) by introducing two different dielectric functions
\begin{equation}
S = \int d^{3}x \;\;\Big[\omega_1(\rho)\Big( i\phi^\dagger \partial_0 \phi
-\frac{1}{2m}|\partial_i \phi|^2\Big) + \lambda \omega_2(\rho)|\phi|^2 \Big]
\label{Ac9}
\end{equation}
As the model (\ref{Ac3}), it is easy to check that (\ref{Ac9}) is also Galilean invariant. Indeed, we can rewrite (\ref{Ac9}) in
three separate actions as in formula (\ref{three})
\begin{eqnarray}
&&S_1 = \int d^{3}x \;\;\omega_1(\rho) i\phi^\dagger \partial_0 \phi
\nonumber \\[3mm]
&&S_2 = -\int d^{3}x \;\;\omega_1(\rho)\frac{1}{2m}|\partial_i \phi|^2
\nonumber \\[3mm]
&&S_3 = \int d^{3}x \;\;\omega_2(\rho) \lambda |\phi|^2
\label{three1}
\end{eqnarray}
and as we showed in section (\ref{4v}) each of the three actions are Galilean invariant for an arbitrary dielectric function. So,
the actions written in formula (\ref{three1}) are Galilean invariant for arbitrary $\omega_1$ and $\omega_2$.
\\
Again the model do not satisfies the Galilean algebra for an arbitrary $\omega_1$ and $\omega_2$. However, if we choose $\omega_1
=\rho^n$ and $\omega_2 = \rho^h$, with $n$ and $h$ arbitrary positive real numbers, we can rewrite (\ref{Ac9}) as
\begin{equation}
S = \int d^{3}x \;\;\Big( \frac{i}{n+1}(\phi^\dagger)^{n+1} \partial_0 \phi^{n+1}
-\frac{1}{2m(n+1)^2}|\partial_i \phi^{n+1}|^2 + \lambda |\phi|^{2(h+1)} \Big)
\label{Ac10}
\end{equation}
if we define $\psi$ as in (\ref{55}) we have,
\begin{equation}
\phi = \psi^{\frac{1}{n+1}}
\end{equation}
so that
\begin{equation}
S = \int d^{3}x \;\;\Big( \frac{i}{n+1}\psi^\dagger \partial_0 \psi
-\frac{1}{2m(n+1)^2}|\partial_i \phi^{n+1}|^2 + \lambda |\phi|^{2\frac{h+1}{n+1}} \Big)
\label{Ac11}
\end{equation}
Writing in this form it is evident that the model (\ref{Ac9}) satisfies the Galilean algebra. We can proceed in the same way for
the gauged model.
\\
Finally, let us concentrate on the solutions of the deformed model (\ref{Ac3}) . Here, we are interested on the
static field configurations that minimize the energy functional associated to the model (\ref{Ac3}). Thus, for
the model (\ref{Ac3}), we have
\begin{equation}
E= \int d^2x \Big( \frac{1}{2m}|\partial_i \phi|^2 + \lambda |\phi|^2 \Big) \omega(\rho)
\label{Ec2}
\end{equation}
In the particular case that we choose the coupling constant to be
\begin{equation}
\lambda = \frac{1}{2m}
\end{equation}
the theory is governed by the Hamiltonian
\begin{equation}
E= \int d^2x \Big( |\partial_i \phi - \phi|^2 + \partial_i\rho \Big)\frac{\omega(\rho)}{2m}
\label{Ec3}
\end{equation}
The last term of this expression may be written as a total derivative if
\begin{equation}
\omega(\rho) = \frac{\partial f}{\partial \rho}
\label{}
\end{equation}
which, may be supposed without loss of generality. In this way we have,
\begin{equation}
E= \int d^2x \Big( |\partial_i \phi - \phi|^2 \frac{\omega(\rho)}{2m} + \partial_i f(\rho)\Big)
\label{Ec4}
\end{equation}
The total derivative may be dropped with the hypothesis that $f(\rho)$ is well-behaved. Then, the energy is bounded below by
zero, and this lower bound is saturated by solutions to the first-order self-duality equation
\begin{equation}
\partial_i \phi= \phi
\label{7}
\end{equation}
The solution of this equation satisfies, not only, the Euler-Lagrange equation of (\ref{Ec2}), but also Euler-Lagrange equation
of the model (\ref{Ac1}), i.e.
\begin{equation}
\partial_i^2 \phi= \phi
\end{equation}
Here, it is important to note that the Euler-Lagrange equation following from the functional (\ref{Ec2}) is different from that
following from the model (\ref{Ac1}). Thus, we conclude that the solution of the equation (\ref{7}) solve the Euler-Lagrange
equations of an infinitely large family of theories parametrized by the functional $\omega(\rho)$.
This deformation procedure has been used recently by many authors \cite{bp1, bp2, bp3, bp4, bp5, bp6, bp7} to obtain relations
between
similar models and their solutions.
\vspace{0.3cm}
In summary we have proposed a generalization of the nonrelativistic Schr\"{o}dinger-Higgs model. We have shown that this
generalized model admits Galilean invariance and we have also explored its twin models and their solutions. In addition we show
the Galilean invariance of a generalization of the Jackiw-Pi
model.
\vspace{0.6cm}
{\bf Acknowledgements}\\
I would like to thank Department of physics at Univesidad de Buenos Aires for hospitality.
This work is supported by CONICET.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,247 |
12th May, 2015, 12:29 pm
MSP calls for Jim Murphy to resign as leader
Labour MSP Alex Rowley has resigned from his position as the Shadow spokesperson for Local Government and Community Empowerment, and has called on Jim Murphy to stand down as leader of the Scottish Labour Party. Rowley also calls on Murphy's office staff, specifically Chief of Staff John McTernan, to follow suit.
Rowley is a former constituency office manager to Gordon Brown and is considered very close the former PM, but his views are said not to reflect those of Brown. However, Rowley was one of the MSPs to back Neil Findlay in December's leadership contest, and his letter follows a resignation from Findlay (as well as calls for Murphy to go from two unions) over the weekend.
Murphy lost his seat of East Renfrewshire in last week's election, where only one Labour MP was returned in Scotland, having held 41 seats in 2010. On Friday, however, Murphy said he intended to continue as leader into next year's Scottish Parliamentary election.
You can read the full letter below:
Dear Jim
I am writing to you to advise that I am, from this morning, standing down from the position as Labour's spokesperson for Local Government and Community Empowerment in the Scottish Parliament.
I said yesterday at the meeting of Labour MSPs that I thought your speech on Friday stating that you would stay on and lead Labour into the 2016 election was a mistake, and that it would also be a mistake for the team you put in place, including your Chief of Staff, to remain in post.
As you know, I praised the level of hard work and dedication that you brought to the campaign over the last six months and I absolutely agree that the challenges facing Labour in Scotland will not be fixed solely by a change of leadership. However, we have a leader in the Scottish Parliament and much of the focus of the next year will be on the Scottish Parliament and the performance of the SNP government over the last 8 years in Scotland. I sincerely hold the view that you continuing as leader whilst not in the Scottish Parliament, and not in an elected position holding a democratic mandate, means you will become an unhelpful distraction from the real issues that Scottish Labour must focus on.
Over the coming weeks rank and file Labour Party members must have their say on the way forward for Labour in Scotland and I want to be part of that discussion. It is clear from the discussion yesterday that dissent in public from the leadership view is perceived as disloyalty, but I am convinced we need a fundamental change in direction and strategy and therefore cannot sign up to your leadership as one of your shadow team.
From an early age my memories are of my parents talking about politics and the need for working people to organise and fight for a better and fairer society and that is why I joined the Labour Party. It was suggested to me at the weekend that it would be disloyal to the Labour Party if I were to speak publicly on these issues. I have given that a lot of thought and consideration and I concluded that it would be disloyal and damaging to Labour were I not to speak out. I believe now, more than ever, that we in Scotland need a strong relevant Labour Party and we will not achieve this under your leadership therefore I have no choice but to speak out.
Alex Rowley MSP
Cowdenbeath Constituency
UPDATE: Another Labour MSP joins in, saying the party needs a "new direction". Smith also voted for Findlay in the leadership election.
Well done @Neil_FindlayMSP and @Alex_RowleyMSP Time for a new direction for Scottish Labour.
— Elaine Smith MSP (@elainesmithmsp) May 12, 2015
UPDATE II: Unison Scotland have not quite called for Murphy to resign, according to the Sunday Herald's Paul Hutcheon. They say Murphy is not regarded "as a credible messenger of Scottish Labour values" by voters, and it "is unprecedented for a party leader not to stand down after such a defeat." However, they fall short of demanding a resignation, saying instead they "not oppose" a change in leadership.
Tags: Scottish Labour / Scotland / Alex Rowley / Neil Findlay / Jim Murphy / | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,039 |
\section*{}}
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\begin{document}
\title{Counteracting dephasing in Molecular Nanomagnets by optimised qudit encodings}
\author{F. Petiziol}
\affiliation{Universit\`a di Parma, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, I-43124 Parma, Italy}
\affiliation{UdR Parma, INSTM, I-43124 Parma, Italy}
\author{A. Chiesa}
\affiliation{Universit\`a di Parma, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, I-43124 Parma, Italy}
\affiliation{UdR Parma, INSTM, I-43124 Parma, Italy}
\author{S. Wimberger}
\affiliation{Universit\`a di Parma, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, I-43124 Parma, Italy}
\affiliation{INFN, Sezione di Milano Bicocca, Gruppo Collegato di Parma, I-43124 Parma, Italy}
\author{P. Santini}
\affiliation{Universit\`a di Parma, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, I-43124 Parma, Italy}
\affiliation{UdR Parma, INSTM, I-43124 Parma, Italy}
\author{S. Carretta}
\email{stefano.carretta@unipr.it}
\affiliation{Universit\`a di Parma, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, I-43124 Parma, Italy}
\affiliation{UdR Parma, INSTM, I-43124 Parma, Italy}
\begin{abstract}
Molecular Nanomagnets may enable the implementation of qudit-based quantum error-correction codes which exploit the many spin levels naturally embedded in a single molecule, a promising step towards scalable quantum processors.
To fully realize the potential of this approach, a microscopic understanding of the errors corrupting the quantum information encoded in a molecular qudit is essential, together with the development of tailor-made quantum error correction strategies.
We address these central points by first studying dephasing effects on the molecular spin qudit produced by the interaction with surrounding nuclear spins, which are the dominant source of errors at low temperatures.
Numerical quantum error correction codes are then constructed, by means of a systematic optimisation procedure based on simulations of the coupled system-bath dynamics, that provide a striking enhancement of the coherence time of the molecular computational unit. The sequence of pulses needed for the experimental implementation of the codes is finally proposed.
\end{abstract}
\maketitle
\noindent{\bf INTRODUCTION} \vspace{0.1cm} \\
Reliable quantum computation demands the adoption of strategies to protect the information being processed from external noise, \emph{i.e.}, of quantum error correction (QEC) schemes \cite{Terhal2015}. At the same time, while the ultimate quantum computer is expected to host QEC routines based on abstract, system-independent error models, the modern pioneering era of noisy intermediate-scale quantum devices \cite{Preskill2018, Tacchino2020} calls for strategies tailored for the specific physical hardware utilised.
In their essence, QEC algorithms achieve information protection by suitably encoding an elementary two-state computational unit, a logical qubit, into a larger Hilbert space. This permits one to distinguish, and thus detect, different error occurrences without corrupting the information, so that it is then possible to retrieve it \cite{NielsenChuang2000}.
While traditional QEC approaches realize the extra space by exploiting registers of many physical two-level systems, alternative routes to QEC have emerged wherein a logical qubit is hosted by a single many-level system (\emph{multi-level} or \emph{qudit} encodings) \cite{Gottesman2001,Troiani2012,Leghtas2013,Mirrahimi2014,Vlastakis2013,Linshu2017,Hu2019,Michael2016,Chiesa2020, Pirandola2008,Cafaro2012,Hussain2018}.
The first advantage of the latter strategy is to prevent an overhead of physical units necessary to implement the code. Also, the manipulation of single or of pairs of logical qubits can be much easier, since they do not require controlling multiple physical objects at once \cite{Chiesa2021}. Moreover, the same multi-level object can provide protection against different types of errors \cite{Cafaro2012,Michael2016}.
A very promising architecture for the implementation of multi-level encodings is given by molecular nanomagnets \cite{Grover,Hussain2018,Chiesa2020}. Indeed, these highly coherent systems \cite{Bader2014,Zadrozny,Hill,Atzori2016,Atzori2017,Atzori2018,Atzori_JACS,Freedman_JACS,Freedman2014} offer many accessible spin levels \cite{Freedman_Ni,Freedman_Cr,Mn19powell,giant}, which can be manipulated with high accuracy through electromagnetic pulses, and they can be chemically engineered to meet desired requirements \cite{PRLLuis2011,PRLWedge,Aromi2012,Aguila2014,SciRepNi,Ardavan2015,modules,Chem,SIMqubit,VO2,Sessoli2019,Gaitarev,ErCeEr}.
The most important error in molecular spin systems at low temperature is given by pure dephasing, that is, the suppression of coherence between different spin states.
Such a decoherence mechanism originates principally from the hyperfine interaction of the central (electronic or nuclear) molecular spin with the bath of surrounding nuclear spins \cite{Troiani2008,Ghirri2015,Chen2020}. Except from specific situations \cite{Coish2008}, decoherence of a central spin induced by a nuclear spin bath is known to produce non-exponential decay behaviour \cite{Klauder1962,Abe2004,Witzel2005,Ardavan2007,Troiani2008,Bader2014,Graham2017,Chen2020}. This is due to many factors, such as non-negligible entanglement building up between the spin and an evolving bath, the limited number of nuclear spins ($\sim 10^2$) usually surrounding the molecular spin $\mathcal{S}$, the slow relaxation timescales of the bath relative to the motion of the central spin.
Although mandatory to design targeted codes, a QEC framework which takes into account both the multi-level nature of a spin $S$ larger than 1/2 and the explicit structure and dynamics of the nuclear spin bath is still missing.
In this work, we develop a class of numerical spin qudit codes which are designed based on a the detailed microscopic structure of the environment responsible for errors {and which provide a strongly enhanced correction efficiency}. Moreover, the sequence of control pulses and measurements needed for an experimental implementation of such codes is discussed. While the advantage of these codes as compared to a simple spin 1/2 is evident already using small $S$, the performance strikingly improves as qudits with larger spin are used, thus positively exploiting more and more available levels in the molecular spectrum.
The codes are derived by first analysing the decoherence effects experienced by a qudit spin $S>1/2$ embedded into a realiztic nuclear spin bath, by means of numerical simulations of the coupled qudit-bath dynamics through a cluster-correlation expansion \cite{Yang2008}.
A systematic procedure is then put forward to capture the spin-dephasing process by means of error operators acting on the system, which are then used to derive optimised code words for QEC.
Thanks to the flexibility of the procedure, the numerical codes can be optimised taking into account the specific timescale of free evolution admitted between two subsequent QEC cycles, thus allowing one to largely reduce the number of correction steps sufficient to ensure a desired fidelity. As such, they are an optimal candidate for realizations in near-term devices, in which the implementation of the QEC can be noisy and can take relatively long times. \\
\begin{figure}[b]
\includegraphics[width=0.8\linewidth]{Figure1.pdf}
\caption{ \textbf{Model system}. A spin $\mathcal{S}$ larger than 1/2, whose many-level structure is exploited for performing multi-level encodings, interacts with the bath $\mathcal{B}$ of surrounding nuclear spins. Entanglement between $\mathcal{S}$ and $\mathcal{B}$ induces spin dephasing, which is counteracted through quantum error correction. Nuclear spins are plotted from a sample configuration of 100 nuclear spins used in the simulations, generated randomly within a sphere of radius $15 \, \AA$ and with a minimal distance of $3 \,\AA$ between spins (see `Methods').}
\label{fig0}
\end{figure}
\noindent {\bf RESULTS} \vspace{0.1cm}
\noindent \textbf{Physical system and decoherence mechanisms} \\%
The system analyzed in the following is a molecular electronic spin $\mathcal{S}$ (sketched in Fig. \ref{fig0}), described by the Hamiltonian $\hat{H}_{\mathcal{S}}= D \hat{S}_z^2 + \Omega \hat{S}_z$. Here, $\{\hat{S}_x, \hat{S}_y, \hat{S}_z\}$ are spin $S>1$ operators, with eigenstates of $\hat{S}_z$ being defined by $\hat{S}_z\ket{m} = m \ket{m}$. The parameter $D$ indicates the zero-field splitting (assumed to be axial, for simplicity) and $\Omega = g_z \mu_\mathrm{B} B_z$, with $g_z$ the longitudinal $g$-factor and $\mu_\mathrm{B}$ the Bohr magneton, characterises the Zeeman interaction with a static magnetic field along the $z$ direction.
The analysis developed here can apply both to a qudit given by a single spin-$S$ ion and to a giant spin originating from the strong exchange interactions between different ions \cite{Chiesa2017}. Also, while we focus here on the case of an electronic qudit, the same treatment can also be applied to a nuclear qudit with small adaptations commented in `Methods'.
For a molecular electronic spin $\mathcal{S}$ at low temperature, the hyperfine coupling with the surrounding nuclear spin bath is the dominant source of decoherence. Indeed, as typically done in quantum computing platforms, we assume to work at temperatures much smaller than the relevant energy scales of the qudit ($\Omega,D k_\mathrm{B}^{-1} \sim$ K), such that thermal population of the excited states is negligible. In these conditions, phonon absorption is very unlikely. At the same time, the qudit energy gaps are much smaller than the Debye energy (typically in the $\gtrsim 50$ K range), thus making also resonant phonon emission (whose probability scales as the third power of the gap) negligible \cite{Wurger}.
In general, the effect of spin-phonon coupling on the system dynamics can be calculated from diagonalisation of the rate matrix \cite{Wurger}, yielding a decay of both diagonal and off-diagonal elements of the system density matrix (associated to relaxation and dephasing, respectively) on related time-scales. In particular, phonon-induced dephasing is limited by the relaxation time $T_1$. A body of experiments on molecular spin qubits and qudits \cite{SIMqubit,BaderChemComm,Freedman_Cr} demonstrate that this is not the case at low temperature, where $T_1$ increases exponentially and becomes several orders of magnitude longer than the dephasing time. Hence, at low temperature phonons are not responsible for pure dephasing on the spin system and other mechanisms come into play.
Spin dipole-dipole interactions between electronic spins can have an important effect in concentrated samples, but this is strongly reduced if magnetic centres are diluted in a diamagnetic matrix~\cite{Hill} and is not relevant here, because we consider a perspective architecture consisting of a single (or a few) molecular unit(s) \cite{Takahashi,Ghirri2015}.
We therefore focus on the bath $\mathcal{B}$ of nuclear spins surrounding $\mathcal{S}$. The number of nuclear spins in the bath may range from a few tens to a few hundreds in realiztic molecular complexes, thus being rather far from the infinite-bath limit underpinning typical Markovianity approximations.
By working in the so-called `coherence window' \cite{Stamp}, in which the system energy gaps are much larger than the gaps of the nuclear spin bath, off-diagonal operators inducing population transfer on the system can be neglected. The system-bath dynamics can be described in this regime by effective spin Hamiltonians featuring only a diagonal coupling between $\mathcal{S}$ and the bath, which are derived via perturbation theory. This type of Hamiltonians has been studied in the context of a (pseudo)spin $S=1/2$ interacting with a nuclear spin bath \cite{Yao2006,Troiani2008, Troiani2012}.
In `Methods', we deduce an effective Hamiltonian for the dynamics of a generic spin $S>1/2$. In interaction picture with respect to $\hat{H}_\mathcal{S}$ and to first order in $\Omega^{-1}$, this Hamiltonian is of the form
\begin{equation}\label{eq:ham_eff}
\hat{H} = \hat{H}_{\mathcal{B}}^{(0)} + \hat{S}_z \hat{H}_{\mathcal{B}}^{(1)} + (\hat{S}^2 - \hat{S}_z^2) \hat{H}_{\mathcal{B}}^{(2)}.
\end{equation}
Both the intrinsic and the qudit-conditioned Hamiltonians $H_{\mathcal{B}}^{(k)}$ of the bath can be written in the general form
\begin{align}\label{eq:ham_det}
H_{\mathcal{B}}^{(k)} = & \sum_{n=1}^{N} \left(a_n^{(k)} \hat{I}_n^z + b_{n}^{(k)} (\hat{I}_n^z)^2\right) \nonumber \\
&+ \sum_{ \substack{n,m=1\\m\ne n}}^N \left( c_{n,m}^{(k)} \hat{I}_n^+ \hat{I}_m^- + d^{(k)}_{n,m} \hat{I}_n^z \hat{I}_m^z \right),
\end{align}
where $ \{\hat{I}_k^x, \hat{I}_k^y, \hat{I}_k^z\}$ are spin operators for the $k$-th nuclear spin of the bath and $\hat{I}_k^{\pm} = \hat{I}_k^x \pm \mathrm{i} \hat{I}_k^y$. The coefficients $a_{n,m}^{(k)}, b_{n,m}^{(k)}, c_{n,m}^{(k)}, d_{n,m}^{(k)}$ are a function of the hyperfine couplings between $\mathcal{S}$ and $\mathcal{B}$, the nuclear spin-spin dipolar couplings, and $\Omega$. In the following, nuclear spins are assumed to be protons ($I=1/2$), since hydrogen nuclei typically represent the major decoherence source. Other relevant parameters and details on the simulations are given in `Methods'.
System-bath entanglement, generated by the Hamiltonian of Eq. \eqref{eq:ham_eff}, can be interpreted in terms of `which-way information' accumulated in the state of the nuclear spins: depending on the state $\ket{m}$ of $\mathcal{S}$, the bath undergoes different interacting evolutions described by Hamiltonians $\hat{H}_{\mathcal{B},m} = \bra{m} \hat{H} \ket{m}$.
These conditioned bath evolutions result in a decay of coherences in the system density matrix $\rho_{\mathcal{S}}(t)$, according to $\bra{n} \rho_{\mathcal{S}}(t) \ket{m} = L_{nm}(t) \bra{n} \rho_{\mathcal{S}} (0) \ket{m}$. The function $L_{nm}(t)=\text{tr}_\mathcal{B} \left[ \mathrm{e}^{- \mathrm{i} \hat{H}_{\mathcal{B},n} t} \rho_\mathcal{B}(0) \mathrm{e}^{\mathrm{i} \hat{H}_{\mathcal{B},m} t} \right]$, with $\rho_\mathcal{B}(0)$ the initial bath state, is computed numerically in the following through a cluster-correlation expansion (CCE) \cite{Yang2008,Yang2009}.
In a free-decay experiment, the main decoherence process is given by inhomogeneous broadening \cite{Klauder1962}. The system-bath diagonal coupling $a_n^{(1)} \hat{S}_z \hat{I}_k^z$ has the effect of a classical random magnetic field --- the Overhauser field --- on $\mathcal{S}$. Uncertainty in the actual bath state then produces, for the density matrix $\rho_{\mathcal{S}}(t)$ of the qudit, a Gaussian decay for the single transition coherence,
\begin{equation}
\bra{m} \rho_{\mathcal{S}}(t) \ket{n} \approx \mathrm{e}^{-(n-m)^2 \Gamma^2 t^2 } \bra{m} \rho_{\mathcal{S}}(0)\ket{n},
\end{equation}
with $\Gamma^2 = \sum_k (a_n^{(1)})^2/4$ (see `Methods'), over timescales much faster than those set by the nuclear spin-spin interaction. This is shown in Fig. \ref{fig:IB_echo}, where the squared fidelity $\mathcal{F}^2_S(t)$ with respect to the initial state, with $\mathcal{F}_S(t) =\text{tr}_\mathcal{S} \sqrt{\sqrt{\rho_\mathcal{S}(t)} \rho_\mathcal{S}(0) \sqrt{\rho_\mathcal{S}(t)}}$ \cite{NielsenChuang2000}, is depicted for different spin $S$. The fidelity decays over timescales of hundreds of nanoseconds.
The dramatic effect of inhomogeneous broadening on the spin coherence is routinely compensated for in experiments by means of spin echo/refocusing schemes, whose basic example is the Hahn echo. For different qudit spins, the echo dynamics is shown in Fig. \ref{fig:IB_echo}(b). The realization of the echo transformations is further described in `Practical Implementation'. The coherence decays over timescales of tens-to-hundreds of microseconds, signalling that the effect of inhomogeneous broadening is removed to large extent. The decay is now due to the quantum dynamics of the bath, and is mainly determined by the contribution given by intra-bath interactions in $H_{\mathcal{B}}^{(0)}$, of the form $c_{n,m}^{(0)} \hat{I}_n^+ \hat{I}_m^-$. If the latter were absent, the echo would recover unit fidelity independently from $S$ over timescales of hundreds of microseconds, until virtual flip-flops described by terms of type $c_{n,m}^{(1)} \hat{S}_z\hat{I}_n^+ \hat{I}_m^{-}$ set in. This effect is still partially visible in Fig. \ref{fig:IB_echo}(b) in the fact that, for short timescales with respect to interactions in $H_{\mathcal{B}}^{(0)}$, the fidelity exhibits almost overlapping decay for different $S$.\\
\begin{figure}[t]
\includegraphics[width=\linewidth]{Figure2.pdf}
\caption{ {\bf Inhomogeneous broadening and spin echo. (a)} Decay of the squared fidelity with respect to the initial state under inhomogeneous broadening for different spins $S$ initialised in a state $\ket{\psi_L} = (\ket{0_L}+ \mathrm{i} \ket{1_L})/\sqrt(2)$, where $\ket{0_L}$ and $\ket{1_L}$ are spin-binomial code words corresponding to spin $S$ \cite{Chiesa2020} (see also `Methods'); the results have been averaged over $2^6$ initial spatial configurations of the nuclear spins, as explained in `Methods'. {\bf(b)} Decay of the echo squared fidelity for the same initialisation of (a). For each time point $t$, a generalised echo transformation of the form $\mathrm{e}^{-\mathrm{i} \pi \hat{S}_x}$ at $t/2$ is understood.}
\label{fig:IB_echo}
\end{figure}
\noindent\textbf{Optimised qudit encoding}
\noindent While increasingly sophisticated echo pulse sequences can recover the effect of inhomogeneous broadening to a better and better degree, the spin coherence remains irremediably affected by the interacting quantum dynamics of the bath. We derive in the following qudit QEC codes as a means to protect the system from these effects.
In particular, we develop a framework for designing optimal numerical codes which are based on the detailed description of the system-bath dynamics adopted in this work.
A quantum error correction code can be defined by following two fundamental steps. The first step is to identify error operators $\hat{E}_k$ which describe the effect of the noise source on the system, \emph{i.e.}, such that the state of $\mathcal{S}$ at time $t$ can be related to the initial state through
\begin{equation}\label{eq:err_ops}
\hat{\rho}_{\mathcal{S}}(t) = \sum_k \hat{E}_k \hat{\rho}_{\mathcal{S}}(0) \hat{E}_k^\dagger.
\end{equation}
The second step is the derivation of computational states $\ket{0_L}$ and $\ket{1_L}$ that satisfy Knill-Laflamme conditions for quantum error correction \cite{Knill1997}, namely, for all $k$ and $j$,
\begin{eqnarray}
& \bra{0_L} \hat{E}_k^\dagger \hat{E}_j \ket{0_L} = \bra{1_L} \hat{E}_k^\dagger \hat{E}_j \ket{1_L}, \label{eq:kl}\\
& \bra{0_L} \hat{E}_k^\dagger \hat{E}_j \ket{1_L} = 0. \label{eq:kl2}
\end{eqnarray}
These conditions demand that, when errors $\hat{E}_k$ affect an initial state $\ket{\psi_L}=\alpha \ket{0_L} + \beta \ket{1_L}$, the code words are modified but the corresponding coefficients $\alpha$ and $\beta$ are not, and the information they carry is thus preserved. Moreover, errors do not create ambiguity between $\ket{0_L}$ and $\ket{1_L}$: error words $\hat{E}_k \ket{w_L}$ span subspaces that are orthogonal to each other for different $w=0,1$.\\
If these conditions are fulfilled, then different errors $\hat{E}_k$ can be distinguished, detected and corrected. In practice, a measurement is devised whose outcome discriminates the error occurrence and a recovery operation restores the initial state. Importantly, the $2S+1$ levels of a spin $S$ offer enough state space to detect and correct a number $N_\textrm{corr} = \lfloor S \rfloor$ of error operators $\hat{E}_k$ \cite{Chiesa2020}, where $\lfloor S \rfloor$ indicates the largest integer smaller or equal than $S$.
Given that a decomposition of the form \eqref{eq:err_ops} involves in general a larger number of $\hat{E}_k$, it is essential to identify the $N_\textrm{corr}$ error operators which have stronger effect, such that the code can be tailored for them ensuring optimal correction.
\begin{figure}[t]
\includegraphics[width=\linewidth]{Figure3}
\caption{ \textsc{Figure 3.} {\bf Numerically optimised qudit codes}. \textbf{(a)} Absolute value of the overlap of the optimised code-words $\ket{0_L}$ (blue) and $\ket{1_L}$ (orange) with each state $\ket{m}$ for different spins $S$. \textbf{(b)} Squared average fidelity (over nuclear configurations) for different qudit spins $S$, for initial state $\ket{\psi_L} = [\ket{0_L} + \mathrm{i} \ket{1_L}]\sqrt{2}$ and numerical code words $\ket{0_L}$ and $\ket{1_L}$ optimised at $t=5~\mu$s. \textbf{(b)} Infidelity $1-\mathcal{F}_S^2$ for an inset of panel (b) representing the region of $\mathcal{F}_S^2\ge 0.9$.}
\label{fig:opt1}
\end{figure}
For the spin-dephasing scenario considered here, a decomposition of the form \eqref{eq:err_ops} with exact error operators is not known, thus preventing a derivation of adequate code words for this type of noise. In order to overcome this limitation, we introduce an iterative numerical optimisation procedure which, given $\hat{\rho}_\mathcal{S}(0)$ and $\hat{\rho}_\mathcal{S}(t)$ computed through CCE, aims at determining a number $N_\textrm{corr}$ of operators $\hat{E}_k$ by decreasing contribution to $\hat{\rho}_\mathcal{S}(t)$.
Starting from $\hat{\rho}_\mathcal{S}^{(0)} \equiv \hat{E}_0 \hat{\rho}_{\mathcal{S}}(0) \hat{E}_0^\dagger$, the $n$-step estimate $\hat{\rho}_{\mathcal{S}}^{(n)}$ to $\hat{\rho}_{\mathcal{S}}(t)$ of the iteration is defined according to
\begin{equation}\label{eq:iteration}
\hat{\rho}_{\mathcal{S}}^{(n)} = \hat{\rho}_\mathcal{S}^{(n-1)} + \hat{E}_n \hat{\rho}_\mathcal{S}(0) \hat{E}_n^\dagger.
\end{equation}
At the $n$-th step, the distance $\lVert \rho_\mathcal{S}(t)-\rho_\mathcal{S}^{(n-1)} \lVert$ (here, $\Vert \cdot \Vert$ is the Hilbert-Schmidt norm) is numerically minimised in order to find optimal parameters for a parametrised form or $\hat{E}_n$ (specified in the following), and the outcome is used for the subsequent step of the iteration. If a hierarchy of $\hat{E}_k$ exists, a successful optimisation will return error operators which give less and less contribution, in norm, to the density matrix. In this sense, the numerical procedure then leads to an optimal decomposition of $\hat{\rho}_\mathcal{S}(t)$ in the form \eqref{eq:err_ops}.
From the structure of the system-bath Hamiltonian $\hat{H}$ of Eq. \eqref{eq:ham_eff}, it follows that $\rho_{\mathcal{S}}(t)$ can be generically written in the form \eqref{eq:err_ops} with error operators that are diagonal in the basis of states $\ket{m}$.
Given that the Hilbert space of $\mathcal{S}$ is finite with dimension $2S+1$, the error operators $\hat{E}_k$ can be expanded onto a basis $\{ \hat{D}_m \}$ of $2S+1$ diagonal operators. Indeed, a diagonal matrix in this space can have at most $2S+1$ linearly independent entries.
This justifies an expansion for the error operators of the form
\begin{equation}\label{eq:err_ansatz}
\hat{E}_k = \sum_{m=0}^{2S} E_{k,m} \hat{D}_m.
\end{equation}
\begin{figure}[t]
\includegraphics[width=\linewidth]{Figure4.pdf}
\caption{\textbf{Gain}, defined in Eq. \eqref{eq:gain}, with respect to non-corrected spin 1/2 for the data in Fig. \ref{fig:opt1}. The curves indicate the average of $\mathcal{G}_S(t)$ over the different nuclear spin spatial configurations (generated as explained in `Methods'), while the shaded areas mark the region included between $\mathcal{G}_S(t) \pm \sigma_{S}(t)$, where $\sigma_S(t) = \sqrt{\langle \mathcal{G}_S^2(t)\rangle - \langle \mathcal{G}_S(t)\rangle^2}$ is the standard deviation and $\langle \cdot \rangle$ denotes averaging. The shaded areas show that a large gain is obtained for all spatial configurations at given spin $S$. The standard deviation $\sigma_S(t)$ increases with $S$ since the denominators in Eq. \eqref{eq:gain} become smaller and smaller for increasing $S$. A comparison with spin-binomial codes is further given in `Methods'.}
\label{fig:gainopt}
\end{figure}
The coefficients $E_{k,m}$ are the free parameters for the numerical optimisation.
The basis $\{ \hat{D}_m \}$ is chosen in the following to be given by the projectors $\hat{D}_m = \ket{m}\!\bra{m}$ over the $\ket{m}$ states.
Once the error operators are found, the code words enabling their quantum error correction are determined by imposing Knill-Laflamme's conditions \eqref{eq:kl} numerically, as detailed in `Methods'. The code words obtained from this procedure are depicted in Fig. \ref{fig:opt1}(a) for values of spin from $S=3/2$ to $S=9/2$. By construction, $\ket{0_L}$ and $\ket{1_L}$ have support on different subsets of $\ket{m}$ states in an alternate fashion. This automatically guarantees the fulfilment of Knill-Laflamme's condition \eqref{eq:kl2}, reducing the number of free parameters for the numerical search required to impose condition \eqref{eq:kl}.
The performance of the optimised qudit codes is remarkable, as shown in Fig. \ref{fig:opt1}, where the fidelity after the QEC is reported, for a set of codewords corresponding to different qudit spin $S$. In particular, for each time $t$, Fig. \ref{fig:opt1}(b) represents the squared fidelity of the recovered state with respect to the encoded state, achieved by performing an instantaneous QEC at time $t$. In panel \ref{fig:opt1}(c), we report instead the infidelity $1-\mathcal{F}_S^2(t)$ in log-log scale for an inset of panel (b). One can observe that, while a squared fidelity above 0.9 is maintained for a spin 1/2 only up to $\sim 30~\mu s$, the recovered fidelity is above that value for as long as $\sim$40, 65, 240 and 300 $\mu$s for qudit spin $S=3/2$, 5/2, 7/2 and 9/2, respectively. Similarly, the same spin values guarantee a recovered fidelity above 0.99 for up to $\sim$15, 20, 29 and 37 $\mu$s, well longer than the spin 1/2, $\sim10$ $\mu$s. The possibility to recover high fidelity even after rather long evolution times is a crucial resource for near-term implementations. Indeed, if the rate at which subsequent QEC cycles need to be done is too large, the advantage of the correction may get lost in a realiztic implementation because of the non-negligible time necessary to implement all the measurements and recovery operations of the QEC step.
The substantial advantage in increasing the spin of the qudit is further emphasised in Fig. \ref{fig:gainopt}, where the gain with respect to the spin 1/2,
\begin{equation}\label{eq:gain}
\mathcal{G}_S(t) = \frac{1-\mathcal{F}_{1/2}^2(t)}{1 - \mathcal{F}_S^2(t)},
\end{equation}
is reported as a function of time, for different values of the spin $S$. A remarkable maximal gain, larger than $10$, is attained, \emph{e.g.}, for a spin $7/2$ at around $60$ $\mu$s, and a maximal gain around $15$ is attained for $S=9/2$.
\begin{figure}[t]
\includegraphics[width=\linewidth]{Figure5}
\caption{{\bf Optimisation time and initial states. (a)} Gain as a function of time for a logical state $\ket{\psi_L}=(\ket{0_L} + \mathrm{i} \ket{1_L})/\sqrt{2}$ for numerical codewords optimised at three different times $t_\textrm{opt}= 10, 50, 100\, \mu$s and $S=3/2$.
{\bf(b)} Fidelity (color scale) as a function of time (radial scale) for different initial superposition states $\cos\theta\ket{0_L} + \mathrm{i} \sin\theta\ket{1_L}$ ($\theta$ depicted in angular scale) of a set of optimised numerical code words for $S=3/2$. }
\label{fig:opt2}
\end{figure}
Depending on the time at which the optimisation is performed, rather different numerical code words can be obtained, reflecting the interplay of different interaction scales in the system-bath Hamiltonian. However, broad temporal windows can be recognised, in which the code words maintain essentially the same structure, while being quite different in two different regimes. Therefore, a given set of code words maintains a stable performance if the QEC is implemented in a rather broad time interval around the optimisation time. These features can be observed in Fig. \ref{fig:opt2}(a), where the gain as a function of time is shown for three different code word pairs obtained by optimising at times $t_{\textrm{opt}}=10,50,100$ $\mu$s for $S=3/2$. As expected, while a unique pair giving largest gain at all times cannot be found, code words optimised at a given time provide very good performance in a rather large region around that time.
We have finally checked that the performance of the numerically optimised code words does not critically depend on the initial state chosen for our procedure, as demonstrated in Fig. \ref{fig:opt2}(b) for the exemplary case of $S=3/2$.
The squared fidelity (color scale) as a function of time (radial scale) is reported for different values of the angle $\theta$ (angular scale) characterising an initial state of the form $\cos(\theta) \ket{0_L} + \mathrm{i} \sin(\theta) \ket{1_L}$. Large fidelities are attained for all initial states, with fidelity increasing as one departs from the state with equal weights [$\theta=\pi/4$], that is, the most decoherence-prone state, used in the other reported simulations. \\
%
\noindent \textbf{Practical implementation}
\begin{figure*}[t]
\includegraphics[width=0.8\linewidth]{Figure6}
\caption{{\bf Pulse sequence for $\bm{S=5/2}$}. {\bf (a)} Explicit sequence of pulses to implement the QEC for a spin $S=5/2$. Each horizontal line represents a spin state $\ket{m}$ with time flowing from left to right. Each grey box indicates a pulse implementing a rotation $Y_m(\theta)$ as described in the text, with the values of the angles $\theta_k$ and $\phi_k$ given in Table \ref{tab:1} in `Methods'. {\bf (b)} Time evolution of the absolute value of the amplitude $\lvert\braket{m \vert \psi(t) }\!\lvert$ of the qudit state $\ket{\psi(t)}$ over each $\ket{m}$ state at different stages of the control sequence, corresponding to the control pulses depicted in (a). The sequence holds for any choice of $\alpha$ and $\beta$, though panel (b) shows an example with $\alpha=1/\sqrt{2}$, $\beta=\mathrm{i}/\sqrt{2}$. Blue and orange colours represent amplitudes associated to $\alpha$ and $\beta$ ({\it i.e.}, $\ket{0_L}$ and $\ket{1_L}$), respectively. To exemplify the effect of the basis rotation and the outcome of the measurement during the detection, the first-error case (related to the error operator $\hat{E}_1$ of Eq. \eqref{eq:err_ansatz}) is shown in (b). }
\label{fig:pulses}
\end{figure*}
%
%
\noindent Having analysed the ideal efficiency of the optimised qudit codes, we now turn to a discussion of how to implement in practice all the steps of the quantum error correction procedure, namely encoding, detection and recovery, whose formal description is given in `Methods'.
The manipulation of a spin $S>1/2$ system requires of course more complex control sequences compared to a spin $S=1/2$. Nonetheless, it can be realized in a total time sufficiently short so that it does not significantly impact the efficiency of the ideal QEC, as detailed in the following.
The population transfers required among spin states can be realized using sequences of resonant microwave/radiofrequency pulses. These are described in the Hamiltonian by time-dependent control fields of the form $g_y \mu_\mathrm{B} B_y(t) \cos(\omega t) \hat{S}_y$ where the envelope $B_y(t)$ is typically rectangular or Gaussian. These pulses induce transitions from a spin state $\ket{m}$ to $\ket{m\pm 1}$ when $\omega$ is set at the corresponding transition frequency, implementing a two-state unitary rotation $Y_m(\theta) = \exp[\theta/2(\ket{m+1}\bra{m} - \ket{m}\bra{m+1})]$ between states $\ket{m}$ and $\ket{m+1}$ of an arbitrary angle $\theta$.
All the steps of the QEC are illustrated in the following, and the explicit realization for a $S=5/2$ qudit code is depicted in Fig. \ref{fig:pulses}. The implementation proposed here generalises the one proposed for spin-binomial codes \cite{Chiesa2020}. \\
{\it Encoding.} We assume that the information, {\it i.e.} the coefficients $\alpha$ and $\beta$, is initially encoded in a simple state such as $\alpha \ket{-1/2} + \beta\ket{1/2}$. The preparation of the logical state $\alpha \ket{0_L} + \beta \ket{1_L}$ for arbitrary $\alpha$ and $\beta$ is then realized by alternating pulses $Y_m(\theta)$ distributing population among the different $\ket{m}$ states and $\pi$-pulses $Y_m(\pm\pi)$ which rearrange the different populations in the correct order. The angles $\theta$ of the two-level rotations are fully determined by the components of the code-words on each $\ket{m}$ state. The explicit sequence for $S=5/2$ is given in panel `encoding' of Fig \ref{fig:pulses}(a) with the angles given in Table \ref{tab:1} of `Methods'. Once the state has been codified, the system is left to decay freely for a time $t/2$, then a spin echo pulse sequence is performed, and the quantum error correction finally takes place at time $t$ starting with the error detection. These pulse sequences can also be used to perform single-qubit gates between the code-words, for instance by first re-mapping the code-words to, e.g. $\ket{\pm1/2}$ states, performing the desired two-state operation, and re-encoding.
{\it Spin echo.} The Hahn echo for a spin 1/2 can be understood as a magnetic pulse along $x$ or $y$ which effectively flips the spin. Then, the spin can be viewed as effectively evolving with $\hat{H}$ for a time $t/2$ and
with the same Hamiltonian but with $\hat{S}_z$ changed to $-\hat{S_z}$ for an equal time $t/2$, where $t$ is the time at which the QEC is performed. Similarly, the echo scheme is extended here to a larger spin $S>1/2$ by considering a `generalised-pulse' transformation of the form $U_{\mathrm{echo}} = \mathrm{e}^{-\mathrm{i} \pi \protect \hat {S}_x}$ at time $t/2$ which inverts the spin, sending state $\ket{m}$ to $\ket{-m}$. This transformation can be realized with a sequence of $S+1/2$ (for half-integer $S$) resonant $\pi$-rotations along $x$ or $y$, coupling pairs of $|m\rangle \leftrightarrow |-m\rangle$ states, followed by $Y_m(\pm \pi)$ pulses to rearrange populations. For instance, in the case of a spin $5/2$, $U_{\mathrm{echo}}$ is obtained by three independent rotations $|5/2\rangle \leftrightarrow |-5/2\rangle$, $|3/2\rangle \leftrightarrow |-3/2\rangle$ and $|1/2\rangle \leftrightarrow |-1/2\rangle$. Due to the lack of a direct matrix element between $\ket{m}$ and $\ket{-m}$ in the architecture considered here, each of these $\Delta m > 1$ transformations needs to be decomposed into $\Delta m = \pm 1$ transitions. This is done, for instance, using the strategy discussed in `Methods-Pulse sequences'. \\
{\it Detection.}
To realize the error detection, two additional ingredients are introduced in the system considered until now. The first one is a weak coupling of the qudit to a spin $s_A=1/2$ ancilla. The ancilla is described by adding to the Hamiltonian of Eq. \eqref{eq:Htot} the terms
\begin{equation}\label{eq:ancilla}
\Omega_A \hat{s}_A^z + \sum_{k=x,y,z} \mathbb{J}_k \hat{s}_A^k \hat{S}^k,
\end{equation}
where $\{\hat{s}_A^x, \hat{s}_A^y, \hat{s}_A^z\}$ are spin-1/2 operators for the ancilla.
The first term in Eq. \eqref{eq:ancilla} describes the Zeeman coupling of the ancilla to the static magnetic field whilst the second one describes the ancilla-qubit coupling parametrised by the tensor $\mathbb{J}$. For $\mathbb{J}_{x,y} \ll |\Omega-\Omega_A|$, such that states of qudit and ancilla remain essentially factorised, the excitation energies of the ancilla $\Delta_A^{(m)}$ depend on the state $\ket{m}$ of the qudit via the diagonal coupling $\mathbb{J}_z \hat{s}_A^z \hat{S}_z$ only, i.e.
\begin{equation}
\Delta_A^{(m)} = g_A \mu_\mathrm{B} B_z + \mathbb{J}_z m.
\end{equation}
By irradiating the ancilla with a resonant magnetic pulse at angular frequency $\Delta_A^{(m)}$ it is thus possible to selectively excite the ancilla only if the qudit is in state $\ket{m}$. A subsequent measurement of the state of the ancilla then reveals whether the qudit state has support on $\ket{m}$ or not. This mechanism will be exploited in the following to detect the different possible errors without corrupting $\alpha$ and $\beta$. Apart from this selective excitation immediately followed by a measurement, the ancilla is always in its ground state, and thus it does not affect the previously developed treatment of the qudit incoherent dynamics. For this reason, its coupling to the nuclear spins is also irrelevant for the present discussion.
The second ingredient is a coupling of the magnetic molecule to a microwave resonator. Crucial steps towards achieving the strong coupling between magnetic molecules and a resonator have been experimentally demonstrated recently \cite{Gimeno2020}. This coupling can then be exploited to measure the ancilla, building on techniques well developed in the field of circuit quantum-electrodynamics \cite{Blais2004,Blais2021,Krantz2019} and adapted to Molecular Nanomagnets \cite{Jenkins2016,Carretta2021}. The coupling of the molecule to the resonator induces a shift of the resonance frequency of the resonator which depends on the ancilla-qudit state. As explained below, this can be exploited to measure the state of the ancilla without collapsing the qudit state.
%
The error detection is described in an abstract setting by a projective measurement with the projector operators given in Eq. \eqref{eq:projectors} of `Methods'. Given the difficulty to implement similar operators, that project into complex superpositions of system eigenstates, the detection is divided into two steps. In the first step, a sequence of pulses is performed which rotates the full basis of error words into the basis of $\ket{m}$ states in both error spaces corresponding to $\ket{0_L}$ and $\ket{1_L}$ (see panel `basis rotation' of Fig. \ref{fig:pulses} for $S=5/2$) \cite{Chiesa2020}. This operation thus converts the detection of the projectors \eqref{eq:projectors} into an easier-to-implement measurement in the $\ket{m}$ basis, and the unitarity of the transformation ensures that $\alpha$ and $\beta$ are preserved. At this point, every possible post-error state is of the form $\alpha\ket{m}+\beta\ket{m'}$ for different pairs of states $(\ket{m},\ket{m'})$.
We now aim to induce an excitation of the ancilla only for a superposition state of the qudit with components on $(\ket{m},\ket{m'})$, without collapsing the superposition. We achieve this by applying a two-tone two-photon drive at frequencies $\Delta_A^{(m)}$, $\Delta_A^{(m')}$ (panel `ancilla measurement' of Fig. \ref{fig:pulses}) \cite{Royer2018}.
Then, in the dispersive limit, the coupling $G$ between resonator and ancilla induces a shift of the cavity angular frequency $\omega_c$ of $\pm G^2/\delta^m$, with $\delta^m=\Delta_A^{(m)}-\omega_c$ and the sign of the shift depending on the state of the ancilla \cite{Blais2004}. Since here we need to measure the ancilla irrespective of the state of the qudit in the subspace $(\ket{m},\ket{m'})$, a frequency-independent measurement of the state of the ancilla must be performed. Two different approaches to solve this same issue, by detecting the amplitude (but not the frequency) of the output field, are described in \cite{Royer2018}.
The ancilla is then measured by exploiting its coupling to the resonator and the qudit wavefunction is projected onto such states.
The sequence of measurements is then repeated probing each (mutually exclusive) pair of $(\ket{m},\ket{m'})$ states sequentially, returning a yes/no answer at each step if the system is found in the corresponding error state, and stopping if a positive outcome is obtained. Hence, there will be at most $\lfloor S \rfloor$ measurements given that the number of possible errors for a qudit of spin $S$ is $\lceil S \rceil$, where $\lceil S \rceil$ ($\lfloor S \rfloor$) indicates the smallest integer larger (largest integer smaller) or equal than $S$. \\
%
{\it Recovery.} After detection, the system has been projected into a superposition state of the form $\alpha \ket{m} + \beta \ket{m'}$ with known $m$ and $m'$. The simplest way to restore the encoded state $\alpha \ket{0_L} + \beta \ket{1_L}$ is then to first use a few $\pi$-pulses to send $\ket{m}\to \ket{1/2}$ and $\ket{m'}\to \ket{-1/2}$, and then to repeat the pulse sequence which implements the encoding (panel `recovery' in Fig. \ref{fig:pulses}). Alternatively, one can save a few pulses by redesigning the encoding sequence starting from each possible pair $\ket{m}$, $\ket{m'}$ resulting from detection. \\
{\it Impact on performance.} The non-instantaneous duration of the QEC procedure in a realiztic implementation (during which information is {\it not protected}), together with related potential imperfections, may cause a loss of efficiency in the correction. We thus here discuss to what extent such effects can reduce the expected performance.
The operations described to implement the quantum error correction involve sequences of resonant pulses (and ancilla measurements) only. In electronic spin systems, a single $\pi$-pulse requires less than 10 ns for achieving a state transfer with high fidelity, and this time could be further reduced by pulse-shaping techniques \cite{Motzoi2009,Theis2018,Werninghaus2021}. The measurement time for the ancilla readout through a microwave resonator can be roughly estimated from the field of circuit QED to be of 40-100 ns with fidelity above 0.98 \cite{Walter2017,Blais2021}. Then, for the spins $S\le 9/2$ considered here the QEC procedure requires a total time ranging from a few hundreds of nanoseconds to few microseconds at most, and is hence much shorter than the decay time that can be allowed by the optimised code-words while ensuring a recovery fidelity above 0.99. Indeed, the latter can be of tens of microseconds, as visible from Fig. \ref{fig:opt1} and \ref{fig:opt2}.
The practical implementation of the QEC is thus expected not to significantly reduce the correction performance for the qudits studied here. However, one could also predict that the growth of the complexity of the implementation for very large spins will eventually set a tradeoff between gain and duration of the QEC favouring the use of moderately large spins, similarly to what observed for spin-binomial codes \cite{Chiesa2020}.
Importantly, this limitation can be mitigated in the present scheme by optimising the code words at larger times. Moreover, it should be noted that the bottleneck in the specific experimental implementation proposed here for large spins is related to the rapid scaling of the number of pulses required with $S$. This, in turn, is due to the low connectivity of the $2S+1$ spin levels that permits resonant state transfers only between states with $\Delta m = \pm 1$. A possible way around this problem is to consider magnetic molecules with competing interactions \cite{vanSlageren2006,Schnack2010, Adelnia2015, Ghirri2015, Baker2016}, for which the multi-level structure used for the qudit encoding is given by low-energy states belonging to different multiplets that can provide larger state connectivity. \\
\noindent {\bf DISCUSSION}\vspace{0.1cm}
\noindent We have investigated decoherence effects produced by a realiztic nuclear spin bath on a spin qudit $S>1/2$ in Molecular Nanomagnets, by simulating the coupled system-bath dynamics via a cluster-correlation expansion. Building on this analysis, we have developed a versatile numerical strategy to construct optimal quantum error correction codes tailored for the specific spin-dephasing errors induced by the bath, thus bridging the gap between traditional general-purpose correction algorithms and the necessity of hardware-specific strategies meeting current experimental capabilities. The resulting qudit codes achieve a remarkable performance, and can be optimised by taking into account constraints on the time interval between subsequent QECs. Moreover, the increase in performance with the increase of the qudit spin is striking, signalling that the codes exploit the available levels of the molecular system as a resource very efficiently. The proposed codes can be implemented experimentally using standard sequences of resonant control pulses. Such sequences are explicitly designed and discussed, and their practical realization is predicted not to set important limits on the efficiency of the codes. Given these results, the proposed codes are a promising candidate for realizing error-protected quantum computational units embedded at the single molecule level, a central building block for implementing reliable quantum information processing on short-term molecular devices.
Recent works \cite{Chen2020} point out that the CCE method used here correctly reproduces the phenomenology of coherence enhancement due to the existence of a nuclear diffusion barrier \cite{Graham2017}.
An interesting perspective is the integration of the framework developed in this work with chemical engineering techniques for achieving an even longer lifetime of the error-corrected logical qubit through this mechanism. The synergy of tailored quantum error correction codes as investigated here with engineered nuclear spin distributions may pave the way towards a class of highly coherent molecular qubits.
The framework developed in this work is general, and can be applied to a wide landscape of molecular systems and also to other individual spin systems such as impurities in semiconductors, in order to design a proof-of-principle experiment to demonstrate the effectiveness of the QEC code. In addition, it can be extended, in the future, to investigate decoherence effects affecting superpositions of spin states belonging to different spin multiplets \cite{vanSlageren2006,Schnack2010}. This would be interesting since, on the one hand, the use of many low-$m$ spin states belonging to different multiplets may allow one to increase the number of levels available for error correction without exasperating dephasing effects given by large $m-m'$ transitions. On the other hand, it would enable a thorough study of standard block encodings embedded in a single molecule, wherein a register of qubits is achieved through many effective spin-1/2 systems selected from different spin multiplets. \\
\noindent {\bf METHODS}\vspace{0.1cm} \\
\noindent \textbf{Derivation of the effective Hamiltonian}.
The spin Hamiltonian describing the interacting evolution of the molecular spin $\mathcal{S}$ and the bath $\mathcal{B}$ of $N$ nuclear spins is
\begin{equation}\label{eq:Htot}
\hat{H}_{\mathcal{S}\mathcal{B}} = \hat{H}_{\mathcal{S}} + \sum_{n=1}^N \omega_n \hat{I}_n^z + \sum_{n=1}^N \hat{\bm{S}} \cdot \mathbb{D}_{n} \cdot \hat{\bm{I}}_n + \sum_{n\ne m} \hat{\bm{I}}_n \cdot \mathbb{E}_{n,m}\cdot \hat{\bm{I}}_m,
\end{equation}
where $\hat{H}_{\mathcal{S}}= D \hat{S}_z^2 + \Omega \hat{S}_z$, $\bm{\hat{S}} = \{\hat{S}^z, \hat{S}^+, \hat{S}^- \}$, $\bm{\hat{I}}_n = \{\hat{I}_n^z, \hat{I}_n^+, \hat{I}_n^- \}$. The tensors $\mathbb{D}_n$ contain dipole-dipole couplings between $\mathcal{S}$ and $\mathcal{B}$, while the tensors $\mathbb{E}_{n,m}$ contain nuclear-nuclear dipolar couplings. The elements of $\mathbb{D}_n$ satisfy
\begin{equation}
\mathbb{D}_n^{\scriptscriptstyle ++} = (\mathbb{D}_n^{\scriptscriptstyle --})^*, \quad \mathbb{D}_n^{\scriptscriptstyle +-} = (\mathbb{D}_n^{\scriptscriptstyle +-})^* = \mathbb{D}_n^{\scriptscriptstyle -+}, \mathbb{D}_n^{+z} = (\mathbb{D}_n^{-z})^*,
\end{equation}
and the same holds for the corresponding elements of $\mathbb{E}_{n,m}$.
A canonical perturbative (Schrieffer-Wolff) transformation generated by
\begin{equation}
G=\sum_{\beta=\pm,z} [G_{k}^{+\beta} \hat{S}^+ \hat{I}_n^\beta - \text{h.c.}],
\end{equation}
with h.c. indicating the hermitian conjugate, is constructed such that $\hat{H}_G = \mathrm{e}^{G} \hat{H} \mathrm{e}^{-G}$, within first order in $\Omega^{-1}$, does not contain off-diagonal couplings between $\mathcal{S}$ and $\mathcal{B}$ with respect to the states $\ket{m}$ \cite{Yao2006,Coish2008,Troiani2008,Bravyi2011}. As detailed in the following, $G$ is proportional to $\Omega^{-1}$ and leading orders in $\hat{H}_G$ can thus be computed from the Baker-Campbell-Hausdorf expansion
\begin{equation}\label{eq:bkh}
\hat{H}_G = \hat{H} + [G,\hat{H}] + \frac{1}{2!}[G,[G,\hat{H}]] + \dots \ .
\end{equation}
The coefficients $G_k^{\alpha\beta}$ are determined explicitly by imposing the cancellation of the off-diagonal couplings between $\mathcal{S}$ and $\mathcal{B}$ to first order. This results in the relation
\begin{equation}
[G,\hat{H}_{\mathcal{S}} + \sum_{n=1}^N \omega_n \hat{I}_n^z] = - \sum_{\substack{\alpha=\pm,\\ \beta=\pm,z}}\sum_{n=1}^N \mathbb{D}_n^{ \alpha\beta} \hat{S}^\alpha \hat{I}_n^\beta.
\end{equation}
The coefficients $G_k^{\alpha\beta}$ then read
\begin{align}
& G_k^{\scriptscriptstyle ++} = \frac{\mathbb{D}_k^{\scriptscriptstyle ++}}{\Omega +D + \omega_k }, \quad G_k^{\scriptscriptstyle +-} = \frac{\mathbb{D}_k^{\scriptscriptstyle +-}}{\Omega+D - \omega_k } , \nonumber \\
& G_k^{\scriptscriptstyle +z} = \frac{\mathbb{D}_k^{\scriptscriptstyle +z}}{\Omega +D }.
\end{align}
These expressions are indeed of order $\Omega^{-1}$ and the transformation generated by $G$ is thus perturbative, such that its effect on the initial factorized qudit-nuclei state is neglected.
By keeping terms in Eq. \eqref{eq:bkh} to first order in $\Omega^{-1}$ only and neglecting energy-non-conserving terms, the effective Hamiltonian of Eq.s \eqref{eq:ham_eff} and \eqref{eq:ham_det} is obtained with coefficients
\begin{align}
a_n^{(0)} = & \omega_n, \qquad b_n^{(0)} = 0, \qquad c_{n,m}^{(0)} = \mathbb{E}_{n,m}^{\scriptscriptstyle +-}, \nonumber \\
d_{n,m}^{(0)} = & \mathbb{E}_{n,m}^{zz}/2, \qquad a_n^{(1)} = \mathbb{D}_n^{\scriptsize zz}, \nonumber \\
b_n^{(1)} = & \frac{2}{\Omega} \left[ |\mathbb{D}_n^{\scriptstyle +z}|^2 - |\mathbb{D}_n^{\scriptscriptstyle{++}}|^2 - (\mathbb{D}_n^{\scriptscriptstyle{+-}})^2 \right], \nonumber \\
c_{n,m}^{(1)} = & \frac{2}{\Omega} (\mathbb{D}_n^{\scriptscriptstyle ++} \mathbb{D}_m^{\scriptscriptstyle --} + \mathbb{D}_n^{\scriptscriptstyle -+} \mathbb{D}_m^{\scriptscriptstyle +-}), \nonumber \\
d_{n,m}^{(1)} = & \frac{2}{\Omega} \mathbb{D}_n^{+z} \mathbb{D}_m^{-z},\nonumber \\
a_n^{(2)} = & \frac{2}{\Omega} \left[ |\mathbb{D}_n^{\scriptscriptstyle ++}|^2 - (\mathbb{D}_n^{\scriptscriptstyle +-})^2 \right] ,
\end{align}
and $b_n^{(2)} = c_n^{(2)} = d_n^{(2)} = 0$. The energy of $\mathcal{S}$ is also renormalized according to
\begin{equation}
\tilde{\Omega} = \Omega + \frac{2 I(I+1)}{\Omega} \sum_{n=1}^N \left[|\mathbb{D}_n^{\scriptscriptstyle ++}|^2 + (\mathbb{D}_n^{\scriptscriptstyle +-})^2 \right],
\end{equation}
but this is absorbed into the interaction picture in Eq. \eqref{eq:ham_eff}.\\
\noindent \textbf{Simulations and dephasing timescales.} The configuration of nuclear spins in space is generated randomly within a sphere of radius $15 \, \AA$ from the spin $\mathcal{S}$, as sketched in Fig. \ref{fig0}. Further, a minimal distance of $3 \,\AA$ is considered between nuclear spins and between each nuclear spin and $\mathcal{S}$. In all the simulations presented in this work, configurations of $N=100$ nuclear spins are considered, whose initial state is taken to be thermal at temperature $T=2$ K. Moreover, a static magnetic field $B_z=1$ T along $z$ is assumed, for achieving the regime of large Zeeman energy of $\mathcal{S}$.
The decoherence function,
\begin{equation}
L_{nm}(t)=\text{tr}_\mathcal{B} \left[ \mathrm{e}^{- \mathrm{i} \hat{H}_{\mathcal{B},n} t} \rho_\mathcal{B}(0) \mathrm{e}^{\mathrm{i} \hat{H}_{\mathcal{B},m} t} \right] \ ,
\end{equation}
is computed by means of a cluster-correlation expansion (CCE) \cite{Yang2008,Yang2009}.
This expansion decomposes $L_{nm}(t)$ as a product of contributions originating from clusters involving an increasing number of nuclear spins, and is formally equivalent to a perturbative expansion in small intra-bath effective couplings. Clusters involving more and more spins contribute smaller and smaller corrections to $L_{nm}(t)$, justifying a truncation of the expansion to few-spin clusters for practical applications. For inclusion up to $n$-size clusters, we call this truncation CCE-$n$, which yields the truncated function $L^{(n)}_{nm}(t)$.
The effect of inhomogeneous broadening is well captured by CCE-1 \cite{Yang2009,Maze2008}. Given that nuclear gaps are of the order of millikelvin in magnitude, an initial thermal state of the nuclear spin-bath at temperatures $T$ of a few kelvins is to a good approximation a fully unpolarized state
\begin{equation}
\rho_{\mathcal{B}}(0) = \frac{\mathrm{e}^{-H_\mathcal{B}^{(0)}/k_B T}}{\text{tr}_\mathcal{B} [ \mathrm{e}^{-H_\mathcal{B}^{(0)}/k_B T}] }\simeq \mathbb{1}_{\mathcal{B}}/2^N.\end{equation}
Under this approximation the CCE-1 can be solved analytically also for $S>1/2$ giving
\begin{equation}
L_{nm}^{(1)}(t) = \prod_{k=1}^N \cos\left[\frac{(n-m) \mathbb{D}_k^{zz} }{2} t \right].
\end{equation}
Here, $\mathbb{D}_k^{zz}$ is the hyperfine coupling $\propto\hat{S}_z \hat{I}_k^z$ between $\mathcal{S}$ and the $k$-th nuclear spin.
For small $\mathbb{D}_k^{zz} t$, this is well approximated by $\mathrm{e}^{-(n-m)^2 \Gamma^2 t^2}$ with $\Gamma^2 = \sum_k (D_k^{zz})^2/4$ as given in the main text.
For the echo dynamics, we find that CCE-2 gives essentially converged results, as tested by including also larger clusters. For this reason, the numerical results reported here are obtained using CCE-2. This convergence confirms that intrinsic nuclear flip-flop processes give the dominant contribution to spin dephasing after echo. Indeed, inspection of the magnitude of the coefficients $a_{n,m}^{(k)}$, $b_{n,m}^{(k)}$, $c_{n,m}^{(k)}$, $d_{n,m}^{(k)}$ reveals three fundamental interaction scales at play, which, ordered by decreasing strength, are associated to \emph{(i)} the diagonal coupling between $\mathcal{S}$ and nuclear spins (terms $\propto \hat{S}_z \hat{I}_k^z$ which are compensated for by the echo),
\emph{(ii)} the intrinsic interacting evolution (terms $\propto \hat{I}_n^{\alpha} \hat{I}_m^{\beta}$), \emph{(iii)} the $\mathcal{S}$-conditioned interacting evolution (terms $\propto \hat{S}_z \hat{I}_m^\alpha \hat{I}_n^\beta$). These different energy scales are responsible for contributions to decoherence over different timescales, with {\it (ii)} being dominant in the echo decay. \\
\noindent \textbf{Nuclear spin qudit.} \label{app:nuclear}
The quantitative analyses presented in this work are focused on the case of an electronic spin qudit. Nevertheless, the theoretical framework applies also to the case of a nuclear spin qudit. The crucial approximation underlying the physical model studied here is that the qudit energy gaps are much larger than the gaps of surrounding nuclear spins of the bath. For an electronic qudit, these energy differences can be made intrinsically large by using a sufficiently large static magnetic field. For a nuclear spin qudit of a magnetic ion (coupled to an electronic spin), whose interaction with surrounding nuclear spins is mediated by virtual excitations of the electronic spin, the necessary energy difference mainly originates from contact hyperfine interaction between the nuclear and electronic spin. The construction of the effective
Hamiltonian, and the hierarchy of the interaction scales at play, then follows as discussed above. \\
\noindent \textbf{Derivation of numerical qudit codes.} \label{app:numcodes}
The iteration defined by Eq. \eqref{eq:iteration} is first used to determine the error operators given in Eq. \eqref{eq:err_ansatz}.
The numerical code words $\ket{0_L}$ and $\ket{1_L}$ are found by starting from the following \emph{ansatz}, inspired by spin-binomial codes \cite{Chiesa2020},
\begin{equation}
\ket{0_L/1_L} = \sum_{\stackrel{\ell=0}{\ell \textrm{ odd/even}}}^{S+1/2} \gamma_\ell^{(0)/(1)} \ket{-S + \ell}.
\end{equation}
This \emph{ansatz} permits one to impose Eq. \eqref{eq:kl2} by construction and hence to reduce the number of free parameters, thanks to $\ket{0_L}$ and $\ket{1_L}$ having non-zero components on different sets of $\ket{m}$ states.
Knill-Laflamme conditions \eqref{eq:kl} are finally enforced on the coefficients $\gamma_{\ell}^{(0)/(1)}$ by numerically minimizing the function
\begin{equation}
\sum_{k,j=0}^{2S} \left\lvert \bra{0_L} \hat{E}_k^\dagger \hat{E}_j \ket{0_L} -\bra{1_L} \hat{E}_k^\dagger \hat{E}_j \ket{1_L} \right\lvert.
\end{equation}
\noindent \textbf{Detection and recovery.} \label{app:qec}
The abstract detection and recovery operations follow from the general quantum error correction theory of Ref. \cite{Knill1997}. Once a set of operators $\{\hat{E}_k\}_{k=0,\dots, \lfloor S \rfloor}$ to be corrected with a spin $S$ is identified, the two error subspaces corresponding to $\ket{0_L}$ and $\ket{1_L}$ are first determined. These two subspace are defined as the linear span of the states $\hat{E}_k\ket{0_L}$ and $\hat{E}_k\ket{1_L}$ for all $k$, respectively. For each of these subspaces, a basis $\{\ket{e_k^{(0)}}\}$ ($\{\ket{e_k^{(1)}}\}$) is selected. This is chosen here to be given by a Gram-Schmidt orthonormalization of states $\hat{E}_k\ket{0_L}$ ($\hat{E}_k\ket{1_L}$). The detection measurement is then described by the projectors
\begin{equation}\label{eq:projectors}
\hat{P}_k = \ket{e_k^{(0)}}\bra{e_k^{(0)} }+ \ket{e_k^{(1)}}\bra{e_k^{(1)}},
\end{equation}
with $k=0,\ldots,\lfloor S \rfloor$. Finally, the recovery operation, given an outcome $j$ of the measurement, maps back the states $\ket{e_j^{(0)}}$ and $\ket{e_j^{(1)}}$ corresponding to that outcome to the computational states $\ket{0_L}$ and $\ket{1_L}$, respectively. Since the coefficients $\alpha$ and $\beta$ of the encoded state have been preserved, this operation then fully restores the logical state $\ket{\psi_L}$. The recovery is formalised by a set of transformations $\{\hat{R}_k\}$ such that
\begin{equation}
\hat{R}_k \ket{e_k^{(c)}} = \ket{c_L},
\end{equation}
for $c=0,1$. Here, each transformation $\hat{R}_k$ is constructed as a rotation in the two-dimensional space spanned by $\ket{e_k^{(c)}}$ and $\ket{c_L}$. \\
\noindent {\bf Pulse sequences}. To systematically convert a generic transformation $U$ acting on the state space of a spin $S>1/2$ into a sequence of resonant $Y_m(\theta)$ pulses, one can exploit known decomposition strategies from quantum control theory \cite{Dalessandro2007, Chiesa2020}. In a first step, the unitary $U$ can be decomposed into a sequence of two-state planar rotations using the algorithm given in Ref. \cite{Dalessandro2007}. This gives a sequence of two-state transfers which involve states $\ket{m}$ and $\ket{m'}$ also with $|m-m'|>1$. To further convert such two-state rotations into rotations $Y_m(\theta)$, {\it i.e.} with $|m-m'|=1$, one finally exploits $\pi$-pulses to bring the population of $m'$ close to $m$ and back. For instance, defining
\begin{equation}
Y_{m,m'}(\theta) = \exp\big[\theta/2 (-\ket{m}\bra{m'} +\ket{m'}\bra{m} \big]
\end{equation}
for $m'>m$, one can iteratively exploit the properties
\begin{align}
Y_{m,m+2} (\theta) & = Y_{m,m+1}(\pi) Y_{m+1,m+2} (-\theta) Y_{m,m+1}(-\pi). \\
& = Y_{m,m+1}(-\pi) Y_{m+1,m+2} (\theta) Y_{m,m+1}(\pi).
\end{align}
As an example, the pulse sequence depicted in panel `basis rotation' of Fig. \ref{fig:pulses}a, which realizes a transformation mapping the basis of error words $\ket{e_k^{(c)}}$ to the basis of $\ket{m}$ states, is obtained with the procedure sketched here. The resulting angles are given in Table \ref{tab:1}.\\
\begin{table}
\caption{ Angles for the pulse sequence realizing the QEC for $S=5/2$ as depicted in Fig. \ref{fig:pulses}.
}
\label{tab:1}
\begin{ruledtabular}
\begin{tabular}{p{0.20\linewidth} p{0.20\linewidth} | p{0.20\linewidth} p{0.20\linewidth}}
\multicolumn{4}{c}{\bf Angles for the pulses in Fig. 6}\\
\hline
\multicolumn{2}{c |}{$\theta$ angles} & \multicolumn{2}{c}{$\phi$ angles} \\
\hline
\hline
$\theta_1$ & 0.342 $\pi$ & $\phi_2$ & 0.339 $\pi$ \\
$\theta_2$ & 0.560 $\pi$ & $\phi_2$ & 0.562 $\pi$ \\
$\theta_3$ & 0.552 $\pi$ & $\phi_3$ & 0.888 $\pi$ \\
$\theta_4$ & $-$0.085 $\pi$ & $\phi_4$ & $-$0.440 $\pi$ \\
$\theta_5$ & 1.531 $\pi$ & $\phi_5$ & 1.538 $\pi$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Figure7.pdf}
\caption{{\bf Performance of spin-binomial codes}. \textbf{(a)} Average squared fidelity as a function of time for different values of the qudit spin $S$, for initial state $\ket{\psi_L} = (\ket{0_L} + \mathrm{i} \ket{1_L})/\sqrt{2}$ and spin-binomial code words $\ket{0_L}$ and $\ket{1_L}$. The results are averaged over $2^6$ nuclear spin configurations. \textbf{(b)} Gain, defined in Eq. \eqref{eq:gain}, with respect to a spin 1/2, for the same data of panel (a). The shaded area are constructed as in Fig. \ref{fig:gainopt}.}
\label{fig:gain_spin_binomial}
\end{figure}
\noindent \textbf{Spin-binomial codes.} \label{app:spinbin}
In order to compare the numerical codes with other qudit approaches to spin dephasing, we test recently-proposed spin-binomial codes \cite{Chiesa2020}. These codes are based on a description of spin dephasing as produced by a Markovian bath which couples to the system via operator $\sqrt{\gamma} \hat{S}_z$. The latter model can only describe an exponential decay of coherence with rate $\gamma$ and contributions of order $(\gamma t)^n$ to the density matrix can be computed exactly and are determined by powers up to $\hat{S}_z^n$.
We find that, while spin-binomial codes still give interesting performance, they are largely overwhelmed by the numerical codes. This can be appreciated by comparing Fig. \ref{fig:opt1} and \ref{fig:gainopt} with Fig. \ref{fig:gain_spin_binomial}, both in terms of fidelity and gain.
The fact that spin-binomial codes still give an increasing gain for increasing qudit spin despite being designed for a simpler dephasing mechanism, suggests that small powers of the coupling operators $\hat{S}_z$ play a fundamental role in the decoherence process also in the present scenario, over short timescales with respect to the intra-bath interaction strength. \\
\noindent{\bf \small \small DATA AVAILABILITY}\vspace{0.1cm}
\noindent The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.\\
\noindent {\bf \small CODE AVAILABILITY}\vspace{0.1cm}
\noindent The codes developed for the numerical simulations are available from the corresponding author on reasonable request. \\
\noindent {\bf \small ACKNOWLEDGMENTS} \vspace{0.1cm}
\noindent This work received financial support from the European Project `Scaling Up quantum computation with MOlecular spins' (SUMO) of the call QuantERA, cofunded by the Italian MUR, and the European Union's Horizon 2020 program under Grant Agreement No. 862893 (FET-OPEN project FATMOLS).\\
\noindent {\bf \small AUTHOR CONTRIBUTIONS}\vspace{0.1cm}
\noindent S.W., P.S. and S.C. conceived the research. F.P. derived the theoretical model and produced the numerical data. F.P., A.C. and S.C. analyzed the data. All authors discussed the results. F.P. and A.C. wrote the manuscript, with input from all the co-authors.\\
\noindent {\bf \small COMPETING INTERESTS}\vspace{0.1cm}
\noindent The authors declare no competing interests. \\
\noindent{\bf REFERENCES}\\
\vspace{-2cm}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,731 |
\section{Introduction}
\label{S:Introduction}
LISA is a proposed joint ESA-NASA mission which will be launched around 2025 and which will be sensitive to gravitational waves (GWs) of low-frequency (between $10^{-4}$ and $0.1$ Hz). Astrophysical sources that emit gravitational waves in this frequency range are significantly more numerous than those that emit in the high frequency range (above a few Hz) which is accessible to ground-based detectors. In our own Galaxy there will be $\sim 10^6-10^7$ ultra-compact white dwarf binaries which are emitting gravitational waves in the LISA band. The majority of these will not be individually resolved by LISA but will constitute a gravitational wave foreground below $\sim3$mHz \cite{Nelemans:2009hy, Nelemans:2009ea}. LISA will detect coalescing massive black hole (MBH) binaries throughout the Universe if the masses are in the right range ($\sim 10^4M_\odot$--$10^7M_\odot$). The parameters of these binaries will be determined to an unprecedented accuracy and the set of events LISA detects will provide valuable constraints on the processes driving galaxy formation and evolution~\cite{Gair:2010bx}. At lower redshift, $z\lesssim 1$, LISA will observe extreme-mass-ratio inspirals (EMRIs), which follow from the capture of a compact stellar mass object (a white dwarf, neutron star or black hole) by a MBH from a surrounding cusp of stars in a galactic nucleus. These sources will offer an insight into the stellar dynamics and evolution of the central parsec of galaxies. All of this astrophysical information is very difficult or impossible to obtain in any other way but through GW observations with LISA \cite{Prince:2009uc}.
Besides astrophysics, LISA will be a laboratory for fundamental physics, and it is on this application of LISA observations that we will focus on in this review. LISA will be able to detect
gravitational wave bursts coming from cusps forming on cosmic strings if they exist. Strings are one dimensional topological defects that could be produced as a result of the breaking of a $U(1)$ symmetry. Such a symmetry breaking (and therefore cosmic string formation) is predicted by grand unification theory (GUT) \cite{stringsBook} and many string-inspired models of inflation~\cite{Sarangi:2002yt, HenryTye:2006uv}. The low frequency operational band of LISA makes it more sensitive to these bursts than ground based GW detectors.
In a similar way to the Cosmic Microwave Background, a stochastic gravitational wave background might have been formed in the early Universe as a result of the parametric amplification of ``zero-point'' quantum oscillations \cite{Grishchuk}, during the first order phase transition
during or at the end of inflation (\cite{Apreda:2001us, Kaminkowski1994, Baccigalupi1997}, see also \cite{Durrer:2010xc} and references therein) or during reheating
(\cite{Easther:2006vd, GarciaBellido:2007af, Dufaux:2007pt}, see also \cite{GarciaBellido:2010jp} and references therein).
The present energy density of the stochastic GW background is very uncertain, as it depends on poorly constrained parameters that enter models describing its production. However, if the
density is above $\Omega_{GW} \approx 10^{-10}$
then it could be detected by LISA. Even if the GW background is strong enough, it will need to be distinguished from the GW foreground discussed above. This might be possible by using the cyclo-stationarity of the Galactic foreground, i.e., the fact that the level of the galactic foreground varies over a year as LISA becomes more or less sensitive to gravitational waves arriving from the direction of the galactic centre~\cite{Edlund:2005ye}.
Coalescing MBH binaries observed with LISA can be used as standard sirens (as opposed to the ``standard candles'' of electromagnetic astronomy) to probe the expansion history of the Universe~\cite{Schutz:1986gp, Holz:2005df}, see also \cite{VanDenBroeck:2010fp} and references therein. GW measurements provide an accurate estimate of the luminosity distance to a MBH merger. If an electromagnetic counterpart to a GW event can be identified, it will provide the redshift of the source and hence a point on the luminosity-distance/redshift relation that can be used to constrain cosmological parameters (like the cosmological constant, matter density, Hubble constant, spatial curvature etc.). In fact it is not necessary to assume that electromagnetic (e/m) identification of the host is possible. Instead, since LISA will observe about 30 events per year, it will be possible to use statistical techniques to derive constraints on the cosmological parameters without redshift measurements. This was the basis of the original standard siren proposal in~\cite{Schutz:1986gp}, and in this paper we will describe new results for LISA that appear here for the first time. In particular, we will assume that all the cosmological parameters are known besides the effective equation of state of the dark energy and then focus on the precision with which LISA could measure that.
The study of stellar dynamics (especially the dynamics of the S-stars) in the Galactic center tells us that there is a very compact dark object with a mass $\sim 4\times 10^6 M_{\odot}$ in the Galactic nucleus
\cite{Genzel:2010zy}. Observations and simulations lead us to expect the presence of a dark massive object (DMO) in the nuclei of almost all galaxies. The common assumption is that these DMOs will be mass black holes described by the Kerr metric, but we do not yet have direct observational evidence for the presence of an event horizon in these systems. I should be possible to use LISA observations of GWs generated during extreme-mass-ratio inspirals to directly probe the nature of the DMOs for the first time. A compact object captured by a DMO with mass in the LISA range
will generate $\sim10^4-10^6$ cycles of gravitational radiation while it is orbiting in the strong gravitational field of the DMO. The emitted GWs encode an imprint of the spacetime surrounding the DMO and the hope is that we will be able to extract information from the GW signal which will allow us to test the nature of the DMO. The most promising approach described to date is to measure the multipole moments characterising the DMO (mass, spin and quadrupole moment) and compare them with the corresponding values for the Kerr metric.
In this article we will review our current understanding of these topics. We will start, in Section~\ref{S:strings}, with a description of LISA's ability to detect GW bursts from cosmic strings before moving on to a discussion of how to detect a cosmological stochastic GW background in Section~\ref{S:stochastic}. In Section~\ref{S:cosmo} we will present new results on constraining cosmological parameters with LISA in the absence of electromagnetic counterparts using a statistical method. We then review the possibilities for ``mapping spacetime'' using EMRIs in Section~\ref{S:mapping} before concluding the article with a summary in Section~\ref{S:summary}.
\section{Detecting gravitational wave bursts from cosmic strings}
\label{S:strings}
As mentioned above, a string soliton solution will arise in any gauge theory that has a broken $U(1)$ symmetry or in any phase transition in which $U(1)$ becomes broken during the evolution.
Interest in cosmic strings originated in the context of grand unification theories (see \cite{stringsBook} for a review on the subject), but it was later suggested that strings would also be produced by $D$-brane annihilation at the end of brane inflation~\cite{Sarangi:2002yt, HenryTye:2006uv, Tye:2002yb}. Some fraction of these strings will be in the form of finite size loops, and other strings will be infinite. The latter are particularly important as they stretch with the expansion of the Universe while the small loops quickly decay. When two strings collide they may reconnect with a probability $P_{rec}$ which is a property of the strings. This probability is almost unity for GUT-strings and has a range of values $[10^{-3}, 1]$ for brane-strings. The main parameter characterizing an individual string is its tension $\mu$. The two parameters ($\mu, P_{rec}$) characterize the properties of a network of strings
\cite{AllenShellard}. We refer readers to \cite{Polchinski:2004ia, Polchinski:2007qc} for nice reviews on the mechanisms for production of cosmic strings and to~\cite{CurtLISA9} for a review on prospects for their detection.
When a long string intersects itself, a closed loop will break off. If the size of the loop formed is smaller than the horizon, it will not expand and behaves like a massive object. The closed loops decay through emission of gravitational radiation and a gravitational wave background is produced from the incoherent superposition of all the signals generated by the decaying loops produced by the network over time. The spectrum of this GW background is expected to be approximately flat over LISA's operational frequency band and can be characterized in terms of the energy density of the gravitational waves (GW) $\rho_{GW}$:
\begin{equation}
\Omega_{GW}(f) \equiv \frac{d\rho_{GW}/d \ln{f}}{\rho_c},
\end{equation}
where $\rho_c$ is the critical density of the Universe. This quantity depends on the tension of the strings as $\Omega_{GW} \sim \sqrt{G\mu}$ \cite{Polchinski:2007qc, stringsBook, Polchinski:2004ia}. The proportionality coefficient depends on the scaling factor at the moment of loop formation $\alpha$ and on $\Gamma$, the scaling factor for the decay time of a loop (of length $l$) $t_d = l/\Gamma G \mu$.
At present, there is no observational evidence that cosmic strings exist, but assuming they do, current observations can set up an upper limit on the string tension. The most recent published limit on $\Omega_{GW}$ from the ground-based gravitational wave detectors, LIGO and Virgo, is $\Omega_{GW} (f\sim 100 Hz) \le 6.9\times 10^{-6}$~\cite{LVCNature}. A somewhat better limit has been derived from pulsar timing observations. A signature in the timing residuals of pulsar observations arises from the effect of propagation of electromagnetic pulses in a stochastic gravitational field. Analysis of current pulsar timing data has given an upper limit $\Omega_{GW}(f \sim 10^{-8} Hz) \le 4\times 10^{-8} $, which corresponds to a limit on the tension of $G\mu < 1.5 \times 10^{-8}$ \cite{Jenet:2006sv, DePies:2007bm}. This limit will improve further as the observation time increases. However, the prospects with LISA are even better. It was shown in \cite{DePies:2007bm} that LISA would be able to detect a background from cosmic strings
with a tension $G\mu > 10^{-16}$.
When two long strings reconnect, two long kinked strings are produced. These kinks propagate along the strings and tend to straighten and diminish in strength over time as energy is emitted in gravitational wave radiation. Kinks thus produce short GW bursts, which could also potentially be observable by gravitational wave detectors. A second mechanism for the production of GW bursts from strings are cusps, which are created when a tiny part of the string propagates with a speed close
to the speed of light. It was shown in \cite{Damour:2001bk} that GW bursts generated by cusps will be more detectable in GW observations than those generated by kinks, since the latter fall off more rapidly with frequency. The GW radiation from cusps is linearly polarized and highly beamed in the direction of propagation of the cusp. The shape of the burst in the time domain takes a very simple form, $h(t) \sim |t- t_c|^{1/3}$, where $t_c$ is the time of arrival of the burst at an observer. However, the cusp at $t=t_c$ exists only for an observer who lies \emph{exactly} along the direction of the cusp's velocity. For an observer in a direction at an angle $\theta$ to the center of the radiation cone, there is an exponential decay in the radiation spectrum at frequencies above $f_{max} \sim 1/(L\theta^3)$ where $L$ is a characteristic lengthscale of the cosmic string. This upper frequency cut off smoothes $h(t)$ at $t=t_c$~\cite{Damour:2001bk, Siemens:2003ra}. The GW waveform in the frequency domain scales as $\tilde{h}(f) \sim f^{-4/3}$ (as compared to the GW bursts from the kinks which scales as $\tilde{h}(f) \sim f^{-5/3}$). It was demonstrated in \cite{Cohen:2010xd} that LISA will be a factor of ten more sensitive to GWs from cusps than the advanced ground based detectors. The square of the signal-to-noise ratio (SNR) per logarithmic frequency interval ($d\ln{f}$) is approximately $|f \tilde{h}(f)|^2/fS_h(f)$, where $S_h(f)$ is the noise power spectral density. The SNR for a GW burst from a cusp therefore scales roughly as $f_b^{-1/3}/(f_b S_h(f_b))^{1/2}$, where $f_b$ is the frequency where the detector has the best sensitivity (this is $\sim 100$Hz for advLIGO and $\sim 0.003$Hz for LISA). The denominator ($(f_b S_h(f_b))^{1/2}$) for LISA is a few times higher than the corresponding value for advanced LIGO while the numerator, $f_b^{-1/3}$, is approximately 30 times larger for LISA. This means that, for a given SNR threshold, the volume in which LISA could detect bursts from cusps is a factor of $\sim 10^3$ larger than the observable volume for advLIGO. The majority of the bursts will be rather weak and will contribute to a gravitational wave background, however we would expect that a string network would also produce a few strong bursts which stand above the background.
The detection of cosmic strings would have a great impact on fundamental physics and on our understanding of the early Universe. As mentioned above the string network can be characterized by two parameters $\mu, P_{rec}$, therefore if those two parameters can be determined we will learn about the nature of strings and about the network. An individual burst has no characteristic frequency scale and is characterized primarily by its amplitude, which is therefore not enough to determine the network parameters. However, it was argued by Polchinski \cite{Polchinski:2004ia} that the observation of $\gtrsim10$ bursts would provide the distribution of amplitudes, $dN \sim A h^{-B}dh$, and in principle the parameters $A$ and $B$ would constrain $\mu, P_{rec}$. However, any individual bursts that are observed will be at distances much closer than the Hubble distance and so the burst distribution
is likely to be close to the Euclidean, $dN/dh \sim h^{-4}$ \cite{Cohen:2010xd}, which does not constrain the network parameters. To place constraints on the string network, it may therefore be necessary to measure both the gravitational wave background ($\Omega_{GW}$) and the distribution of individual bursts $dN/dh$.
We will finish with a discussion of the detectability of bursts from cosmic string cusps using LISA. Since we know the form of the GW signal from a cosmic string cusp, we will be able to use matched filtering techniques to search for them in the LISA data stream. In order to assess existing data analysis techniques there have been a sequence of Mock LISA data challenges, and the last two of these have included bursts from cosmic string cusps in the data sets. These challenges involve blind searches for multiple signals in simulated LISA data \cite{Babak:2008sn, Babak:2009cj}. For the cosmic string cusps, the exact number of bursts in the data was not known (it was drawn from a Poisson distribution with mean 5).The injections used the following waveform model for the GW signal \cite{Babak:2009cj}:
\begin{eqnarray}
\tilde{h}_{ij}(f) = (A^{+}_{ij} + A^{\times}_{ij})A(f)e^{2\pi i ft_c},\\
A^{+}_{ij} + A^{\times}_{ij} = \mathcal{A}\Lambda(f) (e^{+}_{ij}cos(\psi) + e^{\times}_{ij}sin(\psi))
\end{eqnarray}
\begin{equation}
\Lambda(f) = \left\{\begin{array}{cc}
f^{-4/3} & f< f_{max} \\
f^{-4/3} e^{1-f/f_{max}} & f>f_{fmax}
\end{array}\right.
\end{equation}
in which the overall amplitude
$
\mathcal{A} \sim G\mu l^{2/3}/D_L
$ is dependent on the luminosity distance to the source, $D_L$, the string tension and the characteristic scale of the string. The polarization angle $\psi$ and the polarization tensors are defined with respect to the fixed direction of GW propagation.
The results of the blind search for the cosmic string bursts~\cite{Cohen:2010xd, Feroz:2009eb, Key:2008tt} (see also \cite{Babak:2009cj} for a summary of all the results for that round of the data challlenge) showed that current methods can be used to find GW bursts from cusps rather easily. Parameter estimation for these systems is a rather different issue, however, as the short duration of the signal means there are strong degeneracies in the parameter space. In particular the sky localization was rather poorly constrained (the error is more than one radian in both right ascension and declination) even if we are at the right maximum of the likelihood. Other parameters like the polarization angle and the amplitude strongly correlate with the sky location and were therefore also poorly determined. The parameter which will be determined most precisely is the time that the burst passes through the detector ($t_c$). Based on these results, we have very good prospects of detecting GW bursts from cosmic cusps with LISA but the parameters of the bursts will be rather poorly recovered.
\section{Stochastic gravitational wave background}
\label{S:stochastic}
There are different mechanisms which could lead to the production of a cosmological gravitational wave background. The primary mechanism is the adiabatic amplification of quantum fluctuations~\cite{Grishchuk}, which is most efficient during inflation. The resulting background is often referred to as the inflationary ``relic'' GW background. The other mechanisms involve classical (as opposed to quantum) sources and have the potential to be much stronger in certain frequency bands. One such mechanism is the decay of cosmic string loops, as we discussed in the previous section. There is a vast literature reviewing this topic \cite{Hogan:2006va, Maggiore:1999vm, Chongchitnan:2006pe, Buonanno:2003th,Hughes:2002yy} and so here we will just mention the various possible sources and the prospects for their detection with LISA.
A phase transition corresponding to symmetry breaking of the fundamental interactions could have happened in the early Universe. In the first order phase transition the Universe is initially trapped in a metastable phase (with unbroken symmetries). The transition from the metastable phase to the ground state occurs by the quantum tunneling of a scalar field across a potential energy barrier. This transition nucleates randomly in bubbles. The size of these bubbles increases as the temperature of the Universe drops and large bubbles then collide bringing the Universe to a broken symmetry phase. Gravitational radiation is produced as soon as the spherical symmetry of an
individual bubble is broken during the collisions. The spectrum of GWs from the first order phase transition is peaked around a frequency determined by the typical temperature at which the transition takes place:
$f_{peak} \sim 100\, \rm{Hz} (T/10^5 TeV)$ \cite{Kosowsky:2000rq, Kaminkowski1994, Baccigalupi1997}. The electroweak phase transition happened at an energy scale of 100 GeV, for which the peak frequency is $\sim0.1$mHz, in LISA's sensitive frequency band. In the standard inflationary model, the electroweak transition is not of first order, although there are still Higgs field fluctuations (the vacuum expectation value undergoes a crossover from zero to a finite value), but these do not produce significant amounts of gravitational waves. If the standard inflation model is modified~\cite{Durrer:2010xc} (for example through the (next-to-)minimal supersymmetric extension of the standard model \cite{Apreda:2001us}) the electroweak phase transition could become strongly first order with a GW background of strength $\Omega_{GW} \le 10^{-12}-10^{-11}$ being generated through bubble nucleation.
In addition, if the Reynolds number of the cosmic plasma is high and the growing bubbles convert internal energy into relativistic flows, this can generate high bulk turbulent velocities which accelerates the fluid leading to further production of GWs \cite{Kosowsky:2001xp, Gogoberidze:2007an, Durrer:2010xc}. The strength of the resulting GW background depends on the temperature of the Universe at the time of the turbulence, the duration of this phase, the stirring scale and on the Reynolds and Mach numbers of the flows. If there is a small electromagnetic field (which should be present during the broken phase of the electroweak transition), this field is amplified by the turbulence leading to a magnito-hydrodynamic turbulent plasma (MHD turbulence), which in turn generates further GWs. Under favorable conditions the level of this background could be comparable or even higher than that coming from the bubble collisions and reach a level of $\Omega_{GW} \approx few \times 10^{-11}$~\cite{Durrer:2010xc}.
Other possible sources of stochastic GWs are reheating and the dynamics of extra dimensions. The presence of higher dimensions in, for example, brane-world scenarios, boosts the ratio of tensor-to-scalar perturbations in the early Universe and correspondingly increases the expected strength of the GW background. Inflation must end with a transition to a thermalized Universe. This transition is called reheating, when the inflaton field decays into other degrees of freedom which also include the standard model particles. This process could generate significant relativistic inhomogeneities which would lead to the production of a GW background. However, the peak of the spectrum of such a background would lie at GHz frequencies for GUT-scale models of inflation. In the case of hybrid inflation, in which gauge fields associated with the breaking of some symmetries are present, the GW spectrum will have extra peaks associated with the mass scales of the corresponding gauge fields \cite{Dufaux:2010cf}. Unfortunately the predicted strength of this background is too weak to be observed by LISA, but it could be detected by BBO. This model also predicts an angular anisotropy of the GW background which potentially could be tested with the second generation of space-based GW observatories. If inflation ended with a global $O(N)$ transition, then the produced GW background would be at the level $\Omega_{GW} = 10^{-15} - 10^{-11}$~\cite{Fenu:2009qf}.
Detecting a stochastic GW background and measuring its spectral shape will provide us information about the mechanism leading to its formation and the epoch in which this occurred, but how do we detect a GW background with LISA? In order to cancel laser phase noise, the six phase measurements at the spacecraft will be combined using a technique known as Time Delay Interferometry \cite{lrr-2005-4} to produce three TDI streams ($A,E,T$) in which the noise is uncorrelated (even for an unequal arm LISA \cite{Adams:2010vc}). The stochastic GW signal will be present in the $A$ and $E$ channels, but it will be strongly suppressed at low frequencies in the $T$ channel, which means a correlated stochastic signal will be present in $A$ and $E$, while only instrumental noise will be present in $T$. Using the correlation between $A$ and $E$ and the ``null'' signal in $T$ at low frequencies allows the detection of a stochastic GW signal. The authors in \cite{Adams:2010vc} demonstrated using Bayesian techniques that LISA should be able to detect a GW background as weak as $\Omega_{GW} = 6\times 10^{-13}$. However, this estimate was obtained assuming instrumental noise only, i.e., the Galactic foreground from white-dwarf binaries was absent. The capability of LISA to detect a cosmological GW background with other sources in the data stream is yet to be fully quantified, although one early estimation~\cite{HoganBender2001} indicated $\Omega_{GW} \ge 10^{-11}- 10^{-10}$ as a detection threshold.
\section{Constraining cosmological parameters with LISA}
\label{S:cosmo}
The gravitational wave signals from coalescing MBH binaries that we expect to be present in the LISA data will, in general, be quite loud which will allow very precise measurements to be made of the source parameters. In particular, we expect to be able to determine the luminosity distance of a binary at redshift $z=1$ to a precision better than 1\% and to localize it on the sky to within
$10 - 100$ arcminutes \cite{Lang:2008gh}. If the binary is embedded in a circumnuclear disk there might be a transient electromagnetic signal associated with the GW event and, for some nearby events, this e/m counterpart could be detectable (if the GW measurement provides sufficiently precise localisation of the source) which would provide a measurement of the redshift of the source in addition to the luminosity distance $D_L$ that can be measured from the GW data. Measurements of $D_L(z)$ tell us about the cosmological parameters describing the expansion history of the Universe. For this reason, coalescing MBH binaries are referred to as standard sirens~\cite{Schutz:1986gp,Holz:2005df, VanDenBroeck:2010fp} and much effort is currently being put into modelling the possible e/m signals which could accompany the merger of two MBHs (see for example~\cite{Schnittman:2010wy} and references therein).
In the following we will describe results that do not rely on making redshift measurements using e/m observations, but are instead based on a statistical approach to the problem. The motivation for
this analysis is that e/m counterparts may be very weak or indeed not exist at all. We must then ask whether it is still possible to constrain the cosmological parameters by combining GW measurements from multiple sources. While the other sections of this article are reviewing previous work on the capabilities of LISA, the results in this section are new and appear here for the first time.
We can simulate the set of events that LISA will detect by using the model for the hierarchical formation of MBHs described in~\cite{Volonteri:2010py}. The model predicts the number of MBH binary merger events that LISA will see, and the SNR and MBH mass distribution of those events. We assume three years of LISA observations and generate many realizations of the set of events detected by LISA during its lifetime. Typically, we expect there to be about thirty events observed in three years of observation, up to a redshift of $z\le3$. We assume that the MBHs are spinning and that the spins point in random directions and have arbitrary magnitude. We adopt the model for the gravitational wave signal described in \cite{Petiteau:2010zu} which is a restricted waveform with the phase up to second post-Newtonian order. This model includes modulation of the amplitude and the phase due to spin-orbital and spin-spin coupling but neglects all corrections in the amplitude (higher harmonics).
We assume that the errors in the parameters determined from the GW observation have a multivariate Gaussian distribution with variance-covariance as predicted by the Fisher information matrix for this signal model. This is a reasonable assumption due to the expected high strength of the GW signals. It was shown in~\cite{Petiteau:2010zu} that the likelihood surface for this waveform model contains several local maxima and some of these are comparable in likelihood to the global maximum. Here we will assume that the search algorithm used does manage to identify the true maximum. This will not be a trivial task, but we believe that it should be possible if we observe the signal for a reasonably long time (more than half a year) and observe the merger as well as the inspiral (some sky location degeneracies that exist at low frequencies could be broken as the GW signal propagates into frequencies above a few milli-Hertz).
We are interested in the errors that are made in the estimation of the luminosity distance and the sky location. To each gravitational wave event we may then associate an error box (a cylinder~\footnote{In fact, the error box should be an ellipsoid, but for simplicity we approximate it as a cylinder with
the cross-section equal to the $2\sigma$ error ellipse for the sky position and with the length equal to the $2\sigma$ error associated with the uncertainty in $D_L$}). The radius of the
error cylinder is determined by the accuracy with which the GW source can be localized on the sky. The length of the cylinder along the line of sight is determined by the error associated with the measurement of $D_L$ and the uncertainties in the cosmological parameters, which are taken as priors. The error in $D_L$ arising from the GW measurement alone is quite small and dominates the error only at low redshift ($z<0.5$). At higher redshifts, the main source of distance error is weak lensing. Dark and bright matter along the line of sight to the source (de)magnifies the GW signal making it appear
(further) closer than it is in reality. We model weak lensing using the mean errors quoted in~\cite{Shapiro:2009sr} and assume these are normally distributed, although strictly speaking this is not correct~\footnote{The fact
that the error in the luminosity distance associated with weak lensing is not gaussian could potentially
improve our results. This will be explored in a follow-up publication.}. In Figure \ref{F:DLerr} we show
the (median) error in the GW measurements due to instrumental noise as a black (solid) line, the errors due to weak lensing as (red) circles and the combined error as a (blue) circle-solid line.
\begin{figure}[ht]
\center{
\includegraphics[height=0.3\textheight,keepaspectratio=true]{DLerror_GR19}
}
\caption{Relative error in the luminosity distance due to instrumental noise (black solid line),
due to weak lensing (red circles) and the combined error (blue circle-solid line). The weak lensing error
is taken from \cite{Shapiro:2009sr}.}
\label{F:DLerr}
\end{figure}
The next ingredient required is a model for the Universe. We have used the Millennium simulation \cite{Lemson:2006ee} to represent the large scale structure of the Universe at different redshifts. The Millennium simulation assumes a lambda cold dark matter
model of the Universe and so do we. In particular, we model the evolution of the Hubble parameter as
\begin{equation}
H^2 = H_0^2 \left[ \Omega_m^0 (1+z)^3 +
\Omega_{de}^0 \exp \left(
3\int_0^z dz \frac{1+\omega(z)}{1+z}\right) \right].
\end{equation}
with $H_0 = 73.0\,\rm{km/(sec\, Mpc)},\; \Omega_m^0 = 0.25, \;\Omega^0_{de} =
0.75,\; \omega(z) = -1$.
Using the semi-analytic model described in \cite{De_Lucia:2006vua}, one can attach to each dark-matter halo the masses of the components of each galaxy and their luminosity.
Galaxy evolution is followed `self-consistently' by including in each protohalo identified in the simulation at high redshift a $10\%$ mass fraction in baryons, in the form of cooling gas. Baryon evolution is then followed taking into account a whole series of physical processes including cooling, star formation, heating by supernova feedback, bulge formation triggered by mergers, secular disk formation, etc. The parameters defining the recipes describing each of the physical processes are tuned in order to reproduce a broad range of observations (galaxy luminosity functions at different redshift, color distributions, etc)~\cite{DeLucia:2005yk}.
For each GW event in our realisation of the LISA data, we choose the nearest snap shot of the Millennium simulation and associate one of the directions in the comoving volume as the direction of the line of sight (redshift). There are a total of 63 snapshots which are logarithmically spaced in redshift. We have used snapshots corresponding to the redshifts $\{z = 0.51, 1.08, 1.50, 2.07, 2.62, 3.06\}$. Then we chose the galaxy that is the host of the merging MBHs. The probability of the galaxy to be a host is taken to be proportional to the local density. We note, however, that the probability that the host is in a low density region is not small (about 50\%), since although the probability that the merger happens in a low density part of the universe is low, there are many such parts with low local density. Once the host is chosen it is surrounded by the error box (cylinder) described above. In practice we need to place an error box in the sky coordinates and in the redshift and in order to go from the measured luminosity distance to the redshift we must assume a cosmological model. As a result the size of the error box is determined not only by the errors associated with the determination of the luminosity distance but also by the cosmological uncertainty in translating the $D_L$ to a redshift. We assume that all cosmological parameters are known except for the effective equation of state of dark energy, which is modelled as $\omega(z) = -1 + w$. We use the currently estimated range for $w$, which is $w \in [-0.3:0.3]$.
Our main objective is to show that by using all GW observations together it is possible to tighten these constraints on $w$. A similar statistical approach was
employed in \cite{MacLeod:2007jd} to reduce the uncertainty in $H_0$ as derived from GW observations of extreme-mass-ratio inspirals with LISA. The basic idea is to measure the redshift of all galaxies which are potential hosts of the merger, i.e., which lie within the error box, combine these statistically into an estimate of the source redshift, and correlate the results across all GW events. We emphasize several important points:
(i) we assume that, by the time LISA flies, we will be able to measure the redshift to all galaxies with an apparent magnitude $m \le 24$, which means there is a strong selection effect at high redshifts, at which we will see only very massive galaxies, whereas LISA mergers are most likely to be associated with
moderate mass $\sim 10^{10} M_{\odot}$ merging galaxies. The consequence of this assumption is that the host may not be included in the set of galaxies used, but this was also true in~\cite{MacLeod:2007jd} and it was seen not to be a problem in that case. We investigate the importance of this selection effect for the current work in a separate publication~\cite{Babak_inPrep2010}. (ii) The method relies on the fact that the distribution density within the error box is not uniform. The larger the contrast in density along the line of sight the better the method works. For this reason we do not consider GW events at redshifts $z\ge 3$ (iii) Since we might not be able to observe hosts at moderate to high redshifts, we rely on the similarity in the density profile across different mass ranges, i.e., that the spacial distribution of the lighter galaxies which host LISA sources traces the distribution of the more massive galaxies that we can observe. We verified the similarity of the different mass distributions using Millennium data. In our simulations we choose the host from all galaxies in the snapshot but for measuring the redshift we use only the galaxies with apparent magnitude $m\le 24$. We also use only the bright galaxies to estimate the local density. If the chosen host is close to the boundary of the snap shot box, we assume periodic boundary conditions for the data (as was done in the Millennium simulation itself).
The posterior (marginalized) density distribution for $w$ from an observation ($s$) of a single GW event is
\begin{eqnarray}
P_j(w | s) &=& \frac{p_0(w) P_j(s | w)}{E_j} , \\
P_j(s | w) &=& \int \Lambda_j(D_L(z,w), \theta, \phi) p_j(\theta, \phi, z)\; d\theta\; d\phi\; dz; \;\;\;\;
E_j = \int p_0(w) P_j(s | w) dw
\label{poster}
\end{eqnarray}
where $p_0(w)$ is the prior on $w$, $\Lambda_j(D_L(z,w), \theta, \phi)$ is the likelihood
marginalized over the other parameters of the MBH binary so it depends only on $D_L$ and the sky
position ($\theta, \phi$), and $p_j(\theta, \phi, z)$ is the astrophysical prior for a given galaxy to be the host, which is proportional to the square of the local density. The final distribution for $w$ (assuming that the individual events are independent) is constructed from the products of the individual $P_j(s|w)$:
\begin{equation}
P(w) = \frac{p_0(w) \prod_j P_j(s| w)} {\int p_0(w) \prod_j P_j(s | w) dw}.
\end{equation}
We considered 30 realisations of the LISA data and used a flat prior on $w$. The final probability density
$P(w)$ could be well approximated in most of the realisations by a single gaussian and in all other cases by a sum of a small number of gaussian profiles. The peak of the distribution was usually found to be close to zero (which was the true value of $w$ used to generate the data set) and the ($1\sigma$)width of the gaussian was a factor 2-8 smaller than the prior range. A summary of the results of the simulations is shown in Figure~\ref{F:summary}.
\begin{figure}[ht]
\center{
\includegraphics[height=0.25\textheight,keepaspectratio=true]{Sample_n}
}
\caption{Parameters of the gaussian fit for the final probability density $P(w)$ for each of the 30 realizations.}
\label{F:summary}
\end{figure}
To produce this figure, we have fit the posterior in each realisation using a Gaussian and the figure shows the mean (as a circle) and the variance (as an error bar) of the fitted Gaussian for each realisation. In a few cases there was a nearby merger which could be very well localized on the sky, and there was then only one cluster of galaxies in the error box, which is essentially equivalent to having an e/m counterpart. The GW events for which we could identify the e/m counterpart in this way were removed from the analysis to allow us to show how well we can constrain $w$ without an e/m identification of the host. The figure shows that we have very good prospects for constraining the effective equation of state of dark energy provided that we know all the other cosmological parameters (which could be the case by the time LISA is launched). In a separate publication~\cite{Babak_inPrep2010} we will present a more detailed description of our simulations and an analysis of a larger number of realizations with varying parameters (including the effect of using a different weak lensing model or a deeper spectroscopic survey, the effect of including or omitting events with an identified e/m counterpart in the cases where these are present, etc).
\section{Mapping spacetime around compact massive objects in galactic nuclei}
\label{S:mapping}
In this final section of the review, we turn our attention to probing the nature of the DMOs that are observed to be present in galactic nuclei. The common belief is that these DMOs are Kerr BHs, but can we test this assumption? One of the ways to answer this question is through observations, with LISA, of GWs from extreme-mass-ratio inspirals. In such systems, the inspiralling compact object (CO) can spend a significant amount of time in the strong field region of spacetime close to the central object before it plunges. Detailed information about the structure of the spacetime that the CO is orbiting in will be encoded in the GW signals that are received by LISA. If a strong EMRI signal is detected, then we will first need to extract this information, and, if a deviation from ``Kerriness'' is detected, we will need to interpret it. We refer readers to~\cite{Drasco:2006ws} for a review on EMRI waveforms and to~\cite{AmaroSeoane:2007aw} for a review of related astrophysics and concentrate here only on the use of the EMRI signals for mapping spacetime.
The question is, what kind of information in a GW signal would indicate a deviation from the Kerr spacetime in the host system? GW measurements with LISA will rely on matched filtering and will therefore be sensitive to effects that lead to a mismatch in the phase between the signal and the model. For EMRIs, we will detect GWs over $10^4-10^6$ orbits of the CO, and so we will be sensitive to relatively small changes in the orbital motion. Most of the research on this question to date has focussed on how the orbits of test particles differ in spacetimes that deviate from Kerr. We review below the various attempts that have been made to construct and characterize such spacetimes.
The first approach to spacetime mapping was proposed in~\cite{Ryan:1995wh}. This relied on using the multipole decomposition of the metric outside a stationary and axisymmetric central body in general relativity, which can be expressed in terms of mass, $M_l$, and current, $S_l$, multipole moments. For a Kerr BH these moments are determined by two parameters --- $M_0 = M_{BH}$, the total mass of the BH and $S_1 = M_0 a$, the spin --- and all higher moments are determined from these through the relation $M_l + iS_l = M_{BH} (ia)^l$. If more than two moments of a spacetime were measured using GW observations, these could then be tested for consistency with the Kerr solution. It was shown that the different multipole moments enter at different orders in a decomposition of the vertical and radial epicyclic frequencies as a function of orbital frequency for circular-equatorial orbits~\cite{Ryan:1995wh}. These epicyclic frequencies can, in principle, be determined from a gravitational wave observation, allowing the multipole moments to be measured. However, the higher multipoles enter the phasing of the GWs waves with a strength that decreases with multipole number, so in practice we would expect only to be able to measure the lowest few moments. Nonetheless, for an EMRI with masses $10\, M_{\odot}, 10^6\, M_{\odot}$ and with SNR $\sim 100$ it is estimated that we will be able to measure the quadrupole moment of the central object with a precision of $\sim10^{-3}$~\cite{Barack:2006pq}, while simultaneously measuring the mass and spin of the black hole to $\sim10^{-4}$.
The multipole decomposition is not a convenient way to characterize nearly-Kerr spacetimes, since the Kerr spacetime has an infinite number of non-zero multipoles. For that reason, a number of authors have considered various ``bumpy'' BHs, which are solutions to the Einstein field equations that are close to Kerr but depend on a deviation parameter, $\epsilon$, such that for $\epsilon=0$ the spacetime reduces to the Kerr metric. This deviation parameter can in principle be measured or bounded by a GW observation.
\begin{itemize} \item The first such analysis was described in~\cite{Collins:2004ex}. They constructed a bumpy BH using a perturbation of the Schwarzschild metric. This was later extended to rotating objects using an elegant new algorithm in~\cite{Vigeland:2009pr}. In these papers the authors analyzed how the deviations from the Kerr solution modified the fundamental orbital frequencies of the geodesics in the spacetime.
\item A different approach was taken in \cite{Glampedakis:2005cf} in which the bumpy BH was inspired by the Hartle-Thorne metric for slowly rotating bodies. A kludge waveform (without radiation reaction) was constructed and for the first time the confusion problem was considered, i.e., that there could exist a Kerr waveform which is a near-perfect match to the waveform of a quasi-Kerr spacetime with slightly different parameters, if radiation reaction is not taken into account. This ``quasi-Kerr'' metric was used in \cite{Johannsen:2010xs} and follow up papers to analyze the propagation of null geodesics and to consider the imprint of non-Kerrness on electromagnetic observations.
\item In~\cite{Gair:2007kr} the authors considered an exact solution in GR (Manko-Novikov) describing an arbitrary axisymmetric body which can deviate from Kerr at any multipole moment. They again studied the frequencies of geodesic motion and the existence of a third integral of the motion (which is the Carter constant in the Kerr spacetime) as well as precession rates in this general spacetime. It was found that the third integral could be lost under certain circumstances, leading to ergodic motion, which would be a ``smoking-gun'' for a deviation from Kerr if it were observed.
\end{itemize}
In each of the above papers, the main observables considered were the frequencies of geodesic orbits, deviations in which which would show up in GW observations through small phase-shifts over a long observation. The loss of the third integral considered in~\cite{Gair:2007kr} is a more robust signature of a deviation from the Kerr metric as that behaviour is qualitatively different from what is expected in the Kerr spacetime. Another example of a feature that is qualitatively different in a perturbed spacetime was suggested in \cite{LukesGerakopoulos:2010rc}. When an integrable system is perturbed, resonant points become smeared out into resonant chains of islands (the Poincar\'{e}-Birkhoff theorem). Such perturbations would manifest themselves as a persistent resonance in the observed GWs, i.e., by a persistent period of time in which two of the fundamental frequencies of the orbit were commensurate.
In all the above cases, the authors dealt with vacuum solutions of GR, but it is clear that astrophysical black holes will not be pure vacuum solutions. There have also been several attempts to describe a deviation from Kerr due to the presence of matter, for instance by a scalar field with considerable self-interaction (a so-called massive Boson star)~\cite{Kesden:2004qx}. The non-rotating Boson star has no horizon and is not as compact as a BH, so the orbit of an inspiralling compact object could pass inside the Boson star, where the metric deviates from Schwarzschild. This will leave an imprint on the GW signal, as the GW emission will persist after the plunge should have been observed. Another possibility is that the Kerr BH is surrounded by an accretion disk. The authors in~\cite{Barausse:2006vt} considered the gravitational influence of a massive self-gravitating torus around a Kerr MBH on the orbit of a CO that did not intersect the disc. If radiation reaction is not taken into account, one can readjust the orbital parameters, e.g., the mass and spin of the MBH, in the presence of the torus so that the confusion problem (i.e., that we cannot distinguish the waveform from a pure Kerr waveform) discussed earlier is seen. The amount of material required to leave an imprint on the signal was found to be unreasonably high, suggesting that the gravitational influence of an accretion disc would be undetectable even when radiation reaction was taken into account. However, in a follow up paper~\cite{Barausse:2007dy}, the effect of the hydrodynamical drag force on the particle when the CO orbit intersected the disc was considered. In that case, a small amount of energy would be dissipated due to the interaction of the body with the gas on each passage through the disc (possibly several times per orbit). This interaction would lead to small changes in the orbital radius and eccentricity and to a decrease in the orbital inclination, which is qualitatively different to the pure-GR case in which radiation-reaction drives an increasing inclination. However, for the drag force effect to be comparable to the radiation reaction force (self-force) one would again need a very massive accretion disk and a relatively low-mass MBH ($\sim 10^5\, M_{\odot}$). Recently it was shown \cite{Yunes:2010sm} that the presence of a second MBH within a few tenths of a parsec could also leave a measurable imprint on the EMRI waveform. In that case, the effect arises from the fact that the center of mass of the EMRI system is no longer an inertial frame due to the acceleration exerted by the perturber which this leads to a Doppler phase shift of the GW signal.
In all of the proceeding analyses, the calculations were done within the theory of GR. However, deviations in EMRI observations might also arise if the true theory of gravity is \emph{not} GR. In \cite{Sopuerta:2009iy}, the authors considered BHs in Chern-Simons theory, which deviate from Kerr BHs in the fourth multipole moment. This affects geodesic motion and correspondingly the phase of the GW signal. The authors found that the deviations in geodesic orbits could be significant, but more work is required to estimate the precision with which EMRI observations will be able to constrain a Chern-Simons modification to GR. EMRI observations can also be used to test theories of massive gravity. In these theories the GWs have additional polarizations~\cite{Babak:2002uz}, and propagate with a speed different from the speed of light. One would thus expect to see a dispersion when comparing different harmonics of an EMRI signal. If we are lucky enough to see a disruption of a WD by a $\sim 10^5\, M_{\odot}$ MBH \cite{Sesana:2008zc}, then we could also compare the time delay between the GW and electromagnetic signals. EMRIs can also be used to constrain scalar-tensor theories, but this requires the inspiralling object to be a neutron star rather than a black hole. In \cite{Berti:2005qd} it was estimated that observations of a neutron star inspiralling into a $10^4\, M_{\odot}$ spinning BH with SNR $\sim 10$ could bound the Brans-Dicke parameter, $\omega_{BD}$, to $\omega_{BD}>$few$\times 10^3$, which is slightly better than current Solar System bounds from observations of the Cassini satellite.
While most research to date on testing GR using LISA has focussed on EMRIs, test will also be possible using observations of coalescing MBH binaries of comparable mass. In such systems, the deviation from GR will again leave an imprint on the inspiral and correspondingly on the phase of the GWs. For instance, in \cite{Berti:2005qd} the authors estimated the precision with which effects coming from massive gravity could be measured with LISA. They showed that by observing a spinning $M \sim 10^6\, M_{\odot}$ comparable mass MBH binary LISA would be able to constrain the Compton wavelength of the graviton down to $\sim 10^{-16}$ (in units $M$). In addition, the post merger GW signal (quasi normal mode ring-down) depends on the final MBH which, in GR, is characterized only by its mass and spin. If we were able to detect two or more harmonics of the ringdown waveform, we would be able to test the no-hair theorem by making a consistency check~\cite{Berti:2005ys} that the mass and spin measured from each of the harmonics were consistent. This test will only be practical for comparable-mass systems, as the amplitude of ringdown radiation after an EMRI will be too small to allow detection.
\section{Summary}
\label{S:summary}
In this article we have discussed the capability of LISA as a laboratory for testing fundamental
physics. LISA, operating in a low GW frequency band, will be sensitive enough to detect or to set
upper bounds on cosmic string networks and the stochastic gravitational wave background that are much better than those currently available. By analyzing the phasing of the GWs emitted during extreme-mass-ratio inspirals and the coalescence of MBHs we will be able to extract information about the deviation of the spacetime in the host systems from the pure (vaccum) Kerr solution of relativity. Possible sources of such deviations include the presence of a non-Kerr object in the spacetime, perturbing material around the black hole or a deviation of the theory of gravity from general relativity.
We have also presented new results on the capability of LISA to constrain cosmological parameters through GW observations of MBH binaries out to high redshift ($z\le3$). In particular, if we assume that the only unknown parameter is the effective equation of state for the dark energy, $w$, then, using a statistical method, if LISA observes approximately 30 MBH binaries we will be able to reduce the current uncertainty $|\delta w | < 0.3$ by a factor of 2 to 8.
\begin{acknowledgments}
S.B. would like to thank Curt Cutler for help in preparing section on cosmic strings.
\end{acknowledgments}
\section*{References}
\bibliographystyle{apsrev4-1}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,656 |
Q: Can I use Kotlin Arrow-lib with Quarkus in native builds I started a new Kotlin project and i want to use the arrow-kt core Lib in combination with Quarkus (1.12.2). I want to use the native compilation feature of Quarkus with the GraalVM. My first thought was that arrow is a simple lib without reflection but then i read that.
Since GraalVm has a problem with reflection in native executables at runtime, will that be a problem with Arrow? If it is a problem, can i bypass the problem by simply avoiding some features of Arrow?
I know that i can mark classes for reflection in Quarkus/GraalVM.
Which classes are inspected by reflection? Can i simply add reflection information for a few classes or do i need to that for the whole lib or my whole code?
A: Starting in 0.12.0, which is about to release, Arrow does not use reflection. Previously it did in monad comprehensions for all inheritors of MonadContinuation in their bind operation accessing the ContinuationUtils class. In this class, we used reflection to read and write private fields related to the continuation stack labels.
A: As another answer states a newer release might not use reflection, making the question about the particular library not that important. However, for completeness, here are some answers to these questions in general.
Since GraalVm has a problem with reflection in native executables at
runtime, will that be a problem with Arrow?
GraalVM native image uses static analysis during building the executable out of your program. This means that dynamic features of the langauge require explicit configuration to help the analysis to include the necessary classes / methods into the binary. For example, static analysis cannot predict which classes will be accessed through reflection or proxied when these are referenced through strings only which can sometimes are constructed only at runtime.
Can i simply add reflection information for a few classes or do i need to that for the whole lib or my whole code?
You do need to configure all the accesses through the reflection API. The libraries can provide the config for their use of reflection, resources, etc. But if they need refletive access to your application classes then they cannot do that.
The configuration required is in the form of json files, for example a reflection configuration to include a class might look like:
[
{
"name" : "java.lang.String",
"fields" : [
{ "name" : "value", "allowWrite" : true },
{ "name" : "hash" }
],
"methods" : [
{ "name" : "<init>", "parameterTypes" : [] },
{ "name" : "<init>", "parameterTypes" : ["char[]"] },
{ "name" : "charAt" },
{ "name" : "format", "parameterTypes" : ["java.lang.String", "java.lang.Object[]"] }
]
}
]
The example above specifies that the program would like to be able to use java.lang.String reflectively, have access to the fields value and hash and the methods listed.
It might be a bit tedious, however rather straightforward to create config like that. Some frameworks help you by providing annotations to mark classes with and then generate the config themselves.
But if you want to create the config for the library that you don't know, so it's hard to manually create the config, you can use and it's recommended to use the assisted configuration agent.
This means you execute your program enabling a javaagent, which will trace and write down config for all necessary features: resource access, serialization/deserialization, proxies, JNI, reflection, etc.
So you run the application like this and execute the codepaths you're interested in (maybe through your tests) and the output dir will contain the config.
java -agentlib:native-image-agent=config-output-dir=/path/to/config-dir/ -jar myjar.jar
You can then edit the config if needed manually to, for example, extrapolate to the code paths you didn't run with the tracing agent.
Then you run the native image build process passing the config options, for example, for the reflection file config specify:
-H:ReflectionConfigurationFiles=/path/to/reflectconfig.
You can also use the fact that META-INF/native-image directory is the default location for the configuration files, so you don't have to specify the options. For example if you generate the config in the config/META-INF/native-image directory, then you can place it on the classpath for the native image and the files will be picked up automatically:
native-image -cp config -jar myjar.jar
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,472 |
Q: Angular: How to keep $scope properties in sync with controller functions? I have a controller like so:
angular.module('myApp')
.controller('MyCtrl', function ($scope, items, UserFactory) {
$scope.items = items;
$scope.changeItems = function (item) {
// Do something with items, in the UserFactory
UserFactory.removeItem(items);
}
}
Where items is returned by a resolve in the routeProvider.
If UserFactory.removeItem operates on something like cookies or localStorage, or changes something on a model, what is the best way to also update $scope.items
Should I have the UserFactory.removeItem return the new items, or should I do something directly in the $scope.changeItems function?
A: Have you tried observing the changes and doing a callback? Here is a good example how to use them https://stackoverflow.com/a/17558885/5136207
So every time the value in the service is changed you can call the observer function which will lunch a callback to the controllers that are registered and update the values for them.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,137 |
M-production («М-продакшн», «М-продакшн Медиа», «М-продакшн Групп», Телекомпания «Штрихкод») — производитель разнообразных программ для российских телеканалов. Образована в 1993 году на базе телекомпании «ВИD».
Деятельность
Компания была основана в 1993 году для производства игрового шоу «L-клуб» его сотрудниками: директором Маргаритой Житницкой (в телевизионном мире известна как Марго Кржижевская), автором Владиславом Листьевым и ведущим Леонидом Ярмольником. Помимо производства передачи студия занималась выпуском видеокассет с российскими и зарубежными кинофильмами, реже видеопрограммами.
В 1997 году новым партнёром Кржижевской становится бывший редактор «L-клуб» Наталья Билан, после чего студия меняет название на «Я&Co» (Ярмольник и компания) и отделяется от телекомпании «ВИD». В 1999—2002 годах она называлась «Промоушэн Групп», в 2002—2005 годах — «Меридиан холдинг». В 2005 году компания расширяется и приобретает название «МБ-групп».
Являлась основным производителем программ телеканала «Домашний» в 2005—2014 годах.
В 2009 году компанию покидает Наталья Билан в связи с получением должности генерального директора телеканала «Домашний», однако продолжает продюсировать проекты студии. В 2013 году, после ухода Билан с «Домашнего», студия меняет название на «М-продакшн».
«М-продакшн» специализируется на производстве тематических программ, программ lifestyle, развлекательных шоу, а также авторских проектах и ток-шоу, документальных фильмах сериалах и адаптаций зарубежных форматов. Сотрудничала с телеканалами «ОРТ», «СТС», «Карусель». Сейчас сотрудничает с телеканалами: «Россия-1», «Россия К» («Россия-Культура»), «ТВ Центр», «Москва Доверие», «Москва 24».
Ежегодно продюсерский/авторский коллектив «М-продакшн» разрабатывает оригинальные телевизионные форматы, производит контент для российских телеканалов, а общий объем телевизионного производства превышает 3 000 часов.
С 2012 года компания участвует в производстве телевизионных сериалов и фильмов студий «Стори Фильм» и «Свэлл Фильм». Их основатели — Евгения Вильшанская и Кирилл Нерсесян — параллельно занимают в «М-продакшн» должности креативного и технического продюсеров соответственно.
Программы
Сериалы
Телевизионные фильмы (1—2 серии)
Документальные фильмы
Художественные фильмы
Награды
В 2001 году программа «И дольше века...» получила премию ТЭФИ в номинации «Интервьюер» (Владимир Молчанов).
В 2003 году программа «О.С.П.-студия» получила премию ТЭФИ в номинации «Развлекательная программа».
В 2006 году ток-шоу «Детали» получило премию ТЭФИ в номинации «Ведущий ток-шоу» (Тина Канделаки).
В 2008 году шоу «Хорошие шутки» получило премию ТЭФИ в номинации «Звукорежиссёр телевизионной программы» (Александр Зеленов, Александр Белоусов).
В 2012 году программа «Сати. Нескучная классика» получила премию ТЭФИ в номинации «Музыкальная программа. Классика».
В 2012 году ток-шоу «Тем временем» получило премию ТЭФИ в номинации «Ведущий информационно-аналитической программы» (Александр Архангельский).
В 2014 году программа «Полиглот. Немецкий с нуля за 16 часов!» получила премию ТЭФИ в номинации «Просветительская программа».
В 2015 году ток-шоу «Белая студия» получило премию ТЭФИ в номинации «Просветительская программа».
В 2017 году программа «Правила жизни» получила премию ТЭФИ в номинации «Развлекательная программа. Образ жизни».
В 2017 году сериал «Отель последней надежды» получил премию ТЭФИ в номинации «Дневной телевизионный сериал».
В 2017 году ток-шоу «Вечер с Владимиром Соловьёвым» получило премию ТЭФИ в номинации «Интервьюер» (Владимир Соловьёв).
В 2017 году конкурс «Синяя птица» получил премию ТЭФИ в номинации «Развлекательная программа».
В 2019 году ток-шоу «Агора» получило премию ТЭФИ в номинациях «Информационно-аналитическая итоговая программа» и «Ведущий информационно-аналитической итоговой программы» (Михаил Швыдкой).
В 2019 году ток-шоу «Право знать!» получило премию ТЭФИ в номинации «Ведущий общественно-политического ток-шоу прайм-тайма» (Дмитрий Куликов).
Примечания
Ссылки
Официальный сайт
Список проектов, произведённых телекомпанией
Бизнес и немножко для души
Биография Марго Кржижевской на сайте Академии российского телевидения
Первый канал
Россия-1
НТВ
ВГТРК
Телекомпании России
Телекомпании, производящие телепередачи
Телевизионные продакшн компании
ВИD | {
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} | 965 |
A new train storage yard for regional trains, known as stabling, is now operational just north of Wyndham Vale Station.
The new facility will help to replace existing stabling at the E-Gate facility near Footscray — making way for an extension of Wurundjeri Way as part of the West Gate Tunnel Project.
It is essential to meet interpeak stabling needs for V/Line trains operating on the regional rail network, while also ensuring there is capacity to house additional trains in the future.
The project involved construction of a stabling yard, driver facilities and a bypass track connected to the Geelong line, which will allow trains to access the facility without delaying passenger services.
The design of the facility – housing up to six V/Line 'VLocity' trains – also caters for a further stabling expansion and a maintenance facility if needed in the future.
The project commenced in late 2018 and was completed in April 2020.
Regional trains now rolling into new Wyndham Vale stabling yard
V/Line's regional network will now benefit from greater flexibility thanks to the opening of the Wyndham Vale stabling yard on the Geelong line.
Spring-ing into action at Wyndham Vale
Wyndham Vale's new storage facility for V/Line trains is edging closer to completion to provide immediate and long-term benefits to the regional network.
Wyndham Vale Stabling Facility July - October 2019
Progress at the Wyndham Vale Stability Facility from July until October 2019.
Wyndham Vale Stabling Facility - June 2019
Progress pictures of works at Wyndham Vale Stability Facility.
Wyndham Vale - Frequently asked questions
Wyndham Vale Stabling Facility - Frequently asked questions | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,131 |
Dāl (د) – ósma litera alfabetu arabskiego. Używana jest do oznaczenia dźwięku [], tj. spółgłoski zwartej dziąsłowej dźwięcznej. Pochodzi od fenickiej litery Dalet.
W języku polskim litera Dāl jest transkrybowana za pomocą litery D.
W arabskim systemie liczbowym literze Dāl odpowiada cyfra 4.
Postacie litery
Kodowanie
Zobacz też
Alfabet arabski
Alfabet białoruski (arabski)
Przypisy
Litery alfabetu arabskiego | {
"redpajama_set_name": "RedPajamaWikipedia"
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\section{Introduction} \label{sec:intro}
\section{ANTARES and Diffuse Cosmic Neutrinos} \label{sec:main}
The ANTARES neutrino telescope~\cite{antares} is taking data in its final configuration since 2008, see Fig. \ref{fig:lambda} (left). The main physics subject is the search for cosmic sources of neutrinos, even though several results have been obtained in other topics, e.g. neutrino oscillations, searches for dark matter and exotics (monopoles, nuclearites), oceanography and marine biology~\cite{mangano}.
An update
of the analysis on the search for a diffuse flux of cosmic neutrinos~\cite{diffusi09}
with two years of data is presented.
Two years more of data are added to the measurement.
In the meanwhile, our knowledge of the detector has improved and better Monte Carlo (MC) simulations have been made available, allowing to use a larger data fraction for analyses, with less requirements on the data quality.
The equivalent live-time is 885 days now, about a factor three larger than for previous analysis.
\begin{figure}[t!]
\centering
\psfig{figure=antares_fig,width=0.45\textwidth}
\psfig{figure=FLC_lambda_fit,width=0.54\textwidth}
\caption{Left: A schematic view of ANTARES. Right: Data and MC events with an equivalent live-time of 885 days as a function of $\Lambda$, selected with $\theta > 90\ensuremath{^\circ}$, $\beta < 0.5$, $N_{hit} > 35$. $\Lambda$ is negative and takes values closer to zero for well reconstructed tracks.
Red represents atmospheric muons, blue the conventional atmospheric neutrinos, green the cosmic signal $\Phi_{test}\propto E^{-2}$, points are data.
The cut at $\Lambda > 4.9$ (pink vertical line) is adjusted according to an exponential fit of the muon distribution
that reduces the muon background to 0.15\% (1.3 $\mu$) of the total. }
\label{fig:lambda}
\end{figure}
An isotropic diffuse flux of neutrinos generated by extragalactic sources in the Universe is expected.
Atmospheric neutrinos with an energy spectrum $\propto E^{-3.7}_\nu$ represent an irreducible background
in the search for a diffuse flux of cosmic $\nu$ and the signal
might be seen as an excess in the high energy region of the spectrum.
A harder spectrum, proportional to E$^{-2}$, is expected for $\nu$ of astrophysical origin; above an unknown value of the critical energy E$_\nu^c$ (which depends on the absolute normalization of the cosmic $\nu$ flux) astrophysical neutrinos should exceed those of atmospheric origin. The discrimination can be done on the basis of the visible energy of muons generated by neutrinos.
In the following we refer to $\nu_\mu$ and $\overline{\nu}_\mu$ as ``muon neutrinos'', because the sign of the charged muon is indistinguishable.
IceCube results on the search for diffuse neutrinos
using IC59 data~\cite{sullivan} show
no significant excess with respect to the background expectations, hence an upper limit at 90\% confidence level was derived:
\begin{equation}\label{eq:ic59_lim}
E^2 \Phi_{\nu_\mu} = 1.44 \cdot 10^{-8} \ \mathrm{GeV\ cm^{-2}\ s^{-1}\ sr^{-1}} .
\end{equation}
The IC59 sensitivity to diffuse fluxes is a factor two lower than the limit. Even if this result is statistically compatible with zero at 1.8 $\sigma$, there is a soft indication of the presence of high energy astrophysical neutrinos.
A test $\nu_\mu$ signal with a flux proportional to E$^{-2}$ and normalized at
\begin{equation}\label{eq:signal}
E^2 \Phi_{test} = 10^{-8} \ \mathrm{GeV\ cm^{-2}\ s^{-1}\ sr^{-1}}
\end{equation}
is used in this analysis. The normalization is arbitrary and does not affect the result of the following cut optimization.
The entire procedure and the optimization of all the cuts was done using only the 10\% of available data (blind analysis) and the full MC set, in order to avoid any bias.
Down-going atmospheric muons reconstructed as up-going can mimic high-energy neutrino induced muons.
In fact, atmospheric muons reach the detector in ``bundles'' of particles with a large multiplicity.
The main effect is the production of a large amount of hits on the PhotoMultiplier Tubes (PMTs) -- the signature of high energy events.
A cut on the quality of reconstructed tracks is defined to keep under control the atmospheric muon background.
The algorithm that reconstructs the muon direction~\cite{lambda} uses as input the time and position information of hits produced by the photons recorded by the PMTs, and gives as output the direction of the muon (zenith and azimuth), a quality parameter ($\Lambda$), the estimation of the angular resolution ($\beta$), and the number of hits correlated with the track ($N_{hit}$).
An a priori cut on the reconstructed zenith angle is applied
($\theta > 90 \ensuremath{^\circ}$).
A combined cut on the three parameters given by the reconstruction algorithm ($\Lambda$, $\beta$, $N_{hit}$) has been optimized to reduce the muon background
maximizing the total number of signal events.
First level cuts are defined as the combination of:
\begin{equation}\label{eq:FLC}
\Lambda > -4.9 , \ \
\beta < 0.5 , \ \
N_{hit} > 35 .
\end{equation}
Due to the reduced statistics in the muon MC sample (one third of the equivalent live-time in data) the cut on $\Lambda$ is adjusted according to the fit shown in Fig. \ref{fig:lambda} (right). The first level cuts allow to reduce the muon background at the level of 1.3 event (885 days).
After the cuts of eq. \ref{eq:FLC}
the
prevailing
background for cosmic neutrinos is due to atmospheric neutrinos, which are expected to dominate below the critical energy E$_\nu^c$.
The neutrino energy cannot be directly measured
and the neutrino induced muons are observed in a limited interval of their range,
due to the limited size of ANTARES.
An estimate of the muon energy can be done through the measurement of some observables related with the muon energy loss in water. In fact, at energies higher than $\sim500$ GeV, the energy loss is proportional to the energy of the muon.
An energy estimator~\cite{fabian}, $\rho$, is defined through an approximation of the total muon energy deposited in the detector along its path,
$\Delta E / \Delta x \propto \rho(Q_i, \overrightarrow{x}, L_\mu, \epsilon)$, and it is a function of the total number of photoelectrons recorded by PMTs ($Q_i$), the reconstructed muon direction ($\overrightarrow{x}$), the geometrical track length within the sensitive volume ($L_\mu$), and the detector efficiency ($\epsilon$).
The atmospheric neutrinos are simulated according to the conventional ``Bartol'' flux~\cite{bartol},
while the signal is taken assuming the flux of eq. \ref{eq:signal}.
The inverse cumulative distributions of expected neutrinos as a function of $\log \rho$ is shown in Fig. \ref{fig:energy} (left).
The energy estimator $\rho$ is used in the Model Rejection Factor (MRF) procedure~\cite{mrf} to define the cut which gives the best sensitivity.
The MRF as a function of $\rho$ was computed through pseudo-experiments using the Feldman\&Cousins statistics~\cite{feldman} at 90\% confidence level.
The minimum
occurs selecting events with $\log \rho > 3.1$ and corresponds to a sensitivity:
\begin{equation}\label{eq:sensitivity}
\Phi_{sens} = 3.0 \cdot 10^{-8} E^{-2} \ \mathrm{GeV\ cm^{-2}\ s^{-1}\ sr^{-1}} .
\end{equation}
\begin{figure}[t!]
\centering
\psfig{figure=FLC_dEdX_cum_norm,width=0.496\textwidth}
\psfig{figure=Energy_MC,width=0.496\textwidth}
\caption{Left: Inverse cumulative distributions of data (crosses) and Monte Carlo as a function of the energy estimator $\rho$. The blue histogram represents the atmospheric Bartol $\nu_\mu$ normalized to the total number of data events, without any prompt contributions; the green histogram the cosmic neutrinos with the test flux of eq. \ref{eq:signal}; the pink vertical line shows the position of the cut which minimizes the MRF.
Right: Distribution of atmospheric and signal $\nu_\mu$ as a function of the true energy obtained from MC, before and after the energy cut $\log \rho > 3.1$. The region that contains the 90\% of signal is highlighted. }
\label{fig:energy}
\end{figure}
\section{Results} \label{sec:results}
Applying the first level cuts to the data sample, the conventional atmospheric neutrinos from MC simulations show a 27\% deficit with respect to the observed data events.
This is well within the systematic uncertainties on the theoretical expectation at these energies; the $\nu$ background is then normalized to the data.
After normalization, 7.9 atmospheric events are expected for $\log\rho > 3.1$, and 1.8 signal events from the test flux (eq. \ref{eq:signal}).
After unblinding the high energy region, 7 neutrino events are found in the full data sample.
Fig. \ref{fig:energy} (left) shows the inverse cumulative distribution as a function of the energy estimator $\rho$ for
data and MC.
The pink line at $\log\rho = 3.1$ shows the cut value which minimizes the MRF.
The effects of systematics is considered in the calculation of the upper limit.
The evaluation of the systematic errors on the background is taken from the normalization factor applied to Monte Carlo, giving a $\pm27\%$ effect on the predicted 7.9 events. This factor includes the systematics about the knowledge of the detector plus theoretical uncertainties on the conventional neutrino flux.
Concerning the signal, an assumption is done on the flux shape and the absolute normalization does not influence the resulting upper limit.
The variation in the number of expected signal events depends on the detector efficiencies only.
Some critical parameters were changed in the MC simulations to evaluate the signal expectations: absorption length of light in water ($\pm 10\%$), PMT quantum efficiency ($\pm 10\%$), PMT angular acceptance.
The effect of the systematic uncertainties is to change the expected 1.8 signal events by $\pm 14\%$.
Using the method described in Conrad {\it et al.}~\cite{conrad}, the upper limit at 90\% confidence level is:
\begin{equation} \label{eq:limit}
E^2 \Phi_{90\%} = 3.2 \cdot 10^{-8} \ \mathrm{GeV\ cm^{-2}\ s^{-1}\ sr^{-1}}
\end{equation}
The central 90\% of the signal is found in the neutrino energy range from 45 TeV to 6.3 PeV --~see Fig. \ref{fig:energy} (right).
The interval is the region containing 90\% of the signal from MC simulations.
\begin{figure}[t!]
\centering
\psfig{figure=flux_limits,width=0.496\textwidth}
\psfig{figure=NeutrinoFlux_diffuse,width=0.496\textwidth}
\caption{Left: The ANTARES 08-11 upper limit is compared with limits from other experiments and theoretical models. The gray band represents the conventional Bartol flux from vertical to horizontal events.
Right: The upper limit on diffuse fluxes and the unfolded atmospheric neutrino energy spectrum are shown for comparison in the same picture, together with the expectations from the Bartol flux.}
\label{fig:limit}
\end{figure}
\section{Conclusions} \label{sec:conclusions}
ANTARES data taken in the years 2008-2011 were analyzed to search for a diffuse cosmic neutrino signal. The whole period corresponds to 855 days of equivalent live-time. Using an estimator of the muon energy loss in sea water, no excess of events is found with respect to the atmospheric neutrino flux hence an upper limit at 90\% c.l. is obtained. This result is compared with other experiments and theoretical expectations in Fig. \ref{fig:limit} (left) -- see~\cite{diffusi09} for references. The same data set was analyzed to unfold the atmospheric neutrino energy spectrum~\cite{fusco}; Fig. \ref{fig:limit} (right) shows the combination of both results together with the conventional Bartol flux.
\section*{References}
| {
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{"url":"http:\/\/www.physicsforums.com\/showthread.php?p=3632627","text":"# Linear Algebra (showing if a vector spans a vector space)\n\nby JameB\nTags: algebra, linear, showing, space, spans, vector\n P: 25 1. The problem statement, all variables and given\/known data I was wondering if someone could explain the easiest way to determine if a set S spans V? some example questions would be: show that S = {v1, v2, v3, v4} spans R4 where v1 = [1 0 +1 0] v2 = [0 1 -1 2] v3 = [0 2 +2 1] v4 = [1 0 0 1] 2. Relevant equations 3. The attempt at a solution I know that you need to let x = [a, b, c, d] be any vector in R4 and form an eqn like a1v1 + a2v2 + a3v3 + a4v4 = x but now I'm lost... please help!\nMentor\nP: 20,962\n Quote by JameB 1. The problem statement, all variables and given\/known data I was wondering if someone could explain the easiest way to determine if a set S spans V? some example questions would be: show that S = {v1, v2, v3, v4} spans R4 where v1 = [1 0 +1 0] v2 = [0 1 -1 2] v3 = [0 2 +2 1] v4 = [1 0 0 1] 2. Relevant equations 3. The attempt at a solution I know that you need to let x = [a, b, c, d] be any vector in R4 and form an eqn like a1v1 + a2v2 + a3v3 + a4v4 = x\nThis is a good start. You know v1, v2, v3, and v4, so you have essentially a system of four equations in four unknowns (a, b, c, and d). Write the system as an augmented matrix [A|x] and row reduce.\nP: 25\n Quote by Mark44 This is a good start. You know v1, v2, v3, and v4, so you have essentially a system of four equations in four unknowns (a, b, c, and d). Write the system as an augmented matrix [A|x] and row reduce.\nThen what do I do?\n\nMentor\nP: 20,962\n\n## Linear Algebra (showing if a vector spans a vector space)\n\n P: 25 I get a1v1 + a2v2 + a3v3 + a4v4 = x => a1 00 00 a4 = a 00 a2 2a3 00 = b a1 -a2 2a3 00 = c 00 2a2 00 a4 = d => 1 0 0 1 | a 0 1 2 0 | b 1 -1 2 0 | c 0 2 0 1 | d => RRE form 1 0 0 .... OOooohhh I see!! Now when I turn it into RREF, I'll get something like a1 = a+b-c etc.. correct?! => RREF 1 0 0 1 | a 0 1 2 0 | b 0 0 1 (-1\/4) | d-2b\/-4 0 0 0 0 | c-a+b+(d-2b) Does that look like I'm on the right track?\nP: 703\n Quote by JameB 1. The problem statement, all variables and given\/known data I was wondering if someone could explain the easiest way to determine if a set S spans V? some example questions would be: show that S = {v1, v2, v3, v4} spans R4 where v1 = [1 0 +1 0] v2 = [0 1 -1 2] v3 = [0 2 +2 1] v4 = [1 0 0 1]\nThe easiest way is to calculate the determinant of the 4x4 matrix formed by v1, v2, v3 and v4. If the determinant isn't zero, then the vectors span R4.\nP: 25\n Quote by The Electrician The easiest way is to calculate the determinant of the 4x4 matrix formed by v1, v2, v3 and v4. If the determinant isn't zero, then the vectors span R4.\nRight, but what happens if it's not a square matrix? I'm pretty sure my prof won't make it that easy on a test :(\nEmeritus\nHW Helper\nThanks\nPF Gold\nP: 11,521\n Quote by JameB OOooohhh I see!! Now when I turn it into RREF, I'll get something like a1 = a+b-c etc.. correct?!\nRight.\n => RREF 1 0 0 1 | a 0 1 2 0 | b 0 0 1 (-1\/4) | d-2b\/-4 0 0 0 0 | c-a+b+(d-2b) Does that look like I'm on the right track?\nI think you made a mistake somewhere (or I did). I didn't get all 0s on the last row.\nP: 2,568\n Quote by vela Right. I think you made a mistake somewhere (or I did). I didn't get all 0s on the last row.\nYou didn't, I did it with Maple and I got all pivots except last column.\n\nNow @OP, what does this mean? Having no non-pivot columns? How does this relate to Span?\n Mentor P: 20,962 And to continue with what vela and flyingpig said, think about what system of equations you're reduced augmented matrix represents. If is any arbitrary vector in R4, is there a specific linear combination of the vi vectors that forms that vector ? If so, those vi vectors span R4.\n P: 25 Umm I don't remember learning about pivots. and I did the thing again and I still got 0 0 0 0 for the last row :O Here are my steps: -r1 + r3 -> r3 r2 + r3 -> r3 r4-2r2 -> r4 1\/4 r3 -> r3 r4 + 4r3 -> r4 this gave me a final matrix of 1 0 0 1 a 0 1 2 0 b 0 0 1 -1\/4 [(c-a+b)\/4] 0 0 0 0 d-2b+4c Now what do I do? Or even better, could someone please explain to me the general steps of finding if a vector spans the v.s. so I can complete these problems? Thank you! EDIT: Mark I just saw your response, how would I know if there is a linear combination that forms ?\n P: 703 It's not clear to me whether you want to demonstrate a knowledge of basic methods of determining whether a set of vectors spans a space, or whether you just want to know how to use functions available in modern mathematical software such as Maple, Mathematica or Matlab to make the determination. If it's the latter, proceed as follows: If you have a bunch of N element vectors, let's say you have M of them, such that M\u2265N, let the vectors form the rows of an MxN matrix. Then you don't need to augment that matrix to determine if there are at least N linearly independent vectors in the set. Simply row reduce the MxN matrix. The number of 1's on the main diagonal of the result is the number of linearly independent vectors in the set. Only if that number is N does the set span the space R(N). The number of linearly independent rows (your vectors) is also equal to the rank of the MxN matrix of vectors, and you can use the \"rank\" function to determine this. You could also calculate the singular values of the MxN matrix; the rank is equal to the number of non-zero singular values (that's what the \"rank\" function does). See the attached image. Attached Thumbnails\n P: 25 Sorry, I don't mean to be rude but I'm just kinda lost. This is my first course in linear algebra and that's why I'm pretty confused. What I was trying to do was to find a basis. The theorem states that vectors are said to form basis for V if a) if they span V b) they are linearly independent I know that to prove b) I need to put it in a matrix, reduce and if I get a matrix with trivial solution meaning everything equals to 0 then it's a trivial solution and it's linearly independant. But I'm having trouble with checking if they span or not... that's the whole point of this post :) The examples in my book don't show step by step solution so that's why I'm lost. e.g. Show that S = {v1, v2, v3, v4} where v1 = [1 0 0 1], v2 = [0 1 -1 2], v3 = [0 2 2 1], v4 = [1 0 0 1] is a basis for R4. (R4 is written as R subscript 4 meaning it's referring to the row spaces?...) to show that S is linearly independant, I formed the eqn a1v1 + a2v2 + a3v3 + a4v4 = 0 turned it into a augmented matrix and got its RREF. This means that it's linearly independent. NOW, to show that S spans R4, I let x = [a, b, c, d] be a vector in R4 what should I do next? thanks!\nMath\nEmeritus\nThanks\nPF Gold\nP: 38,879\n Quote by JameB Sorry, I don't mean to be rude but I'm just kinda lost. This is my first course in linear algebra and that's why I'm pretty confused. What I was trying to do was to find a basis.\nI don't mean to be rude but this is your 6th post in this thread and that is the first mention of finding a basis!\n\n The theorem states that vectors are said to form basis for V if a) if they span V b) they are linearly independent\nThat's not a theorem, that is the definition of \"basis\". You can add a \"c\" to this- the number of vectors in the set is the dimension of the space. And then there is a theorem that says any two implies the third.\n\n I know that to prove b) I need to put it in a matrix, reduce and if I get a matrix with trivial solution meaning everything equals to 0 then it's a trivial solution and it's linearly independant.\nWell, no, you don't need to do that- that's one method. What you need to do is think about the definition: a set of vectors spans a space if and only if, any vector, y, in the space can be written as a linear combination of the vectors in the set. If the set is $\\{v_1, v_2, \\cdot\\cdot\\cdot, v_n\\}$ then there exist scalars, $a_1, a_2, \\cdot\\cdot\\cdot, a_n$ such that $y= a_1v_1+ a_2v_2+ \\cdot\\cdot\\cdot+ a_nv_n$.\n\n But I'm having trouble with checking if they span or not... that's the whole point of this post :) The examples in my book don't show step by step solution so that's why I'm lost. e.g. Show that S = {v1, v2, v3, v4} where v1 = [1 0 0 1], v2 = [0 1 -1 2], v3 = [0 2 2 1], v4 = [1 0 0 1] is a basis for R4. (R4 is written as R subscript 4 meaning it's referring to the row spaces?...) to show that S is linearly independant, I formed the eqn a1v1 + a2v2 + a3v3 + a4v4 = 0 turned it into a augmented matrix and got its RREF. This means that it's linearly independent.\nThen you have a problem- either you did the problem wrong or you gave the vectors incorrectly here. Those vector are NOT independent because $v_1$ and $v_2$ are the same: $(1)v_1+ (-1)v_2= 0$.\n\n NOW, to show that S spans R4, I let x = [a, b, c, d] be a vector in R4 what should I do next? thanks!\nIf those vectors were actually the ones you were given, you do nothing now. Since there are 4 vectors and the are NOT indpendent, they cannot span R4. If you gave the fouth vector incorrectly and the ones you were given were independent, you still would not have to do anything more- four independent vectors must span R4.\n\nIn either case, showing linear independence or spanning, you are, in effect, solving a system of equations. You are either trying to solve $a_1v_1+ a_2v_2+ a_3v_3= 0$ (for linear independence) or $a_1v_1+ a_2v_2+ a_3v_3+ a_4v_4= y$ for any y in the space (for spanning). You could do either by setting up the \"augmented\" matrix with the right side as the \"fifth column\". Of course, with all \"0\"s in the fifth column, no row operations will change those: $a_1= a_2= a_3= a_4= 0$ is an obvious solution. Showing that you can reduce the first four columns to the identity matrix is enough to show that is the only solution. Similarly, to show that the set spans the space you add the components of the vector y as the fifth column. You would row-reduce the matrix in exactly the same way, just doing the row operations on the fifth column on the y- components rather than \"0\"s. But the actual value of the coefficients is not important, only that you can find them. So the values you wind up with in the fifth column is not important, only that you can reduce the first four columns to the identity matrix- exactly what you did to show independence. They really are exactly the same thing.\n\nWhen someone said you could look at the determinant of the matrix, earlier, you asked what to do if the matrix is not square. In that case, there is not work to do- such a set of vectors cannot be a basis (though it may still span the space or be independent). The number of columns of the matrix, the number of components in each vector, is the dimension of the vector space. The number of rows is the number of vectors in the set. And those must be equal in order to have a basis. If there are fewer vectors than the dimension- if there are more columns than row- they might be independent but cannot span the space. If there are more vectors than the dimension- if there are more rows than columns, they might span the space but cannot be independent.\n P: 25 Thank you all for being patient and helping me, I do understand it now! I really appreciate your help guys! :) and as a side note to anyone who comes across this, yes I did make a mistake in the vectors (late night mistakes...) v1 was actually [1 0 1 0]\n\n Related Discussions Calculus & Beyond Homework 3 Calculus & Beyond Homework 5 Calculus & Beyond Homework 2 Calculus & Beyond Homework 4 Calculus & Beyond Homework 1","date":"2014-04-18 08:14:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.6222982406616211, \"perplexity\": 271.31626076838165}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-15\/segments\/1397609533121.28\/warc\/CC-MAIN-20140416005213-00233-ip-10-147-4-33.ec2.internal.warc.gz\"}"} | null | null |
Авторы: Lys E.V. (ИНГГ СО РАН) Lisitsa V.V. (ИНГГ СО РАН) Reshetova G.V.
The paper presents an original approach to numerical simulation of sonic logging for tilted transversely isotropic media with attenuation. We suggest a new way to introduce attenuation to the stiffness tensor components for anisotropic media, so that the quality factors of the elastic waves are of a given quantity. The numerical approach itself is based on the Lebedev finite difference scheme on staggered grid applied to cylindrical coordinate system which is more efficient than well-known rotated staggered grid scheme. The numerical experiments, presented in the paper, were done to illustrate an impact of anisotropy and attenuation on a sonic logging, for example appearance of azimuthal component of the displacement velocity even in case of volumetric source acting at the axis of a fluid-filled borehole. | {
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John Millard Tawes (* 8. April 1894 in Crisfield, Somerset County, Maryland; † 25. Juni 1979 ebenda) war ein US-amerikanischer Politiker (Demokratische Partei) und von 1959 bis 1967 Gouverneur des Bundesstaates Maryland.
Leben
Millard Tawes besuchte die öffentlichen Schulen im Somerset County und dann bis 1912 die Wilmington Conference Academy in Dover (Delaware). Danach studierte er das Buchhaltungswesen und das Bankgeschäft am Sadlers, Bryant and Stratton Business College. Nach der Schulzeit war Tawes zunächst im Holzhandel, im Bankwesen und im Schiffbau tätig, ehe er in die Politik ging.
Von 1930 bis 1938 war er in der Verwaltung des Bezirksgerichtes im Somerset County angestellt. Zwischen 1938 und 1947 und nochmals von 1950 bis 1959 war er Leiter des Rechnungshofes von Maryland. Von 1947 bis 1950 war er Bankenbeauftragter (State Bank Commissioner) in diesem Staat. Im Jahr 1946 scheiterte er in den Vorwahlen seiner Partei, als er deren Nominierung als Spitzenkandidat für die Gouverneurswahlen anstrebte.
Gouverneur von Maryland
Am 4. November 1958 wurde Millard Tawes zum neuen Gouverneur von Maryland gewählt. Er trat sein neues Amt am 14. Januar 1959 an und konnte nach einer Wiederwahl im Jahr 1962 bis zum 25. Januar 1967 im Amt bleiben. In dieser Zeit wurden in Maryland unter anderem die Ministerien für wirtschaftliche und industrielle Entwicklung geschaffen. Außerdem entstand eine landwirtschaftliche Beratungsstelle (Agricultural Advisory Board). Das Schulsystem des Staates wurde reformiert und verbessert und der Ausbau der Straßen weiter vorangetrieben. Damals wurde mit einem Gesetz die Rassentrennung in öffentlichen Einrichtungen aufgehoben. Glücksspielautomaten wurden unter Gouverneur Tawes in Maryland verboten. Als Gouverneur förderte Tawes auch den Umweltschutz. Die Anzahl der staatlichen Parks wurde verdoppelt. In seiner Amtszeit wurden auch die Wahlkreise im Staat neu eingeteilt.
Umweltpolitisches Engagement
Nach dem Ende seiner Amtszeit war Tawes von 1967 bis 1968 Delegierter auf einer Konferenz zur Überarbeitung der Staatsverfassung. Im Jahr 1969 wurde er erster Leiter einer Dienststelle, die sich mit den Bodenschätzen von Maryland befasste (Department of Natural Resources). In diesem Amt setzte sich Tawes erneut für den Umweltschutz bei der Ausbeutung der Bodenschätze ein. Zwischen 1973 und 1975 amtierte Tawes als Finanzminister (State treasurer) von Maryland. Das war sein letztes öffentliches Amt. Er starb im Juni 1979. Mit seiner Frau Helen Avalynne Gibson hatte er zwei Kinder.
Weblinks
Tawes in der National Governors Association (englisch)
Gouverneur (Maryland)
Mitglied der Demokratischen Partei (Vereinigte Staaten)
US-Amerikaner
Geboren 1894
Gestorben 1979
Mann | {
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We belongs to Upper middle class, Nuclear family background with MODERATE values MY DAUGHTER IS ANGSHIK MANGLIK .
My daughter is Banking Professional (GOVERMENT/ PSU) SECTOR ) after BE/Btech . She is Intelligent, Loyal,Honest, Well Mannered and well cultured, Moderate with traditional value.We are not paid member at this site.We are members at I AM PRACTICING CA , MY WIFE IS HOMEMAKER. I HAVE ONE SON WHO IS ALSO BANKING PROFESSIONAL . We come from an upper middle class, nuclear family background with moderate values. | {
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Oregon offensive coordinator Scott Frost reportedly interviewed for Boise State job
Updated: Jan. 10, 2019, 5:22 a.m. | Published: Dec. 13, 2013, 3:42 p.m.
By Seth Prince | The Oregonian/OregonLive
Scott Frost.JPG
Scott Frost
(Thomas Boyd/The Oregonian)
Some more details out of Boise today via Chadd Cripe that shed light on how the Broncos went about their coaching search -- including news that Oregon offensive coordinator Scott Frost interviewed:
That means all six of Boise State's finalists had ties to the state of Oregon, and four to the Oregon Ducks specifically:
1. Bryan Harsin, who got the job and was previously Arkansas State's head coach, started out as
running backs and receivers coach at Eastern Oregon in 2000.
2. Justin Wilcox, a Junction City native who is now Washington's defensive coordinator, played for the Ducks in the late 1990s.
3. Dirk Koetter, who is the Atlanta Falcons' offensive coordinator, was Oregon's offensive coordinator from 1996-97.
4. Scott Frost, who is currently Oregon's offensive coordinator and quarterbacks coach, was the Ducks' assistant overseeing wide receivers from 2009-12.
5. Bob Gregory, who is currently Boise State's linebackers coach and will lead the Broncos in their Hawaii Bowl matchup against Oregon State, was previously defensive backs coach at Oregon from 1998-2000 and defensive coordinator at Willamette from 1992-97.
6. Chris Strausser, who is currently Boise State's associate head coach, was Portland State's
offensive line coach and run-game coordinator in 2000, and the Vikings' off
ensive line coach and recruiting coordinator in 1993-94. He was an assistant at Oregon State from 1990-91, coaching
running backs the first year and offensive tackles and tight ends his second year. | {
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{"url":"https:\/\/www.physicsforums.com\/threads\/how-i-solve-this-indefinite-integral.809292\/","text":"# How i solve this indefinite integral?\n\nTags:\n1. Apr 18, 2015\n\n### sleepwalker27\n\n1.\nhttp:\/\/www.imageurlhost.com\/images\/cnj1t05jh6e4fxqy4i5_integral.png\nI know that this integral is solved by the sustitution method\n\n3. The attempt at a solution\nI tried converting the integral to the form of Arctanx, but that x2 on the numerator ruined everything. Thanks\n\n2. Apr 18, 2015\n\n### Staff: Mentor\n\nPlease show us the substitution you used, which sounds like a trig substitution.\n\n3. Apr 18, 2015\n\n### sleepwalker27\n\nI tried to made the integral to the form \u222bdx\/x2+1 so that their solution is something like arctanx\n\nLast edited by a moderator: Apr 18, 2015\n4. Apr 18, 2015\n\n### Staff: Mentor\n\nI figured you did something like that, but that isn't what I asked you for. Please show me your substitution.\n\n5. Apr 19, 2015\n\n### HallsofIvy\n\nDid you not notice that the numerator and denominator have the same degree?\n\n$$\\frac{2x^2}{2x^2+ 1}= 1- \\frac{1}{2x^2+ 1}$$\n\nKnow someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook\n\nHave something to add?\nDraft saved Draft deleted","date":"2018-02-23 13:14:13","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\": 0.7591388821601868, \"perplexity\": 3622.3659417199906}, \"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-2018-09\/segments\/1518891814700.55\/warc\/CC-MAIN-20180223115053-20180223135053-00331.warc.gz\"}"} | null | null |
'A Woman's correct to tradition' is a brand new and insightful research of the standard meme that cultural rights in foreign legislations are at odds with the rights of girls in affected societies. instead of seeing those ideas as jointly specific, Linda Veazey frames cultural rights — via exact case reviews and research of legislations — in a fashion that comes with and enriches the very gender-protective norms they can be notion to defeat.
A new and critical booklet in foreign human rights, and gender experiences, from the autonomous educational press Quid seasoned Books.
Lecturers and policymakers usually talk about international governance yet they deal with governance as a constitution or procedure, hardly ever contemplating who truly does the governing. This quantity makes a speciality of the brokers of worldwide governance: 'global governors'. the worldwide coverage enviornment is full of a wide selection of actors similar to overseas agencies, firms, expert institutions, and advocacy teams, all trying to 'govern' task surrounding their problems with hindrance.
Peter Yearwood reconsiders the League of countries, no longer as an try to become aware of an idea yet as a component within the day by day behavior of Britain's international coverage and household politics through the interval 1914-25. He demanding situations the standard view that London reluctantly followed the belief in line with strain from Woodrow Wilson and from household public opinion, and that it was once really cautious of principles of collective safeguard.
This quantity discusses the influence of human rights legislation on different fields of overseas legislations. Does foreign human rights legislations alter different fields of foreign legislations? Contributions concentrate on attainable spillover results of human rights on foreign monetary or foreign legal legislation. Does overseas human rights legislation have a streamlining impression on overseas legislation as a complete?
Over 100,000 United international locations uniformed peacekeeping group of workers are deployed on missions with authority from the safety Council to guard civilians in clash zones. bankruptcy VII of the UN constitution allows using strength on UN missions, yet doesn't checklist the principles governing the use; they're present in both the jus in bello provisions of overseas humanitarian legislations (IHL) or the laws at the use of strength in foreign human rights legislation. | {
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// $Id: Server_Logging_Handler.cpp 81085 2008-03-25 10:43:41Z johnnyw $
#if !defined (ACE_SERVER_LOGGING_HANDLER_C)
#define ACE_SERVER_LOGGING_HANDLER_C
#include "Server_Logging_Handler.h"
ACE_RCSID(lib, Server_Logging_Handler, "$Id: Server_Logging_Handler.cpp 81085 2008-03-25 10:43:41Z johnnyw $")
// The following are "Factories" used by the ACE_Service_Config and
// svc.conf file to dynamically initialize the state of the
// single-threaded and multi-threaded logging server.
ACE_SVC_FACTORY_DEFINE (ACE_Server_Logging_Acceptor)
ACE_SVC_FACTORY_DEFINE (ACE_Thr_Server_Logging_Acceptor)
#if defined (ACE_HAS_EXPLICIT_STATIC_TEMPLATE_MEMBER_INSTANTIATION)
template u_long
ACE_Server_Logging_Handler_T<LOGGING_PEER_STREAM,
u_long,
ACE_NULL_SYNCH,
Null_Synch_Static_Receiver>::request_count_;
#endif /* ACE_HAS_EXPLICIT_STATIC_TEMPLATE_MEMBER_INSTANTIATION */
#endif /* ACE_SERVER_LOGGING_HANDLER_C */
| {
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I work full time as a writer and a friend and long time collaborator of mine has asked whether I can help co author their book.
This job would entail two or three evenings a week on top of my job. I forsee that I would spend around 3-4 hours each evening on the task which would see me produce upwards of 2000 words.
I also need to consider how much to charge for a speech/blog post written that came to over 3000 words.
Considering that it's a friend and that I will be a listed author, I am imagining that I should and will offer a lower rate than if those two facts weren't the case.
How much would be appropriate to charge in this instance, preferably as an evening / hourly rate?
It would be good to consider both scenarios, one if it were not a friend and I were not a co author and the other as I have described.
I've faced this issue from both sides (as a writer and co-writer/editor) so hopefully I can offer some useful advice.
First off, be clear as to what your role/status is: to be blunt, is it a partnership or are you more like an employee? If you're effectively working for your friend, then an hourly rate (as others have suggested). The upside: you definitely get paid; the downside: it's unlikely to be a lot. In the UK, a typical advance for fiction from an unknown author will be £5,000-£10,000, so your friend won't have a lot of money to play with.
If it's more collaborative, then you need to agree on what percentage contribution you're making to the book, and write that into a contract. So if you decide on one-third, you get 33% of the advance and 33% of future royalties.
It's not clear how much work you're doing - do you mean 2,000 words in total? If so, then that's a tiny proportion of the total, and the 'collaborator' aspect isn't appropriate.
I think this is VERY opinion based due to the situation, so here are my opinions.
First, I agree with @Cyn, put the deal in writing and in detail and signed and dated. Contracts are simple, and if either the best or worst happens, you will want to refer to what you agreed upon to start, before the work went to hell or broke records.
Personally, I would not do this without some share of the income; and my rate would vary from "minimal" at a 50/50 split, to "no thanks" if there is no split.
I would insist on co-authorship (second name is fine).
My personality is my own, but if I wrote half of a best-seller and my friend got $1,000,000 dollars and I got $50,000: I would resent that scenario. Fair is fair, and I don't think either "hours" or "words finished" are a good measure of creative contribution. They simply do not capture the difference between 300 pages of crap and 300 pages readers cannot put down.
For that reason, I would not go in as less than a 40% partner, and even then I'd have to truly love their plot or universe or characters or something about what they have already accomplished. Don't forget your opportunity cost, in those evenings you could be writing your own stuff, or hanging with people you love, or reading books (or books on writing), or watching TV.
And I can make exceptions to that rule if the expenses of marketing or selling the work are being shouldered by my friend; any significant financial risk is worth some % too.
Doing contracts for some company, I typically get 2.5x my "daily job" rate to give up that idle time, and I never take on the role of employee with a friend. With friends, it's a joint venture or nothing. If they want anything else, then I assume they don't think my contribution is really necessary to the success of their project, in which case I am not interested in the participation.
In short, working on this project will almost certainly change your relationship with your friend (same goes for working with family), and you need to engineer things so whether the project succeeds or fails miserably, you have not created resentment by either of you. So you won't resent your share if there is great success, and your friend won't resent what they paid you if the result is a terrible failure.
Something in the range of $50/hour (assuming US dollars within the US or the equivalent in industrial countries) is reasonable for professionals. It's about what artists charge to do illustrations and the like. Some charge more, some charge less. My guess is anything from $20-70/hour is what people might charge, but it really depends. I have not seen polls or all that many rates, so it's completely a guess, just to give you a ballpark.
It is also reasonable to charge less per hour when you have a longer assignment. If it's a 1-2 hour job you should charge more than if it's a 10 hour job, since the parts you don't charge for (negotiating, setup, etc) will be proportionately less as you do more work.
If you give your friend a discount, put the full price in the contract then state the discount, so it's on the record.
If you decide to trade some of your fee in exchange for a byline or royalties, put that in the contract too.
If you weren't thinking about a contract, I urge you to rethink it. Don't do this verbally. Even if you're best friends, it's still easy to misremember a detail (which can turn out to be vital) or have a misunderstanding. Make it super clear. Especially in regards to royalties, credit, and reprint rights.
Not the answer you're looking for? Browse other questions tagged book collaboration authorship freelance or ask your own question.
How does one go about self-publishing their first book? | {
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\section{ \bf Definitions, Berge $F$ subhypergraphs}
A hypergraph ${\mathcal H}$ is $r$-\emph{uniform} or simply an {\em $r$-graph} if it is a family of $r$-element subsets of a finite set $V({\mathcal H})$.
If the \emph{vertex set} $V({\mathcal H})$ is clear from the text, then we associate an $r$-graph ${\mathcal H}$ with its edge set $E({\mathcal H})$.
Usually we take $V({\mathcal H})=[n]$, where $[n]$ is the set of first $n$ integers, $[n]:=\{ 1, 2, 3,\dots, n\}$.
We also use the notation ${\mathcal H}\subseteq \binom{[n]}{r}$. For a set of vertices $S \subseteq V({\mathcal H})$ define the \emph{codegree} of $S$, denoted as $\deg(S)$, to be the number of edges of ${\mathcal H}$ containing $S$.
The $s$-\emph{shadow}, $\partial_s{\mathcal H}$, is the family of $s$-sets contained in the edges of ${\mathcal H}$. So $\partial_1{\mathcal H}$ is the set of non-isolated vertices, and $\partial_2{\mathcal H}$ is the graph whose edges are the pairs with positive co-degree in ${\mathcal H}$.
\begin{definition}
For a graph $F$ with vertex set $\{v_1, \ldots, v_p\}$ and edge set $\{e_1, \ldots, e_q\}$, a hypergraph $\mathcal H$ contains a {\bf Berge $F$} if there exist distinct vertices $\{w_1, \ldots, w_p\} \subseteq V(\mathcal H)$ and distinct edges $\{f_1, \ldots, f_q\} \subseteq E(\mathcal H)$, such that if $e_i = v_{\alpha} v_{\beta}$, then $\{w_{\alpha}, w_{\beta}\} \subseteq f_i$. The vertices $\{w_1, \ldots, w_p\}$ are called the {\bf base vertices} of the Berge $F$.
\end{definition}
\begin{definition} For a graph $F$ with vertex set $\{v_1, \ldots, v_p\}$ and edge set $\{e_1, \ldots, e_q\}$, a hypergraph $\mathcal H$ contains an {\bf induced Berge $F$} if there exists a set of distinct vertices $W:=\{w_1, \ldots, w_p\} \subseteq V(\mathcal H)$ and distinct edges $\{f_1, \ldots, f_q\} \subseteq E(\mathcal H)$, such that if $e_i = v_{\alpha} v_{\beta}$, then $\{w_{\alpha}, w_{\beta}\} = f_i\cap W$.
\end{definition}
In particular, in the case that $\mathcal H$ is a graph (2-uniform), an induced Berge $F$ is just any copy of $F$ in $\mathcal H$, not to be confused with the notion of induced subgraphs.
If the two hypergraphs have the same number of edges, $e({\mathcal H})=e({\mathcal F})$, then we say that ${\mathcal H}$ itself is a(n induced) Berge $F$ hypergraph.
The set of $r$-uniform (induced) Berge $F$ hypergraphs is denoted by $\{{{\rm Berge \,}} (F)\}_{r}$ ($\{{\rm ind'd \, Berge \,} (F)\}_{r}$, resp.).
For example, if $F$ is a triangle, $E(F)= \{ 12, 13, 23\}$, then $\{B(F)\}_3$ contains four triple systems:
$\{12a, 13a, 23a \}$, $\{12a, 13a, 23b \}$, $\{12a, 13b, 23c \}$, and $\{123, 13a, 23b \}$. The first three of them contains an induced $C_3$, the fourth does not.
Parenthesis and indices are omitted when it does not cause ambiguities.
\subsection{Three types of extremal numbers}
Given a set of $r$-graphs ${\mathcal F}$ the hypergraph ${\mathcal H}$ is called ${\mathcal F}$-\emph{free} if it does not have any subgraph isomorphic to any member of ${\mathcal F}$.
The \emph{Tur\'an number} of ${\mathcal F}$, denoted by ${\rm{ex}}_{r}(n, {\mathcal F})$, is the maximum size of an ${\mathcal F}$-free ${\mathcal H}\subseteq \binom{[n]}{r}$.
Usually it is assumed that $|{\mathcal F}|$ is finite, so the well-known fact ${\rm{ex}}_{2}(n, \{ C_3, C_4, C_5, \dots\})= n-1$ usually is not considered a Tur\'an type result because the set of forbidden graphs ${\mathcal F}$, the set of all cycles, is infinite.
If $r=2$ then the index is usually omitted.
Also if ${\mathcal F}$ has only one member, ${\mathcal F}=\{ F\}$, then we write ${\rm{ex}}_{r}(n,F)$ instead of ${\rm{ex}}_{r}(n, \{ F\})$.
The {\em generalized Tur\'an number} for graphs, pioneered by
Erd\H os~\cite{Erdos} and recently systematically investigated by Alon and Shikhelman~\cite{alon-shik}, is the following extremal problem.
We only formulate the case relevant to this paper.
Given a graph $F$, let ${\rm{ex}}(n,K_r,F)$ denote the maximum possible number of copies of $K_r$'s in an $F$-free, $n$-vertex graph, i.e.,
\[
{\rm{ex}}(n,K_r, F) := \max \left\{ |{\mathcal N}_r(H)|: H \text{ is }F\text{-free }, H \subseteq {[n] \choose 2} \right\},
\]
where ${\mathcal N}_r(H)\subseteq \binom{[n]}{r}$ is the family of $r$-element vertex sets that span a $K_r$ in $H$.
In particular ${\mathcal N}_2(H)= E(H)$ and ${\rm{ex}}(n, K_2, F) = {\rm{ex}}(n, F)$ is the regular Tur\'an number of $F$.
For a graph $F$ and positive integer $r$, let
\[{\rm{ex}}_{r}(n, {{\rm Berge \,}} F) := \max \{e(\mathcal H): \mathcal H \subseteq {[n] \choose r} \text{ and } \mathcal H \text{ is Berge } F\text{-free}\}.
\]
Ever since Gy\H ori, G.~Y.~Katona, and Lemons~\cite{GKL} investigated hypergraphs without long Berge paths there is a renewed interest concerning extremal Berge type problems.
Here we define a related function, the \emph{induced Berge Tur\'an number} of $F$.
Special cases were studied earlier, especially the 3-uniform case (e.g., Maherani and Shahsiah~\cite{MS3}, Gy\'arf\'as~\cite{Gy}, Sali and Spiro~\cite{SS}).
\[ {\rm{ex}}_{r}(n,{\rm ind'd \, Berge \,} F) := \max \{e(\mathcal H): \mathcal H \subseteq {[n] \choose r} \text{ and } \mathcal H \text{ is induced Berge } F\text{-free}\}.\]
We consider the relationship between these three functions.
Obviously,
\begin{equation}\label{eqindF}
{\rm{ex}}(n,K_r, F) \leq {\rm{ex}}_{r}(n,{{\rm Berge \,}} F) \leq {\rm{ex}}_{r}(n, {\rm ind'd \, Berge \,} F).
\end{equation}
Indeed, consider a graph $G$ with $|{\mathcal N}_r(G)|={\rm{ex}}(n,K_r, F)$.
Since $G$ is $F$-free, the $r$-graph ${\mathcal N}_r(G)$ is Berge $F$-free, implying $|{\mathcal N}_r(G)|\leq {\rm{ex}}_{r}(n, {{\rm Berge \,}} F)$.
The second inequality holds because if a hypergraph contains no Berge $F$ then it also contains no induced Berge $F$.
The induced Berge $F$ problem is motivated by the forbidden configuration problem for matrices (see Anstee~\cite{ans} for a survey).
It can also be reformulated as a hypergraph trace problem (see, e.g., Mubayi and Zhao~\cite{MZ}).
Few results are known for the induced Berge Tur\'an problem. In~\cite{MZ}, the value of ${\rm{ex}}_r(n, {\rm ind'd \, Berge \,} K_t)$ is determined asymptotically for $K_3$ and $K_4$, as well as $K_t$ when $t$ is close to the uniformity $r$.
A special case of induced Berge hypergraphs, so called \emph{expansions} were intensively studied,
see, e.g., Pikhurko~\cite{P}, Kostochka, Mubayi, and Verstra\"ete~\cite{KMV}, and the survey by Mubayi and Verstra\"ete~\cite{MV}.
There are also other areas of research in extremal graph theory which are called `induced' Tur\'an type results. E.g.,
Pr\"omel and Steger~\cite{PS} investigated the extremal properties of graphs not containing an induced copy of a given graph $F$. A more recent version is by Loh, Tait, Timmons, and Zhou~\cite{LTTZ}. But most of these are only distant relatives of our induced Berge question.
\section{Main results, bounds for ${\rm{ex}}_r(n, {\rm ind'd \, Berge \,} F)$}
\subsection{The order of magnitude}
Let $F$ be a graph, $r\geq 2$.
Our aim is to determine the order of magnitude of the induced Berge Tur\'an number of $F$ as $n\to \infty$, or to reduce it to known problems.
Then in the next subsection we define a large class of 3-chromatic graphs $\GT$ which contains, e.g., all outerplanar graphs, and
apply our results and methods to determine their induced Berge Tur\'an number more precisely.
\begin{thm}\label{mainbigr}
Let $r\geq 2$, and fix a graph $F$ such that $E(F)\neq \emptyset$.
Then, as $n\to \infty$
\[{\rm{ex}}_{r}(n, {\rm ind'd \, Berge \,} F) = \Theta(\max_{2 \leq s \leq r}\{{\rm{ex}}(n,K_{s}, F) \}).\]
\end{thm}
This theorem shows that the order of magnitudes of the three functions in~\eqref{eqindF}
behave differently as $r$ changes.
For small $r$, in the range $r \leq \chi(F) - 1$, all the three, ${\rm{ex}}_{r}(n, F)$, ${\rm{ex}}_{r}(n,{{\rm Berge \,}} F)$, and ${\rm{ex}}_{r}(n,{\rm ind'd \, Berge \,} F)$, are of order $\Theta(n^r)$ because
the balanced complete $(\chi(F) - 1)$-partite $r$-graph contains no Berge $F$ (so its 2-shadow, the $r$-partite Tur\'an graph is $r$-chromatic).
If $r \geq |V(F)|$ then ${\rm{ex}}(n, K_r, F) = 0$ (since a $K_r$ contains a copy of $F$).
For general graphs $F$, the behavior of the three functions in the range $\chi(F) \leq r \leq |V(F)|-1$ is still unknown.
Determining the order of ${\rm{ex}}(n,K_r, F)$ for $r$ in this range would give an answer for the growth of ${\rm{ex}}_{r}(n,{\rm ind'd \, Berge \,} F)$.
Concerning the Berge Tur\'an function Gerbner and Palmer~\cite{GP} showed that
\[{\rm{ex}}_{r}(n,{{\rm Berge \,}} F) \leq {\rm{ex}}(n,F)\]
for $r\geq |V(F)|$.
So in this range ${\rm{ex}}_{r}(n,{{\rm Berge \,}} F) = O(n^2)$.
For the complete graphs the two sides have the same order: ${\rm{ex}}_{r}(n, {{\rm Berge \,}} K_r) = \Theta(n^2)$ if $r\geq 3$.
However this does not hold if $r$ is large compared to $|V(F)|$.
Gr\'osz, Methuku, and Tompkins~\cite{GMT} proved that for any non-bipartite $F$ and sufficiently large $r$, the order of ${\rm{ex}}_{r}(n,F)$ differs from that of ${\rm{ex}}(n,F)$: there exists some number $th(F)$ such that if $r \geq th(F)$ then ${\rm{ex}}_{r}(n,F)= o(n^2)$.
In contrast, the order of the induced Berge Tur\'an function ${\rm{ex}}_{r}(n,{\rm ind'd \, Berge \,} F)$ is non-decreasing in $r$. Moreover, it is basically monotone. If $\bigcap E(F) =\emptyset$, i.e., $F$ is not a star, then we will see later by Lemma~\ref{lb} that
\begin{equation}\label{eq21}
\left(1-\frac{r-1}{n}\right){\rm{ex}}_{r-1}(n, {\rm ind'd \, Berge \,} F) \leq {\rm{ex}}_{r}(n, {\rm ind'd \, Berge \,} F).
\end{equation}
\subsection{Outerplanar graphs and more}
We define the class of $t$-vertex graphs $\GT^{(t)}$ by induction on $t$ as follows.
The class $\GT^{(2)}$ has only a single member, $K_2$.
For $t>2$ one obtains each member $G$ of $\GT^{(t)}$ by taking a $G^{(t-1)}\in \GT^{(t-1)}$, taking an edge $xy\in G^{(t-1)}$, adding a new vertex $z\notin V(G^{(t-1)})$, and joining $z$ to $x$ and to $y$.
Each $G\in \GT^{(t)}$ has exactly $t$ vertices and $2t-3$ edges.
Finally, let $\GT$ be the family of all non-empty subgraphs of the members of $\cup_{t\geq 2} \GT^{(t)}$.
Note that $\GT$ contains all outerplanar graphs, particuarly cycles, $C_t$, and forests. Each $G\in \GT$ has chromatic number at most 3 and are obviously planar.
\begin{thm}\label{maincycle}
Let $r\geq 2$ be a positive integer.
Fix a graph $F\in \GT$.
As $n\to \infty$ we have ${\rm{ex}}_{r}(n, {\rm ind'd \, Berge \,} F) = \Theta({\rm{ex}}(n, F))$.
\end{thm}
This theorem reveals further gaps between ${\rm{ex}}_{r}(n, {{\rm Berge \,}} F)$ and ${\rm{ex}}_{r}(n, {\rm ind'd \, Berge \,} F)$.
Gy\H{o}ri and Lemons~\cite{GL3,GL} proved that for $r\geq 3$ an $r$-uniform hypergraph avoiding a Berge cycle $C_{2t+1}$
has at most $O({\rm{ex}}(n, C_{2t}))$ edges, which is known to be $O(n^{1 + (1/t)})$.
On the other hand, in the same range, we have ${\rm{ex}}_{r}(n, {\rm ind'd \, Berge \,} C_{2t+1})=\Theta(n^2)$.
Together, Theorems~\ref{mainbigr} and~\ref{maincycle} show that ${\rm{ex}}(n,C_t)$ has the same order as $\max_{2 \leq s \leq r}\{{\rm{ex}}(n,K_{s}, F) \}$. We obtain the following (known) corollary.
For any $r\geq 2$ and $t \geq 3$
\[{\rm{ex}}(n, K_r, C_t) = O({\rm{ex}}(n, C_t)).\]
We also state the case of trees.
\begin{cor}\label{maintree}
Let $r \geq 2$ and $T$ be a forest with at least two edges. Then ${\rm{ex}}_{r}(n,{\rm ind'd \, Berge \,} T) = \Theta({\rm{ex}}(n,T))= \Theta(n)$.
\end{cor}
Finally, we get better bounds for stars, $F = K_{1,t-1}$.
\begin{thm}\label{star}For any $r\geq 2$, $t \geq 3$, if $n = a(r+t-3) + b$ with $b \leq r+t-4$ then \[a{r+t-3 \choose r} + {b \choose r} \leq {\rm{ex}}_{r}(n, {\rm ind'd \, Berge \,} K_{1,t-1}) \leq \frac{n}{r}{r + t - 3 \choose r-1} .\] In particular, if $n$ is divisible by $r+t-3$, the lower bound is $\frac{n}{r}{r+t-4 \choose r-1}$.
\end{thm}
\section{Constructions and proofs}
\subsection{Simple constructions and a monotonicity of the induced Berge Tur\'an function}
If $E(F)$ has a single edge then for $n\geq |V(F)|+r-2$ we have
${\rm{ex}}(n, F)={\rm{ex}}(n,K_r, F) ={\rm{ex}}_{r}(n,{{\rm Berge \,}} F) = {\rm{ex}}_{r}(n, {\rm ind'd \, Berge \,} F)=0$, so there is nothing to prove, all of our statements trivially hold.
In all other cases we have ${\rm{ex}}_{r}(n, {\rm ind'd \, Berge \,} F)=\Omega(n)$ as one can see from the following constructions.
If $F$ has two non-disjoint edges then a matching of $r$-sets gives
${\rm{ex}}_{r}(n, {\rm ind'd \, Berge \,} F)\geq \lfloor n/r \rfloor$.
If $F$ has two disjoint edges then the hypergraph consisting of $n-r+1$ sets sharing a common $(r-1)$-set yields ${\rm{ex}}_{r}(n, {\rm ind'd \, Berge \,} F)\geq n-r+1$.
If $x\in V(F)$ is an isolated vertex then
${\rm{ex}}_{r}(n, {\rm ind'd \, Berge \,} F) = {\rm{ex}}_{r}(n, {\rm ind'd \, Berge \,} (F\setminus \{ x\}))$ for all $n> (r-2)|E(F)|+|V(F)|$.
So we may delete isolated vertices and asymptotically get the same Tur\'an number.
From now on, we suppose that $F$ has no isolated vertex and $|E(F)|\geq 2$.
\begin{lem}\label{lb}Fix integers $r,t \geq 2$. If $F$ is a graph on $t$ vertices such that $F \neq K_{1, t-1}$ (and $e(F)\geq 2$ and $F$ has no islated vertex), then ${\rm{ex}}_{r}(n,{\rm ind'd \, Berge \,} F) \geq {\rm{ex}}_{(r-1)}(n-1,{\rm ind'd \, Berge \,} F)$. In particular, ${\rm{ex}}_{r}(n, {\rm ind'd \, Berge \,} F) = \Omega({\rm{ex}}(n,F))$.
\end{lem}
\begin{proof}Let $\mathcal H$ be an $(r-1)$-uniform hypergraph on $n-1$ vertices with ${\rm{ex}}_{r}(n-1, {\rm ind'd \, Berge \,} F)$ edges and no induced Berge $F$. Construct an $r$-uniform hypergraph $\mathcal H'$ with $V(\mathcal H') = V(\mathcal H) \cup \{v\}$ such that the edges of $\mathcal H'$ are obtained by extending every edge of $\mathcal H$ to include the new vertex $v$. Suppose $\mathcal H'$ contains an induced Berge $F$. Since $\mathcal H$ was induced Berge $F$-free, $v$ must be a base vertex. Because $v$ is contained in every edge of $\mathcal H'$, there is a fixed vertex contained in every edge of $F$. I.e., $F = K_{1, t-1}$, a contradiction.
Inductively, we obtain ${\rm{ex}}_{2}(n-r+2, {\rm ind'd \, Berge \,} F) \leq {\rm{ex}}_r(n, {\rm ind'd \, Berge \,} F)$. But ${\rm{ex}}_2 (n-r+2, {\rm ind'd \, Berge \,} F) = {\rm{ex}}(n-r+2, F) = \Theta({\rm{ex}}(n, F))$.
\end{proof}
To show~\eqref{eq21} let ${\mathcal H}$ be an induced Berge $F$-free $(r-1)$-uniform hypergraph on $n$ vertices, $|{\mathcal H}|={\rm{ex}}_{(r-1)}(n,{\rm ind'd \, Berge \,} F)$.
For $x\in V:=V({\mathcal H})$ let ${\mathcal H}_x:=\{ e\in {\mathcal H}: e\subset V\setminus \{x\}\}$.
Since each ${\mathcal H}_x$ is also induced Berge $F$-free we get
\[ (n-r+1){\rm{ex}}_{(r-1)}(n,{\rm ind'd \, Berge \,} F) = (n-r+1)|{\mathcal H}|=\sum_{x\in V}|{\mathcal H}_x|\leq n\times
{\rm{ex}}_{(r-1)}(n-1,{\rm ind'd \, Berge \,} F).
\]
By Lemma~\ref{lb} the right hand side is at most $n \times
{\rm{ex}}_{r}(n,{\rm ind'd \, Berge \,} F)$. Rearranging yields~\eqref{eq21}. \hfill\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else\hfill$\Box$\fi
\subsection{The $\alpha$-core of a hypergraph}
Let ${\mathcal H}$ be an $r$-partite, $r$-uniform hypergraph with parts $V({\mathcal H}) = V_1 \cup \ldots \cup V_r$. For some $1\leq s \leq r$ and edge $e \in {\mathcal H}$, define $e[\overline{s}]$ to be the trace
of $e$ onto all parts other than $V_s$. That is, $e[\overline{s}] = e \setminus V_s$. Let ${\mathcal H}[\overline{s}] = \{e[\overline{s}]: e \in E({\mathcal H})\}$.
\begin{thm}\label{core}
For positive integers $\alpha, r$, any $r$-uniform $r$-partite hypergraph ${\mathcal H}$ contains edge-disjoint subhypergraphs ${\mathcal A}$ and ${\mathcal B}$ such that
\begin{enumerate}
\item[{\rm (a)}] For any $S \subseteq V({\mathcal H})$, with $|S| = r-1$, either $\deg_{{\mathcal A}}(S) = 0$ or $\deg_{{\mathcal A}}(S) \geq \alpha$.
\item[{\rm (b)}] $|{\mathcal B}| \geq \frac{|{\mathcal H} \setminus {\mathcal A}|}{\alpha - 1}$ and $|{\mathcal B}| \leq \sum_{s=1}^r |{\mathcal B}[\overline{s}]|$.
\end{enumerate}
\end{thm}
\begin{proof}
We build ${\mathcal A}$ and ${\mathcal B}$ inductively. Initially set ${\mathcal H}_0 := {\mathcal H}$, ${\mathcal B}_0 := \{\emptyset\}$.
At step $i$, if there exists an $S \subseteq V({\mathcal H}_{i-1})$ with $|S| = r-1$ and $1\leq \deg_{{\mathcal H}_{i-1}}(S) \leq \alpha -1 $, then let $E_S$ be the edges of ${\mathcal H}_{i-1}$ containing $S$. Set ${\mathcal H}_i = {\mathcal H}_{i-1} \setminus E_S$. Pick any edge, say $B_i \in E_S$, and set ${\mathcal B}_i = {\mathcal B}_{i-1} \cup \{B_i\}$.
The process ends after $k$ steps when for every $S\subseteq V({\mathcal H}_k)$ with $|S|=r-1$, either $\deg_{{\mathcal H}_k}(S) =0$ or $\deg_{{\mathcal H}_k}(S) \geq \alpha$. Let ${\mathcal A} := {\mathcal H}_k$ and ${\mathcal B} := {\mathcal B}_k = \{B_1 , \ldots , B_k\}$. Then ${\mathcal A}$ satisfies $(a)$.
To see that ${\mathcal B}$ satisfies $(b)$, at each step $i$ when we choose $B_i \in E_S$, $|E_S| \leq \alpha - 1$, so we obtain that $|{\mathcal B}|$ is at least a $1/(\alpha - 1)$ portion of the deleted edges. Next, at each step, we associated with $B_i$ a distinct set $S_i$ of $r-1$ vertices. If $B_i$ and $B_j$ are associated with sets $S_i$ and $S_j$ respectively such that both sets are contained in $(V_1 \cup \ldots \cup V_r) \setminus V_s$, then in ${\mathcal B}[\overline{s}]$, $B_i[\overline{s}]= S_i$ and $B_j[\overline{s}]=S_j$ are distinct. Hence $\sum_{s=1}^r {\mathcal B}[\overline{s}] \geq |\{S_1, \ldots , S_k\}| = |{\mathcal B}|$.
\end{proof}
Let any ${\mathcal A}\subseteq {\mathcal H}$ satisfying $(a)$ be called an {\bf $\alpha$-core} of ${\mathcal H}$.
\begin{lem}\label{coreF}
Let $\alpha, r$ be positive integers, and let $F$ be a graph with $|V(F)|-1
\leq \alpha$. Let ${\mathcal H}$ be an $r$-uniform, $r$-partite hypergraph with an $\alpha$-core ${\mathcal A}$. If the 2-shadow $\partial_2{\mathcal A}$ of ${\mathcal A}$ contains a copy of $F$ then ${\mathcal A}$
(and therefore ${\mathcal H}$) contains an induced Berge $F$.
\end{lem}
\begin{proof}
We will find an induced Berge $F$ on the same base vertex set $V(F)$.
Let $xy$ be an edge in the copy of $F$, and let $e_{xy}$ be an edge of ${\mathcal A}$ containing $\{x,y\}$ with minimum $|e_{x
y} \cap V(F)|$. Such an edge $e_{xy}$ exists by the definition of the 2-shadow. If $e_{xy}$ contains some vertex $z \in V(F) \setminus \{x,y\}$, then the $(r-1)$-set $e_{xy} \setminus \{z\}$ is contained in at least $\alpha-1$ other edges in ${\mathcal A}$. Since there are $|V(F)| - 3 \leq \alpha - 2$ vertices in $V(F) \setminus \{x,y,z\}$, we may find some $z' \not\in V(F) - \{x,y,z\}$ such that $e_{xy} \setminus \{z\} \cup \{z'\} \in E({\mathcal A})$, contradicting the choice of $e_{xy}$. Therefore $e_{xy} \cap V(F) =\{x,y\}$. We find such an edge of ${\mathcal A}$ for each edge of $F$.
\end{proof}
If $\alpha \geq e(F) +|V(F)|$, then with the same method one can find an induced Berge $F$ in ${\mathcal A}$ such that each pair of hyperedges $e_{xy}$ and $e_{uv}$ intersect only at $\{x,y\} \cap \{u,v\}$. This is called an $F$-\emph{expansion}.
But this observation does not seem to help our purposes here.
\begin{claim}\label{cl:34}
Suppose that $r\geq 3$ and ${\mathcal A}$ contains an induced Berge $F$, where $|V(F)|\leq \alpha$ (and $E(F)\neq \emptyset$).
Define a new graph $F^+:=F^+_{xy}$ by adding a new vertex $z\notin V(F)$, taking an edge $xy\in E(F)$, and joining $z$ to $x$ and to $y$.
Then ${\mathcal A}$ also contains an induced Berge $F^+$.\end{claim}
\begin{proof}
By Lemma~\ref{coreF}, there exists a hyperedge $e_{xy}\in {\mathcal A}$ such that $e_{xy} \cap V(F) =\{x,y\}$.
Then for any $z'\in e_{xy} \setminus \{x,y\}$ we have that $xz'$ and $yz'\in \partial_2{\mathcal A}$, so $F^+$ is a subgraph of $\partial_2{\mathcal A}$.
Then Lemma~\ref{coreF} completes the Claim. \end{proof}
\begin{lem}\label{le:35}
Suppose that $G\in \GT$ with $t=|V(T)|\geq 3$. Then $G\in \GT^{(t)}$.
\end{lem}
\begin{proof}
This statement seems to be evident, but still needs a proof.
By definition, there exists an $s\geq t$ such that $G\in \GT^{(s)}$.
Let $s=s(G)$ be the smallest such $s$. We will show by induction on $t$ that $s(G)=t$.
The base case $t=3$ is obvious. Suppose $t>3$ and that $G$ is a subgraph of $H\in \GT^{(s)}$, where the vertices of $H$ are $\{ v_1, \dots, v_s\}$ and each $v_i$ (with $i\geq 3$) has exactly two $H$-neighbors in $\{ v_1,\dots, v_{i-1}\}$. Moreover, these two neighbors (call them $v_{\alpha(i)}$ and $v_{\beta(i)}$) are joined by an edge in $H$.
Let $I\subseteq [s]$, $I:= \{ i_1, \dots, i_t\}$, $1\leq i_1 < \dots < i_t\leq s$, $V_I:=\{ v_i: i\in I\}$, and suppose that
$G$ is a spanning subgraph of $H[V_I]$. Since $s$ is minimal, we have $i_t=s$ and
$N_H(v_s)=\{v_{\alpha(s)}, v_{\beta(s)}\}$.
$G':= H[V_I]\setminus \{ v_s \}$ has $t-1$ vertices, and it belongs to $\GT$.
By our induction hypothesis there exists a $H'\in \GT^{t-1}$ such that $G'$ is a subgraph of $H'$ on the same vertex set $V_I\setminus \{ v_s\}$.
If $\{v_{\alpha(s)}, v_{\beta(s)}\}\subseteq V(H')$ then by adjoining a new vertex $z'$ to $H'$ and connecting it to $v_{\alpha(s)}$ and $v_{\beta(s)}$ we obtain a $t$-vertex graph $H''$ from $\GT^{(t)}$ containing $G$.
If $|N_H(v_s)\cap V(H')|\leq 1$ then it is even simpler to find such a graph $H''$.
\end{proof}
\subsection{Proofs of the upper bounds for induced Berge $F$ problems}
We prove a version of Theorem~\ref{mainbigr} with more precise bounds. For positive integers $a$ and $b$, $(a)_b = (a)(a-1) \cdots (a-b+1)$ denotes the falling factorial.
\begin{thm}\label{rpartite}
Let $t, r, n$ be positive integers, and let $F$ be any graph with $|V(F)| = t$. Let $\mathcal H$ be an $n$-vertex $r$-uniform hypergraph with no induced Berge $F$. If ${\mathcal H}$ is $r$-partite, then \[e({\mathcal H}) \leq \sum_{i=2}^r (t-2)^{r-i}(r)_{r-i}{\rm{ex}}(n,K_i, F).\]
\end{thm}
\begin{proof}
We proceed by induction on $r$. The base case $r=2$ is trivial since an induced Berge $F$ is just a copy of $F$. Thus ${\rm{ex}}_2(n,{\rm ind'd \, Berge \,} F) = {\rm{ex}}(n, K_2, F) = {\rm{ex}}(n,F)$. Now let $r \geq 3$. Let ${\mathcal A}$ and ${\mathcal B}$ be subhypergraphs of ${\mathcal H}$ obtained from Theorem~\ref{core} with $\alpha = t - 1$. So we have \[|{\mathcal H}| = |{\mathcal A}| + |{\mathcal H} \setminus {\mathcal A}| \leq |{\mathcal A}| + (t-2)\sum_{s=1}^r|B[\overline{s}]| \leq |{\mathcal A}| + (t-2)(r){\rm{ex}}_{r-1}(n,{\rm ind'd \, Berge \,} F),\]
where the last inequality holds because each $B[\overline{s}]$ is $(r-1)$-uniform, $(r-1)$-partite and does not contain an induced Berge $F$.
By Lemma~\ref{coreF}, $\partial_2{\mathcal A}$ contains no copy of $F$. Furthermore, since each edge in ${\mathcal A}$ creates a $K_r$ in $\partial_2{\mathcal A}$, $|{\mathcal A}| \leq {\rm{ex}}(n, K_r, F)$. Applying the induction hypothesis, we obtain
\[|{\mathcal H}| \leq {\rm{ex}}(n,K_r, F) + (t-2)r\sum_{i=2}^{r-1} (t-2)^{r-1-i}(r-1)_{r-1-i}{\rm{ex}}(n,K_i, F)\]
and we are done.
\end{proof}
\begin{cor}
Let $t, r, n$ be positive integers, and let $F$ be any graph with $V(F) = t$. Then \[\max_{2 \leq s\leq r} \{{\rm{ex}}(n-(r-s), K_{s}, F)\} \leq {\rm{ex}}_{r}(n,{\rm ind'd \, Berge \,} F) \leq \frac{r^r}{r!} \sum_{i=2}^r (t-2)^{r-i}(r)_{r-i}{\rm{ex}}(n,K_i, F).\]
In particular, ${\rm{ex}}_{r}(n,{\rm ind'd \, Berge \,} F) = \Theta(\max_{s \leq r}\{{\rm{ex}}(n, K_{s},F)\})$.
\end{cor}
\begin{proof}
The lower bound follows from Lemma~\ref{lb} and~\eqref{eqindF}. For the upper bound, we use the fact that any $r$-uniform hypergraph ${\mathcal H}$ has an $r$-partite subhypergraph with at least $\frac{r!}{r^r}e({\mathcal H})$ edges. Apply Theorem~\ref{rpartite} to any such subhypergraph.
\end{proof}
{\em Proof of Theorem~\ref{maincycle}}. The lower bound comes from Lemma~\ref{lb}.
For the upper bound, we proceed by induction on $r$.
First we show that if ${\mathcal H}$ is $r$-partite with no induced Berge $F\in \GT$ then
\begin{equation}\label{eq3}
|{\mathcal H}| \leq (t-2)^{r-2}\frac{r!}{2} {\rm{ex}}(n,F).
\end{equation}
The base case $r=2$ is trivial, so let $r \geq 3$.
Let ${\mathcal A}$ and ${\mathcal B}$ be subhypergraphs of ${\mathcal H}$ obtained from Theorem~\ref{core} with $\alpha = t-1$. Again we have
\begin{equation}\label{eqcycle}
|{\mathcal H}| \leq |{\mathcal A}| + (t-2)\sum_{s=1}^r|B[\overline{s}]| \leq |{\mathcal A}| + (t-2)(r) {\rm{ex}}_{r-1}(n,{\rm ind'd \, Berge \,} F).
\end{equation}
Observe that ${\mathcal A}$ is empty. Indeed, if ${\mathcal A}$ contains at least one edge, then the 2-shadow $\partial_2 {\mathcal A}$ contains a $K_r$. So
Claim~\ref{cl:34} and Lemma~\ref{le:35} imply that $\partial_2 {\mathcal A}$ contains a copy of $F$. Then we apply Lemma~\ref{coreF} to find an induced Berge $F$, a contradiction. Hence $|{\mathcal A}| = 0$.
Applying induction hypothesis,~\eqref{eqcycle} yields~\eqref{eq3}.
Finally, if ${\mathcal H}$ is not $r$-partite, then we apply the previous proof to an $r$-partite subgraph ${\mathcal H}'$ of ${\mathcal H}$ with at least $\frac{r!}{r^r}|{\mathcal H}|$ edges to obtain $|{\mathcal H}| \leq \frac{1}{2} r^{r}(t-2)^{r-2} {\rm{ex}}(n, F)$. \hfill\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else\hfill$\Box$\fi
{\em Proof of Theorem~\ref{star}.}
For the lower bound, let each component of ${\mathcal H}$ be a clique such that there are as many cliques of size $r+t-3$ as possible. If $n = a(r+t-3) + b$ where $0 \leq b < r+t-3$, then $|{\mathcal H}| = a{r+t-3 \choose r} + {b \choose r}.$
Suppose ${\mathcal H}$ contains an induced Berge $K_{1,t-1}$. Then its base vertices, say $\{v_1, \ldots, v_t\}$ must be contained in a single component of ${\mathcal H}$. But each edge in a component contains at least 3 base vertices, a contradiction.
For the upper bound, let ${\mathcal H}$ be an $n$-vertex, $r$-uniform hypergraph with no induced Berge $K_{1,t-1}$. We say that a set system $\{f_1, \ldots, f_s\}$ is {\em strongly representable} if for every $f_i \in {\mathcal F}$, there exists a $v_i \in f_i$ such that $v_i \notin f_j$ for all $j \neq i$. F\"uredi and Tuza~\cite{FT} proved that if a set system ${\mathcal F}$ with $|f| \leq r$ for all $f \in {\mathcal F}$ does not contain a strongly representable subfamily of size $s$ then $|{\mathcal F}| \leq {r + s-1 \choose r}$. For any vertex $v \in V({\mathcal H})$, let $E_v:=\{e \setminus \{v\}: v \in e \in {\mathcal H}\}$. The $(r-1)$-uniform set system $E_v$ cannot contain a strongly representable subfamily of size $t-1$, otherwise the corresponding edges in ${\mathcal H}$ and their representative vertices would yield an induced Berge $K_{1,t-1}$ in ${\mathcal H}$ with vertex $v$ as the center vertex. Therefore $\deg(v) \leq {(r-1)+(t-2) \choose r-1}$ so $|{\mathcal H}| \leq \frac{n}{r} {r+t-3 \choose r-1}$.
\hfill\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else\hfill$\Box$\fi
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,945 |
If he makes a mistake when playing for his team the coach tells him to forget it and try to get it right next time - the crowd don't even notice most times. If he makes a mistake umpiring, half the crowd yells at him, making him think twice before blowing the whistle next time, possibly not paying a free that is there and upsetting the crowd more. The post then asked which approach to improving his skills and confidence works best do you think? No prize for the correct answer - it's a no brainer! So just remember that one of the umpires running around at your game is a son, daughter, brother, sister, cousin or friend who is out there giving their all, just like the players. So give them encouragement and reinforce positive behaviours during your match, you never know, they positive words and attitudes you show may help that umpire to become one of the best umpires the Geelong Umpires have.
In the Lead up to Finals I will be highlighting some of our umpires who have either just got in to the game or been long serving members. This week's profile umpire is on Molly McKenzie. Molly is a first year umpire who is show great promise and will feature on the boundary during this year's finals series.
What would be your advice to anyone thinking about getting in to umpiring? It's a great way to make new friends, get fit and get paid. It's an awesome opportunity and it is a really great club that makes you feel so welcome and that you belong with their club even on the first night that you turn up. An amazing experience and you learn valuable life skills from it.
What would be the funniest moment you have experienced since you started umpiring? Probably when I was too close to the play, and two players collided resulted in falling on top of me in front of the crowd.
A full back is kicking in after a behind after being instructed by the field umpire to play on, he is tackled by the full forward inside the goal square and fails to dispose of the football correctly. What is your Decision?
A St Joseph's player runs with the football for 14 metres then handballs it over his opponents head and regains possession, without the ball touching the ground, 3 metres past where he handballed it. What is your Decision?
Answer is a free kick paid for running too far.
What a great question and story! When can do finals Dave? | {
"redpajama_set_name": "RedPajamaC4"
} | 9,168 |
Q: Chartjs and datalabels : automatic y axis (and dynamic data) hide datalabels I have a chart (chartjs) with labels (datalabels).
When my data changes, my chart updates automatically. However, the largest datalabels are most of the time hidden by the automatic resizing of the y axis. Do you have any idea to fix that ?
It should be a 3 up there :)
Here is my code:
const completionOptions = {
responsive: true,
plugins: {
legend: {
display: false,
},
tooltip: {
backgroundColor: 'rgb(85, 85, 85, 1)',
displayColors: false,
// https://www.chartjs.org/docs/latest/configuration/tooltip.html
},
datalabels: {
color: 'rgb(203, 203, 203)',
anchor: 'end',
align: 'end',
labels: {
title: {
font: {
family: 'karla',
weight: '600',
size: 12,
},
}
}
}
},
scales: {
display: false,
y: {
beginAtZero: true,
display: false,
grid: {
display: false,
},
ticks: {
padding: 10
}
},
x: {
grid: {
display: false,
},
ticks: {
autoSkip: false,
maxRotation: 0,
minRotation: 0,
font: {
size: 20,
}
}
},
}
}
A: You can use the grace option to add extra space to the y axes:
Chart.register(ChartDataLabels)
const options = {
type: 'bar',
data: {
labels: ["Red", "Blue", "Yellow", "Green", "Purple", "Orange"],
datasets: [{
label: '# of Votes',
data: [12, 20, 3, 5, 2, 3],
backgroundColor: 'pink'
}]
},
options: {
scales: {
y: {
grace: '10%'
}
},
plugins: {
legend: {
display: false
},
datalabels: {
anchor: 'end',
align: 'end'
}
}
}
}
const ctx = document.getElementById('chartJSContainer').getContext('2d');
new Chart(ctx, options);
<body>
<canvas id="chartJSContainer" width="600" height="400"></canvas>
<script src="https://cdnjs.cloudflare.com/ajax/libs/Chart.js/3.8.0/chart.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/chartjs-plugin-datalabels/2.0.0/chartjs-plugin-datalabels.js"></script>
</body>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,931 |
Q: How to get the second word in a string using t-sql? I would like to extract the second word from a string or the phrase between the first and third spaces.
for example
'The Boeing Corporation'
I would like 'Boeing'
A:
declare @sentence nvarchar(264);
set @sentence = 'The Boeing Corporation';
select ltrim(substring(@sentence,charindex(' ',@sentence), CHARINDEX(' ',ltrim(SUBSTRING(@sentence,charindex(' ',@sentence),LEN(@sentence)-charindex(' ',@sentence)))) ))
| (No column name) |
| :--------------- |
| Boeing |
db<>fiddle here
A: Just another option using a bit of XML
Example
Declare @YourTable table (SomeCol varchar(500))
Insert Into @YourTable values
('The Boeing Corporation')
Select SomeCol
,Pos2 = cast('<x>' + replace(A.SomeCol,' ','</x><x>')+'</x>' as xml).value('/x[2]','varchar(50)')
From @YourTable A
Returns
SomeCol Pos2
The Boeing Corporation Boeing
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,044 |
import React from 'react';
import { mount } from 'enzyme';
import VirtualizedCheckbox from './VirtualizedCheckbox';
describe('VirtualizedCheckbox', () => {
const names = [];
for (var i = 0; i < 100; i++) {
names.push({ name: `Name ${i}`, checked: true });
}
const fixture = props =>
<VirtualizedCheckbox
items={names}
labelKey={'name'}
onOk={(checked, all, textFilter) => {
allFlag = all;
checkedBoxes = checked;
canceledFlag = null;
}}
onCancel={() => {
allFlag = null;
checkedBoxes = null;
canceledFlag = true;
}}
rowHeight={30}
overscanRowCount={0}
height={360}
width={300}
{...props}
/>;
let allFlag = null;
let checkedBoxes = null;
let canceledFlag = null;
// have to insert a div with a height attribute as parent for VirtualizedCheckbox
// otherwise boxes are not rendered
let node = null;
beforeEach(() => {
node = document.createElement('div');
document.body.appendChild(node);
});
afterEach(() => {
document.body.removeChild(node);
allFlag = null;
checkedBoxes = null;
canceledFlag = null;
});
describe('rendered children', () => {
it('should fill the view', () => {
const wrapper = mount(fixture(), { attachTo: node });
// 10 in the view + 10 overscan
expect(wrapper.find('[type="checkbox"]').length).toEqual(10 + 10);
});
it('should work if they do not fill the view', () => {
const wrapper = mount(fixture({ items: names.slice(0, 5) }), {
attachTo: node
});
expect(wrapper.find('[type="checkbox"]').length).toEqual(1 + 5);
});
it('should conform to rowHeight prop', () => {
const wrapper = mount(fixture({ rowHeight: 60 }), { attachTo: node });
// 4 in the view + 10 overscan
expect(wrapper.find('[type="checkbox"]').length).toEqual(4 + 10);
});
it('should work without a filter box', () => {
const wrapper = mount(fixture({ rowHeight: 30, hasFilterBox: false }), {
attachTo: node
});
// 11 in the view + 10 overscan
expect(wrapper.find('[type="checkbox"]').length).toEqual(11 + 10);
});
it('should work without buttons', () => {
const wrapper = mount(
fixture({ rowHeight: 30, hasOkButton: false, hasCancelButton: false }),
{
attachTo: node
}
);
// 11 in the view + 10 overscan
expect(wrapper.find('[type="checkbox"]').length).toEqual(11 + 10);
});
it('should work without buttons and filter box', () => {
const wrapper = mount(
fixture({
rowHeight: 30,
hasOkButton: false,
hasCancelButton: false,
hasFilterBox: false
}),
{ attachTo: node }
);
// 12 in the view + 10 overscan
expect(wrapper.find('[type="checkbox"]').length).toEqual(12 + 10);
});
});
describe('onOk callback', () => {
it('called with true and checked boxes if all boxes are checked', () => {
const wrapper = mount(fixture(), { attachTo: node });
wrapper.find('[value="Ok"]').simulate('click');
expect(allFlag).toBe(true);
expect(checkedBoxes.length).toEqual(100);
});
it('called with false and checked boxes if not all boxes are checked', () => {
const wrapper = mount(
fixture({
items: names.map((name, i) => ({
...name,
checked: [33, 44].indexOf(i) > -1
}))
}),
{ attachTo: node }
);
wrapper.find('[value="Ok"]').simulate('click');
expect(allFlag).toBe(false);
expect(checkedBoxes.length).toEqual(2);
});
});
describe('onCancel callback', () => {
it('is called upon click on cancel Button', () => {
const wrapper = mount(fixture(), { attachTo: node });
wrapper.find('[value="Cancel"]').simulate('click');
expect(canceledFlag).toBe(true);
});
});
describe('filter', () => {
it('works when passed as prop', () => {
const wrapper = mount(fixture({ textFilter: 'Name 95' }), {
attachTo: node
});
// the Name 95 checkbox and the (Select all)
expect(wrapper.find({ type: 'checkbox' }).length).toEqual(1 + 1);
});
it('works when changed in the text box', () => {
const wrapper = mount(fixture(), { attachTo: node });
wrapper
.find('#filter')
.simulate('change', { target: { value: 'Name 95' } });
console.log(wrapper.debug());
// the Name 95 checkbox and the (Select all)
expect(wrapper.find({ type: 'checkbox' }).length).toEqual(1 + 1);
});
it('show (Select all search results) box when filter set', () => {
const wrapper = mount(fixture(), { attachTo: node });
wrapper
.find('#filter')
.simulate('change', { target: { value: 'Name 9' } });
expect(
wrapper.find({ value: '(Select all search results)' }).length
).toEqual(1);
});
it('shows (Select All) box when filter reset to empty string', () => {
const wrapper = mount(fixture(), { attachTo: node });
wrapper
.find('#filter')
.simulate('change', { target: { value: 'Name 95' } });
wrapper.find('#filter').simulate('change', { target: { value: '' } });
expect(wrapper.find({ value: '(Select all)' }).length).toEqual(1);
});
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,407 |
using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Web;
using System.Collections.Specialized;
using System.Web.SessionState;
using Http.Shared.Contexts;
namespace Http.Contexts
{
public class SimpleHttpSessionState : HttpSessionStateBase, IHttpSession
{
public override void Abandon()
{
_isChanged = true;
}
public void SetIsChanged(bool val)
{
_isChanged = false;
}
public override void Add(String name, Object value)
{
_isChanged = true;
_item[name] = value;
}
public override void Clear()
{
_isChanged = true;
_item.Clear();
}
public override void Remove(String name)
{
_isChanged = true;
_item.Remove(name);
}
public override void RemoveAll()
{
_isChanged = true;
_item.Clear();
}
public override void RemoveAt(Int32 index)
{
var key = _item.Keys.ToArray()[index];
_isChanged = true;
_item.Remove(key);
}
public override void CopyTo(Array array, Int32 index)
{
throw new NotImplementedException();
}
public override IEnumerator GetEnumerator()
{
return null;
}
public override Int32 CodePage { get; set; }
private HttpSessionStateBase _contents;
public override HttpSessionStateBase Contents { get { return _contents; } }
public void SetContents(HttpSessionStateBase val)
{
_contents = val;
}
private HttpCookieMode _cookieMode;
public override HttpCookieMode CookieMode { get { return _cookieMode; } }
public void SetCookieMode(HttpCookieMode val)
{
_cookieMode = val;
}
private Boolean _isCookieless;
public override Boolean IsCookieless { get { return _isCookieless; } }
public void SetIsCookieless(Boolean val)
{
_isCookieless = val;
}
private Boolean _isNewSession;
public override Boolean IsNewSession { get { return _isNewSession; } }
public void SetIsNewSession(Boolean val)
{
_isNewSession = val;
}
private Boolean _isReadOnly;
public override Boolean IsReadOnly { get { return _isReadOnly; } }
public void SetIsReadOnly(Boolean val)
{
_isReadOnly = val;
}
public override NameObjectCollectionBase.KeysCollection Keys { get { throw new NotImplementedException(); } }
public override Int32 LCID { get; set; }
private SessionStateMode _mode;
public override SessionStateMode Mode { get { return _mode; } }
public void SetMode(SessionStateMode val)
{
_mode = val;
}
private String _sessionID = "";
public override String SessionID { get { return _sessionID; } }
public void SetSessionID(String val)
{
_sessionID = val;
}
private HttpStaticObjectsCollectionBase _staticObjects;
public override HttpStaticObjectsCollectionBase StaticObjects { get { return _staticObjects; } }
public void SetStaticObjects(HttpStaticObjectsCollectionBase val)
{
_staticObjects = val;
}
public override Int32 Timeout { get; set; }
private readonly Dictionary<string, object> _item = new Dictionary<string, object>();
public override object this[string key]
{
get
{
if (!_item.ContainsKey(key)) return null;
return _item[key];
}
set
{
_isChanged = true;
_item[key] = value;
}
}
public override object this[int key]
{
get
{
if (_item.Count() >= key) return null;
return _item.Keys.ToArray()[key];
}
}
public Dictionary<string, object> ItemDictionary
{
get { return new Dictionary<string, object>(_item); }
}
public override Int32 Count { get { return _item.Count; } }
public void SetCount(Int32 val)
{
}
private Boolean _isSynchronized;
public override Boolean IsSynchronized { get { return _isSynchronized; } }
public void SetIsSynchronized(Boolean val)
{
_isSynchronized = val;
}
private Object _syncRoot = new Object();
public override Object SyncRoot { get { return _syncRoot; } }
public void SetSyncRoot(Object val)
{
_syncRoot = val;
}
private bool _isChanged;
public bool IsChanged
{
get { return _isChanged; }
}
public void Initialize(System.Collections.Generic.Dictionary<string, object> initVals)
{
foreach (var kvp in initVals)
{
_item.Add(kvp.Key, kvp.Value);
}
_isChanged = false;
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,332 |
\section{Introduction}
Audio-visual automatic speech recognition (AV-ASR) is a case of multi-modal analysis in which two modalities (audio and visual) complement each other to recognize speech. Incorporating visual features, such as speaker's lip movements and facial expressions, into automatic speech recognition (ASR) systems has been shown to improve their performances especially under noisy conditions. To this end several methods have been proposed which traditionally include variants of GMM/HMM models\cite{dupont2000audio}\cite{brand1997coupled}. More recently AV-ASR methods based on deep neural networks (DNN) models\cite{huang2013audio}\cite{mroueh2015deep}\cite{noda2015audio} have been proposed.
End-to-end speech recognition methods based on RNNs trained with CTC objective function\cite{graves2014towards}\cite{miao2015eesen}\cite{hannun2014deep} have come to the fore recently and have been shown to give performances comparable to that of DNN/HMM. The RNN trained with CTC directly learns a mapping between audio feature frames and character/phoneme sequences. This method eliminates the need for intermediate GMM/HMM training thereby simplifying the training procedure. To our knowledge, so far AV-ASR systems based on RNN trained with CTC have not been explored.
In this work, we design and evaluate an audio-visual ASR (AV-ASR) system using deep recurrent neural network (RNN) and CTC objective function. The design of an AV-ASR system includes the tasks of visual feature engineering, and audio-visual information fusion. Figure \ref{fig:avasr} shows the AV-ASR pipeline at test time. This work mainly deals with the visual feature extraction and processing steps and training protocol for the fusion model. Proper visual features are important especially in the case of RNNs as RNNs are difficult to train. Bottleneck features used in tandem with audio features are known to improve ASR performance \cite{gehring2013extracting}\cite{hermansky2000tandem}\cite{yu2011improved}. We employ a similar idea to improve the discriminatory power of video features. We show that this helps the RNN to converge properly when compared with raw features. Finally, we compare the performances of feature fusion and decision fusion methods.
The paper is organized as follows: Section \ref{sssec:num2} presents the prior work on AV-ASR. Bi-directional RNN and its training using CTC objective function are discussed in Section \ref{sssec:num3}. Section \ref{sssec:num4} describes the feature extraction steps for audio and visual modalities. In section \ref{sssec:num5} different fusion models are explained. Section \ref{sssec:num6} explains the training protocols and experimental results. Finally, we summarize our work in \ref{sssec:num7}.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{avasr.jpg}
\caption{Pipeline of AV-ASR system using feature fusion method}
\label{fig:avasr}
\end{figure}
\section{Related Work} \label{sssec:num2}
The differences between various AV-ASR systems lie chiefly in the methods employed for visual feature extraction and audio-visual information fusion. Visual feature extraction methods can be of 3 types\cite{potamianos2003recent} : 1. Appearance based features where each pixel in the mouth region of the speaker (ROI) is considered to be informative. Usually a transformation such as DCT or PCA is applied to the ROI to reduce the dimensions. Additional feature processing such as mean normalization, intra-frame and inter-frame LDA may be applied \cite{huang2003improving}\cite{potamianos2003recent}. 2. Shape based features utilize the geometric features such as height, width and area of the lip region or build a statistical model of the lip contours whose parameters are used as features. 3. Combination of appearance and shape based features.
Fusion methods can be broadly divided into two types\cite{potamianos2003recent}\cite{katsaggelos2015audiovisual}: 1. Feature fusion 2. Decision fusion. Feature fusion models perform a low level integration of audio and visual features and this involves a single model which is trained on the fused features. Feature fusion may include a simple concatenation of features or feature weighting and is usually followed by a dimensionality reduction transformation like LDA. Decision fusion is applied in cases where the output classes for the two modalities are same. Various decision fusion methods based on variants of HMMs have been proposed\cite{dupont2000audio}\cite{brand1997coupled}. In Multistream HMM the emission probability of a state of audio-visual system is obtained by a linear combination of log-likelihoods of individual streams for that state. The parameters of HMMs for individual streams can be estimated separately or jointly. While multistream HMM assumes state level synchrony between the two streams, some methods\cite{bengio2004multimodal}\cite{brand1997coupled} such as coupled HMM\cite{brand1997coupled} allow for asynchrony between two streams. For a detailed survey on HMM based AV-ASR systems we refer the readers to \cite{potamianos2003recent}\cite{katsaggelos2015audiovisual}
Application of deep learning to multi-modal analyses was presented in \cite{ngiam2011multimodal} which describes multi-modal, cross-modal and shared representation learning and their applications to AV-ASR. In \cite{huang2013audio}, Deep Belief Networks(DBN) are explored. In \cite{mroueh2015deep} the authors train separate networks for audio and visual inputs and fuse the final layers of two networks, and then build a third DNN with the fused features. In addition, \cite{mroueh2015deep} presents a new DNN architecture with a bilinear soft-max layer which further improves the performance. In \cite{noda2015audio} a deep de-noising auto-encoder is used to learn noise robust speech features. The auto-encoder is trained with MFCC features of noisy speech as input and reconstructs clean features. The outputs of final layer of the auto-encoder are used as audio features. A CNN is trained with images from the mouth region as input and phoneme labels as output. The final layers of the two networks are then combined to train a multi-stream HMM.
\section{Sequence Labeling Using RNN}\label{sssec:num3}
The following notations are adopted in this paper. For an utterance $u$ of length $T_{u}$, $\textbf{O}_{a}^{u}=(\overline{O}_{a,1}^{u}, \overline{O}_{a,2}^{u}, ... , \overline{O}_{a,T_{u}}^{u})$ and $\textbf{O}_{v}^{u}=(\overline{O}_{v,1}^{u}, \overline{O}_{v,2}^{u}, ... , \overline{O}_{v,T_{u}}^{u})$ denote the observation sequences of audio and visual frames where $\overline{O}_{a,t} \in \mathbb{R}^{d_{a}}$ and $\overline{O}_{v,t} \in \mathbb{R}^{d_{v}} $. We assume equal frame rates for audio and visual inputs which is ensured in experiments by means of interpolation. $\textbf{O}_{av}^{u}=(\overline{O}_{av,1}^{u}, \overline{O}_{av,2}^{u}, ... , \overline{O}_{av,T_{u}}^{u})$ where $\overline{O}_{av,t}^{u} = [\overline{O}_{a,t}^{u}, \overline{O}_{v,t}^{u}] \in \mathbb{R}^{d_{av}}$ where $d_{av}=d_{a}+d_{v}$ denotes the concatenated features at time t for utterance u. The corresponding label sequence is given by $l=(l_{1}, l_{2}, ..., l_{S_{u}})$ where $S_{u}\le T_{u}$ and $l_{i}\in L$ where $L$ is the set of English letters and an additional element representing a space. For ease of representation, we drop the utterance index $u$. All the models described in this paper are character based.
\subsection{Bi-directional RNN}
RNNs are a class of neural networks used to map sequences to sequences. This is possible because of the feedback connections between hidden nodes. In a bi-directional RNN, the hidden layer has two components each corresponding to forward(past) and backward(future) connections. For a given input sequence $\textbf{O}=(\overline{O}_{1}, \overline{O}_{2}, ... , \overline{O}_{T})$, the output of the network is calculated as follows:
forward pass through forward component of the hidden layer at a given instant $t$ is given by
\begin{equation}
\label{eq:eq8}
\overline{h}_{t}^{f} = g(\textbf{W}_{ho}^{f}\overline{O}_{t}+\textbf{W}_{hh}^{f}\overline{h}_{t-1}^{f}+\overline{b}_{h}^{f})
\end{equation}
where $\textbf{W}_{ho}^{f}$ is the input-to-hidden weights for forward component, $\textbf{W}_{hh}^{f}$ corresponds to hidden-to-hidden weights between forward components, and $\overline{b}_{h}^{f}$ is the forward component bias. $g$ is a non-linearity depending on the choice of the hidden layer unit. Similarly, forward pass through the backward component of the hidden layer is given by
\begin{equation}
\label{eq:eq9}
\overline{h}_{t}^{b} = g(\textbf{W}_{ho}^{b}\overline{O}_{t}+\textbf{W}_{hh}^{b}\overline{h}_{t-1}^{b}+\overline{b}_{h}^{b})
\end{equation}
where $\textbf{W}_{ho}^{b}$, $\textbf{W}_{hh}^{b}$, $\overline{b}_{h}^{b}$ are the corresponding parameters for the backward component. The input to next layer is the concatenated vector $[\textbf{h}_{t}^{f},\textbf{h}_{t}^{b}]$. In a deep RNN multiple such bidirectional hidden layers are stacked.
RNNs are trained using Back-Propagation Through Time (BPTT) algorithm. The training algorithm suffers from vanishing gradients problem which is overcome by using a special unit in hidden layer called the Long Short Term Memory(LSTM)\cite{hochreiter1997long}\cite{graves2012neural}.
\subsection{Connectionist Temporal Classification}
DNNs used in ASR systems are frame-level classifiers i.e., each frame of the input sequence is requires a class label in order for the DNN to be trained. The frame-level labels are usually HMM states, obtained by first training a GMM/HMM model and then by forced alignment of input sequences to the HMM states. CTC objective function\cite{graves2006connectionist}\cite{graves2014towards} obviates the need for such alignments as it enables the network to learn over all possible alignments.
Let the input sequence be $\textbf{O}=(\overline{O}_{1}, \overline{O}_{2}, ... , \overline{O}_{T})$ and a corresponding label sequence $\textbf{l}=(l_{1}, l_{2}, ..., l_{S})$ where $S\le T$. The RNN employs a soft-max output layer containing one node for each element in $L'$ where $L' = L \cup \{\phi\}$. The number of output units is $|L'|=|L|+1$. The additional symbol $\phi$ represents a blank label meaning that the network has not produced an output for that input frame. The additional blank label at the output allows us to define an alignment $\pi$ of length T containing elements of $L'$. For example, $(A \phi \phi M \phi), (\phi A \phi \phi M)$ are both alignments of length 5 for the label sequence $AM$. Accordingly, a many to one map $B: L'^{T} \longmapsto L^{\le T}$ can be defined which generates the label sequence from an alignment.
Assuming that the posterior probabilities obtained at soft-max layer, at each instant are independent we get
\begin{equation} \label{eq:eq1} P(\pi | \textbf{O}) = \prod_{t=1}^{T} P(k_{t}| \overline{O}_{t}) \end{equation} where $k \in L'$ and
\begin{equation} \label{eq:eq2} P(k_{t} | \overline{O}_{t}) = \frac{\exp(y_{t}^{k})}{\Sigma_{k'}\exp(y_{t}^{k'})} \end{equation}
where $y_{t}^{k}$ is the input to node $k$ of the
soft-max layer at time $t$
The likelihood of the label sequence given an observation sequence can be calculated by summing (\ref{eq:eq1}) over all possible alignments.
\begin{equation} \label{eq:eq3} P(\textbf{l}|\textbf{O}) = \sum_{\pi \in B^{-1}(\textbf{l})} P(\pi | \textbf{O}) \end{equation}
The goal is to maximize the log-likelihood $\log P(\textbf{l}|\textbf{O})$ estimation of a label sequence given an observation sequence. Equation \ref{eq:eq3} is computationally intractable since the number of alignments increases exponentially with the number of labels.
For efficient computation of (\ref{eq:eq3}), forward-backward algorithm is used.
\section{Feature Extraction}\label{sssec:num4}
\subsection{Audio Features}
The sampling rate of audio data is converted to 16kHz. For each frame of speech signal of 25ms duration, filter-bank features of 40 dimensions are extracted. The filter-bank features are mean normalized and $\Delta$ and $\Delta \Delta$ features are appended. The final 120 dimensional features are used as audio features.
\subsection{Visual Features}
The video frame rate is increased to match the rate of audio frames through interpolation. For AV-ASR, the ROI for visual features is the region surrounding the speaker's mouth. Each frame is converted to gray scale and face detection is performed using Viola-Jones algorithm. The 64x64 lip region is extracted by detecting 68 landmark points\cite{kazemi2014one} on the speakers face, and cropping the ROI surrounding speakers mouth and chin. 100 dimensional DCT features are extracted from the ROI.
After several experiments of training with DCT features, we found that RNN training either exploded or converged poorly. In order improve the discriminatory power of the visual features, we perform non-linear dimensionality reduction of the features using a deep bottleneck network. Bottleneck features are obtained by training a neural network in which one of the hidden layers has relatively small dimensions. The DNN is trained using cross-entropy cost function with character labels as output. The frame-level character labels required for DNN training are obtained by first training an acoustic model ($RNN_{a}$) and then obtaining the outputs from the final soft-max layer of $RNN_{a}$.
The DNN configuration is given by $dim-1024-1024-40-1024-opdim$ where $dim=1100$ and is obtained by splicing each 100 dimensional video frame with a context of 10 frames - 5 on each side. $opdim=|L'|$. After training, the last 2 layers are discarded and 40-dimensional outputs are used as visual features. The final dimension of visual feature vector is 120 including the $\Delta$ and $\Delta \Delta$ features.
\section{Fusion models}\label{sssec:num5}
In this work, the fusion models are character based RNNs trained using CTC objective function i.e. $L'$ is the set of English alphabet including a blank label. The two fusion models are shown in Figure \ref{fig:fusion}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{fusion_correct_2.png}
\caption{Fusion models (a) Featue fusion (b) Decision fusion. The bottleneck network for visual feature extraction is enclosed in the dotted box.}
\label{fig:fusion}
\end{figure}
\subsection{Feature Fusion}
In feature fusion technique, a single $RNN_{av}$ is trained by concatenating the audio and visual features using the CTC objective function. In the test phase, at each instant the concatenated features are forward propagated through the network. In the CTC decoding step, the posterior probabilities obtained at the soft-max layer are converted to pseudo log-likelihoods\cite{vesely2013sequence} as
\begin{equation}\label{eq:eq4} \log P_{av}(\overline{O}_{av,t} | k) = \log P_{av}(k | \overline{O}_{av,t}) - \log P(k)\end{equation}
where $ k \in L' $ and $P(k)$ is the prior probability of class $k$ obtained from the training data \cite{miao2015eesen}.
\subsection{Decision Fusion}
In decision fusion technique the audio and visual modalities are modeled by separate networks, $RNN_{a}$ and $RNN_{v}$ respectively.
$RNN_{v}$ is a lip-reading system. The networks are trained separately. In the test phase, for a given utterance the frame level,
the pseudo log-likelihoods of $RNN_{a}$ and $RNN_{v}$ are combined as
\begin{equation}\label{eq:eq5}\log P_{av}(\overline{O}_{a,t}, \overline{O}_{v,t} | k) = \gamma \log P_{a}(k | \overline{O}_{a,t}) + (1 - \gamma) \log P_{v}(k | \overline{O}_{v,t}) - \log P(k)\end{equation}
where $0 \le \gamma \le 1$ is a parameter dependent on the noise level and the reliability of each modality\cite{dupont2000audio}. For example, at higher levels of noise in audio input, a low value of $\gamma$ is preferred. In this work, we adapt the parameter $\gamma$ for each utterance based on KL-divergence measure between the posterior probability distributions of $RNN_{a}$ and $RNN_{v}$. The divergence between the posterior probability distributions is expected to vary as the noise in the audio modality increases. The KL-divergence is scaled to a value in $[0,1]$ using logistic sigmoid. The parameter $b$ was determined empirically from validation dataset.
\begin{equation}\label{eq:eq6} D_{KL}(P_{v}||P_{a}) = \sum_{i} P_{v}log P_{a}\end{equation}
where we consider the posteriors of $RNN_{v}$ as the true distribution based on the assumption that video input is always free from noise.
\begin{equation}\label{eq:eq7} \gamma= \frac{1}{1+exp(-D_{KL}+b)}\end{equation}
\section{Experiments}\label{sssec:num6}
The system was trained and tested on GRID audio-visual corpus\cite{cooke2006audio}. GRID corpus is a collection of audio and video recordings of 34 speakers (18 male, 16 female) each uttering a 1000 sentences. Each utterance has a fixed length of approximately 3 seconds. The total number of words in the vocabulary is 51. The syntactic structures of all sentences are similar as shown below. \newline \newline
$< command >$ \space $<color>$ \space $<preposition>$ \space $<letter>$ \space $<digit>$ \space $<adverb>$ \newline
Ex. PLACE RED AT M ZERO PLEASE\newline
\subsection{Training}
In the corpus obtained, the video recordings for speaker 21 were not available. In addition, 308 utterances by various speakers could not be processed due to various errors. The dataset in effect consisted of 32692 utterances 90\% of the which (containing 29423 utterances) was used for training and cross validation while the remaining (10\%) data was used as test set. Both training and test data contain utterances from all of the speakers. Models were trained and tested using Kaldi speech recognition tool kit\cite{povey2011kaldi}, Kaldi+PDNN\cite{miao2014kaldi+} and EESEN framework\cite{miao2015eesen}.
\subsubsection{$RNN_{a}$-Acoustic model}
$RNN_{a}$ contains 2 bi-directional LSTM hidden layers.Input to the network is 120-dimensional vector containing filter-bank coefficients along with $\Delta$ and $\Delta \Delta$ features. The model parameters are randomly initialized within the range [-0.1,0.1]. The initial learning rate is set to 0.00004. Learning rate adaption is performed as follows: when the improvement in accuracy on the cross-validation set between two successive epochs falls below 0.5\%, the learning rate is halved.The halving continues for each subsequent epoch until the training stops when the increase in frame level accuracy is less than 0.1\%.
\subsubsection{Deep Bottleneck Network}
The training protocol similar to \cite{vesely2013sequence} was followed to train the bottleneck network. Input video features are mean normalized and spliced. Cross-entropy loss function is minimized using mini-batch Stochastic Gradient Descent (SGD). The frames are shuffled randomly before each epoch. Batch size is set to 256 and initial learning rate is set to 0.008. Learning rate adaptation similar to acoustic model is employed.
\subsubsection{$RNN_{v}$-Lip Reader}
$RNN_{v}$ is trained with bottleneck network features as input. The network architecture and training procedure is same as $RNN_{a}$. Figure \ref{fig:dctvsbn} depicts the learning curves when trained with bottleneck features and DCT features. The figure shows that bottleneck features are helpful in proper convergence of the model.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{learning_curves.png}
\caption{Learning curves for bottleneck(bn) features and DCT features for training(tr) and validation(cv) data sets.}
\label{fig:dctvsbn}
\end{figure}
\subsubsection{$RNN_{av}$}
The feature fusion model $RNN_{av}$ consists of 3 bi-directional LSTM hidden layers. The input dimension is 240, corresponding to filter-bank coefficients of audio modality, bottleneck features of visual modality and their respective $\Delta$ features. The initialization and learning rate adaption are similar to acoustic model training. However, the learning rate adaptation is employed only after a minimum number of(in this case 20) epochs are completed.
During each utterance in an epoch we first present the fused audio-visual fused input sequence followed by the input sequence with audio input set to very low values. This prevents the $RNN_{av}$ from over-fitting to audio only inputs. Thus the effective number of sequences presented to the network in a given epoch is twice the total number of training utterances (AV and V features). After the training with AV and V features we train the network once again with two epochs of audio only utterances obtained by turning off the visual modality.
\subsection{Results}
The audio-visual model is tested with three levels of babble noise 0dB SNR, 10dB SNR and clean audio. Noise was added to test data artificially by mixing babble noise with clean audio .wav files. In order to show the importance of visual modality under noisy environment, the model is tested with either audio or video inputs turned off. A token WFST\cite{miao2015eesen} is used to map the paths to their corresponding label sequences. The token WFST obtains this mapping by removing all the blanks and repeated labels. Character Error Rate(CER) is obtained from the decoded and expected label sequences by calculating the Edit distance between them. The CER results are shown in Table \ref{table:FUSIONCOMPARE}.\begin{center}
\begin{table}
\centering
\caption[short text]{\% CER comparison for decision fusion($RNN_{a},RNN_{v}$) and feature fusion ($RNN_{av}$) models. $RNN_{a}$
is the acoustic model and $RNN_{v}$ is the lip reader}
\begin{tabular}{| l | l | l | l |}
\hline
Model & \multicolumn{2}{ c| }{Input} & CER \% \\ \hline
& Audio & Visual & \\ \hline
$RNN_{av}$ & Clean & OFF & 7.35 \\ \hline
$RNN_{av}$ & Clean & ON & 5.74 \\ \hline
$RNN_{av}$ & OFF & ON & 11.42 \\ \hline
$RNN_{av}$ & 10 SNR dB & OFF & 38.31 \\ \hline
$RNN_{av}$ & 10 SNR dB & ON & 10.24 \\ \hline
$RNN_{av}$ & 0 SNR dB & OFF & 59.65 \\ \hline
$RNN_{av}$ & 0 SNR dB & ON & 11.57 \\ \hline
$RNN_{a},RNN_{v}$ & Clean & OFF & 2.45 \\ \hline
$RNN_{a},RNN_{v}$ & Clean & ON & 8.46 \\ \hline
$RNN_{a},RNN_{v}$ & OFF & ON & 11.06 \\ \hline
$RNN_{a},RNN_{v}$ & 10 SNR dB & OFF & 23.83 \\ \hline
$RNN_{a},RNN_{v}$ & 10 SNR dB & ON & 14.83 \\ \hline
$RNN_{a},RNN_{v}$ & 0 SNR dB & OFF & 59.27 \\ \hline
$RNN_{a},RNN_{v}$ & 0 SNR dB & ON & 16.84 \\ \hline
\end{tabular}
\label{table:FUSIONCOMPARE}
\end{table}
\end{center}
We observe that with clean audio input, audio only $RNN_{a}$ performs significantly better (CER 2.45\%) compared to audio-visual $RNN_{av}$ (CER 5.74\%). However as audio becomes noisy, the performance of $RNN_{a}$ deteriorates significantly whereas the performance of $RNN_{av}$ remains relatively stable. Under noisy conditions the feature fusion model behaves as if it is not receiving any input from the audio modality.
Table \ref{table:FUSIONCOMPARE} also gives a comparison between feature fusion model and decision fusion model. We find that feature fusion model performs better than decision fusion model in all cases except under clean audio conditions. The poor CER of $RNN_{a},RNN_{v}$ model indicates that the frame level predictions between $RNN_{a}$ and $RNN_{v}$ are not synchronous. However, both the fusion models provide significant gains under noisy audio inputs. While there is large difference between $RNN_{a}$ and other models with clean inputs, we believe this difference is due to the nature of dataset and will reduce with larger datasets.
\section{Conclusions And Future Work}\label{sssec:num7}
In this work we presented an audio-visual ASR system using deep RNNs trained with CTC objective function. We described a feature processing step for visual features using deep bottleneck layer and showed that it helps in faster convergence of RNN model during training. We presented a training protocol in which either of the modalities is turned off during training in order to avoid dependency on a single modality. Our results indicate that the trained model is robust to noise. In addition, we compared fusion strategies at the feature level and at the decision level.
While the use of bottleneck features for visual modality helps in training, it requires frame level labels which involves an additional step of training audio RNN. Therefore, our system is not yet end-to-end. Our experiments in visual feature engineering with unsupervised methods like multi-modal auto-encoder\cite{ngiam2011multimodal} did not produce remarkable results. In future work we intend to explore other unsupervised methods for visual feature extraction such as canonical correlation analysis.
\bibliographystyle{splncs03}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,284 |
package com.cyanogenmod.filemanager.util;
import android.app.Activity;
import android.app.AlertDialog;
import android.content.ActivityNotFoundException;
import android.content.Context;
import android.content.DialogInterface;
import android.os.AsyncTask;
import android.util.Log;
import android.widget.Toast;
import com.cyanogenmod.filemanager.FileManagerApplication;
import com.cyanogenmod.filemanager.R;
import com.cyanogenmod.filemanager.commands.SyncResultExecutable;
import com.cyanogenmod.filemanager.commands.shell.InvalidCommandDefinitionException;
import com.cyanogenmod.filemanager.console.CommandNotFoundException;
import com.cyanogenmod.filemanager.console.ConsoleAllocException;
import com.cyanogenmod.filemanager.console.ConsoleBuilder;
import com.cyanogenmod.filemanager.console.ExecutionException;
import com.cyanogenmod.filemanager.console.InsufficientPermissionsException;
import com.cyanogenmod.filemanager.console.NoSuchFileOrDirectory;
import com.cyanogenmod.filemanager.console.OperationTimeoutException;
import com.cyanogenmod.filemanager.console.ReadOnlyFilesystemException;
import com.cyanogenmod.filemanager.console.RelaunchableException;
import com.cyanogenmod.filemanager.preferences.AccessMode;
import java.io.FileNotFoundException;
import java.io.IOException;
import java.io.PrintWriter;
import java.io.StringWriter;
import java.text.ParseException;
import java.util.List;
/**
* A helper class with useful methods for deal with exceptions.
*/
public final class ExceptionUtil {
/**
* An interface to communicate events related with the result of a command relaunch.
*/
public interface OnRelaunchCommandResult {
/**
* Method invoked when the relaunch operation was success
*/
void onSuccess();
/**
* Method invoked when the relaunch operation was cancelled by the user
*/
void onCancelled();
/**
* Method invoked when the relaunch operation was failed
*
* @param cause The cause of the failed operation
*/
void onFailed(Throwable cause);
}
/**
* Constructor of <code>ExceptionUtil</code>.
*/
private ExceptionUtil() {
super();
}
//Definition of known exceptions and his representation mode and resource identifiers
private static final Class<?>[] KNOWN_EXCEPTIONS = {
FileNotFoundException.class,
IOException.class,
InvalidCommandDefinitionException.class,
ConsoleAllocException.class,
NoSuchFileOrDirectory.class,
ReadOnlyFilesystemException.class,
InsufficientPermissionsException.class,
CommandNotFoundException.class,
OperationTimeoutException.class,
ExecutionException.class,
ParseException.class,
ActivityNotFoundException.class
};
private static final int[] KNOWN_EXCEPTIONS_IDS = {
R.string.msgs_file_not_found,
R.string.msgs_io_failed,
R.string.msgs_command_not_found,
R.string.msgs_console_alloc_failure,
R.string.msgs_file_not_found,
R.string.msgs_read_only_filesystem,
R.string.msgs_insufficient_permissions,
R.string.msgs_command_not_found,
R.string.msgs_operation_timeout,
R.string.msgs_operation_failure,
R.string.msgs_operation_failure,
R.string.msgs_not_registered_app
};
private static final boolean[] KNOWN_EXCEPTIONS_TOAST = {
false,
false,
false,
false,
false,
true,
true,
false,
true,
true,
true,
false
};
/**
* Method that attach a asynchronous task for executing when exception need
* to be re-executed.
*
* @param ex The exception
* @param task The task
* @see RelaunchableException
*/
public static void attachAsyncTask(Throwable ex, AsyncTask<Object, Integer, Boolean> task) {
if (ex instanceof RelaunchableException) {
((RelaunchableException)ex).setTask(task);
}
}
/**
* Method that captures and translate an exception, showing a
* toast or a alert, according to the importance.
*
* @param context The current context
* @param ex The exception
*/
public static synchronized void translateException(
final Context context, Throwable ex) {
translateException(context, ex, false, true);
}
/**
* Method that captures and translate an exception, showing a
* toast or a alert, according to the importance.
*
* @param context The current context.
* @param ex The exception
* @param quiet Don't show UI messages
* @param askUser Ask the user when if the exception could be relaunched with other privileged
*/
public static synchronized void translateException(
final Context context, final Throwable ex,
final boolean quiet, final boolean askUser) {
translateException(context, ex, quiet, askUser, null);
}
/**
* Method that captures and translate an exception, showing a
* toast or a alert, according to the importance.
*
* @param context The current context.
* @param ex The exception
* @param quiet Don't show UI messages
* @param askUser Ask the user when if the exception could be relaunched with other privileged
* @param listener The listener where return the relaunch result
*/
public static synchronized void translateException(
final Context context, final Throwable ex,
final boolean quiet, final boolean askUser,
final OnRelaunchCommandResult listener) {
//Get the appropriate message for the exception
int msgResId = R.string.msgs_unknown;
boolean toast = true;
int cc = KNOWN_EXCEPTIONS.length;
for (int i = 0; i < cc; i++) {
if (KNOWN_EXCEPTIONS[i].getCanonicalName().compareTo(
ex.getClass().getCanonicalName()) == 0) {
msgResId = KNOWN_EXCEPTIONS_IDS[i];
toast = KNOWN_EXCEPTIONS_TOAST[i];
break;
}
}
//Check exceptions that can be asked to user
if (ex instanceof RelaunchableException && askUser) {
((Activity)context).runOnUiThread(new Runnable() {
@Override
public void run() {
askUser(context, (RelaunchableException)ex, quiet, listener);
}
});
return;
}
//Audit the exception
Log.e(context.getClass().getSimpleName(), "Error detected", ex); //$NON-NLS-1$
//Build the alert
final int fMsgResId = msgResId;
final boolean fToast = toast;
if (!quiet) {
((Activity)context).runOnUiThread(new Runnable() {
@Override
public void run() {
try {
if (fToast) {
DialogHelper.showToast(context, fMsgResId, Toast.LENGTH_SHORT);
} else {
AlertDialog dialog =
DialogHelper.createErrorDialog(
context, R.string.error_title, fMsgResId);
DialogHelper.delegateDialogShow(context, dialog);
}
} catch (Exception e) {
Log.e(context.getClass().getSimpleName(),
"ExceptionUtil. Failed to show dialog", ex); //$NON-NLS-1$
}
}
});
}
}
/**
* Method that ask the user for an operation and re-execution of the command.
*
* @param context The current context
* @param relaunchable The exception that contains the command that must be re-executed.
* @param listener The listener where return the relaunch result
* @hide
*/
static void askUser(
final Context context,
final RelaunchableException relaunchable,
final boolean quiet,
final OnRelaunchCommandResult listener) {
//Is privileged?
boolean isPrivileged = false;
try {
isPrivileged = ConsoleBuilder.getConsole(context).isPrivileged();
} catch (Throwable ex) {
/**NON BLOCK**/
}
// If console is privileged there is not need to change
// If we are in a ChRooted environment, resolve the error without doing anymore
if (relaunchable instanceof InsufficientPermissionsException &&
(isPrivileged ||
FileManagerApplication.getAccessMode().compareTo(AccessMode.SAFE) == 0)) {
translateException(
context, relaunchable, quiet, false, null);
// Operation failed
if (listener != null) {
listener.onFailed(relaunchable);
}
return;
}
//Create a yes/no dialog and ask the user
final DialogInterface.OnClickListener clickListener =
new DialogInterface.OnClickListener() {
@Override
public void onClick(DialogInterface dialog, int which) {
if (which == DialogInterface.BUTTON_POSITIVE) {
//Run the executable again
try {
//Prepare the system before re-launch the command
prepare(context, relaunchable);
//Re-execute the command
List<SyncResultExecutable> executables = relaunchable.getExecutables();
int cc = executables.size();
for (int i = 0; i < cc; i++) {
SyncResultExecutable executable = executables.get(i);
Object result = CommandHelper.reexecute(context, executable, null);
AsyncTask<Object, Integer, Boolean> task = relaunchable.getTask();
if (task != null && task.getStatus() != AsyncTask.Status.RUNNING) {
task.execute(result);
}
}
// Operation complete
if (listener != null) {
listener.onSuccess();
}
} catch (Throwable ex) {
//Capture the exception, this time in quiet mode, if the
//exception is the same
boolean ask = ex.getClass().getName().compareTo(
relaunchable.getClass().getName()) == 0;
translateException(context, ex, quiet, !ask, listener);
// Operation failed
if (listener != null) {
listener.onFailed(ex);
}
}
} else {
// Operation cancelled
if (listener != null) {
listener.onCancelled();
}
}
}
};
AlertDialog alert = DialogHelper.createYesNoDialog(
context,
R.string.confirm_operation,
relaunchable.getQuestionResourceId(),
clickListener);
alert.setOnDismissListener(new DialogInterface.OnDismissListener() {
@Override
public void onDismiss(DialogInterface dialog) {
// Operation cancelled
if (listener != null) {
listener.onCancelled();
}
}
});
DialogHelper.delegateDialogShow(context, alert);
}
/**
* Method that prepares the system for re-execute the command.
*
* @param context The current context
* @param relaunchable The {@link RelaunchableException} reference
* @hide
*/
static void prepare(final Context context, final RelaunchableException relaunchable) {
//- This exception need change the console before re-execute
if (relaunchable instanceof InsufficientPermissionsException) {
ConsoleBuilder.changeToPrivilegedConsole(context);
}
}
/**
* Method that prints the exception to an string
*
* @param cause The exception
* @return String The stack trace in an string
*/
public static String toStackTrace(Exception cause) {
StringWriter sw = new StringWriter();
PrintWriter pw = new PrintWriter(sw);
try {
cause.printStackTrace(pw);
return sw.toString();
} finally {
try {
pw.close();
} catch (Exception e) {/**NON BLOCK**/}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,496 |
\section{Introduction}
Suppose that $(L_t: t \ge 0)$ is a real-valued L\'evy process defined on some
probability space $(\Omega, \mathcal A, \Pr)$ and we observe $n$ of its
increments
\begin{equation} \label{data}
X_k = L_{k \Delta} - L_{(k-1)\Delta},\quad k =1, \dots, n,
\end{equation}
sampled at frequency $1/\Delta>0$. Equivalently the $X_k$'s are drawn i.i.d.~from some
infinitely divisible distribution $\PP_\Delta$, with corresponding empirical measures $\PP_{\Delta, n} = \frac{1}{n} \sum_{k=1}^n \delta_{X_k}$.
\smallskip
L\'evy processes are increasingly popular in stochastic modelling. A question of key importance is how the structure of the L\'evy process, particularly its jump behaviour, can be
recovered from these observed increments.
From a statistical point of view it is natural to consider a growing observation
horizon $n \Delta \to \infty$. If simultaneously $\Delta = \Delta_n$ approaches
zero one speaks of a `high-frequency' sampling regime, as opposed to
`low-frequency' sampling where $\Delta$ remains fixed.
Inference problems of this kind have recently gained increased attention.
\cite{JMV05} studied nonparametric inference for L\'evy-driven
Ornstein--Uhlenbeck processes. \cite{BR06} treat nonparametric estimation of
L\'evy processes in a financial model.
Low-frequency observations were considered, e.g., by
\cite{neumannReiss2009}, \cite{B10}, \cite{G12} as well as
\cite{NicklReiss2012}, whereas
\cite{figueroa-lopez2009,Figueroa-Lopez2011} treats high-frequency
observations.
Nonparametric estimation of L\'evy processes in a model selection context was
studied by
\cite{CGC11} and
\cite{kappus2014}. A general discussion of the literature
and further references can be found in the recent survey paper \cite{R13}.
\smallskip
By the L\'evy--Khintchine representation (\cite{sato1999}) the L\'evy process $(L_t: t \ge 0)$ is entirely
characterised by three parameters: the \textit{diffusion coefficient} $\sigma^2$
describing the Brownian motion component, the \textit{centring} or
\textit{drift} parameter $\gamma$, and the \textit{L\'evy measure} $\nu$.
Recovering the L\'evy process can thus be reduced to recovering the L\'evy
triplet $(\sigma^2, \gamma, \nu)$. Statistical inference for the one-dimensional
parameters $\sigma^2, \gamma$ can be based on standard statistics such as
the quadratic variation and the sample average of the increments, or on spectral
estimators, see Section~\ref{discussion} for discussion and references.
\smallskip
An intrinsically more complex problem than inference on $\sigma^2$ and $\gamma$
is the recovery of the L\'evy measure $\nu$, which describes the jump behaviour
of the L\'evy process. We recall that there is a bijection between the set of
L\'evy measures $\nu$ and all positive Borel measures $\nu$ on $\R$ s.t.
\[\int_{\R} (1 \wedge x^2) \nu(\d x) <\infty, \quad \nu(\{0\})=0.\]
Thus a natural target is to recover the cumulative distribution function
\begin{equation}\label{eq:cdf}
N(t)
= \int_{-\infty}^t (1 \wedge x^2) \nu(\d x), ~~t \in \R,
\end{equation}
from the observed increments; it encodes both
local and global information about $\nu$.
The presence of $(1\wedge x^2)$
smooths the singularity
that $\nu$ may possess at the origin.
Other possibilities to smooth the singularity exist and our results will cover
functions from a general
class (see Section~\ref{secProc}).
In particular this will include recovery of the
distribution function
\begin{equation}
\mathcal N(t) =\int_{-\infty}^t \nu(\d x), ~t < 0,\quad \text{and} \quad
\mathcal N(t) =\int_{t}^{\infty} \nu(\d x), ~t > 0,
\end{equation}
of the L\'evy measure at any point $t\neq0$.
For statistical applications, inference on the functions $N, \mathcal N$ in the
uniform norm $\|\cdot\|_\infty$ on the real line is of particular interest,
paralleling the classical Donsker-Kolmogorov-Smirnov central limit theorems
$$\sqrt n (F_n -F) \to^{\mathcal L} \mathbb G_F $$ in the space of bounded functions on $\R$, where $F_n$ is the empirical
distribution function of a random sample from distribution $F$, and where
$\mathbb G_F$ is the $F$-Brownian bridge (\cite{dudley1999,
vanderVaartWellner1996}). In the L\'evy setting, \cite{NicklReiss2012} considered an estimator for the distribution function
$\mathcal N(t)$, $|t|\ge\zeta,$ based on low-frequency observations ($\Delta$ fixed) and proved such a Donsker-Kolmogorov-Smirnov theorem.
The purpose of the present article is to derive such results when also $\Delta
\to 0$. The main message is that high-frequency observations reveal much finer
statistical properties of the L\'evy measure, and inference is possible for a
much larger class of L\'evy processes than considered in \cite{NicklReiss2012},
including processes with a nonzero Gaussian component. Moreover, the theory does
not only cover nonlinear `inversion' estimators based on the L\'evy-Khintchine
formula, but also `linear' estimators based on elementary counting statistics.
At the heart of these results is a general purpose uniform central limit theorem
for a basic `smoothed empirical process' arising from the $X_k$'s in
(\ref{data}), see Theorem \ref{main1} below.
\smallskip
In the next section we introduce the estimators and give the main results as
well as some statistical applications. In
Section~\ref{secProc} we show how to reduce the proofs to the study of a unified smoothed empirical process, and in Section~\ref{discussion} we discuss
our conditions and their interpretation in a variety of concrete examples of
L\'evy processes. The remainder of the article is then devoted to the proofs of
our results.
\section{Main results: Asymptotic inference on the L\'evy measure $\nu$}\label{secEstimators}
In this section we study two approaches to estimate the distribution
functions $N, \mathcal N$ of a L\'evy measure, based on discrete observations (\ref{data}). The first estimator is constructed by a direct
approach and counts the number of increments below a certain threshold, where
increments are weighted by $1\wedge X_k^2$. The second approach
relies on the L\'evy--Khintchine representation and a
spectral regularisation step.
\subsection{Basic notation and assumptions}
The symbol $\ell^\infty(T)$ denotes the space of bounded functions on a set $T$ normed by the usual supremum norm $\|\cdot\|_\infty$. We will measure the smoothness of functions in a local H\"older
norm: Denoting by $C(U)=C^0(U)$ the set of all functions on an open
set $U\subset\R$ which are bounded, continuous and real-valued, we
define for $s>0$ the H\"older spaces
\[
C^s(U):=\Big\{f\in C(U): \|f\|_{C^s(U)}:=\sum_{k=0}^{\lfloor
s\rfloor}\sup_{x\in U}|f^{(k)}(x)|+\sup_{x,y\in U:x\neq y} \frac{|f^{(\lfloor
s\rfloor)}(x)-f^{(\lfloor s\rfloor)}(y)|} {|x-y|^{ s-\lfloor
s\rfloor}}<\infty\Big\}
\]
where $\lfloor s \rfloor$ denotes the largest integer strictly smaller than
$s$.
\smallskip
We assume throughout this article that the L\'evy measure has finite second moments,
\begin{equation}\label{eqSecMom}
\int_{\R} x^2 \nu(\d x) <\infty.
\end{equation}
This is equivalent to $\PP_\Delta$ having finite second moments $\forall \Delta>0$ (\cite{sato1999}).
For our main results we will rely on the following stronger assumption on $\nu$. Slightly abusing notation we shall use the same symbol for a measure and its Lebesgue density, if the latter exists. Also we use
$\lesssim, \gtrsim, (\sim)$ to denote (two-sided)
inequalities up to a multiplicative constant.
\begin{assumption}\label{assProc}
\begin{enumerate}
\item[(a)] For some $\eps>0$ we have $$\int_{\R}|x|^{4+\eps}\nu (d
x)<\infty.$$
\item[(b)] The L\'evy measure $\nu$ has a Lebesgue density, also denoted by $\nu$, and $$(1\wedge x^4)\nu\in \ell^\infty(\R).$$
\item[(c)] The measure $x^3\PP_{\Delta}$ admits a Lebesgue density, also denoted by $x^3\PP_\Delta$, satisfying, as $\Delta \to 0$,
$$\norm{x^3\PP_{\Delta}}_\infty\lesssim \Delta.$$
\item[(d)] Let $U$ be a neighbourhood of the origin and $V \subset \R$. For some $s>0$ and some finite constants $c_t>0$, $t\in V$, we have
\begin{align*}
\|g_t(-\cdot) \ast
(x^2\nu)\|_{C^s(U)} \le c_t \qquad \text{ with }
g_{t}(x):
(1\wedge x^{-2})\1_{(-\infty,t]}(x).
\end{align*}
\end{enumerate}
\end{assumption}
\medskip
Assumptions~(a) and (b) are a moment condition and a mild regularity condition
on the L\'evy measure, respectively. Assumption~(c) is
the key condition and will be discussed in detail in Section~\ref{sec:Boundx3P}. Here we just remark that for
instance under the assumption $x^3\nu\in \ell^\infty(\R)$, this condition will be shown
to be satisfied whenever the diffusion coefficient is positive ($\sigma>0$).
Assumption~(d) is used to control approximation theoretic properties of the distribution function of $x^2\nu$. For global results ($V=\R$) we notice that it is easily seen that (d)
is satisfied with a uniform constant $c>0$ if $x^2\nu\in C^{s-1}(\R), s \ge 1$.
Recall that a function $l$ defined on $(0,\infty)$ is \emph{slowly varying} at
the origin if
\[\frac{l(tx)}{l(t)}\to1,\qquad \text{as } t\to0,~~\forall x>0.\]
A function $f$ is \emph{regularly varying} at the origin with exponent
$p\in\R$ if $f$ is of the form
\begin{align*}
f(x)=x^p l(x)
\end{align*}
with $l$ slowly varying at the origin.
We denote the
symmetrised L\'evy density by $\tilde\nu(x):=\nu^+(x)+\nu^-(-x)$,
where $\nu^+=\nu\1_{\R^+}$ and
$\nu^-=\nu\1_{\R^-}$.
Throughout the paper we write $\to^\mathcal L$
to denote convergence in distribution of random elements in a
metric space as in Chapter 1 in \cite{vanderVaartWellner1996}.
\subsection{The direct estimation approach}
In the
high-frequency regime $\Delta \to 0$ inference on $\nu$ can be based on the
following simple observation.
\begin{lemma} \label{weakcon}
If the L\'evy measure $\nu$ satisfies \eqref{eqSecMom}, then we have weak
convergence
\begin{equation} \label{eqWeakConvNu}
x^2 \frac{\PP_\Delta}{\Delta} \to \sigma^2 \delta_0 + x^2 \nu
\end{equation}
as $\Delta \to 0$ in the sense that
\begin{equation} \label{prettyweak}
\int_{\R} f(x) x^2 \frac{\PP_\Delta(\d x)}{\Delta} \to \sigma^2 f(0) + \int_{\R}
f(x) x^2 \nu(\d x)
\end{equation}
for every bounded continuous function $f: \R \to \R $.
\end{lemma}
Starting with L\'evy processes without
diffusion component, that is, with $\sigma=0$, the asymptotic identification
\eqref{eqWeakConvNu} motivates a linear estimator of
$N(t)$ given by
\begin{equation}\label{eq:naiveEst}
\tilde N_n(t):=\int_{-\infty}^t(1\wedge x^2)\frac{\PP_{\Delta,n}(\d
x)}{\Delta}
=\frac1{n\Delta}\sum_{k=1}^{n}
(1\wedge X_k^2)\1_{(-\infty,t]}(X_k),\quad t \in \R,
\end{equation}
where $\PP_{\Delta,n}=\frac{1}{n}\sum_{k=1}^{n}\delta_{X_k}$ is the empirical
measure of the increments from (\ref{data}).
Similarly, and including the case $\sigma \neq 0$, one can estimate the function $\mathcal N$ by
\begin{align*}
\tilde{\mathcal N}_n(t):=\int_ {\R} f_t(x)\frac{\PP_{\Delta,n}(\d
x)}{\Delta}\quad\text{with}\quad f_t(x):=\begin{cases}
\1_{(-\infty,t]},\quad t<0\\
\1_{[t,\infty)},\quad t>0.
\end{cases}
\end{align*}
\smallskip
We start with a theorem for the basic estimator $\tilde N_n$.
\begin{theorem}
\label{thm:UCLTnaive}
Let $\sigma=0$ and grant Assumption~\ref{assProc} for $V=\R$ for some
$s\in (0,2]$ and with uniform constant $\sup_{t\in\R}c_t=c<\infty$.
Assume either that
a) the density of $x \nu$ exists and is of bounded variation, and
the drift $\gamma_0 := \gamma-\int x \nu (\d x)=0$;~ or that
b) $\tilde\nu(x)=\nu^+(x)+\nu^-(-x)$ is regularly varying at
zero with exponent $-(\beta+1)$, $\beta\in(0,2), s \in (0,2-\beta)$.
If $n\to\infty$ and
$\Delta_{n}\to0$ such that
\begin{align*}
n\Delta_{n}\to\infty,\quad \Delta_{n}=o(n^{-1/(s+1)})\quad\text{ and }\quad
\log^4(1/\Delta_n)=o(n\Delta_n),
\end{align*}
then
\[
\sqrt{n\Delta_n}\big(\tilde{N}_{n}-N\big)\to^{\mathcal{L}}\mathbb{G}
\quad\text{in}\quad\ell^{\infty}(\R),
\]
where $\mathbb{G}$ is a tight Gaussian random variable arising from the
centred Gaussian process
$\{\mathbb{G}(t):t\in\R\}$
with covariance
\[
\E[\mathbb{G}(t)\mathbb{G}(t')]=\int_{-\infty}^{t \wedge t'}(1\wedge
x^4)\nu(\d x),\quad t,t'\in\R.
\]
\end{theorem}
Since estimation at the origin $t=0$ is included in the last theorem, the assumption $\sigma=0$ is natural -- the simple linear estimator $\tilde N_n$ cannot distinguish between arbitrarily small jumps and a Brownian diffusion component. Moreover, setting the drift $\gamma_0 =0$ in a) rules out situations where the measure $\PP_\Delta$ has a discrete component $\delta_{\Delta \gamma_0}$, which causes complications in the analysis. Simultaneous estimation of all parameters of the L\'evy triplet without restrictions on $\gamma$ and $\sigma$ will be considered by non-linear methods in the next subsection.
The conditions a) and b) are required to show that the deterministic `bias' term arising from the basic
linear estimator is negligible in the limit distribution (Proposition
\ref{prop:bias}). The case a) covers many examples of finite activity L\'evy processes as well as some limiting cases where the singularity of $\nu$ at the origin behaves like $|x|^{-1}$ (see Subsection \ref{secExamples} for examples). In contrast case b) covers infinite activity processes with a singularity of the form $|x|^{-1-\beta}, \beta \in (0,2)$. The assumption of regular variation of $\tilde \nu$ at zero is natural in all
key examples considered in Subsection \ref{secExamples} below -- typically the variation exponent will be closely related to the regularity $s$ of $N$, and we discuss in Section \ref{sec:discussion} how our
parameter constraints on $\beta$ and $s$ are compatible.
When the origin is \textit{excluded} from consideration, an argument of \cite{Figueroa-Lopez2011} can be used to obtain the following result for the linear estimator $\tilde {\mathcal N}$, allowing also for $\sigma \neq 0$:
\begin{theorem}\label{thm:UCLTnaive2}
Grant Assumptions~\ref{assProc}(a)-(c).
Let $\zeta>0$ and suppose that the L\'evy density $\nu$ is Lipschitz
continuous in an open set $V_0$ containing
$V=(-\infty,-\zeta]\cup[\zeta,\infty)$.
If $n\to\infty$ and $\Delta_{n}\to0$ such that
\begin{align*}
n\Delta_{n}\to\infty,\quad \Delta_{n}=o(n^{-1/3})\quad\text{ and }\quad
\log^4(1/\Delta_n)=o(n\Delta_n).
\end{align*}
Then
\[
\sqrt{n\Delta_n}\big(\tilde{\mathcal
N}_{n}-\mathcal N\big)\to^{\mathcal{L}}\mathbb{W}
\quad\text{in}\quad\ell^{\infty}(V),
\]
where $\mathbb{W}$ is a tight Gaussian random variable arising from the
centred Gaussian process
$\{\mathbb{W}(t):t\in\R\}$
with covariance, for $f_t=\1_{(-\infty,t]}$ for $t<0$ and $f_t=\1_{[t,\infty)}$ for $t>0$,
\[
\E[\mathbb{W}(t)\mathbb{W}(t')]=\int_{\R}f_t(x)f_{t'}(x)\nu(\d
x),\quad t,t'\in V.
\]
\end{theorem}
\smallskip
The estimators $\tilde N_n,\tilde{\mathcal{N}}_n$ are `linear' in the
observations $\PP_{\Delta, n}$, and their consistency relies on the assumption
that $\Delta_n$ tends
to zero fast enough, in Theorems~\ref{thm:UCLTnaive} and \ref{thm:UCLTnaive2} at least of order $\Delta_{n}=o(n^{-1/(s+1)})$ for $s\in(0,2]$. In both theorems a weaker assumption than
$\Delta_{n}=o(n^{-1/3})$ cannot be expected in general: In typical situations the function $\PP(X_{\Delta_n}\le t)$, $t<0$, can be
expressed in terms of $\Delta_n$ as a series expansion
\[\PP(X_{\Delta_n}\le t)=\nu((-\infty,t])\Delta_n+b_t\Delta_n^2+O(\Delta_n^3),~~b_t \in (0,\infty).\]
For a compound Poisson process this follows
by conditioning on the number of jumps but it also holds in more general
infinite activity cases \cite[see][]{FLH09}. From the expansion
we see that the approximation error
$\Delta_n^{-1}\PP(X_{\Delta_n}\le t)-\nu((-\infty,t])$ will not decay
faster than $\Delta_n$, and the assumption $\Delta_n=o(1/\sqrt{n\Delta_n})$ is expressed
equivalently as $\Delta_{n}=o(n^{-1/3})$.
\subsection{The spectral estimation approach}
Instead of relying on $\Delta\to0$ one can identify the L\'evy measure by the
L\'evy--Khintchine
formula
\begin{align}\label{eqLevyKhintchine}
\phi_\Delta(u):=\E[e^{iuX_k}]=e^{\Delta\psi(u)}, \quad
\psi(u)=-\frac{\sigma^2u^2}{2}+i\gamma u+\int_{\R}\big(e^{iux}-1-iux\big)\nu(\d x),~u \in \R,
\end{align}
which we give here in Kolmogorov's version (valid under \eqref{eqSecMom}, see (8.8) in \cite{sato1999}).
Differentiating the characteristic exponent $\psi(u)=\Delta^{-1}\log\phi_\Delta(u)$, one sees
\begin{align}\label{eqIdent}
\psi''(u)&=\frac{\phi_\Delta''(u)\phi_\Delta(u)-(\phi_\Delta')^2(u)}{\Delta\phi_\Delta^2(u)}
=-\sigma^2-\F[x^2\nu](u),
\end{align}
where $\F f(u):=\int e^{iux}f(x)\d x$ and $\F \mu(u) := \int e^{iux}\mu(\d x)$ for any $f\in L^1(\R)\cup L^2(\R)$ and any finite measure $\mu$, respectively, denotes the Fourier transform. If $\F^{-1}$ is the inverse Fourier transform we hence have
\begin{equation}\label{eqident2}
-\mathcal F^{-1}[\psi''] = \sigma^2 \delta_0 + x^2 \nu.
\end{equation}
In contrast to (\ref{eqWeakConvNu}) this identification of $\nu$ is nonlinear in
$\varphi_\Delta = \F \PP_{\Delta}$, but has the remarkable advantage of being
nonasymptotic and valid for all $\Delta>0$, without relying on a high-frequency
approximation $\Delta \to 0$. This was exploited in \cite{NicklReiss2012} to
show that a plug-in of the empirical characteristic function $\mathcal F
\PP_{\Delta, n}$ into (\ref{eqIdent}) can result, for a (naturally)
restricted class of L\'evy processes, in efficient recovery of $\mathcal N(t), t
\neq 0,$ \textit{without} the requirement $\Delta \to 0$.
In the low-frequency case only
L\'evy processes without diffusion
component can be covered. Our high-frequency setting allows us to drop
this (otherwise necessary) restriction and to treat L\'evy processes with diffusion
component and with L\'evy measures from a much wider class.
Replacing $\phi_\Delta(u)$ in (\ref{eqIdent}) by the
empirical characteristic function of the observed increments,
\[\phi_{\Delta,n}(u):=\F \PP_{\Delta, n} (u) = \frac{1}{n}\sum_{k=1}^ne^{iuX_k},\]
(and its derivatives $\phi_{\Delta, n}^{(i)}, i=1,2$, respectively), we obtain an empirical plug-in estimate
$\hat\psi_n''$ of $\psi''$. Recalling the definitions of $g_t, f_t$
in Assumption~\ref{assProc}(d) and in Theorem~\ref{thm:UCLTnaive2}, respectively, the
resulting estimators of $N,\mathcal
N$ are given by
\begin{align}
\hat N_n(t)&:=\int_{\R}g_t(x)\F^{-1}\left[(-\hat\psi''_n-\hat\sigma^2)\mathcal
F K_h \right](x)\d x,\label{eq:kernelEst}\\
\hat{\mathcal{N}}_n(t)&:=\int_{\R}x^{-2}f_t(x)\F^{-1}\left[
(-\hat\psi''_n-\hat\sigma^2)\mathcal
F K_h \right](x)\d x.\notag
\end{align}
Here $K_h$ is a kernel such that $\F K_h$ has compact support, specified in detail below, ensuring in particular that $\hat N_n,\hat{\mathcal{N}}_n$ are well-defined (on sets of probability
approaching one). Moreover, $\hat\sigma^2$ is any pilot estimate of $\sigma^2$.
We can estimate $\sigma^2$ for instance as in
\cite{JacodReiss2013} by
\begin{align}\label{eq:sigmahat}
\hat\sigma^2:=\frac{2}{\Delta u_n^2}\log(|\phi_{\Delta,n}(u_n)|)\qquad \text{
with }u_n:=\sqrt{\frac{2c_0\log(n)}{\Delta \sigma_{\max}^2}},
\end{align}
where $c_0>0$ is a suitable numerical constant, and if we assume a lower bound
on the characteristic function determined by $\sigma_{\max}>0$.
Under suitable conditions Proposition \ref{sigmaest} below entails that the
estimator~$\hat\sigma^2$ satisfies
\begin{align}\label{eq:sigmarate}
\hat\sigma^2-\sigma^2=o_P((n\Delta)^{-1/2}),
\end{align}
and hence is negligible in the limit process $\mathbb G$ in the next theorem.
While the construction of an optimal estimator of $\sigma$ in the setting
considered here is a topic of independent interest, Theorem~\ref{thmUniform}
below will hold for \textit{any} plug-in estimator that satisfies
(\ref{eq:sigmarate}).
We regularise with a band-limited kernel $K_h:=h^{-1}K(h^{-1}\bull)$ of
bandwidth $h>0$. The following properties of $K$ are supposed:
\begin{gather}\label{propKernel}
\begin{split}
\int_{\R} K(x)\d x=1,\qquad \int x^lK(x)\d x=0 \quad\text{ for
}l=1,\dots,p,\\
\supp \mathcal F K\subset[-1,1],\quad x^{p+1}K(x)\in L^1(\R), ~~p \in \mathbb N.
\end{split}
\end{gather}
The main result for the spectral estimators is the following theorem,
where $\mathbb{G}$ and $\mathbb{W}$ are tight Gaussian random variables arising
from the same Gaussian processes as in
Theorems~\ref{thm:UCLTnaive} and \ref{thm:UCLTnaive2}, respectively. For
Part~(ii) we recall the definition $f_t=\1_{(-\infty,t]}$ for $t<0$ and
$f_t=\1_{[t,\infty)}$ for $t>0$.
\begin{theorem}\label{thmUniform}
Grant Assumptions~\ref{assProc}(a)-(c) and let $s>0$.
Let the kernel satisfy \eqref{propKernel}
with $p\ge s\vee 2$ and choose
$h_n\sim\Delta_n^{1/2}$.
Let either $\sigma^2$ be known (in which case $\hat \sigma^2 := \sigma^2$), or
let $\hat\sigma^2$ be any estimator satisfying \eqref{eq:sigmarate}.
Suppose $n\to\infty$ and $\Delta_{n}\to0$ such that
\begin{align*}
n\Delta_{n}\to\infty,\quad \Delta_{n}=o(n^{-1/(s+1)})\quad\text{ and }\quad
\log^4(1/\Delta_n)=o(n\Delta_n).
\end{align*}
\begin{enumerate}
\item\label{global}
Grant Assumption~\ref{assProc}(d) for $s>0$, for $V=\R$ and
for constants $c_t$ with $\sup_{t\in\R}c_t=c<\infty$.
Then
\[
\sqrt{n\Delta_n}\big(\hat{N}_{n}-N\big)\to^{\mathcal{L}}\mathbb{G}
\quad\text{in}\quad\ell^{\infty}(\R).
\]
\item\label{away}
Grant Assumption~\ref{assProc}(d) for $s>0$, for $g_t(x)=x^{-2}f_t(x)$,
for $V=(-\infty,-\zeta]\cup[\zeta,\infty)$, $\zeta>0$, and for constants $c_t$
with $\sup_{t\in
V}c_t=c'<\infty$.
Then
\[
\sqrt{n\Delta_n}\big(\hat{\mathcal{N}}_{n}-\mathcal{N}\big)\to^{\mathcal{L}}
\mathbb{W}
\quad\text{in}\quad\ell^{\infty}(V).
\]
\end{enumerate}
\end{theorem}
\subsection{Limit process and statistical applications}
The continuous mapping theorem with the usual sup-norm
$\|\cdot\|_\infty$ combined with Theorems~\ref{thm:UCLTnaive} and \ref{thmUniform} yields
in particular the limit theorems, as $n \to \infty$,
\begin{equation}
\sqrt{n \Delta} \|\tilde N_n -N\|_\infty \to^{\mathcal L} \|\mathbb
G\|_\infty
\quad
\text{ and }
\quad
\sqrt{n \Delta} \|\hat N_n -N\|_\infty \to^{\mathcal L} \|\mathbb
G\|_\infty.
\end{equation}
This can be used to construct Kolmogorov--Smirnov tests for L\'evy measures and
global confidence bands for the function $N$, as we explain now.
For absolutely continuous L\'evy measures $\nu$ the Gaussian random function $(\mathbb G(t):t\in\R)$ can be realised as a version of
\begin{align}
\mathbb
G(t)&=\mathbb B \left(\int_{-\infty}^t(1\wedge x^4)\nu(\d
x)\right), ~t \in \R,
\end{align}
where $\mathbb B$ is a standard Brownian motion.
An alternative representation is given by $\mathbb
G(t)=\int_{-\infty}^{t}(1\wedge x^2) \nu(x)^{1/2}\d \mathbb B(x)$, where $\mathbb B$ is a
two-sided Brownian motion.
We have
\begin{align*}
\PP\left(\sup_{s\le t}|\mathbb G(s)|\ge
a\right)
=\PP\left(\left(\int_{-\infty}^t(1\wedge x^4)\nu(\d
x)\right)^{1/2}\max_{s\in[0,1]}|\mathbb B(s)|\ge a\right),~~~ t \in \R \cup
\{\infty\},
\end{align*}
so that quantiles of the distribution of $\|\mathbb G\|_\infty$ can
be calculated. For example a global asymptotic confidence band
for $N$ can be constructed in the setting of Theorems~\ref{thm:UCLTnaive}
and~\ref{thmUniform} by defining
\begin{align*}
\tilde C_n(t)&:=\left[\tilde N_n(t)-\frac{\tilde d q_{\alpha}}{\sqrt {n \Delta}}, \tilde N_n(t)+\frac{\tilde d
q_{\alpha}}{\sqrt {n \Delta}}\right],~~
\hat C_n(t):=\left[\hat N_n(t)-\frac{\hat d q_{\alpha}}{\sqrt {n \Delta}}, \hat N_n(t)+\frac{\hat d q_{\alpha}}{\sqrt {n\Delta}}\right], ~~ t \in \R,
\end{align*}
with consistent estimators
\begin{align*}
\tilde d&:=\left(\frac{1}{n\Delta}\sum_{k=1}^{n}(1\wedge
X_k^4)\right)^{1/2},\\
\hat d &:=
\left(\int_{\R}(x^{-2}\wedge
x^2)\F^{-1}\left[(-\hat\psi''_n-\hat\sigma^2)\mathcal
F K_h \right](x)\d x\right)^{1/2}
\end{align*}
of the standard deviation $(\int_{\R}(1\wedge x^4)\nu(\d x))^{1/2}$,
and with $q_\alpha$ the upper $\alpha$--quantile, $0<\alpha<1$,
of the distribution of $\max_{s\in[0,1]}|\mathbb B(s)|$ (see Example~X.5(c) in
\cite{feller1971} for its well-known formula).
For the confidence band $C_n$ equal to either
$$\tilde C_n := \left\{f: f(t) \in \tilde C_n(t) ~ \forall t \in \R\right\},
~\text{ or }~\hat C_n := \left\{f: f(t) \in \hat C_n(t) ~ \forall t \in
\R\right\},$$
Theorems~\ref{thm:UCLTnaive} and~\ref{thmUniform} imply, under the respective assumptions, that the asymptotic coverage probability of $C_n$ equals
\begin{align*}
\lim_{n\to\infty}\PP\left(N(t)\in C_n(t) \; \forall
t\in\R\right)=1-\alpha.
\end{align*}
Theorems~\ref{thm:UCLTnaive} and~\ref{thmUniform} allow likewise the
construction of tests: If $H_0$ is a set of L\'evy measures, let $\mathcal D$ be the set of the corresponding cumulative distribution functions
of the form~\eqref{eq:cdf}. We define $T_n=\1\{\mathcal D\cap
C_n=\varnothing\}$ to reject $H_0$ and accept $H_0$ when $T_n=0$. This test has
asymptotic level $\alpha$: if $\PP_\theta$ is the law of a L\'evy process from $\theta \in H_0$ then we have
\begin{align*}
\lim_{n\to\infty}\PP_\theta(T_n\neq0)\le\alpha,
\end{align*}
assuming $H_0$ satisfies the assumptions of Theorem~\ref{thm:UCLTnaive} or
\ref{thmUniform}.
\subsection{Numerical example}
Let us briefly illustrate the finite sample performance of the two estimation approaches and their corresponding confidence bands. We apply the procedures to two standard examples of pure jump L\'evy processes: a Gamma process and a normal inverse Gaussian (NIG) process. The empirical coverage of the confidence bands reveals the finite sample level of the associated Kolmogorov--Smirnov test and the size of the bands indicates the power of the test.
The Gamma process has infinite, but relatively small jump activity (its Blumenthal-Getoor index equals zero). Its L\'evy measure is given by the Lebesgue density
$ \nu(x)=\frac{c}{x}e^{-\lambda x}, x>0,$
and we choose $c=30$ and $\lambda=1$ here. The NIG process can be constructed by subordinating a diffusion with volatility $s>0$ and drift $\theta\in\R$ by an inverse Gaussian process with variance $\kappa>0$. The resulting infinite variation process has Blumenthal-Getoor index equal to one. The NIG process admits an explicit formula for the jump measure and for its law we apply the simulation algorithm from \cite{contTankov2004}, choosing $s=1.5, \theta=0.1$ and $\kappa=0.5$. Both processes satisfy the assumptions of Theorems~\ref{thm:UCLTnaive} and \ref{thmUniform}, cf. Section~\ref{secExamples}.
\begin{figure}[t]\centering
\includegraphics[width=7.5cm,height=4cm]{GammaNaiv
\includegraphics[width=7.5cm,height=4cm]{GammaSpektral
\hfill
\includegraphics[width=7.5cm,height=4cm]{NIGnaiv
\includegraphics[width=7.5cm,height=4cm]{NIGspektral
\caption{Direct estimator $\tilde N_n$ (\emph{left}) and spectral estimator $\hat N_n$ (\emph{right}) for the Gamma (\emph{top}) and NIG process (\emph{bottom}). Each time 50 estimators (\emph{light blue}) and the true distribution function (\emph{black}) are shown. One estimator (\emph{blue, solid}) with its asymptotic 0.9-confidence band (\emph{blue, dashed}) is highlighted.}\label{fig:sim}
\end{figure}
We simulate $n=2000$ increments with observation distance $\Delta=0.01$. For the spectral estimator we apply a flat top kernel and the universal bandwidth choice $h=\sqrt \Delta$ which turned out to perform well in a variety of settings. Figure~\ref{fig:sim} shows the true distribution-type function $N$, the direct estimator $\tilde N_n$ from \eqref{eq:naiveEst} and the spectral estimator $\hat N_n$ from \eqref{eq:kernelEst} for 50 simulations. In each setting the confidence band for level $\alpha=0.9$, as constructed in the previous section, is plotted for the first simulation result. We clearly see the higher activity of small jumps of the NIG process from the linear growth of $N$ at zero. On the other hand, the choice of our process parameters yields more pronounced tails of the jump measure for the gamma process.
By construction, the direct estimator is not smooth. For the Gamma process it possesses a significant bias. The intensity of the small jumps is systematically underestimated which results in an overestimation of the larger jumps and thus too large values of $\tilde N_n(t)$ for $t$ large. For the choice $\Delta=0.001$ this bias of the direct estimator is already negligible. In the simulations of the NIG process, $\tilde N_n$ achieves good results that coincide with the asymptotic theory. In the simulations for the Gamma process the empirical coverage of the $\alpha=0.9$ confidence bands in 500 Monte Carlo iterations is 0.86 for the Gamma process. The direct estimator has an empirical coverage of 0.59, reflecting the bias problem mentioned above. For the NIG process both estimators yield bands covering the true $N$ uniformly in $92\%$ of cases.
\section{Unifying empirical process} \label{secProc}
The key probabilistic challenge in the proofs of Theorems \ref{thm:UCLTnaive} - \ref{thmUniform} is a uniform central limit theorem for certain smoothed empirical processes arising from the sampled increments (\ref{data}). We show in this section how these processes arise naturally for both estimation approaches considered here.
\smallskip
We will consider slightly more general objects than the distribution function $N(t) = \int_{-\infty}^t (1 \wedge x^2) \nu(\d x)$ -- the truncation at one in $(1 \wedge x^2)$ is somewhat arbitrary and, in
particular, not smooth. Other truncations such as $x^2/(1+x^2)$, or
variations thereof can be of interest. To accommodate such examples we thus consider recovery of the functionals
\begin{equation}
N_\rho(t) = \int_{-\infty}^t \rho(x) x^2\nu(\d x), ~~ t \in \R,
\end{equation}
where the `clipping function' $\rho$ satisfies the following condition:
\begin{assumption} \label{clip}
The function $\rho$ satisfies $0< \rho(x) \le C(1 \wedge x^{-2})$ for all $x
\in \R$ and some constant $0<C<\infty$. Moreover, $\rho, x\rho$ are Lipschitz
continuous functions of bounded variation (i.e., their weak derivative is equal
to a finite signed measure).
\end{assumption}
This covers the above examples (with either $\rho(x) = 1 \wedge x^{-2}$ or
$\rho(x) = 1/(1+x^2)$). In the definition of the basic estimator
\eqref{eq:naiveEst} and the kernel estimator~\eqref{eq:kernelEst}, we only need to replace
$(1\wedge x^2)\1_{(-\infty,t]}(x)$ by $x^2g_t(x)$ where now
\begin{equation} \label{grho}
g_t(x):=\rho(x)\1_{(-\infty,t]}(x),
\end{equation}
replacing also $g_t$ in Assumption~\ref{assProc}. The covariance of the limit process in Theorems~\ref{thm:UCLTnaive}
and~\ref{thmUniform} then changes to
\[\E[\mathbb{G}(s)\mathbb{G}(t)]=\int_{\R}x^{4}g_s(x)g_t(x)\nu(\d x)\]
and the according representation of $\mathbb G$ in terms of a reparametrised
Brownian motion is
\begin{align}\label{eqBtransformed}
\mathbb
G(t)=\mathbb B\left(\int_{\R}x^4g_t^2(x)\nu(\d
x)\right)=\mathbb B\left(\int_{\R}x^4\rho^2(x)\1_{(-\infty,t]}(x)\nu(\d
x)\right).
\end{align}
Let us turn to the main purpose of this section: We start with the direct estimator $\tilde N_n$, which is easier to analyse.
The estimation error of $\tilde{N}_{n}$ can be decomposed as follows
\begin{align}
\tilde{N}_{n}(t)-N_{\rho}(t)&=
\int_{\R}x^{2}g_t(x)\big(\Delta^{-1}\PP_{
\Delta,n}
(\mathrm{d}x)-\nu(\mathrm{d}x)\big)\nonumber \\
&= \int_{\R}x^2g_t(x)
\big(\Delta^{-1}\PP_{\Delta}(\mathrm{d}x)-
\nu(\mathrm{d}x)\big
+\int_{\R}g_t(x)
\frac{x^{2}}{\Delta}(\PP_{\Delta,n}-\PP_{\Delta})(\d
x)\nonumber \\
&=: B(t)+S(t),\label{eq:decomp}
\end{align}
for any $t\in\R$. The first term $B$ is a deterministic approximation error
and the rough idea for controlling it is to view $\PP_\Delta$ as an approximate
identity and to use
similar arguments as for the approximation error of a kernel estimator.
The second term $S$ is the main stochastic error term driven by the empirical
process
\begin{equation} \label{naivproc}
\sqrt{n\Delta}\left(\frac{x^{2}}{\Delta}\PP_{\Delta,n}-\frac{x^{2}}{\Delta}\PP_
{
\Delta }
\right)=\sqrt{\frac{n}{\Delta}}x^{2}\big(\PP_{\Delta,n}-\PP_{\Delta}\big),
\end{equation}
where the scaling follows from the intuitive observation that the $X_k$'s are
drawn i.i.d.~from law $\PP_{\Delta}$ and hence satisfy, using that $\PP_\Delta$
is an infinitely divisible distribution, $$\Var\left(\sum_{k=1}^n X_k\right) =
\Var (L_{n \Delta}) = n \Delta \Var (L_1).$$
Turning our attention to the second estimator we decompose $\hat N_n-N_\rho$
into three error terms, using \eqref{eqIdent}:
\begin{align}
\hat N_n(t)-N_\rho(t)
&=
\int_{\R}(g_t(x)\F^{-1}\left[(-\hat\psi''_n-\hat\sigma^2)\mathcal
F K_h \right](x)\d x-x^2 g_t(x)\nu(\d x))\notag\\
&=\int_{\R} g_t(x)\big(K_h\ast\big(y^2\nu(\d y)\big)-x^2\nu\big)(\d
x)\label{eqErrorDecomp}\\
&\quad+\int_{\R} g_t(x)\F^{-1}\Big[\mathcal F
K_h(u)\big(\psi''(u)-\hat\psi_n''(u)\big)\Big](x)\d x\notag\\
&\quad+(\sigma^2-\hat\sigma^2)\int_{\R}g_t(x)K_h(x)\d x.\notag
\end{align}
The first term is a deterministic approximation error, which can be bounded by
Assumption~\ref{assProc}(d) on the smoothness. The last term will be negligible since we assume
that $\hat\sigma^2$ converges to $\sigma^2$ with a faster rate than $1/\sqrt{n\Delta}$.
The key stochastic term is the second one. Compared to the basic estimator
$\tilde N_n$ we face the additional difficulty that $\hat \psi_n''$ depends
nonlinearly on $\PP_{\Delta,n}$. The following result shows that even after
linearisation the resulting term is still different from the basic process
$\tilde N_n -\E \tilde N_n$ in that it performs a division by $\phi_\Delta$ in
the spectral domain.
\begin{proposition}\label{propRem}
Grant Assumptions~\ref{assProc}(a) and assume $\sup_{u\in[-1/h_n,1/h_n]}|\phi_{\Delta_n}(u)|^{-1}\lesssim1$ for some $h_n \to 0, \Delta_n \to 0$. Let the function
$$m(u):=\frac{\mathcal F K(h_nu)}{\phi_{\Delta_n}(u)}$$ satisfy uniformly for
$h_n,\Delta_n\to0$, $\|m\|_\infty\lesssim 1$ (valid for $K$ as in
(\ref{propKernel}) and $h_n \sim \sqrt {\Delta_n}$). If $n\Delta_n\to\infty$ and
$h_n\to 0$ with $h_n\gtrsim \Delta_n^{1/2}$, then we
have
\[\int_{\R} g_t(x)\F^{-1}\Big[\mathcal F
K_{h_n}(u)(\psi-\hat\psi_n)''(u)\Big](x)\d
x=M_{\Delta,n}+o_P(1/\sqrt{n\Delta_n}),
\]
where
\begin{align}
M_{\Delta,n}&:=-\Delta_n^{-1}\int_{\R} g_t(x){\cal
F}^{-1}[\phi_{\Delta_n}^{-1}(\phi_{\Delta_n,n}''-\phi_{\Delta_n}''){\cal
F}K_{h_n}](x)\d x\notag\\
&\phantom{:}= \int_{\R} g_t(x)
\left(\frac{x^2}{\Delta}(\PP_{\Delta,n}-\PP_{\Delta})\right)*\F^{-1}[m](\d
x).\label{Mproc}
\end{align}
\end{proposition}
We refer to $M_{\Delta,n}$ as the \emph{main
stochastic term}. To accommodate both (\ref{naivproc}) and (\ref{Mproc}) we now study empirical processes
\begin{equation} \label{genproc0}
\sqrt{\frac{n}{\Delta}}( x^2 (\PP_{\Delta, n}-\PP_{\Delta})) \ast \mathcal
F^{-1} m.
\end{equation}
for general $(n,\Delta)$-dependent Fourier multipliers $m: \R \to \C$ satisfying the following condition.
\begin{assumption} \label{multass}
For every $n, \Delta$ the twice differentiable functions $m = m_{n, \Delta}: \R
\to \C$ are
either such that
(a) $\F^{-1}[m_{n, \Delta}]$, $\F^{-1}[m_{n, \Delta}']$ are
finite signed measures with uniformly bounded total variations, \newline
or such that
(b) $\F^{-1}[m_{n, \Delta}]$ is real-valued
and $m_{n, \Delta}$ is supported in $[-C\Delta^{-1/2}, C \Delta^{-1/2}]$ for
some fixed constant $C>0$.
\medskip
Moreover, letting $\Delta=\Delta_n \to 0$ as $n
\to
\infty$ we assume that $m_{n,\Delta} \to 1$ pointwise on $\R$, that
\[\|(1+|u|)^{k}m_{n, \Delta}^{(k)}\|_\infty\le c,
\quad k\in\{0,1,2\},\]
for some $0<c<\infty$ independent of $n, \Delta$ and that
\[\|m'_{n, \Delta}\|_{L^2}\to 0,\quad \Delta^{-1/2}\|m''_{n,
\Delta}\|_{L^2}\to0.\]
\end{assumption}
The above assumption is an adaptation of the usual Mikhlin-type Fourier
multiplier conditions to the situation
relevant here \citep[see][Cor. 4.11]{girardiWeis2003}.
It ensures that $m$, $m'$ act as norm-continuous Fourier multipliers on
suitable function spaces, which will be a key tool in our
proofs. Obviously Assumption \ref{multass} covers the case $m=1$ relevant in
(\ref{naivproc}) above. Moreover, we show in Proposition~\ref{multver} below
that it
also covers
$m=\mathcal F K_{(\Delta)}/\phi_\Delta$ under our conditions on $\phi_\Delta$
and $K_{(\Delta)}$,
where $K_{(\Delta)}$ denotes a kernel as in (\ref{propKernel}) with bandwidth depending on $\Delta$.
It includes other situations not studied further here, too, such as
smoothed empirical processes
based on $\PP_{\Delta, n}$ convolved with an approximate identity $K_h= h^{-1}
K(\cdot/h), h:=h_n \to 0, \int K=1,$ upon setting $m=\mathcal F K_h$.
With the definition of general $m=m_{n,\Delta}$ at hand we can now
unify the second term $S(t)$ in~\eqref{eq:decomp} and the main stochastic error
\eqref{Mproc}, and study the
smoothed
empirical process
\begin{align}
\mathbb{G}_n(t)&:= \sqrt{n\Delta}\int_{\R} g_t(x)
\left(\frac{x^2}{\Delta}(\PP_{\Delta,n}-\PP_{\Delta})\right)*\F^{-1}[m](\d
x),\label{genproc}
\\
& = \sqrt{n \Delta}\int_{\R}\F^{-1}[m(-u)\F [g_t](u)](x)
\frac{x^2}{\Delta}(\PP_{\Delta,n}-\PP_\Delta)(\d x), ~t \in \R,\notag
\end{align}
the identity following from Fubini's theorem and standard properties of
Fourier transforms.
When $t$ is a fixed point in $\R$ and $m=1$ one shows without difficulty that, as $n \to \infty$,
$$\Var(\mathbb G_n(t)) \to \int_{\R} x^4 g_t^2(x) \nu(\d x)$$ whenever
$\nu(\{t\})=0$. More generally one can show
convergence
of the finite-dimensional distributions of the process $(\mathbb G_n(t): t \in
\R)$ to the process $(\mathbb G(t): t \in
\R)$ from Theorem~\ref{thm:UCLTnaive}.
\begin{proposition} \label{fidi0}
Let $\Delta= \Delta_n \to 0$ in such a way that $n \Delta_n \to \infty$.
Suppose the L\'evy process satisfies Assumption~\ref{assProc}, that $\rho$ satisfies Assumption \ref{clip}, and that $m$ satisfies Assumption \ref{multass}.
Then as $n \to \infty$ we have, for any $t_1, \dots, t_k \in \R$, that
\[
\left[\mathbb G_n(t_1), \dots, \mathbb G_n(t_k) \right]
\rightarrow^{\mathcal L} \left[\mathbb G(t_1), \dots \mathbb G(t_k)\right].
\]
\end{proposition}
We remark that in this proposition we can omit $(1\wedge x^4)\nu\in
\ell^\infty(\R)$ from Assumption~\ref{assProc} as it is only needed
later in the proof of the tightness of the process $\mathbb{G}_n$.
By sample-continuity of Brownian motion, and
since the integral in \eqref{eqBtransformed} takes values in a fixed compact
set, we deduce that there exists a version of $(\mathbb G(t):t\in\R)$ with
uniformly
continuous sample paths for the intrinsic covariance metric
\[d^2(s,t)=\int_{\R} x^4(g_t(x)-g_s(x))^2\nu(\d
x)=\int_{\R} x^4\rho^2(x)\1_{(s\wedge t,s\vee t]}\nu(\d
x),\]
and that, moreover, $\R$ is totally bounded with respect to $d$. As a consequence we
obtain:
\begin{lemma} \label{levybrown}
Grant Assumption~\ref{clip}. For $g_t$ as in (\ref{grho}) and any L\'evy measure $\nu$, the law of the
centred Gaussian process
$\{\mathbb{G}(t):t\in\R\}$
with covariance
\[
\E[\mathbb{G}(t)\mathbb{G}(t')]=\int_{\R}x^4 g_t(x) g_{t'}(x) \nu(\d
x),\quad t,t'\in\R
\]
defines a tight Gaussian Borel random variable in $\ell^\infty(\R)$. In particular, there exists a
version of the process $(\mathbb G(t): t \in \R)$ such that $$\sup_{t \in \R} |\mathbb G(t)|<\infty ~a.s.$$
\end{lemma}
The most difficult part in the proofs of Theorem~\ref{thm:UCLTnaive}
and~\ref{thmUniform} is to show
that $\mathbb G_n$ converges in law to $\mathbb G$ in the space
$\ell^\infty(\R)$ of bounded functions on the real line. Given that convergence
of the finite-dimensional distributions and tightness of the limit process have
already been established, this can be reduced to showing asymptotic
equicontinuity of the process $(\mathbb G_n(t): t \in \R)$, or equivalently,
uniform tightness of the random variables $\mathbb G_n$ in the Banach space
$\ell^\infty(\R)$ (see Section 1.5 in \cite{vanderVaartWellner1996} and
(\ref{asymptotic_equicontinuity}) below for precise definitions).
\begin{theorem} \label{main1}
Let $\Delta= \Delta_n \to 0$ in such a way that $n \Delta_n \to \infty$ and
$\log^4(1/\Delta_n)=o(n\Delta_n)$.
Suppose the L\'evy process satisfies Assumption~\ref{assProc}, that $\rho$ satisfies Assumption \ref{clip}, and that $m$ satisfies Assumption \ref{multass}. Then the
process $\mathbb G_n$ from~(\ref{genproc}) is asymptotically equicontinuous in
$\ell^\infty(\R)$. In particular, $\mathbb G_n$ is uniformly tight in
$\ell^\infty(\R)$,
\begin{equation*}
\mathbb G_n \to^{\mathcal L} \mathbb G \quad\text{in} \quad\ell^\infty(\R),
\end{equation*}
and
\begin{equation}
\sup_{t \in \R} \left|\int_{\R}g_t(x)
\left(\frac{x^2}{\Delta}(\PP_{\Delta,n}-\PP_{\Delta})\right)*\F^{-1}[m](\d x)
\right| = O_P\left(\frac{1}{\sqrt {n \Delta}} \right).
\end{equation}
\end{theorem}
\smallskip
The proof is based on ideas from the theory of
smoothed empirical processes \citep[in particular from][]{GineNickl2008}.
The main mathematical
challenges consist in dealing with
envelopes of the empirical process that can be as large as $1/\sqrt
\Delta\to\infty$ in the high-frequency setting,
and in accommodating the presence of an $n$-dependent Fourier multiplier $m$
that needs to be general enough to allow for $m=\mathcal F K_{h}/\phi_\Delta$. The latter
requires the treatment of empirical processes that cannot be controlled with the
standard bracketing or uniform metric entropy techniques.
Our proofs rely on direct arguments for symmetrised empirical processes
inspired by \cite{GineZinn1984}
and on sharp bounds on certain covering numbers based on a suitable
Fourier integral operator inequality for $\F^{-1}[m]$ in
$L^2(\PP_\Delta)$-norms.
\section{Discussion and examples} \label{discussion}
\subsection{Regularity of $x^2 \nu$ and the Blumenthal--Getoor
index}\label{sec:discussion}
The regularity index $s>0$ in Assumption \ref{assProc} measures the smoothness of the function $g_t(-\cdot)\ast(x^2\nu)$. When $x^2\nu$ is sufficiently regular away from the origin, this will equivalently measure the
smoothness of the function $t \mapsto \int_{-\infty}^t x^2\nu(\d x)$, and hence is effectively driven by the singularity that $\nu$ possesses at zero. The latter can be quantitatively measured by the
\cite{blumenthalGetoor1961}-index
\begin{align}
\beta: & =\inf\left\{\alpha>0:\int_{|x|<1}|x|^{\alpha}\nu(\d
x)<\infty\right\}=\inf\left\{\alpha>0:\lim_{r\downarrow0}r^{\alpha}\int_{
\R\setminus[-r,r]}\d\nu=0\right\}.\label{eq:blumenthalGetoor}
\end{align}
The Blumenthal--Getoor index $\beta$ takes values in $[0,2]$ and we have
$\int_{\R}|x|^\alpha\nu(\d x)=c<\infty$ for all $\alpha\in(\beta,2]$ ($\alpha=2$ if $\beta=2$).
In fact, for such $\alpha$ and for all intervals $[a,b]$ containing the origin
\begin{align*}
\int_a^b |x|^2\nu(\d x) \le \int_a^b|x|^\alpha\nu(\d x) (b-a)^{2-\alpha}\le c
(b-a)^{2-\alpha}.
\end{align*}
Provided $\nu$ is smooth away from zero this shows that the H\"older smoothness
of $\int_{-\infty}^t x^2\nu(\d x)$ is at least $2-\beta^+$, where
$\beta^+>\beta$ and $\beta^+\ge1$.
For a singularity of the from $\nu(x)=|x|^{-\beta-1}$, $\beta\in(1,2)$, which
corresponds to Blumenthal--Getoor index $\beta$, we have $\int_{|x|<t}x^2\nu(\d
x)=2(2-\beta)^{-1}t^{2-\beta}$ showing that the H\"older smoothness is at most
$(2-\beta)$.
This argument can be extended
to the case where the symmetrised L\'evy density $\tilde\nu$ is
regularly
varying:
If $\tilde\nu$ is regularly varying with exponent
$-(\beta+1)$ at zero then
$\int_{|x|<t}|x|^2\nu(\d x)$ is regularly varying of exponent $(2-\beta)$
at zero by a Tauberian theorem \citep[see e.g.][Thm. VIII.9.1]{feller1971}.
For Blumenthal--Getoor index $\beta\in(1,2]$ this means that the H\"older
regularity of $\int_{-\infty}^t x^2\nu(\d x)$ is at most $(2-\beta)$.
\subsection{The drift parameter $\gamma$}
None of the above estimators $\tilde N, \hat N, \tilde {\mathcal N}, \hat {\mathcal N}$ require knowledge, or estimation, of the drift parameter $\gamma$, which, at any rate, can be naturally estimated by $L_{n\Delta}/(n\Delta)$. It is interesting to note that the `nonlinear' estimator $\hat N_n$ is even invariant under a change of the drift parameter $\gamma$, as the following lemma shows.
\begin{lemma}\label{lemDrift}
Let $Y_k:=X_k-\Delta\gamma,k=1,\dots,n,$ which are increments of a L\'evy
process with characteristic triplet $(\sigma,0,\nu)$. Denoting the estimators
\eqref{eq:kernelEst} based on $(X_k)$ and $(Y_k)$ as $\hat N_{X,n}$ and $\hat
N_{Y,n}$, respectively, we obtain
\[
\forall t\in\R: \hat N_{X,n}(t)=\hat N_{Y,n}(t).
\]
\end{lemma}
\begin{proof}
The drift causes a factor $e^{-i\Delta\gamma u}$ in the empirical characteristic function $\phi_{\Delta,n,Y}$ such that
\[ \hat\psi_{n,Y}''(u)=\Delta^{-1} (\log(\phi_{\Delta,n,X}(u))-i\Delta\gamma u)''=\hat\psi_{n,X}''(u).
\]
$\hat N_n$ only depends via $\hat\psi_{n}''$ on the observations.
\end{proof}
Consequently, without loss of generality
a specific value of $\gamma$ can be assumed in the proofs
for
the estimator~$\hat N_n$ based on the L\'evy--Khintchine representation. In
particular, the conditions on $\PP_\Delta$ need to be verified only for one
$\gamma$.
\subsection{A pilot estimate of the diffusion coefficient $\sigma$}
\begin{proposition} \label{sigmaest}
Suppose the L\'evy measure satisfies $\int \abs{x}^\alpha\nu(dx)<\infty$ for some $\alpha\in[0,2]$ and the characteristic function is bounded from below via
\[ \abs{\phi_\Delta(u)}\ge \exp(-\Delta\sigma_{max}^2u^2/2)\text{ for all } u\ge 0.\]
Let $\hat \sigma^2$ be as in (\ref{eq:sigmahat}). Then we have, for $c_0$ small enough, as $n \to\infty$, and uniformly in $\Delta\le 1$,
\[\abs{\hat\sigma^2-\sigma^2}=O_P\Big((\log
n)^{(\alpha-2)/2}\Delta^{1-\alpha/2}+(\log n)^{-1}n^{c_0-1/2}\Big).\]
\end{proposition}
The proof follows along the lines of \cite{JacodReiss2013} and is omitted.
The previous discussion and the examples in Section~\ref{secExamples}
below show that the natural connection between smoothness $s$ and
Blumenthal--Getoor index $\beta$ is given by $s=2-\beta$. For such $s$ and with
the choice $c_0=1/6$ the conditions of Theorem~\ref{thmUniform} ensure that
(\ref{eq:sigmarate}) is satisfied provided the infimum in the definition of the
Blumenthal--Getoor index is attained. Otherwise it suffices to replace the
condition $\Delta_n=o(n^{-1/(s+1)})$ by the slightly stronger condition
$\Delta_n=o(n^{-1/(s^-+1)})$ for some $s^-<s$ in order to guarantee
(\ref{eq:sigmarate}).
Other estimators, based for instance on the truncated quadratic variations of
the process, can be considered, and different sets of conditions are possible.
As this is beyond the scope of the present paper, we refer to
\cite{JacodReiss2013} for discussion and references.
\subsection{Bounding $\|x^3\PP_\Delta\|_\infty$}\label{sec:Boundx3P}
A key condition in all results above is a uniform bound on
$\|x^3\PP_{\Delta}\|_\infty$ of order $\Delta$. The following proposition shows
that this condition follows already from $\|x\PP_\Delta\|_\infty\lesssim1$ and
$x^3\nu\in \ell^\infty(\R)$.
We recall that we always assume $\int_{\R}x^2\nu(\d x)<\infty$.
\begin{proposition}\label{Propx3PDelta}
For any L\'evy process $(L_t:t\ge 0)$ with $\norm{x\PP_\Delta}_\infty\lesssim 1$ and $x^3\nu\in \ell^\infty(\R)$ we have $\|x^3\PP_\Delta\|_\infty\lesssim\Delta$ (with constants uniform in $\Delta$).
\end{proposition}
\begin{proof}
From $\phi_\Delta''=(\Delta
\psi''+(\Delta\psi')^2)\phi_\Delta=\Delta\psi''\phi_\Delta+(\Delta\psi'\phi_{
\Delta/2})^2$ by the infinite divisibility, we conclude
\begin{equation}\label{eqX2P}
x^2\PP_\Delta=\Delta \nu_\sigma \ast \PP_\Delta+4(x\PP_{\Delta/2})\ast(x\PP_{\Delta/2}),
\end{equation}
where $\nu_\sigma=\sigma^2\delta_0+x^2\nu$. Using $x(P\ast Q)=(xP)\ast Q+P\ast(xQ)$, we infer further
\[ x^3\PP_\Delta=\Delta\Big( (x\nu_\sigma)\ast \PP_\Delta+\nu_\sigma\ast(x\PP_\Delta)\Big)+8(x\PP_{\Delta/2})\ast (x^2\PP_{\Delta/2}).
\]
By assumption and properties of L\'evy processes, we have
$\norm{x\nu_\sigma}_\infty<\infty$, $\PP_\Delta(\R)=1$,
$\nu_\sigma(\R)<\infty$ and $\norm{x^2\PP_\Delta}_{L^1}\lesssim\Delta$.
This yields
\[ \norm{x^3\PP_\Delta}_\infty \lesssim \Delta(1+\norm{x\PP_{\Delta}}_\infty + \norm{x\PP_{\Delta/2}}_\infty)\lesssim\Delta.\qedhere
\]
\end{proof}
The condition $\|x\PP_{\Delta_n}\|_\infty\lesssim 1$ is satisfied for all basic
examples of L\'evy processes like Brownian motion, compound Poisson, Gamma and
symmetric (tempered) $\alpha$-stable processes. For the latter
processes it is interesting to compare the resulting bounds to the small time
estimates by \cite{Picard1997}. The conjecture
that the bound $\norm{x\PP_\Delta}_\infty\lesssim 1$ is universal for arbitrary
jump behaviour near zero, however, is wrong as the case of a completely
asymmetric (tempered) 1-stable process shows where
$\PP_\Delta(-\Delta\log(1/\Delta))\thicksim \Delta^{-1}$ holds, see the
exceptional case in Example~4.5 of \cite{Picard1997}.
\medskip
If $\int_{\R}|x|\nu(\d x)<\infty$ we can define the drift parameter $\gamma_0 :=
\gamma - \int x \nu(\d x)$.
\begin{assumption}\label{AssPDelta}
Let $(\sigma^2,\gamma,\nu)$ be a L\'evy triplet and $\nu^+=\nu{\1}_{\R^+}$, $\nu^-=\nu{\1}_{\R^-}$. Consider the following conditions for the two triplets $(\sigma^2,\gamma,\nu^\pm)$:
\begin{enumerate}
\item (diffusive case) $\sigma>0$
\item (small intensity case) $\sigma=0$, $\gamma_0=0$,
$\norm{x\nu^\pm}_\infty<\infty$
\item (finite variation case) $\sigma=0$, $\gamma_0=0$, $x\nu^\pm$ admits a
Lebesgue density in $\ell^\infty(\R\setminus[-\eps,\eps])$ for all $\eps>0$,
$\eps^{-1}\int_{-\eps}^\eps \abs{x}\nu^\pm(\d
x)\lesssim \eps\nu(\pm\eps)$ for $\eps\downarrow 0$ and
\[ \liminf_{\eps\downarrow 0}\inf_{t\in(0,1]}
\frac{(t\eps)^{-1}\int_{\abs{x}\le t\eps} x^2\nu^\pm(\d x)} {\eps^2\nu(\pm
\eps)t\log(t^{-1})}>0
\]
\item (infinite variation case) $\sigma=0$, $\nu^{\pm}$~admits a Lebesgue
density,
\begin{align*}
\frac1{\eps}\int_{-\eps}^\eps
x^2\nu^\pm(\d x)\gtrsim \int_{\abs{x}>\eps}\abs{x}\nu^\pm(\d x)+1\quad\text{ for
}
\eps\in(0,1)
\end{align*}
\end{enumerate}
\end{assumption}
\begin{proposition}\label{Propx1PDelta}
If each of the triplets $(\sigma^2,\gamma,\nu^\pm)$ of the L\'evy process
satisfies one of the Assumptions \ref{AssPDelta}(i)-(iv), then
$\norm{x\PP_\Delta}_\infty\lesssim 1$ holds uniformly in $\Delta$.
\end{proposition}
\subsection{Examples}\label{secExamples}
Let us discuss the applicability of Proposition~\ref{Propx1PDelta} together with the smoothness conditions on the jump measure from Theorem \ref{thmUniform} in a few examples.
\begin{enumerate}
\item {\it Diffusion plus compound Poisson process.}\\
Let $\nu$ be a finite measure on $\R$ with a Lebesgue density. Suppose
$\int_{\R}|x|^{4+\eps}\nu(\d x)<\infty$ for some $\eps>0$ and
$\|x^3\nu\|_\infty<\infty$. Proposition~\ref{Propx1PDelta} yields
$\|x\PP_\Delta\|_\infty\lesssim1$ if either $\gamma_0=0$ and
$\nu(x)\lesssim|x|^{-1}$ as $x\to0$, or if $\sigma>0$.
For $x^2\nu\in \ell^\infty(\R)$ the global H\"older regularity in Assumption
\ref{assProc}(d) is $s=1$, and for smooth compounding measure $x^2\nu\in
C^r(\R)$ it is satisfied with $s=r+1$.
\item {\it Self-decomposable L\'evy process.}\\
The jump measures of self-decomposable L\'evy processes are characterised by
$\nu(\d x)=\frac{k(x)}{|x|}\d x$ for a function $k:\R\to\R_+$ which is
monotonically increasing on the negative half line and decreasing on the
positive one. An explicit example is given by the Gamma process where
$k(x)=ce^{-\lambda x}\1_{\R_+}(x)$ for $c,\lambda>0$. Note that nontrivial
self-decomposable processes have an infinite jump activity. If $k$ is a bounded
function, then Assumption~\ref{AssPDelta}(ii) is fulfilled. The smoothness is
determined by the H\"older regularity of $|x|k(x)$, for instance, Gamma
processes induce regularity $s=2$ at $t=0$ and $C^\infty$ away from the origin.
\item {\it Tempered stable L\'evy process.}\\
Let $L$ be a tempered stable process, that is a pure jump process with L\'evy measure given by the Lebesgue density
\[
\nu(x)=|x|^{-1-\alpha}\left(c_-e^{-\lambda_-|x|}\1_{(-\infty,0)}(x)+c_+e^{-\lambda_+|x|}\1_{(0,\infty)}(x)\right)
\]
with parameters $c_\pm\ge 0,\lambda_\pm>0$ and stability index $\alpha\in(0,2)$. By the exponential tails of~$\nu$ the moment assumptions are satisfied. For the finite variation case $\alpha\in(0,1)$ Assumption~\ref{AssPDelta}(iii) can be verified since $\eps^{-1}\int_{-\eps}^\eps|x|\nu^\pm(x)\d x\sim\eps^{-\alpha}\sim \eps\nu(\pm\eps)$ and the second condition simplifies to $t^{-\alpha}/\log(t^{-1})>0$. In the infinite variation case $\alpha\in(1,2)$ Assumption \ref{AssPDelta}(iv) is satisfied owing to $$\eps^{-1}\int_{-\eps}^\eps x^2\nu^\pm(x)\d x\sim\eps^{1-\alpha}\sim\int_{|x|>\eps}|x|\nu^\pm(x)\d x, ~~\eps\in(0,1).$$
Outside of a neighbourhood of zero the L\'evy measure is arbitrarily smooth. Due to the cusp of $x^2\nu(x)$ at the origin the global H\"older regularity is in general given by $s=2-\alpha$. In the case $\alpha=1$ and $c_+=c_-$, $x^2\nu$ is already Lipschitz continuous at zero and so $s=2$.
\item {\it Jump densities regularly varying at zero.}\\
The first condition in Assumption~\ref{AssPDelta}(iii) holds for regularly varying $\nu$ with $\alpha<1$, that is $\nu^\pm(x)=|x|^{-1-\alpha}l(x)$ with slowly varying $l$ at zero, by a classical Tauberian theorem \citep[see e.g.][Thm. VIII.9.1]{feller1971}. The second condition then reduces to
\[ l(t\eps)\ge C_\alpha t^\alpha\log(t^{-1})l(\eps),\quad C_\alpha>0,\]
uniformly over $t\in(0,1]$ for small $\eps>0$, which is always satisfied for $\alpha>0$. Similarly, Assumption \ref{AssPDelta}(iv) is satisfied if $\nu^\pm(x)=|x|^{-1-\alpha}l(x)$ holds with $\alpha\in(1,2)$ and a slowly varying function $l$ at zero.
\end{enumerate}
\section{Proofs}
We collect the proofs for Sections~\ref{secEstimators}, \ref{secProc}, \ref{discussion}. Theorem~\ref{main1} is proved in the next section.
\subsection{Proof of Lemma \ref{weakcon}}
The result is a standard -- for convenience of the reader we include a short proof. Using
the L\'evy--Khintchine formula (\ref{eqLevyKhintchine}) we see
\begin{equation} \label{var}
c_\Delta:=\Delta^{-1}\E[X_1^2]= -\Delta^{-1}\phi_{\Delta}''(0) = -\psi''(0)- \Delta (\psi'(0))^2 \to\sigma^2+\int x^2 \nu(\d x)=:c
\end{equation}
as $\Delta\to0$. The characteristic function of the probability measure
$c_\Delta^{-1}x^2\Delta^{-1}\PP_\Delta(\d x)$ converges pointwise to the
characteristic function of $c^{-1}(\sigma^2 \delta_0+ x^2\nu(\d x))$ as
$\Delta\to0$ since
\begin{align*}
\frac{1}{c_\Delta \Delta}\int e^{iux}x^2 \PP_\Delta(\d x)=-\frac{1}{c_\Delta
\Delta}\phi_\Delta''(u)&=-\frac{1}{c_\Delta \Delta}(\Delta
\psi''(u)+\Delta^2(\psi'(u))^2)e^{\Delta\psi(u)}\\
&\to \frac{1}{c}(\sigma^2+\int
e^{iux}x^2 \nu(\d x)).
\end{align*}
Therefore, we obtain \eqref{eqWeakConvNu} from L\'evy's continuity theorem.
\subsection{Proof of Theorem \ref{thm:UCLTnaive}}
Using decomposition (\ref{eq:decomp}), Theorem \ref{thm:UCLTnaive} follows from Theorem \ref{main1} with $m=1$ (which trivially satisfies Assumption \ref{multass}(a)), if we can show that the `bias' term $B(t)$ is asymptotically negligible uniformly in $t \in \R$. This is achieved in the following proposition.
\begin{prop}
\label{prop:bias} Grant the assumptions of Theorem~\ref{thm:UCLTnaive}. Then it
holds
\[
\sup_{t\in\R}|B(t)|=\sup_{t\in\R}\Big|\int
g_{t}(x)\big(\Delta^{-1}x^{2}\PP_{\Delta}(\mathrm{d}x)-x^{2}
\nu(\mathrm{d}x)\big)\Big|=\mathcal{O}(\Delta^{s/2}).
\]
\end{prop}
\begin{proof}
We decompose the bias into
\begin{align}
B(t)&= \int
g_{t}(x)\big(\Delta^{-1}x^{2}\PP_{\Delta}(\mathrm{d}x)-((x^2\nu)\ast\PP_{
\Delta})(\d x)\big)\nonumber \\
& \qquad+\int g_{t}(x)\big(((x^{2}\nu)\ast\PP_{\Delta})(\d x)-x^{2}\nu(\d
x)\big)\nonumber \\
&=: B_{1}(t)+B_{2}(t).\label{eq:BiasDecomp}
\end{align}
We start with the first term $B_1(t)$: Using $-(\F f)''=\F[x^{2}f]$ for any
function $f$ satisfying $(1\vee x^{2})f\in
L^{1}(\R)$,
we have
\[
-\F\big[x^{2}\PP_{\Delta}(\mathrm{d}x)]=\phi_{\Delta}
''=(\Delta\psi''+(\Delta\psi')^{2})\phi_{\Delta}.
\]
Plancherel's identity and $\psi''=-\F[x^2\nu]$ then gives
\begin{align*}
B_{1}(t)&= \frac{1}{2\pi}\int\F
g_{t}(-u)\F\big[\Delta^{-1}x^{2}\PP_{\Delta}(\mathrm{d}x)-((x^2\nu)\ast\PP_{
\Delta})(\d x)\big](u)\,\d u\\
&= \frac{1}{2\pi}\int\F
g_{t}(-u)\big(-\Delta^{-1}\phi_{\Delta}''(u)-\F[x^2\nu](u)\phi_{\Delta}
(u)\big)\,\d u.\\
&= -\frac{\Delta}{2\pi}\int\F g_{t}(-u)\psi'(u)^{2}\phi_{\Delta}(u)\,\d u.
\end{align*}
The proofs below will imply that the last integral exists, which in particular justifies the preceding manipulations. We shall repeatedly use that $\sup_{t}\|g_{t}\|_{L^{1}}\le \|\rho\|_{L^{1}}$ and
$\sup_{t}\|g_{t}\|_{BV}\le\|\rho\|_{BV}$ imply
\begin{equation} \label{boringbound}
|\F g_{t}(u)|\lesssim(1+|u|)^{-1},u\in\R,
\end{equation}
uniformly in $t \in \R$. In case a) we can use (\ref{boringbound}),
$\|\phi_\Delta\|_\infty =1$, $|\F[x\nu](u)| \lesssim(1+|u|)^{-1}$, the
hypothesis $\gamma_0=0$ and the resulting identity
$$\psi'(u) = i \F[x\nu](u) - i \left(\int x \nu(\d x) - \gamma \right) = i \F[x \nu](u)$$
to bound
$$|B_1(t)| \le \frac{\Delta}{2\pi}\int |\F g_{t}(-u)||\F[x \nu](u)|^{2}|\phi_{\Delta}(u)|\,\d u \lesssim \Delta \int \frac{1}{(1+|u|)^3} \d u =O(\Delta).$$
For case b) we will show that
\begin{equation}
\sup_{t\in\R}|B_{1}(t)|\lesssim\Delta^{p}\quad\text{for any}\quad
p\in\big(0,\frac{2-\beta}{\beta}\vee1\big).\label{eq:b1}
\end{equation}
By assumption $\tilde\nu(x)=\nu^+(x)+\nu^-(-x)$ is regularly varying at zero
with
exponent $-(\beta+1)$ and so the function $H(r):=\int_{|x|<r}x^2\nu(x)\d
x=\int_0^r x^2 \tilde\nu(x)\d x$ is regularly varying with exponent $(2-\beta)$
by a Tauberian theorem \citep[Thm. VIII.9.1]{feller1971}. Especially we can
bound $H(r)$ from below, more precisely for any $\beta^-\in(0,\beta)$ there
exists $r_0>0$ such that $H(r)\gtrsim r^{2-\beta^-}$ for all $r\in(0,r_0)$.
By~\citet{orey1968}
there is a constant $c>0$ such that
\begin{equation}\label{eq:phibound}
|\phi_{\Delta}(u)|\lesssim\exp(-c\Delta|u|^{\beta^{-}})
\end{equation}
for $|u|$ sufficiently large.
On the other hand, it is easily seen that
\begin{equation}\label{eq:psibound}
|\psi'(u)|\lesssim1+|u|^{(\beta^{+}-1)\vee0}
\end{equation}
for any $\beta^{+}\in(\beta,2)$ and that $|\psi''(u)|$ is bounded.
Especially we have
$\phi_{\Delta},\phi_{\Delta}''\in L^{2}(\R)$.
Collecting the above and using (\ref{boringbound}) implies
\[
\sup_{t\in\R}|B_{1}(t)|\lesssim\Delta\int(1+|u|)^{-1}|\psi'(u)|^{2}|\phi_{\Delta
}(u)|\,\d u.
\]
Let us distinguish the cases $\beta\ge1$
and $\beta<1$, which will yield together (\ref{eq:b1}). We will be using the
bounds for $\phi_\Delta$ and $\psi'$ in \eqref{eq:phibound} and
\eqref{eq:psibound}, respectively.
\begin{enumerate}
\item For $\beta\ge1$ substituting $u=\Delta^{-1/\beta^{-}}z$ yields
\begin{align*}
\sup_{t\in\R}|B_{1}(t)|&\lesssim
\Delta\int(1+|u|)^{(2\beta^{+}-3)}\exp(-c\Delta|u|^{\beta^{-}})\,\d u\\
&\lesssim
\Delta^{(\beta^{-}-2\beta^{+}+2)/\beta^{-}}\int((1+|z|)^{2\beta^{+}-3}\vee|z|^{
2\beta^{+}-3})\exp(-c|z|^{\beta^{-}})\,\d z,
\end{align*}
where the integral in the last display is finite owing to $\beta^{+}>1$.
Noting that $2\beta^{+}-\beta^{-}>\beta$, we conclude that
$|B_{1}|\lesssim\Delta^{p}$
for any $p<(2-\beta)/\beta$.
\item For $0<\beta<1$ boundedness of $|\psi'|$ and the same substitution
yields for any $\delta>0$
\begin{align*}
\sup_{t\in\R}|B_{1}(t)|&\lesssim
\Delta\int(1+|u|)^{-1}\exp(-c\Delta|u|^{\beta^{-}})\,\d u\\
&\le
\Delta^{1-1/\beta^{-}}\int(1+\Delta^{-1/\beta^{-}}|z|)^{-1+\delta}\exp(-c|z|^{
\beta^{-}})\,\d z\\
&\le \Delta^{1-\delta/\beta^{-}}\int|z|^{-1+\delta}\exp(-c|z|^{\beta^{-}})\,\d
z.
\end{align*}
By choosing $\delta$ sufficiently small, we obtain $|B_{1}|\lesssim\Delta^{p}$
for any $p<1$.
\end{enumerate}
Let us now consider $B_{2}$ in (\ref{eq:BiasDecomp}) which we can
write as
\begin{align*}
B_{2}(t)= & \int\int\big(g_{t}(x+y)-g_{t}(x)\big)x^{2}\nu(\d x)\PP_{\Delta}(\d
y)\\
= &
\int\big((g_{t}(-\bull)\ast(x^{2}\nu))(-y)-(g_{t}(-\bull)\ast(x^{2}
\nu))(0)\big)\PP_{\Delta}(\d y).
\end{align*}
For the sake of brevity we define $h_{t}(y):=(g_{t}(-\bull)\ast(x^{2}\nu))(y)$.
We decompose the integration domain into the neighbourhood of the
origin $(-U,U)$ and the tails $\{y:|y|\ge U\}$. For small $y$ the
uniform H\"older regularity of $h_{t}(y)$, for $|y|<U$, as well as
$\E[|X_{1}|^{2}]\lesssim\Delta$ and Jensen's inequality yield for $s\le1$
\[
\sup_{t\in\R}\Big|\int_{|y|<U}\big(h_{t}(-y)-h_{t}(0)\big)\PP_{\Delta}(\d
y)\Big|\lesssim\int_{\R}|y|^{s}\PP_{\Delta}(\d
y)=\E[|X_{1}|^{s}]\lesssim\Delta^{s/2}.
\]
and for $s>1$ with $x_y\in[-y,0]$ an intermediate point from the mean value
theorem
\begin{align*}
&\sup_{t\in\R}\Big|\int_{|y|<U}\big(h_{t}(-y)-h_{t}(0)\big)\PP_{\Delta}(\d
y)\Big|\\
&\le
\sup_{t\in\R}\Big|\int_{|y|<U}\big(h_{t}'(x_y)-h_{t}'(0)\big)y\PP_{\Delta}(\d
y)\Big|
+
\sup_{t\in\R}\Big|\int_{|y|<U}h_{t}'(0)y\PP_{\Delta}(\d
y)\Big|\\
&\le
\sup_{t\in\R}\Big|\int_{|y|<U}|y|^s\PP_{\Delta}(\d
y)\Big|
+
\Big|\int_{\R}y\PP_{\Delta}(\d
y)\Big|\sup_{t\in\R}\left|h_{t}'(0)\right|
+
\Big|\int_{|y|\ge U}y^2 U^{-1}\PP_{\Delta}(\d
y)\Big|\sup_{t\in\R}\left|h_{t}'(0)\right|\\
&\lesssim\Delta^{s/2}+\Delta+\Delta\lesssim\Delta^{s/2}.
\end{align*}
For the tails we conclude from
$\sup_{t}\|h_{t}\|_{\infty}\le\|\rho\|_{\infty}\int x^{2}\nu(\d x)$
and Markov's inequality
\[
\sup_{t\in\R}\Big|\int_{|y|\ge U}\big(h_{t}(-y)-h_{t}(0)\big)\PP_{\Delta}(\d
y)\Big|\lesssim\PP_{\Delta}(|X_{1}|\ge U)\le
U^{-2}\E[|X_{1}|^{2}]\lesssim\Delta.
\]
The previous two estimates finally yield
$\sup_{t}|B_{2}(t)|\lesssim\Delta^{s/2}$.
\end{proof}
\subsection{Proof of Theorem \ref{thm:UCLTnaive2}}
We only prove the case $V=(-\infty, -\zeta]$, the general case follows from
symmetry arguments that are left to the reader. We use decomposition
(\ref{eq:decomp}) and apply Theorem \ref{main1} -- with $m=1$ and $\rho$
suitably chosen such that $\rho(x)=x^{-2}$ for all $x \in (-\infty, -\zeta]$ --
to the stochastic term $S(t)$. For our choice of $\Delta_n$ the bias term $B(t)$
is negligible in the asymptotic distribution in view of Proposition 2.1 in
\cite{Figueroa-Lopez2011} (which holds also for unbounded $V$ separated away
from the origin, as inspection of that proof shows).
\subsection{Proof of Theorem \ref{thmUniform}}
For Theorem~\ref{thmUniform}\ref{away} we choose a suitable $\rho$ such that
$\rho(x)=x^{-2}$ on $V$ and we restrict to the case $(-\infty,-\zeta]$ since
the proof can be easily extended to cover the general case by symmetry
arguments.
We use the decomposition (\ref{eqErrorDecomp}). The third term is negligible in
view (\ref{eq:sigmarate}). Recalling
\[
g_t(x)=\rho(x)\1_{(-\infty,t]}(x),\quad t\in\R.
\]
the following result shows that the deterministic approximation error is negligible in the asymptotic distribution of
$\sqrt{n\Delta_n}(\hat N_n-N_\rho)$ whenever $h^{s}=o(1/\sqrt{n\Delta_n})$, valid for our choice of $\Delta_n$.
\begin{proposition}\label{propBias}
Suppose $x^2\nu$ is a finite measure satisfying Assumption~\ref{assProc}(d).
If the kernel satisfies \eqref{propKernel} with order $p\ge s$, then
\[
\Big|\int_{\R} g_{t}(x)\big(K_h\ast\big(y^2\nu(\d y)\big)-x^2\nu\big)(\d
x)\Big|\lesssim c_t h^{s},
\]
with constants independent of $t$.
\end{proposition}
\begin{proof}
Using Fubini's theorem,
\begin{align}\label{eqRewrBias}
\int_{\R} g_{t}(x)\big(K_h\ast\big(y^2\nu(\d y)\big)-x^2\nu\big)(\d x)
=K_h\ast g_{t}(-\bull)\ast(x^2\nu)(0)-g_{t}(-\bull)\ast(x^2\nu)(0).
\end{align}
The result now follows from Assumption \ref{assProc}(d) and a standard Taylor
expansion argument using the order $p$ of the kernel.
\end{proof}
\smallskip
The second, stochastic, term in (\ref{eqErrorDecomp}) can be reduced to the linear term from Proposition \ref{propRem}, which is proved as follows:
\begin{proof}[Proof of Proposition~\ref{propRem}]
To linearise
$\psi''-\hat\psi_n''=-\Delta^{-1}\log(\phi_{\Delta,n}/\phi_\Delta)''$, we set
$F(y)=\log(1+y)$, $\eta=(\phi_{\Delta,n}-\phi_\Delta)/\phi_\Delta$, and use
\begin{align*}
(F\circ \eta)''(u)&=F'(\eta(u))\eta''(u)+F''(\eta(u))\eta'(u)^2\\
&=F'(0)\eta''(u)+O\Big(\norm{F''}_\infty\Big(\norm{\eta}_\infty\norm{\eta''}
_\infty
+\norm{\eta'}_\infty^2\Big)\Big).
\end{align*}
On the event $\Omega_n:=\{\sup_{|u|\le
1/h}\abs{(\phi_{\Delta,n}-\phi_\Delta)(u)/\phi_\Delta(u)}\le 1/2\}$ we thus
obtain
\begin{align*}
&\sup_{\abs{u}\le h^{-1}}
\big|\log(\phi_{\Delta,n}/\phi_\Delta)''(u)-(\phi_\Delta^{-1}(\phi_{\Delta,n}
-\phi_\Delta))''(u)\big|\\
&\qquad=O\Big(\norm{\eta}_{\ell^\infty[-h^{-1},h^{-1}]}\norm{\eta''}_{\ell^\infty[-h^{
-1},h^{-1}]} +\norm{\eta'}_{\ell^\infty[-h^{-1},h^{-1}]}^2\Big).
\end{align*}
To estimate $\norm{\eta^{(k)}}_{\ell^\infty[-h^{-1},h^{-1}]},k=0,1,2$, we note
$\abs{\psi'(u)}\lesssim 1+\abs{u}$, $\abs{\psi''(u)}\lesssim 1$ and $h\gtrsim
\Delta^{1/2}$
\[ \sup_{u\in[-h^{-1},h^{-1}]}\abs{(\phi_\Delta^{-1})'(u)}\lesssim \Delta
h^{-1}\lesssim\Delta^{1/2},\quad
\sup_{u\in[-h^{-1},h^{-1}]}\abs{(\phi_\Delta^{-1})''(u)}\lesssim
\Delta^2h^{-2}+\Delta\lesssim\Delta.
\]
Moreover, from Theorem 1 by \cite{kappusReiss2010} we know that under our moment
assumption on $\nu$ (for $k=0,1,2$ and any $\delta>0$)
\begin{equation}\label{eqJM}
\norm{(\phi_{\Delta,n}-\phi_\Delta)^{(k)}}_{\ell^\infty[-h^{-1},h^{-1}]}
=O_P(n^{-1/2}\Delta^{(k\wedge 1)/2}(\log h^{-1})^{(1+\delta)/2}).
\end{equation}
This yields for $k=0,1,2$
\begin{align*}
\|\eta^{(k)}\|_{\ell^\infty[-h^{-1},h^{-1}]}&=O_P\big(n^{-1/2}\Delta^{k/4}(\log
h^{-1})^{(1+\delta)/2}\big).
\end{align*}
In combination with $n(\log h^{-1})^{-1-\delta}\gtrsim
n\Delta^{3(1+\delta)/4}\to\infty$ for $\delta\in(0,1/3)$ and $|1/\varphi_\Delta|
\lesssim 1$ on $[-1/h_n, 1/h_n]$ the bound \eqref{eqJM} shows also
$\PP(\Omega_n)\to 1$ and then
\[ \sup_{\abs{u}\le h^{-1}}
\abs{\hat\psi_n''(u)-\psi''(u)-\Delta^{-1}(\phi_\Delta^{-1}(\phi_{\Delta,n}
-\phi_\Delta))''(u)}
=O_P(n^{-1}\Delta^{-1/2}\log(h^{-1})^{1+\delta}).
\]
We decompose the linearised stochastic error into
\[
(\phi_\Delta^{-1}(\phi_{\Delta,n}-\phi_\Delta))''
=\phi_\Delta^{-1}(\phi_{\Delta,n}-\phi_\Delta)''
+2(\phi_\Delta^{-1})'(\phi_{\Delta,n}-\phi_\Delta)'
+(\phi_\Delta^{-1})''(\phi_{\Delta,n}-\phi_\Delta).
\]
By the previous estimates we have
\begin{align*}
\sup_{\abs{u}\le
h^{-1}}\abs{(\phi_\Delta^{-1})'(\phi_{\Delta,n}-\phi_\Delta)'}(u)&=O_P(
\Delta h^{-1}n^{-1/2}\Delta^{1/2}(\log h^{-1})^{(1+\delta)/2}),\\
\sup_{\abs{u}\le
h^{-1}}\abs{(\phi_\Delta^{-1})''(\phi_{\Delta,n}-\phi_\Delta)}(u)&=O_P(
(\Delta^2 h^{-2}+\Delta)n^{-1/2}(\log h^{-1})^{(1+\delta)/2}).
\end{align*}
Inserting the asymptotics in $h$, we conclude
\begin{align*}
&\quad\sup_{\abs{u}\le h^{-1}}
\abs{\hat\psi_n''(u)-\psi''(u)-\Delta^{-1}\phi_\Delta^{-1}(\phi_{\Delta,n}
-\phi_\Delta)''(u)}\\
&\le\sup_{\abs{u}\le
h^{-1}}2\Delta^{-1}\abs{(\phi_\Delta^{-1})'(\phi_{\Delta,n}-\phi_\Delta)'}(u)
+\sup_{\abs{u}\le
h^{-1}}\Delta^{-1}\abs{(\phi_\Delta^{-1})''(\phi_{\Delta,n}-\phi_\Delta)}(u)\\
&\qquad+O_P\big(n^{-1}\Delta^{-1/2}\log(h^{-1})^{1+\delta}\big)\\
&=O_P\left(n^{-1/2}\Delta^{-1/2}h^{1/2}\left(\Delta
h^{-3/2}+\Delta^{3/2}h^{-5/2}+\Delta^{1/2}h^{-1/2}\right)(\log
h^{-1})^{(1+\delta)/2}\right)\\
&\qquad+O_P\big(n^{-1}\Delta^{-1/2}\log(h^{-1})^{1+\delta}\big)\\
&=o_P\big(n^{-1/2}\Delta^{-1/2}h^{1/2}\big).
\end{align*}
By the Plancherel formula and Cauchy-Schwarz inequality we have
\begin{align*}
&\babs{\int g_t(x)\F^{-1}\Big[\mathcal F
K_h(u)\big(\hat\psi_n''(u)-\psi''(u)-\Delta^{-1}\phi_\Delta(u)^{-1}(\phi_{\Delta
,n}-\phi_\Delta)''(u)\big)\Big]
(x)\d x}\\
&\le
\norm{\mathcal F g_t}_{L^2}\norm{\mathcal F K_h}_{L^2}\sup_{\abs{u}\le
h^{-1}}\abs{\hat\psi_n''(u)-\psi''(u)-\Delta^{-1}\phi_\Delta(u)^{-1}(\phi_{
\Delta,n}-\phi_\Delta)''(u)}\\
&=o_P(n^{-1/2}\Delta^{-1/2}).\qedhere
\end{align*}
\end{proof}
\smallskip
Finally, to the main stochastic term
\begin{align*}
M_{\Delta,n} &=\Delta_n^{-1}\int g_t(x)
\big(\F^{-1}[\phi_{\Delta_n}^{-1}\mathcal
F K_{h_n}]\ast (x^2(\PP_{\Delta_n,n}-\PP_{\Delta_n}))\big)(\d x)\\
&=\Delta_n^{-1}\int {\cal F}^{-1}[\phi_{\Delta_n}^{-1}(-u)\mathcal F
K_{h_n}(-u){\cal F}g_t(u)](x)x^2(\PP_{\Delta_n,n}-\PP_{\Delta_n})(\d x),
\end{align*}
we apply Theorem \ref{main1}. The proof of Theorem \ref{thmUniform} is thus
complete upon verification of Assumption~\ref{multass} for the present choice of
$m$. This is achieved in the following proposition.
\begin{proposition}\label{multver} Assume that $K$ satisfies (\ref{propKernel}) for
$p \ge 2$ and that $\nu$ satisfies $\int_{\R}|x|^3\nu(\d x)<\infty$.
Let $h=h_n\to0$ and $\Delta=\Delta_n\to0$ as $n\to\infty$ with $h^3=o(\Delta)$, $h^{-1} =
O(\Delta^{-1/2})$.
Then
$m_{n,\Delta}(u):=\mathcal F K_h(u)/\phi_\Delta(u)$, $u \in \R$, satisfies Assumption~\ref{multass}.
\end{proposition}
\begin{proof}
We have $m(-u)=\overline{m(u)}$ so that $\F^{-1}m$ is real-valued.
By the compact support of $\mathcal F K$ and the assumption on $h^{-1}$ the support
assumption on $m$ is satisfied. Since $\phi_{\Delta} = e^{\Delta \psi}$, we have
$m_{\Delta, n} \to 1$ pointwise as $\Delta \to 0$, $h\to0$. Moreover, by
\eqref{eqIdent} we have $|\psi''(u)|\lesssim 1$ hence for $|u| \le
C\Delta^{-1/2}$ we have $$|\phi_\Delta(u)|= |e^{\Delta \psi(u)}| \ge e^{-\Delta
cu^2} \ge c'>0$$ uniformly in $\Delta$, and thus $m \in \ell^\infty(\R)$, using
also
$\sup_h\|\mathcal F K_h\|_\infty \le \|K\|_{L^1}$. Next
\[m' = h \frac{i\F
[xK](h\bull)}{\phi_\Delta} + \mathcal F K_h \frac{\Delta \psi' \phi_\Delta
}{\phi^2_\Delta}\]
so that using $xK \in L^1, |\psi'(u)| \lesssim 1+|u|,
|u| \le h^{-1}=O(\Delta^{-1/2})$ and the bound for $m$ above we see
$$|m'(u)| \lesssim (h + \sqrt{\Delta}) \lesssim h.$$
Using $|\psi''(u)|\lesssim 1$ we further obtain
\begin{align*}
|m''(u)| &\lesssim (h^2+h\sqrt{\Delta}+\Delta)
\lesssim h^2.
\end{align*}
On the support of $m$ we have $|u|\le h^{-1}$ so that
$\|(1+|u|)^{k}m^{(k)}\|_\infty\le c$, $k\in\{0,1,2\}$, follows.
Likewise by the support of $m$ we have $\|m'\|_{L^2}\lesssim h^{1/2}\to0$
and $\Delta^{-1/2}\|m''\|_{L^2}\lesssim \Delta^{-1/2}h^{3/2}\to0$.
\end{proof}
\subsection{Convergence of finite-dimensional distributions}
We next turn to the proof of Proposition~\ref{fidi0}.
\begin{definition} \label{admis}
A function $g$ is called \textit{admissible} if it is of bounded variation and satisfies for all $x, u \in \R$,
\[
|g(x)| \lesssim 1 \wedge x^{-2},\quad |\mathcal F
g(u)|\lesssim (1+|u|)^{-1} \quad\text{and}\quad u\F[xg](u)\in \ell^\infty(\R).
\]
\end{definition}
Note that the bound on $\mathcal F g$ follows from the bounded variation of
$g$, and that $x^2 g^2(x)$ is
of bounded variation whenever $g$ is admissible.
\begin{proposition} \label{fidi1}
Let $g$ be admissible and suppose the conditions of Proposition \ref{fidi0} are satisfied. Then
\[
\sqrt{n\Delta_n} \int \F^{-1}[m(-\bull)\mathcal F
g](x)\frac{x^2}{\Delta_n}(\PP_{\Delta_n,n}-\PP_{\Delta_n})(\d x)\to^{\mathcal L}
\mathcal N(0,\sigma_g^2)
\]
with variance $\sigma_g^2=\int_{\R} x^4g(x)^2\nu(\d x)$.
\end{proposition}
The functions $g_t=\rho\1_{(-\infty,t]}$ are uniformly bounded in bounded
variation and are admissible with constants
independent of $t\in\R$.
The convergence of the finite dimensional distributions in Proposition
\ref{fidi0} hence follows from the Cram\'er-Wold device since linear combinations
of the functions $g_{t_1},\dots,g_{t_k}$ for $t_1,\dots,t_k\in\R$
are admissible.
For the proof of Proposition~\ref{fidi1} we will use the following lemma, whose
assumptions are in particular fulfilled for $m_{n,\Delta}$ satisfying
Assumption~\ref{multass} and for classes of functions with uniform constants in
the admissibility definition.
\begin{lemma}\label{lemMultCon}
Let $\|x^3\PP_\Delta\|\lesssim\Delta$. For $\Delta\to0$ as $n\to\infty$ let $\|m_{n,\Delta}\|_\infty$
and $\|m_{n,\Delta}'\|_\infty$ be uniformly bounded and $m_{n,\Delta}\to1$
pointwise.
If $\G$ is a class of functions such that for all $u\in\R$
\[\sup_{g\in\G}|\mathcal{F}g(u)|\lesssim (1+|u|)^{-1}, \quad
\sup_{g\in\G}\|xg(x)\|_{L^2}\lesssim 1,\]
then
\begin{align*}
\lim_{n\to\infty}\sup_{g\in\G}\int_{\R}\Big(x^2\F^{-1}[m_{n,\Delta}
(-u)\mathcal F
g(u)](x)-x^2g(x)\Big)^2\frac{\PP_\Delta}{\Delta}(\d x)=0.
\end{align*}
\end{lemma}
\begin{proof}
We rewrite the term with $m=m_{n,\Delta}$ as
\begin{align}
&\Delta^{-1}\int_{\R}\F^{-1}[\mathcal F g(u)(m(-u)-1)](x)\F^{-1}[\F
g(u)(m(-u)-1)](x)x^4\PP_\Delta(\d x)\notag\\
=&\frac{-i}{\Delta}\int_{\R}\F^{-1}\big[\mathcal F
g(u)(m(-u)-1)\big](x)\label{eqVarRem}\\
&\qquad\quad\times\F^{-1}\big[i\F[xg](u)(m(-u)-1)-\mathcal F
g(u)m'(-u)\big](x)x^3\PP_\Delta(\d x)\notag.
\end{align}
Using $\|x^3\PP_\Delta\|_\infty\lesssim \Delta$, the term \eqref{eqVarRem} can be estimated by the Cauchy-Schwarz inequality and Plancherel's identity yielding the bound
\begin{align*}
&\int_{\R}\Big|\F^{-1}\big[\mathcal F
g(u)(m(-u)-1)\big](x)\F^{-1}\big[i\F[xg](u)(m(-u)-1)-\F
g(u)m'(-u)\big](x)\Big|\d x\\
\le&\frac{1}{2\pi}\big\|\mathcal F
g(u)(m(-u)-1)\big\|_{L^2}\big\|\big(i\F[xg](u)(m(-u)-1)-\F
g(u)m'(-u)\big)\big\|_{L^2}.
\end{align*}
The first factor converges to zero by the dominated convergence theorem because $m$ is uniformly bounded and converges pointwise to one while $|\mathcal{F}g(u)|\le C(1+|u|)^{-1}$ for all $g\in\G$. For the second factor we estimate, using that $g$ and $xg$ are uniformly bounded in $L^2(\R)$ and that $\|m\|_\infty$ and $\|m'\|_\infty$ are uniformly bounded,
\begin{align*}
\big\|i\F[xg](u)(m(-u)-1)-\mathcal F g(u)m'(-u)\big\|_{L^2}
&\lesssim\|\F[xg](u)\|_{L^2}+\|\mathcal F g(u)\|_{L^2}<\infty,
\end{align*}
which completes the proof of the lemma.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{fidi1}]
We define
\begin{equation}
S_n - \E S_n := \frac{1}{n}\sum_{k=1}^n(
Y_{n,k}-\E[Y_{n,k}])\quad\text{with}\quad
Y_{n,k}:=\Delta^{-1/2}\F^{-1}[m(-\bull)\mathcal F g](X_k)X_k^2.
\end{equation}
We will prove the proposition for general Fourier multipliers satisfying Assumption~\ref{multass}(b), the case where $\F^{-1} m$ is a finite signed measure is similar (in fact easier) and is omitted. We will verify the conditions of Lyapunov's central limit theorem, see, e.g., \cite{Bauer1996}, Theorem 28.3 and (28.8).
\textit{Step 1:}
We will show that
$\lim_{n\to\infty}\Var(Y_{n,k})=\int_{\R} x^4 g(x)^2\nu(\d x)$, noting that $Y_{n,k}$ are real valued. We estimate
\begin{align*}
\abs{\E[Y_{n,k}]}&=\Delta^{-1/2}\babs{\int_{\R}{\cal F}^{-1}\big[m(-u)\F
g(u)\big](x)x^2\PP_\Delta(\d x)}\\
&\le \Delta^{-1/2}\norm{{\cal F}^{-1}[m(-u)\F
g(u)]}_\infty\norm{x^2\PP_\Delta}_{L^1}\\
& \lesssim \Delta^{-1/2} \int_{-C \Delta^{-1/2}}^{C \Delta^{-1/2}}
(1+|u|)^{-1}\d u ~\E[X_1^2] \\
&\lesssim \Delta^{-1/2}\log(\Delta^{-1})\Delta\to 0
\end{align*}
where we have used that $\E[X_1^2]= O(\Delta)$.
Consequently,
$\lim_{n\to\infty}\Var(Y_{n,k})=\lim_{n\to\infty}\E[Y_{n,k}^2]$,
which we decompose in the following way:
\begin{align*}
\lim_{n\to\infty}\E[Y_{n,k}^2]
&=\lim_{n\to\infty}\Delta^{-1}\int_{\R}(\F^{-1}[m(-u)\mathcal F
g(u)](x)x^2)^2\PP_\Delta(\d x)\notag\\
&= \lim_{n\to\infty}\Delta^{-1}\int_{\R}\Big((\F^{-1}[m(-u)\mathcal F
g(u)](x)x^2)^2-(x^2g(x))^2\Big)\PP_\Delta(\d x)\\
&\quad+\lim_{n\to\infty}\left(\Delta^{-1}\int_{\R}(x^2g(x))^2\PP_\Delta(\d
x)-\int_{\R}(xg(x))^2 x^2\nu(\d
x)\right)\notag\\
&\quad+\int_{\R}(xg(x))^2x^2\nu(\d x).\notag
\end{align*}
The last term is the claimed limit. The first limit is zero by Lemma~\ref{lemMultCon}.
For the second limit we deduce by Lemma~\ref{weakcon}
that $(x^2 \wedge x^4) \PP_\Delta/\Delta$ converges weakly to the absolutely
continuous measure $(x^2 \wedge x^4) \nu $, and thus in particular
by the Portmanteau lemma when
integrating against the function $(x^2 \vee 1) g(x)^2$, which is of bounded
variation. This implies convergence to zero of the
second term. This shows
$\lim_{n\to\infty}\Var(Y_{n,k})=\int x^4 g(x)^2\nu(\d x)$.
\textit{Step 2:} We verify Lyapunov's moment condition: For some $\eps\in(0,1)$ and $S_n=\sum_{k=1}^nY_{n,k}$
\begin{align*}
\lim_{n\to\infty}\frac{1}{\Var(S_n)^{1+\eps/2}}\sum_{k=1}^{n}\E[|Y_{n,k}|^{
2+\eps}]=0.
\end{align*}
From the previous step we know $n^{-1}\Var(S_n)=\Var(Y_{n,k})\to\sigma_g^2$
as $n\to\infty$. Moreover, by $|x|^{4+2\eps}\lesssim|1+ix|^{2+\eps}|x|^3$,
$\|x^3\PP_\Delta\|_\infty\lesssim\Delta$ and the Hausdorff--Young inequality
\cite[e.g., 8.30 on p.~253 in][]{folland1999}
\begin{align*}
\E[|Y_{n,k}|^{2+\eps}]&\lesssim
\Delta^{-1-\eps/2}\int_{\R}\big|\F^{-1}\big[m(-u)\mathcal F
g(u)\big](x)x^2\big|^{2+\eps}\PP_{\Delta}(\d x)\\
&\lesssim\Delta^{-\eps/2}\int_{\R}\big|\F^{-1}\big[m(-u)\mathcal F
g(u)\big](x)(1+ix)\big|^{2+\eps}\d x\\
&=\Delta^{-\eps/2}\Big\|\F^{-1}\big[m(-u)\mathcal F g(u)-m'(-u)\F
g(u)+im(-u)\F[xg](u)\big]\Big\|_{L^{2+\eps}}^{2+\eps}\\
&\lesssim\Delta^{-\eps/2}\Big\|m(-u)\mathcal F g(u)-m'(-u)\F
g(u)+im(-u)\F[xg](u)\Big\|_{L^{(2+\eps)/(1+\eps)}}^{2+\eps}.
\end{align*}
By Assumption~\ref{multass}, $m$ and $m'$ are uniformly bounded, $\|\mathcal F
g(u)\|_{L^{(2+\eps)/(1+\eps)}}$ is bounded by $|{\F}g|\lesssim (1+|u|)^{-1}$
and
\begin{align*}
\quad\big\|\F[xg](u)\big\|_{L^{(2+\eps)/(1+\eps)}}
\lesssim\big\|\F[xg](u)\big\|_{L^{(2+\eps)/(1+\eps)}([-1,1])}+\big\|\F[xg]
(u)\big\|_ { L^
{ (2+\eps)/(1+\eps)}([-1,1]^c)},
\end{align*}
which are finite by $xg\in L^2(\R)$ and by $u\F[xg](u)\in \ell^\infty(\R)$,
respectively.
Consequently, $\E[|Y_{n,k}|^{
2+\eps}]\lesssim \Delta^{-\eps/2}$, implying
\begin{align*}
\lim_{n\to\infty}\frac{1}{\Var(S_n)^{1+\eps/2}}\sum_{k=1}^{n}\E[|Y_{n,k}|^{
2+\eps}]\lesssim\lim_{n\to\infty}\frac{n\Delta_n^{-\eps/2}}{n^{1+\eps/2}}
=\lim_{n\to\infty}(n\Delta_n)^{-\eps/2}=0.
\end{align*}
\end{proof}
\subsection{Proof of Proposition \ref{Propx1PDelta}}
\begin{proof}
For (ii) and (iii) we have $\int |x| \nu (\d x)<\infty$ and will use that the function $\psi$ in the exponent of the
L\'evy--Khintchine formula \eqref{eqLevyKhintchine} may be
written as
\[\psi(u)=-\frac{\sigma^2u^2}{2}+i\gamma_0 u+\int_{\R}(e^{iux}-1)\nu(\d
x)\quad\text{ with }\gamma_0:=\gamma-\int_{\R}x\nu(\d x).\] For (iii) note
$x\PP_\Delta=(x\PP_\Delta^+)\ast\PP_\Delta^-+(x\PP_\Delta^-)\ast\PP_\Delta^+$
with the corresponding laws for $\nu^+,\nu^-$. It thus suffices to prove
$\norm{x\PP_\Delta^+}_\infty+ \norm{x\PP_\Delta^-}_\infty\lesssim 1$ and without
loss of generality we only consider $\PP_\Delta^+$ in the proof of case~(iii).
For~(iv) we use the same decomposition but this time the
law $\PP_\Delta^+$ corresponds to the L\'evy triplet $(0,\gamma,\nu^+)$ so that
it also incorporates the drift.
\begin{enumerate}
\item If $\sigma>0$ holds, then $\abs{\psi'(u)}\lesssim 1+\abs{u}$ implies
\[ \norm{x\PP_\Delta}_\infty \le \norm{\phi_\Delta'}_{L^1}\lesssim \int \Delta(1+\abs{u})e^{-\Delta\sigma^2u^2/2}\d u\lesssim 1.
\]
\item On the assumptions the measure $x\nu$ is finite yielding the
identity $x\PP_\Delta=\Delta (x\nu)\ast\PP_\Delta$, which implies that even
\[ \norm{x\PP_\Delta}_\infty\le \Delta\norm{x\nu}_\infty.\]
\item Without loss of generality we suppose $\|x\nu^\pm\|_\infty=\infty$. Denote
the limit inferior in condition (iii) by $\delta>0$ and define
\[
a_\Delta:=\inf\left\{a>0:\sup_{x>a}\Delta x\nu(x)\le\frac{4}{\delta}\right\},
\]
where $a_\Delta>0$ follows from
$\lim_{a\to0}\sup_{x>a}x\nu(x)=\|x\nu^+\|_\infty=\infty$. Since
$\|x\nu\|_{\ell^\infty(\R\setminus[-\eps,\eps])}$ is bounded for any $\eps>0$ we
deduce that $a_\Delta\downarrow0$ as $\Delta\to0$.
Let us introduce $\nu_\Delta^s:=\nu{\1}_{[0,a_\Delta]}$ and
$\nu_\Delta^c:=\nu^+-\nu_\Delta^s$. By
$\|x\nu_\Delta^c\|_\infty\le\frac{4}{\Delta\delta} $ and the argument in (ii),
applied to $\nu_\Delta^c$, the corresponding law $\PP_\Delta^c$ satisfies
$\norm{x\PP_\Delta^c}_\infty\lesssim 1$. Because of
\[x\PP_\Delta=(x\PP_\Delta^c)\ast\PP_\Delta^s+(x\PP_\Delta^s)\ast\PP_\Delta^c
=(x\PP_\Delta^c)\ast\PP_\Delta^s+(\Delta x\nu_\Delta^s)\ast\PP_\Delta
\]
we shall bound $\norm{\Delta x\nu_\Delta^s}_{L^1}$ and
$\norm{\PP_\Delta}_\infty$.
From the assumptions we infer $\norm{\Delta x\nu_\Delta^s}_{L^1}\lesssim
a_{\Delta}$ via
\[
\int_0^{a_\Delta} \Delta x\nu(\d x)
=\lim_{a\downarrow a_\Delta}\int_0^a\Delta x\nu(\d x)
\lesssim \limsup_{a\downarrow a_\Delta}a\Delta
a\nu(a)\le\frac{4a_\Delta}{\delta}.
\]
On the other hand, by construction there is some
$a_\Delta^-\in[\frac{1}{2}a_\Delta,a_\Delta]$ such that $\Delta
a_\Delta^-\nu(a_\Delta^-)\ge4/\delta$. Together with the assumptions, and
$\|\PP_\Delta\|_\infty \le \|\phi_\Delta\|_1$, we see that for
$\eps:=a_\Delta^-$ sufficiently small, that is for $\Delta$ small, and for some
$\kappa\in(2,4)$
\begin{align*}
a_\Delta\norm{\PP_\Delta}_\infty &\le 2a_\Delta^-\int_{-\infty}^\infty
e^{-\frac{\Delta }{\kappa} u^2\int_0^{1/\abs{u}} x^2\nu(\d x)}\d u\\
&=2\int_{-\infty}^\infty e^{-\frac{\Delta}{\kappa}
(v/a_\Delta^-)^{2}\int_0^{a_\Delta^-/\abs{v}} x^2\nu(\d x)}\d v\\
&\le 4+2\int_{\abs{v}>1} e^{-\frac{\delta}{\kappa}\Delta
a_\Delta^-\nu(a_\Delta^-)\log(\abs{v})}\d v\\
&\le4+4\int_{1}^\infty v^{-4/\kappa}\d v\sim 1,
\end{align*}
which together with the bound on $\norm{\Delta x\nu_\Delta^s}_{L^1}$ yields the
result.
\item By Theorem 27.7 in \cite{sato1999} $\PP_\Delta$ admits a Lebesgue density, hence by Fourier inversion $\norm{x\PP_\Delta}_\infty\le \norm{\phi_\Delta'}_{L^1}$ and by the
hypothesis on $\nu^+$,
we
estimate for some $\kappa>0$ and for some small $c>0$
\begin{align*}
\norm{x\PP_\Delta^+}_\infty &\le \int_{-\infty}^\infty \Delta
\babs{\int_0^\infty(e^{iux}-1)x\nu(\d x)+\gamma}e^{-\Delta \int_0^\infty
(1-\cos(ux))\nu(\d x)}\d u\\
&\lesssim \int_{-\infty}^\infty \Delta\Big(1+\int_0^\infty (\abs{u}x^2\wedge
x)\nu(\d x)\Big) e^{-\frac{\Delta}{\kappa} u^2\int_0^{1/\abs{u}}x^2\nu(\d x)}\d
u\\
&\le \int_{-\infty}^\infty \Delta\Big(1+\int_0^\infty (\abs{u}x^2\wedge x)\nu(\d
x)\Big) e^{-c\Delta \int_0^\infty (\tfrac{u^2}2x^2\wedge\abs{u}x)\nu(\d x)}\d u.
\end{align*}
The derivative of the exponent is given by
$-c\Delta\sgn(u)\int_0^\infty (\abs{u}x^2\wedge x)\nu(\d x)$ such that the last
line of the display is bounded by
\[ \int_{-\infty}^\infty \Delta e^{-c\Delta \int_0^\infty (\tfrac{u^2}2 x^2\wedge\abs{u}x)\nu(\d x)}\d u +2/c.
\]
From $\abs{u}\int_0^{1/u}x^2\nu(\d x)\gtrsim 1$ we infer that the integral is
at most of order $\int \Delta e^{-\Delta \abs{u}}\d u\thicksim 1$ and the result
follows.\qedhere
\end{enumerate}
\end{proof}
\section{Proof of Theorem \ref{main1}} \label{mainproof}
We recall $g_t(x)=\rho(x)\1_{(-\infty,t]}(x)$ and hence
\begin{align*}
\mathbb{G}_n(t)=\sqrt{n}\int_{\R}\Delta^{-1/2}x^2\F^{-1}[m(-u)\mathcal F
g_t(u)](x)(\PP_{\Delta,n}-\PP_\Delta)(\d x), \quad t\in\R.
\end{align*}
By Proposition~\ref{fidi0} and Theorem~1.5.7 in \cite{vanderVaartWellner1996} it suffices to show that there
is a semimetric $d$ such that $(\R,d)$ is totally bounded and for every
$\gamma>0$ we have
\begin{align}\label{asymptotic_equicontinuity}
\lim_{\delta\to0}\limsup_{n\to\infty}\Pr \left(\sup_{s,t \in \R: d(s,t)\le\delta}
|\mathbb{G}_n(s)-\mathbb{G}_n(t)|>\gamma\right)=0.
\end{align}
We note that $\mathbb G_n$ equals a triangular array of empirical processes $\sqrt n (\PP_{\Delta, n}-\PP_{\Delta})$ indexed by the class
\begin{align*}
\tilde \G_n&:=\{\tilde g_t(x):t\in\R\},\\
\tilde g_t(x)&:=\Delta^{-1/2}x^2\F^{-1}[m(-\bull)\mathcal F g_t(\bull)](x).
\end{align*}
\subsection{Equicontinuity and a change of metric}
For $t\le0$ we decompose $\tilde g_t$ into the three terms
\begin{align}
\tilde g_t^{(1)}(x)&:=\Delta^{-1/2}x^2\F^{-1}[m(-u)\F [(\rho(\bull)
-e^{\bull-t}\rho(t)
)\1_{(-\infty,t]}(\bull)](u)](x),\label{eqtildeg1}\\
\tilde g_t^{(2)}(x)&:=\Delta^{-1/2}x\F^{-1}[m(-u)\F[te^{\bull-t}\rho(t)
\1_{(-\infty,t]}(\bull)](u)](x),\label{eqtildeg2}\\
\tilde g_t^{(3)}(x)&:=\tilde g_t(x)-\tilde g_t^{(1)}(x)-\tilde
g_t^{(2)}(x).\label{eqtildeg3}
\end{align}
Heuristically speaking the main difficulties arise from the fact that $\1_{(-\infty, t]}$ is nonintegrable on $\R$ and discontinuous at $t$. The above decomposition separates the jump-discontinuity from the non-integrable part, and the third term collects the remainder without discontinuity or integrability issues. We refer to the second term as the `critical term' since it is not regular
enough to be treated by the usual metric entropy techniques.
For $t>0$ we replace $e^{y-t}\rho(t)\1_{(-\infty,t]}(y)$ by
$-e^{t-y}\rho(t)\1_{(t, \infty)}(y)$, and the proof below proceeds
with only notational changes. We thus restrict to $t \in (-\infty,0]$.
By the triangle inequality it suffices to show asymptotic
equicontinuity for the empirical processes
indexed by the three terms in the above decomposition
separately with appropriate metrics $d^{(i)}$, and then
\eqref{asymptotic_equicontinuity} holds with the overall
metric $d=\max _i d^{(i)}$ equal to the maximum
of the three metrics $d^{(i)}, i =1,2, 3$. In view of the variance structure
of the limiting process $\mathbb G$ it is natural to choose the semimetrics
\[d^{(i)}(s,t) = \sqrt{\int_{\R} (g_s^{(i)}-g^{(i)}_t )^2(x) \nu(\d x) },\quad
i=1,2,3,\]
where
\begin{align}
g_t^{(1)}(x)&:=x^2(\rho(x)-e^{x-t}\rho(t)
)\1_{(-\infty,t]}(x)\label{eqg1},\\
g_t^{(2)}(x)&:=xte^{x-t}\rho(t)
\1_{(-\infty,t]}(x)\label{eqg2},\\
g_t^{(3)}(x)&:=x(x-t)e^{x-t}\rho(t)
\1_{(-\infty,t]}(x),\label{eqg3}
\end{align}
and we note $x^2g_t=g_t^{(1)}+g_t^{(2)}+g_t^{(3)}$.
On the other hand the covariance metric compatible with the distribution
$\PP_\Delta$ of the $X_k$'s driving the empirical process is given by the
$L^2(\PP_\Delta)$-distance. In the following we will show that a
$\delta$-increment for the limiting metric $d^{(i)}$ corresponds, for $n$ large
enough, to a $\delta$-increment in the $L^2(\PP_\Delta)$-metric on the functions
$\tilde g_t^{(i)}$. Verifying asymptotic equicontinuity for the whole process
then reduces to showing total boundedness of each subclass and that, for each
$i=1, 2, 3$, and every $\gamma>0$,
\begin{align}\label{eqEquContF}
\lim_{\delta\to0}\limsup_{n\to\infty}\Pr \left(\sup_{\|\tilde
g_s^{(i)}-\tilde g_t^{(i)}\|_{2,\PP_\Delta}\le\delta}
\left|\sqrt n\int_{\R} (\tilde g_s^{(i)}-\tilde
g_t^{(i)})(\PP_{\Delta,n}-\PP_\Delta)(\d x)\right|>\gamma\right)=0,
\end{align}
where $\|f\|_{2,P}:=(\int |f|^2 \d P)^{1/2}$.
This will permit the application of powerful tools from empirical process theory
to control the last probabilities. Before we do this, we demonstrate the
reduction
to (\ref{eqEquContF}) for all three terms in the above decomposition
separately. We note that total boundedness of the
classes $\mathcal G^{(i)} = \{g_t^{(i)}: t \in \R\}$ for the $d^{(i)}$-metric
follows from entropy computations given in the following subsections.
\medskip
Starting with $\{\tilde g_t^{(1)}: t \le0\}$,
we note that the functions
\begin{align*}
x^{-1}g_t^{(1)}(x)&=x(\rho(x)-e^{x-t}\rho(t))\1_{(-\infty,t]}(x)\\
&=(x\rho(x)-(x-t)e^{x-t}\rho(t)-e^{x-t}t\rho(t))\1_{(-\infty,t]}(x),\quad t\le0,
\end{align*}
are uniformly bounded and
uniformly Lipschitz
continuous.
In order to compare $d^{(1)}$ to the $L^2(\PP_\Delta)$-norm on $\{\tilde
g_t^{(1)}:t\le0\}$, we
claim
\begin{align}\label{metric_comparison}
\sup_{s,t\le0}
\left|\int(\tilde g_s^{(1)}(x)-\tilde
g_t^{(1)}(x))^2{\PP_\Delta(\d x)}-\int(g_s^{(1)}(x)-g_t^{(1)}(x))^2\nu(\d
x)\right| \to 0
\end{align}
as $n\to\infty$. Any class of
functions that is uniformly bounded and uniformly Lipschitz continuous is a
uniformity class for weak convergence using either Theorem~1 in
\cite{BillingsleyTopsoe1967}, or the well-known fact that the BL-metric
metrises weak convergence.
So, the weak convergence in Lemma~\ref{weakcon}
yields
\begin{align*}
\sup_{s,t\le0}\left|\int(x^{-1}g_s^{(1)}(x)-x^{-1}g_t^{(1)}(x))^2x^2\Delta^{
-1 } \PP_\Delta(\d
x)-\int(x^{-1}g_s^{(1)}(x)-x^{-1}g_t^{(1)}(x))^2x^2\nu(\d x)\right|\to0
\end{align*}
as $n\to\infty$. Next using $0<\rho(x)\le C (1\wedge x^{-2})$ and the bounded
variation of $\rho$, we see that
$\G:=\{x^{-2}(g_s^{(1)}(x)-g_t^{(1)}(x)):s,t\le0\}$ satisfies the assumption of
Lemma~\ref{lemMultCon} and hence
\begin{align}
\sup_{s,t\le0}\int\left((\tilde g_s^{(1)}(x)-\tilde
g_t^{(1)}(x))-\Delta^{-1/2}(g_s^{(1)}(x)-g_t^{(1)}(x))\right)^2{\PP_\Delta(\d
x)}\to0\label{eqConvTildef}
\end{align}
as $n\to\infty$.
We conclude that \eqref{metric_comparison} and then also the reduction to
(\ref{eqEquContF}) holds for $\{\tilde g_t^{(1)}: t \in \R\}$.
A similar reduction for $\tilde g_t^{(2)}$ defined in \eqref{eqtildeg2} is
achieved as follows. As in \eqref{metric_comparison} we claim that
\begin{align}
&\sup_{s,t\le0}\left|\int(\tilde g_s^{(2)}-\tilde g_t^{(2)})^2\d\PP_\Delta
-\int(g_s^{(2)}-g_t^{(2)})^2\d\nu\right|\notag\\
&\le\sup_{s,t\le0}\left|\int(\tilde g_s^{(2)}-\tilde g_t^{(2)})^2\d\PP_\Delta
-\int(g_s^{(2)}-g_t^{(2)})^2\frac{\d\PP_\Delta}{\Delta}\right|\notag\\
&+\sup_{s,t\le0}\left|\int(g_s^{(2)}-g_t^{(2)})^2\frac{\d\PP_\Delta}{\Delta
}
-\int(g_s^{(2)}-g_t^{(2)})^2\d\nu\right|\label{convergence_to_Levy_measure}
\end{align}
converges to zero as $n\to\infty$. To see this we observe that by
Lemma~\ref{weakcon} the
measures $(1\wedge
x^4)\Delta^{-1}\PP_\Delta(\d x)$ converge weakly to $(1\wedge x^4)\nu(\d x)$.
The limit is absolutely continuous with respect to Lebesgue measure and thus the
functions $g_t^{(2)}(x)/(1\wedge x^2)$, $t<0$, are $(1\wedge x^4)\nu(\d
x)$-almost everywhere continuous. Moreover, the functions
\begin{align}
\frac{g_t^{(2)}(x)}{1\wedge x^2}=xte^{x-t}\rho(t)\1_{(-\infty,t]}(x)\vee
\frac{t}{x}e^{x-t}\rho(t)\1_{(-\infty,t]}(x),\quad t<0,\label{eqg2x2}
\end{align}
are all contained in a bounded set of the space
of bounded variation functions and hence $\{(g_s^{(2)}(x)/(1\wedge
x^{2})-g_t^{(2)}(x)/(1\wedge x^{2}))^2:~s,t<0\}$ forms a uniformity class for
weak convergence towards $(1 \wedge x^4)\nu(\d x) \in \ell^\infty(\R)$ (after
renormalising the
measures
involved to have mass one and by Theorem 1 in
\cite{BillingsleyTopsoe1967}).
Consequently
\begin{align}
\sup_{s,t\le0}\left|\int(g_s^{(2)}-g_t^{(2)})^2\frac{\d \PP_\Delta}{\Delta}
-\int(g_s^{(2)}-g_t^{(2)})^2\d\nu\right|\to0\label{eqUniformityClassQ}
\end{align}
as $n\to\infty$, where we recall that $g_0^{(2)}=0$.
To deal with the first term in
\eqref{convergence_to_Levy_measure} we define
\begin{align}
\bar
g_t^{(2)}(x)&:=\Delta^{-1/2}x^2\F^{-1}[m(-u)\F[y^{-1}e^{y-t}t\rho(t)
\1_{(-\infty,t]}(y)]
(u)](x).\label{eqbarg2}
\end{align}
Lemma~\ref{lemMultCon} can be applied to the class
$\G:=\{y^{-2}(g_s^{(2)}(y)-g_t^{(2)}(y)):s,t\le0\}$ using that $y^{-2}g_t(y)$
is uniformly bounded in the space of bounded variation functions, as observed
after \eqref{eqg2x2}.
This yields
\begin{align}
\sup_{s,t\le0}\int\left((\bar g_s^{(2)}-\bar g_t^{(2)})
-\Delta^{-1/2}(g_s^{(2)}-g_t^{(2)})\right)^2\d
\PP_\Delta\to0\label{eqComparisonqtobarq}
\end{align}
as $n\to\infty$. Therefore, \eqref{convergence_to_Levy_measure} follows from
\eqref{eqUniformityClassQ} and \eqref{eqComparisonqtobarq} if
\begin{equation}\label{eqBarTilde}
\| \bar g_t^{(2)} -
\tilde
g_t^{(2)}\|_{L^2(\PP_\Delta)}\to0
\end{equation}
uniformly in $t\le0$. To show this,
note that
\begin{align}
\bar g_t^{(2)}(x)-\tilde g_t^{(2)}(x)&=i\Delta^{-1/2}x
\F^{-1}[m'(-u)\F [y^{-2}g_t^{(2)}(y)](u)](x),\label{eqDifference}
\end{align}
for $t<0$ and $\bar g_0^{(2)}(x)=\tilde g_0^{(2)}$.
We will use the following proposition, which is an adaptation of the
pseudo-differential operator inequality Proposition~10 in \cite{NicklReiss2012}.
We denote the $L^q$-Sobolev space for $q\in(0,\infty)$ and $s\in\N$
by $W^s_q(\R):=\{f\in L^q(\R):\sum_{k=0}^s\|f^{(k)}\|_{L^q}<\infty\}$ and
define $\|f\|_{L^2(P)}:=(\int |f|^2 \d P)^{1/2}$.
\begin{prop}\label{propPseudoLocality}
Let $P$ be a probability measure with Lebesgue density $P$ and such that
$\|x^{2j+k} P\|_\infty<\infty$ for some $j,k\in\N$. Let $f\in L^2(\R)$ with
$\supp(f)\cap(-\delta,\delta)=\varnothing$ for some
$\delta>0$. Then for any $p,q\in[1,2]$, $s\in\{1,2\},$ and any
compactly supported function $\mu\in W^s_q(\R)$
\begin{align*}
\|x^{j}(\F^{-1}[\mu]*f)\|_{L^2(P)}
\lesssim \frac{\|x^{2j+k}
P\|_\infty^{1/2}}{\delta^{k/2}}\|\mu\|_{L^{2p/(2-p)}}\|f\|_{L^{p}}
+\delta^j\|\mu^{(s)}\|_{L^q}\left\|\frac{f(y)}{y^s}\right\|_{L^q}
\end{align*}
provided that the right-hand side is finite. The constant does not depend on $\mu$, $\delta$ or $f$.
\end{prop}
\begin{proof}
For $f\in L^2(\R)$ and $s=1,2$ we can show, as in \cite{NicklReiss2012}, the pseudo-differential operator identity
\begin{align*}
(\F^{-1}[\mu]*f)(x)=\left(\left(\frac{1}{(i\bull)^s}\F^{-1}\big[\mu^{(s)}\big]
\right)*f\right)(x), \quad x\notin\supp(f).
\end{align*}
Let $\delta':=\delta/2$. We use H\"older's inequality, Plancherel's identity and the Hausdorff-Young inequality to conclude
\begin{align*}
&\quad\int |x|^{2j}|\F^{-1}[\mu]*f|^2P(\d x)\\
&\le\|\F^{-1}[\mu]*f\|^2_{L^{2}} \|x^{2j}\d
P\|_{\ell^\infty([-\delta',\delta']^c
) }
+\|\F^{-1}[\mu]*f\|^2_{\ell^\infty([-\delta',\delta'])}\int_{-\delta'}^{\delta'}
|x|^{2j}P(\d x)\\
&\lesssim\|\mu\mathcal F f\|_{L^2}^2\|x^{2j+k}
P\|_{\infty}(\delta')^{-k}
+\|(x^{-s}\F^{-1}[\mu^{(s)}](x))*f\|^2_{ L^ { \infty } ( [ -\delta',\delta']
) }(\delta')^{2j}\\
&\lesssim(\delta')^{-k}\|x^{2j+k}
P\|_{\infty}\|\mu\|_{L^{2p/(2-p)}}^2\|\mathcal F f\|^2_{L^{p/(p-1)}} \\
&\qquad+(\delta')^{2j}\|\F^{-1}[\mu^{(s)}]\|_{L^{q/(q-1)}}^2\sup_{x\in[
-\delta',\delta']} \left(\int_{\R}\frac{|f(y)|^q}{|x-y|^{sq}}\d y\right)^{2/q}\\
&\lesssim (\delta')^{-k}\|x^{2j+k}
P\|_{\infty}\|\mu\|_{L^{2p/(2-p)}}^2\|f\|^2_{L^{p}}
+(\delta')^{2j}\|\mu^{(s)}\|_{L^q}^2\|f(y)/y^s\|_{L^q}^2.
\end{align*}
The result follows by taking the square root.
\end{proof}
We apply
Proposition~\ref{propPseudoLocality} with $P=\PP_\Delta$,
$\mu=m'(-\bull)$,
$f(y)=g_t^{(2)}(y)/y^2$, $\delta= |t|$, $p=1$, $q=2$, $k=1$, $j=1$ and $s=1$.
Using
$\|x^3\PP_\Delta\|_\infty\lesssim\Delta$ we estimate
\eqref{eqDifference} for $t<0$ by
\begin{align*}
&\quad\Delta^{-1}\|x\F^{-1}[m'(-u)\F
[y^{-2}g_t^{(2)}(y)](u)]\|_{L^2(\PP_\Delta)}^2\\
&\lesssim
|t|^{-1}\|m'(-\bull)\|_{L^2}^2\|y^{-2}g_t^{(2)}(y)\|_{L^1}^2+t^2\Delta^{
-1}
\|m''(-\bull)\|_{L^2}^2\|y^{-3}g_t^{(2)}(y)\|_{L^2}^2\displaybreak[0]\\
&\lesssim
|t|^{-1}\|m'(-\bull)\|_{L^2}^2\left(\int_{-\infty}^{t}y^{-1}e^{y-t}t\rho(t)
\d
y\right)^2+\Delta^{-1}
\|m''(-\bull)\|_{L^2}^2\|e^{y-t}\1_{(-\infty,t]}(y)\|_{L^2}^2\displaybreak[0
]\\
&\lesssim
\|m'(-\bull)\|_{L^2}^2\left(\int_{-\infty}^{t}|y|^{-1}e^{y-t}|t|^{1/2}
\rho(t)
\d
y\right)^2+\Delta^{-1}
\|m''(-\bull)\|_{L^2}^2\displaybreak[0]\\
&\lesssim
\|m'(-\bull)\|_{L^2}^2\left(\int_{-\infty}^{-1}e^{y+1}\d
y+|t|^{1/2}\int_{-1}^{t}|y|^{-1}\d y\1_{\{t>-1\}}\right)^2+\Delta^{-1}
\|m''(-\bull)\|_{L^2}^2\\
&\lesssim
(1+\max_{t\in[-1,0)
} |t|\log(1/|t|)^2)\|m'(-\bull)\|_{L^2}^2+\Delta^{-1}
\|m''(-\bull)\|_{L^2}^2\\
&\lesssim
\|m'(-\bull)\|_{L^2}^2+\Delta^{-1}
\|m''(-\bull)\|_{L^2}^2,
\end{align*}
which converges to zero uniformly for $t<0$ by Assumption~\ref{multass}.
Hence, tightness of the
empirical processes indexed by $\tilde g_t^{(2)}$ can be verified by
\eqref{eqEquContF} with $i=2$.
Finally, we discuss the remaining $\tilde g_t^{(3)}=\tilde g_t-\tilde
g_t^{(1)}-\tilde g_t^{(2)}$. We have
\[
g_t^{(3)}(x)=x^2g_t(x)-g_t^{(1)}(x)-g_t^{(2)}(x).
\]
We combine Lemma~\ref{lemMultCon} for $\G:=\{g_s-g_t:s,t\le0\}$, with \eqref{eqConvTildef}, \eqref{eqComparisonqtobarq} and
\eqref{eqBarTilde}
and obtain
\[
\sup_{s,t\le0}\left|\left\|\tilde g_s^{(3)}-\tilde
g_t^{(3)}\right\|_{L^2(\PP_\Delta)}-\Delta^{-1/2}\left\|g_s^{(3)}-g_t^{(3)}
\right\|_{ L^2(\PP_\Delta)}\right|\to0.
\]
Exactly as in \eqref{eqUniformityClassQ} we infer
\begin{align*}
\sup_{s,t\le0}\left|\int(g_s^{(3)}-g_t^{(3)})^2\frac{\d \PP_\Delta}{\Delta}
-\int(g_s^{(3)}-g_t^{(3)})^2\d\nu\right|\to0
\end{align*}
and thus we obtain the counterpart to \eqref{metric_comparison}
\begin{align*}
\sup_{s,t\le0}\left|\int(\tilde g_s^{(3)}-\tilde g_t^{(3)})^2\PP_\Delta
(\d x)-\int(g_s^{(3)}-g_t^{(3)})^2\d\nu\right|\to0.
\end{align*}
\subsection{Asymptotic equicontinuity for the `non-critical terms'}
We next turn to verifying the asymptotic equicontinuity condition
(\ref{eqEquContF}) for the terms $\tilde g_t^{(i)}, i \in \{1,3\}$. We refer to
them as non-critical since uniform tightness of these processes can be deduced
directly from existing bracketing metric entropy inequalities for the empirical
process.
We recall standard empirical process notation such as $\|G\|_{\mathfrak
F}:=\sup_{f\in\mathfrak F}|G(f)|$ and $\|f\|_{2,P}:=(\int |f|^2 \d P)^{1/2}$.
We denote by
$H(\eps,\mathfrak{F},\|\cdot\|)$ the logarithm of the covering number
$N(\eps,\mathfrak{F},\|\cdot\|)$ and by
$H_{[\,]}(\eps,\mathfrak{F},\|\cdot\|)$ the logarithm of the covering number
under bracketing $N_{[\,]}(\eps,\mathfrak{F},\|\cdot\|)$ (see
\cite{vanderVaartWellner1996} for definitions). For a
class of functions $\mathfrak F$ we define
\begin{align*}
\mathfrak{F}_\delta':=\{f-g:f,g\in\mathfrak{F},\|f-g\|_{2,\PP_\Delta}\le\delta
\}.
\end{align*}
We define the functions $f_t(x):=x^{-2}g_t^{(1)}(x)=(\rho(x)-e^{x-t}\rho(t)
)\1_{(-\infty,t]}(x)$ and
recall $\tilde
g_t^{(1)}(x)=\Delta^{-1/2}x^2\F^{-1}[m(-u)\mathcal F f_t(u)](x)$.
In order to show the equicontinuity condition~\eqref{eqEquContF} for $\tilde
g_t^{(1)}$ we define the corresponding classes
\begin{align*}
\tilde{\mathfrak{F}}:=\{\tilde
g_t^{(1)}:t\le0\}.
\end{align*}
We suppress in the notation the implicit dependence on $n$ through $\Delta$.
The weak derivative $D\rho$ is in $\ell^\infty(\R)$ by the Lipschitz continuity of
$\rho$. Since $\rho$ is also of bounded variation we have $D\rho\in
L^1(\R)\cap \ell^\infty(\R)\subset L^2(\R)$.
The class $\{f_t:t\le0\}$
is contained in a bounded set of the Sobolev space $W_2^1(\R)$
since the $L^2(\R)$-norms of $f_t$ and $Df_t$ are bounded.
By
boundedness of $m$ we conclude that $\F^{-1}[m(-u)\F
f_t(u)](x)$, $t\le0$, are contained in bounded subset of $W_{2}^1(\R)$,
which embeds continuously into $\ell^\infty(\R)$.
As an envelope of the class $(\tilde{\mathfrak{F}})_\delta'$ we can thus take
$F(x):=c\Delta^{-1/2}x^2$ for some $c>0$.
By Lemma~19.34 in \cite{vanderVaart1998} we have
\begin{align}\label{BracketingBound}
\E\|\sqrt{n}(\PP_{\Delta,n}-\PP_\Delta)\|_{(\tilde{\mathfrak{F}})_\delta'}
\lesssim J_{[\,]}(\delta,(\tilde{\mathfrak{F}})_\delta',L^2(\PP_\Delta)) +
\sqrt{n}\PP_\Delta F\{F>\sqrt{n}a(\delta)\},
\end{align}
where $a(\delta):=\delta/\sqrt{\log
N_{[\,]}(\delta,(\tilde{\mathfrak{F}})_\delta',L^2(\PP_\Delta)}$ and
\begin{align*}
J_{[\,]}(\delta,(\tilde{\mathfrak{F}})_\delta',L^2(\PP_\Delta)):=\int_0^\delta
\sqrt{\log
N_{[\,]}(\eps,(\tilde{\mathfrak{F}})_\delta',L^2(\PP_\Delta))}\d\eps.
\end{align*}
$\Delta^{1/2}x^{-2}(\tilde{\mathfrak{F}})_\delta'$ is contained in a bounded set
of the
Besov space $B^{s}_{22}(\R)$ for $s\le1$, which does not depend on $\Delta$ or
$\delta$.
Let $\gamma>0$ be such that $\int| x |^{4+2\gamma}\nu(\d x)<\infty$. We take
$s\in(1/2,1/2+\gamma)$.
The proof of Theorem~1 in \cite{NicklPoetscher2007} with $p=2$, $q=2$ and
$\beta=0$ yields
\begin{align*}
H(\eps,\Delta^{1/2}x^{-2}(\tilde{\mathfrak{F}})_\delta',\|\cdot\langle x
\rangle^{-\gamma}\|_\infty)\lesssim \eps^{-1/s},
\end{align*}
where $\langle
x \rangle:=(1+x^2)^{1/2}$.
The entropy can be rewritten as
$H(\eps,\Delta^{1/2}x^{-2}(\tilde{\mathfrak{F}})_\delta',\|\cdot\langle x
\rangle^{-\gamma}\|_\infty)=H(\eps,\Delta^{1/2}(\tilde{\mathfrak{F}}
)_\delta',\|\cdot x^{-2}\langle x
\rangle^{-\gamma}\|_\infty)$.
A ball in the $\|\cdot x^{-2}\langle x
\rangle^{-\gamma}\|_\infty$-norm with centre $f$ and radius $\eps$ is a
bracket
\begin{align*}
[f-\eps x^2 \langle x \rangle^{\gamma},f+\eps x^2\langle x \rangle^{\gamma}],
\end{align*}
whose $L^2(\PP_\Delta)$-size is given by $\|2\eps x^2\langle x
\rangle^{\gamma}\|_{2,\PP_\Delta}$. Consequently we have
\begin{align*}
H_{[\,]}(\eps\|2x^2\langle x
\rangle^{\gamma}\|_{2,\PP_\Delta},\Delta^{1/2}(\tilde{\mathfrak{F}})_\delta',
L^2(\PP_\Delta))\le
H(\eps,\Delta^{1/2}(\tilde{\mathfrak{F}})_\delta',\|\cdot x^{-2}\langle x
\rangle^{-\gamma}\|_\infty)\lesssim \eps^{-1/s}.
\end{align*}
By Theorem~1.1 in \cite{Figueroa-Lopez2008} (see also \cite{FLH09}) we have
$\Delta^{-1/2}2\eps\|x^2\langle
x
\rangle^{\gamma}\|_{2,\PP_\Delta}\to2\eps\|x^2\langle x
\rangle^{\gamma}\|_{2,\nu}$ as $n\to\infty$. We obtain by a rescaling that
\begin{align*}
H_{[\,]}(\eps,(\tilde{\mathfrak{F}})_\delta',
L^2(\PP_\Delta))
\lesssim \eps^{-1/s}.
\end{align*}
Taking $s>1/2$ we conclude that the entropy integral
$J_{[\,]}(\delta,(\tilde{\mathfrak{F}})_\delta',L^2(\PP_\Delta))$ is finite and
tends to zero as $\delta\to0$.
To show that the left hand side of \eqref{BracketingBound} tends to zero, we
first ensure that the entropy integral is small by choosing $\delta>0$. Upon
fixing $\delta$ and thus for fixed $a(\delta)$ bounded away from zero uniformly
in $\Delta$, we choose $n$ large enough
such that the second term is small.
We recall that we have taken the
envelopes to be $F(x)=c\Delta^{-1/2}x^2$.
We bound
\begin{align*}
\sqrt{n}\PP_\Delta F\{F>\sqrt{n}a(\delta)\}
&\lesssim
\sqrt{n}\Delta^{-1/2}\int
x^2\1_{\{x^2>\sqrt{n\Delta}\,a(\delta)/c\}}\PP_\Delta(\d x)\\
&\lesssim \Delta^{-1}\int
x^4\1_{\{x^2>\sqrt{n\Delta}\,a(\delta)/c\}}\PP_\Delta(\d x),
\end{align*}
where we multiplied by $cx^2/(\sqrt{n\Delta}\,a(\delta))>1$.
For $M$ large enough
$\int x^4 \1_{\{x^2>M\}}\nu (\d x)$ is small.
Since $\Delta^{-1}\int x^4 \1_{\{x^2>M\}}\PP_\Delta(\d x)\to\int x^4
\1_{\{x^2>M\}}\nu(\d x)$ by Theorem~1.1 in \cite{Figueroa-Lopez2008},
$n\Delta\to\infty$ as $n\to\infty$ and $a(\delta)$ is bounded away from zero, we
have that
$\Delta^{-1}\int x^4\1_{\{x^2>\sqrt{n\Delta}\,a(\delta)/c\}}\PP_\Delta(\d x)$
is small for $n$ large enough.
So indeed the left hand of \eqref{BracketingBound} tends to zero as
$\delta\to0$ and $n\to\infty$ and we have shown tightness of the empirical
process indexed by $\{\tilde g^{(1)}_t:t\le0\}$.
Let us now consider the terms associated to
\begin{align*}
\tilde g_t^{(3)}(x)&=\tilde g_t(x)-\tilde g_t^{(1)}(x)-\tilde
g_t^{(2)}(x)\\
&=\Delta^{-1/2}x^2\F^{-1}[m(-u)\F [e^{y-t}\rho(t)
\1_{(-\infty,t]}(y)](u)](x)\\
&\quad-\Delta^{-1/2}x\F^{-1}[m(-u)\F[te^{y-t}
\rho(t)\1_{(-\infty,t]}(y)](u)](x)\\
&=i\Delta^{-1/2}x\F^{-1}[m'(-u)\F [e^{y-t}\rho(t)\1_{(-\infty,t]}(y)](u)](x)\\
&\quad+\Delta^{-1/2}x\F^{-1}[m(-u)\F[(y-t)e^{y-t}
\rho(t)\1_{(-\infty,t]}(y)](u)](x).
\end{align*}
The functions $(y-t)e^{y-t}
\rho(t)\1_{(-\infty,t]}(y)$ are uniformly for all $t\le0$ bounded in $L^2(\R)$
and likewise are their weak derivatives. We conclude that they are contained in
a bounded set of $B^1_{22}(\R)$. The functions
$e^{y-t}\rho(t)\1_{(-\infty,t]}(y)$,
$t\le0$, are contained in a bounded set of
$L^2(\R)$.
Assumption \ref{multass} implies, together with the Mikhlin Fourier multiplier
theorem (e.g., Corollary 4.11 in \cite{girardiWeis2003}), that $m$ is a Fourier
multiplier on every Besov space $B^s_{pq}(\R)$, $s\in\R$, $p,q\in[1,\infty]$,
and, moreover, that
$m'$ is a Fourier multiplier mapping $B^s_{pq}(\R)$ into
$B^{s+1}_{pq}(\R)$.
We see that $\Delta^{1/2}x^{-1}\tilde g^{(3)}_t(x)$, $t\le0$, are contained in a
bounded set of $B^1_{22}(\R)$. We define the class $\tilde{\mathcal
G}:=\{\tilde g^{(3)}_t:t\le0\}$.
As an envelope of the class $(\mathcal {\tilde G})_{\delta}'$ we
can take $G(x):=c\Delta^{-1/2}x$ for some constant $c>0$.
Lemma~19.34 in \cite{vanderVaart1998} yields
\begin{align}\label{BracketingBoundH}
\E\|\sqrt{n}(\PP_{\Delta,n}-\PP_\Delta)\|_{(\tilde{\mathcal{G}})_\delta'}
\lesssim J_{[\,]}(\delta,(\tilde{\mathcal{G}})_\delta',L^2(\PP_\Delta)) +
\sqrt{n}\PP_\Delta G\{G>\sqrt{n}a(\delta)\}.
\end{align}
Again by the proof of
Theorem~1 in \cite{NicklPoetscher2007} with $s=1$, $p=2$, $q=2$,
$\beta=0$ and $\gamma=1$ we have
\begin{align*}
H(\eps,\Delta^{1/2}x^{-1}(\tilde{\mathcal G})_{\delta}',\|\cdot
\langle x\rangle^{-1}\|_\infty)\lesssim \eps^{-1}.
\end{align*}
The entropy can be rewritten as $H(\eps,\Delta^{1/2}(\tilde{\mathcal
G})_{\delta}',\|\cdot
x^{-1}\langle x\rangle^{-1}\|_\infty)$.
A corresponding $\eps$ ball is in the $L^2(\PP_\Delta)$-norm of size
$2\eps\|x\langle x \rangle\|_{2,\PP_\Delta}$.
By Theorem~1.1 in
\cite{Figueroa-Lopez2008} we have
$\Delta^{-1/2}\|x\langle x\rangle\|_{2,\PP_\Delta}\to\|x\langle x
\rangle\|_{2,\nu}+\sigma^2$ as $n\to\infty$.
Arguing as for $\tilde{\mathfrak{F}}$ we obtain
\begin{align*}
H_{[\,]}(\eps,(\tilde{\mathcal G})_\delta',L^2(\PP_\Delta))\lesssim
\eps^{-1}.
\end{align*}
The entropy integral in \eqref{BracketingBoundH} is finite and converges to
zero
as $\delta\to0$. The second term $\sqrt{n}\PP_\Delta G\{G>\sqrt{n}a(\delta)\}$
can be treated exactly as the second term in \eqref{BracketingBound} with
$x^2$ replaced by $x$.
So the
$\lim_{\delta\to0}\limsup_{n\to\infty}$ of \eqref{BracketingBoundH} is zero and
thus
\eqref{eqEquContF} follows for the functions $\tilde g^{(3)}_t$.
\subsection{Asymptotic equicontinuity of the `critical term'}
It remains to show asymptotic equicontinuity of the empirical process indexed by
the class
\begin{align*}
Q_n&:=\{\tilde g_t^{(2)}:t\le0\},
\end{align*}
where we recall from \eqref{eqtildeg2} that
\begin{equation}
\tilde
g_t^{(2)}(x)=\Delta^{-1/2}x(\F^{-1}[m(-u)]\ast q_t)(x),~~
q_t(y):=t\rho(t)e^{y-t}\1_{(-\infty,t]}(y).\label{eqQ}
\end{equation}
We refer to this term as `critical': the functions $q_t$ contain a
step-discontinuity at $t$ and controlling its interaction with the operator
$\F^{-1}[m(-\cdot)]$ needs some more elaborate techniques than in the previous
section.
We will rely on the following auxiliary result, which is a modification of
Theorem 3 in \cite{GineNickl2008}, which in itself goes back to fundamental
ideas in \cite{GineZinn1984}. It is designed to allow for
maximally growing envelopes of the empirical process, which is crucial in our
setting to allow for minimal conditions on $\Delta$. Note that indeed
Condition~\eqref{envelopes} only requires $M_n/n^{1/2}\to0$ instead of the more
stringent condition $M_n/n^{1/4} \to 0$ which was required in Theorem~3 in
\cite{GineNickl2008}.
\begin{proposition}
For every $n \in \N,$ let $X_{n,j}, j=1, \dots, n,$ be i.i.d.~from law $P_n$ on a
measurable space $(S, \mathcal B)$ and let $\eps_j$, $j=1,\dots,n,$ be i.i.d.
Rademacher random variables independent of the $X_{n,j}$'s, all defined on a
common probability space $(\Omega, \mathcal A, \Pr)$. For any sequence
$(\mathcal Q_n)_{n\ge 1}$ of classes of measurable functions $q: S \to \R$ and
\begin{equation*}
(\mathcal Q_n)'_{r}:=\{q-q':q,q'\in\mathcal Q_n,
\|q-q'\|_{2,P_n}\le r_n\}, n \in \N,
\end{equation*}
suppose the following conditions are satisfied for some sequence $r_n\to0$ as
$n\to\infty$
\begin{enumerate}[(a)]
\item\label{envelopes} $\sup_{q\in\mathcal Q_n}\|q\|_\infty\le M_n$ for a sequence $M_n$ such that $n r_n^{2}{M_n}^{-2}\to\infty$.
\item\label{r_increments} \begin{equation*}
\left\|\frac{1}{\sqrt{n}}\sum_{j=1}^n\eps_j q(X_{n,j})\right\|_{(\mathcal
Q_n)'_{r}} = o_P(1)
\end{equation*}
as $n\to\infty$.
\item\label{entropy_bound} There exists $n_0\in\N$ such that for all $n\ge n_0$
\begin{equation*}
23 H(r_n,\mathcal Q_n,L^2(P_n))\le n r_n^{2}{M_n}^{-2}.
\end{equation*}
\item\label{uniform_entropy_integrals} \begin{equation*}
\lim_{\delta\to0}\limsup_{n\to\infty}\int_0^\delta
\sqrt{H( \eps,\mathcal Q_n, L^2(P_n))}\d\eps=0
\end{equation*}
\end{enumerate}
Then for all $\gamma>0$
\begin{align*}
\lim_{\delta\to0}\limsup_{n\to\infty}\Pr \left(\sup_{q,q' \in \mathcal Q_n: \|q-q'\|_{2,P_n}\le\delta}
\left|\sqrt n\int_S (q- q')\left(\frac{1}{n}\sum_{j=1}^n \delta_{X_{n,j}}-P_n\right)(\d x)\right|>\gamma\right)=0.
\end{align*}
\end{proposition}
\begin{proof}
Let $\gamma >0$ be given. We sometimes omit to mention $q,q' \in \mathcal Q$ to
expedite notation. By Lemma~11.2.6 in \cite{dudley1999} we have for
$\delta\in(0,\gamma/\sqrt{2})$
\begin{align*}
&{\Pr}\left(\sup_{\|q-q'\|_{2,P_n}\le\delta}
\frac{1}{\sqrt{n}}\left|\sum_{j=1}^{n}
q(X_{n,j})-q'(X_{n,j})-\E[q(X_{n,j})]+\E[q'(X_{n,j})]\right|>\gamma\right)\\
&\le4{\Pr}\left(\sup_{ \|q-q'\|_{2,P_n}\le\delta}
\frac{1}{\sqrt{n}}\left|\sum_{j=1}^{n}\eps_j(q(X_{n,j})-q'(X_{n,j}
))\right|>\frac{\gamma-\sqrt{2}\delta}{2}\right),
\end{align*}
where $\eps_j$ are Rademacher random variables independent of the $(X_{n,j})$,
all defined on a large product probability space.
Since $\gamma$ is given and
$\delta$ tends to zero, we can choose $\delta$ small enough such that
$\delta<\gamma/2$. Hence it suffices to show for all $\gamma>0$ that
\begin{align*}
\lim_{\delta\to0}\limsup_{n\to\infty}{\Pr}\left(\sup_{ q\in(\mathcal
Q_n)'_\delta}
\frac{1}{\sqrt{n}}\left|\sum_{j=1}^{n}\eps_j
q(X_{n,j})\right|>3\gamma\right)=0.
\end{align*}
Let $\mathcal H=\mathcal H_n$ be a maximal collection of functions
$h_1,\dots,h_m$ in $\mathcal Q_n$ such that $\|h_j-h_k\|_{2,\PP_\Delta}>r_n$ if
$j\neq k$. The closed balls with centres $h_1,\dots,h_m$ of radius $r_n$ cover
$\mathcal Q_n$.
We define \[\mathcal H'_\delta:=\{g-h:g,h\in \mathcal H,
\|g-h\|_{2,P_n}\le \delta\}.\] For $n$ large enough such that
$r_n<\delta/2$ we have
\begin{align}
&{{\Pr}}\left(\sup_{q\in(\mathcal Q_n)'_\delta}
\frac{1}{\sqrt{n}}\left|\sum_{j=1}^{n}\eps_j
q(X_{n,j})\right|>3\gamma\right)\notag\\
&\le 2{\Pr}\left(\sup_{q\in(\mathcal Q_n)'_r}
\frac{1}{\sqrt{n}}\left|\sum_{j=1}^{n}\eps_j
q(X_{n,j})\right|>\gamma\right)+{\Pr}\left(\max_{h\in \mathcal H_{2\delta}'}
\frac{1}{\sqrt{n}}\left|\sum_{j=1}^{n}\eps_j
h(X_{n,j})\right|>\gamma\right).\label{eqGridH}
\end{align}
By condition~\eqref{r_increments} the first term tends to zero.
To control the
second term we define the event
\begin{equation*}
A_n:=\left\{\max_{h\in\mathcal
H_{2\delta}'\backslash\{0\}}\frac{\sum_{j=1}^n
h^2(X_{n,j})}{nP_n h^2}<2\right\},
\end{equation*}
where we used the notation $P_n f:=\int_S f \d P_n$ for functions $f:S\to\R$.
Using Markov's inequality the second term in \eqref{eqGridH} can be bounded by
\begin{align}
&{\Pr}\left(\max_{h\in \mathcal H_{2\delta}'}
\frac{1}{\sqrt{n}}\left|\sum_{j=1}^{n}\eps_j
h(X_{n,j})\right|>\gamma\right)\notag\\
&\le
{\Pr}(A_n^c)+\frac{1}{\gamma}\E_X\E_\eps\left[\left\|\frac{\sum_{j=1}^{n}\eps_j
h(X_{n,j})}{\sqrt{n}} \right\|_{\mathcal H_{2\delta}'}\1_ { A_n } \right
].\label{conditioned_on_A}
\end{align}
The number of elements in $\mathcal H'_{2\delta}$ is bounded by
\begin{equation}\label{size_H}
\#\mathcal H'_{2\delta}\le \exp(2 H(r_n,\mathcal Q_n,L^2(P_n))).
\end{equation}
For a single $h\in\mathcal H_{2\delta}'\backslash\{0\}$ we have, using
Bernstein's inequality,
\begin{align*}
{\Pr}\left(\frac{\sum_{j=1}^n
h^2(X_{n,j})}{nP_n h^2}\ge2\right)
&={\Pr}\left(\sum_{j=1}^{n}(h^2(X_{n,j})-P_n h^2)\ge n P_n
h^2\right)\\
&\le \exp\left(-\frac{n^2 (P_n h^2)^2}{2n P_n h^4+8M_n^2
n P_n h^2/3}\right)\\
&\le\exp\left(-\frac{nP_n h^2}{11 M_n^2}\right).
\end{align*}
Combining the last bound and \eqref{size_H} we obtain
\begin{align*}
\Pr(A_n^c)\le\exp\left(2 H(r_n,\mathcal Q_n,L^2(P_n))-\frac{n r_n^2}{11
M_n^2}\right)\to0
\end{align*}
by condition~\eqref{envelopes} and~\eqref{entropy_bound}.
It remains to show that the second term in~\eqref{conditioned_on_A} converges
to zero. Conditional on the $X_{n,j}$'s the process
\begin{equation*}
Z(h):=\frac{1}{\sqrt{n}}\sum_{j=1}^{n}\eps_j h(X_{n,j}), \quad h\in\mathcal H,
\end{equation*}
is subgaussian. Let $h,h'\in\mathcal H$ such that $h-h'\in\mathcal
H_{2\delta}'\backslash\{0\}$. On the event $A_n$ we have
\begin{align*}
d_Z(h,h')^2&:=\E_\eps\left[\left(\frac{1}{\sqrt{n}}\sum_{j=1}
^n\eps_j h(X_{n,j})-\frac{1}{\sqrt{n}}\sum_{j=1}
^n\eps_j
h'(X_{n,j})\right)^2\right]\\
&=\frac{1}{n}\sum_{j=1}^n(h(X_{n,j})-h'(X_{n,j}
))^2<2 P_n(h-h')^2.
\end{align*}
Especially we have on $A_n$ that $\|h-h'\|_{2,P_n}<\eps$ implies
$d_Z(h,h')<\sqrt{2}\eps$ for all $\eps>0$ with $\eps\le2\delta$ and for all
$h,h'\in\mathcal H$.
We define
$\psi_2(x):=\exp(x^2)-1$ and the norm
$\|\xi\|_{\psi_2}:=\inf\{c>0:\E[\psi_2(|\xi|/c)]\le1\}$. By~(4.3.3) in
\cite{delaPenaGine1999} there is a constant $c>0$ such that
$\E[|\xi|]\le c \|\xi\|_{\psi_2}$. So we obtain the bound
\begin{align*}
\E_\eps\left[\left\|\frac{\sum_{j=1}^{n}\eps_j
h(X_{n,j})}{\sqrt{n}} \right\|_{\mathcal H_{2\delta}'}\1_ { A_n } \right
]\le c \left\|\sup_{d_Z(h,h')<2\sqrt{2}\delta}\left|\frac{\sum_{j=1}^{n}\eps_j
(h(X_{n,j})-h'(X_{n,j}))}{\sqrt{n}} \right| \1_{A_n} \right\|_{\psi_2}.
\end{align*}
Next we apply Dudley's theorem in the form of Corollary~5.1.6 and Remark~5.1.7
in \cite{delaPenaGine1999} to the process $Z$. This yields a constant $K$ such
that
\begin{align*}
\E_\eps\left[\left\|\frac{\sum_{j=1}^{n}\eps_j
h(X_{n,j})}{\sqrt{n}} \right\|_{\mathcal H_{2\delta}'}\right
]\1_ { A_n }
&\le
K \int_0^{2\sqrt{2}\delta}\left(\log(N(\eps,\mathcal
H_n,d_Z))\right)^{1/2}\d\eps \1_{A_n}\\
&\le
K \int_0^{2\sqrt{2}\delta}\left(\log(N(\eps/\sqrt{2},\mathcal
H_n,L^2(P_n)))\right)^{1/2}\d\eps \1_{A_n},\\
&\le\sqrt{2}
K \int_0^{2\delta}\left(\log(N(\eps,\mathcal Q_n,L^2(P_n)))\right)^{1/2}\d\eps,
\end{align*}
a bound independent of $X$. In order to complete the proof we take expectation with respect to $X$,
consider
the limit $\lim_{\delta\to0}\limsup_{n\to\infty}$ of the expression and
apply condition~\eqref{uniform_entropy_integrals}.
\end{proof}
\medskip
To proceed with the tightness proof for the critical term we will show conditions~\eqref{envelopes}
to~\eqref{uniform_entropy_integrals} for
$r_n:=\log(1/\Delta_n)^{-\alpha}$, $\alpha\in(1/2,1)$, and for the class $
Q_n=\{\tilde
g_t^{(2)}:t\le0\}$ defined above.
\eqref{envelopes}
We rewrite
\begin{align}
\tilde
g^{(2)}_t&=\Delta^{-1/2}i\F^{-1}[m'(-u)\F[t\rho(t)e^{y-t}\1_{(-\infty,t]}(y)](u)
] \notag\\
&\quad+\Delta^{-1/2}\F^{-1}[m(-u)\F[t\rho(t)ye^{y-t}\1_{(-\infty,t]}(y)](u)]
\notag\displaybreak[0]\\
&=\Delta^{-1/2}i\F^{-1}[m'(-u)\F[t\rho(t)e^{y-t}\1_{(-\infty,t]}(y)](u)]\notag\\
&\quad+\Delta^{-1/2}\F^{-1}[m(-u)\F[t\rho(t)(y-t)e^{y-t}\1_{(-\infty,t]}(y)](u)]
\label{eqDecomposition}\\
&\quad+\Delta^{-1/2}\F^{-1}[m(-u)\F[t^2\rho(t)e^{y-t}\1_{(-\infty,t]}(y)](u)],
\notag
\end{align}
where the last step also shows that the bounded variation norm of
\(
t\rho(t)ye^{y-t}\1_{(-\infty,t]}(y)
\)
is bounded uniformly in $t\le0$.
If $\F^{-1}[m]$, $\F^{-1}[m']$ are finite signed measures as in
Assumption~\ref{multass}(a), then the bounded
variation norms of
$\F^{-1}[m'(-\bull)]*(t\rho(t)e^{y-t}\1_{(-\infty,t]}(y))$ and
$\F^{-1}[m(-\bull)]*(t\rho(t)ye^{y-t}\1_{(-\infty,t]}(y))$ are bounded uniformly
in
$t\le0$
and
\begin{align*}
\|\tilde g^{(2)}_t\|_\infty\le\|\tilde g^{(2)}_t\|_{BV}\lesssim \Delta^{-1/2},
\end{align*}
where $\|f\|_{BV}$ denotes the bounded variation
norm equal to the sum of the $\ell^\infty$-norm of $f$ and the usual total
variation norm of the weak derivative $Df$.
For $m$ supported in $[-C\Delta^{-1/2},C\Delta^{-1/2}]$ as in
Assumption~\ref{multass}(b),
we have
\begin{equation*}
\|\tilde g^{(2)}_t\|_\infty\lesssim \|\tilde
g^{(2)}_t\|_{B^0_{\infty,1}}\lesssim
\|\tilde g^{(2)}_t\|_{B^1_{1,1}}
\end{equation*}
and
the Fourier transform of $\tilde g^{(2)}_t$ is supported on
$[-C\Delta^{-1/2},C\Delta^{-1/2}]$. In view of the
Littlewood-Paley definition of Besov spaces we can
estimate the $B^1_{11}(\R)$-norm of $\tilde g^{(2)}_t$ by
$\log(C/\Delta^{1/2})$-times its $B^1_{1\infty}(\R)$-norm. With the Fourier
multiplier
property of $m$ and $m'$ this yields
\begin{align*}
\|\tilde g^{(2)}_t\|_\infty &\lesssim
\log(C/\Delta^{1/2})\|\tilde g^{(2)}_t\|_{B^1_{1\infty}}\\
&\lesssim
\Delta^{-1/2}\log(1/\Delta)(\|t\rho(t)e^{y-t}\1_{(-\infty,t]}(y)\|_{B^1_{1\infty
}}+\|t\rho(t)ye^{y-t}\1_{(-\infty,t]}(y)\|_{B^1_{1\infty}})\\
&\lesssim\Delta^{-1/2}\log(1/\Delta),
\end{align*}
since the $B^1_{1\infty}(\R)$-norm of
$t\rho(t)e^{y-t}\1_{(-\infty,t]}(y)$ and $t\rho(t)ye^{y-t}\1_{(-\infty,t]}(y)$
are uniformly in $t$ bounded by
integrability and
bounded variation.
So $M_n$ can be chosen proportional to $\Delta^{-1/2}\log(1/\Delta)$ and
$nr_n^2M_n^{-2}\to\infty$ by $\log^4(1/\Delta)=o(n\Delta)$.
\eqref{r_increments} We will show condition~\eqref{r_increments} by applying a moment inequality for empirical processes under uniform
entropy bounds for $Q_n$.
We decompose $\tilde g^{(2)}_t$ according to \eqref{eqDecomposition}. Using
that $B_{11}^{1}(\R)$ embeds continuously into the space $\text{BV}$ of bounded
variation functions,
the bounds in \eqref{envelopes} show that
\begin{align}
\|\Delta^{-1/2}\F^{-1}[m'(-u)\F[e^{y-t}\1_{(-\infty,t]}(y)](u)]\|_{BV}
&\lesssim
\Delta^{-1/2}\log(1/\Delta),\notag\\
\|\Delta^{-1/2}\F^{-1}[m(-u)\F[(y-t)e^{y-t}\1_{(-\infty,t]}(y)](u)]\|_{BV}
&\lesssim
\Delta^{-1/2}\log(1/\Delta),\label{BV_bound}\\
\|\Delta^{-1/2}\F^{-1}[m(-u)\F[e^{y-t}\1_{(-\infty,t]}(y)](u)]\|_{BV}
&\lesssim
\Delta^{-1/2}\log(1/\Delta),\notag
\end{align}
where we omitted the factors $t\rho(t)$ and $t^2\rho(t)$ to obtain
translation invariant
classes. Since the functions in the class
\begin{align}
\mathfrak{F}_n:=\{\Delta^{-1/2}\F^{-1}[m'(-u)\F[e^{y-t}\1_{(-\infty,t]}(y)](u)]
:t\le0\}
\end{align}
are of bounded variation, we can write
them as the composition of a 1-Lipschitz function after a nondecreasing
function.
The class of all translates of a nondecreasing function has VC index~2 and thus
polynomial $L^2(\QQ)$-covering numbers uniformly in all probability measures
$\QQ$ by Theorem~5.1.15 in \cite{delaPenaGine1999}. The $\eps$-covering numbers
are preserved under 1-Lipschitz transformations and thus the covering
numbers of $\mathfrak F_n$ are polynomial in
$M_n/\eps$. The $\eps$-covering numbers of $\{t\rho(t):t\in\R\}$
are
polynomial in $1/\eps$. To obtain an $\eps$-covering of the functions in the
first term of \eqref{eqDecomposition} we cover
the class
$\mathfrak{F}_n$ by balls of size $\eps/2$ and the class
$\{t\rho(t):t\in\R\}$ by balls of size $\eps/(2M_n)$. We see that the
covering numbers can be bounded by a product of two
polynomial
covering numbers and thus are polynomial in $M_n/\eps$.
Arguing in the same way for the two other terms in \eqref{eqDecomposition}
yields polynomial covering numbers for them, too.
Using that the covering numbers of $Q_n$ can be bounded by the product of the
covering numbers for the respective terms we see that the covering numbers of
$Q_n$ are polynomial in $M_n/\eps$.
By Proposition~3 in
\cite{GineNickl2009} there exists a universal constant $L>0$ such that
\begin{align*}
\E \left\|\frac{1}{\sqrt{n}}\sum_{j=1}^n\eps_j
q(X_{n,j})\right\|_{(Q_n)'_{r}}\le L
\max\left(r_n\sqrt{\log\left(\frac{M_n}{r_n}\right)},\frac{M_n}{\sqrt{n}}
\log\left(\frac{M_n}{r_n}\right)\right).
\end{align*}
Condition \eqref{r_increments} is satisfied if this maximum tends to zero.
We have
\begin{align*}
r_n\sqrt{\log\left(\frac{M_n}{r_n}\right)}
\lesssim\frac{\sqrt{\log\left(\log(1/\Delta)^{1+\alpha}/\Delta^{1/2}\right)}}{
\log(1/\Delta)^\alpha}
\lesssim\frac{\sqrt{\log(\log(1/\Delta))}}{\log(1/\Delta)^\alpha}+\frac{
\sqrt { \log(1/\Delta)}}{\log(1/\Delta)^\alpha}\to0,
\end{align*}
and
\begin{align*}
\frac{M_n}{\sqrt{n}}
\log\left(\frac{M_n}{r_n}\right)
&\lesssim\frac{\log(1/\Delta)}{\sqrt{\Delta n}}
\log\left(\frac{\log(1/\Delta)^{1+\alpha}}{\Delta^{1/2}}\right)\\
&\lesssim\frac{\log(1/\Delta)\log(\log(1/\Delta))}{\sqrt{\Delta n}}
+\frac{\log(1/\Delta)^2}{\sqrt{\Delta n}},
\end{align*}
which tends to zero by $\log^4(1/\Delta)=o(n\Delta)$.
\eqref{entropy_bound}
In order to verify \eqref{entropy_bound}, we will show that
$H(\eps,Q_n,L^2(\PP_\Delta)\lesssim
\log(\eps^{-1})$ uniformly in $n$.
Applying Proposition~\ref{propPseudoLocality}
with $j=k=1$, $p=q=2$ and $s=2$ yields
that for $\mu=m(-\bull)$
and for
all $f\in L^2(\R)$ with $\supp (f)\cap (-\delta,\delta)=\varnothing$ for some
$\delta>0$
\begin{align}
\Delta^{-1/2}\|x(\F^{-1}[m(-u)]*f)\|_{2,\PP_\Delta}
&\lesssim\left(
\delta^{-1/2}\|m\|_\infty+\delta^{-1}\Delta^{-1/2}\|m''\|_{L^2}\right)\|f\|_
{L^2}\notag\\
&\lesssim \left(\delta^{-1/2}\vee
\delta^{-1}\right)\|f\|_{L^2}\label{eqApplProp}
\end{align}
where we used $\|m\|_\infty\le C$ and $\Delta^{-1/2}\|m''\|_{L^2}\to0$ by
Assumption~\ref{multass}.
Let $M\ge1$ and $\eta\in[0,1]$.
We will distinguish the three cases $s,t\le-M$, $s,t\in[-M,-\eta]$ and
$s,t\in[-\eta,0]$.
\noindent
\textbf{Case 1:}
Let $s,t\le-M$. We apply \eqref{eqApplProp} to $\delta=M$ and
$f(y):=q_s(y)-q_t(y)$ with $q_t$ defined in \eqref{eqQ}. Noting that we can
bound $\|q_t\|_{L^2}\lesssim
M^{-1}$ uniformly in $t\le0$, we obtain for $s,t\le-M$
\begin{align*}
\|\tilde g_s^{(2)} -\tilde g_t^{(2)}\|_{2,\PP_\Delta}\lesssim M^{-3/2}.
\end{align*}
\noindent
\textbf{Case 2:}
For the second case let $-M\le s,t\le-\eta$.
We apply \eqref{eqApplProp} with $\delta=\eta$ to $f(y):=q_s(y)-q_t(y)$.
Without loss of generality we assume $s\le t$. We estimate
\begin{align*}
& \int_{-\infty}^{t}(q_s(y)-q_t(y))^2\d y\notag\\
&=\int_{-\infty}^{0}\left(s\rho(s)e^{y}-t\rho(t)e^{y+s-t}\right)^2\d
y+\int_{s}^{t}t^2\rho(t)^2e^{2(y-t)}\d
y\notag\\
&\le 2\left(s\rho(s)-t\rho(t)\right)^2\int_{-\infty}^0e^{2y}\d
y+2t^2\rho(t)^2\int_{-\infty}^0(1-e^{s-t})^2e^{2y}\d y
+t^2\rho(t)^2|s-t|\notag\\
&\lesssim |s-t|^2+|s-t|
\end{align*}
by the Lipschitz continuity of $x\rho$ and
obtain for $s,t\in[-M,-\eta]$ with $|s-t|\le1$
\begin{align*}
\|\tilde g_s^{(2)} -\tilde g_t^{(2)}\|_{2,\PP_\Delta}\lesssim
|s-t|^{1/2}/\eta.
\end{align*}
\noindent
\textbf{Case 3:}
Let $-\eta\le s,t\le0$. We have $\|\tilde g_s^{(2)} -\tilde
g_t^{(2)}\|_{2,\PP_\Delta}\le2\sup_{t\in[-\eta,0]}\|\tilde
g_t^{(2)}\|_{2,\PP_\Delta}$. We apply Proposition~\ref{propPseudoLocality} with
$f=q_t$, $\mu=m(-\bull)$,
$\delta=|t|$, $k=1$, $j=1$, $p=2$, $q=2$ and $s=2$. We have
\begin{align*}
|t|^{-1}\|q_t\|_{L^2}^2&=|t|\rho(t)^2\int_{-\infty}^{0}e^{2y}\d y\lesssim
|t|\quad\text{and}\\
t^2\left\|q_t(y)/y^2\right\|_{L^2}^2&\le\int_{-\infty}^t\frac{t^4}{y^4}\d
y=|t|\int_{-\infty}^{-1}x^{-4}\d x\lesssim |t|.
\end{align*}
and consequently
$\|\tilde g_s^{(2)} -\tilde
g_t^{(2)}\|_{2,\PP_\Delta}\lesssim\eta^{1/2}$ for $-\eta\le s,t\le0$.
Having treated these three cases we can show $N(\eps,
Q_n,L^2(\PP_\Delta))\lesssim \eps^{-7}$. For an integer $J>0$ we
consider the grid of points $t_j=-j J^{-6}$ with $j=J^4, J^4+1, J^4+2,\dots,
J^{7}$. We take $\eta=J^{-2}$. By Case~3 we see that $\|\tilde
g_s^{(2)}-\tilde
g_t^{(2)}\|_{2,\PP_\Delta}\lesssim J^{-1}$ for all $s,t\in[-J^{-2},0]$. By
Case~2 we
have
$\|\tilde g_s^{(2)}-\tilde
g_t^{(2)}\|_{2,\PP_\Delta}\lesssim|s-t|^{1/2}/\eta\le J^{-1}$ for
$s,t\in[-(j+1)J^{-6},-jJ^{-6}]$. And by Case~1 $\|\tilde g_s^{(2)}-\tilde
g_t^{(2)}\|_{2,\PP_\Delta}\lesssim J^{-1}$ for $s,t\le-J$.
We have polynomial covering numbers and it suffices
for condition~\eqref{entropy_bound} that
\begin{align*}
\frac{nr_n^2}{M_n^2\log\left({r_n}^{-1}\right)}\to\infty.
\end{align*}
In \eqref{envelopes} we have seen that $M_n\lesssim
\Delta^{-1/2}\log(1/\Delta)$.
For the choice $r_n=\log(1/\Delta)^{-\alpha}$ we obtain
\begin{align*}
\frac{n
r_n^2}{M_n^2\log(1/r_n)}\gtrsim\frac{n\Delta}{
\log(1/\Delta)^{2+2\alpha}\log(\log(1/\Delta)) },
\end{align*}
which tends to infinity by $\log^4(1/\Delta)=o(n\Delta)$.
\eqref{uniform_entropy_integrals} In \eqref{entropy_bound} we have seen that
the covering numbers $N(\eps,Q_n,L^2(\PP_\Delta))$ are uniformly in
$n$ polynomial in $\eps^{-1}$ so that the condition is satisfied.
\smallskip
\textbf{Acknowledgement.} The authors acknowledge insightful remarks from the
Associate Editor and two anonymous referees that helped to improve the
presentation of the paper.
Financial Support by the Deutsche
Forschungsgemeinschaft via FOR 1735 Structural Inference in Statistics is
gratefully acknowledged.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,884 |
Q: animation not positioned correctly I'm trying to do an animation with the next code:
.slogan {
width: 73.5%;
color: black;
}
.left-slogan {
font-size: 7vw;
font-weight: bold;
}
.left-slogan > p {
line-height: 0;
}
.left-slogan {
text-align: right;
margin-right: 24px;
}
.right-slogan span {
position: absolute;
opacity: 0;
overflow: hidden;
color: black;
font-size: 7vw;
font-weight: bold;
-webkit-animation: rotateWord 15s linear infinite 0s;
-ms-animation: rotateWord 15s linear infinite 0s;
animation: rotateWord 15s linear infinite 0s;
}
.right-slogan span:nth-child(1) {
-webkit-animation-delay: 0s;
-ms-animation-delay: 0s;
animation-delay: 0s;
}
.right-slogan span:nth-child(2) {
-webkit-animation-delay: 3s;
-ms-animation-delay: 3s;
animation-delay: 3s;
}
.right-slogan span:nth-child(3) {
-webkit-animation-delay: 6s;
-ms-animation-delay: 6s;
animation-delay: 6s;
}
.right-slogan span:nth-child(4) {
-webkit-animation-delay: 9s;
-ms-animation-delay: 9s;
animation-delay: 9s;
}
.right-slogan span:nth-child(5) {
-webkit-animation-delay: 12s;
-ms-animation-delay: 12s;
animation-delay: 12s;
}
@-webkit-keyframes rotateWord {
0% {
opacity: 0;
}
2% {
opacity: 0;
-webkit-transform: translateY(-30px);
}
5% {
opacity: 1;
-webkit-transform: translateY(0px);
}
17% {
opacity: 1;
-webkit-transform: translateY(0px);
}
20% {
opacity: 0;
-webkit-transform: translateY(30px);
}
80% {
opacity: 0;
}
100% {
opacity: 0;
}
}
@-ms-keyframes rotateWord {
0% {
opacity: 0;
}
2% {
opacity: 0;
-ms-transform: translateY(-30px);
}
5% {
opacity: 1;
-ms-transform: translateY(0px);
}
17% {
opacity: 1;
-ms-transform: translateY(0px);
}
20% {
opacity: 0;
-ms-transform: translateY(30px);
}
80% {
opacity: 0;
}
100% {
opacity: 0;
}
}
@keyframes rotateWord {
0% {
opacity: 0;
}
2% {
opacity: 0;
-webkit-transform: translateY(-30px);
transform: translateY(-30px);
}
5% {
opacity: 1;
-webkit-transform: translateY(0px);
transform: translateY(0px);
}
17% {
opacity: 1;
-webkit-transform: translateY(0px);
transform: translateY(0px);
}
20% {
opacity: 0;
-webkit-transform: translateY(30px);
transform: translateY(30px);
}
80% {
opacity: 0;
}
100% {
opacity: 0;
}
}
<div class="slogan">
<div class="left-slogan">
<p>We are
<div class="right-slogan">
<span id="aux-word">testing</span>
<span id="aux-word">experiencing</span>
<span id="aux-word">checking</span>
<span id="aux-word">solving</span>
</div>
</p>
<p>a bug</p>
</div>
</div>
For any reason, the animated text is not positioned at the right side of the words "We are" even though it's animated.
Could anybody make me know how to solve it?
Thanks in advice
A: Make some changes as follows:
@-webkit-keyframes rotateWord {
0% {
opacity: 0;
}
2% {
opacity: 0;
-webkit-transform: translateY(-200%);
}
5% {
opacity: 1;
-webkit-transform: translateY(-180%);
}
17% {
opacity: 1;
-webkit-transform: translateY(-140%);
}
20% {
opacity: 0;
-webkit-transform: translateY(-140%);
}
80% {
opacity: 0;
}
100% {
opacity: 0;
}
}
Add a white space between "We are" and the animated text:
<span id="aux-word"> testing</span>
<span id="aux-word"> experiencing</span>
<span id="aux-word"> checking</span>
<span id="aux-word"> solving</span>
A: The right-slogan div is acting as a block element. If you can make this a span instead, it will layout inline with the other text. A couple other CSS tweaks below to get things laid out nicely (removed the overflow:hidden and added a margin-left).
.slogan {
width: 73.5%;
color: black;
}
.left-slogan {
font-size: 7vw;
font-weight: bold;
}
.left-slogan > p {
line-height: 0;
}
.left-slogan {
text-align: right;
margin-right: 24px;
}
.right-slogan span {
position: absolute;
opacity: 0;
margin-left: 10px;
color: black;
font-size: 7vw;
font-weight: bold;
-webkit-animation: rotateWord 15s linear infinite 0s;
-ms-animation: rotateWord 15s linear infinite 0s;
animation: rotateWord 15s linear infinite 0s;
}
.right-slogan span:nth-child(1) {
-webkit-animation-delay: 0s;
-ms-animation-delay: 0s;
animation-delay: 0s;
}
.right-slogan span:nth-child(2) {
-webkit-animation-delay: 3s;
-ms-animation-delay: 3s;
animation-delay: 3s;
}
.right-slogan span:nth-child(3) {
-webkit-animation-delay: 6s;
-ms-animation-delay: 6s;
animation-delay: 6s;
}
.right-slogan span:nth-child(4) {
-webkit-animation-delay: 9s;
-ms-animation-delay: 9s;
animation-delay: 9s;
}
.right-slogan span:nth-child(5) {
-webkit-animation-delay: 12s;
-ms-animation-delay: 12s;
animation-delay: 12s;
}
@-webkit-keyframes rotateWord {
0% {
opacity: 0;
}
2% {
opacity: 0;
-webkit-transform: translateY(-30px);
}
5% {
opacity: 1;
-webkit-transform: translateY(0px);
}
17% {
opacity: 1;
-webkit-transform: translateY(0px);
}
20% {
opacity: 0;
-webkit-transform: translateY(30px);
}
80% {
opacity: 0;
}
100% {
opacity: 0;
}
}
@-ms-keyframes rotateWord {
0% {
opacity: 0;
}
2% {
opacity: 0;
-ms-transform: translateY(-30px);
}
5% {
opacity: 1;
-ms-transform: translateY(0px);
}
17% {
opacity: 1;
-ms-transform: translateY(0px);
}
20% {
opacity: 0;
-ms-transform: translateY(30px);
}
80% {
opacity: 0;
}
100% {
opacity: 0;
}
}
@keyframes rotateWord {
0% {
opacity: 0;
}
2% {
opacity: 0;
-webkit-transform: translateY(-30px);
transform: translateY(-30px);
}
5% {
opacity: 1;
-webkit-transform: translateY(0px);
transform: translateY(0px);
}
17% {
opacity: 1;
-webkit-transform: translateY(0px);
transform: translateY(0px);
}
20% {
opacity: 0;
-webkit-transform: translateY(30px);
transform: translateY(30px);
}
80% {
opacity: 0;
}
100% {
opacity: 0;
}
}
<div class="slogan">
<div class="left-slogan">
<p>We are
<span class="right-slogan">
<span id="aux-word">testing</span>
<span id="aux-word">experiencing</span>
<span id="aux-word">checking</span>
<span id="aux-word">solving</span>
</span>
</p>
<p>a bug</p>
</div>
</div>
A: Make following changes in CSS to overwrite default browser css
body {
margin: 0px; // to overwrite any margin given by browsers
}
.left-slogan > p { // removed line-height style
-webkit-margin-after: 0px; // to overwrite any margin given by webkit browsers
-webkit-margin-before: 0px;
}
.right-slogan span {
position: absolute;
opacity: 0;
overflow: hidden;
color: black;
font-size: 7vw;
font-weight: bold;
-webkit-animation: rotateWord 15s linear infinite 0s;
-ms-animation: rotateWord 15s linear infinite 0s;
animation: rotateWord 15s linear infinite 0s;
top: 0; // added top position to zero
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 543 |
– Oh God of our fathers give me supernatural release and provision to meet all my needs.
– This year 2017 hear the word of God I ……(Mention your names)……shall be fruitful in all my undertakings.
As God of Obadare liveth it shall be well with my soul.
– In this year Oh Lord replace every weak and tired angels with strong ones to minister in all area of my life.
– Merciful God may I not be a cast away in the name of our Lord Jesus Christ.
– Oh God of KOSEUNTI, you are KOSEUNTI yourself manifest mightily in our midst in this year 2017.
– Oluwa Olorun Koseunti release fresh fire on your servant pastor (Dr.) Paul Obadare and all ministers at Koseunti prayer meetings.
– Arise Oluwa Olorun KOSEUNTI send kingdom helpers into your work.
– In 2017 God as of old refresh the angels working at KOSEUNTI for greater works.
– Oh God proclaim KOSEUNTI afresh in all corners of the world.
– Let the blood of Jesus overcome every enemy fighting against KOSEUNTI prayer meetings.
Sing this song aloud to the King of Kings- Do something new in KOSEUNTI…(x3)…..oh Lord. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,003 |
\chapter{Introduction}
A theory like QCD with massless quarks in four dimensions has no explicit
mass parameters in its classical Lagrangian; instead a mass scale $\Lambda$ is
generated dynamically at the quantum level.
The quantity $\Lambda$ sets the scale of low-energy physics
so that the masses of all states in the theory, glue-balls, protons,
{\fam\itfam\eightit etc.}, are simply numbers times $\Lambda$.
These numbers are notoriously difficult to extract in QCD, either on
the lattice or analytically.
At energies much greater than $\Lambda$, on the other hand, the theory is
asymptotically free and perturbation theory can be used reliably.
In the language of the renormalization group (RG),
QCD is described by a trajectory
emanating from a fixed point which corresponds to a free theory of gluons and
quarks, the direction of the trajectory being determined by an operator which
is {\fam\itfam\eightit marginally relevant\/}, by which we mean that it is marginal but not
truly marginal.
The fact that the operator is marginal means that
no explicit mass scale is introduced at the fixed point itself,
whilst the fact that it is not truly marginal means that conformal invariance
is broken by the dynamical generation of the scale $\Lambda$ as one moves
away from the fixed point.
The RG trajectory is specified by a running coupling
constant $e(h)$
which depends upon the mass scale scale $h$ being probed and which
in the ultra-violet regime (large $h$) behaves like
$$
{1\over e(h)}=\beta_0\ln(h/\Lambda)+{\beta_1\over\beta_0}
\ln\ln(h/\Lambda)+{\cal O}\left({\ln\ln(h/\Lambda)\over\ln(h/\Lambda)}\right),
\nameformula} \def\numali{\numero{BET}
which in fact serves to define $\Lambda$ precisely. In the above
$\beta_0$ and $\beta_1$ are universal numbers which appear as the
first two coefficients of the beta-function in perturbation theory.
A more general situation can be envisaged for a theory with dynamical
mass generation, namely the ultra-violet fixed point of the theory,
while necessarily conformally invariant, need not be
free. The purpose of this paper is to analyse such a situation in
two dimensions in which the ultra-violet fixed point is a non-trivial
Conformal Field Theory
(CFT) -- in fact a WZW model. The direction of the RG trajectory is again
determined by some marginally relevant
operator and we say that the theory is ``asymptotically CFT''.
In the case of QCD the main difficulty is the
absence of non-perturbative calculational techniques which can be applied
in the low-energy regime.
In two dimensions, however, there is a rich class of
asymptotically-free theories
which are integrable: the O($N$) sigma models; the
principal chiral models; and the Gross-Neveu models. In these theories
the existence of higher spin conserved charges means that
the S-matrix factorizes, a property which allows in some cases for
its complete determination (see [\Ref{SG}], [\Ref{ORW}]
and [\Ref{KT}] respectively)
yielding an exact description of the low-energy physics.
These integrable theories are therefore particularly
interesting from a theoretical point of view since they provide an
arena in which one can attempt to understand the connection between the
infra-red and ultra-violet regimes.
Such a connection is also important in order to confirm the S-matrices
written down for these models. This is because the S-matrices must,
in the first instance, be regarded as conjectures which
should be tested; in
particular the question of CDD ambiguities
must be resolved.\note{In the case of
the SU$(N)$ chiral
Gross-Neveu model there is a derivation of the S-matrix from first
principles via the Bethe Ansatz [\Ref{AL}].}
In a series of papers ([\Ref{HMN},\Ref{HN}] for the O($N$) sigma
model, [\Ref{BNNW}] for the SU($N$) principal chiral model,
[\Ref{THIII}] for the SO($N$) and Sp($N$) principal chiral models and
[\Ref{FNW}] for the O($N$) Gross-Neveu model) various
authors have used a technique relying on
integrability to relate the infra-red and ultra-violet physics of
families of integrable models, building on the original work of
[\Ref{PW},\Ref{W}].
The idea is to compute a particular physical
quantity -- the free-energy in the presence of a coupling to a conserved
charge -- in two ways: firstly from the S-matrix using
a technique known as the Thermodynamic Bethe Ansatz (TBA) and secondly
from the lagrangian via perturbation theory. For the cases
mentioned above the results of the two calculations are found to be in
perfect agreement in the ultra-violet regime,
thus resolving the problem of CDD ambiguities and,
as a bonus, yielding an exact expression for
the mass gap (the ratio of the physical mass to the
$\Lambda$-parameter). As well as providing a very stringent test of
the form the S-matrix, knowing the mass-gap ratios is interesting in its
own right as they can be compared directly
with the results of lattice simulations.
In this paper we analyse a class of theories which are
asymptotically WZW models
based on the group SU(2) at level $k$ (see {\fam\itfam\eightit e.g.}~[\Ref{EW},\Ref{GO}]).
In accordance with the general situation described above,
the model will correspond to
an RG trajectory defined by the marginally relevant operator
Tr$(J_LJ_R)$ in the WZW theory, where $J_L$ and $J_R$ are the usual
left/right conserved Kac-Moody currents.
This family of examples fits into the general scheme of
``massive current algebras'' set out in [\Ref{DB}].
It is crucial that the theories we consider can also be described
explicitly at the lagrangian level: they are in fact
sigma models with a Wess-Zumino (WZ) term defined on the group
manifold SU(2). This family of lagrangians was first written
down by Balog {\fam\itfam\eightit et al\/} in
[\Ref{BFHP}] who argued further that the resulting theories should
be quantum integrable. What
was not so clear in their work was whether the models would lie in the
class of massive current algebras at the quantum level.
Our strategy for showing that these models do lie in that class, and
in particular that they are quantum integrable,
is to use the exact S-matrices that has been proposed by Ahn {\fam\itfam\eightit et al\/}
[\Ref{LB}] to describe perturbations of WZW models.
We shall then use the
ideas of [\Ref{HMN}--\Ref{FNW}] to carry out a highly non-trivial consistency
check between the lagrangian formulation of [\Ref{BFHP}] and the
S-matrix written down in [\Ref{LB}] in the manner we have already outlined
above. As a by-product we will
extract an expression for the mass gap valid to leading order in $1/k$.
The paper is organized as follows.
In section 2 we discuss the lagrangian for the model,
its current algebra, and its renormalization to one-loop.
In section 3 we write down the S-matrix conjectured to describe the quantum
scattering and in section 4 this is used in conjunction with TBA techniques to
calculate the response of the free-energy to an external field.
Section 5 contains a calculation
of this same quantity in perturbation theory, after which we compare
the expressions to confirm the choice of S-matrix and extract the mass gap
of the model. We conclude with some further remarks in section 6.
\chapter{The lagrangian, current algebra and one-loop renormalization}
The integrable field theories that we shall
investigate are described in two-dimensional Minkowski space-time
(with coordinates $\xi^\mu = (\tau, \sigma)$)
by the lagrangian density [\Ref{BFHP}]
$$\eqalign{
{\cal L}_0={1\over2e^2}&\left\{{1\over x^2-1}\left(\partial_\mu w
\right)^2+
{\beta(w)\over x+1}\left(\partial_\mu
n_a\right)^2+\right.\cr
&\qquad\left.+{1\over
x+1}\left[{1\over\sqrt{x^2-1}}\left({\pi\over2}-w\right)-
\alpha(w)\right]\epsilon_{abc}\epsilon^{\mu\nu}
n_a\partial_\mu n_b\partial_\nu n_c\right\},\cr}
\nameformula} \def\numali{\numero{LAG}
with
$$
\beta(w)={\cos^2w\over x+\cos2w},\qquad \alpha(w)=\sqrt{x-1\over
x+1}{\sin w\cos w\over x+\cos2w} .
\nameformula} \def\numali{\numero{AB}
The fields $(w,n_a)$ parameterize the SU(2) group manifold in such a way
that a general group element can be written $g=\cos w+in_a\sigma_a\sin w$ where
the $\sigma_a$'s are the Pauli matrices and
the fields $n_a$ are constrained via $n_1^2+n_2^2+n_3^2=1$.
$e$ and $x$ are coupling constants with $x>1$.
The complicated form of
${\cal L}_0$ requires some explanation.
The most important point is that it ensures that the resulting theory is
classically integrable -- in fact it ensures the existence of
a canonical structure
consisting of two commuting current algebras [\Ref{BFHP}] -- precisely
the structure studied in [\Ref{LB}]. We shall elaborate on this point
below.
The theory has an SU(2) global symmetry generated by transformations
$n_a\mapsto n_a+\epsilon_{abc}q_bn_c$ for parameters $q_b$.
Finite symmetry transformations are given by the adjoint action
$g\mapsto hgh^{-1}$, using some $h\in\,$SU(2).
This is to be contrasted with the principal sigma-model and WZW model
which both have chiral SU(2) $\times$ SU(2) global symmetries.
Our models are invariant under just the diagonal subgroup.
The antisymmetric term in \LAG\ is an example of a Wess-Zumino
(WZ) term and as usual its presence leads to a quantization condition on
coupling constants which is essential in order to obtain a consistent
quantum theory.
In the present case this condition is
$$
{2\pi\over e^2(x+1)\sqrt{x^2-1}}=k\in{\Bbb N}.
\nameformula} \def\numali{\numero{QK}
One way to derive this is to consider the integral of the curl or exterior
derivative of the WZ term over an arbitrary three-sphere, as in [\Ref{EW}],
and to demand that this always be a multiple of $2 \pi$.
Alternatively one can require that the WZ term itself, although not globally
well-defined, is ambiguous only up to multiples of $2 \pi$.
In our case we can choose the ranges of our coordinates
to be, for example, $0 \leq w < \pi $ with $n_a$ labelling any point on a
two-sphere, which covers SU(2) except for one point.
Then we demand that the integral of
the WZ term should be changed by $2 \pi$ on sending $w \to w + \pi$ which
gives exactly the condition above.\note{The precise relationship
between this criterion and the previous one is quite subtle in the general
case; see {\fam\itfam\eightit e.g.}~[\Ref{A}].}
Yet a third possibility is to appeal to the general representation theory
of Kac-Moody algebras because, as we shall see below, the combination
in \QK\ appears as a central term in the current algebras which
are responsible for the integrability of this model.
All the information concerning the model \LAG\ that
we shall need in the remainder of this paper has now been
set down.
However, in view of the brevity of the presentation in [\Ref{BFHP}]
(and because there appear to be a number of numerical misprints
in the relevant equations which can only be detected after long calculations)
we shall, before proceeding, elaborate on the current algebra structure
which is responsible for the particular form of the Lagrangian \LAG .
We shall also supply some details of the one-loop renormalizability of the
model which were left implicit in [\Ref{BFHP}].
The theory \LAG\ is of the general form
$$
{\cal L}_0={1\over2e^2}\left \{
G_{ij}(\phi)\partial_\mu\phi^i\partial^\mu\phi^j+
\epsilon^{\mu\nu}B_{ij}(\phi)\partial_\mu\phi^i\partial_\nu\phi^j\right \}
\, .
\nameformula} \def\numali{\numero{GLAG}
where the fields $\phi^i (\xi^\mu)$ describe a map from two-dimensional
Minkowski space-time to some target manifold.
Motivated by the example of WZW models, one can ask
when such a general sigma-model exhibits a classical current algebra.
We restrict attention to the case in which the target manifold is the group
SU(2) and it is convenient for this part of our discussion
to choose antihermitian generators normalized so that
$$
\lambda_a = - {i \over 2} \sigma_a \, , \qquad
[ \lambda_a , \lambda_b ] = \epsilon_{abc} \lambda_c\,,
\efr
which corresponds to choosing the single simple root of SU(2) to
have length one. We shall make no distinction between upper and lower
SU(2) indices.
A natural Ansatz for the light-cone components
$I^a_\pm = I^a_0 \pm I^a_1$ of a current in the SU(2) Lie algebra is
$$
I^a_+ = - {1\over e^2} L^a_i \partial} \def\alfa{\alpha} \def\ro{\rho_+ \phi^i \, ,
\qquad
I^a_- = - {1\over e^2} R^a_i \partial} \def\alfa{\alpha} \def\ro{\rho_- \phi^i,
\efr
where $L^a_i$ and $R^a_i$ are vielbeins for the sigma-model metric:
$$
L^a_i L^a_j = R^a_i R^a_j = G_{ij},
\nameformula} \def\numali{\numero{VB}
The equations of motion following from \GLAG\ ensure that these currents are
conserved $\partial} \def\alfa{\alpha} \def\ro{\rho_\mu I^{a \mu} = 0$.
It can also be shown by tedious calculation that, with the canonical structure
defined by \GLAG ,
these currents obey a classical (equal-$\tau$) Poisson bracket algebra
$$\eqalign{
\{ I^a_{\pm} (\sigma) , I^b_{\pm} (\sigma^\prime) \}
& = \epsilon^{abc} \, ( \, a I^c_{\pm} (\sigma) + b I^c_{\mp} (\sigma) \, )
\delta (\sigma - \sigma^\prime) \pm {2\over e^2} \delta^\prime (\sigma - \sigma^\prime)
\cr
\{ I^a_+ (\sigma) , I^b_- (\sigma^\prime) \} & =
- b\, \epsilon^{abc} \, (\, I^c_+ (\sigma) + I^c_- (\sigma) \, )
\delta(\sigma - \sigma^\prime),
\cr}
\nameformula} \def\numali{\numero{calg}
with $a$ and $b$ constants,
provided that the quantities $L^a_i$ and $R^a_i$ satisfy certain conditions.
To express these conditions compactly it is convenient to introduce
differential forms on the group manifold:
$$
L = \lambda_a L^a_i {\rm d} \phi^i \, , \qquad
R = \lambda_a R^a_i {\rm d} \phi^i.
\efr
Then the current algebra above will hold provided
$$\eqalign{
hL + Rh & = 0 \cr
{\rm d} L + a L^2 - b h^{-1} L^2 h & = 0\cr
{\rm d} R + a R^2 - b h R^2 h^{-1} & = 0\cr
3 H = -a {\rm Tr} L^3 - 3b {\rm Tr} R L^2 & = a {\rm Tr} R^3 + 3b {\rm Tr} L R^2,
\cr}
\nameformula} \def\numali{\numero{ceqn}
where $h$ is some group-valued function on SU(2) and
$2H = {\rm d} B$ is the field-strength three-form corresponding to $B$.
We use the conventions $B = {1 \over 2!} B_{ij} {\rm d} \phi^i {\wedge} {\rm d} \phi^j$
and $H = {1 \over 3!} H_{ijk} {\rm d} \phi^i {\wedge} {\rm d} \phi^j {\wedge} {\rm d} \phi^k$
for the components of two-forms and three-forms respectively;
as a result
$2H_{ijk} = \partial} \def\alfa{\alpha} \def\ro{\rho_i B_{jk} + \partial} \def\alfa{\alpha} \def\ro{\rho_j B_{ki} + \partial} \def\alfa{\alpha} \def\ro{\rho_k B_{ij}$.
The SU(2) WZW model corresponds to a special solution
of the current algebra conditions above in which
$$
L = g^{-1} {\rm d} g
\, , \quad
R = - {\rm d} g g^{-1}
\, , \quad
h = g
\, , \quad
a = 1
\, , \quad
b = 0 ,
\efr
and in this case \calg\ clearly collapses to two commuting Kac-Moody algebras.
The action written in \LAG\ corresponds to a slightly more complicated
solution of \ceqn\ which can be motivated as follows.
First consider the Ansatz
$$
L = c (h^{-1} {\rm d} h) + \lambda_a n_a f(\rho) {\rm d} \rho \, , \quad
R = c ( - {\rm d} h h^{-1} ) - \lambda_a n_a f(\rho) {\rm d} \rho \, ,
\efr
where $h = \exp (- \lambda_a n_a \rho )$,
$c$ is some constant, $f(\rho)$ is a function to be determined,
and the variable $\rho(w)$ is itself some function of our SU(2) coordinate $w$
which we will fix in a convenient way at the end, after finding a solution.
This seems on the face of it to be a rather redundant
procedure, but it turns out to simplify some technical aspects of the
discussion.
The Ansatz above is clearly a straightforward
modification of the WZW case, and
it is chosen so as to satisfy the first equation in \ceqn\ automatically.
It is not difficult to check that the remaining conditions
in \ceqn\ hold if
$$a = 2x + 1 \, , \quad b = -1 \, , \quad
c = {1 \over 2 (x + 1)} \, , \quad
f(\rho) = {1 \over x+1} {\cos \rho - 1 \over x + \cos \rho}.
\nameformula} \def\numali{\numero{SOL}
This solution can also be expressed in the form
$$\eqalign{
L & = \phantom{-} \lambda_a
( \alpha {\rm d} n_a - \beta \epsilon_{abc} n_b {\rm d} n_c + \gamma n_a {\rm d} \rho )
\cr
R & = - \lambda_a ( \alpha {\rm d} n_a + \beta \epsilon_{abc} n_b {\rm d} n_c
+ \gamma n_a {\rm d} \rho ),
\cr
}
\efr
where the functions $\alpha$, $\beta$ and $\gamma$ are given by
$$
\alpha
= - {\sin \rho \over 2(x+1)} \, , \quad
\beta
= {1 - \cos \rho \over 2(x+1)} \, , \quad
\gamma = - {1 \over 2(x + \cos \rho)} \, .
\efr
The sigma-model metric and WZ term
are now determined as functions of $\rho$ and $n_a$
by the equations \VB\ and \ceqn\ respectively.
The final step is to relate $\rho$ to $w$, which can be done in such a way
that the expression for the WZ term can be written in closed form.
This is achieved by choosing
$$
{\alpha \over \beta} = - \cot{\rho \over 2} = \sqrt{x-1 \over x+1} \tan w.
\efr
Using this one can deduce the expressions $\alpha (w)$ and $\beta (w)$ given
in \AB\ and, after some effort, one then recovers \LAG .
The current algebra corresponding to the solution \SOL\ above was first
considered by Rajeev [\Ref{R}], who showed that it could be decomposed into two
commuting Kac-Moody algebras. The combinations which achieve this are
$$
J^a_\pm = {1\over 4} \left \{ \,
\left ( {1 \over x+1} + {1 \over \sqrt{x^2 -1} } \right ) I^a_\pm
+ \left ( {1 \over x+1} - {1 \over \sqrt{x^2-1 } } \right ) I^a_\mp
\, \right \},
\efr
obeying
$$\eqalign{
\{ J^a_\pm (\sigma) , J^b_\pm (\sigma^\prime) \, \}
& = \epsilon^{abc} J^c_\pm (\sigma) \delta(\sigma - \sigma^\prime)
\pm {1 \over 2 e^2 (x+1)\sqrt{x^2 -1} } \delta^\prime (\sigma - \sigma^\prime) \cr
\{ J^a_\pm (\sigma) , J^b_\mp (\sigma^\prime) \, \}
& = 0. \cr
}
\efr
The $\pm$ signs occur in the central terms because these are {\fam\itfam\eightit classical\/}
Kac-Moody algebras, and the
quantization condition \QK\ can now be recovered by comparison
with some standard reference ({\fam\itfam\eightit e.g.}~equation (2.3.14) of [\Ref{GO}]).
Unlike the WZW case, the components of these Kac-Moody currents are
not chirally conserved (although the original current $I^a_\mu$ is
conserved by construction).
Notice, however, that the WZW case can be recovered by taking the limit
$x \to \infty$, $k$ fixed, provided
we rescale the fields appropriately.
Since the theory \LAG\ is a generalized sigma model (a sigma model
with a WZ term) its renormalization group flow can be analysed using the
background field method (see for example [\Ref{RG}]). We can simply
quote the well-known results for the way that the metric and WZ term
in \GLAG\ run with the renormalization scale to one-loop, but we must then
ensure that these equations are indeed consistent with the specific Ansatz
of \LAG . In our discussion of the current algbera, it was convenient
to keep the coupling constant dependence explicit, but to apply
the general renormalization results of sigma-models it is better
to absorb the coupling constant $e$ into our definitions
of the metric and WZ term by defining
$g_{ij}=G_{ij}/e^2$,
$b_{ij}=B_{ij}/e^2$ and $h_{ijk}=H_{ijk}/e^2$.
The coefficients of the beta-function are calculated in terms of the
generalized curvature ${\hat R}_{ijkl}$ corresponding to the connection
$$
\hat \Gamma_j{}^i{}_k = \Gamma_j{}^i{}_k + h^i{}_{jk}
\efr
which involves the usual Christoffel connection $\Gamma_j{}^i{}_k$
(constructed from the metric $g_{ij}$) modified
by a torsion term.
To one loop one finds [\Ref{RG}] that under the renormalization group
transformation
of the subtraction point $\mu$ the metric and
anti-symmetric field satisfy
$$\mu{\partial g_{ij}\over\partial\mu}=-{1\over2\pi}{\hat R}_{(ij)},\qquad
\mu{\partial b_{ij}\over\partial\mu}=-{1\over2\pi}{\hat R}_{[ij]},
\nameformula} \def\numali{\numero{FLO}
where ${\hat R}_{ij}={\hat R}^k_{\ ijk}$ is the generalized Ricci tensor.
We now apply these formulae to the theory \LAG. First of all, we
define the coordinates $\theta$ and $\psi$ via
$n_a=(\cos\theta,\sin\theta\cos\psi,\sin\theta\sin\psi)$.
In these coordinates the metric has non-zero components
$$
g_{ww}={1\over e^2(x^2-1)},\qquad g_{\theta\t}={\beta\over e^2(x+1)},\qquad
g_{\psi\psi}={\beta\over e^2(x+1)}\sin^2\theta,
\efr
and the anti-symmetric field has non-zero components
$$
b_{\theta\psi}=-b_{\psi\theta}={1\over
e^2(x+1)}\left[{1\over\sqrt{x^2-1}}\left({\pi\over2}-w\right)-\alpha\right]
\sin\theta.
\efr
{}From these we find that the non-zero components of the generalized
Ricci tensor are
$$\eqalign{
{\hat R}_{ww}&={2\over x^2-1}+4\alpha'\sqrt{x+1\over x-1},\cr
{\hat R}_{\theta\t}&=-{2x\over x+1}\beta+2\beta\alpha'\sqrt{x^2-1},\cr
{\hat R}_{\psi\psi}&={\hat R}_{\theta\t}\sin^2\theta,\cr
{\hat R}_{\theta\psi}&=-{\hat
R}_{\psi\theta}=2\beta\beta'\sqrt{x^2-1}\sin\theta,\cr}
\nameformula} \def\numali{\numero{CURV}
where the prime denotes a derivative with respect to $w$ at constant $x$.
Using the expression for
the Ricci tensor in \CURV\ in the equations \FLO\
shows that under renormalization group flow the form of the lagrangian
is preserved up to a renormalization of the
coupling constants $e$ and $x$:
$$
\mu{\partial e\over\partial\mu}={1\over2\pi}(1-2x)e^3+{\cal O}(e^5),\qquad
\mu{\partial x\over\partial\mu}={1\over\pi}(x^2-1)e^2+{\cal O}(e^4),
\nameformula} \def\numali{\numero{RGCC}
and a diffeomorphism of the field $w$ given by
$$
\mu{\partial w\over\partial\mu}=-{1\over\pi}(x^2-1){\cos w\sin w\over
x+\cos2w}e^2+{\cal O}(e^4).
\nameformula} \def\numali{\numero{DIFF}
These one-loop results \RGCC\ agree with the analysis of
[\Ref{BFHP}]. Notice that to this order $k$ defined in \QK\ is
constant under renormalization group flow as we expect. In
the ultra-violet, $\mu\rightarrow\infty$, $e$ runs to zero and $x$
runs to infinity. In this limit one can easily show that
$$
{\cal L}_0={\cal L}_{\fam0\eightrm WZW}+{k\over8\pi x}{\fam0\eightrm Tr}\left(J_LJ_R\right)+
{\cal O}(1/x^2),
\efr
where ${\cal L}_{\fam0\eightrm WZW}$ is the usual SU(2) WZW lagrangian at level $k$,
and $J_L=g^{-1}\partial_+g$ and $J_R=-(\partial_-g)g^{-1}$ are its
left and right conserved Kac-Moody currents. This expression justifies
our earlier statement that the theories \LAG\ are SU(2) WZW models perturbed
by the operator Tr$(J_LJ_R)$.
It is also easy to see that this perturbation
breaks the chiral SU(2) $\times$ SU(2) symmetry of ${\cal L}_{\fam0\eightrm WZW}$
to the diagonal, or adjoint, SU(2) subgroup mentioned above.
Assuming that $k$ is indeed constant we can eliminate $x$ from \RGCC\
to get the flow equation just involving $e$. Later we shall be
interested in this flow equation for large but finite values of $k$. In this
regime we deduce from \RGCC\ that
$$
\mu{\partial e\over\partial\mu}=-\beta_0e^2-\beta_1e^3-{\cal O}(e^4),
\efr
where
$$
\beta_0=\sqrt{2\over\pi k}\left(1+{\cal O}(1/k)\right),\qquad
\beta_1=-{1\over\pi}\left(1+{\cal O}(1/k)\right).
\nameformula} \def\numali{\numero{BCO}
Notice that terms coming from higher loops can produce corrections of
lower order in $1/k$ assuming that the coefficient of $e^p$ in
$\mu(\partial e/\partial\mu)$
is polynomial in $x$. It is important to remember that the expressions
\BCO\ are universal.
We have now described in some detail the lagrangian field theory
we wish to study, and in the next section we conjecture an S-matrix
to describe the scattering of states in this theory. We will subsequently
undertake a non-trivial check of the form of the S-matrix
by using the ideas of [\Ref{HMN}-\Ref{FNW}].
To do this we need to
couple the theory to a conserved charge. The idea is to modify the
hamiltonian $H\rightarrow H-hQ$, where $Q$ is a conserved charge
corresponding to some generator of the SU(2) symmetry of the model.
In the Minkowski space lagrangian picture this is
achieved simply by replacing the derivative of $n_a$ in the
time-direction by the ``covariant derivative'':
$$
\partial_0n_a\rightarrow\partial_0n_a+2h\epsilon_{abc}q_bn_c,
\nameformula} \def\numali{\numero{CCP}
where the $q_a$ are a set of parameters which later we take to be
$q=(1,0,0)$ without loss of generality (due to the SU(2) symmetry).
We will then compute the response of the free-energy per unit volume
$\delta f(h)=f(h)-f(0)$ in the ultra-violet regime in two ways: using
the S-matrix along with thermodynamic Bethe Ansatz techniques and using
conventional perturbation theory.
\chapter{The S-matrix}
Consider, for a moment, the most general way of associating
S-matrices to the Lie algebra
SU($N$). The particles form multiplets associated to the fundamental,
or completely anti-symmetric, representations of the algebra, and each
particle carries in general say $m$
copies of the quantum numbers of that
representation. The general two-body
S-matrix element -- from which all the others may be deduced by
factorization -- has the block form [\Ref{THI},\Ref{THII}]
$$
S^{ab}_{(k_1,k_2,\ldots,k_m)}(\theta)=
X^{ab}(\theta)S^{ab}_{(k_1)}(\theta)\otimes S^{ab}_{(k_2)}(\theta)\otimes\cdots
\otimes S^{ab}_{(k_m)}(\theta),
\nameformula} \def\numali{\numero{SGEN}
where factor $S^{ab}_{(k_j)}(\theta)$ acts between the $j^{\fam0\eightrm th}$ copies
of the fundamental representations $a$ and $b$ ($a$,
$b=1,2,\ldots,N-1$) and the $k_j$'s are parameters or coupling constants.
The part $X^{ab}(\theta)$ is a scalar factor which
ensures that the overall S-matrix has the right analytic structure.
Each block $S^{ab}_{(k)}(\theta)$ is invariant under the action of the
quantum loop-group $U_q({\fam0\eightrm SU}(N)\otimes{\Bbb C}[\theta,\theta^{-1}])$ where
the deformation parameter is $q=-\exp(-i\pi/(N+k))$. In the limit
$k=\infty$ the quantum loop-group reduces to the ordinary loop-group and the
block $S^{ab}_{(\infty)}(\theta)$ is invariant under the action of the
group SU($N$) itself. When $k$ is a natural number the blocks are
proportional to RSOS solutions of the Yang-Baxter equation and
$S_{(1)}^{ab}(\theta)=1$.
The S-matrix that was proposed in [\Ref{LB}] to describe the
perturbation of the
WZW model of level $k$ is $S^{ab}_{(k,\infty)}(\theta)$. For the case of
SU(2) there is only one particle and $X^{11}(\theta)=-1$. It is worth
pointing out that this general form subsumes the well-known
S-matrices of the SU($N$) chiral Gross-Neveu model, given by
$S^{ab}_{(\infty)}(\theta)\equiv S^{ab}_{(\infty,1)}(\theta)$,
and the SU($N$) principal chiral model, given by
$S^{ab}_{(\infty,\infty)}(\theta)$. Remarkably, this implies
that the model \LAG\ is
equivalent at the quantum level to the SU$(2)$ chiral
Gross-Neveu and principal
chiral models, for $k=1$ and $k=\infty$, respectively. We shall make a
comment about these equivalences at the end of the paper.
We now write down the S-matrix that is proposed to describe
the theory \LAG. As is conventional, we take the kinematic variable to
be the rapidity difference $\theta$ of the incoming
particles.\note{The velocity and rapidity of a particle are related by
$v={\fam0\eightrm tanh}\theta$.} The
S-matrix describes one massive particle with internal quantum
numbers and for the two-body process
it has the product form mentioned above [\Ref{LB}]:
$$
S(\theta)=S_{\fam0\eightrm SU(2)}(\theta)\otimes S^{\fam0\eightrm kink}_{(k)}(\theta),
\nameformula} \def\numali{\numero{FF}
with $k\in{\Bbb N}$ being identified with \QK.
The product form means that the particle carries
two sets of quantum numbers and
each factor acts on one of the sets only. The first factor $S_{\fam0\eightrm
SU(2)}(\theta)$ is the S-matrix of the SU(2) chiral Gross-Neveu model; hence
the particle transforms in the two-dimensional representation of SU(2)
and the two-body S-matrix elements may be written [\Ref{KT}]
$$
S_{\fam0\eightrm SU(2)}(\theta)={\Gamma\left(1-{\theta\over2\pi i}\right)
\Gamma\left({1\over2}+{\theta\over2\pi i}\right)\over
\Gamma\left(1+{\theta\over2\pi i}\right)
\Gamma\left({1\over2}-{\theta\over2\pi i}\right)}\left[{\Bbb P}_{\fam0\eightrm
t}+
\left({\theta+2\pi i\over\theta-2\pi i}\right){\Bbb
P}_{\fam0\eightrm s}\right],
\efr
where ${\Bbb P}_{\fam0\eightrm t,s}$ indicate the triplet and singlet channels.
This part is equal to $-S^{11}_{(\infty)}(\theta)$ as written above
and is invariant under SU(2).
The other factor in \FF\ describes the scattering of kink degrees-of-freedom
carried by the particle. The
particle can either be a kink or an anti-kink which interpolates
between a set $k+1$ vacua $\{1,2,\ldots,k+1\}$ with the following
selection rule: a kink can
connect vacuum $a$ with $a+1$ and an anti-kink $a$ with $a-1$.
The S-matrix is that of soliton scattering in the
restricted sine-Gordon model [\Ref{RSG}] so the S-matrix element
for the process $K_{da}(\theta_1)+K_{ab}(\theta_2)\rightarrow
K_{dc}(\theta_2)+K_{cb}(\theta_1)$ is
$$\eqalign{
S_{(k)}^{\fam0\eightrm kink}&\pmatrix{a&b\cr d&c\cr}(\theta)={u(\theta)\over2\pi
i}\left({\sinh(\pi a/p)\sinh(\pi c/p)\over
\sinh(\pi d/p)\sinh(\pi b/p)}\right)^{-\theta/2\pi i}\cr
&\times\left\{
\sinh\left({\theta\over p}\right)\delta_{db}\left({
\sinh(\pi a/p)\sinh(\pi c/p)\over\sinh(\pi d/p)\sinh(\pi b/p)
}\right)^{1/2}+\sinh\left({i\pi -\theta\over p}
\right)\delta_{ac}\right\},\cr}
\efr
where $p=k+2$ and
$$\eqalign{
u(\theta)=&\Gamma\left({1\over p}\right)\Gamma\left(1+{i\theta\over p
}\right)\Gamma\left(1-{\pi+i\theta\over p}\right)\prod_{n=1}^\infty
{R_n(\theta)R_n(i\pi-\theta)\over R_n(0)R_n(i\pi)}\cr
R_n(\theta)=&{\Gamma\left({2n\over p}+{i\theta\over\pi p}\right)
\Gamma\left(1+{2n\over p}+{i\theta\over\pi p}\right)\over
\Gamma\left({2n+1\over p}+{i\theta\over\pi p}\right)
\Gamma\left(1+{2n-1\over p}+{i\theta\over\pi p}\right)}.\cr}
\efr
We remark that the form of the S-matrix reflects the form of
the lagrangian: the SU(2) part manifests the SU(2) symmetry of the
model and the kink part describes degrees-of-freedom associated to the
periodic field $w$.
\chapter{The free-energy from the S-matrix}
In this section we evaluate the response of the free-energy
$\delta f(h)$ to the coupling with the charge
directly from the S-matrix. The
technique we use is known as the thermodynamic Bethe Ansatz
[\Ref{TBA}]. In its most general form it leads to an expression for
the free-energy in a cylindrical geometry coupled to a
conserved charge which plays the r\^ole of a chemical potential.
In our case we wish to evaluate the free-energy in the plane,
{\fam\itfam\eightit i.e.\/} at zero temperature.
Consider the (one-dimensional) statistical mechanics of a
gas of particles described by the
S-matrix \FF. Since this theory is integrable, the
number of particles is conserved under interaction
and it is meaningful to consider single
particle energy levels. In a free-theory the energy of these levels
would simply be $\epsilon(\theta)=m\cosh\theta-h$, where $h$ is the
chemical potential, and the free-energy (per unit volume) at zero temperature
would be that of a free one-dimensional relativistic fermion gas:\note{The
fact that particles should be treated as fermions in the thermodynamic Bethe
Ansatz results from that fact that the S-matrix satisfies $S(0)=-1$.}
$$
f(h)={m\over2\pi}\int_{-\theta_{\fam0\eightrm F}}^{\theta_{\fam0\eightrm F}}d\theta\epsilon(\theta)\cosh \theta ,
\nameformula} \def\numali{\numero{FE}
where $\theta_{\fam0\eightrm F}$ the Fermi rapidity is determined by the condition
that $\epsilon(\pm\theta_{\fam0\eightrm F})=0$.
In our case, the complications are two-fold: the theory is, after all,
not free and furthermore the particles carry internal quantum numbers. As a
result of the
former $\epsilon(\theta)$ now satisfies an integral equation involving
kernels related to the S-matrix of the theory. The
effect of the internal quantum numbers is to couple the energy
$\epsilon(\theta)$ to the ``magnon'' energy levels $\xi_p(\theta)$,
$p=1,2,\ldots,k-1$, and $\zeta_q(\theta)$, $q=1,2,\dots,\infty$; where
the former result from the kink part of the S-matrix and the
latter from the SU(2) part. The free-energy is then still given by
\FE. The equations are known as the
TBA equations and they have been derived at finite
temperature and zero chemical potential for our
S-matrix in [\Ref{ZAM}].\note{The TBA equations for the more general
S-matrices \SGEN\
was considered in [\Ref{THII}].} At $T=0$ and in the presence of a
chemical potential coupling to the charge of the SU(2) symmetry,
the TBA equations adopt the form
$$\eqalign{
&\epsilon^+(\theta)+R\ast\epsilon^-(\theta)+\sum_{p=1}^{k-1}
a_p^{(k)}\ast\xi_p^+(\theta)
+\sum_{q=1}^\infty a_q^{(\infty)}\ast
\zeta_q^+(\theta)=m\cosh\theta-h,\cr
&\xi_p^-(\theta)+\sum_{q=1}^{k-1}
A^{(k)}_{pq}\ast\xi_q^+(\theta)=a_p^{(k)}\ast\epsilon^-(\theta),\cr
&\zeta_p^-(\theta)+\sum_{q=1}^\infty A^{(\infty)}_{pq}\ast
\zeta_q^+(\theta)=a_p^{(\infty)}\ast\epsilon^-(\theta)-2hp,
}\nameformula} \def\numali{\numero{TBA}
where star denotes the convolution $f\ast g(\theta)=\int d\theta'
f(\theta-\theta')g(\theta')$ and $f^\pm(\theta)$ denote the positive/negative
decomposition of $f(\theta)=f^+(\theta)+f^-(\theta)$, {\fam\itfam\eightit i.e.}
$f^{\pm}(\theta)=f(\theta)$ if $f(\theta)>0$ or $f(\theta)<0$,
respectively, being otherwise zero. The kernels in \TBA\ are given by
$$\eqalign{
R(\theta)=&\int_0^{\infty}{dx\over\pi}\cos(\theta x){
\sinh^2(\pi x/2)\over\sinh(\pi kx/2)\sinh(\pi x)}\exp(k\pi
x/2),\cr
A^{(k)}_{pq}(\theta)=&\int_0^\infty{dx\over\pi}\cos(\theta x)
{2\sinh(\pi
px/2)\sinh(\pi(k-q)x/2)\cosh(\pi x/2)\over\sinh(\pi x)
\sinh(\pi x/2)},\cr
a^{(k)}_p(\theta)=&{1\over\pi k}\cdot{\sin(\pi
p/k)\over\cosh(2\theta/k)-\cos(\pi p/k)},
}\efr
for $q\geq p$ ($A_{pq}^{(k)}(\theta)=A_{qp}^{(k)}(\theta)$).
The dependence of the free-energy (per unit volume) on the chemical
potential is then given by
$$
\delta f(h)=f(h)-f(0)={m\over2\pi}\int_{-\infty}^{\infty}d\theta
\epsilon^-(\theta)\cosh\theta.
\efr
The problem before us is to solve the coupled integral equations \TBA.
Our strategy will implicitly assume that the solution of the equations
\TBA\ is unique. Given this the crucial
observation is that $a^{(k)}_p(\theta)$ is a positive kernel; hence
the solution of the TBA equations is
$\xi_p^+(\theta)=\zeta_q^+(\theta)=0$ with
$\xi_p^-(\theta)=a_p^{(k)}\ast\epsilon^-(\theta)$,
$\zeta_p^-(\theta)=a_p^{(\infty)}\ast\epsilon^-(\theta)-2hp$ and
$$
\epsilon^+(\theta)+R\ast\epsilon^-(\theta)=m\cosh\theta-h.
\nameformula} \def\numali{\numero{STBA}
The solution $\epsilon(\theta)$ to \STBA\ is some symmetric convex
function with zeros at the Fermi rapidity $\pm\theta_{\fam0\eightrm F}(h)$.
When $h<m$ the system is
below threshold; the external field is too weak to excite any particle states
and hence $\delta f(h)$ is
zero. Beyond the threshold $h=m$ the chemical potential forces
the system into a state where the particles line up their spins with
the external field. From the form of the TBA equations we see that the
external field does not couple to the kink
number and it turns out that the ground-state has total kink number zero.
Notice that the reasoning which led to identifying the solution of the
TBA equations in terms of a particular configuration of the quantum numbers
of the particles was arrived at
from studying the full TBA equations. This is to be contrasted with
the more
heuristic arguments used in [\Ref{HMN}-\Ref{FNW}] leading to the hypothesis
that only one particle-state contributed to the ground-state.
We have arrived at an expression for the free-energy in terms of a
single function $\epsilon(\theta)$ which satisfies the single integral
equation \STBA. This equations are of the same form, but with a
different kernel, as those
of the O($N$) sigma model [\Ref{HMN},\Ref{HN}],
principal chiral models [\Ref{BNNW},\Ref{THIII}] and Gross-Neveu
models [\Ref{FNW}].
It is not possible to solve the equation \STBA\ in closed form;
however, we will be interested in the solution only in the deep ultra-violet
$h\gg m$ for which one can develop a series solution using generalized
Wiener-Hopf techniques [\Ref{HMN},\Ref{JNW}] (see the appendix of
[\Ref{FNW}] for a clear
summary). Rather than explain these techniques we simply follow the
manipulations of [\Ref{FNW}] required to extract the series solution.
The method starts by decomposing the Fourier transform of the kernel
$R(\theta)$ in the following way:
$$
{\sinh^2(\pi x/2)\over\sinh(\pi kx/2)\sinh(\pi
x)}\exp\left({k\pi x/2}\right)={1\over G_+(x)G_-(x)},
\efr
where $G_-(x)=G_+(-x)$ and $G_\pm(x)$ are analytic in the upper/lower
half-planes, respectively. So
$$
G_+(x)=\sqrt{2k}{\Gamma^2\left(1-i{x/2}\right)\over\Gamma\left(
1-i{kx/2}\right)\Gamma\left(1-ix\right)}\exp\left(ixb-i{kx\over2}\ln(-ix)
\right),
\efr
where
$$
b={k\over2}-\ln2-{k\over2}\ln{k\over2}.
\efr
Following [\Ref{FNW}] we now define the function
$\alpha(x)=\exp(2ix\theta_{\fam0\eightrm F})G_-(x)/G_+(x)$, where
$\epsilon(\pm\theta_{\fam0\eightrm F})=0$. $\alpha(x)$ has a cut along the
positive imaginary axis and we define $\gamma(\xi)$ in terms of the
discontinuity:
$$
\alpha(i\xi+0)-\alpha(i\xi-0)=-2ie^{-2\xi\theta_{\fam0\eightrm F}}\gamma(\xi),
\efr
giving in this case
$$
\gamma(\xi)=\exp\left(-k\xi\ln\xi+2\xi
b\right){\Gamma^2\left(1-{\xi/2}\right)\Gamma\left(1+{k\xi/2}\right)
\Gamma(1+\xi)\over\Gamma^2\left(1+{\xi/2}\right)\Gamma\left(1-{k\xi/
2}\right)\Gamma(1-\xi)}\sin\left({\pi k\xi/2}\right).
\nameformula} \def\numali{\numero{GEX}
If one consults [\Ref{FNW}] then it soon becomes apparent that
$\gamma(\xi)$ for our model has the same functional form as
a fermion model, rather than a sigma model, namely
$$
\gamma(\xi)=\pi e^{-k\xi\ln\xi}\sum_{n=1}^\infty d_n\xi^n.
\efr
The expansion of the free-energy is given in terms of the quantities
$d_j$. It turns out that $\delta f(h)/h^2$ is power series in the
effective coupling $u=u(h)$ defined through
$$
{1\over u}-k\ln u={1\over z},
\efr
where
$$
{1\over z}=\ln\left[{h^2\over m^2}\left({2G_+(0)\over G_+(i)}\right)^2\right].
\efr
Putting these expression together with the results of [\Ref{FNW}] allows us
to extract the
first few terms in the expansion of the free-energy as a function of $h/m$
$$\eqalign{
&\delta f(h)=\cr
&-{h^2\over2\pi}G_+(0)^2\left\{{1-2d_1 z+2kd_1z^2\ln
z}-2\left[2d_1-\Gamma'(2)kd_1-d_1^2+d_2\right]z^2\right.\cr
&\left.-2k^2d_1z^3\ln^2z+2k\left[4d_1-\Gamma'(3)kd_1-2d_1^2+2d_2\right]z^3\ln
z+{\cal O}(z^3)\right\}.\cr
}\efr
In the above, $\delta f(h)$ is an expansion in terms of the form
$z^m\ln^n z$ where $m>n$. From \GEX\ we find
$$
d_1={k\over2},\quad
d_2=-k\ln2-{k^2\over2}\ln\left({k\over2}\right)+{k^2\over2}\Gamma'(2).
\efr
So putting everything together for the models that we are considering
we find that the free-energy extracted from the S-matrix yields the following
expansion in the ultra-violet:
$$\eqalign{
\delta f(h)=&-{h^2k\over\pi}\left\{1-{k\over2}s+{k^2\over4}s^2\ln s
-{k\over2}\left[1-{1\over2}\ln{64k\over\pi}-{k\over4}-{k\over2}\ln{k\over4}
\right] s^2\right.\cr
&\qquad\left.-{k^3\over8}s^3\ln^2s+{k^2\over2}\left[1-{1\over2}\ln{64k\over
\pi}-{k\over2}-{k\over2}\ln
{k\over4}\right]s^3\ln s+{\cal O}(s^3)\right\},\cr
}\nameformula} \def\numali{\numero{FES}
where $s^{-1}=\ln(h/m)$.
\chapter{The free-energy from perturbation theory}
In this section, we develop the expansion of the free-energy $\delta f(h)$ in
perturbation theory. We assume, following the discussion to one-loop
in section 2 and [\Ref{BFHP}], that under
renormalization group flow $k$ is constant. Therefore, we may express $1/x$
in terms of $e$:
$$
{1\over x}=\sqrt{k\over2\pi}e+{k\over4\pi}e^2+{\cal O}(e^3),
\nameformula} \def\numali{\numero{XEXP}
and we shall find that in the ultra-violet $e$ runs to zero and hence
$x$ runs to infinity. The loop expansion parameter is $e^2(x+1)^2$
which is expressed in terms of $e$ as
$$
e^2(x+1)^2={2\pi\over
k}\left(1+\sqrt{k\over2\pi}e+{k\over2\pi}e^2+{\cal O}(e^3)\right).
\nameformula} \def\numali{\numero{XIE}
This means that
the contributions from higher loops can lead to terms of the same
order in $e$, but their coefficients will be suppressed by
higher powers of $1/k$. Our result to one-loop will therefore be valid in the
large but finite $k$ limit.
The Minkowski space lagrangian in the presence of the chemical potential is
given by substituting \CCP\ in \LAG\ which on Wick rotating to Euclidean space
becomes
$$\eqalign{
&{\cal L}={\cal L}_0
-{2h^2\beta(w)\over e^2(x+1)}\left(1-n_1^2\right)\cr
&+{2h\over e(x+1)}\left\{\beta(w)(n_2\partial_0n_3-n_3\partial_0n_2)
+\left[{1\over\sqrt{x^2-1}}\left({\pi\over2}-w\right)-\alpha(w)\right]
\partial_1n_1\right\}.\cr
}\efr
The ground-state of the system is given by $n_1= w=0$ (modulo some discrete
ambiguity depending on the precise way we parametrize SU(2)).
We wish to calculate the change in the free-energy per-unit-volume
$\delta f(h) $ to one-loop, so it suffices to
expand the lagrangian to quadratic order around the ground
state with $(n_2,n_3)=\sqrt{1-n_1^2}(\cos\psi,\sin\psi)$.
On suitably re-scaling the field $w$ we find
$$\eqalign{
{\cal L}={1\over2e^2(x+1)^2}&\left\{(\partial_\mu w)^2+(\partial_\mu
n_1)^2+(\partial_\mu\psi)^2-{8 hx\over x+1}w\partial_1n_1\right.\cr
&\left.-4h^2+4h^2n_1^2+4h^2w^2\left({x-1\over
x+1}\right)^2+\cdots\right\},\cr
}\efr
where the ellipsis represent the interaction.
We can simply read off the
tree-level contribution to $\delta f(h)$:
$$
\delta f(h)_0=-{2h^2\over
e^2(x+1)^2}=-{h^2k\over\pi}\left\{1-\sqrt{k\over2\pi}e+{\cal O}(e^3)\right\}.
\efr
The one-loop contribution is, from the quadratic Euclidean lagrangian
written above,
$$
\delta f(h)_1 = {1 \over 2} \ln {\rm Det} \left \{ {M \over e^2 (x+1)^2} \right \},
\nameformula} \def\numali{\numero{ONEL}
where the operator $M$ acts on $(n_1 , w)$ according to
$$
M ={1 \over \mu^2} \pmatrix{ -\partial} \def\alfa{\alpha} \def\ro{\rho^2+4h^2 & { 4 h x\over x+1} \partial} \def\alfa{\alpha} \def\ro{\rho_1 \cr
-{4hx \over x+1} \partial} \def\alfa{\alpha} \def\ro{\rho_1 & -\partial} \def\alfa{\alpha} \def\ro{\rho^2+4h^2
\left({x-1\over x+1}\right)^2\cr} \, .
\efr
Here $\mu$ is a mass-scale introduced to make the eigenvalues
of $M$ dimensionless.
Note also that the field $\psi$ is completely decoupled to this order
in the loop expansion.
The operator above is rather unconventional in nature and it proves convenient
to evaluate its determinant using zeta-function techniques.
Some basic facts concerning these methods, together with their application to
operators of the above type, are summarized in the appendix.
The important point for our purposes is that
$$
\ln {\rm Det} \, M = - \zeta_M ' (0),
\nameformula} \def\numali{\numero{ZEQN}
where the zeta-function $ \zeta_M (s)$ corresponding to the operator $M$ can be
represented by
$$
\zeta_M (s) = { 2 V \mu^{2s} \over \Gamma(s)} \int_0^\infty dt \, t^{s-1}
\int {d^2 p \over (2 \pi)^2}
e^{ -t (p^2 + \lambda)} \cosh\left( t ( \eta^2 p_1^2 + \rho^2
)^{1/2}\right)\, ,
\nameformula} \def\numali{\numero{INT}
with
$$
\lambda = { 4h^2(x^2 + 1) \over (x+1)^2} \, , \qquad
\rho = { 8 h^2 x \over (x + 1)^2} \, , \qquad
\eta = {4hx \over x+1}.
\nameformula} \def\numali{\numero{CONSTS}
The factor $V$ denotes formally the volume of two-dimensional
spacetime (strictly speaking
this should be dealt with using some explicit infra-red
regularization but these details are irrelevant for our purposes).
Unfortunately the integral above cannot be evaluated in closed form,
but we shall sketch below how it can be successfully expanded in powers of
$1/x$ to the order which we need.
(Recall that the ultra-violet limit of our models is $x \to \infty$ with $k$
fixed.)
The strategy is to expand the cosh factor in \INT\
as a series and collect terms of a given power in $p_1$ so that the momentum
integrals can be evaluated.
The $t$ integrals can then be expressed in terms of $\Gamma$-functions,
and one obtains the result
$$
\zeta_M (s) = {V \mu^{2s} \over 2 \pi \Gamma (s)}
\sum_{m = -1}^\infty C_m \lambda^{-(s+m)} \Gamma (s + m) \, ,
\efr
where the coefficients are given by the rather complicated expressions
$$
C_m = \sum_{ {m=2n-r-1 \atop 0\leq r\leq n\leq \infty } }
{ (2r)! n! \over (2n)! (r!)^2 (n-r)! 2^{2r}} \eta^{2r} \rho^{2(n-r)} \, .
\nameformula} \def\numali{\numero{UGH}
It is not obvious how to sum the series above completely, but we note
that $\rho$ in these expressions is
${\cal O}(1/x)$ which will enable us to simplify things presently.
On differentiating and setting $s=0$ we find
$$
\zeta_M '(0) = {V\over 2 \pi} \left \{
- \lambda + \left (\lambda - {\eta^2 \over 4} \right )
\ln {\lambda \over \mu^2}
+ \sum_{m=1}^\infty C_m \lambda^{-m} \Gamma (m)
\right \}.
\nameformula} \def\numali{\numero{ANS}
At this stage our result is still exact, but we must now consider its
behaviour as a power series in $\rho$ (and hence $1/x$) to make
further progress. The leading contribution is ${\cal O} (\rho^0)$;
extracting all such terms from the expressions \UGH\ for $C_m$ with $m \geq 1$
gives us an infinite series which can be summed\note{
$\sum_{n=2}^{\infty} {1 \over n (n-1)} y^n = (1 -y) ( \ln(1-y) - 1) + 1$
for suitable $y$.}
to yield
$$
{ V \over 2 \pi} \lambda \left \{
\left( 1 - {\eta^2 \over 4 \lambda }\right)
\left( \ln \left( 1 - {\eta^2 \over 4 \lambda} \right) - 1 \right )
+ 1 \right \}.
\nameformula} \def\numali{\numero{SUMA}
The next contributions are ${\cal O} (\rho^2)$ and again the series resulting
from the coefficients $C_m$ with $m\geq1$ can be summed\note{
$\sum_{n=1}^\infty {1 \over 2n -1} z^{2n -1} = {1 \over 2} ( \ln (1 + z ) -
\ln (1 - z ) )$ for suitable $z$.}
to give
$$
{V \over 2 \pi} {\rho^2 \over 2 \eta \sqrt{\lambda} }
\left \{ \ln \left( 1 + {\eta \over 2 \sqrt \lambda} \right )
- \ln \left( 1 - {\eta \over 2 \sqrt \lambda} \right ) \right \}.
\nameformula} \def\numali{\numero{SUMB}
The other contributions to \ANS\ are ${\cal O}(\rho^4) = {\cal O}(1/x^4)$
and we neglect them.
Now we substitute \SUMA\ and \SUMB\ in \ANS\ and expand each of
$\lambda$, $\rho$ and $\eta$ in powers of $1/x$ using \CONSTS .
It turns out that although \SUMA\ is ${\cal O} (\rho^0)$ and
$\rho = {\cal O} (1/x)$ the particular
form of the expressions for $\lambda$ and $\eta$ imply that \SUMA\ is,
more precisely, ${\cal O} (1/x^2)$. The final result is
$$
\zeta_M ' (0) = { V \over 2 \pi} {4 h^2 \over x^2}
\left [ \ln {16 h^2 \over \mu^2} - 1 \right] + {\cal O} (1/x^3).
\efr
Now to obtain the required determinant in \ONEL\ all we need do is restore
the factor $e^2 (x+1)^2$
(which we dropped in \ZEQN\
for simplicity) by rescaling $\mu$
and use the consequence \XEXP\ of the
quantization condition \QK\ to eliminate
$x$ in favour of $e$ and $k$.
This gives a final result
for the change in the one-loop contribution to the free-energy per unit
volume of
$$
\delta f(h)_1={h^2k\over\pi}\left\{{e^2\over2\pi}
\left(1-\ln{8kh^2\over\pi\mu^2}\right)+{\cal O}(e^3)\right\}.
\efr
The change in the free-energy $\delta f_0+\delta f_1$
which we have just calculated
is a renormalization group invariant quantity,
{\fam\itfam\eightit i.e.} it is independent of $\mu$ when the
coupling constant runs with $\mu$.
This fact allows us to determine the running of the coupling, as expressed
by the beta-function
$$
\mu{\partial e\over\partial\mu}=-\sqrt{2\over\pi
k}e^2-\beta_1e^3+{\cal O}(e^4),
\nameformula} \def\numali{\numero{BF}
although the coefficient $\beta_1$ is not determined to the order that
we are working. Since the first two coefficients of the beta-function
are universal, it is comforting that \BF\ is consistent with the
calculation of the beta-function via the background field method
\BCO. It is convenient to use the fact that
the free-energy is independent of $\mu$ to set $\mu=h$
and then, integrating \BF , one obtains
$$
e(h)=\sqrt{\pi k\over2}\xi+\beta_1\left({\pi k\over2}\right)^{3/2}\xi^2\ln\xi
+\beta_1^2\left({\pi
k\over2}\right)^{5/2}\xi^3\left(\ln^2\xi+\ln\xi\right)
+{\cal O}(\xi^3),
\efr
where $\xi^{-1}=\ln(h/\Lambda_\zeta)$ is defined in terms of
the $\Lambda$-parameter of the zeta-function regularization scheme
and the expansion is in terms of the form $\xi^m\ln^n\xi$ with $m>n$.
Hence to one-loop we deduce
$$\eqalign{
\delta f(h)=&-{h^2k\over\pi}\left\{1-{k\over2}\xi-{\pi
k^2\beta_1\over4}\xi^2\ln\xi-{k\over4}\left(1-\ln{8k\over\pi}\right)\xi^2
\right.\cr &\quad\left.-{\pi^2k^3\beta_1^2\over8}\xi^3\ln^2\xi-{\pi
k^2\beta_1\over4}
\left({\pi k\beta_1\over 2}+1-\ln{8k\over\pi}\right)\xi^3\ln\xi
+{\cal O}(\xi^3)\right\}
}\nameformula} \def\numali{\numero{FEP}
The effect of higher loops would be to introduce corrections at the same
order in $\xi$ but with coefficients which are suppressed by powers of $1/k$.
We now reach our main result: at the order to which we are working
the two expansions \FES\ and \FEP\ are consistent, a fact which provides a
highly non-trivial check of the conjectured S-matrix.
Furthermore, by comparing the
two expressions we can extract the mass-gap ratio for large $k$:
$$
\ln {m\over\Lambda_\zeta}=
-{1\over2}+{3\over2}\ln2+{k\over4}+{k\over2}\ln{k\over4} +
{\cal O}(1/k).
\nameformula} \def\numali{\numero{MG}
We also deduce that the second coefficient of the beta-function is, for
large $k$, simply $\beta_1=-1/\pi+{\cal O}(1/k)$, a result which is in
perfect agreement with the second coefficient of the beta-function
computed directly using the background field method \BCO.
\chapter{Conclusions}
We have investigated a series of theories that generate their mass
dynamically but which are asymptotically (in the ultra-violet limit)
non-trivial CFTs. The
theories are in addition integrable, a property
implying that their S-matrices factorize, allowing us to
conjecture a form for these S-matrices.
The calculations of the free energy which we carried out provide
a highly non-trivial check on the form of the S-matrices since,
as pointed out in [\Ref{BNNW}], the addition of any CDD factors
would drastically alter the thermodynamics of the system and
consequently destroy the remarkable consistency between the TBA calculation
and the perturbative result. We can conclude with some confidence therefore
that the proposed S-matrices are correct and that the classical integrability
of these models extends to the quantum regime. Notice that the
S-matrix has a quantum group symmetry, an invariance which does not
seem to be manifested at the lagrangian level in any simple fashion.
It is worth pointing out that the leading order behaviour
$-h^2k/\pi$
of the free-energy near the ultra-violet fixed-point
can easily be deduced from the knowledge that
in this limit the theory is an SU(2) WZW model of level $k$. At the fixed
point the chemical potential couples to
the combination ${\tilde J}={\tilde J}_L+{\tilde J}_R$, where ${\tilde
J}_L$ and ${\tilde J}_R$ are the left and
right currents of a U(1) subalgebra of the SU(2) current algebra.
Following [\Ref{FI}], the response of the free-energy in a finite
volume $V$ is given by
$$
\delta f(h)=-{h^2\over2V}\int{d^2z\over2\pi}\int{d^2w\over2\pi}\langle\tilde
J(z)\tilde J(w)\rangle,
\efr
where the expectation value is evaluated in the WZW model.
The final result is proportional to the anomaly in the U(1) currents, however
one must take careful account of the form of the operator products in
a finite volume [\Ref{FI}]:
$$\eqalign{
\langle\tilde J_L(z)\tilde J_L(w)\rangle={k\over(z-w)^2}+{2\pi
k\over V},&\qquad
\langle\tilde J_R(z)\tilde J_R(w)\rangle={k\over(\overline z-\overline
w)^2}+{2\pi k\over V},\cr
\langle\tilde J_L(z)\tilde J_R(w)\rangle&=2\pi k\delta^{(2)}
(z-w)-{2\pi k\over V}.\cr}
\efr
It is then straightforward to to extract [\Ref{FI}]
$$
\delta f(h)=-{h^2 k\over\pi},
\efr
in agreement with the leading order term in \FES\ and \FEP.
Obviously one could
extend this calculation away from the fixed-point by perturbation
theory and hope to reproduce the series \FEP; a calculation which
would be interesting since it would be valid not just in the large $k$
regime.
We should also emphasize a remarkable consequence of the equivalence
of the lagrangian and S-matrix descriptions we have established in this paper:
namely that the
field theory \LAG\ for $k=1$ and $k=\infty$ is quantum equivalent to
the SU$(2)$ chiral Gross-Neveu model and principal chiral model,
respectively (since our Ansatz for the S-matrix then reduces to these
well-known cases).
Let us consider the latter equivalence in more detail. In
taking the limit $k\rightarrow\infty$, at fixed $e$,
it is necessary to introduce the field
$r=((\pi/2)-w)/\sqrt{x^2-1}$. The lagrangian then has a well-defined limit
$$
{\cal L}_0={1\over2 e^2}\left\{\left(\partial_\mu
r\right)^2+{r^2\over1+4r^2}\left(\partial_\mu n_a\right)^2+
{2r^3\over1+4r^2}\epsilon_{abc}\epsilon^{\mu\nu}n_a\partial_\mu n_b
\partial_\nu n_c\right\}.
\efr
With a simple change of variables $\phi_a=r n_a$, this may be written
$$
{\cal L}_0={1\over2e^2}\left({1\over1+4\phi^2}\right)
\left\{\left(\delta_{ab}+4\phi_a\phi_b\right)\partial_\mu\phi_a
\partial^\mu\phi_b+2\epsilon_{abc}\epsilon^{\mu\nu}\phi_a\partial_\mu
\phi_b\partial_\nu\phi_c\right\}.
\efr
This lagrangian is actually the non-abelian dual of the SU$(2)$
principal chiral model [\Ref{NAD}] and hence it is indeed known to be quantum
equivalent to it. It would be interesting to show, in a similar spirit,
that \LAG\ with $k=1$ was a bosonized form of the SU$(2)$
chiral Gross-Neveu model.
Finally, it would clearly be interesting to extend the various results
above to larger groups and
also to S-matrices of the more general form \SGEN.
TJH would like to thank Michel Bauer for many interesting discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 245 |
Northern Technologies International Corporation (NTIC) is in the business of converting unique environmentally beneficial materials science into value added products and services for industrial and consumer applications.
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Search our current openings to see if there is a fit based on your skills and interests. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,604 |
Q: How to do Text classification using tensorflow? I am new to tensorflow and machine learning. I am facing issues with writing a tensorflow code which does the text classification similar to one I tried using sklearn libraries. I am facing major issues with vectorising the dataset and providing the input to tensorflow layers.
I do remember being successful in one hot encoding the labels but the tensorflow layer ahead did not accept the created array.
Please note, I have read majority of text clasification answered questions on stackoverflow but they are too specific or have complex needs to resolve.
My problem case is too narrow and requires very basic solution.
It would be great help if anyone could tell me the steps or tensorflow code similar to my sklearn machine learning algorithm.
Dataset used is avaialable at : https://www.kaggle.com/virajgala/classifying-text
from sklearn.feature_extraction.text import TfidfVectorizer
from sklearn.model_selection import train_test_split, GridSearchCV
from sklearn.linear_model import SGDClassifier
from sklearn.pipeline import Pipeline
#Reading the csv dataset
df = pd.read_csv(('/Classifyimg_text.csv'), index_col=False).sample(frac=1)
#Splitting the dataset
train_data, test_data, train_labels, test_labels = train_test_split(df['sentence'], df['label'], test_size=0.2)
#Vectorization and Classification
streamline = Pipeline([('vect', TfidfVectorizer(max_features=int(1e8))),
('clf', SGDClassifier())]).fit(train_data, train_labels)
#Prediction
Output = streamline.predict(["This is my action to classify the text."])
A: If you want to achieve seminal scores I'd rather use some embedder. Natural language is rather quite hyper-dimensional. Nowadays there's a lot of pretrained architectures. So, you simply encode your text to latent space and later train your model on those features. It's also much easier to apply resampling techniques, once you have numerical feature vector.
Myself, I mostly use LASER embedder from Facebook. Read more about it here. There's unofficial pypi package, which works just fine. Additionally, your model will be working on dozens of languages out-of-the-box, which is quite cute.
There's also BERT from Google, but the pretrained model is rather bare, so you have to push it a bit further first.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,496 |
local lock = require "resty.lock"
local setmetatable = setmetatable
local tonumber = tonumber
local concat = table.concat
local now = ngx.now
local shared = ngx.shared
local function enabled(val)
if val == nil then return nil end
return val == true or (val == "1" or val == "true" or val == "on")
end
local defaults = {
store = ngx.var.session_shm_store or "sessions",
uselocking = enabled(ngx.var.session_shm_uselocking or true),
lock = {
exptime = tonumber(ngx.var.session_shm_lock_exptime) or 30,
timeout = tonumber(ngx.var.session_shm_lock_timeout) or 5,
step = tonumber(ngx.var.session_shm_lock_step) or 0.001,
ratio = tonumber(ngx.var.session_shm_lock_ratio) or 2,
max_step = tonumber(ngx.var.session_shm_lock_max_step) or 0.5,
}
}
local shm = {}
shm.__index = shm
function shm.new(config)
local c = config.shm or defaults
local l = enabled(c.uselocking)
if l == nil then
l = defaults.uselocking
end
local m = c.store or defaults.store
local self = {
store = shared[m],
encode = config.encoder.encode,
decode = config.encoder.decode,
delimiter = config.cookie.delimiter,
uselocking = l
}
if l then
local x = c.lock or defaults.lock
local s = {
exptime = tonumber(x.exptime) or defaults.exptime,
timeout = tonumber(x.timeout) or defaults.timeout,
step = tonumber(x.step) or defaults.step,
ratio = tonumber(x.ratio) or defaults.ratio,
max_step = tonumber(x.max_step) or defaults.max_step
}
self.lock = lock:new(m, s)
end
return setmetatable(self, shm)
end
function shm:key(i)
return self.encode(i)
end
function shm:cookie(c)
local r, d = {}, self.delimiter
local i, p, s, e = 1, 1, c:find(d, 1, true)
while s do
if i > 2 then
return nil
end
r[i] = c:sub(p, e - 1)
i, p = i + 1, e + 1
s, e = c:find(d, p, true)
end
if i ~= 3 then
return nil
end
r[3] = c:sub(p)
return r
end
function shm:open(cookie, lifetime)
local r = self:cookie(cookie)
if r and r[1] and r[2] and r[3] then
local i, e, h = self.decode(r[1]), tonumber(r[2]), self.decode(r[3])
local k = self:key(i)
if self.uselocking then
local l = self.lock
local ok, err = l:lock(concat{k, ".lock"})
if ok then
local s = self.store
local d = s:get(k)
s:set(k, d, lifetime)
l:unlock()
return i, e, d, h
end
return nil, err
else
local s = self.store
local d = s:get(k)
s:set(k, d, lifetime)
return i, e, d, h
end
end
return nil, "invalid"
end
function shm:start(i)
if self.uselocking then
return self.lock:lock(concat{self:key(i), ".lock"})
end
return true, nil
end
function shm:save(i, e, d, h, close)
local l = e - now()
if l > 0 then
local k = self:key(i)
local ok, err = self.store:set(k, d, l)
if self.uselocking and close then
self.lock:unlock()
end
if ok then
return concat({ k, e, self.encode(h) }, self.delimiter)
end
return nil, err
end
if self.uselocking and close then
self.lock:unlock()
end
return nil, "expired"
end
function shm:destroy(i)
self.store:delete(self:key(i))
if self.uselocking then
self.lock:unlock()
end
return true, nil
end
return shm | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,264 |
What went wrong in Indy, and can the Bears beat the Jaguars?
It's our sleepy podcast coming after the late-night Cubs game, where we recap Bears-Colts and discuss Brian Hoyer vs. Jay Cutler (2:58) and what went wrong in Indianapolis. We turn the page to the Jaguars and go Behind Enemy Lines with Ryan O'Halloran (@ryanohalloran) of the Florida Times-Union to learn about Jacksonville (12:48), talking Blake Bortles, Myles Jack and Jalen Ramsey. Is this a must-win for the Bears? The guys close up shop with their pick for Sunday's game, plus ... See More some hot dog talk for John Fox. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,125 |
Advancing Localization One Click at a Time
Photo credit: USAID/India
by: Colleen Allen
Colleen Allen is the Assistant Administrator for USAID's Bureau for Management.
USAID Assistant Administrator Colleen Allen
Big things are happening these days at USAID's headquarters in Washington, D.C., and at our Missions around the world—and many of them have to do with localization.
Just two weeks ago, on October 19, the Agency launched our first-ever Local Capacity Strengthening (LCS) Policy, which encapsulates USAID's approach to engaging with local partners in the countries where we work. Sitting in the audience for that event, I was reminded of another important localization event, held one year ago today at Georgetown University.
In her speech entitled "A New Vision for Global Development," Administrator Samantha Power outlined her localization priorities and announced the launch of WorkwithUSAID.org—a free, global one-stop shop for partnership tools and resources.
The site, which is celebrating its one-year anniversary today, has changed the way organizations engage with the Agency.
WorkwithUSAID.org home page
The site lowers barriers to partnership for new and nontraditional partners by proactively offering tools and resources designed to be welcoming, approachable, and understandable. And it brings USAID together with both new and established partners to share knowledge and enhance transparency. Finally, it provides a common platform for raising visibility so that partnership can be about "what you know" not about "who you know."
Over the past year, the site's performance has been wonderful to watch:
More than 3,100 organizations from over 140 countries have registered in the Partner Directory, and more than 60 percent of these are local partners;
More than 2,000 organizations have taken the site's Pre-Engagement Assessment to help them understand their partnership readiness; and
More than 90 pieces of content have been shared on the News & Insights blog, ensuring a steady supply of fresh guidance and encouragement, from inspirational success stories to insightful Q&A-style interviews with USAID staff.
Even better, the WorkwithUSAID.org website is a key pillar of USAID's designation as a federal High-Impact Service Provider (HISP). The site's focus on the customer experience represents a new approach to engaging with the partner ecosystem and the public overall. I am proud of how, as a HISP, USAID is demonstrating its leadership among federal agencies in "doing business differently" in order to become more open, responsive, and streamlined.
So, on this one-year anniversary of WorkwithUSAID.org, I am feeling encouraged by the progress we have made in fostering a shared localization mindset at USAID and taking real steps to work hand-in-hand with local partners to find solutions to the world's pressing development challenges. I look forward to the upcoming year and all the progress we will make now that both the policies, like the LCS Policy, and the tools, like WorkwithUSAID.org, are in place and gaining momentum. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,215 |
After the war was lost, Ferraro persevered with another war, until the final defeat in 1945. In this war, he didn't kill anyone, never fired against other Italians. On the contrary, in agreement with the partisans who, in theory, were his enemies, he saved men and important industrial estates from the Nazi retaliation.
The "Gamma" Group "Licio Visintini", of which Ferraro was part, did not dissolve on September 8th 1943. Instead, it was fully handed over to the R.S.I. From November 1943, he was in Valdagno (Vicenza). In January 1945, the "Gamma Group" was split into various Squads, which would have had to act behind enemy lines to carry out sabotage as the front line advanced. In April 1945, some of these Squads were already stationed in the areas where they were scheduled to operate. The Command stayed in Valdagno with Ferraro and about forty men. The Commanding Officer Wolk was given another assignment and was transferred to Venice.
"On April 26th, the Council Secretary and two representatives of the C.L.N. came here and said: "The authority passed from the Social Republic over to the C.L.N., we need your help [ ... ] there is a German column that wants to blow up the bridge and the premises". I answered that I was and remained the highest military authority in the country. Therefore, in order to save the bridge and the Marzotto premises, I would have to intervene in person by leading the German column out of Valdagno, without causing any difficulty to the partisans.
I went to negotiate. The column wanted to pass through Valdagno and I needed some guarantees on the part of the C.L.N in order to make an agreement with the Germans.
Dressed as the "X Mas" Commander, on the day following the liberation I went into the village. I was looking for valid partisan leaders, but each one of them claimed to be a leader. Finally, I found two guys in the centre of Valdagno, whom I felt I could rely upon more. Same discussions, same promises.
Then, when I returned to the column's Commander, a German captain, and after I gave him the assurances I received, he said: "You have to follow me". I was on my motorbike. The Captain, pointing his gun at me, followed me together with the column. Everything went smoothly and we went through Valdagno without any problems. Outside the village, I said goodbye and came back.
This episode lasted around fifteen days. On one occasion, a German column, instead of stopping near Valdagno, decided to stop at Cernedo, 5 kilometers before the town. An official arrived with an interpreter from Trieste, saying that they intended to take a number of hostages. The discussion lasted for many hours. In the afternoon, a crackle of firearms came from the undergrowth. There was great alarm among the Germans. A marshal pointed a machine-gun at my chest, I snatched it away from him and started shouting curses at him and at everyone else. An officer rushed towards us, a Major, who, in a low tone, told me he was prepared to believe me and to accept my suggestions. "Head to Vicenza and go through the Schio valley".
They all followed me, the Major with the entire column. I led the way as usual, on my motorbike. Then we said goodbye. At the last moment, the Major said: "You should stop here, we have to go [ ... ] give me your motorbike, we need it more than you do". I gave him the motorbike, then I realised that I was 5 kilometres away from Valdagno and I was alone with a sailor. In Valdagno I was protected by the C.L.N., but there I definitely wasn't. So we went in the river and made our way back that way.
In the meanwhile, on a daily basis, I was discharging the sailors of the Group one by one, with safe conduct by the "Stella" Brigade. Eventually I was left alone with some non-commissioned officers. One day, to prevent the Germans from passing through, the partisans told me they intended to blow up the bridge, but I objected. It would have been a terrible mistake. "I will mine it myself" I told them "But I'll only blow it up if I'm forced to". I went there with my non-commissioned officers and I mined the bridge, standing by to detonate the explosive. "If the Germans execute me, go ahead with the blast, otherwise everything should stay as it is" I ordered the C.L.N. The Germans crossed over again and the bridge was saved.
On one particular day, during the second half of the month of May, something changed in their attitude towards me. The Command of the "Stella" brigade, informed me that they decided to transfer me to Valdagno di Sopra, where a dangerous mob was based. "No chance" I answered. "I have nothing in common with them. You approached me, asked me to cooperate and so I did. If that's how things stand, I'll go back to the barracks". Then another change in attitude. They started talking of safe conduct: it had to be the only logical solution.
"I sent all the Navy material to La Spezia, where it was delivered on a regular basis." As per the rest of the material, I sent it to the Municipal Administration. After gradually discharging all the personnel, I was left completely alone. At the end of May, when everything came to an end, we embraced and said goodbye and I headed to Bergamo where my family lived.
Once all the men were back home, I started hearing that some of them were arrested here and there. So I went to Venice to the allied Command to protest, and the Command sent dispatches to all police departments, so that those arrested were released immediately".
On May 27, Lieutenant Commander Lionel Crabb, a famous Royal Navy "frogman" and Major Antony Marzullo from the U.S. Navy, arrived in Valdagno and offered Ferraro cooperation with the allied navy forces in the war against Japan. Ferraro thanked them for the proposal but refused. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,068 |
\section{Interpolators}
The single particle interpolating operators for $\Sigma_c$, $\bar{D}$ and $\bar{D}^*$ are:
\begin{eqnarray}
\Sigma_{c,\alpha}^{++}&=&\epsilon^{ijk}({u^i}^T C\gamma_5 c^j) u^k_\alpha \\
\Sigma_{c, \alpha}^{+} &=& \frac{1}{2}\epsilon^{ijk}[({u^i}^T C\gamma_5 c^j) d^k_\alpha + ({d^i}^T C\gamma_5 c^j) u^k_\alpha] \\
D^- &=& \bar{c} \gamma_5 d, \quad \bar{D}^0 = \bar{c} \gamma_5 u \\
D_k^{*-} &=& \bar{c} \gamma_k d, \quad \bar{D}_k^{*0} = \bar{c} \gamma_k u, \quad k = 1,2,3, \
\end{eqnarray}
where $C$ is the charge conjugation matrix.
The two-particle operators for $\Sigma_c \bar{D}$ and $\Sigma_c\bar{D}^*$ with $I(J^P) = \frac{1}{2}({\frac{1}{2}}^-)$ are
\begin{eqnarray}
\mathcal{O}^{\Sigma_c\bar{D}}_{\mathbf{p_1}, \mathbf{p_2}} &=&\sum_{\alpha, \mathbf{p_1}, \mathbf{p_2}} C_{\alpha, \mathbf{p_1}, \mathbf{p_2}} \big( \sqrt{\frac{2}{3}} \Sigma_{c, \alpha}^{++} (\mathbf{p_{1}})D^{-} (\mathbf{p_{2}}) \nonumber \\
&& - \sqrt{\frac{1}{3}}\Sigma_{c, \alpha}^{+}(\mathbf{p_{1}}) \bar{D}^0(\mathbf{p_{2}}) \big),\\
\mathcal{O}^{\Sigma_c\bar{D}^*}_{\mathbf{p_1}, \mathbf{p_2}}&=& \sum_{\alpha, k, \mathbf{p_1}, \mathbf{p_2}} C_{\alpha, k, \mathbf{p_1}, \mathbf{p_2}} \big( \sqrt{\frac{2}{3}} \Sigma_{c, \alpha}^{++}(\mathbf{p_1}) D_k^{*-}(\mathbf{p_2}) \nonumber \\
&& - \sqrt{\frac{1}{3}}\Sigma_{c, \alpha}^{+}(\mathbf{p_1}) \bar{D}_k^{*0}(\mathbf{p_2})\big).
\end{eqnarray}
We use three $\Sigma_c\bar{D}$ operators with $|\mathbf{p_{1,2}}| = 0, 1$ and $\sqrt{2}$(in units of $2\pi/L$) and two $\Sigma_c\bar{D}^*$ operators with $|\mathbf{p_{1,2}}| = 0$ and $1$. The coefficients $C_{\alpha, \mathbf{p_1}, \mathbf{p_2}}$ and $C_{\alpha, k, \mathbf{p_1}, \mathbf{p_2}}$ are chosen so that these operators transform in the $G_1^-$ irrep of the cubic group. $G_1$ is a two-dimensional representation. We use only the first row which is sufficient for the calculation. The coefficients are listed in TABLE~\ref{Table:SigmacDOps} and TABLE~\ref{Table:SigmacDstarOps} for $\Sigma_c\bar{D}$ and $\Sigma_c\bar{D}^*$ respectively. Note that these coefficients are worked out using the Dirac-Pauli representation for Dirac $\gamma$ matrices.
\begin{table}
\begin{tabular}{|c|c|c|c|c|}
\hline
& $\alpha$ &$\mathbf{p_1}$ &$\mathbf{p_2}$ & $C_{\alpha, \mathbf{p_1}, \mathbf{p_2}}$ \\
\hline
\hline
$|\mathbf{p_{1,2}}| = 0$ & 1 & (0,0,0) & (0,0,0) & 1 \\
\hline
\hline
\multirow{6}{*}{$|\mathbf{p_{1,2}}| = 1$} &1 & (-1,0,0) & (1,0,0) & 1\\
\cline{2-5}
&1 & (1,0,0) &(-1,0,0) & 1 \\
\cline{2-5}
&1 & (0,-1,0) &(0,1,0) & 1 \\
\cline{2-5}
&1 & (0,1,0) &(0,-1,0) & 1 \\
\cline{2-5}
&1 & (0,0,-1) &(0,0,1) & 1 \\
\cline{2-5}
&1 & (0,0,1) &(0,0,-1) & 1 \\
\hline
\hline
\multirow{12}{*}{$|\mathbf{p_{1,2}}| = \sqrt{2}$} &1 & (-1,-1,0) & (1,1,0) & 1\\
\cline{2-5}
&1 & (1,1,0) &(-1,-1,0) & 1 \\
\cline{2-5}
&1 & (-1,0,-1) &(1,0,1) & 1 \\
\cline{2-5}
&1 & (1,0,1) &(-1,0,-1) & 1 \\
\cline{2-5}
&1 & (0,-1,-1) &(0,1,1) & 1 \\
\cline{2-5}
&1 & (0,1,1) &(0,-1,-1) & 1 \\
\cline{2-5}
&1 & (-1,1,0) &(1,-1,0) & 1\\
\cline{2-5}
&1 & (1,-1,0) &(-1,1,0) &1 \\
\cline{2-5}
&1 &(-1,0,1) &(1,0,-1) &1 \\
\cline{2-5}
&1 &(1,0,-1) &(-1,0,1) &1 \\
\cline{2-5}
&1 &(0,1,-1) &(0,-1,1) &1 \\
\cline{2-5}
&1 &(0,-1,1) &(0,1,-1) & 1 \\
\hline
\end{tabular}
\caption{The coefficients of the $\Sigma_c\bar{D}$ operators.}
\label{Table:SigmacDOps}
\end{table}
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
&$\alpha$ & $k$ &$\mathbf{p_1}$ &$\mathbf{p_2}$ & $C_{\alpha, k, \mathbf{p_1}, \mathbf{p_2}}$ \\
\hline
\hline
\multirow{3}{*}{$|\mathbf{p_{1,2}}| = 0$} &1 &3 &(0,0,0) & (0,0,0) & 1\\
\cline{2-6}
&2 &1 &(0,0,0) & (0,0,0) & $1$\\
\cline{2-6}
&2 &2 &(0,0,0) & (0,0,0) & $-i$\\
\hline
\hline
\multirow{6}{*}{$|\mathbf{p_{1,2}}| = 1$} &1 &3 &(0,0,-1 & (0,0,1) & 1\\
\cline{2-6}
&1 &3 &(0,0,1) & (0,0,-1) & $1$\\
\cline{2-6}
&2 &1 &(1,0,0) & (-1,0,0) & $1$\\
\cline{2-6}
&2 &1 &(-1,0,0) & (1,0,0) & $1$\\
\cline{2-6}
&2 &2 &(0,-1,0) & (0,1,0) & $-i$\\
\cline{2-6}
&2 &2 &(0,1,0) & (0,-1,0) & $-i$\\
\hline
\end{tabular}
\caption{The coefficients of the $\Sigma_c\bar{D}^*$ operators.}
\label{Table:SigmacDstarOps}
\end{table}
\section{Computation and analysis of the correlation functions}
The distillation quark smearing method is used to compute the quark propagators. The quark smearing operator is composed of a small number($N_{ev}$) of the eigenvectors of the three-dimensional Laplacian that correspond to the $N_{ev}$ lowest eigenvalues. We compute the propagators with $N_{ev} = 100$ for the L32 ensemble and $N_{ev} = 200$ for the L48 ensemble.
The single particle energies are extracted from the two-point correlation functions of the pertinent single particle operators. In FIG.~\ref{Figure:em_Sigmac},
we present the effective energies of $D$, $D^*$ and $\Sigma_c$ at the lowest five momenta for the ensemble L48. The fit of the five energies to the dispersion relation for each particle is shown in FIG.~\ref{Figure:Dispersion}.
\begin{figure*}[tb]
\includegraphics[width =0.33 \textwidth]{em_pion_uc_2pt_all.pdf}\includegraphics[width =0.33 \textwidth]{em_rho_uc_2pt_all.pdf}\includegraphics[width =0.33 \textwidth]{em_Sigmac_pp_all.pdf}
\caption{Effective energies of $D$, $D^*$ and $\Sigma_c$ at the five lowest momenta for the ensemble L48.}
\label{Figure:em_Sigmac}
\end{figure*}
\begin{figure*}[tb]
\includegraphics[width =0.4 \textwidth]{D_Dstar_dispersion.pdf}\includegraphics[width =0.4 \textwidth]{corr_Sigmac_pp_dispersion.pdf}
\caption{Fits of the energies of $D$, $D^*$ and $\Sigma_c$ to the dispersion relation for the ensemble L48. The values of $\chi^2$ of the fits are shown in the plots.}
\label{Figure:Dispersion}
\end{figure*}
The finite volume two-particle energies are obtained from the matrix of the correlation functions of the five operators described in the last section. The charm quark annihilation diagrams are ignored in the calculation of the correlation functions. Solving the generalized eigenvalue problem(GEVP)
\begin{equation}
C(t) v^n(t) = \lambda^n(t) C(t_0) v^n(t),
\end{equation}
the energies are determined by fitting the eigenvalues $\lambda^n(t)$ to the form
\begin{equation}
\lambda^n(t) = (1-A_n)e^{-E_n(t-t_0)} + A_ne^{-E_n^\prime(t-t_0)},
\end{equation}
where the fit parameters are $A_n$, $E_n$ and $E_n^\prime$. This form allows for a second exponential to capture the residual contaminations from the excited states. We tried four different values of $t_0$: 4, 6, 8 and 10, and did not observe differences in the fitted energies. The fits of the five eigenvalues for $t_0=4$ are shown in FIG.~\ref{Figure:Fits_Eigvals}
for the ensemble L48. The fitted energies are collected in TABLE~\ref{Table:energies}
for both ensembles. We also presented the three energies extracted from the GEVP analysis using only the $\Sigma_c\bar{D}$ operators and the two energies using only the $\Sigma_c\bar{D}^*$ operators. They agree perfectly with the values using all five operators, indicating negligible mixing between the $\Sigma_c\bar{D}$ and $\Sigma_c\bar{D}^*$ operators.
\begin{figure*}[tb]
\includegraphics[width =0.33 \textwidth]{eigvals_0_t0_4_2exp_L48.pdf}\includegraphics[width =0.33 \textwidth]{eigvals_1_t0_4_2exp_L48.pdf}\includegraphics[width =0.33 \textwidth]{eigvals_2_t0_4_2exp_L48.pdf}
\includegraphics[width =0.33 \textwidth]{eigvals_3_t0_4_2exp_L48.pdf}\includegraphics[width =0.33 \textwidth]{eigvals_4_t0_4_2exp_L48.pdf}
\caption{Fits of the eigenvalues $\lambda_n(t)$. Plotted are the data $\lambda_n(t) e^{E_n(t-t_0)}$ and the fits. The blue points are those included in the fits.}
\label{Figure:Fits_Eigvals}
\end{figure*}
\begin{table}
\begin{tabular}{|c|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{} & all ops. & $\mathcal{O}^{\Sigma_c \bar{D}}$ & $\mathcal{O}^{\Sigma_c \bar{D}^*}$ \\
\hline
\hline
\multirow{5}{*}{L48} &$aE_0$ &1.7738(09) &1.7738(09) &1.8160(10) \\
\cline{2-5}
&$aE_1$ &1.7845(11) &1.7845(11) &1.8326(12) \\
\cline{2-5}
&$aE_2$ &1.8051(11) &1.8051(11) & -- \\
\cline{2-5}
&$aE_3$ &1.8160(10) & -- & -- \\
\cline{2-5}
&$aE_4$ &1.8326(12) &-- & -- \\
\hline
\hline
\multirow{5}{*}{L32} &$aE_0$ &1.7747(12) &1.7747(12) &1.8167(20) \\
\cline{2-5}
&$aE_1$ &1.8025(19) &1.8025(20) &1.8535(16) \\
\cline{2-5}
&$aE_2$ &1.8166(20) &1.8389(21) & -- \\
\cline{2-5}
&$aE_3$ &1.8389(21) &-- & -- \\
\cline{2-5}
&$aE_4$ &1.8535(16) &-- &-- \\
\hline
\end{tabular}
\caption{The finite volume two-particle energies. For each ensemble, we list the five energies extracted from the GEVP analysis using all five operators(all ops.). The three energies using only the $\Sigma_c \bar{D}$ operators and the two energies using only the $\Sigma_c\bar{D}^*$ operators are also presented for comparison.}
\label{Table:energies}
\end{table}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,438 |
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"redpajama_set_name": "RedPajamaC4"
} | 6,436 |
{"url":"https:\/\/math.stackexchange.com\/questions\/404570\/how-to-prove-this-function-is-surjective","text":"# How to prove this function is surjective\n\nI'm trying to solve this question:\n\nIn order to solve this question above, I found this function: $r\/w\\mapsto (r\/s)\/(w\/s)$ such that $w\/s\\in T$, I almost proved this map is an isomorphism, I'm stuck just in the surjectivity part.\n\nIf we get an element $(r\/s)\/(w\/s)$ of $T^{-1}(S^{-1}R)$, ok! However, an element of $T{^1}(S^{-1}R)$ can be for example $(r\/s_1)\/(w\/s_2)$ with $s_1\\neq s_2$\n\nI need help in this part.\n\nHint: Note that in $S^{-1}R$ we have $(ws_1)\/(s_1s_2) = w\/s_2 \\in T$, so $ws_1 \\in S_*$. Now consider $rs_2\/ws_1 \\in S_*^{-1}R$.\n\nConsider an element $z$ of $T{^1}(S^{-1}R)$, we can write it in the form\n\n$$z=\\frac{\\frac{r}{s_1}}{\\frac{w}{s_2}}$$\n\nwith $r,w\\in R$, $s_1,s_2\\in S$ and further $\\frac{w}{s_2} \\in T$.\n\nLet $w'=s_1\\frac{w}{s_2}$. Then $\\frac{w'}{s_1}=\\frac{w}{s_2} \\in T$, so we deduce that $w' \\in S_{*}$. Finally $z=\\frac{r}{w'}$ with $r\\in R,w'\\in S_{*}$.\n\n\u2022 How do you define this multiplication $w'=s_1\\frac{w}{s_2}$? \u2013\u00a0user42912 May 28 '13 at 11:33\n\u2022 Second, are you saying that an element $w'$ of $S_*$ is equal than other element $s_1(w\/s_2)$ of $T$? \u2013\u00a0user42912 May 28 '13 at 11:35\n\u2022 @user42912 $s_1$ and $\\frac{w}{s_2}$ are both elements of the ring $S^{-1}R$, so multiplying them makes sense. \u2013\u00a0Ewan Delanoy May 28 '13 at 11:40\n\u2022 @user42912 The element in $T$ is $w\/s_2$, not $s_1(w\/s_2)$. \u2013\u00a0Ewan Delanoy May 28 '13 at 11:41","date":"2019-10-19 09:17:04","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\": 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\": 0.9695016741752625, \"perplexity\": 218.35726992859668}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986692723.54\/warc\/CC-MAIN-20191019090937-20191019114437-00195.warc.gz\"}"} | null | null |
The Gereg – debiutancki album mongolskiego zespołu folkmetalowego The HU, wydany w 2019 roku.
Przypisy
Albumy muzyczne wydane w roku 2019 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,981 |
Q: Mutt Notify on Finish Sending I'm looking for a way in .muttrc to tell it to invoke notify-send when it's done sending an email. I use Mutt inside a Guake window. So when I hit y to send I immediately hit F12 to close the terminal and get back to work.
I need a way for Mutt to invoke notify-send after it's done sending so I know it's done sending.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,146 |
{"url":"https:\/\/pdfhall.com\/mathematics-and-statistics-fr_5b6b4010097c47cc6e8b4571.html","text":"## Mathematics and Statistics .fr\n\np^\/a = a]lp a\u00c2\u00b0=\\ 0\" = 0. 1.1.2 Solutions of equations in one ..... DEH = COA = b also COB = a, AOB = c, and since. OB = OA = OC = radius r of sphere, OF= rcos a,\u00a0...\n1\n\nMathematics and Statistics B C Best BSc Consultant\n\nContents 1.4\n\nMathematics 1.1\n\n1.2\n\n1.3\n\nAlgebra 1.1.1 Powers and roots 1.1.2 Solutions of equations in one unknown 1.1.3 Newton\u2019s method 1.1.4 Progressions 1.1.5 Logarithms 1.1.6 Permutations and combinations 1.1.7 The binomial theorem\n\n1\/3 1\/3 1\/3 1\/3 1\/4 1\/4 1\/4 1\/5\n\nTrigonometry 1\/5 1.2.1 Positive and negative lines 1\/5 1.2.2 Positive and negative angles 1\/5 1.2.3 Trigonometrical ratios of positive and negative angles 1\/6 1.2.4 Measurement of angles 1\/6 1.2.5 Complementary and supplementary angles 1\/7 1.2.6 Graphical interpretation of the trigonometric functions 1\/7 1.2.7 Functions of the sum and difference of two angles 1\/7 1.2.8 Sums and differences of functions 1\/7 1.2.9 Functions of multiples of angles 1\/7 1.2.10 Functions of half angles 1\/7 1.2.11 Relations between sides and angles of a triangle 1\/7 1.2.12 Solution of trigonometric equations 1\/10 1.2.13 General solutions of trigonometric equations 1\/10 1.2.14 Inverse trigonometric functions 1\/10 Spherical trigonometry 1.3.1 Definitions 1.3.2 Properties of spherical triangles\n\n1\/10 1\/10 1\/11\n\nHyperbolic trigonometry 1.4.1 Relation of hyperbolic to circular functions 1.4.2 Properties of hyperbolic functions 1.4.3 Inverse hyperbolic functions\n\n1\/12 1\/12 1\/12\n\n1.5\n\nCoordinate geometry 1.5.1 Straight-line equations 1.5.2 Change of axes 1.5.3 Polar coordinates 1.5.4 Lengths of curves 1.5.5 Plane areas by integration 1.5.6 Plane area by approximate methods 1.5.7 Conic sections 1.5.8 Properties of conic sections\n\n1\/13 1\/13 1\/14 1\/14 1\/15 1\/16 1\/16 1\/17 1\/17\n\n1.6\n\nThree-dimensional analytical geometry 1.6.1 Sign convention 1.6.2 Equation of a plane 1.6.3 Distance between two points in space 1.6.4 Equations of a straight line\n\n1\/20 1\/20 1\/21 1\/21 1\/21\n\n1.7\n\nCalculus 1.7.1 Differentiation 1.7.2 Partial differentiation 1.7.3 Maxima and minima 1.7.4 Integration 1.7.5 Successive integration 1.7.6 Integration by substitution 1.7.7 Integration by transformation 1.7.8 Integration by parts 1.7.9 Integration of fractions\n\n1\/22 1\/22 1\/23 1\/23 1\/23 1\/23 1\/24 1\/24 1\/25 1\/26\n\n1.8\n\nMatrix algebra 1.8.1 Addition of matrices 1.8.2 Multiplication of matrices\n\n1\/27 1\/27 1\/27\n\nThis page has been reformatted by Knovel to provide easier navigation.\n\n1\/12\n\n1.8.3 1.8.4 1.8.5 1.8.6\n\nThe unit matrix The reciprocal of a matrix Determinants Simultaneous linear equations\n\n1\/27 1\/27 1\/27 1\/27\n\nStatistics\n\n1.15\n\n1.9\n\nIntroduction\n\n1\/28\n\n1.10\n\nDefinitions of elementary statistical concepts 1.10.1 Statistical unit or item 1.10.2 Observation \u2013 observed value\n\n1\/28 1\/28 1\/28\n\n1.11\n\nLocation 1.11.1 Measures\n\n1\/29 1\/29\n\n1.12\n\nDispersion 1.12.1 Measures\n\n1\/29 1\/29\n\n1.13\n\nSamples and population 1.13.1 Representations\n\n1\/30 1\/30\n\n1.14\n\nThe use of statistics in industrial experimentation 1.14.1 Confidence limits for a mean value\n\n1\/30 1\/30\n\n1.14.2 The difference between two mean values 1.14.3 The ratio between two standard deviations 1.14.4 Analysis of variance 1.14.5 Straight-line fitting and regression\n\n1\/33\n\nTolerance and quality control\n\n1\/34\n\n1\/33 1\/33 1\/34\n\nComputers 1.16\n\nHardware and software\n\n1\/35\n\n1.17\n\nComputers 1.17.1 The use of computers by civil engineers 1.17.2 Nontechnical computing 1.17.3 Specific vs. general-purpose software 1.17.4 Computers and information 1.17.5 Computers and management\n\n1\/35 1\/36 1\/37 1\/38 1\/38 1\/38\n\nReferences\n\n1\/39\n\nBibliography\n\n1\/39\n\nThis page has been reformatted by Knovel to provide easier navigation.\n\nMATHEMATICS\n\nl + B = L\\ - i2- ( fU\n\n1.1 Algebra\n\nand the three roots, in terms of y are:\n\n1.1.1 Powers and roots The following are true for all values of indices, whether positive, negative or fractional: ap*aq = ap+q (a\"}q = apq (alb)p = aplbp (ab)p = apbp aplaq = ap q a p = (\\\/a)p = \\\/ap p^\/a = a]lp a\u00b0=\\ 0\" = 0\n\n= [A + B]\n\ny2^ = (-(A + B)\/2\u00b1J-3(A-B)\/2]\n\nb X\n\n\\, 2, l~y\\,2, 3~~\n\nT\n\n1.1.2.4 Equations of higher degree Equations of degree higher than the second (quadratic equations) are not solvable directly as the method of solving the cubic equation above shows. Generally recourse must be had to either graphical or numerical techniques. If the equation be of the form: F(x) = Q\n\n1.1.2.1 Linear equations Generally ax + b = Q of which there is one solution or root x= \u2014b\/a\n\ne.g. anx\" + an^x\" ' . . . + 00 = 0\n\n1.1.2.2 Quadratic equations Generally ax2 + bx + c = 0 of which there are two solutions or roots\n\n-b\u00b1J(b2-4ac) 2a\n\ny]\n\nand in terms of x the three roots are:\n\n1.1.2 Solutions of equations in one unknown\n\nx_\n\n*L\\T 27; J\n\n^ Aj\n\nwhere, if b2 > 4ac, the roots are real and unequal, b2 = 4ac, the roots are real and equal, and b2, = \u00ab(\u00ab- I X \u00bb - 2 ) . . . ( \u00ab - r + . ) = ^L_\n\n(I8)\n\nwhere n\\ = n(n- l)(n \u2014 2), ... 3.2.1 is called factorial n. It is clear that: \"Pn = n\\\n\nand that: 1.1.5 Logarithms Logarithms, which, short of calculating machinery of some form, are probably the greatest aid to computation are based on the properties of indices.\n\n\"P, = n\n\nIf, of n things taken r at a time p things, are to occupy fixed positions then the number of permutations is given by:\n\n\"-\"Pr-p\n\n(1.9)\n\nIf in the set of n things, there are g groups each group containing H 1 , \/I2 . . . ng things which are identical then the number of permutations of all n things is: n\\ H1In2I.. .ngl\n\n1.1.6.2 Combinations The number of combinations of n different things, into groups of r things at a time is given by: \u00ab!__= \"Pr^ r\\(n-r}\\ r\\\n\n(1.10)\n\nIt is important to note that, whereas in permutations the order of the things does matter, in combinations the order does not matter. From the general expression above, it is clear that: \"Cn= 1\n\n\"C, = n\n\n(1.11)\n\nIf, of n different things taken r at a time p are always to be taken then the number of combinations is: \"-\"Cr-p\n\nFigure 1.1 Trigonometric functions These functions satisfy the following identities: sin2a -I- COS2Gi = I 1 + tan2a = sec2a 1 + cot2a = cosec2a\n\n1.2.1 Positive and negative lines In trigonometry, lines are considered positive or negative according to their location relative to the coordinate axes xOx', yOy', (see Figure 1.2).\n\n(1.12)\n\nIf, of n different things taken r at a time p are never to occur the number of combinations is:\n\n\"-\"Cr\n\n(1.13)\n\nNote that combinations from an increasing number of available things are related by: \"+ 1 Cr = \"O+ \"Cr-I also\n\nn\n\n(1.14)\n\nn\n\nCr = Cn-r\n\n(1.15)\n\n1.1.7 The binomial theorem The general form of expansion of (x + a)n is given by: (* +OT=^C 0 Jt\"+ \"C1 x \"->ar +nC2x\"--2a2 ...\n\n(1.16)\n\nFigure 1.2 Positive and negative lines\n\nAlternatively this may be written as:\n\n(j +tf-X-+ !!*'-'*+^*^^ (1.17) It should be noted that the coefficients of terms equidistant from the end are equal (since \"Cr = nCn \u2014 r).\n\n1.2.1.3 Negative lines\n\n1.2 Trigonometry The trigonometric functions of the angle a (see Figure 1.1) are defined as follows: sin a = y\/r cos a = xjr tan a = y\/x\n\n1.2.1.2 Positive lines Radial: any direction. Horizontal: to right of yOy'. Vertical: above xOx'.\n\ncosec a = r\/y sec a = r\/x cot a = x\/y\n\nHorizontal: to left of yOy'. Vertical: below xOx'. 1.2.2 Positive and negative angles Figure 1.3 shows the convention for signs in measuring angles. Angles are positive if the line OP revolves anti-clockwise from\n\nOx as in Figure 1.3a and are negative when OP revolves clockwise from Ox. Signs of trigonometrical ratios are shown in Figure 1.4 and in Table 1.1.\n\nTable 1.1 Sign of ratio Quadrant positive First\n\nsin cos tan cosec sec cot sin cosec\n\nSecond\n\nFigure 1.3 (a) Positive (b) negative angle\n\nThird\n\ntan cot\n\nFourth\n\ncos sec\n\nnegative\n\ncos sec tan cot sin cosec cos sec sin cosec tan cot\n\n1.2.4 Measurement of angles 1.2.4.1 English or sexagesimal method 1 right angle = 90\u00b0 (degrees) 1\u00b0 (degree) = 60' (minutes) 1' (minute) = 60\" (seconds) This convention is universal. 1.2.4.2 French or centesimal method This splits angles, degrees and minutes into 100th divisions but is not used in practice. 1.2.4.3 The radian\n\nFigure 1.4 (a) Angle in first quadrant; (b) angle in second quadrant; (c) angle in third quadrant; (d) angle in fourth quadrant\n\nThis is a constant angular measurement equal to the angle subtended at the centre of any circle by an arc equal in length to the radius of the circle as shown in Figure 1.5. n radians= 180\u00b0 1 OQ\n\n1.2.3 Trigonometrical ratios of positive and negative angles\n\nn\n\nJ Of\\\n\n=\n\n- =51\u00b0 17'44\" approximately 3.141 6\n\nTable 1.2\n\nsin ( - a) = - sin a cos ( - a) = cos a sin (90\u00b0 - a) = cos a cos (90\u00b0 - a) = sin a sin (90\u00b0 -I- a) = cos a cos (90\u00b0 -fa) = - sin a sin (180\u00b0 -a) = sin a cos (180\u00b0 -a)= -cos a sin (180\u00b0 4- a) = -sin a cos (180\u00b0 -I- a)= -cos a\n\ntan ( - a) = - tan a cot ( - a) = - cot a tan (90\u00b0 -a) = cot a cot (90\u00b0 -a) = tana tan (90\u00b0 + a) = -cot a cot (90\u00b0 4- a) = - t a n a tan (180\u00b0 -a)= -tan a cot (180\u00b0-a) = -cota tan (180\u00b0 + a)= tana cot (180\u00b0 + a)= cot a\n\nsec (-a) cosec ( - a) sec (90\u00b0 - a) cosec (90\u00b0 \u2014 a) sec (90\u00b0 -H a) cosec (90\u00b0 -I- a) sec (1 80\u00b0 - a) cosec (180\u00b0- a) sec (180\u00b0 4- a) cosec (180\u00b0 + a)\n\n= sec a = - cosec a = cosec a = sec a = - cosec a = sec a = - sec a = cosec a = -sec a = -cosec a\n\n1.2.8 Sums and differences of functions sin A + sin B=2 sin \\(A + E) cos \u00b1(A - B) sin A - sin 5=2 cos J(^ + B) sin H^ ~ ^) cos A + cos 5=2 cos i(y4 + 5) cos iC4 - B) cos A - cos 5= - 2 sin J(^ 4- 5) sin J(X - 5) sin2 A - sin2 \u00a3=sin (A + E) sin (A - B) cos2,4 - cos2 B= - sin (A + B) sin (,4 - B) cos2,4 - sin2 B=cos (\/4 -I- B) cos (\/4 - \u00a3)\n\n1.2.9 Functions of multiples of angles sin 2A = 2sin A cos A cos 2\/1 = cos2 A - sin2 A = 2 cos2 \/1-1 = 1-2 sin2,4 tan 2\/4 = 2 tan A\/(I - tan 2 .4) sin 3A = 3 sin ,4 - 4 sin3,4 cos 3 A \u2014 4 cos3 \/1-3 cos \/* tan 3\/4 = (3 tan A - tan3 A)I(I - 3 tan2 ^) sin pA = 2 sin (p- l)\/lcos\/l-sin(\/?-2)\/f cos \/7\/4 = 2 cos (p\u2014 \\)AcosA \u2014 cos(p \u2014 2)A\n\nFigure 1.5 The radian\n\n1.2.10 Functions of half angles\n\n1.2.4.4 Trigonometrical ratios expressed as surds\n\ns i n \/ 4 \/ 2 = y ( 1 - C f - 4 ) = ^ 1 - f 2 sin ^-- \/(1 7 in - 4)\n\nTable 1.3 TT\n\nTT\n\nTl\n\nTT\n\n^\n\n6\n\n4\n\n1\n\n2\n\n\/4ngte \/\/i Agrees\n\n30\u00b0\n\n45\u00b0\n\n60\u00b0\n\n90\u00b0\n\n^\n\n'\n\n\\\n\n^\n\n\\$\n\n'\n\n'\n\nf\n\nJl\n\n3\n\n\u00bb\n\n\u2122 c j n - \/Y 1+cos^ \\ _ V(I+sin^) cos AIZ v I ^ ) 2\n\nV(I -sin.4) 2\n\nAn= 1 ~ cos ^4 sin\/l \/ \/ 1 - cos ,4 \\ tan ^- sin ^ ~ \\+cosA~v {\\+cosA J 1.2.11 Relations between sides and angles of a triangle (Figures 1.10 and 1 . 1 1 ) a _ b _ c sin A sin B sin C a = bcos C+ccos B\n\ntan\n\nO\n\n-JT V-J\n\n!\n\nx\/3\n\noo\n\nTable 1.3 gives these ratios for certain angles. 1.2.5 Complementary and supplementary angles Two angles are complementary when their sum is a right angle; then either is the complement of the other, e.g. the sine of an angle equals the cosine of its complement. Two angles are supplementary when their sum is two right angles.\n\nc2 = a2 + b2-2abcosC\n\n(1.18)\n\nsin A = \u00a3 V W^ ~ *)(-* ~ W(^ - c)}\n\n( l \u2022{9>\n\nwhere 2s = a + b + c Area of triangle A = \\ab sin C= V7W-5 ~ a)(s ~ b)(s ~ c)l A 2\n\nv\n\n\/I (5-6X^-CJI I .v(^-\u00ab) I\n\n1.2.6 Graphical interpretation of the trigonometric functions\n\n^M^l\n\nFigures 1.6 to 1.9 show the variation with a of sin a, cos a, tan a and coseca respectively. All the trigonometric functions are periodic with period 2n radians (or 360\u00b0).\n\nsinf=^^?^}\n\n1.2.7 Functions of the sum and difference of two angles sin (A \u00b1 B) = sin A cos B \u00b1 cos A sin B cos (A \u00b1 B) = cos A cos B + sin A sin B \/ A , m tan ^ \u00b1 tan B tanM\u00b1H)=1\u00b1tany'2 = \\ COS X\n\n60\n\n30\n\n0.5 0.333\n\n0.866 0.289\n\n210\n\n240\n\n270\n\n300\n\n-0.866 -0.289\n\n-1.0 -0.1667\n\n-0.866 O\n\n-0.5 0.1667\n\n1.0 0.1667\n\n120\n\n150\n\n180\n\n0.5 -0.1667\n\nO -0.289\n\n-0.5 -0.333\n\n330\n\n360\n\n90\n\nO\n\n0.866 O\n\n+ 0.5 + 0.333\n\nO \u2022 0.289\n\nB'C on great circle B'CA'C and edge CB on great circle ACBD'. The angles of a spherical triangle are equal to the angles between the planes of the great circles or, alternatively, the angles between the tangents to the great circles at their points of intersection. They are denoted by the letters C, B', B for the triangle CB'. Area of spherical triangle CB'B = (B' + B+ C- n)r\\ Spherical excess Comparing a plane triangle with a spherical triangle the sum of the angles of the former is n and the spherical excess E of a spherical triangle is given by E=B' + \u00a3 + C-TT; hence, area of a spherical triangle can be expressed as (E\/4n) x surface of sphere.\n\nFigure 1.16 Sphere illustrating spherical trigonometry definitions\n\nSpherical polygon A spherical polygon of n sides can be divided into (H-2) spherical triangles by joining opposite angular points by the arcs of great circles. Area of spherical polygon = [sum of angles -(H- 2)n]r = j- x surface of sphere. Note that (H - 2)n is the sum of the angles of a plane polygon of n sides. 1.3.2 Properties of spherical triangles\n\nFigure 1.17 Spherical triangles Great circle The section of a sphere cut by a plane through any diameter, e.g. ACBC'. Poles Poles of any circular section of a sphere are the ends of a diameter at right angles to the section, e.g. D and D' are the poles of the great circle ACBC'. Lunes The surface areas of that part of the sphere between two great circles; there are two pairs of congruent areas, e.g. ACA'C'A; CBC'B'C and ACB'C'A; A'CBC'A'. Area oflune If the angle between the planes of two great circles forming the lune is B (radians), its surface area is equal to 26r2. Spherical triangle A curved surface included by the arcs of three great circles, e.g. CB'B is a spherical triangle formed by one edge BB' on part of the great circle DB'BA the second edge\n\nLet ABC, in Figure 1.17, be a spherical triangle; BD is a perpendicular from B on plane OAC and OED, OFD, OEB, OFB, OGE, DHG are right angles; then BED = A and BFD = C are the angles between the planes OBA, OAC and OBC, OAC respectively. DEH = COA = b also COB = a, AOB = c, and since OB = OA = OC = radius r of sphere, OF = r cos a, OE = r cose; then cos a = cos b cos c + sin b sin c cos A cos b = cos a cos c + sin a sin c cos B cos c = cos a cos b + sin a sin b cos C\n\nAlso the sine formulae are: sin A _ sin B _ sin C sin a sin b sin c\n\nand the cotangent formulae are: sin a cot c \u2014 cos a cos B + sin B cot C sin b cot c = cos b cos A + sin A cot C sin b cot a = cos b cos C + sin C cot A sin c cot a = cos c cos B + sin B cot A sin c cot b = cos c cos A + sin A cot B sin a cot b = cos a cos C+sin C cot B\n\n1.4.1 Relation of hyperbolic to circular functions\n\nFigure 1.18 Polar triangles\n\nsin 9= \u2014i sinh i 9 cos 9 = cosh \/ 9 tan 0 = \/ tanh j 9 cosec 9 = \/ cosech \/ 9 sec 0 = sech i 9 cot0 = icothi0 sinh9= \u2014isin 10 cosh 0 = cos \/ 9 tanh Q= -i tan \/ 0 cosech 9 = i cosec \/ 9 sech 0 = z sec 10 coth 0 = zcot id\n\nIn Figure 1.18, ABC, A1B1C, are two spherical triangles in 1.4.2 Properties of hyperbolic functions which A1, B1, C1 are the poles of the great circles BC, CA, AB cosh2 0- sinh2 0=1 respectively; then A1B1C1 is termed the polar triangle of ABC sech2 0=1- tanh 2 9 and vice versa. Now OA1, OD are perpendicular to the planes sinh 20 = 2 sinh 9 cosh 0 BOC and AOC respectively; hence A1OD = angle between cosh 29 = cosh2 0+ sinh2 (9 planes BOC and AOC = C. Let sides of triangle A1B1C1 be denoted by a\\b\\c\\ then C1 = A1OB1 = Tr-C also Ci1 = n-A and bl = n \u2014 B;c = n-C];a = n-A^b = n-Bl and from these we get cosech2 9 = coth2 0-1 ,\n\n2 tanh 9 + toffi^\n\ncos B+cos A cos C : , . sm A sin C\n\nn om (1-20)\n\ncosa= cos A.+ cos _ . Bcos _ C sin B sin C\n\nn on (1.21)\n\nsinh (jc \u00b1 x) = sinh Jt cosh y \u00b1 cosh jc sinh y cosh (x \u00b1 y) = cosh Jt cosh y \u00b1 sinh jc sinh y\n\ncos C + cos A cos B :\u2014\u2014\u2014 sin A sin B\n\n(1.22)\n\n, , tanh x \u00b1 tanh y x tanh v(jc \u00b1 \" v) = T-T~:\u2014r r\u2014>> 1 \u00b1 tanh jc tanh\n\nCOS 6 =\n\ncose=\n\ntonh2 =\n\n*T\n\nsinh x + sinh y = 2 sinh J(jc -f y) cosh J(AC - >>) sinh x - sinh _y = 2 cosh J(jc -I- y) sinh J(x - ^) cosh jc -I- cosh y = 2 cosh J(jc + y) cosh J(;t - _y) cosh jc - cosh y = 2 sinh }(jc + y) sinh \u00b1(x - y)\n\n1.3.2.1 Right-angled triangles If one angle A of a spherical triangle ABC is 90\u00b0 then cos a = cos b cos c = cot B cot C _. tan c \u201e tan b . _ sin b cos B= : cos C= : smB=-.\u2014: tan a tan a sin c . _, sine _ tan b _ tanc sin C = -\u2014: tan B= \u2014.\u2014: tan C= -.\u2014r: sin a sin c sin b cos B = cos b sin C; cos C = cose sin B.\n\n1.4.3 Inverse hyperbolic functions As with trigonometric functions, we define the inverse hyperbolic functions by ^y = sinh\" 1 x where jc = sinh^: Therefore:\n\nx = (ey \u2014 e - l ')\/2\n\nRearranging and adding jc2 to each side:\n\n1.4 Hyperbolic trigonometry The hyperbolic functions are related to a rectangular hyperbola in a manner similar to the relationship between the ordinary trigonometric functions and the circle. They are denned by the following exponential equivalents:\n\ne 2v -2jc.e v + jc2 = jc2-f 1 or: ey-x = J(x2+\\) and therefore: y = sinh ' .x = logc [jc + J(x2 +I)]\n\nsinh 0= ^f-\n\ncosech","date":"2021-04-22 23:58:23","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\": 0.8021674752235413, \"perplexity\": 5379.929015493017}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618039563095.86\/warc\/CC-MAIN-20210422221531-20210423011531-00002.warc.gz\"}"} | null | null |
Helmut Engler (* 14. April 1926 in Freiburg im Breisgau; † 25. Oktober 2015 ebenda) war ein deutscher Jurist und Politiker der CDU.
Ausbildung und Beruf
Engler wuchs in Freiburg auf und machte am Berthold-Gymnasium 1944 das Abitur. Nach seinem Kriegsdienst und einer einjährigen Tätigkeit als Landarbeiter studierte er ab 1946 Jura und legte 1949 seine erste und 1953 seine zweite juristische Staatsprüfung ab. 1968 wurde er an der Albert-Ludwigs-Universität Freiburg mit dem Dissertationsthema Annahme an Kindes Statt (§§ 1741–1772 BGB) zum Dr. jur. promoviert.
Ab 1953 war Engler Gerichtsassessor und Landgerichtsrat am Landgericht Freiburg, dazwischen zwei Jahre wissenschaftlicher Hilfsarbeiter im Bundesjustizministerium, ab 1959 Persönlicher Referent des Präsidenten des Bundesverfassungsgerichts und Präsidialrat des Ersten Senats dieses Gerichts, ab 1963 Oberlandesgerichtsrat am Oberlandesgericht Karlsruhe (Zivilsenate in Freiburg). Zum 1. August 1968 wurde er als ordentlicher Professor auf einen Lehrstuhl für Bürgerliches Recht und Zivilprozessrecht an der Rechtswissenschaftlichen Fakultät der Albert-Ludwigs-Universität Freiburg berufen, der er ab 1973 vier Jahre lang als Rektor vorstand, seit 1975 zugleich als Vorsitzender der Landesrektorenkonferenz der baden-württembergischen Universitäten.
Politische Tätigkeit
1977 wurde Engler Nachfolger von Kurt Rebmann als Ministerialdirektor beim Justizministerium Baden-Württemberg unter Justizminister Guntram Palm. Als Ministerpräsident Hans Filbinger im Mai 1978 sein Kabinett umbildete und das Kultusministerium in zwei Ministerien aufteilte, wurde Engler zum ersten Minister für Wissenschaft und Kunst Baden-Württembergs ernannt. Er trat dann auch in die CDU ein.
Engler konnte sich auch unter Ministerpräsident Lothar Späth als Wissenschaftsminister behaupten, musste dann aber 1991 sein Amt aufgeben, als Späth zurücktrat. In seine Amtszeit fiel unter anderem der Entzug der kirchlichen Lehrbefugnis für Hans Küng an der Universität Tübingen sowie die Schließung zweier Pädagogischer Hochschulen.
Nach seinem Ausscheiden als Minister nahm Engler seine Professur an der Rechtswissenschaftlichen Fakultät der Universität Freiburg wieder auf.
Engler starb mit 89 Jahren im Oktober 2015. Er hinterließ seine Frau Gertrud geb. Schmukle und drei Kinder.
Ehrungen
1983: Großes Verdienstkreuz der Bundesrepublik Deutschland
1986: Großes Verdienstkreuz mit Stern der Bundesrepublik Deutschland
Einzelnachweise
Wissenschaftsminister (Baden-Württemberg)
Rektor (Albert-Ludwigs-Universität Freiburg)
Kunstminister (Baden-Württemberg)
Ministerialdirektor (Baden-Württemberg)
Richter (Oberlandesgericht Karlsruhe)
CDU-Mitglied
Träger des Großen Bundesverdienstkreuzes mit Stern
Träger des Verdienstordens des Landes Baden-Württemberg
Absolvent der Albert-Ludwigs-Universität Freiburg
Deutscher
Geboren 1926
Gestorben 2015
Mann | {
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Ekanem Bálint (Pápa, 1990. március 17. –) nigériai származású magyar énekes, színész.
Élete
Az édesapja révén nigériai származású Ekanem Bálint Emota fiatalon kezdett el énekléssel és színészettel foglalkozni. Szülővárosában, Pápán több helyi vagy iskolás fellépés után végzős gimnazistaként 2010-ben lett a Madách Színház tagja. A Madách mellett játszik a Szegedi Szabadtéri Játékokon, ahol fontosabb szerepei közé tartozik az Egy bohém rapszódiája ifjú Freddieje vagy a Szegény gazdagok öreg betyárja és rendszeresen fellép a Margitszigeti Szabadtéri Színpadon is. Az X-Faktor énekes tehetségkutató harmadik évadában a mentorok házáig jutott 2012-ben, de Geszti Péter nem juttatta be az élő showba, majd egy év múlva, 2013-ban saját, Speak up! nevű együttesével jutott el ugyanoda.
2015-ben Puskás Petivel kezdett zenélni az akkor alapított hét fős The Biebers elnevezésű formációban. 2017 májusában ők készítették a győri 2017. évi nyári európai ifjúsági olimpiai fesztivál hivatalos videóklipjét. 2017 őszén indult a TV2 Sztárban sztár című zenés műsor ötödik évadában, ahol az első élő adásban a Bánk bánból adta elő a Hazám, hazám című áriát, amit a nézők a legjobb produkciónak választottak. 2022-ben Hideg szél című dalával elindult a MTV (Duna Televízió) által megrendezett A dal 2022 c. dalmustrán. A dal sikeresen szerepelt az élő, televíziós válogatóadásban, közönségszavazattal jutott az elődöntőig. Szerzőtársai: Horváth Szabolcs, Győry Zoltán, Miklós Csongor, Holb Henrik András, Puskás Ádám Dániel, Gerendás Dániel voltak. A dalszöveget Csukárdi Sándor jegyzi.
Színházi szerepei
Madách Színház
Az operaház fantomja (Joseph Bouquet)
József és a színes szélesvásznú álomkabát (Simeon)
Mamma Miaǃ (Pepper)
Vuk (Gúnár)
A tizenötödik (Klapka György)
Aranyborjú (Adam Kozlevics)
Fekete Péter (Badilla / Sofőr / Boy / Virág alt.)
A nyomorultak
Macskák (Ben Mickering)
Pannon Várszínház
Hair (Hud)
Filmes és televíziós szerepei
Tóth János (2017)
Sikítófrász (2017)
Jegyzetek
Források
A Madách Színház honlapján
1990-ben született személyek
Élő személyek
Magyar könnyűzenei előadók
Magyar énekesek
X-faktor-versenyzők
Pápaiak
Magyar színészek | {
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Gary Bannister (ur. 22 lipca 1960 w Warrington) – angielski piłkarz występujący na pozycji napastnika.
Kariera klubowa
Bannister karierę rozpoczynał w 1978 roku w zespole Coventry City. W Division One zadebiutował 26 sierpnia 1978 w wygranym 4:1 meczu z Norwich City. 5 maja 1979 w wygranym 3:0 spotkaniu z Wolverhampton strzelił pierwszego gola w Division One. W 1980 roku przebywał na wypożyczeniu w amerykańskim Detroit Express, grającym w NASL.
W 1981 roku Bannister odszedł do Sheffield Wednesday z Division Two. Występował tam przez trzy sezony, a potem przeszedł do Queens Park Rangers z Division One. W sezonie 1984/1985 został królem strzelców Pucharu UEFA. W QPR spędził cztery sezony. Potem wrócił do Coventry, nadal grającego w Division One. Tym razem grał tam przez trzy sezony.
Na początku 1990 roku Bannister przeniósł się do drużyny Division Two, West Bromwich Albion. W sezonie 1990/1991 spadł z nim do Division Three. W kolejnym sezonie grał na wypożyczeniu w Oxford United (Division Two).
W 1992 roku został graczem Nottingham Forest, występującego w nowo utworzonej Premier League. Spędził tam sezon 1992/1993. Następnie grał w zespołach Stoke City (Division One), Hong Kong Rangers z Hongkongu, Lincoln City (Division Three) oraz Darlington (Division Three). W 1996 roku zakończył karierę.
Kariera reprezentacyjna
W 1982 roku Bannister wystąpił jeden raz w reprezentacji Anglii U-21.
Bibliografia
Gary Bannister w bazie Worldfootball
Angielscy piłkarze
Piłkarze Coventry City F.C.
Piłkarze Sheffield Wednesday F.C.
Piłkarze Queens Park Rangers F.C.
Piłkarze West Bromwich Albion F.C.
Piłkarze Oxford United F.C.
Piłkarze Nottingham Forest F.C.
Piłkarze Stoke City F.C.
Piłkarze Hong Kong Rangers FC
Piłkarze Lincoln City F.C.
Piłkarze Darlington F.C.
Królowie strzelców Pucharu UEFA i Ligi Europy
Urodzeni w 1960
Ludzie urodzeni w Warrington | {
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Chloronia bogotana är en insektsart som beskrevs av Van der Weele 1909. Chloronia bogotana ingår i släktet Chloronia och familjen Corydalidae. Inga underarter finns listade i Catalogue of Life.
Källor
Vattennätvingar
bogotana | {
"redpajama_set_name": "RedPajamaWikipedia"
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Acantholycosa plumalis är en spindelart som beskrevs av Marusik, Azarkina och Koponen 2004. Acantholycosa plumalis ingår i släktet Acantholycosa och familjen vargspindlar. Inga underarter finns listade i Catalogue of Life.
Källor
Vargspindlar
plumalis | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,515 |
\section{Introduction}
In deep learning, various architectures have been designed such as convolutional neural networks(CNN), recurrent neural networks(RNN) and graph neural networks(GNN) and recursive neural networks of tree(TNN)\citep{ lecun1989backpropagation,hopfield1982neural, scarselli2008graph, wu2020comprehensive}.
These networks show good performance in various fields such as image, text, and sound etc.
However, most of the existing studies are focused on a specific dataset or task. And there are many tasks that humans can do but are difficult to perform with neural networks.
\begin{figure}[h]
\centering
\hspace{-20px}
\subfloat[Human brain]{{\includegraphics[width=0.45\textwidth ]{figure/human_brain.pdf} }}
\subfloat[Neuro Tree]{{\includegraphics[width=0.38\textwidth ]{figure/neurotree.png}}}
\caption{Comparison of the human brain and artificial association neural network}
\label{fig:network_comparison}
\end{figure}
First, some of the difficulties with imitating human neural networks are introduced below.
\begin{itemize}
\item[C-\romannumeral 1] -- Sensory organs that process information for each data type exist at several starting points.
\item[C-\romannumeral 2] -- A tree cannot express the relationship between sibling nodes, and there is only one parent node, and it is difficult to define the graph's starting point and ending point, and it's hard to handle.
\item[C-\romannumeral 3] -- The structure that receives information is hierarchical and freely combined in various structures. But existing networks are designed to learn only specific tasks or datasets because layers are fixed.
\item[C-\romannumeral 4] -- The number of activated neurons and the processing depths differ depending on the data type and complexity. And sometimes the information is entered and sometimes not.
\item[C-\romannumeral 5] -- Human neurons are very numerous and process various types of data.
\end{itemize}
C-$[\cdot]$ denotes the characteristics.
(C-\romannumeral 1) Humans have various sensory organs from which they receive information such as sight, hearing, and smell, and transfer it to the cerebral cortex using different information-processing organs.
The optic nerve also begins with the visual cortex of the occipital lobe, and the auditory nerve begins with the auditory cortex of the temporal lobe; this means that the two paths have different starting points.
(C-\romannumeral 2) A graph can describe the relationship depending on whether all nodes are connected, but it is not easy to define the start and end points. And when expressing the propagation path of the tree structure, it is not easy to describe the path that starts in parallel and propagates to the root node in a graph structure. A tree is a data structure that can define direction with the leaf and root nodes. On the other hand, a tree structure cannot express the relationship between sibling nodes. And the existing tree data structure cannot express the process of delivering to multiple nodes because only one parent node exists.
(C-\romannumeral 3) Information from different sensory organs is gathered in the association area.
Typically, the posterior parietal cortex\citep{malach1995object} is located at the top of the head as part of the parietal lobe, and sensory information such as vision and hearing is fused and interpreted; Then, information is coordinated in the frontal lobe and humans act.
(C-\romannumeral 4) There is also an over-fitting problem in DNN\citep{widrow1960adaptive}, and various attempts have been made to solve this in \citep{srivastava2014dropout}. In DNN, simply deeper layers lead to an over-fitting problem\citep{dai2017very}.
In particular, NasNet\citep{zoph2018learning} has attempted to design a layer automatically through reinforcement learning and RNN. This means that there is an appropriate network depth depending on the complexity of the dataset; we want to design a network that could adjust the depth of the layers according to the complexity of each data, not the data set.
(C-\romannumeral 5) Also, the number of human neurons is about 85 billion\citep{herculano2009human}, which is difficult to express in a network.
Therefore, AANs are designed to solve these problems.
\begin{itemize}
\item[A-\romannumeral 1] -- We express these sensory organs as multi-feature-extraction processes(sec.\ref{sec:extraction}).
\item[A-\romannumeral 2] -- We propose a new data structure called neuro node(NN) and neuro tree(NT)(sec.\ref{subsec:gtnode}).
\item[A-\romannumeral 3] -- We propose a recursive propagation method called depth-first convolution(DFC) and depth-first deconvolution(DFD). It can learn about relationships, hierarchical, multi-feature extraction networks, and propagate to several parent nodes(sec.\ref{sec:Depthfirst}).
\item[A-\romannumeral 4] -- We do not design layers but express the flow of information delivery in a neuro tree(sec.\ref{appendix:graphtree_architecture}).
\item[A-\romannumeral 5] -- Various neuro tree structures can be learned with one network cell (association networks)(sec.\ref{sec:network}).
\end{itemize}
A-$[\cdot]$ denotes our approach of C-$[\cdot]$.
(A-\romannumeral 1)
This network can perform information processing, such as that done by sensory organs before their signals are entered into the cerebral cortex.
In this paper, we express this sensory organ as a feature-extraction process(sec.\ref{sec:extraction}). Type information is stored with the data and converted into an input vector by different feature extraction processes for each type.
(A-\romannumeral 2)
We propose a new data structure called neuro node(NN) and neuro tree(NT). An neuro tree can be expressed using leaf and root nodes as the starting and ending points, as well as the relationship among sibling nodes. If a neuro node does not reverse the propagation path, it is possible to have multiple parent nodes; This allows propagation to multiple nodes. In addition, we can modify the existing tree dataset at no high cost.
(A-\romannumeral 3)
subtrees of different types are merged and make a final decision at the root node; this structure is difficult to express with the existing forward-learning method. Therefore, we used a recursive-learning method\citep{goller1996learning}. This methodology has been effective in analyzing the semantics of programming source code or natural language\citep{socher-etal-2013-recursive, mou2014tbcnn}.
We call the recursive-convolution methodology used in AANs the depth-first convolution(DFC) and depth-first deconvolution(DFD) methodology(sec.\ref{sec:Depthfirst}).
DFC is a convolution method in which subtrees originating from different leaf nodes are integrated into a bottom-up approach and represent hierarchical and relational information.
artificial association networks can simultaneously learn various types of datasets through the recursive learning method because the neuro tree data structure can express various model structures.
(A-\romannumeral 4) graph tree neural networks are data-driven learning in which the number of convolutions varies according to the depth of the tree. Instead of using fixed sequence layers, we create a neuro tree for each data and learn according to the tree's structure. If we modify this network further, we can adjust the depth according to the type and complexity of each data.
In addition, we introduce two models of artificial association networks called level association networks(LAN) and recursive association networks(RAN), and these models are mathematically related to MLP\citep{hinton2006fast} and RNN. The fully connected layer(FC layer) in MLP as Level Layer of LAN, and time in RNN as depth of RAN on a special case. In addition, the some structures of graph neural networks and recursive neural networks can be expressed by graph tree.
(A-\romannumeral 5) The number of human neurons is enormous, and it is difficult to express all of their structures in a network; on the other hand, this network can represent multiple neurons because it can perform recursive end-to-end learning according to the amount of information.
This network theory holds that "units of information have a relationship in the form of a graph, then become a bigger unit of information, and have a relationship with other units of information. At this point, the unit of information is a set of neurons, and we can express it as a vector with artificial association networks."
To demonstrate the performance of this network, we conducted four experiments.
And we used several benchmark datasets (image(MNIST\citep{lecun1998gradient}), sound(Speech Commands\citep{warden2018speech}), text(IMDB\citep{maas2011learning},SST\citep{socher-etal-2013-recursive}), graph\citep{dou2021user}).
In the first experiment, It verifies whether feature extraction networks and association networks can be learned together.
we compared the performance of existing networks that separately learned image(LeNet-5\citep{lecun1989backpropagation}, sound(M5\citep{dai2017very}), text(CNN\citep{kim-2014-convolutional}) with the performance simultaneously learned by connecting these networks for feature extraction into artificial association networks.
In the second experiment, we verifies whether data of various structures(image, sound, tree, graph) can be learned and checked the plot of the learning process in sec \ref{fig:exp3feplot}.
In the third experiment, we verifies whether data of deep time series structures can be learned and checked the plot of the learning process in sec \ref{fig:gru_exp_plot}.
The fourth experiment contained one or three types of data into a neuro tree, and artificial association networks learned these neuro trees.
Then, we verified whether the output contained all the information inside the neuro tree.
As a result, the network was learned without significant degradation in performance compared to when learning existing networks separately, and all information on the tree could be embedded as a vector.
\section{Related Works}
\paragraph{Graph Neural Networks}
GNNs have recently shown good performance in various fields\cite{scarselli2008graph, wu2020comprehensive, velivckovic2017graph,gilmer2017neural}.
GNN has a relationship term and a structure more similar to that of human neural networks than other networks. It was used as a study to embed knowledge with GNN. But it isn't easy to understand the propagation path with the graph data structure, although the tree-structured graph is trained through graph neural networks.
\paragraph{Recursive Neural Networks}
Recursive neural networks are used to learn data of tree structures. Recently, it has been useful in the field of natural language processing.
I think this network is relatively unnoticed because it needs tree-structured data to learn.
This methodology has been effective in analyzing the semantics of programming source code or natural language\citep{socher-etal-2013-recursive, mou2014tbcnn}. Recursive neural networks are easier to understand the propagation path than graph neural networks because the starting and the ending point are clear in a tree structure.
\subsection{Multi Deep Learning}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.30\textwidth}
\centering
\includegraphics[height=3.5cm]{figure/multi-domain.png}
\caption{Multi-domain}
\label{fig:multi-domain}
\end{subfigure}
\hfill
\begin{subfigure}{0.30\textwidth}
\centering
\includegraphics[height=3.8cm]{figure/multi-modal.png}
\caption{Multi-modal}
\label{fig:multi-modal}
\end{subfigure}
\hfill
\begin{subfigure}{0.30\textwidth}
\centering
\includegraphics[height=3.8cm]{figure/multi-task.png}
\caption{Multi-task}
\label{fig:multi-task}
\end{subfigure}
\caption{Multi Deep Learning}
\label{fig:figures}
\end{figure}
\paragraph{Multi-domain Deep Learning}
First, a multi-domain refers to a structure in which different domain datasets are learned using the same neural network as Fig.\ref{fig:multi-domain}.
Related studies include Parallel Residual Adapters\cite{rebuffi2018efficient}, which learns different types of image data. In this study, 10 different domain image datasets(MNIST, CIFAR-100, VGG-Flower etc..) are learned in one neural network structure. In particular, this study can be a very effective study in reducing parameters because it learns with a single model when learning datasets.
\paragraph{Multi-modal Deep Learning}
Multi-modal deep learning means combining data from various domains and converging them into information, and then solving the task using the fused information\cite{baltruvsaitis2018multimodal} as Fig.\ref{fig:multi-modal}. Representatively, there is an image captioning field\cite{yu2019multimodal}.
The structure may be slightly different from Fig.\ref{fig:multi-modal}, but using a variety of information is the key to this study.
\paragraph{Multi-task Deep Learning}
This refers to a neural network structure that performs various tasks using the output of a network as Fig.\ref{fig:multi-task}. The advantage of this process is that the network's output performs multi-task, providing the data characteristics to be more diverse and apparent.
Representatively, there is an MT-DNN\cite{liu2019multi} model, and it has a good effect.
\subsection{Multi-Domain \& Multi-Modal \& Multi-Task simultaneously deep learning}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.31\textwidth}
\centering
\includegraphics[height=3.5cm]{figure/association_node_1.png}
\caption{Multi-Domain}
\label{fig:first}
\end{subfigure}
\hfill
\begin{subfigure}{0.31\textwidth}
\centering
\includegraphics[height=3.5cm]{figure/association_tree_1.png}
\caption{Multi-Modal}
\label{fig:second}
\end{subfigure}
\hfill
\begin{subfigure}{0.31\textwidth}
\centering
\includegraphics[height=3.5cm]{figure/multi-task-net.png}
\caption{Multi-Task}
\label{fig:third}
\end{subfigure}
\caption{Multi Deep learning}
\label{fig:dataset_GT1}
\end{figure}
So, how can multi-domain, modal, and task be performed simultaneously?
Humans perform multi-domain, modal, and task at the same time.
To perform multi-domain, multi-modal, and multi-task learning using relatively few parameters for all data, it should not be structurally fixed.
In this study, we introduced association networks capable of multi-domain deep learning.
And it shows the possibility of expansion to multi-modal deep learning.
And as a series of future works, multi-task learning is performed using root vectors($\overrightarrow{h}_{root}$) from association networks.
In addition, we design to perform multi-task in a human-like structure rather than simply performing multi-task learning in future studies\cite{kim2021deductive, kim2021memory, kim2021imagine}.
\section{Artificial Association Networks}
\label{sec:pre}
In this section, we proposes artificial association architecture.
The architecture consists of five parts.
\begin{itemize}
\item[$(\romannumeral 1)\mathbf{NN}:$] (Neuro Node) -- Defining the data structure of neuro node.
\item[$(\romannumeral 2)\mathbf{NT}:$] (Neuro Tree) -- Designing the neuro tree structure.
\item[$(\romannumeral 3)\mathbf{AAN}:$] (Artificial Association Networks) -- Defining the artificial association networks.
\item[$(\romannumeral 4)\Psi:$] (Multi-Feature Extraction Networks) -- Defining the feature-extraction process.
\item[$(\romannumeral 5)\Phi:$] (Multi-Task Networks) -- Defining the task network to perform tasks.
\end{itemize}
\subsection{Association Data Structure : Neuro Node}
\label{subsec:gtnode}
\begin{figure}[h]
\centering
\includegraphics[height=3.5cm]{figure/association_node.png}
\hfil
\caption{ Relational(Graph) + Hierarchical(Tree) = Neuro Node }
\label{fig:dfcdfd}
\end{figure}
The neuro tree($\mathbf{NT}$) is a data structure that can express relational, hierarchical, domain and task information. And the node of $\mathbf{NT}$ is called the neuro node($\mathbf{NN}$).
\begin{itemize}
\item[$x:$](Input) -- we can store the data such as images, sound, text, or tabular to each node.
\item[$\tau_{d}:$] (Domain) --
This refers to the domain of information matching the input $x$.
the feature-extraction function($\psi$) is selected by $\tau_{d}$ like Eqn.\ref{eq:feature_extraction}.
\item[$\tau_{t}:$] (Task) --
This refers to the task of information.
the task networks ($\phi$) is selected by $\tau_{t}$ like Eqn.\ref{eq:multi-task}. If there is no task to be performed at the node level, it may exist only in the root.
\item[$\mathbf{A}_c:$](Children Adjacency Matrix) -- This refers to the relationship information that exists in a neuro node. the number of children is $N$ and we can express it as $\mathbf{A}_c\in\mathbb{R}^{N \times N}$.
\item[$\mathbf{C}:$] (Children) -- This refers to the child nodes of neuro node($\mathbf{NN}$) matching to the node of $\mathbf{A}_{c}$. If there is no matching child, it is replaced with $\mathbf{NN}_{\emptyset}$, which carries the initial hidden state $\overrightarrow{0}$ instead. We can express this as $\mathbf{C}=\{\mathbf{NN}_{1},\mathbf{NN}_{2},..\mathbf{NN}_{N}\}$.
This $\mathbf{C}$ is different from the existing tree data structure.
In general, when creating a tree, There is only one parent node in the existing tree data structure.
However, Neuro nodes can have multiple parent nodes. Therefore, this neuro node can be a child node of several nodes.
There is one condition to become a child node.
"\textbf{Higher layer nodes that use information from the current node cannot be child nodes of the current node.}"
This condition is not to deliver the information results from one node only to the direct parent node, as shown in Fig.\ref{fig:graphtree1}, but to connect to other nodes.
At this time, the propagation path does not reverse.
\end{itemize}
We can express $\mathbf{NN} = \{x,\tau_{d}, \tau_{t},\mathbf{A}_c,\mathbf{C}\}$, $\mathbf{NN}_{i} \in \mathbf{NT}$.
$\mathbf{NN}_{root}$ denotes the root node of $\mathbf{NT}$.
The reason for defining the relationship among child nodes is the convenience of implementation. And by combining graphs and trees, information delivery processes such as fully connected neural networks(FCNN), multi-layer perceptron(MLP), recurrent neural networks(RNN), recursive neural networks(TNN), convolutional neural networks(CNN), and graph convolutional networks(GCN) can be expressed as data. And it is easier to understand than learning the graph of the tree structure with GNN(Sec \ref{appendix:graphtree_architecture}).
If we define $\mathbf{NN}$ in the above way, we can convert the tree dataset into the $\mathbf{NT}$ dataset without significantly modifying the tree structure that has previously been useful.
And in this model, $\tau_{d}$ is information to become multi-domain, and building a neuro tree means multi-modal. And $\tau_{t}$ is information to become multi-task.
\subsection{ Feature Extraction networks \& Type Bias }
\begin{figure}[H]
\centering
{\includegraphics[width=0.30\textwidth]{figure/fe.pdf}}
\caption{ Feature Extraction Networks (Multi-domain) }
\label{fig:feature-extraction}
\end{figure}
\label{sec:extraction}
\begin{equation}
\label{eq:feature_extraction}
\sigma(W\overrightarrow{x} + b) = \overrightarrow{h}
\end{equation}
\begin{equation}
\label{eq:feature_extraction}
\sigma(W[\overrightarrow{x},1]) = \overrightarrow{h}
\end{equation}
\begin{equation}
\label{eq:feature_extraction}
\sigma(W[\overrightarrow{x},onehot(\tau_{d})]) = \overrightarrow{h}
\end{equation}
\begin{equation}
\label{eq:feature_extraction}
\sigma(W[\psi_{\tau}(x),onehot(\tau_{d})]) = \overrightarrow{h}
\end{equation}
\begin{equation}
\label{eq:feature_extraction}
\therefore \overrightarrow{x} = \Psi(x, \tau_{d}) = [\psi_{\tau_{d}}(x),onehot(\tau_{d})]
\end{equation}
The $x$(input) and $\tau_{d}$(domain) exist together in $\mathbf{NN}$, and the feature extraction process of $x$ is selected by $\tau_{d}$ as Eqn \ref{eq:feature_extraction}.
We expressed the function of feature extraction as $\Psi$, and this function converts $x$ to $\overrightarrow{x}$.
And the one-hot vector has the effect of having a different bias value for each domain($\tau_{d}$). The weight parameter corresponding to the type one-hot vector means the bias value for the corresponding type, and the activation threshold value is adjusted for each type-bias. It can represent data type information and that it is transmitted from which nerve.
In addition, we do not need to perform adding operations(+) for bias. Therefore, the empty space of $\mathbf{NT}$ can be used as a zero-vector when batch learning is performed. And the dictionary structure is very useful for batch learning; we attached the methodology to sec.\ref{algorithm:batch_type_embedding}.
\subsection{How do we design the structure of the neuro tree?}
\label{appendix:graphtree_architecture}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.30\textwidth}
\centering
\includegraphics[height=5.0cm]{figure/MLPGT.png}
\caption{MLP \& NT}
\end{subfigure}
\hfill
\begin{subfigure}{0.30\textwidth}
\centering
\includegraphics[height=5.0cm]{figure/CNNGT.png}
\caption{CNN \& NT}
\end{subfigure}
\hfill
\begin{subfigure}{0.30\textwidth}
\centering
\includegraphics[height=5.0cm]{figure/RNNGT.png}
\caption{RNN \& NT}
\end{subfigure}
\caption{Model structure \& Data structure 1 (Multi-domain)}
\label{fig:graphtree1}
\end{figure}
These architectures are the models underlying existing networks.
First, Fig.\ref{fig:graphtree1}(a) means a multi-layer perceptron model.
If we express this architecture as NT and learn with $\mathbf{LAN}$(sec \ref{sec:gtlan}), which has input only in leaf node, it becomes the same operation process.
Second, Fig.\ref{fig:graphtree1}(b) means a CNN model.
Using the CNN network for the feature extraction network part with $\mathbf{LAN}$ will be the same operation.
Third, Fig.\ref{fig:graphtree1}(c) means a recurrent neural network model(RNN).
Using a tree with only one child to express sequence and learn with $\mathbf{RAN}$(sec \ref{sec:gtran}) will be the same operation.
\begin{figure}[H]
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[height=3.0cm]{figure/GNNGT.png}
\caption{GNN \& NT}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[height=3.0cm]{figure/RNN2GT.png}
\caption{RNN(recursive) \& NT}
\end{subfigure}
\caption{Model structure \& Data structure 2 (Multi-domain)}
\label{fig:graphtree2}
\end{figure}
These models are architectures that learn relational and hierarchical information.
First, Fig.\ref{fig:graphtree2}(a) means a GNN model.
The same delivery process is obtained if the relationship and inputs are expressed as relationship and inputs among sibling nodes of the neuro tree with $\mathbf{LAN}$.
Second, Fig.\ref{fig:graphtree2}(b) means a recursive neural network model(RNN).
If we learn using an identity matrix with $\mathbf{RAN}$, it will be the same delivery process.
\begin{figure}[H]
\centering
\begin{subfigure}{0.55\textwidth}
\centering
\includegraphics[height=5.0cm]{figure/2streamGT.pdf}
\caption{2-stream CNN \& NT}
\end{subfigure}
\hfill
\begin{subfigure}{0.30\textwidth}
\centering
\includegraphics[height=5.0cm]{figure/GTFES.png}
\caption{NT contained various data}
\end{subfigure}
\caption{ Model structure \& Data structure 3 (Multi-modal)}
\label{fig:graphtree3}
\end{figure}
These networks are structures in which each feature extraction process is combined.
If we design feature extraction networks for each type and construct a neuro tree with two or three children Fig.\ref{fig:graphtree3}(a,b), it will be the same delivery process.
\begin{figure}[h!]
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[height=2.5cm]{figure/e_models.png}
\caption{A network that can express only one fixed path.}
\label{fig:path1}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[height=2.5cm]{figure/a_models.png}
\caption{Artificial association networks can express various paths using neuro trees.}
\label{fig:path2}
\end{subfigure}
\caption{Artificial association networks that learn according to the tree structure.}
\label{fig:aan_model_path}
\end{figure}
Fig.\ref{fig:aan_model_path} shows the advantages of artificial association networks.
Most existing neural network models are fixed in a specific structure because they use fixed layers.
Fig.\ref{fig:path1} represents three models that perform two CNN models, and one concatenate process.
On the other hand, the neuro tree used by $\mathbf{AAN}$ can express these models as data(Fig.\ref{fig:path2}), so they all become one network model.
\begin{figure}[h!]
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[height=5.0cm]{figure/neurotree1.png}
\caption{ Ideal neuro tree (Multi-modal)}
\label{fig:idealtree}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[height=5.0cm]{figure/neurotree.png}
\caption{A neuro tree with multiple parent nodes}
\label{fig:multi-parent}
\end{subfigure}
\caption{A neuro tree that the propagation path doesn't go backward with multiple parent nodes}
\label{fig:graphtree1}
\end{figure}
Finally, this structure is an ideal neuro tree structure (Fig.\ref{fig:idealtree}).
The feature extraction network corresponds to the sensory organ.
And the feature extraction process($\Psi$) is selected according to the domain($\tau_{d}$) of data x($x$), and x becomes vector $\overrightarrow{x}$.
And information integration occurs. We call it an association network.
This network serves to embed various extracted information into one vector($\overrightarrow{h}_{root}$) by applying relationship and hierarchical information.
At this time, expressing a tree structure that learns well for each type of data or task is the same as designing the architecture of the network.
Therefore, instead of using fixed layers to learn according to the flow of information, build a neuro tree to learn according to the flow of information.
It will enable the network to learn networks that can handle multiple tasks or multiple datasets simultaneously, not just specific tasks or specific datasets.
Because various architectures can be expressed by data.
Visual information is input when we open our eyes, and information is not input when we close our eyes. It can be expressed when there is or does not have image information in the neuro tree.
In addition, it could express a structure such as V1, V2, V3, V4, V5/MT in visual cortex, and a neuro tree has hierarchical information and relational information.
The grammar of natural languages can also be expressed as a tree parser\cite{socher-etal-2013-recursive}.
Voice information can also be extracted from MFCC\cite{muda2010voice} algorithms etc.
And all information is integrated and embedded in association networks.
The integrated structure of these information can be used for multi-modal deep learning.
\paragraph{ Multiple parent nodes }
Fig.\ref{fig:multi-parent} shows the characteristics of a neuro tree that can have multiple parent nodes and connections with sibling nodes, not a general tree structure. And this structure becomes a propagation path doesn't go backward.
\textbf{a parent node or ancestor that uses the information of the current node should not be the child node} (Sec \ref{subsec:gtnode}).
This condition is not to deliver the information results from one node only to the direct parent node, as shown in Fig.\ref{fig:multi-parent}, but to connect to other nodes.
Therefore, This information has been added to express more routes.
Hidden information($\overrightarrow{h}$) corresponding to each neuro node($\mathbf{NN}_{p}$) is stored in the neuro node($\mathbf{NN}_{p}$) and recycle it when the information on the current node is needed again.
\subsection{ Propagation Method : Depth-first Convolution \& Deconvolution }
\label{sec:Depthfirst}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.47\textwidth}
\centering
\includegraphics[height=3.0cm]{figure/dfc.png}
\caption{Depth-First Convolution (post-order \& left-first)}
\label{fig:dfc}
\end{subfigure}
\hfill
\begin{subfigure}{0.47\textwidth}
\centering
\includegraphics[height=3.0cm]{figure/dfd.png}
\caption{Depth-First Deconvolution (pre-order \& right-first)}
\label{fig:dfd}
\end{subfigure}
\caption{Depth First Convolution \& Deconvolution}
\label{fig:dfcdfd}
\end{figure}
This section introduces two propagation methods of depth-first convolution (DFC) (a convolution method for traversing all nodes of NT) and depth-first deconvolution (DFD).
Depth-first search (DFS) is a search algorithm for the tree data structure. Since this methodology is transferred from the nodes at the deepest level, various paths can be expressed, and the convolution path is simplified. Unlike general recursive convolution, DFC and DFD are methodologies for learning relationships among the sibling nodes and feature extraction networks together.
\begin{minipage}{1.02\textwidth}
\begin{algorithm}[H]
\caption{Depth First Convolution}
\label{algorithm:dfc}
\begin{multicols}{2}
\begin{algorithmic}[1]
\Function{DFC}{$\mathbf{NN}_{p},lv$}
\If{$\mathbf{NN}_{p}.isCalculated$} \Comment{stored hidden}
\State \Return{$\mathbf{NN}_{p}.\overrightarrow{h'}$}
\EndIf
\State {$x_{p},\tau_{dp},\mathbf{A}_{cp},\mathbf{C}_{p}$ $\gets$ $\mathbf{NN}_{p}.items()$}
\State {$\overrightarrow{x}_{p}, *i_{1}$ $\gets$ $\Psi[\tau_{dp}]({x}_{p})$} \Comment{feature}
\State {$ChildrenList$ $\gets$ {$[\ ]$}}
\For{$q \gets 1...N_{p}$} \Comment{left-first}
\State {$\overrightarrow{h'}_{pq}$ $\gets$ \Call{DFC}{$\mathbf{C}_{p}[q],lv+1$}}
\State {$ChildrenList$.append($\overrightarrow{h'}_{pq}$)}
\EndFor
\State
\If{len($\mathbf{C}_{p}$) is $0$} \Comment{leaf node}
\State {$\mathbf{NN}_{p}.more$ $\gets$ $(*i_{1})$}
\State {$\overrightarrow{h'}_{p}$ $\gets$ $\mathbf{RNN}$($\overrightarrow{x}_{p},\overrightarrow{0}$)} \Comment{$\mathbf{W}[\overrightarrow{x}, \overrightarrow{0}]$}
\State {$\mathbf{NN}_{p}.\overrightarrow{h'} \gets \overrightarrow{h'}_{p}$}
\State \Return {$\overrightarrow{h'}_{p}$}
\EndIf
\State {$\mathbf{h'}_{p}$ $\gets$ Stack($ChildrenList$)}
\State {$\overrightarrow{h}_{p}, i_{2}$ $\gets$ $g(\mathbf{GNN}(\mathbf{A}_c, \mathbf{h'}_{p}))$} \Comment{$g(\mathbf{A}_{c}\mathbf{h}'_{p})$}
\State {$\overrightarrow{h'}_{p}$ $\gets$ $\mathbf{RNN}$($\overrightarrow{x}_{p},\overrightarrow{h}_{p}$)} \Comment{$\mathbf{W}[\overrightarrow{x}_{p}, \overrightarrow{h}_{p}]$}
\State {$\mathbf{NN}_{p}.\overrightarrow{h'} \gets \overrightarrow{h'}_{p}$}
\State {$\mathbf{NN}_{p}.isCalculated \gets True$} \Comment{storing hidden}
\State {$\mathbf{NN}_{p}.more$ $\gets$ $(*i_{1}, *i_{2})$}
\State \Return {$\overrightarrow{h'}_{p}$}
\EndFunction
\end{algorithmic}
\end{multicols}
\end{algorithm}
\end{minipage}
\paragraph{Post-order Depth-first convolution (Left-First)} If DFS is post-order, iteration starts from the leaf node. Likewise, DFC is a recursive convolution that propagates from the deepest nodes.
artificial association networks utilize the hidden vector expressing the current level and state.
First, check whether the current node has already been calculated. This process is because one node can exist as a child node of multiple nodes; This allows one node to have multiple parent nodes and if it's already been calculated, use the results from the previous calculation.
And then, we need to bring five items($x_{p}, \tau_{dp}, \tau_{tp}, \mathbf{A}_{cp}, \mathbf{C}_{p}$) of the current $\mathbf{NN}_{p}$.
\textbf{the $p$ means the convolution order} in the propagation path, and $x_{p}$ is embedded by the feature extraction network($\Psi$) according to the type($\tau_{dp}$) and becomes the feature vector($\overrightarrow{x}_{p}$).
artificial association network models perform the convolution by receiving information($\overrightarrow{h'}_{pq}$) from \textbf{the current node's children($\mathbf{C}_{p}$)} with \textbf{the children's relationship($\mathbf{A}_{cp}$) among children}.
We can use this process \textbf{like learning the relationship among the siblings nodes.}
the $\overrightarrow{h'}_{p}$ means the convolution output of the child nodes. This process is similar to the process of graph convolution and then the readout process.
And perform convolution using these $\overrightarrow{h}_{p}$ and $\overrightarrow{x}_{p}$ of the current node. This process is similar to the recurrent neural network(RNN).
And store the convolution output($\overrightarrow{h'}_{p}$) on the current node and recycle it when the information on the current node is needed again.
If there is a task to be performed at the node level, it goes through task networks($\Phi$).
Finally, $\overrightarrow{h'}_{p}$ is delivered to the parent node.
If the child node does not exist($\mathbf{C}_{p}$), the current node means a leaf node. Therefore, the hidden vector( $\overrightarrow{h}_{p}$) is the initial hidden vector($\overrightarrow{0}$).
If we repeat this process, we finally get $\overrightarrow{h'}_{root}$.
Additionally, we can store information($*i_{1}, *i_{2}$) generated during the DFC process in $\mathbf{NN}.more$ and use it during the DFD process (ex. If the aggregate function is a maxpool, we can use the indices information during deconvolution).
\begin{minipage}{1.01\textwidth}
\begin{algorithm}[H]
\caption{Depth First Deconvolution}
\label{algorithm:dfd}
\begin{multicols}{2}
\begin{algorithmic}[1]
\Function{DFD}{$\protect\tilde{h}_{p},\mathbf{NN}_{p},*lv$}
\State {$\tilde{x}_{p}, \tilde{h}'_{p}$ $\gets$ $\mathbf{RNN}^{-1}$($\tilde{h}_{p}$)} \Comment{$\mathbf{W}\overrightarrow{h}_{p}$}
\State $\tau_{dp},\mathbf{A}_{cp},\mathbf{C}_{p}$ $\gets$ $\mathbf{NN}_{p}.dconv\_items()$
\State $*i_{1}, *i_{2}$ $\gets$ $\mathbf{NN}_{p}.more$
\State $\mathbf{NN}_{p}.\hat{x}_{p}$ $\gets$ $\Psi^{-1}[\tau_{dp}](\tilde{x}_{p}, *i_{1})$
\If{len($\mathbf{C}_{p}$) is $0$} \Comment{leaf node}
\State \Return \label{algoline:return}
\EndIf
\State $\mathbf{\hat{A}}_{cp}, \mathbf{\tilde{h}'}_{p}$ $\gets$ $g^{-1}(\mathbf{GNN}^{-1}(\mathbf{A}_{cp},\tilde{h}'_{p},lv),*i_{2})$
\State {$\mathbf{NN}_{p}.\hat{\mathbf{A}}_{c}$ $\gets$ {$\mathbf{\hat{A}}_{cp}$}}
\For{$q \gets N_{p}...1$} \label{algoline:rightfirst} \Comment{right-first}
\State {$\tilde{h}_{pq}$ $\gets$ $ \mathbf{\tilde{h}}_{p}'[q]$}
\State {\Call{DFD}{$\protect\tilde{h}_{pq},\mathbf{NN}_{p}.\mathbf{C}[q],lv+1$}}
\EndFor
\State \Return
\EndFunction
\end{algorithmic}
\end{multicols}
\end{algorithm}
\end{minipage}
\paragraph{Pre-order Depth-first Deconvolution (Right-First) }
DFD is a decoding methodology for artificial association networks.
The DFC propagates from the leaf node to the root node, whereas the DFD propagates from the root node to the leaf node in the order of pre-order depth first to decode this. And the convolution paths are in reverse order with each other.
First, The hidden state($\tilde{h'}_{p}$) is restored to $\tilde{x}_{p},\tilde{h}'_{p}$ through $\mathbf{RNN}^{-1}$. And we need to bring three items($\tau_{dp}, A_{cp}, C_{p}$) of the current $\mathbf{NN}_{p}$.
$\tilde{x}_{p}$ is decoded by the feature decoder network($\Psi^{-1}$) according to the type($\tau_{dp}$) and the decoded output is stored in the $\mathbf{NN}_{p}.\hat{x}_{p}$.
The hidden state($\tilde{h}'_{p}$) is restored to $\mathbf{\tilde{h}}'_{p}$ through $\mathbf{AAN}^{-1}$; and $\mathbf{\tilde{h}}'_{p}$ is transferred to child nodes.
If the child does not exist, this means a leaf node and move to a different path by return(line \ref{algoline:return} in algo \ref{algorithm:dfd}).
Also, if DFC proceeds from left to right as left first ($1...N$), DFD proceeds from right to left as right first ($N...1$)(line \ref{algoline:rightfirst} in algo \ref{algorithm:dfd}). Please see Fig.\ref{fig:dfcdfd}.
\begin{minipage}{1.01\textwidth}
\begin{algorithm}[H]
\caption{Propagate (for AutoEncoder Models)}
\label{algorithm:propagate}
\begin{algorithmic}[1]
\Function{propagate}{$\mathbf{NN}_{root}$}
\State {$\overrightarrow{h'}_{root}$ $\gets$ {\Call{DFC}{$\mathbf{NN}_{root},0$}}} \Comment{$Convolution (Neuro Tree, level)$}
\State {\Call{DFD}{$\protect\overrightarrow{h'}_{root},\mathbf{NN}_{root},0$}} \Comment{$Deconvolution (Final Hidden, Neuro Tree, level)$}
\State \Return $\overrightarrow{h}_{root}, \mathbf{NN}_{root}$
\EndFunction
\end{algorithmic}
\end{algorithm}
\end{minipage}
Like the propagate function (algo.\ref{algorithm:propagate}), we can freely encode and decode through DFC and DFD.
We want to show the relationship between DFC and DFD and the possibility of expanding to the widely used autoencoder\citep{hinton2006reducing} models.
In addition, Implementing this recursive function as a loop(while) will speed up.
\subsection{Multi-Mini-Batch Training : Multi-Batch Feature Extraction Method}
\label{appendix:type_embedding}
\begin{algorithm}[H]
\caption{Multi-Batch Feature Extraction Method}
\label{algorithm:batch_type_embedding}
\begin{algorithmic}[1]
\Function{multi-batch-convolution}{$NT_{batch}$}
\State $dict_{x}$ = \{\} \Comment{$key(\tau)$ : $value(list_{x})$}
\State $dict_{idx}$ = \{\} \Comment{$key(batch\_idx)$ : $value(\tau_{d}$, $index_{dict_{x}})$}
\For{$idx_{batch}, \tau_{d}, x \gets enumerate(NT_{batch})$}
\State $dict_{idx}$[$idx_{batch}$] = ($\tau_{d}$, len($dict_{x}$[$\tau_{d}$]))
\State $dict_{x}$[$\tau_{d}$].append($x$) \Comment{How to combine data of the same type}
\EndFor
\For{$\tau_{d}, X_{\tau_{d}}$ $\gets$ $dict_{x}.items()$} \Comment{$\tau_{d}(key),X(values)$}
\State {$\mathbf{x}_{\tau_{d}}$ = [$\Psi$[$\tau_{d}$]($X_{\tau_{d}}$), onehot($\tau_{d}$).repeat(len($X_{\tau_{d}}$))]}
\Comment{Type Batch Convolution}
\State {$dict_{x}$[$\tau_{d}$] = $\mathbf{x}_{\tau_{d}}$}
\EndFor
\State $output_{batch}$ = []
\For{$idx_{batch}$ $\gets$ $1...N$} \Comment{$Restoration$}
\State {$\tau_{d}, idx_{\tau_{d}}$ $\gets$ $dict_{idx}$[$idx_{batch}$]}
\State {$\mathbf{x}_{\tau_{d}}$ = $dict_{x}$[$\tau_{d}$]}
\State $output_{batch}$.append($\mathbf{x}_{\tau_{d}}[idx_{\tau_{d}}]$)
\EndFor
\State \Return $Stack(output_{batch})$
\EndFunction
\end{algorithmic}
\end{algorithm}
First, make two dictionary data structures.
One is a dictionary($dict_{x}$) that stores data for each data type.
The other is a dictionary($dict_{idx}$) that preserves the batch-index information of the data.
We tour the mini-batch data in order and store the key is the domain($\tau_{d}$) of data, and the value is the input data($x$). And the other dictionary stores the type and index.
Therefore, the input data are collected for each type, which is called $X_{\tau}$.
And the input data become type-mini-batch data for each type and becomes batch-inputs for the feature extraction network of that type.
Let $\mathbf{x}_{\tau}$ denote the output of the feature extraction network.
Finally, the $\mathbf{x}_{\tau}$ moves to the original batch index through the previously-stored batch index($dict_{idx}$).
\begin{figure}[h]
\centering
\includegraphics[height=5cm]{figure/fe_batch.png}
\hfil
\caption{ Multi-feature extraction networks for multi-mini-batch training }
\label{fig:febatch}
\end{figure}
This methodology divides mini-batch samples by type, and the domain-mini-batch size varies each time(fig.\ref{fig:febatch}). Therefore, we recommend using a weight standardization\citep{qiao2019micro} or group normalization\citep{wu2018group} to avoid batch normalization\citep{ioffe2015batch} affected by the batch size. We described the details of the network we used in sec \ref{appendix:usedmodels}.
\subsection{ Artificial Association Network Models : RAN \& LAN Series }
\label{sec:network}
This section proposes recursive and level association networks series that perform inductive learning and recognition part in graph tree neural networks.
And we describe \textbf{the convolution process in the p-th order of the depth-first convolution path in a neuro tree}.
These networks can be used as an association cell or layer, and we explained the difference between the two models in sec \ref{appendix:compare}.
These association models do not use any fixed architecture and integrate information according to the neuro tree structure. And by fusing graph and tree data structures, we can learn more diverse data structures with one cell and various architectures can be expressed by data(please see the sec \ref{appendix:graphtree_architecture}).
Accordingly, these models are freer to learn various datasets using fewer parameters.
artificial association models are as follows: The feature extraction models create the information from data; It is the "smallest unit" of information in the $\mathbf{NT}$.
The parent $\mathbf{NN}$ receives information from their children and perform the convolution operation with the relation term; the convolution output is a "bigger unit" of information.
Finally, a root $\mathbf{NN}_{root}$ has all $\mathbf{NT}$ information and perform the convolution operation; the output is "the biggest unit" of information.
\subsubsection{ Association Cell : Recursive Association Networks }
\label{sec:gtran}
We will describe a $\mathbf{RAN}$ cell that trains the $\mathbf{NT}$ described previously in Sec \ref{subsec:gtnode}.
recursive association models can be expressed as $\mathbf{RAN}=\{\mathbf{W}, \Psi, g, \sigma \}$ $\mathbf{W}\in\mathbb{R}^{F' \times (F+B+F')}$.
$F$ is the node-feature size, $B$ is the type-bias size, and $F'$ is the hidden size, which is the information received from the child node.
$\Psi$ denotes the feature-extraction(Eqn \ref{eq:feature_extraction}) process described above, \(g\) denotes the aggregate function, and $\sigma$ denotes the activation function.
The propagation is proceeding from the leaf nodes at the deepest level to the root node at the top level(sec \ref{sec:Depthfirst}).
$p$ is the order of depth-first convolution, $q$ is the child node number, $N_{p}$ is the number of children of $\mathbf{NN}_{p}$, $x_{p}$ is the node-input of $\mathbf{NN}_{p}$.
\begin{equation}
\label{eq:gtran1}
\overrightarrow{x}_{p} = [\psi_{\tau_{dp}}(x_{p}),onehot(\tau_{dp})] = \Psi({x}_{p},\tau_{dp}), \overrightarrow{x}_{p} \in \mathbb{R}^{F+B}
\end{equation}
First, The $\Psi$ extracts the feature vector($\overrightarrow{x}_{p}$) from $x_{p}$ of $\mathbf{NN}_{p}$ through the feature extraction process(Eqn.\ref{eq:gtran1}) considering the type information($\tau_{dp}$).
The first location to start is leaf nodes by DFC(algo.\ref{algorithm:dfc}).
\begin{equation}
\label{eq:gtran2}
\overrightarrow{h'}_{pq} = DepthFirstConvolution(NN_{p}.C_{q}, *lv + 1)
\end{equation}
the current node($\mathbf{NN}_{p}$) receives the child's hidden states($\overrightarrow{h'}_{pq}$) from the child nodes.
the current node in the p-th order receive $\overrightarrow{h'_{pq}}$ from the q-th child node(Eqn \ref{eq:gtran2}).
\begin{equation}
\label{eq:gtran3}
\mathbf{h'}_{p} = \mathbin\Vert_{q=0}^{N_{p}}\overrightarrow{h'}_{pq}
\end{equation}
\begin{equation}
\label{eq:gtran4}
\overrightarrow{h}_{p} = g(\tilde{\mathbf{D}}_{cp}^{-\frac{1}{2}}\tilde{\mathbf{A}}_{cp}\tilde{\mathbf{D}}_{cp}^{-\frac{1}{2}}(\mathbf{h'}_{p}))
\end{equation}
The child's hidden states($\overrightarrow{h'}_{pq}$) and $\mathbf{A}_{c}$ perform graph convolution and readout to become a children's hidden state($\overrightarrow{h}_{p}$) containing all the information of the child nodes.
$\mathbin\Vert_{q=0}^{N_{p}}\overrightarrow{h'}_{pq}$ means that $\overrightarrow{h'}_{pq}$ matching the graph nodes($\mathbf{A}_{cp}$) existing in the p-th node are stacked as much as $N_{p}$, which is $\mathbin\Vert_{q=0}^{N_{p}}\overrightarrow{h}_{pq}\in\mathbb{R}^{N_{p} \times F'}$.
We applied the GCN methodology\citep{kipf2016semi}, which is useful in the GNN field.
Therefore we expressed as $\tilde{\mathbf{A}}_{cp}=\mathbf{A}_{cp} + \mathbf{I}$, where $\mathbf{I}$ is the Identity matrix.
If we express the connection value as 1, just the more connected nodes becames the scale value larger than others.
Therefore, $\tilde{\mathbf{D}}_{cp}^{-\frac{1}{2}}\tilde{\mathbf{A}}_{cp}\tilde{\mathbf{D}}_{cp}^{-\frac{1}{2}}$ is applied using the order matrix $\tilde{\mathbf{D}}_{cp}$ of $\tilde{\mathbf{A}}_{cp}$ as a method of normalizing the relationship matrix in Eqn.\ref{eq:gtran4}.
\begin{equation}
\label{eq:gtran5}
\overrightarrow{h}'_{p} = \sigma([\overrightarrow{x}_{p},\overrightarrow{h}_{p}]\mathbf{W}^{T})
\end{equation}
In the leaf node, the children's hidden state($\overrightarrow{h}_{p}$) is a initial hidden state($\overrightarrow{0}$).
the hidden state($\overrightarrow{h}_{p}$) is concatenated with $\overrightarrow{x}_{p}$ to become $[\overrightarrow{x}_{p}, \overrightarrow{h}_{p}] \in \mathbb{R}^{F+B+F'}$ through the concatenation process as $[,]$.
The children's hidden state($\overrightarrow{h}_{p}$) and the ($\overrightarrow{x}_{p}$) of the current node perform convolution to become the current node's hidden state($\overrightarrow{h'}_{p}$).
\begin{equation}
\label{eq:gtran6}
\overrightarrow{h'}_{p} = \sigma([\overrightarrow{x}_{p}, g((\tilde{\mathbf{D}}_{cp}^{-\frac{1}{2}}\tilde{\mathbf{A}}_{cp}\tilde{\mathbf{D}}_{cp}^{-\frac{1}{2}})\mathbf{h'}_{p})]\mathbf{W}^{T})
\end{equation}
Finally, we can express these processes(Eqn \ref{eq:gtran1}, \ref{eq:gtran2}, \ref{eq:gtran3}, \ref{eq:gtran4}, \ref{eq:gtran5}) as Eqn \ref{eq:gtran6}.
we used $F$ as 128, $B$ as 3(image, sound, language), and $F'$ as 128, and $\sigma$ as ReLU\citep{nair2010rectified}, and g was the readout of Max.
\begin{equation}
\label{eq:spcase1}
RNN = \sigma([\overrightarrow{x}_{t},\overrightarrow{h}_{t-1}]\mathbf{W}^{T}) = \sigma\{[\overrightarrow{x}_{lv},g(\mathbf{I}\overrightarrow{h}_{lv+1})]\mathbf{W}^{T}\} = \sigma([\overrightarrow{x}_{lv},\overrightarrow{h}_{lv+1}]\mathbf{W}^{T})
\end{equation}
If there is no sibling node of the current node, $\mathbf{RAN}$ is mathematically identical to the RNN(Eqn.\ref{eq:spcase1}).
RNN can be a special case of $\mathbf{RAN}$, and the depth of the neuro tree means times of the RNN.
\subsubsection{Association Layers : Level Association Networks}
\label{sec:gtlan}
level association networks($\mathbf{LAN}$) is composed of $\mathbf{LAN}=\{\{\mathbf{W}_{0},...\mathbf{W}_{m}\},\Psi, g, \sigma\}$.
\begin{equation}
\label{eq:gtlan}
\overrightarrow{h'}_{p} = \sigma([\overrightarrow{x}_{p}, g((\tilde{\mathbf{D}}_{cp}^{-\frac{1}{2}}\tilde{\mathbf{A}}_{cp}\tilde{\mathbf{D}}_{cp}^{-\frac{1}{2}})\mathbf{h'}_{p})]\mathbf{W}^{T}_{lv})
\end{equation}
The mathematical expression(Eqn \ref{eq:gtlan}) of $\mathbf{LAN}$ is similar to Eqn \ref{eq:gtran4}.
There is $\mathbf{W}_{lv}\in\mathbb{R}^{F'_{lv} \times (F_{lv}+B+F'_{lv+1})}$ in charge of each level, and the network in charge of each level is called the level layer.
The input size of the level is $F_{lv}$, its output size is $F'_{lv}$, and the output size of the child is $F'_{lv+1}$.
Therefore, it is possible to adjust the input, hidden size; and it can learn the depth-limited tree, $m$ means maximum-depth. We set $F_{lv}$ and $F'_{lv}$ to 128 for all lvs in the same way as $\mathbf{RAN}$ in the experiment.
\begin{equation}
\label{eq:spcase2}
FC\ layer = fc(\overrightarrow{h}) = \sigma(\overrightarrow{h}\mathbf{W}^{T}) = \sigma(g(\mathbf{I}\overrightarrow{h})\mathbf{W}^{T}_{lv}) = \sigma(\overrightarrow{h}\mathbf{W}^{T}_{lv})
\end{equation}
\begin{equation}
\label{eq:spcase3}
\mathbf{NN}_{m} = \{\overrightarrow{x},t,\mathbf{I},\{\}\}
\end{equation}
\begin{equation}
\label{eq:spcase4}
\mathbf{NN}_{1} = \{\overrightarrow{0},layer,\mathbf{I},\mathbf{NN}_{2}\}
\end{equation}
\begin{equation}
\label{eq:spcase5}
\mathbf{NN}_{0} = \{\overrightarrow{0},layer,\mathbf{I},\mathbf{NN}_{1}\}
\end{equation}
\begin{equation}
\label{eq:spcase6}
Multi\ layer = fc_{N}(fc_{2}(fc_{1}(\overrightarrow{x}))) = \mathbf{LAN}(\mathbf{NN}_{0})
\end{equation}
If the input size of $F_{lv}$ is 0 and there is no sibling node of the current node, this network is mathematically identical to the FC layer as Eqn \ref{eq:spcase2} and the MLP as Eqn \ref{eq:spcase3}, \ref{eq:spcase4}, \ref{eq:spcase5}, \ref{eq:spcase6}. $\mathbf{I}$ is the Identity matrix.
The depth of the tree indicates the number of layers of the MLP. Therefore, the MLP and FC layer can be a special case of $\mathbf{LAN}$.
\subsection{Compare Architecture: LAN \& RAN }
\label{appendix:compare}
\begin{table}[H]
\caption{Compare Architecture}
\centering
\begin{tabular}{lll}
\toprule
Name & LAN & RAN \\
\midrule
Level layer & O & X \\
The number of $\mathbf{W}$ & The number of levels & 1 \\
Depth-limited & Fixed & Not fixed \\
The number of parameters & Use more & Use less than \\
Input size by level & Adjustable & Fixed \\
The special cases & FCNN, MLP & RNN(recurrent, recursive) \\
\bottomrule
\end{tabular}
\label{table:Comparison}
\end{table}
In this section, we compare the network characteristics of $\mathbf{LAN}$-series of level layers and $\mathbf{RAN}$-series of recursive cell.
In the case of level association networks($\mathbf{LAN}$), there is a different $\mathbf{W}_{lv}$ for each level, expressing each level layer and the convolution of all information.
On the other hand, in the case of recursive association networks($\mathbf{RAN}$), the main difference is the recursive convolution with only one $\mathbf{W}$ as a cell. This is the most significant difference when comparing the networks.
Therefore, level association networks can adjust the number of features by level;
for example, the size of input features can differ between Levels 1 and 0. It is possible for level association networks to train datasets by adjusting parameters at any level.
On the other hand, recursive association networks must be trained with the same input and hidden size.
Thus, we divided artificial association networks into two groups.
Since level association networks have a network in charge of each level, more parameters are needed, and it is appropriate for depth-limited neuro tree.
if the $F_{lv}$ of input size is 0 and the number of nodes is 1 in neuro node, FC layer and MLP can be special cases of level association networks.
On the other hand, recursive association networks traverse all neuro nodes with a recursive association cell.
Therefore it is possible to train even if the maximum depth is not fixed and fewer parameters are used.
in addition, If there is no sibling node in $\mathbf{NN}_{p}$, recurrent neural networks can be a special cases of recursive association networks.
And If there's no relationship($\mathbf{A}_{cp}$) among the sibling nodes, recursive neural networks can be a special cases of recursive association networks.
Consequently, the compare result in Table \ref{table:Comparison}.
\subsection{ Attention Models }
There is an attention process in the human brain. Therefore, GATs\citep{velivckovic2017graph}, a model that has been useful recently, was combined.
\subsubsection{ Recursive Attentional Association Networks }
\label{subsec:RAANn}
We introduce $\mathbf{RAAN}$ that learn the importance through attention by slightly modifying the expression of the $\mathbf{RAN}$.
It is composed of $\mathbf{RAANs}=\{\mathbf{W}, \Psi,g, \sigma, \sigma_{a}, \overrightarrow{a} \}$ that added $\{\sigma_{a}, \overrightarrow{a}\}$ in $\mathbf{RAN}$. A parameter for attention mechanism is added ($\mathbb{R}^{2F'} \times \mathbb{R}^{2F'} \rightarrow \mathbb{R}$) and the attention's activation function used LeakyReLU\citep{xu2015empirical} in the same way as GATs.
$\mathcal{N}_{pq}$ is a set of nodes connected to the q-th child's node in the $A_{cp}$ of $\mathbf{NN}_{p}$, and we can express this as:
\begin{equation}
\label{eq:attention_matrix}
\alpha_{pqr} = \frac{\mathbf{exp}(LeakyReLU(\overrightarrow{\mathbf{a}}^{T}[\overrightarrow{h'}_{pq},\overrightarrow{h'}_{pr}]))}{\sum_{k\in \mathcal{N}_{pq}}\mathbf{exp}(LeakyReLU(\overrightarrow{\mathbf{a}}^{T}[\overrightarrow{h'}_{pq},\overrightarrow{h'}_{pk}]))}
\end{equation}
With the attention methodology introduced in GATs\citep{velivckovic2017graph}, it learns how the r-th node is of importance to the q-th node.
This information is replaced with the part to which the adjacency matrix is connected.
Therefore, the $\mathbf{RAAN}$ are as follows:
\begin{equation}
\label{eqn:attention1}
\overrightarrow{h'}_{p} = \sigma([\overrightarrow{x}_{p}, g((\mathbf{A}_{cp}\odot\mathbf{\alpha}_{p})\mathbf{h'}_{p})]\mathbf{W}^{T})
\end{equation}
In Eqn.\ref{eqn:attention1}, $\odot$ indicates point-wise and this process is similar to Eqn.\ref{eq:gtran6}.
To further stabilize the self-attention process, we introduce a multi-head attention mechanism:
\begin{equation}
\label{eqn:attention3}
\overrightarrow{h'}_{p} = ||_{k=1}^{K}\sigma([\overrightarrow{x}_{p}, g((\mathbf{A}_{cp}\odot\mathbf{\alpha}^{k}_{p})\mathbf{h'}_{p})]\mathbf{W}^{kT})
\end{equation}
where $K$ is the number of multi-heads. The results from multiple heads are concatenated and delivered to a parent node.
We set K to 8 and set output dim for each head to 16(=$F'_{lv}/K$). therefore we set $F'_{lv}$ to 128.
This process becomes a cell and delivers the result from the leaf node to the root node.
\subsubsection{Level Attentional Association Networks}
The $\mathbf{LAAN}$ model is modified from $\mathbf{LAN}$, and GATs' attention mechanism was applied.
this model can be expressed as $\mathbf{LAAN}=\{\{\mathbf{W}_{0},...\mathbf{W}_{m}\},\{\overrightarrow{a}_{0},...\overrightarrow{a}_{m}\},\Psi, g, \sigma, \sigma_{a}\}$, and $\{\sigma_{a}, \overrightarrow{a}_{lv}\}$ are added in LAN($\overrightarrow{a_{lv}}\in\mathbb{R}^{2F^{'}_{lv}}$).
Unlike $\mathbf{RAAN}$, $\mathbf{LAAN}$ can design different sizes of $F, F'$ to match the lv.
\begin{equation}
\label{eq:level_attention_matrix}
\alpha_{pqr} = \frac{\mathbf{exp}(LeakyReLU(\overrightarrow{\mathbf{a}}^{T}_{lv}[\overrightarrow{h'}_{pq},\overrightarrow{h'}_{pr}]))}{\sum_{k\in \mathcal{N}_{pq}}\mathbf{exp}(LeakyReLU(\overrightarrow{\mathbf{a}}^{T}_{lv}[\overrightarrow{h'}_{pq},\overrightarrow{h'}_{pk}]))}
\end{equation}
\begin{equation}
\label{eqn:levelattention1}
\overrightarrow{h'}_{p} = \sigma([\overrightarrow{x}_{p}, g((\mathbf{A}_{cp}\odot\mathbf{\alpha}_{p})\mathbf{h'}_{p})]\mathbf{W}^{T}_{lv})
\end{equation}
this model is designed to select critical features, and it is similar to Sec.\ref{subsec:RAANn}
\subsection{ Gated Models }
Basically, the longer the depth of time, the more difficult the RNN is to propagate the error. Therefore, we add the gated recurrent unit(GRU\cite{chung2014empirical}) model.
We designed Gated association unit($\mathbf{GAU}$) by slightly modifying $\mathbf{RAN}$.
\subsubsection{ Gated Association Unit }
\label{subsec:GAU}
\begin{equation}
\label{eqn:GGTAN}
\overrightarrow{h}_{p} = g((\tilde{\mathbf{D}}_{cp}^{-\frac{1}{2}}\tilde{\mathbf{A}}_{cp}\tilde{\mathbf{D}}_{cp}^{-\frac{1}{2}})\mathbf{h'}_{p})
\end{equation}
\begin{equation}
\label{eqn:gatedgcn1}
\overrightarrow{r}_{p} = \sigma(\mathbf{W}_{r}\overrightarrow{h}_{p} + \mathbf{U}_{r}\overrightarrow{x}_{p})
\end{equation}
\begin{equation}
\label{eqn:gatedgcn2}
\overrightarrow{u}_{p} = \sigma(\mathbf{W}_{u}\overrightarrow{h}_{p} + \mathbf{U}_{u}\overrightarrow{x}_{p})
\end{equation}
\begin{equation}
\label{eqn:gatedgcn3}
\tilde{h}_{p} = tanh(\mathbf{W}(\overrightarrow{h}_{p} \odot \overrightarrow{r}_{p}) + \mathbf{U}\overrightarrow{x}_{p})
\end{equation}
\begin{equation}
\label{eqn:GAU}
\overrightarrow{h'}_{p} = (1-\overrightarrow{u}_{p}) \odot \overrightarrow{h}_{p} + \overrightarrow{u}_{p} \odot \tilde{h}_{p}
\end{equation}
\subsubsection{ Gated Attentional Association Unit }
\label{subsec:GAAU}
\begin{equation}
\label{eqn:GTAAN}
\overrightarrow{h}_{p} = g((\mathbf{A}_{cp}\odot\mathbf{\alpha}_{p})\mathbf{h'}_{p})
\end{equation}
\begin{equation}
\label{eqn:gatedgat1}
\overrightarrow{r}_{p} = \sigma(\mathbf{W}_{r}\overrightarrow{h}_{p} + \mathbf{U}_{r}\overrightarrow{x}_{p})
\end{equation}
\begin{equation}
\label{eqn:gatedgat2}
\overrightarrow{u}_{p} = \sigma(\mathbf{W}_{u}\overrightarrow{h}_{p} + \mathbf{U}_{u}\overrightarrow{x}_{p})
\end{equation}
\begin{equation}
\label{eqn:gatedgat3}
\tilde{h}_{p} = tanh(\mathbf{W}(\overrightarrow{h}_{p} \odot \overrightarrow{r}_{p}) + \mathbf{U}\overrightarrow{x}_{p})
\end{equation}
\begin{equation}
\label{eqn:GAAU}
\overrightarrow{h'}_{p} = (1-\overrightarrow{u}_{p}) \odot \overrightarrow{h}_{p} + \overrightarrow{u}_{p} \odot \tilde{h}_{p}
\end{equation}
We designed Gated attentional association unit($\mathbf{GAAU}$) by slightly modifying $\mathbf{RAAN}$.
It combined the attention matrix and the GRU to improve learning in time serial data as deep neuro tree data.
\subsection{ Task Networks }
\begin{figure}[h]
\centering
\includegraphics[height=3.9cm]{figure/multi-task-net.png}
\hfil
\caption{ Task Networks (Multi-Task)}
\label{fig:multi-task-net}
\end{figure}
\begin{equation}
\hat{y}_{p} = \Phi(\overrightarrow{h}_{p}, \tau_{tp})
\label{eq:multi-task}
\end{equation}
This process means performing a task at each node level or root level. Performing multi-task learning in this process means that all multi-domain, multi-modal, and multi-task can be processed simultaneously, and it is an end-to-end learning model.
So, how will we perform multi-task?
The author of this paper does not simply perform multi-tasks but wants to define the structure like humans.
Therefore, future works divided the processes imagined by the author and separated tasks into future works\cite{kim2021deductive, kim2021memory, kim2021imagine}.
\paragraph{End-to-End Multi-Deep learning}
\begin{equation}
\hat{y}_{p} = \mathbf{AAN}(NTBuilder(\mathbf{A},\mathbf{X}, \mathbf{\tau_{d}}, \mathbf{\tau_{t}}),\Psi,\Phi)
\label{eq:end-to-end}
\end{equation}
To express the structure of this neural network as an end-to-end structure, it is as follows.
Also, in the author's opinion, the structure of the human brain is quite complex.
Therefore, it is very important for neural networks to structure themselves.
This $\mathbf{AAN}$ has changed the structure of the network to be modified by expressing the information delivery structure as a tree structure.
Therefore, Neuro Tree Builder becomes the future works of this study.
\textbf{ Supervised learning \;\;}
Supervised learning uses datasets with labels.
In this network, the $\overrightarrow{h}_{root}$ was mapped to the dimension of the class using a fully connected layer($F'$ to class count) and we calculated the log softmax and negative log likelihood loss for supervised learning as:
\begin{equation}
\overrightarrow{h}_{root} = propagate(\mathbf{NT})
\end{equation}
\begin{equation}
\hat{y} = log\_softmax(\Phi(\overrightarrow{h}_{root},\tau_{t}))
\end{equation}
\begin{equation}
loss = negative\_log\_likelihood\_loss(y, \hat{y})
\end{equation}
\section{Experimental Results}
\label{sec:experimental}
\textbf{Our goal is to express the information delivery process of existing models as a neuro tree of data.} This section introduces the contents of the experiment and the neuro tree dataset consisting of the image, sound, natural language and relation. How to design neuro tree is introduced in sec.\ref{appendix:graphtree_architecture}.
\subsection{Exp 1 : Are feature extraction networks and association networks well learned together? (Multi-Domain)}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[height=3.5cm]{figure/sound_gt.png}
\caption{ NT of sound}
\label{fig:first}
\end{subfigure}
\hfill
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[height=3.5cm]{figure/image_gt.png}
\caption{NT of image}
\label{fig:second}
\end{subfigure}
\hfill
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[height=3.5cm]{figure/lang_gt.png}
\caption{NT of language}
\label{fig:third}
\end{subfigure}
\hfill
\begin{subfigure}{0.27\textwidth}
\centering
\includegraphics[width=\textwidth, height=3.5cm]{figure/exp1_fe.png}
\caption{Feature extraction net}
\label{fig:third}
\end{subfigure}
\caption{Combining feature extraction networks and association networks}
\label{fig:dataset_GT1}
\end{figure}
\paragraph{Datasets}
The first experiment is whether various datasets can be simultaneously learned by combining $\mathbf{AAN}$ and feature extraction networks.
the neuro tree we used in the experiment is illustrated by Fig.\ref{fig:dataset_GT1}(a,b,c).
Layer node means that it has no input value, only performs convolution. If there is no sibling node, $\mathbf{LAN}$ performs the same operation as one FC layer.
Sound and natural language NT have two nodes, and image NT has three nodes because It was configured similarly to figure (likes this fig.\ref{fig:network_comparison}).
The important point is not limited to any specific architecture; later, this NT could be modified according to the task or complexity. Existing networks learn only about specific datasets.
We trained without using a pre-trained model to check whether various feature extraction models can be learned simultaneously. Only word embedding network\citep{pennington2014glove} was used as a pretrained model.
Therefore, LeNet-5\citep{lecun1998gradient}, M5\citep{dai2017very}, and CNN\citep{kim-2014-convolutional} were selected for feature extraction networks of the image(MNIST\citep{lecun1998gradient}), sound(Speech Command\citep{warden2018speech}), and natural language(IMDB\citep{maas2011learning}). We described more details in sec \ref{appendix:usedmodels}.
The reason for choosing these networks is based on Convolutional Neural Networks specialized extract features.
Since the number of classes is different for each data, the test was divided into two cases.
(Task Layers): The last layer was placed differently for each dataset and mapped to the class dimension of the dataset.
(47 class): The one last layer was used for all datasets, and the number of class dimensions is equal to the sum of class dimensions of all datasets. Because if the last layer is placed differently for each dataset, information on the input type may be informed.
All models were learned during 30 epochs, and the results were as Table.\ref{table:task1}(Task layers, 47 class), and the learning process is (Fig.\ref{fig:accuracy_plot}(a)).
The learning processes of artificial association networks are similar to the process of individually learning feature extraction networks; if the model learns like this, several problems are found.
LeNet-5 is well trained, but M5 needs to be more trained, and CNN has an overfitting problem.
The reason is that each network has different epochs to have optimal performance, and setting learning parameters is more complex than learning individually.
\begin{table}[h]
\begin{minipage}{1.0\linewidth}
\centering
\caption{the result of Experiment 1}
\label{table:task1}
\medskip
{\tiny
\begin{tabular}{l
c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
}
\toprule
& \multicolumn{3}{c}{ \scriptsize{Task layers}}
& \multicolumn{3}{c}{ \scriptsize{47(=10+35+2) class}}
& \multicolumn{3}{c}{ \scriptsize{Transfer learning} }
& \multicolumn{3}{c}{ \scriptsize{Fine tuning} }
\\
\cmidrule(lr){2-4}\cmidrule(lr){5-7}\cmidrule(lr){8-10}\cmidrule(lr){11-13}
Model & MNIST & SC & IMDB & MNIST & SC & IMDB & MNIST & SC & IMDB & MNIST & SC & IMDB\\
\midrule
LeNet-5 & 98.46 & - & -
& 98.46 & - & -
& 98.49 & - & -
& 98.49 & - & - \\
M5 (Group Norm) & - & \textbf{97.63} & -
& - & \textbf{97.63} & -
& - & \textbf{100.0} & -
& - & \textbf{100.0} & - \\
CNN & - & - & \textbf{87.13}
& - & - & \textbf{87.13}
& - & - & \textbf{87.48}
& - & - & \textbf{87.48}\\
LAN & 98.84 & 97.14 & 86.10
& 98.58 & 96.87 & 85.88
& 98.39 & 98.81 & 82.27
& 98.87 & 99.07 & 86.99 \\
LAAN & \textbf{98.96} & 96.91 & 85.38
& 98.77 & 96.83 & 85.76
& \textbf{98.65} & 98.91 & 82.20
& 98.64 & 97.39 & 86.62 \\
RAN & 98.53 & 96.54 & 85.62
& \textbf{98.79} & 97.07 & 86.24
& 98.48 & 99.66 & 81.91
& \textbf{98.94} & 99.17 & 87.18\\
RAANs & 98.79 & 96.66 & 86.28
& 98.78 & 96.68 & 85.79
& 98.47 & 98.64 & 82.18
& 98.92 & 98.89 & 87.27\\
\midrule
FE Epochs & 30 & 30 & 30
& 30 & 30 & 30
& 26 & 95 & 3
& 26 & 95 & 3\\
\bottomrule
\end{tabular}
}
\end{minipage}
\end{table}
\begin{figure}[]
\centering
\subfloat[Simultaneous-learning(47cls)]{{\includegraphics[width=0.33\textwidth]{figure/gtnn_test_47classnet.png}}}
\subfloat[Transfer-learning]{{\includegraphics[width=0.33\textwidth]{figure/gtnn_test_transfer_task.png}}}
\subfloat[Fine-Tuning]{{\includegraphics[width=0.33\textwidth]{figure/gtnn_test_fine_task.png}}}
\caption{ Test Accuracy (The Experiment 1) }
\label{fig:accuracy_plot}
\end{figure}
\paragraph{transfer learning \& fine-tuning}
Therefore, after learning individually, we combined the association model to perform transfer-learning and fine-tuning\citep{pan2009survey}. The result is Table.\ref{table:task1}(Transfer-learning, Fine-tuning).
In transfer learning, the parameters of feature extraction networks are not modified, and in fine-tuning, the parameters are modified.
We used the validation set to learn each feature extraction network and adopt the lowest value of the validation loss to use the model.
As a result, the LeNet-5, M5, CNN were pre-trained with 26, 95, 3 epochs. These models were combined with artificial association networks to perform fine-tuning and transfer learning. Then, the network was re-trained, and learning was stopped at epochs with the lowest value of validation loss using each of the same validation datasets.
As in the results of Table.\ref{table:task1}(Transfer-learning, Fine-tuning) and Fig.\ref{fig:accuracy_plot}(b,c), in the case of transfer learning, performance was poor in the IMDB dataset. We thought it was overfitting because the IMDB dataset was relatively small. We reduced the learning rate from 0.001 to 0.0001, but the results were similar. While fine-tuning was similar to the performance learned individually in all datasets. And the performance improved compared to when fine-tuning was not performed. Consequently, If transfer learning and fine tuning are used by setting parameters well according to the network, the performance of the network can be improved.
\subsection{Exp 2 : Could various data structures be learned? (Multi-domain)}
\label{appendix:exp2}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[height=3.8cm]{figure/sound.png}
\caption{Sound structure}
\label{fig:first}
\end{subfigure}
\hfill
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[height=3.8cm]{figure/image.png}
\caption{Image structure}
\label{fig:second}
\end{subfigure}
\hfill
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[height=3.8cm]{figure/graph.png}
\caption{Graph structure}
\label{fig:third}
\end{subfigure}
\hfill
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[height=3.8cm]{figure/tree.png}
\caption{Tree structure}
\label{fig:forth}
\end{subfigure}
\caption{(a) Speech commands, (b) MNIST, (c) UPFD-GOS , (d) Stanford Sentiment Treebank }
\label{fig:figures}
\end{figure}
\paragraph{Datasets}
As a second experiment, we learned datasets of various structures with artificial association networks.
Data in the form of images, sound data, language data\citep{socher-etal-2013-recursive} in the tree structure, and data in the graph structure\citep{dou2021user} were learned at the same time, and the results are as follows.
All of the performances at this time were similar to those of existing networks.
\begin{figure}[H]
\centering
\subfloat[Feature Extraction Networks]{{\includegraphics[height=3cm ]{figure/exp3_fe.png}}}
\hspace{10mm}
\subfloat[Learning plot (49class-Test acc)]{{\includegraphics[height=5cm ]{figure/49_class_test_acc.png}}}
\caption{Feature Extraction Networks \& Learning plot}
\label{fig:exp3feplot}
\end{figure}
In addition, in the case of recursive association networks and recursive attentional association networks, it can be seen that the learning speed for language data is lower than recursive neural networks.
The reason is that the size of language dataset is relatively less than others.
Therefore, since the language model has a relatively less influence on loss, it seems to have been pushed back in the optimization process.
In the case of level association networks and level attentional association networks, the result of overfitting in the language model is shown.
This is a natural result because the layers of tree data are learned differently for each level.
As a result, we proved that learning to express various information delivery structures with one network cell does not significantly affect performance.
\begin{table}[h]
\begin{minipage}{1.0\linewidth}
\centering
\caption{the result of Experiment 3}
\label{table:task3}
\medskip
{\begin{tabular}{lccccccccc}
\toprule
& \multicolumn{4}{c}{ 49(=10+35+2+2) class}
\\
\cmidrule(lr){2-5}
Model & MNIST & SC & SST & UPFD-GOS \\
\midrule
LeNet-5 & 98.52 & - & - & - \\
M5 (Group Norm) & - & \textbf{98.37} & - & - \\
RNN(Recursive) & - & - & \textbf{77.69} & -\\
GCN & - & - & - & 93.67\\
GATs & - & - & - & 93.93\\
\midrule
LAN & \textbf{98.88} & 97.18 & 73.30 & 94.74 \\
LAAN & 98.80 & 96.67 & 73.39 & \textbf{94.85} \\
RAN & 98.68 & 97.46 & 76.74 & 93.15 \\
RAAN & 98.35 & 97.44 & 76.79 & 94.67 \\
\midrule
class count & 10 & 35 & 2 & 2\\
\midrule
type & image & sound & tree & graph\\
\bottomrule
\end{tabular}
}
\end{minipage}
\end{table}
When we stop learning this network at 30 epochs, the performance in test dataset is in the table\ref{table:task3}.
\subsection{Exp 3 : Can artificial association networks learn "deep" neuro trees? (multi-domain)
}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[height=3.8cm]{figure/sound_deep.png}
\caption{sound structure}
\label{fig:first}
\end{subfigure}
\hfill
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[height=3.8cm]{figure/image.png}
\caption{Image structure}
\label{fig:second}
\end{subfigure}
\hfill
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[height=3.8cm]{figure/graph.png}
\caption{Graph structure}
\label{fig:third}
\end{subfigure}
\hfill
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[height=3.8cm]{figure/tree.png}
\caption{Tree structure}
\label{fig:third}
\end{subfigure}
\caption{(a) Speech commands, (b) MNIST, (c) UPFD-GOS , (d) Stanford Sentiment Treebank }
\label{fig:figures}
\end{figure}
\begin{figure}[h!]
\centering
\subfloat[Feature Extraction Networks]{{\includegraphics[height=3cm ]{figure/feature-extraction-mfcc.png}}}
\hspace{10mm}
\subfloat[Learning plot (49class-Test acc)]{{\includegraphics[height=5cm ]{figure/test-task-49-gru.png}}}
\caption{We only need to look at the sound data, so the dataset of the other domain was drawn with dotted lines and the sound data was drawn with solid lines. the Light green line is $\mathbf{GRU}$ and the pink line is $\mathbf{GAAU}$.
}
\label{fig:gru_exp_plot}
\end{figure}
The third experiment is about whether the networks can be trained when need to train deep neuro tree.
So, we utilized the MFCC algorithm for the speech command dataset. A $\mathbf{GAAU}$ is added; And the models that become baseline are recurrent neural network(RNN) and gated recurrent unit(GRU) models that learn only a single dataset.
this experiment is important because, generally, the basic recurrent neural network model isn't well trained with very long-time serial data.
By slightly changing exp3, feature is extracted through MFCC, the sound-sample-rate is 16000, and the feature size is 40 and time-size is 81.
And we separated this data by time dimension to create a deep neuro tree with only one child on each node and a maximum depth of 81.
At this time, 1 FC layer and MFCC are used together in the feature extraction process so that the FC layer is added so that feature size is from 40 to 128.
In addition, MFCC and one FC layer are used together to change the feature size from 40 to 128 during the feature extraction network process of the sound.
We proceeded with learning at 30 epochs, and the process is Fig.\ref{fig:gru_exp_plot}(b) and result is Table.\ref{table:exp:task3}.
\begin{table}[h!]
\begin{minipage}{1.0\linewidth}
\centering
\caption{the result of Experiment 3}
\label{table:exp:task3}
\medskip
{\begin{tabular}{lccccccccc}
\toprule
& \multicolumn{4}{c}{ 49(=10+35+2+2) class}
\\
\cmidrule(lr){2-5}
Model & MNIST & SC & SST & UPFD-GOS \\
\midrule
LeNet-5 & \textbf{98.81} & - & - & - \\
RNN(Recurrent) & - & 44.03 & - & -\\
GRU & - & \textbf{93.34} & - & -\\
RNN(Recursive) & - & - & 77.42 & -\\
GCN & - & - & - & 93.52\\
GATs & - & - & - & 93.57\\
\midrule
RAN & 98.68 & 4.87 & 75.97 & 92.84 \\
RAAN & 98.35 & 4.12 & \textbf{77.69} & \textbf{94.98} \\
GAAU & 98.35 & 91.50 & 77.05 & 94.59 \\
\midrule
class count & 10 & 35 & 2 & 2\\
\midrule
type & image & sound & tree & graph\\
\bottomrule
\end{tabular}
}
\end{minipage}
\end{table}
The models of $\mathbf{RAN}$, and $\mathbf{RAAN}$ were all well learned in different domain datasets, but when the neuro tree was deep, they could not even catch up with the performance of recurrent neural network.
On the other hand, it can be seen that the performance of the $\mathbf{GAAU}$ model gradually becomes similar to the $\mathbf{GRU}$ model even when the tree is deep.
\subsection{Exp 4 : Could all data information be contained? (Multi-domain and Multi-Modal)}
\paragraph{Datasets} The fourth experiment is about whether the output($\overrightarrow{h}_{root}$) of artificial association networks is embedded information on all data contained in the neuro tree(association).
Therefore, we performed image, sound, and text classification by putting image, sound, text data in one neuro tree, and we also learned the neuro tree used in the first experiment simultaneously.
One problem is that when the three types of data are combined, The number of combinations of the dataset becomes too huge.
For example, there are 50,000 images, 105,000 sounds, and 17,500 natural language samples, and when these data are combined, we need to generate 50,000$\times$105,000$\times$17,500 GT samples.
For this problem, we used a sampling method.
When loading each data, two data of different types are sampled randomly.
Thus, when we load one data, two neuro trees are generated of the one type neuro tree and the three type neuro tree.
The test accuracy is the result of averaging 5-fold.
the neuro tree dataset of the second experiment is illustrated by Fig.\ref{fig:dataset_GT}(a,b,c,d).
\begin{figure}[h]
\centering
\subfloat[Expt 1,2: One type]{{\includegraphics[width=0.25\textwidth, height=3.7cm]{figure/task1_gt.png}}}
\subfloat[Expt 2: All to sound]{{\includegraphics[width=0.235\textwidth, height=3.8cm]{figure/sound_all_gt.png}}}
\subfloat[Expt 2: All to image]{{\includegraphics[width=0.235\textwidth, height=3.7cm]{figure/image_all_gt.png}}}
\subfloat[Expt 2: All to language]{{\includegraphics[width=0.235\textwidth, height=3.8cm ]{figure/language_all_gt.png}}}
\caption{Association tree datasets of the experiments}
\label{fig:dataset_GT}
\end{figure}
\begin{table}[h]
\begin{minipage}{1.0\linewidth}
\centering
\caption{the result of association test}
\label{exp:table2}
\medskip
{\tiny
\begin{tabular}{l
c
@{\hspace*{2.0mm}}c
@{\hspace*{2.0mm}}c
@{\hspace*{2.0mm}}c
@{\hspace*{2.0mm}}c
@{\hspace*{2.0mm}}c
c
@{\hspace*{2.0mm}}c
@{\hspace*{2.0mm}}c
@{\hspace*{2.0mm}}c
@{\hspace*{2.0mm}}c
@{\hspace*{2.0mm}}c
}
\toprule
& \multicolumn{6}{c}{\scriptsize Task 2 + task layer (\%)}
& \multicolumn{6}{c}{\scriptsize task 2 + fine-tuning (\%)} \\
\cmidrule(lr){2-7}\cmidrule(lr){8-13}
\scriptsize Model
& MNIST & MNIST\&All & SC & SC\&All & IMDB & IMDB\&All
& MNIST & MNIST\&All & SC & SC\&All & IMDB & IMDB\&All \\
\midrule
LAN & 98.90 & 98.88 & 96.83 & 95.71 & \textbf{87.20} & 87.18
& 98.63 & 98.62 & 98.83 & 97.66 & 86.76 & 86.61 \\
LAAN & 98.87 & 98.88 & 97.30 & 96.37 & 86.73 & \textbf{87.41}
& 98.93 & \textbf{98.87} & 98.67 & 98.52 & 87.43 & 87.30 \\
RAN & \textbf{99.05} & \textbf{99.04} & 96.46 & 97.03 & 87.16 & 87.21
& \textbf{98.97} & 98.84 & 98.83 & 98.58 & 87.00 & 87.14 \\
RAANs & 98.95 & 98.96 & 97.36 & 95.76 & 86.55 & 87.08
& 98.82 & 98.73 & 98.60 & 98.22 & 87.30 & 87.26 \\
\midrule
FE Baseline & 98.46 & 98.46 & \textbf{97.63} & \textbf{97.63} & 87.13 & 87.13
& 98.49 & 98.49 & \textbf{100.0} & \textbf{100.0} & \textbf{87.48} & \textbf{87.48} \\
\bottomrule
\end{tabular}
}
\end{minipage}
\end{table}
We constructed the neuro tree dataset described above(Fig.\ref{fig:dataset_GT}(a,b,c,d)) to validate if the output vector can contain all the information in the neuro tree, and the results are as follows Table.\ref{exp:table2}.
The meaning of FE Baseline is the performance of networks of LeNet-5, M5, and CNN.
In these experiments, it has been verified that it is possible to learn various types of datasets using one network cell and that information can be embedded together.
Consequently, we can share an association cell or layers to learn without being limited to the dataset type and embed all information inside the neuro tree into a vector.
\section{Used Models}
\label{appendix:usedmodels}
\begin{table}[h]
\begin{minipage}{1.0\linewidth}
\centering
\caption{ Feature Extraction Networks for Image dataset }
\label{exp:image_FE}
\medskip
{\tiny
\begin{tabular}{l
c
@{\hspace*{5.0mm}}c
@{\hspace*{5.0mm}}c
@{\hspace*{5.0mm}}c
@{\hspace*{5.0mm}}c
c
@{\hspace*{5.0mm}}c
@{\hspace*{5.0mm}}c
@{\hspace*{5.0mm}}c
@{\hspace*{5.0mm}}c
}
\toprule
& \multicolumn{5}{c}{ LeNet-5 }
& \multicolumn{5}{c}{ LeNet-5 for AAN } \\
\cmidrule(lr){2-6}\cmidrule(lr){7-11}
\scriptsize Model
& In & Out & kernel & stride & activation
& In & Out & kernel & stride & activation \\
\midrule
Conv2D & 1 & 6 & (5,5) & 1 & tanh
& 1 & 6 & (5,5) & 1 & tanh \\
AvgPool2D & - & - & (2,2) & 1 & -
& - & - & (2,2) & 1 & - \\
Conv2D & 6 & 16 & (5,5) & 1 & tanh
& 6 & 16 & (5,5) & 1 & tanh \\
AvgPool2D & - & - & (2,2) & 1 & -
& - & - & (2,2) & 1 & - \\
Conv2D & 16 & 120 & (5,5) & 1 & tanh
& 16 & 120 & (5,5) & 1 & tanh \\
FC layer 1 & 120 & 84 & - & - & tanh
& - & - & - & - & - \\
FC layer 2 & 84 & 10 & - & - & softmax
& - & - & - & - & - \\
Zero padding & - & - & - & - & -
& 120 & 128 & - & - & - \\
\midrule
Final & - & 10 & - & - & -
& - & 128 & - & - & - \\
\bottomrule
\end{tabular}
}
\end{minipage}
\end{table}
LeNet-5\citep{lecun1989backpropagation} was used as the image feature extraction network.
We create a dimension of 128 by applying zero padding to the extracted features without using the affine-layer(FC layers) and then forward it to artificial association networks.
\begin{table}[h]
\begin{minipage}{1.0\linewidth}
\centering
\caption{ Feature Extraction Networks for Sound dataset }
\label{exp:sound_FE}
\medskip
{\tiny
\begin{tabular}{l
c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
}
\toprule
& \multicolumn{6}{c}{ M5(Group Norm) }
& \multicolumn{6}{c}{ M5(Group Norm) for AAN } \\
\cmidrule(lr){2-7}\cmidrule(lr){8-13}
\scriptsize Model
& In & Out & kernel & stride & norm & activation
& In & Out & kernel & stride & norm & activation \\
\midrule
Conv1D & 1 & 128 & 80 & 4 & group 16 & relu
& 1 & 128 & 80 & 4 & group 16 & relu \\
MaxPool1D & - & - & 4 & 1 & - & -
& - & - & 4 & 1 & - & - \\
Conv1D & 128 & 128 & 3 & 1 & group 16 & relu
& 128 & 128 & 3 & 1 & group 16 & relu \\
MaxPool1D & - & - & 4 & 1 & - & -
& - & - & 4 & 1 & - & - \\
Conv1D & 128 & 256 & 3 & 1 & group 16 & relu
& 128 & 256 & 3 & 1 & group 16 & relu \\
MaxPool1D & - & - & 4 & 1 & - & -
& - & - & 4 & 1 & - & - \\
Conv1D & 256 & 512 & 3 & 1 & group 16 & relu
& 256 & 512 & 3 & 1 & group 16 & relu \\
MaxPool1D & - & - & 4 & 1 & - & -
& - & - & 4 & 1 & - & - \\
AdaptiveAvgPool1d & - & 1 & - & - & - & -
& - & 1 & - & - & - & - \\
FC layer & 512 & 35 & - & - & - & log softmax
& 512 & 128 & - & - & - & leaky relu \\
\midrule
Final & - & 35 & - & - & - & -
& - & 128 & - & - & - & - \\
\bottomrule
\end{tabular}
}
\end{minipage}
\end{table}
M5\citep{dai2017very} was used as the sound feature extraction network.
As described above(Sec.\ref{sec:extraction}), we used group normalization\citep{wu2018group} without using Batch norm\citep{ioffe2015batch}.
We use the affine layer(FC layer) and deliver it to the artificial association networks in 128 dimensions.
It could be implemented in torchaudio\footnote{\url{https://pytorch.org/tutorials/intermediate/speech_command_recognition_with_torchaudio_tutorial.html}}.
\begin{table}[H]
\begin{minipage}{1.0\linewidth}
\centering
\caption{ Feature Extraction Networks for Natural language dataset }
\label{exp:language_FE}
\medskip
{\tiny
\begin{tabular}{l
c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
@{\hspace*{3.0mm}}c
}
\toprule
& \multicolumn{5}{c}{ CNN }
& \multicolumn{5}{c}{ CNN for AAN } \\
\cmidrule(lr){2-6}\cmidrule(lr){7-11}
\scriptsize Model
& In & Out & kernel & stride & activation
& In & Out & kernel & stride & activation \\
\midrule
Conv2D & 1 & 100 & (3,3) & 1 & relu
& 1 & 100 & (3,3) & 1 & relu \\
AvgPool2D & - & - & (2,2) & 1 & -
& - & - & (2,2) & 1 & - \\
Conv2D & 1 & 100 & (4,4) & 1 & relu
& 1 & 100 & (4,4) & 1 & relu \\
AvgPool2D & - & - & (2,2) & 1 & -
& - & - & (2,2) & 1 & - \\
Conv2D & 1 & 100 & (5,5) & 1 & relu
& 1 & 100 & (5,5) & 1 & relu \\
Concat & (300,400,500) & 1200 & - & - & -
& (300,400,500) & 1200 & - & - & - \\
Dropout & - & - & - & - & -
& - & - & - & - & - \\
FC layer & 1200 & 1 & - & - & -
& 1200 & 128 & - & - & - \\
\midrule
Final & - & 1 & - & - & -
& - & 128 & - & - & - \\
\bottomrule
\end{tabular}
}
\end{minipage}
\end{table}
CNN\citep{kim-2014-convolutional} was used as the feature extraction network for natural language processing. We slightly modified the contents in this paper and combined them with artificial association networks.
We did not use drop out because we wanted to emphasize the difference between transfer learning and general learning in the association model.
The glove\citep{pennington2014glove} was used for the pre-trained word embedding network with 100 dimensions.
We used an adam optimizer\citep{kingma2014adam}, a learning rate of 0.001, a cosine annealing(T max=2, eta min=1e-05) for schedulers\citep{loshchilov2016sgdr}.
the batch-size is 32 and the MNIST, Speech Command and IMDB classes are 10, 35, 2.
\section{Discuss : Why is it important to express various routes?}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[height=3.9cm]{figure/path1.png}
\caption{A network that can express only one fixed path and a specific task.}
\label{fig:music_path1}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[height=3.9cm]{figure/path2.png}
\caption{Artificial association networks that can express various paths and tasks}
\label{fig:music_path2}
\end{subfigure}
\caption{Neural networks seem like musical instruments.}
\end{figure}
Do we need a model that can handle all domains? Do we need the structure of the human brain and artificial neural networks to be similar? You may question that.
However, it is important to process various domains and perform various tasks with one neural network model. I would like to explain why I think so and introduce some future works.
The author thinks that the human brain is like a musical instrument. The human brain is playing music with a variety of tasks. But, since most existing neural networks represent only one path, And this makes it limited to performing only specific tasks.
Suppose that there is a neural network that performs only a classification task.
This network is like having only one key on the keyboard.
However, we cannot play music with one key.
On the other hand, because an artificial association network can express various paths, it can express various models and perform more diverse tasks. It means that a wide variety of keys are created.
Therefore, we can play the music that we haven't done before.
And the music means \textbf{a human thinking ability}, various data will constitute a variety of paths, and each path and the task will be a key. And these keys gather to become a keyboard.
So how do we express human thinking ability? There are a lot of contents, so to introduce some of future works, there are deductive networks\cite{kim2021deductive}, memory networks\cite{kim2021memory}, and imagine networks\cite{kim2021imagine}.
\section{Conclusion}
\label{sec:conclusion}
We introduced a data-driven network that can jointly learn relationships and hierarchical information. This study is to be free from architecture and has been developed to connect various types of information being developed. And The author of this paper believes that one day a neural network will emerge that can perform all human tasks.
In addition, this is an association model that behaves like human sensory organs and can be described as similar to a human neural network.
We think this network can be used as a deep neural network part of DQN\citep{mnih2013playing}.
We will try to leverage this network to approach problems that have not been solved before. This paper is part of a series; in the next paper, we introduce Deductive Association Networks(DANs).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,304 |
\section{Theme}
Some numerical schemes sampling space with finite elements for interpolation or integration
may employ mesh refinement techniques which start from
a coarse grid of mesh points and build finer grids with denser point sets
at latter stages. The requirements of (i) efficient use of the values samples
at previous stages, and (ii) maximum independent information contributed by the
refined stages will be embodied in this manuscript by the following recipe
and guideline:
Place points of the new, refined stages at interstitial positions of the previous grid such
that they have maximum Euclidean distance to the points of the previous grid.
Other criteria based on Fourier amplitudes are possible and have been applied to
symmetry-adapted sampling of the reciprocal lattice \cite{CunninghamPRB10,PackPRB16,EntezariIEEEvis04}.
This idea will be worked out by starting from the square or hexagonal
grid in two
dimensions and as warm-up, then the simple cubic lattice in three dimensions.
These grids are the basic ones because coding the lattice points
in Cartesian coordinates is as simple as running with integer
lattice coordinates independently through the base vectors of their lattices.
\section{Two-dimensional Templates} \label{sec.2d}
\subsection{Square Grid}
The simplest coverage of the plane by a point grid is the square grid with
lattice points placed at
\begin{equation}
\mathbf{p} = i \mathbf{e}^{(0)}_1 + j\mathbf{e}^{(0)}_2,
\end{equation}
where the two unit vectors $\mathbf{e}_{1,2}^{(0)}$ point from
the Cartesian coordinates $(0,0)$ at
the origin to $(a,0)$ and $(0,a)$. $a$ is the lattice constant,
and $i$ and $j$ run through all integers of both signs.
There are two evident choices for the unit cell:
\begin{itemize}
\item
The \emph{primitive}
unit cell is a square
and stretches from one lattice point to the two nearest
neighbors at distance $a$ to the right and up, and along
diagonals of the square to the second nearest neighbors at distance $\sqrt{2}a$.
\item
The \emph{Wigner-Seitz} (WS) (or Voronoi) unit cell uses the Brillouin zone construction
of solid state physics. Between each
pair of points in the grid, a line is drawn that cuts mid-way orthogonally through the
finite line that connects the two points, and which divides space
into
two half-spaces. The closest polygon around a point
build from the finite pieces of these lines of separation encompasses
the unit cell. It is the convex polytope defined by intersection
of the half-spaces. For the square grid, this unit cell is also a square of edge
length $a$, but with a lattice point in the square center.
\end{itemize}
These unit cells have area $a^2$.
The obvious choice for an additional set of points towards a denser mesh
is to add one new point at the center of each primitive unit cell. This meets the
requirement of maximum distance formulated above, because these points are
$a/\sqrt{2}$ away from any of the points of the original mesh. So around
these we can draw the largest circles that contain no point of the original
mesh.
These points could also be found by putting them at the
vertices of the WS unit cells with maximum degree (the degree being the
number of WS unit cells that meet there). By construction, these vertices
are as far away from as many as possible points of the original grid. The
reasoning with the Voronoi cells is more satisfactory from a conceptional
point of view: finding a point in the plane with largest free circumcircle
tastes like a minimization problem that needs numerical treatment.
On the other hand,
given a point $\mathbf{o}$ in the plane with surface normal $\mathbf{n}$,
$\mathbf{o}$ has a unique representation
$\mathbf{o}=\alpha\mathbf{n}+\mathbf{r}$ with $\mathbf{r}\perp\mathbf{n}$,
$\alpha=\mathbf{o}\cdot \mathbf{n}$, and with normal form $Ax+By+Cz+D=0$, $D=-\alpha$
of the plane, where $A$, $B$, $C$ are the Cartesian components of $\mathbf{n}$.
Finding the common vertex of an intersection of three planes is therefore
equivalent to solving a $3\times 3$ inhomogeneous linear system of equations
for three unknowns $x,y$ and $z$. (The equivalent statements appear in the 2-dimensional problem.)
With some knowledge of which three lattice points nearby a pivotal lattice point
define a vertex of the Voronoi polytope, the explicit determination of the Cartesian
components of the vertex is therefore easy.
The additional grid points are obviously a copy of the original grid
points translated by half a diagonal. They are located at positions
\begin{equation}
(i+\frac{1}{2})\mathbf{e}^{(0)}_1+(j+\frac{1}{2})\mathbf{e}^{(0)}_2
\end{equation}
with integer $i$ and $j$. A pleasant observation is that the union
of the old and new grid points \emph{again} represents a simple square
grid---with smaller lattice constant $a/\sqrt{2}$ and rotated by $45^\circ$
relative to the original one. This could be noted by defining the unit vectors
\begin{equation}
\mathbf{e}^{(1)}_1\equiv \frac{1}{2}(\mathbf{e}^{(0)}_1+\mathbf{e}^{(0)}_2),\quad
\mathbf{e}^{(1)}_2\equiv \frac{1}{2}(-\mathbf{e}^{(0)}_1+\mathbf{e}^{(0)}_2)
\end{equation}
and accessing the union of these lattice points of two levels by
\begin{equation}
i\mathbf{e}^{(1)}_1+j\mathbf{e}^{(1)}_2
.
\end{equation}
This leads to a simple recursive mesh refinement: iteratively
the lattice constant is divided by $\sqrt 2$, and the
two orthogonal unit vectors are rotated alternatingly by $45^\circ$ to the left
and to the right.
\subsection{Triangular (Hexagonal) Grid}
The fundamental areal element of the triangular grid is the isosceles triangle
with side length $a$, and two unit vectors with Cartesian coordinates
\begin{equation}
\mathbf{e}^{(0)}_1= (a,0),\quad \mathbf{e}^{(0)}_2 = (-a/2,\sqrt{3}a/2)
\end{equation}
with an angle of $120^\circ$ between.
The unit cell has an area of $\sqrt{3}a^2/4$. The primitive unit cell
is a kite shaped quadrangle formed by two of these triangular elements glued by one side.
The WS unit cell is a hexagon centered at a lattice point.
The refinement of this triangular lattice is again obvious. By any of the two
methods proposed above (maximum free distance or vertices of the WS unit cell)
the additional points are placed in the middle (mid-point of the circumcircle)
of each the two triangles of the primitive unit cell, called the $K$ point
in solid state plane groups \cite{TerzibaschPSS133}.
The first has Cartesian
coordinates $(a/2,a/(2\sqrt{3}))$, the second $(0,a/\sqrt{3})$.
The union of the grid points
of the original lattice and the additional points at the interstitial locations
establishes another triangular lattice, rotated by $30^\circ$
from the original lattice. The lattice constant is shrunk
to $a/\sqrt 3$, the nearest neighbor distance in the refined grid.
So the area of the unit cell of the refined grid has shrunk by a factor $1/3$
compared to the area of the original cell. The factor of three is essentially
indicating that the density of the mesh points has triplicated because
we added \emph{two} grid points into each unit cell of the original mesh,
which contained \emph{one} grid point.
Similar to the finding with the square grid, this refinement preserves the grid structure, and recursive
refinement is therefore a rather easy task from a programmer's point of view.
At each step, the previous unit vectors are shrunk by a factor $1/\sqrt 3$
and rotated by $30^\circ$.
\section{Three-dimensional Grid}
\subsection{Level 0: Simple Cubic}
The starting point for an unbiased sampling
in three dimensions is the Simple Cubic (SC) lattice with lattice constant $a$,
unit vectors $\mathbf{e}^{(0)}_1=(a,0,0)$
$\mathbf{e}^{(0)}_2=(0,a,0)$
$\mathbf{e}^{(0)}_3=(0,0,a)$.
The volume of the unit cell is
\begin{equation}
V^{(0)}=a^3.
\label{eq.scv}
\end{equation}
Points are located at positions
\begin{equation}
i \mathbf{e}^{(0)}_1
+ j \mathbf{e}^{(0)}_2
+ k \mathbf{e}^{(0)}_3
\label{eq.scp}
\end{equation}
with integer coordinates $i$, $j$ and $k$.
Counting neighbors in shells of common distance to the grid points is
a useful digital signature of the grid structure. For the SC
lattice this statistics starts as in Table \ref{tab.nl0} for the
smallest distances. Top to bottom,
the 6 nearest neighbors are in the directions of the unit
vectors $\mathbf{e}^{(0)}_i$ and their opposites,
the 12 second nearest neighbors are found at positions along the face diagonals, and
the 8 third nearest neighbors are
in the directions of the space diagonals \cite[A005875]{EIS}.
\begin{table}
\caption{Frequencies of distances to neighbors in the SC lattice.}
\begin{tabular}{r|r}
\hline
count & squared distance\\
\hline
6 & $a^2$ \\
12 & $2a^2$ \\
8 & $3a^2$ \\
6 & $4a^2$ \\
24 & $5a^2$ \\
24 & $6a^2$ \\
\hline
\end{tabular}
\label{tab.nl0}
\end{table}
\subsection{Level 1: Body-centered Cubic} \label{sec.bcc}
The obvious first refinement places interstitial points half way
along the space diagonal for another copy with points at
\begin{equation}
(i+\frac{1}{2})
\mathbf{e}^{(0)}_1
+
(j+\frac{1}{2})
\mathbf{e}^{(0)}_2
+
(k+\frac{1}{2})
\mathbf{e}^{(0)}_3
\label{eq.bccp}
\end{equation}
with integer triples $i$, $j$ and $k$.
The major difference in comparison with the two-dimensional examples
is that the union of these grid points does not define another
simple cubic lattice but a body-centered cubic (BCC) lattice.
The WS unit cell of this lattice is characterized by six
squares in the directions of the six next nearest neighbors
of the original unit cell plus eight hexagons in the directions
of the eight new lattice points in the centers of the eight
primitive cells with common vertex at $(0,0)$.
In Brillouin zones of the corresponding space group,
the mid points of the squares are labeled $X$
and the mid points of the hexagons are labeled $L$, and
the vertices of the squares and hexagons are labeled $W$
\cite{Bouckaert,HerringPR52,ElliottPR96}.
The statistics of neighbors around the lattice points
(at both levels) is indicated in Table \ref{tab.nl1}.
\begin{table}
\caption{Statistics of distances for points in the BCC lattice. \cite[A004013]{EIS}}
\begin{tabular}{r|r}
\hline
count & squared distance \\
\hline
8 & $\frac34 a^2$\\
6 & $a^2$ \\
12 & $2a^2$ \\
24 & $\frac{11}{4}a^2$ \\
8 & $3a^2$ \\
6 & $4a^2$ \\
24 & $\frac{19}{4}a^2$ \\
24 & $5a^2$ \\
24 & $6a^2$\\
\hline
\end{tabular}
\label{tab.nl1}
\end{table}
The frequencies of neighbors around any of the new
points at the positions (\ref{eq.bccp})
is the same as for the points of the
zeroth level of refinement (\ref{eq.scp}).
\begin{figure}
\includegraphics[width=0.99\columnwidth]{WSbcc}
\caption{Points on a BCC lattice (dotted squares). Three of them are surrounded
by the truncated octahedra of the Wigner-Seitz cell.}
\label{fig.WSbcc}
\end{figure}
\begin{figure}
\includegraphics[width=0.99\columnwidth]{bcc}
\caption{WS cells of the BCC lattice as in Figure \ref{fig.WSbcc}, but
with back surfaces hidden.}
\label{fig.bcc}
\end{figure}
The weights associated with numerical integration are the volumes
of the Voronoi cells around each lattice point.
The two Voronoi cells
around the points at level 0---enumerated by (\ref{eq.scp})---
and around the points at level 1---enumerated by (\ref{eq.bccp})---have
the same shape, illustrated in Figure \ref{fig.WSbcc} and \ref{fig.bcc}.
The polyhedron of the Wigner-Seitz cell
of that BCC lattice is a Truncated Octahedron with six quadratic and eight regular hexagonal faces.
Figure \ref{fig.WSbcc} shows three of these unit cells. The two cells at the back
share a common square face; the one at the front is attached to the one right at the back
by a hexagon.
The unit box is the primitive unit cell of the SC lattice.
The volume of each Truncated Octahedron is half of the primitive unit cell,
\begin{equation}
V_\Gamma^{(1)}=a^3/2.
\label{eq.bccv}
\end{equation}
The upper index indicates that the volume is defined at level 1, after
one refinement, and the lower label is the standard symbol for
the type of symmetry of the center point of the cell.
\subsection{Level 2: body-centered with W or X}
\subsubsection{Level 2: body-centered with W} \label{sec.bcW}
The next refinement step adds points at $W$ positions
(the name of the points at the 24 vertices of the truncated octahedra
in the associated Brillouin zone of the
face-centered cubic lattice).
The coordinate triples of $W$ on the left, bottom and front faces of the
primitive unit cell of the SC are
\begin{multline}
i\mathbf{e}_1^{(0)}
+(j+\frac{1}{4}) \mathbf{e}_2^{(0)}
+(k+\frac{1}{2}) \mathbf{e}_3^{(0)},\\
i\mathbf{e}_1^{(0)}
+(j+\frac{3}{4}) \mathbf{e}_2^{(0)}
+(k+\frac{1}{2}) \mathbf{e}_3^{(0)},\\
i\mathbf{e}_1^{(0)}
+(j+\frac{1}{2}) \mathbf{e}_2^{(0)}
+(k+\frac{1}{4}) \mathbf{e}_3^{(0)},\\
i\mathbf{e}_1^{(0)}
+(j+\frac{1}{2}) \mathbf{e}_2^{(0)}
+(k+\frac{3}{4}) \mathbf{e}_3^{(0)},
\\
(i+\frac{1}{4})\mathbf{e}_1^{(0)}
+(j+\frac{1}{2}) \mathbf{e}_2^{(0)}
+k \mathbf{e}_3^{(0)},\\
(i+\frac{3}{4})\mathbf{e}_1^{(0)}
+(j+\frac{1}{2}) \mathbf{e}_2^{(0)}
+k \mathbf{e}_3^{(0)},\\
(i+\frac{1}{2})\mathbf{e}_1^{(0)}
+(j+\frac{1}{4}) \mathbf{e}_2^{(0)}
+k \mathbf{e}_3^{(0)},\\
(i+\frac{1}{2})\mathbf{e}_1^{(0)}
+(j+\frac{3}{4}) \mathbf{e}_2^{(0)}
+k \mathbf{e}_3^{(0)},
\\
(i+\frac{1}{4})\mathbf{e}_1^{(0)}
+j\mathbf{e}_2^{(0)}
+(k+\frac{1}{2}) \mathbf{e}_3^{(0)},\\
(i+\frac{3}{4})\mathbf{e}_1^{(0)}
+j\mathbf{e}_2^{(0)}
+(k+\frac{1}{2}) \mathbf{e}_3^{(0)},\\
(i+\frac{1}{2})\mathbf{e}_1^{(0)}
+j\mathbf{e}_2^{(0)}
+(k+\frac{1}{4}) \mathbf{e}_3^{(0)},\\
(i+\frac{1}{2})\mathbf{e}_1^{(0)}
+j\mathbf{e}_2^{(0)}
+(k+\frac{3}{4}) \mathbf{e}_3^{(0)},
\label{eq.bccW}
\end{multline}
with integer $i$, $j$ and $k$.
A $W$ point is also a point of maximum free range in the BCC lattice. The common distance
between $(0,a/4,a/2)$ and its four nearest neighbors at $(0,0,0)$, $(-a/2,a/2,a/2)$,
$(a/2,a/2,a/2)$ and $(0,0,a)$ is $\sqrt{5}a/4\approx 0.55902a$.
Since each of the 24 $W$ points is shared by two primitive unit cells of the
SC lattice, adding the $W$ points adds 12 points to the primitive unit cell,
for a total of 14 once the 2 points already present at the previous levels
of refinement are included.
An alternative way of counting what is inside the WS cell of the BCC lattice looks as follows:
each of the 24 $W$ points is shared by 4 Wigner-Seitz cells that meet at each $W$.
Adding $W$ points therefore adds $24/4=6$ points to the Wigner-Seitz cell of the
BCC lattice for a total of 7.
After these points at $W$ positions have been added to BCC lattice,
the statistics of shells of neighbors around the $\Gamma$-points
is gathered in Table \ref{tab.nl2Gamma}.
The statistics of shells of neighbors around the $W$-points
is obviously different: Table \ref{tab.nl2W}.
\begin{table}
\caption{Statistics of distances for points with $\Gamma$-symmetry after the $W$ points
have been added to the BCC lattice.
}
\begin{tabular}{r|r}
\hline
count & squared distance \\
\hline
24 & $\frac{5}{16} a^2$ \\
8 & $\frac34 a^2$ \\
24 & $\frac{13}{16}a^2$ \\
6 & $a^2$\\
48 & $\frac{21}{16}a^2$\\
72 & $\frac{29}{16}a^2$ \\
\hline
\end{tabular}
\label{tab.nl2Gamma}
\end{table}
\begin{table}
\caption{Statistics of distances for points with $W$-symmetry after the $W$ points
have been added to the BCC lattice.
}
\begin{tabular}{r|r}
\hline
count & squared distance \\
\hline
4 & $\frac18 a^2$ \\
2 & $\frac14 a^2$ \\
4 & $\frac{5}{16}a^2$ \\
8 & $\frac38 a^2$ \\
4 & $\frac12 a^2$ \\
8 & $\frac58 a^2$ \\
8 & $\frac34 a^2$ \\
\hline
\end{tabular}
\label{tab.nl2W}
\end{table}
The weights of the two kinds of points in numerical integration
are the volumes of their Voronoi cells,
computed in Appendix \ref{app.volW}:
\begin{equation}
V_\Gamma^{(2)} = \frac{125}{1152}a^3;\quad
V_W^{(2)} = \frac{451}{6912}a^3 ;\quad
2V_\Gamma^{(2)} + 12 V_W^{(2)} = V^{(0)}.
\label{eq.bccWv}
\end{equation}
The generic algorithm to determine the volumes of polyhedra is to
acquire the Cartesian coordinates of all vertices, to define the
face set as a set of co-planar triangles and to sum the contribution
of each triangle (with outwards orientation of the face normal) to the
volume with the divergence theorem. The contribution of each triangle
is a sixth of the scalar triple product of the three vectors
from the origin of coordinates to the triangle's
vertices.
\subsubsection{Level 2: body-centered plus $X$}
That growth
proposed in Section \ref{sec.bcW}
for the number of points by a factor 7 compared to the BCC level
may be too drastic for some applications. So in engineering practise one could
as well add the $X$ points at mid-points of the square faces of the BCC lattice, although their
free range to their 2 nearest neighbors (the cube centers) is only $a/2$, smaller than
the free range of the $W$. The $X$ are shared
between two WS cells of the BCC lattice, so the total number of points
in the WS unit cell of the BCC lattice raises by 3 to a total of 4.
With the simplified setup proposed above, the Level 2 grid points contain
the SC points of (\ref{eq.scp}), the body-centered points of
(\ref{eq.bccp}) and---after a glance at Figure \ref{fig.WSbcc}---also the face-centered
and
edge-centered grid points.
They have distance $a/2$ to their next nearest neighbors.
In the language of Brillouin zones
one could call these points $M$ points
of the (level 0) Wigner-Seitz cell.
At that level, the full set of refined grid points is a SC lattice
with lattice constant $a/2$, volume $a^3/8$ in the unit cell,
and the same directions of its unit vectors
as the host grid of level zero. In the SC unit cell of level 0 we started
with one point per unit cell, added one point per unit cell at level 1,
added three $X$ and three $M$ points per unit cell at
level 2, such that there are now 8 lattice points per level 0 unit cell.
The benefit of that choice of adding points is equivalent to
the one illustrated two refinements of the plane in Section \ref{sec.2d}:
Adding more points at finer levels onwards is a procedure
continuing recursively with the procedure in Section \ref{sec.bcc},
because the levels zero and three contain congruential sets of lattice points.
The analysis is in that sense complete if points at $X$ had been
added at level 2.
\subsection{Level 3: body-centered with $W$ and $\Lambda$} \label{sec.WL}
If the points at $W$ have been added in level 2, the points with maximum
minimum distance to be added in the next refinement of level 3 are
located on the $\Lambda$-line of the space diagonal at $(\frac{5}{24}a,\frac{5}{24}a,\frac{5}{24}a)$. [This is close to but not
precisely at the $L$ which is at $(\frac14,\frac14,\frac14)$.]
There are three replicas of this point with the same point group
symmetry along the space
diagonal at $(\lambda a,\lambda a,\lambda a)$ with
$\lambda=1/2-5/24=7/24$, $1/2+5/24=17/24$ and $1-5/24=19/24$.
Each point of these quartets adds equivalent positions on the other
three space diagonals; so there are $4\times 4=16$ new lattice
points added here in level 3 for a combined
total of 30 in the primitive SC unit cell.
\begin{figure}
\includegraphics[width=0.8\columnwidth]{tOct}
\caption{
Geometry of the Tetrakis Hexahedra encapsulating the $\Gamma$ points of the level 2
lattice.
The edge length $e-e'$ of the cube is $5a/12$.
}
\label{fig.tOct}
\end{figure}
\begin{figure}
\includegraphics[width=0.6\columnwidth]{WSbccWaroundW}
\caption{Voronoi Octahedron encapsulating the $W$ points
of the level 2 lattice. The set of faces contains 4 regular triangles
and 4 hexahedra. Each face normal of a triangle center is a 3-fold rotation axis.
}
\label{fig.WSbccWaroundW}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\columnwidth]{WSbccW}
\caption{
Level 2 dissection of the unit cell with lattice points at $\Gamma$
(the BCC lattice points) and at $W$.
The Voronoi cells around the $\Gamma$ points have the shape of Figure \ref{fig.tOct};
five of them are shown. The cells around the $W$ points have the shape of Figure \ref{fig.WSbccWaroundW};
only one of these is shown.
Each pair of cells around $\Gamma$ at a mutual distance $a$ are connected by 4
cells around the four intermediate $W$, which share the 4-fold rotation
axis that runs through $X$.
}
\label{fig.WSbccW}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\columnwidth]{WSbccWalt2}
\caption{
The Voronoi cells of Figure \ref{fig.WSbccW} from a
perspective
rotated by 90 degrees.
}
\label{fig.WSbccWalt2}
\end{figure}
The distance of this point $(5a/24,5a/24,5a/24)$ to its 7 nearest
neighbors
is $5\sqrt{3}/24a\approx 0.3608439a$.
The set of 7 nearest neighbors contains $(0,0,0)$ and 6 $W$-points of
the list (\ref{eq.bccW}),
$(0,a/4,a/2)$,
$(0,a/2,a/4)$,
$(a/4,a/2,0)$,
$(a/2,a/4,0)$,
$(a/4,0,a/2)$,
$(a/2,0,a/4)$.
The geometric interpretation of these coordinates looks as follows:
After the $W$-points have been added to the lattice,
the Voronoi cell around the $\Gamma$-points are 24-hedrons
(Tetrakis Hexahedra)
with flat square pyramids glued to each side of a cube
around the $\Gamma$-point (Figure \ref{fig.tOct}).
The cube inside has an edge length of $5a/12$, so its eight vertices
have coordinates of $\pm 5a/24$.
The Voronoi cells around the $W$-points at level 2
are 8-hedrons (Figure \ref{fig.WSbccWaroundW})
with
4 triangles that touch the triangles
of the truncated octahedra and 4 irregular hexagons that touch 8-hedrons
of adjacent $W$-points.
The short sides of the hexagons are the residual left from the distance
of $\sqrt{3} a/2$ between two truncated octahedra along the space diagonal
after removal of the diagonals inside the two cubes of the truncated
octahedra, $\sqrt{3} a/2-2\times \sqrt{3} 5a/24 = \sqrt{3}a/12\approx 0.1443376 a$.
Two such edges are the two rightmost edges of the 8-hedron
in Figure \ref{fig.WSbccWalt2} that each join two of the Tetrakis Hexahedra.
The points added here at level 3
are those vertices around the $W$ Voronoi cell
where a triangle meets a \emph{short} edge of
the hexagon. At these points the Tetrakis Hexahedron of the
Voronoi cell around the $\Gamma$-points meets 6 Voronoi cells
of $W$ points. They lie on the axes with 3m symmetry (along the
space diagonal) of the SC lattice.
The factor $5/24$ that fixes the position of
the new $\Lambda$ points is then found by noticing that these
short edges run along a space diagonal, that the projection
of the new point on the $xy$ plane runs along the plane diagonal,
and to solve the planar Voronoi cell problem for
the triple of points $(0,0)$, $(a/2,a/4)$ and $(a/4,a/2)$
in that plane.
The Voronoi cell around a $\Gamma$ point at level 3
is an Octahedron with
space diagonal $5a/8$ (all bounding box coordinates are $\pm 5a/16$),
so the edge length at the square base of the pyramid is $5a/(8\sqrt 2)$,
and the volume is \cite{ConwayITIT28}
\begin{equation}
V_\Gamma^{(3)}=\frac{125}{3072}a^3\approx 0.04069010 a^3.
\label{eq.gamma3v}
\end{equation}
The degeneracies of the various points in the SC lattice require
\begin{equation}
2V_\Gamma^{(3)}+16V_\Lambda^{(3)}+12V_W^{(3)}=a^3.
\end{equation}
Taking the volume $V_\Lambda^{(3)}$ from Eq. \ (\ref{eq.Vlam3})
and solving for $V_W^{(3)}$ we obtain
\begin{equation}
V_W^{(3)} = \frac{24505}{663552}a^3\approx 0.03693003a^3
\label{eq.W3v}
\end{equation}
for the volumes associated with the $W$ points at level 3.
Finally, Figures \ref{fig.gammWLlam1}--\ref{fig.gammWLall}
try to give an impression of how the three
different types of Voronoi cells at level 3 subdivide the
region near two points of the BCC WS cell.
\begin{figure}
\includegraphics[width=0.95\columnwidth]{gammWLlam1}
\caption{Level 3 Voronoi cells around $\Gamma$ (Octahedron) and around
a neighboring $\Lambda$. The $\Lambda$ cell has 11 faces: 1 regular hexagonal
side where it points along the space diagonal (shared with another $\Lambda$ cell),
1 regular triangular side
where it attaches to the Octahedron, and 6 quadrangles (shared with $W$ cells)
and 3 triangles (shared with $\Lambda$ cells)
at the interface. The symmetry is the 3-fold rotation axis along the space
diagonal.
}
\label{fig.gammWLlam1}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\columnwidth]{gammWLcross}
\caption{Illustration of the bridging between the Octahedra of the
level 3 lattice along each space diagonal of the SC lattice by pairs
of Voronoi cells around the $\Lambda$ points. This follows by extension
of Figure \ref{fig.gammWLlam1} with mirror symmetry at the hexagonal face
of the $\Lambda$ cell.
}
\label{fig.gammWLcross}
\end{figure}
\begin{figure}
\includegraphics[width=0.4\columnwidth]{gammWLWup}
\includegraphics[width=0.4\columnwidth]{gammWLWtilt}
\caption{Level 3 Voronoi cell around any of the $W$ points after $W$ and $\Lambda$
points have been added to the BCC lattice.
It has 12 faces:
two pairs of pentagons, each pair with a common long edge (right dihedral
angle), plus a band
of eight quadrangles
that runs in between.
}
\label{fig.gammWLWup}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\columnwidth]{gammWLW3}
\caption{Voronoi cells at a two $\Gamma$ points plus three $W$ cells
of the shape of Figure \ref{fig.gammWLWup} around
$W$ points after $W$ and $\Lambda$
points have been added to the BCC lattice.
Each of the long edges of the pentagons of Figure \ref{fig.gammWLWup} touches
one point of an Octahedron.
}
\label{fig.gammWLW3}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\columnwidth]{gammWLlam1Wup}
\caption{The configuration if a single cell of Figure \ref{fig.gammWLWup}
is added to Figure \ref{fig.gammWLlam1}.
}
\label{fig.gammWLlam1Wup}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\columnwidth]{gammWLall2}
\caption{
A composite of Figure \ref{fig.gammWLcross}, Figure \ref{fig.gammWLlam1Wup}
and two further cells
(around a $\Lambda$ point on a different space diagonal and around
a second $W$ point).
}
\label{fig.gammWLall}
\end{figure}
\section{Summary}
Regular dissection of space with Voronoi cells on the points of
the simple cubic grid assigns the volume (\ref{eq.scv}) to each cell.
If points are added at the body centers (\ref{eq.bccp}),
the volume (\ref{eq.bccv}) is assigned to the Voronoi cells around the grid points.
If another set of points is added at all $W$ positions (\ref{eq.bccW}),
Voronoi cells of two different shapes appear, characterized by
volumes defined in Eq.\ (\ref{eq.bccWv}).
If finally that set is augmented by points on the $\Lambda$ line,
space is divided by three different types of Voronoi cells with volumes
of (\ref{eq.Vlam3}), (\ref{eq.gamma3v}) and (\ref{eq.W3v}), respectively.
\begin{acknowledgments}
Many images of Voronoi cells shown here have been created by loading
stereolithography data into \texttt{MeshLab} \cite{meshlab}.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,127 |
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