text:
// Copyright 2021 Esri.
//
// Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License.
// You may obtain a copy of the License at: http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an
// "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific
// language governing permissions and limitations under the License.

using Esri.ArcGISRuntime.Data;
using Esri.ArcGISRuntime.Geometry;
using Esri.ArcGISRuntime.Mapping;
using Esri.ArcGISRuntime.Portal;
using Esri.ArcGISRuntime.Symbology;
using Esri.ArcGISRuntime.Tasks;
using Esri.ArcGISRuntime.Tasks.Offline;
using Esri.ArcGISRuntime.UI;
using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Threading.Tasks;
using Windows.UI.Popups;
using Windows.UI.Xaml.Controls;

namespace ArcGISRuntime.UWP.Samples.GenerateOfflineMapWithOverrides
{
    [ArcGISRuntime.Samples.Shared.Attributes.Sample(
        name: "Generate offline map (overrides)",
        category: "Map",
        description: "Take a web map offline with additional options for each layer.",
        instructions: "Modify the overrides parameters:",
        tags: new[] { "LOD", "adjust", "download", "extent", "filter", "offline", "override", "parameters", "reduce", "scale range", "setting" })]
    public partial class GenerateOfflineMapWithOverrides
    {
        // The job to generate an offline map.
        private GenerateOfflineMapJob _generateOfflineMapJob;

        // The extent of the data to take offline.
        private Envelope _areaOfInterest = new Envelope(-88.1541, 41.7690, -88.1471, 41.7720, SpatialReferences.Wgs84);

        // The ID for a web map item hosted on the server (water network map of Naperville IL).
        private const string WebMapId = "acc027394bc84c2fb04d1ed317aac674";

        public GenerateOfflineMapWithOverrides()
        {
            InitializeComponent();

            // Load the web map, show area of interest, restrict map interaction, and set up authorization.
            Initialize();
        }

        private async void Initialize()
        {
            try
            {
                // Create the ArcGIS Online portal.
                ArcGISPortal portal = await ArcGISPortal.CreateAsync();

                // Get the Naperville water web map item using its ID.
                PortalItem webmapItem = await PortalItem.CreateAsync(portal, WebMapId);

                // Create a map from the web map item.
                Map onlineMap = new Map(webmapItem);

                // Display the map in the MapView.
                MyMapView.Map = onlineMap;

                // Disable user interactions on the map (no panning or zooming from the initial extent).
                MyMapView.InteractionOptions = new MapViewInteractionOptions
                {
                    IsEnabled = false
                };

                // Create a graphics overlay for the extent graphic and apply a renderer.
                SimpleLineSymbol aoiOutlineSymbol = new SimpleLineSymbol(SimpleLineSymbolStyle.Solid, System.Drawing.Color.Red, 3);
                GraphicsOverlay extentOverlay = new GraphicsOverlay
                {
                    Renderer = new SimpleRenderer(aoiOutlineSymbol)
                };
                MyMapView.GraphicsOverlays.Add(extentOverlay);

                // Add a graphic to show the area of interest (extent) that will be taken offline.
                Graphic aoiGraphic = new Graphic(_areaOfInterest);
                extentOverlay.Graphics.Add(aoiGraphic);

                // Hide the map loading progress indicator.
                LoadingIndicator.Visibility = Windows.UI.Xaml.Visibility.Collapsed;

                // Clean up any existing output data folders that might exist from running this sample previously.
                // The output data folder is where the results of the taking the web map offline get stored on the device.
                MyMapView.Unloaded += (s, e) =>
                {
                    // Find output mobile map folders in the temp directory.
                    string[] outputFolders = Directory.GetDirectories(Environment.ExpandEnvironmentVariables("%TEMP%"), "NapervilleWaterNetwork*");

                    // Loop through the folder names and delete them.
                    foreach (string dir in outputFolders)
                    {
                        try
                        {
                            // Delete the folder.
                            Directory.Delete(dir, true);
                        }
                        catch (Exception)
                        {
                            // Ignore exceptions (files might be locked, for example).
                        }
                    }
                };
            }
            catch (Exception ex)
            {
                MessageDialog dialog = new MessageDialog(ex.ToString(), "Error loading map");
                await dialog.ShowAsync();
            }
        }

        private async void TakeMapOfflineButton_Click(object sender, Windows.UI.Xaml.RoutedEventArgs e)
        {
            // Create a new folder for the output mobile map.
            string packagePath = Path.Combine(Environment.ExpandEnvironmentVariables("%TEMP%"), @"NapervilleWaterNetwork");
            int num = 1;
            while (Directory.Exists(packagePath))
            {
                packagePath = Path.Combine(Environment.ExpandEnvironmentVariables("%TEMP%"), @"NapervilleWaterNetwork" + num.ToString());
                num++;
            }

            // Create the output directory.
            Directory.CreateDirectory(packagePath);

            try
            {
                // Show the progress indicator while the job is running.
                BusyIndicator.Visibility = Windows.UI.Xaml.Visibility.Visible;

                // Create an offline map task with the current (online) map.
                OfflineMapTask takeMapOfflineTask = await OfflineMapTask.CreateAsync(MyMapView.Map);

                // Create the default parameters for the task, pass in the area of interest.
                GenerateOfflineMapParameters parameters = await takeMapOfflineTask.CreateDefaultGenerateOfflineMapParametersAsync(_areaOfInterest);

                #region overrides

                // Generate parameter overrides for more in-depth control of the job.
                GenerateOfflineMapParameterOverrides overrides = await takeMapOfflineTask.CreateGenerateOfflineMapParameterOverridesAsync(parameters);

                // Configure the overrides using helper methods.
                ConfigureTileLayerOverrides(overrides);
                ConfigureLayerExclusion(overrides);
                CropWaterPipes(overrides);
                ApplyFeatureFilter(overrides);

                // Create the job with the parameters and output location.
                _generateOfflineMapJob = takeMapOfflineTask.GenerateOfflineMap(parameters, packagePath, overrides);

                #endregion overrides

                // Handle the progress changed event for the job.
                _generateOfflineMapJob.ProgressChanged += OfflineMapJob_ProgressChanged;

                // Await the job to generate geodatabases, export tile packages, and create the mobile map package.
                GenerateOfflineMapResult results = await _generateOfflineMapJob.GetResultAsync();

                // Check for job failure (writing the output was denied, e.g.).
                if (_generateOfflineMapJob.Status != JobStatus.Succeeded)
                {
                    MessageDialog dialog = new MessageDialog("Generate offline map package failed.", "Job status");
                    await dialog.ShowAsync();
                    BusyIndicator.Visibility = Windows.UI.Xaml.Visibility.Collapsed;
                }

                // Check for errors with individual layers.
                if (results.LayerErrors.Any())
                {
                    // Build a string to show all layer errors.
                    System.Text.StringBuilder errorBuilder = new System.Text.StringBuilder();
                    foreach (KeyValuePair<Layer, Exception> layerError in results.LayerErrors)
                    {
                        errorBuilder.AppendLine(string.Format("{0} : {1}", layerError.Key.Id, layerError.Value.Message));
                    }

                    // Show layer errors.
                    string errorText = errorBuilder.ToString();
                    MessageDialog dialog = new MessageDialog(errorText, "Layer errors");
                    await dialog.ShowAsync();
                }

                // Display the offline map.
                MyMapView.Map = results.OfflineMap;

                // Apply the original viewpoint for the offline map.
                MyMapView.SetViewpoint(new Viewpoint(_areaOfInterest));

                // Enable map interaction so the user can explore the offline data.
                MyMapView.InteractionOptions.IsEnabled = true;

                // Hide the "Take map offline" button.
                TakeOfflineArea.Visibility = Windows.UI.Xaml.Visibility.Collapsed;

                // Show a message that the map is offline.
                MessageArea.Visibility = Windows.UI.Xaml.Visibility.Visible;
            }
            catch (TaskCanceledException)
            {
                // Generate offline map task was canceled.
                MessageDialog dialog = new MessageDialog("Taking map offline was canceled");
                await dialog.ShowAsync();
            }
            catch (Exception ex)
            {
                // Exception while taking the map offline.
                MessageDialog dialog = new MessageDialog(ex.Message, "Offline map error");
                await dialog.ShowAsync();
            }
            finally
            {
                // Hide the activity indicator when the job is done.
                BusyIndicator.Visibility = Windows.UI.Xaml.Visibility.Collapsed;
            }
        }

        #region overrides

        private void ConfigureTileLayerOverrides(GenerateOfflineMapParameterOverrides overrides)
        {
            // Create a parameter key for the first basemap layer.
            OfflineMapParametersKey basemapKey = new OfflineMapParametersKey(MyMapView.Map.Basemap.BaseLayers.First());

            // Get the export tile cache parameters for the layer key.
            ExportTileCacheParameters basemapParams = overrides.ExportTileCacheParameters[basemapKey];

            // Clear the existing level IDs.
            basemapParams.LevelIds.Clear();

            // Get the min and max scale from the UI.
            int minScale = (int)MinScaleEntry.Value;
            int maxScale = (int)MaxScaleEntry.Value;

            // Re-add selected scales.
            for (int i = minScale; i < maxScale; i++)
            {
                basemapParams.LevelIds.Add(i);
            }

            // Expand the area of interest based on the specified buffer distance.
            int bufferDistance = (int)ExtentBufferEntry.Value;
            basemapParams.AreaOfInterest = GeometryEngine.BufferGeodetic(_areaOfInterest, bufferDistance, LinearUnits.Meters);
        }

        private void ConfigureLayerExclusion(GenerateOfflineMapParameterOverrides overrides)
        {
            // Apply layer exclusions as specified in the UI.
            if (ServiceConnCheckbox.IsChecked == false)
            {
                ExcludeLayerByName("Service Connection", overrides);
            }

            if (SysValvesLayerCheckbox.IsChecked == false)
            {
                ExcludeLayerByName("System Valve", overrides);
            }
        }

        private void CropWaterPipes(GenerateOfflineMapParameterOverrides overrides)
        {
            if (CropLayerCheckbox.IsChecked == true)
            {
                // Get the ID of the water pipes layer.
                long targetLayerId = GetServiceLayerId(GetLayerByName("Main"));

                // For each layer option.
                foreach (GenerateLayerOption layerOption in GetAllLayerOptions(overrides))
                {
                    // If the option's LayerId matches the selected layer's ID.
                    if (layerOption.LayerId == targetLayerId)
                    {
                        layerOption.UseGeometry = true;
                    }
                }
            }
        }

        private void ApplyFeatureFilter(GenerateOfflineMapParameterOverrides overrides)
        {
            // For each layer option.
            foreach (GenerateLayerOption option in GetAllLayerOptions(overrides))
            {
                // If the option's LayerId matches the selected layer's ID.
                if (option.LayerId == GetServiceLayerId(GetLayerByName("Hydrant")))
                {
                    // Apply the where clause.
                    option.WhereClause = "FLOW >= " + (int)FlowRateFilterEntry.Value;

                    // Configure the option to use the where clause.
                    option.QueryOption = GenerateLayerQueryOption.UseFilter;
                }
            }
        }

        private IList<GenerateLayerOption> GetAllLayerOptions(GenerateOfflineMapParameterOverrides overrides)
        {
            // Find the first feature layer.
            FeatureLayer targetLayer = MyMapView.Map.OperationalLayers.OfType<FeatureLayer>().First();

            // Get the key for the layer.
            OfflineMapParametersKey layerKey = new OfflineMapParametersKey(targetLayer);

            // Use that key to get the generate options for the layer.
            GenerateGeodatabaseParameters generateParams = overrides.GenerateGeodatabaseParameters[layerKey];

            // Return the layer options.
            return generateParams.LayerOptions;
        }

        private void ExcludeLayerByName(string layerName, GenerateOfflineMapParameterOverrides overrides)
        {
            // Get the feature layer with the specified name.
            FeatureLayer targetLayer = GetLayerByName(layerName);

            // Get the layer's ID.
            long targetLayerId = GetServiceLayerId(targetLayer);

            // Create a layer key for the selected layer.
            OfflineMapParametersKey layerKey = new OfflineMapParametersKey(targetLayer);

            // Get the parameters for the layer.
            GenerateGeodatabaseParameters generateParams = overrides.GenerateGeodatabaseParameters[layerKey];

            // Get the layer options for the layer.
            IList<GenerateLayerOption> layerOptions = generateParams.LayerOptions;

            // Find the layer option matching the ID.
            GenerateLayerOption targetLayerOption = layerOptions.First(layer => layer.LayerId == targetLayerId);

            // Remove the layer option.
            layerOptions.Remove(targetLayerOption);
        }

        private FeatureLayer GetLayerByName(string layerName)
        {
            // Get the first map in the operational layers collection that is a feature layer with name matching layerName
            return MyMapView.Map.OperationalLayers.OfType<FeatureLayer>().First(layer => layer.Name == layerName);
        }

        private long GetServiceLayerId(FeatureLayer layer)
        {
            // Find the service feature table for the layer; this assumes the layer is backed by a service feature table.
            ServiceFeatureTable serviceTable = (ServiceFeatureTable)layer.FeatureTable;

            // Return the layer ID.
            return serviceTable.LayerInfo.ServiceLayerId;
        }

        #endregion overrides

        // Show changes in job progress.
        private async void OfflineMapJob_ProgressChanged(object sender, EventArgs e)
        {
            // Get the job.
            GenerateOfflineMapJob job = sender as GenerateOfflineMapJob;

            // Dispatch to the UI thread.
            await Dispatcher.RunAsync(Windows.UI.Core.CoreDispatcherPriority.Normal, () =>
           {
                // Show the percent complete and update the progress bar.
                Percentage.Text = job.Progress > 0 ? job.Progress.ToString() + " %" : string.Empty;
               ProgressBar.Value = job.Progress;
           });
        }

        private void CancelJobButton_Click(object sender, Windows.UI.Xaml.RoutedEventArgs e)
        {
            // The user canceled the job.
            _generateOfflineMapJob.CancelAsync();
        }
    }
}

redpajama_set_name: RedPajamaGithub


text:
Christopher C. Rants (born September 16, 1967) is a former Iowa State Representative. He served in the Iowa House of Representatives from 1992 to 2010. He received his BA from Morningside College. His work experience includes managing environmental compliance projects for Metz Baking Company, and Pierce & Associates, a Sioux City consulting firm.  After leaving the legislature, Rants started inSight Communication as government affairs and public relations firm.  In 2015 Rants partnered his firm with PolicyWorks LLC, another Iowa government affairs firm.

Rants served on several committees in the Iowa House - Commerce, Education, Ways & Means, Labor, and Rules committees.  His political experience includes serving as Assistant Majority Leader beginning in 1994, serving as the Speaker Pro Tempore, serving as House Majority Leader beginning in 1999, and serving as Speaker of the Iowa House from 2003 through 2006.

Rants was re-elected in 2008 by a margin of 57% to 43%.

In June 2009, Rants announced his intention to seek his party's nomination for the 2010 election for Governor of Iowa. Rants announced that he was pulling out of the race for governor in February 2010.

In 2006 and 2008 Rants backed Mitt Romney in the Iowa caucuses, and in 2016 Rants was the state chairman for the Carly Fiorina for President campaign.

Christopher is an Aspen Institute Rodel Fellow in Public Leadership. Rants serves as Vice Chairman  of the Republican State Leadership Committee.

Rants is an avid golfer, who has played the 2004 Golf Magazine's Top 100 Public Courses You Can Play.

References

External links
 Representative Christopher Rants - official Iowa General Assembly site
 Rants' website

Profile at iowahouserepublicans.com

1967 births
Living people
People from Grosse Pointe, Michigan
Methodists from Iowa
Republican Party members of the Iowa House of Representatives
Speakers of the Iowa House of Representatives
Morningside University alumni
Politicians from Sioux City, Iowa

redpajama_set_name: RedPajamaWikipedia


text:
Palestinian amputee athlete: I want to represent my country at international level
Global Stringer
Asia;Palestine
To coincide with International Day for People with Disabilities, Palestine's first national amputee football team was launched in Gaza on December 3, supported by the International Committee of the Red Cross (ICRC). The squad of 20 young Palestinians who lost parts of their bodies, either in previous wars or in accidents, hopes to play in the Amputee Football World Cup next year. This video shows how the team trains with crutches and prosthetic legs at a stadium in Gaza City. CGTN stringer interviewed the consultant for ICRC, Simon Baker, and one of the amputee athletes, Hassan Karim, to talk about their hopes and ambition for this team. Hassan said, "I have dreams and ambitions, and my goal is to represent Palestine in international forums."

redpajama_set_name: RedPajamaCommonCrawl


text:
Vedika Pawan
Because of the pandemic situation, the situation has become quite weird. Even though the most important part of the situation is to be safe and be inside the house, people do need some kind of entertainment to distract their minds off. And, what can be the better way to do that other than watching movies from the endless collection of OTT platforms.
On the one hand, there are superhit and commercially recognized movies. While on the other hand, there are some underrated Bollywood movies that are not yet recognized by the people. However, trust me, they are worth watching and promise great entertainment as well. So, to make things easier for you, we have gathered a list of must-watch underrated movies on the OTT platform that you will surely not regret watching.
1) Kaamyaab
Photo Credit: IMDb
Undoubtedly, Sanjay Mishra is one of the best actors in the Bollywood industry. No matter if he does one of the flop Bollywood movies or the superhit Bollywood movies, his work is always remarkable. And, that is what "Kaamyaab" is all about. Even though it is one of the underrated Hindi movies, this movie is an ode to the actor who is on the verge to get his retirement from his acting career.
The movie shows the real picture of the Bollywood industry and the supporting actors in it. "Kaamyaab," tells that how important supporting actors really are and despite that, they do not get the name and fame they deserve. Without a doubt, Sanjay Mishra was in the limelight of the movie.
Watch On: Netflix
2) Bhonsle
"Bhonsle" is one of the underrated Hindi movies that feature Manoj Bajpayee in the lead role as a retired cop, Ganpat Bhonsle. In the movie, Bhonsle is living in a dark shabby room in Churchill Chawl of Mumbai and struggling to get a service extension. This is one of the latest movies on OTT platform in India that shows the reality of human emotion. And, the audience will surely fall in love with Manoj Bajpayee after watching his brilliant performance.
Watch On: Sony Liv
3) Pareeksha
"Pareeksha" is a movie on OTT platform that comes under one of the flop Bollywood movies. Even though this is a simple movie where a poor father wants to give proper education to his son in an English medium school, it shows the importance of education, honesty, and the consequences of dishonesty in life. It is a well-intentional film that is directed by director Prakash Jha in the year 2020.
Watch On: Zee5
4) Chippa
Even though "Chippa" is one of the underrated Bollywood movies, it is an amazing film that revolves around an orphan boy. He wanders around the streets of Kolkata to find his father whom unfortunately, he never met. This movie showcases his wild childhood imagination and experiences while searching for his father.
5) Axone
"Axone" is one of the unique OTT platform movies that showcase how Northeastern Indians are being treated throughout the country. The movie talks about a group of Northeastern Indians who are living in Delhi and make their traditional dish "Axone" at their friend's wedding. Unfortunately, things get messed up and they face a lot of difficulties. With a strong message and amazing storyline, it is a must-watch movie.
6) Bahut Hua Samman
"Bahut Hua Samman" is yet another amazing movie by Sanjay Mishra that is one of the latest movies on OTT platform in India. The story revolves around the two friends of engineering students who are looking for ways to make instant money. However, they rather get trapped in a messy situation. This movie is the perfect example of satire and toxic political references.
Watch On: Disney Hotstar
Genre: Comedy, Crime, Drama
7) Bulbbul
Photo Credit: Heaven Of Horror
"Bulbbul" is an Anushka Sharma production and Anvita Dutta's directorial film that was liked and appreciated by a section of the society. It is a movie on OTT platform that created a buzz even before getting released. This movie showcases the story of a young girl who turns into an enigmatic woman. Unfortunately, she is tortured by her aged husband because he has the power.
Genre: Drama, Horror, Mystery
8) Chaman Bahaar
"Chaman Bahaar" is one of the best OTT platform movies that feature Jitendra Kumar. With some of the notable performances, he is the gem of his field. The story of this movie revolves around a paan-wala and his love story.
9) Choked: Paisa Bolta Hai
"Choked: Paisa Bolta Hai" is a story of a middle-class family with a working mother and a stay-at-home father. The family faces constant financial problems and then, the mother gets a strange solution to their problems. Even though this movie is one of the underrated Bollywood movies, it is a must-watch.
10) Khuda Hafiz
When it comes to martial arts and action sequences, nobody can be compared to the actor Vidyut Jammwal. Even though his movie "Khuda Hafiz" is focused on action, it is a bit different than his other movies. His acting is quite remarkable in this movie and that makes it worth watching.
underrated bollywood movies
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text:
Latest K-Pop News: Latest K-Pop News: TWICE, TXT and ATEEZ work on their upcoming comebacks, while BTS' V creates havoc among fans in Mexico
Francisca Santos Dec 21, 2022 12:00 am Dec 20, 2022 4:57 pm
The week's just starting and the K-Pop side of the world continues to be the gift that keeps on giving. With a relentless Monday overwhelmed with announcements, it's only natural that Tuesday (or Wednesday!) follow through with this eventful week among all the upcoming holiday events and celebrations. This time, it appears that everyone's favorite fourth-generation groups are tirelessly working to bring everyone a little joy just in time for Christmas and all its variations, but that's not all.
K-Pop veterans (well, more or less) Monsta X are also teasing a little track list for their 12th mini album. MAMAMOO's Hwa Sa and Moon Byul are following through with all the comeback momentum as they keep promoting their solo work outside of the group. Of course, no day would be complete without BTS stealing hearts everywhere they go, so leave it to V to completely swoon fans even in Mexico, and for We Got This Covered to tell you all about it.
Twice, TXT, ATEEZ, and Monsta X prepare to wow fans with their future releases
Screenshot via HYBE Labels
Everyone's favorite up-and-coming girl group is getting ready for their comeback in 2023, and you can start writing it in your calendars as TWICE return with even more songs to get stuck in your head. On Dec.21 at midnight (KST), JYP Entertainment and TWICE finally released a teaser poster announcing the dates for their upcoming English single and mini-album. According to the poster, the pre-release is set for Jan. 2023, and keep your calendars free in March to fully get enthralled in their 12th mini-album.
As for the world of male groups, KQ Entertainment's ATEEZ is also preparing a lil' something up its sleeve. The 8-member group just dropped their tracklist for the upcoming mini-album SPIN-OFF: FROM THE WITNESS. The album is set to drop on Dec.30, and it will be comprised of the title song "Halazia," and remixes of their past songs "WIN," "I'm The One" and "Take Me Home."
Also with an upcoming mini-album, is MONSTA X, who just revealed a teaser for their mini-album Reason with the extra drop of a tracklist. Their 12th mini album will feature the songs "Beautiful Liar", "Daydream", "Crescendo", "Lone Ranger", "Deny", and "It's Okay," and you can hear all about it on Jan.9. TXT, on the other hand, has also released the teaser video for The Name Chapter: TEMPTATION, with an enigmatic and mysterious silhouette accompanied by a butterfly. TXT will be marking their comeback on Jan.27. Needless to say, 2023 is absolutely packed.
MAMAMOO's Hwa Sa and Moonbyul work on promoting their upcoming singles and solo albums
Screenshot via RBW Entertainment
MAMAMOO, MAMAMOO! The girls are back and more powerful than ever, introducing an entirely new side as soloists. As Moonbyul keeps reintroducing herself as the most credited female idol, the singer and rapper is ceaselessly working on her individualism with her latest single album, to be released on Dec. 22. Just ahead of the release, the rapper from MAMAMOO will be holding a V Live stream, as announced by a teaser photo saying: "I remember spending this time last year too. Let's be together for my birthday, 3 days before Christmas." The live event will occur on Dec. 21 at 8 pm, KST.
That's not all from the MAMAMOO girls. Hwa Sa has also taken part in the most recent comebacks in the K-Pop world. The MAMAMOO vocalist revealed her softest side through a special holiday single music video for "Grey Christmas." The seemingly homemade video shows Hwa Sa in cozy Homer Simpson pajama pants and a large grey sweater, perfectly embodying the title of her holiday single. MAMAMOO fans are definitely being gifted in the best possible way.
BTS' V will appear in the reality show "Seojin's" and the shooting in Mexico made fans go wild
Screenshot via BANGTANTV
Everyone, behold! BTS' V has graced everyone with his presence in Mexico, solidifying his role as one of the group's visuals. Although some rumors had been flying here and there about the possible appearance by Kim Taehyung in the reality show Seojin's, nothing was confirmed- until now. According to recent reports, V will be joining the reality show alongside his friends Choi Woo Shik and Park Seo Joon, with whom he also filmed In The Soop: Friendcation. Following the success of their cozy time in the woods, the three biggest stars in South Korea will be once again together on screen, much to our contentment.
The trio filmed the show overseas, all the way in Mexico. They will be working to operate a Korean restaurant overseas, and clearly, fans were more than excited to find V just casually strolling around their city. While they were fully respectful of the singer's privacy, some fans came forward after the announcement, to reveal some details about when they came to face with the handsome baritone. "We couldn't say 'Hi' to him, still he greeted all of us," one fan said "I literally saw him mop up the water from the rain and he looked so beautiful." Oh to meet V casually on the streets of Mexico.
That's all we have for you today, K-Pop fans. As always, annyeonghi gaseyo (go peacefully), and until the next time.


text:
\section{Introduction}
Suppose we are interested in the physics of events localized in a region $V$ of the space. The Hilbert space of states can be decomposed accordingly as a tensor product $
 {\cal H}={\cal H}_{V}\otimes {\cal H}_{-V}$
 of the spaces of the states localized in $V$ and  in the complementary region $-V$. Take now the vacuum $\left| \Psi \right>$ as a global state  of the system, with density matrix $
\rho_0= \left| \Psi \right> \left< \Psi \right|$.
 The state $\rho_V$ relevant to the algebra of operators acting on ${\cal H}_{V}$
  follows from the partial trace of $\rho_0$ over the complementary Hilbert space ${\cal H}_{-V}$. This gives the local reduced density matrix
\begin{equation}
 \rho_{V}=\textrm{tr}_{{\cal H}_{-V}}\left| \Psi \right> \left< \Psi \right|\,.
\end{equation}
The global state $\rho_0$ is generally entangled in the bipartite system ${\cal H}_{V}\otimes {\cal H}_{-V}$ and in consequence this matrix is mixed.
The corresponding entropy
 \begin{equation}
 S(V)=-\textrm{tr}(\rho_V\log \rho_V)\,
\end{equation}
is usually called entanglement or geometric entropy.

The entanglement entropy is one of the most prominent candidates to explain the intriguing entropy of the black holes \cite{bombelli}. However, in this proposal the role of quantum gravity is fundamental to produce a finite entropy, and the whole subject is still controversial.
On the other hand, this and other measures of entanglement have also been extensively studied in condensed matter and low dimensional systems, partially motivated by advances in quantum information theory and the density matrix renormalization group method.  As a result it was uncovered that a variety of phenomena such as quantum phase transitions  have an interesting  correlate in the entanglement properties of fundamental states \cite{phase}.
From the point of view of quantum field theory (QFT) the function $S(V)$ can be considered as a non local variable with interesting non perturbative properties \cite{hg}, which can be defined for any theory disregarding the field content. In this context, we can mention among the applications the description of topological order \cite{topological}, and the renormalization group irreversibility in two dimensions \cite{cteor,rg}.

In the continuum limit described by a quantum field theory the entanglement entropy is divergent due to the presence of an unbounded number of local degrees of freedom. The divergent terms must be proportional to quantities which are local and extensive on the boundary of $V$.
This can be seen as a consequence of the local nature of the ultraviolet divergences, and that the boundary is shared between $V$ and $-V$ which have the same entropy (for any pure global state)
\begin{equation}
S(V)=S(-V)\,.
\end{equation}
On a technical level this characteristic of the divergent terms is due the fact that the entanglement entropy is the variation of the euclidean free energy with respect to conical singularities located on the boundary of $V$ \cite{tech}. Thus, in $d$ spatial dimensions, we expect to have an expansion of the form
 \begin{equation}
 S(V)=g_{d-1}[\partial V] \,\epsilon^{-(d-1)} + g_{d-2}[\partial V]\,\epsilon^{-(d-2)} +...+ g_0[\partial V]\,\log (\epsilon \Lambda)+ S_0(V)\,,   \label{div}
 \end{equation}
 where $S_0(V)$ is a finite part, $\epsilon$ is a short distance cutoff, and the $g_i$ are local and extensive  functions on the boundary $\partial V$, which are homogeneous of degree $i$ (in a different context a similar expansion was used in \cite{ryu}). The leading divergent term coefficient  $g_{d-1}[\partial V]$ is proportional to the $d-1$ power of the size of $V$. This was noted since the earliest papers on the subject \cite{bombelli,sred} and is usually referred to as the area law for the entanglement entropy. However, strictly speaking, $g_{d-1}$  depend on the regularization procedure and it is not proportional to the area if this later is not rotational invariant.
 For example, if we draw a square of side $L$ on a two dimensional lattice,  we would have $g_1(\partial V)\sim C_1\frac{L}{\epsilon}$, where $\epsilon$ is the lattice spacing, but where the dimensionless constant $C_1$ strongly depends on the relative angle between the square sides and the lattice symmetry axes.
 Moreover, all the terms $g_i$ with $i\ge 1$ are not physical within QFT since they are not related to continuum quantities.

On the contrary, the dimensionless coefficient $g_0(V)$ of the logarithmic term is expected to be universal.
Logarithmic divergent terms in the entropy have been previously found in four dimensional black hole space-times \cite{ss}. They are present generically in even dimensions for sets with smooth curved boundaries. This follows from the heat kernel expansion for conical manifolds with smooth singularity surface \cite{conical}.

In this work we show that there is also a logarithmic term in three dimensions for sets $V$ with non-smooth boundary.
In particular, we consider the case of spatial polygonal sets  (in a different scenario, a logarithmic contribution to the entanglement entropy was also reported in \cite{diferent}). Since $g_0(V)$ is dimensionless, extensive and local on the boundary we conclude that for $V$ a polygon it must be of the form
\begin{equation}
g_0(V)=\sum_{v_i} s(x_i)\,,
\end{equation}
where the sum is over all vertices $v_i$ and  $x_i$ is the vertex angle. On general grounds one also expects point-like vertex induced logarithmic  terms in any dimensions.

All these considerations apply to the alpha-entropies (or R\'enyi entropies) as well
\begin{equation}
S_\alpha(V)=\frac{1}{1-\alpha}\log(\textrm{tr}\rho_V^\alpha),\,\,\,\,\,\, S(V)=\lim_{\alpha\rightarrow 1} S_\alpha (V) \,.\label{cuatro}
\end{equation}
 These measures of information behave very much like the entanglement entropy, as discussed in several papers \cite{fermion,cahu}. For integer $\alpha=n$ they are also more suitable for explicit calculation, since the traces $\textrm{tr}\rho_V^n$ involved can be represented by a functional integral on an n-sheeted space with conical singularities located at the boundary of $V$ \cite{tech}. The entanglement entropy follows from $S_n$ by analytical continuation.
Thus, in 2+1 dimensional theories the alpha-entropies contain a logarithmic term
\begin{equation}
\left.S_n\right|_{\log}=\sum_{v_i} s_n(x_i) \log(\epsilon \Lambda)\,, \label{cinco}
\end{equation}
analogous to the one in (\ref{div}) for the entropy. We also have $s(x)=\lim_{\alpha\rightarrow 1} s_\alpha(x)$.
In eq. (\ref{cinco}) $\Lambda$ is a parameter with the dimensions of an energy, depending on $V$ and on the particular  theory. For a massless field it is the inverse of any typical dimension $R$ of $V$ and when the mass dominates, $MR\gg 1$, it can be taken  as $\Lambda=M$. We find the analytic expression of $s_n(x)$ for a free scalar field, and check our results by numerical simulations on a two dimensional lattice.

The outline of the paper is the following. In Section 2, we obtain the analytic expression for $\textrm{tr}\rho_V^n$ for the local density matrix associated to a plane angular sector for a massive scalar field in $2+1$ dimensions. The calculation is reduced to the one of the trace of the Green function on a two dimensional sphere, where boundary conditions are specified on a segment of a great circle.
We use the method introduced in \cite{cahu} to obtain this trace exactly.
 Finally, we identify the logarithmic coefficients $s_n(x)$ of (\ref{cinco}). We also compare them  with the results given by simulations in a square lattice finding a perfect accord.
In Section 3, we find the small angle limit of $s(x)$ for free fields and relate it to the two dimensional entropic c-functions. Finally, in Section 4 we present our conclusions. We have also included in the Appendix A a detailed derivation of the formulas concerning the Green function in the sphere with a cut presented in Section 2.1, and in Appendix B, the method we have used to compute numerically the geometric entropy in a two dimensional lattice.

\section{Vertex induced logarithmic terms in the entropy}

To compute the coefficients $s_n(x)$ of the logarithmic term in (\ref{cinco}) for a scalar field we choose for convenience to study the entropy associated to a set $V$ given by a plane angular sector of angle $x$.

The traces $\textrm{tr}\rho_V^{\alpha }$ involved in (\ref{cuatro}), with $\alpha=n\in
{\mathbb Z}$, can be represented by a functional integral on an $n$-sheeted three-dimensional Euclidean space with conical singularities
located at the boundary of the set $V$ \cite{tech,fermion}. To be explicit, calling $u$ and $v$ to the upper and lower faces of the plane angular sector, the replicated space is obtained considering $n$ copies of the three-space cut along the angular sector, and sewing together the upper side of the cut $u^{k}$ with
the lower one $v^{k+1}$, for the different copies $ k=1,...,n$, and where the copy $n+1$ coincides with the first one.
The trace of $\rho^{n}$ is then given by the functional integral
$Z[n]$ for the field in this manifold,
\begin{equation}
\textrm{tr}\rho^{n}= \frac{Z[n]}{Z[1]^{n}}\,.
\label{dd}
\end{equation}

Then, following \cite{fermion, cahu} we consider a free massive complex scalar field on this manifold, and map the problem to an equivalent one in which we deal with $n$ decoupled and multivalued free complex scalar fields. First we arrange the values of the field in the different copies in a single vector field living in a three dimensional space,
\begin{equation}
\vec{\Phi}=\left(\begin{array}{c}
\phi _{1}(\vec{x}) \\
\vdots \\
\phi _{n}(\vec{x}) \end{array}
\right) \,,
\end{equation}
where $\phi _{l}(\vec{x})$ is the field on the $l^{\textrm{th}}$ copy.
Note that in this way the space is simply connected but the singularities at the boundaries of $V$ are still there since the vector $\vec{\Phi}$ is not singled valued.  In fact, crossing the plane angular sector from above (side $u$) or from below (side $v$), the field gets multiplied by a matrix $T$ or $T^{-1}$ respectively. Here
\begin{equation}
\begin{array}{c}
T=\left(
\begin{array}{lllll}
0 & 1 &  &  &  \\
& 0 & 1 &  &  \\
&  & . & . &  \\
&  &  & 0 & 1 \\
1 &  &  &  & 0
\end{array}
\right)
\end{array}\,,
\end{equation}
which has eigenvalues\footnote{There is an important difference with the free fermion case studied in \cite{fermion}, where the eigenvalues of $T$ are $e^{i\frac{k}{n}2\pi }$, with $k=-(n-1)/2\,,...,\,(n-1)/2$. This is because in the fermionic case it is $T^n=-1$.} $e^{i\frac{k}{n}2\pi }$, with
$k=0\,,...,\,(n-1)$.
Then, changing basis by a unitary transformation in the replica space,
we can diagonalize $T$, and the problem is reduced to $n$ decoupled
fields $\tilde{\phi}_{k}$ living on a single three dimensional space. These fields are multivalued and defined on the euclidean three dimensional space with boundary conditions imposed on the two dimensional set $V$ given by
\begin{equation}
\tilde{\phi}^u_k(\vec{r})=e^{i\frac{2 \pi k}{n}}\tilde{\phi}^v_k(\vec{r})\,\,\,\,\,\,\,\,\,\,k=0,\,\,...\,n-1, \,\,\,\,\,\,\, \, \vec{r}\in V\,.
\label{bc}
\end{equation}
Here $\tilde{\phi}^{u}_k$ and $\tilde{\phi}^{v}_k$ are the limits of the field as the variable approaches $V$ from each of its two opposite sides in three dimensions.
In this formulation we have
\begin{equation}
\log(\textrm{tr}\rho_V^n)=\sum_{k=0}^{n-1}\textrm{log}Z[k/n]\,,\label{sumz}
\end{equation}
where $Z[a]$ is the partition function corresponding to a field which acquires a phase $e^{i2\pi a}$ when the variable crosses $V$. We note that the coefficient of the logarithmic divergent term in $\log Z[k/n]$ corresponds to the integrated trace anomaly  $\int dr^3 T^\mu_\mu(\vec{r})$, or equivalently the $a_3=c_{3/2}/(4\pi)^{3/2}$ coefficient of the heat kernel expansion \cite{vassi}, for a massless scalar in three dimensions with the boundary conditions (\ref{bc}).

Our calculation of $Z[a]$ is based on the relation between the free energy and the Green function for a free massive scalar
\begin{equation}
\partial_{M^2}\textrm{log}Z[a]=-\int dr^3 G_a(\vec{r},\vec{r})\,.
\label{green}
\end{equation}
Here $G_a(\vec{r}_1,\vec{r}_2)$ is the Green function for a complex scalar of mass $M$ in three dimensions subject to the boundary conditions (\ref{bc}). To be explicit, we have
\begin{eqnarray}
(-\Delta_{\vec{r}_1}+M^2) \,G_a(\vec{r}_1,\vec{r}_2)&=&\delta(\vec{r}_1-\vec{r}_2)\,,\\
\lim_{\varepsilon\rightarrow 0^+} G_a(\vec{r}_1+\varepsilon \hat{\eta},\vec{r}_2)&=&e^{i 2 \pi a}  \,\lim_{\varepsilon\rightarrow 0^+} G_a(\vec{r}_1-\varepsilon \hat{\eta},\vec{r}_2)\,, \hspace{1cm} r_1\in V\, ,\label{cator}
\end{eqnarray}
 where  $\hat{\eta}$ is orthogonal to the plane of $V$.

The Laplacian and the boundary conditions allow the separation of angular and radial equations in polar coordinates. Using standard methods we arrive at the expression (see for example the similar calculation in \cite{cc})
\begin{equation}
G_a(\vec{r}_1,\vec{r}_2)=\sum_\nu \int d\lambda \frac{\lambda}{\lambda^2+M^2}\psi_{\nu}( \theta_1 , \varphi_1)\psi^*_{\nu}(\theta_2 , \varphi_2)\frac{J_{\frac{1}{2}+\nu}(\lambda r_1)J_{\frac{1}{2}+\nu}(\lambda r_2)}{\sqrt{r_1 r_2}}\,,
\label{ga}
\end{equation}
where $J$ is the standard Bessel function. Here the sum is over the normalized eigenvectors $\psi_{\nu}(\theta , \varphi)$ of the angular equation
\begin{equation}
\Delta_{\Omega}\psi_{\nu}=-\nu(\nu+1)\psi_{\nu}\,,
\end{equation}
where $\Delta_{\Omega}$ is the Laplacian on the sphere with domain given by the functions satisfiyng the boundary conditions inherited from (\ref{cator}). Specifically we can choose the cut on the equatorial plane
\begin{equation}
\lim_{\varepsilon\rightarrow 0^+} \psi_{\nu}(\pi/2+\varepsilon , \varphi)= e^{i 2 \pi a}  \lim_{\varepsilon\rightarrow 0^+} \psi_{\nu}(\pi/2-\varepsilon , \varphi)\,,  \hspace{1.3cm} \varphi \in [-x/2,x/2]\,.\label{boun}
\end{equation}
The functions $\psi_{\nu}$ are the spherical harmonics if $a=0$, otherwise they are known to satisfy Lam\'e differential equations \cite{lame}. Also, the values of $\nu$ are not integer if $a\neq 0$. The precise expressions for $\psi_{\nu}$ and $\nu$ will not be relevant in what follows.

Taking the trace $\int dr^3 G_a(\vec{r},\vec{r})$ in eq. (\ref{ga}) gives
\begin{equation}
\partial_{M^2}\log Z[a]=-\frac{1}{2M^2}\sum_{\nu}(\nu+1/2)=-\frac{1}{2M^2}\textrm{tr}\sqrt{-\Delta_{\Omega}+\frac{1}{4}}\,.
\label{trsqrt}
\end{equation}
Though this expression is divergent, the piece we are interested in, which is the one dependent on the angle $x$, is finite.

To proceed, we find convenient to express the trace of the square root of the operator in (\ref{trsqrt}) in terms of the corresponding Green function. This is done by using the expression for the powers of an elliptic operator $O$ in terms of the resolvent given in \cite{seeley},
\begin{equation}
O^\gamma=\frac{i}{2 \pi}\int_{\Gamma}\lambda^\gamma(O-\lambda)^{-1}d\lambda\,,
\end{equation}
where $\Gamma$ is a curve depending on the particular operator $O$. In the present case $\Gamma$ begins at infinity, pass  along the negative real axes on the upper complex plane, encircles the origin and goes back to infinity on the lower half of the complex plane along the negative real axes.  This gives for the trace
\begin{equation}
\textrm{tr}  \sqrt{-\Delta_{\Omega}+\frac{1}{4}}=-\frac{1}{\pi}\,\int_0^{\infty}\lambda^{\frac{1}{2}}\,\,\textrm{tr}\frac{1}{\Delta_{\Omega}-\frac{1}{4}-\lambda}\, d\lambda
\,.
\label{tr}
\end{equation}

\subsection{Green function on a sphere with a cut}
The problem is then reduced to the calculation of the trace of the two dimensional Green function on a sphere with a cut of angle $x$, where the boundary conditions (\ref{boun}) are imposed. In a previous paper \cite{cahu} we have solved the analogous problem on the plane with boundary conditions imposed on an interval,  using a generalization of a method originally developed in \cite{myers}. It essentially consists in exploiting the symmetries of the Helmholtz equation even in the presence of symmetry breaking boundary conditions by analyzing the behavior of the Green function at the singular points.

Following the recipe of \cite{cahu} step by step (with the additional algebraic complications corresponding to the spherical case) we find the analytic expression for the trace of the Green function as a solution of a system of ordinary differential equations. The details of the derivation are given in the appendix A. Explicitly we find
\begin{equation}
\textrm{tr}\frac{1}{\Delta_{\Omega}-m^2}=-8\pi (1-a)a\int_{x}^{\pi}H_a(y)dy\,.\label{quince}
\end{equation}
The function $H_a(x)$ is the solution of the following set of ordinary non linear differential equations (we omit the subscript $a$ and the dependence on $x$ of the variables for notational convenience)
\begin{eqnarray}
H' &=& - \frac{m}{2}\,\left( b\,B_2 +c\, B_1 + 2\,u\,B_{12} \right)\label{hprima}\,,\\
X_1' &=& - m\,\left( b\,B_{12}+ u\,B_1 \right)\label{x1prima} \,,\\
X_2' &=& - m\,\left( c\,B_{12} + u\,B_2 \right) \label{x2prima}\,,\\
c' &=&- 2\,m\,\beta_2 \,u\, \csc(x)\,\sin(x/2) - c\,(1 - a)\,\csc(x)\,(1 + \cos(x))\label{cprima}\,, \\
b' &=& -2\,m\,\beta_1\,u\,\csc(x)\,\sin(x/2) - b\,a\,\csc(x)\,(1 + \cos(x)) \label{bprima} \,,\\
u'&=&-\frac{m}{2}\,\sec(x/2)\,(b\,\beta_2+ c\,\beta_1) + \frac{1}{2}\,u\,\tan \left(x/2 \right)\label{uprima} \,,
\end{eqnarray}
where $B_1$, $B_2$, $B_{12}$, $\beta_1$, $\beta_2$ are functions of $x$ given in terms of $H$, $X_1$, $X_2$, $c$, $b$, and $u$ by the following set of algebraic equations
\begin{eqnarray}
\frac{\cos(x/2)}{8
\pi a(1 - a)}&=&\sin(x/2)\,H - m \left(\beta_1\, X_2 + \beta_2\, X_1\right) + 2\,m\,\cos(x/2)\, u\, B_{12}\,,\label{29} \\
\frac{\sin(x/2)}{8 \pi a(1 - a)}&=&-\cos(x/2)\,H - m\, \tan(x/2)\left(\beta_1 X_2 + \beta_2 X_1 \right) +
  m\,\sin(x/2)(b B_2 + c B_1) \,,\label{30}\\
0&=& -m \sin(x/2)(c X_1
 -b X_2) + m \tan(x/2)(\beta_2 B_1- \beta_1 B_2)+(1-2a)\cos(x/2) B_{12}\,,\label{31}\\
0&=&-4a(a - 1) - m^2(4 - 8\beta_1\beta_2 + b c + 3u^2) \nonumber\\&&\hspace{2cm}-
      4\cos(x)\left(a(a - 1) + m^2(u^2 + 1)\right)
      + m^2\cos(2x)(b\,c - u^2) \label{4a}\,,\\
0&=&(2a - 1)u\cos(x/2)+m \tan(x/2)(\beta_1 c - b \beta_2)\,.\label{minuscula}
\end{eqnarray}
The boundary conditions at $x\rightarrow\pi$ are
\begin{eqnarray}
H(\pi)&=&0\,,\label{hache}\\
X_1(\pi)&=&\frac{ \Gamma(-a) \left( \cosh \left( \frac{\pi \mu}{2} \right) \textrm{Im} \left[ \psi \left( \frac{1}{2} + a + \frac{i\mu}{2} \right) \right] - \frac{\pi}{2} \sinh \left( \frac{\pi\mu}{2} \right)   \right)}{2^{2a}\mu  \left( \cos \left( 2 a \pi\right)+\cosh (\pi \mu)\right) \Gamma (1+a) \left| \Gamma \left( \frac{1}{2}-a+\frac{i\mu}{2}\right) \right|^2 }\,,\label{xx1}\\
X_2(\pi)&=&X_1(\pi)\left|_{\,a\rightarrow (1-a)} \right. \,,\label{xx2} \\
u(\pi)&=&0\,, \label{upi}\\
b	(\pi)&=&\frac{ 2^{1-2a} a (1-a) \left| \Gamma\left( \frac{1}{2} +a+\frac{i\mu}{2}\right)\right|^2}{m\Gamma^2 (1+a)}\,,\label{treintaytres}\\
c(\pi)&=&b(\pi)\left|_{\,a\rightarrow (1-a)}\right.  \,,\label{treintaydos}
\end{eqnarray}
where $\mu=\sqrt{4\,m^2-1}$ and $\psi$ is the digamma function. The meaning of the extra variables $B_1$, $B_2$, $B_{12}$, $X_1$, $X_2$, $u$, $b$, $c$, $\beta_1$ and $\beta_2$ is the same as in \cite{cahu} and is given in appendix A. The trace in (\ref{quince}) is regularized such that it vanishes when $x=\pi$, where there is no vertex point and no logarithmic term is present in the entropies.

In  \cite{doyon} the partition function for a Dirac fermion on the Poincar\'e disk with boundary conditions analogous to (\ref{boun}) imposed on a geodesic segment  has been written in terms of a solution of the Painlev\'e VI differential equation. Though we were not able to find an explicit relation, it is likely that our results for $H(x)$ could have also an expression in terms of solutions of these type of equations.

\begin{figure} [tbp]
\centering
\leavevmode
\epsfysize=6cm
\bigskip
\epsfbox{coefnuevo.eps}
\caption{The solid curves are the functions $s_2(x)$ (top) and $s_3(x)$ (bottom). The points are obtained by numerical simulations in a lattice, and correspond to, from top to bottom, $s=s_1$, $s_2$ and $s_3$ evaluated for the angles $x=\pi / 4$, $\pi /2$ and $3/4 \,\pi$.}
\end{figure}

\subsection{The coefficient of the logarithmic term}
 Gathering all the results together, using eqs. (\ref{cuatro}), (\ref{cinco}) with $\Lambda =M$, (\ref{sumz}), (\ref{trsqrt}), (\ref{tr}), and (\ref{quince}), we arrive at the following result for a real scalar
\begin{equation}
s_n(x)= \sum_{k=1}^{n-1}\frac{8\,k\,(n-k)}{n^2\,(n-1)}\int_{1/2}^{\infty}dm\,m\,(m^2-1/4)^{\frac{1}{2}}\int_{x}^{\pi}dy \,H_\frac{k}{n}(y,m)\,,
\label{final}
\end{equation}
where we have made explicit the $m$ dependence of $H$.
This equation, together with (\ref{hprima}-\ref{treintaytres}) gives our final expression for the coefficient of the logarithmic term in $S_n$ for a real scalar field (half the complex scalar result), which is in position to be evaluated by solving the ordinary differential equations numerically.

The functions $s_n(x)$ satisfy  $s_n(x)=s_n(2 \pi-x)$, which is a consequence of the symmetry in the entropies $S_n(V)=S_n(-V)$ due to the purity of the vacuum state. In the figure (1) we have plotted $s_2(x)$ and $s_3(x)$ for $x\in[0,\pi]$. The values of $s_1=s$, $s_2$ and $s_3$ for $x=\pi / 4$, $\pi /2$ and $3/4 \,\pi$ obtained by lattice simulations are also plotted. They  show a perfect accord (less than one percent error) with the analytical results. These particular values of the angle are the ones for which the coefficient can be calculated in absolute terms with very small error on a square lattice of limited size (in the present case it was $100\times 200$ points). The numerical methods consist of evaluating the entropy for a massless real scalar  (see Appendix B and  \cite{cahu,peschel}) for a given shape (square, triangle, etc.) and different overall size $\lambda$, and then fitting the result as $S_n=C_0+C_1\, \lambda+C_{-1}\, \lambda^{-1} +C_{-2} \,\lambda^{-2}- s_n \log(\lambda)$. It is also possible to evaluate  very accurately $s_n$ for specific combinations of angles using rectangular triangles. In this way we have computed in the lattice $s_n (y)+s_n (\pi/2-y)$ with $y=\arctan(p/q)$, where $p$ and $q$ are small integers. We also obtain in this case a perfect accord with the analytical results.  We have also checked that $s_n$ does not depend on the orientation of the polygon with respect to the lattice simmetry axes.

\section{The small angle limit  and the two dimensional entropic c-functions}
From  (\ref{final}) and the differential equations for $H$, it is possible to derive expansions for $s_n(x)$ and $s(x)$ for small $x$ or $\pi-x$. In particular, in the small $x$ limit the Green function on the cut sphere is related to the corresponding one for the flat space problem with boundary conditions imposed on an interval. Taking into account the results of \cite{cahu}, which deal with this later problem, we can show directly from the differential equations that for small $x$
\begin{equation}
s(x)\sim \frac{\int_0^\infty dt \, c_S(t)}{\pi x}\,,
\label{35}
\end{equation}
where $c_S(t)$ is the one dimensional entropic c-function for a free scalar field \cite{cteor,fermion,cahu}. The c-function for a theory in $1+1$ dimensions is defined as
\begin{equation}
c(r)=r\frac{dS(r)}{dr}\,,
\end{equation}
where $S(r)$ is the entanglement entropy corresponding to an interval of length $r$ in $1+1$ dimensions. For free fields  $c(r)\equiv c(t)$ where $t=m r$, and $m$ is the field mass.

We can obtain the formula (\ref{35}) (and the analog one which is valid for free fermions) with a less technical derivation. This also sheds light on the origin and the necessity of the logarithmic term. In a previous paper we have shown that the entropy corresponding to a spatial rectangle in $2+1$ dimensions with a short side $L$ and long side $R$, $R/L\gg 1$, contains the universal term (included in the finite term $S_0$ in eq.(\ref{div})) \cite{cahu}
\begin{equation}
S_{\textrm{univ}}\simeq -k R/L\,, \label{tete}
\end{equation}
where $k$ is a dimensionless function of $L$ and the renormalized parameters of the theory. For a free massless theory it is
\begin{equation}
k=\frac{1}{\pi}\int_0^\infty dt \, (n_S c_S(t)+n_F c_F(t))\,,
\end{equation}
where $c_S$ and $c_F$ are the one dimensional entropic c-functions for a real scalar and a majorana fermion, and $n_S$ and $n_F$ are the multiplicity of the boson and fermion degree of freedom.
This gives $k\simeq 0.039$ for a real scalar and $k\simeq 0.072$ for a $2+1$ dimensional Dirac fermion. Now, we may approximately decompose a plane angular sector with small angle $x$ as a union of thin and long rectangles with bigger size as we move further from the angle vertex. Thus, there is a term in the entropy for the angular sector which is due to the sum of the terms (\ref{tete}) for the rectangles (the term (\ref{tete}) is extensive in the direction in which lie the different rectangles). In the limit of an infinite partition this gives the desired result
\begin{equation}
\left.S\right|_{\log}\sim -k\int_\epsilon \frac{dR}{\tan(x)R} \sim \frac{\int_0^\infty dt \, c(t)}{\pi \,x} \log(\epsilon)\,.
\label{38}
\end{equation}

\begin{figure} [tb]
\centering
\leavevmode
\epsfysize=4cm
\bigskip
\epsfbox{starshaped.eps}
\caption{The dashed sets $A$ and $B$ are anti-starshaped, in the sense that any ray from the origin has no intersection with them or the intersection is a half line which does not contain the origin. A contraction of the space transforms them into $A^\prime$ and $B^\prime$. In the limit of an infinite contraction they are mapped to two angular sectors with common vertex.}
\end{figure}

\section{Final remarks}
We have shown that there is a logarithmic divergent term with universal coefficient for the entanglement entropy corresponding to spatial polygonal sets in 2+1 dimensions. We have found the analytic expression for the coefficient in the alpha entropies for integer $\alpha=n$ and a free scalar field, and the small angle limit for the corresponding coefficient on the geometric entropy for general free fields.

Interestingly, since this paper first appeared as a preprint, a vertex induced logarithmically divergent term in three dimensions has also been found in the conjectured geometrical form of the entanglement entropy for some conformal field theories  arising in the AdS/CFT duality context \cite{ryu,ht}. The overall form of the logarithmic coefficient as a function of the angle is similar to our exact results, and the authors have also been able to show the relation between $k$ and $s$ (eqs. (\ref{tete}) and (\ref{38})) for their entropy functions, which presumably correspond  to some interacting theory.

Technically, the  calculation in this paper amounts to the one of the conformal anomaly for a free scalar on a three-dimensional manifold with a conical singularity located on the boundary of a plane angular sector. This is a non-trivial result, since there is no known general method applicable to this case involving conical singularity surfaces which are themselves non-smooth.
We have shown that this particular problem can be mapped to the one of the calculation of the trace of the Green function for a massive scalar field on a two dimensional sphere where boundary conditions are imposed on a segment of a great circle. We have found this trace analytically by a method introduced in \cite{cahu,myers} which allows one to exploit the rotational symmetries of the sphere even in the presence of the symmetry breaking boundary conditions.

Our results for the Green function on the cut sphere may find different uses beyond the present one. One possible application is the problem of scattering of waves by a plane angular sector in three dimensions \cite{lame}.
Remarkably, it also gives the exact entropy functions for a spatial segment in $1+1$ dimensional de Sitter space. This space is equivalent to the surface
\begin{equation}
\vec{r}^2-t^2=R^2
\end{equation}
in $2+1$ Minkowski space. The euclidean de Sitter space then corresponds to a sphere with radius $R$. The formulas corresponding to (\ref{sumz}) and (\ref{green}) in this case take us again to the Green function on the cut sphere. The result can then be readily written off,
\begin{equation}
S_n^{\textrm{de Sitter}}(r/R)=- \sum_{k=1}^{n-1} \frac{8 \pi k (n-k)}{n^2 (n-1)} \int_{(MR)^2}^\infty dm^2\,   \int_{r/R}^\pi H_{\frac{k}{n}} (y,m)\, dy  + \textrm{cons} \,,
 \end{equation}
where  $r$ is the physical size of the geodesic interval, and $M$ is the field mass. The additive constant is logarithmically divergent with the cutoff, in order to keep $S_n$ positive for small values of $r$.

The formulas (\ref{35}), (\ref{tete}) and (\ref{38}) show a remarkable relation between universal terms in the entropies for different dimensions. The function $c(r)=r\, dS(r)/dr$, with $S(r)$ the one dimensional entropy corresponding to an interval of length $r$, plays the role of the Zamolodchikov's c-function in the entanglement entropy c-theorem \cite{cteor}. This theorem states that there is a universal dimensionless quantity in two dimensions which is decreasing under scaling and has a well defined value at the fixed points (proportional to the Virasoro central charge)\cite{zamo}. The c-function introduced by Zamolodchikov is constructed from a correlator of stress tensor traces, and we have shown that it can be taken to be $c(r)$ as well. There has been a substantial effort to extend the c-theorem to higher dimensions, but a definitive result in this direction is still missing \cite{sofar}.
At fixed points the function $c(r)$ is a constant given by the coefficient of the term proportional to $\log(\epsilon)$ in the two dimensional entropy. Then, one would be tempted to speculate that a running function which takes the value $s(x)$ at fixed points could be a good candidate for a c-function in three dimensions.

On the other hand, the logarithmic term in the entropy induced by the vertices is, remarkably, the only obstacle in the following simple argument attempting to prove the c-theorem in any dimensions.
Consider the mutual information
\begin{equation}
I(A,B)=S(A)+S(B)-S(A\cup B) \label{mar}
\end{equation}
 between two non intersecting sets $A$ and $B$. This is dimensionless and universal (all boundary terms get subtracted), and it is also positive and increasing with the size of each set separately \cite{cteor}. Thus, if we take $A$ and $B$ as anti-starshaped (see figure (2)) $I(A,B)$ is decreasing under dilatations. However, it fails to have a well-defined value at the ultraviolet fixed point, since in that limit $A$ and $B$ go to angular sectors with a common vertex, and $I(A,B)$ has a logarithmic divergence due to the mismatch of the vertex induced terms in (\ref{mar}).

\section*{Acknowledgments}
We warmly thank C.D.Fosco for very useful discussions and important input in the early phase of this project,
and R.Trinchero for pointing us to ref. \cite{seeley}.

\section{Appendix A: Green function on the cut sphere}
In this appendix we derive the set of non linear differential equations (\ref{hprima}-\ref{treintaytres}) which give the trace of the Green function on a sphere with a cut of angle $x$ (see figure 3), where the boundary conditions (\ref{boun}) are imposed. We will follow here the same steps as in \cite{cahu} where we have solved the analogous problem on the plane with boundary conditions imposed on a finite interval.  We use a generalization of a method originally developed in \cite{myers}.

The Green function $G(z,z^\prime)=(-\Delta_{\Omega} +m^{2})^{-1}_{z,z^\prime}$, where $z$ and $z^{\prime}$ are points on the sphere, is uniquely defined  by the following three requirements:
\bigskip

\noindent a.- It satisfies the Helmholtz equation on the sphere
\begin{equation}
\left( -\Delta_{\Omega} +m^{2}\right) G(z,z^{\prime})=\delta (
z-z^{\prime})\,, \label{equ1}
\end{equation}
where, in polar coordinates,
\begin{equation}
\Delta_{\Omega}=\frac{1}{\sin \theta}\frac{\partial}{\partial \theta}(\sin\theta\frac{\partial}{\partial \theta})+\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \varphi^2}\,.
\end{equation}
\bigskip

\noindent b.- The boundary condition is (it also holds for the Green function derivatives)
\begin{equation}
\lim_{\epsilon \rightarrow 0^{+}}G((\pi+\varepsilon,\varphi ),z^{\prime})=e^{i2\pi a }\lim_{\epsilon \rightarrow 0^{+}}G((\pi-\varepsilon,\varphi ),z^{\prime}) \hspace{1cm} \textrm{for}\,\, \varphi \in [\varphi_1,\varphi_2]\,.
\end{equation}
 Here the points with polar coordinates $(\pi/2,\varphi)$, $\varphi \in [\varphi_1,\varphi_2]$, form the cut location. We need to specify the cut through its two endpoints with angle $\varphi_1$ and $\varphi_2$. At the end of the calculation we can set as in Section 2 $\varphi_1=-\varphi_2=x/2$. From now on we also choose $a \in [0,1]$.

\bigskip

\noindent c.- $G(z,z^{\prime})$ is bounded everywhere (including the cut) except at $z=z^{\prime}$.
\bigskip

\noindent Several properties of the Green function can be derived from its definition. First it is Hermitian
\begin{equation}
G(z,z^{\prime})=G(z^{\prime},z)^{*}\,,
\end{equation}
and the two reflection symmetries of the problem give
\begin{eqnarray}
G(z,z^{\prime})&=&G(Tz,Tz^{\prime})^{*}\,,\\
G(z,z^{\prime})&=&G(Rz,Rz^{\prime})\,,
\label{sime1}
\end{eqnarray}
where $R$ and $T$ are the reflexion operations given by
\begin{eqnarray}
R\,\,(\theta,\varphi)&=&(\theta,\varphi_1+\varphi_2-\varphi)\,,\\
T\,\,(\theta,\varphi)&=&(\pi-\theta,\varphi)\,.
\end{eqnarray}

In what follows we study the structure of singularities of some functions related to $G(z,z^\prime)$ and use repeatedly the fact that a nonsingular solution of the homogeneous Helmholtz equation must be zero due to the uniqueness theorem.

According to the boundary conditions, near the end points of the cut the Green function must
have branch cut singularities. The requirement that
the function must remain bounded on the cut and the equation (\ref{equ1})
imply that the leading terms of $G(z,z^{\prime})$ for $(\theta,\varphi)$ near $
(\pi/2,\varphi_1)$ (and fixed $z^{\prime}$) have to be of the form
\begin{equation}
G(z,z^{\prime})\sim [(\varphi-\varphi_1)-i(\theta-\pi/2)]^{a }S_{1}(z^{\prime})+[(\varphi-\varphi_1)+i(\theta-\pi/2)]^{1-a }S_{2}(z^{\prime}) \,, \label{equ2}
\end{equation}
for some functions $S_1(z)$ and $S_2(z)$.
We have written explicitly only the terms with powers of $[(\varphi-\varphi_1)-i(\theta-\pi/2)]$ with exponent smaller than one. These are the ones of interest in the sequel. Note that the contributions at this order must be
analytic or anti-analytic in $z$ in
order to cancel the Laplacian term in (\ref{equ1}).

The most singular contributions to $\partial _{\varphi_{1}}G(z,z^{\prime})$
for $z\rightarrow (\pi/2,\varphi_{1})$ follow from the derivative of (\ref{equ2})
\begin{equation}
\partial _{\varphi_{1}}G(z,z^{\prime})\sim -a [(\varphi-\varphi_1)-i(\theta-\pi/2)]^{a
-1}S_{1}(z^{\prime})+(a -1)[(\varphi-\varphi_1)-i(\theta-\pi/2)]^{-a
}S_{2}(z^{\prime})  \,.\label{deri}
\end{equation}
The function $\partial _{\varphi_{1}}G(z,z^{\prime})$ satisfies the homogeneous Helmholtz equation and the boundary conditions, and it is not singular at $
z\rightarrow z^{\prime}$. It has only one singularity located at $\varphi_1$ whose expression is given by (\ref{deri}). Thus, an adequate linear combination of this function for different values of $z^\prime$ must be a nonsingular solution of the Helmholtz equation, and therefore identically zero.
Following the same argument as in the flat case \cite{cahu}, this leads to the fundamental relation
\begin{equation}
\partial _{\varphi_{1}}G(z,z^{\prime})=S(z)^{\dagger}AS(z^{\prime})  \label{c}\,,
\end{equation}
where we are using vectorial notation
\begin{equation}
S(z)=\left(
\begin{array}{l}
S_1(z)  \\
 S_2(z)
\end{array}\right)
\end{equation}
and $A$ is an hermitian matrix.

The function $S(z)$ satisfies the homogeneous Helmholtz equation with the
same boundary conditions as $G(z^{\prime},z)$ with $z^{\prime}$
fixed, but it is unbounded around $z=(\pi/2,\varphi_{1})$. In fact, it follows from (\ref{deri}) and (\ref{c}) that it has singular terms proportional to
$[(\varphi-\varphi_1)-i(\theta-\pi/2)]^{-a }$ and $[(\varphi-\varphi_1)+i(\theta-\pi/2)]^{a -1}$.
 However, these
disappear if we derive with respect to $\varphi_{2}$.
On the other hand $S(z)$ behaves around $(\pi/2,\varphi_{2})$ as an ordinary wave, that
is, it vanishes proportionally to $[(\varphi-\varphi_2)-i(\theta-\pi/2)]^{a }$ and $
[(\varphi-\varphi_2)+i(\theta-\pi/2)]^{1-a }$. Then $\partial _{\varphi_{2}}S(z)$ has singular
terms around $(\pi/2,\varphi_{2})$ which are proportional to $[(\varphi-\varphi_2)-i(\theta-\pi/2)]^{a -1}$ and $ [(\varphi-\varphi_1)+i(\theta-\pi/2)]^{-a }$.
This structure of singularities  and the relation (\ref{sime1}) gives place to
\begin{eqnarray}
\partial _{\varphi_{2}}S(z)&=&\gamma \,S(Rz) \,, \label{a}\\
\partial _{\varphi_{1}}S(Rz) &=&-\gamma \,S(z)\,, \label{b}
\end{eqnarray}
where the matrix $\gamma $ is a function of $\varphi_{1}-\,\varphi_{2}$.

\begin{figure} [tb]
\centering
\leavevmode
\epsfysize=5cm
\bigskip
\epsfbox{esfera.eps}
\caption{The plane angular sector of angle $x=\varphi_1-\varphi_2$ and the sphere with a cut of angle $x$}
\end{figure}

Using (\ref{c}), (\ref{a}) and (\ref{b}) to calculate $\partial
_{\varphi_{1}}\partial _{\varphi_{2}}G(z,z^{\prime})-\partial _{\varphi_{2}}\partial
_{\varphi_{1}}G(z,z^{\prime})=0$ we get
\begin{equation}
\gamma ^{\dagger }=A\gamma A^{-1}\,, \label{alge1}
\end{equation}
and that the matrix $A$ must be a constant, $\partial A / \partial_{\varphi_1}=\partial A/ \partial_{\varphi_2}= 0$. Thus, it can be evaluated by the knowledge of the solutions for $x=\pi$ or using the flat space  limit $x\rightarrow 0$ \cite{cahu}. It is given by
\begin{equation}
A=-4\pi (1-a )a \,\sigma _{1}\,, \label{algea}
\end{equation}
where $\sigma _{1}$ is the Pauli matrix.

The equation (\ref{deri}) leads to the behavior
\begin{equation}
S(z)\sim \frac{1}{4\pi }\left(
\begin{array}{l}
\frac{1}{a }[(\varphi-\varphi_1)-i(\theta-\pi/2)]^{-a } \\
\frac{1}{(1-a )}[(\varphi-\varphi_1)+i(\theta-\pi/2)]^{a -1}
\end{array}
\right)  \label{wr}
\end{equation}
for $z$ in the vicinity of $(\pi/2,\varphi_{1})$. Using (\ref{b}) we get a similar expression for S(Rz)

\begin{equation}
S(Rz)\sim \frac{\gamma}{4\pi a(1-a)}\left(
\begin{array}{l}
\left[(\varphi -\varphi_1)-i(\theta-\pi/2)\right]^{1-a } \\
\left[(\varphi -\varphi_1)+i(\theta-\pi/2)\right]^{a}
\end{array}
\right) \,. \label{wrr}
\end{equation}

\subsection{Equations for $S(z)$ from rotation symmetries}
In order to use eq. (\ref{c}) to compute the trace of the Green function we need more information on $S(z)$. With this aim we exploit the symmetries of the problem as follows.
We consider two rotations around the axes lying on the equator which cut the sphere at the points $(\theta=\pi/2, \varphi=\frac{\varphi_1+\varphi_2}{2})$ and $(\theta=\pi/2, \varphi=\frac{\varphi_1+\varphi_2}{2}+\frac{\pi}{2})$. The associated differential operators are given respectively by
\begin{eqnarray}
L_1&=&-\cot\theta\sin(\varphi-\varphi^{+})\partial\varphi+\cos(\varphi-\varphi^{+})\partial\theta \label{defl1}\,, \\
L_2&=&-\cot\theta\cos(\varphi-\varphi^{+})\partial\varphi-\sin(\varphi-\varphi^{+})\partial\theta\,. \label{defl2}
\end{eqnarray}
 We use the notation $\varphi^{\pm}=\frac{\varphi_1\pm\varphi_2}{2}$. From (\ref{defl1}) and (\ref{defl2}) we check that they satisfy the algebra of angular momentum operators
\begin{eqnarray}
\left[L_1,L_2\right]&=&\partial_{\varphi}\,,\\
\left[L_1,\partial_{\varphi}\right]&=&-L_2\,,\\
\left[L_2,\partial_{\varphi}\right]&=&L_1\,,\\
L_1^2+L_2^2+\partial^2_{\varphi}&=&\Delta_{\Omega}\,,
\end{eqnarray}
and the following relations
\begin{eqnarray}
\frac{\partial L_1}{\partial\varphi_1}&=&-\frac{1}{2}L_2\,,\\
\frac{\partial L_2}{\partial\varphi_1}&=&\frac{1}{2}L_1\,.
\end{eqnarray}

They commute with the Laplacian operator and
thus the functions $(L_i+L_i^{\prime})\,G(z,z^{\prime})$ with $i=1,2$
are non singular at $z=z^{\prime}$ and satisfy the
homogeneous Helmholtz equation and the boundary conditions.
However, due to the $L_i G(z,z^{\prime})$ terms, they are singular at $(\pi/2,\varphi_{1})$ and $(\pi/2,\varphi_{2})$ as a function of $z$. Near $(\pi/2,\varphi_{1})$ we have from (\ref{equ2}), (\ref{wr}), (\ref{defl1}) and (\ref{defl2})
\begin{eqnarray}
L_1 G(z,z^{\prime})&\sim& \cos\varphi^{-}\left[i a [(\varphi-\varphi_1)-i(\theta-\pi/2)]^{a
-1}S_{1}(z^{\prime})-i(1-a )[(\varphi-\varphi_1)+i(\theta-\pi/2)]^{-a
}S_{2}(z^{\prime})\right] \nonumber \\
&\sim& -i\cos(\varphi^{-}) S(z^{\prime})A \sigma_3 S(Tz)\,,\\
L_2 G(z,z^{\prime})&\sim& i \sin(\varphi^{-})S(z^{\prime})A \sigma_3 S(Tz)\,.
\end{eqnarray}
This gives from the uniqueness of the solution of the homogeneous Helmholtz equation for non-singular functions
\begin{eqnarray}
L_1 G(z,z^{\prime})+L_1^{\prime} G(z,z^{\prime})&=& -i\cos(\varphi^{-}) \left[S^{\dagger}(z) \sigma_3A S(z^{\prime})+ S^{\dagger}(Rz) \sigma_3A S(Rz^{\prime})\right]\,,\label{dL1}\\
L_2 G(z,z^{\prime})+L_2^{\prime} G(z,z^{\prime})&=& -i\sin(\varphi^{-}) \left[S^{\dagger}(z) \sigma_3A S(z^{\prime})- S^{\dagger}(Rz) \sigma_3A S(Rz^{\prime})\right]\,.\label{dL2}
\end{eqnarray}

From (\ref{wr})  we see that the most divergent terms of $L_1 S^{\dagger}(z)A- i\cos(\varphi^{-})\partial_{\varphi}S^{\dagger}(z)\sigma_3 A$ exactly cancel. But this combination should also have a contribution to order $[(\varphi-\varphi_2)-i(\theta-\pi/2)]^{a -1}$ and $ [(\varphi-\varphi_1)+i(\theta-\pi/2)]^{-a }$ for $z$ in the vicinity of $(\pi/2,\varphi_{1})$. Thus we have
\begin{equation}
L_1 S^{\dagger}(z)A-i \cos(\varphi^{-})\partial_{\varphi}S^{\dagger}(z)\sigma_3 A=-S^{\dagger}(z)A \xi \label{chi}\,,
\end{equation}
where $\xi$ is a matrix which depends on $x=\varphi_1-\varphi_2$.
A similar argument hold for $L_2 S^{\dagger}(z)A  + i \sin(\varphi^{-}
) $ $\partial  \varphi S^{\dagger}(z) \sigma_3  A$ giving place to
\begin{equation}
L_2 S^{\dagger}(z)A +i \sin(\varphi^{-})\partial\varphi S^{\dagger}(z)\sigma_3 A=S^{\dagger}(z)A \xi_1 \label{chi1}\,,
\end{equation}
with $\xi_1$ depending on $x=\varphi_1-\varphi_2$.

Taking the derivative with respect to $\varphi_1$ of (\ref{dL1}) and (\ref{dL2}), and extracting the coefficients of the divergent terms, we get
\begin{eqnarray}
\label{bata}
L_1 S(z)&=&\xi S(z)-i \sin(\varphi^{-})\sigma_3 S(z)-i\cos\varphi^{-}\left[\left\{ \gamma ,\sigma _{3}\right\}S(Rz)-\sigma_3 \partial_{\varphi}S(z)\right]\,,\\
\label{tata}
L_2 S(z)&=&-\xi_1S(z)-i\cos(\varphi^{-})\sigma_3 S(z)+i\sin(\varphi^{-})\left[\left[\sigma_3,\gamma\right]S(Rz)+\sigma_3\partial_{\varphi}S(z)\right]\,.
\end{eqnarray}
We can also write the reflected equation for (\ref{bata}) and (\ref{tata})
\begin{eqnarray}
L_1 S(Rz)=\xi S(Rz)-i \sin\varphi^{-}\sigma_3 S(Rz)-i\cos\varphi^{-}\left[\left\{ \gamma ,\sigma _{3}\right\}S(z)-\sigma_3 \partial_{\varphi}S(Rz)\right]  \,,\label{bata1}\\
L_2 S(Rz)=\xi_1S(Rz)+i\cos(\varphi^{-})\sigma_3 S(Rz)-i\sin(\varphi^{-})\left[\left[\sigma_3,\gamma\right]S(Rz)-\sigma_3\partial_{\varphi}S(z)\right]\,.\label{tata1}
\end{eqnarray}
Subtracting the hermitian conjugate of the equation (\ref{bata}) and (\ref{bata1}) and  using $\left\{ A,\sigma
_{3}\right\} =0$, $\left\{ \gamma ,\sigma _{3}\right\} ^{\dagger }A+A\left\{
\gamma ,\sigma _{3}\right\} =0$, we obtain
\begin{equation}
\xi ^{\dagger }A+A\xi =-i \sin \varphi^{-} \sigma_3 A \,. \label{alge2}\\
\end{equation}
A similar relation for $\xi_1$ follows by subtracting the hermitian conjugate of (\ref{tata}) and (\ref{tata1}),
\begin{equation}
\xi_1 ^{\dagger }A+A\xi_1=i \cos \varphi^{-} \sigma_3 A \,. \label{alge2p}
\end{equation}
\subsection{Parametrization}
The expansion of $S(z)$ can be extended to the following order around the singular point $(\pi/2,\varphi_1)$ introducing a real matrix $N$
\begin{equation}
S(z)\sim \frac{1}{4\pi }\left(
\begin{array}{l}
\frac{1}{a }[(\varphi-\varphi_1)-i(\theta-\pi/2)]^{-a } \\
\frac{1}{(1-a )}[(\varphi-\varphi_1)+i(\theta-\pi/2)]^{a -1}
\end{array}
\right) +N  \left(
\begin{array}{l}
\left[(\varphi-\varphi_1)-i(\theta-\pi/2)\right]^{1-a } \\
\left[(\varphi-\varphi_1)+i(\theta-\pi/2)\right]^{a}
\end{array}
\right)\,.\label{2orden}
\end{equation}
This general expansion for $S$ inserted in (\ref{chi}) and  (\ref{chi1}) allows us to solve for some components of $N$ and obtain a relation between $\xi$ and $\xi_1$
\begin{equation}
\xi_1=\tan (\varphi^{-})\xi+i q \sec(\varphi^{-})\,,
\end{equation}
with
\begin{equation}
q=\left(
\begin{array}{ll}
a-1 &  \\
& a
\end{array}
\right)\,.
\end{equation}
It also gives additional relations which, together with the algebraic equations (\ref{alge1}), (\ref{alge2}), give the general form of $\gamma$ and $\xi$ in terms of the
following parametrization
\begin{equation}
\gamma =\frac{m}{2}\left(
\begin{array}{ll}
u & b \\
c & u
\end{array}
\right)
\hspace{1.5cm};\hspace{1.5cm}\xi
=\left(
\begin{array}{ll}
-i(a-1)\sin(\varphi^{-}) & im\beta _{1} \\
-im\beta _{2} & -ia\sin(\varphi^{-})
\end{array}
\right)\,,
\end{equation}
where $u$, $b$, $c$, $\beta _{1}$, and $\beta _{2}$ are real functions of $m$ and $x$.

Taking derivatives respect to $\theta$ and $\varphi$ of (\ref{bata}) and (\ref{tata}) and combining them to reconstruct the Helmholtz equation for $S(z)$, and using the expansions of $S(z)$ and $S(Rz)$, we get from the series around the singular points
 the differential equations (\ref{cprima}), (\ref{bprima}), (\ref{uprima}) for $b$, $c$ and $u$, and the algebraic equations (\ref{4a}) and  (\ref{minuscula}) which determine $\beta_1$ and $\beta_2$.
\subsection{Integrated quantities}
The equation (\ref{c}) gives for the trace of the Green function
\begin{equation}
\frac{d}{dx}\int d\Omega\, G(z,z)=\int d\Omega\,  S^{\dagger
}A\;S=-8\pi a \left( 1-a \right) H \,,\nonumber
\end{equation}
where
\begin{equation}
H(x) =\int d\Omega \,S_{1}^{*}(z)S_{2}(z) \,, \label{hh}
\end{equation}
and $d\Omega=d\theta d\varphi \sin(\theta)$.

To find this quantity, we use the information obtained in 5.1 and 5.2. We first define the following auxiliary integrals
\begin{eqnarray}
B_{1}(x) &=&\int d\Omega \,S_{1}^{*}(z)S_{1}(Rz)\,, \\
B_{2}(x) &=&\int d\Omega\,S_{2}^{*}(z)S_{2}(Rz) \,,\\
B_{12}(x) &=&\int d\Omega\,S_{2}^{*}(z)S_{1}(Rz) \,,\\
X_{1}(x) &=&\int d\Omega\,S_{1}^{*}(z)S_{1}(z) \,,\label{x1}\\
X_{2}(x) &=&\int d\Omega\,S_{2}^{*}(z)S_{2}(z)\,.\label{x2}
\end{eqnarray}
These are convergent. They are also real since the relation (\ref{sime1}) implies that $S_{i}^{*}(z)=S_{i}(Tz)$.

The equations for these quantities are obtained basically combining conveniently the components of (\ref{bata}) and (\ref{tata}) multiplied by the components of $S(z)$ and $S(Rz)$, and integrating on the sphere. In this way the combination $\left(L_1\,S_1(z)\right)S_2^{*}(z)+S_1(z)\left(L_1\,S_2(z)\right)^{*}$ integrated on the sphere gives place to the equation (\ref{29}).
Similarly, from the combination $\left(L_2S_1(z)\right)S_2^{*}(z)+S_1(z)\left(L_2S_2(z)\right)^{*}$ we get equation (\ref{30}) and from $\left(L_2S_2(Rz)^{*}\right)S_1(z)+S_2(Rz)^{*}\left(L_2S_1(z)\right)^{*}$ we get equation (\ref{31}).

Differential equations for the integrated variables follow by taking the derivative of (\ref{hh}), (\ref{x1}) and (\ref{x2}) with respect to $\varphi_{2}$.
These correspond to (\ref{hprima}), (\ref{x1prima}) and (\ref{x2prima}) of Section  2.1.

In the flat space analog of the present calculation it is possible to give a closed algebraic expression of $H$ in terms of $u$ \cite{cahu}. This simplification seems to be lost in the sphere.

\subsection{Boundary conditions at $x=\pi$}
The homogeneous Helmholtz equation on the cut sphere for a function $f(z)$ can be solved exactly by separation of variables when $x=\pi$. The solution has the general form
\begin{equation}
f(z)=\sum_{n=-\infty}^{\infty} f_n(\theta)e^{i\varphi(a+n)}\,,
\end{equation}
where we have put now the end points of the cut at the poles $\theta=0$ and $\theta=\pi$ of the polar coordinates.  Here $f_n$ is given by
\begin{equation}
f_n(\theta)=C_1 P_{\frac{1}{2}(-1+i\mu)}^{a+n}(w)+C_2 Q_{\frac{1}{2}(-1+i\mu)}^{a+n}(w)\,,
\end{equation}
with $Q$ and $P$ the standard Legendre functions, $w=\cos\theta$ and $\mu=\sqrt{4m^2-1}$.
Evaluating the limits $w\rightarrow1$ and $w\rightarrow-1$ and comparing with the expansion (\ref{wr}) for $S(z)$  we get for $x=\pi$
\begin{eqnarray}
S_1&=&\frac{2^{-a/2}}{4\pi a}\frac{1}{2^{\frac{a}{2}-1}\cos(a\pi-i\frac{\pi}{2}\mu)\csc (a\pi)\Gamma(a)} \left[\frac{\pi}{2}\cot(a\pi)P_{\frac{1}{2}(-1+i\mu)}^{a}(w) \label{s1pi}
-Q_{\frac{1}{2}(-1+i\mu)}^{a}(w)\right]e^{i\varphi a} \,,  \\
S_2&=&\frac{2^{(a-1)/2}}{4\pi (1-a)}\frac{1}{2^{\frac{-(a+1)}{2}}\cos(a\pi+i\frac{\pi}{2}\mu)
\csc (a\pi)\Gamma(1-a)}\left[\frac{\pi}{2}\cot((1-a)\pi)P_{\frac{1}{2}(-1+i\mu)}^{a-1}(w)\right.\nonumber \\ &-&\left.Q_{\frac{1}{2}(-1+i\mu)}^{a-1}(w)\right]e^{i\varphi (a-1)} \,. \label{s2pi}
\end{eqnarray}
Using these expressions, the eqs. (\ref{a}) and (\ref{b}), and the expansion of $S(Rz)$ we find $u(\pi)$, $b(\pi)$ and $c(\pi)$ given in (\ref{upi}), (\ref{treintaytres}) and (\ref{treintaydos}).
The results (\ref{hache}), (\ref{xx1}) and (\ref{xx2}) for $H(\pi)$, $X_1(\pi)$ and $X_2(\pi)$ follow directly from (\ref{hh}), (\ref{x1}) and  (\ref{x2}), and the explicit form of $S_1$  and $S_2$, where we have used some formulae for the integrals of a product of two Legendre functions given in \cite{higher}.

\section{Appendix B: numerical entropy in a two dimensional lattice}
We use the method presented in \cite{peschel} to give an expression for $\rho _{V}$  in
terms of correlators for free bosonic discrete systems.
Take a free Hamiltonian for bosonic degrees of freedom with the form
\begin{equation}
H=\frac{1}{2}\sum_{\vec{r}} \pi _{\vec{r}}^{2}+\frac{1}{2}\sum_{\vec{r}\vec{r}^\prime}\phi _{\vec{r}}M_{\vec{r}\vec{r}^\prime}\,\phi
_{\vec{r}^\prime}\,\,,
\end{equation}
where $\phi _{\vec{r}}$ and $\pi _{\vec{r}}$ obey the canonical commutation relations $
[\phi _{\vec{r}},\pi _{\vec{r}^\prime}]=i\delta _{\vec{r}\vec{r}^\prime}$, $M$ is a Hermitian positive
definite matrix, and the sums are over the lattice sites $\vec{r}$. The vacuum (ground state) correlators are given by
\begin{eqnarray}
X_{\vec{r}\vec{r}^\prime} &=&\left\langle \phi _{\vec{r}}\phi _{\vec{r}^\prime}\right\rangle =\frac{1}{2}(M^{-
\frac{1}{2}})_{\vec{r}\vec{r}^\prime}\,,  \label{x} \\
P_{\vec{r}\vec{r}^\prime} &=&\left\langle \pi _{\vec{r}}\pi _{\vec{r}^\prime}\right\rangle =
\frac{1}{2}(M^{\frac{1
}{2}})_{\vec{r}\vec{r}^\prime}\,.  \label{p}
\end{eqnarray}
Let $X_{\vec{r}\vec{r}^\prime}^{V}$ and $P_{\vec{r}\vec{r}^\prime}^{V}\,$ be the correlator matrices restricted to
the region $V$, that is $\vec{r},\vec{r}^\prime\in V$.
The entropies  can be evaluated as
\begin{eqnarray}
S&=&\sum_k \left((\nu _{k}+1/2)\log (\nu _{k}+1/2)-(\nu
_{k}-1/2)\log (\nu _{k}-1/2) \right)\,,  \label{for}  \\
S_n&=&\frac{1}{n-1}\sum_k \log\left( \left( \nu_k+1/2 \right)^n - \left( \nu_k-1/2 \right)^n\right)\,,
\end{eqnarray}
where $\nu_k$ are the eigenvalues of  $\sqrt{X^{V}.P^{V}}$.
Thus, to compute the entropy numerically we need to diagonalize the matrix $
X^{V}.P^{V}$.

We took the lattice Hamiltonian for a real massless scalar in three dimensions as
\begin{equation}
{\cal H}=\frac{1}{2}\sum_{n,m=-\infty}^{\infty}( \pi _{nm}^{2}+(\phi _{n+1,m}-\phi_{n,m})^{2}+(\phi _{n,m+1}-\phi_{n,m})^{2} )\,.
\end{equation}
We have set the lattice spacing to one.
The correlators (\ref{x}) and (\ref{p}) are
\begin{eqnarray}
\langle \phi_{00}\phi_{ij}\rangle &=&\frac{1}{8\pi^2}\int_{-\pi}^{\pi} \int_{-\pi}^{\pi}dxdy\frac{\cos(x(i-1))\cos(y(j-1))}{\sqrt{2
(1-\cos(x))+2
(1-\cos(y))}}\,, \\
\langle \pi_{00}\pi_{ij}\rangle &=& \frac{1}{8\pi^2}\int_{-\pi}^{\pi} \int_{-\pi}^{\pi}dxdy \cos(x(i-1))\cos(y(j-1))}{\sqrt{2
(1-\cos(x))+2
(1-\cos(y))}\,.
\end{eqnarray}
It is relevant to the accuracy of the entropy calculation to evaluate the correlators with enough precision. This can be very time-consuming. We have found it is much faster to evaluate one of the two integrals in the correlators analytically (in terms of polynomials times elliptic functions) using a program for analytic mathematical manipulations, and then doing the last integral numerically.

redpajama_set_name: RedPajamaArXiv


Dirk COETSEE was born estimated 1735. He married 2 times. The first time he married Adriana Martha MULDER on April 20, 1767. She was born estimated 1747. The second time he married Helena de KLERK on December 19, 1784. She was born about 1764.
Susanna Johanna COETSEE, daughter of Dirk COETSEE and Adriana Martha MULDER, was born estimated 1774. She married Johannes Christoffel GREYLING, son of Jan Christoffel GREYLING and Hendrina BOTMA, on October 5, 1788. He was born estimated 1765.


text:
en Mobile Marketing Trends Programmatic
Beyond the Walled Gardens: Creating a Fairer Ad Tech Ecosystem
By Satya Vinnakota
The largest ad networks in existence are self-attributing – they perform attribution on their network independently without the need for third-party trackers. This lack of transparency has led them to be known among marketers as 'Walled Gardens' or 'Blackbox Ecosystems'.
There are a great many reasons for marketers to believe that this system is not only unfair on smaller players but harmful to the overall ad tech ecosystem. We must push to heighten transparency and open up this closed system.
Problem 1: Overly Generous Attribution Windows
While many large networks do run Mobile Measurement Partnership (MMP) programs, closed super-networks Google and Facebook have an effective duopoly on data within their networks and have almost complete control over the flow of that data. Rather than marketers measuring app install data, the app install data is sent to the closed networks who perform the measurement themselves before sending those results to marketers.
In this sense, these massive networks hold all the aces. They are able to self-attribute conversion success without meaningful regulation or industry oversight. The system is akin to marking one's own homework without having to show the workings.
Take the example of Facebook whose default attribution window is 24 hours post-view and 28 days post-click. That allows the social media giant to credit itself for conversions that occur a day after a user may (or may not) have seen an ad and credit itself for conversions that occur a full month after an ad click.
While walled gardens are often thought of as all-seeing Big Brother figures, there is still a great deal of data that they do not have access to. In reality, those conversions could have come from an ad on another platform (or indeed via a non-digital ad or word of mouth). It is true that there is no official standard in place to determine the appropriate length of an attribution window, yet the vast majority of vendors use no more than a 7-day window. Trusting a self-attributing ecosystem creates a reliance on walled gardens to provide due credit, with no third parties in place to oversee it.
In short, these closed networks, because of their size, are able to credit themselves for organic installs and thereby present their campaigns as being more successful than they are. While they do provide ways for advertisers to shorten these windows, few take them up on it.
Long lookback windows are problematic, though they are also only emblematic of a larger issue.
Problem 2: Black Box
Typically, publishers share their impression and click data with their MMPs whose role it is to assign credit. With self-attributing networks (SANs), that system is reversed – MMPs send campaign outcome data to them to compare with their own ad-serving data. If they find exposure, based on their own self-arbitrating rulebook, they alert the MMP that they are making a claim.
If an MMP believes that it is due unreceived credit, it faces a significant problem – it cannot independently verify the self-attributing network's claims, leading to disparity in reported numbers.
When the largest publishers in the world self-attribute they set a precedent for smaller networks to adopt similar strategies. This fosters an opaque, chaotic, and ultimately self-serving environment with few restraints to prevent the exploitation of loopholes.
In spite of this, it is still possible to have a fairer system where the effectiveness of running performance-based campaigns can be measured, with credit sent where credit is due. Though this is unlikely to be realised without collective industry action.
A Healthier Ecosystem is Possible
Facebook and Google undeniably remain great places to run campaigns, and MMPs do fantastic work. Nonetheless, there are a great many reasons for attribution partners and advertisers (the ones who hold the budget) to band together and pressure large, self-attributing networks to be more transparent on both supply and attribution.
MMPs have advanced technical means such as Device ID Matching and Mobile Device Fingerprinting to attribute which events came from which media vendors. They have the ability to provide an ideal ecosystem wherein every advertiser can create tailored attribution models that suit their needs. These models could be based on industry-agreed, performance-based pricing and standards for attribution windows and KPI thresholds. The MMPs could post the attributed event to the media vendor for optimization and billing.
If media vendors start a call for standards and accept third-party neutral attribution, then later MMPs could help advertisers to standardize a multi-touch attribution model based on individual advertiser needs. This system would benefit both advertisers and publishers by providing a holistic view of when exactly a conversion takes place.
While many conversations surrounding the creation of universal standards are dismissed as pie-in-the-sky, there is momentum growing around ideas such as universal IDs for attribution purposes. In addition, the incrementality model of attribution (as opposed to the last-touch model) is gaining favour and followers. With more support, it would form a significant first step in the battle for transparency and fairness. And at the very least, it would remove a few key bricks from the walled gardens.
As part of Verve Group, we strive to create an open mobile ecosystem with our brand-safe and transparent data-first ad platform. Contact us below to learn about Verve Group's extensive tech stack, which includes an Open Source SDK, and solutions for brand and performance campaigns.
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attribution windowsblackbox ecosystemsMMPsMobile attributionself-attributionwalled gardens
Satya Vinnakota
Satya Vinnakota is Head of Publishers at Applift and Director of M&A Operations for the MGI Group. Apart from his interests in mobile advertising, interactive media and mobile gaming, he is an avid football fan and loves Manchester United. Armed with an engineering degree, Satya has experience in product and business development, business operations, market research and technical sales. Follow him on all major social channels: @satyamanojv.
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text:
Green Garden (en inglés: Green Garden Precinct) es un distrito electoral ubicado en el condado de Madison en el estado estadounidense de Nebraska. En el Censo de 2010 tenía una población de 160 habitantes y una densidad poblacional de 1,72 personas por km².

Geografía
Green Garden se encuentra ubicado en las coordenadas . Según la Oficina del Censo de los Estados Unidos, Green Garden tiene una superficie total de 93.07 km², de la cual 93.01 km² corresponden a tierra firme y (0.06%) 0.06 km² es agua.

Demografía
Según el censo de 2010, había 160 personas residiendo en Green Garden. La densidad de población era de 1,72 hab./km². De los 160 habitantes, Green Garden estaba compuesto por el 96.88% blancos, el 0% eran afroamericanos, el 0% eran amerindios, el 0% eran asiáticos, el 0% eran isleños del Pacífico, el 0% eran de otras razas y el 3.13% pertenecían a dos o más razas. Del total de la población el 3.75% eran hispanos o latinos de cualquier raza.

Referencias

Enlaces externos

Distritos electorales de Nebraska
Localidades del condado de Madison (Nebraska)


text:
Which iPhone?
Top Android phones
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Which mid-range phone?
Phone news
Essential Phone's new colours are gorgeous, but also very limited
Elyse Betters | 15 February 2018
1/1 Essential
What is the Pocket-lint daily and how do you get it for free?
- Essential said it's only making a "small batch"
Essential is trying to lure more people into buying its phone.
The company has launched three limited-edition colours - and they're gorgeous. Essential is giving them a staggered release, starting with the "Ocean Depths" model on 15 February at 12pm PST, followed by "Stellar Gray" on 20 February at 12pm PST and "Copper Black" on 22 February at 12pm PST. All three models will be available for $599 each - about $100 more than the standard black or white model.
In a blog post, Essential explained the new Essential Phone models feature a custom ceramic colour, which the company described as a "feat". Apparently, it is "much more complex and challenging" than altering colours on standard materials like plastic. As a result, Essential said it's only making a "small batch" of these new colours, because "colored ceramic is extremely difficult to produce consistently."
1/3Essential
Upcoming phones: The future smartphones of 2018
Best smartphones 2018: The best phones available to buy
Essential Phone 2 (PH-2): What's the story so far?
The company hasn't revealed how many units will be available of each. Keep in mind, according to research firm IDC, Essential only sold less than 90,000 units of its flagship phone in the six months following launch. Initial reviews of the modular device were highly critical of its steep price tag and so-so camera, but the company has pushed on, rolling out firmware updates to help improve the Essential Phone.
Essential has steadily optimised the camera, for instance. However, it's worth noting that the Essential Phone was initially pitched as a modular smartphone, and yet, at this time, only the 360-degree camera accessory is available to buy. Also, the charging dock and the headphone jack add-on listed on Essential's website still aren't available to buy. The startup certainly seems to be on shaky ground.
Nevertheless, it is currently working on a follow-up smartphone, thought to be unveiled in late May or early June.
Ocean Depths: style as unique as you. This #EssentialPhone is our most fashion-forward design yet, featuring a vibrant teal ceramic back and copper frame that will make waves among a sea of lookalike devices. Ocean Depths will be available in limited quantity today at 12pm PT. pic.twitter.com/8GRcA8bM3Y
— Essential (@essential) February 15, 2018
Source: Essential
The Samsung Galaxy S10+ is now available on EE who have been awarded the UK's best network for the fifth year running. RootMetrics tested the four UK networks and EE was faster and more reliable than all of them, with better data performance. Their network has come a long way since they launched in 2012. Back then they had 11 UK cities covered by 4G. Today they cover most of the UK's land mass, thanks to 19,000 state-of-the-art 4G sites. They've got faster, too – from 50Mbps to a maximum speed of 400Mbps. And they're soon to experience even greater possibilities with the launch of 5G.
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