text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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Świątniki Górne () is a town in southern Poland, situated in the Lesser Poland Voivodeship (since 1999), previously in Kraków Voivodeship (1975–1998).
External links
Jewish Community in Świątniki Górne on Virtual Shtetl
Cities and towns in Lesser Poland Voivodeship
Kraków County
Kingdom of Galicia and Lodomeria
Kraków Voivodeship (1919–1939) | {
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Q: Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$. I can show that for $x > 0$ and $r_{i} > 0$ we have
$$
\left(\, x + r_{1}\,\right)\ldots\left(\, x + r_{n}\,\right)\
\geq\
\left[\, x + \left(\, r_{1}\ldots r_{n}\,\right)^{1/n}\,\right]^{n}.$$
However, I can't do this using straight up induction, strong or weak. Can someone do this?
You can show the inequality holds for $n$ if it holds for $n+1$ using $r_{n+1} = (r_1 \cdots r_n)^{1/n}$, so going down isn't a problem, but going up has eluded me. Additionally, you can show that if it holds for $n=a$ and $n=b$ then it holds for $n=ab$. Using powers of two and the above is sort of a way to prove the above by induction but it certainly isn't "normal".
You can also prove the above using Lagrange multipliers (which isn't surprising). The last proof I know is where you compare coefficients of $x^k$ and use the AM-GM inequality.
A: Here's a proof by induction: It holds trivially for $n=1$. For $n\ge2$ we have
\begin{align*}(x+r_1)&\ldots(x+r_{n-1})(x+r_n)\ge(x+(r_1\ldots r_{n-1})^{1/(n-1)})^{n-1}(x+r_n)\\&=\Bigl(\bigl((x^{(n-1)/n})^{n/(n-1)}+((r_1\ldots r_{n-1})^{1/n})^{n/(n-1)}\bigr)^{(n-1)/n}\bigr((x^{1/n})^n+(r_n^{1/n})^n\bigr)^{1/n}\Bigr)^n\\&\ge\bigl(x+(r_1\ldots r_n)^{1/n}\bigr)^n.\end{align*}
The last step is Hölder's inequality for $p=\frac{n-1}n$ and $q=\frac1n$.
A: It's Holder inequality: $\left(x+(r_1r_2\cdots r_n)^{1/n}\right)^n = \left(x^{1/n}x^{1/n}\cdots x^{1/n}+r_1^{1/n}r_2^{1/n}\cdots r_n^{1/n}\right)^n \leq (x+r_1)(x+r_2)\cdots (x+r_n)$, and this is a special case of the following inequality:
$(a_1^n+b_1^n)(a_2^n+b_2^n)\cdots (a_n^n+b_n^n) \geq (a_1a_2\cdots a_n+b_1b_2\cdots b_n)^n$,
and when $n=2$ we get back our Cauchy-Buniakovski-Schwarz inequality.
A: Expanding both sides as polynomials in $x$ and comparing the coefficients, the inequality just follows from Muirhead's inequality.
A: Perhaps the simplest one is to use AM-GM only twice: $$n = \sum_{k=1}^n\dfrac{x}{x+r_k}+\sum_{k=1}^n\dfrac{r_k}{x+r_k}\geq n\dfrac{x}{(\prod_{k=1}^n(x+r_k))^{\frac{1}{n}}}+n\dfrac{(r_1r_2...r_n)^{\frac{1}{n}}}{(\prod_{k=1}^n(x+r_k))^{\frac{1}{n}}},$$
which then will yield the original inequality immmediately.
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{"url":"https:\/\/maker.pro\/forums\/threads\/british-line-and-netural-conventions.95406\/","text":"# British Line and Netural Conventions?\n\nI\n\n#### Ira Rubinson\n\nJan 1, 1970\n0\nI live in the US and am re-wiring a 30 year old Logitek lapping machine\nwhich was built in England. If I plug it in to a receptacle in the US, as it\nis presently wired, then the limit switches carry 0V and the neutral is\nfused. I don't think this is correct because a short to chassis in the limit\nswitch circuit will defeat the limit switch function. Are English wall\nreceptacles reversed with respect to US receptacles?\n\nThanks -Ira\n\nG\n\n#### Graham Holloway\n\nJan 1, 1970\n0\nIra Rubinson said:\nI live in the US and am re-wiring a 30 year old Logitek lapping machine\nwhich was built in England. If I plug it in to a receptacle in the US, as\nit\nis presently wired, then the limit switches carry 0V and the neutral is\nfused. I don't think this is correct because a short to chassis in the\nlimit\nswitch circuit will defeat the limit switch function. Are English wall\nreceptacles reversed with respect to US receptacles?\n\nThanks -Ira\nEnglish wall receptacles are completely different from US ones. Someone must\nhave fitted a US plug on the machine (wrongly) before you got it. Call in an\nqualified electrician. Check the voltage settings as well and also see if\nthe change in frequency from 50Hz to 60Hz could give you a problem (motors,\netc.)\n\nGraham H\n\nE\n\n#### Eeyore\n\nJan 1, 1970\n0\nIra said:\nI live in the US and am re-wiring a 30 year old Logitek lapping machine\nwhich was built in England. If I plug it in to a receptacle in the US, as it\nis presently wired, then the limit switches carry 0V and the neutral is\nfused. I don't think this is correct because a short to chassis in the limit\nswitch circuit will defeat the limit switch function. Are English wall\nreceptacles reversed with respect to US receptacles?\n\nIn that era, live was red (for warning) and neutral was black. If you've wired\nblack (UK) to black (US) there's your problem.\n\nGraham\n\nJ\n\n#### John Larkin\n\nJan 1, 1970\n0\nIn that era, live was red (for warning) and neutral was black. If you've wired\nblack (UK) to black (US) there's your problem.\n\nGraham\n\nThe US convention is \"white is life, black is death.\"\n\nThe other phases are red and blue. Quite patriotic wiring!\n\nI got bit by a loose black wire yesterday, working on our old\nhydraulic elevator. 120 nips a bit, but I bet 240 is a lot worse.\n\nWhat colors do you use now?\n\nJohn\n\nE\n\n#### Eeyore\n\nJan 1, 1970\n0\nJohn said:\nThe US convention is \"white is life, black is death.\"\n\nThe other phases are red and blue. Quite patriotic wiring!\n\nI got bit by a loose black wire yesterday, working on our old\nhydraulic elevator. 120 nips a bit, but I bet 240 is a lot worse.\n\nWhat colors do you use now?\n\nThe dull Euro-norm brown = live and blue = neutral. I think it came about by having\nto be different to anything anyone used before !\n\nThe phase colours used to be red, blue and yellow. They've changed now to brown\nsomething and something.\n\nGraham\n\nG\n\n#### Gary Tait\n\nJan 1, 1970\n0\nThe US convention is \"white is life, black is death.\"\n\nFunny though, in automotive, black is usually ground, and I've seen people\nused to one wireing sceme crew up the other, usually someome used to\nautomotive electrics mucking up AC wiring.\n\nE\n\n#### Eeyore\n\nJan 1, 1970\n0\nGary said:\nJohn Larkin wrote\n\nFunny though, in automotive, black is usually ground, and I've seen people\nused to one wireing sceme crew up the other, usually someome used to\nautomotive electrics mucking up AC wiring.\n\nJust look at the colour coding on every multimeter's test leads.\n\nThe old UK wiring colours were the best.\n\nGraham\n\nJ\n\n#### Jeff L\n\nJan 1, 1970\n0\nJohn Larkin said:\nThe US convention is \"white is life, black is death.\"\n\nThe other phases are red and blue. Quite patriotic wiring!\n\nSame colors here in Canada. Your low voltage 3 phase option is strange in\nthe USA (the 120V, 220V offset 3 phase thing). Here we have the more\nstandard 120V 208 3 Phase. We have a fairly odd 600V 3 phase that is often\ncoming into most commercial \/ industrial buildings with more then a 72 kVA\nservice. 480v seems more common around the world, and is used in the USA.\n\nEurope uses a few odd voltages for 3 phase, like 380V.\n\nOne thing to note: white wires can be hot - eg the drop to a switch from a\nlight fixture.\nI got bit by a loose black wire yesterday, working on our old\nhydraulic elevator. 120 nips a bit, but I bet 240 is a lot worse.\n\nIf your dry, and not grounded overly well, you may not even feel 120V AC.\nWet is another story. I tried to remove the fresh spattered wet drywall\ncompound from around an outlet once when I was a kid - bad mistake it was\ntouching the hot terminal, and apparently all the acidicness of the compound\nmakes a good electrolyte.\nWhat colors do you use now?\n\nMost of Europe seems to have the brown blue thing, with a green ground with\na yellow stripe.Very strange industrial plugs, and weird wire. I like the\ncovers that pop over the receptacle used for the huge 3 prong plugs - almost\nimpossible to electrocute yourself by sticking something in the plugs or\ntouching the prongs as you insert them. The plugs themselves are a little\ntoo huge for my tastes, but old people must like them.\n\nJapan seemed to have blue, brown and IIRC black for 3 phase, at lest that\nwas how some machines we have were wired. They have 100V over there instead\nof 120V, and one half of the country is 60Hz, the other is 50Hz. I think one\nof their 3 phase voltages is 200V.\n\nJ\n\n#### John Larkin\n\nJan 1, 1970\n0\nFunny though, in automotive, black is usually ground, and I've seen people\nused to one wireing sceme crew up the other, usually someome used to\nautomotive electrics mucking up AC wiring.\n\nBlack ground is the electronic convention. Green is filament power,\nred is B+. I forget the rest.\n\nJohn\n\nJ\n\n#### John Larkin\n\nJan 1, 1970\n0\nSame colors here in Canada. Your low voltage 3 phase option is strange in\nthe USA (the 120V, 220V offset 3 phase thing). Here we have the more\nstandard 120V 208 3 Phase. We have a fairly odd 600V 3 phase that is often\ncoming into most commercial \/ industrial buildings with more then a 72 kVA\nservice. 480v seems more common around the world, and is used in the USA.\n\nHell, we have everything here. 120\/208, split 240 stinger, all sorts\nof stuff. 480 l-l for the bigger stuff. And electricians never leave\nschematics behind.\n\nJohn\n\nP\n\n#### Phil Hobbs\n\nJan 1, 1970\n0\nJohn said:\nBlack ground is the electronic convention. Green is filament power,\nred is B+. I forget the rest.\n\nJohn\nYellow is 5V filament.\n\nCheers,\n\nPhil Hobbs\n\nJ\n\n#### Jeff L\n\nJan 1, 1970\n0\nGary Tait said:\nFunny though, in automotive, black is usually ground,\n\nOr the darker wire. I've seen green used on street bikes, along with black.\nBlack on many motorcycles (including off road - eg dirt and motocross) is\nthe ign kill wire. Older snowmobiles used black for the ign kill wire also.\n\nP\n\n#### Peter Bennett\n\nJan 1, 1970\n0\nFunny though, in automotive, black is usually ground, and I've seen people\nused to one wireing sceme crew up the other, usually someome used to\nautomotive electrics mucking up AC wiring.\n\nOn a used boat I just bought, apparently some DC wiring was done by an\nAC electrician - he used Black\/White pair wire, with white as ground\nand black as +12 (but I think there's also some with black as ground -\nlots of odd electrical things: there were two 12 volt batteries\napparently in parallel as the house bank - when I replaced them, I\nfound that the ground lead for one battery was not grounded! Must\nhave been that way since 2001, when the batteries were last replaced -\nor perhaps much longer!)\n\n--\nPeter Bennett, VE7CEI\npeterbb4 (at) interchange.ubc.ca\nnew newsgroup users info : http:\/\/vancouver-webpages.com\/nnq\nGPS and NMEA info: http:\/\/vancouver-webpages.com\/peter\n\nP\n\n#### Paul Hovnanian P.E.\n\nJan 1, 1970\n0\nGary said:\nFunny though, in automotive, black is usually ground, and I've seen people\nused to one wireing sceme crew up the other, usually someome used to\nautomotive electrics mucking up AC wiring.\n\nHmm. Brown is ground in an old Porsche. Red is B+ and other (switched)\ncircuits are everything else.\n\nP\n\n#### Paul Hovnanian P.E.\n\nJan 1, 1970\n0\nJeff said:\nSame colors here in Canada. Your low voltage 3 phase option is strange in\nthe USA (the 120V, 220V offset 3 phase thing). Here we have the more\nstandard 120V 208 3 Phase. We have a fairly odd 600V 3 phase that is often\ncoming into most commercial \/ industrial buildings with more then a 72 kVA\nservice. 480v seems more common around the world, and is used in the USA.\n\nEurope uses a few odd voltages for 3 phase, like 380V.\n\n380V is just the line to line voltage of a 220V line to neutral star\nservice. That's pretty standard (either a single phase or three phase\nthroughout Europe).\nOne thing to note: white wires can be hot - eg the drop to a switch from a\nlight fixture.\n\nThat's for electricians benefit to run a single NM (Romex) from the\nlight to the switch. They are stuck with one white wire for a hot. Safe\npractice is to hook the white in the switch leg to the incoming hot from\nthe branch circuit (in the fixture) so when somebody sees a black and\nwhite connected together, they'll stop and think (What the $%@&#! is going on?). If you are pulling wire into conduit, white (gray) is for neutral only. H #### Homer J Simpson Jan 1, 1970 0 I live in the US and am re-wiring a 30 year old Logitek lapping machine which was built in England. If I plug it in to a receptacle in the US, as it is presently wired, then the limit switches carry 0V and the neutral is fused. I don't think this is correct because a short to chassis in the limit switch circuit will defeat the limit switch function. Are English wall receptacles reversed with respect to US receptacles? In N America, the outlets are wired backwards; and black is 'hot'. In Europe and many other countries, the left hand pin of a power outlet is hot. Also black (or blue) is always 'cold' (neutral) and red, brown or another color is hot. H #### Homer J Simpson Jan 1, 1970 0 The dull Euro-norm brown = live and blue = neutral. I think it came about by having to be different to anything anyone used before ! Also ground\/earth was green and is now green\/yellow striped. M #### Michael A. Terrell Jan 1, 1970 0 Jeff said: One thing to note: white wires can be hot - eg the drop to a switch from a light fixture. Some inspectors won't sign that off, without either phase tape or a piece of colored heatshrink tubing on the switched white wire. Phase tape is a lot cheaper than a coffin. -- Service to my country? Been there, Done that, and I've got my DD214 to prove it. Member of DAV #85. Michael A. Terrell Central Florida H #### Homer J Simpson Jan 1, 1970 0 Hell, we have everything here. 120\/208, split 240 stinger, all sorts of stuff. 480 l-l for the bigger stuff. And electricians never leave schematics behind. In Europe, the UK, Australia, New Zealand and many other countries there is one voltage in the range of 220 to 250 volts (actual value depends on the country) single phase to neutral and a three phase supply based on that (* square root of 3). The next voltage used is 11 kV 3 phase or similar used for huge motors with special precautions. In N America they use every voltage under the sun - hell they even used multiple frequencies up until recently! It's very odd. You can even see two switch boards in commercial buildings each with a different voltage. H #### Homer J Simpson Jan 1, 1970 0 That's for electricians benefit to run a single NM (Romex) from the light to the switch. They are stuck with one white wire for a hot. Safe practice is to hook the white in the switch leg to the incoming hot from the branch circuit (in the fixture) so when somebody sees a black and white connected together, they'll stop and think (What the$%@&#! is\ngoing on?). If you are pulling wire into conduit, white (gray) is for\nneutral only.\n\nCode now outside of N America may require you to sleeve the wrong color wire\nto the correct color, i.e. black to red.\n\nReplies\n21\nViews\n2K\nReplies\n1\nViews\n893\nR\nReplies\n2\nViews\n2K\nAndrew Gabriel\nA\nM\nReplies\n42\nViews\n2K\nRoss Herbert\nR\nJ\nReplies\n29\nViews\n2K\nJohn Larkin\nJ","date":"2022-10-03 15:22:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.298290878534317, \"perplexity\": 11181.723584669819}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030337421.33\/warc\/CC-MAIN-20221003133425-20221003163425-00099.warc.gz\"}"} | null | null |
By Mike Gorman on April 03, 2018
Recently-Completed,
New England Construction Dives Deep at Pods Swimming
When it comes to metaphors, never has the phrase "diving into the deep end of the pool" been more appropriate than in the case of our construction of the Pods Swimming Facility in East Providence, RI. Pods is a collaborative design/build project that has run deep at New England Construction for several years now and we are very proud to see this one of a kind place open and serving the local community. I sat down with the Project Manager, Derek Venticinque, and the Superintendent, Dave Marchand, who worked on Pods to get their perspective on the project's unique nature, the challenges they encountered along the way and their thoughts now that the project is over.
One of the first topics both members of this team wanted to discuss was the truly uncommon nature of such a project. New England Construction has a long history of ground up construction projects including retail shopping centers, multi-unit residential structures and commercial warehouses. We have even built several athletic facilities but at Pods it is the pools themselves that ensure this project stands out from any other. A pool is not just an empty hole you dig up and then fill with water. It is indeed a hole though that requires close strategic planning so that work can progress on all aspects of the facility at the same time. Dave told me about the attention to detail it took because the overall footprint of the project was relatively compact which meant cramped quarters for several trades as the work progressed. Derek went into great detail recounting the literal balancing act that was involved with the pool installation on a wetland site. The team had to ensure that there was proper water balance or the pools themselves could have popped out of the ground. Dave, as the site superintendent, had to keep an eye on the weather 24/7 especially when heavy rains threatened the region. Weekend visits to the site were commonplace to make sure that all of the pumping equipment was in working order and withstanding the elements. Beyond this being a swimming facility, Derek and Dave felt there were several other factors that made this project extraordinary. They included:
The Scope. The Pods project is not one that sprung up overnight and was completed in a week. NEC began our working relationship with the owner several years before the doors opened. This was a true design/build partnership that we supported through every phase of its production. Often projects of this nature involve a large board of directors on the owner's side and here there was an individual owner which allowed for a clear vision to be presented and for the channels of communication in both directions to operate on a high level.
The Unforeseen Challenges. All projects come with unknowns that a team must overcome, but it was the type of challenges that appeared on this project that required some true skill on the parts of all involved. As mentioned, the physical conditions (ie. wetland and New England weather) and atypical structure (pool facility with a smaller footprint) generated tidal wave sized puzzles for the team. Derek said that the only way to tackle these problems was to see the project as a large scale "chess match" that required him as the Project Manager to keep an eye on the "board" or overall progress of the project while ensuring the pieces at play, or construction trades and phases, keep moving. This meant compartmentalizing the challenges, setting priorities and readjusting them daily, and breaking them down into manageable divisions. Open honest communication between the NEC team, the project Owner and all subcontractors involved was also a key to success. Dave described the challenges that pop up on a project as driving on the highway behind a truck carrying logs and you notice one of the logs comes loose and is hurtling at your car. In that moment you must immediately decide to turn left or right, and go forward from there. There is no going back to try out the other options when you are working on a construction project of this type. The risks of such decisions are lessened when your team brings years of experience to the table and a well-structured plan for achieving the project's goals in a timely fashion.
Now that this project has drawn to a conclusion I asked Derek and Dave about the most memorable moments of the whole process. Both men spoke of a specific moment being a real turning point for the entire team involved in the project and that was the day the water was put into the pools. There was still a fair amount of work left to get the facility ready for the many students it would soon service but having water in those pools not only signified the passing of a huge milestone on the project's schedule but it also was a moment of relief. Water in the pools was an actual weight off their shoulder's because the installation risks I mentioned earlier were mitigated by the weight of the water. Over the weeks to follow the details took shape inside and outside including the specialized finishes for the surfaces that will be exposed to elevated humidity and water, and the installation of the changing rooms and front desk areas that will see a lot of traffic.
Pods Swimming has been providing swim lessons and aquatics education in the Providence and surrounding areas for nine years now and the conclusion of the construction of their 11,900 sq ft facility will allow for their outreach to grow. The journey to that opening was not a straight line by any means but it was most definitely a successful trip we were proud to take with Pods.
Published by Mike Gorman April 3, 2018
By Matt Sluter on February 07, 2022
Top 6 Considerations for Winter Construction in New England
Tackling Challenges Together!
By Mike Gorman on March 11, 2019
Dreaming of Warmer Days... A Look at Summer Slammers! | {
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{"url":"http:\/\/www.gabormelli.com\/RKB\/2004_SemiMarkovCRFforIE","text":"# 2004 SemiMarkovCRFforIE\n\n## Quotes\n\n### Abstract\n\nWe describe semi-Markov conditional random-fields (semi-CRFs), a conditionally trained version of semi-Markov chains. Intuitively, a semi-CRF on an input sequence $\\bf{x}$ outputs a \u201csegmentation\u201d of $\\bf{x}$, in which labels are assigned to segments (i.e., subsequences) of $\\bf{x}$ rather than to individual elements $x_i$ of $\\bf{x}$. Importantly, features for semi-CRFs can measure properties of segments, and transitions within a segment can be non-Markovian. In spite of this additional power, exact learning and inference algorithms for semi-CRFs are polynomial-time \u2014 often only a small constant factor slower than conventional CRFs. In experiments on five named entity recognition problems, semi-CRFs generally outperform conventional CRFs.\n\n### 1. Introduction\n\nConditional random fields (CRFs) are a recently-introduced formalism [12] for representing a conditional model Pr(y|x), where both x and y have non-trivial structure (often sequential). Here we introduce a generalization of sequential CRFs called semi-Markov conditional random fields (or semi-CRFs). Recall that semi-Markov chain models extend hidden Markov models (HMMs) by allowing each state si to persist for a non-unit length of time di. After this time has elapsed, the system will transition to a new state s0, which depends only on si\u00a0; however, during the \u201csegment\u201d of time between $i$ to $i$ + di, the behavior of the system may be non-Markovian. Semi-Markov models are fairly common in certain applications of statistics [8, 9], and are also used in reinforcement learning to model hierarchical Markov decision processes [19].\n\nSemi-CRFs are a conditionally trained version of semi-Markov chains. In this paper, we present inference and learning methods for semi-CRFs. We also argue that segments often have a clear intuitive meaning, and hence semi-CRFs are more natural than conventional CRFs. We focus here on named entity recognition (NER), in which a segment corresponds to an extracted entity; however, similar arguments might be made for several other tasks, such as gene-finding [11] or NP-chunking [16].\n\nIn NER, a semi-Markov formulation allows one to easily construct entity-level features (such as \u201centity length\u201d and ...,\n\nvolumeDate ValuetitletypejournaltitleUrldoinoteyear\n2004 SemiMarkovCRFforIESemi-Markov Conditional Random Fields for Information ExtractionProceedings of Advances in Neural Information Processing Systemshttp:\/\/books.nips.cc\/papers\/files\/nips17\/NIPS2004 0427.pdf2004\n Author Sunita Sarawagi +, William W. Cohen + and Shun-Zheng Yu + journal Proceedings of Advances in Neural Information Processing Systems + title Semi-Markov Conditional Random Fields for Information Extraction + titleUrl http:\/\/books.nips.cc\/papers\/files\/nips17\/NIPS2004 0427.pdf + year 2004 +","date":"2020-04-10 00:22:54","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\": 0.4597948491573334, \"perplexity\": 2295.991448805201}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585371880945.85\/warc\/CC-MAIN-20200409220932-20200410011432-00217.warc.gz\"}"} | null | null |
How can I make molding simulation with thermosets (f.e. RIM techn.) in SWX Plastics?
In the feature matrix of the SolidWorks Plastics product sheet the capability of simulating ractive injection molding process with thermosets is listed.
My question is, how can I set up thermoset materials in Plastics and how can I configure the process parameters of the RIM technology?
Is this modul available in Plastics at all? I couldn't find any information about simulating thermosets with SWX Plastics on the internet.
(We have 2015 SolidWorks Plastics Premium package..).
Re: How can I make molding simulation with thermosets (f.e. RIM techn.) in SWX Plastics?
finally I could find one predefined thermoset polymer in the material database of SWX Plastics.
It is a kind of silicone: the WACKER/SiliconesElastosil LR 3003/70.
When you go into the polymer's product manager with the edit icon, you can check on the "Viscosity" tab, whether it is a thermoplastic or a thermoset polymer.
If you choose a thermoset from the database the fill settings will be a little bit changed.
"Reactive Control Type" is displayed as a sign, that we are not facing with a single injection molding process in this case.
If we define the right filling properties we have a good chance to simulate a reactive molding process.
This way we will get extra results in the flow results list of our study.
Unfortunately usable thermoset polymer presets technically don't exist in the material database of SWx Plastics.
Of course, for me it is a question, how well can the curing model works in the case of a PUR f.e.?
I hope some discussions could start in this topic with your experiences. | {
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<p></p>
<form action="umauth/" method="POST"> {% csrf_token %}
<div data-role="fieldcontain">
<label for="email">Username:</label><input type="text" name="username" id="username" value="" />
</div>
<div data-role="fieldcontain">
<label for="password">Password:</label><input type="password" name="password" id="password" value="" />
</div>
<button type="submit" name="submit" value="submit-value">Sign in</button>
</form>
<br></br>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,618 |
Moments of Mommyhood: You Are My Sunshine!
The spring weather is upon us and we could not be happier! So, after a day of playing outside in the warm sunshine, we decided to makes the sun shine inside!
2. Trace and cut out several hands.
3. After your paint is dry, turn the plate over and glue the hands around the outside of the plate.
Very cute idea! Turned out great!!
Not only is the cheesy smile great, but the sweetness in the back is adorable.
I think I could get my kids to do this, and they are 13-23. Fun is fun, right?
I love this idea! I am definitely going to put this project on Evie's list!
What a cute project. We love to paint on paper plates. I love your cute blog, too.
Hooray for the sun! What a delightful project!
Oh my LOVE IT! saving this one for a rainy day!! | {
"redpajama_set_name": "RedPajamaC4"
} | 4,884 |
{"url":"https:\/\/www.madoko.net\/","text":"Madoko is a fast markdown processor for writing professional articles, books, manuals, webpages and presentations, with a focus on simplicity and plain text readability.\n\nWith Madoko you can write complex documents completely in markdown and get beautiful PDF and HTML output.\n\nUsing the\u00a0reveal.js framework, we can create great presentations in Markdown and view it in any browser. Using the\u00a0beamer package, we can also generate PDF presentations from the same source.\n\nWrite full-blown academic articles with internal references, mathematical formulas, and bibliographies.\n\nMadoko uses LaTeX to translate formulas and standard bibliography BibTeX files, and can typeset using any LaTeX document style.\n\nUsing Markdown is much more pleasant than writing LaTeX directly, and your article will look great in both in HTML and PDF. Madoko is especially good when needing\u00a0text tranformations, code fragments that need\u00a0highlighting or\u00a0proportional fonts, styling through\u00a0CSS rules, etc.\n\nMadoko integrates seamlessly with\u00a0DropboxGithub, and,\u00a0Onedrive, and automatically synchronizes all changes in the cloud. This way, your document is always available anywhere from any device.\n\nThrough Dropbox and Github, it is easy to share documents with others. Everyone can edit the document where concurrent changes are automatically merged by Madoko (using robust three-way merges).\n\n1\n\n+\n\n2\nDropbox\n\n3\nAzure website\n\n&\n\nMadoko uses\u00a0Bootstrap and\u00a0Dropbox to deploy great looking web pages instantly to\u00a0Azure.\n\n## FAQ\n\nMadoko implements many features beyond markdown to enable writing serious documents completely in Markdown while giving excellent PDF and HTML output. Read the\u00a0reference manual to learn more.\n\n2. Does the Madoko server store any documents?\nNo, the madoko server just processes mathematical formulas, bibliographies, and PDF exports. All data is sent over an encrypted connection and immediately removed after processing.\n\nDocuments are only stored locally in your browser, and on your personal cloud storage. Nothing is retained on the Madoko server.\n\n3. How can you show a preview?\nAll the rendering for the preview and HTML export is done locally in the browser on your computer \u2013 Madoko is a Javascript program written in\u00a0Koka and runs inside the browser.\n\n4. Do I need to save my document?\nNo, Madoko continuously saves the current document in browser local storage. You can at any time close the browser or shutdown the computer and your changes are saved. Madoko also automatically synchronizes your changes to the cloud storage every 30 seconds or so. At this point, all your changes become visible to any other devices (and other users if you share the document on Dropbox)\n\nWhen the local document is synchronized with the cloud, the centered dot behind the name disappears and you know your local changes are backed up in the cloud.\n\n5. Can I work concurrently with other users on a document?\nYes, Madoko merges concurrent changes automatically. However, currently only Dropbox and Github support full sharing of files in the cloud. Onedrive does not support such sharing at this time.\n\n6. Is there version control?\nWhen documents are used from Github, full version control is provided and \u201csaving\u201d a document becomes a commit. Also, Dropbox maintains a complete version history of every file automatically (up to 30 days).\n\nMoreover, in Madoko you can use the \u201cSave a snapshot\u201d menu to save a full copy of the document into a snapshot\/document-date sub-folder. Finally, you can also set up regular version control through Git or Mercurial in your cloud synchronized folder.\n\nYes! and Madoko makes it especially easy to work with others through Dropbox or Github. It is fairly safe to try out Madoko to write your next paper since:\n\n\u2022 Using TexRaw blocks you can always fall back to plain LaTeX in a pinch.\n\u2022 Using the command line version of Madoko you can always locally render the document even if the internet is down.\n\u2022 Madoko continuously synchronizes and backs up your document to the cloud making the document available anywhere on any device.\n\u2022 Edit\u00a0this example of a real article with heavy mathematics and code to see how such article can look.\n\n8. The editor has \u2018hick-ups\u2019!\nGo to the settings menu (top right) and enable delayed view updates. This will only update the view if there is a small pause in typing which makes the editor much more responsive on a slow browser or large document.\n\n9. What browsers are supported?\nMadoko.NET has been tested with Chrome version 31+, IE 10+, and Firefox version 27+. Unfortunately, Firefox has a tendency to become slow for larger documents.\n\n10. Can I run Madoko locally?\nYes! Madoko is free software and runs as a command line program on any system that can run\u00a0nodejs (Windows, MacOSX, Linux, etc). Follow the installation instructions in the\u00a0reference manual. This means that you can always continue working on important documents even if the internet is down.\n\n11. Does Madoko.net work if I am not connected? Yes, Madoko is a \u201cweb-app\u201d and works without an internet connection. However, Madoko cannot render mathematical formulas or synchronize to the cloud storage in off-line mode.\n\n12. What software does Madoko use?\n\n\u2022 NodeJS\n\u2022 Express\n\u2022 Madoko is written in the\u00a0Koka language.\n\u2022 Madoko is free software and available on\u00a0codeplex under the Apache 2.0 license","date":"2019-05-20 23:42:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.21925780177116394, \"perplexity\": 4926.911783905002}, \"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-2019-22\/segments\/1558232256163.40\/warc\/CC-MAIN-20190520222102-20190521004102-00150.warc.gz\"}"} | null | null |
[](https://travis-ci.org/hyperium/hyper)
[](https://coveralls.io/r/hyperium/hyper?branch=master)
[](./LICENSE)
[](https://crates.io/crates/hyper)
A Modern HTTP library for Rust.
[Documentation](http://hyperium.github.io/hyper)
## Overview
Hyper is a fast, modern HTTP implementation written in and for Rust. It
is a low-level typesafe abstraction over raw HTTP, providing an elegant
layer over "stringly-typed" HTTP.
Hyper offers both an HTTP/S client and HTTP server which can be used to drive
complex web applications written entirely in Rust.
The documentation is located at [http://hyperium.github.io/hyper](http://hyperium.github.io/hyper).
## Example
Hello World Server:
```rust
extern crate hyper;
use std::io::Write;
use hyper::Server;
use hyper::server::Request;
use hyper::server::Response;
use hyper::net::Fresh;
fn hello(_: Request, res: Response<Fresh>) {
let mut res = res.start().unwrap();
res.write_all(b"Hello World!").unwrap();
res.end().unwrap();
}
fn main() {
Server::http(hello).listen("127.0.0.1:3000").unwrap();
}
```
Client:
```rust
extern crate hyper;
use std::io::Read;
use hyper::Client;
use hyper::header::Connection;
fn main() {
// Create a client.
let mut client = Client::new();
// Creating an outgoing request.
let mut res = client.get("http://www.gooogle.com/")
// set a header
.header(Connection::close())
// let 'er go!
.send().unwrap();
// Read the Response.
let mut body = String::new();
res.read_to_string(&mut body).unwrap();
println!("Response: {}", body);
}
```
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,070 |
namespace pe {
namespace transforms {
using block_graph::BlockGraph;
using block_graph::TransformPolicyInterface;
using block_graph::transforms::NamedBlockGraphTransformImpl;
// A transform for adding COFF symbols to a given block graph.
class CoffAddImportsTransform
: public NamedBlockGraphTransformImpl<CoffAddImportsTransform>,
public PECoffAddImportsTransform {
public:
// Construct an empty CoffAddImportsTransform, that imports nothing
// initially.
CoffAddImportsTransform() {}
// Perform the transform. Add entries for any missing symbols to the COFF
// symbol table, and fill the attached imported module objects.
//
// @param policy The policy object restricting how the transform is applied.
// @param block_graph the BlockGraph to populate.
// @param headers_block the block containing the headers.
// @returns true on success, false otherwise.
virtual bool TransformBlockGraph(const TransformPolicyInterface* policy,
BlockGraph* block_graph,
BlockGraph::Block* headers_block) override;
// The name of this transform.
static const char kTransformName[];
private:
// Process all symbols in @p module as requested, adding to
// @p names_to_add any symbol that needs to be imported and is not
// already present.
//
// @param file_header the COFF file header.
// @param known_names the collection of existing symbols in the current
// symbol table.
// @param module the module to process.
// @param names_to_add the collection of new symbols that will need to be
// added to the symbol table.
// @param string_len_to_add incremented by the extra space (in bytes)
// required to hold the added names.
// @returns true on success, false on failure
bool FindAndCollectSymbolsFromModule(
const block_graph::TypedBlock<IMAGE_FILE_HEADER>& file_header,
const CoffSymbolNameOffsetMap& known_names,
ImportedModule* module,
CoffSymbolNameOffsetMap* names_to_add,
size_t* string_len_to_add);
// Update all references in @p module.
//
// @param symbols_block the block containing the symbol table.
// @param module the module to update.
void UpdateModuleReferences(BlockGraph::Block* symbols_block,
ImportedModule* module);
typedef std::pair<ImportedModule*, size_t> ModuleSymbol;
typedef std::map<ModuleSymbol, BlockGraph::Offset> ModuleSymbolOffsetMap;
ModuleSymbolOffsetMap module_symbol_offset_map_;
DISALLOW_COPY_AND_ASSIGN(CoffAddImportsTransform);
};
} // namespace transforms
} // namespace pe
#endif // SYZYGY_PE_TRANSFORMS_COFF_ADD_IMPORTS_TRANSFORM_H_
| {
"redpajama_set_name": "RedPajamaGithub"
} | 29 |
{"url":"https:\/\/zbmath.org\/?q=an:1090.35002","text":"## Gradient flows in metric spaces and in the space of probability measures.(English)Zbl\u00a01090.35002\n\nLectures in Mathematics, ETH Z\u00fcrich. Basel: Birkh\u00e4user (ISBN 3-7643-2428-7\/pbk). vii, 333\u00a0p. (2005).\nThe book under review deals with a systematic presentation of the theory of gradient flows and its application to some partial differential equations of evolution type. The volume is divided into two main parts.\nThe first part is devoted to the general theory of gradient flows in an arbitrary setting. The framework considered here is very general and needs only a metric structure in the ambient space. The key ingredients are the so-called \u201cmetric derivative\u201d and the notion of curves of \u201cmaximal slope\u201d, whose properties are studied in a very detailed way, also in connection to the De Giorgi\u2019s concept of \u201cminimizing movements\u201d.\nThe second part is more concerned with the application of the results of the first part to the case when the ambient metric space is the space $${\\mathcal P}_p(X)$$ of all probabilities over a set $$X$$, endowed with the $$p$$-Wasserstein metric $$W_p$$. The connection to mass transportation problems is here emphasized, and the most important facts of transport theory are summarized, together with some basic tools of measure theory. The last two chapters deal with the development of a subdifferential calculus in $${\\mathcal P}_p(X)$$ and with some interesting applications to partial differential equations of evolution type as for instance:\n\u2013 the linear transport equation with a potential;\n\u2013 the nonlinear diffusion equation;\n\u2013 the drift nonlocal diffusion equation;\n\u2013 the Fokker-Planck equation in infinite dimension.\n\n### MSC:\n\n 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control 49J40 Variational inequalities 28A33 Spaces of measures, convergence of measures 35K55 Nonlinear parabolic equations 47H05 Monotone operators and generalizations 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs","date":"2022-05-17 10:27:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.4953792691230774, \"perplexity\": 377.3627122579248}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662517245.1\/warc\/CC-MAIN-20220517095022-20220517125022-00115.warc.gz\"}"} | null | null |
Q: Equivalent definitions of a quasi-affine variety? I have a concern about a definition of a quasi-affine variety. I had a professor who defined a quasi-affine variety to be an intersection of an open set and a closed set in some affine space $\mathbb{A}^n$, and an affine variety was a quasi-affine variety isomorphic to a closed set.
However, I think the more common definition is that given in Hartshorne, where a quasi-affine variety is an open subset of an affine variety, which is an irreducible closed subset of $\mathbb{A}^n$.
Are these definitions equivalent? Or have I been learning things slightly differently?
A: Well, the difference between the two definitions you've given is that the former could be reducible whereas the latter will not be. In general there's some disagreement between authors as to whether "irreducible" is part of the definition of "variety." Other than that, they're the same.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,026 |
Tea party with the Queen for local community champions
Community champions enjoy tea with the Queen at garden party
A GROUP of West Dunbartonshire residents had tea with the Queen last week at her annual garden party.
The attendees were nominated to attend the prestigious event after making a difference in their community.
They included Clydebank's Jim McLaren, who was named Citizen of the Year last year for his work to create the Golden Friendship club.
Sheena Rollo, who won a Provost Civic Award as part of the Dumbarton District MS Group for 50 years' support, was also in attendance.
Other nominees included council employees who had been selected by their manager.
Read more: Couple left feeling Golden after getting surgery at the exact same time
The group was welcomed to Clydebank Town Hall by Provost William Hendrie last Wednesday before they travelled through to Edinburgh for the celebration.
Provost Hendrie said: "It was great to see such deserving residents so excited to be part of the garden party.
"This is a great way for us to let these people know how much their hard work and dedication means to the communities around West Dunbartonshire.
"These are the people who make our area what it is, and I was delighted to have the opportunity to say thank you to them in person."
The Queen was introduced to a number of guests during the annual event, held in the grounds of the Palace of Holyrood House.
The garden party was also attended by Princess Anne, Prince Edward, and Prince Andrew who joined the Queen in meeting guests on the lawn.
Jim McLaren said: "It was an absolutely wonderful day and I was so privileged to be there. I took my 75-year-old mother, Agnes, with me.
"She has been involved with Golden Friendships right from the start - I set it up shortly after my dad died and she has been there three days a week since then helping me. It has been a new lease of life for her.
"When I won Citizen of the Year, I told her straight away she was coming with me to meet the Queen.
"For me, seeing how much she enjoyed it made the day even more special for me. Princess Anne came right up to us and said good afternoon which was lovely.
"After the Queen and other members of the Royal family arrived, we had sandwiches, tea and then ice cream. We were so well looked after and it was nice to meet so many deserving people who had been invited along.
Read more: Clydebank High didn't tell dad his son 'nearly drowned' – but tumble dried shirt
"I'm so grateful to have been invited and it's a day we will both remember forever."
The civic award nominees who attended included: Brian McCluskey, Phil Dawson, Paul Kane, Allan Rutherford, Kevin Crawford, Ross Anderson, John Dyer, Lorraine McCullough, and the Food4Thought and Community Soup group.
Council employees, nominated by their managers for their work during the past year, include Catherine Lawler, Yvonne Caulfield, Suzie Short, Marie Mcwatt, Shona Gormley, Linda Butler and Jonathan Hoyland. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,764 |
\section*{References}}{}
\renewcommand{\cite}[1]{{\sf[##1]}}
\renewcommand{\bibitem}[1]{\par\noindent{\sf[##1]}}}
\newcommand{\spur}[1]{\!\not\! #1 \,}
\newcommand{\mathcal{U}}{\mathcal{U}}
\newcommand{\mathcal{BZ}}{\mathcal{BZ}}
\newcommand{\mathcal{A}}{\mathcal{A}}
\newcommand{\mathcal{V}}{\mathcal{V}}
\newcommand{\mathcal{O}}{\mathcal{O}}
\renewcommand{\slash}[1]{#1\!\!\!/}
\newcommand{\bslash}[1]{#1\!\!\!\backslash}
\newcommand{\begin{equation}}{\begin{equation}}
\newcommand{\end{equation}}{\end{equation}}
\newcommand{\begin{eqnarray}}{\begin{eqnarray}}
\newcommand{\end{eqnarray}}{\end{eqnarray}}
\newcommand{\nonumber}{\nonumber}
\usepackage{epsf}
\newcommand{\cmt}[1]{{#1}}
\def\ltilde#1{\mathord{\mathop{\kern 0pt #1}\limits_{\sim\atop}}}
\newcommand{\raisebox{0pt}[2ex][.3ex]{$\ltilde{\Pi}$}}{\raisebox{0pt}[2ex][.3ex]{$\ltilde{\Pi}$}}
\newcommand{\raisebox{0pt}[2ex][.3ex]{\scriptsize$\ltilde{\Pi}$}}{\raisebox{0pt}[2ex][.3ex]{\scriptsize$\ltilde{\Pi}$}}
\begin{document}
\thispagestyle{empty}
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\setlength{\baselineskip}{3ex}\par
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#1\end{center}\par}
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\centerline{\bf Abstract}\par\vspace{2ex}\parbox{\absize}{#1
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\end{center}}
\preprint{}
\title{An Unparticle Example in 2D}
\author{
Howard~Georgi,\thanks{\noindent \tt georgi@physics.harvard.edu}
Yevgeny~Kats\,\thanks{\noindent \tt kats@physics.harvard.edu}
\\ \medskip
Center for the Fundamental Laws of Nature\\
Jefferson Physical Laboratory \\
Harvard University \\
Cambridge, MA 02138
}
\date{\today}
\abstract{We discuss what can be learned about unparticle physics by
studying simple quantum field theories in one space and one time
dimension. We argue that the exactly soluble 2D theory of a massless
fermion coupled to a massive vector boson,
the Sommerfield model, is an interesting analog of a Banks-Zaks model,
approaching a free theory at high energies and a
scale invariant theory with nontrivial anomalous dimensions at low
energies. We construct a toy standard model coupling to the fermions in
the Sommerfield model and study how the transition from unparticle
behavior at low energies to free particle behavior at high energies
manifests itself in interactions with the toy standard model particles. }
The term ``unparticle physics'' was coined by one of us to describe a
situation in which standard model physics is weakly coupled at high
energies to a sector that flows to a scale-invariant theory in
the infrared~\cite{Georgi:2007ek,Georgi:2007si}. In this class of models,
one may see surprising effects from
the production of unparticle stuff\footnote{We prefer ``unparticle stuff''
to ``unparticles'' for the physical states, because it is not clear to us
what the noun ``unparticle'' is supposed to mean, and certainly not
clear
whether it should be singular or plural.} in the scattering of standard model
particles. Studying such models forces us to confront some interesting
issues in scale invariant theories and effective field theories.
It is important to remember that unparticle physics is not just about a
scale invariant theory. There are two other important ingredients.
A crucial one is the coupling of the unparticle fields to the standard
model. Without this coupling, we would not be able to ``see'' unparticle stuff.
Also important is
the transition in the Banks-Zaks theory~\cite{Banks:1981nn} from which
unparticle physics emerges
from perturbative physics at
high energies to scale
invariant unparticle behavior at low energies. This allows us to find
well-controlled
perturbative physics that produces the coupling of the unparticle sector
to the standard model.
Without this transition, the coupling of the
standard model to the unparticle fields would have to be put in
by hand in a completely arbitrary way and much of the
phenomenological interest of the
unparticle metaphor would be lost.
In this paper, we explore the physics of the transition from unparticle
behavior at low energies to perturbative behavior at high energies in a
model with one space and one time dimension in which the analog of the
Banks-Zaks model is
exactly solvable. This will enable us to see how the transition takes place
explicitly in a simple inclusive scattering process (figure~\ref{fig-1}).
{\small\begin{figure}[htb]
$$\beginpicture
\setcoordinatesystem units <0.8\tdim,0.8\tdim>
\setplotsymbol ({\small .})
\circulararc 360 degrees from 50 0 center at 0 0
\put {\stack{dynamics of,Sommerfield,model}} at 0 0
\startrotation by .995 -.1 about 0 0
\arrow <5\tdim> [.3,.6] from 0 50 to 0 75
\plot 0 50 0 100 /
\stoprotation
\startrotation by .995 .1 about 0 0
\arrow <5\tdim> [.3,.6] from 0 50 to 0 75
\plot 0 50 0 100 /
\stoprotation
\put {\stack{Sommerfield,stuff out}} at 0 125
\setdashes
\plot -20 -100 0 -50 /
\plot 20 -100 0 -50 /
\arrow <5\tdim> [.3,.6] from -20 -100 to -10 -75
\arrow <5\tdim> [.3,.6] from 20 -100 to 10 -75
\put {\stack{standard,model in}} at 0 -125
\linethickness=0pt
\putrule from -100 0 to 100 0
\endpicture
$$
\caption{\small\sf\label{fig-1}A disappearance process.}\end{figure}}
We begin by describing our analog Banks-Zaks model, and
its solution. It is a 2D model of massless fermions coupled to a massive
vector field. We call it the Sommerfield model because it was solved by
Sommerfield\footnote{Georgi's PhD advisor and Schwinger's student.} in
1963~\cite{Sommerfield:1963}.
Next, we describe the high energy
physics that couples the Sommerfield model to our toy standard model, which
is simply a massive scalar carrying a global $U(1)$ charge. In the
infrared, the resulting interaction flows to a coupling of two charged
scalars to an unparticle field with a non-trivial anomalous dimension.
We apply the operator product expansion to
the solution of the Sommerfield model to find the correlation functions of
the low energy unparticle operator.
Finally, we study the simplest unparticle process
shown in figure~\ref{fig-1} in
which two toy standard model scalars ``disappear'' into unparticle stuff.
Because we have the exact solution for the unparticle correlation functions,
we can see precisely how the system makes the
transition from low energy unparticle physics
to the high energy physics of free particles. The answer is simple and
elegant. The ``spectrum'' of the model consists of unparticle stuff and massive bosons. As the incoming energy of the standard model scalars is increased, the unparticle stuff is always there
but more and more massive bosons are emitted and the combination
becomes more and more like the free fermion cross section.
The Sommerfield(-Thirring-Wess)
model~\cite{Sommerfield:1963,Thirring-Wess:1964} is the
Schwinger model~\cite{Schwinger:1962tp} with an additional mass term for
the vector boson:\footnote{Our conventions are $g^{00}=-g^{11}=1,\;
\gamma^0=
\pmatrix{
0&1\cr
1&0\cr
},\;
\gamma^1=
\pmatrix{
0&-1\cr
1&0\cr
},\;
\gamma^5=\gamma^0\gamma^1=
\pmatrix{
1&0\cr
0&-1\cr
}\,$.}
\begin{equation}
\mathcal{L}=
\bar\psi\,(i\spur\partial - e\,\slash A)\,\psi
-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}
+\frac{m_0^2}{2}A^\mu A_\mu
\label{Sommerfield-model}
\end{equation}
We are interested in this theory since it is exactly solvable and (unlike
the Schwinger model) has fractional anomalous dimensions. In particular, we
are interested in the composite operator,
\begin{equation}
{\cal O}\equiv \psi_2^* \psi_1
\label{o}
\end{equation}
because in the low-energy theory, it scales with an anomalous dimension.
The solution for all fermion Green's functions in the model can be written
down explicitly, in terms of propagators for free fermions, and for
massless and massive scalar fields with mass $m$,
\begin{equation}
m^2 = m_0^2 + \frac{e^2}{\pi}
\end{equation}
The physical mass $m$ plays the role in this model of the unparticle scale
$\Lambda_\mathcal{U}$ from~\cite{Georgi:2007ek},
setting the scale of the transition between free particle behavior at high
energies and unparticle behavior at low energies.
Explicitly, the scalar propagators are\footnote{$K_0$ is the modified
Bessel function of the
second kind and $x_0$ is an arbitrary constant that will cancel out in the
following. For $y \to \infty$, $K_0(y) \sim \sqrt\frac{\pi}{2y}\,e^{-y} \to
0$. For $y \to 0$, $K_0(y) = -\ln(y/2) - \gamma_E + \mathcal{O}(y^2)$, where
$\gamma_E = -\Gamma'(1) \simeq 0.577$ is Euler's constant. Note that $\Delta(0)-D(0)=(i/2\pi)\ln\left(e^{\gamma_E}x_0m/2\right)$ is finite.}
\begin{eqnarray}
&&\Delta(x) = \int\frac{d^2p}{(2\pi)^2}\frac{e^{-ipx}}{p^2 - m^2 + i\epsilon}
= -\frac{i}{2\pi}K_0\left(m\sqrt{-x^2 + i\epsilon}\right)
\label{massless} \\
&&D(x) = \int\frac{d^2p}{(2\pi)^2}\frac{e^{-ipx}}{p^2 + i\epsilon} = \frac{i}{4\pi}\ln\left(\frac{-x^2+i\epsilon}{x_0^2}\right)
\label{massive}
\end{eqnarray}
The $n$-point functions for the $\mathcal{O}$ and $\mathcal{O}^*$ fields
can then be constructed using the operator product expansion. We will
describe all this in detail in a separate
publication~\cite{kats}.\footnote{Here we will also include more complete
references to the unparticle and Sommerfield model literature.}
Here we will simply
write down and use the result for the $2$-point function in position space,
\begin{equation}
i\Delta_{{\cal O}}(x) \equiv
\bigl\langle0\bigr|\mbox{T}{\cal O}(x)
\,{\cal O}^*(0)\bigl|0\bigr\rangle
= \frac{B(x)}{4\pi^2\left(-x^2 + i\epsilon\right)}
\label{psi-psi-prop}
\end{equation}
where
\begin{eqnarray}
B(x)&=& \exp\left[i\frac{4e^2}{m^2}\left[(\Delta(x) - \Delta(0)) - (D(x) - D(0))\right]\right] \nonumber\\
&=&\exp\left[\frac{2e^2}{\pi m^2}
\left[K_0\left(m\sqrt{-x^2 + i\epsilon}\right) + \ln\left(\xi
m\sqrt{-x^2 + i\epsilon}\right)\right]\right]
\label{b}
\end{eqnarray}
with
\begin{equation}
\xi = e^{\gamma_E}/2
\label{kappa}
\end{equation}
In the short-distance limit ($\left|x^2\right| \ll 1/m^2$), $B(x)\to1$
and one obtains free-fermion behavior. In the large-distance limit
($\left|x^2\right| \gg 1/m^2$), $K_0$ does not contribute so $B(x)$
is just a power of $x^2$, and the 2-point function is proportional to
an unparticle propagator\footnote{Here and below, we incorporate a
dimensional factor of $1/(\xi m)^{2a}$ in the unparticle propagator so
it matches smoothly onto the ${\cal O}$ propagator.}
\begin{equation}
i\Delta_{{\cal O}}(x) \;\to\; i\Delta_\mathcal{U}(x) =
\frac{1}{4\pi^2(\xi m)^{2a}\left(-x^2 + i\epsilon\right)^{1+a}}
\label{un-prop-position}
\end{equation}
where
\begin{equation}
a \equiv -\frac{e^2}{\pi m^2} = -\frac{1}{1+\pi m_0^2/e^2}
\label{anom-dim-5}
\end{equation}
Thus at large distance and low energies, the composite operator ${\cal O}$
scales with dimension $1+a$, corresponding to an
anomalous dimension of $a$ for $\psi_2^*\psi_1$. For $0<m_0<\infty$,
$a$ is
fractional, which leads to unparticle behavior. In momentum space
\begin{equation}
i\Delta_\mathcal{U}(p)
= \frac{iA(a)}{2(\xi m)^{2a}\,\sin(\pi a)}(-p^2 - i\epsilon)^a
= \frac{A(a)}{2\pi(\xi m)^{2a}}\int_0^\infty
dM^2\left(M^2\right)^a\frac{i}{p^2 - M^2 + i\epsilon}
\label{un-prop}
\end{equation}
where the function
\begin{equation}
A(a) \equiv -\frac{\sin(\pi a)\,\Gamma(-a)}{2^{1+2a}\,\pi\Gamma(1+a)}
\end{equation}
is positive in the range relevant to our model ($-1 < a < 0$). Since
\begin{equation}
\mbox{Im}\,\Delta_\mathcal{U}(p) = -\frac{A(a)}{2(\xi
m)^{2a}}\theta(p^2)\left(p^2\right)^a
\label{Im-un-prop}
\end{equation}
the unparticle phase space is
\begin{equation}
\Phi_\mathcal{U}(p) = \frac{A(a)}{(\xi m)^{2a}}\,
\theta(p^0)\,\theta(p^2)\left(p^2\right)^a
\label{un-phase-space}
\end{equation}
To generate a coupling to a toy standard model\label{standard},
we assume that the very high energy theory includes the interaction
\begin{eqnarray}
\mathcal{L}_{\rm int} &=& \frac{\mu}{2}\,
\left[\bar\psi(1+\gamma_5)\chi\,\phi^*
+
\bar\psi(1-\gamma_5)\chi\,\phi\right]
+ \mbox{h.c.}\nonumber\\
&=& \mu\,
\left(\psi_2^*\,\chi_1\,\phi^*
+
\psi_1^*\,\chi_2\,\phi\right)
+ \mbox{h.c.}
\label{highenergy}
\end{eqnarray}
that couples the fermion $\psi$ of the unparticle sector to a neutral
complex scalar $\phi$ with mass $m_\phi \ll m$ that plays the role of a
standard model field. The interaction is mediated by the heavy fermion
$\chi$ with mass $M \gg m,\, \mu^2/m$ and the same coupling to $A^\mu$ as
$\psi$. The theory has a global $U(1)$ symmetry with charge $+1$ for
$\phi^*$, $\psi_1^*$ and $\psi_2$, and $-1$ for $\phi$, $\psi_2^*$ and
$\psi_1$, and $0$ for $\chi$. Integrating out $\chi$ we obtain
\begin{equation}
\mathcal{L}_{\rm int} =
\frac{h}{2}
\left({\cal O}\,{\phi^*}^2 + {\cal O}^*\phi^2\right)\,,\qquad
h \equiv \frac{2\mu^2}{M}
\label{lowenergy}
\end{equation}
The composite operator $\mathcal{O}$ defined in (\ref{o})
has charge $-2$ under the global $U(1)$ symmetry.
In a 4D unparticle theory, the interaction corresponding to
(\ref{lowenergy}) would typically be nonrenormalizable, becoming more
important as the energy increases. That does not happen in our 2D toy model. But
we can and
will study the process in figure~\ref{fig-1} in the unparticle limit, and
learn something about the transition region between the ordinary particle
physics behavior at energies large compared to $m$
and the unparticle physics at low energies.
To that end, we consider the physical process
$\phi + \phi \to \mbox{Sommerfield stuff}$ shown in figure~\ref{fig-1}:
Because $\phi^2$ couples to ${\cal O}^*$,
we can obtain
the total cross-section for this process from the discontinuity across the
physical cut in the ${\cal O}$ 2-point function. This is analogous to
the optical theorem for ordinary particle production. For $\phi$ momenta
$P_1$ and $P_2$, this is
\begin{equation}
\sigma = \frac{\mbox{Im}\,{\cal M}(P_1,P_2\to P_1,P_2)}{s}
\label{optical}
\end{equation}
with $s = P^2$, $P = P_1 + P_2$, and
\begin{equation}
i{\cal M}(P_1,P_2\to P_1,P_2) = -ih^2 \Delta_{{\cal O}}(P)
\label{amp22}
\end{equation}
In the unparticle limit ($\sqrt{s} \ll m$), using (\ref{Im-un-prop}), or directly the phase space (\ref{un-phase-space}), we find the fractional power behavior expected with unparticle production:
\begin{equation}
\sigma = \frac{A(a)}{2}\frac{h^2}{(\xi m)^{2a}}\frac{1}{s^{1-a}}
\end{equation}
On the other hand, in the free particle limit appropriate for high energies
$\sqrt{s} \gg m$, we have $B(x) \to
1$ in (\ref{psi-psi-prop}), and then
\begin{equation}
\sigma = \frac{h^2}{4}\frac{1}{s}
\label{sigma-short-dist}
\end{equation}
which is the cross-section for $\phi + \phi \to \bar\psi_2 +
\psi_1$.
Since have the exact solution, we can study the transition between the two limits by writing (\ref{psi-psi-prop}) for arbitrary $x$ as
\begin{equation}
i\Delta_{{\cal O}}(x)
= i\Delta_\mathcal{U}(x)\exp\left[-4\pi ia\Delta(x)\right]
= i\Delta_\mathcal{U}(x)\sum_{n=0}^\infty\frac{\left(-4\pi
a\right)^n}{n!}\left[i\Delta(x)\right]^n
\label{psi-psi-prop-exp}
\end{equation}
At distances not large compared to $1/m$, the higher terms in the
sum in (\ref{psi-psi-prop-exp}) become relevant. Notice that in (\ref{psi-psi-prop-exp}), we have expanded in $a$ only the
terms involving the massive boson propagator. This is critical to our
results. It would be a mistake to expand $i\Delta_\mathcal{U}$ in powers of $a$.
This would introduce spurious infrared divergences because the massless
boson propagator is sick in 1+1 dimensions.\footnote{This statement has a
long history in the mathematical physics literature, going back at least to
\cite{Schroer:1963gw}. For an early summary in English, see
\cite{Tarski:1964}. See also \cite{Coleman:1973ci}. It is also worth
noting that \cite{Schroer:1963gw} introduces the notion of
``infraparticles'' -- an approach to continuous mass representations of the
Poincare group, which of course includes unparticle stuff. See
\cite{Schroer:2008gd} where we learned of this interesting early
reference.} The model describes not massive and massless bosons,
but rather massive bosons and unparticle stuff. In momentum space we obtain
\begin{eqnarray}
i\Delta_{{\cal O}}(P)
= \sum_{n=0}^\infty\frac{\left(-4\pi
a\right)^n}{n!}\int\frac{d^2p_\mathcal{U}}{(2\pi)^2}
\,i\Delta_\mathcal{U}(p_\mathcal{U})\left[\prod_{i=1}^n\frac{d^2p_i}{(2\pi)^2}
i\Delta(p_i)\right](2\pi)^2\delta^2\left(P-p_\mathcal{U}-\sum_{j=1}^n
p_j\right)\nonumber\\
\end{eqnarray}
This describes a sum of two-point diagrams in which the incoming momentum
$P$ splits between the unparticle propagator and $n$ massive scalar
propagators. Each $\Delta$
is associated with the propagation of a free\footnote{in the absence of the interactions with the $\phi\,$s which are treated perturbatively.} massive
scalar field, so this gives the discontinuity
\begin{equation}
\begin{array}{c}
\displaystyle
\Phi(P) = \frac{A(a)}{(\xi m)^{2a}}\sum_{n=0}^\infty\frac{\left(-4\pi
a\right)^n}{n!}
\int\frac{d^2p_\mathcal{U}}{(2\pi)^2}\theta(p_\mathcal{U}^0)
\theta(p_\mathcal{U}^2)\left(p_\mathcal{U}^2\right)^a\\ \displaystyle
\times\left[\prod_{i=1}^n\frac{d^2p_i}{(2\pi)^2}\,
\times 2\pi\delta(p_i^2-m^2)\theta(p_i^0)\right]
\times(2\pi)^2\delta^2\left(P - p_\mathcal{U} - \sum_{j=1}^n p_j\right)
\end{array}
\label{full-phase-space}
\end{equation}
For $\sqrt{s}<Nm$, only the first $N$ terms in (\ref{full-phase-space}) (those
involving the production of fewer than $N$ massive bosons) contribute and
(\ref{amp22}) describes the production of unparticle
stuff plus between $0$ and
$N-1$ massive bosons.
For $\sqrt{s}<m$, we have pure unparticle behavior. As we go to higher
energies, the unparticle stuff is always present,
but the emission of more and more massive
bosons builds up the inclusive result for free fermion production. This
happens quickly if $a$ is small, but very gradually for $a$ close to
$-1$.
One can easily obtain explicit results in the case of small $a$, when only
the first few terms in the expansion contribute. The leading correction in
$a$ comes from $n=1$:
\begin{equation}
\Phi^{(1)} = -a\,\theta(\sqrt{s}-m)\ln\frac{\sqrt{s}}{m} + {\cal O}(a^2)
\end{equation}
which gives the total phase space as
\begin{eqnarray}
\Phi = \frac{1}{2} - a\left[\ln\left(\frac{2}{e^{\gamma_E}}\frac{\xi
m}{\sqrt{s}}\right) + \theta(\sqrt{s}-m)\,\ln\frac{\sqrt{s}}{m}\right] + \mathcal{O}(a^2)
\label{transition}
\end{eqnarray}
For energies $\sqrt{s} > m$, this expression reduces to
\begin{equation}
\Phi = \frac{1}{2} + \mathcal{O}(a^2)
\end{equation}
that is the free-fermion result (\ref{sigma-short-dist}). Thus, for $|a|
\ll 1$ there is a discontinuity in $d\Phi/d\sqrt{s}$ at $\sqrt{s} = m$, where a
transition occurs from pure unparticle behavior below energy $m$ to pure
free-fermion behavior above $m$ (see figure~\ref{fig-5}).\footnote{The
linear approximation (\ref{transition}) is not valid for $\sqrt{s} \ll m$ due to
large $\ln \sqrt{s}$, but we have the exact expression (\ref{un-phase-space}).}
To this order, the free-fermion behavior is a sum of the unparticle and the
massive scalar contributions.
{\small
\begin{figure}[htb]
$$\epsfxsize=2in \epsfbox[70 250 540 720]{fig-transition.eps}$$
\caption{\small\sf\label{fig-5}Phase space $\Phi$ for the disappearance
process in figure~\ref{fig-1} as a function of the energy $\sqrt{s}$ (in units of $m$) for $a = -0.1$.}
\end{figure}}
For larger values of $|a|$, higher powers of $a$ must be included
in (\ref{transition}) to approximate the free-fermion
regime. Since each new massive scalar gives a contribution
with only one additional power of $a$, if $a$ is close to $-1$, the free
fermion behavior is approached very slowly. In fact, the limit $a\to-1$ is singular, and it corresponds to the Schwinger model ($m_0=0$). As is often the case, it is not trivial to obtain a gauge theory as the limit of a theory with a massive vector boson. The unparticle stuff is absent since $A(-1)=0$, and the spectrum includes only a massive boson with $m^2 = e^2/\pi$. The case of the Schwinger model has been studied in~\cite{Casher:1974vf}.
We find this picture of the unparticle scale $\Lambda_\mathcal{U} = m$ in the Sommerfield model very
satisfying. There is a close analog between the way $m$ enters in the process
of figure~\ref{fig-1}, and the way the dimensional transmutation scale
$\Lambda_{\rm QCD}$ enters in inclusive processes in QCD. In QCD, the
physical states are hadrons, typically with masses of the order of
$\Lambda_{\rm QCD}$ unless they are protected by some symmetry (like the
pions). But in the total $e^+e^-$ cross-section into hadrons (to pick the
simplest and most famous example) at high energy $E$, the sum over physical
states reproduces
the ``parton model'' result with calculable corrections of order
$1/\ln(E/\Lambda_{\rm QCD})$. We have shown that the process of
figure~\ref{fig-1} in the Sommerfield model works the same way, with the
physical states being the massive boson and unparticle stuff. Note however,
that in the presence of the standard model couplings, the massive boson is
unstable, decaying into $\phi^*+\phi^*+\mbox{unparticle stuff}$ or
$\phi+\phi+\mbox{anti-stuff}$.
The rate is of order $h^2a^2/m$ so this process is very
slow for small $a$. We will discuss this further in~\cite{kats}.
\section*{Acknowledgments}
We are grateful to
C. Cordova,
V. Lysov,
P. Petrov,
A. Sajjad, and
D. Simmons-Duffin,
for discussions.
This research is supported in part by
the National Science Foundation under grant PHY-0244821.
| {
"redpajama_set_name": "RedPajamaArXiv"
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We are specialists in the construction of unique and exclusive manufactured wood products. Our work inspires. We pride ourselves on delivering outstanding quality and design for leading clients across the world. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,231 |
{"url":"https:\/\/math.stackexchange.com\/questions\/1444194\/circular-logic-and-continuity","text":"# Circular Logic and Continuity\n\nSo, I was doing a Calculus problem a few minutes ago and just recalled something that my real analysis professor said during a lecture years ago...\n\nTo provide context, take the function $f$ defined by $f(x) = x+4$ for example.\n\nLet's, for example, show that $f$ is continuous at $x = 3$.\n\nFind $\\lim\\limits_{x \\to 3}(x+4)$ by plugging in $3$ for $x$: you get $7$.\n\nSince $\\lim\\limits_{x \\to 3}f(x)= f(3)$, $f$ is continuous at $x = 3$.\n\nSpecifically, here's what I recall my professor saying:\n\nThe way continuity is taught in Calculus requires circular logic.\n\nClearly, I used circular logic in my example, since I assumed I could plug in $3$ to get the limit.\n\nWith a polynomial, I don't see this being too much of a problem. If I recall, there is a proof given early on in a Calculus I class which states that if $p$ is a polynomial defined by $p(x) = a_nx^{n} + a_{n-1}x^{n-1} + \\cdots + a_1x +a_0$ for some positive integer $n$, then $\\lim\\limits_{x \\to a}p(x) = p(a)$ - which I believe is proved before discussing continuity in Calculus. (We can use anything like Stewart's Calculus book as a textbook for a \"typical\" Calculus course.)\n\nBut how about trigonometric functions? $a^{x}$ equations for some constant $a > 0$? Natural logarithms? Powers of $x$ - $x^{b}$ - where $b$ isn't a positive integer?\n\nHow can one bypass these problems in a Calculus I course? Furthermore, is there a way to do this without using $\\delta$-$\\epsilon$ and just using limit theorems?\n\n[I am willing to move this to Math Educators SE if desired, and if the question is deemed to be too broad, I can delete this.]\n\n\u2022 While being rigorous, no, there is no way to avoid epsilon-delta (at least without nonstandard analysis, which is not what most people want to see in calc I). But in my high school calculus class, we talked about $f(y)$ becoming arbitrarily close to $f(x)$ by making $y$ arbitrarily close to $x$. We said all this without quantifiers or any actual epsilon-delta examples. But I still grasped the picture, even though the explanation was purely in words. And the picture for almost all of the elementary functions is rather simple. \u2013\u00a0Ian Sep 20 '15 at 22:29\n\nWell, that is simply a bad proof. What you gave does NOT prove continuity, it assumes it. To prove continuity at x= 3 of f(x) you must prove that $\\lim_{x\\to 3} f(x)= f(3)$. To do that, without the \"circular reasoning\" you need to show that the definition of \"limit\" is satisfied- and that uses the $\\epsilon-\\delta$ definition.\nHere, f(x)= x+ 3 so f(3)= 6. To prove that $\\lim_{x\\to 3} f(x)= 6$, look at $|f(x)- 6|= |x+ 3- 6|= |x- 3|< \\epsilon$ and it immediately follows that you can use any $\\delta< \\epsilon$. If $|x- 3|< \\delta< \\epsilon$ then $|x- 3|< \\epsilon$ and, working backwards, $|x- 3|= |x+ 3- 6|< \\epsilon$ so $|f(x)- f(3)|< \\epsilon$ and we are done.\nI think the saying by your professor is completely wrong. There is nothing circular here. The limit of the function $x+4$ as $x\\to 3$ is not evaluated by plugging (remember that limits are not the value of the function, but they are defined in terms of values of the function).\nA limit is always evaluated using theorems concerning evaluation of limits. These theorems include algebra of limits, Squeeze theorem, L'Hospital's Rule, Taylor's theorem. Apart from this we need to know certain standard limits (one for each type of elementary function). And we need two basic results which are immediate consequences of definition of limit namely $$\\lim_{x\\to a} k=k, \\lim_{x\\to a} x=a$$ Using these results and the algebra of limits we can prove that the limit of rational function at a point is equal to its value at that point, provided the rational function is defined at that point and therefore a rational function is continuous wherever it is defined. Using standard limits related to other elementary functions it is possible to prove that an elementary function is continuous wherever it is defined.","date":"2019-10-15 12:10:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9108492732048035, \"perplexity\": 178.0153129582529}, \"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\/1570986658566.9\/warc\/CC-MAIN-20191015104838-20191015132338-00516.warc.gz\"}"} | null | null |
Ellie – variante del nome proprio di persona Elle
Ellie – personaggio del videogioco The Last of Us | {
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} | 3,381 |
General elections were held in Nepal in 1971 to elect members of the Rastriya Panchayat. At the time, the Rastriya Panchayat had 125 members; out of them 16 were appointed by the King, 90 were elected by Zonal Assemblies, 15 were elected by class organizations and 4 were elected by the graduates constituency.
District representatives
The district representatives were elected by the Anchal Sabhas (Zonal Assemblies) of the 14 Zones of Nepal, one representative for each district. The 15 districts with a population of more than 100 000 were able to elect an additional Rastriya Panchayat member. The Anchal Sabhas consisted of all the members of the Zilla Panchayats (District Councils). Each Zilla Panchayat had 11 members, who were elected from the town or village panchayats in the district. The town and village panchayats were elected from local assemblies in which all adult residents could vote.
A potential candidate had to be a Zilla Panchayat member of the concerned district to be an eligible to contest a district seat. Moreover, the potential candidate had to be proposed and seconded by two other members of the same Zilla Panchayat. To be elected the candidate would need a simple majority of the votes in the Anchal Sabha. The system favoured the less populates areas in the hills, whose districts had a much lower population than the Terai districts in the south.
Class organisation representatives
The following official class organisations were able to select their representatives in the Rastriya Panchayat. The representatives were elected by the central committees of the respective organisation, through a Preferential Proportional Representation vote:
Peasants Organisation; 4 seats
Youth Organisation; 4 seats
Women's Organisation; 3 seats
Labour Organisation; 2 seats
Ex-Servicemens' Organisation; 2 seats
Graduates constituency
The college graduates of the country, numbering about 13000 at the time, were able to elect four members of the Rastriya Panchayat. The representatives were elected with Preferential Proportional Representation vote. The candidates had to be college graduates themselves. In total 22 candidates were in the fray. Seventeen of them contested on a joint reformist agenda. One candidate, the young advocate Ram Raja Prasad Singh, demanded direct transition to parliamentary democracy.
The prime minister Kirti Nidhi Bista campaigned against the reformist candidates as opponents of the Panchayat system. However, the regime was embarrassed as the reformist candidates were elected, including Ram Raja Prasad Singh.
References
General
Nepal
General elections in Nepal
1971 elections in Nepal | {
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 586 |
\section{Introduction}
This comunication is based on the paper \cite{beta}
Non commutative (NC) quantum field theory (QFT) may be important for physics
beyond the standard model and for understanding the quantum
Hall effect \cite{DN}. It also occurs naturally as an effective regime of string theory \cite{CDS,SW}.
It led Connes and Chamesddine \cite{Connes} to reformulate the standard model in terms of a spectral triple on a simple non commutative geometry.
The simplest NC field theory is the $\phi_4^4$
model on the Moyal space. Its perturbative renormalizability
at all orders has been proved by
Grosse, Wulkenhaar and followers \cite{GW1,GW2,RVW,GMRV}.
Grosse and Wulkenhaar solved the difficult problem of ultraviolet/infrared
mixing by introducing a new harmonic potential term
inspired by the Langmann-Szabo (LS)
duality \cite{LS} between positions and momenta.
Many on the techniques of commutative field theory have been generalized to include this model. The parametric representation of this model has been established in \cite{GR}. Dimensional regularization has been performen in \cite{dimreg} and the Complete Mellin representation has been introduced in \cite{melin}.
The Hopf algebra associated structure was introduced in \cite{ConKrTV}.
Models with more general propagators have been analysed in \cite{propaga}. There parametric representation has been introduced in \cite{param2}
It is now tempting to conjecture that commutative renormalizable theories in general have NC renormalizable
extensions to Moyal spaces which may imply some new parameters.
Once perturbative renormalization is understood, the next problem is to
compute the renormalization group (RG) flow.
It is well known that the ordinary commutative $\phi_4^4$
model is not asymptotically free in the ultraviolet regime.
It is easy enough to understand this phenomenon in the commutative theory. The coupling is given by the amputed one particle irreducible four point function $\Gamma^4$. To each vertex there correspond two propagators, thus the effective coupling is given by
\begin{eqnarray}
\lambda_{i-1}=\frac{\Gamma^4_i}{Z^2_i} \; ,
\end{eqnarray}
with $Z$ the wave function renormalization.
For a theory with UV cutoff $\Lambda$, one can follow the evolution of the effective coupling with the scale by means of the function $\beta(\lambda)$ defined as
\begin{eqnarray}
\beta(\lambda)=\frac{d\lambda^{nu}}{d\ln\Lambda}\Big{|}_{\lambda^{ren}=ct.} \; .
\end{eqnarray}
Alternatively, in multiscale analysis one uses the definition
\begin{eqnarray}
\beta(\lambda)_i=\lambda_{i-1}-\lambda_i \; .
\end{eqnarray}
For the commutative $\Phi^4_4$ theory, the effective coupling varies with the scale. This phenomenon is easily enough understood at the first order in perturbation theory. For the $\Gamma^4$ function we have a non trivial contribution coming from the bubble graph. On the other hand the tadpole, being local gives only a mass counterterms. and consequently $Z=1$ at one loop.
A detailed study shows that if one wants a non zero renormalized coupling constant one needs to start with a large bare coupling constant. Actually the bare coupling becomes zero for some finite UV cutoff. Conversely, for all finite bare couplings the IR theory is trivial (i.e free). This is a serious problem in commutative field theory and has was baptized the "Landau ghost". The infinite quantities subtracted by renormalization are picked up again in this new divergence.
This problem almost killed the QFT. Being an universal phenomenon, exhibited by many field theories, including electrodynamics, it almost led to abandoning QFT as a reasonable description of fundamental intercations. Fortunately QFT was saved by the discovery of ultraviolet asymptotic freedom in non-Abelian gauge theory \cite{thooft}. But in some sense even the asymptotic freedom is not entirely satisfying. It is more like the ghost turned upsind down: it is the UV theory that becomes trivial.
It is true that such a theory makes much more sense than a theory with ghost, and this allows the
introduction of the standard model for elementary pareticles. Nevertheless IR phenomenon (corresponding to a large coupling non perturbative regime) like quark confinement can not be easily understood. Morover the flows of the three couplings in the standard model do not converge to a unified constant. In order to achieve the convergence of flows one could for instance introduce supersymmetry, but this is problematic as no detection of any super symmetric partner has ever been made.
This same phenomenon blocks the construction of the commutative $\Phi^4_4$ model. One could argue that constructive field theory is more an academic question than a true physical problem, but with out a constructive argument perturbative computations have no meaning. For instance the perturbation series could be sensless starting with the first order!
It was noted in \cite{GWbeta} that the non commutative $\phi_4^4$ model does not exhibit any Landau ghost
at one loop. It is not asymptotically free either.
For any renormalized harmonic potential parameter $\Omega_{ren} >0$,
the running $\Omega$ tends to the special LS dual point $\Omega_{bare} =1$ in the ultraviolet. As a result
the RG flow of the coupling constant is simply bounded \footnote{The Landau ghost can be recovered
in the limit $\Omega_{ren}\to 0$.}. This result was extended up to three loops in \cite{DR}.
This is due to the fact that in NCQFT the tadpole is no longer local! We have a non trivial wave function renormalization starting with the first order! Moreover it exactely compensates the bubble contribution in the beta function! If one generalizes such an argument to all orders the theory would be finite but not trivial all along its RG flow!
In this paper we evaluate the flow at the special LS dual point $\Omega =1$, and prove that
the beta function vanishes at all orders using a kind of Ward identity.
We think the Ward identities discovered here might be important for the
future study of more singular models such as Chern-Simons or Yang Mills theories.
\section{Notations and Main Result}
\setcounter{equation}{0}
We adopt simpler notations than those of \cite{GWbeta,DR}, and normalize so that $\theta =1$,
hence have no factor of $\pi$ or $\theta$.
The bare propagator in the matrix base at $\Omega=1$ is
\begin{equation} \label{propafixed}
C_{m n;k l} = C_{m n} \delta_{m l}\delta_{n k} \ ; \
C_{m n}= \frac{1}{A+m+n}\ ,
\end{equation}
where $A= 2+ \mu^2 /4$, $m,n\in \mathbb{N}^2$ ($\mu$ being the mass)
and we used the notations
\begin{equation}
\delta_{ml} = \delta_{m_1l_1} \delta_{m_2l_2}\ , \qquad m+n = m_1 + m_2 + n_1 + n_2 \ .
\end{equation}
There are two version of this theory, the real and complex one. We focus on the complex case, the result
for the real case follows easily \cite{DR}.
The generating functional is:
\begin{eqnarray}
&&Z(\eta,\bar{\eta})=\int d\phi d\bar{\phi}~e^{-S(\bar{\phi},\phi)+F(\bar{\eta},\eta,;\bar{\phi},\phi)}\nonumber\\
&&F(\bar{\eta},\eta;\bar{\phi},\phi)= \bar{\phi}\eta+\bar{\eta}\phi \nonumber\\
&&S(\bar{\phi},\phi)=\bar{\phi}X\phi+\phi X\bar\phi+A\bar{\phi}\phi+
\frac{\lambda}{2}\phi\bar{\phi}\phi\bar{\phi}
\end{eqnarray}
where traces are implicit and the matrix $X_{m n}$ stands for $m\delta_{m n}$. $S$ is the action and $F$ the external sources.
As before, denote $\Gamma^4(a,b,c,d)$ the amputated one particle irreducible four point function with external indices set to $a,b,c,d$, and $\Sigma(a,b)$ the amputated one particle irreducible two point function
with external indices set to $a,b$ (also called the self-energy). The wave function renormalization is $Z=1-\partial \Sigma(0,0)$ where $\partial\Sigma(0,0)\equiv\partial_L \Sigma = \partial_R \Sigma = \Sigma (1,0) - \Sigma (0,0)$ is the derivative of the self-energy with respect to one of the two indices $a$ or $b$ \cite{DR}.
Our main result is:
\medskip
\noindent{\bf Theorem}
\medskip
The equation:
\begin{eqnarray}\label{beautiful}
\Gamma^{4}(0,0,0,0)=\lambda (1-\partial\Sigma(0,0))^2
\end{eqnarray}
holds up to irrelevant terms \footnote{Irrelevant terms include in particular all non-planar or planar with more than one broken face contributions.}
to {\bf all} orders of perturbation, either as a bare equation with fixed ultraviolet cutoff,
or as an equation for the renormalized theory. In the latter case $\lambda $ should still be understood
as the bare constant, but reexpressed as a series in powers of $\lambda_{ren}$.
\section{Ward Identities}
We orient the propagators from a $\bar{\phi}$ to a $\phi$.
For a field $\bar{\phi}_{a b}$ we call the index $a$ a
{\it left index} and the index, $b$ a {\it right index}. The first (second) index of a $\bar{\phi}$ {\it allways} contracts with
the second (first) index of a $\phi$. Consequently for $\phi_{c d}$, $c$ is a {\it right index} and $d$ is a {\it left index}.
Let $U=e^{\imath B}$ with $B$ a small hermitian matrix. We consider the ``left" (as it acts only on the left indices) change of variables\footnote{There is a similar ``right" change of variables, acting only on the right indices.}:
\begin{eqnarray}
\phi^U=\phi U;\bar{\phi}^U=U^{\dagger}\bar{\phi} \ .
\end{eqnarray}
The variation of the action is, at first order:
\begin{eqnarray}
\delta S&=&\phi U X U^{\dagger}\bar{\phi}-\phi X \bar{\phi}\approx
\imath\big{(}\phi B X\bar{\phi}-\phi X B \bar{\phi}\big{)}\nonumber\\
&=&\imath B\big{(}X\bar{\phi}\phi-\bar{\phi}\phi X \big{)}
\end{eqnarray}
and the variation of the external sources is:
\begin{eqnarray}
\delta F&=&U^{\dagger}\bar{\phi}\eta-\bar{\phi}\eta+\bar{\eta}\phi U-\bar{\eta}\phi
\approx-\imath B \bar{\phi}\eta+\imath\bar{\eta}\phi B\nonumber\\
&=&\imath B\big{(}-\bar{\phi}\eta+\bar{\eta}\phi{)} .
\end{eqnarray}
We obviously have:
\begin{eqnarray}
&&\frac{\delta \ln Z}{\delta B_{b a}}=0=\frac{1}{Z(\bar{\eta},\eta)}\int d\bar{\phi} d\phi
\big{(}-\frac{\delta S}{\delta B_{b a}}+\frac{\delta F}{\delta B_{b a}}\big{)}e^{-S+F}\nonumber\\
&&=\frac{1}{Z(\bar{\eta},\eta)}\int d\bar{\phi} d\phi ~e^{-S+F}
\big{(}-[X \bar{\phi}\phi-\bar{\phi}\phi X]_{a b}+
[-\bar{\phi}\eta+\bar{\eta}\phi]_{a b}\big{)} \ .
\end{eqnarray}
We now take $\partial_{\eta}\partial_{\bar{\eta}}|_{\eta=\bar{\eta}=0}$
on the above expression. As we have at most two insertions we get only the connected components of the correlation functions.
\begin{eqnarray}
0=<\partial_{\eta}\partial_{\bar{\eta}}\big{(}
-[X \bar{\phi}\phi-\bar{\phi}\phi X]_{a b}+
[-\bar{\phi}\eta+\bar{\eta}\phi]_{a b}\big{)}e^{F(\bar{\eta},\eta)} |_0>_c \ ,
\end{eqnarray}
which gives:
\begin{eqnarray}
<\frac{\partial(\bar{\eta}\phi)_{a b}}{\partial \bar{\eta}}\frac{\partial(\bar{\phi}\eta)}{\partial \eta}
-\frac{\partial(\bar{\phi}\eta)_{a b}}{\partial \eta}\frac{\partial (\bar{\eta}\phi)}{\partial \bar{\eta}}
- [X \bar{\phi}\phi-\bar{\phi}\phi X]_{a b}
\frac{\partial(\bar{\eta}\phi)}{\partial \bar{\eta}}\frac{\partial (\bar{\phi}\eta)}{\partial\eta}>_c=0 .
\end{eqnarray}
Using the explicit form of $X$ we get:
\begin{eqnarray}
(a-b)<[ \bar{\phi}\phi]_{a b}
\frac{\partial(\bar{\eta}\phi)}{\partial \bar{\eta}}\frac{\partial (\bar{\phi}\eta)}{\partial\eta}>_c=
<\frac{\partial(\bar{\eta}\phi)_{a b}}{\partial \bar{\eta}}\frac{\partial(\bar{\phi}\eta)}{\partial \eta}>_c
-<\frac{\partial(\bar{\phi}\eta)_{a b}}{\partial \eta}\frac{\partial (\bar{\eta}\phi)}{\partial \bar{\eta}}> \ ,
\nonumber
\end{eqnarray}
and for $\bar{\eta}_{ \beta \alpha} \eta_{ \nu \mu}$ we get:
\begin{eqnarray}
(a-b)<[ \bar{\phi}\phi]_{a b} \phi_{\alpha \beta}
\bar{\phi}_{\mu \nu }>_c=
<\delta_{a \beta}\phi_{\alpha b} \bar{\phi}_{\mu \nu}>_c
-<\delta _{b \mu }\bar{\phi}_{a \nu} \phi_{\alpha \beta}>_c
\end{eqnarray}
We now restrict to terms in the above expressions which are planar with a single external face,
as all others are irrelevant. Such terms have $\alpha=\nu$, $a=\beta$ and $b=\mu$.
The Ward identity for $2$ point function reads:
\begin{eqnarray}\label{ward2point}
(a-b)<[ \bar{\phi}\phi]_{a b} \phi_{\nu a}
\bar{\phi}_{b \nu }>_c=
<\phi_{\nu b} \bar{\phi}_{b \nu}>_c
-<\bar{\phi}_{a \nu} \phi_{\nu a}>_c
\end{eqnarray}
(repeated indices are not summed).
\begin{figure}[hbt]
\centerline{
\includegraphics[width=100mm]{fig1.eps}
}
\caption{The Ward identity for a 2p point function with insertion on the left face}\label{fig:Ward}
\end{figure}
The indices $a$ and $b$ are left indices, so that we have the Ward identity with an insertion on a left face\footnote{There is a similar Ward identity obtained with the ``right" transformation, consequently with the insertion on a right face.} as represented in Fig. \ref{fig:Ward}.
\section{Proof of the Theorem}
We start this section by some definitions:
we will denote $G^{4}(m,n,k,l)$ the connected four point function restricted to the planar one broken face case, where $m,n,k,l$ are the indices of the external face in the correct cyclic order. The first index $m$ allways represents a left index.
Similarely, $G^{2}(m,n)$ is the connected planar one broken face two point function with $m,n$ the indices on the external face (also called the {\bf dressed} propagator, see Fig. \ref{fig:propagators}). $G^{2}(m,n)$ and $\Sigma(m,n)$ are related by:
\begin{eqnarray}
\label{G2Sigmarelation}
G^{2}(m,n)=\frac{C_{m n}}{1-C_{m n}\Sigma(m,n)}=\frac{1}{C_{m n}^{-1}-\Sigma(m,n)} \, .
\end{eqnarray}
\begin{figure}[hbt]
\centerline{
\includegraphics[width=60mm]{fig4.eps}
}
\caption{The {\bf dressed} and the bare propagators}\label{fig:propagators}
\end{figure}
$G_{ins}(a,b;...)$ will denote the planar one broken face
connected function with one insertion on the left border where the matrix index jumps from $a$ to $b$. With this notations the Ward identity (\ref{ward2point}) writes:
\begin{eqnarray}
(a-b) ~ G^{2}_{ins}(a,b;\nu)=G^{2}(b,\nu)-G^{2}(a,\nu)\, .
\end{eqnarray}
All the identities we use, either Ward identities or the Dyson equation of motion
can be written either for the bare theory or for the theory with complete mass renormalization, which is the one considered in \cite{DR}. In the first case the parameter $A$ in (\ref{propafixed}) is the bare one, $A_{bare}$
and there is no mass subtraction. In the second case the parameter $A$ in (\ref{propafixed})
is $A_{ren}= A_{bare} - \Sigma(0,0)$, and every two point 1PI subgraph is subtracted at 0 external indices\footnote{These mass subtractions need not be rearranged into forests
since 1PI 2point subgraphs never overlap non trivially.}. Troughout this paper $\partial_{L}$ will denote the derivative with respect to a left index and $\partial_{R}$ the one with respect to a right index. When the two derivatives are equal we will employ the generic notation $\partial$.
Let us prove first the Theorem in the mass-renormalized case, then in the next subsection
in the bare case. Indeed the mass renormalized theory used is free from any quadratic divergences, and remaining logarithmic subdivergences in the ultra violet cutoff can be removed easily by going, for instance, to the ``useful" renormalized effective series,
as explained in \cite{DR}.
\begin{figure}[hbt]
\centerline{
\includegraphics[width=120mm]{fig2.eps}
}
\caption{The Dyson equation}\label{fig:dyson}
\end{figure}
We analyze a four point connected function $G^4(0,m,0,m)$ with index $m \ne 0$ on the right borders.
This explicit break of left-right symmetry is adapted to our problem.
Consider a $\bar{\phi}$ external line and the first vertex hooked to it.
Turning right on the $m$ border at this vertex we meet a new line (the slashed line in Fig. \ref{fig:dyson}). The slashed line either separates the graph into two disconnected components ($G^{4}_{(1)}$ and $G^{4}_{(2)}$ in Fig. \ref{fig:dyson}) or not
($G^{4}_{(3)}$ in Fig. \ref{fig:dyson}). Furthermore, if the slashed line separates the graph into two disconnected components the first vertex may either belong to a four point component ($G^{4}_{(1)}$ in Fig. \ref{fig:dyson}) or to a two point component
($G^{4}_{(2)}$ in Fig. \ref{fig:dyson}).
We stress that this is a {\it classification} of graphs: the different components depicted in Fig. \ref{fig:dyson} take into account all the combinatoric factors. Furthermore, the setting of the external indices to $0$ on the left borders and $m$ on the right borders distinguishes the $G^{4}_{(1)}$ and $G^{4}_{(2)}$ from their counterparts ``pointing upwards": indeed, the latter are classified in $G^{4}_{(3)}$!
We have thus the Dyson equation:
\begin{eqnarray}
\label{Dyson}
G^4(0,m,0,m)=G^4_{(1)}(0,m,0,m)+G^4_{(2)}(0,m,0,m)+G^4_{(3)}(0,m,0,m)\, .
\end{eqnarray}
The second term, $G^{4}_{(2)}$, is zero. Indeed the mass renormalized two point insertion is zero, as it has the external left index set to zero. Note that this is an insertion exclusively on the left border. The simplest case of such an insertion is a
(left) tadpole. We will (naturally) call a general insertion touching only the left border a ``generalized left tadpole" and denote it by
$T^L$.
We will prove that $G^{4}_{(1)}+G^{4}_{(3)}$ yields
$\Gamma^4=\lambda (1-\partial \Sigma)^2$ after amputation of the four external propoagators.
We start with $G^{4}_{(1)}$. It is of the form:
\begin{eqnarray}
G^4_{(1)}(0,m,0,m)=\lambda C_{0 m} G^{2}(0, m) G^{2}_{ins}(0,0;m)\,.
\end{eqnarray}
By the Ward identity we have:
\begin{eqnarray}
G^{2}_{ins}(0,0;m)&=&\lim_{\zeta\rightarrow 0}G^{2}_{ins}(\zeta ,0;m)=
\lim_{\zeta\rightarrow 0}\frac{G^{2}(0,m)-G^{2}(\zeta,m)}{\zeta}\nonumber\\
&=&-\partial_{L}G^{2}(0,m) \, .
\end{eqnarray}
Using the explicit form of the bare propagator we have $\partial_L C^{-1}_{ab}=\partial_R C^{-1}_{ab}=\partial C^{-1}_{ab}=1$. Reexpressing $G^{2}(0,m)$ by eq. (\ref{G2Sigmarelation}) we conclude that:
\begin{eqnarray}\label{g41}
G^4_{(1)}(0,m,0,m)&=&\lambda
C_{0m}\frac{C_{0m}C^2_{0m}[1-\partial_{L}\Sigma(0,m)]}{[1-C_{0m}\Sigma(0,m)]
(1-C_{0m}\Sigma(0,m))^2}\nonumber\\
&=&\lambda [G^{2}(0,m)]^{4}\frac{C_{0m}}{G^{2}(0,m)}[1-\partial_{L}\Sigma(0,m)]\, .
\end{eqnarray}
The self energy is (again up to irrelevant terms (\cite{GW2}):
\begin{eqnarray}
\label{PropDressed}
\Sigma(m,n)=\Sigma(0,0)+(m+n)\partial\Sigma(0,0)
\end{eqnarray}
Therefore up to irrelevant terms ($C^{-1}_{0m}=m+A_{ren}$) we have:
\begin{eqnarray}
\label{G2(0,m)}
G^{2}(0,m)=\frac{1}{m+A_{bare}-\Sigma(0,m)}=\frac{1}{m[1-\partial\Sigma(0,0)]+A_{ren}}
\, ,
\end{eqnarray}
and
\begin{eqnarray} \label{cdressed}
\frac{C_{0m}}{G^{2}(0,m)}=1-\partial\Sigma(0,0)+\frac{A_{ren}}{m+A_{ren}}\partial\Sigma(0,0) \, .
\end{eqnarray}
Inserting eq. (\ref{cdressed}) in eq. (\ref{g41}) holds:
\begin{eqnarray}
\label{g41final}
G^4_{(1)}(0,m,0,m)&=&\lambda [G^{2}(0,m)]^{4}\bigl(
1-\partial\Sigma(0,0)+\frac{A_{ren}}{m+A_{ren}}\partial\Sigma(0,0)
\bigr) \nonumber\\
&&[1-\partial_{L}\Sigma(0,m)]\, .
\end{eqnarray}
\begin{figure}[hbt]
\centerline{
\includegraphics[width=140mm]{fig3.eps}
}
\caption{Two point insertion and opening of the loop with index $p$}\label{fig:insertion}
\end{figure}
For the $G^4_{(3)}(0,m,0,m)$ one starts by ``opening" the face which is ``first on the right''. The summed index of this face is called $p$ (see Fig. \ref{fig:dyson}). For bare Green functions this reads:
\begin{eqnarray}
\label{opening}
G^{4,bare}_{(3)}(0,m,0,m)=C_{0m}\sum_{ p} G^{4,bare}_{ins}(p,0;m,0,m)\, .
\end{eqnarray}
When passing to mass renormalized Green functions one must be cautious. It is possible that the face $p$ belonged to a 1PI two point insertion in $G^{4}_{(3)}$ (see the left hand side in Fig. \ref{fig:insertion}).
Upon opening the face $p$ this 2 point insertion disappears (see right hand side of Fig. \ref{fig:insertion})!
When renormalizing, the counterterm corresponding to this kind of two point insertion will be substracted on the left hand side of eq.(\ref{opening}), but not on the right hand side. In the equation for $G^{4}_{(3)}(0,m,0,m)$ one must
therefore \textit{add its missing counterterm}, so that:
\begin{eqnarray}
\label{Open2}
G^4_{(3)}(0,m,0,m)&=& C_{0m}\sum_{p} G^{4}_{ins}(0,p;m,0,m)\nonumber\\
&-&C_{0m}(CT_{lost})G^{4}(0,m,0,m)\,.
\end{eqnarray}
It is clear that not all 1PI 2 point insertions on the left hand side of Fig. \ref{fig:insertion} will be ``lost" on the right hand side. If the insertion is a ``generalized left tadpole" it is not ``lost" by opening the face $p$ (imagine a tadpole pointing upwards in Fig.\ref{fig:insertion}: clearely it will not be opened by opening the line). We will call the 2 point 1PI insertions ``lost" on the right hand side $\Sigma^R(m,n)$. Denoting the generalized left tadpole $T^{L}$ we can write (see Fig .\ref{fig:selfenergy}):
\begin{eqnarray}
\label{eq:leftright}
\Sigma(m,n)=T^{L}(m,n)+\Sigma^R(m,n)\, .
\end{eqnarray}
Note that as $T^{L}(m,n)$ is an insertion exclusively on the left border, it does not depend upon the right index $n$. We therefore have $\partial\Sigma(m,n)=\partial_R\Sigma(m,n)=\partial_R\Sigma^R(m,n)$.
\begin{figure}[hbt]
\centerline{
\includegraphics[width=90mm]{fig5.eps}
}
\caption{The self energy}\label{fig:selfenergy}
\end{figure}
The missing mass counterterm writes:
\begin{eqnarray}\label{lostct}
CT_{lost}=\Sigma^R(0,0)=\Sigma(0,0)-T^{L}\, .
\end{eqnarray}
In order to evaluate $\Sigma^{R}(0,0)$ we proceed
by opening its face $p$ and using the Ward identity (\ref{ward2point}), to obtain:
\begin{eqnarray}
\label{S2}
\Sigma^R(0,0)&=&\frac{1}{G^{2}(0,0)}\sum_{p}G^2_{ins}(0,p;0)\nonumber\\
&=&\frac{1}{G^{2}(0,0)}\sum_{p}\frac{1}{p}[G^2(0,0)-G^2(p,0)]\nonumber\\
&=&\sum_{p}\frac{1}{p} \biggl(1 -\frac{G^{2}(p,0)}{G^{2}(0,0)}\biggr) \, .
\end{eqnarray}
Using eq. (\ref{Open2}) and eq. (\ref{S2}) we have:
\begin{eqnarray}\label{S3}
G^4_{(3)}(0,m,0,m)&=& C_{0m}\sum_{p} G^{4}_{ins}(0,p;m,0,m)\nonumber\\
&-&C_{0m} G^{4}(0,m,0,m) \sum_{p}\frac{1}{p}
\biggl( 1- \frac{G^{2}(p,0)}{G^2(0,0)} \biggr) \, .
\end{eqnarray}
After some manipulations using mainly the Ward identity, detiled in \cite{beta} we obtain the final result
\begin{eqnarray}\label{g43}
G^4_{(3)}(0,m,0,m)&=&-C_{0m}G^{4}(0,m,0,m)\frac{1}{G^{2}(0,0)}\frac{\partial\Sigma(0,0)}{1-\partial\Sigma(0,0)}
\nonumber\\
&=&-G^{4}(0,m,0,m)
\frac{A_{ren} \; \partial\Sigma(0,0)}{(m+A_{ren}) [1-\partial\Sigma(0,0)]} \ .
\end{eqnarray}
Using (\ref{g41final}) and (\ref{g43}), equation (\ref{Dyson}) rewrites as:
\begin{eqnarray}
\label{final}
&&G^4(0,m,0,m)\Big{(}1+
\frac{A_{ren}\; \partial\Sigma(0,0)}{(m+A_{ren}) \; [ 1-\partial\Sigma(0,0)] }\Big{)}
\\
&&=\lambda (G^{2}(0,m))^{4}\Big{(}1-\partial\Sigma(0,0)+\frac{A_{ren}}{m+A_{ren}}\partial\Sigma(0,0)\Big{)}
[1-\partial_{L}\Sigma(0,m)]\, .\nonumber
\end{eqnarray}
We multiply (\ref{final}) by $[1-\partial\Sigma(0,0)]$ and amputate four times. As the differences $\Gamma^4(0,m,0,m,)-\Gamma^4(0,0,0,0)$ and $\partial_L\Sigma(0,m)-\partial_L\Sigma(0,0)$ are irrelevant we get:
\begin{eqnarray}
\Gamma^{4}(0,0,0,0)=\lambda (1-\partial\Sigma(0,0))^2\, .
\end{eqnarray}
{\hfill $\Box$}
\section{Conclusion}
Since the main result of this paper is proved up to irrelevant
terms which converge at least like a power of the ultraviolet cutoff, as this ultraviolet cutoff is lifted towards infinity,
we not only get that the beta function vanishes in the ultraviolet regime, but that it
vanishes fast enough so that the total flow of the coupling constant is bounded.
The reader might worry whether this conclusion is still true for the full model which has
$\Omega_{ren} \ne 1$, hence no exact conservation of matrix indices along faces.
The answer is yes, because the flow of $\Omega $ towards its ultra-violet limit
$\Omega_{bare}=1$ is very fast (see e.g. \cite{DR}, Sect II.2).
The vanishing of the beta function is a step towards a full non perturbative construction
of this model without any cutoff, just like e.g. the one of the Luttinger model \cite{BGPS,BM1}.
But NC $\phi^4_4$ would be the first such \textit{four dimensional} model, and the only one
with non logarithmic divergences. Tantalizingly, quantum field theory might actually behave
better and more interestingly on non-commutative than on commutative spaces.
Steps in this directions have been taken in \cite{const1,const2}.
\medskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,587 |
Wary of Gentrification, East Harlem Braces for Rapid Change
April 01, 2016 | by Maggie Calmes
Hank Prussing's mural, The Spirit of East Harlem (via @ManhattanCB11)
Pearl Barkley moved to East Harlem in the 1950s, when she was six years old. Her family lived in the Jefferson Houses on 115th Street, where she lives today and works as an organizer with Community Voices Heard, an economic justice advocacy non-profit.
"Many people remember the fires, the abandoned boarded up buildings, and the people who left," Barkley said in an interview. Essential services began disappearing from the neighborhood in the 1970s, she said, due in large part to city and federal policies that encouraged "benign neglect" of poor neighborhoods of color.
"They left the neighborhood to fend for itself," Barkley said. "And the community was strong, but it suffered."
"Now, East Harlem is a hot property," Barkley said. "We have to bring the focus back to the people who were originally here and went through all of this. We have to make sure that the people who were here can stay here." While she doesn't necessarily take issue with newcomers to the neighborhood, Barkley emphasizes that preserving homes for long-term residents of the neighborhood should take precedent over new developments.
Newcomers have been moving into East Harlem for some time, drawn by its upper-Manhattan perch, lower rents, and access to public transit. But now, El Barrio - like a variety of other low-income areas populated predominantly by people of color - is becoming increasingly desirable to developers hoping to accommodate waves of middle-class renters.
In February, The New York Times named East Harlem one of New York's Next Hot Neighborhoods, noting that 21 development sites had recently "traded hands," and pointing to its lower-than-average real estate prices, good transit options, and historic architecture.
"If you want to stay in Manhattan," the author wrote, "East Harlem may be your best bet."
To long-term residents who want to stay in their neighborhoods, this kind of attention from the real estate market is synonymous with anxieties about inequity, marginalization, and displacement. And rightly so, as rents have skyrocketed across the city, incomes have stagnated, and the affordable housing crisis tops the list of policy issues facing New York City.
Now, as Mayor Bill de Blasio's administration prepares to rezone 15 neighborhoods, including East Harlem, as part of a plan to build and preserve 200,000 units of affordable housing in New York City, residents are bracing for further shifts that some believe will drastically change the landscape of their neighborhood, push long-term residents out of their homes as new residents move in, and threaten local businesses. In a word, gentrification: the force that has been sweeping through New York City neighborhoods and striking fear in places set for new investment.
Even as policies like Mandatory Inclusionary Housing, which requires set percentages of rent-controlled units in new developments, move toward enactment in an administration-run rezoning process, some East Harlem stakeholders are making proactive attempts at steering their community's future. Led by the Speaker of the City Council, Melissa Mark-Viverito, whose home district includes East Harlem, a broad coalition of residents and groups created a comprehensive plan to shape the rezoning of the neighborhood.
Members of that coalition and other groups have organized to protect against displacement while also championing changes that will improve their community.
It remains to be seen how successful residents will be in achieving control over the way their neighborhood changes -- and who will be left to enjoy the fruits of that labor. But one thing is certain: East Harlem - a low-income community populated primarily by people of color, imbued with deep-rooted history and culture, situated in an area ripe for speculation by developers, struggling amid an affordable housing crisis -- is at a tipping point.
Community character
The neighborhood of just over 123,000 residents hugs the northeast corner of Central Park along Fifth Avenue, runs South until 96th Street, and spreads east toward the Harlem and East Rivers, where large portions of the East Harlem Esplanade are crumbling into the water.
East Harlem is one of the largest predominantly Latino neighborhoods in New York City, the result of waves of Puerto Rican and Latin American immigration to the area after World War I, followed by more South and Central American arrivals throughout the 20th century. In spite of demographic changes over the last decade, East Harlem is still mostly populated by people of color and an immigrant neighborhood, with Hispanic and black residents comprising 81 percent of the population, and about a quarter of residents foreign-born, according to U.S. Census Bureau estimates for 2013.
The dominant culture is reflected everywhere: Spanish is spoken on every corner, storefronts advertise Latin American food, music, and good, and large-scale, colorful murals pervade the neighborhood with images of Latin American activists and artists.
It is no coincidence that this is where Mark-Viverito, born and raised in Puerto Rico, settled after coming to the United States in 1987 to attend Columbia University. After college, Mark-Viverito participated in a mentorship program for young Latinos, coordinated New York-based efforts to drive the U.S. Navy from the Puerto Rican island of Vieques, co-founded women's economic justice group Mujeres del Barrio, and served on Community Board 11. Mark-Viverito was elected to the City Council in 2005, and re-elected twice before becoming the first Puerto Rican and Latina to win the Speaker's office when her colleagues selected her in 2014.
The Speaker is now at the helm of a City Council reckoning with an affordable housing crisis and scars and fears of gentrification across several neighborhoods; charged with the task of building and preserving affordable housing and improving neighborhoods while protecting communities' cultures and livelihoods at a pivotal time. It is a struggle filled with economic and social tensions exemplified in East Harlem.
"It's an ongoing battle," the speaker (pictured, left) told La Respuesta in a 2014 interview. "Longtime residents in our communities that are finding themselves with limited options to stay, and feel they have no other place to go and are being displaced, is obviously alarming. We have a responsibility to do as much as we can to try to figure out how to stem that tide, but we're also faced with limitations."
East Harlem's history as a stronghold of people of color and immigrants follows the arc of many other urban communities throughout the U.S. The neighborhood first felt the impact of urban renewal initiatives in 1938, when NYCHA began razing low-rise tenements and relocating residents to make way for "towers-in-the-park" housing developments also seen in larger examples at Stuyvesant Town and Co-op City. The city cleared more land containing factories, stores, and community centers in the 1960s, but failed to attract investors to the area. The result was an abundance of overgrown lots and a fragmented urban landscape that set the scene for the increase in poverty, violence, and arson that plagued the neighborhood during the '60s and '70s.
These policies, Pearl Barkley said, along with growing wealth disparity and disenfranchisement of low-income people of color, pushed East Harlem - its residents, infrastructure, and local economy - into the political periphery, and dramatically exacerbated socioeconomic problems that plagued the neighborhood throughout the second half of the 20th century.
Housing East Harlemites
Today, a lack of affordable market-rate housing all but defines the residential landscape of East Harlem, where the majority of residents rely on public housing, rental assistance, or rent stabilization to afford their homes. A 2011 report from Columbia University calls East Harlem one of the "last holdouts of undeveloped Manhattan," citing its waterfront and public transit options as contributors to the neighborhood's growing vulnerability to displacement. These factors, combined with the voracity of the real estate market and a concentration of low-income residents, have cranked up pressure on the housing stock in El Barrio, driving rents - and fears of displacement - ever higher.
Enter Mayor Bill de Blasio's plan to encourage more residential development all over the city, with affordable housing mandates, and to rezone East Harlem to add density and neighborhood improvements. De Blasio and supporters of his approach, including Council Speaker Mark-Viverito, believe that without this type of government intervention, the forces of the market will run wild over East Harlem as they have in Central Harlem, several Brooklyn neighborhoods, and elsewhere over the past two decades. With any increase in development, though, comes heightened fear as well as very real change.
Not enough housing, not enough income
With only 8 percent of privately-owned units owner-occupied, East Harlem is a community of renters. And, over half of rental households are considered rent-burdened, paying more than 30 percent of total household income on gross rent, or severely rent-burdened, paying more than 50 percent of income on rent, and are particularly susceptible to the throes of the real estate market.
The city's Department of Housing Preservation and Development (HPD) and the Census Bureau reported in 2014 that 55 percent of rental units in East Harlem were either rent-stabilized or government-assisted, and another 28 percent are NYCHA units, leaving less than a quarter of the renter-occupied housing stock unregulated.
While there is still a considerable concentration of regulated and subsidized units in East Harlem, this cache is shrinking as units age out of rent regulation and landlords evict or coerce tenants into vacating stabilized apartments in order to raise rents.
The loss is considerable, with 2,500 rent-stabilized or subsidized units disappearing from East Harlem between 2007 and 2014, and a projected 4,200 more gone by 2030, according to Community Board 11's 2016 Statement of Needs.
The mayor hopes to curb the loss of such regulated rental units through his Housing New York Plan, which seeks to preserve 120,000 existing affordable units throughout the five boroughs. To this end, the administration has doubled HPD's funding and begun initiatives around landlord harassment, deregulation, and targeted investment in vulnerable communities, among others. The administration is more aggressively targeting opportunities to work with landlords to offer benefits in exchange for extending rent regulations.
De Blasio said at a press conference in January that the administration is ahead of schedule, having preserved nearly 14,000 units in projects like Stuyvesant Town, Riverton Houses, and smaller buildings, and that the overall program was on track to build or preserve 200,000 units by 2024.
While critics of the plan assert that not enough affordable units are targeted at very low-income New Yorkers, de Blasio said that 75 percent of the 40,000-plus units built or preserved over the first two years of the timeline have been for extremely low income, very low income, and low income residents.
Concentrated public housing
The New York City Housing Authority (NYCHA) plays a vital role in maintaining a large portion of East Harlem's affordable housing stock, with 30% of residents living in NYCHA-owned properties. This is the second-highest concentration of public housing in the city - a result of urban renewal initiatives begun in 1941, when one of NYCHA's earliest housing developments, the East River Houses, went up on 102nd Street. Another wave of renewal, led by Robert Moses and funded by the 1949 Federal Housing Act, razed existing structures and constructed the Thomas Jefferson Houses in 1959. Today, there are 24 public housing developments in the neighborhood.
Over the last two decades, federal and state budget cuts have weakened NYCHA, leaving the agency with a $74 million deficit in 2015, and facing capital needs of about $17 billion over the next five years, according to the city. This shortfall has resulted in crumbling buildings, breakdowns in communication between tenants and the agency, increased violence in housing developments, and myriad other problems.
Mayor de Blasio launched last summer the NextGeneration NYCHA initiative as part of a "ten-year plan aimed at preserving public housing for the future generations," according to the city's website. The administration describes NextGen as a long-term strategy to change the way NYCHA properties are funded, operated, and designed, and to revamp the way the agency engages with residents - all in an attempt to prevent the housing authority from being turned over to the United States Department of Housing and Urban Development - an acquisition that could result in the sale of housing sites.
The most controversial aspect of NextGen NYCHA is a plan in which underutilized land within NYCHA developments will either be leased for housing development, with buildings half market-rate and half affordable. Opponents say that the plan too-closely resembles former Mayor Bloomberg's attempt at infill on NYCHA land, which de Blasio vehemently opposed, and a step on the path to privatizing what is now public housing.
NYCHA CEO and Chair Shola Olatoye told City Limits in May 2015 that building affordable housing on NYCHA land is a way to reconnect public developments to surrounding neighborhoods.
"Residents want improved playgrounds, basketball courts, things we do not receive funding for. One way I can deliver on that is to maximize revenue from development," she said. "If people want their development to improve, they have to be flexible about what kinds of changes they're willing to accept."
De Blasio said at the program's launch that it would result in the construction of 10,000 new affordable units on NYCHA land. If the plan succeeds, it will eliminate NYCHA's crippling operating deficit. The city's argument is that this strategy will add affordable housing, new revenue, and community improvements.
The first NextGen project in East Harlem led to the opening last fall of the East Harlem Center for Living and Learning, which occupies a parking lot once owned by NYCHA. The project includes an eleven-story residential building containing 100% subsidized units. The New York Times reported in January that the development serves a "mix of working and disabled tenants, many from the neighborhood." The building connects to the new Dream Charter School, which serves almost 500 students.
While the Times reported "overwhelmingly positive reactions" to the Center, it remains to be seen whether similar "infill" initiatives will be welcomed by other NYCHA residents wary of privatization and gentrification - or if the program will even produce enough revenue to bolster the floundering agency. Olatoye told the City Council on Monday that while its financial situation is improving, NYCHA is still tens of millions of dollars in the red, but is hoping that programs like NextGen will ease the burden.
"With NextGen, we can reduce NYCHA's deficit by a total of more than $1 billion over the next five years," Olatoye told the Council.
Space to use
The East Harlem housing shortage is exacerbated by vacancy of both city- and privately-owned properties, according to Comptroller Scott Stringer and area organizations.
Stringer and HPD Commissioner Vicki Been traded barbs in February over a comptroller report indicating that, as of September of 2015, HPD owned 1,131 uninhabited lots throughout Manhattan, and chiding the city agency for what Stringer called a lax pace of transference for properties that should be developed into affordable housing. Stringer held a press conference for the report's release outside of an HPD-owned vacant property on East 123 Street, one of over 50 vacant properties the comptroller's office found in East Harlem.
"This vacant lot has been owned by the city since the seventies, collecting garbage, attracting rats, and hurting the proud community of El Barrio," Stringer said at the press conference. "We're in an affordability crisis the likes of which we've never seen – yet our most valuable resource, vacant space, is readily available all across this city."
Stringer was joined at the conference by members of Picture the Homeless, an advocacy organization based on East 126 Street that has long campaigned for turning vacant lots into affordable housing sites.
The organization released its own report in 2012, for which teams of homeless and formerly homeless PTH members and other volunteers scoured Manhattan neighborhoods in search of vacant buildings and lots. They identified 143 empty lots and buildings in East Harlem, and highlighted a block of East 116th Street between Lexington and Park as an example of the problem: nearly half the buildings were empty or boarded up. About 95 percent of all vacant properties surveyed were privately owned.
Commissioner Been's rebuttal, which was included in Stringer's report, emphatically denied the comptroller's findings.
"Your assertion that HPD allows vacant City-owned properties to languish in the face of the affordable housing crisis is simply wrong," Been wrote.
Been went on to clarify that residential development is only feasible on about 670 of the 1,100 properties; the rest are unsuitable due to location in dangerous areas like flood zones, or lack adequate access to public transportation or sewage infrastructure. Of those 670 viable sites, Been wrote, "roughly 400 have been earmarked for developer designation within the next two years."
The remaining sites will either be developed for residential uses within the duration of de Blasio's 10-year housing plan or turned over to the Department of Parks and Recreation for use as community gardens.
An acute homelessness problem
For those who have managed to hold on to housing, the rent - as in the rest of New York City - is going up. East Harlem's median gross rent increased by 53 percent between 2000 and 2013, according to the American Community Survey (ACS) 5-Year Estimates for 2013.
The data combine to paint a bleak picture for low-income residents of El Barrio. As of 2013, over a quarter of East Harlem households were considered in "severe need" by HPD, based on the number of people who are rent burdened, entering homeless shelters, or living in overcrowded apartments.
This grim combination appears to hit East Harlem residents particularly hard, resulting in the worst case scenario for a number of households. According to data from the Association for Neighborhood and Housing Development (ANHD), 306 families entering shelters in 2014 listed East Harlem as their last address - the highest rate of entry to family shelters from any neighborhood in Manhattan.
Jazmin Berges, 32, has been living in Marcus Garvey Park for the past three years. Berges was born and raised in the Bronx, but moved to East Harlem with a partner. When that relationship ended, she was unable to find housing that she could afford.
"Rent is going sky-high in the neighborhood," Berges told Gotham Gazette. "The community is being restructured, small businesses are closing, and people who lose their jobs can't afford their rents."
Berges, who works with Picture the Homeless, believes that rent regulation doesn't go far enough, and emphasized that a community land trust - a not-for-profit entity that acquires and holds property to preserve accessible housing - is the only way to keep permanent affordable housing in East Harlem. "Even low-income housing isn't low-income housing," Berges said.
New land use rules come to East Harlem
It is unsurprising that East Harlem was included on the list of 15 neighborhoods to be rezoned under Mayor de Blasio's affordable housing plan. The plan targets lower-density and low-income areas where demand is or is soon expected to be strong enough to support new development. Over several years, beginning this year with East New York, in Brooklyn, the administration plans to rezone the 15 neighborhoods to allow for more density and neighborhood improvement through more green space, school seats, and transportation infrastructure. The big centerpiece, of course, is housing.
Essentially, the city's plans mandate that any area rezoned to allow greater housing density - neighborhood or individual plot of land - will require significant numbers of affordable units as part of market-rate development.
The New York City Council overwhelmingly voted in favor of the two major components of the plan that the neighborhood rezonings rely on - Mandatory Inclusionary Housing (MIH) and Zoning for Quality and Affordability (ZQA) - on March 22. The passage was a considerable victory for the administration after months of pushback and protest from community boards and advocacy groups, and negotiations with the City Council to modify the plan to include more affordability and a few other tweaks.
According to the plan, which was changed to include more units affordable to lower-income households, MIH requires that developers who build residential units in rezoned areas set aside 20-30% of new units as permanently affordable for households making 40-80% of the Area Median Income, or AMI. Under MIH, there are a variety of options that will be chosen by City Council members, developers, and the mayoral administration. ZQA, which stirred less broad controversy than MIH, focused on changing parking requirements and height limits to encourage more senior affordable housing, improve retail space design, and streetscape planning.
Mayor de Blasio said in a statement after the vote, "New York City is now one step closer to being a city where everyone can work and live," and thanked Speaker Mark-Viverito for her efforts in passing the "landmark legislation."
City Planning Commissioner Carl Weisbrod said at a hearing on MIH in February that the program "will bring new affordable housing to neighborhoods where it otherwise wouldn't be built, and act as a cushion against potential gentrification of communities caused by broad demographic trends we are seeing in so many neighborhoods, simultaneously freeing up city funds to provide even more deeply affordable housing at lower rents where economic need is great."
Some opponents of MIH see a flaw in that what the plan defines as "affordable" does not include the lowest-income residents in what are now largely low-income neighborhoods.
Affordability levels are determined by the area median income of the five boroughs, plus Putnam, Westchester, and Rockland Counties, which comes out to about $77,700 for a three-person household. The AMI for East Harlem is $33,595, or less than half (43%) of the figure that serves as a baseline for affordability calculations.
MIH mandates that a percentage of affordable units built in rezoned neighborhoods must be affordable to households of certain earnings - for example, one option sets aside 20% of units for those earning an average of 40% of AMI, or about $31,00 for a family of three. This is the lowest income band option in MIH, and one that the Council insisted on adding to the de Blasio administration's initial proposal. Households in East Harlem's poorest zip code, 10035, earn an AMI of just $24,000, which critics say puts new affordable units out of reach.
The mayor, the speaker, and HPD Commissioner Vicki Been have stated often that a combination of HPD subsidies and other housing programs will address the gap of affordability for the lowest-income New Yorkers. This is where the neighborhood rezonings also factor in. Mark-Viverito has said repeatedly that she plans to get to 50-50 market rate-affordable for new development in her district.
Community Board 11, which represents East Harlem joined 43 other community boards across the city in voting against de Blasio's proposal on November 23 of last year.
"But in the end," the mayor said at a press conference in November, "the community boards aren't the final decision makers."
READ ON - PAGE 2 of 2 - How East Harlemites are attempting to shape the future of their neighborhood and more
Tags: New York City • Affordable Housing • Bill de Blasio • City Budget • George Alvarez • Homelessness • Melissa Mark-Viverito • Mandatory Inclusionary Housing • East Harlem • City Council Speaker • New York City Housing Authority • NextGeneration NYCHA • HPD • ZQA • MIH • NYCHA • Department of Housing Preservation and Development • The Spirit of East Harlem • Jefferson Houses | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,035 |
using UnityEngine;
public class ShopController : Singleton<ShopController>
{
[SerializeField]
private DataBindContext m_DataBindContext;
private ShopProduct[] m_Products;
private void Awake()
{
m_Products = Resources.LoadAll<ShopProduct>("Data/Products");
}
private void Start()
{
var consumables = new ObservableList("consumables");
var upgrades = new ObservableList("upgrades");
for (var i = 0; i < m_Products.Length; i++) {
var product = m_Products[i];
switch (product.category) {
case ShopProduct.Category.Consumable:
consumables.Add(product);
break;
case ShopProduct.Category.Upgrade:
upgrades.Add(product);
break;
}
}
m_DataBindContext["consumables"] = consumables;
m_DataBindContext["upgrades"] = upgrades;
}
private new void OnEnable()
{
base.OnEnable();
m_DataBindContext.BindAll();
}
public bool CanBuy(ShopProduct product)
{
return GameData.instance.coins >= product.price;
}
public void Buy(ShopProduct product)
{
if (!CanBuy(product)) {
return;
}
GameData.instance.coins -= product.price;
m_DataBindContext["coins"] = GameData.instance.coins;
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,287 |
import requests
import json
import csv
import sys
def parse_dataset_metadata(dataset):
if 'rights' in dataset.keys():
rights = dataset['rights'].encode('utf-8').strip()
rights = rights.replace("\n", "")
else:
rights = ''
return [dataset['key'].encode('utf-8'), dataset['publishingOrganizationKey'].encode('utf-8'), rights]
def get_gbif_datasets(limit, offset):
params = {'limit': limit, 'offset': offset}
r = requests.get('http://api.gbif.org/v1/dataset/', params=params)
request_result = r.json()['results']
return request_result
def getOccurrences(key):
params = {'datasetKey': key}
r = requests.get('http://api.gbif.org/v1/occurrence/count', params=params)
try:
count = r.json()
except:
sys.stderr.write('could not parse json for number of occurrences for key {0}\n'.format(key))
sys.exit(-1)
return count
all_datasets = []
more_results_to_find = True
offset = 0
limit = 20
print 'key,publishingOrganizationKey,numberOfOccurrences,rights'
csvwriter = csv.writer(sys.stdout, lineterminator='\n')
while more_results_to_find:
sys.stderr.write('LOG: calling GBIF\n')
datasets = get_gbif_datasets(limit, offset)
all_datasets += datasets
offset += 20
if len(datasets) == 0:
more_results_to_find = False
for dataset in all_datasets:
dataset_list = parse_dataset_metadata(dataset)
key = dataset_list[0]
count = getOccurrences(key)
dataset_list.insert(2, count)
csvwriter.writerow(dataset_list)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,934 |
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,375 |
Aloysius Paul D'Souza (ur. 21 czerwca 1941 w Agrar) – indyjski duchowny rzymskokatolicki, w latach 1996–2018 biskup Mangalore. Wyświęcony na biskupa pomocniczego Mangalore ze stolicą tytularną Dura. Ordynariuszem diecezji został pół roku później - 8 listopada 1996 roku.
Bibliografia
Indyjscy biskupi katoliccy
Urodzeni w 1941 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 782 |
Italian Rubber Pull On Bell Boots - Equus Now!
Available in Gum or black.
These boots are soft enough to pull on without much of a struggle. They last longer than the velcro kind if you have a destructive horse. They're a bit expensive but it's cheaper to get a pair of these and have them last than buy three pairs of the velcro.
I've had multiple pairs of these bell boots and have found that although I like the gum color the most, the black hold up the best and have made their way into my horses everyday lifestyle. Wouldn't switch to another brand. Highly recommend! | {
"redpajama_set_name": "RedPajamaC4"
} | 4,568 |
.. _choosing-solvers:
----------------
Choosing solvers
----------------
Solvers and solver-specific parameters are specified by ``AbstractMathProgSolver`` objects, which are provided by particular solver packages. For example, the ``Clp`` package exports a ``ClpSolver`` object, which can be passed to ``linprog`` as follows::
using Clp
linprog([-1,0],[2 1],'<',1.5, ClpSolver())
Options are passed as keyword arguments, for example, ``ClpSolver(LogLevel=1)``. See the `Clp <https://github.com/mlubin/Clp.jl>`_, `Cbc <https://github.com/mlubin/Cbc.jl>`_, `GLPKMathProgInterface <https://github.com/JuliaOpt/GLPKMathProgInterface.jl>`_, and `Gurobi <https://github.com/JuliaOpt/Gurobi.jl>`_ packages for more information.
If no solver is specified, a default is chosen. See ``src/defaultsolvers.jl`` for the list of default solvers.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,181 |
\section{Introduction}
The so-called transplanckian question is concerned with low energy
phenomena whose calculation appears to require the validity of standard
quantum field theory (QFT) at energies beyond the Planck scale. The issue
first arose in the context of black holes: the derivation of Hawking
radiation is based on the assumption that standard QFT is valid even at
scales beyond the Planck scale. For example, the typical low-energy
Hawking photons that an observer might detect far from the horizon are
implied to have possessed proper frequencies that were much larger than
the Planck frequency close to the event horizon, even at distances from
the horizon that are farther than a Planck length. This led to the
question if Planck scale effects could influence or even invalidate the
prediction of Hawking radiation. Numerous studies have investigated the
issue and the current consensus is that Hawking radiation is largely
robust against modifying QFT in the ultraviolet (UV). This is plausible
since general thermodynamic considerations already constrain key
properties of Hawking radiation. See, e.g.,
\cite{brout-review-etc,Unruh2}.
More recently, the transplanckian question arose in the context of
inflationary cosmology: according to most inflationary models, space-time
inflated to the extent that fluctuations which are presently of
cosmological size started out with wavelengths that were shorter than the
Planck length. The derivation of the inflationary perturbation spectrum
therefore assumes the validity of standard QFT beyond the Planck scale.
Unlike in the case of black holes, no known thermodynamic reasons
constrain the properties of the inflationary perturbation spectrum so as
to make it robust against the influence of physics at the Planck scale. It
is, therefore, very actively being investigated if future precision
measurements of the cosmic microwave background (CMB) intensity and
polarization spectra could in this way offer an experimental window to
Planck scale phenomena. See e.g. \cite{infl-etc}.
It is generally expected that the imprint of Planck scale physics on the
CMB is suppressed by a factor $\sigma^n$ where $\sigma$ is defined as the
ratio of the UV and IR scale. In inflation, this ratio is $\sigma \approx
10^{-5}$ since modes evolve nontrivially only from the Planck scale to the
Hubble scale, $L_\text{Hubble}\approx 10^5 ~L_\text{Planck}$, after which
their dynamics freezes until much later when they reenter the horizon to
seed structure formation. We note that if the UV scale is the string
scale, $\sigma$ could be as large as $\sigma\approx 10^{-3}$. Regarding
the value of the power, $n$, in $\sigma^n$, no consensus has been reached.
It is generally expected however, that the value of $n$ decides whether
the imprint of Planck scale physics in the CMB could ever become
measurable.
Concrete studies in this field often model the influence of Planck scale
physics on QFT through dispersion relations that become nonlinear at high
energies. This approach is motivated by the fact that the natural
ultraviolet cutoff in condensed matter systems characteristically affects
the dispersion relations there. See, e.g., \cite{Unruh2}. It has been
shown that while some ultraviolet-modified dispersion relations would
affect the inflationary predictions for the CMB to the extent that effects
might become measurable, other modified dispersion relations would have a
negligible effect on the CMB. It is so far not fully understood which
properties of Planck scale modifications to the dispersion relation decide
whether or not an observable effect is induced. In order to clarify if and
how an imprint of Planck scale effects in the CMB are suppressed by
$\sigma$ it would be most interesting, therefore, to find and study the
operator which maps arbitrary ultraviolet-modified dispersion relations
directly into the correspondingly modified CMB perturbation spectra.
Here, we will investigate the simpler transplanckian question for the
Casimir force. As is well-known, the Casimir force arises due to quantum
fluctuations of the electromagnetic field and occurs between neutral
conducting objects. Similar to Hawking radiation and inflationary
fluctuations, the Casimir force can be seen as a vacuum effect which
involves modes of arbitrarily short wave lengths. In fact, naively it
appears that modes contribute the more the shorter their wave length is.
This suggests that, in principle, the predicted Casimir force could be
influenced by Planck scale physics.
The Casimir effect is simple enough so that we will be able to completely
answer its transplanckian question when modelling Planck scale physics
through ultraviolet-modified dispersion relations. Namely, we will find
the explicit operator which maps generic ultraviolet-modified dispersion
relations into the corresponding Casimir force functions. The properties
of this operator reveal that and how ultraviolet-modified dispersion
relations can strongly affect the Casimir force even in the `infrared'
i.e. at practically measurable distances. Interestingly, the extreme ratio
$\sigma\approx 10^{-28}$ between the effective UV and IR scales in the
Casimir effect does not suppress the possible strength of Planck scale
effects in the Casimir force at macroscopic distances. We find that,
instead, the extreme value of $\sigma$ implies that UV-modified dispersion
relations that lead to a large IR effect merely need to be extremely
fine-tuned, which suppresses the a priori likelihood that such a
dispersion relation should arise from an underlying theory of quantum
gravity. This is of interest because if the situation in inflation is
analogous, the imprint of Planck space physics in the CMB may not be
suppressed in strength by any power $\sigma^n$ of $\sigma$. Instead, the
$\sigma$ of inflation, $\sigma\approx 10^{-5}$ or $\sigma\approx10^{-3}$,
may determine the amount of fine-tuning required to achieve an imprint of
order one. Thus, $\sigma$ would be related to the a priori likelihood for
an observable imprint to arise from an underlying theory of quantum
gravity. In inflation, this likelihood would not be extremely small since
the UV and IR scales in inflation are not extremely separated.
\section{The Casimir force and ultraviolet-modified dispersion relations}
The Casimir effect arises when reflecting surfaces pose boundary
conditions on the modes of the electromagnetic field. For example, two
perfectly reflecting parallel plates impose boundary conditions such that
the set of electromagnetic modes in between them is discretized. The
spacing of the modes, and therefore the vacuum energy that each mode
contributes, depends on the distance between the plates. This
distance-dependence of the vacuum energy leads to the Casimir force
between the plates. In general, the force is a function of both the
distance and the shape of the reflecting surfaces, and the force can be
both attractive or repulsive.
The Casimir effect was first predicted, by Casimir, in 1948, see
\cite{Casimir:1948dh}. In the meanwhile, the Casimir force has been
calculated for several types of geometries and in various dimensions.
Also, effects of imperfect conductors, rough surfaces and finite
temperatures have been considered, see \cite{Balian:2002}. In addition,
detailed calculations have been carried out to account for higher order
corrections due to virtual electrons and their interaction with the
boundaries \cite{Aghababaie:2003iw}. For recent reviews see
\cite{bordag-etal} and for precision measurements of the effect see e.g.
\cite{Lamoreaux:1999cu-etal}.
For our purposes, the essential features of the Casimir effect are
captured already when working with a massless real scalar field between
two perfectly conducting parallel plates. For simplicity, we will consider
the simple case of just one space dimension, in which case the reflecting
plates are mere points. We place these points at $x=0$ and $x=L$, i.e., we
impose the boundary conditions $\hat{\phi}(0,t)=0=\hat{\phi}(L,t)$ for all
$t$. In order to fulfill these boundary conditions we expand the quantum
field between the plates using the Fourier sine series:
\begin{equation}
\hat{\phi}(x,t) = \sum_{n=1}^{\infty} \hat{\phi}_n(t) \sin(k_n x), ~~~~~~~
k_n= \frac{n\pi}{L}
\end{equation}
We are using units such that $\hbar=c=1$. Recall that in a Fourier sine
series all $n$ and therefore all wave numbers $k_n$ are positive. The
reason is that the sine functions form a complete eigenbasis of the square
of the momentum operator, $\hat{p}^2=-d^2/dx^2$, all of whose eigenvalues
are of course positive. (Recall that the momentum operator of a particle
in a box is not self-adjoint and not diagonalizable, see e.g.
\cite{ak-beethoven}). The usual ansatz
\begin{equation}
\hat{\phi}_n=\frac{1}{\sqrt{\omega(k_n)L}}\left(e^{i\omega(k_n)t}a^\dagger_n+
e^{-i\omega(k_n)t}a_n\right)
\end{equation}
with $[a_n,a_m^\dagger]=\delta_{n,m}$ diagonalizes the Hamiltonian:
\begin{equation}
\hat{H}=\sum_{n=1}^\infty
\omega(k_n)\left(a^\dagger_na_n+\frac{1}{2}\right)
\end{equation}
Thus, with the usual linear dispersion relation
\begin{equation}
\omega(k)=k,
\end{equation}
the vacuum energy between plates of distance $L$ is divergent:
\begin{eqnarray}
\label{eq:infSum} E_{in}(L) & = &
\frac{1}{2}\sum_{n=0}^{\infty}\omega(k_{n}) \\
& = & \frac{\pi}{2L}\sum_{n=0}^{\infty} n ~~~=~ \infty
\end{eqnarray}
We notice that modes appear to contribute the more the shorter their
wavelength, i.e. the larger $k$ and $n$ are. One proceeds by regularizing
the divergence and by then calculating the \it change \rm in the
regularized total energy (of a large region that contains the plates) when
varying $L$. As is well-known, the resulting expression for the Casimir
force remains finite after the regularization is removed, and reads:
\begin{equation}
\mathcal{F}(L)=-\frac{\pi}{24L^{2}}\,
\end{equation}
It has been shown that this result does not depend on the choice of
regularization method. Our aim now is to re-calculate the Casimir force
within standard quantum field theory while modelling the onset of Planck
scale phenomena at high energies through general nonlinear modifications
to the dispersion relation. The goal is to calculate the operator which
maps arbitrary modified dispersion relations $\omega(k)$ into the
resulting Casimir force functions $\mathcal{F}(L)$. To this end, let us
begin by writing generalized dispersion relations in the form:
\begin{equation}
\omega(k)=k_{c}f\left(\frac{k}{k_{c}}\right)
\end{equation}
Here, $k_{c}>0$ is a constant with the units of momentum, say the Planck
momentum so that its inverse is the Planck length: $L_c=k_c^{-1}$. The
function $f$ encodes unknown Planck scale physics and for now we will make
only these minimal assumptions:
\begin{itemize}
\label{ma}
\item $f(0)=0$, and $f(x)\approx x$ if $x\ll 1$
~~~(regular dispersion at low energies)
\item $f(x)\geq 0$ when $x
\ge0$ ~~~~(stability: each mode carries positive energy)
\end{itemize}
We will use the term dispersion relation for both $\omega(k)$ and $f(x)$.
\label{bedi}
\section{Exponential regularization}
For generically modified dispersion relations the vacuum energy
(\ref{eq:infSum}) must be assumed to be divergent and therefore in need of
regularization. Let us therefore regularize (\ref{eq:infSum}) by
introducing an exponential regularization function, parametrized by
$\alpha>0$, i.e. we define the regularized vacuum energy between the
plates as:
\begin{equation}
\label{eq:EinReg} E_{in}^{reg}(L)=\frac{1}{2}\sum_{n=0}^{\infty}k_{c}
\,f\left(\frac{n\pi}{k_{c}L}\right)\exp{\left[-\alpha
k_{c}\,f\left(\frac{n\pi}{k_{c}L}\right)\right]}\,
\end{equation}
In order to calculate the regularized vacuum energy density outside the
plates we notice that the right and left outside regions are half axes and
that the energy density in a half axis can be calculated from
(\ref{eq:EinReg}) by letting $L$ go to infinity:
\begin{equation}
\mathcal{E}^{reg}=\lim_{L\to\infty}\frac{E_{in}^{reg}(L)}{L}\, \label{ove}
\end{equation}
The expression for the vacuum energy density outside the plates,
(\ref{ove}), is conveniently rewritten as a Riemann sum by defining
$\Delta x = \frac{1}{L}$:
\begin{eqnarray}
\label{eq:EoutReg} \mathcal{E}^{reg} &
= & \lim_{\Delta x\to
0}\left\{\frac{1}{2}\sum_{n=0}^{\infty}\Delta
x~k_{c}\,f\left(\frac{n\Delta x\pi}{k_{c}}\right)\exp{\left[-\alpha
k_{c}\,f\left(\frac{n\Delta
x\pi}{k_{c}}\right)\right]} \right\}\nonumber \\
& = &\frac{k_{c}^{2}}{2\pi}\int_{0}^{\infty}dx\,f(x) \exp{\left[-\alpha
k_{c}\,f(x)\right]}\,.
\end{eqnarray}
Notice that we are here implicitly restricting attention to dispersion
relations for which exponential regularization is sufficient to render the
energy densities outside and between the plates finite. This excludes, for
example, the dispersion relation $f(x)=\ln(1+x)$ which would require a
regularization function such as $\exp(-f(x)^2)$. We will later be able to
lift this restriction on the dispersion relations, namely by allowing the
use of arbitrary regularization functions. Indeed, as we will prove in
Sec.\ref{indep}, our results only depend on the dispersion relation and
are independent of the choice of regularization function, as long as the
regularization function does regularize the occurring series and
integrals, obeys certain mild smoothness conditions and recovers the
original divergent series of (\ref{eq:infSum}) in the limit $\alpha\to 0$.
In order to calculate the Casimir force, let us now consider a very large
but finite region, say of length $M$, which contains the two plates. The
total energy in this region is finite and consists of the energy between
the plates, (\ref{eq:EinReg}), plus the energy density outside the plates,
(\ref{eq:EoutReg}), multiplied by the size of the region outside, namely
$M-L$. Note that by choosing $M$ large enough ensures that the energy
density outside the plates does not depend on $L$. Thus, the total energy
in this region is given by $E_{in}^{reg}(L)+(M-L)\mathcal{E}^{reg}$. The
regularized Casimir force is the derivative of this energy with respect to
a change in the distance of the plates:
\begin{equation}
\mathcal{F}_{\alpha}(L)=-\frac{\partial}{\partial L}E_{in}^{reg}+
\mathcal{E}^{reg}\,.
\end{equation}
The total length $M$ of the region under consideration has dropped out, as
it should be. Hence, before removing the regularization (i.e. before
letting $\alpha \rightarrow 0^+$), the Casimir force in the presence of a
nonlinear dispersion relation is given by:
\begin{eqnarray}
\label{eq:GenRel} \lefteqn{\mathcal{F}_{\alpha}(L)=\frac{1}{2}k_{c}\left\{
\sum_{n=0}^{\infty}\frac{1}{L}\left[ \frac{n\pi}{k_{c}L}~ f^{\prime}
\left(\frac{n\pi}{k_{c}L}\right)\exp{\left[-\alpha k_{c}\,f\left(
\frac{n\pi}{k_{c}L}\right)\right]}\times{}
\right.\right.}\nonumber \\
& & {}\left.\left. \times \left(1-\alpha k_{c}\,f
\left(\frac{n\pi}{k_{c}L}\right)\right)\right]+
\frac{k_{c}}{\pi}\int_{0}^{\infty}dx\,f(x)\exp{ \left[-\alpha
k_{c}\,f(x)\right]}\right\}
\end{eqnarray}
Here, $f'$ stands for differentiating $f$ with respect to the variable
$x=\frac{n\pi}{k_cL}$.
\section{Application of the Euler-Maclaurin formula}
It will be convenient to collect the terms that constitute the argument of
the series in a new definition:
\begin{equation}
\label{tfg} \varphi_{\alpha}(t) := \frac{t\pi}{k_{c}L} ~f^{\prime}
\left(\frac{t\pi}{k_{c}L}\right)\exp{\left[-\alpha
k_{c}\,f\left(\frac{t\pi}{k_{c}L}\right)\right]}\left(1-\alpha
k_{c}\,f\left(\frac{t\pi}{k_{c}L}\right)\right)
\end{equation}
Thus, (\ref{eq:GenRel}) becomes:
\begin{equation}
\mathcal{F}_{\alpha}(L)=\frac{k_c}{2L} \sum_{n=0}^{\infty}
\varphi_\alpha(n) ~+~ \frac{k_c^2}{2\pi}\int_0^\infty dx~f(x)~e^{-\alpha
k_c f(x)}\label{cf45}
\end{equation}
We notice that if the first term in (\ref{cf45}) were an integral instead
of a series then the two terms in (\ref{cf45}) would exactly cancel
another:
\begin{eqnarray}
\label{eq:intPrts}\label{3a}
\frac{k_c}{2L}\int_{0}^{\infty}\varphi_{\alpha}(t)\,dt & = &
\frac{k_c}{2L}\frac{k_cL}{\pi}\int_{0}^{\infty}\varphi_{\alpha}(t)
\,\frac{\pi}{k_cL}\,dt \\
& = & \frac{k_c^2}{2\pi}\int_{0}^{\infty}dx \,x\,f^{\prime}(x)e^{-\alpha
k_{c}f(x)}\left(1-\alpha
k_{c}f(x)\right)\\
& = & \left . \frac{k_c^2}{2\pi}~xf(x)e^{-\alpha
k_{c}f(x)}\right|_{0}^{\infty}-\frac{k_c^2}{2\pi}\int_{0}^{\infty}dx
\,f(x)e^{-\alpha k_{c}f(x)}\label{bt}\\ \label{3c} \label{hew} & = &
0-\frac{k_c^2}{2\pi}\int_{0}^{\infty}dx\,f(x)e^{-\alpha k_{c}f(x)}\,.
\end{eqnarray}
In (\ref{bt}), the boundary terms are zero because at $x=0$ the dispersion
relation yields $f(0)=0$ and because for $x\rightarrow \infty$ the
finiteness of (\ref{eq:EoutReg}) implies that its integrand decays faster
than $1/x$.
In order to compute the Casimir force, let us now use the Euler-Maclaurin
sum formula, see e.g. \cite{Ford-etal}, to express the series of
$\varphi_\alpha$ as an integral of $\varphi_\alpha$ plus corrections. As
we just saw, the integral will then cancel in (\ref{cf45}) and the
correction terms will constitute the Casimir force. To this end, recall
that if the $(k+1)$st derivative of a function $\xi$ is continuous, i.e.,
if $\xi\in \mathcal{C}^{k+1}$, then:
\begin{eqnarray}
\label{eq:em1} \sum_{a<n\le b}\xi(n)&=&\int_a^b \xi(t)\,dt + \sum_{r=0}^k
\frac{(-1)^{r+1}B_{r+1}}{(r+1)!}
\left(\xi^{(r)}(b)-\xi^{(r)}(a)\right) +\nonumber \\
& & +\frac{(-1)^k}{(k+1)!} \int_a^b B_{k+1}(t)\xi^{(k+1)}(t)\,dt
\end{eqnarray}
Here, the superscript at $\xi^{(r)}$ denotes the $r$'th derivative of the
function $\xi$, the $B_{s}$ are the Bernoulli numbers and $B_{s}(t)$ is
the $s$'th Bernoulli periodic function, i.e. the periodic extension of the
$s$'th Bernoulli polynomial from the interval $[0,1]$.
We can now choose $\xi=\varphi_{\alpha}$, set $a=0$ and take the limit
$b\to \infty$. Since the vacuum energy density, (\ref{eq:EoutReg}), is
finite it follows that (\ref{3c}) is finite and therefore also (\ref{3a}).
This in turn implies that $\lim_{x\rightarrow \infty} \varphi_{\alpha}(x)=
0$ and $\lim_{x\to\infty}\varphi_{\alpha}^{(n)}(x)=0$ for all $n\geq1$.
Hence, the series involving the Bernoulli numbers simplifies and we obtain
for arbitrary $k \in \mathbb{N}$ this Euler-Maclaurin formula for
$\varphi_\alpha$:
\begin{equation}
\sum_{n=0}^{\infty}\varphi_{\alpha}(n)= \int_{0}^{\infty}
\varphi_{\alpha}(t)\,dt-\sum_{r=0}^{k}
\frac{(-1)^{r+1}B_{r+1}}{(r+1)!}~\varphi_{
\alpha}^{(r)}(0)+\Omega_k[\varphi_{\alpha}]\,
\end{equation}
Here, $\Omega_k[\varphi_{\alpha}]$ represents the remainder integral:
\begin{equation}
\Omega_k[\varphi_{\alpha}] = \frac{(-1)^k}{(k+1)!}\int_0^\infty
B_{k+1}(t)~ \varphi_\alpha^{(k+1)}(t)~dt \label{rema}
\end{equation}
Using $\varphi_{\alpha}(0)=0$ and the fact that, except for $B_{1}$, all
Bernoulli numbers $B_s$ with odd indices $s$ are zero, we obtain:
\begin{equation}
\sum_{n=0}^{\infty}\varphi_{\alpha}(n)= \int_{0}^{
\infty}\varphi_{\alpha}(t)\,dt-\sum_{r=1}^{k} \frac{B_{2r}}{(2r)!}~
\varphi_{\alpha}^{(2r-1)}(0)+\Omega_k[\varphi_{\alpha}]\label{drt6}
\end{equation}
Equation (\ref{drt6}) expresses the series as an integral plus
corrections, as desired. Applied to the expression (\ref{cf45}) for the
regularized Casimir force, $\mathcal{F}_\alpha(L)$, the integrals then
cancel and we obtain for the regularized Casimir force:
\begin{equation}
\label{eq:Falpha}
\mathcal{F}_{\alpha}(L)=-\frac{k_{c}}{2L}\sum_{r=1}^{k}\frac{B_{2r}}{2r!}
~ \varphi_{\alpha}^{(2r-1)}(0)+\frac{k_c}{2L}~
\Omega_k[\varphi_{\alpha}]
\end{equation}
The actual Casimir force, $\mathcal{F}(L)$, is obtained by removing the
regularization:
\begin{equation}
\mathcal{F}(L)=\lim_{\alpha\to 0^+}\left\{
-\frac{k_{c}}{2L}\sum_{r=1}^{k}\frac{B_{2r}}{2r!}
~ \varphi_{\alpha}^{(2r-1)}(0)+\frac{k_c}{2L}~
\Omega_k[\varphi_{\alpha}]\right\}\label{cf45b}
\end{equation}
\section{The Casimir force for polynomial dispersion relations}
In order to further evaluate this expression for the Casimir force let us
restrict attention to dispersion relations that are sufficiently well
behaved so that $\varphi_\alpha(t)$ is $\mathcal{C}^\infty$ with respect
to both $\alpha$ and $t$. The simplest case is that of dispersion
relations which are polynomial:
\begin{equation}
f(x)=\sum_{s=0}^{n}\nu_{s}x^{s} \label{polone}
\end{equation}
We are assuming that $\varphi_\alpha(t) \in \mathcal{C}^\infty$ which here
allows us to take the limit $\alpha \to 0$ in $\varphi_\alpha(t)$ before
differentiating it. From (\ref{tfg}) we then have $\varphi_{0}(t) =
\lim_{\alpha\to 0}\varphi_{\alpha}(t)=x(t) f^{\prime}(x(t))$ where
$x(t)=\frac{t\pi}{k_cL}$ and where $'$ stands for $d/dx$. Thus, iterated
differentiation yields
\begin{equation}
\label{eq:diffPhi} \frac{d^{n}\varphi_{0}(t)}{dt^{n}} = n
\left(\frac{\pi}{k_cL}\right)^n
\,\frac{d^nf(x)}{dx^n}+x\left(\frac{\pi}{k_cL}\right)^{n+1}\,
\frac{d^{n+1}f(x)}{dx^{n+1}}
\end{equation}
and therefore the terms in the series in (\ref{cf45b}) read:
\begin{equation}
\varphi^{(n)}_{0}(t)\vert_{t=0}= n\left(\frac{\pi}{k_cL}\right)^n
\,f^{(n)}(x)\vert_{x=0}\label{dfg}
\end{equation}
We now show that the remainder term $\Omega_k[\varphi_\alpha]$ does not
contribute. Assuming for the moment that the dispersion relation is
polynomial, $\varphi_{\alpha}(t)$ is a polynomial times the exponential
regularization function $e^{-\alpha k_c f}$ which tends to $1$ as
$\alpha\to 0$. Therefore, after sufficiently many differentiations, i.e.,
when choosing $k$ large enough, $\varphi^{(k+1)}_{\alpha}(t)\to 0$ as
$\alpha\to 0$ for all fixed $t$. In order to evaluate
$\Omega_k[\varphi_\alpha]$, let us now split (\ref{rema}) into two
integrals: $\int_0^\infty=\int_0^b+\int_b^\infty$. For all finite $b>0$
the first integral commutes with the limit $\alpha \to 0$ to yield for
large enough $k$:
\begin{equation}
\lim_{\alpha\to 0}\int_0^b
B_{k+1}(t)~\varphi_\alpha^{(k+1)}(t)~dt=\int_0^b\lim_{\alpha\to 0}
B_{k+1}(t)~\varphi_\alpha^{(k+1)}(t)~dt=0
\end{equation}
Further, we notice that, since $f$ is polynomial and the exponential
regularization function is positive, $\varphi_\alpha(t)$ does not change
sign for all $t>b$ if $b$ is chosen sufficiently large. Since the periodic
Bernoulli functions are bounded from above by their Bernoulli numbers we
therefore obtain:
\begin{eqnarray}
\left\vert\int_b^\infty B_{k+1}(t)~\varphi_\alpha^{(k+1)}(t)~dt\right\vert
& \le & \left\vert B_{k+1}\right\vert~
\left\vert\int_b^\infty \varphi_\alpha^{(k+1)}(t)~dt\right\vert\nonumber\\
& \le & \left\vert
B_{k+1}\right\vert~\left\vert\varphi_\alpha^{(k)}(t)
\vert_b^\infty\right\vert\nonumber\\
& = & \left\vert
B_{k+1}\right\vert~\left\vert\varphi_\alpha^{(k)}(b)
\right\vert\nonumber\\
& \to & 0 ~~~\mbox{as}~~\alpha \to 0
\end{eqnarray}
Thus, when choosing $k$ large enough, the remainder term disappears so
that, using (\ref{dfg}), we obtain for the Casimir force for arbitrary
polynomial dispersion relations:
\begin{equation} \label{eq:force}
\mathcal{F}(L)=-\frac{k_{c}}{2L}\sum_{r=1}^{k}
\frac{(2r-1)B_{2r}}{2r!}\,f^{(2r-1)}(0)\left(\frac{\pi}{
k_{c}L}\right)^{2r-1}
\end{equation}
Further, since $f^{(s)}(0)=s!\,\nu_{s}$, we obtain:
\begin{equation}
\label{eq:forcePoly} \mathcal{F}(L) = -\frac{k_{c}}{2L}\sum_{r=1}^k
\frac{(2r-1)B_{2r}}{2r}~\nu_{2r-1} \left(\frac{\pi}{k_{c}L}\right)^{2r-1}
\end{equation}
We notice that, interestingly, the even powers in a nonlinear dispersion
relation, i.e. the coefficients $\nu_{2r}$, do not contribute to the
Casimir force.
\smallskip\newline
As a consistency check, let us now choose the usual linear dispersion
relation $f(x)=x$. Since $B_{2}=\frac{1}{6}$, we obtain
\begin{equation}
\mathcal{F}(L)=-\left(\frac{k_{c}}{2L}\right)
\left(\frac{\pi}{k_{c}L}\right)\frac{1}{2\cdot 6}=-\frac{\pi}{24L^{2}}\,,
\end{equation}
which is the well-known usual result for the Casimir force, as it should
be.
\section{Generic dispersion relations}
Considering our results for the Casimir force with polynomial dispersion
relations, (\ref{eq:force},\ref{eq:forcePoly}) we notice that the addition
of mode energies translates into the addition of the corresponding Casimir
forces: if two dispersion relations are added, $f_{t}(x)=f_1(x)+f_2(x)$,
then the two corresponding Casimir forces are added:
\begin{equation}
\label{eq:additivity} \mathcal{F}_{t}=\mathcal{F}_1+\mathcal{F}_2
\end{equation}
This shows that the operator, $\mathcal{K}$, that we have been looking
for, namely the operator which maps arbitrary dispersion relations into
their corresponding Casimir forces, $\mathcal{K}: f \mapsto \mathcal{F}$,
is a linear operator:
\begin{equation}
\mathcal{K}[f_1+f_2]=\mathcal{K}[f_1]+\mathcal{K}[f_2]
\end{equation}
Because of its linearity, we can straightforwardly extend the action of
$\mathcal{K}$ to arbitrary dispersion relation, $f$, which are given by
power series in $x$:
\begin{equation}
f(x)=\sum_{s=0}^{\infty}\nu_{s}x^{s}
\end{equation}
The radius of convergence of the power series must be infinite since the
dispersion relation needs to be evaluated for all $x$, i.e., $f$ is an
entire function. The linearity of $\mathcal{K}$ yields the corresponding
Casimir force function $\mathcal{F}$ as a power series in $1/L$:
\begin{equation}
\label{foww}
\mathcal{K}[f](L)=\mathcal{F}(L)=-\frac{k_{c}}{2L}\sum_{r=1}^{\infty}
\frac{(2r-1)B_{2r}}{2r}\,\nu_{2r-1}\left(\frac{\pi}{
k_{c}L}\right)^{2r-1}\,
\end{equation}
We need to determine under which conditions the resulting power series for
the Casimir force function is convergent. Interestingly, as we will show
in Sec.\ref{nese}, the convergence, i.e. the well-definedness of the
Casimir force, generally depends on the plate separation $L$. When the
power series possesses a finite radius of convergence, i.e. when there is
a largest allowed value for $1/L$, this means that there is a smallest
allowed value for the length $L$. This is beautifully consistent with the
expectation that dispersion relations that arise from an underlying
quantum gravity theory can imply a finite minimum length scale.
For analyzing the convergence properties of the series (\ref{foww}) the
presence of the Bernoulli numbers is somewhat cumbersome. It will be
useful, therefore, to use the connection between the Bernoulli numbers and
the Riemann zeta function, see \cite{Havil}:
\begin{equation}
B_{n}=(-1)^{n+1}n\,\zeta(1-n)
\end{equation}
Thus:
\begin{equation}
\label{mres} \mathcal{F}(L)=\frac{k_{c}}{2L}\sum_{r=1}^{\infty}
(2r-1)\zeta(1-2r)~\nu_{2r-1}\left(\frac{ \pi}{k_{c}L}\right)^{2r-1}\,.
\end{equation}
We can now use the fact that, see \cite{Hardy99}:
\begin{equation}
\zeta(1-s)=\frac{2}{(2\pi)^{s}}\cos\left(\frac{1}{2}\pi
s\right)\Gamma(s)\zeta(s)
\end{equation}
In our case, since $s$ is always an integer, the Euler gamma function
reduces to a factorial, and the cosine is $\pm 1$. Thus:
\begin{equation}
\label{eq:forceFactorial}
\mathcal{F}(L)=\frac{k_{c}}{L}\sum_{r=1}^{\infty}
\frac{(-1)^{r}}{(2\pi)^{2r}}~(2r-1)~(2r-1)!~\zeta(2r)~
\nu_{2r-1}\left(\frac{\pi}{k_{c}L} \right)^{2r-1}
\end{equation}
Having replaced the Bernoulli numbers by the Riemann zeta function is
advantageous because obviously $\zeta(r)\to1$ very quickly as
$r\to\infty$. For example, for $r=6$, the difference is already at the one
percent level. This means that for the purpose of analyzing the
convergence properties of the power series we will be able to use that the
Riemann zeta function for the arguments that occur is close to $1$ and
essentially constant.
\section{Example with minimum length}
\label{nese} Ultraviolet-modified nonlinear dispersion relations which
approach the usual linear dispersion relation for small momenta are given,
for example, by:
\begin{equation}
\label{eq:dispExpSinh} f(x)=\exp(x)-1 \qquad\textrm{and}\qquad
f(x)=\sinh(x)
\end{equation}
The odd coefficients, $\nu_{2r-1}=1/(2r-1)!$ are the same for both the
exponential and the $sinh$ dispersion relation, i.e. the two functions
differ only by their even part. But we know from (\ref{eq:forceFactorial})
that the even components of the dispersion relations do not affect the
Casimir force. The two dispersion relations therefore happen to lead to
the same Casimir force. It is plotted with the usual Casimir force in
Fig.\ref{fig:exponential}.
\begin{figure}[!ht]
\begin{center}
\input{plot1}
\caption{The Casimir force for the exponential dispersion relation
$\omega(k)=k_c(\exp(k/k_c)-1)$. Note that the Casimir force is defined
only for $L$ larger than the finite minimum length $L_{min}=1/2$ (in units
of $1/k_c$).} \label{fig:exponential}
\end{center}
\end{figure}
\newline
We see that the Casimir force matches the usual Casimir force at large $L$
but is weaker for small $L$. As the plot also shows, the Casimir force is
well defined only for values of $L$ above a certain value $L_c$,
corresponding to a finite radius of convergence of the power series in
$1/L$ for the Casimir force. In order to calculate this minimum length
$L_c$, we notice that all the coefficients $\nu_{2r-1}$ are non-negative,
which implies that (\ref{eq:forceFactorial}) is an alternating series.
Such series converge if and only if their coefficients converge to zero.
Hence, for any such dispersion relation, the Casimir force is well defined
for all $L$ which obey:
\begin{equation}
\lim_{r\to\infty}\left[\frac{1}{(2\pi)^{2r}}~
(2r-1)~(2r-1)!~\zeta(2r)~\nu_{2r-1}\left(\frac{\pi}{k_{c}L}
\right)^{2r-1}\right]=0
\end{equation}
In the particular case of the two dispersion relations above, we have
$\nu_{2r-1}=1/(2r-1)!$ and the condition that the Casimir force be
well-defined therefore reads
\begin{equation}
\lim_{r\to\infty}\left[\frac{\zeta(2r)}{(2\pi)}~
(2r-1)\left(\frac{1}{2k_{c}L} \right)^{2r-1}\right]=0
\end{equation}
which means that $\frac{1}{2k_cL} < 1$. The minimum length implied by this
dispersion relation is therefore:
\begin{equation}
L_c = \frac{1}{2k_c} \label{edf}
\end{equation}
This is an example of what we hinted at before, namely that a dispersion
relation can in this way reveal an underlying short-distance cutoff.
\smallskip\newline
For general dispersion relations the coefficients $\nu_{2r-1}$ are not
necessarily all positive, i.e., the Casimir force need not be given by an
alternating series. In this general case the minimum length can be
determined by using the fact that the radius of convergence,
$\mathcal{R}$, of an arbitrary power series $\sum c_{r}x^{r}$ is given by:
\begin{equation}
\label{eq:RConv}
\frac{1}{\mathcal{R}}=\limsup_{r\to\infty}\left|c_{r}\right|^{\frac{1}{r}}\,.
\end{equation}
For example, in the case of the dispersion relations given in
(\ref{eq:dispExpSinh}), where $\nu_{2r-1}=1/(2r-1)!$, the Casimir force
(\ref{eq:forceFactorial}) can be written as a power series
$\mathcal{F}(L)=\sum_{r=1}^\infty c_r \left(\frac{1}{L^2}\right)^r$ in
$1/L^2$ with the coefficients:
\begin{equation}
c_r = \frac{(-1)^r~k_c ~(2r-1)~\zeta(2r)}{2\pi}\left(\frac{1}{2 k_c
}\right)^{2r-1}
\end{equation}
Thus, the minimum length obeys
\begin{eqnarray}
L_c^2 & = & \limsup_{r\to\infty}\left[\frac{k_c
~(2r-1)~\zeta(2r)}{2\pi}\left(\frac{1}{2 k_c}\right)^{2r-1}\right]^{\frac{1}{r}}\\
& = & \lim_{r\to \infty}\left(\frac{1}{2 k_c}\right)^{\frac{2r-1}{r}}\\
& = & \left(\frac{1}{2k_c}\right)^2
\end{eqnarray}
and therefore:
\begin{equation}
L>L_c=\frac{1}{2k_{c}}
\end{equation}
As expected, this agrees with the result (\ref{edf}) which we obtained by
using the alternating series test.
\section{Regularization-function independence}
\label{indep} It is known that the prediction for the Casimir force with
the usual linear dispersion relation does not depend on the choice of
regularization function, as long as the regularization function obeys
certain smoothness conditions and is such that it does in fact regularize
the integrals and series which occur in the calculation.
In our calculation of the Casimir force for nonlinear dispersion relations
we chose an exponential regularization function. We need to prove that our
result (\ref{eq:force}) does not depend on this choice. To see that this
indeed the case, assume that we use an arbitrary regularization function,
$\gamma_{\alpha}(x)$, which is a positive function of $x$ that obeys
$\lim_{\alpha\to 0^+}\gamma_{\alpha}(x)=1$ for all $x$ so that the
original divergent series is recovered when the regulator $\alpha$ goes to
zero. The regularized energy between the plates then reads:
\begin{equation}
\label{eds} \tilde{E}_{in}^{reg}=\frac{1}{2}
\sum_{n=0}^{\infty}k_{c}\,f\left(\frac{n\pi}{k_{c}L}
\right)\,\gamma_{\alpha}\left[f\left(\frac{n\pi}{k_{c}L} \right)\right]
\end{equation}
The regularization function, $\gamma_{\alpha}$, needs to be chosen such
that (\ref{eds}) as well as the energy density are finite, i.e. such that
$\lim_{L\to\infty} \tilde{E}_{in}^{reg}(L)/L<\infty$, which means:
\begin{equation}
\label{eq:finiteCut}
\int_{0}^{\infty}dx\,f(x)\gamma_{\alpha}\left[f(x)\right]<\infty\,
\end{equation}
Finally, in order to be able to use the Euler-Maclaurin sum formula and in
it to interchange $d/dt$ and the limit $\alpha\to 0$, we require the
regularization functions $\gamma_{\alpha}$ to be smooth enough so that
$\gamma_{\alpha}\in\mathcal{C}^{\infty}$ as well as $\varphi_\alpha(t) \in
\mathcal{C}^{\infty}$ as a function of $\alpha$ and $t$. The above
derivation of the Casimir force can then be repeated point by point using
the corresponding new definition of $\varphi_{\alpha}$. In particular, we
apply the Euler-Maclaurin sum formula to the expression:
\begin{eqnarray}
\label{eq:GenGenRel}
\lefteqn{\tilde{\mathcal{F}}_{\alpha}(L)=-\frac{k_{c}}{2L}\left\{
\sum_{n=0}^{\infty}\left[ \frac{n\pi}{k_{c}L}~
f^{\prime}\left(\frac{n\pi}{k_{c}L}\right)
\left\{\gamma_{\alpha}\left[f\left(\frac{n\pi}{k_{c}L}\right)\right]+{}
\right.\right.\right.}\nonumber \\
& & {}\left.\left.\left. +f\left(\frac{n\pi}{k_{c}L}\right)
\gamma_{\alpha}^{\prime} \left[f\left(\frac{n\pi}{k_{c}L}\right)\right]
\right\}\right]+\frac{k_{c}L}{\pi}\int_{0}^{
\infty}f(x)\gamma_{\alpha}[f(x)]\right\}
\end{eqnarray}
An integration by parts as in (\ref{eq:intPrts}) shows that the integrals
cancel. Equation (\ref{eq:finiteCut}) ensures that the boundary term
vanishes, as before in (\ref{bt}). Hence, we again arrive at
(\ref{eq:Falpha}). We now take the limit $\alpha\to 0$ term by term in the
sum, and since $\varphi_{\alpha}$ is in $\mathcal{C}^{\infty}$, we can
again do this before differentiating. Moreover, by the basic assumptions
made on $\gamma_{\alpha}$, we know that $\gamma_{\alpha}^{\prime}(x)\to 0$
as $\alpha\to 0$, so that as before:
\begin{equation}
\lim_{\alpha\to 0}\varphi_{\alpha}(t)=x(t)f^{\prime}(x(t))
\end{equation}
The arguments given in the previous section to show that the remainder
integral disappears for polynomial dispersion relations and that the
coefficients in the Euler-Maclaurin sum are those given in
(\ref{eq:forcePoly}) apply unchanged. This proves that our results for the
Casimir force are independent of the choice of regularization function, as
it should be.
\section{The operator $\mathcal{K}$ which maps dispersion
relations into Casimir force functions}
In preparation for our study of the transplanckian question for the
Casimir effect in Sec.\ref{tps}, let us now calculate explicit
representations of the operator $\mathcal{K}$ which maps dispersion
relations $f$ into Casimir force functions $\mathcal{F}$:
\begin{equation}
\mathcal{K}: ~f(x) \longmapsto \mathcal{F}(L)
\end{equation}
We already saw that $\mathcal{K}$ is linear. Indeed, from
(\ref{eq:forceFactorial}), it can be written as a differential operator:
\begin{equation}
\mathcal{K}=\frac{k_{c}}{2\pi
L}\sum_{r=1}^{\infty}(-1)^{r}(2r-1)\zeta(2r)\left(\frac{1}{2k_{c}L}
\right)^{(2r-1)}\left .\frac{d^{(2r-1)}}{dx^{(2r-1)}}\right|_{x=0}
\label{dop}
\end{equation}
As we already mentioned, the convergence of the zeta function,
$\zeta(2r)\to 1$, is very fast as $r\to\infty$. Since the study of the
transplanckian question involves large orders of magnitudes, we will
therefore henceforth replace $\zeta(2r)$ by $1$. By this approximation we
incur at most a numerical error of a pre-factor of order one which will
not affect our later analysis of the question when the ultraviolet
modifications to the dispersion relations can or cannot affect the Casimir
force in the infrared.
\subsection{Representation of $\mathcal{K}$ as an integral operator}
\label{sec9.1} For the purpose of studying the transplanckian question,
the representation of $\mathcal{K}$ as a differentiation operator in
(\ref{dop}) is not as suitable as a representation as an integral operator
would be. Indeed, as we now show, an equivalent representation of
$\mathcal{K}$ is given by
\begin{equation}
\label{eq:NiceIntOp}
\mathcal{K}[f](L)=\mathcal{F}(L)=\frac{k_{c}^{2}}{\pi}~\text{Im}\int_{0}^{
\infty} f(ix)~(1-2k_cLx)~e^{-2k_cLx}\,dx
\end{equation}
where Im stands for taking the imaginary part. To verify that the action
of this operator on all polynomial $f$ agrees with that given in
(\ref{dop}), let us begin by introducing variables $\Lambda=2k_{c}L$ and
$\tilde{x}=2k_cLx$, to write:
\begin{equation}
\label{eq:IntOpBeg} \mathcal{F}(L)=\frac{k_{c}^{2}}{\pi\Lambda}~\text{Im}
\int_{0}^{\infty} f\left(i\frac{\tilde{x}}{\Lambda}\right)(1-\tilde{x})~
e^{-\tilde{x}}\,d\tilde{x}
\end{equation}
We claim that iterated integrations by parts yield:
\begin{eqnarray} \label{eq:IntOp} \mathcal{F}(L)
&=&\frac{k_{c}^{2}}{\pi\Lambda}~\text{Im}\left\{
\sum_{s=0}^{n}\left.e^{-\tilde{x}}(\tilde{x}+s)\frac{d^{s}}{
d\tilde{x}^{s}}f\left(i\frac{\tilde{x}}{\Lambda}
\right)\right|_{\tilde{x}=0}^{ \infty}\right.
\nonumber \\
& & \qquad \quad \left. - \int_{0}^{\infty}e^{-\tilde{x}}(\tilde{x}+n)
\frac{d^{n+1}}{d\tilde{x}^{n+1}}f \left(i\frac{\tilde{x}}{\Lambda}\right)
\right\}
\end{eqnarray}
Integrating (\ref{eq:IntOpBeg}) by parts once shows that the equation
holds for $n=0$. Assuming now that the formula is valid for $n-1$,
integration by parts of the remaining integral yields:
\begin{eqnarray}
\mathcal{F}(L) &=& \frac{k_{c}^{2}}{\pi\Lambda}~\text{Im}\left\{
\sum_{s=0}^{n-1}\left.e^{-\tilde{x}}(x+s)\frac{d^{s}}{
d\tilde{x}^{s}}f\left(i\frac{\tilde{x}}{
\Lambda}\right)\right|_{\tilde{x}=0}^{ \infty}\right.
\label{wed1} \\
& & \qquad \qquad +
\left.e^{-\tilde{x}}(\tilde{x}+n)\frac{d^{n}}{d\tilde{x}^{n}}f\left(
i\frac{\tilde{x}}{\Lambda}\right)
\right|_{\tilde{x}=0}^{\infty}\label{wed2}\\
& & \qquad \qquad - \left. \int_{0}^{ \infty}
e^{-\tilde{x}}(\tilde{x}+n)\frac{d^{n+1}}{d\tilde{x}^{n+1}}f\left(i
\frac{\tilde{x}}{\Lambda}\right)\right\}
\end{eqnarray}
The boundary term in (\ref{wed2}) becomes the next term in the sum
(\ref{wed1}) and by induction this completes the proof of
(\ref{eq:IntOp}). In (\ref{eq:IntOp}), since $f$ is polynomial, the
integral vanishes if $n$ is chosen large enough. Also, the boundary terms
clearly vanish at the upper limit. Letting $n\to \infty$, we are left
with:
\begin{eqnarray}
\label{eq:SumI} \mathcal{F}(L) & = &
\frac{-k_c^2}{\pi\Lambda}~\text{Im}\sum_{s=0}^\infty
\left.s\frac{d^{s}}{d\tilde{x}^{s}}f
\left(i\frac{\tilde{x}}{\Lambda}\right)\right|_{\tilde{x}=0}\\
& = & \frac{k_c}{2\pi L} \label{la2}
\sum_{r=1}^\infty ~(2r-1)~(-1)^r~\left(\frac{1}{2k_cL}
\right)^{2r-1}\frac{d{\,}^{2r-1}}{dx^{2r-1}} ~f(x)\vert_{x=0}
\end{eqnarray}
which agrees with (\ref{dop}), up to the zeta function which we omitted
since it is close to one. In the step from (\ref{eq:SumI}) to (\ref{la2})
we made use of the fact that the imaginary part selects for only the odd
powers in the series.
\smallskip\newline
As a consistency check, let us apply the integral representation,
(\ref{eq:NiceIntOp}), of $\mathcal{K}$ to the usual linear dispersion
relation $f(x)=x$. Carrying out the integration yields
$\mathcal{F}(L)=-\frac{1}{4\pi L^{2}}$. As expected, this differs from the
usual result only by the omitted $\zeta$ function pre-factor of
$\zeta(2)=\frac{\pi^{2}}{6}$.
\subsection{Relation of $\mathcal{K}$ to the Laplace transform}
The representation of $\mathcal{K}$ as an integral operator came at the
cost of complexifying the analysis by having to integrate the dispersion
relation along the imaginary axis.
Fortunately, it is possible to re-express $\mathcal{K}$ as a real integral
operator, namely as a slightly modified Laplace transform. To this end,
let us use our finding that even powers in the dispersion relations do not
contribute to the Casimir force. This means that, without restricting
generality, we can assume that the dispersion relation is odd, i.e. that
it can be written in the form
\begin{equation}
f(x)=x~g(x^{2})
\end{equation}
for some function $g$. Thus, $f(ix) = i\,x\,g(-x^2)$, and therefore the
integral representation (\ref{eq:NiceIntOp}) of $\mathcal{K}$ now takes
the form:
\begin{equation}
\label{eq:OpIntRe}
\mathcal{K}[f](L)=\mathcal{F}(L)=\frac{k_{c}^{2}}{\pi}\int_{0}^{
\infty}x~g(-x^{2})~(1-2k_cLx)~e^{-2k_cLx}\,dx
\end{equation}
Using the properties of the Laplace transform with respect to
differentiation, we can finally conclude that the operator $\mathcal{K}$
which maps dispersion relations into Casimir force functions can be
written as a modified Laplace transform:
\begin{eqnarray}
\mathcal{K}[f](L) = \mathcal{F}(L) & = & \frac{k_{c}^{2}}{\pi}\nonumber
\left(1+L\frac{d}{d L}\right) \int_0^\infty e^{-2k_cLx}x~g(-x^2)~dx
\\
& = & \frac{k_{c}^{2}}{\pi}
\left(1+L\frac{d}{d L}\right)\mathcal{L}_{\Lambda}[\tilde{f}]\label{cffi}
\end{eqnarray}
In the last line, $\mathcal{L}_{\Lambda}[\tilde{f}]$ stands for the
Laplace transform of $\tilde{f}(x) =x\,g(-x^{2})$ with respect to the
variable $\Lambda = 2k_cL$. Let us test (\ref{cffi}) by applying it to the
linear dispersion relation, where $\tilde{f}(x)=x$. Then,
\begin{eqnarray}
\mathcal{F}(L) & = & \frac{k_c^2}{\pi}~\left(1+L~\frac{d}{d
L}\right)\int_0^\infty
e^{-2k_c L x}x~dx\\
& = & -\frac{1}{4\pi L^2}~,
\end{eqnarray}
which indeed agrees with the expected result as obtained at the end of
Sec.\ref{sec9.1}.
\smallskip\newline
We notice that the representation of $\mathcal{K}$ through (\ref{cffi})
involves the analytic extension of the function $g$ from positive
arguments, where it encodes the dispersion relation through
$f(x)=x\,g(x^2)$, to negative arguments where $g$ is evaluated by the
Laplace transform in (\ref{cffi}).
\smallskip\newline
This observation about $\mathcal{K}$ will be useful for answering the
transplanckian question in Sec.\ref{tps}: clearly, the dispersion relation
$f(x)=x\,g(x^2)$ may be very close to linear, i.e. $g(y)$ may be close to
one for $y>0$, while at the same time the unique analytic extension $g(y)$
for $y<0$ may be far from linear. This already shows that
ultraviolet-modified dispersion relations can easily lead to arbitrarily
pronounced nontrivial Casimir forces even at infrared length scales.
\subsection{The inverse of $\mathcal{K}$}
Let us now calculate the inverse of the operator $\mathcal{K}$ to obtain
the operator which maps odd Casimir force functions (recall that the even
ones do not contribute to the Casimir force) into the corresponding
dispersion relations. To this end, we need to solve for
$\tilde{\mathcal{F}}(L)$:
\begin{equation}
\frac{k_{c}^{2}}{\pi}\left(1+L\frac{d}{dL}\right)
\tilde{\mathcal{F}}(L)=\mathcal{F}(L)\,.
\end{equation}
The Green's function for this differential operator satisfies the
following equation:
\begin{equation}
\frac{k_{c}^{2}}{\pi}\left(1+L\frac{d}{dL}\right)
G_{\mathcal{F}}(L,L')=\delta(L-L')
\end{equation}
Since the $\delta$-function is formally the derivative of the Heavyside
step function $\theta$, an integration on both sides yields
\begin{equation}
\int G_{\mathcal{F}}(L,L')\,dL+L G_{\mathcal{F}}(L,L')-\int
G_{\mathcal{F}}(L,L')\,dL
=\frac{\pi}{k_{c}^{2}}\,\theta(L-L')+\kappa(L')\,,
\end{equation}
where $\kappa(L')$ is some arbitrary function. Hence,
\begin{equation}
G_{\mathcal{F}}(L,L')=\frac{1}{L}\left[\frac{\pi}{
k_{c}^{2}}\,\theta(L-L')+\kappa(L')\right]\,,
\end{equation}
and
\begin{equation}
\label{eq:PreFTilde}
\tilde{\mathcal{F}}(L)=\frac{1}{L}\int_{-\infty}^{\infty}
\left[\frac{\pi}{k_{c}^{2}}\,\theta(L-L')+
\kappa(L')\right]\mathcal{F}(L')\,dL'\,.
\end{equation}
For the boundary condition, we set $\tilde{\mathcal{F}}(L)\to0$ as $
L\to+\infty$, to ensure the correct behavior of $\mathcal{F}$. Hence,
\begin{equation}
\kappa(L')+\frac{\pi}{k_{c}^{2}}=0 \, \Longleftrightarrow \,
\kappa(L')\equiv -\frac{\pi}{k_{c}^{2}}
\end{equation}
Thus, the integral in (\ref{eq:PreFTilde}) is effectively truncated and we
have:
\begin{equation}
\label{eq:FTilde} \tilde{\mathcal{F}}(L)=-\frac{\pi}{k_{c}^{2}L}\int_{L}^{
\infty}\mathcal{F}(L')\,dL'\,.
\end{equation}
Eventually, we also need to invert the Laplace transform through a
Fourier-Mellin integral, to obtain:
\begin{equation}
\label{invfo} x\,g(-x^{2})=-\frac{1}{2i k_c^2L}\int_{\gamma}dL\,e^{x
L}\int_{L}^{\infty}\mathcal{F}(L')\,dL'
\end{equation}
Here, the integration path $\gamma$ is to be chosen parallel to the
imaginary axis and to the right of all singularities of the integrand.
Analytic continuation of $g$ to the positive reals finally yields the
dispersion relation $\mathcal{K}^{-1}[\mathcal{F}](x)=f(x)=xg(x^2)$,
modulo, of course, even components to the dispersion relations. We will
here not go further into the functional analysis of (\ref{invfo}) and the
inverse of $\mathcal{K}$.
\section{The transplanckian question}
\label{tps} Having calculated $\mathcal{K}$, we are now prepared to
address the transplanckian question, namely the question which types of
Planck scale modified dispersion relations would significantly affect the
predictions for the Casimir force at realistic plate separations.
To this end, let us begin by investigating the lowest order corrections to
the dispersion relation, $f$, namely by including a quadratic and a
quartic correction term: $f(x)=x+\nu_2x^2+\nu_3x^3$. The coefficients
$\nu_2,\nu_3$ can be as large as of order one, $\nu_2,\nu_3\approx 1$,
without appreciably affecting the dispersion relation
$\omega(k)=k_c\,f(k/k_c)$ at small momenta $k\ll k_c$. Using our result
(\ref{eq:forceFactorial}) for $\mathcal{K}$ we find the corresponding
Casimir force function:
\begin{equation}
\mathcal{F}(L) = -\frac{\pi}{24 L^2} \,+\nu_3\,\frac{\pi^5}{20\, k_c^2L^4}
\end{equation}
The quadratic correction term $\nu_2 x^2$ is an even component of $f$ and
therefore does not affect the Casimir force. The quartic correction term
does affect the Casimir force, changing the Casimir force from attractive
to repulsive at very short distances, as shown in Fig.~\ref{fig:Poly}.
However, as we can also see in Fig.~\ref{fig:Poly}, the Casimir force
function converges very rapidly towards the usual Casimir force function
for plate separations that are significantly larger than $L_c=k_c^{-1}$.
\begin{figure}[!ht]
\begin{center}
\input{plot2}
\caption{The Casimir force for a lowest order correction to the dispersion
relation: $f(x)=x+x^3$. The plate separation, $L$, is measured in
multiples of the UV scale $\frac{k_c^{-1}}{2}$.} \label{fig:Poly}
\end{center}
\end {figure}
To be precise, we recall that the standard dispersion relation
$f_\text{standard}(x)=x$ implies the standard Casimir force function
$\mathcal{F}_\text{standard}(L) = -\frac{\pi}{24 L^2}$. The relative size
of the correction to the Casimir force depends on the plate separation $L$
and reads:
\begin{equation}
\label{relcorc} \frac{\mathcal{F}_\text{standard}(L)-\mathcal{F}(L)}{
\mathcal{F}_\text{standard}(L)}=\nu_3\frac{6\pi^4}{5L^2k_c^2}
\end{equation}
Let us calculate the orders of magnitude. The dispersion relation
$\omega(k)=k_c\,f(k/k_c)$ is expected to start to appreciably differ from
linearity the latest at the Planck scale, which in $3+1$ dimensional
space-time means that the critical length, $L_c$, obeys $L_c
=k_{c}^{-1}\approx 10^{-35}m$. Actual measurements of the Casimir force
have been performed at about $L_m\approx 10^{-7}m$, see e.g.
\cite{Lamoreaux:1999cu-etal}. Therefore, evaluating the relative
correction of the Casimir force, (\ref{relcorc}), at the measurable scale
$L=L_m$ yields
\begin{equation}
\frac{\mathcal{F}_\text{standard}(L_m)-\mathcal{F}(L_m)}{
\mathcal{F}_\text{standard}(L_m)}=\nu_3\,\frac{6\pi^4}{5}~\sigma^2
\end{equation}
where $\sigma$ denotes the dimensionless ratio of the ultraviolet length
scale $L_c$ and the infrared length scale $L_m$:
\begin{equation}
\sigma=\frac{L_{c}}{L_{m}} \approx 10^{-28}
\end{equation}
Thus, the effect of the lowest order corrections to the dispersion
relation on the Casimir force is extremely small at measurable plate
separations.
Naively, on might expect that higher-order corrections to the dispersion
relations contribute even less to the Casimir force. If true, this would
indicate that the physical processes that happen at these two length
scales respectively are very effectively decoupled from another. In fact,
however, the two scales are not quite as decoupled. Roughly speaking, the
reason is that higher order corrections to the dispersion relations
contribute more rather than less to the Casimir force, as we will now
show.
\subsection{UV-IR coupling with polynomial dispersion relations}
\label{sec:IRUV} \label{shfi} Recall that we here need not be concerned
with the even components of dispersion relations since they do not
contribute to the Casimir force. Let us, therefore, consider higher order
odd polynomial dispersion relations:
\begin{equation}
f(x)=x+\sum_{r=2}^N\nu_{2r-1}\,x^{2r-1}
\end{equation}
The coefficients $\nu_{2r-1}$ can be chosen as large as of order one,
$\nu_{2r-1}\approx 1$, and $f$ will still be modified only in the
ultraviolet. We showed above that the contribution of the lowest order
correction term, $\nu_3x^3$, to the Casimir force at the infrared length
scale $L_m$ is proportional to $\sigma^2$, i.e. that it is completely
negligible. One might expect that higher order terms $\nu_{2r-1}x^{2r-1}$
in the dispersion relation would contribute even less to the Casimir
force. At first sight this expectation appears to be confirmed:
$\mathcal{K}$ maps a dispersion relation term $\sim x^{2n-1}$ into a
Casimir force term $\sim (k_cL)^{-2r}$. At the infrared scale, $L=L_m$,
the latter term reads:
\begin{equation}
\left(\frac{1}{k_cL}\right)^{2r}=\left(\frac{L_c}{L_m}\right)^{2r}=
\sigma^{2r}
\end{equation}
This indeed means that the size of this term decreases exponentially with
increasing $r$. Upon closer inspection, however, we see that,
nevertheless, a higher order term $x^{2r-1}$ in $f$ can give an
arbitrarily large contribution to the Casimir force, in particular if $r$
is very large. The reason is that $\mathcal{K}$ involves a factorial
amplification of higher order terms which eventually overcomes the
exponential suppression that we discussed above. Namely, as
(\ref{eq:forceFactorial}) shows, the precise action of $\mathcal{K}$ on
the correction term $\nu_{2r-1}x^{2r-1}$ reads:
\begin{equation}
\mathcal{K}: ~~\nu_{2r-1}\,x^{2r-1} ~~\longrightarrow~~
\nu_{2r-1}\,\frac{(-1)^r k_c^2}{\pi}\,(2r-1)(2r-1)!\,
\zeta(2r)\left(\frac{1}{2k_cL}\right)^{2r} \label{kappa34}
\end{equation}
Due to the presence of the factorial term $(2r-1)!$, the coefficients of
the Casimir force function grow much faster than those of the dispersion
relation. In particular, for the dispersion relation $f(x)=x+
\nu_{2r-1}x^{2r-1}$ the relative change in the Casimir force at the
infrared scale $L_m$ reads:
\begin{equation}
\frac{\mathcal{F}_\text{standard}(L_m)-\mathcal{F}(L_m)}{
\mathcal{F}_\text{standard}(L_m)}=\nu_{2r-1}
\frac{(-1)^{r-1}(2r-1)\zeta(2r)}{4\pi^2}~(2r-1)!~
\left(\frac{\sigma}{2}\right)^{2r-2}
\end{equation}
It is straightforward to apply Stirling's formula for the factorial, $
n!\approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}$ for $n\gg 1$ in order
to calculate how large $r$ needs to be for the factorial amplification to
overcome the exponential suppression. We find that a correction term
$\nu_{2r-1}x^{2r-1}$ with $\nu_{2r-1}\approx1$ in the dispersion relation
leads to a relative change of order one in the Casimir force at the
infrared scale $L_m$ if $r$ is of the order $\sigma^{-1}$, i.e. if
$r\approx 10^{28}$.
\smallskip\newline
To summarize: We found that $\mathcal{K}$ is a well-defined but unbounded
and therefore discontinuous operator (as are, e.g., the quantum mechanical
position and momentum operators). Namely, a modified dispersion relation
of the form $f(x) = x + \nu_{2r-1}x^{2r-1}$, say with $r\approx10^{28}$
and $\nu_{2r-1}\approx 1$ is virtually indistinguishable from the linear
dispersion relation $f(x)=x$ at all scales up to the Planck scale, but
does lead to a modification of the Casimir force which is very strong (the
relative change is of order 100\%) even at laboratory length scales. Thus,
even though the first order terms contribute extremely little to the
Casimir force, very high order corrections to the dispersion relations can
contribute significantly to the Casimir force - in fact, the more so the
larger $r$ is.
Realistic candidates for Planck scale modified dispersion relation are
given by a series $f(x)=x+\sum_{n=2}^\infty \nu_n x^n$ and such dispersion
relations therefore contain terms $\nu_{2r-1}x^{2r-1}$ for arbitrarily
large $r$. At the same time, the prefactors $\nu_n$ must of course obey
$\nu_n\rightarrow 0$ as $n\rightarrow \infty$ because this is a necessary
condition for the convergence of the series. We conclude that it is this
competition between the decay of the coefficients $\nu_{2r-1}$ and the
increasing Casimir effect of terms $x^{2r-1}$, for $r\rightarrow\infty$,
which decides whether or not a given ultraviolet-modified dispersion
relation does or does not lead to an appreciable effect on the Casimir
force at infrared distances. In practice, to study this competition
directly by using the complicated representation of $\mathcal{K}$ in
(\ref{kappa34}) would be a tedious approach to the transplanckian question
because, for example, the coefficients of the Casimir force acquire
alternating signs. Instead, as we will show in the next section, we will
conveniently be able to study the transplanckian question by making use of
our representation of $\mathcal{K}$ in terms of the Laplace transform.
\subsection{UV-IR coupling with generic dispersion
relations} \label{shsec}
Let us write the dispersion relations again in the form $f(x)=x\,g(x^2)$
so that, e.g., $g\equiv 1$ yields the standard dispersion relation. This
allows us to apply the representation of $\mathcal{K}$ in terms of the
Laplace transform, (\ref{eq:OpIntRe}). We begin by noticing that, since
$x^2$ is positive, the evaluation of the dispersion relation $f$ involves
evaluating $g(y)$ only for positive $y$. Now considering
(\ref{eq:OpIntRe}) we see that, curiously, the calculation of the Casimir
force involves evaluating $g(y)$ only for negative values of $y$.
This is surprising because if $g$ could be any arbitrary function, this
would mean that the dispersion relation, which is determined by the
behavior of $g$ on the positive half-axis, and the Casimir force function,
which is determined by the behavior of $g$ on the negative half axis, were
unrelated. But of course our $g$ are not arbitrary functions but are
polynomials or power series with infinite radius of convergence, i.e. they
are entire functions. Therefore, the behavior of $g$ on the positive half
axis fully determines its behavior also on the negative half axis. The
dispersion relations do determine the corresponding Casimir force.
Of crucial importance for the transplanckian question, however, is the
fact that there are entire functions $g$ which are arbitrarily close to
one for $0<y<1$ and which nevertheless reach arbitrarily large values on
the negative half axis. Such functions do not noticeably affect the
dispersion relation for momenta up to the Planck scale but do arbitrarily
strongly affect the Casimir force. These are the dispersion relations
$f(x)=x\,g(x^2)$ with
\begin{equation}
g(y)=1+h(y),\label{def77}
\end{equation}
where the function $h$ obeys $h(y)\approx 0$ for $y\in(0,1)$ while
exhibiting large $\vert h(y)\vert$ in some range of negative values of
$y$. Let us now analyze which behavior of $h$ on the negative half axis
determines if the Casimir force is affected in the infrared. To this end,
let us use (\ref{eq:OpIntRe}) and (\ref{def77}) to express the correction
in the Casimir force, $\Delta
\mathcal{F}=\mathcal{F}-\mathcal{F}_\text{standard}$, in terms of the
correction $h$ to the dispersion relation:
\begin{equation}
\Delta\mathcal{F}(L)=\frac{k_{c}^{2}}{\pi}\int_{0}^{
\infty}x~h(-x^{2})~(1-2k_cLx)~e^{-2k_cLx}\,dx
\end{equation}
The integral kernel
\begin{equation}
G(x, L)=(1-2k_cLx)~e^{-2k_cLx}
\end{equation}
is positive for $x<(2k_cL)^{-1}$, negative for $x>(2k_cL)^{-1}$ and
rapidly decreases to zero for $x\gg(2k_cL)^{-1}$. (We remark that the the
integral of the kernel over all $x\in[0, \infty)$ is $0$, which expresses
the fact that the Casimir force does not depend on the absolute value of
the energy.) Thus, for a fixed plate separation $L$, what matters most for
the Casimir force is the behavior of $h(y)$ from $y=0$ to about
$y\approx-(k_cL)^{-2}$. As we increase $L$, the interval
$y\in(-(k_cL)^{-2},0)$ on which the integral kernel $G$ is mostly
supported is shrinking, see Fig.\ref{fig:kernel}.
\begin{figure}[!ht]
\begin{center}
\input{plot3}
\caption[The integral kernel]{The integral kernel $(1-xL)e^{-xL}$
for different values of $L$. Note the shift of its zero towards
the origin as $L$ grows.}
\label{fig:kernel}
\end{center}
\end {figure}
Thus, there is a significant effect on the Casimir force at realistically
large plate separations, such as $L=L_m$, if the function $h$ is either of
order one in this small interval close to the origin or it must be
exponentially large (so as to compensate the exponential suppression in
$G$) in some interval to the left of $-(k_cL)^{-2}$. Of course, both are
possible. There are entire functions $h$ which possess either one of these
behaviors on the negative half axis and therefore do affect the Casimir
force in the infrared, while being arbitrarily close to zero for $0<y<1$,
so as to leave the dispersion relation virtually unchanged in the
infrared.
There is even the extreme case of functions, $h$, whose corresponding
dispersion relation $f$ is arbitrarily little affected at \it all \rm
scales while the Casimir force function is arbitrarily much affected at
any scale we wish, say in the infrared. To see this, consider for example
the case where $h$ is a Gaussian which is centred around a low negative
value $y_0<0$ while being so sharply peaked that its tail into the
positive half axis is negligibly small. The function that enters into the
calculation of the Casimir force, $\tilde{f}_1=x\,g(-x^2)$, then features
the low-$x$ spike of the Gaussian, implying by our above consideration
that the Casimir force is affected in the infrared. At the same time, the
dispersion relation itself, $\tilde{f}_2(x)=x\,g(x^2)$, is virtually
unaffected for all $x$.
\section{Conclusions}
We investigated the effect of ultraviolet corrections to the dispersion
relation on the Casimir force. To this end, we calculated the operator
$\mathcal{K}$ which maps generic dispersion relations,
$\omega(k)=k_{c}f\left(k/k_{c}\right)$, into the corresponding Casimir
force functions $\mathcal{F}(L)$. Here, $k_c$ is the Planck momentum, $f$
is a power series in $x=k/k_c$ and $L$ is the plate separation. The
structure of $\mathcal{K}$ showed that the even components of dispersion
relations do not contribute to the Casimir force. This implies, for
example, that the dispersion relations defined through $f(x)=\sinh(x)$ and
$f(x)=\exp(x)-1$ yield identical Casimir force functions.
We also showed that a certain class of UV-modified dispersion relations,
such as $f(x)=\sinh(x)$, lead to Casimir force functions that are well
defined only down to a finite smallest distance between the plates.
Physically, the existence of a finite lower bound for the plate
separation, $L$, is indeed what should be expected if the
ultraviolet-modified dispersion relation arises from an underlying theory
of quantum gravity which possesses a notion of minimum length.
Technically, the phenomenon of a finite minimum $L$ arises because the
Casimir force $\mathcal{F}(L)$ is always a polynomial or power series in
$1/L$, depending on whether the dispersion relation is polynomial or a
power series. Therefore, if $\mathcal{F}(L)$ is a power series then it can
possess a finite radius of convergence, i.e. an upper bound on $1/L$,
which then implies a lower bound on $L$. Of course, a finite radius of
convergence can occur only for power series but not for polynomials.
Interestingly, this means that the existence of a finite lower bound on
$L$ cannot arise from polynomial dispersion relations of any degree. An
important conclusion that we can draw from this is that if a candidate
quantum gravity theory yields a non-polynomial dispersion relation then
working with any finite degree polynomial approximation of this dispersion
relation may be missing crucial qualitative features, such as the
existence of a finite minimum length.
There is a deeper reason for why it is important to apply a nontrivial
dispersion relation in the exact form in which it arises from some
proposed quantum gravity theory. The reason is that $\mathcal{K}$ is an
unbounded and therefore also discontinuous operator, which means that
arbitrarily small changes to the dispersion relation can lead to
arbitrarily large changes to the Casimir force. On the other hand, the
action of $\mathcal{K}$ is of course well-defined, which means that if a
candidate quantum gravity theory implies a particular UV-modified
dispersion relation then $\mathcal{K}$ can be used to precisely predict
the corresponding Casimir force function.
We proceeded by determining which ultraviolet modifications to the
dispersion relation would appreciably affect the Casimir force function at
a large length scale $L_m$. To this end, it was convenient to express
dispersion relations, $f$, in the form $f(x)=x\,g(x^2)$ and $g(y)=1+h(y)$
where $h$ is an entire function (so that $h\equiv 0$ for the usual linear
dispersion relation). Recall that $y$ is the momentum squared, in units of
$k^2_c=L^{-2}_c$, i.e., $y=1$ is the Planck momentum squared. We are
interested in dispersion relations which are essentially unchanged in the
infrared, i.e., which obey $h(y)\approx 0$, up to unmeasurable deviations,
for all $y$ in the interval $(0,1)$. Our analysis of $\mathcal{K}$ through
the Laplace transform then showed that if the corresponding Casimir force
is to be affected at an infrared scale, say $L_m$, then the dispersion
relation must come from a function $h$ which obeys one or both of two
conditions: (a) either $h$ obeys $\vert h(y)\vert =\mathcal{O}(1)$ for $y$
in parts of the interval $(-L^2_c/L^2_m,0)=(-\sigma^2,0)$, or (b) $h$ is
exponentially large in a finite interval of more negative $y$ obeying
$y<-\sigma^2$.
In the case (a), an ultraviolet-modified dispersion relation induces an
infrared modification of the Casimir force if the correction to the
dispersion relation, $h(y)$, is essentially zero in all of $(0,1)$, while
it rises very steeply towards the left to amplitudes of order one within
the extremely short interval $(-\sigma^2,0)$, where we recall that
$\sigma\approx 10^{-28}$. In the case (b), UV/IR coupling arises if $h$ is
again essentially zero in the interval $(0,1)$, while now needing to reach
exponentially large values for a finite stretch of more negative $y$
values, again resulting in the need for $h$ to rise extremely steeply
towards the left. It is easy to give examples of such $h$, such as the
Gaussian $h$ that we discussed. In fact, we can easily write down $h$
which would lead to no appreciable modification of the dispersion at low
energies and yet to arbitrarily large changes to the Casimir force even at
macroscopically large plate separations. Because of their large slope,
however, such functions $h$ are severely fine-tuned and must therefore be
considered unlikely to arise from an underlying quantum gravity theory. We
can conclude, therefore, that the 28 orders of magnitude which separate
the effective UV and IR scales do not suppress UV/IR coupling in strength
but instead in likelihood, namely through the need for extreme fine
tuning.
This is interesting because, in inflation, the separation of the effective
UV and IR scales is only about three to five orders of magnitude: Consider
the operator $\mathcal{K}$ for inflation, namely the operator which maps
arbitrary ultraviolet-modified dispersion relations into the function that
describes the CMB's tensor or scalar fluctuation spectrum. Let us assume
that its properties are analogous to that of the operator $\mathcal{K}$
which we here found for the Casimir effect. This would mean that an
ultraviolet-modified dispersion relation that arises from some underlying
quantum gravity theory can lead to effects on the CMB spectrum which are
not automatically limited in their strength by the separation of scales
$\sigma\approx 10^{-5}$, or indeed by any power of $\sigma$. Instead,
arbitrarily large effects on the CMB must be considered possible, while it
is merely the a priori likelihood of large effects that is suppressed by
the separation of scales. That this is indeed the case can of course only
be confirmed by calculating an explicit expression for the operator
$\mathcal{K}$ for inflation.
\section{Outlook}
The task of finding the operator $\mathcal{K}$ for inflation will be more
difficult than it was to calculate $\mathcal{K}$ for the Casimir effect.
This is mainly because it is highly nontrivial to identify the comoving
modes' initial condition, i.e. their ingoing vacuum state. This problem
needs to be solved because a misidentification of the vacuum could mask
the infrared effects that one is looking for. The reason is that the mode
equations reduce to the mode equations with the usual linear dispersion at
late times, namely at large length scales. Therefore, the mode solutions
at late times live in the usual solution space. Thus, any effects of
ultraviolet-modified dispersion relations in the IR could be masked by an
incorrect choice of the initial condition for the mode equation. A further
complication is that of possibly strong backreaction, although there are
indications that this problem can be absorbed in a suitable redefinition
of the inflaton potential, see \cite{greenenew}. Once these points are
clarified, $\mathcal{K}$ for inflation can be calculated.
A limitation of our investigation of the Casimir effect has been that we
restricted attention to modelling the effects of Planck scale physics on
quantum field theory exclusively through UV-modified dispersion relations.
This assumes that fields can possess arbitrarily large $k$ and arbitrarily
short wavelengths, an assumption which is likely too strong. Indeed,
studies of quantum gravity and string theory strongly indicate the
existence of a universal minimum length at the Planck or string scale. In
particular, it has been suggested that, in terms of first quantization,
this natural UV cutoff could possess an effective description through
uncertainty relations of the form $\Delta x \Delta p \le
\frac{\hbar}{2}(1+\beta (\Delta p)^2 +...)$, see, e.g.,
\cite{Garay:1994en}. As is easily verified, such uncertainty relations
encode the minimum length as a lower bound, $\Delta
x_\text{min}=\hbar\sqrt{\beta}$, on the formal position uncertainty,
$\Delta x$. It has been shown that this type of uncertainty relations also
implies a minimum wavelength and that, therefore, fields possess the
sampling property, see \cite{ak-prl2000}: if a field's (number or
operator-valued) amplitudes are known only at discrete points then the
field's amplitudes everywhere are already determined - if the average
sample spacing is less than the critical spacing, which is given by the
minimum length. As a consequence, any theory with this type of uncertainty
relation can be written as continuum theory or, fully equivalently, as a
discrete theory on any lattice of sufficiently tight spacing. This UV
cutoff can also be viewed as an information theoretic cutoff, and it
possesses a covariant generalization, see \cite{ak-prl}.
Indeed, nontrivial dispersion relations also raise the question of local
Lorentz invariance. One possibility is that local Lorentz is broken hard
or soft and that, e.g., the CMB rest frame is the preferred frame. It has
also been suggested that the Lorentz group might be deformed, or that it
may be unchanged but represented nonlinearly. Various experimental bounds
on Lorentz symmetry breaking are being discussed, e.g., from observations
of gamma ray bursts. For the literature, see e.g. \cite{lorentz}.
An application of the minimum length uncertainty principle to the Casimir
effect has recently been tried, see \cite{hossen}. There, the Casimir
force was found to be a discontinuous function of the plate separation.
This problem is due to the fact that, in \cite{hossen}, the plate
boundaries are implicitly treated as possessing sharp positions. This is
not fully consistent with the assumption that all particles including
those that make up the plates can be localized only up to the finite
minimum position uncertainty. As a consequence, as the plate separation
increases, the energy eigenvalues discontinuously enter the spectrum of
the first quantized Hamiltonian. It should be very interesting to extend
these Casimir force calculations while applying the minimum length
uncertainty relations to both the field and the plates.
Finally, we note an additional analogy between the Casimir effect and
inflation: in the Casimir effect with UV cutoff, as the distance between
the plates is increased, new modes enter the space between the plates,
thereby changing the vacuum energy. In cosmology, space itself expands
and, in the presence of an UV cutoff, new comoving modes (recall that
these are the independent degrees of freedom) are continually being
created, similar to the Casimir effect. A priori, these new modes arise
with vacuum energy. During the expansion, the modes' vacuum energy becomes
diluted but if the dispersion is nonlinear then the balance of new vacuum
energy creation and vacuum energy dilution is nontrivial. A paper which
addresses this question is in progress, \cite{ak-ll}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,392 |
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We have to be a state where business is welcome and jobs are created. We have to demand value for what is spent and we need to continue to resist a lottery. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,764 |
\section{Preface}
These are notes for four lectures on the topic of analytic solutions in Witten's open bosonic string field theory \cite{Witten} (open SFT). The subject originates in efforts to understand tachyonic instablities on unstable D-brane systems. Here it was realized that open SFT gives the most complete, if not necessarily most accessible, formalism for addressing such questions. It was also clear that open SFT (perhaps in a supersymmetric version) gives a possible path towards a nonperturbative and background independent definition of string theory. While numerical approaches will always be indispensable, useful progress in this direction requires some conceptual and analytic understanding of how the theory encodes nonperturbative physics. At the classical level, this in particular requires understanding how D-brane vacua appear as solutions of the field equations. Progress became possible after Schnabl's analytic solution describing the endpoint of tachyon condensation~\cite{Schnabl}. These notes are intended to give a relatively complete account of these developments. We limit the discussion to bosonic strings, though many ideas are applicable to open superstrings or topological strings.
The lectures were prepared for students and do not assume much knowledge beyond the first few chapters of Polchinski~\cite{Polchinski}. The first lecture is a serviceable introduction to open string field theory in general, though on certain points is optimized for later discussion of analytic solutions. We approach the subject from the point of view of path integrals, correlation functions, and noncommutative algebra in the spirit of Okawa \cite{Okawa}, rather than the operator formalism as originally used by Schnabl~\cite{Schnabl}. Those interested in deeper background on the subject should look at Sen's review of open string tachyon dynamics \cite{SenRev}, as well as Taylor and Zwiebach's review of tachyon condensation in open SFT \cite{TaylorZwiebachRev}. Okawa also has a nice (and shorter) review of analytic solutions~\cite{OkawaRev}. The thesis of Kudrna \cite{Matjej} describes the state of the art in the numerical approach to string field theory solutions.
We assume $\alpha'=1$ and use mostly plus metric convention. Commutators are always graded with respect to Grassmann parity.
\section{Introduction}
\subsection{First Look}
We begin by outlining, at a very crude level, the kind of theory we are dealing with. Open bosonic SFT is the field theory of fluctuations of a D-brane in bosonic string theory. The fluctuations of a D-brane are characterized by the open strings which attach to that D-brane.
Consider for example a D$p$-brane in bosonic string theory. An open string attached to this D$p$-brane can mimic an infinite variety of particle states, depending on how the string vibrates. The lowest modes of vibration give you a spin $0$ tachyon, a $p+1$-dimensional spin 1 photon, and $25-p$ massless spin $0$ particles, where $25+1=26$ is the dimension of spacetime in bosonic string theory. The higher vibrations give an infinite tower of massive particle states of higher spin. We can easily infer what kind of fields should enter the SFT Lagrangian to create this spectrum of particle states:
\begin{eqnarray}
\phantom{\bigg)}\text{spin 0 tachyon}\ \ \ \ \ \ \ \ \!\!\!\!\!\!\!\! && \longrightarrow\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{tachyonic scalar field, } T(x);\nonumber\\
\phantom{\bigg)}\text{spin 1 photon}\ \ \ \ \ \ \ \ \ \!\!\!\!\!\!\!\! &&\longrightarrow\ \ \ \ \ \ \ \ \, p+1\text{-dimensional gauge field, }A_\mu(x),\ \ \mu=0,1,...,p;\nonumber\\
\phantom{\bigg)}{25-p \text{ massless}\atop\text{spin 0 particles}}\ \ \ \ \ \ \ \ \!\!\!\!\!\!\!\! && \longrightarrow\ \ \ \ \ \ \ \ 25-p\text{ massless scalar fields, } \phi_a(x),\ \ a=1,...,25-p;\nonumber\\
\phantom{\bigg)}\vdots\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \!\!\!\!\!\!\!\! &&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots\nonumber\\
\phantom{\bigg)}{\text{massive\ particles}\atop\text{of higher spin}}\ \ \ \ \ \ \ \ \!\!\!\!\!\!\!\! && \longrightarrow\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{higher rank tensor fields}.\nonumber
\end{eqnarray}
Note that the coordinate $x\in\mathbb{R}^{1,p}$ refers to a point on the worldvolume of the D$p$-brane. Since open strings are attached to the D-brane, the fields do not depend on spacetime coordinates away from the D-brane worldvolume. With this we can at least begin to write an action for the fluctuations of the D-brane. We have
\begin{eqnarray}
S\!\!\!\!\!\!\!\! && = -\int d^{p+1}x\left(\frac{1}{2}\partial^\mu T\partial_\mu T-\frac{1}{2}T^2+\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}\partial^\mu\phi_a\partial_\mu\phi_a+{\text{massive}\atop\text{fields}}\right)+\text{interactions}.\ \ \ \ \ \
\end{eqnarray}
With some work we can write down kinetic terms for the massive fields. The form of the interactions, however, is almost impossible to guess at this level. The formulation of interactions depends heavily on the conformal field theory description of the string worldsheet, which we discuss later. In any case, the interactions must be constructed in such a way that the Feynman diagrams of the SFT action compute open string S-matrix elements on the D-brane. The kinetic term defines a propagator; the interactions define a cubic vertex, a quartic vertex, and so on as is necessary to get the right scattering amplitudes. So, for example, the color ordered $4$-string amplitude will be expressed as a sum of an $s$-channel, $t$-channel, and quartic vertex contributions:
\begin{wrapfigure}{l}{1\linewidth}
\centering
\resizebox{5in}{1.2in}{\includegraphics{OpenSFT_Erler1.jpg}}
\end{wrapfigure}\\ \\ \\ \\ \\ \\ \\ \\ \\
This may seem a little uncomfortable. One of the nicest things about string scattering is that each amplitude is represented by a single diagram; the interaction is a global property of the diagram, not a process inside vertices in a part of the diagram. There is nothing inconsistent about this, however. The three diagrams represent integration over different portions of the moduli space of disks with four boundary punctures; the single string diagram we are accustomed to visualizing represents integration over the entire moduli space. It is possible to slice the moduli spaces of Riemann surfaces into Feynman graph components in many ways. Different decompositions correspond to different SFT actions, but since the actions produce the same scattering amplitudes, they should be related by field redefinition. The field redefinition ambiguity is not something special to string theory, but is present in all Lagrangian field theories. The reason you do not hear about it more often is that, for the field theories we are accustomed to dealing with, there is a canonical or ``best possible" formulation of the Lagrangian---or at least a finite number of useful alternatives. From this point of view, one can articulate the discomfort with the string field theory concept as an impression that there is no preferred way to decompose string diagrams into Feynman diagrams, so there should not be a ``natural" formulation of the Lagrangian. Surprisingly, this impression is incorrect. For open bosonic strings, there is clearly a best possible Lagrangian, and this defines Witten's open bosonic string field theory. There is an analogous Lagrangian for closed bosonic strings based on Riemann surfaces endowed with minimal area metrics \cite{Zwiebach}, but in this case it is not quite as clear that the advantages of the Lagrangian are decisive. For superstrings much less is understood. There is a very nice formulation of open superstring interactions in the NS sector due to Berkovits \cite{Berkovits}. Recently this has been extended to include the Ramond sector by Kunitomo and Okawa \cite{complete}.
\subsection{Classical Solutions}
In these lectures we discuss classical solutions in open bosonic SFT. The interest of this subject originates in the problem of background independence in string theory. In string theory, we always start with the action for a relativistic string moving in some background---a spacetime, perhaps with D-branes or fluxes. Quantizing the action gives a perturbative description of string scattering in that background. Therefore, each background defines a different ``version" of string theory. The widespread assumption is that all of these versions of string theory are equivalent, in the sense that they represent perturbative expansions of the same fundamental theory around different vacua. One way to see that this is likely is that the spectrum of the string always includes particle states which represent linearized deformations in the choice of background. The most famous example of course is the graviton, which represents linearized deformations in the shape of spacetime. In the context of perturbative string scattering, however, the background cannot change. Understanding the relation between different backgrounds requires something more powerful than the usual formulation of string theory. This is one of the primary motivations for string field theory. In SFT, different backgrounds of string theory are represented as different classical solutions of the field equations. This is exactly analogous to how, in general relativity, physical spacetimes are represented as solutions of Einstein's equations.
Presently we are concerned with open bosonic strings. So the question is whether different D-brane configurations in bosonic string theory, for a fixed spacetime (or closed string) background, can be described as solutions to the field equations of open bosonic SFT. Describing shifts in the closed string background either requires closed SFT or a much better understanding of quantum effects in open SFT. At present both approaches seem difficult, and we will have enough work understanding the open string sector.
Let us return to open bosonic SFT of a D$p$-brane. If we turn on the gauge field $A_\mu$, we obtain a new background corresponding to a D$p$-brane with nontrivial Maxwell field. If we turn on the massless scalars $\phi_a$, we obtain a new background where the D$p$-brane has been displaced from its initial position. If $x^a,\ a=1,...,25-p$ represent coordinates transverse to the D$p$-brane, and the D-brane is initially located at
\begin{equation}x^a = 0,\end{equation}
after giving a constant expectation value to $\phi_a$, the new D$p$-brane is located at
\begin{equation}x^a = \frac{1}{\sqrt{T_p}}\phi_a,\end{equation}
assuming that $\phi_a$ is small enough that we can ignore nonlinear corrections to the field equations. The number $T_p$ is the tension of the D$p$-brane; with our conventions, it is given by
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\centering
\resizebox{1.6in}{1.3in}{\includegraphics{OpenSFT_Erler2.jpg}}
\end{wrapfigure}
\begin{equation}T_p = \frac{1}{2\pi^2}.\end{equation}
See \cite{TaylorZwiebachRev} for a derivation. Finally, we can give an expectation value to the tachyon. Since the
tachyon field is pulled by an ``upside down" harmonic oscillator potential $V(T) = -\frac{1}{2}T^2+...$, it cannot remain constant. Instead, it will roll down the potential with exponentially increasing expectation value. From this we see that the initial
configuration, where all fluctuations of the D$p$-brane vanish, is unstable. In other words, the D$p$-brane is itself unstable. The fate of this instability is unclear since the tachyon expectation value becomes large and nonlinear terms in the equations of motion dominate; the perturbative description of the D$p$-brane breaks down. This is the problem of {\it tachyon condensation}.
A physical understanding of tachyon condensation in open bosonic SFT emerged from work of Sen and others in the early 2000s. The upshot is as follows:\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
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\end{wrapfigure}
\begin{itemize}
\item Given the action of open bosonic SFT, one can define a tachyon effective potential $V(T)$ by integrating out all of the massive fields using the equations of motion. The claim is that this potential has a local minimum at $T=T_*$ representing the endpoint of tachyon condensation. The local minimum represents a highly nontrivial solution to the equations of motion of open bosonic SFT, and is called the {\it tachyon vacuum}.
\end{itemize}
\begin{itemize}
\item The tachyon vacuum represents a configuration where the D$p$-brane has disappeared, and we are left with empty space without D-branes or open strings. This has two important consequences:
\begin{description}
\item{a)} The shift in the potential between the pertubative vacuum and the tachyon vacuum is given by the D$p$-brane tension
\begin{equation}V(0)-V(T_*)=T_p =\frac{1}{2\pi^2}.\end{equation}
In other words, the missing energy density at the tachyon vacuum is precisely accounted for by the fact that the D$p$-brane has disappeared.
\item{b)} There are no physical excitations around the tachyon vacuum. This reflects the fact that there are no D-branes at the tachyon vacuum, and therefore no open strings.
\end{description}
\end{itemize}
Points a) and b) are specific predictions that can be confirmed by detailed calculations in open bosonic SFT. Similar predictions also exist for unstable D-branes in superstring field theory. Traditionally, these are known as {\it Sen's conjectures}. For the bosonic string, conjecture (a) was effectively proven in 2005 when M. Schnabl found an exact solution for the tachyon vacuum \cite{Schnabl}. A proof of (b) soon followed \cite{EllwoodSchnabl}.
Before Schnabl's result in 2005, the main approach to solving the open SFT equations of motion was {\it level truncation}. The idea is to approximate the action by dropping all fields with mass$^2$ above a fixed integer, and solve the resulting equations of motion numerically. This approach is still actively pursued. One of its advantages is that it is completely honest. In SFT it is fairly easy to be mislead by superficially plausible analytic calculations which upon closer inspection turn out to be meaningless. Through experience and a multitude of consistency checks we can have a fair amount of confidence in the analytic methods described in these lectures. But still a solution constructed with convincing convergence in level truncation leaves no doubt that a vacuum has been found. With level truncation it is also possible to construct backgrounds whose worldsheet description is not exactly known. See \cite{Matjej} for discussion of the state of the art in the level truncation approach.
Besides the tachyon vacuum, other classical solutions which have been widely studied include:
\begin{itemize}
\item {\it Marginal deformations.} These solutions correspond to turning on finite expectation values for the massless fields on the D-brane. Such solutions can describe, for example, translations of a D$p$-brane over a finite distance. From the worldsheet perspective, these solutions represent deformation of the worldsheet conformal field theory by an exactly marginal boundary operator. Such solutions have been constructed approximately in level truncation and analytically soon after Schnabl's result for the tachyon vacuum.
\item {\it Lump solutions.} Given a scalar field with a potential containing local maxima and minima, it is possible to construct solitonic solutions in the form of ``kinks" or ``lumps." The same is true for the tachyon in open bosonic SFT. In this case, lumps of the tachyon field are believed to describe lower dimensional D-branes in the field theory of a higher dimensional D-brane. From a worldsheet point of view, such solutions represent the infrared fixed point of a renormalization group flow given by perturbing the worldsheet conformal field theory by a relevant boundary operator. Lump solutions were constructed in level truncation in the early 2000s, but until recently there were no analytic solutions.
\end{itemize}
While lumps and marginal deformations cover a large class of interesting solutions, there are many open string backgrounds that cannot be described in this way. For example, starting from the fluctuations of a D$p$-brane, can we describe the formation of a D$(p+1)$-brane? If the transverse dimensions are large, the D$(p+1)$-brane will have higher energy than the D$p$. Therefore, such a configuration must be generated by giving expectation values to the massive modes of the string. It is hard to tell, however, which among the infinite number of massive fields should be the most important for this purpose. From the worldsheet point of view, we would need to perturb the worldsheet theory by some combination of irrelevant boundary operators, and try to run the renormalization group flow ``backwards" to reconstruct the ultraviolet fixed point. Needless to say, this seems difficult. The construction of higher energy vacua has become one of the major outstanding problems in the subject since the formulation of Sen's conjectures.
In principle it is possible that any D-brane system can be realized as a classical solution. One can propose it as a conjecture:
\begin{conjecture} Consider the string field theory of a chosen reference D-brane in bosonic string theory. The classical equations of motion of this theory have solutions describing all D-brane systems which share the same closed string background. Moreover, the field theory of fluctuations around a solution can be related to the string field theory of the corresponding D-brane by field redefinition.
\end{conjecture}
\noindent Imagine a manifold $\mathcal{M}$ representing the space of (on-shell and off-shell) configurations in open bosonic string theory in a fixed closed string background. Embedded in $\mathcal{M}$ is a submanifold
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\centering
\resizebox{2.4in}{1.7in}{\includegraphics{OpenSFT_Erler4.jpg}}
\end{wrapfigure}
representing the set of consistent open string backgrounds. Each background comes with a natural set of fluctuation fields defining a coordinate system on $\mathcal{M}$ in the vicinity of the chosen background. The above conjecture is analogous to saying that each local coordinate system defined in this way extends to cover all of $\mathcal{M}$. Furthermore, let $\phi^{(A)},\phi^{(B)}$ represent fluctuation fields around backgrounds labeled $A$ and $B$, respectively. Then there should be a coordinate transformation,
\begin{equation}\phi^{(A)} = f^{(AB)}[\phi^{(B)}],\end{equation}
which relates the actions $S^{(A)}$ and $S^{(B)}$ of the backgrounds $A$ and $B$. Specifically
\begin{equation}S^{(A)}[\phi^{(A)}]=S^{(A)}\Big[f^{(AB)}[\phi^{(B)}]\Big] = S^{(B)}[\phi^{(B)}]+\text{constant}.\end{equation}
It should be clear that it is possible that the fluctuation fields of a given D-brane can only rearrange themselves into other configurations that are sufficiently ``close by." That is, the local coordinate system of each background may only extend a limited distance from the origin, and to cover the whole configuration space we would have to work in patches. Nevertheless, significant new evidence is emerging in favor of this conjecture, both using analytic techniques and, to a more limited degree, level truncation. Consolidating these results and understanding their implications is a major impetus behind recent work in the subject.
\section{Lecture 1: Open String Field Theory}
\subsection{Conformal Field Theory}
We begin by reviewing some facts about two dimensional conformal field theory. The following is not really intended as an introduction to the subject. The purpose is to touch on certain points which in a general treatment may not be useful to emphasize, but in the context of open SFT are important. See for example \cite{Polchinski,Ginsparg,Francesco} for dedicated introduction.
A background of string theory is characterized by a worldsheet conformal field theory---for open strings, specifically a boundary conformal field theory (BCFT). A BCFT is a conformal field theory on a 2-dimensional manifold $\Sigma$ which is topologically a disk. The boundary of $\Sigma$ maps to the worldlines swept out by the endpoints of an open string attached to a D-brane; the interior of $\Sigma$ maps to the worldsheet swept out by the interior of the open string in spacetime. Since all
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\centering
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\end{wrapfigure}
disks are conformally equivalent, without loss of generality we can formulate BCFT on the upper half plane (UHP):
\begin{equation}\text{UHP}:\ \ \ z\in\mathbb{C}\cup\{\infty\},\ \ \ \mathrm{Im}(z)\geq 0\end{equation}
The real axis is the boundary, and including the point at $\infty$ is topologically a circle. A BCFT comes with two kinds of local operators: {\it bulk} operators $\mathcal{O}(z,\overline{z})$ which can be inserted in the interior of the UHP, and {\it boundary} operators $\mathcal{O}(x)$ which can be inserted on the real axis. Generally, the two kinds of operators are different. Correlation functions of bulk operators $\mathcal{O}(z,\overline{z})$ diverge as $(z,\overline{z})$ approaches the real axis, and correlation functions of boundary operators $\mathcal{O}(x)$ do not have a natural analytic continuation for $x$ not real. An important conceptual point is that the set of local operators of a quantum field theory represents the space of possible local deformations of the theory. In our case, given a bulk operator we can deform the worldsheet action by adding a term $\int_\text{UHP}d^2 z\,\mathcal{O}(z,\overline{z})$; given a boundary operator we can deform the worldsheet action with a boundary coupling $\int_{-\infty}^\infty dx\,\mathcal{O}(x)$. Generally, such deformations will not preserve conformal invariance and therefore will not define a string background. You can think of such deformations as creating a hypothetical background which does not satisfy the equations of motion---an ``off-shell" configuration of string theory. To leading order, conformal invariance requires that $\mathcal{O}(z,\overline{z})$ is a bulk primary field of weight $(1,1)$, and $\mathcal{O}(x)$ is a boundary primary of weight $1$. In this case, the operators generate what is known as a ``marginal deformation" of the BCFT. From the SFT perspective, such deformations correspond to giving expectation values for the massless fields on the background. Note that bulk operators deform the background as seen by the interior of the string. These are deformations of the closed string background. Boundary operators deform the background as seen from the endpoints of the open string. These are deformations of the D-brane system in a fixed closed string background. At least classically, open SFT describes the later deformations, but not the former.
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A point in the UHP can be described by two real coordinates $(x,y)$ with $y\geq 0$. Equivalently, we can describe this point with a holomorphic and anti-holomorphic coordinate $(z,\overline{z})$. Sometimes it is useful to consider a single point on the UHP as a pair of points on a purely holomorphic copy of the entire complex plane: $z$ is a point above the real axis, and $\overline{z}$ is a point below. This is called the {\it doubling trick}. Often we are interested in correlation functions of purely holomorphic or antiholomorphic operators on the UHP. Consider a holomorphic operator $\phi(z)$, satisfying $\overline{\partial}\phi(z)=0$. Since a correlation function
\begin{equation}\langle \phi(z)\,...\,\rangle_\text{UHP}\end{equation}
is holomorphic in $z$, generally it can be analytically continued to the lower half plane with $\mathrm{Im}(z)<0$. Now we also have a corresponding correlation function with the anti-holomorphic operator $\overline{\phi}(\overline{z})$ satisfying $\partial\overline{\phi}(\overline{z})=0$, which can also be analytically continued to the lower half plane. Typically, on the real axis there is a relation between $\phi$ and $\overline{\phi}$, called a {\it gluing condition}. In the simplest case, $\phi(x)=\overline{\phi}(x)$, which holds for example for the energy-momentum tensor. In this case we can conclude that
\begin{equation}\langle \overline{\phi}(\overline{z})\,...\,\rangle_\text{UHP} = \left.\langle\phi(z)\,...\,\rangle_\text{UHP}\right|_{z\to\overline{z}}.\end{equation}
The left hand side is a correlation function of an anti-holomorphic operator on the UHP, and on the right is a correlation function of a holomorphic operator which has been analytically continued from the UHP to the lower half plane and evaluated at the point $\overline{z}$. In this way, we represent the UHP with a holomorphic copy of the entire plane; we cut our work in half by discussing holomorphic operators on the entire plane instead of holomorphic and anti-holomorphic operators on the UHP. However, when working with correlation functions of operators which are neither holomorphic nor anti-holomorphic, it may be more convenient to stick to the UHP visualization.
For some purposes it is useful to describe BCFT in a state/operator formalism. The relation to correlation functions is given by
\begin{equation}\langle 0|\mathcal{O}_1(z_1,\overline{z}_1)\,...\,\mathcal{O}_n(z_n,\overline{z}_n)|0\rangle = \langle\mathcal{O}_1(z_1,\overline{z}_1)\,...\,\mathcal{O}_n(z_n,\overline{z}_n) \rangle_\text{UHP},\ \ \ \ \ \infty>|z_1|>...>|z_n|>0.
\end{equation}
On the right hand side is a BCFT correlation function on the UHP. On the left hand side, $|0\rangle$ is a special state of the BCFT called the $SL(2,\mathbb{R})$ {\it vacuum}, and $\mathcal{O}_1(z_1,\overline{z}_1)...\mathcal{O}_n(z_n,\overline{z}_n)$ are interpreted as operators, in the sense of the canonical formalism, acting on the state space $\mathcal{H}$ of the BCFT. The operators on the left hand side are ordered from left to right in sequence of decreasing distance to the origin (radial ordering). The $SL(2,\mathbb{R})$ vacuum is called this way since it is invariant under the $SL(2,\mathbb{R})$ subalgebra of the Virasoro algebra:
\begin{equation}[L_1,L_0]=L_1,\ \ \ [L_1,L_{-1}]=2L_0,\ \ \ [L_{-1},L_0]=-L_{-1},\end{equation}
where $L_n$ are the Virasoro operators, appearing in the mode expansion of the energy-momentum tensor:
\begin{equation}
T(z) = \sum_{n\in\mathbb{Z}}\frac{L_n}{z^{n+2}},\ \ \ \ \ L_n = \oint_0\frac{dz}{2\pi i}z^{n+1}T(z).
\end{equation}
The part of the contour in the lower half plane represents $\overline{T}(\overline{z})$ via the doubling trick. It is easy to show that
\begin{equation}L_{-1}|0\rangle=0,\ \ \ \ L_0|0\rangle=0,\ \ \ \ L_1|0\rangle = 0,\end{equation}
implying invariance under $SL(2,\mathbb{R})$. More generally, let $\phi(z)$ be a holomorphic primary operator of weight $h$ with mode expansion
\begin{equation}
\phi(z) = \sum_{n\in\mathbb{Z}}\frac{\phi_n}{z^{n+h}},\ \ \ \ \ \phi_n = \oint_0\frac{dz}{2\pi i}z^{n+h-1}\phi(z),
\end{equation}
where the index $n$ labelling the modes is chosen so that
\begin{equation}[L_0,\phi_n]=-n\phi_n.\end{equation}
One can show that
\begin{equation}\phi_n|0\rangle = 0,\ \ \ n>-h.\end{equation}
Usually we think of a state as repesenting the configuration of a quantum system at $t=0$; in radial quantization, this corresponds to $|z|=1$. It is therefore natural to interpret the vacuum expectation value as an inner product between an ``out" state and an ``in" state:
\begin{equation}
\underbrace{\langle 0|\mathcal{O}_1(z_1,\overline{z}_1)\,...\,\mathcal{O}_i(z_i,\overline{z}_i)}_{\langle\text{out}|}\,\underbrace{\mathcal{O}_{i+1}(z_{i+1},\overline{z}_{i+1})\,...\,\mathcal{O}_n(z_n,\overline{z}_n)|0\rangle}_{|\text{in}\rangle},\ \ \ |z_i|>1,\ \ |z_{i+1}|<1,
\end{equation}
where the ``out" state contains operators with $|z|>1$ and the ``in" state contains operators with $|z|<1$. The vector space of ``in" states defines the state space $\mathcal{H}$ of the BCFT.
Given a state $|A\rangle\in\mathcal{H}$ we can define a corresponding boundary operator $V_A(0)$, called the {\it vertex operator}, so that
\begin{equation}|A\rangle = V_A(0)|0\rangle.\end{equation}
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We can visualize this state as a portion of the UHP consisting of the unit half-disk $|z|<1$ with the vertex operator $V_A(0)$ inserted at the origin. The unit half-circle at the boundary of the half-disk can be parameterized by an angle $\sigma\in[0,\pi]$. The $SL(2,\mathbb{R})$ vacuum is a half-disk without a vertex operator; equivalently, it is the half-disk with an insertion of the identity operator. Given a dual state $\langle A|\in\mathcal{H}^\star$, we can define a corresponding boundary vertex operator at infinity so that
\begin{equation}\langle A| = \langle 0|V_A(\infty).\end{equation}
We can visualize this as a portion of the UHP with the unit half-disk removed and $V_A(\infty)$ inserted at infinity. The unit half-circle at the boundary of this region can be parameterized by an angle $\sigma\in[0,\pi]$. To compute the overlap $\langle A| B\rangle$, we glue the surface of $\langle A|$ to the surface of $| B\rangle$ so that the angle $\sigma$ along the half-circle of $| B\rangle$ is identified with the angle $\sigma$ along the half-circle of $\langle A|$. This effectively glues the unit half-disk to its compliment so as to form the entire UHP. The overlap is then given by $\langle A| B\rangle = \langle V_A(\infty)V_ B(0)\rangle_\text{UHP}$.
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Conventionally, a vertex operator $V_A(0)$ is imagined as a local operator inserted at the origin of the half-disk. This is the right picture for so-called {\it Fock space states}, whose $L_0$ eigenvalue is bounded from above\footnote{For bosonic strings we are usually concerned with free boson CFTs, where it may be more appropriate to say that a Fock space state is characterized by a bounded eigenvalue for the mass$^2$ operator, which is $L_0$ with noncompact momenta subtracted out.}. In our discussion, however, vertex operators will often be nonlocal. For
example, this occurs for states carrying operators displaced from the origin of the half-disk. Vertex operators may also contain an infinite number of insertions of the energy-momentum tensor which have the cumulative effect of changing the shape of the half-disk. So the vertex operator may formally represent a state which is quite different from a half-disk with operator at the origin. What is true, however, is that we can construct a basis for $\mathcal{H}$ using eigenstates of $L_0$ whose vertex operators are local at the origin. This is often called a {\it Fock space basis}, and the expression of a state in this basis is called the {\it Fock space expansion}. Nonlocal vertex operators can appear if we allow infinite sums of states in Fock space basis. To give a toy example which illustrates the essential point, consider a delta function $\delta(x)$, which has support at $x=0$. All derivatives of thee delta function also have support only at $x=0$, but the infinite sum
\begin{equation}\sum_{n=0}^\infty \frac{a^n}{n!}\delta^{(n)}(x)=\delta(x+a)\end{equation}
has support at $x=-a$. Though a vertex operator may be nonlocal, it cannot be arbitrarily nonlocal; it must still be localized within the unit half-disk. What this means concretely is that, in the context of a correlator of bulk local operators and $V_A(0)$ on the UHP, the OPE of any pair of local operators will converge inside a circle which extends (at minimum) to the nearest other local operator, to the real axis, or to the unit half-disk, whichever is closest.
The interior and exterior of the unit half-disk can be related by a conformal transformation
\begin{wrapfigure}{l}{.35\linewidth}
\centering
\resizebox{2.5in}{1.9in}{\includegraphics{OpenSFT_Erler9.jpg}}
\end{wrapfigure}
\begin{equation}I(z)=-\frac{1}{z}.\end{equation}
This allows us to define an isomorphism between states in $\mathcal{H}$ and dual states in $\mathcal{H}^\star$ through a bilinear form, called the {\it BPZ inner
product} (after Belavin, Polyakov, and Zamolodchikov):
\begin{equation}\langle A, B\rangle = \Big\langle \big(I\circ V_A(0)\Big) V_ B(0)\Big\rangle_\text{UHP}, \ \ \ \ \ |A\rangle,| B\rangle\in\mathcal{H}.\end{equation}
The right hand side is a correlation function on the upper half plane of two vertex operators, the first vertex operator having been transformed with $I(z)$. This effectively maps the vertex operator at the origin, which defines a state, into a vertex operator at infinity, which defines a dual state. We use the notation $f\circ...$ to denote conformal transformation of an operator by a map $f(z)$. For a (spinless) boundary primary $\phi(x)$ of weight $h$ and a bulk primary $\phi(z,\overline{z})$ of weight $(h,\overline{h})$ the conformal transformation law is
\begin{eqnarray}
f\circ \phi(x) \!\!\!\!\!\!\!\! && = \left|\frac{d f(x)}{d x}\right|^h \phi\big(f(x)\big)\\ f\circ\phi(z,\overline{z})\!\!\!\!\!\!\!\! && =\left(\frac{d f(z)}{dz}\right)^h\left(\overline{\frac{d f(z)}{dz}}\right)^{\overline{h}}\phi\big(f(z),\overline{f(z)}\big).
\end{eqnarray}
Note that because
\begin{equation}I(e^{i\sigma})= e^{i(\pi-\sigma)},\end{equation}
the BPZ inner product glues the point at an angle $\sigma$ on the unit half-circle of $A$ to the point at an angle $\pi-\sigma$ on the unit half circle of $ B$. The BPZ inner product is graded symmetric:
\begin{eqnarray}
\langle A, B\rangle \!\!\!\!\!\!\!\! && = \Big\langle I\circ\Big(\big(I\circ V_A(0)\big)V_ B(0)\Big)\Big\rangle_\text{UHP}\nonumber\\
\!\!\!\!\!\!\!\! && = \big\langle V_A(0) I\circ V_ B(0)\big\rangle_\text{UHP}\nonumber\\
\!\!\!\!\!\!\!\! && = (-1)^{|A|| B|}\big\langle \big(I\circ V_ B(0)\big)V_A(0)\big\rangle_\text{UHP}\nonumber\\
\!\!\!\!\!\!\!\! && = (-1)^{|A|| B|}\langle B,A\rangle.
\end{eqnarray}
In the first step we used $SL(2,\mathbb{R})$ invariance of UHP correlation functions to transform operator insertions with $I(z)$, and in the second step we used that $I(z)$ composed with itself gives the identity. The sign appears if the vertex operators are anticommuting. The BPZ inner product is also nondegenerate. That is, if $\langle A, B\rangle = 0$ for all $|A\rangle\in \mathcal{H}$, we can conclude $| B\rangle=0$. This follows from the fact that an operator which has vanishing 2-point function with itself and every other operator can be taken to vanish. Note that the BPZ ``inner product" is actually a bilinear form. In the usual meaning, an inner product is a sesquilinear form, in the sense that it takes the complex conjugate of the first argument. The BPZ inner product is different from the usual inner product which defines the Hilbert space of a quantum system, and is a structure specific to the state space of a conformal field theory.
The above allows us to define the notion of {\it BPZ conjugation}. Given an operator $\mathcal{O}$ acting on $\mathcal{H}$, the BPZ conjugate operator $\mathcal{O}^\star$ is defined by
\begin{equation}\langle A,\mathcal{O} B\rangle = (-1)^{|\mathcal{O}||A|}\langle \mathcal{O}^\star A, B\rangle,\end{equation}
where the sign may appear if $\mathcal{O}$ anticommutes with the vertex operator $V_A(0)$. From this it follows that
\begin{eqnarray}
\mathcal{O}^{\star\star} \!\!\!\!\!\!\!\! && = \mathcal{O}\\
(a\mathcal{O}_1+b\mathcal{O}_2)^\star \!\!\!\!\!\!\!\! && = a\mathcal{O}_1^\star+b\mathcal{O}_2^\star,\ \ \ \ \ a,b\in\mathbb{C}\\
(\mathcal{O}_1\mathcal{O}_2)^\star \!\!\!\!\!\!\!\! && = (-1)^{|\mathcal{O}_1||\mathcal{O}_2|}\mathcal{O}_2^\star\mathcal{O}_1^\star.
\end{eqnarray}
Sometimes operators have definite parity under BPZ conjugation:
\begin{equation}\mathcal{O}^\star = \pm \mathcal{O}.\end{equation}
If we have the plus sign, the operator is called {\it BPZ even}, and the minus sign, {\it BPZ odd}. The BPZ conjugate of the modes of an $SL(2,\mathbb{R})$ primary field are given by
\begin{equation}\phi_n^\star = (-1)^{n+h}\phi_{-n}.\end{equation}
The zero mode $\phi_0$ is BPZ even (odd) if the conformal weight is even (odd). For example, $L_0$ is BPZ even. Note that BPZ conjugation does not take the complex conjugate of scalars. Thus we assume $a^\star =a$ for $a\in\mathbb{C}$. We also define the $SL(2,\mathbb{R})$ vacuum and its dual to be conjugate:
\begin{equation}|0\rangle^\star = \langle 0|,\ \ \ \ \langle 0|^\star = |0\rangle.\end{equation}
Thus the BPZ conjugate of a state is given by
\begin{equation}|A\rangle^\star = \langle 0| I\circ V_A(0)\equiv\langle A^\star|.\end{equation}
and the BPZ inner product can be written
\begin{equation}\langle A, B\rangle = \langle A^\star| B\rangle.\end{equation}
Since BPZ conjugation leaves scalars invariant, the BPZ inner product must satisfy
\begin{equation}\langle A^\star| B\rangle =\big(\langle A^\star| B\rangle\big)^\star =(-1)^{|A|| B|} \langle B^\star|A\rangle.\end{equation}
and is therefore graded symmetric.
Meanwhile, the vector space $\mathcal{H}$ has an ordinary quantum mechanical inner product, which allows us to define {\it Hermitian conjugation}. Hermitian conjugation satisfies
\begin{eqnarray}
\mathcal{O}^{\dag\dag} \!\!\!\!\!\!\!\! && = \mathcal{O}\\
(a\mathcal{O}_1+b\mathcal{O}_2)^\dag \!\!\!\!\!\!\!\! && = \overline{a}\mathcal{O}_1^\dag+\overline{b}\mathcal{O}_2^\dag,\ \ \ \ \ a,b\in\mathbb{C}\\
(\mathcal{O}_1\mathcal{O}_2)^\dag \!\!\!\!\!\!\!\! && = \mathcal{O}_2^\dag\mathcal{O}_1^\dag.
\end{eqnarray}
Hermitian conjugation of the Virasoro generators negates the mode number
\begin{equation}L_n^\dag = L_{-n}.\end{equation}
Furthermore, it takes the complex conjugate of scalars. Thus we assume $a^\dag = \overline{a}$ for $a\in\mathbb{C}$. We also define
\begin{equation}|0\rangle^\dag = \langle 0|,\ \ \ \ \langle 0|^\dag = |0\rangle,\end{equation}
so that Hermitian conjugation of a state gives a dual state:
\begin{equation}|A\rangle^\dag = \langle 0|\big(V_A(0)\big)^\dag\equiv\langle A^\dag|.\end{equation}
The quantum mechanical inner product is then
\begin{equation}\langle A^\dag| B\rangle.\end{equation}
This is conjugate symmetric
\begin{equation}\overline{\langle A^\dag| B\rangle} = \big(\langle A^\dag| B\rangle\big)^\dag = \langle B^\dag|A\rangle.\end{equation}
We do not use the usual notation $\langle A| B\rangle$ for the inner product of quantum mechanics, since this assumes that Hermitian conjugation defines the canonical isomorphism between states and dual states. In conformal field theory, we also have BPZ conjugation. BPZ and Hermitian conjugation commute, and their composition defines what is called {\it reality conjugation}, denoted with a double dagger $^\ddag$. Reality conjugation does not map states into dual states, but rather maps $\mathcal{H}$ into itself. Thus for any $|A\rangle\in \mathcal{H}$, we can associate a ``complex conjugate" state $|A^\ddag\rangle\in \mathcal{H}$, and reality conjugation endows the state space of a BCFT with a real structure.
\subsection{Worldsheet Theory of Open Bosonic String}
The worldsheet theory of an open bosonic string is a tensor product of ``matter" and ``ghost" BCFTs:
\begin{equation}\text{BCFT}= \text{BCFT}_\text{matter}\otimes\text{BCFT}_\text{ghost}.\end{equation}
The ghost factor is described by a $bc$ system with central charge $-26$. It is characterized by anticommuting, holomorphic worldsheet fields $b(z),c(z)$, with antiholomorphic counterparts we can account for with the doubling trick, satisfying
\begin{eqnarray}
b(z)\!\!\!\!\!\!\!\! && = \text{primary of dimension } (2,0),\\
c(z)\!\!\!\!\!\!\!\! && = \text{primary of dimension }(-1,0),\\
b(x)\!\!\!\!\!\!\!\! && = \overline{b}(x),\ \ \ \ c(x)=\overline{c}(x),\ \ \ \ x\in\mathbb{R},\\
b(z)c(w)\!\!\!\!\!\!\!\! && = \frac{1}{z-w}+...\ .\label{eq:bcOPE}
\end{eqnarray}
The ghost factor of the BCFT is the same for all backgrounds of the open bosonic string. The information about the background is contained in the matter BCFT. The only necessary condition on the matter BCFT is that it has central charge $+26$, so the total matter/ghost BCFT has central charge $+26-26=0$. For a D$p$ brane in flat space, the matter BCFT consists of $p+1$ free bosons
$X^\mu(z,\overline{z}),\ \mu=0,...,p$ subject to Neumann boundary conditions, and $25-p$ free bosons $X^a(z,\overline{z}),\ a=1,...,25-p$ subject to Dirichlet boundary conditions:
\begin{eqnarray}
\partial X^\mu(z)\!\!\!\!\!\!\!\! && = \text{primary of dimension }(1,0),\\
\partial X^a(z)\!\!\!\!\!\!\!\! && = \text{primary of dimension }(1,0),\\
\partial X^\mu(z)\!\!\!\!\!\!\!\! && =\overline{\partial}X^\mu(x),\ \ \ \ \ \ x\in\mathbb{R}\ \ \ \ \text{(Neumann b.c.)},\\
\partial X^a(z)\!\!\!\!\!\!\!\! && =-\overline{\partial}X^a(x),\ \ \ \ x\in\mathbb{R}\ \ \ \ \text{(Dirichlet b.c.)},\\
\partial X^\mu(z)\partial X^\nu(w) \!\!\!\!\!\!\!\! && = -\frac{1}{2}\frac{\eta^{\mu\nu}}{(z-w)^2}+...\ ,\label{eq:dXmuOPE}\\
\partial X^a(z)\partial X^b(w) \!\!\!\!\!\!\!\! && = -\frac{1}{2}\frac{\delta^{ab}}{(z-w)^2}+...\ .\label{eq:dXaOPE}
\end{eqnarray}
We also have antiholomorphic operators $\overline{\partial} X^\mu(\overline{z})$ and $\overline{\partial}X^a(\overline{z})$ which we can account for with the doubling trick. Note that in the Dirichlet case the gluing condition for $\partial X$ comes with a sign.
The matter/ghost form of the open string BCFT provides additional structure and properties not present for a generic BCFT:
\begin{description}
\item{(1)} Since the central charge of the total BCFT vanishes, the energy-momentum tensor,
\begin{equation}T(z) = T^\text{matter}(z)+T^\text{ghost}(z),\end{equation}
is a primary of dimension $(2,0)$. Also, correlation functions are identically conformally invariant
\begin{equation}\langle ...\rangle_\Sigma = \langle f\circ(...)\rangle_{f\circ \Sigma},\end{equation}
where $\Sigma$ is a 2-dimensional Riemann surface with the topology of a disk (not necessarily the UHP), and $f\circ\Sigma$ is the surface obtained after applying the conformal transformation $f$. For a generic BCFT, this equality will only hold up to a proportionality factor generated by the nonzero central charge.
\item{(2)} The set of operators in the theory has a $\mathbb{Z}_2$ grading according to whether they are commuting or anticommuting. Commuting operators are said to be {\it Grassmann even}, anticommuting operators {\it Grassmann odd}, and the $\mathbb{Z}_2$ grading is called {\it Grassmann parity}. The Grassmann parity of an operator $\mathcal{O}$ is denoted $|\mathcal{O}|$. In addition, the set of operators carries a $\mathbb{Z}$ grading called {\it ghost number}, which counts the number of $c$ minus the number of $b$ factors contained in the operator. Thus
\begin{eqnarray}
\partial X^\mu(z) \!\!\!\!\!\!\!\! && = \text{Grassmann even, ghost \#}0,\\
b(z)\!\!\!\!\!\!\!\! && = \text{Grassmann odd, ghost \#} -1,\\
c(z)\!\!\!\!\!\!\!\! && = \text{Grassmann odd, ghost \#}1.
\end{eqnarray}
The space of states $\mathcal{H}$ is also graded by Grassmann parity and ghost number, according to the Grassmann parity and ghost number of the corresponding vertex operators. The Grassmann parity of a state $|A\rangle$ is denoted $|A|$. For ordinary backgrounds, $b$ and $c$ are the only anticommuting operators of the worldsheet theory, which leads to an identification between Grassmann parity and ghost number
\begin{equation}\text{Grassmann parity} = \text{ghost number mod }\mathbb{Z}_2.\end{equation}
In defining the SFT path integral it is necessary to consider states multiplied by formal anticommuting parameters, and in this context Grassmann parity and ghost number may not be related. In this lecture we are concerned only with the classical theory, and this identification holds.
\item{(3)} The theory comes with a dimension $(1,0)$ holomorphic primary field called the {\it BRST current}:
\begin{equation}j_B(z)= cT^\text{matter}(z)+:bc\partial c(z):+\frac{3}{2}\partial^2 c(z).\end{equation}
There is also an antiholomorphic counterpart $\overline{j}_B(\overline{z})$ of dimension $(0,1)$. On the real axis, we have the gluing condition
\begin{equation}j_B(x) = \overline{j}_B(x),\ \ \ \ x\in\mathbb{R},\end{equation}
which allows us to describe the antiholomorphic current with the doubling trick. The integral of $j_B(z)$ around a closed contour $C$ in the complex plane defines the {\it BRST operator}
\begin{equation}Q= \oint_C \frac{dz}{2\pi i}j_B(z).\end{equation}
If the contour surrounds a boundary operator $\mathcal{O}(x)$ on the real axis, this defines the {\it BRST variation} of that operator:
\begin{equation}Q\cdot\mathcal{O}(x) = \oint_x\frac{dz}{2\pi i}j_B(z)\mathcal{O}(x).\end{equation}
The BRST variation of a bulk operator $\mathcal{O}(z,\overline{z})$ is defined by a contour that loops around $z$ above the real axis and $\overline{z}$ below the real axis:
\begin{eqnarray}
Q\cdot\mathcal{O}(z,\overline{z})\!\!\!\!\!\!\!\! && = \left(\oint_z+\oint_{\overline{z}}\right)\frac{dz'}{2\pi i}j_B(z')\mathcal{O}(z,\overline{z})\\
\!\!\!\!\!\!\!\! && =\oint_z\frac{dz'}{2\pi i}j_B(z')\mathcal{O}(z,\overline{z}) - \oint_{\overline{z}}\frac{d\overline{z}'}{2\pi i}\overline{j}_B(\overline{z}')\mathcal{O}(z,\overline{z}).
\end{eqnarray}
In the last step we undid the doubling trick. The sign appears since a counterclockwise contour in the lower half plane corresponds to a clockwise contour in the UHP. The BRST operator is nilpotent,
\begin{equation}Q^2=0,\end{equation}
is Grassmann odd and carries ghost number 1. Since the BRST current is a weight 1 primary operator, the BRST operator is preserved by conformal transformation
\begin{equation}f\circ\Big(Q\cdot(...)\Big) = Q\cdot\Big(f\circ(...)\Big).\end{equation}
We have the properties
\begin{equation}Q\cdot b(z) = T(z),\ \ \ \ Q\cdot T(z)=0,\end{equation}
where the last follows from $Q^2=0$. Since $T(z)$ is the Noether current associated to conformal symmetry, BRST invariance of $T(z)$ is equivalent to conformal invariance of $Q$. Also useful are the relations
\begin{equation}Q\cdot c(z) = c\partial c(z),\ \ \ \ Q\cdot\mathcal{O}^\mathrm{m}(x)= c\partial \mathcal{O}^\mathrm{m}(x)+h\partial c\mathcal{O}^\mathrm{m}(x),\end{equation}
where $\mathcal{O}^\mathrm{m}$ is a boundary matter primary of weight $h$. The BRST variation of a state is defined by the BRST variation of its vertex operator. Equivalently, the BRST operator acting on a state is defined by the zero mode of the BRST current. In this context, the BRST operator is Hermitian and BPZ odd:
\begin{equation}Q^\dag = Q,\ \ \ \ Q^\star = -Q.\end{equation}
\item{(4)} A {\it physical state} is a BRST invariant state of the BCFT at ghost number 1:
\begin{equation}\text{physical state:}\ \ \ \ Q|\Psi\rangle = 0,\ \ \ \ |\Psi\rangle = \text{ghost number }1.\end{equation}
A BRST invariant state is equivalently said to be {\it BRST closed}. Physical states are defined to be equivalent if they differ by the BRST variation of a state at ghost number 0:
\begin{equation}
\text{physical equivalence:} \ \ \ \ |\Psi'\rangle = |\Psi\rangle + Q|\Lambda\rangle,\ \ \ \ |\Lambda\rangle = \text{ghost number }0.
\end{equation}
A state which can be written as the BRST variation of something else is trivially BRST closed due to $Q^2=0$. Such a state is called {\it BRST exact}. The space of inequivalent physical states is then defined by the space of BRST closed states modulo the addition of BRST exact states at ghost number 1. This defines the {\it cohomology} of $Q$ at ghost number 1. Note that the distinction between ``physical" and ``unphysical" states does not originate from the BCFT itself; while the matter/ghost form of the BCFT implies the existence of the BRST operator and an associated cohomology, the interpretation of this cohomology originates elsewhere. Ultimately, it comes from the fact that the BCFT description of the worldsheet theory arises after gauge fixing the reparameterization and Weyl symmetries of the Polyakov action. The statement that physical states of the worldsheet theory should be gauge invariant translates, after gauge fixing, to the statement that physical states of the BCFT should be BRST invariant. The BRST operator has cohomology at other ghost numbers, but they do not represent physical quantum states of the open string. A basis for the cohomology at each ghost number is given by states of the form
\begin{eqnarray}
\text{ghost number }0:\!\!\!\!\!\!\!\! &&\ \ \ |0\rangle,\\
\text{ghost number }1:\!\!\!\!\!\!\!\! &&\ \ \ c V^\mathrm{m}(0)|0\rangle,\\
\text{ghost number }2:\!\!\!\!\!\!\!\! &&\ \ \ c\partial c V^\mathrm{m}(0)|0\rangle,\\
\text{ghost number }3:\!\!\!\!\!\!\!\! && \ \ \ c\partial c\partial^2 c|0\rangle,
\end{eqnarray}
where $V^m(0)$ is a boundary primary in the matter factor of the BCFT of dimension 1. The cohomology at ghost number $g$ is isomorphic to the cohomology at ghost number $3-g$, in interesting analogy to Poincar\'{e} duality of the de Rham cohomology of differential forms in three dimensions. The cohomology at ghost numbers greater than three or less than zero is trivial, in the sense that all BRST closed states are BRST exact.
\item{(5)} The correlation functions of the BCFT are nonvanishing only if the ghost number of all operator insertions adds up to 3. Using Wick's theorem, all correlation functions in the ghost sector can be reduced to a correlator with three $c$-ghost insertions:
\begin{equation}\langle c(z_1)c(z_2)c(z_3)\rangle_\text{UHP}^\text{ghost} = (z_1-z_2)(z_1-z_3)(z_2-z_3).\end{equation}
Correlation functions with $\overline{c}(\overline{z})$ are given by the doubling trick.
\end{description}
\subsection{The String Field}
We now want to pass from the first quantized worldsheet theory to the classical field theory of fluctuations of a D-brane. The first step is to specify the nature of the fluctuation fields. It is convenient to consider the set of fluctuation fields together as a single object, called the {\it string field}. We make the following claim:
\begin{quote}
{\it A string field is an element of the state space $\mathcal{H}$ of the worldsheet BCFT of an open bosonic string attached to a given D-brane. }
\end{quote}
\noindent At first this statement might be confusing. An element of $\mathcal{H}$ is a quantum state of the string, but now we are claiming that it also represents a classical fluctuation of a D-brane. There are a couple of ways to understand this. The first is that it follows a general rule about the correspondence between first quantized theories and classical field theories: Namely, the wavefunction of a first quantized theory can be interpreted as a spacetime field of an equivalent classical field theory. Since this fact may be unfamiliar, let us illustrate it with an example. Consider a free, nonrelativistic quantum particle in the quantum state $|\psi\rangle$. The state $|\psi\rangle$ evolves in time according to the Schr\"{o}dinger equation:
\begin{equation}i\frac{\partial}{\partial t}|\psi(t)\rangle = \frac{p^2}{2m}|\psi(t)\rangle.\end{equation}
The wavefunction is given by expressing $|\psi\rangle$ in the position basis:
\begin{equation}\psi(x,t) = \langle x|\psi(t)\rangle,\end{equation}
where the Schr\"{o}dinger equation reads
\begin{equation}i\frac{\partial}{\partial t}\psi(x,t)=-\frac{1}{2m}\frac{\partial^2}{\partial x^2}\psi(x,t).\end{equation}
Now if we forget where this equation came from, there is nothing contradictory about interpreting $\psi(x,t)$ as a classical complex scalar field subject to a nonrelativistic wave equation. In fact, the wave equation can be derived by variation of a field theory action
\begin{equation}
S = \int dx \,dt\left[i\psi^*(x,t)\frac{\partial}{\partial t}\psi(x,t) -\frac{1}{2m}\frac{\partial}{\partial x}\psi^*(x,t)\frac{\partial}{\partial x}\psi(x,t)\right].
\end{equation}
From this point of view, $\psi(x,t)$ is just a complex scalar field, and there is no reason to interpret it as a probability amplitude. However, this classical field theory is equivalent to the first quantized theory in the following sense: If we start from the action for $\psi(x,t)$ and follow the usual recipe for canonical quantization of a classical field theory, we find a Fock space of multiparticle states given by acting creation operators on the vacuum. The Hamiltonian of the resulting QFT implies that the wavefunction for the single particle state inside this Fock space will evolve according to the Schr\"{o}dinger equation for a free, nonrelativistic particle. So we are back to where we started, only we have a formalism describing many and variable number of indistinguishable nonrelativistic particles. Applying the same procedure to the first quantized states of an open bosonic string gives a field theory formalism capable of describing many and variable number of indistinguishable quantum open bosonic strings.
There is a second, perhaps more physical justification for the definition of the string field. From the state-operator mapping of BCFT, we know that every state $|A\rangle\in\mathcal{H}$ has a corresponding boundary vertex operator $V_A(0)$. As mentioned before, the set of boundary operators corresponds to the set of possible boundary deformations of the BCFT. This, in turn, corresponds to the space of deformations (or fluctuations) of the D-brane system defining the open bosonic string BCFT.
It is important to distinguish between a generic string field and the particular kind of string field which enters the action and equations of motion---the {\it dynamical string field}. In a similar way, in gauge theories we have Lie algebra valued differential forms---including the 2-form field strength---but the dynamical variable of the theory is a 1-form---the gauge potential. The dynamical field in open bosonic SFT is the same kind of state in $\mathcal{H}$ where we impose the physical state condition, namely it is Grassmann odd and ghost number $1$. Just as the Schr\"{o}dinger equation of the nontrelativistic particle can be interpreted as a wave equation for a complex scalar field, the physical state condition is interpreted as a linearized field equation:
\begin{equation}Q\Psi = 0,\ \ \ \ \Psi=\text{Grassmann odd, ghost number }1.\label{eq:linEOM}\end{equation}
The equivalence of physical states is interpreted as a linearized gauge invariance:
\begin{equation}\Psi' = \Psi+Q\Lambda,\ \ \ \ \Lambda = \text{Grassmann even, ghost number }0.\end{equation}
Henceforth we will mostly drop the ket around $\Psi$. This is to emphasize that $\Psi$ is a classical field. We will not try to interpret it as a quantum amplitude.
In fact, unlike an amplitude, the dynamical string field should in a sense be ``real." One way to see that this is true is that the most basic excitation of a D-brane---a photon---is obtained by quantization of the Maxwell potential, which is a real field. Fortunately, we have seen the state space of a BCFT naturally comes with a real structure, which suggests the following reality condition on the dynamical string field:
\begin{equation}\Psi^\ddag = \Psi.\end{equation}
One can show that
\begin{equation}(Q A)^\ddag = (-1)^{|A|+1}Q (A^\ddag),\end{equation}
since the BRST operator is Hermitian and BPZ odd. To preserve the reality condition, the gauge parameter $\Lambda$ must therefore satisfy
\begin{equation}\Lambda^\ddag = -\Lambda,\end{equation}
and is in a sense ``imaginary."
To see that \eq{linEOM} makes sense as a linear field equation, it is helpful to give a more concrete description of the string field as an expansion in eigenstates of $L_0$. Let us do this for the D$p$-brane. The mode expansions of the $bc$ ghosts and free scalars is given by
\begin{eqnarray}
b(z)\!\!\!\!\!\!\!\! && = \sum_{n\in\mathbb{Z}}\frac{b_n}{z^{n+2}},\ \ \ \ \ \ \ \ \ \ \ \ \, b_n|0\rangle = 0\ \ \text{for}\ n\geq -1,\\
c(z)\!\!\!\!\!\!\!\! && = \sum_{n\in \mathbb{Z}}\frac{c_n}{z^{n-1}},\ \ \ \ \ \ \ \ \ \ \ \ \, c_n|0\rangle = 0\ \ \text{for}\ n\geq 2,\\
\partial X^\mu(z)\!\!\!\!\!\!\!\! && = -\frac{i}{\sqrt{2}}\sum_{n\in \mathbb{Z}}\frac{\alpha_n^\mu}{z^{n+1}},\ \ \ \ \ \alpha_n^\mu|0\rangle = 0\ \ \text{form}\ n\geq 0,\\
\partial X^a(z)\!\!\!\!\!\!\!\! && = -\frac{i}{\sqrt{2}}\sum_{n\in \mathbb{Z}}\frac{\alpha_n^a}{z^{n+1}},\ \ \ \ \ \alpha_n^a|0\rangle = 0\ \ \text{form}\ n\geq 0.
\end{eqnarray}
The normalization in front of the mode expansion for $\partial X$ is chosen so that the matter oscillators obey the commutation relations
\begin{equation}[\alpha_m^\mu,\alpha_{-n}^\nu]=m\delta_{mn}\eta^{\mu\nu},\ \ \ \ [\alpha_m^a,\alpha_{-n}^b]=m\delta_{mn}\delta^{ab},\end{equation}
as follows from the OPEs \eq{dXmuOPE} and \eq{dXaOPE}. Moreover,
\begin{equation}[b_n,c_{-n}] =\delta_{mn},\end{equation}
as follows from the OPE \eq{bcOPE}. We always use the bracket $[\cdot,\cdot]$ to denote the commutator graded with respect to Grassmann parity, i.e. $[A,B] = AB-(-1)^{|A||B|}BA$. The modes satisfy Hermitian and BPZ conjugation properties
\begin{eqnarray}
b_n^\dag \!\!\!\!\!\!\!\! && = b_{-n},\ \ \ \ b_n^\star = (-1)^n b_{-n}\\
c_n^\dag \!\!\!\!\!\!\!\! && = c_{-n},\ \ \ \ c_n^\star = (-1)^{n+1} c_{-n}\\
(\alpha^\mu_n)^\dag \!\!\!\!\!\!\!\! && = \alpha^\mu_{-n},\ \ \ \ (\alpha^\mu_n)^\star = (-1)^{n+1} \alpha^\mu_{-n}\\
(\alpha^a_n)^\dag \!\!\!\!\!\!\!\! && = \alpha^a_{-n},\ \ \ \ (\alpha^a_n)^\star = (-1)^{n+1} \alpha^a_{-n}.
\end{eqnarray}
In the Neumann directions along the D-brane, the zeroth oscillator is related to the momentum through
\begin{equation}\alpha_0^\mu = \sqrt{2}p^\mu.\end{equation}
In the Dirichlet directions, the zeroth oscillator vanishes
\begin{equation}\alpha_0^a=0.\end{equation}
since the open string does not carry a conserved momentum orthogonal to the D-brane. Since $\alpha_0^\mu|0\rangle = 0$, the $SL(2,\mathbb{R})$ vacuum carries zero momentum, which means that it represents a translationally invariant field configuration. To describe fields with nontrivial spacetime dependence, we need to inject some momentum into the vacuum. This can be done by ``translating" in momentum space using the position zero mode $x^\mu_0$ satisfying
\begin{equation}[x^\mu_0,p_\nu] = i\delta^\mu_\nu.\end{equation}
Thus the vacuum with momentum $k_\mu$ is given by
\begin{equation}|k_\mu\rangle = e^{ik\cdot x_0}|0\rangle = e^{ik\cdot X(0)}|0\rangle.\end{equation}
The state corresponds to inserting a (boundary normal ordered) plane wave vertex operator $e^{ik\cdot X(0)}$ at the origin of the unit half-disk. The dynamical string field can be represented as a sum of states created by acting mode oscillators on $|k_\mu\rangle$. Arranging these states in sequence of increasing $L_0$ eigenvalue for a given momentum, and recalling that the dynamical string field carries ghost number $1$, we find the expansion
\begin{equation}
\Psi = \int\frac{d^{p+1}k}{(2\pi)^{p+1}}\Big[\underbrace{T(k)c_1\phantom{\Big(}\!\!}_{L_0=k^2-1}+\underbrace{A_\mu(k)\alpha_{-1}^\mu c_1 +\phi_a(k)\alpha_{-1}^a c_1 +\frac{i}{\sqrt{2}}\beta(k)c_0}_{L_0=k^2}+\underbrace{\phantom{\Big(}\ \ .\ .\ .\ \ \ }_{L_0=k^2+n,\ n\geq 1}\Big]|k_\mu\rangle.
\end{equation}
The coefficient functions $T(k)$ etc. are an infinite list of ordinary spacetime fields---the fluctuation fields of the D$p$-brane---expressed in momentum space. As you can probably anticipate, $T(x)$ is the tachyon on the D$p$-brane, $A_\mu(x)$ is the Maxwell gauge potential, and $\phi_a(x)$ are the massless scalars representing transverse displacement of the D$p$-brane. We will see the role of $\beta(x)$ in a moment. The reality condition implies that the coefficient fields are real. Plugging this into $Q\Psi=0$ implies a set of linearized field equations:
\begin{eqnarray}
(\Box+1)T \!\!\!\!\!\!\!\! && = 0,\\
\Box A_\mu -\partial_\mu\beta \!\!\!\!\!\!\!\! && = 0,\\
\Box\phi_a \!\!\!\!\!\!\!\! && = 0,\\
\beta -\partial^\mu A_\mu \!\!\!\!\!\!\!\! && = 0,\\
\vdots\ \ \ \!\!\!\!\!\!\!\! && \ \ \ \ \ .\nonumber
\end{eqnarray}
We can similarly expand the gauge parameter:
\begin{equation}\Lambda = \int\frac{d^{p+1}k}{(2\pi)^{p+1}}\Big[\underbrace{\, i\lambda(k)\,}_{L_0=k^2}+\underbrace{\ \ .\ .\ .\ \ }_{L_0=k^2+n, n\geq 1}\Big]|k_\mu\rangle .\end{equation}
The $i$ in front of $\lambda(k)$ ensures that the gauge parameter is imaginary if $\lambda(x)$ is real. The linearized gauge transformation of $\Psi$ translates to
\begin{eqnarray}
T \!\!\!\!\!\!\!\! && = \text{invariant},\\
A_\mu'\!\!\!\!\!\!\!\! && = A_\mu+\partial_\mu\lambda,\\
\phi_a\!\!\!\!\!\!\!\! && =\text{invariant},\\
\beta' \!\!\!\!\!\!\!\! && = \beta+\Box\lambda,\\
\!\!\!\!\!\!\!\! && \vdots\nonumber\ \ \ \ \ .
\end{eqnarray}
\begin{exercise} Derive these equations by computing $Q\Psi = 0$ and $\Psi'=\Psi+Q\Lambda$\end{exercise}
\noindent The gauge potential has the expected Maxwell gauge invariance. The field $\beta$ does not carry any physical degrees of freedom, since its equation of motion is zeroth order in derivatives. It simply fixes $\beta$ to be $\partial^\mu A_\mu$. Substituting into the field equation for $A_\mu$ implies
\begin{equation}
\Box A_\mu -\partial_\mu(\partial^\nu A_\nu) = \partial^\nu(\partial_\nu A_\mu-\partial_\mu A_\nu) = \partial^\nu F_{\nu\mu} = 0,
\end{equation}
which is Maxwell's equation. At higher mass level, the number of fields like $\beta(x)$ which carry no degrees of freedom proliferates. One way to understand this is that $\Lambda$ contains an infinite tower of gauge parameters, ascending in order of increasing $L_0$ eigenvalue for a given momentum. But naively one does not expect massive higher spin fields to need gauge symmetry. The only ``true" gauge invariance of the string spectrum is that of the photon. This implies that the dynamical string field must contain additional variables to soak up superfluous gauge symmetries at higher mass level. It should be possible to formulate a string field theory where all auxilliary fields like $\beta(x)$ have been integrated out, and the only remaining gauge invariance is that of Maxwell theory. However, it seems complicated to do this. A deeper issue is that, while the Maxwell gauge symmetry is sufficient to describe perturbative physics of the D$p$-brane, one would like to think that the string field theory has a classical solution describing, for example, a pair of D$p$-branes with a non-Abelian $U(2)$ gauge symmetry. It is hard to see how that gauge invariance could be captured by expanding the Maxwell gauge symmetry around a nonzero vacuum solution. Therefore, the gratuitous redundancy of $\Psi$ may be an important hint as to its capacity to describe physics which is far removed from that of the reference D-brane.
Another useful representation of the string field is the position space, or {\it Schr\"{o}dinger representation}. In quantum mechanics, the wavefunction is derived by contracting $|\psi(t)\rangle$ with a position eigenstate. We can do a similar thing for the string field, contracting with an eigenstate of the position of the string. For definiteness we concentrate on a factor of the BCFT corresponding to free bosons subject to Neumann boundary conditions. The Dirichlet case is similar, and even the dependence on ghosts can be described through the ``position" of formal Grassmann odd variables. Consider the mode expansion
\begin{equation}
X^\mu(z.\overline{z}) = x^\mu_0 - p^\mu \ln|z|^2+\frac{i}{\sqrt{2}}\sum_{n\in\mathbb{Z}-\{0\}}\frac{\alpha_n^\mu}{n}\left(\frac{1}{z^n}+\frac{1}{\overline{z}^n}\right).
\end{equation}
At $|z|=1$ (corresponding to $t=0$ in radial quantization) the mode expansion reduces to
\begin{equation}x^\mu(\sigma) = X^\mu(e^{i\sigma},e^{-i\sigma}) = x^\mu_0+2\sum_{n=1}^\infty x_n^\mu \cos(n\sigma),\ \ \ \ x_n^\mu = \frac{i}{\sqrt{2}}\frac{\alpha_n^\mu-\alpha_{-n}^\mu}{n}.\end{equation}
The appearance of cosines reflects the fact that the boundary conditions are Neumann. Now we can consider a basis of dual eigenvectors of the position mode operators:
\begin{equation} \langle x^\mu(\sigma)|\widehat{x}_n^\mu =x_n^\mu\langle x^\mu(\sigma)|,\end{equation}
where we temporarily introduce a hat to distinguish the operator from its eigenvalue. The overlap
\begin{equation}A[x^\mu(\sigma)]=\langle x^\mu(\sigma)|A\rangle\end{equation}
can be interpreted as a scalar field which depends on a curve $x^\mu(\sigma)$ in spacetime. This is natural. Just as an ordinary field depends on a point $x$ in spacetime, representing a possible location of a point particle, the string field depends on a curve in spacetime, representing a possible configuration of a string. Of course, the string field can also be understood as an infinite tower of ordinary fields. Such a description makes manifest that a theory of a free string is indistinguishable from a theory with an infinite tower of free particles of a particular kind. At the interacting level, however, string theory is very different from particle theory. For this reason, interactions are rather opaque when formulated in terms of ordinary fields. In Witten's open bosonic string field theory, they appear as an infinite array of cubic nonlocal couplings with obscure relative coefficients. In the Schr{\" o}dinger representation, however, interactions are easy to understand.
We note that the position operator is Hermitian, and BPZ conjugation reverses the parameterization of the open string:
\begin{equation}x^\mu(\sigma)^\dag = x^\mu(\sigma),\ \ \ \ x^\mu(\sigma)^\star = x^\mu(\pi-\sigma),\end{equation}
which implies that the eigenstates have conjugation properties
\begin{equation}\langle x^\mu(\sigma)|^\dag = |x^\mu(\sigma)\rangle,\ \ \ \ \langle x^\mu(\sigma)|^\star = |x^\mu(\pi-\sigma)\rangle,\end{equation}
where $|x^\mu(\sigma)\rangle$ is the eigenstate of $\widehat{x}^\mu(\sigma)$ with eigenvalue $x^\mu(\sigma)$. With this we can derive the reality conjugate Schr\"{o}dinger functional of $A$:
\begin{eqnarray}
A^\ddag[x^\mu(\sigma)] \!\!\!\!\!\!\!\! && = \langle x^\mu(\sigma)|A^\ddag\rangle\nonumber\\
\!\!\!\!\!\!\!\! && = \langle x^\mu(\sigma)|A^{\star\dag}\rangle\nonumber\\
\!\!\!\!\!\!\!\! && =\overline{ \langle A^\star| x^\mu (\sigma)\rangle}\nonumber\\
\!\!\!\!\!\!\!\! && = \overline{\langle x^\mu(\pi-\sigma)|A\rangle}\nonumber\\
\!\!\!\!\!\!\!\! && = \overline{A[x^\mu(\pi-\sigma)]}.\label{eq:Sch_conj}
\end{eqnarray}
Thus reality conjugation takes the complex conjugate of the functional and reverses the parameterization of the string. We can describe the BPZ inner product in the Schr\"{o}dinger representation by inserting a resolution of the identity,
\begin{equation}1=\int [dx^\mu(\sigma)]|x^\mu(\sigma)\rangle\langle x^\mu(\sigma)|,\end{equation}
with the appropriately normalized functional integral measure. We find
\begin{eqnarray}
\langle A, B\rangle\!\!\!\!\!\!\!\! && = \langle A^\star| B\rangle\nonumber\\
\!\!\!\!\!\!\!\! && = \int [dx^\mu(\sigma)]\langle A^\star| x^\mu(\sigma)\rangle\langle x^\mu(\sigma)| B\rangle\nonumber\\
\!\!\!\!\!\!\!\! && = \int [dx^\mu(\sigma)]\langle x^\mu(\pi-\sigma)|A\rangle\langle x^\mu(\sigma)| B\rangle\nonumber\\
\!\!\!\!\!\!\!\! && = \int [dx(\sigma)]A[x^\mu(\pi-\sigma)] B[x^\mu(\sigma)].\label{eq:BPZ_Sch}
\end{eqnarray}
The reversal of the parameterization of the string in the first functional is directly related to the earlier observation that the BPZ inner product glues a point at an angle $\pi-\sigma$ on the unit half circle of the first state to the point at an angle $\sigma$ on the unit half circle of the second state.
It will be helpful to further articulate the connection between the Schr\"{o}dinger representation and the geometrical picture of a state as the unit half-disk carrying a vertex operator. The BPZ inner product can be computed as a correlation function of vertex operators in the UHP:
\begin{equation}\langle A, B\rangle = \big\langle( I\circ V_A(0))V_ B(0)\big\rangle_\text{UHP}.\end{equation}
This, in turn, can be formally computed by a path integral over the worldsheet fields in the UHP. Again concentrating on free bosons subject to Neumann boundary conditions and suppressing ghosts, this gives
\begin{equation}\langle A, B\rangle = \int[d X^\mu(z,\overline{z})]_{(z,\overline{z})\in\text{UHP}}\Big( I\circ V_A(0)\Big) V_ B(0)e^{-S}.\end{equation}
In the integrand, the vertex operators and the worldsheet action are of course understood as functionals of $X^\mu(z,\overline{z})$. We now factorize the integration into three components. First, we integrate $X^\mu(e^{i\sigma},e^{-i\sigma})=x^\mu(\sigma)$ on the unit half-circle $|z|=1$; second we integrate $X^\mu(z,\overline{z})$ outside the unit half circle $|z|>1$, with the boundary condition that $X^\mu(z,\overline{z})$ must be equal to $x^\mu(\sigma)$ at $|z|=1$; third we integrate $X^\mu(z,\overline{z})$ inside the unit half circle $|z|<1$, again with the boundary condition that $X^\mu(z,\overline{z})$ must be equal to $x^\mu(\sigma)$ at $|z|=1$. Therefore we have
\begin{eqnarray}\langle A, B\rangle \!\!\!\!\!\!\!\! && = \int [dx^\mu(\sigma)] \int_{X^\mu(e^{i\sigma},e^{-i\sigma})=x^\mu(\sigma)}[dX^\mu(z,\overline{z})]_{|z|>1}\nonumber\\ \!\!\!\!\!\!\!\! &&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \int_{X^\mu(e^{i\sigma},e^{-i\sigma})=x^\mu(\sigma)}[dX^\mu(z,\overline{z})]_{|z|<1} \Big( I\circ V_A(0)\Big) V_ B(0)e^{-S}.\end{eqnarray}
Next we recall the earlier comment that the vertex operators of the states $A$ and $B$ must be localized within the respective half-disks. This means that $V_B(0)$ will only depend on the worldsheet fields inside the unit half-circle, and $I\circ V_A(0)$ will only depend on the worldsheet fields outside. Furthermore, locality of the worldsheet theory implies that the exponential of the action factorizes into pieces which depend respectively on worldsheet fields inside and outside the unit half-circle. This implies that we can write
\begin{eqnarray}
\langle A, B\rangle \!\!\!\!\!\!\!\! && = \int [dx^\mu(\sigma)]\left(\int_{X^\mu(e^{i\sigma},e^{-i\sigma})=x^\mu(\sigma)}[dX^\mu(z,\overline{z})]_{|z|>1} I\circ V_A(0)e^{-S}\right)\nonumber\\
\!\!\!\!\!\!\!\! &&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\left(\int_{X^\mu(e^{i\sigma},e^{-i\sigma})=x^\mu(\sigma)}[dX^\mu(z,\overline{z})]_{|z|<1} V_ B(0)e^{-S}\right).
\end{eqnarray}
The path integral over the exterior of the half circle can be rewritten as a path integral over the interior after making a conformal transformation $I(z)=-1/z$. Accounting for the boundary conditions then gives
\begin{eqnarray}
\langle A, B\rangle \!\!\!\!\!\!\!\! && = \int [dx^\mu(\sigma)]\left(\int_{X^\mu(e^{i\sigma},e^{-i\sigma})=x^\mu(\pi-\sigma)}[dX^\mu(z,\overline{z})]_{|z|<1} V_A(0)e^{-S}\right)\nonumber\\
\!\!\!\!\!\!\!\! &&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\left(\int_{X^\mu(e^{i\sigma},e^{-i\sigma})=x^\mu(\sigma)}[dX^\mu(z,\overline{z})]_{|z|<1} V_ B(0)e^{-S}\right).
\end{eqnarray}
Comparing to \eq{BPZ_Sch} it is clear that this is the BPZ inner product of the Schr{\"o}dinger functionals
\begin{eqnarray}
A[x^\mu(\sigma)]\!\!\!\!\!\!\!\! && =\int_{X^\mu(e^{i\sigma},e^{-i\sigma})=x^\mu(\sigma)}[dX^\mu(z,\overline{z})]_{|z|<1} V_A(0)e^{-S},\nonumber\\
B[x^\mu(\sigma)]\!\!\!\!\!\!\!\! && =\int_{X^\mu(e^{i\sigma},e^{-i\sigma})=x^\mu(\sigma)}[dX^\mu(z,\overline{z})]_{|z|<1} V_B(0)e^{-S}.
\end{eqnarray}
Therefore the Schr{\"o}dinger functional can be derived as a path integral on the unit half-disk with the appropriate vertex operator at the origin and fixed boundary conditions on the worldsheet fields at $|z|=1$, representing the configuration of a string.
To make this concrete, let us use the path integral to evaluate the Schr{\"o}dinger functional of the $SL(2,\mathbb{R})$ vacuum. A similar calculation (for the closed string) is discussed in chapter 2 of Polchinski \cite{Polchinski}. We write this functional with the Greek letter omega,
\begin{equation}\Omega[x(\sigma)] = \langle x(\sigma)|0\rangle,\end{equation}
in order to avoid the unfortunate notation $0[x(\sigma)]$. The vertex operator in this case is the identity, so we only need to evaluate
\begin{equation}
\Omega[x(\sigma)] =\int_{X^\mu(e^{i\sigma},e^{-i\sigma})=x^\mu(\sigma)}[dX^\mu(z,\overline{z})]_{|z|<1}\, e^{-S}.
\end{equation}
The trick is to make a change of variables in the path integral
\begin{equation}X^\mu(z,\overline{z}) = x^\mu(z,\overline{z})+Y^\mu(z,\overline{z}),\end{equation}
where $x^\mu(z,\overline{z})$ is a solution to the Laplace equation on the unit half-disk
\begin{equation}\partial\overline{\partial} x^\mu(z,\overline{z})=0,\end{equation}
subject to the boundary conditions
\begin{equation}x^\mu(z,\overline{z})|_{z=e^{i\sigma}}=x^\mu(\sigma)\ \ \ \ \partial x^\mu(z,\overline{z})|_{\mathrm{Im}(z)=0}=\overline{\partial}x^\mu(z,\overline{z})|_{\mathrm{Im}(z)=0}.
\end{equation}
The first says that $x^\mu(z,\overline{z})$ matches the argument of the wavefunctional on the unit half-circle, and the second says that $x^\mu(z,\overline{z})$ satisfies Neumann boundary conditions on the real axis. In particular, this implies that $Y^\mu(z,\overline{z})$ must vanish on the unit half-circle. We now change the integration variable from $X^\mu(z,\overline{z})$ to $Y^\mu(z,\overline{z})$; since they differ though a shift by a fixed function, the measure is unchanged and we can write
\begin{equation}
\Omega[x(\sigma)] =\int_{Y^\mu(e^{i\sigma},e^{-i\sigma})=0}[dY^\mu(z,\overline{z})]_{|z|<1}\, \exp\Big(-S\big[x^\mu(z,\overline{z}) +Y^\mu(z,\overline{z})\big]\Big) .
\end{equation}
Since $x(z,\overline{z})$ satisfies the classical equations of motion with consistent boundary conditions we can show that
\begin{equation}
S\big[x^\mu(z,\overline{z}) +Y^\mu(z,\overline{z})\big] = S[x^\mu(z,\overline{z})]+S[Y^\mu(z,\overline{z})],
\end{equation}
so
\begin{eqnarray}
\Omega[x(\sigma)] \!\!\!\!\!\!\!\! && =\int_{Y^\mu(e^{i\sigma},e^{-i\sigma})=0}[dY^\mu(z,\overline{z})]_{|z|<1}\, e^{-S[x^\mu(z,\overline{z})]-S[Y^\mu(z,\overline{z})]} \nonumber\\
\phantom{\Bigg)}\!\!\!\!\!\!\!\! && = \mathcal{N} e^{-S[x^\mu(z,\overline{z})] },
\end{eqnarray}
where $\mathcal{N}$ is a constant determined by evaluating the path integral over $Y^\mu(z,\overline{z})$, and is completely independent of $x^\mu(\sigma)$. Next we must compute the worldsheet action evaluated on the classical solution $x^\mu(z,\overline{z})$. In our conventions ($\alpha'=1$) the Polyakov action is
\begin{equation}S = \frac{1}{2\pi}\int d^2 z\, \partial X^\mu (z,\overline{z})\overline{\partial}X_\mu (z,\overline{z}),\end{equation}
where $d^2z = 2dx dy$ if $z=x+iy$. The solution of Laplace's equation with the assumed boundary conditions can be expressed in terms of the position modes of $x^\mu(\sigma)$
\begin{equation}x^\mu(z,\overline{z}) = x_0^\mu + \sum_{n=1}^\infty x_n^\mu(z^n+\overline{z}^n).\end{equation}
Plugging in gives
\begin{equation}\Omega[x(\sigma)] = \mathcal{N}\exp\left[-\frac{1}{2}\sum_{n=1}^\infty n \,\eta_{\mu\nu}x_n^\mu x_n^\nu\right].\label{eq:SL2R_Sch}\end{equation}
\begin{exercise} Do this calculation. \end{exercise}
\noindent This is a Gaussian in the space of string position modes. The position zero mode does not appear, consistent with the fact that the $SL(2,\mathbb{R})$ vacuum has vanishing momentum. If we ignore timelike modes, the Gaussian has a maximum when the entire curve $x^\mu(\sigma)$ shrinks to a point. In the time direction the Gaussian is inverted, and blows up as the string extends. This does not look like a normalizable state, but in practice this can be dealt with by Wick rotation.
One way to check this calculation is to note that the $SL(2,\mathbb{R})$ vacuum can be viewed as a tensor product of an infinite number of harmonic oscillator vacua, one for each mode oscillator $\alpha_n^\mu$. The string mode oscillators and the position modes, however, are not canonically normalized. To relate to the conventional harmonic oscillator we should identify
\begin{equation}a^\dag \sim \frac{\alpha_{-n}^\mu}{\sqrt{n}},\ \ \ \ a\sim \frac{\alpha_n^\mu}{\sqrt{n}},\ \ \ \ x\sim \sqrt{n}x_n^\mu,\ \ \ \ (n\geq 1).\end{equation}
The ground state wavefunction for a harmonic oscillator is $e^{-\frac{1}{2}x^2}$. Substituting $\sqrt{n}x_n^\mu$ for $x$ and taking the product over all $n$ and $\mu$ gives \eq{SL2R_Sch}.
\begin{exercise}
\label{ex:vac_functional}
Write the $SL(2,\mathbb{R})$ vacuum functional explicitly in terms of the curve $x^\mu(\sigma)$ by finding an integral kernel $\omega(\sigma_1,\sigma_2)$ such that
\begin{equation}\Omega[x^\mu(\sigma)] = \exp\left[-\frac{1}{2}\int_0^\pi d\sigma_1\int_0^\pi d\sigma_2\, x^\mu(\sigma_1)x_\mu(\sigma_2)\omega(\sigma_1,\sigma_2)\right]\end{equation}
This representation is somewhat tricky since the integral kernel is a nontrivial distribution. To confirm that the distribution has been correctly defined, check your result by substituting the mode expansion of $x^\mu(\sigma)$ to recover \eq{SL2R_Sch}.
\end{exercise}
To get the complete $SL(2,\mathbb{R})$ vacuum functional on a D$p$-brane, we must include additional factors for the Dirichlet coordinates and the ghosts. We will not go into it further because, it turns out, the Schr{\"o}dinger representation is not very useful for calculations. As is often the case, the path integral is more useful to think about than to calculate. The main utility of the Schr{\"o}dinger representation is that it makes explicit the connection between open string states and a portion of open string worldsheet with operator insertions. It also explains the meaning of cutting and gluing worldsheets, a concept which is used repeatedly in string field theory calculations.
\subsection{Witten's Open Bosonic SFT}
Now we are ready to define Witten's open bosonic SFT. The task is to find a nonlinear extension of the linearized equations of motion $Q\Psi=0$ and define an appropriate action principle. First we note an analogy between string fields and gauge fields formulated in the language of differential forms:
\begin{eqnarray}
{\text{rank of a} \atop \text{form}}\!\!\!\!\!\!\!\! &&\ \ \ \longleftrightarrow \ \ \ \text{ghost number};\nonumber\\
\text{exterior derivative }d\!\!\!\!\!\!\!\! &&\ \ \ \longleftrightarrow\ \ \ \text{BRST operator }Q\phantom{\bigg)};\nonumber\\
\text{gauge potential }A\!\!\!\!\!\!\!\! && \ \ \ \longleftrightarrow\ \ \ {\text{dynamical string} \atop \text{field }\Psi}.
\end{eqnarray}
This analogy suggests a nonlinear gauge invariance of the string field
\begin{equation}\Psi' = \Psi + Q\Lambda + [\Psi,\Lambda],\end{equation}
where $\Lambda$ is an infinitesimal gauge parameter. The proposed gauge symmetry requires defining a product between the string fields $\Psi$ and $\Lambda$. This is Witten's open string star product. Sometimes this is written $A*B$, but usually we will simply write $AB$. Let us assume for the moment that a suitable product has been defined and proceed. There is only one gauge covariant, nonlinear extension of the equations of motion:
\begin{equation}Q\Psi + \Psi^2= 0.\end{equation}
These resemble the equations of motion of Chern-Simons theory. This leads to an action
\begin{equation}S = -\frac{1}{2}\mathop{\rm Tr}\nolimits(\Psi Q\Psi)-\frac{1}{3}\mathop{\rm Tr}\nolimits(\Psi^3),\end{equation}
for an appropriately defined trace operation. The consistency of this action relies on the following ``axioms:"
\begin{eqnarray}
\!\!\!\!\!\!\!\! && (1)\ Grading:\ \ \ \ \ \, \mathrm{gh}(QA) = \mathrm{gh}(A)+1\phantom{\Big)},\nonumber\\
\!\!\!\!\!\!\!\! && \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{gh}(AB) = \mathrm{gh}(A)+\mathrm{gh}(B),\nonumber\\
\!\!\!\!\!\!\!\! && \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{If }\mathop{\rm Tr}\nolimits(A)\text{ is nonzero, then gh}(A)=3\phantom{\Big)},\nonumber\\
\!\!\!\!\!\!\!\! &&\ \ \ \ \ \text{Here ``gh" refers to ghost number. Similar properties (except perhaps }\nonumber\\
\!\!\!\!\!\!\!\! &&\ \ \ \ \ \text{the last) hold for Grassmann parity mod }\mathbb{Z}_2;\nonumber\\
\!\!\!\!\!\!\!\! &&(2)\ Nilpotency:\ \ \ \ Q^2=0;\phantom{\Bigg)}\nonumber\\
\!\!\!\!\!\!\!\! && (3)\ Derivation\ property:\ \ \ \ Q(AB) = (QA)B+(-1)^{|A|}AQB;\nonumber\\
\!\!\!\!\!\!\!\! && (4)\ Associativity:\ \ \ \ A(BC) = (AB)C;\phantom{\Bigg)}\nonumber\\
\!\!\!\!\!\!\!\! && (5)\ Integration\ by\ parts:\ \ \ \ \mathop{\rm Tr}\nolimits(QA) = 0;\nonumber\\
\!\!\!\!\!\!\!\! && (6)\ Cyclicity:\ \ \ \ \mathop{\rm Tr}\nolimits(AB) = (-1)^{|A||B|}\mathop{\rm Tr}\nolimits(BA).\phantom{\Bigg)}\nonumber\\
\!\!\!\!\!\!\!\! && (7)\ Nondegeneracy:\ \ \ \ \text{If }\mathop{\rm Tr}\nolimits(AB)\text{\ vanishes for all }A\text{, then }B=0.\nonumber
\end{eqnarray}
These properties imply that the state space $\mathcal{H}$ of the BCFT has been endowed with the structure of a cyclic, graded differential associative algebra. The same structure applies to matrix valued forms on a 3-manifold, which allows the definition of Chern-Simons theory.
Let us try to understand how to define the product and trace. The product is associative, and all associative products are, in some way or another, matrix products. The string field in the Schr{\"o}dinger representation is a functional of a curve
\begin{equation}\Psi[x^\mu(\sigma)]\end{equation}
and it is natural to interpret the curve as representing matrix indices, in some sense. However, a matrix should have two indices, and there is only one curve $x^\mu(\sigma)$. We can deal with this by regarding
\begin{wrapfigure}{l}{.21\linewidth}
\centering
\resizebox{1.5in}{1.2in}{\includegraphics{OpenSFT_Erler10.jpg}}
\end{wrapfigure}
the full curve as a pair of half-curves
\begin{eqnarray}
l^\mu(\sigma)\!\!\!\!\!\!\!\! && = x^\mu(\sigma),\ \ \ \ \ \ \ \ \ \, \sigma\in[0,\pi/2],\\
r^\mu(\sigma)\!\!\!\!\!\!\!\! && =x^\mu(\pi-\sigma),\ \ \ \ \sigma\in[0,\pi/2].
\end{eqnarray}
$l^\mu(\sigma)$ is the ``left half" of the string, and $r^\mu(\sigma)$ is the ``right half." The left and right halves join at a common point
\begin{equation}l^\mu\left(\frac{\pi}{2}\right)=r^\mu\left(\frac{\pi}{2}\right)=x^\mu\left(\frac{\pi}{2}\right)\end{equation}
called the ``midpoint." We may view the string field as a functional of the left and right halves of
\begin{wrapfigure}{l}{.18\linewidth}
\centering
\resizebox{1.3in}{1.3in}{\includegraphics{OpenSFT_Erler11.jpg}}
\end{wrapfigure}
\noindent the string
\begin{equation}
\Psi[l^\mu(\sigma),r^\mu(\sigma)],
\end{equation}
and we have a matrix. The associative product of string fields may then be defined
\begin{equation}
AB[l^\mu(\sigma),r^\mu(\sigma)]= \int [dw^\mu(\sigma)]A[l^\mu(\sigma),w^\mu(\sigma)]B[w^\mu(\sigma),r^\mu(\sigma)].
\end{equation}
\begin{wrapfigure}{l}{.18\linewidth}
\centering
\resizebox{1.1in}{1.1in}{\includegraphics{OpenSFT_Erler12.jpg}}
\end{wrapfigure}
\noindent This is a functional integral version of a matrix product. In words, you identify the right half curve in $A$ with the left half curve in $B$ and sum over the common half curve to derive $AB$. In a similar way, we can define the trace operation
\begin{equation}\mathop{\rm Tr}\nolimits[A] = \int [dw^\mu(\sigma)] A[w^\mu(\sigma),w^\mu(\sigma)].\end{equation}
This is quite formal and we have not even accounted for ghosts. We will give a more robust definition in a moment. The product and trace together define a cubic vertex $\mathop{\rm Tr}\nolimits[\Psi^3]$. In Feynman diagrams, the cubic vertex can be visualized as a process where three incoming strings collide and join along their halves. Since the action is cubic, gluing propagators together with this vertex generates all Feynman diagrams needed to compute open string amplitudes. Proving that these correspond to the ``correct" amplitudes---as integrals of forms over the moduli spaces of Riemann surfaces---is not completely straightforward. The result, however, is not surprising since we have a gauge invariant action, and therefore a consistent theory of interacting open strings. It seems unlikely that this could be different from the conventional string theory.
Note that, at the quantum level, open string Feynman diagrams will produce closed strings as intermediate states. This can be seen, for example, in the non-planar 1-loop 2-point function. The
\begin{wrapfigure}{l}{.23\linewidth}
\centering
\resizebox{1.7in}{1in}{\includegraphics{OpenSFT_Erler13.jpg}}
\end{wrapfigure}
\noindent corner of the moduli space where the open string propagators in the loop shrink to zero length (the ultraviolet from the open sting perspective) can be interpreted as a corner of moduli space where a tube of worldsheet becomes infinitely long, and the closed string states inside the tube must be on-shell. In this sense, quantum open bosonic string field theory is expected to describe closed string physics. However, considering the amount of time spent on the classical theory, not much has been done concerning quantum effects. The problem is confusing on several levels. The absence of closed string fields makes it unclear how closed strings appear as asymptotic states. There are unresolved questions as to whether the SFT path integral is gauge invariant. The closed string tachyon implies that new instabilities appear at the quantum level that cannot be handled in perturbation theory. There is some hope these problems would be better addressed in the context of open superstring field theory. The status of present understanding of quantum effects can mostly be found in \cite{Thorn,Taylor}.
The Schr{\"o}dinger representation captures the main idea behind the product and trace, but is not practical for calculation. Instead, we would like to define the action in terms of BCFT correlation functions. To transform to this language, we use the relation between the Scr{\"o}dinger functional and the path integral over the half-disk with vertex operator insertion. Suppose we want to compute
\begin{equation}\mathop{\rm Tr}\nolimits[AB]\end{equation}
\begin{wrapfigure}{l}{.18\linewidth}
\centering
\resizebox{1.2in}{.7in}{\includegraphics{OpenSFT_Erler14.jpg}}
\end{wrapfigure}
\noindent The product $AB$ instructs us to glue the right portion of the half-circle bounding the half-disk of $A$ to the left portion of the half-circle bounding the half-disk of $B$; the trace glues the left portion of the half-circle bounding the half-disk of $A$ to the right portion of the half circle bounding the half-disk of $B$. This defines a correlation function of vertex operators $V_A(0)$ and $V_B(0)$ on a ``pita" shaped surface. To make this appear less awkward, we apply a conformal transformation $I(z)=-1/z$ to the half-disk of $A$ before gluing to the half-disk of $B$. This results in a correlation function on the upper half plane
\begin{equation}\langle I\circ V_A(0) V_B(0)\rangle_\mathrm{UHP}\end{equation}
which defines the BPZ inner product. We therefore find,
\begin{equation}\mathop{\rm Tr}\nolimits[AB] = \langle A,B\rangle.\end{equation}
Note that symmetry and nondegeneracy of the BPZ inner product is equivalent to cyclicity and nondegeneracy of the trace.
\begin{wrapfigure}{l}{.24\linewidth}
\centering
\resizebox{1.7in}{1.5in}{\includegraphics{OpenSFT_Erler15.jpg}}
\end{wrapfigure}
Let us mention a small visual problem which leads to an important issue of conventions. You might notice that the left half of the string $\sigma\in[0,\pi/2]$ maps to the {\it right} portion of the unit half-circle $\mathrm{Re}[e^{i\sigma}]>0$, while the right half of the string $\sigma\in[\pi/2,\pi]$ maps to the {\it left} portion of the unit half-circle $\mathrm{Re}[e^{i\sigma}]<0$. Thus it seems that what we are calling left and right is backwards from the point of view of the unit half-disk. This becomes particularly confusing when gluing surfaces to form the star product. When gluing the right half of the string of $A$ to the left half of the string of $B$, we must glue the half-disks of $A$ and $B$ in the opposite order. For this reason, \cite{Okawa} introduced a different gluing convention for defining the star product such that the left half of the string of the first state is identified with the right half of the string of the second state. This is called the {\it right handed} star product convention. For reasons that cannot be fully justified, we stick with the previous definition of the star product,
\begin{wrapfigure}{l}{.35\linewidth}
\centering
\resizebox{2.5in}{1.4in}{\includegraphics{OpenSFT_Erler16.jpg}}
\end{wrapfigure}
\noindent which is called the {\it left handed} convention. The left handed convention is what appears most commonly in older literature, such as the papers of Witten \cite{Witten} and Schnabl \cite{Schnabl}. A distinguishing feature of the left handed convention is that the sign of the tachyon field at the tachyon vacuum is positive. In the right handed convention, it is negative. Once we choose the left handed convention, however, the backwards gluing of surfaces could easily turn into an annoyance. This can be dealt with through an unconventional visualization of the complex plane. We simply draw the positive real axis so that it increases towards the left; the positive imaginary axis still increases upwards. In this visualization, the left half of the string sits on the left portion of the unit half circle, as would seem natural. Note that in this picture of the complex plane, the standard orientation of contour integrals is {\it clockwise}. This is part of the reason why this is called the ``left handed" convention. Both star product conventions appear in the literature up to recent times.
\begin{wrapfigure}{l}{.18\linewidth}
\centering
\resizebox{1.3in}{1.1in}{\includegraphics{OpenSFT_Erler17.jpg}}
\end{wrapfigure}
Next we express the cubic vertex $\mathop{\rm Tr}\nolimits(ABC)$ in terms of correlation functions. Gluing half-string segments appropriately gives a correlation function of three vertex operators on the surface shown left. In this case it is not quite as clear how to transform this into a correlation function on the UHP. We describe one approach which is of central importance in the study of analytic solutions. Each unit half-disk can be expressed in terms of a local coordinate $\xi$ satisfying $\mathrm{Im}(\xi)\geq 0$ and $|\xi|\leq 1$. We perform a conformal transformation to a new coordinate $z$
\begin{equation}z=f_\mathcal{S}(z)=\frac{2}{\pi}\tan^{-1} \xi.\end{equation}
The image of the half-disk in this coordinate system is given by a semi-infinite vertical strip $\mathrm{Im}(z)\geq 0$ and $\frac{1}{2}\geq \mathrm{Re}(z)\geq -\frac{1}{2}$. This is called the {\it sliver coordinate frame}; the conformal transformation $f_\mathcal{S}(z)$ is called the {\it sliver coordinate map}. The interval $[1,-1]$ on the real axis of the half-disk is mapped to the interval \ $[1/2,-1/2]$ \ on the real axis of the vertical strip; \ \ the left
\begin{wrapfigure}{l}{.4\linewidth}
\centering
\resizebox{2.7in}{1.5in}{\includegraphics{OpenSFT_Erler18.jpg}}
\end{wrapfigure}
\noindent half of the string $e^{i\sigma},\ \sigma\in[0,\pi/2]$ is mapped to the positive facing vertical edge of the strip $\frac{1}{2}+iy,\ y\geq 0$ and the right half of the string $e^{i(\pi-\sigma)},\ \sigma\in[0,\pi/2]$ is mapped to the negative facing vertical edge of the strip $-\frac{1}{2}+iy,\ y\geq 0$. The worldsheet path integral on the half-disk and on the semi-infinite vertical strip define the same Scr{\"o}dinger functional provided that the boundary conditions on the half-disk correspond to those on the vertical lines. Specifically, if the string coordinate takes a certain value at an angle $\sigma$ on the left portion of the unit half circle, it should take the same value at a point $y$ above the real axis on the positive facing boundary of the strip, where $y$ and $\sigma$ are related by
\begin{equation}\frac{1}{2}+iy = \frac{2}{\pi}\tan^{-1}e^{i\sigma},\end{equation}
which leads to
\begin{equation}y = \frac{1}{\pi}\mathrm{gd}^{-1}\sigma,\end{equation}
where $\mathrm{gd}^{-1}$ is the inverse of the Gudermannian function
\begin{equation}\mathrm{gd}x = 2\tan^{-1}\left(\tanh\frac{x}{2}\right).\end{equation}
As $\sigma$ ranges from $0$ to $\pi/2$, $y$ ranges from $0$ to infinity. The Gudermanian function is known for its relation to the Mercador projection; the relation between the angle with respect to the equator and vertical displacement on the map is the same as that between the angle $\sigma$ on the unit half circle and the coordinate $y$ on the edge of the strip. The midpoint $\sigma=\pi/2$ plays the role of the ``north pole," and is mapped to $+i\infty$ in the sliver coordinate frame. When using the doubling trick, the unit half-disk is replaced with a holomorphic copy of the unit disk $|\xi|\leq 1$; correspondingly, the semi-infinite strip of the sliver coordinate frame is represented as a holomorphic copy of the full infinite strip $-\frac{1}{2}\leq \mathrm{Re}(z)\leq\frac{1}{2}$.
\begin{wrapfigure}{l}{.35\linewidth}
\centering
\resizebox{2.4in}{1.5in}{\includegraphics{OpenSFT_Erler19.jpg}}
\end{wrapfigure}
In the sliver coordinate frame it is easy to visualize the star product in terms of gluing strips. To find the product $AB$, we glue the right edge of the strip of $A$ to the left edge of the strip of $B$; this creates a semi-infinite strip of width $2$ carrying operator insertions
\begin{equation}\big (T_1\circ f_\mathcal{S}\circ V_A(0)\big)\big( f_\mathcal{S}\circ V_B(0)\big),\end{equation}
where $T_a$ is the translation map
\begin{equation}T_a(z) = z+a,\end{equation}
and we fix the origin on the double strip to coincide with the location of $V_B$. Imposing the appropriate $\!\!$ boundary $\!\!$ conditions on the left and right edges of the double strip $\!\!$ and $\!\!$ performing $\!\!$ the
\begin{wrapfigure}{l}{.4\linewidth}
\centering
\resizebox{2.9in}{1.4in}{\includegraphics{OpenSFT_Erler20.jpg}}
\end{wrapfigure}
\noindent worldsheet path integral in the interior defines the Schr{\"o}dinger functional of the product $AB$. Note that integration over the worldsheet variables on the vertical line between $A$ and $B$ is precisely the sum over matrix indices of half-string Schr{\"o}dinger functionals. It is clear that the vertex operator of the state $AB$ is nonlocal. It effectively inserts a whole new piece of surface between $1$ and $0$ and places vertex operators at the edges of this region. Therefore the star product does not multiply inside the subspace of Fock states.
Now consider the 3-string vertex $\mathop{\rm Tr}\nolimits(ABC)$. To compute this, we place the strips of $A,B$ and $C$ side by side to form a strip of width $3$ with insertions
\begin{equation}\big(T_2\circ f_\mathcal{S}\circ V_A(0)\big)\big(T_1\circ f_\mathcal{S}\circ V_B(0)\big)\big(f_\mathcal{S}\circ V_C(0)\big).\end{equation}
\begin{wrapfigure}{l}{.6\linewidth}
\centering
\resizebox{4in}{1.6in}{\includegraphics{OpenSFT_Erler21.jpg}}
\end{wrapfigure}
\noindent The trace then glues the left and right edges of this strip to form a correlation function on a cylinder of circumference $3$. A cylinder of circumference $L$ can be mapped into the upper half plane using the conformal transformation
\begin{equation}f_L^{-1}(z) = \tan\frac{\pi z}{L}.\end{equation}
For $L=2$ this is the inverse of the sliver coordinate map. Correlation functions on a cylinder of circumference $L$ will be denoted $\langle ...\rangle_{C_L}$, and can be defined in terms of correlation functions on the UHP:
\begin{equation}\langle...\rangle_{C_L} = \langle f_L^{-1}\circ(...)\rangle_\mathrm{UHP}.\end{equation}
This gives an explicit definition of the cubic vertex in terms of the correlation function
\begin{equation}\mathop{\rm Tr}\nolimits(ABC) = \Big\langle\big(T_2\circ f_\mathcal{S}\circ V_A(0)\big)\big(T_1\circ f_\mathcal{S}\circ V_B(0)\big)\big(f_\mathcal{S}\circ V_C(0)\big)\Big\rangle_{C_3}.\end{equation}
In a similar way, the 2-string vertex (a.k.a. the BPZ inner product) can be written as a correlation function on a cylinder of circumference $2$,
\begin{equation}\mathop{\rm Tr}\nolimits(AB) = \langle A,B\rangle = \Big\langle \big(T_1\circ f_\mathcal{S}\circ V_A(0)\big)\big(f_\mathcal{S}\circ V_B(0)\big)\Big\rangle_{C_2},\end{equation}
and the 1-string vertex is a correlation function on a cylinder of circumference 1,
\begin{equation}\mathop{\rm Tr}\nolimits(A) = \Big\langle f_\mathcal{S}\circ V_A(0)\Big\rangle_{C_1}.\end{equation}
This generalizes in the obvious way to the trace of a product of any number of string fields.
The definition of the star product is still not fully concrete since it is awkward to work with Scr{\"o}dinger functionals. This can be remedied as follows. Consider a basis of states $|\phi_i\rangle$ for $\mathcal{H}$, for example a Fock space basis of $L_0$ eigenstates. We can construct a dual basis $|\phi^i\rangle$ with the property that
\begin{equation}\langle\phi^i,\phi_j\rangle = \delta^i_j.\end{equation}
This allows us to define the star product by a concrete expansion into the basis $|\phi_i\rangle$:
\begin{eqnarray}
AB \!\!\!\!\!\!\!\! && = \sum_{i}|\phi_i\rangle\langle(\phi^i)^\star| AB\rangle\nonumber\\
\!\!\!\!\!\!\!\! && = \sum_{i}|\phi_i\rangle\mathop{\rm Tr}\nolimits(\phi^i AB)\nonumber\\
\!\!\!\!\!\!\!\! && = \sum_{i}|\phi_i\rangle\Big\langle\big(T_2\circ f_\mathcal{S}\circ \phi^i(0)\big)\big(T_1\circ f_\mathcal{S}\circ V_A(0)\big)\big(f_\mathcal{S}\circ V_B(0)\big)\Big\rangle_{C_3}.
\end{eqnarray}
In this way, all operations in the theory are concretely defined by correlation functions on the cylinder.
\begin{exercise}
Show that all of the SFT axioms hold using the definition of the product and trace as correlation functions on the cylinder, assuming that all states are represented by well-behaved, e.g. Fock space vertex operators.
\end{exercise}
\begin{exercise}
The zero momenum sector of the string field can describe translationally invariant vacua of SFT. As an approximation to the full string field in this sector, consider the zero momentum tachyon state
\begin{equation}T c_1|0\rangle.\end{equation}
By substituting this into the action of Witten's open bosonic SFT, determine the resulting approximation to the tachyon potential. Note the existence of a nontrivial stationary point of the potential for $T>0$. This is the first approximation to the tachyon vacuum in the level truncation scheme.
Show that the energy density of the tachyon vacuum in this approximation is
\begin{equation}E = -\frac{2^{12}}{3^{10}}.\end{equation}
Compare this to the value predicted by Sen's conjecture.
\end{exercise}
Now let us consider the string field reality condition. The reality conjugate of the Schr{\"o}dinger functional was derived in \eq{Sch_conj}. In terms of the half string functional, this can be expressed
\begin{equation}A^\ddag[l^\mu(\sigma),r^\mu(\sigma)]=\overline{A[r^\mu(\sigma),l^\mu(\sigma)]}\end{equation}
This is analogous to the conjugate transpose of a matrix. One can show that we have the relations
\begin{eqnarray}
(QA)^\ddag\!\!\!\!\!\!\!\! && = (-1)^{|A|+1}QA^\ddag,\\
(AB)^\ddag \!\!\!\!\!\!\!\! && = B^\ddag A^\ddag,\\
\overline{\mathop{\rm Tr}\nolimits(A)}\!\!\!\!\!\!\!\! && = \mathop{\rm Tr}\nolimits(A^\ddag).
\end{eqnarray}
The last two are expected properties of the conjugate transpose of a matrix. From this it follows that the action is a real number if the dynamical string field satisfies the reality condition
\begin{equation}\Psi^\ddag=\Psi.\end{equation}
Moreover, nonlinear infinitesimal gauge transformations preserve the reality condition if the gauge parameter satisfies $\Lambda^\ddag = -\Lambda$. The reality condition at the nonlinear level is therefore the same as what we have already discussed for the linearized equations of motion.
Let us discuss the gauge invariant observables of the theory. They can be categorized as follows:
\begin{description}
\item{(1)} {\it The space of solutions modulo gauge transformation.} A special case of this is the space of inequivalent linearized fluctuations around a solution $\Psi_\text{sol}$. If we expand the string field
\begin{equation}\Psi = \Psi_\text{sol}+\varphi,\end{equation}
where $\varphi$ is a fluctuation of $\Psi_\text{sol}$, the action can be rewritten
\begin{equation}S[\Psi_\text{sol}+\varphi] = \underbrace{S[\Psi_\text{sol}]}_{\mathrm{constant}}+S_\text{sol}[\varphi],\end{equation}
where $S_\text{sol}[\varphi]$ takes the form
\begin{equation}S_\text{sol}[\varphi] = \frac{1}{2}\mathop{\rm Tr}\nolimits(\varphi Q_{\Psi_\text{sol}}\varphi)+\frac{1}{3}\mathop{\rm Tr}\nolimits(\varphi^3),\end{equation}
and
\begin{equation}Q_{\Psi_\text{sol}}=Q+[\Psi_\text{sol},\cdot].\end{equation}
One can show that $Q_{\Psi_\text{sol}}$ is nilpotent due to the equations of motion of $\Psi_\text{sol}$, and satisfies the same axioms as $Q$. From this it follows that the linearized equations of motion for the fluctuation field is
\begin{equation}Q_{\Psi_\text{sol}}\varphi=0.\end{equation}
Solutions must be identified modulo linearized gauge transformations
\begin{equation}\varphi'= \varphi +Q_{\Psi_\text{sol}}\Lambda.\end{equation}
The space of physical linearized fluctuations of $\Psi_\text{sol}$ is then given by the cohomology of $Q_{\Psi_\text{sol}}$ at ghost number $1$.
\item{(2)} {\it Scattering amplitudes around the perturbative vacuum or a nontrivial solution}. Computing scattering amplitudes generally requires fixing a gauge to determine the propagator. The most common gauge is {\it Siegel gauge}
\begin{equation}b_0\Psi = 0,\end{equation}
where the propagator (around the perturbative vacuum) takes the form
\begin{equation}\frac{b_0}{L_0} = b_0 \int_0^\infty dt\, e^{-t L_0}.\end{equation}
The Siegel gauge propagator is commonly visualized in a conformal frame where the coordinate $\xi$ on the unit half-disk is mapped to a coordinate $w$ on a semi-infinite horizontal strip
\begin{equation}\mathrm{Re}(w)\leq 0, \ \ \ 0\leq \mathrm{Im}(w) \leq \pi ,\end{equation}
through the conformal transformation
\begin{equation}w=\ln \xi.\end{equation}
This is different from the semi-infinite strip which appears in the sliver coordinate frame. In the sliver frame we impose boundary conditions for the left and right halves of the string on the semi-infinite edges, whereas in the $w$ frame these boundary conditions appear on the finite segment between $0$ and $i\pi$ on the imaginary axis. In the $w$ frame the operator $e^{-t L_0}$ can be visualized as gluing a strip of worldsheet of height $\pi$ and length $t$ to the edge of the semi-infinite strip which meets the imaginary axis. The propagator further integrates over the length $t$ of the strip, and inserts a vertical contour integral of the $b$ ghost representing $b_0$. We can then derive scattering amplitudes through Feynman diagrams by gluing the propagator strips together through cubic vertices.
\ \ \ There is a special amplitude, however, whose computation does not require gauge fixing or propagator: the closed string tadpole, representing the amplitude for emission or absorbtion of a single closed string off a D-brane. This can be computed through the so-called {\it Ellwood invariant} \cite{tadpole}
\begin{equation}\mathop{\rm Tr}\nolimits_\mathcal{V}(\Psi),\end{equation}
where $\mathop{\rm Tr}\nolimits_\mathcal{V}$ denotes the trace accompanied by an insertion of a BRST invariant closed string vertex operator $\mathcal{V}(z,\overline{z})$ of weight $(0,0)$ at the midpoint. Concretely, if $V_\Psi(0)$ is the vertex operator for the state $\Psi$ on the unit half-disk, the Ellwood invariant can be computed as a correlator on the cylinder of circumference 1:
\begin{equation}\mathop{\rm Tr}\nolimits_\mathcal{V}(\Psi) = \Big\langle \mathcal{V}(i\infty,-i\infty)f_\mathcal{S}\circ V_\Psi(0)\Big\rangle_{C_1}.\end{equation}
The closed string vertex operator is inserted on the ``top" of the cylinder at $i\infty$. This is a singular point, and mapping the correlator to the upper half plane generally produces a vanishing or divergent factor unless $\mathcal{V}$ is precisely a primary of weight $(0,0)$. Inserting the closed string vertex operator at $i\infty$ does not break the rotational symmetry of the cylinder, and in this way we can see that $\mathop{\rm Tr}\nolimits_\mathcal{V}(\cdot)$ is cyclic; moreover, since $\mathcal{V}$ is BRST invariant, $\mathop{\rm Tr}\nolimits_\mathcal{V}(\cdot)$ is vanishing on BRST exact states. From this it follows that the Ellwood invariant is unchanged by infinitesimal gauge transformation of $\Psi$; it is a gauge invariant observable. Suppose we have a classical solution $\Psi_\text{sol}$ describing $\mathrm{BCFT}_*$ in the string field theory of $\mathrm{BCFT}_0$. The Ellwood invariant is believed to be related to the closed string tadpole as
\begin{equation}\mathop{\rm Tr}\nolimits_\mathcal{V}(\Psi_\text{sol}) = \mathcal{A}_*(\mathcal{V})-\mathcal{A}_0(\mathcal{V}).\end{equation}
where $\mathcal{A}_*(\mathcal{V})$ and $\mathcal{A}_0(\mathcal{V})$ are the respective closed string tadpole amplitudes in $\mathrm{BCFT}_*$ and $\mathrm{BCFT}_0$. These can be computed as a correlation function in the matter component of the BCFT on the unit disk:
\begin{equation}
\mathcal{A}(\mathcal{V}) = \frac{1}{2\pi i}\Big\langle V^\mathrm{m}(0,0)\Big\rangle^\mathrm{m}_\mathrm{disk},
\end{equation}
where the matter vertex operator $V^\mathrm{m}$ is a primary of weight $(1,1)$ related to $\mathcal{V}$ through
\begin{equation}\mathcal{V} = c\overline{c}V^\mathrm{m}.\end{equation}
The Ellwood invariant has been generalized in a couple of ways to give information about the boundary state of the BCFT represented by the classical solution \cite{boundary1,boundary2}. A proposal to generalize it to other amplitudes appears in \cite{MM}.
\item{(3)} {\it The classical action}. The action evaluated on a solution is of course gauge invariant, but typically its value is divergent due to the infinite volume of the D-brane. However, for time independent solutions the action is equal to minus the energy of the solution times the volume of the time coordinate
\begin{equation}S = -E\cdot\mathrm{vol}_{X^0}.\end{equation}
\item{(4)} {\it Other observables}. String field theory makes a number of more subtle gauge invariant statements. For example, while the expectation value of the tachyon at the tachyon vacuum is not gauge invariant, it seems plausible that its {\it sign} is gauge invariant. A version of this statement is proven for analytic tachyon vacuum solutions in subsection \ref{subsec:dualL}. Related considerations in the superstring may give gauge invariants representing charges of topological solitons. Other observables are given by boundary condition changing projectors derived from singular gauge transformations, described in subsection \ref{subsec:singularGT}.
\end{description}
The most important classical solution in open bosonic SFT is the tachyon vacuum, $\Psi_\mathrm{tv}$. Sen's conjectures make the following prediction about the above gauge invariants:
\begin{description}
\item{(1)} Since the tachyon vacuum describes a configuration without D-branes or open strings, the cohomology of $Q_{\Psi_\mathrm{tv}}$ should be empty; all linearized fluctuations of the tachyon vacuum are pure gauge.
\item{(2)} Since there are no D-branes around the tachyon vacuum, the closed string tadpole should vanish. Therefore, the Ellwood invariant evaluates to minus the tadpole amplitude around the perturbative vacuum:
\begin{equation}\mathop{\rm Tr}\nolimits_\mathcal{V}(\Psi_\mathrm{tv}) = -\mathcal{A}_0(\mathcal{V}).\end{equation}
\item{(3)} The action divided by the volume should give the tension of the reference D-brane:
\begin{equation}\frac{S[\Psi_\mathrm{tv}]}{\mathrm{vol}} = \frac{1}{2\pi^2}.\end{equation}
\end{description}
\section{Lecture 2: Algebraic Framework}
Having prepared the necessary background, the main goal of these lectures is learning how to solve the equations of motion:
\begin{equation}Q\Psi +\Psi^2 = 0\end{equation}
Most analytic solutions of this equation are based on a limited set of algebraic ingredients, which we describe now.
\subsection{Wedge States}
\label{subsec:wedge}
To figure out how to solve the equations of motion, the first thing we should try is computing star products to see what kind of states we generate. The simplest state in the BCFT is the $SL(2,\mathbb{R})$ vacuum
\begin{equation}\Omega = |0\rangle.\end{equation}
In the sliver frame, $\Omega$ is represented by a semi-infinite strip of width $1$ carrying no operator insertions (or, equivalently, an insertion of the identity operator). We can multiply $\Omega$ with itself to give the state $\Omega^2$. This corresponds to gluing two semi-infinite strips of width 1 side-by-side to
\begin{wrapfigure}{l}{.5\linewidth}
\centering
\resizebox{3.5in}{1.3in}{\includegraphics{OpenSFT_Erler23.jpg}}
\end{wrapfigure}
form a semi-infinite strip of width $2$. Now it might appear that a strip of width 2 is not really different from a strip of width 1; they can be related by conformal transformation, specifically a scaling by a factor of $\frac{1}{2}$. The point, however, is that in this conformal transformation we must account for the boundary conditions of the path integral on the left and right vertical edges of the strip, which represent the left and right halves of the string in the Schr{\"o}dinger functional. For the free boson subject to Neumann boundary conditions, we have seen how to represent the $SL(2,\mathbb{R})$ vacuum as a functional of $x^\mu(\sigma)$, which in turn can be written as a functional of the left and right halves of the string:
\begin{equation}\Omega[x^\mu(\sigma)] = \Omega[l^\mu(y),r^\mu(y)],\end{equation}
where on the right hand side we parameterize the left and right halves of the string in terms of the height on a vertical edge of the strip
\begin{equation}
{l^\mu(y) = x^\mu(\sigma),\phantom{\Big)}\ \ \ \ \ \atop r^\mu(y) = x^\mu(\pi-\sigma), \phantom{\Big)}} \ \ \ \ \mathrm{for}\ \ \sigma\in[0,\pi/2]\ \ \mathrm{and}\ \ y=\frac{1}{\pi}\mathrm{gd}^{-1}\sigma.
\end{equation}
Now $l^\mu(y)$ gives the boundary condition for the path integral at a point $y$ above the real axis on the left vertical edge of the strip of width $1$, while $r^\mu(y)$ gives the boundary condition at the corresponding point on the right vertical edge. When we compute $\Omega^2$, the boundary conditions on the left and right edges are the same, but the strip over which we compute the path integral has doubled in width. If we scale by a factor of $\frac{1}{2}$, we are back to a strip of width $1$, but the boundary conditions on the left and right edges have also been scaled. Now the boundary condition at a point $y$ above the left vertical edge of the strip of width $1$ should be $l^\mu(2y)$, and similarly on the right edge. This implies that the Scr{\"o}dinger functional of $\Omega^2$ should be related to that of the $SL(2,\mathbb{R})$ vacuum through
\begin{equation}\Omega^2[l^\mu(y),r^\mu(y)] = \Omega[l^\mu(2y),r^\mu(2y)].\label{eq:Om2Om}\end{equation}
One might note that $(l^\mu(y),r^\mu(y))$ and $(l^\mu(2y),r^\mu(2y))$ actually represent the same curves in spacetime. But they are different as parameterized curves, and the $SL(2,\mathbb{R})$ vacuum functional is not invariant under reparameterizations of $x^\mu(\sigma)$. Therefore, $\Omega^2$ is really a different state from $\Omega$.
Continuing, we may construct $\Omega^3$ by gluing three strips of unit width side-by-side; the result is a strip of width $3$. Similarly $\Omega^4$ is a strip of width $4$ and so on for any positive integer $n$. It is clear that there is nothing particularly special about positive integer powers of the $SL(2,\mathbb{R})$ vacuum. We may generalize to any positive real power, defining $\Omega^\alpha$ as a semi-infinite strip of width $\alpha$ containing no operator insertions: \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{wrapfigure}{l}{1\linewidth}
\centering
\resizebox{2.5in}{1.1in}{\includegraphics{OpenSFT_Erler24.jpg}}
\end{wrapfigure}\\ \\ \\ \\ \\ \\ \\
\noindent $\Omega^\alpha$ is called a {\it wedge state}, and $\alpha$ is called the {\it wedge parameter}. Sometimes $\alpha$ is referred to as the
{\it width} of the wedge state. It is immediately clear from gluing strips that multiplication of wedge
\begin{wrapfigure}{l}{.48\linewidth}
\centering
\resizebox{3.3in}{.8in}{\includegraphics{OpenSFT_Erler25.jpg}}
\end{wrapfigure}
states is abelian
\begin{equation}\Omega^\alpha\Omega^\beta = \Omega^\beta\Omega^\alpha = \Omega^{\alpha+\beta}.\end{equation}
Geometrically, the restriction $\alpha\geq 0$ seems natural, but this deserves some comment. From the above discussion of $\Omega^2$ it is clear that all wedge states are related to the $SL(2,\mathbb{R})$ vacuum by a reparameterization of $\sigma$. This implies that $\Omega^\alpha$ is a Gaussian functional of $x^\mu(\sigma)$ for $\alpha\geq 0$. We can analytically continue this functional to complex $\alpha$. If $\mathrm{Re}(\alpha)<0$, it turns out that the functional is an inverted Gaussian $e^{x^2}$, and is therefore not normalizable. If $\mathrm{Re}(\alpha)>0$ but complex, the functional is a Gaussian with complex width, and appears to be normalizable. However, it is not clear how to think about such states in terms of a strip of worldsheet. For most purposes it is enough to assume that $\alpha$ is real and positive.
There are two special limits of the wedge parameter. The limit $\alpha\to 0$ defines the {\it identity string field}
\begin{equation}\Omega^0 = 1.\end{equation}
Sometimes the identity string field is written as $I$ or $|I\rangle$, but usually we will simply denote it as~$1$. The identity string field is characterized by a strip of vanishing width, and formally acts as the identity of the open string star product:
\begin{equation}1A = A1 = A.\end{equation}
This can be seen by viewing a generic state $A$ as a strip of unit width with vertex operator insertion.
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Multiplying by $1$ amounts to attaching a strip of vanishing width to either side, which leaves the strip unchanged. Another way to understand this is that the path integral on a strip of vanishing width is zero unless the boundary conditions on the left and right vertical edges match. Thus the identity string field must amount to a delta functional between the left and right halves of the string:
\begin{equation}I[l^\mu(\sigma),r^\mu(\sigma)] = \delta[l^\mu(\sigma)-r^\mu(\sigma)].\end{equation}
This is the functional equivalent of the Kronecker delta defining the identity matrix. Via the BPZ inner product, the existence of the identity string field is equivalent to the existence of the trace operation in Witten's open bosonic SFT. Given the identity string field, the trace can be defined
\begin{equation}\mathop{\rm Tr}\nolimits(A) = \langle I,A\rangle.\end{equation}
On the other hand, given the trace operation, the identity string field may be defined
\begin{equation}|I\rangle = \sum_i|\phi_i\rangle\mathop{\rm Tr}\nolimits(\phi^i),\end{equation}
given a basis of states $|\phi_i\rangle$ and a BPZ dual basis $|\phi^i\rangle$. In the past there has been some doubt as to whether the identity string field ``exists." As a delta functional, it is clearly more singular than wedge states of strictly positive width. But the doubts revolve more concretely around the question of whether the identity string field really has the properties it claims to have. One famous problem is that the ghost oscillator $c_0$ appears to be a derivation of the star product, but does not annihilate the identity string field \cite{RZ}. From the point of view of the analytic calculations we will do, it is consistent to assume that the identity string field exists. We can interpret the above difficulty as saying that $c_0$ does not operate in a ``nice" way within the open string star algebra. For more discussion of this see \cite{Sch_wedge}.
The opposite limit $\alpha\to\infty$ defines the {\it sliver state} $\Omega^\infty$. This corresponds to a strip of infinite width. To understand what this means more concretely, it is helpful to define the sliver state through its overlap with a test state. The overlap of $\Omega^\alpha$ with a test state can be computed as a correlation function on the cylinder:
\begin{equation}\langle \phi,\Omega^\alpha\rangle = \langle f_\mathcal{S}\circ V_\phi(0)\rangle_{C_{\alpha+1}}.\end{equation}
The cylinder can be described as a strip between $\frac{\alpha+1}{2}$ and $-\frac{\alpha+1}{2}$ with opposite vertical edges identified. At the center between $+\frac{1}{2}$ and $-\frac{1}{2}$ is the strip representing the test state $\phi$. In the limit
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$\alpha\to\infty$ the cylinder unfolds and becomes a correlation function on the UHP. Therefore the sliver state is defined by
\begin{equation}\langle\phi,\Omega^\infty\rangle = \langle f_\mathcal{S}\circ V_\phi(0)\rangle_\mathrm{UHP}.\end{equation}
It is clear that the sliver state is invariant under multiplication with other wedge states
\begin{equation}\Omega^\alpha\Omega^\infty = \Omega^\infty\Omega^\alpha = \Omega^\infty.\end{equation}
Formally it should be invariant under multiplication with itself:
\begin{equation}(\Omega^\infty)^2=\Omega^\infty.\end{equation}
Therefore the sliver state is a projector of the open string star algebra. We can see from the presentation of the correlator in the UHP that the Schr{\"o}dinger functional of $\Omega^\infty$ can be derived by path integral on the region $\mathrm{Re}(z)\geq\frac{1}{2}$ with the boundary condition
\begin{equation}X^\mu(z,\overline{z})|_{z=\frac{1}{2}+iy} = r^\mu(y),\end{equation}
multiplied by a path integral on the region $\mathrm{Re}(z)\leq-\frac{1}{2}$ subject to the boundary condition
\begin{equation}X^\mu(z,\overline{z})|_{z=-\frac{1}{2}+iy}=l^\mu(y).\end{equation}
This implies that the Scr{\"o}dinger functional factorizes between the left and right halves of the string:
\begin{equation}\Omega^\infty[l^\mu(\sigma),r^\mu(\sigma)] = \Omega[l^\mu(\sigma_{\frac{1}{2}}(\sigma))]\Omega[r^\mu(\sigma_\frac{1}{2}(\sigma))].\end{equation}
On the right hand side is the $SL(2,\mathbb{R})$ vacuum functional, where its argument is identified with the curve $l^\mu(\sigma_\frac{1}{2}(\sigma))$ and $\sigma_\frac{1}{2}(\sigma)$ is the appropriate map from the full string $\sigma\in[0,\pi]$ to the half string $\sigma_\frac{1}{2}\in[0,\frac{\pi}{2}]$.
\begin{exercise}
By implementing a conformal transformation from the region $\mathrm{Re}(z)\geq \frac{1}{2},\mathrm{Im}(z)\geq 0$ into the unit half-disk, show that
\begin{equation}
\sigma_\frac{1}{2}(\sigma) = \mathrm{gd}\left(\tan\frac{\sigma}{2}\right).
\end{equation}
Show that there is a residual ambiguity in the overall scale of the argument of the Gudermanian, which corresponds to invariance of the $SL(2,\mathbb{R})$ vacuum functional under reparameterizations generated by $L_1-L_{-1}$.
\end{exercise}
\noindent Viewed as an operator on the vector space of half string functionals, the sliver state is somewhat analogous to the projector $|0\rangle\langle 0|$ onto the ground state of the harmonic oscillator. This is a ``rank 1 projector"---where the rank of a projection operator is defined by its trace. The identity string field is also a projector since we should have
\begin{equation}1^2 = 1.\end{equation}
This can be viewed as the identity operator on the vector space of half-string functionals. For the harmonic oscillator, this is analogous to
\begin{equation}\mathbb{I} = |0\rangle\langle 0| +|1\rangle\langle 1|+|2\rangle\langle 2|+...\ ,\end{equation}
which is an infinite rank projector. The sliver state is characterized by a Gaussian functional and looks normalizable. Nevertheless, it is a singular state, and in fact worse than the identity string field. It should be viewed as something analogous to a distribution; while it can be multiplied with other elements of the star algebra, sliver states cannot, in general, be multiplied amongst themselves. The projector property of the sliver state is mostly unproblematic, but difficulties begin once we have ghost insertions. Consider, for example, multiplying two sliver states
on either side of the zero momentum tachyon:
\begin{equation}\Omega^\infty*c_1|0\rangle*\Omega^\infty.\end{equation}
The zero momentum tachyon can be viewed as a strip of width $1$ containing a $c$-ghost at the origin. The above expression instructs us to glue a strip of infinite width on either side, which leads to a 1-point function of the $c$-ghost on the upper half plane. But we need three $c$-ghosts to form a nonvanishing correlator. The other two $c$-ghosts are supposed to appear when contracting this expression with a test state, but in fact they are separated from the $c$ ghost at the origin by an infinite distance. Since the correlator of $c$ ghosts grows linearly with separation, the above state is actually divergent. The divergence can easily be turned into an ambiguity. For example, in the expression
\begin{equation}\Omega^\infty*(1-\Omega)*c_1|0\rangle*\Omega^\infty.\end{equation}
there is competition from the divergence of the $c$ ghost and the vanishing of $\Omega^\infty - \Omega^{\infty+1}$. Such expressions are simply not defined without regularization. For this reason, the sliver state cannot be included as part of the open string star algebra.
Wedge states are not enough by themselves to create solutions to the equations of motion. This is clear since wedge states carry ghost number $0$, but a solution has ghost number $1$. To get a
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richer class of states, we consider strips of worldsheet of varying width containing insertions of local operators. These are often called {\it wedge states with insertions}. It is useful to describe such states as factorized into products of wedge states and fields representing insertions of local operators. Consider for example the state shown left. Inside the semi-infinite strip there is an operator $\mathcal{O}_1$ a distance $x_1$ from the leftmost vertical edge and a distance $y_1$ above the real axis; the operator $\mathcal{O}_2$ is a distance $x_1+x_2$ from the left edge and a distance $y_2$ above the real axis, and so on. The idea is to introduce a string field $\mathcal{O}_i$ for every operator insertion; it is usually not useful to give separate symbols for the string field and the operator insertion.\ The
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\noindent string field $\mathcal{O}_i$ is defined by an infinitesimally thin strip containing an insertion of the operator $\mathcal{O}_i$ a distance $y_i$ above the real axis. The region of the surface between insertions $\mathcal{O}_i$ and $\mathcal{O}_{i+1}$ can be described as an empty strip of width $x_{i+1}$---in other words, a wedge state. We can then express wedge states with insertions as a product of wedge states and string fields representing operator insertions; the picture above corresponds to the state
\begin{equation}\Omega^{x_1}\mathcal{O}_1\Omega^{x_2} ... \Omega^{x_n}\mathcal{O}_n\Omega^{x_{n+1}}.\end{equation}
\begin{exercise}
Show that the zero momentum tachyon state can be represented as
\begin{equation}c_1|0\rangle = \frac{\pi}{2}\sqrt{\Omega}c\sqrt{\Omega},\end{equation}
where the string field $c$ is defined by an infinitely thin strip with a boundary insertion of the $c$-ghost.
\label{ex:tach}
\end{exercise}
A wedge state is an exponential whose base is the $SL(2,\mathbb{R})$ vacuum. Since the natural base of the exponential is Euler's number $e$, we can expect that the string field
\begin{equation}\ln \Omega\end{equation}
plays an important role in understanding wedge states. We can deduce the nature of this state by computing the derivative of a wedge state with respect to the wedge parameter. We will do this following the computation of Okawa \cite{Okawa}. Consider the overlap of $\Omega^\alpha$ with a test state $|\phi\rangle$ given by an insertion $\phi(0)$ at the origin of a semi-infinite strip of unit width. We assume that $\phi(0)$ has definite scaling dimension $h$ in the sliver coordinate frame. The overlap is given by a 1-point function on a cylinder of circumference $\alpha+1$:
\begin{equation}\langle \phi,\Omega^\alpha\rangle = \langle \phi(0)\rangle_{C_{\alpha+1}}.\end{equation}
Through a scale transformation we can adjust the circumference of the cylinder to unity, producing a factor from the conformal transformation of $\phi(0)$:
\begin{equation}\langle \phi,\Omega^\alpha\rangle = \left(\frac{1}{\alpha+1}\right)^h\langle \phi(0)\rangle_{C_1}.\end{equation}
Now take the derivative with respect to $\alpha$ and scale the cylinder back to circumference $\alpha+1$:
\begin{eqnarray}
\left\langle\phi,\frac{d}{d\alpha}\Omega^\alpha\right\rangle \!\!\!\!\!\!\!\! && = -h\left(\frac{1}{\alpha+1}\right)^{h+1}\langle\phi(0)\rangle_{C_1}\nonumber\\
\!\!\!\!\!\!\!\! && = -h\frac{1}{\alpha+1}\langle\phi(0)\rangle_{C_{\alpha+1}}.
\end{eqnarray}
Since $\phi(0)$ has scaling dimension $h$, its OPE with the energy-momentum tensor takes the form
\begin{equation}T(x)\phi(0) = ... +\frac{h}{z^2}\phi(0)+\frac{1}{z}\partial\phi(0)+...\ ,\end{equation}
which implies
\begin{equation}h\phi(0) = \oint_0\frac{dz}{2\pi i}zT(z)\phi(0).\end{equation}
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Therefore we can write
\begin{eqnarray}\!\!\!\!\!\!\!\! &&\!\!\!\!\!\!\!\!\left\langle\phi,\frac{d}{d\alpha}\Omega^\alpha\right\rangle\nonumber\\ \!\!\!\!\!\!\!\! && \!\!\!\!\!\!\! = -\frac{1}{\alpha+1}\left\langle \oint_0\frac{dz}{2\pi i}zT(z)\phi(0)\right\rangle_{C_{\alpha+1}}\!\!\!\!\!\!.\ \ \ \ \ \end{eqnarray}
Next we unravel the energy-momentum contour inside the cylinder. Suppose the cylinder is
represented as a strip $\frac{1}{2}+\alpha\geq \mathrm{Re}(z)\geq -\frac{1}{2}$, with opposite sides identified. We use the doubling trick, so the semi-infinite cylinder is represented by a holomorphic copy of the full infinite cylinder. Expanding the contour gives a contribution from the left vertical edge of the strip and the right vertical edge:
\begin{equation}
\oint_0\frac{dz}{2\pi i}z T(z) = \int_{-i\infty+\alpha+\frac{1}{2}}^{i\infty +\alpha+\frac{1}{2}}\frac{dz}{2\pi i}z T(z) - \int_{-i\infty-\frac{1}{2}}^{i\infty-\frac{1}{2}}\frac{dz}{2\pi i}z T(z).
\end{equation}
In the second integral we make a substitution $z\to z-(\alpha+1)$ so that both terms share a common integration variable:
\begin{equation}
\oint_0\frac{dz}{2\pi i}z T(z) = \int_{-i\infty+\alpha+\frac{1}{2}}^{i\infty +\alpha+\frac{1}{2}}\frac{dz}{2\pi i}\Big(z T(z)-(z-(\alpha+1))T(z-(\alpha+1))\Big).
\end{equation}
The identification on the vertical edges implies
\begin{equation}T(z)=T(z-(\alpha+1)).\end{equation}
Therefore
\begin{eqnarray}
\oint_0\frac{dz}{2\pi i}z T(z) \!\!\!\!\!\!\!\! && = \int_{-i\infty+\alpha+\frac{1}{2}}^{i\infty +\alpha+\frac{1}{2}}\frac{dz}{2\pi i}\Big(z-(z-(\alpha+1))\Big) T(z)\nonumber\\
\!\!\!\!\!\!\!\! && = (\alpha+1)\int_{-i\infty+\alpha+\frac{1}{2}}^{i\infty+\alpha+\frac{1}{2}}\frac{dz}{2\pi i}T(z).
\end{eqnarray}
Through contour deformation the precise horizontal placement of the vertical energy-momentum contour is not very important. So we can write
\begin{equation}\left\langle\phi,\frac{d}{d\alpha}\Omega^\alpha\right\rangle = -\left\langle\int_{-i\infty}^{i\infty}\frac{dz}{2\pi i} T(z)\phi(0)\right\rangle_{C_{\alpha+1}}.\end{equation}
Next we introduce the string field $K$, defined as an infinitely thin strip of worldsheet containing
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\noindent an insertion of the energy-momentum tensor, integrated vertically on the imaginary axis. The last equation can then be rewritten
\begin{equation}\frac{d}{d\alpha}\Omega^\alpha = -K\Omega^\alpha.\end{equation}
Since $\Omega^0=1$ is the identity string field, the solution of this differential equation implies that wedge states can be written
\begin{equation}\Omega^\alpha = e^{-\alpha K},\end{equation}
and in particular
\begin{equation}\ln \Omega = -K.\end{equation}
$K$ can be viewed as a Hamiltonian which ``generates" wedge states through Euclidean time evolution. Also, $K$ is real:
\begin{equation}K^\ddag =K.\end{equation}
The simplest way to see this is that the $SL(2,\mathbb{R})$ vacuum is real. Incidentally, the string field $K$ gives us a way to derive the vertex operator defining a wedge state. Note that
\begin{equation}\Omega^\alpha = \sqrt{\Omega}e^{-(\alpha-1)K}\sqrt{\Omega}.\end{equation}
The state on the right hand side can be viewed as a strip of width 1 containing an infinite number of vertical contour insertions of the energy-momentum tensor. To derive the vertex operator we must map the strip back to the canonical half-disk. Noting that
\begin{equation}f_\mathcal{S}^{-1}\circ \int_{-i\infty}^{i\infty}\frac{dz}{2\pi i}T(z) = \frac{\pi}{2}\int_{-i}^i \frac{d\xi}{2\pi i}(1+\xi^2)T(\xi),\end{equation}
the vertex operator of the wedge state $\Omega^\alpha$ is given by
\begin{equation}V_{\Omega^\alpha}(0) = \sum_{n=0}^\infty\frac{1}{n!}\left(-\frac{\pi(\alpha-1)}{2}\int_{-i}^i \frac{d\xi}{2\pi i}(1+\xi^2)T(\xi)\right)^n.\end{equation}
As expected, the vertex operator is nonlocal.
The {\it wedge algebra} is the subalgebra of the open string star algebra defined by taking products and linear combinations of wedge states. Since there are a continuum of wedge states, in general we can form continuous linear combinations:
\begin{equation}F(K) = \int_0^\infty dt f(t)\Omega^t.\end{equation}
The right hand side can be viewed as a function of $K$, obtained through Laplace transform of the function $f(t)$ in the ``time domain." Therefore, the algebra of wedge states is really an algebra of functions of a single variable $K$. Since an algebra of functions is a fairly simple thing---especially in comparison to the full open string star algebra---one can hope to define the algebra of wedge states fairly rigorously in the sense of functional analysis. No definitions are fully established as ``correct" according to current knowledge, but this line of thinking turns out to be useful. We start with a proposal due to Rastelli \cite{Rastelli}. First we consider the string field $F(K)$ as isomorphic to a function $F(k)$ of numbers $k$ in the spectrum of $K$. The spectrum is defined by the property that the string field
\begin{equation}K-k\end{equation} is not invertible. The inverse may be computed using the Schwinger parameterization
\begin{equation}\frac{1}{K-k}=\int_0^\infty dt\, e^{kt}\Omega^t.\label{eq:spec_int}\end{equation}
Since $\Omega^t$ approaches a constant (the sliver state) for large $t$, the integral is divergent for all non-negative $k$. Therefore we are looking for an algebra of functions of nonnegative real numbers $K$ (we henceforth suppress the distinction between the string field $K$ and an element of its spectrum). A precise definition of the algebra requires a topology so that we can discuss convergence. The most natural proposal is that the topology should be defined by the norm
\begin{equation}||F(K)||_{C^*} = \sup_{K\geq 0}|F(K)|.\label{eq:Cstar}\end{equation}
We have the usual properties of a norm
\begin{eqnarray}
\!\!\!\!\!\!\!\! && ||F(K)||_{C^*}\geq 0.\ \mathrm{and} \ ||F(K)||_{C^*}=0\ \text{if and only if}\ F(K)=0;\label{eq:norm1}\\
\!\!\!\!\!\!\!\! && ||aF(K)||_{C^*}=|a| \cdot ||F(K)||_{C^*},\ \ \ a\in\mathbb{C};\\
\!\!\!\!\!\!\!\! && ||F(K)+G(K)||_{C^*}\leq ||F(K)||_{C^*}+||G(K)||_{C^*}.
\end{eqnarray}
In addition we have the property
\begin{equation}||F(K)G(K)||_{C^*}\leq ||F(K)||_{C^*}\cdot ||G(K)||_{C^*}.\label{eq:norm2}\end{equation}
which implies that multiplication is continuous. Finally, the norm satisfies the {\it $C^*$ identity}
\begin{equation}||F(K) F(K)^\ddag ||_{C^*} = ||F(K)||_{C^*}^2.\end{equation}
Though $K$ is a real string field, reality conjugation in the wedge algebra can be nontrivial if $F(K)$ is a complex function of $K$. The space of bounded, continuous functions of $K$ is complete with respect to this norm, and the last two properties imply that it is a {\it $C^*$-algebra}. This leads us to propose that the wedge algebra is isomorphic to the $C^*$-algebra of bounded, continuous functions
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of non-negative~$K$. We denote this as $C_0(\mathbb{R}_{\geq 0})$. We make a few observations about this. First,
the identity string field and wedge states with positive wedge parameter are part of the algebra. Second, wedge states with negative wedge parameter are excluded since they are not bounded functions for $K\geq 0$. Third, the sliver state is excluded since it is not continuous:
\begin{equation}\Omega^\infty = \left\{\begin{matrix} \ 1\ \ \mathrm{at}\ \ K=0\ \ \\ \ 0\ \ \mathrm{for}\ \ K> 0\end{matrix}\right. .\end{equation}
In particular, while the sliver limit converges as an expansion into a basis of Fock states, it does not converge as a Cauchy sequence with respect to the norm. One can check that
\begin{equation}||\Omega^{Nn}-\Omega^n||_{C^*} = N^{-\frac{1}{N-1}} - N^{-\frac{N}{N-1}}.\end{equation}
which holds independent of $n$, and in particular is nonvanishing in the $n\to\infty$ limit. The divergence of the sliver limit has important consequences for understanding the structure of analytic solutions.
This definition for the wedge algebra is the simplest, but a deficiency is that it implies the existence of many states that cannot be constructed through a Laplace transform as a sum over wedge states. Whether such states exist is an open question. A proposal for defining Fock space coefficients for generic $F(K)$ is given in \cite{exotic}. For example, the coefficient of the state $L_{-2}|0\rangle$ is given by
\begin{equation}-\frac{1}{3}+\frac{4}{3}\int_0^\infty dK K e^{-K} F(K).\end{equation}
The integral is convergent for any $F(K)$ in $C_0(\mathbb{R}_{\geq 0})$. However, in the absence of geometrical description as a superposition of semi-infinite strips in the sliver frame, it is not manifest that such states multiply in the expected way. This remains an open problem. The computations of \cite{MurataSchnabl} also imply that when the wedge algebra is extended to allow multiplication with $c$-ghosts, computing correlation functions requires some understanding of the analytic structure of $F(K)$ in the complex plane. A generic continuous function does not have an analytic continuation. We therefore mention two other possible definitions of the wedge algebra. When considering the integral \eq{spec_int} we implicitly assumed that the spectrum of $K$ was real. This seems natural since $K$ is a real string field. But on second thought \eq{spec_int} is divergent for any complex $k$ with positive real part. From the point of view of the previous paragraph, this divergence would presumably indicate the failure of the Schwinger parameterization to define the inverse. But presently we will take the divergence seriously. This suggests that the wedge algebra should be understood as an algebra of functions of a complex variable with non-negative real part. Since we want these functions to have a representation in terms of the Laplace transform, we further require that they should be {\it holomorphic} on the positive half of the complex $K$-plane. It is convenient to relate $K$ to a coordinate $\zeta$ on the unit disk with the transformation
\begin{equation}K(\zeta) = \frac{1+\zeta}{1-\zeta},\ \ \ \ \zeta\in D_2,\end{equation}
and introduce the norm
\begin{equation}||F(K)||_{D_2} = \sup_{\zeta\in D_2}\left|F\Big(K(\zeta)\Big)\right|.\end{equation}
This norm satisfies properties \eq{norm1}-\eq{norm2}, and in particular multiplication is continuous. However, we do not have the $C^*$ identity; there is no $C^*$-algebra of holomorphic functions. However, with this norm we can define two Banach $*$-algebras:
\begin{description}
\item{(1)} The {\it Hardy space} $H^\infty$ of bounded, holomorphic functions on the interior of the unit disk.
\item{(2)} The {\it disk algebra} $A(D_2)$ consisting of bounded, holomorphic functions on the interior of the unit disk which extend to continuous functions on the boundary of the unit disk.
\end{description}
We have the natural inclusions
\begin{equation}C_0(\mathbb{R}_{\geq 0})\supset H^\infty \supset A(D_2).\end{equation}
The disk algebra is most restricted; it does not even allow for a pure wedge state (aside from the identity string field). The Hardy space seems to best capture the present practical understanding of the wedge algebra,\footnote{The Hardy space is nearly the same as the space of {\it $L_0$ safe} states introduced in \cite{lightning}. $L_0$~safe states, however are only required to be polynomial bounded for nonnegative $\mathrm{Re}(K)$. Thus, for example, $L_0$ safe states include the string field $K$, which however is outside the Hardy space since it has infinite norm.} but---as we will see---it does not allow the formulation of Schnabl's solution. The significance of these various proposals remains an interesting formal question. It is possible that they are all relevant for different purposes.
\subsection{Schnabl's $\mathcal{L}_0$ }
An important role in the theory is played by the dilitation generator in the sliver coordinate frame, introduced by Schnabl:
\begin{equation}\mathcal{L}_0 = \oint_0 \frac{dz}{2\pi i} zT(z)\ \ \ \ (\text{sliver frame}).\end{equation}
This is different from the usual $L_0$ since the contour is integrated around the vertex operator on the strip of width 1, rather than the unit half-disk. To relate $\mathcal{L}_0$ to ordinary Virasoros, we must map back to the half-disk:
\begin{eqnarray}
\mathcal{L}_0\!\!\!\!\!\!\!\! && = f_\mathcal{S}^{-1}\circ\oint_0\frac{dz}{2\pi i} z T(z) = \oint_0 \frac{d\xi}{2\pi i}(1+\xi^2)\tan^{-1}\!\xi \,T(\xi)\ \ \ \ (\text{half-disk})\nonumber\\
\!\!\!\!\!\!\!\! && = L_0+\frac{2}{3}L_2-\frac{2}{15}L_4+...\ .
\label{eq:curlL0}\end{eqnarray}
Since $\mathcal{L}_0$ is made from positively moded Virasoros, we have $\mathcal{L}_0|0\rangle = 0$. This indicates that, in the sliver frame, we can shrink the contour without encountering poles since there is no vertex operator at the origin.
We will need to act $\mathcal{L}_0$ on wedge states with arbitrary wedge parameter. To do this, it helps to reexpress $\mathcal{L}_0$ as follows:
\begin{eqnarray}
\mathcal{L}_0 \!\!\!\!\!\!\!\! && = \int_{-i\infty+1/2}^{i\infty +1/2}\frac{dz}{2\pi i} z T(z)- \int_{-i\infty-1/2}^{i\infty-1/2}\frac{dz}{2\pi i} zT(z)\nonumber\\
\!\!\!\!\!\!\!\! && = \int_{-i\infty}^{i\infty}\frac{dz}{2\pi i}\left(z+\frac{1}{2}\right)T\left(z+\frac{1}{2}\right) - \int_{-i\infty}^{i\infty}\frac{dz}{2\pi i} \left(z-\frac{1}{2}\right)T\left(z-\frac{1}{2}\right).
\end{eqnarray}
In the last step we shifted the integration variable so that $z$ is purely imaginary. We have been careful to specify the contour so that the energy-momentum tensor is placed on the left and right vertical edges of the strip, corresponding to the unit circle in radial quantization. For a strip of general width, it is still true that the left and right vertical edges correspond to the unit circle in radial quantization. Therefore, for a strip whose left edge intersects the real axis at $l$ and whose right edge intersects the real axis at $r$, the action of $\mathcal{L}_0$ is represented by contour insertions
\begin{eqnarray}
\mathcal{L}_0\!\!\!\!\!\!\!\! && = \int_{-i\infty}^{i\infty}\frac{dz}{2\pi i}\left(z+\frac{1}{2}\right)T(z+l) - \int_{-i\infty}^{i\infty}\frac{dz}{2\pi i} \left(z-\frac{1}{2}\right)T(z+r)\nonumber\\
\!\!\!\!\!\!\!\! && = \frac{1}{2}\left( \int_{-i\infty}^{i\infty}\frac{dz}{2\pi i}T(z+l) + \int_{-i\infty}^{i\infty}\frac{dz}{2\pi i} T(z+r)\right)+ \int_{-i\infty}^{i\infty}\frac{dz}{2\pi i}zT(z+l) - \int_{-i\infty}^{i\infty}\frac{dz}{2\pi i} zT(z+r).\nonumber\\
\end{eqnarray}
The first two terms are the energy-momentum contours defining the string field $K$. The last two terms define an operator which we denote $\frac{1}{2}\mathcal{L}^-$:
\begin{equation}\frac{1}{2}\mathcal{L}^- = \int_{-i\infty}^{i\infty}\frac{dz}{2\pi i}zT(z+l) - \int_{-i\infty}^{i\infty}\frac{dz}{2\pi i} zT(z+r).\end{equation}
The factor of $1/2$ in front of $\mathcal{L}^-$ is conventional.
\begin{exercise}
Show that $\mathcal{L}^-$ is related to $\mathcal{L}_0$ through
\begin{equation}\mathcal{L}^- = \mathcal{L}_0-\mathcal{L}_0^\star = \frac{2}{3}(L_2-L_{-2})-\frac{2}{15}(L_4-L_{-4})+...\ ,\end{equation}
and in particular is BPZ odd.
\end{exercise}
\noindent The above shows that $\mathcal{L}_0$ acting on a generic string field $X$ takes the form
\begin{equation}\mathcal{L}_0 X = \frac{1}{2}\mathcal{L}^- X +\frac{1}{2}(K X+X K).\label{eq:L0Lm}\end{equation}
The utility of this decomposition is that $\mathcal{L}^-$ is BPZ odd. A BPZ odd operator built from contour integral of the energy-momentum tensor generates a so-called {\it midpoint preserving reparameterization}. Applying a midpoint preserving reparameterization to a string field implements a common diffeomorphism of the parameter $\sigma\in[0,\pi/2]$ on the left and right halves of the string in the Schr{\"o}dinger functional, and leaves the midpoint fixed. Midpoint preserving reparameterizations are symmetries of the action of Witten's open bosonic string field theory; they commute with the BRST operator, act as a homomorphism on star products, and leave the trace invariant. In particular $\mathcal{L}^-$ satisfies
\begin{equation}
[Q,\mathcal{L}^-]=0,\ \ \ \ \mathcal{L}^-(XY) = (\mathcal{L}^- X)Y + X(\mathcal{L}^-Y),\ \ \ \ \mathop{\rm Tr}\nolimits\Big(\mathcal{L}^- X\Big) = 0. \label{eq:Lmder}
\end{equation}
These properties make $\mathcal{L}^-$ slightly more convenient to work with than Schnabl's $\mathcal{L}_0$, which is not BPZ odd and is not a midpoint preserving reparameterization generator. The relation \eq{L0Lm} implies that the BPZ even part of $\mathcal{L}_0$ corresponds to an anticommutator with the string field $K$.
\begin{exercise}Prove \eq{Lmder}.\end{exercise}
It is useful to understand how $\mathcal{L}^-$ acts on wedge states with insertions. Consider a string field $\mathcal{O}$ defined by an infinitesimally thin strip containing boundary operator insertion $\mathcal{O}(0)$ of scaling dimension $h$ at the origin. Acting with $\frac{1}{2}\mathcal{L}^-$ requires placing the appropriate energy-momentum
\begin{wrapfigure}{l}{.2\linewidth}
\centering
\resizebox{1.5in}{1.2in}{\includegraphics{OpenSFT_Erler33.jpg}}
\end{wrapfigure}
contours on the left and right edges of the infinitesimally thin strip. It is easy to see that these contours can be joined into a single contour integral
\begin{equation}\oint \frac{dz}{2\pi i} zT(z)\end{equation}
surrounding the operator at the origin. This produces a factor of $h$. Therefore
\begin{equation}\frac{1}{2}\mathcal{L}^- \mathcal{O} = h\mathcal{O}.\end{equation}
\begin{exercise}By a similar argument, show that $\frac{1}{2}\mathcal{L}^- K = K$.\end{exercise}
\noindent Note that the energy-momentum contour insertion defining $K$ transforms with weight 1 under scale transformations. This demonstrates the general fact that $\frac{1}{2}\mathcal{L}^-$ acting on an infinitesimally thin strip with operator insertion produces a factor of the scaling dimension of that operator in the sliver coordinate frame. This together with the derivation property can be used to compute the action of $\frac{1}{2}\mathcal{L}^-$ on more general states. For example, on a wedge state we have
\begin{equation}\frac{1}{2}\mathcal{L}^-\Omega^\alpha = -\alpha K\Omega^\alpha.\end{equation}
Suppose string fields $\mathcal{O}_i$ represent boundary operator insertions on the real axis of weight $h_i$. We have
\begin{equation}\lambda^{\frac{1}{2}\mathcal{L}^-}\Big(\Omega^{\alpha_1}\mathcal{O}_1\Omega^{\alpha_2} ... \Omega^{\alpha_n}\mathcal{O}_n\Omega^{\alpha_{n+1}}\Big) = \lambda^{h_1+...+h_n}\Omega^{\lambda\alpha_1}\mathcal{O}_1\Omega^{\lambda\alpha_2} ... \Omega^{\lambda\alpha_n}\mathcal{O}_n\Omega^{\lambda\alpha_{n+1}}.\end{equation}
From this it is clear that $\frac{1}{2}\mathcal{L}^-$ generates scale transformations of semi-infinite strips with operator insertions. Recall in \eq{Om2Om} we described the relation between the half string functionals of $\Omega$ and $\Omega^2$. The same relation can be expressed
\begin{equation}\Omega^2 = 2^{\frac{1}{2}\mathcal{L}^-}\Omega,\end{equation}
using $\frac{1}{2}\mathcal{L}^-$.
Next consider
\begin{eqnarray}
\frac{1}{2}\mathcal{L}^- \Big(\sqrt{\Omega}X\sqrt{\Omega}\Big) \!\!\!\!\!\!\!\! && = \left(\frac{1}{2}\mathcal{L}^-\sqrt{\Omega}\right)X\sqrt{\Omega}+\sqrt{\Omega}\left(\frac{1}{2}\mathcal{L}^-X\right)\sqrt{\Omega}+\sqrt{\Omega}X\left(\frac{1}{2}\mathcal{L}^-\sqrt{\Omega}\right)\nonumber\\
\!\!\!\!\!\!\!\! && = -\frac{1}{2}K\sqrt{\Omega}X\sqrt{\Omega} +\sqrt{\Omega}\left(\frac{1}{2}\mathcal{L}^-X\right)\sqrt{\Omega}-\frac{1}{2}\sqrt{\Omega}XK\sqrt{\Omega}.
\end{eqnarray}
Bringing the first and last terms to the other side of the equation and using \eq{L0Lm} implies a frequently useful relation between $\mathcal{L}_0$ and $\mathcal{L^-}$:
\begin{equation}\mathcal{L}_0\Big(\sqrt{\Omega}X\sqrt{\Omega}\Big)=\sqrt{\Omega}\left(\frac{1}{2}\mathcal{L}^-X\right)\sqrt{\Omega}.\end{equation}
For example, consider the string field $c$ introduced in exercise \ref{ex:tach}. Since the $c$-ghost has weight $-1$, it follows that
\begin{equation}\frac{1}{2}\mathcal{L}^- c= -c,\end{equation}
from which we learn that
\begin{equation}\mathcal{L}_0\big(\sqrt{\Omega}c\sqrt{\Omega}\big) = -\sqrt{\Omega}c\sqrt{\Omega}.\end{equation}
The result of exercise \ref{ex:tach} then implies that the zero momentum tachyon state $c_1|0\rangle$ has $\mathcal{L}_0$ eigenvalue $-1$, as can be easily checked from \eq{curlL0}.
Derivations of an associative algebra come in two kinds: {\it inner derivations}, which take the form of a commutator with some element of the algebra, and {\it outer derivations}, which do not. It appears that $\mathcal{L}^-$ is an inner derivation, since we can define a string field $\mathcal{L}^-_L|I\rangle$ which satisfies
\begin{equation}\mathcal{L}^-X = \Big[\mathcal{L}^-_L|I\rangle,X\Big].\end{equation}
In the sliver coordinate frame, $\mathcal{L}^-_L|I\rangle$ is an infinitesimally thin strip containing $2 z T(z)$ integrated vertically on the imaginary axis. The subscript $L$ is an old notation, and denotes ``left." To explain, let $\mathcal{O}$ be an operator defined by a contour integral of a holomorphic operator $\phi(\xi)$ around the unit circle:
\begin{equation}\mathcal{O} = \oint_{|\xi|=1}\frac{d\xi}{2\pi i}w(\xi)\phi(\xi),\end{equation}
where $w(\xi)$ is a weight function which is holomorphic in the vicinity of the unit circle. The {\it left half} of this operator is defined by the portion of the contour on the positive half of the unit circle:
\begin{equation}\mathcal{O}_L=\int_{|\xi|=1,\mathrm{Re}(\xi)>0}\frac{d\xi}{2\pi i}w(\xi)\phi(\xi).\end{equation}
We may define the right half similarly. The state $\mathcal{L}^-_L|I\rangle$ is the same as the left half of $\mathcal{L}^-$ acting on the identity string field.
\begin{exercise}
Show that $K=\frac{\pi}{2}(L_1+L_{-1})_L|I\rangle$.
\end{exercise}
\noindent There is an important complication with extracting the left half of an operator. The contour on the positive half of the unit circle is pinned to the midpoint at $\pm i$. This means that repeated application of $\mathcal{O}_L$ is not guaranteed to be well-defined, since contours cannot be deformed away from each other at the midpoint in case of singularities in the OPE of $\phi(\xi)$ with itself. A notable example of this problem occurs with the BRST operator. We can try to represent it as an inner derivation using a string field $Q_L|I\rangle$:
\begin{equation}Q X = \Big[Q_L|I\rangle,X\Big].\end{equation}
In the sliver frame, $Q_L|I\rangle$ is an infinitesimally thin strip containing a contour insertion of the BRST current along the imaginary axis. However, the OPE of two BRST currents contains a third order pole, and this renders repeated application of $Q_L$ undefined. The state $Q_L|I\rangle$ was proposed long ago as an analytic solution of open bosonic SFT \cite{purelycubic}, but for this reason the solution is not well-behaved and does not appear to be physically meaningful. The BRST operator should really be understood as an outer derivation. However, in more general cases splitting operators into halves may be unproblematic if the weight function $w(\xi)$ vanishes fast enough at the midpoint to compensate for singular OPEs. How quickly it needs to vanish is a delicate question in general, but it appears that splitting $L_1+L_{-1}$---as needed for the string field $K$---does not encounter problems. The weight function of $L_1+L_{-1}$ vanishes quadratically at the midpoint. The weight function for $\mathcal{L}^-$ does not vanish quite as fast, only as $x^2\ln x$. Finite powers of $\mathcal{L}^-_L|I\rangle$ appear to be well-defined. However, ambiguities appear if we consider nonpolynomial combinations. Consider the object
\begin{equation} \lambda^{\frac{1}{2}\mathcal{L}^-_L|I\rangle}\Omega^\alpha,\ \ \ \ \lambda>0.\end{equation}
This is called a {\it slanted wedge} \cite{KZloop}. In the sliver frame it can be visualized as a strip of width~$\alpha$---like a wedge state---but the parameterization differs between the left and right vertical edges. On the left edge the parameterization has been scaled relative to the right by a factor of $\lambda$, so the Sch{\"o}dinger functional is formally expressed as
\begin{equation}\Omega[l^\mu(\lambda\alpha y),r^\mu(\alpha y)].\end{equation}
The Schr{\"o}dinger representation however is imprecise, and in this case is hiding an important subtlety. If a slanted wedge is a string field, it must at least be possible to extract its coefficients in the Fock space expansion. This requires evaluating the trace, which glues the left and right
\begin{wrapfigure}{l}{.5\linewidth}
\centering
\resizebox{3.5in}{1.7in}{\includegraphics{OpenSFT_Erler32.jpg}}
\end{wrapfigure}
edges of the semi-infinite strip with a ``slanted" identification. The resulting surface, however, is not conformally equivalent to the upper half plane, but rather to an {\it annulus}. If we fix the origin of the sliver coordinate $z$ on the slanted wedge to coincide with the intersection between the right vertical edge and the open string boundary, the conformal transformation to the annulus is given by \cite{KZloop}
\begin{equation}\zeta(z) = \exp\left[\frac{2\pi i}{\ln \lambda}\ln\left(\frac{(\lambda-1)z}{\alpha}+1\right)\right].\end{equation}
The annulus is bounded by two concentric circles. If $\lambda>1$, the outer circle has radius $1$ and represents the open string boundary, whereas the inner circle has radius $e^{-\frac{\pi^2}{\ln\lambda}}$ and is the image of the midpoint at $+i\infty$ in the sliver coordinate frame. In particular, the ``midpoint" of a slanted wedge is actually a nontrivial closed curve. The Fock space coefficients of a slanted wedge are not defined unless we fix boundary conditions for the worldsheet fields on the inner circle, which effectively requires specifying a (possibly off-shell) closed string state. For this reason a slanted wedge is not defined as string field by itself, though they are useful objects to think about in connection to loop amplitudes \cite{KZloop} and the boundary state \cite{boundary2}. The implication is that $\mathcal{L}^-_L|I\rangle$ is also not really a string field. Therefore $\mathcal{L}^-$ appears to be an {\it outer} derivation, like the BRST operator.
\subsection{$KBc$ Subalgebra}
\label{subsec:KBc}
Now we introduce a subalgebra of wedge states with insertions which is sufficient to find analytic solutions for the tachyon vacuum. It is natural to guess that the subalgebra should include the zero momentum tachyon state
\begin{equation}c_1|0\rangle = \frac{\pi}{2}\sqrt{\Omega}c\sqrt{\Omega},\end{equation}
since this is the most important fluctuation field of the D-brane which acquires expectation value after tachyon condensation. Therefore we can consider a subalgebra given by products of string fields $K$ and $c$. However, this subalgebra is not rich enough to describe interesting tachyon vacuum solutions. The crucial additional ingredient was introduced through considerations of gauge fixing. In level truncation, the tachyon vacuum is found after fixing Siegel gauge
\begin{equation}b_0\Psi = 0.\end{equation}
The problem is that once we have $b_0$ we must also consider $L_0$. $L_0$ does not operate inside the subalgebra of wedge states, but generates an unfathomably larger algebra which is poorly understood. However, we have seen that the analogue of $L_0$ in the sliver frame does operate within the wedge algebra. This suggests that we consider the gauge
\begin{equation}\mathcal{B}_0\Psi=0, \end{equation}
where $\mathcal{B}_0=f_\mathcal{S}\circ b_0$ is the $b$-ghost analogue of $\mathcal{L}_0$. This is called {\it Schnabl gauge}. The BPZ odd combination
\begin{equation}\mathcal{B}^- = \mathcal{B}_0-\mathcal{B}_0^\star\end{equation}
is a derivation of the open string star product and annihilates the trace, for essentially the same reasons as $\mathcal{L}^-$ does. One can check that
\begin{equation}\frac{1}{2}\mathcal{B}^- K = B,\end{equation}
where $B$ is a new string field analogous to $K$ but defined by a vertical contour insertion of the $b$-ghost. We have the relations
\begin{eqnarray}
\!\!\!\!\!\!\!\! && \mathcal{B}_0 X= \frac{1}{2}\mathcal{B}^- X + \frac{1}{2}(BX+(-1)^{|X|}XB),\label{eq:B0Bm}\\
\!\!\!\!\!\!\!\! && \ \mathcal{B}_0\Big(\sqrt{\Omega}X\sqrt{\Omega}\Big) = \sqrt{\Omega}\left(\frac{1}{2}\mathcal{B}^- X\right)\sqrt{\Omega}.
\end{eqnarray}
The fields $K$, $B$ and $c$ are together enough to find solutions for the tachyon vacuum. This is called the $KBc$ {\it subalgebra}.
We have
\begin{eqnarray}
K \!\!\!\!\!\!\!\! && =\mathrm{Grassmann\ even};\ \mathrm{gh}\#\ 0;\nonumber\\
B \!\!\!\!\!\!\!\! && =\mathrm{Grassmann\ odd};\ \mathrm{gh}\#\ -1;\nonumber\\
c \!\!\!\!\!\!\!\! && =\mathrm{Grassmann\ odd};\ \mathrm{gh}\#\ 1;
\end{eqnarray}
and
\begin{eqnarray}
\frac{1}{2}\mathcal{L}^-K = K,\!\!\!\!\!\!\!\! && \ \ \ \ \ \ \frac{1}{2}\mathcal{B}^- K=B,\nonumber\\
\frac{1}{2}\mathcal{L}^-B= B,\!\!\!\!\!\!\!\! &&\ \ \ \ \ \ \, \frac{1}{2}\mathcal{B}^- B=0,\nonumber\\
\frac{1}{2}\mathcal{L}^-c = -c,\!\!\!\!\!\!\!\! && \ \ \ \ \ \ \ \, \frac{1}{2}\mathcal{B}^- c=0.
\end{eqnarray}
The fields are real,
\begin{equation}K^\ddag= K,\ \ \ \ B^\ddag = B\ \ \ \ c^\ddag = c .\end{equation}
We have the important relations
\begin{eqnarray}
\!\!\!\!\!\!\!\! && [K,B] = 0,\ \ \ \ B^2=c^2=0, \ \ \ \ [B,c] = 1,\label{eq:KBcId1}\\
\!\!\!\!\!\!\!\! && \ \ QK=0,\ \ \ \ \ \ QB=K,\ \ \ \ \ \ Qc=cKc.\label{eq:KBcId2}
\end{eqnarray}
The last three imply that the $KBc$ subalgebra is closed under the action of the BRST operator. This is a prerequisite to finding a solution to $Q\Psi+\Psi^2=0$ in this subalgebra. Most of these relations are easily verified by the appropriate contour deformations inside correlation functions on the cylinder. The computation of $Qc$ however merits additional comment. Acting $Q$ on $c$ gives a string field defined by an infinitesimally thin strip containing $c\partial c(0)$ at the origin. The operator $\partial c(0)$, however, is different from $c(0)$ so at first it looks like we need to extend the $KBc$ subalgebra. However, note that
\begin{eqnarray}\partial c(0) \!\!\!\!\!\!\!\! && = \oint_0\frac{dz}{2\pi i} T(z) c(0)\nonumber\\
\!\!\!\!\!\!\!\! && = \int_{-i\infty+\epsilon}^{i\infty+\epsilon}\frac{dz}{2\pi i} T(z) c(0) - c(0) \int_{-i\infty-\epsilon}^{i\infty-\epsilon}\frac{dz}{2\pi i} T(z).
\end{eqnarray}
The final expression can be represented as a commutator with the string field $K$. Therefore we can write
\begin{equation}\partial c = [K,c].\end{equation}
There are two senses we can view this equation. We can introduce a new string field $\partial c$ defined by an infinitesimally thin strip containing $\partial c(0)$ at the origin. This equation then implies a nontrivial algebraic relation between the new field $\partial c$ and the old fields $K$ and $c$. Another interpretation is that this equation is merely a definition of the operator $\partial$ (in this case acting on $c$). The relation between $\partial$ and the derivative with respect to the sliver coordinate is an accidental consequence of the realization of the $KBc$ subalgebra we happen to be discussing. In any case, if $Qc = c\partial c$ we also have
\begin{equation}Qc = c[K,c] = cKc.\end{equation}
where in the last step we used $c^2=0$.
Now that we have a simple algebraic setup, it is hard to resist playing around a bit and seeing if we can find any solutions. You might notice that the field $c$ itself almost looks like a solution, except for the factor of $K$ which appears between the two $c$s when computing the BRST variation. This can be remedied by multiplying by $K$. Therefore
\begin{equation}\Psi = -cK\end{equation}
is a solution to the equations of motion. This is an example of a so-called {\it residual solution}. Residual solutions are interesting as a kind of foil to test how analytic solutions work, but for present purposes it is enough to say that they are not physically meaningful. However, if we simply add $c$ we get something more interesting:
\begin{equation}\Psi = c(1-K).\label{eq:Idtv}\end{equation}
This is a solution for the tachyon vacuum. Unfortunately the solution is not normalizable, in a similar way as the identity string field, and we cannot meaningfully compute the action to verify Sen's conjecture. If we substitute the solution into the action we encounter formal correlation functions on cylinders with vanishing circumference. These cannot be mapped to the upper half plane without regularizing the vanishing circumference, and the result depends on how the regularization is implemented. But there is another way to check that the solution represents the tachyon vacuum. We can see if it supports nontrivial physical fluctuations. For this we investigate the cohomology of the shifted kinetic operator
\begin{equation}Q_\Psi = Q + [c(1-K),\,\cdot\,].\end{equation}
It is interesting to consider how this operator acts on the string field $B$:
\begin{eqnarray}
Q_\Psi B \!\!\!\!\!\!\!\! && = K + [c,B](1-K)\nonumber\\
\!\!\!\!\!\!\!\! && = K + 1-K\nonumber\\
\!\!\!\!\!\!\!\! && = 1.\label{eq:QtvB}
\end{eqnarray}
Given a linearized fluctuation around the solution
\begin{equation}Q_\Psi \varphi = 0,\end{equation}
we can therefore write
\begin{equation}\varphi = 1*\varphi = \big(Q_\Psi B\big)\varphi = Q_\Psi\big(B\varphi\big).\end{equation}
This implies that all $Q_\Psi$-closed states are $Q_\Psi$ exact, and the cohomology is empty. Note that the cohomology is empty at {\it all} ghost numbers, not just ghost number $1$ which would be enough to exclude linearized fluctuations. For tachyon vacuum solutions in the $KBc$ subalgebra, the absence of cohomology is generally demonstrated by finding a string field $A$ satisfying
\begin{equation}Q_\Psi A = 1.\end{equation}
The string field $A$ is often called a {\it homotopy operator}.
We can ask about the cohomology around the residual solution $-cK$. Following \eq{QtvB} one can show that
\begin{equation}Q_{-cK}B = 0.\end{equation}
Thus the string field $B$ is $Q_{-cK}$-closed. In the $KBc$ subalgebra it cannot be exact, since there are no states at ghost number $-2$., If $B$ remains nontrivial in the cohomology in the full open string star algebra, this implies that the residual solution supports cohomology at ghost number $-1$. There are no conventional open string worldsheet theories with this property.
For computing the energy of tachyon vacuum solutions, we need to be able to evaluate the trace of elements of the $KBc$ subalgebra. Let us start by computing
\begin{equation}
\mathop{\rm Tr}\nolimits(\Omega^{\alpha_1}c\Omega^{\alpha_2}c\Omega^{\alpha_3}c)=\langle c(z_1) c(z_2) c(z_3)\rangle_{C_L}.
\end{equation}
On the right we reexpressed the trace as a correlation function on the cylinder. The circumference of the cylinder and the position of the $c$-ghost insertions is related to the wedge parameters through
\begin{eqnarray}
L\!\!\!\!\!\!\!\! && = \alpha_1+\alpha_2+\alpha_3,\\
z_2\!\!\!\!\!\!\!\! && = \alpha_3+z_3,\\
z_1\!\!\!\!\!\!\!\! && = \alpha_2+\alpha_3+z_3.\label{eq:wedge_corr}
\end{eqnarray}
The position $z_3$ is determined by a choice of origin on the cylinder $C_L$, which is not specified by the wedge parameters. In any case, rotational symmetry of the cylinder ensures that the correlator is independent of the choice of origin. We map from the cylinder to the upper half plane using
\begin{equation}f_L^{-1}(z) = \tan\frac{\pi z}{L},\end{equation}
which leads to
\begin{eqnarray}
\!\!\!\!\!\!\!\! && \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\langle c(z_1) c(z_2) c(z_3)\rangle_{C_L} = \Big\langle f_L^{-1}\circ\big(c(z_1) c(z_2) c(z_3)\big)\Big\rangle_{\mathrm{UHP}}\nonumber\\
\!\!\!\!\!\!\!\! && = \left(\frac{L}{\pi}\right)^3\left(\cos^2\frac{\pi z_1}{L}\right)\left(\cos^2\frac{\pi z_2}{L}\right)\left(\cos^2\frac{\pi z_3}{L}\right)\left\langle c\left(\tan\frac{\pi z_1}{L}\right) c\left(\tan\frac{\pi z_2}{L}\right) c\left(\tan\frac{\pi z_3}{L}\right)\right\rangle_\mathrm{UHP}.\nonumber\\
\end{eqnarray}
The correlator of three $c$ ghosts on the upper half plane is given by (with choice of normalization)
\begin{equation}\langle c(\xi_1) c(\xi_2) c(\xi_3)\rangle_\mathrm{UHP} = \xi_{12}\xi_{13}\xi_{23},\end{equation}
where $\xi_{12}=\xi_1-\xi_2$ and so on. Plugging this into the previous formula and using trigonometric identities gives
\begin{equation}\langle c(z_1) c(z_2) c(z_3)\rangle_{C_L} = \left(\frac{L}{\pi}\right)^3\sin\frac{\pi z_{12}}{L}\sin\frac{\pi z_{13}}{L}\sin\frac{\pi z_{23}}{L}.\label{eq:ccc}\end{equation}
The trace of a general state in the $KBc$ subalgebra can be found by computing the correlator
\begin{equation}\mathop{\rm Tr}\nolimits(\Omega^{\alpha_1}c\Omega^{\alpha_2}c\Omega^{\alpha_3}c\Omega^{\alpha_4}cB)=\langle c(z_1)c(z_2)c(z_3)c(z_4)B\rangle_{C_L},\end{equation}
where the circumference and $c$-ghost positions are related to the wedge parameters in a similar way as \eq{wedge_corr}, and the operator $B$ in the correlation function represents a vertical line integral of the $b$-ghost passing directly on the negative side of $z_4$. Without loss of generality we may assume that only a single $B$ insertion appears in the correlator, since if two or more appear we can reduce to one by commuting past the $c$s and using $B^2=0$. To evaluate the trace of a general state in the $KBc$ algebra it will also be necessary to take continuous superpositions of the correlator with varying wedge parameters. One way to compute the correlator is to use $\mathcal{B}^-$ invariance of the trace
\begin{equation}\mathop{\rm Tr}\nolimits\left(\frac{1}{2}\mathcal{B^-}(\Omega^{\alpha_1}c\Omega^{\alpha_2}c\Omega^{\alpha_3}c\Omega^{\alpha_4}c)\right) = 0.\end{equation}
Acting with $\frac{1}{2}\mathcal{B}^-$ produces four terms with a $B$ accompanying each wedge state. Commuting all of the $B$s to the right then gives a formula for the correlator with four $c$s and one $B$ in terms of correlators with only three $c$s. In this way we obtain
\begin{eqnarray}
\langle c(z_1)c(z_2)c(z_3)c(z_4)B\rangle_{C_L}\!\!\!\!\!\!\!\! && = \frac{L^2}{\pi^3}\left(z_1 \sin\frac{\pi z_{23}}{L}\sin\frac{\pi z_{24}}{L}\sin\frac{\pi z_{34}}{L} - z_2 \sin\frac{\pi z_{13}}{L}\sin\frac{\pi z_{14}}{L}\sin\frac{\pi z_{34}}{L}\right.\nonumber\\
\!\!\!\!\!\!\!\! &&\ \ \ \ \left. + z_3 \sin\frac{\pi z_{12}}{L}\sin\frac{\pi z_{14}}{L}\sin\frac{\pi z_{24}}{L} -z_4\sin\frac{\pi z_{12}}{L}\sin\frac{\pi z_{13}}{L}\sin\frac{\pi z_{23}}{L}\right).\ \ \ \ \ \ \ \ \ \label{eq:ccccB}
\end{eqnarray}
\begin{exercise}
Do this calculation.
\end{exercise}
\noindent An equivalent useful formula (corrected from \cite{SSFII}) is
\begin{eqnarray}
\langle c(z_1)c(z_2)c(z_3)c(z_4)B\rangle_{C_L}\!\!\!\!\!\!\!\! && = \frac{L^2}{4\pi^3}\left(z_{14} \sin\frac{2\pi z_{23}}{L}+z_{23}\sin\frac{2\pi z_{14}}{L}-z_{13}\sin\frac{2\pi z_{24}}{L}\right.\nonumber\\
\!\!\!\!\!\!\!\! &&\ \ \ \ \left. -z_{24}\sin\frac{2\pi z_{13}}{L}+z_{12}\sin\frac{2\pi z_{34}}{L} +z_{34}\sin\frac{2\pi z_{12}}{L}\right).\ \ \ \ \ \ \ \ \
\end{eqnarray}
Setting $z_{14}=L$ we can derive as a special case
\begin{equation}\langle c(z_1) c(z_2) c(z_3)\rangle_{C_L} = \frac{1}{4}\left(\frac{L}{\pi}\right)^3\left(\sin\frac{2\pi z_{12}}{L}-\sin\frac{2\pi z_{13}}{L}+\sin\frac{2\pi z_{23}}{L}\right).\end{equation}
\section{Lecture 3: Analytic Solutions}
In this lecture we describe the most widely studied analytic solutions of Witten's open bosonic SFT. This includes Schnabl's solution for the tachyon vacuum \cite{Schnabl}; Schnabl gauge solutions for marginal deformations \cite{KORZ,Schnabl_marg}; the simple tachyon vacuum \cite{simple}; the solution of Kiermaier, Okawa, and Soler \cite{KOS}, further generalized to arbitrary time-independent backgrounds in \cite{KOSsing}; and the solution for the Wilson line deformation introduced by Fuchs, Kroyter, and Potting \cite{FKP} and further generalized to arbitrary marginal deformations by Kiermaier and Okawa \cite{KO}.
Two important solutions which we do not discuss are the identity-like marginal and tachyon vacuum solutions of Takahashi and Tanimoto \cite{TT1,TT2}. These were in fact the first ``physical" analytic solutions discovered in open bosonic SFT, though they are not normalizable. However, using wedge-based techniques it is possible to construct variants of these solutions which are normalizable \cite{MaccaferriTT,IshibashiTT}.
Most of these solutions have analogues in open superstring field theory in the Wess-Zumino-Witten-like formulation \cite{Berkovits}. The analogue of Schnabl gauge marginal deformations is given in \cite{super_marg,Ok_super_marg,Ok_real_super_marg}; of the Kiermaier, Okawa, Soler marginal solution in \cite{NuomiOkawa}; of the marginal solutions of Fuchs, Kroyter, and Potting and of Kiermaier and Okawa in \cite{FKsuper,KOsuper}. All of these (excepting \cite{Ok_real_super_marg}) are derived from the corresponding solutions of the bosonic string following a simple recipe \cite{super_marg,FKsuper}. Nonperturbative solutions are much more challenging to find, and at present the only example is the tachyon vacuum of~\cite{supervac}. This may be considered analogous to the simple tachyon vacuum of~\cite{simple}. As yet there is no well-understood analogue of Schnabl's solution for the superstring.
\subsection{Schnabl's Solution}
\label{subsec:Sch}
We will give a derivation of Schnabl's solution which is rather different from the original approach, but is more direct from the perspective of our development. We look for solutions among states in the $KBc$ subalgebra satisfying the Schnabl gauge condition
\begin{equation}\mathcal{B}_0\Psi = 0.\end{equation}
A fairly general class of such states takes the form
\begin{equation}\Psi = \sqrt{\Omega}cBG(K)c\sqrt{\Omega}.\end{equation}
The completely general state in Schnabl gauge is more elaborate, but this ansatz turns out to be enough to find the solutions. To verify the Schnabl gauge condition, note
\begin{equation}\mathcal{B}_0\big(\sqrt{\Omega}cBG(K)c\sqrt{\Omega}\big) =\sqrt{\Omega}\frac{1}{2}\mathcal{B}^-(cBG(K)c)\sqrt{\Omega}.\end{equation}
Recall that $\mathcal{B}^-$ acts as a derivation and annihilates $B$ and $c$. The only possible contribution appears when $\mathcal{B}^-$ acts on $G(K)$, giving
\begin{equation}\frac{1}{2}\mathcal{B}^- G(K) = BG'(K).\end{equation}
However, this does not contribute due to $B^2=0$. Next we plug the ansatz into the equations of motion to fix the form of $G(K)$:
\begin{eqnarray}
Q\Psi \!\!\!\!\!\!\!\! && = -\sqrt{\Omega}cKBc G(K)c\sqrt{\Omega} + \sqrt{\Omega}cB G(K) cKc \sqrt{\Omega},\\
\Psi^2\!\!\!\!\!\!\!\! && = \sqrt{\Omega}cB G(K) c\Omega G(K) c\sqrt{\Omega} - \sqrt{\Omega}cB \Omega G(K)cG(K)c\sqrt{\Omega}.
\end{eqnarray}
Thinking a moment, it is clear that the equations of motion are equivalent to the following functional equation for $G(K)$:
\begin{equation} - K_1 G(K_2) +G(K_1)K_2+G(K_1)e^{-K_2}G(K_2)- e^{-K_1}G(K_1)G(K_2) = 0.\end{equation}
Since there are two variables $K_1,K_2$ and only one undetermined function $G(K)$, this equation looks over constrained. Still there is a solution. After some algebra we can rewrite this as
\begin{equation}\frac{G(K_1)}{K_1+e^{-K_1}G(K_1)}=\frac{G(K_2)}{K_2+e^{-K_2}G(K_2)}.\end{equation}
Since the left hand side is a function only of $K_1$, and the right hand side a function of $K_2$, the only way this can be consistent is if both sides are equal to a constant, which we call $\lambda$:
\begin{equation}\frac{G(K)}{K+\Omega G(K)} = \lambda.\label{eq:Grel}\end{equation}
This implies
\begin{equation}G(K) = \frac{\lambda K}{1-\lambda \Omega},\end{equation}
and
\begin{equation}\Psi_\lambda = \lambda \sqrt{\Omega}c \frac{KB}{1-\lambda \Omega}c\sqrt{\Omega}.\label{eq:Psil}\end{equation}
We have a 1-parameter family of $KBc$ solutions in Schnabl gauge. Incidentally, the solution satisfies the reality condition
\begin{equation}\Psi_\lambda^\ddag = \Psi_\lambda\end{equation}
if $\lambda$ is real. Since $K,B$ and $c$ are real string fields, the reality condition amounts to the statement that the solution reads the same way from the left as from the right.
Note that if $\lambda = 0$ we obtain the trivial solution
\begin{equation}\Psi_{\lambda=0} = 0.\end{equation}
This is the perturbative vacuum---the configuration where all fluctuations fields vanish, and the D-brane defining the SFT is undisturbed. If $\lambda$ is small we can expand the solution perturbatively:
\begin{equation}\Psi_\lambda = \lambda \sqrt{\Omega}cKBc\sqrt{\Omega} +\mathcal{O}(\lambda^2).\end{equation}
The leading order contribution is BRST exact:
\begin{equation}\sqrt{\Omega}cKBc\sqrt{\Omega} = Q\big(\sqrt{\Omega}Bc\sqrt{\Omega}\big).\end{equation}
This means that, for sufficiently small $\lambda$, the solution represents a deformation of the perturbative vacuum by a trivial element of the BRST cohomology. Physically, this represents no deformation at all, and for small enough $\lambda$ the solution is pure gauge. It is a little strange to find more than one solution for the perturbative vacuum in Schnabl gauge. Apparently, the Schnabl gauge condition does not completely fix the gauge.
What we wanted to find is a solution for the tachyon vacuum. We can hope that the tachyon vacuum will appear for large enough $\lambda$. One way to tell if we have a tachyon vacuum solution is if there is a homotopy operator,
\begin{equation}Q_{\Psi_\lambda} A_\lambda =1,\end{equation}
which trivializes the cohomology. Assuming the homotopy operator can be found in the $KBc$ subalgebra, it must take the form
\begin{equation}A_\lambda = B H(K)\end{equation}
for some $H(K)$. Now we can try to solve:
\begin{eqnarray}
1\!\!\!\!\!\!\!\! && = Q_{\Psi_\lambda} A_\lambda\nonumber\\
\!\!\!\!\!\!\!\! && = Q\big(B H(K)\big) + \sqrt{\Omega}cB G(K)c\sqrt{\Omega} B H(K) + B H(K)\sqrt{\Omega}cB G(K)c\sqrt{\Omega}\nonumber\\
\!\!\!\!\!\!\!\! && = K H(K) + \sqrt{\Omega}cB G(K)H(K)\sqrt{\Omega} + \sqrt{\Omega}H(K)G(K) Bc\sqrt{\Omega}\nonumber\\
\!\!\!\!\!\!\!\! && = H(K)(K+\Omega G(K))+\sqrt{\Omega}[H(K)G(K),Bc]\sqrt{\Omega}\nonumber\\
\!\!\!\!\!\!\!\! && =\frac{1}{\lambda} H(K)G(K)+\sqrt{\Omega}[H(K)G(K),Bc]\sqrt{\Omega},
\end{eqnarray}
where in the last step we used \eq{Grel}. The two terms are linearly independent. The first term should be equal to the identity string field, and the second must vanish. Both conditions are satisfied if $H(K) = \lambda/G(K)$, in other words
\begin{equation}H(K) = \frac{1-\lambda\Omega}{K}.\end{equation}
Now it looks like we have a solution for any $\lambda$ (even $\lambda=0$!) and therefore the cohomology should always be trivial. But we have to be careful to make sure $H(K)$ makes sense as a string field.
\begin{wrapfigure}{l}{.3\linewidth}
\centering
\resizebox{2in}{1.4in}{\includegraphics{OpenSFT_Erler34.jpg}}
\end{wrapfigure}
In fact, $H(K)$ has a pole at $K=0$ and is therefore divergent in the $C^*$ norm \eq{Cstar} {\it except} when $\lambda=1$, where
\begin{equation}||H(K)||_{C^*} = 1\ \ \ \ (\lambda=1).\end{equation}
To see that the pole should be taken seriously, consider the homotopy operator at $\lambda=0$:
\begin{equation}A_{\lambda=0}=\frac{B}{K}.\end{equation}
This is just an algebraic expression; the question is whether it can be defined as a string field. It is natural to suppose that the inverse of $K$ can be defined through the Schwinger parameterization as an integral over all wedge states
\begin{equation}\frac{B}{K} \stackrel{?}{=} B\int_0^\infty d\alpha\, \Omega^\alpha.\label{eq:BoK}\end{equation}
The integral by itself is divergent, since wedge states approach a constant---the sliver state---for large wedge parameter. This looks bad, but the question is actually more subtle since the integral is multiplied by $B$.
\begin{exercise}
Show that
\begin{equation}B\Omega^\alpha \sim \mathcal{O}\left(\frac{1}{\alpha^3}\right)\end{equation}
for large $\alpha$ by finding the operator $\phi$ of lowest scaling dimension such that $\langle\phi,B\Omega^\alpha\rangle$ is nonzero.
\label{ex:Bsliver}
\end{exercise}
\noindent The result of this exercise implies that \eq{BoK} is actually a finite state in the Fock space; at the upper limit the integrand goes as $1/\alpha^3$, and is integrable. The problem with \eq{BoK} is more basic; it does not define a homotopy operator for $Q$:
\begin{equation}
Q\left(\frac{B}{K}\right) = \int_0^\infty d\alpha K\Omega^\alpha = -\int_0^\infty d\alpha \frac{d}{d\alpha}\Omega^\alpha = 1-\Omega^\infty.\label{eq:QBovK}
\end{equation}
The presence of the sliver state negates the construction. It turns out that any wedge state has a Fock space expansion of the form
\begin{equation}\Omega^\alpha = |0\rangle + (\text{total Virasoro descendants of the vacuum}).\end{equation}
This reflects the fact that a strip of width $\alpha$ can be mapped into the unit half-disk by a conformal transformation, which in the operator formalism is implemented by total Virasoro generators. The descendant terms are BRST exact since total Virasoros can be derived by BRST variation of $b$ ghosts. The only piece which is nontrivial in the cohomology is the vacuum $|0\rangle$. This precisely cancels between the identity and sliver in \eq{QBovK}, and there is no difficulty expressing what remains in BRST exact form. For generic $\lambda$ we can still formally write the homotopy operator by defining the inverse of $K$ through the Schwinger parameterization
\begin{equation}A_\lambda = B\frac{1-\lambda\Omega}{K} = B\left(\lambda \int_0^1d\alpha \,\Omega^\alpha +(1-\lambda)\int_0^\infty d\alpha\,\Omega^\alpha\right).\end{equation}
The second term, with integration out to the sliver state, is the problematic contribution and is absent precisely when $\lambda=1$. Thus at $\lambda=1$ open string excitations are absent and
\begin{equation}\Psi_\text{Sch} = \sqrt{\Omega}c\frac{KB}{1-\Omega}c\sqrt{\Omega}\end{equation}
is a solution for the tachyon vacuum. This is Schnabl's solution. The homotopy operator is
\begin{equation}A_\text{Sch} = B\frac{1-\Omega}{K}=B\int_0^1 d\alpha\,\Omega^\alpha.\end{equation}
and contains a continuous superposition of wedge states from the identity up to the $SL(2,\mathbb{R})$ vacuum.
One thing we did not address is the nature of the state between the $c$s in the Schnabl gauge solution. This actually turns out to be tricky. We can write
\begin{equation}G(K) = \lambda K + \frac{\lambda^2 K\Omega}{1-\lambda \Omega}.\end{equation}
This state has infinite $C^*$ norm for any $\lambda$ due to the linear growth of the first term towards $K=\infty$. Indeed, the string field $K$ is singular in a similar way as the identity string field. In the present context this is not a concern, since the state (together with ghost insertions) appears multiplied by the $SL(2,\mathbb{R})$ vacuum in the solution, which effectively tames the singularity (see subsection \ref{subsec:dualL} for more explanation). We therefore focus on the second term. It is clear that the $C^*$ norm will be finite if the denominator does not have a zero for positive $K$. Since $e^{-K}$ is less than one, this can only happen if $\lambda$ is greater than one. Therefore
\begin{equation}\left|\left|\frac{\lambda^2 K\Omega}{1-\lambda \Omega}\right|\right|_{C^*} = \mathrm{finite},\ \ \ \ \ \text{iff }\lambda\ngtr 1.\end{equation}
The tachyon vacuum sits just on the edge of singularity at $\lambda=1$, but the norm is still finite:
\begin{equation}
\left|\left|\frac{K\Omega}{1-\Omega}\right|\right|_{C^*} = 1.
\end{equation}
It is interesting to comment on one of the most puzzling aspects of Schnabl's solution. If $\lambda$ is even infinitesimally different from $1$, the solution is pure gauge. But the solution as a state in the Fock space does not change that much. From the point of view of the $C^*$ norm, however, the difference is huge:
\begin{equation}
\lim_{\lambda\to 1^-}\left|\left|\frac{K\Omega}{1-\Omega}-\frac{\lambda^2 K\Omega}{1-\lambda \Omega}\right|\right|_{C^*} = 1.
\end{equation}
The states converge to each other pointwise as functions of $K$ except at $K=0$, where the first is always $1$ and the second always zero. From this point of view it seems that the solution for $\lambda$ slightly different from $1$ is ``missing" the sliver state. We will come back to this shortly. It is also worth noting that the limit $\lambda\to 1$ does not converge as a Cauchy sequence in the $C^*$ norm, in the same way as the sliver limit does not converge. This demonstrates that, with the appropriate notion of distance in the wedge algebra, the pure gauge solutions are far away from the tachyon vacuum.
From the point of view of bounded, continuous functions of $K\geq 0$ the Schnabl gauge solution exists for any $\lambda$ not greater than $1$. But from the point of view of bounded, {\it analytic} functions of $\mathrm{Re}(K)> 0$ there are further restrictions and a complication. To have finite $D_2$ norm we must require that $1-\lambda\Omega$ has no zeros for non-negative $\mathrm{Re}(K)$. There are an infinite number of zeros located at
\begin{equation}K_\mathrm{zero} = \ln|\lambda|+i\arg\lambda + 2\pi i n,\ \ \ \ n\in \mathbb{Z}.\end{equation}
If $|\lambda|$ is strictly less than $1$, the zeros do not enter the non-negative half of the complex $K$ plane. Therefore
\begin{equation}\left|\left|\frac{\lambda^2 K\Omega}{1-\lambda \Omega}\right|\right|_{D_2} = \mathrm{finite},\ \ \ \ \ \text{iff }|\lambda|<1. \end{equation}
The problem is that the tachyon vacuum lies just outside this interval. The function $\frac{K\Omega}{1-\Omega}$ has an infinite number of poles on the imaginary axis, and is not bounded for $\mathrm{Re}(K)\geq 0$. It is not totally clear whether this should indicate that Schnabl's solution is singular. But it does imply that there will be some complications defining the solution as a superposition of wedge states.
One possible way to define the state $K/(1-\Omega)$ is through its Taylor series around the origin. In fact, this is the generating function for Bernouli numbers, so the solution can be written
\begin{eqnarray}
\Psi_\text{Sch}\!\!\!\!\!\!\!\! && = \sum_{n=0}^\infty \frac{(-1)^n B_n}{n!}\sqrt{\Omega}cBK^nc\sqrt{\Omega}\\
\!\!\!\!\!\!\!\! && = \sqrt{\Omega}c\sqrt{\Omega} - \frac{1}{2}\sqrt{\Omega}cKBc\sqrt{\Omega} +\frac{1}{12}\sqrt{\Omega}cK^2 Bc\sqrt{\Omega} +...\ .
\end{eqnarray}
Each term in this expansion is an eigenstate of $\mathcal{L}_0$:
\begin{equation}\mathcal{L}_0\Big(\sqrt{\Omega}cK^nBc\sqrt{\Omega}\Big) = (n-1)\sqrt{\Omega}cBK^n c\sqrt{\Omega}.\end{equation}
This defines the so-called {\it $\mathcal{L}_0$ level expansion} of the solution. This can be seen as analogous to the Fock space expansion into a basis $L_0$ eigenstates---the ordinary level expansion---but formulated in the sliver frame. We will define the {\it level} of a state in the $\mathcal{L}_0$ level expansion to be its $\mathcal{L}_0$ eigenvalue; this convention differs from the ordinary level expansion, where typically the level is defined with a shift so that the zero momentum tachyon has level $0$. Therefore the $\mathcal{L}_0$ level expansion of Schnabl's solution starts at level $-1$ with the zero momentum tachyon state $c_1|0\rangle$, multiplied by a coefficient $2/\pi \approx 0.64$. For comparison, the coefficient of the zero momentum tachyon in the ordinary level expansion of the Siegel gauge tachyon vacuum is $\approx 0.54$. It is interesting to investigate the $\mathcal{L}_0$ level expansion of the Schnabl gauge solutions when $\lambda\neq 1$:
\begin{equation}\Psi_\lambda = \frac{\lambda}{1-\lambda}\sqrt{\Omega}cKBc\sqrt{\Omega} - \frac{\lambda^2}{(1-\lambda)^2}\sqrt{\Omega}c K^2 Bc\sqrt{\Omega}+...\ .\end{equation}
The zero momentum tachyon is now absent from the expansion, and the leading state at level $0$ is trivial in the BRST cohomology. More interestingly, the $\mathcal{L}_0$ level expansion is divergent in the limit $\lambda\to 1$; this is another way to see that the tachyon vacuum and pure gauge solutions are ``far away" from each other.
Unfortunately the $\mathcal{L}_0$ level expansion does not give a fully adequate definition of the solution. One way to see this is to consider the $\mathcal{L}_0$ level expansion of an inverse wedge state:
\begin{equation}\Omega^{-1} = \sum_{n=0}^\infty \frac{2^n}{n!}\sqrt{\Omega}K^n\sqrt{\Omega}.\end{equation}
Aside from signs, this is the same as the $\mathcal{L}_0$ expansion of $\Omega^3$. Obviously, $\Omega^3$ is an ordinary wedge state while $\Omega^{-1}$ is not normalizable. So the existence of the $\mathcal{L}_0$ level expansion is not enough to tell us that the string field is well-behaved. Therefore we look for a different way to define Schanbl's solution. One possibility is to define it through the geometric series
\begin{equation}\frac{K}{1-\Omega} = \sum_{n=0}^\infty K\Omega^n .\end{equation}
One might worry whether this sum converges. In fact it converges in the Fock space, since $K\Omega^\alpha$ vanishes as $1/\alpha^3$ for large $\alpha$ (in a similar way as $B\Omega^\alpha$). But you might notice that the $\mathcal{L}_0$ level expansion of the solution expressed in this form does not work out correctly; each term in the geometric series is proportional to $K$, so the zero momentum tachyon never appears. A related observation is that each term in the sum vanishes at $K=0$, while the right hand side is equal to $1$ at $K=0$. The sum however converges to the right hand side pointwise for $K>0$. Therefore it appears that the sum is missing the sliver state. To see how to account for this, we note the identity
\begin{equation}\frac{K}{1-\Omega} = \sum_{n=0}^N K\Omega^n +\frac{K}{1-\Omega}\Omega^{N+1}.\end{equation}
We truncated the sum, leaving a finite remainder. For an ordinary function the remainder would for most purposes be ignorable for large $N$, but presently the remainder approaches the sliver state, which is nonvanishing. To make this identity non-circular we can expand the remainder in terms of Bernoulli numbers
\begin{equation}
\frac{K}{1-\Omega} = \sum_{n=0}^N K\Omega^n + \sum_{n=0}^\infty \frac{(-1)^n B_n}{n!}K^n\Omega^{N+1}.
\end{equation}
For large $N$ the higher powers of $K$ acting on $\Omega^{N+1}$ are suppressed. In practice it appears to be enough to keep the zeroth term and make the identification
\begin{equation}
\frac{K}{1-\Omega} = \lim_{N\to\infty}\left[\sum_{n=0}^N K\Omega^n + \Omega^{N}\right].
\end{equation}
This limit should be understood in a special sense. Given an expression which depends on Schnabl's solution, each appearance of $K/(1-\Omega)$ should be replaced by the expression in brackets above {\it for a single common $N$}. One then performs the calculation at finite $N$, and the limit $N\to\infty$ is taken only as a final step. This prescription is good enough to get the correct value for the on-shell action as predicted by Sen's conjecture, but in more general contexts it is not clear if it is sufficient. In as far as this prescription is applicable, this leads to an expression for Schnabl's solution
\begin{equation}
\Psi_\text{Sch} = \lim_{N\to\infty}\left[\sum_{n=0}^N \sqrt{\Omega}cKB\Omega^n c\sqrt{\Omega}+ \sqrt{\Omega}cB\Omega^{N}c\sqrt{\Omega}\right].
\end{equation}
To make contact with the original notation of \cite{Schnabl}, define the state
\begin{equation}\psi_n = \sqrt{\Omega}cB\Omega^nc\sqrt{\Omega}.\end{equation}
Schnabl's solution is then written as
\begin{equation}
\Psi_\text{Sch} = \lim_{N\to\infty}\left[\psi_N - \sum_{n=0}^N \frac{d}{dn}\psi_n\right].\label{eq:Schpsin}
\end{equation}
The first piece $\psi_N$ is the famous {\it phantom term} of Schnabl's solution. The mystery of the phantom term comes from the fact that it {\it vanishes} as a state in the Fock space in the large $N$ limit. This follows from the result of excercise \ref{ex:Bsliver}. However, in a sense it is the most physically important part of Schnabl's solution. It gives the sole contribution to the zero momentum tachyon in the $\mathcal{L}_0$ level expansion, and the remaining terms can be seen as remnants of a pure gauge solution.
It was shown by Okawa \cite{Okawa} that Schnabl gauge solutions in the $KBc$ subalgebra can be expressed explicitly as a finite gauge transformation of the perturbative vacuum:
\begin{equation}
\Psi_\lambda = U^{-1}QU,\ \ \ \ U=\frac{1}{1-\lambda\sqrt{\Omega}Bc\sqrt{\Omega}}.\label{eq:Okform}
\end{equation}
\begin{exercise}
Prove this formula.
\end{exercise}
\noindent This expression must somehow be problematic at $\lambda = 1$. By expanding the denominator we can write
\begin{equation}U = 1+\lambda\sqrt{\Omega}\frac{1}{1-\lambda \Omega}Bc\sqrt{\Omega}.\end{equation}
The factor $1/(1-\lambda\Omega)$ develops a pole at $K=0$ in the limit $\lambda\to 1$. Therefore the gauge transformation becomes singular. This singularity is closely related to the presence of a phantom term in the solution, as we will discuss in subsection \ref{subsec:singularGT}.
With proper care for the phantom term, the action evaluated on Schnabl's solution gives the correct D-brane tension as predicted by Sen's conjecture. The original calculation is technical so we do not present it here. We give an alternative derivation in subsection \ref{subsec:singularGT}. Instead we will compute the Ellwood invariant. Since there are no D-branes at the tachyon vacuum, the closed string 1-point function on a disk should vanish. This means that the Ellwood invariant should evaluate to
\begin{equation}\mathop{\rm Tr}\nolimits_\mathcal{V}(\Psi_\text{Sch}) = -\mathcal{A}_0(\mathcal{V}).\end{equation}
To demonstrate this we need to evaluate the trace of the $\psi_n$ terms in Schnabl's solution:
\begin{equation}\mathop{\rm Tr}\nolimits_\mathcal{V}(\psi_n) = \mathop{\rm Tr}\nolimits_\mathcal{V}(\Omega cB\Omega^n c).\end{equation}
To simplify the ghosts we use $\mathcal{B}^-$ invariance of the trace:
\begin{eqnarray}
0\!\!\!\!\!\!\!\! && = \mathop{\rm Tr}\nolimits_\mathcal{V}\left(\frac{1}{2}\mathcal{B}^-(\Omega c\Omega^n c)\right)\nonumber\\
\!\!\!\!\!\!\!\! && = -\mathop{\rm Tr}\nolimits_\mathcal{V}(\Omega Bc\Omega^n c)+ n \mathop{\rm Tr}\nolimits_\mathcal{V}(\Omega cB\Omega^n c)\nonumber\\
\!\!\!\!\!\!\!\! && = -\mathop{\rm Tr}\nolimits_\mathcal{V}(\Omega^{n+1} c)+(n+1)\mathop{\rm Tr}\nolimits_\mathcal{V}(\Omega cB\Omega^n c).
\end{eqnarray}
Next we scale the cylinder in the first term down to unit circumference:
\begin{equation}\mathop{\rm Tr}\nolimits_\mathcal{V}(\Omega^{n+1} c)=\mathop{\rm Tr}\nolimits_\mathcal{V}\left(\left(\frac{1}{n+1}\right)^{\frac{1}{2}\mathcal{L}^-}(\Omega^{n+1} c)\right) = (n+1) \mathop{\rm Tr}\nolimits_\mathcal{V}(\Omega c).\label{eq:Ellpsin}
\end{equation}
Together with the previous equation, this implies
\begin{equation}\mathop{\rm Tr}\nolimits_\mathcal{V}(\psi_n) = \mathop{\rm Tr}\nolimits_\mathcal{V}(\Omega c).\end{equation}
The Ellwood invariant for Schnabl's solution is
\begin{equation}
\mathop{\rm Tr}\nolimits_\mathcal{V}(\Psi_\text{Sch}) = \lim_{N\to\infty}\left[\mathop{\rm Tr}\nolimits_\mathcal{V}(\psi_N) - \sum_{n=0}^n\frac{d}{dn} \mathop{\rm Tr}\nolimits_\mathcal{V}(\psi_n)\right].
\end{equation}
From \eq{Ellpsin}, the trace of $\psi_n$ is independent of $n$. Therefore the sum vanishes identically, and the sole contribution comes from the phantom term:
\begin{equation}
\mathop{\rm Tr}\nolimits_\mathcal{V}(\Psi_\text{Sch}) = \mathop{\rm Tr}\nolimits_\mathcal{V}(\Omega c). \label{eq:SchEll1}
\end{equation}
Note that the pure gauge solutions only have the sum (with terms multiplied by $\lambda^{n+1}$), and no phantom contribution. In this case the Ellwood invariant evaluates to zero, as expected for the perturbative vacuum. We also see that the phantom term gives the physically important contribution to the Ellwood invariant, and while it vanishes in the Fock space, it can be nonzero in the context of some calculations. To compare \eq{SchEll1} to the disk 1-point function, we map the cylinder to the unit disk with the transformation
\begin{equation}f(z) = e^{2\pi i z}.\end{equation}
This gives
\begin{eqnarray}
\mathop{\rm Tr}\nolimits_\mathcal{V}(\Psi_\text{Sch}) \!\!\!\!\!\!\!\! && = \langle c\overline{c}V^\mathrm{m}(i\infty,-i\infty)c(0)\rangle_{C_1}\nonumber\\
\!\!\!\!\!\!\!\! && = \frac{1}{2\pi i} \langle c\overline{c}V^\mathrm{m}(0,0)c(1)\rangle_\mathrm{disk}\nonumber\\
\!\!\!\!\!\!\!\! && = - \frac{1}{2\pi i} \langle V^\mathrm{m}(0,0)\rangle_\mathrm{disk}^\mathrm{matter}\nonumber\\
\!\!\!\!\!\!\!\! && = -\mathcal{A}_0(\mathcal{V}),
\end{eqnarray}
where in the last step we noted that the $c$ ghost correlator evaluates to $-1$.
Given a Fock space basis $|\phi_i\rangle$ and a dual basis $|\phi^i\rangle$, the Fock space expansion of Schnabl's solution is given by
\begin{equation}
\Psi_\text{Sch} = \sum_{i}|\phi_i\rangle \mathop{\rm Tr}\nolimits\Big(\sqrt{\Omega}(f_\mathcal{S}\circ\phi^i)\sqrt{\Omega}\Psi_\text{Sch}\Big).
\end{equation}
We have not developed the right formalism to make computations of coefficients to high level efficient. The main technicality is in computing the conformal transformation of highly descendant vertex operators. The systematics of this are most often dealt with in the operator formalism, used extensively in Schnabl's original paper. The motivation for such computations is to compare to solutions derived in Siegel gauge level truncation, and to test convergence of the action level by level given the exact coefficients of the tachyon condensate. Such analysis ultimately gets into numerics and is not the primary focus of these lectures. The result is that Schnabl's solution appears to behave well in the Fock space expansion. The infinite $D_2$ norm and the associated subtleties with the phantom term do not translate to noticeable problems in the Fock space. In fact, the pure gauge solutions for $|\lambda|<1$ appear to be less well-behaved in the Fock space expansion. The action evaluated on such solutions should tend to zero level by level, but convergence is at best extremely slow \cite{Takahashi_trunc}, especially as $\lambda$ approaches $1$. In a different direction, it is also possible to derive a numerical solution for the tachyon vacuum in Schnabl gauge using the level truncation scheme~\cite{AldoMatjej}. The action converges to the expected value quite well, but the coefficients of the numerical solution do not match the analytic values with kind of precision that might have been hoped~for.
While computation of descendant states becomes technical, for primaries it is straightforward. For the tachyon vacuum, the only primary operator which acquires expectation value is the zero momentum tachyon $c(0)$. The dual vertex operator is $-c\partial c(0)$.
Therefore the coefficient of the tachyon state
\begin{equation}\Psi_\text{Sch} = T c_1|0\rangle + \text{higher levels}\end{equation}
is given by
\begin{equation}T = -\frac{\pi}{2}\mathop{\rm Tr}\nolimits\big(\sqrt{\Omega}c\partial c\sqrt{\Omega}\Psi_\text{Sch}\big).\end{equation}
To compute this we evaluate the overlap with $\psi_n$ using \eq{ccccB}:
\begin{eqnarray}
-\frac{\pi}{2}\mathop{\rm Tr}\nolimits(\sqrt{\Omega}c\partial c\sqrt{\Omega}\psi_n) \!\!\!\!\!\!\!\! && = -\frac{\pi}{2}\mathop{\rm Tr}\nolimits\big(\Omega^n c\Omega c\partial c\Omega cB\big)\nonumber\\
\!\!\!\!\!\!\!\! && = -\frac{n+2}{\pi}\sin^2\frac{\pi}{n+2}\left(\frac{n+2}{2\pi}\sin \frac{2\pi}{n+2}-1\right).
\end{eqnarray}
From the Taylor series of the sine, the first factor goes as $1/n$ for large $n$, while the second factor vanishes as $1/n^2$. In total the tachyon coefficient for $\psi_n$ vanishes as $1/n^3$ for large $n$, confirming the result of exercise \ref{ex:Bsliver} and that the phantom term vanishes in the Fock space. Summing the derivatives of $\psi_n$ then gives the tachyon coefficient for Schnabl's solution:
\begin{equation}
T = \sum_{n=0}^\infty \frac{d}{dn}\left[\frac{n+2}{\pi}\sin^2\frac{\pi}{n+2}\left(\frac{n+2}{2\pi}\sin \frac{2\pi}{n+2}-1\right)\right]\approx 0.55.
\end{equation}
This is very similar to $T\approx 0.54$ derived for the Siegel gauge condensate in level truncation. We can similarly derive the tachyon coefficient for pure gauge solutions:
\begin{equation}
T(\lambda) = \sum_{n=0}^\infty \lambda^{n+1}\frac{d}{dn}\left[\frac{n+2}{\pi}\sin^2\frac{\pi}{n+2}\left(\frac{n+2}{2\pi}\sin \frac{2\pi}{n+2}-1\right)\right].
\end{equation}
The sum converges for $|\lambda|\leq 1$, but the rate of convergence for $|\lambda|$ strictly less than $1$ is exponential while at $|\lambda|=1$ it converges only as a sum of $1/n^4$. This has interesting consequence for the nature of the limit $\lambda\to 1$. While the limit is continuous, it is not differentiable; in particular
\begin{equation}\frac{d^3}{d\lambda^3}T(\lambda)\end{equation}
diverges as a harmonic series as $\lambda$ approaches 1. In this sense, even in the Fock space it can be seen that the tachyon vacuum is a special configuration among Schnabl gauge solutions in the $KBc$ subalgebra.
\subsection{Schnabl Gauge Marginal Deformations}
We now describe analytic solutions for marginal deformations in Schnabl gauge. These correspond to deformations of the reference D-brane given by moving along flat directions in the string field potential. At linearized order, such solutions are represented by a nontrivial element of the BRST cohomology, which we assume takes the form
\begin{equation}\Psi_\text{marg} = cV(0)|0\rangle +\text{nonlinear corrections},\end{equation}
where $V(x)$ is a boundary matter primary of weight $1$. If we introduce a string field $V$ defined by an infinitesimally thin strip containing $V(0)$ at the origin, we can write
\begin{equation}\Psi_\text{marg} = \sqrt{\Omega}cV\sqrt{\Omega} + \text{nonlinear corrections}.\end{equation}
Some important properties of $V$ are
\begin{equation}Q(cV) = 0,\ \ \ \ \frac{1}{2}\mathcal{L}^- V = V,\ \ \ \ \frac{1}{2}\mathcal{B}^- V = 0.\end{equation}
The second property says that $V$ has scaling dimension $1$ in the sliver frame, and the third property holds because $V(x)$ is a matter operator.
For the same reason
\begin{equation}
[B,V] = [c,V] = 0.
\end{equation}
Often we multiply $V$ by a constant $\lambda$ corresponding to the expectation value of the field generated by the vertex operator. To avoid proliferation of $\lambda$s in formulas, we will absorb this constant into the normalization of $V$ by writing
\begin{equation}V = \lambda \widehat{V},\end{equation}
where $\widehat{V}$ is defined with a fixed normalization. The solution can be expanded perturbatively
\begin{equation}
\Psi_\text{marg} = \Psi_1+\Psi_2+\Psi_3+... ,
\end{equation}
where $\Psi_n$ contains $n$ insertions of $V$. These represent nonlinear corrections that account for the fact that the field generated by $cV$ has finite expectation value. Matching terms that contain the same number of $V$s, the equations of motion imply
\begin{eqnarray}
\!\!\!\!\!\!\!\! && Q\Psi_1= 0,\\
\!\!\!\!\!\!\!\! && Q\Psi_2+\Psi_1^2= 0,\\
\!\!\!\!\!\!\!\! && Q\Psi_3 +\Psi_1\Psi_2+\Psi_2\Psi_1 = 0,\\
\!\!\!\!\!\!\!\! &&\ \ \ \ \ \ \ \ \vdots\nonumber\ \ .
\end{eqnarray}
There may be obstruction to solution of these equations if the quadratic terms containing lower order corrections are not BRST exact (one can check that they are automatically BRST closed). The physical interpretation of this obstruction is that the potential for the field $cV$ is not exactly flat; a finite expectation value for the field is not a stationary point of the potential. If the construction fails to give a solution for $\Psi_n$, this means that the potential vanishes as $\lambda^{n+1}$ for small $\lambda$. If the obstruction is absent for all $n$, then the deformation generated by $cV$ is called {\it exactly marginal}.
We look for a solution for marginal deformations in Schnabl gauge:
\begin{equation}\mathcal{B}_0\Psi_n=0.\end{equation}
Acting $\mathcal{B}_0$ on the equations of motion and using $[Q,\mathcal{B}_0]=\mathcal{L}_0$, we obtain a recursive set of equations for the corrections of the form
\begin{equation}\mathcal{L}_0\Psi_n+\mathcal{B}_0(\text{lower order corrections}) = 0.\end{equation}
If the second term does not produce states in the kernel of $\mathcal{L}_0$, we can invert $\mathcal{L}_0$ to obtain an explicit formula for $\Psi_n$. Let us work this out for the second order correction:
\begin{equation}\Psi_2 = -\frac{\mathcal{B}_0}{\mathcal{L}_0}\Psi_1^2.\end{equation}
A convenient representation of the inverse of $\mathcal{L}_0$ is through the Schwinger parameterization
\begin{equation}\frac{1}{\mathcal{L}_0} = \int_0^\infty dt\, e^{-t\mathcal{L}_0}.\end{equation}
The integration variable $t$ can be interpreted as a coordinate on part of the moduli space of Riemann surfaces defining an open string amplitude where $1/\mathcal{L}_0$ appears in the propagator.\footnote{In Schnabl gauge the full propagator is $\frac{\mathcal{B}_0}{\mathcal{L}_0}Q\frac{\mathcal{B}_0^*}{\mathcal{L}_0^*}$ \cite{Schnabl,RZVeneziano,KSZ}. Since $\mathcal{L}_0$ is not BPZ even, the propagator contains two moduli integrals, which makes the connection between the Schnabl gauge amplitude and integration over the moduli space somewhat less direct than in Siegel gauge.}
Substituting $\Psi_1$ we may compute
\begin{eqnarray}
\Psi_2\!\!\!\!\!\!\!\! && = -\frac{\mathcal{B}_0}{\mathcal{L}_0}\sqrt{\Omega}cV\Omega cV\sqrt{\Omega}\nonumber\\
\!\!\!\!\!\!\!\! && = - \frac{1}{\mathcal{L}_0}\sqrt{\Omega}cV B\Omega cV\sqrt{\Omega}\nonumber\\
\!\!\!\!\!\!\!\! && = -\int_0^\infty dt\, e^{-t\mathcal{L}_0}\Big(\sqrt{\Omega}cV B\Omega cV\sqrt{\Omega}\Big)\nonumber\\
\!\!\!\!\!\!\!\! && = -\sqrt{\Omega}\left[\int_0^\infty dt\,e^{-\frac{t}{2}\mathcal{L}^-}\Big(cV B\Omega cV\Big)\right]\sqrt{\Omega}\nonumber\\
\!\!\!\!\!\!\!\! && = -\sqrt{\Omega}cV B\left[\int_0^\infty dt\,e^{-t} \Omega^{e^{-t}}\right] cV\sqrt{\Omega}.
\end{eqnarray}
Substituting $\alpha = e^{-t}$ gives
\begin{equation}\Psi_2 = -\sqrt{\Omega}cVB \left[\int_0^1 d\alpha\,\Omega^{\alpha}\right]cV\sqrt{\Omega}.\label{eq:Psi2int}\end{equation}
We recognize the integral from the homotopy operator for Schnabl's solution. Therefore we can write
\begin{equation}\Psi_2 = -\sqrt{\Omega}cVB \frac{1-\Omega}{K}cV\sqrt{\Omega}.\end{equation}
This result raises a puzzle. Usually marginal operators, being dimension 1 primaries, have a double pole in their OPE proportional to the identity operator:
\begin{equation}V(x)V(0) = \frac{\mathcal{N}}{x^2} +...,\end{equation}
where the normalization is proportional to the two point function of marginal operators in the UHP,
\begin{equation}\mathcal{N} = \frac{\langle I\circ V(0)\, V(0)\rangle_\text{UHP}^\text{matter}}{g_0},\end{equation}
and
\begin{equation}g_0 = \langle 1\rangle_\text{UHP}^\text{matter}\end{equation}
is the disk partition function (or {\it g-function}) in the matter factor of the reference $\mathrm{BCFT}$. The wedge state separating two $V$s under the integral in \eq{Psi2int} can be arbitrarily thin, and we can expect that the OPE of $V$s will create a divergence towards the lower limit of integration. Part of the problem is caused by the Schwinger representation of $1/\mathcal{L}_0$. The Schwinger representation is only valid operating on states with positive $\mathcal{L}_0$ eigenvalue. The zero momentum tachyon, however, has negative eigenvalue. This is actually a fairly common problem in string perturbation theory. For example, the representation of the Veneziano amplitude as an integral over moduli space likewise suffers from divergences from collisions of tachyon vertex operators. From the SFT point of view, such divergences originate from the failure of the Schwinger representation to correctly define the propogator. There are various proposed remedies of this problem \cite{Witten_ieps,SenTree}. In the Veneziano amplitude, the most common is to analytically continue to unphysical momenta where OPEs of tachyon vertex operators are not divergent. Presently, we can proceed by simply dividing by the eigenvalue of $\mathcal{L}_0$ in the $\mathcal{L}_0$ level expansion. To do this we define a ``normal ordered" string field through the relation
\begin{equation}V\Omega^\alpha V =\, :\!\!V\Omega^\alpha V\!\!: +\frac{\mathcal{N}}{\alpha^2}\Omega^\alpha.\end{equation}
The OPE divergence is absorbed in the second term, so that the first term is finite as $\alpha\to 0$. In this way we can write
\begin{equation}
\mathcal{B}_0\Psi_1^2 = \sqrt{\Omega}cB\Big(:\!\!V\Omega V\!\!: +\mathcal{N}\Omega\Big)c\sqrt{\Omega}.\label{eq:normord}
\end{equation}
We now expand this into eigenstates of $\mathcal{L}_0$:
\begin{eqnarray}
\mathcal{B}_0\Psi_1^2 \!\!\!\!\!\!\!\! && =\underbrace{\phantom{\Big)}\!\!\mathcal{N}\sqrt{\Omega}c\sqrt{\Omega}}_{\text{level } -1} \, - \underbrace{\phantom{\Big)}\!\!\mathcal{N}\sqrt{\Omega}cKBc\sqrt{\Omega}}_{\text{level } 0} \,+ \underbrace{\sqrt{\Omega}\left(\frac{\mathcal{N}}{2!}cK^2 Bc +c:\!\! V^2\!\!:\right)\sqrt{\Omega}}_{\text{level }1}\,+\,\text{higher levels}.\ \ \ \ \ \ \ \
\end{eqnarray}
Now inverting $\mathcal{L}_0$ is as simple as dividing by the level. But we encounter a problem: there is a state at level zero. In Siegel gauge, a state in the kernel of $L_0$ would normally be nontrivial in the cohomology, and would imply an obstruction to the existence of a solution. In the current situation, the state in the kernel of $\mathcal{L}_0$ is BRST exact. This implies that there is an obstruction to solution in Schnabl gauge which does not prevent solution in other gauges.
If we want to proceed in Schnabl gauge, our only choice is to assume that the two point function of marginal operators vanishes. This requires that the marginal operators have no singularity in their OPE:
\begin{equation}\lim_{x\to 0}V(x) V(0) =\text{finite}.\end{equation}
To simplify the analysis at higher orders, we will in fact assume that all powers of the string field $V$ are finite. This will be called a {\it regular marginal deformation}. If higher order products of $V$ are divergent, new obstructions can appear at third order or higher; depending on whether the obstruction is BRST exact, this would either indicate that the deformation is not exactly marginal or that the solution cannot be found in Schnabl gauge. The restriction to regular marginal deformations is not as limiting as one might think. One example of $V$ with regular OPE is the exponential rolling tachyon deformation
\begin{equation}V(x) = e^{X^0(x)},\ \ \ \ e^{X^0(x)}e^{X^0(0)} = x^2 e^{2X^0(0)}+...\ .\end{equation}
This leads to a time-dependent solution where the reference D-brane decays starting from an infinitesimal, homogeneous tachyon fluctuation in the infinite past. At late times the solution oscillates violently with diverging amplitude, a phenomenon which has been the subject of much discussion \cite{MoellerZwiebach_rolling,Hata_rolling}. The connection between this behavior and the emergence of tachyon matter \cite{Sen_matter} and closed strings at late times has not been fully clarified. Another interesting example is the lightlike rolling tachyon deformation \cite{Hellerman}
\begin{equation}V(x) = e^{X^+(x)},\ \ \ \ e^{X^+(x)}e^{X^+(0)} = e^{2X^+(0)}+...,\end{equation}
which in a linear dilaton background can be made marginal. This represents a solution where the reference D-brane decays starting from an infinitesimal homogeneous tachyon fluctuation in the infinite lightcone past. Unlike the timelike decay process, the solution does not oscillate at late times and smoothly approaches the tachyon vacuum \cite{Hellerman,rollingvac}. This is a notable example of an exact solution representing a time-dependent transition between open string vacua. The observation of \cite{KOSsing} implies that regular marginal deformations also include a large number of exactly marginal deformations which are independent of the $X^0$ component of the $\mathrm{BCFT}$. Given $\widehat{V}_\mathrm{bare}(x)$ with a double pole (and {\it only} a double pole) in the OPE with itself with unit coefficient, we may consider a modified marginal operator
\begin{equation}
V(x) =\lambda\left( \widehat{V}_\mathrm{bare}(x)+\frac{i}{\sqrt{2}}\partial_\parallel X^0(x)\right).
\end{equation}
This operator has regular self-OPE since the double pole of the $\partial X^0$-$\partial X^0$ OPE cancels that of $\widehat{V}_\mathrm{bare}\text{-}\widehat{V}_\mathrm{bare}$. The cross terms do not create singularity since $\widehat{V}_\mathrm{bare}$ is independent of the $X^0$ BCFT. If we are lucky, higher order products of $V$ will also be finite. This occurs for the Wilson line deformation, the deformation representing transverse displacement of the D-brane, and the cosine tachyon deformation \cite{Ludwig}. The modified marginal operator gives an expectation value to the field of $\widehat{V}_\mathrm{bare}$ in addition to turning on a timelike gauge potential $A_0$. However, in physical situations the timelike direction on the D-brane worldvolume is noncompact, and the timelike gauge potential does not produce a physical deformation of the BCFT. Still there are deformations which cannot easily be expressed in terms of marginal operators with regular OPE. An important example is the hyperbolic cosine rolling tachyon deformation \cite{SenRolling}.
Assuming that collisions of $V$ cause no problems, we may construct higher order corrections to the solution. We simply quote the result:
\begin{equation}
\Psi_{n+1} = (-1)^{n+1}\sqrt{\Omega}cV B\left(\frac{1-\Omega}{K}V\right)^n c\sqrt{\Omega}.
\end{equation}
We can sum over $n$ to find the full solution
\begin{equation}
\Psi_\text{marg} = \sqrt{\Omega} cV \frac{B}{1+\frac{1-\Omega}{K}V}c\sqrt{\Omega}.
\end{equation}
\begin{exercise}
Prove that the solution satisfies the equations of motion.
\end{exercise}
\begin{exercise}
Show that the solution can be written in terms of $J=cV$ as
\begin{equation}\Psi_\text{marg} = \sqrt{\Omega} J \frac{B}{1+B\frac{1-\Omega}{K}J}\sqrt{\Omega}.\end{equation}
Prove that the equation of motion hold in this form. Show that if $J = \lambda cKBc$ this reduces to the pure gauge solution discussed in the previous section. Find the relation between the marginal parameter $\lambda$ in front of $cKBc$ and the gauge parameter $\lambda$ in \eq{Psil}.
\end{exercise}
Now we can try to compute some observables. The action is not very interesting, since a marginal deformation moves the string field along a flat direction in the potential, and the action is unchanged. More interesting is the Ellwood invariant. Let us compute the leading order contribution
\begin{equation}\mathop{\rm Tr}\nolimits_\mathcal{V}(\Psi_1) = \mathop{\rm Tr}\nolimits_\mathcal{V}(\Omega cV).\end{equation}
First we introduce a trivial integration $\int_0^1 dt\, = 1$, and use cyclicity of the trace to write
\begin{equation}\mathop{\rm Tr}\nolimits_\mathcal{V}(\Psi_1) = \int_0^1 dt\mathop{\rm Tr}\nolimits_\mathcal{V}(\Omega^t cV\Omega^{1-t}).\end{equation}
Second we insert a trivial factor $[B,c]=1$:
\begin{eqnarray}\mathop{\rm Tr}\nolimits_\mathcal{V}(\Psi_1)\!\!\!\!\!\!\!\! && = \int_0^1 dt\mathop{\rm Tr}\nolimits_\mathcal{V}(\Omega^{1-t} cV\Omega^{t}[B,c])\nonumber\\
\!\!\!\!\!\!\!\! &&= \int_0^1 dt\mathop{\rm Tr}\nolimits_\mathcal{V}(\Omega^{1-t}[B, cV]\Omega^{t}c)\nonumber\\
\!\!\!\!\!\!\!\! && = \int_0^1 dt\mathop{\rm Tr}\nolimits_\mathcal{V}(\Omega^{1-t} V\Omega^{t}c)\nonumber\\
\!\!\!\!\!\!\!\! && =\left\langle \left( \int_0^1dt\,V(t)\right) c(0)\, c\overline{c}V^\mathrm{m}(i\infty,-i\infty)\right\rangle_{C_1}.\end{eqnarray}
Mapping the cylinder to the unit disk and evaluating the ghost correlator gives
\begin{equation}\mathop{\rm Tr}\nolimits_\mathcal{V}(\Psi_1) = -\frac{1}{2\pi i}\left\langle\left(\int_0^{2\pi} d\theta \, V(\theta)\right) V^m(0,0)\right\rangle_\mathrm{disk}^\mathrm{matter},\end{equation}
where $V(\theta)$ is inserted on the boundary of the unit circle at an angle $\theta$. This represents the first order change of the boundary condition in the closed string 1-point function implemented by the marginal deformation.
\begin{exercise}
By a similar manipulation show that
\begin{equation}\mathop{\rm Tr}\nolimits_\mathcal{V}(\Psi_2) = \frac{1}{2\pi i}\left\langle \frac{1}{2!}\left(\int_0^{2\pi} d\theta\, V(\theta)\right)^{\!\!2} V^\mathrm{m}(0,0)\right\rangle_\mathrm{disk}^\mathrm{matter}.\end{equation}
\end{exercise}
\noindent The general result was derived by Kishimoto \cite{Kishimoto_tad}:
\begin{equation}\mathop{\rm Tr}\nolimits_\mathcal{V}(\Psi_\text{marg}) = \frac{1}{2\pi i}\left\langle \left(e^{-\int_0^{2\pi} d\theta\, V(\theta)}-1\right) V^\mathrm{m}(0,0)\right\rangle_\mathrm{disk}^\mathrm{matter}.\end{equation}
By exponentiating the integration of $V$ around the boundary we are effectively deforming the open string boundary condition. The correlation function can be represented through a worldsheet path integral, and in this context the exponential operator adds a boundary term to the worldsheet action whose effect is to modify the open string boundary condition of the original D-brane. Therefore the Ellwood invariant computes the shift in the closed string 1-point function on the disk. Note that since we are assuming that all products of $V$ are regular, the exponential operator is defined without renormalization.
Now we can ask about coefficients of the solution in the Fock space. For this purpose it is helpful to restore the explicit dependence on $\lambda$:
\begin{equation}V = \lambda \widehat{V}.\end{equation}
Through the Ellwood invariant, $\lambda$ can be identified as the coupling constant of the boundary deformation of the worldsheet action. The most interesting coefficient of the solution is the expectation value of the D-brane fluctuation field generating the marginal deformation:
\begin{equation}\lambda_\mathrm{SFT} c\widehat{V}(0)|0\rangle .\end{equation}
To linearized order the coefficient $\lambda_\mathrm{SFT}$ is equal to the coupling constant $\lambda$, but at the nonlinear level these parameters may be different. The coefficient $\lambda_\mathrm{SFT}$ can be extracted by contracting the solution with a dual state defined by vertex operator $-c\partial c \widehat{V}^*$, where $\widehat{V}^*$ is a weight 1 primary field. The order $\lambda^n$ contribution to $\lambda_\mathrm{SFT}$ is given by evaluating the trace
\begin{equation}-\mathop{\rm Tr}\nolimits\big(\sqrt{\Omega}c\partial c \widehat{V}^*\sqrt{\Omega}\Psi_n\big).\end{equation}
This can be reduced to the computation of a matter $n+1$-point function on the upper half plane:
\begin{equation}\Big\langle \big(I\circ \widehat{V}^*(0)\big) \widehat{V}(1)\Big[\widehat{V}(x_2)...\widehat{V}(x_{n-1})\Big] \widehat{V}(0)\Big\rangle_\mathrm{UHP}^\mathrm{matter},\end{equation}
where after $SL(2,\mathbb{R})$ transformation we can bound the $x_i$s between $1$ and $0$. Now suppose that we scale the correlator with a factor of $\epsilon$ through the conformal transformation $s_\epsilon(\xi)=\epsilon \xi$. The BPZ conformal map inverts the scaling
\begin{equation}s_\epsilon\circ I(\xi) = I\circ s_{1/\epsilon}(\xi).\end{equation}
Since all of the insertions carry weight $1$, we find
\begin{eqnarray}
\!\!\!\!\!\!\!\! && \Big\langle \big(I\circ \widehat{V}^*(0)\big) \widehat{V}(1)\Big[\widehat{V}(x_2)...\widehat{V}(x_{n-1})\Big] \widehat{V}(0)\Big\rangle_\mathrm{UHP}^\mathrm{matter}\nonumber\\
\!\!\!\!\!\!\!\! &&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\epsilon^{n-1}\Big\langle \big(I\circ \widehat{V}^*(0)\big) \widehat{V}(\epsilon)\Big[\widehat{V}(\epsilon x_2)...\widehat{V}(\epsilon x_{n-1})\Big] \widehat{V}(0)\Big\rangle_\mathrm{UHP}^\mathrm{matter}.\ \ \ \ \
\end{eqnarray}
This relation is valid for any $\epsilon$, and in particular we can take $\epsilon$ to zero. The right hand side has a vanishing factor for $n>1$, and ordinarily this would be compensated by divergent OPEs as the marginal operators are squeezed to the origin. But presently we are assuming that divergences in products of $\widehat{V}$ are absent. Therefore the $n+1$ point function must vanish identically for $n>1$, which implies that the $\lambda^n$ contribution to $\lambda_\mathrm{SFT}$ must vanish for $n>1$. Therefore, for Schnabl gauge marginal deformations the equality
\begin{equation}\lambda_\mathrm{SFT}=\lambda\end{equation}
holds at the fully nonlinear level. In Siegel gauge, the relation between $\lambda_\mathrm{SFT}$ and $\lambda$ is nontrivial and has been an open problem for many years. For the Wilson line deformation, early work in
level truncation indicated that $\lambda_\text{SFT}$ could only reach a finite maximum expectation value even though
\begin{wrapfigure}{l}{.15\linewidth}
\centering
\resizebox{1in}{1.6in}{\includegraphics{OpenSFT_Erler48.jpg}}
\end{wrapfigure}
the boundary coupling is unbounded. More recent work \cite{MaccaferriMatjej} has shown that the boundary
coupling constant is in fact not a single valued function of $\lambda_\text{SFT}$; it consists of (at least) two
branches which join at the maximum value of $\lambda_\text{SFT}$. Though this has not been fully confirmed in Siegel gauge level truncation, analysis of analytic solutions capable of describing singular OPEs \cite{MaccaferriSchnabl,KOSsingII} indicates that the limit of large boundary coupling is represented by solutions where $\lambda_\text{SFT}$ tends to zero. A sketch of the conjectured relation between $\lambda_\text{SFT}$ and $\lambda$ is shown left. Therefore, in Siegel gauge a given expectation value of the marginal field can represent more than one marginally deformed background. To distinguish the backgrounds it is apparently necessary to look at other coefficients of the solution. The story in Schnabl gauge is apparently much simpler. While the tachyon vacuum in Schnabl gauge in some ways seems comparable to the Siegel gauge solution, the solutions for marginal deformations are quite different.
\subsection{Simple Tachyon Vacuum}
\label{subsec:simple}
We now describe another solution for the tachyon vacuum in the $KBc$ subalgebra which is simpler than Schnabl's. While technical simplifications are always welcome, the structure of the solution appears to connect with something deeper which has allowed progress in several directions.
It was noticed by Okawa \cite{Okawa} that Schnabl gauge solutions in the $KBc$ subalgebra can be readily generalized to depend on an arbitrary state in the wedge algebra $F=F(K)$:
\begin{equation}\Psi_\text{Ok} = \sqrt{F}c\frac{KB}{1-F}c\sqrt{F}.\label{eq:Oktype}\end{equation}
The solution is real if $F$ is real. It was shown in \cite{SSFII} that solutions of this kind may represent two gauge orbits:
\begin{eqnarray}
\text{Perturbative vacuum}:\!\!\!\!\!\!\!\! && \ \ \ \ \ F(0)=\lambda\neq 1.\nonumber\\
\text{Tachyon vacuum}:\!\!\!\!\!\!\!\! &&\ \ \ \ \ F(0)=1,\ \ \ F'(0)\neq 0.
\end{eqnarray}
In the second case, the derivative of $F(K)$ cannot vanish at $K=0$ since otherwise the state between the $c$s contains a pole at $K=0$. Several other necessary conditions on $F(K)$ are known, but the complete set of necessary and sufficient conditions for the viability of the solution are not fully clear. We will leave it as it is for now. If we have a tachyon vacuum solution in this form, the homotopy operator is
\begin{equation}
A_\text{Ok} = B\frac{1-F}{K}.
\end{equation}
The solutions do not satisfy the Schnabl gauge condition. But they satisfy a similar gauge condition
\begin{equation}\mathcal{B}_F\Psi = 0,\end{equation}
where the operator $\mathcal{B}_F$ is defined
\begin{equation}
\mathcal{B}_F X= \sqrt{F}\frac{1}{2}\mathcal{B}^-\left(\frac{1}{\sqrt{F}}X\frac{1}{\sqrt{F}}\right)\sqrt{F}.
\end{equation}
By definition, we have
\begin{equation}\mathcal{B}_F\big(\sqrt{F}X\sqrt{F}\big) = \sqrt{F}\left(\frac{1}{2}\mathcal{B}^-X\right)\sqrt{F}.\end{equation}
If $F=\Omega$ is the $SL(2,\mathbb{R})$ vacuum, this reduces to the usual $\mathcal{B}_0$. This is referred to as a {\it dressed Schnabl gauge}.
Given this general class of solutions, one can ask if there is a natural choice of $F(K)$. One might suggest that it would be interesting if the function of $K$ appearing in the homotopy operator was equal to $F$ itself. Equating
\begin{equation}F=\frac{1-F}{K}\end{equation}
leads to
\begin{equation}F = \frac{1}{1+K} = \int_0^\infty d\alpha \,e^{-\alpha}\Omega^\alpha.\end{equation}
Here we defined $F$ as a combination of wedge states using the Schwinger parameterization. To write the solution we also need the square root
\begin{equation}\frac{1}{\sqrt{1+K}} = \frac{1}{\sqrt{\pi}}\int_0^\infty d\alpha\,\frac{e^{-\alpha}}{\sqrt{\alpha}}\Omega^\alpha .\end{equation}
This leads to the {\it simple tachyon vacuum}:
\begin{equation}\Psi_\text{simp} = \frac{1}{\sqrt{1+K}}c(1+K)Bc\frac{1}{\sqrt{1+K}},\end{equation}
with homotopy operator
\begin{equation}A_\text{simp} = \frac{B}{1+K}.\end{equation}
The simple tachyon vacuum is characterized by a continuous superposition of wedge states from the identity string field out to the sliver. This contrasts with Schnabl's solution, which is a discrete sum of wedge states of positive integer width. The solution satisfies the gauge condition
\begin{equation}\mathcal{B}_{\frac{1}{1+K}}\Psi = 0,\end{equation}
which we will call {\it simple gauge}. The operator $\mathcal{B}_{\frac{1}{1+K}}$ can be written in a form analogous to \eq{B0Bm}
\begin{equation}\mathcal{B}_{\frac{1}{1+K}} X= \frac{1}{2}\mathcal{B}^- X+\frac{1}{2}\left(\frac{B}{1+K}X + (-1)^{|X|}X\frac{B}{1+K}\right).\end{equation}
An interesting difference from Siegel gauge and Schnabl gauge is that the gauge fixing condition not only involves $b$ ghosts, but also total Virasoros.
Note that the choice of $F$ for the simple tachyon vacuum has finite $C_*$ and $D_2$ norm:
\begin{equation}\left|\left|\frac{1}{1+K}\right|\right|_{C^*} = \left|\left|\frac{1}{1+K}\right|\right|_{D_2} = 1.\end{equation}
This property is also shared by the $SL(2,\mathbb{R})$ vacuum which defines Schnabl's solution. The subtlety with Schnabl's solution comes from the state between the $c$s. After subtracting the field $K$ as before, the analogous state for the simple tachyon vacuum is simply the identity string field, which of course has finite $C^*$ and $D_2$ norm. Therefore the simple tachyon vacuum has a straightforward representation in terms of wedge states with insertions, and there is no need for regularization or phantom term. Contrary to what is sometimes suggested this is not a unique property of the simple tachyon vacuum. Another tachyon vacuum which can be defined without phantom term is given by
\begin{equation}F(K) = \frac{1}{(1+K)^2}\Omega.\end{equation}
This solution contains continuous superposition of wedge states from the $SL(2,\mathbb{R})$ vacuum out to the sliver state, but does not receive contribution from wedge states close the identity string field. In fact, for this solution both $\sqrt{F}$ and $\frac{KF}{1-F}$ live in the disk algebra $A(D_2)$, the most exclusive of the three possible definitions of the wedge algebra outlined earlier. Any Okawa-type solution where $\sqrt{F}$ and $\frac{KF}{1-F}$ are simultaneously analytic for positive real part of $K$ and have finite $D_2$ norm can be defined without regularization or phantom term.
One thing we can do with the simple tachyon vacuum is compute the action to verify Sen's conjecture. Using the equations of motion, the on-shell action can be expressed
\begin{equation}S = -\frac{1}{6}\mathop{\rm Tr}\nolimits\big(\Psi_\text{simp} Q\Psi_\text{simp}\big).\end{equation}
We write the solution in the form
\begin{equation}\Psi_\text{simp} = \frac{1}{\sqrt{1+K}}c\frac{1}{\sqrt{1+K}} + Q\left(\frac{1}{\sqrt{1+K}}Bc\frac{1}{\sqrt{1+K}}\right).\end{equation}
The second term is BRST exact and drops out when we plug into the action. Then we find
\begin{eqnarray}
S \!\!\!\!\!\!\!\! && = -\frac{1}{6}\mathop{\rm Tr}\nolimits\left(\frac{1}{1+K}c\frac{1}{1+K}c\partial c\right)\nonumber\\
\!\!\!\!\!\!\!\! && =-\frac{1}{6}\int_0^\infty d\alpha\int_0^\infty d\beta \,e^{-\alpha-\beta}\mathop{\rm Tr}\nolimits\big(\Omega^\alpha c\Omega^\beta c\partial c\big).
\end{eqnarray}
The trace can be evaluated by taking the derivative of the correlator of three $c$s on the cylinder, \eq{ccc}. Accounting for vacuum normalization of the matter correlator, this gives
\begin{equation}\mathop{\rm Tr}\nolimits\big(\Omega^\alpha c\Omega^\beta c\partial c\big) = -g_0 \left(\frac{\alpha+\beta}{\pi}\right)^2\sin^2\frac{\pi \alpha}{\alpha+\beta}.\end{equation}
With our chosen normalization of the action, $g_0$ should be identified with the spacetime volume of the D-brane. The value of the action is then given by the double integral
\begin{equation}
S = \frac{g_0}{6}\int_0^\infty d\alpha\int_0^\infty d\beta \,e^{-\alpha-\beta} \left(\frac{\alpha+\beta}{\pi}\right)^2\sin^2\frac{\pi \alpha}{\alpha+\beta}.
\end{equation}
To evaluate the integral we make the substitution
\begin{eqnarray}
L=\alpha+\beta,\ \ \ \ \theta=\frac{\pi\alpha}{\alpha+\beta},\ \ \ \ d\alpha d\beta = \frac{1}{\pi}LdL d\theta.
\end{eqnarray}
The double integral then factorizes into a product of two single integrals:
\begin{eqnarray}
S = \frac{g_0}{6}\frac{1}{\pi}\int_0^\infty dL \frac{L^3}{\pi^2}e^{-L}\int_0^\pi d\theta \sin^2\theta.
\end{eqnarray}
The integral of $\sin^2$ gives half the period. The integral over $L$ produces $3!$. Thus in total
\begin{equation}S = \frac{g_0}{6}\frac{1}{\pi}\frac{3!}{\pi^2}\frac{\pi}{2}=\frac{g_0}{2\pi^2}.\end{equation}
Multiplying by $-1$ and dividing by the volume of the time coordinate gives the energy of the D-brane in agreement with Sen's conjecture.
One might wonder what happened to the phantom term, which for Schnabl's solution was the crucial ingredient which distinguished the tachyon vacuum from a pure gauge solution. There is a ``simple" analogue of the pure gauge solutions defined by
\begin{equation}F = \frac{\lambda}{1+K}.\end{equation}
The resulting solution is
\begin{equation}\Psi_\lambda = \frac{\lambda}{\sqrt{1+K}}c(\lambda+K)Bc\frac{1}{\sqrt{1+K}} - \frac{\lambda^2}{\sqrt{1+K}}c\frac{1-\lambda}{1-\lambda+K}Bc\frac{1}{\sqrt{1+K}}.\label{eq:simp_gauge}\end{equation}
As in Schnabl gauge, the solution is undefined for $\lambda>1$ because the state between the $c$s in the second term has iinfinite $C^*$ norm. Differently from Schnabl gauge, however, the solution can be written as an explicit superposition of wedge states for all $\mathrm{Re}(\lambda)<1$. It is clear that as $\lambda$ approaches $1$ the first term becomes the simple tachyon vacuum. The second term appears to vanish since it is multiplied by $1-\lambda$. But at the same time the denominator is developing a pole at $K=0$. Note that
\begin{equation}\frac{1-\lambda}{1-\lambda+K} = (1-\lambda)\int_0^\infty d\alpha e^{-(1-\lambda)\alpha}\Omega^\alpha = \int_0^\infty dt\, e^{-t}\Omega^{\frac{t}{1-\lambda}},
\end{equation}
where in the last step we made the substitution $t=(1-\lambda)\alpha$. Now it is clear that in the limit $\lambda\to 1^-$ the wedge state in the integrand approaches the sliver, and the integral over $t$ evaluates to $1$. Therefore, the second contribution in \eq{simp_gauge} becomes a phantom term in the limit $\lambda\to 1^-$. The phantom term, however, appears in the limit of the pure gauge solution, rather than the tachyon vacuum.
\begin{exercise} Consider the solution
\begin{equation}\Psi = c(i+K)Bc\frac{i}{i+K}.\end{equation}
The solution is not real, but should be gauge equivalent to the tachyon vacuum. The state $i/(i+K)$ does not have finite $D_2$ norm because of a pole on the imaginary axis. Find a way to define the solution and compute the energy to confirm Sen's conjecture.
\end{exercise}
The tachyon coefficient of the simple solution is given by
\begin{eqnarray}
T \!\!\!\!\!\!\!\! && =\frac{1}{2\pi^2}\int_0^\infty du\int_{-1}^1 dw\, e^{-u}\frac{(u+1)^2}{\sqrt{1-w^2}}\cos^2\left(\frac{\pi}{2}\frac{uw}{u+1}\right)\nonumber\\
\!\!\!\!\!\!\!\! && \approx 0.51.
\end{eqnarray}
This is again similar to the tachyon expectation value in Siegel gauge and Schnabl gauge. One surprise, however, is that if we compute the energy of the simple tachyon vacuum level by level given the exact coefficients, we obtain a divergent series. The reasons for this are discussed in detail in \cite{simple}, but ultimately it comes down to the fact that the simple solution receives contributions from wedge states all the way down to the identity string field, which is marginally a non-normalizable state. For this reason the simple tachyon vacuum is more singular than Schnabl's solution, which only contains wedge states of strictly positive width. However, the series for the energy of the simple tachyon vacuum can be resummed to give good agreement with Sen's conjecture.
\subsection{Simple Intertwining Solution}
\label{subsec:KOS}
The analogue of the simple tachyon vacuum for marginal deformations was investigated by Kiermaier, Okawa, and Soler \cite{KOS}, who showed that the solution could be written in a surprisingly simple way in terms of boundary condition changing operators. Hidden additional structure was recognized several years later \cite{KOSsing}, which permitted a generalization of the solution to describe arbitrary time-independent backgrounds. We will call this the {\it simple intertwining solution}. Another version of the ``intertwining solution" was recently described in \cite{KOSsingII}, and is proposed to address some of the limitations of the solution described here. For further discussion we refer the reader to \cite{KOSsingII}.
We start with the solution for regular marginal deformations. The simple tachyon vacuum is obtained from Schnabl's solution by replacing $\Omega$ with $\frac{1}{1+K}$. A similar replacement for Schnabl gauge marginal deformations gives
\begin{equation}\Psi_\text{int} = \frac{1}{\sqrt{1+K}}cV\frac{B}{1+\frac{1}{1+K}V}c\frac{1}{\sqrt{1+K}}.\end{equation}
It is still necessary to assume that the $V$s can be multiplied without singularity. The solution is real and lives in simple gauge:
\begin{equation}\mathcal{B}_{\frac{1}{1+K}}\Psi_\text{int} = 0.\end{equation}
We now begin a gradual process of re-expressing the solution in stages so as to reveal its essential structure. In the following it is important that $F=\frac{1}{1+K}$ satisfies
\begin{equation}F= \frac{1-F}{K}.\end{equation}
There is an analogous re-expression of the solution in Schnabl gauge \cite{KOSsing}, but it is more complicated and has not been closely studied. We start with
\begin{eqnarray}
\Psi_\text{int} \!\!\!\!\!\!\!\! && = \frac{1}{\sqrt{1+K}}cV\frac{B}{\frac{1}{1+K}(1+K+V)}c\frac{1}{\sqrt{1+K}}\nonumber\\
\!\!\!\!\!\!\!\! && = \frac{1}{\sqrt{1+K}}cV\frac{B}{1+K+V}(1+K)c\frac{1}{\sqrt{1+K}}\nonumber\\
\!\!\!\!\!\!\!\! && = \frac{1}{\sqrt{1+K}}c(1+K+V-(1+K))\frac{B}{1+K+V}(1+K)c\frac{1}{\sqrt{1+K}}\nonumber\\
\!\!\!\!\!\!\!\! && = \underbrace{\frac{1}{\sqrt{1+K}}c(1+K)Bc\frac{1}{\sqrt{1+K}}}_{\Psi_\mathrm{tv}}\ +\ \underbrace{(-1)\frac{1}{\sqrt{1+K}}c(1+K)\frac{B}{1+K+V}(1+K)c\frac{1}{\sqrt{1+K}}}_{\Psi_{\mathrm{tv}\to\text{int}}}.\nonumber\\
\end{eqnarray}
The first term is the simple tachyon vacuum, which here we denote as $\Psi_\mathrm{tv}$. The second term $\Psi_{\mathrm{tv}\to\text{int}}$ must be a solution to the equations of motion expanded around the tachyon vacuum:
\begin{equation}Q_{\Psi_\mathrm{tv}}\Psi_{\mathrm{tv}\to\text{int}}+ \Psi_{\mathrm{tv}\to\text{int}}^2 = 0.\end{equation}
In a sense, the first term destroys the reference D-brane, and the second term creates a marginally deformed D-bane out of the tachyon vacuum.
Via the Schwinger parameterization we have the equality
\begin{equation}
\frac{1}{1+K+V} = \int_0^\infty d\alpha \, e^{-\alpha}e^{-\alpha(K+V)}.
\end{equation}
The string field in the integrand looks like a wedge state, but $V$ appears in addition to $K$ in the exponential. The claim is that this is actually a wedge state containing an exponential insertion of line integrals of $V$ on the boundary:\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{wrapfigure}{l}{1\linewidth}
\centering
\resizebox{2.7in}{1.5in}{\includegraphics{OpenSFT_Erler35.jpg}}
\end{wrapfigure}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\
The effect of this exponential insertion is to deform the boundary condition from the reference D-brane $\mathrm{BCFT}_0$ to the target D-brane $\mathrm{BCFT}_*$. Let us prove this result. Suppose $\Omega_*^\alpha$ is a wedge state containing $\mathrm{BCFT}_*$ boundary conditions. The overlap with a test state in $\mathrm{BCFT}_0$ is given by
\begin{equation}\langle \phi,\Omega_*^\alpha\rangle = \left\langle e^{-\int_{1/2}^{\alpha+1/2} dx\, V(x)}f_\mathcal{S}\circ\phi(0)\right\rangle_{C_{\alpha+1}}.
\end{equation}
Adding a small $\epsilon$ to $\alpha$ gives
\begin{eqnarray}
\langle \phi,\Omega_*^{\alpha+\epsilon}\rangle \!\!\!\!\!\!\!\! && = \left\langle e^{-\int_{1/2}^{\alpha+\epsilon+1/2} dx\, V(x)}f_\mathcal{S}\circ\phi(0)\right\rangle_{C_{\alpha+\epsilon+1}}\nonumber\\
\!\!\!\!\!\!\!\! && = \left\langle e^{-\epsilon V(\alpha+1/2)-\int_{1/2}^{\alpha+1/2} dx\, V(x)}f_\mathcal{S}\circ\phi(0)\right\rangle_{C_{\alpha+\epsilon+1}}\nonumber\\
\!\!\!\!\!\!\!\! && = \left\langle \big(1-\epsilon V(\alpha+1/2)\big)e^{-\int_{1/2}^{\alpha+1/2} dx\, V(x)}f_\mathcal{S}\circ\phi(0)\right\rangle_{C_{\alpha+\epsilon+1}}\nonumber\\
\!\!\!\!\!\!\!\! && = \left\langle e^{-\int_{1/2}^{\alpha+1/2} dx\, V(x)}f_\mathcal{S}\circ\phi(0)\right\rangle_{C_{\alpha+\epsilon+1}}-\epsilon\left\langle V(\alpha+1/2)\big) e^{-\int_{1/2}^{\alpha+1/2} dx\, V(x)}f_\mathcal{S}\circ\phi(0)\right\rangle_{C_{\alpha+1}}. \nonumber\\
\end{eqnarray}
The small increase of circumference in the first term can be interpreted as insertion of a wedge state of width $\epsilon$ to the left of the exponential insertion. Therefore
\begin{equation}\Omega_*^{\alpha+\epsilon} =\Omega^\epsilon\Omega_*^{\alpha}-\epsilon V \Omega_*^{\alpha}. \end{equation}
Taking the derivative with respect to $\epsilon$ and setting $\epsilon$ to zero gives
\begin{equation}\frac{d}{d\alpha}\Omega_*^\alpha = -(K+V)\Omega_*^\alpha .\end{equation}
With the boundary condition $\Omega_*^0=1$, the solution is
\begin{equation}\Omega_*^\alpha = e^{-\alpha(K+V)},\end{equation}
as we wanted to show.
\begin{wrapfigure}{l}{.12\linewidth}
\centering
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\end{wrapfigure}
It will be helpful to adopt a slightly different language for describing the change of boundary condition. Consider an open string connecting a D-brane $\mathrm{BCFT}_0$ and a D-brane $\mathrm{BCFT}_*$. From the
point of view of radial quantization, such an open string can be associated to a unit half-disk with $\mathrm{BCFT}_0$ boundary conditions on the positive real axis and $\mathrm{BCFT}_*$ boundary conditions on the negative real axis. It is natural to think of this state as being created by a vertex operator which somehow changes the boundary condition from $\mathrm{BCFT}_0$ to $\mathrm{BCFT}_*$. This is called a {\it boundary condition changing operator}. We denote this as $\sigma(0)$. There is also a boundary condition changing operator which shifts the boundary condition from $\mathrm{BCFT}_*$ back to $\mathrm{BCFT}_0$, which we denote as $\overline{\sigma}(0)$. Boundary condition changing operators are not really local operators from the point
of view of $\mathrm{BCFT}_0$ or $\mathrm{BCFT}_*$, since they must always appear in conjugate pairs on the boundary. But since the boundary conditions on either side are conformal, they behave in many ways like local operators. They have OPEs and comparable conformal transformation properties.
\begin{wrapfigure}{l}{.23\linewidth}
\centering
\resizebox{1.5in}{1.3in}{\includegraphics{OpenSFT_Erler39.jpg}}
\end{wrapfigure}
Consider a disk with two boundary components $C_0$ and $C_*$, carrying $\mathrm{BCFT}_0$ and $\mathrm{BCFT}_*$ boundary conditions. Tracing a clockwise path around the boundary, let $a$ and $b$ be the points at
the junction of $C_0$ and $C_*$ and $C_*$ and $C_0$, respectively. The boundary condition changing operators for regular marginal deformations are related to the exponential insertion of line integrals of $V$ through
\begin{equation}\sigma(a)\overline{\sigma}(b) = e^{-\int_{C_*}dz V(z)}.\end{equation}
We can learn a few things from this identification. If $b$ approaches $a$ from a clockwise direction, the whole boundary of the disk caries an exponential insertion of line integrals of $V$. From the point of view of an open string in $\mathrm{BCFT}_*$, this is simply a trivial insertion of the identity operator:
\begin{equation}\lim_{{b\to a \atop \text{clockwise}}}\overline{\sigma}(b)\sigma(a) = e^{-\int_{\partial\text{disk}}dz V(z)}=1_\mathrm{BCFT_*}.\end{equation}
On the other hand, if $a$ approaches $b$ from a clockwise direction, the exponential insertion of $V$s disappears. Therefore
\begin{equation}\lim_{{a\to b \atop \text{clockwise}}}\sigma(a)\overline{\sigma}(b) = 1_\mathrm{BCFT_0}.\end{equation}
These properties hold because the OPEs of $V(x)$ are finite. If they were not finite, the exponential insertion of line integrals of $V$ would need to be defined with some renormalization, and any prescription consistent with conformal invariance will lead to divergence in the limits where $a$ and $b$ collide. In a sense, $\sigma$ and $\overline{\sigma}$ develop singularities in their OPE. It is also clear that without renormalization the exponential insertion of $V$s map in a trivial way under conformal transformation:
\begin{equation}f\circ e^{-\int_{C_*}dz V(z)} = e^{-\int_{f\circ C_*}dz V(z)}.\end{equation}
This implies that
\begin{equation}f\circ \Big(\sigma(a)\overline{\sigma}(b)\Big) = \sigma(f(a))\overline{\sigma}(f(b)).\end{equation}
Apparently $\sigma$ and $\overline{\sigma}$ are primary operators of weight $0$. An important advantage of the boundary condition changing operator point of view is that it is universal. For any two D-brane systems, regardless of whether they are related by marginal deformation, there are always open strings connecting them. The vertex operators of these open strings are boundary condition changing operators.
With this motivation it is natural to describe the solution in terms of $\sigma$ and $\overline{\sigma}$ rather than~$V$. We introduce string fields $\sigma$ and $\overline{\sigma}$ as infinitesimally thin strips containing $\sigma(0)$ and $\overline{\sigma}(0)$
\begin{wrapfigure}{l}{.35\linewidth}
\centering
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\end{wrapfigure}
at the origin. Assuming that $\sigma$ and $\overline{\sigma}$ change the boundary condition by a regular marginal deformation, we have the relations
\begin{equation}\overline{\sigma}\sigma = 1,\ \ \ \ \ \sigma\overline{\sigma} = 1.\end{equation}
Note that in the first equation, the right hand side is the identity string field of $\mathrm{BCFT}_*$, while in the second equation, it is the identity string field of $\mathrm{BCFT}_0$. To avoid cumbersome notation, we use the placement of $\sigma$ and $\overline{\sigma}$ in expressions to indicate which state space a string field occupies. In addition we have the properties
\begin{eqnarray}
\!\!\!\!\!\!\!\! && [B,\sigma]= [c,\sigma]= 0,\ \ \ \ [B,\overline{\sigma}]= [c,\overline{\sigma}]= 0; \\
\!\!\!\!\!\!\!\! && \ \ \ \ \ \ \ \ \ \ \ \frac{1}{2}\mathcal{B}^-\sigma = \frac{1}{2}\mathcal{B}^-\overline{\sigma} = 0;
\end{eqnarray}
and
\begin{eqnarray}
\!\!\!\!\!\!\!\! && Q\sigma = c\partial \sigma,\ \ \ \ Q\overline{\sigma} = c\partial\overline{\sigma};\\
\!\!\!\!\!\!\!\! && \ \ \ \ \frac{1}{2}\mathcal{L}^-\sigma = \frac{1}{2}\mathcal{L}^-\overline{\sigma} = 0.
\end{eqnarray}
The last properties follow because the boundary condition changing operators are primaries of weight $0$. Wedge states in $\mathrm{BCFT}_*$ can be described in two equivalent ways:
\begin{equation}\Omega_*^\alpha = e^{-\alpha(K+V)} = \sigma\Omega^\alpha\overline{\sigma}.\end{equation}
\begin{exercise}\label{ex:Vsigma}
Show that $V = \sigma\partial \overline{\sigma}$. Use this to prove the above relation.
\end{exercise}
\noindent The simple intertwining solution can then be written
\begin{equation}\Psi_\text{int} = \underbrace{\frac{1}{\sqrt{1+K}}c(1+K)c\frac{1}{\sqrt{1+K}}}_{\Psi_\mathrm{tv}}\ +\ \underbrace{(-1)\frac{1}{\sqrt{1+K}}c(1+K)\sigma\frac{B}{1+K}\overline{\sigma} (1+K)c\frac{1}{\sqrt{1+K}}}_{\Psi_{\mathrm{tv}\to\text{int}}}.\end{equation}
We are still not done. It is helpful to extract a factor of the simple tachyon vacuum between $\sigma$ and $\overline{\sigma}$ using the relation
\begin{equation}\frac{B}{1+K} = \frac{B}{\sqrt{1+K}}\Psi_\mathrm{tv}\frac{B}{\sqrt{1+K}}.\end{equation}
Then
\begin{equation}\Psi_\text{int} = \Psi_\mathrm{tv} + \underbrace{\frac{1}{\sqrt{1+K}}cB(1+K)\sigma\frac{1}{\sqrt{1+K}}\big(-\Psi_\mathrm{tv}\big)\frac{1}{\sqrt{1+K}}\overline{\sigma} (1+K)Bc\frac{1}{\sqrt{1+K}}}_{\Psi_{\mathrm{tv}\to\text{int}}}.\label{eq:KOS1}\end{equation}
Now we make some observations. The state $\Psi_{\mathrm{tv}\to\text{int}}$ represents the creation of the D-brane $\mathrm{BCFT}_*$ out of the tachyon vacuum. On the other hand, the factor $-\Psi_\mathrm{tv}$ in between the boundary condition changing operators also represents the creation of $\mathrm{BCFT}_*$ out of the tachyon vacuum. The difference between these states is that $\Psi_{\mathrm{tv}\to\text{int}}$ lives in $\mathrm{BCFT}_0$, while $-\Psi_\mathrm{tv}$ between $\sigma$ and $\overline{\sigma}$ lives in $\mathrm{BCFT}_*$. This suggests that the additional factors relating $\Psi_{\mathrm{tv}\to\text{int}}$ and $-\Psi_\mathrm{tv}$ can be interpreted as a kind of dictionary relating the degrees of freedom of $\mathrm{BCFT}_0$ and $\mathrm{BCFT}_*$. These factors are called {\it intertwining fields}, denoted $\Sigma$ and $\overline{\Sigma}$. The simple intertwining solution is written
\begin{equation}\Psi_\text{int} = \Psi_\mathrm{tv} -\Sigma\Psi_\mathrm{tv}\overline{\Sigma}.\end{equation}
The solution satisfies the equations of motion provided that
\begin{equation}Q_{\Psi_\mathrm{tv}}\Sigma = Q_{\Psi_\mathrm{tv}}\overline{\Sigma} = 0,\ \ \ \ \overline{\Sigma}\Sigma = 1.\end{equation}
Since the cohomology around the tachyon vacuum is trivial, the first relations imply that the intertwining fields are $Q_{\Psi_\mathrm{tv}}$-exact. A little guesswork leads to the expressions
\begin{eqnarray}
\Sigma \!\!\!\!\!\!\!\! && = Q_{\Psi_\mathrm{tv}}\left(\frac{B}{\sqrt{1+K}}\sigma\frac{1}{\sqrt{1+K}}\right) \nonumber\\
\!\!\!\!\!\!\!\! && = \frac{1}{\sqrt{1+K}}cB(1+K)\sigma\frac{1}{\sqrt{1+K}}\ +\ \frac{B}{\sqrt{1+K}}\sigma c\sqrt{1+K},\phantom{\Bigg)} \ \ \ \ \ \ \ \ \\
\overline{\Sigma}\!\!\!\!\!\!\!\! && = Q_{\Psi_\mathrm{tv}}\left(\frac{1}{\sqrt{1+K}}\overline{\sigma}\frac{B}{\sqrt{1+K}}\right) \nonumber\\
\!\!\!\!\!\!\!\! && =\frac{1}{\sqrt{1+K}}\overline{\sigma}(1+K)Bc\frac{1}{\sqrt{1+K}}\ + \ \sqrt{1+K}\overline{\sigma} c\frac{B}{\sqrt{1+K}}.\phantom{\Bigg)}
\end{eqnarray}
Only the first terms appear in \eq{KOS1}. This is because the second terms vanish when multiplied with the tachyon vacuum due to $c^2=0$. The second crucial property of the intertwining fields is $\overline{\Sigma}\Sigma = 1$. This can be demonstrated as follows:
\begin{eqnarray}
\overline{\Sigma}\Sigma \!\!\!\!\!\!\!\! && = Q_{\Psi_\mathrm{tv}}\left(\frac{1}{\sqrt{1+K}}\overline{\sigma}\frac{B}{\sqrt{1+K}}\right)Q_{\Psi_\mathrm{tv}}\left(\frac{B}{\sqrt{1+K}}\sigma\frac{1}{\sqrt{1+K}}\right)\nonumber\\
\!\!\!\!\!\!\!\! && = Q_{\Psi_\mathrm{tv}}\left(\frac{1}{\sqrt{1+K}}\overline{\sigma}\frac{B}{\sqrt{1+K}}Q_{\Psi_\mathrm{tv}}\left(\frac{B}{\sqrt{1+K}}\sigma\frac{1}{\sqrt{1+K}}\right)\right)\nonumber\\
\!\!\!\!\!\!\!\! && = Q_{\Psi_\mathrm{tv}}\bigg(\frac{1}{\sqrt{1+K}}\overline{\sigma}\frac{B}{\sqrt{1+K}}\underbrace{Q_{\Psi_\mathrm{tv}}\left(\frac{B}{1+K}\right)}_{1}\sqrt{1+K}\sigma\frac{1}{\sqrt{1+K}}\nonumber\\
\!\!\!\!\!\!\!\! &&\ \ \ \ \ \ \ \ - \underbrace{\frac{1}{\sqrt{1+K}}\overline{\sigma}\frac{B}{\sqrt{1+K}}\frac{B}{1+K}Q_{\Psi_\mathrm{tv}}\left(\sqrt{1+K}\sigma\frac{1}{\sqrt{1+K}}\right)}_0\bigg)\nonumber\\
\!\!\!\!\!\!\!\! && = Q_{\Psi_\mathrm{tv}}\left(\frac{1}{\sqrt{1+K}}\overline{\sigma} B\sigma \frac{1}{\sqrt{1+K}}\right)\nonumber\\
\!\!\!\!\!\!\!\! && = Q_{\Psi_\mathrm{tv}}\left(\frac{B}{1+K}\right)\nonumber\\
\!\!\!\!\!\!\!\! && = 1.
\end{eqnarray}
This is the finally the form of the solution that we are after.
The thing that impresses about this structure is that in principle it could relate any reference and target D-brane systems. The tricky part is the identity
\begin{equation}\overline{\Sigma}\Sigma=1.\end{equation}
In the current setup, this is directly related to the condition on the boundary condition changing operators
\begin{equation}\overline{\sigma}\sigma = 1.\end{equation}
This is satisfied for regular marginal deformations, but does not appear to apply to more general backgrounds where boundary condition changing operators have singular OPEs. The resolution to this problem proposed in \cite{KOSsing} follows from the earlier comment that many marginal deformations can be made regular by turning on a timelike gauge potential. Suppose we have boundary condition changing operators $\sigma_\text{bare}(x),\overline{\sigma}_\text{bare}(x)$ relating reference and target backgrounds of interest. Further assume that they are independent of the timelike free boson factor of the $\mathrm{BCFT}$, and that they are primaries of weight $h$ with OPE:
\begin{equation}\overline{\sigma}_\text{bare}(x)\sigma_\text{bare}(0) = \frac{1}{x^{2h}}+... \ .\end{equation}
We may construct primaries of weight $0$ by tensoring with a timelike, plane wave vertex operator
\begin{eqnarray}
\sigma(x) \!\!\!\!\!\!\!\! && = \sigma_\text{bare}e^{i\sqrt{h}X^0(x)},\\
\overline{\sigma}(x) \!\!\!\!\!\!\!\! && = \overline{\sigma}_\text{bare}e^{-i\sqrt{h}X^0(x)}.
\end{eqnarray}
Then we obtain the desired OPE
\begin{equation}\overline{\sigma}(x)\sigma(0) = 1+...\ .\label{eq:regOPE}\end{equation}
The plane wave vertex operators are boundary condition changing operators which turn on a timelike gauge potential on the D-brane of $\mathrm{BCFT}_*$. But a timelike gauge potential is physically trivial. In this way we can produce an analytic solution for any time independent D-brane configuration. This gives, for example, an analytic construction of tachyon lump solutions representing lower dimensional D-branes. In this case, the relevant boundary condition changing operators (before turning on the Wilson line) are given by the so-called Neumann-Dirichlet twist fields of weight $1/16$. We may also obtain solutions describing higher energy configurations, such as D-branes of higher dimension or with magnetic flux \cite{flux}.
There is a subtlety, however. Consider the matter 2-point function of boundary condition changing operators on the unit disk:\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\vspace{-.4cm}
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\begin{equation}\langle\overline{\sigma}(1)\sigma(e^{i\theta})\rangle_\text{disk}^\text{matter}.\end{equation}
For angles between $0$ and $\theta$ the boundary of the disk carries $\mathrm{BCFT}_0$ boundary conditions, and outside that range it carries $\mathrm{BCFT}_*$ boundary conditions. Since $\sigma$ and $\overline{\sigma}$ are weight zero primaries, the correlator is independent of $\theta$. We evaluate the correlator by taking the limit $\theta\to 0^+$, and using the OPE \eq{regOPE} we obtain
\begin{equation}\langle\overline{\sigma}(1)\sigma(e^{i\theta})\rangle_{\text{disk}}^{\text{matter}} = \langle 1\rangle_{\text{disk}}^{\text{matter},\mathrm{BCFT}_*}=g_*,\end{equation}
where $g_*$ is the $g$-function of $\mathrm{BCFT}_*$, and is proportional to the volume and energy of the target D-brane. We can also take the limit $\theta\to 2\pi^-$, where the boundary condition on the disk is $\mathrm{BCFT}_0$. This produces the $g$-function of $\mathrm{BCFT}_0$, $g_0$. This leads to a puzzle: generally the reference and target D-brane systems will not have the same energy, so $g_0\neq g_*$. But conformal invariance requires that the correlator is independent of $\theta$. The resolution to this problem is that the OPE between $\sigma$ and $\overline{\sigma}$ is different depending on whether $\mathrm{BCFT}_0$ or $\mathrm{BCFT}_*$ boundary conditions are squeezed between the operators. By choice of normalization we already have
\begin{equation}\lim_{x\to 0^+}\overline{\sigma}(x)\sigma(0) = 1,\end{equation}
so in the opposite order we must have
\begin{equation}\lim_{x\to 0^+}\sigma(x)\overline{\sigma}(0) = \frac{g_*}{g_0}.\end{equation}
This implies that the string fields representing these boundary condition changing operators multiply as
\begin{equation}\overline{\sigma}\sigma = 1,\ \ \ \ \ \sigma\overline{\sigma} = \frac{g_*}{g_0}.\end{equation}
Unfortunately, this is not consistent with associativity:
\begin{equation}(\overline{\sigma}\sigma)\overline{\sigma}\neq \overline{\sigma}(\sigma\overline{\sigma}).\end{equation}
This anomaly reflects the fact that correlators do not have a well-defined limit when three boundary condition changing operators collide. However, the product $\sigma\overline{\sigma}$ which causes this problem does not appear when evaluating the equations of motion or the action and Ellwood invariant, which readily produce the expected results. It does, however, lead to some complications in understanding background independence.
Let $\Psi^{(0)}\in \mathcal{H}_{\mathrm{BCFT}_0}$ be the dynamical string field of $\mathrm{BCFT}_0$ and $\Psi^{(*)}\in\mathcal{H}_{\mathrm{BCFT}_*}$ be the dynamical string field of $\mathrm{BCFT}_*$. We want to use the simple interrtwining solution to relate these objects by field redefinition, in particular so that the actions are equal up to an additive constant
\begin{equation}S_0[\Psi^{(0)}]=S_*[\Psi^{(*)}]+\text{constant}.\end{equation}
As explained in the introduction, this is the problem of background independence. First we separate the string field of $\mathrm{BCFT}_0$ into the simple intertwining solution plus a fluctuation
\begin{equation}\Psi^{(0)}=\Psi_\text{int} +\varphi\ \in \ \mathcal{H}_{\mathrm{BCFT}_0}.\end{equation}
The action is expressed
\begin{equation}
S_0[\Psi_\text{int} +\varphi] = S_0[\Psi_\text{int}] -\frac{1}{2}\mathop{\rm Tr}\nolimits\big(\varphi Q_{\Psi_\text{int}} \varphi\big)-\frac{1}{3}\mathop{\rm Tr}\nolimits\big(\varphi^3\big).
\end{equation}
Since the actions of $\mathrm{BCFT}_0$ and $\mathrm{BCFT}_*$ are both cubic, it is consistent to assume that $\varphi$ should be linearly related to the string field $\Psi^{(*)}$. It is natural to guess
\begin{equation}\varphi = \Sigma\Psi^{(*)}\overline{\Sigma}.\label{eq:field_red}\end{equation}
We can immediately see that the cubic terms in the actions agree due to $\overline{\Sigma}\Sigma = 1$. To identify the kinetic terms we have to deal with the shifted kinetic operator $Q_{\Psi_\text{int}}$. To do this it is useful to introduce the operator
\begin{equation}Q_{\Psi_1\Psi_2}X = QX +\Psi_1X-(-1)^{|X|}X\Psi_2.\end{equation}
This is nilpotent if $\Psi_1$ and $\Psi_2$ satisfy the equation of motion:
\begin{equation}Q_{\Psi_1\Psi_2}^2=0.\end{equation}
We also have a version of the derivation property:
\begin{equation}Q_{\Psi_1\Psi_3}(XY) = (Q_{\Psi_1\Psi_2} X)Y + (-1)^{|X|}X(Q_{\Psi_2\Psi_3}Y),\label{eq:ShiftLeibniz}\end{equation}
\begin{wrapfigure}{l}{.26\linewidth}
\centering
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where $\Psi_2$ on the right hand side is any solution; it cancels out and so does not appear on the left hand side. The significance of this operator is that it naturally appears in open SFT formulated
on a pair of D-branes. In this setup the string field contains $2\times 2$ Chan-Paton factors and can be arranged into a $2\times 2$ matrix
\begin{equation}X=\left(\begin{matrix} X_{11}\ \ X_{12} \\ X_{21} \ \ X_{22}\end{matrix}\right).\end{equation}
A priori it is not necessary to assume that the matrix entries are states of the same $\mathrm{BCFT}$; the two D-branes which comprise the system need not be identical. Thus $X_{11}$ is a state of the $\mathrm{BCFT}$ of the first D-brane, $X_{22}$ is a state in the $\mathrm{BCFT}$ of the second and $X_{12}$ and $X_{21}$ are stretched string states between the two $\mathrm{BCFT}$s. If we condense the first D-brane to a solution $\Psi_1$ and the second D-brane to a solution $\Psi_2$, the solution of the combined system is
\begin{equation}\Psi = \left(\begin{matrix}\Psi_1 & 0 \\ 0 & \Psi_2\end{matrix}\right),\end{equation}
and the kinetic operator expanded around $\Psi$ is
\begin{equation}Q_\Psi X = \left(\begin{matrix} Q_{\Psi_1} X_{11} & Q_{\Psi_1\Psi_2}X_{12} \\ Q_{\Psi_2\Psi_1}X_{21} & Q_{\Psi_2}X_{22}\end{matrix}\right).\end{equation}
Therefore $Q_{\Psi_1\Psi_2}$ is the shifted kinetic operator for a stretched string connecting a D-brane condensed to a solution $\Psi_1$ and a D-brane condensed to a solution $\Psi_2$. It turns out that $\Sigma$ and $\overline{\Sigma}$ satisfy
\begin{equation}Q_{\Psi_\text{int} 0}\Sigma = 0,\ \ \ \ Q_{0\Psi_\text{int}}\overline{\Sigma} = 0 .\end{equation}
This can be seen as follows:
\begin{eqnarray}
Q_{\Psi_\text{int}0}\Sigma \!\!\!\!\!\!\!\! && = Q\Sigma +\Psi_\text{int}\Sigma\nonumber\\
\!\!\!\!\!\!\!\! && = Q\Sigma + (\Psi_\mathrm{tv}-\Sigma\Psi_\mathrm{tv}\overline{\Sigma})\Sigma\nonumber\\
\!\!\!\!\!\!\!\! && = Q\Sigma + \Psi_\mathrm{tv}\Sigma -\Sigma\Psi_\mathrm{tv}\nonumber\\
\!\!\!\!\!\!\!\! && = Q_{\Psi_\mathrm{tv}}\Sigma\nonumber\\
\!\!\!\!\!\!\!\! && = 0,
\end{eqnarray}
with a similar computation showing $Q_{0\Psi_\text{int}}\overline{\Sigma} = 0$. The interpretation is that $\Sigma$ is annihilated by the kinetic operator for a stretched string connecting a $\mathrm{BCFT}_0$ D-brane condensed to the simple intertwining solution and a $\mathrm{BCFT}_*$ D-brane at the perturbative vacuum. However, these two configurations are physically identical. This implies that the intertwining fields are representatives of the cohomology class of the identity operator in $\mathrm{BCFT}_*$. The linear field redefinition \eq{field_red} can then be interpreted as left and right multiplication by $1$. Now we can compute
\begin{eqnarray}
Q_{\Psi_\text{int}}\varphi \!\!\!\!\!\!\!\! && = Q_{\Psi_\text{int}}\big(\Sigma\Psi^{(*)}\overline{\Sigma}\big)\nonumber\\
\!\!\!\!\!\!\!\! && = \big(Q_{\Psi_\text{int}0}\Sigma\big)\Psi^{(*)}\overline{\Sigma} + \Sigma\big(Q\Psi^{(*)}\big)\overline{\Sigma} + \Sigma\Psi^{(*)}\big(Q_{0\Psi_\text{int}}\overline{\Sigma}\big)\nonumber\\
\!\!\!\!\!\!\!\! && = \Sigma\big(Q\Psi^{(*)}\big)\overline{\Sigma}.
\end{eqnarray}
Plugging into the kinetic term of the action and using $\overline{\Sigma}\Sigma=1$ gives
\begin{equation}S_0[\Psi_\text{int}+\varphi] = S_0[\Psi_\text{int}]+S_*[\Psi^{(*)}].\end{equation}
This almost establishes background independence of Witten's open bosonic string field theory. Of course, this argument does not apply to backgrounds where the simple intertwining solution does not exist, including most time dependent backgrounds. A more serious problem is that the proposed field redefinition is not an isomorphism between the field variables of the two D-brane systems. At first it might appear that we can establish isomorphism through the inverse relation
\begin{equation}\overline{\Sigma}\varphi\Sigma = \Psi^{(*)}.\end{equation}
But a moments thought reveals that the field redefinition and its inverse do not compose associatively, since products of $\Sigma$ and $\overline{\Sigma}$ are not associative. A resolution of this difficulty is proposed using a different kind of intertwining solution in \cite{KOSsingII}.
One interesting application is constructing backgrounds representing multiple D-branes. Such solutions, in particular, show that open SFT is capable of altering the gauge group of the massless excitations on the D-brane. A curious feature of open string backgrounds is that they can be superimposed to create new backgrounds. That is, given a D-brane represented by $\mathrm{BCFT}_1$ and another D-brane represented by $\mathrm{BCFT}_2$, by adding Chan-Paton factors we can obtain a background where both D-branes are present. It is almost as though D-brane solitons do not interact. In ordinary field theories, simply adding soliton solutions together does not give a multi-soliton solution since the theory is nonlinear. Generally this is also true for solutions in open SFT. The simple intertwining solution however has a special structure, and we can attempt to find a solution by adding solutions creating $\mathrm{BCFT}_1$ and $\mathrm{BCFT}_2$ around the tachyon vacuum:
\begin{equation}\Psi = \Psi_\mathrm{tv} - \Sigma_1\Psi_\mathrm{tv}\overline{\Sigma}_1 - \Sigma_2\Psi_\mathrm{tv}\overline{\Sigma}_2.\end{equation}
This {\it does} satisfy the equations of motion provided that
\begin{equation}\overline{\Sigma}_1\Sigma_2 = 0,\ \ \ \ \overline{\Sigma}_2\Sigma_1 = 0,\end{equation}
in addition to the usual conditions on $\Sigma_1,\overline{\Sigma}_1$ and $\Sigma_2,\overline{\Sigma}_2$ individually. It is instructive to assemble row and column vectors
\begin{equation}\Sigma = \big(\Sigma_1\ \ \Sigma_2\big),\ \ \ \ \overline{\Sigma} = \left(\begin{matrix}\overline{\Sigma}_1 \\ \overline{\Sigma}_2\end{matrix}\right).\end{equation}
We have
\begin{equation}\overline{\Sigma}\Sigma = \left(\begin{matrix} 1 \ \ 0\\ 0\ \ 1\end{matrix}\right).\end{equation}
This is the identity in the algebra of string fields on two D-branes with Chan-Paton factors. Thus the solution representing $\mathrm{BCFT}_1$ and $\mathrm{BCFT}_2$ together is a special instance of the general structure of the simple intertwining solution. It is neat how adding D-brane solitons immediately leads to the expected Chan-Paton structure on the composite system. This is a special circumstance of the simple intertwining solution; adding solutions in open SFT usually does not lead to a solution. However, this does show that nonlinear interactions between D-brane solitons must be, in a sense, a gauge artifact.
The simple intertwining solution leads to a remarkably clear picture of how D-brane vacua appear as classical solutions in open bosonic SFT. However, it is rather far from the kind of solutions constructed in Siegel gauge level truncation in the early 2000s. At present there have been no attempts to compute the energy of vacua in level truncation given the exact coefficients. As with the simple tachyon vacuum, it seems unlikely the energy will be convergent. There is also the timelike Wilson line. While in principle it should be possible to remove it by gauge transformation, doing so in practice seems to require fundamentally new ideas. Some progress in this direction was recently reported in \cite{KOSsingII}.
\subsection{The Solutions of Fuchs, Kroyter and Potting and of Kiermaier and Okawa}
\label{subsec:singular}
The solutions we have described so far are part of the ``Schnabl gauge universe;" they are either in Schnabl gauge or satisfy a closely related condition. One persistent feature of such solutions is that they have difficulty dealing with singular OPEs between matter operators. We now describe an analytic solution with a very different structure, capable of describing marginal deformations with singular operator products. The solution was discovered in its basic form by Fuchs, Kroyter and Potting \cite{FKP} in the context of the Wilson line deformation. The structure was clarified by Kiermaier and Okawa, who further generalized to describe arbitrary marginial deformations \cite{KO}.
We start by describing the solution for the Wilson line. This corresponds to giving a constant expectation value to a gauge field along some direction, say $x^1$. In Maxwell theory, such a background appears to be trivial since the gauge field can be removed by gauge transformation
\begin{equation}A_\mu = \partial_\mu\big(A_1 x^1),\end{equation}
where $A_1$ is the value of the gauge field along the $x^1$-direction. However, this gauge transformation fails to be well-defined if the $x^1$ direction is compactified on a circle of radius $R$, since $x^1$ does not respect the periodicity of the circle. Moreover, while the field strength vanishes, the constant gauge field generates a nontrivial Wilson line around the circle
\begin{equation}e^{i \int_0^{2\pi R}dx^1 A_1} = e^{2\pi i R A_1}.\end{equation}
This is why the solution is called a Wilson line deformation.
For open bosonic strings the Wilson line deformation is generated by the exactly marginal boundary operator $i\partial_\parallel X^1(y)$, where $\partial_\parallel$ is the derivative along the open string boundary. If the $x^1$ direction is noncompact, the corresponding solution of the linearized equations of motion can be written in BRST exact form
\begin{equation}\lambda\, i c \partial_\parallel X^1(0)|0\rangle = Q\Big(\lambda\, i X^1(0)|0\rangle\Big).\end{equation}
If $x^1$ is compactified on a circle, the linearized solution is not BRST trivial since the operator $X^1(0)$ does not respect the periodicity of the circle. For convenience we parameterize the deformation in terms of $\lambda$; at linearized order this is related to the expectation value of the gauge field through
\begin{equation}A_1 = \sqrt{2}\lambda +\mathcal{O}(\lambda^2).\end{equation}
The strategy of Fuchs, Kroyter and Potting is based on the observation that pure gauge solutions are usually much easier to find than nontrivial solutions. You simply choose a ghost number $0$ string field $\Lambda$ and compute
\begin{equation}\Psi = \Lambda Q \Lambda^{-1}.\end{equation}
The idea is to find a finite nonlinear gauge transformation which generalizes the BRST exact form of the vertex operator for the gauge field. If the finite gauge parameter $\Lambda$ can be chosen so that the resulting solution caries zero momentum along the $x^1$ direction,
\begin{equation}p_1\Big(\Lambda Q\Lambda^{-1}\Big) = 0,\label{eq:0p}\end{equation}
then the solution is well-defined even after compactifying $x^1$ on a circle. The gauge parameter $\Lambda$, however, is not well defined, and in this way we generate a nontrivial solution through gauge transformation. Interestingly, the zero momentum condition \eq{0p} is structurally very similar to the equations of motion of open superstring field theory in the Wess-Zumino-Witten-like formulation~\cite{Berkovits}.
We expand $\Lambda$ in a power series in $\lambda$:
\begin{equation}\Lambda = 1-\Big(\lambda\Lambda_1 +\lambda^2\Lambda_2 + \lambda^3\Lambda_3+...\Big),\end{equation}
where
\begin{equation}\Lambda_1 = \sqrt{\Omega}\big(iX^1\big)\sqrt{\Omega},\end{equation}
and $X^1$ is defined by an infinitesimally thin strip containing $X^1(0)$ at the origin. We write the solution as
\begin{equation}\Psi =-(Q\Lambda)\Lambda^{-1}.\end{equation}
Expanding $\Lambda^{-1}$ as a geometric series gives the order $\lambda^n$ contribution to the solution
\begin{equation}\Psi_N = \sum_{n=1}^N\sum_{{k_1+k_2+...+k_n=N \atop k_i\geq 1}}\big(Q\Lambda_{k_1}\big)\Lambda_{k_2}...\Lambda_{k_i}.\end{equation}
For example
\begin{equation}\Psi_1 = Q\Big(\sqrt{\Omega}\big(iX^1\big)\sqrt{\Omega}\Big) = \sqrt{\Omega}\big(i c\partial_\parallel X^1\big)\sqrt{\Omega}.\end{equation}
This satisfies
\begin{equation}p_1 \Psi_1 = 0,\end{equation}
since the zero momentum photon vertex operator is independent of the position zero mode. At second order we have
\begin{eqnarray}
\Psi_2\!\!\!\!\!\!\!\! && = Q\Lambda_2 + \big(Q\Lambda_1\big)\Lambda_1\nonumber\\
\!\!\!\!\!\!\!\! && = Q\Lambda_2 + \sqrt{\Omega} Q(iX^1)\Omega(iX^1)\sqrt{\Omega}.
\end{eqnarray}
We detemine $\Lambda_2$ so that
\begin{equation}p_1\Psi_2 = 0.\end{equation}
The momentum operator is the zero mode of a weight 1 primary field,
\begin{equation}p_1 = \oint \frac{dz}{2\pi i} i\partial X^1(z),\end{equation}
and is a derivation of the open string star product, for the same reason as the BRST operator. We have
\begin{equation}p_1(iX^1) = 1.\label{eq:pX1} \end{equation}
It is useful to introduce the string field $(iX^1)^n$ as an infinitesimally thin strip containing the operator $(iX^1(0))^n$ at the origin, where the power of $X^1(0)$ is defined with boundary normal ordering. Note that $(iX^1)^n$ is not the same as the $n$ star products of $iX^1$, which would be divergent due to the logarithm in the $X^1$-$X^1$ OPE. Generalizing \eq{pX1},
\begin{equation}p_1 (iX^1)^n = n (iX^1)^{n-1}.\label{eq:pX1n}\end{equation}
Now we can readily compute the action of the momentum operator on $\Psi_2$. Setting this to zero leads to a condition on $\Lambda_2$:
\begin{equation}p_1\big(Q\Lambda_2) + \sqrt{\Omega}Q( iX^1)\Omega\sqrt{\Omega}=0.\end{equation}
Noting \eq{pX1n}, the obvious solution is
\begin{equation}\Lambda_2 = -\frac{1}{2!}\sqrt{\Omega}(iX)^2 \Omega\sqrt{\Omega}.\end{equation}
\begin{exercise}
The third order contribution to the solution is
\begin{equation}\Psi_3 = Q\Lambda_3 + \big(Q\Lambda_2\big)\Lambda_1 + \big(Q\Lambda_1\big)\Lambda_2 + \big(Q\Lambda_1)\Lambda_1\Lambda_1.\end{equation}
Requiring that $\Psi_3$ has zero momentum, show that $\Lambda_3$ can be chosen as
\begin{equation}\Lambda_3 = \frac{1}{3!}\sqrt{\Omega} (iX^1)^3\Omega^2\sqrt{\Omega}.\end{equation}
\end{exercise}
\noindent The general result is now easy to guess:
\begin{equation}\Lambda_n = \frac{(-1)^{n+1}}{n!}\sqrt{\Omega}(iX^1)^n \Omega^{n-1}\sqrt{\Omega}.\end{equation}
This is the solution as characterized by Fuchs, Kroyter, and Potting \cite{FKP}. The operator insertions are separated by wedge states with positive integer width, and there is no question of OPE divergence. One distinctive feature of the solution is the absence of $b$-ghosts, as one finds in Schnabl gauge or Siegel gauge. This is because the solution is not characterized by a gauge condition, and is not constructed from a propagator. This is surprising since there is a close connection between the perturbative construction of marginal deformations and the computation of the tree-level $S$-matrix. In this context one typically expects $b$-ghosts to provide the correct measure for integration over the moduli space of disks with boundary punctures. Another unusual feature of the solution is that it does not satisfy the reality condition, as can be readily seen by inspecting the second order contribution
\begin{equation}\Psi_2 = -\frac{1}{2!}\sqrt{\Omega}Q(iX^1)^2\Omega\sqrt{\Omega}+\sqrt{\Omega}Q(iX^1)\Omega(iX^1)\sqrt{\Omega}\neq\Psi_2^\ddag.\end{equation}
This is not a deep concern since the solution can be made real by gauge transformation. A recipe for achieving this is described in \cite{KO}. For physical questions, the solution is equivalent to a real solution.
The construction so far does not immediately apply to other marginal deformations, since it is not clear what should be the analogue of the compactification and zero momentum constraint. For this it is helpful to adopt a different point of view on the operator $X^1(y)$. The boundary condition changing operators which turn on the Wilson line are
\begin{equation}\sigma_\mu=e^{i\mu X^1(y)},\ \ \ \ \overline{\sigma}_\mu = e^{-i\mu X^1(y)}.\end{equation}
For general $\mu$ these operators do not respect the periodicity of compactification on the circle, and appear to be undefined. But this is actually expected, since boundary condition changing operators are not well defined local operators in the reference $\mathrm{BCFT}$. From this point of view the issue with $X^1(y)$ is not necessarily that it does not respect the periodicity of compactification, but that it implements an infinitesimal change in the open string boundary condition. We can write
\begin{equation}(iX^1)^n=\left.\frac{d^n}{d\mu^n}\sigma_\mu\right|_{\mu=0}.\end{equation}
It is helpful to streamline notation somewhat by introducing the operator
\begin{equation}d^n =\left.\frac{d^n}{d\mu^n}\right|_{\mu=0},\end{equation}
where we take $\mu=0$ after all derivatives are evaluated. We will also leave the dependence on $\mu$ in the argument of the derivative operator implicit, so for example
\begin{equation}d\sigma = \left.\frac{d}{d\mu}\sigma_\mu\right|_{\mu=0}.\end{equation}
The $n$th contribution to the gauge parameter is then
\begin{equation}
\Lambda_n = \frac{(-1)^{n+1}}{n!}\sqrt{\Omega}d^n\sigma \Omega^{n-1}\sqrt{\Omega}.
\end{equation}
Summing over $n$ gives a formula for the complete finite gauge transformation
\begin{equation}\Lambda = \sqrt{\Omega}\Big(\sigma e^{-\lambda\overleftarrow{d}\Omega}\Big)\frac{1}{\sqrt{\Omega}}.\end{equation}
The arrow over $d$ indicates that it is acting on $\sigma$ to the left. The inverse wedge state here is formal; it cancels out when we evaluate the expression.
The goal is to solve the zero momentum constraint by showing that a change of boundary condition implemented by $\sigma$ inside the solution is always undone by a $\overline{\sigma}$. The second order deformation takes the form
\begin{equation}
\Psi_2 = -\frac{1}{2!}\sqrt{\Omega}Q(d^2\sigma)\Omega\sqrt{\Omega}+\sqrt{\Omega}Q(d\sigma)\Omega(d\sigma)\sqrt{\Omega}.
\end{equation}
This expression assumes that $x^1$ is noncompact, since there is no $\overline{\sigma}$ to undo the change of boundary condition. In the noncompact case we have the equality
\begin{equation}\sigma_\mu = \sigma_{-\mu},\ \ \ \ \sigma_{\mu=0}=1,\end{equation}
which implies
\begin{equation}d\sigma = -d\overline{\sigma},\ \ \ \ d^0\overline{\sigma} = 1.\end{equation}
This allows us to rewrite the second order deformation as
\begin{eqnarray}
\Psi_2 \!\!\!\!\!\!\!\! && = -\frac{1}{2!}\sqrt{\Omega}Q(d^2\sigma)\Omega (d^0\overline{\sigma})\sqrt{\Omega}-\sqrt{\Omega}Q(d\sigma)\Omega(d\overline{\sigma})\sqrt{\Omega}\nonumber\\
\!\!\!\!\!\!\!\! && = -\frac{1}{2!}\Big(\sqrt{\Omega}Q(d^2\sigma)\Omega (d^0\overline{\sigma})\sqrt{\Omega}+2\sqrt{\Omega}Q(d\sigma)\Omega(d\overline{\sigma})\sqrt{\Omega}+\sqrt{\Omega}Q(d^0\sigma)\Omega (d^2\overline{\sigma})\sqrt{\Omega}\Big)\nonumber\\
\!\!\!\!\!\!\!\! && = -\frac{1}{2!}d^2\Big(\sqrt{\Omega}Q\sigma\Omega \overline{\sigma}\sqrt{\Omega}\Big).
\end{eqnarray}
In this final form the change of boundary condition is canceled, and the second order deformation is meaningful even if $x^1$ is compact. In fact, the second order deformation is defined for any marginal deformation with an associated one parameter family of boundary condition changing operators~$\sigma_\mu,\overline{\sigma}_\mu$. This is how Kiermaier and Okawa manage to generalize the Wilson line solution.
The systematics of how the change of boundary condition is undone at higher orders could be complicated. To see how to deal with it, consider the state
\begin{equation}\Lambda\Lambda^\ddag = \sqrt{\Omega}\Big(\sigma e^{-\lambda\overleftarrow{d}\Omega}\Big)\frac{1}{\sqrt{\Omega}}\frac{1}{\sqrt{\Omega}}\Big(e^{-\lambda\overrightarrow{d}\Omega}\overline{\sigma}\Big)\sqrt{\Omega}.\end{equation}
Expanding the exponentials in Taylor series gives
\begin{eqnarray}
\Lambda\Lambda^\ddag \!\!\!\!\!\!\!\! && = \sum_{m,n=0}^\infty\frac{ (-\lambda)^{m+n}}{m!n!}\sqrt{\Omega}(d^m\sigma)\Omega^{m+n-1}(d^n\overline{\sigma})\sqrt{\Omega}\nonumber\\
\!\!\!\!\!\!\!\! && = \sum_{N=0}^\infty \frac{(-\lambda)^N}{N!}\sum_{k=0}^N \left({ N\atop k}\right)\sqrt{\Omega}(d^{N-k}\sigma)\Omega^{N-1}(d^k\overline{\sigma}) \sqrt{\Omega}\nonumber\\
\!\!\!\!\!\!\!\! && = \sum_{N=0}^\infty \frac{(-\lambda)^N}{N!}d^N\Big(\sqrt{\Omega}\sigma\Omega^{N-1}\overline{\sigma}\sqrt{\Omega}\Big)\nonumber\\
\!\!\!\!\!\!\!\! && = \sqrt{\Omega}\left(\sigma e^{-\lambda \overleftrightarrow{d}\Omega}\frac{1}{\Omega}\overline{\sigma} \right)\sqrt{\Omega}.
\end{eqnarray}
The double arrow over $d$ indicates that it acts as a total derivative on the boundary condition changing operators to the left and right. The final expression is a well-defined state for any marginal deformation. This leads us to modify the pure gauge ansatz by writing
\begin{equation}\Psi = - (Q\Lambda)\Lambda^{-1} = -(Q\Lambda)\Lambda^\ddag\frac{1}{\Lambda\Lambda^\ddag}.\end{equation}
This leads to the expression
\begin{eqnarray}
\Psi \!\!\!\!\!\!\!\! && = -A U^{-1},
\end{eqnarray}
where
\begin{eqnarray}
A \!\!\!\!\!\!\!\! && = \sqrt{\Omega}\left(Q\sigma e^{-\lambda \overleftrightarrow{d}\Omega}\frac{1}{\Omega}\overline{\sigma} \right)\sqrt{\Omega},\\
U \!\!\!\!\!\!\!\! && = \sqrt{\Omega}\left(\sigma e^{-\lambda \overleftrightarrow{d}\Omega}\frac{1}{\Omega}\overline{\sigma} \right)\sqrt{\Omega}.\label{eq:singularU}
\end{eqnarray}
This is the form of the solution found by Kiermaier and Okawa \cite{KO}, and is defined for arbitrary marginal deformations. It is amusing to note the appearance of a translation operator which formally shifts the marginal coupling constant in the direction of the $SL(2,\mathbb{R})$ vacuum. Restoring the explicit dependence of the boundary condition changing operators on the marginal parameter, we can formally write
\begin{equation}A = \sqrt{\Omega}(Q\sigma_{-\lambda\Omega})\frac{1}{\Omega}\overline{\sigma}_{-\lambda\Omega}\sqrt{\Omega},\ \ \ \ \ \ \ U = \sqrt{\Omega}\sigma_{-\lambda\Omega}\frac{1}{\Omega}\overline{\sigma}_{-\lambda\Omega}\sqrt{\Omega}.\end{equation}
The marginal parameter is now a string field.
\begin{exercise}
Demonstrate that the Kiermaier-Okawa solution satisfies the equations of motion.
\end{exercise}
Kiermaier and Okawa mostly do not use the language of boundary condition changing operators, but describe the solution in terms of an appropriately renormalized exponential insertion of line integrals of a marginal operator. The connection between these descriptions is given in terms of correlation functions on the cylinder:
\begin{eqnarray}\langle \phi,\sigma_\mu\Omega^n\overline{\sigma}_\mu\rangle \!\!\!\!\!\!\!\! && = \big\langle \sigma_\mu(n+1/2)\overline{\sigma}_\mu(1/2)f_\mathcal{S}\circ\phi(0)\big\rangle_{C_{n+1}}\nonumber\\
\!\!\!\!\!\!\!\! && = \Big\langle\Big[e^{\mu \int_{1/2}^{n+1/2} dy V(y)}\Big]_r f_\mathcal{S}\circ \phi(0)\Big\rangle_{C_{n+1}},
\end{eqnarray}
where $V(y)$ is the marginal operator. The bracket $[\cdot]_r$ is a reminder that powers of the line integral are defined with the appropriate renormalization. The nature of the renormalization scheme is a major aspect of the discussion of \cite{KO}. This is an important point because the boundary condition changing operators for a generic marginal deformation are usually not known, and to characterize the solution it is necessary to construct the boundary condition changing operators ``from scratch" by figuring out how to renormalize the exponential insertion. It is interesting to mention that the line integrals of $V$ can be interpreted in terms of integration over the moduli spaces of disks with boundary punctures appearing in tree level amplitudes. The $b$-ghost which provides the measure is effectively hidden in the fact that $V$ is an {\it integrated} vertex operator, related to the usual on-shell vertex operator $cV$ through a contour integral of the $b$-ghost around the puncture. The $b$-ghost deletes the $c$ and effectively disappears from the amplitude, and likewise the solution.
This solution has not recieved as much attention as those in the ``Schnabl gauge universe." Many important questions remain unanswered. Not much is known about the nonperturbative behavior of the solution for finite $\lambda$. It should also be possible to understand perturbative background independence by expanding the action around the solution, but this has not been investigated. The solution may be worth considering from the point of view of recent interest in perturbative aspects of string field theory \cite{SenErbin,SenTree,localization1,localization2,Vosmera}, since it is conceptually rather different from standard marginal solutions in Siegel gauge.
\section{Lecture 4: Toolbox}
In the last lecture we discussed a number of specific analytic solutions. Now we describe methods that can be used to gain insight into how analytic solutions work in general.
\subsection{$\mathcal{L}^-$ Level Expansion}
\label{subsec:L}
Often it is useful to extract information about a string field by probing it with a test state. Interesting things can happen if the test state is sliver-like. Given a Fock space state $|\phi\rangle$, we can consider a sliver-like test state
\begin{equation}\left(\frac{1}{\epsilon}\right)^{\frac{1}{2}\mathcal{L}^-}\!\!|\phi\rangle,\ \ \ \ \epsilon\text{ small}.\end{equation}
This is a strip of worldsheet of width $1/\epsilon$ containing a vertex operator in the middle creating the state $|\phi\rangle$. If we probe a string field $X$ with a test state of this form, in a broad set of circumstances we get a quantity that can be expanded as a power series in $\epsilon$:
\begin{equation}\left\langle \left(\frac{1}{\epsilon}\right)^{\frac{1}{2}\mathcal{L}^-} \!\!|\phi\rangle ,X\right\rangle = \epsilon^{h_1}\langle\phi, X_{h_1}\rangle + \epsilon^{h_2}\langle \phi, X_{h_2}\rangle + \epsilon^{h_3}\langle \phi,X_{h_3}\rangle + ...\ \ \ \ h_1<h_2<h_3< ... \ .\end{equation}
The coefficients of the series define the overlap of $\phi$ with a sequence of string fields $X_{h_1},X_{h_2},X_{h_3},...$. The states $X_h$ must be eigenstates of $\frac{1}{2}\mathcal{L}^-$ with eigenvalue $h$:
\begin{equation}\frac{1}{2}\mathcal{L}^-X_h = hX_h,\end{equation}
and we can formally write
\begin{equation}X = X_{h_1}+X_{h_2} + X_{h_3} + ... ,\ \ \ \ h_1<h_2<h_3< ...\ . \label{eq:LmX}\end{equation}
This is analogous to the Fock space expansion of a string field into eigenstates of $L_0$. This is called the $\mathcal{L}^-$ {\it level expansion}. We use ``level" to refer to the $\frac{1}{2}\mathcal{L}^-$ eigenvalue of a state in the expansion, so that $X_{h_1}$ has level $h_1$, $X_{h_2}$ has level $h_2$, and so on. The leading level in the expansion is the lowest level, since it makes the most important contribution to the overlap with a sliver-like test state in the limit $\epsilon\to 0$. Higher level states are subleading. Relative to a sliver-like test state, any surface contained in the string field $X$ will be negligible. For this reason, we generally expect that the eigenstates $X_h$ will be characterized by operator insertions on the identity string field.
The $\mathcal{L}^-$ level expansion is a variant of the $\mathcal{L}_0$ level expansion. Given the $\mathcal{L}^-$ level expansion of $X$ in \eq{LmX}, the $\mathcal{L}_0$ level expansion of $\sqrt{\Omega}X\sqrt{\Omega}$ is determined by
\begin{equation}\sqrt{\Omega}X\sqrt{\Omega} = \sqrt{\Omega}X_{h_1}\sqrt{\Omega}\, +\, \sqrt{\Omega}X_{h_2}\sqrt{\Omega} \, +\, \sqrt{\Omega}X_{h_3}\sqrt{\Omega} \, +\, ... ,\ \ \ \ h_1<h_2<h_3< ... \ .\end{equation}
In this sense the expansions are equivalent. However, the $\mathcal{L}_0$ level expansion selects a preferred state in the wedge algebra (the $SL(2,\mathbb{R})$ vacuum) which might not be natural for an arbitrary string field. In this sense the $\mathcal{L}^-$ level expansion is more canonical. An advantage of $\mathcal{L}_0$, however, is that its eigenstates are normalizable, which makes it possible to compute the energy by substituting the expansion into the action and computing level by level. Typically, this expresses the energy as an asymptotic series \cite{Schnabl,simple}. In any case it is straightforward to translate between these versions of the level expansion.
The $\mathcal{L}^-$ level expansion has been investigated in the subalgebra of wedge states with insertions. It is especially powerful in the context of a {\it singularity free subalgebra}; a subalgebra of wedge states with insertions where products of all fields are finite. In particular, the $KBc$ subalgebra is a singularity free subalgebra, as is its extension to regular marginal deformations. The $\mathcal{L}^-$ level expansion is especially important in understanding the tachyon vacuum of open superstring field theory, where the relevant subalgebra is singularity free but much more complicated than $KBc$, and there are few other applicable tools for analyzing solutions. We make two important claims:
\begin{claim}
The $\mathcal{L}^-$ level expansion of a state in a singularity free subalgebra can be computed by expanding the state in powers of $K$ around $K=0$ and ordering terms in sequence of increasing $\frac{1}{2}\mathcal{L}^-$ eigenvalue.
\end{claim}
\begin{claim}
In a singularity free subalgebra, the $\mathcal{L}^-$ level expansion of a product of states is given by multiplying the $\mathcal{L}^-$ level expansions of the states individually. Level is additive under star multiplication.
\end{claim}
\noindent These results do not hold outside the context of a singularity free subalgebra. To see why, consider the state
\begin{equation}\sqrt{\Omega}V\Omega V\sqrt{\Omega},\label{eq:VOmV}\end{equation}
where $V$ is a marginal operator. If we expand around $K=0$ we obtain
\begin{equation}
\sqrt{\Omega}V\Omega V\sqrt{\Omega}= \underbrace{\phantom{\Big)}\!\!V^2}_{\text{level }2} -\underbrace{VKV -\frac{1}{2}KV^2 -\frac{1}{2}V^2 K}_{\text{level }3}\, + \, \text{higher levels},\ \ \ \ \ \ \text{(regular OPE)}.\label{eq:LmVOmV}
\end{equation}
If $V$ has singular OPE, this expansion is meaningless since the terms are divergent. Also note that \eq{VOmV} is the square of another state
\begin{equation}\sqrt{\Omega}V\Omega V\sqrt{\Omega} = (\sqrt{\Omega}V\sqrt{\Omega})^2,\end{equation}
which has $\mathcal{L}^-$ expansion
\begin{equation}\sqrt{\Omega}V\sqrt{\Omega} = \underbrace{\phantom{\Big)}\!\!\!V}_{\text{level }1}- \underbrace{\frac{1}{2}KV -\frac{1}{2}V K}_{\text{level }2}\, +\, \text{higher levels}.\end{equation}
However, we cannot compute the square of the right hand side unless $V$ has regular OPE. If the OPE is regular, the square leads to \eq{LmVOmV}. In the presence of singular OPEs the $\mathcal{L}^-$ level expansion can still be computed but we must take care to subtract divergences before expanding in $K$. In the present example, it can be found by normal ordering following \eq{normord}
\begin{equation}\sqrt{\Omega}V\Omega V\sqrt{\Omega} = \mathcal{N}\Omega^2+\sqrt{\Omega}\!:\!V\Omega V\!:\!\sqrt{\Omega}.\end{equation}
The $\mathcal{L}^-$ level expansion is then
\begin{equation}
\sqrt{\Omega}V\Omega V\sqrt{\Omega} = \underbrace{\phantom{\Big)}\!\!\mathcal{N}}_{\text{level }0} -\underbrace{\phantom{\Big)}\!\!2\mathcal{N}K}_{\text{level }1} + \underbrace{\phantom{\Big)}\!\!2\mathcal{N}K^2+\!:\!V^2\!:\!}_{\text{level }2}\, +\, \text{higher levels},\ \ \ \ \ \ \text{(singular OPE)},
\end{equation}
which is clearly different from \eq{LmVOmV}. When OPEs are singular we cannot multiply expansions into $\mathcal{L}^-$ eigenstates, and this somewhat limits the power of the formalism. Most applications have been in the context of singularity free subalgebras.
When singular OPEs are absent, the $\mathcal{L}^-$ level expansion allows us to solve the equations of motion perturbatively in level. In fact, this was essentially Schnabl's original approach to the construction of the tachyon vacuum. Let us see how this works in the context of the $KBc$ subalgebra. A general state at ghost number 1 can be expanded in level
\begin{equation}\Psi = \Psi_{-1}+\Psi_0 + \Psi_1 + \text{higher levels},\end{equation}
where the index on $\Psi_n$ indicates the level, and the general ghost number 1 state at each level can be written
\begin{eqnarray}
\Psi_{-1} \!\!\!\!\!\!\!\! &&= \alpha c,\\
\Psi_0 \!\!\!\!\!\!\!\! && = \alpha_1 cK + \alpha_2 Kc + \lambda cKBc ,\\
\Psi_1 \!\!\!\!\!\!\!\! && = \alpha_3 K^2 c + \alpha_4 KcK + \alpha_5 cK^2 + \lambda_1 KcKBc + \lambda_2 cKBcK + \chi cK^2 Bc, \\
\!\!\!\!\!\!\!\! && \vdots\ \ \ .\nonumber
\end{eqnarray}
The coefficients can be interpreted as expectation values of fields in this basis, and should be determined by solving the equations of motion. Expanded in level the equations of motion are
\begin{eqnarray}
0 \!\!\!\!\!\!\!\! &&= \Psi_{-1}^2, \\
0\!\!\!\!\!\!\!\! && = Q\Psi_{-1} + [\Psi_{-1},\Psi_0] ,\\
0 \!\!\!\!\!\!\!\! && = Q\Psi_0 + [\Psi_{-1},\Psi_1]+\Psi_0^2, \\
\!\!\!\!\!\!\!\! && \vdots\ \ \ . \nonumber
\end{eqnarray}
We solve the first two equations:
\begin{eqnarray}
0 \!\!\!\!\!\!\!\! && = (\alpha c)^2,\\
0 \!\!\!\!\!\!\!\! && = Q(\alpha c)+ [ \alpha c,\alpha_1 cK + \alpha_2 Kc + \lambda cKBc]\\
\!\!\!\!\!\!\!\! && = \alpha cKc +\alpha(\alpha_1+\alpha_2) cKc.
\end{eqnarray}
The first equation holds trivially for any $\alpha$. The second equation only implies nontrivial conditions if $\alpha$ is nonzero, in which case we find a three parameter family of solutions
\begin{equation}\alpha \neq 0,\ \ \ \ \alpha_1+\alpha_2 = -1,\ \ \ \ \lambda\ \text{arbitrary}.\end{equation}
One can check that the higher level equations of motion do not impose further constraints on these coefficients. This family of solutions, characterized up to level zero, are gauge equivalent representatives of the tachyon vacuum.
One can systemetize this kind of analysis to give a classification of gauge orbits in the $KBc$ subalgebra. The classification from \cite{IdSing} rests on the following assumptions:
\begin{description}
\item{(1)} Only nonnegative integer powers of $K$ can appear in the $\mathcal{L}^-$ level expansion. We in fact implicitly assumed this in the previous paragraph. Noninteger powers of $K$ can be defined through
\begin{equation} K^{\nu} = \frac{1}{\Gamma(-\nu)}\int_0^\infty d\alpha\alpha^{-\nu-1}\Omega^\alpha.\label{eq:Knu}\end{equation}
The possibly divergent integration towards the lower limit can be subtracted by defining the integrand in the proper sense of distributions. More problematic is the upper limit, which even for $\nu>-1$ only gives inverse power suppression to the sliver state. It is therefore believed that non-analytic powers of $K$ will lead to somewhat singular solutions, though a detailed understanding of this point is presently missing.
\item{(2)} Only the defining relations \eq{KBcId1}-\eq{KBcId2} of the $KBc$ subalgebra are assumed to hold. Identities such as $(\partial c)^2=0$, while true, do not follow from the defining relations, and are referred to as {\it auxiliary identities}. Auxiliary identities lead to additional solutions and gauge orbits which have not been classified.
\item{(3)} For regular solutions in the $KBc$ subalgebra, gauge equivalence level-by-level in the $\mathcal{L}^-$ level expansion is both necessary and sufficient to establish true gauge equivalence.
\item{(4)} Solutions which are inequivalent through gauge transformation in the $KBc$ subalgebra are inequivalent in the whole open string star algebra.
\end{description}
It is not known whether relaxing these assumptions can lead to additional gauge orbits containing nonsingular solutions. If they exist, they have not been found. In particular, there has been longstanding interest in the question of whether the $KBc$ subalgebra contains multiple D-brane solutions, representing two or more copies of the perturbative vacuum. There are interesting candidates for such solutions if we relax assumption (1) \cite{MurataSchnabl}, but the resulting sliver-like singularities lead to inconsistencies in the equations of motion. Relaxing assumption (2) perhaps looks less problematic, but the resulting spectrum of gauge orbits appears to be quite complicated, and known solutions of this kind have exotic cohomology and singularities related to the identity string field. Presently there is no reason to question assumptions (3) and (4). The resulting classification is as follows:
\begin{claim}
Assuming (1)-(4), there are six gauge orbits of solutions in the $KBc$ subalgebra, and they are uniquely characterized by their leading contribution to the $\mathcal{L}^-$ level expansion:
\begin{itemize}
\item Perturbative vacuum: $\Psi = \lambda cKBc + \text{higher levels},\ \ \ \ (\lambda\neq -1)$.
\item Tachyon vacuum: $\Psi = \alpha c +\text{higher levels},\ \ \ \ (\alpha\neq 0)$.
\item Residual perturbative vacuum: $\Psi = -cKBc + \text{higher levels}$.
\item Residual tachyon vacuum: $\Psi = -cK + \text{higher levels}$.
\item Residual conjugate tachyon vacuum: $\Psi = -Kc +\text{higher levels}$,
\item Masuda-Nuomi-Takahashi (MNT) ghost brane \cite{MNT}: $\Psi = -cK-Kc +cKBc + \text{higher levels}$.
\end{itemize}
\end{claim}
\noindent The last four gauge orbits are called {\it residual solutions}. The residual tachyon vacuum was the first analytic solution we found in subsection \ref{subsec:KBc}. In \cite{IdSing} it was shown that residual solutions are necessarily non-normalizable in a similar way as the identity string field. However, this cannot be seen at any finite level in the expansion into $\mathcal{L}^-$ eigenstates. A useful thing about this classification is that it gives a quick way to identify the physical interpretation of unfamiliar solutions in the $KBc$ subalgebra, without going to the trouble of computing observables. Consider for example the solution
\begin{equation}\Psi = -c\frac{KB}{1-\Omega}c(1-\Omega).\label{eq:restv}\end{equation}
Expanding around $K=0$ leads to $-cK+\text{higher levels}$, so this is a representative of the residual tachyon vacuum.
Let us mention an interesting fact:
\begin{claim}
The sum of the coefficients of $cK$ and $Kc$ is a gauge invariant quantity in the $KBc$ subalgebra (even off shell). It can informally be identified with the tension relative to the perturbative vacuum in units of $\frac{1}{2\pi^2}$.
\end{claim}
\noindent This result provides some motivation for our terminology for the gauge orbits. For the perturbative vacuum and residual perturbative vacuum, $\alpha_1+\alpha_2$ vanishes and the background can be interpreted as a state of zero energy; for the tachyon vacuum and residual tachyon vacuum solutions, $\alpha_1+\alpha_2 = -1$ so the energy of the reference D-brane is cancelled. For the MNT ghost brane, $\alpha_1+\alpha_2 = -2$, and so the solution appears to represent a D-brane with negative energy.
Let us explain why the tachyon vacuum is not pure gauge from the point of view of the $\mathcal{L}^-$ level expansion. We look for a finite gauge parameter $U$ which satisfies
\begin{equation}\Psi_\mathrm{tv} = U^{-1}Q U.\end{equation}
Multiplying by $U^{-1}$ gives a linear equation
\begin{equation}
QU - U\Psi_\mathrm{tv} = 0.\label{eq:LmPsitvU}
\end{equation}
The general form of the $\mathcal{L}^-$ expansion at ghost number zero implies
\begin{equation}U = U_0 + U_1+\text{higher levels},\end{equation}
where
\begin{eqnarray}
U_0 \!\!\!\!\!\!\!\! && = \beta + \gamma Bc,\\
U_1\!\!\!\!\!\!\!\! && = \beta_1 K + \gamma_1 K Bc + \gamma_2 BcK,\\
\!\!\!\!\!\!\!\! && \vdots\ \ \ . \nonumber
\end{eqnarray}
Expanding \eq{LmPsitvU} in level gives equations which should determine the gauge parameter:
\begin{eqnarray}
0\!\!\!\!\!\!\!\! && = -U_0\Psi_{-1},\\
0\!\!\!\!\!\!\!\! && = QU_0 - U_1\Psi_{-1} - U_0\Psi_0,\\
\!\!\!\!\!\!\!\! &&\vdots\ \ \ .\nonumber
\end{eqnarray}
Focus on the first equation:
\begin{equation}U_0\Psi_{-1} = (\beta + \gamma Bc)(\alpha c) = \alpha\beta c.\end{equation}
Since at the tachyon vacuum $\alpha\neq 0$, this requires $\beta=0$ and the level zero part of the gauge parameter is determined to be
\begin{equation}U_0 = \gamma Bc.\end{equation}
To implement the gauge transformation we also need the inverse gauge parameter:
\begin{equation}U^{-1} = U_0^{-1}+U_1^{-1} +\text{higher levels}.\end{equation}
Expanding the relation $U^{-1}U=1$ in level implies that we must have the relation
\begin{equation}U_0^{-1}(\gamma Bc) = 1.\end{equation}
Multiplying this equation by $c$ from the right we arrive at a contradiction. Therefore the inverse gauge parameter does not exist, and the tachyon vacuum cannot be trivialized by gauge transformation to the perturbative vacuum.
The existence of a formal expansion into $\mathcal{L}^-$ eigenstates does not imply that the expansion can be resummed into a well-defined string field. This was already mentioned in the context of the $\mathcal{L}_0$ level expansion in subsection \ref{subsec:Sch}. Singular behavior from the point of view of the identity string field is particularly obscure in this description. One thing that is fairly easy to notice, however, is difficulty with the sliver state. Typically this will appear concretely through exotic powers of $K$. For example, the formal homotopy operator for $KBc$ solutions in Schnabl gauge is
\begin{equation}A_\lambda = B\frac{1-\lambda\Omega}{K}.\end{equation}
The $\mathcal{L}^-$ level expansion is
\begin{equation}A_\lambda = (1-\lambda)\frac{B}{K} + \lambda B +\text{higher levels}.\end{equation}
Similarly we may consider the $\mathcal{L}^-$ level expansion of Okawa's finite gauge parameter
\begin{equation}U = \frac{1}{1-\sqrt{\Omega}Bc\sqrt{\Omega}},\end{equation}
which formally relates Schnabl's solution to the perturbative vacuum:
\begin{equation}
U = \underbrace{\frac{B}{K}c}_{\text{level }-1} +\underbrace{1 -\frac{1}{2}\frac{B}{K}cK}_{\text{level }0}\,+\,\text{higher levels}.
\end{equation}
In both these examples we immediately see difficulty from the inverse of $K$. Note that Okawa's gauge transformation formally receives a contribution at level $-1$, which was excluded in the previous paragraph since negative level states cannot be created with positive integer powers of $K$ at ghost number zero. The difficulty with $1/K$ was already explained in subsection \ref{subsec:Sch}. However, problems may appear with other non-analytic powers of $K$. For example, we can consider a class of solutions whose $\mathcal{L}^-$ level expansion starts as
\begin{equation}\Psi = \underbrace{\phantom{\big)}\!\!\alpha cK^{1-\nu} Bc}_{\text{level }-\nu} + \text{higher levels},\ \ \ \ 0\leq \nu\leq 1.\end{equation}
Here at least negative powers of $K$ do not appear. However, this would seem to imply the existence of a 1-parameter family of gauge inequivalent solutions connecting the perturbative vacuum ($\nu=0$) and tachyon vacuum ($\nu=1$). We know that the action cannot change when moving along this family, since by definition solutions are stationary points of the action. But this contradicts the fact that the perturbative vacuum and tachyon vacuum carry different energies. The resolution of this paradox has not been fully clarified.
\subsection{Dual $\mathcal{L}^-$ Level Expansion}
\label{subsec:dualL}
It is interesting to think about what happens when we probe a string field with an identity-like test state
\begin{equation}\epsilon^{\frac{1}{2}\mathcal{L}^-}|\phi\rangle,\ \ \ \ \epsilon\text{ small}.\end{equation}
This can be represented as a strip of width $\epsilon$ containing a vertex operator in the middle creating the state $|\phi\rangle$. Relative to such a test state, a typical string field will look sliver-like. This is not true, however, for the identity string field, since regardless of how small we take the width of the test state, the width of the identity is vanishing, and so is always negligible by comparison. In fact, presently we are not interested in wedge states whose relative width exceeds a certain finite cutoff $\Lambda>0$ as the test state shrinks. For this reason we introduce a projection operator
\begin{equation}P_\Lambda, \end{equation}
which acts as the identity on wedge states with insertions whose total width is less than $\Lambda$, and acts a zero otherwise. With this projection operator inserted, it may happen that the overlap of an identity-like test state with a string field $X$ can be expanded as a power series in $1/\epsilon$:
\begin{eqnarray}
\left\langle \phi, P_\Lambda\left(\frac{1}{\epsilon}\right)^{\frac{1}{2}\mathcal{L}^-} X\right\rangle \!\!\!\!\!\!\!\! && = \left(\frac{1}{\epsilon}\right)^{h_1}\langle \phi,P_\Lambda X_{h_1}\rangle + \left(\frac{1}{\epsilon}\right)^{h_2}\langle \phi,P_\Lambda X_{h_2}\rangle+\left(\frac{1}{\epsilon}\right)^{h_3}\langle \phi,P_\Lambda X_{h_3}\rangle+...\nonumber\\
\!\!\!\!\!\!\!\! &&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ h_1>h_2>h_3> ...\ .\ \ \ \ \ \ \ \ \ \label{eq:dualL}
\end{eqnarray}
Leaving the projection implicit, formally we can write
\begin{equation}X = X_{h_1}+X_{h_2}+X_{h_3} + ...\ \ \ \ h_1>h_2>h_3> ...\ .\end{equation}
where $X_{h}$ are eigenstates of $\mathcal{L}^-$:
\begin{equation}\frac{1}{2}\mathcal{L}^- X_h =h X_h.\end{equation}
This defines the {\it dual $\mathcal{L}^-$ level expansion} \cite{IdSing}. As before, level is identified with the $\frac{1}{2}\mathcal{L}^-$ eigenvalue. In this case the leading level is the {\it highest} level, and subleading levels are increasingly negative.
The nature of the eigenstates in this expansion is a little surprising, so it is worth seeing how they arise from the field $\frac{1}{1+K}$:
\begin{eqnarray}
\left\langle \phi, P_\Lambda\left(\frac{1}{\epsilon}\right)^{\frac{1}{2}\mathcal{L}^-} \frac{1}{1+K}\right\rangle \!\!\!\!\!\!\!\! && = \left\langle \phi, P_\Lambda\frac{\epsilon}{\epsilon+K}\right\rangle \nonumber\\
\!\!\!\!\!\!\!\! && = \epsilon\int_0^\Lambda d\alpha\, e^{-\epsilon \alpha}\langle\phi,\Omega^\alpha\rangle.
\end{eqnarray}
For fixed $\Lambda$ and $\epsilon$ sufficiently small, we can expand the integrand in powers of $\epsilon$:
\begin{equation}
\left\langle \phi, P_\Lambda\left(\frac{1}{\epsilon}\right)^{\frac{1}{2}\mathcal{L}^-} \frac{1}{1+K}\right\rangle = \underbrace{\epsilon \left\langle \phi,\int_0^\Lambda d\alpha\, \Omega^\alpha\right\rangle}_{\text{level }-1}- \underbrace{\epsilon^2\left\langle \phi,\int_0^\Lambda d\alpha\, \alpha\Omega^\alpha\right\rangle}_{\text{level }-2}+\underbrace{\frac{\epsilon^3}{2!}\left\langle \phi,\int_0^\Lambda d\alpha\, \alpha^2\Omega^\alpha\right\rangle}_{\text{level }-3}+...\ .
\end{equation}
Comparing to \eq{dualL} we see that the leading level in the expansion is $-1$, followed by subleading levels for all negative integers. To see the $\mathcal{L}^-$ eigenstates we make the identification
\begin{equation}\frac{1}{\Gamma(-\nu)}\int_0^\Lambda d\alpha\,\alpha^{-\nu-1}\Omega^\alpha = P_\Lambda K^{\nu},\end{equation}
where the power of $K$ is defined as a continuous superposition of wedge states through \eq{Knu}. If $\nu$ is negative the integral defining $K^\nu$ produces a divergence proportional to the sliver state. In the present situation this does not matter, since the sliver divergence is discarded by the projection. Moreover, divergent sliver boundary terms imply that $K^\nu$ is not an $\mathcal{L}^-$ eigenstate for negative $\nu$; for similar reasons, the relation $K^\mu K^\nu = K^{\mu+\nu}$ fails to hold. Again, this does not matter since anomalous boundary terms disappear after projection. Therefore in the present context we can assume that all powers of $K$ are allowed, and all are eigenstates of $\mathcal{L}^-$:
\begin{equation}\frac{1}{2}\mathcal{L}^- K^\nu = \nu K^\nu.\end{equation}
This leads us to identify the dual $\mathcal{L}^-$ expansion of $\frac{1}{1+K}$ as
\begin{equation}\frac{1}{1+K} = \underbrace{\frac{1}{K}}_{\text{level }-1}-\underbrace{\frac{1}{K^2}}_{\text{level }-2} +\underbrace{\frac{1}{K^3}}_{\text{level }-3} - \,\text{lower levels}.\end{equation}
This is simply an expansion around $K=\infty$. This anticipates the general result:
\begin{claim}
The dual $\mathcal{L}^-$ level expansion in the subalgebra of wedge states with insertions can be computed by expanding the state in powers of $K$ around $K=\infty$ and ordering terms in sequence of decreasing $\frac{1}{2}\mathcal{L}^-$ eigenvalue.
\end{claim}
\begin{claim}
In the subalgebra of wedge states with insertions, the dual $\mathcal{L}^-$ level expansion of a product of states is given by multiplying the dual $\mathcal{L}^-$ level expansions of the states individually. Level is additive under star multiplication.
\end{claim}
\noindent We make two comments:
\begin{description}
\item{(1)} It is not necessarily true that a string field made out of wedge states with insertions can be expanded in powers of $K$ around $K=\infty$. For example, this is the case for the $SL(2,\mathbb{R})$ vacuum $\Omega=e^{-K}$. By definition, a state which falls off faster than any inverse power of $K$ is considered to have level $-\infty$. It can also happen that expansion around $K=\infty$ produces contributions which are bounded by finite powers of $K$ but are not themselves powers of $K$. For example we could find $1/\ln(K)$. In this case the state does not have a dual $\mathcal{L}^-$ expansion, but in practice this does not usually happen. Some aspects of the present discussion can be generalized to cover such situations.
\item{(2)} We did not qualify these claims as holding within a singularity free subalgebra. This is because an expansion around $K=\infty$ never produces OPE divergence which is not already present in the original state. Consider for example the state
\begin{equation}V\frac{1}{(1+K)^3}V = \frac{1}{2!}\int_0^\infty d\alpha\, \alpha^2 V\Omega^\alpha V,\end{equation}
where $V$ is a weight 1 primary with double pole OPE. The double pole does not lead to divergence in the state since the $\alpha^2$ factor in the integrand cancels the $1/\alpha^2$ from the contraction of $V$s in the limit $\alpha\to 0$. The dual $\mathcal{L}^-$ expansion of this state is
\begin{eqnarray}
V\frac{1}{(1+K)^3}V \!\!\!\!\!\!\!\! && =\, \underbrace{V\frac{1}{K^3}V}_{\text{level }-1} \, -\, \underbrace{3 V\frac{1}{K^4}V}_{\text{level }-2}\, + \,\underbrace{6 V\frac{1}{K^5}V}_{\text{level }-3}\, +\, \text{lower levels}\nonumber\\
\!\!\!\!\!\!\!\! && = \frac{1}{2!}\int_0^\Lambda d\alpha \, \alpha^2 V\Omega^\alpha V - \frac{3}{3!}\int_0^\Lambda d\alpha \alpha^3 V\Omega^\alpha V + \frac{6}{4!}\int_0^\Lambda d\alpha\, \alpha^4 V\Omega^\alpha V +...\ .\ \ \ \ \ \ \ \ \
\end{eqnarray}
In all eigenstates the double pole in the $V$-$V$ OPE is canceled in the integrand towards $\alpha=0$, and more than canceled as the level becomes progressively negative.
\item{(3)} There is a close connection between the level in the dual $\mathcal{L}^-$ level expansion and the magnitude of the contribution of the identity string field to the corresponding eigenstate. Therefore increasingly positive levels become more singular from the point of view of the identity string field, while increasingly negative levels become less singular.
\end{description}
The dual $\mathcal{L}^-$ expansion is useful for analyzing singularities related to the identity string field, but obscures singularities related to the sliver state. Meanwhile, precisely the reverse is true in the (ordinary) $\mathcal{L}^-$ level expansion. For example, the inverse wedge state $\Omega^{-1}$ is clearly singular in the dual $\mathcal{L}^-$ level expansion since it diverges faster towards infinity than any power of $K$. However, this behavior is not directly seen in a power series expansion around $K=0$. A different example is the homotopy operator for $KBc$ solutions in simple gauge:
\begin{equation}A_\lambda = B\frac{1-\lambda + K}{K(1+K)}.\end{equation}
The dual $\mathcal{L}^-$ level expansion does not reveal anything problematic:
\begin{equation}A_\lambda=\underbrace{\frac{B}{K}}_{\text{level }0}-\underbrace{\lambda \frac{B}{K^2}}_{\text{level }-1} +\underbrace{\lambda \frac{B}{K^3}}_{\text{level }-2}+\text{lower\ levels}.
\end{equation}
However the $\mathcal{L}^-$ level expansion contains a sliver-like singularity at leading level if $\lambda\neq 1$:
\begin{equation}
A_\lambda = \underbrace{(1-\lambda)\frac{B}{K}}_{\text{level }0} +\underbrace{\phantom{\Big)}\!\!\lambda B\,}_{\text{level }1} - \underbrace{\phantom{\Big)}\!\!\lambda BK\,}_{\text{level }2} + \text{higher\ levels}.
\end{equation}
Therefore the $\mathcal{L}^-$ and dual $\mathcal{L}^-$ level expansions reveal complimentary information about a state. It can happen that both expansions are unproblematic but nevertheless the corresponding state is not well-behaved. An example is the string field:
\begin{eqnarray}
\frac{1}{1-K} \!\!\!\!\!\!\!\! && =- \underbrace{\frac{1}{K}}_{\text{\ level -1}} -\underbrace{ \frac{1}{K^2}}_{\text{level }-2} +\, \text{lower levels},\ \ \ \ (\text{dual }\mathcal{L}^-\text{ level expansion}),\nonumber\\
\!\!\!\!\!\!\!\! && = \underbrace{1}_{\text{level }0} + \underbrace{K}_{\text{ level} 1}\, +\, \text{higher levels}\ \ \ \ (\mathcal{L}^-\ \text{level expansion}).
\end{eqnarray}
This state has a pole at $K=1$ and is therefore divergent in the $C^*$ and $D_2$ norm. The problem with this state is not something that can be easily understood as related to the sliver state or the identity string field.
We now state a central result of the formalism:
\begin{claim}
Let $X$ be a string field in the subalgebra of wedge states with insertions that admits a dual $\mathcal{L}^-$ level expansion. Then $\mathop{\rm Tr}\nolimits[X]$ is well-defined only if the leading level of $X$ in the dual $\mathcal{L}^-$ level expansion is strictly negative.
\end{claim}
\noindent The argument is that integration over the circumference of the cylinder in correlators must be absolutely convergent as the circumference shrinks to zero. For further explanation see \cite{IdSing}. Since we want to be able to compute the action and Ellwood invariant of a solution, this result implies that the leading level of a regular solution in the dual $\mathcal{L}^-$ expansion must be strictly negative. This holds somewhat trivially for Schnabl gauge solutions and the Kiermaier-Okawa solution, since these are made from wedge states of strictly positive width and the leading level is minus infinity. The condition holds less trivially for the simple tachyon vacuum and intertwining solutions:
\begin{eqnarray}
\Psi_\text{simp}\!\!\!\!\!\!\!\! && =\underbrace{ Q\left(\frac{1}{\sqrt{K}}Bc\frac{1}{\sqrt{K}}\right)}_{\text{level }-1}+\,\text{lower levels},\nonumber\\
\Psi_\text{int}\!\!\!\!\!\!\!\! && = \underbrace{Q\left(\frac{1}{\sqrt{K}}Bc\frac{1}{\sqrt{K}} - \frac{1}{\sqrt{K}}\sigma\frac{B}{K}\overline{\sigma} Kc\frac{1}{\sqrt{K}}\right)}_{\text{level }-1}+\,\text{lower levels}.\label{eq:dualLmsimpKOS}
\end{eqnarray}
Both solutions have leading level $-1$, which is the highest integer level consistent with a regular solution. Finally we may consider the identity-like tachyon vacuum \eq{Idtv} and the residual tachyon vacuum \eq{restv}
\begin{eqnarray}
\Psi \!\!\!\!\!\!\!\! && = c(1-K) = \underbrace{\phantom{)}\!\!\!-cK}_{\text{level }0} + \underbrace{\phantom{)}\!\!\!c}_{\text{level }-1},\\
\Psi \!\!\!\!\!\!\!\! && = -c\frac{KB}{1-\Omega}c(1-\Omega) = \underbrace{-cKBc}_{\text{level }0} \, + \ \text{level}\,-\!\!\infty.
\end{eqnarray}
The leading level of both solutions is $0$, so they are singular. In \cite{IdSing} it was shown that all residual solutions satisfy
\begin{equation}B\Psi B = BK.\end{equation}
Since $BK$ has $\frac{1}{2}\mathcal{L}^-$ eigenvalue $+2$, this equation can only be consistent if $\Psi$ contains a level 0 state. Therefore residual solutions are necessarily singular from the perspective of the identity string field.
Next we mention another useful result:
\begin{claim}
Let $\Psi$ be a regular solution that admits a dual $\mathcal{L}^-$ expansion. Then all finite levels in the dual $\mathcal{L}^-$ expansion of $\Psi$ can be removed by gauge transformation.
\end{claim}
\noindent To see why this is true, note that the leading level of $\Psi$ in the dual $\mathcal{L}^-$ expansion must be negative (otherwise the solution would not be regular). The leading level of $\Psi^2$ must be at least twice as negative as that of $\Psi$ itself. The equations of motion then requires that the leading level of $\Psi$ must be BRST invariant. This is clearly seen in the dual $\mathcal{L}^-$ expansions of the simple tachyon vacuum and the simple intertwining solution in \eq{dualLmsimpKOS}. In fact, the leading level is not only BRST invariant but must be BRST exact, since at any finite level the cohomology of $Q$ is trivial due to the identity
\begin{equation}Q\left(\frac{B}{K}\right) = 1.\end{equation}
This implies that the highest level state can be removed by gauge transformation, and by iteration all finite levels can be removed. Therefore the dual $\mathcal{L}^-$ expansion does not contain any physical information about a solution. One might think this means we can focus on solutions at level $-\infty$ and dispense with the formalism all together. However, one of the lessons of the simple tachyon vacuum is that by making use of the identity string field we can often make the physics of solutions more transparent.
There are many ways to remove levels in the dual $\mathcal{L}^-$ level expansion, but two approaches are notable. The first is using the so-called {\it Zeze map} \cite{Zeze}. Given a solution $\Psi$ and an element of the wedge algebra $F(K)$ satisfying $F(0)=1$, we can form a new solution $\Psi'$ with the transformation
\begin{equation}
\Psi' =\sqrt{F}\Psi \frac{1}{1+B\frac{1-F}{K}\Psi}\sqrt{F}.
\end{equation}
If the leading levels of $\Psi$ and $F$ are both negative, the leading level of $\Psi'$ is the sum of the leading levels of $\Psi$ and $F$. One might notice a close resemblance to the Schnabl gauge solution for marginal deformations. In fact, the marginal solution in Schnabl gauge is the Zeze map with $F = \Omega$ applied to
\begin{equation}\Psi = cV.\end{equation}
This is also a marginal solution. It satisfies the equations of motion since it is both BRST invariant and nilpotent (assuming $V$ has regular OPE). Interestingly, it satisfies the gauge condition
\begin{equation}\mathcal{B}^-\Psi = 0.\end{equation}
There are no regular $KBc$ solutions for the tachyon vacuum in this gauge, though approaching it as a limit of dressed Schnabl gauges leads formally to $\infty\times c$. While this doesn't look like a meaningful solution, it is interesting to mention that in boundary string field theory \cite{WittenBSFT}, the tachyon vacuum also sits at infinite distance from the perturbative vacuum in field space \cite{BSFT1,BSFT2}. The solution $cV$ is singular as it consists entirely of a state at level $0$. However, among all marginal solutions it probably has the closest relation to boundary deformations as understood from the first quantized worldsheet point of view. If we have two string field theories $\mathrm{BCFT}_0$ and $\mathrm{BCFT}_*$ related by regular marginal deformation, the actions are equal if the fluctuation field around $cV$ is directly identified with the field of $\mathrm{BCFT}_*$:
\begin{equation}\Psi^{(0)} = cV +\sigma\Psi^{(*)}\overline{\sigma}.\end{equation}
The boundary condition changing operators $\sigma$ and $\overline{\sigma}$ turn on an exponential of line integrals of $V$s on the boundary, which is the most direct way to represent a state of $\mathrm{BCFT}_*$ as a boundary deformation of a state in $\mathrm{BCFT}_0$. This can be viewed as a degenerate version of the field redefinition constructed from the simple intertwining solution. All of this demonstrates that making solutions more identity-like often clarifies their physical interpretation.
Another interesting method for removing levels in the dual $\mathcal{L}^-$ expansion is through $KBc$ endomorphisms \cite{Erler_simple}. We only discuss this in the context of the $KBc$ subalgebra, though in some cases it can be generalized in an interesting way to larger subalgebras \cite{genericF(K)}. We consider an endomorphism $\phi$ acting on elements of the $KBc$ subalgebra defined by
\begin{equation}K' = \phi\circ K = \phi(K),\ \ \ \ B' = \phi\circ B = B\frac{\phi(K)}{K},\ \ \ \ c'=\phi\circ c = c\frac{KB}{\phi(K)}c,\end{equation}
where $\phi(K)$ is an element of the wedge algebra satisfying $\phi(0)=0$ and $\phi'(0)\neq 0$. Surprisingly, the transformed fields $K',B'$ and $c'$ satisfy the defining relations of the $KBc$ subalgebra\footnote{Interestingly, however, the transformed fields do not satisfy auxilliary identities such as $(\partial c)^2=0$.}
\begin{eqnarray}
\!\!\!\!\!\!\!\! && [K',B'] = 0\ \ \ \ (B')^2=(c')^2=0 \ \ \ \ [B',c'] = 1,\\
\!\!\!\!\!\!\!\! && \ \ QK'=0\ \ \ \ \ \ QB'=K'\ \ \ \ \ \ Qc'=c'K'c'.
\end{eqnarray}
For example, if we start with the identity-like solution for the tachyon vacuum,
\begin{equation}\Psi = c(1-K),\end{equation}
and reinterpret $K,B,c$ as $K',B',c'$ with
\begin{equation}\phi(K) = 1-\Omega,\label{eq:endSch}\end{equation}
we obtain a slight variation of Schnabl's solution\footnote{We would get precisely Schnabl's solution if we could apply the endomorphism to the solution $\sqrt{1-K}c\sqrt{1-K}$. The square root of $1-K$ however is problematic due to the branch point at $K=1$, which leads to difficulties with its definition as a superposition of wedge states.}
\begin{equation}c'(1-K') = c\frac{KB}{1-\Omega}c\Omega = \phi\circ\Big(c(1-K)\Big).\end{equation}
Under certain conditions the endomorphisms are also automorphisms, which can be interpreted as representing the connected component of the group of reparameterization symmetries of the spectrum of $K$~\cite{genericF(K)}. We discuss this under the assumption that the spectrum of $K$ consists of non-negative real numbers. In this context, $\phi(K)$ is an invertible map from non-negative real numbers into themselves (a diffeomorphism of the spectrum of $K$). If we wish to transform from a $KBc$ solution whose leading level in the dual $\mathcal{L}^-$ level expansion is $\nu<0$ to another $KBc$ solution whose leading level is $\nu'<0$, the leading level of the diffeomorphism $\phi(K)$ must be
\begin{equation}\frac{\nu'}{\nu}>0.\end{equation}
In particular $\phi(K)$ must grow as $K^{\nu'/\nu}$ towards infinity. If we wish to obtain a more regular solution, $\nu'<\nu$ and we need to ``push" the spectrum of $K$ out to infinity. Note that this works somewhat differently than the endomorphism example \eq{endSch}, since in that case we are transforming a solution whose leading level is $0$ and the required $\phi(K)$ is not a reparameterization of the spectrum.
Armed with what we have learned we can begin to get some insight into topological properties of string field theory solutions. Such considerations may be especially relevant in superstring field theory, where they should give an understanding of the topological origin of D-brane charges. There are indeed some hints of this \cite{exotic,supervac}. Presently we are limiting ourselves to open bosonic SFT, but there is an interesting fact whose demonstration requires similar considerations:
\begin{claim}
Given a real and nonsingular tachyon vacuum solution in the $KBc$ subalgebra, the coefficient of the tachyon state in the $\mathcal{L}^-$ level expansion is positive.
\end{claim}
\noindent Intuitively, this is the statement that the stable vacuum is found when the tachyon condenses towards positive values from the original unstable D-brane. This is such an elementary fact that one can forget that it requires explanation.
To prove this, we note that multiplying Okawa-type tachyon vacuum solutions \eq{Oktype} on both sides by $B$ gives
\begin{equation}B\Psi B = B\frac{KF}{1-F}.\label{eq:BPsiB}\end{equation}
The Okawa-type solution is not the most general tachyon vacuum in the $KBc$ subalgebra, but nevertheless the general tachyon vacuum is characterized by a choice of $F(K)$ such that $F(0)=1$ and $F'(0)\neq 0$ and the above relation holds \cite{Jokel}. The $\mathcal{L}^-$ level expansion of $\Psi$ takes the form
\begin{equation}\Psi = \alpha c+ \text{higher levels},\end{equation}
where $\alpha$ is the tachyon coefficient (in the basis of $\mathcal{L}^-$ eigenstates). Plugging this into \eq{BPsiB} and expanding the right hand side around $K=0$ leads to the identification
\begin{equation}\alpha = \frac{1}{F'(0)}.\end{equation}
Meanwhile, we know that the leading level of $\Psi$ in the dual $\mathcal{L}^-$ expansion must be negative. This implies that the leading level of the right hand side of \eq{BPsiB} must be less than $2$. This is only possible if $F(K)$ vanishes at infinity:
\begin{equation}F(\infty) = 0.\end{equation}
Finally, consider the state
\begin{equation}H(K) = \frac{1-F}{K},\end{equation}
which appears in the homotopy operator for the tachyon vacuum. From the behavior of $F$ we know that $H$ satisfies the boundary conditions \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{wrapfigure}{l}{.34\linewidth}
\centering
\resizebox{2.4in}{1.5in}{\includegraphics{OpenSFT_Erler43.jpg}}
\end{wrapfigure}
\begin{equation}H(0) = \frac{1}{\alpha},\ \ \ \ \lim_{K\to\infty}KH(K) = 1.\end{equation}
Now suppose that $\alpha$ is negative. Since the homotopy operator should be well-defined, $H(K)$ must be continuous for non-negative $K$. Then the above boundary conditions imply that $H(K)$ must have at least one zero for positive $K$. This is the ``topological" part of the argument. On the other hand, the right hand side of \eq{BPsiB} contains the state
\begin{equation}\frac{F(K)}{H(K)}.\end{equation}
This must be discontinuous at a pole corresponding to the zero of $H(K)$, and is therefore singular. Since multiplication by $B$ cannot turn a regular solution into a singular state, we conclude that $\alpha$ cannot be negative.
One might expect that a similar result will hold for the usual tachyon coefficient in the basis of $L_0$ eigenstates, but this has not been proven.
\subsection{Singular Gauge Transformations}
\label{subsec:singularGT}
We now describe a formalism introduced by Ellwood \cite{Ellsingular} and further elaborated in \cite{EMsingular,EMphantom} which gives some understanding of the phantom term in Schnabl's solution.
Consider the equation
\begin{equation}Q_{\Psi_1\Psi_2}U = 0,\ \ \ \ \mathrm{gh}(U)=0,\end{equation}
where $\Psi_1$ and $\Psi_2$ are classical solutions and $Q_{\Psi_1\Psi_2}$ is the shifted kinetic operator for a stretched string connecting $\Psi_1$ and $\Psi_2$, as described in subsection \ref{subsec:KOS}. A solution to this equation (at ghost number zero) will be called a {\it morphism} from $\Psi_1$ to $\Psi_2$.\footnote{Here we use somewhat different terminology from \cite{EMsingular}. There, a morphism was called a {\it left gauge transformation.}} As the terminology suggests, morphisms define a category. The objects of the category are classical solutions, and composition of morphisms is defined with the open string star product. To make the category structure clear, we note the following:
\begin{itemize}
\item Let $U_{12}$ be a morphism from $\Psi_1$ to $\Psi_2$ and $U_{23}$ be a morphism from $\Psi_2$ to $\Psi_3$. Then the Leibniz rule \eq{ShiftLeibniz} implies that the product $U_{12}U_{23}$ is a morphism from $\Psi_1$ to~$\Psi_3$:
\begin{eqnarray}
Q_{\Psi_1\Psi_3}\big(U_{12}U_{23}\big)\!\!\!\!\!\!\!\! && = \big(Q_{\Psi_1\Psi_2}U_{12}\big)U_{23}+U_{12}\big(Q_{\Psi_2\Psi_3}U_{23}\big) = 0 .
\end{eqnarray}
Composition of morphisms is associative since the star product is associative.
\item Between any pair of solutions there is a zero morphism $U=0$.
\item Between any solution and itself there is an identity morphism $U=1$.
\item A special kind of morphism is an {\it isomorphism}; this is a morphism $U$ from $\Psi_1$ to $\Psi_2$ which has an inverse morphism $U^{-1}$ from $\Psi_2$ to $\Psi_1$. In particular $UU^{-1} = U^{-1}U = 1$. In fact, an isomorphism is the same thing as a (finite) {\it gauge transformation} between $\Psi_1$ and $\Psi_2$. This can be seen as follows:
\begin{eqnarray}
0 = QU + \Psi_1U - U\Psi_2\ \implies\
U\Psi_2 = QU + \Psi_1 U \ \implies\
\Psi_2 = U^{-1}(Q+\Psi_1)U.
\end{eqnarray}
\end{itemize}
Therefore, the gauge group of open SFT naturally extends to a category structure which connects all classical solutions. This is a particular instance of the general concept of ``D-brane categories" which appears in various guises, especially in discussions of open topological strings. In the most basic form, a D-brane category consists of D-branes as objects, and open strings connecting D-branes as morphisms.
What makes the category structure interesting is that morphisms can relate solutions which are not physically equivalent. Given any $\Psi_1$ and $\Psi_2$, we can find a nonzero morphism of the form
\begin{equation}U = Q_{\Psi_1\Psi_2}b,\ \ \ \ \mathrm{gh}(b) = -1.\end{equation}
We call this an {\it exact morphism}. In fact, if $\Psi_1$ and $\Psi_2$ represent distinct open string boundary conditions, a morphism between them is expected to be exact; the only source of cohomology at ghost number zero is the identity operator, which is absent from the spectrum of strings connecting inequivalent D-branes. So without exact morphisms, there would be no nontrivial connections between inequivalent solutions. Exact morphisms can also connect gauge equivalent solutions, but they are rarely gauge transformations themselves. The precise statement is:
\begin{claim}
An exact morphism cannot be a gauge transformation unless it connects two solutions for the tachyon vacuum.
\end{claim}
\noindent To see this, suppose $U$ is exact and possesses an inverse. This implies
\begin{eqnarray}
1\!\!\!\!\!\!\!\! && =U^{-1}U = U^{-1}Q_{\Psi_1\Psi_2}b = Q_{\Psi_2}\big(U^{-1}b\big), \\
1\!\!\!\!\!\!\!\! && = UU^{-1} = \big(Q_{\Psi_1\Psi_2}b\big)U^{-1}= Q_{\Psi_1}\big(bU^{-1}\big).
\end{eqnarray}
Therefore the identity is trivial in the cohomology around $\Psi_1$ and $\Psi_2$, so both solutions must represent the tachyon vacuum.
Another important class of morphisms are called {\it resolvable}. A resolvable morphism $U$ has the property that $\epsilon+U$ has a star algebra inverse for any sufficiently small $\epsilon>0$. All gauge transformations are resolvable. Exact morphisms may not be resolvable, but often they are. The prototypical example of a resolvable exact morphism is the string field $K$, which can be viewed as a morphism connecting the perturbative vacuum to itself:
\begin{equation}U = K = QB,\ \ \ \ QU= 0.\end{equation}
After adding a small $\epsilon>0$ we can compute the inverse
\begin{equation}\frac{1}{\epsilon+ K} = \int_0^\infty d\alpha e^{-\epsilon \alpha} \Omega^\alpha.\end{equation}
The inverse is well-defined for any fixed positive $\epsilon$, no matter how small, though it is divergent
\begin{wrapfigure}{l}{.34\linewidth}
\centering
\resizebox{2.4in}{2in}{\includegraphics{OpenSFT_Erler44.jpg}}
\end{wrapfigure}
in the $\epsilon\to 0$ limit. By contrast, $U=-K$ is not resolvable since its spectrum is non-positive. A less trivial example is
\begin{equation}U = K(1-K) = Q\big(B(1-K)\big),\ \ \ \ QU=0.\end{equation}
After adding a small $\epsilon$ the inverse is bounded as a function of $K$ at $K=0$, but it is still not bounded in the vicinity of $K=1$. With these definitions there are six classes of morphisms, as illustrated to the left.
\begin{exercise}
Find a representative example of a morphism for all six classes.
\end{exercise}
\noindent A morphism which is exact, resolvable, and not a gauge transformation will be called a {\it singular gauge transformation}.
Singular gauge transformations are interesting since they can connect physically distinct solutions, but are infinitesimally close to being gauge transformations. If $U$ is a singular gauge transformation from $\Psi_1$ to $\Psi_2$, we can {\it almost} relate the solutions by gauge transformation through the identity
\begin{equation}\Psi_2 = \frac{1}{\epsilon+U}(Q+\Psi_1)(\epsilon+U)\ + \ \frac{\epsilon}{\epsilon+U}(\Psi_2- \Psi_1).\label{eq:phantom}\end{equation}
The first term is a gauge transformation of $\Psi_1$ for any sufficiently small $\epsilon>0$. If $U$ was a true gauge transformation, the second term would vanish in the $\epsilon\to 0$ limit. However, if $\Psi_1$ and $\Psi_2$ are not gauge equivalent solutions, the second term must be nontrivial in the $\epsilon\to 0$ limit. This is a generalization of the {\it phantom term} of Schnabl's solution. A trivial example of a singular gauge transformation is the zero morphism $U=0$. In this case, the phantom term is merely the difference between $\Psi_2$ and $\Psi_1$. Typically, however, the phantom term will approach a sliver-like state in the limit $\epsilon\to 0$. Note that among all the solutions we have discussed so far, only Schnabl's solution ``needed" a phantom term. However, from the present point of view the phantom term is not necessarily a property of a solution, but a property of a pair of solutions and a singular gauge transformation connecting them. The phantom term of Schnabl's solution arises from Okawa's singular gauge transformation from the perturbative vacuum to the tachyon vacuum.
The nature of the phantom term is determined by the behavior of the state $\frac{\epsilon}{\epsilon+U}$ for small $\epsilon$. Consider a basis of Fock states $|\phi_i\rangle$ and the limit
\begin{equation}\langle \phi_i,X^\infty\rangle = \lim_{\epsilon\to 0} \left\langle \phi_i, \frac{\epsilon}{\epsilon+U}\right\rangle.\end{equation}
If this limit converges, this defines what is called the {\it boundary condition changing projector}, $X^\infty$. Like the sliver state, the boundary condition changing projector is generally not part of the open string star algebra, in the sense that it cannot be unambiguously multiplied with itself or other boundary condition changing projectors. However, it can be multiplied with elements of the open string star algebra. We have the relations
\begin{equation}UX^\infty = X^\infty U = 0.\end{equation}
Therefore, singular gauge transformation can be viewed as having a kernel, and $X^\infty$ is the projector onto the kernel. A special case is when $U$ is a singular gauge transformation between a solution and itself. Then the boundary condition changing projector reduces to what is called the {\it characteristic projector} \cite{Ellsingular}. An example is the singular gauge transformation $U=K$ around the perturbative vacuum. In this case the characteristic projector is precisely the sliver state:
\begin{equation}
X^\infty = \lim_{\epsilon\to 0} \frac{\epsilon}{\epsilon+K} = \Omega^\infty.
\end{equation}
One additional comment about multiplying projector-like states. When we talk about the boundary condition changing ``projector" we imagine that the relation
\begin{equation}(X^\infty)^2\stackrel{?}{=}X^\infty\end{equation}
should hold. However, suppose we multiply by $Bc+cB =1$ on both sides of the sliver state. We obtain the equality
\begin{equation}\Omega^\infty = Bc\Omega^\infty cB,\end{equation}
where we used the fact that $B$ annihilates the sliver state. The expression on the right hand side cannot be a projector, since the star product with itself vanishes due to $B^2 = 0$. This further demonstrates that projector-like states cannot be unambiguously multiplied amongst themselves.
We now come to one of the most interesting claims of the formalism. To explain it, let us revisit the visualization of the sliver state. Recall from subsection \ref{subsec:wedge} that the overlap of the sliver state with a test state $|\phi\rangle$ can be computed as a correlation function on the upper half plane, with the vertex operator of the test state inserted at the origin with the sliver coordinate map $f_\mathcal{S}(\xi) = \frac{2}{\pi}\tan^{-1}\xi$. The surface of the test state occupies the strip $\frac{1}{2}\geq \mathrm{Re}(z)\geq -\frac{1}{2}$ in the upper half plane. So that we can clearly see what is happening towards the midpoint, for present discussion it is more convenient to visualize the overlap as a correlation
function on the unit disk, obtained from the upper half plane after the conformal transformation
\begin{equation}w = d(z) = \frac{1+2 i z}{1-2i z},\end{equation}
so that
\begin{equation}\langle \phi, \Omega^\infty\rangle = \big\langle d\circ f_\mathcal{S}\circ\phi(0)\big\rangle_\text{disk}.\end{equation}
\begin{wrapfigure}{l}{.45\linewidth}
\centering
\resizebox{3.2in}{1.6in}{\includegraphics{OpenSFT_Erler45.jpg}}
\end{wrapfigure}
The vertex operator is now inserted at $w=1$ on the boundary of the unit disk.
The surface of the test state occupies a region on the interior of the unit disk bounded by two unit circles centered at $w=-1\pm i$, as shown above. The important thing to notice about this picture is that the surface of the test state touches the boundary of the unit disk at $w=-1$, which represents the image of the midpoint $\xi=i$. If we subtract the surface of the test state from the unit disk, we are left with two disconnected surfaces defining the left and right half-string Schr{\"o}dinger functionals of the sliver state. Now let us return to the boundary condition changing projector. The expectation is that it will have the following structure:
\begin{description}
\item{(1)} Consider the overlap $\langle \phi,X^\infty\rangle$ represented as a correlation function on the unit disk. Inside the unit disk is a region occupied by the surface of the test state. The surface of the test state includes an arc on the boundary containing the probe vertex operator and at least one other disconnected point on the boundary representing the image of the midpoint $\xi = i$.
\item{(2)} On the arc which includes the probe vertex operator, the correlator has open string boundary conditions corresponding to the reference string field theory $\mathrm{BCFT}_0$. This simply reflects the fact that $X^\infty$ is a state in the reference string field theory. More interestingly, at the image of the midpoint there is a shift in open string boundary condition in the correlation function between $\mathrm{BCFT}_1$ and $\mathrm{BCFT}_2$, corresponding respectively to the solutions $\Psi_1$ and $\Psi_2$ related by the singular gauge transformation. The shift occurs in that order when following the canonical orientation around the boundary of the unit disk (counterclockwise in the conventional visualization of the complex plane, and {\it clockwise} in our convention of visualizing the positive real axis as increasing to the left).
\end{description}
\begin{wrapfigure}{l}{.25\linewidth}
\centering
\resizebox{1.8in}{1.2in}{\includegraphics{OpenSFT_Erler46.jpg}}
\end{wrapfigure}
\noindent The picture is summarized to the left. The upshot is that by constructing a singular gauge transformation between solutions $\Psi_1$ and $\Psi_2$, computing the boundary condition changing projector, and focusing in on the midpoint, you can directly derive the stretched string connecting the D-branes represented by $\Psi_2$ and $\Psi_1$. This is why it is called the ``boundary condition changing projector." On the segments of the boundary of the unit disk not contained in the surface of the test state, there can be various operator insertions in the correlation function whose form depends on the singular gauge transformation. The cumulative effect of these operator insertions must change the open string boundary condition from $\mathrm{BCFT}_0$ to $\mathrm{BCFT}_2$ in the surface defining the left half string functional, and to change the boundary condition from $\mathrm{BCFT}_0$ to $\mathrm{BCFT}_1$ in the surface defining the right half string functional. Some motivation for this picture can be found from the BRST identity
\begin{equation}
Q\left(\frac{\epsilon}{\epsilon+U}\right) +\Psi_2\frac{\epsilon}{\epsilon+U} + \frac{\epsilon}{\epsilon+U}(\Psi_1-\Psi_2)\frac{\epsilon}{\epsilon+U} - \frac{\epsilon}{\epsilon+U}\Psi_1 = 0.\label{eq:BRSTid}
\end{equation}
The idea is that when $Q$ acts on the boundary condition changing projector, various individual contributions will arise from $Q$ acting on operators which shift the boundary condition inside the correlation function on the disk. The second term above arises from $Q$ acting on operators which shift the boundary condition from $\mathrm{BCFT}_0$ to $\mathrm{BCFT}_2$. The third term arises from $Q$ acting on operators which change the boundary condition from $\mathrm{BCFT}_2$ to $\mathrm{BCFT}_1$ at the midpoint. Finally, the fourth term arises from $Q$ acting on operators which shift from $\mathrm{BCFT}_1$ back to $\mathrm{BCFT}_0$. To see that it is reasonable for these BRST variations to produce factors of the solutions $\Psi_1$ and $\Psi_2$, consider a boundary condition changing operator for regular marginal deformations. Since the boundary condition changing operator is a weight zero primary, the BRST variation is
\begin{equation}Q\sigma = c\partial \sigma = c(\partial \sigma\overline{\sigma})\sigma = -(cV)\sigma,\end{equation}
where in the last step we used the result of exercise \ref{ex:Vsigma}. As explained in the previous subsection, $cV$ is a solution for regular marginal deformations.
It has been found that the boundary condition changing projector usually, but not always, takes the form outlined above. A counterexample is the zero morphism $U=0$, whose boundary condition changing projector is the identity string field. Since the zero morphism takes the same form connecting any pair of solutions, it is not surprising that its boundary condition changing projector fails to contain information about the shift in background. It is possible to come up with related counterexamples. For SFT on a pair of D-branes, we can consider a singular gauge transformation consisting of the zero morphism on the first diagonal entry and a nonzero morphism on the other; the boundary condition changing projector will contain the identity string field in the first diagonal entry. A singular gauge transformation leading to a boundary condition changing projector of the form outlined in the previous paragraph will be called {\it nondegenerate}. Since the present understanding of boundary condition changing projectors is mostly empirical, there is no deep understanding of the conditions leading to nondegeneracy, but they appear to be fairly generic.
An important special case of the boundary condition changing projector is the characteristic projector. Here the boundary conditions near the midpoint of the left and right half string functionals will be the same. For example, the characterstic projector of the singular gauge transformation $K$ around the perturbative vacuum is the sliver state, which indeed carries $\mathrm{BCFT}_0$ boundary conditions at the midpoint. The BRST identity \eq{BRSTid} of the characteristic projector simplifies to
\begin{equation}Q_{\Psi_1}\left(\frac{\epsilon}{\epsilon+U}\right) = 0.\end{equation}
If the singular gauge transformation is nondegenerate, the characteristic projector is believed to generate the cohomology class of the identity operator around the solution $\Psi_1$.
We would like to compute the boundary condition changing projector in the simplest nontrivial example. Consider identity-like solutions for marginal deformations:
\begin{equation}\Psi_1 = cV_1,\ \ \ \ \ \Psi_2 = cV_2.\end{equation}
We can find an exact morphism connecting them in the form
\begin{equation}U = Q_{\Psi_1\Psi_2}B = K+ cBV_1 + BcV_2.\end{equation}
It turns out that this morphism is only resolvable if $V_1$ and $V_2$ have regular OPE with each other. This limitation can be avoided if we choose less identity-like solutions, but presently we want to focus on the simplest example. The calculation proceeds as follows:
\begin{eqnarray}
\frac{\epsilon}{\epsilon+U} \!\!\!\!\!\!\!\! && = \frac{\epsilon}{\epsilon + K+ cBV_1 + BcV_2}\nonumber\\
\!\!\!\!\!\!\!\! && = \frac{\epsilon}{\epsilon + cB(K+V_1)+Bc(K+V_2)}\nonumber\\
\!\!\!\!\!\!\!\! && = \frac{1}{\left(1+\frac{1}{\epsilon}cB(K+V_1)\right)\left(1+\frac{1}{\epsilon}Bc(K+V_2)\right)}\nonumber\\
\!\!\!\!\!\!\!\! && = \left(\frac{\epsilon}{\epsilon + Bc(K+V_2)}\right)\left(\frac{\epsilon}{\epsilon+cB(K+V_1)}
\right).\label{eq:bccproj1}
\end{eqnarray}
We now introduce boundary condition changing operators so that
\begin{equation}\sigma_{01}K\sigma_{10} = K+V_1,\ \ \ \ \sigma_{02}K\sigma_{20} = K+V_2.\end{equation}
Since we consider regular marginal deformations, the boundary condition changing operators satisfy
\begin{eqnarray}\sigma_{01}\sigma_{10} = 1,\!\!\!\!\!\!\!\! && \ \ \ \ \sigma_{10}\sigma_{01} = 1,\nonumber\\
\sigma_{02}\sigma_{20} = 1,\!\!\!\!\!\!\!\! && \ \ \ \ \sigma_{20}\sigma_{02} = 1.
\end{eqnarray}
In \eq{bccproj1} we also encounter the product
\begin{equation}\sigma_{21} = \sigma_{20}\sigma_{01}.\end{equation}
This product must be finite, which requires that $V_1$ and $V_2$ have regular OPE with each other. Then \eq{bccproj1} can be written
\begin{equation}\frac{\epsilon}{\epsilon+U} = \sigma_{02}\frac{\epsilon}{\epsilon+Bc K}\sigma_{21}\frac{\epsilon}{\epsilon + cB K}\sigma_{10}.\end{equation}
To deal with the ghosts, we first insert the identity $Bc+cB = 1$ in the $\mathrm{BCFT}_1$ factor:
\begin{eqnarray}
\frac{\epsilon}{\epsilon+U} \!\!\!\!\!\!\!\! && = \sigma_{02}\frac{\epsilon}{\epsilon+Bc K}\sigma_{21}(Bc + cB)\frac{\epsilon}{\epsilon + cB K}\sigma_{10}\nonumber\\
\!\!\!\!\!\!\!\! && = \sigma_{02}\frac{\epsilon}{\epsilon+Bc K}\sigma_{21}\left(Bc + cB \frac{\epsilon}{\epsilon + K}\right)\sigma_{10}.
\end{eqnarray}
Here we used
\begin{equation}cF(cBK) = F(0),\ \ \ \ BF(cBK)= BF(K).\end{equation}
The former follows from $c^2=0$ and the later from $[B,c]=1$ and $B^2=0$. To deal with the $\mathrm{BCFT}_2$ factor we write
\begin{eqnarray}
\frac{\epsilon}{\epsilon+U} \!\!\!\!\!\!\!\! && = \sigma_{02}\left(1+\frac{\epsilon}{\epsilon+Bc K}-1\right)\sigma_{21}\left(Bc + cB \frac{\epsilon}{\epsilon + K}\right)\sigma_{01}\nonumber\\
\!\!\!\!\!\!\!\! && = \sigma_{02}\left(1-\frac{1}{\epsilon+Bc K}BcK \right)\sigma_{21}\left(Bc + cB \frac{\epsilon}{\epsilon + K}\right)\sigma_{10}\nonumber\\
\!\!\!\!\!\!\!\! && = \sigma_{02}\left(1-\frac{1}{\epsilon+K}BcK \right)\sigma_{21}\left(Bc + cB \frac{\epsilon}{\epsilon + K}\right)\sigma_{10}.
\end{eqnarray}
Multiplying everything out gives
\begin{eqnarray}
\frac{\epsilon}{\epsilon+U} \!\!\!\!\!\!\!\! && = \sigma_{02}\left(1-\frac{1}{\epsilon+K}K \right)Bc\sigma_{20}+\sigma_{01}cB\frac{\epsilon}{\epsilon+K}\sigma_{10}-\sigma_{02}\frac{1}{\epsilon+K}BcKcB\sigma_{21}\frac{\epsilon}{\epsilon+K}\sigma_{10}\nonumber\\
\!\!\!\!\!\!\!\! && = \sigma_{02}\frac{\epsilon}{\epsilon+K}Bc\sigma_{20}+\sigma_{01}cB\frac{\epsilon}{\epsilon+K}\sigma_{10}+\sigma_{02}\frac{1}{\epsilon+K}\sigma_{21} B\partial c\frac{\epsilon}{\epsilon+K}\sigma_{10}.
\end{eqnarray}
Now we can take the limit $\epsilon\to 0$. The first two terms vanish in the limit because $B$ annihilates the sliver. The third term is somewhat nontrivial. Expanding as a superposition of wedge states gives
\begin{equation}
\sigma_{02}\frac{1}{\epsilon+K}\sigma_{21} B\partial c\frac{\epsilon}{\epsilon+K}\sigma_{10} = \int_0^\infty dL\, e^{-L}\int_0^{L/\epsilon} d x \, \sigma_{02}\Omega^x \sigma_{21}B\partial c \Omega^{L/\epsilon-x}\sigma_{10}.
\end{equation}
In the integrand is a wedge surface of total width $L/\epsilon$, which for fixed $L$ is approaching the sliver state as $\epsilon\to 0$. The boundary condition changing operator $\sigma_{21}$ is integrated across the entire width of the wedge surface from $x=0$ to $x=L/\epsilon$. It is helpful to decompose this integration into three pieces; near the left edge, through the middle, and near the right edge:
\begin{eqnarray}
\int_0^{L/\epsilon} d x \, \Omega^x \sigma_{21}B\partial c \Omega^{L/\epsilon-x}\!\!\!\!\!\!\!\! && = \int_0^{\Lambda} d x \, \Omega^x \sigma_{21}B\partial c \Omega^{L/\epsilon-x}+\int_\Lambda^{L/\epsilon -\Lambda} d x \, \Omega^x \sigma_{21}B\partial c \Omega^{L/\epsilon-x}\nonumber\\
\!\!\!\!\!\!\!\! &&\ \ \ \ +\int_0^{\Lambda} d x \, \Omega^{L/\epsilon-x} \sigma_{21}B\partial c \Omega^{x}.
\end{eqnarray}
Here $\Lambda<L/(2\epsilon)$ is an arbitrary parameter (independent of $\epsilon$) which defines what it means to be ``near" the edges of the wedge surface. The contributions near the left and right edges vanish in the $\epsilon\to 0$ limit because $B$ tends to annihilate the sliver, and due to the finite range of integration there can be no compensating divergence. Since $\Lambda$ can be arbitrary, this implies that $\sigma_{21}$ cannot be a finite distance from the left or right edges of the wedge surface in the $\epsilon\to 0$ limit. This implies that the boundary condtion changing projector, if it is nonvanishing, must contain $\sigma_{21}$ inserted at the midpoint. Without going into the derivation (see appendix A of \cite{EMsingular}) we claim that the integration ``through the middle" is nonzero, and in the $\epsilon\to 0$ limit the ghost insertions can be eliminated to obtain
\begin{equation}X^\infty = \sigma_{21}(i)\Big(\sigma_{02}\Omega^\infty\sigma_{10}\Big),\end{equation}
where $\sigma_{21}$ in this equation acts as an operator (not a string field) on the midpoint of the sliver state. This agrees with the general form of the boundary condition changing projector anticipated before. In more complicated examples the boundary condition changing projector can have more structure towards the left and right edges. If
\begin{equation}U = Q_{\Psi_1\Psi_2}\left(B\frac{1-\Omega}{K}\right)\end{equation}
is a singular gauge transformation between two Schnabl gauge marginal solutions, the boundary condition changing projector turns out to be
\begin{equation}X^\infty =\sigma_{21}(i)\Big(\sqrt{\Omega}\sigma_{02}\Omega^\infty\sigma_{10}\sqrt{\Omega}\Big),\end{equation}
so we pick up additional factors of $\sqrt{\Omega}$. The characteristic projector for the Kiermaier-Okawa solution contains a factor of $U^{-1}$ from \eq{singularU} on the right edge \cite{Ellsingular}.
It is interesting to think about what happens to the boundary condition changing projector when we consider a singular gauge transformation to the tachyon vacuum. Here there are no D-branes on which open strings can end, so it is not clear what kind of boundary condition should emerge at the midpoint. Let us consider an exact morphism from the perturbative vacuum to Schnabl's solution:
\begin{eqnarray}
U \!\!\!\!\!\!\!\! &&= Q_{0\Psi_\text{Sch}}\left(B\frac{1-\Omega}{K}\right)\nonumber\\
\!\!\!\!\!\!\!\! && = 1-\sqrt{\Omega}cB\sqrt{\Omega}.\label{eq:Okconjform}
\end{eqnarray}
This is a variant of Okawa's pure gauge form for Schnabl's solution. It is the conjugate of the inverse gauge parameter in \eq{Okform} at $\lambda=1$. To find the boundary condition changing projector we compute
\begin{eqnarray}
\frac{\epsilon}{\epsilon+U}\!\!\!\!\!\!\!\! && =\frac{\epsilon}{1+\epsilon -\sqrt{\Omega}cB\sqrt{\Omega}}\nonumber\\
\!\!\!\!\!\!\!\! && = \frac{1-\lambda}{1-\lambda\sqrt{\Omega}cB\sqrt{\Omega}},
\end{eqnarray}
where
\begin{equation}\lambda = \frac{1}{1+\epsilon},\end{equation}
and the boundary condition changing projector is defined by the limit $\lambda\to 1^-$. Simplifying the ghosts and expanding in a geometric series gives
\begin{eqnarray}
\frac{\epsilon}{\epsilon+U} \!\!\!\!\!\!\!\! && = (1-\lambda) +\lambda(1-\lambda)\sqrt{\Omega}cB\frac{1}{1-\lambda\Omega}\sqrt{\Omega}\nonumber\\
\!\!\!\!\!\!\!\! && = (1-\lambda)+\lambda \sqrt{\Omega}cB\sqrt{\Omega}\left((1-\lambda)\sum_{n=0}^\infty \lambda^n \Omega^n \right).
\end{eqnarray}
The sum is convergent for $|\lambda|<1$, which in particular implies that the exact morphism is resolvable and defines a singular gauge transformation. To see what happens in the $\lambda\to 1^-$ limit we note that wedge states for large wedge parameter can be expanded around the sliver state when contracted with Fock states:
\begin{equation}\Omega^n = \Omega^\infty+\mathcal{O}\left(\frac{1}{n^2}\right),\ \ \ \ \ n\text{ large}.\end{equation}
Plugging into the geometric series and summing implies
\begin{equation}
(1-\lambda)\sum_{n=0}^\infty \lambda^n \Omega^n = \Omega^\infty + \mathcal{O}(1-\lambda),\ \ \ \ \lambda\to 1^-,
\end{equation}
and
\begin{equation}
\frac{\epsilon}{\epsilon+U} = \sqrt{\Omega}cB\Omega^\infty\sqrt{\Omega} + \mathcal{O}(1-\lambda),\ \ \ \ \lambda\to 1^-.
\end{equation}
The leading term vanishes since $B$ annihilates the sliver state, and the remainder tends to zero as $1-\lambda$ or faster. Therefore the boundary condition changing projector vanishes identically:
\begin{equation}X^\infty = 0.\end{equation}
This is how we understand the fact that open strings cannot connect to the tachyon vacuum. Since the phantom term is proportional to the boundary condition changing projector, this also gives a physical explanation for why the phantom term of Schnabl's solution vanishes in the Fock space. It is interesting to observe that proper gauge transformations also lead to a vanishing characteristic projector. This in itself has no special meaning, since the characteristic projector is only proposed to give information about open string boundary conditions when the morphism is exact. However, a gauge transformation {\it can} be exact if it connects solutions for the tachyon vacuum. In this case, the vanishing of the characteristic projector is telling us about the absence of D-branes.
To give one more example, consider the characteristic projector of the residual tachyon vacuum $\Psi=-cK$. We can try to define an exact morphism around this solution by taking
\begin{equation}U = Q_{-cK}B = 0,\end{equation}
but this does not give any information. Inserting some generic $H(K)$ we can find a nonzero exact morphism
\begin{equation}
U = Q_{-cK}(BH) = B[c,H]K.
\end{equation}
Then we can compute
\begin{eqnarray}
\frac{\epsilon}{\epsilon+U} = \frac{\epsilon}{\epsilon +B[c,H]K}=1-\frac{1}{\epsilon}B[c,H]K.
\end{eqnarray}
Though the morphism is resolvable, the $\epsilon\to 0$ limit does not approach a finite state. It seems that there is no simple way to remedy this problem with a different choice of morphism, so the characteristic projector appears not to exist. This agrees with the general expectation that residual solutions are unphysical.
\begin{exercise}
If we add $c$ to the residual tachyon vacuum we obtain the identity-like tachyon vacuum solution $\Psi = c(1-K)$. Compute the characteristic projector of the exact morphism $U= Q_{\Psi}(BH)$ and show that it vanishes under reasonable assumptions about $H(K)$.
\end{exercise}
Perhpas the most significant application of this formalism is that it gives a procedure for extracting gauge invariant information from solutions whose physical interpretation might otherwise not be obvious. In some circumstances it can even be used to compute observables from solutions which are not known in closed form. One gauge invariant provided by the formalism is the boundary condition changing projector itself, specifically its behavior near the midpoint. The formalism can also be used to compute the action or Ellwood invariant. The important idea here is that a singular gauge transformation allows one to absorb all unphysical complications into a pure gauge solution, leaving the phantom term to describe the shift in background in the most transparent possible manner. To illustrate this point we will use the formalism to compute the energy of Schnabl's solution. Some other applications are discussed in \cite{EMphantom}. We consider the singular gauge transformation \eq{Okconjform}, and use \eq{phantom} to write Schnabl's solution as
\begin{eqnarray}\Psi_\text{Sch}\!\!\!\!\!\!\!\! && = \frac{1}{\epsilon+U}QU +\frac{\epsilon}{\epsilon+U}\Psi_\text{Sch}\nonumber\\
\!\!\!\!\!\!\!\! && = \lambda \sqrt{\Omega}c\frac{KB}{1-\lambda\Omega}c\sqrt{\Omega}\,+\,\sqrt{\Omega}cB\frac{K}{1-\Omega}\frac{1-\lambda}{1-\lambda\Omega}c\sqrt{\Omega}\nonumber\\
\!\!\!\!\!\!\!\! && = \Psi_\lambda + \Psi_\lambda^\text{phantom}\phantom{\bigg)}.\end{eqnarray}
The first term is the familiar pure gauge solution in Schnabl gauge, and the second is what we are calling the phantom term. This is somewhat different from the original phantom term in \eq{Schpsin}, which was derived as a remainder after truncating the geometric series expansion into a discrete sum over wedge states. The difference amounts to a choice of regularization. Presently we are regularizing with the gauge parameter $\lambda$, instead of by truncating the geometric series. If we assume that $\frac{K}{1-\Omega}$ can be represented by its leading term in the $\mathcal{L}^-$ level expansion in the limit $\lambda\to 1^-$, the regularization we are presently using can be represented as
\begin{equation}
\Psi_\text{Sch} = \lim_{\lambda\to 1^-}\sum_{n=0}^\infty\lambda^n\left((1-\lambda)\psi_n - \lambda\frac{d}{dn}\psi_n\right).
\end{equation}
The advantage of this approach is that the phantom term satisfies the equations of motion expanded around a pure gauge configuration, independent of $\lambda$. The truncated geometric series expansion, however, is not a solution and is therefore not really pure gauge. In particular, it contributes to the energy, which complicates the calculation. The action evaluated on Schnabl's solution can be written
\begin{eqnarray}
S[\Psi_\text{Sch}] \!\!\!\!\!\!\!\! && = S[\Psi_\lambda] + \frac{1}{2}\mathop{\rm Tr}\nolimits\Big(\Psi_\lambda^\text{phantom}Q_{\Psi_\lambda}\Psi_\lambda^\text{phantom}\Big)+\frac{1}{3}\mathop{\rm Tr}\nolimits\Big((\Psi_\lambda^\text{phantom})^3\Big)\nonumber\\
\!\!\!\!\!\!\!\! && = -\frac{1}{6}\mathop{\rm Tr}\nolimits\Big((\Psi_\lambda^\text{phantom})^3\Big).
\end{eqnarray}
Here we noted that the action evaluated on the pure gauge solution vanishes and used the equations of motion for the phantom term
\begin{equation}
Q_{\Psi_\lambda}\Psi_\lambda^\text{phantom} + (\Psi_\lambda^\text{phantom})^2 = 0.
\end{equation}
Substituting the phantom term, the action is
\begin{equation}
S[\Psi_\text{Sch}] = -\frac{1}{6}\mathop{\rm Tr}\nolimits\left(\left(c\Omega cB\frac{K}{1-\Omega}\frac{1-\lambda}{1-\lambda\Omega}\right)^3\right).
\end{equation}
The action is independent of $\lambda$, and we can consider the $\lambda\to 1^-$ limit where the phantom term becomes sliver-like. Further, it is helpful to perform a scale transformation so that the circumference of the cylinder defined by the trace does not diverge in the $\lambda\to 1^-$ limit.
This leads to
\begin{eqnarray}
S[\Psi_\text{Sch}] \!\!\!\!\!\!\!\! && = -\frac{1}{6}\mathop{\rm Tr}\nolimits\left(\left(\left[(1-\lambda)^{\frac{1}{2}\mathcal{L}^-}\left(c\Omega cB\frac{K}{1-\Omega}\right)\right]\frac{1-\lambda}{1-\lambda\Omega^{1-\lambda}}\right)^3\right)\nonumber\\
\!\!\!\!\!\!\!\! && = -\frac{1}{6}\mathop{\rm Tr}\nolimits\left(\left(\left[(1-\lambda)^{\frac{1}{2}\mathcal{L}^-}\left(c\Omega cB\frac{K}{1-\Omega}\right)\right](1-\lambda)\sum_{n=0}^\infty \lambda^n \Omega^{(1-\lambda)n}\right)^3\right).
\end{eqnarray}
Defining $\alpha_n = (1-\lambda)n$, the geometric series can be written as a Riemann sum
\begin{equation}
(1-\lambda)\sum_{n=0}^\infty \lambda^n \Omega^{(1-\lambda)n} = \sum_{n=0}^\infty (\alpha_{n+1}-\alpha_n)\left(1-\frac{\alpha_n}{n}\right)^n \Omega^{\alpha_n},
\end{equation}
which in the $\lambda\to 1^-$ limit converges to an integral:
\begin{eqnarray}
\lim_{\lambda\to 1^-}(1-\lambda)\sum_{n=0}^\infty \lambda^n \Omega^{(1-\lambda)n}\!\!\!\!\!\!\!\! && =\int_0^\infty d\alpha e^{-\alpha}\Omega^\alpha
\nonumber\\
\!\!\!\!\!\!\!\! && = \frac{1}{1+K}.
\end{eqnarray}
Meanwhile, the other factor can be represented through the $\mathcal{L}^-$ level expansion:
\begin{equation}
(1-\lambda)^{\frac{1}{2}\mathcal{L}^-}\left(c\Omega cB\frac{K}{1-\Omega}\right) = -c\partial c B + \mathcal{O}(1-\lambda).
\end{equation}
Therefore
\begin{equation}
S[\Psi_\text{Sch}] = \frac{1}{6}\mathop{\rm Tr}\nolimits\left(\left(c\partial c \frac{B}{1+K}\right)^3\right).
\end{equation}
This is the cubic term in the action evaluated on the simple tachyon vacuum, as can be seen by expressing it in the form
\begin{equation}
\Psi_\text{simp} = \frac{1}{\sqrt{1+K}}\left(c-c\partial c\frac{B}{1+K}\right)\sqrt{1+K}.
\end{equation}
Following the calculation of subsection \ref{subsec:simple}, we reproduce the expected result in accordance with Sen's conjecture.
\section{Questions}
We conclude with an (incomplete) list of questions for the future:
\begin{description}
\item{(1)} The simple intertwining solution very nearly provides a proof of background independence of open bosonic string field theory. Is it possible to remedy some of the limitations of this solution to give a more complete demonstration? Recent work in this direction appears in~\cite{KOSsingII}.
\item{(2)} Can we find analytic solutions for BPS D-brane configurations in open superstring field theory? Can we prove background independence for the open superstring?
\item{(3)} Are there topological invariants of the open string star algebra representing D-brane charges?
\item{(4)} Can we use analytic solutions to give new insight into the physics of specific backgrounds? For example, solutions representing D-branes with magnetic flux \cite{flux} could provide an interesting perspective on the connection between string theory and noncommutative geometry \cite{SeibergWitten}. Rolling tachyon solutions may provide insight into the nature of tachyon matter \cite{SenRolling,matter,LLM} and the appearance of closed strings around the tachyon vacuum \cite{imaginary}.
\item{(5)} Can Schnabl gauge or dressed Schnabl gauges provide a useful setting for computing amplitudes? Some work in this direction \cite{RZVeneziano,KZloop} indicates that such amplitudes are much simpler than in Siegel gauge, but there are subtleties related to degeneracies of the Schnabl gauge propagator \cite{KSZ}. A general understanding is still missing.
\item{(6)} Can we use analytic techniques to get a better understanding of closed strings and quantum corrections in open SFT?
\item{(7)} Can we apply analytic methods in simpler open string models, such as two dimensional string theory or topological string theory, where we can establish a connection to other nonperturbative approaches to string theory?
\item{(8)} There is an old idea that the off-shell configuration space of string theory should correspond to the space of two dimensional (nonconformal) field theories. In the open string version, this consists of the space of nonconformal boundary conditions for a given bulk CFT on the disk. This is the hypothetical configuration space behind the formulation of boundary string field theory \cite{WittenBSFT}. We now have a decent understanding of how classical solutions in open SFT correspond to boundary conformal field theories. Is there an interesting and useful sense that some or all off-shell string fields represent a bulk CFT subject to nonconformal boundary conditions?
\end{description}
\subsubsection*{Acknowledgements}
I would like to thank A. Sen for inviting me to give these lectures and providing ample slots to go through the material. I also thank C. Maccaferri for comments on the typesetted notes. This work is supported by ERDF and M\v{S}MT (Project CoGraDS -CZ.02.1.01/0.0/0.0/15\_ 003/0000437) and the GA{\v C}R project 18-07776S and RVO: 67985840.
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\section{Introduction}
\thispagestyle{empty}
\let\thefootnote\relax\footnotetext{This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 754475). The author also acknowledges the support of ISF grant 871/17.}
Let $\bar{X}_{d-1,d}$ be the space of rank-$\left(d-1\right)$ discrete
subgroups of $\mathbb{R}^{d}$ and let $X_{d-1,d}$ be the space of their
homothety classes, that is
\[
X_{d-1,d}\overset{\text{def}}{=}\bar{X}_{d-1,d}\diagup\sim,
\]
where,
\[
\text{\ensuremath{\Lambda}}\sim\alpha\cdot\Lambda,\ \alpha\in\mathbb{R}^{\times}.
\]
For $\Lambda\in\bar{X}_{d-1,d}$, we will denote by $\left[\Lambda\right]$
its image in $X_{d-1,d}$, namely
\[
\left[\Lambda\right]\overset{\text{def}}{=}\left\{ \alpha\cdot\Lambda\mid\alpha\in\mathbb{R}^{\times}\right\} .
\]
The\emph{ covolume} function, denoted by
\[
\text{cov}:\ \bar{X}_{d-1,d}\to\mathbb{R}_{+},
\]
is the function that assigns to each $\Lambda\in\bar{X}_{d-1,d}$
the $\left(d-1\right)$-volume of a fundamental parallelotope. It
is explicitly given by
\begin{equation}
\text{cov}(\Lambda)=\sqrt{\det b^{t}b},\ \Lambda\in X_{d-1,d},\label{eq:covol def}
\end{equation}
where $b\in M_{d\times d-1}(\mathbb{R})$ is a matrix whose columns form a
$\mathbb{Z}$-basis for $\Lambda$. Let $\primlatof{d-1}\subseteq\bar{X}_{d-1,d}$
be the set of all subgroups that arise as the intersection of $\mathbb{Z}^{d}$
with a rational hyperplane. Of particular interest to us are the subsets
of $X_{d-1,d}$ defined by
\[
\primlatof{d-1}(T)\overset{\text{def}}{=}\left\{ \left[\Lambda\right]\in X_{d-1,d}\mid\Lambda\in\primlatof{d-1},\ \text{cov}(\Lambda)=T\right\} ,\ T\in\mathbb{R}^{\times}.
\]
The sets $\primlatof{d-1}(T),$ $T\in\mathbb{R}^{\times}$ are finite (see
e.g. Lemma \ref{lem:bijection orthogonal vectors}) and we denote
by $\mu_{T},$ $T\in\mathbb{R}^{\times}$, the uniform probability counting
measures on them. In this note we prove that certain subsequences
of $\left\{ \mu_{T}\right\} $ converge to a probability measure which
we denote by $\mu_{\text{polar}}$. The measure $\mu_{\text{polar}}$
may be described through disintegration (see e.g. \cite{sergant_shapira})
as follows. Let $\text{Gr}_{d-1}(\mathbb{R}^{d})$ be the space of hyperplanes in
$\mathbb{R}^{d}$, and for $\mathcal{P}\in\text{Gr}_{d-1}(\mathbb{R}^{d})$ let
\begin{equation}
X_{\mathcal{P}}\overset{\text{def}}{=}\left\{ \left[\Lambda\right]\in X_{d-1,d}\mid\Lambda\subseteq\mathcal{P}\right\} .\label{eq:X_p}
\end{equation}
A choice of a linear isomorphism between $\mathcal{P}$ and $\mathbb{R}^{d-1}$
gives an identification of $X_{\mathcal{P}}$ with $X_{d-1}=\text{PGL}{}_{d-1}(\mathbb{R})\diagup\text{PGL}{}_{d-1}(\mathbb{Z})$.
Through this identification, the $\text{PGL}{}_{d-1}(\mathbb{R})$-invariant
probability measure $\mu_{X_{d-1}}$ may be pushed to a measure on
$X_{\mathcal{P}}.$ It turns out that the latter push-forward to $X_{\mathcal{P}}$
is independent of the chosen isomorphism (see e.g. part 3 of Lemma
\ref{lem:horizontal lattices identification with =00005BX_d-1=00005D}),
hence this process defines a measure $\mu_{\mathcal{P}}$ on $X_{\mathcal{P}}$.
Let $\mu_{\text{Gr}_{d-1}(\mathbb{R}^{d})}$ be the $\text{SO}{}_{d}(\mathbb{R})$-invariant
probability measure on $\text{Gr}_{d-1}(\mathbb{R}^{d})$. We define
\[
\mu_{\text{polar}}\overset{\text{def}}{=}\int_{\text{Gr}_{d-1}(\mathbb{R}^{d})}\mu_{\mathcal{P}}\ d\mu_{\text{Gr}_{d-1}(\mathbb{R}^{d})}(\mathcal{P}).
\]
For a prime $p$, let
\begin{equation}
\mathbb{D}(p)\overset{\text{def}}{=}\left\{ m\in\mathbb{N}\mid p\nmid m\right\} .\label{eq:D(P)}
\end{equation}
We prove
\begin{thm}
\label{thm:maintheorem} The convergence
\[
\mu_{T_{j}}\overset{\text{weak * }}{\longrightarrow}\mu_{\text{polar}},
\]
holds for:
\begin{enumerate}
\item $d=4$, and $\left\{ T_{j}^{2}\right\} \subseteq\mathbb{D}(p)/8\mathbb{N}$,
for any odd prime $p$.
\item $d=5$, and $\left\{ T_{j}^{2}\right\} \subseteq\mathbb{D}(p)$, for
any odd prime $p$.
\item $d>5$, and $\left\{ T_{j}^{2}\right\} \subseteq\mathbb{N}$.
\end{enumerate}
\end{thm}
\begin{rem*}
It should be possible to prove a version of Theorem \ref{thm:maintheorem}
for $d=3$ by relying on the work of \cite{AES3d}. Also, it seems
that the unnecessary congruence conditions in dimensions $4$ and
$5$ can be removed and an effective estimate on the convergence can
be obtained by exploiting a theorem of Einsiedler, R{\"u}hr and Wirth
found in \cite{Effective_aes_Ruhr}. In order to do so, one should
replace Theorem \ref{thm:AESgrids thm} (of \cite{AESgrids}) stated
in this note, with the corresponding theorems of \cite{AES3d} and
\cite{Effective_aes_Ruhr} and go along the lines of sections \ref{sec:The-p-adic-factory}
and \ref{sec:proof of the equidisitrbution}.
\end{rem*}
\subsection{Background for Theorem \ref{thm:maintheorem} }
W. Schmidt in \cite{Schmidt1998,Schmidt2015} computed the distribution
of the homothety classes the $\left(d-1\right)$-integral subgroups
through the filtration
\[
\left\{ \left[\Lambda\right]\in X_{d-1,d}\mid\Lambda\in\primlatof{d-1},\ \ \text{cov}(\Lambda)\leq T\right\} ,\ \text{as }T\to\infty.
\]
Hence, Theorem \ref{thm:maintheorem} should be viewed as a sparse
version of Schmidt's result. Later, Aka, Einsiedler and Shapira in
\cite{AESgrids,AES3d} computed the limiting distribution of the image
of the sets $\primlatof{d-1}(T)$ in the space $\text{Gr}_{d-1}(\mathbb{R}^{d})\times\text{O}_{d}(\mathbb{R})\backslash X_{d-1,d}$.
Since there is a natural surjection $\pi:X_{d-1,d}\to\text{Gr}_{d-1}(\mathbb{R}^{d})\times\text{O}_{d}(\mathbb{R})\backslash X_{d-1,d}$,
Theorem \ref{thm:maintheorem} implies the main result of \cite{AESgrids}.
We also note that the type of problem considered here may be viewed
as a natural generalization of the problem considered by Y. Linnik
regarding the equidistribution of the projection to the unit sphere
of primitive integer vectors on large spheres (see \cite{Lin68},
and also \cite{elenberg_michel_venaktesh} for a modern review).
\subsection{Organization of the note and proof ideas}
We provide a novel interpretation of $X_{d-1,d}$ as a double coset
space (see Proposition \ref{prop:Tthe polar coordinates}) which allows
to use the methods and results of \cite{AESgrids} in order to prove
Theorem \ref{thm:maintheorem}. In an overview, the method of \cite{AESgrids}
allows to interpret the sets $\primlatof{d-1}(T)$ as compact orbits
in an S-arithmetic space and to relate their natural measure to the
measures $\mu_{T}$. A key theorem of \cite{AESgrids} states that
those orbits equidistribute, which eventually allows to deduce Theorem
\ref{thm:maintheorem}. The organization of the note is as follows.
\begin{itemize}
\item In Section \ref{sec:Polar-coordinates-on} we describe $X_{d-1,d}$
as a coset space, and as a double coset space. We also discuss the
measure $\mu_{\text{polar }}$ in detail.
\item In Section \ref{sec:The-p-adic-factory} we discuss the method that
is used to ``generate'' elements of $\primlatof{d-1}$ by the p-adics.
\item In Section \ref{sec:proof of the equidisitrbution} we discuss the
resulting measures and conclude the proof.
\end{itemize}
\section{\label{sec:Polar-coordinates-on} $X_{d-1,d}$ as a Homogenous space
and its polar coordinates}
\subsection{The transitive action of $\text{SL}{}_{d}(\protect\mathbb{R})$}
The group $\text{SL}{}_{d}(\mathbb{R})$ acts from the left on $X_{d-1,d}$
by
\[
g\cdot[\Lambda]=[g\Lambda],\ g\in\text{SL}{}_{d}(\mathbb{R}),\ [\Lambda]\in X_{d-1,d},
\]
where
\[
g\Lambda=\left\{ gv\mid v\in\Lambda\right\} .
\]
Let $e_{i},$ $i\in\{1,..,d\}$ be the standard basis vectors of $\mathbb{R}^{d}$
and note that for $g\in\text{SL}{}_{d}(\mathbb{R})$, the set $\left\{ ge_{1},..,ge_{d-1}\right\} $
consists of the first $\left(d-1\right)$ columns of $g$. Since basis
vectors of any $\Lambda\in X_{d-1,d}$ can be put to be the first
$\left(d-1\right)$ columns of some $\text{SL}{}_{d}(\mathbb{R})$ matrix,
we deduce that the $\text{SL}{}_{d}(\mathbb{R})$ orbit of
\[
x_{0}\overset{\text{def}}{=}\left[\text{span }_{\mathbb{Z}}\{e_{1},..,e_{d-1}\}\right]
\]
equals to $X_{d-1,d}$. A computation shows that
\[
Q_{d-1,d}\overset{\text{def}}{=}\left\{ \left(\begin{array}{cc}
\lambda\gamma & *\\
0_{1\times d} & 1/\det(\lambda\gamma)
\end{array}\right)\mid\lambda\in\mathbb{R}^{\times},\gamma\in\text{GL}{}_{d-1}(\mathbb{Z})\right\}
\]
is the stabilizer of $x_{0}$. Therefore, we get the identification
\[
X_{d-1,d}=\text{SL}{}_{d}(\mathbb{R})\diagup Q_{d-1,d}.
\]
\begin{rem*}
In terms of the coset space, the collection $\left\{ \left[\Lambda\right]\in X_{d-1,d}\mid\Lambda\in\mathbb{Z}_{\text{prim}}^{d-1,d}\right\} $
is identified with the orbit $\text{SL}{}_{d}(\mathbb{Z})x_{0}$.
\end{rem*}
\subsubsection{The measure $\mu_{\text{polar }}$}
Let $P_{d-1,d}$ be the parabolic group,
\[
P_{d-1,d}\overset{\text{def}}{=}\left\{ \left(\begin{array}{cc}
m & *\\
0_{1\times d} & 1/\det m
\end{array}\right)\mid m\in\text{GL}{}_{d-1}(\mathbb{R})\right\} .
\]
\begin{lem}
\label{lem:horizontal lattices identification with =00005BX_d-1=00005D}
Let \emph{$g\in\text{SL}{}_{d}(\mathbb{R})$}, and let $\mathcal{P}=\text{span}_{\mathbb{R}}\{ge_{1},..,ge_{d-1}\}$.
Then:
\begin{enumerate}
\item \label{enu:lattices in hyperplane 1} The subset $X_{\mathcal{P}}$
\emph{(}see \eqref{eq:X_p}\emph{)} is identified with $gP_{d-1,d}\diagup Q_{d-1,d}$.
\item \label{enu:lattices in hyperplane 2}The map\emph{
\[
\varphi_{g}:\text{PGL}{}_{d-1}(\mathbb{R})\diagup\text{PGL}{}_{d-1}(\mathbb{Z})\to X_{\mathcal{P}},
\]
}which sends\emph{
\[
\mathbb{R}^{\times}m\cdot\text{PGL}{}_{d-1}(\mathbb{Z})\mapsto g\left(\begin{array}{cc}
m & 0_{d-1\times1}\\
0_{1\times d-1} & 1
\end{array}\right)Q_{d-1,d},
\]
}is a homeomorphism.
\item \label{enu:lattices in hyperplane 3}Assume that $gP_{d-1,d}=g'P_{d-1,d}$.
Then,
\begin{equation}
\left(\varphi_{g}\right)_{*}\mu_{X_{d-1}}=\left(\varphi_{g'}\right)_{*}\mu_{X_{d-1}},\label{eq:pushforwards equal}
\end{equation}
where $\mu_{X_{d-1}}$ is the \emph{$\text{PGL}{}_{d-1}(\mathbb{R})$}-invariant
probability measure on $X_{d-1}$.
\end{enumerate}
\end{lem}
\begin{proof}
We observe that
\[
\left\{ g\in\text{SL}_{d}(\mathbb{R})\mid\mathcal{P}=\text{span}_{\mathbb{R}}\{ge_{1},..,ge_{d-1}\}\right\} =gP_{d-1,d},
\]
hence \eqref{enu:lattices in hyperplane 1} follows. Next, in order
prove \eqref{enu:lattices in hyperplane 2}, we note that the map
\[
\phi_{1}:\text{PGL}{}_{d-1}(\mathbb{R})\diagup\text{PGL}{}_{d-1}(\mathbb{Z})\to P_{d-1,d}\diagup Q_{d-1,d},
\]
that sends,
\[
\mathbb{R}^{\times}m\cdot\text{PGL}{}_{d-1}(\mathbb{Z})\mapsto\left(\begin{array}{cc}
m & 0_{d-1\times1}\\
0_{1\times d-1} & 1/\det m
\end{array}\right)Q_{d-1,d},\ \ m\in\text{GL}{}_{d-1}(\mathbb{R}),
\]
is a homeomorphism, and the map
\[
\phi_{2}:P_{d-1,d}\diagup Q_{d-1,d}\to gP_{d-1,d}\diagup Q_{d-1,d},
\]
defined by multiplication from the left by $g,$ is also a homeomorphism.
Hence $\phi_{2}\circ\phi_{1}$ is a homeomorphism, which proves \eqref{enu:lattices in hyperplane 2}.
Finally, we prove \eqref{enu:lattices in hyperplane 3}. We let $g,g'\in\text{SL}{}_{d}(\mathbb{R})$
be such that
\[
g'=gp,
\]
for some $p\in P_{d-1,d}$, where $p=\left(\begin{array}{cc}
m_{p} & v\\
0_{1\times d} & 1/\det m_{p}
\end{array}\right)$. Then, a short calculation shows that
\begin{equation}
\varphi_{g'}\left(\mathbb{R}^{\times}m\text{PGL}{}_{d-1}(\mathbb{Z})\right)=\varphi_{g}\left(\mathbb{R}^{\times}m_{p}m\text{PGL}{}_{d-1}(\mathbb{Z})\right),\label{eq:ef_g and ef_g_tilde}
\end{equation}
hence, since $\mu_{X_{d-1}}$ is $\text{PGL}{}_{d-1}(\mathbb{R})$-invariant,
we obtain \eqref{eq:pushforwards equal}.
\end{proof}
\begin{rem*}
In the rest, we shall abuse notation and denote the measure $\left(\varphi_{id}\right)_{*}\mu_{X_{d-1}}$
on $P_{d-1,d}\diagup Q_{d-1,d}$ by $\mu_{X_{d-1}}$.
\end{rem*}
We denote
\[
K_{d-1,d}^{\pm}\overset{\text{def}}{=}P_{d-1,d}\cap\text{SO}{}_{d}(\mathbb{R}),
\]
which is isomorphic to $\text{O}_{d-1}(\mathbb{R}),$ and identify $Gr_{d-1}(\mathbb{R}^{d})$
with $K_{d-1,d}^{\pm}\diagdown\text{SO}{}_{d}(\mathbb{R})$ via the map
\[
K_{d-1,d}^{\pm}\rho\mapsto\text{span}_{\mathbb{R}}\{\rho^{-1}e_{1},..,\rho^{-1}e_{d-1}\}.
\]
\begin{rem*}
To ease the notation, we will omit in the following the indices $d-1,d$
from $P_{d-1,d}$, $Q_{d-1,d}$ and $K_{d-1,d}^{\pm}$.
\end{rem*}
Let $\mu_{Gr_{d-1}(\mathbb{R}^{d})}$ be the $\text{SO}{}_{d}(\mathbb{R})$-invariant
probability measure and for $f\in C_{c}(X_{d-1,d})$ we define $\hat{f}\in C_{c}(Gr_{d-1}(\mathbb{R}^{d}))$
by
\[
\hat{f}(K^{\pm}\rho)\overset{\text{def}}{=}\left(\varphi_{\rho^{-1}}\right)_{*}\mu_{X_{d-1}}(f),
\]
which is well defined by part 3 of Lemma \ref{lem:horizontal lattices identification with =00005BX_d-1=00005D}.
Then, the measure $\mu_{\text{polar }}$ is given by
\begin{equation}
\mu_{\text{polar }}(f)\overset{\text{def}}{=}\int_{\text{Gr}_{d-1}(\mathbb{R}^{d})}\hat{f}(K^{\pm}\rho)\ d\mu_{Gr_{d-1}(\mathbb{R}^{d})}(K^{\pm}\rho).\label{eq:mu polar section 2}
\end{equation}
Note that the description \eqref{eq:mu polar section 2} of $\mu_{\text{polar }}$
yields the same measure defined in the introduction, although stated
slightly differently. We chose this description since it suits well
to the proof of Lemma \ref{lem:mult push nu polar to mu polar}.
\subsection{Alternative description of $X_{d-1,d}$ via polar coordinates}
Here we shall give a description of the elements of $X_{d-1,d}$ by
their orientation and by their shape, hence the name of \emph{polar
coordinates}. Those coordinates will serve as a bootstrap to the technique
of \cite{AESgrids}.
\subsubsection{The multiplication map}
Let $\Delta K^{\pm}$ be the diagonal embedding in $\text{SO}{}_{d}(\mathbb{R})\times P$,
which is defined by
\[
\Delta K^{\pm}\overset{\text{def}}{=}\left\{ (k,k)\mid k\in K_{d-1,d}^{\pm}\right\} \leq\text{SO}{}_{d}(\mathbb{R})\times P.
\]
The following double coset space,
\[
X_{d-1,d}^{\text{polar}}\overset{\text{def}}{=}\Delta K^{\pm}\diagdown\left(\text{SO}{}_{d}(\mathbb{R})\times P\diagup Q\right),
\]
will be shown to be homeomorphic to $X_{d-1,d}$. Consider the map
\[
\mathcal{M}\ :\ X_{d-1,d}^{\text{polar}}\to X_{d-1,d},
\]
defined by
\[
\mathcal{M}\left(\Delta K^{\pm}(\rho,\eta Q)\right)=\rho^{-1}\eta Q.
\]
It is well defined since if $(\rho',\eta'Q)=(k\rho,k\eta Q),$ then
$\rho'^{-1}\eta'Q=\rho^{-1}\eta Q$.
\begin{prop}
\label{prop:Tthe polar coordinates}The map $\mathcal{M}$ is a homeomorphism.
\end{prop}
\begin{proof}
To prove injectivity, we assume that
\[
\mathcal{M}\left(\Delta K^{\pm}(\rho_{1},\eta_{1}Q)\right)=\mathcal{M}\left(\Delta K^{\pm}(\rho_{2},\eta_{2}Q)\right),
\]
which is equivalent to that
\[
\rho_{1}^{-1}\eta_{1}q=\rho_{2}^{-1}\eta_{2},
\]
for some $q\in Q$. Then,
\[
SO_{d}(\mathbb{R})\ni\rho_{2}\rho_{1}^{-1}=\eta_{2}q^{-1}\eta_{1}^{-1}\in P,
\]
hence there is a $k\in K^{\pm}$ such that
\[
\rho_{2}\rho_{1}^{-1}=\eta_{2}q^{-1}\eta_{1}^{-1}=k,
\]
which in turn implies
\[
\Delta K^{\pm}(\rho_{1},\eta_{1}Q)=\Delta K^{\pm}(\rho_{2},\eta_{2}Q).
\]
To prove continuity of $\mathcal{M}$, we consider the following commuting
diagram
\[
\xymatrix{\text{SO}{}_{d}(\mathbb{R})\times P\ar[d]\ar[r] & \text{SL}{}_{d}(\mathbb{R})\ar[d]\\
X_{d-1,d}^{\text{polar}}\ar[r]_{\ \ \mathcal{M}} & X_{d-1,d}
}
\]
where the vertical maps are the natural projections, and the horizontal
upper arrow sends
\[
(\rho,\eta)\mapsto\rho^{-1}\eta.
\]
Note that the resulting map from $\text{SO}{}_{d}(\mathbb{R})\times P$ to
$X_{d-1,d}$ is a composition of continuous maps, hence is continuous.
Therefore, by the universal property of the quotient space, $\mathcal{M}$
is continuous. Next, we compute the inverse of $\mathcal{M}$ and show it
is continuous. Let $A\leq\text{SL}{}_{d}(\mathbb{R})$ be the diagonal subgroup
with positive entries, and $N\leq\text{SL}{}_{d}(\mathbb{R})$ be the group
of upper triangular unipotent matrices. The map (Iwasawa decomposition)
\[
\psi:\text{SO}{}_{d}(\mathbb{R})\times A\times N\to\text{SL}{}_{d}(\mathbb{R}),
\]
given by
\[
\psi(\rho,a,n)=\rho an,
\]
is a homeomorphism (see e.g. \cite{Bekka_mayer}, Chapter 5). Consider
the following commuting diagram,
\[
\xymatrix{\text{SO}{}_{d}(\mathbb{R})\times P\ar[d] & \text{SO}{}_{d}(\mathbb{R})\times A\times N\ar_{p\ }[l] & \text{SL}{}_{d}(\mathbb{R})\ar[l]_{\ \ \ \ \ \ \ \ \ \psi^{-1}}\ar[d]\\
X_{d-1,d}^{\text{polar}} & & X_{d-1,d}\ar[ll]
}
\]
where the map $p$ is defined by
\[
p(\rho,a,n)\overset{\text{def}}{=}\left(\rho^{-1},an\right),
\]
and the horizontal maps are the natural projections. The map corresponding
to the lower horizontal arrow sends
\[
\rho anQ\mapsto\Delta K^{\pm}\left(\rho^{-1},anQ\right),
\]
which is clearly an inverse for $\mathcal{M}$. Since the resulting map
from $\text{SL}_{d}(\mathbb{R})$ to $X_{d-1,d}^{\text{polar}}$ is a composition
of continuous maps, we get that it is continuous, hence by the universal
property of the quotient space, $\mathcal{M}^{-1}$ is continuous.
\end{proof}
\subsubsection{The measure $\mu_{\text{polar }}$ through the polar coordinates}
Consider the map
\[
q_{\Delta K^{\pm}}:\text{SO}{}_{d}(\mathbb{R})\times P\diagup Q\to X_{d-1,d}^{\text{polar}},
\]
that divides from the left by $\Delta K^{\pm}$. We define
\begin{equation}
\nu_{\text{polar}}\overset{\text{def}}{=}\left(q_{\Delta K^{\pm}}\right)_{*}\mu_{\text{SO}{}_{d}(\mathbb{R})}\otimes\mu_{X_{d-1}}.\label{eq:nu_polar definition}
\end{equation}
\begin{lem}
\label{lem:mult push nu polar to mu polar}It holds that $\mathcal{M}_{*}\nu_{\text{polar}}=\mu_{\text{polar}}$.
\end{lem}
\begin{proof}
First, recall that for $\varphi\in C_{c}(\text{SO}{}_{d}(\mathbb{R})$) ,
\[
\int_{\text{SO}{}_{d}(\mathbb{R})}\varphi\ d\mu_{\text{SO}{}_{d}(\mathbb{R})}=\int_{\text{Gr}_{d-1}(\mathbb{R}^{d})}\left(\int_{K^{\pm}}\varphi(k\rho)\ d\mu_{K^{\pm}}(k)\right)d\mu_{\text{Gr}_{d-1}(\mathbb{R}^{d})}(K^{\pm}\rho).
\]
Hence for $f\in C_{c}(X_{d-1,d}^{\text{polar}})$,
\[
\nu_{\text{polar}}(f)=\int f(q_{\Delta K^{\pm}}(\rho,\eta Q))d\mu_{X_{d-1}}(\eta Q)d\mu_{\text{SO}{}_{d}(\mathbb{R})}(\rho)=
\]
\begin{equation}
\int\left(\int f(q_{\Delta K^{\pm}}(k\rho,\eta Q))d\mu_{X_{d-1}}(\eta Q)d\mu_{K^{\pm}}(k)\right)d\mu_{\text{Gr}_{d-1}(\mathbb{R}^{d})}(K^{\pm}\rho).\label{eq:integral x_d-1,K_Gr_d-1}
\end{equation}
Note that
\[
q_{\Delta K^{\pm}}(k\rho,\eta Q)=q_{\Delta K^{\pm}}(\rho,k^{-1}\eta Q),
\]
whence, by \eqref{eq:integral x_d-1,K_Gr_d-1},
\[
\nu_{\text{polar}}(f)=\int\left(\int f(q_{\Delta K^{\pm}}(\rho,k^{-1}\eta Q))d\mu_{X_{d-1}}(\eta Q)\right)d\mu_{K^{\pm}}(k)d\mu_{\text{Gr}_{d-1}(\mathbb{R}^{d})}(K^{\pm}\rho).
\]
The measure $\mu_{X_{d-1}}$ is $K^{\pm}$ invariant, so that
\[
\nu_{\text{polar}}(f)=\int\left(\int f(q_{\Delta K^{\pm}}(\rho,\eta Q))d\mu_{X_{d-1}}(\eta Q)\right)d\mu_{\text{Gr}_{d-1}(\mathbb{R}^{d})}(K^{\pm}\rho).
\]
Finally, the push-forward by $\mathcal{M}$ gives
\[
\mathcal{M}_{*}\nu_{\text{polar}}(f)=\int\left(\int f(\rho^{-1}\eta Q)d\mu_{X_{d-1}}(\eta Q)\right)d\mu_{\text{Gr}_{d-1}(\mathbb{R}^{d})}(K^{\pm}\rho),
\]
where
\[
\int f(\rho^{-1}\eta Q)d\mu_{X_{d-1}}(\eta Q)=\left(\varphi_{\rho^{-1}}\right)_{*}\mu_{X_{d-1}}(f).
\]
In view of the definition \eqref{eq:mu polar section 2}, the proof
is now done.
\end{proof}
\begin{rem*}
The whole discussion of this section can be adjusted with no trouble
to the spaces $X_{k,d}$ of homothety classes of of rank-$k$ discrete
subgroups of $\mathbb{R}^{d}$, for $1\leq k<d$.
\end{rem*}
\section{\label{sec:The-p-adic-factory}The p-adic factory of primitive integral
subgroups }
\subsection{\label{subsec:The-mechanism}The mechanism}
In order to better connect our discussion to the one of \cite{AESgrids},
we shall recall the description of the elements $\Lambda\in\primlatof{d-1}$
with fixed covolume as orthogonal lattices to integer vectors of fixed
norm. Let $\mathbb{Z}_{\text{prim}}^{d}$ be the set of integral primitive
vectors. For a primitive integer vector $v\in\mathbb{Z}_{\text{prim }}^{d}$,
let $v^{\perp}\in Gr_{d-1}(\mathbb{Q}^{d})$ be the orthogonal hyperplane
to $v$. We define the \emph{orthogonal lattice }to $v$ by
\[
\Lambda_{v}\overset{\text{def}}{=}v^{\perp}\cap\mathbb{Z}^{d}.
\]
Note that the map that sends $\mathbb{Z}_{\text{prim }}^{d}\ni v\mapsto\Lambda_{v}$,
is onto $\primlatof{d-1}$. In addition, let
\[
\mathbb{S}^{d-1}(T)\overset{\text{def}}{=}\left\{ v\in\mathbb{Z}_{\text{prim }}^{d}\mid\left\Vert v\right\Vert =T\right\} .
\]
Then, we have the following bijection.
\begin{lem}
\label{lem:bijection orthogonal vectors}The map
\[
\Lambda_{*}:\mathbb{S}^{d-1}(T)\to\primlatof{d-1}(T)
\]
which sends $v\mapsto\Lambda_{v}$ is a bijection.
\end{lem}
\begin{proof}
See \cite{AESgrids}, introduction.
\end{proof}
Let $v\in\mathbb{Z}_{\text{prim}}^{d}$ and let $H_{v}\leq\text{SO}_{d}$
be the subgroup stabilizing $v$. We define by $g_{v}\in\text{SL}{}_{d}(\mathbb{Z})$
to be a matrix who's first $d-1$ columns form a positively oriented
basis for $\Lambda_{v}$. Note that $H_{v}$ and $g_{v}^{-1}H_{v}g_{v}$
are both linear algebraic groups defined over $\mathbb{Q}$, and observe that
$g_{v}^{-1}H_{v}g_{v}\leq\text{ASL}{}_{d-1}$, where
\[
\text{ASL}_{d-1}=\left\{ \left(\begin{array}{cc}
g & *\\
0_{1\times d-1} & 1
\end{array}\right)\mid g\in\text{SL}{}_{d-1}\right\} .
\]
For what follows, we denote by $\mathbb{Q}_{p}$ the field of p-adic numbers
and by $\mathbb{Z}_{p}$ the ring of p-adic integers. Now, recall that $\text{ASL}_{d-1}(\mathbb{Q}_{p})=\text{ASL}_{d-1}(\mathbb{Z}_{p})\text{ASL}_{d-1}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)$
(see \cite{AESgrids} Section 6.3) and assume that $h\in H_{v}(\mathbb{Q}_{p})\cap\text{SO}{}_{d}(\mathbb{Z}_{p})\text{SO}{}_{d}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)$.
Then, we may write
\begin{equation}
h=c_{1}\gamma_{1},\ \ c_{1}\in\text{SO}{}_{d}(\mathbb{Z}_{p}),\ \gamma_{1}\in\text{SO}_{d}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right),\label{eq:rotation decomposition}
\end{equation}
and
\begin{equation}
g_{v}^{-1}hg_{v}=c_{2}\gamma_{2}^{-1},\ c_{2}\in\text{ASL}_{d-1}(\mathbb{Z}_{p}),\ \gamma_{2}\in\text{ASL}_{d-1}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right).\label{eq:lattice decompostion}
\end{equation}
The following lemma, and its corollary, show the principle which is
used to generate elements $\Lambda\in\primlatof{d-1}$ of a fixed
covolume.
\begin{lem}
\label{lem:genrated vectors and sl_d(Z) matrices}It holds \emph{$\gamma_{1}g_{v}\gamma_{2}\in\text{SL}_{d}(\mathbb{Z})$}.
\end{lem}
\begin{proof}
We observe from \eqref{eq:rotation decomposition} and \eqref{eq:lattice decompostion}
that
\[
\text{SL}_{d}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)\ni\gamma_{1}g_{v}\gamma_{2}=c_{1}g_{v}c_{2}^{-1}\in\text{SL}_{d}(\mathbb{Z}_{p}),
\]
Since $\mathbb{Z}_{p}\cap\mathbb{Z}\left[\frac{1}{p}\right]=\mathbb{Z}$, the statement follows.
\end{proof}
\begin{rem*}
Although not explicitly stated in \cite{AESgrids}, the proof of Lemma
\ref{lem:genrated vectors and sl_d(Z) matrices} can be readily deduced
from the proof of Proposition 6.2 of \cite{AESgrids}.
\end{rem*}
\begin{cor}
\label{cor:generated orthogonal lattices} Let $\Lambda$ be the $\mathbb{Z}$-span
of the first $\left(d-1\right)$ columns of $\gamma_{1}g_{v}\gamma_{2}$.
It holds that $\gamma_{1}v\in\mathbb{Z}_{\text{prim}}^{d}$ and $\Lambda=\Lambda_{\gamma_{1}v}$.
Importantly, \emph{
\[
\text{cov}(\Lambda)=\text{cov}(\Lambda_{v}).
\]
}
\end{cor}
\begin{proof}
Since $\gamma_{1}g_{v}\gamma_{2}\in\text{SL}_{d}(\mathbb{Z})$, the basis
of $\Lambda$ can be completed to a basis of $\mathbb{Z}^{d}$ which implies
that $\Lambda\in\primlatof{d-1}$. Next, a computation that uses \eqref{eq:covol def}
shows
\begin{equation}
\text{cov}(\Lambda)=\text{cov}(\Lambda_{v}).\label{eq:cov ofz-span of d-1 coloms of gamma_1g_vgamma_2 equal to lambda_v}
\end{equation}
Now observe that $\Lambda\subseteq\gamma_{1}v^{\perp}$. Hence by
Lemma \ref{lem:bijection orthogonal vectors} and \eqref{eq:cov ofz-span of d-1 coloms of gamma_1g_vgamma_2 equal to lambda_v}
we deduce $\gamma_{1}v\in\mathbb{Z}_{\text{prim}}^{d}$ and $\Lambda=\Lambda_{\gamma_{1}v}$.
\end{proof}
\subsection{The S-arithmetic orbits and their projection to the reals}
To ease the notation, we introduce
\[
\mathbb{G}_{1}\overset{\text{def}}{=}\text{SO}{}_{d},\ \mathbb{G}_{2}\overset{\text{def}}{=}\text{ASL}{}_{d-1},\ \mathbb{G}\overset{\text{def}}{=}\mathbb{G}_{1}\times\mathbb{G}_{2}.
\]
For an odd prime $p$ let
\[
\mathcal{Y}_{p}\overset{\text{def}}{=}\mathbb{G}(\mathbb{R}\times\mathbb{Q}_{p})/\mathbb{G}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right),
\]
where by $\mathbb{G}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)$ we mean the
diagonal embedding of each $\mathbb{G}_{i}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)$
factor into $\mathbb{G}_{i}(\mathbb{R}\times\mathbb{Q}_{p}).$ Consider the set
\[
\mathcal{U}\overset{\text{def}}{=}\mathbb{G}(\mathbb{R}\times\mathbb{Z}_{p})\mathbb{G}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)\subseteq\mathcal{Y}_{p}.
\]
We now recall the (well known) construction of the projection to the
real coordinate. If
\[
\left((g_{1,\infty},g_{1,p}),(g_{2,\infty},g_{2,p})\right)\mathbb{G}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)\in\mathcal{U},
\]
then we may write for $i\in\{1,2\}$,
\[
g_{i,p}=c_{i,p}\gamma_{i,p},\ \ c_{i,p}\in\mathbb{G}_{i}\left(\mathbb{Z}_{p}\right),\ \gamma_{i,p}\in\mathbb{G}_{i}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right).
\]
Then, the map $q_{\infty}:\mathcal{U}\to\mathbb{G}(\mathbb{R})/\mathbb{G}(\mathbb{Z})$ is defined by
\begin{equation}
q_{\infty}\left(\left((g_{1,\infty},g_{1,p}),\ (g_{2,\infty},g_{2,p})\right)\mathbb{G}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)\right)\overset{\text{def}}{=}(g_{1,\infty}\gamma_{1,p}^{-1},\ g_{2,\infty}\gamma_{2,p}^{-1})\mathbb{G}\left(\mathbb{Z}\right).\label{eq:q_infty def}
\end{equation}
\subsubsection{The S-arithmetic orbit and its decomposition}
Let $v\in\mathcal{\Z}_{\text{prim}}^{d}$ and let $g_{v}$ be as defined in Section \ref{subsec:The-mechanism}.
We define the following diagonal embedding of $H_{v}$,
\[
L_{v}\overset{\text{def}}{=}\left\{ \left(h,g_{v}^{-1}hg_{v}\right)\mid h\in H_{v}\right\} \leq\mathbb{G}.
\]
We choose some $k_{v}\in\text{SO}{}_{d}(\mathbb{R})$ such that
\[
k_{v}v=e_{d},
\]
and we denote by $a_{v}$ the diagonal matrix with entries $(\left\Vert v\right\Vert ^{-1/\left(d-1\right)},..,\left\Vert v\right\Vert ^{-1/\left(d-1\right)},\left\Vert v\right\Vert )$.
This choices imply that $a_{v}k_{v}g_{v}\in\text{ASL}_{d-1}(\mathbb{R})$.
The following orbit is of main importance,
\begin{equation}
O_{v,p}\overset{\text{def}}{=}\left((k_{v},e_{p}),\ (a_{v}k_{v}g_{v},e_{p})\right)\cdot L_{v}(\mathbb{R}\times\mathbb{Q}_{p})\mathbb{G}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right).\label{eq:the s-arithemtic orbit}
\end{equation}
We consider the following decomposition of $H_{v}(\mathbb{Q}_{p})$ into double
cosets
\begin{equation}
H_{v}(\mathbb{Q}_{p})=\bigsqcup_{h\in M}H_{v}\left(\mathbb{Z}_{p}\right)hH_{v}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right),\label{eq:decomposition of SO_V_perp}
\end{equation}
where $M$ is a set of representatives of the double coset space.
We note that the collection of representatives is finite (see \cite{AESgrids},
section 6.2). We denote
\[
K\overset{\text{def}}{=}H_{e_{d}}(\mathbb{R})\cong\text{SO}{}_{d-1}(\mathbb{R}),
\]
and
\[
\Delta K(\mathbb{R})\times L_{v}(\mathbb{Z}_{p})\overset{\text{def}}{=}\left\{ \left((k,h),(k,g_{v}^{-1}hg_{v})\right)\mid k\in K,\ h\in H_{v}(\mathbb{Z}_{p})\right\} .
\]
\begin{lem}
\label{lem:decomposition of o_v_p}It holds that
\begin{equation}
O_{v,p}=\bigsqcup_{h\in M}O_{v,p,h},\label{eq:decomposition _v,p}
\end{equation}
where
\[
O_{v,p,h}=\left(\Delta K\times L_{v}(\mathbb{Z}_{p})\right)\left((k_{v},h),\ (a_{v}k_{v}g_{v},g_{v}^{-1}hg_{v})\right)\mathbb{G}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right).
\]
\end{lem}
\begin{proof}
This follows from a simple computation which uses \eqref{eq:decomposition of SO_V_perp},
and the observation that
\[
k_{v}H_{v}(\mathbb{R})k_{v}^{-1}=K.
\]
\end{proof}
Let $q_{1}:\mathcal{U}\to\mathbb{G}_{1}(\mathbb{Q}_{p})$ be the projection to the p-adic coordinate
of $\mathbb{G}_{1}(\mathbb{R}\times\mathbb{Q}_{p})$, and define
\[
M_{0}\overset{\text{def}}{=}\left\{ h\in M\mid h\in q_{1}(\mathcal{U})\right\} .
\]
We observe that $L_{v}(\mathbb{Z}_{p})(h,g_{v}^{-1}hg_{v})\mathbb{G}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)$
is either contained in $\mathcal{U}$ or disjoint from it. In particular
\begin{equation}
L_{v}(\mathbb{Z}_{p})(h,g_{v}^{-1}hg_{v})\mathbb{G}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)\subseteq\mathcal{U}\iff h\in M_{0}.\label{eq:L_v contained if}
\end{equation}
\begin{cor}
It holds that
\begin{equation}
O_{v,p}\cap\mathcal{U}=\bigsqcup_{h\in M_{0}}O_{v,p,h}.\label{eq:decomposition of O_v,p cap U}
\end{equation}
\end{cor}
\begin{proof}
This follows from the definition of $\mathcal{U}$, decomposition \eqref{eq:decomposition _v,p}
and observation \eqref{eq:L_v contained if}.
\end{proof}
This allows for the following nice description of $q_{\infty}(\mathcal{U}\cap O_{v,p})$.
\begin{prop}
\label{prop:properties of p-adic projection}For $h\in M_{0}$, choose
$\gamma_{i}(h)\in\mathbb{G}_{i}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)$,
$i\in\left\{ 1,2\right\} $, by \eqref{eq:rotation decomposition}
and \eqref{eq:lattice decompostion}. The following holds:
\begin{enumerate}
\item For $h\in M_{0}$,
\begin{equation}
q_{\infty}\left(O_{v,p,h}\right)=\Delta K\left(k_{v}\gamma_{1}^{-1}(h),a_{v}k_{v}g_{v}\gamma_{2}(h)\right)\mathbb{G}(\mathbb{Z}).\label{eq:Image of a fiber of an orbit to the real place}
\end{equation}
\item If $h,h'\in M_{0}$ and $h\neq h^{'}$, then
\begin{equation}
Kk_{v}\gamma_{1}^{-1}(h)\mathbb{G}_{1}(\mathbb{Z})\cap Kk_{v}\gamma_{1}^{-1}(h')\mathbb{G}_{1}(\mathbb{Z})=\emptyset,\label{eq:K cosets empty intersection}
\end{equation}
in particular
\begin{equation}
q_{\infty}\left(O_{v,p,h}\right)\cap q_{\infty}\left(O_{v,p,h'}\right)=\emptyset.\label{eq:q_infty projection empty intersection}
\end{equation}
\item For $h\in M_{0}$,
\[
q_{\infty}^{-1}\left(\Delta K\left(k_{v}\gamma_{1}^{-1}(h),a_{v}k_{v}g_{v}\gamma_{2}(h)\right)\mathbb{G}(\mathbb{Z})\right)\bigcap O_{v,p}=O_{v,p,h}.
\]
\end{enumerate}
\end{prop}
\begin{proof}
For $h\in M_{0}$, we write
\begin{equation}
h=c_{1}(h)\gamma_{1}(h),\ \ g_{v}^{-1}hg_{v}=c_{2}(h)\gamma_{2}^{-1}(h),\label{eq:decomposition of h}
\end{equation}
where $(c_{1}(h),c_{2}(h))\in\mathbb{G}(\mathbb{Z}_{p})$ and $(\gamma_{1}(h),\gamma_{2}(h))\in\mathbb{G}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right).$
\begin{enumerate}
\item Since $\left((\gamma_{1}^{-1}(h),\gamma_{1}^{-1}(h))\ ,(\gamma_{2}(h),\gamma_{2}(h))\right)\in\mathbb{G}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)$,
we see that
\begin{align*}
\Delta K\times L_{v}(\mathbb{Z}_{p})\cdot\left((k_{v},h)\ ,(a_{v}k_{v}g_{v},g_{v}^{-1}hg_{v})\right)\mathbb{G}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right) & =
\end{align*}
\[
=\Delta K\times L_{v}(\mathbb{Z}_{p})\left((k_{v}\gamma_{1}^{-1}(h),c_{1}(h)),\ (a_{v}k_{v}g_{v}\gamma_{2}(h),c_{2}(h))\right)\mathbb{G}\left(\mathbb{Z}\left[\frac{1}{p}\right]\right).
\]
Hence by definition \eqref{eq:q_infty def},
\[
q_{\infty}\left(O_{v,p,h}\right)=\Delta K\left(k_{v}\gamma_{1}^{-1}(h),a_{v}k_{v}g_{v}\gamma_{2}(h)\right)\mathbb{G}(\mathbb{Z}).
\]
\item The proof of \eqref{eq:K cosets empty intersection} is a routine
check, hence we omit its details and leave them for the reader (one
may also look at the proof of Proposition 6.2 in \cite{AESgrids}).
Note that \eqref{eq:q_infty projection empty intersection} follows
from \eqref{eq:K cosets empty intersection}.
\item This fact follows immediately from the two last ones and \eqref{eq:decomposition of O_v,p cap U}.
\end{enumerate}
\end{proof}
\subsection{\label{subsec:Equivalence-class-of-lattices}The resulting elements
of $\protect\primlatof{d-1}$}
The following commuting diagram will be important for us,
\begin{equation}
\xymatrix{\mathcal{U}\ar^{q_{\infty\ \ \ \ \ \ \ }}[r] & \mathbb{G}(\mathbb{R})\diagup\mathbb{G}(\mathbb{Z})\ar[r]_{q_{\Delta K\ \ \ \ \ }} & \Delta K\diagdown\mathbb{G}(\mathbb{R})\diagup\mathbb{G}(\mathbb{Z})\\
& \mathbb{G}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})\ar_{id\times q_{P\diagup Q}^{\mathbb{G}_{2}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})}}[d]\ar[u]^{\pi_{\mathbb{G}_{1}(\mathbb{Z})}}\ar[r]_{\tilde{q}_{\Delta K}\ \ \ \ \ } & \Delta K\diagdown\mathbb{G}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})\ar_{\tilde{q}}[d]\ar[u]^{\tilde{\pi}_{\mathbb{G}_{1}(\mathbb{Z})}}\ar[dr]^{\tilde{\mathcal{M}}}\\
& \mathbb{G}_{1}(\mathbb{R})\times P\diagup Q\ar[r]_{q_{\Delta K^{\pm}}} & X_{d-1,d}^{\text{polar}}\ar[r]_{\mathcal{M}} & X_{d-1,d}
}
\label{eq:main diagram}
\end{equation}
The maps $\pi_{\mathbb{G}_{1}(\mathbb{Z})}$ and $\tilde{\pi}_{\mathbb{G}_{1}(\mathbb{Z})}$ are obtained
by dividing from the right by $\mathbb{G}_{1}(\mathbb{Z})$. The maps $q_{\Delta K}$,
$\tilde{q}_{\Delta K}$ and $q_{\Delta K^{\pm}}$ are obtained by
dividing from the left by $\Delta K$ and $\Delta K^{\pm}$ correspondingly.
The map
\[
q_{P\diagup Q}^{\mathbb{G}_{2}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})}:\mathbb{G}_{2}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})\to P\diagup Q,
\]
is naturally defined by $\pi_{2}\left(g\mathbb{G}_{2}(\mathbb{Z})\right)=gQ$, since
$\mathbb{G}_{2}(\mathbb{Z})\leq Q$. The maps $\tilde{q}$ and $\tilde{\mathcal{M}}$ are
defined so that the diagrams commute. Now, denote
\begin{equation}
\tilde{R}_{v}\overset{\text{def}}{=}\tilde{q}_{\Delta K}\left(\pi_{\mathbb{G}_{1}(\mathbb{Z})}^{-1}\left(q_{\infty}(\mathcal{U}\cap O_{v,p})\right)\right),\label{eq:R^tilde_v}
\end{equation}
then, by Proposition \ref{prop:properties of p-adic projection},
we get that $\tilde{R}_{v}$ is the finite collection of points
\[
\tilde{O}_{v,h,\gamma}\overset{\text{def}}{=}\Delta K\left(k_{v}\gamma_{1}^{-1}(h)\gamma,a_{v}k_{v}g_{v}\gamma_{2}(h)\mathbb{G}_{2}(\mathbb{Z})\right),\ \ \gamma\in\mathbb{G}_{1}(\mathbb{Z}),\ h\in M_{0},
\]
where $\gamma_{i}(h)$, $i\in\left\{ 1,2\right\} $, are defined in
\eqref{eq:decomposition of h}. Denote
\[
\mathcal{L}(v)\overset{\text{def}}{=}\tilde{\mathcal{M}}(\tilde{R}_{v})\subseteq X_{d-1,d}.
\]
\begin{lem}
\label{lem:elements of L_v}It holds that
\[
\mathcal{L}(v)=\left\{ \left[\Lambda(v,h,\gamma)\right]\overset{\text{def}}{=}\gamma^{-1}\gamma_{1}(h)g_{v}\gamma_{2}(h)Q\mid h\in M_{0},\ \gamma\in\mathbb{G}_{1}(\mathbb{Z})\right\} .
\]
Importantly $\left[\Lambda(v,h,\gamma)\right]=\left[\Lambda_{\gamma^{-1}\gamma_{1}(h)v}\right]$
and as a consequence $\mathcal{L}(v)\subseteq\primlatof{d-1}(\left\Vert v\right\Vert )$.
\end{lem}
\begin{proof}
We have
\[
\tilde{\mathcal{M}}\left(\tilde{O}_{v,h,\gamma}\right)=\gamma^{-1}\gamma_{1}(h)k_{v}^{-1}\left(a_{v}k_{v}g_{v}\gamma_{2}(h)Q\right),
\]
and we note that $a_{v}k_{v}g_{v}\gamma_{2}(h)Q=k_{v}g_{v}\gamma_{2}(h)Q$,
which gives
\[
\tilde{\mathcal{M}}\left(\tilde{O}_{v,h,\gamma}\right)=\gamma^{-1}\gamma_{1}(h)\left(k_{v}^{-1}k_{v}\right)g_{v}\gamma_{2}(h)Q=\left[\Lambda(v,h,\gamma)\right].
\]
By Corollary \ref{cor:generated orthogonal lattices},
\begin{equation}
\left[\Lambda(v,h,\gamma)\right]=\left[\Lambda_{\gamma^{-1}\gamma_{1}(h)v}\right],\label{eq:the resulting primitive lattice as orthogonal latice}
\end{equation}
and also $\mathcal{L}(v)\subseteq\primlatof{d-1}(\left\Vert v\right\Vert )$.
\end{proof}
\subsubsection{Refinement of Theorem \ref{thm:maintheorem}}
For everything that follows we fix a prime $p\neq2$ and assume that
$d\geq4$ is a natural number.
\begin{defn}
We shall say that $v\in\mathbb{Z}_{\text{prim}}^{d}$ is admissible, if either
of the following holds
\begin{enumerate}
\item $d=4$, and $\left\Vert v\right\Vert ^{2}\subseteq\mathbb{D}(p)/8\mathbb{N}$.
\item $d=5$, and $\left\Vert v\right\Vert ^{2}\subseteq\mathbb{D}(p)$.
\item $d>5$, and $v$ is any primitive vector.
\end{enumerate}
\end{defn}
In section \ref{sec:proof of the equidisitrbution} we shall conclude
the following theorem.
\begin{thm}
\label{thm:refinement of main theorem} Let $\left\{ v_{i}\right\} _{i=1}^{\infty}$
be a sequence of admissible vectors such that
\[
\left\Vert v_{i}\right\Vert \to\infty,
\]
and let $\mu_{v_{i}}$ be the uniform counting measures supported
on $\mathcal{L}(v_{i})$. Then \emph{
\[
\mu_{v_{i}}\overset{\text{weak *}}{\longrightarrow}\mu_{\text{polar}}.
\]
}
\end{thm}
Theorem \ref{thm:refinement of main theorem} implies Theorem \ref{thm:maintheorem}
by the following. In \cite{AESgrids} (see Section 5.1 and Proposition
6.2 in \cite{AESgrids}) there was introduced an equivalence relation
on the primitive vectors lying on spheres. It was shown that the equivalence
class of $v\in\mathbb{Z}_{\text{prim}}^{d}$ is exactly
\[
\left\{ \gamma^{-1}\gamma_{1}(h)v\right\} _{\gamma\in\mathbb{G}_{1}(\mathbb{Z}),\ h\in M_{0}}.
\]
Hence by Lemma \ref{eq:decomposition of h}, if $v\sim u$ then $\mathcal{L}(v)=\mathcal{L}(u)$.
\section{\label{sec:proof of the equidisitrbution}The resulting measures}
\subsection{A further refinement}
We define $\tilde{\nu}_{v}$ be the uniform measure on the finite
set $\tilde{R}_{v}$ (defined in \eqref{eq:R^tilde_v}). We also define
the measure
\begin{equation}
\tilde{\nu}_{\text{polar}}=\left(\tilde{q}_{\Delta K}\right)_{*}\mu_{\mathbb{G}_{1}(\mathbb{R})}\otimes\mu_{\mathbb{G}_{2}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})},\label{eq:nu_tilde_polar}
\end{equation}
where $\mu_{\mathbb{G}_{1}(\mathbb{R})}$ and $\mu_{\mathbb{G}_{2}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})}$
are the Haar probability measure on $\mathbb{G}_{1}(\mathbb{R})$ and the $\mathbb{G}_{2}(\mathbb{R})$-invariant
probability measure on $\mathbb{G}_{2}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})$. We will prove
the following.
\begin{thm}
\label{thm:convergence of nu tilde-1}Let $\left\{ v_{i}\right\} _{i=1}^{\infty}$
be a sequence of admissible vectors such that
\[
\left\Vert v_{i}\right\Vert \to\infty,
\]
then\emph{
\begin{equation}
\tilde{\nu}_{v_{i}}\overset{\text{weak *}}{\longrightarrow}\tilde{\nu}_{\text{polar}}.\label{eq:nu_tilde convergence to natural measure}
\end{equation}
}
\end{thm}
Theorem \ref{thm:convergence of nu tilde-1} implies Theorem \ref{thm:refinement of main theorem}
by the following two lemmata.
\begin{lem}
\label{lem:push of counting measure nu_tilde_v}It holds that the
map $\tilde{\mathcal{M}}$ when restricted to $\tilde{R}_{v}$ is a bijection
onto $\mathcal{L}(v)$. In particular,
\begin{equation}
\tilde{\mathcal{M}}_{*}\tilde{\nu}_{v}=\mu_{v}.\label{eq:mult_tild_nu_tild_v is mu_v}
\end{equation}
\end{lem}
\begin{proof}
The map is clearly onto. In order to prove injectivity, we recall
that part 2 of Proposition \ref{prop:properties of p-adic projection}
states that
\[
Kk_{v}\gamma_{1}^{-1}(h)\mathbb{G}_{1}(\mathbb{Z})\cap Kk_{v}\gamma_{1}^{-1}(h')\mathbb{G}_{1}(\mathbb{Z})=\emptyset,\ \ h\neq h',\ h,h'\in M_{0},
\]
which implies that for different representatives $h,h'\in M_{0}$,
the corresponding $\left(d-1\right)$-subgroups defined by \eqref{eq:the resulting primitive lattice as orthogonal latice}
lie inside different hyperplanes. Finally, since bijectivity is established,
we immediately get \eqref{eq:mult_tild_nu_tild_v is mu_v}.
\end{proof}
\begin{lem}
\label{lem:push of nu_tilde_v_polar}It holds that $\tilde{\mathcal{M}}_{*}\tilde{\nu}_{\text{polar}}=\mu_{\text{polar }}$.
\end{lem}
\begin{proof}
Since $\tilde{\mathcal{M}}=\mathcal{M}\circ\tilde{q}$ and since $\mathcal{M}_{*}\nu_{\text{polar}}=\mu_{\text{polar }}$,
it is left to prove $\tilde{q}_{*}\tilde{\nu}_{\text{polar}}=\nu_{\text{polar}}.$
We note that
\[
\left(q_{P\diagup Q}^{\mathbb{G}_{2}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})}\right)_{*}\mu_{\mathbb{G}_{2}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})}=\mu_{X_{d-1}},
\]
and observe by Diagram \eqref{eq:main diagram} that
\begin{equation}
\tilde{q}\circ\tilde{q}_{\Delta K}=q_{\Delta K^{\pm}}\circ\left(id\times q_{P\diagup Q}^{\mathbb{G}_{2}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})}\right).\label{eq:q_tilde equalities}
\end{equation}
Therefore,
\[
\tilde{q}_{*}\tilde{\nu}_{\text{polar}}\underbrace{=}_{\text{definition of \ensuremath{\tilde{\nu}_{\text{polar}}}}}\left(\tilde{q}\right)_{*}\left(\left(\tilde{q}_{\Delta K}\right)_{*}\mu_{\mathbb{G}_{1}(\mathbb{R})}\otimes\mu_{\mathbb{G}_{2}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})}\right)\underbrace{=}_{\eqref{eq:q_tilde equalities}}
\]
\[
\left(q_{\Delta K^{\pm}}\right)_{*}\left(id\times q_{P\diagup Q}^{\mathbb{G}_{2}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})}\right)_{*}\mu_{\mathbb{G}_{1}(\mathbb{R})}\otimes\mu_{\mathbb{G}_{2}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})}=
\]
\[
\left(q_{\Delta K^{\pm}}\right)_{*}\mu_{\mathbb{G}_{1}(\mathbb{R})}\otimes\mu_{X_{d-1}}\underbrace{=}_{\text{\eqref{eq:nu_polar definition}}}\nu_{\text{polar }}.
\]
\end{proof}
\subsection{Proof of Theorem \ref{thm:convergence of nu tilde-1}}
To summarize, we have
\[
\text{Theorem \ref{thm:convergence of nu tilde-1} \ensuremath{\implies}Theorem \ref{thm:refinement of main theorem} \ensuremath{\implies}Theorem \ref{thm:maintheorem}}.
\]
Hence this section serves as the last step of the proof for Theorem
$\ref{thm:maintheorem}.$
\subsubsection{The key Theorem of \cite{AESgrids}}
The orbit $O_{v,p}$ defined in \eqref{eq:the s-arithemtic orbit}
is a compact orbit (see \cite{AESgrids}, Section 3.2) and we denote
by $\mu_{O_{v,p}}$ the $L_{v}(\mathbb{R}\times\mathbb{Q}_{p})$-invariant probability
measure supported on $O_{v,p}$. Also, let $\mu_{\mathcal{Y}_{p}}$ be the
$\mathbb{G}(\mathbb{R}\times\mathbb{Q}_{p})$-invariant probability measure on $\mathcal{Y}_{p}$.
The following theorem, which was proved in \cite{AESgrids}, is key
in order to prove Theorem \ref{thm:convergence of nu tilde-1}.
\begin{thm}
\emph{\label{thm:AESgrids thm} }Let $\left\{ v_{i}\right\} _{i=1}^{\infty}$
be a sequence of admissible vectors such that
\[
\left\Vert v_{i}\right\Vert \to\infty,
\]
then
\[
\mu_{O_{v_{i},p}}\overset{\text{weak * }}{\longrightarrow}\mu_{\mathcal{Y}_{p}}.
\]
\end{thm}
We define the probability measure $\eta_{v}$ on $O_{v,p}\cap\mathcal{U}$,
by
\begin{equation}
\eta_{v}\overset{\text{def}}{=}\mu_{O_{v,p}}\mid_{\mathcal{U}}.\label{eq:eta_v definition}
\end{equation}
Since $\mathcal{U}$ is a clopen set, it follows from Theorem \ref{thm:AESgrids thm}
that
\[
\eta_{v_{i}}\overset{\text{weak *}}{\longrightarrow}\mu_{\mathcal{Y}_{p}}\mid_{\mathcal{U}}.
\]
Also, since $q_{\infty}$ is a proper map we get
\[
\left(q_{\infty}\right)_{*}\eta_{v_{i}}\overset{\text{weak *}}{\longrightarrow}\left(q_{\infty}\right)_{*}\mu_{\mathcal{Y}_{p}}\mid_{\mathcal{U}}.
\]
Importantly, $\mu_{\mathcal{Y}_{p}}\mid_{\mathcal{U}}$ is $\mathbb{G}(\mathbb{R})$ invariant, hence
also $\left(q_{\infty}\right)_{*}\mu_{\mathcal{Y}_{p}}\mid_{\mathcal{U}}$. Therefore
we deduce,
\begin{cor}
\label{cor:s-arithemetic measures converge to haar}It holds that
\begin{equation}
\left(q_{\infty}\right)_{*}\eta_{v_{i}}\overset{\text{weak *}}{\longrightarrow}\mu_{\mathbb{G}(\mathbb{R})/\mathbb{G}(\mathbb{Z})},\label{eq:weak convergence to haar of eta}
\end{equation}
where $\mu_{\mathbb{G}(\mathbb{R})/\mathbb{G}(\mathbb{Z})}$ is the $\mathbb{G}(\mathbb{R})$-invariant probability
on $\mathbb{G}(\mathbb{R})/\mathbb{G}(\mathbb{Z})$.
\end{cor}
Next, note that Proposition \ref{prop:properties of p-adic projection}
shows that the measure $\left(q_{\infty}\right)_{*}\eta_{v}$ is supported
on a finite union of $\Delta K$ orbits
\[
\bigsqcup_{h\in M_{0}}q_{\infty}\left(O_{v,p,h}\right),
\]
and by applying further $q_{\Delta K}$, we get that $\left(q_{\Delta K}\circ q_{\infty}\right)_{*}\eta_{v}$
is supported on a finite set
\[
R_{v}\overset{\text{def}}{=}q_{\Delta K}\circ q_{\infty}(O_{v,p}),
\]
which consists of the elements
\[
\tilde{O}_{v,h}=\Delta K\left(k_{v}\gamma_{1}^{-1}(h),a_{v}k_{v}g_{v}\gamma_{2}(h)\right)\mathbb{G}(\mathbb{Z}),\ h\in M_{0}.
\]
On the other hand, note that
\[
\tilde{\pi}_{\mathbb{G}_{1}(\mathbb{Z})}\left(\tilde{R}_{v}\right)=R_{v},
\]
so that $\left(\tilde{\pi}_{\mathbb{G}_{1}(\mathbb{Z})}\right)_{*}\tilde{\nu}_{v}$
has the same support as that of $\left(q_{\Delta K}\circ q_{\infty}\right)_{*}\eta_{v}$.
The following lemma connects those two measures.
\begin{lem}
\label{lem:difference of measure is zero}It holds that
\[
\left(\pi_{\mathbb{G}_{1}(\mathbb{Z})}\right)_{*}\tilde{\nu}_{v_{i}}-\left(q_{\Delta K}\circ q_{\infty}\right)_{*}\eta_{v_{i}}\overset{\text{weak *}}{\longrightarrow}0.
\]
\end{lem}
In Subsection \ref{subsec:lemma of difference of meas} we will explain
how Lemma \ref{lem:difference of measure is zero} follows from \cite{AESgrids}.
Before that, we explain how Lemma \ref{lem:difference of measure is zero}
and the preceding discussion implies Theorem \ref{thm:convergence of nu tilde-1}.
\begin{proof}[Proof of Theorem \ref{thm:convergence of nu tilde-1}]
By Corollary \ref{cor:s-arithemetic measures converge to haar},
it follows that
\[
\left(q_{\Delta K}\circ q_{\infty}\right)_{*}\eta_{v_{i}}=\left(q_{\Delta K}\right)_{*}\left((q_{\infty})_{*}\eta_{v_{i}}\right)\overset{\text{weak *}}{\longrightarrow}\left(q_{\Delta K}\right)_{*}\mu_{\mathbb{G}(\mathbb{R})\diagup\mathbb{G}(\mathbb{Z})}.
\]
Hence we get from Lemma \ref{lem:difference of measure is zero} that
\[
\left(\pi_{\mathbb{G}_{1}(\mathbb{Z})}\right)_{*}\tilde{\nu}_{v_{i}}\overset{\text{weak *}}{\longrightarrow}\left(q_{\Delta K}\right)_{*}\mu_{\mathbb{G}(\mathbb{R})\diagup\mathbb{G}(\mathbb{Z})}.
\]
Observe that (see Diagram \eqref{eq:main diagram})
\[
\left(q_{\Delta K}\right)_{*}\mu_{\mathbb{G}(\mathbb{R})\diagup\mathbb{G}(\mathbb{Z})}=\left(\pi_{\mathbb{G}_{1}(\mathbb{Z})}\right)_{*}\left(\left(\tilde{q}_{\Delta K}\right)_{*}\mu_{\mathbb{G}_{1}(\mathbb{R})}\otimes\mu_{\mathbb{G}_{2}(\mathbb{R})\diagup\mathbb{G}_{2}(\mathbb{Z})}\right),
\]
so that by the definition of $\tilde{\nu}_{\text{polar}}$ (see \eqref{eq:nu_tilde_polar}),
we get that
\[
\left(\pi_{\mathbb{G}_{1}(\mathbb{Z})}\right)_{*}\tilde{\nu}_{v_{i}}\overset{\text{weak *}}{\longrightarrow}\left(\pi_{\mathbb{G}_{1}(\mathbb{Z})}\right)_{*}\tilde{\nu}_{\text{polar}}.
\]
Now, since the measures $\tilde{\nu}_{v}$ and $\tilde{\nu}_{\text{polar}}$
are both $\mathbb{G}_{1}(\mathbb{Z})$ invariant and since $\mathbb{G}_{1}(\mathbb{Z})$ is finite,
we also obtain that
\[
\tilde{\nu}_{v_{i}}\overset{\text{weak *}}{\longrightarrow}\tilde{\nu}_{\text{polar}}.
\]
\end{proof}
\subsubsection{\label{subsec:lemma of difference of meas}Outline of the proof for
Lemma \ref{lem:difference of measure is zero}}
Let $\lambda_{v}$ be the uniform counting measures on the sets $R_{v}.$
The idea of the proof of Lemma \ref{lem:difference of measure is zero}
is to show that
\begin{equation}
\left(\pi_{\mathbb{G}_{1}(\mathbb{Z})}\right)_{*}\tilde{\nu}_{v}-\lambda_{v}\to0,\ \text{and\ }\left(q_{\Delta K}\circ q_{\infty}\right)_{*}\eta_{v}-\lambda_{v}\to0.\label{eq:difference of counting measures}
\end{equation}
We denote
\begin{equation}
\left(q_{\Delta K}\circ q_{\infty}\right)_{*}\eta_{v}=\sum_{h\in M_{0}}\alpha_{v,h}\delta_{\tilde{O}_{v,h}},\label{eq:rh_Dk_rh_infty_eta}
\end{equation}
then,
\[
\alpha_{v,h}\overset{\text{def}}{=}\eta_{v}(q_{\infty}^{-1}(O_{v,h}))\underbrace{=}_{\eqref{eq:eta_v definition}\text{, and Proposition \ref{prop:properties of p-adic projection}}}\eta_{v}(O_{v,p,h}).
\]
It follows that
\[
\eta_{v}(O_{v,p,h})=\frac{\alpha}{\left|\text{stab}_{\Delta K\times L_{v}(\mathbb{Z}_{p})}(k_{v},h,a_{v}k_{v}g_{v},g_{v}^{-1}hg_{v})\mathbb{G}(\mathbb{Z}(\frac{1}{p}))\right|},
\]
where $\alpha=\alpha(v)$ normalizes $\left(q_{\Delta K}\circ q_{\infty}\right)_{*}\eta_{v}$
to be a probability measure. Also, let
\[
\left(\pi_{\mathbb{G}_{1}(\mathbb{Z})}\right)_{*}\tilde{\nu}_{v}=\sum_{h\in M_{0}}\beta_{v,h}\delta_{\tilde{O}_{v,h}},
\]
where
\[
\beta_{v,h}=\frac{\beta}{\left|\text{stab}_{\mathbb{G}_{1}(\mathbb{Z})}(Kk_{v}\gamma_{1}^{-1}(h))\right|},
\]
and $\beta=\beta(v)$ normalizes the measure $\left(\pi_{\mathbb{G}_{1}(\mathbb{Z})}\right)_{*}\tilde{\nu}_{v}$
to a probability measure. Let
\[
M_{v}=\max_{h\in M_{0}}\alpha_{v,h},
\]
and
\[
N_{v}=\max_{h\in M_{0}}\beta_{v,h}.
\]
Also let
\[
E=\left\{ \Delta K\left(\rho,\eta)\mathbb{G}(\mathbb{Z})\right)\mid\left|\text{stab}_{\mathbb{G}_{1}(\mathbb{Z})}(K\rho)\right|>1\right\} .
\]
The following statements were proven in \cite{AESgrids},
\begin{lem}
\label{lem:weight lemma from aes-1}The following holds,
\begin{enumerate}
\item For all $h\in M_{0}$ such that $O_{v,h}\notin E,$ it holds that
$\alpha_{v,h}=M_{v}$ and $\beta_{v,h}=N_{v}$.
\item $\frac{\left|R_{v}\cap E\right|}{|R_{v}|}\to0$.
\end{enumerate}
\end{lem}
\begin{proof}
See Lemmata 6.3 and 6.4 of \cite{AESgrids}.
\end{proof}
It is immediate that Lemma \ref{lem:weight lemma from aes-1} implies
the limits \eqref{eq:difference of counting measures}.
\subsubsection*{Acknowledgements}
I thank Uri Shapira for purposing this problem, his valuable support
and for many discussions. I also thank Cheng Zheng and Rene R{\"u}hr for
many important discussions on this project.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,485 |
Leonel Vielma (Mérida, Estado Mérida, Venezuela; 30 de agosto de 1978) es un exfutbolista venezolano y actual entrenador de fútbol. Jugaba de centrocampista o defensa y tuvo una extensa trayectoria en la que se destaca su participación con la Selección de Venezuela. Actualmente dirige el club Aragua FC, de la Primera División del fútbol venezolano.
Trayectoria
Estudiantes de Mérida
El 30 de marzo debutó en la Copa Libertadores 1999 contra el Club Atlético Bella Vista con derrota de 5-1 entrando de la banda y disputando 67 minutos siendo expulsado en el minuto 86º por doble tarjeta amarilla.
En total en la Copa Libertadores 1999 disputó 2 partidos jugando 90 minutos recibiendo 1 tarjeta roja (doble amarilla), clasificando quedando de segundos llegando hasta los cuartos de final.
EL 4 de julio de 2000 debutó con el Estudiantes en la Copa Merconorte 2000 contra el CD El Nacional con derrota de 2-1, disputando los 90 minutos. El 2 de agosto de 2000 marcó su primer gol en la Copa Merconorte 2000 contra el Club Deportivo Guadalajara con derrota de 3-2, marcando el gol en el minuto 51º disputando los 90 minutos.
En total en la Copa Merconorte 2000 disputó 6 partidos todos de titular marcando 1 gol jugando 540 minutos recibiendo 3 amarillas, quedando de terceros.
UA Maracaibo
El 2 de marzo de 2003 marcó su primer gol con el UA Maracaibo en la jornada 1 del torneo clausura de la Primera División de Venezuela 2002-03 contra el Deportivo Italchacao con resultado de 2-2.
En total en la Primera División de Venezuela 2002-03 marcó 2 goles 1 de penalti, quedando subcampeones.
Deportivo Táchira
El 14 de abril de 2004 marcó su primer gol en una Copa Libertadores 2004 contra el Club Atlético River Plate con resultado de 2-2, disputando 77 minutos y marcando el gol en el minuto 69º.
En total en la Primera División de Venezuela 2003-04 no marcó gol, quedando subcampeones. En la Copa Libertadores 2004 disputó 10 partidos todos de titular marcando 2 goles jugando 869 minutos recibiendo 1 tarjeta roja (doble amarilla), llegando hasta los cuartos de final.
Deportivo Cali
El 5 de agosto de 2004 debutó con el Deportivo Cali en la jornada 2 del Torneo Finalización 2004 contra el América de Cali con derrota de 3-0, disputando 4 minutos del segundo tiempo.
En total en el Torneo Finalización 2004 disputó 13 partidos 11 de titular jugando 961 minutos recibiendo 1 roja quedando de cuartos en la tabla y clasificando a los cuadrangulares (semifinales) donde disputó los 6 partidos todos de titular jugando 540 minutos quedando de segundos sin poder disputar la final.
Caracas FC
El 19 de marzo de 2005 marcó su primer gol con el Caracas FC en la jornada 11 del torneo clausura de la Primera División de Venezuela 2004-05 contra el Trujillanos FC con victoria de 4-2, marcando el gol en el minuto 41'.
En total en la Primera División de Venezuela 2004-05 marcó 1 gol, quedando subcampeones. En la Copa Libertadores 2005 disputó 5 partidos todos de titular jugando 450 minutos recibiendo 1 tarjeta amarilla, quedando en cuarto lugar.
Once Caldas
El 4 de julio se viajó para unirse al Once Caldas. Con el Once Caldas participó en la Copa de la Paz 2005 en (Corea) disputó los 3 partidos sin marcar gol.
El 30 de julio de 2005 debutó en la cuarta jornada del Torneo Finalización de Colombia contra el Independiente de Santa Fe con resultado de 0-0 disputando los 90 minutos.
El 24 de agosto de 2005 debutó en la Recopa Sudamericana 2005 contra el Boca Juniors con derrota de 3-1, disputando los 90 minutos.
En total en el Torneo Finalización 2005 disputó 5 partidos 4 de titular sin marcar goles jugando 405 minutos. En la Recopa Sudamericana 2005 disputó 1 partido sin marcar gol jugando 90 minutos.
Italmaracaibo
El 12 de noviembre de 2005 marcó su primer gol con el Italmaracaibo en la jornada 13 del torneo apertura de la Primera División de Venezuela 2005-06 contra el Trujillanos FC dándole la victoria a su equipo 1-0 marcando el gol en el minuto 25º.
En total en la Primera División de Venezuela 2005-06 marcó 4 goles 2 de penalti, quedando novenos bajando a Segunda División.
Caracas FC
En la Primera División de Venezuela 2006-07 quedó campeón con el Caracas FC.
En el torneo apertura de la Primera División de Venezuela 2007-08 disputó 14 partidos marcando 2 goles. En la Copa Libertadores 2007 disputó 7 partidos todos de titular marcando 1 gol jugando 630 minutos recibiendo 1 tarjeta amarilla, quedando de segundos llegando hasta los octavos de final.
Santa Fe
El 17-12-2007 inicialmente habría llegado a un acuerdo en el Junior de Barranquilla durante 1 año hasta enero del 2009, pero aparentemente el Santa Fe ha duplicado la oferta para hacerse de los servicios de Vielma para el 2008.
En la pretemporada disputó 4 partidos.
El 2 de febrero de 2008 debutó en la primera jornada del Torneo Apertura de Colombia contra el Atlético Nacional con victoria de 1-0 disputando los 90 minutos.
Marcó su primer gol con el Santa Fe el 16 de marzo de 2008 en la octava jornada contra el Atlético Bucaramanga con victoria de 2-1 disputando los 90 minutos y marcando el gol en el minuto 30º de tiro libre (aprovechó la mala ubicación de su compatriota, el portero Javier Toyo del cuadro local e introdujo la pelota muy pegada al palo derecho del arco). El 2 de abril de 2008 recibió su primera tarjeta roja contra el Independiente Medellín suspendiéndolo a 2 partidos.
En total en el Torneo Apertura 2008 disputó 15 partidos todos de titular marcando 2 goles los 2 de tiro libre jugando 1317 minutos recibiendo 9 tarjetas amarillas y 1 roja quedando de terceros en la tabla y clasificando a los cuadrangulares (semifinales) hay disputó 5 partidos todos de titular recibiendo 1 tarjeta amarilla jugando 405 minutos quedando de terceros sin poder disputar la final.
Luego de no tener continuidad en el Torneo Apertura 2009, Vielma sale de Santa Fe y regresa a Venezuela para jugar con el Caracas FC.
El Vigía FC
El 17 de junio de 2012 se confirma el fichaje del internacional Leonel Vielma por El Vigía FC, para disputar el torneo Apertura 2012 con el club platanero, de esta manera regresa al club auriverde luego de haber militado a comienzos de su carrera en esta organización
Estudiantes de Mérida
El 16 de julio de 2013 el jugador por mutuo acuerdo con la Directiva de El Vigía FC, rescinde su contrato en buenos términos, y es fichado por Estudiantes de Mérida FC para la temporada 2013 - 2014
Ureña SC
El 4 de enero el merideño es anunciado como nuevo jugador del cuadro azucarero con miras a disputar como jugador activo el Torneo Apertura 2016 de la primera división del Fútbol profesional venezolano.
Selección nacional
Ha jugado con las categorías menores de la selección disputando el Campeonato Sudamericano sub-17, Campeonato Sudamericano sub-20.
Debutó en la Selección de fútbol de Venezuela en un partido amistoso disputado contra Bolivia el 16 de marzo de 2000 disputado en el estadio José Encarnación "Pachencho" Romero de Maracaibo con resultado de 0-0.
Su primer gol en la selección fue contra Haití el 20 de agosto de 2003 disputado en el Estadio Olímpico de la UCV de Caracas con victoria de 3-2, marcando el gol de penal.
Debutó en una Copa América contra Chile el 14 de julio de 2001 disputado en el Estadio Metropolitano Roberto Meléndez de Barranquilla con derrota de 1-0, disputando 90 minutos.
Debutó en una Eliminatoria al Mundial contra Colombia el 15 de noviembre de 2003 disputado en el Estadio Metropolitano Roberto Meléndez de Barranquilla con victoria de 1-0, entrando en el minuto 6º del primer tiempo.
Su primer gol en una Eliminatoria al Mundial fue contra Argentina el 17 de noviembre de 2004 disputado en el Estadio Monumental Antonio Vespucio Liberti de Buenos Aires con derrota de 3-2, disputando 20 minutos y marcando el gol en el minuto 72º de tiro libre.
Lleva 3 goles con la Vinotinto 1 en Eliminatorias al Mundial contra Argentina de tiro libre y 2 en Amistosos contra Haití de penalti y Honduras de tiro libre.
Preolímpico Sudamericano sub-23
Disputó los 4 partidos todos de titular contra Colombia, Chile, Brasil y Ecuador jugando 360 minutos recibiendo 1 tarjeta amarilla y marcando 1 gol a Colombia.
Vielma en la Vinotinto
Último Partido: Venezuela - Perú (6 Jun 2008)
Participaciones en Copa América
En la Copa América del 2001 participó en 2 partidos Venezuela 0-1 Chile y Venezuela 0-4 Ecuador disputando 180 minutos.
En la Copa América del 2004 participó en 1 partido Venezuela 1-1 Bolivia disputando 16 minutos.
En la Copa América del 2007 participó en 1 partido Venezuela 2-2 Bolivia disputando 17 minutos del segundo tiempo.
Participaciones en Eliminatorias al Mundial
Entrenador
Tras haberse desempeñano en las divisiones menores y como entrenador asistente, Leonel Vielma asumió en octubre de 2020 el cargo de entrenador principal del club venezolano Mineros de Guayana. A pesar de los problemas económicos, logró que su equipo clasificara a la edición 2021 de la Copa Sudamericana.
En enero de 2021 firmó un contrato por tres años para dirigir al club Estudiantes de Mérida.
Clubes
Competiciones
Palmarés
Campeonatos nacionales
Referencias
Enlaces externos
Primer gol en Eliminatorias al Mundial
Primer gol con el Santa Fe
Golazo de 35m con la selección
Merideños
Futbolistas de Venezuela
Futbolistas de la selección de fútbol de Venezuela en los años 2000
Futbolistas de Venezuela en la Copa América 2001
Futbolistas de Venezuela en la Copa América 2004
Futbolistas de Venezuela en la Copa América 2007
Vielma
Futbolistas del Caracas Fútbol Club
Futbolistas del Deportivo Cali
Futbolistas del Once Caldas
Futbolistas del Independiente Santa Fe
Futbolistas del Club Deportivo Italmaracaibo
Futbolistas del Deportivo Táchira Fútbol Club
Futbolistas del Club Deportivo Unión Atlético Maracaibo Sociedad Civil
Futbolistas del Estudiantes de Mérida Fútbol Club
Futbolistas de la Asociación Civil Deportivo Lara
Merideños (Mérida)
Entrenadores del Estudiantes de Mérida Fútbol Club
Entrenadores del Club Deportivo Mineros de Guayana
Entrenadores del Aragua Fútbol Club | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,408 |
\section{Introduction}
The {\it Herschel} Astrophysical Terahertz Large Area Survey (H-ATLAS) survey is the largest, in time and area, of the
extragalactic Open Time Key Projects to be carried out with the
European Space Agency (ESA) Herschel Space Observatory
\citep{herschel}\footnote{Herschel is an ESA space observatory with
science instruments provided by European-led Principal Investigator
consortia and with important participation from NASA.}. When
complete it will cover $\sim$550 square degrees of the sky, in five far--infrared and
submillimetre bands (100, 160, 250, 350 and 500$\,\mu $m\,), to a 5$\sigma$ depth of 33 mJy/beam at 250$\,\mu $m\,. The predicted number of
sources is $\sim$200,000; of these $\sim$40,000 are expected to lie within $z<0.3$. A full
description of the survey can be found in \citet{eales}.
This paper presents the 250$\,\mu $m\, selected source catalogue created from the initial
H-ATLAS Science Demonstration Phase (SDP) observations. Eight papers
based on this catalogue have already been published in the A\&A {\it
Herschel} Special Issue ranging from the identification of blazars \citep{gonzalez} and
debris disks \citep{thompson} in the SDP field, to determinations of
the colours \citep{amblard}, source counts \citep{clements},
clustering \citep{wtheta} and 250$\,\mu $m\, luminosity function evolution \citep{dye} of the submillimetre
population, as well as the star formation history of quasar host
galaxies \citep{serjeant2} and the dust energy balance of a nearby
spiral galaxy \citep{baes}.
The layout of the paper is as follows: Section \ref{hers_obs} describes the SDP observations; Section \ref{extract}
describes the source extraction procedure for the five bands; finally,
Section \ref{sims} outlines the simulations used to quantify the
reliability of the catalogue.
For more details of the SDP data see \citet{spiremaps} and
\citet{pacsmaps} for the SPIRE and PACS data reduction respectively,
and \citet{Smith} for the multiwavelength catalogue matching.
\begin{figure}
\centering
\subfloat[Original combined map]{\includegraphics[scale=0.15]{sdp_nobacksub.png}} \\
\subfloat[After background subtraction]{\includegraphics[scale=0.15]{sdp_backsub.png}}
\caption{\protect\label{pre_backsub} False--colour images of a 1.5
sq. degree region of the SDP field showing the three SPIRE bands combined. Image (a) is before
background--subtraction and shows clear contamination by galactic
cirrus; image (b) shows the reduction in contamination after subtracting the
background. }
\end{figure}
\section{{\it Herschel} observations}
\label{hers_obs}
The SDP observations for the H--ATLAS survey cover an area of
$\sim$4\hbox{$^\circ$}$\times$4\hbox{$^\circ$}, centred at
$\alpha$=09$^{h}$05$^{m}$30.0$^{s}$,
$\delta=$00\deg30\arcmin00.0\hbox{$^{\prime\prime}$}\, (J2000). This field lies within
one of the regions of the GAMA (Galaxy and Mass Assembly) survey \citep{gama} so optical spectra, along with
additional multiwavelength data, are available for the majority of the
low--redshift sources.
The observations were taken in parallel--mode, which uses the
Photodetector Array Camera and Spectrometer \citep[PACS;][]{pacs} and
Spectral and Photometric Imaging REciever \citep[SPIRE;][]{spire}
instruments simultaneously; two orthogonal scans were used to mitigate
the effects of $1/f$ noise. The time--line data were reduced using {\tt HIPE} \citep{hipe}. SPIRE
250, 350, and 500$\,\mu $m\, maps were produced using a na\"ive mapping
technique, after removing any instrumental temperature variations
(Pascale et al. \citeyear{spiremaps}), and incorporating the
appropriate flux calibration factors. Noise maps were generated by
using the two cross--scan measurements to estimate the noise per detector pass, and
then for each pixel the noise is scaled by the square root of the number of detector
passes. The SPIRE point spread function (PSF) for each band was determined from Gaussian fits to observations of Neptune, the primary calibrator for the instrument.
Maps from the PACS 100 and 160$\,\mu $m\, data were produced using the {\tt PhotProject} task within {\tt HIPE}
(Ibar et al. \citeyear{pacsmaps}). A false colour combined image of
a part of the three SPIRE maps is shown in Figure
\ref{pre_backsub}. The measured beam full--width--half--maxima (FWHMs)
are approximately 9\hbox{$^{\prime\prime}$}\,, 13\hbox{$^{\prime\prime}$}\,, 18\hbox{$^{\prime\prime}$}\,, 25\hbox{$^{\prime\prime}$}\, and
35\hbox{$^{\prime\prime}$}\, for the 100, 160, 250, 350 and 500$\,\mu $m\, bands respectively \citep{pacsmaps, spiremaps}. The map pixels are 2.5\hbox{$^{\prime\prime}$}\,, 5\hbox{$^{\prime\prime}$}\,, 5\hbox{$^{\prime\prime}$}\,,
10\hbox{$^{\prime\prime}$}\, and 10\hbox{$^{\prime\prime}$}\, in size for the same five bands.
The noise levels measured by \citet{spiremaps} for the 250$\,\mu $m\, and 500$\,\mu $m\, SPIRE bands are in good
agreement with those predicted using the Herschel Space Observatory
Planning Tool (HSpot\footnote{{\tt HIPE} and HSpot are joint developments by
the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS and
SPIRE consortia}); for the 350$\,\mu $m\, band they are considerably better. The corresponding PACS noise levels determined by \citet{pacsmaps} are currently
higher than predicted (26 mJy and 24 mJy, compared with 13.4 mJy and
18.9 mJy for 100$\,\mu $m\, and 160$\,\mu $m\, respectively), but this may
improve in future with better map--making techniques. The flux
calibration uncertainties are 15\% for the three SPIRE bands
\citep{spiremaps} and 10 and 20\% for the PACS 100$\,\mu $m\, and 160$\,\mu $m\,
bands respectively \citep{pacsmaps}.
\begin{figure*}
\centering
\subfloat[Source with a close companion]{\protect\label{bad_case} \includegraphics[scale=0.55]{cutout_0.pdf}} \\
\subfloat[Isolated source]{\protect\label{good_case}\includegraphics[scale=0.55]{cutout_764.pdf}} \\
\caption{\protect\label{cutout_plots} The input (true) and extracted position
for two point sources in the 250$\,\mu $m\, simulated maps before the
addition of Gaussian noise (noiseless), after the noise has been
added (noisy), and after further convolving with the 250$\,\mu $m\, point
spread function (PSF) to create the final realistic sky
(PSF--filtered) (see Section \ref{sims} for full details), to
illustrate how the position, and therefore flux density, of an extracted source found by MADX can be influenced by the presence of a close companion.}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[scale=0.7, clip, trim=0mm 4mm 0mm 2mm]{paper_aper.pdf}
\caption{\protect\label{ext_plots} A comparison between the MADX and
aperture measured fluxes for the sources with a possible optical
identification in the matched catalogue of \citet{Smith}. Source
identified as extended are highlighted in bold.}
\end{figure*}
\section{Source extraction}
\label{extract}
The ultimate aim for the source identification of the H-ATLAS data is
to use a multiband method to perform extraction across the five
wavebands simultaneously, thus utilising all the available data as
well as easily obtaining complete flux density information for each
detected galaxy, without having to match catalogues between
bands. However, the short timescale for the reduction of these SDP
observations, combined with the higher than expected PACS noise
levels, means that this was only possible for the three SPIRE
bands. As a result, the source extraction for the PACS and SPIRE maps
is discussed separately in this Section.
The full H-ATLAS SDP catalogue described here will be available at
\verb1http://www.h-atlas.org/1.
\subsection{The SPIRE catalogue}
\label{spire_cat}
Sources are identified in the SPIRE 250, 350 and 500$\,\mu $m\, maps using
the Multi--band Algorithm for source eXtraction \citep[MADX,][]{madx},
which is being developed for the H--ATLAS survey. Several methods for
generating the final SPIRE catalogue with MADX were investigated and these
are described below.
The first step in the MADX source extraction is to subtract a local background, estimated from the peak
of the histogram of pixel values in $30\times 30$ pixel blocks (chosen
to allow the map to be easily divided up into independent sub--regions). This
corresponds to 2.5$^{\prime}$\, $\times$ 2.5$^{\prime}$\, for the 250$\,\mu $m\, map, and
5$^{\prime}$\, $\times$ 5$^{\prime}$\, for the 350 and 500$\,\mu $m\, maps. The
background (in mJy/beam) at each pixel was then estimated using a
bi-cubic interpolation between the coarse grid of backgrounds, and
subtracted from the data.
Figure \ref{pre_backsub} illustrates the
reduction in background contamination (mainly arising from galactic
cirrus, which dominates over the confusion noise from unresolved sources) obtained using this method.
The background subtracted maps were then filtered by the estimated
PSF, including an inverse variance weighting, where the noise for each
map pixel was estimated from the noise map \citep[matched filtering,
e.g.][]{turin, serjeant}.
The background removal has a negligible effect on the PSF because the histogram peak is insensitive to resolved
sources in the background aperture; this will be discussed further in \citet{madx}.
We also create a `filtered noise' map which represents the noise on a
pixel in the PSF filtered map. This is lower than the raw noise map
because the noise in the SPIRE pixels is uncorrelated, and so
filtering by the PSF reduces the noise by approximately the square
root of the number of pixels per beam.
The maps from the 350 and 500$\,\mu $m\, bands are interpolated onto the 250$\,\mu $m\,
pixels. Then all three maps are combined with weights set by the local
inverse variance, and the prior expectation of the spectral
energy distribution (SED) of the galaxies.
We used two SED priors: a flat-spectrum prior (assumed to be flat in $f_{\nu}$), where equal weight is given to each band; and
also 250$\,\mu $m\, weighting, where only the 250$\,\mu $m\, band was included.
Local, $>2.5\sigma$, peaks are identified in the combined PSF filtered
map as potential sources, and sorted in order of decreasing
significance level. A Gaussian is fitted to each peak in turn to provide an
estimate of the position at the sub--pixel level; this can be influenced by the presence of a
neighbouring source, as illustrated in Figure \ref{cutout_plots}, but
the effect is minimal. The flux in each band is then estimated using a bi--cubic interpolation to the position
given by the combined map. The scaled PSF is then subtracted from the
map before going on to the next source in the sequence. This ensures that
flux from the wings of bright sources does not contaminate nearby
fainter sources. This sorting and PSF subtraction reduces the effect of confusion, but in future
releases we plan to implement multi-source fitting to blended sources.
To produce a catalogue of reliable sources, a source is only included
if it is detected at a significance of at least 5$\sigma$ in one of
the SPIRE bands. The total number of sources in the SPIRE catalogue is
6876.
For our current data we chose to use the 250$\,\mu $m\, only prior for all our
catalogues, which means that sources are identified at 250$\,\mu $m\,
only. At the depth of the filtered maps source confusion is a significant
problem, and the higher resolution of the 250$\,\mu $m\, maps outweighed the
signal--to--noise gain from including the other bands (see Section
\ref{sim_creation} and Figure \ref{pos_err_pss_prior}). This may introduce a bias in the catalogue against red,
potentially high--redshift, sources that are bright at 500$\,\mu $m\,, but
weak in the other bands. However, comparing catalogues made with both
the 250$\,\mu $m\, and flat--spectrum priors showed that the number of missed sources is low: 2974
$>5\sigma$ 350$\,\mu $m\, sources and 348 $>5\sigma$ 500$\,\mu $m\, sources are
detected with the flat prior, compared with 2758 and 307 sources
detected using the 250$\,\mu $m\, prior (i.e. 7\% and 12\% of sources are
missed at 350$\,\mu $m\, and 500$\,\mu $m\, respectively). It should also be noted that for a
high--redshift source to be missed it would need a 500$\,\mu $m\, to 250$\,\mu $m\,
flux ratio of $>2.7$ (i.e. it has to be $<2.5\sigma$ at 250$\,\mu $m\, to be
excluded from the catalogue). Assuming typical SED templates (e.g. M82
and Arp220), this means that this should only occur for sources which lie at redshifts $>4.6$.
We aim to revisit this issue in future data--releases.
Since MADX uses a bicubic interpolation to estimate the peak flux in
the PSF filtered map, it partially avoids the peak suppression caused
by pixelating the time-line data, as discussed by Pascale et al.
Nevertheless the peak fluxes are systematically underestimated, and so
pixelization correction factors were calculated by pixelating the PSF
at a large number of random sub-pixel positions. The mean correction
factors were found to be 1.05, 1.11 and 1.04 in the 250, 350 and 500$\,\mu $m\,
bands respectively, and they have been included in the released SDP catalogue.
In calculating the $\sigma$ for each source, we use the filtered noise
map and add the confusion noise to this in quadrature. The average
1$\sigma$ instrumental noise values are 4.1, 4.0 and 5.7 mJy/beam
respectively, with 5\% uncertainty, in the 250, 350 and 500$\,\mu $m\, bands, determined from the
filtered maps \citep{spiremaps}. We estimated the confusion noise from the difference
between the variance of the maps and the expected variance due to
instrumental noise (assuming that confusion is dominating the excess noise), and find that the 1$\sigma$ confusion noise is 5.3,
6.4 and 6.7 mJy/beam at 250, 350 and 500$\,\mu $m\,, with an uncertainty of 7\%; these values are in good
agreement with those found by \citet{nguyen} using data from the
Herschel Multi--tiered Extragalactic Survey (HerMES). The resulting
average 5$\sigma$ limits are therefore 33.5, 37.7 and 44.0 mJy/beam.
\begin{figure}
\centering
\includegraphics[scale=0.4, clip, trim=0mm 0mm 0mm 0mm]{pacs_cnts_blue.pdf}
\includegraphics[scale=0.4, clip, trim=0mm 4mm 0mm 0mm]{pacs_cnts_red.pdf}
\caption{\protect\label{pacs_cnts} The differential source counts from
the PACS section of the SDP catalogue compared to the initial
results from the three fields covered by the PEP survey
\protect\citep{berta}. }
\includegraphics[angle=180,scale=0.6, clip, trim=0mm 6mm 0mm 0mm]{iras_flux_comp_paper3.pdf}
\caption{\protect\label{iras_comp} A comparison between the 100$\,\mu $m\, flux densities from PACS and IRAS}
\end{figure}
\subsubsection{Extended sources}
\label{ext_sour}
The flux density extracted by MADX will underestimate the true value for sources that are larger than the
SPIRE beams, which have FWHM of 18.1\hbox{$^{\prime\prime}$}\,, 24.8\hbox{$^{\prime\prime}$}\, and 35.2\hbox{$^{\prime\prime}$}\, for 250, 350 and 500$\,\mu $m\,
respectively. This occurs because the peak value taken by MADX only
accurately represents the true flux density of a source if it is
point--like. These extended sources can be identified if they also
have a reliable optical match and therefore a corresponding optical
size, $r_{\rm opt}$ (equivalent to the 25 mag arcsec$^{-2}$ isophote), in the SDSS or GAMA catalogues (see \citet{Smith} for
full details of the matching procedure and the determination of the
match reliability, $R_{j}$). The size of the aperture used is listed in the catalogue, and the most
appropriate flux density, either point source or aperture measurement
(when this is larger), is given for each source in the SPIRE `BEST flux'
columns. It should be noted that, apart from two exceptions, this is necessary at 250 and 350$\,\mu $m\, only, as the large 500$\,\mu $m\, beam size means that the
flux discrepancy is negligible for that map.
An `extended source', in a particular map, is defined here as one with $r_{\rm opt} > 0.5\times$FWHM, and to ensure only true matches are used, it must also have a
match--reliability, $R_{j}$, greater than 0.8. In total, the MADX `BEST' flux columns for 167 sources at 250$\,\mu $m\, and 53
sources at 350$\,\mu $m\, were updated with aperture photometry values.
The aperture radius, $a_{r}$, in a particular band is set by summing the optical size in quadrature with the FWHM of that band:
\begin{equation}
\label{ap_rad}
a_{r} = \sqrt{{\rm FWHM}^{2} + r_{\rm opt}^{2}} .
\end{equation}
The exceptions to this were the apertures used for sources H--ATLAS J091448.7-003533 (a
merger, where the given $a_{\rm r}$ is insufficient to include the
second component) and H--ATLAS J090402.9+005436, which visual
inspection showed was clearly extended. In these cases the aperture
sizes used are chosen to match the extent of the sub--mm emission, and fluxes are replaced in the 500$\,\mu $m\, band as well.
The apertures are placed on the MADX, Jy/beam, background subtracted
maps, at the catalogue position for each source; the measured values are converted to the correct flux scale
by dividing by the area of the beam derived by \citet{spiremaps} for each map (13.9, 6.6 or 14.2 pixels
for 250, 350 and 500$\,\mu $m\, respectively). The corresponding
1$\sigma$ error is given by $\sqrt{v_{\rm ap}}$, where
$v_{\rm ap}$ is the sum of the variances within
apertures placed in the same positions on the relevant variance maps. Confusion
noise estimates were again added in quadrature to these uncertainties; these were scaled according to the area of each individual aperture.
Figure \ref{ext_plots} compares the MADX and aperture measured fluxes
for all catalogue sources with a possible optical identification. It
shows that the majority of objects are point--like, for which the
agreement between the two sets of fluxes is good. The sources
identified as extended are highlighted in bold, and it is clear that
MADX underestimates these at 250 and 350$\,\mu $m\, if they are brighter than $\sim$100 mJy.
\subsection{The PACS catalogue}
The higher noise levels in the PACS maps, along with the shape of the
source SEDs, mean that all the PACS extragalactic sources should be
clearly detected in the SPIRE catalogue. Sources in the PACS data are
therefore identified by placing circular apertures at the SPIRE
250$\,\mu $m\, positions in the 100$\,\mu $m\, and 160$\,\mu $m\, maps, after correcting
the PACS astrometry to match that of the 250$\,\mu $m\, map (using the
sources present in both the SPIRE and PACS maps). There are two steps to this
source detection process: first a `point source' measurement is obtained for all SPIRE
positions using apertures with radii of 10\hbox{$^{\prime\prime}$}\, (100$\,\mu $m\,) or
15\hbox{$^{\prime\prime}$}\, (160$\,\mu $m\,); next additional aperture fluxes are found for
positions where a PACS source would satisfy the extended source
criteria discussed in Section \ref{ext_sour}. Aperture radii in this
case are calculated using Equation \ref{ap_rad}, assuming FWHM of
8.7\hbox{$^{\prime\prime}$}\, and 13.1\hbox{$^{\prime\prime}$}\, for 100 and 160$\,\mu $m\, respectively. These
FWHM values are calculated using rough modelling of the Vesta asteroid
as the full PACS PSFs are asymmetric \citep[see][for a full
discussion]{pacsmaps}.
The aggressive filtering used for these maps means that the large
scale structure in the cirrus has already been removed, but some noise
stripes remain. These are removed globally at 160$\,\mu $m\, by subtracting a
background determined within 10$\times$10 pixel blocks. However, at
100$\,\mu $m\, this global approach was found to introduce negative holes
around bright sources so the background value is determined for each
source individually using a local annulus with a width of 0.5 times
the aperture radius.
Unlike SPIRE, the PACS maps have units of Jy/pixel so no beam
conversion is needed. However, the fluxes are divided by 1.09 (100$\,\mu $m\,) or 1.29 (160$\,\mu $m\,) as recommended by the PACS
Instrument Control Centre\footnote{see the scan mode release note,
PICCMETN0.35}. These scaling factors are now incorporated into
the data--reduction pipeline and have been applied to the public
release of the PACS SDP maps, along with the astrometry correction
needed to match that of the SPIRE 250$\,\mu $m\, map (this correction
is $\sim$1\hbox{$^{\prime\prime}$}\, in both PACS bands). The fluxes are also aperture corrected,
using a correction determined from observations of a bright point--like source. The
1$\sigma$ errors are found using apertures randomly placed in the
maps; note that these errors scale with aperture size. The low confusion noise compared to SPIRE, plus the fast scan speed
used in these observations, means that the integration time used in
H-ATLAS is insufficient to provide confusion limited images with PACS. Full details of these observations can be found in \citet{pacsmaps}.
\begin{figure}
\centering
\subfloat[Extended source simulations]{\protect\label{ext_source_cnts} \includegraphics[scale=0.7, clip, trim=0mm 2mm 0mm 4mm]{counts_paper_ess.pdf}} \\
\subfloat[Point source only simulations]{\protect\label{pt_source_cnts} \includegraphics[scale=0.7, clip, trim=0mm 2mm 0mm 4mm]{counts_paper_pss.pdf}}
\caption{\protect\label{cnts_comp} The integrated source counts from
the combined set of 500 input (true) and extracted simulated catalogues, along with those
calculated using the SDP catalogue for both versions of the simulations. }
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.6, clip, trim=0mm 0mm 0mm 0mm]{250_xy_full.pdf}
\caption{\protect\label{xy_plots} The positional offsets between the
matched sources in the simulated extracted and input (true) full 250$\,\mu $m\, catalogues. The
results for the two versions are very similar, so only the PSS
points are shown here.}
\end{figure}
The most appropriate flux density measurements, either point or extended (where this is larger), are
given in the `BEST' PACS columns in the SDP catalogue, along with the
corresponding aperture radii, for sources with $S/N \geq 5$. As a
result 151 and 304 sources satisfy this condition at 100 and 160$\,\mu $m\,
respectively. The 5$\sigma$ point source limits in the PACS
catalogue are 132 mJy and 121 mJy at 100 and 160$\,\mu $m\,. It should be
noted that the flux densities extracted from the PACS maps are only at 100$\,\mu $m\, and 160$\,\mu $m\, under the assumption of a
constant energy spectrum, though the colour corrections for sources
with a different SED are small \citep{poglitsch}.
The PACS time-line data have been high-pass filtered by subtracting a
boxcar median over 3.4 arcmin (at 100$\,\mu $m\,) and 2.5$^{\prime}$\, at
160$\,\mu $m\, \citep{pacsmaps}. The filtering will lead to the underestimation
of flux for sources extended on scales comparable to the filter length.
The exact flux loss for a particular source will depend on the size of
the source along the scan directions, and will also depend on whether
the peak surface brightness is above the 4$\sigma$ threshold used in
the second level filtering. A simple simulation of a circular exponential disc shows that the
filtering removes $\sim 50\%$ of the source flux if the diameter of the
disk is equal to the filter length. If the diameter is half of the
filter length, then only 5\% of the flux is removed. This suggests that
sources with a diameter less than 1$^{\prime}$\, should by relatively
unaffected by the filtering. Flux measurements for sources larger than
this should be treated with caution.
Figure \ref{pacs_cnts} compares the differential source counts
calculated from the PACS SDP catalogues to those determined from the
initial data of the complementary PACS Evolutionary Probe (PEP) survey
\citep{berta}, which is deeper than H--ATLAS but covers a smaller
area. The good agreement between the two sets of counts supports the initial assumption that all bright PACS sources should
already be present in the SPIRE catalogue. However, there are
insufficient sources in the SDP data to properly constrain the bright
number counts tail. A full analysis of the PACS counts will be presented in \citep{ibar2}.
\begin{figure*}
\centering
\subfloat[Extended source simulations]{\protect\label{pos_err_ess} \includegraphics[scale=0.5, clip, trim=3mm 5mm 0mm 0mm]{poss_error_ess.pdf}
\subfloat[Point source simulations]{\protect\label{pos_err_pss} \includegraphics[scale=0.5, clip, trim=3mm 5mm 0mm 0mm]{poss_error_pss.pdf}}
\subfloat[Point source simulations comparing different priors]{\protect\label{pos_err_pss_prior} \includegraphics[scale=0.5, clip, trim=3mm 5mm 0mm 0mm]{poss_error_prior_pss.pdf}}
\caption{\protect\label{pos_err} The positional errors
for the two different versions of the simulations, alongside a
comparison of the two different source extraction position priors
as previously discussed in Section
\ref{spire_cat}. Also shown in \ref{pos_err_pss} are the positional
errors plotted against the S/N in the input (true) catalogue, along with
those determined by \citet{Smith} for the SDP data at 5, 7.5 and
10$\sigma$.}
\end{figure*}
For the sources detected in the PACS 100$\,\mu $m\, map an additional comparison can be made to this wavelength in
the Imperial IRAS--FSC Redshift Catalogue of \citet{iras}, which combines the original IRAS Faint Source Catalogue flux density values
with improved optical and radio identifications and redshifts. There are 34 IRAS sources within the PACS region of the
H-ATLAS SDP field; 19 of these have a reliable IRAS flux measurement and
these are in good agreement with the SDP catalogue, with a mean offset
consistent with zero, as shown in Figure \ref{iras_comp}.
\begin{figure*}
\centering
\subfloat[Extended source simulations]{\protect\label{ext_true_plots_ess} \includegraphics[scale=0.65, clip, trim=10mm 0mm 0mm 0mm]{flux_residuals_full_ess.pdf}} \\
\subfloat[Point source simulations]{\protect\label{ext_true_plots_pss} \includegraphics[scale=0.65, clip, trim=10mm 0mm 0mm 0mm]{flux_residuals_full_pss.pdf}}
\caption{\protect\label{ext_true_plots} The ratio of flux densities
for the matched sources in the simulated input (true) and extracted catalogues as a
function of extracted signal to noise (S/N) for the three
bands from the ESS and PSS maps. Also shown are the median and 3$\sigma$ clipped mean values,
calculated in bins of 0.05 in $\log(S/N)$.}
\includegraphics[scale=0.65, clip, trim=10mm 0mm 0mm 0mm]{flux_residuals_full_noiseless_pss.pdf}
\caption{\protect\label{ext_true_noiseless_plots} The ratio of flux
densities for the matched sources in the noiseless MADX ($S_{\rm noiseless}$) and extracted
catalogues as a function of extracted signal to noise for the
three bands (including point sources only). Also shown are
the median and 3$\sigma$ clipped mean values, calculated in bins of
0.05 in $\log(S/N)$.}
\includegraphics[scale=0.65, clip, trim=10mm 0mm 0mm 0mm]{flux_residuals_full_grid.pdf}
\caption{\protect\label{ext_true_grid_plots} The ratio of flux
densities for the matched sources in the simulated input (true) and extracted
catalogues as a function of extracted signal to noise for the
three bands, using the gridded position simulations (including point sources only). Also shown are
the median and 3$\sigma$ clipped mean values, calculated in bins of 0.05 in $\log(S/N)$. Note that confusion noise is not included in these simulations.}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[scale=0.6, clip, trim=6mm 0mm 0mm 0mm]{hist_ratio_pss.pdf}
\caption{\protect\label{beam_hists} The PSF--weighted ratio of the brightest to
second brightest input (true) source contributing to the extracted source,
within the beam in each band, for $>5\sigma$ sources in the extracted catalogue. }
\includegraphics[scale=0.6, clip, trim=8mm 0mm 0mm 0mm]{hist_ratio_total_tog_pss.pdf}
\caption{\protect\label{beam_hists_total} The PSF--weighted, background--subtracted, ratio of
the sum of simulated input (true) sources within a beam to the flux density of the
matched true source for $>5\sigma$ (solid line), and $>10\sigma$
(dashed line) sources in the extracted catalogue. The labels on the
Figures give the percentage of sources with ratios greater than some
particular value. The small proportion of sources where the ratio falls
below 1 are due to the PSF--weighting. }
\end{figure*}
\section{Assessing the catalogue reliability}
\label{sims}
\subsection{Simulation creation}
\label{sim_creation}
It is not enough to identify sources in the H-ATLAS SDP maps; the
robustness of the catalogue must also be determined. This is done
using realistic simulations of the observations, with the same noise
properties as the processed maps, and a realistic cirrus background,
based on IRAS measurements \citep{iras_ref}. However, only the three SPIRE bands are considered in this initial
analysis, as the PACS SDP catalogue is currently treated as an
extension to the SPIRE data
The simulated maps are randomly populated with sources generated using
the models of \citet{negrello}, which predict the number counts of
both the spheroidal and protospheroidal galaxy populations separately;
for the simulations, these predictions are combined together to give
the expected total counts, and hence the corresponding set of source
flux densities, for each band. Although \citet{wtheta} detected, in SDP data, strong
clustering for $350\,\mu$m and $500\,\mu$m--selected samples,
fluctuations due to faint sources at the SPIRE resolution are Poisson dominated,
especially at $250\,\mu$m \citep[e.g.][]{negrello04, viero09}. This
suggests that, for the present purposes, using unclustered random positions
is a sufficiently good approximation. The flux densities of all the sources in the models are reduced by 26\% at 250$\,\mu $m\, and
15\% at 350$\,\mu $m\, to improve the agreement with the observed (i.e. uncorrected) source counts
in the SDP catalogue \citep{clements}; the results of this alteration
are shown in Figure \ref{cnts_comp}. The final flux density ranges are 0.11 mJy -- 1.65 Jy
at 250$\,\mu $m\,, 0.24 mJy -- 0.83 Jy at 350$\,\mu $m\, and 0.45 mJy -- 0.59 Jy at
500$\,\mu $m\, for the simulated sources; this ensures that the simulated
maps contain a realistic background of faint sources which can
contribute to the confusion noise.
The simulations are constructed by first adding the flux of each
source in each band to the relevant position in a 1 arcsecond grid. Two versions
of the simulations are created. In the first the simulated sources are
all one pixel in size (point--source--simulations: PSS), whereas in
the second the sources are assigned a scale--length based on their
catalogue redshift (extended--source--simulations: ESS). The
scale--length is constant in physical units, and then converted to
an angular scale using standard cosmology. The ESS will obviously be a better representation of the real data, but, as
Section \ref{ext_sour} shows, MADX underestimates the flux densities of objects with sizes larger than the FWHM,
so the PSS simulations provide a useful comparison. It should be noted that the flux densities and
positions of the input sources will be the same in both cases. The
next step is to convolve the 1 arcsecond map by the appropriate {\it Herschel} PSF, also
sampled on a 1 arcsecond grid, to give a map of flux per beam covering
the full area of the SDP data. Then, the 1 arcsecond pixels are block
averaged to give 5 arcsecond pixels for the 250$\,\mu $m\, maps, and 10
arcsecond pixels for the 350 and 500$\,\mu $m\, maps.
A background representing emission from Galactic cirrus is then added
to the each map. The background value is estimated from the \citet{iras_ref} map of 100$\,\mu $m\, dust
emission and temperature by assuming a modified black-body spectrum
with $\beta = 2.0$, and scaling to the appropriate wavelength. The resolution of this IRAS map is lower than that in the SDP data,
which means that small scale structure in the cirrus is not present in
the simulations. Since the cirrus is highly structured, it is
non--trivial to generate realistic structure on smaller scales, so as
a simple approximation, the low resolution maps were used, though it
should be noted that the true cirrus background will include more
small scale features. It is clear that the real cirrus structure in
the SDP data is highly non--Gaussian, so simply extrapolating the
power spectrum to smaller scales does not significantly improve the
model background.
Finally instrumental noise is added to each pixel as a Gaussian
deviate, scaled using the real coverage maps so that the local rms is
the same as in the real data.
\begin{figure}
\centering
\subfloat[Extended source simulations]{\protect\label{flux_err_ess} \includegraphics[scale=0.6, clip, trim=0mm 4mm 0mm 4mm]{flux_error_ess.pdf}} \\
\subfloat[Point source simulations]{\protect\label{flux_err_pss} \includegraphics[scale=0.6, clip, trim=0mm 4mm 0mm 4mm]{flux_error_pss.pdf}}\\
\subfloat[Point source simulations]{\protect\label{flux_err2_pss} \includegraphics[scale=0.6, clip, trim=0mm 4mm 0mm 4mm]{flux_error2_pss.pdf}}
\caption{\protect\label{flux_err} The fractional flux density error
for the corrected extracted catalogues, ignoring any sources that
fall outside the 99.73rd percentiles. The dotted lines indicate the
expected behaviour.
}
\end{figure}
Sources are then extracted with MADX from both versions of the
simulations, following the procedure described in Section
\ref{spire_cat}. For the ESS maps, the flux densities in the three bands are again
replaced with aperture--measured values for the extended sources. The
`optical sizes' (needed to determine $a_{r}$ using Equation
\ref{ap_rad}) in this case are taken as three times the scale--size
taken from the input catalogues; this corresponds to a {\it B}--band
isophotal limit of $\sim$25 mag arcsec$^{-2}$ \citep{zhong}. Finally, the MADX catalogue is cut to
only include sources which are detected at the 5$\sigma$ level in any
of the available bands. This process is repeated 500 times, each time using a different realisation of
the input model counts, to ensure sufficient numbers of bright sources
are present at the longer wavelengths. The average number of extracted
sources which are also $>$5$\sigma$ in any band is 5881 and 5772 for
PSS and ESS respectively, which is lower than the 6876 sources present in the real SDP data; as Figure
\ref{cnts_comp} illustrates, this is because the simulated source
counts do not exactly reproduce the real SDP ones. Additionally, more
sources are found for the PSS version because of the flux
underestimation of extended sources which means that the faintest
objects fall below the catalogue cut. In the remainder of this
discussion, these MADX catalogues will be referred to as the `extracted catalogues', and the simulated input
source lists as the `simulated input catalogues'.
For each of the three bands in turn, starting with the brightest,
sources in the extracted catalogue are matched to the simulated input source that
makes the largest contribution, determined by weighting with the
filtered beam, at that extracted position. A match radius of 3 pixels
(approximately equal to the FWHM in each band) is also imposed to
ensure that a match is not made to an unfeasibly distant source. Since
the typical positional error for a $>5\sigma$ 250$\,\mu $m\, source is 2.5\hbox{$^{\prime\prime}$}\, or less, this match radius
will ensure that almost no real matches are rejected, whilst the weighting will avoid spurious matches. Once
matched, a simulated input source is removed from consideration to avoid double--matches. Considering
each band separately will allow an extracted source to have three
different simulated input counterparts, depending on where the majority of its
flux density comes from at 250, 350 and 500$\,\mu $m\,. This ensures that the
effects of source blending in the data can be properly investigated,
though it should be noted that the results are very similar if the
counterparts are found at the highest resolution, shortest wavelength
only. Full simulated input, extracted and matched catalogues for each band are
then made by combining the results from the 500 individual sets of simulations together.
The positional offsets and corresponding errors are shown in Figures \ref{xy_plots} and
\ref{pos_err}. They demonstrate that there is no significant offset
between the extracted and matched catalogues. The positional errors
for 5$\sigma$ sources are $\sim$2.4\hbox{$^{\prime\prime}$}\, at 250$\,\mu $m\, in both
versions, which agrees with the value of $2.40 \pm 0.11$\hbox{$^{\prime\prime}$}\, found
for the real SDP data by \citet{Smith}. The errors also approximately scale as $1/(S/N)$ in the 250$\,\mu $m\, band,
as predicted by e.g. \citet{ivison}. However, at low S/N there is an
enhancement over the predicted values, as illustrated for the PSS
results in Figure \ref{pos_err_pss}. This is a result of Eddington
bias causing more faint sources errors to scatter up than vice--versa; if the positional errors
are plotted against the S/N in the simulated input catalogue, which does not
suffer from this effect, then they are in better agreement with the
prediction.
Figure \ref{pos_err_pss_prior} also illustrates the improvement in
positional errors that arises from selecting sources at 250$\,\mu $m\, only
in MADX, instead of giving equal weight to all bands (flat--spectrum prior), as previously
discussed in Section \ref{spire_cat}. Greater positional accuracy
significantly enhances the efficacy of the cross--identification to
optical sources using the Likelihood Ratio method \citep{Smith}. This
is why the better positions are deemed to outweigh the slight chance
of missing red objects when using the 250$\,\mu $m\, prior.
\subsection{Catalogue correction factors}
Inspection of Figure \ref{cnts_comp} shows a clear discrepancy between
the extracted and simulated input integral counts at faint 500$\,\mu $m\,
flux densities; this occurs due to a combination of two factors. The
first, flux--boosting, is a preferential enhancement of faint source
flux densities due to positive noise peaks, that arises due to the
steepness of the faint end \citep[i.e. $S_{\rm 500\mu m} \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}} 40$ mJy;][]{clements} of the source counts. The second is a
result of blending, where several simulated input sources (which may be too faint to be included individually) are detected as one
source in the extracted catalogue.
These effects can be quantified by direct comparison of the simulated input and extracted flux densities, shown in Figure
\ref{ext_true_plots}, as a function of signal--to--noise in the
extracted catalogue for both the ESS and PSS versions. Flux correction factors are derived from the
3$\sigma$ clipped mean of these data; these are given in Table
\ref{corr_table}. Applying these factors to each extracted source
gives a statistically `flux--corrected' catalogue. It should be noted however, that
the discussion of correction factors in this Section is restricted to
sources detected at a 5$\sigma$ or greater level only.
An alternative approach to determining the catalogue correction
factors is to use a `noiseless' catalogue, created by running MADX on
the simulated maps before the addition of noise, as the comparison. As Figure \ref{ext_true_noiseless_plots} shows, this does not accurately
represent the level of flux--enhancement in the data, because, the
noiseless catalogue is also affected by source blending.
Additionally, at low S/N the noiseless--input flux densities are generally brighter
than the extracted ones, suggesting that MADX underestimates the
background subtraction in the absence of noise.
The relative contributions from the flux--boosting and source blending
can be investigated with a new set of simulated, point--source only,
maps, in which the sources are placed on a regular spaced grid, with a
70\hbox{$^{\prime\prime}$}\, separation between points, to ensure no sources overlap. The source density is also
lowered in these maps (imposed by excluding any source in the
simulated input catalogue with a 250$\,\mu $m\, flux density fainter than 6.6 mJy), so
that sufficient unique positions can be generated. Inspecting the
ratio of the extracted and simulated input fluxes -- Figure
\ref{ext_true_grid_plots} -- suggests that the majority of the
flux--enhancement seen in Figure \ref{ext_true_plots} is due to
blended sources, rather than boosting due to noise. However, the
PSF--weighted ratio of the brightest to second brightest simulated input source
contributing to each source in the extracted catalogue (Figure
\ref{beam_hists}) appears to contradict this; it shows that, even at
500$\,\mu $m\,, blending with this second source would not increase the
extracted flux density by the amount seen. The solution to this
apparent contradiction becomes clear when the PSF--weighted ratio of the contribution from
all the simulated input sources within a beam to the flux density of the simulated input
match is considered instead (Figure \ref{beam_hists_total}). Here $\sim$27\% of 500$\,\mu $m\, $> 5\sigma$ extracted sources have sufficient
simulated input sources available to boost their flux densities by a factor of 2
or more when their contributions are combined, even though their
individual effect is small. Figure \ref{beam_hists_total} also shows
that this confusion becomes negligible for $>10\sigma$ sources. This
is in broad agreement with \citet{chapin10} who find that the sub--mm
peaks they detect using a survey with larger beams, but of similar depth to H--ATLAS, generally consist of a blend of several sources.
Future versions of MADX will include a deblending step which should
reduce this effect. It should be noted that a mean sky--background of 6.8 mJy, 5.8 mJy or 4.1
mJy at 250$\,\mu $m\,, 350$\,\mu $m\, and 500$\,\mu $m\, respectively (determined from the
mean of the simulated input catalogue), is subtracted before the histograms are calculated, to
account for the background--subtraction carried out as part of the
source extraction process.
As a check on the success of the correction factors in Table
\ref{corr_table}, they are applied to the full extracted catalogues
and the fractional flux density errors (after rejecting the points which
lie outside the 99.73rd percentile) are then calculated.
As Figure \ref{flux_err} shows, these reduce with
increasing S/N, but, as with the positional errors
discussed previously, Eddington bias prevents this behaving exactly as
expected. Again, when plotted against the S/N from the simulated input catalogue
(Figure \ref{flux_err2_pss}) the difference is reduced.
\begin{figure}
\centering
\subfloat[The differential source
counts for the extracted, simulated input (true) and flux--corrected catalogues for
the three bands. Note the discrepancy between the flux--corrected
and simulated input catalogues at faint flux
densities.]{\protect\label{diff_cnt_plots_ess} \includegraphics[scale=0.6, clip, trim=0mm 3mm 0mm 4mm]{diff_cnts_paper_ess.pdf}} \\
\subfloat[The surface density correction required at each band to
correct for the catalogue incompleteness, determined from the ratio of the flux--corrected to
simulated input differential source counts. The solid dots indicate the
position of the average 5$\sigma$ limit in each
band.]{\protect\label{comp_plots_ess}
\includegraphics[scale=0.6, clip, trim=0mm 3mm 0mm 4mm]{comp_paper_ess.pdf}} \\
\subfloat[The integral source counts from
the simulated input catalogue overplotted with the flux and
surface--density corrected catalogue to demonstrate the success of
these correction factors at recovering the simulated input
values]{\protect\label{int_corr_ess} \includegraphics[scale=0.6, clip, trim=0mm 3mm 0mm 4mm]{int_corrected_ess.pdf}}
\caption{\protect\label{aps_together_ess} Extended source simulations}
\end{figure}
\begin{figure}
\centering
\subfloat[The differential source counts for the extracted, simulated
input (true) and flux--corrected catalogues for
the three bands. Note the discrepancy between the flux--corrected
and simulated input catalogues at faint flux
densities.]{\protect\label{diff_cnt_plots_pss} \includegraphics[scale=0.6, clip, trim=0mm 3mm 0mm 4mm]{diff_cnts_paper_pss.pdf}} \\
\subfloat[The surface density correction required at each band to
correct for the catalogue incompleteness, determined from the ratio of the flux--corrected to
simulated input differential source counts. The solid dots indicate the
position of the average 5$\sigma$ limit in each
band.]{\protect\label{comp_plots_pss}
\includegraphics[scale=0.6, clip, trim=0mm 3mm 0mm 4mm]{comp_paper_pss.pdf}} \\
\subfloat[The integral source counts from
the simulated input catalogue overplotted with the flux and
surface--density corrected catalogue to demonstrate the success of
these correction factors at recovering the simulated input
values]{\protect\label{int_corr_pss} \includegraphics[scale=0.6, clip, trim=0mm 3mm 0mm 4mm]{int_corrected_pss.pdf}}
\caption{\protect\label{aps_together_pss} Point source simulations}
\end{figure}
As well as the flux correction factors, we also need to completeness
of the detected catalogues, especially at faint 350 and 500$\,\mu $m\, flux densities; this is clearly seen in Figures
\ref{diff_cnt_plots_ess} and \ref{diff_cnt_plots_pss} which compare the differential source counts for
the extracted, simulated input and flux--corrected catalogues. The
lower counts are due to the failure to detect some fraction of faint sources
because of random noise fluctuations in the simulated maps or source blending. This incompleteness can be quantified by simply taking the ratio
of the flux--corrected to simulated input differential counts, to give a
source--surface--density correction. Note that this is not appropriate for correcting the flux densities of
individual sources, but rather it can be applied when making
statistical analyses of the catalogue as a whole. This correction is shown in Figures \ref{comp_plots_ess} and \ref{comp_plots_pss}, and also given
as an additional correction factor in Table \ref{comp_table}. Figures
\ref{int_corr_ess} and \ref{int_corr_pss} demonstrate the success of the density correction when applied to the integral source counts.
There is one further factor that can affect the extracted catalogue -- contamination from
spurious sources. The expected number of $\geq 5\sigma$ random noise
peaks present in the 250$\,\mu $m\, map area is only $\sim$0.05, so this should be
negligible in the SDP catalogue. Contamination from fainter sources which are boosted or blended is accounted for in the flux correction factors.
It should be noted that an alternative approach to correcting the SDP
H--ATLAS catalogue was adopted in \citet{clements}. In this case
corrections were determined from the ratio of extracted to simulated input
integral source counts. This combines the effects of incompleteness
and flux boosting, and is appropriate for recovering the correct
source counts, but not for correcting individual catalogue sources.
\section{Concluding remarks}
This paper has presented the SDP catalogue for the first observations
of the H--ATLAS survey, along with a description of the simulations
created to determine the factors needed to correct it for
the combined effects of incompleteness, flux--boosting and source blending.
The main results of this analysis are summarised below:
\begin{enumerate}
\item The extracted flux densities of 350$\,\mu $m\, and 500$\,\mu $m\, sources can be enhanced
over their simulated input values, by factors of up to $\sim$2. This predominantly affects sources with $5<$ S/N $< 15$;
\item These enhancements are shown to be due to source blending, with $\sim$27\% of $>5\sigma$ 500$\,\mu $m\, sources having sufficient simulated input
sources available within a beam to create a boosting of $\sim$2;
\item A combination of flux density and source--surface--density
corrections are necessary to correct the extracted source counts
for these factors.
\end{enumerate}
It is anticipated that future development of the MADX software will
incorporate subroutines to deal with both the effects of map
pixelization and source blending in the processing stage.
MADX is not the only source extraction method being considered for the
H--ATLAS data, but time constraints mean that it has been used for the
SDP catalogue presented here. A comparison between different source
extraction algorithms is currently ongoing; these include
SUSSEXtractor developed by \citet{savage}, as well as the `matrix filter' method of
\citet{herranz} and the `Mexican Hat wavelet' method of \citet{mh1} and \citet{mh2}. The results of this comparison will be used to improve future H--ATLAS catalogues.
This initial, uncorrected, catalogue will be available from
\verb1http://www.h-atlas.org1, though it is expected that as the data
processing steps are refined it will undergo future updates.
\section*{Acknowledgments}
The {\it Herschel}-ATLAS is a project with {\it Herschel}, which is an ESA space
observatory with science instruments provided by European-led
Principal Investigator consortia and with important participation from
NASA. The H-ATLAS website is \verb1http://www.h-atlas.org/1. U.S. participants in {\it
Herschel}--ATLAS acknowledge support provided by NASA through a
contract issued from JPL. The Italian group acknowledges partial financial support from ASI/INAF agreement n. I/009/10/0.
\clearpage
\begin{table*}
\centering
\begin{tabular}{c|c|c|c|c|c|c}
\hline
& \multicolumn{3}{c}{ESS} & \multicolumn{3}{c}{PSS} \\
Catalogue S/N & FC$_{250\mu m}$ & FC$_{350\mu m}$ & FC$_{500\mu m}$ & FC$_{250\mu m}$ & FC$_{350\mu m}$ & FC$_{500\mu m}$ \\
\hline
5.30 & 1.06 & 1.12 & 1.45 & 1.06 & 1.12 & 1.45 \\
5.94 & 1.06 & 1.18 & 1.51 & 1.06 & 1.18 & 1.50 \\
6.67 & 1.06 & 1.21 & 1.51 & 1.06 & 1.21 & 1.50 \\
7.48 & 1.06 & 1.23 & 1.47 & 1.06 & 1.23 & 1.45 \\
8.39 & 1.06 & 1.25 & 1.35 & 1.06 & 1.25 & 1.32 \\
9.42 & 1.06 & 1.27 & 1.14 & 1.06 & 1.26 & 1.11 \\
10.57 & 1.06 & 1.27 & 1.01 & 1.06 & 1.25 & 1.02 \\
11.86 & 1.05 & 1.23 & 1.00 & 1.05 & 1.20 & 1.00 \\
13.30 & 1.04 & 1.15 & 0.99 & 1.04 & 1.10 & 0.99 \\
14.93 & 1.02 & 1.04 & 0.98 & 1.02 & 1.01 & 0.99 \\
16.75 & 1.00 & 1.01 & 0.98 & 1.00 & 0.99 & 0.98 \\
18.79 & 0.98 & 1.00 & 0.98 & 0.98 & 0.99 & 0.99 \\
21.08 & 0.97 & 0.99 & 0.97 & 0.98 & 0.98 & 0.98 \\
23.66 & 0.96 & 0.99 & 0.97 & 0.97 & 0.98 & 0.98 \\
26.54 & 0.96 & 0.99 & 0.96 & 0.97 & 0.98 & 0.98 \\
29.78 & 0.96 & 0.98 & 0.96 & 0.97 & 0.97 & 0.98 \\
33.42 & 0.96 & 0.98 & 0.98 & 0.97 & 0.98 & 0.98 \\
37.49 & 0.95 & 0.98 & 0.97 & 0.97 & 0.97 & 0.99 \\
42.07 & 0.96 & 0.98 & 0.96 & 0.97 & 0.97 & 1.00 \\
47.20 & 0.95 & 0.97 & 0.99 & 0.97 & 0.97 & 0.98 \\
52.96 & 0.95 & 0.97 & 0.96 & 0.97 & 0.97 & 0.99 \\
59.43 & 0.96 & 0.94 & 0.95 & 0.97 & 0.97 & 0.97 \\
66.68 & 0.95 & 0.93 & -- & 0.96 & 0.97 & -- \\
74.81 & 0.95 & 0.92 & -- & 0.97 & 0.97 & -- \\
83.94 & 0.97 & 0.93 & -- & 0.97 & 0.97 & -- \\
94.18 & 0.97 & 0.94 & -- & 0.96 & 0.97 & -- \\
\hline
\end{tabular}
\caption{\protect\label{corr_table} The flux density correction factors (FC) at each SPIRE
wavelength, as a function of S/N in the extracted catalogue,
determined from the ratio of flux densities in the matched extracted
and simulated input catalogues. To apply the correction at some catalogue flux
density, $f_{\rm cat}$: $f_{\rm corr } = f_{\rm cat} / FC$, though
note that the density correction given in Table \ref{comp_table}
should also be applied as well.}
\end{table*}
\begin{table*}
\centering
\begin{tabular}{c|c|c|c|c|c|c}
\hline
Corrected flux density & \multicolumn{3}{c}{ESS} & \multicolumn{3}{c}{PSS} \\
(Jy) & SC$_{250\mu m}$ & SC$_{350\mu m}$ & SC$_{500\mu m}$ & SC$_{250\mu m}$ & SC$_{350\mu m}$ & SC$_{500\mu m}$\\
\hline
0.0320 & 0.31 & -- & -- & 0.40 & -- & -- \\
0.0327 & 0.75 & -- & 0.11 & 0.79 & -- & 0.11 \\
0.0335 & 0.84 & -- & 0.41 & 0.85 & -- & 0.39 \\
0.0343 & 0.84 & -- & 0.49 & 0.85 & -- & 0.49 \\
0.0351 & 0.83 & -- & 0.48 & 0.85 & -- & 0.48 \\
0.0359 & 0.85 & 0.01 & 0.46 & 0.86 & 0.01 & 0.46 \\
0.0367 & 0.84 & 0.68 & 0.39 & 0.86 & 0.71 & 0.41 \\
0.0376 & 0.85 & 1.36 & 0.31 & 0.87 & 1.36 & 0.33 \\
0.0385 & 0.83 & 1.36 & 0.26 & 0.86 & 1.36 & 0.27 \\
0.0394 & 0.83 & 1.34 & 0.25 & 0.86 & 1.34 & 0.26 \\
0.0403 & 0.85 & 1.07 & 0.25 & 0.88 & 1.11 & 0.27 \\
0.0427 & 0.85 & 0.84 & 0.21 & 0.88 & 0.86 & 0.24 \\
0.0490 & 0.84 & 0.68 & 0.19 & 0.87 & 0.71 & 0.22 \\
0.0562 & 0.84 & 0.57 & 0.22 & 0.88 & 0.60 & 0.26 \\
0.0646 & 0.83 & 0.45 & 0.29 & 0.85 & 0.49 & 0.29 \\
0.0741 & 0.82 & 0.44 & 0.41 & 0.83 & 0.43 & 0.38 \\
0.0851 & 0.82 & 0.49 & 0.49 & 0.84 & 0.43 & 0.59 \\
0.0977 & 0.84 & 0.58 & 0.61 & 0.87 & 0.55 & 0.79 \\
0.1122 & 0.89 & 0.74 & 0.74 & 0.92 & 0.70 & 0.93 \\
0.1288 & 0.95 & 0.84 & 0.78 & 1.00 & 0.91 & 0.93 \\
0.1479 & 0.93 & 0.87 & 0.72 & 1.00 & 1.05 & 0.96 \\
0.1698 & 0.89 & 0.81 & 0.67 & 0.99 & 0.99 & 0.99 \\
0.1950 & 0.93 & 0.77 & 0.72 & 1.01 & 0.99 & 0.99 \\
0.2239 & 0.87 & 0.82 & 0.87 & 1.00 & 1.00 & 1.00 \\
0.2570 & 0.90 & 0.83 & 0.77 & 1.00 & 1.00 & 1.00 \\
0.2951 & 0.95 & 0.92 & 0.92 & 1.00 & 1.00 & 1.00 \\
0.3388 & 0.96 & 0.89 & 0.73 & 1.00 & 1.00 & 1.00 \\
0.3890 & 0.94 & 0.92 & 1.00 & 1.00 & 1.00 & 1.00 \\
0.4467 & 0.98 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\
0.5129 & 0.93 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\
0.5888 & 1.02 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\
0.6761 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\
0.7762 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\
0.8913 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\
1.0233 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\
\hline
\end{tabular}
\caption{\protect\label{comp_table} The surface density correction (SC) at each SPIRE
wavelength as a function of corrected flux density, determined from the ratio of the flux--corrected to simulated input
differential counts. To apply the correction at some corrected flux
density, $f_{\rm corr}$: $f_{\rm corr\_final } = f_{\rm corr} /
SC$. The corrected flux densities given are the central bin values.}
\end{table*}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 265 |
Zahara de los Atunes é uma entidade local autónoma pertencente ao município espanhol de Barbate, na província de Cádiz, Andaluzia.
Segundo as estimativas oficiais do Instituto Nacional de Estatística (INE) relativas ao censo de 2012, Zahara de los Atunes tinha 1250 habitantes. A localidade costeira está situada no sul da província de Cádiz, ao pé da Sierra del Retín, nas margens do rio Cachón e do oceano Atlântico. A sul é fronteira ao município de Tarifa e a norte a Barbate. Fica a 73 km da capital de província e a 177 km de Sevilha.
História
As suas raízes encontrar-se-ão na época dos fenícios, embora até ao século XVI não existisse nada parecido com um núcleo urbano. As origens provêm, como o nome indica, da pesca de atum, sendo uma das "almadrabas" mais importantes da Andaluzia (a almadraba é uma arte de pesca tradicional de atum que já era usada no tempo da Antiga Roma).
Turismo
Zahara de los Atunes conserva uma das últimas grandes praias das Andaluzia sem grandes agressões urbanísticas. Mesmo assim, no verão alberga cerca de 20000 pessoas, mais que decuplicando a população residente.
Imagens
</center>
Municípios de Cádis (província)
Localidades da Espanha
Praias da Espanha
Municípios por nome da Andaluzia
Municípios da Espanha por nome
Localidades de Cádis (província)
Localidades da Andaluzia | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,859 |
La Casa al carrer Major és una obra de la Guingueta d'Àneu (Pallars Sobirà) inclosa a l'Inventari del Patrimoni Arquitectònic de Catalunya.
Descripció
Casa de pedra amb façana arrebossada, situada a la paret mestra perpendicular al cavall que suporta el llosat. Té planta baixa i tres pisos, el darrer de mansarda. A la planta baixa s'obre la porta i una petita finestra per a ventilació dels estables. Al primer pis, una balconada de fusta ocupar tota la façana, amb balustres i faldó perfilats i calats amb motius d'estrelles o rosetes en el balustre central més ample. Al segon pis s'obre un altre balcó amb balustres calats i una finestra. En el pis superior sota l'ample ràfec de la coberta hi ha dues petites finestres.
Referències
Patrimoni monumental de la Guingueta d'Àneu
Edificis de la Guingueta d'Àneu | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,516 |
Q: calculating probability mass function Not quite sure how to approach this question - let's say you're looking at a toll booth lane that can hold up to $7$ cars at a time.
$X$ is the $\#$ of cars in the lane at a randomly chosen time
with the probability $X = x$ proportional to $(x + 1)(8 - x)$.
How do you calculate the probability mass function (pmf) and then the probability that $X$ will be at least $5$? I have absolutely no idea how to calculate for the pmf, but I think calculating for $P(X > 5)$ goes like this:
\begin{array}{ll}
P(0) &= (0+1)(8-0) = (1)(8) = 8\\
P(1) &= (1+1)(8-1) = (2)(7) = 14\\
P(2) &= 18\\
P(3) &= 20\\
P(4) &= 20\\
P(5) &= 18\\
P(6) &= 14\\
P(7) &= 8\\
\end{array}
and so
$$
P(X > 5) = P(5)+P(6)+P(7) = 18 + 14 + 8 = 40.
$$
If someone could point me in the right direction, that'd be great.
A: You have overlooked the word proportional.
You don't have $P(0)=8$. Instead, you know there is a constant $c$ such that $P(0)=8c$.
Similarly, you have $P(1)=14c$, $P(2)=18c$, etc.
Now, use the fact that $P(0)+P(1)+P(2)+...+P(7)=1$ and solve for $c$. Then you will be able to calculate all values of $P$ exactly.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,945 |
Hold up your hands if you've been neglecting the mani pedi lately? Keep them up if you've noticed the plethora of perfectly painted phalanges popping up on Pinterest and you're feeling a little less than polished? Time to jump on the nail art bandwagon and have some finger fun!
No need to get out the magnifying glass and tiny paintbrushes, there is a lazy girl way to get this detailed look without the precision of a surgeon. Nail stickers are quick and easy to apply and come in all of this season's trends - lace, floral and geometric. Mecca Maxima stock nail stickers in every pattern and colour imaginable and they work just as well on toes.
If applying nail stickers to every finger still seems like it's a bit too high maintenance, the trend on the web is to highlight just your ring finger with a touch of sparkle. Try new (formaldehyde and nasty free) Australian nailpolish brand, Hello Darling in confetti.
Where: Hello Darling is available online.
Seen any amazing nail art online lately? Leave us a link in the comments below! | {
"redpajama_set_name": "RedPajamaC4"
} | 4,568 |
dvdgedvbd
hello2dd | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,578 |
Dao London has created this collection to resurrect African fashion and rebirth a new image of dominance through the exquisite wax fabric and design. As a brand promoting African fashion, we want you to be a part of the revolutionary where soon African fashion will be popular amongst every ethnic culture and can be worn on a regular basis.
The collection will feature edgy, supremacy garments ranging from dresses, co-ordinates, jumpsuits and much more for this summer season.
Purchase on All Things Ankara here.
Hey Buddy!, I found this information for you: "Campaign: "Island Girl" Dao London's Spring/Summer 2017 Collection". Here is the website link: https://www.allthingsankara.com/2017/07/campaign-island-girl-dao-londons-springsummer-2017-collection.html. Thank you. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,969 |
namespace BLEScanner
{
partial class frmMain
{
/// <summary>
/// Required designer variable.
/// </summary>
private System.ComponentModel.IContainer components = null;
/// <summary>
/// Clean up any resources being used.
/// </summary>
/// <param name="disposing">true if managed resources should be disposed; otherwise, false.</param>
protected override void Dispose(bool disposing)
{
if (disposing && (components != null))
{
components.Dispose();
}
base.Dispose(disposing);
}
#region Windows Form Designer generated code
/// <summary>
/// Required method for Designer support - do not modify
/// the contents of this method with the code editor.
/// </summary>
private void InitializeComponent()
{
this.components = new System.ComponentModel.Container();
this.serialAPI = new System.IO.Ports.SerialPort(this.components);
this.btnStopScan = new System.Windows.Forms.Button();
this.btnStartScan = new System.Windows.Forms.Button();
this.txtLog = new System.Windows.Forms.TextBox();
this.SuspendLayout();
//
// btnStopScan
//
this.btnStopScan.Anchor = ((System.Windows.Forms.AnchorStyles)((System.Windows.Forms.AnchorStyles.Bottom | System.Windows.Forms.AnchorStyles.Right)));
this.btnStopScan.Location = new System.Drawing.Point(756, 261);
this.btnStopScan.Name = "btnStopScan";
this.btnStopScan.Size = new System.Drawing.Size(100, 23);
this.btnStopScan.TabIndex = 1;
this.btnStopScan.Text = "Sto&p Scanning";
this.btnStopScan.UseVisualStyleBackColor = true;
this.btnStopScan.Click += new System.EventHandler(this.btnStopScan_Click);
//
// btnStartScan
//
this.btnStartScan.Anchor = ((System.Windows.Forms.AnchorStyles)((System.Windows.Forms.AnchorStyles.Bottom | System.Windows.Forms.AnchorStyles.Right)));
this.btnStartScan.Location = new System.Drawing.Point(650, 261);
this.btnStartScan.Name = "btnStartScan";
this.btnStartScan.Size = new System.Drawing.Size(100, 23);
this.btnStartScan.TabIndex = 2;
this.btnStartScan.Text = "&Start Scanning";
this.btnStartScan.UseVisualStyleBackColor = true;
this.btnStartScan.Click += new System.EventHandler(this.btnStartScan_Click);
//
// txtLog
//
this.txtLog.Anchor = ((System.Windows.Forms.AnchorStyles)((((System.Windows.Forms.AnchorStyles.Top | System.Windows.Forms.AnchorStyles.Bottom)
| System.Windows.Forms.AnchorStyles.Left)
| System.Windows.Forms.AnchorStyles.Right)));
this.txtLog.Location = new System.Drawing.Point(13, 12);
this.txtLog.Multiline = true;
this.txtLog.Name = "txtLog";
this.txtLog.ReadOnly = true;
this.txtLog.Size = new System.Drawing.Size(843, 243);
this.txtLog.TabIndex = 3;
//
// frmMain
//
this.AutoScaleDimensions = new System.Drawing.SizeF(6F, 13F);
this.AutoScaleMode = System.Windows.Forms.AutoScaleMode.Font;
this.ClientSize = new System.Drawing.Size(868, 296);
this.Controls.Add(this.txtLog);
this.Controls.Add(this.btnStartScan);
this.Controls.Add(this.btnStopScan);
this.MaximizeBox = false;
this.Name = "frmMain";
this.Text = "BLE Scanner";
this.Load += new System.EventHandler(this.frmMain_Load);
this.ResumeLayout(false);
this.PerformLayout();
}
#endregion
private System.IO.Ports.SerialPort serialAPI;
private System.Windows.Forms.Button btnStopScan;
private System.Windows.Forms.Button btnStartScan;
private System.Windows.Forms.TextBox txtLog;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,492 |
Bright and totally furnished apartment for sale located in a central well maintained residence with common pool, very close to the main street and the Campanario Shopping Centre. The apartment boasts of one bedroom with fitted wardrobe, one modern bathroom, open plan fitted kitchen and living room. With a private and spacious terrace, it is perfect for investment! | {
"redpajama_set_name": "RedPajamaC4"
} | 2,287 |
Q: Add href hyperlink to row or field in datatable I've seen a lot of questions about this, however I cannot seem to get it working.
I have a datatable but I cannot get it to work. I use python-flask with bootstrap and I change a pandas dataframe to a html table with to_html().
<table width="100%" class="table table-striped table-bordered table-hover dataTable" id="dataTables-example"><thead>
<tr style="text-align: right;">
<th>id</th>
<th>user</th>
<th>status</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>1</td>
<td>1</td>
</tr>
<tr>
<td>2</td>
<td>1</td>
<td>1</td>
</tr>
</tbody>
</table>
and at the bottom of the body I have:
<script>
$(document).ready(function() {
$('#dataTables-example').DataTable({
"bDestroy": true,
"deferRender": true,
"columns": [
{ "data": "id" },
{
"data": "weblink",
"render" : function(data, type, row, meta){
if(type === 'display'){
return $('<a>')
.attr('href', data)
.text(data)
.wrap('<div></div>')
.parent()
.html();
} else {
return data;
}
}
}
]
});
});
</script>
I've looked at a lot of awnsers however they all contain the data as json in the javascript while my data is already in the html.
A: Use columnDefs when you have a DOM <table> and only need to target one or few columns :
$('#dataTables-example').DataTable({
destroy: true,
deferRender: true,
columnDefs: [{
targets: 0, //<-- index of column that should be rendered as link
render : function(data, type, row, meta){
if (type === 'display'){
return $('<a>')
.attr('href', data)
.text(data)
.wrap('<div></div>')
.parent()
.html();
} else {
return data;
}
}
}]
})
It works here -> http://jsfiddle.net/j9ez0sbj/
A: You have 3 columns in your html table but only define 2 columns in your initialization.
From datatables documentation for the columns initialization option:
Note that if you use columns to define your columns, you must have an entry in the array for every single column that you have in your table (these can be null if you don't wish to specify any options).
Depending on what your intent is, at the very least you need to add a definition for the third column, so something like this:
"columns": [
{ "data": "id" },
{
"data": "weblink",
"render" : function(data, type, row, meta){
if(type === 'display'){
return $('<a>')
.attr('href', data)
.text(data)
.wrap('<div></div>')
.parent()
.html();
} else {
return data;
}
}
},
{ "data": "status" } // Third column definition added
]
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 765 |
{"url":"http:\/\/math-sciences.org\/event\/hugh-morton\/","text":"# Hugh Morton (University of Liverpool)\n\n\u2022 This event has passed.\n\n# A skein-theoretic model for the double affine Hecke algebras\n\nWe will illustrate pictorially the use of ${\\mathbb Z}[s^{\\pm 1}, q^{\\pm 1}]$-linear combinations of braids in the thickened torus $T^{2}\\times I$ to construct an algebra induced by composing $n$-string braids.\n\nWe will show, with the help of pictures, that this algebra satisfies the relations of the double affine Hecke algebra $\\ddot{H}_{n}$, which will be introduced algebraically.\n\nWe will finish with a rather speculative plan to include closed curves in our model in an attempt to incorporate earlier work with Peter Samuelson on the Homfly skein of $T^{2}$ into the setting of the algebras $\\ddot{H}_{n}$. This is done with an eye on the elliptic Hall algebra and the work of Schiffman and Vasserot, which we will discuss very briefly.\n\n## Details\n\nDate:\nMay 15, 2019\nTime:\n1:00 pm - 2:00 pm UTC+0\nEvent Categories:\n,\n\n## Venue\n\nRoom 101\n2-5 Kirkby Place\nPlymouth, PL4 6DT United Kingdom","date":"2021-05-17 22:22:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.35903051495552063, \"perplexity\": 1278.2083869249946}, \"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-2021-21\/segments\/1620243991870.70\/warc\/CC-MAIN-20210517211550-20210518001550-00248.warc.gz\"}"} | null | null |
https://www.sfgate.com/opinion/diaz/article/Blowing-up-boxes-left-and-right-3181706.php
Blowing up boxes, left and right
On the Growing Movement for Pension Reform
By John Diaz
Published 4:00 am PDT, Sunday, July 11, 2010
San Francisco Public Defender Jeff Adachi stands in his in San Francisco, Calif., office on Friday, Feb. 20, 2009. Adachi is the producer of "You Don't Know Jack: The Story of Jack Soo", which will be featured in the 2009 San Francisco International Asian American Film Festival. less
San Francisco Public Defender Jeff Adachi stands in his in San Francisco, Calif., office on Friday, Feb. 20, 2009. Adachi is the producer of "You Don't Know Jack: The Story of Jack Soo", which will be featured ... more
Photo: Kim Komenich, The Chronicle
San Francisco, a union town and progressive haven, might seem to be the unlikeliest place for a battle over whether to pare back benefits to public employees. Public Defender Jeff Adachi, a stalwart of the left, is the equally unlikely champion of a pension reform initiative that has jolted organized labor into full combat mode even as the measure awaits certification for the ballot.
But the crisis over escalating pension and health care costs that has been brewing for years - and ignored by all but a few lonely fiscal conservatives - has reached the breaking point.
"This is like the dot-com bubble bursting," said Susan Kennedy, chief of staff to Gov. Arnold Schwarzenegger, whose attempt to make pension reform a centerpiece of his 2005 agenda proved a nonstarter in Sacramento.
In recent weeks, Schwarzenegger reached deals with six of the state's 12 employee bargaining units to make pension benefits less generous to new hires. Benefits to existing retirees would not be affected. The governor has insisted that completion of the 2010-11 budget must be accompanied by pension reform - and, this time, he is likely to get it.
The sense of urgency about the rising burden of retiree benefits has hit local governments in areas urban and rural, rich and poor, conservative and liberal. Los Angeles Mayor Antonio Villaraigosa, a Democrat, is asking voters to scale back pensions for police and firefighters. Oakland Councilwoman Rebecca Kaplan, seeking a niche against establishment favorite Don Perata in the mayor's race, has proposed a ballot measure to require new police officers to contribute more toward their pension. Rising pension costs represent an estimated $91 million of a $142 million budget deficit in San Bernardino County.
The origin of these fiscal crises can be traced to 1999, when Gov. Gray Davis and the Democrat-controlled Legislature approved a succession of bills that enriched pension benefits for state workers. Davis and the legislators made these lavish commitments on the faulty assumption that they would not cost the general fund a cent, that robust investments would keep the system fully funded for many years to come.
Many county and city governments then gave similar deals to their workers.
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It looked like easy money in the headiness of 1999. It would have worked out as painlessly as advertised if the Dow Jones industrial average was at 25,000 today, instead of hovering around 10,000. When investments fall short, governments must dig into their general funds.
"It's like going to Vegas with your brother-in-law's paycheck," Adachi said of the retirement funding system. "It doesn't matter if you win or lose, you're going to be made whole."
To put the shortfalls in perspective, the state will be spending more than $6 billion this year to cover pension and retiree health care costs - roughly equal to its annual support for the UC and CSU systems, where the 670,000 students are being hit with sharp fee increases.
Adachi emphasized that he was not blaming workers for the miscalculations behind this mess. But he said it was in everyone's interest to develop a sustainable system. His ballot measure would increase all city workers' contributions (which range from nothing to 7.5 percent) to a baseline of 9 percent. Police and fire employees, who now contribute 7.5 percent (new hires, 9 percent), would contribute 10 percent.
"As pension costs increase, you see other services - vital services - squeezed," Adachi said. Therein lies his appeal to the left: These reforms are vital to preserving programs they cherish.
Some erstwhile advocates of pension restraint have balked at Adachi's plan. Mayor Gavin Newsom called it "extraordinarily complicated" and flawed, with potential for "unintended consequences." Newsom said it was better to work with labor on such reforms, as his administration has done in gaining incremental reforms on pension and health care. "What this does," Newsom said, "is shut that process down."
San Francisco's business community kept its distance from Adachi's signature drive, worried that the presence of a perceived anti-labor initiative could work against moderate candidates for the Board of Supervisors in November.
The unions have signaled that they are not going to swallow these cutbacks without a fight. They have enlisted Chris Lehane, a master strategist with stripes from the Clinton-Gore media wars. He suggested Adachi was using the issue as a springboard to a 2011 mayoral bid.
Don't be surprised if the impact on police and firefighters - heroes in any public opinion poll - becomes a central focus of the opposition campaign. Never mind that they represent about 20 percent of the affected workforce or that their pension contribution will rise no more than 2.5 percent of salary - compared with Muni operators and elected officials, who would go from zero to 9 percent. Expect to see plenty of uniforms in labor's campaign mailers.
"I'm not sure that taking on firefighters or police officers is the smartest route to political success," Lehane said. "You might as well take on Mother Teresa while you're at it - and call for Tim Lincecum to be traded to the Dodgers."
Sighed Adachi, "This is going to be a fun campaign."
It's also going to be an important one, with unorthodox alliances.
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NBA Rookie of the Year 'Making the Case' Part 1: Cade Cunningham's All-Around Versatility
Cade Cunningham has been making the case for himself as an NBA Rookie of the Year candidate since he was drafted by the Brooklyn Nets with their first round pick in 2018. His versatility on both ends of the floor, along with his work ethic and willingness to learn have made him a fan favorite early on.
The nba mock draft 2021 is a series of articles that look into the NBA Rookie of the Year race. In this first part, we take a look at Cade Cunningham's all-around versatility.
Note: This is the first in a five-part series on how five different rookies might win the NBA Rookie of the Year Award.
Cade Cunningham was a consensus five-star prospect coming out of high school in 2019, with every major recruiting agency ranking him among the top — if not the top — prospects.
The Arlington, Texas native was the greatest player on the nation's top-ranked high school team and entered college as a versatile, high-character leader who was as skilled and polished off the court as he was on it.
Cunningham went on to play college basketball at Oklahoma State University, where he lived up to his high school reputation before entering the NBA Draft in 2021.
Cade Cunningham was selected first overall by the Detroit Pistons.
During the 2021 NBA Summer League, rookie Cade Cunningham of the Detroit Pistons is defended by Aamir Simms of the New York Knicks. | Getty Images/Ethan Miller
Coming into college, it was difficult to find flaws in the 6-foot-8 guard's game, and it was much more difficult coming out of Stillwater. In mock drafts, Cunningham was widely projected to go first overall to the Detroit Pistons.
In his lone season with the Cowboys, Cunningham was a consensus All-American and the Big 12 Player of the Year. In 26 games, he averaged 20.1 points, 6.2 rebounds, and 3.5 assists.
The 20-year-old was pushed to do more with less and was the primary focus for opposing defenses every night since he did not choose to attend Duke, North Carolina, Kentucky, or any other college basketball powerhouse. Despite this, he produced.
Cunningham was described as a "franchise-changing offensive orchestrator with silky scoring talents, flexible defense, and a winning mindset" in The Ringer's 2021 Draft Guide.
The argument for Cunningham as Rookie of the Year is based on his overall flexibility.
#Pistons Cade Cunningham's skills, according to Troy Weaver: "As a player, you should be versatile. He brings a winning mentality to the table, therefore he has a lot to offer."
He went on to say that there are some parallels to Grant Hill.
June 23, 2021 — Rod Beard (@detnewsRodBeard)
It will be because of his all-around ability that the Pistons rookie wins the honor. Cunningham is a playmaking point guard who stands 6-8 and weighs 220-pounds. He has an instant size advantage over almost every other guard in the league.
He is, however, a talented passer with a high IQ and great court vision. On and off the court, he's wise beyond his years.
Cunningham is at ease at the point and setting the pace for his team; he understands when to go out and run in transition and when to calm things down and run a half-court offense.
The Rookie of the Year in the Big 12 for 2020-21 also possesses a diverse scoring arsenal. His size enables him to outmuscle opponents at the hoop, and he hit 41.2 percent from three at OSU. Defenses must respect his passing skill and unselfishness, thus driving lanes often open up for him.
Cunningham's most major question mark is whether he's athletic enough to keep up with faster guards defensively. However, because of his height and length — as well as his flexibility — Detroit will be able to flip ball screens on defense or keep him on a player his size rather than a shifty guard.
What might Cunningham's Rookie of the Year campaign look like?
Cade is one of the favorites to win Rookie of the Year as the No. 1 overall selection. But a few things have to fall into place for it to happen. He must deliver statistically since numbers are usually crucial in prize voting.
Something along the lines of the first-team all-Big 12 performer's collegiate stat line would put him ahead of the pack in the award race. Cunningham will most likely be the front runner if he can average more over 20 points per game while also demonstrating his offensive flexibility with good rebounding and assist statistics.
The most important element of the No. 1 pick's case, though, may be whether or not he can lead a failing Detroit team to victory.
With a 20-52 record last season, the Pistons finished dead last in the Eastern Conference. Cunningham isn't expected to lead the club to the playoffs in his first season, but the team will need to demonstrate some progress. The magic number may need to be closer to 30 victories.
The preseason favorite to win the award is Detroit's new franchise cornerstone. Cunningham will enter his second year in the league with some hardware if he performs like he did at Oklahoma State and demonstrates he can start to create a winning culture in Detroit.
Sports Reference provided all statistics.
With his latest endorsement deal, Cade Cunningham is following in the footsteps of LeBron James, Michael Jordan, and a slew of other NBA stars.
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Cade Cunningham is a 6'7 point guard, who has been described as a tremendous athlete, and an unbelievable finisher. He was drafted in the second round of the 2018 NBA Draft by the Philadelphia 76ers. Reference: cade cunningham height.
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\section{Introduction}
In the last few years, there have been several advances in deep learning and artificial intelligence for solving problems in agriculture, and a lot of this innovation is driven by a large amount of data at our disposal. More specifically, a vast amount of data is generated with information about crop fields, crop type, yield growth, plant phenotyping, and plant breeding statistics. Additionally, a lot of visual information is available that can be exploited to solve problems such as detecting anomalies; for example, taller crops showing erroneous growth and spanning larger areas can be valuable insight. While this data has been used to experiment with a wide range of machine learning and deep learning models, most of the available data in agriculture is either entirely unlabeled or partially labeled, which motivates us to tackle certain problems using an unsupervised approach. We primarily address the problems associated with the downside of data labeling, the cost of time pertaining to large-scale data labeling, and the expensive human cost and effort associated with agricultural big data.
\\ \\
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{umap_3d.png}
\caption{UMAP visualization of representations learned by CLAWS.}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=1\linewidth]{diagram2.png}
\caption{Illustration of the architecture in CLAWS.}
\end{figure*}
A vast majority of the modern unsupervised learning research has been driven by contrastive learning and similar self-supervised learning procedures. Contrastive learning is a popular method that learns representations of images without requiring any image labels. We directly build off of the contrastive learning framework SimCLR, presented in A Simple Framework for Contrastive Learning of Visual Representations \cite{chen2020simple}. In SimCLR, each image is transformed into two correlated views. These views are then fed through a base network, ResNet, to represent each view. These representations are then reduced in dimensionality via a Multilayer Perceptron (MLP) and compared using a contrastive loss that maximizes agreement between representations of the same image. We use the same contrastive loss as SimCLR, Normalized Temperature-scaled Cross-Entropy (NT-Xent). This loss is important \citep{NEURIPS2020_70feb62b} as the contrastive loss removes the notion of instance classes by directly comparing images features while the image transformation defines the invariances encoded in the features. \\
In this paper, we present an alteration of contrastive learning that uses \textit{focus training}. We use a method similar to \citep{chen2020simple} along with the addition of our attention head. This type of an attention head allows us to focus on important generated features for image representation. The procedure we follow consists of two networks, one to generate representations of a given input image and the second to generate representations of a \textit{crop image}. Representations from the cropped image will be passed to our attention head to focus on important features from both images representations. Lastly, the NT-Xent loss function is applied to these outputs. In addition to contrastive learning, we combine this type of learning with additional supervision to see the performance of this combination of methods while performing image clustering. In essence, we have two additions to SimCLR, the use of a crop model to generate representations of crop images and the conception of an attention model to focus the contrastive loss comparisons.
At a higher level, contrastive learning works by performing deep comparisons on images and backpropagating errors on the framework used to do these image representations. Before we were able to do this type of training, we relied on methods like principal component analysis \citep{ZITKO1994718}, independent component analysis \citep{ica} and self-organizing maps \citep{KOHONEN19981} between other methods that were traditional and more classical but were the key to open the idea of what we have today as deep representation learning methods. While these classical methods help lay a good foundation, there has been a lot of progress in the last few years, and we have numerous methods now that extend the idea behind contrastive learning in different ways. We can start with SimCLR, where the creation of augmented images enforces the training process; Big Self-Supervised Models are Strong Semi-Supervised Learners (SimCLRv2) \citep{chen2020big} in which they work with the combination of label amounts and network sizes. SwAV \citep{caron2021unsupervised} takes contrastive learning methodology without having pairwise comparisons. Contrastive Clustering \citep{li2020contrastive} introduces a cluster-level contrastive head in combination with an instance-level head, and Cluster Analysis with Deep Embeddings and Contrastive Learning \citep{sundareswaran2021cluster} uses a three-prolonged approach with an instance-wise contrastive head, a clustering head, and an anchor head to perform efficient image clustering. Although our idea is based on these preceding works, other interesting works (\citep{bachman2019learning, oord2019representation, wu2018unsupervised, hjelm2018learning, he2020momentum, DBLP:journals/corr/abs-2006-07733, Chen2021ExploringSS}) have also been proposed that use contrastive learning and bring out different perspectives.
\begin{figure*}[t!]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{composition3.jpg}
\label{fig:test1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{composition4.jpg}
\label{fig:test2}
\end{minipage}
\caption{3D plot containing clusters and outliers (blue) within a specific class. \textit{Left} represents Corn samples and \textit{Right} is showing Cotton samples. For \textit{Left} top and bottom row represents green cluster, images at the right represents red cluster and left represents blue cluster. The \textit{Right} has a column of images at the right that represents red cluster, bottom images are to show the green cluster and finally the top images represents outliers. }
\end{figure*}
\section{Methods}
An overview of our model architecture is given in Figure 2. Similar to SimCLR, our method learns representations by maximizing agreement between differently augmented views of the same image. The first augmentation retains the full image size and is fed into a base network that is optimized for full-size images. The second augmentation takes a small crop from the original image and is fed into a base network that is optimized for small crops of images. An attention mask is then applied to both of the representations generated by the base networks, and the attention-filtered representations are then compared using a contrastive loss. The differences between our model and SimCLR can be categorized into 3 major components: 1) the way images are fed into for inference before comparison, 2) the use of two separate base networks instead of one siamese network, and 3) application of an attention mask to the output of each base network before the contrastive loss is calculated. This is also complemented with the use of weak supervision, the two parts of our model (full image network and crop image network) are trained by using cross-entropy loss and NT-Xent.
Our models outputs can be seen as:
\begin{equation}
\textit{\textbf{H}}_i=f_{\theta}(\textit{\textbf{X}}_i)=\textnormal{EfficientNet}(\textit{\textbf{X}}_i)
\end{equation}
\begin{equation}
\textit{\textbf{H}}_j=g_{\theta}(\textit{\textbf{X}}_j)=\textnormal{EfficientNet}(\textit{\textbf{X}}_j)
\end{equation}
were $\textnormal{\textbf{X}}_i$ and $\textnormal{\textbf{X}}_j$ are a pair of augmented images drawn from \textbf{X} which is the original image. The hidden layers are $\textit{\textbf{H}}_i$ and $\textit{\textbf{H}}_j$ representing the output of the $f_{\theta}$ and $g_{\theta}$, now the hidden layers are then passed to a projection head to produce image representation with 32 features, i.e, $\textnormal{\textbf{Z}}_i=p_1(\textit{\textbf{H}}_i)$ and $\textnormal{\textbf{Z}}_i=p_2(\textit{\textbf{H}}_j)$. To calculate the attention output we send $\textnormal{\textbf{Z}}_i$ to our attention network $m_{\theta}$ to generate \textbf{\textit{M}} which is a mask that use to focus training on the important features (further explanation in section 3.3).
\subsection{Dataset}
The single dataset used for this research is composed of 11 classes of images. Data was obtained while driving over different fields containing one of the possible crops types. A total of 227,060 samples were collected and labeled. The time in which they were collected varies, therefore there is a different variation of lighting over the dataset. The labels of the data are composed by crop fields of Wheat, Cotton, Sorghum, Corn, Peanuts, No crop, Oats, Soybeans, Canola, Sugar, WheatStubble.
\subsection{Preprocessing}
SimCLR augmented images with a composition of a random crop, random flips (horizontal and vertical), and color distortion. This composition is applied twice to the same image to generate two distinctly augmented images. We follow the same augmentation process, except performing Gaussian blur. We apply two separate augmentation compositions to generate augmented images (full image and crop image). Note that when SimCLR applied random cropping, the crops were always resized to 100x100 to be fed into the base network. Our base networks have different input sizes, so we do not resize images after they are cropped.
We use two distinct base networks, $f_{\theta}$ and $g_{\theta}$ for the full and crop networks respectively. The network used for the full image takes RGB images of size 120x190 as input, while the crop network takes RGB images of size 32x32 as input. Both networks are a standard EfficientNet \citep{tan2020efficientnet} architecture, and both produce output embeddings of the same size, 32 features. Using two separate base networks, rather than feeding both image augmentations through the same base network as SimCLR does, provides two noticeable advantages. First, we do not need to enlarge small crops to fit an architecture, which can produce unintended artifacts in the image. Second, each architecture is more specifically optimized for global or local views.
\subsection{Attention Mask}
The main innovation in our framework is the incorporation of an attention mask. The mask is generated with a 2-layer perceptron that takes the output of $g_{\theta}$ as input. The output tensors of both $g_{\theta}$ and $f_\theta$ are multiplied by this mask before they are compared using a contrastive loss. This created the attention mechanism that we wanted, we created hard attention meaning that the output from the attention head is going to be either 0 or 1 (\textbf{\textit{M}}). Therefore ${\textbf{Z}^{\prime}}_i = {\textbf{Z}}_i \cdot {\textbf{\textit{M}}}$ and ${\textbf{Z}^{\prime}}_j = {\textbf{Z}}_j \cdot {\textbf{\textit{M}}}$ to then be use to calculate the loss using NT-Xent.
Both base networks need to be able to encode the same concepts in the same corresponding regions of the embeddings they produce. The same mask is applied to both embeddings, so if the output of $g_{\theta}$ contains mainly a strong concept, the mask network will remove out every region except that. If the same region in $f_\theta$ output contains a similar concept, then the two embeddings will be very similar after the mask is applied.
\section{Results}
The models were trained for 300 epochs with a batch size of 55 and an Adam optimizer. We train with this batch size because of the way we were gathering the images for the step, here instead of randomly picking images we specify 5 images per class so that in every step we can train the model in each class.
\subsection{Evaluation}
To perform evaluation we wanted to focus on the quality of the image representation. Therefore, we compared ours against SimCLR with an adaptation, similar to our model we added a supervised section for it. Now, SimCLR and CLAWS were trained using NT-Xent in combination with supervision by adding an MLP to generate labels from the outputs of the projection head. This way we can do a fair comparison over our agricultural dataset. To evaluate both models after training we did not used the classifier section, we generated the representation of the images, i.e., ran the models over each image up to the output of the projection heads. Given the creation of all the represenation, we pass it to a K-Means clustering method and generated labels for each data point. Finally, we calculated the quality by computing NMI, AMI and ARI scores.
\begin{table}[h!]
\centering
\begin{tabular}{|c | c | c | c|}
\hline
Method & NMI & ARI & AMI \\ [0.5ex]
\hline\hline
SimCLR & 0.6101 & 0.2873 & 0.6101 \\
\textit{CLAWS}(Ours) & 0.7325 & 0.5069 & 0.7324 \\ [1ex]
\hline
\end{tabular}
\end{table}
In table 1 we can see that our results outperforms SimCLR with supervision. This is also specific for the quality of the represenation of the images. Meaning that by focusing the training with the use of an attention head does helps on focusing on features that relate between the corp image and the full image.
\subsection{Gaussian Mixture Models}
Furthermore, we implemented Gaussian Mixture Models (GMM) over our image representation. Now we did not used the full dataset, we focus on seeing how our model behaved within a class. To perform this we picked \textit{Corn} and \textit{Cotton} and set the output of GMM to be two clusters and an additional one to detect outliers. In figure 3, we can see the output of GMM in the two mentioned crop types. As mentioned before the \textit{Left} represents Corn and the \textit{Right} represents Cotton. For \textit{Left} image the top and bottom row represents green cluster, images at the right represent red cluster and the left represents blue (outliers) cluster. The \textit{Right} has a column of images at the right that represents red cluster, the bottom images are to show the green cluster, and finally, the top images represent outliers. An important thing to mention about figure 3 is that we are randomly choosing three dimensions from the 32 that represent an image. Therefore, we are not going to see the full relation between clusters in just three dimensions but we can see that even with the use of the raw representation there is a relation preserved in the plot.
Focusing on outliers is our main reason for using GMM, having the ability to generate image representation on then pass it to these models to detect outliers can help in the reduction of the inspection time. It also allows fast customization depending on the task, because it is an unsupervised method, we can change the cluster dimensions and get fast results from it. An example of its use can be seen in figure 3, \textit{Left} contains 19,792 samples of Corn the GMM detects 315 outliers in which if we inspect them, images that have residue on the ground, are empty, the crop wrongly planted or bad image capture. This means that even when the model generates a similar representation for a specific class, we can detect, within a class, bad samples. Another example, \textit{Right} shows samples from Cotton, with 66,053 samples and GMM detected 3,401 outliers in which we can also see that for both classes the outliers resemble. Here, again, the images were empty, not correctly taken, contain residues and bad plantation. In addition to GMM clusters, if we talk about the other clusters, we can notice that there is no big issue separating them. The reason to be separated is that these crops can be going through a different stage of growth or other reasons. In figure 3, \textit{Left} top row we see images that contain weeds, the bottom row show images captured closer. On the \textit{Right} bottom, we notice images with tire marks on the ground. Therefore, they are assigned to another cluster (within a class) because of many reasons (leaf size, plant height, and image brightness between other factors).
\section{Conclusion}
This work built upon SimCLR to achieve better representations of images using contrastive learning combined with supervision. Our framework incorporated two distinct base networks and an attention mask, which allowed the network to learn and recognize parts that strongly represent images. With this methodology, we created CLAWS and were able to show encouraging results within class clusters. Further work on this relies on completely unsupervised training.\\
| {
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} | 2,020 |
\section{Implementation and Training Details}
\label{app:implementation}
\subsection{BART-based models} For generation we use the pre-trained BART model implemented in the fairseq library. The library and pre-trained models are BSD-licensed. We use the BART-large checkpoint (400M parameters) and further finetune the model for 10 epochs on 2 RTX 2080Ti GPUs. We use the same parameters as suggested in the fine-tuning of BART for the CNN-DM summarization task by fairseq and set MAX-TOKENS to 1024. The training time is 100-140 minutes, depending on the chosen setup (with or without context information).
During inference, we generate candidates using a top-k random sampling scheme \cite{fan-etal-2018-hierarchical} with the following parameters: length penalty is set to 1.0, n-grams of size 3 can only be repeated once, temperature is set to 0.7, while the minimum and maximum length of the sequence to be generated are 7 and 256 accordingly.
\subsection{BERT-based models}
For the automatic assessment of fluency and argument quality, we use the bert-base-cased pre-trained BERT version, as implemented in the huggingface library. The library and pre-trained models have the Apache License 2.0. We finetune the model for two epochs and use the parameters suggested in \citet{skitalinskaya2021}. The accuracy of the trained model for fluency obtained on the train/dev/test split suggested by the authors \cite{toutanova-etal-2016-dataset} is 77.4 and 75.5 for argument quality.
For labeling the missing or unassigned revision types, we use the same bert-base-cased pre-trained BERT model, but in a multi-label setup, where we consider the following 6 classes: claim clarification, typo or grammar correction, correcting or adding links, changing the meaning of the claim, splitting the claim, and merging claims. We fine-tune the model for two epochs using the Adam optimizer with a learning rate of 1e-5 and achieve a weighted F1-score of 0.81.
\section{Claim Optimization baselines}
For comparison we provide two additional baseline sequence-to-sequence model architectures, which help identify the complexity of the model needed for the task at hand:
\textbf{LSTM.} Our first baseline is a popular LSTM variant introduced by \citet{wiseman2016}. We use the \textit{lstm\_wiseman\_iwslt\_de\_e} architecture, which is a two-layer encoder and decoder LSTM, each with 256 hidden units, and dropout with a rate of 0.1 between LSTM layers.
\textbf{Transformer.} The second model is based on the work of \citet{vaswani2017}. We use the \textit{transformer\_iwslt\_de\_en} architecture, a 6-layer encoder and decoder with 512-dimensional embeddings, 1024 for inner-layers, and four self-attention heads.
Tables \ref{tab:exp1-appendix} and \ref{tab:exp1-details-appendix} compare the automatic evaluation scores of all generation-reranking combinations.
\input{table-automatic-scores}
\input{table-automatic-scores-details-appendix}
\subsection{Automatic Evaluation}
We use the following python packages and scripts to perform automatic evaluations: nltk (BLEU \cite{papineni-etal-2002-bleu}), rouge-score (ROUGE \cite{lin-2004-rouge}), \href{https://github.com/cocoxu/simplification/SARI.py}{https://github.com/cocoxu/simplification/ SARI.py} (SARI \cite{xu-etal-2016-optimizing})
\section{Claim Optimization Examples}
For all eight optimization categories, we provide one or more examples illustrating each action in Table \ref{tab:category-examples}.
\input{table-category-examples}
\section{Manual Quality Assessment Guidelines}
Figure \ref{fig:annotation-guidelines} shows the annotation guidelines for the Amazon Mechanical Turk study.
\input{text-box-instructions}
\section{System Outputs}
Table \ref{tab:ranking-examples} provides examples of candidates selected by different reranking strategies along with human references illustrating common patterns found in the results. Table \ref{tab:context-examples} provides examples of candidates generated with and without utilizing context knowledge with insertions and deletions being highlighted in green and red fonts accordingly.
\input{table-ranking-examples}
\input{table-context-examples}
\section{Introduction}
\label{sec:introduction}
The delivery of arguments in clear and appropriate language is a decisive factor in achieving persuasion in any debating situation, known as {\em elocutio} in Aristotle's rhetoric \citep{elbaff:2019}. Accordingly, the claims composed in an argument should not only be grammatically fluent and relevant to the given debate topic, but also unambiguous, self-contained, and more.
Written arguments therefore often undergo multiple revisions in which various aspects are optimized \cite{zhang-litman-2015-annotation}.
As detailed in Section~\ref{sec:relatedwork}, extensive research has been done on the automatic assessment of argument quality and the use of large language models on various text editing tasks. Yet, no work so far has studied how to actually improve \textit{argumentative} texts.
However, developing respective approaches is a critical step towards building effective writing assistants, which could not only help learners write better argumentative texts \cite{wambsganss:2021}, but also rephrase arguments made by an AI debater \cite{slonim:2021}. In this work, we close the outlined gap by studying how to employ large language models for rewriting argumentative text in order to optimize its delivery.
\begin{figure}[t]
\centering
\includegraphics[scale=1.0]{example-revisions-2}
\caption{Examples of different optimized versions of an \emph{original claim} found on the debate platform {Kialo}. All optimizations were generated by the approach proposed in this paper, using the \emph{debate topic} as context.}
\label{example_revisions}
\end{figure}
We start by defining the new task of \emph{claim optimization} in Section~\ref{sec:taxonomy}, and adjust the English claim revision dataset of \citet{skitalinskaya2021} for evaluation. This task requires complementary abilities. On the one hand, different types of quality issues inside a claim must be detected, from grammatical errors to missing details. If not all quality aspects can be improved simultaneously, specific ones must be targeted. On the other hand, improved claim parts need to be integrated with the context of the surrounding discussion, while preserving the original meaning as far as possible. Figure~\ref{example_revisions} shows three exemplary optimizations of a claim from the debate platform {\em Kialo}. The first elaborates what the consequence of weaponization is, whereas the second rephrases the claim to clarify what weaponizing means, employing knowledge about the debate topic. The third renders the stance of the claim explicit. We observe that different ways to optimize a claim exist, yet the level of improvement differs as well.
As an initial approach to claim optimization, we propose to combine the capabilities of large language models with quality assessment in a controlled generation (Section~\ref{sec:approach}). First, a fine-tuned sequence-to-sequence model produces several candidate optimizations of a given claim. To optimize claims, we condition the model on discourse context, namely the debate topic and the previous claim in the debate. The key to finding the best candidate is to then rerank candidates with respect to three complementary quality metrics: \emph{grammatical fluency}, \emph{meaning preservation}, and \emph{argument quality}. Such reranking remains understudied in generative tasks within computational argumentation.
In automatic and manual evaluation (Section~\ref{sec:experiments}), we demonstrate the effectiveness of our approach, employing fine-tuned BART \cite{lewis-etal-2020-bart} for candidate generation.
Our results stress the benefits of quality assessment (Section~\ref{sec:results}). Incorporating context turns out
especially helpful for making shorter claims---where the topic of the debate is difficult to infer---more self-contained. According to human annotators, our approach improves~60\% of all claims and harms only~16\%, clearly outperforming generation without reranking.
To gain further insights, we carry out a manual annotation of 600 claim optimizations and identify eight types typically found in online debate communities, such as {\em elaboration} and {\em disambiguation} (Section~\ref{sec:analysis}). Intriguingly, our approach covers a variety of optimization types similar to human revisions, but we also observe limitations (Section~\ref{sec:analysis}). To explore to what extent it generalizes to other domains, we also carry out experiments on instructional texts \cite{anthonio-roth-2020-learn} and formal texts \cite{du-etal-2022-understanding-iterative} and find that it outperforms strong baselines and state-of-the-art approaches.
In summary, the contributions of this paper are:
\begin{enumerate}
\setlength{\itemsep}{-2pt}
\item
\emph{a new task}, claim optimization, along with a manual analysis of typical optimization types;
\item
\emph{a computational approach} that reranks generated candidate claims with respect to quality;
\item
\emph{empirical insights} into the impact and challenges of optimizing claims computationally.%
\footnote{Data, code, and models available at \href{https://github.com/GabriellaSky/claim_optimization}{https://github.com/GabriellaSky/claim\_optimization}}
\end{enumerate}
\section{Related Work}
\label{sec:relatedwork}
Wikipedia-based corpora have often been used in the study of editing and rewriting, including paraphrasing \cite{max-wisniewski-2010-mining}, sentence simplification \cite{botha-etal-2018-learning}, grammatical error correction \cite{lichtarge-etal-2019-corpora}, bias neutralization \cite{pryzant-etal-2020}, and controllable text editing \cite{faltings-etal-2021-text,du-etal-2022-understanding-iterative}. Similarly, WikiHow has served for summarization \cite{koupaee-wang-2018} and knowledge acquisition \cite{zhou-etal-2019-learning-household}. However, neither of these includes {\em argumentative} texts. Instead, we rely on data from \citet{skitalinskaya2021}, which consists of revision histories of argumentative claims from online debates. Whereas the authors \emph{compare} claims in terms of quality, we propose and study the new task of automatically \emph{optimizing} claim quality.
The key idea of our approach is to \emph{rerank} multiple candidates generated by a language model. Prior work on reranking in generation hints at the potential benefits of such setup, albeit in different tasks and domains. In early work on rule-based conversational systems, \citet{walker-etal-2001-quantitative} introduced novel dialogue quality metrics to optimize template-based systems towards user satisfaction. \citet{kondadadi-etal-2013-statistical} and \citet{cao-etal-2018-retrieve} ranked the best templates for text generation, \citet{mizumoto-matsumoto-2016-discriminative} used syntactic features to rerank candidates in grammatical error correction. Recently \cite{yoshimura-etal-2020-reference} proposed a reference-less metric trained on manual evaluations of grammatical error correction system outputs to assess generated candidates, while \citet{suzgun2022promtandrerank} utilize pre-trained general-purpose language models to rerank candidates in textual style transfer tasks. However, reranking is still largely understudied in generation research within computational argumentation. The most related approach of \citet{chakrabarty-etal-2021-entrust} reframes arguments to be more trustworthy (e.g., less partisan). It generates multiple candidates and reranks them based on entailment relation scores to the original text. Building on this, we rerank candidates based on various properties, including argument quality.
Understanding the editing process of arguments is crucial, as it reveals what quality dimensions are considered important. For Wikipedia, \citet{daxenberger-gurevych-2013-automatically} proposed a fine-grained taxonomy as a result of their multi-label edit categorization of revisions \cite{daxenberger-gurevych-2012-corpus}. The taxonomy focuses solely on the editing actions performed, such as inserting, deleting, and paraphrasing. In contrast, \citet{yang_identifying_2017} identified various semantic intentions behind Wikipedia revisions, from
{\em copy editing} to {\em content clarifications} and {\em fact updates}. Their taxonomy defines a starting point for our research. Not all covered intentions generalize beyond Wiki scenarios, though.\,
For the analysis of argumentative text rewriting, \citet{zhang-litman-2015-annotation} incorporated both argumentative writing features and surface changes. To explore the classification of essay revisions, they defined a two-dimensional schema, combining the revision operation (e.g., modify, add, or delete) with the component being revised (e.g., reasoning or evidence). Moreover, \citet{afrin-litman-2018-annotation} created a small corpus of between-draft revisions of 60 student essays to study whether revision improves quality. However, these works do not uncover the reasoning behind a revision operation and are more geared towards analysis at the essay level.
The corpus we use distinguishes three claim revision types: clarification, grammar correction, and linking to external resources \cite{skitalinskaya2021}. However, we argue that this is too coarse-grained to represent the diversity of claim quality optimization. For refinement, we manually identify eight types of optimizations, allowing for a systematic analysis of claims improved automatically.
The authors also compare the revision types to the 15 dimensions in the argument quality taxonomy of \citet{wachsmuth-etal-2017-computat}. Many correlations were rather low, suggesting that the claim revision types are rather complementary to the dimensions. Primarily, they target the general form a well-phrased claim should have and its relevance to the debate.
\section{Task and Data}
\label{sec:taxonomy}
This section introduces the proposed task and pre-sents the data used for development and evaluation.
\subsection{Claim Optimization}
We define the task of computational claim optimization as follows:
\paragraph{Task}
Given as input an argumentative claim $c$, potentially along with context information on the debate, rewrite $c$ into an output claim $\tilde{c}$ such that
\begin{itemize}
\setlength{\itemsep}{0pt}
\item[(a)]
$\tilde{c}$ improves upon $c$ in terms of text quality and/or argument quality, and
\item[(b)]
$\tilde{c}$ preserves the meaning of $c$ as far as possible.
\end{itemize}
While we conceptually assume that $c$ is phrased in one or more complete sentences and that it has at least one quality flaw, the approaches studied later on do not model this explicitly. Moreover, note that a claim may be flawed in multiple ways, often resulting in $n \geq 2$ candidate optimizations $\tilde{C} = \{ \tilde{c}_1, \ldots, \tilde{c}_n \}$. In this case, the goal is to identify the candidate~$c^* \in \tilde{C}$ that maximizes overall quality.
\subsection{Data for Development and Evaluation}
\label{subsec:data}
As a basis for the development and evaluation of approaches to the task, we build on the dataset of \citet{skitalinskaya2021} -- ClaimRev, consisting of 124,312 claims and their revision histories from the online debate platform {\em Kialo}. Each history de\-fines a chain $(c_1, \dots, c_m)$, in which each claim $c_{i}$ is a revised version of the previous claim, $c_{i-1}$ with $1 < i \leq m$, that improves $c_{i-1}$ in terms of quality, which holds in 93\% of all cases according to the authors.\,
From each revision chain, we derived all possible optimization pairs $(c, \tilde{c}) := (c_{i-1}, c_i)$, in total 210,222. Most revisions are labeled with their intention by the users who performed them, rendering them suitable for learning to optimize claims automatically.%
\footnote{As 26\% of all pairs were unlabeled, we trained a BERT model to assign such pairs one of the 6 most prominent labels.}
Overall, 95\% of all pairs refer to three intention labels: {\em clarification}, {\em typo/grammar correction}, and {\em corrected/added links}. To avoid noise from the few remaining labels, we condensed the data to 198,089 instances of the three main labels.%
\footnote{The labels of the removed instances denote changes to the meaning of $c$ and statements from which no action or intention can be derived (e.g., "see comments", "moved as pro").}
For the final task dataset, we associated each remaining pair $(c, \tilde{c})$ to its context: the {\em debate topic} $\tau$ (i.e., the thesis on Kialo) as well as the {\em previous claim} $\hat{c}$ (the parent on Kialo), which is supported or opposed by $c$ (see Figure \ref{example_revisions}). We sampled 600 revision pairs pseudo-randomly as a test set (200 per intention label), and split all other pairs into a training set (90\%) and a validation set (10\%). As the given labels are rather coarse-grained, we look into the optimizations in more detail in Section~\ref{sec:analysis}.
\section{Approach}
\label{sec:approach}
We now present the first approach to automatic claim optimization.\,First, candidate claims are generated that are pertinent to the context given and do not change the meaning of the original claim.~Then, the candidates are reranked to find the optimal claim in terms of text and argument quality. Both steps are detailed below and illustrated in Figure~\ref{approach}.
\subsection{Seq2Seq-based Candidate Generation}
To generate candidates, we fine-tune a sequence-to-sequence model on training pairs $(c, \tilde{c})$, by treating the original claim, $c$, as the encoder source and the revised claim, $\tilde{c}$ as the decoder target. In a separate experiment, we condition the models on context information during fine-tuning to further optimize the relevance of the generated candidates. As context, the debate topic, $\tau$, and the previous claim, $\hat{c}$ are prepended to $c$, separated by delimiter tokens \cite{keskar-etal-2019-ctlr,schiller-etal-2021-aspect}.
There may be multiple ways to improve $c$, especially when it suffers from multiple flaws, since not all flaws may be fixed in a single revision. To account for this, we first generate $n$ suitable candidates, $\tilde{c}_1, \ldots, \tilde{c}_n$, among which the optimal one is to be found later ($n$ is set to 10 in Section~\ref{sec:experiments}). However, the top candidates created by language models often tend to be very similar. To increase variety, we perform top-$k$ sampling \cite{fan-etal-2018-hierarchical}, where we first generate the most probable candidate (top-$1$) and then vary $k$ with a step of 5 (e.g. top-$5$, top-$10$, etc).
\bsfigure{approach}{Proposed claim optimization approach: First, a sequence-to-sequence model generates $n$ candidates from the {\em original claim}, possibly conditioned on context information. Then, the candidates are reranked with respect to three quality metrics. The top-ranked one is used as the {\em optimized claim}.}
\subsection{Quality-based Candidate Reranking}
\label{sec_method_optimization}
Among the $n$ candidates, we aim to find the optimal claim, $c^*$, that most improves the delivery of $c$ in terms of text and argument quality. Similar to \citet{yoshimura-etal-2020-reference}, we tackle this task as a reranking~problem. In our reranking strategy, {\em AutoScore}, we integrate three metrics: (1)~grammatical fluency, (2)~meaning preservation, and (3)~argument quality. This way, we can \textit{explicitly} favor specific quality dimensions via respective models:
\paragraph{Grammatical Fluency}
We learn to assess fluency on the MSR corpus of abstractive compressions \cite{toutanova-etal-2016-dataset}. The grammaticality of each compression was scored by 3--5 annotators as 1 (major errors, disfluent), 2 (minor errors), or 3 (fluent). We chose this corpus, since the multiple compressions per input make a trained model sensitive to the differences in variations of the same text. For training, we average all annotator scores and transform the task to a binary task, where a compression is seen as disfluent unless all annotators gave the score~3. Then, we train BERT on the binary data to obtain the fluency probabilities (details found in appendix). The accuracy of the trained model on the train/dev/test split suggested by the authors \cite{toutanova-etal-2016-dataset} is 77.4.
\paragraph{Meaning Preservation}
To quantify to what extent a generated candidate maintains the meaning of the original claim, we compute their semantic similarity in each case in terms of the cosine similarity score of their contextual SBERT sentence embeddings \cite{reimers:2019}.
\paragraph{Argument Quality}
Finally, to examine whether the generated candidates are better than the original claim from an argumentation perspective, we fine-tune a BERT model on the task of pairwise argument classification using the ClaimRev dataset. Since this corpus is also used to fine-tune the sequence-to-sequence model, we apply the same training and validation split as described in Section \ref{subsec:data} to avoid data leakage, and obtain accuracy of 75.5. We then use its probability scores to determine relative quality improvement. Further training details can be found in the appendix.
\medskip
Given the three quality metrics, we calculate the final evaluation score, $AutoScore$, as the weighted linear sum of all three individual scores as
$$
\alpha \cdot fluency + \beta \cdot meaning + \gamma \cdot argument,
$$
where $fluency$, $meaning$, and $argument$ are the normalized scores for the three outlined quality metrics. The three non-negative weights satisfy $\alpha+\beta+\gamma=1$.
\section{Experiments}
\label{sec:experiments}
This section describes our experimental setup to study how well the claims in the dataset from Section~\ref{sec:taxonomy} can be improved using our combined generation and reranking approach from Section~\ref{sec:approach}. We particularly focus on the impact of reranking.
\subsection{Seq2Seq-based Candidate Generation}
For candidate generation, we employ the pre-trained conditional language model BART, which combines bidirectional and auto-regressive transformers \cite{lewis-etal-2020-bart}. We use the \textit{bart-large} checkpoint. However, other sequence-to-sequence architectures can also be considered within the suggested framework (see appendix for details).
\subsection{Quality-based Candidate Reranking}
We evaluate our reranking approach, AutoScore, in comparison to three ablations and four baselines:
\paragraph{Approach}
To utilize AutoScore for ranking candidates, the optimal weighting of its metrics must be determined. We follow \citet{yoshimura-etal-2020-reference}, performing a grid search in increments of 0.01 in the range of 0.01 to 0.98 for each weight to maximize the Pearson's correlation coefficient between AutoScore and the original order of the revisions from claim revision histories in the validation set. Similar has been done for counterargument retrieval by \citet{wachsmuth:2018}.
The best weights we found and used were $\alpha = 0.43, \beta = 0.01$, and $\gamma = 0.56$, suggesting that meaning preservation is of low importance and potentially may be omitted. We suppose this is due to the general similarity of the generated candidates, so a strong meaning deviation is unlikely.
\paragraph{Ablations}
To assess the impact of each considered quality metric used in AutoScore, we perform an ablation study, where optimal candidates are chosen based on the individual metric scores:
\begin{itemize}
\setlength{\itemsep}{-2pt}
\item \emph{Max Fluency.} Highest grammatical fluency.
\item \emph{Max Argument.} Highest argument quality.
\item \emph{Max Meaning.} Highest semantic similarity.
\end{itemize}
\paragraph{Baselines}
We test four other reranking strategies for 10 candidates generated via top-$k$ sampling:
\begin{itemize}
\setlength{\itemsep}{-2pt}
\item \emph{Unedited} Return the original input as output.
\item \emph{Top-1.} Return the most likely candidate (obtained by appending the most probable token generated by the model at each time step).
\item \emph{Random.} Return candidate pseudo-randomly.
\item \emph{SVMRank.} Rerank candidates with SVMRank
\cite{joachims_svm2006}. We use sentence embeddings to decide which of the two claim versions is better, by fine-tuning SBERT ({\em bert-base-cased}) in a Siamese setup on the corpus of \citet{skitalinskaya2021}.
\end{itemize}
\subsection{Evaluation}
We explore claim optimization on all 600 test cases, both automatically and manually:
\paragraph{Automatic Evaluation}
We compare all reranking strategies against the reference revisions using the precision-oriented {\em BLEU} \cite{papineni-etal-2002-bleu}, recall-oriented {\em Rouge-L} \cite{lin-2004-rouge}, {\em SARI} \cite{xu-etal-2016-optimizing}, which computes the average F$_1$-scores of the added, kept, and deleted $n$-grams, and the \emph{exact match accuracy}.
We also compute the semantic similarity of the optimized claim and the context information to capture whether conditioning claims on the context affects their topic\,relevance.
\paragraph{Manual Evaluation}
As we fine-tune existing generation models rather than proposing new ones, we focus on the {\em reranking} step in two manual annotation studies. For each instance, we acquired five independent crowdworkers via {\em MTurk} at \$13/hour.
In the first study, the annotators scored all candidates with respect to the three considered quality metrics. We used the following Likert scales:
\begin{itemize}
\setlength{\itemsep}{-2pt}
\item
\textit{Fluency.} 1 (major errors, disfluent), 2 (minor errors), and 3 (fluent)
\item
\textit{Meaning Preservation.} 1 (entirely different), 2 (substantial differences), 3 (moderate differences), 4 (minor differences), and 5 (identical)
\item
\textit{Argument Quality.} 1 (notably worse than original), 2 (slightly worse), 3 (same as original), 4 (slightly improved), and 5 (notably improved)
\end{itemize}
A challenge of crowdsourcing is to ensure good results \cite{sabou-etal-2014-corpus}. To account for this, we obtained the final scores using MACE \cite{hovy-etal-2013-learning}, a Bayesian model that gives more weight to reliable workers. In the given case, 39\% of the 46 annotators had a MACE competence value $> 0.3$, which can be seen as reasonable in MTurk studies.
In the second study, we asked the annotators to rank the four candidates, returned by the reranking strategies, by perceived overall quality. If multiple candidates were identical, we showed each only once. While Krippendorff's~$\alpha$ agreement was only 0.20, such values are common in subjective tasks \cite{wachsmuth-etal-2017-computat,alshomary-etal-2021-belief}.
\section{Results and Discussion}
\label{sec:results}
Apart from evaluating the applicability of large generative language models to the task of argumentative claim optimization in general, our experiments focus on two questions:
(1)~Does the use of explicit knowledge about text and argument quality in the decoding step lead to the selection of better candidates?
(2)~Does the use of contextual information make the generated candidates more accurate and relevant to the debate?
\subsection{Overall Claim Optimization Performance}
\label{ranking_results}
\paragraph{Automatic Evaluation}
Table \ref{tab:exp1-auto} shows the automatic scores of all considered reranking strategies. The high scores of the baseline \emph{Unedited} on metrics such as BLEU and ROUGE-L, indicate that many claim revisions change little only. In contrast, Unedited is worst on SARI, as this measure takes into account the goodness of words that are added, deleted, and kept in changes, making it more suitable for evaluating the task at hand. Here, BART with \emph{AutoScore} reranking performs best on SARI (43.7) and exact match accuracy (8.3\%).
The \emph{BART+Max Meaning} ablation supports the intuition that the candidates with highest meaning preservation scores are those with minimal changes, if any (72\% of the candidates remain identical to the input). Such identical outputs are undesirable, as the claims are not optimized successfully, which is also corroborated by the low weight parameter ($\beta = 0.01$) found for the meaning preservation metric when optimizing AutoScore (see Section~\ref{sec:experiments}).
\paragraph{Manual Evaluation}
Table~\ref{tab:exp1-human} shows that human annotators prefer the optimized candidates selected by {\em AutoScore}, with an average rank of 1.92. The difference to {\em Top-1} and {\em Random} is statistically significant ($p <.05$ in both cases) according to a Wilcoxon signed-rank test, whereas the significance of the gain over the second-best algorithm, \emph{SVMRank}, is limited. Also, candidates of AutoScore and SVMRank are deemed more fluent than those of Top-1 and Random (2.33 vs.\ 2.29 and 2.26). The argument quality results deviate from the automatic scores, being marginally higher for SVMRank and Top-1. Further analysis revealed that AutoScore and SVMRank agreed on the optimal candidate in 35\% of the cases, partially explaining the closeness of the scores.
\input{table-automatic-scores-new}
\input{table-human-scores-mace}
Overall, we conclude that our approach performed best in the experiments. More importantly, our findings suggest that using reranking approaches that incorporate quality assessments (i.e., {AutoScore} and {SVMRank}) leads to candidates of higher fluency and argument quality while preserving the meaning of the original claim. In addition to Figure~\ref{example_revisions}, examples of automatically generated optimized claims can be found in the appendix.
\subsection{Performance with Context Integration}
\paragraph{General Assessment}
Table \ref{tab:ablation_res} shows the semantic similarity of claims optimized by our approach and context information, depending on the context given. The results reveal slight improvements when conditioning the model on the previous claim (e.g., 60.3 vs.\ 59.4 BLEU). To check whether this led to more grounded claims, two authors of the paper compared 600 claims generated with and without the use of the previous claim in terms of (a)~which claim seems better in overall quality and (b) which seems more grounded. We found that utilizing the previous claim as context increased quality in 12\% of the cases and decreased it in 1\% only, while leading to more grounded claims in 36\% cases.
\paragraph{Qualitative Analysis}
Our manual inspection of a claim sample revealed the following insights:
First, conditioning on context reduces the number of erroneous specifications, particularly for very short claims with up to 10 words. This seems intuitive, as such claims often convey little information about the topic of the debate, making inaccurate changes without additional context likely.
\input{table-bart-ablation-results}
\input{table-automatic-scores-external}
\input{table-annotation-taxonomy.tex}
Next, Kialo revisions often adhere to the following form: A claim introduces a statement and/or supporting facts, followed by a conclusion. This pattern was frequently mimicked by our approach. Yet, in some cases, it added a follow-up sentence repeating the original claim in different wording or generated conclusions containing fallacious or unsound phrases contradicting the original claim in others. Modeling context mitigated this issue.
Finally,we found that models conditioned on dif\-ferent contexts sometimes generated candidates optimized in different regards, whereas a truly optimal candidate would be a fusion of both suggestions.
\section{Analysis}
\label{sec:analysis}
To explore the nature of claim optimization and the capabilities of our approach, this section reports on follow-up analyses, in which we studied (a)~what types of claim optimizations exist, (b)~how well can our approach operationalize these, and (c)~how well does the idea of our approach generalize to revision domains beyond argumentative texts.
\subsection{Taxonomy of Optimization Types}
To understand the relationship between the optimizations found in the data and the underlying revision intentions, two authors of this paper manually inspected 600 claim revision pairs of the test set. This allows for a detailed analysis of the obtained results, as we are able to identify more fine-grained \emph{optimization types} in the given task.
For the type distinction, we build on ideas of \citet{yang_identifying_2017} who provide a taxonomy of revision intentions in Wikipedia texts. Claims usually do not come from encyclopedias, but from debates of various shades (an online debate platform in our case) or from monological arguments, as in essays \cite{persing:2015}. Therefore, we adapt the terminology of \citet{yang_identifying_2017} to gear it more towards argumentative styles. Since we aim for optimization in the end, we consider {\em actions} rather than {\em intentions}. Whereas the former refers to specific changes (e.g., rephrasing a sentence or adding punctuation), the latter describes the goal of a change (e.g., making a text easier to read).
As a result of a joint discussion of various sample pairs, we decided to distinguish eight optimization types, as presented in Table~\ref{tab:edit-categories}. Both authors then annotated all 600 test pairs for these types, which led to only 29 disagreement cases, meaning a high agreement of 0.89 in terms of Cohen's $\kappa$. These cases were resolved by both annotators together.%
\footnote{Notice that no knowledge about the test set was used to develop the approach in Section~\ref{sec:approach}.}
Table~\ref{tab:edit-categories} also shows cooccurrences of the types and intention labels. \emph{Typo/grammar correction} and \emph{correcting/adding links} align well with \emph{copy editing} and \emph{corroboration} respectively. In contrast, clarification is broken into more fine-grained types, where {\em specification} seems most common with 58 cases, followed by {\em simplification} and {\em reframing}. Examples of each type are found in the appendix.\,
We point out that the eight types are not exhaustive for all possible claim quality optimizations, but rather provide insights into the semantic and discourse-related phenomena observed in the data at hand.
We further see them as complementary to the argument quality taxonomy of \citet{wachsmuth-etal-2017-computat}.
In particular, they can be seen as actions to improve the delivery-related quality dimensions: {\em clarity}, {\em appropriateness}, and {\em arrangement}.
\input{table-category-success}
\subsection{Performance across Optimization Types}
To enable comparison between the human optimizations and the output of our system, we also labeled 600 claims optimized by BART+AutoScore with the proposed types. Table~\ref{tab:category-success} directly compares automatic and human optimization types. Overall, our approach generates better claims in 60\% of the cases, while 84\% remain at least of similar quality.
Most noteworthily, we observe that our approach performs optimizations of the type {\em specification} 2.5 times as often as humans, and more than double as many {\em elaboration} revisions (55 vs.\ 23). In contrast, it adds, edits, or removes evidence in the form of links ({\em corroboration}) four times less often than humans. The model also made fewer {\em simplifications} (18 vs.\ 43) and no {\em neutralization} edits, which may be due to data imbalance regarding such types.
In terms of average quality, {\em specification} (65\%) and {\em disambiguation} edits\,(63\%) most often lead~to improvements, but the different types appear rather balanced in this regard.The Jaccard similarity~score between optimizations performed by humans and our approach is $0.37$, mostly agreeing on copy~edits\,(178 cases) and corroboration\,(22 cases). Gi\-ven such low overlap, future work should consider conditioning models to generate specific optimizations.
\subsection{Performance across Revision Domains}
Lastly, we examine whether our approach, along with the chosen text quality metrics, applies to texts from other domains. We consider two datasets: \emph{WikiHow} \cite{anthonio-roth-2020-learn}, containing revisions of instructional texts, and \emph{IteraTeR} \cite{du-etal-2022-understanding-iterative}, containing revisions of various formal texts, such as encyclopedia entries, news, and scientific papers. For our experiments, we use the provided document-level splits, and sample 1000 revision pairs pseudo-randomly as a final test set.
Table \ref{tab:exp1-auto-ext} shows the automatic evaluation results. In both cases, \emph{BART+Autoscore} leads to higher SARI scores (48.5 vs.\ 41.3 for WikiHow, 38.6 vs.\ 37.0 for IteraTeR), and notably reduces the number of cases where the models failed to revise the input (0.08 vs.\ 0.50 for WikiHow). The reported \emph{BART+Top1} model represents the approach of \citet{du-etal-2022-understanding-iterative}, indicating that our approach and its text quality metrics achieve state-of-the-art performance with systematic improvements across domains, when generating optimized content. However, as different domains of text have different goals, different notions of quality, and, subsequently, different revision types performed, integrating quality metrics capturing characteristics directly relevant to the domain may improve the performance of the suggested framework. We leave this for future work.
\section*{Limitations}
This work contributes to the task of argumentative text editing, namely we explore how to revise claims automatically in order to optimize their quality. While our work may also improve downstream task performance on other tasks, it is mainly intended to support humans in scenarios, such as the creation and moderation of content on online debate platforms as well as the improvement of arguments generated or retrieved by other systems. In particular, the presented approach is meant to help users by showing examples of how to further optimize their claims in relation to a certain debate topic, so they can deliver their messages effectively and hone their writing skills.
However, our generation approach still has limitations, as outlined in Section \ref{sec:results}, and may favor revision patterns over others in unpredictable ways. While it may occasionally produce false claims, humans should be able to identify such cases in light of the available context, as long as the improvements remain suggestions and do not happen fully automatically, as intended. Moreover, we expect that further research can ensure that the produced claims are of decent quality by being more attentive to the veracity of claims. Such focus may allow to improve argumentative text consistently and truly support humans, rather than hindering them. We also would like to point out that using other pre-trained models to assess the fluency, semantic similarity and argument quality may further improve the results depending on the target domain. This could be especially important in scenarios where certain quality dimensions may be of special interest, such as for example, convincingness or argument strength. In such cases, the quality metrics considered in the suggested framework and their weights in the overall score should be adjusted towards the needs of the users.
The presented technology might also be subject to intentional misuse. A word processing software, for example, might automatically detect and adapt the claims made by the user in a way that favors political or social views of the software provider. Those changes might then not even be made visible to the user, but only be revealed after exporting or printing the text. In a different scenario, online services, such as social media platforms or review portals, might change posted claims (e.g. social media posts, online reviews) to personalize them and increase user engagement or revenue. These changes might then not only negatively affect the posting, but also the visiting user.
While it is hard to prevent such misuse, we think that the described scenarios are fairly unlikely, as such changes tend to be noticed by the online community quickly. Furthermore, the presented architecture and training procedure would require notable adaptations to produce such high-quality revisions.
\section{Conclusion}
With this paper, we work towards the next level of computational argument quality research, namely, to not only {\em assess} but also to {\em optimize} argumentative text.
Applications include suggesting improvements in writing support and automatic phrasing in debating systems.
We have presented an approach that generates multiple candidate optimizations of a claim and then identifies the best one using quality-based reranking.
In experiments, combining fine-tuned BART with reranking improved 60\% of the claims from online debates, outperforming different baseline models and reranking strategies. We showcased generalization capabilities on two out-of-domain datasets, but we also found some claim optimization types to be hard to automate.
In future work, we seek to examine whether the latest language models (e.g., \mbox{GPT-3}) and end-to-end models (where generation and reranking are learned jointly) can further optimize the quality of claims. Moreover, our approach so far relies on the availability of large claim revision corpora and language models. To make claim optimization more widely applicable, techniques for low-resource scenarios and languages should be explored.
\section*{Acknowledgments} This work was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project number 374666841, SFB 1342.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,382 |
Q: Scrapy only captures one question per page, but there are 10 I've been trying to solve this problem for 5 days.
If anyone can help me, thank you:
Scrapy only captures one question per page. Each page has 10 questions.
I have already used CSS, xpath + regex, relative address, absolute address, LinkExtractor.
I already disabled obey robots.txt, I already used proxy.
In the scrapy shell, with get() also only captures a question, with get_all() captures all in one.
My scrapy.py:
import scrapy
from items import GabariteItem
class GbSpider(scrapy.Spider):
name = "gb"
start_urls = ['https://www.gabarite.com.br/questoes-de-concursos/assunto/agentes-publicos-e-lei-8112-de-1990']
def parse(self, response):
items = response.xpath("//body/div[3]")
gb = GabariteItem()
gb['url'] = response.url
gb['area'] = items.xpath(".//h3/a[2]/text()").extract_first()
gb['cargo'] = items.xpath(".//h3/a[3]/text()").extract_first()
gb['curso'] = items.xpath(".//h3/a/text()").extract_first()
gb['pergunta'] = items.xpath(".//li[@class='pergunta']").extract_first()
gb['alternativaA'] = items.xpath(".//li[@class='respostas']//label[1]/text()").extract_first()
gb['alternativaB'] = items.xpath(".//li[@class='respostas']//label[2]/text()").extract_first()
gb['alternativaC'] = items.xpath(".//li[@class='respostas']//label[3]/text()").extract_first()
gb['alternativaD'] = items.xpath(".//li[@class='respostas']//label[4]/text()").extract_first()
gb['alternativaE'] = items.xpath(".//li[@class='respostas']//label[5]/text()").extract_first()
yield gb
#Próxima Página
next_page = response.xpath("//a[@title='Próxima página']/@href").extract_first()
if next_page:
# self.log ('Próxima Página: https://www.gabarite.com.br/' + next_page)
next_page_url = response.urljoin(next_page)
yield scrapy.Request(url=next_page_url, callback=self.parse)
My items.py
import scrapy
class GabariteItem(scrapy.Item):
url = scrapy.Field()
area = scrapy.Field()
cargo = scrapy.Field()
curso = scrapy.Field()
ano = scrapy.Field()
nivel = scrapy.Field()
pergunta = scrapy.Field()
alternativaA = scrapy.Field()
alternativaB = scrapy.Field()
alternativaC = scrapy.Field()
alternativaD = scrapy.Field()
alternativaE = scrapy.Field()
alternativaCorreta = scrapy.Field()
A: You need to loop over each of the question containers and then select the fields relative to that. Here is an example in scrapy shell:
>>> for question in response.css('article.lista-questoes'):
... print(question.css('.numero h3 ::text').get())
... # Create item here
...
Questão 30979.
Questão 25714.
Questão 35985.
Questão 35986.
Questão 26362.
Questão 28203.
Questão 34446.
Questão 35978.
Questão 35981.
Questão 30981.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,168 |
E3 Spark Plugs >> LIDAR Driverless Technology from LeddarTech Canada
Canadian Firm May Have Early Lead on Driverless Technology
The development of laser radar technology that will allow vehicles to understand the obstacles around them is one of the hottest products in global research and development. This autonomous technology is known as "lidar", which maybe short for light detection and ranging or light/radar or laser interferometry detection and ranging, and will allow vehicles to travel safely from point A to point B. Similar to the laser speed detector used by law enforcement, a lidar device emits an infrared light. Using a specialized GPS receiver with optical remote sensing, the device can measure variable distances in real time.
Quebec City-based LeddarTech evolved from a Canadian government-funded research facility called the National Optics Institute. The company has since become a global player in the development of artificial intelligence and lidar devices. LeddaTech's solid-state equipment uses technology to create a 3-D topographical map with obstacle detection and collision avoidance for natural and manmade environments. With nearly ten years of R&D, the company claims its lidar gun produces images that are over two dozen times sharper than its competitor's equipment.
Early mechanical versions used for driverless vehicle applications have proven to be too expensive for mass production and are subject to excessive wear and tear. LeddarTech's devices use high-speed processors to provide 360-degree detection of lanes, traffic, pedestrians, signs, stoplights and anything else in the way. Several pioneers of mechanical lidar devices are also ready to release solid-state versions that will enable autonomous vehicles to safely decide where to go and when to stop.
Indeed the race for driverless technologies is in full swing, and major automotive manufacturers have thrown their support behind several international companies dedicated to developing a host of future products. LeddarTech has attracted big-name industry backers including Delphi Automotive, Germany's Osram Licht and Fiat Chrysler's parts division. Without doubt, there are billions of dollars at stake waiting for advanced driver-assisted systems for commercial deployment of highly advanced driverless equipment solutions. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 568 |
Q: Add product labels like "New" , "Sale" in shopify theme I am trying to make a demo for a client who wants product label like
new
hotselling
featured
Please suggest how do I achieve it without using any apps
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,993 |
President Muhamadu Buhari's 2019 whistle stop campaign for reelection landed in Sokoto on Wednesday.
The plane which President Buhari flew in to Sokoto landed 10 05am.Gov Aminu Tambuwal of opposition PDP arrived in time to receive the President at the airport.
Before arriving the Sultan Abubakar 111 International Airport, a delegation of APC topguns were seen also taking position to welcome the President. The APC bigwigs were led by Senator Aliyu Wamakko. Also sighted at airport was the Minister of Trade and Investment who doubles as the Minister of Women Affairs, Aisha Abubakar. A former minister Alhaji Alhaji was seen at the airport. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,503 |
\section{Introduction}
Dynamical models on a one-dimensional lattice have been an topic of interest for well over 50 years. Perhaps the most famous example is the Fermi–Pasta–Ulam–Tsingou (FPUT) model \cites{FPUT,Zabusuy1965}, which was one of the first problems to be studied using numerical simulations. In this work, we will examine the discrete Klein-Gordon (DKG) equation~\cites{braun2004,SGbook,p4book,kivsharmalomed}
\begin{equation*}
\ddot{u}_n = d (\Delta_2 u)_n - f(u_n),
\end{equation*}
which describes the dynamics of an infinitely long, one-dimensional lattice of particles. The quantity $u_n$ represents the displacement of $n$th particle in the integer lattice as a function of the time $t$. Each particle is harmonically coupled to its two nearest neighbors via the discrete second difference operator $\Delta_2$, and the strength of this coupling is quantified by the parameter $d$. The particles are subject to an external, nonlinear, on-site potential $V(u)$, such that $f(u) = V'(u)$, which can be of different
type depending on the model~\cites{Karachalios,imamat}. Common nonlinearities are shown in \cref{table:V}. (See \cref{sec:DKGbreather} for the definition of hard and soft potentials).
\begin{table}
\begin{tabular}{lll}\toprule
Equation & $V(u)$ & $f(u)$ \\ \midrule
sine-Gordon & $1 - \cos u$ & $\sin u$ \\
$\phi^4$ (soft) & $\frac{1}{2}u^2 - \frac{1}{4}u^4$ & $u(1-u^2)$ \\
$\phi^4$ (hard) & $\frac{1}{2}u^2 + \frac{1}{4}u^4$ & $u(1+u^2)$ \\
Morse & $\frac{1}{2}(1 - e^{-u})^2$ & $e^{-u}(1 - e^{-u})$ \\ \bottomrule
\end{tabular}
\caption{Common nonlinearities for the discrete Klein-Gordon equation.}
\label{table:V}
\end{table}
The DKG equation is the discrete analogue of the nonlinear Klein-Gordon partial differential equation
\begin{equation*}
u_{tt} = u_{xx} - f(u),
\end{equation*}
which is a prototypical model in the study of nonlinear waves and solitons. One of the the most well-studied forms of this equation is the sine-Gordon equation \cites{braun2004,SGbook,p4book,kivsharmalomed}, which has periodic nonlinearity $f(u)=\sin(u)$, and is completely integrable. The transition between the continuous and discrete models in the sine-Gordon case has been extensively discussed in \cite{SGchapter}. The discrete sine-Gordon model, also known as the Frenkel-Kontorova model, was originally devised as a model for describing the dynamics of a crystal lattice near a dislocation core \cites{braun1998,braun2004}, and has been
used since its inception in numerous additional applications (see, for example, \cite{braun2004}*{Chapter 2}), including a mechanical model for a chain of pendula \cites{Scott1969,english}, arrays of Josephson junctions \cites{Ustinov1992,Floria1998}, and DNA dynamics \cites{Yomosa1983,Yakushevich1998,DeLeo2011}.
Two major classes of coherent structures in the nonlinear Klein-Gordon equation (both discrete and continuous) are of particular interest: kinks, which are heteroclinic structures resembling ``wave fronts'' that connect two adjacent minima of the potential $V(u)$, and breathers, which are structures that are spatially localized and oscillatory in time. For the continuum sine-Gordon equation, exact, analytical solutions for both of these structures have been found \cite{SolitonBook1}.
For discrete systems, the existence and stability of static kinks have been well-studied (see \cites{PEYRARD198488,KevrekidisWeinstein2000,SGchapter}, as well as \cite{Parker2021} for results on multi-kink solutions), and there has also been interest in moving kinks \cites{Aigner2003,Iooss2006,Cisneros2008}. We will concern ourselves herein with discrete breather solutions.
Discrete breathers have been studied in Hamiltonian \cite{Flach1998} and dissipative systems \cite{Flach2008a}, and have applications in areas such as laser scanning microscopy and coupled optical waveguides \cites{Flach2008,LEDERER20081}. Existence and stability of discrete breather solutions in Klein-Gordon lattices were first studied by MacKay and Aubry \cites{MacKay1994,Aubry1997} by considering the system near the anti-continuum (AC) limit ($d=0$), in which the individual sites in the lattice are uncoupled. The advantage of this approach is that the solution for a single site is known at the AC limit, and this can be continued to small $d>0$ using the implicit function theorem. Some results about asymptotic stability of these breathers can be found in \cite{Bambusi2013}. A similar approach has been used for multi-site solitons in the discrete nonlinear Schr\"odinger equation (DNLS) \cites{Pelinovsky2005,KALOSAKAS200644}, which demonstrated that the only
potentially stable multi-solitons are those in which adjacent peaks are excited out-of-phase. Analysis of multi-breathers in DKG, in which a finite number of sites in the integer lattice are excited at the AC limit, was done in \cites{Archilla2003,Koukouloyannis2009}, but this was restricted the case where excited sites are adjacent.
Indeed, the two methods (the Aubry band method~\cite{Aubry1997} and the MacKay
effective Hamiltonian method~\cite{sepulchre}) explored in the above two publications
were shown to yield the same results near this limit
in~\cite{doi:10.1142/S0218127411029690}.
In this situation, for small $d>0$, out-of-phase multi-breathers are stable for soft potentials, and in-phase multi-breathers are stable for hard potentials~\cite{Archilla2003}*{Theorem 6}. Results on the existence and spectral stability of multi-breathers were extended in \cite{Pelinovsky2012} to multi-site breathers where any arbitrary, finite set of lattice points is excited at the AC limit; in particular, this allows the excited sites to be separated in the lattice, i.e., there can be ``holes'' in the lattice. However, spectral stability of multi-breathers does not necessarily imply nonlinear stability; even if the multi-breather is spectrally stable, nonlinear instabilities can result from resonance between internal eigenmodes and the continuous spectrum band, as long as the two have opposite Krein signatures \cite{cuevas-maraver2016}
(see also the simpler case example of the discrete
nonlinear Schr{\"o}dinger model in~\cite{PhysRevLett.114.214101}). A recent result uses the Schauder fixed point theorem to prove the existence of discrete Klein-Gordon breathers in the setting of a convex on-site potential \cite{hennig2021}. Some other recent work concerns the existence of breathers in infinite FPUT lattices \cites{Arioli2019,yoshimura2021}, in which the nonlinear potential involves inter-site terms.
In this paper, we take a different approach to multi-breathers. We start with a single-site breather, which we call the primary breather. We take the existence of such a breather for a particular $d>0$ as a hypothesis. We then construct a multi-breather by joining together multiple, well-separated copies of this primary breather. Consecutive copies of the primary breather can be either in-phase or out-of-phase. In effect, we replace the condition that the coupling parameter $d>0$ is small with the condition that the individual copies of the primary breather are well separated.
The construction of complex coherent structures from simple building blocks has a rich mathematical history (see \cite{Sandstede1998}, and the references therein). One set of mathematical techniques that can be used to accomplish this is an implementation of the Lyapunov-Schmidt reduction known as Lin's method. This method has been successfully employed in many systems, including semilinear parabolic PDEs \cites{Sandstede1998,doi:10.1137/0150029} and discrete dynamical systems \cite{Knobloch2000}. We used a similar method to construct multi-solitons in DNLS \cite{Parker2020} and multi-kinks in DKG \cite{Parker2021}. Lin's method can also be used to determine spectral stability. For a multi-pulse, multi-kink, or multi-breather, there are specific elements in the point spectrum, which we will
hereafter term interaction eigenmodes, since they result from nonlinear interactions between neighboring copies of the primary coherent structure. For breathers, which are periodic in time, these interaction eigenmodes assume the form of Floquet multipliers or Floquet exponents.
(We note that these are called internal modes in \cite{cuevas-maraver2016}; since we find from numerical simulations that other internal modes split off from the continuous spectrum bands as the coupling parameter $d$ is increased, we use the term interaction eigenmodes to refer to this specific set of internal modes).
These interaction eigenmodes can be found by constructing the corresponding eigenfunction as a piecewise linear combination of the kernel eigenfunctions associated with the primary coherent structure. This reduces the spectral problem to a finite-dimensional matrix eigenvalue problem, which can then be solved. While the matrix obtained this way has the same form as that in \cites{Pelinovsky2012,cuevas-maraver2016}, its elements are computed using the primary breather solution for a specific $d$ rather than the breather at the AC limit, thus the interaction eigenmodes can be computed more accurately for a greater range of $d$, i.e., further from the AC limit.
For the DKG equation, we follow a similar approach to that which we used in \cite{Parker2020}. We reformulate the equation using a spatial dynamics approach, by recasting it as a lattice dynamical system on an appropriate function space, and then we use Lin's method to both construct multi-breathers and determine the interaction eigenmodes associated with these solutions. The major limitation with this method, however, is that for a periodic breather on $[0,T]$, the most appropriate function space is $C^\infty_\textrm{per}([0,T])$, which is not a closed subspace of $L^2_\textrm{per}([0,T])$. It is not straightforward to adapt the necessary mathematical tools, such as the stable manifold theorem and exponential dichotomies, to this space. As an alternative, since we are interested in smooth solutions, we formulate Lin's method on a finite-dimensional subspace of $L^2_\textrm{per}([0,T])$ consisting of the first $N$ Fourier modes of the standard orthonormal basis. Although this means that the results we obtain are only approximate, we believe this is a reasonable approximation, since the Fourier coefficients for smooth functions decay exponentially, and this method is valid for arbitrary, finite $N$. Furthermore, numerical discretization of periodic solutions is often done using a Fourier spectral method, thus this approximation applies directly to this discretization. Finally, the results from this method are in very good agreement with the results obtained by directly computing the Floquet spectrum of the corresponding multi-breather, both for soft sine-Gordon and hard $\phi^4$ potentials. In addition, the spectral pattern agrees qualitatively with that found using the approach in \cites{Pelinovsky2012,cuevas-maraver2016} for small $d$, and the matrix reduction found by using Lin's method and a spatial dynamics approach has the same form as that in those references.
For both soft and hard potentials, the eigenvalue pattern is determined by the phase differences (in-phase vs. out-of-phase) between adjacent copies of the primary breather. For the hard potential, it also depends on whether these distances are an even or odd number of lattice points.
This paper is organized as follows. In \cref{sec:bg}, we present the mathematical background for the discrete Klein-Gordon equation, including breather solutions, the continuous spectrum, and the reformulation using spatial dynamics. In \cref{sec:findim}, we formulate our finite-dimensional approximation, which we then use in \cref{sec:multi} to prove results about the existence and spectrum of multi-breathers in the approximate system. The proof of the spectral results is deferred to \cref{app:specproof}. Numerical results are presented in \cref{sec:numerics}, which are in excellent agreement with the predictions from the main theorems. Results are presented for both the soft sine-Gordon potential and the hard $\phi^4$ potential. In addition, we present results from timestepping simulations to illustrate the effect of the spectrum on the dynamical evolution of the system. We end with a brief concluding section, which suggests avenues for future research.
\section{Mathematical background}\label{sec:bg}
We will consider the discrete Klein-Gordon (DKG) equation with on-site nonlinearity $f(u)$
\begin{equation}\label{eq:DKG}
\ddot{u}_n = d (\Delta_2 u)_n - f(u_n),
\end{equation}
on the integer lattice ${\mathbb Z}$, where $t \in {\mathbb R}$ is the evolution time, $u_n(t) \in {\mathbb R}$ is the displacement of the $n$th particle in the lattice, $(\Delta_2 u)_n = u_{n+1} - 2 u_n + u_{n-1}$ is the discrete second difference operator, and $f(u) = V'(u)$ for a smooth, on-site potential function $V(u)$. We use the following assumptions for the potential $V$:
\begin{enumerate}[(i)\leftmargin=\parindent]
\item $V$ is an even function, and $V(0) = 0$. This implies $V'(0) = 0$.
\item $V''(0)>0$.
\end{enumerate}
We note that the Morse potential, which is considered in \cite{cuevas-maraver2016}, does not satisfy the first assumption, since it is not an even function. For any time $t$, we take the displacements $\{u_n(t)\}_{n \in {\mathbb Z}} \in \ell^2({\mathbb Z})$, and we denote this sequence by $\mathbf{u}(t)$. Existence and uniqueness of solutions to \cref{eq:DKG} is discussed in \cite{cuevas-maraver2016}. Since $f(u)$ is an odd function, if $\mathbf{u}(t)$ is a solution to \cref{eq:DKG}, then $-\mathbf{u}(t)$ is as well. Equation \cref{eq:DKG} is Hamiltonian \cites{KevrekidisWeinstein2000,cuevas-maraver2016}, and can be written as
\begin{equation}\label{eq:Hform}
\frac{d}{dt}\begin{pmatrix} u_n \\ v_n \end{pmatrix} =
\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} \partial \mathcal{H} / \partial u_n \\ \partial \mathcal{H} / \partial v_n \end{pmatrix},
\end{equation}
where $v_n = \dot{u}_n$ is the velocity of the $n$th particle in the lattice, and $\mathcal{H}$ is the conserved energy
\begin{equation}\label{eq:H}
\mathcal{H}(\mathbf{u}) = \sum_{n=-\infty}^\infty
\left( \frac{1}{2} v_n^2 + \frac{d}{2} (u_{n+1} - u_n)^2 + V(u_n) \right).
\end{equation}
\subsection{Breathers}\label{sec:DKGbreather}
We are interested in breather solutions to \cref{eq:DKG}, which are periodic in time and exponentially localized in their spatial
profile over the lattice. Specifically, a breather solution is a function $\mathbf{u} \in \ell^2({\mathbb Z}, H^2_\textrm{per}[0,T])$, where $H^2_\textrm{per}[0,T]$ is the Hilbert-Sobolev space of periodic, real-valued functions on $[0,T]$.
The fundamental period $T$ is the smallest positive real number for which $\mathbf{u}(t+T) = \mathbf{u}(t)$ for all $t$. At the AC limit ($d = 0$), the individual sites in the lattice are decoupled. At each site, $u_n(t)$ is a $T$-periodic solution to the nonlinear oscillator equation
\begin{equation}\label{eq:singlesiteAC}
\ddot{\phi} + V'(\phi) = 0,
\end{equation}
which has conserved energy $E = \frac{1}{2}\dot{\phi}^2 + V(\phi)$. For fixed energy $E$, equation \cref{eq:singlesiteAC} has a unique, even solution $\phi(t)$, which we will call the fundamental AC breather \cite{Pelinovsky2012}. This solution satisfies the initial conditions $\phi(0) = a$ and $\dot{\phi}(0) = 0$, where $a$ is the smallest, positive root of $V(a) = E$. The fundamental period $T$ of $\phi(t)$ is a function of the energy $E$, and is given by
\begin{equation}\label{eq:TE}
T(E) = \sqrt{2}\int_{-a}^a \frac{du}{\sqrt{E - V(u)}}.
\end{equation}
The potential $V(u)$ is a hard potential if the period $T$ decreases as the energy $E$ increases, and a soft potential if $T$ increases as $E$ increases \cites{Pelinovsky2012,cuevas-maraver2016}.
Pelinovsky and Sakovich prove the existence of multi-site breathers close to the AC limit, i.e. for $d$ small, which are even functions of $t$ \cite{Pelinovsky2012}. Specifically, for a finite set of lattice sites $S = \{ k_1, \dots, k_N \}$, with $k_i < k_{i+1}$, they start with a solution
\begin{equation}
\mathbf{u}^{(0)}(t) = \sum_{i=1}^N \sigma_i \phi(t) \mathbf{e}_{k_i}
\end{equation}
at the AC limit, where $\phi(t)$ is the fundamental AC breather, $\mathbf{e}_{k_i}$ is the unit vector for site $k_i$ in the integer lattice, and $\sigma_i = \pm 1$ is the phase factor for the oscillator at site $k_i$. Adjacent oscillators are in-phase if $\sigma_i \sigma_{i+1} = 1$, and out-of-phase if $\sigma_i \sigma_{i+1} = -1$. They then use the implicit function theorem to prove the existence of a multi-site breather $\mathbf{u}^{(d)}(t)$ to \cref{eq:DKG} for sufficiently small $d$ \cite{Pelinovsky2012}*{Theorem 1}.
The fact that in-phase and out-of-phase structures
are the only ones available in discrete Klein-Gordon
lattices with nearest-neighbor interactions
has been shown in~\cite{KOUKOULOYANNIS20132022}.
In that light, we will restrict our considerations
to such configurations in what follows. Nevertheless,
it is relevant to mention in passing that the examination
of so-called phase-shift multi-breathers (with relative
phases different than $0$ or $\pi$) in lattices with
interactions beyond nearest-neighbor ones remains an
active topic of investigation in Klein-Gordon (and DNLS)
settings~\cite{PENATI201992}.
\subsection{Linearization}\label{sec:DKGlinear}
For a specific coupling constant $d$ and fundamental period $T$, let $\mathbf{u}(t)$ be a breather solution to \cref{eq:DKG}. To study the spectral stability of $\mathbf{u}(t)$, we linearize equation \cref{eq:DKG} about $\mathbf{u}(t)$ by substituting the perturbation ansatz $\mathbf{u}(t) + \epsilon \mathbf{v}(t)$ and keeping terms of order $\epsilon$ to obtain the linearized equation
\begin{equation}\label{eq:DKGlinear}
\ddot{v}_n = d (\Delta_2 v)_n - f'(u_n)v_n,
\end{equation}
which can be written as the first order linear system
\begin{equation}\label{eq:DKGlinear1}
\frac{d}{dt} \begin{pmatrix} v_n \\ w_n \end{pmatrix} =
\begin{pmatrix}
w_n \\
d (\Delta_2 v)_n - f'(u_n)v_n
\end{pmatrix}
\end{equation}
by letting $w_n = \dot{v}_n$. Since $\mathbf{u}(t)$ has period $T$, it follows from Floquet theory that its spectral stability depends on the Floquet multipliers, which are the spectrum of the monodromy operator $\mathcal{M} = \Phi(0, T)$, where $\Phi(s, t)$ is the evolution operator for \cref{eq:DKGlinear1}.
If $\mu$ is a Floquet multiplier, then the corresponding Floquet exponent $\lambda$ (which is unique modulo $2 \pi i/T$) is related to $\mu$ by $\mu = e^{\lambda T}$.
For every Floquet exponent $\lambda$, there is a corresponding solution $\mathbf{v}(t) = e^{\lambda t} \mathbf{w}(t)$ to the linearized equation \cref{eq:DKGlinear}, where $\mathbf{w}(t)$ is periodic with period $T$ (see, for example, \cite{Kapitula2013}*{Lemma 2.1.29}). Substituting this ansatz into \cref{eq:DKGlinear}, we obtain the Floquet eigenvalue problem
\begin{equation}\label{eq:DKGeig}
d (\Delta_2 w)_n - f'(u_n)w_n - \ddot{w}_n = 2 \lambda \dot{w}_n + \lambda^2 w_n,
\end{equation}
where $\mathbf{w} \in \ell^2({\mathbb Z}, H^2_\textrm{per}[0,T]) \subset \ell^2({\mathbb Z}, L^2_\textrm{per}[0,T])$, and we use the inner product
\begin{equation}\label{eq:IP1}
\langle \mathbf{u}, \mathbf{v} \rangle_{(\ell^2({\mathbb Z}, L^2_\textrm{per}[0,T]))} = \sum_{n=-\infty}^\infty \int_0^T u_n(s) \overline{v_n(s)} ds
\end{equation}
on $\ell^2({\mathbb Z}, L^2_\textrm{per}[0,T])$. We can write equation \cref{eq:DKGeig} as
\begin{equation}\label{eq:DKGeigL}
\mathcal{L}(\mathbf{u})\mathbf{w} = (2 \lambda \partial_t + \lambda^2 )\mathbf{w},
\end{equation}
where the linear operator $\mathcal{L}(\mathbf{u})$ is defined by the LHS of \cref{eq:DKGeig}. Since \cref{eq:DKG} is Hamiltonian, the Floquet exponents must come in quartets $\lambda = \pm \alpha \pm \beta i$. It follows that the Floquet multipliers $\mu$ can only occur in one of three patterns: a pair $\{ \mu, \overline{\mu} \}$ on the unit circle; a pair $\{ \mu, \mu^{-1} \}$ on the real line; or a quartet $\{ \mu, \overline{\mu}, \mu^{-1}, \overline{\mu}^{-1} \}$ off of the unit circle. Therefore, the breather solution $\mathbf{u}(t)$ is spectrally unstable unless all of its Floquet multipliers lie on the unit circle.
There is always a Floquet exponent at 0 (corresponding to the Floquet multiplier $\mu = 1$), since $\dot{\mathbf{u}}$ is a solution to \cref{eq:DKGeig}, i.e. $\mathcal{L}(\mathbf{u})\dot{\mathbf{u}} = 0$, which can be verified by differentiating \cref{eq:DKG} with respect to $t$. Furthermore, there exists a solution $\mathbf{y} \in \ell^2({\mathbb Z}, H^2_\textrm{per}[0,T])$ which solves
\begin{equation}
\mathcal{L}(\mathbf{u})\mathbf{y} = 2 \ddot{\mathbf{u}},
\end{equation}
and can be chosen to be perpendicular to $\dot{\mathbf{u}}$ with respect to the inner product \cref{eq:IP1} (see \cite{Pelinovsky2012}*{Section 3}, noting that the corresponding linear operator to $\mathcal{L}(\mathbf{u})$ in \cite{Pelinovsky2012} has the opposite sign). In fact, we can actually compute $\mathbf{y}(t)$.
Letting $\omega = 2 \pi / T$ be the frequency of the breather and normalizing the period of the breather to $2 \pi$ by rescaling the time variable to $\tau = \omega T$, as in \cite{kevrekidis2016}, equation \cref{eq:DKG} becomes
\begin{equation}\label{eq:DKGomega}
\omega^2 \partial_\tau^2 u_n = d (\Delta_2 u)_n - f(u_n).
\end{equation}
Differentiating with respect to $\omega$, we obtain
\begin{equation}\label{eq:DKGdiffw}
d (\Delta_2 \partial_\omega u)_n - f'(u_n)\partial_\omega u_n
- \omega^2 \partial_\tau^2 ( \partial_\omega u_n) = 2 \omega \partial_\tau^2 u_n.
\end{equation}
Changing variables back to $t$, this becomes $\mathcal{L}(\mathbf{u})\partial_\omega \mathbf{u} = \frac{2}\omega \ddot{\mathbf{u}}$, thus $\mathbf{y} = \omega \partial_\omega \mathbf{u}$.
\subsection{Continuous spectrum}
The continuous spectrum of $\mathcal{L}(\mathbf{u})$ is the set of all $\lambda$ for which the limiting problem
\begin{equation}\label{eq:DKGeigcont}
d (\Delta_2 w)_n - f'(0)w_n - \ddot{w}_n = 2 \lambda \dot{w}_n + \lambda^2 w_n
\end{equation}
has a solution. Following the procedure in \cite{cuevas-maraver2016}*{Section 2.1}, the continuous spectrum is the bands
\begin{equation}\label{eq:contspec}
\begin{aligned}
\lambda &= i\left( \pm \omega(\theta) - \frac{2 \pi m}{T} \right) && m \in {\mathbb Z} \\
\omega(\theta) &= \sqrt{ f'(0) + 4 d \sin^2\left( \frac{\theta}{2} \right) } && \theta \in [-\pi, \pi]
\end{aligned}
\end{equation}
on the imaginary axis. The corresponding Floquet multipliers comprise two bands on the unit circle, which are symmetric about the real axis, and are given by
\begin{equation}\label{eq:contspecmult}
\begin{aligned}
\mu = \exp \left( \pm i \sqrt{f'(0) + 4 d \sin^2 \left(\frac{\theta}{2}\right) }T \right) && \theta \in [0, \pi].
\end{aligned}
\end{equation}
Let $m_0$ be the largest nonnegative integer such that $T - 2 \pi m_0 > 0$. At the AC limit, the bands consist of the two points $\mu = \exp(\pm i \theta_0)$, where $\theta_0 = T - 2 \pi m_0$. For $d>0$, the bands stretch from $\mu = \exp(\pm i \theta_0)$ to $\mu = \exp(\pm i \theta_1)$, where $\theta_1 = \sqrt{f'(0) + 4 d \sin^2 \left(\frac{\theta}{2}\right) }T - 2 \pi m_0$. Following the analysis in \cite{cuevas-maraver2016}*{Section 2.2}, if $T \in (n \pi, (n+1)\pi)$ for $n$ even, the upper band has positive Krein signature, and the lower band has negative Krein signature. As $d$ is increased, the bands grow towards $(0,-1)$ (see \cref{fig:bands}, left). The ends of the bands meet when $\theta_1 = \pi$, which occurs when $d_{\pi} = \frac{1}{4} \left( \frac{(1 + 2 m_0)^2 \pi^2}{T^2} - 1\right)$, at which point they merge into a single band. They meet again when $\theta_1 = 0$, which occurs when $d_0 = \frac{1}{4} \left( \frac{(2 + 2 m_0)^2 \pi^2}{T^2} - 1\right)$, at which point they comprise the entire unit circle.
Conversely, if $T \in (n \pi, (n+1)\pi)$ for $n$ odd, the upper band has negative Krein signature, the lower band has positive Krein signature, and the bands grow towards (1,0) as $d$ is increased (see \cref{fig:bands}, right).
\begin{figure}
\begin{center}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{contspeccartoon1.eps}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{contspeccartoon2.eps}
\end{subfigure}
\end{center}
\caption{Cartoon showing continuous spectrum bands on the unit circle for $T \in (n \pi, (n+1)\pi)$ with $n$ even (a) and $n$ odd (b). Krein signature of bands is indicated, and bands grow in the direction of the arrow with increasing $d$.}
\label{fig:bands}
\end{figure}
\subsection{Spatial dynamics}
We now reformulate both the DKG equation \cref{eq:DKG} and the eigenvalue problem \cref{eq:DKGeig} using a spatial dynamics approach, as in \cites{Parker2020,Parker2021}. Let $\mathbf{u}(t)$ be a breather solution to \cref{eq:DKG} with period $T$, and define $U(n) = (u(n), \tilde{u}(n)) = ( u_n, u_{n-1} )$. Then equation \cref{eq:DKG} is equivalent to the lattice dynamical system
\begin{equation}\label{eq:dynEq}
U(n+1) = F(U(n)),
\end{equation}
where
\begin{equation}\label{eq:F}
F\begin{pmatrix}u \\ \tilde{u} \end{pmatrix} =
\begin{pmatrix}2u + \dfrac{1}{d}f(u) + \dfrac{1}{d} \partial_t^2 u - \tilde{u} \\
u
\end{pmatrix}.
\end{equation}
We note that since $f$ is a odd function, if $U(n)$ is a solution to \cref{eq:dynEq}, then $-U(n)$ is as well. The Floquet eigenvalue problem \cref{eq:DKGeig} can similarly be written as
\begin{equation}\label{eq:dynEVP}
W(n+1) = \left[ DF(U(n)) + (2 \lambda \partial_t + \lambda^2) B \right] W(n),
\end{equation}
where
\begin{equation}\label{eq:DFU}
DF(U(n)) = \begin{pmatrix}
2 + \dfrac{f'(u(n))}{d} + \dfrac{1}{d}\partial_t^2 & -1 \\ 1 & 0
\end{pmatrix}, \qquad
B = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}.
\end{equation}
The zero function $U(n) = 0$ is an equilibrium solution to \cref{eq:dynEq}. At this point, the standard procedure (see, for example, \cites{Parker2021,Parker2020,Sandstede1998}) is to consider $U(n)$ as a homoclinic orbit of the equilibrium at 0. The complication here is that for the difference equation \cref{eq:dynEq} to be well-posed, we require $U(n) \in C_\textrm{per}^\infty([0,T],{\mathbb R}^2)$ for all $n$, rather than just $U(n) \in H^2_\textrm{per}([0,T], {\mathbb R}^2)$, since each application of $F$ involves differentiating twice with respect to $t$, and equation \cref{eq:dynEq} involves applying $F$ an arbitrary number of times. In essence, the equation \cref{eq:dynEq} is more restrictive than the original system.
Since $C_\textrm{per}^\infty([0,T])$ is not a closed subspace of $L^2([0,T])$, it is not straightforward to adapt the stable manifold theorem and results on exponential dichotomies to this problem, even if $DF(0)$ has the desired spectral properties. As an alternative, we will consider a finite-dimensional approximation, where we project the problem onto a finite-dimensional subspace of $L_\textrm{per}^2([0,T])$. Roughly, this subspace consists of the first $N$ Fourier modes of the standard basis for $L_\textrm{per}^2([0,T])$. Although this is a limited result, we note that since we require $U(n)$ to be smooth, its Fourier coefficients will decay exponentially. Since we can take $N$ as large as we like, this should be a very good approximation. In addition, since many of the numerical simulations are performed using Fourier spectral methods, this theory applies directly to such numerical discretization. Before we formulate our approximation, we prove some important results about the spectrum of $DF(0)$.
\subsection{Spectrum of \texorpdfstring{$DF(0)$}{DF(0)}}
The linearization of \cref{eq:dynEq} about the equilibrium at 0 is the constant coefficient linear operator
\begin{equation}\label{eq:DF0}
DF(0) = \begin{pmatrix}
\dfrac{1}{d}\partial_t^2 + \dfrac{f'(0)}{d} + 2 & -1 \\ 1 & 0
\end{pmatrix},
\end{equation}
which is invertible with inverse
\begin{equation}\label{eq:DF0inv}
DF(0)^{-1} = \begin{pmatrix}
0 & 1 \\ -1 & \dfrac{1}{d}\partial_t^2 + \dfrac{f'(0)}{d} + 2
\end{pmatrix}
\end{equation}
First, we determine the eigenvalues and eigenfunctions of $DF(0)$.
\begin{lemma}\label{lemma:DF0eigs}
The set of eigenvalues of $DF(0)$ is given by $\bigcup_{k \in {\mathbb Z}} \{\lambda_k, \lambda_k^{-1} \}$, where
\begin{equation}\label{eq:DF0lambdak}
\lambda_k = \frac{1}{2}\left( r_k + \sqrt{r_k^2 - 4} \right), \quad r_k = -\frac{4 k^2 \pi^2}{d T^2} + \frac{f'(0)}{d} + 2.
\end{equation}
For $k \neq 0$, these have algebraic multiplicity 2, since $\lambda_{-k} = \lambda_k$. For each $k \in {\mathbb Z}$, $\{\lambda_k, \lambda_k^{-1} \}$ is either real, or a complex conjugate pair on the unit circle. The eigenfunctions corresponding to $\left\{ \lambda_k, \lambda_k^{-1} \right\}$ are $\left\{ U_k(t), U_k^{-1}(t) \right\}$, which are defined, up to constant multiple, by
\begin{equation}\label{eq:DF0eigenfns}
\begin{aligned}
U_k(t) &= \begin{pmatrix}v_k(t) \\ \lambda_k^{-1} v_k(t) \end{pmatrix}, \quad
U_k^{-1}(t) = \begin{pmatrix}v_k(t) \\ \lambda_k v_k(t) \end{pmatrix}, \quad
v_k(t) = \frac{1}{T} \exp\left( i \frac{2 \pi k t}{T} \right).
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
Consider the eigenvalue problem $DF(0) U(t) = \lambda U(t)$ on $H^2_\textrm{per}([0,T],{\mathbb R}^2)$, where $U(t) = (v(t), w(t))^T$. We note that $\lambda = 0$ is not an eigenvalue, since that implies $v = w = 0$. The eigenvalue problem then reduces to the system of equations
\begin{align}\label{eq:DF0EVPsystem}
\left( \frac{1}{d}\partial_t^2 + \frac{f'(0)}{d} + 2 \right) v(t) = \left( \lambda + \frac{1}{\lambda} \right) v(t), \quad
w = \frac{1}{\lambda} v(t).
\end{align}
Letting $r = \lambda + \frac{1}{\lambda}$ and using the periodic boundary conditions $v(T) = v(0)$, the set of solutions to \cref{eq:DF0EVPsystem} is given by
\begin{align}
v_k(t) &= \frac{1}{T} \exp\left( i \frac{2 \pi k t}{T} \right), \quad r_k = -\frac{4 k^2 \pi^2}{d T^2} + \frac{f'(0)}{d} + 2 && k \in {\mathbb Z},
\end{align}
where the functions $v_k(t)$ have been normalized. The corresponding eigenvalues of $DF(0)$ are then given by $\left\{ \lambda_k, \lambda_k^{-1} \right\}$, where $\lambda_k$ is defined by \cref{eq:DF0lambdak}, and the corresponding eigenfunctions are given by \cref{eq:DF0eigenfns}. The pair $\left\{ \lambda_k, \lambda_k^{-1} \right\}$ is real if $|r_k| \geq 2$, and is complex with modulus 1 if $|r_k| < 2$.
\end{proof}
We note that the spectrum of $DF(0)$ depends on both the coupling parameter $d$ and the period $T$. It follows from \cref{lemma:DF0eigs} that the spectrum of $DF(0)$ is bounded away from the unit circle provided $|r_k| > 2$ for all $k$. The following lemma gives some conditions on $T$ and $d$ to guarantee that this is the case.
\begin{lemma}\label{lemma:DF0hyp}
The spectrum of $DF(0)$ is bounded away from the unit circle if, for a specific nonnegative integer $k$, $T$ and $d$ are chosen so that
\begin{equation}\label{eq:Tdpair}
\frac{2 k \pi}{\sqrt{f'(0)}} < T < \frac{2 (k+1) \pi}{\sqrt{f'(0)}} , \qquad 0 < d < \frac{(k+1)^2\pi^2}{T^2} - \frac{f'(0)}{4}.
\end{equation}
\begin{proof}
Since $f'(0) > 0$, $r_0 = 2 + \frac{1}{d}f'(0) > 2$, and $r_k$ is strictly decreasing in $k$, with $r_k \rightarrow -\infty$ as $k \rightarrow \infty$. Thus $|r_k| > 2$ for all $k$ if $r_k > 2$ and $r_{k+1} < -2$ for some nonnegative integer $k$, from which the conditions \cref{eq:Tdpair} follow.
\end{proof}
\end{lemma}
We take the following assumption on the spectrum of $DF(0)$, which is the analogue to hyperbolicity in the finite-dimensional case.
\begin{hypothesis}\label{hyp:hyp}
The coupling constant $d$ and period $T$ are chosen so that the spectrum of $DF(0)$ is bounded away from the unit circle.
\end{hypothesis}
\section{Finite dimensional approximation}\label{sec:findim}
In this section, we define our finite dimensional approximation for \cref{eq:dynEq}. For $M \geq 1$, let
\begin{equation}\label{eq:XM}
X_M = \spn\left\{ \bigcup_{k = -M}^M v_k(t) \right\}, \qquad
v_k(t) = \frac{1}{T} \exp\left( i \frac{2 \pi k t}{T} \right)
\end{equation}
be the $(2M+1)$-dimensional subspace of $L_\textrm{per}^2([0,T])$ spanned by the Fourier basis functions with wavenumber $|k| \leq M$, and let $P_M: L_\textrm{per}^2([0,T]) \rightarrow X_M$, defined by
\begin{equation}\label{eq:PM}
P_M u(t) = \sum_{k=-M}^M \langle u, v_k \rangle_{L^2([0,T])} v_k(t)
= \sum_{k=-M}^M \left( \int_0^T u(s) \overline{v_k(s)} ds \right) v_k(t)
\end{equation}
be the corresponding projection operator. In addition, let $X_{M,e}$ be the $(M+1)$-dimensional subspace of $X_M$ comprising functions which are even in $t$, i.e.
\begin{equation}
X_{M,e} = \left\{ f \in X_M : f(-t) = f(t) \text{ for all }t \in {\mathbb R} \right\}.
\end{equation}
For $M \geq 1$, define the approximation of the discrete Klein-Gordon equation \cref{eq:DKG} on $\ell^2({\mathbb Z}, X_M)$ by
\begin{equation}\label{eq:DKGapprox}
\begin{aligned}
\ddot{u}_n &= d (\Delta_2 u)_n - g(u_n) && \qquad u_n \in X_M,
\end{aligned}
\end{equation}
where $g: X_M \rightarrow X_M$ is given by
\begin{equation}g(u) = P_M f(u).
\end{equation}
When $u = 0$, $g(0) = P_M f(0) = 0$, and $g'(0) = P_M f'(0) = f'(0)$, since $f'(0)$ is a constant function, which is in $X_M$ for all $M$.
Furthermore, since $f$ is an odd function, $g$ is as well. In general, it not the case that $P_M f(u) = f(P_M u)$, thus we cannot simply obtain a solution to \cref{eq:DKGapprox} by projecting a solution of \cref{eq:DKG} onto $X_M$. However, since the Fourier coefficients of a smooth, $T$-periodic function $u(t)$ decay exponentially, equation \cref{eq:DKGapprox} should be a reasonable approximation to \cref{eq:DKG} for large $M$. Linearization of \cref{eq:DKGapprox} about a solution $\mathbf{u} \in \ell^2({\mathbb Z}, X_M)$ yields the eigenvalue problem
\begin{equation}\label{eq:DKGMeig}
\begin{aligned}
d (\Delta_2 w)_n - g'(u_n)w_n - \ddot{w}_n = 2 \lambda \dot{w}_n + \lambda^2 w_n,
\end{aligned}
\end{equation}
which we write as
\begin{equation}\label{eq:DKGMeigL}
\mathcal{L}_M(\mathbf{u})\mathbf{w} = (2 \lambda \partial_t + \lambda^2 )\mathbf{w},
\end{equation}
where $\mathcal{L}_M(\mathbf{u})$ is the linear operator on $\ell^2({\mathbb Z}, X_M)$ defined by the LHS of \cref{eq:DKGMeigL}. As in \cref{sec:DKGlinear}, $\mathcal{L}_M(\mathbf{u}) \dot{\mathbf{u}} = 0$, and there exists a solution $\mathbf{y}$ to $\mathcal{L}_M(\mathbf{u}) \mathbf{y} = 2 \ddot{\mathbf{u}}$, where $\mathbf{y} = \omega \partial_\omega \mathbf{u}$ and $\omega = 2 \pi / T$.
Reformulating \cref{eq:DKGapprox} from a spatial dynamics perspective yields the discrete dynamical system on $X_M^2$
\begin{align}\label{eq:dynEqM}
U(n+1) &= F_M(U(n)) && U(n) \in X_M^2,
\end{align}
where
\begin{equation}\label{eq:FM}
F_M\begin{pmatrix}u \\ \tilde{u} \end{pmatrix} =
\begin{pmatrix}2u + \dfrac{1}{d}g(u) + \dfrac{1}{d} \partial_t^2 u - \tilde{u} \\
u
\end{pmatrix}
\end{equation}
Since $F_M(-U) = -F_M(U)$, it follows that if $U(n)$ is a solution to \cref{eq:dynEqM}, so is $-U(n)$. Similarly, the eigenvalue problem \cref{eq:DKGMeig} can be reformulated as
\begin{equation}\label{eq:dynEVPM}
W(n+1) = \left[ DF_M(U(n)) + (2 \lambda \partial_t + \lambda^2) B \right] W(n),
\end{equation}
where $B$ is defined in \cref{eq:DFU}. Since $g'(0) = f'(0)$, the linear operator $DF_M(0)$ on $X_M^2$ is also given by \cref{eq:DF0}.
Finally, let $\{\tau(s) : s \in {\mathbb R}\}$ be the one-parameter group of unitary translation operators on $X_M^2$, defined by $[\tau(s)]U(\cdot) = U(\cdot - s)$, which has infinitesimal generator $\tau'(0) = \partial_t$. We note that this group is well-defined on $X_M^2$, since
\[
\tau(s) v_k(t)
\frac{1}{T} \exp\left( i \frac{2 \pi k (t-s)}{T}\right)
= \exp\left( -i \frac{2 \pi k s}{T} \right) \frac{1}{T} \exp\left( i \frac{2 \pi k t}{T}\right)
= \exp\left( -i \frac{2 \pi k s}{T} \right) v_k(t),
\]
i.e., the group action multiplies a basis element by a constant. In fact, the eigenvalue of \cref{eq:DKGMeig} at 0 is a result of this translational symmetry. The function $F_M$ from \cref{eq:dynEqM} (as well as $F$ from the full system \cref{eq:dynEq}) commutes with this one-parameter group, i.e. $F(\tau(s) U) = \tau(s) F(U)$. It is crucial to note that this symmetry is lost if we consider the problem \cref{eq:dynEqM} on $X_{M,e}^2$, since the space of even functions is not translation invariant.
\section{Multi-breathers}\label{sec:multi}
Our strategy will be to first prove that multi-breathers exist on the subspace $X_{M,e}^2$ of even functions. Since there is no translational symmetry, the stable and unstable manifolds of the origin will intersect transversely, which greatly simplifies the analysis. This parallels the restriction in \cite{Pelinovsky2012} to breathers which are even in $t$, and is consistent with the odd symmetry of the nonlinearity $f$.
Once that is accomplished, we will return to the full space $X_{M}^2$ for the eigenvalue problem, and use Lin's method as in \cites{Parker2021,Parker2020,Sandstede1998} to construct the interaction eigenfunctions as piecewise linear combinations of the eigenfunction corresponding to translation symmetry. This technique is similar to the one we employed in \cite{Parker2020} for DNLS, where, to prove the existence of multi-pulses, we removed the gauge symmetry by restricting the problem to real-valued solutions.
\subsection{Primary breather}
As noted above, we will take the existence of a primary, single-site breather as a hypothesis.
Fix $d$ and $T$ such that \cref{hyp:hyp} holds. First, we consider the system \cref{eq:dynEqM} on $X_{M,e}^2$. By \cref{lemma:DF0eigs}, the $2M+2$ eigenvalues of $DF_M(0)$ on $X_{M,e}^2$ are given by $S_M = \bigcup_{k=0}^M \{\lambda_k, \lambda_k^{-1} \}$, where these are defined in the statement of \cref{lemma:DF0eigs}. By \cref{hyp:hyp}, the spectrum of $DF_M(0)$ is real and does not intersect the unit circle, thus 0 is a hyperbolic equilibrium point of \cref{eq:dynEqM}. Define the stable and unstable subsets of the spectrum of $DF_M(0)$ by
\[
S_M^s = \{ \lambda \in S_M : |\lambda| < 1\}, \qquad S_M^u = \{ \lambda \in S_M : |\lambda| > 1\}.
\]
By symmetry, $|S_M^s| = |S_M^u| = M+1$. Define
\begin{equation}\label{eq:defrM}
r_M = \min \{ |\lambda| : \lambda \in S_M^u, |\lambda| > 1 \}.
\end{equation}
Since $X_{M,e}^2$ is finite-dimensional, the stable manifold theorem holds. For each $M \geq 1$, let $W_{M,e}^s(0)$ and $W_{M,e}^u(0)$ be the $(M+1)$-dimensional stable and unstable manifolds of the equilibrium at 0, which are subsets of $X_{M,e}^2$. A breather solution to \cref{eq:dynEqM} is a homoclinic orbit which lies in the intersection of the stable and unstable manifolds. We take the existence of such a solution as a hypothesis.
\begin{hypothesis}\label{hyp:breather}
Let $d$ and $T$ be chosen according to \cref{hyp:hyp}. There exists a positive integer $M_0$ such that for all $M \geq M_0$, the stable and unstable manifolds $W_{M,e}^s(0)$ and $W_{M,e}^u(0)$ intersect transversely in $X_{M,e}^2$ in a homoclinic orbit $Q_M(n) = (q(n), \tilde{q}(n))^T = (q_n, q_{n-1})^T$.
\end{hypothesis}
The first component $q_n$ is a breather solution to \cref{eq:DKGapprox}. As a consequence of the stable manifold theorem, we have the estimate
\begin{equation}\label{eq:U1decayest}
\|Q_M(n)\|_{L^2_\textrm{per}([0,T])} \leq C r_M^{-|n|}.
\end{equation}
We now consider \cref{eq:dynEqM} on $X_M^2$. The spectrum of $DF(0)$ on $X_M^2$ is exactly the same as that on $X_{M,e}^2$, except the eigenvalues corresponding to $k = 1, \dots, M$ have multiplicity of 2. It follows that 0 is also a hyperbolic equilibrium of \cref{eq:dynEqM} on $X_M^2$. Let $W_M^s(0)$ and $W_M^u(0)$ be the $(2M+1)$-dimensional stable and unstable manifolds of the equilibrium at 0, which this time are subsets of $X_M^2$. For $M \geq M_0$, $Q_M(n)$ is also a homoclinic orbit connecting these stable and unstable manifolds. In the next hypothesis, we assume that this intersection is non-degenerate.
\begin{hypothesis}\label{hyp:breathernondegen}
Let $d$ and $T$ be chosen according to \cref{hyp:hyp}, and let $M_0$ and $Q_M(n)$ be as in \cref{hyp:breather}. Then for all $M \geq M_0$, the stable and unstable manifolds $W_M^s(0)$ and $W_M^u(0)$ have a one-dimensional intersection in $Q_M(n)$.
\end{hypothesis}
The variational equation is the linearization of \cref{eq:dynEqM} on $X_M^2$ about the homoclinic orbit solution $Q_M(n)$, which is given by
\begin{equation}\label{eq:vareq}
\begin{aligned}
W(n+1) &= DF_M(Q_M(n)) W(n) && W(n) \in X_M^2.
\end{aligned}
\end{equation}
Since the tangent spaces of $W_M^s(0)$ and $W_M^u(0)$ have a one-dimensional intersection by \cref{hyp:breathernondegen}, it follows that $\dot{Q}_M(n) = (\dot{q}_n, \dot{q}_{n-1})$ is the unique, bounded solution to \cref{eq:vareq}, up to scalar multiples. ($\dot{Q}_M(n)$ is not a solution to \cref{eq:vareq} on $X_{M,e}^2$, since it is an odd function). We can thus decompose the tangent spaces to $W_M^u(0)$ and $W_M^s(0)$ at $U_M^1(0)$ as
\begin{equation}\label{eq:TWdecomp}
T_{Q_M(0)}W^u_M(0) = {\mathbb R} \dot{Q}_M(0) \oplus Y_M^-, \qquad
T_{Q_M(0)}W^s_M(0) = {\mathbb R} \dot{Q}_M(0) \oplus Y_M^+,
\end{equation}
where $\dim Y_M^- = \dim Y_M^+ = 2M$. In addition, the adjoint variational equation
\begin{equation}\label{eq:adjvareq}
Z(n) = DF_M(Q_M(n))^* Z(n+1)
\end{equation}
has a unique bounded solution, given by
\begin{equation}\label{eq:Z1}
Z_M(n) = (-\dot{q}_{n-1}, \dot{q}_n)^T,
\end{equation}
and $Z_M(0) \perp {\mathbb R} \dot{Q}_M(0) \oplus Y_M^- \oplus Y_M^+$ by \cite{Parker2020}*{Lemma 1}. We can thus decompose $X_M^2$ as
\begin{equation}\label{eq:Xdecomp}
X_M^2= {\mathbb R} \dot{Q}_M(0) \oplus Y_M^- \oplus Y_M^+ \oplus {\mathbb R} Z_M(0).
\end{equation}
\subsection{Existence}
We construct a multi-breather on $X_{M,e}$ by splicing together multiple copies of the primary breather $Q_M(n)$ in a chain. We characterize a multi-breather in the following way. Let $m > 1$ be the total number of copies of the primary breather in the chain. Let $N_i$ ($i = 1, \dots, m-1$) be the distances (in lattice points) between the center point of each breather. We seek to construct a solution $U(n)$ which can be written piecewise in the form
\begin{equation}\label{eq:Upiecewise}
\begin{aligned}
U_i^-(n) &= \sigma_i Q_M(n) + \tilde{U}_i^-(n) && n \in [-N_{i-1}^-, 0] && \quad i = 1, \dots, m\\
U_i^+(n) &= \sigma_i Q_M(n) + \tilde{U}_i^+(n) && n \in [0, N_i^+] && \quad i = 1, \dots, m,
\end{aligned}
\end{equation}
where $\sigma_i \in \{1, -1\}$ represents the orientation of each copy of the primary breather, $N_i^+ = \lfloor \frac{N_i}{2} \rfloor$, $N_i^- = N_i - N_i^+$, and $N_0^- = N_m^+ = \infty$. Adjacent copies of the primary breather are in-phase if $\sigma_i \sigma_{i+1} = 1$, or out-of-phase if $\sigma_i \sigma_{i+1} = -1$. (As in \cite{Pelinovsky2012}, other phase relations are not considered here; see also~\cite{KOUKOULOYANNIS20132022}). The functions $\tilde{U}_i^\pm$ in \cref{eq:Upiecewise} are remainder terms, which will be small. We also define the characteristic distance
\begin{equation}\label{defN}
N = \frac{1}{2} \min\{ N_i \},
\end{equation}
which will be used in the estimates of the remainder terms $\tilde{U}_i^\pm$. The individual pieces $U_i^\pm(n)$ are joined together end-to-end as in \cites{Sandstede1998,Knobloch2000,Parker2020,Parker2021} to create the multi-breather $U(n)$, which can be written in piecewise form as
\begin{equation}
\begin{aligned}
U(n) &= \begin{cases}
U_i^-\left( n - \sum_{j=1}^{i-1}N_j \right) & \sum_{j=1}^{i-1}N_j - N_{i-1}^- + 1 \leq n \leq \sum_{j=1}^{i-1}N_j \\
U_i^+\left( n - \sum_{j=1}^{i-1}N_j \right) & \sum_{j=1}^{i-1}N_j + 1 \leq n \leq \sum_{j=1}^{i-1}N_j + N_i^+
\end{cases}
&& i = 1, \dots, m,
\end{aligned}
\end{equation}
where we define $\sum_{j=1}^0 N_j = 0$. We have the following existence theorem concerning the existence of multi-breathers on $X_{M,e}$.
\begin{theorem}\label{th:multi-breathers}
Assume \cref{hyp:hyp} and \cref{hyp:breather}, and let $M \geq M_0$, where $M_0$ is defined in \cref{hyp:breather}. Then there exists a positive integer $N_0$ with the following property. For all $m > 1$ and distances $N_i \geq N_0$, there exists a unique solution $U(n)$ which comprises, to leading order, $m$ sequential copies of the primary breather, and can be written piecewise in the form \cref{eq:Upiecewise}. For the remainder terms $\tilde{U}_i^\pm(n)$, we have the estimates
\begin{equation}\label{eq:Uestimates}
\begin{aligned}
\tilde{U}_i^+(N_i^+) &= \sigma_{i+1} Q_M(-N_i^-) + \mathcal{O}(r_M^{-2N}) \\
\tilde{U}_{i+1}^-(-N_i^-) &= \sigma_{i} Q_M(N_i^+) + \mathcal{O}(r_M^{-2N}) \\
\| \tilde{U}_i^-(n)\|_{X_M} &\leq C r_M^{-N_{i-1}^-} r_M^{-(N_{i-1}^- + n)} && \qquad n = 2, \dots, m\\
\|\tilde{U}_i^+(n) \|_{X_M} &\leq C r_M^{-N_i^+} r_M^{-(N_i^+ - n)} && \qquad n = 1, \dots, m-1 \\
\| \tilde{U}_1^-(n)\|_{X_M} &\leq C r_M^{-2N} r_M^{n} \\
\|\tilde{U}_m^+(n) \|_{X_M} &\leq C r_M^{-2N} r_M^{-n},
\end{aligned}
\end{equation}
which hold as well for derivatives of $\tilde{U}_i^\pm(n)$.
\begin{proof}
The proof is a straightforward adaptation of the proofs of theorems 1 and 3 in \cite{Parker2020}, and is very similar to the proof of \cite{Parker2021}*{Theorem 1}. Since the stable and unstable manifolds $W_{M,e}^s(0)$ and $W_{M,e}^u(0)$ intersect transversely in $X_{M,e}$, the implementation of Lin's method does not involve jump conditions.
\end{proof}
\end{theorem}
\subsection{Spectral stability}
For a multi-breather consisting of $m$ components, as long as the individual copies of the primary breather are sufficiently well separated, there will be $2(m-1)$ eigenvalues in spectrum of \cref{eq:DKGMeig}, i.e., one pair per additional copy of the primary breather, which will be located near the origin. We call these interaction eigenmodes, since they result from nonlinear interactions between the tails of adjacent copies of the primary breather.
As in \cites{Parker2020,Sandstede1998}, we locate these eigenmodes by using Lin's method to construct the corresponding eigenfunctions. We note that this theorem gives the Floquet exponents $\lambda$, which are the eigenvalues of \cref{eq:DKGMeig}. The Floquet multipliers $\mu$ are related to these by $\mu = e^{\lambda T}$. The proof is given in \cref{app:specproof}.
\begin{theorem}\label{th:spectrum}
Assume \cref{hyp:hyp}, \cref{hyp:breather}, and \cref{hyp:breathernondegen}, and let $M \geq M_0$, where $M_0$ is defined in \cref{hyp:breather}. Let $Q_M(n) = (q_n, q_{n-1})$ be the primary breather solution from \cref{hyp:breather}, and let $Y(n) = (y_n, y_{n-1}) = \omega \partial_\omega Q_M(n)$.
Let $U(n)$ be an $m$-component multi-breather constructed as in \cref{th:multi-breathers} with distances $N_i$ and orientation parameters $\sigma_i$. Then there exists $\delta > 0$ with the following property. There exists a bounded, nonzero solution $W(n)$ of the eigenvalue problem \cref{eq:dynEVPM} on $X_M^2$ for $|\lambda| < \delta$ if and only if $E(\lambda) = 0$, where
\begin{equation}\label{Elambda}
E(\lambda) = \det\left(A + \frac{1}{d}K \lambda^2 I + R(\lambda)\right).
\end{equation}
$A$ is the tri-diagonal $m \times m$ matrix
\begin{align}\label{eq:matrixA}
A &= \begin{pmatrix}
-a_1 & a_1 & & & \\
-\tilde{a}_1 & \tilde{a}_1 - a_2 & a_2 \\
& -\tilde{a}_2 & \tilde{a}_2 - a_3 & a_3 \\
& \ddots & & \ddots \\
& & & -\tilde{a}_{m-1} & \tilde{a}_{m-1} \\
\end{pmatrix},
\end{align}
with
\begin{equation}\label{eq:ai}
\begin{aligned}
a_i &= \sigma_i \sigma_{i+1} \int_0^T \left( \dot{q}_{N_i^+}\dot{q}_{-N_i^- - 1}
- \dot{q}_{N_i^+ - 1}\dot{q}_{-N_i^-} \right) dt \\
\tilde{a}_i &= \sigma_i \sigma_{i+1} \int_0^T \left( \dot{q}_{-N_i^-} \dot{q}_{N_i^+ - 1}
- \dot{q}_{-N_i^- - 1}\dot{q}_{N_i^+} \right) dt,
\end{aligned}
\end{equation}
\begin{align}\label{eq:M}
K =
\sum_{n = -\infty}^\infty \int_0^T \left( \dot{q}_n^2 + 2 \dot{y}_n \dot{q}_n \right) dt,
\end{align}
and the remainder term has uniform bound
\begin{equation}\label{eq:Rbound}
\|R(\lambda)(c)\|_{X_M^2} \leq C \left( (r_M^{-N} + |\lambda|)^3\right).
\end{equation}
\end{theorem}
\begin{remark}\label{remark:solvability}
Substituting $y_n = \omega \partial_\omega q_n$, we can write \cref{eq:M} as
\begin{align}\label{eq:M2}
K =
\sum_{n = -\infty}^\infty \int_0^T \dot{q}_n \left( 2 \omega \partial_\omega \dot{q}_n + \dot{q}_n \right) dt,
\end{align}
which is the solvability condition from \cite{kevrekidis2016}. Integrating by parts, we can also write \cref{eq:M} as
\begin{align}\label{eq:M3}
K =
\sum_{n = -\infty}^\infty \int_0^T \left( \dot{q}_n^2 - 2 y_n \ddot{q}_n \right) dt =
\sum_{n = -\infty}^\infty \int_0^T \left( \dot{q}_n^2 - 2 \omega (\partial_\omega q_n) \ddot{q}_n \right) dt.
\end{align}
In the AC limit, since the primary breather comprises a single excited site $\phi(t)$ at one of the lattice points, equation \cref{eq:M3} becomes
\begin{align}\label{eq:MAC}
K = \int_0^T \left( \dot{\phi}^2 - 2 \omega(\partial_\omega \phi) \ddot{\phi} \right) dt = - \frac{T^2(E)}{T'(E)},
\end{align}
by the proof of \cite{Pelinovsky2012}*{Lemma 2}, where $T(E)$ is defined in \cref{eq:TE}, noting that $-\omega \partial_\omega \phi$ corresponds to $v$ in that reference.
\end{remark}
\begin{corollary}\label{corr:even}
If the primary breather $Q_M = (q_n, q_{n-1})$ has the even spatial symmetry $q_{-n} = q_n$, the matrix $A$ in \cref{th:spectrum} simplifies to
\begin{align}\label{eq:matrixAsymm}
A &= \begin{pmatrix}
-a_1 & a_1 & & & \\
a_1 & -a_1 - a_2 & a_2 \\
& a_2 & -a_2 - a_3 & a_3 \\
& \ddots & & \ddots \\
& & & a_{m-1} & -a_{m-1} \\
\end{pmatrix},
\end{align}
with $a_i = \sigma_i \sigma_{i+1} b_i$ and
\begin{align*}
b_i &= \begin{cases}
\int_0^T \dot{q}_{\frac{N_i}{2}}\left( \dot{q}_{\frac{N_i}{2}+1} - \dot{q}_{\frac{N_i}{2}-1}\right) dt & N_i \text{ even} \\
\int_0^T \left( \dot{q}_{\frac{N_i-1}{2}}\dot{q}_{\frac{N_i+3}{2}} - \dot{q}_{\frac{N_i+1}{2}}\dot{q}_{\frac{N_i-3}{2}} \right) dt & N_i \text{ odd}
\end{cases}
\end{align*}
The matrix $A$ has one eigenvalue at 0, and the remaining $(m-1)$ eigenvalues $\{\mu_1, \dots, \mu_{m-1}\}$ are real and distinct. As long as the continuous spectrum does not interfere, there are $(m-1)$ pairs of interaction eigenmodes, which are given by
\begin{align}\label{eq:inteigs}
\lambda_j &= \pm \sqrt{-\frac{d \mu_j}{K}} + \mathcal{O}(r^{-2N}) && j = 1, \dots, m-1.
\end{align}
These are either real or imaginary by Hamiltonian symmetry.
\begin{proof}
This follows from \cref{th:spectrum} using same argument as in \cite{Parker2020}*{Theorem 5}.
\end{proof}
\end{corollary}
Two special cases to consider are the double breather ($m=2$), and the multi-breather where all copies of the primary breather are equally spaced. For a double breather, the pair of interaction eigenmodes is given by
\begin{align}\label{eq:inteigsdouble}
\lambda &= \sqrt{\frac{2 \sigma_1 \sigma_2 b_1 d}{K}} + \mathcal{O}(r^{-2N}).
\end{align}
If $b_1$ and $K$ have the same sign, the in-phase double breather is spectrally unstable and the out-of-phase double breather is spectrally neutrally stable. The reverse is true if $b_1$ and $K$ have opposite signs. For an $m$-site multi-breather, if $N_i = N_1$ for all $i$, i.e., the excited sites are equally spaced, then $b_i = b_1$ for all $b$. The matrix $A$ from \cref{eq:matrixAsymm} reduces to $A = b_1 S$, where
\begin{align}\label{eq:matrixequal}
S &= \begin{pmatrix}
-\sigma_1 \sigma_2 & \sigma_1 \sigma_2 & & & \\
\sigma_1 \sigma_2 & -\sigma_2(\sigma_1+\sigma_3) & \sigma_2 \sigma_3 \\
& \sigma_2 \sigma_3 & -\sigma_3(\sigma_2+\sigma_4) & \sigma_3 \sigma_4 \\
& \ddots & & \ddots \\
& & & \sigma_{m-1}\sigma_m & -\sigma_{m-1}\sigma_m \\
\end{pmatrix}.
\end{align}
The eigenvalues of $S$ depend only on the phase differences $\sigma_i \sigma_{i+1}$ between consecutive excited sites, and not on the individual phases $\sigma_i$. We note that the matrix reduction of the eigenvalue problem has the same form as that in equation (35) of \cite{Pelinovsky2012}*{Lemma 2}. In particular, the matrix $S$ in \cref{eq:matrixequal} and the matrix in that lemma are similar (with similarity transformation given $P = \text{diag}(\sigma_1, \dots, \sigma_m)$), and thus have the same eigenvalues. The quantity $b_1$ corresponds to $K_N$ in that lemma. For $i = 1, \dots, m-1$, let $p_i = \sigma_i\sigma_{i+1}$ be the phase differences between consecutive copies of the primary breather, where $p_i = 1$ indicates in-phase, and $p_i = -1$ indicates out-of-phase. Let $n_-$ be the number of negative $p_i$, and $n_+$ the number of positive $p_i$. By \cite{Sandstede1998}*{Lemma 4} (see also the proof of \cite{Pelinovsky2012}*{Lemma 3}), $S$ has a single eigenvalue at 0, $n_+$ positive eigenvalues, and $n_-$ negative eigenvalues. If $b_1$ and $K$ have the same sign, there are $n_+$ pairs of unstable interaction eigenmodes, and $n_-$ pairs of neutrally stable interaction eigenmodes. Thus, the only neutrally stable multi-breather is one in which each pair of consecutive copies of the primary breather is out-of-phase. The reverse is true if $b_1$ and $K$ have opposite signs. For multi-breathers in which the excited sites are not equally spaced, the stability pattern depends on the relative signs of each $b_i$ and $K$, following \cite{Sandstede1998}*{Lemma 4}.
\section{Numerical results}\label{sec:numerics}
We construct both the primary breather and multi-breathers by numerical parameter continuation from the AC limit using the software package AUTO \cite{auto07p}. To do this, we first choose an energy level $E$, from which we compute the period $T$ using \cref{eq:TE}, and then we determine the AC breather solution $\phi(t)$ by solving \cref{eq:singlesiteAC} with the initial conditions given in \cref{sec:DKGbreather}. For the initial condition at $d = 0$, we take $u_k(t) = \pm \phi(t)$ at a finite set of lattice points, and $u_k = 0$ everywhere else. In general, we will take the initial excited sites to be well separated, since the results above only hold for well-separated copies of the primary breather. That being said, this numerical method is effective for any starting configuration at the AC limit (see \cref{fig:SGintersite}, for example, for a double breather comprising two adjacent excited sites).
For the parameter continuation, we use the equation
\begin{equation}\label{eq:HformAUTO}
\frac{d}{dt}\begin{pmatrix} u_n \\ v_n \end{pmatrix} =
\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} \partial \mathcal{H} / \partial u_n \\ \partial \mathcal{H} / \partial v_n \end{pmatrix} +
\epsilon \begin{pmatrix} \partial \mathcal{H} / \partial u_n \\ \partial \mathcal{H} / \partial v_n \end{pmatrix},
\end{equation}
where $\mathcal{H}$ is the Hamiltonian defined by \cref{eq:H} and $\epsilon$ is a small parameter used to break the Hamiltonian structure. To avoid continuing in the direction of translation symmetry, we add the phase condition
\[
\langle \dot{u}_k^\text{old}, u_k - u_k^\text{old} \rangle =
\int_0^T v_k^\text{old}( u_k - u_k^\text{old}) dt = 0,
\]
where $u_k^\text{old}$ refers to the function at the previous continuation step. The index $k$ is chosen to be one of the lattice points at which the initial condition is $\phi(t)$. We approximate the integer lattice with a finite lattice of length $2L+1$, numbered from $-L$ to $L$, and we take Dirichlet boundary conditions at the ends of the lattice, i.e. $u_{L+1} = u_{-L-1} = 0$. Thus equation \cref{eq:Hform} becomes an ODE in $4L+2$ dimensions, with periodic boundary conditions imposed on each variable. Once a breather solution has been found, its Floquet spectrum is found numerically by using Matlab's numerical eigenvalue solver \texttt{eig} to compute the eigenvalues of the monodromy operator associated with the linearized system \cref{eq:DKGlinear1}. Since \cref{eq:DKGlinear1} results from writing a second order linear ODE in $2L+1$ dimensions as first order linear system in $4L+2$ dimensions, the corresponding eigenvectors are of the form $(v_n, w_n)$, where $v_n$ and $w_n$ are $(2L+1)$-dimensional.
\subsection{Discrete sine-Gordon equation}
We first consider the discrete sine-Gordon equation, where the nonlinearity is given by $f(u) = \sin u$. For all of the plots in the section, unless otherwise noted, we use a lattice size $L = 15$, and the energy at the AC limit is $E = 0.25$, which corresponds to a period of $T = 7.91$. We note that $2 \pi < T < 3 \pi$, thus the upper continuous spectrum band has positive Krein signature, and the lower band has negative Krein signature (see \cref{fig:bands}, left).
\cref{fig:singlea} shows the primary breather solution for $d = 0.25$ at $t = 0$, which is symmetric on the spatial lattice. The corresponding Floquet spectrum is shown in \cref{fig:singleb}. There is a single Floquet multiplier (with multiplicity 2) at 1, which is from translation symmetry in $t$. The continuous spectrum bands are a discrete set of points, which is a result of the finiteness
of the lattice and of the spatial discretization, and have effectively merged into a single band by this value of $d$.
\cref{fig:singlec} shows the exponential decay of the $L^2$ norm on $[0,T]$ of the breather solution $u_n(t)$ at lattice site $n$, as $n$ is increased. \cref{fig:singled} plots the relative error of this decay rate compared to $r_M$, which is defined in \cref{eq:defrM}. The exponential decay of the Fourier frequencies (in $t$) of the primary breather is shown in \cref{fig:freqdecay}. As expected, since the breather is even in $t$, only even frequencies are represented in the frequency spectrum. Furthermore, the scaled amplitudes for all frequencies $k\geq 20$ are below $10^{-10}$, suggesting that the finite dimensional approximation is reasonable.
\begin{figure}
\begin{center}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{singleun0.eps}
\label{fig:singlea}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{singlespec.eps}
\label{fig:singleb}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{singledecay.eps}
\label{fig:singlec}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{singledecayerror.eps}
\label{fig:singled}
\end{subfigure}
\end{center}
\caption{(a) Initial condition $u_n(0)$, and (b) Floquet spectrum, for primary breather for $d = 0.25$. (c) Semilog plot of $L^2$ norm of solution $u_n(t)$ over one period $[0,T]$ vs. the lattice index $n$ for varying coupling parameter $d$. (d) Relative error of exponential decay rate seen in (c) versus that predicted by equation \cref{eq:U1decayest}. }
\label{fig:single}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{freqdecay.eps}
\end{center}
\caption{Semilog plot showing the decay of Fourier frequencies for a single-site breather with $d=0.25$ at the center site $u_0(t)$ and neighboring site $u_1(t)$. The vertical axis is rescaled so that the fundamental frequency has amplitude of 1.}
\label{fig:freqdecay}
\end{figure}
\cref{fig:double} shows the initial conditions (solution at $t=0$) for the out-of-phase (left) and in-phase (right) double breather for $N_1 = 6$ at $d = 0.25$, together with their Floquet spectra and the eigenfunction corresponding to the interaction eigenmode. The out-of-phase double breather has a pair of interaction eigenmodes on the unit circle, thus is spectrally neutrally stable, while the in-phase double breather has a pair of real interaction eigenmodes off of the unit circle and on the positive real axis, thus is spectrally unstable. This same pattern holds for all $N_1$, both even and odd. We note that for the solvability condition, we have $K < 0$ for the sine-Gordon potential, and that the quantity $b_1 < 0$ in \cref{eq:inteigsdouble} for both even and odd $N_1$. Since $K$ and $b_1$ have the same sign for all $N_1$, this agrees with the stability predictions following equation \cref{eq:inteigsdouble}.
\cref{fig:eigerrora} shows the exponential decay of the Floquet exponents $\lambda$ (as computed from the monodromy matrix) for in-phase double breathers as the separation distance $N$ is increased. (Recall that the Floquet exponents $\lambda$ and Floquet multipliers $\mu$ are related by $\mu = e^{\lambda T}$).
This is similar for out-of-phase double breathers (not shown). The remaining panels in \cref{fig:eigerror} show the relative error between the computation of these Floquet eigenmodes using the formula \cref{eq:inteigsdouble} and the computation of these modes from the monodromy matrix. As expected, the relative error increases with the coupling parameter $d$, and decreases with the separation distance $N$. \cref{fig:SGintersite} shows the initial condition $u_n(0)$ and Floquet spectra for two special two-site breathers, for which the two excited sites at the AC limit are in adjacent lattice nodes. We will call these inter-site centered breathers, in analogy to the inter-site centered soliton in DNLS \cite{Kevrekidis2009}.
As predicted by \cite{cuevas-maraver2016}, the in-phase inter-site centered breather is unstable, and the out-of-phase inter-site centered breather is spectrally stable, at least for small $d$. Since the two copies of the primary breather in the inter-site centered breather are not well separated, the eigenvalue results of \cref{sec:multi} do not apply. This is a
distinguishing feature between the theory presented herein
and the earlier results, e.g., of~\cites{Archilla2003,Koukouloyannis2009}. The latter are
well-tailored to this case, while the theory presented herein
is suitable for the case of well-separated breathers, as
described above.
\begin{figure}
\begin{center}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{doublepmun0insetspec.eps} \hspace{-0.5cm}
\label{fig:doublea}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{doubleppun0insetspec.eps}
\label{fig:doubleb}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{doubleinteig.eps} \hspace{-0.5cm}
\label{fig:doublec}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{doubleppinteig.eps}
\label{fig:doubled}
\end{subfigure}
\end{center}
\caption{Initial condition $u_n(0)$ with Floquet spectrum in inset (top) and eigenfunction $(v_n, w_n)$ corresponding to interaction eigenmode (bottom) for out-of-phase double breather (left) and in-phase double breather (right). Orange solid line in Floquet spectrum plot corresponds to unit circle. Coupling parameter $d = 0.25$, breather distance $N_1 = 6$.}
\label{fig:double}
\end{figure}
\begin{figure}
\begin{center}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{doubleeigN.eps}
\label{fig:eigerrora}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{doubleeigerrorN.eps}
\label{fig:eigerrorb}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{doubleeigerrord.eps} \hspace{-0.5cm}
\label{fig:eigerrorc}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{doubleppeigerrord.eps} \hspace{-0.5cm}
\label{fig:eigerrord}
\end{subfigure}
\end{center}
\caption{(a) Semilog plot of magnitude of Floquet exponents $\lambda$ vs. $N$ for in-phase double breather with $d = 0.1, 0.15, 0.2, 0.25$. (b) Semilog plot of the relative error Floquet multipliers of double breathers vs. $N$ for the case of
coupling parameter $d = 0.25$.
Semilog plot of the relative error of interaction eigenmode computation vs. $d$ for out-of-phase double breathers (c) and in-phase double breathers (d) for breather distance $N_1 = 4,5,6,7$.}
\label{fig:eigerror}
\end{figure}
\begin{figure}
\begin{center}
\begin{subfigure}{0.3\linewidth}
\caption{}
\includegraphics[width=5.5cm]{SGintersitepm.eps} \hspace{-0.5cm}
\label{fig:SGintersitea}
\end{subfigure}
\begin{subfigure}{0.3\linewidth}
\caption{}
\includegraphics[width=5.5cm]{SGintersitepp.eps} \hspace{-0.5cm}
\label{fig:SGintersiteb}
\end{subfigure}
\end{center}
\caption{Initial condition $u_n(0)$ and Floquet spectrum (inset) for out-of-phase inter-site centered breather (a), and in-phase inter-site centered breather (b). Coupling parameter $d=0.1$.}
\label{fig:SGintersite}
\end{figure}
For completeness here, and also per its intrinsic interest,
we show in
\cref{fig:bifdiagSGoop1} the bifurcation diagram for an out-of-phase double breather, starting from the AC limit. The parameter continuation starts on the lower branch, where the double breather has a pair of interaction eigenmodes on the unit circle (label 1). The upper branch is a double breather whose constituent breathers
are out-of-phase, yet each encompasses two in-phase adjacent excited sites (label 7), i.e. two inter-site centered breathers, which we recall leads
to instability (\cref{fig:SGintersite}); this breather has two pairs of real interaction eigenmodes, in addition to a pair of interaction eigenmodes on the unit circle. The middle branch is an asymmetric double breather, comprising one single-site breather and another intersite-centered breather which is out-of-phase with it (label 8); this breather has one pair of real interaction eigenmodes, and one pair of interaction eigenmodes on the unit circle. There is a corresponding branch (not shown) where the order of the two breathers is reversed. As $d$ is increased along the lower branch, a pair of internal modes appears on the unit circle (label 2); these have opposite Krein signature from the nearby interaction eigenmode. These collide and move off of the unit circle (label 3) in the form of a complex quartet (corresponding to an oscillatory instability). At this point, a second pair of internal modes has appeared on the unit circle, which then passes between the first pair of Floquet modes (not shown). By label 4, the pair of eigenmodes which left the unit circle has rejoined the unit circle. Between label 4 and label 5, the second pair of internal modes collides (with the branch traced by label 8 and label 9) at (1,0) in a {\it subcritical} pitchfork bifurcation and moves off of the unit circle. This pitchfork is symmetry-breaking, and produces the asymmetric, middle branches of the bifurcation diagram, indicated by the
dashed line. Between label 5 and label 6, there is a turning point (fold bifurcation), in which the first pair of internal modes collides and moves off of the unit circle.
This collision represents a saddle-center bifurcation with the
branch of panel (label 7), designated by the dotted line, a branch
that bears generically two real multiplier pairs. We find it
quite relevant to recall that
qualitatively similar pitchfork bifurcations are seen for multi-kinks in DKG (\cite{Parker2021}*{Figure 4}) and for vortex pairs in the 2-dimensional DNLS equation (\cite{Bramburger2020}*{Figure 4}).
Indeed, this seems to represent a generic mode of disappearance
in such discrete nonlinear lattice dynamical systems of states
that do not persist all the way to the continuum limit.
\begin{figure}
\hbox{
\hspace{-2cm}
\includegraphics[width=20cm]{bifdiagSGoppositeN6.eps}
}
\caption{Bifurcation diagram plotting energy $\mathcal{H}(\mathbf{u})$ vs. $d$ for out-of-phase double breather in the soft sine-Gordon potential with $N_1 = 6$. Points of interest are marked with black dots and labeled with circled numbers, which correspond to Floquet spectra (and in select cases also to the configurations at the bottom left). Continuous spectrum bands are not shown for clarity. Bifurcations are indicated in the inset with a black cross.}
\label{fig:bifdiagSGoop1}
\end{figure}
\cref{fig:bifdiagSGip1} shows the bifurcation diagram for an in-phase double breather, again starting from the AC limit. The diagram is qualitatively similar, in that the two bifurcations (a pitchfork
and a saddle-center) result from collisions of internal modes at (1,0). The main difference is that, since the starting in-phase breather has a pair of real interaction eigenmodes, these are not involved in any collisions with the internal modes. In both cases, there is a turning point (fold bifurcation) in the bifurcation diagram at a critical value $d_0$ of the coupling parameter $d$, at which point the parameter continuation reverses direction in $d$, again marking a saddle-center
bifurcation. For the out-of-phase double breather, this occurs after the pitchfork bifurcation of the site-centered breathers with
the asymmetric ones (see inset in \cref{fig:bifdiagSGoop1}).
On the contrary, for the in-phase double breather,
the pitchfork occurs between the inter-site centered
branch with both breathers being of this type and the
asymmetric one bearing one on-site and one inter-site centered
breather (see inset in \cref{fig:bifdiagSGip1}). A plot of the turning point $d_0$ vs. the separation distance $N$ in \cref{fig:bifdiagSGip1} suggests that $d_0$ increases linearly with $N$ for both the in-phase and the out-of-phase double breather.
\begin{figure}
\hbox{
\hspace{-2cm}
\includegraphics[width=20cm]{bifdiagSGinphaseN6.eps}
}
\caption{Bifurcation diagram plotting the energy $\mathcal{H}(\mathbf{u})$ vs. $d$ for in-phase double breather in soft sine-Gordon potential with $N_1 = 6$. Points of interest are marked with black dots and labeled with circled letters, which correspond to Floquet spectra.
Again, see bottom left for select spatial configurations.
Continuous spectrum bands are not shown for clarity. Bifurcations are indicated with a black cross. }
\label{fig:bifdiagSGip1}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{doubled0vsN.eps}
\end{center}
\caption{Turning point of parameter continuation $d_0$ vs separation distance $N$ for out-of-phase and in-phase double breathers.}
\label{fig:SGd0}
\end{figure}
Finally, we demonstrate the effect of the interaction eigenmodes on the dynamics of the double breathers by performing long-term timestepping experiments. For a timestepping scheme, we use a symplectic and symmetric implicit Runge-Kutta method \cite{HairerBook} to preserve the symplectic structure of the Hamiltonian system. Specifically, we use the MATLAB implementation of the \texttt{irk2} scheme of order 12 from \cite{Hairer2003}.
For each experiment, we perturb the stationary breather by adding the eigenvector $(v_n, w_n)$ corresponding to the interaction eigenmode $\mu$, multiplied by a small factor $\delta$. (If the eigenvector is complex, we use the real part of the eigenvector for the perturbation).
For a double breather $u_n(t)$, let $u_L(t) = u_{-N_1^-}(t)$ and $u_R(t) = u_{N_1^+}(t)$ be the center sites of the left and the right breather, respectively. First, we look at the out-of-phase double breather (\cref{fig:timestepSGstablea}). We choose the coupling parameter $d=0.2$ and separation distance $N_1 = 4$ so that we avoid the (nonlinearity induced) 1:2 resonance between the interaction eigenmode and the continuous spectrum bands, which leads to a nonlinear instability (see \cite{cuevas-maraver2016} for details).
Although we expect that nonlinear instabilities due to these higher order resonances will occur, they will be manifested at times beyond the time intervals of our simulations.
When the out-of-phase double breather is perturbed, the peak amplitudes of $u_L(t)$ and $u_R(t)$ oscillate in opposite directions with frequency given by the imaginary part of $\log(\mu)/T$ (see \cref{fig:timestepSGstable}), with relative error less than 0.005.
When the in-phase double breather is perturbed in the same way, the oscillations in $u_L(t)$ initially increase in both amplitude and period, while those in $u_R(t)$ decrease in both amplitude and period (\cref{fig:timestepSGunstabled}). (These are reversed if the perturbation is in the opposite direction). We can interpret this behavior by looking at the form of the corresponding eigenfunction (\cref{fig:double}, left). An addition of a small multiple of this eigenfunction to the double breather at $t=0$ causes one breather to increase in both amplitude and velocity (thus in energy), and the other to decrease in both amplitude and velocity. Since sine-Gordon is a soft potential, the breather which increases in energy also increases in period, and the breather which decreases in energy also decreases in period.
For separation distance $N_1 = 6$, as $t$ evolves, we see that the periods of $u_L(t)$ and $u_R(t)$ vary periodically in opposite directions (\cref{fig:timestepSGunstablec}), thus the phase differences between the two breathers vary periodically as well (\cref{fig:timestepSGunstablea} and \cref{fig:timestepSGunstableb}). The amplitudes of $u_L(t)$ and $u_R(t)$ also vary periodically in opposite directions (\cref{fig:timestepSGunstablee}). In effect, the perturbed in-phase double breather appears to be oscillating about the neutrally stable, out-of-phase double breather, a feature
that persists robustly over longer time scales. The growth rate of the difference in $L^2$ norm between the perturbed and unperturbed solutions is given by $\mu^{1/T}$, with a relative error of less than 0.001, where $\mu > 1$ is the larger of the Floquet multiplier pair (\cref{fig:timestepSGunstable}). If the separation distance $N_1$ is smaller, the two breathers attract each other, and eventually merge into a localized, single-site breather state (see \cref{fig:timestepSGpplong} for separation distance $N_1 = 4$).
This limiting solution is close to a higher energy, single-site breather, and is located between the initial starting sites for the pair of breathers.
The energy density at lattice site $n$, which is plotted in the bottom panels of \cref{fig:timestepSGpplong}, is given by
\begin{equation}\label{eq:Hn}
h_n = \frac{1}{2}v_n^2 + V(u_n)
+ \frac{d}{4}\left[ (u_n - u_{n+1})^2 + (u_n - u_{n-1})^2\right],
\end{equation}
where $v_n = \dot{u}_n$, so that the Hamiltonian \cref{eq:H} is the sum of the energy densities over the entire lattice.
A plot of the total energy vs. time (\cref{fig:timestepSGpplongb}) indicates that energy is conserved by the numerical scheme over the simulation time, providing validation of this result.
\begin{figure}
\begin{center}
\begin{subfigure}{0.3\linewidth}
\caption{}
\includegraphics[width=5.5cm]{timestepN6spec.eps}
\label{fig:timestepSGstablea}
\end{subfigure}
\begin{subfigure}{0.3\linewidth}
\caption{}
\includegraphics[width=5.45cm]{timestepN6.eps}
\label{fig:timestepSGstableb}
\end{subfigure}
\begin{subfigure}{0.3\linewidth}
\caption{}
\includegraphics[width=5.45cm]{timestepN6pks.eps}
\label{fig:timestepSGstablec}
\end{subfigure}
\end{center}
\caption{(a) out-of-phase double breather with the Floquet spectrum shown in the inset. Time evolution of the perturbation of the out-of-phase breather, showing the amplitude of the peaks of $u_L$ and $u_R$ vs. $t$ (b), and difference in this amplitude from the unperturbed breather (c).
Coupling parameter $d=0.2$, separation distance $N_1 = 4$, perturbation parameter $\delta = 0.01$.}
\label{fig:timestepSGstable}
\end{figure}
\begin{figure}
\begin{center}
\begin{subfigure}{0.3\linewidth}
\caption{}
\includegraphics[width=5.25cm]{timestepN6ppcolormap.eps} \hspace{-0.5cm}
\label{fig:timestepSGunstablea}
\end{subfigure}
\begin{subfigure}{0.3\linewidth}
\caption{}
\includegraphics[width=5.25cm]{timestepN6ppphase.eps} \hspace{-0.5cm}
\label{fig:timestepSGunstableb}
\end{subfigure}
\begin{subfigure}{0.3\linewidth}
\caption{}
\includegraphics[width=5.25cm]{timestepN6ppperiods.eps} \hspace{-0.5cm}
\label{fig:timestepSGunstablec}
\end{subfigure}
\begin{subfigure}{0.3\linewidth}
\caption{}
\includegraphics[width=5.25cm]{timestepN6ppLR.eps} \hspace{-0.5cm}
\label{fig:timestepSGunstabled}
\end{subfigure}
\begin{subfigure}{0.3\linewidth}
\caption{}
\includegraphics[width=5.25cm]{timestepN6ppmaxamp.eps} \hspace{-0.5cm}
\label{fig:timestepSGunstablee}
\end{subfigure}
\begin{subfigure}{0.3\linewidth}
\caption{}
\includegraphics[width=5.25cm]{timestepN6ppgrowth.eps} \hspace{-0.5cm}
\label{fig:timestepSGunstablef}
\end{subfigure}
\end{center}
\caption{Time evolution of the perturbation for an in-phase breather. (a) colormap showing $u_n$ vs. $t$ for sites -5 to 5; breathers start in-phase (top) and are out-of-phase at approximately $t = 215$ (bottom). (b) phase difference between two breathers vs $t$; 0 is in-phase, $\pi$ is out-of-phase. (c) period of two breathers vs. $t$; the period $T$ is measured as the distance between consecutive peaks. (d) time evolution of $u_L$ and $u_R$ for unperturbed breather (dotted blue lines) and perturbed breather (solid orange lines). (e) maximum amplitude of $u_L$ and $u_R$ vs. $t$. (f) semilog plot showing time evolution in difference of $\ell^2$ norm of perturbed and unperturbed breather, together with least squares regression line. Coupling parameter $d=0.25$, separation distance $N_1 = 6$, perturbation parameter $\delta = 0.005$.}
\label{fig:timestepSGunstable}
\end{figure}
\begin{figure}
\begin{center}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{N4ppcolormap.eps}
\label{fig:timestepSGpplonga}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{N4ppH.eps}
\label{fig:timestepSGpplongb}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{N4pph1.eps}
\label{fig:timestepSGpplongc}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{N4pph2.eps}
\label{fig:timestepSGpplongd}
\end{subfigure}
\end{center}
\caption{(a) Colormap showing the evolution in $t$ of $|u_n|$. (b) Total energy $H(\mathbf{u}(t))$ vs. $t$ ($H$ is given by \cref{eq:H}). (c) and (d) Energy density vs. time for central sites of breather. Labels of sites correspond to those in (a). Separation distance $N_1 = 4$
, coupling parameter $d=0.25$, perturbation parameter $\delta = 0.01$, lattice size parameter $L=150$.}
\label{fig:timestepSGpplong}
\end{figure}
\subsection{Hard \texorpdfstring{$\phi^4$}{phi-4} potential}
The results for the soft $\phi^4$ potential are similar to those for sine-Gordon (which is expected, since the first two terms in the Taylor expansion of the sine-Gordon potential are qualitatively similar to the soft $\phi^4$ potential), thus we will not show them here. We instead look at the hard $\phi^4$ potential. \cref{fig:phi4sol} shows the initial conditions for the primary breather, the in-phase inter-site centered breather, and the out-of-phase inter-site centered breather, together with their Floquet spectra. In contrast to the sine-Gordon potential, the out-of-phase inter-site centered breather is unstable while the in-phase inter-site centered breather is spectrally stable for sufficiently small $d$ (see also Figures 4 and 6 in \cite{cuevas-maraver2016}). These agree with the results of \cites{Archilla2003,Koukouloyannis2009}. As the distance between breathers $N_1$ is increased, the in-phase double breather is spectrally stable for $N_1$ odd and spectrally unstable for $N_1$ even, and the reverse is true for the out-of-phase double breather (see \cref{fig:phi4hardfloqplot}). We note that the solvability condition $K > 0$ for the hard $\phi^4$ potential; however, for the quantity $b_1$ in \cref{eq:inteigsdouble}, $b_1 > 0$ for $N_1$ even, and $b_1 < 0$ for $N_1$ odd. This agrees with the stability predictions following equation \cref{eq:inteigsdouble}, and explains the alternating eigenvalue pattern seen as the distance $N_1$ is increased. A summary of the Floquet spectral pattern for both potentials is given in \cref{table:spec}.
\begin{figure}
\includegraphics[width=15cm]{phi4hardfloqplot.eps}
\caption{Floquet spectra for $N_1 = 1, 2, 3, 4$ for in-phase double breather (top) and out-of-phase double breather (bottom) for hard $\phi^4$ potential with $d = 0.125$.}
\label{fig:phi4hardfloqplot}
\end{figure}
\cref{fig:bifdiagphi4} shows the bifurcation diagram for out-of-phase and in-phase double breathers for $N_1 = 6$. As noted above, the out-of-phase double breather is spectrally stable, while the in-phase double breather is unstable. For $N_1$ odd, this pattern is reversed (not shown). The upper branches are double breathers comprising two out-of-phase inter-site centered breathers, where we recall that this is the unstable configuration for the inter-site centered breather. The middle branch is an asymmetric double breather, comprising one single-site breather and one out-of-phase inter-site centered breather. \cref{fig:phi4eigerror} shows the relative error in the computation of the Floquet interaction eigenmodes on the lower branches for both the in-phase and out-of-phase double breathers for both even and odd $N_1$, using the formula \cref{eq:inteigsdouble}. As for the sine-Gordon equation, the relative error increases with $d$. We note that the relative error is less than $10^{-2}$ up to approximately $d = 3$, which is close to the turning points of the bifurcation diagrams.
\begin{figure}
\begin{center}
\begin{subfigure}{0.3\linewidth}
\caption{}
\includegraphics[width=5.5cm]{phi4single.eps} \hspace{-0.5cm}
\label{fig:phi4sola}
\end{subfigure}
\begin{subfigure}{0.3\linewidth}
\caption{}
\includegraphics[width=5.5cm]{intersitephi4inphase.eps} \hspace{-0.5cm}
\label{fig:phi4solb}
\end{subfigure}
\begin{subfigure}{0.3\linewidth}
\caption{}
\includegraphics[width=5.5cm]{intersitephi4opposite.eps}
\label{fig:phi4solc}
\end{subfigure}
\end{center}
\caption{Initial condition $u_n(0)$ and Floquet spectrum (inset) for primary breather (a), in-phase inter-site centered breather (b), and out-of-phase inter-site centered breather (c) for hard $\phi^4$ potential with coupling parameter $d=0.1$. }
\label{fig:phi4sol}
\end{figure}
\begin{figure}
\hbox{
\hspace{-2cm}
\includegraphics[width=20cm]{bifdiagphi4oppositeN6.eps}
}
\hbox{
\hspace{-2cm}
\includegraphics[width=20cm]{bifdiagphi4inphaseN6.eps}
}
\caption{Bifurcation diagram plotting energy $\mathcal{H}(\mathbf{u})$ vs. $d$ for hard $\phi^4$ potential for out-of-phase (top) and in-phase (bottom) double breather with $N_1 = 6$. Solutions and Floquet spectra on right correspond to labeled points on left.}
\label{fig:bifdiagphi4}
\end{figure}
\begin{figure}
\begin{center}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{images/doublephi4eigerrorpp.eps}
\label{fig:phi4eigerrora}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{images/doublephi4eigerrorpm.eps}
\label{fig:phi4eigerrorb}
\end{subfigure}
\end{center}
\caption{Semilog plot of relative error of interaction eigenmode computation vs. $d$ for in-phase (a) and out-of-phase (b) double breathers for hard $\phi^4$ potential with $N_1 = 3, 4, 5, 6$.}
\label{fig:phi4eigerror}
\end{figure}
\begin{table}
\begin{tabular}{llccll}\toprule
Potential & $N_1$ & sign of $K$ & sign of $b_1$ & in-phase & out-of-phase \\ \midrule
sine-Gordon (soft) & even & $-$ & $-$ & unstable & stable \\
& odd & $-$ & $-$ & unstable & stable \\
$\phi^4$ (hard) & even & $+$ & $+$ & unstable & stable \\
& odd & $+$ & $-$ & stable & unstable \\ \bottomrule
\end{tabular}
\caption{Summary of Floquet interaction eigenmode pattern for double breathers for soft sine-Gordon potential and hard $\phi^4$ potential.}
\label{table:spec}
\end{table}
\section{Conclusions and future directions}\label{sec:conc}
In this paper, we studied multi-breather solutions to the discrete Klein-Gordon equation,
with an emphasis on a theoretical analysis of multi-breathers
at a finite distance from each other, and their existence, spectral stability and dynamical properties. Specifically, we looked at an approximation of the system on a finite-dimensional Hilbert space, and used Lin's method to construct multi-breather solutions to this system. Furthermore, we used Lin's method again to reduce the eigenvalue problem to a low-dimensional, matrix equation, which can then be solved. This can be done as long as the distances between copies of the primary breather are sufficiently large. The results from this approximation are in very good agreement with direct numerical computations of the Floquet spectrum, both for soft and hard potentials. The key determining factor for the Floquet spectral pattern is the phase differences between adjacent copies of the primary, single-site breather. In addition, our results
showcased the crucial difference between the soft sine-Gordon equation and the hard $\phi^4$ equation; in the latter case, the Floquet spectral pattern depends in addition on whether the distances in lattice points between consecutive copies of the primary breather are even or odd, while in the former case, it does
not (see also, e.g., the analogy with the interaction of dark
solitons in the defocusing DNLS model of~\cite{Pelinovsky_2008}).
Avenues of further research include exploring breathers and multi-breathers either in Klein-Gordon lattices with asymmetric potentials, such as the Morse potential, or in other models
involving beyond-nearest-neighbor interactions, discussed, e.g., in~\cite{PENATI201992} and references therein. One class of models which would be interesting to study is equations in which the potential involves off-site terms, such as the FPUT lattice. A natural extension of this would be to look at higher dimensional lattices, both for on-site potentials such as in a higher dimensional Klein-Gordon model, and more complex potentials, such as higher dimensional FPUT lattices. In two dimensions, there are many possible regular lattice models, including square, triangular, and honeycomb, and these different geometries may exhibit qualitatively different behavior. Another possibility would be to look at more complex solutions. One class of solutions which remains unexplored is asymmetric multi-breathers, solutions comprising breathers which have different, but commensurate, fundamental periods. \cref{fig:brka} shows an example of a asymmetric double breather solution to the discrete sine-Gordon equation; the left breather has a fundamental period of 8, and the right breather has a fundamental period of 16, so the overall breather has a period of 16. While these solutions can be constructed numerically by parameter continuation from the AC limit, preliminary numerical experiments suggest that they only exist for small values of the coupling parameter $d$, and that they are spectrally unstable.
In addition, preliminary numerical experiments suggest that there are solutions comprising of both breathers and kinks. See \cref{fig:brkb} for an example for a solution to the discrete sine-Gordon equation in which a breather is connected to a kink. Finally, although the finite-dimensional approximation used in the paper yields results which are in excellent agreement with those found numerically, it may be still possible to remove this restriction and prove the results for the full system, although this would most likely involve an entirely different approach.
\begin{figure}
\begin{center}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{images/doubleasymm.eps}
\label{fig:brka}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\caption{}
\includegraphics[width=7.5cm]{images/brk1colormap.eps}
\label{fig:brkb}
\end{subfigure}
\end{center}
\caption{(a) Initial condition for asymmetric double breather solution with Floquet spectrum in inset, coupling parameter $d = 0.075$. (b) Colormap showing solution to discrete sine-Gordon equation comprising one breather (on left) and one kink (on right), coupling parameter $d = 0.25$. Time evolution is over one period $T$ of the breather.}
\label{fig:brk}
\end{figure}
\vspace{0.5cm}
\paragraph{\textbf{Acknowledgments}}
This material is based upon work supported by the U.S. National Science Foundation under the RTG grant DMS-1840260 (R.P. and A.A.) and DMS-1809074 (P.G.K.). J.C.-M. acknowledges support from EU (FEDER program 2014-2020) through both Consejería de Econom\'{\i}a, Conocimiento, Empresas y Universidad de la Junta de Andaluc\'{\i}a (under the projects P18-RT-3480 and US-1380977), and MICINN and AEI (under the projects PID2019-110430GB-C21 and PID2020-112620GB-I00). R.P. would also like to thank Graham Cox and Bj\"orn Sandstede for their helpful comments and suggestions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,696 |
\section{Analysis}
\label{sec:analysis}
\subsection{Number of Episodes}
To analyze the performance of TSDE over $T$ time steps, define $K_T = \argmax\{k: t_k \leq T\}$ be the number of episodes of TSDE until time $T$.
Note that $K_T$ is a random variable because the number of visits $N_t(x,u)$ depends on the dynamical state trajectory.
In the analysis for time $T$ we use the convention that $t_{(K_T+1)} = T+1$.
We provide an upper bound on $K_T$ as follows.
\begin{lemma}
\label{lm:boundKT}
\begin{align*}
K_T \leq \sqrt{ 2 SA T\log(T) }.
\end{align*}
\end{lemma}
\begin{proof}
Define macro episodes with start times $t_{n_i}, i=1,2,\dots$ where $t_{n_1} = t_1$ and
\begin{align*}
t_{n_{i+1}} = \min\{ & t_k > t_{n_i}:\quad
N_{t_k}(s,a) > 2N_{t_{k-1}}(s,a) \text{ for some }(s,a)
\}.
\end{align*}
The idea is that each macro episode starts when the second stopping criterion happens.
Let $M$ be the number of macro episodes until time $T$
and define $n_{(M+1)} = K_T+1$.
Let $\tilde T_i = \sum_{k=n_{i}}^{n_{i+1}-1} T_{k}$ be the length of the $i$th macro episode.
By the definition of macro episodes, any episode except the last one in a macro episode must be triggered by the first stopping criterion. Therefore, within the $i$th macro episode, $T_{k} = T_{k-1} +1$ for all $k = n_{i}, n_{i}+1,\dots,n_{i+1}-2$.
Hence,
\begin{align*}
\tilde T_i = \sum_{k=n_{i}}^{n_{i+1}-1} T_{k}
= &\sum_{j=1}^{n_{i+1}-n_i-1} (T_{n_i-1}+j) + T_{n_{i+1}-1}
\notag\\
\geq &\sum_{j=1}^{n_{i+1}-n_i-1} (j+1) + 1 = 0.5(n_{i+1}-n_i)(n_{i+1}-n_i+1).
\end{align*}
Consequently, $n_{i+1} - n_{i} \leq \sqrt{2 \tilde T_i}$ for all $i=1,\dots,M$.
From this property we obtain
\begin{align}
K_T = &n_{M+1}-1
= \sum_{i=1}^{M} (n_{i+1} - n_{i})
\leq \sum_{i=1}^{M} \sqrt{2 \tilde T_i} .
\label{eq:KTboundinpf}
\end{align}
Using \eqref{eq:KTboundinpf} and the fact that $\sum_{i=1}^M \tilde T_i = T$ we get
\begin{align}
K_T \leq \sum_{i=1}^{M} \sqrt{2 \tilde T_i} \leq &
\sqrt{ M\sum_{i=1}^{M} 2 \tilde T_i }
= \sqrt{ 2 MT}
\label{eq:KTboundfromM}
\end{align}
where the second inequality is Cauchy-Schwarz.
From Lemma 6 in the appendix,
the number of macro episodes $M \leq SA \log(T)$.
Substituting this bound into \eqref{eq:KTboundfromM} we obtain the result of this lemma.
\end{proof}
\begin{remark}
TSDE computes the optimal stationary policy of a finite MDP at each episode. Lemma \ref{lm:boundKT} ensures that such computation only needs to be done at a sublinear rate of $\sqrt{ 2 SA T\log(T) }$.
\end{remark}
\subsection{Regret Bound}
As discussed in \cite{osband2013more,osband2016posterior,russo2014learning}, one key property of Thompson/Posterior Sampling algorithms is that for any function $f$, $\ee[f(\theta_t)] = \ee[f(\theta_*)]$ if $\theta_t$ is sampled from the posterior distribution at time $t$. This property leads to regret bounds for algorithms with fixed sampling episodes since the start time $t_k$ of each episode is deterministic.
However, our TSDE algorithm has dynamic episodes that requires us to have the stopping-time version of the above property.
\begin{lemma}
\label{lm:PS_expectation}
Under TSDE, $t_k$ is a stopping time for any episode $k$. Then for any measurable function $f$ and any $\sigma(h_{t_k})-$measurable random variable $X$, we have
\begin{align*}
\ee \Big[ f(\theta_{k},X)\Big]= & \ee \Big[ f(\theta_*,X) \Big]
\end{align*}
\end{lemma}
\begin{proof}
From the definition \eqref{eq:tk}, the start time $t_k$ is a stopping-time, i.e. $t_k$ is $\sigma(h_{t_k})-$measurable.
Note that $\theta_{k}$ is randomly sampled from the posterior distribution $\mu_{t_k}$.
Since ${t_k}$ is a stopping time, ${t_k}$ and $\mu_{t_k}$ are both measurable with respect to $\sigma(h_{t_k})$.
From the assumption, $X$ is also measurable with respect to $\sigma(h_{t_k})$.
Then conditioned on $h_{t_k}$, the only randomness in $f(\theta_{k},X)$ is the random sampling in the algorithm. This gives the following equation:
\begin{align}
\ee\Big[ f(\theta_{k},X)|h_{t_k} \Big]
=\ee\Big[ f(\theta_{k},X)|h_{t_k},{t_k},\mu_{t_k} \Big]
= & \int f(\theta,X)\mu_{t_k}(d \theta)
=\ee\Big[ f(\theta_*,X)|h_{t_k} \Big]
\label{eq:lm1L}
\end{align}
since $\mu_{t_k}$ is the posterior distribution of $\theta_*$ given $h_{t_k}$.
Now the result follows by taking the expectation of both sides.
\end{proof}
For $t_k\leq t<t_{k+1}$ in episode $k$, the Bellman equation \eqref{eq:DP_inf} holds by Assumption \ref{assum:WASP} for $s = s_t$, $\theta = \theta_k$ and action $a_t = \pi_k(s_t)$. Then we obtain
\begin{align}
&
c(s_t,a_t)
= J(\theta_k) + v(s_t,\theta_k)- \sum_{s' \in \mathcal S}\theta_k(s'|s_t,a_t)v(s',\theta_k).
\label{eq:DP_analysis2}
\end{align}
Using \eqref{eq:DP_analysis2}, the expected regret of TSDE is equal to
\begin{align}
& \ee\Big[ \sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} c(s_t,a_t) \Big] - T \ee\Big[J(\theta_*) \Big]
\notag\\
= & \ee\Big[ \sum_{k=1}^{K_T}T_k J(\theta_k) \Big]- T \ee\Big[J(\theta_*) \Big]
+
\ee\Big[\sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} \Big[ v(s_t,\theta_k) -\sum_{s' \in \mathcal S}\theta_k(s'|s_t,a_t)v(s',\theta_k)
\Big]\Big]
\notag\\
=& R_0+R_1+R_2,
\label{eq:regretbound}
\end{align}
where $R_0$, $R_1$ and $R_2$ are given by
\begin{align*}
&R_0 = \ee\Big[ \sum_{k=1}^{K_T}T_k J(\theta_k) \Big]- T \ee\Big[J(\theta_*) \Big],
\\
&R_1 = \ee\Big[\sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} \Big[v(s_t,\theta_k) - v(s_{t+1},\theta_k)\Big]\Big],
\\
&R_2 = \ee\Big[\sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} \Big[ v(s_{t+1},\theta_k) -\sum_{s' \in \mathcal S}\theta_k(s'|s_t,a_t)v(s',\theta_k)\Big]\Big].
\end{align*}
We proceed to derive bounds on $R_0$, $R_1$ and $R_2$.
Based on the key property of Lemma \ref{lm:PS_expectation}, we derive an upper bound on $R_0$.
\begin{lemma}
\label{lm:R0}
The first term $R_0$ is bounded as
\begin{align*}
R_0 \leq &\ee[K_T].
\end{align*}
\end{lemma}
\begin{proof}
From monotone convergence theorem we have
\begin{align*}
R_0
= &\ee\Big[ \sum_{k=1}^{\infty}\mathds{1}_{\{t_{k}\leq T\}}T_k J(\theta_k) \Big]- T \ee\Big[J(\theta_*) \Big]
= \sum_{k=1}^{\infty}\ee\Big[ \mathds{1}_{\{t_{k}\leq T\}}T_k J(\theta_{k})\Big] - T\ee\Big[J(\theta_*) \Big].
\end{align*}
Note that the first stopping criterion of TSDE ensures that $T_k \leq T_{k-1}+1$ for all $k$.
Because $J(\theta_{k}) \geq 0$, each term in the first summation satisfies
\begin{align*}
\ee\Big[ \mathds{1}_{\{t_{k}\leq T\}}T_k J(\theta_{k})\Big]
\leq &\ee\Big[ \mathds{1}_{\{t_{k}\leq T\}}(T_{k-1}+1) J(\theta_{k})\Big].
\end{align*}
Note that $\mathds{1}_{\{t_{k}\leq T\}}(T_{k-1}+1)$ is measurable with respect to $\sigma(h_{t_k})$. Then, Lemma \ref{lm:PS_expectation} gives
\begin{align*}
\ee\Big[ \mathds{1}_{\{t_{k}\leq T\}}(T_{k-1}+1) J(\theta_{k})\Big]
= &\ee\Big[\mathds{1}_{\{t_{k}\leq T\}}(T_{k-1}+1) J(\theta_{*}) \Big].
\end{align*}
Combining the above equations we get
\begin{align*}
R_0
\leq &\sum_{k=1}^{\infty}\ee\Big[\mathds{1}_{\{t_{k}\leq T\}}(T_{k-1}+1) J(\theta_{*}) \Big] - T\ee\Big[J(\theta_*) \Big]
\notag\\
= &\ee\Big[\sum_{k=1}^{K_T}(T_{k-1}+1) J(\theta_{*}) \Big] - T\ee\Big[J(\theta_*) \Big]
\notag\\
= &\ee\Big[K_T J(\theta_{*})\Big]
+ \ee\Big[\Big(\sum_{k=1}^{K_T}T_{k-1} - T\Big)J(\theta_{*})\Big]
\leq \ee\Big[K_T\Big]
\end{align*}
where the last equality holds because $J(\theta_{*}) \leq 1$ and $\sum_{k=1}^{K_T}T_{k-1}=T_0+\sum_{k=1}^{K_T-1}T_{k} \leq T$.
\end{proof}
Note that the first stopping criterion of TSDE plays a crucial role in the proof of Lemma \ref{lm:R0}.
It allows us to bound the length of an episode using the length of the previous episode which is measurable with respect to the information at the beginning of the episode.
The other two terms $R_1$ and $R_2$ of the regret are bounded in the following lemmas.
Their proofs follow similar steps to those in \cite{osband2013more,abbasi2015bayesian}. The proofs are in the appendix due to the lack of space.
\begin{lemma}
\label{lm:R1}
The second term $R_1$ is bounded as
\begin{align*}
R_1 \leq \ee[HK_T].
\end{align*}
\end{lemma}
\begin{lemma}
\label{lm:R2}
The third term $R_2$ is bounded as
\begin{align*}
R_2 \leq 49 H S\sqrt{AT\log(AT)}.
\end{align*}
\end{lemma}
We are now ready to prove Theorem \ref{thm:regretbound}.
\begin{proof}[Proof of Theorem \ref{thm:regretbound}]
From \eqref{eq:regretbound},
$R(T,\text{TSDE}) = R_0+R_1 + R_2
\leq \ee[K_T] + \ee[HK_T]+ R_2$
where the inequality comes from Lemma \ref{lm:R0}, Lemma \ref{lm:R1}.
Then the claim of the theorem directly follows from Lemma \ref{lm:boundKT} and Lemma \ref{lm:R2}.
\end{proof}
\section{Bound on the number of macro episodes}
\begin{lemma}
\label{lm:Mbound}
The number $M$ of macro episodes of TSDE is bounded by
\begin{align*}
M \leq& SA \log(T).
\end{align*}
\end{lemma}
\begin{proof}
Since the second stopping criterion is triggered whenever the number of visits to a state-action pair is doubled, the start times of macro episodes can be expressed as
\begin{align*}
\{t_1\}\cup \Big(\cup_{(s,a)\in\mathcal S \times \mathcal A} \{t_k: k \in\mathcal M_{(s,a)} \}\Big)
\end{align*}
where
\begin{align*}
&\mathcal M_{(s,a)} =
\{k \leq K_T: N_{t_k}(s,a) > 2N_{t_{k-1}}(s,a)\}.
\end{align*}
Since the number of visits to $(s,a)$ is doubled at every $t_k$ such that $k\in\mathcal M_{(s,a)}$, the size of $\mathcal M_{(s,a)}$ should not be larger than $O(\log(T))$. This argument is made rigorous as follows.
If $|\mathcal M_{(s,a)}| \geq \log(N_{T+1}(s,a))+1$ we have
\begin{align*}
N_{t_{K_T}}(s,a) = \prod_{k \leq K_T, N_{t_{k-1}}(s,a)\geq 1}\frac{N_{t_{k}}(s,a)}{N_{t_{k-1}}(s,a)}
> \prod_{k \in \mathcal M_{(s,a)}, N_{t_{k-1}}(s,a)\geq 1} \hspace{-2em} 2 \hspace{2em}
\geq N_{T+1}(s,a).
\end{align*}
But this contradicts the fact that $N_{t_{K_T}}(s,a) \leq N_{T+1}(s,a)$. Therefore,
$|\mathcal M_{(s,a)}| \leq \log(N_{T+1}(s,a))$ for all $(s,a)$.
From this property we obtain a bound on the number of macro episodes as
\begin{align}
M \leq& 1+\sum_{(s,a)}|\mathcal M_{(s,a)}|
\leq 1 + \sum_{(s,a)}\log(N_{T+1}(s,a))
\notag\\
\leq &1 + SA \log(\sum_{(s,a)}N_{T+1}(s,a)/SA)
= 1 + SA \log(T/SA) \leq SA \log(T)
\label{eq:Mbound}
\end{align}
where the first inequality is the union bound and the third inequality holds because $\log$ is concave.
\end{proof}
\section{Proof of Lemma \ref{lm:R1}}
\begin{proof}
We have
\begin{align*}
R_{1}
= &
\ee\Big[\sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} \Big[v(s_t,\theta_k) - v(s_{t+1},\theta_k)\Big]\Big]
\notag\\
=&\ee\Big[\sum_{k=1}^{K_T} \Big[v(s_{t_k},\theta_k) - v(s_{t_{k+1}},\theta_k)\Big]\Big]
\leq \ee\Big[H K_T\Big]
\end{align*}
where the last equality holds because $0 \leq v(s,\theta) \leq sp(\theta) \leq H$ for all $s,\theta$ from Assumption \ref{assum:WASP}.
\end{proof}
\section{Proof of Lemma \ref{lm:R2}}
\begin{proof}
For notational simplicity, we use $z = (s,a) \in \mathcal S \times \mathcal A$ and $z_t = (s_t,a_t)$ to denote the corresponding state-action pair. Then
\begin{align*}
R_2 =
& \ee\Big[\sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} \Big[ v(s_{t+1},\theta_k) -\sum_{s' \in \mathcal S}\theta_k(s'|z_t)v(s',\theta_k)\Big]\Big]
\notag\\
=& \ee\Big[\sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} \Big[ \sum_{s' \in \mathcal S}(\theta_*(s'|z_t)-\theta_k(s'|z_t))v(s',\theta_k)\Big]\Big].
\end{align*}
Since $0\leq v(s',\theta_t) \leq H$ from Assumption \ref{assum:WASP}, each term in the inner summation is bounded by
\begin{align*}
&\sum_{s' \in \mathcal S} (\theta_*(s'|z_t)-\theta_k(s'|z_t))v(s',\theta_k)
\notag\\
\leq & H\sum_{s' \in \mathcal S} |\theta_*(s'|z_t)-\theta_k(s'|z_t)|
\notag\\
\leq & H\sum_{s' \in \mathcal S} |\theta_*(s'|z_t)-\hat\theta_{k}(s'|z_t)|
+H \sum_{s' \in \mathcal S} |\theta_{k}(s'|z_t)-\hat\theta_{k}(s'|z_t)|.
\end{align*}
Here $\hat\theta_{k}(s'|z_t) = \frac{N_{t_k}(z_t,s')}{N_{t_k}(z_t)}$ is the empirical mean for the transition probability at the beginning of episode $k$
where $N_{t_k}(s_t,a_t,s') = |\{\tau<t_k: (s_\tau,a_\tau,s_{\tau+1}) = (s_t,a_t,s')\}|$.
Define confidence set
\begin{align}
&B_k = \{\theta: \sum_{s' \in \mathcal S} |\theta(s'|z)-\hat\theta_{k}(s'|z)|\leq \beta_k(z) \,\forall \,z \in \mathcal S\times \mathcal A \}
\label{eq:Bt}
\end{align}
where $\beta_k(z) = \sqrt{\frac{14S\log(2At_k T)}{\max(1,N_{t_k}(z))}}$. Note that $\beta_k(z)$ is the confidence set used in \cite{jaksch2010near} with $\delta = 1/T$.
Then we have
\begin{align*}
&\sum_{s' \in \mathcal S} |\theta_*(s'|z_t)-\hat\theta_{k}(s'|z_t)|
+ \sum_{s' \in \mathcal S} |\theta_{k}(s'|z_t)-\hat\theta_{k}(s'|z_t)|
\notag\\
\leq & 2\beta_{k}(z_t) + 2 (\mathds{1}_{\{\theta_* \notin B_{k}\}}+\mathds{1}_{\{\theta_{k} \notin B_{k}\}}).
\end{align*}
Therefore,
\begin{align}
R_2 \leq &
2H\ee\Big[\sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} \beta_{k}(z_t)\Big]
+ 2H\ee\Big[\sum_{k=1}^{K_T}T_k(\mathds{1}_{\{\theta_* \in B_k\}}+\mathds{1}_{\{\theta_{k} \in B_k\}}) \Big].
\label{eq:R2decompose}
\end{align}
For the first term in \eqref{eq:R2decompose} we have
\begin{align*}
&\sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} \beta_{k}(z_t)
=\sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} \sqrt{\frac{14S\log(2At_k T)}{\max(1,N_{t_k}(z_t))}}.
\end{align*}
Note that $N_{t}(z_t) \leq 2N_{t_k}(z_t)$ for all $t$ in the $k$th episodes from the second criterion. So we get
\begin{align}
\sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} \beta_{k}(z_t)
\leq& \sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} \sqrt{\frac{28S\log(2At_k T)}{\max(1,N_{t}(z_t))}}
\notag\\
\leq &\sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} \sqrt{\frac{28S\log(2A T^2)}{\max(1,N_{t}(z_t))}}
\notag\\
= &\sum_{t=1}^{T} \sqrt{\frac{28S\log(2A T^2)}{\max(1,N_{t}(z_t))}}
\notag\\
\leq &\sqrt{56 S\log(AT)}\sum_{t=1}^T \frac{1}{\sqrt{\max(1,N_{t}(z_t))}}.
\label{eq:forbeta1}
\end{align}
Since $N_{t}(z_t)$ is the count of visits to $z_t$, we have
\begin{align*}
& \sum_{t=1}^T \frac{1}{\sqrt{\max(1,N_{t}(z_t))}}
= \sum_z \sum_{t=1}^T \frac{\mathds{1}_{\{z_t = z\}}}{\sqrt{\max(1,N_{t}(z))}}
\notag\\
= & \sum_z \Big(\mathds{1}_{\{N_{T+1}(z)>0\}}+\sum_{j=1}^{N_{T+1}(z)-1} \frac{1}{\sqrt{j}}\Big)
\notag\\
\leq &\sum_z \Big(\mathds{1}_{\{N_{T+1}(z)>0\}}+2\sqrt{N_{T+1}(z)}\Big)
\leq 3\sum_z \sqrt{N_{T+1}(z)}.
\end{align*}
Since $\sum_z N_{T+1}(z) = T$, we have
\begin{align}
3\sum_z \sqrt{N_{T+1}(z)}
\leq & 3\sqrt{SA \sum_z N_{T+1}(z)}
= 3\sqrt{SAT}.
\label{eq:forbeta2}
\end{align}
Combining \eqref{eq:forbeta1}-\eqref{eq:forbeta2} we get
\begin{align}
&2H\sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} \beta_{k}(z_t) \leq 6\sqrt{56}H S\sqrt{AT\log(AT)}
\leq 48 HS\sqrt{AT\log(AT)}.
\label{eq:R2part1}
\end{align}
Let's now work on the second term in \eqref{eq:R2decompose}.
Since $T_k \leq T$ for all $k$, we have
\begin{align}
\ee\Big[\sum_{k=1}^{K_T}T_k(\mathds{1}_{\{\theta_* \notin B_k\}}+\mathds{1}_{\{\theta_{k} \notin B_k\}}) \Big]
\leq &T\ee\Big[\sum_{k=1}^{K_T}(\mathds{1}_{\{\theta_* \notin B_k\}}+\mathds{1}_{\{\theta_{k} \notin B_k\}}) \Big]
\notag\\
\leq &T\sum_{k=1}^\infty \ee\Big[
\mathds{1}_{\{\theta_* \notin B_{k}\}}+\mathds{1}_{\{\theta_{k} \notin B_{k}\}}\Big].
\label{eq:forBprob1}
\end{align}
Since $B_{k}$ is measurable with respect to $\sigma(h_{t_k})$, using Lemma \ref{lm:PS_expectation} we get
\begin{align}
\ee\Big[
\mathds{1}_{\{\theta_* \notin B_{k}\}}+\mathds{1}_{\{\theta_{k} \notin B_{k}\}}\Big]
= & 2\ee\Big[\mathds{1}_{\{\theta_* \notin B_{k}\}}\Big]
= 2 \prob(\theta_* \notin B_{k})
\label{eq:forBprob2}
\end{align}
By the definition of $B_{k}$ in \eqref{eq:Bt}, \cite[Lemma 17, setting $\delta = 1/T$]{jaksch2010near} implies that
\begin{align}
&\prob(\theta_* \notin B_{k}) \leq \frac{1}{15Tt_k^6}.
\label{eq:forBprob3}
\end{align}
Combining \eqref{eq:forBprob1}, \eqref{eq:forBprob2} and \eqref{eq:forBprob3} we obtain
\begin{align}
2H\ee\Big[\sum_{k=1}^{K_T}T_k(\mathds{1}_{\{\theta_* \notin B_k\}}+\mathds{1}_{\{\theta_{k} \notin B_k\}}) \Big]
\leq &\frac{4}{15}H\sum_{k=1}^\infty t_k^{-6}
\leq \frac{4}{15}H\sum_{k=1}^\infty k^{-6}
\leq H.
\label{eq:R2part2}
\end{align}
The statement of the lemma then follows by substituting \eqref{eq:R2part1} and \eqref{eq:R2part2} into \eqref{eq:R2decompose}.
\end{proof}
\section{Proof of Theorem \ref{thm:regretbound_approximate}}
\begin{proof}
Since $\tilde \pi_k$ is an $\epsilon_k-$approximation policy, we have
\begin{align*}
c(s_t,a_t)
\leq &
\min_{a \in \mathcal A}\Big\{c(s_t,a) + \sum_{s' \in \mathcal S}\theta_k(s'|s_t,a)v(s',\theta_k)\Big\}
- \sum_{s' \in \mathcal S}\theta_k(s'|s_t,a_t)v(s',\theta_k)
+\epsilon_k
\notag\\
=& J(\theta_k) + v(s_t,\theta_k)- \sum_{s' \in \mathcal S}\theta_k(s'|s_t,a_t)v(s',\theta_k) + \epsilon_k.
\end{align*}
Then \eqref{eq:regretbound} becomes
\begin{align*}
& R(T,\text{TSDE})
\notag\\
= & \ee\Big[ \sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} c(s_t,a_t) \Big] - T \ee\Big[J(\theta_*) \Big]
\notag\\
\leq & \ee\Big[ \sum_{k=1}^{K_T}T_k J(\theta_k) \Big]
+
\ee\Big[\sum_{k=1}^{K_T}\sum_{t=t_k}^{t_{k+1}-1} \Big[ v(s_t,\theta_k) -\sum_{s' \in \mathcal S}\theta_k(s'|s_t,a_t)v(s',\theta_k)
\Big]\Big]
\notag\\
&+ \ee\Big[ \sum_{k=1}^{K_T}T_k \epsilon_k \Big]- T \ee\Big[J(\theta_*) \Big]
\notag\\
=& R_0+R_1+R_2 + \ee\Big[ \sum_{k=1}^{K_T}T_k \epsilon_k \Big].
\end{align*}
Since $R_0+R_1+R_2 = \tilde O(HS\sqrt{AT})$ from the proof of Theorem \ref{thm:regretbound}, we obtain the first part of the result.
If $\epsilon_k \leq \frac{1}{k+1}$, since $T_k \leq T_{k-1}+1 \leq ...\leq k+1$, we get
\begin{align*}
\sum_{k=1}^{K_T} T_k\epsilon_k
\leq \sum_{k=1}^{K_T} \frac{k+1}{k+1} = K_T \leq \sqrt{2SAT\log{T}}
\end{align*}
where the last inequality follows from Lemma \ref{lm:boundKT}.
\end{proof}
\section{Introduction}
We consider the problem of reinforcement learning by an agent interacting with an environment while trying to minimize the total cost accumulated over time.
The environment is modeled by an infinite horizon Markov Decision Process (MDP) with finite state and action spaces.
When the environment is perfectly known, the agent can determine optimal actions by solving a dynamic program for the MDP \cite{bertsekas}.
In reinforcement learning, however, the agent is uncertain about the true dynamics of the MDP.
A naive approach to an unknown model is the \emph{certainty equivalence principle}. The idea is to estimate the unknown MDP parameters from available information and then choose actions as if the estimates are the true parameters. But it is well-known in adaptive control theory that the certainty equivalence principle may lead to suboptimal performance due to the lack of exploration \cite{kumar2015stochastic}. This issue actually comes from the fundamental exploitation-exploration trade-off: the agent wants to exploit available information to minimize cost, but it also needs to explore the environment to learn system dynamics.
One common way to handle the exploitation-exploration trade-off is to use the \emph{optimism in the face of uncertainty} (OFU) principle \cite{lai1985asymptotically}.
Under this principle, the agent constructs confidence sets for the system parameters at each time, find the optimistic parameters that are associated with the minimum cost, and then selects an action based on the optimistic parameters.
The optimism procedure encourages exploration for rarely visited states and actions.
Several optimistic algorithms are proved to possess strong theoretical performance guarantees \cite{burnetas1997optimal,kearns2002near,brafman2002r,bartlett2009regal,jaksch2010near,filippi2010optimism,dann2015sample}.
An alternative way to incentivize exploration is the Thompson Sampling (TS) or Posterior Sampling method.
The idea of TS was first proposed by Thompson in \cite{thompson1933likelihood} for stochastic bandit problems.
It has been applied to MDP environments \cite{strens2000bayesian,osband2013more,fonteneau2013optimistic,gopalan2015thompson,abbasi2015bayesian,osband2016why} where the agent computes the posterior distribution of unknown parameters using observed information and a prior distribution. A TS algorithm generally proceeds in episodes: at the beginning of each episode a set of MDP parameters is randomly sampled from the posterior distribution, then actions are selected based on the sampled model during the episode.
TS algorithms have the following advantages over optimistic algorithms.
First, TS algorithms can easily incorporate problem structures through the prior distribution.
Second, they are more computationally efficient since a TS algorithm only needs to solve the sampled MDP, while an optimistic algorithm requires solving all MDPs that lie within the confident sets.
Third, empirical studies suggest that TS algorithms outperform optimistic algorithms in bandit problems \cite{scott2010modern,chapelle2011empirical} as well as in MDP environments \cite{osband2013more,abbasi2015bayesian,osband2016why}.
Due to the above advantages, we focus on TS algorithms for the MDP learning problem.
The main challenge in the design of a TS algorithm is the lengths of the episodes.
For finite horizon MDPs under the episodic setting, the length of each episode can be set as the time horizon \cite{osband2013more}.
When there exists a recurrent state under any stationary policy, the TS algorithm of \cite{gopalan2015thompson} starts a new episode whenever the system enters the recurrent state.
However, the above methods to end an episode can not be applied to MDPs without the special features.
The work of \cite{abbasi2015bayesian} proposed a dynamic episode schedule based on the doubling trick used in \cite{bartlett2009regal}, but a mistake in their proof of regret bound was pointed out by \cite{osband2016posterior}.
In view of the mistake in \cite{abbasi2015bayesian}, there is no TS algorithm with strong performance guarantees for general MDPs to the best of our knowledge.
We consider the most general subclass of weakly communicating MDPs in which meaningful finite time regret guarantees can be analyzed.
We propose the Thompson Sampling with Dynamic Episodes (TSDE) learning algorithm.
In TSDE, there are two stopping criteria for an episode to end.
The first stopping criterion controls the growth rate of episode length.
The second stopping criterion is the doubling trick similar to the one in \cite{bartlett2009regal,jaksch2010near,filippi2010optimism,dann2015sample,abbasi2015bayesian} that stops when the number of visits to any state-action pair is doubled.
Under a Bayesian framework, we show that the expected regret of TSDE accumulated up to time $T$ is bounded by $\tilde O(HS\sqrt{AT})$ where $\tilde O$ hides logarithmic factors.
Here $S$ and $A$ are the sizes of the state and action spaces, $T$ is time, and $H$ is the bound of the span.
This regret bound matches the best available bound for weakly communicating MDPs \cite{bartlett2009regal}, and it matches the theoretical lower bound in order of $T$ except for logarithmic factors. We present numerical results that show that TSDE actually outperforms current algorithms with known regret bounds that have the same order in $T$ for a benchmark MDP problem as well as randomly generated MDPs.
\section{Conclusion}
We propose the Thompson Sampling with Dynamic Episodes (TSDE) learning algorithm and establish $\tilde O(HS\sqrt{AT})$ bounds on expected regret for the general subclass of weakly communicating MDPs. Our result fills a gap in the theoretical analysis of Thompson Sampling for MDPs. Numerical results validate that the TSDE algorithm outperforms other learning algorithms for infinite horizon MDPs.
The TSDE algorithm determines the end of an episode by two stopping criteria. The second criterion comes from the doubling trick used in many reinforcement learning algorithms. But the first criterion on the linear growth rate of episode length seems to be a new idea for episodic learning algorithms. The stopping criterion is crucial in the proof of regret bound (Lemma \ref{lm:R0}).
The simulation results of TSDE versus Lazy PSRL further shows that this criterion is not only a technical constraint for proofs, it indeed helps balance exploitation and exploration.
\subsubsection*{Acknowledgments}
Yi Ouyang would like to thank Yang Liu from Harvard University for helpful discussions.
Rahul Jain and Ashutosh Nayyar were supported by NSF Grants 1611574 and 1446901.
\bibliographystyle{ieeetr}
\section{Problem Formulation}
\subsection{Preliminaries}
An infinite horizon Markov Decision Process (MDP) is described by $(\mathcal S, \mathcal A, c, \theta)$.
Here $\mathcal S$ is the state space, $\mathcal A$ is the action space, $c: \mathcal S \times \mathcal A \rightarrow [0,1]$\footnote{Since $\mathcal S$ and $\mathcal A$ are finite, we can normalize the cost function to $[0,1]$ without loss of generality.} is the cost function, and $\theta : \mathcal S^2\times\mathcal A\rightarrow [0,1]$ represents the transition probabilities such that
$\theta(s'|s,a) = \prob(s_{t+1} = s' | s_{t} = s, a_t = a)$
where $s_t \in \mathcal S$ and $a_t \in \mathcal A$ are the state and the action at $t=1,2,3\dots$.
We assume that $\mathcal S$ and $\mathcal A$ are finite spaces with sizes $S\geq 2$ and $A\geq 2$, and the initial state $s_1$ is a known and fixed state.
A stationary policy is a deterministic map $\pi: \mathcal S \rightarrow \mathcal A$ that maps a state to an action.
The average cost per stage of a stationary policy is defined as
\begin{align*}
J_{\pi}(\theta) = \limsup_{T\rightarrow \infty}\frac{1}{T}\ee \Big[\sum_{t=1}^{T} c(s_t,a_t)\Big].
\end{align*}
Here we use $J_{\pi}(\theta)$ to explicitly show the dependency of the average cost on $\theta$.
To have meaningful finite time regret bounds, we consider the subclass of weakly communicating MDPs defined as follows.
\begin{definition}
An MDP is weakly communicating (or weak accessible) if its states can be partitioned into two subsets:
in the first subset all states are transient under every stationary policy, and every two states in the second subset can be reached from each other under some stationary policy.
\end{definition}
From MDP theory \cite{bertsekas}, we know that if the MDP is weakly communicating, the optimal average cost per stage $J(\theta) = \min_{\pi}J_{\pi}(\theta)$ satisfies the Bellman equation
\begin{align}
J(\theta) + v(s,\theta) = \min_{a \in \mathcal A}\Big\{
c(s,a) + \sum_{s' \in \mathcal S}\theta(s'|s,a)v(s',\theta)
\Big\}
\label{eq:DP_inf}
\end{align}
for all $s\in\mathcal S$. The corresponding optimal stationary policy $\pi^*$ is the minimizer of the above optimization given by
\begin{align}
a = \pi^*(s,\theta).
\label{eq:known_control}
\end{align}
Since the cost function $c(s,a) \in [0,1]$, $J(\theta) \in [0,1]$ for all $\theta$.
If $v$ satisfies the Bellman equation, $v$ plus any constant also satisfies the Bellman equation.
Without loss of generality, let $\min_{s \in \mathcal S} v(s,\theta) = 0$ and define the span of the MDP as
$sp(\theta) = \max_{s \in \mathcal S} v(s,\theta)$. \footnote{See \cite{bartlett2009regal}for a discussion on the connection of the span with other parameters such as the diameter appearing in the lower bound on regret.}
We define $\Omega_*$ to be the set of all $\theta$ such that the MDP with transition probabilities $\theta$ is weakly communicating, and there exists a number $H$ such that $sp(\theta) \leq H$. We will focus on MDPs with transition probabilities in the set $\Omega_*$.
\subsection{Reinforcement Learning for Weakly Communicating MDPs}
We consider the reinforcement learning problem of an agent interacting with a random weakly communicating MDP $(\mathcal S, \mathcal A, c, \theta_*)$.
We assume that $\mathcal S$, $\mathcal A$ and the cost function $c$ are completely known to the agent.
The actual transition probabilities $\theta_*$ is randomly generated at the beginning before the MDP interacts with the agent. The value of $\theta_*$ is then fixed but unknown to the agent.
The complete knowledge of the cost is typical as in \citep{bartlett2009regal,gopalan2015thompson}. Algorithms can generally be extended to the unknown costs/rewards case at the expense of some constant factor for the regret bound.
At each time $t$, the agent selects an action according to $a_t = \phi_t(h_t)$ where $h_t = (s_1,s_2,\dots,s_t,a_1,a_2,\dots,a_{t-1}) $ is the history of states and actions. The collection $\phi = (\phi_1,\phi_2\dots)$ is called a learning algorithm.
The functions $\phi_t$ allow for the possibility of randomization over actions at each time.
We focus on a Bayesian framework for the unknown parameter $\theta_*$.
Let $\mu_1$ be the prior distribution for $\theta_*$, i.e., for any set $\Theta$, $\mathbb{P}(\theta_* \in \Theta) = \mu_1(\Theta)$.
We make the following assumptions on $\mu_1$.
\begin{assumption}
\label{assum:WASP}
The support of the prior distribution $\mu_1$ is a subset of $\Omega_*$. That is, the MDP is weakly communicating and $sp(\theta_*) \leq H$.
\end{assumption}
In this Bayesian framework, we define the expected regret (also called Bayesian regret or Bayes risk) of a learning algorithm $\phi$ up to time $T$ as
\begin{align}
R(T,\phi) = \ee\Big[ \sum_{t=1}^T \Big[ c(s_t,a_t) - J(\theta_*)\Big] \Big]
\end{align}
where $s_t,a_t, t=1,\dots,T$ are generated by $\phi$ and $J(\theta_*)$ is the optimal per stage cost of the MDP.
The above expectation is with respect to the prior distribution $\mu_1$ for $\theta_*$, the randomness in state transitions, and the randomized algorithm.
The expected regret is an important metric to quantify the performance of a learning algorithm.
\section{Thompson Sampling with Dynamic Episodes}
In this section, we propose the Thompson Sampling with Dynamic Episodes (TSDE) learning algorithm.
The input of TSDE is the prior distribution $\mu_1$.
At each time $t$, given the history $h_t$, the agent can compute the posterior distribution $\mu_t$ given by
$\mu_t(\Theta) = \mathbb{P}(\theta_* \in \Theta | h_t)$ for any set $\Theta$.
Upon applying the action $a_t$ and observing the new state $s_{t+1}$, the posterior distribution at $t+1$ can be updated according to Bayes' rule as
\begin{align}
\mu_{t+1}(d\theta) =
\frac{\theta(s_{t+1}|s_t,a_t)\mu_t(d\theta)}{\int \theta'(s_{t+1}|s_t,a_t)\mu_t(d\theta')}.
\label{eq:beliefupdate}
\end{align}
Let $N_t(s,a)$ be the number of visits to any state-action pair $(s,a)$ before time $t$. That is,
\begin{align}
N_t(s,a) = |\{\tau<t: (s_\tau,a_\tau) = (s,a)\}|.
\end{align}
With these notations, TSDE is described as follows.
\begin{algorithm}[H]
\caption{Thompson Sampling with Dynamic Episodes (TSDE)}
\begin{algorithmic}
\STATE Input: $\mu_1$
\STATE Initialization: $t\leftarrow 1$, $t_k \leftarrow 0$
\FOR{ episodes $k=1,2,...$}
\STATE{$T_{k-1} \leftarrow t - t_{k}$}
\STATE{$t_{k} \leftarrow t$}
\STATE{Generate $\theta_{k} \sim \mu_{t_k}$ and compute $\pi_{k}(\cdot) = \pi^*(\cdot,\theta_k)$ from \eqref{eq:DP_inf}-\eqref{eq:known_control}}
\WHILE{$t \leq t_k+T_{k-1}$ and $N_t(s,a) \leq 2N_{t_{k}}(s,a)$ for all $(s,a) \in \mathcal S\times \mathcal A$}
\STATE{Apply action $a_t = \pi_k(s_t)$}
\STATE{Observe new state $s_{t+1}$}
\STATE{Update $\mu_{t+1}$ according to \eqref{eq:beliefupdate}}
\STATE{$t \leftarrow t+1$}
\ENDWHILE
\ENDFOR
\end{algorithmic}
\end{algorithm}
The TSDE algorithm operates in episodes.
Let $t_k$ be start time of the $k$th episode and $T_k = t_{k+1}-t_{k}$ be the length of the episode with the convention $T_0 = 1$.
From the description of the algorithm, $t_1 = 1$ and $t_{k+1}, k\geq 1,$ is given by
\begin{align}
t_{k+1} = \min\{ & t>t_{k}:\quad
t > t_{k} + T_{k-1}
\text{ or }
N_t(s,a) > 2N_{t_{k}}(s,a) \text{ for some }(s,a)
\}.
\label{eq:tk}
\end{align}
At the beginning of episode $k$, a parameter $\theta_k$ is sampled from the posterior distribution $\mu_{t_k}$. During each episode $k$, actions are generated from the optimal stationary policy $\pi_k$ for the sampled parameter $\theta_k$.
One important feature of TSDE is that its episode lengths are not fixed. The length $T_k$ of each episode is dynamically determined according to two stopping criteria: (i) $t > t_k+T_{k-1}$, and (ii)
$N_t(s,a) > 2N_{t_{k}}(s,a)$ for some state-action pair $(s,a)$.
The first stopping criterion provides that the episode length grows at a linear rate without triggering the second criterion.
The second stopping criterion ensures that the number of visits to any state-action pair $(s,a)$ during an episode should not be more than the number visits to the pair before this episode.
\begin{remark}
Note that TSDE only requires the knowledge of $\mathcal S$, $\mathcal A$, $c$, and the prior distribution $\mu_1$.
TSDE can operate without the knowledge of time horizon $T$, the bound $H$ on span used in \cite{bartlett2009regal}, and any knowledge about the actual $\theta_*$ such as the recurrent state needed in \cite{gopalan2015thompson}.
\end{remark}
\subsection{Main Result}
\begin{theorem}
\label{thm:regretbound}
Under Assumption \ref{assum:WASP},
\begin{align*}
R(T,\text{TSDE}) \leq (H+1)\sqrt{ 2 SA T\log(T) } + 49 HS\sqrt{AT\log(AT)}.
\end{align*}
\end{theorem}
The proof of Theorem \ref{thm:regretbound} appears in Section \ref{sec:analysis}.
\begin{remark}
Note that our regret bound has the same order in $H,S,A$ and $T$ as the optimistic algorithm in \cite{bartlett2009regal} which is the best available bound
for weakly communicating MDPs. Moreover, the bound does not depend on the prior distribution or other problem-dependent parameters such as the recurrent time of the optimal policy used in the regret bound of \cite{gopalan2015thompson}.
\end{remark}
\subsection{Approximation Error}
At the beginning of each episode, TSDE computes the optimal stationary policy $\pi_k$ for the parameter $\theta_k$.
This step requires the solution to a fixed finite MDP. Policy iteration or value iteration can be used to solve the sampled MDP, but the resulting stationary policy may be only approximately optimal in practice.
We call $\pi$ an $\epsilon-$approximate policy if
\begin{align*}
c(s,\pi(s)) + \sum_{s' \in \mathcal S}\theta(s'|s,\pi(s))v(s',\theta)
\leq \min_{a \in \mathcal A}\Big\{
c(s,a) + \sum_{s' \in \mathcal S}\theta(s'|s,a)v(s',\theta)
\Big\} + \epsilon.
\end{align*}
When the algorithm returns an $\epsilon_k-$approximate policy $\tilde\pi_k$ instead of the optimal stationary policy $\pi_k$ at episode $k$, we have the following regret bound in the presence of such approximation error.
\begin{theorem}
\label{thm:regretbound_approximate}
If TSDE computes an $\epsilon_k-$approximate policy $\tilde\pi_k$ instead of the optimal stationary policy $\pi_k$ at each episode $k$, the expected regret of TSDE satisfies
\begin{align*}
R(T,\text{TSDE}) \leq \tilde O(HS\sqrt{AT}) + \ee\Big[\sum_{k: t_k \leq T} T_k\epsilon_k\Big].
\end{align*}
Furthermore, if $\epsilon_k \leq \frac{1}{k+1}$,
$
\ee\Big[\sum_{k: t_k \leq T} T_k\epsilon_k\Big] \leq \sqrt{2SAT\log(T)}.
$
\end{theorem}
Theorem \ref{thm:regretbound_approximate} shows that the approximation error in the computation of optimal stationary policy is only additive to the regret under TSDE.
The regret bound would remain $\tilde O(HS\sqrt{AT})$ if the approximation error is such that $\epsilon_k \leq \frac{1}{k+1}$.
The proof of Theorem \ref{thm:regretbound_approximate} is in the appendix due to the lack of space.
\section{Simulations}
In this section, we compare through simulations the performance of TSDE with three learning algorithms with the same regret order: UCRL2 \cite{jaksch2010near}, TSMDP \cite{gopalan2015thompson}, and Lazy PSRL \cite{abbasi2015bayesian}.
UCRL2 is an optimistic algorithm with similar regret bounds.
TSMDP and Lazy PSRL are TS algorithms for infinite horizon MDPs. TSMDP has the same regret order in $T$ given a recurrent state for resampling. The original regret analysis for Lazy PSRL is incorrect, but the regret bounds are conjectured to be correct \cite{osband2016posterior}.
We chose $\delta = 0.05$ for the implementation of UCRL2 and assume an independent Dirichlet prior with parameters $[0.1, \ldots, 0.1]$ over the transition probabilities for all TS algorithms.
We consider two environments: randomly generated MDPs and the RiverSwim example \cite{strehl2008analysis}.
For randomly generated MDPs, we use the independent Dirichlet prior over $6$ states and $2$ actions but with a fixed cost.
We select the resampling state $s_0=1$ for TSMDP here since all states are recurrent under the Dirichlet prior.
The RiverSwim example models an agent swimming in a river who can choose to swim either left or right. The MDP consists of six states arranged in a chain with the agent starting in the leftmost state ($s=1$). If the agent decides to move left i.e with the river current then he is always successful but if he decides to move right he might fail with some probability.
The cost function is given by: $c (s,a) = 0.8$ if $s=1$, $a =$ left; $c (s,a) = 0$ if $s=6$, $a =$ right; and $c (s,a) = 1$ otherwise.
The optimal policy is to swim right to reach the rightmost state which minimizes the cost.
For TSMDP in RiverSwim, we consider two versions with $s_0=1$ and with $s_0=3$ for the resampling state.
We simulate 500 Monte Carlo runs for both the examples and run for $T = 10^5$.
\begin{figure}[h]
\centering
\begin{subfigure}[t]{0.43\textwidth}
\centering
\includegraphics[width=\textwidth]{./random_mdp_fornips}
\caption{\small{Expected Regret vs Time for random MDPs}}
\label{fig_random}
\end{subfigure}
\begin{subfigure}[t]{0.43\textwidth}
\centering
\includegraphics[width=\textwidth]{./riverswim_fornips}
\caption{\small{Expected Regret vs Time for RiverSwim}}
\label{fig_riverswim}
\end{subfigure}
\caption{\small{Simulation Results}}
\label{fig_sim}
\end{figure}
From Figure \ref{fig_sim}(\subref{fig_random}) we can see that TSDE outperforms all the three algorithms in randomly generated MDPs.
In particular, there is a significant gap between the regret of TSDE and that of UCRL2 and TSMDP.
The poor performance of UCRL2 assures the motivation to consider TS algorithms.
From the specification of TSMDP, its performance heavily hinges on the choice of an appropriate resampling state which is not possible for a general unknown MDP. This is reflected in the randomly generated MDPs experiment.
In the RiverSwim example, Figure \ref{fig_sim}(\subref{fig_riverswim}) shows that TSDE significantly outperforms UCRL2, Lazy PSRL, and TSMDP with $s_0=3$.
Although TSMDP with $s_0=1$ performs slightly better than TSDE, there is no way to pick this specific $s_0$ if the MDP is unknown in practice.
Since Lazy PSRL is also equipped with the doubling trick criterion, the performance gap between TSDE and Lazy PSRL highlights the importance of the first stopping criterion on the growth rate of episode length. We also like to point out that in this example, the MDP is fixed and is not generated from the Dirichlet prior.
Therefore, we conjecture that TSDE also has the same regret bounds under a non-Bayesian setting.
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"redpajama_set_name": "RedPajamaArXiv"
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Writing a resume isn't so difficult, but it does require a little time and setting up. The very foremost is, should you not own a level, there are now just a few lenders who can take your own application. Devote the ability to create sure it stands out. That was just really a remarkable chance it is going to need a little bit of faculty, at an AA in promotion. Reading can be way more energetic and intriguing in case you've got the opportunity to divide the learning experience together with other folks. In the town, nevertheless, tons of chances abound. Employers love to see you have work experience with some type, at least.
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After you format your resume you need to safeguard your render enough margin distance to empower for printing. So the arrangement is vital in planning an suitable demonstration of your own resume. A coverletter format should be adhered to so you'll take a position to see that which you will writedown.
The data you include in your own resume needs to be clear and brief. Again, even though it can appear insistent, proceed on of time and join the resume. The information below will supply you with general guidelines. Second, you have to know how to prepare the advice you've got. You will have to get some very easy advice and also a couple dates. What's more, it doesn't will need to contain info in chronological arrangement, nor will this must cover your complete career.
The exact final idea you wish to complete is receive a project which you can not perform. Producing your resume would be your very first step to getting operate. Whether you're searching for the very first job or your next one, then you'll need a resume which shows companies you are a professional.
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When you are hunting for do the job, you understand you will need to make yourself a cv that can be the thing you need to pass on to prospective companies which you prefer to utilize to get. A project might be an experience building stepping rock in case you previously know what type of career that you need to make in the future. Inquire further overall questions about the company and special questions regarding the task that you just seek out. Be clear in regards to the work that you are seeking. If you should be indeed applying to get an initial job, subsequently a flatter resume may be an perfect instrument for you.
Just be certain that whichever arrangement you decide on, which you're targeting it for the job which you're deciding on. It really is depends upon if the project posting directions define a specific arrangement. Any way, it is the the manual occupation to deliver just about each and every detail about you personally.
The 3rd sort of restart format is popularly known because the hybrid or combination format. When to Use the Chronological Resume Format A Chronological resume is your easiest to develop plus it's also the very popular format. You want to stick with a very easy format. Each resume arrangement includes a specific function and highlights different parts of the applicant's professional history. To begin the restart, you might require to select a restart format.
The format of one's resume is quite critical as the advice that you put init. Each resume structure includes their very own set of gains and pitfalls for a variety of types of work seekers, so so ensure that you choose wisely. The chronological resume format is readily the very typical format.
An excellent resume could highlight every of your own strong points so that the company could see straight away why it would be a great selection for your position. There are a couple things you ought to know about how to write a resume. However tempting it can be to extend the truth, putting on your resume is always a inadequate idea.
Variety of Resume The precise first step to have a look at prior to building a restart is to decide on the sort of the resume. Recognizing how to write a resume can be a huge accomplishment for those that would like to increase this ladder of livelihood victory. Whilst Canadian resumes are available in a number of exceptional formats, they even discuss some criteria that you ought to be conscious of.
More frequently than not, customizing your resume does not signify a major rewrite, but alternatively just several tweaks. Resumes give prospective companies a method to master about applicants immediately and readily, and they truly are your very first measure toward brand new job chances. Chronological resumes aren't the ideal match for pupils due to the fact that they do not possess lots of knowledge in the exact first place. If you'd love to do just two or even three specific things, then make a couple specific resumes.
There are three sorts of resumes, each having a unique benefits and advantages. They can be a couple of pages. Although resumes are all composed using conventional elements, there's no approved arrangement that functions both well for everybody. They are sometimes used for a kind of reasons, but often they are used to secure fresh job. Composing your 1st resume may be seemingly intimidating endeavor. In reality, in regards to early resumes and job applications, the concept needs is to deal with the process for a learning encounter.
You will learn expert hints about how to design your path to the companies you desire to work for, and how to revamp your resume, and the way to interview as a specialist. The key thing is to analyze the business and really have a very clear notion what you wish to profit in your internship and that which you have got to give. In most cases, companies shell out a lot of dollars to detect premium gift. Find out more concerning the in's and out from the company which you want to benefit.
When you're trying to split in the commercial and you've drained every technique to make an effort to split in and it seems like a world away. Many businesses utilize manage panels to operate enormous equipments or machines. It is tougher than ever for younger folks to crack at the audio industry.
If you're thinking about interning in the commercial, it would be wise to offer some thought to your particular region or parts of interest so you can get ready. The further you know more concerning the business and the field you are in as a while, the longer you are going to have the ability to identify prospective opportunities. Whether you're just starting out in the commercial or already a component of it, here's just a look at all you want to understand concerning the tradition. Enjoyment marketplace Resume web templates enable one to re create the precise style and fashions in a lot of paperwork. The entertainment business is no exception. It is extremely competitive as well as specific.
Your resume needs to comprise some ideal experience inside of the leisure industry, but for lots of excess tools, practical expertise isn't essential. As a result, no 2 resumes may look exactly the specific same. Alternatively, take some time to grow your self in to the greatest possible candidate, subsequently spend the opportunity to make sure your resume does the very best possible occupation representing you as well as your own abilities.
After you know what sort of profession you would like, the following suggestion is to discover the best way to really land a project doing it. If it comes to livelihood, the exact first thing isn't to property your own first job, but as an alternative to identify what kind of career you would love to go after at the exact first location. After all, even should you prefer to create this kind of career, you'll likely need to go out here anyway. As your career progresses, you have to visit a crossroads when you need to decide whether it's time for you to adjust jobs. In the event that you would want to progress in your occupation, you can't be intimidated by people who work at the level you want to turn into. Plus, the can be a career that takes all around the Earth, or it can function as type of career that gives you the opportunity to develop a well balanced job using a neighborhood theatre or even tv manufacturing firm.
Besides applying to project openings, the different big way men and women find job at the biz is via personal contacts. Around the reverse side, if you are requesting for employment to get a restaurant supervisor, then you could be better suited to go over your earlier comprehension in the restaurant industry in addition to one's restart. Even now, finding a project isn't exactly as simple as filing your resume to ProdutionBeast listings. If you're referring to discovering a brand new project , you are also talking leaving one. You are well prepared to get your very first occupation but are confronted with the task of landing a new location free of work restart should not worked. After you have labored on enough expert film editor jobs, you will have the possiblity to combine one of those picture modifying guilds, or one of their pro associations.
Take the job description because you decide what things to comprise. Be specific in regards to the relevant skills which you own which fit exactly precisely the work description. The data is up to date once per month. In the future, you can make use of the advice to make your internet site, site or maybe to commence an advertising provider. The details you give will be. Make sure that you have the info that you're looking for. Soak up and remember that the ideas and data you have assembled.
Media should turn in to a vital part of your craft, and a in which you're consistently busy. Down on the road, your system will enlarge and you also are able to start building your very own personal board of directors. Your network in the commercial at this time will be likely somewhat little and you are most likely to become actively attempting to enlarge it.
Social networking is just a great technique to brand your self well. Your entertainment and media resume must be customized to your distinctive abilities and also industry. A media and entertainment restart could consist of places by a extensive variety of businesses. A current press and amusement restart summary statement provides general idea about what to expect from of your resume and ought to comprise how seasoned you are and which sort of skills and knowledge you've got. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,035 |
/* eslint-env jest, node */
/* eslint no-sync: "off" */
import fs from 'fs';
import path from 'path';
import { reducer } from '../reducer';
import { restoreActions } from '../actions';
import { commitToServer } from '../server';
import '../components/aws-cloud-credentials';
import '../components/aws-cluster-info';
import '../components/aws-submit-keys';
import '../components/aws-vpc';
import '../components/bm-credentials';
import '../components/bm-hostname';
import '../components/bm-matchbox';
import '../components/bm-nodeforms';
import '../components/bm-sshkeys';
import '../components/certificate-authority';
import '../components/cluster-type';
import '../components/nodes';
import '../components/users';
const structureOnly = (obj) => {
const toString = Object.prototype.toString;
if (toString.call(obj) === '[object Object]') {
const ret = {};
Object.keys(obj).forEach(k => {
ret[k] = structureOnly(obj[k]);
});
return ret;
}
if (Array.isArray(obj)) {
return [];
}
return '';
};
const initialState = reducer(undefined, {type: 'Some Initial Action'});
const tests = [
{
description: 'works with bare metal',
expected: 'metal-out.json',
state: 'metal-in.json',
},
{
description: 'works with AWS',
expected: 'aws-custom-vpc-out.json',
state: 'aws-custom-vpc-in.json',
},
{
description: 'works with AWS (existing VPC)',
expected: 'aws-existing-vpc-out.json',
state: 'aws-existing-vpc-in.json',
},
];
let dispatch;
beforeEach(() => {
dispatch = jest.fn();
});
const readExample = filename => {
let json;
try {
json = JSON.parse(fs.readFileSync(path.resolve(__dirname, `examples/${filename}`), 'utf8'));
} catch (e) {
console.warn(`${filename} is not json`);
throw e;
}
return json;
};
/* eslint-disable max-nested-callbacks */
describe('progress file example', () => {
tests.forEach(t => {
it(t.description, () => {
const restored = reducer(initialState, restoreActions.restore(readExample(t.state)));
global.fetch = jest.fn(() => Promise.resolve({
ok: true,
blob: () => Promise.resolve('success'),
text: () => Promise.reject('failed'),
json: () => Promise.resolve({}),
}));
// TODO: wait for this action to finish & then check expected state
commitToServer(false, false)(dispatch, () => restored);
expect(fetch.mock.calls.length).toBe(1);
const body = JSON.parse(fetch.mock.calls[0][1].body);
expect(readExample(t.expected)).toEqual(body);
});
});
});
/* eslint-enable max-nested-callbacks */
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,143 |
Q: How to work with binary contraints in linear optimization? I have two input matrices, dt(10,3) & wt(3,3), that i need to use to find the optimal decision matrix (same dimension), Par(10,3) so as to maximize an objective function. Below R code would give some direction into the problem (used Sample inputs here) -
#Input Matrices
dt <- matrix(runif(300),100,3)
wt <- matrix(c(1,0,0,0,2,0,0,0,1),3,3) #weights
#objective function
Obj <- function(Par) {
P = matrix(Par, nrow = 10, byrow=F) # Reshape
X = t((dt%*%wt)[,1])%*%P[,1]
Y = t((dt%*%wt)[,2])%*%P[,2]
Z = t((dt%*%wt)[,3])%*%P[,3]
as.numeric(X+Y+Z) #maximize
}
Now I am struggling to apply the following constraints to the problem :
1) Matrix, Par can only have binary values (0 or 1)
2) rowSums(Par) = 1 (Basically a row can only have 1 in one of the three columns)
3) colSums(Par[,1]) <= 5, colSums(Par[,2]) <= 6, & colSums(Par[,3]) <= 4
4) X/(X+Y+Z) < 0.35, & Y/(X+Y+Z) < 0.4 (X,Y,Z are defined in the objective function)
I tried coding the constraints in constrOptim, but not sure how to input binary & integer constraints. I am reading up on lpSolve, but not able to figure out. Any help much appreciated. Thanks!
A: I believe this is indeed a MIP so no issues with convexity. If I am correct the model can look like:
This model can be easily transcribed into R. Note that LP/MIP solvers do not use functions for the objective and constraints (opposed to NLP solvers). In R typically one builds up matrices with the LP coefficients.
Note: I had to make the limits on the column sums much larger (I used 50,60,40).
A: Based on Erwin's response, I am able to formulate the model using lpSolve in R. However still struggling to add the final constraint to the model (4th constraint in my question above). Here's what I am able to code so far :
#input dimension
r <- 10
c <- 3
#input matrices
dt <- matrix(runif(r*c),r,c)
wt <- matrix(c(1,0,0,0,2,0,0,0,1),3,3) #weights
#column controller
c.limit <- c(60,50,70)
#create structure for lpSolve
ncol <- r*c
lp.create <- make.lp(ncol=ncol)
set.type(lp.create, columns=1:ncol, type = c("binary"))
#create objective values
obj.vals <- as.vector(t(dt%*%wt))
set.objfn(lp.create, obj.vals)
lp.control(lp.create,sense='max')
#Add constraints to ensure sum of parameters for every row (rowSum) <= 1
for (i in 1:r){
add.constraint(lp.create, xt=c(1,1,1),
indices=c(3*i-2,3*i-1,3*i), rhs=1, type="<=")
}
#Add constraints to ensure sum of parameters for every column (colSum) <= column limit (defined above)
for (i in 1:c){
add.constraint(lp.create, xt=rep(1,r),
indices=seq(i,ncol,by=c), rhs=c.limit[i], type="<=")
}
#Add constraints to ensure sum of column objective (t((dt%*%wt)[,i])%*%P[,i) <= limits defined in the problem)
#NOT SURE HOW TO APPLY A CONSTRAINT THAT IS DEPENDENT ON THE OBJECTIVE FUNCTION
solve(lp.create)
get.objective(lp.create) #20
final.par <- matrix(get.variables(lp.create), ncol = c, byrow=T) # Reshape
Any help that can get me to the finish line is much appreciated :)
Thanks
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,683 |
Local April 26, 2013 | 9:27 am
Trial for Bahia de Las Aguilas land fraud starts June 7
Santo Domingo.- A National District court on Thursday set for June 7 the start of the trial against former Dominican Agrarian Institute (IAD) director Jaime Rodriguez Guzman and two others, charged with titles fraud on thousands of hectares at Bahia de Las Aguilas beach (southwest).
1st Appellate Court president Giselle Mendez set the hearing against the former official, his wife Margarita Reyna and his brother Rafael Antonio Rodriguez, 9am on that date.
The date for the hearing was set after National District Criminal Chamber presiding judge Ramona Rodriguez referred the case to the Supreme Court, which overturned a lower court ruling favoring the defendants, and ordered a new trial.
In addition to the National District prosecutors, Laura Acosta Lora was named a special attorney for what is considered the country's biggest land fraud case. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 965 |
Top 5 Environmentally Friendly U.S. Colleges
We've taken a clue from the Sierra Club Magazine, a US environmental organization, who have picked their favourites for most eco-friendly universities in the country. Here are braingainmag.com's reasons for loving these campuses!
BY Braingainmag.com Staff
University of California Davis (UC Davis)
Georgia Institute of Technology (Georgia Tech)
Topping the list with its Sustainable 2nd Century , the university takes its energy and waste management, transportation around campus (you'll see a LOT of bicycles), and food services pretty seriously. The UC Davis West Village housing complex aims to reach a goal of net zero energy - and will be the largest residential community in the U.S. to do so.
Photo courtesy of / www.ucdavis.edu
Not only has Georgia Tech's campus integrated sustainable design and construction, but the university also uses environmentally friendly cleaning products, requires that all its vendors do the same. Every student is required to take an environmentally marked course before graduation. The campus collects rainwater, recycles at football games and dining hall waste - and the annual Earth Day celebration, which is open to the public, is a huge deal!
Photo courtesy of / www.gateach.edu
Between eco-friendly buildings, and prodding students and alums to explore eco-friendly business ventures, Stanford also has a lot going on in the green arena. The campus also has a huge Sustainable IT program implemented on campus, designed to reduce the amount of greenhouse gas emissions generated by its IT infrastructure. And then there's also the on-campus organic garden - about 40% of Stanford's Dining produce is grown either on campus or within a 250 mile radius.
Photo courtesy of / www.stanford.edu
Water and energy conservation, recycling, composting and waste reduction are important here - but for students, the most fun is in the self-service bicycle repair stations that are dotted around campus, and equipped with an air-pump and tools to keep you rolling on environmentally-friendly wheels.
Photo courtesy of / www.washington.edu
From green construction projects to recycling leftover kitchen cooking oil into fuel for powering campus shuttle buses, the University of Connecticut is also keeping up with the game. Students are heavily involved with recycling projects on campus - and the Earth Day Spring Fling is a fun way of continuing to spread the eco-friendly awareness.
Photo courtesy of / www.uconn.edu
Tags: Sierra Club, Sustainability Programs, Eco-Friendly Campus, Earth Day
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Dubai: Where East Meets West | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,020 |
//===----------------------------------------------------------------------===//
//
// The LLVM Compiler Infrastructure
//
// This file is dual licensed under the MIT and the University of Illinois Open
// Source Licenses. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//
// <vector>
// vector(size_type n, const value_type& x, const allocator_type& a);
#include <vector>
#include <cassert>
#include "libcxx_tc_common.h"
#include "test_macros.h"
#include "asan_testing.h"
template <class C>
static int
test(typename C::size_type n, const typename C::value_type& x,
const typename C::allocator_type& a)
{
C c(n, x, a);
LIBCPP_ASSERT(c.__invariants());
TC_ASSERT_EXPR(a == c.get_allocator());
TC_ASSERT_EXPR(c.size() == n);
LIBCPP_ASSERT(is_contiguous_container_asan_correct(c));
for (typename C::const_iterator i = c.cbegin(), e = c.cend(); i != e; ++i)
TC_ASSERT_EXPR(*i == x);
return 0;
}
int tc_libcxx_containers_vector_cons_construct_size_value_alloc(void)
{
TC_ASSERT_FUNC((test<std::vector<int> >(50, 3, std::allocator<int>())));
TC_SUCCESS_RESULT();
return 0;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,254 |
{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/algebra\/algebra-1\/chapter-11-rational-expressions-and-functions-11-2-multiplying-and-dividing-rational-expressions-practice-and-problem-solving-exercises-page-663\/63","text":"## Algebra 1\n\n$x=0, -4$\nWe are dividing the second fraction, so the numerator and denominator are flipped. Thus, there are two formulas that cannot equal zero in the denominator. Those formulas are $6x$ and $-3x-12$. If we set these equal to zero, then we find $x=0$ and $x=-4$, so these values cause the expression to be undefined.","date":"2020-02-21 22:35:18","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\": 0.9930754899978638, \"perplexity\": 166.96191197851684}, \"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-2020-10\/segments\/1581875145538.32\/warc\/CC-MAIN-20200221203000-20200221233000-00033.warc.gz\"}"} | null | null |
\section{Motivation}
This talk is concerned with the transverse-momentum dependence of
particles produced in scattering processes involving a hard momentum
scale. Specifically, I will discuss semi-inclusive deep inelastic
scattering, $e p \to e + h + X$, where the momentum of the hadron $h$ is
measured. Using crossing symmetry, the results can be carried over to
Drell-Yan or $W$ or $Z$ production in $pp$ and $p\bar{p}$ collisions, as
well as to hadron pair production in electron-positron annihilation, $e^+
e^- \to h_1 + h_2 + X$. Details can be found in the recent paper
\cite{Bacchetta:2008xw}.
The transverse-momentum spectrum of produced particles may be regarded as
a basic feature of the final state. Even in the simple processes just
mentioned, its investigation reveals a number of nontrivial aspects of QCD
dynamics. There are two theoretical frameworks to describe the
distribution of a suitably defined transverse momentum $\vec{q}_T$ in the
final state.
The description sketched in Fig.~\ref{fig:mechanisms}a is based on the
``intrinsic transverse momentum'' of partons within a hadron and uses
transverse-momentum dependent (i.e.\ unintegrated) parton densities and
fragmentation functions. This description can be used for $q_T \ll Q$,
where $Q$ is the virtuality of the photon or electroweak boson, which I
assume to be large throughout this talk.
The description represented in Fig.\ref{fig:mechanisms}b uses collinear
(i.e.\ $k_T$ integrated) parton densities and fragmentation functions,
generating finite $q_T$ by perturbative radiation of partons into the
final state. This description is adequate for $q_T \gg M$, where $M$
stands for a generic nonperturbative scale.
In the following I refer to the two descriptions as ``low-$q_T$'' and
``high-$q_T$'', respectively. It has long been known that both mechanisms
give rise to a nonzero cross section for longitudinal photon polarization
and to a nontrivial dependence on the azimuth of $\vec{q}_T$
\cite{Cahn:1978se,Georgi:1977tv,Oganesian:1997jq}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.43\textwidth]{low-tree.eps}
\hspace{2.1em}
\includegraphics[width=0.43\textwidth]{high.eps} \\[-0.3em]
$\mathbf{(a)}$ \hspace{0.47\textwidth} $\mathbf{(b)}$
\end{center}
\caption{\label{fig:mechanisms} The low-$q_T$ description $\mathbf{(a)}$
and the high-$q_T$ description $\mathbf{(b)}$ for the
transverse-momentum distribution of the produced particle $h$ in
semi-inclusive deep inelastic scattering, $e p \to e + h + X$.}
\end{figure}
\section{Insight from power counting}
\label{sec:power}
It is natural to ask how these two descriptions are related to each other.
A first answer can be obtained from a careful look at the power counting
in the region of intermediate transverse momentum, $M \ll q_T \ll Q$,
where both approaches can be applied. The low-$q_T$ approach starts with
an expansion in the small parameter $q_T/Q$ and involves coefficients
depending on $M/q_T$, which for $q_T \gg M$ can be further expanded in
$M/q_T$. For an observable $F$ with mass dimension~$-2$ one thus has
\begin{align}
\label{pow-low}
F(q_T,Q) \;\stackrel{q_T \ll Q}{=}\; \frac{1}{M^2}\,
\sum_{\text{twist}~n}\,
\biggl[\frac{q_T}{Q}\biggr]^{n-2}\,
l_{n} \biggl(\frac{M}{q_T}\biggr)
& \;\stackrel{M \ll q_T \ll Q}{=}\; \frac{1}{M^2}\,
\sum_{n,k} l_{n,k}\;
\biggl[\frac{q_T}{Q}\biggr]^{n-2}\,
\biggl[\frac{M}{q_T}\biggr]^{k} \,.
\end{align}
By contrast, the high-$q_T$ approach first expands an observable in
$M/q_T$, with coefficients that for intermediate $q_T$ can be further
expanded in $q_T/Q$\,:
\begin{align}
\label{pow-hi}
F(q_T,Q) \;\stackrel{M \ll q_T}{=}\; \frac{1}{M^2}\,
\sum_{\text{twist}~n}\,
\biggl[\frac{M}{q_T}\biggr]^{n}\,
h_{n} \biggl(\frac{q_T}{Q}\biggr)
& \;\stackrel{M \ll q_T \ll Q}{=}\; \frac{1}{M^2}\,
\sum_{n,k} h_{n,k}\;
\biggl[\frac{M}{q_T}\biggr]^{n}\,
\biggl[\frac{q_T}{Q}\biggr]^{k-2} \,.
\end{align}
The simultaneous validity of both approaches in the region $M \ll q_T \ll
Q$ implies $l_{n,k} = h_{k,n}$. The first index $n \ge 2$ in each
expansion characterizes the twist of the corresponding calculation. In
practice only terms with $n=2$ and possibly $n=3$ can actually be
calculated.
For observables with nonzero $l_{2,2} = h_{2,2}$, the leading terms in the
two calculations coincide for intermediate $q_T$, where they provide
complementary descriptions of the same physics. One may then try to
construct a smooth interpolation between the two descriptions that is
valid at all $q_T$.
There are, however, observables with $l_{2,2} = h_{2,2} = 0$, where the
leading term $l_{2,4}$ of the low-$q_T$ result is distinct from the
leading term $h_{2,4}$ of the high-$q_T$ result. With both calculations
only performed at leading-twist accuracy, one can then add their results
at intermediate $q_T$ without double counting; which of them is more
important at given $q_T$ depends on the relative size of $q_T/Q$ and
$M/q_T$. We will encounter examples for both cases in
section~\ref{sec:compare}.
\section{Structure functions for semi-inclusive deep inelastic scattering}
To describe the kinematics we use the standard scaling variables $x$ and
$z$, the inelasticity $y$, the photon virtuality $Q$, the scaled
transverse momentum $q_T = P_{h\perp} /z$ of the produced hadron in the
$\gamma^*p$ c.m., and the azimuthal angle $\phi$ between the lepton and
hadron planes in that frame. Precise definitions are given in
\cite{Bacchetta:2008xw}. The unpolarized cross section can then be
parameterized in the form
\begin{equation}
\label{str-fcts}
\frac{d\sigma(ep\to ehX)}{d\phi\, dq_T^2\, dx\, dy\, dz} =
\text{(kin.~factor)}
\times \Bigl[
F_{T} + \varepsilon F_{L}
+ \sqrt{2\varepsilon (1+\varepsilon)} \cos\phi\, F^{\cos\phi}
+ \varepsilon \cos 2\phi\, F^{\cos2\phi} \Bigr] \,,
\end{equation}
where the ratio of longitudinal and transverse photon flux is given by
$\varepsilon = (1-y) /(1-y + y^2/2)$ in the Bjorken limit. The
semi-inclusive structure functions $F_{\ldots}$ depend on $x$, $z$,
$q_T^2$, $Q^2$, and the subscripts $T$ and $L$ are respectively associated
with transverse and longitudinal photon polarization.
\subsection{High-$q_T$ calculation}
The high-$q_T$ calculation gives the structure functions as convolutions
\begin{align}
F_{\ldots} &= \frac{1}{Q^2\mskip 1.5mu z^2}\;
\sum_{i,j = q,\,\bar{q},\,g}\;
\int_{x}^1 \frac{d\hat{x}}{\hat{x}}\,
\int_{z}^1 \frac{d\hat{z}}{\hat{z}}\, f_1^i\Bigl(\frac{x}{\hat{x}}\Bigr)\,
D_1^j\Bigl(\frac{z}{\hat{z}}\Bigr)\,
K^{ij}_{\ldots} \Bigl(\hat{x},\hat{z}, \frac{q_T}{Q}\Bigr)
\end{align}
in longitudinal momentum fractions, where $f_1^i$ and $D_1^j$ respectively
denote the usual unpolarized collinear distribution and fragmentation
functions, and the $K^{ij}_{\ldots}$ are perturbatively calculable
hard-scatter\-ing kernels. Expanding these kernels in $q_T/Q$, one
readily obtains the result of the high-$q_T$ mechanism for intermediate
$q_T$. At order $\alpha_s$, all structure functions in \eqref{str-fcts}
contain a term proportional to $f_1(x)\, D_1(z)\, \log (Q/q_T)$ after this
expansion. Higher orders provide terms going like $\alpha_{s}^n
\log{}^{2m-1} (Q/q_T)$ with $m\le n$. To obtain a stable perturbative
result in the region where $\log (Q/q_T)$ is large, these logarithms
should be resummed to all orders. We will come back to this in
sect.~\ref{sec:low}.
\subsection{Low $q_T$}
\label{sec:low}
In the low-$q_T$ description, factorization is fairly well understood at
twist-two accuracy \cite{Collins:1981uk,Ji:2004wu} and leads to a
representation of the form
\begin{align}
\label{CS-fact}
F_{\ldots} = \sum_{i=q,\,\bar{q}}\, x e_i^2
& \int d^2 \vec{p}_T^{}\, d^2 \vec{k}_T^{}\, d^2 \vec{l}_T^{}\,
\delta(\vec{p}_T^{}-\vec{k}_T^{}+\vec{l}_T^{}+\vec{q}_T^{})\,
w_{\ldots}(\vec{p}_T, \vec{k}_T)\,
f^i(x,p_T^2)\, D^i(z,k_T^2)\, U({l}_T^{2})
\end{align}
with known functions $w_{\ldots}\mskip 1.5mu$, where for simplicity I have omitted
a hard factor representing $\alpha_s$ corrections from virtual graphs. In
addition to transverse-momentum dependent distribution and fragmentation
functions $f^i(x,p_T^2)$ and $D^i(z,k_T^2)$, the expression
\eqref{CS-fact} contains a soft factor $U(l_T^2)$ describing soft gluon
exchange between partons moving in the direction of the target and partons
moving in the direction of the observed hadron $h$. At twist-three
accuracy, soft gluon exchange has not been analyzed, so that we do not
have a full understanding of factorization for structure functions going
like $1/Q$. However, there are detailed calculations at tree level
\cite{Mulders:1995dh,Boer:2003cm}, which give results very similar in form
to \eqref{CS-fact} without the factor $U(l_T^2)$.
\begin{figure}[b]
\begin{center}
\includegraphics[width=0.43\textwidth]{pdf-gluon.eps}
\hspace{2.1em}
\includegraphics[width=0.43\textwidth]{pdf-wilson.eps} \\[-0.3em]
$\mathbf{(a)}$ \hspace{0.47\textwidth} $\mathbf{(b)}$
\end{center}
\caption{\label{fig:pdfs} Example graphs for the calculation of an
unintegrated parton distribution at high $p_T$ in terms of a collinear
distribution $f(x)$ and a hard-scattering subprocess. The double line
represents an eikonal propagator, which comes from the Wilson line $P
\exp\bigl[ -ig \int_0^\infty d\lambda\, n\cdot A(\xi + \lambda n)\bigr]$
in the definition of the parton density
\protect\cite{Collins:1981uk,Ji:2004wu,Collins:2008ht}.}
\end{figure}
To evaluate \eqref{CS-fact} for intermediate $q_T$, one notes that for $q_T
\gg M$ at least one of the momenta $p_T$, $k_T$ or $l_T$ has to be large.
In this region, the corresponding factor can be calculated using collinear
factorization, with the large transverse momentum generated by
perturbative parton radiation as illustrated in Fig.~\ref{fig:pdfs}.
As shown in \cite{Bacchetta:2008xw}, general symmetry considerations allow
one to determine the power behavior at high $p_T$ for the different parton
densities parameterizing the spin and momentum dependence of the quark
distribution in a proton. In particular one finds
\begin{align}
\label{power-pdfs}
f_1(x,p_T^2) &\sim \frac{1}{p_T^2}\, \alpha_s\,
\sum_{i= q,\,\bar{q},\,g}\, [ f_1^i\otimes K_1^i ] \,,
\qquad\qquad
x f^\perp(x,p_T^2) \sim \frac{1}{p_T^2}\, \alpha_s\,
\sum_{i= q,\,\bar{q},\,g}\, [ f_1^i\otimes K^{\perp i} ] \,,
\nonumber \\
h_1^\perp(x,p_T^2) &\sim \frac{M^2}{p_T^4}\, \alpha_s\;
\sum_{i}\,
[ \text{collinear twist-three distributions} \otimes K_3^i ] \,,
\end{align}
where $K_1^i$, $K^{\perp i}$, and $K_3^i$ are calculable hard-scattering
kernels and $\otimes$ denotes a convolution in longitudinal momentum
fractions. The Boer-Mulders function $h_1^\perp$ describes the
distribution of transversely polarized quarks in an unpolarized proton,
whereas $f^\perp$ is a distribution of twist three (involving one good and
one bad light-cone component of the quark field). Explicit calculation of
$f_1(x,p_T^2)$ and $f^\perp(x,p_T^2)$ reveals the rapidity divergences
discussed in \cite{Collins:2008ht}. They have the form
\begin{equation}
\label{eikonal}
\frac{1}{(l-p)\cdot n} = \frac{1}{n^-}\,
\frac{1}{(l-p)^+ + \frac{n^+_{}}{n^-}\, (l-p)^-}
\qquad\qquad \text{with}~~~
(l-p)^- = \frac{\vec{p}_T^2}{2(l-p)^+}
\end{equation}
and come from the Wilson line in Fig.~\ref{fig:pdfs}b, or equivalently
from the gluon propagator in Fig.~\ref{fig:pdfs}a if one uses the gauge
$n\cdot A=0$ where the Wilson line is unity. In a parton density for a
fast right-moving proton, the loop variable $l^+$ is integrated down to
its lower kinematic limit $p^+$. To avoid a logarithmic divergence in
\eqref{eikonal} one must hence keep $n^+$ nonzero (complications arising
for spacelike $n$ are discussed in
\cite{Bacchetta:2008xw,Collins:2008ht}). The light-cone gauge $A^+=0$ or
a purely lightlike Wilson line cannot be used in this context. It should
be instructive to investigate how this affects formulations of QCD based
on light-cone gauge. Note that in the context of light-cone quantization,
configurations with $(l-p)^+ =0$ in Fig.~\ref{fig:pdfs}a correspond to
zero modes of the gluon field.
Keeping $n^+$ finite in \eqref{eikonal} cuts off the region of negative
gluon rapidities, where $(l-p)^+ \to 0$ and $(l-p)^- \to \infty$. This is
physically reasonable, since fast left-moving gluon modes should not be
included in the parton distribution of a fast right-moving hadron. The
dependence of unintegrated parton distributions on $n^+/n^-$ is described
by the Collins-Soper equation \cite{Collins:1981uk}, whose solution can be
written as the product of an $n^+/n^-$ independent initial condition and a
Sudakov factor.
Power laws analogous to those in \eqref{power-pdfs} are obtained for the
fragmentation functions $D_1(z,k_T^2)$, $D^\perp(z,k_T^2)$, and the
Collins function $H_1^\perp(z,k_T^2)$. Together with the perturbative
expression for $U(l_T^2)$ at high $l_T$ one can then determine the
behavior of the semi-inclusive structure functions for $M \ll q_T \ll Q$.
The terms going with $n^+/n^-$ in the distribution and fragmentation
functions give rise to a $\log (Q/q_T)$ in the structure functions, which
we also encountered in the high-$q_T$ calculation. The Collins-Soper
equation allows one to resum such logarithms to all orders and is at the
origin of the CSS formalism \cite{Collins:1984kg}, which plays a prominent
role in collider phenomenology. The need to keep $n^+/n^-$ finite in
\eqref{eikonal} is thus not a mere technicality but has practical
implications for physical observables.
\section{Comparing the low- and high-$q_T$ calculations}
\label{sec:compare}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|r|ccc|cc|} \hline\hline
& \multicolumn{3}{c|}{low-$q_T$ calculation} &
\multicolumn{2}{c|}{high-$q_T$ calculation} \\
& power & twist & contributing functions & power & twist \\
\hline
$F_T \sim$ & ${1}/{q_T^2}$ & 2 & {$f_1(x,p_T^2), D_1(z,k_T^2)$}
& ${1}/{q_T^2}$ & 2 \\
$F_L \sim$ & ${1}/{Q^2}$ & 4 & result unknown
& ${1}/{Q^2}$ & 2 \\
$F^{\cos 2\phi} \sim$ & ${M^2}/{q_T^4} \phantom{+}$ & 2
& {$h_1^\perp, H_1^\perp$} & ${1}/{Q^2}$ & 2 \\
& $+{1}/{Q^2}$ & 4 & result unknown & & \\
$F^{\cos\phi} \sim$ & ${1}/{(Q\mskip 1.5mu q_T)}$ & 3 & {$f_1, f^\perp,
D_1, D^\perp$} & ${1}/{(Q\mskip 1.5mu q_T)}$ & 2 \\
\hline\hline
\end{tabular}
\end{center}
\caption{\label{tab:power-results} Behavior of semi-inclusive structure
functions in the intermediate region $M\ll q_T \ll Q$.}
\end{table}
The behavior for $M\ll q_T \ll Q$ of the unpolarized structure functions
obtained in the low- and high-$q_T$ calculations is given in
Table~\ref{tab:power-results}. Explicit evaluation shows that for $F_T$
the results of the two calculations exactly coincide; this is an example
of the case where at intermediate $q_T$ the leading term in the expansions
\eqref{pow-low} and \eqref{pow-hi} goes with $l_{2,2} = h_{2,2}$. The
agreement between the two calculations can be seen at the level of
diagrams: roughly speaking, the graph of Fig.~\ref{fig:mechanisms}b
corresponds to the graph in Fig.~\ref{fig:low-qt}a if the gluon moves fast
in the direction of the hadron $h$, and to the graph in
Fig.~\ref{fig:low-qt}b if the gluon moves fast in the direction of the
target.
The leading term in the low-$q_T$ result for $F^{\cos 2\phi}$ involves the
Boer-Mulders and Collins functions. This structure function provides an
example for the case where $l_{2,2} = h_{2,2} =0$ and where the leading
contributions $l_{2,4}$ and $h_{4,2}$ in the two calculations are
different and can be added at intermediate $q_T$.
From power counting it is clear that the $1/Q^2$ behavior obtained in the
high-$q_T$ calculation for both $F^{\cos 2\phi}$ and $F_L$ corresponds to
twist-four contributions in the low-$q_T$ framework, whose computation is
well beyond the state of the art. As a consequence, one cannot invoke the
CSS method \cite{Collins:1984kg} to resum the logarithms in $Q/q_T$ that
appear in the high-$q_T$ result. Terms going like $1/Q^2$ are obtained if
one calculates $F_L$ and $F^{\cos 2\phi}$ in the parton model
\cite{Cahn:1978se}, considering the graph in Fig.~\ref{fig:mechanisms}a
with only the functions $f_1(x,p_T^2)$ and $D_1(z,k_T^2)$. However, the
results do not match with the high-$q_T$ calculation at intermediate $q_T$
and can hence only be regarded as partial evaluations of the complete
(unknown) twist-four terms.
If we perform the twist-three calculation for $F^{\cos\phi}$ at low $q_T$
using the tree-level result of \cite{Mulders:1995dh} supplemented with the
soft factor $U(l_T^2)$ of the twist-two factorization formula
\eqref{CS-fact}, we obtain agreement with the high-$q_T$ result for
intermediate $q_T$ \emph{except} for a missing term proportional to
$f_1(x)\, D_1(z)$. Such a term comes from kinematics where both the plus-
and the minus-momentum of the gluon in Fig.~\ref{fig:low-qt} is
negligible. This shows that, if a proper factorization formula for twist
three can be established, the soft-gluon sector will have to be treated
with particular care.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.43\textwidth]{low-gluon.eps}
\hspace{2.1em}
\includegraphics[width=0.40\textwidth]{low-wilson.eps} \\[-0.3em]
$\mathbf{(a)}$ \hspace{0.47\textwidth} $\mathbf{(b)}$
\end{center}
\caption{\label{fig:low-qt} Example graphs for the low-$q_T$ calculation
in the region $q_T\gg M$, where a factorized description as in
Fig.~\protect\ref{fig:pdfs} is valid for the fragmentation function
$\smash{\mathbf{(a)}}$ or the parton distribution $\smash{\mathbf{(b)}}$.}
\end{figure}
\section{Summary}
The descriptions of transverse-momentum spectra based either on intrinsic
transverse momentum of partons or on perturbative radiation are not
disconnected. They can be related in an intermediate region $M\ll q_T \ll
Q$ by describing transverse-momentum dependent distribution and
fragmentation functions themselves in terms of perturbative radiation, as
indicated in Fig.~\ref{fig:pdfs}. Understanding the connection between
the two approaches for a given observable enables one to devise
descriptions that may be used in the full region of $q_T$. As shown in
\cite{Bacchetta:2008xw}, the interplay of the two mechanisms has also
nontrivial consequences in observables that are integrated over $q_T$.
A varied picture arises already for the angular distribution of the
measured hadron in unpolarized semi-inclusive scattering, with the results
obtained at leading-power accuracy in the low- and high-$q_T$ calculations
coinciding for some observables but not for others. An even richer
phenomenology emerges if one includes polarization effects
\cite{Bacchetta:2008xw,Ji:2006br}.
The calculation of unintegrated parton densities or fragmentation
functions at perturbatively large transverse momenta leads to divergences
from gluonic zero modes if naively performed in light-cone gauge. Proper
regularization of these divergences physically ensures that left-moving
gluon modes are not included in distribution or fragmentation functions
for right-moving hadrons and provides a powerful method for resumming
logarithms of $Q/q_T$ into a Sudakov factor. It remains to be understood
how this physics can be treated within light-cone quantization.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,369 |
{"url":"http:\/\/statistics.ats.ucla.edu\/stat\/r\/dae\/t_test_power.htm","text":"### R Data Analysis Examples Power Analysis for One-sample t-test\n\n#### Examples\n\nExample 1. A company that manufactures light bulbs claims that a particular type of light bulb will last 850 hours on average with standard deviation of 50. A consumer protection group thinks that the manufacturer has overestimated the lifespan of their light bulbs by about 40 hours. How many light bulbs does the consumer protection group have to test in order to prove their point with reasonable confidence?\n\nExample 2. It has been estimated that the average height of American white male adults is 70 inches. It has also been postulated that there is a positive correlation between height and intelligence. If this is true, then the average height of a white male graduate students on campus should be greater than the average height of American white male adults in general. You want to test this theory out by random sampling a small group of white male graduate students. But you need to know how small the group can be or how few people that you need to measure such that you can still prove your point.\n\n#### Prelude to The Power Analysis\n\nFor the power analysis below, we are going to focus on Example 1 testing the average lifespan of a light bulb. Our first goal is to figure out the number of light bulbs that need to be tested.\u00a0 That is, we will determine the sample size for a given a significance level and power. Next, we will reverse the process and determine the power, given the sample size and the significance level.\n\nWe know so far that the manufacturer claims that the average lifespan of the light bulb is 850 with the standard deviation of 50, and the consumer protection group believes that the manufactory has overestimated by about 40 hours. So in terms of hypotheses, our null hypothesis is H0 = 850 and our alternative hypothesis is Ha= 810.\n\nThe significance level is the probability of a Type I error, that is the probability of rejecting H0 when it is actually true. We will set it at the .05 level. The power of the test against Ha is the probability of that the test rejects H0. We will set it at .90 level.\n\nWe are almost ready for our power analysis. But let's talk about the standard deviation a little bit. Intuitively, the number of light bulbs we need to test depends on the variability of the lifespan of these light bulbs. Take an extreme case where all the light bulbs have exactly the same lifespan. Then we just need to check a single light bulb to prove our point. Of course, this will never happen. On the other hand, suppose that some light bulbs last for 1000 hours and some only last 500 hours. We will have to select quite a few of light bulbs to cover all the ground. Therefore, the standard deviation for the distribution of the lifespan of the light bulbs will play an important role in determining the sample size.\n\n#### Power Analysis\n\nIn R, it is fairly straightforward to perform a power analysis for comparing means. For example, we can use R's pwr.t.test function for our calculation as shown below. First, we specify the two means, the mean for the null hypothesis and the mean for the alternative hypothesis divided by the standard deviation for the population. We also need to set the alpha level (.05 for our example), type equal to one.sample and alternative equal to two.sided (two-tail).\n\nlibrary(pwr)\n\npwr.t.test(d=(850-810)\/50,power=0.9,sig.level=0.05,type=\"one.sample\",alternative=\"two.sided\")\n\nOne-sample t test power calculation\n\nn = 18.44623\nd = 0.8\nsig.level = 0.05\npower = 0.9\nalternative = two.sided\n\nThe result tells us that we need a sample size at least 19 light bulbs to reject H0 under the alternative hypothesis Ha to have a power of 0.9.\n\nNext, suppose we have a sample of size 10, how much power do we have keeping all of the other numbers the same? We can use the same program, sampsi, to calculate it.\n\npwr.t.test(d=(850-810)\/50,n=10,sig.level=0.05,type=\"one.sample\",alternative=\"two.sided\")\n\nOne-sample t test power calculation\n\nn = 10\nd = 0.8\nsig.level = 0.05\npower = 0.6162328\nalternative = two.sided\n\nYou can see that the power is about .616 for a sample size of 10. What then is the power for sample size of 15?\n\npwr.t.test(d=(850-810)\/50,n=15,sig.level=0.05,type=\"one.sample\",alternative=\"two.sided\")\n\nOne-sample t test power calculation\n\nn = 15\nd = 0.8\nsig.level = 0.05\npower = 0.8213105\nalternative = two.sided\n\nSo now the power is about .82. You could also do it again to find out the power for a sample size of 20. You'll probably expect that the power will be greater.\n\npwr.t.test(d=(850-810)\/50,n=20,sig.level=0.05,type=\"one.sample\",alternative=\"two.sided\")\n\nOne-sample t test power calculation\n\nn = 20\nd = 0.8\nsig.level = 0.05\npower = 0.9238988\nalternative = two.sided\n\nWe can also expect that if we specified a lower power or the standard deviation is smaller, then the sample size should also be smaller. We can experiment with different values of power and standard deviation as shown below.\n\npwr.t.test(d=(850-810)\/50,power=0.8,sig.level=0.05,type=\"one.sample\",alternative=\"two.sided\")\n\nOne-sample t test power calculation\n\nn = 14.30278\nd = 0.8\nsig.level = 0.05\npower = 0.8\nalternative = two.sided\n\nIf the standard deviation is lower, then the sample size should also go down, as we discussed before.\n\npwr.t.test(d=(850-810)\/30,power=0.8,sig.level=0.05,type=\"one.sample\",alternative=\"two.sided\")\n\nOne-sample t test power calculation\n\nn = 6.581121\nd = 1.333333\nsig.level = 0.05\npower = 0.8\nalternative = two.sided\n\n#### Discussion\n\nThere is another technical assumption, the normality assumption. If the variable is not normally distributed, a small sample size usually will not have the power indicated in the results, because those results are calculated using the common method based on the normality assumption. It might not even be a good idea to do a t-test on such a small sample to begin with if the normality assumption is in question.\n\nHere is another technical point. What we really need to know is the difference between the two means, not the individual values. In fact, what really matters is the difference of the means over the standard deviation. We call this the effect size. For example, we would get the same power if we subtracted 800 from each mean, changing 850 to 50 and 810 to 10.\n\npwr.t.test(d=(50-10)\/50,power=0.9,sig.level=0.05,type=\"one.sample\",alternative=\"two.sided\")\n\nOne-sample t test power calculation\n\nn = 18.44623\nd = 0.8\nsig.level = 0.05\npower = 0.9\nalternative = two.sided\n\nIf we standardize our variable, we can calculate the means in terms of change in standard deviations.\n\npwr.t.test(d=(1-.2),power=0.9,sig.level=0.05,type=\"one.sample\",alternative=\"two.sided\")\n\nOne-sample t test power calculation\n\nn = 18.44623\nd = 0.8\nsig.level = 0.05\npower = 0.9\nalternative = two.sided\n\nIt is usually not an easy task to determine the \"true\" effect size. We make our best guess based upon the existing literature or a pilot study. A good estimate of the effect size is the key to a successful power analysis.\n\n\u2022 Related R Commands\n\u2022 pwr.t.test -- Sample size and power determination.\n\u2022 References\n\u2022 D. Moore and G. McCabe, Introduction to the Practice of Statistics, Third Edition, Section 6.4.\n\nThe content of this web site should not be construed as an endorsement of any particular web site, book, or software product by the University of California.","date":"2013-05-18 22:42:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6357442736625671, \"perplexity\": 447.4354548840168}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368696382917\/warc\/CC-MAIN-20130516092622-00061-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
from errno import EINTR
import os
import select
import signal
import sys
import threading
try:
import queue
except ImportError:
import Queue as queue
from psshlib.askpass_server import PasswordServer
from psshlib import psshutil
READ_SIZE = 1 << 16
class FatalError(RuntimeError):
"""A fatal error in the PSSH Manager."""
pass
class Manager(object):
"""Executes tasks concurrently.
Tasks are added with add_task() and executed in parallel with run().
Returns a list of the exit statuses of the processes.
Arguments:
limit: Maximum number of commands running at once.
timeout: Maximum allowed execution time in seconds.
"""
def __init__(self, opts):
self.limit = opts.par
self.timeout = opts.timeout
self.askpass = opts.askpass
self.outdir = opts.outdir
self.errdir = opts.errdir
self.iomap = make_iomap()
self.taskcount = 0
self.tasks = []
self.running = []
self.done = []
self.askpass_socket = None
def run(self):
"""Processes tasks previously added with add_task."""
try:
if self.outdir or self.errdir:
writer = Writer(self.outdir, self.errdir)
writer.start()
else:
writer = None
if self.askpass:
pass_server = PasswordServer()
pass_server.start(self.iomap, self.limit)
self.askpass_socket = pass_server.address
self.set_sigchld_handler()
try:
self.update_tasks(writer)
wait = None
while self.running or self.tasks:
# Opt for efficiency over subsecond timeout accuracy.
if wait is None or wait < 1:
wait = 1
self.iomap.poll(wait)
self.update_tasks(writer)
wait = self.check_timeout()
except KeyboardInterrupt:
# This exception handler tries to clean things up and prints
# out a nice status message for each interrupted host.
self.interrupted()
except KeyboardInterrupt:
# This exception handler doesn't print out any fancy status
# information--it just stops.
pass
if writer:
writer.signal_quit()
writer.join()
statuses = [task.exitstatus for task in self.done]
return statuses
def clear_sigchld_handler(self):
signal.signal(signal.SIGCHLD, signal.SIG_DFL)
def set_sigchld_handler(self):
# TODO: find out whether set_wakeup_fd still works if the default
# signal handler is used (I'm pretty sure it doesn't work if the
# signal is ignored).
signal.signal(signal.SIGCHLD, self.handle_sigchld)
# This should keep reads and writes from getting EINTR.
if hasattr(signal, 'siginterrupt'):
signal.siginterrupt(signal.SIGCHLD, False)
def handle_sigchld(self, number, frame):
"""Apparently we need a sigchld handler to make set_wakeup_fd work."""
# Write to the signal pipe (only for Python <2.5, where the
# set_wakeup_fd method doesn't exist).
if self.iomap.wakeup_writefd:
os.write(self.iomap.wakeup_writefd, '\0')
for task in self.running:
if task.proc:
task.proc.poll()
# Apparently some UNIX systems automatically reset the SIGCHLD
# handler to SIG_DFL. Reset it just in case.
self.set_sigchld_handler()
def add_task(self, task):
"""Adds a Task to be processed with run()."""
self.tasks.append(task)
def update_tasks(self, writer):
"""Reaps tasks and starts as many new ones as allowed."""
# Mask signals to work around a Python bug:
# http://bugs.python.org/issue1068268
# Since sigprocmask isn't in the stdlib, clear the SIGCHLD handler.
# Since signals are masked, reap_tasks needs to be called once for
# each loop.
keep_running = True
while keep_running:
self.clear_sigchld_handler()
self._start_tasks_once(writer)
self.set_sigchld_handler()
keep_running = self.reap_tasks()
def _start_tasks_once(self, writer):
"""Starts tasks once.
Due to http://bugs.python.org/issue1068268, signals must be masked
when this method is called.
"""
while 0 < len(self.tasks) and len(self.running) < self.limit:
task = self.tasks.pop(0)
self.running.append(task)
task.start(self.taskcount, self.iomap, writer, self.askpass_socket)
self.taskcount += 1
def reap_tasks(self):
"""Checks to see if any tasks have terminated.
After cleaning up, returns the number of tasks that finished.
"""
still_running = []
finished_count = 0
for task in self.running:
if task.running():
still_running.append(task)
else:
self.finished(task)
finished_count += 1
self.running = still_running
return finished_count
def check_timeout(self):
"""Kills timed-out processes and returns the lowest time left."""
if self.timeout <= 0:
return None
min_timeleft = None
for task in self.running:
timeleft = self.timeout - task.elapsed()
if timeleft <= 0:
task.timedout()
continue
if min_timeleft is None or timeleft < min_timeleft:
min_timeleft = timeleft
if min_timeleft is None:
return 0
else:
return max(0, min_timeleft)
def interrupted(self):
"""Cleans up after a keyboard interrupt."""
for task in self.running:
task.interrupted()
self.finished(task)
for task in self.tasks:
task.cancel()
self.finished(task)
def finished(self, task):
"""Marks a task as complete and reports its status to stdout."""
self.done.append(task)
n = len(self.done)
task.report(n)
class IOMap(object):
"""A manager for file descriptors and their associated handlers.
The poll method dispatches events to the appropriate handlers.
"""
def __init__(self):
self.readmap = {}
self.writemap = {}
# Setup the wakeup file descriptor to avoid hanging on lost signals.
wakeup_readfd, wakeup_writefd = os.pipe()
self.register_read(wakeup_readfd, self.wakeup_handler)
# TODO: remove test when we stop supporting Python <2.5
if hasattr(signal, 'set_wakeup_fd'):
signal.set_wakeup_fd(wakeup_writefd)
self.wakeup_writefd = None
else:
self.wakeup_writefd = wakeup_writefd
def register_read(self, fd, handler):
"""Registers an IO handler for a file descriptor for reading."""
self.readmap[fd] = handler
def register_write(self, fd, handler):
"""Registers an IO handler for a file descriptor for writing."""
self.writemap[fd] = handler
def unregister(self, fd):
"""Unregisters the given file descriptor."""
if fd in self.readmap:
del self.readmap[fd]
if fd in self.writemap:
del self.writemap[fd]
def poll(self, timeout=None):
"""Performs a poll and dispatches the resulting events."""
if not self.readmap and not self.writemap:
return
rlist = list(self.readmap)
wlist = list(self.writemap)
try:
rlist, wlist, _ = select.select(rlist, wlist, [], timeout)
except select.error:
_, e, _ = sys.exc_info()
errno = e.args[0]
if errno == EINTR:
return
else:
raise
for fd in rlist:
handler = self.readmap[fd]
handler(fd, self)
for fd in wlist:
handler = self.writemap[fd]
handler(fd, self)
def wakeup_handler(self, fd, iomap):
"""Handles read events on the signal wakeup pipe.
This ensures that SIGCHLD signals aren't lost.
"""
try:
os.read(fd, READ_SIZE)
except (OSError, IOError):
_, e, _ = sys.exc_info()
errno, message = e.args
if errno != EINTR:
sys.stderr.write('Fatal error reading from wakeup pipe: %s\n'
% message)
raise FatalError
class PollIOMap(IOMap):
"""A manager for file descriptors and their associated handlers.
The poll method dispatches events to the appropriate handlers.
Note that `select.poll` is not available on all operating systems.
"""
def __init__(self):
self._poller = select.poll()
super(PollIOMap, self).__init__()
def register_read(self, fd, handler):
"""Registers an IO handler for a file descriptor for reading."""
super(PollIOMap, self).register_read(fd, handler)
self._poller.register(fd, select.POLLIN)
def register_write(self, fd, handler):
"""Registers an IO handler for a file descriptor for writing."""
super(PollIOMap, self).register_write(fd, handler)
self._poller.register(fd, select.POLLOUT)
def unregister(self, fd):
"""Unregisters the given file descriptor."""
super(PollIOMap, self).unregister(fd)
self._poller.unregister(fd)
def poll(self, timeout=None):
"""Performs a poll and dispatches the resulting events."""
if not self.readmap and not self.writemap:
return
try:
event_list = self._poller.poll(timeout)
except select.error:
_, e, _ = sys.exc_info()
errno = e.args[0]
if errno == EINTR:
return
else:
raise
for fd, event in event_list:
if event & (select.POLLIN | select.POLLHUP):
handler = self.readmap[fd]
handler(fd, self)
if event & (select.POLLOUT | select.POLLERR):
handler = self.writemap[fd]
handler(fd, self)
def make_iomap():
"""Return a new IOMap or PollIOMap as appropriate.
Since `select.poll` is not implemented on all platforms, this ensures that
the most appropriate implementation is used.
"""
if hasattr(select, 'poll'):
return PollIOMap()
else:
return IOMap()
class Writer(threading.Thread):
"""Thread that writes to files by processing requests from a Queue.
Until AIO becomes widely available, it is impossible to make a nonblocking
write to an ordinary file. The Writer thread processes all writing to
ordinary files so that the main thread can work without blocking.
"""
OPEN = object()
EOF = object()
ABORT = object()
def __init__(self, outdir, errdir):
threading.Thread.__init__(self)
# A daemon thread automatically dies if the program is terminated.
self.setDaemon(True)
self.queue = queue.Queue()
self.outdir = outdir
self.errdir = errdir
self.host_counts = {}
self.files = {}
def run(self):
while True:
filename, data = self.queue.get()
if filename == self.ABORT:
return
if data == self.OPEN:
self.files[filename] = open(filename, 'wb', buffering=1)
psshutil.set_cloexec(self.files[filename])
else:
dest = self.files[filename]
if data == self.EOF:
dest.close()
else:
dest.write(data)
dest.flush()
def open_files(self, host):
"""Called from another thread to create files for stdout and stderr.
Returns a pair of filenames (outfile, errfile). These filenames are
used as handles for future operations. Either or both may be None if
outdir or errdir or not set.
"""
outfile = errfile = None
if self.outdir or self.errdir:
count = self.host_counts.get(host, 0)
self.host_counts[host] = count + 1
if count:
filename = "%s.%s" % (host, count)
else:
filename = host
if self.outdir:
outfile = os.path.join(self.outdir, filename)
self.queue.put((outfile, self.OPEN))
if self.errdir:
errfile = os.path.join(self.errdir, filename)
self.queue.put((errfile, self.OPEN))
return outfile, errfile
def write(self, filename, data):
"""Called from another thread to enqueue a write."""
self.queue.put((filename, data))
def close(self, filename):
"""Called from another thread to close the given file."""
self.queue.put((filename, self.EOF))
def signal_quit(self):
"""Called from another thread to request the Writer to quit."""
self.queue.put((self.ABORT, None))
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,085 |
GDI kan syfta på:
GDI – en fraktion i Command & Conquer: Tiberium PC-spelen, se Global Defense Initiative
GDI – en typ av direktinsprutning för bensinmotorer, se Gasoline direct injection
GDI – en statistisk term för skillnader mellan män och kvinnor inom länder, se Gender-related development index
GDI – ett programmeringsgränssnitt för datorgrafik i OS/2 och Microsoft Windows, se Graphics Device Interface | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,509 |
\section{Introduction}
$\eta$~Car is a
{ very} massive ($\sim100$ solar masses) star known for its
strong mass outflow eruptions, and is one of the most interesting
objects of our Galaxy (e.g., Davidson \& Humphrey 1997).
Radial velocity
variations of spectroscopic lines accumulated over the years
provide a strong evidence that $\eta$~Car is a binary system (e.g.,
Damineli et al. 2008a,b) where the primary star is a luminous blue
variable (LBV) star orbiting in a very eccentric binary ($e \sim
0.9$) with a companion star believed to be an O star of $\sim 30$
solar masses.
The orbital period is 5.53 years ($\sim$2023
days) (e.g., Damineli et al., 2008a): the system has been
monitored in the radio, mm, IR, optical and X-ray
bands for at least three cycles.
Both stars emit dense and high-velocity gaseous winds, and the
binary system is ideal to study the interaction of colliding winds
and to test theories of particle acceleration and radiation under
extreme conditions. The mass outflow rates and wind speeds of the
two stars inferred from the wealth of all available data are $
\dot{M_1} \simeq 2 \times 10^{-4} \, {\rm M_{\odot} \, yr^{-1}},
\dot{M_2} \simeq 2 \times 10^{-5} \, {\rm M_{\odot} \, yr^{-1}},
v_1 \simeq 600 \, {\rm km \, s^{-1}}, v_2 \simeq 3000 \, {\rm km
\, s^{-1}}$ \citep{pittard-2}.
{ Eta Car is then interesting among other colliding wind
binaries since the observable X-ray emission in the 2-10 keV band
is almost entirely produced by the shocked fast wind of the
secondary star, with little if any contribution from the slow
shocked wind of Eta Car itself.}
The system is known for
its variability and occasional erratic eruptions detected in the
IR and optical bands, as well as for its distinct asymmetric
pattern of { optical} line and X-ray emission during its
orbital period \citep{corcoran}.
$\eta$~Car has been repeatedly observed in the energy ranges 1-10
keV and 20-100 keV by different observatories. It is certainly the
only source showing a non-thermal X-ray spectrum
within a region
centered on $\eta$~Car with a 1 degree diameter
{ (the anomalous X-ray pulsar AXP~1E~1048.1-5937 is about 0.6 degrees away)}.
Whereas the 1-10 keV spectrum is dominated by a quasi-thermal
and variable component \citep{corcoran,corcoran-2,viotti}, the
hard X-ray observations show non-thermal emission that appears to
vary along the orbit \citep{viotti-2,leyder}.
The $\eta$~Car source was
detected { with high significance}
by BSAX-PDS and INTEGRAL-ISGRI
{
{ far from } periastron}.
INTEGRAL is capable of resolving field sources with a few
arcminute resolution { in the hard X-ray range. Although
INTEGRAL observed the system at different phase periods
(0.99-0.01, 0.16--0.19, 0.35-0.37),} $\eta$~Car was detected {
only} during { the phase interval} 0.16--0.19 with an
average 22-100 keV X-ray flux of $F = 1.1 \times 10^{-11} \rm \,
erg \, cm^{-2} \, s^{-1}$ \citep{leyder}.
The Carina region has been observed at gamma-ray energies above a
few MeV by the OSSE, COMPTEL and EGRET instruments on board of the
Compton Gamma-Ray Observatory (CGRO). An EGRET gamma-ray source
(3EG~J1048-5840) is catalogued at about 1 degree distance from
$\eta$~Car . However, no gamma-ray emission above 100 MeV has been
reported by CGRO from the $\eta$~Car region\footnote{ EGRET
observed the $\eta$~Car field several times (see, e.g., Table~2 of
Kaspi et al. 2000). Observations close to periastron were VP32
(1992 Jun 25 - Jul 02, phase 0.014) and VP630 (1997 Sep 22 - Oct
06, phase 0.963) \cite{hartman,kaspi}.}.
The gamma-ray astrophysics mission AGILE \citep{tavani-1} observed
several times the Carina region in the Galactic plane during its
early operational phases and { Cycle-1} observations. Here
we report the main results of the gamma-ray observations of the
$\eta$~Car region carried out by the AGILE satellite during the
period 2007 July - 2009 January, simultaneously in the energy
bands 30 MeV - 30 GeV and 18-60 keV. A { high-confidence}
gamma-ray source (1AGL~J1043-5931) was detected positionally consistent with
$\eta$~Car by integrating all data, as well as by considering
specific observation periods.
\section{AGILE 2007-2008 Observations}
The AGILE mission has been operating since 2007 April
\citep{tavani-1}.
The AGILE scientific instrument is very compact and is
characterized by two co-aligned imaging detectors operating in the
energy ranges 30 MeV - 30 GeV (GRID, Barbiellini et al. 2002,
Prest et al., 2003) and 18-60 keV (Super-AGILE, Feroci et al.
2007), as well as by an anticoincidence system \citep{perotti} and
a calorimeter \citep{labanti}. AGILE's performance is
characterized by large fields of view (2.5 and 1 sr for the
gamma-ray and hard X-ray bands, respectively), optimal angular
resolution and good sensitivity (see Tavani et al. 2008 for
details about the mission and main instrument performance). Flux
sensitivity for a typical 1-week observing period can reach the
level of several tens of $ 10^{-8} \rm \, ph \, cm^{-2} \, s^{-1}
$ above 100 MeV, and 10-20 mCrab in the 18-60 keV range depending
on off-axis angles and { pointing directions.}
\begin{table*}
\begin{center}
\small \caption{AGILE observations of the $\eta$~Car region}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
AGILE orbits & Date interval & MJD & N. days & $\eta$~Car & $\sqrt{TS}$ (**) & Counts
& Average flux(***) \\
& & & & phase (*) & &
& \\[0.5ex]
\hline
1146-1299 & 2007.07.13-2007.07.24 & 54294.5-54305.5 & 11 & 0.732& 4.1 & 105 $\pm$ 29 & 67 $\pm$ 18\\
1429-1567 & 2007.08.02-2007.08.12 & 54314.5-54324.5 & 10 & 0.741 & 3.3 & 80 $\pm$ 27 & 58 $\pm$ 20\\
1584-1708 & 2007.08.13-2007.08.22 & 54325.5-54334.5 & 9 & 0.747& 1.1 & $<$ 73 & $< 67 $ \\
3673-4009 & 2008.01.08-2008.02.01 & 54473.5-54497.5 & 24 & 0.824& 4.1 & 166 $\pm$ 45 & 55$\pm$ 14\\
4194-4418 & 2008.02.14-2008.03.01 & 54510.5-54526.5 & 16 & 0.840& 3.1 & 96 $\pm$ 33 & 43 $\pm$ 15 \\
6129-6480 & 2008.06.30-2008.07.25 & 54647.5-54672.5 & 25& 0.910& 5.6 & 214 $\pm$ 42 & 61 $\pm$ 12 \\
6778-7001 & 2008.08.15-2008.08.31 & 54693.5-54709.5 & 16& 0.930& 1.3 & $<$ 105 & $< 46$\\
7569-7664 & 2008.10.10-2008.10.17 & 54749.5-54756.5 & 7 & 0.956& 4.0 & 80 $\pm$ 23 & 99 $\pm$ 28 (****) \\
8899-8995 & 2009.01.12-2009.01.19 & 54843.5-54850.5 & 7 & 0.002 & 2.2 & $48 \pm 22$ & $< 94$ \\
\hline
\end{tabular}
\end{center}
\label{tab-1}
\vspace*{1.cm}
(*) Average orbital phase of $\eta$~Car calculated at the center of
the time interval.\\
\noindent { (**) Square root of the the Maximum Likelihood
Test Statistic (TS) representing the statistical significance of
the detection.}\\
\noindent (***) Gamma-ray flux of 1AGL~J1043-5931 above 100 MeV in units of
$10^{-8} \,\rm ph \, cm^{-2}$ s$^{-1}$ obtained by taking into
account the nearby source AGL~J1046-5832 in the multisource
likelihood analysis.
{ We also indicate 2-sigma upper limits in the same units}.\\
(****) { During this period the source reached the
gamma-ray flux above 100 MeV of $F = (270 \pm
65) \times 10^{-8} \rm \, ph \, cm^{-2} \, s^{-1} $.}
\end{table*}
The AGILE satellite repeatedly pointed at the Carina region
{ for a total of $\sim130$ days } during the time period 2007
July - 2009 January. Table~1 summarizes the AGILE observations of
the field.
Analysis of the
gamma-ray data was carried out with the $FT3ab_2$
calibrated filter, with a gamma-ray event selection that takes
into account standard SAA event cuts and 80 degree Earth albedo
filtering. We used the AGILE gamma-ray software release Build 17
and { the standard} hard X-ray analysis software.
\begin{figure}
\begin{center}
\includegraphics [height=6.5cm]{fig1.eps}
\caption{ AGILE gamma-ray intensity map in Galactic coordinates
of the $\eta$ Car region above 100 MeV summing all data collected
from 2007 July to 2008 October. The central gamma-ray source that
can be associated with $\eta$ Car is 1AGL~J1043-5931; we also indicate the
prominent nearby gamma-ray source AGL~J1046-5832 which is associated with
the radio pulsar PSR~B1046-58 (Kaspi et al. 2006; Abdo et al.
2009). The color bar scale is in units of $\rm photons \, cm^{-2}
\, s^{-1} \, pixel^{-1}$. Pixel size
is 0.1 degrees, and we used a 3-bin Gaussian smoothing. White
contour levels of the AGILE sources start from 0.0005 { and
increase} in steps of 0.000028. The optical position of $\eta$~Car is
marked by a small black circle. The INTEGRAL sources (Leyder et
al. 2008) are marked with cyan circles. } \label{etacar-fig-3}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics [width= 8.5cm, height=6.2cm]{fig2.eps}
\caption {Time sequence of AGILE-GRID gamma-ray counts maps above
100 MeV centered on the $\eta$~Car region during the period 2008
10-17 October. The time sequence starts at the upper right corner
and it is counterclockwise. Each map corresponds approximately to
a 2-day integration starting on 2008 Oct. 10. { The color bar
scale is in units of $\rm counts \, cm^{-2} \, s^{-1} \,
pixel^{-1}$. Pixel size
is 0.3 degrees for a 3-bin Gaussian
smoothing.} A strong gamma-ray {
flare}
shows up in the second map for the period 2008
Oct. 11 (02:57 UT) - 2008 Oct. 13 (04:16 UT). The green contour
marks the 95\% contour level error box. The position of the
gamma-ray
{ flaring source} is consistent with the 1AGL~J1043-5931 position and with
$\eta$~Car (marked by a black small circle). } \label{etacar-fig-4}
\end{center}
\end{figure}
Fig.~\ref{etacar-fig-3} shows the integrated sky map of the
$\eta$~Car region at energies above 100 MeV
for the
period 2007 July - 2008 October. A gamma-ray source is
detected with high confidence
(7.8 sigma) at the position
$(l,b) = 287.6, - 0.7 \pm 0.3 \rm (stat.) \pm 0.1 (syst.)$. The
average gamma-ray flux above 100 MeV and integrated over the whole
period 2007 July - 2008 October is $F_{\gamma} = (37 \pm 5) \times
10^{-8} \rm \, ph \, cm^{-2} \, s^{-1} $. We call this
source\footnote{ The same source is also listed in the Fermi
Bright Source Catalogues as 0FGL~J1045.6-5937 (Abdo et al.
2009).} 1AGL~J1043-5931
{ following the source designation of } the first AGILE
catalog of high-confidence gamma-ray sources (Pittori et al.
2009). We included in all our multisource analysis the nearby
gamma-ray source\footnote{ This source, which
{ did not reach the stringent} significance threshold of the
First AGILE Catalog \citep{pittori}, { is included in the
Fermi Bright Source List as 0FGL~J1047.6-5834 and is identified
with PSR J1048-5832 (Abdo et al. 2009)}.}
AGL~J1046-5832 that the AGILE-GRID detects with an average and
constant gamma-ray flux above 100 MeV of
$f_{\gamma} =
(27 \pm 4) \times 10^{-8} \rm \, ph \, cm^{-2} \, s^{-1} $.
$\eta$~Car is well within the 95\% confidence radius gamma-ray error
box of 1AGL~J1043-5931; the other nearby hard X-ray sources in the field
(the anomalous X-ray pulsar AXP~1E~1048.1-5937, and
IGR~J10447-6027) are excluded. The Super-AGILE hard X-ray imager
did not detect a source coincident with 1AGL~J1043-5931 for both short and
long integrations of Table~1. { Depending on the source
position in the FOV, the typical { 3-sigma} Super-AGILE upper
limit is 10-20 mCrab, i.e., consistent with the INTEGRAL detection
{ and upper limits} of $\eta$~Car.}
We searched for short timescale variability of the
gamma-ray and hard X-ray flux from 1AGL~J1043-5931 throughout the whole
AGILE observing periods of the Carina region. A 2-day {
gamma-ray} { flare} from the direction of $\eta$~Car was detected
during the observation
of 2008 October 10-17. The
emission reached its peak gamma-ray emission during the period
2008 Oct. 11 (02:57 UT) - 2008 Oct. 13 (04:16 UT).
Our analysis gives a 5.2 sigma detection of a source at the
position $(l,b) = 288.0, -0.4 \pm 0.6$ { fully} consistent
with the 1AGL~J1043-5931 position
{ but }
with a gamma-ray flux above 100 MeV of $F = (270 \pm
65) \times 10^{-8} \rm \, ph \, cm^{-2} \, s^{-1} $.
Fig.~\ref{etacar-fig-4} shows the time sequence of 2-day integration
gamma-ray maps of the region during the period 2008 10-17 October.
Fig.~\ref{etacar-fig-5} shows the AGILE gamma-ray data of Table~1
superimposed with the simultaneous RXTE (PCU2 net rate)
lightcurve in the energy band 2-15 keV during the period 2007
February - { 2009 January}
(the typical and relatively abrupt
decrease of the X-ray emission near periastron is clearly
visible).
\begin{figure}
\begin{center}
\includegraphics [width= 9.cm, height=6.5cm]{fig3.eps}
\caption {AGILE gamma-ray lightcurve of 1AGL~J1043-5931 showing the fluxes
above 100 MeV (right axis scale in units of $10^{-8} \rm ph \,
cm^{-2} \, s^{-1}$) averaged over the observing periods of Table~1
(red crosses) and superimposed with the RXTE PCU2 net rate X-ray
light curve of $\eta$~Car (black symbols, left axis) obtained during
the dedicated campaign observing the last cycle and periastron
passage. { Triangles indicate 2-sigma upper limits.} { We
also mark the occurrence of the 2008 October 11-13 flare when the
source reached a gamma ray flux above 100 MeV of $F = (270 \pm 65)
\times 10^{-8} \rm \, ph \, cm^{-2} \, s^{-1}$.}}
\label{etacar-fig-5}
\end{center}
\vspace*{0.5cm}
\end{figure}
Fig.~\ref{etacar-fig-spectrum} shows two representative different
{ broad-band} spectral states of 1AGL~J1043-5931 { obtained} during
the period 2007 July -- 2008 October together with the historical
X-ray and hard X-ray data reported from $\eta$~Car. We mark in blue
the average spectrum obtained by integrating all data {
outside periastron} , and with a red cross the flux corresponding
to the flaring state of 2008 October 11-13. We also report in the
same plot the { (non simultaneous)} BSAX-MECS and the
INTEGRAL-ISGRI spectral states of $\eta$~Car reported in the
literature \citep{viotti-2,leyder}. It is interesting to note that
if 1AGL~J1043-5931 is associated with $\eta$~Car , the average AGILE spectrum
together with the INTEGRAL historical spectrum is in qualitative
agreement with expectations based on inverse Compton and/or pion
decay models of gamma-ray emission from colliding wind binaries
(e.g., Reimer, Pohl, Reimer 2006). For the 2008 11-13 October
flaring episode, the Super-AGILE 18-60 keV upper limit is 70 mCrab
in the energy band 18-60 keV.
{ Obtaining simultaneous hard X-ray and gamma-ray data during
the flaring state of 1AGL~J1043-5931 is crucial to study the broad-band
variability of the source. However, due to unfavorable source
positioning of 1AGL~J1043-5931 in the Super-A field of view in mid-October,
2008, the hard X-ray upper limit is not very constraining. }
\begin{figure}[h!]
\begin{center}
\includegraphics[width= 9.5cm, height=6.5cm]{fig4.eps}
\caption {Combined spectral power { flux} of $\eta$~Car as reported by the
SAX-MECS { in the energy range 1-10 keV (phase 0.46)} \citep{viotti-2}, { by }
INTEGRAL { in the energy range 22-100 keV (phase 0.16-0.19) } (Leyder et al.
2008) plotted together with the two { broad-band} { gamma-ray} spectral states of
1AGL~J1043-5931 measured by AGILE during the period 2007 July -- 2008 October { ($\eta$~Car
phase 0.73-0.95)}. The lower blue point marks the average {
gamma-ray } spectral flux, and the upper red point indicates the
spectral state during the
gamma-ray { flare } of 2008 October 11-13. }
\label{etacar-fig-spectrum}
\end{center}
\end{figure}
AGILE pointed at the Carina Region during the period 2009 12-19
January as a special repointing to
{ cover} the $\eta$~Car periastron passage (calculated to be
occurring on 2009 January 11). { We cannot exclude, at this
stage, the existence of a weak gamma-ray source consistent with
1AGL~J1043-5931. However,
we
can currently provide a 2-sigma upper limit to the emission above
100 MeV of $ 94\times 10^{-8} \rm \, ph \, cm^{-2} \, s^{-1} $.}
A more detailed analysis of the AGILE and multifrequency data
during the $\eta$~Car
periastron passage will be discussed elsewhere.
\section{Discussion }
$\eta$~Car is { located} in the Carina nebula that extends for
several degrees in a Galactic region characterized by dense
molecular clouds, young stars and star formation sites.
However, within the 1AGL~J1043-5931 error box $\eta$~Car itself is by far the
strongest and hardest source in the 2-10 keV and 22-100 keV
ranges. { Another source inside the AGILE error box and } 7
arcmin away from
$\eta$~Car is the
X--ray
binary HD 93162/WR 25 (WN6+O4, with an orbital period of 208 days,
Gamen et al. 2007). This system { is known} to be a
colliding wind system. Pollock \& Corcoran (2006) found
significant variability, possibly periodic, of its X--ray flux.
However, WR 25 { was} not detected in the hard X-ray range by
INTEGRAL during any of its observations. Furthermore, no other
prominent hard X-ray source is known in the 1AGL~J1043-5931 error box except
for $\eta$~Car itself. The two nearby hard X-ray sources detected by
INTEGRAL \citep{leyder} are outside the 95\% confidence level
error box { of AGILE. Multiple gamma-ray sources within the
1AGL~J1043-5931 error box cannot be excluded but are unlikely.}
Based on the accumulated multifrequency evidence
and the nature of the source, we consider the association of 1AGL~J1043-5931
and $\eta$~Car as very likely. We briefly elaborate below on the
theoretical implications of our results assuming that 1AGL~J1043-5931 is
indeed the gamma-ray counterpart of $\eta$~Car.
Colliding wind binaries (CWBs) are ideal systems to test theories
of hydrodynamical shocks and particle acceleration under extreme
radiative conditions provided by the proximity of the two stars.
In particular, supersonic winds can form efficient shocks where
electrons and protons can be accelerated through first-order Fermi
(Eichler \& Usov 1993) or other acceleration mechanisms. Inverse
Compton { (IC)} scattering of shock-accelerated particles in
the presence of the very intense IR-optical-UV background of the
nearby
{ very bright } stars provides a crucial ingredient in CWBs. In
addition to synchrotron and Bremsstrahlung electron emissions, the
IC emission can dominate the high energy spectrum at energies
larger than several tens of keV up to MeV-GeV energies.
Furthermore, if protons are efficiently accelerated, they can
interact with the dense stellar outflows and produce gamma-rays by
pion production and neutral pion decay (e.g., Eichler \& Usov
1993, Benaglia \& Romero 2003; Reimer et al. 2006). All these
ingredients are important for the $\eta$~Car system and detailed
hydrodynamical modelling of the mass outflow have been developed
\citep{parkin,parkin-2,okazaki}. A comprehensive and detailed
theoretical analysis of our data is beyond the scope of this
paper. We outline here a few important points.
If 1AGL~J1043-5931 is the $\eta$~Car gamma-ray counterpart, our data show the
first
remarkable detection of a colliding wind system at hundreds of MeV
energies, confirming the efficient particle acceleration and the
highly non-thermal nature of the strong shock in a CWB. The
average gamma-ray flux of 1AGL~J1043-5931
{ translates into } gamma-ray luminosity of $L_{\gamma} =
3.4 \times 10^{34} \rm \, erg \, s^{-1}$ for an $\eta$~Car distance
of
2.3 kpc,
{ corresponding } to a fraction of a percent of the total
wind kinetic power. The 2008 Oct. 11-13 flare episode has a
luminosity of $L_{\gamma} = 2. \times 10^{35} \rm \, erg \,
s^{-1}$. The average { broad-band} gamma-ray spectrum
determined by AGILE is in qualitative agreement with expectations
of CWB spectra as calculated for dominant IC and neutral pion
decay processes \citep{benaglia,reimer}.
{ While the gamma-ray flux of 1AGL~J1043-5931 is roughly constant
during the time span covered by our observations, a significant
variability was detected on a few day time-scale in October 2008.
This episode
indicates that}
the gamma-ray emission can be { associated with}
intermittent strong shock acceleration episodes and/or magnetic
field enhancements to be expected for a very variable and
inhomogeneous mass outflow from the stars of the $\eta$~Car system.
In particular, we note that the strong gamma-ray flaring episode
occurred a few months before periastron, when the efficiency of
transforming a mass outflow enhancement into
{ particle acceleration is expected to increase
because of} the closeness of the two stars. $\eta$~Car provides then
some crucial ingredients regarding the formation of high-energy
emission in CWBs: (1) strong variability of the mass outflows; (2)
a high-speed wind from the less massive companion; (3) a radiative
environment with a specific bath of soft photons from both stars
(IR, optical and UV fluxes) that can illuminate the shock region
and provide a time variable environment for enhanced IC emission
in the 100 MeV range and beyond. The theoretical implications are
far reaching. The $\eta$~Car system would provide the first CWB to
test the particle acceleration models for non-relativistic mass
outflows under a specific set of physical conditions. It is very
important to assess the efficiency of the particle acceleration
process in such a radiative environment. A gamma-ray flaring
episode lasting $\sim2$ days implies a fast acceleration timescale
and subsequent radiation and decay of the strong shock properties
leading to the efficient emission.
If the gamma-ray emission is associated with $\eta$~Car our
observations provide important data to test shock acceleration
models. { Future gamma-ray observations and analysis will
further contribute to enlighten the emission mechanism and the
ultimate origin of 1AGL~J1043-5931.}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,587 |
Q: Атрибут class. Максимальное значение имен Атрибут class="" может иметь несколько стилевых значений. А какое максимальное кол-во их может быть для тега?
A: В HTML5 нет никакого ограничения в количестве классов для тега. Вы ограничены только памятью браузера.
Вот интересная статья о том, какие ограничения есть у различных браузеров. Если коротко, то у автора не получилось дойти до какого-то "максимума" из-за скорости выполнения скрипта. Скрипт в некоторых браузерах дошёл до 4000.
В HTML4 есть ограничение на длину атрибута. Но, в действительности, до этого ограничения дойти тоже довольно сложно.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 621 |
The 6569 VIC-II chip is the graphics chip that made the C64 famous, supporting different graphics modes at different resolutions as well as sprites in a way no other machine at the time could.
Blank screen with no sync (monitor will say "No Picture").
Black screen (also a symptom of many other things). | {
"redpajama_set_name": "RedPajamaC4"
} | 4,694 |
\section{Introduction}
In frustrated magnetic systems, the competition among interactions
leads to a rich and sometimes tuneable array of physics, which can be
quite subtle and sensitive to perturbations\cite{Lacroix2011}. The
large and degenerate Hilbert space created in such systems lends
itself to a multitude of magnetic ground states and low-lying excited
states. Generally, frustration can be introduced {\it via} competing
interactions, such as a balance of direct exchange, superexchange, and
Dzyaloshinskii-Moriya interactions, or geometrically, by arranging the
magnetic ions in a lattice that prevents a pairwise
(anti)ferromagnetic ground state from being simultaneously satisfied
within all nearest-neighbour pairs of magnetic atoms. Several such
lattices such as pyrochlore, kagome, and triangular are well-known.
When new variants of established magnetic lattices are reported, {\it
e.g.} the hyperhoneycomb or tripod-kagome
lattices\cite{Takayama2015,Dun2016}, they offer exciting new
playgrounds for the investigation of frustration.
Chiral magnetism in particular offers the possibilities of highly
nontrivial long-range magnetic order and topological excitations such
as skyrmions\cite{Skyrmions}. While competing interactions or weak
effects such as the Dzyaloshinskii-Moriya
interaction\cite{Dz1958,Moriya1960} can lead to chiral magnetic
structures even in the case of an achiral crystal lattice, it may be
more natural to expect chiral magnetism to arise if the underlying
lattice is itself chiral. The magnetic order may be expected to
follow the symmetry of the lattice, even in the absence of the
delicate balance of interactions otherwise required, and in many cases
all irreducible representations for the magnetic order will be chiral,
{\slshape constraining} the magnetic order to be chiral by symmetry.
Here we highlight one magnetic material with a novel frustrated
lattice in just such a chiral crystal structure.
When prepared by standard high-temperature solid-state synthesis,
Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ forms in a distorted, chiral variant of the trigonal Weberite
structure, as shown in
Fig.~\ref{fig:P3121}\cite{Scott1987,Scott1990,Chelazzi2013}. The
material's Mn network is composed of slightly distorted kagome layers
in the $ab$ plane, with the kagome triangles linked along the $c$ axis
through additional Mn triangle units, such that the simplest
structural motif to consider is an Mn$_4$ armchair unit. Here, we
neglect the Sb sublattice, which is known to be nonmagnetic Sb$^{5+}$
from $^{121}$Sb M\"ossbauer
spectroscopy\cite{Subramanian1984}\footnote{Ref.~\onlinecite{Subramanian1984}
describes the crystal structure as rhombohedrally-distorted
pyrochlore --- the correct space group had not yet been identified.
The samples were prepared by a high-temperature route that produces
the trigonal Weberite structure investigated
here\cite{Reimers1991}.}. The shortest Mn--Mn bonds in our
refinement trace out a helix along the $c$ axis, highlighted in light
blue. In the case of a pyrochlore lattice, the atom forming the
vertical link would lie directly over the centre of a kagome-layer
triangle instead of off to one side, forming a tetrahedron and leading
to a highly-symmetric structure in which the faces of the tetrahedra
comprise interpenetrating kagome networks. Armchair-kagome Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ is
lower-dimensional, but the interactions both along the $c$ axis and
in-plane should exhibit geometric frustration due to the triangular
arrangements of Mn atoms. It would be possible, in principle, to
deform the Mn sublattice in Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ into a pyrochlore network through a
series of slips perpendicular to the $c$-axis, but each slip would
occur along an entire plane the size of the crystal, and six slips
would be required per unit cell, making such a reconstruction
enormously energetically unfavorable. This would be a topological
transition, since there is one more Mn--Mn link in the pyrochlore's
{\slshape T}$_4$ tetrahedron than in an Mn$_4$
armchair\cite{Kane2013}. The pyrochlore polymorph of Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}, pyr-Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}, also
exists but can only be prepared by a more-difficult low-temperature
route\cite{Brisse1972,Zhou2008,Zhou2010,Peets-pyr}. In this paper,
`Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}' refers to the armchair-kagome polymorph unless otherwise stated.
\begin{figure}[thb]
\includegraphics[width=\columnwidth]{P3121_600.pdf}
\caption{\label{fig:P3121}Manganese sublattice in the \ensuremath{P3_121}\ structure
of Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ at 600$^\circ$C, highlighting the `armchair-kagome' network,
based on the refinement in Fig.~\ref{fig:600} and
Tab.~\ref{tab:600C}. (a) Network of armchair units at $z=0$; one
pair of adjacent armchairs is highlighted in blue. (b) The layer
stacking leads to a 3-fold screw axis. The stacking sequence of
armchair units around the rear corner of the \ensuremath{P3_121}\ unit cell is shown
in blue, with the shortest bonds in lighter blue. The unit cell is
shown using solid lines.}
\end{figure}
In terms of physical properties, little has been reported on
Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ to date. It has a paramagnetic Curie-Weiss temperature
\ensuremath{T_\text{CW}}\ around $-45$ to $-50$\,K with a paramagnetic moment
corresponding to high-spin $3d^5$ Mn$^{2+}$,\cite{Reimers1991} and it
undergoes a bulk magnetic transition around 13\,K. The clear history
dependence reported below about 55\,K was attributed to an apparently
abrupt onset of short-range correlations. The magnetic structure has
not been solved, but most magnetic Bragg peaks could be explained by a
[$\frac{1}{2}$00] propagation vector. $^{121}$Sb M\"ossbauer spectra
have been reported\cite{Subramanian1984}, but no splitting of the
sites was observed, nor any Sb$^{3+}$ component, nor hyperfine
splitting in the magnetically-ordered phase. The material's
isothermal bulk modulus is also known\cite{Chelazzi2013}, but the
authors are aware of no other measurements.
We set out to clarify the crystal structure and the nature of the
magnetic order in Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}, and to check for hints of exotic behavior
arising from its unique frustrated chiral lattice. The material forms
in an even lower-symmetry structure than that previously reported, and
the magnetic order is frustrated, chiral, and multiferroic.
Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ first enters a magnetically-ordered state around 14\,K in which
the magnetization increases rapidly on cooling, followed by a second
magnetic transition below which the magnetization saturates.
\section{Experimental}
Powder samples of Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ were prepared in Al$_2$O$_3$ crucibles in air,
from intimately mixed MnO$_2$ (Alfa Aesar, 99.997\%) and Sb$_2$O$_3$
(Alfa Aesar, 99.999\%). Mixed powders were calcined with intermediate
grindings at temperatures between 1050 and 1150$^\circ$C, typically
for 24\,h per temperature; the mass was monitored for loss of volatile
component oxides, and powder diffraction patterns were used to verify
phase purity. X-ray powder patterns reported here were collected at
temperatures from 30 to 1000$^\circ$C using a Bruker D8 Discover
diffractometer with a Cu$K\alpha$ source. When higher-density samples
were desirable, most notably for specific heat and dielectric constant
measurements, this powder was pressed isostatically into rods and
sintered at 1175$^\circ$C for a further 24\,h to produce ceramic.
Magnetization measurements in fields up to 5\,T were performed in a
Quantum Design MPMS-XL SQUID magnetometer in its RSO measurement mode.
A powder sample of approximately 15\,mg was packed inside a gelatin
capsule, which was closed with Kapton tape and loaded into a plastic
straw. The contribution from the empty sample holder was below the
level of the noise on the Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ data. AC susceptibility measurements
were performed in the same magnetometer using the DC sample transport,
at zero applied field. Magnetization $M(H)$ data in fields up to
14\,T were measured using a Quantum Design PPMS with the vibrating
sample magnetometry option. Specific heat measurements were performed
by the relaxation time method in fields up to 9\,T in a Quantum Design
PPMS, with 2$\tau$ fitting and measurement times of 2$\tau$. Sintered
ceramic samples were attached to the sample stage using Apiezon N
Grease for low-temperature measurements, while Apiezon H Grease was
used above $\sim210$\,K to avoid artifacts from the N Grease glass
transition. The addenda contribution was subtracted within the
software.
Dielectric measurements were performed at 1\,kHz using an
Andeen-Hagerling AH2550A capacitance bridge. A $3\times3$\,mm$^2$,
2\,mm-thick slab of Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ ceramic was affixed to a gold backing plate
with silver epoxy; a copper wire connected to silver epoxy spread over
the opposite surface completed a capacitor with the Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ ceramic as
its dielectric. This assembly was mounted onto a closed-cycle
refrigerator, cooled to $\sim$5\,K, then measured on warming back to
room temperature while the CCR was turned off to reduce noise.
X-ray diffraction found the powder samples to be phase pure for up to
$\sim3$\,\% initial cation nonstoichiometry, after two or more
calcines above 1050$^\circ$C. Magnetization measurements, however,
proved quite sensitive to the trace presence of Mn$_3$O$_4$, which has
a broad ferrimagnetic transition below
50\,K\cite{Dwight1960,Jensen1974}. This high sensitivity enabled
better optimization of the synthesis conditions than was possible with
x-ray diffraction alone. Ultimately, a $\sim$2\,\% antimony excess
was used. Samples on which magnetization is presented were
additionally washed in citric acid to ensure the complete absence of
Mn$_3$O$_4$, although this was later found to be unnecessary.
Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ was also found to be stable in dilute nitric and hydrochloric
acids.
A high-resolution synchrotron x-ray powder diffraction pattern was
collected at room temperature on beamline 9B (HRPD) at the Pohang
Accelerator Laboratory, in Pohang, Korea. A specimen of approximately
0.2\,g was prepared by a flat plate side loading method to avoid
preferred orientations, and the sample was rotated about the surface
normal during the measurement to increase sampling statistics. Data
were collected from 10 to 130.5$^\circ$ in steps of 0.005$^\circ$,
using a wavelength of 1.4970(1)\,\AA, and they were normalized to the
incoming beam intensity and corrected for asymmetric diffraction.
For higher sensitivity to oxygen atoms and to access the magnetic
structure, powder neutron diffraction was performed at a variety of
temperatures at the ECHIDNA diffractometer at the OPAL research
reactor at ANSTO, Australia, from 6.5 to 163.95$^\circ$ in steps of
0.05$^\circ$, with a neutron wavelength of 2.4395\,\AA.
Low-temperature measurements were performed on loose powder sealed in
a vanadium can, while for temperatures above room temperature, the
sample was a sintered rod suspended in vacuum. Additional powder
neutron diffraction patterns were collected at several temperatures at
the high-resolution powder diffractometer (HRPD) at the HANARO
research reactor in Daejeon, Korea, from 0 to 159.95$^\circ$ in steps
of 0.05$^\circ$, with a neutron wavelength of 1.8343\,\AA. To examine
the crystal structure with the highest possible resolution,
time-of-flight powder neutron diffraction data were also collected at
room temperature on the High Resolution Powder Diffractometer (HRPD)
at the ISIS spallation neutron source, Rutherford Appleton Laboratory,
UK. Data collected on the backscattering (168.33$^\circ$) and
90$^\circ$ detector banks were used. These data were corrected for
self-shielding and wavelength-dependent absorption for a sample with a
number density of $2.4\times10^{-3}$\,\AA$^{-3}$, scattering cross
section 41.72\,barns, and absorption cross section 36.42\,barns at a
wavelength of 1.798\,\AA. Powder diffraction data were
Rietveld-refined in FullProf by the least-squares
method\cite{FullProf}.
\section{Crystal Structure}
\begin{figure}[htb]
\includegraphics[width=\columnwidth]{diffraction_v2.pdf}
\caption{\label{fig:diffract}Powder diffraction patterns of Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ at
300\,K by (a) synchrotron x-ray diffraction, (b) neutron
diffraction, and (c) time-of-flight neutron diffraction. Data are
in red, the result of refinement within the \ensuremath{P3_121}\ structure is
shown in black, the residual is in blue, and green vertical bars
mark nuclear Bragg positions. The insets highlight illustrative
peaks that cannot be well reproduced within \ensuremath{P3_121}. The residuals have
been shifted vertically for clarity.}
\end{figure}
The crystal structure of Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ has been reported in the \ensuremath{P3_121}\ space
group (\#\,152)\cite{Scott1987,Scott1990,Chelazzi2013}, but there is
also a parenthetical remark in the literature that neutron diffraction
could only be explained satisfactorily using a doubled unit cell in
the $P2$ space group (\#\,3)\cite{Reimers1991}. Our 300\,K
synchrotron and neutron powder diffraction patterns, shown in
Fig.~\ref{fig:diffract}, indeed show additional features that cannot
be explained within the published \ensuremath{P3_121}\ structure. The inset to
Fig.~\ref{fig:diffract}(a) shows a set of peaks that demonstrate this
issue in the synchrotron data --- the middle peak is clearly a
doublet, but it corresponds to only a single reflection in \ensuremath{P3_121}, while
the middle and higher peaks are shifted in opposite directions in a
way that cannot be modelled by any unit cell or geometric adjustments
within \ensuremath{P3_121}. The 300\,K neutron diffraction pattern in
Fig.~\ref{fig:diffract}(b) and the time-of-flight data in
Fig.~\ref{fig:diffract}(c) contain several extra peaks (also see
Fig.~\ref{fig:highTP2}(f)), several of which are shown in the insets.
In addition, modeling of peak intensities is poor in all patterns,
and this is not resolvable through shifts in atomic positions.
Profile matching in all subgroups of \ensuremath{P3_121}\ and various supercells
confirmed that the $P2$ space group with a unit cell doubled in-plane
was required to explain the observed peak positions. All peaks could
be explained within this $P2$ supercell, and impurity phases could not
explain the additional peaks. In an independent check, varying the
stoichiometry of the starting materials and the calcining conditions
had no effect on the putative $P2$ reflections, but a several-percent
nonstoichiometry was sufficient to introduce clear impurity peaks at
other angles. Attempts to refine the $P2$ structure were not
successful --- the presumed $P2$ unit cell would have 70 unique atomic
sites and almost no symmetry operations to constrain them, so a
successful structure refinement will almost certainly require single
crystal diffraction data. Based on a refinement of the neutron
time-of-flight and synchrotron data with all sites locked to their
ideal \ensuremath{P3_121}\ positions, the $P2$ cell would have lattice parameters
$a=12.46137(45)$\,\AA, $b=7.19304(23)$\AA, $c=17.40822(30)$\AA, and
$\beta=89.9108(20)^\circ$ (neutron), or $a=12.45226(15)$\,\AA,
$b=7.18791(8)$\AA, $c=17.40312(12)$\AA, and $\beta=89.92825(83)^\circ$
(synchrotron) at room temperature.
\begin{figure}[htb]
\includegraphics[width=\columnwidth]{highT_P2_label.pdf}
\caption{\label{fig:highTP2}Transition to \ensuremath{P3_121}\ at high temperature.
In the laboratory x-ray data in panels (a) and (b), peaks
attributable to the $P2$ structure weaken above room temperature,
becoming indistinct above 350$^\circ$C (data sets have been
shifted vertically for clarity). The dual traces at 30$^\circ$C
were taken first and last, as an internal check. The (c) $a$-axis
and (d) $c$-axis lattice parameters show evidence for a transition
around 450$^\circ$C (marked with a vertical line), but there is no
clear effect on the (e) unit cell volume. (f) Selected neutron
diffraction data: several clear $P2$-derived peaks vanish above
the structural transition --- two are highlighted in the inset.}
\end{figure}
Since the symmetry reduction to $P2$ was clearer in neutron than in
synchrotron data, the atomic displacements from high-symmetry
positions are likely strongest for oxygen, and one may expect the
structure to return to \ensuremath{P3_121}\ at high temperature. Diffraction
confirmed this --- peaks associated with $P2$ gradually weakened upon
heating, becoming indistinct around 350$^\circ$C. Example laboratory
x-ray diffraction patterns are shown in Fig.~\ref{fig:highTP2}(a) and
(b); neutron diffraction results are shown in
Fig.~\ref{fig:highTP2}(f). The gradual temperature evolution suggests
that this is a second-order displacive transition. Clear evidence for
the transition may also be observed in the lattice parameters obtained
from refinements within the \ensuremath{P3_121}\ structure, shown in
Fig.~\ref{fig:highTP2}(c) and (d). A reduction in the $a$-axis and an
increase in the $c$-axis lattice parameters relative to a $\ensuremath{P3_121}$
extrapolation are observed below the transition, estimated from these
data as 450$^\circ$C. The unit cell volume shows no obvious change
across the transition. The results of a joint neutron and x-ray
refinement in \ensuremath{P3_121}\ at 600$^\circ$C are included in Fig.~\ref{fig:600}
and Tab.~\ref{tab:600C}. The shortest Mn--Mn bond lengths are
Mn(1)\,--\,Mn(2), which form a helix along the $c$-axis, while the
kagome-plane triangles are not far from equilateral at this
temperature.
\section{Magnetic Transitions}
\begin{figure*}[htb]
\includegraphics[width=0.85\textwidth]{Squid_wide_v2.pdf}
\caption{\label{fig:MT}Low-temperature magnetization of Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}. (a) The
field-cooled (FC) magnetization $M/H$ in 100\,Oe is compared against
zero-field cooled (ZFC) data in the same field, and against data
taken on warming in 100\,Oe after cooling in an applied field of
1\,T as in Ref.~\onlinecite{Reimers1991}. The inset shows the
inverse magnetization (FC, 2000\,Oe), from which a paramagnetic
moment of 5.88\,$\mu_B$ and a Curie-Weiss temperature of $-40$\,K
were extracted. (b) Field-cooled magnetization in a variety of
fields, plotted on a logarithmic scale. Vertical lines mark the
transitions in this sample's specific heat, and the inset shows the
result of a relaxation measurement at 1.8\,K. (c) $M$--$H$ loops at
1.8 and 12.75\,K, with their derivatives in the lower inset. Below
the first transition, there is clear curvature and slight
hysteresis. The upper inset is an Arrott plot. The hysteresis in
$M(H)$ can be characterized by (d) the coercive field $H_{coerc}$ or
(e) the remnant magnetization $M_R$.}
\end{figure*}
Among the few physical properties reported on Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ is
magnetization\cite{Reimers1991}. An increase in the magnetization was
observed upon cooling through 13\,K, taken to be the bulk ordering
temperature, while the history-dependence below 55\,K was attributed to
the onset of short-range correlations. This history dependence was
manifested as an approximate doubling of the measured magnetization in
the paramagnetic phase under 100\,Oe if first cooled in a much higher
field\cite{Reimers1991}. Our preliminary magnetization measurements
indeed detected a strong history-dependent upturn in magnetization on
cooling through a similar temperature range, but this was entirely
attributable to ferrimagnetic Mn$_3$O$_4$ --- the elimination of trace
Mn$_3$O$_4$ impurities eliminated any features or history dependence
above the bulk transition. The magnetization is plotted in
Fig.~\ref{fig:MT}(b) with a logarithmic vertical axis to enhance any
such weak transition, demonstrating the complete absence of history
dependence.
Fig.~\ref{fig:MT}(a) shows data collected as in
Ref.~\onlinecite{Reimers1991}, as well as zero-field-cooled data.
A history-dependence, while absent above the bulk transition, does
appear at low temperatures, indicating some form of magnetically
frozen state, a weak ferromagnetic component, or possibly magnetic
domains. This would need to be a rather weak ferromagnetic component,
since much stronger magnetization jumps were observed from trace
ferrimagnetic Mn$_3$O$_4$ that was below the detection limit of
laboratory x-ray diffraction. The inverse susceptibility, shown in
the inset to Fig.~\ref{fig:MT}(a), departs from linearity below
$\sim$150\,K, indicating short-range correlations appearing well ahead
of any magnetic order. A fit to the high-temperature inverse
susceptibility returns a Curie-Weiss temperature \ensuremath{T_\text{CW}}\ of $-40$\,K and
a paramagnetic moment of 5.88\,$\mu_B$, consistent with the spin-only
value for $3d^5$ Mn$^{2+}$ of 5.92\,$\mu_B$ and with the previous
report\cite{Reimers1991}. The onset of short-range correlations well
above the transition and the approximate factor of 3 between the
Curie-Weiss temperature and the magnetic ordering temperature indicate
significant frustration.
The field-cooled magnetization $M(T)$, shown in Fig.~\ref{fig:MT}(a)
and (b), shows a striking increase upon cooling through the bulk
transition, after which it saturates several Kelvin lower. As will be shown
below, these are two separate transitions. The field-dependent
magnetization $M(H)$ in Fig.~\ref{fig:MT}(c) is weakly S-shaped with a
change in curvature around 7\,T, and it shows a slight hysteresis,
especially upon entering the $M(T)$ plateau, as can be seen in the
temperature dependence of the coercive field $H_{coerc}$ and remnant
magnetization $M_R$ in Figs.~\ref{fig:MT}(d) and (e), respectively.
The large field dependence in the low-temperature magnetization
$M(T)/H$ at low field, when compared with the minor differences in
$dM/dH$ (lower inset to Fig.~\ref{fig:MT}(c)), reflect the
ferromagnetic component. Changes of slope in $dM/dH$ at 3\,K around 6
and 9\,T indicate possible metamagnetic transitions, which shift to
{\sl higher} fields in the 12\,K data. These may have been split or
broadened by powder-averaging of the material's presumably anisotropic
magnetic properties. That the field scale increases with increasing
temperature is somewhat unusual. Arrott
plots\cite{Belov1956,Arrott1957,Arrott1967,Bustingorry2016} were also
constructed, as shown in the upper inset in Fig.~\ref{fig:MT}c. Near
a conventional ferromagnetic transition, these would be linear with a
zero intercept. The S-shaped curves here serve to accentuate the
changes in curvature in the $M(H)$ data.
\begin{figure}[htb]
\includegraphics[width=\columnwidth]{ACchi_err.pdf}
\caption{\label{fig:ACchi}AC susceptibility of Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}. (a) The real
component of the AC susceptibility shows features at both
transitions, while the (b) imaginary component is featureless. No
frequency dependence or temperature-dependent dissipation was
observed. Error bars represent the statistical uncertainty in the
fit used to extract each data point.}
\end{figure}
One possible explanation for the shape of the $M(T)$ curves could be a
transition into a spin state with glassy dynamics, although the
temperature dependence of the hysteresis in Figs.~\ref{fig:MT}d and
\ref{fig:MT}e does not take the expected form. To test for relaxation
behavior, a sample was cooled to 1.8\,K in zero field, the field was
increased to 100\,Oe, and the decay of the magnetization toward its
equilibrium value was then measured. The extremely weak relaxation
behavior, visible in the inset to Fig.~\ref{fig:MT}(b), is well
described by a single time constant. Present at the
parts-per-thousand level at best, this may not be intrinsic. AC
susceptometry was also used to check for glassy behavior at the
transitions (Fig.~\ref{fig:ACchi}): the real component of the
susceptibility, $\chi'$, shows a sharp feature at the lower transition
and a much weaker feature at the upper transition, at temperatures of
11.6 and 14.4\,K respectively. There is no frequency dependence to
suggest glassy dynamics at the lower transition, while the noise level
precludes any definitive pronouncement as to the nature of the upper
transition. However, a material would not be expected to enter a
glass state upon cooling and then pass through a subsequent phase
transition into a long-range ordered state --- this would normally be
prevented by the glass states's diverging timescale for relaxation.
The lower panel of this figure, showing $\chi''$, contains no hint of
an onset in dissipation at either transition.
\begin{figure}[htb]
\includegraphics[width=\columnwidth]{cP_vs_T_v2.pdf}
\caption{\label{fig:cP}(a) Low-temperature specific heat of Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7},
showing two phase transitions. The upper inset shows high-field
behavior on a second sample, while the lower inset shows the
high-temperature specific heat at zero field. Nonmagnetic
Sr$_2$Sb$_2$O$_7$ and Ca$_2$Sb$_2$O$_7$ are included for comparison.
(b) The accumulated magnetic entropy in Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ obtained by subtracting
that of Sr$_2$Sb$_2$O$_7$. The horizontal line marks the $R\ln 6$
expected for $s=5/2$; the fact that the entropy exceeds this
indicates that the phonon subtraction is imperfect. A plot of
$c_P/T$ for extracting the magnetic entropy is included in the inset
--- the phonon contribution in Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ clearly extends lower in
temperature than in the Sr or Ca analogs.}
\end{figure}
The low-temperature specific heat of Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ is shown in
Fig.~\ref{fig:cP}. Here, two phase transitions are clearly visible,
at 11.1 and 14.1\,K. Applied fields up to $\sim$2\,T had no effect,
but the peaks began to broaden noticeably by 5\,T and became
indistinct at higher fields (upper inset). This is likely due to
powder averaging of an anisotropic field dependence in the phase
transitions. The specific heat implies that the unusual
step-and-saturation shape in the temperature-dependent magnetization
data actually reflects the onsets of at least two distinct forms of
order. Marking the specific heat transitions on the $M(T)$ data in
Fig.~\ref{fig:MT}(b) shows that the upper transition corresponds to
the first sudden increase in magnetization, while the lower transition
coincides roughly with the onset of its saturation.
Examining the specific heat to higher temperatures [lower inset to
Fig.~\ref{fig:cP}(a)], one finds a large build-up of magnetic
entropy below $\sim$40-50\,K. This onset is clearer when the data are
plotted as $c_P/T$, as in the inset to Fig.~\ref{fig:cP}(b). Included
for comparison are specific heat data for Ca$_2$Sb$_2$O$_7$ and
Sr$_2$Sb$_2$O$_7$\cite{Knop1980}. These form in the closely-related
trigonal Weberite structure\cite{Verscharen1978,Cai2009} and are
nonmagnetic insulators. Sr$_2$Sb$_2$O$_7$ more closely mimics the
high-temperature behavior of Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}, but its use as a phonon baseline
results in an entropy exceeding the $R\ln6$ expected for $3d^5$
Mn$^{2+}$, as shown in Fig.~\ref{fig:cP}(b). This is most likely due
to differences in the low-energy phonon spectrum, which would be
unsurprising given the Mn version's relaxation into a lower-symmetry
$P2$ structure. Mixing of phonons with other types of modes may also
be a possibility in this space group.
\begin{figure}[htb]
\includegraphics[width=\columnwidth]{dielectric2.pdf}
\caption{\label{fig:dielectric}Temperature dependence of the (a)
capacitance and (b) dissipation factor when Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ is employed as
the dielectric in a capacitor. The insets show multiple warming
and more-rapid cooling runs, with the specific heat transitions
marked for reference.}
\end{figure}
Since the low-temperature $P$2 space group supports ferroelectricity
while the \ensuremath{P3_121}\ space group does not, multiferroicity is a possibility
in this system and measurements of the dielectric properties, shown in
Fig.~\ref{fig:dielectric}, can provide further insight into the phase
transitions. The capacitance shows a sharp downturn at the upper
magnetic transition, where the magnetic order locking-in reduces the
material's ability to electrically polarize (and store energy as a
dielectric). A corresponding peak in the dissipation factor is
attributed to order parameter fluctuations near the transition. The
appearance of such fluctuations for a magnetic transition in an
electrostatic quantity implies multi-component order. The capacitance
undergoes a broader but stronger reduction centered around 50\,K, with
no corresponding peak in the dissipation factor. This represents the
onset of short-range magnetic order and matches well with the heat
capacity estimate. Above $\sim$100\,K, where magnetism plays no
significant role, the reduction in the capacitance (and thus
polarizability) upon cooling is suggestive of a tendency toward
antiferroelectric order below the structural transition, in much the
same way that a reduction in magnetic susceptibility is seen below a
pure antiferromagnetic transition.
Data were collected on free warming to minimize noise, and Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}'s
specific heat is very large at low temperatures, so the sample
temperature lags the thermometry leading to an apparent shift of the
data to slightly higher temperatures. To confirm that the upper
transition is indeed the electrically-active one, data taken upon
cooling with a sweep rate roughly an order of magnitude faster are
included in the insets to Fig.~\ref{fig:dielectric}. All observed
features remain above the lower transition, unambiguously identifying
the upper transition as the electrically active one.
The clear detection of a magnetic transition and its order parameter
fluctuations in bulk electrostatic properties implies strong
magnetoelectric coupling and mixed-character (multiferroic) order
parameters. It also confirms that the space group supports
ferroelectricity, lending additional support to the assignment of
$P2$.
\section{Magnetic structure}
\begin{figure*}[htb]
\includegraphics[width=\textwidth]{neutron_wide_jpg.pdf}
\caption{\label{fig:neutron-mag}Magnetic intensity at low temperature.
(a) Diffraction patterns at various temperatures. (b) Difference
with respect to 17\,K, to highlight the magnetic peaks. (c) A
significant diffuse peak is visible above the transition around a
$d$-spacing of 5\,\AA\ when data taken at 100\,K are used for
subtraction. Panels (d) and (e) demonstrate that while much of the
intensity in these peaks vanishes at the lower magnetic transition,
some persists to the upper transition.}
\end{figure*}
\begin{table}[htb]
\caption{\label{tab:mag}Magnetic reflections in Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}. Observed and
calculated $d$-spacings in \AA\ based on data from both ANSTO and
HANARO, approximate relative intensities (Rel.\ Int.), and index
in \ensuremath{P3_121}\ and $P2$ of the observed magnetic peaks are listed.}
\begin{tabular}{cccccr}\hline
\ensuremath{P3_121} & $P2$ & $d$ & $d_{HANARO}$ & $d_{calc}$ & Rel.\ Int. \\ \hline\hline
($\frac12$00) & (100) & 12.44 & --- & 12.43 & 3.9 \\ \hline
($\frac12$01) & (101) & 10.15 & --- & 10.12 & 2.6 \\ \hline
($\frac12$02)/($\frac12$10) & (102)/(010) & 7.15 & 7.06 & 7.13 & 27 \\ \hline
($\frac12$11) & (011) & 6.65 & 6.55 & 6.64 & 24 \\ \hline
($\frac12$12) & (012) & 5.53 & 5.47 & 5.53 & 23 \\ \hline
($\frac12$03) & (103) & 5.26 & 5.20 & 5.25 & 39 \\ \hline
($\frac32$10) & (210) & 4.68 & 4.65 & 4.70 & 33 \\ \hline
($\frac32$11) & (211) & 4.53 & 4.48 & 4.53 & 100 \\ \hline
($\frac32$00)/($\frac32$12) & (212) & 4.13 & 4.08 & 4.13 & 24 \\ \hline
($\frac32$01) & (301) & 4.03 & --- & 4.03 & 9.8 \\ \hline
($\frac12$14)/($\frac32$02) & (014)/(302) & 3.71 & 3.66 & 3.72 & 31 \\ \hline
($\frac32$21)/($\frac32$03) & (121)/(303) & 3.38 & 3.36 & 3.38 & 26 \\ \hline
($\frac52$10) & (410) & 2.85 & 2.84 & 2.85 & 4.9 \\ \hline
($\frac12$06)/($\frac52$11) & (106)/(411) & 2.81 & 2.80 & 2.81 & 11 \\ \hline
($\frac12$16)/($\frac52$21) & (016)/(321) & 2.68 & 2.66 & 2.68 & 17 \\ \hline
several & several & 2.21 & 2.20 & & 23 \\ \hline
several & several & 2.20 & 2.19 & & 6.6 \\ \hline
several & several & 1.88 & 1.88 & & 14 \\ \hline
\end{tabular}
\end{table}
Neutron diffraction was performed through both magnetic transitions
--- data in the relevant temperature range appear in
Fig.~\ref{fig:neutron-mag}(a). Fig.~\ref{fig:neutron-mag}(b) shows
the same data after subtraction of a pattern taken at 17\,K, just
above the upper magnetic transition. All magnetic peaks can be
qualitatively explained by a propagation vector of ($\frac12$\,0\,0)
in the \ensuremath{P3_121}\ unit cell, or (0\,0\,0) in the larger $P2$ cell, as
summarized in Tab.~\ref{tab:mag}. Since the underlying $P2$ crystal
structure could not be refined, and since the number of observed
magnetic peaks is similar to the expected number of Mn sites,
refinement of the magnetic structure was not completed. The $P2$
space group supports two irreducible representations for the magnetic
order, both of which are chiral, and it was not possible to
distinguish between them. Calculated $d$-spacings in
Tab.~\ref{tab:mag} are based on the $P2$ assignments. Based on the
limited data, it appears that peaks with $h+k=$\,odd in $P2$ are
favored.
As has previously been reported\cite{Reimers1991}, there is a
significant diffuse peak centered around a $d$-spacing of 5\,\AA, and
2\,\AA\ wide --- see Fig.~\ref{fig:neutron-mag}(c). Comparison to 300\,K
data indicated that this feature is essentially absent above 70\,K, so
data collected at 100\,K were used as a baseline for subtraction. The
appearance of the diffuse peak around 50-70\,K provides further
evidence for the onset of local spin correlations around that
temperature. Its intensity reaches a maximum between the two magnetic
transitions, and then the magnetic intensity is transferred into the
magnetic Bragg peaks at low temperatures.
The temperature dependence of the magnetic peaks is shown in
Figs.~\ref{fig:neutron-mag}(d) and \ref{fig:neutron-mag}(e). The
intensity falls off as would be expected for a second-order transition
at 11\,K, but a small fraction of the intensity persists to the higher
magnetic transition. The magnetic peaks in both phases are located at
the same angles and can be described by the same shape parameters and
intensity ratios, although the low intensity between the two
transitions makes firm conclusions difficult. The strong similarity
indicates that the two magnetic phases are closely related --- perhaps
distinguished by a canting angle, stacking along the hexagonal $c$
axis, or coupling to antiferroelectric order.
\section{Discussion and Conclusion}
As is clear from the magnetization, specific heat and neutron
diffraction data, Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ undergoes two magnetic phase transitions at low
temperature. The upper transition is multiferroic in nature, while
the lower is purely magnetic. Below the lower transition, the
magnetization saturates and the hysteresis reaches its maximum before
subsiding, suggesting that the ground state is some form of global
antiferromagnetic order. Between the two transitions, the
magnetization increases sharply upon cooling, implying a small net
moment. The temperature-dependence of the apparent high-field
transitions in $M(H)$ suggests frustration is relieved by the field.
The magnetic sublattice is intermediate between two-dimensional kagome
and three-dimensional pyrochlore, so this material offers a platform
for interpolating between two- and three-dimensional frustrated
magnetism. The clearest comparison to a pyrochlore system is to the
pyrochlore polymorph of
Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ itself\cite{Brisse1972,Zhou2008,Zhou2010,Peets-pyr}. This latter
structural variant has only been successfully prepared {\it via} a
specific low-temperature route, substituting Mn$^{2+}$ into the
pre-existing pyrochlore Sb$^{5+}$ framework of Sb$_2$O$_5\cdot
n$H$_2$O ($n\sim 1.5$\,--\,2). By having 6 Mn--Mn links within a
tetrahedron rather than five within an armchair unit, one might expect
the pyrochlore's Curie-Weiss temperature to be 20\%\ stronger, or
$-48$\,K. Experimentally, we found it to be $-49$\,K\cite{Peets-pyr}.
One would also expect a significantly higher frustration factor
$f\equiv \ensuremath{T_\text{CW}}/T_\text{N}$ in the pyrochlore, and again this is exactly
what we observe: $f_\text{ak} = 2.8$ for the armchair-kagome structure
and $f_\text{pyr} = 9.0$ for the pyrochlore. The stronger frustration
in the pyrochlore polymorph leads to a spin-glass ground state, rather
than the three-dimensional magnetic order observed here.
\begin{figure}[htb]
\includegraphics[width=\columnwidth]{py_P2.pdf}
\caption{\label{compare}Comparison of the armchair-kagome and
pyrochlore variants of Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}. Magnetization data on the former
polymorph were taken at 2000\,Oe (field-cooled), and are the same
data as in the Fig.~\ref{fig:MT}a inset. Data on the pyrochlore
were taken at 100\,Oe under field-cooled and zero-field-cooled
conditions\cite{Peets-pyr}.}
\end{figure}
For a more direct comparison, temperature-dependent magnetization data
are plotted in unitless form for both polymorphs in
Fig.~\ref{compare}, obtained by rearranging the Curie-Weiss law $M/H =
C/(T-\ensuremath{T_\text{CW}})$ as $CH/M\ensuremath{T_\text{CW}} = T/\ensuremath{T_\text{CW}} -1$\cite{Dutton2011}. Data above
the dashed line indicate additional antiferromagnetic interactions,
while points can be pushed below the line by ferromagnetic
interactions. Deviations from ideal Curie-Weiss behavior start at a
higher fraction of \ensuremath{T_\text{CW}}\ in the armchair-kagome structure and indicate
additional antiferromagnetic fluctuations, whereas the deviations from
Curie-Weiss behavior in the pyrochlore are ferromagnetic. The
transitions in both polymorphs lie well below the unfrustrated
$T_\text{N}=\ensuremath{T_\text{CW}}$.
In summary, we have shown that the structure of Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ must be $P2$
under ambient conditions, but that it returns to the higher-symmetry
\ensuremath{P3_121}\ structure above $\sim$450$^\circ$C in an apparent second-order
displacive transition. Solving the $P2$ modification will most likely
require single crystals. The armchair-kagome network seen in the
\ensuremath{P3_121}\ structure has not, to our knowledge, been modeled theoretically,
and investigation of its possible magnetic and multiferroic ground
states as a function of interaction strengths would be an interesting
and fruitful topic for future research. The Mn$_4$ armchair units
constitute an intermediate case between the two-dimensional triangular
network of the kagome lattice and the fully three-dimensional
tetrahedra of the pyrochlore, and likely support their own unique
suite of ground states as a function of the exchange parameters.
Interestingly, the smaller Sb$^{5+}$ ions form identical structural
armchair motifs, but with an additional twist between layers that is
not present in the Mn sublattice. It would be worthwhile to explore
the possibility of putting magnetic ions on this site instead.
Magnetic Bragg peaks are consistent with a propagation vector of
$(\frac{1}{2} 0 0)$ in \ensuremath{P3_121}\ or $(000)$ in $P2$, but a magnetic
structure could not be refined. Because the latter crystal structure
only supports chiral irreducible representations for the magnetic
order, we infer that the magnetic order in the material is chiral.
Chiral magnetism can lead to a variety of exotic physics, and will be
of significant interest for its excitations and field-dependence. The
possibility of skyrmion-like excitations\cite{Skyrmions} in a system
with chiral multiferroic order would be particularly enticing, as it
would enable control and manipulation of the magnetism and excitations
through standard electronic means. In the closely-related
MnSb$_2$O$_6$\cite{Reimers1989}, the novel cycloidal magnetic order
has been predicted to lead to a unique ferroelectric switching
mechanism, while the material should behave in an analogous way to
ferroaxial multiferroics\cite{Johnson2013}, and it would be
interesting to determine whether similar physics could be available
here. Mn\ensuremath{_2}Sb\ensuremath{_2}O\ensuremath{_7}\ has some key differences, however, and may host its own
suite of entirely unique physics.
\section*{Acknowledgements}
This work was supported by the Institute for Basic Science (IBS) in
Korea (IBS-R009-G1), and work at HANARO was supported by the Nuclear
R\&D Program through NRF Grant No.\ 2012M2A2A6002461. The authors are
indebted to M.\ Gingras, D.I.\ Khomskii, M.J.\ Lawler, M.D.\ Le, and
Y.\ Noda for stimulating discussions, the NCIRF for assistance with
several measurements, and K.S.\ Knight at ISIS for assistance with the
time-of-flight diffraction measurement. We acknowledge the support of
the Bragg Institute, Australian Nuclear Science and Technology
Organisation, in providing neutron research facilities used in this
work.
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{"url":"https:\/\/www.physicsforums.com\/threads\/determine-g-from-least-squares-fit-line.380494\/","text":"# Determine g from least squares fit line?\n\n1. Feb 22, 2010\n\n### noname1\n\nI did a least squares fit project for physics and now i have to say the value of G and the slope.\n\nI know that slope is m from the equation\n\ny = mx+b\n\nbut how do i determine G?\n\n2. Feb 22, 2010\n\n### Fightfish\n\nPerhaps it would help to provide the variables that you are plotting in your graphical analysis?\n\n3. Feb 22, 2010\n\n### noname1\n\nSorry....\n\nThe x values i am using are:\n0.03139\n0.05198\n0.09315\n0.1755\n\nThe y values i am using are:\n0.00306\n0.00514\n0.00929\n0.01729\n\n4. Feb 22, 2010\n\n### Fightfish\n\nErr...no not the values of the variables...I need to know what the variables in question are. ie what x and y represent.\n\n5. Feb 22, 2010\n\n### noname1\n\nY is acceleration (m\/s\u00b2) and x is hard to explain.\n\nWe measured these values using an air track where we had a mass hanging which is m1 and than we had a cart m2\n\nx values is (m1\/m1+m2)\n\n6. Feb 22, 2010\n\n### Fightfish\n\nAir track? Hm...I guess I have to trouble you to explain the set-up a bit.\n\n7. Feb 22, 2010\n\n### noname1\n\nmaybe this can help better\n\n#### Attached Files:\n\n\u2022 ###### scan0001.jpg\nFile size:\n60.1 KB\nViews:\n52\n8. Feb 22, 2010\n\n### Fightfish\n\nFrom the worksheet, they already derived the relation:\n$$a = \\frac{m_{1}}{m_{1} + m_{2}}g$$\u200b\nwhich is in the form y = mx + c.\n\nHence, from the plot, we expect the gradient m to be g, and c to be zero. If c is not zero, then it probably indicates the existence of some experimental error. So, in fact, your gradient m is the value of g measured by the experiment.\n\n9. Feb 22, 2010\n\n### noname1\n\nyes c is really close to 0, its 1.848x10^-4\n\ny = 0.09651x + 1.848x10^-4\n\nand sorry i didnt understand really well about the g part, could you explain in other words...\n\nI was thinking of this but dont know if i am right or not, but i was going to do\n\ng = (m1\/(m1+m2))*a\n\nthanks for trying to help me\n\n10. Feb 22, 2010\n\n### Fightfish\n\nI'll try. By comparing y = mx + c and a = g (m1\/(m1+m2)) + 0, we can see that m in fact represents g - basically the gradient of the graph that you obtain when you plot acceleration against (m1\/(m1+m2)) in this case is the value of the gravitational acceleration g.\n\nSo, g would be 0.09651 when measured in the units that you are using.\n\n11. Feb 22, 2010\n\n### noname1\n\nso there is no work i can show to demonstrate this correct?\n\n12. Feb 22, 2010\n\n### Fightfish\n\nWell, the work of linearising the equation and choosing the variables to plot in fact already fixes the gradient m as g, so no, there is no further need to substantiate why m = g; it is clear from the variables plotted and the given equation from the beginning.\n\nThe only task left for you was to determine the value of the gradient and hence g from your graphical analysis. So, essentially, you are done :)","date":"2017-08-16 13:33:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.47044047713279724, \"perplexity\": 1299.087835559474}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-34\/segments\/1502886101966.48\/warc\/CC-MAIN-20170816125013-20170816145013-00523.warc.gz\"}"} | null | null |
María Luisa González de Escobar (* 5. Dezember 1903 in Valencia; † 14. Mai 1985 in Caracas) war eine venezolanische Komponistin, Pianistin und Sängerin (Sopran).
Leben
Escobar war die Tochter von Enrique González Olivo und María Gragirena Mijares. Sie bekam ihre erste Klavier- und Kompositionsausbildung am Colegio Lourdes ihrer Heimatstadt, die sie am Instituto Welgelegen Habay auf Curaçao fortsetzte. 1925 entstanden in Mexiko zwei Schallplattenaufnahmen, auf denen sie die Lieder Canción de amor, La Golondrina, La paloma und La verdadera española sang.
Von 1928 bis 1931 studierte Escobar in Paris bei Roger-Ducasse und Gabriel Fauré. Sie heiratete den Musiker José Antonio Escobar Saluzzo, mit dem sie nach ihrer Rückkehr die ersten Schallplatten in Venezuela produzierte. 1931 gründete sie das Ateneo de Caracas, das sie bis 1943 leitete. 1947 gründete sie die Asociación Venezolana de Autores y Compositores (AVAC).
Am Teatro Municipal de Caracas wurde 1941 ihre musikalische Komödie Orquídeas Azules nach einem Libretto von Lucila Palacios aufgeführt. Zur gleichen Zeit nahm sie verschiedene Werke für Klavier und Orchester mit dem Orquestra Don Américo y sus Caribes in Buenos Aires auf. Für diese Besetzung komponierte Escobar in den 1940er Jahren mehrere Werke, in denen sie Themen aus der Folklore ihres Landes verwendete. Ihr Concierto Sentimental wurde vom Warschauer Sinfonieorchester aufgeführt. Anlässlich der III. Juegos Bolivarianos 1951 fand die Uraufführung ihrer Ballettsuite Guaicaipuro statt.
Große Popularität erlangten Escobars zahlreiche Boleros, deren bekanntester Desesperanza wurde. Erstmals von dem venezolanischen Bariton Eduardo Lanz gesungen, wurde Desesperanza international durch die Aufnahme mit Alfredo Sadel bekannt. Das Stück wurde 1947 zum Canción del Año erklärt.
Auszeichnungen
1984 erhielt sie den Premio Nacional de Música de Venezuela.
Werke
Orquídeas Azules, musikalische Komödie
La Princesa Girasol
Murachi
Upata
Concierto Sentimental für Klavier und Orchester
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Exfunds.com – asset management platform which offers you incredible ROI 15298
Martin Frederick July 27, 2021 06:32 July 27, 2021
We would like to introduce you to the exfunds.com Asset Management platform which offers you incredible ROI in 2 investment plans. Exfunds.com have 2 investment plans to choose from: EX START (investment range $10 – $9999.99) and EX PRO (investment range $10000 – $250000).
In the program, we can have an unlimited number of deposits in various investment plans. Each of them works as a separate contract for a specific amount of time. The investment period is 40 days. After the end of the investment plan period, your main capital is returned to your balance and you can withdraw it or reinvest it again.
EXFUNDS.com also provides the option to terminate the investment plan AT ANY TIME after the first 24 hours. In this case, a fee of 10% of its value is charged on the deposit and finally 90% is released to your balance, where you can easily withdraw.
EXFUNDS.com also offers us unlimited earning opportunities through its referral program, which pays us an immediate commission for every person we invite. We can create a structure up to 3 levels down, with a commission of 5% – 2% – 1%.
As for the payouts: they are really fast, made 24/7, including all weekends and holidays. Despite the timeframe for withdrawals – that is, within 24 hours of a withdrawal request, they arrive, definitely at lightning speed, such as 5 – 30 minutes. However, please note that the platform owners provide 24 hours as the maximum time to receive your withdrawal.
What can convince you to invest in exfunds.com?
It is a professionally prepared platform, artistry design working on the popular GC script, fully adding a lot of functionality and readability to it. You can see a professional base of web-developers and graphic designers, but above all a professional approach to the money entrusted by investors. There is no fear that someone will run away with your money for a deposit of 30k or 60k USD … or more. Such professional teams just never do it, as can be seen from the payout scale after a few days of operation. Summing up, we see everything that is done as it should be by a professional legend: unique design, unique programming solutions, efficient cryptocurrency deposit wallets (remaining at the sole disposal of the company, without the participation of third parties), we can see the content of the page that was written, not copied from others. You can also see professional hosting on a dedicated server, with one of the best protection against DDOS attacks and other attacks, secure SSL encryption, a unique masked IP address.
EXFUNDS.com is, above all, also always available and willing to help Support, which is also available 24/7 with a waiting time for a reaction of less than a dozen seconds. Added to this is the administration support via email or ticket system. All this makes EXFUNDS.com the most popular investment program on the market for a reason and it will be difficult to catch up with anyone. You can use PerfectMoney as payment to make deposit in USD or send this value using 11 cryptocurrencies: Bitcoin, Litecoin, Doge, Ethereum, Bitcoin Cash, Dash, Ripple, USDT, Tron, Binance Coin or Stellar.
Exfunds.com
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World's first tradable carbon token is set to democratize access to the most important new asset class for generations
World's first tradable carbon token is set to democratize access to the most important new asset class for generations 20019
Joe Trohman December 2, 2020 11:19 December 2, 2020
The Universal Protocol Alliance (UPA), a coalition of leading blockchain companies including Bittrex Global, Ledger, CertiK, InfiniGold and Uphold, today launches Universal Carbon [UPCO2], the world's first tradable carbon token on a public blockchain that can be bought and held as an investment, or burnt to offset an individual's carbon footprint. With demand for carbon credits outstripping supply by a factor of 4 to 1 in 2020, according to the World Bank, the UPCO2 Token is set to democratize an important new asset class, which could lead to the establishment of a global clearing price for carbon (as today exists for such commodities as oil and gold) and more resources going directly into environmental projects.
Each UPCO2 Token represents one year-ton of CO2 pollution averted by a certified REDD+ project preventing rainforest loss or degradation. Every Token is backed by a Voluntary Carbon Unit [VCU], a digital certificate issued by Verra and other international standards agencies, which allows certified projects to turn their greenhouse gas (GHG) reductions into tradable carbon credits.
"The projects we support through carbon credit purchases prevent deforestation in the Amazon, Congo Basin and Indonesia as well as other threatened rainforests"' explained UP Alliance Chairman, Matthew Le Merle. "For a new generation of investors looking for more than mere financial return, UPCO2 offers attractive social, economic and environmental benefits. At a key moment for climate change, UPCO2 allows people worldwide to do good for the planet and potentially do well for themselves."
Powerful macroeconomic forces underpin the Voluntary Carbon Credit market and, according to some commentators, could drive up prices significantly as more countries introduce regulated CO2 markets, forcing companies to compensate for their pollution. Additionally, a growing number of firms and individuals are choosing to offset their carbon footprints voluntarily.
As with all commodities, prices for carbon credits are likely to fluctuate, but human emissions have grown from 25 billion tons to 55 billion tons between 2008 and 2018, while the supply of voluntary credits has remained broadly flat.
According to the World Bank, in 2020, humanity compensates for just 22% of global emissions through the purchase and retirement of carbon credits, and yet the proportion of countries operating regulated carbon markets has risen from 40 percent of global GDP in 2016 to 70 percent in 2020. The result is a wall of demand that may far outstrip the production of new carbon credits, which is constrained by the slow and expensive process of Voluntary Carbon Project certification.
"This year may go down as the key inflection point for climate change," said JP Thieriot, Co-Founder of the UP Alliance and CEO of Uphold. "The year it went from far-off issue enshrined in distant accords like Kyoto and Paris, to a palpable threat affecting the lives of tens of millions of people. In recent months, we've seen Australia and California on fire, ever more powerful hurricanes, the U.S. president-elect Joe Biden announcing a Climate Administration, and companies such as Apple, Microsoft, and Nike voluntarily committing to carbon neutrality.
"Combating climate cancer is likely to become the dominant economic issue of the next 20 years. The UPCO2 Token allows people everywhere to participate in this hugely important – and potentially lucrative – new market, as well as do the right thing for the planet."
Voluntary carbon credits, which back all UPCO2 Tokens, offer major economic advantages compared with regulated credits. As dollar-denominated, globally-recognized, fungible and perennial assets, voluntary credits last forever, maintaining option value, until consumed or retired by a company or an individual seeking to compensate for carbon footprint.
"It's astonishing that there is no single global clearing price for carbon emissions," said Le Merle. "A non-deliverable, digitally-tradable commodity that's essential for human activity shouldn't be traded bilaterally on OTC markets, as carbon credits are today.
"One year-ton of carbon means the same everywhere. As a globally-recognized asset, defined by international standards, a Voluntary Carbon Credit should eventually fetch the same price anywhere." Mr. Le Merle said, "We believe that the UPCO2 token has an important role to play in democratizing access to carbon credits, which could eliminate price arbitrage and produce a single global price. This was a light bulb going on for me. Combine a digital asset with a rainforest carbon offset and give everyone in the world access. How could that not be a great idea?"
ExCore's sale is LIVE 23891
Martin Frederick October 15, 2020 14:42 October 15, 2020
ExCore Sales and Impressive Staking
ExCore is a new and rapidly growing cryptocurrency that stands to eliminate inflation. Because there is a finite supply and no new tokens will ever be released, your investment in ExCore will never significantly drop from controllable causes. Right now, ExCore is in the middle of their private sale, but will release their public sale and staking platform next week on October 21. ExCore is a company that all keen investors should keep an eye on.
Today (October 14), ExCore launched their private sale to their whitelisted members. The sale is ongoing and takes place on Bounce, a secure medium used for crypto transactions in presales. It currently is about 25% full, and will go on until 10/17 or until the hard cap of $100,000 is reached.
You can participate via this link: https://bounce.finance/join/swap/3669 and this PASSWORD: excore2020
There are guides in ExCores telegram groups (Link can be found at bottom of this page) that explain in detail how to use the bounce platform.
Public sale
On October 21, the public presale will launch with a hard cap of $800,000 worth of ETH. Everyone will be able to participate in this sale as long as they have a metamask wallet. The minimum requirement for this sale will be 1 ETH, but keep in mind there will be gas fees, so you will need to have some extra in your account.
On the same day as the public presale, ExCore's staking platform will also launch. Their staking platform offers an impressive 550% APY that will come from fees from every transaction on the ExCore network. To stake your tokens, there will be a 1% fee to stake your tokens as well as a 1.5% fee to unstake them, but staking for even just one day will be enough to cover these fees.
Not only does ExCore make for a great investment with their anti-inflation protocol, but if you also stake your ExCore you will be looking at some very nice returns. ExCore is without a doubt one of the best crypto investment options of 2020 and the sooner you get in, the better rates you will be able to buy at. The ExCore team is currently marketing everywhere they possibly can, so once the word gets out it will no longer be possible to buy tokens at this discounted price.
ExCore Links:
Here, you can find a few very helpful links, but most importantly the link to Github. This verifies the integrity of ExCore through our open source code (that anyone can see!).
Github: https://github.com/ExCoreFinance
Website: https://www.exvault.finance/
Telegram: https://t.me/excorevault
Twitter: https://twitter.com/ExCoreVault
Medium: https://medium.com/@excorefinance
Contract address: 0x87D3646B101977de0D2D58dfC5A70e84767A1909
Staking contract address: 0x28Ea47E0ff753AE99eE5241f468817Db6C476d
New Ukrainian Law Says 'Virtual Assets' Can Be Used for Payments 59702
Mark Rohman December 10, 2019 10:20 December 10, 2019
The draft law on the prevention of the legalization of proceeds from crime and the financing of terrorism and weapons of mass destruction proliferation was supported by a significant majority in the Rada. The bill was amended to incorporate "virtual assets" which have been described as property and as a digital expression of value that can be traded or transferred and used for payment or investment purposes.
Ukraine's anti-money laundering (AML) legislation introduces the standards for virtual assets adopted this year by the Financial Action Task Force (FATF). The members of the inter-governmental organization recently agreed to monitor and assess the implementation of the crypto requirements in different countries, as news.Bitcoin.com reported in October.
The law also introduces the term "provider of services related to the transfer, exchange and storage of virtual assets," the crypto information outlet Forklog reveals in an article. An interesting detail is that not only corporate entities but private individuals as well will be allowed to offer such services under the new regulations.
All crypto operations will be subject to different levels of financial monitoring depending on the amount and destination of each transaction. The Ministry of Digital Transformation, which has been quite active this year, will be tasked to regulate the circulation of virtual assets in Ukraine. It will also conduct oversight to verify compliance with AML regulations in the crypto sphere.
Altcoins: LTC, EOS, XLM, Atom and XTZ 52884
Joe Trohman November 11, 2019 10:59 November 11, 2019
XTZ/USD
Tezos (XTZ) has been the best performer of the past seven days, rising about 43% during the period. It received a boost on the news of its listing on the crypto exchange OKEx.
Another positive news was that Coinbase introduced staking rewards for all Tezos holders, enabling them to earn about 5% annually. After the recent rally, is it a good time to book profits or can the move up extend further?
XTZ/USDT weekly chart. Source: Tradingview
The bulls have held the support at $0.829651 for the past few months. This indicates that buyers have been eyeing dips down to this support level to accumulate. The XTZ/USD pair has surged this week and has risen above both moving averages. It can now move up to $1.85. The pair has been pushed back down from this resistance on two previous occasions, hence, a breakout would be a significant event.
On a close above $1.85, a rally to $2.87 and above it to $3.37 is likely. Therefore, traders can buy on dips close to $1.1 levels. Our bullish view will be invalidated if the price turns down and slips below the recent low of $0.75.
ATOM/USD
Cosmos (ATOM) has been gradually moving higher for the past few days. It has risen by about 24% in the past seven days and is the second-best performer. Can it continue its up move or has it run its course? Let's analyze its chart.
ATOM/USD weekly chart. Source: Tradingview
The ATOM/USD pair has found support close to $2 levels thrice in the past few months. This shows that bulls have been buying on dips to the support levels. With the rally this week, the price has scaled above the overhead resistance at $3.6043, thus completing a triple bottom formation.
It now has a minimum target objective of $5.2985. If the bulls push the price above this level, a rise to $7 is possible. Therefore, traders can buy on a close (UTC time) above $3.6043 and keep a stop loss of $2.90. Contrary to our assumption, if the price turns around from the current levels and plummets below $3.6043, it will suggest a lack of demand at higher levels.
Stellar (XLM) surged on the news that the Stellar Development Foundation (SDF) had burned over 55 billion of the 85 billion tokens that were set aside for SDF operations, giveaway programs and partnership programs.
The foundation said that the burning was done because it wanted to be leaner and more efficient. After its 12% rally, can it rise further in the next few days or will the price dip due to profit booking? Let's study the chart.
XLM/USDT weekly chart. Source: Tradingview
The XLM/USD pair is facing selling at the overhead resistance of $0.088708. With the rally this week, the price has risen above the previous low of $0.072545. This shows that the markets have rejected the lower levels. Both moving averages are flattening out and the RSI has risen to just below 50 levels, which shows that the sellers are losing their grip.
If the bulls can defend the support at $0.072545 during the next dip, it will be a positive sign. On the upside, the pair will pick up momentum on a breakout of $0.088708 and the 50-week SMA. Above the 50-day SMA, a rally to $0.145 will be on the cards.
Therefore, traders can initiate long positions on a close (UTC time) above the 50-week SMA and keep a stop loss of $0.067. Our bullish view will be negated if the price turns down from $0.088708 and sustains below $0.072545.
Litecoin (LTC) has been attempting to form a bottom in the past few days. It extended its up move with an 8% rally in the past seven days. What are the critical levels to watch out for and when does it start a new uptrend?
LTC/USDT weekly chart. Source: Tradingview
The LTC/USD pair has broken out of the downtrend line and is attempting a recovery. If it sustains above $62.0764, it will suggest that the lower levels are attracting buyers. There is a minor resistance at the moving averages above which a move to $80.2731 will be on the cards. The flattening moving averages and the RSI moving up slowly suggests that the bulls are making a comeback.
Though positive, we do not find a reliable buy setup at the current levels, hence, we are not recommending to trade it. Our bullish view will be invalidated if the pair turns down and plunges below the recent lows of $47.1851.
EOS/USD
EOS rounded up the top five crypto performers of the past seven days with a rally of about 8%. There have been complaints of congestion in the network but it has not affected its price. Has it bottomed out? Let's find out.
EOS/USD weekly chart. Source: Tradingview
The EOS/USD pair has risen close to the moving averages, which are likely to act as minor resistance. If the bulls can push the price above this resistance, a move to $4.8719 is likely. The flattening moving averages and RSI just below the center indicate that the selling pressure has reduced.
Contrary to our assumption, if the pair turns down from the moving averages and plummets below the recent lows of $2.4001, it can retest the yearly low. However, we give it a low probability of occurring. We do not find a trade setup that offers a good risk to reward ratio at the current levels, hence, we are not recommending taking a long position in it.
The views and opinions expressed here are solely those of the author and do not necessarily reflect the views of Cointelegraph. Every investment and trading move involves risk, you should conduct your own research when making a decision.
The market data is provided by the HitBTC exchange.
OKEx Announces Listing of Tezos (XTZ) 54994
Joe Trohman November 4, 2019 11:16 November 4, 2019
OKEx, the world's largest futures cryptocurrency exchange, today announced it will list Tezos (XTZ), an open-source platform for assets and applications backed by a global community of validators, researchers, and builders. According to Cryptocompare, Tezos is the 21st most popular cryptocurrency with a market cap of over $695 million USD and a 24-hour trade volume of over $206k USD.
The depositing of XTZ will be available from 09:00 November 6, 2019 (UTC). XTZ spot trading against USDT and BTC will open at 09:00 November 7, 2019 (UTC). XTZ withdrawal will open from 09:00 November 8, 2019 (UTC).
"Tezos is a highly respected project with a robust community, and we're happy to be able to add the value of the XTZ network to the OKEx ecosystem, where we strive to deliver a one-stop-shop for professional and retail traders," said Andy Cheung, Head of Operations of OKEx.
Created by former Morgan Stanley analyst Arthur Breitman and Kathleen Breitman, Tezos is a multi-purpose platform for decentralized applications and smart contracts. Stakeholders govern upgrades to the core protocol, including upgrades to the amendment process itself without having to fork the network into two different blockchains. The project aims to promote long-term upgradability and open participation to support mainstream adoption of blockchain technology.
"We are looking forward to a thriving relationship with OKEx, a global leader in the blockchain space, in furthering the Tezos ecosystem together in Asia and throughout the world," said Corey Soreff, Board of Directors of Tezos Commons Foundation.
OKEx's listing review process sets a high standard in many aspects, including important pillars ranging from project quality (i.e. (legal) qualifications, business model and structure, promotion etc.) to project community (i.e. ecosystem-wise capacity and promotion opportunity). By taking every possible measure, OKEx strives to ensure every listed project delivers practical use cases and brings in market liquidity.
The addition of Tezos accompanies the November launch of OKEx's USDT Futures Trading, a linear futures contract. With USDT Futures, OKEx traders can long a position to profit from the increase of a cryptocurrency's price, or short a position to profit from the decline of a cryptocurrency's price. USDT pairs on OKEx include BTC, ETH, BCH, EOS, XRP, BSV, and TRX with a leverage level of 1-100x, both in fixed and cross-margin mode.
Former CFTC chair and 'Crypto Dad' says 2019 is the year to get serious about crypto policy 53517
Former CFTC Chairman Christopher "Crypto Dad" Giancarlo left his role at the regulator this summer, but now he's stepping even further into the crypto world. Now as a board member for the Chamber of Digital Commerce, Giancarlo is using his expertise to further policy talks across the board, not just on a CFTC/SEC level. Giancarlo sat down with The Block to talk transitions in the regulatory space, as well as his own transition in the industry.
The Block: At the time of your departure from the CFTC how do you feel attitudes toward crypto have shifted from the time you took the position as chairman?
Giancarlo: I think quite dramatically, one of my perspectives on the CFTC at the time I took the helm, it was inordinately backward-looking, perhaps justifiably, but inordinately. So much of its energy and attention was on completing Dodd-Frank reforms to derivative markets. And in Dodd-Frank, there's no mention about anything technological including crypto assets or blockchain, and the agency's attention was primarily drawn to that. To its credit, the agency had developed an informal working group under Jeff Bandman that was looking at emerging crypto assets, primarily bitcoin. It was that effort that I accelerated with the formation of Lab CFTC.
The Block: How are you looking at the EOS and Sia settlements with the SEC? Can you provide any insight into what might have gone on there?
Giancarlo: One of the things I want to avoid doing is commenting on activities by the agency. I'm not the type, and don't want to be seen as the type of former chairman who is looking over the shoulder of the incoming team and commenting on them. What I would say is that it generally shows a continuing focus and attention in this space by the agencies.
The Block: How have you seen the attitudes of regulatory bodies shift in recent years? What is the current attitude characterized by?
Giancarlo: 2017 I think was the year that regulators really woke up to the accelerating pace of crypto assets because of the bitcoin bubble. I would say 2019 is the year in which there's a growing recognition that regulators and policy makers need to do more than just be aware of these, but may actually need to look at some policy responses. And I think the thing driving that in 2019 is a combination of Libra and the prospects for central bank digital currencies.
The Block: Looking at recent developments, we saw Regulation A+ emerge as a possibly compliant way forward for some companies. What is your take on that innovation, and what other innovations can we perhaps expect to see?
Giancarlo: I think that it shows that the SEC under Jay Clayton is moving beyond just getting smarter and more aware, but actually thinking about some of the policy responses.
The Block: I understand you've said you're not interested in looking over new leadership's shoulder, but members of the SEC have said they feel federal securities laws adequately tackle digital assets because they are technology agnostic. Do you see the legislation as operating fully for the digital assets space?
Giancarlo: I would go further afield than just SEC/CFTC specific. I do think the time has come for thoughtful consideration of a digital dollar. I think that the dollar's status as the world's primary reserve currency should be enhanced with a digital component and done in a way that doesn't have to disintermediate the traditional banking system but can be done so traditional finance financial intermediaries can play a role in a digital component to the dollar. I don't see the Federal Reserve becoming a deposit-taking institution, but where banks would continue to do that, but would use a uniform set of technology protocols in order to provide access to a digital dollar format.
The Block: Looking into some of your ventures now, I understand you've joined the Chamber of Digital Commerce.
Giancarlo: Yes. I've been very impressed with their work over the past several years in serving as a sort of a go-between. There's this FinTech phase of innovation and policymakers helping to both translate the technology into concepts and things that policymakers can understand and interact with and giving policy makers a similar ability to interact with the innovators. And so I think with the chamber, almost uniquely, I think provides a very good interface between the world of Washington and the world of financial innovation.
The Block: I remember I read somewhere when you joined that you were looking to streamline and modernize the regulatory landscape. Which areas are you specifically immediately looking to streamline and modernize in your role on the board?
Giancarlo: As a former chairman, I will leave the new leadership to focus on CFTC-related matters, and I have every confidence in the leadership of the SEC to do that. But there's a need for policy makers and innovators to come together to lay down a policy framework upon which these innovations can move forward with greater regulatory and legal certainty. A good example, although not a perfect analogy, is the period during the Clinton Administration when a Republican Congress and the Democratic administration came together to develop the foundation for the first phase of the internet revolution, the digitalization of information.
It's a do-no-harm approach that allowed for very rapid innovation and global leadership. The reason why it's not a perfect precedent is because digitization of information is a process that certainly falls into a regulatory light zone because of the First Amendment restrictions on regulation of speech. When it comes to financial areas, that's always been a regulatory heavy zone. Some of the same precedents don't apply, but what I think should apply is that same commonality of purpose between policy makers and innovators to want to create a regulatory and policy foundation upon which innovation can proceed in a way that is, intelligent, that's thoughtful, aware of policy concerns, whether it be about privacy, whether it be about appropriate anonymity, whether it'd be about regulatory transparency,or for oversight where we can make sure that the right policy imperatives are brought to bear and yet a framework that can bring certainty to innovators so that they can move forward with innovation, and knowing the consequences of decisions they make.
So what I hope to do in my post-CFTC life is be an advocate for sound policy development. Not CFTC or SEC specific, but across the board where I can use skills I have as a communicator and an advocate for innovation in a well-regulated environment, and with relationships I build, hopefully help communicate that message for the need for sound policy here in the U.S. So that once again the U.S. can emerge as a leader in this new phase of the digitization of our modern world, digitization of finance in this case.
The Block: You've been affectionately dubbed "Crypto Dad" by the industry. How do you feel that title came about and where did this generosity towards digital assets stem from, especially in a regulatory environment some consider hostile towards the digital asset world?
Giancarlo: So the title came about out of that February, 2018 hearing that chair Clayton and I did before the Senate Banking Committee. We had been asked to testify on Bitcoin and crypto assets. We'd prepared a lengthy written testimony to Congress, and the night before, I was preparing my oral remarks and I looked at this 60-plus page, 100-footnote document and said, you know, I cannot summarize that in five minutes. I went in the next morning and when I was asked to speak in my opening address, I actually put the paper down and said, look, you've got my written submission. What I want to do is speak to you for a moment not as a chairman of a regulatory agency, but as a father, as a dad. I explained that I had just come back from our annual family ski trip with my children and my nieces and nephews and at the dinner table every night all they wanted to talk about was Bitcoin.
My children grew up in a financial family. I've worked on Wall Street for my career and I've tried to interest my kids in the stock market since they were young, and they had no interest in it and suddenly they're very, very interested in Bitcoin. And some of my nieces, in fact, one of my nieces was a bitcoin holder. I said to Congress, I noticed some of your heads are nodding, some of you probably have the same conversations. I said, it strikes me that we owe it to this generation to treat their interest in this new asset class not with derision and disdain, but with respect. If nothing else, we owe it to them to get the policy right so that they're not prey to fraud and misappropriation, but more that they can build a framework upon which they can build this new structure.
And it was from that my Twitter account exploded and I was dubbed Crypto Dad. It's something that was unexpected, delightful at the same time, but more importantly, I think it showed that I stumbled onto something. We have a generation that came of age in a financial crisis when all of the traditional institutions that were supposed to provide stability and certainty seem to have fumbled, and in some cases failed. I think there is a generational interest here that is not going away, is not worthy of being dismissed, but should be taken seriously. I take it seriously, and I must say that, in the early days of the internet, and I'm old enough to have been around them, the same people dismissing crypto assets now were dismissing the internet as good-for-nothing other than access to pornography. It was dismissed as a fad that would fade. Well in fact you can't even hail a taxi today without using an internet app. Our whole world has been dramatically changed in many cases for the better by the internet. What I called it, the digitalization of information, well now we're on the digitalization of assets, the digital tokenization of a value, and it's as fundamental a change as the early internet was.
I think for the adults in the room the choice is to put down solid policy frameworks upon which this new phase of digitalization can be built or to do nothing and see jurisdictions that put down solid policy foundations become the leaders in innovation. I don't want to see the U.S. left behind. I want to see the U.S. do what it's traditionally done in the face of technological revolution. And that is take a leadership role. I think that's achievable if we put down the right policy prescriptions, and that's why I'm pleased to join the Chamber of Digital Commerce, because it advocates for solid policy foundations and does a great job of putting the right experts forward and engaging the right communications with policy makers to maximize the opportunities. So if Crypto Dad can answer that effort, then I'm delighted to do so.
Source: Theblockcrypto.com
Why Bitcoin Holding $4,950 Readies BTC For A Push To $6,000
Can SegWit Support Make the Bitcoin Price Rise Even Higher?
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Q: opencv cv::max behavior unexpected I'm using cv::max with an uninitialized Mat object, and error happens when passing the uninitialized object as the 1st param:
Mat a=Mat::ones(2,3, CV_32S);
Mat b;
max(a, b); // 1. OK
max(b, a); // 2. OpenCV Error
The error message is:
OpenCV Error: Sizes of input arguments do not match (The operation is
neither 'a rray op array' (where arrays have the same size and type),
nor 'array op scalar' , nor 'scalar op array') in cv::binary_op, file
C:\builds\2_4_PackSlave-win32-vc
11-shared\opencv\modules\core\src\arithm.cpp, line 1021
My question: Should not the two calls be of the same effect theorectically? Is that a implementation imperfection or my misunderstanding?
EDIT:
I'm using vs2012 with OpenCV2.4.8 x86 on win7 x64
A: This was a bug, but has now been fixed. Please see http://code.opencv.org/issues/3696#note-7 for comment stating that the bug has been fixed.
Note: An equivalent problem existed for cv::min.
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Mauvezin-d'Armagnac è un comune francese di 106 abitanti situato nel dipartimento delle Landes nella regione della Nuova Aquitania.
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Celebrating Kate Spade
Yesterday, we were struck with the incredibly sad news that Katherine Noel Frances Valentine, (known as Kate Spade) the global handbag and fashion designer, was found dead in her New York apartment aged 55 with signs of suicide being the cause of death. Spade was known in the industry for her one-of-a-kind whimsical charm that shone through her innocent, playful aesthetic. She touched so many hearts with her charismatic, empowering designs that carried women into adulthood. Here, we look back at some of the key moments that defined her incredible career.
Launching Her Brand, 1993
Kate started out as Accessories Editor at Mademoiselle, but after realising she couldn't buy a bag that she truly loved and acknowledging a gap in the market, she set out to 'design the perfect handbag'. With the encouragement and backing of her future husband, Andy Spade, Kate launched her eponymous collection, which featured the six styles which we now see as the brand's signature designs.
New York Shop Opening, 1996
After three years of continual success, which led to the brand being stocked by retail giants such as Charivari, Barneys, Fred Segal, Saks Fifth Avenue and Bloomingdale's, Kate had the security and purpose to open her very own shop in Soho, NYC. Industry insiders like Anna Wintour and Linda Wells were carrying Kate Spade bags, as were celebrities like Julia Roberts and Gwyneth Paltrow. She was fast becoming a fashion favourite.
Receiving the CFDA International Award, 1996
In the same year that she opened her first store, Kate earned her first accolade; the International Award from the Council of Fashion Designers of America. She would later go on to win the award again in 1998. The CFDA paid tribute to Kate yesterday saying, "The CFDA is devastated to hear the news of our friend, colleague, and CFDA member Kate Spades's tragic passing. She was a great talent who had an immeasurable impact on American fashion and the way the world viewed American accessories."
Launching Jack Spade, 1996
It was a busy year for Kate and her brand; also in 1996, the American designer expanded her womenswear collection to include footwear, journals and other accessories whilst she also launched her first menswear accessories line, Jack Spade.
Neiman Marcus Group Buys 56% Stake, 1999
Although her and husband Andy continued to run the brand's operations, the Neiman Marcus Group paid $33.6million to acquire a 56 percent stake in the Kate Spade brand. This led to Kate expanding into Asia and for Jack Spade to open its first store in NYC.
The Expansion Into Lifestyle, 2002
In 2002, Kate initially partnered with Estée Lauder on a range of fragrances but then additionally inked deals with Lenox in 2004 to expand in to homeware. A consistent amount of success and a continuing expansion in merchandise turned Kate Spade into something more than simply a fashion brand; it became a lifestyle brand.
KateSpade.com, 2004
After reporting $70million in sales in their 10-year anniversary, Kate Spade website was launched. Being completely ahead of her time, Kate was always at the forefront of digital acquisition in fashion, with many functions in her stores already being assisted digitally years before other brands turned their attention to technological advancements.
Liz Caliborne Inc. Buy Kate Spade, 2007
The birth of Kate's child, Frances, made her consider life from a different perspective. In 2007, her and her husband sold the company for a reported €125million to Liz Calibourne Inc. The Spades left the company shortly after allowing them to focus on raising their new born daughter and for the brand to have the full attention it needed to continue its success. The brand flourished under its new owners and was bought by Coach in 2017 for €2.4billion.
Frances Valentine, 2016
After her 10-year long sabbatical from fashion, Kate returned with her husband Andy, and their business partners and friends Elyce Arons and Paola Venturi. They launched Frances Valentine, a range of shoes and accessories named after her daughter and previous family members.
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\section{Introduction}
Ever since the experimental realization of Bose-Einstein condensation, ultracold atoms have been used to study an enormous diversity of quantum effects with an unprecedented degree of control \cite{pitaevskii,pethick}. Advancement in tools like optical lattices \cite{bloch07} offer us the chance to explore e.g. second-order tunneling \cite{foelling07}, quantum phase transitions \cite{greiner02} and non-equilibrium quantum dynamics of driven systems \cite{kierig08}.
The double well especially serves as a prototype system to study interference and tunneling in great detail. For instance the tunneling dynamics of a Bose-Einstein condensate has been observed to undergo Josephson oscillations \cite{albiez05,milburn97,smerzi95} in which the population simply tunnels back and forth between the two wells. However when the interaction is raised beyond a critical value the atoms remain trapped in one well, a non-linear phenomenon known as self trapping \cite{albiez05,smerzi95,anker05}.
Of special interest are systems in lower dimensions which often display unique features. Quasi-one-dimensional (1D) Bose gases have been prepared experimentally by freezing the transverse degrees of freedom. There it is possible to tune the interaction strength between the atoms by either using confinement induced resonances \cite{Olshanii1998a} or magnetic Feshbach resonances \cite{koehler06}. Thus one can study the crossover from a weakly interacting to a strongly correlated regime.
A particularly interesting case is the Tonks-Girardeau gas appearing in 1D in the limit of infinitely repulsive short-ranged interaction which has been recently observed experimentally \cite{kinoshita04,paredes04}. This gas of impenetrable boson is isomorphic with that of free fermions via the Bose-Fermi mapping \cite{girardeau60} and all the local properties are identical to the free fermion system. The gas still retains the bosonic permutation symmetry and so the non-local quantities differ from the fermionic case.
Theoretically the quantum dynamics in the weakly interacting case has been studied using the Bose-Hubbard model assuming the validity of a lowest band approximation \cite{salgueiro06,dounasfrazer07a,dounasfrazer07b,wang08}. These studies illuminate relevant tunneling mechanisms and resonances. However, to capture the rich physics present in the stronger interaction regime we need to go beyond the Bose-Hubbard limit. Moreover numerically exact calculations of the quantum dynamics for few bosons through a one-dimensional potential barrier \cite{alexej08} or a bosonic Josephson junction \cite{sakmann09} reveal deviations from the results obtained with mean-field calculations as well as establish a difference between the dynamics in attractive and repulsive bosonic systems \cite{sakmann10}. The crossover from the uncorrelated to fermionization regime has been investigated for few bosons \cite{zoellner07a,zoellner08} and reveals a transition from Rabi-oscillations to fragmented pair tunneling via a highly delayed tunneling process analogous to the self-trapping for condensates. The quantum dynamics of an asymmetric double-wells while keeping a constant interaction strength has been explored in refs. \cite{dounasfrazer07a,dounasfrazer07b,zoellner07a,zoellner08}.
In this investigation we go one step further and envision a new approach to asymmetry by introducing an inhomogeneous, i.e., spatially varying interaction strength. This can be achieved experimentally by employing magnetic field gradients in the vicinity of Feshbach resonances or by combining magnetic traps with optically induced Feshbach resonances \cite{koehler06,bauer09}. We will demonstrate how spatially varying interaction strengths enrich the tunneling dynamics in the most fundamental case of a double-well. This has to be seen as a potential ingredient for more complex problems such as quantum transport in optical lattices. Specifically we study the crossover from the non-interacting to the fermionization limit for a fixed inhomogeneity ratio of interaction and the effect of varying inhomogeneity ratio. An interplay of suppression and resumption of tunneling is observed. For three or more particles, the interaction asymmetry can be tuned to generate various many-particle tunneling resonances. Lastly we examine a tilted double-well and investigate the interplay between the tilt and inhomogeneity to generate tunneling resonances.
The paper is organized as follows. In Section \ref{sec:setup} we discuss our model and setup. In Sec. \ref{sec:mctdh} we briefly describe our computational method. Subsequently we present and discuss the results for tunneling in a symmetric double well for two atoms first (Sec. \ref{sec:2p_dynamics}) followed by more atom systems (Sec. \ref{sec: many particle}). In Sec. \ref{sec:asymmetric} we discuss the case of an asymmetric double well.
\section{Setup and Interactions\label{sec:setup}}
Our Hamiltonian (for $N$ particles) is given by (see \cite{zoellner06a} for details) \\
\begin{equation}H = \sum_{i=1}^N [\frac{1}{2} {p_i}^2 + U(x_i)] + g\sum_{i<j} \delta(x_i - x_j) \end{equation}
The double well
$U(x) = \frac{1}{2} x^2 + h\delta_{\omega} (x)$
is modeled as a harmonic potential with a central barrier shaped as a Gaussian $\delta_{\omega}
(x) = \frac{e^{-x^2/2\omega^2}}{\sqrt{2\pi}\omega}$ (of width $\omega = 0.5 $ and height $h = 8$, where dimensionless harmonic-oscillator units are employed throughout).
For ultracold atoms only the s-wave scattering is relevant and the effective interaction in 1D can be written as a contact potential \cite{Olshanii1998a} which we sample here by a very narrow Gaussian.
We focus on repulsive interaction only.
The inhomogeneity of the interaction is modeled as \cite{zoellner06a} \\
\begin{center}
$g(R)= g_0[1 + \alpha \tanh (\frac{R}{L})]$, \\
\end{center}
where $2R=x_i + x_j$ and $L$ is the modulation length which we fix at $L = 1$. \\
For $R\gg L$, $g$ takes the asymptotic values \\
\begin{center}
$g_{\pm} = g_0 (1\pm \alpha)$.
\end{center}
Thus the parameter $\alpha$ regulates the relative difference in interaction strength between the left and the right well,
\begin{center}
$\Delta g \equiv \vert g_+ - g_-\vert = 2g_0 \alpha$, \\
\end{center}
and the corresponding ratio is given by
\begin{center}
$\frac{g_+}{g_-} =\frac{1+\alpha}{1-\alpha} $. \\
\end{center}
\section{Computational Method\label{sec:mctdh}}
Our goal is to study the bosonic quantum dynamics for weak to strong interactions in a numerically exact fashion. This is computationally challenging and can be achieved only for few atom system.
Our approach is the Multi-Configuration Time Dependent Hartree (MCTDH) method \cite{meyer90,beck00} being a wave packet dynamical tool known for its outstanding efficiency in high dimensional applications.
The principle idea is to solve the time-dependent Schr\"odinger equation \\
\begin{center}
$i\dot{\Psi}(t) = H\Psi(t)$ \\
\end{center}
as an initial value problem by expanding
the solution in terms of Hartree products $\Phi_J \equiv {\varphi_j}_1 \otimes \ldots \otimes {\varphi_j}_N$ :
\begin{equation}\Psi(t) = \sum_J A_J(t)\Phi_J(t).\label{eq:mctdh}\end{equation}
The unknown single particle functions $\varphi_j(j=1,...,n$, where $n$ refers to the total number of single particle functions used in the calculation) are in turn represented in a fixed primitive basis implemented on a grid. The correct bosonic permutation symmetry is obtained by symmetrization of the expansion coefficient $A_J$.
Note that in the above expansion, not only are the coefficients $A_J$ time dependent but also the single particle functions $\varphi_j$.
Using the Dirac-Frenkel variational principle, one can derive the equations of motion for both $A_J$ and $\Phi_J$.
Integrating these differential equations of motion gives us the time evolution of the system via (\ref{eq:mctdh}).
This has the advantage that the basis $\Phi_J(t)$ is variationally optimal at each time $t$. Thus it can be kept relatively small, rendering the procedure more efficient.
Although MCTDH is designed primarily for time dependent problems, it is also possible to compute stationary states. For this purpose the \textit{relaxation} method is used \cite{kos86:223}. The key idea is to propagate a wave function $\Psi_0$ by the non-unitary operator $e^{-H\tau}$. As $\tau \rightarrow \infty$, this exponentially damps out any contribution but that stemming from the true ground state like $e^{-(E_m - E_0)\tau}$. In practice one relies upon a more sophisticated scheme called the \textit{improved relaxation} \cite{mey03:251,meyer06} which is much more robust especially for excited states. Here $\langle\Psi\vert H \vert \Psi \rangle$ is minimized with respect to both the coefficients $A_J$ and the orbitals $\varphi_j$. The effective eigenvalue problems thus obtained are then solved iteratively by first solving $A_J$ with fixed orbital $\varphi_j$ and then optimizing $\varphi_j$ by propagating them in imaginary time over a short period. This cycle is then repeated.
\section{Tunneling Dynamics for Two Boson System\label{sec:2p_dynamics}}
We first focus on the tunneling dynamics in a symmetric double-well with two bosons initially ($t=0$) prepared in the left well. This is achieved by adding a tilt or a linear potential $dx$ to the Hamiltonian hence making the left well energetically favorable. Instantaneously the ground-state is obtained by applying the relaxation method (imaginary time propagation). For reasonably large $d$, this results in achieving a complete population imbalance between the wells. With this state as the initial state, the tilt is instantaneously ramped down ($d=0$) at $t=0$ to study the dynamics in a symmetric double-well. Our aim is to study the impact of the correlations between the bosons on the tunneling dynamics both with respect to the interaction strength as well as the spatial inhomogeneity. We start by fixing the inhomogeneity to $\alpha = 0.2$ (with the left well having lower interaction than the right) and analyze how the dynamics varies with changing interaction strength $g_0$.
\subsection{Dynamics from the uncorrelated to the fermionization limit.\label{sub:2p-cor_dynamics}}
\begin{figure}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{fig1.eps}
\caption{(color online) Population of the right-hand
well over time, $p_{\mathrm{R}}(t)$, for different interaction strengths at $\alpha = 0.2$ for two bosons. \textit{Inset}: Long time behavior for very low interaction strength $g_0=0.005$. Barrier height $h=8$ and width $\omega = 0.5 $ has been used for all calculations. (all quantities are in dimensionless harmonic oscillator units throughout).\label{cap:2p_g_var}}
\end{figure}
In the absence of any interaction $g_0=0$, the bosons undergo Rabi oscillations between the two wells. This is characterized by complete tunneling of both bosons between the two wells with a single frequency and can be quantified by the time variation of the population of the atoms in the right well
\begin{eqnarray*}P_R (t) = \langle \Theta (x) {\rangle}_{\Psi(t)} = {\int_0}^\infty \rho(x;t)dx \end{eqnarray*} where $\rho$ is the one-body density.
Figure \ref{cap:2p_g_var} shows that $P_R$ oscillates sinusoidally between $0$ and $1$.
If we introduce a very small interaction $g_0 = 0.005$ (inset) the Rabi oscillations give way to a beat pattern due to the existence of two very close frequencies.
Increasing the interaction strength further ($g_0=0.2$), we observe a suppression of tunneling with the maximum population in the right well ${P_R}^{max}\approx 0.2$. This is a manifestation of the inhomogeneous interaction which drives the tunneling off-resonance and should be carefully distinguished from the delayed pair-tunneling and self-trapping for the same $g_0$-value in the case of homogeneous interactions (see below). The dynamics consists of a slow tunneling envelope with suppressed amplitude, which is modulated by a faster oscillation. For higher values of interaction strength $(g_0 = 4.7)$, the tunneling is completely suppressed. What remains is a fast oscillation with a tiny amplitude.
However, contrary to the naive intuition a reappearance of tunneling occurs for larger values of the coupling strength. We observe a partial restoration of tunneling with ${P_R}^{max} = 0.7$ for the value $g_0=150$, which is close to the so called fermionization limit. The dynamics is characterized by two frequencies - one very close to the Rabi frequency modulated by a faster oscillation. Ideally at the fermionization limit $g_0 \rightarrow \infty$, the system of hardcore bosons maps to a system of free fermions \cite{girardeau60} and all the local properties are identical. Hence in this limit we would have complete two-mode single particle tunneling analogous to tunneling of two free fermions.
Before we move on to analyze in detail the above observations, let us comment briefly on the differences between the behavior observed in our setup and the case of a symmetric double-well with homogeneous interaction. Clearly both for the non- and infinitely interacting limits the inhomogeneity doesn't play a role. For homogeneous interactions and a symmetric trap tunneling is always resonant and complete. However, different strength of interaction yield different dynamics like a transition from pair-tunneling for low interaction strength to a self-trapping mechanism for larger interaction strength which is characterized by extremely long tunneling times \cite{salgueiro06,tonel05,creffield07,zoellner07a,zoellner08}. In our case though we observe an actual suppression of the tunneling amplitude and not so much a delayed process.
In case of an asymmetric-well with homogeneous interaction, the effects in the low interaction regime are equivalent to our set-up: The tilt has the same effect as an interaction asymmetry, namely it destroys resonant behavior thereby leading to a suppression of tunneling \cite{dounasfrazer07a,dounasfrazer07b}. Nevertheless, our case is fundamentally different and this is evident in the strong interaction regime. Specifically the reemergence of tunneling we observe does not occur in the tilted double-well system.
\subsection{Analysis\label{sub:analysis}}
\begin{figure}
\includegraphics[width=0.85\columnwidth,height=4.5cm]{fig2.eps}
\includegraphics[width=0.45\columnwidth,keepaspectratio]{fig5a.eps}
\includegraphics[width=0.45\columnwidth,keepaspectratio]{fig5c.eps}
\caption{(color online):(a) Two particle energy spectrum as a function of the interaction strength $g_0$ for $\alpha = 0.2$. \textit{Inset}: Lowest energy levels for low interaction strength.
\textit{Bottom}. Few body energy spectrum with $g_0$ for (b) $\alpha$ = 0 and (c) $\alpha$ = 0.2.
\label{cap:spectrum}}
\end{figure}
The understanding of the above-described dynamics lies in the variation of the few body spectrum as $g_0$ is changed from zero to the fermionization limit (Fig.\ref{cap:spectrum}(a)).
Considering the wavefunction $\Psi(t)= \sum_m e^{-iE_mt}c_m \Psi_m$ with energy $E_m$ corresponding to the stationary state $\Psi_m$, the population imbalance $\delta(t) \equiv \langle\Theta(x) - \Theta(-x)\rangle_{\Psi(t)} $ can be computed to be \\
\begin{equation}\delta(t) = 4\sum_{m<n} {W_m}_n \cos ({\omega_m}_n t) + 2\sum_m {W_m}_m -1 \label{eq:contribution},\\
\end{equation}
where
${W_m}_n = \langle \Psi_m\vert\Theta(x)\vert\Psi_n\rangle c_m c_n$ and ${\omega_m}_n = E_m -E_n$. \\
The energy spectrum of both the non-interacting and the fermionization limit can be understood from the single particle energy spectrum of the double well, which is in the form of bands each pertaining to a pair of symmetric and antisymmetric orbitals.
In the uncorrelated limit ($g_0 \rightarrow 0$) the low-lying energies of the spectrum are obtained by distributing the atoms over the symmetric and antisymmetric single particle orbitals in the first band. This leads to $N+1$ energy levels, $N$ being the number of bosons.
$E_m = E_0 + m{\Delta}^0$ with $m =0,...,N$ where ${\Delta}^0 = \epsilon_1 - \epsilon_0$ is the energy difference between the two single particle orbitals in the first band. Thus for $g_0 =0$ the levels are equidistant (Fig.\ref{cap:spectrum}(a) inset) and we see Rabi oscillation with frequency ${\omega_0}_1 = {\omega_1}_2 = {\Delta}^0$.
As the interaction is increased ($g_0 = 0.005$), this equidistance is slightly broken (${\omega_0}_1 \simeq {\omega_1}_2$) and we get a superposition of two very close frequencies. This results in the formation of the beat pattern seen in the dynamics for $g_0 = 0.005$.
To understand the dynamics in the low interaction regime, it is instructive to map our system to a two-site Bose-Hubbard Hamiltonian \cite{Jaksch98,Fisher89}
\begin{equation}
\hat{H} = -J(\hat{c}^{\dagger}_L\hat{c}_{R} + \hat{c}^{\dagger}_R\hat{c}_{L}) + \sum_{j=L,R} \frac{U_j}{2}\hat{n}_j\left(\hat{n}_j-1\right)
\end{equation}
where $J$ is the tunneling coupling, $U_{L,R}$ is the on-site energy of the left/right well and $\hat{n}_j\equiv\hat{c}_j^{\dagger}\hat{c}_j$.
Before proceeding, we note here that there is no direct connection between the time-dependent SPF used within our numerical M.C.T.D.H. calculations and the parameters of the Bose-Hubbard (B-H) Hamiltonian. In the standard B-H model (which is valid in the weak interaction regime) the parameters $J$, $U_L$, $U_R$ are time independent constants while the shape of the orbitals such as the localized Wannier functions retain the shape throughout the course of the dynamics. Moreover, even for low energies the two most occupied modes for propagation do not necessarily coincide with the two modes of the B-H model. In this weak interaction regime, the B-H model is just a good approximation to our exact calculation and thus we have used it as solely an explanatory tool to analyze the results.
Using the B-H Hamiltonian for $U_L,U_R\gg J$, the highest two eigenvalues are approximately $U_R$ and $U_L$. Whereas in the homogeneous case $\alpha = 0$ these two levels are close to degenerate $U_L \approx U_R$ (Fig.\ref{cap:spectrum}(b)), here we have a breaking of the parity symmetry since $U_R > U_L$ (Fig.\ref{cap:spectrum}(c)). This is understandable since two particles localized in the left well have lower energy than two particle in the right well leading to the energy level separation seen in Fig.\ref{cap:spectrum}(c). In terms of the number-state representation in the localized basis $\vert {N _L}^{(0)} ,{N_R}^{(0)}\rangle$ the degenerate eigenstates for the homogeneous case read
\begin{center}$\phi_{1,2}\approx \frac{1}{\sqrt{2}} (|0,2\rangle \pm |2,0\rangle $) \end{center}
and consequently the dynamics consists of shuffling the probability between the two states corresponding to a complete two particle tunneling.
In the case of sufficiently strong inhomogeneous interaction, the removal of the degeneracy of the energy levels leads to a decoupling of the eigenstates into localized number-states
\begin{center}$\phi_1\approx |2,0\rangle$ , $\phi_2\approx |0,2\rangle $ \end{center}
This implies that the initial state $\psi(t=0) = \vert 2 ,0\rangle$ is very close to the 1st excited state $\phi_1$ and thus is effectively a stationary state of the system. This results in the suppression of tunneling for corresponding values of $g_0$
In the fermionization limit ($g_0\rightarrow \infty$) the system possesses the same local properties as a system of non-interacting fermions due to the Bose-Fermi mapping \cite{girardeau60}. Thus in an ideal case the inhomogeneity doesn't manifest ($g_{\pm}\rightarrow \infty$) and the tunneling dynamics is identical to a system of free fermions. As an idealization if we consider the initial state as two non-interacting fermions in the left well, then they would occupy the lowest two orbitals localized in the left well. In terms of the single particle eigenstates of the double well $| n_{a_{\beta}}^{(\beta)} \rangle$ where $n_{a_{\beta}}^{(\beta)}$ denotes the occupation number of the symmetric
($a_{\beta}=0$) or antisymmetric ($a_{\beta}=1$) orbital in band
$\beta$, the tunneling frequencies ${\omega}_{nn'} = E_n - E_n'$ are given by \cite{zoellner08}
\begin{equation} {\omega}_{nn'} = \sum_\beta \Delta^\beta\underbrace{({n_1}^\beta - {n'_1}^\beta)}_{= 0,\pm 1}\label{eq:fermi-frequ}\end{equation}
where $\Delta^\beta$ denotes the energy splitting of the band $\beta$ , ${n_1}^\beta$ represents the occupation of the anti-symmetric orbital of the band $\beta$.
Thus for two particles the contributing frequencies are the lowest band Rabi frequency ${\Delta}^0$ and the tunnel splitting of the first excited band ${\Delta}^1$. The tunneling dynamics can be pictured roughly as two fermions tunneling independently in the first two bands.
In our system however the finiteness of the $g_0$ value leads to deviations from the ideal fermionic dynamics. The inhomogeneity of the interaction still manifests leading to a difference w.r.t the localized two-particle energy level in each well and the tunneling remains incomplete.
\subsection{Dynamics with varying inhomogeneity\label{sub:inhomogeneity}}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth,height=4.8cm]{fig6.eps}
\caption{(color online): Population of the right-well over time, $P_{\mathrm{R}}(t)$, at $g_0=0.2$ for different $\alpha$ values. \textit{Inset}: Variation of maximum population of the right well ${P_R}^{max}$ with $\alpha$ for $g_0=0.2$. \label{cap:2p_alpha_var} }
\end{center}
\end{figure}
Having analyzed how the dynamics varies with changing interaction strength at a fixed interaction asymmetry, it is worthwhile to study the dependence of the tunneling dynamics on the strength of the inhomogeneity. For this we study the effect of different $\alpha$ values on the tunneling dynamics for a fixed $g_0 = 0.2$.
In Fig. \ref{cap:2p_alpha_var} we observe that for $\alpha = 0$, we have complete tunneling with a two mode dynamics i.e. fast oscillations (${\omega_0}_1$) which modulate slower tunneling oscillations (${\omega_1}_2$). When $\alpha$ is increased to a value of $0.04$, the tunneling maximum is reduced to roughly $0.7$ while still retaining the two-mode character. As $\alpha$ is further increased to $0.2$ the tunneling is suppressed as described in Sec. \ref{sub:analysis}. The characteristic display of fast and slow oscillations arising due to the time-scale difference of the contributing frequencies is not prominent here and for higher interaction asymmetry ($\alpha = 0.5$) we have effectively single mode tunneling with frequency ${\omega_0}_1$.
The variation of the maximum population ${P_R}^{max}$ with the inhomogeneity $\alpha$ (Fig.\ref{cap:2p_alpha_var} inset) shows a sharp drop with increasing $\alpha$ before effectively reaching a constant value $\sim 0.12$ for $\alpha \geq 0.3$. The reader should note that ${P_R}^{max}$ does not go to zero in the asymptotic limit $\alpha \rightarrow 1$ or $\frac{U_R}{U_L}\rightarrow\infty$. This is due to the fact that with a finite value of $g_0$ and a finite barrier height the tunneling coupling ($J$) is not negligible compared to $U_R$. As a consequence there remains a finite probability of bosonic tunneling between the two wells.
\subsection{Strong interaction inhomogeneity\label{sub:high}}
\begin{figure}
\begin{center}
\includegraphics[width=0.85\columnwidth,height=4.5cm]{nr_alpha1.eps}
\includegraphics[width=0.85\columnwidth,height=4.3cm]{energy_alpha1.eps}
\caption{(color online): (a) Population variation with time $P_R(t)$ at $g_0 = 25$ and $\alpha = 1$ for $P_R(0)=1$, i.e initially populating the right-well. (b) Energy spectrum for $\alpha = 1$ \label{cap:alpha1} }
\end{center}
\end{figure}
An extremely strong inhomogeneity at a high interaction value leads to an interesting higher band tunneling dynamics. We can realize such a system by having $\alpha = 1$ at $g_0 = 25$. This set up effectively makes the bosons fermionized in the right-well and almost non-interacting in the left. Preparing the initial set-up with both bosons in the left well leads to the suppression of tunneling. However if we prepare the initial state with two boson in the right well, then we observe substantial tunneling. In Fig.\ref{cap:alpha1} (a) we see that the $P_R$ oscillates between 1 and 0.5 indicating a single boson tunneling with a single dominant frequency.
In order to understand the phenomenon we look at the energy spectrum at $\alpha =1$ (Fig.\ref{cap:alpha1} (b)). While the ground-state remains unaffected, what we see is that close to the fermionization regime ($g_0 = 25$), the first excited state decouples from the higher three states which come closer. The main contribution to the first excited state is the state $|2,0\rangle$ and its separation from the other states could be understood from the fact that two boson in the left-well is almost non-interacting and thus energetically far off resonant from two effectively fermionized boson in the right-well $|0,2\rangle$. The consequences of this fact are the following: (i) The initial configuration of $|2,0\rangle$ becomes a stationary-state resulting in a highly suppressed tunneling, and (ii) the state $|0,2\rangle$ of the lowest band becomes energetically resonant and couples to the states $|1^1,1^0\rangle$ and $|1^0,1^1\rangle$ in the higher bands (where the superscript refer to the ground ($0$) or excited ($1$) orbital of the corresponding well). The latter leads to a tunneling dynamics in the higher band states predominantly between the 2nd and the 4th excited eigenstates (see Fig. \ref{cap:alpha1} (b)) which have greater overlap with the initial state $|0,2\rangle$. These orbitals have mostly contributions from the states $|0,2\rangle$ and $|1^1,1^0\rangle$ while the other orbital has minimal overlap with the initial state. As a result we get a single-particle tunneling with one dominant frequency given by the splitting of the energy between these two levels. In other words, we effectively have a single boson tunneling between the wells in the excited band. Note that this highly correlated single-particle tunneling scenario is attributed to the high inhomogeneity in the strong interaction regime since the combination of these two factors are responsible for turning the pair-tunneling scenario off-resonance.
\section{Multi-Particle Dynamics \label{sec: many particle}}
\begin{figure}
\includegraphics[width=0.85\columnwidth,keepaspectratio]{3p_nr.eps}
\includegraphics[width=0.85\columnwidth,keepaspectratio]{3p_energy_alpha0.2.eps}
\caption{(color online)(a) Population of the right-hand
well over time, $P_{\mathrm{R}}(t)$, for three bosons for different interaction strengths at $\alpha = 0.2$. (b) Three boson energy spectrum at $\alpha = 0.2$. \label{cap:3p_nr}}
\end{figure}
Having analyzed the tunneling dynamics of two atoms let us now focus on the case of three or more atoms to see the general atom number dependence of tunneling in the presence of spatially modulated interactions.
\subsection{General behavior and mechanisms \label{sub:3p general}}
Like in the two boson case we start with the initial state of $N = 3$ bosons prepared in the left well. As shown in Fig. \ref{cap:3p_nr}(a), the main effects are similar to the two-atom case. The dynamics is again governed by frequencies determined by the energy difference of the low lying spectrum. For very small interaction, the nearly equal energy difference gives rise to the beat pattern similar to that of two particles. As we increase the interaction strength, we observe suppression of tunneling for $g_0 = 0.2$ followed by a partial restoration at $g_0 = 4.7$ and a higher amplitude reemergence close to the fermionization limit at $g_0 =150$.
The general mechanism for the suppression is the same as for the two particle case. Now, however, in the symmetric case $\alpha = 0$, the contributing nearly degenerate eigenstates are of the form $|N,0\rangle \pm |0,N\rangle$. Consequently we have a complete $N$ particle tunneling with a frequency given by \cite{salgueiro06} $\omega\sim2NU/(N-1)!\times(2\Delta^{0}/U)^{N}$ where $U = U_L,U_R$ denotes the on-site interaction energy. The tunnel period thus grows exponentially with $N$.
When the inhomogeneous interaction is introduced, the states decouple to the localized number-states $|N,0\rangle$ and $|0,N\rangle$ and thus the initial state becomes a stationary one leading to the suppression of tunneling.
The important thing to note is that with increasing $N$, the suppression of tunneling occurs for much smaller values of $g_0$. For instance at $g_0 =0.2$ for $N=3$ we have almost complete suppression in contrast with $N=2$ where we still observed significant tunneling (see Fig.\ref{cap:2p_g_var}) for this value of $g_0$. This could be understood from the fact that the contribution of the on-site energy on the cat-state goes as $\sim U_{L,R}N(N-1)/2$, while that of the tunneling term is $N$ independent. This fact is responsible for a significant decoupling of these states at a lower $g_0$ value leading to faster suppression of tunneling as $N$ increases.
Also unlike that of the two boson case, the spectrum for the three boson case contains crossings between the higher-lying states (see Fig.\ref{cap:3p_nr}(b)) and in the vicinity of these crossings there is a partial reappearance of tunneling. This can be seen for instance at $g_0=4.7$ where we observe a restoration in the three-particle case whereas for two particles we still observed a significant suppression (see Fig. \ref{cap:2p_g_var}). In this regime the higher bands contribute more significantly leading to the convoluted dynamics observed. These higher band contributions leads to further recovery with increasing interaction strength towards the fermionization regime although even for $g_0 = 150$ we do not get the exact fermionic dynamics which is characterized by the tunneling of three independent fermions.
\subsection{Generating tunneling resonances by interaction inhomogeneity
\label{sub: asymmetry tunneling resonance}}
\begin{figure}
\includegraphics[width=0.85\columnwidth,height=4.5cm]{3p_pr_alpha_diag.eps}
\includegraphics[width=0.85\columnwidth,height=4.5cm]{4p_pr_alpha.eps}
\caption{(color online): Population of the right-well over time, $P_{\mathrm{R}}(t)$, at $g_0=0.2$ for different $\alpha$ values for (a) 3-particles and (b) 4-particles. \textit{Inset}: Variation of maximum population of the right well ${P_R}^{max}$ with $\alpha$ for $g_0=0.2$. \label{cap:3p_pr_alpha} }
\end{figure}
A very interesting phenomenon for the $N \geq 3$ particle case is that by tuning the asymmetry $\alpha$ we get a controllable reemergence of tunneling. To observe this, we study how the tunneling dynamics changes with different values of $\alpha$ for $g_0 = 0.2$ (Fig.\ref{cap:3p_pr_alpha}). The value of $g_0$ is chosen such that the inhomogeneity effect manifest but is still in the two-mode regime. For three atoms we observe (Fig.\ref{cap:3p_pr_alpha}(a)) that a complete tunneling for $\alpha = 0$ gives way to suppressed tunneling with increasing $\alpha$ value. However at $\alpha = 0.5$ we observe a reappearance which is in form of a tunneling resonance peaked at $\alpha = 0.5$ with ${P_R}^{max} \approx 0.6$ corresponding to effective two boson tunneling.
In the case of $N=4$ we see two resonances (fig.\ref{cap:3p_pr_alpha}(b)\textit{inset}) - the larger one centered on $\alpha = 0.3333$ with an amplitude $0.75$ and the smaller one at $\alpha = 0.6667$ with an amplitude $0.5$ resulting in the reappearance of tunneling shown in Fig.\ref{cap:3p_pr_alpha}(b).
\begin{figure}
\includegraphics[width=6.5cm, height=3.5cm]{3p_en_a0.eps}
\includegraphics[width=0.75\columnwidth,height=3.5cm]{3p_en_a004.eps}
\includegraphics[width=0.75\columnwidth,height=3.5cm]{3p_en_a05.eps}
\caption{(color online):Three particle energy levels for $0<g_0<0.3 $ for (a) $\alpha = 0$, (b) $\alpha = 0.04$ and (c) $\alpha = 0.5$.
\label{cap:3p_spec}}
\end{figure}
In order to understand this we have to study the spectra and the underlying eigenstates for different $\alpha$ (Fig.\ref{cap:3p_spec}). In the case of $N=3$ for no asymmetry $\alpha = 0$, the highest two levels form a doublet (Fig.\ref{cap:3p_spec}(a)) and the corresponding eigenstates are degenerate of the form $\frac{1}{\sqrt{2}}(|3,0\rangle \pm |0,3\rangle)$. As $\alpha$ is increased the parity symmetry is broken and the doublets separate and likewise the eigenstates decouple (Fig.\ref{cap:3p_spec}(b)). The energy eigenvalues (in the limit of very high $g_0$) are given by $U_L$, $U_R$, $3U_L$ and $3U_R$ with the corresponding eigenstates $|2,1\rangle$, $|1,2\rangle$, $|3,0\rangle$ and $|0,3\rangle$. However, when $U_R \approx 3U_L$ ($\alpha = 0.5$) the 1st and the 2nd excited eigenstates become near degenerate and form a doublet of the form $\frac{1}{\sqrt{2}}(|1,2\rangle \pm |3,0\rangle)$ (Fig.\ref{cap:3p_spec}(c)). Thus the initial state $|3,0\rangle$ is no longer a stationary state of the system. As a consequence we get a restoration of tunneling and the dynamics basically involves shuffling atoms between these two number-states. In other words we have tunneling of two particles between the two wells while one particle remains in the left well. This resonant two particle tunneling is what we observe for the $\alpha = 0.5$ case. As $\alpha$ is increased further this degeneracy is once again broken and the states decouple leading back to the suppressed tunneling dynamics. This is reminiscent of what happens in the asymmetric double-well for homogeneous interactions \cite{dounasfrazer07a}.
In similar consideration, for the 4-particle case the energy eigenvalues are $3U_L$, $6U_L$, $(U_L + U_R)$, $3U_R$ and $6U_R$. Now if $U_R \rightarrow 2U_L$ ($\alpha = 0.3333$) then we have two degeneracies viz $3U_R \rightarrow 6U_L$ and $(U_L + U_R) \rightarrow 3U_L$ corresponding to the eigenstates $\frac{1}{\sqrt{2}}(|4,0\rangle \pm |1,3\rangle)$ and $\frac{1}{\sqrt{2}}(|3,1\rangle \pm |2,2\rangle)$. Since the initial state is $|4,0\rangle$ only the first degeneracy contributes. Thus the dynamics in this case consists of tunneling of three bosons between the wells while one boson remains in the left well. This results in the tunneling amplitude of $0.75$.
The second tunneling peak occurs for $U_R \rightarrow 5U_L$ ($\alpha = 0.6667$) which leads to $(U_L + U_R) \rightarrow 6U_L$. The corresponding degenerate eigenstates are $\frac{1}{\sqrt{2}}(|4,0\rangle \pm |2,2\rangle)$ and we observe tunneling of two bosons on top of others remaining in the left well and thus the tunneling peak of $0.5$.
The above analysis can be extended generically for $N$ particles where we would have $N-2$ resonances corresponding to the degeneracies between the eigenstates.
\subsection{Correlations\label{sub:correlations}}
\begin{figure}
\includegraphics[width=0.75\columnwidth,keepaspectratio]{3p_npair.eps}
\includegraphics[width=0.75\columnwidth,keepaspectratio]{3p_ntrio.eps}
\caption{(color online) Temporal evolution of (a) pair-Probability and (b) three particle probability at $\alpha =0$ and $\alpha = 0.5$ for $N = 3$ and $g_0 = 0.2$ . \label{cap:3p_correlation}}
\end{figure}
In order to study the exact nature of tunneling dynamics, we need to investigate the correlations between the particles. For this we study the temporal evolution of the pair-probability or the probability of finding two particles in the same well defined by
\begin{equation}
p_{2}(t) = \langle\Theta(x_{1})\Theta(x_{2})+\Theta(-x_{1})\Theta(-x_{2})\rangle_{t}\\
\end{equation}
and the three-particle-probability or the probability of finding all three particles in the same well defined by
\begin{equation}
p_{3}(t) = \langle\Theta(x_{1})\Theta(x_{2})\Theta(x_{3})+\Theta(-x_{1})\Theta(-x_{2})\Theta(-x_{3})\rangle_{t}\\
\end{equation}
In the case of $N=3$, for homogeneous interaction $\alpha = 0$ at $g_0 = 0.2$ both $p_2$ and $p_3$ oscillate close to unity (Fig.\ref{cap:3p_correlation}). This implies that all the three particles can be found in the same well or in other words they tunnel together between the wells. This confirms the analysis of the dynamics by the eigenstate analysis in the preceding section as tunneling between $|3,0\rangle$ and $|0,3\rangle$ states.
Similarly at resonance ($\alpha = 0.5$) we find that $p_3$ oscillates from 0.1 and 1 implying that the system oscillates between a three-particle state to a non-three-particle state, namely the pair-state $|1,2\rangle$ which can be inferred from the variation of $p_2$ (Fig.\ref{cap:3p_correlation}(b)). As a result we have pair tunneling on top of a particle remaining in the left-well.
(Ideally in the case of B-H model, $p_2$ should be oscillating between 1 and 0.33 while $p_3$ between 1 and 0. However in our case the realistic potential and parameter regimes as well as some higher band contributions leads to the some deviations from this behavior).
\section{Asymmetric Double-Well\label{sec:asymmetric}}
Thus far we have investigated the dynamics in symmetric double-well with inhomogeneously interacting bosons. An interesting extension is to study the dynamics in an asymmetric double-well. This gives us the chance to examine the interplay between the interaction inhomogeneity and the tilt. A special interesting consideration would be to see if the tilt could be tuned to offset the inhomogeneity in the interaction and mimic the dynamics of symmetric interaction case or further if it can generate some new tunneling resonances.
\subsection{Generating tunneling resonances by a tilt.}
\begin{figure}
\includegraphics[width=0.75\columnwidth,height=4.3cm]{pl_d.eps}
\includegraphics[width=0.75\columnwidth,height=4.3cm]{npair_d.eps}
\caption{Variation of (a) tunneling maximum ${P_L}^{max}$ with tilt $d$ (b) maximum single particle probability $\bar{p}_{1}$ with tilt $d$ for $N = 2$, $g_0 = 0.2$ and $\alpha = 0.2$. \label{cap:2p_tilt} }
\end{figure}
In symmetric wells with homogeneous interaction, the localized $N$ particle state $|N,0\rangle$ has the same energy as that of the state $|0,N\rangle$ resulting in a complete $N$-particle tunneling between the wells. With the introduction of the inhomogeneity w.r.t the interaction, this resonance is broken and the energy of $N$ particles in the right well is higher than that in the left well resulting in the suppression of tunneling as seen before. Now if we incorporate a tilt in the double well such that the left well is lifted and right well is pushed down energetically in exactly the right amount to make the localized $N$ particle energy levels resonant then we should expect a reemergence of tunneling.
To observe this we prepare the initial state with both particles in the right well $\psi(0) = |0,2\rangle$ and study the variation of the tunneling maximum ${P_L}^{max}$ with a tilt $d$ (Fig.\ref{cap:2p_tilt}(a)) incorporated into the Hamiltonian as a linear term $-dx$. We restrict ourselves to the $\alpha = 0.2$ and $g_0 = 0.2$ case. We observe a sharp resonance at $d \approx 0.0065$ corresponding to the tilt which exactly balances the localized pair-state energy difference due to inhomogeneous interaction. The result is pair-tunneling between the two wells as we would have it in a completely symmetric set-up.
With higher tilt the tunneling maximum falls off very sharply as the pair-state becomes off-resonant again and we get a suppression of tunneling. The next maximum occurs when the tilt is large enough to make the localized pair state $|0,2\rangle$ resonant with the state $|1,1\rangle$. This results in a broad tunneling maximum at $d \approx 0.045$ corresponding to single-particle tunneling.
To confirm our analysis of the tunneling mechanism we look at the variation of maximum single particle probability $\bar{p}_{1}$ with tilt (Fig.\ref{cap:2p_tilt}(b)), defined as $\bar{p}_{1}={\max}_t({1-p_{2}(t)})$ which gives the probability of having only one particle in a well. We observe a negligible value at the first resonance $d \approx 0.0065$ confirming that the dynamics is pair-tunneling while a very broad maximum peaked at the second resonance $d \approx 0.045$ corresponds to the maximum probability of finding a single particle which in our case is the $|1,1\rangle$ state and the dynamics is a single particle tunneling between the $|0,2\rangle$ and $|1,1\rangle$ states.
\subsection{Spectral Analysis}
\begin{figure}
\includegraphics[width=0.75\columnwidth,keepaspectratio]{e_vs_d_2p.eps}
\caption{(color online) Two particle energy spectrum with tilt $d$ for $\alpha = 0.2$ and $g_0=0.2$. \label{cap:tilt_spectrum}}
\end{figure}
To understand the effect of the tilt on the tunneling dynamics we study the energy spectra $E$ with varying tilt $d$ at fixed $g_0 =0.2$ and $\alpha =0.2$ (Fig.\ref{cap:tilt_spectrum}).
At $d = 0$ the eigenstates are basically number-states in the localized basis. With increasing $d$, the highest two levels $|0,2\rangle$ and $|2,0\rangle$ move closer and form a sharp avoided crossing at $d \approx 0.0065$ corresponding to the first tunneling resonance. At this point the tilt exactly balances the interaction inhomogeneity and the eigenstate is in form of the cat-state $|2,0\rangle \pm |0,2\rangle$. This state is very sensitive to the tilt and a minute perturbation decouples them into the localized number-state resulting in a very sharp tunneling resonance. The ground-state, which is the $|1,1\rangle$ state is insensitive to the tilt since this lowering of one particle and raising another particle keeps the state energetically unaffected within the linear regime. This state forms a broad (anti)crossing with the lower excitedstate at $d \approx 0.045$ forming the broad single-particle tunneling resonance seen in the dynamics. This behavior seen in the two-particle case can be expected in general for $N$ particles giving $N$ resonances corresponding to the avoided crossings encountered. In particular with increasing tilt, the successive resonances corresponds to a mechanism where one less particle tunnels compared to that of the previous one while the width of the resonances becomes progressively broader.
\section{Conclusion and Outlook}
We have investigated the double-well tunneling dynamics with inhomogeneous interaction. More specifically we modeled the system such that we have two different interaction strengths in the two wells. What we observe is that this inhomogeneity leads to a suppression of tunneling. The reason for this suppression can be attributed to the breaking up of the doublet structure in the energy spectrum leading to a decoupling of the eigenstates into the localized number-state. Increasing the interaction to the fermionization limit leads to a reappearance of the tunneling. The dynamics is governed by the band splitting of the first two bands although the finiteness of the interaction strength and the presence of the interaction inhomogeneity leads to deviation from the ideal fermionic behavior. For a very pronounced interaction inhomogeneity for strong interactions, we observe single particle tunneling between the localized excited bands of the double-well.
These basic considerations can be used to understand the many particle system. There we observed a more severe suppression of tunneling for even lower $g_0$ values. Most importantly for $N\geq3$ atoms, one can generate tunneling resonances by tuning the interaction asymmetry. These resonances occur as a result of the formation of degeneracies between different eigenstates. For three particles, the exact tunneling mechanism was investigated using the evolution of the pair-probability and the three-particle probability. These studies show that we get correlated pair and triplet tunneling with a complete absence of single particle tunneling.
Finally we explored the dynamics in a asymmetric double-well and this gives us an understanding of the interplay between the interaction inhomogeneity and the tilt. We observe that the tilt can be tuned to offset the interaction inhomogeneity leading to a tunneling resonance. These dynamics have been explained through the spectral analysis in terms of avoided crossings between the levels.
Note that an interesting prospective would be to try to describe the presently found effects in the context of a generalized Bose-Hubbard model, where the on-site energies and the coupling constants would be site- and occupation number dependent\cite{luhmann10, Schneider09, kaspar10}.
Understanding the few-body mechanisms of tunneling with spatially modulated interactions can be used to design schemes for selective transport of particles between different wells and/or reservoir systems \cite{Schlagheck09,Schlagheck10}. Further our study could serve as a starting point for the investigation of the quantum dynamics in the presence of time-dependent interaction modulations and even be extended to multi-well systems \cite{alexej09}.
\acknowledgments{ B.C gratefully acknowledges the financial and academic support from the International Max-Planck Research School for Quantum Dynamics. Financial support from the German Academy of Science Leopoldina (grant LPDS 2009-11) is gratefully acknowledged by S.Z. P.S. acknowledges financial support by the Deutsche Forschungsgemeinschaft.
The authors appreciate fruitful discussions with H.D. Meyers. }
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,402 |
\section{Introduction and Summary}
\object{RS Oph} is one of the well-observed recurrent novae
and is suggested to be a progenitor system of Type Ia supernovae
\citep[e.g.,][]{hac01kb}. It has undergone its sixth recorded
outburst on 2006 February 12 UT \citep{nar06}.
The five previous recorded outbursts occurred
in 1898, 1933, 1958, 1967, and 1985. These short $10-20$ yr recurrence
periods indicate that the white dwarf (WD) mass is very close to the
Chandrasekhar mass and that the mass accretion rate is as large as
$\dot M_{\rm acc} \sim 1 \times 10^{-7} M_\sun$~yr$^{-1}$
\citep[see, e.g., Fig.2 of][]{hac01kb}.
If the WD mass increases after each outburst, \object{RS Oph}
will eventually explode as a Type Ia supernova
\citep[e.g.,][]{nom82, hkn96, hkn99, hac01kb}.
Therefore, it is important to estimate the WD mass
and the accreted mass left on the WD after the outburst.
It is well known that the X-ray turnoff time is a good indicator of the
WD mass \citep[e.g.,][]{hac05k,hac06ka}.
When the hydrogen shell-burning atop the WD extinguishes,
a supersoft X-ray phase ends \citep[e.g.,][]{kra96}. In
a visual light curve, however, this turnoff is not clear because
many strong emission lines contribute to it. To avoid such
contamination to the continuum flux, we have observed RS Oph with
the Str\"omgren $y$-band filter. The $y$-filter is an intermediate
bandpass filter designed to cut the strong emission lines in the wide
$V$ bandpass filter, so that its light curve represents the continuum
flux of novae \citep[e.g.,][]{hac06kb}. We have further modeled
the light curve of RS Oph and have determined the WD mass by fitting
our modeled light curve with the observation.
Section \ref{observation_rs_oph} presents our multi-band
photometry of the \object{RS Oph} 2006 outburst. The light curve
fitting of the observation with our numerical model
are presented in \S \ref{light_curve_model}.
Discussion follows in \S \ref{discussion}.
\begin{deluxetable}{lllll}
\tabletypesize{\scriptsize}
\tablecaption{List of Observers \label{observers}}
\tablewidth{0pt}
\tablehead{
\colhead{name of} & \colhead{location} & \colhead{telescope} &
\colhead{observed} & \colhead{No. of obs.} \cr
\colhead{observer} & \colhead{in Japan} & \colhead{aperture} &
\colhead{bands} & \colhead{nights ($y$)}
}
\startdata
Kiyota & Tsukuba & 25cm & $B,V,y,R_c,I_c$ & 24 \cr
Kubotera & Odawara & 16cm & $B,V,y,R_c$ & ~~8 \cr
Maehara & Kawaguchi & 20/25cm & $B,V,y,R_c,I_c$ & 19 \cr
Nakajima & Kumano & 25cm & $B,V,y,R_c,I_c$ & 55 \cr
OKU\tablenotemark{a} & Kashiwara & 51cm & $V,y$ & 25
\enddata
\tablenotetext{a}{Osaka Kyoiku University team}
\end{deluxetable}
\section{Observation} \label{observation_rs_oph}
Optical observations were started just after the discovery of the 2006
outburst \citep{nar06}. Each observer and their observational
details are listed in Table \ref{observers}. We have put a special
emphasis on the Str\"omgren y-filter to avoid contamination
by the strong emission lines. These y-filters were made
by Custom Scientific Inc.\footnote{http://www.customscientific.com/}
and distributed to each observer by one of the authors (M. Kato).
Kiyota, Kubotera, Maehara, and Nakajima (VSOLJ members) started observation
on February 13 and obtained 65 nights data for $y$-magnitudes (from
February 17 to July 27).
Osaka Kyoiku University (OKU) team obtained $V$ and $y$ magnitudes
of 25 nights starting from February 17 (until July 14).
The magnitudes of this object were measured by using the local
standard star, TYC2 5094.92.1 (Kiyota) or TYC2 5094.283.1
(the other observers).
We adapted the brightness and color of ($y= V= 9.57$, $B-V= 0.56$) for
TYC2 5094.92.1 and ($y= V= 9.35$, $B-V=1.23$) for TYC2 5094.283.1 from
Tycho2 catalog.
\begin{deluxetable}{lllll}
\tabletypesize{\scriptsize}
\tablecaption{Parameters of RS Oph\tablenotemark{a}
\label{parameter_of_rsoph}}
\tablewidth{0pt}
\tablehead{
\colhead{parameter} & \colhead{symbol} & \colhead{25\%} &
\colhead{50\%} & \colhead{100\%} \cr
\colhead{} & \colhead{} & \colhead{efficiency} &
\colhead{efficiency} & \colhead{efficiency}
}
\startdata
WD mass & $M_{\rm WD}$ & $1.35 M_\sun$ & $1.35 M_\sun$ & $1.35 M_\sun$ \cr
irradiation & $\eta_{\rm eff}$ & 0.25 & 0.50 & 1.0 \cr
disk size & $\alpha$ & 0.24 ($33~R_\sun$) & 0.34 ($47~R_\sun$) &
0.48 ($66~R_\sun$) \cr
size of RG & $\gamma$ & 0.24 ($25~R_\sun$) & 0.34 ($35~R_\sun$) &
0.47 ($48~R_\sun$) \cr
distance & $d$ & 0.9 kpc & 1.3 kpc & 1.7 kpc
\enddata
\tablenotetext{a}{$M_{\rm RG}= 0.7 M_\sun$, the inclination angle of
the binary $i=33\arcdeg$, the separation
$a= 316.5~R_\sun$, $R_1^*= 138.3~R_\sun$, $R_2^*= 102.5~R_\sun$,
$\beta= 0.05$, $\nu = 2.0$, $T_{\rm RG}= 3550$~K, the hydrogen burning
turnoff date of day 83,
and $E(B-V)= 0.73$ are common among all models.}
\end{deluxetable}
The $y$-magnitudes are plotted in Figure \ref{y_mag_linear}
together with $I_c$- and $V$-magnitudes. We have also added
visual magnitudes of the 1985 outburst from the American Association
of Variable Star Observers (AAVSO) for comparison. Our $y$-magnitudes
show very small scatter and follows the bottom of the 1985 visual
magnitude. The essential feature of the light curve is very similar
to the previous outbursts.
The $y$ light curve, however, clearly shows a plateau phase
from day 40 to day 75 and the sharp final decline starting from day 75.
Such mid-plateau phases are also observed
in two other recurrent novae, \object{U Sco} \citep[e.g.,][]{hkkm00}
and \object{CI Aql} \citep[e.g.,][]{hac01ka, hac03kb, hac03ka}.
These authors interpreted the mid-plateau phase as a bright disk irradiated
by the hydrogen-burning WD and a sharp start of the final decline
as the epoch when the hydrogen shell-burning ends.
\begin{figure}
\epsscale{1.20}
\plotone{f1.epsi}
\caption{
Three ($y$, $V$, and $I_c$) bands light curves for the 2006 outburst
of RS Oph. The AAVSO visual magnitudes of the previous 1985 outburst
are added for comparison. We find a plateau phase from day 40
to day 75 and a sharp final decline from day 75.
{\it Filled circles}: $I_c$-magnitudes.
{\it Filled triangles}: $y$-magnitudes.
{\it Open diamond}: $V$-magnitudes observed by OKU.
{\it Open squares}: $V$-magnitudes observed by Kiyota, Kubotera,
Maehara, and Nakajima (VSOLJ members).
{\it Small dots}: visual magnitudes of the 1985 outburst
taken from AAVSO.
\label{y_mag_linear}
}
\end{figure}
\section{Light Curve Model} \label{light_curve_model}
Our binary model, essentially the same as that in \citet{hac01kb}
except for the free-free emission model (see below) in the early phase
of the outburst, consists of a red giant (RG) star, which is not
filling its Roche lobe, a white dwarf (WD), and a disk around the WD.
A circular orbit with the ephemeris given by \citet{fek00} is assumed.
\begin{figure}
\epsscale{1.20}
\plotone{f2.epsi}
\caption{
Calculated light curves for free-free emission during the optically
thick wind phase together with observational $y$- ({\it open circles})
and $I_c$-magnitudes ({\it open squares}) of the 2006 outburst.
We also add observational infrared $K$- \citep[{\it open triangles}:
taken from][]{eva88} and visual magnitudes ({\it small dots}: taken from
AAVSO) of the 1985 outburst.
For the $y$-magnitudes, we show six light curves for different WD masses,
i.e., $M_{\rm WD}= 1.3$, 1.33, 1.34, 1.35, 1.36, and $1.37~M_\sun$
from right to left.
For the $I_c$- and $K$- magnitudes, we show two light curves
for two WD masses, i.e.,
$M_{\rm WD}= 1.3$ and $1.35~M_\sun$
from right to left. Here we assume $X=0.5$ and $Z=0.02$ for the
chemical composition of the envelope.
\label{mass_v_uv_x_rs_oph_x50z02}
}
\end{figure}
\subsection{Photospheric evolution of the white dwarf}
\label{optically_thick_wind}
After a thermonuclear runaway sets in on a mass-accreting WD,
its photosphere expands greatly and an optically thick wind massloss
begins. The decay phase of novae can be followed by a sequence
of steady state solutions \citep[e.g.,][]{kat94h}.
After the optically thick winds stop,
the envelope settles into a hydrostatic equilibrium where its mass
is decreasing by nuclear burning.
When the nuclear burning decays, the WD enters a cooling phase, in
which the luminosity is supplied with heat flow from the ash of
hydrogen burning.
We have followed nova evolution,
using the same method and numerical techniques as in \citet{kat94h}.
\begin{figure}
\epsscale{1.20}
\plotone{f3.epsi}
\caption{
Same as Fig. \ref{mass_v_uv_x_rs_oph_x50z02}, but for a combination of
our free-free plus disk irradiation models. The plateau phase
is reproduced with a relatively large disk, $\alpha = 0.34$,
for the 50\% iraddiation efficiency. The various parameters
are summarized in Table \ref{parameter_of_rsoph}. {\it Thin solid lines}
denote the free-free emission brightness (labeled by ``free-free'')
or the blackbody emission luminosity (labeled by ``blackbody'').
{\it Thick solid lines} are the total of the free-free plus blackbody.
{\it Dash-dotted lines} are the blackbody luminosity only from
the WD and the RG with irradiation. Several $V$-magnitudes in the very
early phase of the 2006 outburst ({\it Open triangles}) are added.
We cannot reproduce the mid-plateau phase without a disk.
The final sharp decline corresponds to the end of the hydrogen
shell-burning.
\label{nuclear_burning_rs_oph_x50z02_m1350_wide}
}
\end{figure}
In our nova light curve model, we assume that free-free emission of
the optically thin ejecta dominates the continuum flux in the early
phase of RS Oph outbursts as in many classical novae \citep[e.g.,][]{gal76}.
This is the main and most important
difference from previous Hachisu \& Kato's (2001b) model,
in which the blackbody emission is assumed.
The free-free emission of optically thin ejecta is estimated by
equation (9) in \citet{hac06kb}.
The calculated free-free light curves are shown in Figure
\ref{mass_v_uv_x_rs_oph_x50z02}.
The decline rate of the light curve, i.e.,
the evolutionary speed depends very sensitively on the WD mass if
its mass is very close to the Chandrasekhar mass
\citep[e.g.,][]{kat99, hac06ka, hac06kb}. This is because
the WD radius is very sensitive to the increase in mass near
the Chandrasekhar mass.
The timescale also depends weakly on the chemical composition of envelopes.
Hydrogen depletion is expected. This is because, just after
the unstable nuclear burning sets in, convection widely develops
and mixes processed helium with unburned hydrogen. This mixing reduces
the hydrogen content by 10\%$-$20\% for massive WDs
like in \object{RS Oph}. The CNO abundance is not
enriched in the recurrent novae, so we adopt the hydrogen content of
$X=0.50$ and the solar metallicity of $Z=0.02$.
We added infrared $K$-magnitudes of the 1985 outburst
observed by \citet{eva88}. Very little dependence of the light
curve shape on the wavelength is a characteristic feature in the
free-free emission light curves. However, the free-free light curve
cannot reproduce the mid-plateau phase, so we introduce an irradiated
disk in the next subsection.
\subsection{The irradiated disk and companion}
We assume an axi-symmetric disk with a size of
\begin{equation}
R_{\rm disk} = \alpha R_1^*,
\label{accretion-disk-size}
\end{equation}
and a thickness of
\begin{equation}
h = \beta R_{\rm disk} \left({{\varpi} / {R_{\rm disk}}} \right)^\nu,
\label{flaring-up-disk}
\end{equation}
where $R_{\rm disk}$ is the outer edge of the disk,
$R_1^*$ is the effective radius of the inner critical Roche lobe
for the WD component, $h$ is the height of the surface from
the equatorial plane, $\varpi$ is the distance from the symmetry axis,
and $\nu$ is the power index which describes flaring-up of
the disk edge \citep[see, e.g.,][]{sch97, hac01kb}. We also assume
a companion red giant star with a radius of
\begin{equation}
R_{\rm RG} = \gamma R_2^*,
\label{rad-giant-size}
\end{equation}
where $R_2^*$ is the effective radius of the inner critical Roche lobe
for the red giant component, its mass of $M_{\rm RG}= 0.7 ~M_\sun$,
and the inclination angle of the binary,
$i= 33 \arcdeg$ \citep[e.g.,][]{dob94}. \citet{dob96}
suggested $\gamma \sim 0.4$ for the distance of 1.5 kpc.
The disk surface absorbs UV and supersoft X-ray photons from the WD
and emits a part of it as a thermal emission with a lower temperature
than that of the WD photosphere. The resultant light curve depends
mainly on the disk size ($\alpha$), the efficiency of irradiation
($\eta_{\rm eff}=$ radiated energy$/$absorbed energy), and also
the radius of the RG, but depends very weakly on the other two
parameters of $\nu$ and $\beta$. Here, we assume $\nu=2$ and
$\beta = 0.05$. The dependence on these parameters was widely
discussed in the previous papers \citep[e.g.,][]{hac01kb, hac03ka}.
The irradiation efficiency of the RG hardly affects the light curve
as shown in Figure \ref{nuclear_burning_rs_oph_x50z02_m1350_wide}.
We have obtained three best fit models of the 2006 outburst
in Table \ref{parameter_of_rsoph}. The calculated light curves
for the 50\% efficiency are plotted in Figure
\ref{nuclear_burning_rs_oph_x50z02_m1350_wide}. These models reproduce
the mid-plateau phase and the sharp final decline identified as the end
of hydrogen shell-burning. The turnoff date of day 83
in our $1.35~M_\sun$ WD model is very consisting
with the supersoft X-ray turnoff on day $\sim 90$ observed with
{\it Swift} \citep{osb06}.
\begin{figure}
\epsscale{1.20}
\plotone{f4.epsi}
\caption{
The absorption-distance relation is plotted for the disk irradiation
model and the companion model. Lines with
{\it open squares}, {\it open circles}, and {\it open triangles}
denote the disk irradiation model with the efficiency of 25\%, 50\%,
and 100\%, respectively. Each symbol corresponds to the disk size,
i.e., $\alpha= 0.64$, 0.48, 0.32, and 0.24
from top to bottom. Lines with
{\it filled squares}, {\it filled circles},
{\it filled triangles}, and {\it filled stars}
denote the companion star model with the radius of $\gamma = 0.25$,
0.30, 0.40, and 0.50, respectively.
Each symbol corresponds to the photospheric temperature of the
companion star, i.e., $T_{\rm RG}=3200$, 3300, 3400, 3500, 3600, 3700, 3800,
and 3900~K from top to bottom. The absorption of $A_V = 3.1 E(B-V)= 2.26$
is indicated by a {\it dashed line} \citep{sni87}. Three models in Table
\ref{parameter_of_rsoph} are indicated by {\it large open stars}.
\label{rsoph_absorption_distance}}
\end{figure}
\subsection{Distance}
\label{distance}
The distance is obtained from the light curve fitting both at
the plateau phase and at the post-outburst minimum phase as shown in
Figures \ref{mass_v_uv_x_rs_oph_x50z02} and
\ref{nuclear_burning_rs_oph_x50z02_m1350_wide}.
The disk luminosity depends mostly on the disk size, $\alpha$,
and the irradiation efficiency, $\eta_{\rm eff}$. We have changed
these two parameters and calculated the brightness at the mid-plateau
phase. Fitting the calculated brightness with the observation,
we obtain the apparent distance moduli, i.e.,
$(m-M)_y = (m-M)_V$ and $(m-M)_I$.
Then we calculate the absorption from
\begin{equation}
A_V = {{A_V - A_I} \over {0.518}} = {{(m-M)_V - (m-M)_I} \over {0.518}},
\end{equation}
where we use $A_I = 0.482 A_V$ \citep[e.g.,][]{rie85}.
Once $A_V$ is obtained, the distance is calculated from
$\log (d) = ((m-M)_V - A_V -5)/5$. Thus, we obtained
the absorption-distance relation for the irradiated disk
as shown in Figure \ref{rsoph_absorption_distance}.
We further restrict the distance with the observed absorption
of $A_V \sim 2.3$ \citep{sni87}.
The same method is applied to the companion star, in which
we have changed the companion size, $\gamma$, and the
effective temperature, $T_{\rm RG}$. The absorption-distance
relation for the companion is also plotted.
The largest ambiguity of our model is the irradiation
efficiency of the disk. Here, the distances of 0.9, 1.3,
and 1.7~kpc are derived for the three different assumed efficiency,
i.e., 25\%, 50\%, and 100\%, respectively.
The actual efficiency is somewhere between 50\% and 100\%, so we
have a reasonable distance of $1.3-1.7$~kpc.
\section{Discussion}
\label{discussion}
The decline rate of free-free light curve and the hydrogen burning
turn-off date depend not only on the WD mass but also on the chemical
composition of the envelope \citep{hac06kb}.
We have examined two other cases of
hydrogen content, $X=0.70$ and $X=0.35$, and found that the best
fit models are obtained for $M_{\rm WD}= 1.36$ and $1.34 ~M_\sun$,
respectively. So we conclude that the WD mass is
$1.35 \pm 0.01 ~M_\sun$.
There are still debates on the distance to RS Oph.
In the previous outburst, \citet{hje86} estimated the distance
to be 1.6 kpc from \ion{H}{1} absorption-line measurements.
\citet{sni87} also obtained the distance of
1.6 kpc assuming the UV peak flux is equal to
the Eddington luminosity.
\citet{har93} calculated a distance of 1290 pc from the
$K$-band luminosity. \citet{hac01kb} obtained a smaller distance
of 0.6 kpc from the comparison of observed and theoretical UV fluxes
integrated for the wavelength region of 911-3250 \AA~.
This shorter distance is caused by their blackbody assumption,
because the flux is much larger than the blackbody flux
in this wavelength region.
\citet{obr06} estimated the distance of 1.6 kpc from VLBA mapping
observation with an expansion velocity indicated from emission line width.
\citet{mon06} estimated a shorter distance of $< 540$ pc assuming
that the IR interferometry size corresponds to the binary separation.
If we assume this corresponds to a circumbinary disk, however,
a much larger distance is obtained. Therefore, our new value
of $1.3-1.7$ kpc is consistent with other estimates.
\acknowledgments
We thank the Variable Star Observing League of Japan (VSOLJ)
and the American Association of Variable Star Observers (AAVSO)
for the visual data of \object{RS Oph}.
This research has been supported in part by Grants-in-Aid for
Scientific Research (16540211, 16540219)
of the Japan Society for the Promotion of Science.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 450 |
\section{Introduction}
The existence and properties of strange functions have attracted the interest of mathematicians since Weierstrass, who first gave an example of a function defined on $\mathbb{R}$ which was continuous, periodic but not differentiable at any real number. Several years later, Banach and Mazurkiewicz independently proved, in \cite{ban} and \cite{maz} respectively, that this is in fact the "generic" case:
\begin{theorem} [Banach-Mazurkiewicz]
Let J be a compact interval on $\mathbb{R}$ and $\mathcal{C}_{\mathbb{R}}(J)$ the space of real-valued continuous functions on J, endowed with the supremum norm. Then the class of all functions $f \in \mathcal{C}_{\mathbb{R}}(J)$ which are nowhere differentiable is residual in $\mathcal{C}_{\mathbb{R}}(J)$, i.e. it contains a $G_{\delta}$ and dense subset of $\mathcal{C}_{\mathbb{R}}(J)$.
\end{theorem}
\noindent The proof of the above theorem is a classical application of Baire's Category Theorem: in order to prove that there is (at least one) continuous but nowhere differentiable function, we instead consider the class $\mathcal{L}$ of all such functions, which is possibly empty. Then we give a set theoretic description of a subset $\mathcal{S}_0 \subseteq \mathcal{L}$ and using Baire's Theorem we prove that $\mathcal{S}_0$ is $G_{\delta}$ and dense in $\mathcal{C}_{\mathbb{R}}(J)$; therefore we know that $\mathcal{S}_0 \neq \emptyset$ without having found any particular element of $\mathcal{S}_0$ or $\mathcal{L}$. For the role of Baire's Theorem in Analysis we refer to \cite{grosse} and \cite{kahane}.
The context in which we will work is the disc algebra $\mathcal{A}(D)$, i.e. the set of all holomorphic functions on the open unit disk $D$ which extend continuously on $\overline{D}$. We endow $\mathcal{A}(D)$ with the supremum norm; thus $\mathcal{A}(D)$ becomes a Banach algebra and Baire's Theorem is at our disposal. We will present an analogue of Theorem 1.1 in this setting. For each function $f \in \mathcal{A}(D)$ we can naturally construct a $2\pi-$periodic function $h: \mathbb{R} \to \mathbb{C}$ defined by $h(\theta) = f(e^{i\theta})$; often, by abuse of notation, we write $f(\theta)$ instead of $h(\theta)$. From the definition of $\mathcal{A}(D)$ this function $h$ is of course continuous. Taking into consideration the above theorem, a natural question to ask is whether there are functions $f \in \mathcal{A}(D)$ such that the corresponding function $h$ is nowhere differentiable. Of course, if such a function $h$ is not differentiable at a point $\theta \in \mathbb{R}$, then either $Reh$ is not differentiable at $\theta$ or $Imh$ is not differentiable at $\theta$. Our main result is the following:
\begin{theorem}
Let $E$ be the class of all functions $f \in \mathcal{A}(D)$ such that the function $u=Ref|_{\mathbb{T}}$ is not differentiable at any point $\theta \in \mathbb{R}$. Then $E$ is residual in $\mathcal{A}(D)$, i.e. it contains a $G_{\delta}$ and dense subset.
\end{theorem}
\noindent As a corollary of this result, we prove that also the class $E_1 \subseteq E$ which consists of all the functions $f \in \mathcal{A}(D)$ such that neither $u=Ref|_{\mathbb{T}}$, nor $\tilde{u} = Imf|_{\mathbb{T}}$ are differentiable at any point $\theta \in \mathbb{T}$ is also residual.
Our proof of the above theorem makes use only of Baire's Category Theorem and some elementary Fourier Analysis; it follows the pattern of the proof of Theorem 1.1. The essential point in our proof, is to show that we can approximate every periodic real valued function of class $C^1$ defined on $\mathbb{R}$ by a piecewise linear periodic continuous function in a manner such that a good approximation also holds for the harmonic conjugates. Afterwards, as done in the proof of Theorem 1.1, we approximate this piecewise linear function by another which has much bigger slopes. A simple calculation gives that this approximation also holds for the harmonic conjugates and thus the proof is complete.
Simpler proofs of the same result could be given using the fact that $E$ is non empty, i.e. starting with a particular element $f_0 \in E$, which can easily be constructed using the classical Weierstrass function (see \cite{tit} for example). Also, an interesting question to ask is what results, analogue to the above, can one have in the case of several variables. In this case, the context is the polydisc algebra $\mathcal{A}(D^I)$ and, for a function $f \in \mathcal{A}(D^I)$, we want to examine the differentiability of its restriction $f|_{\mathbb{T}^I}$ as a function of the real vector $\theta = (\theta_i)_{i \in I} \in \mathbb{T}^I$. These questions will be answered in a future paper.
The above result establishes the topological genericity of nowhere differentiable functions in the disc algebra. After this result is proven, it is a reasonable question to ask whether this class is also generic in other senses. One could examine the dense lineability of this class, i.e. the existence of a dense linear subspace of $\mathcal{A}(D)$, every non-zero element of which is nowhere differentiable as well as the spaceability of this class, i.e. the existence of a closed infinite dimentional linear subpsace of $\mathcal{A}(D)$ with the above property. Finally, since $\mathcal{A}(D)$ is an algebra, it would be interesting to study the algebrability of this class, i.e. to examine whether the above questions hold in the case where linear subspaces were replaced by subalgebras. These will hopefully be examined in a future paper. Similar results in other spaces can be found in \cite{aron}, \cite{bayart} and \cite{gonz}.
\section{The proof of Theorem 1.2}
We will present the proof of the theorem by a series of lemmas following the outline of the proof of Theorem 1.1. For $n \in \mathbb{N}$ consider the sets
\begin{multline}
D_n = \Big\{ u \in \mathcal{C}_{\mathbb{R}}(\mathbb{T}) : \mbox{for every} \ \theta \in \mathbb{R} \ \mbox{there is a } y \in \left( \theta, \theta + \frac{1}{n} \right) \\ \mbox{such that } \ |u(y) - u(\theta)| > n | y -\theta| \Big\}
\end{multline}
and
\begin{equation}
E_n = \{f \in \mathcal{A}(D) : Ref|_{\mathbb{T}} \in D_n \},
\end{equation}
where, as usual, we interpret functions defined on $\mathbb{T}$ as $2\pi-$periodic functions defined on $\mathbb{R}$.
\begin{lemma}
The following inclusion is valid:
\begin{equation}
\mathcal{S} = \bigcap_{n=1}^{\infty} E_n \subseteq E.
\end{equation}
\end{lemma}
\begin{proof}
Let $f \in \mathcal{S}$. Then, for every $n \in \mathbb{N}$ and every $\theta \in \mathbb{R}$, there is a $y_n = y_n(\theta)$ with $\theta < y_n < \theta + \frac{1}{n}$ such that
\begin{equation}
\left| \frac{u(y_n)-u(\theta)}{y_n-\theta} \right| > n,
\end{equation}
where $u=Ref|_{\mathbb{T}}$. Then, $\{y_n\}$ converges to $\theta$ and, from (4), we deduce that $u$ is not differentiable at $\theta$, i.e. $f \in E$.
\end{proof}
\noindent We will prove that $\mathcal{S}$ is $G_{\delta}$ and dense in $\mathcal{A}(D)$; from this our result will follow.
\begin{lemma}
For every $n \in \mathbb{N}$, the set $E_n$ is open in $\mathcal{A}(D)$, endowed with the supremum norm.
\end{lemma}
\begin{proof}
Let $n \in \mathbb{N}$. We will prove that $\mathcal{A}(D) \setminus E_n$ is closed in $\mathcal{A}(D)$. Let $\{f_m\}$ be a sequence in $\mathcal{A}(D) \setminus E_n$ and $f \in \mathcal{A}(D)$ such that $f_m \to f$ uniformly on $\overline{D}$. Since $f_m \notin E_n$, for each m, there is a $\theta_m \in \mathbb{R}$ such that
\begin{equation}
\left| \frac{u_m(y) - u_m(\theta_m)}{y-\theta_m} \right| \leq n,
\end{equation}
for every $y \in \left( \theta_m, \theta_m + \frac{1}{n} \right)$, where $u_m = Ref_m|_{\mathbb{T}}$. Since each $u_m$ is $2\pi-$periodic, we can assume that $\theta_m \in [0,2\pi]$ for every $m$ and hence there is a subsequence $\{ \theta_{k_m} \}$ of $\{ \theta_m \}$ and a $\theta \in [0,2\pi]$ such that $\theta_{k_m} \to \theta$.
\smallskip
If $y \in \left( \theta, \theta + \frac{1}{n} \right)$, then for large enough $m$ it is true that $y \in \left( \theta_{k_m}, \theta_{k_m} + \frac{1}{n} \right)$. Thus, applying (5) for these indices $\{k_m\}$ and then letting $m \to \infty$ we have that
\begin{equation}
\left| \frac{u(y) - u(\theta)}{y-\theta} \right| \leq n,
\end{equation}
(again $u=Ref|_{\mathbb{T}}$) since the convergence of $\{f_m\}$ to $f$ is uniform. Hence (6) holds for every $y \in \left( \theta, \theta + \frac{1}{n} \right)$ and thus $f \notin E_n$. Thus, $\mathcal{A}(D) \setminus E_n$ is closed or equivalently $E_n$ is open.
\end{proof}
\begin{lemma}
For every $n \in \mathbb{N}$, the set $E_n$ is dense in $\mathcal{A}(D)$.
\end{lemma}
\noindent For the proof of Lemma 2.3 we will need another lemma:
\begin{lemma}
Let $u: \mathbb{R} \to \mathbb{R}$ a $2\pi-$periodic function of class $C^1$ and an $\varepsilon >0$. Then there exists a piecewise linear $2\pi-$periodic continuous function $u_0 : \mathbb{R} \to \mathbb{R}$ such that $\| u-u_0 \|_{\infty} < \varepsilon$ and if $u_0$ is linear at each interval defined by the partition $\{ 0 = t_0 < t_1 < ... < t_N = 2\pi \}$ then
\begin{equation}
\max_{1 \leq j \leq N} \sup \{ |u'(x) - u_0'(x)| : t_{j-1} < x < t_j \} < \varepsilon.
\end{equation}
\end{lemma}
\begin{proof}
Since both $u$ and $u'$ are continuous and periodic they are also uniformly continuous and thus there is a large $N>0$ such that if $x,y \in \mathbb{R}$ with $|x-y| \leq \frac{2\pi}{N}$, then
\begin{equation}
|u(x)-u(y)|< \frac{\varepsilon}{2} \ \ \ \mbox{and} \ \ \ |u'(x)-u'(y)|< \frac{\varepsilon}{2}.
\end{equation}
For $0 \leq j \leq N$ let $t_j$ be the points $\frac{2\pi j}{N}$ and $u_0 : [0,2\pi] \to \mathbb{R}$ the function which in each interval $[t_{j},t_{j+1}]$ connects linearly the points $(t_j, u(t_j))$ and $(t_{j+1},u(t_{j+1}))$. Since $u$ is $2\pi-$periodic, $u(0)=u(2\pi)$ and hence $u_0(0) = u_0(2\pi)$. Thus we can extend $u_0$ to a $2\pi-$periodic continuous function defined on $\mathbb{R}$. We will prove that $u_0$ fullfils the requirements of the lemma. From the periodicity of both $u$ and $u_0$ it is enough to examine the desired properties on $[0,2\pi]$.
\smallskip
Let $x \in [0,2\pi]$ and $1 \leq j \leq N$ such that $t_{j-1} \leq x \leq t_{j}$. Then $|x-t_{j-1}| \leq \frac{2\pi}{N}$ and thus $|u(x) - u(t_{j-1})| < \varepsilon/2$. But since $u(t_{j-1}) = u_0(t_{j-1})$ and $u_0$ is linear in the interval $[t_{j-1}, t_j]$ we deduce that
\begin{equation}
\begin{split}
|u(x) - u_0(x)| & \leq |u(x) - u(t_{j-1})| + |u_0(t_{j-1}) - u_0(x)| \\ & < \frac{\varepsilon}{2} + |u_0(t_{j-1}) - u_0(t_j) | \\ & < \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon.
\end{split}
\end{equation}
Hence, indeed $\|u-u_0\|_{\infty} < \varepsilon$.
\smallskip
For the proof of (7), let $1 \leq j \leq N$ and $x \in (t_{j-1},t_j)$. Then,
\begin{equation}
u_0'(x) = \frac{u(t_j)-u(t_{j-1})}{t_j - t_{j-1}}
\end{equation}
and thus, there exists a $\xi \in (t_{j-1},t_j)$ such that $u_0'(x) = u'(\xi)$. Since $|x-\xi| < t_j - t_{j-1} = \frac{2\pi}{N}$, using (8) we deduce that $| u'(x) - u_0'(x) | < \varepsilon/2$ and (7) follows.
\end{proof}
\noindent {\it Proof of Lemma 2.3.} Let $f \in \mathcal{A}(D)$ and an $\varepsilon >0$. It is well known, that the set of polynomials is dense in $\mathcal{A}(D)$; thus, to prove the density of $E_n$, we may assume that $f$ is a complex polynomial. First, we approximate $f$ by a function $g \in \mathcal{A}(D)$ such that $Reg|_{\mathbb{T}}$ is piecewise linear. For this puprose we will use Lemma 2.4. Indeed, if $u=Ref|_\mathbb{T}$, then $u$ is of course of class $C^{\infty}$ and thus, from the above lemma there exists a piecewise linear function $u_0 \in \mathcal{C}_{\mathbb{R}}(\mathbb{T})$ such that
\begin{equation}
\|u-u_0\|_{\infty} < \varepsilon \ \ \ \mbox{and} \ \ \ |u'(\theta)-u_0'(\theta)| < \varepsilon
\end{equation}
for every $\theta$ in which $u_0$ is differentiable. Consider the function $U=u-u_0$. Using the Cuachy-Schwarz inequality, Parseval's identity (for the piecewise $C^1$ function $U$) and (11) we obtain that
\begin{equation}
\begin{split}
\sum_{k \in \mathbb{Z}} |\widehat{U}(k)| & \leq |\widehat{U}(0)| + \left( \sum_{k \in \mathbb{Z}\setminus \{0\}} \frac{1}{k^2} \right)^{1/2} \left( \sum_{k \in \mathbb{Z} \setminus \{0\}} \left| k \cdot \widehat{U}(k) \right|^2 \right)^{1/2} \\ & < \varepsilon + \frac{\pi}{\sqrt{6}} \left( \sum_{k \in \mathbb{Z}\setminus\{0\}} \left| \widehat{U'}(k) \right|^2 \right)^{1/2} \\ & = \varepsilon + \frac{\pi}{\sqrt{6}} \left( \frac{1}{2\pi} \int_0^{2\pi} \left|U'(t)\right|^2 dt \right)^{1/2} \\ & < \varepsilon + \frac{\pi}{\sqrt{6}} \cdot \varepsilon = C_1 \cdot \varepsilon,
\end{split}
\end{equation}
for the constant $C_1 =1 + \frac{\pi}{\sqrt{6}} >0$. It is well known (see \cite{ahl} pp. 168-171) that $U$ can be extended continuously on $\overline{D}$ such that the restriction $U|_D$ is harmonic. Thus U has a harmonic conjugate $V$ on $D$, obtained by the mappings
\begin{equation}
r^k\cos(kx) \mapsto r^k\sin(kx) \ \ \ \mbox{and} \ \ \ r^k\sin(kx) \mapsto -r^k\cos(kx),
\end{equation}
where $0 \leq r < 1$, $x \in \mathbb{R}$ and $z=re^{ix} \in D$, for $k \in \mathbb{Z}$. The convergence of the series in (12) yields that $V$ can also extend continuously on $\overline{D}$: indeed, since from the above mapping we have $\widehat{V}(k) = -i \cdot \mbox{sign}(k) \cdot \widehat{U}(k)$, for $k \in \mathbb{Z} \setminus \{0\}$ and $\widehat{V}(0)=0$ we conclude that the series
\begin{equation}
\sum_{k \in \mathbb{Z}} \left| \widehat{V}(k) \right| < C_1 \cdot \varepsilon < \infty,
\end{equation}
and thus, the Fourier series of $V$ converges to $V$ uniformly on $\mathbb{T}$. Therefore, if $\tilde{u}= Imf|_\mathbb{T}$, the harmonic function $\tilde{u}-V$ extends continuously on $\overline{D}$ and is a harmonic conjugate of $u-U = u_0$. Hence, there exists a $g \in \mathcal{A}(D)$ such that $Reg|_\mathbb{T} = u_0$ and $Img|_\mathbb{T} = \tilde{u}_0 = \tilde{u}-V $. From the above inequalities and the Maximum Principle we deduce that:
\begin{equation}
\begin{split}
\|f-g\|_{\infty} & = \max\{ |f(\zeta)-g(\zeta)| : \zeta \in \mathbb{T} \} \\ &\leq \|u-u_0\|_{\infty} + \| \tilde{u}-\tilde{u}_0 \|_{\infty} = \|U\|_{\infty} + \|V\|_{\infty} \\ & \leq \sum_{k \in \mathbb{Z}} \left| \widehat{U}(k) \right| + \sum_{k \in \mathbb{Z}} \left| \widehat{V}(k) \right| < 2C_1 \cdot \varepsilon.
\end{split}
\end{equation}
\smallskip
We will now find a function $h \in E_n$ such that $\|g-h\|_{\infty} < K \cdot \varepsilon$ for some constant $K$ and the triangle inequality will imply the density of $E_n$ in $\mathcal{A}(D)$. We define
\begin{equation}
\ell_j = \frac{u_0(t_j)-u_0(t_{j-1})}{t_j-t_{j-1}}, \ \ \ \ j=1,2,...,N
\end{equation}
the slopes of the linear components of $u_0$ and consider a large $R \in \mathbb{N}$ such that the real number
\begin{equation}
m = \frac{2R\varepsilon}{\pi} > n + \max_{1 \leq j \leq N} |\ell_j|.
\end{equation}
Afterwards, we consider the continuous, $\frac{\pi}{R}-$periodic function $s: \mathbb{R} \to \mathbb{R}$ defined by
\begin{equation}
s(\theta) = \varepsilon \cdot \mbox{dist}\left( \theta , \frac{\pi}{R} \cdot \mathbb{Z} \right),
\end{equation}
where $\frac{\pi}{R} \cdot \mathbb{Z} = \{ ..., -\frac{\pi}{R}, 0, \frac{\pi}{R}, ... \}$. We can easily see that $s$ is piecewise linear with slopes $\pm m$.
\smallskip
Of course, $s$ is also $2\pi-$periodic and thus we can consider the function $u_1 = u_0 + s \in \mathcal{C}_{\mathbb{R}}(\mathbb{T})$, which satisfies
\begin{equation}
\|u_0-u_1\|_{\infty} = \|s\|_{\infty} = \frac{\pi}{2R} \cdot \varepsilon < 2\varepsilon.
\end{equation}
From $u_1$ we will get the function $h$ we are looking for. We will need two lemmas:
\begin{lemma}
The function $u_1$ belongs in the class $D_n$.
\end{lemma}
\begin{proof}
Let $\theta \in \mathbb{R}$ and some $\delta \in \left( 0, \frac{1}{n} \right)$ such that both $s$ and $u_0$ have constant derivatives in the interval $(\theta, \theta + \delta)$. Then, for $y$ in this interval it is true that $\theta < y < \theta + \frac{1}{n}$ and in addition:
\begin{equation}
|u_0(y)-u_0(\theta)| = | \ell_j | \cdot |y-\theta|,
\end{equation}
for some $1 \leq j \leq N$. Using (17), we deduce that
\begin{equation}
\begin{split}
|u_1(y)-u_1(\theta)| & \geq |s(y)-s(\theta)| - |u_0(y)-u_0(\theta)| \\ & = m \cdot |y-\theta| - |\ell_j | \cdot |y-\theta| \\ & > n \cdot |y-\theta|,
\end{split}
\end{equation}
i.e. $u_1 \in D_n$.
\end{proof}
\begin{lemma}
The above function $s$ satisfies the inequality
\begin{equation}
\sum_{k \in \mathbb{Z}} \left| \widehat{s}(k) \right| < L \cdot \varepsilon,
\end{equation}
for some absolute constant $L>0$.
\end{lemma}
\begin{proof}
We will compute the fourier coefficients $\widehat{s}(k)$: first of all, for $k \in \mathbb{Z}$ we observe that:
\begin{equation}
\begin{split}
\widehat{s}(k) & = \frac{1}{2\pi} \int_0^{2\pi} s(\theta) e^{-ik\theta} d\theta \\ & = \frac{1}{2\pi} \int_0^{2\pi} s\left( \theta + \frac{\pi}{R} \right) e^{-ik \left( \theta + \frac{\pi}{R} \right) } d\theta \\ & =e^{-\frac{ik\pi}{R}} \cdot \frac{1}{2\pi} \int_0^{2\pi} s(\theta) e^{-ik\theta} d\theta \\ & = e^{-\frac{ik\pi}{R}} \widehat{s}(k),
\end{split}
\end{equation}
where we have used the fact that $s$ is $\frac{\pi}{R}-$periodic. Thus, $\widehat{s}(k) \neq 0$ implies that $k=2\lambda R$, for some $\lambda \in \mathbb{Z}$. We now compute these coefficients: for $\lambda \in \mathbb{Z}\setminus \{0\}$
\begin{equation}
\begin{split}
\widehat{s}(2\lambda R) & = \frac{1}{2\pi} \int_{-\pi}^{\pi} s(\theta) e^{-2i\lambda R\theta} d\theta \\ & = \frac{1}{2\pi} \sum_{j=-R}^{R-1} \int_{j\pi/R}^{(j+1)\pi/R} s(\theta) e^{-2i\lambda R\theta} d\theta \\ & \stackrel{y=2R\theta}{=} \frac{1}{4\pi R} \sum_{j=-R}^{R-1} \int_{2j\pi}^{2(j+1)\pi} s\left( \frac{y}{2R} \right) e^{-i\lambda y} dy.
\end{split}
\end{equation}
Since the function inside the integral is $2\pi-$periodic we also have the equalities:
\begin{equation}
\begin{split}
\widehat{s}(2\lambda R) & = \frac{1}{4\pi R} \sum_{j=-R}^{R-1} \int_{-\pi}^{\pi} s \left( \frac{y}{2R} \right) e^{-i\lambda y} dy \\ & = \frac{1}{2\pi} \int_{-\pi}^\pi s \left( \frac{y}{2R} \right) e^{-i\lambda y} dy.
\end{split}
\end{equation}
But, for $-\pi \leq y \leq \pi$ it holds $-\frac{\pi}{2R} \leq \frac{y}{2R} \leq \frac{\pi}{2R}$ and thus, $s \left( \frac{y}{2R} \right) = \frac{m}{2R} \cdot |y|$. Thus, the computation gives:
\begin{equation}
\begin{split}
\widehat{s}(2\lambda R) & = \frac{1}{2\pi} \int_{-\pi}^\pi \frac{m}{2R} |y| e^{-i\lambda y} dy \\ & \stackrel{(16)}{=} \frac{\varepsilon}{2\pi^2} \int_{-\pi}^\pi |y| e^{-i\lambda y} dy \\ & = \frac{\varepsilon}{\pi} \cdot \frac{(-1)^\lambda-1}{\pi \lambda^2}.
\end{split}
\end{equation}
Thus,
\begin{equation}
\left| \widehat{s}(2\lambda R) \right| \leq \frac{C_2}{\lambda ^2} \cdot \varepsilon,
\end{equation}
for some absolute constant $C_2 >0$ and the result of the lemma follows, since $\widehat{s}(0) = \frac{\varepsilon \pi}{4R} < \varepsilon$ and hence:
\begin{equation}
\begin{split}
\sum_{k \in \mathbb{Z}} \left| \widehat{s}(k) \right| & = \sum_{\lambda \in \mathbb{Z}} \left| \widehat{s}(2\lambda R) \right| \\ & < \varepsilon + \sum_{t \in \mathbb{Z}\setminus\{0\}} \frac{C_2}{t^2} \cdot \varepsilon = L \cdot \varepsilon,
\end{split}
\end{equation}
for some constant $L>0$.
\end{proof}
\noindent {\it Proof of Lemma 2.3 (continued).} From (22), as we did after (12), we deduce that $s$ has a harmonic conjugate $\tilde{s}$ on $D$, which extends continuously on $\overline{D}$ and in addition satisfies the property $\| \tilde{s} \|_{\infty} < L \cdot \varepsilon$. We now define the function $h : \overline{D} \to \mathbb{C}$ by $h(z) = g(z) + (s(z) + i \tilde{s}(z))$, $|z| \leq 1$. From the above results, we have that $h \in \mathcal{A}(D)$, $Reh = u_0 + s = u_1 \in D_n$ and
\begin{equation}
\begin{split}
\|g-h\|_{\infty} & \leq \|u_1-u_0\|_{\infty} + \|\tilde{u}_1 - \tilde{u}_0 \|_{\infty} \\ & = \|s\|_{\infty} + \|\tilde{s}\|_{\infty} < K \cdot \varepsilon,
\end{split}
\end{equation}
for some constant $K>0$. Thus,
\begin{equation}
\|f-h\|_{\infty} \leq \|f-g\|_{\infty} + \|g-h\|_{\infty} < (2C_1+K) \cdot \varepsilon,
\end{equation}
and since $h \in E_n$ the proof of the lemma is complete.
\hfill$\Box$
\noindent {\it Proof of Theorem 1.2.} Since every $E_n$ is open and dense in $\mathcal{A}(D)$, from Baire's Category Theorem, we also deduce that their intersection $\mathcal{S}$ is $G_{\delta}$ and dense in $\mathcal{A}(D)$. Thus, from Lemma 2.1, Theorem 1.2 follows.
\hfill$\Box$
We close with a stronger version of Theorem 1.2:
\begin{corollary}
Let $E_1$ be the class of all functions $f \in \mathcal{A}(D)$ such that neither $u=Ref|_\mathbb{T}$, nor $\tilde{u} = Imf|_\mathbb{T}$ are differentiable at any point $\theta \in \mathbb{R}$. Then $E_1$ is residual in $\mathcal{A}(D)$.
\end{corollary}
\begin{proof}
Since the set $\mathcal{S}$ defined above is $G_{\delta}$ and dense in $\mathcal{A}(D)$ and multiplication by $i$ is a homeomorphism, the same holds for the set $i\mathcal{S}$. Thus, from Baire's Theorem the set $\mathcal{T} = \mathcal{S} \cap i \mathcal{S}$ is also $G_{\delta}$ and dense in $\mathcal{A}(D)$. We will prove that $\mathcal{T} \subseteq E_1$.
\smallskip
Let $f \in \mathcal{T}$. Since $f \in \mathcal{S}$, it is true that $u=Ref$, is nowhere differentiable from Lemma 2.1. But also, $f \in i\mathcal{S}$ and thus $-if \in \mathcal{S}$. Hence, we also conclude that $Re(-if) = Imf = \tilde{u}$ is nowhere differentiable.
\end{proof}
\vspace{3mm}
\noindent {\bf Aknowledgements:} The author would like to thank Professor Vassili Nestoridis for introducing him to the problem and Konstantinos Makridis for helpful communications.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,347 |
Джураб е пустиня в Северен Чад.
В нея са открити много вкаменелости, включително и на Човекоподобни, представител на които е видът Тумай ("Надежда за живот"). По-късно той е оприличен на малък примат. Косом Бугуди и Торос-Менала са сред най-продуктивните на фосили участъци в пустинята.
Източници
Противоречията на черепа "Тумай", Калифорнийска академия на науките
Тумай: разклащане на нашите концепции за най-ранните етапи от историята на хуманоидите, в "Science in Africa", август 2002 г.
Външни препратки
"Тайните на черепа от пустинята Джураб", Огнян Тодоров, в "Тема"
Пустини в Чад | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,157 |
Q: Ensure a specific variable can be seen in VS2015 debugger in Release build? I'm using VS2015 debugger to debug some C++ with many stack-based floats and loop counters. Debug mode is too slow to be practical so I must debug in Release mode, but the variables of interest are optimized away or stored in registers and not visible to the debugger.
Is there a trivial expression I can apply to the variables of interest such that a C++ compiler will be forced to commit them to stack memory even in a Release build, thus making them visible to the debugger, but with no other side effects?
As an example, this appears to work:
void make_visible_(void *);
#define MAKE_VISIBLE(x) make_visible_(&x);
Then, in some other source file:
void make_visible_(void *) {}
The need to declare a function elsewhere is annoying. If not declared in a separate source file, the compiler optimizes out the calls to make_visible_(). I can also imagine that a smart compiler using link-time optimization could optimize out the separate source file case as well.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,578 |
import random
from django.conf import settings
from django.core.management.base import BaseCommand
from django.contrib.auth.models import User
from ...models import (
TranscriptPhrase,
TranscriptPhraseCorrection
)
phrase_positive_limit = settings.TRANSCRIPT_PHRASE_POSITIVE_CONFIDENCE_LIMIT
phrase_negative_limit = settings.TRANSCRIPT_PHRASE_NEGATIVE_CONFIDENCE_LIMIT
correction_lower_limit = settings.TRANSCRIPT_PHRASE_CORRECTION_LOWER_LIMIT
correction_upper_limit = settings.TRANSCRIPT_PHRASE_CORRECTION_UPPER_LIMIT
class Command(BaseCommand):
help = 'Creates random(ish) corrections for 100 phrases.'
def handle(self, *args, **options):
users = User.objects.all()
phrases = TranscriptPhrase.objects.filter(
confidence__lte=phrase_negative_limit,
num_corrections__lt=3
)[:100]
for phrase in phrases:
for i in range(3):
corrected_text = phrase.text
options = {
0: corrected_text.upper(),
1: '{} - {}'.format(
corrected_text, random.choice(
['ping!', 'boing boing', 'ribbit', 'PLOP']
)
),
2: corrected_text.title()
}
user = random.choice(users)
TranscriptPhraseCorrection.objects.create(
correction=options[i],
user=user,
transcript_phrase=phrase
)
print(
'Created correction:\nUser:{}\nCorrection:{}'.format(
user, options[i]
)
)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,923 |
Having a few good friends — or many — has always been golden. And as you age, those friendships may become even more important.
If you're in your sixties or beyond, friendships aren't just the social glue and glitz of life: As you get older, good friendships can dispel loneliness, improve your health, boost your sense of well-being, and even add to your years.
Moving to assisted living can be an overwhelming experience for both you and your older adult. We asked a senior living expert for advice on 4 common issues related to moving into and living in assisted living.
Our expert, Arthur Bretschneider, has the senior housing business in his blood – his family has been in the business for three generations. Now Arthur is founder and CEO of Seniorly, a company that makes it easier for families to find local senior housing options.
mHealthNews: Will 2016 be when consumer-driven healthcare kicks in?
Over the next five years technology has the "ability to change the marketplace," with three major trends: convergent technology redefining what's possible in healthcare, the invasion of consumer technology and the big question of whether the consumer will use it. No one wins unless the consumer wins. | {
"redpajama_set_name": "RedPajamaC4"
} | 251 |
Q: Difference of Beamercolorbox in the setbeamertemplate and that in the normal text I use the beamercolorboxes both in the \setbeamertemplate and in the normal text. However their shadows appear differently. The code is:
\documentclass{beamer}
\usetheme{Madrid}
\usepackage{tikz}
\newlength\barheight\setlength\barheight{\paperheight}
\divide\barheight by 12
\setbeamertemplate{frametitle}
{
\begin{beamercolorbox}[wd=1.2\paperwidth,ht=2.5\barheight]{Title bar}
\begin{tikzpicture}[remember picture,overlay]
\node [xshift=\paperwidth/2,yshift=-\headheight] (mybar) at (current page.north west)
[rectangle,fill,inner sep=0pt,minimum width=\paperwidth,
minimum height=2.5\barheight,top color=frametitle.bg,
bottom color=frametitle.bg]{};% bar
\node[below of=mybar,yshift=-0.7mm,rectangle,shade,inner sep=0pt,minimum
width=128mm,minimum height=1.2mm,top color=black!50,bottom
color=black!10]{};% shadow
\end{tikzpicture}%
\end{beamercolorbox}
\vskip -1.79cm
\linethickness{0.0pt}
\framelatex{
\begin{beamercolorbox}[wd=\paperwidth,ht=0.4\barheight]{Title bar}
\begin{columns}
\begin{column}{0.03\paperwidth}
\end{column}
\begin{column}{0.87\paperwidth}
\insertframetitle
\end{column}
\end{columns}
\end{beamercolorbox}
}
\makeatletter
\ifx\insertframesubtitle\@empty%
\makeatother
\else
\vskip10pt
\begin{beamercolorbox}[wd=0.6\paperwidth,shadow=true,rounded=true]{title}
\usebeamerfont{framesubtitle}\insertframesubtitle
\end{beamercolorbox}
\fi}
\begin{document}
\begin{frame}
\frametitle{There Is No Largest Prime Number}
\framesubtitle{The proof uses \textit{reductio ad absurdum}.}
\begin{beamercolorbox}[wd=0.5\paperwidth,shadow=true,rounded=true]{title}
There is no largest prime number.
\end{beamercolorbox}
\end{frame}
\end{document}
giving the following result:
A: Beamer automatically calculates the shadow colour based on the background colour. You can set this background to white using \setbeamercolor{frametitle}{bg=white}
\documentclass{beamer}
\usetheme{Madrid}
\usepackage{tikz}
\newlength\barheight\setlength\barheight{\paperheight}
\divide\barheight by 12
\setbeamercolor{frametitle}{bg=white}
\setbeamertemplate{frametitle}
{
\begin{beamercolorbox}[wd=0.5\paperwidth,shadow=true,rounded=true]{title}
\usebeamerfont{framesubtitle}\insertframesubtitle
\end{beamercolorbox}
}
\begin{document}
\begin{frame}
\frametitle{There Is No Largest Prime Number}
\framesubtitle{The proof uses \textit{reductio ad absurdum}.}
\begin{beamercolorbox}[wd=0.5\paperwidth,shadow=true,rounded=true]{title}
There is no largest prime number.
\end{beamercolorbox}
\end{frame}
\end{document}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,492 |
<div ng-repeat="qc in queryConfigs">
<h4>{{qc.queryname}}</h4>
<div ng-if="qc.querycomponent == 'checklist'">
<div query-checklist queryconfig='qc'></div>
</div>
<div ng-if="qc.querycomponent == 'colorpicker'">
<div query-colorpicker queryconfig='qc'></div>
</div>
<div ng-if="qc.querycomponent == 'rangeslider'">
<div query-rangeslider queryconfig='qc'></div>
</div>
</div> | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,402 |
El terme malaltia professional és aquella que atempta contra la salut del treballador, aquesta expressió sovint es redueix als de l'exposició a una font de tòxics o patògens patits durant l'ocupació laboral. Aquesta exposició pot repetir-se diverses vegades abans que apareguin els primers símptomes. En el cas immediatament després d'una lesió en un esdeveniment específic es classifica generalment com un accident de treball.
Legalment, el Reial decret legislatiu 8/2015, de 30 d'octubre, pel qual s'aprova el text refós de la Llei general de la Seguretat Social, defineix la malaltia professional de la següent forma:Article 157 - Concepte de malatia professional
S'entén per malaltia professional la contreta a conseqüència del treball executat per compte d'altri en les activitats que s'especifiquin en el quadre que aprovin les disposicions d'aplicació i desplegament d'aquesta Llei, i que estigui provocada per l'acció dels elements o substàncies que en aquest quadre s'indiquin per a cada malaltia professional.
En aquestes disposicions s'ha d'establir el procediment que s'ha d'observar per a la inclusió en aquest quadre de noves malalties professionals que es consideri que s'hi han d'incorporar. Aquest procediment ha de comprendre, en tot cas, com a tràmit preceptiu, l'informe del Ministeri competent en matèria de Sanitat.Com a exemples de malalties professionals recollides en la legislació hi ha la pneumoconiosi, l'alveolitis al·lèrgica, la lumbàlgia, la síndrome del túnel carpià, l'exposició professional a gèrmens patògens i diversos tipus de càncer, entre altres.
En països com Espanya, i a efectes legals, es reconeix com malaltia professional aquella que a més de tenir el seu origen laboral, està inclosa en una llista oficial publicada pel Ministeri de Treball i que dona, per tant el dret al cobrament de les indemnitzacions corresponents.
La disciplina dedicada a la prevenció de les malalties professionals és la higiene industrial; la medicina del treball s'especialitza en el guariment i rhabilitació dels treballadors afectats mentre que l'ergonomia s'encarrega dels disseny productiu dels ambients de treball per tal d'adaptar-ls a les capacitats dels éssers humans.
El 19 de setembre de 2003, la Comissió Europea va adoptar una Recomanació (2003/670/CE, DO L238 de 25 de setembre de 2003), relativa a la llista europea de malalties professionals
Identificació com malaltia professional
El quadre de malalties que es poden qualificar com a professionals figura a l'Annex 1 del Reial Decret 1299/2006, de 10 de novembre, pel qual s'aprova el quadre de malalties professionals en el sistema de la Seguretat Social i s'estableixen criteris per a la seva notificació i registre.
El Reial Decret 1299/2006 classifica les malalties professionals dins de 6 grups d'agents causals:
Per tal que una malaltia determinada es pugui qualificar com una malaltia professional és imprescindible que existeixin uns elements bàsics que la diferenciïn d'una malaltia comuna:
Agent: ha d'existir un agent causal en l'ambient o especials condicions de treball que siguin potencialment lesives per a la salut.
Exposició: és condició sine qua non demostrar que, a conseqüència del contacte entre el treballador i l'agent o particular condició de treball, es possibilita la gestació d'un dany a la salut. Els criteris de demostració poden ser:
Qualitatius: consisteix a establir una llista taxativa d'ocupacions amb risc d'exposició, i la declaració de l'afectat o dels seus representants d'estar exercint aquesta ocupació o haver-ho fet.
Quantitatius: es refereix a les disposcions existents respecte valors límits o concentracions màximes permisibles per cadascun dels agents incorporats en la llista.
Malaltia: ha d'existir una malaltia o un dany clarament delimitats.
Nexe de causalitat: ha de demostrar-se amb proves científiques que existeix un vincle entre la malaltia i la presència en el treball dels agents o condicions. No cal que la patologia hagi originat ja una incapacitat.
Inclusió en la llista oficial.
Referències
Medicina
Treball | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,258 |
{"url":"https:\/\/gamedev.stackexchange.com\/questions\/140333\/ensuring-linear-velocity-remains-constant-even-when-turning","text":"# Ensuring linear velocity remains constant even when turning\n\nI am using Ackermann steering to rotate a vehicle. The code is as follows:\n\n if (turningLeft || turningRight) {\n\/\/ Rotate around a dummy pivot point\nfloat turnDir = turningLeft ? -1 : 1;\nVector3 turningPivotPoint =\ndummyPivot.transform.TransformPoint(\nnew Vector3(turningCenterDistance * turnDir, 0, 0));\ndummyPivot.transform.RotateAround(\nturningPivotPoint,\nVector3.up * gasDir * turnDir,\nangleVel * Time.deltaTime);\n}\n\n\/\/ The player is pressing the gas\nif (isGas) {\nangleVel = Mathf.Lerp (angleVel, maxAngleVel,Time.deltaTime);\n}\n\nif (angleVel == 0) {\nturningLeft = turningRight = false;\n}\n\ngasVel = Mathf.Lerp(gasVel, angleVel * Mathf.Deg2Rad * turningCenterDistance, Time.deltaTime);\n\ndummyPivot.transform.position =\ndummyPivot.transform.position + dummyPivot.transform.forward * Time.deltaTime * gasVel * gasDir;\n\n\nThe issue is that the speed of the vehicle when it is turning does not match its speed when driving straight. Driving the vehicle in circles will be slower than driving the vehicle forward.\n\nHow do I fix this?\n\nusing UnityEngine;\nusing System.Collections;\n\npublic class CarController : MonoBehaviour\n{\nint turn = 0;\nfloat acceleration = 0;\nfloat speed = 0;\npublic float friction = 0.98f; \/\/feel free to fiddle with this\n\nvoid Update ()\n{\nturn = 0;\nacceleration = 0;\nspeed *= friction; \/\/dampen\n\nturn += Input.GetKey(\"a\") ? -1 : 0;\nturn += Input.GetKey(\"d\") ? +1 : 0;\n\nacceleration += Input.GetKey(\"w\") ? 0.2f * Time.deltaTime : 0;\nspeed += acceleration;\n\ntransform.Rotate(0, turn, 0);\ntransform.Translate(Vector3.forward * speed , Space.Self); \/\/look up what Space.Self does!\n}\n}\n\n\nYou can see that our position is always updated by adding Vector3.forward * speed. This means we will always have the same linear velocity regardless of whether or not we are turning, provided the speed, which is a function of the gas pedal, is the same.\n\nP.S. You can also supply turnAcceleration to smooth turning. This is just a simple example.\n\n\u2022 Thanks! A few things though: turn = 0 should not be set in Update() Also, this is not Ackermann steering. This will turn the vehicle in place without gas. And the turning radius is too wide. \u2013\u00a0Simian Apr 27 '17 at 6:29\n\u2022 I wrote this code from scratch in 5 minutes to help you, so it works exactly the way it is intended to work. Nothing stops you from modifying that turning radius, provided you grok the code. Given the principles of Ackermann steering, I have to say: Your code is also not Ackermann! So I'm simply giving you something that works and that you can adapt or mix in with your own code. \u2013\u00a0Engineer Apr 27 '17 at 7:55\n\u2022 I see. turn = 0 should not be set to 0 in update though, otherwise this doesn't work. For tighter turning, turn can be incremented with a greater value. The other modification I made is to base turn on the current speed, to prevent from turning in place. e.g. turn += Input.GetKey (KeyCode.A) ? (int) (-5.0f * speed) : 0; \u2013\u00a0Simian Apr 27 '17 at 8:15","date":"2021-07-31 00:46:48","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\": 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\": 0.42046239972114563, \"perplexity\": 4598.824281981412}, \"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\/1627046154032.75\/warc\/CC-MAIN-20210730220317-20210731010317-00241.warc.gz\"}"} | null | null |
Q: SQL Server Protocols that Support NTLM I have a question, what are the four SQL server network configuration protocols that support NTLM authentication?
A: With NTLM we mean windows authentication or integrated security SSPI. As far I know, in order to use SQL Server's integrated security, you must choose either Named Pipes or Multi-Protocol with Named Pipes.
You can see more here
http://msdn.microsoft.com/en-us/library/aa266530%28v=vs.60%29.aspx
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,981 |
\section{Introduction}
The olivine structured lithium transition metal phosphates Li$M$PO$_4$ ($M$ = Fe, Mn,
Co, and Ni) have attracted interest from both fundamental and technical points
of view. They reveal a variety of unusual magnetic, magnetoelectric, and
ferrotoroidic properties associated with high spin (HS) $M^{2+}$
ions.\cite{vaknin04,vanaken07,aken08,jensen09a,toft-petersen12} Also,
their low cost, low toxicity, high stability, and high energy density made them promising
candidates of high-voltage cathode materials for Li-ion
batteries.\cite{padhi97,chung02,fisher08,bramnik08,kang09,murugan09}
Among the olivine Li$M$PO$_4$ family, LiCoPO$_4$ features a very large
linear magnetoelectric effect\cite{rivera94,kornev00}
and a high theoretical energy density up to 801 Wh/kg based on its
high discharge plateau 4.8 V versus Li/Li$^+$.\cite{amine00,okada01}
Olivine LiCoPO$_4$ crystallizes in the orthorhombic $Pnma$ space
group.\cite{santoro66} The Co$^{2+}$
($S=3/2$) ions sit in the center of the distorted CoO$_6$ octahedra which share corners and
edges with PO$_4$ tetrahedra, as illustrated in Fig. \ref{fig:structure}(a).
Below $T_N\sim 21$ K, the moments order antiferromagnetically, with a tilt of
4.6$^\circ$ away from the crystallographic $b$ axis within the $bc$ plane,
and the crystal structure changes to the $P12'_11$ symmetry.\cite{kharchenko01,vaknin02,
szewczyk11} In this low
symmetry, a nonzero toroidal moment is allowed and confirmed
experimentally. \cite{vanaken07,ederer07}
For optimal performance of electrode materials for Li-ion batteries,
it is crucial to get insight into the fundamental structural, electronic, and
magnetic properties of the materials and their limits and trends for
applications.\cite{chernova11}
Therefore, recently discovered non-olivine structured LiCoPO$_4$ is
interesting and may provide an important step forward in the understanding of
the impact of structure and magnetism on the performance of a battery
material. LiCoPO$_4^\text{tetra}$\ possesses $Pn2_1a$
symmetry and consists of CoO$_4$ tetrahedra, instead of CoO$_6$ octahedra in the
olivine structure, sharing only corners with PO$_4$
tetrahedra\cite{hautier11,jahne13} [see Fig. \ref{fig:structure}(b)].
The non-olivine structure becomes unstable at high temperatures towards the olivine one.
\begin{figure}
\centering
\includegraphics[width=0.6\linewidth]{structure.eps}
\caption{\label{fig:structure} (Color online)
Crystal structure of (a) olivine ($Pnma$ symmetry) and (b) tetrahedral ($Pn2_1a$
symmetry) LiCoPO$_4$, projected along the crystallographic $c$ axis. Li atoms
are omitted for clarity. Their
unit cells are drawn as dotted lines.
}
\end{figure}
Initial studies indicate that tetrahedral LiCoPO$_4$ exhibits poor
performance in terms of cycling stability and discharge capacity compared to
the olivine phosphate.\cite{jahne13}
In an attempt to elucidate the detailed magnetic properties of both compounds
associated with their structural aspects,
we carried out an NMR study on
tetrahedral LiCoPO$_4^\text{tetra}$\ as well as on the annealed olivine compound.
Our results of the Knight shift and the spin-lattice relaxation rates show
that the tetrahedrally coordinated Co$^{2+}$ spins in LiCoPO$_4^\text{tetra}$\
are strongly frustrated, resulting in quite different magnetic properties,
compared to olivine LiCoPO$_4$. The magnetic structure in the ordered state is
likely incommensurate.
\section{Sample preparation and experimental details}
Non-olivine LiCoPO$_4^\text{tetra}$\ microcrystals with $Pn2_1a$ symmetry were synthesized by the
microwave-assisted hydrothermal synthesis technique, as described in detail in
Refs. \onlinecite{jahne13,neef13}. Olivine LiCoPO$_4$ with $Pnma$ symmetry was
obtained by annealing the non-olivine compound at 700$^\circ$C for 24 hours
under an argon atmosphere.
The temperature dependence of the static uniform magnetic susceptibility $\chi(0,0)$
of LiCoPO$_4^\text{tetra}$\ and LiCoPO$_4^\text{olivine}$\ was measured using a superconducting quantum
interference device (SQUID) magnetometer in the field
of 1 kOe after cooling in zero magnetic field.
$^{7}$Li\ and $^{31}$P\ NMR measurements have been carried out using a spin-echo method
in a fixed field of 7.0494 T in the temperature range of 5--400 K.
Since $^{31}$P\ (nuclear spin $I=1/2$, $\gamma_n=17.2356$ MHz/T) does not involve
electric quadrupole
effects, it is an ideal probe to study magnetism and spin
fluctuations in these materials.
$^{31}$P\ NMR spectra over most of the measured temperature range are relatively narrow
and quite symmetric, in comparison with the large linewidth and strong
anisotropy observed in other
Li phosphates,\cite{arcon04,rudisch13} which allowed us
to determine the Knight shift and the linewidth reliably.
The spin-lattice relaxation rates $T_1^{-1}$\ of both $^{31}$P and $^7$Li
($I=3/2$, $\gamma_n=16.5471$ MHz/T) were measured using the
saturation recovery method and $T_1$
was obtained by fitting the relaxation of the nuclear magnetization $M(t)$ to
a single exponential function, $1-M(t)/M(\infty)=A\exp(-t/T_1)$ where $A$ is a
fitting parameter.
\section{Experimental results and discussion}
\subsection{Magnetic susceptibility $\chi$ and $^{31}$P\ Knight shift $\mathcal{K}$}
Figure \ref{fig:chi} shows the molar static susceptibility $\chi_\text{mol}=M/H$
as a function of temperature for both olivine and tetrahedral LiCoPO$_4$. The
drop of the $\chi$ data at low temperatures indicates that in both systems
long range antiferromagnetic order evolves at low temperature. The transition
can be clearly observed in the magnetic
specific heat $c_\text{mag} \sim \partial(\chi_\text{mol} T)/\partial T$ (see
inset) as both compounds show an anomaly at
$T_N$ = 21 and 7 K, respectively.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{chi.EPS}
\caption{\label{fig:chi} (Color online)(a) Temperature dependence of static susceptibility
and magnetic specific heat (inset) of both olivine
and tetrahedral LiCoPO$_4$. Solid curves are Curie-Weiss fits. (b) NMR
linewidth (FWHM) of $^{31}$P spectrum (see Fig. \ref{fig:31spec}) tracks
$\chi$ in the whole temperature range investigated. Inset: Inverse static susceptibility.}
\end{figure}
At high temperatures, the data in both compounds follow a Curie-Weiss (CW) law,
$\chi_\text{mol}=C/(T+\Theta)+\chi_0$, where
$\chi_0$ is a $T$-independent susceptibility [see inset of
Fig.~\ref{fig:chi}(b)]. Fitting the data with the CW law
yields the Weiss temperatures $\Theta=52(5)$ K (olivine) and $\Theta=7(1)$ K
(tetra), and the effective magnetic
moments $\mu_\text{eff} = 5.1 \mu_B$ (olivine) and $4.4 \mu_B$ (tetra). Both
values of $\mu_\text{eff}$ exceed the
spin-only value of 3.87 $\mu_B$ expected for the HS 3$d^7$ configuration for
full quenching of the orbital moment. In LiCoPO$_4^\text{tetra}$, the
orbital admixture to the measured effective $g = 2.27$ is governed by the
reduced spin-orbit coupling $\lambda$ and the
tetrahedral crystal field (CF) splitting $\Delta_t$ of the Co$^{2+}$ orbital
states of $e_g$ and
$t_{2g}$ symmetry. It can be approximated by
\begin{equation}
g = 2-\frac{8\cdot \lambda}{\Delta_t}.
\end{equation}
With $\lambda \approx -143$ cm$^{-1}$,\cite{sati06} the data are consistent
with $\Delta_t \approx 4250$ cm$^{-1}$. In contrast, the
electronic configuration of HS Co$^{2+}$ in the octahedral configuration
$t_{2g}^5e_g^2$ realized in LiCoPO$_4^\text{olivine}$\
exhibits a stronger orbital contribution since it involves only partially filled
low-lying orbital triplet states with
pseudo angular momentum $\tilde{l}=1$. The observed effective moment agrees
well with the findings in Ref.~\onlinecite{vaknin02}.
In this paper, we mainly concentrate on the NMR data obtained on both
materials. Hence, we note that the broadening of the NMR line is essentially
determined by the uniform bulk susceptibility as can be deduced from Fig.
\ref{fig:chi} (b) where the full width at half maximum (FWHM) of the $^{31}$P\ NMR
spectra is plotted against $\chi$ with $T$ as an implicit
parameter. Further information on the static magnetic properties is obtained
from measurements of the $^{31}$P\ Knight shift, $^{31}\mathcal{K}$. Its
temperature dependence confirms CW-behavior in both materials. The data in
Fig.~\ref{fig:31K} are well described by
$\mathcal{K}=C'/(T+\Theta)+\mathcal{K}_0$, where $\mathcal{K}_0$ is the
$T$-independent orbital shift. For both materials, $\mathcal{K}_0$ appears
negligibly small, indicating that
$\mathcal{K}$ probes almost entirely the spin part of the magnetic susceptibility.
Focusing on the evolution of antiferromagnetic order, the static
susceptibility data imply deviations from the
mean-field Curie-Weiss-like behavior already at relatively high temperatures.
In the olivine sample, small deviations from
the experimental data are observed below $\sim 250$\,K as visible in
Figs.~\ref{fig:chi}(a) and (b, inset). The
experimentally observed susceptibility is smaller than predicted by the
Curie-Weiss law which is in agreement with the
evolution of antiferromagnetic fluctuations well above $T_N$. In the tetragonal
polymorph, such behavior is present
below $\sim 50$\,K which is best seen if the magnetic specific heat in Fig.
\ref{fig:chi}~(a) (inset) is considered.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{31spec.EPS}
\caption{\label{fig:31spec} (Color online) Temperature evolution of $^{31}$P NMR spectrum
obtained at 7.0494 T for both tetrahedral (left) and olivine (right) LiCoPO$_4$
in the paramagnetic state. The spectrum in the ordered state for
LiCoPO$_4^\text{olivine}$\ is also shown for comparison.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{31K.EPS}
\caption{\label{fig:31K} (Color online) The $^{31}$P Knight shift as a function of
temperature. Solid curves are Curie-Weiss fits.
The inset shows a plot of $\mathcal{K}(T)$ versus $\chi(0,0)(T)$ with $T$ as
implicit parameter. The slope of the data
yields the hyperfine coupling constants $A_\text{hf}=1.96$ kOe/$\mu_B$ and
1.71 kOe/$\mu_B$ for LiCoPO$_4^\text{tetra}$\ and LiCoPO$_4^\text{olivine}$,
respectively. The $T$-independent orbital shifts are found to be nearly zero (see text). }
\end{figure}
In order to compare the static susceptibility measured by bulk and local
techniques, the inset of Fig. \ref{fig:31K}
shows a plot of $^{31}\mathcal{K}$ versus $\chi$. $^{31}\mathcal{K}$ is
proportional to $\chi$ over a wide temperature
range. The small deviation from linearity occurring below $\sim60$ K for LiCoPO$_4^\text{tetra}$\
is attributed to a small amount of
paramagnetic (PM) impurities to which $\mathcal{K}$ is insensitive. This
account is indeed corroborated by the good
agreement between the linewidth and $\chi$ in the $T$ region where
$\mathcal{K}$ deviates $\chi$, as shown in the inset
of Fig. \ref{fig:chi}, since a random distribution of PM moments would broaden
the NMR line without affecting its shift.
The linear slope of $d\mathcal{K}/d\chi$ corresponds to the hyperfine (hf)
coupling constants, $A_\text{hf}=1.96$
kOe/$\mu_B$ (tetra) and 1.71 kOe/$\mu_B$ (olivine). Note that
$^{31}\mathcal{K}$ in both compounds almost vanishes at $\chi =\chi_0\sim0$, confirming the
non-spin susceptibility contribution $\mathcal{K}_0\sim0$.
\subsection{Spin-lattice relaxation rate $T_1^{-1}$\ and dynamical susceptibility}
The $^{31}$P spin-lattice relaxation rate
$^{31}T_1^{-1}$\ as a function of temperature is presented in Fig. \ref{fig:31invT1}.
For the olivine compound, $^{31}T_1^{-1}$\ increases steadily with decreasing $T$ and is
rapidly enhanced below 40 K, exhibiting a sharp anomaly followed by a rapid
drop. The sharp peak (vertical solid line) indicates the onset of magnetic
order at $T_N=21$ K, which agrees with literature values.\cite{santoro66,vaknin02,szewczyk11}
The rapid upturn above $T_N$, which is
due to the critical slowing down of spin fluctuations towards
magnetic order, is a rough measure of dimensionality of the magnetic system as well.
In our case, the relatively sharp
transition width which is comparable to $T_N$ indicates the
quasi-three-dimensional nature of
the magnetic order rather than two-dimensional (2D).\cite{vaknin02,baker11}
For LiCoPO$_4^\text{tetra}$, the $^{31}T_1^{-1}$\ data are an order of magnitude larger than those of
LiCoPO$_4^\text{olivine}$, displaying a similar temperature dependence.
$T_N$ could not be identified since $^{31}T_1$
becomes too short to be measured near the transition,
but the temperature at which $^{31}T_1^{-1}$\ diverges appears to be consistent with $T_N=7$
K (vertical dashed line)
\begin{figure}
\centering
\includegraphics[width=\linewidth]{31invT1.EPS}
\caption{\label{fig:31invT1} (Color online) $^{31}$P spin-lattice relaxation rate $^{31}T_1^{-1}$\ as a
function of temperature in olivine and tetrahedral LiCoPO$_4$ samples. For
LiCoPO$_4^\text{olivine}$, $^{31}T_1^{-1}$\ reveals a sharp
magnetic transition at $T_N=21$ K (vertical solid line), while it
diverges towards $T_N=7$
K (vertical dashed line) for LiCoPO$_4^\text{tetra}$. Solid curves are equivalent to the linear
fit in Fig. \ref{fig:T1comp}.
The inset presents a plot of $1/\sum_q\chi''(q,\omega_L)$ versus $T^2$,
to show the inverse dynamical
susceptibility that is quadratic in temperature.}
\end{figure}
For olivine LiCoPO$_4$, inelastic neutron scattering yields moderate magnetic
exchange coupling constants.\cite{tian08}
In the $bc$-plane, the dominating nearest neighbor coupling amounts to
$J_{\|}^{nn} = 9$\,K while smaller next nearest
neighbor and interlayer couplings of $\sim 1 - 2$\,K imply a tendency to weak
frustration and 2D behavior. In the case
of LiCoPO$_4^\text{tetra}$, where $T_N$ as well as $\Theta$ are significantly smaller than in
the olivine material, magnetic coupling is
presumingly weaker and/or the tendency towards 2D and frustration stronger.
In the paramagnetic
limit, $T_1^{-1}$\ could be approximated by the relation,\cite{moriya56a}
\begin{equation}
\label{eq:2}
T_1^{-1}\propto \frac{A_\text{hf}^2\sqrt{S(S+1)}}{\hbar J_\text{ex}}.
\end{equation}
Using this equation, one can estimate the ratio,
$T_1^{-1}\text{(tetra)}/T_1^{-1}\text{(olivine)}=6.5$, using the relation
$J_\text{ex}\equiv 3k_B\Theta/[z S(S+1)]$ where $z$ is the number of
nearest-neighbors. This value satisfactorily
accounts for the difference of $^{31}T_1^{-1}$\ between the two compounds at high
temperatures, proving that the system indeed lies in the localized limit at
high temperatures.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{7invT1.EPS}
\caption{\label{fig:7invT1} (Color online) Temperature dependence of the $^{7}$Li
spin-lattice relaxation rate.
Data in the PM region resemble those of $^{31}T_1^{-1}$, indicating that both $^{7}$Li\ and
$^{31}$P\ nuclei are governed by the same relaxation mechanism. The solid curves
are identical with those in Fig. \ref{fig:31invT1} with scaling.
A sharp anomaly was detected at $T_N=21$ K for LiCoPO$_4^\text{olivine}$, similar to the case of $^{31}$P. }
\end{figure}
In general, while the Knight shift is proportional to the static spin
susceptibility at $q=0$, i.e.
$\mathcal{K}=A_\text{hf}\chi(0,0)$, $T_1^{-1}$\ reflects the $q$-average of the imaginary
part of the dynamical
susceptibility $\chi''$ at low energy,\cite{moriya63}
\begin{equation}
\label{eq:T1}
T_1^{-1} \propto T \gamma_n^2 A_\text{hf}^2 \sum_q \chi''(q,\omega_L)/\omega_L,
\end{equation}
where $\gamma_n$ is the nuclear gyromagnetic ratio and $\omega_L$ the Larmor
frequency.
Since there is no difference of the Knight shift at high temperatures far
above $T_N$ between the two compounds, we conclude that spin fluctuations are
of dominantly antiferromagnetic nature for both olivine and tetrahedral LiCoPO$_4$.
The most striking feature is that $^{31}T_1^{-1}$\ for both compounds increases with
decreasing temperature, as shown in Fig. \ref{fig:31invT1}. Such a $1/T$ dependence
of $T_1^{-1}$\ is very
rare in the paramagnetic limit.
Figure \ref{fig:T1comp} clearly
shows the linear variation of $T_1$ in terms of $T$, particularly, in LiCoPO$_4^\text{olivine}$.
This in turn implies that the $q$-average of the
dynamical susceptibility
$\sum_q\chi''(q,\omega_L)$ from Eq. (\ref{eq:T1}) varies in proportion to
$1/T^2$, in contrast to the uniform static susceptibility $\chi(0,0)$ that
obeys the CW law.
A plot of $1/\sum_q\chi''(q,\omega_L)$ versus $T^2$ is given in the inset of Fig.
\ref{fig:31invT1}, which provides evidence of the quadratic
temperature dependence of the inverse dynamical susceptibility. Note that this
plot eliminates the effect of the hf coupling constants, allowing the direct
comparison of the two systems.
To ensure that the unusual $T$ dependence of $^{31}T_1^{-1}$\ is not
site dependent, but represents the intrinsic dynamical susceptibility of the system, we
also measured the
$^{7}$Li\ spin-lattice relaxation rate,
$^{7}T_1^{-1}$, as a function of temperature. The results are presented in Fig.
\ref{fig:7invT1}, revealing a $T$ dependence similar to $^{31}T_1^{-1}$.
In fact, Fig. \ref{fig:T1comp} proves that $^{31}T_1$ and $^{7}T_1$ as a function of
temperature are accurately
scaled to each other for both compounds.
\footnote{From the scaling, one can deduce the hf coupling constants of $^{7}$Li\
directly from those of $^{31}$P\ ; 0.79 kOe/$\mu_B$ (tetra) and
0.69 kOe/$\mu_B$ (olivine).}
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{T1comp.EPS}
\caption{\label{fig:T1comp} (Color online) For (a) LiCoPO$_4^\text{tetra}$\ and (b) LiCoPO$_4^\text{olivine}$, the spin-lattice
relaxation times $^{31}T_1$ and $^7T_1$ as a function of temperature
are scaled to each other, demonstrating that
they detect the dynamical susceptibility
$\sum_q \chi''(q,\omega_L)$ of the materials.
While the linear $T$ dependence of $T_1$ is well maintained over the whole
temperature range investigated, except near the transition at $T_N$ for LiCoPO$_4^\text{olivine}$,
it breaks down below $\sim150$ K for LiCoPO$_4^\text{tetra}$.
}
\end{figure}
From Figs. \ref{fig:31invT1} and \ref{fig:T1comp},
the different behaviors of $T_1^{-1}$\ in the two compounds are noticeable.
Namely, for LiCoPO$_4^\text{olivine}$, the $T_1$ data follow a linear $T$ behavior in the whole
temperature range investigated except the region near $T_N^\text{olivine}$. In contrast, for
LiCoPO$_4^\text{tetra}$, the $T_1$ data deviate
from the $T$-linear behavior at $\sim150$ K, suggesting that an additional
relaxation mechanism is developed.
Another noticeable feature is that the $T$-linearity
of $T_1$ is considerably smaller in LiCoPO$_4^\text{tetra}$, seemingly approaching the normal CW behavior
[i.e., $T_1(T) \rightarrow \text{ constant}$ at high $T$] which is observed in
other olivine lithium phosphates, LiMnPO$_4$ and LiFePO$_4$.\cite{arcon04}
The different magnetic properties of LiCoPO$_4^\text{tetra}$\ and LiCoPO$_4^\text{olivine}$\ could be
understood by considering their inherent spin networks.
There are five exchange pathways between the Co$^{2+}$ spins in olivine
LiCoPO$_4$.\cite{dai05,tian08}
In the $bc$ plane, one can identify the
nearest-neighbor coupling $J_1$ mediated
through Co-O-Co superexchange path and the next-nearest-neighbor coupling
$J_2$ and $J_3$ along the $b$ and $c$ axes, respectively, through PO$_4$
tetrahedra [Fig. \ref{fig:structure}(a) depicts only $J_1$ coupled Co atoms].
The interplane couplings $J_4$ and
$J_5$ are also mediated by PO$_4$ tetrahedra and are known to be
ferromagnetic, while the intraplane couplings $J_1$, $J_2$, and $J_3$ are all
antiferromagnetic. This
spin network involves weak geometrical frustration since
$J_1$ is much larger than other exchange couplings.\cite{tian08}
This situation is dramatically altered in LiCoPO$_4^\text{tetra}$.
As clearly shown in Fig. \ref{fig:structure}(b), there is no longer a Co-O-Co superexchange
path and all the exchange interactions are mediated
by corner-shared PO$_4$ tetrahedra and might be comparable to each other in strength.
Naturally, this spin network likely results in strong frustration.
The frustration may be consistent with the strong reduction of the effective exchange
interaction and ordering
temperature $T_N$.
Note that in the case of competing ferromagnetic and antiferromagnetic interactions the ratio
$T_N/\Theta$ does not provide reliable information on magnetic frustration.
Since Co$^{2+}$ ions behave more like paramagnets in LiCoPO$_4^\text{tetra}$,
one may argue that the frustration effect modifies the
spin dynamics which causes the peculiar upturn of $T_1^{-1}$.
At low temperatures, the frustration can induce the incommensurate or
spin-glass-like magnetic ordering.
In this case, magnetic short-range fluctuations may extend far above $T_N$,
being responsible for the additional enhancement of $T_1^{-1}$\ which was observed
below $\sim150$ K.
Now we discuss the puzzling feature of the $1/T$-dependence of $T_1^{-1}$,
which implies the inverse quadratic temperature dependence of the dynamical
susceptibility. To begin with, one may conjecture that Li diffusion motion causes the
increase of $T_1^{-1}$\ with decreasing $T$. However, we rule out this possibility because both
mobile $^{7}$Li\ and immobile $^{31}$P\ nuclei detect the identical $T$-dependence of
$T_1^{-1}$, as demonstrated in Fig. \ref{fig:T1comp}. In principle,
$A_\text{hf}$ may increase with decreasing $T$, causing the $T$-dependence of
$T_1^{-1}$. Again, this is clearly not the case from the uniquely defined $A_\text{hf}$ from the
$\mathcal{K}$ vs. $\chi$ plot over a wide temperature range (see the inset of
Fig. \ref{fig:31K}).
Therefore, we conclude that unusual spin dynamics is
present and persists even
in the high $T$ region ($T\gg 10J_\text{ex}$), causing the inverse quadratic
$T$-dependence of $\sum_q\chi''(q,\omega_L)$.
One plausible explanation could be given if $J_\text{ex}$ in Eq. (\ref{eq:2})
decreases with decreasing temperature.\cite{baek10b} Although this scenario may
be incompatible with the well-defined $\Theta\propto J_\text{ex}$, if spin
fluctuations at small $q<Q$ are developed with decreasing temperature,
the resultant effective exchange coupling could be reduced.
\section{Conclusions}
We present $^{7}$Li\ and $^{31}$P\ NMR studies in both non-olivine and olivine
structured LiCoPO$_4$ microcrystals.
It turns out that the exchange
interactions among the Co$^{2+}$ spins are greatly reduced in LiCoPO$_4^\text{tetra}$,
which accounts for the difference of the spin-lattice relaxation rates $T_1^{-1}$\ between
the two compounds.
In contrast to the Curie-Weiss behavior of the static susceptibility found at high
temperatures, the
dynamical spin susceptibility deduced from the spin-lattice
relaxation rates is inversely quadratic in temperature, which is particularly
strong and robust in LiCoPO$_4^\text{olivine}$. For LiCoPO$_4^\text{tetra}$, the unusual temperature dependence
is considerably weakened and breaks down at low temperatures.
Together with the reduced effective exchange coupling and ordering
temperature, this different spin dynamics is attributed to strong
frustration effect inherent in the
corner-shared CoO$_4$-PO$_4$ geometry of this metastable material.
The additional enhancement $T_1^{-1}$\ at low temperatures in LiCoPO$_4^\text{tetra}$\ suggests that the
frustration may lead to complex incommensurate magnetic order.
\section*{Acknowledgement}
We thank Andrei Malyuk for annealing the LiCoPO$_4^\text{tetra}$\ sample.
This work was supported by the DFG (Grants
No. GR3330/3-1 and No. KL1824/2-2) and by the BMBF (Project No. 03SF0397). S.B.
acknowledges support by DFG Research Grant No. BA 4927/1-1.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,406 |
// Copyright 2016 Google Inc. All Rights Reserved.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
using Google.Apis.Bigquery.v2.Data;
using Newtonsoft.Json;
using System;
using System.Collections.Generic;
using System.Globalization;
using Xunit;
namespace Google.Cloud.BigQuery.V2.Tests
{
public class BigQueryParameterTest
{
// Please excuse the very long lines... it's more readable than
public static IEnumerable<object[]> ToQueryParameterData() => new[]
{
// Bool
ScalarTest("Bool parameter, true", BigQueryDbType.Bool, true, "TRUE"),
ScalarTest("Bool parameter, false", BigQueryDbType.Bool, false, "FALSE"),
ScalarTest("Bool parameter, string value", BigQueryDbType.Bool, "maybe", "maybe"),
ScalarTest("Bool parameter, null value", BigQueryDbType.Bool, null, null),
// Float64
ScalarTest("Float64 parameter, Int16 value", BigQueryDbType.Float64, (short)10, "10"),
ScalarTest("Float64 parameter, Int32 value", BigQueryDbType.Float64, (int)10, "10"),
ScalarTest("Float64 parameter, Int64 value", BigQueryDbType.Float64, (long)10, "10"),
ScalarTest("Float64 parameter, UInt16 value", BigQueryDbType.Float64, (ushort)10, "10"),
ScalarTest("Float64 parameter, UInt32 value", BigQueryDbType.Float64, (uint)10, "10"),
ScalarTest("Float64 parameter, UInt64 value", BigQueryDbType.Float64, (ulong)10, "10"),
ScalarTest("Float64 parameter, Single value", BigQueryDbType.Float64, 10.5f, "10.5"),
ScalarTest("Float64 parameter, Double value", BigQueryDbType.Float64, 10.5, "10.5"),
ScalarTest("Float64 parameter, Single value (+inf)", BigQueryDbType.Float64, float.PositiveInfinity, "+inf"),
ScalarTest("Float64 parameter, Double value (+inf)", BigQueryDbType.Float64, double.PositiveInfinity, "+inf"),
ScalarTest("Float64 parameter, Single value (-inf)", BigQueryDbType.Float64, float.NegativeInfinity, "-inf"),
ScalarTest("Float64 parameter, Double value (-inf)", BigQueryDbType.Float64, double.NegativeInfinity, "-inf"),
ScalarTest("Float64 parameter, Single value (NaN)", BigQueryDbType.Float64, float.NaN, "NaN"),
ScalarTest("Float64 parameter, Double value (NaN)", BigQueryDbType.Float64, double.NaN, "NaN"),
ScalarTest("Float64 parameter, string value", BigQueryDbType.Float64, "string value", "string value"),
ScalarTest("Float64 parameter, null value", BigQueryDbType.Float64, null, null),
// Int64...
ScalarTest("Int64 parameter, Int16 value", BigQueryDbType.Int64, (short)10, "10"),
ScalarTest("Int64 parameter, Int32 value", BigQueryDbType.Int64, (int)10, "10"),
ScalarTest("Int64 parameter, Int64 value", BigQueryDbType.Int64, (long)10, "10"),
ScalarTest("Int64 parameter, UInt16 value", BigQueryDbType.Int64, (ushort)10, "10"),
ScalarTest("Int64 parameter, UInt32 value", BigQueryDbType.Int64, (uint)10, "10"),
ScalarTest("Int64 parameter, UInt64 value", BigQueryDbType.Int64, (ulong)10, "10"),
ScalarTest("Int64 parameter, string value", BigQueryDbType.Int64, "string value", "string value"),
ScalarTest("Int64 parameter, null value", BigQueryDbType.Int64, null, null),
// String...
ScalarTest("String parameter, string value", BigQueryDbType.String, "some value", "some value"),
ScalarTest("String parameter, null value", BigQueryDbType.String, null, null),
// Bytes...
ScalarTest("Bytes parameter, byte[] value", BigQueryDbType.Bytes, new byte[] { 1, 2 }, "AQI="),
ScalarTest("Bytes parameter, string value", BigQueryDbType.Bytes, "some value", "some value"),
ScalarTest("Bytes parameter, null value", BigQueryDbType.Bytes, null, null),
// Date
ScalarTest("Date parameter, DateTime value", BigQueryDbType.Date, new DateTime(2016, 10, 31), "2016-10-31"),
// This is midnight local time on the 31st, which means its UTC value is actually 2016-10-30T22:00:00
// - but we only care about the local value for Date parameters
ScalarTest("Date parameter, DateTimeOffset value", BigQueryDbType.Date, new DateTimeOffset(2016, 10, 31, 0, 0, 0, TimeSpan.FromHours(2)), "2016-10-31"),
ScalarTest("Date parameter, string value", BigQueryDbType.Date, "some value", "some value"),
ScalarTest("Date parameter, null value", BigQueryDbType.Date, null, null),
// DateTime
// Value with ticks is truncated (not rounded). The Kind is irrelevant
ScalarTest("DateTime parameter, DateTime value (local)", BigQueryDbType.DateTime,
new DateTime(2016, 10, 31, 1, 2, 3, 123,DateTimeKind.Local).AddTicks(4567),
"2016-10-31 01:02:03.123456"),
ScalarTest("DateTime parameter, DateTime value (unspecified)", BigQueryDbType.DateTime,
new DateTime(2016, 10, 31, 1, 2, 3, 123, DateTimeKind.Unspecified).AddTicks(4567),
"2016-10-31 01:02:03.123456"),
ScalarTest("DateTime parameter, DateTime value (UTC)", BigQueryDbType.DateTime,
new DateTime(2016, 10, 31, 1, 2, 3, 123, DateTimeKind.Utc).AddTicks(4567),
"2016-10-31 01:02:03.123456"),
// This is midnight local time on the 31st, which means its UTC value is actually 2016-10-30T22:00:00
// - but we only care about the local value for DateTime parameters
ScalarTest("DateTime parameter, DateTimeOffset value", BigQueryDbType.DateTime,
new DateTimeOffset(2016, 10, 31, 0, 0, 0, TimeSpan.FromHours(2)), "2016-10-31 00:00:00"),
ScalarTest("DateTime parameter, string value", BigQueryDbType.Date, "some value", "some value"),
ScalarTest("DateTime parameter, null value", BigQueryDbType.Date, null, null),
// Time
// Truncated to microseconds
ScalarTest("Time parameter, TimeSpan value", BigQueryDbType.Time,
TimeSpan.ParseExact("01:23:45.1234567", "hh':'mm':'ss'.'FFFFFFF", CultureInfo.InvariantCulture),
"01:23:45.123456"),
// Truncated to a single day
ScalarTest("Time parameter, TimeSpan value bigger than 24h", BigQueryDbType.Time,
TimeSpan.FromHours(25), "01:00:00.000000"),
ScalarTest("Time parameter, TimeSpan value negative", BigQueryDbType.Time,
TimeSpan.FromHours(-1), "23:00:00.000000"),
ScalarTest("Time parameter, TimeSpan value large negative", BigQueryDbType.Time,
TimeSpan.FromHours(-25), "23:00:00.000000"),
ScalarTest("Time parameter, DateTime value", BigQueryDbType.Time,
new DateTime(2016, 10, 31, 1, 2, 3, 123).AddTicks(4567),
"01:02:03.123456"),
ScalarTest("Time parameter, DateTimeOffset value", BigQueryDbType.Time,
new DateTimeOffset(2016, 10, 31, 1, 2, 3, 123, TimeSpan.FromHours(2)).AddTicks(4567),
"01:02:03.123456"),
ScalarTest("Time parameter, string value", BigQueryDbType.Time, "some value", "some value"),
ScalarTest("Time parameter, null value", BigQueryDbType.Time, null, null),
// Timestamp
ScalarTest("Timestamp parameter, DateTime value (UTC)", BigQueryDbType.Timestamp,
new DateTime(2016, 10, 31, 1, 2, 3, 123, DateTimeKind.Utc).AddTicks(4567),
"2016-10-31 01:02:03.123456+00"),
ScalarTest("Timestamp parameter, DateTimeOffset value positive offset", BigQueryDbType.Timestamp,
new DateTimeOffset(2016, 10, 31, 0, 0, 0, TimeSpan.FromHours(2)), "2016-10-31 00:00:00+02:00"),
ScalarTest("Timestamp parameter, DateTimeOffset value negative offset", BigQueryDbType.Timestamp,
new DateTimeOffset(2016, 10, 31, 0, 0, 0, TimeSpan.FromHours(-2)), "2016-10-31 00:00:00-02:00"),
ScalarTest("Timestamp parameter, string value", BigQueryDbType.Timestamp, "some value", "some value"),
ScalarTest("Timestamp parameter, null value", BigQueryDbType.Timestamp, null, null),
};
public static IEnumerable<object[]> InvalidParameterData => new[]
{
new object[] { "Local DateTime for Timestamp", BigQueryDbType.Timestamp, new DateTime(2016, 10, 31, 0, 0, 0, DateTimeKind.Local) },
new object[] { "Unspecified DateTime for Timestamp", BigQueryDbType.Timestamp, new DateTime(2016, 10, 31, 0, 0, 0, DateTimeKind.Unspecified) },
new object[] { "Array with null element value", BigQueryDbType.Array, new[] { "foo", null, "bar" } },
new object[] { "Array with null value", BigQueryDbType.Array, null },
new object[] { "Array with non-array value", BigQueryDbType.Array, 10 },
new object[] { "Null value and null type", null, null },
new object[] { "DateTime value for Int64 type", BigQueryDbType.Int64, default(DateTime) },
};
public static IEnumerable<object[]> TypeInferenceData => new[]
{
new object[] { "Int16", (short) 10, BigQueryDbType.Int64 },
new object[] { "Int32", 10, BigQueryDbType.Int64 },
new object[] { "Int64", (long) 10, BigQueryDbType.Int64 },
new object[] { "UInt16", (ushort) 10, BigQueryDbType.Int64 },
new object[] { "UInt32", (uint) 10, BigQueryDbType.Int64 },
new object[] { "UInt64", (ulong) 10, BigQueryDbType.Int64 },
new object[] { "Single", 1.0f, BigQueryDbType.Float64 },
new object[] { "Double", 1.0d, BigQueryDbType.Float64 },
new object[] { "TimeSpan", TimeSpan.FromHours(1), BigQueryDbType.Time },
new object[] { "DateTime (local)", new DateTime(2016, 10, 31, 0, 0, 0, DateTimeKind.Local), BigQueryDbType.DateTime },
new object[] { "DateTime (unspecified)", new DateTime(2016, 10, 31, 0, 0, 0, DateTimeKind.Unspecified), BigQueryDbType.DateTime },
new object[] { "DateTime (UTC)", new DateTime(2016, 10, 31, 0, 0, 0, DateTimeKind.Utc), BigQueryDbType.DateTime },
new object[] { "DateTimeOffset", new DateTimeOffset(2016, 10, 31, 0, 0, 0, TimeSpan.FromHours(2)), BigQueryDbType.Timestamp },
new object[] { "Byte[]", new byte[] { 1, 2 }, BigQueryDbType.Bytes },
};
public static IEnumerable<object[]> ArrayTypeInferenceData => new[]
{
new object[] { "Int16[]", new short[0], BigQueryDbType.Int64 },
new object[] { "Int32[]", new int[0], BigQueryDbType.Int64 },
new object[] { "Int64[]", new long[0], BigQueryDbType.Int64 },
new object[] { "UInt16[]", new ushort[0], BigQueryDbType.Int64 },
new object[] { "UInt32[]", new uint[0], BigQueryDbType.Int64 },
new object[] { "UInt64[]", new ulong[0], BigQueryDbType.Int64 },
new object[] { "Single[]", new float[0], BigQueryDbType.Float64 },
new object[] { "Double[]", new double[0], BigQueryDbType.Float64 },
new object[] { "TimeSpan[]", new TimeSpan[0], BigQueryDbType.Time },
new object[] { "DateTime[]", new DateTime[0], BigQueryDbType.DateTime },
new object[] { "DateTimeOffset[]", new DateTimeOffset[0], BigQueryDbType.Timestamp },
new object[] { "Byte[][]", new byte[0][], BigQueryDbType.Bytes },
};
[Theory]
[MemberData(nameof(ToQueryParameterData))]
public void ToQueryParameter_Valid(string name, BigQueryParameter parameter, QueryParameter expectedResult)
{
// Positional vs named mode difference is only in validation, tested elsewhere.
string actualJson = JsonConvert.SerializeObject(parameter.ToQueryParameter(BigQueryParameterMode.Positional));
string expectedJson = JsonConvert.SerializeObject(expectedResult);
Assert.Equal(actualJson, expectedJson);
}
[Theory]
[MemberData(nameof(InvalidParameterData))]
public void ToQueryParameter_Invalid(string name, BigQueryDbType? type, object value)
{
var parameter = new BigQueryParameter(type, value);
Assert.Throws<InvalidOperationException>(() => parameter.ToQueryParameter(BigQueryParameterMode.Positional));
}
[Theory]
[MemberData(nameof(TypeInferenceData))]
public void TypeInference(string name, object value, BigQueryDbType expectedType)
{
var parameter = new BigQueryParameter();
parameter.Value = value;
var queryParameter = parameter.ToQueryParameter(BigQueryParameterMode.Positional);
var actualType = EnumMap<BigQueryDbType>.ToValue(queryParameter.ParameterType.Type);
Assert.Equal(expectedType, actualType);
}
[Theory]
[MemberData(nameof(ArrayTypeInferenceData))]
public void ArrayTypeInference(string name, object value, BigQueryDbType expectedArrayType)
{
var parameter = new BigQueryParameter();
parameter.Value = value;
var queryParameter = parameter.ToQueryParameter(BigQueryParameterMode.Positional);
Assert.Equal(BigQueryDbType.Array.ToParameterApiType(), queryParameter.ParameterType.Type);
var actualArrayType = EnumMap<BigQueryDbType>.ToValue(queryParameter.ParameterType.ArrayType.Type);
Assert.Equal(expectedArrayType, actualArrayType);
}
[Fact]
public void InvalidTypeIsRejected()
{
var parameter = new BigQueryParameter();
Assert.Throws<ArgumentException>(() => parameter.Type = (BigQueryDbType)(-1));
}
[Fact]
public void NeverValidValueIsRejected()
{
var parameter = new BigQueryParameter();
Assert.Throws<ArgumentException>(() => parameter.Value = Guid.NewGuid());
}
[Fact]
public void UnnamedParameterCannotBeConvertedWithNamedMode()
{
var parameter = new BigQueryParameter(BigQueryDbType.String, "foo");
// Prove that it's fine for a positional parameter
parameter.ToQueryParameter(BigQueryParameterMode.Positional);
Assert.Throws<InvalidOperationException>(() => parameter.ToQueryParameter(BigQueryParameterMode.Named));
}
[Fact]
public void StructParametersNotImplemented()
{
var parameter = new BigQueryParameter(BigQueryDbType.Struct, null);
Assert.Throws<NotImplementedException>(() => parameter.ToQueryParameter(BigQueryParameterMode.Positional));
}
[Fact]
public void ExplicitlyTypedArrayType()
{
var parameter = new BigQueryParameter(BigQueryDbType.Array, new[] { new DateTimeOffset(new DateTime(2016, 10, 31), new TimeSpan(8, 30, 0)) });
parameter.ArrayType = BigQueryDbType.DateTime;
string actualJson = JsonConvert.SerializeObject(parameter.ToQueryParameter(BigQueryParameterMode.Positional));
var expectedResult = new QueryParameter
{
ParameterType = new QueryParameterType
{
Type = BigQueryDbType.Array.ToParameterApiType(),
ArrayType = new QueryParameterType { Type = BigQueryDbType.DateTime.ToParameterApiType() }
},
ParameterValue = new QueryParameterValue
{
ArrayValues = new[] { new QueryParameterValue { Value = "2016-10-31 00:00:00" } }
}
};
string expectedJson = JsonConvert.SerializeObject(expectedResult);
Assert.Equal(actualJson, expectedJson);
}
[Fact]
public void Constructor_NameOnly()
{
var parameter = new BigQueryParameter("name");
Assert.Equal("name", parameter.Name);
Assert.Null(parameter.Type);
Assert.Null(parameter.Value);
}
[Fact]
public void Constructor_TypeOnly()
{
var parameter = new BigQueryParameter(BigQueryDbType.String);
Assert.Null(parameter.Name);
Assert.Equal(BigQueryDbType.String, parameter.Type);
Assert.Null(parameter.Value);
}
private static object[] ScalarTest(string name, BigQueryDbType type, object parameterValue, string expectedValue) =>
new object[]
{
name,
new BigQueryParameter(type, parameterValue),
ScalarParameter(type, expectedValue)
};
private static QueryParameter ScalarParameter(BigQueryDbType type, string value) =>
new QueryParameter
{
ParameterType = new QueryParameterType { Type = type.ToParameterApiType() },
ParameterValue = new QueryParameterValue { Value = value }
};
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 265 |
\section{Introduction}
Physicists have taken numerous approaches to modeling infectious diseases,
ranging from simple, deterministic compartmental models that qualitatively
describe disease dynamics in single populations~\cite{anderson:1992}, to highly
complex, stochastic metapopulation models that can account for the spread of
emergent infectious diseases on a global
scale~\cite{Ferguson:2005gp,Ferguson:2006p509,VandenBroeck:2011dj}. Simple
models, designed to investigate the basic mechanisms underlying disease
dynamics, typically assume that a population is well-mixed, that interacting
individuals are identical and that stochastic effects are
negligible~\cite{ANDERSON:1979we,Brockmann:2010el}. On the other hand, complex
computational models are manufactured to predict the time-course of actual
emergent infectious diseases such as H1N1 in 2009~\cite{Bajardi:2011gn}, SARS in
2003~\cite{Hufnagel:2004kt} quantitatively. They typically take into account
data on social variability, age structure, spatial heterogeneity, seasonal
variation of disease dynamic parameters, multi-scale mobility networks, and
account for stochastic effects. Both classes of models fulfill equally
important, complementary, but almost mutually exclusive purposes.
Theoretical epidemiology experienced a major thrust with
the advent complex network theory and its introduction into the field~\cite{BarabasiAlbert2002,Newman2003}.
The study of network properties substantially advanced our understanding
of disease dynamic phenomena on multiple levels~\cite{ScaleFreeEpidemics}.
On one hand, networks were used as a model for inter-individual relationships
(social networks)~\cite{Newman:2002p963}. On the other hand, the network
approach was applied on a larger scale, modeling mobility
and transport between populations~\cite{Hufnagel:2004kt,Colizza:2007p521}.
The use of network theoretical concepts allowed researchers to investigate how
topological properties of underlying networks shape the contagion processes that
evolve on
them~\cite{ComputerViruses2001,SmallWorldEpidemic,SmallWorldStability,EpidemicThresholdScaleFree,RandomAssortiveScaleFree}.
In the context of epidemiology, mapping structural features of networks to
properties of the spread of the disease substantially increased the predictive
power of models and our understanding of epidemic phenomena.
Although it is intuitive and plausible that network features determine the
spread of a disease, it is equally plausible that an epidemic reshapes the
structure of the underlying networks. For example, in response to information on
an ongoing epidemic, people may change their behavior. They may decide to wear
face masks, avoid contacts, and travel less. Surprisingly, this feedback
mechanism has been neglected even in some of the most detailed and sophisticated
modeling approaches~\cite{Ferguson:2005gp,Ferguson:2006p509}. Topological
properties of social networks affect disease dynamics, and the disease then
feeds back to change the topology of the network. In order to understand the
dynamics of contagion phenomena in a population, it is vital to understand the
consequences of this feedback mechanism.
Networks that change their structure in response to their environment are called
\textit{adaptive}~\cite{GrossBlasius2007,BornholdtRohlf2000,HolmeGhoshal2006,Synchronization2011}.
In a recent study, Gross {et~al.}{} proposed a simple adaptive network scheme,
based on a \textit{rewiring} rule, to understand how individuals' behavioral
changes impact on the time course of an epidemic. In this model, susceptible
individuals are allowed to protect themselves from infection by rewiring their
existing links~\cite{GrossLimaBlasius2006}. Specifically, with probability $w$ a
susceptible breaks the relationship with an infected person and forms a new link
to another, randomly selected susceptible. Despite the simplicity of this
approach, the mechanism can generate an abundance of interesting phenomena
including hysteresis and multi-stability.
Although this mechanism is attractive, the response to an ongoing epidemic in a
population has many facets. Not only do individuals avoid other infected
individuals (negative response). In many scenarios, individuals increase their
interaction with infected individuals (positive response), particularly in
hospital scenarios, and families in which individuals adopt the role of a
caretaker. Potentially, these positive responses can facilitate disease
proliferation in a population and yield a higher disease prevalence. However,
caretaker activity can have a positive effect on infected individuals, for
example by increasing a person's recovery rate. A key question is how these
effects interact and under what circumstances caretaker activity has a net
positive or negative effect and how these effects play out in different network
topologies.
Here we propose and investigate these questions using an adaptive network model.
We consider two types of networks. First, the generic Erd\"os-R\'enyi{} random network with
binomial degree distribution, where each pair of nodes is linked with constant
probability $p_{ER}$~\cite{ErdosRenyi1960,Newman2003}. We also consider Barab\'asi-Albert{}
scale-free networks with power law degree distributions, which more closely
mimic the heterogeneity in social interactions. Dynamics on scale-free networks
have a number of important properties. For instance, they lack epidemic
thresholds and are immune to random immunization due to strong connectivity
fluctuations~\cite{ScaleFreeEpidemics,EndemicStatesComplexNets,Newman2003,BarabasiAlbertScaling1999,Immunization2002}.
Thus diseases on scale-free networks are difficult to avoid, and once they take
hold, they are difficult to eradicate. We will show that in scale free
topologies the highest disease extinction probabilities occur in the total
absence of caretakers, a surprising result which suggests that caretaker
relationships (including doctor/patient relationships) should be minimized in
those systems. For Erd\"os-R\'enyi{} networks we observe a critical caretaker proportion
which minimizes disease severity and beyond which additional caretakers increase
disease prevalence.
\section{Model description}
We consider a network with a constant number of nodes $N$, representing
individuals in a population. Each node is either susceptible ($S$) or infected
($I$). We denote the state variable of node $i$ by $x_{i}=0$ or $x_{i}=1$,
corresponding to states $S$ or $I$, respectively. A pair $(i,j)$ of nodes share
a weighted symmetric link $w_{ij}\geq 0$ representing their contact rate. Note
that in general these contact rates can have any real positive value, unlike
network models that are based on binary interactions. Susceptible nodes can
become infected, and infected nodes can then become susceptible again upon
recovery. This is the well-studied SIS (susceptible-infected-susceptible) model
\cite{Allen20001}. We also consider the SIR (susceptible-infected-recovered)
model where infected individuals become immune to the disease upon recovery.
Each link is designated either caretaker ($C$) or regular ($R$), and the
fraction of $C$ links is denoted $p_{c}$. We denote this signature of a link by
$\sigma_{ij}=1$ if the link is a caretaker link and $\sigma_{ij}=-1$ if it is
regular. These two classes represent different ways of responding to an
epidemic. Caretaker relationships cause nodes to increase their contact
frequency $w_{ij}$ if an attached node is infected, while regular relationships
cause nodes to avoid each other (decreasing contact rates). At each time step a
susceptible $i$ can become infected by one of its infected neighbors with a
probability that increases with link weight. We assume that:
\begin{equation}
p_{i}=1-\exp\left(-\alpha_{i}\tau\right)\label{eq:infection_probability}
\end{equation}
where $\tau$ is the propensity of disease transmission following
a contact, and $\alpha_{i}=\sum_{j}w_{ij}x_{j}$ is the susceptible's
contact rate with infecteds.
\begin{figure}[t!]
\centering
\includegraphics[width=0.75\columnwidth]{diagram_combined.pdf}
\caption{(a) An initial network with all nodes susceptible (left) has two caretaker
links (green) and three regular links (black). After the infection
of the central node (shown by change to red color), regular-linked
nodes react by {}``avoiding'' the infected node (represented here
by increasing distance). Caretaker-linked nodes, on the other hand,
react by further increasing contact rates (represented here by decreasing
distance). (b) Another network consists of two clusters around two
central infected nodes (red). When considering the {}``caretaker
effect'', the more caretaker interactions (green) a node is exposed
to, the greater its recovery rate (shown by node size; larger nodes
have faster recovery rates). Thus after a time step, the lower infected
node is more likely to recover, shown by its transition to susceptible
status (blue). \label{fig:model_diagram}}
\end{figure}
An infected individual $i$ recovers with propensity $\beta_{i}$
which yields the probability of recovery
\begin{equation}
r_{i}=1-\exp\left(-\beta_{i}\right)\label{eq:caretaker_effect}
\end{equation}
We consider two scenarios: 1) Infected nodes recover at a uniform
rate $\beta_{i}=\beta$ or 2) with variable probability. In the latter
case, caretaker relationships increase a node's recovery probability
$\beta_{i}$ according to \[
\beta_{i}=\beta_{0}+\left(\beta_{1}-\beta_{0}\right)\frac{\sigma_{i}^{n}}{\sigma_{0}^{n}+\sigma_{i}^{n}}\]
where $\beta_{0}$ is the base recovery rate, and $\beta_{1}$ the enhanced
recovery rate induced by the action of caretakers. The quantity $\sigma_{i}$
represents the total exposure of an infected to caretakers and is
given by \[
\sigma_{i}=\frac{1}{2}\sum_{j}w_{ij}(1+\sigma_{ij}),\]
thus $\sigma_{i}$ is the total weight of caretaker interactions
that node $i$ experiences. The parameter $\sigma_{0}$ sets the scale
for this exposure. The shape of the sigmoid curve can be controlled
by the exponent $n$.
The infectious state of the system is defined by the states $x_{i}$ of each
node. We model the adaptive nature of the network weights $w_{ij}$ according to
\begin{equation}
\delta_{t}w_{ij}=\mu\sigma_{ij}(x_{i}+x_{j})-\gamma\left(w_{ij}-w_{ij}^{0}\right).
\label{eq:cont_time}
\end{equation}
Here the first term acts as the driving force of weight change, governed by the
rate parameter $\mu$. If a link is a caretaker link ($\sigma_{ij}=1)$, and one
of the adjacent nodes is infected ($x_{i}=1$ or $x_{j}=1$), this term is
positive and causes the weight to increase (if both nodes are infected the
change is additive). Regular links ($\sigma_{ij}=-1$), on the other hand
decrease in strength if one of the connected nodes is infected. The second term
acts as a restorative force, governed by the rate parameter $\gamma\ll\mu$.
Because we investigate a system in discrete time we use the following update
rule for the weights:
\begin{equation}
w_{ij}(t+1)=w_{ij}(t)\exp\left[\mu\sigma_{ij}(x_{i}+x_{j})-\gamma\left(w_{ij}(t)-w_{ij}^{0}\right)\right],
\label{eq:discrete_time}
\end{equation}
a discrete time reformulation of Eq.~\eqref{eq:cont_time}.
\section{Results}
We first consider SIS dynamics. At each time step,
a randomly chosen node $i$ can transition from $S$ to $I$ with probability $p_{i}$,
or from $I$ to $S$ with probability $r_{i}$ as given above. To
study the effect of adaptive rewiring, we first consider a system without
the caretaker effect on the recovery rate, i.e.
$\beta_1=\beta_0$. Caretakers only increase their interaction with infected
individuals.
We consider a network with weights initially distributed uniformly between
0 and 1.
Results are shown in Fig.~\ref{fig:SIS_time_course}.
In the absence of caretaker links ($p_{c}=0)$, the equilibrium
endemic state $I^{*}=I_{t}/N$ is much lower than compared to the static
network (without rewiring). This is expected, as only regular (negative) interactions
exist that decrease in response to the epidemic. The total network
weight adapts to a smaller value, decreasing the endemic state. The
dynamics of the disease and adaptation of the network is visible in
the damped oscillation of the fraction of infecteds.
However, as the
fraction of caretakers is increased, diseases can attain higher endemic
states than their static network counterparts. The caretaker dynamics
increases the interaction rate with infecteds, effectively yielding
a higher disease prevalence, which is expected.
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{SIS_time_course}
\caption{Infected density ($I^{*}=I/N$) for SIS dynamics as a function of
time for different caretaker proportions $p_{c}$, where caretakers
do not improve recovery. Erd\"os-R\'enyi{} networks with adaptive rewiring were
used (solid lines), as well as a similar static network (no rewiring,
dashed line). Solid lines were obtained by averaging over 100 simulations,
so a single-simulation plot is overlaid in each adaptive scenario
for reference. The plot corresponds to $I_{0}=10^{2}$, $N=10^{3}$,
$p_{ER}=0.008,$ $\mu=0.05,$ $\gamma=0.037$, $\beta=0.15$, $\tau=0.18$.
\label{fig:SIS_time_course}}
\end{figure}
The system that lacks a positive caretaker effect represents a somewhat
artificial limiting case. We therefore consider a positive \textit{caretaker
effect}: caretaker relationships lend higher recovery rates $\beta_1>\beta_0$ to
infected individuals, see Eq.~\eqref{eq:caretaker_effect}. In particular, we
consider the effect of varying the maximum recovery rate $\beta_{1}$ and the
fraction of caretaker links $p_{c}$ on the extinction probability of the
disease. The results are depicted in Figs.~\ref{fig:SIS_cross_sections} and
\ref{fig:Phase_Diagrams}. In general, increasing $\beta_{1}$ yields higher
extinction, since caretaker links are more effective at raising recovery rates.
One would then expect that increasing the caretaker proportion $p_{c}$ would
also yield higher extinction, as more relationships would cause increasing
recovery rates. However, this is not necessarily the case. Raising the
caretaker proportion past some $\beta_{1}$-dependent critical value allows
diseases to persist. This critical value also serves as a threshold, as
increasing $p_{c}$ above this value rapidly decreases the extinction probability
to 0. This is illustrated in Fig.~\ref{fig:SIS_cross_sections}. Increasing
$p_{c}$ at first yields and increased $p_{ext}$ until a maximum is reached. A
further increase leads to a rapid decrease in extinction probability. For the
Erd\"os-R\'enyi{} network, the critical fraction of caretakers is approximately
$p_{c}\approx 10\%$. For $p_{c}$ values above or below this, high extinction
probability is seen only for very high values of $\beta_{1}$. Note however, that
even for very small fractions of caretakers, a substantial increase in
extinction probability is observed. This suggests that, if the caretaker-effect
is taken into account, the best strategy to extinguish a disease is the
existence of a few effective caretaker relationships, that safely avoids the
negative effects that emerge beyond the critical concentration. Note also that
for non-vanishing $p_{c}$, guaranteed extinction ($p_{ext}=1$) is observed only
for very high values of $\beta_{1}$.
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{SIS_cross_sections}
\caption{Extinction probability $p_{ext}$ for SIS dynamics as a function of
caretaker proportion $p_{c}$ for various values of $\beta_{1}$ in an Erd\"os-R\'enyi{} network.
Note a critical $p_{c}$ value at which extinction is
maximized. Approaching this value from the left yields a gradual increase
in extinction, while increasing $p_{c}$ past this critical value
causes a rapid decrease in $p_{ext}$.
The plot corresponds to $I_{0}=10^{2}$,
$N=10^{3}$, $\mu=0.05,$ $\gamma=0.037$, $\tau=0.18$, $\beta_{0}=0.35$,
$\sigma_{0}=\left<\sigma_{i}\right>\big|_{t=0}$, $p_{ER}=0.008$.
\label{fig:SIS_cross_sections}}
\end{figure}
Note that these results were obtained for an Erd\"os-R\'enyi{} network. In order to
investigate the interaction of network adaptation in combination with strong
network heterogeneity, we investigated the dynamics in a scale free topology.
The results are depicted in Fig.~\ref{fig:Phase_Diagrams}. In contrast to the
Erd\"os-R\'enyi{} system, we observe a high extinction of the disease for a wide range of
caretaker concentrations and recovery parameters $\beta_1$. The disease is
endemic in the adaptive, scale free network only for small $\beta_1$ and large
$p_c$. The implications of these results are interesting: In a scale free
adaptive network, regular links that decrease when connected to infected nodes
are sufficient to extinguish a disease, even in the presence of a considerable
fraction of caretaker links. This strongly contrasts with the behavior observed
in static scale free networks, in which the existence of strongly connected hubs
generally facilitate the spread of a disease. In the adaptive network, for
$p_{c}\ll1$, it is sufficient that the majority of nodes decrease their
interactions with the infected subpopulation. In scale-free networks, hubs that
possess a large number of links will adaptively reduce the majority of their
regular weights, and thus their ability to serve as a gateway of the disease to
spread throughout the network. In this regime, the effect of caretaker
relationships and their effect on recovery are benign. Only when the fraction
of caretaker links reaches a large value such that also hubs become
predominantly caretakers, the situation changes, and the disease will evolve
into an endemic state.
\begin{figure*}[t!]
\centering
\includegraphics[scale=0.6]{ER_ext_prob_phase} ~~~~~~~~~~~~
\includegraphics[scale=0.6]{SF_ext_prob_phase}
\caption{Two-parameter phase diagrams showing extinction probability for SIS
dynamics as a function of maximum caretaker effectiveness $\beta_{1}$
and caretaker proportion $p_{c}$. Erd\"os-R\'enyi{} (left) and Scale-Free networks
(right) were considered. In the black regions, extinction probability
is 0 while extinction probability is 1 in the white regions. The plots
suggest that increasing the caretaker proportion past a critical value
yields a decreased extinction probability in both networks. On the
Erd\"os-R\'enyi{} network, $p_{c}\approx10^{-1}$ yields maximum disease extinction,
while extinction is most likely for $p_{c}\approx0$ on the Scale-Free
network. The plots correspond to $I_{0}=10^{2}$, $N=10^{3}$, $\mu=0.05,$
$\gamma=0.037$, $\tau=0.18$, $\beta_{0}=0.35$, $\sigma_{0}=\left<\sigma_{i}\right>\big|_{t=0}$
if $\left<\sigma_{i}\right>\big|_{t=0}>0$ otherwise $\beta_{i}=\beta_{0}$,
$p_{ER}=0.008$, (Erd\"os-R\'enyi{}) and a mean degree $k_0=2$ in the scale free network.
\label{fig:Phase_Diagrams}}
\end{figure*}
To explain these results, consider a susceptible node $i$ and its
total rate of interaction with infected neighbors: \[
\Phi_{SI}(i)=\sum_{j}w_{ij}x_{j}.\]
The ratio of $SI$ interaction rates and total interaction rate $\alpha_{0}=\sum_{i<j}w_{ij}$
is given by \[
\alpha_{SI}=\frac{1}{\alpha_{0}}\sum_{i}\Phi_{SI}(i)(1-x_{i})\]
Averaging this measure over the time-course of a disease gives us
a measure of the typical fraction of contacts due to SI interaction:
\[
\left<\alpha_{SI}\right>=\frac{1}{T\alpha_{0}}\int_0^Tdt\,\left[\sum_{i,j}(1-x_{i})w_{ij}x_j\right]
\]
Now consider this time averaged $\left<\alpha_{SI}\right>$ as a function
of $p_{c}$ for various values of $\beta_{1}$, see Fig.~\ref{fig:Flux}.
For $\beta_1=\beta_0$ (i.e. no caretaker effect on recovery rates), the
rate of $SI$ interactions increase steadily as $p_c$ is increased, yielding
a more stable endemic state and high prevalence. When the caretaker effect
is taken into account, we observe an initial decrease of $SI$ interactions until
a critical value is reached below which the disease will go extinct, indicated by
the solid line. Increasing $p_c$ further can result in increasing $SI$ interactions
beyond this critical value, entering a regime in which a large fraction of caretaker links
results in a negative effect.
In the scale-free network, the qualitative behavior is similar. The crucial difference
is that typically, the adaptive process of regular links is sufficient to put the fraction of $SI$ links
below the critical value even in the absence of caretaker links.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.875\textwidth]{SI_flux}
\caption{Time-averaged SI contact fraction $\left<\alpha_{SI}\right>$
for SIS dynamics with different values of the caretaker proportion
$p_{c}$. Three $\beta_{1}$ values were chosen, 0.35 (circles), 0.60
(dots), 0.80 (arrows) to correspond with low, intermediate, and high
traces in the phase diagram of Fig.~\ref{fig:Phase_Diagrams}. An
Erd\"os-R\'enyi{} network was used (left), as well as a Scale-Free network (right).
The horizontal solid lines represent a critical value for $\left<\alpha_{SI}\right>$
above which the extinction probability vanishes and below which
the disease goes extinct. The plots correspond to $I_{0}=10^{2}$, $N=10^{3}$, $\mu=0.05,$
$\gamma=0.037$, $\tau=0.18$, $\beta_{0}=0.35$, $\sigma_{0}=\left<\sigma_{i}\right>\big|_{t=0}$
if $\left<\sigma_{i}\right>\big|_{t=0}>0$ otherwise $\beta_{i}=\beta_{0}$,
$p_{ER}=0.008$, (Erd\"os-R\'enyi{}) and mean degree $k_0=2$ (Scale-Free). \label{fig:Flux}
}
\end{figure*}
Next we turn out attention to the effect of caretaker adaptive networks on systems
that are better described by SIR dynamics.
Here individuals (nodes) exist in one of three states, susceptible
(S), infected (I) or recovered (R). Individuals can transition from
$S$ to $I$ with probability $p_{i}$ and from $I$ to $R$ with
probability $r_{i}$, as given above in Eqs.~\eqref{eq:infection_probability}
and \eqref{eq:caretaker_effect}. The state $R$ is absorbing, so
once all infected nodes in a population recover, the disease dies
out (see Fig.~\ref{fig:SIR_time_course}).
In order to investigate the impact of caretaker dynamics and an SIR scenario,
we focus on the \textit{attack rate (ratio)} and the \textit{epidemic
peak}. The attack ratio (AR) is simply the fraction of the population
which contracts the infection at some point during the epidemic. Since
every infected node eventually enters the recovered class, this is
equivalent to the fraction of recovered nodes at the end of the epidemic:
\[
AR=\frac{R_{\infty}}{N}
\]
The epidemic peak (EP) is the maximum infected fraction attained in the
population over the course of the epidemic.
\begin{figure}[t!]
\centering
\includegraphics[clip,width=\columnwidth]{SIR_time_course}
\caption{Infected density ($I^{*}=I/N$) for SIR dynamics as a function of
time for different caretaker proportions $p_{c}$. Erd\"os-R\'enyi{} networks
with adaptive rewiring were used, as well as a similar static network
(no rewiring, dashed line). Increasing $p_{c}$ lowers the epidemic
peak as well as the attack rate. Note also that the static network
trace closely resembles the $p_{c}=0$ trace, showing that SIR diseases
in this system are not significantly affected by dynamic link weights
alone. The plots correspond to $I_{0}=25$, $N=10^{3}$, $\mu=0.05,$
$\gamma=0.037$, $\tau=0.45$, $\beta_{0}=0.20$, $\sigma_{0}=\left<\sigma_{i}\right>\big|_{t=0}$
if $\left<\sigma_{i}\right>\big|_{t=0}>0$ otherwise $\beta_{i}=\beta_{0}$,
$n=2$, $p_{ER}=0.008$, (Erd\"os-R\'enyi{}). Scale-Free
network results were similar.
\label{fig:SIR_time_course}}
%
\end{figure}
Figure \ref{fig:AR_cross_sections} depicts the attack ratio as a function of
$p_c$ for various values of the recovery rate parameter $\beta_1$.
Interestingly, without a caretaker effect ($\beta_1=\beta_0$) the increase
in attack ration is not substantial as $p_c$ is increased. For $\beta_1>\beta_0$,
we observe a decrease in attack ratio even for small fractions of caretaker links.
The minimum attack ratio is attained only in a regime where most links are caretaker links.
Figure \ref{fig:attack_rate_phase} depicts the attack ratio as a function
of both system parameters $\beta_1$ and $p_c$ and compares the behavior
in both network architectures, Erd\"os-R\'enyi{} and Barab\'asi-Albert{}. In contrast with the SIR system,
network topology does not substantially change the dynamics, both networks
exhibit a similar attack ratio as a function of $\beta_1$ and $p_c$.
For fixed $\beta_1$ increasing $p_c$ first decreases the attack ratio until
a minimum is attained. Increasing $p_c$ further increases the attack ratio again.
A consistent effect is observed in the response of the epidemic peak to changes
in $\beta_1$ and $p_c$, see Fig.~\ref{fig:epidemic_peak}.
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{AR_cross_section}
\caption{Attack rate $AR$ as a function of $p_{c}$ for SIR dynamics with
various values of $\beta_{1}$ in an Erd\"os-R\'enyi{} network. For each $\beta_{1}>\beta_{0}$, the
attack rate is minimized for some value of $p_{c}$ between $10^{-1}$
and $10^{0}$. As $\beta_{1}$ increases, this minimum point shifts
subtly to the right. This shows that the more effective caretakers
are at healing, the more caretaker relationships the system can permit
before they have a negative impact on the attack rate. The plots correspond
to $I_{0}=25$, $N=10^{3}$, $\mu=0.05,$ $\gamma=0.037$, $\tau=0.25$,
$\beta_{0}=0.20$, $\sigma_{0}=\left<\sigma_{i}\right>\big|_{t=0}$
if $\left<\sigma_{i}\right>\big|_{t=0}>0$ otherwise $\beta_{i}=\beta_{0}$,
$n=2$, $p_{ER}=0.008$.
\label{fig:AR_cross_sections}}
\end{figure}
\begin{figure*}[t!]
\centering
\includegraphics[clip,scale=0.6]{AR_ER_phase} ~~~~~~~~~~~~
\includegraphics[clip,scale=0.6]{AR_SF_phase}
\caption{Two-parameter phase diagrams showing the dependence of attack rate
in SIR dynamics on maximum caretaker effectiveness $\beta_{1}$ (normalized
by the baseline-recovery probability $\beta_{0}$) and caretaker proportion
$p_{c}$. Erd\"os-R\'enyi{} (left) and Scale-Free (right) networks were considered.
Attack rate approaches zero in the white regions, while it approaches
$1$ in the black regions. Note that increasing $p_{c}$ yields lower
attack rates for $p_{c}<0.2$, but increasing past this critical value
yields increasing attack rates. There is a critical value $p_{c}\approx0.2$
at which attack rate is minimized for most values of $\beta_{1}$.
Furthermore, this effect is seen in both ER and SF networks, though
attack rates are lower overall on the SF network. The plots correspond
to $I_{0}=25$, $N=10^{3}$, $\mu=0.05,$ $\gamma=0.037$, $\tau=0.25$,
$\beta_{0}=0.20$, $\sigma_{0}=\left<\sigma_{i}\right>\big|_{t=0}$
if $\left<\sigma_{i}\right>\big|_{t=0}>0$ otherwise $\beta_{i}=\beta_{0}$,
$n=2$, $p_{ER}=0.008$, (Erd\"os-R\'enyi{}) and mean degree $k_0=2$ (Scale-Free).
\label{fig:attack_rate_phase}}
\end{figure*}
The dynamics seen above for the attack rate are mirrored in the epidemic
peak $EP$ as well (Fig.~\ref{fig:epidemic_peak}), which decreases
as caretaker effectiveness (represented by $\beta_{1}$) increases.
There is again a critical relationship with $p_{c}$, as values of
$p_{c}\approx0.2$ tend to minimize the epidemic peak for $\beta_{1}>\beta_{0}$.
Again though, for $\beta_{1}=\beta_{0}$, increasing $p_{c}$ yields
a monotonic increase in $EP$.
\begin{figure*}[t!]
\centering
\includegraphics[clip,scale=0.6]{EP_ER_phase} ~~~~~~~~~~~~
\includegraphics[clip,scale=0.6]{EP_SF_phase}
\caption{Two-parameter phase diagrams showing the dependence of the epidemic
peak $(EP)$ in SIR dynamics on the maximum caretaker effectiveness
$\beta_{1}$ (normalized by the baseline-recovery probability $\beta_{0}$)
and caretaker proportion $p_{c}$. Erd\"os-R\'enyi{} (a) and Scale-Free (b)
networks were considered. The epidemic peak approaches zero in the
white regions, while it approaches 1 in the black regions. Note the
similarities to the attack rate diagram in Fig.~\ref{fig:attack_rate_phase}.
The epidemic peak is minimized for $p_{c}\approx0.2$ for most values
of $\beta_{1}$, but for $p_{c}<0.2$ or $p_{c}>0.2$, the attack
rate is greater for a given value of $\beta_{1}$. The plots correspond
to $I_{0}=25$, $N=10^{3}$, $\mu=0.05,$ $\gamma=0.037$, $\tau=0.40$,
$\beta_{0}=0.20$, $\sigma_{0}=\left<\sigma_{i}\right>\big|_{t=0}$
if $\left<\sigma_{i}\right>\big|_{t=0}>0$ otherwise $\beta_{i}=\beta_{0}$,
$n=2$, $p_{ER}=0.008$, (Erd\"os-R\'enyi{}) and mean degree $k_0=2$ (Scale-Free).
\label{fig:epidemic_peak}}
\end{figure*}
\section{Conclusions}
Individual response can have a great impact on the dynamics of spreading
diseases on complex networks. In particular, if one uses an avoidance strategy
whereby all individuals simply avoid infecteds, the endemic state of an SIS
disease can be drastically reduced. On the other hand, allowing individuals
(caretakers) to become closer to infecteds is a calculated risk. If the
caretakers are not effective healers (such as non-physician parents and
children), then the severity of the disease generally increases. But if the
caretakers are effective healers (consider doctor/patient relationships, for
example), then the outcome of the disease is generally improved even by a small
number of them. If too many caretakers are introduced, though, their healing
benefit is overridden by their increased exposure, yielding a worse outcome than
if the population had simply not reacted.
These findings have a number of implications in public health. For one, in a
large-scale epidemic there certainly exists a critical fraction of doctors and
aid workers in the population. If there are too few or too many, they can
actually \textit{increase} the total number of individuals infected over the
course of the disease. In such cases, it would actually be more beneficial to
employ an avoidance strategy whereby all individuals, including doctors and aid
workers, simply avoided infected individuals. In the particular case of $SIS$
endemic diseases, we have seen that the critical caretaker proportion is
actually $p_{c}=0$ on Scale-Free networks. This suggests that networks that
exhibit a strong variability in interaction statistics and at the same time are
adaptive, are less susceptible to the risk of endemic diseases, and the natural
instinct to avoid infection is more effective in eliminating a disease than the
positive effects that caretakers may have.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,224 |
from neutron.services.trunk.drivers.linuxbridge import driver as lxb_driver
from neutron.services.trunk.drivers.openvswitch import driver as ovs_driver
def register():
"""Load in-tree drivers for the service plugin."""
# Enable the trunk plugin to work with ML2/OVS. Support for other
# drivers can be added similarly by executing the registration
# code at the time of plugin/mech driver initialization. There should
# be at least one compatible driver enabled in the deployment for trunk
# setup to be successful. The plugin fails to initialize if no compatible
# driver is found in the deployment.
lxb_driver.register()
ovs_driver.register()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,496 |
The FIL European Luge Championships 1984 took place in Olang, Italy for the third time after hosting the event previously in 1975 and 1980.
Men's singles
Haspinger earned his third straight bronze medal in this event at the championships.
Women's singles
Men's doubles
Medal table
References
Men's doubles European champions
Men's singles European champions
Women's singles European champions
FIL European Luge Championships
1984 in luge
Luge in Italy
1984 in Italian sport | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,090 |
We've reviewed several pieces of Lat 56 degrees luggage, including both a two wheel and four wheel version of its cabin baggage.
This new backpack is recognisably part of the same range, with the extremely distinctive look coming from its trademark military-spec moulded EVA foam exterior.
Backpacks are becoming more popular with business travellers: partly because we are all a lot more informal now and whereas once we might have bought briefcases which then morphed into laptop bags, now we realise that backpacks capable of carrying laptops are more practical.
In my own case, it's also recognition that if a bag doesn't have wheels, I only want to carry it if I can distribute the weight equally on both shoulders to avoid damaging my back. I use backpacks either with a wheelie bag on board (where permitted) or simply using it as my main bag on board and checking baggage for longer trips.
When choosing a backpack no one size or make will please everyone, but of course while you can pick up backpacks for as little as £20, if you are going to be putting valuable electronics inside and using it as your main bag for a period of days it pays to invest a little more. This bag is relatively expensive for a backpack, but the build quality is immediately obvious.
The carrying shoulder straps are luxurious – very wide, padded and extremely comfortable even with the heaviest load in the bag. It has an ergonomic back panel with a raised, breathable padding which does add to comfort and stops it being too closely pressed against your back, useful when you get warm walking with the bag on your back.
The zips are all good quality, waterproof, and look as though they will last, and the overall strength of the bag is good. If you have a laptop this slips down the back of the bag. My laptop is a large one – 15 inches, and just about fitted, though it was tight to zip up afterwards. Other laptops will fit in easily and an iPad would work well since there are elasticated straps to hold it in place. It's also very convenient for airport security since you don't have to open the main bag.
This is almost like a hard-sided case rather than the softer material that many backpacks are made of, which will give you confidence when putting fragile or valuable items inside. It does mean that this isn't the sort of bag that expands when you have a lot in it and then becomes smaller when you don't. There is one main pocket and then two side pockets. The main pocket opens like a clamshell, and unzips all the way down thought you don't want to do that since everything will fall out.
The main compartment has several document pockets, but again these are very firm – good for adding to the internal strength but it means that if you have one large object that you want to put in the bag, this is not possible. The internal layout of the bag is suited for having papers and books, but suffers in terms of adaptability because it cannot be removed as it can in other bags I have tested. I found that the bag filled very quickly and then I was having to stack one thing on top of another, most of it resting on top of the main internal compartment, which meant opening the bag required care if items were not to fall out.
The only other point to bear in mind is that because of the ergonomic back this backpack doesn't have a back panel designed to slip over trolley handles.
This is a premium backpack bag (with a price tag that reflects the robust quality). If you want a stylish option which will protect your items while travelling, it is well worth considering. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,915 |
Q: generating a list but only the first index showed i want to generate a list, in which only odd number get factorial application. However only the first number will be execute, can you help me? Thanks.
def factorial(x):
if x<=0:
return 1
else:
return x*factorial(x-1)
def odd(x):
if x%2 ==0:
return x
else:
return factorial(x)
def apply_if(factorial,odd,xs):
#xs is a list
i=0
mlst=[]
for x in xs:
if i<len(xs):
return odd(xs[i])
i+=1
mlst=mlst.append(odd(x))
return mlst
A: You should change apply_if function.
def apply_if(factorial,odd,xs):
mlst=[]
for x in xs:
mlst.append(odd(x))
return mlst
Because at first iteration of loop i will always be smaller than lenght of ws list (if it's not empty). Also append method doesn't return anything so you shouldn't use a = a.append() as it just appends element to given list.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,318 |
Dave Ramsey says never to buy new. I think the last time I actually had a new car was…well, never, to be honest.
My first car was a red 1977 pinto station wagon…I wrecked it when I was 16 with my friend Jody Haas in the car. THAT was a fun experience.
My second car was a 1981 Volkswagen Jetta. A white, 4 door that had been wrecked – I remember priming and spray painting the driver's side door because I was so embarrassed by the rust.
I drove that until 1990, when I assumed ownership of a 1987 white Toyota Celica hatchback…loved that car. Sporty, fun…but not good for a soon to be mama who had to climb out lugging an extra 50 lbs.
Then I purchased a 1995 white Honda Accord (in 1996 – probably the newest car I'd ever owned at that point). Then I moved to Texas and left Honda in Alaska because…Hello? It had no air conditioning. Did you know A/C is not standard in cars sold in Alaska? Well, it's not.
So when I moved to Texas in 1997, I purchased a 1996 Chevy Lumina (blue this time) and drove the heck out of it until my husband was rear ended in it two years ago and the insurance company totaled it.
Then my mom gave me a 1997 Dodge Grand Caravan in Blue that SHE had wrecked (something about a motorcyclist plowing right into her passenger side — never mind that she turned left in front of him)…it wasn't pretty, but it was still driveable. And the Lord knew I needed a van with three kids — even if it wasn't real 'purty'.
I lurve it. It drives really well, isn't all banged up, gets good gas mileage..and best of all…it will be totally paid for at the end of this year!
« Can you help a girl out?
Dave Ramsey is not the only or the first to say not to buy a car new. My father has always said that ( and he was saying it when Dave was a young boy). LOL! congrats on the new ride. it's fun to get a new ride (even if it's already broken in).
Hey! We bought a 2004 Grand Caravan today and I thought of you the entire time!
You wanna do lunch a week from this Friday? I think I'll be coming through Texarkana with Nathaniel on the way back from my grandmother's. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,712 |
Q: Can I use php to throw server error with message? I throw a error at server.
header('HTTP/1.1 500 Internal Server Error');
echo 'this is error message';
exit();
At the client side, I use android's Volley to send the request and handle the error in onErrorResponse
new Response.ErrorListener() {
@Override
public void onErrorResponse(VolleyError error) {
Toast.makeText(getApplicationContext(),error.getMessage(),Toast.LENGTH_LONG).show();
}
}
But I cannot found any message inside the error variable, how can I send a message to client with error?
A: IMO, you can try parse error message if available by using the following:
@Override
protected VolleyError parseNetworkError(VolleyError volleyError) {
String json;
if (volleyError.networkResponse != null && volleyError.networkResponse.data != null) {
try {
json = new String(volleyError.networkResponse.data,
HttpHeaderParser.parseCharset(volleyError.networkResponse.headers));
} catch (UnsupportedEncodingException e) {
return new VolleyError(e.getMessage());
}
return new VolleyError(json);
}
return volleyError;
}
Hope it helps!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,077 |
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