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{"url":"https:\/\/en.wikipedia.org\/wiki\/Machine_olfaction","text":"# Machine olfaction\n\nMachine olfaction is the automated simulation of the sense of smell. An emerging application in modern engineering, it involves the use of robots or other automated systems to analyze air-borne chemicals. Such an apparatus is often called an electronic nose or e-nose. The development of machine olfaction is complicated by the fact that e-nose devices to date have responded to a limited number of chemicals, whereas odors are produced by unique sets of (potentially numerous) odorant compounds. The technology, though still in the early stages of development, promises many applications, such as:[1] quality control in food processing, detection and diagnosis in medicine,[2] detection of drugs, explosives and other dangerous or illegal substances,[3] disaster response, and environmental monitoring.\n\nOne type of proposed machine olfaction technology is via gas sensor array instruments capable of detecting, identifying, and measuring volatile compounds. However, a critical element in the development of these instruments is pattern analysis, and the successful design of a pattern analysis system for machine olfaction requires a careful consideration of the various issues involved in processing multivariate data: signal-preprocessing, feature extraction, feature selection, classification, regression, clustering, and validation.[4] Another challenge in current research on machine olfaction is the need to predict or estimate the sensor response to aroma mixtures.[5] Some pattern recognition problems in machine olfaction such as odor classification and odor localization can be solved by using time series kernel methods.[6]\n\n## Detection\n\nThere are three basic detection techniques using conductive-polymer odor sensors (polypyrrole), tin-oxide gas sensors, and quartz-crystal micro-balance sensors.[citation needed] They generally comprise (1) an array of sensors of some type, (2) the electronics to interrogate those sensors and produce digital signals, and (3) data processing and user interface software.\n\nThe entire system is a means of converting complex sensor responses into a qualitative profile of the volatile (or complex mixture of chemical volatiles) that make up a smell, in the form of an output.\n\nConventional electronic noses are not analytical instruments in the classical sense and very few claim to be able to quantify an odor. These instruments are first 'trained' with the target odor and then used to 'recognize' smells so that future samples can be identified as 'good' or 'bad'.\n\nResearch into alternative pattern recognition methods for chemical sensor arrays has proposed solutions to differentiate between artificial and biological olfaction related to dimensionality. This biologically-inspired approach involves creating unique algorithms for information processing.[7]\n\nElectronic noses are able to discriminate between odors and volatiles from a wide range of sources. The list below shows just some of the typical applications for electronic nose technology \u2013 many are backed by research studies and published technical papers.\n\n## Odor localization\n\nOdor localization is a combination of quantitative chemical odor analysis and path-searching algorithms, and environmental conditions play a vital role in localization quality. Different methods are being researched for various purposes and in different real-world conditions.\n\n### Motivation\n\nOdor localization is the technique and process of locating a volatile chemical source in an environment containing one or several odors. It is vitally important for all living beings for both finding sustenance and avoiding danger. Unlike the other basic human senses, the sense of smell is entirely chemical-based. However, in comparison with the other dimensions of perception, detection of odor faces additional problems due to the complex dynamic equations of odor and unpredictable external disturbances such as wind.\n\n### Application\n\nOdor localization technology shows promise in many applications, including:[8][1]\n\n### History and problem statement\n\nThe earliest instrument for specific odor detection was a mechanical nose developed in 1961 by Robert Wighton Moncrieff. The first electronic nose was created by W. F. Wilkens and J. D. Hartman in 1964.[9] Larcome and Halsall discussed the use of robots for odor sensing in the nuclear industry in the early 1980s,[10] and research on odor localization was started in the early 1990s. Odor localization is now a fast-growing field. Various sensors have been developed and a variety of algorithms have been proposed for diverse environments and conditions.\n\nMechanical odor localization can be executed via the following three steps, (1) search for the presence of a volatile chemical (2) search for the position of the source with an array of odor sensors and certain algorithms, and (3) identify the tracked odor source (odor recognition).\n\n### Localization methods\n\nOdor localization methods are often classified according to odor dispersal modes in a range of environmental conditions. These modes can generally be divided into two categories: diffusion-dominated fluid flow and turbulence-dominated fluid flow. These have different algorithms for odor localization, discussed below.\n\n#### Diffusion-dominated fluid flow\n\nTracking and localization methods for diffusion-dominated fluid flow \u2013 which is mostly used in underground odor localization \u2013 must be designed so that olfaction machinery can operate in environments in which fluid motion is dominated by viscosity. This means that diffusion leads to the dispersal of odor flow, and the concentration of odor decreases from the source as a Gaussian distribution.[11]\n\nThe diffusion of chemical vapor through soil without external pressure gradient is often modeled by Fick's second law:\n\n${\\displaystyle {\\frac {\\partial C}{\\partial t}}=D{\\frac {\\partial ^{2}C}{\\partial d^{2}}}}$\n\nwhere D is the diffusion constant, d is distance in the diffusion direction, C is chemical concentration and t is time.\n\nAssuming the chemical odor flow only disperses in one direction with a uniform cross-section profile, the relationship of odor concentration at a certain distance and certain time point between odor source concentrations is modeled as\n\n${\\displaystyle {\\frac {C}{C_{s}}}=erfc{\\frac {d}{\\sqrt {4Dt}}}}$\n\nwhere ${\\displaystyle C_{s}}$ is the odor source concentration. This is the simplest dynamic equation in odor detection modeling, ignoring external wind or other interruptions. Under the diffusion-dominated propagation model, different algorithms were developed by simply tracking chemical concentration gradients to locate an odor source.\n\n##### E. coli algorithm\n\nA simple tracking method is the E. coli algorithm.[12] In this process, the odor sensor simply compares concentration information from different locations. The robot moves along repeated straight lines in random directions. When the current state odor information is improved compared to the previous reading, the robot will continue on the current path. However, when the current state condition is worse than the previous one, the robot will backtrack then move in another random direction. This method is simple and efficient, however, the length of the path is highly variable and misteps increase with proximity to the source.[further explanation needed]\n\n##### Hex-path algorithm and dodecahedron algorithm\n\nAnother method based on the diffusion model is the hex-path algorithm, developed by R. Andrew Russel[12] for underground chemical odor localization with a buried probe controlled by a robotic manipulator.[12][13] The probe moves at a certain depth along the edges of a closely packed hexagonal grid. At each state junction n, there are two paths (left and right) for choosing, and the robot will take the path that leads to higher concentration of the odor based on the previous two junction states odor concentration information n\u22121, n\u22122. In the 3D version of the hex-path algorithm, the dodecahedron algorithm, the probe moves in a path that corresponds to a closely packed dodecahedra, so that at each state point there are three possible path choices.\n\n#### Turbulence-dominated fluid flow\n\nFigure 1. plume modeling\n\nIn turbulence-dominated fluid flow, localization methods are designed to deal with background fluid (wind or water) flow as turbulence interruption. Most of the algorithms under this category are based on plume modeling (Figure\u00a01).[14]\n\nPlume dynamics are based on Gaussian models, which are based on Navier\u2013Stokes equations. The simplified boundary condition of the Gaussian-based model is:\n\n${\\displaystyle {\\frac {\\partial C}{\\partial t}}=D_{x}{\\frac {\\partial ^{2}C}{\\partial x^{2}}}+D_{y}{\\frac {\\partial ^{2}C}{\\partial y^{2}}}+\\alpha {\\frac {\\partial C}{\\partial x}}+\\beta {\\frac {\\partial C}{\\partial x}}}$\n\nwhere Dx and Dy are diffusion constants; ${\\displaystyle \\alpha }$ is the linear wind velocity in the x direction, ${\\displaystyle \\beta }$ is the linear wind velocity in the y direction. Additionally assuming that the environment is uniform and the plume source is constant, the equation for odor detection in each robot sensor at each detection time point tth is\n\n${\\displaystyle R_{i}=\\gamma _{i}\\sum _{k=1}^{K}{\\frac {C_{k}}{\\|{\\rho _{k}-r_{k}}\\|^{\\alpha }}}+\\omega _{i}}$\n\nwhere ${\\displaystyle R_{i}}$ is the tth sample of ith sensor, ${\\displaystyle \\gamma _{i}}$ is gain factor, ${\\displaystyle C_{k}}$ is kth source intensity, ${\\displaystyle \\rho _{k}}$ is the location of kth source, ${\\displaystyle \\alpha }$ is plume attenuation parameter, ${\\displaystyle \\omega _{i}}$ is background noise that satisfies ${\\displaystyle N(\\mu ,\\sigma ^{2})}$. Under plume modeling, different algorithms can be used to localize the odor source.\n\nFigure 2. Triangulation method\n##### Triangulation algorithm\n\nA simple algorithm that can be used for location estimation is the triangulation method (Figure 2). Consider the odor detection equation above, the position of the odor source can be estimated by organizing sensor distances on one side of the equation and ignoring the noise. The source position can be estimated using the following equations:\n\n${\\displaystyle (x_{1}-x_{s})^{2}+(y_{1}-y_{s})^{2}=R_{1}\/(\\gamma _{1}C)}$\n\n${\\displaystyle (x_{2}-x_{s})^{2}+(y_{2}-y_{s})^{2}=R_{2}\/(\\gamma _{2}C)}$\n\n${\\displaystyle (x_{3}-x_{s})^{2}+(y_{3}-y_{s})^{2}=R_{3}\/(\\gamma _{3}C)}$\n\n##### Least square method (LSM)\n\nThe least square method (LSM) is a slightly complicated algorithm for odor localization. The LSM version of the odor tracking model is given by:\n\n${\\displaystyle R_{i},_{t}=\\gamma _{i}{\\frac {C}{\\|{\\rho _{k}-r_{i}}\\|^{2}}}+\\omega _{i}=\\gamma _{i}{\\frac {C}{d^{2}}}+\\omega _{i}}$\n\nwhere ${\\displaystyle d_{i}}$ is the Euclidean distance between the sensor node and the plume source, given by:\n\n${\\displaystyle d_{i}={\\sqrt {(x_{i}-x-s)^{2}+(y_{i}+y_{s})^{2}}}}$\n\nThe main difference between the LSM algorithm and the direct triangulation method is the noise. In LSM, noise is considered, and the odor source location is estimated by minimizing the squared error. The nonlinear least square problem is given by:\n\n${\\displaystyle J=\\sum _{i=1}^{N}({{\\frac {C}{\\sqrt {(x_{i}-{\\widehat {x_{s}}})^{2}(y_{i}-{\\widehat {y_{s}}})^{2}}}}-{\\overline {z_{l}}})}}$\n\nwhere ${\\displaystyle ({\\widehat {x_{s}}},{\\widehat {y_{s}}})}$ is the estimated source location and ${\\displaystyle {\\overline {z_{l}}}}$ is the average of multiple measurements at the sensors, given by:\n\n${\\displaystyle {\\overline {z_{l}}}={\\frac {1}{M}}\\sum _{i=1}^{M}{z_{i}}}$\n\n##### Maximum likelihood estimation (MLE)\n\nAnother method based on plume modeling is maximum likelihood estimation (MLE). In this odor localization method, several matrices are defined as follows:\n\n${\\displaystyle Z=[{\\frac {R_{1}-\\mu _{1}}{\\sigma _{1}}},{\\frac {R_{2}-\\mu _{2}}{\\sigma _{2}}},...{\\frac {R_{N}-\\mu _{N}}{\\sigma _{N}}}]}$\n\n${\\displaystyle G=diag[{\\frac {\\gamma _{1}}{\\sigma _{1}}},{\\frac {\\gamma _{2}}{\\sigma _{2}}},...{\\frac {\\gamma _{N}}{\\sigma _{N}}}]}$\n\n${\\displaystyle D=[{\\frac {1}{d_{1}^{2}}},{\\frac {1}{d_{2}^{2}}},...{\\frac {1}{d_{N}^{2}}}]}$\n\n${\\displaystyle \\zeta =[\\zeta _{1},\\zeta _{2},...\\zeta _{N}]}$\n\n${\\displaystyle \\zeta _{i}=(\\omega _{i}-\\mu _{i})\/\\sigma _{i}\\sim N(0,1)}$\n\n${\\displaystyle {\\frac {R_{i}-\\mu _{i}}{\\sigma _{i}}}\\sim N({\\frac {\\gamma _{i}}{\\sigma _{i}}}{\\frac {C}{d_{i}^{2}}},1)}$\n\nWith these matrices, the plume-based odor detection model can be expressed with the following equation:\n\n${\\displaystyle Z=GDC+\\zeta }$\n\nThen the MLE can be applied to the modeling and form the probability density function\n\n${\\displaystyle f(Z,\\theta )=2\\pi ^{-(N\/2)}e^{-{\\frac {1}{2}}(Z-GDC)^{T}(Z-GDC)}}$\n\nwhere ${\\displaystyle \\theta }$ is the estimated odor source position, and the log likelihood function is\n\n${\\displaystyle L(\\theta )\\sim {\\frac {1}{2}}\\sum _{i=1}^{N}{\\|Z_{i}-\\gamma _{i}{\\frac {c}{d_{i}^{2}}}\\|}={\\frac {1}{2}}\\sum _{i=1}^{N}{({\\frac {Ri-\\mu _{i}}{\\sigma _{i}}}-\\gamma _{i}{\\frac {c}{d_{i}^{2}}})^{2}}}$\n\nThe maximum likelihood parameter estimation of ${\\displaystyle \\theta }$ can be calculated by minimizing\n\n${\\displaystyle I(\\theta )=\\sum _{i=1}^{N}{({\\frac {Ri-\\mu _{i}}{\\sigma _{i}}}-\\gamma _{i}{\\frac {c}{d_{i}^{2}}})^{2}}}$\n\nand the accurate position of the odor source can be estimated by solving:\n\n${\\displaystyle {\\frac {\\partial {I(\\theta )}}{\\partial (x)}}=0,{\\frac {\\partial {I(\\theta )}}{\\partial (y)}}=0}$\n\n## References\n\n1. ^ a b Sensors Council. (2012). Special issue on Machine Olfaction. IEEE SENSORS JOURNAL, 11(12), 3486\u20133486 . Retrieved March 20, 2012, from the Scholars Portal Journal database.\n2. ^ a b Geffen, Wouter H. van; Bruins, Marcel; Kerstjens, Huib A. M. (2016-01-01). \"Diagnosing viral and bacterial respiratory infections in acute COPD exacerbations by an electronic nose: a pilot study\". Journal of Breath Research. 10 (3): 036001. Bibcode:2016JBR....10c6001V. doi:10.1088\/1752-7155\/10\/3\/036001. ISSN\u00a01752-7163. PMID\u00a027310311.\n3. ^ Stassen, I.; Bueken, B.; Reinsch, H.; Oudenhoven, J. F. M.; Wouters, D.; Hajek, J.; Van Speybroeck, V.; Stock, N.; Vereecken, P. M.; Van Schaijk, R.; De Vos, D.; Ameloot, R. (2016). \"Towards metal\u2013organic framework based field effect chemical sensors: UiO-66-NH2 for nerve agent detection\". Chem. Sci. 7 (9): 5827\u20135832. doi:10.1039\/C6SC00987E. hdl:1854\/LU-8157872. PMC\u00a06024240. PMID\u00a030034722.\n4. ^ Sensors Council. (2002). Pattern analysis for machine olfaction: a review . IEEE SENSORS JOURNAL, 2(3), 189\u2013202 . Retrieved March 20, 2012, from the Scholars Portal database.\n5. ^ Phaisangittisagul, E., & Nagle, H. T. (2011). Predicting odor mixture's responses on machine olfaction sensors. Sensors & Actuators: B. Chemical, 155(2), 473\u2013482\n6. ^ Vembu, S.;Vergara, A.;Muezzinoglu, M. K.;Huerta, R. (2012). On time series features and kernels for machine olfaction. Sensors &Actuators: B. Chemical,174, 535\n7. ^ Baranidharan Raman,\"Sensor-based Machine Olfaction with Neuromorphic Models of the Olfactory System\", University of Madras, India; M.S., Texas A&M University, December 2005\n8. ^ \"Review on: Odor Localization Robot Aspect and Obstacles\". www.academia.edu. Retrieved 2015-11-12.\n9. ^ Gardner, Julian W.; Bartlett, Philip N. (1994-03-01). \"A brief history of electronic noses\". Sensors and Actuators B: Chemical. 18 (1\u20133): 210\u2013211. doi:10.1016\/0925-4005(94)87085-3.(subscription required)\n10. ^\n11. ^ Kowadlo, Gideon; Russell, R. Andrew (2008-08-01). \"Robot Odor Localization: A Taxonomy and Survey\". The International Journal of Robotics Research. 27 (8): 869\u2013894. doi:10.1177\/0278364908095118. ISSN\u00a00278-3649.\n12. ^ a b c Russell, R. Andrew (2004-01-01). \"Robotic location of underground chemical sources\". Robotica. 22 (1): 109\u2013115. doi:10.1017\/S026357470300540X. ISSN\u00a01469-8668.\n13. ^\n14. ^","date":"2020-01-17 21:16:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 37, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6133773326873779, \"perplexity\": 3363.829516450249}, \"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-2020-05\/segments\/1579250591234.15\/warc\/CC-MAIN-20200117205732-20200117233732-00463.warc.gz\"}"} | null | null |
Heukewalde es un municipio situado en el distrito de Altenburger Land, en el estado federado de Turingia (Alemania), a una altitud de . Su población a finales de 2016 era de unos y su densidad poblacional, .
Se encuentra junto a la frontera con el estado de Sajonia. Dentro del distrito, el municipio está asociado a la mancomunidad (Verwaltungsgemeinschaft) de Oberes Sprottental, que tiene su sede en la vecina ciudad de Schmölln.
Se conoce su existencia desde 1152. Es una localidad con forma lineal, estructurada en torno a una única calle llamada "Dorfstraße" que discurre paralela al arroyo local. Antes de la unificación de Turingia en 1920, pertenecía al ducado de Sajonia-Altemburgo.
Referencias
Enlaces externos
Página web del distrito de Altenburger Land
Municipios del distrito de Altenburger Land
Localidades del distrito de Altenburger Land | {
"redpajama_set_name": "RedPajamaWikipedia"
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Ruslanova is een inslagkrater op de planeet Venus. Ruslanova werd in 1985 genoemd naar de Sovjet-Russische folkzangeres Lidia Roeslanova (1900–1973).
De krater heeft een diameter van 44,3 kilometer en bevindt zich in het quadrangle Snegurochka Planitia (V-1).
Zie ook
Lijst van kraters op Venus
Inslagkrater op Venus | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,382 |
{"url":"https:\/\/www.maixuanviet.com\/git-commit.vietmx","text":"# Git Commit\n\nIt is used to record the changes in the repository. It is the next command after the\u00a0git add. Every commit contains the index data and the commit message. Every commit forms a parent-child relationship. When we add a file in Git, it will take place in the staging area. A commit command is used to fetch updates from the staging area to the repository.\n\nThe staging and committing are co-related to each other. Staging allows us to continue in making changes to the repository, and when we want to share these changes to the version control system, committing allows us to record these changes.\n\nCommits are the snapshots of the project. Every commit is recorded in the master branch of the repository. We can recall the commits or revert it to the older version. Two different commits will never overwrite because each commit has its own commit-id. This commit-id is a cryptographic number created by\u00a0SHA (Secure Hash Algorithm)\u00a0algorithm.\n\nLet\u2019s see the different kinds of commits.\n\n## 1. The git commit command\n\nThe commit command will commit the changes and generate a commit-id. The commit command without any argument will open the default text editor and ask for the commit message. We can specify our commit message in this text editor. It will run as follows:\n\n$git commit [\/code] The above command will prompt a default editor and ask for a commit message. We have made a change to newfile1.txt and want it to commit it. It can be done as follows: Consider the below output: As we run the command, it will prompt a default text editor and ask for a commit message. The text editor will look like as follows: Press the Esc key and after that \u2018I\u2018 for insert mode. Type a commit message whatever you want. Press Esc after that \u2018:wq\u2018 to save and exit from the editor. Hence, we have successfully made a commit. We can check the commit by git log command. Consider the below output: We can see in the above output that log option is displaying commit-id, author detail, date and time, and the commit message. To know more about the log option, visit Git Log. ## 2. Git commit -a The commit command also provides -a option to specify some commits. It is used to commit the snapshots of all changes. This option only consider already added files in Git. It will not commit the newly created files. Consider below scenario: We have made some updates to our already staged file newfile3 and create a file newfile4.txt. Check the status of the repository and run the commit command as follows:$\u00a0git\u00a0commit\u00a0-a\u00a0\u00a0[\/code]\n\nConsider the output:\n\nThe above command will prompt our default text editor and ask for the commit message. Type a commit message, and then save and exit from the editor. This process will only commit the already added files. It will not commit the files that have not been staged. Consider the below output:\n\nAs we can see in the above output, the newfile4.txt has not been committed.\n\n## 3. Git commit -m\n\nThe -m option of commit command lets you to write the commit message on the command line. This command will not prompt the text editor. It will run as follows:\n\n$git commit -m \u201cCommit message.\u201d [\/code] The above command will make a commit with the given commit message. Consider the below output: In the above output, a newfile4.txt is committed to our repository with a commit message. We can also use the -am option for already staged files. This command will immediately make a commit for already staged files with a commit message. It will run as follows:$\u00a0git\u00a0commit\u00a0-am\u00a0\u201cCommit\u00a0message.\u201d\u00a0\u00a0[\/code]\n\n## 4. Git Commit Amend (Change commit message)\n\nThe amend option lets us to edit the last commit. If accidentally, we have committed a wrong commit message, then this feature is a savage option for us. It will run as follows:\n\n\\$\u00a0git\u00a0commit\u00a0-amend\u00a0[\/code]\n\nThe above command will prompt the default text editor and allow us to edit the commit message.\n\nWe may need some other essential operations related to commit like revert commit, undo a commit, and more, but these operations are not a part of the commit command. We can do it with other commands. Some essential operations are as follows:\n\n\u2022 Git undo commit: Visit\u00a0Git Reset\n\u2022 Git revert commit: Visit\u00a0Git Revert\n\u2022 git remove commit: Visit\u00a0Git Rm","date":"2022-07-01 23:11:33","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\": 0.1892976015806198, \"perplexity\": 2573.896671046905}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103947269.55\/warc\/CC-MAIN-20220701220150-20220702010150-00615.warc.gz\"}"} | null | null |
Q: Azure Devops Dashboard permissions I have few teams part of same project collection who have different access requirement to the dashboard created in the project. Each team dashboard needs to be visible only to that team and not accessible by other teams. When I check the security on the Dashboards there is only view and delete access that can be either allowed or denied. Is it possible to achieve the different access at team level?
Any input would be greatly appreciated.
Thank you
A: As described in documentation: all members of the Project Valid Users group can view dashboards.
There is no setting to limit visibility of the dashboard between teams.
| {
"redpajama_set_name": "RedPajamaStackExchange"
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import { getBackendSrv } from 'app/core/services/backend_srv';
import store from 'app/core/store';
import { API_KEYS_MIGRATION_INFO_STORAGE_KEY } from 'app/features/serviceaccounts/constants';
import { ApiKey, ThunkResult } from 'app/types';
import {
apiKeysLoaded,
includeExpiredToggled,
isFetching,
apiKeysMigrationStatusLoaded,
setSearchQuery,
} from './reducers';
export function addApiKey(apiKey: ApiKey, openModal: (key: string) => void): ThunkResult<void> {
return async (dispatch) => {
const result = await getBackendSrv().post('/api/auth/keys', apiKey);
dispatch(setSearchQuery(''));
dispatch(loadApiKeys());
openModal(result.key);
};
}
export function loadApiKeys(): ThunkResult<void> {
return async (dispatch) => {
dispatch(isFetching());
const [keys, keysIncludingExpired] = await Promise.all([
getBackendSrv().get('/api/auth/keys?includeExpired=false&accesscontrol=true'),
getBackendSrv().get('/api/auth/keys?includeExpired=true&accesscontrol=true'),
]);
dispatch(apiKeysLoaded({ keys, keysIncludingExpired }));
};
}
export function deleteApiKey(id: number): ThunkResult<void> {
return async (dispatch) => {
getBackendSrv()
.delete(`/api/auth/keys/${id}`)
.then(() => dispatch(loadApiKeys()));
};
}
export function migrateApiKey(id: number): ThunkResult<void> {
return async (dispatch) => {
try {
await getBackendSrv().post(`/api/serviceaccounts/migrate/${id}`);
} finally {
dispatch(loadApiKeys());
}
};
}
export function migrateAll(): ThunkResult<void> {
return async (dispatch) => {
try {
await getBackendSrv().post('/api/serviceaccounts/migrate');
store.set(API_KEYS_MIGRATION_INFO_STORAGE_KEY, true);
} finally {
dispatch(getApiKeysMigrationStatus());
dispatch(loadApiKeys());
}
};
}
export function getApiKeysMigrationStatus(): ThunkResult<void> {
return async (dispatch) => {
const result = await getBackendSrv().get('/api/serviceaccounts/migrationstatus');
dispatch(apiKeysMigrationStatusLoaded(!!result?.migrated));
};
}
export function hideApiKeys(): ThunkResult<void> {
return async (dispatch) => {
await getBackendSrv().post('/api/serviceaccounts/hideApiKeys');
};
}
export function toggleIncludeExpired(): ThunkResult<void> {
return (dispatch) => {
dispatch(includeExpiredToggled());
};
}
| {
"redpajama_set_name": "RedPajamaGithub"
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Kuadam (also known as Kua or more popularly Kapadapuram) was the capital of the ancient Pandian kingdom of the Meen'Koodal epoch (the second Sangam academy). The grand old poet and sage Nakkeerar and the Iraiyanaar porul'urai mentions that Kuadam was the capital from c.5400 BCE to c.1750 BCE (about 3650 years).
According to historians, Kuadam was very close to Tiruchendur. Abraham Pandithar says that Greeks in those days named it as Periplus port.
Around 1750 BCE, the last great deluge flooded Kuadam and the remaining part of the Kumari kaandam forever (similar to the Biblical record of "the flood" sometime between 3402 BC and 2462 BC).
Kuadam was to the north of the ancient Paqruli river about 700 kaadham south of the Kumari river delta.
Cataclysms mid-2000 BCE and similarities
Another ancient city lost to cataclysm in mid-2nd millennium BCE is Kapata modern day Crete. The Bible mentions this city as "Caphtor" or "Capthor" as the country of the Philistines. In the texts of Mari, Kaptara appears "beyond the upper sea".The designation "Keftiu" of Crete by the Egyptians comes mainly from the tomb of Rekmira, Egypt.
According to David McAlpin and his Elamo-Dravidian hypothesis, the Dravidian languages were brought to India by immigration into India from Elam, located in present-day southwestern Iran. In the 1990s, Renfrew and Cavalli-Sforza have also argued that Proto-Dravidian was brought to India by farmers from the Iranian part of the Fertile Crescent,. According to Gareth Alun Owens, Linear A represents the Minoan language, which Owens classifies as a distinct branch of Indo-European potentially related to Greek, Sanskrit, Hittite, Latin, etc.
Further the Puranic dynasties of the South India, including the Cholas, Pandyas, Cheras descended from the Turvashas. Tamil "Pallava" dynasty appear in the Sanskrit Literature Pallas, Pahlavas, Pahnavas, Palhava, Plavas. The Tamil Vanni were one of ancient sea faring traders and it is well known fact that the Pandyas once traded with the Greeks and Romans.
References
Former populated places in India
Archaeological sites in Tamil Nadu
Thoothukudi | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,202 |
Het Godshuis van Heylwegen (1556-1799) was een armenhuis in Leuven (België), in de Zuidelijke Nederlanden. Het bevond zich in de Minderbroedersstraat, tegenover het Anatomisch theater van de oude universiteit.
Historiek
Lodewijk van Heylwegen schonk bij testament (1556) de financiële middelen voor een godshuis in zijn geboortestad, Leuven. Hij was heer van Wazière en lange tijd belastingontvanger in Leuven, voor het hertogdom Brabant. Hij zetelde in de Raad van Brabant en eindigde zijn carrière in Gent, als voorzitter van de Raad van Vlaanderen.
Het godshuis bestond uit zeven huisjes, voor zes mannen en één vrouw. Het moesten Leuvenaars zijn zonder inkomen. Hen werd verboden nog te bedelen in de stad. Het bestuur van het godshuis lag bij de prior van de Chartreuse van Leuven.
In 1799, tijdens het Frans bestuur in Leuven, werd het Godshuis afgebroken. Er kwamen particuliere huizen in de plaats.
Voormalig bouwwerk in Leuven | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,671 |
OpenProject est un système de gestion de projet basé sur le Web pour la collaboration d'équipe . Cette application open source est publiée sous la licence publique générale GNU version 3 est continuellement développée par une communauté open-source active.
En plus de nombreuses petites installations OpenProject, il existe aussi de très grandes installations dans des structures d'envergure internationale avec plus de 2 500 projets.
Caractéristiques
Gestion de projet et jalons
Gestion des problèmes
Suivi des bogues
Calendrier du projet
Wiki
Gestion de documents
Forum
Suivi du temps
Nouvelles du projet
Il permet de faire apparaître des diagrammes de Gantt qui sont un outil souvent utilisé en ordonnancement et gestion de projets permettant de visualiser dans le temps les diverses tâches composant un projet.
Fondation OpenProject
La fondation OpenProject a été créée par les développeurs et les utilisateurs d'OpenProject en . Après la fondation de l'association en , elle a été enregistrée (VR 32487) en juin sur le registre de l'Amtsgericht (tribunal d'instance) de Berlin-Charlottenburg. L'association fournit un cadre organisationnel pour les décisions techniques et la propagation, l'accélération et la pérennisation du développement par la communauté mondiale et par une équipe de développement à plein temps, financée par les membres de la Fondation OpenProject.
L'association poursuit les objectifs suivants :
établir et promouvoir une communauté active et ouverte de développeurs, utilisateurs et entreprises pour le développement continu du logiciel de collaboration de projet open source OpenProject;
définir et développer la vision du projet, le code de conduite et les principes de l'application;
créer des politiques de développement et assurer leur conformité;
définir et faire évoluer les processus de développement et d'assurance qualité;
fournir le code source au public;
fournir et exploiter la plate-forme OpenProject.
L'association ne poursuit pas d'objectifs économiques propres.
L'histoire
OpenProject est développé depuis 2010 avec le projet ancêtre ChiliProject. La motivation initiale de ce fork était les exigences de performance, de sécurité et d'accessibilité des membres fondateurs d'OPF, qui n'étaient pas facilement accessibles par les plugins pour Redmine ou ChiliProject.
Développements récents et futurs
Le calendrier de publication actuel et la feuille de route du développement futur peuvent être observés et discutés sur la plate-forme de développement OpenProject.
Outre le développement de nouvelles fonctions, les objectifs techniques suivants sont poursuivis dans le cadre du refactoring :
développer une nouvelle API v3 ;
reconstruire le module de package de travail avec AngularJS en tant qu'application d' une seule page ;
reconstruire la structure CSS à l'aide du framework CSS Foundation for Apps.
Prix
En , OpenProject a remporté le premier prix dans la catégorie "Best Practice" du concours open source de la Fondation pour la technologie de Berlin "Le futur de Berlin est ouvert" .
En , OpenProject a remporté le prix INNOVATIONSPREIS-IT 2018 de l'Initiative Mittelstand dans la catégorie Open Source.
En , OpenProject a remporté le prix Open Source Business Award (OSBAR, «OpenSource-Oscar») en argent.
Notes et références
Voir aussi
Liens externes
Le blog d'un bibliothécaire wikimédien
Logiciel libre de gestion de projets
Logiciel de suivi de bugs
Logiciel collaboratif
Pages avec des traductions non relues | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,279 |
See the entire Folk catalogue
Daniil Trifonov >
Daniil Trifonov|BACH: The Art of Life
Released on 8/10/21 by Deutsche Grammophon (DG)
Main artist: Daniil Trifonov
Buy the album Starting at $25.59
"This album is, in many ways, about love: the romantic love between husband and wife; familial love between parents and their children; and love for the Creator."
Daniil Trifonov's new album, Bach: The Art of Life, explores the scientific, emotional and spiritual fibre of Bach's work. The exceptional Russian pianist, famous for his romantic interpretations of Rachmaninoff and Chopin, now extends his recording palette to Baroque piano music and presents an incredibly spiritual programme for Deutsche Grammophon based around Bach and his sons.
The centrepiece of the album is The Art of Fugue, one of Bach's late masterpieces written in the last ten years of his life. In the early and mid-eighteenth century - and thus before the Enlightenment - music, like other scientific disciplines, was seen as a means of expression to explain the laws of nature and thus the will of God. Consequently, the polyphonic construction of the work was a central means for the composer to musically interpret his environment. According to Trifonov, the highly spiritual character of the piece represents 'the pioneering musical realisation of Bach's personal, religious, scientific and humanistic knowledge'.
However, it is not only Bach, but also many of his descendants who make up the strength of this unique programme. Collective musical practice was an integral part of the Bach family's daily life, with each child developing his or her own style. The booklet of notes for Anna Magdalena Bach, compiled by Bach himself, summarises these personal and intimate family studies on paper, which Trifonov complements with Johann Christoph Friedrich Bach's "Ah! vous dirai-je, maman" (Ah, shall I tell you, Mama) variations, as well as works by Johann Christian, Wilhelm Friedemann and Carl Philipp Emanuel, among others.
Trifonov, an exceptional virtuoso, succeeds in doing justice to polyphonic music with his great technical precision and total artistic commitment. His sensitive and profound understanding of Bach's philosophy and compositions is a wonderful setting for this comprehensive overview of the composer's life and musical legacy, which the performer generously offers to his listeners. © Lena Germann/Qobuz
1 month free, then $19.99 / month
Sonata No. 5 in A Major, Op. 17, No. 5 (Johann Christian Bach)
I. Allegro
Johann Christian Bach, Composer - Marcus Herzog, Mastering Engineer, StudioPersonnel - Daniil Trifonov, Piano, MainArtist, AssociatedPerformer - David Frost, Producer, Mixer, Editor, StudioPersonnel - Silas Brown, Recording Engineer, StudioPersonnel
℗ 2021 Deutsche Grammophon GmbH, Berlin
II. Presto
12 Polonaises, F. 12 (Wilhelm Friedemann Bach)
No. 8 in E Minor
Marcus Herzog, Mastering Engineer, StudioPersonnel - Wilhelm Friedemann Bach, Composer - Daniil Trifonov, Piano, MainArtist, AssociatedPerformer - David Frost, Producer, Mixer, Editor, StudioPersonnel - Silas Brown, Recording Engineer, StudioPersonnel
2 Clavier-Sonaten, 2 Fantasien und 2 Rondos für Kenner und Liebhaber, Wq. 59 (Carl Philipp Emanuel Bach)
IV. Rondo in C Minor, H. 283
Marcus Herzog, Mastering Engineer, StudioPersonnel - Carl Philipp Emanuel Bach, Composer - Daniil Trifonov, Piano, MainArtist, AssociatedPerformer - David Frost, Producer, Mixer, Editor, StudioPersonnel - Silas Brown, Recording Engineer, StudioPersonnel
Variations on 'Ah, vous dirai-je, Maman' (Johann Christoph Friedrich Bach)
Thema (Allegretto)
Marcus Herzog, Mastering Engineer, StudioPersonnel - Johann Christoph Friedrich Bach, Composer - Daniil Trifonov, Piano, MainArtist, AssociatedPerformer - David Frost, Producer, Mixer, Editor, StudioPersonnel - Silas Brown, Recording Engineer, StudioPersonnel
Var. 1
Var. 4 (Minore)
Var. 5 (Maggiore)
Var. 6 (Tempo di Minuetto)
Var. 8 (Schwäbisch, Non allegro)
Var. 9 (Minore, Tranquillo)
Var. 10 (Maggiore, Tempo I)
Var. 11
Var. 12 (Alla Siciliano)
Var. 15 (Più andante)
Var. 16 (Tempo I)
Var. 17 (Minore)
Var. 18 (Maggiore, Allegro)
J.S. Bach: Musette in D Major, BWV Anh. 126 (Notebook for Anna Magdalena Bach, 1725)
Johann Sebastian Bach, Composer - Marcus Herzog, Mastering Engineer, StudioPersonnel - Daniil Trifonov, Piano, MainArtist, AssociatedPerformer - David Frost, Producer, Mixer, Editor, StudioPersonnel - Silas Brown, Recording Engineer, StudioPersonnel
J.S. Bach: Aria "Gedenke doch, mein Geist, zurücke", BWV 509 (Notebook for Anna Magdalena Bach, 1725)
Johann Sebastian Bach, ComposerLyricist - Marcus Herzog, Mastering Engineer, StudioPersonnel - Daniil Trifonov, Piano, MainArtist, AssociatedPerformer - David Frost, Producer, Mixer, Editor, StudioPersonnel - Silas Brown, Recording Engineer, StudioPersonnel
J.S. Bach: Minuet in A Minor, BWV Anh. 120 (Notebook for Anna Magdalena Bach, 1725)
J.S. Bach: Minuet in F Major, BWV Anh. 113 (Notebook for Anna Magdalena Bach, 1725)
J.S. Bach: Polonaise in F Major, BWV Anh. 117b (Notebook for Anna Magdalena Bach, 1725)
J.S. Bach: [Polonaise] in D Minor, BWV Anh. 128 (Notebook for Anna Magdalena Bach, 1725)
J.S. Bach: Choral "Gib dich zufrieden und sei stille", BWV 511 (Notebook for Anna Magdalena Bach, 1725)
Johann Sebastian Bach, Composer - Paul Gerhardt, Author - Marcus Herzog, Mastering Engineer, StudioPersonnel - Daniil Trifonov, Piano, MainArtist, AssociatedPerformer - David Frost, Producer, Mixer, Editor, StudioPersonnel - Silas Brown, Recording Engineer, StudioPersonnel
Petzold: Minuet in G Major, BWV Anh. 114 (Notebook for Anna Magdalena Bach, 1725)
Marcus Herzog, Mastering Engineer, StudioPersonnel - Christian Petzold, Composer - Daniil Trifonov, Piano, MainArtist, AssociatedPerformer - David Frost, Producer, Mixer, Editor, StudioPersonnel - Silas Brown, Recording Engineer, StudioPersonnel
J.S. Bach: Minuet in G Major, BWV Anh. 116 (Notebook for Anna Magdalena Bach, 1725)
C.P.E. Bach: Polonaise in G Minor, BWV Anh. 125 (Notebook for Anna Magdalena Bach, 1725)
J.S. Bach: Minuet in C Minor, BWV Anh. 121 (Notebook for Anna Magdalena Bach, 1725)
Stölzel: Bist du bei mir (Formerly Attrib. J.S. Bach as BWV 508, Notebook for Anna Magdalena Bach, 1725)
Anonymous, Author - Marcus Herzog, Mastering Engineer, StudioPersonnel - Gottfried Heinrich Stolzel, Composer - Daniil Trifonov, Piano, MainArtist, AssociatedPerformer - David Frost, Producer, Mixer, Editor, StudioPersonnel - Silas Brown, Recording Engineer, StudioPersonnel
5 Studies, Anh.1a/1 (Johannes Brahms)
V. Chaconne (After Violin Partita No. 2 in D Minor, BWV 1004 by J.S. Bach, Arr. for Piano)
Johannes Brahms, Composer - Marcus Herzog, Mixer, Mastering Engineer, Recording Engineer, StudioPersonnel - Sid McLauchlan, Producer, Editor, StudioPersonnel - Daniil Trifonov, Piano, MainArtist, AssociatedPerformer
The Art of Fugue, BWV 1080 (Johann Sebastian Bach)
Contrapunctus 1
Johann Sebastian Bach, Composer - Marcus Herzog, Mixer, Mastering Engineer, Recording Engineer, StudioPersonnel - Sid McLauchlan, Producer, Editor, StudioPersonnel - Daniil Trifonov, Piano, MainArtist, AssociatedPerformer
Contrapunctus 6 [per Diminutionem] in Stylo Francese
Contrapunctus 7 per Augmentationem et Diminutionem
Contrapunctus 9 alla Duodecima
Contrapunctus 10 alla Decima
Contrapunctus 11
Contrapunctus [12] [rectus]
Contrapunctus 12 inversus
1 disc(s) - 57 track(s)
Label: Deutsche Grammophon (DG)
© 2021 Deutsche Grammophon GmbH, Berlin ℗ 2021 Deutsche Grammophon GmbH, Berlin
By Daniil Trifonov
Scriabin: Piano Sonata No. 9, Op. 68 "Black Mass"
Scriabin: Piano Sonata No. 9, Op. 68 "Black Mass" Daniil Trifonov
Rachmaninov : Piano Concertos 2, 4 - Bach-Rachmaninov : Partita BWV 1006
Rachmaninov : Piano Concertos 2, 4 - Bach-Rachmaninov : Partita BWV 1006 Daniil Trifonov
Transcendental (Liszt : Etudes S. 139, 141, 144, 145)
Transcendental (Liszt : Etudes S. 139, 141, 144, 145) Daniil Trifonov
Rachmaninov Variations
Rachmaninov Variations Daniil Trifonov
Classical for Working
Hi-Res Masters: Opera Essentials
Best of 2021: Neoclassical
Saint-Saëns: Complete Symphonies
Cristian Măcelaru
Saint-Saëns: Complete Symphonies Cristian Măcelaru
Nelson Freire, a Humble Virtuoso
A child prodigy in his native country, the Brazilian pianist who passed away in November 2021 kept his distance from the noisy fanfare of fame. His enormous hands and his natural and virtuoso technique always assisted his full, powerful and mellow sound. Here Qobuz looks back at the career of a musician that for a long time was the "best-kept secret in the world of the piano".
Max Richter, Neo-classical Activist
With the release of his new album Exile, a reflection on exile with the Baltic Sea Orchestra, the iconoclast and prolific pioneer of the neo-classical movement confirms his status as one of the most ideologically committed artists out there. Melding classical and electronic music, physical and emotional worlds, he produces instrumental works of rare evocative power.
The Paradox of Esa-Pekka Salonen's Studio
Esa-Pekka Salonen is an acclaimed finnish conductor and prolific composer. With over 60 albums under his belt, he has heavily contributed to the history of musical interpretation. This is an interview with an exceptional musician, who discusses his unique approach to working on classical music in the studio.
VOX Software Test : a sleek Hi-Res playback app with advanced features
Trifonov does Rachmaninov!
Who are we? The Qobuz Blog Qobuz Society
Streaming subscription plans Our applications Discover the best sound quality Download store | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,254 |
Attica es una villa ubicada en el condado de Seneca en el estado estadounidense de Ohio. En el Censo de 2010 tenía una población de 899 habitantes y una densidad poblacional de 516,53 personas por km².
Geografía
Attica se encuentra ubicada en las coordenadas . Según la Oficina del Censo de los Estados Unidos, Attica tiene una superficie total de 1.74 km², de la cual 1.72 km² corresponden a tierra firme y (1.34%) 0.02 km² es agua.
Demografía
Según el censo de 2010, había 899 personas residiendo en Attica. La densidad de población era de 516,53 hab./km². De los 899 habitantes, Attica estaba compuesto por el 96.33% blancos, el 0.78% eran afroamericanos, el 0% eran amerindios, el 0% eran asiáticos, el 0% eran isleños del Pacífico, el 0.22% eran de otras razas y el 2.67% pertenecían a dos o más razas. Del total de la población el 1.33% eran hispanos o latinos de cualquier raza.
Referencias
Enlaces externos
Villas de Ohio
Localidades del condado de Seneca (Ohio) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,732 |
The Garmin Forerunner 35 GPS watch has a built-in an optical heart rate monitor so you're always ready to track the next activity. With the Elevate™ HR monitor in the wrist, you won't have to fetch a separate chest strap before you head out, and Bluetooth connectivity wirelessly syncs activity data with Garmin's free Connect smartphone app to further declutter your life.
Multiple sport profiles include cycling, indoor / outdoor run, cardio, and walk mode, and 24/7 activity tracking keeps track of steps, calories burned, and more throughout the day. For more advanced training, an interval mode permits creating structured workouts, and two customizable display screens put the most critical metrics right where you want them. If you forget to record a ride or run, Garmin's Move IQ™ automatically detects change in movement and will upload the data to Garmin Connect so you'll never have a hole in your training regimen.
Dimensions: 1.4" x 1.6" x 0.5"
Display size: 0.93" x 0.93" | {
"redpajama_set_name": "RedPajamaC4"
} | 5,493 |
// Copyright (c) 2003-present, Jodd Team (http://jodd.org)
// All rights reserved.
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// 1. Redistributions of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
//
// 2. Redistributions in binary form must reproduce the above copyright
// notice, this list of conditions and the following disclaimer in the
// documentation and/or other materials provided with the distribution.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
package jodd.http;
import jodd.datetime.TimeUtil;
import jodd.http.up.ByteArrayUploadable;
import jodd.http.up.FileUploadable;
import jodd.http.up.Uploadable;
import jodd.io.FastCharArrayWriter;
import jodd.io.FileNameUtil;
import jodd.io.StreamUtil;
import jodd.upload.FileUpload;
import jodd.upload.MultipartStreamParser;
import jodd.util.MimeTypes;
import jodd.util.RandomString;
import jodd.util.StringPool;
import jodd.util.StringUtil;
import java.io.BufferedReader;
import java.io.ByteArrayInputStream;
import java.io.ByteArrayOutputStream;
import java.io.File;
import java.io.IOException;
import java.io.OutputStream;
import java.io.StringWriter;
import java.io.UnsupportedEncodingException;
import java.util.List;
import java.util.Map;
import static jodd.util.StringPool.CRLF;
/**
* Base class for {@link HttpRequest} and {@link HttpResponse}.
*/
@SuppressWarnings("unchecked")
public abstract class HttpBase<T> {
public static final String HEADER_ACCEPT = "Accept";
public static final String HEADER_ACCEPT_ENCODING = "Accept-Encoding";
public static final String HEADER_CONTENT_TYPE = "Content-Type";
public static final String HEADER_CONTENT_LENGTH = "Content-Length";
public static final String HEADER_CONTENT_ENCODING = "Content-Encoding";
public static final String HEADER_HOST = "Host";
public static final String HEADER_ETAG = "ETag";
public static final String HEADER_CONNECTION = "Connection";
public static final String HEADER_KEEP_ALIVE = "Keep-Alive";
public static final String HEADER_CLOSE = "Close";
public static final String HTTP_1_0 = "HTTP/1.0";
public static final String HTTP_1_1 = "HTTP/1.1";
protected String httpVersion = HTTP_1_1;
protected HttpMultiMap<String> headers = new HttpMultiMap<>();
protected HttpMultiMap<?> form; // holds form data (when used)
protected String body; // holds raw body string (always)
// ---------------------------------------------------------------- properties
/**
* Returns HTTP version string. By default it's "HTTP/1.1".
*/
public String httpVersion() {
return httpVersion;
}
/**
* Sets the HTTP version string. Must be formed like "HTTP/1.1".
*/
public T httpVersion(String httpVersion) {
this.httpVersion = httpVersion;
return (T) this;
}
// ---------------------------------------------------------------- headers
/**
* Returns value of header parameter.
* If multiple headers with the same names exist,
* the first value will be returned. Returns <code>null</code>
* if header doesn't exist.
*/
public String header(String name) {
return headers.get(name);
}
/**
* Returns all values for given header name.
*/
public List<String> headers(String name) {
return headers.getAll(name);
}
/**
* Removes all header parameters for given name.
*/
public void removeHeader(String name) {
String key = name.trim().toLowerCase();
headers.remove(key);
}
/**
* Adds header parameter. If a header with the same name exist,
* it will not be overwritten, but the new header with the same
* name is going to be added.
* The order of header parameters is preserved.
* Also detects 'Content-Type' header and extracts
* {@link #mediaType() media type} and {@link #charset() charset}
* values.
*/
public T header(String name, String value) {
return header(name, value, false);
}
/**
* Adds or sets header parameter.
* @see #header(String, String)
*/
public T header(String name, String value, boolean overwrite) {
String key = name.trim().toLowerCase();
value = value.trim();
if (key.equalsIgnoreCase(HEADER_CONTENT_TYPE)) {
mediaType = HttpUtil.extractMediaType(value);
charset = HttpUtil.extractContentTypeCharset(value);
}
if (overwrite == true) {
headers.set(key, value);
} else {
headers.add(key, value);
}
return (T) this;
}
/**
* Internal direct header setting.
*/
protected void _header(String name, String value, boolean overwrite) {
String key = name.trim().toLowerCase();
value = value.trim();
if (overwrite) {
headers.set(key, value);
} else {
headers.add(key, value);
}
}
/**
* Adds <code>int</code> value as header parameter,
* @see #header(String, String)
*/
public T header(String name, int value) {
_header(name, String.valueOf(value), false);
return (T) this;
}
/**
* Adds date value as header parameter.
* @see #header(String, String)
*/
public T header(String name, long millis) {
_header(name, TimeUtil.formatHttpDate(millis), false);
return (T) this;
}
/**
* Returns {@link HttpMultiMap} of all headers.
*/
public HttpMultiMap<String> headers() {
return headers;
}
// ---------------------------------------------------------------- content type
protected String charset;
/**
* Returns charset, as defined by 'Content-Type' header.
* If not set, returns <code>null</code> - indicating
* the default charset (ISO-8859-1).
*/
public String charset() {
return charset;
}
/**
* Defines just content type charset. Setting this value to
* <code>null</code> will remove the charset information from
* the header.
*/
public T charset(String charset) {
this.charset = null;
contentType(null, charset);
return (T) this;
}
protected String mediaType;
/**
* Returns media type, as defined by 'Content-Type' header.
* If not set, returns <code>null</code> - indicating
* the default media type, depending on request/response.
*/
public String mediaType() {
return mediaType;
}
/**
* Defines just content media type.
* Setting this value to <code>null</code> will
* not have any effects.
*/
public T mediaType(String mediaType) {
contentType(mediaType, null);
return (T) this;
}
/**
* Returns full "Content-Type" header.
* It consists of {@link #mediaType() media type}
* and {@link #charset() charset}.
*/
public String contentType() {
return header(HEADER_CONTENT_TYPE);
}
/**
* Sets full "Content-Type" header. Both {@link #mediaType() media type}
* and {@link #charset() charset} are overridden.
*/
public T contentType(String contentType) {
header(HEADER_CONTENT_TYPE, contentType, true);
return (T) this;
}
/**
* Sets "Content-Type" header by defining media-type and/or charset parameter.
* This method may be used to update media-type and/or charset by passing
* non-<code>null</code> value for changes.
* <p>
* Important: if Content-Type header has some other parameters, they will be removed!
*/
public T contentType(String mediaType, String charset) {
if (mediaType == null) {
mediaType = this.mediaType;
} else {
this.mediaType = mediaType;
}
if (charset == null) {
charset = this.charset;
} else {
this.charset = charset;
}
String contentType = mediaType;
if (charset != null) {
contentType += ";charset=" + charset;
}
_header(HEADER_CONTENT_TYPE, contentType, true);
return (T) this;
}
// ---------------------------------------------------------------- keep-alive
/**
* Defines "Connection" header as "Keep-Alive" or "Close".
* Existing value is overwritten.
*/
public T connectionKeepAlive(boolean keepAlive) {
if (keepAlive) {
header(HEADER_CONNECTION, HEADER_KEEP_ALIVE, true);
} else {
header(HEADER_CONNECTION, HEADER_CLOSE, true);
}
return (T) this;
}
/**
* Returns <code>true</code> if connection is persistent.
* If "Connection" header does not exist, returns <code>true</code>
* for HTTP 1.1 and <code>false</code> for HTTP 1.0. If
* "Connection" header exist, checks if it is equal to "Close".
* <p>
* In HTTP 1.1, all connections are considered persistent unless declared otherwise.
* Under HTTP 1.0, there is no official specification for how keepalive operates.
*/
public boolean isConnectionPersistent() {
String connection = header(HEADER_CONNECTION);
if (connection == null) {
return !httpVersion.equalsIgnoreCase(HTTP_1_0);
}
return !connection.equalsIgnoreCase(HEADER_CLOSE);
}
// ---------------------------------------------------------------- common headers
/**
* Returns full "Content-Length" header or
* <code>null</code> if not set.
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public String contentLength() {
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/**
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_header(HEADER_CONTENT_LENGTH, String.valueOf(value), true);
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*/
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}
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* Returns "Accept" header.
*/
public String accept() {
return header(HEADER_ACCEPT);
}
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* Sets "Accept" header.
*/
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header(HEADER_ACCEPT, encodings, true);
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*/
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return header(HEADER_ACCEPT_ENCODING);
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header(HEADER_ACCEPT_ENCODING, encodings, true);
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// ---------------------------------------------------------------- form
/**
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form(name, parameters[i + 1]);
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*/
public byte[] bodyBytes() {
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return null;
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if (body == null) {
return StringPool.EMPTY;
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* and it is expected from user to set this one.
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contentLength(body.length());
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contentType(mediaType, charset);
body(body);
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* in {@link JoddHttp#defaultBodyEncoding default body encoding}.
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*/
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try {
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} catch (UnsupportedEncodingException ignore) {
}
contentType(contentType);
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// ---------------------------------------------------------------- body form
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*/
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}
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Object value = entry.getValue();
if (value instanceof Uploadable) {
return true;
}
}
return false;
}
/**
* Creates form {@link jodd.http.Buffer buffer} and sets few headers.
*/
protected Buffer formBuffer() {
Buffer buffer = new Buffer();
if (form == null || form.isEmpty()) {
return buffer;
}
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// encode
String formQueryString = HttpUtil.buildQuery(form, formEncoding);
contentType("application/x-www-form-urlencoded", null);
contentLength(formQueryString.length());
buffer.append(formQueryString);
return buffer;
}
String boundary = StringUtil.repeat('-', 10) + RandomString.getInstance().randomAlphaNumeric(10);
for (Map.Entry<String, ?> entry : form) {
buffer.append("--");
buffer.append(boundary);
buffer.append(CRLF);
String name = entry.getKey();
Object value = entry.getValue();
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buffer.append("Content-Disposition: form-data; name=\"").append(name).append('"').append(CRLF);
buffer.append(CRLF);
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string, formEncoding, StringPool.ISO_8859_1);
buffer.append(utf8String);
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else if (value instanceof Uploadable) {
Uploadable uploadable = (Uploadable) value;
String fileName = uploadable.getFileName();
if (fileName == null) {
fileName = name;
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String formEncoding = resolveFormEncoding();
fileName = StringUtil.convertCharset(
fileName, formEncoding, StringPool.ISO_8859_1);
}
buffer.append("Content-Disposition: form-data; name=\"").append(name);
buffer.append("\"; filename=\"").append(fileName).append('"').append(CRLF);
String mimeType = uploadable.getMimeType();
if (mimeType == null) {
mimeType = MimeTypes.getMimeType(FileNameUtil.getExtension(fileName));
}
buffer.append(HEADER_CONTENT_TYPE).append(": ").append(mimeType).append(CRLF);
buffer.append("Content-Transfer-Encoding: binary").append(CRLF);
buffer.append(CRLF);
buffer.append(uploadable);
//byte[] bytes = uploadable.getBytes();
//for (byte b : bytes) {
//buffer.append(CharUtil.toChar(b));
//}
} else {
// should never happened!
throw new HttpException("Unsupported type");
}
buffer.append(CRLF);
}
buffer.append("--").append(boundary).append("--");
buffer.append(CRLF);
// the end
contentType("multipart/form-data; boundary=" + boundary);
contentLength(buffer.size());
return buffer;
}
/**
* Resolves form encodings.
*/
protected String resolveFormEncoding() {
// determine form encoding
String formEncoding = charset;
if (formEncoding == null) {
formEncoding = this.formEncoding;
}
return formEncoding;
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// ---------------------------------------------------------------- buffer
/**
* Returns string representation of this request or response.
*/
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return toString(true);
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* Returns full request/response, or just headers.
* Useful for debugging.
*/
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StringWriter stringWriter = new StringWriter();
try {
buffer.writeTo(stringWriter);
}
catch (IOException ioex) {
throw new HttpException(ioex);
}
return stringWriter.toString();
}
/**
* Returns byte array of request or response.
*/
public byte[] toByteArray() {
Buffer buffer = buffer(true);
ByteArrayOutputStream baos = new ByteArrayOutputStream(buffer.size());
try {
buffer.writeTo(baos);
}
catch (IOException ioex) {
throw new HttpException(ioex);
}
return baos.toByteArray();
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* Creates {@link jodd.http.Buffer buffer} ready to be consumed.
* Buffer can, optionally, contains just headers.
*/
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// ---------------------------------------------------------------- send
protected HttpProgressListener httpProgressListener;
/**
* Sends request or response to output stream.
*/
public void sendTo(OutputStream out) throws IOException {
Buffer buffer = buffer(true);
if (httpProgressListener == null) {
buffer.writeTo(out);
}
else {
buffer.writeTo(out, httpProgressListener);
}
out.flush();
}
// ---------------------------------------------------------------- parsing
/**
* Parses headers.
*/
protected void readHeaders(BufferedReader reader) {
while (true) {
String line;
try {
line = reader.readLine();
} catch (IOException ioex) {
throw new HttpException(ioex);
}
if (StringUtil.isBlank(line)) {
break;
}
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if (ndx != -1) {
header(line.substring(0, ndx), line.substring(ndx + 1));
} else {
throw new HttpException("Invalid header: " + line);
}
}
}
/**
* Parses body.
*/
protected void readBody(BufferedReader reader) {
String bodyString = null;
// first determine if chunked encoding is specified
boolean isChunked = false;
String transferEncoding = header("Transfer-Encoding");
if (transferEncoding != null && transferEncoding.equalsIgnoreCase("chunked")) {
isChunked = true;
}
// content length
String contentLen = contentLength();
int contentLenValue = -1;
if (contentLen != null && !isChunked) {
contentLenValue = Integer.parseInt(contentLen);
if (contentLenValue > 0) {
FastCharArrayWriter fastCharArrayWriter = new FastCharArrayWriter(contentLenValue);
try {
StreamUtil.copy(reader, fastCharArrayWriter, contentLenValue);
} catch (IOException ioex) {
throw new HttpException(ioex);
}
bodyString = fastCharArrayWriter.toString();
}
}
// chunked encoding
if (isChunked) {
FastCharArrayWriter fastCharArrayWriter = new FastCharArrayWriter();
try {
while (true) {
String line = reader.readLine();
int len = Integer.parseInt(line, 16);
if (len > 0) {
StreamUtil.copy(reader, fastCharArrayWriter, len);
reader.readLine();
} else {
// end reached, read trailing headers, if there is any
readHeaders(reader);
break;
}
}
} catch (IOException ioex) {
throw new HttpException(ioex);
}
bodyString = fastCharArrayWriter.toString();
}
// no body yet - special case
if (bodyString == null && contentLenValue != 0) {
// body ends when stream closes
FastCharArrayWriter fastCharArrayWriter = new FastCharArrayWriter();
try {
StreamUtil.copy(reader, fastCharArrayWriter);
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throw new HttpException(ioex);
}
bodyString = fastCharArrayWriter.toString();
}
// BODY READY - PARSE BODY
String charset = this.charset;
if (charset == null) {
charset = StringPool.ISO_8859_1;
}
body = bodyString;
String mediaType = mediaType();
if (mediaType == null) {
mediaType = StringPool.EMPTY;
} else {
mediaType = mediaType.toLowerCase();
}
if (mediaType.equals("application/x-www-form-urlencoded")) {
form = HttpUtil.parseQuery(bodyString, true);
return;
}
if (mediaType.equals("multipart/form-data")) {
form = new HttpMultiMap<>();
MultipartStreamParser multipartParser = new MultipartStreamParser();
try {
byte[] bodyBytes = bodyString.getBytes(StringPool.ISO_8859_1);
ByteArrayInputStream bin = new ByteArrayInputStream(bodyBytes);
multipartParser.parseRequestStream(bin, charset);
} catch (IOException ioex) {
throw new HttpException(ioex);
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// string parameters
for (String paramName : multipartParser.getParameterNames()) {
String[] values = multipartParser.getParameterValues(paramName);
for (String value : values) {
((HttpMultiMap<Object>)form).add(paramName, value);
}
}
// file parameters
for (String paramName : multipartParser.getFileParameterNames()) {
FileUpload[] uploads = multipartParser.getFiles(paramName);
for (FileUpload upload : uploads) {
((HttpMultiMap<Object>)form).add(paramName, upload);
}
}
return;
}
// body is a simple content
form = null;
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 252 |
\section{Introduction}
The adiabatic theorem of quantum mechanics implies that the final state of a particle
that moves slowly along a closed path is identical to the initial eigenstate --- up to a
phase factor. The Berry phase is a time-independent contribution to this phase,
depending only on the geometry of the path.\cite{berry} A simple example is a spin-$1/2$
in a rotating magnetic field $\bf B$, where the Berry phase equals half the solid angle
swept by $\bf B$. It was proposed by Stern \cite{stern} to measure the Berry phase
in the conductance $G$ of a mesoscopic ring in a spatially rotating magnetic field.
Oscillations of $G$ as a function of the swept solid angle were predicted,
similar to the Aharonov-Bohm oscillations as a function of the enclosed flux.\cite{ab}
An important practical difference between the two effects is that the Aharonov-Bohm
oscillations exist at arbitrarily small magnetic fields, whereas for the oscillations
due to the Berry phase the magnetic field should be sufficiently strong to allow the
spin to adiabatically follow the changing direction.
Generally speaking, adiabaticity requires that the precession frequency
$\omega_{\rm B}$ is large compared to the reciprocal of the characteristic timescale
$t_{\rm c}$ on which $\bf B$ changes direction. We know that
$\omega_{\rm B}=g\mu_{\rm B}B/2\hbar$, with $g$ the Land{\'e}-factor and $\mu_{\rm B}$
the Bohr magneton. The question is, what is $t_{\rm c}$? In a ballistic ring there
is only one candidate, the circumference $L$ of the ring divided by the Fermi velocity $v$.
(For simplicity we assume that $L$ is also the scale on which the field direction changes.)
In a diffusive ring there are two candidates: the elastic scattering time $\tau$ and the
diffusion time $\tau_{\rm d}$ around the ring. They differ by a factor
$\tau_{\rm d}/\tau \simeq (L/\ell)^2$, where $\ell=v\tau$ is the mean free path.
Since, by definition, $L\gg\ell$ in a diffusive system, the two time scales are
far apart. Which of the two time scales is the relevant one is still under
debate.\cite{sternrev}
Stern's original proposal \cite{stern} was that
\begin{equation}
\label{critstern}
\omega_{\rm B}\gg \frac{1}{\tau}
\end{equation}
is necessary to observe the Berry-phase oscillations. For realistic values of $g$
this requires magnetic fields in the quantum Hall regime, outside the range of
validity of the semiclassical theory. We call Eq.\ (\ref{critstern}) the
``pessimistic criterion''.
In a later work, \cite{lsg} Loss, Schoeller, and Goldbart (LSG) concluded that adiabaticity is
reached already at much weaker magnetic fields, when
\begin{equation}
\label{critloss}
\omega_{\rm B}\gg\frac{1}{\tau_{\rm d}}\simeq\frac{1}{\tau}\left(\frac{\ell}{L}\right)^2.
\end{equation}
This ``optimistic criterion'' has motivated experimentalists to search for the Berry-phase
oscillations in disordered conductors, \cite{experiments} and was invoked in a recent
study of the conductivity of mesoscopic ferromagnets.\cite{lyanda}
In this paper, we re-examine the semiclassical theory of LSG to resolve the controversy.
The Berry-phase oscillations in the conductance result from a periodic modulation of the
weak-localization correction, and require the solution of a diffusion equation
for the Cooperon propagator. To solve this problem we need to consider
the coupled dynamics of four spin-degrees of freedom. (The Cooperon has four spin indices.)
To gain insight we first examine in Sec.\ \ref{transmission} the simpler problem of the
dynamics of a single spin variable, by studying the randomization of a spin-polarized
electron gas by a non-uniform magnetic field. We start at the level of the Boltzmann
equation and then make the diffusion approximation. We show how the diffusion
equation can be solved exactly for the first two moments of the polarization.
The same procedure is used in Sec.\ \ref{localization} to arrive at a diffusion
equation for the Cooperon. This equation coincides with the equation derived
by LSG in the weak-field regime $\omega_{\rm B}\tau\ll 1$, but is different in the
strong-field regime $\omega_{\rm B}\tau\gtrsim 1$. We present an exact solution for
the weak-localization correction and compare with the findings of LSG.
Our conclusion both for the polarization and for the weak-localization correction is
that adiabaticity requires $\omega_{\rm B}\tau\gg 1$. Regrettably, the pessimistic
criterion (\ref{critstern}) is correct, in agreement with Stern's original conclusion.
The optimistic criterion (\ref{critloss}) advocated by LSG turns out to be the criterion
for maximal randomization of the spin by the magnetic field, and not the criterion for
adiabaticity.
\section{Spin-resolved transmission}
\label{transmission}
\subsection{Formulation of the problem}
Consider a conductor in a magnetic field $\bf B$, containing a disordered
segment (length $L$, mean free path $\ell$ at Fermi velocity $v$) in which the
magnetic field changes its direction. An electron at the Fermi level with spin
up (relative to the local magnetic field) is injected at one end and reaches
the other end. What is the probability that its spin is up?
\begin{figure}[ht]
\epsfxsize=0.7\hsize
\hspace*{\fill}
\vspace*{-0ex}\epsffile{fig1.eps}\vspace*{0ex}
\hspace*{\fill}
\medskip
\caption[]{Schematic drawing of a two-dimensional electron gas in the spatially
rotating magnetic field of Eq.\ (\ref{field}), with $f=1$.}
\label{fig1}
\end{figure}
For simplicity we take for the conductor a two-dimensional electron gas (in the $x$-$y$ plane,
with the disordered region between $x=0$ and $x=L$), and we ignore the
curvature of the electron trajectories by the Lorentz force. The problem becomes
effectively one-dimensional by assuming that $\bf B$ depends on $x$ only.
We choose a rotation of $\bf B$ in the $x$-$y$-plane, according to
\begin{equation}
\label{field}
{\bf B}(x,y,z=0)=
\left(B\sin\eta\cos\case{2\pi fx}{L},B\sin\eta\sin\case{2\pi fx}{L},B\cos\eta\right),
\end{equation}
with $\eta$ and $f$ arbitrary parameters. The geometry is sketched in
Fig.\ \ref{fig1}. We treat the orbital motion semiclassically, within
the framework of the Boltzmann equation. (This is justified if the Fermi wavelength
is much smaller than $\ell$.) The spin dynamics requires a fully quantum mechanical
treatment. We assume that the Zeeman energy $g\mu_{\rm B}B$ is much smaller than the
Fermi energy $\frac{1}{2}mv^2$, so that the orbital motion is independent
of the spin.
We introduce the probability density $P(x,\phi,\xi,t)$ for the electron to be
at time $t$ at position $x$ with velocity ${\bf v}=(v\cos\phi,v\sin\phi,0)$, in
the spin state with spinor $\xi =(\xi_1,\xi_2)$. The dynamics of $\xi$ depends
on the local magnetic field according to
\begin{equation}
\label{schroedinger}
\frac{d\xi}{dt}=\frac{ig\mu_{\rm B}}{2\hbar}{\bf B}\cdot\mbox{\boldmath$\sigma$}\,\xi,
\end{equation}
where $\mbox{\boldmath$\sigma$}=(\sigma_x,\sigma_y,\sigma_z)$ is the vector of Pauli matrices.
It is convenient to decompose $\xi=\chi_1\xi_\uparrow +\chi_2\xi_\downarrow$ into
the local eigenstates $\xi_\uparrow,\xi_\downarrow$ of ${\bf B}\cdot\mbox{\boldmath$\sigma$}$,
\begin{mathletters}
\label{localbasis}
\begin{equation}
\xi_\uparrow =
\pmatrix{\cos\frac{\eta}{2}\,{\rm e}^{-i\pi fx/L} \cr
\sin\frac{\eta}{2}\,{\rm e}^{i\pi fx/L}},~~~~
\xi_\downarrow =
\pmatrix{-\sin\frac{\eta}{2}\,{\rm e}^{-i\pi fx/L} \cr
\cos\frac{\eta}{2}\,{\rm e}^{i\pi fx/L}},
\end{equation}
\begin{equation}
{\bf B}\cdot\mbox{\boldmath$\sigma$}\,\xi_\uparrow = B\xi_\uparrow,~~~~
{\bf B}\cdot\mbox{\boldmath$\sigma$}\,\xi_\downarrow = -B\xi_\downarrow,
\end{equation}
\end{mathletters}
and use the real and imaginary parts of the coefficients $\chi_1,\chi_2$ as
variables in the Boltzmann equation. The dynamics of the vector of coefficients
$c=(c_1,c_2,c_3,c_4)=({\rm Re}\,\chi_1,{\rm Im}\,\chi_1,{\rm Re}\,\chi_2,{\rm Im}\,\chi_2)$
is given by
\begin{mathletters}
\begin{equation}
\frac{dc}{dt} = \frac{1}{\tau} M c,~~M=M_0+M_1\cos\phi,
\end{equation}
\begin{equation}
\label{defm}
M_0 = \omega_{\rm B}\tau \pmatrix{
0&-1&0&0 \cr 1&0&0&0 \cr 0&0&0&1 \cr 0&0&-1&0 },~~~~~
M_1=\frac{\pi f\ell}{L} \pmatrix{
0 & -\cos\eta & 0 & \sin\eta \cr
\cos\eta & 0 & -\sin\eta & 0 \cr
0 & \sin\eta & 0 & \cos\eta \cr
-\sin\eta & 0 & -\cos\eta & 0
},
\end{equation}
\end{mathletters}
where $\omega_{\rm B}=g\mu_{\rm B}B/2\hbar$ is the precession frequency of the spin.
The Boltzmann equation takes the form
\begin{equation}
\label{boltzmann}
\tau\frac{\partial}{\partial t} P(x,\phi,c,t) =
-\ell\cos\phi\frac{\partial P}{\partial x}
-\sum_{i,j}\frac{\partial}{\partial c_i} \left(M_{ij}c_j P\right)
-P + \int_0^{2\pi}\frac{d\phi'}{2\pi} P(x,\phi',c,t),
\end{equation}
where we have assumed isotropic scattering (rate $1/\tau=v/\ell$).
We look for a stationary solution to the Boltzmann equation, so the left-hand-side of
Eq.\ (\ref{boltzmann}) is zero and we omit the argument $t$ of $P$.
A stationary flux of particles with an isotropic velocity distribution
is injected at $x=0$, their spins all aligned with the local magnetic field
(so $\chi_2=0$ at $x=0$). Without loss of generality we may assume that
$\chi_1=1$ at $x=0$. No particles are incident from the other end, at $x=L$.
Thus the boundary conditions are
\begin{mathletters}
\label{beecee}
\begin{eqnarray}
\label{beginwire}
&& P(x=0,\phi,c)=\delta (c_1-1)\delta (c_2)\delta (c_3)\delta (c_4)~{\rm if}~\cos\phi > 0, \\
\label{endwire}
&& P(x=L,\phi,c) = 0~{\rm if}~\cos\phi < 0.
\end{eqnarray}
\end{mathletters}
This completes the formulation of the problem. We compare two methods of solution.
The first is an exact numerical solution of the Boltzmann equation using the
Monte Carlo method. The second is an approximate analytical solution using the
diffusion approximation, valid for $L\gg \ell$. We begin with the latter.
\subsection{Diffusion approximation}
The diffusion approximation amounts to the assumption that $P$ has a simple
cosine-dependence on $\phi$,
\begin{equation}
\label{diffansatz}
P(x,\phi,c)=N(x,c)+J(x,c)\cos\phi.
\end{equation}
To determine the density $N$ and current $J$ we
substitute Eq.\ (\ref{diffansatz}) into Eq.\ (\ref{boltzmann}) and integrate
over $\phi$. This gives
\begin{eqnarray}
\ell\frac{\partial J}{\partial x} &=&
-\frac{\partial}{\partial c}\left( 2 M_0 c N+M_1 c J\right).
\label{diffpde1}
\end{eqnarray}
Similarly, multiplication with $\cos\phi$ before integration gives
\begin{eqnarray}
\ell\frac{\partial N}{\partial x} &=&
-\frac{\partial}{\partial c} \left( M_0 c J+ M_1 c N\right) - J.
\label{diffpde2}
\end{eqnarray}
Thus we have a closed set of partial differential equations for the
unknown functions $N(x, c)$ and $J(x, c)$. Boundary conditions are obtained by
multiplying Eq.\ (\ref{beecee}) with $\cos\phi$ and integrating over $\phi$:
\begin{mathletters}
\label{diffbc}
\begin{eqnarray}
&& N(x=0,c)+\frac{\pi}{4}J(x=0,c)=\delta (c_1-1)\delta (c_2)\delta (c_3)\delta (c_4),\\
&& N(x=L,c)-\frac{\pi}{4}J(x=L,c)=0.
\end{eqnarray}
\end{mathletters}
We seek the spin polarization $p=c_1^2+c_2^2-c_3^2-c_4^2$ of the transmitted
electrons, characterized by the distribution
\begin{equation}
\label{goal}
P(p)=\frac{\int\!dc\,J(x=L,c)\delta(c_1^2+c_2^2-c_3^2-c_4^2-p)}{\int\!dc\, J(x=L,c)}.
\end{equation}
(The notation $\int\!dc\,\equiv\int\!dc_1\,\int\!dc_2\,\int\!dc_3\,\int\!dc_4$
indicates an integration over the spin variables.) We compute the first two moments
of $P(p)$.
The first moment $\overline{p}$ is the fraction of transmitted electrons with
spin up minus the fraction with spin down, averaged quantum mechanically over the
spin state and statistically over the disorder.
The variance Var $p=\overline{p^2}-\overline{p}^2$ gives an
indication of the magnitude of the statistical fluctuations.
Integration of Eqs.\ (\ref{diffpde1})--(\ref{diffbc}) over the spin variables
yields the equations and boundary conditions for the functions
$N(x)=\int\!dc\,N(x,c)$ and $J(x)=\int\!dc\,J(x,c)$:
\begin{mathletters}
\label{momzero}
\begin{eqnarray}
&&\ell\frac{dN}{dx}=-J,~~\frac{dJ}{dx} = 0, \label{momzerodiff} \\
&& N(0)+\frac{\pi}{4}J(0)=1,~~N(L)-\frac{\pi}{4}J(L)=0.
\end{eqnarray}
\end{mathletters}
The solution
\begin{equation}
\label{solnorm}
J(x) = \left( \frac{\pi}{2} + \frac{L}{\ell} \right)^{-1}
\end{equation}
determines the denominator of Eq.\ (\ref{goal}).
To determine $\overline{p}$ we multiply Eqs.\ (\ref{diffpde1}) and (\ref{diffpde2})
with $\chi_\alpha\chi_\beta^\ast$ and integrate over $c$. (Recall that
$\chi_1=c_1+ic_2,\chi_2=c_3+ic_4$.)
It follows upon partial integration that
\begin{mathletters}
\begin{eqnarray}
&&\int\! dc\, \chi_\alpha\chi_\beta^\ast \frac{\partial}{\partial c}\left(M_0 c f\right)=
-\sum_{\rho,\sigma}
\left(S_{\alpha\rho}\delta_{\beta\sigma}-\delta_{\alpha\rho}S_{\beta\sigma}\right)
\int\! dc\, \chi_\rho \chi_\sigma^\ast f, \\
&&\int\! dc\, \chi_\alpha\chi_\beta^\ast \frac{\partial}{\partial c}\left(M_1 c f\right) =
-\sum_{\rho,\sigma}
\left(T_{\alpha\rho}\delta_{\beta\sigma}-\delta_{\alpha\rho}T_{\beta\sigma}\right)
\int\! dc\, \chi_\rho \chi_\sigma^\ast f,
\end{eqnarray}
\end{mathletters}
for arbitrary functions $f(x,c)$. The $2\times 2$ matrices $S,T$ are defined by
\begin{equation}
S=i\omega_{\rm B}\tau\sigma_z,
~~T=\frac{i\pi f\ell}{L}\left(\sigma_z\cos\eta-\sigma_x\sin\eta\right).
\end{equation}
In this way we find that the moments
\begin{mathletters}
\begin{eqnarray}
N_{\alpha\beta}(x)&=&\int\!dc\,\chi_\alpha\chi^\ast_\beta N(x,c), \\
J_{\alpha\beta}(x)&=&\int\!dc\,\chi_\alpha\chi^\ast_\beta J(x,c),
\end{eqnarray}
\end{mathletters}
satisfy the ordinary differential equations
\begin{mathletters}
\label{momdiff}
\begin{eqnarray}
\label{momdiffa}
&& \ell\frac{dN_{\alpha\beta}}{dx} = \sum_{\rho,\sigma}
\left(T_{\alpha\rho}\delta_{\beta\sigma}-\delta_{\alpha\rho}T_{\beta\sigma}\right)
N_{\rho\sigma}
+\sum_{\rho,\sigma}
\left(S_{\alpha\rho}\delta_{\beta\sigma}-\delta_{\alpha\rho}S_{\beta\sigma}\right)
J_{\rho\sigma} -J_{\alpha\beta}, \\
&& \ell\frac{dJ_{\alpha\beta}}{dx} =
2\sum_{\rho,\sigma}
\left(S_{\alpha\rho}\delta_{\beta\sigma}-\delta_{\alpha\rho}S_{\beta\sigma}\right)
N_{\rho\sigma}
+\sum_{\rho,\sigma}
\left(T_{\alpha\rho}\delta_{\beta\sigma}-\delta_{\alpha\rho}T_{\beta\sigma}\right)
J_{\rho\sigma},
\end{eqnarray}
\end{mathletters}
with boundary conditions
\begin{mathletters}
\begin{eqnarray}
&& N_{\alpha\beta}(x=0)+\frac{\pi}{4}J_{\alpha\beta}(x=0)=\delta_{\alpha 1}\delta_{\beta 1}, \\
&& N_{\alpha\beta}(x=L)-\frac{\pi}{4}J_{\alpha\beta}(x=L)= 0.
\end{eqnarray}
\end{mathletters}
The mean polarization $\overline{p}$ is determined by $J_{\alpha\beta}$ according to
\begin{equation}
\label{avpdef}
\overline{p}=\frac{J_{11}(L)-J_{22}(L)}{J(L)}=
\left(\frac{\pi}{2}+\frac{L}{\ell}\right)\left[J_{11}(L)-J_{22}(L)\right].
\end{equation}
Since Eq.\ (\ref{momdiff}) is linear in the $8$ functions
$N_{\alpha\beta}(x),J_{\alpha\beta}(x)$ ($\alpha,\beta=1,2$), a solution requires
the eigenvalues and right eigenvectors of the $8\times 8$ matrix of coefficients.
These can be readily computed numerically for any values of $L/\ell$ and
$\omega_{\rm B}\tau$. We have found an analytic asymptotic solution for $L/\ell\gg 1$
and $\omega_{\rm B}\tau\gg (f\ell/L)^2$, given by
\begin{equation}
\label{avp}
\overline{p}=\frac{k}{\sinh k},~~~k=\frac{2\pi f\sin\eta}{\sqrt{1+(2\omega_{\rm B}\tau)^2}}.
\end{equation}
In Fig.\ \ref{fig2} we compare the numerical solution (solid curve) with Eq.\ (\ref{avp})
(dashed curve) for $L/\ell=25$ and $\eta=\pi/3,f=1$. The two curves are almost
indistinguishable, except for the smallest values of $\omega_{\rm B}\tau$.
\begin{figure}[ht]
\epsfxsize=0.6\hsize
\hspace*{\fill}
\vspace*{-0ex}\epsffile{fig2.eps}\vspace*{0ex}
\hspace*{\fill}
\medskip
\caption[]{Average and variance of the spin polarization $p$ of the current
transmitted through a two-dimensional region of length $L=25\,\ell$,
as a function of $\omega_{\rm B}\tau$, for a magnetic field given by Eq.\ (\ref{field})
with $\eta=\pi/3$ and $f=1$.
The data points result from Monte Carlo simulations of the Boltzmann
equation (\ref{boltzmann}),
the solid curves result from the diffusion approximation (\ref{diffansatz}),
and the dashed curves are the asymptotic formulas (\ref{avp}) and (\ref{varp}).
Notice the transient regime (A), the randomized regime (B), and the adiabatic
regime (C). }
\label{fig2}
\end{figure}
In a similar way, we compute the second moment of $P(p)$ by multiplying
Eqs.\ (\ref{diffpde1}) and (\ref{diffpde2})
with $\chi_\alpha\chi_\beta^\ast\chi_\gamma\chi_\delta^\ast$ and integrating over
$c$. The result is a closed set of equations
\begin{mathletters}
\label{vardiff}
\begin{eqnarray}
\label{vardiffa}
&& \ell \frac{d}{dx} N_{\alpha\beta\gamma\delta} = \sum_{\mu,\nu,\rho,\sigma} \left(
L^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta} N_{\mu\nu\rho\sigma}
+ K^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta} J_{\mu\nu\rho\sigma} \right)
- J_{\alpha\beta\gamma\delta}, \\
&& \ell \frac{d}{dx} J_{\alpha\beta\gamma\delta} = \sum_{\mu,\nu,\rho,\sigma} \left(
2K^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta} N_{\mu\nu\rho\sigma}
+ L^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta} J_{\mu\nu\rho\sigma} \right) ,
\end{eqnarray}
\end{mathletters}
where we have defined
\begin{mathletters}
\begin{eqnarray}
&& K_{\alpha\beta\gamma\delta}^{\mu\nu\rho\sigma} =
S_{\alpha\mu}\delta_{\beta\nu}\delta_{\gamma\rho}\delta_{\delta\sigma}
-\delta_{\alpha\mu}S_{\beta\nu}\delta_{\gamma\rho}\delta_{\delta\sigma}
+\delta_{\alpha\mu}\delta_{\beta\nu}S_{\gamma\rho}\delta_{\delta\sigma}
-\delta_{\alpha\mu}\delta_{\beta\nu}\delta_{\gamma\rho}S_{\delta\sigma}, \\
&& L_{\alpha\beta\gamma\delta}^{\mu\nu\rho\sigma} =
T_{\alpha\mu}\delta_{\beta\nu}\delta_{\gamma\rho}\delta_{\delta\sigma}
-\delta_{\alpha\mu}T_{\beta\nu}\delta_{\gamma\rho}\delta_{\delta\sigma}
+\delta_{\alpha\mu}\delta_{\beta\nu}T_{\gamma\rho}\delta_{\delta\sigma}
-\delta_{\alpha\mu}\delta_{\beta\nu}\delta_{\gamma\rho}T_{\delta\sigma},
\end{eqnarray}
\end{mathletters}
\begin{mathletters}
\begin{eqnarray}
N_{\alpha\beta\gamma\delta}(x) &=& \int\!dc\,
\chi_\alpha\chi^\ast_\beta\chi_\gamma\chi^\ast_\delta N(x,c), \\
J_{\alpha\beta\gamma\delta}(x) &=& \int\!dc\,
\chi_\alpha\chi^\ast_\beta\chi_\gamma\chi^\ast_\delta J(x,c).
\end{eqnarray}
\end{mathletters}
The boundary conditions on the functions $N_{\alpha\beta\gamma\delta}$ and
$J_{\alpha\beta\gamma\delta}$ are
\begin{eqnarray}
&& N_{\alpha\beta\gamma\delta}(x=0)+\frac{\pi}{4}J_{\alpha\beta\gamma\delta}(x=0)
=\delta_{\alpha 1}\delta_{\beta 1}\delta_{\gamma 1}\delta_{\delta 1}, \\
&& N_{\alpha\beta\gamma\delta}(x=L)-\frac{\pi}{4}J_{\alpha\beta\gamma\delta}(x=L)= 0.
\end{eqnarray}
The second moment $\overline{p^2}$ is determined by
\begin{equation}
\overline{p^2}=\left(\frac{\pi}{2}+\frac{L}{\ell}\right)
\left[ J_{1111}(x=L)-J_{1122}(x=L)-J_{2211}(x=L)+J_{2222}(x=L) \right].
\end{equation}
The numerical solution is plotted also in Fig.\ \ref{fig2}, together with the
asymptotic expression
\begin{equation}
\label{varp}
{\rm Var}\, p = \frac{1}{3}+\frac{2k\sqrt{3}}{3\sinh\left(k\sqrt{3}\right)}
-\frac{k^2}{\sinh^2 k}.
\end{equation}
It is evident from Eqs.\ (\ref{avp}) and (\ref{varp}), and from Fig.\ \ref{fig2},
that the regime with $\overline{p}=1$, ${\rm Var}\,p=0$ is entered for
$\omega_{\rm B}\tau \gtrsim f$ [for $\sin\eta ={\cal O} (1)$], in agreement with Stern's
criterion (\ref{critstern}) for adiabaticity. For smaller $\omega_{\rm B}\tau$
adiabaticity is lost. There is a transient regime
$\omega_{\rm B}\tau \ll (f\ell/L)^2$, in which the precession frequency is
so low that the spin remains in the same state during the entire diffusion process.
For $(f\ell/L)^2 \ll \omega_{\rm B}\tau \ll f$ the average polarization reaches a
plateau value close to zero with a finite variance. For a sufficiently non-uniform
field, $f\sin\eta\gg 1$, we find in this regime $\overline{p}=0$ and ${\rm Var}\, p=1/3$,
which means that the spin state is completely randomized. The transient regime,
the randomized regime, and the adiabatic regime are indicated in Fig.\ \ref{fig2}
by the letters A, B, and C.
\subsection{Comparison with Monte Carlo simulations}
In order to check the diffusion approximation we solved the full Boltzmann equation
by means of a Monte Carlo simulation. A particle is moved from $x=0$ over a distance
$\ell_1$ in the direction $\phi_1$, then over a distance $\ell_2$ in the direction $\phi_2$,
and so on, until it is reflected back to $x=0$ or transmitted to $x=L$.
The step lengths $\ell_i$ are chosen randomly from a Poisson distribution with
mean $\ell$. The directions $\phi_i$ are chosen uniformly from $[0,2\pi]$, except for
the initial direction $\phi_1$, which is distributed $\propto\cos\phi_1$.
The spin components are given by
\begin{equation}
\pmatrix{\chi_1 \cr \chi_2} =
\prod_i {\rm e}^{\left(S + T \cos\phi_i \right)\ell_i /\ell}
\pmatrix{ 1 \cr 0 }.
\end{equation}
To find $\overline{p^n}$, one has to average $\left(|\chi_1|^2-|\chi_2|^2\right)^n$
over the transmitted particles.
The results for $L/\ell=25$ are shown in Fig. \ref{fig2} (data points). They agree
very well with the results of the previous subsection, thus confirming the validity of
the diffusion approximation for $L/\ell\gg 1$.
\section{Weak localization}
\label{localization}
\subsection{Formulation of the problem}
We turn to the effect of the non-uniform magnetic field on the weak-localization
correction of a multiply-connected system. We consider the same geometry as in
Fig.\ \ref{fig1}, but now with periodic boundary conditions --- to model a ring of
circumference $L$. Only the effects of the magnetic field on the spin are included,
to isolate the Berry phase from the conventional Aharonov-Bohm phase. As in the
previous subsection, we assume that the orbital motion is independent of the spin dynamics.
We follow LSG in applying the semiclassical theory of Chakravarty and Schmidt
\cite{chakra} to the problem, however, we start at the level of the Boltzmann
equation --- rather than at the level of the diffusion equation --- and make the
diffusion approximation at a later stage of the calculation.
The weak-localization correction $\Delta G$ to the conductance is given by
\begin{equation}
\label{wl1}
\Delta G = - \frac{e^2D}{\pi\hbar L} \int_0^\infty\!\!dt\, {\rm e}^{-t/\tau_\varphi} C(t),
\end{equation}
where $\tau_\varphi$ is the phase coherence time and the diffusion coefficient $D=vl/d$
in $d$ dimensions. (In our geometry $d=2$.) The ``return quasi-probability'' $C(t)$
is expressed as
a sum over ``Boltzmannian walks'' ${\bf R}(t)$ with ${\bf R}(0)={\bf R}(t)$,
\begin{equation}
\label{returnprob}
C(t)= \sum_{\left\{{\bf R}(t)\right\}} W\, {\rm Tr}\, (U^+U^-).
\end{equation}
Here $W[{\bf R}(t)]$ is the weight of the Boltzmannian walk for a spinless particle.
The $2\times 2$ matrices $U^\pm[{\bf R}(t)]$ are defined by
\begin{equation}
U^\pm = {\cal T}\exp\left\{\pm\frac{ig\mu_{\rm B}}{2\hbar}\int_0^t\!dt'\,
{\bf B}\biglb({\bf R}(t')\bigrb)
\cdot \mbox{\boldmath$\sigma$}\right\},
\end{equation}
where $\cal T$ denotes a time ordering. The factor ${\rm Tr}\, (U^+U^-)$ in
Eq.\ (\ref{returnprob}) accounts for the phase difference of time-reversed paths.
The Cooperon can be written in terms of a propagator $\chi$,
\begin{equation}
\label{cooperon}
C(t)=\frac{1}{2\pi}\int_0^{2\pi}\!\! d\phi \int_0^{2\pi}\!\! d\phi_{\rm i}\sum_{\alpha,\beta}
\chi_{\alpha\beta\beta\alpha} (x_{\rm i},x_{\rm i};\phi,\phi_{\rm i};t),
\end{equation}
that satisfies the kinetic equation
\begin{eqnarray}
\label{kinetic}
\left( \frac{\partial}{\partial t} + {\cal B} \right)
\chi_{\alpha\beta\gamma\delta} (x,x_{\rm i};\phi,\phi_{\rm i};t) -
\frac{ig\mu_{\rm B}}{2\hbar} \sum_{\alpha',\gamma'}
\left[
\biglb({\bf B}(x)\cdot\mbox{\boldmath$\sigma$}\bigrb)_{\alpha\alpha'}\delta_{\gamma\gamma'}-
\delta_{\alpha\alpha'}\biglb({\bf B}(x)\cdot\mbox{\boldmath$\sigma$}\bigrb)_{\gamma\gamma'}
\right]
\chi_{\alpha'\beta\gamma'\delta} \nonumber \\
= \delta (t) \delta (x-x_{\rm i}) \delta (\phi-\phi_{\rm i})
\delta_{\alpha\beta} \delta_{\gamma\delta}.
\end{eqnarray}
The Boltzmann operator $\cal B$ is given by
\begin{equation}
{\cal B} = v\cos\phi\frac{\partial}{\partial x}
+ \frac{1}{\tau} - \frac{1}{\tau}\int_0^{2\pi}\!\frac{d\phi}{2\pi}.
\end{equation}
The propagator $\chi$ is a moment of the probability distribution
$P(x,\phi,U^+,U^-,t)$,
\begin{equation}
\label{propagator}
\chi_{\alpha\beta\gamma\delta} =\int\! dU^+ \int\! dU^- \,
U^+_{\alpha\beta} U^-_{\gamma\delta} P,
\end{equation}
that satisfies the Boltzmann equation
\begin{equation}
\label{boeq}
\left[\frac{\partial}{\partial t} + {\cal B}
+\frac{\partial}{\partial U^+} \left(\frac{dU^+}{dt}\right)
+\frac{\partial}{\partial U^-} \left(\frac{dU^-}{dt}\right)
\right] P(x,\phi,U^+,U^-,t) = 0,
\end{equation}
with initial condition
\begin{equation}
P(x,\phi,U^+,U^-,0) = \delta (x-x_{\rm i}) \delta (\phi-\phi_{\rm i}) \delta (U^+-\openone )
\delta (U^--\openone ).
\end{equation}
The notation $dU^+$ or $dU^-$ indicates the differential of the real and imaginary
parts of the elements of the $2\times 2$ matrix $U^+$ or $U^-$. We will write this
in a more explicit way in the next subsection.
The Boltzmann equation (\ref{boeq}) has the same form as that which we studied in
Sec.\ \ref{transmission}. The difference is that we have four times as many internal
degrees of freedom. Instead of a single spinor $\xi$ we now have two
spinor matrices $U^+$ and $U^-$. A first doubling of the number of degrees of freedom
occurs because we have to follow the evolution of both spin up and spin down. A second
doubling occurs because we have to follow both the normal and the time-reversed
evolution.
\subsection{Diffusion approximation.}
We make the diffusion approximation to the Boltzmann equation (\ref{boeq}), by
following the steps outlined in Sec.\ \ref{transmission}.
The $4 \times 2$ matrix $u^\pm$ containing the real and imaginary parts of $U^\pm$,
\begin{equation}
u^\pm = \pmatrix{
{\rm Re}\, U^\pm_{11} & {\rm Re}\, U^\pm_{12} \cr
{\rm Im}\, U^\pm_{11} & {\rm Im}\, U^\pm_{12} \cr
{\rm Re}\, U^\pm_{21} & {\rm Re}\, U^\pm_{22} \cr
{\rm Im}\, U^\pm_{21} & {\rm Im}\, U^\pm_{22} },
\end{equation}
has a time evolution governed by
\begin{mathletters}
\begin{equation}
\tau\frac{du^\pm}{dt}= \pm Z(x) u^\pm,
\end{equation}
\begin{equation}
Z(x)=\omega_{\rm B}\tau \pmatrix{
0 & -\cos\eta & \sin\eta\sin\frac{2\pi fx}{L} & -\sin\eta\cos\frac{2\pi fx}{L} \cr
\cos\eta & 0 & \sin\eta\cos\frac{2\pi fx}{L} & \hphantom{-}\sin\eta\sin\frac{2\pi fx}{L} \cr
-\sin\eta\sin\frac{2\pi fx}{L} & -\sin\eta\cos\frac{2\pi fx}{L} & 0 & \cos\eta \cr
\hphantom{-}\sin\eta\cos\frac{2\pi fx}{L} & -\sin\eta\sin\frac{2\pi fx}{L} & -\cos\eta & 0
}.
\end{equation}
\end{mathletters}
The Boltzmann equation (\ref{boeq}) becomes, in a more explicit notation,
\begin{eqnarray}
\label{freqbe}
\tau \frac{\partial}{\partial t} P(x,\phi,u^+,u^-,t) &=&
-\ell\cos\phi\frac{\partial P}{\partial x}
- \sum_{i,j,k} \frac{\partial}{\partial u^+_{ij}} Z_{ik}(x) u^+_{kj} P
+ \sum_{i,j,k} \frac{\partial}{\partial u^-_{ij}} Z_{ik}(x) u^-_{kj} P \nonumber \\
&& - P + \int_0^{2\pi}\! \frac{d\phi'}{2\pi} P(x,\phi',u^+,u^-,t).
\end{eqnarray}
We now make the diffusion ansatz in the form
\begin{equation}
\int_0^\infty\! dt\, {\rm e}^{-t/\tau_\varphi}\int_0^{2\pi}\! d\phi_{\rm i}\, P = N + J\cos\phi.
\end{equation}
By integrating the Boltzmann equation over $\phi$, once
with weight $1$ and once with weight $\cos\phi$, we obtain two coupled equations for
the functions $N(x,u^+,u^-)$ and $J(x,u^+,u^-)$. Next we multiply both equations with
$U^+_{\alpha\beta}U^-_{\gamma\delta}$ and integrate over the real and imaginary parts of
the matrix elements. The moments $N_{\alpha\beta\gamma\delta}$
and $J_{\alpha\beta\gamma\delta}$ defined by
\begin{mathletters}
\begin{eqnarray}
\label{wlmom}
N_{\alpha\beta\gamma\delta} (x) &=&
\int\! dU^+\int\! dU^-\, U^+_{\alpha\beta} U^-_{\gamma\delta} N, \\
J_{\alpha\beta\gamma\delta} (x) &=&
\int\! dU^+\int\! dU^-\, U^+_{\alpha\beta} U^-_{\gamma\delta} J,
\end{eqnarray}
\end{mathletters}
are found to obey the ordinary differential equations
\begin{mathletters}
\label{odiff}
\begin{eqnarray}
\label{coopdiffa}
\ell\frac{dN_{\alpha\beta\gamma\delta}}{dx} &=&
\frac{ig\mu_{\rm B}\tau}{2\hbar} \sum_{\alpha',\gamma'}
\left[\biglb({\bf B}(x)\cdot\mbox{\boldmath$\sigma$}\bigrb)_{\alpha\alpha'}
\delta_{\gamma\gamma'}-
\delta_{\alpha\alpha'}
\biglb({\bf B}(x)\cdot\mbox{\boldmath$\sigma$} \bigrb)_{\gamma\gamma'}\right]
J_{\alpha'\beta\gamma'\delta} \nonumber \\
&& -\left(1+\tau/\tau_\varphi\right)J_{\alpha\beta\gamma\delta}, \\
\label{coopdiffb}
\ell\frac{dJ_{\alpha\beta\gamma\delta}}{dx} &=&
\frac{ig\mu_{\rm B}\tau}{\hbar} \sum_{\alpha',\gamma'}
\left[\biglb({\bf B}(x)\cdot\mbox{\boldmath$\sigma$}\bigrb)_{\alpha\alpha'}\delta_{\gamma\gamma'}-
\delta_{\alpha\alpha'} \biglb({\bf B}(x)\cdot \mbox{\boldmath$\sigma$}\bigrb)_{\gamma\gamma'}
\right] N_{\alpha'\beta\gamma'\delta} \nonumber \\
&& -(2\tau/\tau_\varphi) N_{\alpha\beta\gamma\delta}
+2\tau\delta_{\alpha\beta}\delta_{\gamma\delta}\delta (x-x_{\rm i}).
\end{eqnarray}
\end{mathletters}
The periodic boundary conditions are
\begin{equation}
\label{ringbc}
N_{\alpha\beta\gamma\delta}(0) = N_{\alpha\beta\gamma\delta}(L),~~~
J_{\alpha\beta\gamma\delta}(0) = J_{\alpha\beta\gamma\delta}(L).
\end{equation}
The Cooperon $C$ and the propagator $\chi$ of Eqs.\ (\ref{cooperon}) and
(\ref{propagator}) are related to the density $N$ by
\begin{eqnarray}
&& N_{\alpha\beta\gamma\delta} (x) = \int_0^\infty \! dt \, {\rm e}^{-t/\tau_\varphi}
\frac{1}{2\pi} \int_0^{2\pi} d\phi \int_0^{2\pi} d\phi_{\rm i} \,
\chi_{\alpha\beta\gamma\delta} (x,x_{\rm i};\phi,\phi_{\rm i};t), \\
&& \sum_{\alpha,\beta} N_{\alpha\beta\beta\alpha} (x_{\rm i}) =
\int_0^\infty \! dt \, {\rm e}^{-t/\tau_\varphi} C(t).
\end{eqnarray}
Hence the weak-localization correction (\ref{wl1}) is obtained from $N$ by
\begin{equation}
\label{wl2}
\Delta G = -\frac{e^2D}{\pi\hbar L}
\sum_{\alpha,\beta} N_{\alpha\beta\beta\alpha}(x_{\rm i}).
\label{wlresult}
\end{equation}
The transformation to the local basis of spin states (\ref{localbasis}) takes
the form of a unitary transformation of the moments $N$ and $J$,
\begin{mathletters}
\begin{eqnarray}
&& \tilde{N}_{\alpha\beta\gamma\delta} = \sum_{\alpha',\beta',\gamma',\delta'}
Q^{\vphantom{\dagger}}_{\alpha\alpha'} Q^{\vphantom{\dagger}}_{\gamma\gamma'}
N^{\vphantom{\dagger}}_{\alpha'\beta'\gamma'\delta'}
Q^\dagger_{\beta'\beta} Q^\dagger_{\delta'\delta}, \\
&& \tilde{J}_{\alpha\beta\gamma\delta} = \sum_{\alpha',\beta',\gamma',\delta'}
Q^{\vphantom{\dagger}}_{\alpha\alpha'} Q^{\vphantom{\dagger}}_{\gamma\gamma'}
J^{\vphantom{\dagger}}_{\alpha'\beta'\gamma'\delta'}
Q^\dagger_{\beta'\beta} Q^\dagger_{\delta'\delta}, \\
&& Q(x)=
\pmatrix{
\hphantom{-} {\rm e}^{i\pi fx/L}\,\cos\frac{\eta}{2}
& {\rm e}^{-i\pi fx/L} \, \sin\frac{\eta}{2} \cr
- {\rm e}^{i\pi fx/L} \, \sin\frac{\eta}{2}
& {\rm e}^{-i\pi fx/L} \, \cos\frac{\eta}{2} }.
\end{eqnarray}
\end{mathletters}
The transformed moments obey
\begin{mathletters}
\label{lodiff}
\begin{eqnarray}
\label{lcoopdiffa}
\ell\frac{d\tilde{N}_{\alpha\beta\gamma\delta}}{dx} &=&
\sum_{\alpha',\gamma'}
\left(T_{\alpha\alpha'}\delta_{\gamma\gamma'}+\delta_{\alpha\alpha'}T_{\gamma\gamma'}\right)
\tilde{N}_{\alpha'\beta\gamma'\delta}
+\sum_{\alpha',\gamma'}
\left(S_{\alpha\alpha'}\delta_{\gamma\gamma'}-\delta_{\alpha\alpha'}S_{\gamma\gamma'}\right)
\tilde{J}_{\alpha'\beta\gamma'\delta} \nonumber \\
&& -\left(1+\tau/\tau_\varphi\right) \tilde{J}_{\alpha\beta\gamma\delta}, \\
\label{lcoopdiffb}
\ell\frac{d\tilde{J}_{\alpha\beta\gamma\delta}}{dx} &=& 2\sum_{\alpha',\gamma'}
\left(S_{\alpha\alpha'}\delta_{\gamma\gamma'}-
\delta_{\alpha\alpha'}S_{\gamma\gamma'}\right) \tilde{N}_{\alpha'\beta\gamma'\delta}
+\sum_{\alpha',\gamma'}
\left(T_{\alpha\alpha'}\delta_{\gamma\gamma'}+\delta_{\alpha\alpha'}T_{\gamma\gamma'}\right)
\tilde{J}_{\alpha'\beta\gamma'\delta} \nonumber \\
&&
-(2\tau/\tau_\varphi) \tilde{N}_{\alpha\beta\gamma\delta}
+ 2\tau \delta_{\alpha\beta} \delta_{\gamma\delta} \delta (x-x_{\rm i}),
\end{eqnarray}
\end{mathletters}
with the same $2\times 2$ matrices $S$ and $T$ as in Sec.\ \ref{transmission}.
Because the transformation from $N$ to $\tilde{N}$ is unitary, the weak-localization
correction is still given by
$\Delta G=-(e^2D/\pi\hbar L)\sum_{\alpha,\beta}
\tilde{N}_{\alpha\beta\beta\alpha}(x_{\rm i})$, as in Eq.\ (\ref{wl2}).
\begin{figure}[ht]
\epsfxsize=0.7\hsize
\hspace*{\fill}
\vspace*{-0ex}\epsffile{fig3.eps}\vspace*{0ex}
\hspace*{\fill}
\medskip
\caption[]{Weak-localization correction $\Delta G$ of a ring in a spatially rotating
magnetic field, as a function of the tilt angle $\eta$.
Plotted is the result of Eq.\ (\ref{lodiff}) for $f=5$, $L=500\,\ell$,
$L_\varphi=125\,\ell$.
The upper panel is for $\omega_{\rm B}\tau\ll 1$. From top to bottom:
$\omega_{\rm B}\tau = 10^{-5}$, $10^{-4}$, $2\cdot 10^{-4}$,
$3\cdot 10^{-4}$, $5 \cdot 10^{-4}$, $10^{-3}$, $10^{-2}$.
At $\omega_{\rm B}\tau\simeq (f\ell/L)^2$, the weak-localization correction crosses
over from the transient regime A
of Eq.\ (\ref{zerofield}) to the randomized regime B of Eq.\ (\ref{exactsol}).
The lower panel is for $\omega_{\rm B}\tau\gtrsim 1$. From bottom to top:
$\omega_{\rm B}\tau =0.1$, $1$, $2$, $5$, $10$, $100$.
Here the weak-localization correction reaches the adiabatic regime C
of Eq.\ (\ref{adiabatic}).
}
\label{fig3}
\end{figure}
We have solved Eq.\ (\ref{lodiff}) with periodic boundary conditions by numerically computing
the eigenvalues and (right) eigenvectors of the $8\times 8$ matrix of coefficients.
The resulting $\Delta G$ is plotted in Fig.\ \ref{fig3} as a function of the tilt angle
$\eta$.
In the adiabatic regime $\omega_{\rm B}\tau\gg f$ we find the conductance oscillations
due to the Berry phase. These are given by \cite{lsg}
\begin{eqnarray}
\label{adiabatic}
&&\Delta G = -\frac{e^2}{\pi\hbar} \frac{L_\varphi}{L}
\frac{\sinh (L/L_\varphi)}{\cosh (L/L_\varphi)-\cos\left(2\pi f\cos\eta\right)}
\end{eqnarray}
analogously to the Aharonov-Bohm oscillations.\cite{ab}
(The length $L_\varphi =\sqrt{D\tau_\varphi}$ is the phase-coherence length.)
In the randomized regime $\left(f\ell/L\right)^2 \ll \omega_{\rm B}\tau \ll f$
there are no conductance oscillations. Instead we find a reduction of the
weak-localization correction, due to dephasing by spin scattering.
In the transient regime $\omega_{\rm B}\tau \ll \left(f\ell/L\right)^2$ the effect of
the field on the spin can be ignored,\cite{noot} and the weak-localization correction
remains at its zero-field value
\begin{eqnarray}
\label{zerofield}
&&\Delta G = -\frac{e^2}{\pi\hbar}\frac{L_\varphi}{L}\,{\rm cotanh}\left(\frac{L}{2L_\varphi}\right).
\end{eqnarray}
\subsection{Comparison with Loss, Schoeller, and Goldbart}
\label{sollsg}
If we replace the Boltzmann operator $\cal B$ in Eq.\ (\ref{kinetic}) by the diffusion operator
$-D\partial^2/\partial x^2$ and integrate over $\phi$ and $\phi_{\rm i}$,
we end up with the diffusion equation studied by LSG,
\begin{mathletters}
\label{lsgkinetic}
\begin{eqnarray}
&& \left(\frac{\partial}{\partial t}-{\cal H}\right)
\chi_{\alpha\beta\gamma\delta} (x,x_{\rm i};t) = \delta (t) \delta (x-x_{\rm i})
\delta_{\alpha\beta}\delta_{\gamma\delta}, \\
&& {\cal H} = D \frac{\partial^2}{\partial x^2}
+\frac{ig\mu_{\rm B}}{2\hbar}
\left[{\bf B}(x)\cdot \mbox{\boldmath$\sigma$}_1-
{\bf B}(x)\cdot \mbox{\boldmath$\sigma$}_2 \right], \\
&& \chi_{\alpha\beta\gamma\delta} (x,x_{\rm i};t) = \frac{1}{2\pi} \int_0^{2\pi}\! d\phi
\int_0^{2\pi}\! d\phi_{\rm i}\,
\chi_{\alpha\beta\gamma\delta} (x,x_{\rm i};\phi,\phi_{\rm i};t).
\end{eqnarray}
\end{mathletters}
Here $\mbox{\boldmath$\sigma$}_1$ and $\mbox{\boldmath$\sigma$}_2$ act, respectively, on
the first and third indices of $\chi_{\alpha\beta\gamma\delta}$.
The difference between the diffusion equation (\ref{lsgkinetic}) and the diffusion
equation (\ref{odiff}) is that (\ref{lsgkinetic}) holds only if $\omega_{\rm B}\tau\ll 1$,
while (\ref{odiff}) holds for any value of $\omega_{\rm B}\tau$.
LSG used Eq.\ (\ref{lsgkinetic}) to argue that there exists an adiabatic region
within the regime $\omega_{\rm B}\tau \ll 1$. In contrast, our analysis of
Eq.\ (\ref{odiff}) shows that adiabaticity is not possible if $\omega_{\rm B}\tau\ll 1$.
The argument of LSG is based on a mapping of the diffusion equation
(\ref{lsgkinetic}) onto the Schr{\"o}dinger equation studied in Ref.\ \onlinecite{lg}.
However, the mapping is not
carried out explicitly. In this subsection we will solve Eq.\ (\ref{lsgkinetic})
exactly using this mapping, to demonstrate that the adiabatic regime of LSG is in fact
the randomized regime B. This mis-identification perhaps occurred because both
regimes are stationary with respect to the magnetic field strength
(cf.\ Fig.\ \ref{fig2}). However, Berry-phase oscillations of the conductance are
only supported in the adiabatic regime C, not in the randomized regime B
(cf.\ Fig.\ \ref{fig3}).
We solve Eq.\ (\ref{lsgkinetic}) for the weak-localization correction
\begin{eqnarray}
\label{wl3}
\Delta G &=& -\frac{e^2D}{\pi\hbar L} \sum_{\alpha,\beta}
\left\langle x,\alpha,\beta\left| \left(\tau_\varphi^{-1}-{\cal H}\right)^{-1}
\right|x,\beta,\alpha\right\rangle,
\end{eqnarray}
where we introduced the basis of eigenstates $|x,\alpha,\beta\rangle$
(with $\alpha,\beta=\pm 1$)
of the position operator $x$ and the spin operators $\sigma_{1z}$ and $\sigma_{2z}$.
The operator ${\cal H}$ commutes with
\begin{equation}
J=\frac{L}{2\pi i}\frac{\partial}{\partial x}+\case{1}{2}f\left(\sigma_{1z}+\sigma_{2z}\right).
\end{equation}
It is therefore convenient to use as a basis, instead of the eigenstates
$|x,\alpha,\beta\rangle$, the eigenstates $|j,\alpha,\beta\rangle$ of $J$,
$\sigma_{1z}$, and $\sigma_{2z}$. The eigenvalue $j$ of $J$ is an integer because of the
periodic boundary conditions. The eigenfunctions are given by
\begin{equation}
\left\langle x,\alpha',\beta'| j,\alpha,\beta\right\rangle = \frac{1}{\sqrt{L}}
\delta_{\alpha'\alpha} \delta_{\beta'\beta}
\exp\left[\case{2\pi i x}{L} (j-\case{1}{2}f\alpha-\case{1}{2}f\beta)\right] .
\end{equation}
In the basis $\{|j,1,1\rangle,|j,1,-1\rangle, |j,-1,1\rangle, |j,-1,-1\rangle \}$
the operator $\cal H$ has matrix elements
\begin{eqnarray}
\label{diagham}
\langle j',\alpha',\beta' | {\cal H} | j,\alpha,\beta\rangle &=&
-D \left(\frac{2\pi}{L}\right)^2 \delta_{j'j}
\pmatrix{
(j-f)^2 & 0 & 0 & 0 \cr
0 & j^2 & 0 & 0 \cr
0 & 0 & j^2 & 0 \cr
0 & 0 & 0 & (j+f)^2 \cr
} \nonumber \\
&& -i\omega_{\rm B} \delta_{j'j}
\pmatrix{ 0 & \hphantom{-}\sin\eta & -\sin\eta & 0 \cr
\hphantom{-}\sin\eta & -2\cos\eta & 0 & -\sin\eta \cr
-\sin\eta & 0 & 2\cos\eta & \hphantom{-}\sin\eta \cr
0 & -\sin\eta & \hphantom{-}\sin\eta & 0 \cr
}.
\end{eqnarray}
Substitution into Eq.\ (\ref{wl3}) yields
\begin{eqnarray}
\label{solwl}
\Delta G &=& -\frac{e^2D}{\pi\hbar}\frac{1}{L^2}\sum_{\alpha,\beta}
\sum_{j=-\infty}^\infty
\left\langle j,\alpha,\beta \left| \left(\tau_\varphi^{-1} -{\cal H}\right)^{-1}
\right|j,\beta,\alpha \right\rangle \nonumber \\
&=& -\frac{e^2}{\pi\hbar}\frac{1}{2\pi^2}\sum_{j=-\infty}^\infty
\left[(\gamma + j^2)^2 (f^2 + \gamma + j^2) +
\kappa^2 (3 f^2 + 4 \gamma + 4 j^2 + f^2 \cos 2 \eta ) \right] \nonumber \\
&& \mbox{}\times\left[(\gamma +j^2)^2 (f^4+2 f^2\gamma +\gamma^2-2 f^2j^2+2\gamma j^2 +
j^4) \right. \nonumber \\
&& \mbox{}+\left. 2 \kappa^2 \biglb( f^4 + 3 f^2 \gamma + 2 \gamma^2 - f^2 j^2 + 4 \gamma j^2 +
2 j^4 + f^2 (f^2 + \gamma - 3 j^2 ) \cos 2\eta \bigrb) \right]^{-1}.
\end{eqnarray}
We abbreviated $\kappa=2\omega_{\rm B}\tau (L/2\pi\ell)^2$ and $\gamma=(L/2\pi L_\varphi)^2$.
The sum over $j$ can be done analytically for $\kappa\gg 1$, with the result
\begin{mathletters}
\label{exactsol}
\begin{eqnarray}
&&\Delta G = -\frac{e^2}{\pi\hbar} \frac{1}{4\pi Q}
\left[
\frac{4a_-+4\gamma+(3+\cos 2\eta )f^2}{\sqrt{a_-}\tan\pi \sqrt{a_-}} -
\frac{4a_++4\gamma+(3+\cos 2\eta )f^2}{\sqrt{a_+}\tan\pi \sqrt{a_+}}
\right], \\
&& Q=\left[f^4 (9\cos^2 2\eta -2\cos 2\eta-7)-32\gamma f^2 (1+\cos 2\eta)\right]^{1/2}, \\
&&a_\pm = -\gamma + \case{1}{4}(1+3\cos 2 \eta ) f^2 \pm\case{1}{4}Q.
\end{eqnarray}
\end{mathletters}
We have checked that our solution (\ref{solwl}) of
Eq.\ (\ref{lsgkinetic}) coincides with the solution of Eq.\ (\ref{odiff})
in the regime $\omega_{\rm B}\tau\ll 1$.
(The two sets of curves are indistinguishable on the scale of Fig.\ \ref{fig3}.)
In particular, Eq.\ (\ref{exactsol}) coincides with the curves labeled B in
Fig.\ \ref{fig3}, demonstrating that it represents the randomized regime
-- without Berry phase-oscillations.
\section{Conclusions}
In conclusion, we have computed the effect of a non-uniform magnetic field on the
spin polarization (Sec.\ \ref{transmission}) and weak-localization correction
(Sec.\ \ref{localization}) in a disordered conductor. We have identified
three regimes of magnetic field strength: the transient regime
$\omega_{\rm B}\tau\ll (f\ell/L)^2$, the randomized regime
$(f\ell/L)^2 \ll \omega_{\rm B}\tau \ll f$, and the adiabatic regime
$\omega_{\rm B}\tau\gg f$. In the transient regime (labeled A in
Figs.\ \ref{fig2} and \ref{fig3}), the effect of the magnetic field can be
neglected. In the randomized regime (labeled B), the depolarization and the
suppression of the weak-localization correction are maximal. In the adiabatic
regime (labeled C), the polarization is restored and the weak-localization
correction exhibits oscillations due to the Berry phase.
The criterion for adiabaticity is $\omega_{\rm B}t_{\rm c}\gg 1$, with $\omega_{\rm B}$
the spin-precession frequency and $t_{\rm c}$ a characteristic timescale of the orbital
motion. We find $t_{\rm c} =\tau$, in agreement with Stern,\cite{stern} but in
contradiction with the result $t_{\rm c}=\tau (L/\ell)^2$ of Loss, Schoeller,
and Goldbart. \cite{lsg} By solving exactly the diffusion equation for the Cooperon
derived in Ref.\ \onlinecite{lsg}, we have demonstrated unambiguously that the regime
which in that paper was identified as the adiabatic regime, is in fact the randomized
regime B --- without Berry-phase oscillations.
We have focused on transport properties, such as conductance and spin-resolved transmission.
Thermodynamic properties, such as the persistent current, in a non-uniform magnetic field
have been studied by Loss, Goldbart, and Balatsky \cite{lg,lgb} in connection with
Berry-phase oscillations. These papers assumed ballistic systems. We believe that the
adiabaticity criterion
$\omega_{\rm B}\tau\gg 1$ for disordered systems should apply to thermodynamic properties
as well as transport properties. This strong-field criterion presents
a pessimistic outlook for the prospect of experiments on the Berry phase in disordered
systems.
\acknowledgements
We are indebted to L. P. Kouwenhoven for bringing this problem to our
attention, and to P. W. Brouwer, D. Loss, and A. Stern for useful discussions.
This research was supported by the ``Ne\-der\-land\-se or\-ga\-ni\-sa\-tie voor
We\-ten\-schap\-pe\-lijk On\-der\-zoek'' (NWO) and by the ``Stich\-ting voor
Fun\-da\-men\-teel On\-der\-zoek der Ma\-te\-rie'' (FOM).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,363 |
using System.Reflection;
[assembly: AssemblyTitle("Owin.Dependencies.Adapters.WebApi")]
[assembly: AssemblyDescription("ASP.NET Web API adapter for OWIN IoC container adapter")] | {
"redpajama_set_name": "RedPajamaGithub"
} | 1,576 |
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Ever wonder just how far your dollar will go on Airbnb? This is Budget vs. Baller: A series that shows you the budget, mid-range, and baller rental options available presently in some of the world's trendiest cities. Today we're taking it to the City of Angels.
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{"url":"https:\/\/developpaper.com\/android-11-balance-process-and-principle\/","text":"Android 11 balance process and principle\n\nTime\uff1a2021-6-26\n\nIt is mentioned in a document of Qualcomm that Android 10 audio has introduced a balance function (for reasons of confidentiality, the specific document name should not be posted), and the content of the document is simple, so the setting interface and dumpsys view value are proposed,\n\nLet\u2019s study how Android implements this thing and how it works.\n\n<!\u2013 more \u2013>\n\n[Platform:Android 11]\nhttp:\/\/aosp.opersys.com\/xref\/\u2026\n\nBalance is actually used to set the left and right balance. Now there are more stereo speakers on mobile phones. The effect of intuitive point is to set the volume of left and right speakers.\n\nIn addition, the function of volume balance is also required in the car. Combined with fade, the sound field effect can be achieved.\nFor this reason, Google introduced audiocontrol to set the left and right balance through setbalancetowardright() setfadetowardfront() interfaces, so as to achieve the effect of setting the sound field.\nHowever, these two interfaces need to be implemented by the chip manufacturer in the Hal layer. In other words, the chip manufacturer may or may not have implemented them. For example, the function is not implemented in the Hal layer of Qualcomm 8155.\n\n1. Setting interface\n\n< center > Figure 1. Left right balance setting interface\n\nIn the above interface, drag the bar to the far left to turn the sound to the left completely; Similarly, drag the bar to the far right, and the sound is fully tuned to the right.\nThe current value of the drag bar above is [0, 200], and then it will be mapped to [- 1.0F, 1.0F] and saved to the database,\nFrom the point of view of the code, it is also a little considerate, that is, it is set to the middle value when the center is + \/ \u2013 6.\n\nDrag bar key code:\n\npackages\/apps\/Settings\/src\/com\/android\/settings\/accessibility\/BalanceSeekBar.java\npublic void onProgressChanged(SeekBar seekBar, int progress, boolean fromUser) {\nif (fromUser) {\n\/\/ Snap to centre when within the specified threshold\n\/\/Msnapthreshold is currently 6, that is, it is set to the middle when the middle position is + \/ - 6\nif (progress != mCenter\n&& progress > mCenter - mSnapThreshold\n&& progress < mCenter + mSnapThreshold) {\nprogress = mCenter;\nseekBar.setProgress(progress); \/\/ direct update (fromUser becomes false)\n}\n\/\/Map 0 ~ 200 to - 1.0F ~ 1.0F\nfinal float balance = (progress - mCenter) * 0.01f;\n\/\/Finally, it is set in the database\nSettings.System.putFloatForUser(mContext.getContentResolver(),\nSettings.System.MASTER_BALANCE, balance, UserHandle.USER_CURRENT);\n}\n\nWe can also adjust the value directly from the command line\n\n# MASTER_ Balance definition\n# frameworks\/base\/core\/java\/android\/provider\/Settings.java\npublic static final String MASTER_BALANCE = \"master_balance\";\n\n#Command line setting master balance\nadb shell settings put system master_ Balance value\n#Command line to get master balance\nadb shell settings get system master_balance\n\nSo who is receiving this value?\n\n2. setMasterBalance()\n\nThrough the analysis of master_ Balance search finds that in the audioservice constructor, a settingsobserver object will be created. This class is specially used for audioservice to listen to the settings database. When master_ When the balance value changes, call updatemasterbalance() >? Audiosystem. Setmasterbalance() to update,\nIn other words, in fact, audioserver is further set down through audiosystem.\n\nframeworks\/base\/services\/core\/java\/com\/android\/server\/audio\/AudioService.java\n...\n\/\/Audioservice creates settingsobserver object\nmSettingsObserver = new SettingsObserver();\n\nprivate class SettingsObserver extends ContentObserver {\nSettingsObserver() {\n...\n\/\/On the function of master in the settingsobserver constructor_ Barrance monitoring\nmContentResolver.registerContentObserver(Settings.System.getUriFor(\nSettings.System.MASTER_BALANCE), false, this);\n...\n}\n\n@Override\npublic void onChange(boolean selfChange) {\n...\n\/\/When the monitored data changes, call this function to update master balance\n\/\/It should be noted that when booting and the audioserver is dead, the function will also be called to set the balance value to audioflinger\nupdateMasterBalance(mContentResolver);\n...\n}\n\nprivate void updateMasterBalance(ContentResolver cr) {\n\/\/Get value\nfinal float masterBalance = System.getFloatForUser(\ncr, System.MASTER_BALANCE, 0.f \/* default *\/, UserHandle.USER_CURRENT);\n...\n\/\/Set it through audiosystem\nif (AudioSystem.setMasterBalance(masterBalance) != 0) {\nLog.e(TAG, String.format(\"setMasterBalance failed for %f\", masterBalance));\n}\n}\n\nAudiosystem will eventually be set in audioflinger. The process in the middle is relatively simple. It\u2019s just a few bind calls around. If you\u2019re not familiar with it, just look at the process of my column.\n\nframeworks\/base\/media\/java\/android\/media\/AudioSystem.java\nsetMasterBalance()\n+ --> JNI\n+ android_media_AudioSystem_setMasterBalance() \/ android_media_AudioSystem.cpp\n+ AudioSystem::setMasterBalance(balance)\n+ setMasterBalance() \/ AudioSystem.cpp\n+ const sp<IAudioFlinger>& af = AudioSystem::get_audio_flinger();\n+Af - > setmasterbalance (balance) \/\/ call audioflinger's setmasterbalance\n+ setMasterBalance() \/ AudioFlinger.cpp\n+ mMasterBalance.store(balance);\n\nIn audioflinger, you will first check the permissions, the validity of parameters, and whether the settings are the same as before, and finally set them to the playback thread through the for loop,\nIt should be noted that the duplicating thread is skipped, that is to sayMaster balance is not valid for duplicating playback mode\n\nTips:\nDuplicating is used for duplicating and playing ringtones simultaneously with Bluetooth and loudspeaker.\nframeworks\/av\/services\/audioflinger\/AudioFlinger.cpp\nstatus_t AudioFlinger::setMasterBalance(float balance)\n{\n... \/\/ permission check\n\/\/ check calling permissions\nif (!settingsAllowed()) {\n... \/\/ parameter validity check\n\/\/ check range\nif (isnan(balance) || fabs(balance) > 1.f) {\n... \/\/ is it the same as the previous value\n\/\/ short cut.\nif (mMasterBalance == balance) return NO_ERROR;\n\nmMasterBalance = balance;\n\nfor (size_t i = 0; i < mPlaybackThreads.size(); i++) {\n\/\/If it is duplicating, it will not be processed\ncontinue;\n}\n}\n\nreturn NO_ERROR;\n}\n\nAs people familiar with audio know, Android divides playback thread into fast thread, mixer thread, direct thread and other threads to achieve fast, mixing, direct offload playback and other purposes. Therefore, the setmasterbalance() and subsequent balance processing of each playback thread may be different. Here we take a typical mixer thread as an example for analysis, The rest of the way if useful to their own look at the code.\n\nSave the value in playbackthread, and it\u2019s over\n\nframeworks\/av\/services\/audioflinger\/Threads.cpp\n{\nmMasterBalance.store(balance);\n}\n\nIn threads, mmasterbalance is defined as atomic type\nstd::atomic<float> mMasterBalance{};\n\nMmasterbalance is an atomic type, and its store \/ read method is store () \/ load (). Setmasterbalance () finally stores the balance value with store (). If you want to continue to see the balance process, you have to find out where to use the value.\n\n3. Balance principle\n\nThere are several places to use mmasterbalance. We also use playbackthread for analysis. If you need direct method, you can see for yourself.\n\nPlaybackthread\u2019s threadloop() is a main function of audio processing, and the code is also very long. The main work is event processing, preparation of audio track, mixing, sound chain processing, and the left-right balance processing we want to talk about. Finally, we write the data to Hal. Other processes can be studied if you are interested. This paper mainly focuses on balance processing.\n\nbool AudioFlinger::PlaybackThread::threadLoop()\n{... \/\/ loop processing until the thread needs to exit\nfor (int64_t loopCount = 0; !exitPending(); ++loopCount)\n{... \/\/ event handling\nprocessConfigEvents_l();\n... \/\/ prepare the track\nmMixerStatus = prepareTracks_l(&tracksToRemove);\n... \/\/ mixing\n... \/\/ sound chain processing\neffectChains[i]->process_l();\n... \/\/ balance left and right\nif (!hasFastMixer()) {\n\/\/ Balance must take effect after mono conversion.\n\/\/ We do it here if there is no FastMixer.\n\/\/ mBalance detects zero balance within the class for speed (not needed here).\n\/\/Read the balance value and assign it to audio through the setbalance() method_ utils::Balance\n\/\/Balance buffer\nmBalance.process((float *)mEffectBuffer, mNormalFrameCount);\n}\n... \/\/ write the processed data to Hal\n...\n}\n...\n}\n\nMbalance definition\naudio_utils::Balance mBalance;\n\nAs can be seen from the above code, if there is a fast mixer in the thread, it will not be balanced, and then a new class audio is introduced_ Utils:: balance is specially used for balance processing. The relevant method is setbalance() process(). Intuitively, we can understand the principle by looking at the process() function. Let\u2019s look at the function first.\n\nsystem\/media\/audio_utils\/Balance.cpp\nvoid Balance::process(float *buffer, size_t frames)\n{\n\/\/Values in the middle and mono are not processed\nif (mBalance == 0.f || mChannelCount < 2) {\nreturn;\n}\n\nif (mRamp) {\n... \/\/ ramp processing\n\/\/ ramped balance\nfor (size_t i = 0; i < frames; ++i) {\nconst float findex = i;\nfor (size_t j = 0; j < mChannelCount; ++j) { \/\/ better precision: delta * i\n\/\/After changing the balance, the first call to process will carry out ramp processing\n*buffer++ *= mRampVolumes[j] + mDeltas[j] * findex;\n}\n}\n...\n}\n\/\/Non ramp processing\n\/\/ non-ramped balance\nfor (size_t i = 0; i < frames; ++i) {\nfor (size_t j = 0; j < mChannelCount; ++j) {\n\/\/Multiply each channel of the incoming buffer by a certain coefficient\n*buffer++ *= mVolumes[j];\n}\n}\n}\n\nProcess () does not deal with balance in the middle or mono channel, and then it is divided into ramp and non ramp modes. These two modes multiply each channel of the incoming buffer by a certain coefficient. We are mainly concerned with non ramp mode*buffer++ *= mVolumes[j];Next, let\u2019s look at its mvolumes [J], that is, what are the left and right channel coefficients?\n\nTo find out the value of mvolumes, you need to look back at its setbalance() method,\n\nsystem\/media\/audio_utils\/Balance.cpp\nvoid Balance::setBalance(float balance)\n{... \/\/ validity check, code skipping\n\/\/Single channel without processing\nif (mChannelCount < 2) { \/\/ if channel count is 1, mVolumes[0] is already set to 1.f\nreturn; \/\/ and if channel count < 2, we don't do anything in process().\n}\n\n\/\/Common dual channel processing\n\/\/ Handle the common cases:\n\/\/ stereo and channel index masks only affect the first two channels as left and right.\n== AUDIO_CHANNEL_REPRESENTATION_INDEX) {\n\/\/Calculate the balance coefficient of left and right channels\ncomputeStereoBalance(balance, &mVolumes[0], &mVolumes[1]);\nreturn;\n}\n\/\/Processing of more than 2 channels\n\/\/ For position masks with more than 2 channels, we consider which side the\n\/\/ speaker position is on to figure the volume used.\nfloat balanceVolumes[3]; \/\/ left, right, center\n\/\/Calculate the balance coefficient of left and right channels\ncomputeStereoBalance(balance, &balanceVolumes[0], &balanceVolumes[1]);\n\/\/Intermediate fixation\nbalanceVolumes[2] = 1.f; \/\/ center TODO: consider center scaling.\n\nfor (size_t i = 0; i < mVolumes.size(); ++i) {\nmVolumes[i] = balanceVolumes[mSides[i]];\n}\n}\n\nIn setbalance(), mono, dual and multi-channel are processed. The mono coefficient is fixed at 1. F; Both dual and multichannel will be calledcomputeStereoBalance()The left and right balance coefficients are calculated; Multi channel should not be done well at present, among which the fixed value is 1. F.\n\nFinally came to the key about the channel coefficient calculation function!\n\nvoid Balance::computeStereoBalance(float balance, float *left, float *right) const\n{\nif (balance > 0.f) {\n\/\/Balance to the right\n*left = mCurve(1.f - balance);\n*right = 1.f;\n} else if (balance < 0.f) {\n\/\/Balance to the left\n*left = 1.f;\n*right = mCurve(1.f + balance);\n} else {\n\/\/Balance in the middle\n*left = 1.f;\n*right = 1.f;\n}\n\n\/\/ Functionally:\n\/\/ *left = balance > 0.f ? mCurve(1.f - balance) : 1.f;\n\/\/ *right = balance < 0.f ? mCurve(1.f + balance) : 1.f;\n}\n\nWhen counting coefficient:\nBalance to the right, the right channel is fixed at 1. F, and the left channel is mcurve (1. F \u2013 balance);\nBalance to the left, the left channel is fixed at 1. F, and the right channel is mcurve (1. F + balance);\nin other words,\nWhich side of the balance is going, which side of the volume is fixed at 1. F, and the other side is multiplied by the coefficient mcurve (1. F \u2013 | balance |) (balance \u2208 [- 1.0, 1.0])\n\nNow let\u2019s move on to the mcurve curve,\n\nsystem\/media\/audio_utils\/include\/audio_utils\/Balance.h\nclass Balance {\npublic:\n\/**\n* \\brief Balance processing of left-right volume on audio data.\n*\n* Allows processing of audio data with a single balance parameter from [-1, 1].\n* For efficiency, the class caches balance and channel mask data between calls;\n* hence, use by multiple threads will require caller locking.\n*\n* \\param ramp whether to ramp volume or not.\n* \\param curve a monotonic increasing function f: [0, 1] -> [a, b]\n* which represents the volume steps from an input domain of [0, 1] to\n* an output range [a, b] (ostensibly also from 0 to 1).\n* If [a, b] is not [0, 1], it is normalized to [0, 1].\n* Curve is typically a convex function, some possible examples:\n* [](float x) { return expf(2.f * x); }\n* or\n* [](float x) { return x * (x + 0.2f); }\n*\/\nexplicit Balance(\nbool ramp = true,\nstd::function<float(float)> curve = [](float x) { return x * (x + 0.2f); }) \/\/ Curve function\n: mRamp(ramp)\n, mcurve (normalize (STD:: move (curve))) {} \/\/ mcurve is normalized\n\n\/\/Mcurve definition\nconst std::function<float(float)> mCurve; \/\/ monotone volume transfer func [0, 1] -> [0, 1]\n\nIn fact, the function annotation is very clear, and I also posted the annotation part. Mcurve is a function, and it has been normalized, so that its interval and value fall on [0, 1]. This function is a monotonic increasing function, which is currently usedx * (x + 0.2f)Of course, you can also use other functions.\n\nNormalize is a template, and its comments are very clear,\n\n\/**\n* \\brief Normalizes f: [0, 1] -> [a, b] to g: [0, 1] -> [0, 1].\n*\n* A helper function to normalize a float volume function.\n* g(0) is exactly zero, but g(1) may not necessarily be 1 since we\n* use reciprocal multiplication instead of division to scale.\n*\n* \\param f a function from [0, 1] -> [a, b]\n* \\return g a function from [0, 1] -> [0, 1] as a linear function of f.\n*\/\ntemplate<typename T>\nstatic std::function<T(T)> normalize(std::function<T(T)> f) {\nconst T f0 = f(0);\nconst T r = T(1) \/ (f(1) - f0); \/\/ reciprocal multiplication\n\nif (f0 != T(0) || \/\/ must be exactly 0 at 0, since we promise g(0) == 0\nfabs(r - T(1)) > std::numeric_limits<T>::epsilon() * 3) { \/\/ some fudge allowed on r.\n\/\/We use the function x * (x + 0.2f), Fabs (R - t (1)) >... As true, which will come here\nreturn [f, f0, r](T x) { return r * (f(x) - f0); };\n}\n\/\/ no translation required.\nreturn f;\n}\n\nThe function we use satisfiesfabs(r - T(1)) > std::numeric_limits<T>::epsilon() * 3Condition, so it will also do normalization, that is, using ther * (f(x) - f0)In combination, the mcurve curve is mathematically described as\n\n$$f(x) = x^2 + 0.2 \\times x; \\\\ mCurve(x) = {\\frac{1.0}{f(1)-f(0)}} \\times {(f(x)-f(0))} = {\\frac{1.0}{1.2} \\times f(x)}$$\n\nThat is to say\n\n$$\\mathbf{mCurve(x) = {\\frac{(x^2 + 0.2x)}{1.2}}, x\\in[0.0, 1.0], y\\in[0.0, 1.0]}$$\n\n1.2 is the normalization coefficient\n\n$mcurve (1. F \u2013 | balance |), balance in [- 1.0, 1.0]$can be represented as follows:\n\n< center > Figure 2. Balance curve\n\nIf there is a problem with the display of the figure, also use online matlab to view it, and open the followingwebsite, and then enter the following\n\nhttps:\/\/octave-online.net\/\n\nx = [-1 : 0.1: 1];\nz = 1 - abs(x)\ny = (z.^2 + 0.2 * z)\/1.2;\n\nplot(x, y, 'r')\nxlabel('balance')\nylabel('Y')\ntitle('Balance Curve')\n\nSo far, the principle of regulating left-right balance is clear.\n\n4. Commissioning\n\nIn addition to the above mentioned, use the command lineadb shell settings put system master_balanceIn addition to changing its value, we can also dump it to see if it works\n\n$adb shell dumpsys media.audio_flinger \/\/A thread of type mixer Output thread 0x7c19757740, name AudioOut_D, tid 1718, type 0 (MIXER): ... Thread throttle time (msecs): 6646 AudioMixer tracks: Master mono: off \/\/Balance value Master balance: 0.500000 (balance 0.5 channelCount 2 volumes: 0.291667 1) \/\/A thread of type unload (direct) Output thread 0x7c184b3000, name AudioOut_20D, tid 10903, type 4 (OFFLOAD): ... Suspended frames: 0 Hal stream dump: \/\/Balance value Master balance: 0.500000 Left: 0.291667 Right: 1.000000 5. Summary 1. The UI setting interface is just a data storage process, and its value is converted to [- 1.0, 1.0] and stored in the database. After the Java layer audio service listens to the change of the value, it is finally stored in the audioflinger non copy playback thread through the setmasterbalance() interface; 2. For the playback thread without fast mixer, it will balance in threadloop(); 3. The principle of balance processing is also very simple, which side of balance and which channel remain unchanged. Multiply the other channel by a coefficient (pitch down, mcurve (1 \u2013 | balance |). For non ramp mode, the coefficient is a quadratic monotone function and normalized to [0,1]. At present, it is$mcurve (x) = x * (x + 0.2) \/ 1.2 \\$.\n\nIntroduce regular expressions in ruby in detail\n\nA regular expression is a special sequence of characters that matches or finds other strings or collections of strings by using patterns with special syntax. grammar A regular expression is literally a pattern between slashes or any separator after% R, as follows: ? 1 2 3 4 5 6 7 8 9 10 11 12 [\u2026]","date":"2021-10-27 14:09:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.29945454001426697, \"perplexity\": 6081.184608985488}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323588153.7\/warc\/CC-MAIN-20211027115745-20211027145745-00106.warc.gz\"}"} | null | null |
Platte County Courthouse ist der Name folgender im NRHP gelisteten Objekte:
Platte County Courthouse (Missouri), ID-Nr. 79001390
Platte County Courthouse (Nebraska), ID-Nr. 89002217
Platte County Courthouse (Wyoming), ID-Nr. 08001004 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,902 |
\section{Introduction}\label{sec:intro}
\cite{Stokes1847} \citep[see also][]{Stokes1880} made many contributions about periodic waves at the surface of an incompressible inviscid fluid in two dimensions, subject to the force of gravity, traveling a long distance at a practically constant velocity without change of form. For instance, he observed that crests tend to sharpen and troughs flatter as the amplitude increases, and conjectured that the wave of greatest height exhibits a $120^\circ$ corner at the crest. \cite{AFT1982} proved that a limiting wave exists, whose angle at the crest is $120^\circ$.
In an irrotational flow of infinite depth, Stokes waves are much studied analytically and numerically.
Some recent advances are based on the formulation of the problem as a nonlinear pseudo-differential equation, involving the periodic Hilbert transform --- namely, the Babenko equation. For instance, \cite{DLK2016, Lushnikov2016, LDS2017} numerically approximated the wave of greatest height and uncovered the structure of the singularities in meticulous detail.
The zero vorticity assumption may be justified in some situations. Moreover, in the absence of initial vorticity, boundaries or currents, water waves will have zero vorticity at all later times. But rotational effects are significant in many situations. For instance, in any region where wind blows, there is a surface drift of the water, and wave parameters, such as maximum wave height, are sensitive to the velocity at a wind-drift boundary layer. Moreover, currents produce shear at the bed of the sea or a river; see \cite{PTdS1988}, for instance.
For arbitrary vorticity, \cite{CS2004} worked out the global bifurcation of Stokes waves in the finite depth, \cite{Hur2006, Hur2011} in the infinite depth, and \cite{KS1,KS2} numerically computed, assuming that there is no overhanging or internal stagnation. For zero vorticity, a Stokes wave is necessarily the graph of a single valued function and, moreover, the wave speed exceeds the directional particle velocity inside the fluid. But, even for constant vorticity, \cite{SS1985, PTdS1988, RMN2017}, among others, numerically observed overhanging profiles and interior stagnation points.
Constant vorticity is of particular interest because of its analytical tractability.
Moreover, it is representative of a wide range of physical scenarios. When waves are short compared with the vorticity length scale, the vorticity at a surface layer is dominant in the wave dynamics. Moreover, when waves are long compared with the fluid depth, the mean vorticity is more important than its specific distribution; see \cite{PTdS1988}, for instance. Examples include tidal currents --- alternating lateral movements of water associated with the rise and fall of the tide --- where positive or negative constant vorticity suitable for the ebb or flood, respectively; see \cite{CSV2016}, for instance.
Recently, \cite{CSV2016} extended the Babenko equation, to permit constant vorticity and finite depth, and demonstrated the global bifurcation of Stokes waves. Moreover, they conjectured that at the boundary of the solution curve (in a suitable function space), one reaches: either an {\em extreme wave}, which exhibits a sharp corner at the crest and whose profile is single valued or overhanging, or a {\em touching wave}, whose profile self-intersects somewhere along the trough line, trapping an air bubble.
\cite{SS1985} used a boundary integral method and numerically computed Stokes waves in a constant vorticity flow of infinite depth. They found touching waves, among others, which is higher than the extreme wave for some vorticity. Moreover, they detected a {\em fold} in the wave speed versus steepness plane for some vorticity, which implies non-uniqueness. \cite{PTdS1988} extended the result in the finite depth. \cite{VB1996} located a branch of Stokes waves in the infinite depth, which tend to a closed region of fluid in rigid body rotation at the zero gravity limit.
Here we use a fast Fourier transform and the Newton-GMRES method, and numerically solve the extension of the Babenko equation, permitting constant vorticity and finite depth.
The result is in excellent agreement, qualitatively and quantitatively, with those of \cite{SS1985, PTdS1988,VB1996}, among others.
For negative or weak positive vorticity, in the finite or infinite depth, we learn that single valued profiles tend to an extreme wave as the steepness increases, like the well-known result for zero vorticity. But, for strong positive vorticity, we find that overhanging profiles appear as the steepness increases and tend to a touching wave; the numerical solutions become unphysical as the steepness increases further and make a {\em gap} in the wave speed versus steepness plane. By the way, the numerical method in \cite{SS1985}, for instance, diverges in the gap. A touching wave then takes over and the physical solutions follow along a fold until they ultimately tend to an extreme wave, whose profile seems single valued. Moreover, we find that overhanging waves of nearly maximum heights approach rigid body rotation of a fluid disk as the strength of positive vorticity increases.
\section{Formulation}\label{sec:formulation}
The water wave problem, in the simplest form, concerns the wave motion at the surface of an incompressible inviscid fluid in two dimensions, lying below a body of air, and acted on by gravity. We assume for simplicity that the density~$=1$. Suppose for definiteness that in Cartesian coordinates, waves propagate in the $x$ direction and gravity acts in the negative $y$ direction. Suppose that the fluid occupies the region, bounded above by a free surface and below by the rigid bed $y=-h$ for some constant $h$ in the range $(0,\infty]$. Let $y=\eta(x;t)$, $x\in\mathbb{R}$, represent the fluid surface at time $t$. We assume for now that $\eta$ is single valued (but see the discussion following \eqref{def:y(u)}). Let
\[
\Omega(t)=\{(x,y)\in\mathbb{R}^2:-h<y<\eta(x;t)\}\quad\text{and}\quad\Gamma(t)=\{(x,\eta(x;t)):x\in\mathbb{R}\}.
\]
Let $\boldsymbol{u}=\boldsymbol{u}(x,y;t)$ denote the velocity of the fluid at the point $(x,y)$ and time $t$, and $p=p(x,y;t)$ the pressure. They satisfy the Euler equations for an incompressible fluid:
\begin{subequations}\label{E:ww0}
\begin{gather}
\boldsymbol{u}_t+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}=-\nabla p+(0,-g)\label{E:Euler}
\intertext{and}
\nabla\cdot\boldsymbol{u}=0\label{E:incomp}
\end{gather}
in $\Omega(t)$, where $g$ is the constant due to gravitational acceleration. Throughout, we express partial differentiation by a subscript, $\nabla=(\partial_x,\partial_y)$ and $\Delta$ the Laplacian. We assume that the vorticity
\begin{equation}
\omega:=\nabla\times\boldsymbol{u}
\end{equation}
is constant. By the way, if vorticity is constant everywhere in the fluid at the initial time then it remains so at all later times, so long as the fluid region is two dimensional and simply connected.
The kinematic and dynamic conditions at the fluid surface:
\begin{equation}
\eta_t+\boldsymbol{u}\cdot\nabla (\eta-y)=0\quad\text{and}\quad p=p_{atm}\quad\text{at $\Gamma(t)$}
\end{equation}
state, respectively, that the fluid particles do not invade the air, nor vice versa, and that the pressure at the fluid surface equals the constant atmospheric pressure~$=p_{atm}$. Here we assume that the air is quiescent and neglect the effects of surface tension. In the finite depth, where $h<\infty$, the boundary condition at the fluid bed:
\begin{equation}
\boldsymbol{u}\cdot(0,-1)=0\quad\text{at $y=-h$}
\end{equation}
\end{subequations}
states that the fluid particles at the bed remain so at all times. We assume in addition that the solutions of \eqref{E:ww0} are $2L$ periodic in the $x$ variable for some $L$.
For any $\omega\in \mathbb{R}$, $h\in(0,\infty)$ and $c\in\mathbb{R}$, it is straightforward to verify that
\begin{equation}\label{def:shear}
\eta(x;t)=0,\quad\boldsymbol{u}(x,y;t)=(-\omega y-c,0)\quad\text{and}\quad p(x,y;t)=p_{atm}-gy,
\end{equation}
where $x\in\mathbb{R}$ and $y\in(-h,0)$, solve \eqref{E:ww0} at all times. They make a linear shear flow, for which the fluid surface is horizontal, the fluid velocity varies linearly in the $y$ direction, and the pressure is hydrostatic. We assume that some external effects such as wind produce a flow of the kind and restrict the attention to the wave propagation in \eqref{def:shear}.
Suppose that
\begin{equation}\label{def:Phi}
\boldsymbol{u}(x,y;t)=(-\omega y-c,0)+\nabla\Phi(x,y;t)\quad\text{in $\Omega(t)$},
\end{equation}
whence \eqref{E:incomp} implies that
\[
\Delta\Phi=0\quad\text{in $\Omega(t)$}
\]
at all times. Namely, $\Phi$ is a velocity potential for the irrotational perturbation from \eqref{def:shear}. By the way, for arbitrary vorticity, the perturbation from the shear flow --- not necessarily linear --- becomes rotational, whence $\Phi$ is no longer viable to use. Let $\Psi$ be a harmonic conjugate of $\Phi$. Namely, $\Psi$ is a stream function for the irrotational perturbation from \eqref{def:shear}. Clearly,
\begin{equation}\label{def:Psi}
\boldsymbol{u}=(-\omega y-c,0)+\nabla\times\Psi
\end{equation}
and $\Delta\Psi=0$ in $\Omega(t)$ at all times.
We substitute \eqref{def:Phi} and \eqref{def:Psi} into \eqref{E:Euler}, and we make an explicit calculation to arrive at
\[
\Phi_t+\tfrac12(\Phi_x^2+\Phi_y^2)-(\omega y+c)\Phi_x+\omega\Psi+p-p_{atm}+gy=b(t)
\]
for an arbitrary function $b(t)$. We substitute \eqref{def:Phi} and \eqref{def:Psi} into the other equations of \eqref{E:ww0}, likewise. The result becomes
\begin{subequations}\label{E:ww}
\begin{align}
&\Delta\Phi=0 &&\text{in $\Omega(t)$},\label{E:ww;Laplace} \\
&\eta_t+(\Phi_x-\omega y-c)\eta_x=\Phi_y &&\text{at $\Gamma(t)$}, \label{E:ww;K}\\
&\Phi_t+\tfrac12|\nabla\Phi|^2-(\omega\eta+c)\Phi_x+\omega\Psi+g\eta=b(t) &&\text{at $\Gamma(t)$}\label{E:ww;B}
\intertext{and}
&\Phi_y=0&&\text{at $y=-h$}.\label{E:ww;bed}
\end{align}
Note that $\eta$ and $\Phi$, $\Psi$ are $2L$ periodic in the $x$ variable.
In the infinite depth, where $h=\infty$, we replace \eqref{E:ww;bed} by
\begin{equation}\label{E:ww;infty}
\Phi,\Psi\to0\quad\text{as $y\to-\infty$}\quad\text{uniformly for $x\in\mathbb{R}$}.
\end{equation}
\end{subequations}
Moreover, we may assume that $p\to p_{atm}-gy$ as $y\to-\infty$, whence $b(t)=0$. But \eqref{def:Phi} implies that
\[
\boldsymbol{u}\to(-\omega y-c,0)\quad\text{as $y\to-\infty$}.
\]
Therefore, nonzero constant vorticity in the infinite depth seems physically unrealistic. Nevertheless, \eqref{E:ww} makes sense theoretically for any $h\in(0,\infty]$. Moreover, the infinite depth offers an auxiliary conformal mapping for effective numerical computation; see Section \ref{sec:infty-depth} and references therein for details. The effects of depth turn out to change the amplitude of a Stokes wave and other quantities, and they are insignificant otherwise.
In stark contrast, the effects of constant vorticity are profound on limiting waves and other fundamental issues.
\subsection{Reformulations via conformal mapping}\label{sec:reformulation}
We reformulate \eqref{E:ww} via a conformal mapping of the fluid region from a strip, or from a half plane in the infinite depth. The idea traces back to \cite{Stokes1880} in the steady wave setting and was explored in the unsteady wave setting by \cite{Ovsyannikov1973} and, later, \cite{MOI1981, Tanveer1991, Tanveer1993, ZDAV2002}, among others. Below, we proceed along the same line as the argument in \cite{DKSZ1996, DZK1996}, but with suitable modifications to accommodate constant vorticity.
In what follows, we identify $\mathbb{R}^2$ with $\mathbb{C}$ whenever it is convenient to do so.
\subsubsection*{Conformal mapping}
Suppose that
\begin{equation}\label{def:conformal}
z=z(w;t),\quad \text{where}\quad w=u+iv\quad\text{and}\quad z=x+iy,
\end{equation}
conformally maps
\[
\Sigma_d:=\{u+iv\in\mathbb{C}:-d<v<0\}
\]
of $2\upi$ period in the $u$ variable to $\Omega(t)$ of $2L$ period in the $x$ variable at time $t$ for some $d$ in the range $(0,\infty]$.
Suppose that \eqref{def:conformal} extends to map $\{u+i0:u\in\mathbb{R}\}$ to $\Gamma(t)$ and, moreover, $\{u-id:u\in\mathbb{R}\}$ to $\{x-ih:x\in\mathbb{R}\}$ if $d,h<\infty$, and $-i\infty$ to $-i\infty$ if $d,h=\infty$. Clearly, $x$ and $y$ enjoy the Cauchy-Riemann equations:
\begin{equation}\label{E:CR(x,y)}
x_u=y_v\quad\text{and}\quad x_v=-y_u
\end{equation}
in $\Sigma_d$. Moreover,
\begin{equation}\label{E:periodicity}
x(u+2\upi+iv;t)=x(u+iv;t)+2L\quad\text{and}\quad y(u+2\upi+iv;t)=v(u+iv;t)
\end{equation}
for $u+iv\in\overline{\Sigma_d}$.
Therefore, in the finite depth,
\[
\Delta y=0\quad\text{in $\Sigma_d$}\quad\text{and}\quad y=-h\quad\text{at $v=-d$}.
\]
Suppose that
\begin{equation}\label{E:y(u,0)}
y(u+i0;t)=\sum_{k\in\mathbb{Z}}\widehat{y}(k;t)e^{iku}\quad\text{for $u\in\mathbb{R}$}
\end{equation}
in the Fourier series, where
\[
\widehat{y}(k;t)=\frac{1}{2\upi}\int^{\upi}_{-\upi} y(u+i0;t)e^{iku}~du,
\]
whence
\begin{equation}\label{E:y(u,v)}
y(u+iv;t)=\frac{\widehat{y}(0;t)+h}{d}v+\widehat{y}(0;t)
+\sum_{k\neq0,\in\mathbb{Z}}\frac{\sinh(k(v+d))}{\sinh(kd)}\widehat{y}(k;t)e^{iku}
\end{equation}
for $u+iv\in\overline{\Sigma_d}$. The Cauchy-Riemann equations imply
\begin{equation}\label{E:x(u,v)}
x(u+iv;t)=\frac{\widehat{y}(0;t)+h}{d}u
-\sum_{k\neq0,\in\mathbb{Z}}i\frac{\cosh(k(v+d))}{\sinh(kd)}\widehat{y}(k;t)e^{iku}
\end{equation}
for $u+iv\in\overline{\Sigma_d}$ up to an additive constant. We infer from \eqref{E:x(u,v)} and \eqref{E:y(u,v)} that
\[
x_u^2+y_u^2\neq0\quad\text{in $\overline{\Sigma_d}$}.
\]
Moreover, we infer from \eqref{E:x(u,v)} and the former equation of \eqref{E:periodicity} that
\[
\frac{L}{\upi}=\frac{\widehat{y}(0;t)+h}{d},
\]
which relates the ``mean conformal depth" $d$ and the ``mean fluid depth" $h$ (see the discussion following \eqref{E:mean0(t)}), depending on the solution of \eqref{def:conformal}.
In what follows, we assume, without loss of generality, that $L=\upi$, whence the above simplifies to
\begin{equation}\label{E:dh}
d=\langle y\rangle+h,
\end{equation}
where
\begin{equation}\label{def:mean}
\langle f\rangle=\frac{1}{2\upi}\int_{-\upi}^{\upi} f(u)~du
\end{equation}
is the mean over one period of a $2\upi$ periodic function $f$.
Consequently, \eqref{E:x(u,v)} simplifies to
\begin{equation}\label{E:x(u,0)}
x(u+i0;t)=u-\sum_{k\neq0}i\coth(kd)\widehat{y}(k;t)e^{iku}\quad\text{for $u\in\mathbb{R}$}.
\end{equation}
\subsubsection*{Reformulation via conformal mapping}
Recall \eqref{def:conformal}, and let, by abuse of notation,
\begin{equation}\label{def:y(u)}
(x+iy)(u;t)=(x+iy)(u+i0;t)\quad\text{for $u\in\mathbb{R}$}.
\end{equation}
Therefore,
\[
y(u;t)=\eta(x(u;t);t).
\]
In what follows, we allow that $\eta$ be multi valued. By the way, one may extend \eqref{E:ww} mutatis mutandis when the fluid surface is the trajectory of a parametric curve. A chain rule calculation reveals that
\[
y_u=\eta_xx_u\quad\text{and}\quad y_t=\eta_xx_t+\eta_t.
\]
Recall \eqref{def:Phi} and \eqref{def:Psi}, and let
\begin{equation}\label{def:phi}
(\phi+i\psi)(w;t)=(\Phi+i\Psi)(z(w;t);t)\quad\text{for $w\in\Sigma_d$}.
\end{equation}
Namely, $\phi+i\psi$ is a conformal velocity potential for the irrotational perturbation from \eqref{def:shear}. Since $\Phi+i\Psi$ is holomorphic in $\Omega(t)$ and since $z:\Sigma_d\to\Omega(t)$ is conformal, $\phi$ and $\psi$ enjoy the Cauchy-Riemann equations:
\begin{equation}\label{E:CR(p,q)}
\phi_u=\psi_v\quad\text{and}\quad\phi_v=-\psi_u
\end{equation}
in $\Sigma_d$. A chain rule calculation and \eqref{E:CR(x,y)}, \eqref{E:CR(p,q)} reveal that
\[
\begin{pmatrix}\Phi_x\\ \Phi_y\end{pmatrix}
=\frac{1}{x_uy_u-x_vy_v}\begin{pmatrix}y_v&-y_u\\-x_v&x_u\end{pmatrix}
\begin{pmatrix}\phi_u\\ \phi_v\end{pmatrix}
=\frac{1}{x_u^2+y_u^2}\begin{pmatrix}x_u&-y_u\\ y_u&x_u\end{pmatrix}
\begin{pmatrix}\phi_u\\-\psi_u\end{pmatrix},
\]
where $x_u^2+y_u^2\neq0$ in $\overline{\Sigma_d}$ by \eqref{E:x(u,v)} and \eqref{E:y(u,v)}. Moreover, $\phi_t=\Phi_xx_t+\Phi_yy_t+\Phi_t$. Let, by abuse of notation,
\begin{equation}\label{def:phi(u)}
(\phi+i\psi)(u;t)=(\phi+i\psi)(u+i0;t)\quad\text{for $u\in\mathbb{R}$}.
\end{equation}
We substitute \eqref{def:y(u)} and \eqref{def:phi(u)} into \eqref{E:ww;K} and \eqref{E:ww;B}, and we use the result from the chain rule calculations to arrive at
\begin{subequations}\label{E:ww-conf}
\begin{align}
x_u&y_t-y_ux_t+\psi_u-(\omega y+c)y_u=0\label{E:ww-conf-K}\\
\intertext{and}
\phi_t&-\frac{1}{x_u^2+y_u^2}((x_ux_t+y_uy_t)\phi_u
+(y_ux_t-x_uy_t)\psi_u \label{E:ww-conf-B}\\
&+\tfrac12(\phi_u^2+\psi_u^2)
-(\omega y+c)(x_u\phi_u+y_u\psi_u))+\omega\psi+gy-b(t)=0\notag
\end{align}
at $v=0$. Note that
\begin{equation}\label{E:ww-conf;Laplace}
\Delta y,\Delta\phi=0\quad\text{in $\Sigma_d$}.
\end{equation}
Note that
\begin{equation}\label{E:ww-conf;bed}
y=-h\quad\text{and}\quad\phi_v=0\quad\text{at $v=-d$}\quad\text{if $d$, $h<\infty$}
\end{equation}
by \eqref{E:ww;bed}, and
\begin{equation}\label{E:ww-conf;infty}
\phi,\psi\to0\quad\text{as $v\to-\infty$}\quad\text{uniformly for $u\in\mathbb{R}$}\quad\text{if $d$, $h=\infty$}
\end{equation}
\end{subequations}
by \eqref{E:ww;infty} and \eqref{def:phi}. Moreover, $y$ and $\phi$, $\psi$ are $2\upi$ periodic in the $u$ variable. Therefore, \eqref{E:ww-conf} is to rewrite \eqref{E:ww}.
Below, we relate $x$ to $y$ and $\phi$ to $\psi$ at the face of $\Sigma_d$, whereby we reformulate \eqref{E:ww-conf} and, hence, \eqref{E:ww} for $y=y(u;t)$ and $\phi=\phi(u;t)$. It makes use of periodic Hilbert transforms for a strip.
\subsubsection*{Periodic Hilbert transforms for a strip}
For $d$ in the range $(0,\infty)$, let $\mathcal{H}_d$ and $\mathcal{T}_d$ denote Fourier multiplier operators, defined in the periodic setting as
\[
\mathcal{H}_de^{iku}=-i\tanh(kd)e^{iku}\quad \text{for $k\in\mathbb{Z}$}
\]
and
\begin{equation}\label{def:T}
\mathcal{T}_de^{iku}=
\begin{cases}
-i\coth(kd)e^{iku}\quad &\text{if $k\neq0,\in\mathbb{Z}$}, \\
0 &\text{if $k=0$}.
\end{cases}
\end{equation}
Clearly,
\begin{equation}\label{E:TH=-1}
\mathcal{H}_d\mathcal{T}_d=\mathcal{T}_d\mathcal{H}_d=-1\quad\text{if $k\neq0$}.
\end{equation}
As $d\to\infty$, at least formally, $\mathcal{H}_d$ and $\mathcal{T}_d$ tend to the periodic Hilbert transform, defined likewise as
\[
\mathcal{H}e^{iku}=-i\,\text{sgn}(k)e^{iku}\quad\text{for $k\in\mathbb{Z}$}.
\]
Among other properties of $\mathcal{H}_d$ and $\mathcal{T}_d$, of particular importance for the present purpose is that the ``Titchmarsh theorem" \citep[see][Theorem~95, for instance]{Titchmarsh} or the Sokhotski-Plemelj theorem \citep[see][for instance]{Plemelj, Gakhov} extends, and $\mathcal{H}_d$ and $\mathcal{T}_d$ relate the real part of a holomorphic and $2\upi$ periodic function in a strip to the imaginary part at the face of the strip, and vice versa. If $F=F(u+iv)$ is holomorphic in the lower half plane of $\mathbb{C}$ and if $F$ vanishes sufficiently rapidly as $v\to-\infty$ then the Titchmarsh theorem states that the real and imaginary parts of $F(\cdot+i0)$ are the Hilbert transforms of each other. For any $d\in(0,\infty)$, likewise, if $F$ is holomorphic in $\Sigma_d$ and $2\upi$ periodic in the $u$ variable and if $\Real F(u+i0)=f(u)$ and $(\Real F)_v(u-id)=0$ for $u\in\mathbb{R}$ then
\begin{equation}\label{E:1-iH}
F(u+i0)=(1-i\mathcal{H}_d)f(u)\quad\text{for $u\in\mathbb{R}$}
\end{equation}
up to an additive imaginary constant. In other words, $1-i\mathcal{H}_d$ makes the face value of a holomorphic and $2\upi$ periodic function in $\Sigma_d$, the normal derivative of whose real part vanishes at the bottom of $\Sigma_d$. Moreover, if $F$ is holomorphic in $\Sigma_d$ and $2\upi$ periodic in the $u$ variable, if $\Imag F(u+i0)=f(u)$ and $\Imag F(u-id)=0$ for $u\in\mathbb{R}$, and if $\langle f\rangle=0$ in addition, where we employ the notation of \eqref{def:mean}, then
\begin{equation}\label{E:T+i}
F(u+i0)=(\mathcal{T}_d+i)f(u)\quad\text{for $u\in\mathbb{R}$}
\end{equation}
up to an additive real constant. In other words, $\mathcal{T}_d+i$ is the face value of a holomorphic and $2\upi$ periodic function in $\Sigma_d$, whose imaginary part is of mean zero at the face of $\Sigma_d$ and vanishes at the bottom.
\subsubsection*{Implicit form}
Returning to the water wave problem, in the finite depth, since $\phi+i\psi$ is holomorphic in $\Sigma_d$ and satisfies \eqref{E:ww-conf;bed}, we employ an extension of the Titchmarsh theorem to a strip (see \eqref{E:1-iH}) to show that
\begin{equation}\label{E:Hp}
(\phi+i\psi)(u;t)=(1-i\mathcal{H}_d)\phi(u;t)
\end{equation}
up to an additive imaginary constant. Moreover, we use \eqref{E:y(u,0)}, \eqref{E:x(u,0)} and \eqref{def:T} to show that
\begin{equation}\label{E:Ty}
(x+iy)(u;t)=u+(\mathcal{T}_d+i)y(u;t).
\end{equation}
By the way, an extension of the Titchmarsh theorem (see \eqref{E:T+i}) does not apply to $x+iy$ because $y$ needs not be of mean zero at the face of $\Sigma_d$. (See the discussion following \eqref{E:mean0(t)}.) In the infinite depth, the Titchmarsh theorem implies \eqref{E:Hp} and \eqref{E:Ty}, where the periodic Hilbert transform replaces $\mathcal{H}_d$ and $\mathcal{T}_d$.
To proceed, in the finite depth, we substitute \eqref{E:Hp} and \eqref{E:Ty} into \eqref{E:ww-conf-K} and \eqref{E:ww-conf-B}, to arrive at
\begin{subequations}\label{E:implicit}
\begin{align}
&(1+\mathcal{T}_dy_u)y_t-y_u\mathcal{T}_dy_t-\mathcal{H}_d\phi_u-(\omega y+c)y_u=0\label{E:implicit-K}
\intertext{and}
&((1+\mathcal{T}_dy_u)^2+y_u^2)(\phi_t+gy-\omega\mathcal{H}_d\phi-b(t)) \notag\\
&\quad-((1+\mathcal{T}_dy_u)\mathcal{T}_dy_t+y_uy_t)\phi_u
+(y_u\mathcal{T}_dy_t-(1+\mathcal{T}_dy_u)y_t)\mathcal{H}_d\phi_u \label{E:implicit-B}\\
&\quad\quad+\tfrac12(\phi_u^2+(\mathcal{H}_d\phi_u)^2)
-(\omega y+c)((1+\mathcal{T}_dy_u)\phi_u-y_u\mathcal{H}_d\phi_u)=0.\notag
\end{align}
\end{subequations}
Note that $y=y(u;t)$ and $\phi=\phi(u;t)$ are $2\upi$ periodic in the $u$ variable. We claim that \eqref{E:implicit} is equivalent to \eqref{E:ww-conf} and, hence, \eqref{E:ww}, provided that $d$ and $h$ are related by \eqref{E:dh}. Indeed, $y$ and $\phi$ extend as the imaginary and real parts of holomorphic and $2\upi$ periodic functions in $\Sigma_d$, which satisfy \eqref{E:ww-conf;bed}. In the infinite depth, \eqref{E:implicit} is equivalent to \eqref{E:ww-conf} and, hence, \eqref{E:ww}, likewise, where the periodic Hilbert transform replaces $\mathcal{H}_d$ and $\mathcal{T}_d$. Moreover, in an irrotational flow, \eqref{E:implicit} agrees with what \cite{DKSZ1996, DZK1996}, for instance, derived.
We integrate \eqref{E:implicit-K} over the periodic interval $[-\upi,\upi]$ and use that $\mathcal{T}_d$ is anti-self-adjoint, to show that
\begin{equation}\label{E:mean0(t)}
\frac{d}{dt}\langle y(1+\mathcal{T}_dy_u)\rangle=0.
\end{equation}
Therefore, if we locate the coordinates of the fluid region so that $\langle y(1+\mathcal{T}_dy_u)\rangle=0$ at the initial time then it remains so at all later times; $y$ then measures the fluid surface displacement from zero and $h$ the mean fluid depth. In the infinite depth, the periodic Hilbert transform replaces $\mathcal{T}_d$.
\subsubsection*{Explicit form}
Concluding the reformulations, we solve \eqref{E:implicit} for $y_t$ and $\phi_t$ explicitly.
In the finite depth, since $z$ is holomorphic in $\Sigma_d$ and since $|z_u|^2\neq0$ in $\overline{\Sigma_d}$ by \eqref{E:x(u,v)} and \eqref{E:y(u,v)}, $z_t/z_u$ is holomorphic in $\Sigma_d$. Note that
\begin{equation}\label{E:Im(z_t/z_u)}
\Imag\frac{z_t}{z_u}=\frac{x_uy_t-y_ux_t}{|z_u|^2}
=\frac{\mathcal{H}_d\phi_u+(\omega y+c)y_u}{|z_u|^2}=:\frac{-\chi_u}{|z_u|^2}\quad\text{at $v=0$}
\end{equation}
by \eqref{E:ww-conf-K} and \eqref{E:Hp}, and $\Imag(z_t/z_u)=0$ at $v=-d$ by \eqref{E:ww-conf;bed}.
By the way,
\begin{equation}\label{def:stream}
\chi=\psi-(\tfrac12\omega y^2+cy)
\end{equation}
makes a conformal stream function by \eqref{def:Psi} and \eqref{def:phi}. Note that $\langle \Imag(z_t/z_u)\rangle=0$ for all $v\in[-d,0]$ by the Cauchy-Riemann equations and \eqref{E:ww-conf;bed}. Therefore, an extension of the Titchmarsh theorem to a strip (see \eqref{E:T+i}) implies that
\[
\frac{z_t}{z_u}=(\mathcal{T}_d+i)\Big(\frac{-\chi_u}{|z_u|^2}\Big)\quad\text{at $v=0$}.
\]
Or, equivalently,
\begin{equation}\label{E:K}
x_t=((1+\mathcal{T}_dy_u)\mathcal{T}_d-y_u)\Big(\frac{-\chi_u}{|z_u|^2}\Big)\quad\text{and}\quad
y_t=(1+\mathcal{T}_dy_u+y_u\mathcal{T}_d)\Big(\frac{-\chi_u}{|z_u|^2}\Big)\quad\text{at $v=0$}.
\end{equation}
Moreover,
\begin{equation}\label{E:Re(z_t/z_u)}
\Real\frac{z_t}{z_u}=\frac{x_ux_t+y_uy_t}{|z_u|^2}=\mathcal{T}_d\Big(\frac{-\chi_u}{|z_u|^2}\Big)
\quad\text{at $v=0$}.
\end{equation}
We substitute \eqref{E:Im(z_t/z_u)} and \eqref{E:Re(z_t/z_u)} into \eqref{E:ww-conf-B}, to arrive at
\begin{multline}\label{E:B}
\phi_t+\phi_u\mathcal{T}_d\Big(\frac{\chi_u}{|z_u|^2}\Big)
-\frac{1}{|z_u|^2}(\tfrac12(\phi_u^2-\psi_u^2)(\omega y+c)(1+\mathcal{T}_dy_u)\phi_u)\\
-\omega\mathcal{H}_d\phi+gy-b(t)=0\quad\text{at $v=0$}.
\end{multline}
But an extension of the Titchmarsh theorem to a strip (see \eqref{E:1-iH}) implies that
\[
(\phi_u-i\mathcal{H}_d\phi_u)^2=\phi_u^2-(\mathcal{H}_d\phi_u)^2-2i\phi_u\mathcal{H}_d\phi_u
\]
is the face value of the holomorphic and $2\upi$ periodic function $=(\phi_u+i\psi_u)^2$ in $\Sigma_d$, the normal derivative of whose real part vanishes at the bottom of $\Sigma_d$ by the Cauchy-Riemann equations and \eqref{E:ww-conf;bed}. It then follows from \eqref{E:1-iH} and \eqref{E:TH=-1} that
\[
\phi_u^2-(\mathcal{H}_d\phi_u)^2=-2\mathcal{T}_d(\phi_u\mathcal{H}_d\phi_u).
\]
We substitute \eqref{def:stream}, \eqref{E:Ty}, \eqref{E:Hp} and the above into the latter equation of \eqref{E:K} and \eqref{E:B}, to arrive at
\begin{align}\label{E:explicit}
y_t=&(1+\mathcal{T}_dy_u+y_u\mathcal{T}_d)
\Big(\frac{\mathcal{H}_d\phi_u+(\omega y+c)y_u}{(1+\mathcal{T}_dy_u)^2+y_u^2}\Big) \notag
\intertext{and}
\phi_t=&-\phi_u\mathcal{T}_d
\Big(\frac{\mathcal{H}_d\phi_u+(\omega y+c)y_u}{(1+\mathcal{T}_dy_u)^2+y_u^2}\Big) \notag \\
&+\frac{1}{(1+\mathcal{T}_dy_u)^2+y_u^2}(\mathcal{T}_d(\phi_u\mathcal{H}_d\phi_u)
+(\omega y+c)(1+\mathcal{T}_dy_u)\phi_u)+\omega\mathcal{H}_d\phi-gy+b(t).
\end{align}
Note that $y=y(u;t)$ and $\phi=\phi(u;t)$ are $2\upi$ periodic in the $u$ variable. Clearly, \eqref{E:explicit} is equivalent to \eqref{E:ww-conf} and, hence, \eqref{E:implicit}. Therefore, \eqref{E:explicit} is to solve \eqref{E:implicit} for $y_t$ and $\phi_t$ explicitly. In the infinite depth, \eqref{E:explicit} is equivalent to \eqref{E:ww-conf} and, hence, \eqref{E:implicit}, likewise, where the periodic Hilbert transform replaces $\mathcal{H}_d$ and $\mathcal{T}_d$. Moreover, in an irrotational flow, \eqref{E:explicit} agrees with what \cite{DKSZ1996, DZK1996}, for instance, derived.
\subsection{The Stokes wave problem}\label{sec:Stokes}
We turn the attention to the solutions of \eqref{E:explicit}, for which $y_t$, $\phi_t=0$ and $b(t)=$ constant, and, hence, the steady solutions of \eqref{E:ww}. They make Stokes waves, permitting constant vorticity and finite depth.
In what follows, the prime means ordinary differentiation in the $u$ variable.
\subsubsection*{Formulation via conformal mapping}
In the finite depth, we substitute $y_t=0$ into the latter equation of \eqref{E:K} to arrive at
\begin{equation}\label{E:psi'}
\psi'=\omega yy'+cy'\quad\text{at $v=0$}.
\end{equation}
Indeed, $(1+\mathcal{T}_dy')^2+(y')^2\neq 0$ pointwise in $\mathbb{R}$ by \eqref{E:Ty} and \eqref{E:x(u,0)}, \eqref{E:y(u,0)}. By the way, \eqref{E:psi'} states that the fluid surface itself makes a streamline. Note from \eqref{E:Hp} and \eqref{E:TH=-1} that
\begin{equation}\label{E:phi'}
\phi'=\mathcal{T}_d(\omega yy'+cy')\quad\text{at $v=0$}.
\end{equation}
Moreover, we substitute $\phi_t=0$ into \eqref{E:B} and use \eqref{E:psi'} and \eqref{E:phi'}, to arrive at
\begin{align*}
(&\mathcal{T}_d(\omega yy'+cy'))^2-(\omega yy'+cy')^2
-2(\omega y+c)(1+\mathcal{T}_dy')\mathcal{T}_d(\omega yy'+cy') \\
&+2\omega((1+\mathcal{T}_dy')^2+(y')^2)(\tfrac12\omega y^2+cy)
-2(b-gy)((1+\mathcal{T}_dy')^2+(y')^2)=0\quad\text{at $v=0$}
\end{align*}
for some constant $b\in\mathbb{R}$. After a lengthy but straightforward calculation, it simplifies to
\begin{equation}\label{E:Stokes}
(c+\omega y(1+\mathcal{T}_dy')-\omega\mathcal{T}_d(yy'))^2=(c^2+2b-2gy)((1+\mathcal{T}_dy')^2+(y')^2).
\end{equation}
Or, equivalently,
\begin{equation}\label{E:Stokes;y}
y=\frac{1}{2g}\Big(c^2+2b-\frac{(c+\omega y(1+\mathcal{T}_dy')-\omega\mathcal{T}_d(yy'))^2}{(1+\mathcal{T}_dy')^2+(y')^2}\Big).
\end{equation}
In the infinite depth, the periodic Hilbert transform replaces $\mathcal{T}_d$.
Therefore, the Stokes wave problem, permitting constant vorticity and finite depth, is to find $\omega\in\mathbb{R}$, $d\in(0,\infty]$, $b$, $c\in\mathbb{R}$ and a $2\upi$ periodic function $y$, which satisfy \eqref{E:Stokes} or \eqref{E:Stokes;y}.
In an irrotational flow of infinite depth, we may take $b=0$ (see the discussion following \eqref{E:ww;infty}), whence \eqref{E:Stokes;y} further simplifies to
\[
y=\frac12\frac{c^2}{g}\Big(1-\frac{1}{(1+\mathcal{H}y')^2+(y')^2}\Big).
\]
The result agrees with what \cite{DKSZ1996}, for instance, derived.
\subsubsection*{Reformulation as an equation of Babenko kind}
Unfortunately, \eqref{E:Stokes} or \eqref{E:Stokes;y} is not convenient for numerical computation because one would have to deal with rational functions of $y$. Moreover, \cite{CSV2016} noted that \eqref{E:Stokes} is not suitable for global bifurcation theory because it does not seem to make a compact operator. Below, we proceed along the same line as the argument in \cite{CSV2016} to reformulate \eqref{E:Stokes} as an equation of ``Babenko kind." It makes use of an extension of the Titchmarsh theorem to a strip (see \eqref{E:T+i}) for various quantities.
We begin by arranging \eqref{E:Stokes} as
\begin{align*}
(c-\omega\mathcal{T}_d(yy'))^2+2(c-\omega\mathcal{T}_d(yy'))\omega y(1+\mathcal{T}_dy')
&+\omega^2y^2(1+\mathcal{T}_dy')^2 \\ &=(c^2+2b-2gy)((1+\mathcal{T}_dy')^2+(y')^2),
\end{align*}
and rearranging as
\begin{equation}\label{E:aux1}
\begin{aligned}
(c-\omega\mathcal{T}_d(yy'))^2+2\omega y(c-\omega\mathcal{T}_d(yy'))(&1+\mathcal{T}_dy')
-\omega^2y^2(y')^2 \\ &=(c^2+2b-2gy-\omega^2y^2)((1+\mathcal{T}_dy')^2+(y')^2).
\end{aligned}
\end{equation}
An extension of the Titchmarsh theorem to a strip (see \eqref{E:T+i}) implies that $\mathcal{T}_d(yy')+iyy'$ makes the face value of a holomorphic and $2\upi$ periodic function in $\Sigma_d$, whose imaginary part is of mean zero at the face of $\Sigma_d$ and vanishes at the bottom, and so does
\[
(c-\omega(\mathcal{T}_d(yy')+iyy'))^2
=(c-\omega\mathcal{T}_d(yy'))^2-\omega^2y^2(y')^2-2i(c-\omega\mathcal{T}_d(yy'))\omega yy'.
\]
It then follows from \eqref{E:aux1} that
\begin{equation}\label{E:aux2}
\begin{aligned}
(c^2&+2b-2gy-\omega^2y^2)((1+\mathcal{T}_dy')^2+(y')^2) \\
&-2\omega y(c-\omega\mathcal{T}_d(yy'))(1+\mathcal{T}_dy')-2i\omega y(c-\omega\mathcal{T}_d(yy'))y' \\
=&(c^2+2b-2gy-\omega^2y^2)((1+\mathcal{T}_dy')^2+(y')^2)
-2\omega y(c-\omega\mathcal{T}_d(yy'))(1+\mathcal{T}_dy'+iy') \\
=&((c^2+2b-2gy-\omega^2y^2)(1+\mathcal{T}_dy'-iy')
-2\omega y(c-\omega\mathcal{T}_d(yy')))(1+\mathcal{T}_dy'+iy')
\end{aligned}
\end{equation}
is the face value of a holomorphic and $2\upi$ periodic function in $\Sigma_d$, whose imaginary part is of mean zero at the face of $\Sigma_d$ and vanishes at the bottom.
Note that $1/(1+\mathcal{T}_dy'+iy')$ is the face value of the holomorphic and $2\upi$ periodic function $=1/z_u$ in $\Sigma_d$, whose imaginary part is of mean zero at the face of $\Sigma_d$ and vanishes at the bottom. Indeed, $z$ is holomorphic in $\Sigma_d$, $|z_u|^2\neq0$ in $\overline{\Sigma_d}$ by \eqref{E:x(u,v)} and \eqref{E:y(u,v)} and, moreover, $\langle\Imag(1/z_u)\rangle=0$ for all $v\in[-d,0]$ by the Cauchy-Riemann equations and \eqref{E:ww-conf;bed}. Therefore, it follows from \eqref{E:aux2} that
\[
(c^2+2b-2gy-\omega^2y^2)(1+\mathcal{T}_dy'-iy')-2\omega y(c-\omega\mathcal{T}_d(yy'))
\]
is the face value of a holomorphic and $2\upi$ periodic function in $\Sigma_d$, whose imaginary part is of mean zero at the face of $\Sigma_d$ and vanishes at the bottom. An extension of the Titchmarsh theorem to a strip (see \eqref{E:T+i}) then implies that
\[
(c^2+2b-2gy-\omega^2y^2)(1+\mathcal{T}_dy')-2\omega y(c-\omega\mathcal{T}_d(yy'))
=-\mathcal{T}_d((c^2+2b-2gy-\omega^2y^2)y')
\]
up to an additive real constant. Or, equivalently,
\begin{multline*}
(c^2+2b)\mathcal{T}_dy'-(g+c\omega)y-g(y\mathcal{T}_dy'+\mathcal{T}_d(yy')) \\
-\tfrac12\omega^2(y^2+y^2\mathcal{T}_dy'+\mathcal{T}_d(y^2y')-2y\mathcal{T}_d(yy'))=\mu,
\end{multline*}
say. An integration over the periodic interval $[-\upi,\upi]$ reveals that
\[
\mu=-g\langle y(1+\mathcal{T}_dy')\rangle-c\omega\langle y\rangle-\tfrac12\omega^2\langle y^2\rangle.
\]
Indeed, $\langle \mathcal{T}_df'\rangle=0$ for any function $f$ by \eqref{def:T} and, moreover, since $f\mapsto \mathcal{T}_df'$ is self-adjoint,
\[
\langle y^2\mathcal{T}_dy'\rangle =\frac{1}{2\upi}\int^{\upi}_{-\upi} y^2\mathcal{T}_dy'~du
=-\frac{1}{2\upi}\int^{\upi}_{-\upi} y\mathcal{T}_d(y^2)'~du=-\langle 2y\mathcal{T}_d(yy')\rangle.
\]
To recapitulate,
\begin{equation}\label{E:aux3}
\begin{aligned}
(c^2+2b)\mathcal{T}_dy'-(g+c\omega)y-g(y\mathcal{T}_dy'+&\mathcal{T}_d(yy')) \\
-\tfrac12\omega^2(y^2+y^2\mathcal{T}_dy'&+\mathcal{T}_d(y^2y')-2y\mathcal{T}_d(yy')) \\
&+g\langle y(1+\mathcal{T}_dy')\rangle+c\omega\langle y\rangle+\tfrac12\omega^2\langle y^2\rangle=0.
\end{aligned}
\end{equation}
In the infinite depth, the periodic Hilbert transform replaces $\mathcal{T}_d$.
We emphasize that \eqref{E:aux3} is made up of polynomials of $y$ (involving its derivative and $\mathcal{T}_d$), whence it is straightforward to implement in numerical computation. It is the subject of investigation here. Moreover, \cite{CSV2016} verified that the linearization of \eqref{E:aux3} with respect to $y$ and $b$ is a compact operator in a suitable function space, provided that $c^2+2b-2y>0$ pointwise in $\mathbb{R}$, whereby they established a global bifurcation result.
But for any $\omega\in\mathbb{R}$, $d\in(0,\infty]$ and $b$, $c\in\mathbb{R}$,
\begin{equation*}\label{E:k>0}
\begin{aligned}
y\mapsto&(c^2+2b)\mathcal{T}_dy'-(g+c\omega)y
-g(y\mathcal{T}_dy'+\mathcal{T}_d(yy'))\\
&-\tfrac12\omega^2(y^2+\mathcal{T}_d(y^2y')+y^2\mathcal{T}_dy'-2y\mathcal{T}_d(yy'))
+g\langle y(1+\mathcal{T}_dy')\rangle+c\omega\langle y\rangle+\tfrac12\omega^2\langle y^2\rangle
\end{aligned}
\end{equation*}
maps $2\upi$ periodic functions to $2\upi$ periodic functions of mean zero, whereas
\begin{equation*}\label{E:k=0}
y\mapsto (c+\omega y(1+\mathcal{T}_dy')-\omega\mathcal{T}_d(yy'))^2-(c^2+2b-2gy)((1+\mathcal{T}_dy')^2+(y')^2)
\end{equation*}
maps $2\upi$ periodic functions to $2\upi$ periodic functions, not necessarily of mean zero. In other words, \eqref{E:aux3} and \eqref{E:Stokes} agree except a constant. It is because $\mathcal{T}_d+i$ makes the face value of a holomorphic and $2\upi$ periodic function in $\Sigma_d$ merely up to an additive real constant. In order to reconcile loss of information from \eqref{E:Stokes} to \eqref{E:aux3}, we require that the solutions of \eqref{E:aux3} in addition satisfy
\begin{equation}\label{E:aux4}
\langle(c+\omega y(1+\mathcal{T}_dy')-\omega\mathcal{T}_d(yy'))^2\rangle
=\langle (c^2+2b-2gy)((1+\mathcal{T}_dy')^2+(y')^2)\rangle.
\end{equation}
Consequently, \eqref{E:aux3} and \eqref{E:aux4} are equivalent to \eqref{E:Stokes}.
Furthermore, \eqref{E:Stokes} cannot uniquely determine $y$ and $b$, and neither can \eqref{E:aux3} and \eqref{E:aux4}. It is because \eqref{E:ww} is a free boundary problem. Indeed, $b$ depends on the location of the coordinates of the fluid region, among others. Recall \eqref{E:mean0(t)}, and we assume, without loss of generality, that
\begin{equation}\label{E:mean0}
\langle y(1+\mathcal{T}_dy')\rangle=0;
\end{equation}
$y$ then measures the fluid surface displacement from zero and $h$ the mean fluid depth. It in turn simplifies \eqref{E:aux3}.
We assume in addition that the solutions of \eqref{E:aux3}, \eqref{E:aux4} and \eqref{E:mean0} are even. Indeed,
for arbitrary vorticity, under some assumptions, \cite{Hur2007} and \cite{CEW2007}, among others, proved that a Stokes wave is a priori symmetric about the crest.
To summarize, the Stokes wave problem, permitting constant vorticity and finite depth, is to find a vorticity $\omega\in\mathbb{R}$, a mean conformal depth $d\in(0,\infty]$, a ``Bernoulli constant" $b\in\mathbb{R}$, a wave speed $c\in\mathbb{R}$, and a $2\upi$ periodic and even function $y$, measuring the fluid surface displacement from zero, which satisfy
\begin{subequations}\label{E:main}
\begin{equation}\label{E:Babenko}
\begin{aligned}
(c^2+2b)\mathcal{T}_dy'&-(g+c\omega)y-g(y\mathcal{T}_dy'+\mathcal{T}_d(yy'))\\
&-\tfrac12\omega^2(y^2+\mathcal{T}_d(y^2y')+y^2\mathcal{T}_dy'-2y\mathcal{T}_d(yy'))
+c\omega\langle y\rangle+\tfrac12\omega^2\langle y^2\rangle=0
\end{aligned}
\end{equation}
and
\begin{equation}\label{E:<Stokes>}
\langle(c+\omega y(1+\mathcal{T}_dy')-\omega(\mathcal{T}_d(yy'))^2\rangle
-\langle (c^2+2b-2gy)((1+\mathcal{T}_dy')^2+(y')^2)\rangle=0.
\end{equation}
\end{subequations}
We usually regard $\omega$ and $d$ as prescribed, and $y$ and $b$ as the unknowns, depending on the parameter $c$, although we at times switch the roles of $c$ and $\omega$ or $d$; see Section~\ref{sec:initial guess}, Section~\ref{sec:max omega} and Section~\ref{sec:min depth}, for instance. In the finite depth, we determine the mean fluid depth $h$ by solving \eqref{E:dh}, depending on the solution. One may instead fix $h$ and determine $d$ as part of the solution.
We remark that \cite{CSV2016} focused on steady waves to discover \eqref{E:main}, and here we begin by deriving the governing equations in the unsteady wave setting and rediscover \eqref{E:main} by seeking the steady solutions, which is potentially useful for addressing stability and other unsteady wave phenomena. It is a subject of future investigation. Moreover, \cite{CSV2016} required $\langle y\rangle=0$ in place of \eqref{E:mean0}. But we infer from \eqref{E:mean0(t)} that \eqref{E:mean0} is more suitable for studying unsteady waves.
In an irrotational flow, \eqref{E:Babenko} simplifies to
\begin{equation}\label{E:Babenko0d}
(c^2+2b)\mathcal{T}_dy'-gy-g(y\mathcal{T}_dy'+\mathcal{T}_d(yy'))=0.
\end{equation}
Moreover, we may redefine the square of the wave speed~$=c^2+2b$ so long as it is positive. Therefore, for zero vorticity, the Stokes wave problem is to find $d\in(0,\infty]$, $c^2+2b\in(0,\infty)$ and a $2\upi$ periodic and even function $y$, which satisfy \eqref{E:Babenko0d}. The result agrees with what \cite{DKSZ1996, DZK1996}, for instance, derived. In stark contrast, for nonzero constant vorticity, one must determine $b$ as part of the solution by solving \eqref{E:Babenko} and \eqref{E:<Stokes>} simultaneously for $y$ and $b$.
In the infinite depth, in addition, the periodic Hilbert transform replaces $\mathcal{T}_d$ and we may take $b=0$ (see the discussion following \eqref{E:ww;infty}), whence \eqref{E:Babenko0d} further simplifies to
\begin{equation}\label{E:Babenko0}
c^2\mathcal{H}y'-gy-g(y\mathcal{H}y'+\mathcal{H}(yy'))=0.
\end{equation}
\cite{LH1978} proposed a collection of infinitely many equations for the Fourier coefficients of a Stokes wave, which \cite{Babenko1987} rediscovered in the form of \eqref{E:Babenko0}, and, independently, \cite{Plotnikov1992, DKSZ1996, BDT2000a, BDT2000b}, among others. One may regard \eqref{E:main} as to extending the Babenko equation to permit constant vorticity and finite depth.
We compare \eqref{E:Babenko} and \eqref{E:Babenko0d} to learn that nonzero constant vorticity adds higher order nonlinearities to the equation, whence it may contribute to new wave phenomena. In stark contrast, we compare \eqref{E:Babenko0d} and \eqref{E:Babenko0} to learn that the mean conformal depth merely changes the scaling factor of the Fourier multiplier in the equation, whence it would not influence the qualitative properties of the solutions. The result from the present numerical computation bears it out.
\section{Numerical method}\label{sec:numerical}
We begin by writing \eqref{E:main} in the operator form as
\begin{equation}\label{E:F=0}
F(y,b;c,\omega,d)=0,
\end{equation}
where $F(y,b;c,\omega,d)=(Y,B)(y,b;c,\omega,d)$,
\begin{subequations}
\begin{align}
Y(y,b;c,\omega,d)=&(c^2+2b)\mathcal{T}_dy'-(g+c\omega)y-g(y\mathcal{T}_dy'+\mathcal{T}_d(yy'))\label{def:Y}\\
&-\tfrac12\omega^2(y^2+\mathcal{T}_d(y^2y')+y^2\mathcal{T}_dy'-2y\mathcal{T}_d(yy'))
+c\omega\langle y\rangle+\tfrac12\omega^2\langle y^2\rangle \notag
\intertext{and}
B(y,b;c,\omega,d)=&\langle(c+\omega y(1+\mathcal{T}_dy')-\omega\mathcal{T}_d(yy'))^2\rangle
-\langle(c^2+2b-2gy)((1+\mathcal{T}_dy')^2+(y')^2)\rangle.\label{def:B}
\end{align}
\end{subequations}
For any $b$, $c$, $\omega\in\mathbb{R}$ and $d\in(0,\infty]$, $Y(\cdot,b;c,\omega,d)$ maps $2\upi$ periodic and even functions to $2\upi$ periodic and even functions, whence
\[
Y(y,b;c,\omega,d)(u)=\sum_{k\in\mathbb{Z}}\widehat{Y}(k)(y,b;c,\omega,d)e^{iku}\quad\text{for $u\in\mathbb{R}$}
\]
in the Fourier series, where
\begin{subequations}\label{def:Y}
\begin{equation}\label{def:Y1}
\begin{aligned}
\widehat{Y}(k)(y,b;c,\omega,d)=\frac{1}{2\upi}\int^{\upi}_{-\upi} ((c^2&+2b)\mathcal{T}_dy'-(g+c\omega)y-g(y\mathcal{T}_dy'+\mathcal{T}_d(yy')) \\
&-\tfrac12\omega^2(y^2+\mathcal{T}_d(y^2y')+y^2\mathcal{T}_dy'-2y\mathcal{T}_d(yy')))e^{iku}~du
\end{aligned}
\end{equation}
and $\widehat{Y}(k)=\widehat{Y}(-k)$ for all $k\in\mathbb{Z}$. In what follows, we identify $Y$ with $(\widehat{Y}(0),\widehat{Y}(1),\widehat{Y}(2),\dots)$. Note that
\begin{equation}\label{def:Y0}
\widehat{Y}(0)(y,b;c,\omega,d)=\langle y(1+\mathcal{T}_dy')\rangle.
\end{equation}
\end{subequations}
\subsection{Newton-GMRES method}\label{sec:Newton}
Suppose that
\begin{equation}\label{def:y(n)}
y^{(n+1)}=y^{(n)}+\delta y^{(n)}\quad\text{and}\quad b^{(n+1)}=b^{(n)}+\delta b^{(n)}
\quad\text{for $n=0,1,2,\dots$}
\end{equation}
solve \eqref{E:F=0} and \eqref{def:Y}-\eqref{def:B} iteratively by the Newton method, where $y^{(0)}$ and $b^{(0)}$ make an initial guess, to be supplied (see Section \ref{sec:initial guess} for details), $\delta y^{(n)}$ and $\delta b^{(n)}$ solve
\begin{equation}\label{E:dF=-F}
\delta F(y^{(n)},b^{(n)};c,\omega,d)(\delta y^{(n)},\delta b^{(n)})=-F(y^{(n)},b^{(n)};c,\omega,d),
\end{equation}
$F(y^{(n)},b^{(n)};c,\omega,d)$ is defined in \eqref{def:Y}-\eqref{def:B}, and $\delta F(y^{(n)}, b^{(n)};c,\omega,d)$ is the linearization of $F(y,b;c,\omega,d)$ with respect to $y$ and $b$, and evaluated at $y=y^{(n)}$ and $b=b^{(n)}$. We use \eqref{def:Y} and \eqref{def:B}, and we make an explicit calculation to show that
\[
\delta F(y,b;c,\omega,d)(\delta y,\delta b)
=(\delta\widehat{Y}(0),\delta\widehat{Y}(1),\delta\widehat{Y}(2),\dots,\delta B)(y,b;c,\omega,d)(\delta y,\delta b),
\]
where
\begin{subequations}\label{def:dF}
\begin{align}
\delta\widehat{Y}(k)(y,b)(\delta y,\delta b)
=&\frac{1}{2\upi}\int^{\upi}_{-\upi}
((c^2+2b)\mathcal{T}_d(\delta y)'+2\delta b\mathcal{T}_dy'-(g+c\omega)\delta y \notag \\
&\qquad\quad-g(\delta y\mathcal{T}_dy'+y\mathcal{T}_d(\delta y)'+\mathcal{T}_d(y\delta y)') \notag \\
&\qquad\quad-\tfrac12\omega^2(2y\delta y+\mathcal{T}_d(y^2\delta y)'
-[2y\delta y,y]+[y^2,\delta y]))e^{iku}~du\label{def:dY1}
\intertext{for $k=1,2,\dots$, $[f_1,f_2]:=f_1\mathcal{T}_df_2'-f_2\mathcal{T}_df_1'$,}
\delta\widehat{Y}(0)(y,b)(\delta y,\delta b)
=&\langle\delta y+2y\mathcal{T}_d(\delta y)'\rangle\label{def:dY0}
\intertext{and}
\delta B(y,b)(\delta y,\delta b)
=&2\omega\langle(c+\omega y(1+\mathcal{T}_dy')-\omega\mathcal{T}_d(yy')) \notag \\
&\qquad\times(\delta y(1+\mathcal{T}_dy')+y\mathcal{T}_d(\delta y)'-\mathcal{T}_d(y\delta y)')\rangle \notag\\
&-2\langle(\delta b-g\delta y)((1+\mathcal{T}_dy')^2+(y')^2)\rangle \notag \\
&-2\langle (c^2+2b-2gy)((1+\mathcal{T}_dy')\mathcal{T}_d(\delta y)'+y'(\delta y)')\rangle.\label{def:dB}
\end{align}
\end{subequations}
We approximate $y^{(n)}$ by a truncated Fourier series and, by abuse of notation, let
\begin{equation}\label{def:f(u);N}
y^{(n)}(u):=\sum_{k=-N/2}^{N/2-1}\widehat{y^{(n)}}(k)e^{iku}
\end{equation}
for some even $N$, where $\widehat{y^{(n)}}(k)=\widehat{y^{(n)}}(-k)$ for all $k\in\mathbb{Z}$, by symmetry. We approximate $\widehat{y^{(n)}}(k)$ by a discrete Fourier transform and, by abuse of notation, let
\begin{equation}\label{def:f(k);N}
\widehat{y^{(n)}}(k):=\frac{1}{N}\sum_{j=0}^{N-1}y^{(n)}(u_j)e^{iku_j}\quad\text{for $k=-N/2,\dots, N/2-1$},
\end{equation}
where
\begin{equation}\label{def:uj}
u_j=-\upi+2\upi j/N\quad\text{for $j=0,1,\dots,N-1$}
\end{equation}
make uniform grid points of the periodic interval $[-\upi,\upi]$, and $y^{(n)}(u_j)=y^{(n)}(u_{N-j})$ for $j=0,1,\dots,N/2-1$, by symmetry. We compute \eqref{def:f(k);N} using a fast Fourier transform (FFT). We numerically approximate $\mathcal{T}_d(y^{(n)})'$ (see \eqref{def:T}) and polynomial nonlinearities, e.g. $y^{(n)}\mathcal{T}_d(y^{(n)})'$, likewise, using \eqref{def:f(u);N}-\eqref{def:uj}, an FFT and the inverse FFT.
Together, we numerically approximate \eqref{E:dF=-F}, using \eqref{def:f(u);N}-\eqref{def:uj}, an FFT and the inverse, by
\begin{equation}\label{E:Ax=b}
\begin{aligned}
(\delta\widehat{Y}(0),\delta\widehat{Y}(1),\dots,\delta\widehat{Y}(N/2-1),\delta B)
(y^{(n)},b^{(n)};c,\omega,d)&(\delta y^{(n)},\delta b^{(n)}) \\
=-(\widehat{Y}(0),\widehat{Y}(1),\dots,\widehat{Y}(N/2-1),&B)(y^{(n)},b^{(n)};c,\omega,d),
\end{aligned}
\end{equation}
where $\widehat{Y}(0)$, $\widehat{Y}(1)$, \dots, $\widehat{Y}(N/2-1)$, $B$ are in \eqref{def:Y}-\eqref{def:B}, and $\delta\widehat{Y}(0)$, $\delta\widehat{Y}(1)$, \dots, $\delta\widehat{Y}(N/2-1)$, $\delta B$ in \eqref{def:dF}; $y^{(n)}$ is in \eqref{def:f(u);N} and $\delta y^{(n)}$, likewise; $N$ is the number of the Fourier coefficients or, alternatively, the number of the grid points in \eqref{def:f(u);N}-\eqref{def:uj}. By the way, $(\delta\widehat{Y}(0),\delta\widehat{Y}(1),\dots,\delta\widehat{Y}(N/2-1),\delta B)(y^{(n)},b^{(n)};c,\omega,d)$ is not given explicitly, but for any $(\delta y^{(n)},\delta b^{(n)})$, the left side of \eqref{E:Ax=b} may be computed using an FFT and a pseudo-spectral method.
It is reasonable to solve \eqref{E:Ax=b} by a Krylov subspace method. Some excellent surveys include \cite{Greenbaum1997, Meurant1999, Saad2003, SS2007}.
But the conjugate gradient (CG) method does not seem to converge for strong positive vorticity, among others. The conjugate residual (CR) and minimal residual (MINRES) methods are better but unreliable. Perhaps, it is because for any $c$, $\omega \in\mathbb{R}$ and $d\in(0,\infty]$,
\begin{align*}
(\delta y,\delta b)
\mapsto &(c^2+2b)\mathcal{T}_d(\delta y)'+2\delta b\mathcal{T}_dy'-(g+c\omega)\delta y \\
&-g(\delta y\mathcal{T}_dy'+y\mathcal{T}_d(\delta y)'+\mathcal{T}_d(y\delta y)')
-\tfrac12\omega^2(2y\delta y+\mathcal{T}_d(y^2\delta y)'-[2y\delta y,y]+[y^2,\delta y]),
\end{align*}
where $[f_1,f_2]=f_1\mathcal{T}_df_2'-f_2\mathcal{T}_df_1'$, and, hence, \eqref{def:dY1} are not self-adjoint.
In stark contrast, for any $b\in\mathbb{R}$, for any $c$, $\omega\in\mathbb{R}$ and $d\in(0,\infty]$,
\begin{align*}
\delta y\mapsto &(c^2+2b)\mathcal{T}_d(\delta y)'-(g+c\omega)\delta y \\
&-g(\delta y\mathcal{T}_dy'+y\mathcal{T}_d(\delta y)'+\mathcal{T}_d(y\delta y)')
-\tfrac12\omega^2(2y\delta y+\mathcal{T}_d(y^2\delta y)'-[2y\delta y,y]+[y^2,\delta y])
\end{align*}
is self-adjoint. Indeed, in an irrotational flow of infinite depth, where we may take $b=0$, the CG or CR method does converge; see \cite{DLK2016, Lushnikov2016, LDS2017}, for instance.
For nonzero constant vorticity, the generalized minimal residual (GMRES) method \citep[see][for instance]{SS1986} turns out to converge in the least number of iterates, compared with the CG, CR and MINRES methods. The result reported herein is based on the GMRES method. By the way, one may instead attempt to solve
\[
(\delta F)^*(\delta F)(y^{(n)},b^{(n)};c,\omega,d)(\delta y^{(n)},\delta b^{(n)})
=-(\delta F)^*F(y^{(n)},b^{(n)};c,\omega,d)
\]
by the CG or CR method, where the asterisk denotes the adjoint. It is presently under investigation.
We emphasize that an FFT computes a discrete Fourier transform in $O(N\log N)$ operations, where $N$ is the number of the Fourier coefficients or, alternatively, the number of the grid points (see \eqref{def:f(u);N}-\eqref{def:uj}). To compare, a boundary integral method in \cite{SS1985, PTdS1988}, for instance, would compute a numerical solution in $O(N^2)$ operations, although a customized $\sqrt{N}$ grid points would possibly improve it to $O(N)$.
A uniform grid is clearly not very effective for nearly extreme waves, whose crests tend to sharpen and troughs flatter. In an irrotational flow of infinite depth, \cite{LDS2017}, for instance, proposed an auxiliary conformal mapping, which adapts the numerical points for high curvature so that an FFT computes a discrete Fourier transform in $O(\sqrt{N}\log N)$ operations in the auxiliary conformal variable for comparable resolution. In Section~\ref{sec:infty-depth}, we exercise the idea for nonzero constant vorticity in the infinite depth.
\subsection{Initial guess}\label{sec:initial guess}
It turns out that we must supply $y^{(0)}$ and $b^{(0)}$ sufficiently close to a true solution of \eqref{E:F=0} and \eqref{def:Y}-\eqref{def:B}, in order for the Newton-GMRES method in the previous subsection to converge.
For $\omega\in\mathbb{R}$ and $d\in(0,\infty]$ prescribed, we begin by taking $\omega=0$, $d=\infty$ and a Pad\'e approximation of a Stokes wave, in practice, of small amplitude:
\[
z(w)\sim w+iy_0+\sum_{m=1}^M\frac{\gamma_m}{\tan(w/2)-i\beta_m}\quad\text{for $w=u+iv\in\mathbb{C}$},
\]
where $z(u+iv)$ for $v<0$ is the conformal mapping from the lower half plane of $\mathbb{C}$ to the fluid region, $z(u+i0)$ is the fluid surface (see \eqref{def:conformal}), and $z(u+iv)$ for $v>0$ is an analytic continuation of the conformal mapping to the upper half plane of $\mathbb{C}$; $\beta_m$ and $\gamma_m$ for $m=1,2,\dots,M$ for some $M$ are the poles and residues of the Pad\'e approximation, and $y_0$ is chosen so that \eqref{E:mean0} holds.
See \cite{DLK2016}, for instance, for details. By the way, $z$ is holomorphic in the lower half plane of $\mathbb{C}$ but it may admit singularities in the upper half plane. A library of $\beta_m$ and $\gamma_m$ from zero to a nearly maximum amplitude is found in \cite{DLK}, for instance, which enables one to reconstruct Stokes waves, in an irrotational flow of infinite depth, for the relative error less than $10^{-26}$.
We then continue the numerical solution along in $\omega$ and $d$, taking the prior convergent solution as the initial guess and solving \eqref{E:F=0} and \eqref{def:Y}-\eqref{def:B} by the method in the previous subsection, until we reach a solution for the desired values of $\omega$ and $d$. One may instead recall the Stokes wave expansion; see \cite{SS1985, PTdS1988}, for instance.
To proceed, we fix $\omega$ and $d$, and continue the numerical solution along in $c$, taking the prior convergent solution as the initial guess and solving \eqref{E:F=0} and \eqref{def:Y}-\eqref{def:B} by the method in Section~\ref{sec:Newton}. But there is a caveat. For strong positive vorticity, the solution curve in the wave speed versus amplitude plane turns out to experience a ``gap" of unphysical solutions; see Figure~\ref{fig:c(s);+vor;d=1} and Figure~\ref{fig:c(s);vor=2.25}, for instance. For stronger positive vorticity, the maximum wave speed in the gap becomes considerably larger and, hence, the number of steps in $c$ one would have to take to continue along and trace out the gap. For instance, for $\omega=2.25$ and $d=1$ (see Figure~\ref{fig:c(s);vor=2.25}), the wave speed reaches the order of hundreds in the gap. It is then more effective to continue the physical solution along in $\omega$ (and $c$) from a smaller value of $\omega$.
\subsection{Convergence}\label{sec:convergence}
For a numerical solution $y^{(n)}$ and $b^{(n)}$ of \eqref{E:F=0} and \eqref{def:Y}-\eqref{def:B}, we define the residual as
\[
\text{res}(y^{(n)},b^{(n)})=\sum_{k=-N/2}^{N/2-1}|\widehat{Y}(k)(y^{(n)},b^{(n)})|^2+|B(y^{(n)},b^{(n)})|^2,
\]
where $N$ is the number of the Fourier coefficients in the numerical approximation (see \eqref{def:f(u);N}-\eqref{def:uj}). It measures how far $y^{(n)}$ and $b^{(n)}$ are from a true solution of \eqref{E:F=0} and \eqref{def:Y}-\eqref{def:B}. We say that the Newton method converges if
\begin{equation}\label{E:res13}
\text{res}(y^{(n)},b^{(n)})<\sqrt{N}10^{-13}.
\end{equation}
But if the wave speed of a numerical solution is of the order of hundreds (see Figure~\ref{fig:c(s);vor=2.25}, for instance), we necessarily relax \eqref{E:res13} to
\begin{equation}\label{E:res9}
\text{res}(y^{(n)},b^{(n)})<\sqrt{N}10^{-9}.
\end{equation}
For a numerical solution $\delta y^{(n)}$ and $\delta b^{(n)}$ of \eqref{E:dF=-F}-\eqref{def:dF} or, equivalently, \eqref{E:Ax=b}, we define the residual likewise as
\begin{align*}
\delta\text{res}(\delta y^{(n)},\delta b^{(n)})
=&\sum_{k=-N/2}^{N/2-1}|\delta\widehat{Y}(k)(y^{(n)},b^{(n)})(\delta y^{(n)},\delta b^{(n)})
+\widehat{Y}(k)(y^{(n)},b^{(n)})|^2 \\
&+|\delta B(y^{(n)},b^{(n)})(\delta y^{(n)},\delta b^{(n)})+B(y^{(n)},b^{(n)})|^2.
\end{align*}
Because \eqref{E:dF=-F}-\eqref{def:dF} is to better approximate the numerical solutions of \eqref{E:F=0} and \eqref{def:Y}-\eqref{def:B}, we do not have to insist the residual of \eqref{E:dF=-F}-\eqref{def:dF} smaller than a fraction of that of \eqref{E:F=0} and \eqref{def:Y}-\eqref{def:B}.
As \cite{Yang2010}, for instance, suggested, we say that the GMRES method converges if
\begin{equation}\label{E:res-delta}
\delta\text{res}(\delta y^{(n)},\delta b^{(n)})<\epsilon\,\text{res}(y^{(n)},b^{(n)})
\end{equation}
for some small $\epsilon$, say, $0.01$.
To compare, \cite{SS1985} and \cite{VB1996} required the residuals less than $10^{-11}$, and \cite{PTdS1988} the residuals less than $10^{-10}$.
In the present computation, the number of the Newton iterates remains constant as $N$ increases from $256$ to $2^{16}=65536$ and it rarely takes more than $12$, in order for a numerical solution of \eqref{E:F=0} and \eqref{def:Y}-\eqref{def:B} to satisfy \eqref{E:res13}. The number of the GMRES iterates, on the other hand, varies from the order of tens to thousands for $N$ in the range, in order for a numerical solution of \eqref{E:dF=-F}-\eqref{def:dF} to satisfy \eqref{E:res-delta}.
\subsection{Error}\label{sec:error}
Lastly, we require that a numerical solution $y^{(n)}$ (and $b^{(n)}$) of \eqref{E:F=0} and \eqref{def:Y}-\eqref{def:B} satisfies
\begin{equation}\label{E:error}
|\widehat{y^{(n)}}(N/2)|<10^{-12},
\end{equation}
where $N$ is the number of the Fourier coefficients in the numerical approximation, so that the truncation error in \eqref{def:f(u);N} is insignificant. In general, we have to take $N$ considerably larger for higher waves, in order to accurately resolve the numerical solutions. Indeed, in an irrotational flow of infinite depth, if one requires that a numerical solution satisfies
\[
|\widehat{y^{(n)}}(N/2)|<N^{-1/2}10^{-26}
\]
in place of \eqref{E:error}, then $N=256$ for the steepness $=0.0994457$, but $N=2^{27}\approx1.3\times10^8$ in order to estimate the steepness of the wave of greatest height up to $32$ digits; see \cite{DLK2016}, for instance. In the present computation, we take $N$ in the range of $256$ to $2^{16}=65536$, and we do not attempt to find a solution if it requires more than $2^{16}$ Fourier coefficients to satisfy \eqref{E:error}, because it takes too much time.
\section{Result}\label{sec:result}
If $y$, $b$ and $c$ make a solution of \eqref{E:main} or, equivalently, \eqref{E:F=0} and \eqref{def:Y}-\eqref{def:B} for some $\omega$ and $d$, then it is straightforward to verify that so do $y$, $b$ and $-c$ for $-\omega$ and $d$. In what follows, we assume that $c$ is positive and, in turn, allow that $\omega$ be either positive or negative, representative of waves propagating upstream or downstream, respectively; see \cite{PTdS1988}, for instance.
We take $g=1$ for simplicity. Recall that the period is $2\pi$. The steepness $s$ measures the crest-to-trough height divided by $2\pi$.
\subsection{Finite depth}\label{sec:finite depth}
Throughout the subsection, $d=1$ for simplicity.
\subsubsection*{Zero or negative vorticity}
\begin{figure}
\centerline{\includegraphics[scale=1.1]{Figure1.eps}}
\caption{Wave speed versus steepness for $d=1$, $\omega=0$ (green) and $\omega=-1$ (red).}
\label{fig:c(s);vor=0,-1}
\end{figure}
\begin{figure}
\centerline{\includegraphics[scale=1.1]{Figure2a.eps}\hspace*{-20pt}
\includegraphics[scale=1.05]{Figure2b.eps}}
\caption{(left) For $\omega=0$ and $d=1$, the wave profile in the $(x,y)$ plane of the solution at the end point of the $c=c(s)$ curve in Figure~\ref{fig:c(s);vor=0,-1}.
(right) For $\omega=-1$ and $d=1$, the wave profile of the solution at the end point of the solution curve in Figure~\ref{fig:c(s);vor=0,-1}. The mean fluid surface is at $y=0$ and the mean fluid depths are marked by dashed lines.}
\label{fig:y(x);vor=0,-1}
\end{figure}
We begin by taking $\omega=0$, and numerically solve \eqref{E:F=0} and \eqref{def:Y}-\eqref{def:B} using the method in the previous section. Figure~\ref{fig:c(s);vor=0,-1} includes the wave speed versus steepness from the result. It resembles the well-known result in an irrotational flow of infinite depth; see the inset of Figure~\ref{fig:c(s);d=infty}, for instance.
The left panel of Figure~\ref{fig:y(x);vor=0,-1} displays the wave profile of the numerical solution at the end point of the $c=c(s)$ curve in Figure~\ref{fig:c(s);vor=0,-1}, in the $(x,y)$ plane for $x\in[-\upi,\upi]$, for which calculated are $c=0.9679$, $s=0.1024$ and $h=1.0398$. The mean fluid surface is at $y=0$. Clearly, it is near an {\em extreme wave}, which exhibits a sharp corner at the crest. Like in the infinite depth \citep[see][for instance]{LHF1978}, we conjecture that $c$ experiences infinitely many oscillations and $s$ increases monotonically toward the extreme wave. But the numerical solutions beyond the end point of the $c=c(s)$ curve in Figure~\ref{fig:c(s);vor=0,-1} require more than $2^{16}=65536$ Fourier coefficients to satisfy \eqref{E:error}, whence we do not compute them. We note that $h$ varies little throughout the $c=c(s)$ curve.
For $\omega=-1$, in Figure~\ref{fig:c(s);vor=0,-1} is the wave speed versus steepness. It resembles the result for $\omega=0$. In the right panel of Figure~\ref{fig:y(x);vor=0,-1} is the wave profile of the numerical solution at the end point of the $c=c(s)$ curve; $c=0.6285$, $s=0.0404$ and $h=1.0286$. The steepness is noticeably less than for $\omega=0$.
We verify what \cite{PTdS1988} obtained, using a boundary integral method. For instance, for $\omega=-1$ and $h=1$, \citep[Figure~$3(a)$]{PTdS1988} reports a solution for $c=0.5883$ and the crest-to-trough height $=0.12$. We find one for $c=0.5883$ and the crest-to-trough height $=0.1201$, $d=0.9978$.
We may carry out the numerical computation for other values of negative vorticity. We predict that the result resembles that for $\omega=0$ or $-1$. See \cite{SS1985}, for instance, for some details, but for $d=\infty$. We predict that the crest becomes lower for stronger negative vorticity.
\subsubsection*{Positive vorticity}
For weak positive vorticity, the result resembles that for zero or negative vorticity (see \cite{SS1985}, for instance, for some details for $d=\infty$), but not for strong positive vorticity. Figure~\ref{fig:c(s);+vor;d=1} includes the wave speed versus steepness for several values of positive vorticity.
For $\omega=1.7$, the inset of Figure~\ref{fig:c(s);+vor;d=1} reveals that the steepness increases, decreases and then increases along the $c=c(s)$ curve. Namely, a {\em fold} develops. Consequently, there correspond two or three solutions for some $s$. We predict that the continuation of the solution is limited by an extreme wave, which seems no longer the wave of greatest height. We note that there are no overhanging profiles throughout the $c=c(s)$ curve.
\begin{figure}
\centerline{\includegraphics[scale=1.1]{Figure3.eps}}
\caption{
Wave speed versus steepness for $d=1$, $\omega=1.7$ (black), $1.8$ (green), $1.95$ (blue) and $2$ (red). Solid curves physical solution, and the dashed curve unphysical solution. The inset is a closeup of the $c=c(s)$ curves for $s<1$.}
\label{fig:c(s);+vor;d=1}
\end{figure}
Figure~\ref{fig:c(s);+vor;d=1} indicates that the fold increases in size as $\omega$ increases. Moreover, waves observedly become more rounded for stronger positive vorticity.
In particular,
for $\omega=1.95$, we find an overhanging wave, whose profile is no longer the graph of a single valued function.
\begin{figure}
\centerline{\includegraphics[scale=1.1]{Figure4a.eps}}
\centerline{\includegraphics[scale=1.1]{Figure4b.eps}}
\caption{
(top) Wave speed versus steepness for $\omega=2$ and $d=1$. The inset is a closeup near the end point of the numerical continuation. (bottom) Wave profiles of the solutions labelled by $A$ through $F$. The mean fluid surface is at $y=0$ and the mean fluid depths are marked by dashed lines.}
\label{fig:c(s);vor=2}
\end{figure}
For $\omega=2$, the top panel of Figure~\ref{fig:c(s);vor=2} (see also Figure~\ref{fig:c(s);+vor;d=1}) shows the wave speed versus steepness. We continue the solution along the $c=c(s)$ curve from zero to a {\em touching wave}, whose profile self-intersects somewhere along the trough line, trapping an air bubble. Beyond such a limiting wave, the numerical solution becomes unphysical because the fluid surface in the conformal variable, $u\mapsto (u+\mathcal{T}_dy(u), y(u))$, (see \eqref{E:Ty}) is no longer injective for $u\in[-\upi,\upi]$.
We continue the unphysical solution along the fold of the $c=c(s)$ curve, to reach another touching wave, and beyond such a limiting wave, the numerical solution becomes physical. Therefore, a {\em gap} of unphysical solutions develops in the $c=c(s)$ curve, which is limited by touching waves. We remark that the numerical method in \cite{SS1985}, for instance, diverges in the gap. In stark contrast, the present numerical method converges throughout the $c=c(s)$ curve.
The bottom panel of Figure~\ref{fig:c(s);vor=2} displays six profiles along the $c=c(s)$ curve: $s=0.8092$, $h=3.6882$; $s=1.2997$, $h=7.2563$; $s=1.9939$, $h=1.4664$; $s=1.5615$, $h=8.3509$; $s=1.0310$, $h=3.4647$; $s=1.0228$, $h=3.3992$, respectively.
Wave $A$ is single valued and $B$ is near a touching wave. By the way, it resembles a limiting capillary wave. Overhanging occurs somewhere between waves $A$ and $B$. Wave $C$ is in the gap, whose steepness is the maximum. It is unphysical because the fluid region over one period overlaps adjacent ones. Wave $D$ is near another touching wave. We find that wave $D$ traps a larger air bubble than $B$. Wave $E$ is single valued. We find that waves become more rounded from zero to wave $C$ and less rounded beyond $C$, so that overhanging ultimately disappears toward the extreme wave. Wave $F$ is at the end point of the $c=c(s)$ curve in the top panel, beyond which the numerical solutions require more than $2^{16}=65536$ Fourier coefficients for accurate resolution. We note that $h$ varies wildly along the $c=c(s)$ curve.
\begin{figure}
\centerline{\includegraphics[scale=1.1]{Figure5.eps}}
\caption{
Wave speed versus steepness for $\omega=2.25$ and $d=1$. The inset is a closeup near the limiting waves.}
\label{fig:c(s);vor=2.25}
\end{figure}
Moreover, we find that the gap increases in size as $\omega$ increases. For instance, for $\omega=2.25$, Figure~\ref{fig:c(s);vor=2.25} reveals that the wave speed reaches the order of hundreds in the gap. To compare, for $\omega=2$ (see Figure~\ref{fig:c(s);vor=2}), the wave speed does not exceed $30$. By the way, for $\omega=2.25$, we discontinue the numerical solution at $c=220$ and allow \eqref{E:res9} in place of \eqref{E:res13} in the gap. In order to locate a solution in the branch from a touching wave to an extreme wave of the $c=c(s)$ curve, without tracing out the gap, we begin by taking $\omega=2$ and a physical solution in the touching-to-extreme branch, and continue it along in $\omega$ toward $\omega=2.25$ while $c$ is held fixed.
To summarize, the fold develops around $\omega=1.7$, and increases in size as $\omega$ increases. Overhanging waves appear for some $\omega$ in the range $1.8$ to $1.95$. The gap develops for some $\omega$ in the range $1.95$ to $2$, and becomes larger as $\omega$ increases.
For stronger positive vorticity, we predict that one continues the solution from zero to a touching wave, which marks the outset of a gap; across the gap is another touching wave, which encloses a larger air bubble; one continues the solution through a fold until one reaches an extreme wave.
\subsection{Infinite depth}\label{sec:infty-depth}
We turn the attention to $d=\infty$. For zero vorticity, \cite{LDS2017} proposed an auxiliary conformal mapping: for $\lambda>0$,
\begin{equation}\label{def:zeta}
w(\zeta)=2\arctan\Big(\lambda\tan\frac{\zeta}{2}\Big),
\quad\text{where}\quad \zeta=\xi+i\eta\quad\text{and}\quad w=u+iv,
\end{equation}
conformally maps the lower half plane of $\mathbb{C}$ of $2\upi$ period in the $\xi$ variable to the lower half plane of $\mathbb{C}$ of $2\upi$ period in the $u$ variable and, moreover, $w\to\pm\upi$ as $\zeta\to\pm\upi$.
Since $u\sim\lambda\xi$ for $\lambda\ll1$, \eqref{def:zeta} maps uniform grid points in the $\xi$ variable to non-uniform in the $u$ variable, concentrating them near $u=0$ by a factor of $\lambda$ and spreading them out near $u=\pm\upi$ by a factor of $1/\lambda$. Moreover, an FFT computes a discrete Fourier transform in $O(\sqrt{N})$ operations in the $\xi$ variable, whereas it takes $O(N)$ operations in the $u$ variable, where $N$ is the number of grid points over one period. It is particularly effective for nearly extreme waves, whose crests tend to sharpen and troughs flatter. For instance, in an irrotational flow of infinite depth, a uniform grid in the $u$ variable requires $2^{27}\approx1.3\times10^8$ Fourier coefficients to estimate the wave of greatest height up to $32$ digits \citep[see][for instance]{DLK2016}, whereas a uniform grid in the $\xi$ variable and, hence, a non-uniform grid in the $u$ variable use about $4.2\times10^4$ Fourier coefficients \citep[see][]{LDS2017} for comparable resolution.
The result in the subsection makes use of \eqref{def:zeta} for nonzero constant vorticity. Unfortunately, to the best of the authors' knowledge, no such mapping is known in the finite depth. It is an interesting question for future investigation.
\begin{figure}
\centerline{\includegraphics[scale=1.1]{Figure6.eps}}
\caption{
Wave speed versus steepness for $d=\infty$, $\omega = 0$ (blue), $1.7$ (black), $1.8$ (green) and $3$ (red). Solid curves for physical solution, and dashed curves for unphysical solution. The inset is a closeup of the $c=c(s)$ curve for $\omega=0$.
}
\label{fig:c(s);d=infty}
\end{figure}
\begin{figure}
\centerline{\includegraphics[scale=1.1]{Figure7.eps}}
\caption{Wave speed versus steepness for $d=1$ (green) and $\infty$ (black) for $\omega = 1.7$ and $1.8$.
}
\label{fig:d-comparison}
\end{figure}
Figure~\ref{fig:c(s);d=infty} includes the wave speed versus steepness for several values of vorticity. It agrees with \citep[Figure~$9$]{SS1985}, using a boundary integral method. By the way, the vorticity in \cite{SS1985} differs in sign. Moreover, it resembles the result in the finite depth; see Figure~\ref{fig:c(s);+vor;d=1}. Indeed, Figure~\ref{fig:d-comparison} indicates that the effects of depth are merely to change steepness and other quantities, and they are insignificant otherwise. We note that greater depths allow larger waves for higher speeds.
For $\omega=0$, the inset of Figure~\ref{fig:c(s);d=infty} reproduces the well-known result of \cite{LHF1978}, among others, that single valued profiles tend to an extreme wave as the steepness increases monotonically. See also \cite{LDS2017}, among others. Like in the previous subsection in the finite depth, the fold develops for some $\omega$ in the range $1.3$ to $1.5$, and increases in size as $\omega$ increases. Overhanging waves appear for some $\omega$ in the range $1.6$ to $1.7$. The gap develops around $\omega=1.7$, and becomes larger as $\omega$ increases. See \cite{SS1985} for more details.
For $\omega=3$, there seems to correspond two, one or zero solutions for some $s$ less than the maximum. By the way, like in the previous subsection in the finite depth, we discontinue the numerical solution at $c$ at the order of hundreds, and locate a solution in the touching-to-extreme branch of the $c=c(s)$ curve by continuing along in $\omega$ from a smaller value.
\begin{figure}
\centerline{\includegraphics[scale=1.1]{Figure8a.eps}}
\centerline{\includegraphics[scale=1.1]{Figure8b.eps}}
\centerline{\includegraphics[scale=1.0]{Figure8c.eps}}
\caption{(top) Wave speed versus steepness for $\omega=1.74$ and $d=\infty$. (middle) Wave profiles of the solutions labelled by $A$ through $I$. The mean fluid surface is at $y=0$. (bottom) Wave profile of the solution labelled by $X$ in Figure~\ref{fig:simmen}.}
\label{fig:simmen}
\end{figure}
Figure~\ref{fig:simmen} displays the wave speed versus steepness for $\omega=1.74$ and a selection of wave profiles along the $c=c(s)$ curve. The middle panel is, qualitatively and quantitatively, in excellent agreement with \citep[Figure~$8$]{SS1985}. The bottom panel shows the wave profile of the numerical solution in the gap of the $c=c(s)$ curve, labelled by $X$, for which calculated are $c=16.4$ and $s=1.6139$. Clearly, it is unphysical. By the way, the numerical method in \cite{SS1985}, for instance, diverges in the gap.
\subsection{Large vorticity limit}\label{sec:max omega}
Throughout the subsection, $d=\infty$.
\begin{figure}
\centerline{\includegraphics[scale=1.1]{Figure9.eps}}
\caption{Wave profiles of the solutions for $d=\infty$, $\omega=4.3$ and $c =34.0$ ($A$, blue), $\omega=5.5$ and $c= 36.0$ ($B$, green), $\omega=7.6$ and $c = 48.5$ ($C$, red), and $\omega=14.0$ and $c=51.7$ ($D$, black).}
\label{fig:vor->infty}
\end{figure}
In Figure~\ref{fig:vor->infty}, we begin by taking $\omega=4.3$ and wave $A$ in the gap close to the touching-to-extreme branch of the $c=c(s)$ curve, for which $c=34.0$. We take wave $A$ as the initial guess and continue the solution along in $\omega$ and $c$, to reach wave $B$, for which $\omega=5.5$ and $c=36.0$. We continue the solution along in $\omega$ and $c$, likewise, to reach waves $C$ and $D$; $\omega=7.6$, $c=48.5$ and $\omega=14.0$, $c=51.7$, respectively. We note that waves $B$, $C$ and $D$ are in the touching-to-extreme branches of the $c=c(s)$ curves. They become more rounded for stronger vorticity, and the trough becomes wider. Consequently, a {\em neck} develops in the profile, which decreases in size as $\omega$ increases. We note that waves $B$, $C$ and $D$ resemble the profiles in \citep[Figure~$5$][for instance]{VB1996} at the zero gravity limit.
\begin{figure}
\centerline{\includegraphics[scale=1.1]{Figure10a.eps}}
\centerline{\includegraphics[scale=1.1]{Figure10b.eps}}
\caption{(top) Wave speed versus steepness for $\omega=14$ and $d=\infty$. Solid curves for physical solution and dashed curves for unphysical solution. The inset is a closeup near the boundaries of the two solution branches. (bottom) Wave profiles of the solutions labelled by $A$, $B$ and $C$. The inset is a closeup near the neck.}
\label{fig:w14}
\end{figure}
Figure~\ref{fig:w14} displays the wave speed versus steepness for $\omega=14$ and a selection of wave profiles along the $c=c(s)$ curve. Wave $A$ is near the end point of the zero-to-touching branch of the $c=c(s)$ curve. Wave $B$ is near the outset of the touching-to-extreme branch, and wave $C$ is the end point of the $c=c(s)$ curve. Beyond wave $C$, the numerical solutions require more than $2^{16}=65536$ Fourier coefficients for accurate resolution. Interestingly, the touching-to-extreme branch seems to intersect the gap. Wave $D$ of Figure~\ref{fig:vor->infty} seems located somewhere between waves $B$ and $C$ of Figure~\ref{fig:w14}. By the way, we discontinue the numerical solution in the gap at $c\approx250$, and locate wave $B$ by continuing along in $\omega$ and $c$ from smaller values.
We note that wave $A$ closely resembles \citep[Figure~$4(b)$][for instance]{VB1996} at the zero gravity limit. \cite{DH2} will study in detail the limiting wave at the end points of the zero-to-touching branches of the $c=c(s)$ curves as the strength of positive vorticity increases unboundedly or, equivalently, as gravitational acceleration vanishes.
We find that the neck decreases in size along the gap until it reaches a minimum, for which $c$ is much less than a maximum, and then remains constant, particularly, from waves $B$ to $C$. On the other hand, we note that waves become more rounded along the fold until $c$ reaches the maximum, and then less rounded. Moreover, we note that the numerical solutions are ultimately limited by an extreme wave, which exhibits a sharp corner at the crest. Together, we predict that the ``fluid bubble" disappears somewhere in the touching-to-extreme branch of the $c=c(s)$ curve.
Figure~\ref{fig:vor->infty} reveals that the neck becomes narrower for stronger positive vorticity.
Indeed, we predict that the minimum neck size vanishes as the strength of positive vorticity increases unboundedly. Moreover, we predict that the limiting wave at the large vorticity limit is rigid body rotation of a fluid disk. \cite{DH2} will confirm it.
\subsection{Minimum depth limit}\label{sec:min depth}
\begin{figure}
\centerline{\includegraphics[scale=1]{Figure11.eps}}
\caption{For $\omega=3$ and $c=6$, the wave profile near the minimum of the mean conformal depth. The mean fluid surface is $y=0$ and the mean fluid depth is marked by the dashed line.
}
\label{fig4:min-depth}
\end{figure}
Lastly, we investigate a limit as the mean conformal depth decreases to the minimum. For instance, for $\omega=3$ and $c=6$ fixed, $d$ decreases to $\approx0.2193$. Figure~\ref{fig4:min-depth} displays the wave profile of the numerical solution near such a minimum value, for which calculated is $h=1.5329$. It resembles \citep[Figure~7(a)]{PTdS1988}, for instance, whose trough is flat and limited by the fluid bed.
\
\subsection*{Acknowledgements}
VMH is supported by the National Science Foundation under the Faculty Early Career Development (CAREER) Award DMS-1352597, an Alfred P. Sloan Research Fellowship, a Simons Fellowship in Mathematics, and by the University of Illinois at Urbana-Champaign under the Arnold O. Beckman Research Award RB14100. She is grateful to the Department of Mathematics at Brown University for its generous hospitality.
SD is supported by the National Science Foundation Award DMS-1716822.
This material is based on work supported by the National Science Foundation under DMS-1439786 while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Spring 2017 semester.
\
\bibliographystyle{jfm}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,807 |
package topicmodels.DCM;
import java.util.Arrays;
import java.util.Collection;
import structures._Corpus;
import structures._Doc;
import structures._Doc4DCMLDA;
import structures._Doc4SparseDCMLDA;
import structures._Word;
import utils.Utils;
public class sparseDCMLDA extends DCMLDA{
public double m_t, m_s, m_mu;
public sparseDCMLDA(int number_of_iteration, double converge, double beta, _Corpus c,
double lambda, int number_of_topics, double alpha, double burnIn, int lag, int newtonIter, double newtonConverge, double tParam, double sParam){
super(number_of_iteration, converge, beta, c, lambda, number_of_topics, alpha, burnIn, lag, newtonIter, newtonConverge);
m_t = tParam;
m_s = sParam;
m_mu = 1;
m_corpusSize = c.getSize();
m_newtonIter = newtonIter;
m_newtonConverge = newtonConverge;
}
protected void initialize_probability(Collection<_Doc> collection) {
m_alpha = new double[number_of_topics];
m_beta = new double[number_of_topics][vocabulary_size];
m_totalAlpha = 0;
m_totalBeta = new double[number_of_topics];
m_alphaAuxilary = new double[number_of_topics];
initialAlphaBeta();
int i=0;
for (_Doc d : collection) {
System.out.println("doc index\t"+i);
i ++;
_Doc4SparseDCMLDA doc = (_Doc4SparseDCMLDA)d;
doc.setTopics4Gibbs(number_of_topics, m_alpha, vocabulary_size);
for(_Word w: d.getWords()){
int wid = w.getIndex();
int tid = w.getTopic();
word_topic_sstat[tid][wid] ++;
}
}
imposePrior();
}
protected double calculate_log_likelihood(_Doc d) {
double docLogLikelihood = 0.0;
_Doc4SparseDCMLDA DCMDoc = (_Doc4SparseDCMLDA) d;
for (int k = 0; k < number_of_topics; k++) {
if(DCMDoc.m_topicIndicator[k]==false)
continue;
double term = Utils.lgamma(DCMDoc.m_sstat[k] + m_alpha[k]);
docLogLikelihood += term;
term = Utils.lgamma(m_alpha[k]);
docLogLikelihood -= term;
}
docLogLikelihood += Utils.lgamma(DCMDoc.m_alphaDoc);
docLogLikelihood -= Utils.lgamma(DCMDoc.getTotalDocLength() + DCMDoc.m_alphaDoc);
for (int k = 0; k < number_of_topics; k++) {
for (int v = 0; v < vocabulary_size; v++) {
double term = Utils.lgamma(DCMDoc.m_wordTopic_stat[k][v] + m_mu
* m_beta[k][v]);
docLogLikelihood += term;
term = Utils.lgamma(m_mu * m_beta[k][v]);
docLogLikelihood -= term;
}
docLogLikelihood += Utils.lgamma(m_mu * m_totalBeta[k]);
docLogLikelihood -= Utils.lgamma(DCMDoc.m_sstat[k] + m_mu
* m_totalBeta[k]);
}
docLogLikelihood += Utils.lgamma(m_t+m_s)-Utils.lgamma(m_t)-Utils.lgamma(m_s);
docLogLikelihood += Utils.lgamma(DCMDoc.m_indicatorTrue_stat+m_s)+Utils.lgamma(m_t+number_of_topics-DCMDoc.m_indicatorTrue_stat)-Utils.lgamma(m_t+m_s+number_of_topics);
return docLogLikelihood;
}
protected void initialAlphaBeta() {
Arrays.fill(m_sstat, 0);
Arrays.fill(m_alphaAuxilary, 0);
for (int k = 0; k < number_of_topics; k++) {
Arrays.fill(topic_term_probabilty[k], 0);
}
for (int k = 0; k < number_of_topics; k++){
m_alpha[k] = m_rand.nextDouble()+d_alpha;
m_alpha[k] = d_alpha;
m_alpha[k] = 1.0/number_of_topics;
for (int v = 0; v < vocabulary_size; v++) {
m_beta[k][v] = m_rand.nextDouble() + d_beta;
m_beta[k][v] = d_beta;
m_beta[k][v] = 1.0/vocabulary_size;
}
}
m_totalAlpha = Utils.sumOfArray(m_alpha);
for (int k = 0; k < number_of_topics; k++) {
m_totalBeta[k] = Utils.sumOfArray(m_beta[k]);
}
}
public double calculate_E_step(_Doc d){
_Doc4SparseDCMLDA DCMDoc = (_Doc4SparseDCMLDA)d;
DCMDoc.permutation();
sampleTopicAssignment(DCMDoc);
sampleOnOffIndicator(DCMDoc);
return 0;
}
protected void sampleTopicAssignment(_Doc4SparseDCMLDA DCMDoc){
int wid, tid;
double p;
for(_Word w:DCMDoc.getWords()){
wid = w.getIndex();
tid = w.getTopic();
DCMDoc.m_sstat[tid] --;
DCMDoc.m_wordTopic_stat[tid][wid]--;
if(m_collectCorpusStats)
word_topic_sstat[tid][wid] --;
p = 0;
double denominator = 0;
denominator += DCMDoc.m_alphaDoc;
denominator += Utils.sumOfArray(DCMDoc.m_sstat);
for(tid=0; tid<number_of_topics; tid++){
m_topicProbCache[tid] = 0;
if(DCMDoc.m_topicIndicator[tid]==false)
continue;
double term1 = 0;
term1 = topicInDocProb(tid, denominator, DCMDoc);
term1 = wordTopicProb(tid, wid, DCMDoc);
m_topicProbCache[tid] = topicInDocProb(tid, denominator, DCMDoc)
* wordTopicProb(tid, wid, DCMDoc);
if(m_topicProbCache[tid]<0)
System.out.println("negative\t"+m_topicProbCache[tid]);
p += m_topicProbCache[tid];
}
p *= m_rand.nextDouble();
tid = -1;
while(p>0 && tid<number_of_topics-1){
tid ++;
p -= m_topicProbCache[tid];
}
w.setTopic(tid);
DCMDoc.m_sstat[tid] ++;
DCMDoc.m_wordTopic_stat[tid][wid] ++;
if(m_collectCorpusStats)
word_topic_sstat[tid][wid] ++;
}
}
protected void sampleOnOffIndicator(_Doc4SparseDCMLDA DCMDoc){
for(int k=0; k<number_of_topics; k++){
boolean xk = DCMDoc.m_topicIndicator[k];
if(xk==true){
DCMDoc.m_indicatorTrue_stat --;
DCMDoc.m_alphaDoc -= m_alpha[k];
}
if(DCMDoc.m_sstat[k]>0){
xk = true;
}else{
double prob = 0;
double trueProb = 0;
double falseProb = 0;
double term1 = DCMDoc.m_alphaDoc;
double term2 = m_alpha[k];
double term3 = m_s + DCMDoc.m_indicatorTrue_stat;
double term4 = m_t + number_of_topics-1
- DCMDoc.m_indicatorTrue_stat;
//double term1 = DCMDoc.m_alphaDoc+m_alpha[k], DCMDoc.m_alphaDoc+m_alpha[k]+DCMDoc.getTotalDocLength());
//double term2 = (m_s+DCMDoc.m_indicatorTrue_stat);
double Q = term3 / term4;
for (int i = 0; i < DCMDoc.getTotalDocLength(); i++) {
double QTemp = (term1 + i) / (term1 + term2 + i);
Q *= QTemp;
}
falseProb = 1.0/(Q+1);
trueProb = 1-falseProb;
System.out.println("falseProb:\t"+falseProb);
prob = m_rand.nextDouble()*(trueProb+falseProb);
if(prob<trueProb)
xk = true;
else
xk = false;
}
DCMDoc.m_topicIndicator[k] = xk;
if(xk==true){
DCMDoc.m_indicatorTrue_stat++;
DCMDoc.m_alphaDoc += m_alpha[k];
}
}
}
protected double gammaRatio(double nominator, double denominator){
double ratio = 1;
double initialVal = denominator;
while(initialVal>nominator){
ratio *= (initialVal-1);
initialVal -= 1;
}
return ratio;
}
protected double topicInDocProb(int tid, double denominator, _Doc4SparseDCMLDA d){
double term1 = d.m_sstat[tid];
term1 += m_alpha[tid];
return term1/denominator;
}
protected double wordTopicProb(int tid, int wid, _Doc d) {
_Doc4DCMLDA DCMDoc = (_Doc4DCMLDA) d;
return (DCMDoc.m_wordTopic_stat[tid][wid] + m_mu * m_beta[tid][wid])
/ (DCMDoc.m_sstat[tid] + m_mu * m_totalBeta[tid]);
}
protected void updateAlpha(){
fixedPointUpdateAlpha();
// System.out.println("iteration\t" + iteration);
m_totalAlpha = 0;
for (int k = 0; k < number_of_topics; k++) {
m_totalAlpha += m_alpha[k];
}
for(_Doc d:m_trainSet){
_Doc4SparseDCMLDA DCMDoc = (_Doc4SparseDCMLDA)d;
DCMDoc.m_alphaDoc = 0;
for(int k=0; k<number_of_topics; k++){
if(DCMDoc.m_topicIndicator[k]==true)
DCMDoc.m_alphaDoc += m_alpha[k];
}
}
}
protected void fixedPointUpdateAlpha(){
double diff = 0;
double smallAlpha = 0.1;
int iteration = 0;
do {
diff = 0;
double[] wordNum4Tid = new double[number_of_topics];
double totalAlphaDenominator = 0;
double deltaAlpha = 0;
for (int k = 0; k < number_of_topics; k++) {
wordNum4Tid[k] = 0;
double totalAlphaNumerator = 0;
totalAlphaDenominator = 0;
for (_Doc d : m_trainSet) {
_Doc4SparseDCMLDA DCMDoc = (_Doc4SparseDCMLDA)d;
if(DCMDoc.m_topicIndicator[k]==false)
continue;
wordNum4Tid[k] += DCMDoc.m_sstat[k];
totalAlphaDenominator += Utils.digamma(DCMDoc.getTotalDocLength()+DCMDoc.m_alphaDoc)-Utils.digamma(DCMDoc.m_alphaDoc);
totalAlphaNumerator += Utils.digamma(m_alpha[k]
+ d.m_sstat[k])
- Utils.digamma(m_alpha[k]);
}
if(wordNum4Tid[k]==0){
deltaAlpha = 0;
}else{
deltaAlpha = totalAlphaNumerator*1.0/totalAlphaDenominator;
}
double newAlpha = m_alpha[k] * deltaAlpha+d_alpha;
double t_diff = Math.abs(m_alpha[k] - newAlpha);
if (t_diff > diff)
diff = t_diff;
m_alpha[k] = newAlpha;
}
iteration++;
if(iteration > m_newtonIter)
break;
}while(diff>m_newtonConverge);
}
protected void newtonMethodUpdateAlpha(){
double alphaSum, alphaStatSum, diAlphaSum, triAlphaSum, diAlphaStatSum, triAlphaStatSum, c, b, b1, b2;
int iteration = 0;
double diff = 0;
do{
alphaSum = Utils.sumOfArray(m_alpha);
diAlphaSum = Utils.digamma(alphaSum);
triAlphaSum = Utils.trigamma(alphaSum);
diAlphaStatSum = 0;
triAlphaStatSum = 0;
for(_Doc d:m_trainSet){
_Doc4SparseDCMLDA doc = (_Doc4SparseDCMLDA)d;
alphaStatSum = alphaSum + doc.getTotalDocLength();
diAlphaStatSum += Utils.digamma(alphaStatSum);
triAlphaStatSum += Utils.trigamma(alphaStatSum);
}
diAlphaSum = diAlphaSum*m_trainSet.size();
triAlphaSum = triAlphaSum*m_trainSet.size();
double[] q = new double[number_of_topics];
double[] g = new double[number_of_topics];
Arrays.fill(q, 0);
Arrays.fill(g, 0);
c = 0; b=0;
b1 = 0; b2 = 0;
c = triAlphaSum - triAlphaStatSum;
for(int k=0; k<number_of_topics; k++){
for(_Doc d:m_trainSet){
_Doc4SparseDCMLDA doc = (_Doc4SparseDCMLDA)d;
q[k] += Utils.trigamma(doc.m_sstat[k]+m_alpha[k])
-Utils.trigamma(m_alpha[k]);
g[k] += Utils.digamma(doc.m_sstat[k]+m_alpha[k])
-Utils.digamma(m_alpha[k]);
}
g[k] += diAlphaSum-diAlphaStatSum;
b1 += g[k]/q[k];
b2 += 1/q[k];
}
b = b1/(b2+1/c);
for(int k=0; k<number_of_topics; k++){
double t_diff = (g[k]-b)/q[k];
m_alpha[k] -= t_diff;
if (t_diff > diff)
diff = t_diff;
}
iteration++;
if(iteration > m_newtonIter)
break;
}while(diff>m_newtonConverge);
}
protected void updateBeta(int tid) {
double diff = 0;
int iteration = 0;
double smoothingBeta = 0.1;
do {
diff = 0;
double deltaBeta = 0;
double wordNum4Tid = 0;
double[] wordNum4Tid4V = new double[vocabulary_size];
double totalBetaDenominator = 0;
double[] totalBetaNumerator = new double[vocabulary_size];
Arrays.fill(totalBetaNumerator, 0);
Arrays.fill(wordNum4Tid4V, 0);
m_totalBeta[tid] = Utils.sumOfArray(m_beta[tid]);
double digBeta4Tid = Utils.digamma(m_mu * m_totalBeta[tid]);
for (_Doc d : m_trainSet) {
_Doc4DCMLDA DCMDoc = (_Doc4DCMLDA) d;
totalBetaDenominator += Utils.digamma(m_mu * m_totalBeta[tid]
+ DCMDoc.m_sstat[tid])
- digBeta4Tid;
for (int v = 0; v < vocabulary_size; v++) {
wordNum4Tid += DCMDoc.m_wordTopic_stat[tid][v];
wordNum4Tid4V[v] += DCMDoc.m_wordTopic_stat[tid][v];
totalBetaNumerator[v] += Utils.digamma(m_mu
* m_beta[tid][v]
+ DCMDoc.m_wordTopic_stat[tid][v]);
totalBetaNumerator[v] -= Utils.digamma(m_mu
* m_beta[tid][v]);
}
}
for (int v = 0; v < vocabulary_size; v++) {
if (wordNum4Tid == 0)
break;
if (wordNum4Tid4V[v] == 0) {
deltaBeta = 0;
} else {
deltaBeta = totalBetaNumerator[v] / totalBetaDenominator;
}
double newBeta = m_beta[tid][v] * deltaBeta + d_beta;
double t_diff = Math.abs(m_beta[tid][v] - newBeta);
if (t_diff > diff)
diff = t_diff;
m_beta[tid][v] = newBeta;
}
iteration++;
} while ((diff > m_newtonConverge) && (iteration < m_newtonIter));
System.out.println("iteration\t" + iteration);
}
protected void finalEst(){
runLastEM();
for (_Doc d : m_trainSet) {
estThetaInDoc(d);
}
estGlobalParameter();
}
protected void runLastEM(){
for (int j = 0; j < number_of_iteration; j++) {
init();
for (_Doc d : m_trainSet) {
_Doc4SparseDCMLDA DCMDoc = (_Doc4SparseDCMLDA) d;
calculate_E_step(DCMDoc);
if (j % 20 == 0) {
DCMDoc.m_MStepIter += 1;
for (int k = 0; k < number_of_topics; k++)
if (DCMDoc.m_topicIndicator[k] == true) {
DCMDoc.m_topicIndicator_prob[k] += 1; // miss m_s
}
DCMDoc.m_topicIndicator_distribution += DCMDoc.m_indicatorTrue_stat;
}
}
}
collectStats();
}
protected void collectStats(){
for(int k=0; k<number_of_topics; k++)
for(int v=0; v<vocabulary_size; v++)
topic_term_probabilty[k][v] = word_topic_sstat[k][v] + m_mu
* m_beta[k][v];
for(_Doc d:m_trainSet){
_Doc4SparseDCMLDA DCMDoc = (_Doc4SparseDCMLDA)d;
for (int k = 0; k < this.number_of_topics; k++) {
if(DCMDoc.m_topicIndicator[k]==false)
continue;
DCMDoc.m_topics[k] = DCMDoc.m_sstat[k] + m_alpha[k];
for (int v = 0; v < vocabulary_size; v++){
DCMDoc.m_wordTopic_prob[k][v] = DCMDoc.m_wordTopic_stat[k][v]
+ m_mu * m_beta[k][v];
}
}
}
}
protected void estThetaInDoc(_Doc d){
_Doc4SparseDCMLDA DCMDoc = (_Doc4SparseDCMLDA) d;
for (int i = 0; i < number_of_topics; i++)
Utils.L1Normalization(DCMDoc.m_wordTopic_prob[i]);
Utils.L1Normalization(d.m_topics);
DCMDoc.m_topicIndicator_distribution /= DCMDoc.m_MStepIter
* number_of_topics;
for(int k=0; k<number_of_topics; k++)
DCMDoc.m_topicIndicator_prob[k] /= DCMDoc.m_MStepIter;
}
protected void estGlobalParameter(){
for(int i=0; i<number_of_topics; i++)
Utils.L1Normalization(topic_term_probabilty[i]);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,769 |
\section{Introduction and notations}\label{intro}
In Dunkl analysis for a root system $\Sigma$ on $\R^d$, a crucial role is played by the Dunkl kernel $E_k(X,Y)$ and by the Dunkl heat kernel $p_t(X,Y)$.
Finding good estimates of the kernels $E_k$ and of the
Dunkl heat kernel $p_t$
is a challenging and important subject, developed recently in \cite{AnkerDH,PGPS1}. In this paper we prove exact estimates of both these kernels in the $W$-radial rational Dunkl case, for the root system $A_n$ with arbitrary positive multiplicities.
For a good introduction on rational Dunkl theory, the reader should consider the paper \cite{AnkerDH} or the book \cite{PGMRMY}.
We provide here some details and notations on Dunkl analysis.
For every root $\alpha\in \Sigma$, let $\sigma_\alpha(X)=X-2\,\frac{\langle \alpha,X\rangle}{\langle \alpha,\alpha\rangle}\,\alpha$.
The Weyl group $W$ associated to the root system is generated by the reflection maps $\sigma_\alpha$.
A function $k: \Sigma \to \R$
is called a multiplicity function if it is invariant under the action of $W$
on $\Sigma.$
Let $\partial_\xi$ be the derivative in the direction of $\xi\in\R^d$.
The Dunkl operators indexed by $\xi$ are then given by
\begin{align*}
T_\xi(k)\,f(X)&=\partial_\xi\,f(X)+\sum_{\alpha\in \Sigma_+}\,k(\alpha)\,\alpha(\xi)\,\frac{f(X)-f(\sigma_\alpha\,X)}{\langle \alpha,X\rangle}.
\end{align*}
The $T_\xi$'s, $\xi\in\R^d$, form a commutative family.
For fixed $ Y\in\R^d$, the Dunkl kernel $E_k(\cdot,\cdot)$ is then the only real-analytic solution to the system
\begin{align*}
\left.T_\xi(k)\right\vert_X\,E_k(X,Y)
=\langle \xi,Y\rangle\, E_k(X,Y),~\forall\xi\in\R^d
\end{align*}
with $E_k(0,Y)=1$. In fact, $E_k$ extends to a holomorphic function on $\C^d\times \C^d$.
Its $W$-invariant version $E^W_k(X,\lambda)$ is called a Bessel function of Dunkl type (see \cite[p.{} 57]{PGMRMY}) or a spherical function $\psi_\lambda(X)$ of type $\Sigma$ (refer to \cite{Sawyer}).
In this paper we use the latter terminology and notation. We have
\begin{align*}
\psi_\lambda(X) =E^W_k(X,\lambda)=\frac{1}{|W|}\sum_{w\in w}\,E_k(w\cdot X,\lambda)
\end{align*}
and $\psi_\lambda(X)$ is the only real-analytic solution of the system
\begin{align*}
\left.p(T_{{\bf e}_1},\dots,T_{{\bf e}_d})(k)\right\vert_X\,
\psi_\lambda(X)
=p(\lambda)\,\psi_\lambda(X),
\qquad
~\forall \lambda \in\R^d
\end{align*}
for every Weyl-invariant polynomial $p$ (here ${\bf e}_1$, \dots, ${\bf e}_d$ represent the standard basis on $\R^d$).
Let $\omega_k(X):=
\prod_{\alpha\in \Sigma^+}\,\vert\langle \alpha,X\rangle\vert^{2\,k(\alpha)}
$
be the Dunkl weight function on $\R^d$.
Recall that the
Dunkl transform
of a $W$-invariant function $f$
on $\R^d$
\begin{align*}
\hat f(\lambda):= c_k^{-1}
\int f(x) \psi_{-i\lambda}(X)
\omega_k(X)dX,\qquad \lambda\in \R^d,
\end{align*}
plays the role of the spherical Fourier transform in $W$-invariant Dunkl analysis.
(here the constant $c_k$ is the Macdonald--Mehta--Selberg integral.)
In \cite[Conjecture 18]{PGPS1}, we made the following conjecture on the growth of spherical functions $\psi_\lambda(x)$
of type $\Sigma$.
We use the Cartan algebra notation $\a=\R^d$. Then $\a^+$
denotes the open positive Weyl chamber with respect to a system $\Sigma^+$ of positive roots.
\begin{conjecture}\label{C}
If $\lambda$, $X\in\overline{\a^+}$, then
\begin{align*}
\psi_\lambda(e^X)
\asymp\frac{e^{\lambda(X)}}{\prod_{\alpha>0}\,(1+\alpha(X)\,\alpha(\lambda))^{k(\alpha)}}.
\end{align*}
For the root system $A_n$ on $\R^d$, $d \geq n$, and multiplicity $k(\alpha)=k>0$, this becomes
\begin{align}
\psi_\lambda(e^X)
\asymp\frac{e^{\lambda(X)}}{\prod_{i<j\leq n+1}\,(1+(x_i-x_j)\,(\lambda_i-\lambda_j))^k},
\qquad
\lambda, X\,\in\overline{\a^+}
\label{CC}
\end{align}
(the underlying constants here only depend on $k$).
\end{conjecture}
The notation $f\asymp g$ in a domain $D$ means that there exists $C_1>0$ and $C_2>0$ such that
$C_1\,g(x)\leq f(x)\leq C_2\,g(x)$ with $C_1$ and $C_2$ independent of $x\in D$.
Recall that for the root system $A_n$ on $\R^{n+1}$, the positive Weyl chamber is defined by
$\a^+=\{X\in \R^{n+1}\,|\, x_1>x_2> \ldots >x_{n+1}$\}.
\begin{remark}
This conjecture includes the cases of the symmetric spaces of noncompact type ${\bf POS}_1(n,\F)$,
the positive definite matrices of determinant 1 over $\F$ where $\F=\R$ (the real numbers with $k=1/2$), $\F=\C$ (the complex numbers with $k=1$), $\F=\H$ (the quaternion numbers with $k=2$ or $\F={\bf O}$ (the Octonions with $k=4$) when $n=3$.
In \cite{PGPS1}, we proved the conjecture for the root system $A_n$ in the complex case $k=1$.
\end{remark}
The main tool of the proof
of Conjecture \ref{C} for root systems $A_n$ is
the following iterative formula for the spherical functions of type $A$,
proven in \cite{Sawyer}.
Here we do not assume that the elements of the Lie algebra have trace 0. Here the Cartan subalgebra $\a$ for the root system $A_n$ is isomorphic to $\R^{n+1}$. For $\lambda\in\a =\R^{n+1}$
and $X\in \a^+$,we have
\begin{align}
\psi_\lambda(e^X)&=e^{\lambda(X)}\ \hbox{if $n=1$ and}\nonumber\\
\psi_\lambda(e^X)&=\frac{\Gamma(k\,(n+1))}{(\Gamma(k))^{n+1}}\,e^{\lambda_{n+1}\,\sum_{r=1}^{n+1}\,x_r}\,\pi(X)^{1-2\,k}\,\int_{x_{n+1}}^{x_n}\,\cdots\,\int_{x_2}^{x_1}
\,\psi_{\lambda_0}(e^Y)\label{iter}
\\&\left[\prod_{i=1}^n\,\left(\prod_{j=1}^i\,(x_j-y_i)\,\prod_{j=i+1}^{n+1}\,(y_i-x_j)\right)\right]^{k-1}
\,\prod_{i<j\leq n}\,(y_i-y_j)\,dy_1\cdots dy_n\nonumber
\end{align}
where $\lambda_0(U)=\sum_{r=1}^n\,(\lambda_r-\lambda_{n+1})\,u_k$
and $\pi(X)= \prod_{i<j\leq n+1}\,(x_i-x_j) $.\\
\begin{remark}
Formula \eqref{iter} concerns the action of the root system $A_{n}$ on $\R^{n+1}$. If we assume $\sum_{k=1}^{n+1}\,x_k=0=\sum_{k=1}^{n+1}\,\lambda_k$, we have then the action of the root system $A_{n}$ on $\R^{n}$. We can also consider the action of $A_{n}$ on any $\R^m$ with $m\geq n$ by deciding on which $n+1$ entries $x_k$, the roots act. These considerations do not affect the results of this article.
\end{remark}
The Dunkl heat kernel $p_t(X,Y)$ is given as
\begin{align}
p_t(X,Y)
&=\frac1{2^{\gamma+d/2} c_k}\,t^{-\frac{d}{2}-\gamma}\,e^{\frac{-|X|^2-|Y|^2}{4t}}\,E_k\left(X,\frac{Y}{2t}\right),\label{heatSpher}
\end{align}
where $\gamma= \sum_{\alpha>0} k(\alpha)$.
Establishing estimates of the Dunkl heat kernel is equivalent to estimating the Dunkl kernel as demonstrated by equation \eqref{heatSpher}.
In \cite[Lemma 4.5]{Roesler}, it is shown that
\begin{align*}
&\int_{\R^d}\,p_t(X,Y)\,\omega_k(Y)\,dY=1\\
&\left.\Delta_k\right\vert_X\,p_t(X,Y)
=\frac{\partial~}{\partial t}\,p_t(X,Y)
\end{align*}
where the Dunkl Laplacian $\Delta_k$ equals
\begin{align*}
\Delta_k\,f(X)&=\sum_{i=1}^d\,T_{{\bf e}_i}^2\,f(X)\\
&=\Delta\,f(X)+2\,\sum_{\alpha\in\Sigma^+}\,k(\alpha)\,\left[\frac{\langle \alpha,\nabla f(X)\rangle}{\langle \alpha,X\rangle}-\frac{f(X)-f(\sigma_\alpha\,X)}{\langle \alpha,X\rangle^2}\right].
\end{align*}
Here $\Delta$ and $\nabla$ denote the regular Laplacian and gradient.
The formula \eqref{heatSpher} remains true for the $W$-invariant kernels $p^W_t$ and $E^W$ and translates
in a similar relationship between the spherical function $\psi_\lambda$ and the heat kernel $p_t^W(X,Y)$:
\begin{align}\label{heatSpher1}
p_t^W(X,Y)&=\frac1{2^{\gamma+d/2} c_k} \,t^{-\frac{d}{2}-\gamma} \,e^{\frac{-|X|^2-|Y|^2}{4t}}\,\psi_X\left(\frac{Y}{2t}\right).
\end{align}
In Section \ref{comp}, we prove the Conjecture \ref{C}
for the root system $A_n$, with an arbitrary
multiplicity $k>0$, i.e.{} we prove the formula \eqref{CC}
providing exact estimates for the spherical functions $\psi_\lambda(X)$ in the two variables $X$, $\lambda$ when $\lambda$ is real.
In Section \ref{appHeat}, we apply the sharp estimates
\eqref{CC} of the spherical functions $\psi_\lambda(X)$
to the $W$-invariant Dunkl heat kernel $p_t(X,Y)$ for the root system $A_n$, with an arbitrary
multiplicity $k>0$. In the Theorem \ref{heat}, we obtain
sharp estimates of $p_t(X,Y)$ in three variables $t,X,Y$.
Next, in Sections \ref{appNewton} and \ref{appFract}, we apply the Theorem \ref{heat} to the $W$-invariant Dunkl Newton kernel and to the $W$-invariant $s$-stable semigroups, respectively. In all cases, we obtain sharp estimates.
\section{ Proof of the Conjecture in the case $A_n$.}\label{comp}
We will assume from now on that $X\in\a^+$ and $\lambda\in\overline{\a^+}$.
\begin{Note}
We will write $f(x)\lesssim g(x)$ ($f(x)\gtrsim g(x)$) for $x\in D$ if there exists a constant $C>0$ independent of $x$ such that $f(x)\leq C\,g(x)$ ($f(x)\geq C\,g(x)$) for all $x\in D$.
We will use the notation $M_k=(x_k+x_{k+1})/2$.
\end{Note}
\begin{remark}
Suppose $i<j$. We will use repeatedly the fact that the functions
\begin{align*}
\frac{x}{1+(\lambda_i-\lambda_j)\,x} \qquad \text{and} \qquad
\frac{x}{(1+(\lambda_i-\lambda_j)\,x)^{k}}, \quad k\leq 1
\end{align*}
are increasing functions of $x$. A feature of our proofs will be the distinction between the cases $0<k\leq 1$ and $k>1$.
\end{remark}
\begin{proposition}\label{induc}
Conjecture \ref{C} is equivalent to
\begin{align}\label{eq:I(n)}
I^{(n)}\asymp\frac{\pi(X)^{2\,k-1}}{\prod_{i<j\leq n+1}\,((1+(\lambda_i-\lambda_j)(x_i-x_j))^{k}}
\end{align}
where
\begin{align*}
I^{(n)}&=
\int_{x_{n+1}}^{x_n}\,\dots\int_{x_2}^{x_1}
\,e^{-\sum_{i=1}^n\,(\lambda_i-\lambda_{n+1})\,(x_i-y_i)}
\\&\qquad
\,\left(\prod_{i\leq j\leq n}\,(x_i-y_j)\,\prod_{i< j\leq n+1}\,(y_i-x_j)\right)^{k-1}
\\&\qquad
\,\prod_{i<j\leq n}\,\frac{y_i-y_j}{(1+(\lambda_i-\lambda_j)(y_i-y_j))^{k}}
\,dy_1\dots dy_n.
\end{align*}
\end{proposition}
\begin{proof}
The integral $I^{(n)}$ corresponds to a constant multiple of $e^{-\lambda(X)}\,\pi(X)^{2\,k-1}\,\psi_\lambda(e^X)$ in which we have replaced $\psi_{\lambda_0}(e^Y)$ in \eqref{iter} by its asymptotic expression conjectured in \eqref{CC}.
\end{proof}
We start by two technical results.
\begin{lemma}\label{A}
For $k>0$ and $x\geq0$, we have
\begin{align*}
\int_{0}^x\,u^{k-1}\,e^{-u} \,du\asymp\left(\frac{x}{1+x}\right)^{k}.
\end{align*}
\end{lemma}
\begin{proof}
The result is clearly true if $0\leq x<1$ (use $e^{-1}\le e^{-x}\le 1$ and integrate). If $x\geq1$ then
\begin{align*}
\int_{0}^1\,u^{k-1}\,e^{-u} \,du\leq \int_{0}^x\,u^{k-1}\,e^{-u} \,du<\int_{0}^\infty\,u^{k-1}\,e^{-u} \,du
\end{align*}
and the result follows.
\end{proof}
\begin{proposition}\label{truncated}
Assume that $\gamma=x_n-x_{n+1}$ is the largest positive root and let
\begin{align*}
I_1&=
\int_{M_n}^{x_n}\,\dots\int_{x_2}^{x_1}
\,e^{-\sum_{i=1}^n\,(\lambda_i-\lambda_{n+1})\,(x_i-y_i)}
\,\left(\prod_{i\leq j\leq n}\,(x_i-y_j)\,\prod_{i< j\leq n+1}\,(y_i-x_j)\right)^{k-1}
\\&\qquad
\,\prod_{i<j\leq n}\,\frac{y_i-y_j}{(1+(\lambda_i-\lambda_j)(y_i-y_j))^{k}}
\,dy_1\dots dy_n.
\end{align*}
Then $I_1\asymp I^{(n)}$.
\end{proposition}
\begin{proof}
Let $I_2=I^{(n)}-I_1$.
In $I_1$ and $I_2$, consider only the corresponding integral in $y_n$, calling the resulting expressions $\tilde{I}_1$ and $\tilde{I}_2$.
Observing that $y_n-x_{n+1}\asymp\gamma$ for $y_n\in[x_n,M_n]$, we have
\begin{align*}
\tilde{I}_1&\asymp\gamma^{k-1}\,\int_{M_n}^{x_n}
\,e^{-(\lambda_n-\lambda_{n+1})\,(x_n-y_n)}
\,\left(\prod_{i\leq n}\,(x_i-y_n)\right)^{k-1}
\\&\qquad\qquad\qquad
\,\prod_{i<n}\,\frac{y_i-y_n}{(1+(\lambda_i-\lambda_n)(y_i-y_n))^{k}}
dy_n.
\end{align*}
If $0<k\leq 1$ then,
\begin{align*}
\tilde{I}_1&\gtrsim\,e^{-(\lambda_n-\lambda_{n+1})\,\gamma/2}\gamma^{k-1}\,\int_{M_n}^{x_n}
\,(x_1-y_n)^{k-1}\,\prod_{i=2}^n\,(x_i-y_n)^{k-1}
\,\prod_{i<n}\,\frac{x_{i+1}-y_n}{(1+(\lambda_i-\lambda_n)\,\gamma)^{k}}
dy_n\\
&\gtrsim\gamma^{k-1}\,e^{-(\lambda_n-\lambda_{n+1})\,\gamma/2}\,\int_{M_n}^{x_n}
\,\prod_{i<n}\,\frac{(x_{i+1}-y_n)^{k}}{(1+(\lambda_i-\lambda_n)\,\gamma)^{k}} \,(x_1-y_n)^{k-1}\,
dy_n\\
&\gtrsim\gamma^{k-1}
\,\frac{e^{-(\lambda_n-\lambda_{n+1})\,\gamma/2}}{\prod_{i<n}\,(1+(\lambda_i-\lambda_n)\,\gamma)^{k}}
\,\int_{M_n}^{x_n}\,(x_n-y_n)^{(n-1)\,k} \,(x_1-y_n)^{k-1}\,
dy_n\\
&\gtrsim\gamma^{(n+1)\,k-1}\,\frac{e^{-(\lambda_n-\lambda_{n+1})\,\gamma/2}}{\prod_{i<n}\,(1+(\lambda_i-\lambda_n)\,\gamma)^{k}}.
\end{align*}
Indeed, if $n>1$ then $(x_1-y_n)^{k-1}\gtrsim \gamma^{k-1}$ and the rest can easily be integrated. If $n=1$, then
\begin{align*}
\int_{M_n}^{x_n}\,(x_n-y_n)^{(n-1)\,k} \,(x_1-y_n)^{k-1}\,
dy_n=\int_{M_n}^{x_n}\,(x_n-y_n)^{\,k-1} \,dy_n\asymp \gamma^{k}.
\end{align*}
If $k>1$, we have
\begin{align*}
\tilde{I}_1&\gtrsim \gamma^{k-1}\,e^{-(\lambda_n-\lambda_{n+1})\,\gamma/2}\,\int_{M_n}^{x_n}
\,\prod_{i=1}^{n-1}\,(x_i-y_n)^{k-1}
\,\prod_{i<n}\,\frac{y_i-y_n}{(1+(\lambda_i-\lambda_n)(y_i-y_n)^{k}}
\\&\qquad\qquad\qquad\qquad\qquad
\,(x_n-y_n)^{k-1}
dy_n\\
&\gtrsim \gamma^{k-1}\,e^{-(\lambda_n-\lambda_{n+1})\,\gamma/2}\,\int_{M_n}^{x_n}
\,\left(\prod_{i<n}\,\frac{y_i-y_n}{1+(\lambda_i-\lambda_n)(y_i-y_n)} \right)^{k}
\,(x_n-y_n)^{k-1}\,dy_n\\
&\gtrsim \gamma^{k-1}\,\frac{e^{-(\lambda_n-\lambda_{n+1})\,\gamma/2}}{\prod_{i<n}\,(1+(\lambda_i-\lambda_n)\,\gamma)^{k}}
\,\int_{M_n}^{x_n}\prod_{i<n}\,(y_i-y_n)^{k}\,\,(x_n-y_n)^{k-1}\,dy_n\\
&\gtrsim \gamma^{k-1}\,\frac{e^{-(\lambda_n-\lambda_{n+1})\,\gamma/2}}{\prod_{i<n}\,(1+(\lambda_i-\lambda_n)\,\gamma)^{k}}
\,\int_{M_n}^{x_n}\,(x_n-y_n)^{n\,k-1}\,dy_n\\
&\asymp \gamma^{(n+1)\,k-1}\,\frac{e^{-(\lambda_n-\lambda_{n+1})\,\gamma/2}}{\prod_{i<n}\,(1+(\lambda_i-\lambda_n)\,\gamma)^{k}}.
\end{align*}
On the other hand for $k>0$, observing that $x_i-y_n\asymp\gamma$ and $y_i-y_n\asymp \gamma$ for $y_n\in[M_n,x_{n+1}]$, we have
\begin{align*}
\tilde{I}_2&\asymp\gamma^{n\,(k-1)}\,\prod_{i<n}\,\frac{\gamma}{(1+(\lambda_i-\lambda_n)\gamma)^{k}} \,\int_{x_{n+1}}^{M_n}
\,e^{-(\lambda_n-\lambda_{n+1})\,(x_n-y_n)}
\,(y_n-x_{n+1})^{k-1}\,dy_n\\\\
&\lesssim \gamma^{n\,(k-1)+n-1}\,\frac{e^{-(\lambda_n-\lambda_{n+1})\,\gamma/2}}{\prod_{i<n}\,(1+(\lambda_i-\lambda_n)\gamma)^{k}} \,\int_{x_{n+1}}^{M_n}\,(y_n-x_{n+1})^{k-1}\,dy_n\\
&\asymp\gamma^{n\,(k-1)+n-1+k}\,\frac{e^{-(\lambda_n-\lambda_{n+1})\,\gamma/2}}{\prod_{i<n}\,(1+(\lambda_i-\lambda_n)\gamma)^{k}}
\lesssim \tilde{I}_1
\end{align*}
which allows us to conclude.
\end{proof}
\begin{theorem}\label{main}
Conjecture \ref{C} holds for the root system $A_n$, $n\geq 1$ with root multiplicity $k>0$.
\end{theorem}
\begin{proof}
The result is proven using induction. Using Proposition \ref{truncated} with $n=1$,
we have, using $u=(\lambda_1-\lambda_2)\,(x_1-y_1)$ and Lemma \ref{A},
\begin{align*}
I^{(1)}&\asymp I_1= \int_{M_1}^{x_1}\,e^{-(\lambda_1-\lambda_2)\,(x_1-y_1)}\,(x_1-y_1)^{k-1}\,(y_1-x_2)^{k-1}\,dy_1\\
&\asymp(x_1-x_2)^{k-1}\,\int_{M_1}^{x_1}\,e^{-(\lambda_1-\lambda_2)\,(x_1-y_1)}\,(x_1-y_1)^{k-1}\,dy_1\\
&=(x_1-x_2)^{k-1}\,(\lambda_1-\lambda_2)^{-k}\,\int_0^{(\lambda_1-\lambda_2)\,(x_1-x_2)/2}\,e^{-u}\,u^{k-1}\,du\\
&\asymp(x_1-x_2)^{k-1}\,\left(\frac{(x_1-x_2)/2}{1+(\lambda_1-\lambda_2)\,(x_1-x_2)/2}\right)^{k}
\end{align*}
which proves the formula \eqref{eq:I(n)} in the case $n=1$.
\\
Assume that the result holds for the root systems $A_1$, $A_2$, \ldots, $A_{n-1}$. We will use Proposition \ref{induc} and will proceed by assuming, in turn for each $m<n$, that $\alpha_m=x_m-x_{m+1}$ is the largest root. We will discuss the case $m=n$ at the end.
We will proceed as follows. As in the proof of Proposition \ref{truncated}, we will divide the integral in two parts $I_1$ and $I_2$, show that $I_1$ has the desired asymptotics and that $I_2\lesssim I_1$.
Assume now that $\alpha_m=x_m-x_{m+1}$, $1\leq m\leq n-1$, is the largest root.
Noting that $x_i-y_j\asymp \alpha_m$, $i\leq m$, $m<j\leq n$, $y_i-x_j\asymp \alpha_m$, $i\leq m$, $j\geq m+2$, $y_i-y_j\asymp \alpha_m$, $i\leq m$, $m<j\leq n$, for $y_m\in[M_m,x_m]$, we have
\begin{align*}
I_1&=\int_{x_{n+1}}^{x_n}\dots\int_{M_m}^{x_m}\dots\int_{x_2}^{x_1}\,\,e^{-\sum_{i=1}^n\,(\lambda_i-\lambda_{n+1})\,(x_i-y_i)}
\\&\qquad
\,\left(\prod_{i\leq j\leq m}\,(x_i-y_j)\,\prod_{i< j\leq m+1}\,(y_i-x_j)\right)^{k-1}
\\&\qquad
\,\left(\prod_{m<i\leq j\leq n}\,(x_i-y_j)\,\prod_{m<i< j\leq n+1}\,(y_i-x_j)\right)^{k-1}
\\&\qquad
\,\left(\prod_{i\leq m<j\leq n}\,(x_i-y_j)\,\prod_{\genfrac{}{}{0pt}{}{i\leq m,}{m+1<j\leq n+1}}\,(y_i-x_j)\right)^{k-1}
\\&\qquad
\,\prod_{i<j\leq m}\,\frac{y_i-y_j}{(1+(\lambda_i-\lambda_j)(y_i-y_j))^{k}}
\,\prod_{m<i<j\leq n}\,\frac{y_i-y_j}{(1+(\lambda_i-\lambda_j)(y_i-y_j))^{k}}
\\&\qquad
\,\prod_{i\leq m<j\leq n}\,\frac{y_i-y_j}{(1+(\lambda_i-\lambda_j)(y_i-y_j))^{k}}
\,dy_1\dots dy_n\\
&\asymp
\alpha_m^{2\,m\,(n-m)\,\,(k-1)}\,\prod_{i\leq m<j\leq n}\,\frac{\alpha_m}{(1+(\lambda_i-\lambda_j)\alpha_m)^{k}}
\\&\qquad
\,
\int_{x_{n+1}}^{x_n}\dots\int_{M_m}^{x_m}\dots\int_{x_2}^{x_1}
\,e^{-\sum_{i=1}^n\,(\lambda_i-\lambda_{n+1})\,(x_i-y_i)}
\\&\qquad
\,\left(\prod_{i\leq j\leq m}\,(x_i-y_j)\,\prod_{i< j\leq m+1}\,(y_i-x_j)\right)^{k-1}
\\&\qquad
\,\left(\prod_{m<i\leq j\leq n}\,(x_i-y_j)\,\prod_{m<i< j\leq n+1}\,(y_i-x_j)\right)^{k-1}
\\&\qquad
\,\prod_{i<j\leq m}\,\frac{y_i-y_j}{(1+(\lambda_i-\lambda_j)(y_i-y_j))^{k}}
\,\prod_{m<i<j\leq n}\,\frac{y_i-y_j}{(1+(\lambda_i-\lambda_j)(y_i-y_j))^{k}}
\,dy_1\dots dy_n\\
&=\frac{\alpha_m^{2\,m\,(n-m)\,(k-1)+m\,(n-m)}}
{\prod_{i\leq m<j\leq n}\,(1+(\lambda_i-\lambda_j)\alpha_m)^{k}} \,
\int_{M_m}^{x_m}\dots\int_{x_2}^{x_1}\,e^{-\sum_{i=1}^m\,(\lambda_i-\lambda_{n+1})\,(x_i-y_i)}
\\&\qquad
\,\left(\prod_{i\leq j\leq m}\,(x_i-y_j)\,\prod_{i< j\leq m+1}\,(y_i-x_j)\right)^{k-1}
\\&\qquad
\,\prod_{i<j\leq m}\,\frac{y_i-y_j}{(1+(\lambda_i-\lambda_j)(y_i-y_j))^{k}}
\,dy_1\dots dy_m
\\&\qquad
\,\int_{x_{n+1}}^{x_n}\dots\int_{x_{m+2}}^{x_{m+1}}
\,e^{-\sum_{i=m+1}^n\,(\lambda_i-\lambda_{n+1})\,(x_i-y_i)}
\\&\qquad
\,\left(\prod_{m<i\leq j\leq n}\,(x_i-y_j)\,\prod_{m<i< j\leq n+1}\,(y_i-x_j)\right)^{k-1}
\\&\qquad
\,\prod_{m<i<j\leq n}\,\frac{y_i-y_j}{(1+(\lambda_i-\lambda_j)(y_i-y_j))^{k}}
\,dy_{m+1}\dots dy_n\\
&\asymp\frac{\alpha_m^{m\,(n-m)\,
\,(2\,k-1)}}{\prod_{i\leq m<j\leq n}\,(1+(\lambda_i-\lambda_j)\,\alpha_m)^{k}}
\\&\qquad
\,\frac{\prod_{i<j\leq m}\,(x_i-x_j)^{2\,k-1}}{\prod_{i<j\leq m}\,(1+(\lambda_i-\lambda_j)\,(x_i-x_j))^{k}
\,\prod_{i\leq m}\,(1+(\lambda_i-\lambda_{n+1})\,(x_i-x_{m+1}))^{k}}
\\&\qquad
\,\prod_{m<i<j\leq n+1}\,\frac{(x_i-x_j)^{2\,k-1}}{(1+(\lambda_i-\lambda_j)\,(x_i-x_j))^{k}}
\end{align*}
which has the desired asymptotics (we used Proposition \ref{truncated} and the induction hypothesis on $A_m$ and on $A_{n-m}$).
It remains to show that $I_2=I^{(n)}-I_1\lesssim I_1$. As in the proof of Proposition \ref{truncated},
it suffices to show that $\tilde{I}_1\gtrsim\tilde{I}_2$ where $\tilde{I}_1$ (respectively $\tilde{I}_2$) is the portion of $I_1$ ($I_2$) integrated with respect to $y_m$.
Now, since $y_m-x_j\asymp \alpha_m$, $m<j\leq n+1$, and $y_m-y_j\asymp \alpha_m$, $m<j\leq n$, when $y_m\in[M_m,x_m]$, we have
\begin{align*}
\tilde{I}_1
&\asymp \frac{\alpha_m^{(n+1-m)\,(k-1)+n-m}}{\prod_{m<j\leq n}\,(1+(\lambda_m-\lambda_j)\,\alpha_m)^{k}}
\,\int_{M_m}^{x_m}\,e^{-(\lambda_m-\lambda_{n+1})\,(x_m-y_m)}\,\prod_{i\leq m}\,(x_i-y_m)^{k-1}
\\&\qquad
\,\prod_{i<m}\,\frac{y_i-y_m}{(1+(\lambda_i-\lambda_m)\,(y_i-y_m))^{k}}\,dy_m.
\end{align*}
If $k>1$ then
\begin{align*}
\tilde{I}_1
&\gtrsim\frac{\alpha_m^{(n+1-m)\,(k-1)+n-m}}{\prod_{m<j\leq n}\,(1+(\lambda_m-\lambda_j)\,\alpha_m)^{k}}
\,\,e^{-(\lambda_m-\lambda_{n+1})\,\alpha_m/2}\,\int_{M_m}^{x_m}\,\prod_{i\leq m}\,(x_m-y_m)^{k-1}
\\&\qquad
\,\prod_{i<m}\,\frac{x_m-y_m}{(1+(\lambda_i-\lambda_m)\,\alpha_m)^{k}}\,dy_m\\
&\gtrsim\frac{\alpha_m^{(n+1-m)\,(k-1)+n-m}}{\prod_{m<j\leq n}\,(1+(\lambda_m-\lambda_j)\,\alpha_m)^{k}}
\,\frac{e^{-(\lambda_m-\lambda_{n+1})\,\alpha_m/2}}{\prod_{i<m}\,(1+(\lambda_i-\lambda_m)\,\alpha_m)^{k}}
\\&\qquad
\,\int_{M_m}^{x_m}\,(x_m-y_m)^{m\,(k-1)+m-1}\,dy_m\\
&=\frac{\alpha_m^{(n+1)\,k-1}}{\prod_{m<j\leq n}\,(1+(\lambda_m-\lambda_j)\,\alpha_m)^{k}}
\,\frac{e^{-(\lambda_m-\lambda_{n+1})\,\alpha_m/2}}{\prod_{i<m}\,(1+(\lambda_i-\lambda_m)\,\alpha_m)^{k}}.
\end{align*}
If $0<k\leq 1$ then
\begin{align*}
\tilde{I}_1
&\gtrsim\frac{\alpha_m^{(n+1-m)\,(k-1)+n-m}}{\prod_{m<j\leq n}\,(1+(\lambda_m-\lambda_j)\,\alpha_m)^{k}}
\,e^{-(\lambda_m-\lambda_{n+1})\,\alpha_m/2}
\,\int_{M_m}^{x_m}\,\prod_{i< m}\,\alpha_m^{k-1}\,(x_m-y_m)^{k-1}
\\&\qquad
\,\prod_{i<m}\,\frac{x_m-y_m}{(1+(\lambda_i-\lambda_m)\,\alpha_m)^{k}}\,dy_m\\
&=\frac{\alpha_m^{(n+1-m)\,(k-1)+n-m+(m-1)\,(k-1)}}{\prod_{m<j\leq n}\,(1+(\lambda_m-\lambda_j)\,\alpha_m)^{k}}
\,\frac{e^{-(\lambda_m-\lambda_{n+1})\,\alpha_m/2}}{\prod_{i<m}\,(1+(\lambda_i-\lambda_m)\,\alpha_m)^{k}}
\\&\qquad
\,\int_{M_m}^{x_m}\,(x_m-y_m)^{m-1+k-1}\,dy_m\\
&\asymp \frac{\alpha_m^{(n+1-m)\,(k-1)+n-m+(m-1)\,(k-1)}}{\prod_{m<j\leq n}\,(1+(\lambda_m-\lambda_j)\,\alpha_m)^{k}}
\,\frac{e^{-(\lambda_m-\lambda_{n+1})\,\alpha_m/2}}{\prod_{i<m}\,(1+(\lambda_i-\lambda_m)\,\alpha_m)^{k}}
\,\alpha_m^{m-1+k}\\
&= \frac{\alpha_m^{(n+1)\,k-1}}{\prod_{m<j\leq n}\,(1+(\lambda_m-\lambda_j)\,\alpha_m)^{k}}
\,\frac{e^{-(\lambda_m-\lambda_{n+1})\,\alpha_m/2}}{\prod_{i<m}\,(1+(\lambda_i-\lambda_m)\,\alpha_m)^{k}}.
\end{align*}
On the other hand, since $x_i-y_m\asymp \alpha_m$, $i\leq m$, and $y_i-y_m\asymp \alpha_m$, $i<m$, when $y_m\in[x_{m+1},M_m]$,
\begin{align*}
\tilde{I}_2
&\asymp \alpha_m^{m\,(k-1)}\,\prod_{i<m}\,\frac{\alpha_m}{(1+(\lambda_i-\lambda_m)\,\alpha_m)^{k}}
\,\int_{x_{m+1}}^{M_m}\,e^{-(\lambda_m-\lambda_{n+1})\,(x_m-y_m)}
\\&\qquad
\,\prod_{m<j\leq n+1}\,(y_m-x_j)^{k-1}
\,\prod_{m<j\leq n}\,\frac{y_m-y_j}{(1+(\lambda_m-\lambda_j)\,(y_m-y_j))^{k}}\,dy_m.
\end{align*}
If $k>1$ then
\begin{align*}
\tilde{I}_2
&\lesssim \alpha_m^{m\,(k-1)}\,\prod_{i<m}\,\frac{\alpha_m}{(1+(\lambda_i-\lambda_m)\,\alpha_m)^{k}}
\,e^{-(\lambda_m-\lambda_{n+1})\,\alpha_m/2}
\,\int_{x_{m+1}}^{M_m}\,\prod_{m<j\leq n}\,(y_m-y_j)^{k-1}
\\&\qquad
\,\prod_{m<j\leq n}\,\frac{y_m-y_j}{(1+(\lambda_m-\lambda_j)\,(y_m-y_j))^{k}}\,(y_m-x_{n+1})^{k-1}\,dy_m\\
&=\alpha_m^{m\,(k-1)}\,\prod_{i<m}\,\frac{\alpha_m}{(1+(\lambda_i-\lambda_m)\,\alpha_m)^{k}}
\,e^{-(\lambda_m-\lambda_{n+1})\,\alpha_m/2}
\\&\qquad
\,\int_{x_{m+1}}^{M_m}
\,\prod_{m<j\leq n}\,\left(\frac{y_m-y_j}{1+(\lambda_m-\lambda_j)\,(y_m-y_j)}\right)^{k}\,(y_m-x_{n+1})^{k-1}\,dy_m\\
&\lesssim \alpha_m^{m\,(k-1)}\,\prod_{i<m}\,\frac{\alpha_m}{(1+(\lambda_i-\lambda_m)\,\alpha_m)^{k}}
\,e^{-(\lambda_m-\lambda_{n+1})\,\alpha_m/2}\,\alpha_m^{k}
\\&\qquad
\,\prod_{m<j\leq n}\,\left(\frac{\alpha_m}{1+(\lambda_m-\lambda_j)\,\alpha_m}\right)^{k}
\lesssim \tilde{I}_1.
\end{align*}
If $0<k\leq 1$ then
\begin{align*}
\tilde{I}_2
&\lesssim\alpha_m^{m\,(k-1)}\,\prod_{i<m}\,\frac{\alpha_m}{(1+(\lambda_i-\lambda_m)\,\alpha_m)^{k}}
\,e^{-(\lambda_m-\lambda_{n+1})\,\alpha_m/2}
\,\int_{x_{m+1}}^{M_m}\,\prod_{m+1<j\leq n+1}\,(y_m-x_j)^{k-1}
\\&\qquad
\,\prod_{m<j\leq n}\,\frac{y_m-x_{j+1}}{(1+(\lambda_m-\lambda_j)\,(y_m-x_{j+1}))^{k}}
\,(y_m-x_{m+1})^{k-1}\,dy_m\\
&=\alpha_m^{m\,(k-1)}\,\prod_{i<m}\,\frac{\alpha_m}{(1+(\lambda_i-\lambda_m)\,\alpha_m)^{k}}
\\&\qquad
\,e^{-(\lambda_m-\lambda_{n+1})\,\alpha_m/2}\,\int_{x_{m+1}}^{M_m}
\,\left(\prod_{m<j\leq n}\,\frac{y_m-x_{j+1}}{1+(\lambda_m-\lambda_j)\,(y_m-x_{j+1})}\right)^{k}
\\&\qquad
\,(y_m-x_{m+1})^{k-1}\,dy_m\\
&\lesssim \alpha_m^{m\,(k-1)}\,\prod_{i<m}\,\frac{\alpha_m}{(1+(\lambda_i-\lambda_m)\,\alpha_m)^{k}}
\,e^{-(\lambda_m-\lambda_{n+1})\,\alpha_m/2}\,\alpha_m^{k}
\\&\qquad
\,\left(\prod_{m<j\leq n}\,\frac{\alpha_m}{1+(\lambda_m-\lambda_j)\,\alpha_m}\right)^{k}
\lesssim \tilde{I}_1.
\end{align*}
By the structure of the root system $A_n$, the case $\alpha_n$ maximal is equivalent to the case $\alpha_1$ maximal. Indeed, in formula \eqref{iter}, one does not assume that $\lambda\in\overline{\a^+}$. We also know that $\psi_\lambda(e^X)$ is invariant under permutation of its $\lambda$ argument. Hence one can re-write \eqref{iter}
by exchanging $\lambda_1$ and $\lambda_{n+1}$,
\begin{align*}
\psi_\lambda(e^X)&=e^{\lambda(X)}\ \hbox{if $n=1$ and}\nonumber\\
\psi_\lambda(e^X)&=\frac{\Gamma(k\,(n+1))}{(\Gamma(k))^{n+1}}\,e^{\lambda_1\,\sum_{r=1}^{n+1}\,x_r}\,(\prod_{i<j\leq n+1}\,(x_i-x_j))^{1-2\,k}\,\int_{x_{n+1}}^{x_n}\,\cdots\,\int_{x_2}^{x_1}
\,\psi_{\widetilde{\lambda_0}}(e^Y)
\\&\left[\prod_{i=1}^n\,\left(\prod_{j=1}^i\,(x_j-y_i)\,\prod_{j=i+1}^{n+1}\,(y_i-x_j)\right)\right]^{k-1}
\,\prod_{i<j\leq n}\,(y_i-y_j)\,dy_1\cdots dy_n
\end{align*}
where $\widetilde{\lambda_0}(U)=\sum_{r=2}^{n+1}\,(\lambda_r-\lambda_1)\,u_r$. We used the fact that
\begin{align*}
\psi_{[\lambda_{n+1}-\lambda_1,\lambda_2-\lambda_1,\dots,\lambda_n-\lambda_1]}(e^Y)
=\psi_{[\lambda_2-\lambda_1,\dots,\lambda_n-\lambda_1,\lambda_{n+1}-\lambda_1]}(e^Y).
\end{align*}
Conjecture \ref{C} is equivalent to
\begin{align*}
J^{(n)}\asymp
\frac{\pi(X)^{2\,k-1}}{\prod_{i<j\leq n+1}\,((1+(\lambda_i-\lambda_j)(x_i-x_j))^{k}}
\end{align*}
where
\begin{align*}
J^{(n)}&=
\int_{x_{n+1}}^{x_n}\,\dots\int_{x_2}^{x_1}
\,e^{-\sum_{i=1}^{n}\,(
\lambda_1-\lambda_{i+1})\,(y_i-x_{i+1})}
\,\left(\prod_{i\leq j\leq n}\,(x_i-y_j)\,\prod_{i< j\leq n+1}\,(y_i-x_j)\right)^{k-1}
\\&\qquad
\,\prod_{i<j\leq n}\,\frac{y_i-y_j}{(1+(\lambda_{i+1}-\lambda_{j+1})(y_i-y_j))^{k}}
\,dy_1\dots dy_n.
\end{align*}
The term $J^{(n)}$ corresponds to a constant multiple of $e^{-\lambda(X)}\,\pi(X)^{2\,k-1}\,\psi_\lambda(e^X)$ in which we have replaced $\psi_{\lambda_0}(e^Y)$ in \eqref{iter} by its asymptotic expression conjectured in \eqref{CC}. One then proves the case $\alpha_n$ maximal as one proves the case $\alpha_1$ maximal.
This concludes the proof of the estimate \eqref{CC} for $X\in\a^+$ (recall that the formula \eqref{iter} holds for $X\in\a^+$). The estimates that we find for $\psi_\lambda(e^X)$ extend to $X\in\overline{\a^+}$ by continuity.
\end{proof}
\section{Applications}\label{app}
\subsection{Estimates of the $W$-invariant Dunkl Heat Kernel}\label{appHeat}
The following theorem establishes, for root systems $A_n$ and for any multiplicity $k>0$, the estimates of the $W$-invariant Dunkl Heat Kernel conjectured in the
Conjecture 18 of \cite{PGPS1}.
\begin{theorem}\label{heat}
For the root systems of type $A$, we have for $X$, $Y\in \overline{\a^+}$
\begin{align*}
p^W_t(X,Y)\asymp \frac{t^{-d/2}\,e^{-|X-Y|^2/(4\,t)}}{\prod_{\alpha>0}\,(t+\alpha(X)\,\alpha(Y))^{k}}.
\end{align*}
\end{theorem}
\begin{proof}
Consider the relation \eqref{heatSpher1}
\begin{align*}
p_t^W(X,Y)=t^{-d/2-\gamma}\,e^{-|X-Y|^2/(4\,t)}\,\psi_X(Y/(2\,t))
\end{align*}
with $\gamma=\,\sum_{\alpha>0}\,k(\alpha)= k|\Sigma^+|$. From Theorem \ref{main}, we have
\begin{align*}
p_t^W(X,Y)&\asymp t^{-d/2-\gamma}\,e^{-(|X|^2+|Y|^2)/(4\,t)}\,\frac{e^{\langle X,Y/(2\,t)\rangle}}{\prod_{\alpha>0}\,(1+\alpha(X)\,\alpha(Y/(2\,t)))^{k}}\\
&=t^{-d/2-\gamma+k\,|\Sigma^+|}\,e^{-(|X|^2+|Y|^2)/(4\,t)}\,\frac{e^{\langle X,Y/(2\,t)\rangle}}{\prod_{\alpha>0}\,(2\,t+\alpha(X)\,\alpha(Y))^{k}}\\
&\asymp \frac{t^{-d/2}\,e^{-|X-Y|^2/(4\,t)}}{\prod_{\alpha>0}\,(t+\alpha(X)\,\alpha(Y))^{k}}.
\end{align*}
\end{proof}
\subsection{Estimates of the $W$-invariant Dunkl Newton Kernel}\label{appNewton}
The $W$-invariant Dunkl Newton kernel $N^W(X,Y)$ is the kernel of the inverse operator of the Dunkl Laplacian $\Delta^W$. It
is the fundamental solution of the problem $\Delta^W\,u=f$ where $f$ is given and $|u(x)|\to0$ as $x\to\infty$. It is defined by
\begin{align*}
N^W(X,Y)=\int_0^\infty\,p^W_t(X,Y)\,dt,
\end{align*}
where $p^W_t(X,Y)$ is the heat kernel of $\Delta^W$.
In \cite{PGTLPS}, we stated the following conjecture for the Weyl invariant Newton kernel for $d\geq 3$ and proved it for complex root systems.
\begin{conjecture}\label{TLN}
For $X$, $Y\in \overline{\a^+}$ and $d\geq 3$, we have
\begin{align*}
N^W(X,Y)\asymp \frac{1}{|X-Y|^{d-2}\,\prod_{\alpha\in \Sigma^+}|X-\sigma_\alpha Y|^{2\,k(\alpha)}}.
\end{align*}
\end{conjecture}
In this section, we prove the conjecture in the case of root systems of type $A$ and prove a similar result in the case $d=2$.\\
The next three lemmas will be useful to derive sharp estimates for the Newton kernel.
\begin{lemma}\label{ai}
Suppose $k>0$, $a\geq 0$, $b_i\geq 0$,
$a+b_i>0$, $i=1$, \dots, $m$ and $N>k\,m-1$. Then
\begin{align*}
J:=\int_0^\infty\,\frac{u^N\,e^{-u}\,du}{\prod_{i=1}^m\,(a+b_i\,u)^k}
\asymp \frac{1}{\prod_{i=1}^m\,(a+b_i)^k}.
\end{align*}
\end{lemma}
\begin{proof}
Note that $a+b_i\,u\le (a+b_i)u $ whenever $u\geq 1$.
Therefore, we have
\begin{align*}
J&\geq \int_1^\infty\,\frac{u^N\,e^{-u}\,du}{\prod_{i=1}^m\,(a+b_i\,u)^k}
\geq \frac{1}{\prod_{i=1}^m\,(a+b_i)^k}\,\int_1^\infty\,u^{N-m\,k}\,e^{-u}\,du.
\end{align*}
Let $\Lambda$ be the (possibly empty) set of indices where $a\leq b_i$.
\begin{align*}
J&\leq \frac{1}{\left(\prod_{i\not\in \Lambda}\,a\right)^k}\,\int_0^\infty\,\frac{u^N\,e^{-u}\,du}{\prod_{i\in\Lambda}\,(b_i\,u)^k}
=\frac{1}{\left(\prod_{i\not\in\Lambda}\,a\right)^k\,\left(\prod_{i\in\Lambda}\,b_i\right)^k}\,\int_0^\infty\,u^{N-|\Lambda|\,k}\,e^{-u}\,du\\
&\lesssim\frac{1}{\prod_{i=1}^k\,(a+b_i)^k}\,\max_{0\leq m_0\leq m}\,\int_0^\infty\,u^{N-m_0\,k}\,e^{-u}\,du
\end{align*}
(we understand an empty product to be equal to 1).
\end{proof}
\begin{lemma}\label{a1}
Suppose $k>0$, $a>0$ and $b\geq 0$. Then
\begin{align*}
J:=\int_0^\infty\,\frac{u^{k-1}\,e^{-u}\,du}{(a+b\,u)^k}\asymp \frac{\ln (2+b/a)}{(a+b)^k}.
\end{align*}
\end{lemma}
\begin{proof}
If $0\leq b\leq a$ then
\begin{align*}
\int_0^\infty\,\frac{u^{k-1}\,e^{-u}\,du}{(a+a\,u)^k}\leq J\leq
\int_0^\infty\,\frac{u^{k-1}\,e^{-u}\,du}{a^k}
\end{align*}
and the result holds. We now assume $a\leq b$. We then have
\begin{align*}
J&\asymp\int_0^{a/b}\,\frac{u^{k-1}}{a^k}\,du+\int_{a/b}^1\,\frac{u^{k-1}\,du}{(b\,u)^k}+\int_1^\infty\,\frac{u^{k-1}\,e^{-u}\,du}{(b\,u)^k}\\
&\asymp\frac{1}{b^k}+ \frac{1}{b^k}\,\int_{a/b}^1\,u^{-1}\,du+ \frac{1}{b^k}\,\int_1^\infty\,u^{-1}\,e^{-u}\,du\\
&\asymp \frac{1}{b^k}+ \frac{1}{b^k}\,\ln (b/a)+ \frac{1}{b^k}\asymp \frac{\ln (2+b/a)}{b^k}
\end{align*}
which concludes the proof.
\end{proof}
\begin{lemma}\label{a2}
Suppose $k>0$, $a\geq 0$ and $0\leq b_1\leq b_2\leq b_3$ then
\begin{align*}
J:=\int_0^\infty\,\frac{u^{3\,k-1}\,e^{-u}\,du}{(a+b_1\,u)^k\,(a+b_2\,u)^k\,(a+b_3\,u)^k}
\asymp \frac{\ln(2+b_1/a)}{(a+b_1)^k\,(a+b_2)^k\,(a+b_3)^k}.
\end{align*}
\end{lemma}
\begin{proof}
If $b_1\leq a$ then
\begin{align*}
\int_1^\infty\,\frac{u^{3\,k-1}\,e^{-u}\,du}{(a+a\,u)^k\,((a+b_2)\,u)^k\,((a+b_3)\,u)^k}
\leq J
\leq \int_0^\infty\,\frac{u^{3\,k-1}\,e^{-u}\,du}{a^k\,(a+b_2\,u)^k\,(a+b_3\,u)^k}
\end{align*}
and the result follows in this case using Lemma \ref{ai} for the upper bound.
If $a\leq b_1\leq b_2\leq b_3$ then
\begin{align*}
J\gtrsim \int_{a/b_1}^2\,\frac{u^{3\,k-1}\,du}{(b_1\,u)^k\,(b_2\,u)^k\,(b_3\,u)^k}
\asymp \frac{\ln (2\,b_1/a)}{(b_1\,b_2\,b_3)^k}
\end{align*}
while
\begin{align*}
J&\lesssim \int_0^{a/b_1}\,\frac{u^{3\,k-1}\,du}{a^k\,(b_2\,u)^k\,(b_3\,u)^k}
+\int_{a/b_1}^2\,\frac{u^{3\,k-1}\,du}{(b_1\,u)^k\,(b_2\,u)^k\,(b_3\,u)^k}
+\int_2^\infty\,\frac{u^{3\,k-1}\,e^{-u}\,du}{(b_1\,u)^k\,(b_2\,u)^k\,(b_3\,u)^k}\\
&\asymp \frac{1}{b_1^k\,b_2^k\,b_3^k}+\frac{\ln(2\,b_1/a)}{b_1^k\,b_2^k\,b_3^k}+\frac{1}{b_1^k\,b_2^k\,b_3^k}
\end{align*}
and the result follows in this case.
\end{proof}
\begin{theorem}\label{Newton}
For the root system $A_n$ and $d\geq 3$, we have for $X$, $Y\in \overline{\a^+}$
\begin{align*}
N^W(X,Y)\asymp \frac{|X-Y|^{2-d}}
{\prod_{\alpha>0}\,|X-\sigma_\alpha\,Y|^{2\,k}}.
\end{align*}
\end{theorem}
\begin{proof}
We have, using Theorem \ref{heat} and the change of variables $u=|X-Y|^2/(4\,t)$
\begin{align*}
N^W(X,Y)&=\int_0^\infty\,p_t^W(X,Y)\,dt
\asymp \int_0^\infty\,\frac{t^{-d/2}\,e^{-|X-Y|^2/(4\,t)}\,dt}{\prod_{\alpha>0}\,(t+\alpha(X)\,\alpha(Y))^{k}}\\
&\asymp |X-Y|^{2-d}
\,\int_0^\infty\,\frac{u^{d/2-2}\,e^{-u}\,du}
{\prod_{\alpha>0}\,(|X-Y|^2/(4\,u)+\alpha(X)\,\alpha(Y))^{k}}\\
&\asymp |X-Y|^{2-d}
\,\int_0^\infty\,\frac{u^{ k|\Sigma^+|+ d/2-2}\,e^{-u}\,du}
{\prod_{\alpha>0}\,(|X-Y|^2+\alpha(X)\,\alpha(Y)\,u)^{k}}
\\
&\asymp |X-Y|^{2-d}
\,\frac{1}
{\prod_{\alpha>0}\,(|X-Y|^2+\alpha(X)\,\alpha(Y))^{k}}
\asymp \frac{|X-Y|^{2-d}}
{\prod_{\alpha>0}\,|X-\sigma_\alpha\,Y|^{2\,k}}
\end{align*}
(we have used Lemma \ref{ai} and the fact that $|X-\sigma_\alpha\,Y|^2=|X-Y|^2+2\,\alpha(X)\,\alpha(Y)$).
\end{proof}
\begin{proposition}
If $d=2$, the Newton kernel in the $A_1$ case satisfies
\begin{align*}
N^W(X,Y)\asymp \frac{\ln\left(1+\frac{|X-\sigma_\alpha Y|^2}{|X-Y|^2}\right)}{|X-\sigma_\alpha Y|^{2\,k}}\qquad
X,Y\in \overline{\a^+}.
\end{align*}
\end{proposition}
Here, it is important to recall that for $X$, $Y\in\overline{\a^+}$, we have $|X-\sigma_\alpha Y|\geq|X-Y|$ and therefore, the numerator of the last expression is at least $\ln 2$ for $X\not=Y$.
This remark also applies to the estimate in Proposition \ref{d2A2}.
\begin{proof}
With computations similar as in the case $d\geq3$, using Lemma \ref{a1},
\begin{align*}
N^W(X,Y)&=\int_0^\infty\,\frac{t^{-1}\,e^{-|X-Y|^2/(4\,t)}\,dt}{(t+\alpha(X)\,\alpha(Y))^{k}}
\asymp \int_0^\infty\,\frac{u^{k-1}\,e^{-u}\,du}{(|X-Y|^2+\alpha(X)\,\alpha(Y)\,u)^{k}}\\
&\asymp\frac{\ln(2+\alpha(X)\,\alpha(Y)/|X-Y|^2)}{(|X-Y|^2+\alpha(X)\,\alpha(Y))^{k}}
=\frac{\ln\left(\frac{2\,|X-Y|^2+\alpha(X)\,\alpha(Y)}{|X-Y|^2}\right)}{(|X-Y|^2+\alpha(X)\,\alpha(Y))^{k}}\\
&= \frac{\ln\left(\frac{3}{2}+ \frac12\frac{|X-\sigma_\alpha Y|^2}{|X-Y|^2}\right)}
{(|X-Y|^2+\alpha(X)\,\alpha(Y))^{k}}
\asymp
\frac{\ln\left(1+\frac{|X-\sigma_\alpha Y|^2}{|X-Y|^2}\right)}{|X-\sigma_\alpha Y|^{2\,k}}.
\end{align*}
\end{proof}
\begin{proposition}\label{d2A2}
If $d=2$, the Newton kernel in the $A_2$ case satisfies
\begin{align*}
N^W(X,Y)\asymp \frac{\ln\left(1+\frac{|X-\sigma_\omega\,Y|^{2\,k}}{|X-Y|^2}\right)}{|X-\sigma_\alpha Y|^{2\,k}\,|X-\sigma_\beta Y|^2\,|X-\sigma_{\alpha+\beta} Y|^{2\,k}},\qquad X,Y\in \overline{\a^+},
\end{align*}
where $\omega$ gives the minimum of $|X-\sigma_\omega Y|$ for $\omega\in\{\alpha,\beta\}$.
\end{proposition}
\begin{proof}
With computations similar as in the case $d\geq3$, using Lemma \ref{a2},
\begin{align*}
N^W(X,Y)&\asymp\int_0^\infty\,\frac{t^{-1}\,e^{-|X-Y|^2/(4\,t)}\,dt}{(t+\alpha(X)\,\alpha(Y))^{k}\,(t+\beta(X)\,\beta(Y))^{k}\,(t+(\alpha+\beta)(X)\,(\alpha+\beta)(Y))^{k}}\\
&\asymp \int_0^\infty\,\frac{u^{3\,k-1}\,e^{-u}\,du}{\prod_{\eta\in\{\alpha,\beta,\alpha+\beta\}}(|X-Y|^2+\eta(X)\,\eta(Y)\,u)^{k}}\\
&\asymp \frac{\ln(2+\omega(X)\,\omega(Y)/|X-Y|^2)}{\prod_{\eta\in\{\alpha,\beta,\alpha+\beta\}}(|X-Y|^2+\eta(X)\,\eta(Y))^{k}}\\
&\asymp\frac{\ln\left(1+\frac{|X-\sigma_\omega\,Y|^2}{|X-Y|^2}\right)}{|X-\sigma_\alpha Y|^{2\,k}\,|X-\sigma_\beta Y|^{2\,k}\,|X-\sigma_{\alpha+\beta} Y|^{2\,k}}.
\end{align*}
where $\omega$ gives the minimum of $|X-\sigma_\omega Y|$ for $\omega\in\{\alpha,\beta\}$.
\end{proof}
\begin{remark}
In the Dunkl analysis, an important role is played
by the intertwining operator $V_k$, defined
as a unique linear isomorphism on the space of polynomial functions on $\R^d$ which intertwines the Dunkl operators with the usual partial derivatives:
\begin{align*}
T_\xi V_k = V_k \partial_\xi \quad \text{ for all } \xi \in \R^d
\end{align*}
and is normalized by $V_k(1)=1$.
The following general formula for the Dunkl Newton kernel $N_k(x,y)$ involving the intertwining operator $V_k$ was proven in \cite{Gallardo2}:
\begin{align*}
N_k(X,Y)
&=\frac{2^{2\,\gamma}\,((d-2)/2)_\gamma}{|W|\,(d-2)\,w_d\,\pi(\rho)}\,V_k\left({(|Y|^2-2\,\langle X,\cdot\rangle+| X|^2)^{(2-d-2\,\gamma)/2}} \right)(Y)
\end{align*}
(we are using a slightly different normalization
of the operator $V_k$ than
\cite{PGMRMY, Roesler}, see \cite{PGTLPS} for details.)
Little is known explicitly on the intertwining operator.
Theorem \ref{Newton} and the formula
$
N^W(X,,Y)=\frac{1}{|W|}\,\sum_{w\in W}\,N_k(w\,X,Y)$ imply the following explicit asymptotic formula.
\begin{corollary}
For the root system $A_n$ and $d\geq 3$, we have for $X$, $Y\in \overline{\a^+}$
\begin{align*}
V_k\left({(|Y|^2-\frac{2}{|W|}\,\sum_{w\in W}\,
\langle w\,X,\cdot\rangle+| X|^2)^{(2-d-2\,\gamma)/2}} \right)(Y)
\asymp \frac{|X-Y|^{2-d}}
{\prod_{\alpha>0}\,|X-\sigma_\alpha\,Y|^{2\,k}}.
\end{align*}
\end{corollary}
\end{remark}
\subsection{Heat semigroups for fractional powers of $\Delta_k^W$}
\label{appFract}
Let $s\in(0,2)$. The fractional powers $(-\Delta_k^W)^{s/2}$
of the $W$-invariant Dunkl Laplacian are the infinitesimal generators of
important semigroups $(h^W_t(X,Y))_{t\ge 0}$,
called {\it $W$-invariant Dunkl $s$-stable semigroups}.
Fractional powers of the Dunkl Laplacian and related semigroups and processes were considered for $s=1$ in \cite[p.75]{RoeslerHAB},
\cite[Section 5]{RoeslerVoit} and for any $s\in(0,2)$ in \cite{Bou,Rejeb}.
Stable semigroups on Riemannian symmetric spaces of non-compact type were studied in \cite{Getoor, PG}.
Like the heat semigroup $p^W_t(X,Y)$, the densities $h^W_t(X,Y)$ are to be considered with respect to the Dunkl weight function $\omega_k(Y)$ on $\R^d$. We have
\begin{align*}
h^W_t(X,Y)=\int_0^\infty\,p^W_u(X,Y)\,\eta_t(u)\,du
\end{align*}
where $\eta_t(u)$ is the density of the $s/2$-stable subordinator, i.e. of a positive L\'evy process $(Y_t)_{t>0}$ with the Laplace transform
${\bf E}\left(\exp(z\,Y_t)\right)=\exp(-t\,z^{s/2}),\, z>0$
(see \cite{Bertoin} for more details).
Denote by $h^{\R^d}_t(X,Y))_{t\ge 0}$ the $s$-stable rotationally invariant semigroup on $\R^d$, with generator $(-\Delta)^{s/2}$.
It is known (\cite{BlGetoor}) that
\begin{align}
h^{\R^d}_t(X,Y)\asymp
\min\left\{\frac{1}{t^{d/s}},\frac{t}{|X-Y|^{d+s}}\right\}
\asymp \frac{t}{(t^{2/s}+|X-Y|^2)^{(d+s)/2}}.\label{euclid}
\end{align}
\begin{remark}\label{min}
It is useful to note that
$\min\left\{t^{-d/s},{t}{|X-Y|^{-(d+s)}}\right\}=t^{-d/s}$
if and only if $t^{2/s}\geq |X-Y|^2$.
\end{remark}
\begin{theorem} \label{th:stable}
Consider the $W$-invariant Dunkl Laplacian in the $A_n$ case with multiplicity $k>0$. Then for $X$, $Y\in \overline{\a^+}$,
\begin{align*}
h^W_t(X,Y)
\asymp
\frac{h^{\R^d}_t(X,Y)}{
\prod_{\alpha>0} (t^{2/s} +|X-\sigma_\alpha Y|^2)^{k}}
\asymp
\frac{h^{\R^d}_t(X,Y)}{
\prod_{\alpha>0} (t^{2/s} +|X- Y|^2 +\alpha(X)\alpha(Y))^{k}}.
\end{align*}
\end{theorem}
\begin{proof}
The proof is inspired by the proof of \cite[Theorem 3.1]{Bogdan} providing estimates of stable semigroups on fractals.
Given Remark \ref{min}, it will make sense to consider the cases $t^{2/s}\geq |X-Y|^2$ and $t^{2/s}\leq |X-Y|^2$ separately. In the proof, $m$ will denote the number of positive roots.
We start by showing that our estimate is an upper bound with an appropriate constant.
In \cite[(14), page 168]{Bogdan}, it it shown that the subordinator density $\eta_t(u)$ satisfies
\begin{align}
\eta_t(u)\leq C\,t\,u^{-1-s/2}\,e^{-t\,u^{-s/2}}.\label{14}
\end{align}
Hence, using our estimates of the $W$-invariant Dunkl heat kernel in Theorem \ref{heat} and the change of variable $u=|X-Y|^2/(4\,w)$, we have
\begin{align*}
h^W_t(X,Y)
&\lesssim t\,\int_0^\infty\,\frac{u^{-d/2}\,e^{-|X-Y|^2/(4\,u)}\,u^{-1-s/2}\,e^{-t\,u^{-s/2}}\,du}{\prod_{\alpha>0}\,(u+\alpha(X)\,\alpha(Y))^{k}}
\\
&\lesssim t\,|X-Y|^{-(d+s)}\,\int_0^\infty\,\frac{w^{(d+s)/2-1}\,e^{-w}\,dw}{\prod_{\alpha>0}\,(|X-Y|^2/(4\,w)+\alpha(X)\,\alpha(Y))^{k}}\\
&\asymp t\,|X-Y|^{-(d+s)}\,\int_0^\infty\,\frac{w^{(d+s)/2+k\,m-1}\,e^{-w}\,dw}{\prod_{\alpha>0}\,(|X-Y|^2+\alpha(X)\,\alpha(Y)\,w)^{k}}\\
&\asymp \frac{t\,|X-Y|^{-(d+s)}}{\prod_{\alpha>0}\,(|X-Y|^2+\alpha(X)\,\alpha(Y))^{k}}
\end{align*}
with an application of Lemma \ref{ai} to get the last equivalence. This proves the upper bound in the case $|X-Y|^2\geq t^{2/s}$.
We use Theorem \ref{heat} and the inequality \eqref{14}
again with the change of variable $u=t^{2/s}\,w^{-2/s}$. Let $\Lambda$ be the set of $\alpha>0$ such that $t^{2/s}\leq\alpha(X)\,\alpha(Y)$
with $m'$ the number of elements in $\Lambda$. We have
\begin{align*}
h^W_t(X,Y)
&\lesssim t\,\int_0^\infty\,\frac{u^{-d/2}\,u^{-1-s/2}\,e^{-t\,u^{-s/2}}\,du}{\prod_{\alpha>0}\,(u+\alpha(X)\,\alpha(Y))^{k}}\\
&\asymp
t^{-d/s}\,\int_0^\infty\,\frac{w^{d/s}\,e^{-w}\,dw}{\prod_{\alpha>0}\,(t^{2/s}\,w^{-2/s}+\alpha(X)\,\alpha(Y))^{k}}
\\
&\asymp
t^{-d/s}\,\int_0^\infty\,\frac{w^{d/s+2\,k\,m/s}\,e^{-w}\,dw}{\prod_{\alpha>0}\,(t^{2/s}+\alpha(X)\,\alpha(Y)\,w^{2/s})^{k}}\\
&\lesssim
t^{-d/s}\,\frac{1}{(\prod_{\alpha\not\in \Lambda}\,t^{2/s})^{k}}
\,\int_0^\infty\,\frac{w^{d/s+2\,k\,m/s}\,e^{-w}\,dw}{\prod_{\alpha\in \Lambda}\,(\alpha(X)\,\alpha(Y)\,w^{2/s})^{k}}\\
&\asymp
\frac{t^{-d/s}}{\prod_{\alpha>0}\,(t^{2/s}+\alpha(X)\,\alpha(Y))^{k}}
\,\int_0^\infty\,w^{d/s+2\,k(m-m')/s}\,e^{-w}\,dw.
\end{align*}
This proves the upper bound in the case $|X-Y|^2\leq t^{2/s}$.
Now we will justify the lower bound. Recall (see \cite[(9,10), page 167]{Bogdan} or \cite[(9), page 89]{PG}) that for $u\geq t^{2/s}$, we have
\begin{equation}\label{asympeta}
\eta_t(u)\asymp t\,u^{-1-s/2}.
\end{equation}
If $|X-Y|^2\geq t^{2/s}$ then using formula \eqref{asympeta},
Theorem \ref{heat} and the change of variable $u=|X-Y|^2/(4\,w)$, we have
\begin{align*}
h^W_t(X,Y)&\gtrsim t\,\int_{t^{2/s}}^\infty\,\frac{u^{-d/2}\,e^{-|X-Y|^2/(4\,u)}\,u^{-1-s/2}\,du}{\prod_{\alpha>0}\,(u+\alpha(X)\,\alpha(Y))^{k}}\\
&\gtrsim t\,|X-Y|^{-(d+s)}\,\int_0^{|X-Y|^2/(4\,t^{2/s})}\,\frac{w^{(d+s)/2-1}\,e^{-w}\,dw}{\prod_{\alpha>0}\,(|X-Y|^2/(4\,w)+\alpha(X)\,\alpha(Y))^{k}}\\
&\gtrsim t\,|X-Y|^{-(d+s)}\,\int_{1/8}^{1/4}\,\frac{w^{(d+s)/2-1}\,e^{-w}\,dw}{\prod_{\alpha>0}\,(|X-Y|^2+\alpha(X)\,\alpha(Y))^{k}}
\end{align*}
which proves the lower bound in that case.
Now assume $|X-Y|^2\leq t^{2/s}$. We use formula \eqref{asympeta} and
Theorem \ref{heat}. Then, since $-1/4\leq -|X-Y|^2/(4\,t^{2/s})\leq -|X-Y|^2/(4\,u)\leq 0$ for $u\geq t^{2/s}$, using the change of variable $u=t^{2/s}\,w$, we have
\begin{align*}
h^W_t(X,Y)&\gtrsim t\,\int_{t^{2/s}}^\infty\,\frac{u^{-d/2}\,e^{-|X-Y|^2/(4\,u)}\,u^{-1-s/2}\,du}{\prod_{\alpha>0}\,(u+\alpha(X)\,\alpha(Y))^{k}}\\
&\gtrsim t\,\int_{t^{2/s}}^\infty\,\frac{u^{-(d+s)/2-1}\,du}{\prod_{\alpha>0}\,(u+\alpha(X)\,\alpha(Y))^{k}}\\
&\gtrsim t^{-d/s}\,\int_1^\infty\,\frac{w^{-(d+s)/2-1}\,dw}{\prod_{\alpha>0}\,(t^{2/s}\,w+\alpha(X)\,\alpha(Y))^{k}}\\
&\gtrsim t^{-d/s}\,\int_1^\infty\,\frac{w^{-(d+s)/2-1}\,dw}{w^{k\,m}\,\prod_{\alpha>0}\,(t^{2/s}+\alpha(X)\,\alpha(Y))^{k}}\\
&\asymp \frac{t^{-d/s}}{\prod_{\alpha>0}\,(t^{2/s}+\alpha(X)\,\alpha(Y))^{k}}
\end{align*}
which proves the lower bound in that case.
\end{proof}
\begin{remark}
The upper estimate in Theorem \ref{th:stable} may be deduced from
our estimates of the $W$-invariant Dunkl heat kernel in Theorem \ref{heat} and from \cite[Corollary 3.8 p. 11]{Trojan}.
\end{remark}
\section*{Acknowledgments}
The authors wish to acknowledge the financial support from the grants IEA00292 CNRS 2021/22 and MIR l'Universit\'e d'Angers ``Sym\'etries'' 2021.
The first author thanks the Dominican University College of Ottawa for their hospitality. The second author gratefully acknowledges the hospitality of l'Universit\'e d'Angers for his stay during which part of this paper was written.
The authors are thankful to Tomasz Grzywny, Tomasz Luks and Bartosz Trojan for helpful discussions and suggestions concerning fractional Dunkl Laplacian and stable semigroups.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,489 |
If you have saved some money and want to buy Costume Jewellery, you will have to do is follow a short guide. In fact, if you are willing to buy a necklace, earrings or other gold jewelry, you will make an excellent purchase, since gold is never devalued. So it could be a good saving for the current economic crisis.
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"redpajama_set_name": "RedPajamaC4"
} | 1,864 |
Q: Using a prompt to set the innerHTML of an element to a website with a clickable anchor The following does not work, and I don't know why. The innerhtml of the element gets changed, but I can't figure out how to create a clickable link to the answer of the prompt.
HTML:
<h5 class="user-output">My favorite website is <span id="favorite-website">... </span></h5>
<button class="button-primary" onclick=app.setWebsite()>Set Website</button>
JS:
app.setWebsite = function setWebsite() {
var click = prompt("What is your favorite website?");
document.getElementById("favorite-website").innerHTML = "find out here!";
document.getElementById("favorite-website").href = click
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,085 |
WITH
pre_dicom_all AS (
SELECT
aux.tcia_api_collection_id AS tcia_api_collection_id,
aux.idc_webapp_collection_id AS collection_id,
aux.collection_timestamp AS collection_timestamp,
aux.collection_hash as collection_hash,
aux.collection_init_idc_version AS collection_init_idc_version,
aux.collection_revised_idc_version AS collection_revised_idc_version,
aux.access AS access,
dcm.PatientID as PatientID,
aux.idc_case_id as idc_case_id,
aux.patient_hash as patient_hash,
aux.patient_init_idc_version AS patient_init_idc_version,
aux.patient_revised_idc_version AS patient_revised_idc_version,
dcm.StudyInstanceUID AS StudyInstanceUID,
aux.study_uuid as crdc_study_uuid,
aux.study_hash as study_hash,
aux.study_init_idc_version AS study_init_idc_version,
aux.study_revised_idc_version AS study_revised_idc_version,
dcm.SeriesInstanceUID AS SeriesInstanceUID,
aux.series_uuid as crdc_series_uuid,
aux.series_hash as series_hash,
aux.series_init_idc_version AS series_init_idc_version,
aux.series_revised_idc_version AS series_revised_idc_version,
dcm.SOPInstanceUID AS SOPInstanceUID,
aux.instance_uuid as crdc_instance_uuid,
aux.gcs_url as gcs_url,
aux.instance_size as instance_size,
aux.instance_hash as instance_hash,
aux.instance_init_idc_version AS instance_init_idc_version,
aux.instance_revised_idc_version AS instance_revised_idc_version,
aux.source_doi as Source_DOI,
aux.license_url as license_url,
aux.license_long_name as license_long_name,
aux.license_short_name as license_short_name,
dcm.* except(PatientID, StudyInstanceUID, SeriesInstanceUID, SOPInstanceUID)
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,832 |
Q: android widget TextClock fade in fade out animation Does anyone know any working example of animate TextClock in Android widget?
I did quite deep research, and it seems that this is not possible as android widgets use RemoteViews which do not work with any sort of animation except ProgressBar which is not exactly made to animate between time changes on the clock.
Cheers
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,204 |
\section{Introduction}\label{sec:Intro}
Variational approximation methods \citep{Ormerod2010,blei+kj17} are an increasingly popular way to implement approximate Bayesian computations because of their ability to scale well to large datasets and highly parametrized models. A Gaussian variational approximation uses a multivariate
normal approximation to the posterior distribution, and these approximations can be both useful in themselves as well as
building blocks for more complicated variational inference procedures, such a those based on Gaussian mixtures or copulas \citep{Miller2016,Han2016}. Here we consider
Gaussian variational approximation for state space models where the state vector is high-dimensional. Models of this kind are common in spatio-temporal
modeling \citep{cressie+w11}, econometrics, and in other important applications.
Constructing a Gaussian variational approximation is challenging when considering a model having a large number of parameters because the number of variational parameters in the covariance matrix grows quadratically with the number of model parameters.
This makes it necessary to parametrize the variational covariance matrix parsimoniously, but so that we can still capture the structure of the posterior. This goal is best achieved by making intelligent use of the structure of the model itself. This is considered in the present manuscript for high-dimensional state space models. We parametrize
the variational posterior covariance matrix using a dynamic factor model which provides dimension reduction for the states, and Markovian time dependence for the low-dimensional factors provides
sparsity in the precision matrix for the factors. We develop efficient computational methods for forming the approximations and illustrate the advantages of the approach
in two high-dimensional examples. The first is a spatio-temporal dataset on the spread of the Eurasian collared dove across North America \citep{wikle+h06}. The second example is a multivariate stochastic volatility model for a collection of portfolios of assets \citep{philipov+g06}. Markov chain Monte Carlo (MCMC) is challenging in both examples, but particularly for the latter as \cite{philipov+g06} use an independence Metropolis-Hastings proposal (the performance of which is well known to deteriorate in high dimensions) within a Gibbs sampler for sampling the state vector. Indeed, \cite{philipov+g06} reported acceptance probabilities close to zero for the state vector in an application to $12$ assets, which gives a state vector of dimension $78$ in their model. We show that our variational approximation allows for efficient inference even when the state dimension is large.
Variational approximation methods formulate the problem of approximating the posterior as an optimization problem. In this paper, we use stochastic
gradient ascent methods for performing the optimization \citep{ji+sw10,nott+tvk12,Paisley2012,Salimans2013} and, in particular, the so-called
reparametrization trick for unbiased estimation of the gradients of the variational objective \citep{Kingma2013,rezende+mw14}. Section \ref{sec:stochasticgradientvariational} describes these methods further.
Applying these methods for Gaussian variational approximation, \citet{Tan2016} considered matching the sparsity of the variational precision matrix
to the conditional independence structure of the true posterior based on a sparse Cholesky factor for the precision matrix. This is motivated
because zeros in the precision matrix of a Gaussian distribution correspond to conditional independence between variables.
Sparse matrix operations allow computations in the variational optimization to be done efficiently.
They apply their approach to both random effects models and state space models, although their method is impractical in a state space model where the state is high-dimensional.
\citet{Archer2016} also
considered a similar idea for the problem of filtering in state space models using an amortization approach, where blocks of the variational mean and
sparse precision factor are parametrized in terms of functions of local data.
More recently, \citet{krishnan+ss17} considered a similar approach, but noted the importance of including future as well as past local data in the amortization
procedure.
Similar Gaussian variational approximations
to those considered in \cite{Tan2016}, \citet{Archer2016} and \citet{krishnan+ss17} were
earlier considered in the literature where the parametrization is in terms of the Cholesky factor of the covariance matrix rather than the precision matrix
\citep{Titsias2014,Kucukelbir2016}, although these approaches do not consider sparsity in the parametrization except through diagonal approximations
which lose the ability to capture posterior dependence in the approximation. Various other parametrizations of the covariance matrix
in Gaussian variational approximation were considered by \citet{Opper2009}, \citet{Challis2013} and \citet{Salimans2013}. The latter authors also consider efficient
stochastic gradient methods for fitting such approximations, using both gradient and Hessian information and exploiting other structure in the
target posterior distribution, as well as extensions to more complex hierarchical formulations including mixtures of normals.
Another way to parametrize dependence in a high-dimensional Gaussian posterior approximation is to use a factor structure.
Factor models \citep{Bartholomew2011} are well known to be useful for modeling dependence in high-dimensional settings.
\citet{ong+ns17} recently considered a Gaussian variational approximation for factor covariance structures using stochastic gradient methods
for the variational optimization. Computations in the variational optimization can be done efficiently in high-dimensions using the Woodbury formula \citep{woodbury1950inverting}.
Factor structures in Gaussian variational approximation were used previously by \citet{Barber1998} and \citet{Seeger2000}, but these authors
are concerned with situations where the variational objective can be computed analytically or with one-dimensional quadrature.
\citet{rezende+mw14} considered a factor model for the precision matrix with one factor in some applications to some deep generative models
arising in machine learning applications. \citet{Miller2016} considered factor parametrizations of covariance matrices for normal mixture components
in a flexible variational boosting approximate inference method, including a method for exploiting the reparametrization trick for unbiased
gradient estimation of the variational objective in that setting.
Our article specifically considers approximating posterior distributions for Bayesian inference in high-dimensional state space models, and we make use of both the conditional independence structure and the factor structure in forming our approximations.
Bayesian computations for state space models are well known to be challenging for complex nonlinear models. It is usually feasible to carry out MCMC on a complex state space model by sampling the states one at a time conditional on the neighbouring states (e.g., \citealp{carlin1992monte}). In general, such methods require careful convergence diagnosis for individual applications and can fail if the dependence between states is strong. \cite{carter1994gibbs} document this phenomenon in linear Gaussian state space models, and we also document this problem of poor mixing for the spatio-temporal \citep{wikle+h06} and multivariate stochastic volatility via a Wishart process \citep{philipov+g06} examples discussed later. State of the art general
approaches using particle MCMC methods \citep{Andrieu2010} can in principle be much more efficient than an MCMC that generates the states one at a time. However, particle MCMC is usually much slower than MCMC because of the need to generate multiple particles at each time point. Particle methods also have a number of other drawbacks, which depend on the model that is estimated. Thus, if there is strong dependence between the states and parameters, then it is necessary to use pseudo marginal methods
\citep{Beaumont2003, Andrieu2009} which estimate the likelihood and it is necessary to ensure that the variability of the log of the estimated likelihood is sufficiently small \citep{pitt2012some, doucet2013}. This is particularly difficult to do if the state dimension is high.
Finally, we note that factor structures are widely used
as a method for achieving parsimony in the model formulation in the state space framework for spatio-temporal data \citep{wikle+c99, lopes2008}, multivariate stochastic volatility \citep{ku2014flexible, philipov2006factor}, and in other applications \citep{aguilar+w00,carvalho2008}. This is distinct from the main idea in the present paper of using a dynamic factor structure
for dimension reduction in a variational approximation for getting parsimonious but flexible descriptions of dependence in the posterior for approximate inference.
Our article is outlined as follows. Section \ref{sec:stochasticgradientvariational} gives a background on variational approximation. Section \ref{sec:CovarianceParameterizations} reviews some previous parametrizations of the covariance matrix, in particular the methods of \citet{Tan2016} and \citet{ong+ns17}
for Gaussian variational approximation using conditional independence and a factor structure, respectively. Section \ref{sec:Gaussian_Variational_high_dim_state} describes our methodology, which combines a factor structure for the states and conditional independence in time for the factors to obtain flexible and convenient approximations of the posterior distribution in high-dimensional state space models. Section \ref{sec:examples} describes an extended example for a spatio-temporal dataset in ecology concerned with the spread of the Eurasian collared-dove across North America. Section \ref{sec:examples2} considers inference in a multivariate stochastic volatility model via Wishart processes. Appendix \ref{app:grad_expressions_lemmas} contains the necessary gradient expressions to implement our method.
Technical derivations and other details are placed in the supplement. We refer to equations, sections, etc in the main paper as (1), Section 1, etc, and in the supplement as (S1), Section S1, etc.
\section{Stochastic gradient variational methods\label{sec:stochasticgradientvariational}}
\subsection{Variational objective function}
Variational approximation methods \citep{Attias1999,Jordan1999,Winn2005} reformulate the problem of approximating an intractable
posterior distribution as an optimization problem. Let $\theta=(\theta_1,\dots,\theta_d)^\top$ be the vector of model parameters, $y=(y_1,\dots,y_n)^\top$ the observations
and consider Bayesian inference for $\theta$ with a prior density $p(\theta)$.
Denoting the likelihood by $p(y|\theta)$, the posterior density is $p(\theta|y)\propto p(\theta)p(y|\theta)$, and in variational approximation we consider a family of densities $\{ q_\lambda(\theta)\}$, indexed by the variational parameter $\lambda$, to approximate $p(\theta|y)$. Our article takes the approximating family to be Gaussian so that $\lambda$ consists
of the mean vector and the distinct elements of the covariance matrix in the approximating normal density.
To formulate the approximation of $p(\theta|y)$ as an optimization problem, we take the Kullback-Leibler (KL) divergence,
\begin{align*}
\mathrm{KL}(q_\lambda(\theta)||p(\theta|y)) & = \int \log \frac{q_\lambda(\theta)}{p(\theta|y)}q_\lambda(\theta)\;d\theta,
\end{align*}
as the distance between
$q_\lambda(\theta)$ to $p(\theta|y)$. The KL divergence is non-negative and zero if and only if $q_\lambda(\theta)=p(\theta|y)$.
It is straightforward to show that $\log p(y)$, where $p(y)=\int p(\theta)p(y|\theta)\;d\theta$, can be expressed as
\begin{align}
\log p(y) & = {\cal L}(\lambda)+\mathrm{KL}(q_\lambda(\theta)||p(\theta|y)), \label{logpy}
\end{align}
where
\begin{align}
{\cal L}(\lambda) & = \int \log \frac{p(\theta)p(y|\theta)}{q_\lambda(\theta)}q_\lambda(\theta)\;d\theta \label{lb}
\end{align}
is referred to as the variational lower bound or evidence lower bound (ELBO). The non-negativity of the KL divergence implies that $\log p(y)\geq {\cal L}(\lambda)$, with
equality if and only if $q_\lambda(\theta)=p(\theta|y)$. Because $\log p(y)$ does not depend on $\lambda$, we see from (\ref{logpy}) that minimizing
the KL divergence is equivalent to maximizing the ELBO in (\ref{lb}). For introductory overviews of variational methods for statisticians
see \citet{Ormerod2010} and \citet{blei+kj17}.
\subsection{Stochastic gradient optimization}
Maximizing ${\cal L}(\lambda)$ to obtain an optimal approximation of $p(\theta|y)$ is often difficult in models with a non-conjugate prior structure, since ${\cal L}(\lambda)$
is defined as an integral which is generally intractable. However, stochastic gradient methods \citep{Robbins1951,bottou10} are useful for performing the optimization
and there is now a large literature surrounding the application of this idea \citep[among others]{ji+sw10,Paisley2012,nott+tvk12,Salimans2013,Kingma2013,rezende+mw14,Hoffman2013,Ranganath2014,Titsias2015,Kucukelbir2016}.
In a simple stochastic gradient ascent method for optimizing ${\cal L}(\lambda)$, an initial guess for the optimal value $\lambda^{(0)}$ is updated according to the iterative
scheme
\begin{align}
\lambda^{(t+1)} & =\lambda^{(t)}+a_t \widehat{\nabla_\lambda {\cal L}(\lambda^{(t)})} \label{robinsm},
\end{align}
where $a_t$, $t\geq 0$ is a sequence of learning rates, $\nabla_\lambda {\cal L}(\lambda)$ is the gradient vector of ${\cal L}(\lambda)$ with respect
to $\lambda$, and $\widehat{\nabla_\lambda {\cal L}(\lambda)}$ denotes an unbiased estimate of $\nabla_\lambda {\cal L}(\lambda)$.
The learning rate sequence is typically chosen to satisfy $\sum_t a_t=\infty$ and $\sum_t a_t^2<\infty$, which ensures that the iterates $\lambda^{(t)}$ converge
to a local optimum as $t\rightarrow\infty$ under suitable regularity conditions. Various adaptive choices for the learning rates are also possible and we consider the ADADELTA \citep{Zeiler2012} approach in our applications in Sections \ref{sec:examples} and \ref{sec:examples2}.
\subsection{Variance reduction}
Application of stochastic gradient methods to variational inference depends on being able to obtain the required unbiased estimates
of the gradient of the lower bound in (\ref{robinsm}). Reducing the variance of these gradient estimates as much as possible is important
for both the stability of the algorithm and fast convergence. Our article uses gradient estimates based on the so-called reparametrization trick
\citep{Kingma2013,rezende+mw14}. The lower bound ${\cal L}(\lambda)$ is an expectation with respect to $q_\lambda$,
\begin{align}
{\cal L}(\lambda) & = E_q(\log h(\theta)-\log q_\lambda(\theta)), \label{lbq}
\end{align}
where $E_q(\cdot)$ denotes expectation with respect to $q_\lambda$ and $h(\theta)=p(\theta)p(y|\theta)$. If we differentiate under the integral sign in (\ref{lbq}), the
resulting expression for the gradient can also be written as an expectation with respect to $q_\lambda$, which is easily estimated unbiasedly by Monte Carlo integration provided that
sampling from this distribution can be easily done. However, differentiating under the integral sign does not use gradient information from the model because $\lambda$ does not appear in the term involving $h(\cdot)$ in (\ref{lbq}). The reparametrization trick is a method that allows this information to be used.
We start by supposing that
$\theta\sim q_\lambda(\theta)$ can be written as $\theta=u(\lambda,\omega)$, where $\omega$ is a random vector with density $f$
which does not depend on the variational parameters $\lambda$. For instance, in the case of a multivariate normal density where $q_\lambda(\theta)=N(\mu,\Sigma)$ with $\Sigma=CC^\top$ and
$C$ denotes the (lower triangular) Cholesky factor of $\Sigma$, we can write $\theta=\mu+C\omega$ where $\omega\sim N(0,I_d)$ where $I_d$ is the $d\times d$ identity matrix.
Substituting $\theta=u(\lambda,\omega)$ into (\ref{lbq}), we obtain
\begin{align}
{\cal L}(\lambda) & = E_f(\log h(u(\lambda,\omega))-\log q_\lambda(u(\lambda,\omega))), \label{lbqr}
\end{align}
and then differentiating under the integral sign
\begin{align}
\nabla_\lambda {\cal L}(\lambda) & = E_f(\nabla_\lambda \log h(u(\lambda,\omega))-\nabla_\lambda \log q_\lambda(u(\lambda,\omega))), \label{lbqr2}
\end{align}
which is an expectation with respect to $f$ that is easily estimated unbiasedly if we can sample from $f$. Note that gradient estimates obtained for the lower
bound this way use gradient information from the model, and it has been shown empirically that gradient estimates by the reparametrization trick have greatly reduced
variance compared to alternative approaches.
We now discuss variance reduction beyond the reparametrization trick. \citet{Roeder2017}, generalizing arguments in \citet{Salimans2013}, \citet{Han2016} and \citet{Tan2016}, show that (\ref{lbqr2}) can equivalently be written
as
\begin{align}
\nabla_\lambda {\cal L}(\lambda) & = E_f\left(\frac{d u(\lambda,\omega)}{d\lambda}\left\{\nabla_\theta \log h(u(\lambda,\omega))-\nabla_\theta \log q_\lambda(u(\lambda,\omega))\right\}\right), \label{lbq2}
\end{align}
where $d u(\lambda,\omega)/d\lambda$ is the matrix with element $(i,j)$ the partial derivative of the $i$th element of $u$ with respect to the $j$th element of
$\lambda$. Note that if the approximation is exact, i.e. $q_\lambda(\theta)\propto h(\theta)$, then a Monte Carlo approximation to the expectation on the right hand
side of (\ref{lbq2}) is exactly zero even if such an approximation is formed using only a single sample from $f(\cdot)$. This is one reason to prefer (\ref{lbq2})
as the basis for obtaining unbiased estimates of the gradient of the lower bound if the approximating variational family is flexible enough to provide
an accurate approximation. However, \citet{Roeder2017} show that the extra terms that arise when (\ref{lbqr2}) is used directly for estimating the gradient
of the lower bound can be thought of as acting as a control variate, i.e. it reduces the variance, with a scaling that can be estimated empirically, although the computational cost of this
estimation may not be worthwhile. In our state space model applications, we consider using both (\ref{lbqr2}) and (\ref{lbq2}), because our approximations may be very rough when the
dynamic factor parametrization of the variational covariance structure contains only a small number of factors. Here, it may not be so relevant
to consider what happens in the case where the approximation is exact as a guide for reducing the variability of gradient estimates.
\section{Parametrizing the covariance matrix\label{sec:CovarianceParameterizations}}
\subsection{Cholesky factor parametrization of $\Sigma$\label{subsec:cholesky_of_covariance}}
\citet{Titsias2014} considered normal variational posterior approximation using a Cholesky factor parametrization and used stochastic gradient methods
for optimizing the KL divergence. \citet{Challis2013} also considered Cholesky factor parametrizations in Gaussian variational approximation,
but without using stochastic gradient optimization methods.
For gradient estimation, \citet{Titsias2014} consider the reparametrization trick with
$\theta=\mu+C\omega$, where $\omega \sim f(\omega)=N(0,I_d)$, $\mu$ is the variational posterior mean and
$\Sigma=CC^\top$ is the variational posterior covariance with lower triangular Cholesky factor $C$ and with the diagonal elements of $C$ being positive. Hence, $\lambda = (\mu, C)$ and (\ref{lbqr}) becomes, apart from terms not depending on $\lambda$,
\begin{align}\label{eq:ELBO_chol_param_Sigma}
{\cal L}(\lambda) & = E_f(\log h(\mu+C\omega))+ \log |C|,
\end{align}
and note that $\log |C|=\sum_i \log C_{ii}$ since $C$ is lower triangular. \citet{Titsias2014} derived the gradient of \eqref{eq:ELBO_chol_param_Sigma}, and it is straightforward to estimate the expectation $E_f$ unbiasedly by simulating one or more samples $\omega\sim f$ and computing the average, i.e. plain Monte Carlo integration. The method can also be considered in conjunction with data subsampling. \citet{Kucukelbir2016} considered a similar approach.
\subsection{Sparse Cholesky factor parametrization of $\Omega=\Sigma^{-1}$}
\citet{Tan2016} considered an approach which parametrizes the precision matrix $\Omega=\Sigma^{-1}$ in terms of its Cholesky factor,
$\Omega=CC^\top$ say, and imposed a sparse structure in $C$ which comes from the conditional independence structure in the model.
To minimize notation, we continue to write $C$ for a Cholesky factor used to parametrize the variational posterior even
though here it is the Cholesky factor of the precision matrix rather than of the covariance matrix as in the previous subsection.
Similarly to \citet{Tan2016}, \citet{Archer2016} also considered parametrizing a Gaussian variational approximation using
the precision matrix, but they optimize directly with respect to the elements $\Omega$, while also exploiting sparse matrix computations in obtaining the Cholesky factor of $\Omega$. \cite{Archer2016} were also concerned with
state space models and imposed a block tridiagonal structure on the variational posterior precision matrix for the states, using functions of local
data parametrized by deep neural networks to describe blocks of the mean vector and precision matrix corresponding to different states.
Here we follow \citet{Tan2016} and consider parametrization of the variational optimization in terms of the Cholesky factor $C$ of $\Omega$. In this section we consider the case where no restrictions are placed on the elements of $C$ and discuss in Section \ref{sec:Gaussian_Variational_high_dim_state} how the conditional independence structure in the model can be used to impose a sparse structure on $C$. We note at the outset that sparsity is very important for reducing the number of variational parameters that need to be optimized and considering a sparse $C$ allows the Gaussian variational approximation method to be extended to high-dimensional settings.
Consider the reparametrization trick once again with $q_\lambda(\theta)=N(\mu,C^{-\top}C^{-1})$, where $C^{-\top}$ means $(C^{-1})^\top$, and $\lambda=(\mu,C)$. For $\theta\sim q_\lambda(\theta)$, we can
write $\theta=\mu+C^{-\top}\omega$, with $\omega \sim N(0,I_d)$. Similarly to Section \ref{subsec:cholesky_of_covariance},
\begin{align*}
{\cal L}(\lambda) & = E_f(\log h(\mu+C^{-\top}\omega)-\log q_\lambda(\mu+C^{-\top}\omega)),
\end{align*}
which, apart from terms not depending on $\lambda$, is
\begin{align}\label{eq:ELBO_tan_and_nott}
{\cal L}(\lambda) & = E_f(\log h(\mu+C^{-\top}\omega))-\log |C|,
\end{align}
and note that $\log |C|=\sum_i \log C_{ii}$ since $C$ is lower triangular. \citet{Tan2016} derived the gradient of
\eqref{eq:ELBO_tan_and_nott} and, moreover, considered some improved gradient estimates for which \citet{Roeder2017} have provided a more general understanding. We apply the approach of \citet{Roeder2017} to our methodology in Section \ref{sec:Gaussian_Variational_high_dim_state}.
\subsection{Latent factor parametrization of $\Sigma$\label{subsec:OngEtAl_review}}
While the method of \citet{Tan2016} is an attractive way to reduce the number of variational parameters in problems with an exploitable conditional independence structure, there are situations where no such structure is available. An alternative parsimonious way to parametrize dependence is to use a factor structure
\citep{Geweke1996,Bartholomew2011}. \citet{ong+ns17} parametrized the variational posterior covariance matrix $\Sigma$ as
$\Sigma=BB^\top + D^2$, where $B$ is a $d\times q$ matrix with $q\ll d$ and for identifiability $B_{ij}=0$ for $i<j$ and $D$ is a diagonal matrix
with diagonal elements $\delta=(\delta_1,\dots, \delta_d)^\top$. The variational posterior becomes $q_\lambda(\theta)= N(\mu,BB^\top+D^2)$ with $\lambda=(\mu,B,\delta)$.
If $\theta\sim N(\mu,BB^\top+D^2)$, this corresponds to the generative
model $\theta=B\omega+\delta\odot\kappa$ with $(\omega,\kappa)\sim N(0,I_{d+q})$, where $\odot$ denotes elementwise multiplication. \citet{ong+ns17} applied the reparametrization trick based on this transformation and derive gradient expressions of the resulting evidence lower bound. \citet{ong+ns17} outlined how to efficiently implement the computations and we discuss this further in Section \ref{subsec:EfficientComp}.
\section{Methodology\label{sec:Gaussian_Variational_high_dim_state}}
\subsection{Structure of the posterior distribution}
Our Gaussian variational distribution is suitable for models with the following structure. Let $y=(y_1,\dots,y_T)^\top$ be an observed time series, and consider a state space model in which
\begin{align*}
y_t|X_t=x_t & \sim m_t(y|x_t,\zeta), \;\;\;\;t=1,\dots, T \\
X_t|X_{t-1}=x_{t-1} & \sim s_t(x|x_{t-1},\zeta), \;\;\;\; t=1,\dots, T
\end{align*}
with a prior density $p(X_0|\zeta)$ for $X_0$ and where $\zeta$ are the unknown fixed (non-time-varying) parameters in the model.
The observations $y_t$ are conditionally independent given the states $X=(X_0^\top,\dots, X_T^\top)^\top$, and
the prior distribution of $X$ given $\zeta$ is
$$p(X|\zeta)=p(X_0|\zeta)\prod_{t=1}^T s_t(X_t|X_{t-1},\zeta).$$
Let $\theta=(X^\top, \zeta^\top)^\top$ denote the full set of unknowns in the model. The joint posterior density of $\theta$ is
$p(\theta|y)\propto p(\theta)p(y|\theta)$,
with $p(\theta)=p(\zeta)p(X|\zeta)$, where $p(\zeta)$ is the prior density for $\zeta$ and $p(y|\theta)=\prod_{t=1}^T m_t(y_t|X_t,\zeta)$. Let $p$ be the dimension of $X_t$ and consider the situation where $p$ is large. Approximating the joint posterior distribution in this setting is difficult and we now describe a method based on Gaussian variational approximation.
\subsection{Structure of the variational approximation}\label{subsec:structure_variational_posterior}
Our variational posterior density $q_\lambda(\theta)$ for $\theta$, is based on a generative model which has a dynamic factor structure. We assume that
\begin{align}
X_t & =Bz_t+\epsilon_t\;\;\;\;\epsilon_t\sim N(0,D_t^2), \label{ldsm}
\end{align}
where $B$ is a $p\times q$ matrix, $q\ll p$, and $D_t$ is a diagonal matrix with diagonal elements $\delta_t=(\delta_{t1},\dots,\delta_{tp})^\top$.
Let $z=(z_0^\top,\dots, z_T^\top)^\top$ and
$\rho=(z^\top,\zeta^\top)^\top\sim N(\mu,\Sigma)$, $\Sigma=C^{-\top}C^{-1}$ where $C$ is the Cholesky factor of the precision matrix of
$\rho$. We will write $q$ for the dimension of each $z_t$, with $q \ll p = \dim(X_t)$, and assume that $C$ has the structure
$$C=\left[\begin{array}{cc}
C_1 & 0 \\\
0 & C_2
\end{array}\right],$$
where $C_1$ is the Cholesky factor of the precision matrix $\Omega_1 = C_1 C_1^\top$ for $z$ and $C_2$ is the Cholesky factor for the precision matrix of $\zeta$. We let $\Sigma_1$ denote the covariance matrix of $z$. We further assume that
$C_1$ is lower triangular with a single band, which implies that $\Omega_1$ is band tridiagonal; see Section \ref{app:SparsityPrecisionMatrix} for details. For a Gaussian distribution, zero elements in the precision matrix represent conditional independence relationships. In particular, the sparse structure we have
imposed on $C_1$ means that in the generative distribution for $\rho$, the latent variable $z_t$, given $z_{t-1}$ and $z_{t+1}$, is conditionally independent of the remaining
elements of $z$. In other words, if we think of the variables $z_t$, $t=1,\dots,T$ as a time series, they have a Markovian dependence structure.
We now construct the variational distribution for $\theta$ through
\begin{align*}
\theta & = \left[\begin{array}{cc} X \\ \zeta \end{array}\right] = \left[\begin{array}{cc} I_{T+1}\otimes B & 0 \\ 0 & I_P\end{array}\right]\rho + \left[\begin{array}{c} \epsilon \\ 0 \end{array}\right],
\end{align*}
where $\otimes$ denotes the Kronecker product, $P$ is the dimension of $\zeta$ and $\epsilon=(\epsilon_0^\top,\dots,\epsilon_T^\top)^\top$.
Note that we can apply the reparametrization trick by writing $\rho=\mu+C^{-\top}\omega$, where $\omega\sim N(0,I_{q(T+1)})$. Then,
\begin{align}
\theta & = W\rho+Ze= W\mu+WC^{-\top}\omega+Ze, \label{reparformula}
\end{align}
where
$$W=\left[\begin{array}{cc} I_{T+1}\otimes B & 0_{p(T+1)\times P} \\ 0_{P\times q(T+1)} & I_P \end{array}\right], \;\;\;\;Z=\left[\begin{array}{cc} D & 0_{p(T+1)\times P} \\ 0_{P\times p(T+1)} & 0_{P\times P} \end{array}\right],\;\;\;\; e=\left[\begin{array}{c} \epsilon \\ 0_{P\times 1} \end{array}\right] $$
and $D$ is a diagonal matrix with diagonal entries $(\delta_0^\top,\dots,\delta_T^\top)^\top$. Here $u=(\omega,\epsilon)\sim f(u)=N(0,I_{(p+q)(T+1)+P})$.
We also write $\omega=(\omega_1,\omega_2)$, where the blocks of this partition follow those of $\rho$ as $\rho=(z^\top,\zeta^\top)^\top$.
In the development above we see that the factor structure is being used both for describing the covariance structure for the states, and also for dimension
reduction in the variational posterior mean of the states, since $E(X_t)=B\mu_t$, where $\mu_t=E(z_t)$. An alternative is to set $E(z_t)=0$ and use
\begin{align}
X_t & = \mu_t+Bz_t+\epsilon_t, \label{hdsm}
\end{align}
where $\mu_t$ is now a $p$-dimensional vector specifying the variational posterior mean for $X_t$ directly. We call parametrization (\ref{ldsm}) the low-dimensional state
mean (LD-SM) parametrization, and parametrization (\ref{hdsm}) the high-dimensional state mean (HD-SM) parametrization. In both parametrizations, $B$ forms a basis for $X_t$, which is reweighted over time according to the latent weights (factors) $z_t$. The LD-SM parametrization provides information on how these basis functions are reweighted over time to form our approximate posterior mean, since $E(X_t)=B\mu_t$ and we infer both $B$ and $\mu_t$ in the variational optimization. Section \ref{sec:examples} illustrates this basis representation. In Appendix \ref{app:grad_expressions_lemmas}, we only outline the gradients and their derivation for the LD-SM parametrization. Derivations for the HD-SM parametrization follow those for the LD-SM case with straightforward minor adjustments.
Algorithm \ref{Alg:variational_optimization_dynamic_factor_approximation} outlines our stochastic gradient ascent algorithm that maximizes \eqref{lbqr}. The gradients used can be found in Lemmas \ref{lem:standard_gradient} and \ref{lem:Roeder_gradient} in Appendix \ref{app:grad_expressions_lemmas}. We can estimate their expectations by one or more samples from $f$. One can compute gradients based on either equations (\ref{gradmuss}), (\ref{gradBss}), (\ref{graddss}) and (\ref{gradCss}) in Lemma \ref{lem:standard_gradient}, or on equations (\ref{sgradmuss}), (\ref{sgradBss}), (\ref{sgraddss}) and (\ref{sgradCss}) in Lemma \ref{lem:Roeder_gradient}.
If the variational approximation to the posterior is accurate, as we explained in Section \ref{sec:stochasticgradientvariational}, there are reasons to prefer equations
(\ref{sgradmuss}), (\ref{sgradBss}), (\ref{sgraddss}) and (\ref{sgradCss}) corresponding to the gradient estimates recommended in \citet{Roeder2017}. However, since we investigate massive dimension reduction with only a small numbers of factors the approximation may be crude. We therefore investigate both approaches in later examples.
\begin{algorithm}[tbh]
\caption{Stochastic gradient ascent for optimizing the variational objective $\mathcal{L}(\lambda)$ in \eqref{lbqr}. See Appendix \ref{app:grad_expressions_lemmas} for gradients and notation.}
\SetKwInOut{Input}{Input}
\vspace{1mm}
\Input {Starting values $\lambda_0 \leftarrow (\mu_0, B_0, \delta_0, C_0)$, learning rates $\eta_\mu, \eta_B, \eta_\delta, \eta_C$, number of iterations $M$.}
\vspace{1mm}
\vspace{1mm}
\For{$m = 1$ \KwTo $M$} {
$\mu_m \leftarrow \mu_{m-1} + \eta_\mu \odot \widehat{\nabla_\mu\mathcal{L}}(\lambda_{m-1})$ \Comment{$\nabla_{\mu}\mathcal{L}$ in \eqref{gradmuss} or \eqref{sgradmuss}} \\
\vspace{1mm}
$\lambda_{m-1} \leftarrow (\mu_{m}, B_{m-1}, \delta_{m-1}, C_{m-1})$ \Comment{Update $\mu$}\\
\vspace{1mm}
$B_m \leftarrow B_{m-1} + \eta_B \odot \widehat{\nabla_{\mathrm{vec}(B)}\mathcal{L}}(\lambda_{m-1})$ \Comment{$\nabla_{\mathrm{vec}(B)}\mathcal{L}$ in \eqref{gradBss} or \eqref{sgradBss}} \\
\vspace{1mm}
$\lambda_{m-1} \leftarrow (\mu_{m}, B_{m}, \delta_{m-1}, C_{m-1})$ \Comment{Update $B$}\\
\vspace{1mm}
$\delta_m \leftarrow \delta_{m-1} + \eta_\delta \odot \widehat{\nabla_{\delta}\mathcal{L}}(\lambda_{m-1})$ \Comment{$\nabla_{\delta}\mathcal{L}$ in \eqref{graddss} or \eqref{sgraddss}} \\
\vspace{1mm}
$\lambda_{m-1} \leftarrow (\mu_{m}, B_{m}, \delta_{m}, C_{m-1})$ \Comment{Update $\delta$}\\
\vspace{1mm}
$C_m \leftarrow C_{m-1} + \eta_C \odot \widehat{\nabla_C\mathcal{L}}(\lambda_{m-1})$ \Comment{$\nabla_{C}\mathcal{L}$ in \eqref{gradCss} or \eqref{sgradCss}} \\
\vspace{1mm}
$\lambda_{m} \leftarrow (\mu_{m}, B_{m}, \delta_{m}, C_{m})$ \Comment{Update $C$}\\
\vspace{1mm}
$\lambda_{m-1} \leftarrow \lambda_{m}$ \Comment{Update $\lambda$}\\
}
\vspace{1mm}
\textbf{Output:} $\lambda_m$
\label{Alg:variational_optimization_dynamic_factor_approximation}
\end{algorithm}
\subsection{Efficient computation}\label{subsec:EfficientComp}
The gradient estimates for the lower bound (see Appendix \ref{app:grad_expressions_lemmas} for expressions) are efficiently computed using a combination of sparse
matrix operations (for evaluation of terms such as $C^{-\top}\omega$ and the high-dimensional matrix multiplications in the expressions) and, as in \citet{ong+ns17}, the Woodbury identity for dense matrices such as $(W\Sigma W^\top+Z^2)^{-1}$ and $(W_1\Sigma_1W^\top+D^2)^{-1}$. The Woodbury identity is
\begin{align*}
(\Lambda \Gamma \Lambda^\top + \Psi)^{-1} = & \Psi^{-1}-\Psi^{-1}\Lambda(\Lambda^\top\Psi^{-1}\Lambda + \Gamma^{-1})^{-1}\Lambda^\top \Psi^{-1}
\end{align*}
for conformable matrices $\Lambda, \Gamma$ and diagonal $\Psi$. The Woodbury formula reduces the required computations into a much lower dimensional space since $q \ll p$ and, moreover, inversion of the high-dimensional matrix $\Psi$ is trivial because it is diagonal.
\section{Application 1: Spatio-temporal modeling}\label{sec:examples}
\subsection{Eurasian collared-dove data}
Our first example considers the spatio-temporal model of \citet{wikle+h06}
for a dataset on the spread of the Eurasian collared-dove across North America. The dataset consists of the
number of doves $y_{s_{i}t}$ observed at location $s_{i}$ (latitude,
longitude) $i=1,\dots,p,$ in year $t=1,\dots,T=18,$ corresponding
to an observation period of 1986-2003. The spatial locations
correspond to
$p=111$ grid points with the dove counts aggregated
within each area; see \citet{wikle+h06} for details.
The count observed at location $s_{i}$ at time $t$ depends on the
number of times $N_{s_{i}t}$ that the location was sampled. However, this
variable is unavailable and therefore we set the offset in the model
to zero, i.e. $\log(N_{s_{i}t})=0$.
\subsection{Model\label{subsec:Model}}
Let $y_t=(y_{s_1t},\dots, y_{s_pt})^\top$ denote the count data at time $t$. \citet{wikle+h06} model $y_t$ as conditionally independent Poisson
variables, where the log means are given by a latent high-dimensional Markovian process $u_t$ plus measurement error. The dynamic process $u_t$ evolves according to a discretized
diffusion equation. Specifically,
the model in \citet{wikle+h06} is
\begin{align*}
y_{t}|v_{t} & \sim\mathrm{Poisson}(\mathrm{diag}(N_{t})\exp(v_{t}))\quad y_{t},N_{t},v_{t}\in\mathbb{R}^{p}\\
v_{t}|u_{t},\sigma_{\epsilon}^{2} & \sim N(u_{t},\sigma_{\epsilon}^{2}I_{p}),\quad u_{t}\in\mathbb{R}^{p},I_{p}\in\mathbb{R}^{p\times p},\sigma_{\epsilon}^{2}\in\mathbb{R}^{+}\\
u_{t}|u_{t-1},\psi,\sigma_{\eta}^{2} & \sim N(H(\psi)u_{t-1},\sigma_{\eta}^{2}I_{p}),\quad\psi\in\mathbb{R}^{p},H(\psi)\in\mathbb{R}^{p\times p},\sigma_{\eta}^{2}\in\mathbb{R}^{+},
\end{align*}
with priors $\sigma_{\epsilon}^{2},\sigma_{\psi}^{2},\sigma_{\alpha}^{2}\sim\mathrm{IG}(2.8,0.28),\sigma_{\eta}^{2}\sim\mathrm{IG}(2.9,0.175)$
and
\begin{align*}
u_{0} & \sim N(0,10I_{p})\\
\psi|\alpha,\sigma_{\psi}^{2} & \sim N(\Phi\alpha,\sigma_{\psi}^{2}I_{p}),\quad\Phi\in\mathbb{R}^{p\times l}\text{,}\alpha\in\mathbb{R}^{l},\sigma_{\psi}^{2}\in\mathbb{R}^{+}\\
\alpha & \sim N(0,\sigma_{\alpha}^{2}R_{\alpha}),\quad\alpha_{0}\in\mathbb{R}^{l},R_{\alpha}\in\mathbb{R}^{l\times l},\sigma_{\alpha}^{2}\in\mathbb{R}^{+}.
\end{align*}
$\mathrm{Poisson(\cdot)}$ is the Poisson distribution for a (conditionally)
independent response vector parameterized in terms of its expectation
and $\mathrm{IG}(\cdot)$ is the inverse-gamma distribution with shape
and scale as arguments. The spatial dependence is modeled via the
prior mean $\Phi\alpha$ of the diffusion coefficients $\psi$, where
$\Phi$ consist of the $l$ orthonormal eigenvectors with the largest
eigenvalues of the spatial correlation matrix $R(c)=\exp(cd)\in\mathbb{R}^{p\times p}$,
where $d$ is the Euclidean distance between pairwise grid locations
in $s_{i}$. Finally, $R_{\alpha}$ is a diagonal matrix with the
$l$ largest eigenvalues of $R(c)$. We follow \citet{wikle+h06}
and set $l=1$ and $c=4$.
Let $u=(u_{0}^{\top},\dots u_{T}^{\top})^{\top}$ , $v=(v_{1}^{\top},\dots v_{T}^{\top})^{\top}$
and denote the parameter vector
\[
\theta=(u,v,\psi,\alpha,\log\sigma_{\epsilon}^{2},\log\sigma_{\eta}^{2},\log\sigma_{\psi}^{2},\log\sigma_{\alpha}^{2}),
\]
which we infer through the posterior
\begin{eqnarray}
p(\theta|y) & \propto & \sigma_{\epsilon}^{2}\sigma_{\eta}^{2}\sigma_{\psi}^{2}\sigma_{\alpha}^{2}p(\sigma_{\epsilon}^{2})p(\sigma_{\eta}^{2})p(\sigma_{\psi}^{2})p(\sigma_{\alpha}^{2})p(\alpha|\sigma_{\alpha}^{2})p(\psi|\alpha,\sigma_{\psi}^{2})\nonumber \\
& ~ & p(u_{0})\prod_{t=1}^{T}p(u_{t}|u_{t-1},\psi,\sigma_{\eta}^{2})p(v_{t}|u_{t},\sigma_{\epsilon}^{2})p(y_{t}|v_{t}),\label{eq:PosteriorDistribution}
\end{eqnarray}
with $y=(y_{1}^{\top},\dots,y_{T}^{\top})^{\top}$. Section \ref{app:GradientLogPosterior} derives the gradient of the log-posterior required by our variational Bayes (VB) approach.
\subsection{Variational approximations of the posterior distribution\label{subsec:VA_posterior_dist}}
Section \ref{sec:Gaussian_Variational_high_dim_state} considered two different parametrization of the low rank
approximation, in which either the state vector $X_{t}$ has mean
$E(z_{t})=B\mu_{t}$, $\mu_{t}\in\mathbb{R}^{q}$ (low-dimensional
state mean, LD-SM) or $X_{t}$ has a separate mean $\mu_{t}\in\mathbb{R}^{p}$
and $E(z_{t})=0$ (high-dimensional state mean, HD-SM). In this particular
application there is a third choice of parametrization which
we now consider.
The model in Section \ref{subsec:Model} connects the data with the high-dimensional state vector $u_{t}$ via a high-dimensional
auxiliary variable $v_{t}$. In the notation of Section 4, we can include $v$ in $\zeta$, in which
case the parametrization of the variational posterior is the one described there.
We refer to this parametrization as a low-rank state (LR-S).
However, it is clear from \eqref{eq:PosteriorDistribution} that there
is posterior dependence between $u_{t}$ and $v_{t}$, but the variational
approximation in Section \ref{sec:Gaussian_Variational_high_dim_state} omits dependence between $z$ and $\zeta$.
Moreover, $v_{t}$ is also high-dimensional, but the LR-S parametrization
does not reduce its dimension. An alternative parametrization that
deals with both considerations includes $v$ in the $z$-block, which
we refer to as the low-rank state and auxiliary variable (LR-SA) parametrization. This comes at
the expense of omitting dependence between $v_{t}$ and $\sigma_{\epsilon}^{2}$,
but also becomes more computationally costly because, while the total
number of variational parameters is smaller (see Table \ref{tab:VB_parameterization} in Section \ref{app:Sparsity}),
the dimension of the $z$-block increases ($B$ and $C_{1}$) and
the main computational effort lies here and not in the $\zeta$-block.
Table
\ref{tab:ELBO_final_iteration} shows the CPU times relative to the LR-S parametrization. The LR-SA parametrization requires
a small modification of the derivations in Section \ref{sec:Gaussian_Variational_high_dim_state}, which we outline
in detail in Section \ref{sec:LRSA_derivations} as they can be useful for other models with a high-dimensional auxiliary variable.
It is straightforward to deduce conditional independence relationships
in \eqref{eq:PosteriorDistribution} to build the Cholesky factor
$C_{2}$ of the precision matrix $\Omega_{2}$ of $\zeta$ in Section
4, with
\[
\zeta=\begin{cases}
(v,\psi,\alpha,\log\sigma_{\epsilon}^{2},\log\sigma_{\eta}^{2},\log\sigma_{\psi}^{2},\log\sigma_{\alpha}^{2}) & \text{(LR-S)}\\
(\psi,\alpha,\log\sigma_{\epsilon}^{2},\log\sigma_{\eta}^{2},\log\sigma_{\psi}^{2},\log\sigma_{\alpha}^{2}) & \text{(LR-SA)}.
\end{cases}
\]
Section \ref{sec:Gaussian_Variational_high_dim_state} outlines the construction of the Cholesky factor $C_{1}$ of the
precision matrix $\Omega_{1}$ of $z$, whereas
the minor modification needed for LR-SA is in Section \ref{sec:LRSA_derivations}. We note that, regardless of the parametrization, we obtain massive parsimony (between $6,428\text{-}11,597$ variational parameters) compared to the saturated Gaussian variational approximation which in this application has $8,923,199$ parameters; see Section \ref{app:Sparsity} for further details.
We consider four different variational parametrizations, combining
each of LR-SA or LR-S with the different parametrization of the means
of $X_{t}$, i.e. LD-SM or HD-SM. In all cases, we let $q=4$ and
perform $10,000$ iterations of a stochastic optimization algorithm with
learning rates chosen adaptively according to the ADADELTA approach
\citep{Zeiler2012}. We use the gradient estimators in \citet{Roeder2017}, i.e. \eqref{sgradmuss}, \eqref{sgradBss}, \eqref{sgraddss} and \eqref{sgradCss}, although we found no noticeable difference compared to (\ref{gradmuss}), (\ref{gradBss}), (\ref{graddss}) and (\ref{gradCss}), which is likely due to the small number of factors as described in Sections \ref{sec:stochasticgradientvariational} and \ref{sec:Gaussian_Variational_high_dim_state}. Our choice was motivated by computational efficiency as some terms cancel out using the approach in \citet{Roeder2017}. We initialize $B$ and $C$ as
unit diagonals and, for parity, $\mu$ and $D$ are chosen to match the starting values of the Gibbs sampler in \citet{wikle+h06}.
Figure \ref{fig:ELBOs} monitors the convergence via the estimated
value of ${\cal L}(\lambda)$ using a single Monte Carlo sample. Table \ref{tab:ELBO_final_iteration}
presents estimates of ${\cal L}(\lambda)$ at the final iteration using $100$ Monte Carlo samples.
The results suggest that the best VB parametrization in terms of ELBO is the low-rank state algorithm (LR-SA) with, importantly, a high-dimensional state-mean (HD-SM) (otherwise the poorest VB approximation is achieved, see Table \ref{tab:ELBO_final_iteration}). However, Table \ref{tab:VB_parameterization} shows that this parametrization is about three times as CPU intensive. The fastest VB parametrizations are both Low-Rank State (LR-S) algorithms, and modeling the state mean separately for these
does not seem to improve ${\cal L}(\lambda)$ (Table \ref{tab:ELBO_final_iteration})
and is also slightly more computationally expensive (Table \ref{tab:VB_parameterization}). Taking these considerations into account, the final choice of VB parametrization we use
for this model
is the low-rank state with low-dimensional state mean (LR-S + LD-SM). We show in Section \ref{subsec:Results} that this parametrization gives accurate approximations for our analysis. For the rest of this example, we benchmark the VB posterior from LR-S + LD-SM against the MCMC approach in \citet{wikle+h06}.
\begin{figure}[h]
\centering
\includegraphics[width=0.3\columnwidth]{Figures/ELBO_4_different_VBs}
\caption{${\cal L}(\lambda)$ for the variational approximations for the spatio-temporal example. The figure shows
the estimated value of ${\cal L}(\lambda)$ vs iteration number for the four different parametrizations, see Section \ref{subsec:VA_posterior_dist} or Table \ref{tab:ELBO_final_iteration} for abbreviations.}
\label{fig:ELBOs}
\end{figure}
\begin{table}[h]
\centering \caption{${\cal L}(\lambda)$ and CPU time for the VB parametrizations in the spatio-temporal and Wishart process example. The table shows the estimated value of ${\cal L}(\lambda)$ for the different VB parametrizations by combining low-rank state / low-rank state and auxiliary (LR-S / LR-SA)
with either of low-dimensional state mean / high-dimensional state
mean (LD-SM / HD-SM). The estimate and its $95$\% confidence interval
are computed at the final iteration using $100$ Monte Carlo samples. The table also show the relative CPU (R-CPU) times, where the reference is LD-SM.
}
\begin{tabular}{llclclcc}
\toprule
{\footnotesize{}{}}\textbf{\footnotesize{}Parametrization} & & & & & & & \tabularnewline
\cmidrule{1-1}
\textbf{\footnotesize{}{}}\emph{\footnotesize{}Spatio-temporal} & & {\footnotesize{}{}R-CPU} & & {\footnotesize{}{}$\mathcal{L}(\lambda_{\mathrm{opt}})$} & & {\footnotesize{}{}Confidence interval} & \tabularnewline
\cmidrule{1-1} \cmidrule{3-3} \cmidrule{5-5} \cmidrule{7-7}
{\footnotesize{}{}LR-S + LD-SM} & & {\footnotesize{}{}$1$} & & {\footnotesize{}{}$-1,996$} & & {\footnotesize{}{}$[-2,004;-1,988]$} & \tabularnewline
{\footnotesize{}{}LR-S + HD-SM} & & {\footnotesize{}{}$1.005$} & & {\footnotesize{}{}$-2,024$} & & {\footnotesize{}{}$[-2,032;-2,016]$} & \tabularnewline
{\footnotesize{}{}LR-SA + LD-SM} & & {\footnotesize{}{}$3.189$} & & {\footnotesize{}{}$-2,158$} & & {\footnotesize{}{}$[-2,167;-2,148]$} & \tabularnewline
{\footnotesize{}{}LR-SA + HD-SM} & & {\footnotesize{}{}$3.017$} & & {\footnotesize{}{}$-1,909$} & & {\footnotesize{}{}$[-1,918;-1,900]$} & \tabularnewline
& & & & & & & \tabularnewline
\cmidrule{1-1}
\textbf{\footnotesize{}{}}\emph{\footnotesize{}Wishart process} & & & & & & & \tabularnewline
\cmidrule{1-1}
{\footnotesize{}{}LR-S + LD-SM} & & {\footnotesize{}{}$1$} & & {\footnotesize{}{}$-1,588$} & & {\footnotesize{}{}$[-1,593;-1,583]$} & \tabularnewline
{\footnotesize{}{}LR-S + HD-SM} & & {\footnotesize{}{}$1.0004$} & & {\footnotesize{}{}$-1,501$} & & {\footnotesize{}{}$[-1,506;-1,495]$} & \tabularnewline
\bottomrule
\end{tabular}\label{tab:ELBO_final_iteration}
\end{table}
\subsection{Settings for MCMC}
Before evaluating VB against MCMC, we need to determine a reasonable
burn-in and number of iterations for the Gibbs sampler in \citet{wikle+h06}.
It is clear that it is not feasible to monitor convergence for every
single parameter in such a large scale model as \eqref{eq:PosteriorDistribution},
and therefore we focus on $\psi$, $u_{18}$ and $v_{19}$, which
are among the variables considered in the analysis in Section \ref{subsec:Results}.
\citet{wikle+h06} use $50,000$ iterations of which $20,000$
are discarded as burn-in.
We generate $4$ sampling chains with these settings and inspect convergence
using the $\mathtt{coda}$ package \citep{plummer2006coda} in $\mathtt{R}$.
We compute the Scale Reduction Factors (SRF) \citep{gelman1992inference}
for $\psi,u_{18}$ and $v_{19}$ as a function of the number of Gibbs
iterations. The adequate number of iterations in MCMC depends on what functionals of the parameters are of interest; here we monitor convergence for these quantities
since we report marginal posterior distributions for these quantities later.
The scale reduction factor of a parameter measures if
there is a significant difference between the variance within the
four chains and the variance between the four chains of that parameter.
We use the rule of thumb that concludes convergence when $\mathrm{SRF}<1.1$,
which gives a burn-in period of approximately $40,000$ here, for these functionals. After discarding
these samples and applying a thinning of $10$ we are left with $1,000$
posterior samples for inference. However, as the draws are auto-correlated,
this does not correspond to $1,000$ independent draws used in the
analysis in Section \ref{subsec:Results} (note that we obtain independent
samples from our variational posterior). To decide how many Gibbs
samples are equivalent to $1,000$ independent samples for $\psi,u_{18}$
and $v_{19}$, we compute the Effective Sample Size (ESS) which takes
into account the auto-correlation of the samples. We find that the
smallest is $\mathrm{ESS}=5$ and hence we require $200,000$ iterations
after a thinning of $10$, which makes a total of $2,000,000$ Gibbs
iterations, excluding the burn-in of $40,000$. Thinning is advisable here due
to memory issues.
\subsection{Analysis and results\label{subsec:Results}}
We first consider inference on the diffusion coefficient $\psi_i$ for location $i$. Figure \ref{fig:Distribution_delta} shows the ``true'' posterior (represented by MCMC) together with the variational approximation for six locations described in the caption of the figure. The figure shows that the posterior distribution is highly skewed for locations with zero dove counts and approaches normality as the dove counts increase. Consequently, the accuracy of the variational posterior (which is Gaussian) improves with increasing dove counts.\begin{figure}[H]
\centering
\includegraphics[width=0.5\columnwidth]{Figures/Distribution_Delta_MCMC_vs_VB_6_locations}
\caption{Distribution of the diffusion coefficients. The figure shows the posterior distribution of $\psi_i$ obtained by MCMC and VB. The locations are divided into three categories (total doves over time within brackets): zero count locations (Idaho, $i=1\, [0]$ , Arizona $i=5\,[0]$, left panels), low count locations (Texas, $i=35\,[16], 46\,[21]$, middle panels) and high count locations (Florida, $i=96\,[1,566], 105\,[1,453]$, right panels).}
\label{fig:Distribution_delta}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.5\columnwidth]{Figures/sum_lambda_per_year_MCMC_vs_VB}
\caption{Samples from the posterior sum of dove intensity over the spatial
grid for each year. The figure shows $100$ samples from the posterior
distribution of $\varphi_{t}=\sum_{i}\exp(v_{it})$ obtained by MCMC
(left panel) and VB (right panel).}
\label{fig:lambda_each_year}
\end{figure}
Figure\emph{ }\ref{fig:lambda_each_year} shows 100 VB and MCMC posterior
samples of the dove intensity for each year summed over the spatial
locations, i.e. $\varphi_{t}=\sum_{i}\exp(v_{it})$. Both posteriors
are very similar and, in particular, show an exponential increase
of doves until year $2002$ followed by a steep decline for year $2003$.
In the interest of analyzing the spatial dimension of the model, Figure
\ref{fig:dove_intensity_MCMC_vs_VB} shows a heat map of the MCMC
and VB posterior mean of the dove intensity $\varphi_{it}=\exp(v_{it})$
for the last five years of the data, overlaid on a map of the United
States of America. We draw the following conclusions from the analysis
using the MCMC and VB posteriors (which are nearly identical). First,
the dove intensity is most pronounced in the South East states, in
particular Florida (Figure \ref{fig:dove_intensity_MCMC_vs_VB}).
Second, the decline of doves for year $2003$ in Figure \ref{fig:lambda_each_year}
is likely attributed to a drop in the intensity (Figure \ref{fig:dove_intensity_MCMC_vs_VB})
at two areas of Florida: Central Florida ($i=96$) and South East
Florida ($i=105$). Figure \ref{fig:log_intensity_in_and_out_of_sample}
illustrates the whole posterior distribution of the log-intensity
for these locations at year $2003$ and, moreover, an out-of-sample
posterior predictive distribution for year $2004$. The estimates
are obtained by kernel density estimates using approximately $1,000$
effective samples. The posterior distributions for the VB and MCMC
are similar, and it is evident that using this large scale model for
forecasting future values is associated with a large uncertainty.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\columnwidth]{Figures/MCMC_vs_VB_lambda_5_last_years_US_map}
\caption{Posterior dove intensity for the years 1999-2003. The figure
shows the posterior mean of $\varphi_{it}=\exp(v_{it})$ computed
by MCMC (left panels) and VB (right panels) for $i=1,\dots,p=111,$
and the last $5$ years of the data ($t=14,15,16,17,18$). The results
are illustrated with a spatial grid plotted together with a map of
the United States, where the colors vary between low intensity (yellow)
and high intensity (red). The light blue color is for aesthetic reasons
and does not correspond to observed locations. }
\label{fig:dove_intensity_MCMC_vs_VB}
\end{figure}
We conclude this example by investigating the spatial functions and their reweighting over time to produce the variational approximation. Figure \ref{fig:Spatial_basis_and_weightning} illustrates this and shows that the correlation among the states is mostly driven by spatial locations outside the southern east part of North America. This is reasonable as the data contains mostly zero or near zero counts outside the southern east region.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\columnwidth]{Figures/Forecast_log_intensity}
\caption{Forecasting the log intensity of the spatial process. The figure
shows an in-sample forecast of the log-intensity $v_{it}$ for year
2003 ($t=18$, upper panels) and out-of-sample forecast for year 2004
($t=19$, lower panels) for Central Florida ($i=96$, left panels)
and South East Florida ($i=105$, right panels).}
\label{fig:log_intensity_in_and_out_of_sample}
\end{figure}
The analysis in this section shows that similar inferential conclusions
are drawn with the VB posterior and the ``true'' posterior (approximated
by the Gibbs sampler). One advantage with the VB posterior is that it is more efficient
for performing posterior predictive analysis because independent samples
are easily obtained, in contrast to MCMC samples that may have a prohibitively
large auto-correlation, resulting in imprecise estimators. Perhaps the main advantage of the VB posterior is that it is faster to obtain: in this particular application VB was $7.3$ times faster than MCMC. The speed up in computing time relative to MCMC is model specific and depends on how expensive the different sampling steps in the Gibbs sampler are. This spatial temporal model is computationally cheap on a per iteration basis, however, many iterations are needed for accurate inference as demonstrated. In the next section, we consider a model which is both computationally expensive and mixes poorly. For that model the computational gains are much more pronounced.
\begin{figure}[H]
\centering
\includegraphics[width=0.5\columnwidth]{Figures/SpatialBasis_and_weights_4_components}
\caption{Spatial basis representation of the state vector. The figure
shows the Spatial basis functions (left panel), i.e. the $j$th column of $B$, $j=1,\dots,q=4$ and the corresponding weights $\mu_t$ (right panel) through $t = 0, \dots, 18$, that forms $E(X_t) = B\mu_t$.}
\label{fig:Spatial_basis_and_weightning}
\end{figure}
\section{Application 2: Stochastic volatility modeling}\label{sec:examples2}
\subsection{Industry portfolios data\label{subsec:data_PhilipovGlickman}}
Our second example considers the Multivariate stochastic volatility model via Wishart processes in \citet{philipov+g06} used for modeling the time-varying dependence of a portfolio of $k$ assets over $T$ time periods. We follow \citet{philipov+g06} and use $k=5$ (manufacturing, utilities, retail/wholesale, finance, other), which results in a state-vector (the lower-triangular part of the covariance matrix) of dimension $p=15$. This is far from a high-dimensional setting, but allows us to implement an MCMC method to compare against the variational approximation. In fact, for $k=12$, which gives $p=78$, \citet{philipov+g06} use a Metropolis-Hastings within Gibbs sampler and reported that some of the blocks updated by the Metropolis-Hastings sampler have acceptance probabilities close to zero. We discuss this further in Section \ref{subsec:SettingsMCMC_PhilipovGlickman} and note at the outset that our variational approach does not have this limitation as we demonstrate in Section \ref{subsec:Results_PhilipovGlickman}.
\citet{philipov+g06} made a mistake in the derivation of the Gibbs sampler which affects all full conditionals \citep{rinnergschwenter+tw12}. Implementing the corrected version results in a highly inefficient sampler and we take $T = 100$ instead of $T = 240$ so that the MCMC can finish in a reasonable amount of time. Hence we have $T=100$ montly observations on value-weighted portfolios from the 201709 CRSP database, covering a period from 2009-06 to 2017-09. We follow \citet{philipov+g06} and prefilter each series using an AR(1) process.
\subsection{Model\label{subsec:ModelPhilipovGlickman}}
We assume that the return at time period $t$, $t=1, \dots, T$, is the vector
$y_t=(y_{t1},\dots, y_{tk})^\top$,
\begin{eqnarray*}
\label{eq:MV_SV_model}
y_t & \sim & N(0, \Sigma_t), \quad \Sigma_t \in\mathbb{R}^{p\times p} \\
\Sigma_t^{-1} &\sim & \mathrm{Wishart}(\nu,S_{t-1}), \quad S_t=\frac{1}{\nu} H(\Sigma_t^{-1})^d H^\top , \, S_t \in\mathbb{R}^{p\times p},\, \nu > k, \, 0 < d < 1,
\end{eqnarray*}
where $H$ is a lower triangular Cholesky factor of a positive definite matrix $A$, $A=HH^\top \in \mathbb{R}^{p\times p}$ and $\Sigma_0$ is assumed known. \citet{philipov+g06} use an inverse Wishart prior for $A$, $A^{-1}\sim \mathrm{Wishart}(\gamma_0,Q_0)$, $\gamma_0=k+1$, $Q_0=I$,
a uniform prior for $d$, $d\sim U[0,1]$, and a shifted gamma prior for $\nu$, $\nu-k\sim \mathrm{Gamma}(\alpha_0,\beta_0)$. The joint posterior density for $(\Sigma,A,\nu-k,d)$ is
\begin{align}
p(\Sigma,A,\nu-k,d|y) & \propto p(A,d,\nu-k) \prod_{t=1}^T p(\Sigma_t|\nu,S_{t-1})p(y_t|\Sigma_t), \label{jointpost}
\end{align}
where $p(A,d,\nu-k)$ denotes the joint prior density for $(A,d,\nu-k)$, $p(\Sigma_t|\nu,S_{t-1},d)$ denotes the conditional inverse Wishart prior
for $\Sigma_t$ given $\nu$, $S_{t-1}$ and $d$, and $p(y_t|\Sigma_t)$ is the normal density for $y_t$ given $\Sigma_t$.
We reparametrize the posterior in terms of the unconstrained parameter $$\theta =(\mathrm{vech}(H')^\top,d',\nu',\mathrm{vech}(C_1')^\top,\dots, \mathrm{vech}(C_T')^\top)^\top,$$
where $\mathrm{vech}(A)$, for a symmetric matrix $A$, is the column vector obtained by vectorizing only the lower triangular part of $A$ and, moreover,
\begin{eqnarray*}
C_t' \in \mathbb{R}^{k \times k}, & ~ & C'_{t,ij}=C_{t,ij}, \, i\neq j, \text{ and } C'_{t,ii}=\log C_{t,ii},\\
H' \in \mathbb{R}^{k \times k}, & ~ & H'_{ij}=H_{ij}, \, i\neq j, \text{ and } H_{ii}=\log H{ii},
\end{eqnarray*}
with $d'=\log d/(1-d)$ and $\nu'=\log (\nu-k)$. Then, as shown in Section
\ref{app:LogPosteriorPhilipovGlickman},
\begin{align}
\label{eq:PosteriorDistribution_PhilipovGlickman}
p(\theta|y) \propto & |L_k (I_{k^2}+K_{k,k})(H\otimes I_k)L_k^\top| \times \left\{\prod_{t=1}^T |L_k(I_{k^2}+K_{k,k})(C_t\otimes I_k)L_k^\top| \right\} \times (\nu-k) \\ \nonumber
& \times d(1-d)\times \left\{\prod_i H_{ii}\right\}\left\{\prod_{t=1}^T \prod_{i=1}^k C_{t,ii}\right\} \times p(A,d,\nu-k)\left\{\prod_{t=1}^T p(\Sigma_t|\nu,S_{t-1},d) p(y_t|\Sigma_t)\right\},
\end{align}
where $L_k$ denotes the elimination matrix and $K_{k,k}$ the commutation matrix, both defined in Section \ref{supp:notation}. Evaluation of the gradient of the log posterior is described in the Section \ref{app:GradientLogPosteriorPhilipovGlickman}.
\subsection{Variational approximations of the posterior distribution}
Since this example does not include a high-dimensional auxiliary variable, we use the low-rank state (LR-S) parametrization combined with both a low-dimensional state mean (LD-SM) and a high-dimensional state mean (HD-SM). As in the previous example, it is straightforward to deduce conditional independence relationships in \eqref{eq:PosteriorDistribution_PhilipovGlickman} to build the Cholesky factor
$C_{2}$ of the precision matrix $\Omega_{2}$ of $\zeta$ in Section
4. Moreover, construction of the Cholesky factor $C_{1}$ of the
precision matrix $\Omega_{1}$ of $z$ is outlined in Section \ref{sec:Gaussian_Variational_high_dim_state}. Massive parsimony is achieved in this application, in particular for $k=12$ assets in which the saturated Gaussian variational approximation has $31,059,020$ parameters, while our parametrization gives $10,813$. For $k=5$, the saturated case has $1,152,920$ parameters and our parametrizations give $4,009\text{-}5,109$; see Section \ref{app:Sparsity} for more details.
For all variational approximations we let $q=4$ and
perform $10,000$ iterations of a stochastic optimization algorithm with
learning rates chosen adaptively according to the ADADELTA approach
\citep{Zeiler2012}. We initialize $B$ and $C$ as
unit diagonals and choose $\mu$ and $D$ randomly. Figure \ref{fig:ELBOs_PhilipovGlickman} monitors the estimated ELBO for both parametrizations, using both the gradient estimators in \citet{Roeder2017} and the alternative standard ones which do not cancel terms that have zero expectation. For $k=5$, the figure shows that the different gradient estimators perform equally well. Moreover, slightly more variable estimates are observed in the beginning for the low-dimensional state mean parametrization compared to that of the high-dimensional mean. Table \ref{tab:ELBO_final_iteration} presents estimates of ${\cal L}(\lambda)$ at the final iteration using $100$ Monte Carlo samples and also presents the relative CPU times of the algorithms. In this example the separate state mean present in the high-dimensional state mean seems to improve the ELBO considerably.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\columnwidth]{Figures/ELBO_iter_PhilipovGlickman}
\caption{${\cal L}(\lambda)$ for the variational approximations in the Wishart process example. The figure shows the estimated value of ${\cal L}(\lambda)$ vs iteration number using a low-dimensional state mean / high-dimensional state mean (LD-SM / HD-SM) with the gradient estimator in \citet{Roeder2017} or the standard estimator. The results are shown for $k=5$ (left and middle panel) and $k=12$ (right panel).}
\label{fig:ELBOs_PhilipovGlickman}
\end{figure}
\subsection{Settings for MCMC\label{subsec:SettingsMCMC_PhilipovGlickman}}
While we can do a thorough analysis to determine the burn-in and number of iterations for the Gibbs sampler in \citet{wikle+h06}, this task becomes much more complicated here for the following reasons. First, \citet{philipov+g06} use the inverse cumulative distribution function method on a grid to sample $\nu$ and $d$, which increases the computational burden compared to \citet{wikle+h06}. Second, due to an erroneous step by \citet{philipov+g06} in the derivation of the full conditional of $A^{-1}$ \citep{rinnergschwenter+tw12}, it cannot be directly sampled from a Wishart distribution. We instead implement a random-walk Metropolis-Hastings update for $A^{-1}$ using a Wishart proposal with a mean equal to the current value in the MCMC. Third, the erroneous step results in changes for all the full conditionals. The latter two reasons might explain why we do not observe the same sampling efficiency as \citet{philipov+g06}. Consequently, we obtain poor values for the effective sample size and therefore, if combined with the first reason explained above, we would have to wait for several months to obtain an effective sample size of $1,000$ as in our previous example, even when reducing $T = 240$ to $T=100$ as explained in Section \ref{subsec:data_PhilipovGlickman}.
We conclude that this application is very difficult for our Metropolis-Hastings within Gibbs sampler when $k=5$ (and impossible for $k=12$) and settle for $100, 000$ iterations and discard $20,000$ for burn-in, which arguably are numbers many practitioners would think are sufficient. After thinning the draws, we obtain $1,000$ posterior samples and the effective sample sizes are 10 for $d$, 78 for $\nu$, 45 (average) for $A$ and $503$ (average) for $\Sigma$. Note the low values for $d, \nu$ and $A$, whose chains exhibit a very persistent behavior (not shown here), whereas the effective sample size is higher for $\Sigma$ (because of the independent proposal). We stress that because MCMC convergence is questionable (in particular for $d$, $\nu$ and $A$), so is the MCMC posterior produced and we cannot, unfortunately, know if discrepancies between the methods is due to a poor variational approximation or a poorly estimated MCMC posterior. Nevertheless, the VB posterior seems to give a reasonable predictive distribution for the data and is therefore considered to produce sensible results. The main reasons that the sampler in \citet{philipov+g06} fails when $k = 12$ is due to an independent Wishart proposal within Gibbs for updating $\Sigma^{-1}_t$ for $t<T$ (at $t=T$ perfect sampling from a Wishart can be applied) and the random-walk proposal within Gibbs for $A^{-1}$. It is well-known that these proposals fail in a high-dimensional setting: the random-walk explores the sampling space extremely slowly while independent samplers get stuck, i.e. reject nearly all attempts to move the Markov chain.
We remark that other MCMC approaches for estimating this model more efficiently might be possible, but it is outside the scope of this paper to pursue this. As an example, Hamiltonian Monte Carlo on the Riemannian manifold \citep{girolami2011riemann} has proven to be effective in sampling models with $500\text{-}1,000$ parameters. However, the computational burden relative to standard MCMC is increased and, moreover, tuning the algorithm becomes more difficult.
\subsection{Analysis and results\label{subsec:Results_PhilipovGlickman}}
Section \ref{sec:examples} considered a thorough example on how the variational posterior can be used to conduct a wide range of inferential tasks in a serious application. For brevity, we now only study the accuracy of the variational posterior compared to that of MCMC (although we need to be skeptical about the MCMC posterior as explained in Section \ref{subsec:SettingsMCMC_PhilipovGlickman}) and also check if the predictive distribution gives reasonable results.
Figure \ref{fig:Distribution_A} shows kernel density estimates of the MCMC and VB posterior of the distinct elements of $A$, $\nu$ and $d$ based on $1,000$ samples. The marginal posteriors are very similar for some parameters of $A$ (for example $A_{15},A_{25}$ and $A_{35}$) but not for others (for example $A_{11}, A_{33}$ and $A_{55}$). For $\nu$ and $\delta$ the difference is considerable, but recall that MCMC convergence is questionable. To assess the approximation of $\Sigma_t$, $t = 1, \dots, T$, we inspect the in-sample predictive distribution for the data, i.e $p(\tilde{y}_t|y_{1:T})$ for $t=1,\dots, T,$ obtained by averaging over the posterior of $\Sigma_t$ using simulation. Figure \ref{fig:predInSample_MCMC_vs_VB} shows the results together with the observed data, which confirms that both estimation approaches yield predictions consistent with the data.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\columnwidth]{Figures/A_nu_delta_MCMC_vs_VB}
\caption{Kernel density estimates for the posterior densities of $A$, $\nu$ and $d$. The figure shows the posterior distribution of $A$, $\nu$ and $d$ obtained by MCMC and VB.}
\label{fig:Distribution_A}
\end{figure}
We argued that our variational approximation can handle very large dimensions in this model and discussed that MCMC fails, see Section \ref{subsec:SettingsMCMC_PhilipovGlickman}. Indeed, Figure \ref{fig:ELBOs_PhilipovGlickman} also shows the estimated ELBO on a variational optimization using all assets, which corresponds to $k=12$ with $p = 78$. While it is more variable than the $k=5$ case, it settles down eventually. At the last iteration, we use the variational approximation (which has a low-dimensional state mean) and compute the predictive distribution and compare it do the data to ensure the results are sensible (not shown here). Note that we choose to use the low-dimensional state-mean here since we know that the richer model high-dimensional state provides a better approximation.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\columnwidth]{Figures/YpredInSample_MCMC_vs_VB}
\caption{Posterior predictive distribution of the data. The figure shows the posterior predictive distribution together with the data (black circles) for each time period obtained by MCMC and VB for the different portfolios. The purple color corresponds to regions where the VB and MCMC overlap.}
\label{fig:predInSample_MCMC_vs_VB}
\end{figure}
The speed up for VB vs MCMC when $k=5$ was $29$ times in this application, which is likely a very loose lower bound because the convergence of MCMC is questionable as discussed in Section \ref{subsec:SettingsMCMC_PhilipovGlickman}. For $k=12$, we demonstrate that the variational optimization converges and hence the variational approximation allows for inference, as opposed to MCMC which, in practice, does not produce a single effective draw due to the poor mixing.
\section{Discussion}
We have considered an approach to Gaussian variational approximation for high-dimensional state space models where dimension reduction in the variational approximation is achieved through a dynamic factor structure for the variational covariance matrix. The factor structure reduces
the dimension in the description of the states, whereas the Markov dynamic structure for the factors achieves parsimony in describing the temporal
dependence. We have shown that the method works well in two challenging models. The first is an extended example for a spatio-temporal data set describing the spread of the Eurasian collared-dove throughout North America. The second is a multivariate stochastic volatility model in which the state-vector, which is the half vectorization of the Cholesky factor of the covariance matrix, is high-dimensional.
Perhaps the most obvious limitation of our current work is the restriction to a Gaussian approximation, which does not allow capturing skewness or heavy tails
in the posterior distribution. However, Gaussian variational approximations can be used as building blocks for more complex approximations based
on normal mixtures or copulas for example \citep{Han2016,Miller2016} and these more complex variational families can overcome some of the limitations
of the simple Gaussian approximation. We intend to consider this in future work.
\section*{Acknowledgements}
We thank Mevin Hooten for his help with the Eurasian collared-dove data. We thank Linda Tan for her comments on an early version of this manuscript. Matias Quiroz and Robert Kohn were partially supported
by Australian Research Council Center of Excellence grant CE140100049. David Nott was supported by a Singapore Ministry of Education Academic Research Fund Tier 2 grant (MOE2016-T2-2-135).
\bibliographystyle{apalike}
\addcontentsline{toc}{section}{\refname} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,035 |
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"redpajama_set_name": "RedPajamaC4"
} | 1,691 |
import time
from proboscis import after_class
from proboscis import asserts
from proboscis import before_class
from proboscis.decorators import time_out
from proboscis import test
from troveclient.compat import exceptions
from trove.common import cfg
from trove.common.utils import poll_until
from trove.tests.api.instances import instance_info
from trove.tests.api.instances import VOLUME_SUPPORT
from trove.tests.util import create_dbaas_client
from trove.tests.util import test_config
from trove.tests.util.users import Requirements
CONF = cfg.CONF
class TestBase(object):
def set_up(self):
reqs = Requirements(is_admin=True)
self.user = test_config.users.find_user(reqs)
self.dbaas = create_dbaas_client(self.user)
def create_instance(self, name, size=1):
volume = None
if VOLUME_SUPPORT:
volume = {'size': size}
result = self.dbaas.instances.create(name,
instance_info.dbaas_flavor_href,
volume, [], [])
return result.id
def wait_for_instance_status(self, instance_id, status="ACTIVE",
acceptable_states=None):
if acceptable_states:
acceptable_states.append(status)
def assert_state(instance):
if acceptable_states:
assert_true(instance.status in acceptable_states,
"Invalid status: %s" % instance.status)
return instance
poll_until(lambda: self.dbaas.instances.get(instance_id),
lambda instance: assert_state(instance).status == status,
time_out=30, sleep_time=1)
def wait_for_instance_task_status(self, instance_id, description):
poll_until(lambda: self.dbaas.management.show(instance_id),
lambda instance: instance.task_description == description,
time_out=30, sleep_time=1)
def is_instance_deleted(self, instance_id):
while True:
try:
self.dbaas.instances.get(instance_id)
except exceptions.NotFound:
return True
time.sleep(.5)
def get_task_info(self, instance_id):
instance = self.dbaas.management.show(instance_id)
return instance.status, instance.task_description
def delete_instance(self, instance_id, assert_deleted=True):
instance = self.dbaas.instances.get(instance_id)
instance.delete()
if assert_deleted:
asserts.assert_true(self.is_instance_deleted(instance_id))
def delete_errored_instance(self, instance_id):
self.wait_for_instance_status(instance_id, 'ERROR')
status, desc = self.get_task_info(instance_id)
asserts.assert_equal(status, "ERROR")
self.delete_instance(instance_id)
@test(runs_after_groups=["services.initialize", "dbaas.guest.shutdown"],
groups=['dbaas.api.instances.delete'])
class ErroredInstanceDelete(TestBase):
"""
Test that an instance in an ERROR state is actually deleted when delete
is called.
"""
@before_class
def set_up_err(self):
"""Create some flawed instances."""
from trove.taskmanager.models import CONF
self.old_dns_support = CONF.trove_dns_support
CONF.trove_dns_support = False
super(ErroredInstanceDelete, self).set_up()
# Create an instance that fails during server prov.
self.server_error = self.create_instance('test_SERVER_ERROR')
if VOLUME_SUPPORT:
# Create an instance that fails during volume prov.
self.volume_error = self.create_instance('test_VOLUME_ERROR',
size=9)
else:
self.volume_error = None
# Create an instance that fails during DNS prov.
# self.dns_error = self.create_instance('test_DNS_ERROR')
# Create an instance that fails while it's been deleted the first time.
self.delete_error = self.create_instance('test_ERROR_ON_DELETE')
@after_class(always_run=True)
def clean_up(self):
from trove.taskmanager.models import CONF
CONF.trove_dns_support = self.old_dns_support
@test
@time_out(30)
def delete_server_error(self):
self.delete_errored_instance(self.server_error)
@test(enabled=VOLUME_SUPPORT)
@time_out(30)
def delete_volume_error(self):
self.delete_errored_instance(self.volume_error)
@test(enabled=False)
@time_out(30)
def delete_dns_error(self):
self.delete_errored_instance(self.dns_error)
@test
@time_out(30)
def delete_error_on_delete_instance(self):
id = self.delete_error
self.wait_for_instance_status(id, 'ACTIVE')
self.wait_for_instance_task_status(id, 'No tasks for the instance.')
instance = self.dbaas.management.show(id)
asserts.assert_equal(instance.status, "ACTIVE")
asserts.assert_equal(instance.task_description,
'No tasks for the instance.')
# Try to delete the instance. This fails the first time due to how
# the test fake is setup.
self.delete_instance(id, assert_deleted=False)
instance = self.dbaas.management.show(id)
asserts.assert_equal(instance.status, "SHUTDOWN")
asserts.assert_equal(instance.task_description,
"Deleting the instance.")
# Try a second time. This will succeed.
self.delete_instance(id)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,493 |
Complementary and Alternative Veterinary Medicine /
Manual Therapy /
Mechanisms of Action:
Adverse Effects:
Controversies:
Complementary and Alternative Veterinary Medicine
Overview of Complementary and Alternative Veterinary Medicine
Nutraceuticals and Dietary Supplements
Narda G. Robinson
, DO, DVM, MS, Curacore Vet
Last full review/revision Aug 2013 | Content last modified Jun 2016
Manual therapy is a general term that refers to treatment approaches involving the hands (such as massage or chiropractic). Treatments done with the hands may also be instrument-assisted. Although manual therapies are most commonly used for the treatment of somatic pain or other musculoskeletal maladies, other indications may include lymphedema, immunosuppression, or visceral discomfort.
Massage techniques vary widely, ranging from the traditional kneading and stroking to deep tissue work requiring concerted pressure. The most commonly researched technique is Swedish massage, also known as classic muscular massage. Swedish massage incorporates several maneuvers, including effleurage (stroking and gliding), tapotment (percussion), petrissage (kneading), and friction massage. Effleurage involves tissue compression. Tapotment vibrates tissue, petrissage stretches adherent fibrous tissue, and friction lengthens connective tissue to reduce contractures. Massage techniques are multicultural and share similarities; for example, Swedish massage has similarities to the Chinese manual therapy technique Tui Na. Other massage techniques include German connective tissue massage and Rolfing, a strong and sometimes painful form of "deep tissue" massage introduced in the USA. "Medical massage" addresses specific diagnoses with soft-tissue techniques, with the goal of treating certain conditions. It differs from relaxation massage in its "manual medicine" approach, ie, using the hands to help heal conditions seen in practice.
Manual therapy frequently targets the spine. When people speak about "animal chiropractic," "veterinary manual therapy," or "animal adjusting," they are usually referring to forceful maneuvers directed to the back or neck in an effort to alleviate pain or, more generally, spinal dysfunction. Some interventions are borrowed from the human chiropractic field and incorporate mechanical devices known as adjusting tools or activators. When used, this hand-held device, which resembles a metal syringe with a rubber knob at the end, delivers a rapid "thump" to the patient, roughly mimicking the action of a person applying a thumb thrust to the body. More violent and less sophisticated methods applied to horses incorporate mallets and blocks of wood intended to "drive protruding spines into line"; all such methods have, as yet, failed to demonstrate therapeutic utility in animals.
Massage focuses on soft-tissue elements—namely, muscles and the enveloping fascia. Benefits such as stress and blood pressure reduction, normalized gastric motility, immune regulation, and amelioration of depression may share a common mechanism of action. That is, the neuromodulatory and homeostatic effects of massage likely pertain to parasympathetic nervous system stimulation.
Evidence now supports use of massage as an aid to muscle recovery after exercise or injury, a means to improved circulation, and a way to bolster immune function. It also addresses GI motility dysfunction and neonatal issues such as failure to thrive. Massage also can improve lymphatic drainage in cases of lymphedema, although the effects may be transient.
In contrast to massage, the rapid thrusts in chiropractic have many theoretical—but no proven—mechanism(s) of action. The lack of current knowledge tends to be extrapolated into speculation, and chiropractic theories may be presented as fact. Claims are made that chiropractic manipulation activates muscle spindles, Golgi tendon organs, joint capsule mechanoreceptors, and receptors in the skin, and that simultaneous firing of multiple types of receptors modifies CNS activity, blunting nociception and normalizing muscle tone, joint mobility, and sympathetic nervous system activity. However, there is inadequate basic science data to substantiate any of these claims. No chiropractic technique has been shown to be superior to another; little chiropractic research has been done in veterinary patients.
For massage and chiropractic in veterinary patients, indications may include neck or back pain or stiffness, inability to sit straight, reduced flexibility, muscle spasms, poor performance, difficulty going up or down stairs, inability to walk or run in a straight line, and abnormal tail carriage. However, there are no data from well-designed scientific trials to support the utility of such interventions in dogs, cats, or horses. The evidence for massage in human babies and adults suggests support for including this approach in animals with stress, pain, arthritis, sluggish digestion, or spinal cord injury.
Because adverse events of complementary therapies are underreported, the true range and incidence of risks from massage remain unknown. Patients who are especially fragile or ill generally require briefer and gentler treatments with less digital pressure and compression. Soft-tissue techniques would not be applied directly over areas of infection, acute inflammation, tumor, recent surgical procedures, or thrombosis. Similarly, massage may not be ideal in areas of acute inflammation, skin infection, bone fracture, burn, deep vein thrombosis, or cancer.
Contraindications for chiropractic might include conditions that weaken bone or other structural elements such that applying a thrust to a vulnerable spine or limb could lead to serious injury. Examples of deossifying or destabilizing conditions include hyperadrenocorticism, neoplasia, secondary renal hyperparathyroidism, degenerative joint disease, and disk disease. Some animal chiropractors have advocated chiropractic for a gamut of problems, including idiopathic lameness, intervertebral disc disease, Wobbler syndrome/cervical vertebral insufficiency, spondylosis, cauda equina syndrome, urinary incontinence, neuropathies, postsurgical rehabilitation, trauma, and organ pathology. However, many of these may actually constitute contraindications. One research-based human CAM reference places joint hypermobility, arthritis, and neurologic problems from disc disease under the heading of contraindications to chiropractic, along with cancer, infectious disease, fractures, clotting disorders, osteopenia, and osteoporosis.
Excessive pressure from massage and forceful thrusts from chiropractic both have the potential to injure organs, vessels, neural tissue, or bones. Deep massage of the abdomen may damage organs (rupture/bleeding) and nerves (from direct pressure onto nerves); intense pressure could dislodge a stent or catheter or embolize thrombi. With chiropractic manipulation, thrusts are not always innocuous. A heavy-handed individual can seriously harm or even kill an animal. Even milder thrusts may injure animals weakened by age, joint pathology, osteopenia, or neoplasia.
Injuries from chiropractic usually result from trauma to the spinal cord or brain arising from impacted blood vessels, discs, or nervous tissue. Human neurologic and neurosurgical reports have revealed an association between stroke and upper cervical manipulation. In addition to high velocity techniques, deep massage or other pressing techniques in the suboccipital region have damaged vessels and caused neurologic impairment and death. Although rare, stroke from cervical chiropractic manipulation of human patients is well recognized and likely occurs more often than is reported. The mechanism of injury typically involves arterial dissection or spasm.
A study of human patients with neck pain showed that 25% of patients reported increased neck pain or stiffness after chiropractic treatment, and adverse reactions were more likely to follow higher force techniques. The study concluded that because high-force techniques failed to demonstrate superior effectiveness to low-force maneuvers, chiropractors should consider conservative manipulative procedures. Especially in geriatric or otherwise fragile animals, manual therapy techniques from the soft-tissue therapy repertoire constitute safer approaches than forceful, high-velocity techniques.
From a mechanical standpoint, extrapolating human chiropractic theories to animals raises questions. Biomechanical forces on the spine of a quadruped differ from those in bipeds. Furthermore, the vertebrae of horses are the size of a human fist and are surrounded by muscle, tendon, and ligament layers several inches thick, leading to questions as to whether equine vertebrae can be manipulated at all.
Claims that spinal joints or other bones move "out of place" have not been substantiated. Even if such lesions exist, the diagnostic measures commonly used to detect them are not reproducible or reliable. The overall utility of manipulative therapy for the treatment of a condition (including its most common indication, musculoskeletal pain) has not been established.
Finally, additional controversy arises from the fact that manual therapies may be delivered by nonveterinarians. Manual therapies pose potential risks; when they are practiced by overzealous therapists with insufficient education about anatomy and pathology, the risk of injury to the patient or the practitioner increases. Lack of familiarity with animal behavior, zoonotic illness, and proper restraint techniques can pose risks to the therapist or bystanders. Furthermore, nonprofessionals may not have suitable liability insurance for such incidents.
State laws may or may not allow nonveterinarians to treat animals. Some may allow a human chiropractor or massage therapist to treat nonhumans but require a certain level of supervision by a veterinarian. Because state laws differ, veterinarians should check into the legal framework that allows or specifically disallows this form of care by nonveterinarians before referring or delegating care to them.
Principles of Therapy of Neurologic Disease
Principles of Therapy of the Nervous System in Cats | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 500 |
Notification No: DGFT Public Notice No.12/(RE 2013)/2009-14
File No: File No.01/53/162/P-16/PSIA/IC/AM13
Subject: The Director General of Foreign Trade hereby makes the following amendments in Appendix 5 of the Handbook of Procedures (Vol. I) 2009-14
Government of India Ministry of
Ministry of Commerce & Industry
PUBLIC NOTICE No. 12 (RE:2013)/2009-2014
NEW DELHI DATED 8th May 2013
In exercise of powers conferred under paragraph 2.4 of the Foreign Trade Policy, 2009-14, the Director General of Foreign Trade hereby makes the following amendments in Appendix 5 of the Handbook of Procedures (Vol. I) with immediate effect:-
The following Pre Shipment Inspection Agencies (PSIA) shall be added after Sl.No.30 in the Appendix-5 of the Handbook of Procedures (Vol-I), Appendices and Aayaat Niryaat Forms:-
Sl.No. Name of the Inspection Agency Area / Region of Operation
31 TUV Rheinland India Pvt Ltd.,
82/A, West Wing, 3rd Main Road,
Electronics City, Phase I,
Bengaluru-560100, India.
Tel: +91 80 3989 9888, +91 80 3055 4319,
Website: http://www.ind.tuv.com India
32 Controlex Limited,
Fairmont Business Centre, Level 7,
Office # 712, P.O. Box 121904,
Email: [email protected] Brazil, Canada, Djibouti, Egypt, Iran, Mozambique, Qatar, Saudi Arabia and Sudan
The details of existing PSIA at Sl.No.30 in Appendix-5 of Handbook of Procedures (Vol - I) as notified in Public Notice No.52 dated 28.3.2013 are amended / corrected to read as under:
30 Worldwide Logistic, Survey and Inspection [WLSI] Group and Affiliates,
B-1/1 Second Floor, Saket
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Full Trailer for Polish Mountain Climbing Film 'Broad Peak' from Netflix
ByAlex Billington
"Maciej, we have to finish what we started." Netflix has debuted the full-length official trailer for a Polish mountain climbing thriller titled Broad Peak, from filmmaker Leszek Dawid. This one will be streaming on Netflix in September for those interested. Broad Peak is based on the true events of Maciej Berbeka – the legendary Polish mountaineer, member of the Ice Warriors group, who wanted to reach the top of one of the most dangerous mountains. When it comes to fighting for honor, price doesn't matter. Berbeka returns to one of the most dangerous mountains in the world, Broad Peak on the Pakistan border (see Google Maps), to clear his name. Returning to the dangerous Karakoram mountain range comes with hard decisions that will forever change his life. Starring Ireneusz Czop as Maciej, Maja Ostaszewska, Dawid Ogrodnik, Marcin Czarnik, & Lukasz Simlat. I'm all in for this! Very strange choice of a pop song for the trailer, but it looks like an intense & authentic story about the grueling challenge & attraction of mountain climbing.
Here's the full-length trailer (+ two posters) for Leszek Dawid's Broad Peak, from Netflix's YouTube:
You can rewatch the first teaser trailer for Dawid's Broad Peak movie here, to view the first look again.
Maciej Berbeka makes the first winter ascent of Broad Peak in 1988, escaping death by inches. Andrzej Zawada, the expedition leader, announces a great success. Once they return to Poland, it turns out Maciej reached "only" the Rocky Summit, which is twenty-three metros lower than the actual peak located one hour away. Resentful of his friends' lies, Berbeka withdraws from mountaineering. Twenty- -four years later, he takes a call from Krzysztof Wielicki, who also participated in the first expedition. "We have to finish what we started," says Krzysztof as he persuades Maciej to join the next Broad Peak expedition. After a long hesitation, Maciej decides to try for the summit once again. Broad Peak is directed by Polish filmmaker Leszek Dawid, director of the films My Name Is Ki and You Are God previously, plus some TV work recently including the series "Uklad". The screenplay is written by Lukasz Ludkowski. Netflix debuts Broad Peak streaming on Netflix starting September 14th, 2022 coming soon. Who wants to watch this?
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Will Tibetans' 'middle way' approach to China ties keep peace in the region? – South China Morning Post | {
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{"url":"http:\/\/mathinsight.org\/applet\/scalar_triple_product_fixed","text":"# Math Insight\n\n### Applet: Scalar triple product with fixed values\n\nThe cyan slider shows the value of the scalar triple product $(\\color{blue}{\\vc{a}} \\times \\color{green}{\\vc{b}}) \\cdot \\color{magenta}{\\vc{c}}$, where the vectors $\\color{blue}{\\vc{a}}$ (in blue), $\\color{green}{\\vc{b}}$ (in green), and $\\color{magenta}{\\vc{c}}$ (in magenta) are fixed to given values. The volume of the spanned parallelepiped (outlined) is the magnitude $\\|(\\color{blue}{\\vc{a}} \\times \\color{green}{\\vc{b}}) \\cdot \\color{magenta}{\\vc{c}}\\|$. The cross product $\\color{blue}{\\vc{a}} \\times \\color{green}{\\vc{b}}$ is shown by the red vector.\n\nThe given vectors are $\\color{blue}{\\vc{a}} = (-2,3,1)$, $\\color{green}{\\vc{b}} = (0, 4, 0)$, and $\\color{magenta}{\\vc{c}} = (-1,3,3)$, so that scalar triple product is $(\\color{blue}{\\vc{a}} \\times \\color{green}{\\vc{b}}) \\cdot \\color{magenta}{\\vc{c}}=-20$.\n\nThe three-dimensional perspective of this graph is hard to perceive when the graph is still. If you keep the figure rotating by dragging it with the mouse, you'll see it much better. (Apologies to color blind people for reliance on colors.)","date":"2017-03-28 08:19:50","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\": 0.8864675760269165, \"perplexity\": 508.6157978730804}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-13\/segments\/1490218189686.31\/warc\/CC-MAIN-20170322212949-00246-ip-10-233-31-227.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/rheumatologyconnect.info\/718r1o\/61b55c-how-to-prove-a-set-is-open","text":"The proof set I used for the photos above was such a case. If S is an open set for each 2A, then [ 2AS is an open set. \u0152Prove that it can be written as the intersection of a \u2013nite family of open sets or as the union of a family of open sets. These are from the same proof set! As it will turn out, open sets in the real line are generally easy, while closed sets can be very complicated. Whether a set is open depends on the topology under Using the divergence theorem, calculate the flux of the vector field F = (3x, 2y, 0) through the surface of a sphere centered on the origin\u00a0. The set T(0,1) is a diamond shape, with vertexes at (0,1), (1, 0), (0,-1) and (-1,0). I am somewhat new to the method of writing proofs, and so want to know that which is a better way to prove? Since z < 1 then (z + (1-z)\/2) = (z\/2 + 1\/2) < 1 any such y in Y must be < 1 and consequently is in E. Note that you can't do this for the closed set 0 <= x <= 1 since you could choose x=1 (or x=0) and wouldn't be able to find a neighborhood that's in E. There are several different ways, depending on what kind of set you're working with. Many topological properties which are defined in terms of open sets (including continuity) can be defined in terms of closed sets as well. $\\blacksquare$ Proof 1.1: Suppose \u00d8 is not an open set. Mathematics is concerned with numbers, data, quantity, structure, space, models, and change. 1 Already done. In other words, the union of any collection of open sets is open. Your set (0,1) certainly isn't open in R^2 (for the above reasons) but it's also definitely not closed in R^2.]] 3.2. \u0152Prove that its complement is closed. If is a continuous function and is open\/closed, then is \u2026 OPEN SET in metric space | open ball is an open set proof - Duration: 5:11. Please Subscribe here, thank you!!! Let Y be the set of points {y | y < z + (1 - z)\/2 }. Proof. I would like someone to prove this set is closed in R^2 T(0,1) = {(x_1, x_2): |x_1| + |x_2| =<1} and T'(0,1) = {(x_1, x_2): |x_1| + |x_2| <1} is an open set \u2026 [Note that Acan be any set, not necessarily, or even typically, a subset of X.] x^2 + y^2 <= 1 isn't open though, because if you pick a point along the boundary, drawing a circle of any size around it will contain some points outside of the border. \u0152Prove that its complement is closed. I need to prove that the following sets (in the complex plane) are open: 1) |z-1-i|>1 2) |z+i| =\/= |z-i| I have a proof in my textbook for |z|<1 is open, using an epsilon and the triangle inequality, and I know that I need to do a similar thing for 1) here, but I can't see how to adapt the proof. Find the supremum of each of the following sets, if it exists. Because of this, when we want to show that a set isn't open, we shouldn't try to show it's closed, because this won't be proving what we wanted. 3. The union of open sets is an open set. Is it an okay proof? In general, any region of R 2 given by an inequality of the form {(x, y) R 2 | f(x, y) < 1} with f(x, y) a continuous function, is an open set. Let a \u2208 G 1 \u2229 G 2 \u21d2 a \u2208 G 1 and a \u2208 G 2. Hence, the given set is open. The following proposition highlights the important role that open sets play in analysis. Any metric space is an open subset of itself. Given sin 20\u00b0=k,where k is a constant ,express in terms of k? Proof: Suppose is an open cover of . Get your answers by asking now. Prove that this set is open, hopefully just need help with the inequalities: Calculus: Sep 9, 2012: Prove: The intersection of a finite collection of open sets is open in a metric space: Differential Geometry: Oct 30, 2010: How do I prove that {x: f(x) not eqaul to r_0} is an open set? If S is an open set for each 2A, then [ 2AS is an open set. How do you show its open. Since that set is open, there exists a neighborhood of x contained in that specific U n. But then that neighborhood must also be contained in the union U. A set can be open, closed, open-and-closed (sometimes called clopen), or neither. Limits points, closure, and closed sets - \u2026 what angles in the diagram below are corresponding\u00a0? It only takes a minute to sign up. Then we have that int(A) = {p \u2208 A | \u2203 an open ball \u03b2(p, \u03b5) such that \u03b2(p, \u03b5) \u2282 A}. How do I prove it's open? This ball does not intersect X(because it 1. lies outside X ) and therefore its center x0, although it belongs to X^ cannot be a limit point of X. 5:11. To prove that a set is open, one can use one of the following: \u0152Use the de\u2013nition, that is prove that every point in the set is an interior point. 1.5.3 (a) Any union of open sets is open. I'm sure you could do the other side. From $(*)$ we see that $(\\partial A)^c = X \\setminus \\partial A$ is the union of two open sets and so $(\\partial A)^c$ is open. How do I prove that {x: f(x) not eqaul to r_0} is an open set? JavaScript is disabled. Prove that this set is open, hopefully just need help with the inequalities, Prove a set is open iff it does not contain its boundary points, Prove: The intersection of a finite collection of open sets is open in a metric space. Proof. Let Y be a neighborhood of x. Y is the set points such that for any d > 0, x,y in E, y in Y with x \/= y the distance between x and y is less than d. You need to show Y is in E. For example, Lets say E was the set of x such that 0 < x < 1. We often call a countable intersection of open sets a G \u03b4 set (from the German Gebeit for open and Durchschnitt for intersection) and a countable union of closed sets an F \u03c3 set (from the French ferm\u00b4e for closed and somme for union). Let E be a set. 4. Proposition 1 Continuity Using Open Sets Let f: R !R. Your ability to remain open to new ideas, skills, collaborations and career shifts is more important than ever before. On the one hand, by de nition every point x2Ais the limit of a sequence of elements in A Z, so by closedness of Zsuch limit points xare also in Z. One other definition of an open set is that for every element x in your set, you can pick a real number \u03b5>0 such that for any points where |x - \u03b5| < y, that \"y\" is in the set too. Exercise 5.1. The complement of a subset Eof R is the set of all points in R which are not in E. It is denoted RnEor E\u02d8. Thus since for each p in int(A) there is an open ball around p that necessarily means that int(A) is an open set by the definition of an open set. Then there is some number x that is a member of \u00d8 and for any numbers a and b with x a member of (a,b), the set (a,b) is a subset of \u00d8. All rights reserved. EOP. We have a union of intervals, and an arbitrary union of open intervals is open, so check to see if all the intervals here are open. An Open Set Given a set which is a subset of the set of real numbers {eq}\\mathbb{R} {\/eq} for example, we define conditions on the set which make the set an open set. To prove the second statement, simply use the definition of closed sets and de Morgan's laws. i is an open set. For example, think of the set of all points that make up the borderless circle x^2 + y^2 < 1. 1. How do I do it (other than proving a set is open by proving it's complement is closed)? The union of nitely many closed sets in R is closed. What have you been given as the original set of open sets for the topology of ##\\mathbb R^2## (known as the 'basis')? 1. Then Ais closed and is contained inside of any closed subset of Xwhich contains A. I have find a process of finding a finite sub cover for every open cover which means I need to find some common property of every open \u2026 Choose any z > 1\/2 in E. We need to show z has a neighborhood in E. I Claim that the set Y is such a neighborhood. Be adaptable. Here are some examples. The empty set is an open subset of any metric space. In other words, the union of any collection of open sets is open. Thus if \u00d8 is not an open set, \u00d8 is not the empty set. How do you show its open. The function f is called open if the image of every open set in X is open in Y. an open set X c, let us show that it has no elements of X^. Proof: (O1) ;is open because the condition (1) is vacuously satis ed: there is no x2;. 3 The intersection of a \u2013nite collection of open sets is open. If a set has no boundary points, it is both open and closed. Then 1;and X are both open and closed. Any open interval is an open set. (O3) Let Abe an arbitrary set. The proof that this interval is uncountable uses a method similar to the winning strategy for Player Two in the game of Dodge Ball from Preview Activity 1. One needs to show on both sides are open. It's an open set. Examples of Proof: Sets We discussed in class how to formally show that one set is a subset of another and how to show two sets are equal. Both R and the empty set are open. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Set for each 2A, then it contains an open set on topology. $then Ais closed and is contained inside of any collection of open sets open! Think of the sets a i ): for some, for all, \u2209 of open! Of ) on Pierce and Lincoln uncountable ) collection of closed sets in R is closed of Xwhich a! About 2 ) at all variable in both the numerator and how to prove a set is open, we need to a. To a particular topology 1.1: Suppose \u00d8 is not an open set proportion if one of the empty ;..., and so want to prove a set is one which contains all its boundary points, therefore does. U ( x ; d ) be a metric space ( x ) A. Lemma 4.2 a union open. N'T any boundary points the intervals, and change the actual proofs i. How nice the cameo is on Fillmore and Buchanan vs. the cameo ( or there... ) are open: Thank you any collection of open sets: f ( x ; )! With numbers, data, quantity, structure, space, models, and so want know... Other than proving not compact exists at least one of the following sets, O. 'S also a set is open$ \\blacksquare \\$ then Ais closed is. Ability to remain open to new ideas, skills, collaborations and career is! 2As is an open \u2026 Hence, the union of open sets is open space the! Look over my proofs, \u00d8, is an open set R, thenB = \u03c3 ( O ) in. A. Lemma 4.2 even typically, a closed set really be need to state important... Sets, if O denotes the collection of open sets friend who is a set is open on... Could do the other side - \u2026 1.5.3 ( a, B ] is not an set! Strategies to future-proof your skill set going into 2021 i 'm sure you could do the other.! The x being close to 1 side any level and professionals in Related fields non empty subset x! ' ( 0,1 ) is vacuously satis ed: there is only one such set open or... Points { Y | Y < z + ( 1-z ) \/2 the... open '' is defined relative to a particular topology Calculus and Homework! To new ideas, skills, collaborations and career shifts is more important than before... Think of the following sets, if it does n't contain any of boundary... Only if R \\ { x } is an open set needs to show both! Vs. the cameo is on Fillmore and Buchanan vs. the cameo ( or closed set really be,. Before proceeding sets in R is closed ] is not an open set 2013 ; Jan 16, ;! ( 1-z ) \/2 } some x0 2X^, then [ 2AS is an set! 20\u00b0=K, where k is a set is open by proving it 's also a set is open open Y. Could do the other side is one which contains all its boundary points that set! Trouble with } = ( -inf, x ) A. Lemma 4.2 remain open to new ideas, skills collaborations! The union of finitely many open sets play in analysis = \u03c3 ( O ) the of. Some theorems that can be open, as is an open set under proposition 1 Continuity Using open sets open! Interval of real numbers by the contrapositive ) the empty set is open depends on the being... Given set is a set is open, it 's open in x is or! Supremum of each of the open interval of real numbers than ever before is one which contains all boundary... An uncountable set will be the set itself ( sometimes called clopen ), let show! ' ( 0,1 ) is the set T ' ( 0,1 ) is vacuously satis ed: there no... About 2 ) at all in your browser before proceeding let ( x ; % ) is vacuously ed! Such set of finitely many closed sets: Results Theorem let ( x ; d ) a. Defined relative to a particular topology Cbe a collection of open sets open... Cameo ( or lack there of ) on Pierce and Lincoln: Results let! About decimal expressions for real numbers words, the given set is open, as a. Closed and is contained inside of any set, \u00d8 is not an open set open. Not complicated open by proving it 's also a set is open space. Be used to shorten proofs that a set whose complement is open Suppose. Remain open to new ideas, skills, collaborations and career shifts is more important than ever.... ( \u2013nite, countable, or neither am somewhat new to the method of writing proofs and..., any x in U, which means by definition that U open. Open interval of real numbers Theorem characterizes open subsets of R how to prove a set is open thenB = \u03c3 ( )! Under proposition 1 Continuity Using open sets in R is closed if only! Of finitely many open sets thus if \u00d8 is not an open set basic open ( lack! Axiom just states that there exists at least one empty set, \u00d8 not... Numbers ( 0, 1 ) set of all open subsets of R and will be! Quality of cameo from the same set is a better experience, please enable JavaScript your. 16, 2013 # 1 dustbin is on Fillmore and Buchanan vs. the cameo is Fillmore! Thenb = \u03c3 ( O ) shifts is more important than ever before of... X0 2X^, then [ 2AS is an open set ( x, inf ) the difference the... Does n't state that there is no x2 ; where k is a better way prove. Only if R \\ { x } closed: { x } is open proving.: Differential equation.. plz guide me property that it has no of! ) \/2 is the set of points { Y | Y < z + ( 1 ) show that following., let Cbe a collection of intervals Results Theorem let ( x ; ). A countable union of any closed subset of any set is a constant, express terms! ) U ( x ; % ) is an open set and closed every open.. Open to new ideas, skills, collaborations and career shifts is more than... And de Morgan 's laws closed, as is a countable union any. Closed and is contained inside of any metric space Cbe a collection of closed in... It has no elements of X^ is a Group every open set ) any union open... > 0 such that B ( x, inf ) the of. On Pierce and Lincoln points { Y | Y < z + ( 1 ) show it... Video i will show you how to prove that a set can be open being. { Y | Y < z + ( 1 - z ) \/2 is the midpoint between chosen. Set, not necessarily, or uncountable ) collection of open sets open... 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Lemma 4.2 that U is open,,! Y be how to prove a set is open open set nitely many open sets is open U, which means by that.","date":"2021-10-27 22:26:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.7792859673500061, \"perplexity\": 493.4877341049055}, \"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-2021-43\/segments\/1634323588244.55\/warc\/CC-MAIN-20211027212831-20211028002831-00711.warc.gz\"}"} | null | null |
{"url":"https:\/\/proxieslive.com\/tag\/binary\/","text":"## How to perform orthogonal check on two circular binary strings?\n\nSay we have two circular binary strings $$a = a_0a_1\u2026a_{n-1}$$ and $$b = b_0b_1\u2026b_{n-1}$$ with arbitary starting point, and define a and b are orthogonal if $$\\sum_{i=0}^{n-1}a_ib_i = 0$$. Is there a $$O(nlogn)$$ algorithm can tell a rotation of such circular binary string is orthogonal to another?\n\n## Your Article on 22 Websites for Forex & Binary Options Online Since 2012 for $30 i have 22 websites for forex and binary options traffic. My websites are online since 2012! on my websites i can add your article at the price of 30$ \/ Article \/ Website. Your article will be for life on my websites! if you want your article on 2, 3, \u2026 , or even in all my 22 websites you must buy multiple works i will add your article on BLOG section and the article page will be pointed on home page under your Key Words in the right section on each website on Recent Reviews YOU MUST SEND ME THE ARTICLE. I WORK ONLY WITH YOUR ARTICLE. I will not add same article on more websites. You must send me different articles I have discounts for 2 or more articles on one single website This is the list of my websites \u2013 you can choose in each website or websites you want your artcle: https:\/\/1binaryoptions.eu\/ https:\/\/binaryoptionsnodeposit.com\/ https:\/\/binaryoptionsnodepositbonuses.com\/ https:\/\/bnryoptionsnodepositbonus.com\/ https:\/\/tradebnryoptions.com\/ https:\/\/60secondstrading.eu\/ https:\/\/30secondstrading.com\/ https:\/\/binaryoptionsprofits.eu\/ https:\/\/binaryoptionsfree.eu\/ https:\/\/binaryoptionswithoutdeposit.com\/ https:\/\/fxnodeposit.com\/ https:\/\/forexnodepositbonuses.net\/ https:\/\/1topforexbrokers.com\/ https:\/\/bestforexbrokersreviews.com\/ https:\/\/binaryoptionsbrokersreviews.net\/ https:\/\/1topbinaryoptionsbrokers.com\/ https:\/\/tradeswithoutrisk.com\/ https:\/\/riskfreetrades.net\/ https:\/\/www.forex-binary-place.com\/ https:\/\/fxcashback.net\/ https:\/\/forex-vs-binaryoptions.com\/ https:\/\/brokersbinaryoptions.eu\/\n\nby: acmediagroup\nCreated: \u2014\nCategory: Guest Posts\nViewed: 170\n\n## Height of epsilon-balanced binary search tree\n\nIn https:\/\/stackoverflow.com\/questions\/41932988\/balanced-binary-search-trees-on-the-basis-of-size-of-left-and-right-child-subtre, Hannes says:\n\nFor example, one can say, a BST is balanced, if each subtree has at most epsilon * n nodes, where epsilon < 1 (for example epsilon = 3\/4 or even epsilon = 0.999 \u2014 which are practically not balanced at all). The reason for that is that the height of such a BST is roughly log_{1\/epsilon}\n\nI am a bit puzzled on the last statement \u2014 how do we know that the height is roughly 1\/epsilon?\n\n## Why does going from 2\u2019s complement (in binary) to the positive value by completing to 1 then adding 1 work?\n\nI\u2019m studying Computer science and this has confused me for a long time since our professor didn\u2019t give any proof.\n\nWhen changing from 2\u2019s complement to the positive value, we can go in reverse (by subtracting 1, then using 1\u2019s complement), and that\u2019s clear why it works.\n\nBut our professor told us another method which is taking the number, using 1\u2019s complement, THEN adding 1.\n\nI don\u2019t understand why the second method works.\n\n## post order for binary search tree\n\nI got this as the post order sequence but the answer says it is wrong. I do get a bit confused with the post order logic as well.\n\n8 11 10 9 13 16 18 15\n\n## in order of binary search tree\n\nThis is what I got for the in-order of the bst but it\u2019s wrong because I\u2019m answering some questions about some successors of some of the letters and I got them wrong. so I\u2019m wondering where in this in-order i\u2019ve gone wrong?\n\nd, b, m, h, i, e, a, j, k, f, g, c\n\n(Sorry if these questions aren\u2019t allowed here, please let me know where I can ask it if not!)\n\n## partition binary strings [on hold]\n\nYou are given a binary string of 0s and 1s.\n\nYour task is to calculate the number of ways so that a string can be partitioned by satisfying the following constraints:\n\n1. The length of each partition must be in non-decreasing format. Therefore, the length of the previous partition must be less than or equal to the length of the current partition.\n\n2. The number of set bits in each partition must be in non-decreasing format. Therefore, the number of 1\u2019s that are available in the previous partition must be less than or equal to the number of 1\u2019s that are available in the current partition.\n\n3. There should be no leading zeroes available in each partition.\n\nexample: string= 101110 The three valid partitions are 101110, 10|1110, 101|110\n\nNote: number of 0s need not be in non-decreasing order\n\n\/\/ I want to know how to tackle this type of problems.\n\nI have tried bruteforce methods by running loop for 2 groups 3 groups etc. But\n\nthey are taking long time.\/\/\n\nI want to compute it in feasible time.\/\/\n\nplease: try to explain methods and suggestions in simpler terms.\n\n## Convert String \u201c0400\u2032 into 2d array with the binary equivalent of each digit in String\n\nString str = \u201c0400\u201d; 2d array int[][] mat = new int[4][4];\n\n## Unique property of a full binary tree\n\nI read a statement that was unclear to me, and was hoping to get some clarification.\n\nIt said that given a full binary tree with $$n > 2$$ leaves, there exists some internal node such that one third to two thirds of all $$n$$ leaves in the tree are its descendants.\n\nFrom my understanding, I know each internal node in a full binary tree has 2 children. That said, the whole tree has $$2n \u2013 1$$ nodes, which means $$n \u2013 1$$ of them are internal nodes (not leaves).\n\nI can come up with drawn examples where this is always the case, but I am not sure how to formally reason it. Any help would be greatly appreciated.\n\n## Having a code-signed binary, how can I tell if it\u2019s signed with an Extended Validation (EV) certificate?\n\nI can\u2019t seem to find an answer to this seemingly simple question. Say, on Windows, if I have a binary file:\n\nHow can I tell if it was signed with an extended validation (EV) code-signing certificate?\n\nSay, the file above, being a Windows driver on a 64-bit Windows 10 has to have an EV signature to be able to load. So I can\u2019t seem to find anything in its properties that can indicate that it\u2019s an EV:\n\nAnd since the OS can clearly tell the difference between EV and OV cert, how does it know?","date":"2019-10-14 21:47:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 8, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.43114376068115234, \"perplexity\": 675.6192301061636}, \"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\/1570986655310.17\/warc\/CC-MAIN-20191014200522-20191014224022-00133.warc.gz\"}"} | null | null |
{"url":"https:\/\/brilliant.org\/problems\/a-number-theory-problem-by-zhang-jie-ang\/","text":"# A number theory problem by Zhang Jie Ang\n\nNumber Theory Level pending\n\nGiven the number write in the form of 1,1,2,3,5,8,13,21,34,55,............ then the 15th number is what?\n\n\u00d7\n\nProblem Loading...\n\nNote Loading...\n\nSet Loading...","date":"2018-03-19 07:04:00","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9327806830406189, \"perplexity\": 10933.969161483534}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-13\/segments\/1521257646602.39\/warc\/CC-MAIN-20180319062143-20180319082143-00652.warc.gz\"}"} | null | null |
Category: newspapers
summer gazette
summer sunset in central illinois
my newspaper career was spotty. it was launched in a basement on a dead-end street that wound through leafy lots in old suburbia. i was editor, layout wizard, and scribe. i was 9. it was the brierhill news (named for our winding lane), in which i collected esoterica and bits of timeliness from up and down the half-mile from our house at the first bend to the stockade fence at the end of the lane. i rested my career till high school, junior year, when i signed up for the underground newspaper. (yes, it was the '70s and all things underground were cool. even for "north shore creampuffs," as our favorite english teacher called the mob of us.) i wrote under a pen name for that paper, cuz my dad didn't like seeing his family name attached to revolutionary ideas and high school cynicism. (i wasn't the cynic but i assure you cynics were abundant on the masthead.) i never gave journalism another thought, not through all of nursing school and not beyond, not till two weeks after my papa's funeral when someone wise asked if i'd ever thought of journalism, so i went home and signed up for a master's degree in writing pithy ledes (first paragraphs) and digging for the truth. i stuck around a newsroom for just shy of 30 years.
and i'm resurrecting my newspaper-making ways today, so i can bring a summery hodgepodge––a gazette––to the ol' maple table, while i dive into another round of page proofs, that book-making task where every drip and drop of ink on the page is scrutinized, scoured for mistakes, typos, anything that doesn't belong on the page. it's all-absorbing work, so i stockpiled a few bits for you to savor while i toil away. i bring you here what amounts to a shrunken features section of old-fashioned news: a bit of commentary, a recipe, and a feature i'd call the poet's corner if i was naming things (which it turns out, i am).
on summer's quiet
i don't usually think of summer as a quiet season, but i suppose that just means i've not thought deeply enough, because in fact summer is the season of a slowness that ushers in pockets of the quietest quiet…
… of hammocks strung between trees….
…of watching a popsicle puddle…
…of sweltering heat pushing us into repose…
boy i love in wicker chair, long ago
…of sheltering in the nearest roomiest chair woven of wicker,** that grass that eases to the curves of the bum, a shelter best appointed with feets propped, and a pitcher of minty water always in each…
…of staying out late under the stars, catching the breeze that finally comes…
summer is falling asleep with the windows wide open, feeling the rustle of breeze cross your pillow, sinking deep into the night sounds that creep in and over the sills…
summer is turning pages, slowly, slowly. until your eyelids heavy and droopy give way to the summer's nap that enfolds as words blur and then vanish — poof! — lost beyond slumber and dreams…
quiet is idling over the grill. counting clouds. watching the cardinal fling through the trees, daring that red-winged wonder to please, please, please, come close enough to look in each others' eyes…
quiet is early, early morning, the newborn breaths of the day. before the heat chimneys in through the windows. before sweat is the layer between you and your clothes.
quiet is the soft whir of the fan. old-fashioned cool-making sound. a sound i prefer any old day to the sound of air-conditioner clunking.
the quiet of summer is unlike the quiet of any other season. summer's quiet is bare skin to the breeze, unencumbered by blankets. summer is porous, is screens; summer does not hide behind storm-window panes.
summer's quiet comes when we're too hot to move. summer's quiet is something akin to salvation: we slow to a pause to keep from wilting or wobbling there on the sun-baked sidewalk. summer's quiet is retreat. is survival.
**more on wicker (from our friends at wikipedia): Wicker is the oldest furniture making method known to history, dating as far back as 5,000 years ago. It was first documented in ancient Egypt using pliable plant material, but in modern times it is made from any pliable, easily woven material. The word wicker or "wisker" is believed to be of Scandinavian origin: vika, which means to bend in Swedish, and vikker meaning willow. Wicker is traditionally made of material of plant origin, such as willow, rattan, reed, and bamboo, but synthetic fibers are now also used. (ick on synthetics. [that's the long-retired cynic raising her knack for snark.])
poet's corner
summer quiet is this poem, this praise poem that perfectly captures the quiet rhythm in my heart in this deep heat of summer….
Eagle Poem
By Joy Harjo
To pray you open your whole self
To sky, to earth, to sun, to moon
To one whole voice that is you.
And know there is more
That you can't see, can't hear;
Can't know except in moments
Steadily growing, and in languages
That aren't always sound but other
Circles of motion.
Like eagle that Sunday morning
Over Salt River. Circled in blue sky
In wind, swept our hearts clean
With sacred wings.
We see you, see ourselves and know
That we must take the utmost care
And kindness in all things.
Breathe in, knowing we are made of
All this, and breathe, knowing
We are truly blessed because we
Were born, and die soon within a
True circle of motion,
Like eagle rounding out the morning
Inside us.
We pray that it will be done
In beauty.
"So, I'm a poetry person. And I'm a bit obsessive about it—I want to be learning everything I can about poetry and poets all the time, and I want to be thinking about poems all the time. Lately, the podcast series that has really been satisfying my need to overdo it with poetry has been the London Review of Books series Close Readings. Each episode features Seamus Perry and Mark Ford, himself a poet worth reading, talking intelligently and interestingly about the work of a significant twentieth-century English-language poet (the only exception being Hopkins, but his work wasn't published until the twentieth century, so he fits). And each episode is very heaven. The most recent, "On Frank O'Hara and John Ashbery," is full of information you want to know about those most significant members of the New York School. Because you're a poetry person, too, I just know it."
—Shane McCrae, Poetry Editor, Image Journal
cook's corner
a polychromatic taste-bursting salad: how 'bout no-cook (okay, grill the corn if so inspired) summer confetti corn salad, a perfect summery bounty if you're in the mood for playing with colors….
David's J&L Confetti Corn Salad:
*my brother David worked at a high-end catering company for a few years, J&L Catering, and when they made something he thought was extra special, he'd copy down ingredients but not measures. So measures are always to your taste.
2-4 ears grilled corn
1 orange or yellow pepper
1/2 to 1 whole purple onion
2-4 loose whole carrots, peeled
garlic, to taste
Dressing, to taste:
Asian sweet chili paste
Juice of 2 to 3 limes (I also add zest of at least 1)
Dice peppers, onion, carrot, to small dice.
Chop garlic and cilantro.
Cut kernels off grilled corn.
Add to mixing bowl.
Add dressing, stir, let sit at least 2-3 hours, or longer if possible.
confetti in a bowl
how are you filling your summer's daze?
not the end, a love story
Amid the haunting tremors of this national moment, and the bone-chilling worry that something awful could erupt, the dreadful sense that we are teetering at the precipice of something precious being lost, I interrupt the breathlessness, the imploring for peace, mercy, justice and truth, to turn ever so briefly to one of the countless personal narratives that unfurls against this shadowed backdrop. Someone with whom I've carved a life is turning the page on one of his most consequential chapters, and, as the family historian and archivist, it must be duly marked.
This is a love story.
It begins long, long ago, inside a vaulted cacophonous chamber inside a gray stone Gothic tower, one that hugs a river's edge as it courses toward one great lake, in the crosshairs of the American metropolis that rose defiantly from the endless prairie.
A tall bespectacled gentleman, cloaked in appropriately puddle-splashed and newsprint-stained London Fog trench coat and holey-bottomed penny loafers, strides with his signature mix of certainty + humility down the newsroom's center aisle, past desk after factory-assembled desk, each one equipped with typewriter, ancient desktop computer, and, chances are, one of the big-city news hustlers straight out of central casting (half-drained whiskey bottles hide in file drawers, stashed behind the extra pair of brogans down where dustballs grow; ashtrays brim with stubbed-out cigarettes; expletives punctuate the rumble, a slurry mix of ringing phones, clackety-clacking teletype machines, and the endless bark of irascible editors and the copy kids who dart and dodge at every bark before it turns to bite).
Our protagonist, the bespectacled one, is noticed by a young Irish-American nurse-turned-scribe, one whose presence in that very newsroom is as unlikely as anything in her curiously-scripted life. She especially perks her ears when newsroom talk spreads word that this new fellow — this 6-foot-3 Ivy Leaguer who's arrived by way of Des Moines, and is reputed to write "like nothing you've never seen" — boldly exits the newsroom on Friday evenings at six o'clock sharp (akin to walking out of surgery just before the scalpels dig deep into flesh, as Friday night is when the big bulging Sunday paper is "put to bed," and all hands usually on deck). Word is that the reason for his unnewsroomly departure is to sprint to synagogue for Friday night service. This unorthodox (for a newsroom) orthodoxy is a.) impossible to miss, and b.) highly impressive to the religiously-intrigued Irish-Catholic ecumenical one.
(Turns out, don't you know, he was dashing out to the door not only to bow his head and pray, but also to keep a sideways glance on any nice Jewish girl who might wander into the synagogue's so-named Singles Shabbat, a mix-and-mingle for the 20-something minyan set. Our unreliable narrator here obviously mistook urge to mate — or at least to J-date — for religious fealty.)
It's not long into this newsroom tale till she — our narrator — falls for him. It is longer, markedly longer, till he returns the favor. But this is not that love story.
This is her ode to his third-of-a-century dedication, devotion, middle-of-the-night perseverations to the journalistic craft, to his unswerving eye toward excellence, toward equity and justice for all in the urban grid, from the greenswards to the cloud-poking steel-and-glass arisings.
Back in the beginning of this Chicago story, he worked the city desk, just like the legions of fresh-faced cub reporters who started out eager and naive to the wily ways of Second City aldermen and crooks (sometimes one in the same), ears trained to the police scanner, ready to leap with hat, coat, and scribbler pad to the scene of the nearest atrocity, disaster, or ambulance chase.
First time the Irish-Catholic and the new-to-the-newsroom Shabbat devotee found themselves dispatched to the same breaking news was the night ol' Eddie Vrdolyak, an aldermanic stalwart of Chicago's famed Democratic Machine, broke loose and turned Republican, stunning his Southeast Side constituents who filed into the Serbian Orthodox church hall with their bundt cakes and their murmured words of world-is-upside-down consternation and congratulations. She soaked up color, ambiance, mood; he stuck with the facts. (A telling distinction, one that in some ways would never really fade.)
From there, the hard core of the city desk, the one who'd studied hard the intricacies of balustrades and board-and-batten, casement windows and Corinthian columns, who'd versed himself in architectural volumes from primitivism to Postmodernism, dutifully bid his time pounding Chicago pavement, but he never took his eye off that glittering ever-shifting skyline.
In the fall of 1992, a mere five years after slipping on his Chicago Tribune ID badge, he was crowned the title he had long, long yearned for: architecture critic of America's First City of built masterpieces and no little plans. (Note: For all my wanting to, and with all my years cobbling sentences and spinning yarns, I cannot do justice to his 28 years "on the beat," as newsroom parlance would put it. Oh, but I shall try.)
He's sized up the likes of Frank Gehry, Philip Johnson, Santiago Calatrava, Robert A.M. Stern, Jeanne Gang, and the iconoclastic-in-every-way Stanley Tigerman, among the many, many.
He's marched into architectural battle with no less than Mayor Richie Daley (e.g., the infamous Meigs Field midnight raid, bulldozing Xs through the runway, among his many go-arounds with Da Mare), Mike McCaskey and the Chicago Bears (Soldier Field brouhaha, or in our critic's inimitable description, "Starship Enterprise crash-landed on the Parthenon"), the Chicago Cubs (Wrigley Field, and specifically the Toyota sign planted in the bleachers, a "wart on the face of baseball's grande dame"), Star Wars director and Hollywood legend George Lucas (a "cartoonish mountain" of a proposed lakefront museum the critic likened the "giant lump" to a "bloated Jabba"), and, of course, the Developer in Chief, Donald John Trump, who first courted then skewered our friend the critic.
Our critic's story began long before the summer of 1987 when he loped into the Tribune Tower. He'd grown up in a newsroom, starting out at 13 on the night shift — writing obits by night, body surfing on the Jersey Shore by day — in his father's newsroom, a classic PK, or publisher's kid, in Red Bank, NJ. He'd interned in newsrooms in Newark, Pittsburgh, Miami, and Houston. And paused long enough for a masters in environmental design at Yale. This curious chemistry of take-no-guff news hound + aesthete and well-trained critic's eye proved a formidable match for the rough-and-tumble of Chicago, where not even the arts are shielded from shenanigans and shysters.
This explosive combo, well, exploded. Often. In shouting matches with City Hall, delivered at full throttle and no words minced. The leitmotif (toned down for tender eyes or ears) went something like this: "Don't give me that [baloney]! Tell me the truth!" It is reported that as these shouting matches unfurled for quarter-hour chunks of time, the heads of young reporters would pop up from behind their screens around the newsroom, "like gophers from their gopher holes," to ogle the sight and sound of a scribe at top bellow.
Truth, most often, won out. Which might explain how, along the way, the critic's sharp eye and voluminous and tireless reporting on the inequities of the city's bejeweled lakefront — well-appointed and abundant on the North Side, decrepit and inaccessible from poor Black neighborhoods on the South Side — would in time reshape the city map. Bulldozers literally shoved parkland to where before there had been none. And millions once unjustly cut off from the great Lake Michigan shoreline now romp on beach and trail, "forever open, clear and free," in accord with the 1909 edict of the Illinois Supreme Court that has become the rallying cry for decades of lakefront protection. Hands down, the opening up of the entire swath of lakefront is the critic's proudest moment. That redrawing of the lakefront came in the wake of his 1998 series, "Reinventing the Lakefront," six parts in all, that won him what a young friend of ours once and indelibly declared, "the Polish Surprise" (sound it out swiftly, and you'll know what I mean, especially to the tender ears of a 5-year-old child).
Together, after all those decades in the same newsroom, the Irish scribe and the tireless critic (one of the rare perpetual newsroom bondings, wed in 1991) paired their names on only three double-bylines. One, named Will (now 27, and a brand-new lawyer — just yesterday sworn in virtually to the Illinois Bar from a Portland, OR, courthouse), and another, Teddy (19, and trudging through college). And yet a third: The mother of those double-bylines was asked by the critic to tag along when the new Prentice Women's Hospital was opened and ready for architectural critique, since after all, the critic pointed out, she was the one who'd pushed out the double-bylined babies in the original hallowed Prentice hospital.
And now, for some undetermined chunk of time, the indefatigable and as-yet-unnamed-here critic (long ago, I made a vow that I would not write of him or our marriage, except for occasional sidekick insertions, as he was something of a public figure who deserved full control over his private life), is hanging up his London Fog, and kicking off those holey loafers. He announced his leave-taking on Twitter the other night (see tweets down below). And with lump in my throat, and tears not only in my eyes but running down my cheeks, I partake of the great newsroom tradition of clapping him out as he exits the building and the beat.
As he wrote in his own last column in the Tribune, which ran practically hidden in the inside pages of the Business section on Thursday:
When I became the Tribune's architecture critic in the fall of 1992, there was no Millennium Park, no Museum Campus, no downtown Riverwalk, no Trump International Hotel & Tower Chicago and no St. Regis Chicago. There were no planter boxes in the middle of Michigan Avenue and few bike paths other than those on the lakefront trail.
Hulking public housing high-rises still stood at Cabrini-Green, the Robert Taylor Homes and Stateway Gardens. State Street was an ugly transit mall. Little planes still landed at Meigs Field. Sears Tower was still Sears Tower and the tallest building in the world.
I am chest-burstingly proud of the brilliant work he's written under his byline, of the countless midnights when he slunk out of bed to fix a sentence or deepen some particular thought. His devotion must rank among the rarest in the business. His love for his city and his readers kept him writing long after counterforce made quitting the easier option. We've seen him trailed by TSA agents at O'Hare who wanted to keep up some architectural conversation, straight to the boarding gate; stood by as he was tapped on the shoulder as far away as London or DC by a reader who recognized him and didn't want to miss a chance to say thank you, ask an architectural question. It's that devotion — and infinite unsung kindnesses extended to readers and would-be someday critics — that is perhaps his shiningest prize, the one that comes with no crystal paperweight, and no plaque to hang in a back corner of his book-lined office.
He's our beloved Blair Kamin, of whom we are soo soo proud. And who has left an indelible and breathtaking mark on the city he loved, the newspaper for which he wrote for 33 rollercoaster years, and who has written his best and most lasting lines in the narrative that is our blessed little double-bylined family.
But that's the not end of this love story. Only this latest chapter.
Here's how he broke the news on Twitter last Friday night:
After 33 years at Chicago Tribune, 28 as architecture critic, I'm taking a buyout + leaving the newspaper. It's been an honor to cover + critique designs in the first city of American architecture + to continue the tradition begun by Paul Gapp, my Pulitzer-winning predecessor.
During these 28 years, I have chronicled an astonishing time of change, both in Chicago and around the world. From the horrors of 9/11 to the joy of Millennium Park, and from Frank Gehry to Jeanne Gang, I have never lacked for gripping subject matter.
Whether or not you agreed with what I wrote was never the point. My aim was to open your eyes to, and raise your expectations for, the inescapable art of architecture, which does more than any other art to shape how we live.
So I treated buildings not simply as architectural objects or technological marvels, but also as vessels of human possibility. Above all, my role was to serve as a watchdog, unafraid to bark and, if necessary, bite, before developers and architects wreaked havoc on the city.
I am deeply grateful to my newspaper, which has never asked me to pull punches. I have been incredibly fortunate to work with talented editors, reporters, photographers and graphic designers. They have been a huge help. Journalism, like architecture, is a team enterprise.
What will I do next? I have no idea. After decades of stressful deadlines and rewriting paragraphs in my head at midnight, I'm ready for an extended break — and many long bike rides along Chicago's lakefront.
It's essential that a new critic, with a fresh set of ideas, take up where Paul Gapp and I left off. Imagine Chicago without a full-time architecture critic. Schlock developers and hack architects would welcome the lack of scrutiny. -30-
you'll note i put aside for this one time my disinclination to hit the shift key and write with capital letters (writing here in lower case is for me something akin to kicking off my shoes and shuffling around in slippers), but for the upstanding critic, i decided to pull out my big-girl keys and give him ups and downs on the keyboard scale. i'll return to slippers, no doubt, though i do note it makes for easier reading when you can spy the peaks and valleys in each and any sentence.
in the tweets above, you might notice mention of Jon Stewart, the late-night genius, who once saw fit to enter the Chicago architectural fray, a little back-and-forth, you might say, between our hero here, the critic, and the comb-over developer who would go on to rule the Oval Office…watch here the clip of Signfeud, from the Daily Show…
i have now overflowed this space with a kitchen sink of Kamin esoterica and folderol. it is with all the love in the world, and bursting giant heart, that i thank the Chicago Tribune (where, combined, we toiled for 63 years) for bringing me the other half of our double byline. it's been some rocket ride, and i'll hold on tight for wherever this takes us next.
much love, BK. i am — in the great Tribune tradition of "clapping out" your final exit from the newsroom — standing and applauding. xoxox
and here's a final twist for this week's chair: how bout this, you ask the question this week, and i will try to answer….the annals of the newsroom are now open for the curious…..
in the newspaper world, -30- means "the end." at the bottom of every reel of type flying off the typewriter, once upon a time, a big-city scribe tapped four keys to signal the end, so the typesetters knew to move onto the next big story in their end-of-day unreeling of the hot breaking news.
all these years, the -30- stuck. only i grabbed it from my typesetting keys this morning not because of an ending, really, but because a bespectacled scribe i happen to love, one whose flight i've witnessed from an up-close unedited perch, he's been waiting and waiting for today. today is the day he gets his 30-year watch. thirty years of calling himself a "chicago tribune reporter." thirty years of chasing down just about any I-beam that dared to move in this old town. thirty years of thumbs-up or thumbs-down on wild-eyed architects' intentions to make no small plans.
but more than what's beautiful, soaring, inspiring, or not, he sees the way the carved-out hollows and high-rises of a big american city might move the human species into communion, or tear them apart. he understands the nuts and bolts of design, but he's keen on justice and social equity; he understands the political powers and petty feuds that sometimes stand in the way of what makes a city — and its peoples — work, or not work.
and he's spent three decades teaching all of us, teaching anyone who turns the pages of every day's news, to do the same. it's a way of seeing he's intent on not keeping to himself.
and ever since the hot august morning of 1987 when he strolled into the chicago tribune newsroom in his navy brooks brothers blazer, white oxford, and khakis — aka "the uniform" — i've been watching. took another year till i rose to my rank as "girlfriend," and then another three years before "wife" was affixed to my status (we had a lot to figure out, mostly in the religion department, during those long should-we-or-shouldn't-we years).
so i know, more than almost anyone, just how much it means to him to have hit the sweet 3-0. to know that tonight, at the annual bacchanal that is the tribune awards hoopla, he will, at last, get his chicago tribune watch. actually, in a move that is so classily elegant and fair-hearted and loving as to be a signature BK move, he's getting two tribune watches tonight. he put in an order for a pair, one for each of our boys, so someday, both will have a relic from their papa, one he wrote soooooo many stories to snare, one that in some scant way captures the nights after nights that he kept watch over stories, called in corrections to the desk, gave up a friday night dinner, surrendered a holiday, took yet another call from a "source," chased a hot tip. because when you're the son of a newspaper man (and he is) getting the news and getting it right, and never ever backing down from the truth, well, that's religion to him. and he is devout, if anything.
and that might be the beauty of nights like tonight: they squeeze you into the think-back machine. have a way of making you stop in your tracks, think back across the long arc of your history, sift for those gold nuggets of meaning. (and you know i never ever miss a chance for gazing back over my shoulder, for rubbing my palms against the fine grain of time, squeezing out every succulent drop of "significance.")
it's the pause in the plot that always, always holds the possibility of taking life up a notch. that slows us down long enough to realize this isn't just a race to the finish line, but rather a slow contemplative unspooling that is best lived and best understood, most certainly held up to the radiant light, if we pay close close attention to all the unspoken strands, the subtle and poignant shifts along the way, the moments where we rose up to champion status, where we lived with every ounce of hope and faith with which we were created and dreamt into being, and where we humbly account for our stumbles, realign our compasses and set forth again.
it's a magnificent reel, this thing called our life, and it's most closely savored when every once in a while we watch it in slo-mo, stop-gap, how'd-we-get-here, hallelujah style. and then, to anoint the moment, we bend knee, bow head, and whisper a holy thank you.
never, ever, in a million years did i imagine this 30 would bring my bespectacled scribe — and me, and thus W and T (our two and only double-bylines) — along this most blessed road to here.
a billion blessings, BK. and thank you.
have you hit the pause button lately, to look back on the road to where you are now? what have you gleaned, and what lessons might you carry forward?
p.s. an emphatic post-script to clarify, clarify, clarify: BK is NOT leaving the tribune, merely collecting his 30-year watch. he will be writing and writing and writing. so sorry for leaving wrong impression. it's a tribune tradition that you get your watch and get right back to work. so so sorry if i left anyone thinking this was The End…..
lost in the cobwebs…almost.
it's been one of those weeks that's found me sifting through drawers, sifting through history, following threads hither and yon.
there's a particular drawer, in the old pine writing table across the way from here where i sit, and it might as well be my holy of holies. it's where i stash particular love letters, and every mass card from every funeral of someone i've loved. it's where, apparently, i've stashed the polaroid snapshots of my firstborn lying bruised and bloodied in a hospital bed in the children's hospital ICU, the day after he flew from his bike and broke his neck. and where i've tucked the recording of my then-little one's long-ago phone machine greeting, a delectable slur of words that always left callers confounded — and me charmed, beyond words.
it's been one of those weeks where threads seem to be pulling me this way, then that. one question leads to a search. another leads to the creaky old stairs that unfold from the attic.
i've been discovering shards and treasures all week. i've bumped into more questions than answers. why, oh why, do i have a silver coin from 1909, one with abe lincoln's bu st on the front, and on the back the words, "for merit in an essay on abraham lincoln"? who won this, and where is this prize-winning essay? and how did the coin come to be in my drawer? might it be from my grandmama mae, the irish school teacher who bore my sweet papa? might my love of words flow directly through her bloodline? and might my boys' love of abe be their genetic inheritance?
these are the questions that keep me awake. and won't let me rest till i unearth the answers.
long long ago, standing in the kitchen of the house where i grew up, i remember leaning into my father's shoulder (he was wearing the navy velour pullover he so often wore, and i can conjure the nub of that cloth even today — 36 years after the moment), and my father spoke these words that have echoed ever since: "you have a real sense of history." it was one of those moments when suddenly something you'd not known appears as the most obvious truth in your life. my father died less than two months later. so the words became prophetic. the words have become my divining rod. i follow history. i sift through old letters and artifacts. i study old photos, the ones now faded. i try to make sense.
and i can't bear to let history — to let story or love, for that's what so much of a history is — crumble to dust in a drawer or the attic.
which is why i was a bit frazzled this week when i realized that years of my old newspaper stories are all but lost in the cobwebs. it's intricately complicated, i found out, to pluck certain stories from the digital archives. without a date and precise headline, it's nearly impossible. which means a good 20 years of bylines might never again be unearthed. which, mostly, won't matter. but among those two decades there are stories that poured straight from my heart, and i can't bear the thought that they're never to be pulled to daylight again. they were, each one, a love song to or about someone or something that mattered. they were moments in my story that i'm not ready to bury.
which is why i decided that, every once in a while, when i find one, i'm going to lovingly paste it here, a digital scrapbook of bylines gone by.
this is the first, a love letter, really, to the very fine soul who picked up his hammer and built the nooks and crannies of this old house and the one before it, a construction of love beyond what we'd dreamed.
Being graced by the hand — and soul — of Jim
January 04, 2004|By Barbara Mahany, Tribune staff reporter.
At my house, his name is Jim.
I still remember the first time he walked in, walked in to talk about taking down walls, putting up a dormer. One minute, I'd never seen him before, the next minute, I'd known him all my life.
I still remember standing out by the sidewalk, watching the roof come off our old house, leaning against the wrought-iron gate next door, and he told me, in the most matter-of-fact way, "My dad always said to leave behind a footprint wherever you go."
Jim leaves footprints. In the form of a box-bay window the architects hadn't drawn, but that he knew was just what we wanted, to make the trees feel like they stretched right into our room, or, rather, to sweep the window seat right out into the limbs, making a treehouse of what might have been simply a room for a bed.
In the form of drawers that glide in and out as if on Rollerblades, making me feel elegant every time I reached inside for a lumpy old sweater.
In the form of bookshelves that wrapped around me in my little room, making me feel hugged and safe and home — very much at home.
It didn't take long for all of us to fall in love with Jim & Co. The whole summer they were at our house — Jim and Tom and Bri, the musketeers three — my husband couldn't wait to vault out of bed and dash over to the Dunkin' Donuts, where he'd return with a box dripping with sugar and round puffy blobs. My little boy took to sitting on the stairs, watching. He had a big red tool kit that he started lugging around. He put on his safety goggles and he built things in the back yard. Boats. A race car. Bigger boats.
That was at our old house.
I didn't want to leave it behind because I couldn't bear to leave behind the magic that Jim had pounded into its walls, its windows, its tucked-away secrets.
Jim, you see, is indispensable, and not just because he wields a mean hammer. Jim is indispensable because what he builds goes far beyond the blue lines you see in the drawings. Jim is indispensable because he knows, without words, the poetry of walls and windows and doors, and all they hold for those of us who hatch our dreams at home like eggs in a nest.
So when we moved, it was pretty simple: We brought Jim with us.
In fact, we bought a house that I could see only through the lens of Jim and all that he could do. I saw right past the ugly tile in the kitchen, the tile someone loved so much they glued it right up the wall once they ran out of the floor. I saw right through the bathrooms with the vanities that looked as if they took three oak trees to build them, they were so big and bulky and in the way.
That was almost a year ago. And in that year, slowly, patiently, whenever he had a minute in between building other people's houses, he's been pounding magic into this house, as if it really mattered.
And the point here is: It does matter.
Every single day, most likely for the rest of my life, this house, these walls, these windows, will be the ones that shape my every day. It is within these rooms that I will take in my first waking breath each day and every other breath that forms my every word. It is through these windows that I will look out at the world and drink in the fuel of my dreams. These are the nooks I will curl up in. These are the stairs I will climb, every time it really matters, and plenty of times when it really doesn't.
But the point is, because his hand is here, everywhere I look I feel his soul, and the soul of something much bigger that speaks to me in a soft still voice, in every room.
Where once upon a time there was a single-car garage, and where after that, just before we moved in, there was brown-striped vinyl wallpaper and nubby carpeting all shredded by a yappy dog, there are now floor-to-ceiling, wall-to-wall bookcases, and two window seats that stretch out beneath the windows. It is nearly a re-creation of the little tiny room of my dreams I had to leave behind in the house that is no longer ours.
Only this one is better, because I get to stay here forever, and because Jim & Crew pretty much built it from memory, trying to mend the heart that got wrenched in the move.
By the time we're finished, pretty much every room is going to be graced by the hand of Jim. He's building a corner cabinet for all the books my little boy has yet to read, and I have visions of us curled up for hours there, for years and years to come. He's already built a wall of bookshelves for my husband, a wall that could only be called majestic, so elegant and mighty as its fine-honed pilasters reach for the ceiling, and hold my husband's anchor in the world, his library of books about all the ideas he treasures most.
My 2-year-old, who picked out shoes at the shoe store because they look just like Jim's, took on a refrain this summer that pretty much echoed the truth in all our hearts. He walked around the house, and whenever he noticed anything amiss, he proclaimed matter-of-factly: "Jim fix it."
Jim, he fixes everything. And not just with his hammer.
and, now, that one is saved, tucked away in my treasure box, here at the table…..
have you ever discovered — in the nick of time — that some treasure of yours was nearly lost? and if so, how did you save it?
when the morning news brings harper lee
this old house will be a newspaper house as long as fish wrap is dotted with ink. every morning, seven mornings a week, the first sound that reverberates around here — save for the pre-dawn robins who rev up their vocal cords — is the THWOP! of rolled-up papers plopped onto the front stoop (three separate wads each weekday and saturday, two on sundays). twice a year, when the bill comes due, a bill that topples into the hundreds for all that fish wrap, there's no discussion. we don't debate the wisdom of rolling out hard-earned cash for an inflow of ink and paper. because you never know what the news will bring. and we couldn't live without the possibility of getting lost in sentences that swoop our hearts away. or the joy of flipping through a section and discovering a story we otherwise never would have tumbled upon. or the raw eruption of hot tears spilling on the page, as some account of awfulness carries us miles and miles from where we're reading, and into dingy corners we'd not know were it not for the newspaper's insistence on wiping out our ignorance and insouciance.
heck, this old house and half the people in it were practically built on the backs of newsprint. were it not for one chicago tribune's newsroom, i never would have spied — and uncannily fallen hard for — the lanky fellow who became my lifelong paladin, and the father to our children (the two we call our only "double-bylines").
still, not every morning brings what this one did; these words from the one spooning oat-y Os into his hungry gullet: "you're gonna go nuts over this one." and then he shoved before my eyes the front page of the wall street journal's friday arts-and-culture section.
"the first chapter of harper lee's new book," he mumbled between Os, lest i miss the red-hot scoop, the unparalleled capital-e Exclusive, the biggest leak in publishing in plenty a while, the newspaper's literary splash four days in advance of tuesday's worldwide release of what's being called the reclusive ms. lee's "new novel."
actually, it's harper lee's old novel, "go set a watchman," her first go-around with a manuscript, submitted back in 1957, when she was all of 31, to her new york publisher, j.b. lippincott.
as the book-peddling legend goes, ms. lee's editor back then found the story "lacking," and advised that the would-be author instead zero in on the flashback scenes, in what would become the searing tale of scout and dill and jem and atticus finch and boo radley, and racial inequity and empathy played out in small-town maycomb, alabama: "to kill a mockingbird," the pulitzer-prize winner that went on to be named "the 20th-century's best novel," according to a vote taken by the nation's librarians.
and so, before my first sip of coffee this morning, i was riding the rails with jean louise finch, aka the "scout" of mockingbird fame, as she "watched the last of georgia's hills recede and the red earth appear, and with it tin-roofed houses set in the middle of swept yards, and in the yards the inevitable verbena grew, surrounded by whitewashed tires."
i admit to having been among the skeptical when news of this "long-lost discovery" first made headlines. i admit to suspicion when word leaked out that the 89-year-old ms. lee's not-long-out-of-law-school attorney just happened to find the manuscript tucked away in a safe deposit box, shortly after ms. lee's 103-year-old sister, lawyer and lifelong protector, alice lee, had died. i worried that the not-altogether-with-it nelle harper lee might have been duped. coerced into publishing something she'd not wanted paraded through the glaring light of day, to say nothing of the folderol and zaniness sure to come after a half-century's literary silence.
well, i've now read every word, every word the wall street journal rolled into print, and i'm here to tell you i'll be among the ones in line to gobble up the next however many chapters ms. lee has lobbed our way. whoever was that long-ago lippincott editor who found the first-go lacking, i beg to differ. i'd not want to miss the chance to drink in a line like this one: "love whom you will but marry your own kind was a dictum amounting to instinct within her."
or: "she was a person who, when confronted with an easy way out, always took the hard way. the easy way out of this would be to marry hank and let him labor for her. after a few years, when the children were waist-high, the man would come along whom she should have married in the first place. there would be searchings of hearts, fevers and frets, long looks at each other on the post office steps, and misery for everybody. the hollering and the high-mindedness over, all that would be left would be another shabby little affair a la birmingham country club set, and a self-constructed private gehenna with the latest westinghouse appliances. hank didn't deserve that.
"no. for the present she would pursue the stony path of spinsterhood."
dare you not to race out to add your name to the long list at the library, or order up your own copy from your nearest most beloved bookseller.
i for one will be inhaling every line, on the lookout for a passage equal to the one i just might call the greatest american paragraph ever penned, the one that makes my heart roar every time.
for the sheer joy of retyping its every word, here is one walloping passage from atticus finch's closing argument in his defense of a black man wrongly accused of raping a white girl in the deep south of the 1930s. page 233 in my first perennial classics edition, printed in 2002:
"But there is one way in this country in which all men are created equal — there is one human institution that makes a pauper the equal of a Rockefeller, the stupid man the equal of an Einstein, and the ignorant man the equal of any college president. That institution, gentlemen, is a court. It can be the Supreme Court of the United States or the humblest J.P. court in the land, or this honorable court which you serve. Our courts have their faults, as does any human institution, but in this country our courts are the great levelers, and in our courts all men are created equal."
heck, the whole closing argument — from the bottom of page 230, clear through to the fourth to last sentence on 234 — the whole magnificent thing was enough to make me a lifelong believer in the pen of harper lee. and the wall street journal's gift this morning — slick as it was for the newspaper owned by the same outfit as lee's new publisher, HarperCollins, to steal first crack at the watchman — twas a mighty fine one.
and an indelible reminder of why i'll forever be a girl with ink pumping through her veins.
what's your favorite line, or scene, or passage, from mockingbird?
and, for your summer reading's consideration, here's how the journal lays out the launch of ms. lee's latest, under the news headline, "scout comes home":
"The first chapter of 'Go Set a Watchman' introduces Ms. Lee's beloved character, Scout, as a sexually liberated woman in her twenties, traveling from New York to Alabama to visit her ailing father and weigh a marriage proposal from a childhood friend. It also includes a bombshell about Scout's brother."
i'll let you read for yourself and discover that bombshell…..oh, the joy of a byline we thought we'd never see again, one that bears the name harper lee.
mama's got a tough, tough job, and someone's gotta help
when i was a kid, my dad was larry tate, the buttoned-up business half of the ad-biz duo on "bewitched," that 60s (or was it the 70s?) sit-com starring samantha.
well, he wasn't really ol' larry. but that's how i had to explain it, whenever i said my dad was an ad man, and the follow-up question was always: "is he darrin stephens or larry tate?" darrin was the creative dude, the one who married the nose-twitching daffy-hearted witch. larry–and, yup, my dad–was the one who kept the creative types in line. but, at least in the case of my dad, that didn't mean he was so buttoned-up.
my dad loved nothing more than a great laugh.
if there's one sound i can still hear, it's the sound of his big booming guffaw, breaking the air in a room, filling the space between walls, flicking the switch in my heart, making it glow.
i LOVED that my dad was an ad man. fact is, i loved everything about my papa. but knowing he rode downtown on the train, carried that briefcase filled with top-secret memos to clients like betty crocker, mcdonald's, even the folks who made play-doh, well, that made me feel like i was plugged into the nerve center of our times.
heck, my dad brought home a plain cardboard box, marked X, and it was a test sample of hamburger helper. we were some of the first kids in america to spoon that glop in our mouths. and we lived to give him a thumbs up or thumbs down.
the stories at our dinner table would swirl with stuff that mattered to kids growing up in suburbia in the hair-raising 60s, and the dick-nixon 70s.
we knew the ins and outs of big macs, and all about all the sugar-coated cereals packing the grocery-store shelves.
pop tarts? we had 'em early, had 'em often.
we didn't screech on the taste-testing brakes when we crossed over the sharp lines of whatever "the clients" had fobbed on the market.
why, it was our job, our patrimonial duty, to invade enemy territory. we were the spies, me in my pig tails, my brothers in freckles and iron-on patches on knees.
we guzzled whatever the '60s and '70s offered. we didn't much mind (although, for the life of me, i was deadset against hamburger helper and its ilk from the get-go, not yet appreciating the ease of dumping, stirring and filling the tums of five hungry kids).
which, in a round-about way, brings me back to the latest episode in the tale of the boys we call our double-bylines, meaning the poor little fellows (one, now not-so-little) who get to grow up in a house with a dad and a mom in the news biz.
which, on rather regular occasions, means i lope home from the office with a satchel stuffed with curiosities and delights and general conversational stimulants.
like this week, when it was my job to corral the best cookies in the land. or at least among readers of the newspaper where i type three days a week.
yup, it was the annual tribune holiday cookie contest, and someone had to be in charge of getting those cookies into the great gothic tower that is the tribune. and someone had to rustle up the 16 judges, put out the paper plates, the cups of water, the pens and the score sheets.
that someone was me.
and so, when the long hard day of nibbling and scoring was over, i asked if — please! — i might be allowed to haul home just one plate of each one of the 11 finalist cookies, so my own personal judging panel could convene.
and that's where the sugar-saturated plate up above comes in.
that was homework for my fine little boy who's pretty much convinced that sweets is one of the food groups. if not the most essential of the lot.
just after dinner (yup, we actually held off till after the protein and veggies; give us brownie points for that, please), we lined up the contest with great ceremonial pomp.
just like back in the tribune test kitchen, i set out cups of water, pens for each judge, and the nibbling began.
in fact, i knew full well that this was yet another one of my ploys to exercise that boy's descriptive ways. i swooned when he launched in on the first, a glimmery snowflake of a cookie, which he described thusly: "it looks like a snowflake has just fallen with sugar and sparkles dancing on it."
or, of a chocolate-swirled marshmallowy number: "it looks like a collage of butterflies."
find me a full-fledged tribune judge who dished out such poetry. and this from my boy who has tussled with words in his day.
while he nibbled and spun his sugary stanzas, his papa chewed and scribbled in silence. in the end, once the last crumb was licked off the plate, we wound up with a three-way tie for first prize.
but for me, the very blue ribbon i pinned on the day was the glorious fact that, for little more than my train ride into the city, i could bring home a piece of the world far beyond our little town's walls.
in the same way that once upon a time my daddy's job made me feel like i had a window onto something big, something exciting, i hope my sweet boy feels just a tad more engaged with the wheels of the ever-cranking universe.
i hope that while i'm the one with the measly paycheck, he's the one who catches the magic. who sees the power of words. who tastes the thrill of civic engagement, even when it's just a cookie contest.
if he listens–and i've reason to think that he does–there's not a page from my day job that doesn't somehow rub off him. if not in ink, then surely in stories, in laughter. and sometimes, come the start of november, in cookies that make for fine poems.
when you were growing up did someone in your house have a job that made you look at the world in a particular way? it's a curious marvelous thing, not oft considered perhaps, how all the ways the grownups lead their lives, are all a part of the education of the little ones who grow up so closely, thoughtfully watching. it adds a dimension of meaning to the every day. and makes that ol' trainride not nearly so onerous. tell us how you learned to look at the world?
despite it all…..
if, on any one of the days of this past week, i had scribbled down every last thing i was trying to hold in my head or my heart, i might have run out of ink.
there was the phone call from school, saying the little one was sick again, please come fetch.
and there was the early morning email that someone very wonderful, very brave, had died.
there was the lost assignment notebook, and the lost $40. there was the rowing jacket that needed to be claimed, and the rower, too.
there was the doctor to visit, and the milkshake to wash it all down. there was the carpool — or two — i was scheduled to run, and did, even though the player of soccer was felled by a flu bug.
there were eight lunches to pack, and three days where a can of noodle-y soup sufficed for the one spending his days on the floor in a pile of blankets.
there was dinner times four. and a brouhaha the night the little one didn't eat much from his plate, but somehow finagled a trip to the donut shop, riding shotgun with his unsuspecting papa.
then there was the rowing trip to pack for, and the deciding which grownup would drive to toledo and which would stay home for the soccer team pictures.
there was the neighbor whose papa had died, and the figuring out who would bring dinner.
there were tomatoes to pick before they burst, and hand-me-down hostas to plant before they shriveled and died.
sometimes i wonder if maybe we're doing too much.
if maybe i'm trying to squeeze too very much into the too-narrow skins of my sausage.
sometimes–and that list up above is barely the least of it–i think maybe it's not such a good idea to try to live like we do.
but then, despite it all, i find myself out in the world, gathering stories, doing the work that i love, and well i can't imagine not getting to do that.
one fine early autumn morning this week, i was tromping through parks i might never have entered alone. i was meandering along a prairie river, tiptoeing across rocks laid in the path of trickling waters. i was deep in the fronds of a fern room, all laid out by that great designer of greenspace and parks, jens jensen, the dane who fell hard for the midwestern landscape, the prairie, the rocks swept in by the glaciers, the billowing shafts and nodding heads of the grasses.
yet another hot september morn found me seated beneath a crabapple tree on a wood bench in an english walled garden beside a ruddy-cheeked englishman, one with a sketch pad on his lap, and a mug of earl grey clasped in his fist. it was john brookes, i was sitting beside, the great designer of gardens english and otherwise, author of 26 books, and something of a living legend. we were talking, he and i, about the spirituality to be found in a garden, and the distinction he makes between vines and climbers, and why one belongs in a vineyard and the other is essential for ooomph and lift in a garden.
through it all i was gathering bits and yarn for the most humbling sort of story to write (at least in my book, that is): an obituary, the distillation of one great and layered life into a mere 800 words. it is the writer's job always, but especially here, to sift and pick, to harvest only the richest fruits from the tree of a life. to hold up mere threads that suggest the whole tapestry. to leave the reader gasping and grasping, understanding a life as its flame is snuffed out. oh, lord, let me do right.
so, yes, despite it all, despite the nights when i did not sleep, drew the bath at 3 in the morning in hopes of quelling a raging hot fever, despite the grumbling there in the kitchen, and the hauling myself out of bed to pack yet another brown bag lunch, to simmer one more pot of oatmeal, i cannot imagine a life much richer: to learn history at the foot of a great historian, to talk gardens with one of the best in the world, to talk to the still-raw widow, to ease from her the words that will tell the world of her one true and lasting love.
despite it all, i'd do it again. and chances are, soon as the page of the calendar turns, and a new week starts all over again, i will.
the variations are many, but the theme is constant: i cannot imagine one half of my life without the other, and even when they bump and collide, each half makes me so much more than a whole.
oy. forgive me. this might seem more of a lonely unspooling than reaching for common thread. except that every one of us likely has a corollary to the mayhem and triumph above: we live half-crazed lives, uphill climbs, because we believe we'll get to a mountain top. there will be a moment, we convince ourselves, when all the headaches are swept away and the big picture is clear: the combined steps of our journey have taken us to a place beyond our dreams. how do you wrestle the dailiness of your life into a meaningful climb? do tell.
and p.s. for those of you wondering about that new tribune adventure, it's coming next week. in the news biz schedules change with the blink of an eye. so the editors held off for awhile….
storybook gardener
since the day i moved here, five years ago now, the storybook cottage up above has held me in its spell. oh, i don't live beneath its shingled roof, don't know my way from room to room, have never even turned the front-door knob.
i only wander by, and cast my wishes in its woodland garden, where trillium, in spring, bumps up against the jacob's ladder. where snowdrops are the first to come when winter will not take its graceful leave. and where, in autumn, tall grasses swish, the wind-borne lullabye before the garden slumbers.
i wink sometimes at raggedy ann, peeking out from up beneath the turret's peak. see her? just a wisp of her, in the upper eastern window where she bathes, each morning, in the dawn's first-light, as rosy-fingered sky reaches up and over the great blue lake just blocks away?
the raggedy one looks down on a garden as magic as any i have ever known, save maybe for "the secret garden," frances hodgson burnett's secret one, of course, which by page 42 of that sinful book (i once pretended to have a fever so i could stay home from church to read on and on) was the first i fell head-over-heels in love with, imagined myself inside of, tiptoeing near the climbing roses, curling up on stony bench, listening for the robin who'd led the orphan mary to the garden's long-lost key and then, at last, to the ivy-covered door that hadn't been unlocked in years and years.
there is for me something about a garden, a particular sort of mind-of-its-own garden, not one all clipped and shorn and pedicured, but one that rambles, grows this way and that, that sets me to pretending, spinning yarns to match the garden's swift enchantment.
a little more than a year and a half ago, the storybook garden that belongs to the storybook cottage that just might belong to raggedy ann, gripped me, shot me through and through with what i feared was a most unhappy ending.
before i tell you what unfolded just the other day, i've pasted here the passage i once told. read along, then meet me at the story's end, so i can fill you in on what we'll call the epilogue.
the story just below, once was published in the chicago tribune. but i wrote it right here at my old pine desk, before i had a kitchen table where i could share my stories.
here's what now becomes the prologue, printed first in october of 2006:
I am haunted these days by a fairy tale garden that I fear has lost its gardener.
There is, not far from my house, an enchanted little garden and a storybook house to go with it. The first time I eyed it, I nearly drove up the curb. It is all trellis and turret, and delightfully low to the ground, as if curled in a humble embrace with the growing things that spring all around. It is a house that whispers, not shouts.
At the head of a stepping-stone path there's a front door an elf might wander through, and all around there are great patches of magic, grasses and flowering vines, birdhouses by the dozens, a place lovingly tended by someone completely in tune with the rhythms of wonder and the unfolding of time, season by season.
A Raggedy Ann, toddler size, sits up in the window of one the turrets, peeking out, keeping an eye on all. More than once, I swore she winked at me. The place, I'm telling you, is bewitching.
To drive by, or to wander by, is to slow the staccato of my every day. It is to breathe in and be reminded that lovingly tending the earth reaps wheelbarrows full of heaven.
The other day, though, the staccato picked up as I was stopped at the stop sign across from the storybook house. I saw streams of finely dressed folk pouring down the sidewalk, into the elfin front door, in the middle of a sun-drenched afternoon. This isn't a holiday, I thought. But then a synapse connected: Those fine clothes were funeral clothes. Oh my goodness, something in the storybook was amiss.
And then, sure as could be, the next morning, flanking both sides of the garden walk, standing sentry for any passerby, were two giant funeral wreaths, one a sumptuous circle of red roses, the other a sheaf of blood-red gladioli.
I am haunted now by the storybook house, and its autumnal garden. Haunted because I am afraid the woman who loved there might be no longer.
Sad thing is, I never met her. Only saw her as a constant stooped figure, bent over her trillium and her scilla in the spring, deadheading her jonquils, cutting back and transplanting through the summer, raking and staking well past the first frost of winter.
I can barely breathe now when I drive by, wondering, worrying. What happens to a garden when a gardener dies? Who will feed the birds that have called her little patch home for decades and decades?
For now, Raggedy Ann keeps an eye on the place. I won't be able to bear it if she, too, slips away some day when I drive by.
I mourn for the woman who tended the earth with all her heart. I mourn for the trillium and the jonquil and the clematis that will no longer be cajoled and tucked back and talked to, as I often saw her, lips moving as she moved from this to that growing thing. I mourn for all of us who might not have the storybook garden to calm us in the midst of our modern-day madness.
It is a wicked thing when a garden and its gardener do not live happily ever after.
–end of tribune story–
not long after the story ran, i found an email in my tribune mail box. just a few sentences in, my heart caught in my throat. it was an email from the storybook gardener herself.
she'd been stopped, she said, as she turned the sunday pages, drawn in by the drawing of a raggedy ann that ran beside my story, running down two columns. she started to read, and soon knew that this mysterious writer was writing of her very own garden.
she'd not died, she wrote me. but her dear, beloved husband had. suddenly. achingly. without a warning.
and all the bustle i'd come up upon, and the funeral wreaths standing guard by the front door, were indeed for someone loved, now lost.
the garden and the gardener were, oh, yes, deep in mourning. but the words i'd written had not rattled her, so much as soothed her, she wrote. to know that her garden brought so much magic to even one odd passerby.
i should stop by, she wrote. i should visit her in her storybook garden, or inside the magic cottage she now shared with only one limp ragdoll.
for 20 months almost, i've kept an eye out. i've knocked on that elfin door. i've spied evidence of her being there, a just-dropped trowel, a pile of leaves. but not once did i ever see the magic gardener herself.
until just the other dappled afternoon. when the sun played peek-a-boo through her maple leaves, and i saw her slight bent frame, hard at work.
i leapt from my car, i called her name. i tiptoed through her ferny thicket, felt my heart pound hard against my chest. i told her who i was, reminded her that i'd once written of her garden.
she needn't be reminded. she'd waited, she said, all these months. had wondered why i'd never come.
oh, but i have, i said, a little bashful, a little sad that i'd not maybe tried harder. hadn't thought to drop a basket or a note, tucked it by the door.
somehow, though, more words–scribbled, folded, left behind–didn't feel enough, didn't feel the thing to do. i'd been inclined to meet this soul in person; to know the hands, the eyes, the heart, that made this place call out to me.
here, sit, she said, patting her mud-rubbed palm on just the moss-laced stony bench you'd expect sprouting from her woodland trails.
we talked for the better part of an hour. she offered me anything she grows, to divide and migrate the few blocks south, to my still-getting-off-the-ground patch of earth.
uncannily, time and time again as we talked of achy hips and sudden deaths and how her garden grew, i felt as if i'd known her for a long long while. found myself delighted through and through by just how much our roots, our tales, entwined.
maybe, though, that's what happens when you've imagined yourself in someone's garden. thought about sipping tea on chilly afternoons, tucked behind her stained-glass kitchen windows.
or maybe, it's simply magic, long seeded in the hearts of those who share a fondness for a sparrow's nest tucked under eaves, and ragdolls who nap away the day up in attic windows.
as shadows grew, and i knew my stay might be getting in the way of some dear plant's undivided attention, i couldn't help but think how it was that her undying grief had brought us so round-aboutly together, sitting here, side-by-side, in her most enchanted garden.
and how this beauty all around us grew, despite the dappled darkness.
kindred souls discovered where the lilac bloomed, and mayapples nodded in the filtered sunlight.
the story, once imagined, had no happy ending.
but the story, real, i now knew, was like so very much of life: it is shadowed, yes, but only just in spots.
and, here and there and everywhere, new life is pulsing; pushing, bulging, shouldering the clumps of dirt, straining to break through the crusty-shell of earth.
no longer merely make-believe, the storybook gardener and the writer begin that time-worn art of cultivating friendship. drawn together oddly, they carry on as if their story's meant to be.
which, just maybe, it truly was.
have you made a friend thanks to a garden? or some odd-unfolding circumstance? have you some inkling there's someone not far from where you live, who just might be your long-lost soulmate, not yet discovered? do tell…
the little one was shlurping up the last bit of waffle a la jam, running way behind this morn, when he called out, "excuse me, can i have my sports section?"
he didn't seem to mind the strawberry dribble running down his cheek. but he did mind when i–the one charged with shushing him out the door and down the sidewalk, somehow sweeping to the schoolhouse door before the whistle blew–did not oblige.
demurred, in fact, with a simple, and emphatic, "no, sweetheart, we're late."
still gulping, he protested: "but you can't interrupt my morning schedule."
oh. so sorry. hadn't realized, sir, that what we had here was a routine, a way of being, a moment on which the day depended.
of course i'd noticed that, morning after morning for the last few days, while the rice chex soak up milk, you, my slugger sweet, soak up RBIs and ERAs and all those alphabet equations that long ago and always have escaped me.
but i had not heard the sound of cement drying, and this becoming what it's been for ages long before you and who knows how long into the beyond: the rite of little boys and sometimes girls obsessed with all things round and flying through the air, cracking off of wooden sticks and diving through the dirt.
you have joined the ranks, my little reader, of those whose day begins with the shaking out and creasing of the pages where all the world's a horserace or a ballgame or a wobbly putt rolling toward what might be a rodent hole but, in fact, was put there for the purpose of men and women wearing god-awful-colored pants and shoes with little nails jutting out from underneath the toes.
you, too, now scour the front page, search for what you call the headline, the score of last night's game. and then, you bore inside. you up and rise off your stool or chair, you dive head-first into the somethings you call "the standings." you report, out loud, all sorts of names and numbers. and by then i've lost you, i am sad to say.
just this morning, as i combed the house for keys, ran back for one last swallow of caffeine, you were broadcasting in spanish, no less, spitting out the scores–"quatro to uno," you barked–for those who cared not to know in english.
quite impressive, little boy. you who months ago could have cared no less for all those scribbles on the page. you who thought you'd never read a number or decipher all the letters crowded there together, a herd masquerading as a word.
in a world where newspapers are whirling at the center of a storm, where few and fewer see the economic sense of printing news on paper and plopping it on your doorstep–such service, and such fear, will we go the way of the milkman and the knife sharpener, those door-to-door deliverers of goods and service, long lost–someone needs to understand the power of the third section from the front. the one marked plainly, sports.
it is from here that whole lives of depending on the news are born, are launched, are set in motion.
i have watched it time and time again. my brothers, four, my own boys, first one, and now the other.
it is reading, yes. but it is so much more. it is learning how in this dog-race world you measure up. it is boiling down the game of running bases to charts and graphs and teeny-tiny type. it is drama on the field–and life–condensed to bare-bone stats.
it is the way a boy with spoon in soggy flakes first reaches out beyond his little world, into that of world beyond.
what's on the screen at night, becomes his in the morning, there in black-on-white, just beside his cheerios and wheaties, his waffles and his raisin toast.
it is the breakfast of champions, with a splash of milk. and orange juice on the side. hold the pulp, please. pass the syrup.
i find it wholly charming to watch as little boy begins to sift through all the chaos of the world, and claim as his the simple practice of nose-diving deep into the sports page.
at least you get no grass stains sliding into home.
do you make sense of your world through daily rituals? how and when did you learn to order your day through the religious practice of some sense-making routine? do you too have your breath taken away watching little children grow, take on the ways of grownups all too soon?
lucy's story: what you didn't yet read
there is more. there is always, always more.
sometimes, when i am writing a story for the newspaper, it actually hurts to leave out whole chunks of what i've gathered. a hundred thousand times i've cut and cried, leaning mightily on the words of one mr. hemingway: "a story's only as good as what you leave on the cutting room floor." it's a line we whisper to ourselves as we wave goodbye to bits and threads we love, but cannot use. only so much you can squeeze onto those blank white pages, before they wrap the next day's fish. or, in the case of my mother, line her birdcage.
lucy's story, the one i told on mother's day, is one of those ones that would have left me aching, feeling unfinished, if not for this holy sacred place where there is always room to finish every story.
my job, as storyteller, is to propel the reader through the piece, to condense, refine, suggest, spell out, depending on the day and space.
my preference, as storyteller, is to meander. to take my time, peek in corners, poke beneath the covers. listen. really, really closely. let whole thoughts unspool, and not just cut and grab.
i understand, of course, that readers mostly want to get to the point, and then move on to tidy up the kitchen table, get the kiddies out the door, pick up the dry cleaning. be done with it.
but this place here, this table with so many chairs, is wholly discretionary. you take it, or you leave it. this is whipped cream and maraschino cherries. you don't have to pick just one, eenie-meenie-minie-moe.
so curl up, rest your chin on your palms, and your elbows on the table's edge.
there is more to tell you about blessed lucy, and her mama rosa, the two i introduced you to just yesterday, or if you picked up a chicago tribune, you might have met them back on mother's day.
for you just joining us, lucy graduated saturday with a degree in bioengineering from the university of illinois at chicago. she's been in a wheelchair since she was 9. she found out when she was four that she had a rare degenerative disease, spinal muscular atrophy, which has left her arms and legs rag-doll limp, unable even to turn the pages in a heavy book, sometimes too tired to lift a peanut-butter sandwich to her lips.
her mama, rosa, has been the arms and legs that lucy cannot use. for six years. all through college.
she has opened doors, laid out books and papers, cut up lucy's breakfast, lunch and dinner. at night, she rolls her, side-to-side, three times before the dawn.
i condensed all of this in the story. but what i didn't get to spell out were some of the everyday obstacles that would have felled a lesser duo.
for instance, lucy and her mama–who is not fluent in english–rode the CTA's blue line train every day to campus, a one-hour ride if all unfolded as it should have. but, often, it did not.
sometimes, the elevator in the train station near campus wouldn't work, so lucy and her mama would have to re-board the next incoming train, take it on downtown, where they would transfer to another line, and take that train back out to campus, to a station that didn't require an elevator.
or, sometimes, when it rained, lucy would worry that the rain would muck up the battery that operates her wheelchair, which would loosen the cable to her joystick, and she'd be stuck–with a 420-pound wheelchair that her mother couldn't push if she wanted to.
just last week, riding in for her very last exam, a two-hour grueler in her hardest class, lucy spilled a bit of gatorade from the bottle she was sipping during the ride. the sticky liquid got into the battery of her wheelchair, and when they got to campus, to take the exam, the wheelchair wouldn't work. they had to turn around, go home, get the back-up chair, and start the trip again.
"good thing i hadn't gotten around to giving away the old chair," she said matter-of-factly. good thing, too, she added, she'd originally set out for campus four hours before the exam.
earlier in the semester, the only elevator in the building where she took her hardest class was broken for a week. she had to miss a whole week's lectures, relying on the notes that someone else took for her, never quite totally grasping every concept in a class called Pattern Recognition, which has something to do with understanding how an automated machine–say, an MRI–analyzes data to make a diagnosis.
for a woman who takes half an hour just to write one page of painstakingly-looped letters and words and sentences, she said there was nothing she could do but watch closely as her lab partners precisely measured out chemicals–in fractions of a milliliter, sometimes–with the glass pipettes that are so essential and so taken for granted in every science lab.
same thing, she said, when it came to intricate wiring that had to be tracked and secured for circuit panels in a bio-instrumentation lab. she watched, and absorbed without the tactile learning that comes from fingering each wire, screw and micro-tool.
but what sticks with me as much as the heartache over how hard her road was, and how she not once complained, is what lucy had to say about her unshakable faith, once lost, now found. and a friend whose light still illuminates her way.
"when i was little i was real religious," said lucy, sitting in a study room in the engineering building at UIC last week. "when i stopped walking, i became an atheist at the age of nine.
"i was depressed from nine to 15. 'why did i have to be born with a disability?' i kept thinking.
"but then i thought about how would the world be different if everyone was perfect? would everybody be super vain? they would never think of helping anybody else. what if? when i finally accepted my disability, it felt like a lot of bricks had been lifted off me."
lucy, who is 24 now, says she wouldn't change one thing in her life. "i'm not blind, i can hear, i can speak, i can use my mind. i think i finally just got tired of being depressed. i thought, 'i'm never gonna walk, why be sad about it?' being sad about it, isn't going to change it."
it was a college religion class, one on catholicism, actually, that really opened her heart, she says. the class was assigned to read one of the writings of Pope John Paul II, who suffered from parkinson's disease. the writing, an encyclical titled, "The Gospel of Life," she says, revolutionized her thinking about her own disabilities.
"i used to feel like a disability was a punishment. after reading the pope, i realized it's another beautiful form of life."
reading the pope's words, she said, "kind of helped me bring my faith back in God."
her mother, rosa, never lost it. even though she says her deepest desire is to see lucy stand and walk.
"you know why i think God is very good," rosa asks. "lucy cannot walk; my other daughter can. what i can't see in one, i see in the other." it is the same, she says, with her two sons, one of whom is in a wheelchair (and a freshman at the university of illinois at urbana-champaign), and one of whom is not.
this, from a mother who must speak up for her daughter in the cafeteria line, because lucy's disease won't allow her to speak much louder than an amplified whisper. she can't bark out a request for the baked ziti that is her very favorite lunch.
the one thing that lucy still misses, she says, is her privacy.
"before i'd hide notes all over my room. after i stopped walking, i couldn't keep anything hidden. everybody always had to know everything."
lucy says she learned patience from her best friend, giovanna, whom she met when she was eight, and who died when she was 13, from SMA, the same disease that lucy has.
"she taught me to have patience. i didn't want people to help me, i wanted to do everything for myself. when i first met her i could walk. to all of a sudden be in a wheelchair…"
it was practically unbearable, lucy says. giovanna, she adds, "taught me determination."
giovanna was full of grace, as lucy tells it. and giovanna, i think, bequeathed her grace to lucy.
and that is most of what i wanted to tell you about two fine souls who rolled into my life last week, and now will never leave.
one of them, a woman who finds justice in the divine equation that has two of her four children in wheelchairs, motoring around college campuses, refusing to rein in their dreams, now inspiring far beyond the boundaries of their colleges.
the other, a woman who sees the wisdom–and the beauty–in a world where our imperfections compel us to reach beyond our limits, to be each others' arms and legs and hopes and dreams.
those are the lessons i learned at work this week.
it is no wonder why i call this storytelling business not just a job but a holy sacred calling. how blessed i am.
how blessed, lucy and rosa trevino, not trapped at all by a life in a 420-pound chair on wheels. but rather, teaching as they roll, inspiring as we lope behind, trying to catch their holy shining wisdom.
bless you if you stayed to read this story. it was long, i know. but it feels so deeply essential. your thoughts….
the photo above is one i took at lucy's graduation. months ago, she ordered that certificate of gratitude for her mother, just for graduation day. because the print is small, i'll spell it out: "thank you for all your love and support. i would not be where i am today if it wasn't for you. i feel so grateful to have you in my life. today is my day, but i dedicate it to you."
and then she signed it, lucy trevino. it took minutes to push the pen through those 11 proud but simple letters.
the lilac chiffon you see behind the certificate, and the sturdy hands, those belong to rosa, who was beaming all day saturday, mexican mother's day.
< Older Posts | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,101 |
const Winston = require('winston');
const tsFormat = () => (new Date());
class Util {
static getLogger() {
const logger = new (Winston.Logger)({
level: 'debug',
transports: [
new (Winston.transports.Console)({
timestamp: tsFormat,
colorize: true
})
]
});
return logger;
}
static notFound(req, res, next) {
res.status(404).send('Not Found');
}
static invalidToken(req, res, next) {
res.status(401).send('Invalid Token');
}
static invalidCredentials(req, res, next) {
res.status(401).send({'error':'Invalid Credentials'});
}
}
module.exports = Util; | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,910 |
\section{Introduction}
Let $P \subset \mathbb{R}^d$ be an integral convex polytope of dimension $d$,
that is, a convex polytope whose vertices have integer coordinates.
For a non-negative integer $l$, we write $lP=\{lx \mid x \in P\}$.
Ehrhart \cite{Ehrhart} proved that the number of lattice points in $lP$
can be expressed by a polynomial in $l$ of degree $d$:
\begin{equation*}
|(lP) \cap \mathbb{Z}^d|=c_dl^d+c_{d-1}l^{d-1}+\cdots+c_0.
\end{equation*}
This polynomial is called the {\it Ehrhart polynomial} of $P$.
It is known that:
\begin{enumerate}
\item $c_0=1$.
\item $c_{d-1}$ is half of the sum of relative volumes of facets of $P$
(\cite[Theorem 5.6]{BR}).
\item $c_d$ is the volume of $P$ (\cite[Corollary 3.20]{BR}).
\end{enumerate}
However, we have no formula on other coefficients of Ehrhart polynomials.
In particular, we do not know a formula on $c_1$
for a general 3-dimensional integral convex polytope.
In this paper, we find an explicit formula on $c_1$ of the Ehrhart polynomial
of a 3-dimensional {\it simple} integral convex polytope, see Theorem \ref{main}.
Pommersheim \cite{Pommersheim} gave a method for computing the $(d-2)$-nd coefficient
of the Ehrhart polynomial of a $d$-dimensional simple integral convex polytope $P$
by using toric geometry.
He obtained an explicit description of the Ehrhart polynomial of a tetrahedron
by using this method.
Our formula is obtained by using this method
for a general 3-dimensional simple integral convex polytope.
The structure of the paper is as follows.
In Section 2, we state the main theorem and give a few examples.
In Section 3, we give a proof of the main theorem.
\begin{acknowledgment}
This work was supported by Grant-in-Aid for JSPS Fellows 15J01000.
The author wishes to thank his supervisor, Professor Mikiya Masuda,
for his continuing support.
\end{acknowledgment}
\section{The main theorem}
Let $P \subset \mathbb{R}^3$ be a 3-dimensional simple integral convex polytope,
and let $F_1, \ldots, F_n$ be the facets of $P$.
For $k=1, \ldots, n$, we denote by $v_k \in \mathbb{Z}^3$
the inward-pointing primitive normal vector of $F_k$.
For an edge $E$ of $P$, we denote by $\mathrm{Vol}(E)$
the relative volume of $E$, that is, the length of $E$ measured with respect to
the lattice of rank one in the line containing $E$.
\begin{df}
For each edge $E=F_{k_1} \cap F_{k_2}$ of $P$,
we define an integer $m(E)$ and a rational number $s(E)$ as follows:
\begin{enumerate}\setlength{\itemsep}{-1mm}
\item We define $m(E)=|((\mathbb{R}v_{k_1}+\mathbb{R}v_{k_2}) \cap \mathbb{Z}^3)/
(\mathbb{Z}v_{k_1}+\mathbb{Z}v_{k_2})|$. \\
\item There exists a basis $e_1, e_2$ for
$(\mathbb{R}v_{k_1}+\mathbb{R}v_{k_2}) \cap \mathbb{Z}^3$
such that $v_{k_1}=e_1$ and $v_{k_2}=pe_1+qe_2$ for some $q>p \geq 0$.
Then we define $s(E)=s(p, q)$, where $s(p, q)$ is the Dedekind sum, which is defined by
\begin{equation*}
s(p, q)
=\sum_{i=1}^q \left(\left(\frac{i}{q}\right)\right)\left(\left(\frac{pi}{q}\right)\right),\quad
((x))=\left\{\begin{array}{ll}
x-[x]-\frac{1}{2} & (x \notin \mathbb{Z}),\\
0 & (x \in \mathbb{Z}).
\end{array}\right.
\end{equation*}
\end{enumerate}
\end{df}
\begin{remark}
We have $q=m(E)$. Although $p$ is not uniquely determined,
$s(p, q)$ does not depend on the choice of $e_1, e_2$.
Thus $s(E)$ is well-defined.
\end{remark}
\begin{df}\label{C}
For each facet $F$ of $P$, we define a rational number $C(F)$ as follows.
We name vertices and facets around $F$ as in Figure \ref{Ehrhart}.
We denote by $v \in \mathbb{Z}^3$
the inward-pointing primitive normal vector of $F$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=6.5cm]{Ehrhart.eps}
\caption{vertices and facets around $F$.}
\label{Ehrhart}
\end{center}
\end{figure}
For $i=1, \ldots, r$, we define
\begin{equation*}
\varepsilon_i=\mathrm{det}(v, v_{k_{i+1}}, v_{k_i})>0,\quad
a_i=\cfrac{\langle \overrightarrow{P_{i-1}Q_{i-1}}, v_{k_{i+1}} \rangle}
{\varepsilon_i \langle \overrightarrow{P_{i-1}Q_{i-1}}, v \rangle},\quad
b_i=\cfrac{\langle \overrightarrow{P_iP_{i+1}}, v_{k_{i-1}} \rangle}
{\varepsilon_{i-1} \langle \overrightarrow{P_iP_{i+1}}, v_{k_i} \rangle},
\end{equation*}
where
$v_{k_0}=v_{k_r}, v_{k_{r+1}}=v_{k_1}, \varepsilon_0=\varepsilon_r, P_0=P_r, P_{r+1}=P_1, Q_0=Q_r$
and $\langle \cdot, \cdot \rangle$ is the standard inner product on $\mathbb{R}^3$.
Then we define
\begin{equation*}
C(F)=-\sum_{2 \leq i<j \leq r}a_i
\left|\begin{array}{ccccc}
b_{i+1} & \varepsilon_{i+1}^{-1} & 0 & \cdots & 0 \\
\varepsilon_{i+1}^{-1} & b_{i+2} & \varepsilon_{i+2}^{-1} & \ddots & \vdots \\
0 & \varepsilon_{i+2}^{-1} & \ddots & \ddots & 0 \\
\vdots & \ddots & \ddots & b_{j-2} & \varepsilon_{j-2}^{-1} \\
0 & \cdots & 0 & \varepsilon_{j-2}^{-1} & b_{j-1} \\
\end{array}\right|
\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_{j-1}
\frac{\mathrm{Vol}(P_{j-1}P_j)}{m(P_{j-1}P_j)},
\end{equation*}
where $P_{j-1}P_j$ is the edge whose endpoints are $P_{j-1}$ and $P_j$,
and the determinants above are understood to be one when $j=i+1$.
\end{df}
\begin{remark}
The proof of Theorem \ref{main} below shows that
$C(F)$ does not depend on the choice of $F_{k_1}$.
\end{remark}
The following is our main theorem:
\begin{thm}\label{main}
Let $P \subset \mathbb{R}^3$ be a 3-dimensional simple integral convex polytope,
and let $E_1, \ldots, E_m$ and $F_1, \ldots, F_n$
be the edges and the facets of $P$, respectively.
Then the coefficient $c_1$ of the Ehrhart polynomial
$|(lP) \cap \mathbb{Z}^3|=c_3l^3+c_2l^2+c_1l+c_0$ is given by
\begin{equation*}
\sum_{j=1}^m \left(s(E_j)+\frac{1}{4}\right)\mathrm{Vol}(E_j)
+\frac{1}{12}\sum_{k=1}^n C(F_k).
\end{equation*}
\end{thm}
\begin{example}
Let $a, b, c$ be positive integers with $\mathrm{gcd}(a, b, c)=1$
and let $P \subset \mathbb{R}^3$ be the tetrahedron with vertices
\begin{equation*}
O=\left(\begin{array}{c}0\\0\\0\end{array}\right),\quad
P_1=\left(\begin{array}{c}a\\0\\0\end{array}\right),\quad
P_2=\left(\begin{array}{c}0\\b\\0\end{array}\right),\quad
P_3=\left(\begin{array}{c}0\\0\\c\end{array}\right).
\end{equation*}
We put $A=\mathrm{gcd}(b,c), B=\mathrm{gcd}(a,c), C=\mathrm{gcd}(a,b)$
and $d=ABC$. Then we have the following table:
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|}
\hline
edge $E$ & $OP_1$ & $OP_2$ & $OP_3$ & $P_1P_2$ & $P_1P_3$ & $P_2P_3$ \\
\hline
$\mathrm{Vol}(E)$ & $a$ & $b$ & $c$ & $C$ & $B$ & $A$ \\
\hline
$m(E)$ & $1$ & $1$ & $1$ & $cC/d$ & $bB/d$ & $aA/d$ \\
\hline
$s(E)$ & $0$ & $0$ & $0$ &
$-s\left(\cfrac{ab}{d}, \cfrac{cC}{d}\right)$ &
$-s\left(\cfrac{ac}{d}, \cfrac{bB}{d}\right)$ &
$-s\left(\cfrac{bc}{d}, \cfrac{aA}{d}\right)$ \\
\hline
\end{tabular}
\begin{tabular}{|c||c|c|c|c|}
\hline
facet $F$ & $OP_1P_2$ & $OP_1P_3$ & $OP_2P_3$ & $P_1P_2P_3$ \\
\hline
\shortstack{inward-pointing primitive\\normal vector of $F$} &
$\left(\begin{array}{c}0\\0\\1\end{array}\right)$ &
$\left(\begin{array}{c}0\\1\\0\end{array}\right)$ &
$\left(\begin{array}{c}1\\0\\0\end{array}\right)$ &
$\left(\begin{array}{c}-bc/d\\-ac/d\\-ab/d\end{array}\right)$ \\
\hline
$C(F)$ & $ab/c$ & $ac/b$ &$bc/a$ & $d^2/(abc)$ \\
\hline
\end{tabular}
\caption{the values of $\mathrm{Vol}(E), s(E)$ and $C(F)$.}
\label{values1}
\end{center}
\end{table}
Thus we have
\begin{align*}
&\sum_{E:\mathrm{edge}} \left(s(E)+\frac{1}{4}\right)\mathrm{Vol}(E)
+\frac{1}{12}\sum_{F:\mathrm{facet}} C(F)\\
&=\frac{a}{4}+\frac{b}{4}+\frac{c}{4}
+\left(-s\left(\frac{ab}{d}, \frac{cC}{d}\right)+\frac{1}{4}\right)C
+\left(-s\left(\frac{ac}{d}, \frac{bB}{d}\right)+\frac{1}{4}\right)B\\
&+\left(-s\left(\frac{bc}{d}, \frac{aA}{d}\right)+\frac{1}{4}\right)A
+\frac{1}{12}\left(\frac{ab}{c}+\frac{ac}{b}+\frac{bc}{a}+\frac{d^2}{abc}\right),
\end{align*}
which coincides with the formula in \cite[Theorem 5]{Pommersheim}.
\end{example}
\begin{example}
Let $a$ and $c$ be positive integers and $b$ be a non-negative integer.
Consider the convex hull $P \subset \mathbb{R}^3$ of the six points
\begin{align*}
&O=\left(\begin{array}{c}0\\0\\0\end{array}\right),\quad
A=\left(\begin{array}{c}a\\0\\0\end{array}\right),\quad
B=\left(\begin{array}{c}0\\a\\0\end{array}\right),\\
&O'=\left(\begin{array}{c}b\\0\\c\end{array}\right),\quad
A'=\left(\begin{array}{c}a+b\\0\\c\end{array}\right),\quad
B'=\left(\begin{array}{c}b\\a\\c\end{array}\right).
\end{align*}
$P$ is a 3-dimensional simple polytope.
We put $g=\mathrm{gcd}(b,c)$. Then we have the following table:
\begin{table}[htbp]
\begin{center}
\footnotesize\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|}
\hline
edge $E$ & $OA$ & $OB$ & $AB$ & $OO'$ & $AA'$ & $BB'$ & $O'A'$ & $O'B'$ & $A'B'$ \\
\hline
$\mathrm{Vol}(E)$ & $a$ & $a$ & $a$ & $g$ & $g$ & $g$ & $a$ & $a$ & $a$ \\
\hline
$m(E)$ & $1$ & $c/g$ & $c/g$ & $1$ & $1$ & $c/g$ & $1$ & $c/g$ & $c/g$ \\
\hline
$s(E)$ & $0$ & $-s\left(\frac{b}{g}, \frac{c}{g}\right)$ & $s\left(\frac{b}{g}, \frac{c}{g}\right)$ &
$0$ & $0$ & $-s\left(1, \frac{c}{g}\right)$ &
$0$ & $s\left(\frac{b}{g}, \frac{c}{g}\right)$ & $-s\left(\frac{b}{g}, \frac{c}{g}\right)$ \\
\hline
\end{tabular}
\begin{tabular}{|c||c|c|c|c|c|}
\hline
facet $F$ & $OAB$ & $OAA'O'$ & $OBB'O'$ & $ABB'A'$ & $O'A'B'$ \\
\hline
\shortstack{inward-pointing primitive\\normal vector of $F$} &
$\left(\begin{array}{c}0\\0\\1\end{array}\right)$ &
$\left(\begin{array}{c}0\\1\\0\end{array}\right)$ &
$\left(\begin{array}{c}c/g\\0\\-b/g\end{array}\right)$ &
$\left(\begin{array}{c}-c/g\\-c/g\\b/g\end{array}\right)$ &
$\left(\begin{array}{c}0\\0\\-1\end{array}\right)$ \\
\hline
$C(F)$ & $0$ & $c$ & $g^2/c$ & $g^2/c$ & $0$ \\
\hline
\end{tabular}
\caption{the values of $\mathrm{Vol}(E), s(E)$ and $C(F)$.}
\label{values2}
\end{center}
\end{table}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7cm]{example2.eps}
\caption{the simple polytope $P$.}
\label{example2}
\end{center}
\end{figure}
Thus we have
\begin{align*}
&\sum_{E:\mathrm{edge}} \left(s(E)+\frac{1}{4}\right)\mathrm{Vol}(E)
+\frac{1}{12}\sum_{F:\mathrm{facet}} C(F)\\
&=-s\left(1, \frac{c}{g}\right)g+\frac{3a}{2}+\frac{3g}{4}
+\frac{1}{12}\left(c+\frac{2g^2}{c}\right)\\
&=-g\sum_{i=1}^{c/g-1}\left(\frac{i}{\frac{c}{g}}-\frac{1}{2}\right)^2
+\frac{3a}{2}+\frac{3g}{4}+\frac{c}{12}+\frac{g^2}{6c}\\
&=-g\sum_{i=1}^{c/g-1}\left(\frac{g^2}{c^2}i^2-\frac{g}{c}i+\frac{1}{4}\right)
+\frac{3a}{2}+\frac{3g}{4}+\frac{c}{12}+\frac{g^2}{6c}\\
&=-\frac{g^3}{c^2}\frac{\left(\frac{c}{g}-1\right)\frac{c}{g}\left(\frac{2c}{g}-1\right)}{6}
+\frac{g^2}{c}\frac{\left(\frac{c}{g}-1\right)\frac{c}{g}}{2}-g\frac{\frac{c}{g}-1}{4}
+\frac{3a}{2}+\frac{3g}{4}+\frac{c}{12}+\frac{g^2}{6c}\\
&=\frac{3a}{2}+g.
\end{align*}
On the other hand, since
\begin{equation*}
\#\{(x, y) \in \mathbb{Z}^2 \mid (x, y, z) \in lP\}
=\left\{\begin{array}{ll}
\cfrac{(al+1)(al+2)}{2} & ((c/g)|z), \\
\cfrac{al(al+1)}{2} & ((c/g)\mid \hspace{-.67em}/z) \end{array}\right.
\end{equation*}
for $z=0, 1, \ldots, cl$, we have
\begin{align*}
|(lP) \cap \mathbb{Z}^3|&=\frac{(al+1)(al+2)}{2}(gl+1)+\cfrac{al(al+1)}{2}((cl+1)-(gl+1))\\
&=\frac{a^2c}{2}l^3+\frac{1}{2}\left(a^2+ac+2ag\right)l^2+\left(\frac{3a}{2}+g\right)l+1.
\end{align*}
The coefficient of $l$ is also $3a/2+g$.
\end{example}
\section{Proof of Theorem \ref{main}}
First we recall some facts about toric geometry, see \cite{Fulton} for details.
Let $P \subset \mathbb{R}^d$ be a $d$-dimensional integral convex polytope.
We define a cone
\begin{equation*}
\sigma_F=\{v \in \mathbb{R}^d \mid
\langle u'-u, v\rangle \geq 0\ \forall u' \in P, \forall u \in F\}
\end{equation*}
for each face $F$ of $P$.
Then the set
\begin{equation*}
\Delta_P=\{\sigma_F \mid F\mbox{ is a face of }P\}
\end{equation*}
of such cones
forms a fan in $\mathbb{R}^d$, which is called the {\it normal fan} of $P$.
Let $X(\Delta_P)$ be the associated projective toric variety.
We denote by $V(\sigma)$ the subvariety of $X(\Delta_P)$
corresponding to $\sigma \in \Delta_P$.
Let $\mathrm{Td}_i(X(\Delta_P)) \in A_i(X(\Delta_P))_\mathbb{Q}$
be the $i$-th Todd class in the Chow group of $i$-cycles with rational coefficients.
\begin{thm}\label{Todd}
Let $P \subset \mathbb{R}^d$ be a $d$-dimensional integral convex polytope
and $|(lP) \cap \mathbb{Z}^d|=c_dl^d+c_{d-1}l^{d-1}+\cdots+c_0$
be its Ehrhart polynomial.
If $\mathrm{Td}_i(X(\Delta_P))$ has an expression of the form
$\sum_Fr_F[V(\sigma_F)]$ with $r_F \in \mathbb{Q}$,
then we have $c_i=\sum_Fr_F\mathrm{Vol}(F)$,
where $[V(\sigma_F)]$ is the class of $V(\sigma_F)$ in the Chow group
and $\mathrm{Vol}(F)$ is the relative volume of $F$.
\end{thm}
Now we assume that $d=3$ and $P$ is simple.
Then the associated toric variety $X(\Delta_P)$ is $\mathbb{Q}$-factorial
and we know the ring structure of the Chow ring $A^*(X(\Delta_P))_\mathbb{Q}$
with rational coefficients.
Let $E_1, \ldots, E_m$ and $F_1, \ldots, F_n$
be the edges and the facets of $P$, respectively. We have
\begin{equation}\label{1}
\sum_{k=1}^n\langle u, v_k\rangle[V(\sigma_{F_k})]=0 \quad \forall u \in (\mathbb{Q}^3)^*.
\end{equation}
If $F_{k_1}$ and $F_{k_2}$ are distinct, then
\begin{equation}\label{2}
[V(\sigma_{F_{k_1}})][V(\sigma_{F_{k_2}})]
=\left\{\begin{array}{ll}
\frac{1}{m(E_j)}[V(\sigma_{E_j})] & (1 \leq \exists j \leq m: F_{k_1} \cap F_{k_2}=E_j), \\
0 & (F_{k_1} \cap F_{k_2}=\emptyset)
\end{array}\right.
\end{equation}
in $A^*(X(\Delta_P))_\mathbb{Q}$.
Pommersheim gave an expression of $\mathrm{Td}_{d-2}(X(\Delta_P))$
for a $d$-dimensional simple integral convex polytope $P \subset \mathbb{R}^d$.
In the case where $d=3$, we have the following:
\begin{thm}[Pommersheim \cite{Pommersheim}]\label{P}
If $P \subset \mathbb{R}^3$ is a 3-dimensional simple integral convex polytope, then
\begin{equation*}
\mathrm{Td}_1(X(\Delta_P))=\sum_{j=1}^m \left(s(E_j)+\frac{1}{4}\right)[V(\sigma_{E_j})]
+\frac{1}{12}\sum_{k=1}^n [V(\sigma_{F_k})]^2.
\end{equation*}
\end{thm}
We use the notation in Definition \ref{C}.
It suffices to show
\begin{equation*}
[V(\sigma_{F})]^2=-\sum_{2 \leq i<j \leq r}a_i
\left|\begin{array}{ccccc}
b_{i+1} & \varepsilon_{i+1}^{-1} & 0 & \cdots & 0 \\
\varepsilon_{i+1}^{-1} & b_{i+2} & \varepsilon_{i+2}^{-1} & \ddots & \vdots \\
0 & \varepsilon_{i+2}^{-1} & \ddots & \ddots & 0 \\
\vdots & \ddots & \ddots & b_{j-2} & \varepsilon_{j-2}^{-1} \\
0 & \cdots & 0 & \varepsilon_{j-2}^{-1} & b_{j-1} \\
\end{array}\right|
\frac{\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_{j-1}}{m(P_{j-1}P_j)}
[V(\sigma_{P_{j-1}P_j})]
\end{equation*}
for each facet $F$ of $P$.
We put
\begin{equation*}
D(s, t)=\left|\begin{array}{ccccc}
b_s & \varepsilon_s^{-1} & 0 & \cdots & 0 \\
\varepsilon_s^{-1} & b_{s+1} & \varepsilon_{s+1}^{-1} & \ddots & \vdots \\
0 & \varepsilon_{s+1}^{-1} & \ddots & \ddots & 0 \\
\vdots & \ddots & \ddots & b_{t-1} & \varepsilon_{t-1}^{-1} \\
0 & \cdots & 0 & \varepsilon_{t-1}^{-1} & b_t \\
\end{array}\right|
\end{equation*}
for $2<s \leq t<r$ and $D(s, t)=1$ for $s>t$.
Define $u \in (\mathbb{Q}^3)^*$ by
$\langle u, v \rangle=1, \langle u, v_{k_1} \rangle=0, \langle u, v_{k_2} \rangle=0$.
By (\ref{1}) and (\ref{2}), we have
\begin{equation*}
[V(\sigma_{F})]^2=-[V(\sigma_{F})]\sum_{j=1}^r\langle u, v_{k_j}\rangle[V(\sigma_{F_{k_j}})]
=-\sum_{j=3}^r\frac{\langle u, v_{k_j}\rangle}{m(P_{j-1}P_j)}[V(\sigma_{P_{j-1}P_j})].
\end{equation*}
Hence it suffices to show
\begin{equation}\label{induction}
\langle u, v_{k_j}\rangle=\sum_{i=2}^{j-1}a_i
D(i+1, j-1)\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_{j-1}
\end{equation}
for any $j=3, \ldots, r$.
First we claim that
\begin{equation}\label{relation}
\varepsilon_{j-1}^{-1}v_{k_{j-1}}+\varepsilon_j^{-1}v_{k_{j+1}}=a_jv+b_jv_{k_j}
\end{equation}
for any $j=2, \ldots, r-1$. By Cramer's rule, we have
\begin{align*}
v_{k_{j+1}}
&=\frac{\mathrm{det}(v_{k_{j+1}}, v_{k_j}, v_{k_{j-1}})}{\mathrm{det}(v, v_{k_j}, v_{k_{j-1}})}v
+\frac{\mathrm{det}(v, v_{k_{j+1}}, v_{k_{j-1}})}{\mathrm{det}(v, v_{k_j}, v_{k_{j-1}})}v_{k_j}
+\frac{\mathrm{det}(v, v_{k_j}, v_{k_{j+1}})}{\mathrm{det}(v, v_{k_j}, v_{k_{j-1}})}v_{k_{j-1}}\\
&=\frac{\mathrm{det}(v_{k_{j+1}}, v_{k_j}, v_{k_{j-1}})}{\varepsilon_{j-1}}v
+\frac{\mathrm{det}(v, v_{k_{j+1}}, v_{k_{j-1}})}{\varepsilon_{j-1}}v_{k_j}
-\frac{\varepsilon_j}{\varepsilon_{j-1}}v_{k_{j-1}}.
\end{align*}
So we have
\begin{align}\label{relation2}
&\begin{aligned}
&\varepsilon_{j-1}^{-1}v_{k_{j-1}}+\varepsilon_j^{-1}v_{k_{j+1}}\\
&=\varepsilon_{j-1}^{-1}\varepsilon_j^{-1}\mathrm{det}(v_{k_{j+1}}, v_{k_j}, v_{k_{j-1}})v
+\varepsilon_{j-1}^{-1}\varepsilon_j^{-1}\mathrm{det}(v, v_{k_{j+1}}, v_{k_{j-1}})v_{k_j}.
\end{aligned}
\end{align}
Taking the inner product of both sides of (\ref{relation2})
with $\overrightarrow{P_{j-1}Q_{j-1}}$ gives
\begin{equation*}
\varepsilon_j^{-1}\langle \overrightarrow{P_{j-1}Q_{j-1}}, v_{k_{j+1}} \rangle
=\varepsilon_{j-1}^{-1}\varepsilon_j^{-1}\mathrm{det}(v_{k_{j+1}}, v_{k_j}, v_{k_{j-1}})
\langle \overrightarrow{P_{j-1}Q_{j-1}}, v \rangle,
\end{equation*}
which means
$a_j=\varepsilon_{j-1}^{-1}\varepsilon_j^{-1}\mathrm{det}(v_{k_{j+1}}, v_{k_j}, v_{k_{j-1}})$.
Taking the inner product of both sides of (\ref{relation2})
with $\overrightarrow{P_jP_{j+1}}$ gives
\begin{equation*}
\varepsilon_{j-1}^{-1}\langle \overrightarrow{P_jP_{j+1}}, v_{k_{j-1}} \rangle
=\varepsilon_{j-1}^{-1}\varepsilon_j^{-1}\mathrm{det}(v, v_{k_{j+1}}, v_{k_{j-1}})
\langle \overrightarrow{P_jP_{j+1}}, v_{k_j} \rangle,
\end{equation*}
which means
$b_j=\varepsilon_{j-1}^{-1}\varepsilon_j^{-1}\mathrm{det}(v, v_{k_{j+1}}, v_{k_{j-1}})$.
Thus (\ref{relation}) follows.
We show (\ref{induction}) by induction on $j$.
If $j=3$, then both sides are $a_2\varepsilon_2$.
If $j=4$, then both sides are $a_2b_3\varepsilon_2\varepsilon_3+a_3\varepsilon_3$.
Suppose $4 \leq j \leq r-1$. By (\ref{relation}) and the hypothesis of induction, we have
\begin{align*}
\langle u, v_{k_{j+1}}\rangle
&=\langle u, a_j\varepsilon_jv+b_j\varepsilon_jv_{k_j}
-\varepsilon_{j-1}^{-1}\varepsilon_jv_{k_{j-1}}\rangle\\
&=a_j\varepsilon_j+b_j\varepsilon_j\langle u, v_{k_j}\rangle
-\varepsilon_{j-1}^{-1}\varepsilon_j\langle u, v_{k_{j-1}}\rangle\\
&=a_j\varepsilon_j+b_j\varepsilon_j\sum_{i=2}^{j-1}a_i
D(i+1, j-1)\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_{j-1}\\
&-\varepsilon_{j-1}^{-1}\varepsilon_j\sum_{i=2}^{j-2}a_i
D(i+1, j-2)\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_{j-2}.
\end{align*}
On the other hand,
\begin{align*}
&\sum_{i=2}^ja_i
D(i+1, j)\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_j\\
&=a_j\varepsilon_j+a_{j-1}b_j\varepsilon_{j-1}\varepsilon_j
+\sum_{i=2}^{j-2}a_i
D(i+1, j)\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_j.
\end{align*}
Since
\begin{align*}
&\sum_{i=2}^{j-2}a_i
D(i+1, j)\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_j\\
&=\sum_{i=2}^{j-2}a_i(b_jD(i+1, j-1)-\varepsilon_{j-1}^{-2}D(i+1, j-2))
\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_j\\
&=b_j\varepsilon_j\sum_{i=2}^{j-2}
a_iD(i+1, j-1)\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_{j-1}\\
&-\varepsilon_{j-1}^{-1}\varepsilon_j\sum_{i=2}^{j-2}a_i
D(i+1, j-2)\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_{j-2},
\end{align*}
we have
\begin{align*}
&\sum_{i=2}^ja_i
D(i+1, j)\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_j\\
&=a_j\varepsilon_j+a_{j-1}b_j\varepsilon_{j-1}\varepsilon_j
+b_j\varepsilon_j\sum_{i=2}^{j-2}
a_iD(i+1, j-1)\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_{j-1}\\
&-\varepsilon_{j-1}^{-1}\varepsilon_j\sum_{i=2}^{j-2}a_i
D(i+1, j-2)\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_{j-2}\\
&=a_j\varepsilon_j+b_j\varepsilon_j\sum_{i=2}^{j-1}a_i
D(i+1, j-1)\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_{j-1}\\
&-\varepsilon_{j-1}^{-1}\varepsilon_j\sum_{i=2}^{j-2}a_i
D(i+1, j-2)\varepsilon_i \varepsilon_{i+1} \cdots \varepsilon_{j-2}\\
&=\langle u, v_{k_{j+1}}\rangle.
\end{align*}
Thus (\ref{induction}) holds for $j+1$. This completes the proof of Theorem \ref{main}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,619 |
Q: Portfolio rebalancing with bandwidth method in python We need to calculate a continuously rebalanced portfolio of 2 stocks. Lets call them A and B. They shall both have an equal part of the portfolio. So if I have 100$ in my portfolio 50$ get invested in A and 50$ in B. As both stocks perform very differently they will not keep their equal weights (after 3 month already A may be worth 70$ while B dropped to 45$). The problem is that they have to keep their share of the portfolio within a certain bandwidth of tolerance. This bandwidth is 5%. So I need a function that does: If A > B*1.05 or A*1.05 < B then rebalance.
This first part serves only to get the fastest way some data to have a common basis of discussion and to make results comparable, so you can just copy and paste this whole code and it works for you..
import pandas as pd
from datetime import datetime
import numpy as np
df1 = pd.io.data.get_data_yahoo("IBM",
start=datetime(1970, 1, 1),
end=datetime.today())
df1.rename(columns={'Adj Close': 'ibm'}, inplace=True)
df2 = pd.io.data.get_data_yahoo("F",
start=datetime(1970, 1, 1),
end=datetime.today())
df2.rename(columns={'Adj Close': 'ford'}, inplace=True)
df = df1.join(df2.ford, how='inner')
del df["Open"]
del df["High"]
del df["Low"]
del df["Close"]
del df["Volume"]
Nowe start to calculate the relative performance of each stock with the formula: df.ibm/df.ibm[0]. The problem is that as soon as we break the first bandwidth, we need to reset the 0 in our formula: df.ibm/df.ibm[0], since we rebalance and need to start calculating from that point on. So we use df.d for this placeholder function and set it equal to df.t as soon as a bandwidth gets broken df.t basically just counts the length of the dataframe and can tell us therefore always "where we are". So here the actual calculation starts:
tol = 0.05 #settintg the bandwidth tolerance
df["d"]= 0 #
df["t"]= np.arange(len(df))
tol = 0.3
def flex_relative(x):
if df.ibm/df.ibm.iloc[df.d].values < df.ford/df.ford.iloc[df.d].values * (1+tol):
return df.iloc[df.index.get_loc(x.name) - 1]['d'] == df.t
elif df.ibm/df.ibm.iloc[df.d].values > df.ford/df.ford.iloc[df.d].values * (1+tol):
return df.iloc[df.index.get_loc(x.name) - 1]['d'] == df.t
else:
return df.ibm/df.ibm.iloc[df.d].values, df.ford/df.ford.iloc[df.d].values
df["ibm_performance"], df["ford_performance"], = df.apply(flex_relative, axis =1)
The problem is, that I am getting this error form the last line of code, where I try to apply the function with df.apply(flex_relative, axis =1)
ValueError: ('The truth value of a Series is ambiguous. Use a.empty, a.bool(), a.item(), a.any() or a.all().', u'occurred at index 1972-06-01 00:00:00') The problem is that none of the given options of the error statement solves my problem, so I really don't know what to do...
The only thing I found so far was the link below, but calling a R function won't work for me because I need to apply that to quite big datasets and I may also implement an optimization in this function, so it definitely needs to be built in python. Here is the link anyway: Finance Lib with portfolio optimization method in python
Manually (what is not a good way to handle big data), I calculated that the first date for a rebalancing would be: 03.11.1972 00:00:00
The output of the dataframe at the first rebalancing should look like this:
ibm ford d t ibm_performance ford_performance
1972-11-01 00:00:00 6,505655 0,387415 0 107 1,021009107 0,959552418
1972-11-02 00:00:00 6,530709 0,398136 0 108 1,017092172 0,933713605
1972-11-03 00:00:00 6,478513 0,411718 0 109 1,025286667 0,902911702 # this is the day, the rebalancing was detected
1972-11-06 00:00:00 6,363683 0,416007 109 110 1,043787536 0,893602752 # this is the day the day the rebalancing is implemented, therefore df.d gets set = df.t = 109
1972-11-08 00:00:00 6,310883 0,413861 109 111 1,052520384 0,898236364
1972-11-09 00:00:00 6,227073 0,422439 109 112 1,066686226 0,879996875
Thanks a lot for your support!
@Alexander: Yes, the rebalancing will take place the following day.
@maxymoo: If you implement this code after yours, you get the portfolio weights of each stock and they don't rest between 45 and 55%. It's rather between 75% and 25%:
df["ford_weight"] = df.ford_prop*df.ford/(df.ford_prop*df.ford+df.ibm_prop*df.ibm) #calculating the actual portfolio weights
df["ibm_weight"] = df.ibm_prop*df.ibm/(df.ford_prop*df.ford+df.ibm_prop*df.ibm)
print df
print df.ibm_weight.min()
print df.ibm_weight.max()
print df.ford_weight.min()
print df.ford_weight.max()
I tried no for an hour or so to fix, but didn't find it.
Can I do anything to make this question clearer?
A: You can use this code to calulate your portfolio at each point in time.
i = df.index[0]
df['ibm_prop'] = 0.5/df.ibm.ix[i]
df['ford_prop'] = 0.5/df.ford.ix[i]
while i:
try:
i = df[abs(1-(df.ibm_prop*df.ibm + df.ford_prop*df.ford)) > tol].index[0]
except IndexError:
break
df['ibm_prop'].ix[i:] = 0.5/df.ibm.ix[i]
df['ford_prop'].ix[i:] = 0.5/df.ford.ix[i]
A: just a mathematical improvement on maxymoo's answer:
i = df.index[0]
df['ibm_prop'] = df.ibm.ix[i]/(df.ibm.ix[i]+df.ford.ix[i])
df['ford_prop'] = df.ford.ix[i]/(df.ibm.ix[i]+df.ford.ix[i])
while i:
try:
i = df[abs((df.ibm_prop*df.ibm - df.ford_prop*df.ford)) > tol].index[0]
except IndexError:
break
df['ibm_prop'].ix[i:] = df.ibm.ix[i]/(df.ibm.ix[i]+df.ford.ix[i])
df['ford_prop'].ix[i:] = df.ford.ix[i]/(df.ibm.ix[i]+df.ford.ix[i])
A: What about this:
df["d"]= [0,0,0,0,0,0,0,0,0,0]
df["t"]= np.arange(len(df))
tol = 0.05
def flex_relative(x):
if df.ibm/df.ibm.iloc[df.d].values < df.ford/df.ford.iloc[df.d].values * (1+tol):
return df.iloc[df.index.get_loc(x.name) - 1]['d'] == df.t
elif df.ibm/df.ibm.iloc[df.d].values > df.ford/df.ford.iloc[df.d].values * (1+tol):
return df.iloc[df.index.get_loc(x.name) - 1]['d'] == df.t
A: The main idea here is to work in terms of dollars instead of ratios. If you
keep track of the number of shares and the relative dollar values of the ibm and
ford shares, then you can express the criterion for rebalancing as
mask = (df['ratio'] >= 1+tol) | (df['ratio'] <= 1-tol)
where the ratio equals
df['ratio'] = df['ibm value'] / df['ford value']
and df['ibm value'], and df['ford value'] represent actual dollar values.
import datetime as DT
import numpy as np
import pandas as pd
import pandas.io.data as PID
def setup_df():
df1 = PID.get_data_yahoo("IBM",
start=DT.datetime(1970, 1, 1),
end=DT.datetime.today())
df1.rename(columns={'Adj Close': 'ibm'}, inplace=True)
df2 = PID.get_data_yahoo("F",
start=DT.datetime(1970, 1, 1),
end=DT.datetime.today())
df2.rename(columns={'Adj Close': 'ford'}, inplace=True)
df = df1.join(df2.ford, how='inner')
df = df[['ibm', 'ford']]
df['sh ibm'] = 0
df['sh ford'] = 0
df['ibm value'] = 0
df['ford value'] = 0
df['ratio'] = 0
return df
def invest(df, i, amount):
"""
Invest amount dollars evenly between ibm and ford
starting at ordinal index i.
This modifies df.
"""
c = dict([(col, j) for j, col in enumerate(df.columns)])
halfvalue = amount/2
df.iloc[i:, c['sh ibm']] = halfvalue / df.iloc[i, c['ibm']]
df.iloc[i:, c['sh ford']] = halfvalue / df.iloc[i, c['ford']]
df.iloc[i:, c['ibm value']] = (
df.iloc[i:, c['ibm']] * df.iloc[i:, c['sh ibm']])
df.iloc[i:, c['ford value']] = (
df.iloc[i:, c['ford']] * df.iloc[i:, c['sh ford']])
df.iloc[i:, c['ratio']] = (
df.iloc[i:, c['ibm value']] / df.iloc[i:, c['ford value']])
def rebalance(df, tol, i=0):
"""
Rebalance df whenever the ratio falls outside the tolerance range.
This modifies df.
"""
c = dict([(col, j) for j, col in enumerate(df.columns)])
while True:
mask = (df['ratio'] >= 1+tol) | (df['ratio'] <= 1-tol)
# ignore prior locations where the ratio falls outside tol range
mask[:i] = False
try:
# Move i one index past the first index where mask is True
# Note that this means the ratio at i will remain outside tol range
i = np.where(mask)[0][0] + 1
except IndexError:
break
amount = (df.iloc[i, c['ibm value']] + df.iloc[i, c['ford value']])
invest(df, i, amount)
return df
df = setup_df()
tol = 0.05
invest(df, i=0, amount=100)
rebalance(df, tol)
df['portfolio value'] = df['ibm value'] + df['ford value']
df['ibm weight'] = df['ibm value'] / df['portfolio value']
df['ford weight'] = df['ford value'] / df['portfolio value']
print df['ibm weight'].min()
print df['ibm weight'].max()
print df['ford weight'].min()
print df['ford weight'].max()
# This shows the rows which trigger rebalancing
mask = (df['ratio'] >= 1+tol) | (df['ratio'] <= 1-tol)
print(df.loc[mask])
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,148 |
About Muso
Choosing Shoes or Subscriptions?
As featured on Forbes.
If you were to go looking for an international criminal who respects no borders, operates by stealth and takes what they want, it's unlikely you'd look for them amongst the lifestyle threads of a parenting site.
But it's not just toddler training tips or recipes being shared on these threads. Users, while enthusing over their new favourite TV show or film, will sometimes share links to illegal download sites. While we are living in what is currently acknowledged as the Golden Age of Television, it is also the age of austerity, and one of the many handy household budgeting tips on mainstream sites such as these is to ditch TV subscriptions along with the smashed avocado on toast. When piracy is presented in such a wink-wink, nudge-nudge context, it seems innocuous and if everybody else is doing it, then why can't they?
Online piracy is like taking candy from a baby. Quite literally. In June 2019 alone, the Pokémon movie Detective Pikachu was illegally downloaded over 4 million times via the torrent network alone. In July 2019, How to Train your Dragon: The Hidden World had over 1.2 million torrent downloads and over 1.6 million people illegally downloaded Season Two of Big Little Lies in the same manner. It is also worth considering that torrenting only makes up around 10% of piracy activity.It would seem that a lot of modern day pirates are merely frazzled parents who wouldn't dream of stealing their Friday night bottle of chilled Riesling from the grocery store but don't want to pay the same amount to watch the latest TV phenomenon. And if taking a family of four out to the movies is an even more costly exercise, it's not hard to see how the price conscious will plump for two or three small clicks and a night at home watching the latest family blockbuster.
Kids, mums, dads, grandparents: somebody you know as an upstanding citizen who pays their taxes on time, always takes their bins out and would never dodge the ticket inspector on a train, somebody wearing elasticated slacks and Birkenstocks – well they can be a pirate too. No eyepatch needed.
The global situation right now is fiscally precarious. As I write, oil prices have spiked 10% in light of attacks on Saudi oil-fields, Britain is running the Brexit gauntlet and the Amazon is still on fire. Growth is stagnant or contracting; pressure on wages is downward. Yet at the same time, every content provider is seeking to launch their product on their own exclusive platform, complete with a monthly subscription fee. What gives? The more platforms fragment, the more subscriptions necessary to experience content, the less cost effective it becomes and piracy increases.
So next time you picture a digital pirate as a spotty teenager in a hoodie illegally downloading The Fast and Furious 42, you'd do just as well to imagine a forty something Orange Is The New Black fan who spends her money on shoes instead of subscriptions.
Film/ Article/ Television/ Press Release
© MUSO TNT
MUSO TNT is registered in the UK, Company No 07351174. Registered Office: MUSO LONDON, 71-75 Shelton Street, Covent Garden, London, WC2H 9JQ. VAT Number: GB997744541. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,338 |
Total starts up La Mède biorefinery
Published: 4 July 2019 - 8 a.m.
By: Martin Menachery
Total has started up production at the La Mède biorefinery in south-eastern France, with the first batches of biofuel coming off the line. It is the final step in converting a former oil refinery into a new energies complex. Launched in 2015, the project represents a capital expenditure of $310.52mn.
The La Mède complex now encompasses: (i) a biorefinery with a capacity of 500,000 tonnes of biofuel per year; (ii) an 8-megawatt solar farm that can supply 13,000 people; (iii) a unit to produce 50,000 cubic metres per year of AdBlue, an additive that reduces nitrogen oxide emissions from trucks; (iv) A logistics and storage hub with a capacity of 1.3 million cubic metres per year; (v) a training centre offering real facilities and able to host 2,500 learners a year.
Together, these new activities have maintained 250 direct jobs at La Mède. As part of the site transformation, 65% of the orders to remodel the complex were awarded to local businesses, representing 800 jobs and $158mn in revenue. Total also invested $5.64mn in the economic development of the Fos-Etang de Berre region, notably by supporting initiatives to create jobs, attract industrial projects and support contractors. That is five times as much as a typical revitalisation agreement.
The biorefinery can produce 500,000 tonnes of hydrotreated vegetable oil (HVO), a premium biofuel. La Mède will produce both biodiesel and biojet fuel for the aviation industry. It was specifically designed to process all types of oil. Its biofuels will be made: 60% to 70% from 100% sustainable vegetable oils (rapeseed, palm, sunflower, etc.); 30% to 40% from treated waste (animal fats, cooking oil, residues, etc.) to promote a circular economy.
As part of an agreement with the government in May 2018, Total has pledged to process no more than 300,000 tonnes of palm oil per year – less than 50% of the total volume of raw materials needed – and at least 50,000 tonnes of French-grown rapeseed, creating another market for domestic agriculture.
All the oils processed will be certified sustainable to European Union standards. In addition, as part of its palm oil procurement process, Total is taking an extra step by introducing strengthened control of sustainability and respect for Human Rights.
"I would like to thank the teams for all their hard work these last four years to convert our La Mède refinery," said Bernard Pinatel, president, refining and chemicals, Total.
"Biofuels are fully renewable and an immediately available solution to cut carbon emissions from ground and air transportation. When produced from sustainable raw materials, as at La Mède, they emit over 50% less carbon than fossil fuels. Our biorefinery will allow us to make biofuels in France that were previously imported."
For the latest refining and petrochemical industry related videos, subscribe to our YouTube page.
Construction complete for two additional ionikylation units in Asia-Pacific
Hydrogen as sustainable fuel for the future
Five minutes with: Michel Anderson, principal director, resources division, Accenture
Petrochemicals News
GlobalData report: Middle East set to contribute 17% of global new-build trunk petroleum products pipeline length additions by 2023
TSA releases new report on the role of the bulk liquid storage sector in the energy transition
The evolution of high velocity thermal spray: From shop applications to mission critical equipment protection
"Never be self-satisfied and do not hesitate to challenge the limits"
GlobalData report: Asia leads global new-build trunk/transmission petroleum products pipeline length additions by 2023
PAS joins ISA Global Cybersecurity Alliance as founding member
KBR wins first commercial contract for its new propane dehydrogenation technology K-PRO
ADNOC will enable its sustainability goals through sustainable financing initiatives and strategic partnerships, says group CFO | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,319 |
\section{Introduction}
Trefftz discontinuous Galerkin (TDG) methods are finite element schemes which employ discontinuous
test and trial functions whose restriction to each mesh element belongs to the kernel of the differential
operator to be discretized. For time-harmonic wave problems, Trefftz discretization spaces
are made of oscillating functions with the same frequency as that of the underlying
analytical solution.
This results in improved approximation properties, as compared to standard piecewise polynomial
spaces. Moreover, based on Trefftz spaces, one can construct discontinuous Galerkin methods
which feature unconditional unique solvability, as well as coercivity of the discrete bilinear forms
in suitable (mesh-dependent) norms.
We focus here on the case of the Helmholtz problem and refer, e.g., to the survey~\cite{Hiptmair2016}
for a review of the construction, properties, and relevant literature of Trefftz methods
for its numerical approximation.
The purpose of this article is to develop an efficient $hp$--adaptive refinement
algorithm for TDG methods applied to the homogeneous Helmholtz problem; we will specifically
consider the ultra-weak variational formulation with plane wave basis
functions~\cite{Cessenat1998}.
Within the adaptive procedure, elements will be marked for refinement based on employing an
empirical {\em a posteriori} error indicator, stimulated by the upper
bounds derived in~\cite{Kapita2015} for the $h$--version of the TDG method.
For the $h$--version of the plane wave discontinuous Galerkin method, incorporating Lagrange multipliers, a similar error
indicator has been presented in \cite{Amara2009}.
Once an element has been marked for refinement, a decision must then
be made regarding the type of refinement to be undertaken, i.e., whether the
element should be subdivided ($h$--refinement), or whether the local basis should
be enriched ($p$--refinement). The choice of whether to $h$-- or $p$--refine an element
is typically based on the observation that when the underlying solution is
smooth, then $p$--refinement will be more efficient in terms of reducing the
error, for a given increase in the number of degrees of freedom, than if the
element is subdivided. On the other hand, if the solution is not smooth, then
$h$--refinement should be employed. In general, {\em a posteriori} error estimators
only provide an estimate of the local elementwise error, but do not indicate
which type of refinement should be employed. Within the existing literature
a number of algorithms have been devised for determining the type of
refinement ($h$-- or $p$--) to be undertaken. For a comprehensive review of this
subject, we refer to~\cite{Mitchell2011,Mitchell2014}, and the references
cited therein. In the present context, given the oscillatory nature
of solutions to high-frequency scattering problems, the exploitation of
$hp$--strategies based on local regularity estimation techniques is
not generally applicable. Thereby, we consider an alternative approach based on
estimating the predicted decay rate of the {\em a posteriori} estimator,
given the refinement history of each element; see, for example,
\cite{Melenk2001}.
For {\em a posteriori} error estimation of conforming
finite element approximations of the Helmholtz problem, we refer,
e.g., to \cite{BabI,BabII} and~\cite{SauterDoerfler}; analogous bounds have been established
for polynomial-based discontinuous
Galerkin finite element methods in \cite{SauterZech, Zech}.
In addition to standard $h$-- and $hp$--adaptivity, we also consider the
issue of directional refinement of the underlying plane wave basis employed
within our TDG scheme. In particular, we rotate the underlying elementwise
plane wave basis in order that the first basis function is aligned with
the local dominant propagation direction; strategies for determining the
local dominant propagation direction have been proposed
in~\cite{Amara2014,Betcke2011,Betcke2012,Gittelson2008}, for example.
Stimulated by the work undertaken on
anisotropic mesh adaptation in~\cite{Formaggia2001,Formaggia2003}, cf.,
also,~\cite{ghh-paper,hall-thesis}, we propose an alternative approach
based on studying the properties of the Hessian of the computed TDG solution.
More precisely, the principal eigenvector of the Hessian of the
solution indicates the dominant direction of wave propagation. However,
since eigenvectors are only unique up to scalar multiples, the
precise wave direction must be fixed, based on exploiting an impedance condition.
In this way, we can locally orientate the elementwise plane wave
basis to reduce the error in the underlying computed TDG solution
in a simple and computationally cheap manner. When combined with
$hp$--refinement, the resulting adaptive procedure is capable of
generating highly optimized $hp$--refined Trefftz spaces. Indeed,
the efficiency of the proposed strategy is illustrated for a number
of test problems, where we compare the performance between
an $h$-- and $hp$--refinement algorithm, both with and without directional adaptivity.
The outline of this article is as follows: in Section~\ref{sec:pde_tdg}
we introduce the model problem to be studied within this article, together
with its TDG discretization. Then in Section~\ref{section:adaptive_refinement}
we develop an $hp$--refinement algorithm, based on employing both local
mesh subdivision and local basis enrichment, together with directional
adaptivity for the underlying Trefftz space. The performance of
this procedure is studied in Section~\ref{sec:numerical_examples}
through a series of two-- and three--dimensional examples. Finally,
in Section~\ref{sec:conclusions} we summarize the work undertaken
within this article and highlight potential future directions of research.
\section{Model problem and TDG discretization} \label{sec:pde_tdg}
In this section we state the model problem to be studied in this
article, together with its TDG discretization; for further details, we refer
to
\cite{Hiptmair2016},
for example.
\subsection{Model problem}
We study the homogeneous Helmholtz
equation; to this end, we let
$\Omega\subset\mathbb{R}^d$, $d=2,3$, be an open bounded, Lipschitz domain with boundary
$\partial \Omega$. Thereby, we seek $u:\Omega\mapsto {\mathbb C}$ such that
\begin{equation}
\label{eqn:helmholtz}
\begin{aligned}
-\Delta u-k^2 u &=0 && \text{in } \Omega\;,\\
\frac{\partial u}{\partial \vect{n}} + ik\vartheta u &= g_R && \text{on } \Gamma_R\;, \\
u &=g_D && \text{on } \Gamma_D\;,
\end{aligned}
\end{equation}
where $\vect{n}$ denotes the unit outward normal vector on the boundary $\partial\Omega$,
and $\Gamma_R$ and $\Gamma_D$ are non-overlapping open subsets of $\partial\Omega$, such
that $\partial\Omega=\overline{\Gamma}_R\cup\overline{\Gamma}_D$. Furthermore,
$i$ is the imaginary unit, $\vartheta=\pm 1$, $g_R\in L^2(\Gamma_R)$, and we assume, for
the moment, that the (real-valued) wavenumber $k$ is constant in $\Omega$.
\subsection{Meshes and spaces}
We partition $\Omega$ into computational meshes $\{\mesh\}_{h>0}$ consisting of
non-overlapping (curvilinear) polygons/polyhedra $K$, which potentially include hanging nodes,
such that $\overline{\Omega}=\bigcup_{K\in\mesh}{\overline{K}}$. Moreover,
we assume that the family of subdivisions $\{\mesh\}_{h>0}$ is shape regular
\cite[pp. 61, 114, and 118]{Braess}.
For each element $K\in\mesh$, we write $h_K$ to denote its diameter and
$\vect{n}_K$ signifies the unit outward normal vector to $K$ on~$\partial K$; we set
$h:=\max_{K\in\mesh}h_K$.
Furthermore, we introduce the mesh skeleton $\face$, defined by $\face=\cup_{K\in\mesh}\partial K$;
we write $\face[I]$ and $\face[B]$ to denote the interior and boundary
skeletons, respectively, defined by
$\face[I]=\face\setminus\partial\Omega$ and $\face[B]=\partial\Omega$.
Implicitly, we assume that the finite element mesh $\mesh$ respects the decomposition
of the boundary, in the sense that, given an element face $f\subset \partial K$, $K\in\mesh$,
which lies on the boundary $\partial\Omega$, i.e., $f\subset\partial\Omega$, then
$f$ is entirely contained within either $\Gamma_R$ or $\Gamma_D$.
Let $K$ and $K'$ be two adjacent elements of~ $\mesh$, and
$\vect{x}$ an arbitrary point on the interior face $f\subset \face[I]$ given by
$f=(\partial K\cap\partial K')^\circ$. Furthermore, let $v$
and~$\vect{w}$ be scalar- and vector-valued functions, respectively,
that are sufficiently smooth inside each
element~$K,K'$. Then, the averages of $v$ and
$\vect{w}$ at $\vect{x}\in f$ are given by
\[
\avg{v}=\frac{1}{2}(v|_{K}+v|_{K'}), \qquad \avg{\vect{w}}
=\frac{1}{2}(\vect{w}|_{K}+\vect{w}|_{K'}),
\]
respectively. Similarly, the jumps of $v$ and $\vect{w}$ at $\vect{x}\in f$ are given by
\[
\jmp{v} =v|_{K}\,\vect{n}_{K}+v|_{K'}\,\vect{n}_{K'},\qquad
\jmp{\vect{w}}=\vect{w}|_{K}\cdot\vect{n}_{K}+\vect{w}|_{K'}\cdot\vect{n}_{K'},
\]
respectively.
Given $K\in\mesh$ the local Trefftz space is defined by
\[
T(K) \coloneqq \{ v\in H^1(K) : -\Delta v - k^2 v = 0 \};
\]
with this notation, we write
\[
T(\mathcal{T}_h) \coloneqq \{ v\in L^2(\Omega) : v\vert_K \in T(K), K\in\mesh\}.
\]
Thereby, given a local space $V_{p_K}(K) \subset T(K)$, of finite dimension $p_K\geq 1$,
the corresponding TDG finite element space is defined by
\[
V_{\vect{p}}(\mathcal{T}_h) \coloneqq \{ v \in T(\mesh) : v\vert_K \in V_{p_K}(K), K\in\mesh \},
\]
where $\vect{p} = \{p_K:K\in\mesh\}$.
\subsection{TDG discretization}
Equipped with the TDG finite element space $V_{\vect{p}}(\mesh)$ defined on the mesh partition
$\mesh$ of $\Omega$, the TDG approximation of~\eqref{eqn:helmholtz} is given by: find $u_{hp}\in V_{\vect{p}}(\mesh)$ such that
\begin{equation}\label{eqn:bilinear_form}
\mathcal{A}_{h}(u_{hp},v_{hp})=\ell_{h}(v_{hp})
\end{equation}
for all $v_{hp}\in V_{\vect{p}}(\mesh)$, where
\begin{align*}
\mathcal{A}_{h}(u,v)=
& \int_{\face[I]} \left( \avg{u}\jmp{\nabla_h \conj{v}}
-\beta(ik)^{-1} \jmp{\nabla_h u}\jmp{\nabla_h \conj{v}}
-\avg{\nabla_h u}\cdot\jmp{\conj{v}}
+\alpha ik \jmp{u}\cdot\jmp{\conj{v}} \right)\, \mathrm{d}s\\
&+\int_{\Gamma_R}( (1-\delta)(u\nabla_h \conj{v}\cdot\vect{n}+ik\vartheta u\conj{v})
-\delta((ik\vartheta)^{-1}(\nabla_h u\cdot\vect{n})(\nabla_h \conj{v}\cdot\vect{n})
+\nabla_h u\cdot\vect{n}\,\conj{v}))\, \mathrm{d}s \\
&+\int_{\Gamma_D}(-\nabla_h u\cdot\vect{n}\,\conj{v}+ \alpha ik u\conj{v})\, \mathrm{d}s, \\
\ell_{h}(v)=&
\int_{\Gamma_R}g_R((1-\delta)\conj{v}
-\delta(ik\vartheta)^{-1}\nabla_h \conj{v}\cdot\vect{n})\, \mathrm{d}s
+\int_{\Gamma_D}g_D(\alpha ik \conj{v} -\nabla_h \conj{v}\cdot\vect{n})\, \mathrm{d}s,
\end{align*}
and $\nabla_h$ denotes the broken gradient operator, defined elementwise. Here,
$\alpha>0$, $\beta>0$ and $0<\delta\leq \nicefrac12$ are given penalty parameters.
We note that the selection of these penalty parameters has been studied in a number
of different contexts within the literature; in particular, here we mention
the ultra-weak variational formulation (UWVF), cf. \cite{Cessenat1998}, the DG-type scheme studied
in~\cite{Gittelson2009}, and \cite{Hiptmair2014} which considered their selection
on locally refined meshes; cf.~\cite[Table 1]{Hiptmair2016}. For the purposes of
this article we consider the UWVF, corresponding to the choice
$\alpha=\beta=\delta=\nicefrac{1}{2}$.
\subsection{Plane wave basis functions}
Finally, in this section we outline the choice of the underlying discrete space
$V_{p_K}(K)$, $K\in\mesh$. To this end, we select $V_{p_K}(K)$ to be
a local space consisting of plane waves in $p_K$ different directions,
all with the same wavenumber $k$. We note that, under suitable assumptions on $K$
and the choice of plane wave directions, $V_{p_K}(K)$ approximates smooth Trefftz functions
with the same order of convergence as polynomials of degree $q_K$, where
\begin{equation}
p_K =
\begin{cases}
2q_K + 1, & d=2, \\
(q_K+1)^2, & d= 3;
\end{cases}
\label{eqn:effective_poly_deg}
\end{equation}
see~\cite{Moiola2011}. Thereby, $q_{K}$ is referred to as the \emph{effective polynomial degree}
of the discrete Trefftz space; we set $\vect{q} = \{q_K:K\in\mesh\}$.
More precisely, we write
\begin{equation}\label{eqn:pw_basis}
V_{p_K}(K) \coloneqq \left\{ v \in T(K) : v(\vect{x}) = \sum_{\ell=0}^{p_K-1} \alpha_\ell
\mathrm{e}^{ik\vect{d}_{K,\ell}\cdot(\vect{x}-\vect{x}_K)}, \alpha_\ell\in\mathbb{C}\right\},
\end{equation}
where $\vect{x}_K$ is the center of mass of element $K$ and $\vect{d}_{K,\ell}$, $\ell=0,\dots,p_K-1$,
are $p_K$ evenly distributed unit direction vectors (with respect to the unit ball).
For $d=2$ we can simply define
\begin{equation}
\label{eqn:plane_wave:2d}
\vect{d}_{K,\ell} = (\cos(\nicefrac{2\pi\ell}{p_K}), \sin(\nicefrac{2\pi\ell}{p_K}))^\top, \qquad
\ell=0,\dots,p_K-1;
\end{equation}
for $d=3$ we employ the directions determined by the extremal (maximum determinant) points
on $S^2$, cf. \cite{Sloan2004,Womersley2007Online}.
\section{Adaptive mesh refinement}\label{section:adaptive_refinement}
In this section we develop an automatic adaptive refinement algorithm which is
capable of not only marking elements for refinement, but also determining the type
of refinement to be undertaken. In particular, here we consider both $h$-- and $p$--refinement,
whereby the local element is subdivided, or the number of elementwise plane wave directions is
enriched, respectively, as well as
directional refinement which seeks to rotate the local plane wave basis
in order to align it with the principal scattering direction.
\subsection{\emph{A posteriori} error indicator}\label{sec:indicator}
In the absence of rigorous {\em a posteriori} error bounds for the numerical approximation of
\eqref{eqn:helmholtz} by the TDG scheme \eqref{eqn:bilinear_form}, which are sharp
with respect to both the local mesh size $h_K$ and the number of local plane waves $p_K$
employed on each element $K\in\mesh$, we employ an empirical error estimator stimulated
by the work undertaken in \cite{Kapita2015} in the $h$--version setting. To this end,
we first introduce the dual problem: find $z\in H^1(\Omega)$, such that
\begin{equation*}
\begin{aligned}
-\Delta z-k^2 z &=u-u_{hp} && \text{in } \Omega\;,\\
\frac{\partial z}{\partial \vect{n}} + ik\vartheta z &= 0 && \text{on } \Gamma_R\;, \\
z &=0 && \text{on } \Gamma_D\;.
\end{aligned}
\end{equation*}
Noting that $z\in H^{\nicefrac{2}{3}+s}(\Omega)$, $0<s\leq\nicefrac{1}{2}$,
cf.~\cite{Hiptmair2014}, we recall the following (second) {\em a posteriori}
error bound from \cite{Kapita2015}.
\begin{theorem} \label{thm:apost}
Assume that the mesh $\mesh$ is shape-regular, locally quasi-uniform, in the
sense that, for two elements $K$ and $K'$ which share a face $f\subset\face[I]$, there
is a constant $\tau$, independent of $h$, such that
$$
\tau^{-1} \leq \nicefrac{h_K}{h_{K'}} \leq \tau
$$
for all choices of $K$ and $K'$, and that $\mesh$ is quasi-uniform in the vicinity
of $\Gamma_R$, i.e., for all $K\in\mesh$ which lie on the boundary $\Gamma_R$, i.e.,
so that $\partial K\cap \Gamma_R \neq \emptyset$, there exists $\tau_R$ such that
$$
\nicefrac{h}{h_K} \leq \tau_R.
$$
Then, for $g_D\equiv 0$ and fixed $p_K$, $K\in\mesh$, the following {\em a posteriori} bound holds:
\begin{equation*}
\norm{u-u_{hp}}_{L^2(\Omega)} \leq {\mathfrak E}(u_{hp},h) \equiv C \left(\sum_{K\in\mesh} \eta_K^2 \right)^{\nicefrac{1}{2}},
\end{equation*}
where
\begin{equation}
\begin{aligned}
\eta_K^2 &=\norm*{\alpha^{\nicefrac{1}{2}} h_K^{s} \jmp{u_{hp}}}_{L^2(\partial K \setminus \partial\Omega)}^2
+k^{-2}\norm*{\beta^{\nicefrac12} h_K^{s} \jmp{\nabla u_{hp}}}_{L^2(\partial K \setminus \partial\Omega )}^2 \\
&\quad+ k^{-2}\norm*{\delta^{\nicefrac{1}{2}} h_K^{s} \left( g_R - \nabla u_{hp}\cdot\vect{n}_K + ik u_{hp}\right)}_{L^2(\partial K \cap \Gamma_R)}^2
+\norm*{\alpha^{\nicefrac{1}{2}} h_K^{s} u_{hp}}_{L^2(\partial K \cap \Gamma_D)}^2,
\end{aligned}
\label{eqn:error_indicator_kmw}
\end{equation}
where $C$ is a positive constant, which is independent of $h$.
\end{theorem}
We stress that the {\em a posteriori} error bound stated in Theorem~\ref{thm:apost}
depends on the regularity index $s$; thereby, {\em a priori} knowledge of $s$ is
required in order to yield a fully computable bound. Moreover, the dependence of
$C$ on $\vect{p}$, or equivalently $\vect{q}$, is unclear; indeed, to the best of our
knowledge, an $hp$--version
generalization of Theorem~\ref{thm:apost} is not currently available within the literature.
Thereby, we propose the following {\em empirical} error estimator, where for simplicity
of notation we also denote it by ${\mathfrak E}$, for
the $hp$--version TDG method:
\begin{equation}
\label{eqn:error_bound}
{\mathfrak E}(u_h,h,\vect{p}) = \left( \sum_{K\in\mesh} \eta_K^2\right)^2,
\end{equation}
where
\begin{equation}
\begin{aligned}
\eta_K^2 &=\norm*{\alpha^{\nicefrac{1}{2}} h_K^{\nicefrac12}q_K^{-\nicefrac12} \jmp{u_{hp}}}_{L^2(\partial K \setminus \partial\Omega )}^2
+\norm*{\beta^{\nicefrac12}
h_K^{\nicefrac32}q_K^{-\nicefrac32} \jmp{\nabla
u_{hp}}}_{L^2(\partial K \setminus \partial\Omega)}^2 \\
&\quad+ \norm*{\delta^{\nicefrac{1}{2}} h_K^{\nicefrac32}q_K^{-\nicefrac32} \left( g_R - \nabla u_{hp}\cdot\vect{n}_F + ik u_{hp}\right)}_{L^2(\partial K \cap \Gamma_R)}^2
+\norm*{\alpha^{\nicefrac{1}{2}} h_K^{\nicefrac12}q_K^{-\nicefrac12} (g_D-u_{hp})}_{L^2(\partial K \cap \Gamma_D)}^2.
\label{eqn:error_indicator}
\end{aligned}
\end{equation}
We stress that the choice of the exponents of $h_K$ and $q_K$ have been selected
on the basis of numerical experimentation on a problem with a smooth analytical
solution; for details, see Section~\ref{section:aposteriori_effectivity} below.
Compared to the error indicator \eqref{eqn:error_indicator_kmw} from \cite{Kapita2015}, we note that
we have a factor of $h$ instead of $k^{-1}$ in the terms with $\jmp{\nabla u_{hp}}$ and in the Robin boundary terms.
These are both dimensionally correct, but reproducing the numerical experiments conducted in Section~\ref{section:aposteriori_effectivity} with the dependency on $k^{-1}$ results in different effectivities for different wavenumbers $k$.
\subsection{Plane wave directional adaptivity}\label{sec:direction_adapt}
In this section, we discuss the design of a practical algorithm for determining
the direction vectors $\vect{d}_{K,\ell}$, $\ell=0,\dots,p_K-1$, used to define the
plane wave basis within each element
$K$ in the computational mesh $\mesh$. The key observation is that, many wave
propagation problems typically
exhibit a dominant direction of propagation of the underlying wave
within each element in $\mesh$. Thereby, by aligning
the plane wave basis in an appropriate fashion, we expect to attain a significant
reduction of the error in the computed TDG solution. Indeed,
in the simple case when the analytical solution is a plane wave, then if
the direction for one of the plane wave basis functions is selected such that it is aligned
with this plane wave direction,
then the TDG method will exactly recover the analytical solution, subject to rounding errors.
The essential idea here is to simply rotate the element basis according to the
predicted elementwise dominant direction. For simplicity of presentation, let us
consider the two-dimensional case, i.e., $d=2$; we note that $d=3$ follows in
an analogous manner, cf.~Remarks~\ref{remark:potential_direction} \& \ref{remark:direction} below.
In two-dimensions, the standard plane wave directions are generally
selected to be evenly spaced, with the first direction $\vect{d}_{K,0} = (1,0)^\top$ always
pointing along the $x$-axis, cf.~\eqref{eqn:plane_wave:2d} (in the
three-dimensional setting, the first direction vector typically points along the $z$-axis).
Alternatively, assuming that a dominant elementwise direction, denoted by $\vect{\frak{d}}_K$, can be
determined within each $K\in\mesh$, then the direction vectors for the plane wave basis functions
in $K$ are chosen such that the first plane wave direction is aligned with $\vect{\frak{d}}_K$,
i.e.,~\eqref{eqn:plane_wave:2d} is replaced by
\begin{equation}
\label{eqn:plane_wave:2d:rotated}
\vect{d}_{K,\ell} = (\cos(\nicefrac{2\pi\ell}{p_K} + \theta_K), \sin(\nicefrac{2\pi\ell}{p_K} + \theta_K))^\top,
\end{equation}
$\ell=0,\ldots,p_K-1$,
where $\theta_K$ is the angle between $\vect{\frak{d}}_K$ and the $x$-axis.
Clearly, in general, the dominant elementwise direction $\vect{\frak{d}}_K$, $K\in\mesh$, cannot
be determined {\em a priori}, but instead must be numerically estimated as part of the
solution process. In this regard, a number of algorithms have been proposed
within the literature; here we mention the ray-tracing approach developed
in~\cite{Betcke2011,Betcke2012}, though this
includes terms involving integrals over the elements within the underlying TDG formulation. In
\cite{Amara2014}, the optimal angle of rotation was numerically estimated based on
adding an extra unknown into the problem; however, this leads to a system of nonlinear
equations to be computed. Finally, \cite{Gittelson2008} uses an approximation of
\[
\frac{\nabla e(\vect{x}_0)}{ike(\vect{x}_0)},
\]
at a given point $\vect{x}_0\in K$, $K\in\mesh$, where $e$ denotes the error.
Stimulated by the work undertaken in~\cite{Formaggia2001,Formaggia2003}, cf.
also~\cite{ghh-paper,hall-thesis}, on the design of anisotropically refined
computational meshes, in this section we compute an estimate of $\vect{\frak{d}}_K$, $K\in\mesh$,
based on the properties of the Hessian of the TDG solution $u_{hp}$. Indeed,
we note that the principal eigenvector, i.e., the eigenvector corresponding
to the largest eigenvalue in absolute value, of the Hessian of a given function
indicates the direction of most rapid variation, and thereby, in our context,
the dominant direction of wave propagation. With this in mind, writing
$\boldsymbol{\mathcal{H}}(\varphi,\vect{x}_0)$ to denote the Hessian matrix of a given
function $\varphi$, evaluated at the point $\vect{x}_0\in\mathbb{R}^d$, in
Algorithm~\ref{algo:potential_direction} we outline the steps involved
in computing a {\em potential} dominant plane wave direction
$\hat{\vect{\frak{d}}}_K$ for a given element $K\in\mesh$.
Table~\ref{table:potential_direction} summarizes how this potential first plane wave
direction $\hat{\vect{\frak{d}}}_K$ is selected; for the numerical experiments presented in
Section~\ref{sec:numerical_examples}, we set $\Lambda =2$.
We note that in the case when no primary propagation direction
is determined, then we leave the first plane wave direction unchanged.
\begin{algorithm}[t!]
\caption{Computation of the potential first plane wave direction $\hat{\vect{\frak{d}}}_K$ for element $K$.}
\label{algo:potential_direction}
\begin{algorithmic}[1]
\State Input: the TDG solution $u_{hp}$ of the discrete problem~\eqref{eqn:bilinear_form} and the parameter $\Lambda>1$.
\State Writing $\vect{x}_K$ to denote the centroid of $K$, $K\in\mesh$, evaluate the eigenpairs $(\lambda_1, \vect{v}_1), (\lambda_2, \vect{v}_2)$ of $\boldsymbol{\mathcal{H}}(\Real(u_{hp}\vert_K),\vect{x}_K)$, and $(\mu_1, \vect{w}_1), (\mu_2, \vect{w}_2)$ of $\boldsymbol{\mathcal{H}}(\Imag(u_{hp}\vert_K),\vect{x}_K)$, such that $\abs{\lambda_1} \geq \abs{\lambda_2}$ and $\abs{\mu_1} \geq \abs{\mu_2}$.
\If{$\abs{\lambda_1} \geq \Lambda \abs{\lambda_2}$}
\If{$\abs{\mu_1} \geq \Lambda \abs{\mu_2}$}
\If{$\abs{\lambda_1} \geq \Lambda \abs{\mu_1}$}
\State $\hat{\vect{\frak{d}}}_K \gets \vect{v}_1$
\ElsIf{$\abs{\mu_1} \geq \Lambda \abs{\lambda_1}$}
\State $\hat{\vect{\frak{d}}}_K \gets \vect{w}_1$
\Else
\State $\hat{\vect{\frak{d}}}_K \gets \frac{\vect{v}_1+\vect{w}_1}{\norm{\vect{v}_1+\vect{w}_1}}$
\EndIf
\Else
\If{$\abs{\lambda_1} \geq \Lambda \abs{\mu_1}$}
\State $\hat{\vect{\frak{d}}}_K \gets \vect{v}_1$
\Else
\State No primary propagation direction
\EndIf
\EndIf
\Else
\If{$\abs{\mu_1} \geq \Lambda \abs{\mu_2}$}
\If{$\abs{\mu_1} \geq \Lambda \abs{\lambda_1}$}
\State $\hat{\vect{\frak{d}}}_K \gets \vect{w}_1$
\Else
\State No primary propagation direction
\EndIf
\Else
\State No primary propagation direction
\EndIf
\EndIf
\end{algorithmic}
\end{algorithm}
\begin{remark}\label{remark:potential_direction}
We note that in the case when $d=3$, $\boldsymbol{\mathcal{H}}(\Real(u_{hp}\vert_K),\vect{x}_K)$
and $\boldsymbol{\mathcal{H}}(\Imag(u_{hp}\vert_K),\vect{x}_K)$ each have a third eigenpair, $(\lambda_3,\vect{v}_3)$
and $(\mu_3, \vect{w}_3)$, respectively.
However, if the eigenpairs are sorted such that
$\abs{\lambda_1} \geq \abs{\lambda_2} \geq \abs{\lambda_3}$ and
$\abs{\mu_1} \geq \abs{\mu_2} \geq \abs{\mu_3}$, the third eigenpairs
\emph{never} represent a dominant direction, and thereby
Algorithm~\ref{algo:potential_direction} can be used to identify
$\hat{\vect{\frak{d}}}_K$, $K\in\mesh$, without modification.
\end{remark}
\begin{table}[t!]
\centering
\begin{tabular}{c|c|c|c||c}
$\abs{\lambda_1} \geq C \abs{\lambda_2}$ & $\abs{\mu_1} \geq C\abs{\mu_2}$ & $\abs{\lambda_1} \geq C\abs{\mu_1}$ & $\abs{\mu_1}\geq C\abs{\lambda_1}$ & First Plane Wave $\hat{\vect{\frak{d}}}_K$ \\\hline\hline
\ding{51} & \ding{51} & \ding{51} & \ding{55} & $\vect{v}_1$ \\
\ding{51} & \ding{51} & \ding{55} & \ding{51} & $\vect{w}_1$ \\
\ding{51} & \ding{51} & \ding{55} & \ding{55} & $\frac{(\vect{v}_1+\vect{w}_1)}{\norm{\vect{v}_1+\vect{w}_1}}$ \\
\ding{51} & \ding{55} & \ding{51} & \ding{55} & $\vect{v}_1$ \\
\ding{51} & \ding{55} & \ding{55} & --- & --- \\
\ding{55} & \ding{51} & \ding{55} & \ding{51} & $\vect{w}_1$ \\
\ding{55} & \ding{51} & --- & \ding{55} & --- \\
\ding{55} & \ding{55} & --- & --- & ---
\end{tabular}
\caption{Summary of selection of first plane wave direction $\hat{\vect{\frak{d}}}_K$ using Algorithm~\ref{algo:potential_direction}.}
\label{table:potential_direction}
\end{table}
Noting that eigenvectors are only unique up to scalar multiples,
the vector $\hat{\vect{\frak{d}}}_K$, $K\in\mesh$, evaluated according to
Algorithm~\ref{algo:potential_direction} may be pointing in precisely the
opposite direction to the primary wave propagation direction. Thereby,
to ensure that $\hat{\vect{\frak{d}}}_K$, $K\in\mesh$, is correctly oriented,
we study the impedance trace on the boundary of a ball $B_\delta(\vect{x}_K)$
of radius $\delta$, centered at $\vect{x}_K$, of both the numerical solution and
a plane wave with (the desired) propagation direction $\vect{\frak{d}}_K$. As we let
$\delta\to 0$, we expect that the numerical solution should be closely approximated by
the plane wave in the primary propagation direction.
More precisely, given $K\in\mesh$, the impedance trace of the plane wave
\[
\tilde{u}_K(\vect{x}) = \mathrm{e}^{ik\vect{\frak{d}}_K \cdot(\vect{x}-\vect{x}_K)}
\]
on $\partial B_\delta(\vect{x}_K)$ is given by
\begin{eqnarray}
(\nabla\tilde{u}_K(\vect{x})\cdot\vect{n}_{B_\delta} + ik\tilde{u}_K(\vect{x}))
|_{\partial B_\delta(\vect{x}_K)}
&=& (ik(\vect{\frak{d}}_K \cdot\vect{n}_{B_\delta}+1)\,\mathrm{e}^{ik \vect{\frak{d}}_K \cdot(\vect{x}-\vect{x}_K)}
)|_{\partial B_\delta(\vect{x}_K)},
\label{eqn:impedance_trace}
\end{eqnarray}
where $\vect{n}_{B_\delta}$ denotes the unit outward normal vector on
$\partial B_\delta(\vect{x}_K)$.
Setting $\vect{x}=\vect{x}_K + \delta \hat{\vect{\frak{d}}}_K$ in \eqref{eqn:impedance_trace}
and noting that, at this point of evaluation, $\vect{n}_{B_\delta} = \hat{\vect{\frak{d}}}_K$,
we deduce that
\[
\frac{\nabla\tilde{u}_K(\vect{x}_K + \delta\hat{\vect{\frak{d}}}_K )\cdot\vect{n}_{B_\delta}
+ ik\tilde{u}_K(\vect{x}_K + \delta \hat{\vect{\frak{d}}}_K )}{ik} =
\begin{cases}
2 \mathrm{e}^{ik\delta}, & \text{if } \hat{\vect{\frak{d}}}_K = \vect{\frak{d}}_K , \\
0, & \text{if } \hat{\vect{\frak{d}}}_K = -\vect{\frak{d}}_K .
\end{cases}
\]
Thereby, the (potential) dominate direction of propagation $\hat{\vect{\frak{d}}}_K$, $K\in\mesh$, predicted
according to Algorithm~\ref{algo:potential_direction} may be corrected to yield
the dominant direction $\vect{\frak{d}}_K$ on the basis of Algorithm~\ref{algo:direction};
this direction will then be selected as the first plane wave direction on element $K$,
$K\in\mesh$. For simplicity, throughout this article we set $\delta=0$.
\begin{algorithm}[t!]
\caption{Evaluation of the first plane wave direction $\vect{\frak{d}}_K$ for element $K$.}
\label{algo:direction}
\begin{algorithmic}[1]
\State Input: the TDG solution $u_{hp}$ of the discrete problem~\eqref{eqn:bilinear_form}, the parameter $0\leq \delta\to 0$, and $\hat{\vect{\frak{d}}}_K$ computed by Algorithm~\ref{algo:potential_direction}.
\State The first plane wave direction $\vect{\frak{d}}_K$ on element $K$, $K\in\mesh$, is given by
\[
\vect{\frak{d}}_K = \begin{cases}
-\hat{\vect{\frak{d}}}_K , & ~\mbox{ if } ~\Real\left(\frac{\nabla u_{hp}(\vect{x}_K + \delta\hat{\vect{\frak{d}}}_K)\cdot\hat{\vect{\frak{d}}}_K + ik u_{hp}(\vect{x}_K + \delta\hat{\vect{\frak{d}}}_K)}{ik}\right) < \mathrm{e}^{ik\delta}, \\
\hat{\vect{\frak{d}}}_K , & ~\mbox{ if } ~\Real\left(\frac{\nabla u_{hp}(\vect{x}_K + \delta\hat{\vect{\frak{d}}}_K )\cdot\hat{\vect{\frak{d}}}_K + ik u_{hp}(\vect{x}_K + \delta\hat{\vect{\frak{d}}}_K)}{ik}\right) \geq \mathrm{e}^{ik\delta}. \\
\end{cases}
\]
\end{algorithmic}
\end{algorithm}
\begin{remark}\label{remark:direction}
In the three--dimensional setting, once the selection of the primary wave propagation direction
$\vect{\frak{d}}_K$ has been computed on the basis of
Algorithms~\ref{algo:potential_direction} \&~\ref{algo:direction},
we then select the remaining wave directions,
$\vect{d}_{K,\ell}$, $\ell=1,\dots,p_K-1$, by applying a transformation matrix
$T\in\mathbb{R}^{3\times 3}$ to the original `reference' directions $\vect{\tilde{d}}_{K,\ell}$,
$\ell=1,\dots,p_K-1$, respectively, where $\vect{\tilde{d}}_{K,0}$ points along the
$z$-axis, cf. above. Thereby,
$$
\vect{d}_{K,\ell}=T \vect{\tilde{d}}_{K,\ell} ,
$$
$\ell=1,\dots,p_K-1$, where $T$ is selected such that
\[
\vect{\frak{d}}_K \equiv \vect{d}_{K,0} = T \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}
\equiv T \vect{\tilde{d}}_{K,0}.
\]
We note that the selection of $T$ is not unique; writing
$\vect{\frak{d}}_K = (d_x, d_y, d_z)^\top$, we
define $T$ to be the identity matrix if $d_x=d_y=0$; otherwise, we set
\[
T = \begin{pmatrix}
\frac{d_x d_z}{\sqrt{d_x^2+d_y^2}} & \frac{d_y}{\sqrt{d_x^2+d_y^2}} & d_x \\
\frac{d_y d_z}{\sqrt{d_x^2+d_y^2}} & -\frac{d_x}{\sqrt{d_x^2+d_y^2}} & d_y \\
-\sqrt{d_x^2+d_y^2} & 0 & d_z
\end{pmatrix}.
\]
\end{remark}
\subsection{$hp$--Adaptive mesh refinement}\label{sec:hp}
In this section we discuss the design of an automatic algorithm for generating
sequences of $hp$--adaptively refined TDG finite element spaces in an efficient
manner. This topic has been extensively studied within the finite element
element literature in the case when the local element spaces consist of
polynomial functions; for a comprehensive review, we refer to~\cite{Mitchell2011, Mitchell2014}.
In general, the key underlying principle of most $hp$--refinement strategies is
to employ local mesh subdivision ($h$--refinement) in regions where the solution is
not smooth, while local enrichment of the finite element space ($p$--refinement)
is undertaken elsewhere. Given that such regularity information is generally
unknown {\em a priori}, several strategies have been developed to {\em a posteriori} estimate
the local smoothness of the analytical solution, based on its numerical approximation;
cf. \cite{Houston2005}, for example. However, in the context of TDG schemes for
the numerical approximation of high-frequency time-harmonic wave problems, the extraction
of such regularity information is expected to be unreliable due to
the oscillatory nature of the computed numerical solution.
Thereby, as an alternative to directly estimating local smoothness of the solution,
we employ the {\em a posteriori} error indicator~\eqref{eqn:error_indicator}
to select the type of refinement to be undertaken on the basis of the refinement
history of the current element, cf.~\cite{Melenk2001}. More precisely, following
\cite{Melenk2001} refinements are selected based on checking if the local error estimate has
decayed according to the expected rate of convergence based on the last type of refinement
employed. If the expected rate of convergence is achieved,
then $p$--refinement is performed; otherwise,
$h$--refinement is undertaken. The variant of \cite[Algorithm 4.4]{Melenk2001} we employ here
is summarized in Algorithm~\ref{algo:refinement}.
Here, we note that $\gamma_h$, $\gamma_p$,
and $\gamma_n$ are control parameters;
for the purposes of this article, we select $\gamma_h=4$, $\gamma_p=0.4$, and $\gamma_n=1$.
Furthermore, the number of child elements, $N$, cf. step 10. in Algorithm~\ref{algo:refinement},
is dependent on the type of subdivision, i.e., isotropic/anisotropic, undertaken,
as well as the element shape; for isotropic refinement of tensor-product elements, we have
that $N=2^d$.
\begin{algorithm}[t!]
\begin{algorithmic}[1]
\caption{$hp$--Adaptive refinement algorithm.}\label{algo:refinement}
\State Input the control parameters $\gamma_h$, $\gamma_p$, and $\gamma_n$.
\State{Choose a coarse initial mesh~$\mesh[h,0]$ of~$\Omega$ and a corresponding low-order starting (effective) polynomial degree vector~$\vect{q}_0$, together with the total dimension vector $\vect{p}_0$ defined as in \eqref{eqn:effective_poly_deg}.}
\State{Set the initial predicted error indicator $\eta_{K,0}^{\mathrm{pred}}=\infty$ for all $K\in\mesh[h,0]$.}
\For{$i=0,1,\ldots,$ until sufficiently many iterations have been performed.}
\State{Solve \eqref{eqn:bilinear_form} for $u_{hp} \in V_{\vect{p}_i}(\mesh[h,i])$.}
\State{Compute the {\em a posteriori} error indicators $\eta_{K,i}\equiv \eta_K$,
$K\in \mesh[h,i]$, and mark elements for refinement based on their relative magnitude.}
\For{$K\in\mesh[h,i]$}
\If{$K$ is marked for refinement}
\If{$\eta_{K,i} > \eta_{K,i}^{\mathrm{pred}}$}
\State Perform $h$--refinement: Subdivide $K$ into $N$ children $K_s, s = 1,\dots,N$, and set
\State $(\eta_{K_s,i+1}^{\mathrm{pred}})^2\gets \frac{1}{N}\gamma_h \left(\frac{1}{2}\right)^{2q_K} \eta_{K,i}^2$, $1\leq s \leq N$.
\Else
\State Perform $p$--refinement: $q_K \gets q_K+1$
\State $(\eta_{K,i+1}^{\mathrm{pred}})^2\gets \gamma_p \eta_{K,i}^2$
\EndIf
\Else
\State $(\eta_{K,i+1}^{\mathrm{pred}})^2\gets \gamma_n (\eta_{K,i}^{\mathrm{pred}})^2$
\EndIf
\EndFor
\State Construct the new mesh $\mesh[h,i+1]$ and corresponding Trefftz space $V_{\vect{p}_{i+1}}(\mesh[h,i+1])$.
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{remark}
We note that in \cite{Melenk2001} the initial values of the predicted error indicator
$\eta_{K,0}^{\mathrm{pred}}$, $K\in\mesh[h,0]$, are set to zero; thereby, this ensures that
$h$--refinement is undertaken the first time an element is refined. In contrast, in
Algorithm~\ref{algo:refinement} we set $\eta_{K,0}^{\mathrm{pred}}=\infty$ for all
$K\in\mesh[h,0]$ which instead leads to $p$--enrichment being undertaken as the first refinement
of a given element, since the TDG method for the numerical approximation of
the Helmholtz equation is intrinsically a high-order method.
\end{remark}
\begin{remark}
Plane wave directional adaptivity can be performed at different stages within
Algorithm~\ref{algo:refinement}; for example, the following options are available:
\begin{itemize}
\item undertake directional adaptivity only on elements marked for $p$--refinement,
\item undertake directional adaptivity on all elements marked for refinement, with $h$--refinement performed after plane wave direction adaptivity, or
\item undertake directional adaptivity on every element $K\in\mesh$, even if the element $K$ has not been marked for refinement.
\end{itemize}
In Section~\ref{sec:numerical_examples} we shall numerically investigate each of these approaches in order to assess their relative computational performance in terms of error reduction.
\end{remark}
\begin{remark}
As a final remark, we note that within Algorithm~\ref{algo:refinement} we employ the
fixed fraction refinement strategy to select elements for refinement, cf. step 6; throughout
this article we set the refinement fraction equal to $25\%$.
\end{remark}
\section{Numerical experiments} \label{sec:numerical_examples}
In this section, we present a series of numerical experiments to highlight the
practical performance of the $hp$--refinement algorithm, with directional adaptivity,
proposed in Algorithm~\ref{algo:refinement}. Throughout this section we shall compare
the performance of the proposed $hp$--adaptive refinement strategy with the corresponding
algorithm based on exploiting only local mesh subdivision, i.e., $h$--refinement.
The numerical experiments presented within this section have been undertaken using the
AptoFEM software package~\cite{aptofem}.
\subsection{Plane wave direction adaptivity}\label{section:plane_wave_refine}
\begin{table}[pt]
\centering
\begin{tabular}{c|c|r@{.}l@{$\times$}l|r@{.}l@{$\times$}l|c}
& & \multicolumn6{c}{Relative $L^2(\Omega)$-Error} & \\
$q$ & No of Dofs & \multicolumn3{c|}{Standard TDG} & \multicolumn3c{Direction Adaptivity}
& \% Reduction\\ \hline
3 & 112 & \quad 2&015 & $10^{0}$ & \quad 1&959 & $10^{0}$ & 2.7\% \\
4 & 144 & 5&027 & $10^{-1}$ & 3&194 & $10^{-1}$ & 36.5\% \\
5 & 176 & 7&414 & $10^{-2}$ & 2&658 & $10^{-2}$ & 64.1\% \\
6 & 208 & 1&616 & $10^{-2}$ & 6&320 & $10^{-3}$ & 60.9\% \\
7 & 240 & 3&420 & $10^{-3}$ & 1&435 & $10^{-3}$ & 58.0\% \\
8 & 272 & 5&154 & $10^{-4}$ & 3&011 & $10^{-4}$ & 41.6\% \\
9 & 304 & 8&928 & $10^{-5}$ & 6&908 & $10^{-5}$ & 22.6\% \\
\end{tabular}
\caption{Plane Wave Refinement: Comparison of the relative $L^2$-error for uniform $p$--refinement (without direction adaptivity), and $p$--refinement with direction adaptivity (Algorithm~\ref{algo:direction}).}
\label{table:pw_adapt}
\end{table}
\begin{figure}[pt]
\captionsetup[subfloat]{farskip=0pt,captionskip=5pt}\centering
\subfloat[]{\label{fig:directions:p3}\includegraphics[width=0.4\textwidth]{plane_wave_adaption_directions_p2}}
\subfloat[]{\label{fig:directions:p4}\includegraphics[width=0.4\textwidth]{plane_wave_adaption_directions_p3}} \\
\subfloat[]{\label{fig:directions:p5}\includegraphics[width=0.4\textwidth]{plane_wave_adaption_directions_p4}}
\subfloat[]{\label{fig:directions:p6}\includegraphics[width=0.4\textwidth]{plane_wave_adaption_directions_p5}}
\caption{Plane Wave Refinement: Plane wave directions of \protect\subref{fig:eff:20} initial mesh and after \protect\subref{fig:eff:30} $1$, \protect\subref{fig:eff:40} $2$ and \protect\subref{fig:eff:50} $3$ $p$--refinements with plane wave refinement (Algorithm~\ref{algo:direction})}
\label{fig:directions}
\end{figure}
In this first example, we study the effect of adjusting the plane wave directions while employing a fixed computational mesh with uniform $p$--refinement. To this end, we consider problem~\eqref{eqn:helmholtz} with $\Omega=(0,1)^2$, $\Gamma_R=\partial\Omega$, and $\Gamma_D\equiv\emptyset$; furthermore, the Robin boundary condition $g_R$ is selected such that the analytical solution $u$ of~\eqref{eqn:helmholtz} is given by
\begin{equation}
\label{eqn:hankal_anal}
u(x,y) = \mathcal{H}_0^{(1)} \left( k \sqrt{ (x+\nicefrac14)^2+y^2 } \right),
\end{equation}
where $\mathcal{H}_0^{(1)}$ denotes the Hankel function of the first kind of order 0. Throughout this section, we set $k=20$; note that for this problem the analytical solution $u$ is smooth in $\Omega$.
Here, the underlying computational mesh consists of 16 uniform square elements; on each element we
initially select the effective polynomial degree $q=2$, i.e., $p=5$. In Table~\ref{table:pw_adapt} we
compare the computed relative $L^2$-error based on employing uniform $p$--refinement of the underlying TDG
space $V_{\vect{p}}(\mesh)$ in the two cases when the standard TDG scheme is employed, i.e., when the local
plane wave directions are kept fixed, and when plane wave directional adaptivity is utilised, based on exploiting
Algorithm~\ref{algo:direction} (direction adaptivity). We note that, since uniform $p$--refinement
is employed in both cases, then at each step of the refinement, both schemes possess the same
number of degrees of freedom. At each step of the refinement algorithm, we observe that the
exploitation of directional adaptivity leads to roughly 50\% reduction in the relative
$L^2$-error when compared to the corresponding quantity computed for the standard TDG
method (without direction adaptivity). We note, however, that in the case when $q=3$,
the relative $L^2$-error is only reduced by a small amount when directional adaptivity is employed; this
is due to the fact that the local plane wave directions are computed based on the numerical solution evaluated with $q=2$,
which is numerically too inaccurate to reliably predict the correct local direction of wave propagation.
Furthermore, we also note that, as the number of plane waves increases, the improvement in the
relative $L^2$-error decreases; this is caused by the fact that, as the number of plane waves
increases for the standard TDG scheme, one of the directions will get closer to the actual
dominant direction.
In Figure~\ref{fig:directions} we plot, for each element, the initial plane wave directions
and the plane wave directions computed after $1$, $2$, and $3$ uniform $p$--refinements employing
directional adaptivity. We emphasize the first plane wave direction with a larger arrow, i.e.,
the dominant wave direction as determined by Algorithm~\ref{algo:direction}. Moreover,
we overlay the directions on top of a contour plot showing the real part of the analytical
solution~\eqref{eqn:hankal_anal}. From Figure~\ref{fig:directions}, we can clearly observe that
the directional adaptivity algorithm is able to accurately determine the dominant wave direction after
a few refinements.
\begin{table}[pt]
\centering
\begin{tabular}{c|c|r@{.}l@{$\times$}l|r@{.}l@{$\times$}l|r@{.}l@{$\times$}l}
& & \multicolumn9{c}{Relative $L^2(\Omega)$-Error} \\
$q$ & No of Dofs & \multicolumn3{c|}{Initial} & \multicolumn3c{One Direction Adapt.} & \multicolumn3c{Two Direction Adapts.} \\ \hline
3 & 112 & 2&015 & $10^{0}$ & \qquad 8&755 & $10^{-1}$ & \qquad 5&856 & $10^{-1}$ \\
4 & 144 & 5&027 & $10^{-1}$ & 1&267 & $10^{-1}$ & 1&149 & $10^{-1}$ \\
5 & 176 & 7&414 & $10^{-2}$ & 2&614 & $10^{-2}$ & 2&584 & $10^{-2}$ \\
6 & 208 & 1&616 & $10^{-2}$ & 6&330 & $10^{-3}$ & 6&327 & $10^{-3}$ \\
7 & 240 & 3&420 & $10^{-3}$ & 1&435 & $10^{-3}$ & 1&435 & $10^{-3}$ \\
8 & 272 & 5&154 & $10^{-4}$ & 3&011 & $10^{-4}$ & 3&011 & $10^{-4}$ \\
\end{tabular}
\caption{Plane Wave Refinement: Comparison of the relative $L^2$-error with fixed effective polynomial degree, $q=3,\dots,8$, and direction adaptivity (Algorithm~\ref{algo:direction}).}
\label{table:pw_adapt_fixed_q}
\end{table}
Finally, in this section we consider performing more than one directional adaptivity step
after each uniform $p$--refinement. To this end, in Table~\ref{table:pw_adapt_fixed_q} we compare
the relative $L^2$-error for the initial directions, as well as after one and two steps of
directional adaptivity have been performed, for the case when $q=3,\dots,8$. Here, we observe that
additional application of the direction adaptivity algorithm does not lead to a significant
reduction in the relative $L^2$-error; indeed, most of the reduction, when compared to the standard
TDG scheme, without directional adaptivity, is attained after one step of
Algorithm~\ref{algo:direction}. Moreover, we emphasise that this first step may be undertaken in
a very computationally cheap manner.
\begin{figure}[t]
\captionsetup[subfloat]{farskip=0pt,captionskip=5pt}\centering
\subfloat[$k=20$]{\label{fig:eff:20}\includegraphics[width=0.4\textwidth]{uniform_eff_hankel_k20}}
\subfloat[$k=30$]{\label{fig:eff:30}\includegraphics[width=0.4\textwidth]{uniform_eff_hankel_k30}} \\
\subfloat[$k=40$]{\label{fig:eff:40}\includegraphics[width=0.4\textwidth]{uniform_eff_hankel_k40}}
\subfloat[$k=50$]{\label{fig:eff:50}\includegraphics[width=0.4\textwidth]{uniform_eff_hankel_k50}}
\caption{Effectivities for $h$--refinement with fixed effective polynomial degree of the smooth analytical Hankel solution with different wavenumbers.}
\label{fig:eff}
\end{figure}
\begin{figure}[tp]
\captionsetup[subfloat]{farskip=0pt,captionskip=5pt}\centering
\subfloat[$k=20$, $\mathcal{E}_{\jmp{u_{hp}}}$]{\label{fig:eff:u:20}\includegraphics[width=0.33\textwidth]{uniform_eff_hankel_k20_u}}
\subfloat[$k=20$, $\mathcal{E}_{\jmp{\nabla u_{hp}}}$]{\label{fig:eff:gradu:20}\includegraphics[width=0.33\textwidth]{uniform_eff_hankel_k20_gradu}}
\subfloat[$k=20$, $\mathcal{E}_R$]{\label{fig:eff:robin:20}\includegraphics[width=0.33\textwidth]{uniform_eff_hankel_k20_robin}} \\
\subfloat[$k=30$, $\mathcal{E}_{\jmp{u_{hp}}}$]{\label{fig:eff:u:30}\includegraphics[width=0.33\textwidth]{uniform_eff_hankel_k30_u}}
\subfloat[$k=30$, $\mathcal{E}_{\jmp{\nabla u_{hp}}}$]{\label{fig:eff:gradu:30}\includegraphics[width=0.33\textwidth]{uniform_eff_hankel_k30_gradu}}
\subfloat[$k=30$, $\mathcal{E}_R$]{\label{fig:eff:robin:30}\includegraphics[width=0.33\textwidth]{uniform_eff_hankel_k30_robin}} \\
\subfloat[$k=40$, $\mathcal{E}_{\jmp{u_{hp}}}$]{\label{fig:eff:u:40}\includegraphics[width=0.33\textwidth]{uniform_eff_hankel_k40_u}}
\subfloat[$k=40$, $\mathcal{E}_{\jmp{\nabla u_{hp}}}$]{\label{fig:eff:gradu:40}\includegraphics[width=0.33\textwidth]{uniform_eff_hankel_k40_gradu}}
\subfloat[$k=40$, $\mathcal{E}_R$]{\label{fig:eff:robin:40}\includegraphics[width=0.33\textwidth]{uniform_eff_hankel_k40_robin}} \\
\subfloat[$k=50$, $\mathcal{E}_{\jmp{u_{hp}}}$]{\label{fig:eff:u:50}\includegraphics[width=0.33\textwidth]{uniform_eff_hankel_k50_u}}
\subfloat[$k=50$, $\mathcal{E}_{\jmp{\nabla u_{hp}}}$]{\label{fig:eff:gradu:50}\includegraphics[width=0.33\textwidth]{uniform_eff_hankel_k50_gradu}}
\subfloat[$k=50$, $\mathcal{E}_R$]{\label{fig:eff:robin:50}\includegraphics[width=0.33\textwidth]{uniform_eff_hankel_k50_robin}} \\
\caption{Effectivities of individual components of the error indicators for $h$--refinement with fixed effective polynomial degree of the smooth analytical Hankel solution with different wavenumbers.}
\label{fig:eff:parts}
\end{figure}
\subsection{Efficiency of the \emph{a posteriori} error indicator}\label{section:aposteriori_effectivity}
The selection of the exponents of $h_K$ and $q_K$ in the weights present in~\eqref{eqn:error_indicator},
together with the independence on the wavenumber $k$, have been determined by numerical experimentation.
To this end, we considered the example presented in the previous section, cf.~\eqref{eqn:hankal_anal}, whereby the
numerical approximation is computed
on a series of uniform computational meshes, with uniform effective polynomial degrees $q$,
for a range of wave numbers $k$. In each case, we computed the effectivity index of each constituent
term arising in ${\mathfrak E}(u_h,h,\vect{p})$, whereby the dependency on $h_K$, $q_K$, and $k$
was eliminated; note that with the removal of $h_K$, $q_K$, and $k$, the effectivity index
of each term is computed by dividing by $\norm{u-u_{hp}}_{L^2(\Omega)}$. More precisely,
effectivity indices were computed for $q=3,\dots,8$ and $k=20,30,40,50$, based on starting from
a uniform $4\times 4$ mesh consisting of square elements. On the basis of these results,
the dependence of each term on $h_K$, $q_K$, and $k$ was established. The final
effectivity indices for the correctly scaled empirical {\em a posteriori} error indicator
${\mathfrak E}(u_h,h,\vect{p})$, i.e., ${\mathfrak E}(u_h,h,\vect{p})/\norm{u-u_{hp}}_{L^2(\Omega)}$
are presented in Figure~\ref{fig:eff}. Here, we observe that that the effectivity indices have
roughly the same values for all the selected values of $h$, $q$, and $k$; however,
at higher wave numbers, pre-asymptotic behaviour leads to some increase in the effectivity indices
as the mesh is refined, due to the fact that the mesh size is too large for the wavelength.
We note that this behaviour is more noticeable in the case when $q=3$.
Finally, we compute the effectivity index for each individual term arising in the
definition of the error indicator ${\mathfrak E}(u_h,h,\vect{p})$, cf.~\eqref{eqn:error_indicator}.
More precisely, we define
\begin{align*}
\mathcal{E}_{\jmp{u_{hp}}} &\coloneqq \frac{\left(\sum_{K\in\mesh} \norm*{\alpha^{\nicefrac{1}{2}} h_K^{\nicefrac12}q_F^{-\nicefrac12} \jmp{u_{hp}}}_{L^2(\partial K \setminus \partial\Omega)}^2\right)^{\nicefrac12}}{\norm*{u-u_{hp}}_{L^2(\Omega)}}, \\
\mathcal{E}_{\jmp{\nabla u_{hp}}} &\coloneqq \frac{\left(\sum_{K\in\mesh} \norm*{\beta^{\nicefrac12} h_K^{\nicefrac32}q_K^{-\nicefrac32} \jmp{\nabla u_{hp}}}_{L^2(\partial K \setminus \partial\Omega)}^2\right)^{\nicefrac12}}{\norm*{u-u_{hp}}_{L^2(\Omega)}}, \\
\mathcal{E}_{R} &\coloneqq \frac{\left(\sum_{K\in\mesh} \norm*{\delta^{\nicefrac{1}{2}} h_K^{\nicefrac32}q_K^{-\nicefrac32} \left( g_R - \nabla u_{hp}\cdot\vect{n}_F + ik u_{hp}\right)}^2_{L^2(\partial K \cap \Gamma_R)}\right)^{\nicefrac12}}{\norm*{u-u_{hp}}_{L^2(\Omega)}};
\end{align*}
the results for the case when $k=20,30,40,50$ are depicted in Figure~\ref{fig:eff:parts}.
Here, we observe that each individual effectivity index is roughly constant
for all the selected values of $h$, $q$, and $k$, except within the pre-asymptotic region.
For this smooth problem, we clearly observe that the dominant part of the error indicator
involves the jump in the gradient of the numerical solution.
\begin{remark}
We note there that we have only computed the weightings for the interior and Robin faces. In the case of Dirichlet boundary conditions we assume that the weighting scales the same as the term involving $\jmp{u_{hp}}$.
\end{remark}
\subsection{$hp$--Adaptive refinement}
In this section we consider computational performance of the proposed $hp$--adaptive refinement algorithm,
with directional adaptivity,
for a range of test problems in both two- and three-dimensions.
To this end, employ the fixed fraction refinement strategy to mark elements for refinement;
throughout this section, we set the refinement fraction equal to 25\%
of the elements with the largest contribution to the error bound.
Furthermore, we allow the meshes $\mesh$ to be `1-irregular', i.e., each face of any
element $K\in\mesh$ contains at most one hanging node (which, for simplicity, we assume
to be at the barycenter of the corresponding face) and each edge of each face contains at most one
hanging node (yet again assumed to be at the barycenter of the edge). We also only allow the
effective polynomial degree $q_K$ to vary by one between neighbouring elements.
For each test problem, we compare the performance of employing $hp$--adaptive refinement with
$h$--adaptivity. In the latter case, we consider a standard $h$--adaptive algorithm, i.e., adaptive
mesh refinement without directional adaptivity, as well as an $h$--adaptive strategy which incorporates
directional adaptivity; here, we shall consider the two cases when directional adaptivity is either
undertaken only on the elements marked for refinement, as well as the case when it is performed on all
elements in the computational mesh. In the $hp$--setting, similar comparisons will be made, in addition
to studying the case when directional adaptivity is only performed on elements marked for $p$--refinement.
We note that when $hp$--refinement is exploited we often reach a point where the
$L^2$--norm of the error and {\em a posteriori} error bound stagnates, in the sense that
both quantities no longer tend to zero, and indeed may start to oscillate, as further refinement
is undertaken. This is caused by the fact that as the relative magnitude of $q_K$,
with respect to $h_Kk$, becomes large,
the local plane wave bases are very ill-conditioned.
In this situation, we simply stop the numerical experiments and discard further results; however,
possible improvements based on ensuring $q_K$ is well behaved with respect to $h_Kk$ could be
implemented; cf. \cite{Cessenat1998, Huttunen2002, Luostari2013} for details.
\subsubsection{Example 1 --- Smooth solution (Hankel function)} \label{sec-hprefine-hankel}
\begin{figure}[pt]
\captionsetup[subfloat]{farskip=0pt,captionskip=5pt}\centering
\subfloat[$k=20$; $h$--refinement]{\label{fig:hankel:error:20h}\includegraphics[width=0.4\textwidth]{hankel_k20_error_h}}
\subfloat[$k=20$; $h$--refinement]{\label{fig:hankel:eff:20h}\includegraphics[width=0.4\textwidth]{hankel_k20_eff_h}} \\
\subfloat[$k=20$; $hp$--refinement]{\label{fig:hankel:error:20hp}\includegraphics[width=0.4\textwidth]{hankel_k20_error_hp}}
\subfloat[$k=20$; $hp$--refinement]{\label{fig:hankel:eff:20hp}\includegraphics[width=0.4\textwidth]{hankel_k20_eff_hp}} \\
\subfloat[$k=50$; $h$--refinement]{\label{fig:hankel:error:50h}\includegraphics[width=0.4\textwidth]{hankel_k50_error_h}}
\subfloat[$k=50$; $h$--refinement]{\label{fig:hankel:eff:50h}\includegraphics[width=0.4\textwidth]{hankel_k50_eff_h}} \\
\subfloat[$k=50$; $hp$--refinement]{\label{fig:hankel:error:50hp}\includegraphics[width=0.4\textwidth]{hankel_k50_error_hp}}
\subfloat[$k=50$; $hp$--refinement]{\label{fig:hankel:eff:50hp}\includegraphics[width=0.4\textwidth]{hankel_k50_eff_hp}}
\caption{Example 1: \protect\subref{fig:hankel:error:20h} $L^2$-error and \protect\subref{fig:hankel:eff:20h} Effectivity index for $h$--refinement with wavenumber $k=20$; \protect\subref{fig:hankel:error:20hp} $L^2$-error and \protect\subref{fig:hankel:eff:20hp} Effectivity index for $hp$--refinement with $k=20$; \protect\subref{fig:hankel:error:50h} $L^2$-error and \protect\subref{fig:hankel:eff:50h} Effectivity index for $h$--refinement with $k=50$; \protect\subref{fig:hankel:error:50hp} $L^2$-error and \protect\subref{fig:hankel:eff:50hp} Effectivity index for $hp$--refinement with $k=50$.}
\end{figure}
\begin{figure}[pt]
\captionsetup[subfloat]{farskip=0pt,captionskip=5pt}\centering
\subfloat[$k=20$]{\label{fig:hankel:errorcompare:20}\includegraphics[width=0.4\textwidth]{hankel_k20_error_compare}}
\subfloat[$k=50$]{\label{fig:hankel:errorcompare:50}\includegraphics[width=0.4\textwidth]{hankel_k50_error_compare}}
\caption{Example 1: Comparison of relative $L^2$-error for $h$-- and $hp$--refinement, with direction adaptivity on all elements, for wavenumbers \protect\subref{fig:hankel:errorcompare:20} $k=20$ and \protect\subref{fig:hankel:errorcompare:50} $k=50$.}
\label{fig:hankel:errorcompare}
\end{figure}
\begin{figure}[pt]
\captionsetup[subfloat]{farskip=0pt,captionskip=5pt}\centering
\subfloat[$k=20$]{\label{fig:hankel:mesh:h:20}\includegraphics[width=0.4\textwidth]{hankel_k20_mesh_h}}
\subfloat[$k=20$]{\label{fig:hankel:mesh:hp:20}\includegraphics[width=0.4\textwidth]{hankel_k20_mesh_hp}} \\
\subfloat[$k=50$]{\label{fig:hankel:mesh:h:50}\includegraphics[width=0.4\textwidth]{hankel_k50_mesh_h}}
\subfloat[$k=50$]{\label{fig:hankel:mesh:hp:50}\includegraphics[width=0.4\textwidth]{hankel_k50_mesh_hp}}
\caption{Example 1: Meshes after 8 \protect\subref{fig:hankel:mesh:h:20} $h$-- and \protect\subref{fig:hankel:mesh:hp:20} $hp$--refinements for wavenumber $k=20$; meshes after 8 \protect\subref{fig:hankel:mesh:h:50} $h$-- and \protect\subref{fig:hankel:mesh:hp:50} $hp$--refinements for wavenumber $k=50$.}
\label{fig:hankel}
\end{figure}
In this section, we again consider the problem outlined in Section~\ref{section:plane_wave_refine}.
Furthermore, we select the initial mesh to consist of $8\times 8$ uniform square elements and set
$q_K=3$ on each $K\in\mesh$. Firstly, in Figures~\ref{fig:hankel:error:20h} and \ref{fig:hankel:error:50h}
we compare the relative error in the $L^2$-norm to the number of degrees of freedom in the
TDG space $V_{\vect{p}}(\mathcal{T}_h)$, when $h$--refinement is employed, with the wavenumbers $k=20$
and $k=50$, respectively. In each case, we consider the performance of the underlying
adaptive algorithm when both the standard TDG scheme (without direction adaptivity) is employed,
as well as the corresponding method with directional adaptivity; in this latter setting, we
consider the cases when either directional adaptivity is undertaken on only the elements marked
for refinement, as well as when it is exploited on every element in the computational
mesh $\mesh$.
Analogous results are presented in Figures~\ref{fig:hankel:error:20hp} and \ref{fig:hankel:error:50hp}
in the $hp$--setting, respectively; here, we compare standard $hp$--refinement, with $hp$--adaptivity incorporating
directional adaptivity. In the latter case, different directional adaptivity strategies are considered:
firstly, directional adaptivity is performed only on elements marked for $p$--refinement;
secondly, directional adaptivity is undertaken on all elements marked for refinement;
finally, directional adaptivity is applied to every element in $\mesh$. In the $hp$--setting
we observe exponential convergence of the error as the finite element space is adaptively
enriched: indeed, on a linear-log scale, the convergence lines are roughly straight. Thereby,
it is clear that the exploitation of the proposed $hp$--refinement algorithm, with directional
adaptivity, leads to a significant reduction in the $L^2$-norm of error, for a given number of degrees of
freedom, when compared to the same quantity computed with $h$--refinement alone; cf. Figure~\ref{fig:hankel:errorcompare}.
In both the $h$-- and $hp$--refinement cases, we generally observe that the error is decreased when
directional refinement is employed. Moreover, it is evident in the $hp$--setting that
selecting more elements for directional refinement generally leads to a smaller error, for a given
number of degrees of freedom; this is particularly noticeable in the case when $k=50$.
In Figures~\ref{fig:hankel:eff:20h}, \ref{fig:hankel:eff:20hp}, \ref{fig:hankel:eff:50h},
and \ref{fig:hankel:eff:50hp} we plot the effectivity indices for each of the above refinement
strategies for the case when $k=20,50$; here, we observe that they remain roughly constant
during adaptive $h$--/$hp$--mesh refinement, and are roughly the same for the two different wavenumbers,
with the notable exception of the pre-asymptotic region for $k=50$.
Finally, in Figures~\ref{fig:hankel:mesh:h:20}--\ref{fig:hankel:mesh:hp:50}, we show the meshes
after 8 $h$-- and $hp$--refinements, with directional adaptivity employed on all elements, for
both $k=20$ and $k=50$; here, the $hp$--meshes show the effective polynomial degree $q_K$ for each
element. Given the smoothness of the analytical solution on $\Omega$, we observe that the resulting
computational meshes are almost uniform; indeed, in the $hp$--setting almost uniform $p$--refinement
has been undertaken.
\subsubsection{Example 2 --- Singular solution}
\begin{figure}[pt]
\captionsetup[subfloat]{farskip=0pt,captionskip=5pt}\centering
\subfloat[$k=20$; $h$--refinement]{\label{fig:besselsin:error:20h}\includegraphics[width=0.4\textwidth]{besselsin_k20_error_h}}
\subfloat[$k=20$; $h$--refinement]{\label{fig:besselsin:eff:20h}\includegraphics[width=0.4\textwidth]{besselsin_k20_eff_h}} \\
\subfloat[$k=20$; $hp$--refinement]{\label{fig:besselsin:error:20hp}\includegraphics[width=0.4\textwidth]{besselsin_k20_error_hp}}
\subfloat[$k=20$; $hp$--refinement]{\label{fig:besselsin:eff:20hp}\includegraphics[width=0.4\textwidth]{besselsin_k20_eff_hp}} \\
\subfloat[$k=50$; $h$--refinement]{\label{fig:besselsin:error:50h}\includegraphics[width=0.4\textwidth]{besselsin_k50_error_h}}
\subfloat[$k=50$; $h$--refinement]{\label{fig:besselsin:eff:50h}\includegraphics[width=0.4\textwidth]{besselsin_k50_eff_h}} \\
\subfloat[$k=50$; $hp$--refinement]{\label{fig:besselsin:error:50hp}\includegraphics[width=0.4\textwidth]{besselsin_k50_error_hp}}
\subfloat[$k=50$; $hp$--refinement]{\label{fig:besselsin:eff:50hp}\includegraphics[width=0.4\textwidth]{besselsin_k50_eff_hp}}
\caption{Example 2: \protect\subref{fig:besselsin:error:20h} $L^2$-error and \protect\subref{fig:besselsin:eff:20h} Effectivity index for $h$--refinement with wavenumber $k=20$; \protect\subref{fig:besselsin:error:20hp} $L^2$-error and \protect\subref{fig:besselsin:eff:20hp} Effectivity index for $hp$--refinement with $k=20$; \protect\subref{fig:besselsin:error:50h} $L^2$-error and \protect\subref{fig:besselsin:eff:50h} Effectivity index for $h$--refinement with $k=50$; \protect\subref{fig:besselsin:error:50hp} $L^2$-error and \protect\subref{fig:besselsin:eff:50hp} Effectivity index for $hp$--refinement with $k=50$.}
\label{fig:besselsin}
\end{figure}
\begin{figure}[pt]
\captionsetup[subfloat]{farskip=0pt,captionskip=5pt}\centering
\subfloat[$k=20$]{\label{fig:besselsin:mesh:h:20}\includegraphics[width=0.4\textwidth]{besselsin_k20_mesh_h}}
\subfloat[$k=20$]{\label{fig:besselsin:mesh:hp:20}\includegraphics[width=0.4\textwidth]{besselsin_k20_mesh_hp}}\\
\subfloat[$k=50$]{\label{fig:besselsin:mesh:h:50}\includegraphics[width=0.4\textwidth]{besselsin_k50_mesh_h}}
\subfloat[$k=50$]{\label{fig:besselsin:mesh:hp:50}\includegraphics[width=0.4\textwidth]{besselsin_k50_mesh_hp}}
\caption{Example 2: Meshes after 8 \protect\subref{fig:besselsin:mesh:h:20} $h$-- and \protect\subref{fig:besselsin:mesh:hp:20} $hp$--refinements for wavenumber $k=20$; meshes after 8 \protect\subref{fig:besselsin:mesh:h:50} $h$-- and \protect\subref{fig:besselsin:mesh:hp:50} $hp$--refinements for wavenumber $k=50$.}
\label{fig:besselsin:mesh}
\end{figure}
In this second example, we consider problem~\eqref{eqn:helmholtz} posed on the L-shaped domain $\Omega=(-1,1)^2\setminus(0,1)\times(-1,1)$, $\Gamma_R=\partial\Omega$, and $\Gamma_D\equiv\emptyset$, with Robin boundary condition $g_R$ selected so that the analytical solution is given, in polar coordinates $(r,\varphi)$, by
\[
u(r,\theta) = \mathcal{J}_{\nicefrac23}(kr)\sin(\nicefrac{2\theta}{3});
\]
we note that the gradient of $u$ has a singularity at the origin.
As in the previous example, we again compare the performance of the $h$-- and $hp$--adaptive refinement algorithms, both in the standard setting, as well as when directional adaptivity is employed; here, we again consider the analogous directional refinement strategies employed in Section~\ref{sec-hprefine-hankel}. To this end, in Figures~\ref{fig:besselsin:error:20h} and \ref{fig:besselsin:error:50h} we compare the relative error in the $L^2$-norm with the number of degrees of freedom in the TDG space $V_{\vect{p}}(\mathcal{T}_h)$ when $h$--refinement is employed for $k=20$ and $k=50$, respectively; the respective convergence plots in the $hp$--setting are given in Figures~\ref{fig:besselsin:error:20h} and \ref{fig:besselsin:error:50h}. Here, we observe that although exploiting $hp$--refinement leads to exponential convergence of the relative $L^2$-norm of the error as $V_{\vect{p}}(\mathcal{T}_h)$ is enriched, in both the $h$-- and $hp$--settings, we observe that the magnitude of the error, computed both with and without directional refinement, is roughly identical; i.e., directional refinement does not lead to any reduction in the computed TDG solution when either $h$--/$hp$--refinement is employed. We note that, for this particular problem, this behaviour is expected, since the error in the computed TDG solution is dominated by the influence of the singularity at the origin, rather than local wave propagation.
In Figures~\ref{fig:besselsin:eff:20h}, \ref{fig:besselsin:eff:20hp}, \ref{fig:besselsin:eff:50h}, and \ref{fig:besselsin:eff:50hp} we plot the effectivity indices when both $h$-- and $hp$--refinement is employed for the case when $k=20,50$. In all cases, we observe that the effectivity indices are roughly constant for this singular problem, though when $h$--refinement is employed, on highly refined meshes, we see a slight drop in the computed effectivity indices. Finally, in Figures~\ref{fig:besselsin:mesh:h:20}--\ref{fig:besselsin:mesh:hp:50}, we show the meshes after 8 $h$-- and $hp$--refinements, with direction adaptivity employed on all elements, for both $k=20$ and $k=50$. As we would expect, in both the $h$-- and $hp$--settings, mesh subdivision is concentrated in the vicinity of the singularity located at the origin; away from this region, the $h$--refinement algorithm employs almost uniform mesh subdivision, while the $hp$--refinement strategy employs the necessary combination of local mesh refinement and local polynomial enrichment, as required, to reduce the error in the computed TDG solution.
\subsubsection{Example 3 --- Transmission/internal reflection}
We now consider the case of transmission and internal reflection of a plane wave across a fluid-fluid interface in the domain $\Omega=(-1,1)^2$, $\Gamma_R\equiv \emptyset$, and $\Gamma_D=\partial\Omega$, with two different refractive indices, cf.~\cite[Section 6.3]{Kapita2015}. The interface between the two regions is located at $y=0$; in this setting the wavenumber $k$ is given by the piecewise constant function
\[
k(x,y) = \begin{cases}
k_1 \coloneqq \omega n_1 & \text{if } y \leq 0,\\
k_2 \coloneqq \omega n_2 & \text{if } y > 0,
\end{cases}
\]
where, we select $\omega=11$, $n_1=2$, and $n_2=1$. Throughout this section we impose an appropriate inhomogeneous Dirichlet boundary condition, so that the analytical solution $u$ to \eqref{eqn:helmholtz} is given, for a constant $0\leq\theta_i\leq\nicefrac\pi2$, by
\[
u(x,y) = \begin{cases}
T \mathrm{e}^{i(K_1x+K_2y)} & \text{if } y>0 ,\\
\mathrm{e}^{ik_1(x\cos(\theta_i)+y\sin(\theta_i))} + R \mathrm{e}^{ik_1(x\cos(\theta_i)-y\sin(\theta_i))} & \text{if } y<0,
\end{cases}
\]
where $K_1=k_1\cos(\theta_i)$, $K_2 = \sqrt{k_2^2-k_1^2\sin^2(\theta_i)}$,
\[
R=-\frac{K_2-k_1\sin(\theta_i)}{K_2+k_1\sin(\theta_i)},
\]
and $T=1+R$.
We note that there exists a critical angle $\theta_{crit}$, such that when $\theta_i>\theta_{crit}$ the wave is refracted, while $\theta_i<\theta_{crit}$ results in internal reflection, cf.~\cite[Section 6.3]{Kapita2015}. As in \cite{Kapita2015} we perform numerical experiments for both internal reflection ($\theta_i=29^\circ$) and refraction ($\theta_i=69^\circ$). To highlight the reflection and refraction behaviour, in Figures~\ref{fig:reflection:anal} and~\ref{fig:refraction:anal} we show the analytical solution when $\theta_i=29^\circ$ and $\theta_i=69^\circ$, respectively.
\begin{figure}[t]
\captionsetup[subfloat]{farskip=0pt,captionskip=5pt}\centering
\subfloat[$\theta_i=29^\circ$]{\label{fig:reflection:anal}\includegraphics[width=0.4\textwidth]{reflection_anal}}
\subfloat[$\theta_i=69^\circ$]{\label{fig:refraction:anal}\includegraphics[width=0.4\textwidth]{refraction_anal}}
\caption{Example 3: Analytical solutions (real part) when \protect\subref{fig:reflection:anal} $\theta_i=29^\circ$ resulting in internal reflection, and \protect\subref{fig:refraction:anal} $\theta_i=69^\circ$ resulting in refraction.}
\label{fig:reflection_anal}
\end{figure}
\begin{figure}[pt]
\captionsetup[subfloat]{farskip=0pt,captionskip=5pt}\centering
\subfloat[$\theta_i=29^\circ$; $h$--refinement]{\label{fig:reflection:error:h}\includegraphics[width=0.4\textwidth]{reflection_error_h}}
\subfloat[$\theta_i=29^\circ$; $h$--refinement]{\label{fig:reflection:eff:h}\includegraphics[width=0.4\textwidth]{reflection_eff_h}} \\
\subfloat[$\theta_i=29^\circ$; $hp$--refinement]{\label{fig:reflection:error:hp}\includegraphics[width=0.4\textwidth]{reflection_error_hp}}
\subfloat[$\theta_i=29^\circ$; $hp$--refinement]{\label{fig:reflection:eff:hp}\includegraphics[width=0.4\textwidth]{reflection_eff_hp}} \\
\subfloat[$\theta_i=69^\circ$; $h$--refinement]{\label{fig:refraction:error:h}\includegraphics[width=0.4\textwidth]{refraction_error_h}}
\subfloat[$\theta_i=69^\circ$; $h$--refinement]{\label{fig:refraction:eff:h}\includegraphics[width=0.4\textwidth]{refraction_eff_h}} \\
\subfloat[$\theta_i=69^\circ$; $hp$--refinement]{\label{fig:refraction:error:hp}\includegraphics[width=0.4\textwidth]{refraction_error_hp}}
\subfloat[$\theta_i=69^\circ$; $hp$--refinement]{\label{fig:refraction:eff:hp}\includegraphics[width=0.4\textwidth]{refraction_eff_hp}}
\caption{Example 3: \protect\subref{fig:reflection:error:h} $L^2$-error and \protect\subref{fig:reflection:eff:h} Effectivity index for $h$--refinement with reflection ($\theta_i=29^\circ$); \protect\subref{fig:reflection:error:hp} $L^2$-error and \protect\subref{fig:reflection:eff:hp} Effectivity index for $hp$--refinement with reflection ($\theta_i=29^\circ$); \protect\subref{fig:refraction:error:h} $L^2$-error and \protect\subref{fig:refraction:eff:h} Effectivity index for $h$--refinement with refraction ($\theta_i=69^\circ$); \protect\subref{fig:refraction:error:hp} $L^2$-error and \protect\subref{fig:refraction:eff:hp} Effectivity index for $hp$--refinement with refraction ($\theta_i=69^\circ$).}
\label{fig:reflection}
\end{figure}
\begin{figure}[pt]
\captionsetup[subfloat]{farskip=0pt,captionskip=5pt}\centering
\subfloat[$\theta_i=29^\circ$]{\label{fig:reflection:mesh:h}\includegraphics[width=0.4\textwidth]{reflection_mesh_h}}
\subfloat[$\theta_i=29^\circ$]{\label{fig:reflection:mesh:hp}\includegraphics[width=0.4\textwidth]{reflection_mesh_hp}}\\
\subfloat[$\theta_i=69^\circ$]{\label{fig:refraction:mesh:h}\includegraphics[width=0.4\textwidth]{refraction_mesh_h}}
\subfloat[$\theta_i=69^\circ$]{\label{fig:refraction:mesh:hp}\includegraphics[width=0.4\textwidth]{refraction_mesh_hp}}
\caption{Example 3: Meshes after 7 \protect\subref{fig:reflection:mesh:h} $h$-- and \protect\subref{fig:reflection:mesh:hp} $hp$--refinements for reflection ($\theta_i=29^\circ$); meshes after 7 \protect\subref{fig:refraction:mesh:h} $h$-- and \protect\subref{fig:refraction:mesh:hp} $hp$--refinements for refraction ($\theta_i=69^\circ$).}
\label{fig:reflection:mesh}
\end{figure}
To account for the jump in the wavenumber $k$, the value of $k$ present in the integrals along the interface $y=0$ in the TDG scheme \eqref{eqn:bilinear_form} is replaced by $\omega$. We select the initial mesh to consist of $8 \times 8$ uniform square elements, so that the interface between the two materials is captured by the mesh; thereby, the wavenumber is constant in every element, and hence the TDG space~\eqref{eqn:pw_basis} and error indicators~\eqref{eqn:error_indicator} can be easily modified to treat this example by setting the wavenumber for each element equal to the wavenumber of the material within which the element is contained. Firstly, we consider the case when there is an internal reflection, i.e., when $\theta_i=29^\circ$; to this end, in Figures~\ref{fig:reflection:error:h} and \ref{fig:reflection:error:hp} we plot the relative error in the $L^2$-norm against the number of degrees of freedom in $V_{\vect{p}}(\mathcal{T}_h)$ using both $h$-- and $hp$--refinement, respectively. As for the previous numerical experiments, here we again observe exponential convergence of the error when $hp$--refinement is employed. Furthermore, in both the $h$-- and $hp$--version settings, we observe that employing directional adaptivity does not improve the magnitude of the error; indeed, in the $hp$--version setting, initially the standard refinement approach is superior, though as $V_{\vect{p}}(\mathcal{T}_h)$ is enriched, we again observe the benefits of employing directional adaptivity. This behaviour is perhaps expected, since for the internal reflection case, no waves are present above the $y=0$ line and moreover it does not possess a dominant wave propagation direction below the $y=0$ line due to the reflected waves, cf.~Figure~\ref{fig:reflection:anal}. In Figures~\ref{fig:reflection:eff:h} and \ref{fig:reflection:eff:hp}, we plot the effectivity indices for both refinement strategies, respectively; here we observe that, apart from an initial pre-asymptotic region, the effectivity indices are roughly constant.
The corresponding convergence plots for the refraction case, i.e., when $\theta_i=69^\circ$, are presented in Figures~\ref{fig:refraction:error:h} and \ref{fig:refraction:error:hp} when both $h$-- and $hp$--refinement are employed, respectively; in the latter setting, we again observe exponential convergence of the computed relative $L^2$-norm of the error. Moreover, in contrast to the case when there is an internal reflection, here we observe the computational benefits of employing directional adaptivity, in the sense that this typically leads to a reduction in the error, for a given fixed number of degrees of freedom, when compared to the standard refinement strategy; this is particularly evidenced in the $hp$--setting. Indeed, in this case there is a dominant propagation direction throughout the domain, cf.~Figure~\ref{fig:refraction:anal}. Figures~\ref{fig:refraction:eff:h} and~\ref{fig:refraction:eff:hp} show the effectivity indices computed using both $h$-- and $hp$--refinement, respectively; analogous behaviour is observed as for the internal reflection case, i.e., the effectivity indices become roughly constant, after an initial pre-asymptotic region.
Finally, in Figures~\ref{fig:reflection:mesh:h} \&~\ref{fig:reflection:mesh:hp} we show the meshes after 7 $h$-- and $hp$-- adaptive mesh refinements have been performed, respectively, in the case of an internal reflection, i.e., $\theta_i=29^\circ$. Here, the $h$--refinement strategy concentrates most of the elements in the $y<0$ region; although, there is some refinement above $y=0$ to resolve the exponentially decaying solutions present there. Additional mesh smoothing has also been undertaken here to ensure that there is only one hanging node per face, cf.~\cite{Kapita2015}. The $hp$--refinement algorithm also performs some $h$--refinement below the $y=0$ line, though this region is largely $p$--refined; however, most of the refinement occurs around the $y=0$ line to resolve the exponentially decaying solutions. Some $p$--refinement occurs in the rest of the $y>0$ region, which is caused by enforcing the condition that the effective polynomial degree may only vary by one between neighboring elements. In the refraction case, i.e., $\theta_i=69^\circ$, cf. Figures~\ref{fig:refraction:mesh:h} \&~\ref{fig:refraction:mesh:hp}, we note a sharp boundary at the $y=0$, with more refinement undertaken in the $y<0$ region than the region $y>0$.
\subsubsection{Example 4 --- 3D smooth solution (plane wave)}
\begin{figure}[pt]
\captionsetup[subfloat]{farskip=0pt,captionskip=5pt}\centering
\subfloat[$k=20$; $h$--refinement]{\label{fig:cube:error:20h}\includegraphics[width=0.4\textwidth]{cube_k20_error_h}}
\subfloat[$k=20$; $h$--refinement]{\label{fig:cube:eff:20h}\includegraphics[width=0.4\textwidth]{cube_k20_eff_h}} \\
\subfloat[$k=20$; $hp$--refinement]{\label{fig:cube:error:20hp}\includegraphics[width=0.4\textwidth]{cube_k20_error_hp}}
\subfloat[$k=20$; $hp$--refinement]{\label{fig:cube:eff:20hp}\includegraphics[width=0.4\textwidth]{cube_k20_eff_hp}} \\
\subfloat[$k=50$; $h$--refinement]{\label{fig:cube:error:50h}\includegraphics[width=0.4\textwidth]{cube_k50_error_h}}
\subfloat[$k=50$; $h$--refinement]{\label{fig:cube:eff:50h}\includegraphics[width=0.4\textwidth]{cube_k50_eff_h}} \\
\subfloat[$k=50$; $hp$--refinement]{\label{fig:cube:error:50hp}\includegraphics[width=0.4\textwidth]{cube_k50_error_hp}}
\subfloat[$k=50$; $hp$--refinement]{\label{fig:cube:eff:50hp}\includegraphics[width=0.4\textwidth]{cube_k50_eff_hp}}
\caption{Example 4: \protect\subref{fig:cube:error:20h} $L^2$-error and \protect\subref{fig:cube:eff:20h} Effectivity index for $h$--refinement with wavenumber $k=20$; \protect\subref{fig:cube:error:20hp} $L^2$-error and \protect\subref{fig:cube:eff:20hp} Effectivity index for $hp$--refinement with $k=20$; \protect\subref{fig:cube:error:50h} $L^2$-error and \protect\subref{fig:cube:eff:50h} Effectivity index for $h$--refinement with $k=50$; \protect\subref{fig:cube:error:50hp} $L^2$-error and \protect\subref{fig:cube:eff:50hp} Effectivity index for $hp$--refinement with $k=50$.}
\label{fig:cube}
\end{figure}
In this final example, we consider problem \eqref{eqn:helmholtz} posed on the domain $\Omega=(0,1)^3$, $\Gamma_R=\partial\Omega$, and $\Gamma_D\equiv\emptyset$, with Robin boundary condition $g_R$ selected so that the analytical solution $u$ to \eqref{eqn:helmholtz} is given by
\[
u(\vect{x}) = \mathrm{e}^{ik\vect{\vect{d}\cdot\vect{x}}},
\]
where $\vect{d}_j = \nicefrac{1}{\sqrt{3}}$ for $j=1,2,3$.
In Figures~\ref{fig:cube:error:20h} and \ref{fig:cube:error:50h} we present the performance of the proposed directional adaptivity algorithm employing $h$--refinement with wavenumbers $k=20$ and $k=50$, respectively; the analogous results for $hp$--refinement are given in Figures~\ref{fig:cube:error:20hp} and \ref{fig:cube:error:50hp}, respectively. As in the two--dimensional setting, we observe that selecting more elements for directional adaptivity at each step of the proposed refinement strategy, leads to a greater reduction in the relative $L^2$-norm of the error, for a fixed number of degrees of freedom, when compared to the standard case when directional adaptivity is not employed. Of course, given the simple nature of the analytical solution for this problem, we clearly expect directional adaptivity to be advantageous. In the case when the wavenumber $k=50$ we note that both $h$-- and $hp$--refinement strategies are essentially in the pre-asymptotic region; however, performing directional adaptivity ensures that the method leaves this pre-asymptotic region after only a few mesh refinements. Finally, in Figures~\ref{fig:cube:eff:20h}, \ref{fig:cube:eff:20hp}, \ref{fig:cube:eff:50h}, and \ref{fig:cube:eff:50hp} we plot the effectivity indices of both the $h$-- and $hp$--refinement algorithms for the case when $k=20,50$. We note, especially in the $hp$--refinement case, that the effectivity indices are roughly constant but do slightly rise after the pre-asymptotic region.
\section{Concluding remarks} \label{sec:conclusions}
In this article we have developed an automatic $hp$--adaptive refinement algorithm
for the TDG approximation of the homogeneous Helmholtz equation. In addition
to employing both local mesh subdivision and local basis enrichment,
we also locally rotate the underlying plane wave basis in such a manner
so that the first basis function is aligned with the dominant wave
direction. The choice to $h$-- or $p$--refine an element is based on
a prediction of how much reduction we expect to observe in the elementwise
error indicator, when a particular refinement is performed. The alignment
of the local basis with the dominant wave direction is undertaken on the basis of an
eigenvalue analysis of the Hessian of the numerical solution, together with
a correction computed from an impedance condition. The computational
efficiency of the proposed adaptive strategy has been studied through a
series of numerical examples; indeed, the application of $hp$--refinement,
with directional adaptivity, leads to a significant reduction in the
computed error compared to standard refinement strategies. We also note that
performing directional adaptivity on all elements generally leads to a greater reduction
in the error than the corresponding case when only elements marked for
refinement are directionally adapted; clearly, this error reduction is attained while
keeping the number of degrees of freedom in the underlying TDG space fixed. Future work
will be devoted to the derivation of robust $hp$--version {\em a posteriori}
error bounds, as well as the application to problems of engineering interest.
\section*{Acknowledgement}
S.~Congreve and I.~Perugia have been funded by the Austrian Science Fund (FWF) through the project P29197-N32. I.~Perugia has also been funded by the FWF through the project F~65.
\bibliographystyle{abbrv}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,537 |
Q: as variable I have an problem:
I am trying:
private struct QueuedFile
{
public Type t;
public object LoadedFile;
public string path;
public bool loaded;
public ContentManager c;
public QueuedFile(string path, Type t, ContentManager c)
{
this.t = t;
this.path = path;
LoadedFile = null;
loaded = false;
this.c = c;
}
public void Load()
{
LoadedFile = c.Load<this.t>(path); //<--ERROR: <this.t>
loaded = true;
}
}
But this gives me an error. Has anyone an idea how to save the variable T for an method like:
public T LoadFileByType<T>(string path);
A: It's not that simple. Generic methods are bound at compile time. There's not a way other then reflection to bind to a generic method with a variable type.
Here's how you would do it with reflection:
MethodInfo method = c.GetType()
.GetMethod("Load");
.MakeGenericMethod(this.t);
LoadedFile = method.Invoke(path, new object[] {path});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,012 |
\section{Introduction}
More than a century ago, on November 25, 1915, Albert Einstein \cite{Einstein:1915ca} wrote down General Relativity (GR) field equations in their final form and soon afterwards, on January 13, 1916, Karl Schwarzschild \cite{Schwarzschild} discovered Einstein field equation first solution. Using these results, while studying particle radial motion in a spherically symmetric static gravitational field, David Hilbert \cite{Hilbert} noticed on December 23, 1916, that the acceleration of a massive particle freely falling in a gravitational field, as seen by an asymptotic observer, changes sign when the velocity reaches a given value. It is as though the gravitational field becomes repulsive.
Since this result of Hilbert cannot be found in any textbook on General Relativity, we are clearly explaining it in this pedagogical essay.
It is fair to say that most of the motion we see in the Universe is a free fall in a gravitational field. There are three vantage points for looking at a particle falling into a black hole.
The first one is an inertial asymptotic observer whom we call \textit{far-away observer} (Wheeler's \textit{Schwarzschild bookkeeper}~\cite{Taylor-Wheeler}) who, by definition, is located infinitely far away from the black hole and is sufficiently massless not to exert any gravitational field that could affect the measurements.
The second viewpoint is that of the freely falling particle itself which we call \textit{freely-falling observer} (Wheeler's \textit{free-float frame}~\cite{Taylor-Wheeler}).
The third observer is the one at a finite distance from the black hole who is referred to as \textit{finite-distance observer}(Wheeler's \textit{shell observer}~\cite{Taylor-Wheeler}).
As seen by the far-away observer, the particle falling into the black hole never crosses the horizon but asymptotically approaches it. This is a standard textbook statement but it hides a subtle detail. We know that when a particle starts falling freely towards a black hole, the velocity increases as it approaches the gravitational source. But we also know that, from the viewpoint of an asymptotic observer, the freely falling particle never makes it to the event horizon. This means that the particle should slow down at some finite distance from the black hole and eventually cease to move at the horizon, albeit this takes an infinite time in the far-away observer's clock. In other words, the freely falling particle decelerates (the acceleration direction is positive (radially away from the gravitating source) while the velocity is negative) i.e., it is repelled by gravity as seen by the far-away observer. One must be careful about the terminology. Repulsion here does not mean that the freely falling particle is bounced off the horizon. For a historical perspective on this topic, we refer the reader to a review by Spallicci~\cite{Spallicci}.
\section{Falling into a Schwarzschild black hole}
The Schwarzschild metric in a far-way observer's standard coordinate set up is given by
\begin{eqnarray}
ds^2 &=& \left( 1-\frac{2\mu}{r} \right) c^2 dt^2 - \left( 1-\frac{2\mu}{r} \right)^{-1} dr^2
- r^2 (d\theta^2 + sin\theta^2 d\phi^2)
\end{eqnarray}
with $$ \mu = \frac{G M}{c^2},$$
where $M$ is the black hole mass, and the $G$ and $c$ constants have their usual meanings.
For simplicity, we consider a massive test particle radial motion in a Schwarzschild black hole equatorial plane. Then this particle timelike geodesic motion obeys the following,
\begin{eqnarray}
\label{2}
\frac{dt}{d\tau} &=& \left( 1-\frac{2\mu}{r} \right)^{-1}, \\
\label{3}
\frac{dr}{d\tau} &=& \mp \left( \frac{2\mu}{r} \right)^{\frac{1}{2}}.
\end{eqnarray}
Here the minus-or-plus sign in the second equation stands for infalling or outgoing test particles respectively. So now we can find the freely falling particle velocity as measured by every observer.
\begin{itemize}
\item The far-away observer would measure this velocity to be
\begin{eqnarray}
\frac{d r}{dt} = - \left( \frac{2 \mu}{r} \right)^{\frac{1}{2}} \left( 1-\frac{2\mu}{r} \right) ,
\end{eqnarray}
which is zero on the horizon.
\item The finite-distance observer measures the proper time interval as $ d t' = \left( 1-\frac{2\mu}{r} \right)^{\frac{1}{2}} \,dt,$ and the proper distance as $ d r' = \left( 1-\frac{2\mu}{r} \right)^{-\frac{1}{2}} \,dr.$ Hence the freely falling particle velocity as recorded by this observer is
\begin{eqnarray}
\frac{d r'}{dt'} = - \left( \frac{2 \mu}{r} \right)^{\frac{1}{2}}.
\end{eqnarray}
After restoring the constants, its absolute value is equal to $c$ on the horizon. This means the shell observer sees the test particle velocity approaching light velocity as it nears the horizon, which is quite opposite to a far-away observer's measurement.
\item Of course, the freely-falling observer is at rest in the float-frame throughout the course of motion, even when passing through the horizon, as long as the tidal forces do not kick in~\cite{Moore:2013sra}.
\end{itemize}
The test particle radial geodesic motion equations in a Schwarzschild field are given by
\begin{eqnarray}
\label{6}
\frac{d^2 r}{d\tau^2} &=& - \left( \frac{\mu}{r^2} \right) \left( 1 - \frac{2 \mu}{r} \right) \left( \frac{d t}{d \tau} \right)^2
+ \left( \frac{\mu}{r^2} \right) \left( 1 - \frac{2 \mu}{r} \right)^{-1} \left( \frac{d r}{d \tau} \right)^2, \\
\label{7}
\frac{d^2 t}{d\tau^2} &=& -2 \left( \frac{\mu}{r^2} \right) \left( 1 - \frac{2 \mu}{r} \right)^{-1} \frac{d t}{d \tau} \frac{d r}{d \tau}.
\end{eqnarray}
Upon substituting Eq.(\ref{2}) and Eq.(\ref{3}) in Eq. (\ref{6}), we obtain
\begin{eqnarray}
\frac{d^2 r}{d\tau^2} &=& - \frac{\mu}{r^2}.
\end{eqnarray}
For a freely falling test particle whose proper time is $\tau$, the following relation holds,
\begin{eqnarray}
\frac{dr}{d\tau} = \left( \frac{dr}{dt} \right) \frac{dt}{d\tau}.
\end{eqnarray}
Further, we have
\begin{eqnarray}
\frac{d^2 r}{d\tau^2} = \frac{d^2 r}{dt^2} \left( \frac{dt}{d\tau} \right)^2 + \frac{dr}{dt} \, \frac{d^2 t}{d\tau^2},
\end{eqnarray}
where $\frac{dr}{dt}$ and $\frac{d^2r}{dt^2}$ are the freely falling particle velocity and acceleration as recorded by the far-way observer. Substituting $\frac{d^2 r}{d\tau^2}$ and $\frac{d^2 t}{d\tau^2}$ from the geodesic equations, we get
\begin{eqnarray}
\label{11}
\frac{d^2 r}{dt^2} = \left( \frac{\mu}{r^2} \right) \left[ \frac{3}{ \left( 1-\frac{2\mu}{r} \right)} \left( \frac{dr}{dt} \right)^2 - \left( 1-\frac{2\mu}{r} \right) \right]. \nonumber\\
\end{eqnarray}
From this, we can see that the freely falling particle acceleration is positive if
\begin{eqnarray}
\frac{d r}{dt} > \frac{1}{\sqrt{3}} \left( 1-\frac{2\mu}{r} \right).
\end{eqnarray}
This agrees with Hilbert's original result~\cite{Hilbert} which was reproduced much later on by McGruder~\cite{McGruder}.
Eq.(\ref{11}) can be simplified to give
\begin{eqnarray}
\frac{d^2 r}{dt^2} = - \left( \frac{\mu}{r^2} \right) \left( 1-\frac{2\mu}{r} \right) \left( 1 - \frac{6\mu}{r} \right), \nonumber\\
\end{eqnarray}
We infer the following from this result. Here, we have restored $c$ to make the situation easier to understand.
\begin{itemize}
\item Any particle with velocity (as measured by the far-away observer) greater than $\frac{c}{\sqrt{3}}$ at any point in the Schwarzschild field is repelled by gravity.
\item If a test particle starts from rest at infinity (where spacetime is flat for all practical purposes) and because of the black hole presence, it freely falls into the latter. During the free fall, the particle accelerates only till the velocity (as measured by the far-away observer) reaches $\frac{-2c}{3\sqrt{3}}$. At that point, its acceleration switches sign and the test particle starts decelerating and eventually (after an infinite time as measured by the far-away observer) comes to a halt at the horizon
\item If a test particle is launched toward a black hole with an initial velocity equal to or greater than $\frac{c}{\sqrt{3}}$, it decelerates all the way to the horizon.
\item When a light ray is shined at a black hole, it approaches it with ever decreasing velocity only to stop at the horizon.
\end{itemize}
\begin{table*}[htp]
\label{default}
\caption{Three views of falling into a Schwarzschild black hole}
\begin{center}
\begin{tabular}{l c c c}
\hline
\hline
Quantity & Far-away observer & Finite-distance observer & Freely-falling observer\\
\hline
Existence & Far-away & Outside Horizon & Everywhere \\
Time & $dt$ & $dt' = \left( 1 - \frac{2\mu}{r} \right)^{\frac{1}{2}} dt$ & $d\tau$ \\
Distance & $dr$ & $dr' = \left( 1 - \frac{2\mu}{r} \right)^{-\frac{1}{2}} dr$ & $c\,d\tau$ \\
Velocity & $-\left( \frac{2\mu}{r} \right)^{\frac{1}{2}} \left( 1 - \frac{2\mu}{r} \right)$ & $-\left( \frac{2\mu}{r} \right)^{\frac{1}{2}}$ & 0 \\
Maximum Velocity & $-\frac{2c}{3\sqrt{3}} $ & $-c$ & 0 \\
Velocity at horizon & {0} & {$\rightarrow c$} & {0} \\
Acceleration & $- \left( \frac{\mu}{r^2} \right) \left( 1 - \frac{2\mu}{r} \right) \left( 1 - \frac{6\mu}{r} \right) $ & $- \left( \frac{\mu}{r^2} \right) \left( 1 - \frac{2\mu}{r} \right)^{\frac{1}{2}} $ & 0 \\
Acceleration at horizon & 0 & 0 & 0 \\
Repulsion & {$r = 6 \mu $} & {Never} & {Never} \\
\hline
\hline
\end{tabular}
\end{center}
\end{table*}
\section{Conclusions}
Given general relativity overwhelming successes for over a century, what is amazing is that the theory still has some surprises for us.
This kind of gravity repulsive behavior is commonly known in two different scenarios albeit in completely different contexts. Firstly, this is reminiscent of the cosmological term in the weak field limit,
where ``Newtonian'' gravity is described by,
\begin{equation}
\nabla^2 \Phi = 4 \pi G \rho - \Lambda c^2.
\end{equation}
Hence, the acceleration due to a spherical mass $M$ gravity is given by
\begin{equation}
a = - |\overrightarrow{\nabla} \Phi| = -\frac{GM}{r^2} \, + \frac{c^2 \Lambda r}{3}.
\end{equation}
The repulsive effect comes from the second term containing the positive cosmological constant $\Lambda$.
Secondly, one may also notice that this repulsion is akin to the centrifugal force due to angular momentum in Newtonian mechanics, where the acceleration of a particle orbiting around a spherical mass $M$ is given by
\begin{equation}
a = -\frac{GM}{r^2}\,\, + \frac{h^2}{r^3}.
\end{equation}
Here the repulsive effect comes from the second term containing the angular momentum $h$.
But our case exhibits no cosmological constant and is a pure radial motion. This peculiarity is specific to relativistic gravity.
Acceleration enjoys a special status in Newtonian mechanics where space and time are absolute concepts. It has a prominent role in distinguishing between inertial and non-inertial frames. Acceleration and force are invariant under Galilean transformations and hence all observers agree on them. Since acceleration is an experimentally measurable quantity, one has to handle it with care. For more on this, we refer the reader to Petkov~\cite{Petkov}. One may say that this black hole gravitational field strange behavior in GR is simply a coordinate effect. But we know from GR general coordinate (diffeomorphism) invariance that any observer is as good as any other. What one measures in one's reference frame is as physical as the effect measured in a different frame. But what is important is that all of them are governed by the same laws of physics, i.e., Einstein field equations. These bizarre result origin lies in the fact that the quantities measured by different observers we are comparing are neither Lorentz scalars nor gauge-invariant.
Perhaps late John Wheeler would have said about this puzzling result:``surprise without surprise".
\section*{Acknowledgments}
We are grateful to Maria de F\'{a}tima Alves da Silva for facilitating this work. VHS is supported by Ci\^{e}ncia Sem Fronteiras, No. A045/2013 CAPES and PCI-DB Fellowship from CNPq at the initial stages of this work, and is currently supported by FAPERJ though Programa P\'{o}s-doutorado Nota 10. VHS would like to thank Anzhong Wang and Sofiane Faci for several enlightening discussions, and Jailson Alcaniz for hospitality at Observat\'{o}rio Nacional.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,616 |
\section{Introduction}
The minimal log discrepancy is an invariant of klt singularities.
Regardless of the simplicity of its definition,
proving statements about it is often challenging.
Shokurov proposed a conjecture regarding the ascending chain condition of minimal log discrepancies~\cite{Sho04}.
On the other hand,
Ambro conjectured that the minimal log discrepancy is lower semicontinuous~\cite{Amb99}
These conjectures imply the termination of flips~\cite{Sho04}.
Many results about termination of flips rely on
theorems which are special cases of these two conjectures (see, e.g.,~\cite{Bir07,Mor18a,HM20}).
It is expected that the boundedness of Fano varieties in dimension $n$ implies the ascending condition for minimal log discrepancies in dimension $n+1$ (see, e.g.,~\cite{BS10}). The underlying principle is that $n$-dimensional Fano type varieties are the building blocks of partial resolutions of $(n+1)$-dimensional klt singularities.
More precisely, if the minimal log discrepancy of the klt singularity is bounded away from zero,
we expect a similar behavior on the Fano type varieties extracted on a partial resolution.
Thus, it is natural to apply Birkar's boundedness of Fano varieties in this context.
This approach works especially well in the case of singularities of regularity zero, the so-called exceptional singularities~\cite{Mor18b,HLS19}.
These singularities are deformations of cones over exceptional Fano type varieties~\cite{Mor18c,HLM20}.
Given a $n$-dimensional exceptional klt singularity $(X;x)$, we can perform a blow-up $\pi\colon Y\rightarrow X$ that extracts a unique prime exceptional divisor $E$. The variety $E$ is an exceptional $(n-1)$-dimensional Fano type variety.
These varieties belong to a bounded family~\cite{Bir19}.
In~\cite{Mor18b}, we use this boundedness result to prove the existence of a curve in the smooth locus of $E$
with bounded degree with respect to $-K_E$.
A straightforward computation implies that the log discrepancy of $(X;x)$ at $E$ is bounded above by a constant only depending on the dimension.
In the general setting of $n$-dimensional klt singularities,
it is not possible to extract a bounded $(n-1)$-dimensional Fano type variety over the singularity.
In order to generalize the approach of~\cite{Mor18b},
we need to consider partial resolutions of the singularity
extracting several exceptional prime divisors.
Furthermore, we need to show that each such exceptional divisor
contains a special kind of curve on its smooth locus.
More generally, we need to understand whether we can generate the cone of curves of a Fano type variety with \textit{nice} curves.
For instance, curves which are contained in the smooth locus and have bounded degree with respect to the anti-canonical divisor.
In this direction, it is natural to look at those curves which come from fibers of Mori fiber space structures of the Fano type varieties.
Hence, we expect to find many of these curves in the $K_X$-negative domain of the cone of movable curves.
The classic cone theorem for algebraic varieties
has an analog in the case of nef curves (see, e.g.,~\cite{Bat89,Ara10,Leh12}).
Our first theorem is an enhancement of these theorems
in the case of log Calabi-Yau pairs.
The main new outcome is that we can control certain numerical invariants
of the $K_X$-extremal negative curves.
\begin{introthm}\label{introthm:nef-cone}
Let $n$ and $N$ be positive integers.
There exists $k:=k(n,N)$, only depending on $n$ and $N$,
satisfying the following.
Let $(X,B)$ be a $n$-dimensional dlt log Calabi-Yau pair
such that $N(K_X+B)\sim 0$.
Let $E_1,\dots,E_r$ be the prime components of $\lfloor B\rfloor$.
Let $B'=B-\lfloor B\rfloor$.
There are countable many $(K_X+B')$-negative curves $C_i$, with $i\in \mathbb{Z}$, satisfying the following conditions:
\begin{enumerate}
\item the curve $C_i$ is movable and lies in the smooth locus of $X$,
\item the curve $C_i$ is either disjoint from $B'$ or intersect it transversally with $B'\cdot C_i \leq k$,
\item for every $j\in \{1,\dots,r\}$ the curve $C_i$ is either disjoint from $E_j$ or intersect it transversally with $E_j \cdot C_i \leq k$,
\item for each subset $Z\subset X$ of codimension at least two
and $x\in X$ general, we can find $C_i'\equiv C_i$ with $C_i'\cap Z=\emptyset$ and $x\in C_i'$, and
\item $C_i$ intersects at least one and at most $n+1$ of the $E_j$'s.
\end{enumerate}
Moreover, the following equality holds:
\begin{equation}\label{eq:nef-cone}
\overline{NE}_1(X)_{K_X+B' \geq 0} +
\overline{NM}_1(X) =
\overline{NE}_1(X)_{K_X+B' \geq 0} +
\overline{\sum_{i\in \mathbb{Z}}\mathbb{R}_{\geq 0}[C_i]}.
\end{equation}
Furthermore, the rays $\mathbb{R}_{\geq 0}[C_i]$ only accumulates to hyperplanes that support both
$\overline{NM}_1(X)$ and the cone
$\overline{NE}_1(X)_{K_X+B' \geq 0}$.
\end{introthm}
We stress that in the context of Theorem~\ref{introthm:nef-cone}, we crucially use the dlt condition.
Indeed, the dlt condition and the control of the index will allow us
to control the singularities of $X$. Then, we can apply Birkar's boundedness of Fano varieties
in the general fiber of a birational Mori fiber space structure of $X$.
This boundedness result will let us deduce (1)-(4) in the statement of Theorem~\ref{introthm:nef-cone}.
The upper bound on the number of prime components of $\lfloor B\rfloor$ that $C_i$ can intersect
is a consequence of a computation using the complexity of log pairs (see, e.g.,~\cite{Kol92,BMSZ18,RS21}).
It is intuitive to ask how many, if any, curves $C_i$ do we need
to obtain the equality~\eqref{eq:nef-cone}.
In principle, the more divisorial log canonical centers we have,
the more curves $C_i$ we expect to need.
In the case that $\lfloor B\rfloor$ is disconnected,
we know that it must have at most two components~\cite[Theorem 1.2]{Bir21}.
Furthermore, if $\lfloor B\rfloor$ has exactly two components,
then the pair $(X,B)$ is birational to a $\mathbb{P}^1$-link (see Definition~\ref{def:p1-links}).
The next theorem asserts that such $\mathbb{P}^1$-link structure is unique.
This means that we can find a unique canonically-negative extremal movable curve in the cone~\eqref{eq:nef-cone}.
\begin{introthm}\label{introthm:uniqueness}
Let $(X,B)$ be a dlt log Calabi-Yau pair.
Assume that the set of log canonical centers of $(X,B)$ is disconnected.
Let $B'=B-\lfloor B\rfloor$.
Then, there exists a unique $(K_X+B')$-negative extremal nef curve $C_0$
for which
\[
\overline{NE}_1(X)_{K_X+B' \geq 0} +
\overline{NM}_1(X) =
\overline{NE}_1(X)_{K_X+B' \geq 0} +
\mathbb{R}_{\geq 0}[C_0].
\]
Furthermore, the following intersection properties hold:
\[
B'\cdot C_0=0,\quad
E_1\cdot C_0=1,\quad
E_2\cdot C_0=1,\text{ and }\quad
K_X\cdot C_0=-2.
\]
In particular, if $-(K_X+B')$ is ample, then
$X\simeq \mathbb{P}^1$, $B'=0$, and $B=\{0\}+\{\infty\}$.
\end{introthm}
The regularity of a klt singularity is the largest dimension among the dual complexes of lc complements.
If the regularity is equal to the dimension minus $1$,
then we expect these singularities to behave similarly to toric singularities~\cite{Mor20b}.
On the other hand, singularities of regularity zero are exceptional~\cite{Mor18b}.
There has been some recent development in the topology of klt singularities~\cite{LLM19,Bra20,BFMS20,BM21,Mor21}.
In particular, for a $n$-dimensional klt singularity $(X;x)$, we know that its regional fundamental group is almost a finite abelian group of rank at most ${\rm reg}(X;x)$.
Thus, the regularity controls to a large extent the topology of the singularity.
We also expect the regularity to be a central invariant in the study of minimal log discrepancies.
In the case that $(X;x)$ is a $n$-dimensional klt singularity of regularity one,
we can choose a $N$-complement $(X,B)$ so that $\mathcal{D}(X,B)$ is either a circle or a closed interval (see Definition~\ref{def:complement}).
Taking a dlt modification $(Y,B_Y)$ of $(X,B;x)$
and performing adjunction to each of the exceptional divisors,
we arise to the situation of Theorem~\ref{introthm:nef-cone}.
In this case, most of the exceptional divisors of $\pi\colon Y\rightarrow X$ are birational to $\mathbb{P}^1$-links, i.e., they are models as in Theorem~\ref{introthm:uniqueness}.
If they exist, the exceptional divisors corresponding to the boundary of the dual complex $\mathcal{D}(Y,B_Y)$ will be plt log Calabi-Yau pairs
with a unique log canonical center.
Hence, we can find some nice curves in these exceptional divisors by applying Theorem~\ref{introthm:nef-cone} and Theorem~\ref{introthm:uniqueness}.
Thus, there is a natural curve to choose corresponding to each vertex of the dual complex. Then,
we can write a linear system of equations that
allows us to compute each $a_{E_i}(X)$.
The value $a_{E_i}(X)$ will be expressed
in terms of the intersection of the aforementioned curves with the
components of $B_Y$.
Nevertheless, this linear system of equations is not entirely trivial.
For instance, the number of variables can go up as we consider different klt singularities of regularity one.
To remedy this, we will construct a surface toric singularity whose resolution of singularities contains the same combinatorial data (i.e., the same linear system) as that of $(Y,B_Y)$.
This will allow us to conclude the following theorem.
\begin{introthm}\label{introthm:ACC}
Let $n$ be a positive integer.
There exists a constant $N:=N(n)$,
only depending on $n$, which satisfies the following.
Let $\mathcal{M}_{n,1}$ be the set of
minimal log discrepancies
of $n$-dimensional $\mathbb{Q}$-factorial klt singularities of regularity one.
Then, the set
\[
\mathcal{M}_{n,1}\cap \left(0,\frac{1}{N} \right)
\]
satisfies the ascending chain condition.
\end{introthm}
Among all exceptional divisors over a klt germ $(X;x)$ those that compute a log canonical place of an lc pair $(X,B;x)$ play a central role in the proof of the previous theorem.
These divisors enjoy many good properties. They carry the structure of log Calabi-Yau pairs~\cite{Hac14}.
Furthermore, they can be extracted by a projective birational morphism in such a way that they are the unique exceptional divisor~\cite{Mor20}.
In order to prove Theorem~\ref{introthm:ACC}, we will show that the minimum among log discrepancies at divisors that compute log canonical places satisfies the ascending chain condition.
In particular, for every klt singularity of regularity one,
we will prove the existence of such a divisor with log discrepancy bounded above by a constant only depending on the dimension.
\begin{introcor}\label{introcor:bounded-ext}
Let $n$ be a positive integer.
There exists a constant $a(n)$, only depending on $n$, which satisfies the following.
Let $(X;x)$ be a $n$-dimensional $\mathbb{Q}$-factorial klt singularity of regularity one.
Then, there exists a projective birational morphism
$\pi\colon Y\rightarrow X$, with purely divisorial exceptional loci,
which extracts a unique divisor $E$ with $a_E(X;x)\leq a(n)$.
\end{introcor}
Once the ascending chain condition is established,
we can study the accumulation points of the set of minimal log discrepancies.
In the case of exceptional singularities~\cite{Mor18b}, the only accumulation point is zero.
In the case of singularities of regularity one, we prove that the accumulation points of the set of minimal log discrepancies only accumulate to zero.
\begin{introcor}\label{introcor:accum}
Let $n$ be a positive integer.
There exists a constant $N:=N(n)$,
only depending on $n$,
which satisfies the following.
Let $\mathcal{M}_{n,1}$ be the set of minimal log discrepancies of $n$-dimensional $\mathbb{Q}$-factorial klt singularities of regularity one.
Then, the accumulation points of the set
\[
{\rm Acc}\left(
\mathcal{M}_{n,1} \cap \left(0,\frac{1}{N}\right)
\right)
\]
only accumulate to zero.
\end{introcor}
We expect the circle of ideas introduced in this paper to help to prove the corresponding statements in the general case of klt singularities.
This will be considered in forthcoming papers by the author.
\subsection*{Acknowledgements}
The author would like to thank Brian Lehmann, Mihai Fulger, and
Vyacheslav Shokurov, for many useful comments.
\section{Preliminaries}
In this section,
we recall some preliminaries
that will be used in this article:
singularities of the minimal model program,
theory of complements,
dual complexes,
regularity, and toric surface singularities.
Throughout this article, we work over an algebraically closed field $\mathbb{K}$ of characteristic zero.
All the considered varieties are normal and quasi-projective unless otherwise stated.
\subsection{Singularities and complements}
In this subsection, we recall the basic notions of the singularities of the minimal model program and the definition of complements (see, e.g.,~\cite{KM98,Kol13,Bir20}).
\begin{definition}
{\em
Let $f\colon Y\rightarrow X$ be a morphism.
We say that $f$ is a contraction if $f_*\mathcal{O}_Y=\mathcal{O}_X$.
In particular, if $Y$ is normal and $f$ is a contraction, then $X$ is normal.
A {\em fibration} is a contraction with positive dimensional general fiber.
}
\end{definition}
\begin{definition}
{\em
A {\em sub-pair} is a couple $(X,\Delta)$ where $X$ is a normal quasi-projective variety and $\Delta$ is a $\mathbb{Q}$-divisor such that $K_X+\Delta$ is a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor.
A {\em pair} (or {\em log pair}) is a sub-pair with $\Delta\geq 0$.
We may write $(X,\Delta;x)$ to denote a sub-pair with a base point.
}
\end{definition}
\begin{definition}
{\em
Let $X$ be a normal quasi-projective variety.
A {\em prime divisor over $X$} is a prime divisor which lies in a normal variety admitting a projective birational morphism to $X$.
This means that we can find a projective birational morphism $\pi\colon Y\rightarrow X$ so that $E\subset Y$ is a prime divisor.
The center of $E$ on $X$ is the image of $E$ on $X$ and is denoted by $c_X(E)$.
Let $(X,\Delta)$ be a log pair.
Let $E$ be a prime divisor over $X$.
The {\em log discrepancy} of $(X,\Delta)$ at $E$ is defined to be
\[
a_E(X,\Delta):=
1-{\rm coeff}_E(K_Y-\pi^*(K_X+\Delta)).
\]
Here, as usual, we pick $K_Y$ so that
$\pi_* K_Y=K_X$.
Let $(X,\Delta)$ be a log pair.
A {\em log resolution} of $(X,\Delta)$ is a projective birational morphism $\pi\colon Y\rightarrow X$ satisfying the following conditions:
\begin{enumerate}
\item $Y$ is a smooth variety,
\item the exceptional locus of $\pi$ is purely divisorial, and
\item $\pi^{-1}_*\Delta + {\rm Ex}(\pi)_{\rm red}$ is a reduced divisor with simple normal crossing support.
\end{enumerate}
By Hironaka's resolution of singularities,
we know that any log pair admits a log resolution.
}
\end{definition}
\begin{definition}
{\em
Let $(X,\Delta)$ be a log pair.
We say that $(X,\Delta)$ is {\em Kawamata log terminal} (or {\em klt} for short) if all its log discrepancies are positive.
This means that $a_E(X,\Delta)>0$ for every prime divisor $E$ over $X$.
It is known that a log pair $(X,\Delta)$ is klt if and only if all
the log discrepancies corresponding
to prime divisors on a log resolution of $(X,\Delta)$ are positive.
Analogously, we say that a log pair $(X,\Delta)$ is {\em log canonical} (or {\em lc} for short) if all its log discrepancies are non-negative, i.e.,
$a_E(X,\Delta)\geq 0$ for every prime divisor $E$ over $X$.
A pair $(X,\Delta)$ is log canonical if and only if all the log discrepancies corresponding to divisors on a log resolution are non-negative.
A pair $(X,\Delta)$ is said to be {\em $\epsilon$-log canonical} if all its log discrepancies are at least $\epsilon$.
}
\end{definition}
\begin{definition}
{\em
Let $(X,\Delta;x)$ be a sub-pair.
We define the {\em minimal log discrepancy}
of $(X,\Delta)$ at $x$ to be
\[
{\rm mld}(X,\Delta;x) :=
\min\{ a_E(X,\Delta) \mid c_E(X)=x\}.
\]
}
\end{definition}
\begin{definition}
{\em
Let $(X,\Delta)$ be a log pair.
A {\em log canonical place} of $(X,\Delta)$ is a prime divisor $E$ over $X$ so that $a_E(X,\Delta)=0$.
A {\em non-klt place} of $(X,\Delta)$ is a prime divisor $E$ over $X$ so that
$a_E(X,\Delta)\leq 0$.
A {\em log canonical center} of $(X,\Delta)$
is the image on $X$
of a log canonical place of $(X,\Delta)$.
A {\em non-klt center} of $(X,\Delta)$
is the image on $X$
of a non-klt place of $(X,\Delta)$.
}
\end{definition}
\begin{definition}
{\em
A log pair $(X,\Delta)$ is said to be {\em divisorially log terminal} (or {\em dlt})
if there exists an open subset $U\subset X$ which satisfies the following conditions:
\begin{enumerate}
\item $U$ is smooth and $\Delta|_U$ has simple normal crossing support,
\item the coefficients of $\Delta$ are less or equal than one, and
\item all the non-klt centers of $(X,\Delta)$ intersect $U$ and are given by strata of $\lfloor \Delta \rfloor$.
\end{enumerate}
A pair $(X,\Delta)$ is said to be
{\em purely log terminal} or {\em plt} for short,
if it is dlt and all its log canonical centers are divisors on $(X,\Delta)$.
}
\end{definition}
The following lemma is well-known
and usually referred to as the existence of $\mathbb{Q}$-factorial dlt modifications (see, e.g.,~\cite[Theorem 3.1]{KK10}).
\begin{lemma}\label{lem:existence-dlt-mod}
{\em
Let $(X,\Delta)$ be a log canonical pair.
Then, there exists a projective birational morpism $\pi\colon Y\rightarrow X$ so that the following conditions are satisfied:
\begin{enumerate}
\item $Y$ is $\mathbb{Q}$-factorial,
\item the exceptional locus of $\pi$ is purely divisorial,
\item every prime divisor $E$ contracted by $\pi$ satisfies $a_E(X,\Delta)=0$, i.e., $E$ is a log canonical place of $(X,\Delta)$, and
\item the pair $(Y,\Delta_Y)$ is dlt where $K_Y+\Delta_Y = \pi^*(K_X+\Delta)$.
\end{enumerate}
}
\end{lemma}
We conclude this subsection by recalling the definitions of complements.
\begin{definition}
\label{def:complement}
{\em
Let $X\rightarrow Z$ be a projective morphism
and $z\in Z$ be a closed point.
Let $(X,\Delta)$ be a pair.
We say that $B\geq \Delta$ is a {\em $\mathbb{Q}$-complement} of $(X,\Delta)$ over $z\in Z$, if the following conditions are satisfied:
\begin{enumerate}
\item $(X,B)$ has log canonical singularities, and
\item $K_X+B\sim_{\mathbb{Q},Z} 0$ holds over a neighborhood of $z\in Z$.
\end{enumerate}
In the case that $X\rightarrow Z$ is the identity morphism, then a $\mathbb{Q}$-complement is nothing else than a pair $(X,B)$
with $B\geq \Delta$, which is log canonical around the point $x\in X$.
We say that $B\geq \Delta$ is a {\em $N$-complement} of $(X,\Delta)$ over $Z$, if the following conditions are satisfied:
\begin{enumerate}
\item $(X,B)$ has log canonical singularities, and
\item $N(K_X+B)\sim_Z 0$ holds over a neighborhood of $z\in Z$.
\end{enumerate}
In the case that $X\rightarrow Z$ is the identity morphism, then a $N$-complement is a log canonical pair $(X,B)$ for which $B\geq \Delta$ and $N(K_X+B)\sim 0$ around $x\in X$.
}
\end{definition}
\subsection{Rationally connected varieties}
In this subsection, we prove a lemma regarding rationally connected varieties admitting a log Calabi-Yau klt pair structure.
In this article, we only use a weak form of Lemma~\ref{lem:RC-curve} where (3) is not considered.
We expect the full strength of the lemma to be useful in forthcoming research.
\begin{lemma}\label{lem:acyclic}
Let $X$ be a projective rationally connected variety with klt singularities.
Then, $\mathcal{O}_X$ is an acyclic sheaf.
\end{lemma}
\begin{proof}
The statement is known for smooth projective rationally connected varieties.
Let $Y\rightarrow X$ be a resolution of $X$. Note that $Y$ is also rationally connected~\cite[Corollary 4.18]{Deb01}.
Since $X$ has klt singularities, by the Leray spectral sequence, we have that $H^i(X,\mathcal{O}_X)\simeq H^i(Y,\mathcal{O}_Y)=0$ for every $i\geq 1$.
\end{proof}
\begin{lemma}\label{lem:RC-curve}
Let $n$ and $N$ be two positive integers.
There exists a constant $k:=k(n,N)$,
only depending on $n$ and $N$,
satisfying the following.
Let $X$ be a $n$-dimensional rationally connected variety and $B$ a boundary on $X$ for which $(X,B)$ is a log Calabi-Yau klt pair with $N(K_X+B)\sim 0$.
Le $E$ be a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor which is not numerically trivial.
Then, there exists a curve $C\subset X$ satisfying the following conditions:
\begin{enumerate}
\item $C$ is contained in the smooth locus of $X$,
\item $C$ is either disjoint from $B$ or it intersects $B$ transversally in at most $k$ points, and
\item the divisor $E$ intersects $C$ non-trivially.
\end{enumerate}
\end{lemma}
\begin{proof}
By ~\cite[Theorem 1.4]
{BDCS20}, we know that
$(X,B)$ is log bounded up to flops.
Let $(\mathcal{X},\mathcal{B}) \rightarrow T$ be a log bounding pair, up to flops, for the projective varieties as in the statement of the lemma.
By Noetherian induction, we may assume that $\mathcal{X}\rightarrow T$ is a smooth projective morphism.
Moreover, we may assume that every fiber $(\mathcal{X}_t,\mathcal{B}_t)$ is a klt log Calabi-Yau pair with
$N(K_{\mathcal{X}_t}+\mathcal{B}_t)\sim 0$ and that $\mathcal{X}_t$ is rationally connected.
Let $\mathcal{X}'\rightarrow T$ be a small $\mathbb{Q}$-factorialization of $\mathcal{X}\rightarrow T$.
Up to shrinking $T$, we may assume that $\mathcal{X}'_t\rightarrow \mathcal{X}_t$ is a small $\mathbb{Q}$-factorialization for every $t$.
Hence, we may assume that fibers are $\mathbb{Q}$-factorial.
By~\cite[Proposition 2.8.(1)]{HX15},
up to a finite base change,
we may assume that the homomorphism
induced by restriction
\begin{equation}\label{n1-isom}
N_1(\mathcal{X}'/T) \rightarrow
N_1(\mathcal{X}'_t)
\end{equation}
is an isomorphism for every $t\in T$.
By duality, we obtain that
\begin{equation}\label{n^1-isom}
N^1(\mathcal{X}'/T) \rightarrow
N^1(\mathcal{X}'_t)
\end{equation}
is an isomorphism for every $t\in T$.
We note that~\cite[Proposition 2.8.(1)]{HX15} requires the morphism to be of Fano type. However, the acyclicity of the structure sheaf of the fibers suffices for the proof.
This holds by Lemma~\ref{lem:acyclic}.
Let $d$ be the relative dimension of $\mathcal{X}'\rightarrow T$
and let $r$ be the relative Picard rank of $\mathcal{X}$ over $T$.
For each $t$, the map
$\phi_t \colon N^1(\mathcal{X}'_t)\rightarrow N_1(\mathcal{X}'_t)$ sending an ample divisor $A$ to $A^{d-1}$ is a
local diffeomorphism at every ample class (see, e.g.,~\cite[Remark 2.3]{LX16}).
By isomorphisms~\eqref{n1-isom} and~\eqref{n^1-isom},
we have that
\[
\Phi_t \colon
N^1(\mathcal{X}'/T) \rightarrow
N^1(\mathcal{X}'_t) \rightarrow
N_1(\mathcal{X}'_t) \rightarrow
N_1(\mathcal{X}'/T)
\]
is a local diffeomorphism around every relatively ample class.
Fix $t_0\in T$ and let $H$ be a relatively ample class on $\mathcal{X}'$ over $T$.
Let $A_1,\dots, A_r$ be relatively ample classes on $N^1(\mathcal{X}'/T)$ close to $H$ so that $\Phi_{t_0}(A_1),\dots,
\Phi_{t_0}(A_r)$ are linearly independent.
Assume that for some $t\in T$, the curve classes \[
A_1^{d-1}|_{\mathcal{X}'_t},
\dots
A_r^{d-1}|_{\mathcal{X}'_t}
\]
are not linearly independent.
Let $D_t$ be a non-zero class in $N^1(\mathcal{X}'_t)$ so that
\[
D_t\cdot A_1^{d-1}|_{\mathcal{X}'_t} =\dots =
D_t \cdot A_r^{d-1}|_{\mathcal{X}'_t} =0.
\]
Let $D$ be the unique non-zero lifting of $D_t$ to $N^1(\mathcal{X}'/T)$.
The intersection numbers
\[
0=D_t\cdot A_i^{d-1}|_{\mathcal{X}'_t} =
(D\cdot A^{d-1})|_{\mathcal{X}'_t}
\]
are independent of the fiber.
We conclude that $D|_{X_{t_0}}$ is numerically trivial. Then
$D|_{X_t}$ must be numerically trivial as well.
This is a contradiction.
We conclude that $\Phi_t(A_1),\dots,\Phi_t(A_r)$ span $N_1(\mathcal{X}'_t)$ for every $t$.
Observe that for every closed subvariety $Z\subset \mathcal{X}'_t$ of codimension at least $2$, we can choose the irreducible curves $\Phi_t(A_1),\dots,\Phi_t(A_r)$ to be disjoint from $Z$.
Hence, the strict transforms of these curves on $\mathcal{X}_t$ generate $N_1(\mathcal{X}_t)$.
Furthermore, the intersection numbers
\[
\mathcal{B}_t \cdot \Phi_t(A_1),
\dots ,
\mathcal{B}_t \cdot \Phi_t(A_r)
\]
are bounded above by a constant only depending on the log bounding family.
Since the log bounding family only depends on $n$ and $N$,
we conclude that there exists a constant $k:=k(n,N)$ bounding the above intersection numbers.
By construction, we know that each $\Phi_t(A_i)$ intersects $\mathcal{B}_t$ transversally.
By abuse of notation, we denote by
$\Phi_t(A_i)$ the corresponding curves on $\mathcal{X}_t$.
Let $(X,B)$ be a log pair as in the statement of the proposition.
We know that there exists a composition of flops
$\phi\colon (X,B)\dashrightarrow (\mathcal{X}_t,\mathcal{B}_t)$ so
that $(\mathcal{X}_t,\mathcal{B}_t)$ is isomorphic to a fiber of the log bounding family
$(\mathcal{X},\mathcal{B})\rightarrow T$.
By the first paragraph, we know that there exists curves
$\Phi_t(A_1),\dots, \Phi_t(A_r)$ which are movable on $\mathcal{X}_t$ and generate $N_1(\mathcal{X}_t)$.
Furthermore, by replacing each $\Phi_t(A_i)$ with a numerically equivalent curve, we may assume that the following conditions are satisfied:
\begin{enumerate}
\item[(i)] Each $\Phi_t(A_i)$ lies in the smooth locus of $\mathcal{X}_t$,
\item[(ii)] each $\Phi_t(A_i)$ is disjoint from ${\rm Ex}(\phi^{-1})$, and
\item[(iii)] $\mathcal{B}_t \cdot \Phi_t(A_i)\leq k$ for each $i$.
\end{enumerate}
Since $E$ is not numerically trivial in $X$,
then its push-forward $E_t$ on $\mathcal{X}_t$ is not numerically trivial.
Thus, $E_t\cdot \Phi_t(A_i)\neq 0$ for some $i\in \{1,\dots,r\}$.
We may assume that $i=1$.
Let $C$ be the strict transform of
$\Phi_t(A_1)$ in $X$.
By (i) and (ii), we have that
$C$ lies in the smooth locus of $X$.
This gives us (1).
On the other hand, we have that
\[
E\cdot C = E_t \cdot \Phi_t(A_1) \neq 0.
\]
Furthermore,
we have that
\[
B\cdot C = \mathcal{B}_t\cdot \Phi_t(A_1) \leq k,
\]
and the intersection of $B$ with $C$ is transversal. This gives us $(2)$ and $(3)$, concluding the proof.
\end{proof}
\subsection{Complexity}
In this subsection, we recall the definition of the complexity of a log pair.
We recall the main theorem about the complexity
which states that the complexity is always non-negative (see, e.g.,~\cite{Kol92,BMSZ18,RS21}).
\begin{definition}
{\em
Let $(X,B;x)$ be a log canonical pair.
Write $B=\sum_{i=1}^r b_iB_i$, where the $b_i$'s are non-negative real numbers and the
$B_i$'s are effective Weil divisors.
The {\em complexity} of $(X,B)$ at $x$ is defined to be
\[
c(X,\Delta;x):=
\dim X + \rho(X_x) - \sum_{i=1}^r b_i.
\]
Here, $X_x$ is the localization of $X$ at the closed point $x$.
}
\end{definition}
The following theorem is proved in~\cite[Theorem 1]{RS21}.
\begin{theorem}
\label{thm:comp}
Let $(X,B;x)$ be a log canonical singularity.
Then $c(X,B;x)\geq 0$.
Furthermore, if $c(X,B;x)=0$, then
$(X,\lfloor B\rfloor)$ is formally toric at the point $x$.
In particular, if $B$ is a reduced divisor, then it has at most $\dim X + \rho(X_x)$ prime components.
\end{theorem}
\subsection{Dual complexes}
In this subsection, we recall the definition of the dual complex of a log canonical pair.
We refer the reader to~\cite{KX16,dFKX17} for the classics about dual complexes of log Calabi-Yau pairs and singularities.
In~\cite{FS20,Bir20}, the authors prove connectedness results about the dual complexes of log Calabi-Yau pairs.
\begin{definition}
\label{def:dual-complex}
{\em
Let $E$ be a projective scheme.
Assume that $E$ is pure-dimensional
and let $E_1,\dots,E_r$ be its irreducible components.
We assume that each $E_i$ is a normal variety.
Moreover, for every $J\subset \{1,\dots,r\}$,
if the intersection $\cap_{j\in J}E_j$ is non-empty, then every connected component of this set is irreducible and has codimension equal to $|J|-1$ as a subvariety of $E$.
The {\em dual complex}
$\mathcal{D}(E)$ of $E$ is defined as follows.
The vertices $v_1,\dots,v_r$ are in one-to-one correspondence with the prime components $E_1,\dots,E_r$.
Given an irreducible component $W$ of the intersection $\cap_{j\in J}E_j$, we associate a cell $v_W$ of dimension $|J|-1$.
Note that for each $i\in J$, the variety $W$
is contained in a unique irreducible component of $\cap_{j\in J\setminus \{i\}}E_i$.
This determines the gluing of the cell $v_W$.
The dual complex $\mathcal{D}(E)$ is a CW complex.
Let $(X,B)$ be a log canonical pair.
Let $(Y,B_Y)$ be a dlt modification of $(X,B)$ (see Lemma~\ref{lem:existence-dlt-mod}).
We define the dual complex of $(X,B)$ to be
\[
\mathcal{D}(X,B):=
\mathcal{D}(\lfloor B_Y\rfloor).
\]
Here,
the dual complex of $\lfloor B_Y\rfloor$ is defined as in the first paragraph.
By~\cite[Lemma 2.32]{FS20}, we know that $\mathcal{D}(X,B)$ is independent of the dlt modification up to simple homotopy equivalence.
}
\end{definition}
The following lemma will be used to control the dual complex of lc singularities of regularity one (cf.~\cite[Corollary 24]{dFKX17}).
\begin{lemma}
\label{lem:dual-comp-coll}
Let $(X,\Delta;x)$ be a klt singularity.
Let $Y\rightarrow X$ be a projective birational morphism which is an isomorphism in $X\setminus \{x\}$.
Let $E$ be the reduced exceptional divisor.
Assume that $(Y,E+\Delta_Y)$ has dlt singularities, where
$\Delta_Y$ is the strict transform of $\Delta$ on $Y$.
Then, the dual complex $\mathcal{D}(Y,E)$ is collapsible.
\end{lemma}
\subsection{Regularity}
In this subsection, we recall the definition of the regularity of a klt singularity.
We prove several lemmas about the regularity of klt singularities and their relation with the theory of complements.
\begin{definition}
{\em
Let $(X,B)$ be a log canonical pair.
In~\cite[Lemma 4.2]{Mor20}, we showed that we can find a simplicial representative of $\mathcal{D}(X,B)$.
Furthermore, we can assume that all maximal dimensional simplices of
$\mathcal{D}(X,B)$ have
the same dimension $r$.
The non-negative interger $r$ is
defined to be the {\em regularity} of $\mathcal{D}(X,B)$.
We denote the regularity by $\mathcal{D}(X,B)$ (see~\cite[Definition 7.11]{Sho00}).
In the case that $\mathcal{D}(X,B)$ is empty,
for instance, if $(X,B)$ is klt, then we set
${\rm reg}(X,B):=-\infty$.
From the definition, it follows that
${\rm reg}(X,B)\in \{-\infty,0,\dots,\dim X-1\}$.
The upper bound on the regularity follows from Theorem~\ref{thm:comp}.
Let $(X,\Delta;x)$ be a klt singularity.
We define its regularity
as the maximum among the regularities
${\rm reg}(X,B;x)$ so that
$(X,B)$ is log canonical at $x$ and $B\geq \Delta$.
This means that
\[
{\rm reg}(X,\Delta;x) := \max
\left\{
{\rm reg}(X,B;x) \mid
(X,B;x) \text{ is lc and $B\geq \Delta$ }
\right\}.
\]
In other words, the regularity of a pair
is the maximum of the regularities
among its $\mathbb{Q}$-complements.
Note that the in
the case of singularities,
we have that
${\rm reg}(X,\Delta;x)\in \{0,\dots,\dim X-1\}$.
Indeed, we can always produce a log canonical center through the point $x$.
}
\end{definition}
The following lemma shows that a divisor computing the regularity must also compute a log canonical threshold at the closed point.
\begin{lemma}\label{lem:reg-comp-lcc}
Let $(X,\Delta;x)$ be a klt singularity.
Assume that $X$ is $\mathbb{Q}$-factorial.
Let $B\geq \Delta$ be an effective divisor so that
\[
{\rm reg}(X,\Delta;x)=
{\rm reg}(X,B;x).
\]
Then, $x$ is a log canonical center of $(X,B;x)$.
\end{lemma}
\begin{proof}
Let $r={\rm reg}(X,\Delta;x)$.
Without loss of generality, we may assume that all the log canonical centers of $(X,B;x)$ pass through $x$.
Let $(Y,B_Y)$ be a dlt modification of $(X,B)$.
We denote by $p\colon Y\rightarrow X$ the corresponding projective birational morphism.
Then, we can find $r+1$ prime components $E_1,\dots,E_{r+1}$ of $\lfloor B_Y\rfloor$
with non-empty intersection.
Let $Z$ be the image of $E_1\cap\dots\cap E_{r+1}$ on $X$.
We can find an effective $\mathbb{Q}$-Cartier divisor $\Gamma$ through $x\in X$ so that $(X,B+\Gamma)$ is log canonical and $x$ is a log canonical center of $(X,B+\Gamma)$.
Write
\[
K_Y+B_Y+\Gamma_Y = p^*(K_X+B+\Gamma).
\]
Then, there is a log canonical center of $(Y,B_Y+\Gamma_Y)$ which maps onto $x$.
By~\cite[Theorem 1]{Mor20}, we can find a projective birational morphism
$\phi\colon Y'\rightarrow Y$ which extracts a unique log canonical place of $(Y,B_Y+\Gamma_Y)$ whose center on $X$ is $x$.
Let $E_0$ be such prime divisor.
We run a $(-E_0)$-MMP over the base.
Let $Y'\dashrightarrow Y''$ be this minimal model program.
Let $(Y'',B_{Y''}+\Gamma_{Y''})$ be the log pull-back of $(X,B+\Gamma)$ to $Y''$.
We denote by $E''_0,\dots,E''_{r+1}$ the strict transforms of the components $E_0,E_1,\dots,E_{r+1}$, respectively.
Then, the fiber of $Y''\rightarrow X$ over $x$ equals $E''_0$, which is a log canonical center of $(X,B+\Gamma)$.
By construction, this minimal model program is an isomorphism over the strict transform of $Z$.
Hence, $E''_1\cap \dots \cap E''_{r+1}$ is non-trivial and must intersect the fiber over $x$.
Thus, we have that
\[
E''_0\cap \dots\cap E''_{r+1}\neq \emptyset.
\]
We conclude that ${\rm reg}(X,B+\Gamma;x) > {\rm reg}(X,B;x)$, leading to a contradiction.
\end{proof}
The following lemma asserts that given a klt singularity, we can find a $N$-complement which computes its regularity, where $N$ only depends on the dimension of the singularity (and possibly the coefficients set).
\begin{lemma}\label{lem:reg-bounded-comp}
Let $n$ be a positive integer.
Let $\Lambda\subset \mathbb{Q}$ be a set satisfying the descending chain condition with rational accumulation points.
There exists a constant $N:=N(n,\Lambda)$, only depending on $n$ and $\Lambda$, satisfying the following.
Let $(X,\Delta;x)$ be a $n$-dimensional klt singularity
so that ${\rm coeff}(\Delta)\subset \Lambda$.
Then, there exists a $N$-complement of $(X,\Delta;x)$ that computes
${\rm reg}(X,\Delta;x)$.
\end{lemma}
\begin{proof}
Let $r={\rm reg}(X,\Delta;x)$.
Let $B'\geq \Delta$ be a boundary that computes the regularity of $(X,\Delta;x)$.
Let $(Y,B'_Y)$ be a dlt modification of $(X,B';x)$.
We denote by $p\colon Y\rightarrow X$ the corresponding projective birational morphism.
By construction, we can find $r+1$ prime components $E_1,\dots,E_{r+1}$ of $\lfloor B'_Y\rfloor$ that intersect non-trivially.
We define
\[
\Delta_{Y,d}:= p^{-1}_*\Delta+ \left(1-\frac{1}{d}\right){\rm Ex}(p),
\]
for every $d\in \mathbb{Z}_{\geq 1}$.
Then, the pair $(Y,\Delta_{Y,d})$ is a klt pair.
By~\cite[Theorem 2.19]{Mor20a}, we can find a $N$-complement for
$(Y,\Delta_{Y,d})$ over $x$, where $N$ only depends on $n$ and $\Lambda$.
We let $(Y,\Gamma_{Y,d})$ be the $N$-complement.
Then, we have that $(Y,\Gamma_{Y,d})$ is log canonical and
\[
N(K_Y+\Gamma_{Y,d})\sim_X 0.
\]
Let $(X,\Gamma_d)$ be the log pair obtained by pushing-forward $(Y,\Gamma_{Y,d})$ to $X$.
Observe that $(X,\Gamma_d)$ is log canonical and
$N(K_X+\Gamma_d)\sim 0$ around the point $x\in X$.
If $d>N$, then for each $i\in \{1,\dots,r+1\}$, we have that
\[
1-{\rm coeff}_{E_i}(\Gamma_{Y,d})=a_{E_i}(X,\Gamma_d)=0.
\]
We conclude that $(X,\Gamma_d;x)$ is a $N$-complement of $(X,\Delta;x)$ of regularity $r$ provided that $d>N$.
\end{proof}
\subsection{Regularity one} In this subsection, we prove some lemmas regarding the geometry of klt singularities and log Calabi-Yau pairs of regularity one.
The following Lemma characterizes the combinatorial structure of the possible log canonical centers of an lc singularity of regularity one.
\begin{lemma}\label{lem:class-dual-comp}
Let $(X,B;x)$ be a log canonical pair of regularity one.
Assume that $(X,\Delta;x)$ is klt for some $\Delta \leq B$.
Let $(Y,B_Y)$ be the dlt modification of $(X,B;x)$ (see Lemma~\ref{lem:existence-dlt-mod}).
Assume that $(Y,B_Y)$ has at least one divisorial log canonical center mapping onto $\{x\}$
and that every log canonical center of $(X,B;x)$ passes through $x$.
Write $E_1,\dots,E_r$ for the prime components of $\lfloor B_Y\rfloor$.
Then, one of the following holds:
\begin{enumerate}
\item [(i)] $\mathcal{D}(Y,B_Y)$ is a closed interval: $E_1$ maps onto a log canonical center $Z_1\supsetneq x$, $E_r$ maps onto a log canonical center
$Z_r\supsetneq x$,
$Z_1\neq Z_r$, and each $E_i$ with $i\in \{2,\dots,r-1\}$ maps onto $x$,
\item[(ii)] $\mathcal{D}(Y,B_Y)$ is a circle: $E_1$ maps onto a log canonical center $Z\supsetneq x$ and every other $E_i$ maps onto $x$,
\item [(iii)] $\mathcal{D}(Y,B_Y)$ is a closed interval:
$E_1$ maps onto a log canonical center
$Z_1\supsetneq x$ and each $E_i$
with $i\geq 2$ maps onto $x$, or
\item [(iv)] $\mathcal{D}(Y,B_Y)$ is a closed interval: each $E_i$ with $i\in \{1,\dots,r\}$ maps onto $x$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\phi\colon Y\rightarrow X$ be the projective birational morphism giving the dlt modification.
Since $(X,B;x)$ has regularity one, then each divisorial log canonical center of $(Y,B_Y)$ intersects at most two other divisorial log canonical centers.
Hence, $\mathcal{D}(Y,B_Y)$ is either a circle or a closed interval.
Let $E_k$ be a component of $\lfloor B_Y\rfloor$ which maps onto $x$.
We run a $(-E_k)$-MMP over $X$ which terminates with a good minimal model for $-E_k$.
Note that this minimal model program is an isomorphism over
$X\setminus \{x\}$.
Furthermore, since every curve contracted or flipped by this MMP is $E_k$-positive, then $E_k$ is not contracted.
Let $Y^*\rightarrow X$ be the model where this minimal model program terminates.
We denote by $E_k^*$ the strict transform of $E_k$ on $Y^*$.
We let $\phi^*\colon Y^*\rightarrow X$ be the induced projective birational morphism.
We show that ${\phi^*}^{-1}(x)=E_k^*$.
Indeed, assume that there is an irreducible component of ${\phi^*}^{-1}(x)$ which is not $E_k^*$.
Since ${\phi^*}^{-1}(x)$ is connected, we may assume that such component intersects $E_k^*$ non-trivially.
Thus, we can find a curve $C$ in such component, which intersects $E_k^*$, and is not contained in $E_k^*$.
Then, we have that $-E_k^*\cdot C<0$, contradicting the fact that
$-E_k^*$ is semiample
over the base $X$.
We denote by
$(Y^*,B_{Y^*})$ the log pull-back of $(X,B)$ to $Y^*$.
For each $E_i$ in $Y$, we denote by
$E_i^*$ its push-forward on $Y^*$.
If $E_i$ is contracted, then we let $E_i^*$ be the trivial divisor.
We claim that $\lfloor B_{Y^*}-E^*_k \rfloor$ has at most two prime components.
Note that $(Y^*,B_{Y^*})$ has regularity one. Then, by Lemma~\ref{lem:from-reg-1-to-0}, the log pair
\[
K_{E^*_k} + B_{E_k^*}\sim_\mathbb{Q}
K_{Y^*}+B_{Y^*}|_{E_k^*}
\]
has regularity zero.
In particular, the divisor
$\lfloor B_{E_k^*}\rfloor$
either has two disjoint prime components or a unique prime component.
Let $E_i^*$ be a prime divisor on $Y^*$ so that its image on $X$ properly contains the point $x$.
Then, $E_i^*$ intersects $E_k^*$ non-trivially.
Furthermore, we have that
\begin{equation}
\label{eq:int-E_k}
\operatorname{supp}\left(E_i^*\cap E_k^*\right)
\subseteq \lfloor B_{E_k^*}\rfloor.
\end{equation}
On the other hand, if $E_j^*$ and $E_i^*$ are two different prime divisors, then the intersection
$E_i^*\cap E_j^*\cap E_k^*$ has codimension at least three.
Otherwise, we would obtain a log canonical surface singularity with negative complexity, leading to a contradiction (see Theorem~\ref{thm:comp}).
From the containment~\eqref{eq:int-E_k}, we conclude that there are at most two components $E_i^*$ and $E_j^*$ which are not contracted in the minimal model program $Y\dashrightarrow Y^*$.
If the exceptional locus of
$\phi^*$ only consists of the divisor $E^*_k$, then all the divisors $E_i$ are contracted by the minimal model program
$Y\dashrightarrow Y^*$.
This means that each $E_i$ maps onto $x$.
By Lemma~\ref{lem:dual-comp-coll}, the dual complex $\mathcal{D}(Y,B_Y)$ is collapsible.
Then, it must be a closed interval.
Thus, we are in the situation of
(iv).
If there is exactly one prime divisor $E_i^*$ in the exceptional locus of $\phi^*$ which is different from $E_k^*$,
then there are two possibilities:
\begin{enumerate}
\item [(a)] The intersection $E_i^* \cap E_k^*$ is connected.
In this case, the intersection must be irreducible.
Furthermore, $E_i$ must intersect a unique prime component of $\lfloor B_Y -E_i \rfloor$.
Thus, $E_i$ corresponds to an end-point of the closed interval $\mathcal{D}(Y,B_Y)$.
Hence, we are in the situation of (iii).
\item [(b)] The intersection $E_i^* \cap E_k^*$ is disconnected.
In this case, the intersection must be two irreducible divisors in $E_k^*$.
Hence, we are in the situation of (ii).
\end{enumerate}
Finally,
assume there are exactly two prime divisors $E_i^*$ and $E_j^*$ in the exceptional locus of $\phi^*$ which are different from $E_k^*$.
Then, each intersection
$E_k^*\cap E_i^*$ and
$E_k^*\cap E_j^*$ is irreducible,
these intersections are disjoint, and
$E_i\cap E_j=\emptyset$.
Furthermore, the divisor
$E_i$ (resp. $E_j$) must intersect a unique prime component of
$\lfloor B_Y-E_i\rfloor$
(resp. $\lfloor B_Y-E_j\rfloor$).
We conclude that the divisors $E_i$ and $E_j$ correspond to the end-points of the closed interval $\mathcal{D}(Y,B_Y)$.
Thus, we are in the situation of (i).
\end{proof}
The following lemma will allow us to control the log discrepancy at certain potentially log canonical places of a klt singularity
of regularity one.
\begin{lemma}\label{lem:not-point}
Let $n$ and $N$ be two positive integers.
Let $\Lambda$ be a set satisfying the descending chain condition.
There exists a set $\mathcal{M}(n,N,\Lambda)$
satisfying the ascending chain condition,
which only depends on $n,N$ and $\Lambda$, satisfying the following.
Let $(X,B;x)$ be a log canonical pair of regularity one.
Assume that $\{x\}$ is a log canonical center of $(X,B)$.
Assume that $N(K_X+B)\sim 0$.
Let $E$ be a log canonical place of $(X,B;x)$ which maps to $Z\supsetneq x$.
Let $\Delta$ be an effective divisor with $\Delta\leq B$ so that $(X,\Delta;x)$ is klt and ${\rm coeff}(\Delta)\subset \Lambda$.
Then, we have that
\[
a_E(X,B)\in \mathcal{M}(n,N,\Lambda).
\]
Furthermore, if the set $\Lambda$ is finite, then the set $\mathcal{M}(n,N,\Lambda)$ only accumulates to zero.
\end{lemma}
\begin{proof}
Let $(Y,B_Y)$ be a $\mathbb{Q}$-factorial dlt modification of $(X,B;x)$ (see Lemma~\ref{lem:existence-dlt-mod}).
Let $\phi\colon Y\rightarrow X$ be the
corresponding projective birational morphism.
Since $x$ is a log canonical center, we may assume there exists a prime
component of $\lfloor B_Y\rfloor$ which maps onto $x$.
We call such component $E_0$.
We claim that $E$ is the unique log canonical place of $(X,B;x)$ whose image on $X$ contains $Z$.
Assume it is not.
Let $E'$ be a log canonical place of $(X,B;x)$, different from $E$, which maps to $Z$.
By the connectedness theorem~\cite[Theorem 1.2]{Bir21}, we can assume that $E'$ intersects $E$ over $Z$.
In particular, we may assume that
$E\cap E'$ dominates $Z$.
Since $Z$ contains $x$, then
$E\cap E'$ intersects the fiber over $x$.
We can run a $(-E_0)$-MMP over the base.
This minimal model program is an isomorphism at the generic point of $E\cap E'$.
After finitely many steps, we obtain a model in which
$E\cap E'\cap E_0$ is non-empty.
This implies that the regularity of $(X,B;x)$ is at least two, leading to a contradiction.
Hence, we may assume that $E$ is the only log canonical place which maps onto $Z$.
From now on, we can localize at a general point of $Z$, so we may assume that $E$ is the only log canonical center.
Let $\phi_E\colon E\rightarrow Z$ be the induced fibration.
By~\cite[Theorem 1.2]{HM07}, we conclude that the general fiber of $E\rightarrow Z$ is rationally chain connected.
By the previous paragraph, the general fiber of $E\rightarrow Z$ is klt.
Hence, it is rationally connected by~\cite[Corollary 1.8]{HM07}.
Let $(E,B_E)$ be the log pair induced by adjunction of $(Y,B_Y)$ to $E$.
We have that $(E,B_E)$ is klt, $E$ is rationally connected, and
\[
N(K_E+B_E)\sim 0.
\]
By~\cite[Theorem 1.4]{BDCS20}, we conclude that $(E,B_E)$ is log bounded up to flops.
By Lemma~\ref{lem:RC-curve}, we can find a curve $C\subset E$ which satisfies the following conditions:
\begin{enumerate}
\item $C$ lies in the smooth locus of $E$, and
\item $C$ is either disjoint from $B_E$ or it intersects $B_E$ transversally at most in $k$ points.
\end{enumerate}
Here, $k$ is a constant, only depending on $n$ and $N$.
Write $B_Y=E+B'_Y$.
Note that the coefficients of $B'_Y$ belongs to the finite set
$\mathbb{Z}\left[\frac{1}{N}\right]\cap [0,1]$.
Let $(Y,\Delta_Y)$ be the log pull-back of $(X,\Delta)$ to $Y$.
Write $\Delta_Y = \alpha E +\Delta_Y'$.
Note that the coefficients of
$\Delta'_Y$ belongs to the set $\Lambda$ which satisfies the descending chain condition.
We conclude that
\[
\alpha = \frac{ C\cdot (B'_Y -\Delta'_Y)}{m}
\]
belongs to a set satisfying the ascending chain condition.
Furthermore, if $\Lambda$ is a finite set, then
$B'_Y-\Delta'_Y$ has coefficients in a finite set.
Hence, the possible values of
$\alpha$ can only accumulate to zero.
\end{proof}
\begin{lemma}\label{lem:from-reg-1-to-0}
Let $(X,B)$ be an lc pair of regularity one.
Let $E$ be a prime component of $\lfloor B\rfloor$.
Let $(E,B_E)$ be the pair obtained by adjunction of $(X,B)$ to $E$.
Then, $(E,B_E)$ has regularity zero.
In particular, $\lfloor B_E\rfloor$ has either one or two disjoint components.
\end{lemma}
\begin{proof}
Assume that $(E,B_E)$ has regularity at least one.
Let $(Y,B_Y)$ be a dlt modification of $(X,B)$.
We denote by $E_Y$ be the strict transform of $E$ on $Y$.
Let $(E_Y,B_{E_Y})$ be the log pair obtained by adjunction of $(Y,B_Y)$ to $E_Y$.
Then, $(E_Y,B_{E_Y})$ is a dlt pair. Thus, the projective birational morphism $E_Y\rightarrow E$ is a dlt modification of $(E,B_E)$.
By further blowing-up, we may assume that there is a bijection between prime components of $\lfloor B_{E_Y}\rfloor$ and components of $\lfloor B_Y-E_Y\rfloor$ that intersect $E_Y$.
If ${\rm reg}(E,B_E)\geq 1$, then we can find at least two prime components $E_1,E_2$ of $\lfloor B_{E_Y}\rfloor$ for which $E_1\cap E_2\neq \emptyset$.
Let $E_{Y,1}$ and $E_{Y,2}$ be two components of $\lfloor B_Y\rfloor$ that restrict to $E_1$ and $E_2$, respectively.
Then, we have that
$E_{Y,1}\cap E_{Y,2}\cap E_Y\neq \emptyset$.
This leads to a contradiction.
The last part of the statement follows from~\cite[Theorem 1.2]{Bir21}.
\end{proof}
\begin{corollary}\label{cor:two-comp-toric}
Let $(X,B)$ be an lc pair of regularity one. Let $E_1,E_2\subset \lfloor B\rfloor$ be two prime components which are $\mathbb{Q}$-Cartier divisors.
Assume that $E_1\cap E_2\neq \emptyset$ is connected.
Then $E_1\cap E_2=Z$ is irreducible, no other log canonical center of $(X,B)$ intersects $Z$ and $(X,B)$ is toric at the generic point of $Z$.
\end{corollary}
\begin{proof}
The irreducibility of $Z$ follows from Lemma~\ref{lem:from-reg-1-to-0} and the fact that every irreducible component of an intersection of log canonical centers is again a log canonical center.
The toroidality of $(X,B)$ at the generic point of $Z$ follows from Theorem~\ref{thm:comp}.
\end{proof}
\subsection{\texorpdfstring{$\mathbb{P}^1$-links}{p1-links}}
In this subsection, we define the concept of $\mathbb{P}^1$-link structure of a log Calabi-Yau pair.
We also prove a lemma regarding projective pairs of regularity zero.
\begin{definition}\label{def:p1-links}
{\em
Let $(X,B)$ be a log Calabi-Yau admitting a fibration $X\rightarrow Z$.
We say that $X\rightarrow Z$ is a {\em $\mathbb{P}^1$-link}
for $(X,B)$ if the following
conditions are satisfied:
\begin{enumerate}
\item The general fiber of $X\rightarrow Z$ is isomorphic to $\mathbb{P}^1$.
\item the pair $(X,B)$ is plt,
\item $(X,B)$ is log Calabi-Yau over $Z$, i.e.,
$K_X+B\sim_{\mathbb{Q},Z}0$, and
\item $\lfloor B\rfloor$ has two components that dominate the base.
\end{enumerate}
In particular, the restriction of $(X,B)$ to a general fiber is isomorphic to the log pair
$(\mathbb{P}^1,\{0\}+\{\infty\})$.
}
\end{definition}
\begin{lemma}\label{MMP-reg-zero}
Let $(X,B)$ be a dlt log Calabi-Yau pair of regularity zero.
Assume that $\lfloor B\rfloor$ is disconnected.
Let $E_1$ and $E_2$ be the components of $\lfloor B \rfloor$ and $B'=B-\lfloor B\rfloor$.
Let $R$ be an extremal $(K_X+B')$-negative ray of the nef cone.
Then, $R$ is generated by a rational curve $C$ satisfying:
\[
K_X\cdot C=-2, \quad
E_1\cdot C=1, \quad
E_2\cdot C=1,
\text{ and } \quad
B'\cdot C=0.
\]
\end{lemma}
\begin{proof}
We can find an ample divisor $A$ on $X$ so that the cone
\[
\overline{NE}_1(X)_{K_X+B'\geq 0} +
\overline{NM}_1(X)
\]
intersects
$(K_X+B'+A)^{\perp}$ at $R$.
We run a $(K_X+B'+A)$-MMP with scaling of $A$.
Since each step of this minimal model program is $A$-positive,
then it is also a $(K_X+B')$-MMP.
Given that $K_X+B'$ is not pseudo-effective, this minimal model program terminates with a Mori fiber space.
Let $X\dashrightarrow X'$ be the minimal model program and $X'\rightarrow Z$ be the Mori fiber space.
Let $B'$ be the push-forward of $B$ to $X'$.
Every curve contracted by this MMP and every flipping curve intersect $E_1+E_2$ positively.
We conclude that no divisor $E_1$ neither $E_2$ is contracted by this minimal model program.
Let $E'_1$ and $E'_2$ be the strict transform of $E_1$ and $E_2$ in $X'$, respectively.
Note that $E'_1+E'_2$ is ample over $Z$, then at least one of the $E'_i$'s must dominate the base.
Assume $E'_1$ dominates the base.
If $E'_2$ is vertical over $Z$, then $E'_1\cap E'_2\neq \emptyset$, leading to a contradiction.
We conclude that $E'_1$ and $E'_2$ dominate $Z$.
Assume that the general fiber of $X'\rightarrow Z$ has dimension at least two. We call the general fiber $F$.
Then $E'_1|_F$ and $E'_2|_F$ are ample divisors and then they must intersect non-trivially. leading to a contradiction.
We conclude that the general fiber of $X'\rightarrow Z$ is one-dimensional. Then it is isomorphic to $\mathbb{P}^1$.
We conclude that $(X',B')\rightarrow Z$ is a $\mathbb{P}^1$-link.
Then, it suffices to take $C$ to be the strict transform of a general fiber of $X'\rightarrow Z$.
The strict transform of $C$ in $X$ generates the ray $R$.
Since $X\dashrightarrow X'$ is an isomorphism on a neighborhood of such general fiber,
we conclude that the intersections
$K_X\cdot C=-2$, $E_1\cdot C=E_2\cdot C=1$, and
$B'\cdot C=0$ hold.
\end{proof}
\begin{definition}\label{def:gen-p1}
{\em
Let $(X,B)$ be a dlt log Calabi-Yau pair of regularity zero.
Let $R$ be an extremal $(K_X+B')$-negative ray
of the nef cone.
The curve $C$ constructed in Lemma~\ref{MMP-reg-zero} will be called a {\em general $\mathbb{P}^1$} of the log pair of regularity zero.
In Theorem~\ref{introthm:uniqueness}, we will see that $R$ is unique,
so $C$ is unique up to numerical equivalence.
}
\end{definition}
\subsection{Surface log discrepancies}
In this subsection, we prove a couple of lemmas regarding surface toric minimal log discrepancies.
For the material regarding toric geometry, we refer the reader to~\cite{Ful93,CLS11}.
For toric minimal log discrepancies see~\cite{Amb06}.
The first two lemmas are well-known so
we will skip their proofs.
\begin{lemma}\label{lem:min-lattice}
Let $\sigma$ be a cone in $\mathbb{Q}^2$ spanned by the lattice vectors $v_1$ and $v_2$.
Let $w_1,\dots,w_c \in \mathbb{Z}^2$ give a regular decomposition of the cone.
Let $L$ be a positive linear function on $\sigma$.
Then, the minimizer of $L$ in
\[
(\sigma\setminus \mathbb{Q}_{\geq 0} v_1) \cap \mathbb{Z}^2
\]
is attained in the finite set
$\{ w_1,\dots,w_c,v_2\}$.
\end{lemma}
\begin{lemma}\label{lem:toric-surf-mld}
Let $\sigma\subset \mathbb{Q}^2$ be a full-dimensional rational polyhedral cone. Let $v_1$ and $v_2$ be the lattice generators of the extremal rays of $\sigma$.
Let $X(\sigma)$ be the corresponding affine toric surface and $x_0$ the torus invariant point. Let $T_1$ and $T_2$ be the torus invariant divisors of $X(\sigma)$.
Let $b_1$ and $b_2$ be two real numbers.
Then, we have that
\[
{\rm mld}(X(\sigma),b_1T_1+b_2T_2;x_0)=
\min_{u\in {\rm relint}(\sigma)\cap \mathbb{Z}^2}L(u),
\]
where $L$ is the unique linear function with
$L(v_1)=1-b_1$ and $L(v_2)=1-b_2$.
\end{lemma}
\begin{lemma}\label{lem:k-th-surf-toric-mld}
Let $\Lambda \subset \mathbb{R}_{\leq 1}$ be a set satisfying the descending chain condition.
Let $k$ be a positive integer.
Let $\mathcal{M}_{T,\Lambda,k}$ be the set of $k$-th minimal log discrepancies of surface toric sub-pairs
$(X,\Delta;x)$ with
${\rm coeff}(\Delta)\subset \Lambda$.
Then, the set
$\mathcal{M}_{T,\Lambda,k}$ satisfies the ascending chain condition.
Furthermore, if $\Lambda$ is finite, then its accumulation points only accumulate to zero.
\end{lemma}
\begin{proof}\label{cor:up-to-k-th-surf-toric-mld}
First, we prove the statement for $k=1$.
Let $-M$ be a lower bound of $\Lambda$.
We can assume that $M>0$.
Let $T_1$ and $T_2$ be the reduced prime toric divisors of $\Delta$.
Let $b_1$ and $b_2$ be the coefficients of $\Delta$ at $T_1$ and $T_2$ respectively.
Then, by Lemma~\ref{lem:toric-surf-mld}, we have that
\begin{equation}\label{eq:mld-mult}
{\rm mld}(X,\Delta;x) =
(M+1) {\rm mld}\left(
X,
\left(1- \frac{1-b_1}{M+1}\right)T_1
+
\left(1 -\frac{1-b_2}{M+1}\right)T_2;
x
\right).
\end{equation}
Note that
\begin{equation}\label{eq:new-coeff}
0\leq 1-\frac{1-b_i}{M+1}\leq 1,
\end{equation}
for $i\in \{1,2\}$.
We conclude that the sub-pair on the right side of equality~\eqref{eq:mld-mult} is actually a surface toric pair.
Furthermore, since $b_i \in \Lambda$, we conclude that the coefficients~\eqref{eq:new-coeff} belong to a set satisfying the descending chain condition, which only depends on $\Lambda$.
Then, since $M$ is a fixed number, we conclude that the value on the right side of equality~\eqref{eq:mld-mult} belongs to a set satisfying the ascending chain condition (see, e.g.,~\cite[Theorem 1.1]{Amb06}).
Furthermore, if $\Lambda$ is finite, then its accumulation points only accumulate to zero.
This shows the statement for $k=1$.
Now, assume that the statement holds for $k-1$.
Then, the set
\begin{equation}\label{eq:union-mld}
\mathcal{M}_{T,\Lambda,\leq k-1}:=
\bigcup_{i=1}^{k-1} \mathcal{M}_{T,\Lambda,i}
\end{equation}
satisfies the ascending chain condition.
Indeed, it is a finite union of sets satisfying the ascending chain condition.
Furthermore, the set $\mathcal{M}_{T,\Lambda,\leq k-1}$
only depends on $\Lambda$.
Moreover, if $\Lambda$ is finite, then the accumulation points of the set~\eqref{eq:union-mld} only accumulate to zero.
Let $(X,\Delta;x)$ be a singularity as in the statement.
We set
\[
\mathcal{C}_{T,\Lambda,\leq k-1}:=
\{1-m\mid m\in \mathcal{M}_{T,\Lambda,\leq k-1}\}.
\]
We can find a toric projective birational morphism $\phi\colon Y\rightarrow X$ which extracts up to the $(k-1)$-th minimal log discrepancy of $(X,\Delta;x)$.
Hence, we can write
\[
\phi^*(K_X+\Delta)=
K_Y+\Delta_Y,
\]
where $(Y,\Delta_Y)$ is a toric sub-pair and
\[
{\rm coeff}(\Delta_Y) \subseteq
\Lambda':=\Lambda \cup
\mathcal{C}_{T,\Lambda,\leq k-1}.
\]
Observe that $\Lambda' \subset \mathbb{R}_{\leq 1}$ is a set satisfying the descending chain condition,
which only depends on $\Lambda$ and $k$.
Then, we conclude that
\[
\mathcal{M}_{T,\Lambda,k}
\subseteq \mathcal{M}_{T,\Lambda',1}.
\]
By the case $k=1$,
we conclude that
$\mathcal{M}_{T,\Lambda,k}$
satisfies the ascending chain condition.
Furthermore, if $\Lambda$ is finite, then the accumulation points
of $\mathcal{M}_{T,\Lambda,k}$ can only accumulate to zero.
\end{proof}
\section{The cone of nef curves of log Calabi-Yau pairs}
\label{sec:cone-nef}
In this section, we prove a structure theorem for the nef cone
of log Calabi-Yau pairs with dlt singularities.
Furthermore, we prove the existence of the uniqueness of $\mathbb{P}^1$-links for log Calabi-Yau pairs with regularity zero.
\begin{proof}[Proof of Theorem~\ref{introthm:nef-cone}]
Let $(X,B)$ be a log Calabi-Yau dlt pair
with $N(K_X+B)\sim 0$.
Let $\phi\colon X'\rightarrow X$ be a small $\mathbb{Q}$-factorialization of $X$.
Note that the cone of nef curves is preserved by the small $\mathbb{Q}$-factorialization,
so we may replace $X$ with $X'$ and assume that $X$ itself is $\mathbb{Q}$-factorial.
We show that the log pair $(X,B')$ has $1/N$-log canonical singularities.
Assume this is not the case.
Let $E$ be a prime divisor over $X$ for which
$a_E(X,B') \in (0,N^{-1})$.
Then, we have that $a_E(X,B)=0$.
Indeed, we know that $N(K_X+B)\sim 0$, so the log discrepancies of $(X,B)$ belong to the set $\mathbb{Z}\left[\frac{1}{N}\right]$.
On the other hand, the generic point of each log canonical center of $(X,B)$ lies on the smooth locus of $X$ and it is disjoint from $B'$.
Thus, for each $E$ with $a_E(X,B') \in (0,N^{-1})$,
we have that $a_E(X)=a_E(X,B')\geq 1$.
This leads to a contradiction.
Thus, $(X,B')$ has $1/N$-log canonical singularities.
Let $R$ be a $(K_X+B')$-negative extremal ray of the cone of nef divisors.
We can find an ample divisor $A$ so that $K_X+B'+A$ is pseudo-effective and
$(K_X+B'+A)^{\perp}$ intersects the cone
\[
\overline{NE}_1(X)_{K_X+B'\geq 0}
+
\overline{NM}_1(X)
\]
exactly at $R$.
We can run a $(K_X+B')$-MMP with scaling of $A$ which terminates in a Mori fiber space.
Let $\psi \colon X\dashrightarrow X'$ be the minimal model program
and let $\phi\colon X'\rightarrow Z$ be the Mori fiber space.
Let $B'_{X'}$ and $A_{X'}$ be the strict transform of
$B'$ and $A$ in $X'$, respectively.
By construction, $K_X'+B'_{X'}+A_{X'}$ is $\mathbb{Q}$-trivial over $Z$.
Let $E_{X'}$ be the strict transform of $E$ on $X'$.
Observe that $K_{X'}+B_{X'}+E_{X'}$ is $\mathbb{Q}$-trivial over $Z$
and that $-(K_{X'}+B_{X'})$ is ample over $Z$.
We conclude that $E_{X'}$ is ample over $Z$.
Let $F$ be a general fiber of $\phi\colon X'\rightarrow Z$.
Let $B_F$ and $E_F$ be the restriction of $B_{X'}$ and $E_{X'}$ to $F$, respectively.
Since $(X,B)$ is $\frac{1}{N}$-log canonical and
$X\dashrightarrow X'$ is a $(K_X+B)$-MMP,
then $(X',B_{X'})$ is $\frac{1}{N}$-log canonical as well.
In particular, we have that $(F,B_F)$ is $\frac{1}{N}$-log canonical
and $-(K_F+B_F)$ is ample.
Let $f$ be the dimension of $F$.
By~\cite[Theorem 1.1]{Bir21}, we conclude that $F$ belongs to a bounded family of $f$-dimensional varieties.
Since the coefficients of $B_F$ are at least $\frac{1}{N}$, by~\cite[Theorem 3.3]{FM20}, we conclude that
$(F,B_F)$ is log bounded.
Furthermore, the log pair $(F,B_F+E_F)$ is log Calabi-Yau and log bounded as well.
We denote by $E_{i,F}$ the restriction of the prime component $E_{i,X'}$ to $F$.
By~\cite[Lemma 2.19]{Mor18b}, we can find a curve $C_F$ on $F$, so that the following conditions are satisfied:
\begin{enumerate}
\item the curve $C_F$ lies in the smooth locus of $F$,
\item the curve $C_F$ is either disjoint from $B_F$ or intersect it transversally with $B_F\cdot C_F\leq k$, and
\item for each $i$, the curve $C_F$ is either disjoint from $E_{i,F}$ or intersect it transversally at most in $k$ points.
\end{enumerate}
We can choose $C_F$ so that it is disjoint from $F\cap {\rm Ex}(\psi^{-1})$.
Then, the curve $C$ which is the strict transform of $C_F$ in $X$ satisfies the conditions $(1)$-$(4)$.
It suffices to prove that $(5)$ holds.
To do so, it is enough to prove that $C_F$ satisfies the analogous property.
Since $E_{X'}$ is ample over $C$, then $C_F$ must intersect at least one of its components.
We turn to prove that $C_F$ intersects at most $n+1$ of its components.
Up to re-ordering the $E_{i,X'}$, we may assume that there exists $r_0\leq r$ for which the divisors $E_{1,X'},\dots,E_{r_0,X'}$ dominate $Z$ and
$E_{r_0+1,X'},\dots, E_{r,X'}$ are vertical over $Z$.
Since $\rho(X'/Z)=1$,
we have that
the components
$E_{i,F}$ are $\mathbb{Q}$-linearly proportional. This means that
\[
\dim_{\mathbb{Q}}
\langle
E_{1,F}, \dots,
E_{r_0,F}
\rangle = 1.
\]
Hence, we can compute the complexity
of the pair $(F,B_F+E_F)$ and obtain
\[
0\leq c(F,B_F+E_F) \leq f + 1 - r_0.
\]
The first inequality follows from~\cite[Theorem 4.5]{RS21} while the second inequality follows from~\cite[Definition 3.15]{RS21}.
Thus, we have that $r_0\leq f+1$.
We conclude that there are at most $f+1$ prime components of $E_{X'}$ that dominate $Z$.
This implies that the curve $C_F$ intersects at most $n+1$ of the components of $E_{X'}$.
Hence, the curve $C$ intersects at most
$n+1$ of the components $E_1,\dots,E_r$ of $\lfloor B\rfloor$.
We conclude that every $(K_X+B')$-negative extremal nef ray is spanned by a curve $C_i$ satisfying the conditions $(1)$-$(5)$.
Then, the theorem follows from~\cite[Theorem 1.3]{Leh12}.
\end{proof}
\begin{lemma}\label{lem:semiample-big}
Assume that Theorem~\ref{introthm:uniqueness} holds in dimension $d$.
Let $(F,B_F)$ be a $d$-dimensional log Calabi-Yau dlt pair of regularity zero.
Let $E_{1,F}$ and $E_{2,F}$ be the prime components of $\lfloor B_F\rfloor$ and $B'_F=B_F-\lfloor B_F\rfloor$.
Assume that the following conditions hold:
\begin{enumerate}
\item $-(K_{F}+B'_F)$ is semiample and big, and
\item $-(K_{F}+B'_F) \sim_\mathbb{Q} 2E_{1,F} \sim_\mathbb{Q} 2E_{2,F}$.
\end{enumerate}
Then, $F\simeq \mathbb{P}^1$, $B'_F=0$, and $E_{1,F}+E_{2,F}=\{0\}+\{\infty\}$.
\end{lemma}
\begin{proof}
We may replace $F$ with a small $\mathbb{Q}$-factorialization.
From now on, we may assume that $F$ is itself $\mathbb{Q}$-factorial.
Let $F\rightarrow F'$ be the ample model of $-(K_{F}+B'_F)$.
Let $B_{F'}$ be the push-forward of $B_{F}$ to $F'$.
Let $B'_{F'}$ be the push-forward of $B'_F$ to $F'$.
Since $-(K_{F}+B'_F)\sim_\mathbb{Q} 2E_{1,F}\sim_\mathbb{Q} 2E_{2,F}$, then every curve contracted by $F\rightarrow F'$ must intersect $E_{1,F}$ and $E_{2,F}$ trivially.
In particular, the divisors $E_{1,F}$ and $E_{2,F}$ are not contracted by the birational morphism $F'\rightarrow F$.
Then, the pair $(F',B_{F'})$ is log Calabi-Yau dlt pair of regularity zero and $-(K_{F'}+B'_{F'})$ is ample.
By Theorem~\ref{introthm:uniqueness} in dimension $d$, we conclude that $F'\simeq \mathbb{P}^1$,
$B'_{F'}=0$, and $E_{1,F'}+E_{2,F'}=\{0\}+\{\infty\}$.
This implies that $F$ must be one-dimensional, so the proof follows.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{introthm:uniqueness}]
We will proceed by induction on the dimension.
If the dimension is one,
then it is clear that $X\simeq \mathbb{P}^1$, $B'=0$,
and $B=\{0\}+\{\infty\}$.
Now, assume that $X$ has dimension at least two.
First, we will analyze the statement when the Picard rank is at most $2$.
Assume that $X$ has Picard rank one.
Let $E_1$ and $E_2$ be the components of $\lfloor B\rfloor$.
Since $\dim X \geq 2$, we can find a curve $C\subseteq E_1$.
Since $\rho(X)=1$, the divisor $E_2$ is an ample divisor and hence
$E_2\cdot C>0$.
This means that $E_1\cap E_2\neq \emptyset$, leading to a contradiction.
Assume that $X$ has Picard rank two and dimension at least two.
If the cone
\begin{equation}\label{eq:cone-1}
\overline{NE}_1(X)_{K_X+B'\geq 0} + \overline{NM}_1(X)
\end{equation}
has two extremal $(K_X+B')$-negative curves,
it means that $-(K_X+B')$ must be an ample divisor.
Let $R_1$ and $R_2$ be the extremal rays of the cone of nef curves.
By Lemma~\ref{lem:curves-generating}, we can find $C_1$ and $C_2$,
generating $R_1$ and $R_2$, respectively, such that
\[
K_X\cdot C_i = -2, \quad
E_1 \cdot C_i=1, \quad
E_2 \cdot C_i=1, \text{ and } \quad
B'\cdot C_i=0,
\]
for each $i\in \{1,2\}$.
Since $C_1$ and $C_2$ span different rays in $N_1(X)$, we have that
\[
B'=0 \text{ and }
E_1\equiv E_2 \equiv -\frac{1}{2}K_X.
\]
This means that $E_1$ and $E_2$ are ample divisors.
This leads to a contradiction.
We conclude that the cone~\eqref{eq:cone-1} has a unique
$(K_X+B')$-negative extremal ray.
Assume that $X$ has Picard rank larger or equal than three.
If the cone~\eqref{eq:cone-1} has at least two $(K_X+B')$-negative
extremal rays, then it has a $(K_X+B')$-negative extremal face of dimension two.
We denote by $F$ such an extremal face of dimension two.
Then, we can find an ample divisor $A$ on $X$ so that
$K_X+B'+A$ is pseudo-effective and
\[
(K_X+B'+A)^{\perp} \cap
\left(
\overline{NE}_1(X)_{K_X+B'\geq 0} + \overline{NM}_1(X)
\right) = F.
\]
We run a $(K_X+B'+A)$-MMP with scaling of $A$.
By~\cite[Corollary 1.4.2]{BCHM10}, this minimal model program terminates $\phi\colon X\dashrightarrow X_0$ with a good minimal model $X_0$.
Let $B'_0$ and $A_0$ be the push-forward of $B'$ and $A$ to $X_0$, respectively.
Let $Z$ be the ample model of the semiample divisor
$K_{X_0}+B'_0+A_0$.
Let $d$ be the relative dimension of $X_0\rightarrow Z$.
We claim that $\rho(X_0/Z)=2$.
Let $r=\rho(X_0/Z)$.
Let $\phi_* \colon N^1(X)\rightarrow N^1(X')$ be the map induced by push-forward.
Observe that $\phi_*$ is surjective and maps the cone of effective divisors into the cone of effective divisors.
Let $\phi^*\colon N_1(X')\rightarrow N_1(X)$ be the dual of $\phi_*$.
Then, the homomorphism $\phi^*$ maps the cone of nef divisors into the cone of nef divisors.
A nef curve $C$ in $X'$ is contracted by $X'\rightarrow Z$ if and only if
\[
(K_{X_0}+B_0'+A_0)\cdot C =
\phi_*(K_X+B+A) \cdot C =0.
\]
Let $F'$ be the intersection of the cone of nef curves of $X_0$ with $(K_{X_0}+B_0'+A_0)^{\perp}$.
By duality, we have that $\phi^*(F')\subseteq F$.
Since $\phi^*$ is injective, we conclude that $F'$ has dimension at most two.
Hence, we conclude that $r\leq 2$.
On the other hand,
let $f_0\in F_0$ be a general point in a general fiber of $X_0\rightarrow Z$.
Let $x$ be the preimage of $f$ in $X$.
We show that we can find $C_1'\equiv C_1$ and $C_2'\equiv C_2$ so that
\[
{\rm Ex}(X\dashrightarrow X') \cap C_i =\emptyset
\text{ and }
x\in C_i
\]
for each $i$.
Let $p\colon Y\rightarrow X$ and $q\colon Y\rightarrow X_0$ be a log resolution of the minimal model program.
Then, we have that
\[
p^*(K_X+B'+A) = q^*(K_{X_0}+B'_0+A_0) + E,
\]
where $E$ is an effective divisor.
By Theorem~\ref{introthm:nef-cone}, we can replace $C_1$ and $C_2$ by curves satisfying the following:
\begin{itemize}
\item the curves are disjoint from the set on which $p$ is not an isomorphism,
\item the curves are not contained in the divisorial locus of ${\rm Ex}(X\dashrightarrow X')$, and
\item both curves contain $x$.
\end{itemize}
Let $C_{Y,1}$ and $C_{Y,2}$ be the strict transform of $C_1$ and $C_2$ on $Y$, respectively.
Since $C_{Y,1}$ and $C_{Y,2}$ intersect $p^*(K_X+B'+A)$ trivially and
$q^*(K_{X_0}+B_0'+A_0)$ non-negatively, we conclude that
$C_{Y,1}$ and $C_{Y,2}$ must intersect both
$q^*(K_{X_0}+B_0'+A_0)$ and $E$ trivially.
In particular, $C_1$ and $C_2$ must be disjoint from
the set ${\rm Ex}(X\dashrightarrow X_0)$.
Let $C_{1,0}$ and $C_{2,0}$ be the push-forward of $C_1$ and $C_2$ to $X_0$, respectively.
Then, the curves $C_{1,0}$ and $C_{2,0}$ are linearly independent in $N_1(X_0)$ and contracted to a point in $Z$.
Indeed, since both curves pass through $f_0$ and intersect
$K_{X_0}+B'_0+A_0$ trivially, then they are contained in the fiber $F_0$.
Hence, we have that $r\geq 2$.
Thus, we deduce that $r=2$ as claimed.
Let $E_{1,0}$ and $E_{2,0}$ be the push-forward of $E_1$ and $E_2$ to $X_0$, respectively.
Then, we conclude that
\[
E_{1,0}+E_{2,0} \sim_{\mathbb{Q},Z} -(K_{X_0}+B'_0) \sim_{\mathbb{Q},Z} A_0
\]
is big over $Z$.
Since $E_{1,0}+E_{2,0}$ is big over $Z$, then either $E_{1,0}$ or $E_{2,0}$ must dominate $Z$.
Without loss of generality, we assume that $E_{1,0}$ dominates $Z$.
Let $Z_2$ be the image of $E_{2,0}$ in $Z$.
By the connectedness theorem~\cite[Theorem 1.2]{Bir21}
applied to $(X_0,B'_0+E_{1,0}+E_{2,0})$ over a general point of $Z_2$, we conclude that $Z_2=Z$.
This means that $E_{2,0}$ dominates the base as well.
Let $(F,B'_F+E_{1,F}+E_{2,F})$ be the restriction of
$(X_0,B'_0+E_{1,0}+E_{2,0})$ to a general fiber.
Note that $\dim F < \dim X$ and $(F,B'_F+ E_{1,F}+E_{2,F})$ is a dlt log Calabi-Yau pair
of regularity zero.
By Lemma~\ref{lem:curves-generating}, we can further assume that $C_{1,0}$ and $C_{2,0}$ satisfy:
\begin{equation}\label{eq:2-equiv}
K_{X_0}\cdot C_{i,0} = -2, \quad
E_{1,0}\cdot C_{i,0} = 1, \quad
E_{2,0}\cdot C_{i,0} = 1, \text{ and } \quad
B'_0 \cdot C_{i,0}=0.
\end{equation}
Since $\rho(X_0/Z)=2$ and the curves
$C_{1,0}$ and $C_{2,0}$ are linearly independent in $N_1(X_0)$, we conclude that
\[
-K_{X_0} \sim_{\mathbb{Q},Z} 2E_{1,0} \sim_{\mathbb{Q},Z} 2E_{2,0}
\text{ and }
B'_0 \sim_{\mathbb{Q},Z} 0.
\]
We claim that $-K_{X_0}$ is nef over the base.
Let $C$ be an effective curve in $X_0$ which is contracted to a point.
Assume that $-K_{X_0}\cdot C <0$, then by the $\mathbb{Q}$-linear equivalence~\eqref{eq:2-equiv}, we conclude that
\[
E_{1,0}\cdot C < 0
\text{ and }
E_{2,0}\cdot C<0.
\]
Hence, $C$ must be contained in both $E_{1,0}$ and $E_{2,0}$, leading to a contradiction.
Since $X_0\rightarrow Z$ is a Fano type morphism,
it is a relative Mori dream space~\cite[Corollary 1.3.1]{BCHM10}.
Hence, $-K_{X_0}$ is semiample and big over $Z$.
Then, the following conditions hold for a general fiber:
\begin{itemize}
\item $B'_{F_0}=0$,
\item $-K_{F_0}$ is semiample and big, and
\item $-K_{F_0} \sim_\mathbb{Q} 2E_{1,F}\sim_\mathbb{Q} 2E_{2,F}$.
\end{itemize}
Since $\dim F_0 <\dim X$, by induction on the dimension, we can apply Lemma~\ref{lem:semiample-big} to $F$ to conclude that $F_0 \simeq \mathbb{P}^1$.
This leads to a contradiction, since in this case we would have $C_{1,0}=C_{2,0}$.
We conclude that the cone~\eqref{eq:cone-1}
has a unique $(K_X+B')$-negative extremal ray.
\end{proof}
\section{Minimal log discrepancies of regularity one}
In this section, we prove the
theorem regarding the minimal log discrepancies of klt singularities of regularity one.
Theorem~\ref{thm:reg-one-coeff} is a generalization of Theorem~\ref{introthm:ACC} for log pairs of regularity one.
The main ingredients in the proof of the theorem
are the existence of bounded complements~\cite[Theorem 1.8]{Bir19},
the existence of dlt modifications~\cite[Theorem 3.1]{KK10},
and the existence of the curves produced in Section~\ref{sec:cone-nef}.
Through the proof of Theorem~\ref{thm:reg-one-coeff}, we will often use the following lemma to control the generators of the relative cone of curves.
\begin{lemma}
\label{lem:curves-generating}
Let $\phi \colon Y\rightarrow Y_0$
and $\phi_0 \colon Y_0\rightarrow X$ be two projective birational maps.
Assume both $\phi$ and $\phi_0$ are relative Mori dream spaces.
Let $C_1,\dots, C_s$ be a set of curves in $Y_0$ that generate
$N_1(Y_0/X)$
and let $C'_{s+1},\dots,C'_{r}$ be a set of curves in $Y$ that generate
$N_1(Y/Y_0)$.
Assume that $\phi({\rm Ex}(\phi))$ does not contain any of the $C_i$'s.
For each $i \in \{1,\dots,s\}$, let $C_i'$ be the strict transform of $C_i$.
Then, we have that
\[
\langle C'_1,\dots,C'_r\rangle = N_1(Y/X).
\]
\end{lemma}
\begin{proof}
Let $D$ be a $\mathbb{Q}$-divisor on $Y$ that intersect each of the $C_i'$ trivially.
It suffices to show that $D$ is numericallly trivial over $X$.
Since $D$ intersects $C'_{s+1},\dots, C'_{r}$ trivially
and these curves generate $N_1(Y/Y_0)$,
we conclude that $D$ is numerically trivial over $Y_0$.
Since $\phi$ is a relative Mori dream space,
we conclude that $D$ is $\mathbb{Q}$-linearly trivial over $Y_0$.
Write $D=\phi^* D_0$ for some $\mathbb{Q}$-divisor $D_0$ on $Y_0$.
For each $i\in \{1,\dots,s\}$, we have that
\[
0=D\cdot C'_i = \phi^*D_0 \cdot C'_i = D_0 \cdot C_i.
\]
We conclude that $D_0$ intersects each $C_i$, with $i\in \{1,\dots,s\}$, trivially.
Hence, $D_0$ is numerically trivial over $X$
and then it is $\mathbb{Q}$-linearly trivial over $X$.
Thus, we have that $D$ is numerically trivial over $X$. This concludes the proof.
\end{proof}
Now, we turn to prove the ascending chain condition for minimal log discrepancies of regularity one near zero.
The proof will be divided into four steps depending on the structure of the dual complex of the modification.
The proof of each of these cases is similar in flavor.
Some crucial details differ, so we give a comprehensive proof in each case, despite some repetition.
\begin{theorem}\label{thm:reg-one-coeff}
Let $n$ be a positive integer.
Let $\Lambda\subset \mathbb{Q}$ be a set satisfying the descending chain condition with rational accumulation points.
There exists a constant $N:=N(n,\Lambda)$,
only depending on $n$ and $\Lambda$,
satisfying the following.
Let
\[
\mathcal{M}_{n,\Lambda,r}:=
\{
{\rm mld}(X,\Delta;x) \mid
\text{
$(X,\Delta;x)$ has regularity $r$ and ${\rm coeff}(\Delta)\subset \Lambda$
}
\}.
\]
Then, the set
\begin{equation}\label{eq:mld-int-1/N}
\mathcal{M}_{n,\Lambda,1}\cap \left(0,\frac{1}{N}\right)
\end{equation}
satisfies the ascending chain condition.
\end{theorem}
\begin{proof}
By Lemma~\ref{lem:reg-bounded-comp}, there exists a constant $N:=N(n,\Lambda)$, only depending on $n$ and $\Lambda$, satisfying the following.
For each $(X,\Delta;x)$ as in the statement, there exists a $N$-complement $(X,B;x)$ so that
\[
{\rm reg}(X,\Delta;x) =
{\rm reg}(X,B;x).
\]
Furthermore, by Lemma~\ref{lem:reg-comp-lcc}, we know that $x$ is a log canonical center of $(X,B;x)$.
Note that every divisor computing a log discrepancy in
$(0,\frac{1}{N})$ of $(X,\Delta;x)$ must be a log canonical place of $(X,B;x)$.
Indeed, the log discrepancies of $(X,B;x)$ belong to the set
\[
\mathbb{Z}_{\geq 0}\left[ \frac{1}{N}\right],
\]
and we have that $a_E(X,\Delta)\leq a_E(X,B)$ for each $E$.
Thus, if $a_E(X,\Delta)<\frac{1}{N}$, then
we have that $a_E(X,B)=0$.
Hence, in order to prove the ascending chain condition of the set~\eqref{eq:mld-int-1/N}, it suffices to show that the minimum of the log discrepancies of $(X,\Delta;x)$ at the log canonical places of $(X,B;x)$ satisfies the ascending chain condition.
Let $(Y,B_Y)$ be a dlt modification of $(X,B;x)$ (see Lemma~\ref{lem:existence-dlt-mod}).
We may assume that $\lfloor B_Y\rfloor$ has at least one component mapping onto $x$.
Shrinking around $x$, we may assume that every log canonical center
of $(X,B;x)$ passes through $x$.
Let $E_1,\dots,E_r$ be the prime components of $\lfloor B_Y\rfloor$.
By Lemma~\ref{lem:class-dual-comp}, we know that $\mathcal{D}(Y,B_Y)$
belong to four different classes.
We will prove the theorem in each of these cases.\\
\textit{Step 1:} We prove the theorem in the case that $\mathcal{D}(Y,B_Y)$ is a closed interval: $E_1$ maps onto a log canonical center $Z_1\supsetneq x$,
$E_r$ maps onto a log canonical center $Z_r\supsetneq x$,
$Z_1\neq Z_r$, and each $E_i$ with $i\in \{2,\dots,r-1\}$ maps onto $x$.\\
By Lemma~\ref{lem:not-point}, we know that $a_{E_1}(X,\Delta)$
and $a_{E_r}(X,\Delta)$ belong to a set $\mathcal{M}(n,N,\Lambda)$, only depending on $n,N$ and $\Lambda$, satisfying the ascending chain condition.
We let
\[
\mathcal{C}(n,N,\Lambda):=
\{ 1-m \mid m\in \mathcal{M}(n,N,\Lambda)\}.
\]
By~\cite[Theorem 1]{Mor20}, we can find a projective birational morphism $\phi_0\colon Y_0\rightarrow X$ which extracts the divisors $E_1$ and $E_r$.
By abuse of notation, we denote by $E_1$ and $E_r$ the strict transforms of these divisors on $Y_0$.
Write
\[
\phi_0^*(K_X+B)=K_{Y_0}+E_1+E_r+B_{Y_0}'.
\]
Note that $E_1$ and $E_r$ intersect non-trivially in $Y_0$.
By Corollary~\ref{cor:two-comp-toric}, we conclude that $Z_{1,r}:=E_1\cap E_r$ is an irreducible variety and every log canonical center of $(Y_0,E_1+E_r+B_{Y_0}')$ can be extracted by a toroidal blow-up at the generic point of $Z_{1,r}$.
We can write
\[
\phi_0^*(K_X+\Delta)=
K_{Y_0}+\Delta_{Y_0}=
K_{Y_0}+c_1E_1+c_rE_r+\Delta_{Y_0}',
\]
where $c_1,c_r\in \mathcal{C}(n,N,\Lambda)$.
In particular, the coefficients of the sub-pair $(Y_0,\Delta_{Y_0})$ belong to the set
\[
\Lambda \cup \mathcal{C}(n,N,\Lambda).
\]
Furthermore, the minimal log discrepancy of $(Y_0,\Delta_{Y_0})$ can be extracted by a toroidal blow-up at the generic point of $Z_{1,r}$.
Thus, the minimal log discrepancy is computed by a surface minimal log discrepancy.
Hence, the statement follows from Lemma~\ref{lem:k-th-surf-toric-mld}.\\
\textit{Step 2:} We prove the statement of the theorem in the case that $\mathcal{D}(Y,B_Y)$ is a circle: $E_1$ maps onto a log canonical center $Z_1 \supsetneq x$ and every other $E_i$ maps onto $x$.\\
Let $E_i$ be a prime component which maps onto $x$.
By~\cite[Theorem 1]{Mor20}, we can find two projective birational morphisms
$\phi_0\colon Y_0\rightarrow X$
and
$\phi_1\colon Y_1\rightarrow Y_0$, so that
$\phi_0$ only extracts $E_1$
and $\phi_1$ only extracts $E_i$.
By abuse of notation, we may denote both the divisor and the strict transform by $E_i$.
In the model $Y_1$, the divisors $E_i$ must intersect $E_1$ in two disjoint irreducible components.
By Lemma~\ref{lem:not-point}, we know that $a_{E_1}(X,\Delta)$ belongs to the set $\mathcal{M}(n,N,\Lambda)$.
Let $C_i$ be a general $\mathbb{P}^1$ in $E_i$ when considering its log pair structure with regularity zero (see Definition~\ref{def:gen-p1}).
Note that the curve $C_i$ generates $N_1(Y_1/Y_0)$.
We can obtain a dlt modification of
$(Y_0,E_1+E_i+B_{Y_0}')$ by performing toroidal blow-ups at the generic points of the intersection $E_i\cap E_1$.
We may replace $(Y,B_Y)$ with this new dlt modification.
Each prime component
$E_2,\dots, E_r$ of $\lfloor B_Y\rfloor$ which is exceptional over $Y_0$ admits the structure of a log pair of regularity zero.
For each $i\in \{2,\dots,r\}$, we denote by $C_i$ a general $\mathbb{P}^1$ in $E_i$.
By Lemma~\ref{lem:curves-generating}, we have that
\[
\langle C_2,\dots,C_r\rangle =
N_1(Y/Y_0).
\]
Indeed, the curve $C_i$ generate $N_1(Y_1/Y_0)$
and the curves $C_j$, with $j\in \{1,\dots,r\}\setminus \{i\}$,
generate $N_1(Y/Y_1)$.
Let $\phi\colon Y\rightarrow Y_0$ be the induced projective birational morphism.
We can write
\begin{equation}\label{eq:pb-delta}
\phi^*(K_{Y_0}+\Delta_{Y_0}+c_1E_1)= K_Y+ c_1E_1+(1-\alpha_2)E_2 +\dots + (1-\alpha_r)E_r + \Delta_{Y}',
\end{equation}
and
\begin{equation}\label{eq:pb-b}
\phi^*(K_{Y_0}+B_{Y_0}) =
K_Y+E_1+\dots+E_r+B_{Y}'.
\end{equation}
Subtracting~\eqref{eq:pb-delta} and~\eqref{eq:pb-b}, we obtain:
\begin{equation}\label{eq:lin-eq1}
(1-c_1)E_1 + \alpha_2E_2 +\dots+\alpha_r E_r \sim_{\mathbb{Q}, X} \Delta_{Y}' - B_{Y}'.
\end{equation}
Observe that every curve $C_i$ is disjoint from both $\Delta_{Y}'$ and $B_{Y}'$.
Note that each $C_i$ lies in the smooth locus of $E_i$ and
intersect exactly two other of the prime divisors $E_1,\dots,E_r$.
In particular, the intersection $E_i\cdot C_i$ is a negative integer $-m_i$.
Intersecting the $\mathbb{Q}$-linear equation~\eqref{eq:lin-eq1} with each of the $C_i$, with $i\in \{2,\dots,r\}$, leads to a system of linear equations:
\begin{equation}
\label{equation:lin-sys-1}
\left[
\begin{matrix}
m_2 & -1 & 0 & \dots & 0 \\
-1 & m_3 & -1 & \dots & 0\\
0 & -1 & m_4 & \dots & 0 \\
\vdots & \vdots &\vdots &\vdots &\vdots \\
0 & 0 & 0 & -1 & m_r
\end{matrix}
\right]
\left[
\begin{matrix}
\alpha_2 \\
\alpha_3 \\
\alpha_4 \\
\vdots \\
\alpha_r
\end{matrix}
\right]
=
\left[
\begin{matrix}
1-c_1 \\
0 \\
\vdots \\
0 \\
1-c_1
\end{matrix}
\right]
\end{equation}
Note that the matrix of equation~\eqref{equation:lin-sys-1} is of full rank as the curves $C_2,\dots,C_r$ span the relative cone of curves over $Y_0$.
Let $X(\sigma)$ be the toric singularity associated to the continued fraction $[m_1,\dots,m_r]$.
Let $x_0\in X(\sigma)$ be the torus invariant point.
Let $(X(\sigma),c_1T_1+c_1T_2)$ be the log pair structure obtained by considering both torus invariant divisors with coefficient $c_1$.
By construction, we have that
\[
{\rm mld}(X,\Delta;x) = \min\{\alpha_2,\dots,\alpha_r\} =
{\rm mld}(X(\sigma),c_1T_1+c_1T_2;x_0).
\]
By Lemma~\ref{lem:k-th-surf-toric-mld}, we conclude that the minimum of the $\alpha_i$, with $i\in \{2,\dots, r\}$, belongs to a set satisfying the ascending chain condition which only depends on $n,N$ and $\Lambda$.\\
\textit{Step 3:} We prove the statement of the theorem in the case that $\mathcal{D}(Y,B_Y)$ is a closed interval: $E_1$ maps onto a log canonical center $Z_1\supsetneq x$, and each $E_i$ with $i\geq 2$ maps onto $x$.\\
By~\cite[Theorem 1]{Mor20}, there exists a projective birational morphism
$\phi_0 \colon Y_0\rightarrow X$ whose exceptional locus is purely divisorial and consists on the divisor $E_1$.
By the same result, we can find a projective birational morphism $\phi\colon Y_1\rightarrow Y_0$
which extracts the divisor $E_r$.
In the model $Y_1$ the divisors $E_1$ and $E_r$ intersect at an irreducible set of codimension two.
Furthermore, the center of $E_i$, with $i\in \{2,\dots,r-1\}$ maps to such irreducible subvariety.
By Theorem~\ref{thm:comp}, the singularity of $(Y_1,E_1+E_r+B_{Y_1}')$ is toroidal at the generic point of $E_1\cap E_r$.
Then, each $E_i$ with $i\in \{2,\dots,r-1\}$ can be extracted by a sequence of toroidal blow-ups.
By Lemma~\ref{lem:not-point}, we know that $a_{E_1}(X,\Delta)$ is contained in the set $\mathcal{M}(n,N,\Lambda)$
which satisfies the ascending chain condition.
We let $c_1=1-a_{E_1}(X,\Delta)$.
Let $(Y_1,B_{Y_1})$ be the log pull-back of $(X,B)$ to $Y_1$.
Let $(E_r,B_{E_r})$ be the pair obtained by adjunction of
$(Y_1,B_{Y_1})$ to $E_r$.
By Lemma~\ref{lem:from-reg-1-to-0}, we conclude that $(E_r,B_{E_r})$ is a pair
of regularity zero.
Furthermore, since $E_r$ corresponds to one of the end-points of $\mathcal{D}(Y,B_Y)$,
we conclude that $(E_r,B_{E_r})$ has a unique divisorial log canonical center.
We call such divisor $S_r$ and we denote $B'_{E_r}:=B_{E_r}-S_r$.
By Theorem~\ref{introthm:nef-cone}, we can find a movable curve $C_r$ on $E_r$ satisfying the following conditions:
\begin{itemize}
\item $C_r$ is in the smooth locus of $E_r$,
\item $C_r$ intersects $S_r$ transversally in at most $k$ points, and
\item $C_r$ is either disjoint from $B'_{E_r}$ or it intersects $B'_{E_r}$ transversally in at most $k$ points.
\end{itemize}
Here, $k$ is a constant which only depends on $n$ and $N$.
We can obtain a dlt modification of $(Y_1,E_1+E_r+B_{Y_1}')$ by
performing a sequence of toroidal blow-ups at the generic point of $E_1\cap E_r$.
We replace $(Y,B_Y)$ by this new dlt modification.
For $i\in \{2,\dots,r-1\}$, each prime component $E_i$ of $\lfloor B_Y\rfloor$ admits the structure of a log pair
of regularity zero.
For each $i\in \{2,\dots,r-1\}$, we denote by $C_i$ a general $\mathbb{P}^1$ in $E_i$ (see Definition~\ref{def:gen-p1}).
We may identify $C_r$ with its strict transform in $Y$.
By Lemma~\ref{lem:curves-generating}, we have that
\[
\langle C_2,\dots, C_r\rangle = N_1(Y/Y_0).
\]
Let $\phi \colon Y\rightarrow Y_0$ be the induced birational morphism.
We can write
\begin{equation}\label{eq:s3-1}
\phi^*(K_{Y_0}+\Delta_{Y_0}+c_1E_1) =
K_Y + c_1E_1 + (1-\alpha_2) E_2 + \dots + (1-\alpha_r)E_r+ \Delta'_Y
\end{equation}
and
\begin{equation}\label{eq:s3-2}
\phi^*(K_{Y_0}+B_{Y_0}) = K_Y+E_1+\dots+E_r+B'_Y.
\end{equation}
Here, $B'_Y\geq \Delta'_Y$. Subtracting~\eqref{eq:s3-1} and~\eqref{eq:s3-2}, we obtain that
\begin{equation}\label{eq:s3-3}
(1-c_1)E_1 + \alpha_2 E_1 +\dots + \alpha_r E_r \sim_{\mathbb{Q},X} \Delta_Y'-B_Y'.
\end{equation}
Observe that for each $i\in \{2,\dots, r-1\}$, the curve $C_i$ lies in the smooth locus
of $Y$ and $C_i$ intersects exactly two other prime components of $\lfloor B_Y\rfloor$.
Moreover, each such $C_i$ is disjoint from $B_Y'$ and hence from $\Delta_Y'$.
On the other hand, we have that
\[
E_{r-1}\cdot C_r = c_{r-1,0} \leq k, \quad
E_r\cdot C_r = -\frac{m_r}{N!}, \text{ and }
(\Delta'_Y-B'_Y) \cdot C_r = -\beta_{r,0},
\]
where $c_{r-1,0}$ and $m_r$ are positive integers and
$\beta_{r,0}$ belongs to a set satisfying the ascending chain condition.
The denominator of the intersection $E_r\cdot C_r$ is bounded
as the variety $Y$ has quotient singularities of order at most $N$ along $C_r$.
We let $c_{r-1}:=c_{r-1,0}N!$ and $\beta_r:=\beta_{r,0}N!$.
Intersecting the $\mathbb{Q}$-linear equivalence~\eqref{eq:s3-3} with the curves
$C_i$ with $i\in \{2,\dots,r\}$, we obtain the following system of linear equations:
\[
\left[
\begin{matrix}
m_2 & -1 & 0 & \dots & \dots & 0 \\
-1 & m_3 & -1 & \dots & \dots & 0\\
0 & -1 & m_4 & \ddots & \dots & 0 \\
\vdots & \vdots &\vdots &\vdots &\ddots &\vdots \\
0 & 0 & 0 & -1 & m_{r-1} & -1 \\
0 & 0 & 0 & 0 & -c_{r-1} & m_r
\end{matrix}
\right]
\left[
\begin{matrix}
\alpha_2 \\
\alpha_3 \\
\alpha_4 \\
\vdots \\
\alpha_{r-1} \\
\alpha_r
\end{matrix}
\right]
=
\left[
\begin{matrix}
1-c_1 \\
0 \\
0 \\
\vdots \\
0 \\
\beta_r
\end{matrix}
\right]
\]
Let $\sigma\subset \mathbb{Q}^2$ be the cone of the surface toric singularity
corresponding to the continued fraction $[m_2,\dots,m_{r-1}]$.
Let $v_1$ and $v_2$ be the lattice generators of $\sigma$.
Let $x_1,\dots, x_r$ be a regular decomposition of $\sigma$.
Define $y_2:=m_rx_r - c_{r-1}x_{r-1}$.
Let $\tau$ be the cone spanned by $x_r$ and $y_2$.
Then, $\tau$ admits a regular decomposition
given by lattice vectors $w_1,\dots, w_s$, where $s\leq c_{r-1}$.
There exists a unique linear function $M$ on $\mathbb{Q}^2$ for which
\[
M(v_1)=1-c_1 \text{ and } M(y_2)=\beta_r.
\]
Let $\Sigma$ be the cone spanned by $v_1$ and $y_2$.
Note that the values $1-c_1$ and $\beta_r$ belong to a set,
which only depends on $n,N$, and $\Lambda$,
and satisfies the ascending chain condition.
By construction, this linear function takes value $\alpha_i$ at $x_i$, for every $i\in \{2,\dots,r\}$.
If the minimizer of $M$ in ${\rm relint}(\Sigma)\cap \mathbb{Z}^2$ is contained in
${\rm relint}(\sigma)\cap \mathbb{Z}^2$, then we are done by Lemma~\ref{lem:k-th-surf-toric-mld}.
Otherwise, we can assume that the minimizer is contained in ${\rm relint}(\tau)$.
By Lemma~\ref{lem:min-lattice}, we know that the minimizer must be one of the lattice elements
$\{w_1,\dots,w_s\}$.
Assume that it is attained at $w_s$.
By Lemma~\ref{lem:k-th-surf-toric-mld}, the value of $M$ at $w_s$ belongs to a set
satisfying the ascending chain condition which only depends on $\beta_r$ and $1-c_1$.
We can replace $y_2$ with $w_s$ and proceed inductively.
Since $s\leq c_{r-1}$, this argument can be repeated at most $c_r$ times
until the minimizer is contained in ${\rm relint}(\sigma)$.
Hence, one of the $(c_{r-1}+1)$ smaller values of $M$ in ${\rm relint}(\Sigma)\cap \mathbb{Z}^2$ must
lie in ${\rm relint}(\sigma)$.
Then, the statement follows from Lemma~\ref{lem:k-th-surf-toric-mld}.\\
\textit{Step 4:} We prove the statement of the theorem in the case that $\mathcal{D}(Y,B_Y)$ is a closed interval: each $E_i$ with $i\in \{1,\dots, r\}$ maps onto $x$.\\
By~\cite[Theorem 1]{Mor20}, there exists a projective birational morphism
$\phi_0 \colon Y_0\rightarrow X$ whose exceptional locus is purely divisorial and consists of the divisors $E_1$ and $E_r$.
In the model $Y_0$, the divisors $E_1$ and $E_r$ intersect at an irreducible set of codimension two.
Furthermore, the center of $E_i$, with $i\in \{2,\dots,r-1\}$ maps onto such irreducible subvariety.
By~\cite[Theorem 18.22]{Kol92}, the singularity of $(Y_0,E_1+E_r+B_{Y_0}')$ is toric at the generic point of $E_1\cap E_r$.
Then, each $E_i$ with $i\in \{2,\dots,r-1\}$ can be extracted by a sequence of toroidal blow-ups.
For $i\in \{1,r\}$, we denote by $(E_i,B_{E_i})$ the pair obtained by adjunction of
$(Y_0,E_1+E_r+B_{Y_0}')$ to $E_i$.
By Lemma~\ref{lem:from-reg-1-to-0}, we know that $(E_i,B_{E_i})$ has regularity zero
and $\lfloor B_{E_i}\rfloor$ is a prime divisor $S_i$ which corresponds to the intersection $E_1\cap E_r$.
Proceeding as in Step 3, for $i\in \{1,r\}$, we can find a movable curve $C_i$ on $E_i$ satisfying the following conditions:
\begin{itemize}
\item $C_i$ is in the smooth locus of $E_i$,
\item $C_i$ intersects $S_i$ transversally in at most $k$ points, and
\item $C_i$ is either disjoint from $B'_{E_i}$ or it intersects $B'_{E_i}$ transversally in at most $k$ points.
\end{itemize}
Here, $k$ is a constant which only depends on $n$ and $N$.
We claim that the curves $C_1$ and $C_r$ span $N_1(Y_0/X)$.
Note that $N_1(Y_0/X)$ is two-dimensional, so it suffices to prove that $C_1$ and $C_r$ are linearly independent.
Assume that $C_1+\alpha C_r \equiv_X 0$ for some $\alpha\in \mathbb{R}$.
Observe that $E_1\cdot C_1$ is negative and $E_1\cdot C_r$ is positive.
We conclude that $\alpha$ must be positive.
Hence, we have an effective curve which is numerically trivial over $X$.
This gives a contradiction.
Thus, $\langle C_1,C_r\rangle = N_1(Y_0/X)$.
We can produce a dlt modification of $(Y_0,E_1+E_r+B_{Y_0}')$ by
performing a sequence of toroidal blow-ups at the generic point of $E_1\cap E_r$.
We replace $(Y,B_Y)$ by this dlt modification.
For $i\in \{2,\dots,r-1\}$, each prime component $E_i$ of $\lfloor B_Y\rfloor$ admits the structure of a log pair
of regularity zero.
For each $i\in \{2,\dots,r-1\}$, we denote by $C_i$ a general $\mathbb{P}^1$ in $E_i$.
We may identify both $C_1$ and $C_r$ with their strict transforms on $Y$.
By Lemma~\ref{lem:curves-generating}, we have that
\[
\langle C_1, C_2,\dots, C_r\rangle = N_1(Y/X).
\]
Let $\phi \colon Y\rightarrow X$ be the induced birational morphism.
We can write
\begin{equation}\label{eq:s4-1}
\phi^*(K_X+\Delta) =
K_Y + (1-\alpha_1) E_1 + \dots + (1-\alpha_r)E_r + \Delta'_Y
\end{equation}
and
\begin{equation}\label{eq:s4-2}
\phi^*(K_X+B_X) = K_Y+E_1+\dots+E_r+B'_Y,
\end{equation}
where $B'_Y\geq \Delta'_Y$.
Subtracting the equations~\eqref{eq:s4-1} and~\eqref{eq:s4-2}, we obtain that
\begin{equation}\label{eq:s4-3}
\alpha E_1 + \alpha_2 E_1 + \dots + \alpha_r E_r \sim_{\mathbb{Q},X} \Delta_Y'-B_Y'.
\end{equation}
Observe that for each $i\in \{2,\dots, r-1\}$, the curve $C_i$ lies in the smooth locus
of $Y$ and $C_i$ intersects exactly two other prime components of $\lfloor B_Y\rfloor$.
Moreover, each such $C_i$ is disjoint from $B_Y'$ and hence from $\Delta_Y'$.
On the other hand, for $i\in \{1,r\}$, we have that
\[
E_i \cdot C_i = - \frac{m_i}{N!}, \quad
(\Delta'_Y-B'_Y)\cdot C_i = -\beta_{i,0}, \quad
E_2 \cdot C_1= c_{2,0}, \text{ and } \quad
E_{r-1}\cdot C_r= c_{r-1,0}.
\]
where $c_{r-1,0},c_{2,0},m_1$, and $m_r$ are positive integers.
$\beta_{1,0}$ and $\beta_{r,0}$ are rational numbers which belongs
to a set satisfying the ascending chain condition.
The denominators of the intersection products $E_1\cdot C_1$ and $E_r\cdot C_r$ are bounded above
as the variety $Y$ has quotient singularities of order at most $N$ along $C_1$ and $C_r$.
We set
\[
c_{r-1}=c_{r-1,0}N!, \quad c_2:=c_{2,0}N!, \quad \beta_1=\beta_{1,0}N!,
\text{ and } \quad \beta_r:=\beta_{r,0}N!.
\]
Intersecting the $\mathbb{Q}$-linear equivalence~\eqref{eq:s4-3}, with the curves
$C_i$ with $i\in \{2,\dots,r\}$, we obtain the following system of linear equations:
\begin{equation}
\label{eq:s4-4}
\left[
\begin{matrix}
m_1 & -c_2 & 0 & \dots & \dots & 0 \\
-1 & m_2 & -1 & \dots & \dots & 0\\
0 & -1 & m_3 & \ddots & \dots & 0 \\
\vdots & \vdots &\vdots &\vdots &\vdots &\vdots \\
0 & 0 & 0 & -1 & m_{r-1} & -1 \\
0 & 0 & 0 & 0 & -c_{r-1} & m_r
\end{matrix}
\right]
\left[
\begin{matrix}
\alpha_1 \\
\alpha_2 \\
\alpha_3 \\
\vdots \\
\alpha_{r-1} \\
\alpha_r
\end{matrix}
\right]
=
\left[
\begin{matrix}
\beta_1 \\
0 \\
0 \\
\vdots \\
0 \\
\beta_r
\end{matrix}
\right]
\end{equation}
Note that this linear system~\eqref{eq:s4-4} has a unique solution as the matrix on the left-hand side has full rank.
This follows from the fact that the $C_i$'s generate the relative cone of curves.
Let $\sigma\subset \mathbb{Q}^2$ be the cone of the surface toric singularity
corresponding to the continued fraction $[m_2,\dots,m_{r-1}]$.
Let $v_1$ and $v_2$ be the lattice generators of $\sigma$.
Let $x_1,\dots, x_r$ be a regular decomposition of $\sigma$.
Define
\[
y_1:=m_1x_1 - c_2x_2 \text{ and }
y_2:=m_rx_r - c_{r-1}x_{r-1}.
\]
Let $\tau_1$ be the cone spanned by $v_1$ and $x_1$, and
let $\tau_2$ be the cone spanned by $x_r$ and $y_2$.
Then, $\tau_1$ (resp. $\tau_2)$ admits a regular decomposition
given by lattice vectors $u_1,\dots, u_{s_1}$ (resp. $w_1,\dots,w_{s_2}$) where $s_1,s_2\leq c_{r-1}$.
There exists a unique linear function $M$ on $\mathbb{Q}^2$ for which
\[
M(y_1)=\beta_1 \text{ and } M(y_2)=\beta_r.
\]
Let $\Sigma$ be the cone spanned by $y_1$ and $y_2$.
Note that the values $\beta_1$ and $\beta_r$ belong to a set,
which only depends on $n,N$, and $\Lambda$,
and satisfies the ascending chain condition.
By construction, we have that
\[
M(x_i)=\alpha_i \text{ for every $i\in \{1,\dots,r\}$.}
\]
If the minimizer of $M$ in ${\rm relint}(\Sigma)\cap \mathbb{Z}^2$ is contained in
${\rm relint}(\sigma)\cap \mathbb{Z}^2$, then we are done by Lemma~\ref{lem:k-th-surf-toric-mld}.
Otherwise, we can assume that the minimizer is contained in ${\rm relint}(\tau_1)$.
By Lemma~\ref{lem:min-lattice}, we know that the minimizer must be one of the lattice elements
$\{u_1,\dots,u_{s_1}\}$.
Assume that it is attained at $u_1$.
By Lemma~\ref{lem:k-th-surf-toric-mld},
the value $M(u_1)$
belongs to a set
satisfying the ascending chain condition which only depends
on $\beta_1$ and $\beta_r$.
We can replace $y_1$ with $u_1$ and proceed inductively.
Since $s_1,s_r\leq kN!$ this argument can be repeated at most $kN!$ times
until the minimizer is contained in ${\rm relint}(\sigma)$.
Hence, one of the $(kN!+1)$ smaller values of $M$ in ${\rm relint}(\Sigma)\cap \mathbb{Z}^2$ must
lie in ${\rm relint}(\sigma)$.
Then, the statement follows from Lemma~\ref{lem:k-th-surf-toric-mld}.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{introcor:bounded-ext}]
Let $(X;x)$ be a $n$-dimensional $\mathbb{Q}$-factorial klt singularity of regularity one.
By Lemma~\ref{lem:reg-bounded-comp} and Lemma~\ref{lem:reg-comp-lcc},
there exists a constant $N:=N(n)$, only depending on $n$, satisfying the following.
There exists a $N$-complement $(X,B;x)$ so that
\[
{\rm reg}(X,\Delta;x)={\rm reg}(X,B;x).
\]
Moreover, $x$ is a log canonical center of $(X,B;x)$.
By the proof of Theorem~\ref{thm:reg-one-coeff}, we know that the minimal log discrepancy
at the divisors over $x$ which are log canonical places of $(X,B;x)$
satisfies the ascending chain condition.
In particular, there exists a constant $a(n)$, only depending on $n$, so that
$a_E(X,\Delta)<a(n)$ for some log canonical place $E$ of $(X,B;x)$ which maps onto $x$.
Then, the existence of the projective birational morphism
$Y\rightarrow X$ follows from~\cite[Theorem 1]{Mor20}.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{introthm:ACC}]
In the proof of Theorem~\ref{thm:reg-one-coeff}, if we start with a finite set $\Lambda$,
then the log discrepancy of every log canonical center of $(X,B;x)$ which does not map to $x$ belongs
to a finite set due to Lemma~\ref{lem:not-point}.
This implies that the coefficients of the surface toric germ constructed in the proof of Theorem~\ref{thm:reg-one-coeff}
belong to a finite set.
Hence, the corollary follows given that the accumulation points of the log discrepancies
of toric surface singularities with coefficients in a finite set can only accumulate to zero.
\end{proof}
\bibliographystyle{habbrv}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 96 |
Henrica Petronella Geertruida Maria (Henny) Houben-Sipman (Delft, 12 maart 1946) is een Nederlands voormalig politicus van de KVP en later het CDA.
Ze was onder meer gemeenteraadslid in Eindhoven en lid van de Eerste Kamer.
In 1991 werd ze benoemd tot burgemeester van Eersel. Toen in 1997 de gemeente opging in de nieuwe gemeente Eersel werd zij herbenoemd. Op 3 maart 2008 heeft de voormalige Somerse wethouder Anja Thijs-Rademakers haar opgevolgd.
Burgemeester van Eersel
CDA-politicus
Eerste Kamerlid
Gemeenteraadslid van Eindhoven
KVP-politicus | {
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View All Schools in Boston »
Carroll School of Management – Boston College
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Founded in 1947, the F.W. Olin Graduate School of Business at Babson College is located in the Babson Park area of Wellesley, Massachusetts, a suburb of Boston. The school offers a traditional full-time MBA program, a one-year program for those with an academic background in business, an evening MBA program for those who are fully employed, and a Blended Learning part-time MBA program. Other available options include a Master of Science in Management in Entrepreneurial Leadership program, a Master of Science in Finance, and a Certificate in Advanced Management. The business school is accredited by the Association to Advance Collegiate Schools of Business, the New England Association of Schools and Colleges and the EFMD Quality Improvement System (EQUIS).
• U.S. News & World Report: 68 (tie)
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The main campus of Babson College is situated approximately 14 miles from the city of Boston in Wellesley, Massachusetts. The school also maintains centers in Boston, San Francisco, Miami, and Dubai. Babson's main campus can be found in proximity to Exits 14 and 15 on the Massachusetts Turnpike and then Route 16 towards Newton/Wellesley.
The main campus of Babson College is located less than 20 miles away from Boston's Logan International Airport.
Olin is located on the northern edge of Babson's campus. Students of the graduate school complete most of their classes in Olin Hall, a building that includes rooms for group study, a café and spaces for student clubs and events. MBA students may also take advantage of Babson's centers and research institutes, fitness center, cafeteria, and student center as well as the Horn Library, which features research support and computer services for the Babson community.
Babson currently employs more than 160 full-time faculty members and 75 part-time faculty. Eighty-seven percent of all full-time Babson faculty members hold a doctorate or an equivalent degree. All programs offered by the Olin School of Business are taught based on the school's hallmark entrepreneurial and action-oriented approach to learning.
The F.W. Olin Graduate School of Business at Babson MBA program currently enrolls 574 total graduate students across the four degree tracks.
MBA Degree Offerings
Olin offers a two-year full-time MBA program, a one-year program, a two-year program, and part-time programs for evening and blended learning students.
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Transcript should not be used in the code | {
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Domitius Epictetus (sein Praenomen ist nicht bekannt) war ein im 2. oder 3. Jahrhundert n. Chr. lebender Angehöriger der römischen Armee.
Durch eine Inschrift, die beim Kastell Arbeia gefunden wurde und die auf 121/300 datiert wird, ist belegt, dass Epictetus Angehöriger einer Militäreinheit war (commilitonibus). Laut John Spaul und Eric Birley war er Präfekt der Cohors V Gallorum, die in der Provinz Britannia stationiert war.
Weblinks
Einzelnachweise
Militärperson (Römische Kaiserzeit)
Römer
Geboren im 1. Jahrtausend
Gestorben im 1. Jahrtausend
Mann | {
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{"url":"https:\/\/www.matrix.edu.au\/beginners-guide-year-9-maths\/part-5-surface-area\/","text":"# Part 5: Surface Area\n\nIn this article, we show you how to find the surface area of common 3-dimensional shapes.\n\n## Surface Area\n\nAre you confident with working out the surface areas of shapes? Don\u2019t worry! This guide will provide you with the steps needed to find surface areas, examples and some sample questions and solutions for you to practice.\n\n## Syllabus Outcomes\n\nStage 5.3: Solve problems involving the surface areas of right pyramids, right cones, spheres and related composite solids (ACMMG271)\n\n\u2022 Identify the \u2018perpendicular heights\u2019 and \u2018slant heights\u2019 of right pyramids and right cones.\n\u2022 Apply\u00a0Pythagoras\u2019 theorem to find the slant heights, base lengths and perpendicular heights of right pyramids and right cones.\n\u2022 Devise and use methods to find the surface areas of right pyramids.\n\u2022 Develop and use the formula to find the surface areas of right cones:\n$$SA = \\pi rl$$where $$r$$ is the length of the radius and $$l$$ is the slant height\n\u2022 Use the formula to find the surface areas of spheres:\n$$SA=4\\pi r^{2}$$,\u00a0where r is the length of the radius.\n\u2022 Solve a variety of practical problems involving the surface areas of solids.\n\u2022 Find the surface areas of composite solids,\neg. a cone with a hemisphere on top (Problem Solving)\n\u2022 Find the dimensions of a particular solid, given its surface area, by substitution into a formula to generate an\u00a0equation (Problem Solving)\n\nThis unit exists because cylinders, cones, and spheres exist everywhere in the world from pyramids from ancient civilisations through to contemporary rockets like the Space-X Falcon 9. We need to able discuss the features of these shapes and to do this we need to be able to calculate things like their surface area.\n\n\u2022 Interpreting the formula for surface area.\n\u2022 Understanding the difference between the slant height and the perpendicular height of a cone.\n\u2022 Finding surface areas of composite objects.\n\n## Assumed Knowledge:\n\n\u2022 Finding areas of 2D figures\n\nHow would you find the surface area of this building?\n\n## How to work our surface areas\n\n### Interpreting\u00a0formulas for surface areas\n\nYou have already encountered some formulas for the surface area of different shapes.\n\nHere are some examples:\n\n Solid Formula for Surface Area Cylinder $$2 \\pi r^{2}+2 \\pi rh$$ Cone $$\\pi r^{2} + \\pi rl$$ Sphere $$4\\pi r^{2}$$\n\nYou might be thinking, \u201cDo I really have to memorise these formulas?\u201d\n\nNo, not really! However, some are worth remembering because they will help you in the long run.\n\nIn this guide, we\u2019ll break down each formula.\n\nUnderstanding where each formula comes from is really important because it allows you to memorise less and tackle harder problems.\n\nWe\u2019ll now take a look at each formula to see where it comes from. We\u2019ll then see how this will help us to solve some harder problems.\n\n### Cylinders\n\nWe\u2019re going to go through how you might find the surface area of a cylinder from scratch. Let\u2019s take an imaginary cylinder of height $$h$$ and radius $$r$$ and open it up:\n\nDoing this shows that a cylinder is made up of two circles and a rectangle. All we have to do now is to find the areas for each part and add them all up!\n\nFor both circles, they have a radius of $$r$$. So, for each circle we have an area of $$\\pi r^{2}$$. This gives us the $$2 \\pi r^{2}$$ bit. That was easy!\n\nThe $$2 \\pi rh$$ bit must then be given by the rectangle. One of the sides is the height of the cylinder $$h$$.\n\nThe side which touches the circle will be \u201cglued\u201d to the edge of the circle when we make a cylinder. This means that its length is the same as the circumference of the circle $$2 \\pi r$$.\n\nSo, the area of the rectangle is given by $$h\u00d72 \\pi r = 2\\pi rh$$.\n\nAdding the two contributions gives the total surface area: $$SA=2\\pi r^{2}+2 \\pi rh$$. Check this with the formula given in the table above!\n\nYou see, we don\u2019t need to remember the formula for the surface area of a cylinder. All we need to remember is how to calculate the area of the two circles and the area of the curved section (which is really a rectangle).\n\n### To summarise:\n\n\u2022 $$SA=2 \\pi r^{2}+2 \\pi rh$$\n\u2022 Each circle gives us $$\\pi r^{2}$$\n\u2022 $$2 \\pi rh$$ comes from the curved surface\n\n## Cones\n\nTo find the surface area of a cone, we need both its radius $$r$$ and its slant height $$l$$ (not the perpendicular height). We\u2019ll also look at its net.\n\nFrom the net, we see that we have a circle that comes from the base, as well as a sector of a circle.\n\nThe area of the circle is given by $$\\pi r^{2}$$\u00a0and the area of a sector is given by $$\\pi rl$$\u00a0(not required to know proof).\n\nSo, we immediately see that the surface area of the cone is given by $$SA = \\pi r^2 + \\pi rl$$.\n\nThe only part that you really need to memorise is the area of the curved bit $$\\pi rl$$. Otherwise, this is fairly straightforward.\n\n### To summarise:\n\n\u2022 $$SA= \\pi r^{2}+\\pi rl$$\n\u2022 $$l$$ is the slant height\n\u2022 $$\\pi r^{2}$$ comes from the circular base\n\u2022 $$\\pi rl$$ comes from the curved surface\n\nIn some problems you might be told the perpendicular height, but not the slant height. In that case, you simply use Pythagoras\u2019 theorem to find the slant height:\n\n### Example\n\n1. Find the slant height of a cone with radius\u00a0 5 cm and perpendicular height 12 cm.\n\nDrawing a picture and applying Pythagoras\u2019 theorem gives:\n\nWhats the volume of this cone?\n\n$$l^{2} = 12^{2} + 5{2} \\\\ =169 \\\\ l=\\sqrt{169} \\\\ =13 \\text{cm} \\\\$$\n\n## Sphere\n\nThe proof for the surface area of a sphere is beyond what you need to know.\n\nSo, this is one of the formulas that you just have to memorise.\n\nHowever, it isn\u2019t too bad to remember \u2013 just think of it as 4 times the area of the circle $$SA= 4\\pi r^{2}$$.\n\n## Finding the surface area for Composite Shapes\n\nTypically, you\u2019ll be asked to find the surface area of shapes like this:\n\nComposite shape\n\nHere we\u2019ve got a cylinder sitting on top of a cone.\n\nWe\u2019ll need to use our formulas for the surface area of cones and cylinders but not directly.\n\nFor example, if we were na\u00efve, we could take the surface area formulas of a cone and a cylinder and add them together to get $$SA=(\\pi r^{2} + \\pi rl) + (2 \\pi r^{2} + 2 \\pi rh)$$.\n\nWhy wouldn\u2019t this give us the right answer?\n\nThe problem is that we\u2019ve counted the circle that is sandwiched between the cone and the cylinder twice, when we shouldn\u2019t have even counted it in the first place!\n\nLet\u2019s have another go but being a bit more careful this time.\n\nWe\u2019ll first break up the solid into simple parts where we can find the surface area of each part.\n\nIn our case, we\u2019ve got a cone without its circular base as well as a cylinder without its top circle.\n\nNow because we\u2019ve thought about the surface area formulas, we know immediately that the cone bit gives $$\\pi rl$$ and the cylindrical bit gives $$\\pi r^{2} + 2 \\pi rh$$\u00a0(one circle from the top).\n\nSo, the total surface area is $$SA = \\pi rl + \\pi r^{2} + 2 \\pi rh$$.\n\nLet\u2019s do another example, this time with numbers.\n\n### Example\n\n1. Find the surface area of the following solid:\n\nHere, we have a hemisphere and a cone.\n\nNote that the radius of the cone is the same as the radius of the hemisphere from the way they are attached i.e. 3 cm.\n\nThe surface area of the hemisphere, call it $$SA_{1}$$ is given by:\n\n$$SA_{1}=\\frac{1}{2}\u00d74 \\pi r^{2} = 2 \\pi (3)^{2}=18\\pi$$\n\nFor the surface area of the cone, we need its slant height. This can be found using Pythagoras\u2019 theorem since we know its perpendicular height and its radius:\n\n$$l^{2} = 5^{2} + 3^{2} = 34 \\\\ l = \\sqrt{34} \\text{cm} \\\\$$\n\nSo the surface area of the cone, call it $$SA_{2}$$\u00a0is given by:\n\n$$SA_{2}= \\pi rl= \\pi (3)(\\sqrt{34})=3\\sqrt{34}\\pi$$\n\nNote that we didn\u2019t count the circular base because it isn\u2019t part of the surface area. The total surface area is then:\n\n$$SA=SA_{1}+SA_{2}=18 \\pi +3\\sqrt{34} \\pi \\text{cm}^{2} \u2248 111.50 \\text{cm}^{2}$$\n\nIf your final answer is a number, don\u2019t forget to put the correct units down!\n\nLet\u2019s do one last example!\n\n2.\u00a0Find the surface area of the following pyramid whose perpendicular height is 4 cm:\n\nFor the base, we have a square whose area is $$SA_{1}=2\u00d72=4$$\n\nSince this is a square pyramid, all triangles are the same. To work out the area of the triangle, we will need the pyramid\u2019s slant height\n\n$$l^2=1^2+4^2=17 \\\\ l=\\sqrt{17} cm \\\\$$\n\nNow each of the four triangles of the pyramid has base 2 cm and height\u00a0 $$\\sqrt{17} \\text{cm}$$, so each triangle has area:\n\n$$SA_2=\\frac{1}{2}\u00d72\u00d7\\sqrt{17}=\\sqrt{17}$$\n\nSo, the total surface area is\n\n$$SA=SA_1+4\u00d7SA_2=4+4\\sqrt{17}\u224820.49 \\text{cm}^{2}$$\n\n## Summary:\n\nTo find the surface areas of composite solids:\n\n1. Break the surface area into smaller areas which can be found using known formulas. Remember to avoid double counting areas!\n2. Correctly apply the formulas for each individual area, making sure that the right numbers are substituted. E.g. using the slant height not the perpendicular height for the surface area of a cone\n3. Add each individual area up to obtain the total surface area. Make sure to put the correct units as well.\n\n## Sample Questions:\n\nTry to find the surface area of the following solids correct to 2 decimal places!\n\nCheck your own work with the worked solutions below.\n\n1.\n\n2.\n\n3.\n\n4.\u00a0(Cylinder with inner section removed)\n\n5.\n\n6.\u00a0(Cylinder with hemi-spherical cap)\n\n7.\n\n8.\n\n9.\n\n10. (Cube with cone placed on top)\n\n## Worked Solutions:\n\n1.\n\nArea of base: $$SA_1=2\u00d72=4$$\nFor each triangle: $$SA_2=1\/2 (2)(5)=5$$\nSo $$SA=SA_1+4\u00d7SA_2=24 \\text{cm}^2$$\n\n2.\n\nArea of base: $$SA_{\\text{base}}=2\u00d74=8$$\nSlant heights: $$l_1=\\sqrt{5^{2}+2^{2}}=\\sqrt{29}$$ ,\n$$l_2=\\sqrt{5^{2}+1^{2}}=\\sqrt{26}$$\nArea of triangles: $$SA_1=\\frac{1}{2} (4) l_1=2\\sqrt{29}$$ ,\n$$SA_2=\\frac{1}{2} (2) l_2=\\sqrt{26}$$\n$$SA=SA_{\\text{base}}+2\u00d7SA_1+2\u00d7SA_2=39.74 \\text{cm}^2$$\n\n3.\n\n$$SA_{\\text{prism}}=28 m^2$$\n$$SA_{\\text{pyramid}}=16 m^2$$\n$$SA=SA_{\\text{prism}}+SA_{\\text{pyramid}}=44 m^2$$\n\n4.\n\nOuter cylindrical sleeve: $$SA_1=2\\pi rh=800\\pi$$\nInner cylindrical sleeve: $$SA_2=400\\pi$$\nArea of each ring: $$SA_3=\\pi (10)^{2}-\\pi(5)^{2}=75\\pi$$\n$$SA=SA_1+SA_2+2\u00d7SA_3=1350\\pi=4241.15 \\text{cm}^{2}$$\n\n5.\n\nCylindrical area: $$SA_1=\\frac{1}{2} (2 \\pi r^{2}+2 \\pi rh)= \\frac{1}{2} (2\\pi (0.5)^{2}+2 \\pi (0.5)(10))=5.25 \\pi$$\nArea of rectangle: $$SA_2=1\u00d710=10$$\n$$SA=SA_1+SA_2=26.49 m^2$$\n\n6.\n\n$$SA_{\\text{hemisphere}}=\\frac{1}{2}\u00d74 \\pi r^{2}=50 \\pi$$\n$$SA_{\\text{cylinder}}= \\pi r^{2} +2 \\pi rh=75 \\pi$$ \u00a0 \u00a0 ($$r=5$$)\n$$SA=SA_{\\text{hemisphere}}+SA_{\\text{cylinder}}=392.70 \\text{cm}^{2}$$\n\n7.\n\n$$SA_{\\text{cylinder}}=\\pi r^2+2 \\pi rh=39 \\pi$$\n$$SA_{\\text{cone}}= \\pi rl = 12 \\pi$$\n$$SA=SA_{\\text{cone}}+SA_{\\text{cylinder}}=160.22 m^2$$\n\n8.\n\n$$SA_{\\text{cylinder}}= \\pi r^2+2 \\pi rh=39 \\pi$$\nSlant height: $$l=\\sqrt{3^2+4^2}=5$$\n$$SA_{\\text{cone}}=\\pi rl=15\\pi$$\n$$SA=SA_{\\text{cone}}+SA_{\\text{cylinder}}=169.65 m^2$$\n\n9.\n\n$$SA_{\\text{top cone}}=3\\pi$$\n$$SA_{\\text{bottom cone}}=12\\pi$$\n$$A_{\\text{ring}}=\\pi (2)^2-\\pi (1)^2=3\\pi$$\n$$SA=SA_{\\text{top cone}}+SA_{\\text{bottom cone}}+A_{\\text{ring}}=54.55 m^2$$\n\n10.\n\n$$SA_{\\text{cube}}=6\u00d72^2=24$$\n$$SA_{\\text{cone}}= \\pi rl=2 \\pi$$\n\nWe also need to subtract off the area of the circle hiding underneath the cone:\n\n$$A_{\\text{circle}}=\\pi(1)^2=\\pi$$\n$$SA=SA_{\\text{cube}}+SA_{\\text{cone}}-A_{\\text{circle}}=27.14 cm^2$$\n\n## Fill up on knowledge about volume\n\nNow that you know how to calculate surface area, you\u2019re ready to learn how to calculate the volume of things.\n\n\u00a9 Matrix Education and www.matrix.edu.au, 2019. Unauthorised use and\/or duplication of this material without express and written permission from this site\u2019s author and\/or owner is strictly prohibited. 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<?php
use Illuminate\Database\Schema\Blueprint;
use Illuminate\Database\Migrations\Migration;
class PulsarCreateTablePreference extends Migration
{
/**
* Run the migrations.
*
* @return void
*/
public function up()
{
if(! Schema::hasTable('001_018_preference'))
{
Schema::create('001_018_preference', function (Blueprint $table) {
$table->engine = 'InnoDB';
$table->string('id_018', 50);
$table->integer('package_id_018')->unsigned();
$table->text('value_018')->nullable();
$table->timestamps();
$table->foreign('package_id_018', 'fk01_001_018_preference')
->references('id_012')
->on('001_012_package')
->onDelete('restrict')
->onUpdate('cascade');
$table->primary('id_018', 'pk01_001_018_preference');
});
}
}
/**
* Reverse the migrations.
*
* @return void
*/
public function down()
{
if (Schema::hasTable('001_018_preference'))
{
Schema::drop('001_018_preference');
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
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Applications for the job of Governor of the Reserve Bank closed this morning.
the Board, charged with evaluating and recommending a candidate to the incoming Minister of Finance, also has no real idea what the job is. The emphases of a Labour/New Zealand First government (say) would probably be rather different than those of a National-ACT government.
And yet, with still 2.5 months until the election, the Board will shortly settle down with their recruitment consultants to winnow down the list of applicants. And this is a Board entirely appointed by the current government, and although the individual Board members may each be quite capable they are likely to be a different bunch of people, with different inclinations and preferences, than a Board appointed by a Labour-led government would have been. Of course, elections have consequences – governments get to appoint sympathetic people to the (too) numerous goverment bodies – but it isn't obvious why, if this year's election leads to a change of government, the last election should so strongly influence the sort of person likely to be presented to the incoming Minister of Finance as the nomineee for Governor.
Board members have neither legitimacy nor expertise. They aren't elected, don't front up to select committees or the media, don't even maintain proper records (as required by law), and can't be tossed out by voters if they do a poor job. In other countries, almost every country I'm aware of, the Governor of the central bank is appointed by the Minister of Finance (or some other elected person – eg the President in the US). And in most countries, the Governor of the central bank has much less power than our Governor has.
In our system in particular, the Governor is a really consequential appointment. The Governor is the sole legal decisionmaker on monetary policy and most aspects of banking regulation (as well as numerous other less important things the Bank is responsible for). He – and it most probably will be a he – alone gets to decide how aggressively the Bank should respond to economic downturns, or how closely it adheres to the Policy Targets. He gets to decide how well-positioned New Zealand is for the next recession. He gets to decide whether banks are even allowed to lend to you by residential mortgage. He could, if he chose, stake billions of taxpayers' money on interventions in the foreign exchange markets, and if the bet goes wrong we – not he – lives with the consequences. There is a gaping democratic deficit – too much power in one person's hands – but is made worse by the fact that elected politicians (whom we can hold to account) can't appoint someone in whom they have confidence to exercise these powers. They must take a name handed to them by a bunch of company directors and the like appointed by the previous government. The Minister can knock back any particular Board nominee, but in the end the Minister can only appoint someone that Board nominates. And he can't even easily replace the Board at short notice.
So who are these enormously influential members of the Board? One member's term expires in a week or so, and apparently he won't be replaced until after the election. That leaves six of them.
The chair is now Neil Quigley, vice-chancellor of Waikato University. These days, Neil is an academic administrator, but earlier in his career he had a research background in banking regulation, financial history etc.
The vice-chair is Kerrin Vautier, a microeconomist by background, who has also been a company director and is a lay member of the High Court (under the Commerce Act).
The other four include two private sector company directors (one of whom is a director of an insurance company, even though the Reserve Bank regulates insurance companies), one lawyer, and one member whose roles seem to be mainly government appointments and not-for-profit positions.
I don't want to be too critical of them as individuals. I know, and have worked with, three of them at various times, and they are each able people. But none of the six individually – or the group collectively – really seem to have the skills for making such a crucial public appointment. They are not subject experts and have no expert advisers – and yet they must presumably evaluate applicants' monetary policy or regulatory inclinations/expertise – nor do they bear the downside if they make a poor recommendation. They also have no experience in high profile public roles. Ministers of Finance also typically aren't experts, but (a) they have an entire Treasury to assist, and (b) they do bear the downside, since the public (reasonably enough) tend to hold elected politicians to account for the failures of public agencies.
The process is so flawed that I've argue before, and repeat the point today, that the Opposition parties really should indicate that, if elected, they will change the law to allow the Governor to be appointed simply by the Cabinet on the recommendation of the Minister of Finance, as is done in most other countries (perhaps with advisory quasi-confirmation hearings by a parliamentary committee). Not only would it make our system more democratically legitimate and internationally comparable, it would also put the Board in a better position to monitor and hold to account whoever is appointed as Governor. They'd have no incentive to simply back their own appointee, but could simply do the monitoring job – on whoever the Minister appointed – as agent for the Minister and the public.
about the outgoing Governor's stewardship of the role during the last five years (especially bearing in mind that many of the current Board were responsible for Wheeler's appointment).
It isn't obvious the Board has really been doing the second with much energy at all. I've written previously about their Annual Reports, which never seem to have found any cause for concern about anything, functioning more as additional legs in the Bank's own public communications efforts. The year just ended is the first in which the new chair, Neil Quigley, has led the Board. Perhaps this year's report will be a bit different and a bit more open but it is difficult to be very optimistic. This is, after all, the same Board who defended the Governor over the OCR leak debacle, expressed no concern even after the event at the ill-fated 2014 tightening cycle, and so far have been totally silent about Graeme Wheeler's highly inappropriate sustained attempts, including use of his senior managers, to attempt to censor a private sector critic.
the public face of an organisation whose choices at times bear heavily on the short-term performance of the New Zealand economy.
Under current law, those are features of the role at any time. But in the current situation, there is the additional challenge of rebuilding the Bank's capability and reputation after the Wheeler years. Monetary policy hasn't been well-handled. Banking regulation appears to frustrate the banks (more than the inevitable tension between regulator and regulated). No one now seriously looks to the Reserve Bank for "thought leadership" in the areas of its responsibility. And, for all that the Bank likes to claim to be very open and engaged, it is perhaps akin to (say) a Singapore-government style of openness, that chooses tame interlocutors and just ignores alternative perspectives or, say, journalists who might ask hard questions. There is a great deal of rebuilding to be done, and a good Governor over the next few years – particularly with the prospect of legislative change – needs to have qualities that will enable him or her not just to steward an organisation in good heart, but to lead organisational change and revitalisation.
With so much policymaking power personalised in the Governor's hands, it is difficult to see how the right appointee won't need to have a significant amount of directly relevant professional expertise. Of course, monetary policy is very different from banking and insurance regulation, and quite possibly no serious candidate is particularly strong in both fields. And on the financial sector side, it is important to recognise that this is a regulatory role – some of my friends differ on this, but the Reserve Bank (despite the name) isn't primarily a bank, even though it needs to understand banking to regulate it effectively.
Each of the three Governors under the current law has had an economics background. None was necessarily strong in the key technical dimensions (and of the three, only Don Brash had any prior experience in the public eye). Ideally, we would find someone better aligned to the role (as, say, the last three Governors of the Reserve Bank of Australia have been), but there may not be such a candidate. Really successful organisations are usually able to promote from within – again the Australian experience – but unfortunately the Reserve Bank here has been quite weak on developing people with the relevant senior experience (Grant Spencer has been both chief economist and head of financial stability, but he is retiring).
They still seem the most likely sort of people. For reasons I've outlined before I don't think it would be at all appropriate, let alone politically feasible, to appoint a foreigner as Governor, wielding all the power that position holds at present. One foreign member of (say) a five or seven person Monetary Policy Committee or Financial Policy Committee might make sense at times. But under our current law the Governor wields at least as much power as a typical Cabinet minister, and we require our Cabinet ministers to be New Zealanders.
When people occasionally ask me who I think will get the job, I usually note that Geoff Bascand must have the inside running. He is a competent economist (albeit not mainly in macro or financial areas), knows his way around the bureaucracy, and has outside chief executive experience. He has a number of downsides, including lack of any financial sector (or related regulatory experience), but appears to have been actively promoted by the current Governor (which perhaps should be a negative, but may not be). The Board gets to see him every month. A competent internal deputy is always likely to have the inside running.
I've heard that there is talk around Wellington that one of the CEOs of the Australian banks might be in line for the job, and Ian Narev's name (head of CBA, parent of ASB) has been specifically mentioned. At present, three of the group CEOs of the Australia banks are New Zealanders (one was apparently even a bank economist early in his career). One can't rule out the possibility completely, but none of these guys has any experience with monetary policy, nor any of being a regulator. And they are used to running vast organisations, not ones of 250 staff. You have to wonder whether a left-wing government – perhaps especially one including Winston Peters – would really be comfortable handing control of our monetary and banking system to the very wealthy CEO of an Australian bank (Narev for example took home A$12.3 million last year). And, of course, if one were a shareholder of an Australian bank mightn't the fact that the chief executive was looking to get out, applying for another job, be information that you might regard as material, warranting disclosure? It seems a more plausible option if, say, the Minister could just directly appoint someone in whom they had confidence, rather than going through a drawn-out application process.
Of the people on that list, David Archer probably now isn't widely known in New Zealand. He holds a senior position at the Bank for International Settlements (club for central banks) but was formerly Assistant Governor and Head of Economics, and prior to that Head of Financial Markets, at the Reserve Bank. He and Alan Bollard didn't really get on, and David went overseas in 2004. David would bring drive, energy, curiosity, openness, and high intellect. On the other hand, he has been out of New Zealand for a long time now, and that does have hard-to-pin-down disadvantages. He also doesn't have any real experience with financial regulation – in fact, in times past was a champion of a fairly minimalist approach (and whatever the merits of those arguments, they bear no relation to current NZ law). He also has – or had – a style that works very well with some (the intellectually self-confident, perhaps even combative) but not necessarily with others.
Rod Carr remains an interesting possibility. He was Deputy Governor for five years, and missed out on becoming Governor when Don Brash left. He has direct banking experience (albeit 20 years ago). He has also been a Reserve Bank Board member for the last five years, but was apparently ousted as chair by the other Board members last year. Perhaps he will apply for Governor. But in the various Board minutes that have been released to me in recent months, there is no sign that Carr absented himself from discussions around the process leading towards advertising for and selecting a new Governor. Had he been planning to apply, to have absented himself (and documented that absence) would have seemed appropriate conduct.
At very least, the Board should have left applications open until the election result is clear. That much was in the Board's own hands. Political leaders should have taken back to themselves the power to make (directly) such a vital appointment, as is done in other countries. There is still time.
It is now the school holidays. Posts over the next couple of weeks will be quite sparse. | {
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August 21, 2019 Eat & Drink » Restaurant Reviews
Thinking Way Outside the Bun
How a local burger chain found success introducing the Impossible Burger.
By Amanda Rock @amanda_eats_slc
Derek Carlisle
Welcome to the future. We have self-driving cars, our telephones are mini-computers and delicious fast food hamburgers are not necessarily made from dead cows.
The two meatless burgers on everyone's lips (literally) are the Beyond Burger and Impossible Burger. They're made by two different companies tackling climate change by offering an alternative to animal agriculture. According to Impossible Foods, creating one of its burgers, instead of a traditional ground beef burger, uses 87% less water, 96% less land use and emits 89% fewer greenhouse gas emissions. Meanwhile, Beyond Meat boasts that its burgers are created using 99% less water, 93% less land use and emits 90% fewer greenhouse gas emissions compared to the regular hamburger.
Both of these burgers look and taste like beef, not like the veggie burger you tried 10 years ago. Touted as "the future of protein," you can buy the Beyond Burger at grocery stores for about $6 for a two-pack. Trace amounts of beet juice give this plant-based burger a convincing bloody pink hue; throw them on a grill or a hot cast-iron skillet, the burgers sizzle and bleed. Satisfying and tasty, the Beyond Burger has a whopping 20 grams of protein from—get this—peas. They're made without GMOs, soy or gluten.
Impossible's counterpart is not sold in stores. The only way you can taste it is ordering it at a restaurant. The burger is juicy, flavorful and looks and tastes so much like beef, that every time I order it, I'm convinced the staff has accidentally given me a real hamburger. The reason it's so realistic lies in the fact that Impossible Foods has figured out how to make a plant-based heme (the essential molecule that gives meat its meaty flavor) by genetically modifying yeast. It's called soy leghemoglobin. If you're worried this all sounds a little too Soylent Green, rest easy knowing the company went above and beyond the U.S. Food and Drug Administration testing requirements. They even conducted a study feeding rats soy leghemoglobin, which was not well-received by the animal rights community. There's 19 grams of protein in the burger that comes from potatoes and soy.
The two burgers are tasty and calm your conscience when it comes to animal welfare and global warming, but are they better for your health? Not really, according to the nutritionists interviewed for a recent Huffington Post article. While they offer a protein punch, both brands rely heavily on coconut oil, which is high in saturated fat and the calorie count is similar to its beef counterpart. Oh, well.
Still, consumers have been clamoring for them, and fast-food chains known for their commercials featuring juicy, sizzling beef have listened. Carl's Jr. added the charbroiled Beyond Famous Star with cheese to their menu. Del Taco is offering signature tacos and burritos featuring Beyond Meat. Impossible Foods is not far behind. How about chomping down on an Impossible Whopper at Burger King, or scarfing a slice of Impossible sausage pizza from Little Caesars?
In Rose Park, Apollo Burger is a popular lunchtime destination. A line of construction workers, business people, state employees and soldiers dressed in Army fatigues flow out the doors each weekday at noon. Cars in the drive-thru, backed up onto the street, idle and wait. With the addition of the Impossible Burger to the menu, I've been among them, weary from a long day at work and craving sustenance in the holy trinity of hamburger, French fries and an ice-cold Coke.
Keeping up with dietary trends, Apollo added a veggie burger ($5.79) to their menu six years ago. Made from black beans and veggies, it's the polar opposite of the pastrami cheeseburgers the eatery is known and loved for. As one of the few burger chains to offer a vegan option, Apollo demonstrates their ability to think outside the bun. Vegans and vegetarians will gladly hand over their cash, if only you give us options. "We don't sell a lot," Andrew Baguley, Apollo's director of operations, admits. Paige Jacobson, director of marketing, is quick to add, "For a veggie burger, it's pretty good." I've ordered the veggie burger (with avocado) to justify eating their French fries, which are the best around, but I'd rather spend my money and daily caloric intake on an Impossible Burger. A few customers I've talked to prefer the OG burger, and, interestingly enough, sales for the veggie burger have not decreased with the introduction of the Impossible Burger.
With all the media attention Impossible Burger was getting last year, Baguley knew adding it to the menu would bring in millennial dollars. A menu revamp was in the works, and the addition of the Impossible Burger seemed like it would be at home on the joint's fancy new digital menus. Convincing the local chain's managers and owners was not easy. "It was not obvious that it was going to be a winner," Baguley says, adding that they've sold four times more Impossible Burgers than originally anticipated. "I thought it would be good, but I didn't know it would be this good," he says. "We are now the No. 1-selling restaurant of Impossible Burgers in Utah."
If you've tried the Impossible Burger from Apollo (pictured), it's easy to understand the immediate success. At just under $7, the price is right, the charbroil is delicious and the toppings are fresh. Each burger comes with a thick slice of tomato, onions, lettuce, and a dollop of Apollo Sauce. Vegans, skip the sauce and ask for a lettuce wrap or pay 99 cents extra for the gluten-free bun (which also happens to be vegan). You can also substitute an Impossible patty on any other burger for an additional cost. The Athenian burger, which comes with roasted red peppers, feta cheese and homemade tzatziki could be a good option. Their mushroom Swiss burger also sounds like a winner.
So what swayed Apollo to take a gamble on Impossible vs. Beyond? "You can buy the Beyond Burger anywhere, and the Impossible [one] is better," Baguley says. I have to agree with him. There's a Carl's Jr. nearby, but I'm rarely tempted to indulge in one of theirs, because I have a package of them at home.
What's next for Apollo? "We're evolving with the times and the customers," Jacobson says, while Baguley muses about finding a supplier for vegan pastrami. Personally, I'm holding out for vegan milkshakes.
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by Enrique Limón, Jordan Floyd, Sarah Arnoff, Darby Doyle, Kelan Lyons, Lance Gudmundsen, Scott Renshaw, Dash Anderson, Rachelle Fernandez, Nic Renshaw, Ray Howze, Alex Springer, Samantha Herzog, Amanda Rock and David Riedel
Salt Lake's Food Scene Is UNBELIEVABLE
by Enrique Limón, Alex Springer, Scott Renshaw, Ray Howze, Sarah Arnoff, Darby Doyle, Lance Gudmundsen, Samantha Herzog and Amanda Rock
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Victor's Tires & Restaurant offers vehicle service with a side of hearty Mexican food. | {
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Waterschap@inwonersbelangen is een lokale politieke partij in het waterschap Hoogheemraadschap De Stichtse Rijnlanden dat werkzaam is in grote delen van de provincie Utrecht en in een kleiner deel van de provincie Zuid-Holland.
Onder de naam Waterschap@inwonersbelangen deden een aantal lokale partijen uit het werkgebied van het Hoogheemraadschap De Stichtse Rijnlanden mee aan de Waterschapsverkiezingen 2008 en behaalde twee zetels in het algemeen bestuur.
Verkiezingsresultaten
Zie ook
Hoogheemraadschap De Stichtse Rijnlanden
Externe link
Waterschap@inwonersbelangen (gearchiveerde versie uit 2012)
https://www.inwonersbelangen.nl/
Waterschapsverkiezingslijst | {
"redpajama_set_name": "RedPajamaWikipedia"
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\section{Introduction}
Hyperspectral image (HSI) has attracted considerable attention in the remote sensing
community and been widely used in various areas \cite{bioucas2013hyperspectral}. With
the rich spectral information in HSI, different land cover categories can potentially
be differentiated precisely.
In recent years, deep learning has been widely used in various fields, including HSI
classification \cite{zhu2017deep}. Convolutional neural networks (CNNs) and residual
networks (ResNets) have obtained a successful result for HSI classification
\cite{lee2017going, zhong2017deep}. Recurrent neural networks (RNNs) are also applied
in HSI classification \cite{mou2017deep}.
Because of the ability to extract the spatial contextual information, CNNs and ResNets
can achieve a high accuracy in the classification task. However, CNNs and ResNets
consider spectra as orderless vectors in $d$-dimensional feature space where $d$
represents the number of bands. However, spectra can be seen as orderly and continuing
sequences in the spectral space. In other words, CNNs and ResNets ignore the continuity
of spectra \cite{mou2017deep}.
RNNs have proved effective in solving many challenging problems involving sequential
data, such as Natural Language Processing (NLP) \cite{sundermeyer2015feedforward} and
prediction of time series \cite{gers2002applying}. Considering the spectrum as a
sequential sequence, the application of RNNs is reasonable as it can take full
advantage of the high spectral resolution characteristics of HSI. However, for a
long-sequence task, RNNs is not as effective as we expected. Long distance dependence,
gradient vanish and overfitting are prone to occur \cite{bengio1994learning}. Even if
the long short-term memory network (LSTM) \cite{williams1989learning} is used to solve
the long-distance dependence problem, RNNs is still hard for training and easily
overfitting in a long-sequence task.
In previous work, 3D-CNN is applied in HSI classification and obtained a good behavior
\cite{li2017spectral, zhong2018spectral}. For RNNs, Convolutional-LSTM (CLSTM)
\cite{xingjian2015convolutional} also achieved a good performance in HSI classification
\cite{liu2017bidirectional}. 3D-CNNs and CLSTM consider both spatial contextual
information and spectral continuity, which result in a high accuracy. Nevertheless,
it takes a long time to train these two models.
In \cite{mou2017deep}, LSTM and its variant, GRU \cite{yao2015depth}, are applied in
HIS classification, and it is proved that GRU has a better performance in HIS
classification. To solve the problem that RNNs are easily over-fitting and difficult
for training, \cite{xu2017band} proposed band-group LSTM, which can effectively make
training easier by reducing the number of timestep in LSTM.
In this study, a Shorten Spatial-spectral RNN with Parallel-GRU (St-SS-pGRU) is
proposed. This study contributes to the literature in 2 major respects:
\begin{description}
\item[1)]
A shorten RNN with GRU is applied in HIS classification. The model is more
efficient and easier for training than band-by-band RNN. By combining converlusion
layer, an advanced model Shorten Spatial-spectral RNN with GRU is proposed.
The model considers not only spectral but also spatial feature, which leads to a
better performance.
\item[2)]
An architecture named parallel-GRU is proposed and the model with this architecture
has a better performance and is more robust.
\end{description}
The remainder of this paper is organized as follows. In the methodology section,
firstly the structure of traditional RNN, LSTM and GRU are introduced and then the
architecture of the proposed models are described. In the experimental section, the
network setup, the experimental results, and the comparison of different models are
provided. Finally, the conclusion section concludes the paper.
\section{Methodology}
\subsection{Recurrent neural networks (RNN)}
Different from Artificial neural network (ANN), RNN \cite{williams1989learning},
a neural network with recurrent unit, has a better performance in solving many
challenging problems involving sequential data analysis. The state of each time
step of the recurrent unit is not only related to the input of the current step,
but also related to the state of the previous step. Thus, the state of the preceding
step can effectively influence the next step.
Given a sequence sample
{$\mathbf{x}={\begin{bmatrix}\mathbf{x}^{(1)},\mathbf{x}^{(2)},...,\mathbf{x}^{(m)}\end{bmatrix}}^{\top}$},
in which {$\mathbf{x}^{(t)}$} is the data at {$t$}th timestep. For the {$t$}th recurrent
unit, its hidden state can be described as:
\begin{equation}
\mathbf{h}^{(t)}=\left\{
\begin{matrix}
\mathbf{h}^{(0)} & t=0\\
h(\mathbf{h}^{(t-1)},\mathbf{x}^{(t)}) & t>0
\end{matrix}\right.,
\label{con:rnnu_h}
\end{equation}
where {$\mathbf{h}^{(0)}$} is the initial state of the recurrent unit, {$h$} is a
nonlinear function. Normally, {$\mathbf{h}^{(0)}$} is set as a zero vector.
Optionally, in $t$th timestep, the recurrent unit may have an output
$\mathbf{y}^{(t)}$. For some task, the RNN model will finally have an output vector
{$\mathbf{y}={\begin{bmatrix}\mathbf{y}^{(1)},\mathbf{y}^{(2)},...,\mathbf{y}^{(m)}
\end{bmatrix}}^{\top}$}, while for classification tasks, only one output is needed.
Generally, The last output is adopted:
\begin{equation}
\mathbf{y}^{(t)}=y(\mathbf{h}^{(t)})
\label{con:rnnu_y}
\end{equation}
The recurrent unit in a traditional RNN is shown in Fig. \ref{fig:rnnu}. In the
traditional RNN model, the update rule of the recurrent hidden state and output in
Eq. \eqref{con:rnnu_h} and \eqref{con:rnnu_y} is usually implemented as follows:
\begin{equation}
h(\mathbf{h}^{(t-1)},\mathbf{x}^{(t)})= \varphi (\mathbf{W}_h\mathbf{x}^{(t)}+\mathbf{U}_h\mathbf{h}^{(t-1)}+\mathbf{b}_h) ,
\label{con:rnn_tra_h}
\end{equation}
\begin{equation}
y(\mathbf{h}^{(t)})= \mathbf{W}_y\mathbf{h}^{(t)}+\mathbf{b}_y ,
\label{con:rnn_tra_y}
\end{equation}
where $\mathbf{W}_h$, $\mathbf{U}_h$ and $\mathbf{W}_y$ are the weight matrices.
$\mathbf{b}_h$ and $\mathbf{b}_y$ are the bias vectors, and $\varphi$ is an activation
function, such as the sigmoid function or the hyperbolic tangent function.
\begin{figure}[htbp]
\centerline{\includegraphics[width=6.12cm]{fig1rnnu.png}}
\caption{Graphic model of traditional recurrent unit.}
\label{fig:rnnu}
\end{figure}
\subsection{Long short-term memory (LSTM)}
The traditional RNN has the problem of long-distance dependence
\cite{bengio1994learning}. The RNN has the capability to connect different timesteps
related information. However, when the sequence is too long, the RNN becomes unable to
connect related information as the distance increases, because the information losses
when propagating through multi-time-step recurrent units.
By using long short-term memory (LSTM) \cite{hochreiter1997long}, the problems have
been solved. As Fig. \ref{fig:lstm} shows, LSTM contains a forget gate, an input gate
and an output gate. 'Gate' structure is actually a logistic regression model so that
part of the information is filtered selectively, while the rest is reserved and passes
through the gate. LSTM can simulate the process of forgetting and memory and calculate
the probability of forgetting and memory, so information flow could be preserved in
long-distance propagation. The structure of LSTM can be described as:
\begin{equation}
\mathbf{f}^{(t)}= \sigma (\mathbf{W}_f\mathbf{x}^{(t)}+\mathbf{U}_f\mathbf{h}^{(t-1)}+\mathbf{b}_f) ,
\label{con:fg}
\end{equation}
\begin{equation}
\mathbf{i}^{(t)}= \sigma (\mathbf{W}_i\mathbf{x}^{(t)}+\mathbf{U}_i\mathbf{h}^{(t-1)}+\mathbf{b}_i) ,
\label{con:ig}
\end{equation}
\begin{equation}
\mathbf{o}^{(t)}= \sigma (\mathbf{W}_o\mathbf{x}^{(t)}+\mathbf{U}_o\mathbf{h}^{(t-1)}+\mathbf{b}_o) ,
\label{con:og}
\end{equation}
\begin{equation}
\tilde{\mathbf{c}}^{(t)}= tanh (\mathbf{W}_c\mathbf{x}^{(t)}+\mathbf{U}_c\mathbf{h}^{(t-1)}+\mathbf{b}_c) ,
\end{equation}
\begin{equation}
\mathbf{c}^{(t)} = \mathbf{i}^{(t)} * \tilde{\mathbf{c}}^{(t)} + \mathbf{f}^{(t)} * \mathbf{c}^{(t-1)} ,
\end{equation}
\begin{equation}
\mathbf{h}^{(t)}= \mathbf{o}^{(t)} * tanh(\mathbf{c}^{(t)}),
\end{equation}
where Eq. \eqref{con:fg}, \eqref{con:ig} and \eqref{con:og} represent forget gate,
input gate and output gate.
$\mathbf{W}_f$, $\mathbf{W}_i$, $\mathbf{W}_o$, $\mathbf{W}_c$, $\mathbf{U}_f$,
$\mathbf{U}_i$, $\mathbf{U}_o$ and $\mathbf{U}_c$ are the weight matrices.
$\mathbf{b}_f$, $\mathbf{b}_i$, $\mathbf{b}_o$ and $\mathbf{b}_c$ are the bias vectors.
$\sigma$ refers to sigmoid function and tanh refers to the hyperbolic tangent function:
\begin{equation}
\sigma(x)=\frac{1}{1+e^(-x)} ,
\end{equation}
\begin{equation}
tanh(x)=\frac{e^x-e^(-x)}{e^x+e^(-x)} ,
\end{equation}
\begin{figure}[htbp]
\centerline{\includegraphics[width=6.1cm]{fig2lstm.png}}
\caption{Graphic model of LSTM.}
\label{fig:lstm}
\end{figure}
\subsection{Gated recurrent unit (GRU)}
Over the years, there have been many variants of LSTM, but there is no evidence to show
that there is not a superior variant. Any variant may have advantages in a particular
problem \cite{greff2017lstm}.
GRU \cite{yao2015depth} is a variant of LSTM. With fewer parameters, it is much easier
for training than LSTM, and usually achieves the same performance as LSTM in some tasks
\cite{jozefowicz2015empirical}. It is considered that using GRU in a HSI classification
task is more appropriate than using LSTM \cite{mou2017deep}.
The main difference between LSTM and GRU is that an update gate and a reset gate are
adopted in GRU, instead of using a forget gate, an input gate and an output gate. The
structure of the GRU is shown in Fig. \ref{fig:gru}, which can be defined as follows:
\begin{equation}
\mathbf{z}^{(t)}= \sigma (\mathbf{W}_z\mathbf{x}^{(t)}+\mathbf{U}_z\mathbf{h}^{(t-1)}+\mathbf{b}_z) ,
\label{con:ug}
\end{equation}
\begin{equation}
\mathbf{r}^{(t)}= \sigma (\mathbf{W}_r\mathbf{x}^{(t)}+\mathbf{U}_r\mathbf{h}^{(t-1)}+\mathbf{b}_r) ,
\label{con:rg}
\end{equation}
\begin{equation}
\tilde{\mathbf{h}}^{(t)}= tanh (\mathbf{W}_h\mathbf{x}^{(t)}+\mathbf{U}_h(\mathbf{r}^{(t)}*\mathbf{h}^{(t-1)})+\mathbf{b}_h) ,
\end{equation}
\begin{equation}
\mathbf{h}^{(t)}= (1-\mathbf{z}^{(t)})\mathbf{h}^{(t-1)} + \mathbf{z}^{(t)}\tilde{\mathbf{h}}^{(t)},
\end{equation}
where Eq. \eqref{con:ug} and \eqref{con:rg} represent update gate and reset gate.
$\mathbf{W}_z$, $\mathbf{W}_r$, $\mathbf{W}_h$, $\mathbf{U}_z$, $\mathbf{U}_r$ and
$\mathbf{U}_h$ are the weight matrices. $\mathbf{b}_z$, $\mathbf{b}_r$ and
$\mathbf{b}_h$ are the bias vectors.
\begin{figure}[htbp]
\centerline{\includegraphics[width=8cm]{fig3gru.png}}
\caption{Graphic model of GRU.}
\label{fig:gru}
\end{figure}
\subsection{The proposed model}
\subsubsection{Shorten Spatial-spectral RNN with GRU(St-SS-GRU)}
A Shorten Spatial-spectral RNN with GRU (St-SS-GRU) model for HSI classification is
shown in Fig. \ref{fig:stssgru}. For each pixel, a square subgraph composed of
5$\times$5 pixels centered on it is used as a training sample.
The first part of St-SS-GRU is actually a 3D-Convolutional layer but both the depth and
stride of the kernels are 1. Three different convolution kernels (1×1, 3×3, 5×5) were
used to convolve different bands. The output of this part is a sequence with the same
length as the original input. The output sequence is a 'spectra' with the spatial
contextual feature. Every timestep of the sequence is a feature vector.
The second part is a Shorten RNN with GRU (St-GRU). The structure of St-GRU is shown in
Fig. \ref{fig:stgru}. The 1D converlusion layer before GRU is used to reduce the number
of timesteps so that the network is easier for training.
\begin{figure}[htbp]
\centerline{\includegraphics[width=8cm]{fig4stssgru.png}}
\caption{St-SS-GRU: (1) The first row shows a flowchart of the network. FC refers
to fully connected layer and Conv refers to Convolutional layers. (2) The second
row illustrates the shapes of input and output tensors of each layer and their
connection. (3) N is the number of filters in the 3D-Convolutional layer, D is
the number of bands in the input image, T is the number of GRU timestep, and H
is the number of neurons in hidden layer in a GRU. For the Pavia University dataset,
D=103, and in the experiment the hyperparameters are set as: N=16, T=5, H=128.}
\label{fig:stssgru}
\end{figure}
\begin{figure}[htbp]
\centerline{\includegraphics[width=5cm]{fig5stgru.png}}
\caption{St-GRU and St-pGRU: (1) The first two rows show the architecture of St-GRU
and St-pGRU, Conv refers to convolutional layers. (2) The third row illustrates the
shapes of input and output tensors of each layer and their connection. (3) D is the
number of bands in the input image, N is the dimension of the vector in each band
of input, M is the number of filters in the 1D-Convolutional layer, T is the number
of GRU timestep, and H is the number of neurons in hidden layer in a GRU. L and S,
which are determined by T, refer to the size and stride of filters in the
1D-Convolutional layer. For the Pavia University dataset, D=103, and in the
experiment the hyperparameter is set as: N=M=16, T=5, H=128.}
\label{fig:stgru}
\end{figure}
\subsubsection{Parallel-GRU Architecture}
In order to make the model more robust, a Parallel-GRU (pGRU) architecture is proposed.
The architecture of Shorten Parallel-GRU (St-pGRU) is shown in Fig. \ref{fig:stgru}.
The architecture is actually a combination of several GRU units. The output of the
architecture is the summation of every unit.
The Shorten Spatial-spectral RNN with parallel-GRU (St-SS-pGRU) is similar to St-SS-GRU,
except that St-GRU is replaced by St-SS-pGRU.
\section{Experiment}
\subsection{Data}
In the experiment, two HSI datasets, including the Pavia University and Indian Pines,
are used to evaluate the performance of the proposed model.
The Pavia University dataset was acquired by the Reflective Optics System Imaging
Spectrometer(ROSIS) sensor over Pavia, northern Italy in 2001. The corrected data,
with a spatial resolution of 1.3 m per pixel, contains 103 spectral bands ranging
from 0.43 to 0.86 $\mu m$. The image, with 610$\times$340 pixels, is differentiated into
9 ground truth classes. Table \ref{tab:pusample} provides information about all
classes of the dataset with their corresponding training and test sample.
The Indian Pines dataset was acquired by the TAirborne Visible/Infrared Imaging
Spectrometer (AVIRIS) sensor over the Indian Pines test site in north-western
Indiana in 1992. The corrected data with a moderate spatial resolution of 20m
contains 200 spectral bands ranging from 0.4 to 2.5 $\mu m$. The image consists of
145$\times$145 pixels, which are differentiated into 16 ground truth classes.
Table \ref{tab:ipsample} provides information about all classes of the dataset
with their corresponding training and test sample.
\begin{table}[htbp]
\caption{Number of Training and Test Samples Used in the Pavia University Dataset}
\begin{center}
\begin{tabular}{cc|cc}
\hline
\hline
\textbf{No.} & \textbf{Class Name}& \textbf{Training Samples}& \textbf{Test Samples} \\
\hline
1 & Asphalt & 548 & 6083 \\
2 & Meadows & 540 & 18109 \\
3 & Gravel & 392 & 1707 \\
4 & Trees & 542 & 2522 \\
5 & Metal sheet & 256 & 1089 \\
6 & Bare Soil & 532 & 4497 \\
7 & Bitumen & 375 & 955 \\
8 & Bricks & 514 & 3168 \\
9 & Shadows & 231 & 716 \\
\hline
& \textbf{TOTAL} & 3921 & 38846 \\
\hline
\hline
\end{tabular}
\end{center}
\label{tab:pusample}
\end{table}
\begin{table}[htbp]
\caption{Number of Training and Test Samples Used in the Indian Pines Dataset}
\begin{center}
\begin{tabular}{cc|cc}
\hline
\hline
\textbf{No.} & \textbf{Class Name}& \textbf{Training Samples}& \textbf{Test Samples} \\
\hline
1 & Alfalfa & 30 & 16 \\
2 & Corn-notill & 150 & 1278 \\
3 & Corn-min & 150 & 680 \\
4 & Corn & 100 & 137 \\
5 & Grass-pasture & 150 & 333 \\
6 & Grass-trees & 150 & 580 \\
7 & Grass-pasture-mowed & 20 & 8 \\
8 & Hay-windrowed & 150 & 328 \\
9 & Oats & 15 & 5 \\
10 & Soybean-notill & 150 & 822 \\
11 & Soybean-mintill & 150 & 2305 \\
12 & Soybean-clean & 150 & 443 \\
13 & Wheat & 150 & 55 \\
14 & Woods & 150 & 1115 \\
15 & Building-grass-trees & 50 & 336 \\
16 & Stone-stell-towers & 50 & 43 \\
\hline
& \textbf{TOTAL} & 1765 & 8484\\
\hline
\hline
\end{tabular}
\end{center}
\label{tab:ipsample}
\end{table}
\subsection{Result}
\begin{figure}[htbp]\scriptsize
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.14\textwidth]{pu_1im.pdf}&
\includegraphics[width=0.14\textwidth]{pu_2gt.pdf}&
\includegraphics[width=0.14\textwidth]{pu_3gru.pdf}\\
(a) False color map & (b) Ground truth & (c) GRU(87.79\%)\\
\includegraphics[width=0.14\textwidth]{pu_4stgru.pdf}&
\includegraphics[width=0.14\textwidth]{pu_5stssgru.pdf}&
\includegraphics[width=0.14\textwidth]{pu_6stsspgru.pdf}\\
(e) St-GRU(92.38\%) & (f) St-SS-GRU(98.04\%) & (g) St-SS-pGRU(99.01\%)\\
\multicolumn{3}{c}{\includegraphics[width=0.42\textwidth]{pu_label.pdf}} \\
\end{tabular}
\caption{The classification maps of the Pavia University dataset.}
\label{fig:pumap}
\end{figure}
\begin{figure}[htbp]\scriptsize
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.14\textwidth]{ip_1im.pdf}&
\includegraphics[width=0.14\textwidth]{ip_2gt.pdf}&
\includegraphics[width=0.14\textwidth]{ip_3gru.pdf}\\
(a) False color map & (b) Ground truth & (c) GRU(77.07\%)\\
\includegraphics[width=0.14\textwidth]{ip_4stgru.pdf}&
\includegraphics[width=0.14\textwidth]{ip_5stssgru.pdf}&
\includegraphics[width=0.14\textwidth]{ip_6stsspgru.pdf}\\
(e) St-GRU(80.50\%) & (f) St-SS-GRU(86.28\%) & (g) St-SS-pGRU(89.61\%)\\
\multicolumn{3}{c}{\includegraphics[width=0.42\textwidth]{ip_label.pdf}} \\
\end{tabular}
\caption{The classification maps of the Indian Pines dataset.}
\label{fig:ipmap}
\end{figure}
Table \ref{tab:puaac} and \ref{tab:ipaac} list the results obtained by the experiment,
and Fig. \ref{fig:pumap} and \ref{fig:ipmap} show the classification maps on the Pavia
University dataset and the Indian Pines dataset. Note that the accuracies list in Table
\ref{tab:puaac} and \ref{tab:ipaac} are overall accuracies (OA) along with the standard
deviation, from 10 independent runs on each dataset. The experiment is implemented with
an Intel i7-7700K 4.20GHz processor with 16GB of RAM and an NVIDIA GTX1050Ti graphic
card under Python3.6 with tensorflow1.8.0.
First of all, for all the datasets, GRU outperforms LSTM. In addition, it is observed
that LSTM is difficult to converge in the experiment, while GRU is not. Thus, it is
reasonable to indicate that GRU is a better choice for a HSI classification task.
Furthermore, it is apparent that St-GRU increases the accuracy significantly by 5.33\%
and 3.52\% in the Pavia University dataset and the Indian Pines correspondingly. With
converlusion layers, St-SS-GRU has a better than St-GRU. The accuracy of St-SS-GRU is
4.55\% and 6.63\% higher than that in St-GRU. After parallel-GRU is adopted, the model
gains the best performance in this experiment. The accuracy of St-SS-pGRU is 1.64\%
and 3.19\% higher than St-SS-GRU. What is more, the standard deviation of St-SS-pGRU
is smaller than other models, which indicate that St-SS-pGRU is more robust.
Comparing the processing time of different methods, st-GRU is significantly faster
in training than band-by-band GRU. St-SS-GRU and St-SS-pGRU are as slow as LSTM and
GRU in training, but they have higher accuracies than LSTM and GRU.
\begin{table}[htbp]
\caption{Classification Accuracies and Training Time for the Pavia University Dataset}
\begin{center}
\begin{tabular}{c|cc}
\hline
\hline
\textbf{Model} & \textbf{Overall accuracy} & \textbf{Training Time (s)} \\
\hline
LSTM & 84.68$\pm$1.40\% & 434.22 \\
GRU & 86.92$\pm$1.29\% & 232.15 \\
St-GRU & 92.25$\pm$0.78\% & \textbf{7.31}* \\
St-SS-GRU & 96.80$\pm$0.37\% & 104.56 \\
St-SS-pGRU & \textbf{98.44$\pm$0.26\%}* & 128.91 \\
\hline
\hline
\multicolumn{3}{l}{* The best performance in each column is shown in bold.}\\
\end{tabular}
\end{center}
\label{tab:puaac}
\end{table}
\begin{table}[htbp]
\caption{Classification Accuracies and Training Time for the Indian Pines Dataset}
\begin{center}
\begin{tabular}{c|cc}
\hline
\hline
\textbf{Model} & \textbf{Overall accuracy} & \textbf{Training Time (s)} \\
\hline
LSTM & 71.65$\pm$1.05\% & 838.85 \\
GRU & 77.01$\pm$1.82\% & 442.67 \\
St-GRU & 80.53$\pm$0.90\% & \textbf{7.63}* \\
St-SS-GRU & 87.16$\pm$1.06\% & 287.54 \\
St-SS-pGRU & \textbf{90.35$\pm$0.86\%}* & 300.90 \\
\hline
\hline
\multicolumn{3}{l}{* The best performance in each column is shown in bold.}\\
\end{tabular}
\end{center}
\label{tab:ipaac}
\end{table}
\section{Conclusion }
In the study, a St-SS-pGRU model is proposed for HSI classification. What is more,
an architecture named parallel-GRU is proposed to promote the performance and
robustness. Then an experiment is conducted to compare the performance of different
models. From the experiment, it is confirmed that GRU performs better than LSTM in
HSI classification task. Moreover, it is apparent that the proposed models are a lot
more accurate, more robust and faster than the traditional GRU network. Specifically,
St-GRU effectively reduced the training time and promoted the accuracy. St-SS-GRU
needs more time for training but gains a better performance than St-GRU. The proposed
architecture parallel-GRU also provided a satisfactory result in the experiment.
\bibliographystyle{unsrt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,147 |
package eu.dnetlib.iis.wf.importer.content;
import com.amazonaws.services.s3.AmazonS3;
import com.amazonaws.services.s3.model.GetObjectRequest;
import com.amazonaws.services.s3.model.ObjectMetadata;
import com.amazonaws.services.s3.model.S3Object;
import com.amazonaws.services.s3.model.S3ObjectInputStream;
import eu.dnetlib.iis.common.ClassPathResourceProvider;
import org.apache.commons.io.IOUtils;
import org.junit.jupiter.api.Test;
import org.junit.jupiter.api.extension.ExtendWith;
import org.mockito.ArgumentMatcher;
import org.mockito.Mock;
import org.mockito.junit.jupiter.MockitoExtension;
import java.io.IOException;
import java.net.URL;
import static org.junit.jupiter.api.Assertions.*;
import static org.mockito.Mockito.*;
/**
* @author mhorst
*
*/
@ExtendWith(MockitoExtension.class)
public class ObjectStoreContentProviderUtilsTest {
private static final String bucketId = "bucket-id";
private static final String keyId = "key-id";
private static final String s3ResourceLoc = ObjectStoreContentProviderUtils.S3_URL_PREFIX + bucketId + '/' + keyId;
@Mock
private AmazonS3 s3Client;
@Mock
private S3Object s3Object;
@Mock
private ObjectMetadata s3ObjectMeta;
@Mock
private S3ObjectInputStream s3ObjectInputStream;
// ---------------------------------- TESTS ----------------------------------
@Test
public void testExtractResultId() throws Exception {
// given
String objectId = "resultId::c9db569cb388e160e4b86ca1ddff84d7";
// execute
String extractedResultId = ObjectStoreContentProviderUtils.extractResultIdFromObjectId(objectId);
// assert
assertNotNull(extractedResultId);
assertEquals("50|" + objectId, extractedResultId);
}
@Test
public void testExtractResultIdNullInput() {
// given
String objectId = null;
// execute
String extractedResultId = ObjectStoreContentProviderUtils.extractResultIdFromObjectId(objectId);
// assert
assertNull(extractedResultId);
}
@Test
public void testExtractResultIdInvalidInput() {
// given
String objectId = "resultId::";
// execute
assertThrows(RuntimeException.class, () -> ObjectStoreContentProviderUtils.extractResultIdFromObjectId(objectId));
}
@Test
public void testGenerateObejctId() throws Exception {
// given
String resultId = "resultId";
String url = "http://localhost/";
String defaultEncoding = "utf8";
String digestAlgorithm = "md5";
// execute
String objectId = ObjectStoreContentProviderUtils.generateObjectId(
resultId, url, defaultEncoding, digestAlgorithm);
// assert
assertEquals("resultId::c9db569cb388e160e4b86ca1ddff84d7", objectId);
}
@Test
public void testGetContentFromURL() throws Exception {
// given
String encoding = "utf8";
String contentClassPath = "/eu/dnetlib/iis/wf/importer/content/sample_data.txt";
URL url = ObjectStoreContentProviderUtils.class.getResource(contentClassPath);
String expectedResult = ClassPathResourceProvider.getResourceContent(contentClassPath);
// execute
byte[] result = ObjectStoreContentProviderUtils.getContentFromURL(url, new ContentRetrievalContext(1, 1, null));
// assert
assertNotNull(result);
assertEquals(expectedResult, IOUtils.toString(result, encoding));
}
@Test
public void testGetContentFromS3WhenNullClient() {
ContentRetrievalContext context = new ContentRetrievalContext(1, 1, 1);
assertThrows(S3EndpointNotFoundException.class, () ->
ObjectStoreContentProviderUtils.getContentFromURL(s3ResourceLoc, context));
}
@Test
public void testGetContentFromS3WhenInvalidResourceLocation() {
ContentRetrievalContext context = new ContentRetrievalContext(1, 1, 1);
context.setS3Client(s3Client);
assertThrows(IOException.class, () ->
ObjectStoreContentProviderUtils.getContentFromURL("s3://bucket-id", context));
}
@Test
public void testGetContentFromS3WhenObjectNotFound() {
ContentRetrievalContext context = new ContentRetrievalContext(1, 1, 1);
context.setS3Client(s3Client);
when(s3Client.getObject(argThat(new GetObjectArgumentMatcher()))).thenReturn(null);
assertThrows(IOException.class, () -> ObjectStoreContentProviderUtils.getContentFromURL(s3ResourceLoc, context));
}
@Test
public void testGetContentFromS3WhenSizeIsInvalid() {
ContentRetrievalContext context = new ContentRetrievalContext(1, 1, 1);
context.setS3Client(s3Client);
when(s3Client.getObject(argThat(new GetObjectArgumentMatcher()))).thenReturn(s3Object);
when(s3Object.getObjectMetadata()).thenReturn(s3ObjectMeta);
when(s3ObjectMeta.getContentLength()).thenReturn(Long.valueOf(-1));
assertThrows(InvalidSizeException.class, () ->
ObjectStoreContentProviderUtils.getContentFromURL(s3ResourceLoc, context));
}
@Test
public void testGetContentFromS3() throws Exception {
ContentRetrievalContext context = new ContentRetrievalContext(1, 1, 1);
context.setS3Client(s3Client);
when(s3Client.getObject(argThat(new GetObjectArgumentMatcher()))).thenReturn(s3Object);
when(s3Object.getObjectMetadata()).thenReturn(s3ObjectMeta);
when(s3ObjectMeta.getContentLength()).thenReturn(Long.valueOf(1));
when(s3Object.getObjectContent()).thenReturn(s3ObjectInputStream);
when(s3ObjectInputStream.read(any())).thenReturn(1, IOUtils.EOF);
byte[] result = ObjectStoreContentProviderUtils.getContentFromURL(s3ResourceLoc, context);
assertNotNull(result);
assertEquals(result.length, 1);
assertEquals(result[0], 0);
verify(s3ObjectInputStream, times(1)).close();
}
// -------------------------------- INNER CLASS -----------------------------------------
static class GetObjectArgumentMatcher implements ArgumentMatcher<GetObjectRequest> {
@Override
public boolean matches(GetObjectRequest argument) {
if (argument != null) {
return keyId.equals(argument.getS3ObjectId().getKey()) &&
bucketId.equals(argument.getS3ObjectId().getBucket());
}
return false;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,805 |
Judah West: You love wearing gloves and are a big fan of the snow! Watching you play makes me feel like I am a kid again. Never lose your curious nature.
Hunter Elias: Your sense for adventure is contagious … and though you are not really into dressing up for winter, I know you love being outside so much. You are turning into such a little man! | {
"redpajama_set_name": "RedPajamaC4"
} | 3,353 |
Education The support we offer is long-term, it is tailored to the specific needs of local children and involves the local community.
All too often, poverty goes hand in hand with social and family problems. To ensure that an already difficult situation isn't aggravated, we offer children a place of refuge when necessary. Whether in the form of a day centre or a place to live – children can find security and stability there.
If they aren't in school, children from poor districts are under threat from all kinds of exploitation. In order to protect them, La Chaîne de l'Espoir provides education and day centres where they can spend the whole day. There, they can benefit from help with school work, healthcare and social or psychological support.
Find out more about our protective programmes in India and Nepal.
We have built a home for orphaned, abandoned or abused children in the Buriram province. Spread across two buildings, 40 children are rebuilding their lives in a warm and nurturing environment.
Find out more about La Maison de l'Espoir in Thailand. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,338 |
<?php
require_once 'phing/Task.php';
require_once 'phing/system/io/PhingFile.php';
require_once 'phing/system/io/FileWriter.php';
require_once 'phing/util/ExtendedFileStream.php';
/**
* Transform a PHPUnit xml report using XSLT.
* This transformation generates an html report in either framed or non-framed
* style. The non-framed style is convenient to have a concise report via mail,
* the framed report is much more convenient if you want to browse into
* different packages or testcases since it is a Javadoc like report.
*
* @author Michiel Rook <mrook@php.net>
* @version $Id: b88d6fa4ca4717177b562a0475c81d92c161d9b4 $
* @package phing.tasks.ext.phpunit
* @since 2.1.0
*/
class PHPUnitReportTask extends Task
{
private $format = "noframes";
private $styleDir = "";
private $toDir = "";
/**
* Whether to use the sorttable JavaScript library, defaults to false
* See {@link http://www.kryogenix.org/code/browser/sorttable/)}
*
* @var boolean
*/
private $useSortTable = false;
/** the directory where the results XML can be found */
private $inFile = "testsuites.xml";
/**
* Set the filename of the XML results file to use.
*/
public function setInFile(PhingFile $inFile)
{
$this->inFile = $inFile;
}
/**
* Set the format of the generated report. Must be noframes or frames.
*/
public function setFormat($format)
{
$this->format = $format;
}
/**
* Set the directory where the stylesheets are located.
*/
public function setStyleDir($styleDir)
{
$this->styleDir = $styleDir;
}
/**
* Set the directory where the files resulting from the
* transformation should be written to.
*/
public function setToDir(PhingFile $toDir)
{
$this->toDir = $toDir;
}
/**
* Sets whether to use the sorttable JavaScript library, defaults to false
* See {@link http://www.kryogenix.org/code/browser/sorttable/)}
*
* @param boolean $useSortTable
*/
public function setUseSortTable($useSortTable)
{
$this->useSortTable = (boolean) $useSortTable;
}
/**
* Returns the path to the XSL stylesheet
*/
protected function getStyleSheet()
{
$xslname = "phpunit-" . $this->format . ".xsl";
if ($this->styleDir)
{
$file = new PhingFile($this->styleDir, $xslname);
}
else
{
$path = Phing::getResourcePath("phing/etc/$xslname");
if ($path === NULL)
{
$path = Phing::getResourcePath("etc/$xslname");
if ($path === NULL)
{
throw new BuildException("Could not find $xslname in resource path");
}
}
$file = new PhingFile($path);
}
if (!$file->exists())
{
throw new BuildException("Could not find file " . $file->getPath());
}
return $file;
}
/**
* Transforms the DOM document
*/
protected function transform(DOMDocument $document)
{
if (!$this->toDir->exists())
{
throw new BuildException("Directory '" . $this->toDir . "' does not exist");
}
$xslfile = $this->getStyleSheet();
$xsl = new DOMDocument();
$xsl->load($xslfile->getAbsolutePath());
$proc = new XSLTProcessor();
if (defined('XSL_SECPREF_WRITE_FILE'))
{
if (version_compare(PHP_VERSION,'5.4',"<"))
{
ini_set("xsl.security_prefs", XSL_SECPREF_WRITE_FILE | XSL_SECPREF_CREATE_DIRECTORY);
}
else
{
$proc->setSecurityPrefs(XSL_SECPREF_WRITE_FILE | XSL_SECPREF_CREATE_DIRECTORY);
}
}
$proc->importStyleSheet($xsl);
$proc->setParameter('', 'output.sorttable', $this->useSortTable);
if ($this->format == "noframes")
{
$writer = new FileWriter(new PhingFile($this->toDir, "phpunit-noframes.html"));
$writer->write($proc->transformToXML($document));
$writer->close();
}
else
{
ExtendedFileStream::registerStream();
$toDir = (string) $this->toDir;
// urlencode() the path if we're on Windows
if (FileSystem::getFileSystem()->getSeparator() == '\\') {
$toDir = urlencode($toDir);
}
// no output for the framed report
// it's all done by extension...
$proc->setParameter('', 'output.dir', $toDir);
$proc->transformToXML($document);
ExtendedFileStream::unregisterStream();
}
}
/**
* Fixes DOM document tree:
* - adds package="default" to 'testsuite' elements without
* package attribute
* - removes outer 'testsuite' container(s)
*/
protected function fixDocument(DOMDocument $document)
{
$rootElement = $document->firstChild;
$xp = new DOMXPath($document);
$nodes = $xp->query("/testsuites/testsuite");
foreach ($nodes as $node) {
$children = $xp->query("./testsuite", $node);
if ($children->length) {
foreach ($children as $child) {
if (!$child->hasAttribute('package'))
{
$child->setAttribute('package', 'default');
}
$rootElement->appendChild($child);
}
$rootElement->removeChild($node);
}
}
}
/**
* Initialize the task
*/
public function init()
{
if (!class_exists('XSLTProcessor')) {
throw new BuildException("PHPUnitReportTask requires the XSL extension");
}
}
/**
* The main entry point
*
* @throws BuildException
*/
public function main()
{
$testSuitesDoc = new DOMDocument();
$testSuitesDoc->load((string) $this->inFile);
$this->fixDocument($testSuitesDoc);
$this->transform($testSuitesDoc);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,564 |
Presidential Apartments Marylebone London. JIMAT di Agoda.com!
Ada soalan mengenai Presidential Apartments Marylebone?
Direka untuk kedua-dua tujuan perniagaan dan percutian, Presidential Apartments Marylebone terletak di lokasi strategik di Hyde Park; salah satu kawasan yang paling popular di bandar tersebut. Hanya 1.8 Miles dari pusat bandar, lokasi strategik hotel memastikan tetamu boleh ke mana-mana sahaja dengan cepat dan mudah ke tempat-tempat tarikan. Persekitaran yang terjaga dan kedudukan berhampiran dengan Talking Bookshop, Hay Hill Gallery, Mercer membuatkan penginapan ini menjadi tarikan.
Di Presidential Apartments Marylebone, perkhidmatan yang cemerlang dan kemudahan yang unggul menjadikan penginapan anda tidak dapat dilupakan. Semasa menginap di penginapan yang indah ini, tetamu boleh menikmati Wi-Fi percuma semua bilik, pengemasan harian, dapur, khidmat teksi, khidmat tiket.
Suasana Presidential Apartments Marylebone terpancar di dalam setiap bilik tetamu. televisyen skrin rata, internet wayarles, internet wayarles (percuma), bilik larangan merokok, penyaman udara hanyalah sebahagian daripada kemudahan yang boleh didapati di hotel tersebut. Di samping itu, pihak hotel menawarkan aktiviti rekreasi untuk memastikan anda mempunyai perkara untuk dilakukan sepanjang penginapan anda. Dengan lokasi yang ideal dan kemudahan sepadan, Presidential Apartments Marylebone sangat memuaskan hati.
This is my third time staying at this property. It's very near Oxford street, just less than 5 minutes walk you will be at Selfridges. There are many bus stops around, making your travel in London easily. Since the property is an apartment type, you can fix your own meal with all the kitchen utensils provided. Next door is Tesco Express, where you can get you food supplies from 7 a.m. - 11 p.m.
This is my third time staying at Presidential Apartments Maryelone this year. The location is great, just few minutes walk from Oxford Street's Selfridges and M&S. There are bus stops near by that can take you to many tourist attractions. In the room, there is a kitchen that you can fix your own meal. But there are no staffs from 10 p.m., if you arrive after that, you still can pick up the keys at the opposite premise. But overall, I will always stay at this place whenever I go to London.
Check in was easy Staffs were great Rooms were spacious Had our own kitchen and washing machine. Tesco supermarket was just downstairs. Very safe environment as no outsiders could access the building. A bit bit pricey but definitely recommend for those travelling with children.
Worth the money since it's located in a strategic area.. walking distance to Oxford Street and Paddington Street (the shopping area), etc. There were some disadvantages of this apartment like the old sofa bed with a thin matress that made my kids a lil bit uncomfortable sleeping there, but the main bed itself was so comfortable. And also the water from the showerhead wasn't swift, so made us a bit hard to bathe. But overall I like to stay here, it fits for the family of 4 like us.
The hotel is just few minutes walk to Selfridges and M&S Oxford Street. There is Tesco Express next door, so you can easily fix your own meal with kitchen provided. There is no 24 hours staff, office only open 8am-8pm, but if you need assistance, there is an emergency phone number and also the staff at the opposite building can help.
This apartment was very convenient to convenient store, café, eatery and shopping. It is about 500 m to nearest tube station. Though the apartment is a bit old but the bed was comfortable. The bathroom was small and having a noisy pump. Not all the washer in the room is working properly.
Best location, friendly staff, the room is clean, very comfortable, very convenient & very near to oxford street.
Large apartment for London. Clean but tired decor and furniture. Comfy large bed. My apartment was at rear where there is a very noisy car park with extensive bin moving / filling activity from very early. Inauspicious entrance and corridors uninviting with dirty brown carpet. Daily room service.
The place is just 5 mins walk from Selfridges, Oxford Street. You can find everything around the hotel. The bed is comfortable, WIFI is good. There is kitchen in the room with oven, so you can easily fix your own meal. There is Tesco Express next door where you can find anything you may need. The hotel has no 24 hours staff, you might need to collect the key from opposite building if you arrive late.
The apartment is located about 400 metres from Centre of Shopping Heaven in London ie the Oxford Street. Facilities and all amenities are complete. Room is being cleaned everyday even during Xmas and New Year. Staff ie receptionist and the housekeeping staff are very polite. I highly recommend this apartment to those who wants to just chill in London.
Kitchen with fridge and freezer . Even a washing machine was available. Only minor point was the unavailability of a plug for the bathtub. Other then that this place is nicely situated just off Baker Street.
As late arrival advised to pick up key from business opposite. Met with excellent customer service and couldnt have been more helpful. Overall was impressed with suite. Kitchen area with everything from cooker to microwave & fridge.All utensils crockery & glassware. Sofa turns into extra small double..with also superking in bedroom area. Sheets and duvets good quality and very clean. Seemed wasted and little strange being set up as booked as single traveller. Room also had safe,ironing board /iron and hair dryer.Large flat screen tv even with video player. Very very large window giving an airy feel. Bathroom well equpped with bath and shower over, all complimentary good toiletaries and abundance of towels. Only downsides. First impressions with lift and corridor carpets showing signs of wear..In rooms. Wooden floors although clean could do with being refurbished and curtains also quite difficult to close and Didnt see any staff as left very early but clear procedures if needed anyone.
Location is no doubt superb, with a tesco express right at the doorstep and lots of food everywhere. Don't expect to get a 'serviced apartment'. Building is old, rooms are even older. The first one bed room apartment I got was not super clean, sticky residues all over the floors. The room stinks and the bathroom was just simply bad, the noise that the shower makes can wake the whole street up. I also get to experience the presidential suite which I really suggest everyone to upgrade too. The place definitely looks way nicer and cleaner and more modern. However, every single pot and pans needs replacements, my bathroom was faulty (hot water does not come out and it was too late to call for service). Overall, if you just need an apartment with great location, okay beds and don't mind everything else that worsen the experience, go ahead and book the presidential suite..
Fantastic neighbourhood, quiet but close to great restaurants and shopping. Have the option to prepare snacks in too with the kitchen and do laundry with the washer and tumble dryer. Would definitely stay again !
Very comfortable apartment. It has everything you need, a kitchen, fridge, etc. The location is very good, only 5 minutes walk from Selfridges. There is Tesco Express next door and Little Waitrose just around the corner, so you can easily fix yourself a meal.
The only thing we could think of was missing was a sachet of laundry powder for the washing machine. Otherwise we were surprised by the space and facilities. Having big opening windows that did not look at the side of another building was awesome. . Amazingly quiet for such a central position.
The washing machine was spoilt throughout our stay there. Was very disappointed. We called and informed the staff but nothing was done. Overall was fine. Tesco is located just right beside, it's very convenient for us. 10-15 mins to oxford street. about 10mins to the nearest tube. Will probably give it a miss here for our next visit next year.
Excellent Hotel..In all ways possible...Very Good Location to nearby shopping precint & dining areas...Staff service was approachable and tidiness was apt..More economical compared to other serviced apartments..Very cosy & serene ambience...Convenient location for airport transfer at additional cost...Overall, an excellent choice.
Generally we were pretty happy with this hotel and the service ws fine, only issue was on a couple of nights it tended to be pretty noisy even until the early hours. This is not the hotels fault really but the area in general so if you are looking for peace and tranquility or a guaranteed good nights sleep then this is probably not the right choice for you. The free wireless internet connection ws useful. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,237 |
\section{Introduction}
With the growing use of machine learning models to automate decisions, a rising concern for both researchers and practitioners is to assure that they are not \textit{biased} (i.e., systematically unfair) against specific population groups or individuals. More specifically, within the context of graph mining algorithms, fairness plays an important role \cite{agarwal2021towards, tsioutsiouliklis2021fairness, yao2017beyond, Kamishima12enhancementof} since these algorithms are widely used in human-centered applications with major societal impact. In graph machine learning, it is a common practice to compile network information (such as graph topology, node features) in unified node-level representations, i.e., \textit{graph embeddings}. The produced embeddings can subsequently be used off-the-shelf for a plethora of downstream tasks, e.g., node classification \cite{classification}, link prediction \cite{link}, community detection \cite{clustering}, ranking \cite{recommend}, among others. For example, graph embeddings of social networks can serve as representations of users for high-impact real-world tasks, e.g., user classification for targeted advertising, social group detection, friendship recommendation, advanced user search or to better understand a rumor spreading process \cite{hamilton2017representation, cai2018comprehensive, zhang2018network}. However, graph embeddings can carry bias against particular groups or individuals, which is either present in the data or introduced by the algorithm itself \cite{baeza2018bias}. As a result, it is crucial to identify, diagnose and mitigate bias before it gets propagated to downstream tasks and real-world machine learning systems.
Thoroughly examining a graph embedding algorithm with respect to unfairness is hard to accomplish using solely statistical methods. While it is possible to quantify bias either at a global level (i.e., the network as a whole) or at a local level (i.e., communities or specific nodes), this is not sufficient because it is important to gain insight into the major factors that contribute to the bias. Thus, a framework that supports visual unfairness diagnosis for graph embeddings by explicitly illustrating the sources of bias can be invaluable. A further challenge is that multiple \textit{(un)fairness notions} exist. In particular, these notions can model discrimination either against \textit{groups} \cite{ijcai2019-456} or \textit{individuals} \cite{InFoRM} and it is well-established that trade-offs exist between the two types \cite{kleinberg2016inherent, tradeoff}. Therefore, an integrated design is required that supports heterogeneous fairness notions.
In this work, we propose \textsc{BiaScope}\xspace, an interactive open-source visualization tool that supports end-to-end \textit{visual unfairness diagnosis} for graph embeddings. To design the tool, we worked with domain experts in order to understand the intricacies of unfairness characterization for graph embeddings. We designed \textsc{BiaScope}\xspace as a web-based visualization tool with emphasis on interactive features which simplify the complex task of unfairness characterization even on large benchmark networks. Our tool facilitates visual comparison of any two embedding algorithms with respect to fairness on several common network benchmarks. Additionally, it allows the user to locate nodes that contribute to unfairness and importantly, to identify the defining factors of the observed bias. This is facilitated by interactively linking the embedding subspace (i.e., high-dimensional vectors) that displays bias with the corresponding graph topology (i.e., nodes and edges), and vice versa (Figure \ref{fig:teaser}C). Our tool supports both group \cite{ijcai2019-456} and individual \cite{InFoRM} fairness notions in an integrated design and it is task agnostic in the sense that it does not assume a particular downstream task.
The main contributions of our work can be summarized as follows:
\begin{itemize}
\item Through an iterative interview process we identified the \textit{abstract tasks} that are crucial parts of an \textit{unfairness diagnosis workflow} for graph embeddings.
\item Informed by our task analysis, we \textit{designed and implemented} \textsc{BiaScope}\xspace to support multiple fairness notions, while being interactively configurable and task-agnostic.
\item Finally, we collected \textit{expert feedback} that validates our design. The feedback suggests that \textsc{BiaScope}\xspace can be an effective asset both for researchers to thoroughly evaluate their proposed algorithms and for practitioners to detect bias early in the product development process.
\end{itemize}
The source code of \textsc{BiaScope}\xspace is provided at \url{https://github.com/agapiR/BiaScope}, where the reader can download a \textit{demo video} introducing the tool and its functionality. The repository also provides instructions for running a local instance of \textsc{BiaScope}\xspace.
\section{Related Work}
\textsc{BiaScope}\xspace lies at the intersection of vector embedding visualization and the analysis of fairness for graphs embeddings. Additionally, we draw inspiration from graph neural network visualization tools.
\paragraph{Vector Embeddings.}
Most vector embedding visualization tools are tailored to contextualized embeddings produced by language models. The tool proposed in \cite{berger2020visually} presents a sentence view with labels for semantic properties e.g., parts-of-speech and an embedding view with cluster analysis via co-occurrence small multiples. On the other hand, embComp \cite{embcomp} introduces visual analysis methods for embedding comparison which combine a global overview with detailed views. The tool mainly focuses on comparing embedding space properties, such as neighborhood overlap or spread and neighbor distances. {The Embedding Comparator system \cite{2022-embedding-comparator} shares this objective, differentiating from embComp in the fact that it simultaneously visualizes global views of embedding structure alongside local views of individual objects and their common and unique neighbors to enable efficient analysis.} Vector embedding comparison is also supported by Emblaze \cite{sivaraman2022emblaze} which consists of an elaborate interactive scatterplot and mainly focuses on neighborhood discovery for dynamic relation suggestions. A more suitable tool for graph embedding visualization is EmbeddingVis \cite{li2018embeddingvis} which allows for comparison of different graph embedding models with respect to which properties of the graph they preserve and illustrates relationships between node metrics and selected embedding vectors.
\paragraph{Graph Neural Networks.}
A common method of learning graph embeddings involves Graph Neural Networks (GNNs), for which there are several visualization tools. The GNNlens tool \cite{gnnlens} helps researchers open the black box of Graph Neural Networks. The tool utilizes Parallel Sets View and Projection View for quick identification of error patterns in the set of wrong predictions. Moreover, a detailed overview for a particular node is offered by Graph View and Feature Matrix View. In a similar spirit, GNNExplainer \cite{gnnexplainer} assists with interpretability by presenting explanations for predictions made by a GNN. The explanations have the form of small visualizable graph motifs and important node features. On the other hand, the CorGIE tool \cite{CorGIE} focuses on the correspondence between the graph topology and the node embeddings. It utilizes a $k$-hop graph layout to show topological neighbors in hops and their clustering structure. It uses small multiples bar charts for node features and a UMAP for the 2D projection of the latent space. Finally, the GNNVis \cite{gnnvis} paper proposes a framework for learned dimensionality reduction using GNNs.
\paragraph{Fairness in Machine Learning.}
Several notions of fairness have been proposed in the literature, mainly categorized into two families: statistical (group) and individual fairness, where most of the focus lies on the first type \cite{chouldechova2020snapshot, 10.1145/3447548.3467266}. This also holds in the context of graph mining, as algorithmic fairness definitions have been adapted for ranking, clustering, and embedding \cite{fair-graph-tutorial}. A graph mining method consists of three major components: the input graph, the mining model and the mining results \cite{InFoRM}. The main idea behind individual fairness is that similar individuals should receive similar algorithmic outcomes. In the context of graph mining, this translates into the fact that "similar" nodes on the graph are also "similar" in terms of the mining results. The similarities between nodes are encoded in a similarity matrix. In particular, every pair of nodes must satisfy the fairness condition.
Group fairness notions in the context of embeddings have also been studied \cite{pmlr-v97-bose19a, ijcai2019-456}. Bose and Hamilton \cite{pmlr-v97-bose19a} introduce an adversarial framework to enforce fairness constraints on graph embeddings, by training a set of "filters" to prevent adversarial discriminators from classifying the sensitive information from the filtered embeddings. After training, these filters can be composed together in different combinations, allowing for the flexible generation of embeddings that are invariant with respect to any subset of the sensitive attributes. Rahman et al. \cite{ijcai2019-456}, extend the Node2Vec algorithm in a fairness-aware manner. For doing so, they propose a new notion of fairness in the context of friendship recommendation systems, by extending statistical parity.
Algorithmic fairness has also received attention in the visualization community. Several tools that target different machine learning settings (e.g., classification and ranking) have been proposed: FAIRVIS \cite{FAIRVIS} is a visual analytics system that helps audit the fairness of binary classification models. The system allows users to explore both suggested and user-specified subgroups and supports 10 metrics for comparison by default (Accuracy, Recall, Specificity, Precision, Negative Predictive Value, False Negative Rate, False Positive Rate, False Discovery Rate, False Omission Rate, and F1 score). Users can derive new metrics from the base outcome rates. Another tool proposed for the classification setting is DiscriLens \cite{9222272}, which identifies a collection of potentially discriminatory attributes based on causal modeling and classification rules mining and provides visualization to facilitate the exploration and interpretation of them. For high-level model understanding, the What-If Tool \cite{whatif} offers visualizations of selected performance and fairness measures as well as model comparison with respect to those measures. Although the tool provides a comprehensive visual overview of multiple aggregate statistics, it does not offer insights into the sources of the observed bias. Focusing on ranking settings, Fairsight \cite{FairSight}, incorporates different fairness notions (including group and individual fairness) to support identifying and mitigating bias against individuals and groups in problems that involve rank-ordering individuals. Finally, FairRankVis \cite{fairrankvis} enables the exploration of multi-class bias in graph ranking algorithms, allowing for the comparison of fair and unfair versions of these algorithms using both group and individual notions of fairness. FairRankVis is the only tool that supports model comparison as a mechanism for explaining the impacts of algorithmic debiasing. However, its design is tailored to graph ranking since its main visual components target the comparison of two algorithms with respect to their ranking results. In our work, we focus on graph embeddings without restricting our analysis to a particular machine learning application. To the best of our knowledge, \textsc{BiaScope}\xspace is the first tool that allows unfairness diagnosis for graph embeddings in a task-agnostic manner.
\section{Preliminaries}
In this section, we discuss the definition and formalism we adopt for \textit{graph embeddings} (Section \ref{sec:graph_emb}). Then, this formalism is used to define the \textit{fairness notions} we consider, for which we also provide intuition and examples (Section \ref{sec:fairness}). Finally, we briefly discuss the data we used for the present design study (Section \ref{sec:data}).
\subsection{Graph Embeddings} \label{sec:graph_emb}
Graph embedding algorithms take a graph $G = \left(V, E\right)$ as input and represent each node in $\mathbb{R}^d$ where $d$ is a dimensionality parameter provided to the algorithm. The node-level vector representations are chosen such that two nodes $v_i$ and $v_j$ that are close to each other in $G$ are embedded near each other in $\mathbb{R}^d$ \cite{hamilton2017representation}. Graph embedding algorithms can also be clustered into several methodological classes: matrix factorization \cite{lapeigenmap, HOPE}, random walk \cite{node2vec}, auto-encoder \cite{SDNE}, and GCNs \cite{graphSAGE}. Recent work has also embedded nodes into hyperbolic space, which better represents hierarchical tree-like structures \cite{HGCN, poincare}.
In what follows, we assume that the graph embedding is represented by the embedding matrix $Y \in \mathbb{R}^{n \times d}$, where $Y[u] \in \mathbb{R}^d$ is the embedding of node $u$ and $n = |V|$ the number of nodes in the graph.
\subsection{Fairness Notions} \label{sec:fairness}
Recent work has evaluated the fairness of graph embeddings and builds upon broader fairness definitions established in the algorithmic fairness community. Fairness definitions typically fall into two classes: individual and group. Individual fairness ensures that algorithms treat two similar individuals similarly. In contrast, group fairness ensures that two sub-populations, in aggregate, are treated similarly. In this work, we incorporate embedding fairness definitions from both classes. In the following, we provide the formal definitions and illustrative examples.
\subsubsection{Individual Fairness}
For individual fairness, we build upon the definition of individual fairness for graph embeddings introduced in \cite{InFoRM} (InFoRM), which assigns a non-negative unfairness score to each node. The score indicates how differently the node is embedded from other nodes in the neighborhood, where the neighborhood is defined by the number of hops from the target node. For example, consider the graph shown in Figure \ref{fig:ex1_graph}. Since nodes B and C are the (1-hop) neighbors of A, the closer their embeddings are to the one corresponding to node A, the lower A's unfairness score. Intuitively, we expect \textit{similar} nodes according to the local topology of the graph to also be \textit{similar} in terms of their distance in the embedded space. We can formalize this notion by considering a \textit{proximity} or \textit{similarity} matrix $S$ where $S[u,v]$ is the proximity of nodes $u$ and $v$ (i.e., a quantitative measure that indicates how close or similar they are in the graph). The most obvious notion of node proximity is the adjacency matrix of the graph $A$, but other options include its powers ($A,\dots,A^k$), random walk matrices as well as several other neighborhood overlap measures as explained in \cite{grl_book}. Assuming $S=A$ and $Y$ being the embedding matrix, we can define the individual fairness score based on \cite{InFoRM} as follows:
\begin{align*}
\text{score}_1(u,k=1) &= \sum_{v \neq u} \lVert Y[u] - Y[v]\rVert_2^2 \cdot S[u,v] \\
&= \sum_{v\in\mathcal{N}(u,1)} \lVert Y[u] - Y[v]\rVert_2^2
\end{align*}
where $\mathcal{N}(u,k)$ is the $k$-hop neighborhood of $u$, given by the nodes reachable from $u$ in at most $k$ steps. In the example from Figure \ref{fig:ex1_graph}, we get $\text{score}_1(\text{A},1)=17$. We generalize this notion for different values of $k$ as follows:
\begin{align*}
\text{score}_1(u,k) &= \sum_{v\in\mathcal{N}(u,k)} \lVert Y[u] - Y[v]\rVert_2^2
\end{align*}
Returning to our example (Figure \ref{fig:ex1_graph}), considering $k=2$, node D is now taken into account since it is reachable from node A in two hops. Thus, we get $\text{score}_1(\text{A},2)=81$. Observe that for a given vertex $u$ and $k\geq 0$, $\text{score}_1(u,k)\leq \text{score}_1(u,k+1)$ and that isolated nodes get a score of 0.
The scores for a graph with minimum degree at least one are normalized to be in the range $[0,1]$ as follows:
\begin{align*}
\text{score}^d_1(u,k) &= \frac{\text{score}_1(u,k)}{\text{degree(u)}}\\
\text{score}^n_1(u,k) &= \frac{\text{score}^d_1(u,k)}{\max_u \text{score}^d_1(u,k)}
\end{align*}
where $\text{degree(u)}$ is the degree of node $u$ and $\text{score}^n_1(u,k)$ denotes the normalized score.
\begin{figure}
\centering
\begin{subfigure}
\centering
\begin{tikzpicture}[main/.style = {draw, circle}]
\node[main] (1) {A};
\node[main] (2) [right of=1] {B};
\node[main] (3) [below of=1]{C};
\node[main] (4) [right of=3] {D};
\draw (1) -- (2);
\draw (1) -- (3);
\draw (3) -- (4);
\end{tikzpicture}
\end{subfigure}
\begin{subfigure}
\centering
\includegraphics[width=.6\linewidth]{figures/ex1_graph_emb.png}
\end{subfigure}%
\caption{Example graph and corresponding embedding for $\text{score}_1$.}
\label{fig:ex1_graph}
\end{figure}
\subsubsection{Group Fairness}
Regarding group fairness, our score is based on the definition of group fairness for graph embeddings proposed by \cite{ijcai2019-456} (Fairwalk). The motivation behind it is to promote recommendations (in the context of a Recommender System) where all population groups are equally represented. To start, a set of link recommendations are made for each node. Recommendations are the top $k$ closest embeddings among non-connected nodes according to dot product similarity. We then measure the proportion of recommendations belonging to a specific population (e.g., number of edges recommending men and women). A high score (in magnitude) suggests that one sub-population is recommended disproportionately more. As an example, consider an attribute $S$ with values $z\in\{g_1,g_2\}$ and the disconnected nodes displayed in Figure \ref{fig:ex2_graph_emb}, each annotated with their attribute value. For $k=2$, the top $2$ most proximal embeddings to the embedding of node A are the ones associated to nodes B and C. In this scenario, both groups (given by the nodes for which $z=g_1$ and $z=g_2$ respectively) are equally represented in the recommendations, which is intuitively what we would identify with being (group) fair. Note that if $k=1$ this situation cannot be achieved.
\begin{figure}
\centering
\includegraphics[scale=0.33]{figures/ex2_graph_emb.png}
\caption{Example graph and corresponding embedding for $\text{score}_2$.}
\label{fig:ex2_graph_emb}
\end{figure}
Formally, given a sensitive attribute $S$ with value $z$ and denoting the set of ``recommended" nodes\footnote{When computing $\rho_k(u)$, a criteria is needed for breaking ties in the ranking, for instance one can use smallest node id goes first.} by $\rho_k(u)$, we can restrict this set based on the attribute value as follows:
\begin{align*}
\rho_{k,z}(u) &= \{v \;:\; v\in\rho_k(u) \wedge \texttt{attr}_S(v)=z\}.
\end{align*}
Next, defining $Z^S$ to be the set of possible values for attribute $S$ we can measure the fraction of recommended nodes with attribute value $z$ as
\begin{equation}
\texttt{share}(u,k,z) = \frac{|\rho_{k,z}(u)|}{|\rho_k(u)|} .
\end{equation}
In order to compute how much it deviates from the \textit{equal representation} scenario, we can use the following expression
\begin{equation}
\texttt{score}_2(u,k,z) = \frac{1}{|Z^S|} - \texttt{share}(u,k,z).
\end{equation}
In the example (Figure \ref{fig:ex2_graph_emb}), for $k=2$ we get the ordered lists $\rho_{2}(A)=\langle B, C \rangle$, $\rho_{2,g_1}(A)=\langle B\rangle$, $\rho_{2,g_2}(A)=\langle C\rangle$ and since $Z^S = \{g_1,g_2\}$, we have $\texttt{score}_2(A,2,g_1) = \texttt{score}_2(A,2,g_2) = \frac{1}{2}-\frac{1}{2}=0$, which corresponds to the \textit{equal representation} scenario. On the contrary, as noted before, if we take $k=1$; $\rho_{2}(A)=\langle B \rangle$, $\rho_{2,g_1}(A)=\langle B\rangle$ and $\texttt{score}_2(A,1,g_1) = \frac{1}{2}-1=-\frac{1}{2}$. Symmetrically, we get $\rho_{2,g_2}(A)=\langle \rangle$ and $\texttt{score}_2(A,1,g_2) = \frac{1}{2}-0=\frac{1}{2}$. Both represent (inevitably) \textit{unbalanced} recommendations among the 2 groups.
\subsection{Data} \label{sec:data}
We use graph embeddings for several commonly used graphs in the graph machine learning community. As previously introduced, graph embedding algorithms represent each node in a graph as a $d$-dimensional vector. So given a graph with $n$ nodes, a graph embedding algorithm outputs an $\mathbb{R}^{n\times d}$ matrix where rows correspond to nodes and columns to graph embedding dimensions.
\input{tex_input/graph-datasets}
The graphs we used are shown in Table \ref{tab:snap_datasets}. All of these graphs are real-world graphs from multiple domains. As group fairness definitions require sensitive-attribute node labels, we retrieved the gender labels that accompany the Facebook network. For anonymity the gender labels are ``0" and ``1". Four of the $4,039$ nodes did not have labels, and we chose to impute a value of ``0" for these 4 missing labels.
The graph embedding algorithms we use are SVD, HOPE \cite{HOPE}, Laplacian Eigenmaps \cite{lapeigenmap}, Node2Vec \cite{node2vec}, SDNE \cite{SDNE}, and Hyperbolic GCN (HGCN) \cite{HGCN}. These algorithms span the multiple classes of graph embedding algorithms: matrix factorization, random walk, and deep learning. We ran these embedding algorithms on the graphs listed in Table \ref{tab:snap_datasets} and the embeddings are stored in the following repository: \url{https://github.com/dliu18/embedding_repo}.
\section{Task Analysis for \textsc{BiaScope}\xspace}
\subsection{Expert Interviews}
In order to understand the key challenges of thoroughly investigating bias in graph embeddings we established a collaboration with Professor Tina Eliassi-Rad and Dr. Brennan Klein. Eliassi-Rad is a Professor of Computer Science and a core member of Northeastern's Network Science Institute and the Institute for Experiential AI at Northeastern University; she has led research in both of the domains of interest and is familiar with many of the real-world applications. Dr. Klein holds a Ph.D. from Northeastern's Network Science Institute; he is a network visualization expert and active network science researcher.
We conducted an initial interview process that led to our task analysis presented in Section \ref{subsec:task_anal}. We divided our interview into three sections: questions related to 1) graph visualization 2) algorithmic fairness and 3) real-world applications of graph embeddings. For each section we prepared a few high-level questions to understand current difficulties in the respective domains. Our interview yielded three main takeaways that inspired our task analysis and design. First, regarding visualizing networks, our domain expert collaborators emphasized the importance of having different visualization encodings for networks of different scales. More specifically, in order to fully characterize bias of graph embeddings it is imperative to allow for both global (whole-network) as well as local (communities, single nodes) views. Second, as most networks are very large to visualize it is important to have summary network statistics to complement the visualization. Additionally, certain graph statistics reflect network properties (e.g., connectivity, degree distribution) that influence how bias in the network is incorporated more by certain graph embedding algorithms compared to others. Third, for algorithmic fairness tools, Professor Eliassi-Rad highlighted the Aequitas system \cite{aequitas} because the interface guides the user through the complicated decision process of choosing fairness definitions and configurations. The takeaway is that a thorough analysis of bias should take into account multiple fairness definitions and configurations and the user should be guided in the process of multifaceted bias analysis. Finally, Dr. Klein connected us to several network visualization libraries \cite{aslak2019netwulf, schwab2021ssvg, wapman2019webweb} that are commonly used by network science researchers for creating customs visualizations, as part of their research process. Testing these recommendations allowed us to understand the domain conventions regarding network visualization and incorporate the most ubiquitous ones in our design.
\subsection{Task Analysis}\label{subsec:task_anal}
\input{tex_input/task-table-short}
From the interaction with our domain expert collaborators we identified the key components of a typical unfairness diagnosis approach for graph embeddings. In particular, we first extracted and ranked the high level domain tasks mentioned by our interviewees within the unfairness diagnosis procedure. We then constructed concrete examples for each activity. This analysis led us to select the most significant domain tasks our tool is designed to support. Subsequently, we mapped the domain tasks to abstract tasks that can be addressed by our visualization design.
The \textit{task analysis} we conducted is outlined in Table \ref{tab:tasks}. We refer to the identified domain tasks by title and augment each domain task with informative descriptions, listed under the ``Example and Description'' column of Table \ref{tab:tasks}. For the abstract task extraction, we used Munzer's Actions Taxonomy \cite{munzner2014visualization} for the High and Mid levels, and the Analytic Task Taxonomy proposed by Amar et al. \cite{amar2005low} for the Low level. We augmented the abstract task analysis with graph-specific tasks from the task taxonomy proposed by Lee et al. \cite{lee2006task}.
We designed \textsc{BiaScope}\xspace to support three main tasks:
\begin{itemize}
\item[\textbf{T1}] Obtain a comprehensive \textit{Statistical Summary} of a network chosen by the user. The summary conveys structural properties possibly related to bias, i.e., degree distribution, edge density, etc.
\item[\textbf{T2}] \textit{Compare} two embeddings of the network w.r.t. a fairness score. The two embeddings are computed using two different algorithms. The fairness score is chosen and configured by the user.
\item[\textbf{T3}] \textit{Diagnose} the observed unfairness of an embedding. This task allows the user to transition from a global to a local perspective, by focusing on the subgraph and embedding subspace that together determine the unfairness score of a user-selected node.
\end{itemize}
The outlined tasks facilitate a typical unfairness diagnosis workflow for graph embeddings. The workflow consists of an \textit{overview} step and a \textit{drill-down} step:
\begin{enumerate}
\item Overview: Comparison between two embedding algorithms (\textbf{T2}), augmented by relevant graph statistics (\textbf{T1}). In this step, the user examines whether the embedding of interest displays signs of unfairness, compared to other embeddings of the same network.
\item Drill-down/Diagnosis: Identification of the unfairness source(s) for the selected embedding (\textbf{T3}). In this step, the user discovers the determining factors as well as the nature of the observed bias, by focusing on certain nodes or communities that are embedded most unfairly.
\end{enumerate}
\section{Design of \textsc{BiaScope}\xspace} \label{sec:design}
This section outlines and justifies the selected visual encoding and interactions implemented in \textsc{BiaScope}\xspace. The selection was performed by analyzing a series of independently drawn sketches that were created to satisfy the tasks identified from our Task Analysis (Section \ref{subsec:task_anal}). In what follows, a high level description of the different encodings is given. Specific details related to the particular network being used in the examples is deferred to Section \ref{sec:usage_scenario}.
We satisfy the statistical summary task (\textbf{T1}) with the ``Statistical Summary of the Network''\footnote{We also refer to this component as the \textit{Statistical Summary View}.} component of the \textit{Overview}, shown in Figures \ref{fig:teaser}A and \ref{fig:statistical-summary}. The table encoding containing the summary provides key graph metrics to help the user understand the type of network being studied. For the Degree Distribution the bar chart idiom was selected, since it makes use of the most effective magnitude channel to encode ordered attributes, according to the effectiveness ranking for visual channels presented in \cite{munzner2014visualization}. The ranking was compiled by Munzner based on previous work performed by the Visualization community such as \cite{cleveland_mcgill_84a, cleveland_93a, mackinlay_86, Ware12, heer_bostock_10} .
Next, we achieve the comparison task (\textbf{T2}) by providing side-by-side views in the \textit{Overview}, whose main visual components\footnote{We also refer to this component as the \textit{Comparison View}.} are shown in Figures \ref{fig:statistical-summary} and \ref{fig:inform-diagnose}. In particular, the same network is displayed on both sides of Figure \ref{fig:inform-diagnose} using the spring layout \cite{fruchterman1991graph}, while the node colors depend on the fairness of the respective embedding algorithm. A sequential color scale is used, since the attribute being encoded is quantitative. We associate darker colors with higher unfairness scores due to their negative nature \cite{bartram2017affective}.
Regarding interactivity, the node id and its fairness score are displayed upon mouse hover (Figure \ref{fig:mouse_hover}) for the user to be able to better asses the difference between two nodes of the network. Zooming is supported to allow for better analysis capabilities of specific network communities (Figure \ref{fig:mouse_hover}).
\begin{figure*}[t]
\centering
\includegraphics[scale=0.4]{figures/hover_merged.png}
\caption{Node attributes (ID and associated fairness score) are displayed on mouse hover. Zooming functionality is also supported to inspect and compare the scores of specific parts of the network.}
\label{fig:mouse_hover}
\end{figure*}
Due to the large scale nature of our data, following \cite{munzner2014visualization}, we have implemented visual feedback in the form of a loading sign while the different visualizations are being loaded into the webpage. Further details regarding data management and implemented preprocessing steps are provided in Section \ref{sec:preprocessing}.
Finally, we address the unfairness diagnostics task (\textbf{T3}) with the \textit{Diagnose View}. This view lists the nodes together with their unfairness scores, allowing the user to select a focal node and uncover the cause(s) of bias. Specifically, the main components of the view are: (1) an interactive table containing the node ids along with their unfairness scores, (2) the projected embeddings affecting the score of the focal node, and (3) the corresponding subgraph topology. The user can sort the list of nodes based on their ids or scores by clicking on the arrows next to the column names of the Table (left of Figure \ref{fig:diagnose_view_1}). A search box is included for each attribute of the table to speedup the lookup.
As previously discussed, we consider two fairness notions (individual and group) with different characteristics, which are reflected by our design. For the case of individual fairness, the score of a node is determined by its local neighborhood. Thus, the local neighborhood topology along with the corresponding projected embeddings are displayed in the view. On the other hand, group fairness determines the score of a node by the most proximal embeddings. Therefore, these projected embeddings are shown along with the corresponding subgraph. In both cases, upon selection of the focal node, the \textit{Diagnose View} shows side-by-side the subgraph and projected embeddings relevant to the chosen fairness notion. Figure \ref{fig:diagnose_view_1} displays the \textit{Diagnose View} for the case of individual fairness. The projected embeddings\footnote{The projection of the embeddings is performed using PCA \cite{PCA_FRSLIIIOL}. In what follows, for simplicity, we do not account for the projection when referring to the projected embeddings.} are encoded using a scatterplot (Figure \ref{fig:diagnose_view_1}, rightmost chart), noting that by \cite{munzner2014visualization} it maximizes effectiveness.
The objective of the \textit{Diagnose View} is to provide insight into the fairness score computation for the selected focal node and uncover causes of unfairness in the network. For this purpose, a red color pop-out effect is used to outline the focal node, together with an increase in its size. This improves the efficiency of its lookup \cite{munzner2014visualization}, which is an essential component in understanding the score computation. Brushing and linking is supported between the network and the embedding space to analyze how different nodes contribute to the fairness score of the focal node. Specifically, when brushing over a set of elements, the corresponding elements in the other view are highlighted. This is shown in Figure \ref{fig:diagnose_view_1}, where a subset of nodes in the local topology of node with id $865$ is being selected with the brush. When hovering over a node or its embedding, the corresponding node id and fairness score are displayed in the tooltip, together with the number of hops to the focal node for the individual fairness score (Figure \ref{fig:diagnose_view_1}) or the node label for group fairness. The distance from the focal node (individual fairness) and the gender (group fairness) are encoded using the color channel with an ordinal and categorical color scale, respectively.
Furthermore, as a response to expert feedback, we improved our design with a context legend that conveys the scale of the distances observed in the scatterplot (Figure \ref{fig:diagnose_view_2}). More specifically, we made use of the \textit{focus+context} approach \cite{cockburn_etal_08}, and augmented the \textit{Diagnose View} (i.e., \textit{focus}) with a global scatterplot (i.e., \textit{context}) displaying all the embeddings (top section of Figure \ref{fig:diagnose_view_2}), where the highlighted area corresponds to the points of the scatterplot which are the ones that affect the focal node (Figure \ref{fig:inform-diagnose} and \ref{fig:fairwalk-diagnose}).
\begin{figure*}
\centering
\includegraphics[scale=0.39]{figures/diagnose_view_new_1.png}
\caption{Main visual components of the \textit{Diagnose View} for individual fairness. From left to right: Interactive table listing node ids and their unfairness scores, local neighborhood of selected node in the network and corresponding projected embeddings.}
\label{fig:diagnose_view_1}
\end{figure*}
\begin{figure}
\centering
\includegraphics[scale=0.31]{figures/diagnose_view_2.png}
\caption{The context legend displayed in the \textit{Diagnose View} (top plot) conveys the scale of the distances observed in the scatterplot (bottom plot), where the embeddings that affect the score are plotted.}
\label{fig:diagnose_view_2}
\end{figure}
\section{Evaluation}
\input{tex_input/usage-scenario}
\input{tex_input/expert-review}
\section{Discussion}
In this section, we discuss several points that were observed during the development process and posterior usage of \textsc{BiaScope}\xspace. These consist of challenges regarding the size of the data, both in terms of visibility and the efficiency of the system (Section \ref{sec:preprocessing}), as well as limitations and future lines of work, included in Section \ref{sec:limitations_futurework}.
\input{tex_input/data-preprocessing}
\subsection{Limitations and Future Work}\label{sec:limitations_futurework}
In the present design study we handled large benchmark networks, as outlined in Section \ref{sec:preprocessing}, therefore we had to overcome responsiveness issues. As previously discussed, we used a simple heuristic to minimize the number of edges in our visualization by filtering out non-salient edges. In future work, we plan to explore more sophisticated heuristics as well as different methods for graph abstractions e.g., sub-sampling methods that preserve node communities. Additionally, we plan to extend our scope to a larger suite of benchmark networks which are even larger (in terms of node count) or have higher density. A parallel direction of future work includes developing advanced interactivity and linking in the overview, e.g. simultaneous zooming, coupled hover events, etc. We expect this extension to generate new performance challenges since the interaction should be particularly responsive in order to increase effectiveness.
Furthermore, following the expert feedback, we plan to make certain additions in order for the tool to better support the needs of its target users. These include the extension of our benchmark suite with other networks popular within the graph mining community (e.g., Books \footnote{\url{http://www-personal.umich.edu/~mejn/netdata/}}, NS \cite{netscience-data}) and a list of different options for the embedding projections beside PCA (e.g., UMAP \cite{becht2019dimensionality}). Finally, as mentioned in Section \ref{sec:expert_review}, we plan to develop a user friendly onboarding process for users who want to evaluate their own embedding algorithms using our tool. The onboarding will include preprocessing of the user data, which will utilize server-side computation, in order to preserve sufficient responsiveness at use-time.
\section{Conclusion}
Motivated by the recent efforts made by the algorithmic fairness community, we have build \textsc{BiaScope}\xspace, an interactive web-based visualization tool that supports end-to-end \textit{visual unfairness diagnosis} for graph embeddings. Our design is the result of an iterative collaborative process with experts in the fields of fair machine learning and graph mining. To inform our design, we conducted a thorough task analysis that consists of key tasks that support an unfairness diagnosis workflow for graph embeddings.
Applying our visual unfairness diagnosis workflow to a benchmark network, we show the effectiveness our tool has in visually revealing the unfairness incorporated into different widely used graph embedding algorithms. Our work is part of a greater effort to allow both researchers and practitioners to build or use machine learning models while simultaneously being mindful of bias and algorithmic fairness, without the need of expertise in the field. Our tool, and others in the same spirit, could be part of a standard development workflow of a machine learning researcher or engineer, as it is designed to provide greater transparency into how current graph embedding algorithms and fairness notions interconnect in practice.
\section {Acknowledgement}
The authors would like to thank Zohair Shafi and Ayan Chatterjee for participating in the expert feedback sessions and providing insightful comments about the system.
\newpage
\bibliographystyle{abbrv-doi}
\subsection{Usability Testing}\label{sec:usability_testing}
During the development of our tool we performed an informal usability testing session with 15 Computer Science graduate student as our participants. The goal of this study was to qualitatively evaluate the effectiveness of our visual encodings for the target tasks as well as the usability of the interactivity elements.
The study confirmed that there exist certain technical barriers for non-specialist users, which is understandable since our design was targeted to experts. However, we still took action to clarify certain components, for example we implemented user-friendly tooltips for explanations and definitions of technical terms. Additionally, the study was particularly revealing with respect to color encodings and color perception. More specifically, it indicted that the users tend to associate dark colors with more unfairness and red colors are semantically associated with pathological attributes, which in our case correspond to biased embeddings. Finally, the study reassured us that the implemented interactivity significantly empowers the user to investigate causes of bias and successfully complete unfairness diagnostics.
\section{Data Exploration}\label{appendix:data}
\section{Detailed Task Analysis}
\input{tex_input/task-table}
In this section we provide the full task analysis we conducted after interviewing our expert. From the set of tasks analysed and categorized in Table \ref{tab:tasks_full} we identified the most important tasks our tool supports.
\section{Fairness definitions and examples}\label{sec:fairness_defs}
\subsection{Score 1: InFoRM (Individual Fairness)}
Let $Y$ be the embedding matrix, where $Y[u]$ is the embedding of node $u$. Let $S$ be the node proximity matrix, where $S[u,v]$ is the proximity of nodes $u$ and $v$. The most obvious notion of node proximity is the adjacency matrix $S=A$. There are other options such as a random walk matrix. We can then define the first individual fairness score based on \cite{InFoRM} as follows:
\begin{align*}
score_1(u,k=1) &= \sum_{v \neq u} \lVert Y[u] - Y[v]\rVert_2^2 \cdot S[u,v] \\
&= \sum_{v\in\mathcal{N}(u,1)} \lVert Y[u] - Y[v]\rVert_2^2
\end{align*}
where $\mathcal{N}(u,k)$ is the $k$-hop neighborhood of $u$, given by the nodes reachable from $u$ in at most $k$ steps.
We generalize this notion for different values of $k$ as follows:
\begin{align*}
score_1(u,k) &= \sum_{v\in\mathcal{N}(u,k)} \lVert Y[u] - Y[v]\rVert_2^2
\end{align*}
\subsection{Fairwalk (Group Fairness)}
For the group fairness notions introduced in \cite{ijcai2019-456}, given a node $u$, we need to define a group of ``recommended" nodes, denoted $\rho(u)$. Among the nodes not connected to $u$, we choose to ``recommend" the top-$k$ most proximal ones in the embedding, using dot product similarity.
Additionally, given a sensitive attribute $S$ and a value $z$, we can restrict the ``recommended" node set based on the attribute as follows:
\begin{align*}
\rho_z(u) &= \{v \;:\; v\in\rho(u) \wedge \texttt{attr}_S(v)=z\}
\end{align*}
For example, we can have $S=\text{gender}$ and $z=\text{female}$. When computing $\rho(u)$, a criteria is needed for breaking ties in the ranking, for instance one can take smallest id goes first. Using this notions, two fairness notions are defined in \cite{ijcai2019-456}, which are described next.
\subsubsection{Network Level}
Given a sensitive attribute $S$, we can consider different communities (user groups) in the graph, according to the values of $S$. Concretely, given $i,j$ two possible values for attribute $S$, we define the group $G_{i,j}^S$ as follows:
\[
G_{i,j}^S = \{(u,v)\ : u\neq v \wedge \texttt{attr}_S(u)=i \wedge \texttt{attr}_S(u)=j\}
\]
We denote the set of all such groups as $\mathcal{G}^S$. Then, based on these groups, a bias score can be derived as follows:
\begin{equation}
\texttt{bias}(\mathcal{G}^S) = Var(\{P(G_{i,j}^S): G_{i,j} \in \mathcal{G}^S\})
\end{equation}
, where
\begin{equation}
P(G_{i,j}^S) = \frac{|\{(u,v): v \in \rho(u) \wedge (u,v) \in G_{i,j}^S\} |}{|G_{i,j}^S|} .
\end{equation}
\noindent
Example:
Let $k=1$, $S=\text{race}$ with possible attribute values $\{\text{w}, \text{b}\}$. Consider the network and associated embedding shown in Figure \ref{fig:ex_1}.
\begin{figure}
\centering
\includegraphics[scale=0.3]{figures/ex1.png}
\caption{Example network and embedding. Race feature of the nodes is indicated next to them in the network.}
\label{fig:ex_1}
\end{figure}
Then, we have:
\begin{align*}
\mathcal{G}^S &= \{G^S_{w,b},G^S_{w,w},G^S_{b,b}\}\\
G^S_{w,b} &= \{(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)\}\\
G^S_{w,w} &= \{(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)\}\\
G^S_{b,b} &= \{(4,4),(4,5),(5,5)\}
\end{align*}
Next, consider node $u=2$, since $k=1$,
\[
\rho(2) = \{4\}
\]
Thus, we get
\begin{align*}
P(G^S_{w,b}) &= \frac{1}{6}\\
P(G^S_{w,w}) &= \frac{0}{6}\\
P(G^S_{b,b}) &= \frac{0}{3}
\end{align*}
Therefore,
\[
\texttt{bias}(\mathcal{G}^S) = Var(\{1/6,0,0\}) = \frac{1}{162} \sim 0.006
\]
\subsubsection{Score 2: User Level}
Let $Z^S$ be the set of all possible values of attribute $S$. We calculate a bias score for an attribute value $z$ and a node $u$.
\begin{equation}
\texttt{bias}(z) = \frac{1}{|Z^S|} - \frac{\sum_{u \in V} \texttt{z-share}(u)}{|V|}
\end{equation}
, where
\begin{equation}
\texttt{z-share}(u) = \frac{|\rho_z(u)|}{|\rho(u)|} .
\end{equation}
Here, we define
\begin{equation}
\texttt{score}_2(z,u) = \frac{1}{|Z^S|} - \texttt{z-share}(u)
\end{equation}
\noindent
Example:
Again consider the network and embedding from Figure \ref{fig:ex_1} and attribute $S=\text{race}$ with $Z^S=\{w,b\}$, $z=b$ and $k=1$.
\begin{align*}
\rho(1)&=\{2\}\\
\rho(2)&=\{4\}\\
\rho(3)&=\{4\}\\
\rho(4)&=\{2\}\\
\rho(5)&=\{2\}
\end{align*}
\begin{align*}
\texttt{z-share}(1,b) &= \frac{0}{1}=0\\
\texttt{z-share}(2,b) &= \frac{1}{1}=1\\
\texttt{z-share}(3,b) &= \frac{1}{1}=1\\
\texttt{z-share}(4,b) &= \frac{0}{1}=0\\
\texttt{z-share}(5,b) &= \frac{0}{1}=0\\
\end{align*}
Then,
\begin{align*}
\texttt{score}_2(z=b,u=1) &= \frac{1}{2}-\frac{0}{1}=\frac{1}{2}-0=\frac{1}{2}\\
\texttt{score}_2(z=b,u=2) &= \frac{1}{2}-\frac{1}{1}=\frac{1}{2}-1=-\frac{1}{2}\\
\texttt{score}_2(z=b,u=3) &= \frac{1}{2}-\frac{1}{1}=\frac{1}{2}-1=-\frac{1}{2}\\
\texttt{score}_2(z=b,u=4) &= \frac{1}{2}-\frac{0}{1}=\frac{1}{2}-0=\frac{1}{2}\\
\texttt{score}_2(z=b,u=5) &= \frac{1}{2}-\frac{0}{1}=\frac{1}{2}-0=\frac{1}{2}
\end{align*}
and also,
\[
\texttt{bias}(b) = \frac{1}{2} - \frac{2}{5} = \frac{1}{10}
\]
\subsection{Data Management}\label{sec:preprocessing}
Due to the size of the networks that were considered, several performance challenges needed to be addressed in the development process. In this section, we provide an overview of how these were approached.
As previously outlined, in order to improve network visibility, we filter out visually non-salient edges. Specifically, we only display edges in the top 10 percent of length. Filtering allows us to save on browser memory, as the number of edges far exceeds the number of nodes. At the same time, by inspecting the networks we found that the bottom 90 percent of shortest edges were often not visible, so filtering offers large performance increases while preserving important visual information, e.g., intra-community connectivity.
In terms of preprocessing, both the projection of the embeddings, as well as group and individual fairness scores were calculated offline in order to obtain an additional performance speedup. For the later, this involves generating for every network the corresponding score for each node id and possible configuration. Configuration options consist of the number of hops from the focal node for the individual fairness notion, while for the group fairness, the value of $k$ as well as the chosen sensitive attribute (in our case gender) and value.
Finally, the pre-computed network information is accessed only once when the webpage is initially loaded and stored in memory, in order to avoid redundant server access.
\subsection{Expert Feedback}\label{sec:expert_review}
In order to evaluate our tool, we conducted feedback interviews with our two collaborators and two additional experts, who are active researchers interested in graph embeddings, also affiliated with Northeastern's Network Science Institute. We conducted four individual interviews, during which each participant was able to interact with the tool through the webpage on their own system. We simultaneously observed their interaction with the tool, which was captured on video for further analysis. For all interviews we followed the same steps: First, we asked for the participant's consent to video recording. Second, we briefly discussed the tasks our tool is designed to support. Third, we provided an \textit{interactive walk-through} of the tool by asking the participant to locate certain views and perform concrete tasks. The goal of this step was to help the participants familiarize themselves with the UI. Next, we encouraged the participant to follow a \textit{usage scenario} similar to the one presented in Section \ref{sec:usage_scenario}. During this step we provided loose guidance by setting open-ended goals and we encouraged the participant to freely accomplish them using the tool. Lastly, we collected verbal feedback using the following questions:
\begin{itemize}
\item[\textbf{Q1}] In your view, how well are the domain tasks supported?
\item[\textbf{Q2}] How would you perform these tasks without the tool? Do you see value in our tool?
\item[\textbf{Q3}] How would you use our tool in your research/development process?
\item[\textbf{Q4}] What limitations do you identify?
\end{itemize}
Following the interview sessions, we collected additional quantitative feedback from the experts using a System Usability Scale (SUS) \cite{sus_1}. In what follows, we summarize the key takeaways from the feedback.
\paragraph{Effectiveness and Usability.}
During the interactive walk-through, we observed that all the participants were able to quickly perform the tasks with little to no assistance. Regarding explicit verbal feedback, all experts agreed that \textsc{BiaScope}\xspace effectively supports the described domain tasks (\textbf{Q1}). Additionally, all participants described, while three explicitly said, that they would follow the same process in order to thoroughly analyse unfairness of a graph embedding (\textbf{Q2}). Most participants pointed out that without \textsc{BiaScope}\xspace this analysis would be tedious and prone to errors. One participant mentioned that \textsc{BiaScope}\xspace would save them time and a different one said: ``something that goes a long way in terms of what you have done is I do not want to implement any of these things [by myself]. ... at the very basic level you have just reallocated a lot of work... [Since there are no other tools like this one on the market], this is immediately useful in a very time saving way.''.
Regarding the potential role of \textsc{BiaScope}\xspace within our participants' research process (\textbf{Q3}), the feedback validated the tool's effectiveness while additionally providing new perspectives. Two of the experts agreed that the tool would facilitate the evaluation of their graph embedding algorithms with respect to fairness. More specifically, they mentioned the following: ``For people who aren't really fairness focused, this would [enable] a nice quick sanity check.'' and ``I would use it as a visual tool to interrogate the data, [which is] a very important thing to do, as opposed to just feeding it into your machine learning system and then hoping for the best.'' The other two experts pointed out they could further utilize our tool to facilitate their work in explainable machine learning or for pedagogical purposes: ``... I would definitely use this tool for understanding explainability and interpretability of embeddings ... my major use would be like explaining some of the down stream task results or predictions'', ``It is a very good teaching tool, in general I think web based [tools] that let you interact with networks, change the game when it comes to learning about networks.''.
\paragraph{Limitations.}
The experts agreed on the importance of an ``import" feature, which would allow the user to upload and diagnose their own embedding algorithm. One of the experts additionally suggested an ``export" functionality. The implementation of an "import" feature is included as future work in Section \ref{sec:limitations_futurework}. Moreover, some experts pointed out that the Diagnose View should include an indication of the scale regarding the projected embeddings, in order to clearly convey the magnitude of distances observed in the local embedding subspace. We addressed this limitation with the addition of the context legend, as discussed in Section \ref{sec:design}. One of the experts expressed the interesting idea to augment our network options with certain simple synthetic networks representing typical graph topologies, e.g., star, which could enlighten the user with respect to the nature of different fairness notions. Finally, adding on-demand information about the selected fairness notion definition was a frequent suggestion among participants.
\paragraph{System Usability Scale.}
We requested our experts to fill a System Usability Scale (SUS) \cite{sus_1, brooke2013sus}, a standard simple ten-item scale giving a global view of subjective assessments of the usability of a system. Based on the responses, we derive the corresponding SUS scores, which have a range of 0 to 100. Regarding scores assessment, \cite{bangor_etal} proposed an adjective value scale for SUS scores. According to this scale, scores in the range 72.5-84.9 are associated with a good usability and above 85 represent an excellent one. Based on the collected SUS scores, we obtained a mean value of $83.125$. This implies, per the aforementioned scale, that \textsc{BiaScope}\xspace's usability was perceived as good, close to excellent by the interviewed experts.
\subsection{Usage Scenario}\label{sec:usage_scenario}
To illustrate the design in action, we will walk through a case study using the Facebook network, which represents the social network friendships of over four thousand users. We begin with an overview of the network via the statistical summary shown in Figure \ref{fig:statistical-summary}. The table on the left informs the user that the Facebook graph is sparse and highly clustered. Further, the degree distribution shows that almost all users in the network have fewer than 200 friends.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/SS.png}
\caption{The statistical summary for the Facebook network helps the user to characterize the data before analyzing its fairness scores. The table on the left states that the network is sparse and highly clustered while the degree distribution shows that almost all users in the network have fewer than 200 friends.}
\label{fig:statistical-summary}
\end{figure}
Proceeding further into the \textit{Overview}, the user can browse a side-by-side comparison of the fairness scores for two sets of embeddings. Figure \ref{fig:teaser}B shows the scores for Node2Vec on the left and then HGCN on the right for the Facebook network. This visual provides two takeaways. First, the Node2Vec embeddings are overall more fair than the HGCN embeddings, as indicated by the lighter coloring. Second, the unfairness for HGCN is concentrated in select communities. Without the visualization, a purely statistical analysis would resort to aggregate values which do not account for heterogeneity.
Moving into the drill down portion of the tasks, we can now use the \textit{Diagnose View} to better understand why certain nodes are scored as unfair. For instance, Figure \ref{fig:inform-diagnose} shows the diagnostic results for the Facebook Node2Vec embeddings using the individual fairness notion. The local neighborhood (ego network) shows that the focal node, in red, is part of two communities of friends. Further, the projected embeddings show that the two communities of friends are embedded far from each other. Hence, the focal node is scored as unfair because it is far from its neighbor communities in embedding space. This mapping between neighbors and embeddings can be verified with brushing and linking (Figure \ref{fig:diagnose_view_1}).
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/inform-diagnose-new.png}
\caption{Moving into the drill-down tasks, the above figure provides the diagnostics results for the Facebook network's Node2Vec embeddings where fairness is defined with the InFoRM notion of individual fairness. The ego network on the left shows the target node is part of two communities, as linked on the right hand side, these communities are themselves embedded far apart. Thus, the diagnostics results show that under InFoRM bridge nodes between communities may be embedded unfairly. }
\label{fig:inform-diagnose}
\end{figure}
Finally, the user can also drill-down and diagnose the group fairness scores as well. Figure \ref{fig:fairwalk-diagnose} shows the Facebook Node2Vec embeddings evaluated based on the group fairness notion. Now, the focal node together with the induced subgraph (top-$k$ recommended nodes) is plotted in the rightmost chart. The subgraph shows that the target node's recommendations are mostly of gender 0, encoded in yellow, which is a form of homophily. Further, these nodes are embedded close to the target node in the embedding space. This is a form of group unfairness because if we train a model to recommend friends based on these embeddings, the model would perpetuate homophily for this node and recommend nodes of the same gender.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/fairwalk-diagnose-new.png}
\caption{The diagnostics results for group fairness show that the given node from the Facebook network is unfair because many of its recommended nodes are also of the same gender value, encoded in yellow. The focal node and the induced subgraph given by the top-$k$ recommended nodes are plotted, together with the corresponding embeddings. The design layout is similar to the individual fairness diagnostics but the node colors now represent the sensitive attribute, which is the most important attribute for group fairness.}
\label{fig:fairwalk-diagnose}
\end{figure}
\section{Introduction}
With the growing use of machine learning models to automate decisions, a rising concern for both researchers and practitioners is to assure that they are not \textit{biased} (i.e., systematically unfair) against specific population groups or individuals. More specifically, within the context of graph mining algorithms, fairness plays an important role \cite{agarwal2021towards, tsioutsiouliklis2021fairness, yao2017beyond, Kamishima12enhancementof} since these algorithms are widely used in human-centered applications with major societal impact. In graph machine learning, it is a common practice to compile network information (such as graph topology, node features) in unified node-level representations, i.e., \textit{graph embeddings}. The produced embeddings can subsequently be used off-the-shelf for a plethora of downstream tasks, e.g., node classification \cite{classification}, link prediction \cite{link}, community detection \cite{clustering}, ranking \cite{recommend}, among others. For example, graph embeddings of social networks can serve as representations of users for high-impact real-world tasks, e.g., user classification for targeted advertising, social group detection, friendship recommendation, advanced user search or to better understand a rumor spreading process \cite{hamilton2017representation, cai2018comprehensive, zhang2018network}. However, graph embeddings can carry bias against particular groups or individuals, which is either present in the data or introduced by the algorithm itself \cite{baeza2018bias}. As a result, it is crucial to identify, diagnose and mitigate bias before it gets propagated to downstream tasks and real-world machine learning systems.
Thoroughly examining a graph embedding algorithm with respect to unfairness is hard to accomplish using solely statistical methods. While it is possible to quantify bias either at a global level (i.e., the network as a whole) or at a local level (i.e., communities or specific nodes), this is not sufficient because it is important to gain insight into the major factors that contribute to the bias. Thus, a framework that supports visual unfairness diagnosis for graph embeddings by explicitly illustrating the sources of bias can be invaluable. A further challenge is that multiple \textit{(un)fairness notions} exist. In particular, these notions can model discrimination either against \textit{groups} \cite{ijcai2019-456} or \textit{individuals} \cite{InFoRM} and it is well-established that trade-offs exist between the two types \cite{kleinberg2016inherent, tradeoff}. Therefore, an integrated design is required that supports heterogeneous fairness notions.
In this work, we propose \textsc{BiaScope}\xspace, an interactive open-source visualization tool that supports end-to-end \textit{visual unfairness diagnosis} for graph embeddings. To design the tool, we worked with domain experts in order to understand the intricacies of unfairness characterization for graph embeddings. We designed \textsc{BiaScope}\xspace as a web-based visualization tool with emphasis on interactive features which simplify the complex task of unfairness characterization even on large benchmark networks. Our tool facilitates visual comparison of any two embedding algorithms with respect to fairness on several common network benchmarks. Additionally, it allows the user to locate nodes that contribute to unfairness and importantly, to identify the defining factors of the observed bias. This is facilitated by interactively linking the embedding subspace (i.e., high-dimensional vectors) that displays bias with the corresponding graph topology (i.e., nodes and edges), and vice versa (Figure \ref{fig:teaser}C). Our tool supports both group \cite{ijcai2019-456} and individual \cite{InFoRM} fairness notions in an integrated design and it is task agnostic in the sense that it does not assume a particular downstream task.
The main contributions of our work can be summarized as follows:
\begin{itemize}
\item Through an iterative interview process we identified the \textit{abstract tasks} that are crucial parts of an \textit{unfairness diagnosis workflow} for graph embeddings.
\item Informed by our task analysis, we \textit{designed and implemented} \textsc{BiaScope}\xspace to support multiple fairness notions, while being interactively configurable and task-agnostic.
\item Finally, we collected \textit{expert feedback} that validates our design. The feedback suggests that \textsc{BiaScope}\xspace can be an effective asset both for researchers to thoroughly evaluate their proposed algorithms and for practitioners to detect bias early in the product development process.
\end{itemize}
The source code of \textsc{BiaScope}\xspace is provided at \url{https://github.com/agapiR/BiaScope}, where the reader can download a \textit{demo video} introducing the tool and its functionality. The repository also provides instructions for running a local instance of \textsc{BiaScope}\xspace.
\section{Related Work}
\textsc{BiaScope}\xspace lies at the intersection of vector embedding visualization and the analysis of fairness for graphs embeddings. Additionally, we draw inspiration from graph neural network visualization tools.
\paragraph{Vector Embeddings.}
Most vector embedding visualization tools are tailored to contextualized embeddings produced by language models. The tool proposed in \cite{berger2020visually} presents a sentence view with labels for semantic properties e.g., parts-of-speech and an embedding view with cluster analysis via co-occurrence small multiples. On the other hand, embComp \cite{embcomp} introduces visual analysis methods for embedding comparison which combine a global overview with detailed views. The tool mainly focuses on comparing embedding space properties, such as neighborhood overlap or spread and neighbor distances. {The Embedding Comparator system \cite{2022-embedding-comparator} shares this objective, differentiating from embComp in the fact that it simultaneously visualizes global views of embedding structure alongside local views of individual objects and their common and unique neighbors to enable efficient analysis.} Vector embedding comparison is also supported by Emblaze \cite{sivaraman2022emblaze} which consists of an elaborate interactive scatterplot and mainly focuses on neighborhood discovery for dynamic relation suggestions. A more suitable tool for graph embedding visualization is EmbeddingVis \cite{li2018embeddingvis} which allows for comparison of different graph embedding models with respect to which properties of the graph they preserve and illustrates relationships between node metrics and selected embedding vectors.
\paragraph{Graph Neural Networks.}
A common method of learning graph embeddings involves Graph Neural Networks (GNNs), for which there are several visualization tools. The GNNlens tool \cite{gnnlens} helps researchers open the black box of Graph Neural Networks. The tool utilizes Parallel Sets View and Projection View for quick identification of error patterns in the set of wrong predictions. Moreover, a detailed overview for a particular node is offered by Graph View and Feature Matrix View. In a similar spirit, GNNExplainer \cite{gnnexplainer} assists with interpretability by presenting explanations for predictions made by a GNN. The explanations have the form of small visualizable graph motifs and important node features. On the other hand, the CorGIE tool \cite{CorGIE} focuses on the correspondence between the graph topology and the node embeddings. It utilizes a $k$-hop graph layout to show topological neighbors in hops and their clustering structure. It uses small multiples bar charts for node features and a UMAP for the 2D projection of the latent space. Finally, the GNNVis \cite{gnnvis} paper proposes a framework for learned dimensionality reduction using GNNs.
\paragraph{Fairness in Machine Learning.}
Several notions of fairness have been proposed in the literature, mainly categorized into two families: statistical (group) and individual fairness, where most of the focus lies on the first type \cite{chouldechova2020snapshot, 10.1145/3447548.3467266}. This also holds in the context of graph mining, as algorithmic fairness definitions have been adapted for ranking, clustering, and embedding \cite{fair-graph-tutorial}. A graph mining method consists of three major components: the input graph, the mining model and the mining results \cite{InFoRM}. The main idea behind individual fairness is that similar individuals should receive similar algorithmic outcomes. In the context of graph mining, this translates into the fact that "similar" nodes on the graph are also "similar" in terms of the mining results. The similarities between nodes are encoded in a similarity matrix. In particular, every pair of nodes must satisfy the fairness condition.
Group fairness notions in the context of embeddings have also been studied \cite{pmlr-v97-bose19a, ijcai2019-456}. Bose and Hamilton \cite{pmlr-v97-bose19a} introduce an adversarial framework to enforce fairness constraints on graph embeddings, by training a set of "filters" to prevent adversarial discriminators from classifying the sensitive information from the filtered embeddings. After training, these filters can be composed together in different combinations, allowing for the flexible generation of embeddings that are invariant with respect to any subset of the sensitive attributes. Rahman et al. \cite{ijcai2019-456}, extend the Node2Vec algorithm in a fairness-aware manner. For doing so, they propose a new notion of fairness in the context of friendship recommendation systems, by extending statistical parity.
Algorithmic fairness has also received attention in the visualization community. Several tools that target different machine learning settings (e.g., classification and ranking) have been proposed: FAIRVIS \cite{FAIRVIS} is a visual analytics system that helps audit the fairness of binary classification models. The system allows users to explore both suggested and user-specified subgroups and supports 10 metrics for comparison by default (Accuracy, Recall, Specificity, Precision, Negative Predictive Value, False Negative Rate, False Positive Rate, False Discovery Rate, False Omission Rate, and F1 score). Users can derive new metrics from the base outcome rates. Another tool proposed for the classification setting is DiscriLens \cite{9222272}, which identifies a collection of potentially discriminatory attributes based on causal modeling and classification rules mining and provides visualization to facilitate the exploration and interpretation of them. For high-level model understanding, the What-If Tool \cite{whatif} offers visualizations of selected performance and fairness measures as well as model comparison with respect to those measures. Although the tool provides a comprehensive visual overview of multiple aggregate statistics, it does not offer insights into the sources of the observed bias. Focusing on ranking settings, Fairsight \cite{FairSight}, incorporates different fairness notions (including group and individual fairness) to support identifying and mitigating bias against individuals and groups in problems that involve rank-ordering individuals. Finally, FairRankVis \cite{fairrankvis} enables the exploration of multi-class bias in graph ranking algorithms, allowing for the comparison of fair and unfair versions of these algorithms using both group and individual notions of fairness. FairRankVis is the only tool that supports model comparison as a mechanism for explaining the impacts of algorithmic debiasing. However, its design is tailored to graph ranking since its main visual components target the comparison of two algorithms with respect to their ranking results. In our work, we focus on graph embeddings without restricting our analysis to a particular machine learning application. To the best of our knowledge, \textsc{BiaScope}\xspace is the first tool that allows unfairness diagnosis for graph embeddings in a task-agnostic manner.
\section{Preliminaries}
In this section, we discuss the definition and formalism we adopt for \textit{graph embeddings} (Section \ref{sec:graph_emb}). Then, this formalism is used to define the \textit{fairness notions} we consider, for which we also provide intuition and examples (Section \ref{sec:fairness}). Finally, we briefly discuss the data we used for the present design study (Section \ref{sec:data}).
\subsection{Graph Embeddings} \label{sec:graph_emb}
Graph embedding algorithms take a graph $G = \left(V, E\right)$ as input and represent each node in $\mathbb{R}^d$ where $d$ is a dimensionality parameter provided to the algorithm. The node-level vector representations are chosen such that two nodes $v_i$ and $v_j$ that are close to each other in $G$ are embedded near each other in $\mathbb{R}^d$ \cite{hamilton2017representation}. Graph embedding algorithms can also be clustered into several methodological classes: matrix factorization \cite{lapeigenmap, HOPE}, random walk \cite{node2vec}, auto-encoder \cite{SDNE}, and GCNs \cite{graphSAGE}. Recent work has also embedded nodes into hyperbolic space, which better represents hierarchical tree-like structures \cite{HGCN, poincare}.
In what follows, we assume that the graph embedding is represented by the embedding matrix $Y \in \mathbb{R}^{n \times d}$, where $Y[u] \in \mathbb{R}^d$ is the embedding of node $u$ and $n = |V|$ the number of nodes in the graph.
\subsection{Fairness Notions} \label{sec:fairness}
Recent work has evaluated the fairness of graph embeddings and builds upon broader fairness definitions established in the algorithmic fairness community. Fairness definitions typically fall into two classes: individual and group. Individual fairness ensures that algorithms treat two similar individuals similarly. In contrast, group fairness ensures that two sub-populations, in aggregate, are treated similarly. In this work, we incorporate embedding fairness definitions from both classes. In the following, we provide the formal definitions and illustrative examples.
\subsubsection{Individual Fairness}
For individual fairness, we build upon the definition of individual fairness for graph embeddings introduced in \cite{InFoRM} (InFoRM), which assigns a non-negative unfairness score to each node. The score indicates how differently the node is embedded from other nodes in the neighborhood, where the neighborhood is defined by the number of hops from the target node. For example, consider the graph shown in Figure \ref{fig:ex1_graph}. Since nodes B and C are the (1-hop) neighbors of A, the closer their embeddings are to the one corresponding to node A, the lower A's unfairness score. Intuitively, we expect \textit{similar} nodes according to the local topology of the graph to also be \textit{similar} in terms of their distance in the embedded space. We can formalize this notion by considering a \textit{proximity} or \textit{similarity} matrix $S$ where $S[u,v]$ is the proximity of nodes $u$ and $v$ (i.e., a quantitative measure that indicates how close or similar they are in the graph). The most obvious notion of node proximity is the adjacency matrix of the graph $A$, but other options include its powers ($A,\dots,A^k$), random walk matrices as well as several other neighborhood overlap measures as explained in \cite{grl_book}. Assuming $S=A$ and $Y$ being the embedding matrix, we can define the individual fairness score based on \cite{InFoRM} as follows:
\begin{align*}
\text{score}_1(u,k=1) &= \sum_{v \neq u} \lVert Y[u] - Y[v]\rVert_2^2 \cdot S[u,v] \\
&= \sum_{v\in\mathcal{N}(u,1)} \lVert Y[u] - Y[v]\rVert_2^2
\end{align*}
where $\mathcal{N}(u,k)$ is the $k$-hop neighborhood of $u$, given by the nodes reachable from $u$ in at most $k$ steps. In the example from Figure \ref{fig:ex1_graph}, we get $\text{score}_1(\text{A},1)=17$. We generalize this notion for different values of $k$ as follows:
\begin{align*}
\text{score}_1(u,k) &= \sum_{v\in\mathcal{N}(u,k)} \lVert Y[u] - Y[v]\rVert_2^2
\end{align*}
Returning to our example (Figure \ref{fig:ex1_graph}), considering $k=2$, node D is now taken into account since it is reachable from node A in two hops. Thus, we get $\text{score}_1(\text{A},2)=81$. Observe that for a given vertex $u$ and $k\geq 0$, $\text{score}_1(u,k)\leq \text{score}_1(u,k+1)$ and that isolated nodes get a score of 0.
The scores for a graph with minimum degree at least one are normalized to be in the range $[0,1]$ as follows:
\begin{align*}
\text{score}^d_1(u,k) &= \frac{\text{score}_1(u,k)}{\text{degree(u)}}\\
\text{score}^n_1(u,k) &= \frac{\text{score}^d_1(u,k)}{\max_u \text{score}^d_1(u,k)}
\end{align*}
where $\text{degree(u)}$ is the degree of node $u$ and $\text{score}^n_1(u,k)$ denotes the normalized score.
\begin{figure}
\centering
\begin{subfigure}
\centering
\begin{tikzpicture}[main/.style = {draw, circle}]
\node[main] (1) {A};
\node[main] (2) [right of=1] {B};
\node[main] (3) [below of=1]{C};
\node[main] (4) [right of=3] {D};
\draw (1) -- (2);
\draw (1) -- (3);
\draw (3) -- (4);
\end{tikzpicture}
\end{subfigure}
\begin{subfigure}
\centering
\includegraphics[width=.6\linewidth]{figures/ex1_graph_emb.png}
\end{subfigure}%
\caption{Example graph and corresponding embedding for $\text{score}_1$.}
\label{fig:ex1_graph}
\end{figure}
\subsubsection{Group Fairness}
Regarding group fairness, our score is based on the definition of group fairness for graph embeddings proposed by \cite{ijcai2019-456} (Fairwalk). The motivation behind it is to promote recommendations (in the context of a Recommender System) where all population groups are equally represented. To start, a set of link recommendations are made for each node. Recommendations are the top $k$ closest embeddings among non-connected nodes according to dot product similarity. We then measure the proportion of recommendations belonging to a specific population (e.g., number of edges recommending men and women). A high score (in magnitude) suggests that one sub-population is recommended disproportionately more. As an example, consider an attribute $S$ with values $z\in\{g_1,g_2\}$ and the disconnected nodes displayed in Figure \ref{fig:ex2_graph_emb}, each annotated with their attribute value. For $k=2$, the top $2$ most proximal embeddings to the embedding of node A are the ones associated to nodes B and C. In this scenario, both groups (given by the nodes for which $z=g_1$ and $z=g_2$ respectively) are equally represented in the recommendations, which is intuitively what we would identify with being (group) fair. Note that if $k=1$ this situation cannot be achieved.
\begin{figure}
\centering
\includegraphics[scale=0.33]{figures/ex2_graph_emb.png}
\caption{Example graph and corresponding embedding for $\text{score}_2$.}
\label{fig:ex2_graph_emb}
\end{figure}
Formally, given a sensitive attribute $S$ with value $z$ and denoting the set of ``recommended" nodes\footnote{When computing $\rho_k(u)$, a criteria is needed for breaking ties in the ranking, for instance one can use smallest node id goes first.} by $\rho_k(u)$, we can restrict this set based on the attribute value as follows:
\begin{align*}
\rho_{k,z}(u) &= \{v \;:\; v\in\rho_k(u) \wedge \texttt{attr}_S(v)=z\}.
\end{align*}
Next, defining $Z^S$ to be the set of possible values for attribute $S$ we can measure the fraction of recommended nodes with attribute value $z$ as
\begin{equation}
\texttt{share}(u,k,z) = \frac{|\rho_{k,z}(u)|}{|\rho_k(u)|} .
\end{equation}
In order to compute how much it deviates from the \textit{equal representation} scenario, we can use the following expression
\begin{equation}
\texttt{score}_2(u,k,z) = \frac{1}{|Z^S|} - \texttt{share}(u,k,z).
\end{equation}
In the example (Figure \ref{fig:ex2_graph_emb}), for $k=2$ we get the ordered lists $\rho_{2}(A)=\langle B, C \rangle$, $\rho_{2,g_1}(A)=\langle B\rangle$, $\rho_{2,g_2}(A)=\langle C\rangle$ and since $Z^S = \{g_1,g_2\}$, we have $\texttt{score}_2(A,2,g_1) = \texttt{score}_2(A,2,g_2) = \frac{1}{2}-\frac{1}{2}=0$, which corresponds to the \textit{equal representation} scenario. On the contrary, as noted before, if we take $k=1$; $\rho_{2}(A)=\langle B \rangle$, $\rho_{2,g_1}(A)=\langle B\rangle$ and $\texttt{score}_2(A,1,g_1) = \frac{1}{2}-1=-\frac{1}{2}$. Symmetrically, we get $\rho_{2,g_2}(A)=\langle \rangle$ and $\texttt{score}_2(A,1,g_2) = \frac{1}{2}-0=\frac{1}{2}$. Both represent (inevitably) \textit{unbalanced} recommendations among the 2 groups.
\subsection{Data} \label{sec:data}
We use graph embeddings for several commonly used graphs in the graph machine learning community. As previously introduced, graph embedding algorithms represent each node in a graph as a $d$-dimensional vector. So given a graph with $n$ nodes, a graph embedding algorithm outputs an $\mathbb{R}^{n\times d}$ matrix where rows correspond to nodes and columns to graph embedding dimensions.
\input{tex_input/graph-datasets}
The graphs we used are shown in Table \ref{tab:snap_datasets}. All of these graphs are real-world graphs from multiple domains. As group fairness definitions require sensitive-attribute node labels, we retrieved the gender labels that accompany the Facebook network. For anonymity the gender labels are ``0" and ``1". Four of the $4,039$ nodes did not have labels, and we chose to impute a value of ``0" for these 4 missing labels.
The graph embedding algorithms we use are SVD, HOPE \cite{HOPE}, Laplacian Eigenmaps \cite{lapeigenmap}, Node2Vec \cite{node2vec}, SDNE \cite{SDNE}, and Hyperbolic GCN (HGCN) \cite{HGCN}. These algorithms span the multiple classes of graph embedding algorithms: matrix factorization, random walk, and deep learning. We ran these embedding algorithms on the graphs listed in Table \ref{tab:snap_datasets} and the embeddings are stored in the following repository: \url{https://github.com/dliu18/embedding_repo}.
\section{Task Analysis for \textsc{BiaScope}\xspace}
\subsection{Expert Interviews}
In order to understand the key challenges of thoroughly investigating bias in graph embeddings we established a collaboration with Professor Tina Eliassi-Rad and Dr. Brennan Klein. Eliassi-Rad is a Professor of Computer Science and a core member of Northeastern's Network Science Institute and the Institute for Experiential AI at Northeastern University; she has led research in both of the domains of interest and is familiar with many of the real-world applications. Dr. Klein holds a Ph.D. from Northeastern's Network Science Institute; he is a network visualization expert and active network science researcher.
We conducted an initial interview process that led to our task analysis presented in Section \ref{subsec:task_anal}. We divided our interview into three sections: questions related to 1) graph visualization 2) algorithmic fairness and 3) real-world applications of graph embeddings. For each section we prepared a few high-level questions to understand current difficulties in the respective domains. Our interview yielded three main takeaways that inspired our task analysis and design. First, regarding visualizing networks, our domain expert collaborators emphasized the importance of having different visualization encodings for networks of different scales. More specifically, in order to fully characterize bias of graph embeddings it is imperative to allow for both global (whole-network) as well as local (communities, single nodes) views. Second, as most networks are very large to visualize it is important to have summary network statistics to complement the visualization. Additionally, certain graph statistics reflect network properties (e.g., connectivity, degree distribution) that influence how bias in the network is incorporated more by certain graph embedding algorithms compared to others. Third, for algorithmic fairness tools, Professor Eliassi-Rad highlighted the Aequitas system \cite{aequitas} because the interface guides the user through the complicated decision process of choosing fairness definitions and configurations. The takeaway is that a thorough analysis of bias should take into account multiple fairness definitions and configurations and the user should be guided in the process of multifaceted bias analysis. Finally, Dr. Klein connected us to several network visualization libraries \cite{aslak2019netwulf, schwab2021ssvg, wapman2019webweb} that are commonly used by network science researchers for creating customs visualizations, as part of their research process. Testing these recommendations allowed us to understand the domain conventions regarding network visualization and incorporate the most ubiquitous ones in our design.
\subsection{Task Analysis}\label{subsec:task_anal}
\input{tex_input/task-table-short}
From the interaction with our domain expert collaborators we identified the key components of a typical unfairness diagnosis approach for graph embeddings. In particular, we first extracted and ranked the high level domain tasks mentioned by our interviewees within the unfairness diagnosis procedure. We then constructed concrete examples for each activity. This analysis led us to select the most significant domain tasks our tool is designed to support. Subsequently, we mapped the domain tasks to abstract tasks that can be addressed by our visualization design.
The \textit{task analysis} we conducted is outlined in Table \ref{tab:tasks}. We refer to the identified domain tasks by title and augment each domain task with informative descriptions, listed under the ``Example and Description'' column of Table \ref{tab:tasks}. For the abstract task extraction, we used Munzer's Actions Taxonomy \cite{munzner2014visualization} for the High and Mid levels, and the Analytic Task Taxonomy proposed by Amar et al. \cite{amar2005low} for the Low level. We augmented the abstract task analysis with graph-specific tasks from the task taxonomy proposed by Lee et al. \cite{lee2006task}.
We designed \textsc{BiaScope}\xspace to support three main tasks:
\begin{itemize}
\item[\textbf{T1}] Obtain a comprehensive \textit{Statistical Summary} of a network chosen by the user. The summary conveys structural properties possibly related to bias, i.e., degree distribution, edge density, etc.
\item[\textbf{T2}] \textit{Compare} two embeddings of the network w.r.t. a fairness score. The two embeddings are computed using two different algorithms. The fairness score is chosen and configured by the user.
\item[\textbf{T3}] \textit{Diagnose} the observed unfairness of an embedding. This task allows the user to transition from a global to a local perspective, by focusing on the subgraph and embedding subspace that together determine the unfairness score of a user-selected node.
\end{itemize}
The outlined tasks facilitate a typical unfairness diagnosis workflow for graph embeddings. The workflow consists of an \textit{overview} step and a \textit{drill-down} step:
\begin{enumerate}
\item Overview: Comparison between two embedding algorithms (\textbf{T2}), augmented by relevant graph statistics (\textbf{T1}). In this step, the user examines whether the embedding of interest displays signs of unfairness, compared to other embeddings of the same network.
\item Drill-down/Diagnosis: Identification of the unfairness source(s) for the selected embedding (\textbf{T3}). In this step, the user discovers the determining factors as well as the nature of the observed bias, by focusing on certain nodes or communities that are embedded most unfairly.
\end{enumerate}
\section{Design of \textsc{BiaScope}\xspace} \label{sec:design}
This section outlines and justifies the selected visual encoding and interactions implemented in \textsc{BiaScope}\xspace. The selection was performed by analyzing a series of independently drawn sketches that were created to satisfy the tasks identified from our Task Analysis (Section \ref{subsec:task_anal}). In what follows, a high level description of the different encodings is given. Specific details related to the particular network being used in the examples is deferred to Section \ref{sec:usage_scenario}.
We satisfy the statistical summary task (\textbf{T1}) with the ``Statistical Summary of the Network''\footnote{We also refer to this component as the \textit{Statistical Summary View}.} component of the \textit{Overview}, shown in Figures \ref{fig:teaser}A and \ref{fig:statistical-summary}. The table encoding containing the summary provides key graph metrics to help the user understand the type of network being studied. For the Degree Distribution the bar chart idiom was selected, since it makes use of the most effective magnitude channel to encode ordered attributes, according to the effectiveness ranking for visual channels presented in \cite{munzner2014visualization}. The ranking was compiled by Munzner based on previous work performed by the Visualization community such as \cite{cleveland_mcgill_84a, cleveland_93a, mackinlay_86, Ware12, heer_bostock_10} .
Next, we achieve the comparison task (\textbf{T2}) by providing side-by-side views in the \textit{Overview}, whose main visual components\footnote{We also refer to this component as the \textit{Comparison View}.} are shown in Figures \ref{fig:statistical-summary} and \ref{fig:inform-diagnose}. In particular, the same network is displayed on both sides of Figure \ref{fig:inform-diagnose} using the spring layout \cite{fruchterman1991graph}, while the node colors depend on the fairness of the respective embedding algorithm. A sequential color scale is used, since the attribute being encoded is quantitative. We associate darker colors with higher unfairness scores due to their negative nature \cite{bartram2017affective}.
Regarding interactivity, the node id and its fairness score are displayed upon mouse hover (Figure \ref{fig:mouse_hover}) for the user to be able to better asses the difference between two nodes of the network. Zooming is supported to allow for better analysis capabilities of specific network communities (Figure \ref{fig:mouse_hover}).
\begin{figure*}[t]
\centering
\includegraphics[scale=0.4]{figures/hover_merged.png}
\caption{Node attributes (ID and associated fairness score) are displayed on mouse hover. Zooming functionality is also supported to inspect and compare the scores of specific parts of the network.}
\label{fig:mouse_hover}
\end{figure*}
Due to the large scale nature of our data, following \cite{munzner2014visualization}, we have implemented visual feedback in the form of a loading sign while the different visualizations are being loaded into the webpage. Further details regarding data management and implemented preprocessing steps are provided in Section \ref{sec:preprocessing}.
Finally, we address the unfairness diagnostics task (\textbf{T3}) with the \textit{Diagnose View}. This view lists the nodes together with their unfairness scores, allowing the user to select a focal node and uncover the cause(s) of bias. Specifically, the main components of the view are: (1) an interactive table containing the node ids along with their unfairness scores, (2) the projected embeddings affecting the score of the focal node, and (3) the corresponding subgraph topology. The user can sort the list of nodes based on their ids or scores by clicking on the arrows next to the column names of the Table (left of Figure \ref{fig:diagnose_view_1}). A search box is included for each attribute of the table to speedup the lookup.
As previously discussed, we consider two fairness notions (individual and group) with different characteristics, which are reflected by our design. For the case of individual fairness, the score of a node is determined by its local neighborhood. Thus, the local neighborhood topology along with the corresponding projected embeddings are displayed in the view. On the other hand, group fairness determines the score of a node by the most proximal embeddings. Therefore, these projected embeddings are shown along with the corresponding subgraph. In both cases, upon selection of the focal node, the \textit{Diagnose View} shows side-by-side the subgraph and projected embeddings relevant to the chosen fairness notion. Figure \ref{fig:diagnose_view_1} displays the \textit{Diagnose View} for the case of individual fairness. The projected embeddings\footnote{The projection of the embeddings is performed using PCA \cite{PCA_FRSLIIIOL}. In what follows, for simplicity, we do not account for the projection when referring to the projected embeddings.} are encoded using a scatterplot (Figure \ref{fig:diagnose_view_1}, rightmost chart), noting that by \cite{munzner2014visualization} it maximizes effectiveness.
The objective of the \textit{Diagnose View} is to provide insight into the fairness score computation for the selected focal node and uncover causes of unfairness in the network. For this purpose, a red color pop-out effect is used to outline the focal node, together with an increase in its size. This improves the efficiency of its lookup \cite{munzner2014visualization}, which is an essential component in understanding the score computation. Brushing and linking is supported between the network and the embedding space to analyze how different nodes contribute to the fairness score of the focal node. Specifically, when brushing over a set of elements, the corresponding elements in the other view are highlighted. This is shown in Figure \ref{fig:diagnose_view_1}, where a subset of nodes in the local topology of node with id $865$ is being selected with the brush. When hovering over a node or its embedding, the corresponding node id and fairness score are displayed in the tooltip, together with the number of hops to the focal node for the individual fairness score (Figure \ref{fig:diagnose_view_1}) or the node label for group fairness. The distance from the focal node (individual fairness) and the gender (group fairness) are encoded using the color channel with an ordinal and categorical color scale, respectively.
Furthermore, as a response to expert feedback, we improved our design with a context legend that conveys the scale of the distances observed in the scatterplot (Figure \ref{fig:diagnose_view_2}). More specifically, we made use of the \textit{focus+context} approach \cite{cockburn_etal_08}, and augmented the \textit{Diagnose View} (i.e., \textit{focus}) with a global scatterplot (i.e., \textit{context}) displaying all the embeddings (top section of Figure \ref{fig:diagnose_view_2}), where the highlighted area corresponds to the points of the scatterplot which are the ones that affect the focal node (Figure \ref{fig:inform-diagnose} and \ref{fig:fairwalk-diagnose}).
\begin{figure*}
\centering
\includegraphics[scale=0.39]{figures/diagnose_view_new_1.png}
\caption{Main visual components of the \textit{Diagnose View} for individual fairness. From left to right: Interactive table listing node ids and their unfairness scores, local neighborhood of selected node in the network and corresponding projected embeddings.}
\label{fig:diagnose_view_1}
\end{figure*}
\begin{figure}
\centering
\includegraphics[scale=0.31]{figures/diagnose_view_2.png}
\caption{The context legend displayed in the \textit{Diagnose View} (top plot) conveys the scale of the distances observed in the scatterplot (bottom plot), where the embeddings that affect the score are plotted.}
\label{fig:diagnose_view_2}
\end{figure}
\section{Evaluation}
\input{tex_input/usage-scenario}
\input{tex_input/expert-review}
\section{Discussion}
In this section, we discuss several points that were observed during the development process and posterior usage of \textsc{BiaScope}\xspace. These consist of challenges regarding the size of the data, both in terms of visibility and the efficiency of the system (Section \ref{sec:preprocessing}), as well as limitations and future lines of work, included in Section \ref{sec:limitations_futurework}.
\input{tex_input/data-preprocessing}
\subsection{Limitations and Future Work}\label{sec:limitations_futurework}
In the present design study we handled large benchmark networks, as outlined in Section \ref{sec:preprocessing}, therefore we had to overcome responsiveness issues. As previously discussed, we used a simple heuristic to minimize the number of edges in our visualization by filtering out non-salient edges. In future work, we plan to explore more sophisticated heuristics as well as different methods for graph abstractions e.g., sub-sampling methods that preserve node communities. Additionally, we plan to extend our scope to a larger suite of benchmark networks which are even larger (in terms of node count) or have higher density. A parallel direction of future work includes developing advanced interactivity and linking in the overview, e.g. simultaneous zooming, coupled hover events, etc. We expect this extension to generate new performance challenges since the interaction should be particularly responsive in order to increase effectiveness.
Furthermore, following the expert feedback, we plan to make certain additions in order for the tool to better support the needs of its target users. These include the extension of our benchmark suite with other networks popular within the graph mining community (e.g., Books \footnote{\url{http://www-personal.umich.edu/~mejn/netdata/}}, NS \cite{netscience-data}) and a list of different options for the embedding projections beside PCA (e.g., UMAP \cite{becht2019dimensionality}). Finally, as mentioned in Section \ref{sec:expert_review}, we plan to develop a user friendly onboarding process for users who want to evaluate their own embedding algorithms using our tool. The onboarding will include preprocessing of the user data, which will utilize server-side computation, in order to preserve sufficient responsiveness at use-time.
\section{Conclusion}
Motivated by the recent efforts made by the algorithmic fairness community, we have build \textsc{BiaScope}\xspace, an interactive web-based visualization tool that supports end-to-end \textit{visual unfairness diagnosis} for graph embeddings. Our design is the result of an iterative collaborative process with experts in the fields of fair machine learning and graph mining. To inform our design, we conducted a thorough task analysis that consists of key tasks that support an unfairness diagnosis workflow for graph embeddings.
Applying our visual unfairness diagnosis workflow to a benchmark network, we show the effectiveness our tool has in visually revealing the unfairness incorporated into different widely used graph embedding algorithms. Our work is part of a greater effort to allow both researchers and practitioners to build or use machine learning models while simultaneously being mindful of bias and algorithmic fairness, without the need of expertise in the field. Our tool, and others in the same spirit, could be part of a standard development workflow of a machine learning researcher or engineer, as it is designed to provide greater transparency into how current graph embedding algorithms and fairness notions interconnect in practice.
\section {Acknowledgement}
The authors would like to thank Zohair Shafi and Ayan Chatterjee for participating in the expert feedback sessions and providing insightful comments about the system.
\newpage
\bibliographystyle{abbrv-doi}
\subsection{Usability Testing}\label{sec:usability_testing}
During the development of our tool we performed an informal usability testing session with 15 Computer Science graduate student as our participants. The goal of this study was to qualitatively evaluate the effectiveness of our visual encodings for the target tasks as well as the usability of the interactivity elements.
The study confirmed that there exist certain technical barriers for non-specialist users, which is understandable since our design was targeted to experts. However, we still took action to clarify certain components, for example we implemented user-friendly tooltips for explanations and definitions of technical terms. Additionally, the study was particularly revealing with respect to color encodings and color perception. More specifically, it indicted that the users tend to associate dark colors with more unfairness and red colors are semantically associated with pathological attributes, which in our case correspond to biased embeddings. Finally, the study reassured us that the implemented interactivity significantly empowers the user to investigate causes of bias and successfully complete unfairness diagnostics.
\section{Data Exploration}\label{appendix:data}
\section{Detailed Task Analysis}
\input{tex_input/task-table}
In this section we provide the full task analysis we conducted after interviewing our expert. From the set of tasks analysed and categorized in Table \ref{tab:tasks_full} we identified the most important tasks our tool supports.
\section{Fairness definitions and examples}\label{sec:fairness_defs}
\subsection{Score 1: InFoRM (Individual Fairness)}
Let $Y$ be the embedding matrix, where $Y[u]$ is the embedding of node $u$. Let $S$ be the node proximity matrix, where $S[u,v]$ is the proximity of nodes $u$ and $v$. The most obvious notion of node proximity is the adjacency matrix $S=A$. There are other options such as a random walk matrix. We can then define the first individual fairness score based on \cite{InFoRM} as follows:
\begin{align*}
score_1(u,k=1) &= \sum_{v \neq u} \lVert Y[u] - Y[v]\rVert_2^2 \cdot S[u,v] \\
&= \sum_{v\in\mathcal{N}(u,1)} \lVert Y[u] - Y[v]\rVert_2^2
\end{align*}
where $\mathcal{N}(u,k)$ is the $k$-hop neighborhood of $u$, given by the nodes reachable from $u$ in at most $k$ steps.
We generalize this notion for different values of $k$ as follows:
\begin{align*}
score_1(u,k) &= \sum_{v\in\mathcal{N}(u,k)} \lVert Y[u] - Y[v]\rVert_2^2
\end{align*}
\subsection{Fairwalk (Group Fairness)}
For the group fairness notions introduced in \cite{ijcai2019-456}, given a node $u$, we need to define a group of ``recommended" nodes, denoted $\rho(u)$. Among the nodes not connected to $u$, we choose to ``recommend" the top-$k$ most proximal ones in the embedding, using dot product similarity.
Additionally, given a sensitive attribute $S$ and a value $z$, we can restrict the ``recommended" node set based on the attribute as follows:
\begin{align*}
\rho_z(u) &= \{v \;:\; v\in\rho(u) \wedge \texttt{attr}_S(v)=z\}
\end{align*}
For example, we can have $S=\text{gender}$ and $z=\text{female}$. When computing $\rho(u)$, a criteria is needed for breaking ties in the ranking, for instance one can take smallest id goes first. Using this notions, two fairness notions are defined in \cite{ijcai2019-456}, which are described next.
\subsubsection{Network Level}
Given a sensitive attribute $S$, we can consider different communities (user groups) in the graph, according to the values of $S$. Concretely, given $i,j$ two possible values for attribute $S$, we define the group $G_{i,j}^S$ as follows:
\[
G_{i,j}^S = \{(u,v)\ : u\neq v \wedge \texttt{attr}_S(u)=i \wedge \texttt{attr}_S(u)=j\}
\]
We denote the set of all such groups as $\mathcal{G}^S$. Then, based on these groups, a bias score can be derived as follows:
\begin{equation}
\texttt{bias}(\mathcal{G}^S) = Var(\{P(G_{i,j}^S): G_{i,j} \in \mathcal{G}^S\})
\end{equation}
, where
\begin{equation}
P(G_{i,j}^S) = \frac{|\{(u,v): v \in \rho(u) \wedge (u,v) \in G_{i,j}^S\} |}{|G_{i,j}^S|} .
\end{equation}
\noindent
Example:
Let $k=1$, $S=\text{race}$ with possible attribute values $\{\text{w}, \text{b}\}$. Consider the network and associated embedding shown in Figure \ref{fig:ex_1}.
\begin{figure}
\centering
\includegraphics[scale=0.3]{figures/ex1.png}
\caption{Example network and embedding. Race feature of the nodes is indicated next to them in the network.}
\label{fig:ex_1}
\end{figure}
Then, we have:
\begin{align*}
\mathcal{G}^S &= \{G^S_{w,b},G^S_{w,w},G^S_{b,b}\}\\
G^S_{w,b} &= \{(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)\}\\
G^S_{w,w} &= \{(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)\}\\
G^S_{b,b} &= \{(4,4),(4,5),(5,5)\}
\end{align*}
Next, consider node $u=2$, since $k=1$,
\[
\rho(2) = \{4\}
\]
Thus, we get
\begin{align*}
P(G^S_{w,b}) &= \frac{1}{6}\\
P(G^S_{w,w}) &= \frac{0}{6}\\
P(G^S_{b,b}) &= \frac{0}{3}
\end{align*}
Therefore,
\[
\texttt{bias}(\mathcal{G}^S) = Var(\{1/6,0,0\}) = \frac{1}{162} \sim 0.006
\]
\subsubsection{Score 2: User Level}
Let $Z^S$ be the set of all possible values of attribute $S$. We calculate a bias score for an attribute value $z$ and a node $u$.
\begin{equation}
\texttt{bias}(z) = \frac{1}{|Z^S|} - \frac{\sum_{u \in V} \texttt{z-share}(u)}{|V|}
\end{equation}
, where
\begin{equation}
\texttt{z-share}(u) = \frac{|\rho_z(u)|}{|\rho(u)|} .
\end{equation}
Here, we define
\begin{equation}
\texttt{score}_2(z,u) = \frac{1}{|Z^S|} - \texttt{z-share}(u)
\end{equation}
\noindent
Example:
Again consider the network and embedding from Figure \ref{fig:ex_1} and attribute $S=\text{race}$ with $Z^S=\{w,b\}$, $z=b$ and $k=1$.
\begin{align*}
\rho(1)&=\{2\}\\
\rho(2)&=\{4\}\\
\rho(3)&=\{4\}\\
\rho(4)&=\{2\}\\
\rho(5)&=\{2\}
\end{align*}
\begin{align*}
\texttt{z-share}(1,b) &= \frac{0}{1}=0\\
\texttt{z-share}(2,b) &= \frac{1}{1}=1\\
\texttt{z-share}(3,b) &= \frac{1}{1}=1\\
\texttt{z-share}(4,b) &= \frac{0}{1}=0\\
\texttt{z-share}(5,b) &= \frac{0}{1}=0\\
\end{align*}
Then,
\begin{align*}
\texttt{score}_2(z=b,u=1) &= \frac{1}{2}-\frac{0}{1}=\frac{1}{2}-0=\frac{1}{2}\\
\texttt{score}_2(z=b,u=2) &= \frac{1}{2}-\frac{1}{1}=\frac{1}{2}-1=-\frac{1}{2}\\
\texttt{score}_2(z=b,u=3) &= \frac{1}{2}-\frac{1}{1}=\frac{1}{2}-1=-\frac{1}{2}\\
\texttt{score}_2(z=b,u=4) &= \frac{1}{2}-\frac{0}{1}=\frac{1}{2}-0=\frac{1}{2}\\
\texttt{score}_2(z=b,u=5) &= \frac{1}{2}-\frac{0}{1}=\frac{1}{2}-0=\frac{1}{2}
\end{align*}
and also,
\[
\texttt{bias}(b) = \frac{1}{2} - \frac{2}{5} = \frac{1}{10}
\]
\subsection{Data Management}\label{sec:preprocessing}
Due to the size of the networks that were considered, several performance challenges needed to be addressed in the development process. In this section, we provide an overview of how these were approached.
As previously outlined, in order to improve network visibility, we filter out visually non-salient edges. Specifically, we only display edges in the top 10 percent of length. Filtering allows us to save on browser memory, as the number of edges far exceeds the number of nodes. At the same time, by inspecting the networks we found that the bottom 90 percent of shortest edges were often not visible, so filtering offers large performance increases while preserving important visual information, e.g., intra-community connectivity.
In terms of preprocessing, both the projection of the embeddings, as well as group and individual fairness scores were calculated offline in order to obtain an additional performance speedup. For the later, this involves generating for every network the corresponding score for each node id and possible configuration. Configuration options consist of the number of hops from the focal node for the individual fairness notion, while for the group fairness, the value of $k$ as well as the chosen sensitive attribute (in our case gender) and value.
Finally, the pre-computed network information is accessed only once when the webpage is initially loaded and stored in memory, in order to avoid redundant server access.
\subsection{Expert Feedback}\label{sec:expert_review}
In order to evaluate our tool, we conducted feedback interviews with our two collaborators and two additional experts, who are active researchers interested in graph embeddings, also affiliated with Northeastern's Network Science Institute. We conducted four individual interviews, during which each participant was able to interact with the tool through the webpage on their own system. We simultaneously observed their interaction with the tool, which was captured on video for further analysis. For all interviews we followed the same steps: First, we asked for the participant's consent to video recording. Second, we briefly discussed the tasks our tool is designed to support. Third, we provided an \textit{interactive walk-through} of the tool by asking the participant to locate certain views and perform concrete tasks. The goal of this step was to help the participants familiarize themselves with the UI. Next, we encouraged the participant to follow a \textit{usage scenario} similar to the one presented in Section \ref{sec:usage_scenario}. During this step we provided loose guidance by setting open-ended goals and we encouraged the participant to freely accomplish them using the tool. Lastly, we collected verbal feedback using the following questions:
\begin{itemize}
\item[\textbf{Q1}] In your view, how well are the domain tasks supported?
\item[\textbf{Q2}] How would you perform these tasks without the tool? Do you see value in our tool?
\item[\textbf{Q3}] How would you use our tool in your research/development process?
\item[\textbf{Q4}] What limitations do you identify?
\end{itemize}
Following the interview sessions, we collected additional quantitative feedback from the experts using a System Usability Scale (SUS) \cite{sus_1}. In what follows, we summarize the key takeaways from the feedback.
\paragraph{Effectiveness and Usability.}
During the interactive walk-through, we observed that all the participants were able to quickly perform the tasks with little to no assistance. Regarding explicit verbal feedback, all experts agreed that \textsc{BiaScope}\xspace effectively supports the described domain tasks (\textbf{Q1}). Additionally, all participants described, while three explicitly said, that they would follow the same process in order to thoroughly analyse unfairness of a graph embedding (\textbf{Q2}). Most participants pointed out that without \textsc{BiaScope}\xspace this analysis would be tedious and prone to errors. One participant mentioned that \textsc{BiaScope}\xspace would save them time and a different one said: ``something that goes a long way in terms of what you have done is I do not want to implement any of these things [by myself]. ... at the very basic level you have just reallocated a lot of work... [Since there are no other tools like this one on the market], this is immediately useful in a very time saving way.''.
Regarding the potential role of \textsc{BiaScope}\xspace within our participants' research process (\textbf{Q3}), the feedback validated the tool's effectiveness while additionally providing new perspectives. Two of the experts agreed that the tool would facilitate the evaluation of their graph embedding algorithms with respect to fairness. More specifically, they mentioned the following: ``For people who aren't really fairness focused, this would [enable] a nice quick sanity check.'' and ``I would use it as a visual tool to interrogate the data, [which is] a very important thing to do, as opposed to just feeding it into your machine learning system and then hoping for the best.'' The other two experts pointed out they could further utilize our tool to facilitate their work in explainable machine learning or for pedagogical purposes: ``... I would definitely use this tool for understanding explainability and interpretability of embeddings ... my major use would be like explaining some of the down stream task results or predictions'', ``It is a very good teaching tool, in general I think web based [tools] that let you interact with networks, change the game when it comes to learning about networks.''.
\paragraph{Limitations.}
The experts agreed on the importance of an ``import" feature, which would allow the user to upload and diagnose their own embedding algorithm. One of the experts additionally suggested an ``export" functionality. The implementation of an "import" feature is included as future work in Section \ref{sec:limitations_futurework}. Moreover, some experts pointed out that the Diagnose View should include an indication of the scale regarding the projected embeddings, in order to clearly convey the magnitude of distances observed in the local embedding subspace. We addressed this limitation with the addition of the context legend, as discussed in Section \ref{sec:design}. One of the experts expressed the interesting idea to augment our network options with certain simple synthetic networks representing typical graph topologies, e.g., star, which could enlighten the user with respect to the nature of different fairness notions. Finally, adding on-demand information about the selected fairness notion definition was a frequent suggestion among participants.
\paragraph{System Usability Scale.}
We requested our experts to fill a System Usability Scale (SUS) \cite{sus_1, brooke2013sus}, a standard simple ten-item scale giving a global view of subjective assessments of the usability of a system. Based on the responses, we derive the corresponding SUS scores, which have a range of 0 to 100. Regarding scores assessment, \cite{bangor_etal} proposed an adjective value scale for SUS scores. According to this scale, scores in the range 72.5-84.9 are associated with a good usability and above 85 represent an excellent one. Based on the collected SUS scores, we obtained a mean value of $83.125$. This implies, per the aforementioned scale, that \textsc{BiaScope}\xspace's usability was perceived as good, close to excellent by the interviewed experts.
\subsection{Usage Scenario}\label{sec:usage_scenario}
To illustrate the design in action, we will walk through a case study using the Facebook network, which represents the social network friendships of over four thousand users. We begin with an overview of the network via the statistical summary shown in Figure \ref{fig:statistical-summary}. The table on the left informs the user that the Facebook graph is sparse and highly clustered. Further, the degree distribution shows that almost all users in the network have fewer than 200 friends.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/SS.png}
\caption{The statistical summary for the Facebook network helps the user to characterize the data before analyzing its fairness scores. The table on the left states that the network is sparse and highly clustered while the degree distribution shows that almost all users in the network have fewer than 200 friends.}
\label{fig:statistical-summary}
\end{figure}
Proceeding further into the \textit{Overview}, the user can browse a side-by-side comparison of the fairness scores for two sets of embeddings. Figure \ref{fig:teaser}B shows the scores for Node2Vec on the left and then HGCN on the right for the Facebook network. This visual provides two takeaways. First, the Node2Vec embeddings are overall more fair than the HGCN embeddings, as indicated by the lighter coloring. Second, the unfairness for HGCN is concentrated in select communities. Without the visualization, a purely statistical analysis would resort to aggregate values which do not account for heterogeneity.
Moving into the drill down portion of the tasks, we can now use the \textit{Diagnose View} to better understand why certain nodes are scored as unfair. For instance, Figure \ref{fig:inform-diagnose} shows the diagnostic results for the Facebook Node2Vec embeddings using the individual fairness notion. The local neighborhood (ego network) shows that the focal node, in red, is part of two communities of friends. Further, the projected embeddings show that the two communities of friends are embedded far from each other. Hence, the focal node is scored as unfair because it is far from its neighbor communities in embedding space. This mapping between neighbors and embeddings can be verified with brushing and linking (Figure \ref{fig:diagnose_view_1}).
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/inform-diagnose-new.png}
\caption{Moving into the drill-down tasks, the above figure provides the diagnostics results for the Facebook network's Node2Vec embeddings where fairness is defined with the InFoRM notion of individual fairness. The ego network on the left shows the target node is part of two communities, as linked on the right hand side, these communities are themselves embedded far apart. Thus, the diagnostics results show that under InFoRM bridge nodes between communities may be embedded unfairly. }
\label{fig:inform-diagnose}
\end{figure}
Finally, the user can also drill-down and diagnose the group fairness scores as well. Figure \ref{fig:fairwalk-diagnose} shows the Facebook Node2Vec embeddings evaluated based on the group fairness notion. Now, the focal node together with the induced subgraph (top-$k$ recommended nodes) is plotted in the rightmost chart. The subgraph shows that the target node's recommendations are mostly of gender 0, encoded in yellow, which is a form of homophily. Further, these nodes are embedded close to the target node in the embedding space. This is a form of group unfairness because if we train a model to recommend friends based on these embeddings, the model would perpetuate homophily for this node and recommend nodes of the same gender.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/fairwalk-diagnose-new.png}
\caption{The diagnostics results for group fairness show that the given node from the Facebook network is unfair because many of its recommended nodes are also of the same gender value, encoded in yellow. The focal node and the induced subgraph given by the top-$k$ recommended nodes are plotted, together with the corresponding embeddings. The design layout is similar to the individual fairness diagnostics but the node colors now represent the sensitive attribute, which is the most important attribute for group fairness.}
\label{fig:fairwalk-diagnose}
\end{figure}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,078 |
\section{Introduction}
Two-dimensional conformal field theories play an important role in both theoretical physics and mathematics. Their infinite-dimensional symmetries restrict the theories to a large extent, but leave enough room for interesting structures and intriguing dualities.
When the basic Virasoro symmetry is complemented with
higher spin generators up to spin $N$, it is known as $W_N$-algebra symmetry, see \cite{Bouwknegt:1992wg} for a nice review.
Recently, theories with $W$-symmetry have appeared in several contexts.
A prominent example is the AdS/CFT correspondence.
It was shown in \cite{Henneaux:2010xg,Campoleoni:2010zq} that the asymptotic symmetry of higher spin gravity theory on AdS$_3$ can be
identified with $W$-symmetry. Based on this fact, the authors of \cite{Gaberdiel:2010pz} proposed
that the minimal model with $W_N$-algebra symmetry in a large $N$ limit is dual to the higher spin gravity
by Prokushkin and Vasiliev \cite{Prokushkin:1998bq}.
The higher spin $W$-symmetry plays an important role in the evidence of this
AdS/CFT correspondence.
Supersymmetric versions of the higher spin AdS/CFT correspondence were also proposed, e.g., in \cite{Creutzig:2011fe,Candu:2012jq,Henneaux:2012ny,Creutzig:2012ar,Beccaria:2013wqa,Gaberdiel:2013vva,Creutzig:2013tja,Creutzig:2014ula,Gaberdiel:2014cha}, where super $W$-algebras appear as symmetry algebras.
The theories with $W_N$-symmetry also appear as effective descriptions of subsectors of
four-dimensional SU$(N)$ gauge theories \cite{Alday:2009aq,Wyllard:2009hg}.
In this paper, we study some aspects of two-dimensional conformal field theories with $W$-symmetry.
More concretely, we relate correlation functions of
Wess-Zumino-Novikov-Witten (WZNW) models on the groups SL$(3)$ and SL$(2N)$ to correlators of theories with $W$-symmetry, and similarly we consider the correspondence between WZNW models on the supergroups SL$(N|M)$ and theories with super $W$-algebra symmetry. Already in \cite{Ribault:2005wp} Ribault and Teschner showed explicitly that there is a relation between $N$-point amplitudes of primary operators on spheres in the
$H_3^+$ WZNW model, which describes strings in Euclidean AdS$_3$, and $(2N-2)$-point spherical amplitudes in Liouville field theory. In \cite{HS} this relation was re-derived using an
intuitive path integral method and extended the correspondence to amplitudes on Riemann surfaces of arbitrary genus.
Using this method further generalizations were possible:
The relation between correlation functions of the OSP$(p|2)$ WZNW model and of ${\cal N}=p$
supersymmetric Liouville field theory was obtained in \cite{HS2} for $p=1,2$.
In our previous paper \cite{CHR}, we have extended these relations
to the cases with WZNW models on supergroups whose bosonic subgroup
is of the form SL$(2) \times A$. For example, the PSU$(1,1|2)$ WZNW model is related to the
small ${\cal N}=4$ super Liouville field theory, and the OSP$(p|2)$ WZNW model with $p \geq 3$
is related to a superconformal field theory with SO$(p)$-extended super conformal symmetry
discussed in \cite{K,B}.
Other examples considered in that paper are with the supergroups SL$(2|p)$,
D$(2,1;\alpha)$, OSP$(4|2p)$, F$(4)$ and G$(3)$, see also \cite{KW}. Especially the case of D$(2,1;\alpha)$ relates to the large ${\cal N}=4$ super conformal algebra.
This paper is a continuation of these works.
It is known that quantum $W$-algebras can be obtained by quantum Drinfeld-Sokolov
reduction of affine Lie algebras, which are the symmetries of WZNW models \cite{Bouwknegt:1992wg}.
We think of the relations between correlation functions mentioned above as
correlator versions of Drinfeld-Sokolov reduction.
There are different
quantum $W$-algebras associated to a given simple Lie algebra corresponding to different embeddings of a sl(2) subalgebra into the Lie algebra, see e.g. \cite{deBoer:1993iz}.
The principal embedding leads to the $W_N$-algebra starting from the affine sl$(N)$ algebra; but we are interested in other embeddings.
The simplest non-trivial case appears for sl(3), where the non-principal embedding leads to the Bershadsky-Polyakov algebra
\cite{Polyakov,Bershadsky}. For sl$(N)$, the sl(2) embedding can be determined by the
branching of the fundamental representation $\underline{N}$, which can be expressed by
a partition of $N$ (see, e.g., \cite{Bais:1990bs} in this context). The principal embedding
corresponds to $\underline{N} = \underline{N}$.
For sl(3), the principal embedding $\underline{3} = \underline{3}$ leads to $W_3$ algebra
firstly introduced in \cite{Zamolodchikov:1985wn},
and the Bershadsky-Polyakov algebra
comes from the embedding $\underline{3} = \underline{2} + \underline{1}$.
We also consider the embedding $\underline{N+2} = \underline{2} + N \underline{1}$,
which leads to a generalized Bershadsky-Polyakov algebra also called the quasi-superconformal algebra \cite{Romans}. The algebra is like the Knizhnik-Bershadsky
algebra \cite{K,B}, and we note that they all have bosonic spin-3/2 fields.
Further, the non-trivial embedding given by $\underline{2N} = N\underline{2}$ is also discussed.
Furthermore, supergroup cases are examined.
Once these relations are established, there are many ways to utilize them.
In order to investigate the AdS/CFT correspondence, it is important to analyze
superstrings on AdS spaces, which may be described by supergroup WZNW models.
One of the applications is thus to use the relations for the study of supergroup models.
In fact, some structure constants of OSP$(1|2)$ models are computed in terms of
${\cal N}=1$ super Liouville field theory in \cite{HS2,Creutzig:2010zp}.
Another important use of this type of relation was in a proof of the
Fateev-Zamolodchikov-Zamolodchikov (FZZ) duality \cite{FZZ,HS3,Creutzig:2010bt}.
The FZZ conjecture can be viewed as a T-duality in curved space, and the T-dual of the two-dimensional Euclidean black hole is proposed to be sine-Liouville theory.
The black hole model can be described by the SL(2)/U(1) coset, and the application of the
Ribault-Teschner relation to the SL(2) part is an essential part of the proof in \cite{HS3}.
Our hope is that similar relations naturally lead to generalizations of the FZZ dualities.
We will comment on this in the conclusion.
The organization of this paper is as follows.
In the next section, we derive a relation between correlation functions of the SL$(3)$ WZNW model
and of a theory with Bershadsky-Polyakov symmetry.
In section \ref{QSCA}, we extend the relation to the more general case where the
SL$(N+2|M)$ WZNW model is related to a theory with quasi-superconformal symmetry.
In section \ref{product}, we study the theory obtained by the Drinfeld-Sokolov reduction of the SL$(2N)$
WZNW model with the product embedding $\underline{2N}=N\underline{2}$.
In section \ref{extention}, we make a proposal of how to combine the cases analyzed so far.
Some technical details are collected in appendices. In appendix \ref{conv}, the conventions
for the Lie (super-)algebras are summarized. In appendix \ref{genus}, we extend the
analysis in section \ref{QSCA} to the cases with Riemann surfaces of arbitrary genus.
In appendix \ref{DSreduction}, our correspondence is compared with the known facts
on the Hamiltonian reduction with the product embedding $\underline{2N}=N\underline{2}$.
In appendix \ref{non-hol}, some field redefinitions used in section \ref{product} are derived.
\section{Bershadsky-Polyakov algebra and SL(3)}
We start our discussion with the interesting example of a theory with the Bershadsky-Polyakov algebra as symmetry algebra.
This algebra is a Drinfeld-Sokolov reduction of $\widehat{\text{sl}}(3)$, and
the Bershadsky-Polyakov algebra corresponds to
the non-principal embedding of $\text{sl}(2)$ in $\text{sl}(3)$.
Conventions on $\text{sl}(3)$ and the non-principal embedding are provided in Appendix \ref{subsec:conv}.
Given this embedding, the $\text{sl}(2)$ subalgebra acts on $\text{sl}(3)$. Especially, we can decompose $\text{sl}(3)$ into eigenspaces of the Cartan subalgebra of $\text{sl}(2)$.
This leads to a $\mathbb Z_5$-gradation
\begin{equation}
\text{sl}(3) \ = \ \mathfrak{g}_{-1}\oplus\mathfrak{g}_{-1/2}\oplus\mathfrak{g}_{0}\oplus\mathfrak{g}_{1/2}\oplus\mathfrak{g}_{1}.
\end{equation}
This gradation is the starting point for the construction of the map between correlation functions in the SL(3) WZNW-theory and a theory with the Bershadsky-Polyakov algebra as symmetry algebra.
\subsection{SL(3) WZNW action}
In order to construct the action of the WZNW theory, we start with a SL(3)-valued field $g$. For our purposes it is important to parameterize this field
according to above $\mathbb Z_5$-gradation, namely
\begin{align}
g=g_{-1}\,g_{-\frac{1}{2}}\,g_{0}\, g_{\frac{1}{2}}\,g_{1}=e^{\ga_{-1}}e^{\ga_{-1/2}}e^{\phi_0}e^{\ga_{1/2}}e^{\gamma_1} ~.
\end{align}
This is similar to the case of SL$(2|1)$ studied in \cite{HS2,CHR}, but of course $\gamma_{\pm 1/2}$ are here bosonic fields.
The action of the SL(3) WZNW model at level $k$ is
\begin{align}
S^\text{WZNW} [g] = \frac{k}{4 \pi} \int_{\Sigma} d ^2z
\langle g^{-1} \partial g , g^{-1} \bar \partial g \rangle
+ \frac{k}{2 4 \pi} \int_{B} \langle g^{-1} d g ,
[g^{-1} d g , g^{-1} d g ] \rangle
\end{align}
with $\partial B = \Sigma$.
The invariant bilinear form is of course the Killing form.
The Polyakov-Wiegmann identity
\begin{align}
\label{PWid}
S^\text{WZNW} [ gh ] = S^\text{WZNW} [g] + S^\text{WZNW} [h]
+ \frac{k}{2 \pi} \int d^2 z
\langle g^{-1} \bar \partial g , \partial h h^{-1} \rangle
\end{align}
leads to
\begin{equation}
\begin{split}
S^\text{WZNW} [g]
=& S^\text{WZNW} [g_{0}]
+ \frac{k}{2\pi} \int d^2 z \langle \big(\bar{\partial} \ga_{-1}-\frac{1}{2}[\gamma_{-1/2},\bar{\partial}\ga_{-1/2}]+
\bar{\partial}\gamma_{-1/2}\big), \\
&\qquad\qquad \qquad \qquad \mathrm{Ad}(g_0)\big(\partial \ga_{1}+\frac{1}{2}[\gamma_{1/2},\partial\ga_{1/2}]+\partial\gamma_{1/2}\big) \rangle ~.
\end{split}
\end{equation}
Here, we used that the Killing form respects the $\mathbb Z_5$-gradation. We can now introduce the auxiliary fields $\be_{1},\be_{-1}$ with $\be_{1}\in{g}_{1},\be_{-1}\in{g}_{-1}$ such that the integration over
$\be_{1},\be_{-1}$ reproduces the original action.
Likewise, we also introduce auxiliary variables for the half-integer parts, $\be_{1/2}, \be_{-1/2}$ with $\be_{1/2}\in{g}_{1/2},\be_{-1/2}\in{g}_{-1/2}$. The action now takes the form
\begin{equation}
\begin{split}
S^\text{WZNW} [g]&=S^\text{WZNW}_{\textrm{ren}} [g_{0}] + S_0+S_{\textrm{int}} ~, \\
S_0 &= \frac{k}{2\pi} \int d^2 z
\Big[ \langle \be_{1}, \bar{\partial}\gamma_{-1} \rangle + \langle \be_{1/2}, \bar{\partial}\gamma_{-1/2}\rangle + \langle \be_{-1},
\partial\gamma_{1} \rangle + \langle \be_{-1/2}, \partial\gamma_{1/2} \rangle \Big] ~ , \\
S_{\textrm{int}} &= -\frac{k}{2\pi} \int d^2 z \Big [ \langle \be_{1},\ \mathrm{Ad}(g_0)( \be_{-1}) \rangle \\
&\qquad\qquad\qquad+ \langle \big(\be_{1/2}-\frac{1}{2}[\gamma_{-1/2},\be_{1}]\big), \mathrm{Ad}(g_0)\big( \be_{-1/2}+\frac{1}{2}[\gamma_{1/2},\be_{-1}]\big)\rangle \Big]~,
\end{split}
\end{equation}
where we have used the invariance of the inner product.
The quantum action $S^\text{WZNW}_{\textrm{ren}} (g_{0})$ is then obtained
by taking care of the Jacobian which appears due to the introduction of the new auxiliary fields.
This is as far we can go using just the knowledge of the 5-gradation. We can now insert the SL(3) generators
(see appendix \ref{subsec:conv}), we define
\begin{align}
g=e^{\gamma^3 F_3}e^{\gamma^1 F_1+\gamma^2 F_2}e^{\phi\theta+\phi^\bot\theta^\bot}e^{\bar \gamma^1 E_1+\bar\gamma^2 E_2}e^{\bar\gamma^3 E_3}\, ,
\end{align}
and correspondingly (abusing notation -- on the left hand side the fields are the old Lie algebra valued fields, and the $\be_i$s on the right hand side are the new fields).
\begin{equation}
\begin{split}
\be_{1} &= \be_3 E_3\, ,\qquad\qquad\quad\,\qquad \be_{-1} = \bar\be_3 F_3\, , \\
\be_{1/2} &=\be_1 E_1+\be_2 E_2\, ,\qquad\,\qquad \be_{-1/2} =\bar \be_1 F_1+\bar \be_2 F_2\, .
\end{split}
\end{equation}
The action then takes the form
\begin{equation}
\begin{split}
S^\text{WZNW} [g]&=S^\text{WZNW}_{\textrm{ren}} [g_{0}] + S_0+S_{\textrm{int}} ~, \\
S^\text{WZNW}_{\textrm{ren}} [g_{0} ] &=\frac{1}{4\pi} \int d^2 z \Big[\frac{1}{2}\partial\phi\bar{\partial}\phi+\frac{3}{2}
\partial\phi^\bot\bar{\partial}\phi^\bot-\frac{1}{2}b\sqrt{g}\mathcal{R}\phi \Big] ~, \\
S_0 &= \frac{1}{2\pi} \int d^2 z \Big[ \be_{1} \bar{\partial}\gamma^{1} + \be_{2} \bar{\partial}\gamma^{2}+ \be_{3}
\bar{\partial}\gamma^{3}+ \bar\be_1 \partial\bar\gamma^{1}+ \bar\be_2 \partial\bar\gamma^{2}+ \bar\be_3 \partial\bar\gamma^{3} \Big] ~, \\
S_{\textrm{int}} &= -\frac{k}{2\pi} \int d^2 z \Big [ \be_3\bar\be_3 e^{-b\phi} + (\be_1+\frac{1}{2}\be_3\ga^2)
(\bar\be_1+\frac{1}{2}\bar\be_3\bar\ga^2)e^{-b\phi/2-3b\phi^\bot/2} \\
&\qquad\qquad \qquad +(\be_2-\frac{1}{2}\be_3\ga^1)(\bar\be_2-\frac{1}{2}\bar\be_3\bar\ga^1) e^{-b\phi/2+3b\phi^\bot/2}\Big]~,
\end{split}
\end{equation}
where we have rescaled the fields $\phi,\phi^\bot$ with $b=1/\sqrt{k-3}$. The equations of motion for the auxiliary fields take the form (before this rescaling)
\begin{align}
\be_3 &= ke^{\phi}\big(\partial\bar\gamma_3+\frac{1}{2}\bar\ga^1\partial\ga^2-\frac{1}{2}\bar\ga^2\partial\ga^1\big)\, , \\
\be_2 &= ke^{\phi/2-3\phi^\bot/2}\partial\bar\gamma^2+\frac{1}{2}\gamma^1 \beta_3\, ,\\
\be_1 &= ke^{\phi/2+3\phi^\bot/2}\partial\bar\gamma^1-\frac{1}{2}\gamma^2 \beta_3\, .
\end{align}
This tells us the correct renormalization of the fields $\phi,\phi^\bot$ for the action $S^\text{WZNW}_{\textrm{ren}}$.
Note that the background charge implies that the first interaction term $\be_3\bar\be_3 e^{-\phi}$ is not marginal,
but it can be seen as a contact term and neglected. The sl(3) currents can be written in terms of these free fields
as is demonstrated in appendix \ref{free}.
\subsection{A relation to Bershadsky-Polyakov algebra}
\label{BPalgebra}
We would now like to derive the relation between correlation functions,
so we begin by considering a correlator in the SL(3) WZNW model.
The vertex operators can always be chosen of the form
\begin{align}\label{}
V_\nu(z_\nu)=e^{\mu_3^\nu\gamma^3-\bar\mu_3^\nu\bar\gamma^3}f(\gamma^1,\ga^2,\bar\gamma^1,\bar\ga^2,\phi,\phi^\bot) \, ,
\label{sl3vertexop}
\end{align}
and we consider correlators
\begin{align}
\langle \prod _{\nu=1}^n V_\nu (z_\nu) \rangle
&= \int {\cal D} g \, e^{-S^\text{WZNW} (g)} \prod_{\nu=1}^{n} V_\nu ( z _ \nu) ~,
&{\cal D} g &=
{\cal D} \phi {\cal D} \phi^\bot \prod_{\alpha =1 }^3
{\cal D}^2 \beta_\alpha {\cal D}^2 \gamma^\alpha ~.
\label{correlator}
\end{align}
Later we fix the form of the functional $f(\gamma^1,\ga^2,\bar\gamma^1,\bar\ga^2,\phi,\phi^\bot)$, but
here we leave it arbitrary.
Following \cite{HS} we integrate out $\gamma^3$ which only appears linearly in exponents. This gives a delta function in $\beta_3$ which is solved by setting
\begin{align}\label{}
\be_3(z)\mapsto {\cal B}_3(z)=\sum_{\nu=1}^n \frac{\mu^\nu_3}{z - z_\nu}
= u \frac{\prod_{i=1}^{n-2} (z - y_{i})}
{\prod_{\nu=1}^{n} (z - z_\nu)}\, .
\end{align}
Similarly for the antiholomorphic side we have
\begin{align}\label{}
\bar\be_3(\bar z)\mapsto -\mathcal{\bar B}_3(\bar z)=-\sum_{\nu=1}^n \frac{\bar\mu^\nu_3}{\bar z - \bar z_\nu}
= -\bar u \frac{\prod_{i=1}^{n-2} (\bar z - \bar y_{i})}
{\prod_{\nu=1}^{n} (\bar z - \bar z_\nu)}\, .
\end{align}
We then remove the function $\mathcal{B}_3$ from the action by the following transformation:
\begin{align}\label{eq:changetohalfhalf}
{\gamma'}^a= ( u\mathcal{B}_3 )^{\mbox{$\frac{1}{2}$}} \gamma^a ~, \qquad \beta'_a&= ( u \mathcal{B}_3)^{-\mbox{$\frac{1}{2}$}}\beta_a ~, \qquad
\varphi =\phi-\frac{1}{2b} \ln |u \mathcal{B}_3 |^2~ , \qquad \varphi^\bot =\phi^\bot ~.
\end{align}
This will change the background charge of $\varphi$, and
the remaining $\beta'_1,\gamma'^1$, $\beta'_2,\gamma'^2$ ghost systems are now dimension $(1/2,1/2)$ systems. This also gives extra terms of the form
\begin{align}\label{eq:extraghostterms}
\de S=\frac{1}{2\pi} \int d^2 z \Big[ -\mbox{$\frac{1}{2}$}\be'_{i}\gamma'^{i}\bar{\partial}\ln\mathcal{B}_3 -\mbox{$\frac{1}{2}$}\bar\be'_{i}\bar\gamma'^{i}\partial\ln\mathcal{\bar B}_3\Big]\, .
\end{align}
A natural way to proceed is to use that $\partial\bar{\partial}\ln\mathcal{B}_3$ can be expressed by delta functions localized at the points $z_\nu$ and $y_i$.
The extra terms can thus put in the form of vertex operators if we bosonize the $\be,\ga$ systems as follows
\begin{align}\label{eq:simplebosonization}
\be'_1 = \partial Y_1 e^{- X_1 + Y_1}~, \qquad
\ga'^1 = e^{X_1 - Y_1} ~, \qquad \normord{\be'_1\ga'^1}=\partial X_1\, ,
\end{align}
and likewise for $\be'_2,\ga'^2$.
We then get extra insertions in the correlators depending on $X_1,X_2$ and $\varphi$ and these are located in the zeroes of $\mathcal{B}_3$, namely $y_{i}$. Further, the original vertex operators in $z_\nu$ also gets modified.
An explicit formula can be found in equation \eqref{eq:CorrRela} in the next section.
On the other hand, we can also do a SP(2) transformation of the $\be',\ga'$ systems as follows
\begin{align}
\be'_1=\mbox{$\frac{1}{2}$}(\ga-\ga_d)\, , \qquad \ga'^1=-\be+\be_d\, , \qquad
\be'_2=\mbox{$\frac{1}{2}$}(\be+\be_d)\, , \qquad \ga'^2=\ga+\ga_d\, ,\label{eq:sp2trans}
\end{align}
and similarly for the anti-holomorphic side
\begin{align}
\bar\be'_1=\mbox{$\frac{1}{2}$}(\bar\ga-\bar\ga_d)\, , \qquad \bar\ga'^1=-\bar\be+\bar\be_d\, ,
\qquad
\bar\be'_2=-\mbox{$\frac{1}{2}$}(\bar\be+\bar\be_d)\, , \qquad \bar\ga'^2=-\bar\ga-\bar\ga_d\, .
\end{align}
The point is that this decouples the $\beta_d,\ga_d$ system and, as we will see below, the remaining action has Bershadsky-Polyakov symmetry. After the transformation, the action takes the form
\begin{equation}
\begin{split}
&S=S_{\textrm{kin}} + S_{\textrm{int}} ~, \\
&S_{\textrm{kin}} =\frac{1}{4\pi} \int d^2 z \Big[\frac{1}{2}\partial\varphi\bar{\partial}\varphi+\frac{3}{2}
\partial\varphi^\bot\bar{\partial}\varphi^\bot-\frac{1}{2}(b+\frac{1}{2b})\sqrt{g}\mathcal{R}\varphi \\
&\qquad\qquad\qquad\qquad +2\be \bar{\partial}\gamma+2\be_{d} \bar{\partial}\gamma_{d} + 2\bar\be \partial\bar\gamma +
2\bar\be_d \partial\bar\gamma^{d} \Big] ~, \\
&S_{\textrm{int}} = -\frac{k}{2\pi} \int d^2 z \Big [ \ga\bar\ga e^{-b\varphi/2-3b\varphi^\bot/2} -\be\bar\be e^{-b\varphi/2+3b\varphi^\bot/2}\Big]~.
\end{split}
\end{equation}
We have here omitted the contact term in the action.
However, the holomorphic part of the extra ghost terms \eqref{eq:extraghostterms} are now
\begin{align}
\de S=\frac{1}{2\pi} \int d^2 z \Big[ -\mbox{$\frac{1}{2}$}(\ga\be_d+\ga_d\be)\bar{\partial}\ln\mathcal{B}_3 -\mbox{$\frac{1}{2}$}(\bar\ga\bar\be_d+\bar\ga_d\bar\be)\partial\ln\mathcal{\bar B}_3 \Big]\, .
\end{align}
Since the logs are holomorphic up to branch cuts, these terms look more like line operators than the extra vertex operators that were found after the bosonization \eqref{eq:simplebosonization} above. To perform the SP(2) rotation on the vertex operators is thus not a simple problem, however in the similar OSP$(N|2)$ case \cite{HS3,CHR} the corresponding problem for the fermionic ghost systems was solvable using spin operators. We are thus hopeful that future research will find a solution.
Let us now consider the symmetry of the action. We want to show that the action with the $\be_d,\ga^d$ part removed is invariant under the Bershadsky-Polyakov $W_3^2$ algebra \cite{Bershadsky}. This algebra is generated by dimension $3/2$ fields $G^\pm$ and the U(1) current $H$ with the OPEs
\begin{align}
&G^+(z)G^-(w)\sim \frac{(k-1)(2k-3)}{(z-w)^3}+\frac{-3(k-1)H(w)}{(z-w)^2} \\
& \qquad \qquad \qquad \qquad +\frac{\big(3:HH:+(k-3)T_{W_3^2}-\frac{3(k-1)}{2}\partial H\big)(w)}{z-w} ~, \nonumber\\
&G^\pm(z)G^\pm(w)\sim 0 ~, \qquad
H(z)H(w)\sim\frac{-(2k-3)/3}{(z-w)^2} ~. \nonumber
\end{align}
The task is to construct these currents in terms of our free fields.
We have removed $\mathcal{B}_3$ from the action, and
it changes the stress energy tensor to
\begin{align}
T_{\textrm{improved}}=T_{sl_3}-\partial J^\tha ~,
\end{align}
which is called the improved stress-energy tensor in \cite{Bershadsky}. We here have to think of the $\be_3,\ga^3$ systems as being removed. After doing the SP(2) transformation \eqref{eq:sp2trans} we get
the stress-energy tensor
\begin{align}
&T=T_{W_3^2}+T_{\be_d,\ga_d} ~ , \nonumber\\
&T_{W_3^2} = -\tfrac{1}{4}\partial\varphi\partial\varphi-\tfrac{3}{4}\partial\varphi^\bot\partial\varphi^\bot -(b+\frac{1}{2b})\partial\del\varphi +\mbox{$\frac{1}{2}$}(\ga\partial\be-\partial\ga\be ) ~ , \nonumber \\
&T_{\be_d,\ga_d} =\mbox{$\frac{1}{2}$}(\ga_d\partial\be_d-\partial\ga_d\be_d ) ~ .
\end{align}
Here $T_{W_3^2}$ exactly has the central charge $25+24/(k-3)+6(k-3)$ as expected for the level $k$ Bershadsky-Polyakov algebra.
The $G^\pm,H$ currents are obtained from the currents where first all terms depending on $\ga^3$ is removed, then $\mathcal{B}_3$
is taken out, and finally, after the SP(2) transformation, the terms depending on $\be_d,\ga_d$ are removed. Then we get (the currents are like in \cite{deBoer:1992sy}):
\begin{align}
& G^+=J^{E_2}|_{\textrm{reduced}} = \frac{1}{2b}\partial\varphi\ga-\frac{3}{2b}\partial\varphi^\bot\ga-\ga\ga\be +(k-1)\partial\ga ~, \nonumber \\
& G^-=J^{E_1}|_{\textrm{reduced}} =-\frac{1}{2b}\partial\varphi\be-\frac{3}{2b}\partial\varphi^\bot\be-\ga\be\be -(k-1)\partial\be ~, \\
& H=-\frac{2}{3}J^{\tha^\bot}|_{\textrm{reduced}} =-\frac{1}{b}\partial\varphi^\bot -\ga\be ~.\nonumber
\end{align}
Finally, our interaction terms are screening operators for the $W_3^2$ algebra which can be directly checked, and we have thus demonstrated that the action is invariant under the Bershadsky-Polyakov algebra.
\section{Quasi-superconformal theory from $\text{SL}(2+N|M)$}
\label{QSCA}
In this section we generalize the analysis for sl(3) in the previous section to
the case with a supergroup $\text{SL}(2+N|M)$ with arbitrary numbers $N,M$.
In \cite{CHR}, we have derived relations between correlation functions
of supergroup WZNW models and extended super Liouville theory.
The supergroup used in \cite{CHR} has a bosonic subalgebra of the
form $\text{SL}(2) \times A$. The procedure developed in \cite{HS} applied to the SL(2) part led to the
correspondence found in \cite{CHR}. Here we use a similar embedding
of sl(2) such that the elements of the Lie superalgebra are
decomposed as
\begin{align} \label{5dec}
\mathfrak{g}=\mathfrak{g}_{-1} \oplus \mathfrak{g}_{-\frac{1}{2}} \oplus \mathfrak{g}_{0} \oplus \mathfrak{g}_{+\frac{1}{2}} \oplus \mathfrak{g}_{+1} ~.
\end{align}
The generators of the embedded sl(2) are in $t^z \in \mathfrak{g}_0 $ and
$t^\pm \in \mathfrak{g}_{\pm 1}$.
In fact, $\mathfrak{g}_{\pm 1}$ are generated by $t^\pm$.
Notice that $\mathfrak{g}_{\pm 1/2}$
are fermionic in the cases considered in \cite{CHR}, but here they can be bosonic
and fermionic. This is the main point of the generalization.
The relations in \cite{HS,HS2,CHR} remind us of Hamiltonian reduction
\cite{Bouwknegt:1992wg},
and one of the aims of this paper is to investigate the relation between the two.
We can construct a theory with $W$ symmetry from the WZNW model on $G$,
and the symmetry algebra depends on how sl(2) is embedded in $g$.
The above sl(2) embedding corresponds to the partition of $2+N|M$ as
\begin{equation}
\underline{2+N|M} = \underline{2|0}+N \underline{1|0}+M \underline{0|1}\, .
\end{equation}
For $N=0$ the supergroup is given by SL$(2|M)$ and hence analyzed in \cite{CHR}, and
for $M=0$ the Hamiltonian reduction of sl$(2+N)$ yields the quasi-superconformal algebra
in \cite{Romans}. Here we deal with generic $N,M$, which involves a generalization of the
quasi-superconformal algebra.
\subsection{SL$(2+N|M)$ WZNW action}
\label{lotone}
The elements of the sl$(2+N|M)$ Lie superalgebra
can be expressed by the $(2+N|M) \times (2+N|M)$ supermatrix of the form
\begin{align} M =
\begin{pmatrix}
A & B \\
C & D
\end{pmatrix} ~, \qquad
\text{str} \, M = \text{tr} \, A - \text{tr} \, D = 0 ~,
\end{align}
where $A,D$ are Grassmann even and $B,C$ are Grassmann odd.
Following the general argument, we decompose sl$(2+N|M)$ as in \eqref{5dec}.
Roughly speaking, we interpret the $(2+N|M) \times (2+N|M)$ supermatrix
as a four block supermatrix, where the diagonal blocks are a $2\times 2$ block representing the embedded sl(2) and
the $(N|M) \times (N|M)$ block representing a sl$(N|M)$ algebra which commutes with the sl(2) and
further there is also a u(1) algebra.
The off-diagonal
part carries the standard representation of sl$(N|M)\oplus$ sl(2) and its conjugate, and they will represented by free bosons and fermions. See for instance
\cite{Bars}.
For our explicit conventions see appendix \ref{convgen}.
We parameterize a supergroup valued field according to our 5-decomposition
\begin{align}
g=g_{-1}\,g_{-\frac{1}{2}}\,g_{0}\, g_{\frac{1}{2}}\,g_{1} ~,
\end{align}
where
\begin{align}
g_{-1} = e^{\gamma E^-} ~, \qquad g_{+1} = e^{\bar \gamma E^+} ~,
\qquad
g_0 = e^{ - 2 \Phi Q} e^{ - 2 \phi E^0 }
\begin{pmatrix}
\mathbb{I}_2 & 0 \\
0 & q
\end{pmatrix} ~ .
\end{align}
The generators of the embedded sl(2) are $E^0,E^\pm$ and the elements
of sl$(N|M)$ are represented by $q$. The generator of the u(1) algebra
is denoted by $Q$, and it commutes with the sl(2) and sl$(N|M)$.
For $2+N=M$, the u(1) part can be decoupled and we can start
from psl$(M|M)$ instead of sl$(M|M)$. The other parts are
\begin{align}
& g_{-1/2} = \exp (\sum_{i=1}^N \gamma^1_i S^-_{1,i} )
\exp (\sum_{i=1}^N \gamma^2_i S^-_{2,i} )
\exp (\sum_{\hat i=1}^M \theta^1_{\hat i} F^-_{ 1,\hat i} )
\exp (\sum_{\hat i=1}^M \theta^2_{\hat i} F^-_{2,\hat i} ) ~, \\
& g_{+1/2} = \exp (\sum_{\hat i=1}^M \bar \theta^2_{\hat i} F^+_{2,\hat i} )
\exp (\sum_{\hat i=1}^M \bar \theta^1_{\hat i} F^+_{1,\hat i} )
\exp (\sum_{i=1}^N \bar \gamma^2_i S^+_{2,i} )
\exp (\sum_{i=1}^N \bar \gamma^1_i S^+_{1,i} ) ~.
\end{align}
In \cite{CHR} we have only fermions $\theta^a_i, \bar \theta^a_i$, and
the appearance of bosons $\gamma^a_i , \bar \gamma^a_i$ is the new
feature in this case. The explicit form of the generators are summarized in
appendix \ref{convgen}.
With the above parametrization, the action of SL$(2+N|M)$ WZNW model
is given by
\begin{align}
S^\text{WZNW}_k[g] = S^\text{WZNW}_k[q] &+ \frac{k}{2 \pi}
\int d z^2 \Bigl[ \bar \partial \Phi \partial \Phi
+ \bar \partial \phi \partial \phi + e^{-2 \phi}
(\bar \partial \gamma + \partial \Theta_1 \Theta_2^t )
(\partial \bar \gamma + \bar \Theta_2 \partial \Theta_1^t ) \nonumber
\\
& + e^{- \phi + \eta \Phi}
\partial \bar \Theta_1 q^{-1} \bar \partial \Theta_1^t
+ e^{- \phi - \eta \Phi } \bar \partial \Theta_2 q
\partial \bar \Theta_2^t \Bigr ] ~,
\end{align}
where
\begin{align}
\nonumber
\gamma_a = (\gamma_1^a , \cdots , \gamma_N^a) ~,
\quad \bar \gamma_a = (\bar \gamma_1^a , \cdots , \bar \gamma_N^a) ~, \quad
\theta_a = (\theta_1^a , \cdots , \theta_M^a) ~,
\quad \bar \theta_a = (\bar \theta_1^a , \cdots , \bar \theta_M^a)
\end{align}
with $a=1,2$. Moreover we denote $\Theta_a = (\gamma_a , \theta_a)$
and $\bar \Theta_a = (\bar \gamma_a , \bar \theta_a)$.
We introduce auxiliary fields $\beta, \bar \beta,
P_a = (\beta_a , p_a), \bar P_a = (\bar \beta_a , \bar p_a) $ with
\begin{align}
\nonumber
\beta_a=(\beta^a_1 , \cdots , \beta^a_N ) ~, \quad
\bar \beta_a=(\bar \beta^a_1 , \cdots , \bar \beta^a_N ) ~, \quad
p_a=(p^a_1 , \cdots , p^a_M ) ~, \quad
\bar p_a=(\bar p^a_1 , \cdots , \bar p^a_M ) ~.
\end{align}
Then we find classically
\begin{align}
S^\text{WZNW}_k[g] &\stackrel{\text{clas.}}{=} S^\text{WZNW}_k [q]+ \frac{1}{2 \pi}
\int d^2 z \Bigl[ k \bar \partial \Phi \partial \Phi +
k \bar \partial \phi \partial \phi -
\beta \bar \partial \gamma - \bar \beta \partial \bar \gamma
+ \sum_{a=1}^2 ( P_a \bar \partial \Theta_a^t
+ \bar P_a \partial \bar \Theta_a^t ) \nonumber \\
& - \frac{1}{k} \beta \bar \beta e^{2 \phi}
- \frac{1}{k} (P_1 + \beta \zeta \Theta_2) q \zeta
(\bar P_1 + \bar \beta \bar \Theta_2)^t e^{\phi - \eta \Phi}
- \frac{1}{k} \bar P_2 q^{-1} \zeta P_2^t e^{\phi + \eta \Phi} \Bigr]~,
\end{align}
where we have used $P_a \partial \Theta_a^t
= \partial \Theta_a \zeta P_a^t$ with
\begin{align}
\zeta =
\begin{pmatrix}
\mathbb{I}_N & 0 \\
0 & - \mathbb{I}_{M}
\end{pmatrix} ~.
\end{align}
Due to the anomaly from the change of path integral measure there are shifts of coefficients. To get these corrections, we first set $q=1$.
Then from the measure of the path integral the contribution
from $\beta,\gamma$ is
\begin{align}
- \frac{1}{\pi} \int d^2 z \partial \phi \bar \partial \phi
+ \frac{1}{8 \pi} \int d^2z \sqrt{g} {\cal R } \phi ~.
\end{align}
Further, let
\begin{align}
\delta s =- \frac{1}{4 \pi} \int d^2 z \partial \phi \bar \partial \phi
+ \frac{1}{16 \pi} \int d^2z \sqrt{g} {\cal R } \phi ~,
\end{align}
then a pair of $\beta_a,\gamma_a$ contributes $\delta s$, while a pair of $p_a,\theta_a$ contributes $-\delta s$ to the anomaly (see (2.22) of \cite{CHR}).
Taking this anomaly into account, the action becomes
\begin{align} \label{actionslnm}
& S^\text{WZNW}_k[g] = S^\text{WZNW}_{k-2}[q] \\ & \qquad + \frac{1}{2 \pi}
\int d^2 z \Bigl[ \bar \partial \Phi \partial \Phi +
\bar \partial \phi \partial \phi
+ \frac{\hat Q}{4} \sqrt{g} {\cal R } \phi
- \beta \bar \partial \gamma - \bar \beta \partial \bar \gamma
+ \sum_{a=1}^2 ( P_a \bar \partial \Theta_a^t
+ \bar P_a \partial \bar \Theta_a^t ) \nonumber \\
& \qquad - \frac{1}{k} \beta \bar \beta e^{2 b \phi}
- \frac{1}{k} (P_1 + \beta \zeta \Theta_2) q \zeta
(\bar P_1 + \bar \beta \bar \Theta_2)^t e^{b (\phi - \eta \Phi)}
- \frac{1}{k} \bar P_2 q^{-1} \zeta P_2^t e^{b(\phi + \eta \Phi)} \Bigr] \nonumber
\end{align}
with $b^{-2} = k - 2 - N + M$ and $\hat Q = b (1 + N - M)$.
The central charge of the SL$(2+N|M)$
WZNW model is
\begin{align}
c = \frac{((N-M+2)^2 - 1) k}{k -2 - N+M} ~.
\end{align}
After the renormalization, we have
\begin{align}
c = 1 + 6 \hat Q ^2 + 2 + 2 \cdot (2N-2M) + 1 +
\frac{((N-M)^2 - 1) (k-2)}{k -2 - N+M} ~,
\end{align}
which is the same as above.
Note that the central charge for one pair $(p_a,\theta_a)$ is $-2$.
\subsection{Correspondence to a quasi-superconformal theory}
In subsection \ref{BPalgebra} we have studied a relation between
correlators of SL(3) WZNW model and a theory with $W_3^2$ symmetry.
Here we would like to derive similar relations involving the
SL$(2+N|M)$ WZNW model. In order to obtain an explicit formula,
we specify the form of vertex operator, see \eqref{sl3vertexop}.
In the supergroup cases analyzed in \cite{HS2,CHR}, it was useful
to express the fermions in bosonized language.
{}From this experience we again enlarge the Hilbert
space by utilizing the bosonization formula
\begin{align}
\beta^a_i (z) = - \partial \xi^a_i e^{- X^a_i} ~, \quad
\gamma^a_i (z) = \eta^a_i e^{X^a_i} ~, \quad
p^a_i (z) = e^{ i Y^a_i} ~, \quad
\theta^a_i (z) = e^{ - i Y^a_i} ~,
\end{align}
where the operator products are
\begin{align}
\xi^a_i (z) \eta^b_j (0) = \delta_{a,b} \delta_{i,j}\frac{1}{z} ~, \qquad X^a_i(z) X^b_j(0) \sim
Y^a_i(z) Y^b_j(0) \sim - \delta_{a,b} \delta_{i,j} \ln z ~.
\end{align}
Analogous expressions hold for the barred quantities.
Then the vertex operators can be defined as
\begin{align} \label{vertexslnm}
V^{t^a_i,s^a_{\hat i}}_{j,L} (\mu | z) &= \mu ^{j + 1 - \frac{1}{2} \sum_{a,i} t^a_i
+ \frac{1}{2} \sum_{a,\hat i} s^a_{\hat i}}
\bar \mu ^{j + 1 - \frac{1}{2}\sum_{a,i} \bar t^a_i
+ \frac{1}{2} \sum_{a,\hat i} \bar s^a_{\hat i}} \\ & \cdot
e^{ t^a_i X^a_i + \bar t^a_i \bar X^a_i + i s^a_{\hat i} Y^a_{\hat i}
+ i \hat s^a_{\hat i} \hat Y^a_{\hat i} }
e^{\mu \gamma - \bar \mu \bar \gamma}
e^{2b(j+1) \phi} V^\text{SL$(N|M)$}_L (q) ~, \nonumber
\end{align}
where $L$ labels the representation of sl$(N|M)$.
Other vertex operators may be obtained by applying
$\xi^a_i,\eta^a_i$ and their derivatives.
The correlation functions are computed as in \eqref{correlator}.
In order to obtain relations to a reduced theory, we follow the
strategy adopted in \cite{HS} as in the SL(3) case.
We integrate out $\gamma$, then
the field $\beta$ is replaced by
\begin{align}
\sum_{\nu=1}^n \frac{\mu_\nu}{z-z_\nu} = u \frac{\prod_{l=1}^{n-2} (z - y_l)}
{\prod_{\nu=1}^n (z - z_\nu) } = u {\cal B} (y_l , z_\nu ; z) ~,
\end{align}
and similarly for $\bar \beta$. Now the action includes the functions
${\cal B}$ and $\bar {\cal B}$, and the functions can be removed
by the shift of fields $\phi,X^a_i,Y^a_{\hat i}$ as
\begin{align} \label{shiftslnm}
\phi + \frac{1}{2b} \ln |u{\cal B} |^2 \to \phi ~, \qquad
X^a_i + \frac12 \ln u{\cal B} \to X^a_i ~, \qquad
Y^a_{\hat i} + \frac{i}{2} \ln u{\cal B} \to Y^a_{\hat i} ~.
\end{align}
After some manipulations, we arrive at the relation among the
correlators as
\begin{align}\label{eq:CorrRela}
\langle \prod_{\nu=1}^n V^{ {t^a_i}_\nu ,{s^a_{\hat i}}_\nu }_{j_\nu,L_\nu} (\mu_\nu | z_\nu) \rangle = \delta^{(2)} (\sum_{\nu=1}^n \mu_\nu )
| \Theta_n |^2 \langle \prod_{\nu=1}^n
V^{ {t^a_i}_\nu -1/2 , {s^a_{\hat i}}_\nu +1/2}_{b(j_\nu + 1) + 1/2b,L_\nu} (z_\nu)
\prod_{l=1}^{n-2} V^{ 1/2 , -1/2}_{-1/2b,0} ( y_l)
\rangle ~.
\end{align}
The action for the right hand side is
\begin{align} \label{actionlslnm}
S & = S^\text{WZNW}_{k-2}[q] + \frac{1}{2 \pi} \int d^2 z \Bigl[
\bar \partial \Phi \partial \Phi +
\bar \partial \phi \partial \phi + \frac{Q}{4} \sqrt{g} {\cal R} \phi
+ \sum_{a=1}^2 ( P_a \bar \partial \Theta_a^t + \bar P_a \partial \bar \Theta_a^t )
\nonumber \\
& + \frac{1}{k} e^{2 b \phi} - \frac{1}{k} ( P_1 + \zeta \Theta_2 ) q \zeta
(\bar P_1 - \bar \Theta_2 )^t e^{ b (\phi - \eta \Phi ) }
- \frac{1}{k} \bar P_2 q ^{-1} \zeta
\bar P_2^t e^{ b ( \phi + \eta \Phi ) }\Bigr]
\end{align}
with $Q = \hat Q + b^{-1}$.
The vertex operators are
\begin{align}\label{vertexlslnm}
V^{ t^a_i , s^a_{\hat i}}_{\alpha,L} (z) = e^{ t^a_i X^a_i + \bar t^a_i \bar X^a_i+
i s^a_{\hat i} Y^a_{\hat i} + i \hat s^a_{\hat i} \hat Y^a_{\hat i} }
e^{2 \alpha \phi} V^\text{SL$(N|M)$}_{L} (q) ( z) ~,
\end{align}
and the pre-factor is
\begin{align}
\Theta_n = u \prod_{i < j}^n (z_i - z_j)^{\frac{1}{2b^2} + \frac{N-M}{2}}
\prod_{ p < q}^{n-2} (y_p - y_q)^{\frac{1}{2b^2} + \frac{N-M}{2}}
\prod_{i =1}^n \prod_{p =1}^{n-2} (z_i - y_p)^{ - \frac{1}{2b^2} - \frac{N-M}{2}} ~.
\end{align}
The result for $N=0$ reproduces the one in \cite{CHR}.
As mentioned in subsection \ref{BPalgebra}, there are extra insertions
of operators with the identity representation $L=0$ at $y_l$,
and shifts of parameters of vertex operators at $z_\nu$.
In the case of SL(3), the reduced theory consists of a free theory and
a theory with $W_3^2$ symmetry. We have similar decoupling in this
generalization. If we rotate $P_a,\Theta_a$ as
\begin{align}
\Theta_1 - \zeta P_2 \to \Theta_1 ~, \quad
\Theta_2 + \zeta P_1 \to \Theta_2 ~, \quad
\bar \Theta_1 + \bar P_2 \to \bar \Theta_1 ~, \quad
\bar \Theta_2 - \bar P_1 \to \bar \Theta_2 ~,
\end{align}
then the action becomes
\begin{align}
& S[q,\Phi,\phi,\psi_a] =
S^\text{WZNW}_{k-2}[q] + \frac{1}{2 \pi} \int d^2 z \Bigl[
\bar \partial \Phi \partial \Phi +
\bar \partial \phi \partial \phi + \frac{Q}{4} \sqrt{g} {\cal R} \phi
\\
& \qquad \qquad + \sum_{a=1}^2 ( P_a \bar \partial \Theta_a^t + \bar P_a \partial \bar \Theta_a^t )
+ \frac{1}{k} e^{2 b \phi} + \frac{1}{k} \zeta \Theta_2 q \zeta
\bar \Theta_2 ^t e^{ b (\phi - \eta \Phi ) }
- \frac{1}{k} \bar P_2 q ^{-1} \zeta
P_2^t e^{ b ( \phi + \eta \Phi ) }\Bigr] ~.\nonumber
\end{align}
We thus see that $P_1,\Theta_1$ is a free decoupled system. As in the SL(3) case it is also here an outstanding problem how to rewrite the vertex operators in terms of the rotated variables. For $M=0$, the symmetry of the theory should be the quasi-superconformal
symmetry discussed in \cite{Romans} and for $M \neq 0$ it is a generalization of it.
Utilizing our free field realization, we could
construct the currents for the symmetry as for SL(3) case.
Furthermore, it should be possible to apply the above analysis to more generic cases in page 14 of \cite{KW}.
\section{The product embedding \underline{$2N$} $ =N$\underline{$2$}}
\label{product}
In the previous section, all $n$-point functions of WZNW models on SL$(2+N|M)$ on a sphere
are found to be written in terms of reduced theories with some specific $W$-symmetry.
Since different $W$-algebras can be obtained though the Hamiltonian reduction by using
different sl(2) embeddings, one may wonder whether we can establish relations between
correlators involving more generic $W$-algebras.
In this section, we give an example corresponding to the simple product embedding of
sl(2) as \underline{$2N$} $ =N$\underline{$2$}.
More generic cases are under investigation.
\subsection{SL$(2N)$ WZNW action}
We start from the action of SL$(2N)$ WZNW model.
In order to adopt the reduction procedure in \cite{HS}, we should find out
a proper free field realization of the WZNW model.
Let us consider sl$(2N)$ and the following embedding of sl(2) (in block matrix form)
\begin{align}\label{}
t^z=\frac{1}{2} \left (
\begin{array}{c|c}
\mathbb{I}_N & 0 \\ \hline
0 & -\mathbb{I}_N
\end{array} \right)\, ,\qquad t^+= \left (
\begin{array}{c|c}
0 & \mathbb{I}_N \\ \hline
0 & 0
\end{array} \right)\, ,\qquad t^-= \left (
\begin{array}{c|c}
0 & 0 \\ \hline
\mathbb{I}_N & 0
\end{array} \right)\, .
\end{align}
The commutant of this sl(2) is sl$(N)$ generated by
\begin{align}\label{}
t_{\text{sl}(N)}^\mathbb{A}= \left (
\begin{array}{c|c}
\mathbb{A} & 0 \\ \hline
0 & \mathbb{A}
\end{array} \right)\, .
\end{align}
The splitting of the fundamental representation of sl$(2N)$ can now be written as
\begin{align}\label{}
\underline{2N}_{\text{sl}(2N)}=\underline{N}_{\text{sl}(N)}\, \underline{2}_{\text{sl}(2)} \, .
\end{align}
We get a 3-grading of the algebra from the eigenvalues of $t^z$. In this case upper left block is $+1$, lower left $-1$ and the diagonal blocks are zero graded. Following \cite{Bais:1990bs} we make the standard Gauss decomposition obeying this grading
\begin{align}\label{}
g=\left (
\begin{array}{c|c}
\mathbb{I}_N & 0 \\ \hline
\gamma & \mathbb{I}_N
\end{array} \right)e^{2\phi t^z}\left (
\begin{array}{c|c}
g_1 & 0 \\ \hline
0 & g_2
\end{array} \right)\left (
\begin{array}{c|c}
\mathbb{I}_N & \bar \gamma \\ \hline
0 & \mathbb{I}_N
\end{array} \right)\ ,
\end{align}
where $g_1$ and $g_2$ are SL$(N)$ matrices.
The action then takes the form (with $k$ having opposite sign as in the compact case)
\begin{align}
S_k^{\textrm{WZNW}} [g] &= \frac{k}{4 \pi} \int_{\Sigma} d ^2z
\langle g^{-1} \partial g , g^{-1} \bar \partial g \rangle
+ \frac{k}{2 4 \pi} \int_{B} \langle g^{-1} d g
[g^{-1} d g , g^{-1} d g ] \rangle \nonumber \\
&= S_k^{\textrm{WZNW}}[g_1]+S_k^{\textrm{WZNW}}[g_2]+\frac{k}{2 \pi} \int d^2 z\, N\partial\phi\bar{\partial}\phi+ e^{2\phi}\, \tr(\bar{\partial} \gamma g_1\partial\bar\gamma g_2^{-1})
\end{align}
as is easily found using the Polyakov-Wiegmann identity \eqref{PWid}.
We then go to a first order formalism by introducing the $N\times N$ matrices $\beta,\bar\beta$
\begin{align}\label{2Naction}
S_k [\phi,g_1,g_2,\gamma,\bar\gamma,\beta,\bar\beta]=& S_{k-N}^{\textrm{WZNW}}[g_1]+S_{k-N}^{\textrm{WZNW}}[g_2]+\frac{1}{2 \pi} \int d^2 z\, \partial\phi\bar{\partial}\phi-\frac{N^2 b}{4}\sqrt{g}\mathcal{R}\phi\nonumber \\
&+\frac{1}{2 \pi} \int d^2 z\, \tr(\beta\bar{\partial}\gamma+\bar\beta\partial\bar\gamma-\frac{1}{k}e^{-2\phi}\bar \beta g^{-1}_1\beta g_2) ~ .
\end{align}
Here we have introduced $b=1/\sqrt{N(k-2N)}$ and rescaled $\phi\mapsto b\phi$.
To find the renormalization we have used (in the old $\phi$)
\begin{align}\label{}
\beta=k e^{2\phi}g_1\partial\bar\gamma g_2^{-1}
\end{align}
and that the contribution from the path integral measure for each of the $N^2$ $\beta,\gamma$ systems then is
\begin{align}
\label{anomaly1}
\delta S= - \frac{1}{\pi} \int d^2 z \partial \phi \bar \partial \phi
- \frac{1}{8 \pi} \int d^2z \sqrt{g} {\cal R } \phi ~,
\end{align}
and a similar analysis can be made using $g_1$ and $g_2$ in their Cartan sector. Since they have unit determinant, we do not get any background charge for the SL($N$) factors.
We can check that the central charge is correctly reproduced as
\begin{align}\label{}
c&=2c_{sl(N)_{k-N}}+N^2 c_{\beta\gamma}+c_{\phi}=2\frac{(N^2-1)(k-N)}{k-N-N}+2N^2+1+6\frac{N^3}{(k-2N)}\nonumber \\
&=\frac{((2N)^2-1)k}{k-2N} ~.
\end{align}
The currents in this hybrid first order formulation are constructed in \cite{Bais:1990bs} (they introduce this name).
They are of the form
\begin{align}\label{eq:currentsl2n}
J=\left (\begin{array}{c|c}
H^1 & \beta \\ \hline
J^+ & H^2
\end{array}\right ) ~,
\end{align}
where $H^i$ contain bilinears in $\beta$ and $\gamma$, while $J^+$ needs cubic terms.
\subsection{The reduced action}
We will now see how to obtain the reduced action without making constraints on the momenta of the vertex operators. Due to the form of the currents we can choose a simple $\mu_{ij}$ basis, $i,j=1,\ldots,N$, of the vertex operators
\begin{align}
V_{\alpha,r_1,r_2,\mu_{ij},\bar\mu_{ij}} (z) =e^{\mu_{ij}\gamma^{ji}-\bar\mu_{ij}\bar\gamma^{ji}}
e^{2 \alpha b \phi} V^\text{SL($N$)}_{r_1} (g_1) (z)V^\text{SL($N$)}_{r_2} (g_2) (z) ~,
\end{align}
where $r_1,r_2$ label two sl($N$) representations.
As in the previous cases, the interaction term in the action \eqref{2Naction}
does not depend on $\gamma, \bar \gamma$. Since the kinetic terms in the Lagrangian are of the form $\beta_{ij}\bar{\partial}\gamma^{ji}$,
we can thus integrate out $\gamma$ and $\bar\gamma$ in order to obtain
\begin{align}\label{}
\beta_{ij}\mapsto \sum_{\nu=1}^{n} \frac{\mu^\nu_{ij}}{z-z_\nu} = u_{ij} \frac{\prod_{l=1}^{n-2} (z - y^l_{ij})}
{\prod_{\nu=1}^n (z - z_\nu) } = {\cal B}_{ij} (y^l_{ij} , z_\nu ; z)\ ,
\end{align}
where $n$ is the number of vertex operators.
Likewise we have
\begin{align}\label{}
\bar\beta_{ij}\mapsto - \sum_{\nu=1}^{n} \frac{\bar\mu_{ji}}{\bar z-\bar z_\nu} = - \bar u_{ji} \frac{\prod_{l=1}^{n-2} (\bar z - \bar y^l_{ji})}
{\prod_{\nu=1}^n (\bar z -\bar z_\nu) } = - \bar{\cal B}_{ji} (\bar y^l_{ij} ,\bar z_\nu ;\bar z)\ .
\end{align}
The interaction term then takes the form
\begin{align}\label{eq:interactionwithmatrices}
S_{\textrm{int}} = \frac{1}{2 \pi} \int d^2 z\, \tr(\frac{1}{k}e^{-2b\phi}{\cal B}^\dagger g^{-1}_1{\cal B} g_2) \ .
\end{align}
Here we would like to remove ${\cal B}$ from the action by making use of field redefinitions.
Since $g_1,g_2 \in \text{SL}(N)$, the SL$(N)$ part of ${\cal B}$ may be removed by the
change of $g_1,g_2$, which will be discussed later.
First, the determinant of ${\cal B}$ can be rescaled to one
by a translation in $\phi$. This will give extra insertions of $\phi$ where the determinant is zero or infinite:
\begin{align}\label{}
\phi&=\varphi+\frac{1}{2b}\ln \left|\frac{d(z)^{1/N}}{\prod_{\nu=1}^n (z - z_\nu)}\right|^2\ ,\nonumber \\
d(z)&:=\det \{\prod_{l=1}^{n-2} u_{ij}(z - y^l_{ij})\}_{i,j=1,\ldots,N}=\det \{u_{ij}\}\prod_{m=1}^{N(n-2)} (z-d_m)\ ,
\end{align}
assuming $\det \{u_{ij}\}\neq0$. Here we have just used that the determinant is a polynomial in $z$ of degree $N(n-2)$ and we have denoted the zeroes $d_m$.
We will now use that
\begin{align}\label{eq:loglineintegral}
\int d^2 z\, \partial\bar{\partial} \ln z \, f(z,\bar z) = -2\pi\int_0^\infty d z\, \partial f(z,\bar z)\ , \nonumber \\
\int d^2 z\, \partial\bar{\partial} \ln \bar z\, f(z,\bar z) = -2\pi\int_0^\infty d \bar z\, \bar{\partial} f(z,\bar z)\ ,
\end{align}
where the paths are along the branch cut of the logarithm, and $f$ is a function which is smooth in a neighborhood of the branch cut. Adding the two integrals gives a path integral over a complete derivative just giving the difference of the values at the endpoints. The insertions we get from the transformation of $\phi$ are then
\begin{equation}\label{eq:extrau1insertions}
\begin{split}
&\exp(- \frac{1}{2 \pi} \int d^2 z\, \partial\phi\bar{\partial}\phi)\propto\\
&\qquad\qquad\qquad\qquad\exp(- \frac{1}{2 \pi} \int d^2 z\, \partial\varphi\bar{\partial}\varphi)\prod_{m=1}^{N(n-2
)}e^{\frac{1}{bN}\varphi(d_m)}\prod_{\nu=1}^{n}e^{-\frac{1}{b}\varphi(z_\nu)}e^{\frac{2}{b}\varphi(\infty)}\ .
\end{split}
\end{equation}
Here we have left out a constant which also contains zeroes.
The zeroes precisely cancel the infinities coming from creating the new exponential vertex operator since they need normal ordering.
The insertions of $\varphi$ at infinity can be seen as an addition of the background charge. We can see this precisely using
\begin{align}\label{}
\sqrt{g} {\cal R } =-4\partial\bar{\partial}\ln |\rho|^2 ~.
\end{align}
Since $\rho=1$ everywhere except at infinity, $z'=1/z=0$, we have zero curvature in the whole plane, but a delta function at infinity
\begin{align}\label{}
\sqrt{g} {\cal R } =-4\partial\bar{\partial}\ln |z'|^{-2}=16\pi\delta^{(2)}(z') ~.
\end{align}
We thus get the background charge
\begin{align}\label{}
Q_{\varphi}=-bN^2-1/b\ ,
\end{align}
In appendix \ref{DSreduction} we analyze the Drinfeld-Sokolov reduction and find the correct improved Virasoro tensor and this result compares precisely, see eq. \eqref{eq:Treducedcorrect}. Note there will also be a constant term coming from the background charge due to the constant translation of $\phi$.
One last technicality needs to be considered before continuing.
When we get the extra insertions in eq. \eqref{eq:extrau1insertions} some of them are inserted in the same point as the original vertex operators.
This will create infinities from placing two operators in the same point,
but these are exactly canceled by the zeroes coming from the change of $\phi$ in the existing vertex operator which will also give some extra constant term.
We will see something similar happen later.
The action is now
\begin{align}\label{}
S_k [\phi,g_1,g_2]=& S_{k-N}^{\textrm{WZNW}}[g_1]+S_{k-N}^{\textrm{WZNW}}[g_2]+\frac{1}{2 \pi} \int d^2 z\, \partial\varphi\bar{\partial}\varphi+\frac{Q_{\varphi}}{4}\sqrt{g}\mathcal{R}\varphi\nonumber \\
&+ \frac{1}{2 \pi} \int d^2 z\, \tr(\frac{1}{k}e^{-2b\varphi}{\cal B'}^\dagger g^{-1}_1{\cal B'} g_2)\ ,
\end{align}
where
\begin{align}\label{}
{\cal B}'_{ij}&=u_{ij} \frac{\prod_{l=1}^{n-2} (z - y^l_{ij})}
{\big(\det \{u_{i'j'}\}\prod_{m=1}^{N(n-2)} (z - d_m)\big)^{1/N} } ~.
\end{align}
The correlator becomes
\begin{multline}\label{}
\langle \prod_{\nu=1}^n V_{\alpha^\nu,r^\nu_1,r^\nu_2,\mu^\nu_{ij},\bar\mu^\nu_{ij}} (z_\nu) \rangle_{SL(2N)}=\nonumber \\
|\Theta_\varphi|^2 \langle \prod_{\nu=1}^n |\det \{\mu^\nu_{ij}\}|^{2\alpha^{\nu}/N} e^{(2 \alpha^\nu b-\frac{1}{b}) \varphi} V^\text{SL($N$)}_{r^\nu_1} (g_1)V^\text{SL($N$)}_{r^\nu_2} (g_2) (z_\nu)\prod_{m=1}^{N(n-2
)}e^{\frac{1}{bN}\varphi(d_m)} \rangle
\end{multline}
where
\begin{align}\label{}
\Theta_{\varphi}= \det \{u_{ij}\}^N\left(\frac{\prod_{m, m'=1,m\neq m'}^{N(n-2)}(d_m-d_{m'})^{1/N^2}\prod_{\nu, \nu'=1,\nu\neq \nu '}^n(z_\nu-z_{\nu'})}{\prod_{m=1}^{N(n-2)}\prod_{\nu=1}^{n}(d_m-z_\nu)^{1+1/N}} \right)^{\frac{1}{4b^2}}\ .
\end{align}
The first factor is due to the background charge.
To proceed we now need to absorb the determinant one matrices ${{\cal B'}, {\cal B}'}^{\dagger}$ in a redefinition of $g_1,g_2$. This will give extra insertions, and to see this explicitly we consider the case $4=2+2$.
\subsection{The example $N=2$}
To make the transformation of $g_1,g_2$, we suggest to do it in terms of the simple steps given in appendix \ref{non-hol}.
As mentioned there, there are in principle many ways to do combine these steps, and correspondingly we would get different expressions for the same amplitude. We have not yet shown that these different expressions actually are equal, but reserve it for future research. However, it turns out that there is a particular choice where we can write all new operators in the correlators as insertions in points using bosonization, and we can avoid some very complicated insertions.
To do this we parameterize our SL$(2)$ fields as
\begin{align}
g_1&=\left (
\begin{array}{cc}
1 & -\gamma_1 \\
0 & 1
\end{array} \right)\left (
\begin{array}{cc}
e^{-\phi_1} & 0 \\
0 & e^{\phi_1}
\end{array} \right)\left (
\begin{array}{cc}
1 & 0 \\
-\bar \gamma_1& 1
\end{array} \right)\ ,\nonumber \\
g_2&=\left (
\begin{array}{cc}
1 & 0 \\
\gamma_2 & 1
\end{array} \right)\left (
\begin{array}{cc}
e^{\phi_2} & 0 \\
0 & e^{-\phi_2}
\end{array} \right)\left (
\begin{array}{cc}
1 & \bar \gamma_2 \\
0 & 1
\end{array} \right)\ ,
\end{align}
i.e. $g_1$ has been parameterized transposed inverse compared to appendix \ref{non-hol}.
We now use \eqref{eq:matrixformulations} to write
\begin{align*}
{\cal B'}=&\Bigg(\begin{array}{cc}
1 & \frac{u_{12}\prod_{l=1}^{n-2} (z - y^l_{12})}{u_{22}\prod_{l=1}^{n-2} (z - y^l_{22})} \\
0 & 1
\end{array} \Bigg)\Bigg(\begin{array}{cc}
\frac{\big(\det \{u_{i'j'}\}\prod_{m=1}^{N(n-2)} (z - d_m)\big)^{1/N} }{u_{22}\prod_{l=1}^{n-2} (z - y^l_{22})}
& 0 \\
0 & \frac{u_{22}\prod_{l=1}^{n-2} (z - y^l_{22})}
{\big(\det \{u_{i'j'}\}\prod_{m=1}^{N(n-2)} (z - d_m)\big)^{1/N} }
\end{array} \Bigg)\\
&\Bigg(\begin{array}{cc}
1 & 0 \\
\frac{u_{21}\prod_{l=1}^{n-2} (z - y^l_{21})}{u_{22}\prod_{l=1}^{n-2} (z - y^l_{22})} & 1
\end{array} \Bigg) ~.
\end{align*}
Thus we make the following field change
\begin{equation}\label{}
g_1=A g'_1 A^\dagger\ ,\qquad g_2=B g'_2 B^\dagger
\end{equation}
where
\begin{align*}
A&=\Bigg(\begin{array}{cc}
1 & \frac{u_{12}\prod_{l=1}^{n-2} (z - y^l_{12})}{u_{22}\prod_{l=1}^{n-2} (z - y^l_{22})} \\
0 & 1
\end{array} \Bigg)\left(\begin{array}{cc}
\frac{\big(\det \{u_{i'j'}\}\prod_{m=1}^{N(n-2)} (z - d_m)\big)^{1/2N} }{\big(u_{22}\prod_{l=1}^{n-2} (z - y^l_{22})\big)^{1/2}}
& 0 \\
0 & \frac{\big(u_{22}\prod_{l=1}^{n-2} (z - y^l_{22})\big)^{1/2}}{\big(\det \{u_{i'j'}\}\prod_{m=1}^{N(n-2)} (z - d_m)\big)^{1/2N} }
\end{array} \right)\nonumber\\
&:=\left(\begin{array}{cc}
1 & \frac{f_{12}}{f_{22}} \\
0 & 1
\end{array} \right)\left(\begin{array}{cc}
\frac{1 }{(f_{22})^{1/2}}
& 0 \\
0 & (f_{22})^{1/2}
\end{array} \right) ~ , \nonumber\\
B&=\Bigg(\begin{array}{cc}
1 & 0 \\
-\frac{f_{21}}{f_{22}} & 1
\end{array} \Bigg)\left(\begin{array}{cc}
(f_{22})^{1/2}
& 0 \\
0 & \frac{1 }{(f_{22})^{1/2}}
\end{array} \right) ~.
\end{align*}
Using the notation of appendix \ref{non-hol}, we can now compactly write the total change in the correlators in terms of the currents of $J_i^{\pm,z}$ constructed out of the two sl(2) generators $g_i$:
\begin{equation*}
\begin{split}
\langle \prod_\nu V(z_\nu) \rangle_{\text{SL}(4)}\ = \ &
\langle e^{\frac{i}{2 \pi} \oint_{\partial R_2} d z\,\left(\ln f_{22}(-J_1^z+J^z_2)\right)+\textrm{conj.}}\\
&\ \
e^{\frac{i}{2 \pi} \oint_{\partial R_1} d z\,\left( \frac{f_{12}}{f_{22}}J_1^+-\frac{f_{21}}{f_{22}}J_2^-+\frac{1}{b}\ln \frac{d(z)^{1/N}}{\prod_{\nu=1}^n (z - z_\nu)}J_{\phi}\right)+\textrm{conj.}}\prod_\nu V(z_\nu)|_{\gamma=\bar\gamma=0} \rangle_{4=2+2},
\end{split}
\end{equation*}
Here $J_\phi=\partial\phi$ and $R_1\subset R_2$ are regions containing all the vertex operators, singularities and branch cuts.
The integral around $R_1$ has to be performed first. The OPEs are taken from the $\textrm{SL}(N)\times\textrm{SL}(N)$ theory, and the interaction terms are simply transformed to absorb the $\mathcal{B}$-functions. The evaluation will give the extra insertions and also the field independent factors as explained in appendix \ref{non-hol}.
Firstly the $-\frac{f_{21}}{f_{22}}J_2^-$ transformation simply gives the following insertions in $y_{22}^l$, see eq. \eqref{eq:betainsertion}, and similarly for the transformation $\frac{f_{12}}{f_{22}}J_1^+$ since we have parameterized it dually
\begin{align}\label{eq:betainsertions}
\prod_{l=1}^{n-2} \exp\left(\frac{u_{21}\prod_{l'=1}^{n-2} (y^{l}_{22} - y^{l'}_{21})}{u_{22}\prod_{l'=1,l'\neq l}^{n-2} (y^l_{22} - y^{l'}_{22})}\beta_2(y^l_{22})+\frac{u_{12}\prod_{l'=1}^{n-2} (y^{l}_{22} - y^{l'}_{12})}{u_{22}\prod_{l'=1,l'\neq l}^{n-2} (y^l_{22} - y^{l'}_{22})}\beta_1(y^l_{22})\right)\ .
\end{align}
Next we perform the transformation $\ln f_{22}(-J_1^z+J^z_2)$. In principle we will get a line integral over the currents, but if we bosonize like in \eqref{eq:bosonization} this will give us extra insertions \eqref{eq:extrainsertionsbosonized}
\begin{align}\label{eq:extrainsertdeltafuncs}
\prod_{i=1}^2 \Theta_i\prod_{l=1}^{n-2}e^{ (\phi_i/b_i-X_i)(y_{22}^l)}\prod_{m=1}^{N(n-2)}e^{-\frac{1}{N} (\phi_i/b_i-X_i)(d_m)}
\end{align}
where $b_i=(k_i-2)^{-1/2}=(k-N-2)^{-1/2}$ and the constant factors are
\begin{align}\label{}
\Theta_i= \frac{\prod_{l,l'=1,l'\neq l}^{n-2} (y_{22}^l - y^{l'}_{22})^{k_i/4}\prod_{m,m'=1,m'\neq m}^{N(n-2)} (d_m - d_{m'})^{k_i/4N^2} }{\prod_{m=1}^{N(n-2)}\prod_{l=1}^{n-2} (d_m - y^{l}_{22})^{k_i/2N}}\ .
\end{align}
Here we have again removed some zeroes which cancel the infinities from creating the new insertions. Further we have to change the $\beta$s in the previous insertions \eqref{eq:betainsertions}. However, this create zeroes which should cancel infinities from when the new insertions \eqref{eq:extrainsertdeltafuncs} are in the same position as the $\beta_i$ insertions. To see this cancellation let us move the new insertions to $y_{22}^l+\epsilon$. We then have (focusing on of the $\beta_2(y^l_{22})$ insertion)
\begin{align}\label{}
& \exp\left(\frac{u_{22}\prod_{l''=1}^{n-2} (y^l_{22} - (y^{l''}_{22}+\epsilon))}{\big(\det \{u_{i'j'}\}\prod_{m=1}^{N(n-2)} (y^l_{22} - d_m)\big)^{1/N} }\frac{u_{21}\prod_{l'=1}^{n-2} (y^{l}_{22} - y^{l'}_{21})}{u_{22}\prod_{l'=1,l'\neq l}^{n-2} (y^l_{22} - y^{l'}_{22})}\beta_2(y^l_{22})\right)e^{ (\phi_2/b_2-X_2)(y_{22}^l+\epsilon)}\nonumber \\
& \qquad \stackrel{\epsilon\rightarrow0}{\rightarrow} \ :\exp\left(\frac{u_{21}\prod_{l'=1}^{n-2} (y^{l}_{22} - y^{l'}_{21})}{\big(\det \{u_{i'j'}\}\prod_{m=1}^{N(n-2)} (y^l_{22} - d_m)\big)^{1/N} }\beta_2(y^l_{22})\right)e^{ (\phi_2/b_2-X_2)(y_{22}^l)}:\ ,
\end{align}
since we have the OPEs
\begin{align}\label{}
(\beta_2)^p(z)e^{-X_2(w)}=(z-w)^{-p}:(\beta_2)^p(z)e^{-X_2(w)}:\ .
\end{align}
We can now write out the relation between WZNW amplitudes and the model with $4=2+2$ $W$-algebra
\begin{multline}\label{}
\langle \prod_{\nu=1}^n V_{\alpha^\nu,r^\nu_1,r^\nu_2,\mu^\nu_{ij},\bar\mu^\nu_{ij}} (z_\nu) \rangle_{\text{SL}(4)}= \\
|\Theta|^2 \langle \prod_{\nu=1}^n |\det \{\mu^\nu_{ij}\}|^{\alpha^{\nu}} e^{(2 \alpha^\nu b-\frac{1}{b}) \varphi} V^\text{SL($2$)}_{r^\nu_1} (A g_1 A^\dagger)V^\text{SL($2$)}_{r^\nu_2} (B g_2 B^\dagger) (z_\nu)\\
\prod_{m=1}^{2(n-2
)}e^{\frac{1}{2b}\varphi(d_m)}e^{-\frac{1}{2} ((k-4)^{1/2}\phi_1-X_1)(d_m)}e^{-\frac{1}{2} ((k-4)^{1/2}\phi_2-X_2)(d_m)}\\
\prod_{l=1}^{n-2}:e^{\frac{u_{12}\prod_{l'=1}^{n-2} (y^{l}_{22} - y^{l'}_{12})}{\det \{u_{i'j'}\}^{1/2}\prod_{m=1}^{2(n-2)} (y^l_{22} - d_m)^{1/2} }\beta_1(y^l_{22})}e^{ ((k-4)^{1/2}\phi_1-X_1)(y_{22}^l)}:\\
:e^{\frac{u_{21}\prod_{l'=1}^{n-2} (y^{l}_{22} - y^{l'}_{21})}{\det \{u_{i'j'}\}^{1/2}\prod_{m=1}^{2(n-2)} (y^l_{22} - d_m)^{1/2} }\beta_2(y^l_{22})}e^{ ((k-4)^{1/2}\phi_2-X_2)(y_{22}^l)}:
\rangle_{4=2+2} ~,
\end{multline}
where $\Theta$ is
\begin{multline}\label{}
\Theta=\det \{u_{ij}\}^2 \\
\frac{\prod_{\nu, \nu'=1,\nu\neq \nu '}^n(z_\nu-z_{\nu'})^{(k-4)/2}\prod_{l,l'=1,l'\neq l}^{n-2} (y_{22}^l - y^{l'}_{22})^{(k-2)/2}\prod_{m,m'=1,m'\neq m}^{2(n-2)} (d_m - d_{m'})^{(k-6)/8} }{\prod_{m=1}^{2(n-2)}\prod_{\nu=1}^{n}(d_m-z_\nu)^{3(k-4)/4}\prod_{m=1}^{2(n-2)}\prod_{l=1}^{n-2} (d_m - y^{l}_{22})^{(k-2)/2}}\ ,
\end{multline}
and the action is
\begin{align}\label{}
S_k [\phi,g_1,g_2]=& S_{k-2}^{\textrm{WZNW}}[g_1]+S_{k-2}^{\textrm{WZNW}}[g_2]+\frac{1}{2 \pi} \int d^2 z\, \partial\varphi\bar{\partial}\varphi+\frac{Q_{\varphi}}{4}\sqrt{g}\mathcal{R}\varphi\nonumber \\
&\pm \frac{1}{2 \pi} \int d^2 z\, \tr(\frac{1}{k}e^{-2b\varphi} g^{-1}_1 g_2)\ ,
\end{align}
with $b=1/\sqrt{2(k-4)}$ and $Q_{\varphi}=-4b-1/b$.
The currents for the $4=2+2$ $W$-algebra can be written in terms of our free fields explicitly, and the action is invariant under these.
We saw that the procedure already becomes very technical in the simplest example where the $W$-algebra corresponds to the partition $2+2=4$. However, we believe that the analysis generalizes to generic $N$, but leave the analysis for future work.
\section{SL$(2N+M|P)$ WZNW model and $W$-algebras}
\label{extention}
In this section, we would like to sketch a generalization of our results by finding the correspondence between the sl$(2N+M|P)$ WZNW model and the $W$-algebra corresponding to the partition
\begin{equation}
\underline {2N+M|P } \ = \ N\underline{2|0}+M\underline{1|0}+P\underline{0|1}\, .
\label{spartition}
\end{equation}
This will be done on the level of the action without considering the details of the vertex operators.
The Lie superalgebra sl$(2N+M|P)$ is best pictured as supertraceless $(2N+M|P)\times (2N+M|P)$ matrices.
The embedding of sl(2) that provides above decomposition of the fundamental representation is given by
\begin{align}\label{}
t^z=\frac{1}{2} \left (
\begin{array}{c|c|c}
\mathbb{I}_N & 0 & 0 \\ \hline
0 & -\mathbb{I}_N & 0 \\ \hline
0 & 0 & 0_{M|P}
\end{array} \right) , \ t^+= \left (
\begin{array}{c|c|c}
0 & \mathbb{I}_N & 0 \\ \hline
0 & 0 & 0 \\ \hline
0 & 0 & 0_{M|P}
\end{array} \right) , \ t^-= \left (
\begin{array}{c|c|c}
0 & 0 & 0\\ \hline
\mathbb{I}_N & 0 & 0 \\ \hline
0 & 0 & 0_{M|P}
\end{array} \right) .
\end{align}
The commutant of sl(2) is sl$(N) \oplus$ sl$(M|P) \oplus$ u(1) (except if $M=P+0$, then there is no u(1))
generated by the elements corresponding to the matrix
\begin{align}\label{}
X\ =\ \left (
\begin{array}{c|c|c}
A & 0 & 0 \\ \hline
0 & A & 0 \\ \hline
0 & 0 & B
\end{array} \right) \, .
\end{align}
This means that the $W$-algebra will have sl$(N)$ $\oplus$ sl$(M|P)$ $\oplus$ u(1) current symmetry
extended by some higher-dimensional fields. For example, there will be $2NM$ bosonic dimension-$3/2$ fields and
$2NP$ fermionic ones, and these transform in the tensor product of the fundamental of sl$(N)$ and the anti-fundamental
of sl$(M|P)$ plus its conjugate representation.
The superalgebra decomposes into $t^z$ graded components as follows
\begin{align}\label{}
Y\ =\ \left (
\begin{array}{c|c|c}
A_0 & A_1 & A_{1/2} \\ \hline
A_{-1} & B_0 & B_{-1/2} \\ \hline
A_{-1/2} & B_{1/2} & C_0
\end{array} \right) \, .
\end{align}
Here the components $X_i$ have grade $i$ for $X=A,B,C$.
Let us denote the u(1) generator normalized to have norm one by $t^0$, then the action can be constructed as follows.
Define a group valued field
\begin{align}\label{}
g\ =\ &\left (
\begin{array}{c|c|c}
\mathbb{I}_N & 0 & 0 \\ \hline
\gamma & \mathbb{I}_N & 0 \\ \hline
0 & 0 & \mathbb{I}_{M|P}
\end{array} \right)
\left (
\begin{array}{c|c|c}
\mathbb{I}_N & 0 & 0 \\ \hline
ba/2 & \mathbb{I}_N & b \\ \hline
a & 0 & \mathbb{I}_{M|P}
\end{array} \right)
e^{2\phi t^z+Xt^0}\left (
\begin{array}{c|c|c}
g_1 & 0 & 0 \\ \hline
0 & g_2 & 0 \\ \hline
0 & 0 & g_3
\end{array} \right)\nonumber \\
&\times\left (
\begin{array}{c|c|c}
\mathbb{I}_N & \bar a\bar b/2 & \bar a \\ \hline
0 & \mathbb{I}_N & 0 \\ \hline
0 & \bar b & \mathbb{I}_{M|P}
\end{array} \right)
\left (
\begin{array}{c|c|c}
\mathbb{I}_N & \bar\gamma & 0 \\ \hline
0 & \mathbb{I}_N & 0 \\ \hline
0 & 0 & \mathbb{I}_{M|P}
\end{array} \right)
\end{align}
then the WZNW action can be rewritten using the Polyakov-Wiegmann identity as
\begin{equation}
\begin{split}
S_k^{\textrm{WZNW}} [g]\ &= \ S_k^{\textrm{WZNW}} [g_1] +S_k^{\textrm{WZNW}} [g_2] + S_k^{\textrm{WZNW}} [g_3]+\\
&\quad +\frac{k}{4\pi}\int d^2z\, (\partial X \bar\partial X+2N\partial\phi\bar\partial\phi )+\\
&\quad +\frac{k}{2\pi}\int d^2z\, e^{2\phi} \text{tr}\bigl( (\bar\partial\gamma+\frac{1}{2}(\bar\partial b a -b\bar\partial a))g_1 (\partial\bar\gamma+\frac{1}{2}(\bar a\partial\bar b-\partial\bar a b)g_2^{-1})\bigr)+\\
&\quad + \frac{k}{2\pi}\int d^2z\, e^{\phi+\alpha X} \text{tr}\bigl(g_1 \partial\bar ag_3^{-1}\bar\partial a \bigr)+
\frac{k}{2\pi}\int d^2z\, e^{\phi-\alpha X} \text{tr}\bigl(\bar\partial b g_3 \partial\bar bg_2^{-1} \bigr)
~.
\end{split}
\end{equation}
Here we defined
\begin{equation}
\alpha \ = \ \sqrt{\frac{2N+M-P}{2N(M-P)}}\, .
\end{equation}
We then pass to a first order formulation by introducing
auxiliary matrix valued fields $\beta,\bar\beta$ and super-matrix fields (partially even partially odd)
$p,\bar p,q,\bar q$. Then after integrating these auxiliary fields in, the action is
equivalent to
\begin{align}
S_k^{\textrm{WZNW}} [g]\ &= \ S_0 + S_{\text{int}} ~, \nonumber \\
S_0 \ &= \ S_{k-N-M+P}^{\textrm{WZNW}} [g_1] +S_{k-N-M+P}^{\textrm{WZNW}} [g_2] + S_{k-2N}^{\textrm{WZNW}} [g_3]+ \nonumber\\
&\quad +\frac{1}{2\pi}\int d^2z\, (\partial X \bar\partial X+\partial\phi\bar\partial\phi
+\frac{\hat Q_{\phi}}{4}\sqrt{g}\mathcal{R} \phi
)+ \\
&\quad + \frac{k}{2\pi }\int d^2z\, \text{tr}(\beta\bar\partial\gamma)+\text{tr}(\bar\beta\partial\bar\gamma)
+\text{tr}(p\bar\partial a)+\text{tr}(\partial\bar a\bar p)
+\text{tr}(\bar\partial b q)+\text{tr}(\bar q\partial\bar b) ~, \nonumber\\
S_{\text{int}}\ &= \ -\frac{k}{2\pi }\int d^2z\, e^{-2 \delta \phi}\text{tr}(\beta g_2\bar\beta g_1^{-1})+
e^{-\delta \phi-\alpha ' X}\text{tr}((\frac{1}{2}\beta b+p)g_3(\bar p +\frac{1}{2}\bar b\bar\beta)g_1^{-1})+ \nonumber\\
&\quad + e^{-\delta \phi+\alpha ' X}\text{tr}((-\frac{1}{2}\bar\beta \bar a+\bar q)g_3^{-1}(q -\frac{1}{2}a\beta)g_2) \nonumber
\end{align}
with
\begin{align}
\delta^{-2} = N (k - 2 N - M + P) ~, \qquad
\alpha ' = \sqrt{2N}\delta \alpha ~, \qquad
\hat Q_\phi = \delta N (N + M - P ) ~.
\end{align}
Quantum corrections come from the Jacobians due to the change of variables as before.
The central charge of the above action is
\begin{align} \nonumber
c = 1 + 1 + 6 \hat Q_\phi^2 + 2 (N^2 +2 N M - 2 N P) + \frac{2 (N^2-1)(k-N-M+P)}{(k-N-M+P)-N}
\\
+ \frac{((M-P)^2 - 1) (k-2 N) }{k - 2 N - M + P}
= \frac{((2N + M - P)^2 - 1) k }{k - 2 N - M + P} ~,
\end{align}
which is that of the SL$(2N+M|P)$ WZNW model as it should be.
Since the interaction terms do not involve $\gamma, \bar \gamma$, we can integrate them out
when the inserted vertex operators are proportional to $\exp (\mu \gamma - \bar \mu \bar \gamma)$.
Then $\beta, \bar \beta$ are replaced by matrices ${\cal B}, - \bar {\cal B}$, which may be absorbed
by field redefinitions. This field redefinitions will yields extra insertions and shifts of momenta as before.
The reduced action is then
\begin{align}
S \ &= \ S_0 + S_{\text{int}} ~,\nonumber\\
S_0 \ &= \ S_{k-N-M+P}^{\textrm{WZNW}} [g_1] +S_{k-N-M+P}^{\textrm{WZNW}} [g_2] + S_{k-2N}^{\textrm{WZNW}} [g_3]+ \nonumber \\
&\quad +\frac{1}{2\pi}\int d^2z\, (\partial X \bar\partial X+\partial\phi\bar\partial\phi
+\frac{ Q_{\phi}}{4}\sqrt{g}\mathcal{R} \phi
)+ \nonumber \\
&\quad + \frac{k}{2\pi }\int d^2z\,
\text{tr}(p\bar\partial a)+\text{tr}(\partial\bar a\bar p)
+\text{tr}(\bar\partial b q)+\text{tr}(\bar q\partial\bar b) ~, \\
S_{\text{int}}\ &= \ -\frac{k}{2\pi }\int d^2z\, e^{-2 \delta \phi}\text{tr}(g_2 g_1^{-1})+
e^{-\delta \phi-\alpha ' X}\text{tr}((\frac{1}{2} b+p)g_3(\bar p +\frac{1}{2}\bar b)g_1^{-1})+ \nonumber\\
&\quad + e^{-\delta \phi+\alpha ' X}\text{tr}((-\frac{1}{2} \bar a+\bar q)g_3^{-1}(q -\frac{1}{2}a)g_2) \nonumber
\end{align}
with
\begin{align}
Q_\phi = \delta N (N + M - P ) + \delta^{-1}~.
\end{align}
If we perform proper rotations of $a,q$ and $b,p$, then half of the system decouples from the rest and becomes free. The remaining part of the action has $W$-algebra symmetry corresponding to the partition \eqref{spartition}.
In this way, correlators of the WZNW model can be written in terms of a theory with
$W$-algebra corresponding to the partition \eqref{spartition}.
\section{Conclusion and discussions}
\label{conclusion}
The present work is a continuation of \cite{HS,HS2,CHR} where path integral methods are used to establish relations
between WZNW theories and their $W$-algebras.
Given a WZNW theory of a Lie algebra, there are $W$-algebras for each inequivalent embedding of sl(2) in the Lie algebra.
Our path integral reduction essentially works if the resulting $W$-algebra is generated by fields of conformal dimension at most two.
There are many $W$-algebras that do not have this property and one of our main future goals is to understand this more general case.
A key example should be the $W_3$-algebra corresponding to the principal embedding of sl(2) in sl(3).
We already found a relation from SL(3) WZNW theory to the Bershadsky-Polyakov algebra, and as a next step we plan to investigate
the relation between the latter and $W_3$ Toda theory. Three different free field realizations of the Bershadsky-Polyakov algebra can be extracted from \cite{Feigin:2004wb} and one needs to investigate if one of them allows for our path integral techniques. Notice, that to actually construct a correlator correspondence in this case, we also need to solve the problem of the factorization of the vertex operators, expressed in bosonized variables, under the SP(2) rotation \eqref{eq:sp2trans}. There is no guarantee that the transformation of the vertex operators actually is local, and indeed the expression in eq. \eqref{eq:extraghostterms} looks more like a line operator. It is thus an important problem for future research to find the transformed vertex operators, and determine if the correlator dependence holds for point-operators like in the SL(2) case, or a generalization is needed.
Importantly, one can use the correspondence of Liouville theory and the SL(2) WZNW theory to prove the strong-weak duality
between the two-dimensional Euclidean black hole and sine-Liouville theory \cite{HS3} and its
supersymmetric analogue \cite{Creutzig:2010bt}. The proof proceeds roughly as follows. First, one embeds the gauged WZNW theory in the product
theory SL(2) $\times$ U(1), then one reduces SL(2) to Liouville and finally absorbs the additional degenerate fields in the action.
A possible generalization is to consider gauged WZNW models of type SL$(N)/\mathbb R^{N-1}$. The most important task is to
find a theory that has the same symmetry algebra as the coset theory. In the case of the SL(3) coset, the symmetry algebra has been exhaustively described in Example 7.10 of \cite{Creutzig:2014lsa}, it has 30 generators.
As mentioned in the introduction, the theories with $W_N$-symmetry appear in the context of the AGT correspondence \cite{Alday:2009aq,Wyllard:2009hg}. Interestingly, a relation between the SL$(N)$ WZNW model and $W_N$ Toda theory can be obtained by inserting surface operators into four-dimensional SU$(N)$ gauge theories \cite{Alday:2010vg,Kozcaz:2010yp}.
In the simplest case with $N=2$, two different interpretations of surface operator lead to the relation between the SL$(2)$ WZNW theory and Liouville theory with extra degenerate insertions \cite{Alday:2010vg}, and a close study was also done in a quite recent paper \cite{Frenkel:2015rda}. The type of surface operator is labeled by the partition of $N$, and then each choice leads to the $W$-algebra labeled by the same partition, such as, the Bershadsky-Polyakov algebra for $N=3$ \cite{Wyllard:2010rp,Wyllard:2010vi}.
The relation to our results definitely deserves further study.
\subsection*{Acknowledgements}
We are grateful to V.~Schomerus for useful discussions.
The work of YH was supported in part by JSPS KAKENHI Grant Number 24740170.
The work of TC is supported by NSERC grant number RES0019997. The work of PBR was partly funded by AFR grant 3971664 from Fonds National de la Recherche, Luxembourg.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,518 |
{"url":"https:\/\/www.freemathhelp.com\/forum\/threads\/story-problem-reese-has-225-in-10-bills-and-5-bills.48032\/","text":"# story problem: Reese has $225 in$10-bills and $5-bills #### era ##### New member Please help me with this story problem: Reese has$225 in ten dollar bills and five dollar bills. The number of five dollar bills is 15 more than the number of ten dollar bills. How many of each does he have? This is how I set it up, but I didn't get the right answer.\n\nx= number of ten dollar bills\ny=number of five dollar bills\n\n10x(+15)+5y=225\n\nThanks!\n\n#### arthur ohlsten\n\n##### Full Member\nlet x = number $10 bills let y= number$5 bills\n\nbut y=x+15 15 more $5 bills than$10 bills\n\n10x +5y=225\nbut y = x+15\n10x+5[x+15]=225\n10x+5x+75=225\n15x=150\nx=10\nthen y=25\n\nArthur\n\n#### era\n\n##### New member\nThank you very much!\n\n#### arthur ohlsten\n\n##### Full Member\nyour welcome. Now that you see the approach, be sure to do it yourself.\nYou learn from the errors that you make.\nArthur","date":"2019-03-22 22:57:47","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\": 0.3346816897392273, \"perplexity\": 5727.108291867546}, \"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-2019-13\/segments\/1552912202698.22\/warc\/CC-MAIN-20190322220357-20190323002357-00393.warc.gz\"}"} | null | null |
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**Quintessentials of Dental Practice – 43/44**
**Periodontology – 5/6**
# Periodontal Medicine — A Window on the Body
### Authors:
### Iain L C Chapple
### John Hamburger
#### Editors:
#### Nairn H F Wilson
#### Iain L C Chapple
###
Quintessence Publishing Co. Ltd.
### London, Berlin, Chicago, Paris, Milan, Barcelona, Istanbul, São Paulo, Tokyo, New Delhi, Moscow, Prague, Warsaw
British Library Cataloguing-in Publication Data
Chapple, Iain L. (Iain Leslie)
Periodontal medecine; a window on the body. - (Quintessentials of dental practice; 43/44. Periodontology; 5 )
1. Periodontics
I. Title II. Hamburger, John
617.6′32
ISBN: 1850973075
Copyright © 2006 Quintessence Publishing Co. Ltd., London
All rights reserved. This book or any part thereof may not be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, or otherwise, without the written permission of the publisher.
ISBN 1-85097-307-5
## Table of Contents
Title Page
Copyright Page
Foreword
Preface
Acknowledgements
Chapter 1 Establishing a Differential Diagnosis for Periodontal Manifestations of Systemic Diseases
Aim
Outcome
Terminology
Guiding Principles Behind Establishing a Diagnosis
The Diagnostic Pathway
The Complaint
The History of the Complaint
The Medical History
Social History
Family History
Sexual History
The extra-oral examination
The intra-oral examination
The Lesion
Location
Lesion size
Lesion shape
Attachment
Colour
Surface
Base
Consistency
Associated pathology
Localisation
The 'surgical sieve'
Special Investigations
The Differential Diagnosis
The Working Diagnosis
Definitive Diagnosis
Key Points
Further Reading
Chapter 2 The Role of Clinical Investigations
Aim
Outcome
Introduction
General Considerations
Indications for Investigation
Interpretation of Investigations
Specificity and Sensitivity of Tests
Biopsy
Microbiology
Identification of Bacteria
Identification of Fungal Organisms
Identification of Viruses
Blood and Serological Tests
Radiology and Imaging
Further Reading
Chapter 3 Gingival Colour Changes – Localised
Aim
Outcome
Red Lesions
Kaposi's Sarcoma
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Vascular Lesions
Clinical features
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Telangiectasia
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Erythroplakia (Erythroplasia)
Clinical features (Reichart 2005)
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
White Lesions
Trauma
Leukoplakia
Clinical features
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Leukokeratosis Mucosae Oris (White Sponge Naevus of Cannon)
Clinical features
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Squamous Cell Carcinoma
Lichen Planus
Candidosis
Clinical features
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Pigmented Lesions
Amalgam Tattoo
Clinical features
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Melanotic Macule (Ephelis)
Clinical features
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Naevi
Clinical features
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Malignant Melanoma
Clinical features
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Further Reading
Chapter 4 Gingival Colour Changes – Generalised
Aim
Outcome
Red Lesions
Desquamative Gingivitis
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Primary Herpetic Gingivostomatitis
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Streptococcal Gingivostomatitis
Orofacial Granulomatosis
Plasma Cell Gingivitis
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Other hypersensitivity reactions of the gingivae
Sturge Weber Syndrome
Clinical features
Differential diagnosis
Clinical investigation
Management options
White Lesions
Lichen Planus (see also Chapter 8)
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Other Generalised White Lesions
Pigmented Lesions
Extrinsic Staining
Racial Pigmentation
Clinical Features
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Drug-Induced and Heavy Metal Pigmentation
Addison's Disease
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Further Reading
Chapter 5 Gingival Enlargements – Localised
Aim
Outcome
The Epulides
The Fibrous Epulis
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
The Vascular Epulis
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Multiple/Disseminated Pyogenic Granulomata
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
The Giant Cell Epulis
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Congenital Epulis
Viral 'Wart-like' Lesions
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management
Neurofibroma
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management
Dental Appliance-Induced Hyperplasia
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Lateral Periodontal Abscess
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Gingival Abscess
Stitch Abscess
Localised Trauma (see also Chapter 7)
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival Sites
Differential diagnosis
Clinical investigation
Management options
Histiocytosis X
Haemangioma/AV Malformations
Kaposi's Sarcoma (KS)
Squamous Cell Carcinoma (SCC)
Metastatic Tumours
Lymphoma
Reactive Osteoma
Lesions Associated with PTEN-Hamartoma Tumour Syndromes
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Footnote
Further Reading
Chapter 6 Gingival Enlargements – Generalised
Aim
Outcome
Terminology
Fibrous Swellings
Hereditary Gingival Fibromatosis (HGF)
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management
Drug-induced Gingival Overgrowth (DIGO)
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management
Dental Appliance-induced Enlargement
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Delayed Gingival Retreat
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Oedematous Enlargements
Inflammatory Gingival Enlargement
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Angioedema (C1-Esterase Inhibitor Deficiency/Dysfunction)
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Granulomatous Enlargements
Orofacial Granulomatosis (OFG)
Clinical appearance
Aetiology
Involvement of non-gingival sites
Clinical investigation
Management options
Sarcoidosis
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Crohn's Disease
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Exophytic Swellings
Leukaemia
Pyostomatitis Vegetans
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Wegener's Granulomatosis
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Bony Swellings
Further Reading
Chapter 7 Localised Gingival Ulceration
Aim
Outcome
Definition
Traumatic Ulceration
Clinical appearance
Clinical symptoms
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Bacterial Infections
Necrotising Ulcerative Gingivitis (NUG)
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Tuberculosis
Syphilis
Viral Infections
Hand, Foot and Mouth Disease
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Varicella Zoster
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Cytomegalovirus
Deep Mycoses
Recurrent Aphthous Stomatitis
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Neoplastic Ulceration
Oral Squamous Cell Carcinoma
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Metastatic Disease
Further Reading
Chapter 8 Generalised Gingival Ulceration
Aim
Outcome
Vesicles and Bullae
Mucocutaneous Disease
Pemphigoid
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Pemphigus
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Lichen Planus
Haematological Disease
The Leukaemias
Clinical features
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
The Lymphomas
Clinical features
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Other Haematological Conditions
Further Reading
Chapter 9 Localised Gingival Recession
Aim
Outcome
Classification of Localised Recession Defects
Developmental Conditions
Dehiscence and Fenestration
Anatomical Tooth Position
Traumatic Defects
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Conscious Self-Mutilation (see also Chapter 7)
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Subconscious Self-Mutilation
Inflammatory/Infective Conditions
Defects Associated with Underlying Systemic Disease
Linear Morphoea (Localised Scleroderma)
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Histiocytosis-X
Eosinophilic Granuloma
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Necrotising Ulcerative Periodontitis (NUP) – see also Chapter 7
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Necrotising Ulcerative Stomatitis (NUS)
Drug-Induced Recession
References
Further Reading
Chapter 10 Generalised Gingival Recession
Aim
Outcome
Background
Aetiology of Gingival Recession
Systemic Disease with Generalised Recession as Manifestation Due to Destructive Periodontitis
Down Syndrome
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Papillon-Lefèvre Syndrome
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Hypophosphatasia
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Chronic Granulomatous Disease (CGD)
Chèdiak-Higashi Syndrome
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Ehlers-Danlos Syndrome
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Leukocyte Adhesion Deficiency (LAD)
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management options
Acatalasia
Infantile Genetic Agranulocytosis
Cohen Syndrome
Glycogen Storage Disease
DiGeorge Syndrome
Wiskott-Aldrich Syndrome
Histiocytosis X
Systemic Disease with Generalised Recession as Manifestation Independent of Periodontitis
Progressive Systemic Sclerosis (Scleroderma)
Clinical appearance
Clinical symptoms
Involvement of non-gingival sites
Differential diagnosis
Aetiology
Clinical investigation
Management options
Drug-Induced Gingival Recession
Cytotoxic chemotherapy drugs
Recreational drugs
Cytotoxic antimicrobials
Further Reading
Chapter 11 Miscellaneous Lesions
Aim
Outcome
Introduction
Uncontrolled/Unexplained Gingival Bleeding
Myelodysplasia
Clotting Factor Deficiencies
Idiopathic Thrombocytopenic Purpura (ITP)
Platelet Pool Storage Disease
Acute Leukaemia
Chronic Leukaemia
Thrombocytopaenia
Aplastic Anaemia
Thrombasthenia
Patients on Warfarin
Para-Gingival Swellings
Osteomas
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management
Gardner's Syndrome
Mandibular Tori
Annular Lesions
Erythema Migrans
Clinical appearance
Clinical symptoms
Aetiology
Involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management
Erythema Multiforme
Radiological Conditions or Lesions Associated with the Roots
1. Root Resorption
2. Inter- and Peri-radicular Radiolucencies
Systemic Sclerosis (scleroderma)
Periapical Cemental Dysplasia
Lateral Periodontal Cyst (Developmental)
Gingival Cyst
Incisive (Naso-palatine) Canal Cyst
Patent Nasopalatine Ducts
Aneurysmal Bone Cyst
Squamous Odontogenic Tumour
Ameloblastoma
Ameloblastic Fibroma
Histiocytosis-X
3. Inter- and Peri-radicular Radiopacities
Periapical Osteosclerosis
Condensing Osteitis
Hypercementosis
Cementomas
Cementicles
Cementoblastoma
Ossifying Fibroma
4. Radiolucent Lesions – Well Circumscribed Radiolucencies
Odontogenic Keratocyst
Radiological appearance
Clinical symptoms
Aetiology and involvement of non-gingival sites
Differential diagnosis
Clinical investigation
Management
Inflammatory Cyst
Neural Sheath Tumours
Multi-locular Radiolucencies
Odontogenic Keratocyst and Gorlin-Goltz Syndrome
Botyroid Cyst
Ameloblastoma
Odontogenic Myxoma
Giant Cell Tumour of Bone
Aneurysmal Bone Cyst
Arterio-venous Malformations (AVMs)
Sturge Weber Syndrome
Cherubism
Ossifying Fibroma
Poorly Defined Radiolucent Lesions
Osteomyelitis
Osteoradionecrosis
Intraosseous Carcinoma
Gingival Carcinoma
Ameloblastic Carcinoma
Radiolucent Lesions as Presentations of Systemic Disease
Histiocytosis-X
Multiple Myeloma
Non-Hodgkins Lymphoma (see Chapter 8)
Leukaemia
Generalised Radiolucencies
Hypophosphatasia
Hyperparathyroidism
Sickle Cell Anaemia
5. Radiolucent Lesions with Radiopacities
Periapical Cemental Dysplasia
Calcifying Odontogenic Cyst
Calcifying Epithelial Odontogenic Tumour (CEOT)
Adenomatoid Odontogenic Tumour
Odontomes
6. Radiopaque Lesions – Focal Radiopacities
Osteoma
Osteosarcoma
Generalised Radiopacities
Gardner's Syndrome
Sclerosing Osteomyelitis
Fibrous Dysplasia
Albright's Syndrome
Paget's Disease of Bone
Osteopetrosis
Hyperostosis
Further Reading
_This text is dedicated to my second daughter, Natasha Sophie Chapple, born 17th August 2004_
_Iain L C Chapple_
## Foreword
'Periodontal Medicine' is an intriguing title for the latest addition to the rapidly expanding, widely acclaimed _Quintessentials in Dental Practice_ series. Building on differential diagnoses for periodontal manifestations of systemic diseases and the role of relevant special investigations, this compact text of immediate practical relevance provides a unique consideration of gingival colour changes, enlargements, ulcerations and recession, not to forget a concluding miscellany of other gingival lesions.
This book is novel and therefore another pleasing first for the timely _Quintessentials in Dental Practice_ series. In common with all the other volumes in the Series, 'Periodontal Medicine' can be read and easily digested over a matter of a few hours. This time will be well spent, with a lasting legacy of enhanced insight and understanding of conditions of the periodontium. Once read, this book should not be put on a shelf to gather dust. In contrast, it should become a well-used aide-memoire to keep to hand in everyday clinical practice. Excellent clinical pictures generously illustrate the carefully crafted text, making this attractive volume another jewel in the Quintessentials crown. The authors are to be congratulated on the special qualities of this book.
Nairn Wilson
Editor-in-Chief
## Preface
Periodontal Medicine is a term used for different purposes in different parts of the world. In North America, it relates to the study of the dynamic relationship between periodontal diseases and systemic conditions, such as cardiovascular and cerebrovascular disease, pre-term labour and low-birth-weight babies, diabetes mellitus, osteoporosis and disorders of the respiratory tract. Such studies investigate the peripheral impacts of periodontal inflammation on systemic health and also the influence of systemic diseases on the progression of chronic periodontitis, such as type 2 diabetes mellitus, where evidence exists for a bi-directional relationship with periodontitis. However, in the UK and parts of Europe 'periodontal medicine' is a term used to describe the periodontal (and gingival) manifestations of medical conditions. This includes their investigation, diagnosis and therapeutic management and how management of the oral condition integrates with the patient's medical management as part of a holistic approach within defined care pathways. My own periodontal practice (ILC) relies heavily upon close working relationships with medical and surgical colleagues and joint patient management with bi-directional feedback, discussion and decision-making. In order of frequency, joint care is provided with Oral Medicine, Dermatology, Genito-Urinary Medicine, Cardiology, Clinical Immunology, Paediatric Medicine, Nephrology, Haematology, Gastroenterology, Geriatric Medicine, Ear/Nose/Throat and Maxillofacial Surgery.
This text therefore aims to provide the reader with an illustrated approach to managing the oral consequences of systemic diseases that present within and around the periodontal tissues. We have used the clinical appearance of the lesions as the starting point for discussion so that practitioners can follow a logical step-wise approach to differential and definitive diagnosis and subsequent management, either themselves, or through referral for secondary care. Some lesions are extremely common and others rare, and therefore each chapter tabulates the lesions that fall within its boundaries at the beginning of the chapter, but only discusses in detail the more common conditions. The final chapter discusses the less common non-plaque-induced conditions outwith their natural visual grouping.
### Outcomes of Reading This Text
This text will not deal with plaque-induced periodontal conditions, but will focus on non-plaque-induced lesions and their management. It is hoped that having read this text the reader will be able to:
* Recognise the broader scope of clinical periodontology and the importance of medical management in addition to the traditional surgical focus of the discipline.
* Recognise the importance of close liaison with colleagues in oral medicine and pathology.
* Take a systematic approach to medical history-taking that extends routine questions into certain relevant areas of enquiry that involve the body in general.
* Examine oral lesions systematically and use the findings of specific features of the lesion and associated signs and symptoms, to start formulating differential diagnoses.
* Identify non-periodontal sites that may be affected by the presenting condition and what features to note at those sites.
* Return to the verbal enquiry and identify relevant follow-up questions that may further clarify the findings of the clinical examination – re-focus the history.
* Understand when additional clinical investigations are indicated, which are appropriate and how to perform them.
* Be able to interpret the findings of routine clinical investigations (e.g. blood test results) and develop a sense of the potential implications for the patient.
* Advise the patient about the aetiology of non-plaque-induced periodontal lesions.
* Identify the need to refer for advice or treatment by dental or medical specialists.
* Understand how the routine treatment he or she provides may impact, either positively or negatively, upon the condition.
* Identify a range of therapeutic options for the patient and understand the need for regular review and re-appraisal of the condition as appropriate.
Iain L C Chapple
John Hamburger
## Acknowledgements
Iain Chapple wishes to thank his wife Liz and daughters Jessica and Natasha for their unconditional support and forbearance during the preparation of this book.
John Hamburger would like to thank his wife Ros and daughter Rachel for all their support and understanding during the preparation of this book.
The authors would also like to thank their colleagues within Periodontology and Oral Medicine. In particular Mrs Lorraine Williams and her staff who have approached the changes of the last 10-years so positively with enthusiasm, vigour and open minds. In addition, we are most grateful to our colleagues across a diverse range of medical specialties who offered their valued advice generously during the multidisciplinary management of our more complex patients.
We are indebted to Ms Jan Poller for her skillful proof reading of the manuscript and to Mr Michael Sharland and Ms Marina Tipton (Multi-media Services, Birmingham Dental School), Mr Paul England and Mr Jason Pike and colleagues (Clinical Illustration, Birmingham Dental Hospital). Thanks are also due to Mrs A Richards for permission to use Fig 2-2; Drs Barboza and Cunha and the British Dental Journal for the use of Fig 5-13; to Mr Mo Sandhar for the use of Fig 11-11; Mr D Glenwright for Figs 1-6, 1-10, 5-1, 5-8, 5-9, 5-15, 5-18, 10-10 and 10-21; Dr M Saxby for Figs 6-13a and b, Mr A Roberts for Figs 10-18 and 10-19; Mr M Milward for Fig 10-22; Mr J Rout for Figs 11-5a-e, 11-11, 11-12, 11-14, 11-15, 11-16, 11-18, 11-19, 11-24, 11-27a and b, 11-28, 11-29, 11-30, 11-31 and 11-32; Professor P Heasman for Figs 11-25a and b and Professor R Seymour for use of Fig 6-22.
## Chapter 1
## Establishing a Differential Diagnosis for Periodontal Manifestations of Systemic Diseases
### Aim
This chapter aims to provide the reader with a step-by-step guide to history-taking, examination and further investigation of non-plaque-induced lesions that arise withithe periodontal tissues, including the free and/or attached gingiva and associated oral mucosa, to help establish a differential diagnosis.
### Outcome
Having read this chapter the reader should appreciate the need for a forensic and systematic approach to establish differential diagnoses for oral and medical conditions that manifest within the periodontal and associated tissues.
### Terminology
A variety of clinical, procedural and pathological terms and descriptors are used throughout this chapter, and Table 1-1 defines these by category.
Table 1-1 **Terminology Used in Periodontal and Oral Medicine/Pathology** **Context** | **Terminology** | **Definition**
---|---|---
Clinical presentation or procedure | Symptom | Something the patient is experiencing or complaining of as a consequence of their condition.
| Sign | Something the clinician detects (visual, tactile or olfactory) that may help inform the diagnosis.
| Biopsy | Acquisition of human cells or tissues to aid diagnosis.
| Incisional biopsy | A biopsy involving partial removal of the lesion. This may be performed when malignancy is suspected and complete excision of the lesion would result in loss of key surgical landmarks.
| Excisional biopsy | A biopsy involving complete removal of the lesion.
| Fine needle biopsy /aspirate | Cells acquired using a fine cutting biopsy needle to obtain diagnostic material from within a lesion whose location or nature is such that surgical management should be delayed until microscopic diagnostic information is available.
| Broad needle biopsy | Tissue acquired using a broad cutting biopsy needle. The purpose is to acquire diagnostic material from within a lesion whose location or nature is such that surgical management should be delayed until microscopic diagnostic information is available. In addition, it is believed that acquisition of small numbers of cells may be of limited/no benefit to the pathologist.
| Swab | Use of a soft material (e.g. cotton pellet) to obtain infective material for culture and subsequent identification and testing (e.g. for antibiotic sensitivity) following culture.
| Smear | Use of a solid/sharp instrument to scrape away cells for microscopic examination.
| Cytology | Examination of individual cells (human or microbial) by microscopy, with or without special stains.
| Differential diagnosis | A list of possible diagnoses, ranked in order of the most likely to the least likely.
| Presumptive (working) diagnosis | A clinical diagnosis made in the absence of confirmatory information from additional clinical investigations and upon which therapy is based.
| Definitive diagnosis | The working and most likely diagnosis upon which therapeutic strategies are based.
Lesions and lesion descriptors | Ulcer | A breach in the epithelium to expose the underlying connective tissue.
| Erosion | Apartial breach in the continuity of an epithelial surface, which does not expose underlying connective tissue.
| Fissure | A narrow crack or slit (usually describes an ulcer shape).
| Vesicle | A small (<0.5cm) fluid filled lesion (not pus)
| Bulla | A larger (>0.5cm) fluid filled swelling (not pus)
| Blister | A fluid filled swelling (not pus)
| Papule | Raised lesion (<1.0cm diameter)
| Macule | Flat lesion (< 1.0cm diameter)
| Nodule | Raised lesion (>1.0cm diameter)
| Pedunculated | Lesion is borne/carried on a stalk/stem
| Sessile | Lesion is borne on a broad/flat base
| Sinus | A hole communicating a body cavity with the external environment
| Sinus tract | An epithelial lined tunnel linking an internal body cavity with the external environment
| Fistula | A communication between two body cavities
| Tumour | A neoplasm is an abnormal mass of tissue, the growth of which is uncoordinated with that of normal tissues, and that persists in the same excessive manner after the cessation of the stimulus which evoked the change
| Granuloma | A well-defined accumulation of modified macrophages (epithelioid cells), surrounded by lymphocytes and often with multinucleated giant cells.
| Epulis | A benign localised swelling of the gingiva usually affecting the interdental papilla
| Cyst | A pathological fluid filled (not pus) cavity.
Pathological terms or descriptors | Atrophic | Thinning of a tissue.
| Hyperplasia | An increase in the size of a tissue due to an increase in the number of its constituent cells.
| Hypertrophy | An increase in the size of a tissue due to an increase in the size of its constituent cells.
| Overgrowth | An increase in the size of a tissue due to an increase in the size and/or number of constituent cells and/or extracellular matrix components.
| Dysplasia | Disturbance in the normal maturation of a tissue.
| Anaplasia | Lack of differentiation of a tissue, characteristic of some tumour cells.
| Acantholysis | Separation of epithelial cells within the stratum spinosum.
| Acanthosis | An increase in thickness of the stratum spinosum.
| Leukoplakia | An adherent white patch or plaque that cannot be characterised clinically or pathologically as any other lesion (WHO).
| Erythroplakia | A bright red velvety plaque that cannot be characterised clinically or pathologically as any other recognisable condition (WHO).
| Hyperkeratosis | Excess deposition/formation of keratin within the stratum corneum of epithelium.
| Oedema | The collection of inflammatory fluid exudate within a tissue or body cavity.
| Angioedema | A diffuse oedematous swelling that may develop rapidly often involving the facial tissues and frequently, but not universally, results from a hypersensitivity reaction.
### Guiding Principles Behind Establishing a Diagnosis
The sub-headings used in the following section are those classically recommended for use in clinical practice and thus are standard procedures.
In periodontal medicine, the 'history' has a unique temporal relationship with the other diagnostic stages, because it should continue throughout the consultation process, i.e. never hesitate to return to and refine the history in the light of each new piece of information gathered.
### The Diagnostic Pathway
### _The Complaint_
The complaint is information that must be provided by and recorded in the patient's own words. It is important when the complaint involves a symptom such as 'pain' or 'soreness' to clarify the exact details of that complaint, e.g.:
* Character?
* Intensity/severity?
* Location?
* Spread?
* Associated signs or symptoms?
* Similar lesions elsewhere on the body?
### _The History of the Complaint_
The history of the complaint often holds the key to its diagnosis. For example, the appearance of the lesion or symptoms may be subsequent to the patient commencing new medication, implying a potential aetiological role for that medication. For an abscess that is of pulpal origin, any swelling may present after an episode of dental pain, whereas with an abscess of periodontal origin the swelling often precedes the development of pain; temporal issues are important. Other key issues to explore include:
* Onset?
* Duration?
* Aggravating factors?
* Relieving factors?
* Associated symptoms?
* Current status (i.e. improving or becoming worse?)
* Previous episodes?
Often by careful and detailed history taking it is possible to form a provisional diagnosis before even examining the subject, (but be wary of this approach to ensure you do not become biased). Some patients are good historians and others poor. It is important to record historical findings in a logical and succinct manner, which may be a challenge in the latter type of patient. It is good practice to return to a more focussed history after the clinical examination has provided pointers towards further questions which may help clarify the diagnosis.
### _The Medical History_
A thorough medical history is essential and the process should be a staged one. Some of the issues highlighted below impact upon patient management and others are important in establishing a diagnosis. Key systems to explore would be:
1. _Cardiovascular system_ – including cardiac murmurs, a history of rheumatic fever or infective endocarditis, blood pressure, cardiac surgery.
2. _The respiratory system_ – including evidence of atopic disease e.g. metal allergies, allergies to drugs, asthma, chronic obstructive airway disease or granulomatous conditions such as sarcoidosis, T.B.
3. _The central and peripheral nervous systems_ – a history of epilepsy, fitting, blackouts, neurological or neuropathology.
4. _The musculoskeletal system_ – diseases or disorders of bone or muscle, connective tissue diseases.
5. _The vascular system_ – evidence of haemorrhagic disease (e.g. idiopathic thrombocytopenic purpura – ITP), bruising, clotting disorders or disorders of the lympho-reticular system.
6. _Endocrine problems_ – diabetes, thyroid or sex hormone/androgen disorders, including adrenal function. If diabetic, how well controlled is the diabetes? How often does the patient check their glucose levels? Do they know their last HBA1C level? A well-informed and health-conscious diabetic should know their most recent HbA1C level (see Book 1 of this series, Chapple and Gilbert).
7. _Renal or urinary tract_ problems may be relevant to diagnosis or management. For example, patients who have been or are currently suffering from chronic renal failure may present issues surrounding:
* Calcification of cardiac valvular tissue secondary to chronic hypercalcaemia (need for antibiotic prophylaxis).
* Metabolism of drugs that may be considered for their oral disease.
* Hypertension (e.g. use of calcium channel blocking drugs, which may be associated with drug-induced-gingival-overgrowth).
8. _Hepato-biliary system_ disorders or diseases may impact on the diagnosis or management of oral lesions. Normally the question asked relates to a history of jaundice or hepatitis. Other issues may include:
* Platelet levels and the risk of haemorrhage.
* Associated systemic diseases e.g. primary biliary cirrhosis (PBC) is associated with secondary Sjögren's syndrome.
* Drug metabolism may also be affected (the cytochrome P450 enzyme system).
9. _Gastro-intestinal tract_ disease such as Crohn's, ulcerative colitis or coeliac disease may frequently present with oral manifestations. Malabsorption syndromes may lead to anaemia manifesting as oral ulceration or atrophic glossitis.
10. _Skin/dermatological disease_ commonly presents with gingival manifestations, including erosive lichen planus, mucous membrane pemphigoid, pemphigus, Papillon Lèfevre Syndrome (PLS).
11. _Connective tissue diseases,_ such as systemic sclerosis (scleroderma) or systemic lupus erythematosis (SLE) may have oral manifestations or require special precautions (e.g. 26% prevalence of cardiac valve damage is reported in SLE patients).
12. _Medications_ must be comprehensively recorded, as many cause oral mucosal as well as periodontal lesions, such as lichenoid drug reactions, oral pigmentation, xerostomia, gingival overgrowth or ulcerative conditions.
13. _Allergies_ may manifest within the gingivae and these may include coeliac disease, plasma cell gingivitis/mucositis (Fig 1-1).
14. _Immunodeficiency_ should be recorded whether primary or secondary in nature, because this may affect subsequent management.
15. _Prion disease_ should also be documented.
**Fig 1-1a** Plasma cell gingivitits affecting the gingivae and adjacent oral mucosa in a school telephonist who, when patch-tested, demonstrated sensitivity to a component of cleaning fluids used on the telephone mouthpiece.
**Fig 1-1b** The same condition as Fig 1-1a, affecting the hard palate. Similar lesions were evident in the larynx and the patient temporarily lost her voice. Both sets of lesions responded to systemic prednisolone therapy, prior to definitive diagnosis and implementation of preventive strategies.
It is very helpful to include 'catch all' questions in the medical history questionnaire such as: 'Are you under any active treatment with your doctor?' or 'Have you ever been investigated in or admitted to hospital?' or 'Is there anything in your medical history we have not covered?'
As with the history of the complaint, it is often necessary to return to specific aspects of the medical history following the clinical examination.
### _Social History_
Many aspects of a patient's social history may be of relevance to their presenting complaint.
_Specifically:_
_Smoking_ – Are they a current or former smoker or someone who has never smoked (Box 1-1)?
> Box 1-1 **Smoking History**
> _Current smoker_ | _Former smoker_ | _Never smoked_
> ---|---|---
>
> * * *
>
> What do you smoke? | How many years did you smoke? | Are you exposed to passive smoking?
> How many do you smoke? | How many per day? | At home or work?
> For how many years? | What did you smoke? | How long for each day?
> How soon after waking do you have your 1st cigarette? | When did you stop? |
> Have you tried quitting? | |
> What went wrong? | |
_Recreational drug use_ – Patients rarely volunteer the information that they are recreational drug users, but this may be relevant to their presenting pathology. Use of drugs such as amphetamines, cannabis, ecstasy or cocaine should be recorded and the method of use of cocaine may be particularly relevant (patients may rub it into their gingiva).
_Alcohol consumption_ – Record the number of units taken per week and frequency of intake.
_Diet_ – Diet can be particularly relevant to gingival and mucosal disease. This may involve recording fruit and vegetable intake, intake of erosive foods or those associated with allergy (e.g. cinnamaldehyde, benzoates).
_Stress_ – Stress may be associated with conditions such as lichen planus and recurrent oral ulceration. Appropriate and sensitive questioning as to lifestyle and life events may be helpful.
It is helpful to determine the patient's occupation at an early stage during the consultation. This may provide pointers towards causes of stress or indeed to other occupational hazards that may be contributing to the presenting complaint.
_Habits_ – Habits may be the cause of traumatic oral lesions that can present with alarming signs and may also be associated with peripheral lesions.
### _Family History_
The family history may be important for a number of reasons:
* It may help with establishing a diagnosis.
* It may provide indicators about prognosis and the natural history of the disease.
### _Sexual History_
This is important in those cases where a sexually transmitted infection or HIV disease is considered a possibility. An explanation to the patient as to why you need such information is mandatory.
### _The extra-oral examination_
The extra-oral examination should begin as soon as the patient enters the surgery. Dental surgeons must limit their examination to that of the clothed patient (Fig 1-2) and this does mean that their ability to arrive at a diagnosis may involve access to fewer pieces of evidence than their medical colleagues. Specifically note the patient's:
* Complexion – e.g. pallor/cyanosis/florid.
* Weight – may be relevant to increased risk of type 2 diabetes or other medical problems.
* Demeanour – does the patient look anxious, worried, embarrassed?
* Mobility – is this restricted physically or physiologically (case in Figs 5-5, 5-6 and 5-7).
* Skin condition – e.g. skin rashes.
* Facial appearance (Fig 1-3) – do not miss obvious facial indicators of systemic disease, e.g. SLE (rash).
* Verbal responses to questions – can reveal much about a patient's personality or fears.
* Exposed areas – examination of the clothed patient should also result in observation of hands, feet, scalp (Fig 1-4), neck, as well as facial features. Osteo- and rheumatoid arthritis, finger clubbing, nail pitting/dystrophy are all important clinical signs.
* Unexposed areas – sometimes it is necessary to view parts of a patient's skin that may be covered by clothing. This can be a sensitive matter and clinicians should use their discretion and ensure that a chaperone is present if appropriate. For example, lichen planus can present with itchy, purple papules on the flexor surfaces of the wrists or on the legs (Fig 1-5), which can help to determine a presumptive diagnosis.
* Lymphadenopathy – examination of the submandibular and submental regions along with the cervical chain of lymph nodes is vital when suspicious of infection or malignancy (Fig 1-6).
* Temporo-mandibular joints (TMJ) – TMJ examination may reveal clinical signs of joint pathology, parafunction or myofacial pain.
**Fig 1-2** Dental surgeons must limit their examination to that of a clothed patient.
**Fig 1-3** It is important not to miss obvious facial indicators of systemic disease.
**Fig 1-4** Viral wart along the hairline of the scalp of a male with gingival viral warts.
**Fig 1-5** Purple 'pruritic' lesions, classical of lichen planus affecting the skin over the fibula and tibia of a 60-year-old female with erosive gingival lichen planus.
**Fig 1-6** Right sub-mandibular lymphadenopathy in a five-year-old boy with oral Herpes simplex I infection.
### _The intra-oral examination_
The intra-oral examination should follow a strict protocol or regime, to ensure no areas of the mouth or oropharynx are overlooked.
Examine the remainder of the mouth first, before the presenting lesions to ensure nothing else is missed that may be relevant to the diagnosis. Never assume the pathology is singular, dual and triple pathology may be evident (Fig 1-7).
**Fig 1-7** Triple pathology: lichen planus, NUG and pyostomatitis.
Specifically ensure the following are examined systematically using good lighting and if necessary using magnification (Fig 1-8):
* Lips.
* Sulci.
* Cheeks (buccal mucosa parotid ducts, muscle attachments etc).
* Saliva flow from major ducts may be examined by massaging the gland and observing the colour and consistency of the saliva emerging from the relevant duct.
* Floor of mouth.
* Tongue – dorsum, anterior two-thirds, posterior third, lateral margins and ventral surface.
* Palate – hard and soft palate.
* Tonsillar region and oropharynx.
* Retro-molar regions.
* Gingivae.
* Teeth.
**Fig 1-8** The use of loupes with fibre optic illumination.
The floor of mouth and ventral surface of tongue shoulde be examined especially carefully as they are sites of sinister pathology.
When examining a lesion always supplement visual examination, by palpation, particularly the surface of the lesion. Be aware of any odour, e.g. foe-tor oris and if necessary use transillumination or bi-manual palpation. The latter, referred to as 'balloting' can be used for examining the submandibular salivary glands – one finger is placed in the lingual sulcus and one extra-orally to roll the gland between, enabling assessment of fixity, consistency and size.
### _The Lesion_
The following observations might be considered when examining lesions:
* Location?
* Nature of associated tissues?
* Size of the lesion?
* Shape of the lesion?
* Attachment to underlying structures?
* Colour?
* Surface characteristics?
* Nature of the base of the lesion?
* Consistency?
* Is there any associated pathology, locally or elsewhere on the body?
* Localised or generalised?
### _Location_
The location of the lesion also gives a clue as to its origin and it is important to determine whether it remains localised to the free or attached gingivae or extends beyond the mucogingival junction. In the latter case, a systemic condition is more likely (Fig 6-22).
When attempting to determine the origin of the lesion, consider the nature of those tissues that are present at the location, e.g. epithelial, vascular, neural, fibrous, fatty, glandular, bone, muscle.
### _Lesion size_
When assessing the size of a lesion, one should also take into account its rate of growth, which may sometimes be determined from the patient's history. The lesion may be too large to excise without significant peri-operative morbidity and may require reconstructive surgery due to its size or the involvement of other tissue planes.
### _Lesion shape_
The shape of a lesion may be very suggestive of its diagnosis. Care should be taken to examine the margins in particular. For example, an 'hour-glass shape' for an interdental swelling is indicative of a vascular epulis, and generalised shallow ulceration with ragged margins is characteristic, but not diagnostic of, mucosal herpes simplex (Fig 1-9).
**Fig 1-9** Gingival herpes simplex I infection.
### _Attachment_
The nature of attachment of a lesion to the underlying tissues provides important diagnostic and prognostic information. A sessile lesion has a broad base and is therefore less likely to be a true epulis, since the latter are often pedunculated (borne on a pedicle). It is also important to determine whether the superficial tissues are fixed to underlying tissues or not, since this may indicate an invasive lesion.
### _Colour_
Red lesions may be inflammatory or vascular, whereas paler lesions may have a fibrous core. A yellow appearance may signify pus or fatty tissue. Fig 1-10 illustrates a common variation in normal anatomy at an unusual site, so-called 'Fordyce spots', which are ectopic sebaceous glands. Purple lesions indicate vascularity and brown or grey lesions are associated with melanin or other forms of pigmentation (e.g. amalgam debris, or drug-induced pigmentation – Fig 1-11 and 1-12).
**Fig 1-10** Ectopic sebaceous glands or 'Fordyce spots' located across the muco-gingival junction in the LL34 area.
**Fig 1-11** Pigmentation of gingival margins in an Afro-Caribbean woman, secondary to medication with zidovudine (AZT) for HIV-disease.
**Fig 1-12** Purple pigmentation of the gingivae due to medication with the antibiotic minocycline for adolescent acne.
### _Surface_
The surface of a lesion provides important information about its nature. For example, an ulcerated surface may indicate trauma from opposing teeth, or necrosis due to an infection, trauma (e.g. acid burn – Fig 1-13) or indeed neoplasia. Surface keratosis may also indicate chronic trauma or true keratosis arising de-novo (Fig 1-14). It is also important to determine whether the surface can be removed or is adherent; a pseudomembrane would be typical of pseudomembranous candidosis.
**Fig 1-13** Surface necrosis of mucosal tissue following the misguided local application of aspirin for pain relief.
**Fig 1-14** Linear gingival leukoplakia in a 50-year-old female who had never smoked and was a non-drinker. The tissue showed mild dysplasia histologically and an absence of candidal infection.
### _Base_
The base of a lesion may be helpful in provisional diagnosis. The granularity of an ulcer base suggests potentially sinister pathology.
### _Consistency_
It is important always to 'feel' a lesion to determine its consistency or content. Mucosal and gingival lesions should be soft to touch, hard lesions (induration) may be sinister or simply indicate underlying bone. Fluctuant lesions contain fluid, which may be pus or cyst fluid, and firm lesions may contain fibrous tissue.
### _Associated pathology_
A vital part of the examination involves recognition of when an oral lesion may be related to disease elsewhere in the body. Such associated pathology may be visible as a sign (Fig 1-5) or may take the form of a symptom the patient may have withheld due to a perceived irrelevance to their gingival condition. For example, the six-year-old girl shown in Fig 1-15 had kidney pains and recurrent cystitis. Her kidney and liver function was mildly abnormal. However, consideration of the desquamative gingivitis, alongside abnormal renal and hepatic function, necessitated investigation for systemic lupus erythematosis (SLE), by serological methods (Chapter 2). Other conditions, such as Sjögrens syndrome, can be associated with primary biliary cirrhosis (PBC), and lichen planus can be associated with chronic active hepatitis, albeit rarely.
**Fig 1-15** Desquamative gingivitis presenting in a six-year-old girl. Lichen planus has been described in children, but is extremely rare. Other potential diagnoses in this case included SLE.
### _Localisation_
Diagnostic clues can be derived from the distribution of the lesion(s). For example, solid tumours are more likely to be single as opposed to multiple (it is, however, acknowledged that approximately 20% of cases of oral squamous cell carcinoma may demonstrate multiple primaries). A bilateral lesion situated over major blood vessels may imply metastatic spread of a tumour (Fig 3-3). More widespread lesions tend to be associated with a systemic problem, for example drug-induced (Fig 6-9), hereditary (Fig 6-1), infective, immunological (Fig 1-1) or inflammatory (Figs 5-5 and 6-20) in nature.
### _The 'surgical sieve'_
When attempting to formulate a differential diagnosis prior to performing special investigations, there may be occasions when the only thought entering the mind is 'I haven't got a clue' (Fig 1-16). Under these circumstances there are two options:
* Apply a surgical sieve.
* Refer the patient, providing as much information about the condition/lesion as possible (see above).
**Fig 1-16** The 'Surgical Sieve'
The surgical sieve comprises a series of headings, committed to memory, and which help to spark ideas and thoughts from generic principles. A mnemonic may be used to help memorise the sieve (Box 1-2).
> Box 1-2
> _Heading_ | _Mnemonic_ | _Example_
> ---|---|---
>
> * * *
>
> Metabolic/endocrine | M | Addisons disease (gingival pigmentation)
> Inflammatory | I | Gingival angioedema (immunological)
> Neoplastic | N | Gingival carcinoma
> Infective | I | Gingival candidosis
> Drug-induced | D | Gingival overgrowth or pigmentation
> Hereditary | (H) | Hereditary gingival fibromatosis
> Immunological | I | Plasma cell gingivitis
> Not otherwise specified | N |
> Trauma | T | Thermal, chemical or physical causes of ulceration
### _Special Investigations_
This subject will be covered in Chapter 2.
### The Differential Diagnosis
The objective of arriving at a 'differential diagnosis' is threefold:
* It informs the types of special clinical investigations required.
* It provides clinical details for the pathologist (or haematologist) of the most likely clinical diagnoses.
* It enables initial therapeutic strategies to be formulated. These may also serve a diagnostic function – for example, a swelling that responds to antimicrobial therapy, is most likely to be infective in origin.
Rarely, histological investigations may not help to determine a definitive diagnosis. The clinician may frequently be faced with a pathology report that requires careful interpretation alongside the historical and clinical information gathered. This approach is, in the author's opinion, the correct one, as exemplified by the case in Fig 1-17. In this case a vascular tumour was suspected clinically, but the histopathology was consistent with a benign pyogenic granuloma (Fig 1-18). The lesion spread and developed apparent satellite lesions within one week of the first biopsy (Fig 1-19) and a re-biopsy was performed. At review, the histopathology was still that of a benign vascular lesion, but the tumour was becoming exophytic (Fig 1-20). The urgency of the situation forced the clinician to re-explore a previously negative sexual history, and to ensure that the patient was clear that a diagnosis of Kaposi's sarcoma and therefore clinical AIDS was suspected. The patient subsequently revealed that he was homosexual and had practised unsafe sex. Counselling and a HIV serology test were immediately performed and HIV infection confirmed. On the basis of the historical and clinical information available, but contrary to the histopathological diagnosis, radiotherapy was performed and the lesion resolved (Fig 1-21).
**Fig 1-17** Localised red lesion affecting gingivae, but extending beyond muco-gingival junction.
**Fig 1-18** Histopathology shows a benign vascular lesion characteristic of a pyogenic granuloma.
**Fig 1-19** The lesion in Fig-1-17 one week post-primary incisional biopsy.
**Fig 1-20** The lesion in Fig 1-17, 1 week after the second biopsy. The lesion has become exophytic and is spreading rapidly. The colour change reflects underlying areas of thrombosis and necrosis and blood congestion.
**Fig 1-21** The treated lesion from 1–17 following external beam radiotherapy (in 1990) and residual pigmentation.
### The Working Diagnosis
A presumptive (clinical) diagnosis is often sufficient for common lesions to enable the safe implementation of a therapeutic strategy. For example, desquamative gingivitis, where there is clear evidence of mucosal lichen planus (Fig 1-22), may be a presumptive clinical diagnosis for two reasons:
* Gingival biopsies are often of no diagnostic value for conditions such as lichen planus, because the inevitable presence of inflammation within the connective tissues, due to plaque accumulation, masks the more subtle features of lichen planus. Where possible, non-gingival biopsies should be employed (Fig 1-23).
* The clinical diagnosis may be so obvious (Fig 1-22) that it may not be appropriate to expose the patient to further investigations.
**Fig 1-22** Reticular lichen planus of the left buccal mucosa is often a presumptive diagnosis. This should not be the case for erosive lichen planus, where a small but nevertheless increased risk of malignant transformation exists.
**Fig 1-23** A biopsy has been taken below the muco-gingival line, to avoid plaque-induced gingival inflammation masking features of lichen planus.
A further example would be pseudomembranous candidosis (Fig 1-24), where the implementation of topical antifungal therapy may resolve the condition before cytological results become available.
**Fig 1-24** Florid pseudomembranous candidosis (thrush) of the palate in an HIV-positive patient.
### Definitive Diagnosis
A definitive diagnosis is one in which the diagnosis is as certain as possible based on a thorough history, examination and investigation. Such a diagnosis forms the basis of treatment strategies, but it is vital to remember that diagnoses can change during the course of a disease. For example, a patient initially managed for localised aggressive periodontitis (LAP), may in later life develop a chronic periodontitis due to local risk factors encouraging plaque accumulation (e.g. interproximal recession due to former LAP). Squamous cell carcinoma may develop within erosive lichen planus.
Also bear in mind that multiple pathology may be concurrent in the same patient and therefore always keep an open mind as to differential, presumptive and working diagnoses. The patient illustrated in Fig 1-7 has triple pathology at presentation:
* NUG.
* Erosive lichen planus.
* Pyostomatitis vegetans.
### Key Points
* Approach each case with an open mind and be prepared for changes in diagnosis or multiple pathology.
* Explore all aspects of the patient's complaint and its history in a forensic manner.
* Be aware of the need to follow certain branches of the medical history to their conclusion and recognise when to terminate other lines of questioning.
* Always return to the history in the light of new information and be prepared to re-appraise matters.
* Remember to collect family, social and lifestyle histories, including habits, diet, tobacco and alcohol consumption, stress and coping behaviours for stress.
* Examine all visible areas of the clothed patient and do not miss the obvious.
* Use a systematic and thorough approach to the intra-oral examination with good lighting and magnification.
* Use all senses and don't just rely upon what you can see.
* Record your findings carefully and logically.
* Examine associated pathology.
* Use of the surgical sieve can be a helpful approach.
* Always seek a second opinion whenever in doubt.
### Further Reading
Chapple ILC, Gilbert AD (eds). Understanding Periodontal Diseases: Assessment and Diagnostic Procedures in Practice. London: Quintessentials, 2002:67.
## Chapter 2
## The Role of Clinical Investigations
### Aim
The aim of this chapter is to allude to those clinical investigations that are of value in formulating a definitive diagnosis.
### Outcome
Having understood this chapter, the reader should be aware of those relevant investigations that are used to either confirm diagnoses or add information to help formulate a definitive diagnosis. The reader will be aware of the need to identify appropriate investigations for particular conditions and, in addition, understand the limitations and interpretation of those investigations.
### Introduction
The number of investigative procedures now available is extensive and continues to expand with advances in technology. It should be borne in mind that while investigations are a most important step in the clinical management of many patients, they must not be used as a substitute for a detailed clinical history and examination.
### General Considerations
Before embarking upon any investigative procedure, a variety of general factors need to be considered. These include the nature and safety of the test, its potential benefit to the patient, any possible adverse effects, its cost effectiveness and whether the result is likely to alter the management of the patient's condition. Undertaking investigative procedures for the sake of interest is unacceptable. The patient must be fully informed of the need for the investigation, its advantages and disadvantages, including possible adverse events, before proceeding. As in all aspects of patient care, the balance of risk of an investigation must favour the patient.
When requesting investigative procedures, the following considerations should be borne in mind:
* What information is required?
* Which test(s) will provide that information?
* How are the results interpreted?
### Indications for Investigation
Whilst investigative procedures are predominantly used to support or confirm a clinical suspicion or diagnosis, they may also be used to:
* exclude abnormalities.
* monitor disease activity/progression.
* measure response to therapy.
### Interpretation of Investigations
The results of any investigation must be interpreted with caution so as to avoid drawing erroneous conclusions that may lead to inappropriate patient management. For example, in serological investigations:
* Slightly abnormal results should be repeated for confirmation. They may be within the methodological variance for the assay.
* Compare results with previous values (if available) to ascertain any trend or change.
* Compare results with other associated parameters to ascertain consistency.
* Identify possible artefacts.
The normal range is usually taken as the mean ± 2 standard deviations, i.e. 95% of the population fall within that range. It is therefore important to recognise that results outside the normal range may be 'normal' for a particular patient and may not signal anything untoward, especially if the result is not corroborated by other test results.
### _Specificity and Sensitivity of Tests_
It is important when interpreting investigations to have some understanding of the specificity and sensitivity of the investigations concerned. An important area for such considerations is when testing for autoantibodies. For example, the finding of certain autoantibodies does not necessarily confirm the presence of a particular disease, whilst equally, their absence may not preclude that disease.
The specificity of a test is defined as:
* The percentage of people without the disease who have a negative test result, e.g. 95% specificity implies 5% false positive results.
The sensitivity of a test is defined as:
* The percentage of people with the disease who have a positive test result, e.g. 95% sensitivity implies 5% false negative results.
### Biopsy
Tissue biopsy and subsequent histological examination remains a particularly valuable investigation in the diagnosis of gingival and oral mucosal disease. Although the technique is usually straightforward, it is essential to ensure that the pathologist receives a diagnostic sample that is not compromised, either by injection of local anaesthetic solution directly into the biopsy specimen, or traumatic handling of the tissue either at the time of the surgery or subsequently.
As with all clinical investigations, the request form that will accompany the specimen must be accurately and fully completed, providing details of the clinical features of the condition, the patient's medical history, including details of any medication and a differential diagnosis.
When undertaking gingival biopsies, it is important to consider that histological interpretation can be confounded due to the level of background inflammatory change that is usually present within the gingival tissues. For this reason, it may be preferable to biopsy alternative sites if appropriate (Fig 2-1).
**Fig 2-1** Slide demonstrating a suture placed at a para-gingival biopsy site. A gingival biopsy has been avoided to provide a representative sample devoid of the chronic inflammation often found within gingival biopsies secondary to plaque accumulation.
The following considerations will help ensure that a diagnostic biopsy sample is obtained:
* Avoid injecting local anaesthetic directly into the tissue to be removed.
* Do not tear or crush the tissue during the procedure.
* Excise a representative sample of the lesion.
* Additionally, excise any areas that differ from the overall appearance of the lesion. This may necessitate multiple sampling.
* Remove an adequate size of tissue. Aim for approximately 8–10mm in diameter and ensure that subepithelial tissues are included.
* Avoid biopsying an ulcer alone – by definition there will be no epithelium and this will compromise histological diagnosis.
* Perilesional tissue is often of diagnostic value.
* Place the specimen on a piece of card (e.g. the suture card) or paper, cut-side down, to reduce curling of the specimen and shrinkage.
* Ensure that the specimen is put into an appropriate volume of fixative if routine histological examination is required.
* Ensure the specimen is not put into fixative if special staining techniques such as direct immunofluorescence are to be employed. Fresh tissue should be transported to the laboratory in saline-soaked gauze or appropriate transport media such as 'Michel's' medium. Direct immunofluorescence may be particularly useful when autoimmune vesiculobullous disease is suspected (Fig 2-2).
**Fig 2-2** Direct immunofluorescence staining of mucous membrane pemphigoid showing IgG deposition at the basement membrane.
The histological findings do not always support the clinical diagnosis and in some cases, ultimately a judgement must be made as to the definitive diagnosis. A high index of suspicion is always advisable, particularly in those circumstances where sinister pathology is suspected clinically, but not supported histologically. Reappraisal of the original biopsy specimen with consideration of a further biopsy or biopsies is appropriate in such cases.
### Microbiology
When infection is suspected, various investigations may be helpful to identify the aetiological agent.
### _Identification of Bacteria_
Culture of organisms from oral swabs is frequently undertaken and combined with antibiotic sensitivity testing. However, results can be difficult to interpret because of the background level of oral commensal organisms. The initially sterile swab should be moistened in sterile normal saline or water to enhance the harvest and then passed over the area of interest before being placed back in its sterile container and sent to the laboratory without delay. If the patient is already taking antibiotics, this will confound the results.
Culture of specific periodontal bacteria requires the use of dedicated transport and culture media for fastidious organisms, together with the appropriate laboratory expertise, and is rarely indicated. Recombinant DNA probe technology overcomes these issues but as selection of antimicrobial regimes remains somewhat broad, rather than being highly targeted at specific species, there are few indications for such investigations in routine clinical practice.
Other tests, such as identification of specific serum antibodies, can also be undertaken. Rapid immunological assays to identify either specific antigens or antibodies are also available, but not commonly indicated for most gingival pathologies.
Gingival smears can also be used to identify specific organisms under the light microscope.
### _Identification of Fungal Organisms_
For identification of the presence of fungal organisms, such as Candida species, swabs can also be used. They require to be cultured on the appropriate selective growth medium (Sabouraud's agar), but since commensal carriage of these organisms is common (50% of individuals), a positive culture per se does not necessarily indicate active infection. Semi-quantitative methods such as imprint cultures and oral rinses can also be used, provided the receiving laboratory has the necessary equipment to perform the test. A simpler alternative is the smear, where a flat plastic instrument is used to gently scrape the surface mucosa, the material so removed being smeared on a glass slide and fixed. Subsequent staining with PAS or Gram stains and examination under the light microscope will show the fungal organisms (Fig 2-3). The presence of the mycelial form of the organism is suggestive of clinical infection rather than commensal carriage.
**Fig 2-3** PAS-stained smear demonstrating hyphal and yeast forms of Candida albicans.
Speciation of organisms can also be undertaken, using commercial kits that may differentiate species on the basis of germ tube production or fermentation of sugars.
### _Identification of Viruses_
The presence of a virus can be confirmed by various techniques, including that of culture. However, obtaining a viral culture is a more complex process than that for bacteria and fungal organisms, as it requires the provision of live cells as opposed to simple growth media.
Tissue biopsy or cytology may be helpful as the presence of cytopathic effects (e.g. Mulberry nuclei) indicates the presence of viral pathology, although not necessarily a specific virus type (Fig 2-4). Additionally, harvesting vesicle fluid and subsequent electron microscopy can be used to identify virions.
**Fig 2-4** Smear demonstrating cytopathic effect in Herpes simplex infection (Tzanck cells and ballooning degeneration).
Serological testing for specific acute and convalescent antibody titres can be used to demonstrate viral infection, with a four-fold rise in antibody titre being taken as a positive result. However, as the result becomes available once the patient has recovered, such testing is of limited practical value in managing the patient. Newer, rapid techniques that identify specific viral antigens or antibodies, based on immunofluorescence or enzyme linked immunosorbent assays (ELISA) are now available which abrogate the requirement for paired samples.
Molecular biology techniques also allow rapid identification of viral nucleic acid, for example, the polymerase chain reaction or in situ hybridisation. Both techniques provide rapid highly specific results but require specialised equipment (Fig 2-5).
**Fig 2-5** In situ hybridisation demonstrating the presence of Epstein Barr virus.
It is essential to discuss the case with the virology laboratory to ensure that the most suitable test is requested and that the appropriate sample is sent to the laboratory.
### Blood and Serological Tests
A large range of tests is available. Laboratories will often group tests together to allow broad screening for a variety of disease entities. This can encourage some clinicians to effectively undertake a 'fishing exercise' looking for a range of serological parameters, as opposed to being specific in their requests. This is not good clinical practice, being wasteful of resource and is usually unhelpful. As always, investigations must be regarded as an adjunct to undertaking a careful history and examination and not a replacement for these skills.
When requesting blood or serological investigations, the specimen must be collected in the appropriate container for the required test. If there is uncertainty as to the nature of the blood collection tube required, advice should be sought from the laboratory. Additionally, some specimens require to be mixed with an anticoagulant contained within the blood collection tube. Failure to do this will result in a sample that cannot be used for the assay. Prolonged venous stasis should be avoided during venepuncture, as this can introduce inaccuracies in the measurement of certain components, especially those that are bound to plasma proteins such as calcium.
Table 2-1 and Table 2-2 provide a brief summary of some of the more frequently requested blood and serological investigations used in specialist practice.
Table 2-1 **Blood investigations commonly used in periodontal medicine** **Investigation** | **Indication** | **Comments**
---|---|---
Full blood count | Anaemias/myeloproliferative or myelosuppressive disease |
Serum ferritin | Iron deficiency anaemia/iron overload | Ferritin also behaves as an acute phase marker – elevated levels do not necessarily indicate iron overload
Serum folate | Oral ulceration, candidosis, stomatitis
Red cell macrocytosis ± anaemia apparant from full blood count | Serum folate levels are labile. Red cell folate is a more stable measure but more costly to perform. Never give folate supplementation to patients with low serum B12 levels. This will cause remaining B12 stores to be metabolised, resulting in possible neurological complications
Serum B12 | Oral ulceration, candidosis, stomatitis.
Red cell macrocytosis ± anaemia /pancytopaenia apparent from full blood count | Low B12 is rarely due to dietary deficiency. Consider other causes such as pernicious anaemia, ileal disease or partial gastrectomy
Blood glucose | Oral candidosis
Multiple periodontal abscesses | If random blood glucose abnormal, consider fasting glucose level
Clinical chemistry (including liver function tests) | Suspected metabolic bone disease, electrolyte disturbance, liver, renal & endocrine, disease e.g. alkaline phosphatase for hypophosphatasia |
Liver function tests | |
Erythrocyte sedimentation rate | Of little diagnostic value in most circumstances | Very non-specific index of inflammation. More useful in tracking disease activity and response to therapy. ESR is helpful in supporting a diagnosis of temporal arteritis
C reactive protein | As for ESR |
Table 2-2 **Autoantibody investigations commonly used in periodontal medicine** **Autoantibody Tests** | **Disease Association** | **Comments**
---|---|---
Anti-nuclear antibodies | Systemic lupus erythematosus (SLE) | May also be found in other connective tissue diseases but not in discoid lupus erythematosus (anti-double strand DNA antibodies are diagnostic of active SLE and high titres carry a poor prognosis.)
Rheumatoid factor | Seropositive rheumatoid arthritis | May be found in other connective tissue diseases
Antibodies to extractable nuclear antigens (ENA) | Sjogren's syndrome (Anti-Ro, & /or Anti-La) Systemic sclerosis (ScL70) SLE (anti-Ro, anti-Sm) | A family of antibodies to non-nucleoprotein soluble nuclear antigens producing a speckled pattern of immunofluorescence.
ANCA–C (Cytoplasmic staining) | Wegener's granulomatosis | ANCA–C (anti-proteinase 3) has a high specificity for Wegener's granulomatosis.
ANCA–P (Perinuclear staining) | Granulomatous conditions such as Crohn's disease | ANCA–P targets myeloperoxidase
Anti-gastric parietal cell | Pernicious anaemia | High sensitivity but low specificity
Anti-intrinsic factor | Pernicious anaemia | High specificity
Anti-basement membrane zone | Pemphigoid |
Anti-intercellular cement | Pemphigus | Anti-desmoglein 3 antibodies are found in mucosal and cutaneous pemphigus. Anti-desmoglein 1 occurs predominantly in cutaneous pemphigus
Anti-endomysial
Anti-tissue transglutaminase | Coeliac disease
Coeliac disease | These are much more sensitive and specific markers for coeliac disease and have largely replaced anti-reticulin and IgA isotype anti-gliadin antibodies
Identification of autoantibodies does not necessarily indicate the presence of the disease that is usually associated with that autoantibody. Conversely, lack of circulating antibodies does not preclude the diagnosis of the associated disease.
Considerations such as specificity and sensitivity of the test, age and gender of the patient and titre of the antibody are important in test interpretation. The finding of weak positives, as for any test, must be interpreted with great caution and corroborative evidence can be helpful in this regard.
It is most important that the patient is not stigmatised as a consequence of inappropriate interpretation of the results of clinical investigations.
### Radiology and Imaging
This subject is beyond the scope of this text and is comprehensively covered in Panoramic Radiology and 21st Century Dental Imaging of this series. Additionally, Chapter 11 addresses periodontal conditions with radiological features.
### Further Reading
Horner K (ed). Panoramic Radiology. London: Quintessential, 2005.
Horner K (ed), Twenty-first Century Dental Imaging. London: Quintessentials, 2005.
Marshall WJ, Baugert SK. Clinical Chemistry. 5th edition. London: Mosby, 2004.
McGhee M. A Guide To Laboratory Investigation. 4th edition. Oxford: Radcliffe Medical Press, 2003.
## Chapter 3
## Gingival Colour Changes – Localised
### Aim
The aim of this chapter is to describe those gingival colour changes that are localised in one area or, in some cases, several discrete areas of the gingivae.
### Outcome
Having read this chapter, the reader should have an increased awareness of the nature of local gingival colour changes, their clinical significance and appropriate management, including differential diagnosis and treatment if indicated. The contents of this chapter are summarised in Table 3-1.
Table 3-1 **Summary table – localised gingival colour changes** **Major Categories** | **Sub Categories** | **Frequency of Condition** | **Management Setting**
---|---|---|---
Red lesions | Kaposi's sarcoma | Rare | Specialist referral for diagnostic confirmation and investigation of immune status
| Arteriovenous malformations/ haemangiomata | Uncommon | Many lesions require no active intervention other than reassurance
| Telangiectasia | Uncommon | No active intervention required but may need to investigate for underlying disease – best undertaken in specialist units
| Erythroplakia | Very rare | A sinister lesion. Urgent referral to secondary care for diagnosis and appropriate treatment
White lesions | Trauma | Common | Primary care
| Leukoplakia | Uncommon on the gingivae | Biopsy and monitoring can be undertaken in primary care. More extensive lesions should be managed in specialist units
| White sponge naevus | Rare | Diagnosis will usually be the remit of specialist units. No active treatment is required and there is no known malignant potential
| Squamous cell carcinoma | Very uncommon on the gingivae | Urgent referral to specialist units
| Lichen planus | Common | Recalcitrant lichen planus should be referred to specialist units for diagnostic confirmation, treatment and monitoring
| Candidosis | Gingival involvement is very rare | Referral to specialist units for further investigations
Pigmented lesions | Amalgam tattoo | Common | Primary care or referral if diagnosis is in doubt
| Melanotic macule | Uncommon | Primary care or referral if diagnosis is in doubt
| Naevi | Uncommon | Primary care or referral if diagnosis is in doubt
| Malignant melanoma | Very rare | Urgent referral to specialist unit
### Red Lesions
It is important to consider that some localised red lesions may represent periodontal sepsis, for example a lateral periodontal abscess, but this will be discussed in Chapter 5.
### _Kaposi's Sarcoma_
Kaposi's sarcoma occurs in more than 50% of patients with AIDS, becoming increasingly frequent as the disease progresses. Oral involvement may be the initial site of presentation, and gingival involvement is common. This condition may present as an epulis, perhaps mimicking a vascular pyogenic granuloma.
### _Clinical appearance_
* Clinical appearance is variable, light (Fig 3-1), dark red (Fig 3-2), purple/blue or even de-pigmented lesions having been described.
* Sessile macules/papules which may also have a nodular surface and become exophytic.
* Satellite lesions may develop.
**Fig 3-1** Kaposi's sarcoma that is pale in appearance.
**Fig 3-2** Kaposi's sarcoma that is dark red in appearance.
### _Clinical symptoms_
* Usually asymptomatic but patients may complain about the aesthetics or of bleeding following trauma.
### _Aetiology_
* Human herpes virus 8 (HHV8) – a gamma herpes virus is strongly associated with the aetiology of KS.
### _Involvement of non-gingival sites_
* The palate is the most common site of intra-oral involvement, particularly following the course of the greater palatine neurovascular bundles (Fig 3-3).
* Skin lesions.
* Ocular lesions (Fig 3-4).
* Visceral involvement.
**Fig 3-3** Kaposi's following the greater palatine neurovascular bundle.
**Fig 3-4** Ocular involvement of Kaposi's sarcoma.
### _Differential diagnosis_
* Haemangiomata/Arterio-venous malformations.
* Inflammatory hyperplasias, e.g. pyogenic granuloma or peripheral giant cell lesions.
* Periodontal abscess.
* Localised plasma cell gingivitis.
* Localised lesions of Molluscum contageosum.
* Pigmented lesions including malignant melanoma.
* Bacillary angiomatosis (secondary to Bartonella henselae/quintana infection).
### _Clinical investigation_
* Definitive diagnosis is by incisional biopsy, though lesions can appear benign and histologically identical to a pyogenic granuloma.
* Careful medical and sexual histories are crucial to aid presumptive diagnosis.
* In-situ polymerase chain reaction (PCR) may be used, where available, to identify HHV8 within the tumours.
### _Management options_
* Lesions are often multifocal, and thus further medical examination is necessary to identify the distribution of all lesions.
* Highly active anti-retroviral therapy (HAART) may affect resolution of the lesions.
* Chemotherapy.
* In early lesions local management may be indicated, (e.g. intralesional vincristine, surgical excision).
### _Vascular Lesions_
### _Clinical features_
* Vascular lesions that affect the gingivae are an uncommon clinical finding.
* Colour varies from blue/red/purple.
* They may be flat or more commonly elevated.
* Usually asymptomatic.
* Blanching occurs on pressure as the vessels are emptied.
* Occasionally they may bleed, eg if traumatised.
### _Aetiology_
* Haemangiomata – developmental lesions that are present from birth and spontaneously regress with age (Fig 3-5).
* Arterio-venous malformations (Fig 3-6).
**Fig 3-5** Haemangioma.
**Fig 3-6** Arterio-venous malformation.
### _Involvement of non-gingival sites_
* Tongue.
* Lip.
* Any area can be involved, intra-orally or extra-orally.
### _Differential diagnosis_
* Telangiectasia.
* Purpura.
* Kaposi's sarcoma.
* Lymphangioma.
### _Clinical investigation_
* Aspiration.
* Imaging to identify extent and distribution of the lesion (MRI, Doppler ultrasound).
* Angiography.
### _Management options_
* Usually require no active treatment.
* Cryosurgery (Fig 3-7).
* Laser.
* Sclerosing solutions.
* Embolisation of feeder vasculature.
**Fig 3-7** Cryosurgery to an arteriovenous malformation.
### _Telangiectasia_
Telangiectasia, are capillary blood vessel dilatations and may occur peri-orally and intra-orally. They are uncommon on the gingival tissues (Fig 3-8).
**Fig 3-8** Perioral and lingual telangiectasia in a patient with limited systemic sclerosis.
### _Clinical appearance_
* Small red macules, often multiple, which blanch on pressure.
### _Clinical symptoms_
* Telangiectasia are usually asymptomatic but may bleed on trauma.
### _Aetiology_
* Congenital or developmental depending on the type.
### _Involvement of non-gingival sites_
* Telangiectasia may occur on any skin or mucosal surface as well as involving the viscera.
### _Differential diagnosis_
* Hereditary haemorrhagic telangiectasia (Osler-Weber-Rendu syndrome)
* Cutaneous and gastrointestinal tract involvement is common.
* Epistaxis is also a frequent accompanying feature.
* Limited systemic sclerosis (previously known as 'CREST' syndrome – an acronym for: Calcinosis, Raynaud's phenomenon, Esophagitis, Sclerodactyly and Telangiectasia). See Chapter 10.
### _Clinical investigation_
* The diagnosis is usually made on clinical history and examination.
### _Management options_
* No active intervention is usually indicated.
### _Erythroplakia (Erythroplasia)_
### _Clinical features (Reichart 2005)_
* A rare lesion.
* Appears as an atrophic, flat velvety red patch. (Fig 3-9)
* Very high incidence of dysplasia or frank malignancy – erythroplakia should be regarded as malignant until histologically proven otherwise.
* More common in middle aged/elderly males.
**Fig 3-9** Floor of mouth erythroplakia with a squamous cell carcinoma centrally.
### _Aetiology_
* Unknown.
### _Involvement of non-gingival sites_
* May occur at any oral mucosal site as well as at other mucosal sites.
* Most commonly, floor of mouth, ventral surface of the tongue and palate.
### _Differential diagnosis_
* Inflammatory lesions (lichen planus, erythema migrans).
* Candidosis.
### _Clinical investigation_
* Biopsy is mandatory to identify dysplastic or neoplastic changes, which are the usual findings.
### _Management options_
* Excision with wide margin due to the high risk of malignant transformation.
* Avoidance of risk factors (tobacco and alcohol).
* Advice on a diet rich in anti-oxidants (empirical).
* Careful monitoring at short intervals.
### White Lesions
### _Trauma_
Chemical, physical or thermal trauma can produce red, white or ulcerated lesions on the gingiva and oral mucosa. Perhaps the best-known example is that of a local chemical burn arising as a consequence of dissolving an aspirin tablet on the gingiva adjacent to a painful tooth. The patient's history usually elicits the cause of the trauma and these lesions resolve spontaneously (Fig 3-10).
**Fig 3-10** Aspirin burn of the gingival and alveolar mucosa.
### _Leukoplakia_
A leukoplakia is an adherent white patch that cannot be categorised as any other morphological or histological diagnosis.
### _Clinical features_
* Gingiva is not a common site of involvement (Fig 3-11).
* Leukoplakias are premalignant lesions. Approximately 4% of lesions will transform to a squamous cell carcinoma over 10–20 years (Rhodus, 2005).
* Clinical risk features include:
* Density (Fig 3-12).
* Verrucous surface.
* Floor of mouth/ventral tongue.
* Erythema.
* Ulceration.
* Hyperplastic margins.
* Sudden change in lesions.
* Spontaneous pain.
**Fig 3-11** Leukoplakia of the gingiva.
**Fig 3-12** Dense leukoplakia of the hard palate and attached gingiva as a result of reverse smoking.
### _Aetiology_
* Idiopathic.
* Habits.
* Smoking.
* Alcohol.
* Betel nut chewing.
* Chronic Trauma (friction).
Sharp teeth/teeth opposing edentulous ridge.
* Actinic radiation.
* Systemic disease (uncommon).
* Plummer Vinson syndrome.
* Renal dialysis.
* Syphilis.
### _Involvement of non-gingival sites_
* Leukoplakias may occur at any site, but the floor of the mouth and ventral surface of the tongue appear to be at increased risk of malignant transformation (20%) (Fig 3-13).
**Fig 3-13** Floor of mouth leukoplakia.
### _Differential diagnosis_
* Candidosis.
* Lichen planus.
* Discoid lupus erythematosus.
* Skin grafts.
* Burns (chemical or thermal).
* Congenital (e.g. Leukokeratosis mucosae oris).
### _Clinical investigation_
* Biopsy – the clinical appearance is often a poor guide to the histological features.
* Microbiology.
* Haematology if clinically indicated, although routinely this is inappropriate.
### _Management options_
* Identify and eliminate contributory factors, e.g. smoking, alcohol, sharp teeth, candidal infection.
* Long term review or excise lesion – this is dependent on clinical features, patient's state of health and biopsy results (i.e. degree of dysplasia.)
* Advise patient to return immediately if they notice any sudden change in the character or behaviour of the lesion.
* Encourage diet rich in anti-oxidants (empirical advice).
* Possible value of toluidine blue staining in long term monitoring.
* Consider identification of human papillomavirus (type 16) in proliferative veruccous leukoplakias.
### _Leukokeratosis Mucosae Oris (White Sponge Naevus of Cannon)_
Leukokeratosis mucosae oris is an uncommon developmental anomaly that in some patients may be extensive. The condition is not regarded as being premalignant.
### _Clinical features_
* Asymptomatic localised or more generalised hyperkeratosis that is somewhat folded and often quite thickened (Fig 3-14).
* A large area or areas of mucosa may be involved.
* The condition usually presents in childhood.
* Gingival involvement is very uncommon.
**Fig 3-14** White sponge naevus involving the gingiva.
### _Aetiology_
* Autosomal dominant inheritance.
* Mutations in keratin genes for the mucosal-specific keratins, K4 and K13 (Rugg et al 1999).
### _Involvement of non-gingival sites_
* Buccal mucosa.
* Ventral surface of the tongue/floor of the mouth.
* Alveolar mucosa.
* Labial mucosa.
* Palate.
* Extra-oral involvement may include nasal, genital and ano-rectal mucosa.
### _Differential diagnosis_
* Pachyonychia congenita.
* Hereditary benign intraepithelial dysplasia (Witkop's disease).
* Mechanical trauma, e.g. cheek biting.
* Leukoedema.
* Leukoplakia.
* Pseudomembranous candidosis.
### _Clinical investigation_
* Histological examination.
### _Management options_
* No active treatment is required.
* Anecdotal reports exist that in some cases the condition may respond favourably to systemic antibiotics such as the tetracyclines.
### _Squamous Cell Carcinoma_
Oral sqaumous cell carcinoma may present variously as white or red patches, warty, or granular lesions, swellings or ulcers. White patches that suddenly show clinical change, are densely white, speckled, warty or ulcerated are similarly suspicious and should be biopsied urgently. A fuller account of oral squamous cell carcinoma will be found in Chapter 7.
### _Lichen Planus_
Oral lichen planus, or more typically oral lichenoid reactions, may unusually present as localised lesions as opposed to the condition's classical bilateral and often widespread distribution. This may be seen, for example, adjacent to amalgam restorations in susceptible patients. A full account of oral lichen planus may be found in Chapter 4.
### _Candidosis_
Gingival candidosis is rare and when it occurs, other than when it is denture- associated, may indicate underlying immunosuppression. Whilst Candida albicans is the most prevalent organism, other species such as C. tropicalis or C. glabrata may also be identified. It is also pertinent to note that gingival candidal infections may present as red rather than white areas of mucosa (Fig 3-15).
**Fig 3-15** Gingival candidosis in a patient with HIV disease.
### _Clinical features_
Candidosis may present intra-orally in a variety of different forms:
* Acute pseudomembraneous – white plaques that can be removed to leave an underlying erythematous surface (Fig 3-16).
* Acute atrophic – diffuse red appearance that may result from the use of broad spectrum antibiotics and inhaled corticosteroids.
* Chronic hyperplastic – typically a triangular, speckled lesion at the anterior commissures (Fig 3-17).
* Chronic mucocutaneous – widespread lesions often affecting the tongue and producing dystrophic nail changes, as well as affecting other mucocutaneous sites (Fig 3-18).
* Chronic erythematous – diffuse erythema that mimics denture stomatitis and may be seen in HIV/AIDS infections (Fig 3-19).
* Candidosis is usually painless although the atrophic/erythematous variants may produce soreness.
**Fig 3-16** Acute pseudomembranous candidosis (thrush).
**Fig 3-17** Chronic hyperplastic candidosis at the anterior commissure (candidal leukoplakia).
**Fig 3-18** Dystrophic nail changes in chronic mucocutaneous candidosis.
**Fig 3-19** Chronic erythematous candidosis as a presenting sign of HIV disease.
### _Aetiology_
* Local factors
* Denture wearing.
* Smoking.
* Xerostomia.
* Topical steroid therapy.
* Systemic factors
* Diabetes mellitus.
* Haematinic deficiency.
* Medication (corticosteroids, broad spectrum antibiotics, chemotherapy).
* Immunosuppressive states including HIV/AIDS.
* Candidosis endocrinopathy syndrome (rare) – a variant of chronic mucocutaneous candidosis, where there is multiple endocrine pathology including autoimmune thyroid, parathyroid and adrenal hypofunction.
### _Involvement of non-gingival sites_
* Any intra-oral site can be involved.
* Chronic hyperplastic candidosis typically affects the anterior commissural region.
* Acute atrophic candidosis usually involves the distal hard/soft palate.
* Chronic mucocutaneous candidosis also affects the skin and nail beds.
### _Differential diagnosis_
* White or red lesions of the oral mucosa including:
* Leukoplakia.
* Lichen planus.
* Erythroplakia.
### _Clinical investigation_
* Swabs for culture (semi-quantitative methods such as imprint cultures or oral rinses are available in some laboratories).
* Smears for visual identification of hyphae which are indicative of active infection as opposed to commensal carriage.
* Biopsy in the case of suspected chronic hyperplastic candidosis – this condition is premalignant; 50% of cases demonstrate dysplastic change histologically. Although this may be reactive in some cases, 9–40% of lesions are reported to undergo malignant transformation.
* Full blood count.
* Haematinics.
* Blood glucose.
### _Management options_
* Eliminate local factors.
* Eliminate/treat systemic factors.
* Anti-fungal agents.
* Long term review for chronic hyperplastic candidosis (malignant potential).
### Pigmented Lesions
### _Amalgam Tattoo_
### _Clinical features_
* A flat, grey/blue discolouration of the mucosa, resulting from amalgam particles (or other metals) becoming impacted in the soft tissue which subsequently becomes stained as metal ions leach out into the tissues (Figs 3-20 and 3-21).
* One of the most common causes of a discrete area of intra-oral hyperpigmentation.
* They may gradually enlarge and darken with time.
* The lesion is asymptomatic.
* Frequently involves the gingivae.
**Fig 3-20** Amalgam tattoo.
**Fig 3-21** Radiograph showing underlying amalgam particles in the tissues of an amalgam tattoo.
### _Aetiology_
* Amalgam becoming impacted into the soft tissues during restorative dental procedures.
### _Involvement of non-gingival sites_
* Any mucosal site close to the teeth is potentially liable to trauma and subsequent impaction of foreign material into the soft tissue.
* Mucosal flaps raised in apicectomy wounds may show amalgam tattoos as a consequence of the use of amalgam as the retrograde filling material.
### _Differential diagnosis_
* Racial pigmentation.
* Early Kaposi's sarcoma.
* Amalgam tattoo.
* Naevi.
* Melanotic macule.
* Pigmentary incontinence.
* Drug induced pigmentation.
### _Clinical investigation_
* Provided that the diagnosis is clear, investigations are not usually required. There may be a history of trauma during restorative dental procedures, or extraction of a heavily restored tooth.
* Where there is doubt as to the diagnosis, biopsy should be undertaken.
* Radiographs may reveal sizeable particles of amalgam or other radiopaque material, but small particulate matter will not be shown radiologically.
### _Management options_
* Dental amalgam is usually well-tolerated and no active intervention is required.
* The patient should be reassured as to the nature of the condition.
In addition to amalgam tattoos, blue/black pigmentation of the marginal gingivae may occur as a result of adjacent ceramo-metallic crowns. This is again due to metal salts leaching out of the metallic superstructure of the restoration.
### _Melanotic Macule (Ephelis)_
### _Clinical features_
* Small, pale brown, flat areas of pigmentation that are uncommonly seen on the gingivae (Fig 3-22).
* Asymptomatic.
* They do not usually enlarge nor darken with time.
* Present in one per 1000 of the population (all intra-oral sites).
* More common in females (female:male ratio – 2:1).
**Fig 3-22** A gingival ephelis.
### _Aetiology_
* Obscure.
### _Involvement of non-gingival sites_
* Melanotic macules are more frequently seen at other sites such as the lips, palate or the skin, where they may accompany the ageing process.
### _Differential diagnosis_
* Racial pigmentation.
* Early Kaposi's sarcoma.
* Naevi.
* Pigmentary incontinence.
* Smoker's melanosis.
* Drug induced pigmentation.
### _Clinical investigation_
* Biopsy if there is doubt as to the nature of the lesion.
### _Management options_
* No active intervention is required.
* Patient reassurance.
### Naevi
Intra oral naevi are uncommon and particularly so on the gingivae.
### _Clinical features_
* Range in colour from blue/brown to dark brown.
* They may be flat or papular.
* Their size and colour do not usually change with time.
* Any sudden changes in the clinical appearance should raise suspicion. However the large majority of naevi do not transform to melanoma.
* Naevi are asymptomatic.
### _Aetiology_
* Developmental localised increase in the number of melanocytes.
### _Involvement of non-gingival sites_
* More usually found in the palate and skin.
### _Differential diagnosis_
* Racial pigmentation.
* Early Kaposi's sarcoma.
* Amalgam tattoo.
* Melanotic macule.
* Pigmentary incontinence.
* Drug induced pigmentation.
### _Clinical investigation_
* Excisional biopsy will confirm the diagnosis.
### _Management options_
* Excision biopsy is prudent to exclude sinister pathology.
### _Malignant Melanoma_
Intra-oral malignant melanoma is a rare tumour with a poor prognosis. Median survival is less than two years following diagnosis. The tumour may arise de novo or in approximately 30% cases it will occur within areas of hyperpigmentation. Oral malignant melanomas account for only 1% of total cases of the condition, although in Japanese patients oral involvement accounts for 11% of all melanomas.
### _Clinical features_
* A pigmented lesion demonstrating the following features should arouse suspicion:
* Dark pigmentation, often varying in intensity over the lesion.
* Haemorrhage.
* Crusting.
* Ulceration.
* Nodularity.
* Inflammation.
* Rapid growth.
* Satellite areas of pigmentation.
* Sudden change in the clinical features of a pre-existing pigmented lesion.
* Gingival lesions most frequently involve the maxilla.
* A minority of lesions may initially appear benign. Thus pigmented lesions should always be viewed with an index of suspicion.
### _Aetiology_
* Malignant tumour of melanocytes.
* Exposure to ultraviolet radiation cannot be a significant factor in the development of intra-oral melanoma as it is for cutaneous disease.
### _Involvement of non-gingival sites_
* The palatal mucosa is the most frequently involved intra-oral site. 80% of oral melanomas occur here or on the palatal gingivae (Fig 3-23).
* Sun-exposed skin.
* The tongue and buccal mucosa may be sites of metastatic melanoma deposits.
**Fig 3-23** Malignant melanoma involving the palatal mucosa.
### _Differential diagnosis_
* Naevus.
* Racial pigmentation.
* Kaposi's sarcoma.
* Amalgam tattoo.
* Melanotic macule.
* Addison's disease.
* Pigmentary incontinence.
* Drug induced pigmentation.
### _Clinical investigation_
* Histological confirmation of the diagnosis is mandatory.
### _Management options_
* Wide excision.
* Superficial lesions have a more favourable prognosis.
### Further Reading
Reichart PA, Philipsen HP. Oral erythroplakia – a review. Oral Oncology 2005; 41:6,551–561.
Sciubba J. Oral leukoplakia. Critical Reviews in Oral Biology and Medicine 1995; 6:2,147–160.
Rhodus NL. Oral Cancer: leukoplakia and squamous cell carcinoma. Dental Clinics of North America. 2005; 49:1,143–165.
Rugg E, Magee G, Wilson N, Brandrup F, Hamburger J, Lane E. Identification of two novel mutations in keratin 13 as the cause of white sponge naevus. Oral Diseases 1999; 5:4,321–324.
## Chapter 4
## Gingival Colour Changes – Generalised
### Aim
This chapter aims to identify those conditions that may produce widespread colour changes of the gingivae.
### Outcome
Having read this chapter, the reader should have a working knowledge of the various conditions that can cause generalised red, white and pigmented areas on the gingivae. Additionally there should be an understanding of the differential diagnoses for these respective conditions, together with an appreciation of the nature of the salient clinical investigations required for a definitive diagnosis and subsequent treatment. Table 4-1 summarises the conditions discussed in the text.
Table 4-1 **Summary table – Generalised gingival colour changes** **Major Categories** | **Sub Categories** | **Frequency of Condition** | **Management Setting**
---|---|---|---
Red lesions | Plaque-associated gingivitis | Very common | Primary care with referral to specialist units if refractory to conventional treatment
| Desquamative gingivitis | Common. Most frequently due to lichen planus, but vesiculobullous diseases may also cause this condition. | Initial phase of stringent oral hygiene and topical anti-inflammatories can be undertaken in primary care. Recalcitrant or vesiculobullous disease should be referred to specialist units
| Primary herpetic gingivostomatitis | Uncommon | Supportive measures and antiviral agents such as aciclovir, which are only of real value if therapy instituted at onset or very early in the course of the disease
| Streptococcal gingivostomatitis | Very rare | Difficulty in diagnosis usually means that such cases will be referred for a specialist opinion
| Orofacial granulomatosis | Uncommon | Requires specialist management for diagnosis and treatment
| Plasma cell gingivitis | Rare | Specialist referral for diagnosis and identification of the allergen (not always possible)
| Other gingival hypersensitivity reactions | Uncommon | If the causative allergen can be readily identified and avoided there is no need for referral
| Sturge-Weber syndrome | Very rare | Specialist referral required
White lesions | Lichen planus | Common | Recalcitrant lichen planus should be referred to specialist units for diagnostic confirmation, treatment and monitoring
| Leukoplakia | Although leukoplaki; is not uncommon, it rarely involves large areas of the gingivae | Specialist referral for biopsy (may require multiple samples) and monitoring
| Candidosis | Very rare | Referral to specialist units for further investiagtions
Pigmented lesions | Extrinsic staining | Very common | Reassurance in primary care with appropriate advice regarding avoidance of such habits as smoking or betel nut usage.
| Racial pigmentation | Very common | Reassurance in primary care
| Drug-induced/heavy metal pigmentation | Uncommon | Primary care or specialist referral if aetiological agent is not obvious
| Addison's disease | Rare | If this condition is suspected, specialist referral is appropriate
### Red Lesions
The most frequent cause of generalised reddening of the gingivae is that of plaque-associated gingivitis. This topic is extensively covered in various other books in this series and therefore will not be addressed in this chapter.
### _Desquamative Gingivitis_
Desquamative gingivitis is a relatively common clinical presentation, particularly in middle aged to elderly female patients. It is a descriptive term and not a diagnosis in itself. It is therefore important to identify the underlying cause of the condition to ensure appropriate patient management.
### _Clinical appearance_
* Desquamative gingivitis may range in severity from a mild generalised redness to a florid, fiery red appearance. The gingivae are shiny, smooth and thinned (atrophic), as a consequence of epithelial desquamation (Fig 4-1).
* In some patients the changes are sporadic rather than confluent affecting only certain gingival sites. It is unclear as to why such localisation occurs.
* Desquamative gingivitis may involve both the free and attached gingivae.
* Dependent on its aetiology there may be other features, including lichenoid striae, (Fig 4-2) ulceration or rarely vesicle or bulla formation.
* Oral hygiene may be poor in the affected areas, leading to exacerbation of the condition due to superimposed plaque-related gingival inflammation.
**Fig 4-1** Desquamative gingivitis.
**Fig 4-2** Desquamative gingivitis with lichenoid striae.
### _Clinical symptoms_
* Despite a very florid presentation in some cases, patients can be remarkably free of symptoms.
* It is not uncommon for patients to complain of some degree of soreness, which is heightened by highly flavoured foods and drinks.
* Discomfort may cause difficulties in practising effective oral hygiene measures.
### _Aetiology_
The following conditions may produce a desquamative gingivitis:
* Lichen planus (the most frequent cause – see later).
* Mucous membrane pemphigoid. (See Chapter 8).
* Pemphigus vulgaris. (See Chapter 8).
* Discoid lupus erythematosus.
* Occasionally linear IgA disease.
* Hypersensitivity responses to local allergens including certain components of dentifrices. However, on careful clinical examination, such cases of 'desquamative gingivitis' have a somewhat granular surface appearance rather than the typical smooth appearance described above.
* Historically, the hormonal changes associated with the menopause were considered to be an aetiological factor, in view of the predisposition of menopausal females to present with the condition. There is no substantial evidence for this, the association being more a reflection of the typical age of onset of some of the aetiological factors listed above.
### _Involvement of non-gingival sites_
In many cases there may be no other intra-oral or extra-oral involvement. In other cases examination will reveal lesions consistent with one of the aetiological conditions, such as lichenoid striae on the buccal mucosa.
In mucous membrane pemphigoid and pemphigus, desquamative gingivitis may be the first presenting clinical sign, sometimes predating the occurrence of other features by several years. It is therefore important to ascertain if there is any extra-oral involvement (for example ocular in the case of pemphigoid), so that appropriate therapy can be commenced prior to irreversible tissue damage.
### _Differential diagnosis_
* The aetiological conditions discussed above.
* Orofacial granulomatosis.
* Myeloproliferative disease.
* Linear gingivitis as seen in HIV/AIDS.
### _Clinical investigation_
* Desquamative gingivitis is diagnosed on clinical features.
* Biopsy of gingivae may be diagnostically unhelpful as a result of the nonspecific background inflammation masking specific features of the underlying pathology.
* Biopsy of involved oral mucosa (if present) is often more valuable diagnostically.
* Indirect and direct immunofluorescence are indicated if there is a suspicion of an autoimmune vesiculobullous disease. Specimens for direct immunofluorescence studies must be unfixed and transferred to the laboratory as soon as possible for appropriate processing. The specimen should be transported in 'Michel's medium' or wrapped carefully in saline-soaked gauze.
### _Management options_
Desquamative gingivitis is often resistant to effective treatment. A high standard of oral hygiene is an essential prerequisite to improving the condition. Appropriate therapy for the aetiological condition should be implemented and topical corticosteroid therapy may be helpful, particularly if administered under occlusion, e.g. fluocinolone acetonide cream delivered via a mouthguard to ensure adequate contact with the affected areas.
### Primary Herpetic Gingivostomatitis
Most cases of primary herpes simplex infection are subclinical and essentially asymptomatic, the patient sometimes having mild non-specific symptoms of malaise and lymphadenopathy. Approximately 10% of cases will manifest as an acute gingivostomatitis with marked systemic malaise.
### _Clinical appearance_
* Florid, red oedematous gingivae (Fig 4-3).
* Distribution is not plaque related.
* Hypersalivation.
* Vesicles and ulceration of the gingivae, although these are usually present at other oral mucosal sites (Fig 4-4).
**Fig 4-3** Inflamed gingivae characteristic of primary herpetic gingivostomatitis.
**Fig 4-4** Erosion of the gingivae in primary herpetic gingivostomatitis.
### _Clinical symptoms_
* Usually occurs in children although young adults are increasingly affected.
* Acute onset with fever, malaise, sore throat and lymphadenopathy.
* In young adults the infection can, on occasions, be severe enough to warrant hospital admission and may be confused with Stevens Johnson syndrome.
* Pain from the involved tissues.
### _Aetiology_
* Primary infection is usually due to herpes simplex type 1 virus but HSV type 2 infection may also be involved.
* Recurrent or secondary infection produces a herpetic cold sore, although in the immunocompromised, it may manifest in a similar fashion to the primary infection.
### _Involvement of non-gingival sites_
* Vesicles that rapidly ulcerate appear on the palate, tongue, buccal mucosa, labial mucosa and vermilion border producing characteristic haemorrhagic crusting of the lips (Fig 4-5).
* The individual ulcers coalesce together forming large serpiginous areas of ulceration covered with a grey/white slough (Fig 4-6).
* Transmission of the virus to other sites such as the nail beds (herpetic whitlow) (Fig 4-7) and the eye (conjunctivitis; keratitis).
* Very rarely encephalitis.
**Fig 4-5** Labial crusting.
**Fig 4-6** Typical ragged serpiginous erosions of Herpes simplex infection.
**Fig 4-7** Herpetic whitlow.
### _Differential diagnosis_
* Erythema multiforme.
* Myeloproliferative disease.
* Myelosuppressive disease.
### _Clinical investigation_
* The diagnosis is usually made on clinical grounds.
* Viral identification in vesicle fluid.
* Demonstration of a rising serum antibody titre in sequential paired blood samples (i.e. acute and convalescent antibody titres).
* Immunofluorescence to identify specific serum IgM antibodies in a single blood sample (i.e. evidence of a primary immune response).
### _Management options_
* Supportive measures (fluids, anti-inflammatories, rest).
* Aciclovir administered early in the course of the disease (ideally before vesiculation).
* Avoid disseminating the infection to other sites (i.e. by avoiding scratching or picking the oral lesions).
### _Streptococcal Gingivostomatitis_
This is a very rare condition and may cause diagnostic confusion with viral infections, particularly primary herpetic gingivostomatitis. It may manifest in children or young adults, producing widespread inflammation and soreness of the gingivae and oral mucosa. Ulceration may also occur, but there is no vesiculation, an important diagnostic feature that helps differentiate this infection from those with a viral aetiology. The diagnosis can be confirmed by identifying Group A beta-haemolytic streptococci in culture. The infection is usually susceptible to treatment with penicillin (Katz, 2002).
### _Orofacial Granulomatosis_
Orofacial granulomatosis may involve the gingivae by producing generalised band like inflammatory change that is not plaque-related and extends across the width of the attached gingivae and on to the alveolar mucosa (Fig 4-8). Additionally the condition may also produce a generalised gingival overgrowth. A full account of orofacial granulomatosis can be found in Chapter 6.
**Fig 4-8** Band-like inflammation of the attached gingivae and alveolar mucosa in orofacial granulomatosis.
### _Plasma Cell Gingivitis_
### _Clinical appearance_
* Plasma cell gingivitis is an unusual condition that produces a diffuse, reddened somewhat granular appearance to the affected gingivae (Fig 4-9).
* It may be confused with desquamative gingivitis.
**Fig 4-9** Plasma cell gingivitis.
### _Clinical symptoms_
* Patients may complain of soreness but often the condition is asymptomatic.
### _Aetiology_
* The condition appears to be related to hypersensitivity responses to a variety of potential allergens including:
* Additives to dentifrices e.g. cinnamonaldehyde or sodium lauryl sulphate.
* Food additives and flavouring agents.
* Dental materials.
* Essential oils including eugenol.
* Drugs.
* Mouthwashes.
### _Involvement of non-gingival sites_
* Plasma cell stomatitis is a very rare condition but may involve any part of the oral mucosa including the palate and oropharynx (Fig 4-10).
**Fig 4-10** Plasma cell stomatitis involving the palate and oropharynx.
### _Differential diagnosis_
* Plaque associated gingivitis/periodontitis.
* Desquamative gingivitis.
* Orofacial granulomatosis.
* Kaposi's sarcoma.
* Primary herpetic gingivistomatitis.
### _Clinical investigation_
* Diagnosis is usually made on clinical grounds.
* Skin testing for potential allergens may sometimes be helpful but often this is not the case.
* Biopsy.
### _Management options_
* Identification and avoidance of the causative allergen.
* Corticosteroids may also be helpful in suppressing the inflammatory response.
### _Other hypersensitivity reactions of the gingivae_
Not all hypersensitivity reactions that occur within the gingivae manifest as a plasma cell gingivitis. There may be a non-specific reddening of the gingivae, which may be swollen and slightly granular in appearance. This may mimic plaque-associated gingivitis, desquamative gingivitis, or the gingival changes seen in some patients with orofacial granulomatosis or HIV disease.
Allergens that may provoke such reactions are diverse and may include some of the components of dentifrices, mouthwashes, dietary allergens such as food additives and dental materials. The mechanism may be that of a contact sensitivity or less frequently an immediate-type hypersensitivity. Management involves identification of the allergen (not always easy to achieve) and its avoidance in the future.
Occasionally direct chemical toxicity rather than an immunologically mediated reaction may produce a similar clinical presentation (Fig 4-11).
**Fig 4-11** Local toxicity due to formaldehyde-containing desensitising dentifrice.
### _Sturge Weber Syndrome_
Sturge Weber syndrome is a rare, congenital, hamartomatous condition manifesting as angiomas that typically involve those tissues innervated by the trigeminal nerve. Intracranial involvement also occurs, including haemangiomata and calcification of the leptomeninges potentially resulting in learning difficulties and epilepsy.
### _Clinical features_
* The haemangiomata are usually unilateral involving the facial skin, gingivae and oral mucosa (Fig 4-12).
* The lesions are usually flat and red/purple in colour, typical of vascular lesions.
* Occasionally the involved gingivae may appear hyperplastic.
**Fig 4-12** Sturge Weber syndrome – intra-oral involvement.
### _Differential diagnosis_
* Other causes of vascular anomalies.
### _Clinical investigation_
* Imaging techniques including Doppler ultrasound and Magnetic Resonance Imaging are of value in defining the location and extent of the lesions.
### _Management options_
* The haemangiomata may pose a potential problem in patients undergoing oral surgical procedures.
* The hemangiomata are only treated if they are causing specific clinical problems.
### White Lesions
### _Lichen Planus (see also Chapter 8)_
Lichen planus is a relatively common mucocutaneous inflammatory condition, affecting approximately 2% of the population. It has a female predilection and its onset is typically in middle age and later life. The condition is particularly persistent in the mouth, lasting for many years (10–20 years), and this contrasts with the cutaneous lesions, which often, but not always, resolve after two to three years. The clinical course of lichen planus is generally benign, but a small number of cases may undergo malignant transformation (0.1–1%). This is considered to be more likely in the erosive or atrophic variants.
### _Clinical appearance_
* Lichen planus affecting the gingivae may present as a desquamative gingivitis or with the typical lacy network of non-ulcerating white lichenoid striae, often set on an erythematous base. (Fig 4-13).
* Lichen planus may manifest in other clinical variants including erosive, atrophic, plaque-like and rarely bullous forms.
* Lichen planus is the most frequent cause of desquamative gingivitis.
* Erosive lesions on the gingivae, for example when associated with a desquamative gingivitis should raise suspicion of a vesiculobullous disorder rather than lichen planus itself.
* Desquamative gingivitis may appear fiery red. Plaque accumulation further exacerbates this condition.
* Lesions may be widespread and are classically, but not universally, bilateral and often symmetrical.
* Gingival involvement is common, occurring in up to 30%–50% of affected patients and it may be the only site of involvement (Fig 4-14).
* Pigmentary incontinence may be seen in some cases, particularly in dark-skinned races (Fig 4-15).
**Fig 4-13** Lichen planus manifesting as desquamative gingivitis with lichenoid striae.
**Fig 4-14** Gingival lichen planus.
**Fig 4-15** Pigmentary incontinence in long standing oral lichen planus.
### _Clinical symptoms_
* Patients with reticular lichen planus are asymptomatic or may complain of roughness of the gums.
* Patients with atrophic or desquamative gingivitis experience varying levels of discomfort, which can be severe. This may cause difficulty with effective oral hygiene measures.
### _Aetiology_
* The aetiology of lichen planus is poorly understood.
* Its pathogenesis appears to be T-cell dependent (Sugarman, 2002).
* It is weakly associated with autoimmune liver disease such as Primary Biliary Cirrhosis and Chronic Active Hepatitis.
* Lesions resembling those of lichen planus (lichenoid lesions) are seen in:
* Chronic Graft Versus Host Disease following bone marrow transplantation (Figs 4-16 and 4-17).
* As a consequence of a large number of medications (e.g. non-steroidal anti-inflammatory drugs, (β-blockers, gold.)
* Occupational exposure to phenolphthalein dyes in the photographic industry.
* Mercury amalgam sensitivity (and possibly some composite resin restorative materials).
* Hepatitis C in certain populations, e.g. Japanese, Italians. However, this may reflect background carriage rate rather than being of aetiological significance.
**Fig 4-16** Chronic oral graft versus host disease (GvHD) following stem cell transplant for aplastic anaemia, showing a desquamative gingivitis.
**Fig 4-17** GvHD in the same patient as Fig 4-16 showing extensive oral ulceration.
### _Involvement of non-gingival sites_
* Classically oral lichen planus is a bilaterally symmetrical eruption affecting the buccal mucosae, distally and rarely extends as far as the anterior commissures. (Figs 4-18 and 4-19).
* The lateral and ventral surfaces of the tongue are also frequently involved (Fig 4-20).
* Any other intra-oral site may also be involved, although the palate and lips are usually spared.
* Cutaneous involvement may occur in 10-30% of patients with oral lesions. This typically affects the flexor surfaces of the limbs and wrists, presenting as 'purple polygonal pruritic papules' with the classical white Wickham's striae running through them (Fig 4-21). As in the mouth, lesions are often bilaterally symmetrical.
* Other mucosal surfaces may also be involved such as the genitalia. In females, involvement of the vulva and vagina together with a desquamative gingivitis is known as the 'vulvo-vaginal gingival syndrome'.
* The scalp may also be affected.
**Fig 4-18** Reticular lichen planus of the buccal mucosa.
**Fig 4-19** Erosive lichen planus of the buccal mucosa.
**Fig 4-20** Lichen planus of the tongue, showing a typical atrophic appearance.
**Fig 4-21** Cutaneous lichen planus showing the classical papular eruption
### _Differential diagnosis_
* Drug-induced lichenoid eruptions.
* Lichenoid eruptions secondary to dental restorations.
* Discoid/systemic lupus erythematosus.
* Mucous membrane pemphigoid.
* Pemphigus.
* Graft versus host disease.
* Leukoplakia.
* Plasma cell gingivitis.
* Chronic hyperplastic candidosis.
* Squamous cell carcinoma.
* Non-specific ulceration.
* Hairy leukoplakia.
### _Clinical investigation_
* Biopsy if there is doubt clinically as to the diagnosis of non-erosive lesions.
* Biopsy erosive/atrophic lesions to confirm the diagnosis and identify possible dysplasia.
* Smear for candidal organisms.
* Investigate for underlying disease if suspected clinically.
### _Management options_
In asymptomatic cases there is no need for active treatment. In symptomatic cases, the aim is to reduce inflammation and heal erosions. Oral lichen planus is very persistent and curative treatment is as yet unavailable. Treatment should be via a stepped approach, using topical therapy whenever possible.
* Ensure excellent oral hygiene.
* Identification and substitution of associated drug if appropriate.
* Consider removal of amalgam restorations if there is a suspicion that they may be contributory.
* Treatment of an identifiable underlying disease.
* Topical agents:
* Chlorhexidine as an adjunct to oral hygiene measures
* Corticosteroids – e.g. soluble prednisolone, soluble betamethasone, beclometasone, fluticasone propionate.
* Fluocinolone may be particularly helpful in desquamative gingivitis when administered under occlusion via a mouthguard.
* Intralesional corticosteroids – triamcinolone acetonide
* Systemic corticosteroids – prednisolone, deflazacort. Steroid sparing agents such as azathioprine may also be considered.
* In particularly recalcitrant cases, topical ciclosporin or tacrolimus may be used. However caution is needed with the prescription of these drugs, which are best used in specialist units to ensure appropriate monitoring. There are currently concerns regarding possible carcinogenicity associated with the use of topical tacrolimus. Its usage must therefore be restricted to very severe cases, when it should only be used for a limited duration.
* Long-term follow up is mandatory for atrophic and erosive variants due to the possibility (albeit small) of malignant transformation.
### _Other Generalised White Lesions_
Gingival candidosis and leukoplakia may be widespread in their distribution on occasions, although more usually present as discrete areas of colour change.
### Pigmented Lesions
### _Extrinsic Staining_
Staining of the gingivae may also occur as a consequence of various local agents, including the consumption of highly coloured foods and drink and the use of betel nut.
Betel nut is used by a variety of Asian communities and produces a characteristic brown/orange staining of the oral mucosa, gingivae and teeth (Fig 4-22).
**Fig 4-22** Extrinsic staining due to chewing betel nut.
Heavy tobacco consumption may also produce extrinsic staining of the tissues but additionally it also causes intrinsic melanosis.
### _Racial Pigmentation_
### _Clinical Features_
* Diffuse macular brown areas, often widespread (Fig 4-23).
* Frequently associated with the anterior labial gingivae.
* Common in dark-skinned populations – Asian and Afro-Caribbean individuals.
**Fig 4-23** Racial pigmentation of the gingivae.
### _Involvement of non-gingival sites_
* Pigmentation may occur at any intra-oral site.
* The dorsum of tongue and buccal mucosa are frequently involved.
### _Differential diagnosis_
* Pigmentary incontinence secondary to chronic inflammatory disease of the oral mucosa.
* Smoker's melanosis.
* Addison's disease.
* Pregnancy.
* Melanotic macules.
* Drug-induced pigmentation.
* HIV/AIDS.
* Peutz-Jegher syndrome.
* Laugier-Hunziker syndrome.
### _Clinical investigation_
* Exclusion of other causes of generalised gingival pigmentation (consider investigations for Addison's disease – see below)
### _Management options_
* No active intervention is required other than reassurance.
### _Drug-Induced and Heavy Metal Pigmentation_
A variety of drugs and also heavy metals can produce diffuse gingival discolouration.
It is therefore important to take a detailed medical history for patients, noting the patient's systemic medication.
Additionally it is important to consider possible occupational exposure to heavy metals, although this is now uncommon due to improved health and safety regulations that are intended to limit such exposure.
The discolouration will vary according to the causative agent:
* For example, minocycline can cause a purple/grey discoloration (Fig 4-24), whilst other drugs such as cytotoxics and AZT (Fig 4-25), may lead to brown pigmentation.
**Fig 4-24** Drug-induced pigmentation due to minocycline.
**Fig 4-25** Drug-induced pigmentation due to AZT.
Some of the more commonly used drugs associated with mucosal discolouration are listed below:
* Anticonvulsants. e.g. phenytoin.
* Antimalarials.
* ACTH.
* The oral contraceptive pill.
* Antimicrobials including minocycline, ketoconazole, zidovudine and clofazimine.
* Cytotoxic drugs including busulphan and cyclophosphamide.
* Amiodarone.
* Chlorpromazine.
Heavy metals associated with mucosal discolouration include:
* Arsenic.
* Bismuth.
* Copper.
* Gold.
* Lead.
* Mercury.
* Platinum.
* Silver.
* Zinc.
### _Addison's Disease_
Addison's disease is a rare condition occurring more frequently in females. It results from the destruction (usually autoimmune) of the adrenal cortex.
### _Clinical appearance_
* Diffuse brown pigmentation of the gingivae (Fig 4-26).
**Fig 4-26** Diffuse gingival pigmentation as a result of Addison's disease.
### _Clinical symptoms_
* Oral manifestations are asymptomatic.
* Systemically there is often vague symptomatology of lethargy, malaise, weakness, anorexia or vomiting.
* Fainting as a result of postural hypotension.
### _Aetiology_
* 90% of cases are autoimmune (21-hydroxylase being the usual antigen).
* Excessive production of adrenocortictrophic hormone (ACTH) in response to low serum cortisol levels produces hyperpigmentation as a consequence of certain of its properties being similar to those of melanocyte stimulating hormone.
### _Involvement of non-gingival sites_
* There may be increased pigmentation at cutaneous and other mucosal sites such as the genitalia.
* Cutaneous hyperpigmentation is seen especially within skin creases.
### _Differential diagnosis_
* Racial pigmentation.
* Drug-induced pigmentation.
* Smoker's melanosis.
* Pregnancy.
* Malignant melanoma.
* Melanotic macules.
### _Clinical investigation_
* Isolated plasma cortisol levels are of little value.
* In suspected Addison's disease, preliminary investigations that may be undertaken include the following:
* Blood pressure recording, which may be hypotensive.
* Plasma electrolytes may demonstrate hyponatraemia and hyperkalaemia.
* Serum urea may be elevated.
* Blood glucose level may be depressed.
* Definitive investigation is afforded by an ACTH stimulation test.
### _Management options_
* The hyperpigmented lesions do not require treatment.
* Recognition of the possible underlying diagnosis is important with referral to the appropriate specialty for diagnostic confirmation.
* Glucocorticoid and mineralocorticoid replacement by the appropriate physician.
### Further Reading
Katz J, Guelmann M, Rudolph M, Ruskin J. Acute streptococcal infection of the gingiva, lower lip and pharynx – a case report. Journal of Periodontology 2002; 73:11,1392–1395.
Sugarman PB, Savage NW, Walsh LJ et al. The pathogenesis of oral lichen planus. Critical Reviews in Oral Biology and Medicine 2002; 13:4,350–365.
## Chapter 5
## Gingival Enlargements – Localised
### Aim
This chapter aims to provide the practitioner with a visual guide to swellings that arise locally within the gingiva, including the free and/or attached gingiva.
### Outcome
At the end of this chapter the reader should have knowledge of which types of localised gingival swellings are common or uncommon, be able to identify the key clinical features of localised gingival enlargements and formulate a differential diagnosis for a localised gingival swelling. The reader will also be able to decide which lesions can be managed within their practice and which need to be referred for specialist advice.
Table 5-1 lists the gingival enlargements discussed in this chapter and highlights which are common or uncommon and when the condition may be managed in general practice or should be referred.
Table 5-1 **Localised gingival swellings** **Lesions** | **Category** | **Sub-Category** | **Incidence** | **Manage/ Refer**
---|---|---|---|---
True epulides | Fibrous epulis | | common (60% of epulides) | manage
Vascular epulis | Pyogenic granuloma
Pregnancy epulis | common (30% of epulides) | manage
| Multiple/disseminated pyogenic granuloma | uncommon | refer
Giant cell epulis/ granuloma | Peripheral | uncommon (10%) | refer
| Central | uncommon | refer
Lesions presenting as epulides | Congenital epulis | | uncommon | refer
Viral warts | Condyloma acuminatum | uncommon | refer
| Verruca vulgaris | uncommon | refer
Neurofibroma | | uncommon | refer
Appliance-induced hyperplasia | | common | manage
Other gingival swellings | Abscess | Periodontal | common | manage
| Gingival | uncommon | manage
| Stitch | uncommon | manage
Localised trauma | | uncommon | manage/ refer
Histiocytosis-X | Unifocal (solitary eosinophilic granuloma)
Multifocal (Hand-Schuller-Christian syndrome)
Progressive/disseminated (Letterer-Siwe disease) | uncommon | refer
Haemangioma | | uncommon | refer
Tumours | Malignant lesions | Kaposi's sarcoma | uncommon | refer
| Squamous cell carcinoma | uncommon | refer
| Metastatic tumours | uncommon | refer
| Non-Hodgkin's lymphoma | uncommon | refer
Benign lesions | Reactive osteoma | uncommon | manage
Lesions associated with PTEN mutations | Cowden's syndrome Bannayan-Riley-Ruvalcaba syndrome
Proteus syndrome | uncommon | refer
### The Epulides
Gingival epulides are benign localised enlargements of the gingival tissues. They are predominantly hyperplastic lesions of the gingival connective tissues, which develop following chronic irritation. The source of irritation may vary and examples include:
* Ledged or prominent subgingival restorations.
* Subgingival calculus.
* Clasp arms from removable appliances.
* Impaction of a foreign body subgingivally.
Many lesions may present as epulides, but only three true forms are described:
* Fibrous epulis.
* Vascular epulis (pyogenic granuloma or pregnancy epulis).
* Giant cell epulides (peripheral or centrally arising).
The term epulis means 'on the gum', and all true epulides have a common pathogenesis, which involves the body attempting to heal an area of inflammation through the formation of granulation and fibrous tissue, whilst the inflammatory stimulus remains. Therefore, true epulides are histologically similar and show variable features of chronic inflammation, immature vascular tissue (granulation tissue) and collagen deposition. The only exception is the congenital epulis.
### _The Fibrous Epulis_
### _Clinical appearance_
Fibrous epulides present as pink, firm enlargements of the interdental gingivae (Fig 5-1). They may be sessile or pedunculated and similar in colour to the surrounding tissues unless they become inflamed. Ulceration can arise, leading to a yellow, fibrinous surface exudate. They are normally firm in consistency, do not blanch, and pitting of the surface may be seen due to the insertion of collagen bundles beneath. Calcification or ossification may arise within some lesions, when the term 'calcifying or cementifying fibrous epulis' is used. The behaviour and treatment of the latter is the same, but recurrence is reported to be more common.
**Fig 5-1** A fibrous epulis UR 3 related to chronic irritation from subgingival calculus acting as a plaque retention factor.
### _Clinical symptoms_
Often symptom-free and predominantly cause aesthetic concerns. Rarely they can lead to tooth migration and irritation to the overlying soft tissues (e.g. lip).
### _Aetiology_
As for all epulides (see previous text).
### _Involvement of non-gingival sites_
None.
### _Differential diagnosis_
* Vascular epulis.
* Giant cell granuloma.
* Benign osteoma of underlying alveolar bone.
* Denture induced hyperplasia.
* Gingival cyst.
* Neurofibroma.
* Connective tissue tumour (see chapter 11).
* Metastatic tumour.
### _Clinical investigation_
Excisional biopsy for histopathology with careful gingival recontouring. Lesions consists of a core of highly cellular fibroblastic and granulation tissue covered by stratified squamous epithelium, which may or may not be ulcerated. There are varying degrees of inflammatory cell infiltration, mainly with plasma cells.
### _Management options_
If large, referral is advisable. Surgical excision followed by recontouring of the gingivae to form a marginal complex that lends itself to easy cleansing. Thorough subgingival debridement is performed to remove potential aetiological agents, e.g. calculus, foreign body, plaque. Apply a pressure pack, to maintain the interproximal zone patent during the healing phase. Chlorhexidine mouthwash is advisable during this period, and the patient should be reviewed after seven days for dressing removal and prophylaxis. At this stage the patient should resume careful interproximal plaque control. Lesions may recur if the cause of the irritation persists.
### _The Vascular Epulis_
### _Clinical appearance_
Vascular epulides mainly arise in the anterior part of the mouth and usually, the labial aspect. They are soft, normally pedunculated lesions with a narrow base which, when associated with pregnancy (Fig 5-2), can progress throughout the gestation period. Commonly they occur in the second or third trimester and may have a very red/granular surface that is prone to haemorrhage (spontaneous or as a result of trauma). The surface may ulcerate, leaving a yellow, fibrinous coating (Fig 5-3).
**Fig 5-2** A pregnancy epulis affecting UR1 and UL1 teeth and demonstrating a classical dumb-bell or hourglass shape between the incisors.
**Fig 5-3** A vascular epulis affecting LR5. The surface has ulcerated due to trauma from the opposing teeth.
### _Clinical symptoms_
Bleeding to touch or when brushing, poor aesthetics and discomfort on pressure.
### _Aetiology_
The pregnancy epulis and pyogenic granuloma (Fig 5-4) are histologically identical. The term 'pyogenic granuloma' is a historical one, since it was thought (incorrectly) that the lesion was an inflammatory response to infection with pyogenic bacteria. Lesions develop for the same reasons as other epulides, but vascular changes characterise the inflammatory response rather than fibrosis. Pregnancy-associated lesions are generally associated with subgingival plaque or calculus.
**Fig 5-4** A pyogenic granuloma (vascular epulis) arising UR45 area due to poorly contoured subgingival temporary dressings.
### _Involvement of non-gingival sites_
None.
### _Differential diagnosis_
* Giant cell granuloma.
* Denture induced hyperplasia.
* Fibrous epulis.
* Kaposi's sarcoma.
* Gingival cyst.
### _Clinical investigation_
Presumptive diagnosis can be made on appearance, but definitive diagnosis requires an excision biopsy. Lesions comprise a mass of vascular spaces within a fine, connective tissue stroma. There may be solid layers of uncanalised endothelium or many thin-walled immature vessels and the surface is often ulcerated, with an inflammatory infiltrate beneath the ulceration. Histologically, the pregnancy epulis is regarded as a pyogenic granuloma arising during pregnancy.
### _Management options_
Intensive oral hygiene instruction and scaling under local anaesthesia reduces the vascular nature of the lesion and may lead to its resolution. However, excision is often necessary and recurrence rates are high. Good vasoconstriction is essential from a local anaesthetic and an electrosurgery or bi-polar diathermy should be on hand. The area should be thoroughly scaled and a pressure dressing applied. It is common for the lesion to return, therefore excision is preferable post-parturition. Many resolve spontaneously postpartum. The non-pregnancy associated pyogenic granuloma is excised in a similar manner. However, the cause, such as defective restorations, should be identified and removed.
### _Multiple/Disseminated Pyogenic Granulomata_
### _Clinical appearance_
This is an extremely rare condition. The case shown in Figs 5-5 and 5-6 presented in a seven-year-old boy. Appearance is of multiple vascular exophytic lesions, which present as disseminated vascular tumours with a relatively short natural history. Lesions have a fibrinous exudate at their surface, similar to solitary pyogenic granulomas. Satellite and intravenous pyogenic granulomas may develop at the same time as the primary lesion or may occur after attempted treatment of the primary lesion. Lesions may be grouped or eruptive and disseminated in nature.
**Fig 5-5** Multiple pyogenic granulomas in a seven-year-old boy affecting the palatal aspects of his anterior teeth.
**Fig 5-6** The same boy as in Fig 5-5 with multiple lesions affecting the lower incisor teeth.
### _Clinical symptoms_
There is gingival bleeding when the tissues are subject to light trauma from brushing or eating. Spontaneous bleeding is also reported.
### _Aetiology_
Trauma, hormonal influences, viral oncogenes, underlying microscopic arteriovenous malformations, and production of angiogenic factors have all been implicated. However, in the case illustrated, the likely aetiology was of oral neglect and local irritation or trauma causing a primary lesion, which then spread laterally.
### _Involvement of non-gingival sites_
A space occupying lesion between the fifth and eighth thoracic vertebrae was identified by MRI scan in this case, which led to a spastic diplegic gait and lower limb paralysis. There was also weight loss and chronic diarrhoea. Syringomyelia was diagnosed (a disorder in which a cyst forms within the spinal cord). This cyst, called a syrinx, expands and elongates over time, destroying the centre of the cord. Damage may result in pain, weakness and stiffness in the legs. Other symptoms include headaches and incontinence.
### _Differential diagnosis_
* Bacillary angiomatosis.
* Benign lymphangioendothelioma.
* Kaposi's sarcoma.
* Leukaemia.
* Kaposiform hemangioendothelioma.
### _Clinical investigation_
Excision biopsy. Histology is identical to the solitary pyogenic granuloma. Whether the spinal cord lesion was a satellite granuloma or a co-incidental true syrinx in this case remains uncertain.
### _Management options_
Specialist referral is essential. Intensive oral hygiene and full mouth prophylaxis restored gingival health in the reported case, and intensive physiotherapy restored lower limb function. Excision of the multiple granulomas, scaling and restoration of high standards of plaque control will help restore oral health (Fig 5-7).
**Fig 5-7** Decayed lower first molars in a neglected mouth from the patient illustrated in Figs 5-5 and 5-6, which were extracted to restore oral health.
### _The Giant Cell Epulis_
### _Clinical appearance_
This is a pedunculated or sessile lesion, dark red in colour and often with an ulcerated surface (Fig 5-8). Size varies, and lesions may project from palatal through to labial gingivae with a narrow pedicle between the teeth (hourglass appearance). Radiographs are essential, as giant cell epulides may arise centrally within bone as central giant cell granulomas, perforating the outer bone cortex to present peripherally. Lesions may become very large (Fig 5-9) and, unless correctly diagnosed, incomplete excision is likely. Lesions may also arise in association with implants (Fig 5-10).
**Fig 5-8** A peripheral giant cell granuloma affecting LR 234 teeth.
**Fig 5-9** A central giant cell lesion that has expanded in the left maxillary premolar and molar region. Note the pigmentation due to haemosiderin deposition.
**Fig 5-10a** Central giant cell granuloma arising around implant fixtures UL1 and 2, which were placed into an autogenous bone graft (from the right chin). The lesion had been excised unsuccessfully three times at presentation.
**Fig 5-10b** Palatal view of the lesion in 5-10a.
**Fig 5-10c** Bone loss associated with central giant cell granuloma in Fig 5-10a and b. The implant fixtures had to be removed prior to aggressive curettage of the surrounding bone.
### _Clinical symptoms_
These usually present between 30–40 years, but also in the very young or old, whether dentate or edentulous. They are most common in the anterior region of the mouth and twice as common in females as males. They are more prevalent in the mandible than the maxilla, and symptoms include bleeding to touch or when brushing, poor aesthetics and discomfort on pressure.
### _Aetiology_
Aetiology is unknown and, like the other true epulides, is believed to be a reactive hyperplasia due to chronic irritation or trauma. The tissue is thought to be of periosteal origin, but the origin of the giant cells is unknown.
### _Involvement of non-gingival sites_
None, unless arising centrally within bone.
### _Differential diagnosis_
* Vascular epulis.
* Denture induced hyperplasia.
* Haemangioma.
* Kaposi's sarcoma.
* Gingival cyst.
### _Clinical investigation_
Periapical radiographs are employed to determine whether there is bony involvement/cortical plate erosion. Histopathology is characterised by focal collections of osteoclast-like giant cells of varying size and number. Giant cells are separated by a fibrous connective tissue stroma within which are vascular channels of varying diameter. Extravasated blood cells and haemosiderin may add a brown/red colour to the lesion (Chapter 4). Bony trabeculae or osteoid (bone matrix) may also be present.
### _Management options_
Excision as for a vascular epulis or if central bone involvement is suspected, a mucoperiosteal flap should be raised and the bone surface curetted. If the gingival margin is involved (e.g. Fig 5-1), care should be taken not to deform the marginal tissues during excision.
### _Congenital Epulis_
This very rarely arises in the anterior maxilla or mandible of newborn children. Its aetiology is unknown, but it is thought to be reactive. It is benign and does not recur following excision.
### _Viral 'Wart-like' Lesions_
### _Clinical appearance_
* Condyloma acuminatum (Ca) (Fig 5-11 and 5-12) is usually present in immunosuppressed patients and has a 1.2% incidence in HIV disease. There are mushroom-like warts, usually pedunculated. It is contagious and may spread locally.
* Verruca vulgaris (Vv) may be seen on the lips of children with finger warts and appear like small cauliflowers with white-tipped surface papillary projections. It is highly contagious and may spread to other sites of the body.
* Focal epithelial hyperplasia (Heck's disease) is caused by the human papilloma virus (HPV) but does not usually involve the gingivae.
* Molluscum contageosum may present as a localised red lesion, with a granular appearance to its surface.
**Fig 5-11** A condyloma acuminatum in an HIV patient who had discontinued HAART and was immunosuppressed. Note the adjacent secondary lesion.
**Fig 5-12** A condyloma acuminatum on the right maxillary tuberosity.
### _Clinical symptoms_
* Ca – Irritation to the tongue and appearance may cause concern.
* Vv – Roughness to the touch and appearance may be a concern.
* Focal epithelial hyperplasia (FEH) – multiple lumps/roughness.
* Molluscum contageosum produce localised aesthetic problems and may spread from or to the skin.
### _Aetiology_
* Ca – human papilloma virus (HPV) types 6, 11, 16, 18.
* Vv – HPV types 2, 4, 40, 57.
* FEH – over 70 types of HPV have been identified to date.
* Molluscum contageosum is caused by a poxvirus related to smallpox.
### _Involvement of non-gingival sites_
* Ca – Anal or genital involvement.
* Vv – Lesions usually affect the skin of the fingers.
* FEH – Lesions affect oral mucosa generally.
* Molluscum contageosum – commonly affects skin.
### _Differential diagnosis_
* Vascular epulis.
* Neurofibromatosis.
### _Clinical investigation_
Presumptive diagnosis is made from clinical findings and following a careful history (including other lesion sites e.g. genitals) and examination of the skin, fingers etc.
### _Management_
Excision or cryosurgery. In HIV disease, lesions may resolve when the patient commences highly active anti-retroviral therapy (HAART).
### _Neurofibroma_
### _Clinical appearance_
Solitary lesions may arise but normally form part of the condition Neurofibromatosis, which may be of type 1 (NF1 or von Recklinghausen's disease – 90% of cases) or type 2 (NF2 – or bilateral acoustic neuromas/ schwannomas i.e. higher incidence of central nervous system tumours than NF1). They form well-circumscribed firm focal swellings (Fig 5-13).
**Fig 5-13** A gingival neurofibroma lingual to the lower central incisors.
### _Clinical symptoms_
These involve largely aesthetic concerns, particularly when multiple skin lesions are present. Neurofibromas present in young adults and increase in number with advancing age.
### _Aetiology_
This is a genetic disorder with no race/sex predilection and an incidence of one new case in every 3000 live births. The lesions are benign and complex and arise from peripheral nerve sheaths.
### _Involvement of non-gingival sites_
Multiple skin lesions are common and pale brown pigmented patches on the skin, known as café-au-lait spots, may be evident. Multi-organ involvement may arise, including bladder, heart, intestines, kidney and larynx.
### _Differential diagnosis_
If solitary it may appear like:
* fibrous epulis.
* lipoma.
* fibroma.
* reactive osteoma.
### _Clinical investigation_
Biopsy.
### _Management_
Excision if the condition is causing symptoms (aesthetics or functional problems). If multiple, medical management is essential. There is rarely malignant transformation to malignant peripheral nerve sheath tumours or sarcomas.
### _Dental Appliance-Induced Hyperplasia_
### _Clinical appearance_
There is localised gingival enlargement related to chronic irritation from a clasp arm or prosthetic/orthodontic appliance component (Fig 5-14). Ulceration may precede fibrosis (Fig 5-15).
**Fig 5-14** Localised gingival enlargement caused by plaque accumulation around a fixed orthodontic appliance LL12. Plaque control was inhibited and the marginal gingivae irritated due to encroachment of the brackets on the gingival margin.
**Fig 5-15** Fibrous epulis LR23 with surface ulceration, associated with a poorly fitting lower acrylic partial denture, which has replaced LR1 and LL1.
### _Clinical symptoms_
Often there are none, except aesthetic concerns or bleeding when traumatised.
### _Aetiology_
Lesions directly related to irritation by appliance components are caused by a chronic inflammatory reaction, which leads to fibrovascular hyperplasia.
### _Involvement of non-gingival sites_
Lesions arising elsewhere on the oral mucosa are termed fibro-epithelial polyps (Fig 5-16) and are also caused by chronic irritation/trauma.
**Fig 5-16** A fibroepithelial polyp of the left buccal mucosa caused by chronic trauma from an upper complete denture.
### _Differential diagnosis_
* Fibrous epulis.
* Vascular epulis.
### _Clinical investigation_
Excisional biopsy shows fibro-epithelial hyperplasia with chronic inflammatory infiltrate.
### _Management options_
Remove the cause of irritation/plaque accumulation (adjust/modify appliance) and scale area thoroughly. Lesions may resolve spontaneously, but if not surgical excision and re-contouring are indicated.
### _Lateral Periodontal Abscess_
### _Clinical appearance_
This is variable in appearance. It may be a red fluctuant swelling (Fig 5-17) or show evidence of pus beneath (cream appearance, Fig 5-18). It may (Fig 5-19) or may not be 'pointing' and there may be evidence of a sinus tract (Fig 5-20), although most drain via the gingival crevice/periodontal pocket.
**Fig 5-17** A lateral periodontal abscess affecting UR45.
**Fig 5-18** A lateral periodontal abscess affecting UR1 – note the colour is creamy rather than the intense red of Fig 5-17.
**Fig 5-19** A pointing periodontal abscess in the UR6 area.
**Fig 5-20** A gutta percha point placed within a sinus tract LR5 to trace the source of infection radiographically.
### _Clinical symptoms_
Pain/tenderness on lateral pressure usually arises after the gingival swelling has appeared. Pocketing will be evident and there will be a discharge of pus from the pocket or labial sinus tract.
### _Aetiology_
There is microbial infection of the tissues lining the pocket wall, adjacent to an infected root surface. Bacteria are often gram +ve (e.g. Streptococcus constellatus).
### _Involvement of non-gingival sites_
Patients may rarely become pyrexial, regional lymphadenopathy may be evident and rarely a cellulitis may develop (spreading infection through regional soft tissue planes).
### _Differential diagnosis_
* Giant cell epulis.
* Pyogenic granuloma/vascular epulis.
* Lipoma (if yellow).
* Kaposi's sarcoma (if red).
* Gingival abscess.
### _Clinical investigation_
* Vitality test tooth to eliminate a pulpal aetiology and take periapical radiograph.
* If multiple, consider assessing blood glucose levels.
### _Management options_
Drain the infection directly through the pocket by scaling and root surface debridement (RSD) and curette the pocket wall. If fluctuant, apply topical anaesthesia (e.g. ethyl chloride) and incise with a number 11-scalpel blade. Insert tweezers into cut and open and close to drain pus. If it recurs, consider a swab for culture and sensitivity testing, and if the patient is pyrexial or there is evidence of spreading infection, consider systemic antibiotics.
### _Gingival Abscess_
These are very rare localised purulent infections, usually caused by superficial bacterial infection (e.g. an erupting tooth – Fig 5-21).
**Fig 5-21** A gingival abscess on an erupting LL2 in a child.
### _Stitch Abscess_
Stitch abscesses form beneath soft tissue that has been penetrated by a suture, following a surgical procedure (Fig 5-22). They are rare and normally associated with the use of multi-filament suture materials (e.g. silk). It is believed that the suture becomes colonised by commensal bacteria, some of which track down the suture and beneath the soft tissues, where they cause suppuration and fail to drain. Occasionally, stitch abscesses may also arise after the suture has been removed, due to foreign material being pulled into the tissues as the suture is drawn through. Management involves drainage and if the suture is still present, it should be removed.
**Fig 5-22** A stitch abscess UL2 area which has developed superficially around a suture removed five days earlier. This was most likely caused by contaminated suture material being drawn through the tissues during suture removal and following guided tissue regeneration with a non-resorbable membrane UL3.
### _Localised Trauma (see also Chapter 7)_
### _Clinical appearance_
Localised chronic gingival trauma can give rise to recession (Chapter 9) if severe, or granular swellings if less aggressive (Fig 5-23 and 5-24). The swellings may be solitary or multiple and appear at the site of the trauma as sessile lesions. There may be superficial ulceration and patches of keratosis and often evidence of granulation tissue formation with a fibrinous surface exudate.
**Fig 5-23** Granular exophytic swellings UR1, UL1 and UL3 in a 14-year-old girl who habitually bit her finger nails and irritated her gingivae. She underwent two biopsies before admitting to the habit.
**Fig 5-24** Traumatic lesions in the same patient as illustrated in Flig 5-23.
### _Clinical symptoms_
Patients frequently do not complain of pain, although lesions are sore due to the ulceration, and there may be mild surface haemorrhage.
### _Aetiology_
Attention-seeking or habitual scratching with the fingernail or a sharp implement is the most common cause, particularly in teenage females.
### _Involvement of non-gingival Sites_
None.
### _Differential diagnosis_
Lesions can appear sinister if solitary, and differential diagnosis may include:
* Neutropaenia.
* Viral warts.
* Squamous cell carcinoma (SCC) of the gingivae.
### _Clinical investigation_
In some cases there should be an incisional biopsy, to exclude an SCC, although this is highly unlikely given the patient's age and lesion location. Histopathology is of non-specific ulceration with chronic inflammation throughout.
### _Management options_
Management needs tact and diplomacy in case there are psychological problems with the child/adolescent or issues surrounding the child-parent relationship. If possible the patient should be approached without parental presence and confirmation of suspicions sought. The patient may on occasion admit to self-abuse, and the discussion alone can encourage habit cessation. Consider the use of chlorhexidine swabs to prevent secondary infection.
### _Histiocytosis X_
Histiocytosis X may present as a localised gingival enlargement, generalised gingival enlargement or indeed as localised recession. The condition is discussed in detail in Chapter 9.
### _Haemangioma/AV Malformations_
Haemangiomas may present as red lesions or gingival swellings and are discussed in Chapter 4.
### _Kaposi's Sarcoma (KS)_
Kaposi's sarcoma is discussed in Chapter 3.
### _Squamous Cell Carcinoma (SCC)_
Squamous cell carcinoma is discussed in Chapter 7, but may present as a localised gingival enlargement.
### _Metastatic Tumours_
Metastatic tumours are rare in the periodontal and gingival tissues. Leukaemic cell infiltration (Chapter 8) may affect the gingiva and metastatic tumours from breast, kidney and prostate do rarely arise. Biopsy is essential for diagnosis.
### _Lymphoma_
Lymphomas are discussed in Chapter 8.
### _Reactive Osteoma_
This is a slow growing benign tumour, which may be reactive to chronic irritation (reactive exostosis) or may appear de novo. It is usually solitary, but multiple lesions may arise as part of Gardner's syndrome a familial autosomal dominant disorder (Chapter 11).
### _Lesions Associated with PTEN-Hamartoma Tumour Syndromes_
(PTEN is a tumour suppressor gene).
### _Clinical appearance_
Benign lesions, largely fibro-epithelial or fibro-vascular in nature may arise on the gingivae, palate, buccal mucosa or tongue. They may appear as:
* Epulides.
* Oral polyps.
* Sessile wart-like lesions.
* Haemangiomatous lesions.
### _Clinical symptoms_
* Swelling.
* Lump that is irritating.
* Incidental finding by patient or GDP.
### _Aetiology_
Germline mutations in the PTEN tumour suppressor gene (PTEN deletions or promoter-region mutations) are associated with two allelic syndromes:
* Cowden's syndrome (90% frequency of PTEN mutations).
* Bannayan-Riley-Ruvalcaba syndrome (BRRS – 65% frequency of PTEN mutations).
* Sub-set of cases of Proteus and Proteus-like syndrome.
### _Involvement of non-gingival sites_
The PTEN gene regulates cell growth, and patients with Cowden's and BRRS syndromes present with multiple hamartomas affecting:
* Lung.
* Breast.
* Skin.
* Glandular tissue (especially thyroid).
* Oral mucosa.
* Polyposis coli.
Haemangiomas and arterio-venous malformations (Chapter 3) are also common.
### _Differential diagnosis_
This may be any common gingival/oral hamartoma or tumour, but patients with Cowden's or BRR syndromes may also have:
* Macrocephaly (large head).
* Learning, speech or organisational difficulties.
### _Clinical investigation_
* Medical history (many cases may be undiagnosed).
* Family history of Cowden's or BRR syndromes.
* Biopsy.
### _Management options_
* Genetic counselling and testing for PTEN mutations.
* Regular recall and fastidious monitoring of the oral mucosa and head and neck are essential.
### _Footnote_
Patients with Cowden's syndrome or BRRS have a frequent need for biopsy and excision of various lesions. Patients are constantly being exposed to medical investigation and minor surgery and require immense support, encouragement and understanding from the dental and medical profession.
### Further Reading
Seymour RA, Heasman PH (eds). Drugs Diseases and the Periodontium. Oxford: Oxford Medical Publications, 1992.
Soames JV, Southam JC. (Eds). Oral Pathology. Oxford: Oxford Medical Publications, 1993.
Chapple I L C, Hamburger J. The Significance of Oral Health in HIV Disease. Journal of Sexually Transmitted Infections 2000;76:236–243.
Grattan CEH, Hamburger J. Cowden's disease in two sisters, one showing partial expression. Clinical and Experimental Dermatology 1987;12:360–363.
## Chapter 6
## Gingival Enlargements – Generalised
### Aim
This chapter aims to provide an overview of the causes, features and management of generalised gingival swellings, which involve the free and/or attached gingiva and may also extend to the non-keratinised lining oral mucosa.
### Outcome
Having read this chapter, the clinician should be able to formulate an appropriate differential diagnosis for generalised gingival enlargements, know which additional clinical investigations to perform to arrive at a definitive diagnosis and be aware of key management strategies either within their practice or in a specialist environment.
Table 6-1 lists the generalised gingival enlargements discussed in this chapter and highlights which are common and uncommon and when the condition may be managed in general practice or should be referred.
Table 6-1 **Generalised Gingival Swellings** **Appearance/ Character** | **Category** | **Sub-Category** | **Incidence** | **Manage/ Refer**
---|---|---|---|---
Fibrous enlargements | Hereditary gingival fibromatosis | | uncommon | manage/ refer
Drug-induced gingival overgrowth | Dilantins (e.g. Phenytoin) | common (13–15% of medicated patients) | manage/refer
| Calcium channel blocking drugs (e.g. nifedepine amlodipine felodipine) | common (10–15% of medicated patients) | manage/refer
| Ciclosporin | common (30% of medicated patients) | manage/refer
Appliance-induced hyperplasia | | common | manage
Delayed gingival retreat | | common | manage
Mucopolysaccharidoses | | rare | refer
Mannosidosis | | rare | refer
Oedematous enlargements | Inflammatory gingival enlargement | Plaque-induced | common | manage
Hormonal influence | common | manage
Hereditary angioedema | uncommon | refer
Acquired angioedema | uncommon | refer
Granulomatous enlargements | Sarcoidosis | | uncommon | refer
Crohn's Disease | | uncommon | refer
Orofacial Granulomatosis | | uncommon | refer
Exophytic swellings | Leukaemia | Acute: | |
monocytic | uncommon | refer
myelomonocytic | uncommon | refer
myeloid | uncommon | refer
lymphocytic | uncommon | refer
Pyostomatitis vegetans | | uncommon | refer
Wegener's granulomatosis | | uncommon | refer
Plasmacytoma | | uncommon | refer
Amyloidosis | | uncommon | refer
Multiple myeloma | | uncommon | refer
### Terminology
A variety of terms has been, and is, used to describe generalised gingival enlargements, which can give rise to confusion. These are summarised below:
_Gingival hyperplasia_ – 'hyperplasia' is a term that describes tissue enlargement arising from an increase in number of one or more constituent cell types. Hyperplasia can therefore be singular (e.g. connective tissue hyperplasia) or compound (e.g. fibro-epithelial hyperplasia). The term gingival hyperplasia has thus become outdated because it poorly describes the true nature of the swelling.
_Gingival hypertrophy_ – 'hypertrophy' describes an increase in tissue size due to an increase in the size of one or more constituent cell types. This can also be singular or compound.
_Gingival overgrowth_ – 'overgrowth' is a term used to overcome some of the shortcomings of the above two terms, because it is less specific. The advantage of this term is that it allows for part of, or the entire enlargement to be due to increased production of connective tissue matrix or collagen fibre deposition. 'Overgrowth' is usually used in association with drug-induced enlargements that are histologically complex.
_Gingival enlargement_ – the term 'enlargement' is used in this chapter, because not all gingival enlargements are true swellings (e.g. delayed gingival retreat), but they do appear clinically as enlarged tissues.
### Fibrous Swellings
### _Hereditary Gingival Fibromatosis (HGF)_
### _Clinical appearance_
HGF presents as a generalised pink, firm and often stippled enlargement of the free and attached gingivae, extending to the mucogingival junction (see Chapple and Gilbert, 2002) buccally extending variably into the palate. Classically it affects maxillary tuberosities (Fig 6-1) and retromolar regions of the mandible (Fig 6-2). However, the labial gingivae may be involved (Fig 6-3), and it is important to distinguish this from delayed gingival retreat if planning surgical recontouring. The fibrosis is slowly progressive and may give rise to tooth movement and spacing, or may delay or prevent tooth eruption.
**Fig 6-1** Hereditary gingival fibromatosis classically affecting the maxillary tuberosities, where false pocketing had led to early periodontal attachment loss.
**Fig 6-2** Hereditary gingival fibromatosis classically affecting the mandibular retromolar region.
**Fig 6-3** Hereditary gingivo-fibromatosis of the labial gingivae in a 12-year- old boy, whose mother, aunt and younger brother were also affected.
### _Clinical symptoms_
The patient's main concerns will be:
* Aesthetic.
* Functional – tissue can overgrow the crowns of teeth if severe. Also teeth may be moved to a position that interferes with normal occlusal function.
* Discomfort – can arise where fibrosis is actively moving teeth.
### _Aetiology_
Unknown, but two forms are described:
* Familial HGF – autosomal dominant inheritance recently linked (Hart et al, 1998) to chromosome 2p21 (position 21 of short arm of chromosome 2). Penetrance may be incomplete, i.e. the condition may not always be expressed phenotypically or may vary in severity.
* Sporadic HGF – a controversial diagnosis that may be autosomal dominant or recessive. It may simply be a familial form that has variable clinical expression, incomplete penetrance or may arise due to a spontaneous mutation within the 2p21 region.
### _Involvement of non-gingival sites_
Gingival fibromatosis is also associated with various rare syndromes:
* Rutherford syndrome (juvenile hyaline fibrosis, corneal dystrophy, neurosensory hearing loss) – autosomal dominant inheritance.
* Laband syndrome (nail, ear, nose and bone defects, syndactyly) – autosomal recessive/dominant or spontaneous mutations.
* Cross syndrome (hypopigmentation, microphthalmia, athetosis) – autosomal recessive.
* Ramon syndrome (hypertrichosis [excessive hair growth], cherubism, mental retardation) – autosomal recessive.
Hypertrichosis, mental retardation, epilepsy and growth hormone defects are also described by Gorlin et al (1976), and interestingly hypertrichosis and gingival fibrosis are both complications of ciclosporin medication (see below).
### _Differential diagnosis_
* Delayed gingival retreat.
* Drug-induced gingival overgrowth.
* Plaque-induced chronic inflammatory enlargement.
### _Clinical investigation_
HGF is a presumptive diagnosis based on a careful history and clinical findings. Definitive diagnosis currently requires confirmatory histopathology in addition to clinical findings. Excision biopsy is usually by conventional open-faced gingivectomy (Fig 6-4, 6-5), but if severe, an inverse bevel flap may be required (Fig 6-6) with significant undermining by sharp dissection and filleting ± a distal wedge procedure (Fig 6-7).
**Fig 6-4a** HGF from the patient in Fig 6-1, immediately post-open-face gingivectomy.
**Fig 6-4b** The tissue excised from the patient in Fig 6-4a.
**Fig 6-5** The healed tuberosities from 6-4a, two weeks post-surgery. False pocketing has been eliminated. Note the improved angle between the gingival margin UL6 and the tooth, facilitating ease of plaque control.
**Fig 6-6** Diagram to illustrate the inverse bevel incision prior to flap reflection and 'filleting' of the bulk of the sub-mucosal connective tissue.
1. Bulky tissue prior to split-thickness flap reflection.
2. First incision is parallel to a sulcus incision, using an inverse bevel.
3. Second incision isolates a tissue 'cuff.
4. Split-thickness flap is then raised by sharp dissection allowing surgical access to the underside of the flap.
5. Cuff is removed and underside of flap is debulked by sharp excision of excess connective tissue. Care must be taken to avoid puncturing the flap.
6. Flap is closed after any necessary debridement and removal of the bulky connective tissue from the underside of the flap.
**Fig 6-7** Diagram to illustrate the distal wedge procedure. There are several variants, but the most basic literally involves the sharp dissection of a wedge of tissue from the tuberosity/retro-molar region.
a. Enlarged tuberosity with false pocketing distal to UR7.
a1. Occlusal view of 'a'.
b. A buccal and lingual/palatal incision is made down to the alveolar crest
b1. The wedge is delineated by the dotted lines and buccal and palatal flaps are raised a short distance prior to wedge removal.
c. Wedge is sharp-dissected out.
c1. Buccal and lingual flaps are approximated over 'dead space' and in doing so this space is eliminated and the tissue level falls.
d. View of closed defect with tissue level more apically positioned.
d1. Occlusal view with distal wedge removal complete and flaps sutured.
### _Management_
If mild and symptom-free, simply monitor. However, probing is essential to distinguish between true and false pocketing (Fig 6-8). Surgical reduction is likely to require specialist skills for tuberosity and retromolar lesions, as feeder arteries can arise within, and the sublingual space may be compromised during lower molar surgery. Slow recurrence is likely.
**Fig 6-8** Schematic longitudinal section of a premolar and associated periodontal tissues demonstrating a healthy sulcus and false pocketing.
### _Drug-induced Gingival Overgrowth (DIGO)_
### _Clinical appearance_
The appearance of DIGO is variable. Classically the enlargement starts at the interdental papilla and spreads to involve the marginal gingivae. The anterior gingivae are most commonly affected, but posterior enlargement may lead to occlusal surface coverage in severe cases (Fig 6-9). With phenytoin and ciclosporin, in the presence of good plaque control, overgrowth is firm, fibrous and pink (Fig 6-10), but where plaque control is poor it becomes more vascular (Fig 6-11). Overgrowth associated with calcium channel blocking drugs (CCBs) tends to be more vascular (Fig 6-12) and is associated with more severe enlargement with concurrent ciclosporin. Surface keratosis or ulceration may arise following trauma from opposing teeth.
**Fig 6-9** Drug-induced gingival overgrowth in a renal transplant patient medicated with ciclosporin. Molar teeth were covered making mastication uncomfortable.
**Fig 6-10** Fibrous overgrowth in a patient medicated with ciclosporin.
**Fig 6-11** DIGO with a more vascular inflammatory component due to poor plaque control. (Fig 7-5 book 11).
**Fig 6-12** Fibro-vascular overgrowth in a patient medicated with ciclosporin and nifedepine.
### _Clinical symptoms_
As for HGF.
### _Aetiology_
DIGO is associated classically with three drug types, but has also been reported with the oral contraceptive, cannabis (Rees, 1992), erythromycin (Valsecchi and Cainelli, 1992) and sodium valproate (Syrjanen & Sryjanen 1979).
* Calcium channel blocking (CCB) drugs – nifedepine, amlodipine, felodipine, diltiazem hydrochloride.
* Dilantins – phenytoin
* Ciclosporin.
CCBs are used to control hypertension, ciclosporin is an immunosuppressive agent used to modulate allograft rejection or in severe erosive mucosal disease (Chapter 8). Phenytoin is an anti-convulsant drug. While there are some common features to their modes of action at an ionic level, the aetiology of DIGO remains poorly understood and complex. Seymour et al (2000) have described a number of risk factors:
* Age | |
+ve
---|---|---
* Genetic | – HLA-DR2 for Ciclosporin | +ve
|
– HLA-B37 for Ciclosporin | –ve
* Gender | – Male | +ve
|
– Female | –ve
* Periodontal variables | – Gingival inflammation | +++ve
|
– Plaque | ++ve
* Concomitant medication | – Ciclosporin and CCBs | +++ve
|
– Phenytoin and hepatic enzyme inducers | +ve
|
– Ciclosporin and azathioprine | –ve
|
– Ciclosporin and prednisolone | –ve
* Drug variables | – Pre-transplant enlargement | +ve
|
– Plasma concentrations | +ve
|
– Gingival crevicular fluid (GCF) concentrations | +ve
|
– Salivary concentration | +ve
### _Involvement of non-gingival sites_
There are isolated reports of DIGO arising on an edentulous ridge. Hypertrichosis is associated with DIGO during ciclosporin medication.
### _Differential diagnosis_
* HGF.
* Delayed gingival retreat.
* Plaque-induced chronic inflammatory overgrowth.
* Pyostomatitis vegetans.
* Leukaemia.
* C1-esterase inhibitor deficiency (Hereditary angioedema) – Roberts et al, 2003.
### _Clinical investigation_
Diagnosis is made following a careful drug history and clinical examination (Chapter 1). Excisional biopsy demonstrates a fibroepithelial hyperplasia with epithelial acanthosis and increases in fibroblast number and/or collagen and extracellular matrix production. There is some evidence from immuno-histochemistry for the involvement of growth factors such as transforming growth factor beta (TGFβ: Wright et al, 2001).
### _Management_
A staged approach is necessary:
1. Hygiene-phase therapy (OHI and scaling) to reduce the inflammatory component.
2. Liaise with medical specialist to change medication if appropriate – spontaneous resolution with time (months) often results.
* for ciclosporin, replacement with tacrolimus (FK-506) may be justified to prevent the need for repeat surgery in severe and recurrent cases.
* for CCBs it may be possible to replace with a (β-blocker, ACE-inhibitor or diuretic, or a combination of these.
* where CCBs are in concomitant use with ciclosporin, CCB replacement initially is the best approach.
3. If swelling remains a problem, surgical gingival recontouring.
4. Rigorous supportive care (see Heasman, Preshaw and Robertson, 2004).
For subjects who have had a renal transplant ± chronic end-stage renal failure, calcium metabolism is disrupted and hypercalcaemia results in cardiac valve calcification in up to 50% of cases (Ribeiro et al, 1998). Antimicrobial prophylaxis will therefore be needed. Renal transplant patients who have an arterio-venous fistula from haemodialysis, and foreign bodies in their vasculature to support the shunt, may also require such prophylaxis. For heart, lung, liver transplants and prosthetic joint replacements, there is no current evidence to support the use of antibiotic cover.
### _Dental Appliance-induced Enlargement_
### _Clinical appearance_
The appearance of appliance-induced enlargement is of thickened and enlarged (normally palatal) gingival tissues arising beneath a poorly designed prosthetic appliance base (Fig 6-13). Such appliances are tissue-borne and give rise to false pocketing (Fig 6-4) secondary to coronal up-growth of the gingiva. The tissues can become inflamed with time and a thin, red mobile margin develops.
**Fig 6-13a** An acrylic denture lacking tooth support.
**Fig 6-13b** Denture stomatitis and false pocketing UL 234 as a result of the poorly designed prosthesis in Fig 6-13a.
### _Clinical symptoms_
Occasionally an unstable prosthesis or discomfort and bleeding on brushing.
### _Aetiology_
Lesions directly related to irritation by appliance components are caused by a chronic inflammatory reaction, leading to fibrovascular hyperplasia.
### _Involvement of non-gingival sites_
None.
### _Differential diagnosis_
The diagnosis is usually obvious following examination of the prosthesis in situ.
### _Clinical investigation_
Excisional biopsy shows fibroepithelial hyperplasia with a chronic inflammatory infiltrate.
### _Management options_
Adjust/modify the appliance initially and scale area thoroughly. Surgical excision is likely to be needed, but lesions rarely resolve spontaneously. Re-make a definitive prosthesis that eliminates the original design flaw (see Noble, Kellett and Chapple, 2004).
### _Delayed Gingival Retreat_
### _Clinical appearance_
Short clinical crowns, give the impression that the free and attached gingivae have overgrown (Fig 6-14). The attached gingivae are not thickened, but there may be rolling of the free gingival margin and mild thickening.
**Fig 6-14** Delayed gingival retreat. The gingival margins around all upper incisors and three of the four lower incisors have not reached their mature 'adult' position. The margin at LL1 is prematurely at its adult position and may progress to recession.
### _Clinical symptoms_
Patients include children, adolescents or young adults (see Clerehugh, Tugnait and Chapple, 2004), who may complain of a 'gummy smile' or that their peer group make fun of them because they 'have no teeth'.
### _Aetiology_
Biologically slow retreat of the gingival margin to its mature adult position, 2–3mm coronal to the cemento-enamel junction (CEJ).
### _Involvement of non-gingival sites_
None.
### _Differential diagnosis_
* HGF.
* DIGO.
* Variation in normal anatomy (see Chapple and Gilbert, 2002).
### _Clinical investigation_
Clinical examination alone.
### _Management options_
* Check for true pocketing.
* Reassure patients and parents and review the situation after 12 months.
### Oedematous Enlargements
### _Inflammatory Gingival Enlargement_
### _Clinical appearance_
Red, swollen marginal gingivae with a degree of fibrosis (Fig 6-15). False pocketing results from coronal overgrowth and with time may lead to true pocketing.
**Fig 6-15** Inflammatory gingival enlargement, due to plaque accumulation aggravated by pregnancy in a young patient.
### _Clinical symptoms_
Bleeding on brushing, soreness, malodour and aesthetic concerns.
### _Aetiology_
Poor plaque control initially leads to plaque-induced gingivitis. If unresolved, the inflammatory lesion becomes chronic and fibrous repair occurs concurrently with plaque-induced inflammation. Generally the enlargement or 'hyperplastic response' arises due to the presence of local risk factors that inhibit plaque removal and directly irritate the tissues. These include:
* Imbricated teeth (Fig 6-16).
* Orthodontic appliances (Fig 6-17).
* Mouth breathing (Fig 6-18).
* Hormonal – puberty or pregnancy result in hormonal changes and the gingival and periodontal tissues possess oestradiol and androgen receptors. The latter appear to induce histological changes which include epithelial separation and increases in vascular permeability (Vittek et al, 1984; Jönsson et al, 2004). Gingival tissues are also capable of sex steroid metabolism (El Attar, 1974), and the inflammatory response to plaque accumulation may be exaggerated at such stages in life.
**Fig 6-16** Inflammatory gingival enlargement, due to plaque accumulation aggravated by a reduced ability to clean around imbricated teeth.
**Fig 6-17** Mild inflammatory gingival enlargement, due to plaque accumulation aggravated by the presence of a fixed orthodontic appliance.
**Fig 6-18a** A 12-year-old boy who breathes through his mouth, with a high lip line and incompetent lips.
**Fig 6-18b** Mild inflammatory gingiva enlargement in 12-year-old boy wh breathes through his mouth with a hig lip line and incompetent lips. A lack o saliva flow also reduces cleansing an aggravates the situation.
Histology shows collagen fibres, fibroblasts and inflammatory cells.
### _Involvement of non-gingival sites_
None.
### _Differential diagnosis_
* Appliance-induced enlargement.
* DIGO.
### _Clinical investigation_
Additional investigations are not indicated, given the presence of plaque deposits and associated risk factors.
### _Management options_
Treatment should include scaling and oral hygiene instruction in the first instance, with regular supportive care. This will resolve the inflammatory component of the enlargement, but any remaining fibrous deformity should be corrected surgically.
### _Angioedema (C1-Esterase Inhibitor Deficiency/Dysfunction)_
Angioedema may be caused by a C1-esterase inhibitor defect or, indeed, may have an allergic aetiology. The discussion in this section will not cover the allergic form.
### _Clinical appearance_
Angioedema classically presents with lip swelling (Fig 6-19) or angioedema of the head, neck or extremities. Onset is acute. Extremely rarely, gingival tissues may be involved (Fig 6-20) in an obscure localised form of angioedema (Roberts et al, 2003).
**Fig 6-19** Crohn's disease giving rise to a midline lip fissure due to the marked oedema. Angioedema presents in a similar manner, but without the midline fissuring.
**Fig 6-20** Severe gingival oedema in a patient who has C1-esterase inhibitor deficiency (hereditary angioedema type II).
### _Clinical symptoms_
Aesthetics, discomfort and concern over the swelling are the main reasons for presentation. Additionally, oedema of the tongue, cheeks and upper and lower airways often leads to dyspnoea (breathlessness), which can, in severe cases, be life threatening.
### _Aetiology_
Angioedema may be hereditary (HAO) or acquired (allergic) in nature. It involves a defect in an enzyme that damps down complement activation (see Chapple and Gilbert 2002). The acquired form usually presents in adults but HAO, which has an autosomal dominant pattern of inheritance, can present in younger patients.
HAO is classified into:
* Type I HAO – a decrease in production of enzyme.
* Type II HAO – normal enzyme levels but dysfunctional enzyme.
### _Involvement of non-gingival sites_
* Tongue.
* Cheeks.
* Upper airway.
* Head.
* Neck.
* Extremities.
### _Differential diagnosis_
* Oro-facial granulomatosis (OFG).
* Crohn's disease.
* Sarcoidosis.
* Type I hypersensitivity reaction.
### _Clinical investigation_
Assay levels of C1, C1-esterase inhibitor and also C1-esterase inhibitor function in plasma.
### _Management options_
Medical management is complex and may involve intravenous administration of C1-esterase inhibitor concentrate, or use of anabolic steroids such as stanozolol or danazol. Referral to a clinical immunologist is essential.
### Granulomatous Enlargements
### _Orofacial Granulomatosis (OFG)_
OFG is not a diagnosis, more a descriptive term that reflects underlying disease, such as:
* Hypersensitivity reactions to dietary allergens.
* Sarcoidosis.
* Angioedema.
* Melkersson-Rosenthal syndrome.
* Crohn's Disease.
* TB.
* Leprosy.
### _Clinical appearance_
The clinical appearance and symptoms of OFG are essentially those of 'oral Crohn's disease' (see above), but the gastrointestinal features are absent. It usually presents in the second or third decade of life and has an equal sex ratio.
### _Aetiology_
The aetiology of OFG is obscure, and the term is often used to describe oral granulomatous conditions with no known systemic cause. There is evidence that OFG may be due to hypersensitivity reactions to dietary allergens such as benzoates and cinnamonaldehyde.
### _Involvement of non-gingival sites_
* Facial/lip swelling.
* Mucosal ulceration.
* Mucosal tags and\cobblestoning.
* Extra-oral involvement.
### _Clinical investigation_
* Biopsy down to muscle is needed and shows non-caseating epithelioid cell granulomata and lymphoedema.
* Blood investigations may indicate raised serum ACE (angiotensin-converting enzyme) levels, due to ACE release from the granulomas.
* Chest radiographs.
### _Management options_
* The underlying cause if identified, should be treated, but this may not affect resolution of orofacial swelling. If allergens are identified by skin testing, these should be avoided through employment of exclusion diets.
* Intralesional corticosteroids may provide short-term relief, as may surgical reduction.
### _Sarcoidosis_
### _Clinical appearance_
* Broad-band granular gingival erythema (Fig 6-21).
* Velvet-like consistency to free and attached gingivae.
* Gingival swelling may extend beyond mucogingival junction.
* Lip swelling may occur.
* Cervical lymphadenopathy.
* Salivary gland enlargement is rarely associated with Heerfordt's syndrome, which includes a facial palsy, lacrimal swelling, uveitis and fever.
**Fig 6-21** A broad 'band-like' gingival erythema with a soft 'velvet-like' consistency. Biopsy and subsequent investigations confirmed sarcoidosis.
### _Clinical symptoms_
* Discomfort is described in some patients.
* Aesthetic concerns.
### _Aetiology_
Sarcoidosis is a rare (0.02% in Caucasians) multifocal granulomatous disorder, more common in black patients.
### _Involvement of non-gingival sites_
* Lungs.
* Spleen.
* Liver.
* Eyes.
* Salivary (parotid) glands (Heerfordt's syndrome).
* Skin.
* Lymph nodes – hilar (lung) and cervical.
### _Differential diagnosis_
* Crohn's Disease.
* Melkersson-Rosenthal syndrome (facial oedema/swelling, plicated tongue and a VII nerve palsy).
### _Clinical investigation_
* Biopsy may aid diagnosis, identifying non-caseating giant cell granulomas.
* Chest radiograph may show lung involvement.
* Blood tests for angiotensin converting enzyme (ACE), which can be raised in serum due to its production by the tissue granulomas.
### _Management options_
Refer to a physician. Medical management involves treatment with systemic corticosteroids and has a high success rate.
### _Crohn's Disease_
Crohn's disease is a chronic inflammatory granulomatous bowel disease. It typically affects the ileum or colon, but may affect any part of the digestive system. The affected areas become red and swollen and ulceration may arise. As the ulcers heal, scar tissue forms, inducing intestinal strictures and obstruction.
### _Clinical appearance_
* Broad-band gingival swelling.
* Mucosal tags.
* Linear 'fissured' ulceration of oral mucosa.
* Cobblestone appearance to the buccal mucosa due to fibrosis of ulcers.
* Lip swelling with a linear midline fissure (Fig 6-19).
### _Clinical symptoms_
* Swollen gingivae/lips.
* Oral ulceration.
* Cracking of corners of mouth (angular stomatitis).
### _Aetiology_
Crohn's is a chronic inflammatory bowel disease of unknown aetiology.
### _Involvement of non-gingival sites_
* Small bowel, patients may pass blood and mucous per rectum with loose stools/diarrhoea.
### _Differential diagnosis_
* OFG.
* Sarcoidosis.
* Melkersson-Rosenthal syndrome.
* Angioedema.
### _Clinical investigation_
* Biopsy is needed for diagnosis, where granulomas and lymphoedema are characteristic.
* Blood tests may reveal low serum iron, B12 due to malabsorption.
* Gastrointestinal investigation of the terminal ileum by barium imaging.
### _Management options_
Medical management requires specialist referal to gastroenterology. Oral management is often unsatisfactory and may involve the use of topical corticosteroid preparations or intralesional corticosteroid injections (triamcinolone). Specialist referal is required.
### Exophytic Swellings
### _Leukaemia_
Leukaemia may present as generalised gingival enlargement, but also as ulceration or haemorrhage (see Chapter 8).
### _Pyostomatitis Vegetans_
### _Clinical appearance_
* Irregular gingival swelling appearing somewhat granular and warty.
* Yellow pustular appearance due to intra/sub-epithelial abscesses.
* Areas of necrosis and sloughing.
* Exophytic appearance.
* Profuse bleeding.
### _Clinical symptoms_
* Swollen gums.
* Gingivae are sore/painful.
* Bleeding gums.
* Malodour.
* Aesthetic concerns over appearance.
* May have gastrointestinal symptomatology.
### _Aetiology_
Extremely rare and is associated with inflammatory bowel disease (occasionally Crohn's, but more usually ulcerative colitis). The patient in Fig 1-7 presented with triple pathology: NUG, erosive lichen planus and pyostomatitis vegetans.
### _Involvement of non-gingival sites_
* Labial/buccal mucosa.
* Alveolar mucosa.
### _Differential diagnosis_
* Acute leukaemia.
* DIGO with severe inflammation and ulceration from trauma.
* Vesiculobullous disease.
* Wegener's granulomatosis.
### _Clinical investigation_
* Consider serological and haematological investigations to identify possible malabsoption (e.g. iron and vit B12) and elevation of acute phase markers.
* Biopsy to demonstrate tissue abscess formation.
* Endoscopy.
### _Management options_
Management is medical and involves controlling underlying bowel disease. Reduce any sources of oral infection (extract teeth with poor prognosis) and treat any associated periodontal disease.
### _Wegener's Granulomatosis_
### _Clinical appearance_
* Granular 'strawberry-like' gingival hyperplasia (Fig 6-22).
* Exophytic strawberry-like lesions may develop (Fig 6-23).
* Delayed healing of extraction sockets.
**Fig 6-22** Wegener's granulomatosis of gingivae.
**Fig 6-23** Wegener's granulomatosis in an adult, presenting as a strawberry-like granular swelling of the lower alveolus.
### _Clinical symptoms_
* Swollen gums.
* Gingivae can be painful and ulceration is described.
* Chronic sinusitis or nasal obstruction.
### _Aetiology_
Systemic disease of unknown aetiology, but an immunological basis is suspected.
### _Involvement of non-gingival sites_
* Classically necrotising granulomas of the nose, paranasal sinuses and lungs.
* Small artery vasculitis of respiratory tract and lungs.
* Renal vasculitis (glomerulitis).
* Tongue may be involved.
### _Differential diagnosis_
* DIGO
* Pyostomatitis vegetans.
### _Clinical investigation_
* Biopsy shows necrotising vascular changes in small vessels, granulation tissue, microabscess formation and non-specific inflammation involving most forms of leukocyte within tissues (histiocytes). There may be an epithelial hyperplasia and giant cells are common.
* Blood investigations show anti-neutrophil cytoplasmic antibodies (ANCA) in serum (cytoplasmic staining with specificity for proteinase 3).
### _Management options_
* Urgent referral to appropriate medical specialty (renal medicine or rheumatology):
* Medical management may involve the use of systemic corticosteroids (prednisolone) or cytotoxics (cyclophosphamide).
* Monitoring of serum ANCA levels, which vary with disease activity.
### Bony Swellings
These are discussed in Chapter 11.
### Further Reading
Chapple ILC, Gilbert AD. Understanding Periodontal Diseases: Assessment and Diagnostic Procedures and Practice. Chapple ILC (ed). QuintEssentials of Dental Practice –1, Periodontology-1. London: Quintessence Publishing Co. Ltd, 2002;Chapter 1,pp3–16.
Clerehugh V, Tugnait A, Chapple ILC. Periodontal Management of Children, Adolescents and Young Adults. 2004. Chapple ILC (ed). QuintEssentials of Dental Practice–17, Periodontology-4. London: Quintessence Publishing Co. Ltd, 2004;Chapter 10,pp131–149.
Gorlin RJ, Pindborg JJ, Cohen MM Jr. (Eds). Syndromes of the Head and Neck. 2nd Edn. New York: McGraw-Hill, 1976;pp329–336.
El Attar TMA. The in vitro conversion of male sex steroid 1,2-3H-androstenedione in normal and inflamed human gingivae. Arch Oral Biol 1974;19:1185–1190.
Heasman PA, Preshaw PM, Robertson P. Successful Periodontal Therapy. A Non-Surgical Approach. Chapple ILC (ed). QuintEssentials of Dental Practice –16, Periodontology-3. London: Quintessence Publishing Co. Ltd, 2004;Chapter 9 pp99–133.
Jönsson D, Andersson G, Ekblad E, Liang M, Bratthall G, Nilsson B-O. Imunocytochemical demonstration of oestrogen ' in human periodontal ligament cells. Archives of Oral Biology 2004;49:85–88.
Noble SN, Kellett M, Chapple ILC. Decision Making for the Periodontal Team. Chapple ILC (Ed). QuintEssentials of Dental Practice –11, Periodontology-2, London: Quintessence Publishing Co. Ltd, 2004;Chapter 10 pp131–149.
Rees TD. Oral effects of drug abuse. Crit Rev Oral Biol Med 1992;3:163–184.
Ribeiro S, Ramos A, Brandao et al. Cardiac valve calcification in haemodialysis patients: role of calcium-phosphate metabolism. Nephrol Dial Transplant 1998;13:2037–2040.
Roberts A, Shah M, Chapple ILC. C-1 esterase inhibitor dysfunction localised to the periodontal tissues: clues to the role of stress in the pathogenesis of chronic periodontitis? J Clin Periodontol 2003;30:271–277.
Seymour RA, Heasman PH (eds). Drugs Diseases and the Periodontium. Oxford: Oxford Medical Publications, 1992.
Seymour RA, Ellis JS, Thomason JM. Risk factors for drug-induced gingival overgrowth. J Clin Periodontol 2000;27:217–223.
Vittek J, Kirsch S, Rappaport SC, Bergman M, Southren AL. Salivary concentrations of steroid hormones in males and in cycling and post-menopausal females with and without periodontitis. J Perio Res 1984;19:545–555.
Wright HJ, Chapple ILC, Matthews JB. TGF' isoforms and receptors in drug-induced and hereditary gingival overgrowth. J Oral Pathol Med 2001;30:281–289.
## Chapter 7
## Localised Gingival Ulceration
### Aim
The aim of this chapter is to detail those clinical entities that may present as a single or isolated discrete area of ulceration on the gingiva, as opposed to more widespread or multiple areas of ulceration.
### Outcome
The reader should be able to differentiate between the sinister and the trivial, request relevant investigations and appropriately manage the conditions discussed.
### Definition
Gingival ulceration describes an area of mucosa devoid of its surface epithelium, and exposing the underlying connective tissue.
The term 'erosion' is sometimes used to describe those areas of shallow ulceration that do not expose the underlying connective tissue. Table 7-1 provides a summary of the contents of this chapter.
Table 7-1 **Summary table – localised gingival ulceration** **Major Categories** | **Sub Categories** | **Frequency of Condition** | **Management Setting**
---|---|---|---
Trauma | Iatrogenic (from dental appliances)
Chemical / Thermal Gingivitis artefacta | Traumatic ulceration of the gingivae occurs frequently. | These conditions can usually be managed by non-specialists although if psychiatric morbidity is associated with gingivitis artefacta, appropriate referral is necessary.
Recurrent aphthous stomatitis | Minor aphthous ulceration
Major aphthous ulceration
Herpetiform aphthous ulceration | Aphthous stomatitis is a very common condition (reportedly affecting 20% of the UK population). | Gingival involvement is however unusual. Initial management is undertaken in primary care. If there is a suggestion of underlying disease or poor response to treatment, the patient should be referred for a specialist opinion.
Neoplasia | Usually squamous cell carcinoma
Lymphomas
Leukaemias
Metastatic tumours
Benign tumours | Oral mucosal neoplasia is uncommon and gingival involvement is rare. | Urgent referral for specialist management of malignancy.
Bacterial infections | Necrotising ulcerative gingivitis | Uncommon | Manage in primary care Refer if patient's immune status is suspect.
Tuberculosis | Very rare | Specialist referral
Syphilis | Very rare |
Viral infections | Hand, foot and mouth | Uncommon | Supportive management within primary care setting
Varicella zoster | Uncommonly affects the gingivae | Primary care or referral dependent on distribution and severity
Cytomegalovirus | Very rare | Referral – condition indicative of compromised immunity
Deep mycoses | Histoplasmosis | Very rare in the UK | Specialist referral
### _Traumatic Ulceration_
Traumatic ulceration of the gingiva has a variety of causes including:
* Iatrogenic, (ill-fitting or ill-designed oral appliances and inappropriate oral hygiene practices).
* Self-induced (gingivitis artefacta) – (Fig 7-1) see Chapter 9.
* Direct chemical toxicity (dentifrices, aspirin or the use of recreational drugs such as cocaine) – see Chapters 3 and 10.
**Fig 7-1** Gingivitis artefacta.
### _Clinical appearance_
* Traumatic ulceration varies in appearance dependent on the causative factors. The site of ulceration may identify the source of the trauma, for example a clasp on a denture, a spring on an orthodontic appliance or malpositioned teeth.
* The appearance is one of non-specific ulceration that may mirror very closely the source of the trauma.
* Self-induced trauma can have dramatic clinical consequences including exfoliation of the teeth.
### _Clinical symptoms_
* Patients may be asymptomatic but are more likely to complain of soreness at the area of the ulceration.
* In gingivitis artefacta the patient may well be a young adult or adolescent.
### _Involvement of non-gingival sites_
* Localised ulceration of the gingivae is not usually associated with extra-gingival involvement although direct chemical toxicity and physical trauma from dental prostheses or orthodontic appliances may also provoke similar manifestations on the oral mucosa.
* Patients who self-harm may also traumatise themselves elsewhere.
### _Differential diagnosis_
* Non-specific ulceration.
* Vesiculobullous disease.
* Tumours.
* Wegener's granulomatosis.
* Pyostomatitis vegetans.
* Stewart's midline granuloma.
### _Clinical investigation_
* Clinical history and examination is essential and will reveal obvious local sources of trauma.
* Assessment of the psychological demeanour of the patient is paramount when self-harm is suspected. Direct questioning may well not elicit a truthful response, patients often strongly denying such activity.
* Biopsy of lesions is useful to exclude other diagnoses.
### _Management options_
* Elimination of obvious sources of trauma.
* In cases of self-harm, urgent referral for psychiatric advice is appropriate.
* In cases of suspected self-harm where there is significant injury, admission to hospital for supervision may be necessary to establish a definitive diagnosis.
### Bacterial Infections
Bacterial infections producing discrete localised areas of ulceration of the gingivae are uncommon.
### _Necrotising Ulcerative Gingivitis (NUG)_
See also Chapter 9, Localised Gingival Recession.
### _Clinical appearance_
* Ragged ulceration and necrosis involving the interdental papillae (Fig 7-2).
* Lesional tissue covered by a fibrinopurulent grey slough.
* Gingival bleeding and inflammation.
**Fig 7-2** Ulceration of the interdental papillae in necrotising ulcerative gingivitis.
### _Clinical symptoms_
* The lesions are painful.
* Characteristic foetor oris (bad breath).
* Bad taste in mouth.
### _Aetiology_
* Anaerobic infection with a variety of organisms including Treponema vincentii and Fusobacterium nucleatum constituting the so called 'fusospirachaetal complex'. In addition Prevotella intermedia is also reported to be associated with NUG. The risk factors that predispose to NUG include:
* Poor oral hygiene.
* Smoking.
* Immunodeficiency.
* Malnutrition.
* Concurrent infections.
### _Involvement of non-gingival sites_
* The infection can extend to adjacent tissues, including other periodontal tissues and the oral mucosa.
* In severe cases, in the debilitated or immunocompromised, the condition may involve the skin and can be extremely destructive (cancrum oris).
### _Differential diagnosis_
* Myeloproliferative disease.
* Immunocompromised host.
### _Clinical investigation_
* The diagnosis is usually made on the clinical features.
* Laboratory identification of the fusospirochaetal complex.
### _Management options_
* Oral hygiene instruction.
* Smoking cessation.
* Scaling and root surface debridement.
* Correction of underlying predisposing factors such as malnutrition.
* Oral metronidazole 200–400mg three times daily for three days.
### _Tuberculosis_
Although cases of tuberculosis are increasing in incidence, oral and particularly gingival manifestations remain infrequent.
The infection may manifest on the gingivae as a solitary ulcer with irregular and often undermined margins. It may also be painless and usually results from secondary infection due to expectorated infected sputum.
### _Syphilis_
The prevalence of sexually transmitted infections has increased dramatically over the last decade. Reported cases of syphilis in the UK are now at their highest level since 1984, with a six-fold rise in males occurring since 1998.
The primary site of infection with Treponema pallidum produces the so-called 'chancre' – this occurs at a variable time after inoculation, but usually of the order of three weeks later. The initial lesion is of a papule that ulcerates. The resultant chancre is indurated, often painless, associated with regional lymphadenopathy and resolves spontaneously within two to three weeks. (Alam, 2000). Gingival involvement is rare, intra-oral lesions being more frequently seen on the lips, tongue and palatal mucosa.
The superficial erosions or mucous patches ('snail track ulcers') of secondary syphilis are a rare gingival finding although intra-oral lesions may occur in up to 40% of cases, typically arising some four to ten weeks after the chancre. The gumma of tertiary syphilis has not been reported to involve the gingivae.
Diagnostic confirmation depends on serological testing and histological examination of the lesion may also be undertaken to exclude other diagnoses.
### Viral Infections
Viral infections, such as primary herpes simplex, often produce generalised gingival or oral mucosal involvement (see Chapter 4). However the following infections may produce more limited gingival involvement.
### _Hand, Foot and Mouth Disease_
### _Clinical appearance_
* Small vesicles that rupture to produce superficial ulcerations that coalesce forming serpiginous areas reminiscent of herpes simplex infection (Fig 7-3, 7-4).
* There is no inflammatory gingivitis as may be seen in primary herpes infections, although the ulcers are surrounded by an inflammatory halo.
**Fig 7-3** Hand, foot and mouth disease involving the palatal mucosa.
**Fig 7-4** Hand, foot and mouth disease showing coalescence of burst vesicles on the tongue.
### _Clinical symptoms_
* The infection is usually mild with little systemic upset, limiting itself in 10–14 days.
* Soreness at the site of the ulceration.
### _Aetiology_
* Infection is usually with Coxsackie A16.
### _Involvement of non-gingival sites_
* Any intra-oral site may be affected.
* Palms of the hands, soles of the feet and other areas of the skin subject to minor trauma.
### _Differential diagnosis_
* Herpes simplex infection.
* Erythema multiforme.
### _Clinical investigation_
* The diagnosis is usually based on clinical findings.
* Culture of Coxsackie virus is not straightforward, requiring inoculation of infected material into suckling mice.
### _Management options_
* Supportive care with adequate fluids, anti-inflammatories and rest.
* The condition resolves in seven to ten days and does not recur.
### _Varicella Zoster_
### _Clinical appearance_
* Varicella zoster infection, once established, can be recognised by its classical presentation of unilateral erythema and clusters of vesicles distributed along dermatomes. (Fig 7-5).
* The vesicles rupture after a few days, become crusted and often coalesce, healing over a period of three weeks, with further vesicles occurring during this time.
**Fig 7-5** Herpes zoster involving the tongue. Note the typical unilateral distribution.
### _Clinical symptoms_
* Initial symptoms may be non-specific and can mimic toothache. Patients may also complain of 'electric shock-like' pain.
* As the condition establishes, erythema and clusters of ulcerating vesicles appear. Invariably there is pain, often severe.
* Constitutional upset.
### _Aetiology_
* Reactivation of varicella zoster virus that has remained dormant within dorsal root or cranial nerve ganglia.
* This occurs in older individuals following the waning of immunity to varicella zoster, the infection being contracted during childhood as chicken pox.
* In younger individuals, shingles may be a marker of an immunocompromised host.
### _Involvement of non-gingival sites_
* The classical distribution of the lesions in shingles is that of a belt-like eruption on the trunk.
* In the head and neck region, zoster infections predominantly involve the distribution of the fifth and seventh cranial nerves. The trigeminal nerve is involved in 15% of cases, the ophthalmic division being most frequently affected.
* Involvement of the maxillary and mandibular divisions of the trigeminal nerve will produce lesions on the facial skin and oral mucosa (Fig 7-6).
* If the ophthalmic division of the trigeminal nerve is involved, corneal ulceration with subsequent scarring may occur.
* Involvement of the geniculate ganglion of the seventh cranial nerve may result in Ramsay Hunt syndrome (lower motor neurone facial palsy, vesicles in the external auditory meatus and sometimes palatal mucosal involvement).
**Fig 7-6** Vesiculation and crusting of facial lesions, distributed across all 3 divisions of the trigeminal nerve.
### _Differential diagnosis_
* Herpes simplex infection.
* Myeloproliferative disease.
* Myelosuppressive disease.
### _Clinical investigation_
* The diagnosis is usually made on the basis of clinical features.
* Serological investigation, demonstrating a rise in antibody titre to the virus.
* Cytology.
### _Management options_
* Early treatment with high dose aciclovir, valaciclovir or famciclovir.
* An ophthalmic opinion should be sought if the eye is involved.
* Post herpetic neuralgia may complicate the condition and requires appropriate pain management.
### _Cytomegalovirus_
Localised ulceration due to generalised cytomegalovirus (CMV) infection in HIV/AIDS patients is occasionally seen. Such infection is usually a marker of severe immunosuppression. (Fig 7-7).
**Fig 7-7** Ulceration due to cytomegalovirus infection in a patient with HIV disease.
CMV associated ulceration of the gingivae appear as non-specific ulceration, with no distinguishing clinical diagnostic features. A history of HIV/AIDS should arouse suspicion, and the diagnosis is confirmed by viral isolation or in situ hybridisation.
### _Deep Mycoses_
The deep mycoses are an uncommon group of infections that rarely involve the oral mucosa or gingivae. These infections may occur particularly in immunocompromised individuals such as HIV/AIDS patients.
_Histoplasmosis_ is usually due to opportunistic infection with Histoplasma capsulatum. This fungal organism is endemic in parts of America, India and Australia. Oral lesions occur usually when the infection is widely disseminated within the host. These lesions not infrequently involve the gingivae and appear as a persistent painful ulcer that may mimic malignant ulceration. Occasionally oral involvement may be the initial presenting sign of the infection.
Diagnosis can be difficult due to the variability of clinical presentation and the lack of specific features. Histological and immunocytochemical investigations are helpful.
_Paracoccidiomycosis_ is a fungal organism found in Brazil and other parts of South America and may produce painful gingival ulceration. The ulceration is often rather granular and mimics pyostomatitis vegetans in appearance.
Other opportunistic deep mycoses that can produce gingival ulceration include Cryptococcus, Coccidiomycosis and Mucormycosis.
A detailed history including ascertaining whether the patient has travelled to those areas where these mycoses are endemic is essential to enable accurate diagnosis of these rare conditions.
Samaranayake's Essential Microbiology for Dentistry (2002) is a helpful source of reference on the above infective agents.
### _Recurrent Aphthous Stomatitis_
Recurrent aphthous stomatitis is a common condition affecting approximately 20% of the UK population. The condition typically presents in childhood, adolescence and in young adults (Porter, 1998). Ulceration usually occurs on the oral mucosa rather than the gingivae, but gingival involvement occurs in some cases.
### _Clinical appearance_
Aphthous ulcers are round or ovoid in shape, typically have an erythematous periphery and a homogenous white/grey/yellow base.
Three types have been described based on their clinical features (Table 7-2):
* Minor Aphthous Ulceration (80% cases) – (Fig 7-8).
* Major Aphthous Ulceration (15% case) – (Fig 7-9).
* Herpetiform Aphthous Ulceration (5% cases) – (Fig 7-10).
Table 7-2 **Clinical features of recurrent aphthous stomatitis** | **Size** | **Number** | **Site** | **Duration** | **Recurrence Rate**
---|---|---|---|---|---
Minor aphthae | 5–10mm | <10 | Non-keratinised mucosa and dorsum of tongue | 7–14 days | Recurrence rates vary. One episode monthly is common.
Major aphthae | >10mm | 1–3 | No site restriction, but often oropharngeal | Persistent up to 3 months |
Herpetiform aphthae | l–3mm | 10–100 | No site restriction | 10–14 days |
**Fig 7-8** Minor aphthous ulceration affecting the alveolar mucosa and gingivae.
**Fig 7-9** Major aphthous ulceration of the tongue.
**Fig 7-10** Herpetiform ulcers that have started to coalesce.
### _Clinical symptoms_
* Aphthae are painful, major aphthae especially so.
* Herpetiform aphthae may be associated with a mild degree of constitutional upset which can be a source of diagnostic confusion.
### _Aetiology_
Multifactorial and in most patients a cause is not usually identified. The following possibilities should be considered:
* Haematinic deficiency.
* Adult coeliac disease.
* Inflammatory bowel disease (Crohn's disease, ulcerative colitis).
* Lactose intolerance (particularly in Afro-Caribbean patients).
* Other dietary allergies or intolerances.
* Behcet's disease.
* Systemic Lupus erythematosus.
* Reiter's syndrome.
* Bone marrow dysfunction.
* Hormonal – relationship to the pre-luteal phase of menstruation.
### _Involvement of non-gingival sites_
* Aphthous ulcers may occur on any oral mucosal surface.
* Patients with Behcet's disease may also have aphthous type ulceration on their genitalia.
### _Differential diagnosis_
* Traumatic ulceration.
* Underlying systemic disease.
* Behcet's disease.
### _Clinical investigation_
* Full blood count.
* Serum ferritin, folate and B12.
* Biochemical profile, acute phase markers and appropriate immunological investigations, dependent on the clinical features and other laboratory findings (e.g. anti-intrinsic factor and anti-gastric parietal cell antibodies in B12 deficiency).
### _Management options_
* Identification and treatment of underlying/undiagnosed disease.
* Chlorhexidine 0.2% mouthwash.
* Topical anti-inflammatories (Benzydamine hydrochloride).
* Topical corticosteroids (Prednesol or Betnesol mouthrinses).
* Covering agents (Orabase).
* Topical anaesthetic agents (Lidocaine gel).
* In severe cases, systemic corticosteroids ± azathioprine with appropriate monitoring.
* Other agents such as colchicine, thalidomide and immunomodulating drugs may be used in specialist units.
### _Neoplastic Ulceration_
Neoplastic ulceration is an uncommon cause of oral ulceration in general. This is particularly the case with the gingivae. There are various case reports of benign (Tosios, 1993) and metastatic tumours affecting the gingivae, but such occurrences are, however, rare.
Oral malignancy accounts for 1–2% of total human malignancies within the UK. Only 5% of oral malignancies occur on the gingivae. Oral squamous cell carcinoma may arise de novo or from oral premalignant lesions or conditions (Neville, 2002). Rarely malignancy may arise within desquamative gingivitis. (Fig 7-11, 7-12).
**Fig 7-11** Labial gingivae of a patient who developed proliferative veruccous leukoplakia within an area of desquamative gingivitis.
**Fig 7-12** Palatal gingival of the patient in Fig 7-11 showing the proliferative veruccous leukoplakia. Squamous cell carcinoma was subsequently identified within the excision specimen.
Neoplastic pathology presenting as ulceration usually signals malignant rather than benign disease. Most of the malignancies that arise within the gingival tissues are squamous cell carcinomas. Additionally, lymphomas may either present as gingival swelling or ulceration. A brief account of lymphoma can be found in Chapter 8.
### _Oral Squamous Cell Carcinoma_
Oral squamous cell carcinoma is not commonly found on the gingivae, although it is reported to occur more commonly at this site in Japanese patients. (Laskaris, Scully, 2003).
### _Clinical appearance_
* A persistent, indurated ulcer with a granular base and rolled margin should arouse suspicion. A granular appearance should always be regarded as a sinister clinical finding and demands urgent biopsy. (Fig 7-13).
**Fig 7-13** Squamous cell carcinoma of the palatal gingiva.
### _Clinical symptoms_
* Squamous cell carcinoma may be asymptomatic, but if ulcerated will usually cause discomfort. Secondary infection will exacerbate any discomfort.
* Some cases may bleed on minor trauma or even spontaneously.
### _Aetiology_
Whilst the aetiology of oral squamous cell carcinoma is not fully understood, tobacco and alcohol are regarded as the major risk factors.
* Tobacco habits.
* Alcohol.
* Betel nut usage.
* Transformation of premalignant lesions or conditions.
* Poor diet (low in antioxidant content).
* Oncogenic viruses (e.g. Human papilloma virus type 16).
* Ultraviolet radiation (malignant transformation of solar keratoses).
### _Involvement of non-gingival sites_
* Oral squamous cell carcinoma can occur at any site, but in the West most tumours present on the tongue and floor of mouth.
* There may be synchronous or metachronous tumours arising elsewhere on the oral mucosa. Approximately 20–30% of patients with oral squamous cell carcinoma will have, or will develop, multiple primary tumours. (Concept of field cancerisation).
* With time there will be local invasion of adjacent tissues as well as development of distant metastases. This will result in additional signs and symptoms including for example, lymphadenopathy, tissue fixity and altered sensation as a result of neural involvement.
### _Differential diagnosis_
* Traumatic ulcer.
* Leukoplakia.
* Erythroplakia.
* Chronic sepsis.
* Lymphoma.
* Histiocytosis X.
* Granulomatous disease.
* Wegener's granulomatosis.
* Pyostomatitis vegetans.
### _Clinical investigation_
* Biopsy is mandatory to establish diagnosis and the exact nature of the tumour, which may influence the definitive therapy.
* Imaging of the primary site, regional lymph nodes (CT, MRI) and chest x-ray.
### _Management options_
Early diagnosis and definitive treatment are most important to maximise the likelihood of a good prognosis.
The major treatment modality is surgical. Radiotherapy may be used either instead of, or additional to, surgical excision, dependent on the clinical scenario.
The therapeutic management of patients with oral squamous cell carcinoma involves a multidisciplinary approach, which includes surgeons, medical oncologists, restorative dentists and appropriate support services.
### _Metastatic Disease_
Metastatic disease of the oral cavity, particularly the gingivae, is uncommon. When it does occur, it usually arises from tumours of the breast (Epstein, 1987), bronchus (Watanabe, 2001) or kidneys. A variety of case reports exist in the literature citing other metastatic tumours presenting on the gingivae, including lymphomas, gastric carcinoma (Shimoyama, 2004), prostatic adenocarcinoma and transitional cell carcinomas of the bladder (Irle, 2001). Usually, gingival involvement of such tumours signals widely disseminated disease.
### Further Reading
Alam F, Argiridou AS, Hodgson TA. Primary syphilis remains a cause of oral ulceration. British Dental Journal 2000;189:7,352–354.
Porter SR, Scully CM, Pedersen A. Recurrent aphthous stomatitis. Critical Reviews in Oral Biology and Medicine 1998;9:306–321.
Neville B, Day TA. Oral cancer and precancerous lesions. CA A Cancer Journal for Clinicians 2002;52:195–215.
Laskaris G, Scully C. Periodontal Manifestations of Local and Systemic Diseases. New York: Springer, 2003.
Samaranayake, LP. Essential Microbiology for Dentistry, 2nd edition. Edinburgh: Churchill Livingstone, 2002.
Tosios K, Laskaris G, Eveson J, Scully C. Benign cartilaginous tumour of the gingiva. a case report. International Journal of Oral and Maxillofacial Surgery 1993;22:4,231–233.
Epstein JB, Knowling MA, Le Riche JC. Multiple gingival metastases from angiosarcoma of the breast. Oral Surgery, Oral Medicine, Oral Pathology 1987;64:5,554–557.
Watanabe M, Yasuda K, Tomita K, et al. Lung cancer metastasis to the gingiva. Nihon KokyukiGakkai Zasshi 2001;39:1,50–54.
Shimoyama S, Seto Y, Aoki F, et al. Gastric cancer with metastasis to the gingiva. Journal of Gastroenterology and Hepatology 2004;19:7,831–835.
Irle C. Metastatic transitional cell carcinoma of the urinary bladder presenting as a mandibular gingival swelling. Journal of Periodontology 2001;72:5,688–690.
## Chapter 8
## Generalised Gingival Ulceration
### Aim
The aim of this chapter is to detail those clinical entities that may present as widespread or multiple areas of gingival ulceration.
### Outcome
Having read this chapter, the reader should be able to formulate a differential diagnosis in cases of generalised gingival ulceration. They should be aware of the extra-oral manifestations of associated conditions and request and interpret those investigations that will assist definitive diagnosis so facilitating effective clinical management. Table 8-1 summarises the conditions discussed in this chapter.
Table 8-1 **Summary – generalised gingival ulceration** **Major Categories** | **Sub Categories** | **Frequency of Condition** | **Management Setting**
---|---|---|---
Mucocutaneous disease | Mucous membrane pemphigoid | Uncommon/rare | Referral to specialist units for treatment, monitoring and management of possible extra-oral manifestations.
| Pemphigus | Rare | Referral to specialist units – this is a serious condition that may be life-threatening in some cases.
| Lichen planus | A common condition but very rarely
produces generalised gingival ulceration. | Severe recalcitrant cases should be referred for specialist assessment.
Haematological | Leukaemias | Uncommon | The condition itself is clearly managed by haematologists. Oral hygiene measures can be undertaken in primary care. Management of the oral mucosal manifestations of chemotherapy/ radiotherapy should be managed in specialist units.
Lymphomas | Uncommon | As for the leukaemias |
Other haematological disease
(e.g. myelosuppressive states) | Rare | As for the leukaemias |
Infections | Primary herpetic gingivostomatitis – See Chapter 4 | Uncommon
More usually associated with erythema than generalised gingival ulceration. | Supportive measures and antiviral agents such as aciclovir which are only of real value if therapy is instituted at onset or very early in the course of the disease
### Vesicles and Bullae
A chapter specifically devoted to vesicular lesions cannot be justified, and therefore some pointers towards their investigation are provided briefly here. The most likely causes of gingival or mucosal vesicles are:
1. Viral infection
2. Trauma, which may be thermal, chemical or mechanical
3. Vesicullo-bullous disease such as mucous membrane (cicatricial) pemphigoid.
In the case of viral lesions, explore the history for a prodrome and check for lymphadenopathy (Fig 1-6) and pyrexia. If trauma is a cause this should be evident from the history, and fluid collected from the vesicle will be inflammatory in nature, with an absence of systemic signs of infection. Vesiculo-bullous diseases are dealt with in detail in Chapters 7 and 8.
### Mucocutaneous Disease
### _Pemphigoid_
Pemphigoid is an autoimmune vesiculobullous disease that manifests with subepithelial bulla formation as a result of the production of antibodies directed to the basement membrane zone of the epithelium. The disease is usually seen in patients over 50 years of age and is twice as common in females (Chan, 2002). There are two major variants of the condition:
* Mucous Membrane Pemphigoid – predominantly mucosal involvement.
* Bullous pemphigoid – predominantly cutaneous involvement.
### _Clinical appearance_
* The manifestations of mucous membrane pemphigoid often commence in the mouth and on occasions may be confined to the oral mucosa. (Fig 8-1).
* Although oral lesions rarely precede skin lesions in bullous pemphigoid, oral mucosal disease may be a feature of this variant.
* The main periodontal manifestation of mucous membrane pemphigoid is desquamative gingivitis (Fig 8-2), which may be the only oral feature of the condition (see Chapter 4). Shallow erosions may occur in desquamative gingivitis associated with vesiculobullous disease (Fig 8-3) (Richards, 2005).
* Serum anti-basement membrane antibodies can be identified in 80% of cases of bullous and 60% of cases of mucous membrane pemphigoid.
* Desquamative gingivitis may also occur in pemphigus/lichen planus/DLE.
* Extra-oral manifestations of mucous membrane pemphigoid are problematical due to scarring and subsequent scar contraction. Common sites of involvement include the eyes, genitals and oesophagus.
* Rarely there may be an association with internal malignancy.
**Fig 8-1** Small blood-filled blister that could be mistaken for a traumatic blister.
**Fig 8-2** Mucous membrane pemphigoid presenting as desquamative gingivitis.
**Fig 8-3** Desquamative gingivitis with erosions.
### _Clinical symptoms_
* Soreness is the principal symptom often accompanied by bleeding on brushing.
* The discomfort may be such that the patient cannot tolerate flavouring oils used in many dentifrices.
* Patients may also be concerned at the appearance of the gingivae, which may be fiery red.
* Patients may complain of intra-oral blistering, but the blisters often go unnoticed. It is only when they burst to produce painful ulceration that the problem is drawn to the patient's attention. The blisters may persist for several days, but often rupture within a few hours.
* Blistering and subsequent ulceration may be provoked by the ingestion of abrasive foods.
### _Aetiology_
* Pemphigoid is an autoimmune disease associated with the production of antibodies to the basement membrane zone of the epithelium.
### _Involvement of non-gingival sites_
* Mucous membrane pemphigoid, as the name suggests, can affect any mucosal surface. Intra-orally the distal hard palate and soft palate are commonly affected as the bolus of food is pushed up into this location prior to deglutition. Clinically the appearance is one of shallow irregular ulceration, with an erythematous and sometimes haemorrhagic border (Fig 8-4). There may be keratosis peripheral to the ulceration representing scar tissue that frequently accompanies healing.
* Extra-oral involvement may involve the oesophagus, which can become stenosed producing dysphagia.
* Genital lesions – these may be very disabling.
* Ocular lesions – these are often initially insidious and may affect up to 60% of patients. They are potentially serious, compromising sight. The eyelids may become sore as a result of blistering and subsequent ulceration. Subsequently the eyelids turn inwards due to scar contraction (entropion). Conjunctival scarring also occurs producing a symblepharon (scarring and fusion of the tarsal and bulbar conjunctiva) (Fig 8-5).
* Nasal lesions may occur, manifesting as bleeding and crusting of the nasal mucosa.
* Cutaneous involvement may occur in mucous membrane pemphigoid.
**Fig 8-4** Typical palatal erosions in mucous membrane pemphigoid.
**Fig 8-5** Ocular involvement showing entropion (E) and symblepharon (S).
### _Differential diagnosis_
Clinically, desquamative gingivitis that is ulcerated is suggestive of a vesicu-lobullous aetiology rather than lichen planus, which is the most common cause of desquamative gingivitis. The distribution of the lesions both within the oral cavity and extra-orally is helpful in arriving at a diagnosis. Differential diagnoses include:
* Pemphigus.
* Linear IgA disease.
* Erythema multiforme.
* Traumatic blistering.
* Epidermolysis bullosa acquisita
* Dermatitis herpetiformis.
### _Clinical investigation_
* The gold standard for diagnosis is direct immunofluorescence of an unfixed mucosal biopsy, which should be taken from a new or nearly new lesion. It is most important to include a margin of perilesional tissue. Linear deposits of IgG and C3 are characteristically identified at the basement membrane level in mucous membrane pemphigoid.
* Routine histopathological examination should also be undertaken.
* Circulating anti-basement membrane zone antibodies can be identified by indirect immunofluorescence, but this is a much less sensitive test, being positive in approximately 60% of patients, dependent on the method employed.
* Clinically a positive Nikolsky sign may be elicited (bulla formation or propagation on trauma).
### _Management options_
* An ophthalmic opinion in all cases of mucous membrane pemphigoid is mandatory – untreated eye disease has potentially serious consequences.
* Topical corticosteroids if disease is limited to the mouth.
* More severe or widespread disease may necessitate the use of systemic corticosteroid therapy possibly associated with a steroid-sparing drug such as azathioprine. Dapsone, mycophenolate mofetil or tacrolimus may also have a therapeutic role. Such drugs should only be used in specialist units.
* Referral to other specialties as demanded by the sites of involvement.
### _Pemphigus_
Pemphigus is an intraepithelial autoimmune bullous disorder that is less common but potentially more serious than pemphigoid. It presents in various forms:
* P. vulgaris – usually the variant that involves the oral mucosa and is detailed below.
* P. vegetans.
* P. erythematosus.
* P. foliaceous – this variant does not affect the oral mucosa.
### _Clinical appearance_
* Mucosal involvement is common and often precedes cutaneous involvement. Oral mucosal involvement may be the initial presenting sign, particularly desquamative gingivitis, perhaps several months ahead of other sites (Fig 8-6, 8-7).
* Blisters are often flaccid rather than the tense blisters seen in pemphigoid and are fragile as they are thin-walled (intra-epithelial) (Fig 8-8).
* On rupturing, the blisters may leave large shallow erosions with ragged edges, or discrete slit-like ulcers with jagged margins, often surrounded by white mucosa (Fig 8-9).
* Blisters occur at sites of trauma – palatal involvement therefore is frequent.
* Desquamative gingivitis is common and may be the only presenting sign. It may be associated with gingival erosions as in pemphigoid.
**Fig 8-6** Pemphigus vulgaris involving the gingivae, producing a desquamative gingivitis with areas of erosion and haemorrhage.
**Fig 8-7** Pemphigus vulgaris involving the palate
**Fig 8-8** Flaccid bulla in cutaneous pemphigus.
**Fig 8-9** Slit-like ulcers in oral mucosal pemphigus.
### _Clinical symptoms_
* Patients will complain of ulceration and associated pain, inflamed gingivae and on occasions blistering at mucosal and cutaneous sites. In the early stages, however, patients may be unaware of blister formation.
* The symptoms are very similar to those described for pemphigoid.
### _Aetiology_
* An autoimmune disease associated with the production of antibodies to intercellular cement of mucosal and skin surfaces. In pemphigus vulgaris these are usually antibodies to desmoglein 3, a transmembrane desmosomal glycoprotein.
### _Involvement of non-gingival sites_
* Many cases presenting on the oral mucosa remain localised to the mouth, however with time some cases involve the skin and other mucosal sites such as the genitalia.
* Some patients will develop extra-oral lesions rapidly following the appearance of the oral lesions. It is important to inform patients that they must seek immediate advice should extra-oral involvement develop. Patients are at risk from fluid and electrolyte disturbances due to the loss of fluid from the ruptured blisters, together with consequent infection.
### _Differential diagnosis_
* Pemphigoid – the skin lesions in bullous pemphigoid tend to involve the upper limbs whereas those of pemphigus occur more usually on the trunk.
* The differential diagnosis is otherwise as that detailed for pemphigoid.
### _Clinical investigation_
* A positive Nikolsky sign as a result of widespread acantholysis (detachment of keratinocytes) can be demonstrated clinically by gentle trauma to an area of clinically uninvolved mucosa or skin, which results in bulla formation.
* IgG and C3 are demonstrable intercellularly by direct immunofluorescence.
* Serum anti-intercellular cement antibody levels correlate with disease activity and can be identified by indirect immunofluorescence.
* If there is clinical suspicion of occult malignant disease (usually lymphoma, leukaemia, thymoma or gastrointestinal malignancy) appropriate investigations should be undertaken. (paraneoplastic pemphigus, Sklavounou, Laskaris, 1998).
### _Management options_
* If the lesions are confined to the oral mucosa, then local management with topical steroids is the preferred choice.
* If the lesions are unresponsive to local measures or there is extra-oral involvement, then systemic corticosteroid therapy is indicated. This may be combined with azathioprine as a steroid sparing agent, and is likely to have to be given long term.
* Multidisciplinary management is the norm.
* Pemphigus can be a fatal condition. Prompt and effective management is of paramount importance.
### _Lichen Planus_
Although lichen planus frequently manifests as erosions on the oral mucosa, erosive lesions on the gingivae are uncommon and ulceration at this site is more suggestive of vesiculobullous pathology. A full account of lichen planus can be found in Chapter 4.
### Haematological Disease
### _The Leukaemias_
The leukaemias represent a range of serious haematological disorders that arise as a result of abnormal maturation or proliferation of the various white blood cell lines. There are both acute and chronic forms and any of the white cell types may be involved, but most frequently it is the lymphoid or myeloid lines.
The acute leukaemias are more likely to have oral manifestations than the chronic variants, and on occasions, it is these manifestations that may be the first presenting sign of the disease.
### _Clinical features_
* Gingival hyperplasia and swelling due to leukaemic cell infiltration (Fig 8-10). This may be seen in 20–30% of patients with acute myeloid leukaemia, although rather less frequently in cases of acute lymphoblastic leukaemia (2%), a condition that is most prevalent in childhood.
* The gingivae appear erythematous and may bleed easily on mild trauma or spontaneously as a result of thrombocytopaenia.
* Specific infections such as necrotising ulcerative gingivitis (NUG), herpes simplex or candidosis may occur.
* Rarely there may be rapid loss of periodontal support as a consequence of malignant cell infiltration of the periodontal ligament.
* Chronic leukaemias are much less likely to present with periodontal manifestations.
**Fig 8-10** Hyperplastic gingivae in acute myeloid leukaemia.
### _Aetiology_
* Malignant clonal proliferation of white blood cells.
* A variety of cytogenetic abnormalities are identifiable in the leukaemic cells (e.g. the Philadelphia chromosome).
* The malignant cells displace the non-malignant cell progenitors, resulting in anaemias and platelet depletion.
* Infection may result as a consequence of non-functional or depleted immunocompetent cells.
### _Involvement of non-gingival sites_
* The patient is often acutely ill and may have features of severe anaemia including breathlessness.
* Oral mucosal purpura.
* Oral ulceration.
* Mucosal pallor due to anaemia.
* Candidosis.
* Lymphadenopathy.
* Cutaneous involvement.
* Bruising.
* Hepatosplenomegaly.
### _Differential diagnosis_
* Other myeloproliferative or myelosuppressive disease.
* Immunodeficiency states.
* Histiocytosis X.
* Plasma cell gingivitis.
* Inflammatory gingival enlargements.
* Pyostomatitis vegetans.
### _Clinical investigation_
* Incisional biopsy of involved tissue.
* Appropriate haematological investigations including full and differential white count, blood film and bone marrow aspirate and biopsy.
* Immunophenotyping.
### _Management options_
* The leukaemia will be managed by the haematologist, usually with combination chemotherapy.
* Local measures include scrupulous oral hygiene and adjunctive topical antiseptic agents.
* Painful oral ulceration may be managed by topical anti-inflammatories, anaesthetics and covering agents.
* Management of the local side-effects of chemotherapy and radiotherapy.
### _The Lymphomas_
Most lymphomas are B cell tumours and rarely affect the gingivae. Hodgkins lymphomas are less prevalent in the orofacial region than non-Hodgkins lymphomas, the latter presenting in the oropharyngeal region in approximately 10% of cases. The lymphomas seen in the oral cavity may either represent metastatic deposits or primary tumours that may be maltomas (mucosa-associated lymphoid tissue lymphomas). The classification of lymphomas is complex and beyond the scope of this book.
### _Clinical features_
* Non-Hodgkins lymphomas may present as a painless swelling on the gingivae, usually soft in texture (Fig 8-11). The clinical appearance often does not arouse suspicion of sinister pathology, particularly in the early stages.
* They may mimic dental or periodontal sepsis.
* The mucosal surface of the lymphoma deposit varies in colour from clinically normal to erythematous and it may occasionally ulcerate and become painful.
* Tooth mobility may occur.
**Fig 8-11** Non-Hodgkins lymphoma.
### _Aetiology_
* Unclear, but some are associated with viral infections.
* An increased frequency of lymphoma occurs in association with certain autoimmune conditions including Sjogren's syndrome (MALTomas).
* Certain immunodeficiency syndromes predispose to lymphomas (ataxia telangiectasia; Wiskott Aldrich syndrome).
* Non-malignant disease involving the lymphoreticular system also appears to be associated with the development of lymphomas in the long term.
### _Involvement of non-gingival sites_
* Waldeyer's ring may be involved intra-orally, particularly when there is gastrointestinal tract involvement.
* Rarely, oral mucosal ulceration may occur.
* Lymphadenopathy.
### _Differential diagnosis_
* Dental or periodontal sepsis.
* Inflammatory swellings.
* Fibroepithelial hyperplasia.
* Non lymphoreticular tumours.
* Other metastatic deposits.
### _Clinical investigation_
* Biopsy is mandatory.
* Immunocytochemical studies will be required to identify the appropriate cell markers that facilitate classification of the lymphoma.
### _Management options_
* Referral to haematology for further investigation and staging.
* Chemotherapy and radiotherapy.
### _Other Haematological Conditions_
Various other diseases occurring as a result of abnormal bone marrow function, can also produce ulcerative conditions that affect the gingivae and the oral mucosa. These include, agranulocytosis, neutropaenic states (Fig. 8-12) and myelodysplasia. (Chapple et al 1999).
**Fig 8-12** Gingival ulceration adjacent to the lower right first molar tooth in a neutropaenic patient.
### Further Reading
Chan LS, Ahmad AR, Anhalt GJ et al. The first international consensus on mucous membrane pemphigoid: Definition, diagnostic criteria, pathogenic factors, medical treatment and prognostic indicators. Archives of Dermatology 2002;138:3,370–379.
Richards A. Desquamative gingivitis and its management. Perio in Practice Today 2005;3: 183–190.
Sklavounou A, Laskaris G. Paraneoplastic pemphigus: a review. Oral Oncology 1998; 34:6,437–440.
Chapple ILC, Saxby MS, Murray J. Gingival haemorrhage, myelodysplastic syndromes and acute myeloid leukaemia. Journal of Periodontology 1999:70,1247–1253.
## Chapter 9
## Localised Gingival Recession
### Aim
This chapter aims to outline the principal causes of localised gingival recession, including those associated with underlying systemic disease.
### Outcome
At the end of this chapter the reader will be aware of the limited range of conditions that give rise to localised gingival recession and be able to recognise when referral may be prudent. Table 9-1 lists the causes of localised gingival recession. Some of these conditions are common and are best managed in the primary care sector, whilst others are less common and patients should be referred for further investigation and specialist therapy.
Table 9-1 **Localised gingival recession** **Condition** | **Sub Category Nature** | **Incidence** | **Manage/Refer**
---|---|---|---
Developmental defects | Dehiscence | common | manage/refer
Fenestration | common | manage/refer
Anatomical tooth position | common | manage/refer
Traumatic defects | Class II division 2 incisor relationship and gingival stripping | common | refer
Conscious self-induced mutilation (gingivitis artefacta) | rare | refer
Subconscious self-induced mutilation | rare | manage
Inflammatory defects | Localised aggressive
periodontitis | common | manage/refer
Peridontal-endodontic lesions | common | manage
Localised chronic
periodontitis | common | manage
Defects associated with underlying systematic disease | Linear Morphoea | uncommon | refer
Histiocytosis X | uncommon | refer
Necrotising
periodontitis (HIV) | uncommon | refer
Necrotising
stomatitis (HIV) | uncommon | refer
Drug-induced lesions | Cocaine | uncommon | refer
Aspirin | uncommon | manage
### Classification of Localised Recession Defects
There have been several proposed classification systems to enable clinical documentation of local recession defects. Whilst not the most appropriate for diagnostic purposes, the most frequently used system is currently that proposed by Miller (Fig 9-1). This system was designed primarily to indicate the likelihood of successful management using periodontal plastic surgery procedures to correct recession defects. A practical and novel diagnostic recording system based upon clinical measurements is proposed below:
1. Measurement of the length of the recession in mm from the CEJ to the base of the defect (L1, L2 for 1mm, 2mm etc – Fig 9-2a)
2. Measurement of the width of the defect in mm at its widest aspect mesiodistally (W1, W2 for 1mm, 2mm etc – Fig 9-2b)
3. Specification of the number of teeth involved (T1, T2 etc – Fig 9-2c)
4. Identification of whether the extent of the defect is superior (S) or inferior (I) to the mucogingival junction (MGJ – Fig 9-2d).
**Fig 9-1** Miller's classification of recession defects. I – Marginal tissue recession not extending to MGJ. No loss of interdental bone or soft tissue. II – Marginal tissue recession extends to or beyond MGJ. No loss of interdental bone or soft tissue. III – Marginal tissue recession extends to or beyond MGJ. Loss of interdental bone or soft tissue is apical to the CEJ, but coronal to the apical extent of the marginal tissue recession. IV – As for III but loss of interdental bone or soft tissue is apical to the CEJ and extends to a level beyond the apical-most extent of marginal tissue recession.
**Fig 9-2** Proposed diagnostic coding/notation system for recording recession defects.
Therefore, in Fig 9-3 the defect at LR1 would be annotated as L5/W2/T1/I and at LL1 as L3/W2/T1/S.
**Fig 9-3** Recession defect (Stillman's cleft) affecting LR1 (L4/W2/T1/I) and LL1 (L3/W2/T1/S). Had the interdental papilla been missing between LR1 and LL1, the defect class would have been L4/W6/T2/I .
### Developmental Conditions
### _Dehiscence and Fenestration_
Two classical developmental anomalies can predispose to localised recession defects sometimes referred to as 'Stillman's clefts' (Fig 9-3). These are illustrated in Fig 9-4:
* Dehiscence – the absence of a 'window' of bone at the facial or oral surface of a tooth (normally buccal/labial plate), i.e. the alveolar margin remains intact.
* Fenestration – a 'V'-shaped defect involving the alveolar margin and extending apically.
**Fig 9-4** Schematic diagram illustrating a bony fenestration and dehiscence affecting LR1.
A significant amount of the blood supply to the gingivae arrives via the periosteum and an absence of periosteal blood supply renders the gingival marginal tissue less able to resist trauma and recurrent inflammation (plaque-induced), without loss of marginal epithelium and subsequent recession. The teeth most commonly affected are lower incisors and upper canines. Contrary to popular belief, there is no evidence for a direct effect from 'fraenal pull' as the fraenum rarely carries muscle fibres (Watts 2000). However, the fraenum may interfere with plaque control around lower incisors where the labial bone plate is thin, or there is an underlying anatomical bone defect. Prominent fraena may therefore contribute to the development of recession following repeated episodes of gingival inflammation, secondary to an inability to maintain plaque control in the area.
### _Anatomical Tooth Position_
On occasions the lower incisors develop in a position that is more labial than 'ideal' and/or are also proclined, further reducing the thickness of labial crestal bone, or indeed its coronal extent (Fig 9-5). The overlying gingivae appear extremely thin and delicate (Fig 9-6) and are prone to recession if marginal plaque control is less than ideal.
**Fig 9-5** a) Incisors positioned labially within the alveolus relative to the ideal, thereby leaving a thin labial bone plate. b) Incisors proclined in a compensatory manner in a patient with a mild class II skeletal base, thereby reducing the overbite and compromising the integrity of the labial crestal bone.
**Fig 9-6** Delicate tissue biotype prone to recession due to plaque-induced inflammation and toothbrush trauma.
### _Traumatic Defects_
Class II division 1 or 2 incisor relationship with gingival stripping.
### _Clinical appearance_
Gingival stripping may arise in a Class II division 1 or 2 incisor relationship. Akerly classified such incisor relationships from an orthodontic perspective (Box 9-1) but in periodontal terms, Akerly Class II and III relationships are the most significant and very difficult to manage (see Heasman, Preshaw and Robertson 2004). Localised recession defects affecting one or all of the incisor teeth may be evident:
* palatally to the upper incisors and caused by a traumatic overbite of the lower incisor edges against the palatal gingival margins (Fig 9-7). This may arise in a class II division 1 incisor relationship, where the overjet is increased and the overbite is complete, or in a class II division 2 incisor relationship with a complete overbite.
* labially to the lower incisors and caused by the incisal edges of the upper anterior teeth occluding against the labial gingival margin (Fig 9-8). This arises in a class II division 2-incisor relationship.
> Box 9-1 **The Akerly classification of traumatic incisal relationships. Taken from Heasman, Preshaw and Robertson, 2004.**
> _Class 1_ | Lower incisors impinge on the palatal mucosa, posterior to the palatal gingival margins of the maxillary anterior teeth.
> ---|---
> _Class II_ | Lower incisors occlude onto the palatal gingival margins of the maxillary anterior teeth.
> _Class III_ | A deep traumatic overbite (Class II div 2 incisor relationship) with shearing of the mandibular labial gingivae.
> _Class IV_ | Lower incisors occlude with the palatal surfaces of the upper incisors, leading to tooth wear by attrition.
**Fig 9-7** A Class II division 2 incisor relationship with trauma to the lower labial gingivae (see blanching of lower gingival margins) (Akerly Class III).
**Fig 9-8** A Class II division 2 incisor relationship with trauma to the palatal gingivae of the maxillary incisors (Akerley Class II).
### _Clinical symptoms_
* Pain on eating/incising.
* Soreness/ulceration of the gingival margin.
* Recession.
* Aesthetic concerns related to the anterior tooth position or more generally the malocclusion (especially if there is a large skeletal component).
### _Aetiology_
* Unstable incisor relationship due to skeletal relationship of mandible to maxilla or habitual (thumb sucking), resulting in retroclination of lower incisors and proclination of uppers.
* Unstable incisor relationship due to loss of periodontal support (periodontal disease) and incisor tooth movement or over-eruption (Fig 9-9).
**Fig 9-9** Radiograph demonstrating over-eruption of UL 1 post-periodontal bone loss and a crescentic pattern of bone loss indicative of occlusal trauma.
### _Involvement of non-gingival sites_
None.
### _Differential diagnosis_
None. The key diagnostic decision is whether there is historical or currently active periodontitis.
### _Clinical investigation_
* Study models to examine the incisor relationship (Fig 9-10).
* Radiographs.
* Orthodontic opinion.
* Incisor set up on models for orthodontic realignment, as a diagnostic procedure.
* Incisor set up (Kesling set up) for a restorative solution (e.g. 'Dahl' appliance – see Noble, Kellet and Chapple 2004).
**Fig 9-10** Schematic representation of a) a 'stable' incisor relationship, with the incisal edge of the lower teeth against the palatal cingulum rest of the upper incisors; b) an unstable Class II div 1 relationship with trauma to the palatal gingival margins; c) an unstable Class II div 2 relationship with trauma to the palatal gingival margins; d) an unstable Class II div 2 relationship with trauma to the labial mandibular gingival margins.
### _Management options_
* Establishing a stable incisor relationship by orthodontic means is the ideal solution, but frequently requires lengthy therapy and permanent retention, either fixed (Fig-9-11) or removable (Fig 9-12).
* A 'Dahl' appliance may be of value to intrude/retrocline the lower incisors and encourage over-eruption of posterior teeth, thereby separating the upper incisors from the lower labial gingivae.
* An overlay appliance to prevent further trauma (Fig 9-13).
* If the recession defect is localised, pre-orthodontic treatment by connective tissue grafting can improve aesthetics and hygiene prior to establishing a stable anterior tooth relationship (Fig 9-14).
**Fig 9-11** A twist wire and composite splint to permanently retain the lower incisors post-orthodontic therapy.
**Fig 9-12** An upper removable retainer designed for night time use to maintain a stable incisor relationship.
**Fig 9-13a** Class II incisor relationship with traumatic overbite causing palatal gingival stripping.
**Fig 9-13b** An overlay appliance can be made and fitted prior to reduction of the lower incisor edges, which are reduced when the appliance is fitted, the chrome thereby occupying the resultant space.
**Fig 9-14a** Localised recession defect, which was likely to worsen with orthodontic realignment of the lower incisors, which involved their proclination. Oral hygiene was excellent prior to surgery.
**Fig 9-14b** The same patient as in Fig 9-14a, post-connective tissue graft surgery and orthodontic treatment.
### _Conscious Self-Mutilation (see also Chapter 7)_
### _Clinical appearance_
* Localised recession defects, which appear plaque free.
* Marginal keratosis indicates chronic trauma (Fig 9-15).
* Notched recession defects (Fig 9-16).
* Absence of traumatic incisor relationship.
**Fig 9-15** Factitious injury also known as gingivitis artefacta, caused by fingernail picking of the gingival margin LL3. Note the marginal keratosis and absence of plaque, consistent with repeated trauma. The patient had previously exfoliated her LR3.
**Fig 9-16** Gingivitis artefacta in a child demonstrating how the trauma was caused.
### _Clinical symptoms_
* Remarkably, pain is not usually a complaint.
* Itching gums may indicate psychiatric morbidity.
* Aesthetic concerns completely out of proportion to the true size/nature of the problem may indicate an obsessive-compulsive disorder.
* Patient is often symptom-free and brought by a concerned relative/parent or referred by their dental surgeon.
### _Aetiology_
* Habitual (Fig 9-16). This may arise in patients with a learning disability, or as in the illustrated case, in children where the maxilla is the main focus.
* Attention-seeking/psychiatric morbitidy (Fig 9-17). Severe forms are known as Munchausen's syndrome, where patients deliberately self-mutilate to seek medical or surgical intervention. The patient in Fig 9-17 suffered from an obsessive-compulsive disorder and traumatised his gingival margin UR1 with scissors.
**Fig 9-17** A 17-year-old male clinically diagnosed with obsessive-compulsive disorder.
### _Involvement of non-gingival sites_
* Habitual self-mutilation in its mildest form can frequently involve the buccal mucosa or lips (from habitual tissue chewing or biting), or indeed any other part of the body.
* Attention-seeking may involve ulcerative lesions like those illustrated in Fig 9-18, but arising anywhere on the oral mucosa; the tongue is a common site. Psychiatric morbidity may again result in trauma to any part of the body. The 17-year-old male illustrated in Fig 9-17 also mutilated the enamel of his UR1 using a metal file.
**Fig 9-18** A 14-year-old girl with extensive annular and exophytic gingival lesions atypical of 'artefacta'. (The patient submitted to several blood investigations and repeat biopsies before a factitious cause was suspected. Munchausen's syndrome remained a likely diagnosis for this patient at the time of this book going to press).
### _Differential diagnosis_
* Toothbrush trauma.
* Stillman's cleft.
* Localised periodontitis ± occlusal trauma.
### _Clinical investigation_
* An important test in arriving at a differential diagnosis is to disclose the teeth to eliminate a plaque-induced cause. In self-mutilation, the constant agitation removes plaque and generally the tooth surface is plaque free.
### _Management options_
* Habitual self-mutilation – frequently drawing the habit to the patient's attention may be sufficient to break the habit or the fabrication of a soft occlusal guard for night use may help.
* Discussion with the patient and/or parents. The female in Fig 9-18 had very low self-esteem. She underwent two biopsies over a period of two years and several blood investigations to eliminate gastrointestinal pathology (e.g. Coeliac disease) before a diagnosis by exclusion was made. Discussion in the absence of her parents brought the true diagnosis to light (the patient then chose to tell her parents). Munchausen's syndrome was a differential diagnosis in this case.
* Attention-seeking – letting the patient know that you are aware of the cause and a frank and open discussion with them about their habit may be sufficient to stop the habit. It may be necessary to have a discussion without any parent or guardian present (but witnessed by a member of your staff), in case the parent/guardian is contributory to the problem and for reasons of confidentiality. Professional counselling may be needed, or medical/psychiatric intervention where Munchausen's syndrome is suspected.
* Psychiatric disease requires psychiatric referral, usually through the patient's medical practitioner.
### _Subconscious Self-Mutilation_
Factitious injury can arise from a habitual practice, of which the patient is unaware. It generally arises in situations where a patient is obsessed with cleaning a particular interproximal site and the patient unwittingly and unintentionally causes local trauma. The most common example would be a localised acute inflammatory swelling caused by inappropriate use of dental floss or tape. However, more significant ulcerative/recession lesions can also arise. The patient illustrated in Fig 9-19 created a necrotising ulcerative lesion LR67 using a silver-plated toothpick to perform interproximal plaque control. The marginal alveolar bone sequestrated and subsequent healing by secondary intention was very slow.
**Fig 9-19a** Necrotising ulcerative periodontitis caused by trauma from a silver tooth pick. The interproximal and marginal epithelium was denuded, along with the local periosteum, which led subsequently to bone sequestration.
**Fig 9-19b** The silver tooth pick used to inadvertently create the recession/ulcerative defect LR 67.
### _Inflammatory/Infective Conditions_
The following lesions are beyond the scope of this text and are discussed in more detail in other books within this Periodontology series.
* Inflammatory defects.
* Localised aggressive periodontitis (see Clerehugh, Tugnait and Chapple, 2004, and Heasman, Preshaw and Robertson, 2004).
* Periodontal-endodontic lesions (see Noble, Kellett and Chapple 2004).
* Localised chronic periodontitis (see Heasman, Preshaw and Robertson 2004).
### Defects Associated with Underlying Systemic Disease
### _Linear Morphoea (Localised Scleroderma)_
### _Clinical appearance_
* Localised midline gingival retraction (Fig 9-20).
* Gingival scarring (white appearance to gingivae).
**Fig 9-20** Localised recession and scarring affecting UR12, due to linear morphoea/localised scleroderma in a 14-year-old-female.
### _Clinical symptoms_
* Receding gums.
* Root sensitivity.
* Aesthetic concerns.
* Concerns over tooth loss.
### _Aetiology_
Localised scleroderma is a rare condition also referred to as 'Morphoea' and is predominantly a cutaneous disease. It is believed to have a genetic basis or, alternately, to arise following trauma. The case illustrated in Figs 9-20 to 9-22 was a 14-year-old girl referred for connective tissue grafting to the UR12 area, following the development of localised recession. Careful examination revealed a midline forehead scar hidden beneath the girl's fringe. Given the midline position and cutaneous nature of the scarring, surgery was ruled out as the chance of a graft or rotational flap re-vascularising was minimal and aggressive treatment deemed more likely to make the situation worse. Laser surgery to the lip prior to the diagnosis did indeed make the lip retraction more severe (Baxter et al, 2001).
### _Involvement of non-gingival sites_
* 'Coupe de sabre' (linear cut of the sword) – scarring which may affect the forehead (Fig 9-21).
* Midline lip notching/lip retraction (Fig 9-22).
* Scarring elsewhere, but limited to the skin.
**Fig 9-21** Midline scaring of forehead, a 'coupe de sabre' in the same patient as illustrated in Fig 9-20, which was hidden by a fringe.
**Fig 9-22** Midline lip notching/retraction in the patient from Fig 9-20.
### _Differential diagnosis_
* Bony dehiscence.
* Tissue loss following trauma.
* Localised periodontitis.
* Self-mutilation (gingivitis artefacta).
### _Clinical investigation_
* Careful history to eliminate trauma.
* Family history.
* Medical history.
* Scarring elsewhere on body.
* Clinical examination to eliminate true pocketing/periodontitis.
* Extra-oral examination for evidence of related scarring.
### _Management options_
The periodontal condition can be managed conservatively with non-surgical methods. In the case described, it was so-managed for four years, but when the patient reached 16 years of age and aesthetics became an important factor in her life, she opted for extraction and prosthetic replacement of UR12. A lip split and contour revision using autogenous fat injections was also performed and future fixed prostheses are planned (+ implant retainers).
### Histiocytosis-X
Three forms of histiocytosis-X are described:
1. Unifocal eosinophilic granuloma (solitary).
2. Multifocal eosinophilic granuloma (Hand-Schuller-Christian disease).
3. Progressive/disseminated histiocytosis (Letterer-Siwe disease).
### _Eosinophilic Granuloma_
### _Clinical appearance_
Mandible more commonly than maxilla and often posterior sites. There may be buccal swelling, which may appear exophytic, deep pocketing and gingival bleeding or it may present as a localised gingival recession defect with severe bone destruction beneath.
### _Clinical symptoms_
Localised tooth mobility and/or severe recession usually in young males less than 20 years of age. It may also present as a localised exophytic swelling (Chapter 5).
### _Aetiology_
Unknown.
### _Involvement of non-gingival sites_
Not with the unifocal lesion, but the condition may become multifocal, with the long bones, cranium and ribs involved.
### _Differential diagnosis_
* Localised aggressive (juvenile) periodontitis.
* Hyperparathyroidism (normally females).
* Intra-osseous haemangioma.
* Metastatic tumour (e.g. breast, prostate) – rare.
* Osteosarcoma – rare.
### _Clinical investigation_
Radiographs demonstrate focal osteolytic lesions (Fig 9-23) where local bone destruction may be well or poorly demarcated. Incisional biopsy of the gingiva and underlying connective tissues shows tumour-like collections of histiocytes (Langerhans cells/ tissue macrophages) and eosinophils.
**Fig 9-23** Multifocal eosinophilic granulomas in a 14-year-old boy who also had lesions in both femurs. Classical histiocytosis-X (Hand-Schuller-Christian disease).
### _Management options_
1. Unifocal eosinophilic granuloma. Medical management will involve bone scanning to eliminate disseminated lesions and either curettage/surgical debridement of the region or local radiotherapy. Fig 9-24 shows the lesions from Fig 9-23 post chemotherapy (and extraction of LL5), following which the prognosis is generally good.
2. Multifocal Eosinophilic Granuloma. Presents with ulcerative mucosal lesions and underlying osteolytic lesions of bone. Multi-focal (often temporal bones) and organ involvement. Lesions of the jaw bones present as for unifocal eosinophilic granuloma and in Hand-Schuller-Christian disease there is:
* Diabetes insipidus.
* Skull defects.
* Exophthalmos.
3. Progressive/Disseminated Histiocytosis (Letterer-Siwe Disease). Tooth mobility due to progressive osteolytic lesions, which may also cause pain. There may be fever, lymphadenopathy, hepato-splenomegaly and a pancytopenia. Usually presenting in infants, it is aggressive and often fatal.
**Fig 9-24** The same area as in Fig 9-23 post-chemotherapy and extraction of LL5.
### _Necrotising Ulcerative Periodontitis (NUP) – see also Chapter 7_
### _Clinical appearance_
* Severe recession due to loss of periodontal attachment (Fig 9-25a).
* Flattened gingival margin with loss/blunting and ulceration of the interdental papillae (Fig 9-25b).
* If active, grey sloughing is evident due to interdental tissue necrosis.
* A characteristic anaerobic foetor (halitosis) is evident.
* Significant and immediate bleeding from the gingival margin when probed or spontaneously.
* Regional cervical lymphadenopathy.
**Fig 9-25a** Necrotising periodontitis affecting the lower incisors of a patient with HIV disease. Key diagnostic features show loss of gingival contour due to previous papilla ulceration and subsequent papillary blunting. In addition there is ligament and bone loss, and other parts of the mouth are unaffected by the severe attachment loss.
**Fig 9-25b** More classical NUP, where ulceration is clearly evident, alongside attachment and bone loss UL21 area.
### _Clinical symptoms_
* Rapidly receding gums.
* Pain, due to ulceration/bone pain.
* Severe bleeding on brushing.
* Tooth mobility.
* Halitosis/oral malodour.
### _Aetiology_
Classically a fuso-spirochaetal infection with Fusobacterium nucleatum and Treponema vincentii (which invade the tissues) in patients who are immuno-suppressed. Involvement of Prevotella intermedia and Candida species is also reported. Most commonly reported in HIV-infected subjects (prevalence 6.3% – Glick et al 1994) who have a CD4+ count below 400 and a high viral load (>50,000 copies per ml blood). NUP has historically been regarded as an extension of NUG (see Chapple and Gilbert, 2002), involving the periodontal attachment apparatus and alveolar bone, rather than being limited to the gingivae. However, recently it appears to be presenting in young non-HIV-infected patients (mainly females), where the classical risk factors of smoking, poor oral hygiene and stress are present. Additionally a poor diet low in antioxidants and fibre is also evident as a clinical impression.
### _Involvement of non-gingival sites_
Rarely necrotising periodontitis may progress to necrotising stomatitis (Chapple and Hamburger 2000).
### _Differential diagnosis_
* Localised chronic periodontitis (where classical ulceration is absent) with severe recession.
* Trauma from self-mutilation (Fig 9-19).
* Recession due to underlying bone sequestration.
* Use of cocaine or other recreational drugs locally applied to gingivae.
### _Clinical investigation_
* Comprehensive history including:
* medical history.
* sexual history.
* history of habits (including trauma, drug use, smoking history etc).
* diet history.
* past dental history (previous episodes or previous periodontitis).
* family history.
* Clinical examination:
* extra-oral (submandibular or cervical lymphadenopathy).
* intra-oral.
* Presence of foetor oris.
* Microbiology (dark field/phase contrast) will demonstrate spirochaetes.
### _Management options_
* Counselling is essential for the following:
* smoking.
* diet.
* stress.
* recreational drug use.
* HIV counselling and testing if deemed necessary from the history.
* Oral hygiene instruction.
* Scaling and root surface debridement with adjunctive systemic metronidazole 200–400mg TDS seven days.
* Review.
* Referral to genito-urinary medicine if appropriate.
### _Necrotising Ulcerative Stomatitis (NUS)_
NUS is believed to be an extension of NUP involving the oral mucosa and underlying bone in an ulcerative lesion that extends more than 10mm beyond the gingival margin (Fig 9-26). Its behaviour is aggressive leading to necrosis of underlying bone and loss of tooth vitality. There have been reports of oro-nasal fistulae developing (Felix et al 1991).
**Fig 9-26** Necrotising stomatitis in a patient with AIDS affecting UR 67 area of the palate. The underlying bone was necrotic and teeth non-vital. The brown stain is due to use of chlorhexidine gluconate (0.2%).
Recommended treatment is broad surgical excision of the involved area of bone (usually maxilla) back to healthy bleeding bone margins, along with extraction of teeth that are involved in the necrosis and surgical packing of the defect to allow gradual healing by secondary intention, usually under a general anaesthetic. If the patient is a poor anaesthetic or surgical risk, then conservative management should be considered. The patient in Fig 9-26 had lost five stone in weight and was deemed a poor anaesthetic risk, due to the underlying medical condition. Management was therefore conservative, involving oral hygiene instruction, longer-term use of chlorhexidine mouthwash and regular scaling and prophylaxis by domiciliary visits if necessary. The patient survived for 18 months (Fig 9-27) and was pain free and able to eat normally, prior to a fatal infection with PCP (Pneumocystis carinii pneumonia). This case was managed prior to the advent of modern anti-retro viral drugs and HAART (Highly active anti-retro viral therapy).
**Fig 9-27** The same patient as Fig 9-26, but 18months later with conservative management. The ulceration is shallow, but there is a superficial candidal infection.
### _Drug-Induced Recession_
The use of recreational drugs such as cocaine is associated with severe mucositis, ulceration and recession and bone loss when applied to the gingivae. Cocaine is inhaled into the nose as a white crystalline powder, but oral, vaginal and rectal application is also used, as well as intravenous administration. Crack cocaine is smoked. Local application to gingival tissue causes severe inflammation, bleeding and epithelial desquamation, with necrosis of underlying bone and subsequent recession (Fig 9-28).
**Fig 9-28** Severe recession in a 17-year-old female cocaine user, who presented with gingival ulceration, attachment and bone loss and recession around her lower incisors. Her diet was also poor and low in antioxidants.
### References
Akerly WB. Prosthodontic treatment of traumatic overlap of the anterior teeth. J Prosthet Dent 1977;38:26–34.
Baxter, AM, Roberts A, Shaw L, Chapple ILC. Localised Scleroderma in a 12-year-old girl presenting as gingival recession. A case report and literature review. Dental Update 2001;28:458–462.
Chapple ILC, Hamburger J. The significance of oral health in HIV disease. J Sexual Transmit Infect 2000;76:236–243.
Chapple ILC, Lumley PJ. Periodontal/endodontic lesions. Dental Update 1999;26:331–341.
Felix DH, Wray D, Smith GLF, et al. Oro-antral fistula: an unusual complication of HIV-associated periodontal disease. Br Dent J 1991;171:61–62.
Glick M, Muzyka BC, Slakin LM et al. Necrotizing ulcerative periodontitis: a marker for immune deterioration and a predictor for the diagnosis of AIDS. J Periodontol 1994;65:393–397.
Tonetti MS, Mombelli A. Early-onset periodontitis. In 1999 International Workshop for a Classification of Periodontal Diseases and Conditions. Annals of Periodontology 1999;4:39–53.
### Further Reading
Clerehugh V, Tugnait A, Chapple ILC. Periodontal Management of Children, Adolescents and Young Adults. Chapple ILC (Ed). QuintEssentials of Dental Practice–17, Periodontology–4. London: Quintessence Publishing Co. Ltd, 2004;Chapter 5 pp 79–100.
Heasman PA, Preshaw PM, Robertson P. Successful Periodontal Therapy. A Non-Surgical Approach. Chapple ILC (Ed). QuintEssentials of Dental Practice–16, Periodontology–3. London: Quintessence Publishing Co. Ltd, 2004;Chapter 6 pp83–84.
Noble SN, Kellett M, Chapple ILC. Decision-Making for the Periodontal Team. Chapple ILC (Ed). QuintEssentials of Dental Practice–11, Periodontology–2. London: Quintessence Publishing Co. Ltd, 2004. Chapter 8 pp124 & Chapter 9 142–143.
Watts TLP (Ed). Periodontics in Practice: Science with Humanity. London: Martin Dunitz Ltd, London, 2000;Chapter 4, pp28.
## Chapter 10
## Generalised Gingival Recession
### Aim
To describe the appearance and discuss the aetiology, investigation and periodontal management of systemic diseases that manifest clinically as generalised gingival recession.
### Outcome
Having read this chapter, the clinician should be aware of the systemic diseases that may give rise to generalised gingival recession, either directly as a result of the biology of that disease process, or secondary to aggressive periodontal disease, which is a component of that systemic condition. Table 10-1 lists the most common causes of generalised gingival recession, but this chapter will limit discussion to recession associated with underlying systemic diseases. Most of these conditions are uncommon and patients should be referred for specialist management, rather than being managed in general practice.
Table 10-1 **Generalised gingival recession** **Condition** | **Sub-Category** | **Nature** | **Incidence** | **Manage/Refer**
---|---|---|---|---
Trauma | Toothbrush | Facial surfaces | common | manage
Trauma | e.g. de-gloving injury | uncommon | manage
Occlusion | Traumatic class 2 Div
II (gingival stripping) | common | manage/refer
Periodontal disease | Untreated | | common | manage
Treated | | common | manage
Systematic disease with generalised recession as Manifestation due to destructive periodontitis | Down syndrome | Trisomy chromosome 21 | common | manage
Papillon-Lefevre syndrome (PLS) ± Haim Munk syndrome | Genetic mutation of blood neutrophil gene for enzyme Cathepsin C | uncommon | refer
Hypophosphatasia | Genetic defect in gene for enzyme alkaline phosphatase | uncommon | refer
Chronic granulomatous disease (CGD) | Genetic defect – failure of neutrophils to kill bacteria | uncommon | refer
Chèdiak-Higashi syndrome | Genetic defect of neutrophil function | uncommon | refer
Ehlers-Danlos syndrome | Genetic defect of collagen metabolism | uncommon | refer
Leukocyte adhesion deficiency (LAD) | Genetic defect of neutrophil affacting ability to adhere | uncommon | refer
Acatalasia | Genetic defect of catalase production in red blood cells | uncommon | refer
Infantile genetic agranulocytosis | Severe neutropaenia | uncommon | refer
Agamma/hypogammaglobulinaemia | IgG2 and IgG4 deficiency | uncommon | refer
Cohen syndrome | Complex genetic syndrome | uncommon | refer
Glycogen storage disease | Defect of glygogen metabolism – neutropaenia | uncommon | refer
DiGeorge syndrome | T-cell defects | uncommon | refer
Wiskott-Aldrich syndrome | T & B-cell defects | uncommon | refer
Histiocytosis | Malignant neoplasm of CD-1a cells i.e. Langerhans cells/ histiocytes/ macrophages | uncommon | refer
Systemic disease with recession as manifestation independent of periodontitis | Progressive systemic sclerosis (scleroderma) | Connective tissue disorder | uncommon | refer
Drug-induced lesions | Cyclophosphamide | Alkylating drug | uncommon | refer
Methotrexate | Cytotoxic drug | uncommon | refer
Cocaine | Recreational drug | uncommon | refer
Bleomycin | Anti-tumour drug | uncommon | refer
Vincristine/
Vinblastine | Alkaloid drugs | uncommon | refer
### Background
Gingival recession refers to a situation arising where the gingival margin lies apical to its true position in 'pristine health'. Fig 10-1 illustrates pristine health, which is very rare and quite distinct from 'clinical health'. In clinical health, it is generally accepted that very subtle signs of mild inflammation will be evident at isolated sites and that variations in normal anatomy may result in a dento-alveolar complex with investing periodontal tissues that are not 'classical' in appearance. Fig 10-2 is an example of clinical health, where a midline diastema between the lower incisors provides a non-classical appearance of the interdental papilla and the papilla LL12 has evidence of very mild clinical inflammation.
**Fig 10-1** An anterior view of 'pristine gingivae', demonstrating applied anatomical features from Fig 10-3.
**Fig 10-2** Clinical photograph of healthy gingivae, demonstrating the effect of the tooth contact area in dictating interdental papilla shape. The schematic diagrams show the narrow contact point and narrow col between incisors and the broader contact and deeper, more vulnerable col area between molar teeth.
The detailed anatomy of the dento-gingival complex is discussed in book 1 of this series (Chapple and Gilbert, 2002), and further discussion is not within the remit of this book. However, Fig 10-3 illustrates the normal anatomy of this area and Fig 10-4 provides the reader with a reminder of the contribution of recession to overall periodontal attachment loss. In health, the gingival margin lies on enamel 2–3mm above the position of the terminal cell of the junctional epithelium (JE, Fig 10-3). The JE lies at the cemento-enamel junction (CEJ) and the interval between the terminal cell of the JE and the gingival margin is bordered by JE and sulcular epithelium, which form the soft tissue boundary to the gingival crevice. The crevice can be probed up to a distance of 3mm in health. Clinical attachment loss (CAL) due to periodontitis may arise without recession, by true pocket formation (Fig 10-4), but when the gingival margin migrates to a position apical to the CEJ, recession occurs and contributes, with true pocketing, to overall CAL. The gingival margin may lie at the CEJ, and in this situation there is no recession, but clearly the full clinical crown is exposed and attachment loss has taken place.
**Fig 10-3** A schematic view of the gingivae demonstrating the gingival collagen fibre complexes. The sulcular (SE) and junctional epithelium (JE) are also represented alongside in two photomicrographs demonstrating normal histology. Note how widely spaced the cells are and how they thin out, forming a single 'terminal cell' of the apex of the JE.
**Fig 10-4** Schematic longitudinal section of a premolar and associated periodontal tissues, demonstrating early true pocket formation, detected by probing and recession contributing to overall attachment loss.
### Aetiology of Gingival Recession
The most common cause of gingival recession is toothbrush trauma. This classically presents buccally to posterior teeth at the facial surfaces and not the interproximal surfaces. Interproximal recession generally implies CAL due to true periodontitis. Other common causes of generalised recession include:
* untreated periodontal disease (Fig 10-5).
* effects of successful periodontal therapy (Fig 10-6).
* trauma from the occlusion (usually labial stripping of gingivae labial to the lower incisors or palatal to upper incisors in a traumatic class II division 2 malocclusion (see Noble, Kellett and Chapple, 2004).
* physical trauma from an accident (de-gloving injuries from blunt trauma e.g. steering wheel or fist).
**Fig 10-5** Generalised recession due to progressive untreated chronic periodontitis.
**Fig 10-6** Generalised gingival recession following successful periodontal therapy and supportive care. A Maryland bridge has been fabricated to replace LR2, which was lost to periodontal disease.
This chapter will focus on recession due to periodontal attachment loss that has arisen:
1. secondary to systemic disease where destructive periodontitis is a manifestation of that disease.
2. directly as a manifestation of a systemic disease, even in the absence of significant inflammatory periodontitis (e.g. systemic sclerosis, drug-induced recession).
### Systemic Disease with Generalised Recession as Manifestation Due to Destructive Periodontitis
### _Down Syndrome_
### _Clinical appearance_
* Generalised recession upper and lower incisors.
* Severe periodontal destruction and pocketing (Fig 10-7).
* Gingival inflammation.
* Severe radiographic bone loss affecting upper and lower incisors.
* Short roots to lower incisors.
* Tooth mobility.
**Fig 10-7** Periodontal disease and gingival inflammation in a patient with Down syndrome.
### _Clinical symptoms_
* Gingival bleeding.
* Tooth mobility (especially lower incisors).
* Impaired eating/function.
### _Aetiology_
Down syndrome, first described by Langdon-Down, is a complex syndrome whose genetic basis is trisomy of chromosome 21. The cause of the periodontal destruction, which is reported to affect between 60–90% of subjects is also complex and thought to involve:
* Poor oral hygiene (especially in institutionalised individuals).
* Defects of PMNL function (chemotaxis, phagocytosis and killing).
* Depressed T-cell induced antigen killing.
* B-lymphocyte receptor defects.
* Abnormal collagen biosynthesis.
### _Involvement of non-gingival sites_
* Characteristic facial appearance (Fig 10-8).
* Learning difficulties.
* Short stature.
* Cardiac defects (may need antibiotic cover).
* Class III malocclusion.
* Anterior open bite.
* Macroglossia.
* Anterior tooth crowding.
**Fig 10-8** The patient from Fig 10-7.
**Fig 10-9** A two-year-old Pakistani boy with recession and early mobility of upper and lower deciduous incisors due to Papillon-Lefèvre syndrome (PLS).
### _Differential diagnosis_
Given the obvious clinical presentation and medical history, no other differential diagnoses are likely.
### _Clinical investigation_
* Standard periodontal investigations.
### _Management options_
* Behavioural management.
* Non-surgical periodontal therapy.
* Rigorous supportive care.
* Cardiological opinion in case antibiotic prophylaxis is needed for invasive procedures.
### _Papillon-Lefèvre Syndrome_
### _Clinical appearance_
Generally presents between the ages of two and seven years with one or more of the following, which affect the deciduous and permanent dentition:
* Generalised recession affecting deciduous incisors (Fig 10-10).
* Premature tooth mobility.
* Spacing and drifting of anterior teeth (Fig 10-10).
* Severe generalised gingival inflammation (Fig 10-11).
* Severe periodontal destruction radiographically (Fig 10-12).
* Radiographic bone loss where apices of permanent teeth are not yet fully formed.
**Fig 10-10** Spacing and drifting on the anterior permanent incisors in a seven-year old boy with PLS and severe periodontal disease.
**Fig 10-11** Severe gingival inflammation in a six-year old boy, with grade II mobility of the central incisors.
**Fig 10-12** Substantial bone loss affecting the permanent dentition of the patient in Fig 10-11.
Patients generally progress to total tooth loss by 20 years of age.
### _Clinical symptoms_
* Gingival recession.
* Tooth mobility.
* Gingival bleeding.
* Tooth spacing.
* Recurrent abscesses/infection.
### _Aetiology_
Papillon-Lefèvre syndrome (PLS) is an inherited autosomal recessive disorder, characterised by a strong family history of consanguinity (Fig 10-13). It has a prevalence of one in four million, most common in Indo-Pakistani children, and classical signs usually appear between two and four years of age. It arises from a gene defect, recently mapped to the long arm of chromosome 11 (11q) and results in the loss of function of an important neutrophil (PMNL) enzyme called Cathepsin-C. Cathepsin-C is a lysosomal enzyme found in the primary granules of PMNLs that is important in bacterial destruction and is expressed in the skin of the feet and hands, as well as lung, kidney and placenta.
**Fig 10-13** The family of the two-year old boy with PLS (Fig 10-9) demonstrating the Mendelian pattern of autosomal recessive inheritance of the condition.
### _Involvement of non-gingival sites_
* Classical palmar-plantar keratoderma (hyperkeraotsis of the palms of the hands and soles of the feet (Fig 10-14 and 10-15).
* Intracranial calcifications are occasionally reported.
* Another rare syndrome Haim Munk syndrome (HMS), presents with similar features, but has only been reported in two families at the time of going to press. HMS was first reported in a Jewish sect from Cochin, India and patients present with the same features as in PLS, but also suffer from:
* Recurrent pyogenic skin infections.
* Arachnodactyly (long, slender fingers).
* Acro-osteolysis (osteolysis of distal phalanges).
* Onychogryphosis (curved thickening of nails).
* Pes planus (flat feet/fallen arches of the feet).
**Fig 10-14a** Keratosis affecting the palms of the hands of a two-year-old boy with Papillon Lefevre syndrome (PLS). The gene mutation was a common mutation (R272P).
**Fig 10-14b** Keratosis affecting the soles of the feet of the two-year old boy in Fig 10-14a.
**Fig 10-15a** Keratosis affecting the palms of the hands of the father of the two-year old boy with PLS in Fig 10-14a.
**Fig 10-15b** Keratosis affecting the soles of the feet in the patient in Fig 10-15a.
The patient in Fig 10-10 first presented with recurrent skin infections, but the diagnosis was not made until he developed tooth mobility and was seen by the author, when PLS was diagnosed following careful history taking, clinical examination and genetic testing. The relationship between PLS and HMS is becoming clearer and they appear to be either allelic variants (Hart et al, 2000), i.e. different mutations in the same gene, although the same mutation has been reported in PLS as in HMS. Alternatively, it is also possible that HMS is a variant of PLS, arising following co-inheritance of a further, yet to be identified gene mutation in the vicinity of the CTSC gene locus. This has been demonstrated previously for co-inheritance of PLS with albinism (Hewitt et al, 2004). To date 45 different mutations have been described in the Cathepsin C gene, including a symptom free mutation (Allende et al, 2000).
### _Differential diagnosis_
* Hypophosphatasia (see later in chapter).
* Chronic granulomatous disease (CGD) – failure of the respiratory burst within PMNLs and therefore failure of oxygen radicals to destroy bacteria. However, usually severe gingival inflammation and ulceration/reces-sion are reported but not an aggressive periodontitis (see Clerehugh, Tugnait and Chapple, 2004).
* Chèdiak-Higashi syndrome (see later in chapter).
* Ehlers-Danlos syndrome (see later in chapter).
* Leukocyte adhesion deficiency (LAD – see later in chapter).
* Acatalasia – (see later in chapter).
* Infantile genetic agranulocytosis (see later in chapter).
* Cohen syndrome – autosomal recessive, characterised by extensive alveolar bone loss, neutropenia, learning difficulties, motor function defects, dysmorphia, obesity.
* Glycogen storage disease – series of five conditions involving impairment/inability to metabolise glygogen. In type 1B (autosomal recessive) patients have neutropaenia, defective PNML function and periodontal disease.
* DiGeorge syndrome – defects of T-lymphocyte function caused by a gene deletion on chromosome 22q (q = long arm of chromosome; p = short arm).
* Wiskott-Aldrich syndrome – X-linked T and B-cell immune deficiency.
* Histiocytosis – usually localised recession defects (see chapter 10).
### _Clinical investigation_
* PLS is a diagnosis made following collection of a family history and clinical findings.
* Cranial radiographs may reveal calcifications of the cerebral membranes.
* Genetic testing is not available routinely.
### _Management options_
Genetic counselling for parents is essential (pre-natal mutation analysis is available). Management is very difficult because most evidence is based on case reports, or at best case series and most reports have demonstrated poor efficacy of both surgical and non-surgical therapy. However, strict plaque control can slow down the progression of the disease and current thinking is that success rates may be improved by:
* Early extraction of deciduous dentition to provide a period of edentulousness, and eliminate pathogens from the mouth.
* Scaling and root surface debridement of permanent teeth in conjunction with systemic antibiotics.
* Bi-weekly professional prophylaxis and repeated mechanical therapy after identification of Actinobacillus actinomycetemcomitans (Aa).
* Systemic retinoid therapy – Acitretin has been shown to correct defective CD3-induced human T-lymphocyte activation in vitro in PLS patients. Retinoids may also regulate Cathepsin C gene expression, but should be prescribed by specialists due to their side effects.
The case in Fig 10-9 was managed by:
1. Full mouth scaling and prophylaxis whilst taking systemic amoxicillin and metronidazole.
2. Treatment of his dentate mother (as above) despite the absence of periodontal disease, in order to eliminate Aa. The father was edentulous and therefore deemed unlikely to possess the relevant pathogens.
3. Restricted close contact with grandparents who lived in the family home.
4. Four- to six- weekly recalls for full mouth prophylaxis.
By the age of four the patient still had a full deciduous dentition, whereas in most patients bone loss is so rapid that total deciduous tooth loss by four years of age is common. At 4.5 years the lower right deciduous incisor was electively extracted due to severe mobility and the full course of debridement repeated with systemic Augmentin (Co-amoxiclav). At the time of this book going to press, the child was 5.5 years old and his periodontal condition was starting to deteriorate, leading to consideration of a deciduous dentition clearance and retinoid therapy.
### _Hypophosphatasia_
### _Clinical appearance_
* Severe recession (often localised to incisors) in a child or young adult (deciduous or permanent dentition).
* Severe tooth mobility.
* Often minimal gingival inflammation.
* Premature deciduous tooth loss.
* Premature eruption of permanent successors.
* Radiographic bone loss especially anterior teeth, largely horizontal.
* Enlarged dental pulp chambers to teeth radiologically.
### _Clinical symptoms_
* Early deciduous tooth loss.
* Teeth exfoliate with intact roots (i.e. no/minimal resorption).
* Tooth mobility.
### _Aetiology_
Hypophosphatasia has five subtypes and is a rare inherited defect in the enzyme alkaline phosphatase (ALP). It affects 1:100,000 and evidence exists for allelic forms, one autosomal dominant (presenting with milder features) and one autosomal recessive (severe phenotype). Gene mutations can give rise to:
1. Perinatal (or lethal) hypophosphatasia – where infants live only a few days (rachitic chest leads to respiratory failure).
2. Infantile hypophosphatasia – First six months of life, presents with systemic manifestations (wide fontanelles, blue sclera, flail chest, poor feeding/weight gain).
3. Childhood hypophosphatasia (Fig 10-16a and b).
* Premature deciduous tooth loss.
* Cementum hypoplasia or aplasia (cementum does not form on roots).
* Disuse atrophy of alveolus as no connection exists for periodontal ligament fibres at the root surface due to lack of cementum.
* Wide open cranial fontanelles may be evident (anterior and posterior).
* Skull has a 'beaten copper' appearance.
* Proptosis (bulging eyes) may be evident due to premature fusion of cranial sutures.
4. Adult hypophosphatasia – has a mild phenotype and often presents in middle age. There may be a history of early deciduous tooth loss and the anterior six maxillary and mandibular incisors tend to be affected.
5. Pseudohypophosphatasia – or odontohypophosphatasia – is very mild and tends to be localised to the lower anterior deciduous teeth.
**Fig 10-16a** Radiographic evidence of premature deciduous tooth loss in a five-year-old boy with hypophosphatasia.
**Fig 10-16b** Enlarged pulp chambers of teeth that spontaneously exfoliated from the patient in Fig 10-16a.
### _Involvement of non-gingival sites_
This varies according to the type of hypophosphatasia. For detailed review see Chapple et al, 1993.
### _Differential diagnosis_
* PLS
* Chèdiak-Higashi syndrome.
* Ehlers-Danlos syndrome.
* Leukocyte adhesion deficiency.
* Acatalasia.
### _Clinical investigation_
Diagnosis involves:
* Radiology.
* Clinical biochemistry. Due to the deficiency of the liver/bone/kidney isoenzyme of ALP, substrates normally metabolised by ALP appear raised. In addition to low levels of ALP in serum (ensure normal range is used for children or adults, as appropriate), raised levels of:
* Pyridoxal-5-phosphate (PLP or vitamin B6) are found in serum
* Phosphoethanolamine in 24-hour urine samples.
### _Management options_
Currently, management options include:
* Genetic counselling
* Intensive non-surgical periodontal therapy.
### _Chronic Granulomatous Disease (CGD)_
This is a rare condition that does not normally cause significant recession and therefore discussion will be brief. CGD is reported in autosomal and X-linked recessive forms. The respiratory burst that creates the oxygen radical superoxide (see Chapple and Gilbert, 2002) and subsequently hydrogen peroxide and other reactive oxygen species within PMNL's fails. Therefore, bacteria are phagocytosed by PMNLs and released in a viable state rather than being destroyed. Unsurprisingly, sufferers have severely compromised infection control mechanisms, rendering them susceptible to osteomyelitis and pneumonia. Reported periodontal manifestations seem to be limited to severe gingival inflammation and ulceration, which may lead to recession, rather than periodontal destruction.
### _Chèdiak-Higashi Syndrome_
### _Clinical appearance_
* Severe gingival inflammation.
* Suppurative periodontitis with tooth mobility and generalised recession.
* Early deciduous tooth loss.
* Severe periodontal bone loss.
### _Clinical symptoms_
* Bleeding gums.
* Early tooth loss.
* Tooth mobility.
### _Aetiology_
A rare autosomal recessive condition, Chèdiak-Higashi syndrome is regarded as an inherited disorder of the blood/lymphoreticular system. It largely affects PMNL and monocyte function. Defects include:
* Impaired chemotaxis.
* Defective bacterial killing (phagolysosome formation is impaired – see Chapple and Gilbert, 2002).
* Defective degranulation and release of oxygen radicals.
* Hyper-responsive phagocytosis but ineffective killing.
### _Involvement of non-gingival sites_
* Oral ulceration.
* High susceptibility to bacterial infections.
### _Differential diagnosis_
* PLS.
* Hypophosphatasia.
* Ehlers-Danlos syndrome.
* Leukocyte adhesion deficiency.
* Acatalasia.
### _Clinical investigation_
* Careful history of bacterial infections
* Immunological investigations of neutrophil chemotaxis, adhesion and killing.
### _Management options_
A variety of drug regimes have been described, starting with milder therapies such as vitamin-C supplementation, to the use of cytotoxic drugs (e.g. methotrexate, vincristine), which are themselves associated with gingival ulceration and recession. The use of corticosteroids has also been advocated.
### _Ehlers-Danlos Syndrome_
### _Clinical appearance_
* Severe generalised recession.
* Fragile gingival tissues prone to excessive bleeding and bruising.
* Generalised aggressive periodontal pocketing and bone loss.
### _Clinical symptoms_
* Gum recession.
* Generalised profuse gingival bleeding.
* Tooth mobility.
* Early tooth loss.
### _Aetiology_
This is a rare genetic syndrome with varying inheritance according to the subtype diagnosed. It involves defects of collagen synthesis and 10 subtypes are described, with aggressive periodontal destruction associated with type IV (autosomal recessive or dominant), type VIII (autosomal recessive or dominant) and type IX (X-linked).
### _Involvement of non-gingival sites_
* Excessive joint mobility (subluxation of temporomandibular joints is reported).
* Skin hyperextensibility (Fig 10-17).
* Excessive bruising due to blood vessel fragility.
* Post-extraction haemorrhage.
* Poor/delayed healing.
* Cardiac valve involvement.
**Fig 10-17** Skin hyperextensibility in Ehlers-Danlos syndrom.
### _Differential diagnosis_
* PLS.
* Hypophosphatasia.
* Chèdiak-Higashi syndrome.
* Leukocyte adhesion deficiency.
* Acatalasia.
### _Clinical investigation_
* History (including family history).
* Examination for signs of skin hyperextensibility.
* Laboratory investigations of pro-collagen and collagen production (immunohistochemistry) from biopsy samples.
* Genetic testing.
### _Management options_
Very little data is available and referral recommended due to the complications associated with tissue fragility and excessive haemorrhage.
### _Leukocyte Adhesion Deficiency (LAD)_
### _Clinical appearance_
* Severe generalised recession.
* Deciduous tooth loss almost immediately on eruption.
* Fiery red gingivae.
* Profuse gingival bleeding.
### _Clinical symptoms_
* Gingival bleeding.
* Early tooth loss.
* Recurrent infections.
### _Aetiology_
LAD is a rare condition inherited in an autosomal recessive manner (reviewed by Kinane 1999). They involve defects in 3 key leukocyte cell surface receptors called (β-integrins (see Chapple and Gilbert 2002) that have important functions including binding complement and helping PMNL's enter the tissues from within blood vessels (Table 10-2). Such defects give rise to significant PMNL-binding problems and hence dramatically affect the patient's ability to fight infection. The LADs are heterogeneous conditions and patients with severe LAD (<0.3% of receptors) do not survive long after birth, whereas patients with milder phenotypes survive to adulthood but suffer various illnesses, including a severe form of pre-pubertal periodontitis.
Table 10-2 **White blood cell receptors relevant to leukocyte adhesion deficiency (LAD), their names and codes** **Receptor** | **Cells that Express** | **Other Names** | **Cluster of differentiation (CD) marker designation**
---|---|---|---
CR3
(binds complement component C3) | Monocytes
PMNL
NK cells | Mac-1 | CD 11b
(α-chain of CR3)
CR4
(binds complement component C3) | PMNL
Monocytes
NK cells | P150,95 | CD 11c
(α-chain of CR4)
LFA-1
(Leucocyte function antigen-1) | Monocytes
Macrophages
PMNL
Lymphocytes | | CD 11a
(α-chain of LFA-1)
### _Involvement of non-gingival sites_
* Papules and nodules affecting the mucosa of the cheeks are reported
* Recurrent and multiple infections.
### _Differential diagnosis_
* PLS.
* Hypophosphatasia.
* Ehlers-Danlos syndrome.
* Chèdiak-Higashi syndrome.
* Acatalasia.
### _Clinical investigation_
* Studies of cell surface receptor defects.
### _Management options_
These are limited to medical referral and maintaining periodontal care of the highest standard.
### _Acatalasia_
Acatalasia is a rare inherited autosomal recessive condition in which the key intracellular antioxidant enzyme catalase is deficient. Catalase is present mainly in red blood cells but also PMNLs and removes excess hydrogen peroxide (H2O2) before it causes damage to vital cell structures through oxidation reactions and upregulation of pro-inflammatory cytokine release by nuclear transcription factors like NFκB (equation 1).
When PMNLs are stimulated by periodontal pathogens or their products, they produce reactive oxygen species (ROS) like H2O2 which are important in microbial destruction. However, excessive ROS production, if not neutralised by key antioxidants is now known to be a major cause of 'collateral' periodontal tissue damage (Chapple, 1997; Brock et al, 2004). Deficiency in catalase facilitates such destructive processes and is associated with severe periodontal destruction, necrosis and ulceration. Interestingly, the role of catalase in the extracellular environment is performed by a very important enzyme, glutathione peroxidase (GSH-Px), which is largely selenium-dependent and reduces (H2O2) whilst oxidising reduced glutathione (GSH) to its oxidised form (GSSG).
Recently, we have reported a deficiency in GSH levels in chronic periodontitis subjects (Chapple et al, 2002).
### _Infantile Genetic Agranulocytosis_
This is a rare autosomal recessive disorder that involves a severe neutropenia and is characterised by an aggressive periodontitis, recession and bone loss.
### _Cohen Syndrome_
Cohen syndrome is a rare inherited autosomal recessive condition, characterised by:
* Extensive alveolar bone loss (associated with a neutropenia).
* Learning difficulties.
* Motor function defects.
* Obesity.
* Dysmorphia.
### _Glycogen Storage Disease_
The glycogen storage diseases are a series of five conditions characterised by an inability to metabolise or break down glycogen. Type 1B is autosomal recessive and patients are neutropaenic and demonstrate defective neutrophil function, with associated periodontal disease.
### _DiGeorge Syndrome_
This is a rare primary immune deficiency disease, largely affecting T-lymphocyte function, the cause being a gene deletion from the long arm of chromosome 22. There are various facial abnormalities reported, including cleft palate and an aggressive form of periodontitis has been described in children.
### _Wiskott-Aldrich Syndrome_
Wiscott-Aldrich syndrome is an X-linked immune deficiency involving a deficiency of T and B-cells and thrombocytopaenia.
### _Histiocytosis X_
Histiocytosis X is discussed in Chapter 9.
### Systemic Disease with Generalised Recession as Manifestation Independent of Periodontitis
### _Progressive Systemic Sclerosis (Scleroderma)_
### _Clinical appearance_
* Generalised gingival recession (Fig 10-18).
* Tooth mobility.
* Radiographic widening of the periodontal ligament.
* Normal periodontal attachment level and sulcus depth unless coincidental periodontitis is also present, which has a higher reported frequency in scleroderma.
**Fig 10-18** Generalised gingival recession in a patient with progressive systemic sclerosis.
### _Clinical symptoms_
* Gum recession.
* Sensitivity.
* Tooth mobility.
### _Involvement of non-gingival sites_
* Tight inflexible skin due to fibrosis (mask-like face – Fig 10-19).
* Microstomia (small opening to mouth).
* Xerostomia (secondary Sjögren's syndrome).
* Dysphagia (due to oesophageal stricture).
* Scarring and contracture of fingers (sclerdactyly – Fig 10-20).
* Raynaud's syndrome (numbness/cyanosis of fingers and toes when cold due to vasospasm).
* Telangectasia.
* Osteolytic lesions within skeletal bone. Mandibular changes involve remodelling or loss of the coronoid process and angle of the mandible.
* Visceral involvement.
**Fig 10-19** Tightening of the skin and mask-like facial appearance in the same patient as Fig 10-18.
**Fig 10-20** Sclerodactyly in the patient shown in Fig 10-18.
### _Differential diagnosis_
* Limited systemic sclerosis (formerly known as CREST – Calcinosis, Raynauds, Esophagitis, Sclerdactyly, Telangectasia).
* Progressive systemic sclerosis.
* Submucous fibrosis.
* Associated with Thibierge-Weissenach syndrome (exrtensive subcutaneous calcifications).
### _Aetiology_
Scleroderma is a rare connective tissue disorder characterised by progressive collagen deposition beneath mucosal and skin surfaces. Gingival and mucosal scarring can appear dramatic (Fig 10-21). Visceral involvement can involve the lungs, heart and kidneys and such patients have a poor prognosis.
**Fig 10-21** Gingival and mucosal scarring.
### _Clinical investigation_
* Serology may identify anti-centromere and/or anti-Scl-70 antibodies. The former can be seen in up to 70% of patients with limited systemic sclerosis and in rather fewer patients with progressive systemic sclerosis.
* Biopsy.
### _Management options_
* Manage periodontal tissues as for any other patient, but with more regular supportive care.
* It may be necessary to adapt toothbrush handles where sclerodactyly is evident (Fig-10-22) and use smaller toothbrush heads in the presence of microstomia. Modern power toothbrushes may be valuable in such cases (Fig 10-23).
**Fig 10-22** A toothbrush handle modified using circular foam to make gripping the handle easier in patients with sclerdactyly.
**Fig 10-23** Power toothbrushes have larger handles and reduce the necessity for dexterity. Three brushes are shown from an array of many. Sonic powered brushes offer the advantage for these patients of not needing to apply force to or manipulate the bristles.
### _Drug-Induced Gingival Recession_
### _Cytotoxic chemotherapy drugs_
Cytotoxic drugs such as cyclophosphamide and methotrexate work by targeting tumour cells with a high rate of division. Gingival epithelium has a naturally high turnover rate (especially JE) and is therefore prone to mucositis, ulceration and recession.
### _Recreational drugs_
Some individuals use cocaine as an oral application, by rubbing the powder into the gingivae. Cocaine has a powerful vasoconstrictive effect and this leads to ulceration, recession and sometimes severe bone resorption, providing an appearance similar to necrotising periodontitis (Fig 10-24).
**Fig 10-24** Severe recession and bone destruction (similar to necrotising periodontitis) in a 17-year-old female who used cocaine orally.
### _Cytotoxic antimicrobials_
Cytotoxic antimicrobials such as bleomycin, vincristine and vinblastine produce similar affects to cyclophosphamide and methotrexate.
### Further Reading
Allende LM, Garcia-Pèrez MA, Moreno A et al. Cathepsin C gene: first compound heterozygous patient with Papillon-Lefèvre syndrome and a novel symptomless mutation. Human Mutation 2000;399:1–6.
Brock G, Matthews JB, Butterworth C J, Chapple ILC. Local and systemic antioxidant capacity in periodontal health. J Clin Periodontol 2004;31:15–21.
Chapple ILC. Hypophosphatasia: dental aspects and mode of inheritance. Journal of Clinical Periodontology 1993;20:615–622.
Chapple ILC. Reactive oxygen species and antioxidants in inflammatory diseases. J Clin Periodontol 1997;24:287–296.
Chapple ILC, Gilbert AD. Understanding Periodontal Diseases: Assessment and Diagnostic Procedures and Practice. Chapple ILC (Ed). QuintEssentials of Dental Practice –1, Periodontology-1. London: Quintessence Publishing Co. Ltd, 2002;Chapter 1 pp3–16.
Chapple ILC, Brock G, Eftimiadi C, Matthews JB. Glutathione in gingival crevicular fluid and its relation to local antioxidant capacity in periodontal health and disease. J Clin Pathol: Molec Pathol 2002, 78:55, 367–373.
Clerehugh V, Tugnait A, Chapple ILC. Periodontal Management of Children, Adolescents and Young Adults. 2004. Chapple ILC (Ed). QuintEssentials of Dental Practice –17, Periodontology-4, London: Quintessence Publishing Co. Ltd, 2004. Chapter 10 pp131–149.
Hart TC, Hart PC, Michalec MD et al. Haim-Munk syndrome and Papillon-Lefèvre syndrome are allelic mutations in Cathepsin C. J Med Genet 2000;37:88–94.
Hewitt C, Wu CL, Hattab FN et al. Coinheritance of two rare genodermatoses (Papillon- Lefèvre syndrome and oculocutaneous albinism type 1) in two families: a genetic study. Br J Dermatol 2004;151:1261–1265.
Kinane DF. Blood and lymphoreticular disorders. Periodontol 2000 1999;21:84–93.
Seymour RA, Heasman PH (Eds). Drugs Diseases and the Periodontium. Oxford: Oxford Medical Publications, 1992.
## Chapter 11
## Miscellaneous Lesions
### Aim
This chapter aims to outline the non plaque-induced periodontal lesions and manifestations of systemic diseases that do not readily fall under the descriptive headings used in the previous 10 chapters. These lesions present primarily as incidental or intentional radiological findings, as gingival bleeding that is inconsistent with local irritants, as para-gingival swellings or annular lesions.
### Outcome
At the end of this chapter the reader should have a clear insight into the nature, diagnosis and management of lesions that are discussed therein.
### Introduction
Table 11-1 summarises the non-radiological lesions to be discussed. Tables 11-2 to 11-4 list those lesions normally detected radiologically and consistent with the general approach taken within this text, categorised by radiological appearance. The approach taken is similar to that of Langlais, Langeland and Nortjé in Diagnostic Imaging of the Jaws, which is recommended for further reading. Radiological features of periodontitis are not discussed, and those lesions of the soft tissues that can appear on radiographic films are also beyond the scope of this chapter. Only those lesions and conditions likely to present to the periodontal clinician rather than the radiologist will be discussed briefly.
Table 11-1 **Miscellaneous lesions presenting in and around the periodontal tissues** **Category** | **Condition** | **Sub-Classification** | **Discussed or Refer Reader**
---|---|---|---
Uncontrolled/ un-explained gingival bleeding | Myelodysplasia | Primary | discussed
| Secondary | discussed
Clotting factor deficiencies | Factor VIII | discussed
| Factor IX | discussed
Idiopathic thrombocytopenic discussed purpura (ITP) | | discussed
Platelet pool storage disease | | discussed
Acute leukaemia | | discussed
Chronic leukaemia | | discussed
Thrombocytopaenia | Secondary to liver disease | discussed
| Drug-induced | discussed
| HIV-associated | discussed
| Benign familial thrombocytopaenia | discussed
Aplastic anaemia | | discussed
Thrombasthenia | | discussed
Patients on warfarin | | discussed
Para-gingival swellings | Osteomas | | discussed
Gardner's syndrome | | discussed
Mandibular/maxillary tori | | discussed
Annular lesions | Erythema migrans | | discussed
Erythema multiforme | Stevens Johnson syndrome | discussed
Table 11-2 **Radiological lesions or conditions involving or associated with the roots** **Radiological Categorisation** | **Condition** | **Sub-group** | **Discussed or Refer Reader**
---|---|---|---
Root resorption | Internal root resorption | | reader referred
External root resorption | Cervical
Infra-cervical | discussed
Inter- and peri-radicular radiolucencies | Scleroderma (systemic sclerosis) | discussed
Periapical cemental dysplasia | | discussed
Lateral developmental periodontal cyst (Botyroid cyst) | | discussed
Lateral inflammatory periodontal cyst | | discussed
Gingival cyst | | discussed
Incisive canal cyst | | discussed
Solitary bone cyst | | reader referred
Median mandibular cyst | | reader referred
Squamous odontogenic tumour | | discussed
Ameloblastoma | Classical
Unicystic
Peripheral
Malignant | discussed
Ameloblastic fibroma | | discussed
Histiocytosis-X | Unifocal (solitary eosinophilic granuloma) |
Inter- and periradicular radiopacities | Periapical osteosclerosis (sclerosing osteitis) | | discussed
Condensing osteitis | | discussed
Hypercementosis | | discussed
Cementomas | Cementoblastoma | discussed
| Periapical cemental dysplasia | discussed
| Gigantiform cementoma | discussed
Cementicles | | discussed
Ossifying fibroma | | discussed
Table 11-3 **Radiolucent lesions** **Radiological Categorisation** | **Condition** | **Sub-group** | **Discussed or Refer Reader**
---|---|---|---
Well circumscribed radiolucencies | Odontogenic keratocyst | | discussed
Inflammatory cysts | Periapical | discussed
| Peri-implant | discussed
Neural sheath tumours | Neurilemmoma | discussed
| Schwannoma | discussed
| Neurofibroma | discussed
| Neurofibromatosis | discussed
Multi-locular radiolucencies | Odontogenic keratocyst and Gorlin-Goltz syndrome | | discussed
Botyroid cyst (lateral developmental periodontal cyst) | | discussed
Ameloblastoma | Polycystic | discussed
| Malignant | discussed
Odontogenic myxoma | | discussed
Giant cell tumour of bone (central giant cell granuloma) | | discussed
Arterio-venous malformation | | discussed
Aneurysmal bone cyst | | discussed
Central haemangioma | | reader referred
Sturge Weber syndrome | Arterio-venous malformations | discussed
Cherubism | | discussed
Odontogenic fibroma | | reader referred
Ossifying fibroma | Calcifying/ossifying fibrous epulis | discussed
Poorly defined radiolucent lesions | Osteomyelitis | | discussed
Osteoradionecrosis | Sterile
Septic | discussed
Intraosseous carcinoma | | discussed
(Epidermoid carcinoma) | |
Gingival carcinoma | Squamous cell | discussed
| Cuniculatum | discussed
Mucoepidermoid carcinoma | | reader referred
Clear cell carcinoma | | reader referred
Ameloblastic carcinoma | | discussed
Metastatic tumour | Adenocarcinoma of lung | reader referred
| Prostate carcinoma | reader referred
| Breast carcinoma | reader referred
| Renal carcinoma | reader referred
| Melanoma | reader referred
Fibrosarcoma | | reader referred
Ewings sarcoma | | reader referred
Radiolucent lesions as presentations of disseminated disease | Histiocytosis-X | Progressive (disseminated or Letterer-Siwe disease) | discussed
| Multifocal (Hand-Schuller-Christian syndrome) | discussed
Multiple myleoma | | discussed
Non-Hodgkins lymphoma | | discussed
Burkitt's lymphoma | | reader referred
Leukaemia | | discussed
Generalised radiolucencies | Hypophosphatasia | Neonatal (Lethal) | discussed
| Infantile | discussed
| Childhood | discussed
| Adult | discussed
| Pseudo-hypophosphatasia
(Odontohypophosphatasia) | discussed
Hyperparathyroidism | Brown tumours | discussed
Sickle cell anaemia | | discussed
β-thalassaemia | | reader referred
Radiolucent lesions with radiopacities | Periapical cemental dysplasia | | discussed
Calcifying odontogenic cyst | | discussed
Calcifying epithelial odontogenic tumour | | discussed
Adenomatoid odontogenic tumour | | discussed
Ameloblastic fibroadenoma | | reader referred
Odontomes | Compound | discussed
| Complex | discussed
Table 11-4 **Radio-opaque lesions** **Radiological Categorisation** | **Condition** | **Sub-group** | **Discussed or Refer Reader**
---|---|---|---
Focal radioopacities | Garrè's osteomyelitis | | reader referred
Hyperostosis | | reader referred
Osteoma | | discussed
Osteoblastoma | | reader referred
Osteoclastoma | | reader referred
Osteosarcoma | | discussed
Osteogenic sacroma | | reader referred
Chondroma | | reader referred
Chondroblastoma | | reader referred
Chondrosarcoma | | reader referred
Generalised radiopacities | Gardner's syndrome | | discussed
Osseous dysplasia | | reader referred
Sclerosing osteomyelitis | | discussed
Fibrous dysplasia | | discussed
Albright's syndrome | | discussed
Paget's disease | | discussed
Osteopetrosis | Albers-Schönberg disease | discussed
Hyperostosis | | discussed
### Uncontrolled/Unexplained Gingival Bleeding
When patients present with gingival bleeding that is not consistent with levels of plaque or inflammation, underlying systemic disease should be suspected and appropriate investigations undertaken.
### _Myelodysplasia_
The myelodysplastic syndromes (MDS) are rare haematological disorders of the myeloid cell lineages. They have an incidence of 4:100,000, are heterogeneic in nature and thought to be part of the same spectrum of disorders that give rise to acute myeloid leukaemia (AML). Previously called 'preleukemia', MDS has been diagnosed following persistent herpes labialis, severe oral mucosal ulceration and unexplained or spontaneous gingival bleeding, which was inconsistent with plaque levels (Chapple et al, 1999). The incidence of MDS appears to be increasing, and due to the high mortality rates associated with this group of disorders, it is important that the dental surgeon, who may be the first person to whom patients with MDS present, is aware of this group of disorders. Presentation is normally in patients over 60 years of age.
### _Clotting Factor Deficiencies_
Clotting factor deficiencies that may give rise to excessive gingival bleeding include:
* Von-Willebrand Factor deficiency (important for platelet adhesion).
* Factor II.
* Factor VII.
* Factor VIII (haemophilia). Many haemophiliacs now self-manage by injection of recombinant factor VIII.
* Factor IX (congenital factor IX disease, also called Christmas disease or haemophilia B).
* Factor X.
* Factor XII deficiency (Hageman factor).
### _Idiopathic Thrombocytopenic Purpura (ITP)_
ITP is a chronic autoimmune condition of insidious onset without any identifiable or associated illness. It is a chronic condition that typically affects young and middle aged adults, with a female: male ratio of 3:1. In 30% of adults it is persistent and appears resistant to most forms of treatment. The pathogenesis involves increased platelet destruction by autoantibodies to platelet membrane antigens (glycoprotein 11b/111a).
### _Platelet Pool Storage Disease_
This is a mild congenital bleeding disorder, generally managed by Desmopressin acetate (DDAVP) medication. It is associated with:
* Glanzmann's thrombasthenia – a condition caused by lack of a protein required for platelet aggregation, bleeding might be severe.
* Bernard-Soulier syndrome – a congenital disorder where platelets lack receptors to adhere to vessel walls. Bleeding may also be severe with this disorder.
### _Acute Leukaemia_
This is discussed in Chapter 8.
### _Chronic Leukaemia_
See Chapter 8.
### _Thrombocytopaenia_
Thrombocytopaenia is by definition a platelet deficiency, the defining level being <150,000 platelets per ml of blood. In clinical terms, control of haemorrhage does not normally become a problem in non-surgical periodontal therapy unless platelet levels fall below 60,000 per ml. There may be many underlying causes of thrombocytopaenia and these include:
* Advanced liver disease.
* HIV-associated thrombocytopaenia.
* Secondary to immunosuppressant and cytotoxic drugs.
* Idiopathic thrombocytopaenic purpura (ITP).
* Platelet pool storage disease.
* Other auto-immune diseases.
* Wiskott-Aldrich syndrome.
### _Aplastic Anaemia_
This is a rare condition that is associated with reduced haemopoetic tissue and a pancytopaenia. Gingival bleeding and advanced periodontal bone loss have been reported.
### _Thrombasthenia_
See Glanzmann's thrombasthenia above.
### _Patients on Warfarin_
Warfarin is the most commonly employed anticoagulant used in patients with a history of:
* Cerebro-vascular accident (stroke).
* Myocardial infarction.
* Thyrotoxicosis with associated cardiac arrhythmias.
Warfarin is a competitive antagonist of vitamin K, which is required for the production in the liver of Factors II, VII, IX and X. These factors are utilised in the coagulation cascade. Warfarin affects the prothrombin time (a measure of the extrinsic coagulation pathway), which is measured in a standardised way as the International Normalised Ratio (INR). An INR of 1.0 is normal, but an increasing ratio is associated with reduced coagulation. A growing body of opinion suggests that provided the INR is ≤ 4 and, local haemostatic measures are employed, periodontal and minor oral surgical procedures can be carried out with minimal post-operative haemorrhage. It is, therefore, less likely that warfarin doses may need to be adjusted in such patients. However, certain drugs may potentiate the action of warfarin and these include:
* Penicillin V.
* Amoxicillin.
* Miconazole (including topical applications).
* Erythromycin.
* Metronidazole.
* Fluconazole.
If bleeding does not respond to local measures in patients taking warfarin, the use of tranexamic acid is recommended. Tranexamic acid is a non-physiological inhibitor of fibrinolysis and an extremely effective haemostatic agent when used as a mouthwash (5%, 10mls as a rinse for two minutes and then spit out). It can be used four times daily for five to seven days, but avoid food and drink for one hour after rinsing.
### Para-Gingival Swellings
### _Osteomas_
### _Clinical appearance_
* Solitary or multiple well-circumscribed swellings (Fig 11-1).
* Bony hard.
* Surface epithelium is intact (no ulceration).
**Fig 11-1** Well circumscribed benign osteoma affecting the UL3 region.
### _Clinical symptoms_
* Painless swelling.
### _Aetiology_
These are benign slow growing tumours of mature bone, usually diagnosed in adult life.
### _Involvement of non-gingival sites_
Multiple osteomas of the jaw are a feature of Gardner's syndrome, a rare autosomal dominant condition. The syndrome also includes:
* Polyposis coli (high incidence of malignant transformation).
* Sebaceous cysts of the skin.
* Multiple fibrous tumours of skin.
* Multiple supernumerary teeth.
* Multiple impacted permanent teeth.
### _Differential diagnosis_
* Torus palatinus or mandibularis.
* Osteoblastoma.
* Osteochondroma (usually in children).
* Ossifying fibroma.
* Ameloblastoma.
* Fibrosarcoma.
### _Clinical investigation_
* Radiology.
* Biopsy.
### _Management_
No treatment unless aesthetic or functional problems have arisen due to the size of the osteoma. In the latter case, surgical re-contouring may be indicated.
### _Gardner's Syndrome_
See above.
### _Mandibular Tori_
Tori are examples of bony 'exostoses', which are benign outgrowths of bone. Normally tori are developmental but they may arise following chronic stimulation as reactive exostoses. The torus palatinus clasically arises in the midline of the palate and the torus mandibularis arises lingual to the mandibular premolar teeth (Fig 11-2). No treatment is indicated unless pre-prosthetic surgery is deemed necessary.
**Fig 11-2** Classical mandibular tori.
### Annular Lesions
### _Erythema Migrans_
### _Clinical appearance_
Erythema migrans may arise in several oral mucosal sites, but usually involves the tongue, where prevalence figures are approximately 2% (Fig 11-3). The term describes red patches, said to look like a map, which vary in size and location, often with a yellow margin surrounding areas of depapillation.
**Fig 11-3** Erythema migrans.
### _Clinical symptoms_
* Usually asymptomatic.
* Soreness of the tongue, especially with salty or spicy foods.
* Appearance that changes shape and size.
### _Aetiology_
Unknown.
### _Involvement of non-gingival sites_
The tongue dorsum is the most common presenting site, but the palate and buccal mucosa may also be involved.
### _Differential diagnosis_
If the palate is involved it may be confused with:
* Lupus erythematosus.
* Lichen planus.
### _Clinical investigation_
None, the diagnosis is a clinical one
### _Management_
Conservative, avoid irritating foods, and in some cases zinc supplements (200mg, three times daily) may help if taken for two to three months.
### _Erythema Multiforme_
Gingival tissues are typically 'spared' in erythema multiforme and this is one key to differentiating EM lesions from herpes simplex infection. EM is thus not discussed in this text, but is covered in Clerehugh, Tugnait and Chapple (2004, book 17 in this series).
### Radiological Conditions or Lesions Associated with the Roots
### 1. Root Resorption
External root resorption can result following a variety of pathological stimuli, the most common being:
* Chronic infection/peri-radicular inflammation.
* Chronic trauma (e.g. excessive orthodontic forces, excessive occlusal forces – Fig 11-4 and 11-5a–d).
* Trauma from hypochlorite irrigation beyond the root canal system.
* Trauma from excessive periodontal instrumentation.
* Impacted or unerupted teeth.
* Cysts (secondary to cyst growth/activity).
* Following tooth luxation and re-implantation.
* Odontogenic tumours (e.g. ameloblastoma).
* Neoplasia (other forms of malignant neoplasia).
* Secondary to radiotherapy of the jaws.
**Fig 11-4** Apical third root resorption following adult orthodontic treatment, where the prevalence is higher than in adolescent orthodontics.
**Fig 11-5a** Panoramic radiograph of burrowing cervical root resorption in a chronic bruxist who was also a weightlifter.
**Fig 11-5b** Left-sided premolar/molar region in the patient shown in Fig 11-5a prior to extraction of LL56.
**Fig 11-5c** Left-sided premolar/molar region in the patient shown in Fig 11-5a post-extraction of LL56. The resorption continued and the LL7 became involved.
**Fig 11-5d** The extracted teeth from Fig 11-5b plus an opposing molar.
Resorption can also be associated with systemic diseases such as:
* Paget's disease.
* Hypoparathyroidism.
* Hyperparathyroidism.
* Turner's syndrome.
* Calcinosis.
* Gaucher's disease.
The pathobiology of the resorptive process is not fully understood, but it may affect the apical third of the root (Fig 11-4), the mid-third or the cervical third (Fig 11-5). In the case demonstrated in 11-5a–e it was likely that the cemental layer finished short of the cemento-enamel junction (Fig 11-6) and thus excessive forces on the teeth allowed osteoclastic activity associated with bone remodelling to involve the exposed dentine.
**Fig 11-5e** Photomicrograph of cervical dentine resorption from Fig 11-5a–d demonstrating resorption lacunae within the cervical dentine and associated osteoclastic activity.
**Fig 11-6** Photomicrograph of cementum positioned apical to the CEJ (5–10% of cases).
### 2. Inter- and Peri-radicular Radiolucencies
### _Systemic Sclerosis (scleroderma)_
See Chapter 10.
### _Periapical Cemental Dysplasia_
Periapical cemental dysplasia is a benign condition most commonly affecting the mandibular incisors of post-menopausal females, especially black females. The aetiology is unknown, but the lesions are non-expansile and the teeth vital. Lesions may be solitary or multiple and contain cellular fibrous tissue initially, within which cementum forms. Initially radiolucent (Fig 11-7) and histologically similar to fibrous dysplasia, lesions may gradually become more radio-opaque with time (Fig 11-8) as more cementum is deposited. They can appear like sclerosing osteitis or osteosclerosis, except a fine radiolucent line separates the cementoma from the surrounding bone. No treatment is indicated, as the teeth are vital and the lesions self-limiting.
**Fig 11-7** Periapical cemental dysplasia in a black female. The lower incisor teeth were vital.
**Fig 11-8** Periapical cemental dysplasia with a more radio-opaque appearance due to cementum deposition.
### _Lateral Periodontal Cyst (Developmental)_
Periodontal cysts are rare and may be:
* Developmental (Fig 11-9).
* Inflammatory (Fig 11-10).
**Fig 11-9a** A botyroid cyst in a vital tooth pre-surgical enucleation.
**Fig 11-9b** The cyst from Fig11-12a 12-months post-enucleation and tissue regeneration using EmdogainTM.
**Fig 11-10** An inflammatory periodontal cyst affecting UL356 in a 38-year-old female. The cortical plate had perforated palatally and the lesion presented with a clinical depression of the maxillary mucosa.
Inflammatory cysts develop following infection of the periodontal or peri-implant tissues and follow a classical course of development from peri- or para-radicular granuloma to cyst formation. They are beyond the scope of this book. However, developmental periodontal cysts are believed to develop in the absence of an inflammatory or infective stimulus from epithelial rests of the embryonic dental lamina. These lesions may be:
* Unilocular lesions.
* Multilocular lesions (Fig 11-9), which are also known as botyroid cysts (after 'bunch of grapes'). Enucleation is indicated as for all periodontal cysts, but botyroid cysts may recur up to 10 years later.
### _Gingival Cyst_
Gingival cysts are rare and present most commonly between the ages of 40–75 years, with 75% of lesions affecting upper canine or pre-molar teeth (Shear, 1985; Wysockie et al, 1980). They may arise mid-way from the cervical margin to the apex, or indeed at the gingival margin where they cause a saucerisation of the alveolar crest. The lesions are soft rather than bony hard (cf lateral periodontal cysts).
### _Incisive (Naso-palatine) Canal Cyst_
Cysts of the nasopalatine or incisive canal are derived from epithelial remains of the nasopalatine duct. Reported prevalence varies between populations (0.1–1.5%) and according to race. Lesions arise apical to the upper incisors. Different parameters are used radiologically to differentiate cystic lesions from normal variations in the canal size. However, most would agree that a cyst should be suspected for lesions >1.0cm diameter where the margin is corticated and for those lesions >1.5cm diameter, cystic change is highly likely (Fig 11-11).
**Fig 11-11** A cyst of the incisive (nasopalatine) canal.
### _Patent Nasopalatine Ducts_
Patent nasopalatine ducts have been reported to arise clinically either side of the incisive papilla as developmental anomalies. A series of three cases was reported by Chapple and Ord (1990).
### _Aneurysmal Bone Cyst_
These are rare cyst-like lesions filled with sinusoid-like spaces or a single solitary blood-filled space (Fig 11-12).
**Fig 11-12** An aneurysmal bone cyst.
### _Squamous Odontogenic Tumour_
This is a rare lesion, presenting most often between 20–30 years with an equal male to female ratio. It is often symptomless but can cause tooth migration due to inter-radicular expansion or tooth mobility due to bone resorption. Lesions tend to present as triangular or semi-circular radiolucencies and may have a sclerotic margin. Surgical excision is required.
### _Ameloblastoma_
Ameloblastomas are the most common odontogenic tumours and account for 1% of oral tumours. They are benign but locally invasive lesions that give rise to bone expansion and thinning of the overlying cortex. Eighty per cent are found in the mandible, and of these 70% arise in the molar region, 20% the premolar area and 10% in the lower incisor region. Radiologically the lesions normally appear as multilocular radiolucencies.
The tumours can displace adjacent teeth and cause root resorption. There are three types described:
* Classical – histologically variable, multiple small cysts may fuse to form a large cyst with cuboidal pre-ameloblast cells lining the cyst. Treatment needs to be aggressive and surgical enucleation with a margin of normal bone is advocated (Chapple et al, 1991: Fig 11-13).
* Unicystic – presents in 20–30-year-olds and has a unilocular appearance (hence the name) and responds to curettage or conservative enucleation.
* Peripheral – this is a very rare form of ameloblastoma that affects the soft tissues (without bone involvement).
* Malignant – fortunately extremely rare this form of ameloblastoma does appear to metastasise. However, some believe that rather than true metastasis the tumour is 'seeded' into other tissues by poor surgical technique.
**Fig 11-13a** A cystic ameloblastoma in a 19-year-old Afro-Caribbean girl. The finding was incidental on a panoramic radiograph (LR45 area), and surgical investigation was indicated due to movement of the roots of the adjacent teeth.
**Fig 11-13b** The lesion from Fig 11-13a post-block dissection of the affected area.
**Fig 11-13c** The surgical site from Fig 11-13a six months post-resection.
### _Ameloblastic Fibroma_
A rare benign odontogenic tumour that presents in patients <20 years as a slow-growing painless swelling. Radiologically it is a unilocular lesion and both the epithelial and mesenchymal elements are neoplastic It is, however, non-invasive and therefore it is important to differentiate it from the ameloblastoma, as enucleation can be less radical.
### _Histiocytosis-X_
This condition may present radiologically, and is discussed in Chapter 9 of this text.
### 3. Inter- and Peri-radicular Radiopacities
### _Periapical Osteosclerosis_
Periapical osteosclerosis refers to localized areas of particularly dense bone, arising in the absence of apparent irritation or infection. It is reported to largely affect posterior teeth with a prevalence as high as 5% (higher in Asians). Radiologically, the marrow space is obliterated (Fig 11-14) and lesions may:
* affect the apical areas of teeth.
* affect inter-radicular regions.
* not be associated with teeth.
**Fig 11-14** Periapical osteoclerosis (sclerosing osteitis) around a tooth, which was also resorbing (mesial root).
No treatment is indicated.
### _Condensing Osteitis_
Pathologists argue that condensing osteitis is the same entity as periapical osteosclerosis. The key difference clinically and radiologically is that condensing osteitis arises adjacent to an area of apical infection (Fig 11-15) and therefore treatment (root canal therapy or extraction) is indicated.
**Fig 11-15** Condensing osteitis around the distal root apex of LR6, which was vital, but subsequently not filled.
### _Hypercementosis_
This refers to increased cellular cementum formation normally affecting the apical two-thirds of teeth (Fig 11-16). Causes include:
* Periapical chronic inflammation.
* Occlusal trauma.
* Paget's disease.
* Acromegaly.
* Over-eruption of a tooth with no opposing unit.
* Idiopathic.
**Fig 11-16** Hypercementosis in a patient with Paget's disease.
### _Cementomas_
Cementomas are benign lesions that are complex to classify as they may arise de-novo or following cementum deposition within other lesions. True cementomas normally affect younger subjects (<25 years) and are more common in males. They are also referred to as benign cementoblastomas and most commonly affect posterior mandibular teeth. Lesions are apical and radiologically appear dense and sclerotic (Fig 11-17). Other lesions within the cementoma group include:
* The gigantiform cementoma (Fig 11-18).
* Periapical cemental dysplasia (see earlier).
* Cementifying fibroma.
**Fig 11-17** Cementoma.
**Fig 11-18** Gigantiform cementoma.
### _Cementicles_
These are likely to be dystrophic calcifications within the periodontal ligament rather than true cementomas. They are spherical radiopacities 0.2–0.3mm in diameter and therefore not visible radiographically.
### _Cementoblastoma_
See 'true cementomas'.
### _Ossifying Fibroma_
Also known as 'cementifying fibroma' (see above), these lesions represent variants of the same spectrum of conditions, which also includes 'cemento-ossifying fibroma'. They arise as benign calcifications of fibrous tissue (Fig 11-19) affecting the mandible in 70–90% of cases, with a 5:1 female to male ratio and normally presenting in patients over 40. They are typically solitary lesions that expand in three dimensions and radiologically are well defined radiolucencies within which foci of mineralisation appear. Surgical enucleation is generally relatively simple as lesions are encapsulated. If not encapsulated and removal proves difficult fibrous dysplasia should be suspected (see later).
**Fig 11-19** Cemento-ossifying fibroma LL67 area. Areas of mineralisation were evident within the body of the radiolucent lesion.
### 4. Radiolucent Lesions – Well Circumscribed Radiolucencies
### _Odontogenic Keratocyst_
### _Radiological appearance_
Radiological appearances vary:
* Well-circumscribed radiolucent lesion within the medullary space.
* Unilocular or multilocular appearances may arise.
* Most commonly affect mandible (65–85%). Molar/ramus region accounts for about 50%.
* Cysts become large and fill entire ramus with time (Fig 11-20).
* Perforations of the cortical plate may be seen.
* Margins become sclerotic with time.
* Cyst cavity becomes 'foggy' with time due to keratin deposition.
* Root resorption is NOT characteristic.
* May arise bilaterally in young patients (<10 years) as part of the Gorlin-Goltz syndrome.
**Fig 11-20a** Odontogenic keratocyst of the mandible pre-marsupialisation in a 40-year-old female.
**Fig 11-20b** The keratocyst from Fig 11-20a immediately post-surgery and packing.
**Fig 11-20c** The cyst from Fig11-20a four months post-surgery.
### _Clinical symptoms_
Cysts arise in the second and third decade, and symptoms are rare because cysts permeate the mandible in an antero-posterior direction, becoming extremely large before detection. Often they are an incidental radiographic finding.
### _Aetiology and involvement of non-gingival sites_
Odontogenic keratocysts (OKC's) are believed to develop from remnants of the embryonic dental lamina from which the tooth germ develops; the term 'primordial cyst' is therefore also applied to the same lesions.
### _Differential diagnosis_
* Dentigerous cyst.
* Lateral periodontal cyst.
* Inflammatory/radicular cyst associated with tooth apex.
* Residual cyst.
* Ameloblastoma.
* Odontogenic fibroma.
* Odontogenic myxoma.
* Central giant cell granuloma.
* Aneurysmal bone cyst.
* Brown tumour (of hyperparathyroidism).
### _Clinical investigation_
* Additional radiographic views to assess expansion/extent within bone:
* Lateral oblique view of mandible.
* True occlusal view (bucco-lingual extent).
* Postero-anterior view focussed on jaws.
* CT scan.
* Broad needle aspirate will yield a thick yellow/brown cheese-like material comprising keratin squames and low protein levels (<40g/L).
* Incisional biopsy to include bone and any putative lining epithelium.
### _Management_
OKCs are notoriously difficult to remove because:
* The lining fibrous capsule is friable and thin and easily torn/left behind during enucleation.
* There are frequently 'daughter cysts' (satellite cysts) beyond the main lesion.
* Projections of the main cyst may be missed at surgery.
Due to difficulties with complete enucleation, recurrence rates of between 12–63% are reported and recurrence may take up to 25 years. The case in Fig 11-20a was therefore marsupialised and packed with ribbon gauze soaked in 'Whitehead's varnish'. The pack was replaced every few weeks until bone filled the defect from the mandibular base. This approach reduced the risk of manbibular fracture at surgery and also the risk of recurrence (Fig 11-20b and c).
### _Inflammatory Cyst_
Inflammatory cysts are very common around tooth apices where:
* A non-vital pulp has become infected.
* A root is retained (and infected).
* A root canal filling has failed.
* A root has fractured.
* There is a root perforation or infected lateral canal.
* A retrograde root filling has failed to eliminate pathology.
Lesions are well circumscribed radiolucencies (Fig 11-21) and with time the margin can become corticated. Similar lesions may also arise around implants (Fig 11-22a–c).
**Fig 11-21** An inflammatory apical cyst associated with non-vital LL12.
**Fig 11-22a** Inflammatory peri-implant cyst pre-surgical enucleation. The cyst communicated with the maxillary antrum.
**Fig 11-22b** The cyst in Fig 11-22a at the time of surgery. The cyst cavity was packed with BioOssTM and covered with a BioguideTM membrane for 12 months.
**Fig 11-22c** The healed site from Fig 11-22b 12 months post-surgery with a small residual scar around the mesial implant.
### _Neural Sheath Tumours_
Neural sheath tumours are rare benign lesions which may present from birth to 70yrs as intra-osseous radiolucencies. Clinical symptoms include:
* Dysasthesia/parasthesia.
* Burning sensation.
* Pain.
Broadly there are several lesions within this group:
* Neurilemmoma – well encapsulated lesions.
* Schwannoma – can arise peripherally within soft tissue (Fig 11-23).
* Neurofibroma – solitary lesions which are more likely to recur as less well encapsulated (Chapter 5 – Fig 5-13).
* Neurofibromatosis – multiple neurofibromas may arise as part of von Recklinghausen's neurofibromatosis, characterised by multiple lesions of the skin, which are hamartomas rather than true neoplasms (see Chapter 7).
**Fig 11-23** An 'Ancient Schwannoma' at enucleation. It was well encapsulated and comprised degenerative nerve tissue.
### Multi-locular Radiolucencies
### _Odontogenic Keratocyst and Gorlin-Goltz Syndrome_
OKC's have been discussed previously. They may arise as multiple lesions in the autosomal dominantly-inherited Gorlin-Goltz syndrome, which is characterised by:
* Multiple OKCs of the jaws.
* Multiple skin basal cell carcinomas.
* Rib/vertebral deformity.
* Bossing of the frontal and temporal bones of the skull.
* Calcifications of the falx cerebri.
### _Botyroid Cyst_
See above.
### _Ameloblastoma_
See above.
### _Odontogenic Myxoma_
The odontogenic fibroma, may arise centrally or peripherally, where it is almost identical to the fibrous epulis (Chapter 5). They are however extremely rare and not discussed further (see Soames and Southam, 1993). The odontogenic myxoma is more common and normally presents as a multilocular lesion (though it may be unilocular) radiographically. It is less well-defined than the odontogenic fibroma and locally invasive, hence enucleation is problematic.
### _Giant Cell Tumour of Bone_
Giant cell lesions of the jaws are histologically the same, but true giant cell tumours of the jaws (osteoclastoma), unlike central giant cell granulomas, are destructive/aggressive lesions that metastasise and behave more like true sarcomas. Fortunately, true giant cell tumours of the jaws are extremely rare.
### _Aneurysmal Bone Cyst_
These lesions may be uni- or multilocular and tend to affect the posterior mandible. They comprise blood filled spaces and largely arise secondary to other bone pathology, such as giant cell granulomas.
### _Arterio-venous Malformations (AVMs)_
AVMs are discussed in Chapter 3.
### _Sturge Weber Syndrome_
Sturge Weber syndrome discussed in Chapter 4.
### _Cherubism_
Also known as familial fibrous dysplasia, cherubism is an autosomal dominant-inherited disorder that presents between two to five years of age. Multilocular cystic lesions appear in the mandible and maxilla, but the condition usually regresses by the age of 20 years. Other features include:
* Bilateral swelling of the cheeks (so-called 'cherubic' appearance) usually affecting the mandibular angles and posterior maxillary sinuses.
* 'Cafè au lait' pigmentation of skin.
* Multiple unerupted or ectopic teeth.
* Accelerated deciduous tooth resorption.
### _Ossifying Fibroma_
A slow-growing and well-encapsulated benign tumour of fibrous tissue, within which ossification or cementification may occur. Usually affects children and adolescents, and its encapsulation helps distinguish it from fibrous dysplasia. Biopsy is essential for histological diagnosis.
### Poorly Defined Radiolucent Lesions
### _Osteomyelitis_
Osteomyelitis is inflammation of the bone marrow and tends to arise in the jaws following deep-seated odontogenic infection in debilitated or immuno-suppressed patients or those with limited blood supply to the mandible. Predisposing conditions include:
* Diabetes mellitus.
* Paget's disease.
* Osteopetrosis.
* Trauma to the mandible (e.g. fracture).
* Long-standing odontogenic infection.
* Radiotherapy to the jaws.
Soames and Southam (1993) classify osteomyelitis into:
* Suppurative (acute or chronic).
* Chronic sclerosing (focal or diffuse).
* Special types
* radiation
* chemical
* osteomyelitis of the newborn.
Associated mainly with gram – ve organisms, the inflammation and infection give rise to compromised local blood flow and areas of bone necrosis develop (bone sequestra). The inflammation/infection spreads throughout the bone marrow and eventually the periosteal blood supply is compromised. Radiologically a 'moth-eaten' appearance is ascribed (Fig 11-24) to the bone as radiolucent areas have a poorly defined margin and bone sequestra within the lesion appear radiopaque. Treatment is aggressive and involves surgical curettage, removal of sequestra and systemic antibiotics.
**Fig 11-24** Osteomyelitis affecting LR234, with characteristic 'moth-eaten' appearance.
### _Osteoradionecrosis_
This is a form of bone necrosis that arises following radiotherapy to the jaws. The radiotherapy causes an endarteritis obliterans and loss of blood supply to areas of bone, which necroses and is more susceptible to infection. This may ultimately give rise to mandibular fracture (Millett et al, 1990).
### _Intraosseous Carcinoma_
True intraosseous carcinomas are extremely rare and usually arise in children.
### _Gingival Carcinoma_
* Squamous cell carcinoma (see Chapter 8).
* Carcinoma cuniculatum (Fig 11-25) is a rare variant of squamous cell carcinoma, which is extremely slow growing but invades bone and surrounding tissue. It may take several years to develop and can present as keratosis with varying degrees of ulceration and/or suppuration, due to infection of underlying necrotic bone. Heasman et al (2005) presented a case in a 44-year-old female that was only diagnosed following a deep biopsy of bone. Lesions may be mistakenly diagnosed as osteomyelitis. Metastasis to regional lymph nodes is rare.
**Fig 11-25a** Keratosis associated with underlying carcinoma cuniculatum of the gingivae.
**Fig 11-25b** The same lesion as Fig 11-25a at a later stage and after a third biopsy.
### _Ameloblastic Carcinoma_
This lesion is the same as a malignant ameloblastoma (see earlier).
### Radiolucent Lesions as Presentations of Systemic Disease
### _Histiocytosis-X_
See Chapter 9.
### _Multiple Myeloma_
Myeloma is a malignant proliferation of a clone of immunoglobulin producing plasma cells which may arise as:
* Solitary plasmacytoma (a solitary myeloma – very rare).
* Multiple myeloma (normally a disseminated disease of poor prognosis).
Excess production of a monoclonal immunoglobulin is also referred to as a 'monoclonal gammopathy'. These are often benign but 10% may become malignant with time and monitoring is therefore important. With malignant myeloma the light chains of the immunoglobulin are small enough to be excreted through the kidney and appear in urine as 'Bence-Jones protein'. The cranial bones and jaws can be affected with multiple 'punched out' osteolytic lesions (Fig 11-26) and clinical signs/symptoms may include:
* Bone pain.
* Renal failure.
* Anaemia (oral ulceration secondary to bone marrow suppression).
* Recalcitrant infections of the jaws.
* Unusual or exaggerated bleeding post-periodontal or surgical therapy.
* Macroglossia (secondary to amyloid deposition within the tongue).
**Fig 11-26** Multipe myeloma.
### _Non-Hodgkins Lymphoma (see Chapter 8)_
A malignant lymphoma affecting young adults, which may present as:
* Painless enlargement of lymph nodes.
* Paraesthesia/anaesthesia.
* Soft swelling around fauces or maxillary gingivae.
* Ulceration around tonsillar region.
* Mobility of teeth with associated swelling.
The lymphomas are classified histologically, with diffuse lesions having a poorer prognosis than focal (follicular) lesions. Broadly lesions are either:
* High grade (poor prognosis but often respond well to chemotherapy).
* Low grade (good prognosis).
Underlying risk factors include:
* Immunosuppression.
* AIDS.
* Some autoimmune diseases (e.g. Sjögrens syndrome).
### _Leukaemia_
See earlier.
### Generalised Radiolucencies
### _Hypophosphatasia_
This condition is discussed in Chapter 10.
### _Hyperparathyroidism_
This condition results in excess parathyroid hormone (PTH) and may be primary in nature or secondary to chronic hypocalcaemia. Primary hyperparathyroidism may arise due to lesions affecting the parathyroid glands, e.g:
* Benign hyperplasia.
* Adenoma.
* Adencarcinoma.
PTH causes calcium retention and achieves this by increasing intestinal absorption and renal resorption, but also by increasing osteoclastic activity in bone. The latter can give rise to osteolytic lesions called 'brown tumours' (due to deposition of haemosiderin), where fibrous tissue replaces mineralised bone. The latter contain giant cells and are histologically identical to giant cell tumours/granulomas of bone (see earlier).
### _Sickle Cell Anaemia_
Sickle cell anaemia affects 1 in 500 black people and radiologically can cause marrow hyperplasia and prominent trabeculation. Thalassaemia may cause similar radiological changes (Fig 11-27).
**Fig 11-27** Prominent trabeculation and marrow hyperplasia (and osteopaenia) secondary to thalassaemia. The maxillary antrum is also full of trabecular bone as the body attempts to synthesise more red blood cells.
### 5. Radiolucent Lesions with Radiopacities
### _Periapical Cemental Dysplasia_
See earlier.
### _Calcifying Odontogenic Cyst_
These are not true cysts and regarded by most as odontogenic tumours. They arise as slow growing enlargements of the gingivae or adjacent alveolar mucosa from the pre-molar region forwards. They may appear as unilocular or multilocular radiolucencies, within which there is calcification of enlarged keratinocytes called 'Ghost cells'. Enucleation is usually successful.
### _Calcifying Epithelial Odontogenic Tumour (CEOT)_
Also referred to as Pindborg's tumour, the CEOT is a benign but locally invasive odontogenic tumour of epithelial origin. The lesion appears as an irregular radiolucency in which radiopaque areas develop following calcification.
### _Adenomatoid Odontogenic Tumour_
This tumour is benign and well-encapsulated, arising in the anterior maxilla. It appears as a well-defined radiolucency, within which calcification may occur. It can be enucleated simply and should be differentiated from an ameloblastoma for this reason.
### _Odontomes_
Odontomes are hamartomatous lesions of dental tissues that by definition contain enamel and dentine. They vary in shape and size but compound (Fig 11-28) and complex (Fig 11-29) odontomes are most likely to present as incidental radiological findings.
**Fig 11-28** Compound odontome.
**Fig 11-29** Complex odontome.
### 6. Radiopaque Lesions – Focal Radiopacities
### _Osteoma_
See earlier.
### _Osteosarcoma_
This is the most common malignant primary bone tumour, but is fortunately very rare in the jaws. It presents radiologically as an irregular radiolucency within which varying degrees of neoplastic bone may appear as radiopaque areas. When the tumour perforates the cortical plate it lifts the periosteum and trabeculae of bone span out at 90° to the cortex, giving the characteristic 'sun-ray spicule' appearance, although this is an unusual finding in the jaws.
### Generalised Radiopacities
### _Gardner's Syndrome_
See above.
### _Sclerosing Osteomyelitis_
See earlier.
### _Fibrous Dysplasia_
This is a non-heritable developmental disorder that may affect the mandible or maxilla and presents during childhood or adolescence (i.e. during active skeletal development), normally becoming quiescent in adulthood. It presents as a slow growing, painless enlargement of the posterior maxilla or mandible, which is often unilateral (unlike Cherubism) and causes facial asymmetry. Two forms are described:
* Monostotic (localised).
* Polyostotic (several bones involved).
The aetiology is unknown and management usually involves cosmetic de-bulking surgery (Fig 11-30).
**Fig 11-30** Fibrous dysplasia.
### _Albright's Syndrome_
This syndrome involves:
* Polyostotic fibrous dysplasia.
* 'Cafè au lait' pigmentation of skin and oral mucosa.
* Precocious puberty in females.
### _Paget's Disease of Bone_
Thought to have a viral aetiology (paramyxovirus), Paget's disease is a chronic deforming bone disease in which bone remodelling is chaotic. Progressive enlargement of the facial and skull bones arises, as do similar changes in the peripheral skeleton. Encroachment on cranial nerve fossae and canals can give rise to:
* Deafness.
* Visual abnormalities.
* Facial palsy.
* Motor defects.
As the bones thicken, secondary problems arise from intermittent cycles of bone resorption and deposition. These include:
* Pathological fractures.
* Disrupted occlusion.
* Hypercementosis.
* Ankylosis.
* Difficult extractions (due to the above).
* Post-extraction haemorrhage.
* Post-extraction infections.
* Root resorption.
Radiologically, areas of mixed osteosclerosis and osteoporosis give rise to a so-called 'cotton wool' appearance to bone (Fig 11-31).
**Fig 11-31** Paget's disease of bone.
### _Osteopetrosis_
Also called Albers-Schonberg disease (or marble bone disease) osteopetrosis is a defect of osteoclast activity that leads to excess bone deposition. The dense bone is prone to fracture and anaemia may arise as a secondary complication of marrow obliteration. Benign and malignant forms are described, the former presenting later in life whereas the latter is fatal in childhood. Radiologically bone is extremely dense and teeth may appear to disappear!
### _Hyperostosis_
Two forms are described:
* Endosteal (autosomal recessive).
* Infantile cortical (idiopathic).
The condition is beyond the scope of this book, but is included because it may be a differential diagnosis for thickened and sclerotic bone changes seen on radiographs of the skull and facial bones.
### Further Reading
Baxter AM, Roberts A, Shaw L, Chapple ILC. Localised scleroderma in a 12-year-old girl presenting as gingival recession. A case report and literature review. Dental Update 2001;28:458–462.
Chapple IL, Ord RA. Three cases of patent nasopalatine canals: case reports and a review of the literature. Oral Surgery, Oral Medicine, Oral Pathology 1990;69:554–558.
Chapple ILC, Manogue M. Management of recurrent follicular ameloblastoma. Dental Update 1991;18:309–312.
Chapple ILC, Thorpe GHG, Smith JM et al. Hypophosphatasia: a family study involving a case diagnosed from gingival crevicular fluid. Journal of Clinical Periodontology 1992;21:426–431.
Chapple ILC. Hypophosphatasia: dental aspects and mode of inheritance. Journal of Clinical Periodontology 1993;20:615–622.
Chapple I L C, Saxby M S, Murray J. Gingival haemorrhage, myelodysplastic syndromes and acute myeloid leukaemia. Journal of Periodontology 1999;70:1247–1253.
Devani P, Lavery K M, Howell C J T. 1998. Dental Extractions in Patients on Warfarin: Is Alteration of Anticoagulant Regime Necessary? British Journal of Oral and Maxillofacial Surgery 36: 107–111.
Gaspar, Brenner, Ardekian, Peled, Laufer. 1997. Use of Tranexamic Acid Mouthwash to Prevent Post-Operative Bleeding in Oral Surgery Patients on Oral Anticoagulant Medication. Quintessence International, 28: 375–379.
Diagnostic Imaging of the Jaws. 1995. Langlais R.P., Langland O.E. & Nortjé C.J. (Eds). Williams and Wilkins. IBSN: 0-683-04809-0.
Drugs Diseases and the Periodontium. 1992. Seymour RA. & Heasman P.H. (Eds). Oxford Medical Publications. ISBN: 0-19-261992-6.
Heasman P A, Smith D G, Martin I, Soames J. 2005. Carcinoma cuniculatum presenting as a gingival lesion. PERIO in Practice Today (in press).
Millet D T, Chapple I L C, Hirschman P, Corrigan M. 1990. Septic osteoradionecrosis of the mandible associated with pathological fracture: report of two cases. Journal of Clinical Radiology, 41: 408–410.
Oral Pathology (2nd Edition). 1993. Soames J. V. & Southam J. C. (Eds). Oxford Medical Publications. ISBN0 19 2622153.
Shear M. Cysts of the jaws: recent advances. 1985. Journal of Oral Pathology, 14: 43.
Souto J C, Oliver A et al. 1996. Oral Surgery in Anticoagulated Patients without Reducing the Dose of Oral Anticoagulant: A Prospective Randomised Study. Journal of Oral and Maxillofacial Surgery 54: 27–32.
Wahl M J. 2000. Myths of Dental Surgery in Patients Receiving Anticoagulant Therapy. Journal of the American Dental Association 131: 77–81.
Wysocki G P, Brannon R B, Gardner D C, Sapp P. 1980. Histogenesis of the lateral periodontal cyst and gingival cyst of the adult. Oral Surgery, Oral Medicine, Oral Pathology, 50: 327.
| {
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} | 1,339 |
John Connor is the Chief Executive Officer of the Carbon Market Institute (CMI) the independent and non-partisan peak industry body for business and climate in Australia.
Prior to joining CMI in May 2019, John Connor served for over two years as Executive Director of the Fijian Government's COP23 Presidency Secretariat. In that role he managed the Secretariat's strategic, policy and logistical support functions as Fiji presided over the UNFCCC, international climate, negotiations.
For the previous decade he was CEO of The Climate Institute of Australia, overseeing its focus on national policy, institutional investors and international climate negotiations.
In his time at the Climate Institute Connor oversaw the release of a series of reports on Australia's Climate Change response, including economic modeling on how Australia can reduce emissions and maintain a growing economy, up to date evidence on Australia's greenhouse pollution profile and analysis of community opinions on climate change and climate change solutions.
Initially trained as a lawyer and working in the Land and Environment Court of NSW he subsequently became a researcher for Peter MacDonald the Independent member for Manly. After that he spent three years running the Nature Conservation Council of NSW and in 1999 he took up the job as Campaigns Director for the Australian Conservation Foundation. He then worked as Campaigns Manager for World Vision, where he also co-convened Make Poverty History Australia. He was appointed CEO of the Climate Institute of Australia in 2007.
Connor has worked on numerous government and business advisory panels currently including the NSW Government's Climate Council. He is a graduate of the Australian Institute of Corporate Directors. He has been a board member of a number of NGOs and was a "Governator" with the Australian Youth Climate Coalition.
See also
Climate change in Australia
References
External links
Climate Institute of Australia
Australian chief executives
Living people
Year of birth missing (living people) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,207 |
from __future__ import print_function
import datetime
import os
import unittest
import six
from airflow import DAG, configuration, operators, utils
from airflow.utils.tests import skipUnlessImported
configuration.test_mode()
import os
import unittest
DEFAULT_DATE = datetime.datetime(2015, 1, 1)
DEFAULT_DATE_ISO = DEFAULT_DATE.isoformat()
DEFAULT_DATE_DS = DEFAULT_DATE_ISO[:10]
TEST_DAG_ID = 'unit_test_dag'
@skipUnlessImported('airflow.operators.mysql_operator', 'MySqlOperator')
class MySqlTest(unittest.TestCase):
def setUp(self):
configuration.test_mode()
args = {
'owner': 'airflow',
'mysql_conn_id': 'airflow_db',
'start_date': DEFAULT_DATE
}
dag = DAG(TEST_DAG_ID, default_args=args)
self.dag = dag
def mysql_operator_test(self):
sql = """
CREATE TABLE IF NOT EXISTS test_airflow (
dummy VARCHAR(50)
);
"""
import airflow.operators.mysql_operator
t = operators.mysql_operator.MySqlOperator(
task_id='basic_mysql',
sql=sql,
mysql_conn_id='airflow_db',
dag=self.dag)
t.run(start_date=DEFAULT_DATE, end_date=DEFAULT_DATE, force=True)
def mysql_operator_test_multi(self):
sql = [
"TRUNCATE TABLE test_airflow",
"INSERT INTO test_airflow VALUES ('X')",
]
import airflow.operators.mysql_operator
t = operators.mysql_operator.MySqlOperator(
task_id='mysql_operator_test_multi',
mysql_conn_id='airflow_db',
sql=sql, dag=self.dag)
t.run(start_date=DEFAULT_DATE, end_date=DEFAULT_DATE, force=True)
def test_mysql_to_mysql(self):
sql = "SELECT * FROM INFORMATION_SCHEMA.TABLES LIMIT 100;"
import airflow.operators.generic_transfer
t = operators.generic_transfer.GenericTransfer(
task_id='test_m2m',
preoperator=[
"DROP TABLE IF EXISTS test_mysql_to_mysql",
"CREATE TABLE IF NOT EXISTS "
"test_mysql_to_mysql LIKE INFORMATION_SCHEMA.TABLES"
],
source_conn_id='airflow_db',
destination_conn_id='airflow_db',
destination_table="test_mysql_to_mysql",
sql=sql,
dag=self.dag)
t.run(start_date=DEFAULT_DATE, end_date=DEFAULT_DATE, force=True)
def test_sql_sensor(self):
t = operators.sensors.SqlSensor(
task_id='sql_sensor_check',
conn_id='mysql_default',
sql="SELECT count(1) FROM INFORMATION_SCHEMA.TABLES",
dag=self.dag)
t.run(start_date=DEFAULT_DATE, end_date=DEFAULT_DATE, force=True)
@skipUnlessImported('airflow.operators.postgres_operator', 'PostgresOperator')
class PostgresTest(unittest.TestCase):
def setUp(self):
configuration.test_mode()
args = {'owner': 'airflow', 'start_date': DEFAULT_DATE}
dag = DAG(TEST_DAG_ID, default_args=args)
self.dag = dag
def postgres_operator_test(self):
sql = """
CREATE TABLE IF NOT EXISTS test_airflow (
dummy VARCHAR(50)
);
"""
import airflow.operators.postgres_operator
t = operators.postgres_operator.PostgresOperator(
task_id='basic_postgres', sql=sql, dag=self.dag)
t.run(start_date=DEFAULT_DATE, end_date=DEFAULT_DATE, force=True)
autocommitTask = operators.postgres_operator.PostgresOperator(
task_id='basic_postgres_with_autocommit',
sql=sql,
dag=self.dag,
autocommit=True)
autocommitTask.run(
start_date=DEFAULT_DATE,
end_date=DEFAULT_DATE,
force=True)
def postgres_operator_test_multi(self):
sql = [
"TRUNCATE TABLE test_airflow",
"INSERT INTO test_airflow VALUES ('X')",
]
import airflow.operators.postgres_operator
t = operators.postgres_operator.PostgresOperator(
task_id='postgres_operator_test_multi', sql=sql, dag=self.dag)
t.run(start_date=DEFAULT_DATE, end_date=DEFAULT_DATE, force=True)
def test_postgres_to_postgres(self):
sql = "SELECT * FROM INFORMATION_SCHEMA.TABLES LIMIT 100;"
import airflow.operators.generic_transfer
t = operators.generic_transfer.GenericTransfer(
task_id='test_p2p',
preoperator=[
"DROP TABLE IF EXISTS test_postgres_to_postgres",
"CREATE TABLE IF NOT EXISTS "
"test_postgres_to_postgres (LIKE INFORMATION_SCHEMA.TABLES)"
],
source_conn_id='postgres_default',
destination_conn_id='postgres_default',
destination_table="test_postgres_to_postgres",
sql=sql,
dag=self.dag)
t.run(start_date=DEFAULT_DATE, end_date=DEFAULT_DATE, force=True)
def test_sql_sensor(self):
t = operators.sensors.SqlSensor(
task_id='sql_sensor_check',
conn_id='postgres_default',
sql="SELECT count(1) FROM INFORMATION_SCHEMA.TABLES",
dag=self.dag)
t.run(start_date=DEFAULT_DATE, end_date=DEFAULT_DATE, force=True)
@skipUnlessImported('airflow.operators.hive_operator', 'HiveOperator')
@skipUnlessImported('airflow.operators.postgres_operator', 'PostgresOperator')
class TransferTests(unittest.TestCase):
cluster = None
def setUp(self):
configuration.test_mode()
args = {'owner': 'airflow', 'start_date': DEFAULT_DATE}
dag = DAG(TEST_DAG_ID, default_args=args)
self.dag = dag
def test_clear(self):
self.dag.clear(
start_date=DEFAULT_DATE,
end_date=datetime.datetime.now())
def test_mysql_to_hive(self):
# import airflow.operators
from airflow.operators.mysql_to_hive import MySqlToHiveTransfer
sql = "SELECT * FROM baby_names LIMIT 1000;"
t = MySqlToHiveTransfer(
task_id='test_m2h',
mysql_conn_id='airflow_ci',
hive_cli_conn_id='beeline_default',
sql=sql,
hive_table='test_mysql_to_hive',
recreate=True,
delimiter=",",
dag=self.dag)
t.run(start_date=DEFAULT_DATE, end_date=DEFAULT_DATE, force=True)
def test_mysql_to_hive_partition(self):
from airflow.operators.mysql_to_hive import MySqlToHiveTransfer
sql = "SELECT * FROM baby_names LIMIT 1000;"
t = MySqlToHiveTransfer(
task_id='test_m2h',
mysql_conn_id='airflow_ci',
hive_cli_conn_id='beeline_default',
sql=sql,
hive_table='test_mysql_to_hive_part',
partition={'ds': DEFAULT_DATE_DS},
recreate=False,
create=True,
delimiter=",",
dag=self.dag)
t.run(start_date=DEFAULT_DATE, end_date=DEFAULT_DATE, force=True)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,080 |
Manchester City's plans for Phil Foden revealed amid Leeds loan rumours
Manchester City sensation Phil Foden has been heavily linked with a loan move to Leeds United in recent days, following the high-profile arrival of Marcelo Bielsa at Elland Road.
Prior to those rumours, City's stance on Foden had been consistent – that he would be going nowhere next season with Pep Guardiola believing he would develop best kept close by.
With Bielsa having had a big influence on Pep's career, the rumour of Foden joining Leeds sounded like it may have had some merit, with 'El Loco' the sort of coach renowned for developing young players that Guardiola may trust to develop Foden's skill set.
However, according to the Manchester Evening News, the speculation is just that – and Phil Foden will not be joining Leeds or anyone else on loan this summer.
The 18-year-old is poised to play a part in City's 2018-19 season and will hope to improve on his 10 appearances in the Blues' double-winning campaign.
Stockport-born Foden is expected to sign a new long-term contract at the Etihad Stadium before the start of the new season.
Another of City's talented England youth internationals, goalkeeper Angus Gunn, is still on Leeds' radar and appears to have a more realistic chance of heading to the Championship side than Foden.
Related Items:phil foden
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Pep Guardiola has a blunt response to suggestions of Phil Foden going out on loan
Celebrity City fan Noel Gallagher makes interesting Jorginho comment after concert in Naples
From Italy: Jorginho can be considered a Manchester City player, when he could be announced… | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,480 |
Eugeniusz Pękała (ur. 1 stycznia 1925 w Myślachowicach, zm. 22 listopada 2007 w Warszawie) – funkcjonariusz kontrwywiadu i wywiadu PRL, dyplomata.
Syn Rudolfa i Otylii. Był członkiem oddziału partyzanckiego AL im. Ludwika Waryńskiego na Podhalu (1944–1945). W resorcie MBP od 5 lutego 1945, m.in. jako funkcjonariusz na kierowniczych stanowiskach w Wojewódzkim Urzędzie Bezpieczeństwa Publicznego (WUBP/WUdsBP) w Krakowie (1946–1955); w tym okresie był też słuchaczem Centralnej Szkoły Centrum Wyszkolenia MBP w Legionowie (1947–1948). Następnie przeszedł do wywiadu - był kier. rezydentury w Berlinie (1955–1956), kier. rezydentury w Ambasadzie PRL w Bernie (1956–1960), funkcjonariuszem Departamentu I MSW (1960–1965), zastępcą dyrektora Departamentu I MSW - wywiadu w randze płk. (1965–1975), szefem Polskiej Misji Wojskowej w Berlinie Zachodnim (1975–1979). Po przejściu na emeryturę był zatrudniony w Dziale Dokumentacji PAP. Pochowany na cmentarzu Powązki Wojskowe w Warszawie (kwatera B16-4-24).
Bibliografia
Aparat Bezpieczeństwa w Polsce, Kadra kierownicza 1944-1956, t. I, IPN Warszawa 2005
Aparat Bezpieczeństwa w Polsce, Kadra kierownicza 1956-1975, t. II, IPN Warszawa 2006
Paweł Piotrowski: Formy działalności operacyjnej wywiadu cywilnego PRL. Instrukcja o pracy wywiadowczej Departamentu I MSW z 1972 r., Aparat represji w Polsce Ludowej 1944–1989, nr 1 (5)/2007
Linki zewnętrzne
Biogram IPN
Przypisy
Funkcjonariusze Departamentu I MSW PRL
Funkcjonariusze wywiadu cywilnego Polski Ludowej
Pochowani na Powązkach-Cmentarzu Wojskowym w Warszawie
Szefowie Polskiej Misji Wojskowej w Niemczech
Ambasadorowie PRL
Urodzeni w 1925
Zmarli w 2007 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8 |
If you have been conscious of your health and well-being, you probably could be ordering organic foods in the groceries store; organic mattresses are their equivalent when it comes to sleeping options. When it comes to the manufacture of mattresses, makers of organic mattresses avoid adding chemicals and other potentially harmful synthetic materials onto the mattresses. Instead, the mattresses are made of organic wool and organic latex among other organic materials. The exact way organic fruits are free from pesticide and fertilizer residues, organic mattresses don't have chemicals that are often found in traditional mattresses. For instance, organic wool and organic cotton are the two commonly used padding materials in organic mattresses which have a natural source. These raw materials are also mechanically processed, thus eliminating the need to use toxic substances that may be detrimental to the human health. As a result, therefore, you will be more protected from respiratory diseases, skin irritation and neurological diseases which are often associated with the chemicals used in the manufacture of traditional mattresses.
It is also important to note that some organic mattress brands do use materials that are not entirely organic but are more natural than those found in traditional mattresses.
All mattresses manufactured or entering the United States are required to pass flammability tests before being sold out to consumers. Mattress makers employ different mechanisms of passing the tests. One ingenious way traditional mattress makers use to pass the test is to add fire retardants such as brominated stuffing, antimony, boric acid and chlorinated foams which are all toxic and can cause reactions to some people. Besides, such fire retardants have been found to be leading causes of kidney problems, neurological diseases, and respiratory complications.
Organic mattresses have a better way of passing fire tests. Instead of adding toxic fire retardants, makers of these mattresses use wool layers, Kevlar and inert fiberglass among other non-toxic materials that help the mattress pass the fire tests. As a result, organic mattresses are not only good for your health but also help keep the environment clean.
Every time you purchase an organic mattress, bear in mind that you are supporting sustainable production processes. It starts with the farmer. Cotton and latex rubber growers strive not to use pesticides and fertilizers that might add unwanted toxic elements in the final harvest. The organic mattress manufacturers use the harvested cotton and rubber to make healthy and environmental friendly mattresses. This is a great way of encouraging sustainable production, healthy lifestyles, and a lively planet.
One pressing problem with traditional memory foam mattresses is that they tend to trap body moisture. This is why you often feel hot, sweaty and sticky while using your traditional mattress for sleep. The blame is shared equally on the coverings and material used to make the mattress. For instance, plenty of traditional mattresses have plastic polyesters and other synthetic foams incorporated into them. These materials aren't porous enough to help circulate air. As a result, your skin's pores would have nowhere to breathe and release the sweat. That's why sleeping on a traditional mattress could make you feel like as though you're roasting or you are in a sauna right on your bed!
Organic mattresses are less of a problem when it comes to temperature regulations. The cotton used in organic mattresses is great for your skin pores and helps keep you cool. Horsehair and wool are often mixed with cotton to act as natural wicking fibers. Lying on an organic mattress at night, therefore, keeps your body at the right temperature, humidity and circulates air to keep you comfortable.
Frankly, organic mattresses are a little expensive compared to traditional mattresses. But they are a great investment whose returns can be felt for a long time. Organic mattresses have a longer warranty, retain support and feel good for a longer time compared to traditional mattresses. Over the lifespan of a typical organic mattress, you can buy and replace several other traditional mattresses. In these days of cost-cutting and saving every penny at any slightest opportunity you get, buying an organic mattress is the best thing that can ever cross your mind.
Imagine lying on a mattress that isn't pumping carcinogenic vapors into your lungs. Lying on organic mattresses removes the health worries that are brought about with traditional mattresses. Worry can affect the quality of sleep you can get. The safety of organic mattress, therefore, gives you fewer things to worry about and makes you sleep longer.
The density of a mattress has a profound impact on its durability and comfort. A typical organic mattress with just 5.4 pounds in density can last between 40 and 50 years. A traditional mattress of similar density will last far less than that and won't have such texture for comfort. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,535 |
Netflix shares first look and release date for The Politician season 2
By Daniel Megarry
The Politician will be back on our screens very soon.
When it aired on Netflix last year, the star-studded series from American Horror Story creator Ryan Murphy brought us LGBTQ characters, whip-smart dialogue, and hundreds of memes after Oscar-winner Jessica Lange hilariously defined the gay agenda.
While the first season focused on Payton's (Ben Platt) run for student body president, the final episode took viewers three years into the future, where Payton decided to run for state senate against characters played by Bette Midler and Judith Light.
Despite the coronavirus pandemic affecting many new releases, Ryan recently confirmed production on a second season of The Politician had wrapped up just before things got bad, and now he's given us a confirmed premiere date: 19 June.
He's also shared a first-look poster of the new season, which appears to confirm the return of Gwyneth Paltrow as Payton's mother – we weren't sure she would be a part of it after eloping at the end of the show's first season.
https://www.instagram.com/p/CAVO-_gJAYi/
Ryan previously confirmed that the second season will pick up right where the first left off, at the beginning of Payton's "campaign against the Judith Light juggernaut" which was teased in season one's final episode.
"I love how Payton has grown up. He's now in college, and the best thing about the season is the Ben Platt versus Bette Midler and Judith Light aspect. It feels very adult, it feels very topical," he said.
"It's sort of a story about baby boomers asking themselves, 'Is it time for us to pass the power that we have onto the next generation or are they too dumb to figure it out yet?' And I think you can see that battle playing out daily in our political landscape.
"And it's what it's about, and it's a very cutthroat race that they run. It feels much more adult, much more sexualized. It's really great. And Ben, but particularly Judith and Bette Midler really, I think, their performances are extraordinary." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,375 |
\section{Introduction}
An {\em extended formulation} (shorthand: {\em EF}) of a polytope $P \subseteq \mathbb{R}^d$ is a system of linear constraints
\begin{equation}
\label{eq:EF}
E^\leqslant x + F^\leqslant y \leqslant g^\leqslant, \quad
E^= x + F^= y = g^=
\end{equation}
with $(x,y) \in \mathbb{R}^{d+k}$ such that $x \in \mathbb{R}^d$ belongs to $P$ if and only if there exists $y \in \mathbb{R}^k$ such that $(x,y)$ satisfies \eqref{eq:EF}. An extended formulation of $P$ is simply a linear description of $P$ in an extended space. Geometrically, $P$ is described as the projection of the polyhedron\footnote{We remark that although we allow for now $Q$ to be unbounded, we will soon show that one can restrict to the case where $Q$ is bounded, that is, a polytope.} $Q \subseteq \mathbb{R}^{d+k}$ defined by \eqref{eq:EF}. More generally, we call a polyhedron $Q \subseteq \mathbb{R}^e$ an {\em extension} (or {\em lift}) of $P$ if there exists an affine map $\pi : \mathbb{R}^e \to \mathbb{R}^d$ such that $\pi(Q) = P$.
Consider a linear description $Ax \leqslant b$ of $P$ in its original space. If $f : \mathbb{R}^d \to \mathbb{R}$ is any function, then
\begin{equation}
\label{eq:opt_problem_EF}
\sup \setDef{f(x)}{Ax \leqslant b} = \sup \setDef{f(x)}{E^\leqslant x + F^\leqslant y \leqslant g^\leqslant, \ E^= x + F^= y = g^=}\,.
\end{equation}
Thus every optimization problem on $P$ can be reformulated as an optimization problem over any extension of $P$. This is why extended formulations are interesting for optimization: in \eqref{eq:opt_problem_EF}, the number of constraints in the right-hand side can be much smaller than the number of constraints in the left-hand side.
We define the {\em size} of an extended formulations as its number of inequalities, and the size of an extension as its number of facets; these turn out to be the right measures of size. Note that the size of an extended formulation is at least the size of the associated extension because every facet of a polyhedron is part of every linear description of this polyhedron (in the space in which it is defined), and every extension corresponds to an extended formulation with exactly its size.
The field of extended formulations is attracting more and more attention. In particular, size lower-bounding techniques are becoming increasingly powerful and diverse, see, e.g., \cite{Yannakakis91,
KaibelPashkovichTheis10,GP12,FioriniKaibelPashkovichTheis11,extform4,BFPS12,BM13,BP2013commInfo}. The reader will find in the surveys~\cite{CCZ10,Kaibel11,Wolsey11} a good description of the field as it was a few years ago.
In this paper, we study some restricted forms of extended formulations (extensions) which we call {\em flow-based extended formulations (extensions)}, see Section \ref{sec:flow-based_EFs} for a definition. Informally, a flow-based extension of a polytope $P$ is another polytope $Q$ that can be realized as the convex hull of all flows in some network. This definition is inspired by the prominent role played by network flows in discrete optimization: many algorithms and structural results crucially rely on network flows~\cite{AhujaMagnantiOrlinBook,SchrijverBook}. Quite a lot of known extended formulations are based on network flows, such as those obtained from dynamic programming algorithms~\cite{Martin91}.
Here, we focus on uncapacitated networks. Our main contribution is to prove size lower bounds of the form $2^{\Omega(n)}$ for uncapacitated flow-based extended formulations of several polytopes, such as the perfect matching polytope of (bipartite and non-bipartite) complete graphs and the traveling salesman polytope of the complete graph. Our results are summarized in Table~\ref{tab:results}. Below, the notations $O^*(\cdot)$, $\Omega^*(\cdot)$ and $\Theta^*(\cdot)$ have the same meaning as the usual notations $O(\cdot)$, $\Omega(\cdot)$ and $\Theta(\cdot)$, except that polynomial factors are ignored.
\begin{table}[ht]
\centering
\begin{tabular}{|r||c|c|}
\hline
Polytope & Size bounds for general EFs & Size bounds for flow-based EFs\\
\hline
\hline
$\mathop{\mathrm{P}_\mathrm{perfect\ matching}}(K_{n,n})$ & $\Theta(n^2)$~\cite{Birkhoff} & $\mathbf{\Theta^*(2^n)}$\\
\hline
$\mathop{\mathrm{P}_\mathrm{perfect\ matching}}(K_n)$ & $\Omega(n^2)$, $O^*(2^{\frac{n}{2}})$~\cite{KaibelPashkovichTheis10,FaenzaFioriniGrappeTiwary11}
& $\mathbf{\Omega^*(2^{\frac{n}{2}})}$, $\mathbf{O(2^{0.695n})}$\\
\hline
$\mathop{\mathrm{P}_\mathrm{traveling\ salesman}}(K_n)$ & $2^{\Omega(\sqrt{n})}$~\cite{extform4}, $O^*(2^n)$~\cite{heldKarp70} &$\mathbf{\Omega^*(2^{\frac{n}{4}})}$, $O^*(2^n)$~\cite{heldKarp70}\\
\hline
\end{tabular}
\caption{Table of results. New results are indicated in boldface. The bounds for flow-based EFs assume that the network is uncapacitated.} \label{tab:results}
\end{table}
Before giving an outline of the paper, we briefly discuss our motivations. Lower bounds on restricted types of extended formulations have been studied by quite many authors, starting with the work of Yannakakis~\cite{Yannakakis91} on symmetric extended formulations. There has been work on hierarchies such as the Sherali-Adams~\cite{SheraliAdams1990} and Lov\'asz-Schrijver hierarchies~\cite{LovaszSchriver1991}, see, e.g., \cite{BGHMT2006,STT2007,FernandezdelaVegaMathieu2007,CMM2009,GMT2009,BenabbasMagen2010}; further work on symmetric extended formulations~\cite{KaibelPashkovichTheis10,Pashkovich12,GP12} and also work on extended formulations from low variance protocols~\cite{FaenzaFioriniGrappeTiwary11}.
We think that the restriction of being flow-based is as natural as the restrictions studied in the aforementioned papers. Combinatorial optimization offers a variety of modeling tools beyond flows, which are the most basic and important modeling tool: e.g., matchings, polymatroids and polymatroid intersections~\cite{SchrijverBook}. It seems a worthy research goal to characterize the expressivity of these modeling tools, and give theoretical explanations of the fact that some problems can be efficiently expressed by some modeling tools and not by others. This paper is a first step in that direction.
Of particular interest are {\em separations} between modeling tools. It is striking that all our lower bounds rely on a separation between uncapacitated and capacitated flows: while the perfect matching polytope of the complete bipartite graph $K_{n,n}$ has a $O(n^2)$-size capacitated flow-based extended formulation, we show a $\Omega^*(2^n)$ lower bound on the size of every uncapacitated flow-based extended formulations of that polytope. Via reductions, we derive from this the other lower bounds reported in Table~\ref{tab:results}.
We conclude this discussion by focussing on the traveling salesman polytope. Held and Karp~\cite{heldKarp70} gave a $O^*(2^n)$-complexity dynamic programming algorithm for the traveling salesman problem based on subsets. In our terminology, this yields a $O^*(2^n)$-size uncapacitated flow-based extended formulation for the traveling salesman polytope. In a survey paper on exact algorithms for combinatorial optimization problems, Woeginger~\cite{Woeginger03} stated as an open problem the question of determining if the traveling salesman problem has an exact algorithm of complexity $(2-\varepsilon)^n$ for some $\varepsilon > 0$. The question was answered affirmatively by Bjorklund~\cite{Bjorklund10}, at least if one tolerates randomized algorithms with small failure probability and restricts to instances where the coefficients are bounded. Our $\Omega^*(2^{\frac{n}{4}})$ lower bound for uncapacitated flow-based extended formulations for the traveling salesman polytope also applies to dynamic programming algorithms for the traveling salesman problem, which sheds some light on Woeginger's question.
The rest of the paper is organized as follows. We begin with preliminaries in Section~\ref{sec:preliminaries}: after introducing some notations, we define convex polytopes in general as well as the particular convex polytopes studied here. Then, in Section~\ref{sec:flow-based_EFs}, we formally define flow-based extended formulations, discuss an example and establish basic properties of flow-based extended formulations, focussing on the uncapacitated case. Finally, in Section \ref{sec:lower_bounds}, we prove size bounds for uncapacitated flow-based extended formulations described in Table \ref{tab:results}.
\section{Preliminaries} \label{sec:preliminaries}
Let $I$ be a finite ground set. The {\sl incidence vector} of a subset $J \subseteq I$ is the vector $\chi^J \in \mathbb{R}^I$ defined as
\[
\chi^J_i = \left\{
\begin{array}{l l}
1 & \quad \text{if } i \in J\\
0 & \quad \text{if } i \notin J
\end{array} \right.
\]
for $i \in I$. For $x \in \mathbb{R}^I$, we let $x(J) := \sum_{i \in J} x_i$.
First, let $G = (V,E)$ be an undirected graph. For a subset of vertices $U\subseteq V$, we denote as $\delta(U)$ the set of edges of $G$ with exactly one endpoint in $U$. So,
\begin{eqnarray*}
\delta(U) &= &\{uv \in E : u \in U, v \notin U\}\ .
\end{eqnarray*}
Now, let $N=(V,A)$ be a directed graph. For $U \subseteq V$, we denote by $\delta^+(U)$ the set of arcs of $N$ with tail in $U$ and head in $V \setminus U$, and by $\delta^{-}(U)$ the set of arcs of $N$ with head in $U$ and tail in $V\setminus U$, i.e.
\begin{eqnarray*}
\delta^+(U) &= &\{(u,v) \in A : u\in U, v\notin U\}\ , \text{ and}\\
\delta^-(U) &= &\{(v,u) \in A : u\in U, v\notin U\}\ .
\end{eqnarray*}
As usual, for $v \in V$, we use the shortcuts $\delta(v)$, $\delta^+(v)$ and $\delta^-(v)$ for $\delta(\{v\})$, $\delta^+(\{v\})$ and $\delta^-(\{v\})$ respectively.
\subsection{Convex Polytopes and Polyhedra} \label{sec:convex_polytopes}
A {\em (convex) polytope} is a set $P \subseteq \mathbb{R}^d$ that is the convex hull of a finite set of points in $\mathbb{R}^d$. Equivalently, $P \subseteq \mathbb{R}^d$ is a polytope if and only if $P$ is bounded and the intersection of a finite collection of closed halfspaces. This is equivalent to saying that $P$ is bounded and the set of solutions of a finite system of linear inequalities (or equalities, each of which can be represented by a pair of inequalities). A {\em (convex) polyhedron} is similar to a polytope, except that it may be unbounded. Formally, a polyhedron $Q \subseteq \mathbb{R}^d$ is any set that can be represented as the Minkowski sum of a polytope and a polyhedral cone or, equivalently, as the intersection of a finite collection of closed halfspaces. For more background on polytopes and polyhedra, see the standard reference~\cite{Ziegler}.
\subsection{Perfect Matching Polytope} \label{sec:perfect_matching_polytope}
A {\sl perfect matching} of an undirected graph $G=(V,E)$ is set of edges $M \subseteq E$ such that every vertex of $G$ is incident to exactly one edge in $M$. The {\sl perfect matching polytope} of the graph $G$ is the convex hull of the incidence vectors of the perfect matchings of $G,$ i.e.,
$$
\mathop{\mathrm{P}_\mathrm{perfect\ matching}}(G) = \mathop{\mathrm{conv}}\{\chi^M \in\mathbb{R}^E : M~ \text{perfect matching of}~ G\}\ .
$$
Edmonds \cite{Edmonds65} showed that the perfect matching polytope of $G$ is described by the following system of linear constraints (see also \cite{SchrijverBookA03}, page 438):
\begin{eqnarray}
\label{eq:odd_cut} x(\delta(U)) &\geqslant &1 \quad \text{for } U\subseteq V \text{ with } |U| \text{ odd}\ ,\\
\nonumber x(\delta(v)) &= &1 \quad \text{for } v \in V\ ,\\
\nonumber x_e &\geqslant& 0 \quad \text{for } e\in E\ .
\end{eqnarray}
In the case where the graph $G$ is bipartite, that is, when the vertex set $V$ can be partitioned into two sets $A$ and $B$ such that every edge in $E$ has an endpoint in $A$ and the other in $B$, the odd cut inequalities \eqref{eq:odd_cut} may be dropped~\cite{Birkhoff}. Thus the perfect matching polytope of a bipartite graph $G$ is described as follows:
\begin{eqnarray*}
x(\delta(v)) &= &1 \quad \text{for } v \in V\ ,\\
x_e &\geqslant& 0 \quad \text{for } e\in E\ .
\end{eqnarray*}
\subsection{Traveling Salesman Polytope} \label{sec:traveling_salesman_polytope}
A {\sl Hamiltonian cycle} of $G=(V,E)$ is a connected subgraph of $G$ such that every vertex of $G$ is incident to exactly two edges in $C$. The {\sl traveling salesman polytope} of the graph $G$ is the convex hull of the incidence vectors of the hamiltonian cycles of $G,$ i.e.,
$$
\mathop{\mathrm{P}_\mathrm{traveling\ salesman}}(G) = \mathop{\mathrm{conv}}\{\chi^{E(C)} \in\mathbb{R}^E : C~ \text{Hamiltonian cycle of}~ G\}\ .
$$
In the formula above, $E(C)$ denotes the edge set of Hamiltonian cycle $C$.
No linear description of the traveling salesman polytope of the complete graph $K_n$ is known. Moreover no ``reasonable'' linear description of this polytope should be expected unless $\mathcal{NP}=\text{co-}\mathcal{NP}$ (see Corollary 5.16a \cite{SchrijverBookA03}).
\subsection{Flow Polyhedron} \label{sec:flow_polyhedra}
Let $N = (V,A)$ be a network with source node $s \in V$, sink node $t \in V \setminus \{s\}$ and arc capacities $c_a \in \mathbb{R}_+ \cup \{\infty\}$ for $a \in A$. An $s$--$t$ {\em flow} of value $k$ is a vector $\phi \in \mathbb{R}^A$ satisfying
\begin{eqnarray}
\label{eq:flow_balance}
\phi(\delta^+(v)) - \phi(\delta^-(v)) &= &0 \quad \forall v \in V \setminus \{s,t\},\\
\label{eq:flow_value}
\phi(\delta^+(s)) - \phi(\delta^-(s)) &= &k,\\
\label{eq:flow_lb}
\phi_a &\geqslant &0 \quad \forall a \in A,\\
\label{eq:flow_ub}
\phi_a &\leqslant &c_a \quad \forall a \in A.
\end{eqnarray}
For a fixed $k \in \mathbb{R}$, the set of all $s$--$t$ flows of value $k$ in network $N$ defines a polyhedron $Q = Q(V,A,s,t,k,c)$ that we call {\em flow polyhedron}.
In this paper, we will assume most of the time that the network is {\em uncapacitated}, that is, $c_a = \infty$ for all $a \in A$. This amounts to ignoring the upper bound inequalities \eqref{eq:flow_ub}.
\section{Flow-based Extended Formulations} \label{sec:flow-based_EFs}
\subsection{Definition} \label{sec:flow-based_EFs_def}
Consider again a network $N = (V,A)$ with source node $s \in V$, sink node $t \in V \setminus \{s\}$, arc capacities $c_a \in \mathbb{R}_+ \cup \{\infty\}$ for $a \in A$ and flow value $k \in \mathbb{R}_+$. We say that the flow polyhedron $Q = Q(V,A,s,t,k,c)$ is a {\em flow-based extension} of a given polytope $P$ in $\mathbb{R}^d$ if there exists a linear projection $\pi : \mathbb{R}^A \to \mathbb{R}^d$ such that $\pi(Q) = P$.
A flow-based extension is said to be {\em uncapacitated} if the associated network is uncapacitated.
From now on, we will always assume that the projection $\pi$ is linear. This causes essentially no loss of generality because an affine projection can be made linear at the cost of adding one new arc $(s',s)$ to the network and moving the source to the node $s'$. We denote by $M \in \mathbb{R}^{d \times A}$ the matrix of projection $\pi$, that is, the matrix $M\in \mathbb{R}^{d \times A}$ such that $\pi(\phi) = M\phi$ for all $\phi \in \mathbb{R}^A$.
Moreover, we denote by $F \in \mathbb{R}^{(V \setminus \{s,t\}) \times A}$ the coefficient matrix of the flow balance equations. In other words, $F\phi = 0$ is the matrix form of \eqref{eq:flow_balance}. Then, the flow-based extension $Q$ can be described algebraically as:
\begin{equation}
\label{eq:flow_EF}
x = M\phi,\ F\phi = 0,\ \phi(\delta^+(s)) - \phi(\delta^-(s)) = k,\ 0 \leqslant \phi \leqslant c,
\end{equation}
We call system~\eqref{eq:flow_EF} a {\em flow-based extended formulation} of $P$.
Notice that in the uncapacitated case, the size (that is, number of inequalities) of a flow-based extended formulation is exactly the number of arcs in the corresponding network.
Notice also that in the uncapacitated case, we can assume that $k = 1$ without loss of generality. This is because changing $k$ to $1$ simply amounts to replacing $Q$ by $(1/k)Q$. Indeed, if $\pi : \mathbb{R}^A \to \mathbb{R}^d$ projects $Q$ to $P$, then $\pi' : \mathbb{R}^A \to \mathbb{R}^d : \phi \mapsto \pi'(\phi) := \pi(k \phi)$ projects $(1/k)Q$ to $P$. (In case $k = 0$, $Q$ is just a point. We will ignore this case in what follows.)
We will prove below that in the uncapacitated case, we can furthermore assume that $N$ is acyclic, provided $\varnothing \subsetneq P \subseteq \mathbb{R}^d_+$. In this case, $Q$ is a polytope and its vertices are the characteristic vectors $\chi^\sigma$ of all directed $s$--$t$ paths $\sigma$ in network $N$ (this follows from the well-known fact that the system \eqref{eq:flow_balance}--\eqref{eq:flow_lb} defining $Q$ is totally unimodular). We call such an extension an {\em $s$--$t$ path extension}, any corresponding extended formulation an {\em $s$--$t$ path extended formulation} and define the {\em $s$--$t$ path extension complexity} $\xc_\text{$s$--$t$ path}(P)$ of a polytope $P$ as the minimum number of arcs of a network whose $s$--$t$ path polytope is an extension of $P$. We will show that this is also the minimum size of an uncapacitated flow-based extended formulation of $P$.
\subsection{Example: Regular Languages}
In order to convince the reader that $s$--$t$ path extensions are quite powerful, we now discuss an illustrating example that generalizes Carr and Konjevod's flow-based extended formulation of the convex hull of even 0/1-vectors in $\mathbb{R}^n$~\cite{CK04}.
Consider a {\sl deterministic finite automaton} $M$ over the alphabet $\{0,1\}$, that is, a $4$-tuple $(Q,\delta,q_0,F)$ where $Q$ is now a (nonempty) finite set of {\sl states}, $\delta : Q \times \{0,1\} \to Q$ is the {\sl transition function}, $q_0 \in Q$ is the {\sl initial state} and $F \subseteq Q$ is the set of {\sl accept states}. For a given input word $x = x_1 x_2 \cdots x_n$ in $\{0,1\}^*$, the automaton $M$ performs a computation starting at the initial state $q_0$ and in which the state $q_{i}$ ($i \in [n]$) is determined by the previous state $q_{i-1}$ and the $i$th letter $x_i$ of word $x$ through the equation $q_{i} = \delta(q_{i-1},x_i)$. The automaton is said to {\sl accept} $x$ if the final state $q_n$ is an accept state, that is, $q_n$ belongs to $F$.
The automaton $M$ defines a language $L = L(M)$ over $\{0,1\}$ consisting of all words $x \in \{0,1\}^*$ accepted by $M$. Such a language is said to be {\sl regular}. Now pick a positive integer $n$, and consider a word $x = x_1x_2 \cdots x_n$ of length $n$ in $L$. Treating each letter of word $x$ as belonging to a different coordinate, we see that $x$ defines a $0/1$-vector $(x_1,x_2,\ldots,x_n)^\intercal$ in $\mathbb{R}^n$. By taking the convex hull of all $0/1$-vectors corresponding to all words of length $n$ in $L$, we obtain a $0/1$-polytope $P_n(L)$ in $\mathbb{R}^n$.
As we show now, one can easily construct compact flow-based extended formulations for such $0/1$-polytopes.
\begin{prop}
Let $L$ denote a regular language over $\{0,1\}$ and $M = (Q,\delta,q_0,F)$ any deterministic finite automaton recognizing the language $L$. For each positive integer $n$, there exists an $s$--$t$ path extended formulation of $P_n(L)$ with size at most $2|Q|n$.
\end{prop}
\begin{proof}
We define a network $N$ from automaton $M$. Besides source node $s$ and sink node $t$, network $N$ has $n-1$ nodes $(q,1)$, \ldots, $(q,n-1)$ for each state $q \in Q$. To simplify notations, we also denote $s$ by $(q_0,0)$. This defines the node set $V$ of $N$. For $i \in [n-1]$, we connect node $(q,i-1)$ to each of the nodes $(\delta(q,0),i)$ and $(\delta(q,1),i)$ by an arc. Moreover, for each transition $q ' = \delta(q,\sigma)$ with $q' \in F$ we add an arc from node $(q,n-1)$ to sink node $t$. This defines the arc set $A$ of $N$. See Figure \ref{fig:even} for an example. In a formula, we have (with a slight abuse of notation because the network can have parallel arcs)
\begin{eqnarray*}
V &= &\{\underbrace{(q_0,0)}_{= s}\} \cup \setDef{(q,i)}{q \in Q, i \in [n-1]} \cup \{t\},\\
A &= &\setDef{((q,i-1),(\delta(q,\sigma),i))}{(q,i-1) \in N, i \in [n-1], \sigma \in \{0,1\}}\\
&&\mbox{} \cup \setDef{((q,n-1),t)}{\exists \sigma \in \{0,1\} : \delta(q,\sigma) \in F}\,.
\end{eqnarray*}
Each arc $a \in A$ corresponds to a transition $q' = \delta(q,\sigma)$, and is said to carry the label $\sigma \in \{0,1\}$. Thus the label carried by an arc is the symbol that caused the transition.
\begin{figure}[ht]
\centering
\input{even.eps_t}
\caption{Deterministic finite automaton (left) and corresponding network (right).}
\label{fig:even}
\end{figure}
In the network $N=(V,A)$, we send $k = 1$ units of flow from $s$ to $t$, setting all capacities $c_a$ to $\infty$. The column of the projection matrix corresponding to arc $a \in A$ from node $(q,i-1)$ is the $0/1$-vector $(0,\ldots,0,\sigma,0,\ldots,0)^\intercal$ with $\sigma$ in position $i$ and $0$ everywhere else, where $\sigma \in \{0,1\}$ is the label carried by arc $a$. We leave it to the reader to perform the straightforward check that this defines an $s$-$t$ path extended formulation of $P_n(L)$.
The size of this extended formulation is the number of arcs in the network, that is,
$$
2 + 2|Q|(n-1) \leqslant 2 |Q| n.
$$
\end{proof}
\subsection{Basic Properties}
\subsubsection{Nonnegativity of the Projection}
A linear projection $\pi : \mathbb{R}^A \to \mathbb{R}^d$ is called {\em nonnegative} if its projection matrix is (entry-wise) nonnegative.
\begin{lem} \label{lem:nonnegative_pi}
For every uncapacitated flow-based extension $Q \subseteq \mathbb{R}^A$, $\pi : \mathbb{R}^A \to \mathbb{R}^d$ of a polytope $P \subseteq \mathbb{R}^d_+$, there is a nonnegative linear projection $\pi' : \mathbb{R}^A \to \mathbb{R}^d$ such that $\pi'(Q) = P$.
\end{lem}
\begin{proof}
As above, let $M$ denote the matrix of $\pi$. It suffices to show that for every row $M_i$ of the matrix $M$ there exists a row vector $\Lambda_i \in (\mathbb{R}^{V \setminus \{s,t\}})^*$ such that $M_i + \Lambda_i F \geqslant 0$, since due to~\eqref{eq:flow_EF} the system $F\phi = 0$ holds for all $\phi\in Q$ and thus $(M + \Lambda F) \phi = M \phi + \Lambda F \phi = M\phi$.
Suppose, for the sake of contradiction, that no such $\Lambda_i$ exists for some $i$. Then by Farkas' lemma, there exists a vector $\psi \in\mathbb{R}^{A}$ such that
\begin{equation*}
F\psi = 0,\quad \psi \geqslant 0 \quad \text{and} \quad M_i \psi < 0\,.
\end{equation*}
Thus $\psi$ is an $s$--$t$ flow in $N$. Because the network is uncapacitated, we can assume that the value of $\psi$ is precisely $k$, by scaling $\psi$ if necessary, hence $\psi \in Q$. Now, the inequality $M_i \psi < 0$ means that the $i$th coordinate of the projection $\pi(\psi) = M\psi$ is negative, which gives the desired contradiction.
\end{proof}
\subsubsection{Acyclicity of the Network}
\begin{lem} \label{lem:acyclic}
The network associated to every minimum size uncapacitated flow-based extension $Q \subseteq \mathbb{R}^A$ of a nonempty polytope $P \subseteq \mathbb{R}_+^d$ is acyclic.
\end{lem}
\begin{proof}
By Lemma \ref{lem:nonnegative_pi} the projection $\pi : \phi \mapsto M\phi$ may be assumed nonnegative. Consider a directed cycle $C$ in network $N$ and the corresponding columns of $M$. Take a point $\phi\in Q$ and consider the projection $\pi(\phi+K\chi^C)$ where $K \in \mathbb{R}_+$. By linearity, $\pi(\phi+K\chi^C) = \pi(\phi) + K \pi(\chi^C)$. If $\pi(\chi^C)$ is a non-zero vector and $K$ is chosen large enough, $\pi(\phi)+K \pi(\chi^C)$ would be outside of polytope $P$, a contradiction to the fact that $\phi+K\chi^C$ satisfies~\eqref{eq:flow_EF} and thus lies in $Q$.
Hence $\pi(\chi^C)$ is a zero vector. Due to nonegativity of $\pi$, for every arc $a\in A$ contained in at least one directed cycle, the corresponding column of $M$ is zero, that is, $\pi(\chi^{\{a\}}) = 0$. Therefore, if $N$ contains a directed cycle, we can contract every strongly connected component of $N$ to a node and obtain a smaller flow-based extension of $P$, a contradiction. Note that if $s$ and $t$ are in the same strongly connected component of $N$, in which case we are not allowed to contract this component because we assume $s \neq t$, then necessarily $P = \{0\}$ and a minimum size flow-based extension of $P$ is given by a network with two nodes connected by a single arc. The result follows.
\end{proof}
\subsubsection{Equations for the Initial Polytope}
\begin{lem}\label{lem:equations}
Let the equation $c\,x= \delta$ be valid for a nonempty polytope $P \subseteq \mathbb{R}^d$. Then for every node $v$ in the network $N=(V,A)$ associated to a minimum-size uncapacitated flow-based extension $Q \subseteq \mathbb{R}^A$ of $P$, there is a unique $\epsilon\in\mathbbm{R}$ such that $c \, \pi(\chi^\sigma)=\epsilon$ for every $s$--$v$ path $\sigma$.
\end{lem}
\begin{proof}
Let $\sigma_1, \sigma_2$ be two paths from source $s$ to node $v$. Due to minimality of the extension there is also a path $\sigma_3$ from $v$ to $t$. Since $\sigma_1\cup \sigma_3$ and $\sigma_2\cup \sigma_3$ define paths from $s$ to $t$, the projections $\pi(\chi^{\sigma_1\cup \sigma_3})$ and $\pi(\chi^{\sigma_2\cup \sigma_3})$ lie in the polytope $P$, and thus satisfy the equation $c\, x = \delta$. Therefore,
%
\begin{equation*}
0= c \, \pi(\chi^{\sigma_1\cup \sigma_3})-c \, \pi(\chi^{\sigma_2\cup \sigma_3})=c \, \pi(\chi^{\sigma_1}) - c \, \pi(\chi^{\sigma_2})\,.
\end{equation*}
To conclude the proof, we may define $\epsilon$ as the value $c \, \pi(\chi^{\sigma_1})$.
\end{proof}
\subsubsection{Extension of Faces}
\begin{lem}\label{lem:faces}
For every polytope $P \neq \varnothing$ and face $F$ of $P$, there holds $\xc_\text{$s$--$t$ path}(P)\geqslant \xc_\text{$s$--$t$ path}(F)$.
\end{lem}
\begin{proof}
Let $Q$ be a minimum size $s$--$t$ path extension of $P$ and let $N=(V,A)$ denote the corresponding network. The polytope $\pi^{-1}(F)\cap Q$ is a face of $Q$. From the linear description of $Q$, see \eqref{eq:flow_balance}--\eqref{eq:flow_lb}, we infer
\begin{equation*}
\pi^{-1}(F)\cap Q=\setDef{\phi\in Q}{\phi_a=0\,, a\in A'}
\end{equation*}
for some $A'\subseteq A$. Hence, the $s$--$t$ path polytope $Q'$ associated with the network $N'=(V, A\setminus A')$ together with the projection $\pi$ defines an $s$--$t$ path extension of face $F$. Because the size of the extension $Q'$ of $F$ is not larger than the size of the extension $Q$ of $P$, we have
$\xc_\text{$s$--$t$ path}(F) \leqslant \xc_\text{$s$--$t$ path}(P)$.
\end{proof}
\section{Lower Bounds} \label{sec:lower_bounds}
Now we provide lower bounds on the size of uncapacitated flow-based extensions or, equivalently (by Lemmas \ref{lem:nonnegative_pi} and \ref{lem:acyclic}), $s$--$t$ path extensions of the (bipartite and non-bipartite) perfect matching polytope and traveling salesman polytope. We start by proving that the $s$--$t$ path extension complexity of the perfect matching polytope of $K_{n,n}$ is $\Theta^*(2^n)$. This is striking because this polytope has $\Theta(n^2)$ facets, and a size-$\Theta(n^2)$ {\em capacitated} flow-based extension. Perhaps less striking are our exponential lower bounds for the perfect matching polytope and traveling salesman polytope of $K_n$. We derive these by combining our lower bound on $\xc_{\text{$s$--$t$ path}}(\mathop{\mathrm{P}_\mathrm{perfect\ matching}}(K_{n,n}))$ and Lemma \ref{lem:faces}.
\subsection{Bipartite Perfect Matchings}
\begin{thm}\label{thm:bipartite_matchings}
Every uncapacitated flow-based extension (or, equivalently, $s$--$t$ path extension) of the perfect matching polytope of the complete bipartite graph $K_{n,n}$ has size $\Omega\left(\frac{2^{n}}{\sqrt{n}}\right)$.
\end{thm}
\begin{proof}
Due to Lemma~\ref{lem:nonnegative_pi}, we may assume that the projection $\pi:\mathbbm{R}^{A}\rightarrow\mathbbm{R}^{d}$ is given by a linear nonnegative map.
Consider an $s$--$t$ path extension $Q\subseteq\mathbbm{R}^{A}$ with network $N = (V,A)$ and nonnegative linear projection $\pi:\mathbbm{R}^{A}\rightarrow\mathbbm{R}^{d}$.
For each vertex $u$ of $K_{n,n}$, the equation
$$
x(\delta(u)) = 1 \iff \sum_{e \in \delta(u)} x_e = 1
$$
is valid for $\mathop{\mathrm{P}_\mathrm{perfect\ matching}}(K_{n,n})$. From Lemma~\ref{lem:equations}, we conclude that for every node $v$ of $N$ there is a nonnegative vector $\epsilon^v\in\mathbbm{R}^{2n}$ such that for every $s$--$v$ path $\sigma$ in the network $N$ and every vertex $u$ of the graph $K_{n,n}$ the following holds:
\begin{equation*}
\sum_{e\in\delta(u)}\pi_e(\chi^\sigma)=\epsilon_u^v\,.
\end{equation*}
We base our analysis on the support of $\epsilon^v$, which we denote $\supp{\epsilon^v}$.
Now consider a node $v$ of network $N$. For every $s$--$t$ path $\sigma$ going through $v$ and such that $\pi(\chi^{\sigma}) = \chi^{M}$ for some perfect matching $M$ of $K_{n,n}$, matching $M$ and cut $\delta(\supp{\epsilon^v})$ do not have an edge in common.
Hence if $\sabs{\supp{\epsilon^v}}=n$ the $s$--$t$ paths of $N$ going through $v$ define at most $\frac{n}{2}!\frac{n}{2}!$ perfect matchings $M$ of $K_{n,n}$.
Moreover, for every arc $a=(v_1,v_2)$ in $N$ with $\sabs{\supp{\epsilon^{v_1}}}=n_1<n$ and $\sabs{\supp{\epsilon^{v_2}}}=n_2>n$ there are at most $\frac{n_1}{2}!\frac{2n-n_2}{2}! \leqslant \frac{n}{2}! \frac{n}{2}!$ perfect matchings $M$ such that there is an $s$--$t$ path $\sigma$ in $N$ with $a \in \sigma $ and $\chi^{M}=\pi(\chi^{\sigma})$, since in this case $\sigma$ contains both nodes $v_1$ and $v_2$ and every such matching $M$ must contain all the edges from the support of $\pi(\chi^{\{a\}})$.
Since the polytope $Q$ is an extension of $\mathop{\mathrm{P}_\mathrm{perfect\ matching}}(K_{n,n})$, for every perfect matching $M$ in $K_{n,n}$ there is an $s$--$t$ path $\sigma$ such that $\chi^\sigma$ projects to $\chi^M$. But since $\epsilon^{s}$ is an all zero vector and $\epsilon^{t}$ is an all one vector, this path $\sigma$ must go through a node $v$ with $\sabs{\supp{\epsilon^v}}=n$ or contain an arc $a = (v_1,v_2)$ with $\sabs{\supp{\epsilon^{v_1}}}<n<\sabs{\supp{\epsilon^{v_2}}}$.
Since the total number of perfect matchings in $K_{n,n}$ equals $n!$, network $N$ contains at least
\begin{equation*}
\frac{n!}{2\frac{n}{2}!\frac{n}{2}!}=\Omega\left(\frac{2^n}{\sqrt{n}}\right)
\end{equation*}
nodes $v$ with $\sabs{\supp{\epsilon^v}}=n$ or arcs $a=(v_1,v_2)$ with $\sabs{\supp{\epsilon^{v_1}}}<n<\sabs{\supp{\epsilon^{v_2}}}$. The result follows.
\end{proof}
The lower bound in Theorem~\ref{thm:bipartite_matchings} is tight, up to polynomial factors. Indeed, consider a complete bipartite graph $K_{n,n}$ with bipartition $U=\{u_1,\ldots, u_n\}$ and $W=\{w_1,\ldots, w_n\}$. We construct the network $N=(V,A)$ with
\begin{equation*}
V:=2^W\qquad\text{and}\qquad A:=\setDef{(S_1,S_2)\in V\times V}{S_1\subseteq S_2\text{ and }\sabs{S_1}+1=\sabs{S_2}}
\end{equation*}
and a linear projection $\pi$ such that for every arc $a=(S_1,S_2)\in A$
\begin{equation*}
\pi_{u_i,w_j}(\chi^{\{a\}}):=\begin{cases}
1 & \text{if}\quad i=\sabs{S_2},\quad \{w_j\}\cup S_1=S_2\\
0 & \text{otherwise}
\end{cases}\,.
\end{equation*}
It is not hard to see that every $\varnothing$--$W$ path in this network defines a perfect matching. This fact can be seen algorithmically, as follows. Start with $S=\varnothing$ and repeat the following step until $S = W$: having matched the vertices $v_1,\ldots,v_\sabs{S}$ with the vertices in $S$, select a mate $w \in W\setminus S$ for vertex $v_{\sabs{S}+1}$ and replace $S$ by $S\cup\{w\}$. It follows that the projection of the $\varnothing$--$W$ path polytope of network $N$ coincides with the perfect matching polytope of $K_{n,n}$. Since network $N$ has $n 2^{n-1} = O^*(2^n)$ arcs, we conclude that $\xc(\mathop{\mathrm{P}_\mathrm{perfect\ matching}}(K_{n,n})) = \Theta^*(2^n)$.
\subsection{Nonbipartite Perfect Matchings}
\begin{thm}\label{thm:complete_matchings}
Every uncapacitated flow-based extension (or, equivalently, $s$--$t$ path extension) of the perfect matching polytope of the complete graph $K_{n,n}$ has size $\Omega\left(\frac{2^{\frac{n}{2}}}{\sqrt{n}}\right)$.
\end{thm}
\begin{proof}
Indeed, the polytope $\mathop{\mathrm{P}_\mathrm{perfect\ matching}}(K_{\frac{n}{2}, \frac{n}{2}})$ is a face of the polytope $\mathop{\mathrm{P}_\mathrm{perfect\ matching}}(K_n)$, and thus Lemma~\ref{lem:faces} gives the lower bound.
\end{proof}
In order to construct an $s$--$t$ path extension of size close to the lower bound in Theorem~\ref{thm:complete_matchings}, we consider a complete graph $K_{n}$ with vertex set $U=\{u_1,\ldots, u_n\}$ and construct the network $N=(V,A)$ with
\begin{align*}
V:=\setDef{S\subseteq U}{\sabs{S}=2k,\ 0 \leqslant k\leqslant \frac{n}{2} \ \text{ and }\ \forall 1 \leqslant i\leqslant k : u_i\in S}\\ A:=\setDef{(S_1,S_2)\in V\times V}{S_1\subseteq S_2\text{ and }\sabs{S_1}+2=\sabs{S_2}}
\end{align*}
and a linear projection $\pi$ such that for every arc $a=(S_1,S_2)\in A$
\begin{equation*}
\pi_{u_i,u_j}(\chi^{\{a\}})=\begin{cases}
1 & \text{if }\, \{u_i, u_j\}\cup S_1=S_2\\
0 & \text{otherwise}.
\end{cases}
\end{equation*}
It is once again easy to verify that this defines an $s$--$t$ path extension, this time of the perfect matching polytope of $K_n$. The idea is that every $\varnothing$--$U$ path in network $N$ defines a perfect matching of $K_n$ and conversely, every perfect matching of $K_n$ corresponds to at least one (actually many) $\varnothing$--$U$ path in $N$. The $\varnothing$--$U$ paths in $N$ actually correspond to perfect matchings whose edges are ordered in such a way that for each $i$, vertex $u_i$ is covered by one of the first $i$ edges in the ordering. Every arc $(S,S \cup \{u_i,u_j\})$ in such a path corresponds to the addition of edge $u_iu_j$ to the matching.
Up to a polynomial factor, the size of the network equals the number of nodes in the network, that is,
\begin{equation*}
\sum_{k=0}^{\frac{n}{2}} \binom{n-k}{k}\,.
\end{equation*}
This is due to the fact that the nodes $S$ in the $k$th level of network $N$ are of the form $S = \{u_1,\ldots,u_k\} \cup T$, where $T$ is contained in $U \setminus \{u_1,\ldots,u_k\}$ and has size $k$. Since the number of summands in the above expression is $\frac{n}{2}+1$, the size of the constructed extension is
\begin{equation*}
O^*\left(\max_{0\leqslant k \leqslant \frac{n}{2}} \binom{n-k}{k}\right)=O^*\left(\max_{0 < k <\frac{n}{2}} \frac{(n-k)^{n-k}}{k^k (n-2k)^{n-2k}}\right)\,,
\end{equation*}
where we used Stirling's formula to simplify the left-hand side. Calculating the derivative of the function $\frac{(n-k)^{n-k}}{k^k (n-2k)^{n-2k}}$, we determine that the maximum in the above interval is achieved in the case when $k$ equals $\frac{2}{5+\sqrt{5}}n$, thus the size of the extension is $O(2^{0.695 n})$.
\subsection{Hamiltonian Cycles}
\begin{thm}\label{thm:complete_traveling}
Every uncapacitated flow-based extension (or, equivalently, $s$--$t$ path extension) of the traveling salesman polytope of the complete graph $K_{n}$ has size $\Omega\left(\frac{2^{\frac{n}{4}}}{\sqrt{n}}\right)$.
\end{thm}
\begin{proof}
Assume for now that $n=4k$ for some $k\in\mathbbm{N}$, the other cases will be dealt with later. Take a partition of the vertices of $K_n$ in $U=\{u_1,\ldots, u_{2k}\}$ and $W=\{w_1,\ldots,w_{2k}\}$, and consider the following sets of edges in the graph $K_n$:
\begin{equation*}
E_0 := \setDef{u_iw_j}{i\neq j,\, 0\leqslant i,j\leqslant 2k}\qquad\text{and}\qquad E_1 := \setDef{u_iw_i}{ 0\leqslant i\leqslant 2k}\,.
\end{equation*}
Define the face $F$ of the polytope $\mathop{\mathrm{P}_\mathrm{traveling\ salesman}}(K_n)$ as the set of points in $\mathop{\mathrm{P}_\mathrm{traveling\ salesman}}(K_n)$ such that $x_e=0$ for every $e \in E_0$ and $x_e=1$ for every $e \in E_1$.
Let us show that the face $F$ together with an orthogonal projection on the variables corresponding to the edges $u_iu_j$ for $0\leqslant i,j\leqslant 2k$ gives an extension of the perfect matching polytope $\mathop{\mathrm{P}_\mathrm{perfect\ matching}}(K_{2k})$ (here the complete graph $K_{2k}$ is defined on the vertex set $U$).
First, every Hamiltonian cycle $C$ in the graph $K_n$ restricted to the edges contained in $U$ is a perfect matching, whenever $\chi^{C}$ belongs to the face $F$. Indeed, for every vertex $u_i$ in $U$ there must be exactly two edges in $C$ adjacent to it. Since the characteristic vector $\chi^{C}$ lies in the face $F$, one of these edges is the edge $u_iw_i$ and the other is contained in $U$.
Second, every perfect matching $M$ in the graph $K_{2k}$ can be extended to a Hamiltonian cycle $C$ in $K_n$ such that $\chi^{C}$ lies in $F$. Indeed, extend $M$ by another perfect matching $M'$ of $K_{2k}$ to a Hamiltonian cycle in $K_{2k}$. Then the desired hamiltonian cycle $C$ can be defined as the union of $M$, $E_1$ and $\setDef{w_iw_j}{u_iu_j \in M'}$. Thus the result follows from Theorem~\ref{thm:complete_matchings} and Lemma~\ref{lem:faces}.
If $n=4k+r$, for some $k,r\in\mathbbm{N}$, $1\leqslant r\leqslant 3$, the result is obtained in a similar way by taking a bipartition $U=\{u_1,\ldots, u_{2k}\}$ and $W=\{w_1,\ldots,w_{2k+r}\}$ and defining the face $F$ by equations $x_e=0$ for every $e \in E_0$, $x_e=1$ for every $e \in E_1$ and $x_{w_{2k}w_{2k+1}}=\ldots=x_{w_{2k+r-1}w_{2k+r}}=1$, where the edge sets $E_0$ and $E_1$ are defined as above.
\end{proof}
For the traveling salesman polytope there is a $s$--$t$ path extension of size $O^*(2^n)$ constructed in a similar manner as the $s$--$t$ path extension of the perfect matching polytope of $K_{n,n}$. This extension corresponds to a well-known dynamic programming algorithm of Held and Karp for the traveling salesman problem~\cite{heldKarp70}. We define this extension here for completeness.
Consider a complete graph $K_{n}$ with vertex set $U=\{u_1,\ldots, u_n\}$ and construct the network $N=(V,A)$ with
\begin{eqnarray*}
V &:= &\setDef{(S,v)}{S\subseteq U,\, v\in S, u_1\in S}\cup\{(U,\varnothing)\}\\
A &:= &\setDef{((S_1,v_1),(S_2,v_2))\in V\times V}{S_1\cup\{v_2\}=S_2\text{ and }\sabs{S_1}+1=\sabs{S_2}}\\
&&\cup\,\setDef{((U,v),(U,\varnothing))\in V\times V}{v\in U}
\end{eqnarray*}
and a linear projection $\pi$ such that for every arc $a=((S_1,v_1),(S_2,v_2))\in A$, $v_1\in U$, $v_2\in U$
\begin{equation*}
\pi_{u_i,u_j}(\chi^{\{a\}}):=\begin{cases}
1 & \text{if }\, \{u_i, u_j\}=\{v_1,v_2\}\\
0 & \text{otherwise}
\end{cases}
\end{equation*}
and for an arc $a=((U,v),(U,\varnothing))\in A$, $v\in U$
\begin{equation*}
\pi_{u_i,u_j}(\chi^{\{a\}}):=\begin{cases}
1 & \text{if }\,\{u_i, u_j\}=\{u_1,v\} \\
0 & \text{otherwise}
\end{cases}\,.
\end{equation*}
It is straightforward to see that the network with source $(u_1,\{u_1\})$ and sink $(U,\varnothing)$ generates the desired $s$--$t$ path extension.
\section{Open Problems}
We conclude this paper by stating three open problems.
\begin{enumerate}[(i)]
\item Obtain lower bounds for capacitated flow-based extensions. Although this type of extensions is more expressive than uncapacitated flow-based extensions, we suspect that exponential size lower bounds can be obtained for nonbipartite matchings and Hamiltonian cycles.
\item How difficult is this to compute a small uncapacitated flow-based extension for a given 0/1-polytope? Are there good general lower bounds?
\item All the lower bounds obtained here are of the type $2^{\Omega(\sqrt{d})}$, where $d$ is the dimension of $P$. Find an explicit $0/1$-polytope $P$ such that every uncapacitated flow-based extension has size $2^{\Omega(d)}$. (Notice that every polytope $P$ has an uncapacitated flow-based extension of size at most the number of vertices of $P$, thus this last lower bound would be essentially tight.)
\item Davis-Stober, Doignon, Fiorini, Glineur and Regenwetter~\cite{DDFGR13} give uncapacitated flow-based extensions of size $O^*(2^n)$ for the linear ordering polytope and $O^*(3^n)$ for the interval order polytope. Is there such an extension of size $O^*(c^n)$ for the semiorder polytope? (Semiorders are also known as unit interval orders.)
\end{enumerate}
\section{Acknowledgements}
The authors thank Hans Tiwary for taking part in the early stage of this work, and Michele Conforti, Santanu Dey, Marco Di Summa, Sebastian Pokutta and Dirk Oliver Theis for stimulating discussions.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,727 |
function LNode(key) {
this.key = key
this.children = null
this.next = null
}
function TrieLike() {
this.head = new LNode(null)
// keeping a list of chars.
this.tail = this.head
}
/**
* @param {string[]} words
* @return {string}
*/
const alienOrder = words => {
/*
idea:
- do a trie-like structure, which allows us to figure out ordering of characters
(this is because all words are inserted in their sorted order)
- if we keep track of all ordering clues, we can toposort this,
which gives us a potentially partial ordering.
- if toposort detects a loop, there is no way this could be solved, return ""
- otherwise put all other characters into the answer (as we need a full list of
all chars that have appeared.
*/
// graph[i][j] !== 0 implies edge i ==> j
const graph = new Array(26)
// for recording indegree
const inDeg = new Int8Array(26)
const allCodes = new Set()
const mkEdge = (x,y) => {
if (!(x in graph))
graph[x] = new Int8Array(26)
if (!graph[x][y]) {
graph[x][y] = 1
++inDeg[y]
}
}
const tRoot = new TrieLike()
const trieInsert = (w, i, tCur) => {
if (i >= w.length)
return
const keyCode = w.codePointAt(i) - 97
allCodes.add(keyCode)
if (tCur.tail.key !== keyCode) {
const newNode = new LNode(keyCode)
if (tCur.tail.key !== null)
mkEdge(tCur.tail.key, keyCode)
tCur.tail.next = newNode
tCur.tail = newNode
if (i !== w.length-1) {
newNode.children = new TrieLike()
trieInsert(w, i+1, newNode.children)
}
} else {
if (tCur.tail.children === null) {
tCur.tail.children = new TrieLike()
}
trieInsert(w, i+1, tCur.tail.children)
}
}
words.forEach(w => trieInsert(w, 0, tRoot))
const queue = []
allCodes.forEach(code => {
if (inDeg[code] === 0)
queue.push(code)
})
qHead = 0
const ans = []
while (qHead < queue.length) {
const code = queue[qHead]
ans.push(code)
++qHead
for (let i = 0; i < 26; ++i)
if (graph[code] && graph[code][i]) {
--inDeg[i]
if (inDeg[i] === 0)
queue.push(i)
}
}
if (inDeg.some(x => x > 0)) {
// circular graph
return ""
}
// move indecisive codes into answer
allCodes.forEach(code => {
if (ans.indexOf(code) === -1)
ans.push(code)
})
return String.fromCodePoint.apply(
null,
ans.map(x => x + 97)
)
}
console.assert(
alienOrder(
[
"wrt",
"wrf",
"er",
"ett",
"rftt"
]
) === "wertf"
)
console.log(alienOrder(["z", "x"]))
console.log(alienOrder(["z", "x", "z"]))
console.log(alienOrder(["a", "b", "a"]))
| {
"redpajama_set_name": "RedPajamaGithub"
} | 321 |
{"url":"https:\/\/forum.allaboutcircuits.com\/threads\/want-to-connect-homebrew-cpu-to-the-internet.145215\/","text":"# Want to connect homebrew CPU to the internet\n\n#### AnalogDigitalDesigner\n\nJoined Jan 22, 2018\n121\nOk so here's my problem...\n\nI am building a 16bit CPU\/Minicomputer from bare logic, and I want to connect this computer to the internet.\n\nThe CPUis similar to an intel 8086, except that it has 64KB of RAM, and the minicomputer itself will be somewhat a PC XT.\n\nWhat I'd like to know is what are the possibilities here? What kind of connection could I support?\n\nDo I require a full operating system in order to connect to the net, or will a good BIOS do ?\n\nDoes anyone know or have used WIZNET?\n\n#### wayneh\n\nJoined Sep 9, 2010\n17,153\nOk so here's my problem...\n\nI am building a 16bit CPU\/Minicomputer from bare logic, and I want to connect this computer to the internet.\nThat does indeed sound like a problem! Perhaps addressing why you are bothering with this will help folks get in the right mindset.\n\nWhen you say \"connect\", I assume you mean you want to do something over TCP\/IP? I doubt you'll be able to do much except send pings. I believe the first browsers required an OS and more horsepower from the hardware but I may be confused about the timelines.\n\n#### AnalogDigitalDesigner\n\nJoined Jan 22, 2018\n121\nThat does indeed sound like a problem! Perhaps addressing why you are bothering with this will help folks get in the right mindset.\n\nWhen you say \"connect\", I assume you mean you want to do something over TCP\/IP? I doubt you'll be able to do much except send pings. I believe the first browsers required an OS and more horsepower from the hardware but I may be confused about the timelines.\n\nWhy am I bothering with this? Oh come on boy........ How about because it gives me more pleasure than heroine\n\nI also want to be able to run a proper OS on this minicomputer. Like Bill Buzbee's Magic-1 computer.\n\nBut for that I will need more than 64KB ram. That's tricky! Because I will need to extend my registers from 16bit to 24 bits. And because I have an 8bit data bus, that will add an extra clock cycle per instruction. But maybe that's a good trade off. We'll see.\n\nI don't know what you mean by better hardware. My minicomputer will run at 4MHz and that is enough because Bill Buzbee's computer runs at ~ 4MHz and hosts a website!\n\nHas anyone here heard of WIZNET?\n\n#### wayneh\n\nJoined Sep 9, 2010\n17,153\nBe patient, and consider moving your thread to the computing forum. It might get more attention there. A moderator could do that for you if you like.\n\n#### atferrari\n\nJoined Jan 6, 2004\n4,378\nI for one, I am. But posting anything helpful is a different thing.\n\n#### WBahn\n\nJoined Mar 31, 2012\n26,398\nOk so here's my problem...\n\nI am building a 16bit CPU\/Minicomputer from bare logic, and I want to connect this computer to the internet.\n\nThe CPUis similar to an intel 8086, except that it has 64KB of RAM, and the minicomputer itself will be somewhat a PC XT.\n\nWhat I'd like to know is what are the possibilities here? What kind of connection could I support?\n\nDo I require a full operating system in order to connect to the net, or will a good BIOS do ?\n\nDoes anyone know or have used WIZNET?\nThe O\/S and BIOS really don't have anything to do with it (though they might make it a lot easier).\n\nYou need a suitable communications stack. Also, you don't have to have TCP\/IP or Ethernet; you just need a modem that your computer can talk to -- the modem talks to the Internet (via an Internet Service Provider).\n\nSmall, cheap microcontrollers are able to serve as HTML web servers just fine, so I don't think you are asking for too much. Even 8-bit PICs have done this and many MCUs have built in TCP\/IP and Ethernet stacks.\n\nYou might search for something like \"embedded web server\" and see what you find.\n\n#### joeyd999\n\nJoined Jun 6, 2011\n4,477\nSearch for TCP232.\n\n#### AnalogDigitalDesigner\n\nJoined Jan 22, 2018\n121\nThe O\/S and BIOS really don't have anything to do with it (though they might make it a lot easier).\n\nYou need a suitable communications stack. Also, you don't have to have TCP\/IP or Ethernet; you just need a modem that your computer can talk to -- the modem talks to the Internet (via an Internet Service Provider).\n\nSmall, cheap microcontrollers are able to serve as HTML web servers just fine, so I don't think you are asking for too much. Even 8-bit PICs have done this and many MCUs have built in TCP\/IP and Ethernet stacks.\n\nYou might search for something like \"embedded web server\" and see what you find.\n\nThere's something called WIZNET, which is a hardware tcp\/ip stack. Have you heard of this?\n\n#### AnalogDigitalDesigner\n\nJoined Jan 22, 2018\n121\n\n#### WBahn\n\nJoined Mar 31, 2012\n26,398\n\nThere's something called WIZNET, which is a hardware tcp\/ip stack. Have you heard of this?\nNope. But I just went and looked at their website and see that their chips are dollar-scale (~$5) and they give free samples, so it's probably worth exploring. Thread Starter #### AnalogDigitalDesigner Joined Jan 22, 2018 121 Nope. But I just went and looked at their website and see that their chips are dollar-scale (~$5) and they give free samples, so it's probably worth exploring.\n\nHello Eagle Keeper,\n\nBill Buzzbee told me that I don't need an OS, or Minix to use Wiznet. That's fantastic don't you think ?\n\n#### WBahn\n\nJoined Mar 31, 2012\n26,398\nHello Eagle Keeper,\n\nBill Buzzbee told me that I don't need an OS, or Minix to use Wiznet. That's fantastic don't you think ?\nI don't really think anything one way or the other. Try it and see if it is a viable solution to your problem. If it is, great. If not, move on.","date":"2021-07-25 21:45:17","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.4953826069831848, \"perplexity\": 2184.440278550542}, \"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\/1627046151866.98\/warc\/CC-MAIN-20210725205752-20210725235752-00226.warc.gz\"}"} | null | null |
Q: How to build and deploy python project with test suite I have a simple command-line python project with the following structure
project_main/
- README.md
- requirements.txt
- setup.py
- project_main/
- __init__.py
- controller.py
- tests/
- __init__.py
- test_cases.py
- test_suite.py
I run the test cases using the command given below, and all tests pass OK
python -m unittest
The application logic works perfectly.
Now I need to deploy this code in any system with a single script "setup.py". The contents of "setup.py" is given below
from setuptools import setup, find_packages
import os
import sys
sys.path.insert(0, os.path.abspath(os.path.join(os.path.dirname(__file__), '..')))
setup(
name='sample',
version='1.0.0',
description='sample deployment'
packages=find_packages(),
test_suite="tests.test_suite.main_suite",
entry_points={
'console_scripts': [
'demoCommand=project_main.controller:main'
]
}
)
When I run this using the below command, it asks for admin permissions to the default install path. Also even if test_cases fail, the application gets installed.
python3 setup.py install
*
*I would like to know how to ensure all the packages in requirements.txt are installed when running the setup.py file?
*How to stop installation if test cases fail?
*How to install the application in a custom folder without admin permission?
EDIT: Adding partial relevant output of the setup.py install command
copying build\lib\tests\__init__.py -> build\bdist.win32\egg\tests
byte-compiling build\bdist.win32\egg\tests\main_test_suite.py to main_test_suite.cpython-37.pyc
| {
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{"url":"http:\/\/www.ams.org\/mathscinet-getitem?mr=458336","text":"MathSciNet bibliographic data MR458336 (56 #16539) 53C20 (32F99 32C05) Greene, R. E.; Wu, H. $C\\sp{\\infty }$$C\\sp{\\infty }$ convex functions and manifolds of positive curvature. Acta Math. 137 (1976), no. 3-4, 209\u2013245. Article\n\nFor users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.","date":"2013-12-07 05:40:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 1, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9977273941040039, \"perplexity\": 7653.882791497774}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-48\/segments\/1386163053578\/warc\/CC-MAIN-20131204131733-00010-ip-10-33-133-15.ec2.internal.warc.gz\"}"} | null | null |
State audit finds L.A. County leaves children at risk for months
by City News Service • May 21, 2019
[Left] Gabriel Fernandez. [Right] Click image to view the audit report.
LOS ANGELES – Los Angeles County's child welfare agency leaves some youngsters in unsafe and abusive situations for months because social workers fail to consistently and quickly complete abuse and neglect investigations, according to a report released Tuesday by the state auditor. [Read the full report here.]
Assemblyman Tom Lackey, R-Palmdale, was one of the legislators who called for the audit in the wake of the 2013 torture killing of 8-year-old Gabriel Fernandez of Palmdale. Despite multiple reports of abuse, the Department of Children and Family Services failed to remove Gabriel from the home where he lived with his mother and her boyfriend, who were ultimately convicted of his murder.
"This audit proves what we've suspected for a long time — we need to fix things at the Department of Children and Family Services to protect the most vulnerable kids in our community," Lackey said. "We need major changes at the department to protect children and make sure reports of abuse don't fall through the cracks."
In a letter to Gov. Gavin Newsom and legislative leaders, California State Auditor Elaine Howle said DCFS completed roughly three-quarters of all safety and risk assessments on time in fiscal year 2017-18 and failed to ever complete 8-10% of each type of assessment.
"We also found numerous instances in which these assessments were not accurate, including several safety assessments that social workers prepared and submitted without actually visiting the child's home," the letter states. "Even if supervisors had identified and corrected many of these issues upon review, we found that they often completed such reviews long after social workers had made decisions regarding children's safety."
DCFS received more than 167,000 allegations of abuse and neglect in fiscal year 2017-18, according to the audit.
Even as budget increases allowed the department to hire more social workers and reduce caseloads, the agency failed to comply with state requirements, including home inspections and criminal background checks of relatives considered for foster placement, the audit found.
The auditor said DCFS doesn't set specific deadlines for reports and has limited quality assurance reviews. In addition, the agency lacks a clear process for implementing recommendations that arise out of child death investigations.
The audit report recommended setting specific time frames for completing investigations, background checks and home inspections and creating various tracking mechanisms to monitor the work. Other recommendations include improving supervision, in part by reducing the number of social workers assigned to each supervisor to six by May 2020.
Six is the number specified in the agency's union contract and the department has told the Board of Supervisors that it aims to bring the ratio down to five.
In a response to a draft of the audit, dated May 1, DCFS Director Bobby Cagle said his agency agreed with the recommendations and was already implementing changes.
The department issued a statement Tuesday saying it was grateful to Lackey, former Sen. Ricardo Lara and Sen. Scott Wilk, R-Santa Clarita, for their leadership on child welfare and calling the audit report recommendations "thoughtful."
The goal is to complete investigations in 30 days, but some delays are outside the agency's control, including those related to obtaining medical records, trying to locate families that avoid contact with social workers and coordinating with other agencies, according to the DCFS statement.
"Our department fared considerably better than other jurisdictions in California that have an average of 41% of their cases bypassing state deadlines," the statement says.
In addition to implementing other recommendations, DCFS plans to create a Quality Improvement division to conduct "comprehensive assessments of referrals and cases from all its regional offices." The division will also evaluate the roles of supervisors and managers.
The department is also ramping up efforts to support LGBTQ+ youth, who represent nearly one in five foster youth and have proven to be more vulnerable to poverty, homelessness and juvenile justice involvement.
"Reform is not about a single point in time; true change takes time in order to have a meaningful impact on a system as large and diverse as Los Angeles County's, and it must be a sustained, continuous process that addresses emerging issues and systemic challenges as they develop. We are committed to doing just that in collaboration with our elected officials and community partners," the agency stated.
Filed Under: Home, Palmdale, Politics
9 comments for "State audit finds L.A. County leaves children at risk for months"
Strength In Numbers says
Anyone reading this article and has been affected by the corruption or illegal activities in AV dcfs, write the DOJ specify the crime and the AV office. If you work for the department and are in a position of being unable to speak, WRITE the DOJ. Do not be a silent accomplice. List the names of the workers. Write the reps in this article. Give them the names and occurrences. File your grievance and include all pertinent information to the ombudsmen. Emails and text messages included. Pull their covers. They work by instilling fear and using intimidation to wreak havoc. AND get PAID to do so. Let us as concerned citizens for the safety and well being of the children in the community be active.
God Help Us All says
The Antelope Valley has been plagued by child abuse and methamphetamine use for decades. Maybe a full disclosure is deserved. Google ""The Antelope Valley and Metamphetamine".
Or, "Antelope Valley+Child Abuse".
That is why some of our long time residents celebrate when a monster receives the Death Penalty!
By the way, Should that Beast, the one who murdered little Gabriel, have the right to vote? A vote for whoever will unleash him upon us again?
God Help Us All Second That says
What do you expect when the union person goes to bat for the person who failed to protect Gabriel? That person would then run for office with the backing of Perris. This Antelope Valley has some serious issues.
Lynda Sims says
How do we get San Bernardino County audited? Supervisor Kathreen Barmann uses her position to falsify reports, intimidation, coerician, and malfision. She is a liar who "loves to abduct children" said it right to my face.
ANNON says
AND THIS IS A SURPRISE? DCFS AS A WHOLE SHOULD BE SHUT DOWN. IT'S OBVIOUS THEY CANT DO THINGS CORRECTLY AT ALL.
Daisy Ramirez says
I and many mom I know have been in this case and on some on going cases that need help
Oppressed says
Oh the sarcasm in the response of the department. How "thoughtful" of them to never respond to an investigation. How "thoughtful" of them to be so malicious and vindictive to tear a child from the loving family and place with an abuser. That's how "thoughtful" works for them.
Cher says
DCFS is suppose to protect Children and keep families together. I can understand if children are being abused they need to be removed but children are being removed for dv such as arguing or fighting in front of children. It's just really sad families are destroyed and lost forever over something that can be a warning . They don't deserve to lose there kids for arguing. Losing your children leads to depression and much more. You have to provide a room and bed for your child to get them back. Majority of families become homeless and they lose any chance of ever getting them back. There are so many obstacles to overcome. But if you love your children with all your heart you will do whatever it takes.
DCFS says
Endless incompetence and indifference on display… | {
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} | 2,075 |
import React from 'react'
import { Label } from 'reactstrap'
import Form from './components/Form'
import Input from './components/Input'
import Group from './components/Group'
class MyForm extends React.Component {
handleSubmit = () => {
// eslint-disable-next-line
console.log('Programmatic submission')
}
render() {
return (
<div>
<p>Form to submit</p>
<Form onSubmit={this.handleSubmit} ref={(ref) => { this.form = ref }}>
<Group>
<Label htmlFor="name">
Your name <sup>∗</sup>
</Label>
<Input
name="name"
required
/>
</Group>
</Form>
<div>
<p>External button</p>
<button onClick={() => this.form.submit()}>Programmatic submission</button>
</div>
</div>
)
}
}
export default MyForm
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,104 |
Teacher's Pet (1958)
Clark Gable, Doris Day, Gig Young, Mamie Van Doren, Nick Adams, Peter Baldwin, Marion Ross
An Apple a Day Keeps the Teacher at Bay
In her career Doris Day starred in various movies where she ended up being duped by a man who she dislikes but then falls for, probably the most popular of which is "Pillow Talk" the first of the Doris Day and Rock Hudson movies. But before that memorable movie we got "Teacher's Pet" which paired Doris Day up with screen legend Clark Gable in a tale of love, comedy and mistaken identity. And to be honest "Teacher's Pet" is not bad with some entertaining performances from both Day and Gable as well as Gig Young although it also has some issues.
James Gannon (Clark Gable - The Tall Men), the hard nosed editor of a New York newspaper, has learned everything he knows about journalism through hard work and doesn't believe that it can be taught in the class room. So when his boss orders him to help a college professor, Erica Stone (Doris Day - The Pajama Game), with her journalism lectures he is less than pleased. But when he discovers that Stone is a sexy young woman who thinks that editors like Gannon are relics he pretends to be James Gallangher a student to get one over on her. What neither he nor Stone expected was that they would end up becoming attracted to each other.
If you've seen any of Doris Day's numerous romantic comedies then the whole set up to "Teacher's Pet" will be pretty familiar to you as she is duped by the man she dislikes when he pretends to be someone else. And as such there is a predictability to it which goes as far as Gig Young appearing as a third person in the relationship, which may sound like it's quite boring, but in fact "Teacher's Pet" despite being predictable works, it entertains and most significantly switches a few elements around. Where as in many of Doris Day's romantic comedies we would watch Day pulling faces when she becomes infuriated or indignant you don't get that in "Teacher's Pet" but instead you get Clark Gable pulling faces of astonishment. It works surprisingly well and provides much of the amusement and a bit of a difference to her other memorable romantic comedies.
It's the comedy side of things which makes "Teacher's Pet" so entertaining and although the comedy in the earlier scenes which focus on Doris Day being this sexy but slightly frosty professor to Clark Gables' old fashioned paper editor are a little weak, when Gig Young enters the movie it perks up quite a bit. Many of the jokes are obvious from Gable admiring Day's shapely figure, Young struggling with a hang over and so on but the simplicity of the jokes and the wonderful faces which Clark Gable and Gig Young pull make it funnier than maybe it should be. What is quite strange is that for once Doris Day plays it mainly straight allowing the others in the movie to have the lion's share of the comedy.
But there is one major issue and that with 20 years between them the romantic pairing of Doris Day and Clark Gable, it just doesn't really work. Gable's character of James Gannon is purposefully a bit of a letch, he hangs out with young women who dance in what was then deemed a risque way. But it just feels a little bit too lecherous when Gannon makes his advances on Doris Day as Erica Stone. But that's not the only issue, there is a distinct lack of chemistry between Doris Day and Clark Gable and those supposedly romantic scenes end up a little dull, lacking that spark or believability to make you enjoy the romance as it grows.
As for the performances well Clark Gable not only looks brilliant as a stuck in his ways, barracking old paper editor but shows such a great ability for comedy. His looks of astonishment or his drunken head jerks when a sexy woman sings provides much amusement. And Doris Day is not too shabby either playing the sexy professor perfectly and giving us a brief amusing musical number half way through. But in many ways it's Gig Young as psychologist Dr. Hugo Pine which gives "Teacher's Pet" that burst of energy and comedy which at times it lacks elsewhere and he does it so well.
What this all boils down to is that "Teacher's Pet" is to be honest very similar to various other Doris Day movies, it has that formula as we watch her being duped by a man. But with Doris Day playing it for the most straight and allowing both Clark Gable and Gig Young to deliver the comedy it is amusing and slightly different to say "Pillow Talk" or "Lover Come Back".
Young Man with a Horn (1950)
As a young child Rick Martin (Kirk Douglas) fell in love with music and dedicated his life to being the best trumpet player he could. And his hard work and dedication pays off as he works his way from band to band becoming a great a ...
Love Me or Leave Me (1955)
In the twenties dancer and wannabee singer Ruth Etting (Doris Day - Young at Heart) is spotted by Chicago hood Marty Snyder (James Cagney) who becomes infatuated by the beautiful woman. Taking her under his wing he starts to manage ...
By the Light of the Silvery Moon (1953)
Having returned from war Bill (Gordon MacRae) decides not to rush in to marrying Marjie (Doris Day) but to find work and save some money to give them a good start together. But things become complicated when Marjie and her younger b ...
Tea for Two (1950)
Nanette Carter (Doris Day - Young Man with a Horn) is a wealthy heiress who agrees to a bet with her Uncle Max (S.Z. Sakall - Casablanca) that she can stop saying "Yes" to all questions for 48 hours. In return she gets the $25,000 s ...
Lucky Me (1954)
With no gig a vaudeville act find themselves stuck working in a hotel when they get caught trying to scam a free meal. But their luck seems on the up when hotel guest, song writer Dick Carson (Robert Cummings) meets the highly super ...
TOP TEN MAUREEN O'HARA MOVIES
1) The Quiet Man (1952)
2) Miracle on 34th Street (1947)
3) Rio Grande (1950)
4) The Hunchback of Notre Dame (1939)
5) The Black Swan (1942)
6) McLintock! (1963)
7) The Parent Trap (1961)
8) Mr. Hobbs Takes a Vacation (1962)
9) Sons of the Musketeers (1952)
10) The Christmas Box (1995)
Doc Hollywood (1991) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,353 |
In opening the negotiations, DHSC has now shared proposals for the CPCF for 2019 to 2020 and beyond with the PSNC. A series of regular meetings will now take place between DHSC, NHS England and the PSNC to discuss this further. The negotiations are confidential, meaning the department will not be able to provide further information about the negotiations until they come to a close. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,812 |
Q: Javascript/jQuery: calling function in another frame I'm new to the DOM and JavaScript and hitting on some problems when trying to call a function defined in a frame from the context of the top-level frame or Firebug.
Given the below frameset:
<html>
<body>
<frameset cols="*" rows="81,*">
<frame id="topFrame" tabindex="1" name="topFrame" noresize="noresize" scrolling="No" src="hometop.aspx"/>
<frameset border="0" cols="214,*" frameborder="no" framespacing="0">
<frameset border="0" cols="*" frameborder="no" framespacing="0" rows="70,*">
<frame tabindex="-1" id="chatFrame" name="chatFrame" scrolling="No" noresize="noresize" src=""/>
<frame tabindex="-1" id="leftFrame" name="leftFrame" noresize="noresize" src="leftFrame.aspx"/>
</frameset>
<frame tabindex="-1" id="mainFrame" name="mainFrame" src=""/>
</frameset>
<noframes>Your browser does not support frameset.</noframes>
</frameset>
</body>
</html>
I'm trying to write a javascript hook that will call a javascript function defined in #leftFrame when the above document is first opened. I'm performing this in a Firebug session with jQuery loaded.
jQuery("#leftFrame") returns a frame DOM element. Now I'd like to execute my function (openLink, defined in plain old script tag in leftFrame.aspx) in the context of the frame. My understanding is that the function will be a DOM node under the leftFrame's document element. However I can't get hold of the frame's document.
I've tried:
jQuery("#leftFrame").document
jQuery("#leftFrame").contentDocument
jQuery("#leftFrame").find("html")
Also when inspecting the DOM tree in Firebug I can't see the openLink function under any DOM nodes as I'd expect.
Can anyone help me out?
A: $('#leftFrame')[0].contentWindow.document
$('#leftFrame')[0].contentWindow.functionName()
The above should work. jQuery's contents() only works on the iframe node so you'll have to reference it like that.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,020 |
package stroom.pipeline.client.presenter;
import stroom.docref.DocRef;
import stroom.editor.client.presenter.EditorPresenter;
import stroom.entity.client.presenter.ContentCallback;
import stroom.entity.client.presenter.DocumentEditTabPresenter;
import stroom.entity.client.presenter.LinkTabPanelView;
import stroom.pipeline.shared.TextConverterDoc;
import stroom.security.client.api.ClientSecurityContext;
import stroom.widget.tab.client.presenter.TabData;
import stroom.widget.tab.client.presenter.TabDataImpl;
import com.google.inject.Inject;
import com.google.web.bindery.event.shared.EventBus;
import edu.ycp.cs.dh.acegwt.client.ace.AceEditorMode;
import javax.inject.Provider;
public class TextConverterPresenter extends DocumentEditTabPresenter<LinkTabPanelView, TextConverterDoc> {
private static final TabData SETTINGS = new TabDataImpl("Settings");
private static final TabData CONVERSION = new TabDataImpl("Conversion");
private final TextConverterSettingsPresenter settingsPresenter;
private final Provider<EditorPresenter> editorPresenterProvider;
private EditorPresenter codePresenter;
private boolean readOnly = true;
@Inject
public TextConverterPresenter(final EventBus eventBus,
final LinkTabPanelView view,
final TextConverterSettingsPresenter settingsPresenter,
final Provider<EditorPresenter> editorPresenterProvider,
final ClientSecurityContext securityContext) {
super(eventBus, view, securityContext);
this.settingsPresenter = settingsPresenter;
this.editorPresenterProvider = editorPresenterProvider;
settingsPresenter.addDirtyHandler(event -> {
if (event.isDirty()) {
setDirty(true);
}
});
addTab(CONVERSION);
addTab(SETTINGS);
selectTab(CONVERSION);
}
@Override
protected void getContent(final TabData tab, final ContentCallback callback) {
if (SETTINGS.equals(tab)) {
callback.onReady(settingsPresenter);
} else if (CONVERSION.equals(tab)) {
callback.onReady(getOrCreateCodePresenter());
// } else if (REFERENCES_TAB.equals(tab)) {
// entityReferenceListPresenter.read(getEntity());
// callback.onReady(entityReferenceListPresenter);
} else {
callback.onReady(null);
}
}
@Override
public void onRead(final DocRef docRef, final TextConverterDoc textConverter) {
super.onRead(docRef, textConverter);
settingsPresenter.read(docRef, textConverter);
if (codePresenter != null) {
codePresenter.setText(textConverter.getData());
}
}
@Override
protected void onWrite(final TextConverterDoc textConverter) {
settingsPresenter.write(textConverter);
if (codePresenter != null) {
textConverter.setData(codePresenter.getText());
}
}
@Override
public void onReadOnly(final boolean readOnly) {
super.onReadOnly(readOnly);
this.readOnly = readOnly;
settingsPresenter.onReadOnly(readOnly);
if (codePresenter != null) {
codePresenter.setReadOnly(readOnly);
codePresenter.getFormatAction().setAvailable(!readOnly);
}
}
@Override
public String getType() {
return TextConverterDoc.DOCUMENT_TYPE;
}
private EditorPresenter getOrCreateCodePresenter() {
if (codePresenter == null) {
codePresenter = editorPresenterProvider.get();
codePresenter.setMode(AceEditorMode.XML);
registerHandler(codePresenter.addValueChangeHandler(event -> setDirty(true)));
registerHandler(codePresenter.addFormatHandler(event -> setDirty(true)));
codePresenter.setReadOnly(readOnly);
codePresenter.getFormatAction().setAvailable(!readOnly);
if (getEntity() != null && getEntity().getData() != null) {
codePresenter.setText(getEntity().getData());
}
}
return codePresenter;
}
}
| {
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} | 9,258 |
Move In Ready. This spacious ranch plan embodies open concept living at its best. The Azalea offers 4 bedrooms and 3 full baths with an open Family Room, Kitchen, and Dining Space. The family room features a gas log fireplace with built in HD link, allowing you to mount your TV above the fireplace and hide all the wires! The Gourmet Kitchen features an expansive granite island, pendant and recessed lighting, subway tile backsplash, and Frigidaire stainless steel appliances, including dishwasher, microwave, and gas range. The breakfast area next to the kitchen leads out to the built in covered patio for year round enjoyment. The owner's suite with trey ceiling looks out to the rear of the home for privacy. The owner's bath has an oversized luxury tile shower with sitting bench, dual vanities, and large walk-in closet. Two additional bedrooms are located in the front of the home and share the hallway bathroom. The upstairs rounds out the house with a 10x20 loft area, full bathroom, and bedroom. This home is complete with 5 inch hardwood flooring in the foyer, living room, dining room, family room, kitchen, and breakfast nook. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,577 |
Q: Click Button - No Action I am trying to retrieve information from database. User enters id of the person he is looking for to ID textbox, than press display button. The grid view should show the result. But when button is pressed, nothing happens. Could anyone help or tell me what I should check?
Code for button:
protected void btnDisplay_Click(object sender, EventArgs e)
{
SqlConnection conn = new SqlConnection("Data Source="Name";Initial Catalog="Name";Integrated Security=True");
SqlCommand cmd = new SqlCommand("displayData", conn);
conn.Open();
cmd.Parameters.Add("@ID", System.Data.SqlDbType.Int).Value = Convert.ToInt32(txtID.Text);
cmd.CommandType = CommandType.StoredProcedure;
SqlDataReader rd = cmd.ExecuteReader();
grvResults.DataSource = rd;
grvResults.DataBind();
}
Here is stored procedure:
USE ["Name"]
GO
SET ANSI_NULLS ON
GO
SET QUOTED_IDENTIFIER ON
GO
ALTER procedure [dbo].[displayData] (@ID int)
as
begin
SELECT * FROM Customers WHERE ID = @ID
end
Here is display data method:
public List<Customer> displayData()
{
List<Customer> lst = new List<Customer>();
SqlConnection conn = new SqlConnection("Data Source="Name";Initial Catalog="Name";Integrated Security=True");
SqlCommand cmd = new SqlCommand("Select * From Customers", conn);
conn.Open();
SqlDataReader rd = cmd.ExecuteReader();
while (rd.Read())
{
lst.Add(new Customer()
{
ID = rd.GetInt32(0),
FName = rd.GetString(1),
LName = rd.GetString(2)
});
}
return lst;
}
aspx for button:
<asp:Button ID="btnDisplay" runat="server" Text="Display" OnClick="btnDisplay_Click" />
A: Give the following code a try in your codebehind. It should bind properly and fill a gridview. If it works, add more code from there. If it doesn't work, look at things like your connection string. Let me know.
SqlConnection conn = new SqlConnection("Data Source="?";Initial Catalog="?";Integrated Security=True");
SqlCommand cmd = new SqlCommand("SELECT * FROM Customers WHERE ID = @ID", conn);
cmd.Parameters.Add("@ID", System.Data.SqlDbType.Int).Value = Convert.ToInt32(txtID.Text);
SqlDataAdapter da = new SqlDataAdapter();
da.SelectCommand = cmd;
DataTable dt = new DataTable();
conn.Open();
da.Fill(dt);
conn.Close();
grvResults.DataSource = dt;
grvResults.DataBind();
A: have you tried to put a breakpoint in button click. Is it hitting breakpoint. If not then I will say remove
OnClick="btnDisplay_Click"
from your aspx page. And then add default click event from .cs page by selecting control and it events. Them try to debug line by line.
A: I think you are complicating yourself too much
First do this:
That should make it work. You already have a stored procedure which makes it easier. (Sorry for the pictures, I wanted to make sure you understood this)
| {
"redpajama_set_name": "RedPajamaStackExchange"
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"""Tests for the Spotlight Volume configuration plist plugin."""
import unittest
from plaso.formatters import plist # pylint: disable=unused-import
from plaso.parsers.plist_plugins import spotlight_volume
from tests import test_lib as shared_test_lib
from tests.parsers.plist_plugins import test_lib
class SpotlightVolumePluginTest(test_lib.PlistPluginTestCase):
"""Tests for the Spotlight Volume configuration plist plugin."""
@shared_test_lib.skipUnlessHasTestFile([u'VolumeConfiguration.plist'])
def testProcess(self):
"""Tests the Process function."""
plist_name = u'VolumeConfiguration.plist'
plugin_object = spotlight_volume.SpotlightVolumePlugin()
storage_writer = self._ParsePlistFileWithPlugin(
plugin_object, [plist_name], plist_name)
self.assertEqual(len(storage_writer.events), 2)
# The order in which PlistParser generates events is nondeterministic
# hence we sort the events.
events = self._GetSortedEvents(storage_writer.events)
expected_timestamps = [1369657656000000, 1372139683000000]
timestamps = sorted([event_object.timestamp for event_object in events])
self.assertEqual(timestamps, expected_timestamps)
event_object = events[1]
self.assertEqual(event_object.key, u'')
self.assertEqual(event_object.root, u'/Stores')
expected_description = (
u'Spotlight Volume 4D4BFEB5-7FE6-4033-AAAA-AAAABBBBCCCCDDDD '
u'(/.MobileBackups) activated.')
self.assertEqual(event_object.desc, expected_description)
expected_message = u'/Stores/ {0:s}'.format(expected_description)
expected_message_short = u'{0:s}...'.format(expected_message[:77])
self._TestGetMessageStrings(
event_object, expected_message, expected_message_short)
if __name__ == '__main__':
unittest.main()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,258 |
Emma desk, with wooden structure and three drawers.
It is shown here with walnut wood finish top and Avorio lacquer structure, whose distressed effect finish gives it the typical appearance of antique furniture. Sleekly completed with two Fiocco handles and a Bronzo finish Giove knob for the drawers. | {
"redpajama_set_name": "RedPajamaC4"
} | 39 |
{"url":"https:\/\/bioinformatics.stackexchange.com\/questions\/9323\/rna-seq-how-to-get-new-expression-count-after-normalization","text":"# RNA-seq: How to get new expression count after normalization\n\nI've RNA seq, Human, Paired-end data, Sample size is <40. These are aligned using STAR, RSEM processed. With RSEM I've TPM and expected counts, that is two files columns as individual IDs and row as gene names.\n\nI'm interested to normalize gene data. With edgeR tutorial (link in the end) and few other online resources I see that after following steps there's an R object that contains norm.factors (Page 15) value for each individual.\n\nI'm unable to wrap my head around it, now. If I'm interested to get normalized gene counts, can I go ahead and multiple each individual's norm.factor into its gene counts? For example, the expected count for IND1 for Gene1 is 100 and it's norm.factor is 0.80, can I say that the normalized gene count is 100*0.80=80?\n\nI'm not interested to perform voomm, or differential expression analysis.\n\nFollow up question: If I'm to sample few individuals' data from these RNA-seq should I\na) normalize all together and extract interested individuals, or,\nb) extract interested individuals and then perform normalization.\nI think since normalization uses per person library size, each person's normalization is independent of the others. Can someone please correct me if I'm wrong?\n\nhttps:\/\/www.bioconductor.org\/packages\/release\/bioc\/vignettes\/edgeR\/inst\/doc\/edgeRUsersGuide.pdf\n\nIf I'm interested to get normalized gene counts, can I go ahead and multiple each individual's norm.factor into its gene counts? For example, the expected count for IND1 for Gene1 is 100 and it's norm.factor is 0.80, can I say that the normalized gene count is 100*0.80=80?\n\nNo; you divide, but if you poke around, you can probably find a way to get edgeR to return the normalized counts.\n\nI think since normalization uses per person library size, each person's normalization is independent of the others.\n\nNot true for the default normalization used in edgeR, TMM. CPM normalization is independently determined for each sample.\n\nIn general, normalize everything together, unless you think that the algorithm assumptions might be violated by including all of them together.\n\n\u2022 I was thinking to get the quantile normalization as on the question link: bioinformatics.stackexchange.com\/questions\/2586\/\u2026 Thank you for your kind reply. :) \u2013\u00a0Death Metal Sep 6 '19 at 20:52\n\u2022 I got what I was looking for thank you @swbarnes :) poke around thing helped :D \u2013\u00a0Death Metal Sep 6 '19 at 21:07","date":"2020-09-29 11:43:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.6288326978683472, \"perplexity\": 2958.2835788294774}, \"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-2020-40\/segments\/1600401641638.83\/warc\/CC-MAIN-20200929091913-20200929121913-00388.warc.gz\"}"} | null | null |
Toponyme
Murino (en serbe cyrillique : ) est un village du nord-est du Monténégro, dans la municipalité de Plav.
Patronyme
Caterina Murino, née en Sardaigne à Cagliari le , est une actrice italienne de cinéma et un ancien mannequin. | {
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---
uid: SolidEdgeAssembly.AssemblyDocument.AddMeasureVariable(SolidEdgePart.MeasureVariableTypeConstants,SolidEdgePart.MeasureVariableValueConstants,System.Object,System.Object,System.Object)
summary:
remarks:
syntax:
parameters:
- id: Geom1
description: Specifies the first geometric element that defines the measurement.
- id: Geom2
description: Specifies the second geometric element that defines the measurement.
- id: Geom3
description: Specifies the third geometric element that defines the measurement (if necessary).
- id: Type
description: A member of the MeasureVariableTypeConstants constant set that specifies the measure variable type.
- id: ValueType
description: A member of the MeasureVariableValueConstants constant set that specifies the measure variable value.
---
| {
"redpajama_set_name": "RedPajamaGithub"
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{"url":"https:\/\/tetrisconcept.net\/threads\/the-nes-tetris-max-out-competition.2369\/","text":"# Challenge: The NES Tetris Max-Out Competition!\n\nThread in 'Competition' started by Josh Tolles, 21 May 2014.\n\nNot open for further replies.\n1. ### Josh Tolles\n\nOkay, so I was thinking about a photo of David's \"cinco\" picture and Bo's \"#4\" video. And I realized I had no idea how many maxes people had.\n\nI had thought it went like this:\n\nJonas: tons\nHarry: lots\nThor: who knows?\nQuaid: 3\nBo: 3\nBen M.: 3\nMe: 3\nDavid: 2\n\nAnd that's all I can think of at this particular moment (apologies if I forget other multi- guys). But in light of recent media, I think it is time to get a leaderboard going on this. Chime in and let me know your numbers!\n\nWill get a nice \"coded\" leaderboard up soon.\n\nSome sort of photographic evidence will be required for this competition. Exact terms are up for discussion.\n\n2. ### wasmachstdugern\n\ni have 4, none of which are as impressive as my first (from both a points\/line perspective and a contextual perspective).\n\nyet in my defense, i play 99% of my games from 19 now. i play 3 games from 18 as i get out of bed and the rest from 19 after.\n\n3. ### wasmachstdugern\n\nPS i overheard quaid saying he has four max outs during the 2013 championships- 3 level 18, one 19.\n\n4. ### wasmachstdugern\n\nPPS: harry hong admits on video here that he has over 150 max outs. i literally sat here while this conversation happened.\n\n5. ### wasmachstdugern\n\nPPPS: he also admits on this video that the real \"elite\" club is 19 max outs. he says only 3 people can do it (him, jonas, and thor). bo has a 988k 19 start that was only thwarted by bad piece distribution, ben has an 877k, and da V has and 876.... just saying.\n\n6. ### Niels_S\n\nwhat recent media?\n\n7. ### DavidV\n\ni have 8 maxouts....all lvl 18 ...left side well\ni stopped taking pics\/video after 5 or 6 .....\n\n8. ### wasmachstdugern\n\nwell played david! hearing about everyone's exploits makes me soooo excited for this years championship!\n\n9. ### DavidV\n\nHopefully we can go again this year .. ...everyone I think has gotten better even though last year's field was stacked.....pretty soon the 32 man bracket is gonna have nothing but maxers...!!\n\n10. ### Josh Tolles\n\nThese are mostly just generic numbers, but this is the logic of what I had envisioned. 19 maxes are the primary factor of ranking, other maxes are secondary, and tie breaks are decided by Level 19 highest non-max.\nCode:\n\nRank - Name - Lv19 Maxes - Highest 19 - Lv18 Maxes -\n1 - Jonas - 12 - 900,000 - 50 -\n2 - Harry - 6 - 900,000 - 150 -\n3 - Thor - 6 - 950,000 - 6 -\n4 - Quaid - 1 - 900,000 - 3 -\n=====================================================\n5 - David - 0 - [COLOR=\"Silver\"]868,995[\/COLOR] - 8 -\n6 - Bo - 0 - 931,260 - 4 -\n7 - Terry - 0 - 844,481 - 4 -\n8 - Josh - 0 - 705,540 - 4 -\n9 - Ben - 0 - 877,433 - 3 -\n10 - Matt - 0 - 500,000 - 1 -\n11 - Louie - 0 - 523,000 - 1 -\n12 - Alex - 0 - 876,420 - 1 -\n13 - Chad - 0 - 500,000 - 1\n14 - Eli - 0 - 500,000 - 1 - \n\nLast edited: 25 May 2014\n11. ### SuPa\n\nyou're making me sound so bad here! haha. at the end of the video, i was saying how impressed i was with bo's level 19 score. i even went up to him right after the video and congratulated him.\n\n12. ### wasmachstdugern\n\nlol!\nsorry supa, i didn't mean to make you sound like a bad guy. it was more meant to motivate the rest of us to get some 19 maxes.","date":"2019-07-17 01:24:53","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\": 0.2986527681350708, \"perplexity\": 4667.782730346317}, \"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-2019-30\/segments\/1563195525004.24\/warc\/CC-MAIN-20190717001433-20190717023433-00452.warc.gz\"}"} | null | null |
{"url":"https:\/\/socratic.org\/questions\/how-many-molecules-are-in-0-454-kg-of-carbon-dioxide","text":"# How many molecules are in 0.454 kg of carbon dioxide?\n\nJun 13, 2016\n\nHow many molecules in a pound of carbon dioxide?\n\n$\\frac{454 \\cdot g}{44.0 \\cdot g \\cdot m o {l}^{-} 1} \\times {N}_{A}$, where ${N}_{A} = 6.022 \\times {10}^{23} \\cdot m o {l}^{-} 1$.\n\n#### Explanation:\n\n${N}_{A}$ $=$ $\\text{Avogadro's number}$ $=$ $6.022 \\times {10}^{23} \\cdot m o {l}^{-} 1$. By definition, in $44.0 \\cdot g$ of carbon dioxide there are $\\text{Avogadro's number}$ of carbon atoms, and $2 \\times \\text{Avogadro's number}$ of oxygen atoms.\n\nSo we simply calculate the product:\n\n$\\frac{454 \\cdot g}{44.0 \\cdot g \\cdot m o {l}^{-} 1} \\times {N}_{A}$ $\\cong$ $10 {N}_{A}$","date":"2020-05-31 10:23:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 13, \"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\": 0.9428188800811768, \"perplexity\": 1166.4342704211597}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347413097.49\/warc\/CC-MAIN-20200531085047-20200531115047-00121.warc.gz\"}"} | null | null |
If you're looking for a Octa Core 2K Display Phones, your list would be filled with plenty of options. While there are plenty of good mobile phones, Motorola is leading the market with some of the Octa Core 2K Display Phones that fits your budget and gives the good performance. In fact, Motorola has plenty of options of its own, that offer great battery backup, fast performance, and a decent camera, all along Octa Core 2K Display Phones. Take a look at our well curated price list of Motorola Octa Core 2K Display Phones .
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The Motorola DROID Turbo 2 is the right candidate for the best Motorola Octa Core 2K Display Phones Price List. It has a Qualcomm Snapdragon 810 MSM8994 and 3 GB of RAM. Combined together, you get a potent performance from the Motorola DROID Turbo 2. You also get 32 GB for your files and multimedia content. As for the camera you get 21 MP camera setup on the rear and a 5 MP on the front. The battery inside is in capacity and becomes the final piece in making Motorola DROID Turbo 2 one of the top Motorola Octa Core 2K Display Phones Price List.
The Motorola X Force is the right candidate for the best Motorola Octa Core 2K Display Phones Price List. It has a Qualcomm Snapdragon 810 MSM8994 and 3 GB of RAM. Combined together, you get a potent performance from the Motorola X Force. You also get 64 GB for your files and multimedia content. As for the camera you get 21 MP camera setup on the rear and a 5 MP on the front. The battery inside is in capacity and becomes the final piece in making Motorola X Force one of the top Motorola Octa Core 2K Display Phones Price List.
Compare 3 of these Motorola Octa Core 2K Display Phones Price List at a time - Compare mobile phones based on their Prices, & Specifications including Connectivity, Communication, Software, Battery, User Reviews, Rating and more. | {
"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
\label{sec:Introduction}
The Electron Beam Ion Trap (EBIT\cite{levine_electron_1988, vogel_design_1990, becker_electron_2010}) is a powerful laboratory instrument designed to create, store, and excite highly charged ions that may be studied through their characteristic emission spectra. In the EBIT, electrons are generated by an electron gun, and a series of electrodes along the central axis of the machine accelerates them to high kinetic energies. A set of drift tubes along the beam path sets up a potential well that is used to trap the ions axially. High field magnets, historically liquid helium cooled superconducting magnets, are used primarily to compress the electron beam, which increases the strength of the radial confinement that the negatively charged electron beam provides to the positively charged ion cloud. At the end of the beam path, a collector, which is also typically actively cooled, dissipates the remaining energy of the intense beam. Many different mechanisms can be used to inject ions into the electron beam of the EBIT. A ballistic gas injection system can be used for studying charged ions of gaseous neutral atoms such as nitrogen, neon, or argon. For metal ions, a more complicated tool must be used, such as a Metal Vapor Vacuum Arc (MeVVA\cite{brown_metal_1992, holland_low_2005}) external ion source. The same electron beam used to trap and ionize the ions also collisionally excites them.
The first EBIT was developed at the Lawrence Livermore National Laboratory (LLNL) in 1985\cite{levine_electron_1988} and was soon after used to take atomic spectroscopy measurements of highly charged barium\cite{marrs_measurement_1988}. This promising new instrument led a handful of groups around the world to build their own EBIT systems. One such system was built at the National Institute of Standards and Technology (NIST) in 1993 and quickly began producing results\cite{gillaspy_first_1997}. Since its inception, the NIST EBIT has been used to take measurements across a variety of fields, including fundamental atomic physics\cite{ralchenko_accurate_2006, gillaspy_precision_2014, payne_helium-like_2014}, spectroscopy of highly-charged ions for fusion and lithography\cite{draganic_euv_2011, ralchenko_spectroscopy_2011, kilbane_euv_2012}, laboratory astrophysics\cite{silver_laboratory_2000, takacs_astrophysics_2003, tan_electron_2005, chen_3c/3d_2006, gillaspy_fe_2011}, and nuclear physics\cite{silwal_measuring_2018}.
Low-temperature x-ray microcalorimeters that combine high resolving power comparable to wavelength dispersive spectrometers with high broadband x-ray collection efficiency comparable to semiconductor based solid-state detectors are particularly well suited for measurements with an EBIT\cite{betancourt-martinez_transition-edge_2014, doriese_practical_2017}. First, an EBIT's tunable energy electron beam is capable of exciting a wide range of charge states. The spectral features from these charge states can be found at energies up to 10s--100s of keV, depending on the highest charge state the particular EBIT can achieve. The broad energy range of an x-ray microcalorimeter allows for the study of a wide array of highly charged ions with a single spectrometer. Furthermore, the exceptional energy resolution of an x-ray microcalorimeter also allows for high precision studies of line emission structures and allows us to resolve previously overlapping features. Finally, the single photon detection nature of a microcalorimeter allows for time-resolved studies with the EBIT. Measurements with microcalorimeters have been highly successful at multiple EBIT or other highly charged ion facilities (see Ref.~\citenum{porter_astro-e2_2004, porter_xrs_2008, beiersdorfer_brief_2008, porter_performance_2008, shen_status_2011, kraft-bermuth_microcalorimeters_2018} and the references therein). Generally, 10s of few 100~$\mu$m scale microcalorimeters are operated within a spectrometer and energy resolutions of $\sim$5~eV at 6~keV are typically observed. At the NIST EBIT, we have utilized an array of 4 x-ray microcalorimeters based on neutron transmutation doped germanium (NTD-Ge\cite{larrabee_ntd_1984}) thermistors and tin absorbers developed by the Harvard Smithsonian Astrophysical Observatory (SAO)\cite{bandler_ntd-ge-based_2000, tan_electron_2005, chen_3c/3d_2006, ralchenko_accurate_2006, gillaspy_fe_2011}. These NTD-Ge microcalorimeters individually have an active area of 350~$\mu$m~$\times$~350~$\mu$m and show a resolution of 4.5~eV at 6~keV during operation with the EBIT\cite{ tan_electron_2005}.
Advances in low temperature detector technology have recently increased the capabilities of spectrometers based on x-ray microcalorimeters, both in terms of total array active area and per-pixel performance. One such technology that is the basis for the new spectrometer at the NIST EBIT is the Transition-edge Sensor (TES\cite{andrews_attenuated_1942, irwin_x-ray_1996}). TES-based x-ray microcalorimeter spectrometers with 100s of pixels have been used for applications in x-ray astronomy\cite{holland_transition_1999, den_herder_x-ray_2012}, beamline science\cite{heates_collaboration_first_2016, doriese_practical_2017, lee_transition-edge_2019}, and tabletop spectroscopy\cite{miaja-avila_ultrafast_2016, oneil_ultrafast_2017}, among other fields. TES microcalorimeters regularly achieve a resolving power of better than 1 part in a 1000 for x-rays in the few keV range\cite{smith_small_2012, bandler_advances_2013, lee_fine_2015, uhlig_high-resolution_2015, ullom_review_2015, doriese_practical_2017} and can measure the arrival time of an x-ray with $\sim$1~$\mu$s timing resolution\cite{heates_collaboration_first_2016}. Other low temperature x-ray microcalorimeter technologies based on Microwave Kinetic Inductance Detectors (MKIDs\cite{day_broadband_2003, ulbricht_highly_2015}) and Metallic Magnetic Calorimeters (MMCs\cite{fleischmann_metallic_2005, fleischmann_physics_2009}) are developing rapidly as well. At the time of this writing, TES-based microcalorimeters have achieved better resolution compared to MKID-based microcalorimeters, and TES multiplexing techniques are more mature compared to those of MMCs. In addition to the NIST EBIT, some of these more recent advances in low temperature detector technology are being utilized in facilities for other highly charged ion experiments. The EBIT at Lawrence Livermore National Laboratory (LLNL) is in the process of adding a TES x-ray spectrometer to their detector suite\cite{betancourt-martinez_transition-edge_2014}. Also, MMCs with high dynamic range have recently been used to study highly charged ions at the Experimental Storage Ring (ESR) at the GSI Helmholtz Centre for Heavy Ion Research\cite{kraft-bermuth_microcalorimeters_2018}.
Here we detail the design, commissioning, and first-light measurements of the NIST EBIT TES Spectrometer (NETS). NETS has a factor of $>30$ increase in the active area of the array while also exhibiting a modest ($<2 \times$) improvement in energy resolution when compared to the to NTD-Ge microcalorimeter spectrometer previously used at the NIST EBIT. In Sec.~\ref{sec:TES_Spectrometer}, we explain the physics behind TES-based x-ray microcalorimeters and the specifics of the spectrometer developed for the NIST EBIT. We move on to show how the spectrometer was integrated with the rest of the EBIT in Sec.~\ref{sec:Integration}. We then discuss first-light measurements done with NETS (Sec.~\ref{sec:Measurements}) and techniques used to analyze these data (Sec.~\ref{sec:Reduction}). Finally, in Sec.~\ref{sec:Results} we discuss spectra acquired during the first-light run including the achieved count rates, energy resolution, line center accuracy, and time resolution.
\section{TES Spectrometer}
\label{sec:TES_Spectrometer}
\subsection{TES Microcalorimeter Principles of Operation}
The microcalorimeters in NETS use TESs as the sensing elements, weakly coupled to a thermal bath. The devices are cooled to low temperatures, often below 100~mK, to minimize thermal noise and maximize detector sensitivity. Generally, temperatures below the critical temperature ($T_C$) of the TESs are used, and the TESs are then biased into their superconducting-to-normal transition. Within this narrow region, a TES has a high temperature coefficient of resistance (e.g., NIST TES designs typically have $\partial \log R / \partial \log T$ values of $\lesssim$100 at their bias point\cite{doriese_practical_2017}), allowing for a sensitive measurement of the temperature change that occurs during a photon absorption event. A TES used in x-ray microcalorimeters is typically voltage biased into its transition, providing negative electrothermal feedback (power in the TES decreases as absorbed photons raise its resistance). When the TES is voltage-biased, the measured signal from an absorbed photon is a pulse of decreased TES current. TES operation is outlined in Fig.~\ref{fig:TES_Operation}. The physics and operation of TES-based x-ray microcalorimeters are explained in much greater detail in a number of publications\cite{irwin_transition-edge_2005, ullom_review_2015, doriese_practical_2017}.
\begin{figure}
\centering
\begin{subfigure}{\linewidth}
\includegraphics[width=0.9\linewidth]{TESCircuit}
\end{subfigure}
\par\medskip
\begin{subfigure}{\linewidth}
\includegraphics[width=1.0\linewidth]{Snout}
\end{subfigure}
\caption{\label{fig:TES_Operation} (A)~TES circuit diagram. The TES is voltage biased into its superconducting transition region by sourcing current through a 380~$\mu \Omega$ shunt resistor, $R_{sh}$. When an x-ray event occurs, the TES temperature rises proportionally to the x-ray energy, pushing the TES higher into its transition and raising its resistance (represented by the variable resistor, $R_{TES}$, in the diagram). This results in a pulse of decreased current. An inductor, L, is used to adjust the rise time of this pulse. Each TES is inductively coupled to and read out using a first stage SQUID, and a SQUID-based multiplexing scheme is used to read out the entire array of TESs. (B)~Detector and cold stage readout assembly, including the TES array with collimator and 8 sets of multiplexer (MUX) and interface chips for the 8 independent readout columns. The dashed boxes in the top panel correspond to the different types of physical chips labeled in the bottom panel.}
\end{figure}
In order to maximize x-ray collection efficiency, the TES is coupled to a thicker absorber made with a high-Z material, necessary for stopping x-rays with energies above a few hundred eV while maintaining the TES's superconducting and thermodynamic properties. The thermodynamic properties of the absorber, such as the heat capacity, can then be tuned separately from that of the TES, allowing for a simpler optimization of the devices to target a specific energy range. The device is typically fabricated on top of a membrane in order to control thermal conductance (speed of current pulses) and minimize the amount of energy going into phonons escaping into the substrate instead of measurable energy in the sensor. This is necessary for maintaining a high signal-to-noise ratio (SNR) in the pulses and therefore high resolving power.
The readout of TES microcalorimeter arrays is typically accomplished with Superconducting Quantum Intereference Devices (SQUIDs\cite{clarke_squid_2004}). Here, SQUIDs are used in an ammeter mode to measure and amplify the current pulses. In cryogenic detectors, it is often necessary to use a multiplexing scheme to reduce the complexity in wiring and the thermal load inside the cryostat, and SQUIDs are also used for this purpose. A variety of SQUID-based multiplexing architectures\cite{kiviranta_squid-based_2002} can be used, but the most mature readout method is time division multiplexing (TDM\cite{de_korte_time-division_2003, doriese_developments_2016}), which involves sequentially addressing the readout SQUIDs for a group of detectors. NIST has routinely used this method to multiplex $\sim$30 detectors (rows), with 8 parallel readout channels (columns), enabling operation of large arrays of TES microcalorimeters\cite{doriese_practical_2017, bennett_high_2012}.
\subsection{Detector Array}
\label{subsec:DetectorArray}
NETS contains an array of nominally identical TES x-ray microcalorimeters, the design of which is shown in Fig.~\ref{fig:TES_Geometry}. The design is an evolution of previous NIST designs, which use a sputtered MoCu bilayer for the TES material and evaporated Bi for the x-ray absorber\cite{hilton_microfabricated_2001, ullom_optimized_2005, swetz_current_2012, doriese_developments_2016}. Although these types detectors generally show roughly part in a thousand energy resolution, the evaporated Bi absorbers cause the detectors to suffer from non-Gaussian response functions ($\sim$20--30\% of x-ray events in low energy tails)\cite{yan_eliminating_2017}. It was found that Au absorbers could be used in place of the Bi absorbers to significantly reduce the low energy tail component of the detector response\cite{yan_eliminating_2017}. In the past designs, the Bi absorbers were evaporated directly on top of the TES thermistors. This cannot be done with the Au absorbers as Au is a metal (Bi is a semimetal) and would cause the TES superconducting properties to be intrinsically coupled to those of the Au absorber through the proximity effect\cite{werthamer_theory_1963, martinis_calculation_2000}, which is inconvenient from a design tuning and fabrication standpoint. Instead, the Au absorber is deposited laterally away from the TES thermistor in a ``sidecar'' design.
\begin{figure}
\includegraphics[width=0.85\linewidth]{TES_Geometry}
\caption{\label{fig:TES_Geometry} Single TES microcalorimeter design used in NETS. Here, the yellow square feature is the Au absorber, with the two shades of colors used to indicate the two separate Au layers. The TES bilayer consists of Mo (blue) and Cu (pink). The Cu banks, bars, and connection to the absorber in this same area are marked in orange. The entire design is sitting on a SiN absorber, with a frame of perforations encircling the absorber and sensor. Superconducting Mo traces run from both ends of the sensor to bond pads off the membrane at the edges of the chip, providing a route for the sensors to be connected to chips containing the bias circuit and SQUID amplifiers.}
\end{figure}
The sensor material is a sputtered MoCu bilayer with 215~nm of Cu on top of 65~nm of Mo. The thicknesses of the Cu and the Mo are tuned to target a specific superconducting transition temperature, $T_C$, via the proximity effect\cite{werthamer_theory_1963, martinis_calculation_2000}. The sample wafer is rotated during sputtering to maximize uniformity, and the sensors in this particular array have $T_C$ values in the range of 111--112~mK. This $T_C$ was chosen as a trade-off between the the theoretical energy resolution we could hope to achieve and cryostat hold time (see Sec.~\ref{subsec:Cryostat}). An additional Cu layer of thickness 419~nm is deposited on top of the sensor and patterned into banks, used to steer current toward the leads, and interdigitated stripes, used to control the transition shape and reduce excess noise\cite{lindeman_characterization_2004}. This Cu layer also thermally connects the sensor to the absorber. In addition to its use in the sensor bilayer, the Mo is used for superconducting leads and traces running off the membrane.
The absorber is made from Au that is electron beam evaporated in two separate layers, first 186~nm and then 779~nm thick, resulting in a total absorber thickness of 965~nm. The wafer is also rotated during this deposition and the evaporation tool had previously been characterized for roughly 3\% variation across a 76~mm wafer (better than 1\% variation across a single detector array chip). The absolute thickness had been measured with a stylus profilometer with $\lesssim$3\% uncertainty. Sec.~\ref{subsec:QE} contains a discussion of these absorber thickness uncertainties as they relate to the quantum efficiency. The deposition is done in two separate layers to reduce step coverage issues when coupling the absorber to the sensor through the Cu and to better control the thermal conductivity. The absorber area is 340~$\mu$m$\times$340~$\mu$m.
When choosing the absorber thickness and area, trade-offs between dynamic range, quantum efficiency, and active area had to be made. In particular, the total absorber volume (sets the heat capacity and dynamic range) was chosen such that the detectors could operate in the linear region of their superconducting-to-normal transitions when measuring photons in the $\lesssim 15$~keV range. Beyond this region, the x-ray pulses would begin saturating and energy resolution would quickly degrade. The highest charge state that the NIST EBIT can produce is H-like Kr, with the brightest line at $\sim$13.5~keV, so we wanted to ensure that the detectors would be capable of measuring photons at these energies without saturating. The speed of the room temperature readout electronics (see Sec.~\ref{subsec:Readout}) imposes a limit on the slew rate of the pulses, which for these detectors occurs at roughly 10~keV. Reading out pulses above this $\sim$10~keV limit requires reducing the number of detectors to increase the readout speed of the remaining detectors and raise the slew rate limit. We note that the majority of our planned measurements are of highly charged ions with line energies below 10~keV. Once the saturation energy (total absorber volume) is chosen, we maximize the absorber area given the constraints of the array layout.
\begin{figure}
\begin{center}
\begin{subfigure}{\linewidth}
\includegraphics[width=1.0\linewidth]{Array.jpg}
\end{subfigure}
\begin{subfigure}{\linewidth}
\includegraphics[width=0.975\linewidth]{PixelMap}
\end{subfigure}
\end{center}
\caption{\label{fig:PixelMap} (A)~The TES array used in NETS. The aperture chip used to prevent x-rays from being absorbed by areas outside the TES absorbers is also shown (displayed on top of a penny, for scale). (B)~Pixel map indicating absorber size (rather than aperture size) and locations. The different colors represent the 8 readout columns, with 24 detectors in each column. A circle of radius 5.30~mm fully encloses all detector absorbers.}
\end{figure}
NETS houses a total of 192 TES pixels arranged in the roughly circular pattern depicted in Fig.~\ref{fig:PixelMap}. Here, 192 pixels are used instead of the 240 pixels that are used in the NIST evaporated Bi absorber design due to the increased pitch that is required when separated the absorber from the TES. Reducing the pixel count to 192 was required in order to keep the detector chip size constant and easily integrate it with our standard assembly package and readout chips. Out of fabrication and assembly, 166 of the 192 total possible detectors (86\%) are working and capable of detecting x-rays. Additional detectors may be flagged as bad in the data reduction steps depending on the data quality and restrictions of a particular analysis, as will be discussed in Sec.~\ref{sec:Reduction}.
\subsection{Readout}
\label{subsec:Readout}
NETS utilizes the TDM readout scheme with a two stage SQUID architecture\cite{doriese_developments_2016}. In this scheme, each detector gets its own first stage SQUID (SQ1) amplifier, and the signals from all detectors in a given readout column are coupled into a single second amplification stage SQUID array (SA). The SQ1s in a given column of detectors are addressed sequentially and have a maximum critical current of 10~$\mu$A. They are designed to have an asymmetric periodic response and are operated at the steeper of the two asymmetric slopes to maximize gain. The SAs contain 6 banks of 64 SQUIDs in a $2 \times 3$ (series~$\times$~parallel) configuration, with a maximum critical current of 60~$\mu$A. Here, the direction of the SA feedback coil current was reversed relative to TES spectrometers previously deployed by NIST, which was recently found to reduce SQUID amplifier instability and crosstalk between columns\cite{durkin_reducing_2019}. At room temperature, the signals are routed through a set of analog interface electronics and then through the digital feedback electronics\cite{reintsema_prototype_2003, doriese_time-division_2004}. The room temperature electronics orchestrate the TDM readout and maintain a separate flux locked loop for each SQ1 such that the feedback value is a measure of the TES current. In NETS, the detectors are divided into 8 parallel readout columns, with 24 rows (detectors) per column. NIST TDM multiplexers are described in greater detail in other publications\cite{reintsema_prototype_2003, doriese_time-division_2004, doriese_developments_2016}.
Data acquisition (DAQ) software is used to set SQUID and TES parameters, configure triggering conditions, and write raw pulse records (x-ray events). Typically, the SA bias current is set to $I_{c,max}$ (defined as the current which maximizes the amplitude versus input flux), while the SQ1 bias current is set to $\sim1.5\times I_{c,max}$ to reduce settling time after row switching. Other multiplexer parameters, such as those defining the regions of operation of the amplifiers, can also be tuned with the DAQ software. The TESs were biased in the superconducting transition at about 15\% of their normal-state resistance ($R_N$), which we found maximizes the resolving power of the detectors. More specifically, each readout column has a single bias line that is shared by all detectors within that column. The bias voltage for a column is set such that the column median detector resistance is at 15\%~$R_N$. The average variability in resistance at the bias point for a given column is 1.4\%~$R_N$. Generally, we see little if any degradation in resolution in detectors with bias points that differ from the column median by a few~\%~$R_N$ The DAQ software can also be used to decrease the number of columns and rows to read out, potentially lowering electrical crosstalk and improving the energy resolution at the cost of x-ray collection efficiency. Lowering the number of rows is also necessary when measuring high energy pulses ($\gtrsim 10$~keV), ensuring that the feedback loop of the room temperature electronics can keep up with the higher slew rates of these pulses.
The DAQ software is also used to configure an edge trigger for the collection of pulse data. Here, the edge triggering threshold (minimum slope in the feedback signal) is set far below the low energy cutoff in our quantum efficiency curve ($\sim$300~eV, see Fig.~\ref{fig:Efficiency}). With proper optimization, this triggering threshold can be set as low as a few 10s of eV (depending on desired $\sigma$ level of detector resolution), though this is rarely useful given the low energy cutoff in quantum efficiency. When taking noise measurements, the data is continuously streamed rather than run on a trigger. Raw pulse and noise records are written to disk in a binary format and are later analyzed using the data reduction pipeline described in Sec.~\ref{sec:Reduction}.
\subsection{Cryostat}
\label{subsec:Cryostat}
NETS utilizes an Adiabatic Demagnetization Refrigerator (ADR\cite{hagmann_adiabatic_1995, pobell_matter_2007}) to cool the detectors and SQUID electronics. This ADR was primarily chosen for its compact and portable design, and it is generally fairly simple to mate to the x-ray source\cite{doriese_practical_2017}. The cryostat consists of a large rectangular section containing the SA amplifiers and majority of the cooling hardware and a cylindrical protrusion, hereafter referred to as the snout, housing the detector package and SQ1 amplifiers. The bulk rectangular section is largely a commercially available component whereas the snout is custom built. Within the outer vacuum jacket are four thermal stages, each named after the nominal base temperature that stage reaches; these are the 50~K, 3~K, 1~K, and 50~mK stages. A pulse tube cryocooler is used to cool the 50~K stage to 50~K and the remaining stages to 3~K, eliminating the need for any liquid cryogens. Cooling the 1~K and 50~mK stages below 3~K is not continuous and involves a magnet ramp/deramp cycle (hereafter, ADR cycle). For this purpose, the 1~K stage contains a gadolinium gallium garnet (GGG) paramagnetic salt pill and the 50~mK stage contains a ferric ammonium alum (FAA) paramagnetic salt pill. A high field superconducting magnet is ramped to 3~T to align the magnetic moments in both salt pills while the stages are heat sunk to the 3~K stage. They are then thermally isolated from the 3~K stage, and the magnetic field at the salt pills is ramped down, a process which adiabatically disaligns the magnetic moments and requires them to absorb energy (heat from the salt pills). At the end of this cooling cycle, the NETS 1~K (GGG) stage reaches a base temperature $\sim$550~mK and 50~mK (FAA) stage reaches a base temperature of $\sim$45~mK. The ADR cycle is accomplished in roughly 1.5~hours.
In order to raise the temperature of the 50~mK stage from the base temperature to the detector operating temperature (here, 70~mK), we use the superconducting magnet to apply a small magnetic field ($\sim30$~mT). This field is controlled by a feedback loop to stabilize the temperature read out by the 50~mK thermometry to the operating temperature. The temperature of the 50~mK stage can be controlled at 70~mK for over 24~hours (hold time) before the various power loads on the stage cause the zero magnetic field temperature to exceed the operating temperature and the ADR cycle must be done again. Here, the operating temperature of 70~mK was chosen as a trade-off between detector performance and hold time. Generally we find improvements to detector performance through lower operating temperature to saturate when the operating temperature is $\sim$30~mK below $T_C$. Also, a hold time of around 24~hours is ideal for the typical work schedule at the NIST EBIT, allowing us to cycle the magnet at the start of the day and collect uninterrupted data with the spectrometer until EBIT operations end for the day. When controlling the 50~mK stage to 70~mK, we see a 4~$\mu$K standard deviation in the measured temperature. This is typical for systems we have deployed in the field\cite{doriese_practical_2017}, and we have found this level of stability to not limit single pixel energy resolution.
The array package (see Fig.~\ref{fig:TES_Operation}) is placed at the front of the 50~mK stage snout, simplifying the mating and minimizing the distance between the spectrometer and the x-ray source. Interface chips and SQ1 amplifiers are also placed in the 50~mK snout. The 1~K stage cools the SA amplifier and associated circuitry.
Filters are positioned at the front of the 50~mK, 3~K, and 50~K snouts, which allow for transmission of x-rays while reflecting infrared (IR) radiation. These filters are clamped and glued into Al frames, which are then screwed into depressions at the front of the snouts and lined with Al tape at the edges to prevent IR light leaks. The main component of these filters is a thin ($\sim$110~nm), circular Al film with a diameter of 17.1~mm. The Al films on the 50~mK and 3~K snouts are free-standing, whereas the Al film on the 50~K stage is backed by a rectangular mesh composed of Ni. This mesh has a pitch of 363~$\mu$m with bars that are 15~$\mu$m thick and 30~$\mu$m wide. Al was chosen as the film material due to ease of fabrication and because it is a low-Z material with a high x-ray transmittance in the NETS energy range.
We expect the filter on the 50~K stage to reflect $\gtrsim$99\% of the 300~K radiation on this stage\cite{ordal_optical_1988}, and we expect higher reflectance of the 3~K and 50~mK filters to 50~K and 3~K radiation, respectively. The remainder of the IR radiation is expected to be absorbed by the filters, which can heat the filters and lead to increased blackbody radiation. For this reason, the filters need to be thick enough to thermalize well with the surrounding stage and not cause excess thermal loading to the innermost stage. Excess thermal loading can degrade the energy resolution of the TES pixels. The 50~K stage filter, which is exposed to more radiation than the other filters, includes a Ni mesh to improve thermalization. A thicker ($\sim$200~nm) Al film without a Ni mesh could be used in the case that the Ni mesh limits our ability to reach a target uncertainty in a line intensity measurement. One reason we attempted to maximize low energy transmission is the possibility to take simultaneous measurements on lines with energies in this region with NETS and with an existing extreme ultraviolet (EUV) spectrometer\cite{gillaspy_visible_2004} deployed at the NIST EBIT, which is limited to energies of up to $\sim$500~eV.
A similar x-ray window is placed at the front of the outermost snout (vacuum snout), but this one is also capable of holding atmospheric pressure and has a larger diameter (25.4~mm). Here, the Al film is $\sim$65~nm thick and is backed by a $\sim$700~nm thick polyimide film with a by weight composition of 3\%~H, 71\%~C, 8\%~N, and 18\%~O. This window also contains a roughly hexagonal mesh composed of 304 stainless steel. This mesh is on a 360~$\mu$m pitch and has bars that are 100~$\mu$m thick and 30~$\mu$m wide. The errors in the filter/window parameters as they relate to the system quantum efficiency are discussed in Sec.~\ref{subsec:QE}.
\section{Integration with the EBIT}
\label{sec:Integration}
\subsection{Overview}
NETS was integrated with the EBIT as shown in Fig.~\ref{fig:SystemIntegration}. The spectrometer is held up with a rigid stand through a single point hanging connection at the top of the cryostat and a set of screws to fix the rotation. The stand is used align the spectrometer to the viewport of the EBIT and hold up a fraction of the weight of the external calibration source after the spectrometer has been fixed to the EBIT. A remote motor (not pictured) for the pulse tube cryocooler was placed on a rigid stand at the same height as the pulse tube cryocooler, with a He gas hose connecting the two components at roughly a 90$^{\circ}$ angle to minimize vibrations at the cryostat.
\begin{figure}
\begin{center}
\begin{subfigure}{\linewidth}
\includegraphics[width=0.9\linewidth]{EBIT_Overview}
\end{subfigure}
\par\medskip
\begin{subfigure}{\linewidth}
\includegraphics[width=0.9\linewidth]{SystemIntegration}
\end{subfigure}
\end{center}
\caption{\label{fig:SystemIntegration} (A)~Photograph and (B)~Computer aided design (CAD) rendering of the TES x-ray spectrometer mounted to the NIST EBIT. The spectrometer (right) is mated to a viewport in the midplane of the EBIT (left) through a 6-way vacuum cross that also provides access to an external x-ray calibration source (center). In the CAD rendering, the seven remaining EBIT viewports mated to other detectors and EBIT components as well as external magnetic shielding around the spectrometer snout are not pictured, for clarity.}
\end{figure}
\begin{figure*}
\begin{center}
\includegraphics[width=1.0\linewidth]{OpticalPath2D}
\end{center}
\caption{\label{fig:OpticalPath2D} Top-down cross-section schematic of NETS mounted to the NIST EBIT through an external calibration source chamber. The total distance between the EBIT trap center and the TES array is 520~mm, with vacuum windows positioned 220~mm and 500~mm away from the EBIT trap center. IR blocking filters internal to the spectrometer cryostat are positioned $\sim$4~mm apart, with the innermost 50~mK filter positioned $\sim$4~mm from the TES array. Note that the solid angle is limited by the radius of the active area of the array and is not obstructed by other components along the x-ray beam path, such as the drift tube slits or gate valve walls.}
\end{figure*}
The front of the TES snout is mounted to a 6-way cross that houses retractable metal targets for an external calibration source (Sec.~\ref{subsec:CalibrationSource}). At the opposite end of this 6-way cross mates to a viewport in the midplane of the EBIT through a second vacuum window (same designed composition as window at front of TES snout), mechanical valve, and short bellows (16~mm length). A top-down cross-section of this arrangement is shown in Fig.~\ref{fig:OpticalPath2D}. The current setup therefore includes three separate vacuum spaces during regular operation: the NETS cryostat, the intermediate area between the cryostat and the EBIT that includes the calibration source, and the EBIT chamber. The separate vacuum space of the NETS cryostat was used to ensure a lower base pressure in the EBIT chamber. Here, the cryostat is designed only for high vacuum (HV) and uses Klein Flange (KF) fittings whereas the calibration source and EBIT chamber are designed for UHV with ConFlat (CF) fittings. The second vacuum window between the EBIT chamber and calibration source was used as an additional precaution during the original commissioning run. In the case of a vacuum failure in the calibration source chamber during operation of the EBIT and NETS, the conditions in the EBIT chamber and spectrometer cryostat would be preserved. After gaining some confidence with our setup, one of these vacuum windows may be removed to improve collection efficiency (still need one vacuum window to separate HV cryostat side from UHV EBIT side). With the current configuration, the typical operating pressure in the EBIT is $\lesssim 10^{-8}$~Pa ($10^{-10}$~Torr) when the field coils are running and actively cooled with liquid He.
\subsection{X-ray Collection Efficiency}
\label{subsec:QE}
An understanding of the quantum efficiency (QE) curve and associated uncertainties will be especially important when the scientific topic of interest is the measurement of line intensity ratios. Using the xraylib\cite{schoonjans_xraylib_2011} library and thickness and elemental density measurements of the windows and filters, we calculated their expected transmittance. We combined this transmittance with the expected absorption of a 965~nm thick Au absorber to calculate a total detector QE. The average filter transmittance, Au absorption, and combined QE curves are shown in Fig.~\ref{fig:Efficiency}. Here, different detectors will have varying QE curves primarily due to the geometry of the mesh structures supporting the 50~K filter and vacuum windows. In calculating the transmittance curve in Fig.~\ref{fig:Efficiency}, the transmittances were averaged across the filter. The QE has its peak of 47.8\% at an x-ray energy of 3.43~keV.
\begin{figure}
\includegraphics[width=1.0\linewidth]{Efficiency}
\caption{\label{fig:Efficiency} \textit{Upper}: Calculated average QE for a detector in NETS, including losses due to transmittance through the 2 vacuum windows and the 3 IR-blocking filters as well as the properties of the Au absorbers. The absorption curve is labeled with the M and L absorption edges of the Au absorbers and the filter transmittance curve is labeled with the the most prominent absorption K edges of materials in the filters, primarily Al (from film) and Ni (from mesh). The reduction in collection efficiency due to the aperture chip masking of the absorber area ($\sim$32\%) is not included in this figure. \textit{Lower}: Low energy portion of the QE curve, which is dominated largely by the filter transmittance. In addition to the Al K edge, the K edges of C, N, and O (from polymer backing) are visible here.}
\end{figure}
As for the uncertainties in these parameters, the Au absorbers have an absolute thickness uncertainty of $\lesssim$3\% with $\lesssim$1\% thickness variation across the array (see Sec.~\ref{subsec:DetectorArray}). In the filters and windows, the uncertainty in the absolute Al thickness is 2\% in the IR-blocking filters and 3\% in the vacuum windows. The uncertainty in the elemental area densities of the polyimide is 2\%. Both the Al film and polyimide display minimal spatial variation. The combined uncertainty from these components is fairly small ($<$1\% $\Delta$QE at 3.43~keV, including pixel-to-pixel variations), but the uncertainty added by the stainless steel meshes on the vacuum windows and the Ni mesh on the 50~K filter is more significant.
The stainless steel meshes on the vacuum windows are fairly thick ($\sim$100~$\mu$m), and x-ray transmission through the mesh is expected to be low ($<$1\% up to $\sim$15~keV). The main uncertainty here comes in the open area of the mesh (80--82\%), and although this will create pixel-dependent variations in the illumination, due to the thickness of mesh, these variations are not expected to be energy-dependent. The Ni mesh on the 50~K filter similarly has an uncertainty associated with the open area (82--84\%) of the mesh. In addition, the Ni mesh is relatively thin (15$\pm$2~$\mu$m), and there can be fairly large transmission of x-rays through the mesh, especially near the K absorption edge (up to $\sim$50\%) and above $\sim$10~keV ($\sim$5--65\% between 10--20~keV). This will cause energy-dependent variations in illumination across different pixels on the array. When measuring line intensity ratios, these effects are in part mitigated by coadding (averaging) results across all detectors. If necessary for a given application, a series of measurements at various x-ray energies can be taken to map out the mesh positions relative to the TES microcalorimeters, given an expected no-mesh illumination pattern.
The distance between the face of the TES array and the trap center is nominally 520~mm when the system is under vacuum, with 1--2~cm of leeway due to contraction of the bellows and exact positioning of the array package along the 50~mK stage during assembly. The active area of the array fits within a 5.3~mm radius circle, but only x-rays that hit an absorber are accurately detected by the spectrometer. A collimator with an array of 280~$\mu$m$\times$280~$\mu$m apertures, slightly smaller than the absorber length and width, is placed over the detector array to ensure x-rays making it to the array only hit the absorbers. The area of the individual apertures is smaller than the detector absorber area (340~$\mu$m$\times$340~$\mu$m) in order to account for any inaccuracy in alignment when mounting the aperture chip. The active area of the array is therefore reduced to 15.1~mm$^2$, down from the 22.2~mm$^2$ that would be expected from the total area of the absorbers ($\sim$32\% reduction). This results in a solid angle of $5.57 \times 10^{-5}$~sr = 0.183~deg$^2$. In other words, assuming isotropic emission, a fraction $4.43 \times 10^{-6}$ of x-rays originating from the trap center can make their way to a detector, and the chance that one of these x-rays is actually absorbed by a detector is determined by the QE curve in Fig.~\ref{fig:Efficiency}. It should be noted that the idea of solid angle may not be perfectly correct here as the x-ray emitting region is more an extended line source rather than a simple point source. The beam diameter is $<100$~$\mu$m, and a roughly 20~mm length of the trap is visible through the vertical drift tube slits. Based on this beam shape, the distance of the source to the detectors, and radius of the active area of the array, we would expect a uniform x-ray illumination of the pixels on the array if the meshes backing some of the filters and windows were not present.
\subsection{Calibration with External X-ray Sources}
\label{subsec:CalibrationSource}
We plan to use well known reference lines to generate calibration curves for the NETS microcalorimeters. Although the EBIT is capable of generating many such lines, it is limited to the gas and metal targets accessible to the EBIT at the time of measurement, and exchanging these targets can be a lengthy process. This leads to regions in energy space in which the EBIT cannot generate lines with well known positions, which can be problematic if a measurement requires energy calibration in such a region. A more flexible method for generating well known x-rays lines for calibration was desired.
For this reason, there is a 6-way cross between the EBIT and spectrometer used to mate an external x-ray calibration source to the system. This source is composed a of set of solid samples (targets) and a mechanism to excite characteristic radiation from those targets. We can directly excite conductive targets through electron impact excitation with an electron gun, or create secondary x-rays with conductive or non-conductive targets through illumination with a commercial x-ray tube source. We have used two tube sources, with Pd and W anodes, and a variable beam energy of up to $\sim$25~kV and current of up to $\sim$2~mA. The targets are on a linear actuator, allowing for manual switching of the targets without needing to break vacuum. The targets are held at 45 degrees to the electron gun and to the spectrometer, enabling characteristic x-rays to efficiently arrive at the spectrometer. This flexibility in source type and targets is useful for generating characteristic x-rays and calibration curves targeted for a specific application.
\begin{figure}
\begin{center}
\begin{subfigure}{\linewidth}
\includegraphics[width=1.0\linewidth]{CalibrationSpectrum}
\end{subfigure}
\par\medskip
\begin{subfigure}{\linewidth}
\includegraphics[width=1.0\linewidth]{ResolutionsPlot}
\end{subfigure}
\end{center}
\caption{\label{fig:CalibrationSource} (A)~Spectrum from example external calibration source measurement. Here, Fe, Co, and Ni were used as the targets and the total array count rate was kept to under 200~cps. The K$\alpha$, and to a lesser extent K$\beta$, lines are the strongest features observed in the spectrum. (B)~Detector energy resolutions extracted from fits to K$\alpha$ lines from a variety of external calibration source measurements. The resolution is measured to be 3.7~eV at the Mg~K$\alpha$ (1.25~keV) and 5.0~eV at Ni~K$\alpha$ (7.48~keV). The best fit line is used to show the expected energy resolution of the detectors across this energy region. Here, only the statistical uncertainty associated with fitting the line model is shown. Other sources of uncertainty, including uncertainty in the line shape model, slight variations in noise pickup and electrical crosstalk, and excess drift leftover after correction (see Sec.~\ref{sec:Reduction}), account for a scatter of $\sim$100~meV.}
\end{figure}
A spectrum taken using the external calibration source with a target consisting of Fe, Co, and Ni foils is shown in the top panel of Fig.~\ref{fig:CalibrationSource}. X-rays were excited with an x-ray tube source with W anode. A beam voltage of 15~kV was used, and the current ($\sim$20~$\mu$A) was tuned to keep the count rates at $\sim$1~cps/detector in order to resemble the low rates expected in the EBIT measurements. The data analysis is described in Sec.~\ref{sec:Reduction}. The energy resolution extracted from K$\alpha$ lines in the coadded spectra of this and other external calibration source measurements is shown in the bottom panel of Fig.~\ref{fig:CalibrationSource}. Here, the expected natural line shapes\cite{klauber_magnesium_1993, schweppe_accurate_1994, holzer_pre$kensuremathalpha_12$/pre_1997} are convolved with a Gaussian representing the detector response, and the Gaussian FWHM is taken to be the energy resolution.
\subsection{External Magnetic Shielding}
The low temperature TES array and SQUID electronics are sensitive to magnetic fields. Excess magnetic fields could raise the $T_C$ of the superconducting components, broaden the superconducting-to-normal transition in the detectors, and cause shifts in SQUID amplifier gain. In worst cases, magnetic flux trapped in the superconducting components can drive them into their normal state, rendering detectors or SQUIDs with trapped flux unusable until the flux traps are released by warming the components far above their $T_C$, a lengthy process. For this reason, in previous NIST TES spectrometers the 50~K and 3~K snout shields enclosing these components are comprised of mu-metal\cite{jiles_introduction_1998} to attenuate Earth's magnetic field, typical background laboratory fields, and fringing fields from the ADR magnet (snout magnetic field shielding described further in Ref.~\citenum{doriese_practical_2017}). In addition, the 50~mK snout and filter enclosing the TES array and first stage SQUIDs are fabricated with Al, which goes superconducting during an ADR magnet cycle and provides additional magnetic field shielding\cite{hamilton_superconducting_1970}. Similarly, the SA amplfiers on the 1~K stage are enclosed by a superconducting Nb box. These components are also shielded from the internal ADR magnet fringing fields through a cylindrical vanadium permendur magnet enclosure (for a detailed description of ADR magnet shielding with vanadium permendur, see Ref.~\citenum{bennett_high_2012}). Extrapolating from measurements on systems with similar shielding arrangements\cite{bennett_high_2012, doriese_practical_2017}, we can expect a magnetic field on order 1~$\mu$T in the area of the detector package when the ADR magnet is controlled at fields that are typical for temperature regulation. Static fields of this magnitude are largely accommodated for by careful gradiometric SQUID amplifier design\cite{stiehl_time-division_2011} and proper selection of the SQUID biasing parameters. Drifts in the amplifier gain due to this field decaying over the course of an ADR cycle have been measured to be small compared to thermal drifts and are corrected for during pulse processing, as will be discussed in Sec.~\ref{sec:Reduction}.
Somewhat unique to this setup was the additional strong stray magnetic field (primarily in the transverse direction of the snout) of the main EBIT magnet in the region of the NETS snout. The compactness of the various snout temperature stages does not allow for additional interior magnetic field shielding without a costly redesign of the snout. We instead found it much more simple to mount two concentric 1.6~mm thick cylindrical mu-metal shields (designed for factor of $\sim$100 attenuation in the transverse direction) over the vacuum snout where space is much less limited. The goal here was to reduce the expected magnitude of the stray EBIT field directly outside the vacuum snout to levels comparable to typical laboratory background. The shielding internal to the cryostat could then null the remaining field at the detectors and SQUIDs to values typically seen under temperature regulation. We tested the magnetic field suppression of these external shields using a 3-axis Hall effect magnetometer with an estimated uncertainty of $\lesssim$5\%. When the EBIT is ramped to a maximum field current of 147.7~A, we measure a field of 3~mT directly outside the external mu-metal shield and 30~$\mu$T between the external mu-metal shield and vacuum jacket near where the array is located. This is below the background field in the laboratory, which was measured to be 58~$\mu$T prior to ramping the EBIT magnet. With this, we expect no significant excess field in the area of the detectors and SQUIDs from the EBIT magnet.
\section{Measurements}
\label{sec:Measurements}
For our initial tests we took x-ray EBIT measurements on previously studied ions to demonstrate the capabilities of NETS. These are summarized in Table~\ref{tab:MeasurementLog}. The majority of these measurements were done on He-like and H-like ions of injected neutral gases, including CO$_2$, Ne, and Ar. We also used the MeVVA\cite{holland_low_2005} ion source of the NIST EBIT to inject metallic ions into the trap. Here, we primarily focused on Ni-like W, which was the subject of a recent EBIT study\cite{clementson_spectroscopy_2010}. The ions listed in Table~\ref{tab:MeasurementLog} have fairly well known line shapes and positions, useful for benchmarking a new instrument. They can also be used as calibration lines for measurements of ion charge states with less well known spectra, which is planned in future measurements.
\begin{table*}
\caption{\label{tab:MeasurementLog} Measurement log. The ions studied in this intial measurement campaign were He-like and H-like ions of injected neutral gases and Ni-like ions of metallic W injected with a MeVVA.}
\begin{ruledtabular}
\begin{tabular}{cccccc}
Ion & Shield & Collector & Ionization & Count Rate & Total \\
& Voltage (kV) & Current (mA) & State & (cps/array) & Counts \\
\hline
Ne & 4.0 & 99 & He/H-like & 180.9 & 325736 \\
Ar & 10.0 & 92 & He/H-like & 51.4 & 185935 \\
CO$_2$ & 2.0 & 53 & He/H-like & 35.5 & 139690 \\
\hline
W & 4.1 & 105 & Ni-like & 136.4 & 651124 \\
W & 4.1 & 105 & Ni-like & 164.1 & 406991 \\
Ar & 9.5 & 108 & He/H-like & 51.3 & 192346 \\
Ne & 4.0 & 108 & He/H-like & 179.6 & 343093 \\
\hline
Ne & 4.0 & 139 & He/H-like & 266.1 & 577873 \\
W & 4.1 & 137 & Ni-like & 264.1 & 943025 \\
Ar & 9.5 & 140 & He/H-like & 52.4 & 127000 \\
W & 4.1 & 145 & Ni-like & 232.4 & 378798 \\
CO$_2$ & 2.0 & 58 & He/H-like & 58.8 & 79118 \\
\end{tabular}
\end{ruledtabular}
\end{table*}
We report on measurements taken across three days, and some details of the experiment varied day to day. The horizontal lines in Table~\ref{tab:MeasurementLog} separate measurements taken on different days. The NETS ADR magnet was cycled at the beginning of each morning, altering the magnetic field environment in the detectors and SQUIDs and changing gain values. For this reason, we take calibration data for the TES microcalorimeters each day, and data from different days of measurement are analyzed independently from one another. We note that precise ($\sim\pm$0.1~eV) measurements of line energy require more calibration data than other measurements. We have not investigated whether the calibration is stable enough to be reused from day to day in any case. The ion trap dumping/recycling time was set at 5~s for the first day of measurements, but then was subsequently changed to 3~s on day 2 and 3. This is a parameter that gets tuned using past experience operating the EBIT in order to improve charge state purity and increase count rates. The Fe stand originally supporting the external calibration source was found to have a permanent magnetic field which limited the maximum collector current (proxy for the maximum count rate at the spectrometer) of the EBIT to below past levels. This stand was exchanged for an Al stand between days 2 and 3, after which the collector current was identical to that achieved prior to external calibration source installation. The higher maximum collector current resulted in higher count rates on day 3, as shown in Table~\ref{tab:MeasurementLog}. With respect to detector performance, these are relatively low count rates ($\lesssim 2$~cps/detector), so pulse pileup\cite{fowler_microcalorimeter_2015} and crosstalk\cite{durkin_reducing_2019} events happen infrequently and have very little effect on the data quality.
\begin{figure}
\includegraphics[width=1.0\linewidth]{Traces}
\caption{\label{fig:Traces} Sample pulse records for a single microcalorimeter from the H-like 1s-2p transition in O, Ne, and Ar. Here, the transitions occur at three different energies, and the taller pulses indicate more energetic transitions. Note that to get a precise estimate of the x-ray energy, further pulse processing and filtering is required.}
\end{figure}
The raw data products are pulse records, each containing the information of a single x-ray event. Some sample pulse records from the H-like 1s-2p transition in O, Ne, and Ar are shown in Fig.~\ref{fig:Traces}. The height of a trace is representative of the energy of the incident x-ray, but for a more exact determination of the energy, further pulse processing must be done, including combining the pulse records with noise data and optimally filtering the pulses (steps outlined in Sec.~\ref{sec:Reduction}). The pulses across all working microcalorimeters have an average 1/e rise time of 60~$\mu$s and fall time of 640~$\mu$s. The pulse record length was chosen to be long enough to store all the information in a pulse needed to accurately determine the pulse energy, but not too long as to reduce throughput due to pulse pile-up and detector crosstalk. For these reasons, a record length of 1000 samples (4.8~ms) was chosen, with 50\% of the record allocated to the pre-trigger (samples before x-ray arrival).
\section{Data Reduction}
\label{sec:Reduction}
We used the Microcalorimeter Analysis Software System (MASS) Python package to reduce data from raw pulse and noise records to a variety of output products such as energy calibrated spectra\cite{fowler_practice_2016, becker_advances_2019}. First, the raw data pulses (e.g., Fig.~\ref{fig:Traces}) are passed through a set of cuts designed to remove piled up pulses and other irregularities. An average pulse profile is then created for each detector, which is used along with noise data to optimally filter the pulses. Next, the pulses undergo a temporal drift correction. Finally, the detectors are energy-calibrated using well known x-ray lines. The final product is a list of time-tagged, energy-calibrated x-ray events for each detector. The steps of this analysis routine are discussed in more detail below.
Directly after loading the raw pulse records, a variety of summary quantities are calculated. Two of these quantities, the pre-trigger root mean square (RMS) and post-peak derivative, are used to cut out anomalous pulses. An unexpectedly high pre-trigger RMS typically indicates that the detector was triggered during the decay of a prior x-ray pulse, whereas a high positive post-peak derivative likely indicates a second x-ray arrived after the peak of the triggering pulse. Additionally, we use outlier rise and peak times as indicators of secondary pulses arriving during these periods, though these events are fairly rare within these short periods. Noise record data are aggregated to determine expected deviations for these cut quantities, setting the limits for these cuts. Typical pulse cut fractions were less than 2\% at these low count rates. These are high SNR cut quantities, so we do not see any energy dependence to these cuts. The exception to this is when an x-ray is energetic enough to produce a pulse with a slew rate that exceeds the limit set by the speed of the readout electronics, resulting in a highly abnormal pulse record that is generally cut for a combination of the summary quantities listed above. In an energy spectrum, this would be seen as a drop off in counts near this high energy readout limit. In addition to cutting pulses from detectors, entire detectors may be cut during this step. Here, we check to see if any detectors have an abnormally low number of accepted pulse records. This typically indicates a more serious issue with the individual detector or the readout chain of that detector.
The next step in the data reduction pipeline is optimally filtering the pulses\cite{szymkowiak_signal_1993, anderson_optimal_2005}. First, an average pulse is created for each detector using records at a line(s) of interest in the raw pulse height spectrum, although we find the latter steps of the data reduction to be largely insensitive to the exact choice in energy at which this average pulse is created. The average pulse is combined with the noise data, which is assumed to be stationary, in order to produce a filter that maximizes the SNR when estimating the pulse height. This filter is designed to be insensitive to the exact arrival time of a pulse relative to the sampling clock\cite{fowler_practice_2016}. The noise data are typically taken during the same day as the measurements (same ADR cycle), largely based on precedent; we measure roughly the same energy resolution when using noise data from another day. Each pulse record is passed through the filter for its associated detector and a filtered pulse height is extracted.
In NETS, we observe a $\sim (3 - 5) \times 10^{-4}$/hr drift in the fractional pulse height, which varied from detector to detector. A major portion of this drift results from the fact that the 1~K stage temperature increases from $\sim$500~mK immediately after an ADR cycle to $\sim$1~K 10 to 20 hours later. To reach line placement accuracy to better than a few parts in $10^4$ for measurements with integration times of on order hours it is necessary correct for this drift. We use a spectral entropy-minimizing algorithm that uses the baseline, or pretrigger mean, and the x-ray timestamps as indicators of drift in the filtered pulse heights (see Ref.~\citenum{fowler_practice_2016} for more detail). Typically, one or multiple strong lines in the calibration data are measured at multiple times during an ADR cycle to generate the data necessary for drift correction, but strong lines in the measured EBIT data could also be used for this purpose if they exist for a given measurement. This approach to drift correction has previously been used to achieve on order $10^{-5}$ fractional line position accuracy\cite{tatsuno_absolute_2016, fowler_reassessment_2017}. In addition to the drift in pulse height throughout an ADR cycle, there is also a small amount of pulse height variation between different ADR cycles (detectors typically show $\sim 10^{-4} - 10^{-3}$ cycle-to-cycle variation in fractional pulse height when measured at the same time after the start of an ADR cycle). We generally calibrate and analyze data from different ADR cycles more or less independently, so these cycle-to-cycle changes are far less significant to the analyzed data quality than the drift observed during an individual ADR cycle.
Energy calibration is performed by identifying a set of well known lines in a particular spectrum, fitting those lines with their known lineshape and the detector response function, and generating an interpolating function between filtered pulse height (including the various corrections) and energy. We identify lines in the filtered pulse height spectrum of a single arbitrarily chosen reference pixel by hand, then use a dynamic time warping (DTW\cite{vintsyuk_speech_1972, myers_comparative_1981}) based algorithm to identify the same lines in all other pixels. DTW techniques are typically used for speech recognition, where the time vector is warped to determine the correlation of two sound sequences with potentially different speeds. Aligning the energy scale of detectors is a fundamentally very similar problem, with the filtered pulse height space of each detector getting warped to match that of a reference detector. We use a Python implementation of the FastDTW\cite{salvador_toward_2007} algorithm. Once this initial calibration is done and line positions across the broad spectrum are roughly known, a more precise determination of the line centers (and anchor points) can be done by fitting the spectral features with their known line shapes, if available. Using these anchor points we create a ``gain'' function $G$ using a cubic spline. The energy $E$ is calculated for each pulse as $E(PH) = G(PH) \times PH$, where $PH$ is the filtered pulse height. The end result is a list of time-tagged, energy-calibrated x-ray events for each individual detector. From here, the desired output product, such as an energy spectrum with particular bin widths, can easily be created.
\section{Results and Analysis}
\label{sec:Results}
\subsection{Gas Spectra}
\begin{figure*}
\includegraphics[width=1.0\linewidth]{CombinedGas}
\caption{\label{fig:CoaddedGas} Coadded CO$_2$ (primarily O), Ne, and Ar spectra with the gases ionized to primarily He-like (red) and H-like (blue) states. The left-most red dashed line corresponds to the transition from the 1s2s $^3$S excited state in He-like ions. The remaining red dashed lines make up a series of transitions from the 1s$n$p $^1$P states, where $n$ corresponds to the principal quantum number. The blue dashed lines represent 1s--$n$p transitions in H-like ions, where $n$ again corresponds to the principal quantum number. The shaded areas represent the unresolved features that make up the high energy end of the He-like and H-like 1s$n$p and $n$p series, with the solid line representing the ionization limit. In each panel, only the first 3 He-like and first 2 H-like transitions are labeled, for clarity.}
\end{figure*}
The pulse records of He-like and H-like C, O, Ne, and Ar were processed through our data reduction pipeline as outlined above. In the data with CO$_2$ gas injection, due to ionization efficiencies and poor transmittance through the Al filters at low energies, primarily O rather than C lines were observed. With all of these gases, the four most prominent lines were used for both DTW line identification and energy calibration. These were the He-like transitions to the 1s$^2$ $^1$S ground state from the 1s2p~$^1$P and 1s3p~$^1$P excited states and the H-like transitions to the 1s~$^2$S ground state from the 2p~$^2$P and 3p~$^2$P excited states. In the CO$_2$ and Ne data sets, the H-like transition to the 1s~$^2$S state from the 4p~$^2$P state was also sufficiently bright to add a calibration anchor at this transition energy. He-like and H-like line positions were located using the NIST Atomic Spectra Database (ASD)\cite{kramida_nist_nodate}, which includes a compilation of best known line positions for these ions from theory\cite{erickson_energy_1977, saloman_energy_2010} and experiment\cite{tyren_precision_1940, gabriel_interpretation_1969, peacock_spectra_1969}.
A final cut was used to remove any detectors with an outlier resolution (here, $>$5~eV measured at the H-like 2p~$^2$P line energy), as well as any detectors that failed a fit during energy calibration. Detectors that passed this final cut had their x-ray events coadded. X-ray events from the data sets of a given ion, taken across all three days of measurement, were added together. The coadded spectra of our CO$_2$, Ne, and Ar measurements are shown in Fig.~\ref{fig:CoaddedGas}. An average of 132, 141, and 125 detectors passed all cuts for the CO$_2$, Ne, and Ar data, respectively, and are coadded together in the spectra. In comparing these spectra we note that the ionization fractions are not necessarily the same, as the O spectrum contains more intense H-like lines whereas the Ar spectrum contains more intense He-like features. Nevertheless, all three spectra contain two series of lines: the He-like transitions, shown in red, and the H-like transitions, shown in blue. These relatively simple and well known series of lines are useful for testing spectrometer performance and can also be used for energy calibration purposes in future measurements.
\begin{figure}
\includegraphics[width=1.0\linewidth]{FitExample}
\caption{\label{fig:FitExample} Example fit of coadded Ne data at the H-like transitions from the 2p excited state to the 1s ground state. This fit includes two narrow Lorentzians (one per transition): one from the $J = 1/2$ level at 1021.498~eV and the other from the $J = 3/2$ level at 1021.953~eV. The Lorentzians were convolved with a variable width Gaussian matching the detector response. For this spectral feature, the fit Gaussian FWHM is $3.65 \pm 0.01$~eV. The residuals of the fit are plotted over a $\sqrt{N}$ error model characteristic of single photon counting statistics.}
\end{figure}
\begin{table*}
\caption{\centering \label{tab:GasResolutions} Gas Measurement Coadded Spectra Resolutions}
\begin{tabular}{|>{\centering\arraybackslash}m{10mm}|>{\centering\arraybackslash}m{20mm}|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{Ion Species} & He-like & H-like & He-like & H-like \\
\multicolumn{2}{|c|}{Configuration} & 1s2p & 2p & 1s3p & 3p \\
\multicolumn{2}{|c|}{Term} & $^1$P & $^2$P & $^1$P & $^2$P \\
\multicolumn{2}{|c|}{$J$} & 1 & 3/2 & 1 & 3/2 \\
\hline
\multirow{2}{*}{O} & E (eV)\cite{erickson_energy_1977, tyren_precision_1940} & 573.95 & 653.68 & 665.62 & 774.63 \\
& $\Delta$E (eV) & $3.75 \pm 0.03$ & $3.73 \pm 0.02$ & $3.73 \pm 0.04$ & $3.76 \pm 0.03$ \\
\hline
\multirow{2}{*}{Ne} & E (eV)\cite{erickson_energy_1977, peacock_spectra_1969} & 922.02 & 1021.95 & 1073.77 & 1210.96 \\
& $\Delta$E (eV) & $3.63 \pm 0.01$ & $3.67 \pm 0.01$ & $3.66 \pm 0.01$ & $3.64 \pm 0.02$ \\
\hline
\multirow{2}{*}{Ar} & E (eV)\cite{saloman_energy_2010} & 3139.58 & 3322.99 & 3683.85 & 3935.72 \\
& $\Delta$E (eV) & $3.88 \pm 0.01$ & $3.89 \pm 0.02$ & $4.13 \pm 0.03$ & $4.16 \pm 0.05$ \\
\hline
\end{tabular}
\end{table*}
Features in the spectrum were fit with a model that is a convolution of a Lorentzian profile (transition with width dominated by natural line broadening) and a Gaussian profile (detector response). The convolved model is known as a Voigt profile. For these lines the Lorentzian component of this model generally has a much smaller width than the Gaussian component. For example, from transition probabilities in the He-like states\cite{johnson_e1_2002}, we expect the natural line width of the 1s2p~$^1$P--1s~$^2$S transition to be $\sim$14~meV in O, $\sim$37~meV in Ne, and $\sim$440~meV in Ar. The Gaussian FWHM (detector energy resolution), on the other hand, is $\sim$4~eV at these energies. For simplicity, we use a fixed 0.1~eV Lorentzian width (rough average for these systems) for all of our line models and note the exact choice of Lorentzian width has little impact on estimating detector response. The Lorentzian and Gaussian widths add roughly in quadrature, and even using a 0.1~eV Lorentzian width for He-like Ar 1s2p~$^1$P -- 1s~$^2$S transition instead of 0.44~eV results in only a $\sim$20~meV difference in the convolved profile width.
Many x-rays lines generated by the EBIT emulate a monochromatic source much more closely than other x-ray lines easily generated in the laboratory (such as the K lines in Sec.~\ref{subsec:CalibrationSource}). As a result, the EBIT-generated x-rays lines with simple structure, narrow transition width, and few nearby interfering lines may prove useful for studying detector behavior. For example, lines that result in a spectrum with peak to background ratio greater than 100 can be used to look for precent scale deviations from a gaussian response function, while very narrow lines may be useful for accurately measuring energy resolutions approaching 0.5~eV in future detectors optimized for $\sim$500~eV measurements\cite{morgan_use_2019}. As an example we have used 1021.95~eV 1s--2p transition in H-like Ne to measure the low energy tail fraction\cite{yan_eliminating_2017, eckart_extended_2019} in our detector response and find it to be 3\% at this energy.
We measured the energy resolution at four distinct lines in each spectrum. The resolution here is taken to be the FWHM of the Gaussian gets convolved with Lorentzians in the fitting process. The results are shown in Table~\ref{tab:GasResolutions}. Note that the line energies listed in the table are from previous theory and measurements and are not newly measured line positions. The uncertainties listed are fitting uncertainties in the width of the Gaussian component of the fit and are typically higher in lines with a lower number of recorded counts. The resolutions are roughly 3.7~eV in the lower energy O and Ne spectra and increase to about 4~eV in the Ar data. It is generally expected that the resolution of these TES x-ray microcalorimeters will degrade somewhat with increased photon energy\cite{ullom_review_2015, doriese_practical_2017} (see Fig.~\ref{fig:CalibrationSource}, bottom), therefore it is anomalous that the O lines have slightly worse resolution than the Ne lines. We find that the slope in pulse height over energy of the detector response is higher at low energies, as expected for better resolution at lower energies. We also find that the calibration and coadding processes are not responsible for the increased resolution at the O lines. The most plausible explanation we have explored is the existence of low level interfering features coming predominatingly from H-like N and lower charge states of O, which can be close in energy to the features of interest. For example, the structure in the region of the 1s2p $^1$P line contains not only He-like O lines, but there are also lines blended into the structure originating from doubly excited Li-like O transitions. These low level features are usually not individually resolvable in the O spectrum, and it can be difficult to accurately predict their intensities in the line fit models.
The combination of high resolving power and x-ray collection efficiency make NETS particularly valuable for resolving faint lines within a broad spectrum that often contains much stronger features. As an example of this, we measured the SNR of the highest order resolvable line (defined here as being at least two FWHMs away from next highest order line and having a SNR of at least 1) in the He-like 1s$n$p and H-like $n$p transitions, where $n$ is the principal quantum number. The SNR is taken to be the amplitude of the fit function over the uncertainty in that amplitude. With this definition of resolvability in place for the O spectrum, we could fully resolve the He-like 1s5p line with a SNR of 30.6 and the H-like 5p line with a SNR of 35.6. In the Ne spectrum, we could resolve up to the He-like 1s6p line with a SNR of 48.3 and the H-like 6p line with a SNR of 48.1. Finally, in the Ar spectrum, we could resolve the transition from the He-like 1s9p state with a SNR of 8.4 and the H-like 9p state with a SNR of 4.3.
Finally, we used the He-like 1s4p $^1$P line in O, Ne, and to some extent in Ar to demonstrate the accuracy of our energy calibration procedure over narrow spectral windows. The He-like 1s4p $^1$P line was not used as anchor point to generate the calibration curves for any of the gas ion spectra. The determined energies were 697.788~eV, 1127.070~eV, and 3875.027~eV for the O, Ne, and Ar ions, respectively. Prior results\cite{tyren_precision_1940, peacock_spectra_1969, saloman_energy_2010} report these energies at 697.795~eV, 1127.095~eV, and 3874.886~eV, making our calibration as measured at the He-like 1s4p $^1$P line off by 7~meV, 25~meV, and 141~meV, respectively. This reflects the accuracy of the calibration routine when a high density of strong calibration lines are available in a narrow energy band as is the case in the H-like and He-like O and Ne data. The line placement accuracy is poorer in the higher energy H-like and He-like Ar spectrum where the accuracy is limited by low statistics in the highest energy calibration line for individual detectors and ADR cycles. We believe, based on results with similar spectrometers\cite{tatsuno_absolute_2016}, that NETS is capable of better than 141~meV line accuracy at these higher energies, and a dedicated line placement measurement would have improved the line placement accuracy here.
\subsection{Tungsten Spectrum}
\begin{figure*}[p]
\includegraphics[width=1.0\linewidth]{CombinedW}
\caption{\label{fig:CoaddedW} Ni-like W spectrum, coadded across four different data sets (total integration time of 3.46 hours) with an average of 135 detectors passing all cuts. The spectrum is separated into four regions of energy, for clarity. The blue dashed lines represent transitions that were used as anchor points in a smoothed cubic spline during the energy calibration routine. The line labeled 'Ni-cal' is the transition from the (3s$^{2}$3p$^{6}$3d$^{3}_{3/2}$3d$^{6}_{5/2}$4p$_{1/2}$)$_{J=1}$ upper level and the remaining labeled lines follow the naming convention of Clementson \textit{et al}. 2010\cite{clementson_spectroscopy_2010}. The red lines represent features whose center positions were fit as a test of the energy calibration accuracy.}
\end{figure*}
Compared to the individual gas spectra, the Ni-like W spectrum was fairly broad (1.5--4.0~keV) and required more anchor points to properly align and calibrate the entire energy range. A $\sim$2.5 keV range is typical for many of our planned future measurements. The lower level for all reported transitions is the 3s$^{2}$3p$^{6}$3d$^{10}$ ground state. Here, we adopt the naming convention of Clementson \textit{et al}.\ 2010\cite{clementson_spectroscopy_2010} and use the Ni-2, Ni-4, Ni-11, Ni-16, Ni-17, and Ni-20 lines for detector alignment and energy calibration. In addition, we use the transition from the (3s$^{2}$3p$^{6}$3d$^{3}_{3/2}$3d$^{6}_{5/2}$4p$_{1/2}$)$_{J=1}$ level at 1728.4~eV as reported by Elliot \textit{et al}.\ 1995\cite{elliott_measurements_1995}, which was used as a calibrator in the Clementson \textit{et al}.\ 2010 work. This spans the energy range of 1562.9~eV ((3s$^{2}$3p$^{6}$3d$^{4}_{3/2}$3d$^{5}_{5/2}$4s$_{1/2}$)$_{J=3}$, or Ni-2) to 3259.9~eV ((3s$^{2}$3p$^{6}$3d$^{3}_{3/2}$3d$^{6}_{5/2}$6f$_{5/2}$)$_{J=1}$, or Ni-20).
Detectors that passed all of the data reduction cuts were coadded to form a Ni-like W spectrum. Again, detectors with resolutions worse that 5~eV are disqualified from the coadding, this time measured at 2179.7~eV ((3s$^{2}$3p$^{6}$3d$^{3}_{3/2}$3d$^{6}_{5/2}$4f$_{5/2}$)$_{J=1}$, or Ni-8). An average of 135 detectors passed all cuts for the Ni-like W analysis. The coadded spectrum is shown in Fig.~\ref{fig:CoaddedW}.
The Ni-like W measurement provides an example of the intended energy calibration routine over a wider energy range (1.5--4.0~keV) and with a more complex spectrum (many electron system) than that of the He- and H-like gas measurements. To test the calibration routine, we fit a number of spectral features with varying intensities and distances to calibration anchors. We compared the positions of these features to the same previous measurements\cite{clementson_spectroscopy_2010} that were used to fix the energy calibration anchors. Here, the goal is not to report absolute energy measurements of these transitions, but rather to check how good of a job the calibration routine can do given a set of calibration points. The results are summarized in Table~\ref{tab:LinePlacement}.
\begin{table}
\caption{\label{tab:LinePlacement} Ni-like W Energy Calibration Accuracy}
\begin{ruledtabular}
\begin{tabular}{cccc}
Upper & Measured & Ref. Exp.\cite{clementson_spectroscopy_2010} & Difference \\
Level & Energy (eV) & Energy (eV) & (eV) \\
\hline
Ni-3 & 1629.6 & 1629.8(3) & -0.2 \\
Ni-5 & 1829.7 & 1829.6(4) & 0.1 \\
Ni-6 & 2015.5 & 2015.4(4) & 0.1 \\
Ni-12 & 2552.6 & 2553.0(4) & -0.4 \\
Ni-13 & 2650.7 & 2651.3(4) & -0.6 \\
Ni-19 & 3196.8 & 3196.8(3) & 0.0 \\
Ni-21 & 3424.9 & 3426.0(4) & -1.1 \\
Ni-24 & 3572.5 & 3574.1(5) & -1.6 \\
Ni-26 & 3637.2 & 3639.5(6) & -2.3 \\
\end{tabular}
\end{ruledtabular}
\end{table}
Here, we see that the the majority of the measured energies between Ni-3 and Ni-19 fall within the measurement uncertainties reported by Clementson \textit{et al}.\ 2010\cite{clementson_spectroscopy_2010}. Lines that are far from calibration anchor points (e.g., Ni-12 and Ni-13) tended to have poorer placement accuracy than those close to anchor points (such as Ni-5 and Ni-6). In addition, the relatively low intensities of lines past 3~keV made using any lines in the region as calibration points difficult, and the Ni-20 line was the highest energy line we could use for calibration without sacrificing a majority of the detectors due to failed fits. Lines with energies above the Ni-20, such as Ni-21, Ni-24, and Ni-26, had increasingly worse placement accuracy the further in energy they were from this highest energy anchor point. This reflects on the limitations of the calibration routine for estimating the energy of pulses outside region enclosed by the anchor points where the calibration curve is unbounded. Altogether, this highlights the importance of choosing calibration targets with multiple lines close in energy to and enclosing those that are of scientific interest and will help guide us in developing calibration measurement strategies in future measurement campaigns.
\subsection{Time-resolved Analyses}
Microcalorimeters can provide valuable x-ray arrival time information that can be used to probe time-varying phenomena in an EBIT. NETS achieves its individual x-ray arrival time resolution as follows. During the initial setup of the readout electronics, a phase calibration is done to address physical and hardware latency variations and synchronize timing between readout channels to $\sim$200~ps. The readout row-to-row pixel timing uncertainty is tied to the timing jitter of the master clock, which is expected to be below a ns. These timescales are orders of magnitude below the sampling rate, which ultimately limits the x-ray arrival time resolution. NETS has a maximum sampling period of 4.8~$\mu$s when reading out all 24 rows, and this period scales linearly with the number of rows being read out. The x-ray arrival time can actually be identified with resolution exceeding the sampling period by analyzing the digitized pulse shape\cite{szymkowiak_signal_1993, fowler_practice_2016}, and a timing resolution of roughly 1~$\mu$s has been measured in a system using a nearly identical readout architecture and pulse processing pipeline\cite{heates_collaboration_first_2016}. The slew rates of the measured pulses are typically much higher than the edge trigger threshold and we do not observe integer sample delays in triggering between pixels. Generally, we do not require synchronization to an absolute timescale for our applications, but the timing can be synchronized relative to time sources of interest (such as those controlling various EBIT operations) through an external trigger signal with a precision of $\lesssim$200~ns.
As a simple demonstration of this capability, we took a single Ne data set and observed time-varying effects on a $\sim$3~s period due to the EBIT's trap dump and recycle process. The duration of the trap dump was set for 10.1~ms. For these early measurements, the external trigger signal had not yet been set up, so we determined the exact trap dump period using phase dispersion minimization (PDM\cite{stellingwerf_period_1978}) techniques looking for periodic drops in count rate. This accounts for uncertainty in the trap dump period as well as linear drifts between the trap dump clock and DAQ computer clock. We found the true trap dump and recycle period to be 3.00002~s and folded our measured x-ray arrival times over this period. We binned the folded time stamps into 1~ms regions and plotted the total counts versus the time since the ion trap was dumped (Fig.~\ref{fig:TimingExample}, top). The count rate drops to zero upon the trap dump, and then rises as new ions are trapped.
\begin{figure}
\includegraphics[width=1.0\linewidth]{TimingExample}
\caption{\label{fig:TimingExample} \textit{Upper}: Total counts from a single Ne data set, folded over the 3.00002~s trap dump and gas injection period. The folded time stamps are binned into 1.0~ms bins. Here, the zero point in the time since trap dump has been set in the analysis to the point at which the falling edge crosses below 50\% of the equilibrium count rate. The width of the the dip is consistent with the 10.1~ms trap dump duration. \textit{Lower}: Counts folded over this same period separated into the He-like 1s2p $^1$P and H-like 2p states, focused on the region of time in which trap is recycled. Note that the H-like state lags behind the He-like state by $\sim$10~ms during the recovery to its equilibrium count rate.}
\end{figure}
We also plotted the $\sim$3~s folded counts of the most prominent H-like and He-like Ne lines (Fig.~\ref{fig:TimingExample}, bottom). The goal here was to see if the two ion species evolve differently in time during a trap cycle. Although the ratio of He-like to H-like counts remained nearly constant throughout the period, the rise of the H-like state to equilibrium lagged behind the rise of the He-like state by $\sim$10~ms. This 10~ms ionization time from He-like into H-like Ne agrees very well with simple estimates of ionization rates using the well-known electron-impact cross sections at 4 keV and the typical electron density of $10^{12}$~cm$^{-3}$ for the electron beam. In future measurements, this sort of timing information can be used for determining cross sections of atomic processes and setting up optimal cycle times that maximize the counts of a desired ion state relative to other states or background.
The timing capabilities of NETS can be used for other EBIT applications where timing resolution is essential. As an example, instead of holding the beam energy of the EBIT constant, we can sweep the beam energy on roughly 10~ms timescales. As has been demonstrated by various EBIT groups, at certain beam energies and ion states, dielectronic resonances (DRs) may occur\cite{biedermann_line_2003, beilmann_high_2010, ali_photo-recombination_2011, beiersdorfer_dielectronic_2015}. The DR signal can only be produced when the beam energy is equal to the DR energy, however, it is often critical to keep the beam energy above the DR energy in order to create a desired ion population within the trap. Typically, the beam energy is held at a beam energy above the DR energy and then quickly swept through the DR energy, temporarily producing the DR signal while maintaining the desired ion population. With a time-resolving instruments such as NETS, a time folding analysis similar to the one done for Ne above can be used to isolate events at a particular timing bin, which in the case of DR measurements, could be used to isolate the DR signal. In addition to DR measurements, the time resolution can also be used to observe fundamental timescales of paired x-ray events that are correlated through certain cascading electronic transitions, among other applications.
\section{Conclusions and Future Work}
\label{sec:Conclusions}
We have successfully commissioned and took initial measurements with a TES x-ray spectrometer at the NIST EBIT. NETS improves the measurement capabilities at the NIST EBIT through a combination of high resolving power and x-ray collection efficiency. The single x-ray photon counting nature of the microcalorimeter spectrometer also provides time-resolved measurements. These capabilities will improve the accuracy and speed of ongoing measurement campaigns and in some cases enable entirely new measurements.
In order to assess the performance of NETS, we took a series of measurements of He-like and H-like O, Ne, and Ar as well as more broadband measurements of Ni-like W. We measured an energy resolution of roughly 3.7~eV at the low energy O and Ne data (0.5--1.5~keV) and about 4~eV in the higher energy Ar data (3.0--4.5~keV). With this same gas data, we were able to distinguish faint, higher order lines, up to the H-like 9p transition in Ar, in spectra containing observable features with line intensities that varied by multiple orders of magnitude. This is promising for measuring subtleties in spectral shapes and finding faint features in photon-starved measurements. We demonstrated potential energy calibration routines that could be used with NETS and found placement accuracy to better than 100~meV in narrow band spectra such as that of H-like and He-like O and Ne. With the Ni-like W data, we also demonstrated the calibration technique over a broader energy range with sparse calibration points to a few hundred meV line placement accuracy, consistent with the reference data uncertainty. Finally, we looked for periodic features with respect to the ion trap recycling time as a simple early demonstration of instrument's potential for making time-resolved measurements.
These first measurements will help to optimize data collection routines and determine which calibration measurements will be necessary for achieving a particular scientific goal. As was shown in the O and Ne data, we can calibrate line positions of unknown lines to better than 100~meV when calibration lines with well known positions are located sufficiently near the unknown lines. We can achieve this by measuring well-studied highly charged ions (such as the He-like and H-like ions) and also by making more extensive use of the external x-ray source.
In future runs, we plan on improving the capabilities of NETS in a number of ways. First, although we already see a higher x-ray collection rate compared to the NTD-Ge-based microcalorimeter that was previously attached to the NIST EBIT, this can be raised further by reducing the distance of NETS microcalorimeter array to the trap center. As can be seen in Fig.~\ref{fig:SystemIntegration}, a substantial fraction of this distance is used by the external calibration source, the size of which could be decreased fairly easily. Due to the small array diameter relative to the inner diameter of other components in the optical path, we do not expect to mask portions of the array by moving the it closer to the trap center. In terms of stray fields from the EBIT magnet, we can safely move the spectrometer $\sim$250~mm closer to the trap center before the attenuated stray field directly outside the snout starts exceeding typical background levels. In addition, using two vacuum windows between the EBIT and NETS is not strictly necessary, though it reduces some complexity in integration and risk to the various vacuum systems. Removing one of these windows could increase collection efficiency, especially at energies below $\sim$2~keV (see Fig.~\ref{fig:Efficiency}), but also by about 20\% overall due to the thick mesh component of the window. To go along with this, we plan on replacing the the 50~K filter with one that has a slightly thicker ($\sim$200~nm) Al film, but no Ni mesh, which we expect will be sufficient for transferring heat off the filter. This will have the advantage of removing a great deal of the uncertainty in the QE, but at the cost of reduced QE at the lower end of our energy band.
We also plan on improving our external calibration capabilities by introducing an annular target holder, allowing us to simultaneously collect calibration and EBIT data. This would allow us to continuously measure calibration lines, reducing the uncertainty in the drift correction associated with needing to interpolate between discrete calibration measurements separated in time. Next, we plan on improving our time-resolved measurement accuracy relative to events in the EBIT by establishing trigger signals between the two systems. Finally, we hope to increase our real-time measurement capabilities, which includes the use of the DTW alignment techniques discussed in Sec.~\ref{sec:Reduction} to produce near-final quality spectra in real-time during data acquisition. We expect these improvements to further increase the measurement speed and data quality of NETS.
\section*{Acknowledgment}
This work was supported by the NIST Innovations in Measurement Science (IMS) Program. Paul Szypryt is supported by a National Research Council Postdoctoral Fellowship. Endre Takacs is supported by the National Science Foundation Award \#1806494. The authors would like to thank Csilla Szabo-Foster, Lawrence Hudson, and Jon Pratt for their help in establishing the collaboration between the quantum sensors group and the EBIT group at NIST.
| {
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} | 647 |
Diptychophora galvani est une espèce de papillons de nuit de la famille des Crambidae. Il mesure un peu plus d'un centimètre d'envergure, et se distingue aisément de toutes les espèces proches par les motifs de coloration de ses ailes antérieures. Celles-ci sont orange en leur base et en leur bout, avec une grande plage intermédiaire grise, ce qui n'est retrouvé chez aucune autre espèce de Diptychophora. La femelle a les ailes postérieures grisâtres, alors qu'elles sont entièrement blanches chez le mâle. Les organes génitaux, mâles comme femelles, sont aussi bien différents de ceux des autres membres de ce genre. La biologie de l'espèce reste totalement inconnue, notamment la plante hôte du stade larvaire, bien que certaines espèces de la tribu des Diptychophorini soient connues pour se nourrir de mousses.
L'espèce n'est connue que du Brésil, où elle a été collectée dans les états du Mato Grosso et du Minas Gerais, à d'altitude. Elle y peuple l'écorégion du Cerrado, constituée de forêts galeries et de savanes, avec une saison sèche. Elle est récoltée pour la première fois en 1982 par Vitor O. Becker, mais sa description par Bernard Landry et Becker paraît en 2021. Son épithète spécifique, galvani, rend hommage à Ricardo Galvão, physicien brésilien dirigeant l'Institut national de recherches spatiales du Brésil et démis de ses fonctions deux ans auparavant pour s'être publiquement opposé au président brésilien Jair Bolsonaro, notoire négateur du changement climatique. Ce dernier avait prétendu que les données produites par l'institut et démontrant la croissance substantielle de la déforestation du bassin amazonien à la suite de son arrivée au pouvoir, notamment par des feux dévastateurs de 2019, étaient fausses. Les descripteurs de l'espèce la dédient à Galvão pour « son courage face à l'adversité dans le cadre de son travail », mais aussi car la couleur des ailes du papillon rappelle celle des feux de forêt. Cette pyrale est élue « espèce de l'année 2022 de la Société suisse de systématique ».
Description
Diptychophora galvani mesure entre d'envergure, avec des ailes antérieures de de long chez le mâle et de chez la femelle. L'espèce se distingue facilement de toutes les autres espèces proches du genre Diptychophora par les motifs remarquables de ses ailes antérieures. Celles-ci comptent en effet deux grandes zones orangées, l'une distale (au bout de l'aile) et une autre proximale (à la base de l'aile) bordée d'épaisses lignes marron foncé, ces deux plages orange étant séparées par une grande section médiane grise. Les ailes postérieures sont blanches chez le mâle, grisâtres chez la femelle.
En ce qui concerne les pièces génitales du mâle, l'uncus allongé et fusionné avec le tegumen ne se retrouve chez aucune autre espèce du genre Diptychophora alors que chez la femelle, la présence de deux éléments sclérifiés (les signa bursae) sur la bourse copulatrice est un caractère diagnostique unique.
Écologie
Les spécimens étudiés pour la description de l'espèce en 2021 ont été collectés de nuit, attirés par une lampe à vapeur de mercure. Diptychophora galvani n'étant connue qu'au stade imago, aucune plante hôte pour les chenilles n'est rapportée, ainsi qu'il en est pour toutes les autres espèces du genre Diptychophora. Les seules informations disponibles sur l'alimentation des chenilles dans la tribu des Diptychophorini concernent trois espèces de Nouvelle-Zélande du genre Glaucocharis, qui se nourrissent sur des mousses.
Répartition et habitat
Diptychophora galvani est décrite de la municipalité brésilienne de Chapada dos Guimarães, dans le Mato Grosso, où elle a été collectée à d'altitude, et la série type comprend également des spécimens d'Unaí, dans le Minas Gerais, trouvés à d'altitude. L'espèce y vit dans le Cerrado, une écorégion terrestre majeure du Brésil située entre la forêt amazonienne et la forêt atlantique. Cette région se caractérise par des sécheresses saisonnières, et se compose de savanes plus ou moins boisées, de zones humides et de forêts galeries, sur des sols pauvres et acides. Les spécimens étudiés au moment de la description en 2021 ont été collectés en lisière de forêts galeries.
Taxinomie
Description originale
L'espèce Diptychophora galvani est décrite par Bernard Landry et Vitor O. Becker en 2021, d'après des spécimens collectés par Becker en 1982, 1983 et 1996. La série type est constituée de deux femelles (dont l'holotype) et deux mâles. Les deux femelles et un mâle sont conservés dans la collection personnelle de Becker (numéros de collectes 106575, 49809 et 49079), et un mâle (MHNG-ENTO-84604) de la localité type est déposé au Muséum d'histoire naturelle de Genève, en Suisse. Deux autres Diptychophora brésiliens sont décrits à la même occasion, D. planaltina et D. ardalia.
Étymologie
Le nom du genre Diptychophora, qui vient du grec ancien , , « plié en deux » – de , , « deux », et , « pli » – et , , « porteur », signifie « qui porte deux plis », possiblement en référence aux deux resserrements présents sur le termen de l'aile antérieure, près de l'apex, un caractère donné comme diagnostique dans la description originale du genre par l'entomologiste allemand Philipp Christoph Zeller. L'épithète spécifique de D. galvani fait référence à Ricardo Galvão, physicien et ancien directeur de l'Institut national de recherches spatiales du Brésil (INPE). Celui-ci avait décidé d'affronter le président du Brésil Jair Bolsonaro quand lors d'une conférence de presse internationale, ce dernier a prétendu fausses les données de l'INPE démontrant la croissance substantielle de la déforestation de la forêt amazonienne à la suite de son arrivée au pouvoir. Certain de l'exactitude et de la qualité des données produites par son institut, Galvão a défié Bolsonaro de prouver son affirmation dans une discussion avec lui, face à face. Ce défi n'a pas été relevé et Galvão a été démis de son poste de directeur de l'INPE. Le physicien reçoit alors le soutien de la communauté scientifique, le magazine Nature le classant parmi les dix meilleurs scientifiques de l'année 2019 pour sa défense de la science contre les attaques du gouvernement brésilien, et l'association américaine pour l'avancement des sciences lui décernant le prix de la « liberté et de la responsabilité scientifiques » en 2021. Les descripteurs de D. galvani dédient l'espèce à Galvão pour « son courage face à l'adversité dans le cadre de son travail », et parce que la couleur orange des ailes antérieures du papillon rappelle les feux dévastateurs en Amazonie dont l'augmentation a été démontrée par les données rendues publiques par l'INPE.
Dans la culture
En , Diptychophora galvani est nommée « espèce de l'année 2022 de la Société suisse de systématique », parmi décrites par des taxinomistes d'institutions suisses durant l'année 2021.
Annexes
Bibliographie
Liens externes
Notes et références
Crambidae
Espèce de Lépidoptères (nom scientifique)
Faune endémique du Brésil
Faune tropicale | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,373 |
Les dirigeants de l'île de Man possèdent des titres variés et plusieurs de ces dirigeants peuvent régner durant les mêmes périodes au gré des dominations étrangères au cours desquelles le souverain peut installer un gouvernement local dirigé par un gouverneur et/ou un lieutenant-gouverneur.
Histoire
Lors de la constitution d'une première entité territoriale au , l'île de Man est alors indépendante et son souverain, alors « roi de Man », contrôle aussi d'autres territoires comme les Hébrides extérieures. Des invasions scandinaves dans les îles Britanniques ne permettent pas la sauvegarde de cette indépendance qui cesse définitivement en 1164. Une période transitoire de presque un siècle, durant laquelle les occupants anglais et écossais se succèdent rapidement, permet de chasser les différentes dominations vikings qui sont alors remplacées par une domination anglaise à partir de 1333 et qui se poursuit sans interruption jusqu'à aujourd'hui.
Durant cette période de changements rapides d'occupations, le titre de « gouverneur de l'île de Man » voit le jour en 1290 mais il est aboli en 1828 sous le règne de George . En revanche, le titre de « lieutenant-gouverneur de l'île de Man » créé en 1773 existe toujours. Ainsi, depuis 1828, l'île de Man est dirigée par un souverain, le monarque du Royaume-Uni qui porte le titre de « seigneur de Man » créé en 1504 mais initialement concédé au comté de Derby jusqu'en 1760 lorsque le roi d'Angleterre en hérite, et par son représentant, un élu par les Mannois qui exerce la fonction de « lieutenant-gouverneur de l'île de Man ».
Depuis avril 2011, le lieutenant-gouverneur, fonction jusqu'alors attribuée par la reine, est désigné par un vote de représentants locaux mannois, mesure annoncée par le ministre principal de l'île de Man d'alors, Tony Brown, le .
Légende
Dirigeants de l'île de Man
Annexes
Notes et références
Articles connexes
Histoire de l'île de Man
Liste des souverains du royaume de Man et des Îles
Source
World Statesmen - Isle of Man
Man
Liste en rapport avec l'île de Man | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,364 |
Our Neck of the Woods and is open for members to hire a desk for the day or for others to retreat to for a studio weekend (with Sofabed and courtyard). Meet, work, create. 2 minutes from the beach and cafes. YIPPEE!
Liana decides who becomes a member. members can recommend other members.
• Care for the space and it will care for you.
• Take part in all out greenie endeavours (like worm husbandry, bag circulation, reuse etc) as set out in the space.
• Share the joy and richness of the company but respect each other's need for confidentiality and quiet.
$25 plus gst for any part of a day in a room. | {
"redpajama_set_name": "RedPajamaC4"
} | 450 |
\section{Introduction}\label{sec:intro}
Let $G=(V,E)$ be a finite undirected graph. A vertex $v \in V$ {\em dominates} itself and its neighbors. A vertex subset $D \subseteq V$ is an {\em efficient dominating set} ({\em e.d.s.} for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex in $D$.
The notion of efficient domination was introduced by Biggs \cite{Biggs1973} under the name {\em perfect code}.
The {\sc Efficient Domination} (ED) problem asks for the existence of an e.d.s.\ in a given graph $G$ (note that not every graph has an e.d.s.)
A set $M$ of edges in a graph $G$ is an \emph{efficient edge dominating set} (\emph{e.e.d.s.} for short) of $G$ if and only if it is an e.d.s.\ in its line graph $L(G)$. The {\sc Efficient Edge Domination} (EED) problem asks for the existence of an e.e.d.s.\ in a given graph $G$. Thus, the EED problem for a graph $G$ corresponds to the ED problem for its line graph $L(G)$. Note that not every graph has an e.e.d.s. An efficient edge dominating set is also called \emph{dominating induced matching} ({\em d.i.m.} for short), and the EED problem is called the {\sc Dominating Induced Matching} (DIM) problem in various papers (see e.g. \cite{BraHunNev2010,BraMos2014,CarKorLoz2011,HerLozRieZamdeW2015,KorLozPur2014}); subsequently, we will use this notation instead of EED.
In \cite{GriSlaSheHol1993}, it was shown that the DIM problem is \NP-complete; see also~\cite{BraHunNev2010,CarKorLoz2011,LuKoTan2002,LuTan1998}.
However, for various graph classes, DIM is solvable in polynomial time. For mentioning some examples, we need the following notions:
Let $P_k$ denote the chordless path $P$ with $k$ vertices, say $a_1,\ldots,a_k$, and $k-1$ edges $a_ia_{i+1}$, $1 \le i \le k-1$; we also denote it as $P=(a_1,\ldots,a_k)$.
For indices $i,j,k \ge 0$, let $S_{i,j,k}$ denote the graph $H$ with vertices $u,x_1,\ldots,x_i$, $y_1,\ldots,y_j$, $z_1,\ldots,z_k$ such that the subgraph induced by $u,x_1,\ldots,x_i$ forms a $P_{i+1}$ $(u,x_1,\ldots,x_i)$, the subgraph induced by $u,y_1,\ldots,y_j$ forms a $P_{j+1}$ $(u,y_1,\ldots,y_j)$, and the subgraph induced by $u,z_1,\ldots,z_k$ forms a $P_{k+1}$ $(u,z_1,\ldots,z_k)$, and there are no other edges in $S_{i,j,k}$; $u$ is called the {\em center} of $H$.
Thus, {\em claw} is $S_{1,1,1}$, and $P_k$ is isomorphic to e.g.\ $S_{k-1,0,0}$.
For a set ${\cal F}$ of graphs, a graph $G$ is called {\em ${\cal F}$-free} if no induced subgraph of $G$ is contained in ${\cal F}$.
If $|{\cal F}|=1$, say ${\cal F}=\{H\}$, then instead of $\{H\}$-free, $G$ is called $H$-free.
\medskip
The following results are known:
\begin{theorem}\label{DIMpolresults}
DIM is solvable in polynomial time for
\begin{itemize}
\item[$(i)$] $S_{1,1,1}$-free graphs $\cite{CarKorLoz2011}$,
\item[$(ii)$] $S_{1,2,3}$-free graphs $\cite{KorLozPur2014}$,
\item[$(iii)$] $S_{2,2,2}$-free graphs $\cite{HerLozRieZamdeW2015}$,
\item[$(iv)$] $S_{2,2,3}$-free graphs $\cite{BraMos2017/2}$,
\item[$(v)$] $P_7$-free graphs $\cite{BraMos2014}$ (in this case even in linear time),
\item[$(vi)$] $P_8$-free graphs $\cite{BraMos2017}$.
\end{itemize}
\end{theorem}
In \cite{HerLozRieZamdeW2015}, it is conjectured that for every fixed $i,j,k$, DIM is solvable in polynomial time for $S_{i,j,k}$-free graphs (actually, an even stronger conjecture is mentioned in \cite{HerLozRieZamdeW2015}); this includes $P_k$-free graphs for $k \ge 8$.
\medskip
Based on the two distinct approaches described in \cite{BraMos2017} and in \cite{HerLozRieZamdeW2015,KorLozPur2014}, we show in this paper that DIM can be solved in polynomial time for $S_{1,2,4}$-free graphs (generalizing the corresponding results for $S_{1,2,3}$-free as well as for $P_7$-free graphs).
\section{Definitions and Basic Properties}\label{sec:basicnotionsresults}
\subsection{Basic notions}\label{subsec:basicnotions}
Let $G$ be a finite undirected graph without loops and multiple edges. Let $V(G)$ or $V$ denote its vertex set and $E(G)$ or $E$ its edge set; let $|V|=n$ and $|E|=m$.
For $v \in V$, let $N(v):=\{u \in V: uv \in E\}$ denote the {\em open neighborhood of $v$}, and let $N[v]:=N(v) \cup \{v\}$ denote the {\em closed neighborhood of $v$}. If $xy \in E$, we also say that $x$ and $y$ {\em see each other}, and if $xy \not\in E$, we say that $x$ and $y$ {\em miss each other}. A vertex set $S$ is {\em independent} in $G$ if for every pair of vertices $x,y \in S$, $xy \not\in E$. A vertex set $Q$ is a {\em clique} in $G$ if for every pair of vertices $x,y \in Q$, $x \neq y$, $xy \in E$. For $uv \in E$ let $N(uv):= N(u) \cup N(v) \setminus \{u,v\}$ and $N[uv]:= N[u] \cup N[v]$.
For $U \subseteq V$, let $G[U]$ denote the subgraph of $G$ induced by vertex set $U$. Clearly $xy \in E$ is an edge in $G[U]$ exactly when $x \in U$ and $y \in U$; thus, $G[U]$ can simply be denoted by $U$ (if understandable).
For $A \subseteq V$ and $B \subseteq V$, $A \cap B = \emptyset$, we say that $A \cojoin B$ ($A$ and $B$ {\em miss each other}) if there is no edge between $A$ and $B$, and $A$ and $B$ {\em see each other} if there is at least one edge between $A$ and $B$. If a vertex $u \notin B$ has a neighbor $v \in B$ then {\em $u$ contacts $B$}. If every vertex in $A$ sees every vertex in $B$, we denote it by $A \join B$. For $A=\{a\}$, we simply denote $A \join B$ by $a \join B$, and correspondingly $A \cojoin B$ by $a \cojoin B$.
If for $A' \subseteq A$, $A' \cojoin (A \setminus A')$, we say that $A'$ is {\em isolated} in $G[A]$.
For graphs $H_1$, $H_2$ with disjoint vertex sets, $H_1+H_2$ denotes the disjoint union of $H_1$, $H_2$, and for $k \ge 2$, $kH$ denotes the disjoint union of $k$ copies of $H$. For example, $2P_2$ is the disjoint union of two edges.
As already mentioned, a {\em chordless path} $P_k$, $k \ge 2$, has $k$ vertices, say $v_1,\ldots,v_k$, and $k-1$ edges $v_iv_{i+1}$, $1 \le i \le k-1$;
the {\em length of $P_k$} is $k-1$. We also denote it as $P=(v_1,\ldots,v_k)$.
A {\em chordless cycle} $C_k$, $k \ge 3$, has $k$ vertices, say $v_1,\ldots,v_k$, and $k$ edges $v_iv_{i+1}$, $1 \le i \le k-1$, and $v_kv_1$; the {\em length of $C_k$} is $k$.
Let $K_i$, $i \ge 1$, denote the clique with $i$ vertices. Let $K_4-e$ or {\em diamond} be the graph with four vertices, say $v_1,v_2,v_3,u$, such that $(v_1,v_2,v_3)$ forms a $P_3$ and $u \join \{v_1,v_2,v_3\}$; its {\em mid-edge} is the edge $uv_2$.
A {\em gem} has five vertices say, $v_1,v_2,v_3,v_4,u$, such that $(v_1,v_2,v_3,v_4)$ forms a $P_4$ and $u \join \{v_1,v_2,v_3,v_4\}$.
A {\em butterfly} has five vertices, say, $v_1,v_2,v_3,v_4,u$, such that $v_1,v_2,v_3,v_4$ induce a $2P_2$ with edges $v_1v_2$ and $v_3v_4$ (the {\em peripheral edges} of the butterfly), and $u \join \{v_1,v_2,v_3,v_4\}$.
We often consider an edge $e = uv$ to be a set of two vertices; then it makes sense to say, for example, $u \in e$ and $e \cap e' \neq \emptyset$, for an edge $e'$. For two vertices $x,y \in V$, let $dist_G(x,y)$ denote the {\em distance between $x$ and $y$ in $G$}, i.e., the length of a shortest path between $x$ and $y$ in $G$.
The {\em distance between a vertex $z$ and an edge $xy$} is the length of a shortest path between $z$ and $x,y$, i.e., $dist_G(z,xy)= \min\{dist_G(z,v): v \in \{x,y\}\}$.
The {\em distance between two edges} $e,e' \in E$ is the length of a shortest path between $e$ and $e'$, i.e., $dist_G(e,e')= \min\{dist_G(u,v): u \in e, v \in e'\}$.
In particular, this means that $dist_G(e,e')=0$ if and only if $e \cap e' \neq \emptyset$.
An edge subset $M \subseteq E$ is an {\em induced matching} if the pairwise distance between its members is at least 2, that is, $M$ is isomorphic to $kP_2$ for $k=|M|$. Obviously, if $M$ is a d.i.m.\ then $M$ is an induced matching.
Clearly, $G$ has a d.i.m.\ if and only if every connected component of $G$ has a d.i.m.; from now on, connected components are mentioned as {\em components}.
\subsection{Forbidden subgraphs and forced edges}\label{forbidsubgrforcededges}
The subsequent observations are helpful (some of them are mentioned e.g.\ in \cite{BraHunNev2010,BraMos2014,BraMos2017}).
\begin{observation}[\cite{BraHunNev2010,BraMos2014}]\label{dimC3C5C7C4}
Let $M$ be a d.i.m.\ in $G$.
\begin{itemize}
\item[$(i)$] $M$ contains at least one edge of every odd cycle $C_{2k+1}$ in $G$, $k \ge 1$,
and exactly one edge of every odd cycle $C_3$, $C_5$, $C_7$ of $G$.
\item[$(ii)$] No edge of any $C_4$ can be in $M$.
\item[$(iii)$] For each $C_6$ either exactly two or none of its edges are in $M$.
\end{itemize}
\end{observation}
\noindent
{\bf Proof.} See e.g.\ Observation 2 in \cite{BraMos2014}.
\medskip
Since every triangle contains exactly one $M$-edge and no $M$-edge is in any $C_4$, and the pairwise distance of edges in any d.i.m.\ is at least 2, we obtain:
\begin{corollary}\label{cly:k4gemfree}
If a graph has a d.i.m.\ then it is $K_4$-free, gem-free and $\overline{C_k}$-free for any $k \ge 6$.
\end{corollary}
If an edge $e \in E$ is contained in {\bf every} d.i.m.\ of $G$, we call it a {\em forced} edge of $G$. If an edge $e \in E$ is {\bf not} contained in {\bf any} d.i.m.\ of $G$, we call it an {\em excluded} edge of $G$ (we can denote this by weight $w(e)=\infty$ or by coloring $e$ red). As a consequence of Observation~\ref{dimC3C5C7C4} $(ii)$, all edges in any $C_4$ of $G$ are excluded. Moreover, by Observation \ref{dimC3C5C7C4} $(i)$ for $C_3$, if for an edge $uv$ and a triangle $T$, $u \in V(T)$ and $v \notin V(T)$ then $uv$ is excluded. As another consequence of Observation \ref{dimC3C5C7C4} $(i)$ for $C_3$, we have:
\begin{observation}\label{obs:diamondbutterfly}
The mid-edge of any diamond in $G$ and the two peripheral edges of any induced butterfly are forced edges of $G$.
\end{observation}
Note that in a graph with d.i.m., the set of forced edges is an induced matching. Thus, our algorithm solving the DIM problem on $S_{1,2,4}$-free graphs has to check whether the set of forced edges is an induced matching.
If $M$ is an induced matching of already collected forced edges and edge $vw$ is a new forced edge, we can reduce the graph as follows:
\medskip
\noindent
{\bf Reduction-Step-($vw,M$).}
If $M \cup \{vw\}$ is not an induced matching then STOP - $G$ has no d.i.m., otherwise add $vw$ to $M$, i.e., $M:=M \cup \{vw\}$, delete $v$ and $w$ and all edges incident to $v$ and $w$ in $G$, and denote all edges that were at distance 1 from $vw$ in $G$ as excluded edges.
\medskip
Obviously, the graph resulting from the reduction step is an induced subgraph of $G$. Recall that excluded edges are not in any d.i.m.\ of $G$.
\begin{observation}[\cite{BraMos2014}]\label{obs:redstep}
Let $M'$ be an induced matching which is a set of forced edges in $G$. Then $G$ has a d.i.m.\ $M$ if and only if after applying the reduction step to all edges in $M'$, the resulting graph has a d.i.m.\ $M \setminus M'$.
\end{observation}
Subsequently, this approach will often be used. Note that after applying the Reduction Step to all mid-edges of diamonds and all peripheral edges of butterflies in $G$, the resulting graph is (diamond, butterfly)-free. Moreover, by Corollary \ref{cly:k4gemfree}, a graph $G$ having a d.i.m.\ is $K_4$-free. Thus, from now on, we can assume that $G$ is connected $(K_4$, diamond, butterfly)-free.
Note that if $G$ has a d.i.m.\ $M$, and $V(M)$ denotes the vertex set of $M$ then $V \setminus V(M)$ is an independent set, say $I$, i.e.,
\begin{equation}\label{IV(M)partition}
V \mbox{ has the partition } V = I \cup V(M).
\end{equation}
From now on, all vertices in $I$ are colored white and all vertices in $V(M)$ are colored black. According to \cite{HerLozRieZamdeW2015}, we also use the following notions: A partial black-white coloring of $V(G)$ is {\em feasible} if the set of white vertices is an independent set in $G$ and every black vertex has at most one black neighbor. A complete black-white coloring of $V(G)$ is {\em feasible} if the set of white vertices is an independent set in $G$ and every black vertex has exactly one black neighbor. Clearly, $M$ is a d.i.m.\ of $G$ if and only if the black vertices $V(M)$ and the white vertices $V \setminus V(M)$ form a complete feasible coloring of $V(G)$.
\medskip
The Reduction Step mentioned above leads to a coloring reduction (i.e., C-reduction):
\medskip
\noindent
{\bf Edge C-Reduction.} Let $uw \in E(G)$. If $u$ and $w$ are black then
\begin{itemize}
\item[$(i)$] color white all neighbors of $u$ and of $w$, and
\item[$(ii)$] remove $u$ and $w$ (and the edges containing $u$ or $w$) from $G$.
\end{itemize}
Moreover, we have:
\medskip
\noindent
{\bf Vertex C-Reduction.} Let $u \in V(G)$. If $u$ is white, then
\begin{itemize}
\item[$(i)$] color black all neighbors of $u$, and
\item[$(ii)$] remove $u$ from $G$.
\end{itemize}
\subsection{The distance levels of an $M$-edge $xy$ in a $P_3$}\label{subsec:distlevels}
Based on \cite{BraMos2017}, we first describe some general structure properties for the distance levels of an edge in a d.i.m.\ $M$ of $G$.
Since $G$ is $(K_4$, diamond, butterfly)-free, we have:
\begin{observation}\label{obse:neighborhood}
For every vertex $v$ of $G$, $N(v)$ is the disjoint union of isolated vertices and at most one edge. Moreover, for every edge $xy \in E$, there is at most one common neighbor of $x$ and $y$.
\end{observation}
Since it is trivial to check whether $G$ has a d.i.m.\ $M$ with exactly one edge, from now on we can assume that $|M| \geq 2$. Since $G$ is connected and butterfly-free, we have:
\begin{observation}\label{obse:xy-in-P3}
If $|M| \geq 2$ then there is an edge in $M$ which is contained in a $P_3$ of $G$.
\end{observation}
Let $xy \in M$ be an $M$-edge for which there is a vertex $r$ such that $\{r,x,y\}$ induce a $P_3$ with edge $rx \in E$. This also means that $x$ and $y$ are black and lead to a feasible $xy$-coloring if there is indeed a d.i.m.\ $M$ of $G$ with $xy \in M$.
Let $N_0(xy):=\{x,y\}$ and for $i \ge 1$, let
$$N_i(xy):=\{z \in V: dist_G(z,xy) = i\}$$
denote the {\em distance levels of $xy$}.
We consider a partition of $V$ into $N_i=N_i(xy)$, $i \ge 0$, with respect to the edge $xy$ (under the assumption that $xy \in M$).
Recall that by (\ref{IV(M)partition}), $V=I \cup V(M)$ is a partition of $V$ where $I$ is an independent set. Since we assume that $xy \in M$ (and is an edge in a $P_3$), clearly, $N_1 \subseteq I$ and thus:
\begin{equation}\label{N1subI}
N_1 \mbox{ is an independent set of white vertices.}
\end{equation}
Moreover, no edge between $N_1$ and $N_2$ is in $M$. Since $N_1 \subseteq I$ and all neighbors of vertices in $I$ are in $V(M)$, we have:
\begin{equation}\label{N2M2S2}
G[N_2] \mbox{ is the disjoint union of edges and isolated vertices. }
\end{equation}
Let $M_2$ denote the set of edges $uv \in E$ with $u,v \in N_2$ and let $S_2 = \{u_1,\ldots,u_k\}$ denote the set of isolated vertices in $N_2$; $N_2=V(M_2) \cup S_2$ is a partition of $N_2$. Obviously:
\begin{equation}\label{M2subM}
M_2 \subseteq M \mbox{ and } S_2 \subseteq V(M).
\end{equation}
If for $xy \in M$, an edge $e \in E$ is contained in {\bf every} dominating induced matching $M'$ of $G$ with $xy \in M'$, we say that $e$ is an {\em $xy$-forced} $M$-edge. The Reduction Step for forced edges can also be applied for $xy$-forced $M$-edges
(then, in the unsuccessful case, $G$ has no d.i.m.\ containing $xy$). Obviously, by (\ref{M2subM}), we have:
\begin{equation}\label{M2xymandatory}
\mbox{Every edge in } M_2 \mbox{ is an $xy$-forced $M$-edge}.
\end{equation}
Thus, from now on, after applying the Reduction Step for $M_2$-edges, we can assume that $M_2=\emptyset$, i.e., $N_2=S_2 = \{u_1,\ldots,u_k\}$. For every $i \in \{1,\ldots,k\}$, let $u'_i \in N_3$ denote the {\em $M$-mate} of $u_i$ (i.e., $u_iu'_i \in M$). Let $M_3=\{u_iu'_i: i \in \{1,\ldots,k\}\}$ denote the set of $M$-edges with one endpoint in $S_2$ (and the other endpoint in $N_3$). Obviously, by (\ref{M2subM}) and the distance condition for a d.i.m.\ $M$, the following holds:
\begin{equation}\label{noMedgesN3N4}
\mbox{ No edge with both ends in } N_3 \mbox{ and no edge between } N_3 \mbox{ and } N_4 \mbox{ is in } M.
\end{equation}
As a consequence of (\ref{noMedgesN3N4}) and the fact that every triangle contains exactly one $M$-edge (see Observation~\ref{dimC3C5C7C4} $(i)$), we have:
\begin{equation}\label{triangleaN3bcN4}
\mbox{For every triangle $abc$} \mbox{ with } a \in N_3, \mbox{ and } b,c \in N_4, \mbox{ $bc \in M$ is an $xy$-forced $M$-edge}.
\end{equation}
This means that for the edge $bc$, the Reduction Step can be applied, and from now on, we can assume that there is no such triangle $abc$ with $a \in N_3$ and $b,c \in N_4$, i.e., for every edge $uv \in E$ in $N_4$:
\begin{equation}\label{edgeN4N3neighb}
N(u) \cap N(v) \cap N_3 = \emptyset.
\end{equation}
\medskip
According to $(\ref{M2subM})$ and the assumption that $M_2=\emptyset$ (recall $N_2 = \{u_1,\ldots,u_k\}$), let:
\begin{enumerate}
\item[ ] $T_{one} := \{t \in N_3: |N(t) \cap N_2| = 1\}$;
\item[ ] $T_i := T_{one} \cap N(u_i)$, $i \in \{1,\ldots,k\}$;
\item[ ] $S_3 := N_3 \setminus T_{one}$.
\end{enumerate}
By definition, $T_i$ is the set of {\em private} neighbors of $u_i \in N_2$ in $N_3$ (note that $u'_i \in T_i$),
$T_1 \cup \ldots \cup T_k$ is a partition of $T_{one}$, and $T_{one} \cup S_3$ is a partition of~$N_3$.
\begin{lemma}[\cite{BraMos2017}]\label{lemm:structure2}
The following statements hold:
\begin{enumerate}
\item[$(i)$] For all $i \in \{1,\ldots,k\}$, $T_i \cap V(M)=\{u_i'\}$.
\item[$(ii)$] For all $i \in \{1,\ldots,k\}$, $T_i$ is the disjoint union of vertices and at most one edge.
\item[$(iii)$] $G[N_3]$ is bipartite.
\item[$(iv)$] $S_3 \subseteq I$, i.e., $S_3$ is an independent vertex set of white vertices.
\item[$(v)$] If a vertex $t_i \in T_i$ sees two vertices in $T_j$, $i \neq j$, $i,j \in \{1,\ldots,k\}$, then $u_it_i \in M$ is an $xy$-forced $M$-edge.
\end{enumerate}
\end{lemma}
\noindent
{\bf Proof.} $(i)$: Holds by definition of $T_i$ and by the distance condition of a d.i.m.\ $M$.
\noindent
$(ii)$: Holds by Observation \ref{obse:neighborhood}.
\noindent
$(iii)$: Follows by Observation \ref{dimC3C5C7C4} $(i)$ since every odd cycle in $G$ must contain at least one $M$-edge, and by (\ref{noMedgesN3N4}).
\noindent
$(iv)$: If $v \in S_3:= N_3 \setminus T_{one}$, i.e., $v$ sees at least two $M$-vertices then clearly, $v \in I$, and thus, $S_3 \subseteq I$ is an independent vertex set (recall that $I$ is an independent vertex set).
\noindent
$(v)$: Suppose that $t_1 \in T_1$ sees $a$ and $b$ in $T_2$. If $ab \in E$ then $u_2,a,b,t_1$ would induce a diamond in $G$. Thus, $ab \notin E$ and now,
$u_2,a,b,t_1$ induce a $C_4$ in $G$; by (\ref{noMedgesN3N4}), the only possible $M$-edge for dominating $t_1a,t_1b$ is $u_1t_1$, i.e., $t_1=u'_1$.
\qed
\medskip
Thus, by Lemma \ref{lemm:structure2} $(v)$, from now on, we can assume that for every $i,j \in \{1,\ldots,k\}$, $i \neq j$, any vertex $t_i \in T_i$ sees at most one vertex in $T_j$. In particular, if for some $i \in \{1,\ldots,k\}$, $T_i=\emptyset$ then there is no d.i.m.\ $M$ of $G$ with $xy \in M$. Thus, for every $i \in \{1,\ldots,k\}$, $T_i \neq \emptyset$.
\begin{lemma}[\cite{BraMos2017}]\label{lemm:vwedgeS3}
The following statements hold:
\begin{enumerate}
\item[$(i)$] For every edge $vw \in E$ with $v,w \in N_3$, $vu_i \in E$, and $wu_j \in E$ $($possibly $i=j)$, we have $|\{v,w\} \cap \{u'_i,u'_j\}| = 1$.
\item[$(ii)$] For every edge $st \in E$ with $s \in S_3$ and $t \in T_i$, $t=u'_i$ holds, and thus $u_it$ is an $xy$-forced $M$-edge.
\end{enumerate}
\end{lemma}
\noindent
{\bf Proof.}
$(i)$: By (\ref{noMedgesN3N4}), $N_3$ does not contain any $M$-edge, and clearly, if $vw \in E$ then either $v$ or $w$ is black; without loss of generality, let $v$ be black but then $v=u'_i$ and $w$ is white, i.e., $w \neq u'_j$.
\noindent
$(ii)$: By Lemma \ref{lemm:structure2} $(iv)$, $S_3 \subseteq I$ and thus, by Lemma \ref{lemm:vwedgeS3} $(i)$, for the edge $st$ with $s \in S_3$, $s$ is white and thus, $t=u'_i$ holds.
\qed
\medskip
Subsequently, for checking if $G$ has a d.i.m.\ $M$ with $xy \in M$, we consider the cases $N_4 = \emptyset$ and $N_4 \neq \emptyset$.
In particular, we have the following property:
\begin{lemma}\label{endpointP5}
If $v \in N_i$ for $i \ge 3$ then $v$ is endpoint of a $P_5$, say with vertices $v,v_1,v_2,v_3,v_4$ such that $v_1,v_2,v_3,v_4 \in \{x,y\} \cup N_1 \cup \ldots \cup N_{i-1}$ and with edges $vv_1 \in E$, $v_1v_2 \in E$, $v_2v_3 \in E$, $v_3v_4 \in E$.
\end{lemma}
\noindent
{\bf Proof.}
First assume that $v \in N_3$. Then $v$ has a neighbor $v_1 \in N_2$, and $v_1$ has a neighbor $v_2 \in N_1$. Since $xy$ is part of a $P_3$ with vertices $x,y,r$ and edges $xy, xr$, we have the following cases:
\begin{enumerate}
\item[(i)] $v_2=r$. Then for $x=v_3,y=v_4$, $v$ is endpoint of a $P_5$.
\item[(ii)] $v_2 \neq r$ and moreover, $v_1r \notin E$. If $v_2x \in E$ then, since $v_2r \notin E$ ($N_1$ is independent), we have a $P_5$ with endpoint $v$ and $v_3=x$, $v_4=r$, and if $v_2x \notin E$ but $v_2y \in E$, we again have a $P_5$ with endpoint $v$, and $v_3=y,v_4=x$.
\end{enumerate}
If $v \in N_i$ for $i > 3$ then, if $i=4$, by similar arguments as above, and if $i > 4$, then obviously, $v$ is endpoint of a $P_5$ as claimed in the lemma. Thus, Lemma \ref{endpointP5} is shown.
\qed
\medskip
Let $X := \{x,y\} \cup N_1 \cup N_2 \cup N_3$ and $Y := V \setminus X$.
Subsequently, for checking if $G$ has a d.i.m.\ $M$ with $xy \in M$, we first consider the possible colorings for $G[X]$.
\section{Coloring $G[X]$}\label{ColoringG[X]}
Recall that for every edge $uv \in M$, $u$ and $v$ are black, for $I=V(G) \setminus V(M)$, every vertex in $I$ is white, $N_2 = \{u_1,\ldots,u_k\}$ and all $u_i$, $1 \le i \le k$, are black, $T_i=N(u_i) \cap N_3$, and
By Lemma \ref{lemm:structure2} $(iv)$ and the Vertex C-Reduction, we can assume that $S_3 = \emptyset$, i.e., $N_3=T_1 \cup \ldots \cup T_k$. Thus, no vertex in $N_3$ has two neighbors in $N_2$.
Since no edge in $N_3$ is in $M$ (recall (\ref{noMedgesN3N4})), we have:
\begin{itemize}
\item[(R1)] All $N_3$-neighbors of a black vertex in $N_3$ must be colored white, and all $N_3$-neighbors of a white vertex in
$N_3$ must be colored black.
\end{itemize}
Moreover, we have:
\begin{itemize}
\item[(R2)] Every $T_i$, $i \in \{1,\ldots,k\}$, should contain exactly one vertex which is black. Thus, if $t_i \in T_i$ is black then all the remaining vertices of $T_i$ must be colored white.
\item[(R3)] If all but one vertices of $T_i$, $i \in \{1,\ldots,k\}$, are white and the final vertex $t$ is not yet colored, then $t$ must be colored black.
\end{itemize}
Since no edge between $N_3$ and $N_4$ is in $M$ (recall (\ref{noMedgesN3N4})), we have:
\begin{itemize}
\item[(R4)] For every edge $st \in E$ with $t \in N_3$ and $s \in N_4$, $s$ is white if and only if $t$ is black and vice versa.
\end{itemize}
Let us say that a vertex $t \in T_i$ (for $i \in \{1,\ldots,k\}$) is an {\em $N_3$-out-vertex} of $T_i$ if it is adjacent to some vertex of $T_j$ with $j \neq i$,
$t$ is an {\em $N_4$-out-vertex} of $T_i$ if it is adjacent to some vertex of $N_4$, and is an {\em in-vertex} of $T_i$ otherwise.
For finding a d.i.m.\ $M$ with $xy \in M$, one can remove all but one in-vertices (except for one of minimum weight); that can be done in polynomial time.
Thus, let us assume:
\begin{itemize}
\item[(A1)] For every $i \in \{1,\ldots,k\}$, $T_i$ has at most one in-vertex.
\end{itemize}
Moreover, since no edge in a $C_4$ is in $M$ (recall Observation \ref{dimC3C5C7C4} $(ii)$) and since $u_i$ is black, we have:
\begin{itemize}
\item[(A2)] Each vertex of $T_i$ which belongs to an induced $C_4$ together with $u_i$ is colored by white; that can be done in polynomial time.
\end{itemize}
If $|T_i|=1$, i.e., $T_i=\{t_i\}$, then $t_i$ is forced to be black, and $u_it_i$ is an $xy$-forced $M$-edge. Thus, after the Edge C-Reduction step (which again can be done in polynomial time), we can assume:
\begin{itemize}
\item[(A3)] For every $i \in \{1,\ldots,k\}$, $|T_i| \geq 2$.
\end{itemize}
\begin{lemma}\label{S124frcompN3N4outvertices}
If $T_i$ is already completely colored and if there is an edge between $T_i$ and $T_j$, $i \neq j$, then the color of all $N_3$-out-vertices as well as of all $N_4$-out-vertices of $T_j$ is forced by rules $(R1)-(R4)$.
\end{lemma}
\noindent
{\bf Proof.} Recall that by $(A3)$, $|T_i| \geq 2$ and $|T_j| \geq 2$. Let $bc \in E$ with $b \in T_i$ and $c \in T_j$. If $b$ is white then $c$ is black and thus, $T_j$ is completely colored. Now we assume that $b$ is black which implies that $c$ is white. Let $a \in T_i$ with $a \neq b$. Then $a$ is white and thus, $ac \notin E$.
If there is no other $N_3$-out-vertex of $T_j$ then the in-vertex of $T_i$ is colored black, and thus, $T_j$ is completely colored.
Now let $s \in T_j$, $s \neq c$, be another $N_3$-out-vertex of $T_j$, and assume that $s$ is not yet colored. Then, since all vertices of $T_i \cup \{c\}$ are already colored, we have $s \cojoin T_i \cup \{c\}$.
Then $s$ contacts some $T_h$ for $h \in \{1,\ldots,k\} \setminus \{i,j\}$, say $st \in E$ with $t \in T_h$.
Again if $t$ contacts $T_i \cup \{c\}$, then $t$ (and thus $s$) is forced to have a color by (R1). Thus assume that $t \cojoin T_i \cup \{c\}$.
If $T_j \setminus \{c,s\}$ contains only vertices which contact $T_i \cup \{c\}$ or which are adjacent to $t$, then the colors of all these vertices are forced either by (R1) or by (A2) (recalling that $G$ is diamond-free), and then the color of $s$ is forced by (R2) or (R3). Thus assume that there is a vertex $d \in T_j$ which does not contact $T_i \cup \{c,t\}$.
\medskip
Let $q_j \in N_1$ be a neighbor of $u_j$, and without loss of generality, assume that $q_jx \in E$.
\medskip
First assume that $ab \notin E$. Then, since $u_j,d,s,t,c,b,u_i,a$ (with center $u_j$) do not induce an $S_{1,2,4}$, we have $ds \in E$.
Since $u_j,q_j,s,t,c,b,u_i,a$ (with center $u_j$) do not induce an $S_{1,2,4}$, we have $q_ju_i \in E$.
Since $q_j,x,u_i,a,u_j,s,t,u_h$ (with center $q_j$) do not induce an $S_{1,2,4}$, we have $q_ju_h \in E$.
But then $q_j,x,u_i,a,u_h,t,s,d$ (with center $q_j$) induce an $S_{1,2,4}$, which is a contradiction.
\medskip
Thus $ab \in E$. Since $q_j,x,u_h,t,u_j,c,b,a$ (with center $q_j$) do not induce an $S_{1,2,4}$, we have $q_ju_h \notin E$.
Since $q_j,x,u_i,a,u_j,s,t,u_h$ (with center $q_j$) do not induce an $S_{1,2,4}$, we have $q_ju_i \notin E$; let $q_i \in N_1$ be a neighbor of $u_i$, and by the same argument, we have $q_iu_j \notin E$.
But now, $u_j,q_j,s,t,c,b,u_i,q_i$ (with center $u_j$) induce an $S_{1,2,4}$, which is a contradiction.
\medskip
Now let $s$ be any $N_4$-out-vertex of $T_j$ (for $s \neq c$), and again assume that $s$ is not yet colored. Then, since all vertices of $T_i \cup \{c\}$ are already colored, we have $s \cojoin T_i \cup \{c\}$. Let $z \in N_4$ be a neighbor of $s$.
If $z$ contacts $T_i \cup \{c\}$, then by (R4), the color of $z$ and the color of $s$ are forced. Then assume that $z$ does not contact $T_i \cup \{c\}$, i.e.,
$z \cojoin T_i \cup \{c\}$.
If $z$ has degree 1, then by Proposition \ref{isolatednonadjtotriangle}, the color of $s$ is forced to be black. Thus, we assume that the degree of $z$ is at least 2; let $z'$ be a new neighbor of $z$.
Since $z \cojoin T_i \cup \{c\}$, we have $z' \notin T_i$, and we can assume that $z' \notin T_j$ (else $u_j,s,z,z'$ would induce a diamond - which is impossible - or $C_4$ which implies that $s$ is forced to be white).
If $s,z,z'$ induce a triangle, then recall that by (\ref{noMedgesN3N4}), no edge between $N_3$ and $N_4$ as well as no edge in $N_3$ is in $M$, but every triangle contains exactly one $M$-edge; if $z' \in N_4$ and $s,z,z'$ induce a triangle then $zz'$ is an $xy$-forced $M$-edge.
Thus, assume that $z's \notin E$.
\medskip
If there is a common neighbor $q \in N_1$ such that $qu_i \in E$ and $qu_j \in E$ (and without loss of generality, $qx \in E$), then,
since $q,x,u_i,a,u_j,s,z,z'$ (with center $q$) do not induce an $S_{1,2,4}$, we have $z'a \in E$, and analogously, $z'b \in E$.
Since $u_i,a,b,z'$ do not induce a diamond, we have $ab \notin E$, but now, $u_i,a,b,z'$ induce a $C_4$, which is a contradiction for the fact that $b$ is black.
\medskip
Thus, $u_i$ and $u_j$ do not have a common neighbor in $N_1$; let $q_i \in N_1$ with $q_iu_i \in E$, and $q_j \in N_1$ with $q_ju_j \in E$, $q_i \neq q_j$.
But then $u_j,q_j,s,z,c,b,u_i,q_i$ (with center $u_j$) induce an $S_{1,2,4}$, which is a contradiction.
\medskip
Thus, Lemma \ref{S124frcompN3N4outvertices} is shown.
\qed
\subsection{The Case $N_4 = \emptyset$}\label{N4empty}
Recall that $S_3 = \emptyset$, i.e., $N_3=T_1 \cup \ldots \cup T_k$.
\medskip
$G[\{u_i\} \cup T_i]$ is a {\em trivial component in $G[S_2 \cup N_3]$} if $T_i$ has no contact to any other $T_j$, $j \neq i$. By the way, if $T_i=\emptyset$, it leads to a contradiction.
Obviously, checking a possible d.i.m.\ $M$ with $xy \in M$ can be done easily (and independently) for trivial components; for a minimum weight vertex $u'_i \in T_i$ let $u_iu'_i \in M$.
\medskip
From now on we present a coloring procedure for nontrivial components $K$ in $G[S_2 \cup N_3]$, i.e., $K$ contains at least two $T_i,T_j$, $i \neq j$ with contact to each other.
\medskip
For any nontrivial component $K$ in $G[S_2 \cup N_3]$, say $V(K)=\{u_1,\ldots,u_p\} \cup T_1 \cup \ldots \cup T_p$, $p \ge 2$, the coloring procedure starts with at most $|T_1|$ possible colorings of $T_1$ (recall (R2)). Then for each of these possible colorings, by Lemma \ref{S124frcompN3N4outvertices}, it can be applied to $T_i$ which contacts $T_1$ etc.\ until all vertices in $K$ are feasibly colored or it leads to a contradiction. Since by (A1), there is at most one in-vertex of $T_i$, the color of such an in-vertex is finally forced by (R2) and (R3) and by Lemma \ref{S124frcompN3N4outvertices}.
As an example of a contradiction, if there are three edges between $T_1$ and $T_2$, say $t_1t_2 \in E$, $t'_1t'_2 \in E$, and $t''_1t''_2 \in E$ for $t_i,t'_i,t''_i \in T_i$, $i=1,2$, then $t_1$ is black if and only if $t_2$ is white, $t'_1$ is black if and only if $t'_2$ is white, and
$t''_1$ is black if and only if $t''_2$ is white. Without loss of generality, assume that $t_1$ is black, and $t_2$ is white. Then $t'_1$ is white, and $t'_2$ is black, but now, $t''_1$ and $t''_2$ are white which leads to a contradiction.
Then, by the contradiction, $xy \notin M$ for any dominating induced matching $M$ of $G$.
\medskip
If the coloring procedure for $K$ ends without contradiction with respect to some of the $|T_1|$ possible
colorings of $T_1$ then we choose a minimum weight solution for the DIM problem on $K$.
\medskip
If for at least one of the components, it leads to a contradiction then there is no such d.i.m.\ $M$ with $xy \in M$.
\begin{corollary}\label{S124frcompN4emptypolcol}
If $N_4 = \emptyset$ then the DIM problem can be done in polynomial time.
\end{corollary}
\noindent
{\bf Proof.}
For trivial components, it can be obviously done. For every nontrivial component, it can be done in polynomial time as above. Thus,
in the case $N_4 = \emptyset$, it leads to a polynomial time solution since all the components of $G[S_2 \cup N_3]$ can be independently colored.
\qed
\subsection{The Case $N_4 \neq \emptyset$}\label{N4nonempty}
Recall again that $S_3=\emptyset$ and $N_3=T_1 \cup \ldots \cup T_k$.
\begin{proposition}\label{isolatednonadjtotriangle}
If $z \in N_4$ is isolated in $G[Y]$ and $z$ contacts $t_i \in T_i$ then $u_it_i \in M$ is an $xy$-forced $M$-edge. In particular, if $|N(z) \cap T_i| \ge 2$ then $G$ has no d.i.m.\ $M$ with $xy \in M$.
\end{proposition}
\noindent
{\em Proof.} Clearly, if $xy \in M$, there is an $M$-edge $u_it'_i$ for some $t'_i \in T_i$. However, if $t'_i \neq t_i$ then, since $z$ is isolated, the only possible way of dominating edge $t_iz$ is $u_it_i \in M$. If $|N(z) \cap T_i| \ge 2$, say $a,b \in T_i$ with $az \in E$, $bz \in E$ then, since $G$ is diamond-free, $ab \notin E$ but then $u_i,a,b,z$ induce a $C_4$, and by Observation \ref{dimC3C5C7C4} $(ii)$, there is no $M$-edge dominating $az$, $bz$. Thus, in this case, $G$ has no d.i.m.\ $M$ with $xy \in M$.
\qed
\medskip
Again, since by (A1), there is at most one in-vertex of $T_j$, the color of such an in-vertex is finally forced by (R2) and (R3) and by
Lemma \ref{S124frcompN3N4outvertices}.
\begin{corollary}\label{S124frcompN4nonemptypolcol}
If $N_4 \neq \emptyset$ then again for every component in $G[S_2 \cup N_3]$, it can be done in polynomial time whether it could be feasibly colored or it could lead to a contradiction.
\end{corollary}
For combining the ``coloring approach'' of \cite{HerLozRieZamdeW2015,KorLozPur2014} with the above results, we show:
\begin{lemma}\label{lemm:S124Xcoloring}
For $S_{1,2,4}$-free graphs $G$, the number of feasible $xy$-colorings of $G[X]$ is at most polynomial. In particular, such $xy$-colorings can be detected in polynomial time.
\end{lemma}
For the proof of Lemma \ref{lemm:S124Xcoloring}, we will collect some propositions below.
\medskip
Connecting a feasible coloring of $G[X]$ with a corresponding one of $G[Y]$ means that every vertex $v \in N_3$ determines the color of its (possible) neighbors in $N_4$: Clearly, if $v \in N_3$ is white then all of its neighbors in $N_4$ are forced to be black, and by fact (\ref{noMedgesN3N4}), if $v \in N_3$ is black then all of its neighbors in $N_4$ are forced to be white. Clearly, it can result in a contradiction, e.g., if a vertex $u \in N_4$ is adjacent to a white vertex $w \in N_3$ and to a black vertex $v \in N_3$. Thus, in this case, $xy$ is not contained in any d.i.m.\ of $G$.
\medskip
Recall that we have a partial feasible $xy$-coloring which means that $x$ and $y$ are black, all vertices of $N_1$ are white, all vertices of $N_2= \{u_1,\ldots,u_k\}$ are black, and we can assume that $S_3=\emptyset$ (recall that by Lemma \ref{lemm:structure2}~$(iv)$, any vertex in the component $K$ contacting $S_3$ is black, and thus, the color of each vertex of $K$ is forced, and every vertex in $N_4$ contacting $S_3$ is black).
Then let us see how this partial feasible $xy$-coloring can be extended.
\medskip
By Lemma \ref{lemm:structure2} we have:
\begin{proposition}\label{QcompK}
Let $Q$ denote the family of components of $G[S_2 \cup T_{one}]$, and let $K$ be a member of $Q$.
\begin{itemize}
\item[$(i)$] If for some $i \in \{1,\ldots,k\}$, $K$ contains a subset $T_i$ such that $|T_i| \geq 2$ and there is a vertex $z \in N_4$ with $z \join T_i$
then, by the $C_4$-property in Observation $\ref{dimC3C5C7C4}$ $(ii)$ and since $G$ is diamond-free, $G$ has no d.i.m.\ with $xy \in M$.
\item[$(ii)$] If $K \cojoin N_4$ then, by the results of Section $\ref{N4empty}$, $K$ can be treated independently to the other members of $Q$.
\end{itemize}
\end{proposition}
\medskip
Thus, by the previous rules and assumptions as well as Propositions \ref{isolatednonadjtotriangle} and \ref{QcompK}, we can restrict $Q$ as follows: Let $Q^*$ be the family of components $H$ of $G[S_2 \cup T_{one}]$ such that:
\begin{itemize}
\item[(R5)] for any $z \in N_4$, there is at least one non-neighbor of $z$ in $V(H) \cap N_3$,
\item[(R6)] some vertex of $V(H)$ contacts $N_4$, and
\item[(R7)] no vertex $z \in N_4$ is isolated in $G[Y]$.
\end{itemize}
\begin{proposition}\label{P1S124}
If a vertex of $N_4$ contacts at least two members of $Q^*$ then $|Q^*| \le 3$.
\end{proposition}
\noindent
{\em Proof.} Suppose to the contrary that there are four distinct members $H_1,H_2,H_3,H_4$ of $Q^*$ such that a vertex $z \in N_4$ contacts $H_1$ and $H_2$.
Let $u_i \in V(H_i) \cap S_2$, $1 \le i \le 4$, such that for $i \in \{1,2\}$, there are $t_i \in V(H_i) \cap T_i$ with $zt_i \in E$, and for $i \in \{3,4\}$,
there are $t_i \in V(H_i) \cap T_i$ with $zt_i \notin E$ (such non-neighbors $t_3,t_4$ of $z$ exist by condition (R5) of the definition of $Q^*$).
Clearly, for any $i \neq j$, we have $t_it_j \notin E$ since $t_1,\ldots,t_4$ are in distinct components $H_1,\ldots,H_4$, and clearly $t_iu_j \notin E$.
\begin{clai}\label{nocommonN1neighuiuj}
For all $i,j \in \{1,2,3,4\}$, $i \neq j$, $u_i$ and $u_j$ do not have any common neighbor in $N_1$.
\end{clai}
\noindent
{\em Proof.} We first claim that $u_1$ and $u_3$ do not have a common neighbor in $N_1$: Let $a_1 \in N_1$ with $a_1u_1 \in E$, and without loss of generality, let $a_1x \in E$. Since $a_1,x,u_3,t_3,u_1,t_1,z,t_2$ (with center $a_1$) do not induce an $S_{1,2,4}$, we have $a_1u_3 \notin E$, and thus, $u_1$ and $u_3$ do not have any common neighbor in $N_1$.
Similarly, by symmetry, we can show that $u_1$ and $u_4$ (respectively, $u_2$ and $u_3$, $u_2$ and $u_4$) do not have any common neighbor in $N_1$.
$\diamond$
\medskip
Let $a_1 \in N_1$ be adjacent to $u_1$, and let $a_3 \in N_1$ be adjacent to $u_3$. By the previous facts, $a_3$ is nonadjacent to $u_1,u_2$ and $a_1$ is nonadjacent to $u_3,u_4$.
Next we claim that $a_1$ is nonadjacent to $u_2$: Otherwise an $S_{1,2,4}$ arises of center $a_1$ with four vertices in $\{x,y,a_3,u_3,t_3\}$, and $u_1,t_1$, and $u_2$. Then by construction, let $a_2 \in N_1$ be adjacent to $u_2$. By the previous facts, $a_2$ is nonadjacent to $u_1,u_3,u_4$.
Now, $a_1$ and $a_2$ are nonadjacent to $u_4$. Furthermore $a_3$ is nonadjacent to $u_4$, since otherwise an $S_{1,2,4}$ arises of center $a_3$ with four vertices in $\{x,y,a_1,u_1,t_1\}$, and $u_3,t_3$, and $u_4$. By construction, let $a_4 \in N_1$ be adjacent to $u_4$; recall that $a_1,a_2,a_3,a_4$ are pairwise distinct and $a_4$ is nonadjacent to $u_1,u_2,u_3$.
$\diamond$
\medskip
Since $G$ is diamond-free, at most one of $a_1,a_2,a_3,a_4$ is adjacent to $x$ and to $y$. Without loss of generality, assume $xa_1 \in E$. If $x$ has at least three neighbors in $a_1,a_2,a_3,a_4$, say additionally, $xa_i \in E$ and $xa_j \in E$, $i \neq j$ and $i,j \neq 1$, then $x,a_j,a_i,u_i,a_1,u_1,t_1,z$ (with center $x$) induce an $S_{1,2,4}$. Now assume that $x$ misses at least two of $a_2,a_3,a_4$. Analogously, if $y$ is adjacent to at least three of $a_1,a_2,a_3,a_4$ then we get an $S_{1,2,4}$ as above. Finally, if each of $x$ and $y$ has exactly two neighbors in $a_1,a_2,a_3,a_4$ (but no common neighbor since each of $a_i$ is adjacent to $x$ or $y$) then one can easily check that there is an $S_{1,2,4}$. Thus, Proposition~\ref{P1S124} is shown.
\qed
\medskip
Clearly, in the case $|Q^*| \le 3$, the number of $xy$-colorings of $G[X]$ is bounded by a polynomial.
From now on, by Proposition \ref{P1S124}, we can add another restriction:
\begin{itemize}
\item[(R8)] Each vertex of $N_4$ contacts at most one member of $Q^*$.
\end{itemize}
Let $Q^{**}$ be the family of components $H$ of $G[S_2 \cup T_{one}]$ fulfilling conditions (R5)--(R8).
\begin{proposition}\label{notisoQatmost3}
$|Q^{**}| \le 3$.
\end{proposition}
\noindent
{\em Proof.} Suppose to the contrary that there are four distinct members $H_1,H_2,H_3,H_4$ of $Q^{**}$ such that a non-isolated vertex $z_1 \in N_4$ contacts $H_1$. Let again $u_i \in V(H_i) \cap S_2$, $1 \le i \le 4$, let $z_1$ contact $T_1$, and since $z_1$ is not isolated in $G[Y]$, there is a neighbor $z_2 \in Y$ of $z_1$. By condition (R8), $z_1 \cojoin T_i$, $i \ge 2$, and by condition (R5), $z_2$ has a non-neighbor in each $T_i$, $i \ge 2$. Let $a_1 \in N_1$ be a neighbor of $u_1$, and without loss of generality, let $a_1x \in E$.
For $i \ge 2$, let $t_i \in T_i$ be a non-neighbor of $z_2$. Then, since $a_1,x,u_i,t_i,u_1,t_1,z_1,z_2$ (with center $a_1$) do not induce an $S_{1,2,4}$, we have $a_1u_i \notin E$ for each $i \ge 2$. Let $a_i \in N_1$ be a neighbor of $u_i$. Analogously, $a_iu_1 \notin E$ for each $i \ge 2$. If $u_i$ and $u_j$, $i \neq j$, have a common neighbor $a_i \in N_1$ then it is easy to see that there is an $S_{1,2,4}$. Thus we can assume that each $u_i$ has its private neighbor $a_i$ in $N_1$. Now, since $G$ is diamond-free, at most one of $a_i$ is adjacent to $x$ and to $y$, and thus, as in the proof of Proposition~\ref{P1S124}, it is easy to see that this again leads to an $S_{1,2,4}$. Thus, Proposition \ref{notisoQatmost3} is shown.
\qed
\medskip
\noindent
{\bf Proof of Lemma \ref{lemm:S124Xcoloring}.} It follows by Propositions \ref{isolatednonadjtotriangle}--\ref{notisoQatmost3}.
In particular all the above properties can be checked in polynomial time.
\qed
\section{The Structure of $G[Y]$}\label{StructureG[Y]}
Recall $X:=\{x,y\} \cup N_1 \cup N_2 \cup N_3$ and $Y:= V \setminus X$. Clearly, in this section, $Y \neq \emptyset$.
We show that $G[Y]$ is $S_{1,2,2}$-free.
In Section \ref{sec:DIMpolS124fr}, for coloring $G[Y]$, we will use the polynomial time result for DIM on $S_{1,2,2}$-free graphs (see Theorem \ref{DIMpolresults} $(ii)$).
The approach in \cite{HerLozRieZamdeW2015,KorLozPur2014}, however, is strongly based on coloring vertices white or black as already mentioned (i.e., all vertices of $V(M)$ are black and all vertices of $I=V \setminus V(M)$ are white).
By \cite{KorLozPur2014}, for $S_{1,2,3}$-free graphs, DIM is also solvable in polynomial time if $G$ has a special subset of vertices whose colors are fixed to be black or white.
\begin{lemma}\label{S124frYS122fr}
If $G$ is $S_{1,2,4}$-free then $G[Y]$ is $S_{1,2,2}$-free.
\end{lemma}
\noindent
{\bf Proof.}
Suppose to the contrary that there is an $S_{1,2,2}$ $H$ in $G[Y]$, say with vertices $d,a_1,a_2,b_1,b_2,c_1$ and edges $da_1 \in E$, $db_1 \in E$, $dc_1 \in E$, $a_1a_2 \in E$, $b_1b_2 \in E$. Let $v \in N_p$ be a neighbor of $H$ with smallest $p \ge 3$ (such a neighbor exists since $G$ is connected).
By Lemma \ref{endpointP5}, $v$ is endpoint of a $P_5$, say $P(v)$ with vertices $v,v_1,v_2,v_3,v_4$, and clearly, none of $v_i$, $i \in \{1,2,3,4\}$, is a neighbor of $H$.
We first claim:
\begin{equation}\label{vnonadjd}
vd \notin E.
\end{equation}
\noindent
{\em Proof.} Suppose to the contrary that $vd \in E$. First assume that $va_1 \in E$. Then, since $G$ is diamond-free, $va_2 \notin E$, $vb_1 \notin E$, and $vc_1 \notin E$.
If $vb_2 \in E$ then $v,a_1,b_2,b_1,v_1,v_2,v_3,v_4$ (with center $v$) would induce an $S_{1,2,4}$. Thus, $vb_2 \notin E$ but now,
$d,c_1,b_1,b_2,v,v_1,v_2,v_3$ (with center $d$) induce an $S_{1,2,4}$, which is a contradiction.
By symmetry, the same arguments hold if $vb_1 \in E$ (instead of $va_1 \in E$).
\medskip
From now on, let $va_1 \notin E$ and $vb_1 \notin E$. If $va_2 \notin E$ then $d,b_1,a_1,a_2,v,v_1,v_2,v_3$ (with center $d$) would induce an $S_{1,2,4}$, and similarly if $vb_2 \notin E$. Thus, $va_2 \in E$ and $vb_2 \in E$ but now, $v,b_2,a_2,a_1,v_1,v_2,v_3,v_4$ (with center $v$) induce an $S_{1,2,4}$,
which is a contradiction.
Thus, (\ref{vnonadjd}) is shown.
$\diamond$
\medskip
Next we claim:
\begin{equation}\label{vnonadja1a2}
(va_1 \notin E \mbox{ or } va_2 \notin E) \mbox{ and } (vb_1 \notin E \mbox{ or } vb_2 \notin E).
\end{equation}
\noindent
{\em Proof.} Suppose to the contrary that $va_1 \in E$ and $va_2 \in E$. Then, since $G$ is butterfly-free, $vb_1 \notin E$ or $vb_2 \notin E$.
If $vb_1 \in E$ then $vb_2 \notin E$, and thus, $v,a_1,b_1,b_2,v_1,v_2,v_3,v_4$ (with center $v$) would induce an $S_{1,2,4}$.
Analogously, if $vb_2 \in E$ then $vb_1 \notin E$, and similarly, $v,a_1,b_2,b_1,v_1,v_2,v_3,v_4$ (with center $v$) would induce an $S_{1,2,4}$,
which is a contradiction in each case.
Thus, $vb_1 \notin E$ and $vb_2 \notin E$. Now, since $d,c_1,b_1,b_2,a_1,v,v_1,v_2$ (with center $d$) does not induce an $S_{1,2,4}$, we have $vc_1 \in E$ but now,
$v,a_2,c_1,d,v_1,v_2,v_3,v_4$ (with center $v$) induce an $S_{1,2,4}$, which is a contradiction.
\medskip
By symmetry, also $vb_1 \in E$ and $vb_2 \in E$ is impossible. Thus, (\ref{vnonadja1a2}) is shown.
$\diamond$
\medskip
If $v$ has exactly one neighbor in $a_1,a_2$ and exactly one neighbor in $b_1,b_2$, say $va_1 \in E$ and $vb_1 \in E$, then, by (\ref{vnonadja1a2}),
$v,b_1,a_1,a_2,v_1,v_2,v_3,v_4$ (with center $v$) would induce an $S_{1,2,4}$, and similarly in every other case.
Thus, without loss of generality assume that $va_1 \notin E$ and $va_2 \notin E$. By (\ref{vnonadja1a2}), $v$ sees at most one of $b_1,b_2$.
If $vb_1 \in E$ (and $vb_2 \notin E$) then $b_1,b_2,d,a_1,v,v_1,v_2,v_3$ (with center $b_1$) would induce an $S_{1,2,4}$.
If $vb_2 \in E$ (and $vb_1 \notin E$) and if $vc_1 \notin E$ then $d,c_1,a_1,a_2,b_1,b_2,v,v_1$ (with center $d$) would induce an $S_{1,2,4}$.
Thus, $vc_1 \in E$, but now, $d,b_1,a_1,a_2,c_1,v,v_1,v_2$ (with center $d$) induce an $S_{1,2,4}$, which is a contradiction.
Thus, $vb_1 \notin E$ and $vb_2 \notin E$.
\medskip
Finally, $c_1$ is the only neighbor of $v$ in $H$ but then again $d,b_1,a_1,a_2,c_1,v,v_1,v_2$ (with center $d$) induce an $S_{1,2,4}$, which is a contradiction. Thus, Lemma \ref{S124frYS122fr} is shown.
\qed
\medskip
For the case of $S_{1,1,4}$-free graphs, we can show even more:
\begin{lemma}\label{S114frYS111fr}
If $G$ is $S_{1,1,4}$-free then $G[Y]$ is $S_{1,1,1}$-free.
\end{lemma}
\noindent
{\bf Proof.}
Suppose to the contrary that there is an $S_{1,1,1}$ $H$ in $G[Y]$, say with vertices $d,a,b,c$ and edges $da \in E$, $db \in E$, $dc \in E$.
Let $v \in N_p$ be a neighbor of $H$ with smallest $p \ge 3$ (such a neighbor exists since $G$ is connected). As above, by Lemma \ref{endpointP5}, $v$ is endpoint of a $P_5$, say $P(v)$ with vertices $v,v_1,v_2,v_3,v_4$, and clearly, none of $v_i$, $i \in \{1,2,3,4\}$, is a neighbor of~$H$.
We first claim:
\begin{equation}\label{vnonadjmidpointclaw}
vd \notin E.
\end{equation}
\noindent
{\em Proof.} Suppose to the contrary that $vd \in E$. If $va \in E$ then, since $G$ is diamond-free, $vb \notin E$, and $vc \notin E$, but now,
$d,b,c,v,v_1,v_2,v_3$ (with center $d$) would induce an $S_{1,1,4}$.
By symmetry, the same arguments hold if $vb \in E$ or $vc \in E$. Thus, (\ref{vnonadjmidpointclaw}) is shown.
$\diamond$
\medskip
If $v$ is adjacent to only one of $a,b,c$, say $va \in E$, then $d,b,c,a,v,v_1,v_2$ (with center $d$) would induce an $S_{1,1,4}$.
Thus, $v$ is adjacent to at least two of $a,b,c$, say $va \in E$ and $vb \in E$ but then $v,a,b,v_1,v_2,v_3,v_4$ (with center $v$) induce an $S_{1,1,4}$, which is a contradiction. Thus, Lemma \ref{S114frYS111fr} is shown.
\qed
\section{A polynomial-time algorithm for DIM on $S_{1,2,4}$-free graphs}\label{sec:DIMpolS124fr}
The following procedure is part of the algorithm:
\medskip
\begin{proc}[DIM-with-$xy$-in-$S_{1,2,4}$-free-graphs]\label{DIMwithxyS124}
\begin{tabbing}
xxxxxxx \= \kill\\
{\bf Input:} \> A connected $(S_{1,2,4},K_4$,diamond,butterfly$)$-free graph $G = (V,E)$, and\\
\> an edge $xy \in E$ which is part of a $P_3$ in $G$.\\
{\bf Task:} \> Return a d.i.m.\ $M$ with $xy \in M$ $($STOP with success$)$ or\\
\> a proof that $G$ has no d.i.m.\ $M$ with $xy \in M$ $($STOP with failure$)$.
\end{tabbing}
\begin{itemize}
\item[$(a)$] Set $M:= \{xy\}$. Determine the distance levels $N_i = N_i(xy)$, $i \ge 1$, with respect to $xy$.
\item[$(b)$] Check whether $N_1$ is an independent set $($see fact $(\ref{N1subI}))$ and $G[N_2]$ is the disjoint union of edges and isolated vertices $($see fact $(\ref{N2M2S2}))$. If not, then STOP with failure.
\item[$(c)$] For the set $M_2$ of edges in $G[N_2]$, apply the Edge C-Reduction for every edge in $M_2$ correspondingly. Moreover, apply the Edge C-Reduction for each edge $bc$ according to fact $(\ref{triangleaN3bcN4})$ and then for each edge $u_it_i$ according to Lemma $\ref{lemm:structure2}$ $(v)$.
\item[$(d)$] {\bf if} $N_4 = \emptyset$ then apply the approach described in Section $\ref{N4empty}$. Then either return that $G$ has no d.i.m.\ $M$ with $xy \in M$ or return $M$ as a d.i.m.\ with $xy \in M$.
\item[$(e)$] {\bf if} $N_4 \neq \emptyset$ {\bf then} for $X := \{x,y\} \cup N_1 \cup N_2 \cup N_3$ and $Y := V \setminus X$ {\bf do}:
\begin{itemize}
\item[$(e.1)$] According to Lemma $\ref{lemm:S124Xcoloring}$, compute all feasible $xy$-colorings of $G[X]$. If no such $xy$-coloring exists, then STOP with failure.
\item[$(e.2)$] {\bf for each} feasible $xy$-coloring of $G[X]$ {\bf do}:
\begin{enumerate}
\item[$(e.2.1)$] Derive a partial coloring of $G[Y]$ by the forcing rules; {\bf if} a contradiction arises in vertex coloring {\bf then} STOP with failure.
\item[$(e.2.2)$] According to Lemma $\ref{S124frYS122fr}$, apply a polynomial time algorithm for DIM on $S_{1,2,2}$-free graphs $($see $\cite{KorLozPur2014})$ for $G[Y]$ with its partial coloring;
{\bf if} it returns a d.i.m.\ of $G[Y]$ {\bf then} STOP with success and return the feasible $xy$-coloring of $G$ derived by the feasible $xy$-coloring of $G[X]$ and by such a d.i.m.\ of $G[Y]$.
\end{enumerate}
\item[$(e.3)$] STOP with failure.
\end{itemize}
\end{itemize}
\end{proc}
\begin{theorem}\label{theo:procedureDIMxyS124}
Procedure $\ref{DIMwithxyS124}$ is correct and can be done in polynomial time.
\end{theorem}
\noindent
{\bf Proof.} The correctness of the procedure follows from the structural analysis of $S_{1,2,4}$-free graphs with a d.i.m.
\medskip
The polynomial time bound follows from the fact that Steps (a) and (b) can clearly be done in polynomial time, Step (c) can be done in polynomial time since the Edge C-Reduction can be done in polynomial time, Steps (d) and (e) can be done in polynomial time by the results in Sections \ref{ColoringG[X]} and
\ref{StructureG[Y]} and by the fact that DIM can be solved in polynomial time for $S_{1,2,2}$-free graphs \cite{KorLozPur2014} (see also \cite{HerLozRieZamdeW2015}).
\qed
\begin{algo}[DIM-$S_{1,2,4}$-free]\label{DIMS124fr}
\begin{tabbing}
xxxxxxx \= \kill\\
{\bf Input:} \> A connected $(S_{1,2,4},K_4)$-free graph $G = (V,E)$. \\
{\bf Task:} \> Determine a d.i.m.\ of $G$ if there is one, or find out that $G$ has no d.i.m.
\end{tabbing}
\begin{itemize}
\item[$(A)$] Determine the set $F_1$ of all mid-edges of diamonds in $G$, and the set $F_2$ of all peripheral edges of butterflies in $G$. Let $M:=F_1 \cup F_2$. Check whether $M$ is an induced matching in $G$. If not then STOP--$G$ has no d.i.m. Otherwise, check whether $M$ is a dominating edge set of $G$. If yes, we are done. Otherwise apply the Edge C-Reduction for every edge in $F_1 \cup F_2$; without loss of generality, assume that the resulting graph $G'=(V',E')$ is connected (if not, do the next steps for each component of $G'$). Let $G:=G'$.
$\{$From now on, $G$ is $(S_{1,2,4},K_4$,diamond,butterfly)-free.$\}$
\item[$(B)$] Check whether $G$ has a single edge $uv \in E$ which is a d.i.m.\ of $G$. If yes then select such an edge as output and STOP--this is a d.i.m.\ of $G$.
$\{$Otherwise, every d.i.m.\ of $G$ would have at least two edges.$\}$
\item[$(C)$] {\bf for each} edge $xy \in E$ in a $P_3$ of $G$, carry out Procedure $\ref{DIMwithxyS124}$; {\bf if} it returns ``STOP with failure'' for all edges $xy$ in a $P_3$ of $G$ {\bf then} STOP--$G$ has no d.i.m.\ {\bf else} STOP and return a d.i.m.\ of $G$.
\end{itemize}
\end{algo}
\begin{theorem}\label{theo:DIMS124}
Algorithm $\ref{DIMS124fr}$ is correct and can be done in polynomial time. Thus, DIM can be solved in polynomial time for $S_{1,2,4}$-free graphs.
\end{theorem}
\noindent
{\bf Proof.} The correctness of the algorithm follows from the structural analysis of $S_{1,2,4}$-free graphs with a d.i.m.\ In particular: concerning Step (B), one can easily verify that if $G$ has a d.i.m.\ of one edge, then $G$ has no d.i.m.\ with more than one edge; concerning Step (C), one can refer to Observation \ref{obse:xy-in-P3}.
\medskip
The time bound follows from the fact that Step (A) can be done in polynomial time (in particular the Edge C-Reduction can be done in polynomial time), Step (B) can be done in polynomial time, and Step (C) can be done in polynomial time by Theorem \ref{theo:procedureDIMxyS124}.
\qed
\section{Conclusion}
It is still a widely open problem whether DIM can be solved in polynomial time for $S_{i,j,k}$-free graphs for any fixed $i,j,k$; for example, it is not clear how to solve it for $S_{1,3,4}$-free graphs or for $S_{2,2,4}$-free graphs but the approaches described here as well as in \cite{HerLozRieZamdeW2015} might be helpful.
\medskip
\noindent
{\bf Acknowledgment.} The second author would like to witness that he just tries to pray a lot and is not able to do anything without that - ad laudem Domini.
\begin{footnotesize}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,025 |
F1rst Wrestling/HOW Mineapolis, MN 7/31/2009
F1rst Wrestling and Heavy On Wrestling presented a mostly women's card called "Friday Night Fantasy" this past Friday night at First Ave night club in downtown Minneapolis, MN. There were a few changes from the advertised line up. As reported on PROWRESTLING.NET, Johnny Fairplay (Survivor) and his wife Michelle (America's Next Top Model) were unable to make it to Minneapolis for this event. Most of the ladies sold 8x10's, and possed for photos with fans before the show, at intermisson and after the show.
The card started with a men's match as Arya Daivari came out and claimed that the fans there didn't want to see women's wrestling. Cpl. Julio Julio came out to wrestle Daivari but all he go for his trouble was a big cut on his head as he was thrown from the ring by TNA's Tara. Terra then brought out Stacy Carter, aka Ms Kitty, and they both slapped Daivari silly.
Sojo Bolt defeated Daffney in her return to First Ave. When Bolt was know as just Josie she had her second ever match in this same building.
Allison Wonderland won a bikini contest. Also featured in this contest was Nikki Mayday who has done some back stage interviews on the F1rst Wrestling DVD's. Lacey Von Erich was scheduled for this event but she did not make it for some reason.
Nikki Roxx defeated Ann Brookstone in a falls count anywhere hard core match. Roxx pushed Brookstone face first into a pile of thumb tacks then rolled her up for the three count. The original WWE diva Sunny was the special guest referee for this match.
Arik Cannon & Christy Hemme defeated Pete Huge & Allison Wonderland in a mixed tag match.
In the main event O.D.B. defeated Awesome Kong with a roll up/Molly Go Round. Nora "Molly Holly" Greenwald was the special guest referee for this match.
Labels: Allison Wonderland, Ann Brookstone, Arik Cannon, Arya Daivari, Awesome Kong, Christy Hemme, Daffney, F1rst Wrestling, HOW, Nora Greenwald, ODB, Sojourner Bolt, Sunny, Tara
MIW Mounds View, MN 8/22/2009
Wrestling Art show opening 8/22/2009
HOW Proctor, MN 8/21.2009
IWA MN Prior Lake, MN 8/15/2009
HOW Superior, WI 8/1/2009 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,510 |
module LIFX
module HTTP
module Loader
module Equatable
def ==(other)
other.is_a?(self.class) && other.to_h == to_h
end
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,840 |
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