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Q: Checking a graph for cycles using BFS Suppose we construct a tree on a graph G with breadth-first search (BFS) and determine that there is no edge in the graph that connects nodes that belong to the same layer in the BFS tree. Does that mean that the graph has no cycle?
A: No, it does not. Consider the following directed graph:
If we start BFS from node 1, the search ends at node 3. Each vertex is in a separate layer, so there is no edge in the graph that connects nodes belonging to the same layer. However, the graph contains a cycle.
We can also construct a counterexample for undirected graphs:
The first layer contains node 1. The second layer contains nodes 2 and 4. The third layer contains node 3. The only layer with more than one node is the second layer, and its two nodes are not connected by an edge. Once again, there is no edge in the graph between nodes in the same layer, yet the graph contains a cycle.
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Lamprologus symoensi är en fiskart som beskrevs av Poll, 1976. Lamprologus symoensi ingår i släktet Lamprologus och familjen Cichlidae. IUCN kategoriserar arten globalt som otillräckligt studerad. Inga underarter finns listade i Catalogue of Life.
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{"url":"https:\/\/gmatclub.com\/forum\/for-one-toss-of-a-certain-coin-the-probability-that-the-out-85802.html","text":"GMAT Changed on April 16th - Read about the latest changes here\n\n It is currently 22 Apr 2018, 19:15\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n# Events & Promotions\n\n###### Events & Promotions in June\nOpen Detailed Calendar\n\n# For one toss of a certain coin, the probability that the out\n\nAuthor Message\nTAGS:\n\n### Hide Tags\n\nIntern\nJoined: 02 Oct 2009\nPosts: 8\nFor one toss of a certain coin, the probability that the out\u00a0[#permalink]\n\n### Show Tags\n\nUpdated on: 13 Jul 2013, 13:33\n3\nKUDOS\n8\nThis post was\nBOOKMARKED\n00:00\n\nDifficulty:\n\n65% (hard)\n\nQuestion Stats:\n\n60% (01:28) correct 40% (01:26) wrong based on 241 sessions\n\n### HideShow timer Statistics\n\nFor one toss of a certain coin, the probability that the outcome is heads is 0.6. If this coin is tossed 5 times, which of the following is the probability that the outcome will be heads at least 4 times?\n\nA. (0.6)^5\nB. 2(0.6)^4\nC. 3(0.6)^4\nD. 4(0.6)^4(0.4) + (0.6)^5\nE. 5(0.6)^4(0.4) + (0.6)^5\n[Reveal] Spoiler: OA\n\nOriginally posted by ralucaroman on 25 Oct 2009, 18:38.\nLast edited by Bunuel on 13 Jul 2013, 13:33, edited 2 times in total.\nRenamed the topic, edited the question added the answer choices and OA.\nMath Expert\nJoined: 02 Sep 2009\nPosts: 44599\nRe: Probability Question - GMATPrep\u00a0[#permalink]\n\n### Show Tags\n\n25 Oct 2009, 18:54\nExpert's post\n5\nThis post was\nBOOKMARKED\nFor one toss of a certain coin, the probability that the outcome is heads is 0.6. If this coin is tossed 5 times, which of the following is the probability that the outcome will be heads at least 4 times?\n\nA. (0.6)^5\nB. 2(0.6)^4\nC. 3(0.6)^4\nD. 4(0.6)^4(0.4) + (0.6)^5\nE. 5(0.6)^4(0.4) + (0.6)^5\n\n$$P(h)=0.6$$, so $$P(t)=0.4$$. We want to determine the probability of at least 4 heads in 5 tries.\n\nAt least 4 heads means 4 or 5. Let's calculate each one:\n\n5 heads: $$P(h=5)=0.6^5$$;\n\n4h and 1t: $$P(h=4)=\\frac{5!}{4!}*0.6^4*0.4=5*0.6^4*0.4$$, multiplying by 5 as 4h and 1t may occur in 5 different ways:\nhhhht\nhhhth\nhhthh\nhthhh\nthhhh\n\nSo, $$P(h\\geq{4})=0.6^5+5*0.6^4*0.4$$.\n\n_________________\nIntern\nJoined: 02 Oct 2009\nPosts: 8\nRe: Probability Question - GMATPrep\u00a0[#permalink]\n\n### Show Tags\n\n25 Oct 2009, 19:10\nHi, thank you for bringing light into this problem.\n\nYour solution is correct, this is the gmatprep answer too: 5*(0.6)^4*(0.4) + (0.6)^5.\n\nI had trouble understanding why the first part was multiplied with 5 (when there are 4 heads and 1 tail). I think I get it now (because 4H1T can happen in C5 taken by 4 times and 5H can happen in C5 taken by 5 which is 1). And addition of the 2 is due to fact that Probability(4H1T or 5H) = Probability (4H1T) + Probability (5H).\nSenior Manager\nAffiliations: PMP\nJoined: 13 Oct 2009\nPosts: 276\nRe: Probability Question - GMATPrep\u00a0[#permalink]\n\n### Show Tags\n\n25 Oct 2009, 19:12\n1\nKUDOS\nBunuel - the quant maestro --- nails one more problem... )\n\nI missed the multiplying by 5 too\n_________________\n\nThanks, Sri\n-------------------------------\nkeep uppp...ing the tempo...\n\nPress +1 Kudos, if you think my post gave u a tiny tip\n\nCurrent Student\nJoined: 18 Sep 2014\nPosts: 231\nRe: For one toss of a certain coin, the probability that the out\u00a0[#permalink]\n\n### Show Tags\n\n22 Mar 2015, 13:56\nIt's a straight forward E.\nP (heads occurring 5 times) + P (heads occurring 4 times)P (tails occurring 1 time )\n\nOnly thing since the coin is tossed 5 times, it should be multiple by 5 - the second scenario.\n\nPosted from my mobile device\n_________________\n\nKindly press the Kudos to appreciate my post !!\n\nNon-Human User\nJoined: 09 Sep 2013\nPosts: 6654\nRe: For one toss of a certain coin, the probability that the out\u00a0[#permalink]\n\n### Show Tags\n\n08 Dec 2017, 08:24\nHello from the GMAT Club BumpBot!\n\nThanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).\n\nWant to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.\n_________________\nRe: For one toss of a certain coin, the probability that the out \u00a0 [#permalink] 08 Dec 2017, 08:24\nDisplay posts from previous: Sort by","date":"2018-04-23 02:15:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8481990098953247, \"perplexity\": 3873.9629373165385}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-17\/segments\/1524125945669.54\/warc\/CC-MAIN-20180423011954-20180423031954-00190.warc.gz\"}"} | null | null |
{"url":"https:\/\/docs.rtems.org\/branches\/master\/user\/rsb\/project-sets.html","text":"# 14.2. Project Sets\u00b6\n\nThe RTEMS Source Builder supports project configurations. Project configurations can be public or private and can be contained in the RTEMS Source Builder project if suitable.\n\nThe configuration file loader searches the macro _configdir and by default this is set to %{_topdir}\/config:%{_sbdir}\/config where _topdir is your current working directory, or the directory you invoke the RTEMS Source Builder command in. The macro _sbdir is the directory where the RTEMS Source Builder command resides. Therefore the config directory under each of these is searched so all you need to do is create a config in your project and add your configuration files. They do not need to be under the RTEMS Source Builder source tree. Public projects are included in the main RTEMS Source Builder such as RTEMS.\n\nYou can add your own patches directory next to your config directory as the %patch command searches the _patchdir macro variable and it is by default set to %{_topdir}\/patches:%{_sbdir}\/patches.\n\nThe source-builder\/config directory provides generic scripts for building various tools. You can specialise these in your private configurations to make use of them. If you add new generic configurations please contribute them back to the project\n\nBuild sets can be controlled via the command line to enable (--with-<feature>) and disable (--without-<feature>) various features. There is no definitive list of build options that can be listed because they are implemented with the configuration scripts. The best way to find what is available is to grep the configuration files for with and without.\n\n## 14.2.1. Bare Metal\u00b6\n\nThe RSB contains a bare configuration tree and you can use this to add packages you use on the hosts. For example \u2018qemu\u2019 is supported on a range of hosts. RTEMS tools live in the rtems\/config directory tree. RTEMS packages include tools for use on your host computer as well as packages you can build and run on RTEMS.\n\nThe bare metal support for GNU Tool chains. An example is the lang\/gcc491 build set. You need to provide a target via the command line --target option and this is in the standard 2 or 3 tuple form. For example for an ARM compiler you would use arm-eabi or arm-eabihf, and for SPARC you would use sparc-elf:\n\n$cd rtems-source-builder\/bare$ ..\/source-builder\/sb-set-builder --log=log_arm_eabihf \\\n--prefix=\\$HOME\/development\/bare --target=arm-eabihf lang\/gcc491\nRTEMS Source Builder - Set Builder, v0.3.0\nBuild Set: lang\/gcc491\nconfig: devel\/expat-2.1.0-1.cfg\npackage: expat-2.1.0-x86_64-apple-darwin13.2.0-1\nbuilding: expat-2.1.0-x86_64-apple-darwin13.2.0-1\nconfig: devel\/binutils-2.24-1.cfg\npackage: arm-eabihf-binutils-2.24-1\nbuilding: arm-eabihf-binutils-2.24-1\nconfig: devel\/gcc-4.9.1-newlib-2.1.0-1.cfg\npackage: arm-eabihf-gcc-4.9.1-newlib-2.1.0-1\nbuilding: arm-eabihf-gcc-4.9.1-newlib-2.1.0-1\nconfig: devel\/gdb-7.7-1.cfg\npackage: arm-eabihf-gdb-7.7-1\nbuilding: arm-eabihf-gdb-7.7-1\ninstalling: expat-2.1.0-x86_64-apple-darwin13.2.0-1 -> \/Users\/chris\/development\/bare\ninstalling: arm-eabihf-binutils-2.24-1 -> \/Users\/chris\/development\/bare\ninstalling: arm-eabihf-gcc-4.9.1-newlib-2.1.0-1 -> \/Users\/chris\/development\/bare\ninstalling: arm-eabihf-gdb-7.7-1 -> \/Users\/chris\/development\/bare\ncleaning: expat-2.1.0-x86_64-apple-darwin13.2.0-1\ncleaning: arm-eabihf-binutils-2.24-1\ncleaning: arm-eabihf-gcc-4.9.1-newlib-2.1.0-1\ncleaning: arm-eabihf-gdb-7.7-1\n\n\n## 14.2.2. RTEMS\u00b6\n\nThe RTEMS Configurations are found in the rtems directory. The configurations are grouped by RTEMS version and a release normally only contains the configurations for that release.. In RTEMS the tools are specific to a specific version because of variations between Newlib and RTEMS. Restructuring in RTEMS and Newlib sometimes moves libc functionality between these two parts and this makes existing tools incompatible with RTEMS.\n\nRTEMS allows architectures to have different tool versions and patches. The large number of architectures RTEMS supports can make it difficult to get a common stable version of all the packages. An architecture may require a recent GCC because an existing bug has been fixed, however the more recent version may have a bug in other architecture. Architecture specific patches should only be appliaed when build the related architecture. A patch may fix a problem on one architecture however it could introduce a problem in another architecture. Limiting exposure limits any possible crosstalk between architectures.\n\nIf you have a configuation issue try adding the --dry-run option. This will run through all the configuration files and if any checks fail you will see this quickly rather than waiting for until the build fails a check.\n\nFollowing features can be enabled\/disabled via the command line for the RTEMS build sets:\n\n--without-cxx\n\nDo not build a C++ compiler.\n\n--with-ada\n\nAttempt to build an Ada compiler. You need a native GNAT installed.\n\n--with-fortran\n\nAttempt to build a Fortran compiler.\n\n--with-objc\n\nAttempt to build a C++ compiler.\n\nThe RSB provides build sets for some BSPs. These build sets will build:\n\n\u2022 Compiler, linker, debugger and RTEMS Tools.\n\n\u2022 RTEMS Kernel for the BSP\n\n\u2022 Optionally LibBSD if supported by the BSP.\n\n\u2022 Third party packages if supported by the BSP.\n\n## 14.2.3. Patches\u00b6\n\nPackages being built by the RSB need patches from time to time and the RSB supports patching upstream packages. The patches are held in a seperate directory called patches relative to the configuration directory you are building. For example %{_topdir}\/patches:%{_sbdir}\/patches. Patches are declared in the configuration files in a similar manner to the package\u2019s source so please refer to the %source documentation. Patches, like the source, are to be made publically available for configurations that live in the RSB package and are downloaded on demand.\n\nIf a package has a patch management tool it is recommended you reference the package\u2019s patch management tools directly. If the RSB does not support the specific patch manage tool please contact the mailing list to see if support can be added.\n\nReferenced patches should be placed in a location that is easy to access and download with a stable URL. We recommend attaching a patch to an RTEMS ticket in it\u2019s bug reporting system or posting to a mailing list with online archives.\n\nRTEMS\u2019s former practice of placing patches in the RTEMS Tools Git repository has been stopped.\n\nPatches are added to a component\u2019s name and in the %prep: section the patches can be set up, meaning they are applied to source. The patches are applied in the order they are added. If there is a dependency make sure you order the patches correctly when you add them. You can add any number of patches and the RSB will handle them efficiently.\n\nPatches can have options. These are added before the patch URL. If no options are provided the patch\u2019s setup default options are used.\n\nPatches can be declared in build set up files.\n\nThis examples shows how to declare a patch for gdb in the lm32 architecture:\n\n%patch add <1> gdb <2> %{rtems_gdb_patches}\/lm32\/gdb-sim-lm32uart.diff <3>\n\n\nItems:\n\n1. The patch\u2019s add command.\n\n2. The group of patches this patch belongs too.\n\nPatches require a checksum to avoid a warning. The %hash directive can be used to add a checksum for a patch that is used to verify the patch:\n\n%hash sha512 <1> gdb-sim-lm32uart.diff <2> 77d07087 ... e7db17fb <3>\n\n\nItems:\n\n1. The type of checksum, in the case an SHA512 hash.\n\n2. The patch file the checksum is for.\n\n3. The SHA512 hash.\n\nThe patches are applied when a patch setup command is issued in the %prep: section. All patches in the group are applied. To apply the GDB patch above use:\n\n%patch setup <1> gdb <2> -p1 <3>\n\n\nItems:\n\n1. The patch\u2019s setup command.\n\n2. The group of patches to apply.\n\n3. The patch group\u2019s default options. If no option is given with the patch these options are used.\n\nArchitecture specific patches live in the architecture build set file isolating the patch to that specific architecture. If a patch is common to a tool it resides in the RTEMS tools configuration file. Do not place patches for tools in the source-builder\/config template configuration files.\n\nTo test a patch simply copy it to your local patches directory. The RSB will see the patch is present and will not attempt to download it. Once you are happy with the patch submit it to the project and a core developer will review it and add it to the RTEMS Tools git repository.\n\n### 14.2.3.1. Testing a Newlib Patch\u00b6\n\nTo test a local patch for newlib, you need to add the following two lines to the .cfg file in rsb\/rtems\/config\/tools\/ that is included by the bset you use:\n\nSteps:\n\n1. Create patches for the changes you want to test. (Note: For RSB, before creating Newlib patch, you must run autoreconf -fvi in the required directory after you make changes to the code. This is not required when you create patch to send to newlib-devel. But if you want RSB to address your changes, your patch should also include regenerated files.)\n\n2. Calculate sha512 of your patch.\n\n3. Place the patches in rsb\/rtems\/patches directory.\n\n4. Open the .bset file used by your BSP in rsb\/rtems\/config. For example, for rtems5, SPARC, the file will be rsb\/rtems\/config\/5\/rtems-sparc.bset.\n\n5. Inside it you will find the name of .cfg file for Newlib, used by your BSP. For example, I found tools\/rtems-gcc-7.4.0-newlib-1d35a003f.\n\n6. Edit your .cfg file. In my case it will be, rsb\/rtems\/config\/tools\/rtems-gcc-7.4.0-newlib-1d35a003f.cfg. And add the information about your patch as mentioned below.\n\n%patch add newlib -p1 file:\/\/0001-Port-ndbm.patch <1>\n\n1. The diff file prepended with file:\/\/ to tell RSB this is a local file.","date":"2020-09-29 05:10:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.27937233448028564, \"perplexity\": 4744.743393279638}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600401624636.80\/warc\/CC-MAIN-20200929025239-20200929055239-00259.warc.gz\"}"} | null | null |
namespace chromecast {
namespace media {
SwitchingMediaRenderer::SwitchingMediaRenderer(
scoped_ptr<::media::Renderer> default_renderer,
scoped_ptr<::media::Renderer> cma_renderer)
: default_renderer_(default_renderer.Pass()),
cma_renderer_(cma_renderer.Pass()) {
DCHECK(default_renderer_);
DCHECK(cma_renderer_);
}
SwitchingMediaRenderer::~SwitchingMediaRenderer() {
}
void SwitchingMediaRenderer::Initialize(
::media::DemuxerStreamProvider* demuxer_stream_provider,
const ::media::PipelineStatusCB& init_cb,
const ::media::StatisticsCB& statistics_cb,
const ::media::BufferingStateCB& buffering_state_cb,
const base::Closure& ended_cb,
const ::media::PipelineStatusCB& error_cb,
const base::Closure& waiting_for_decryption_key_cb) {
// At this point the DemuxerStreamProvider should be fully initialized, so we
// have enough information to decide which renderer to use.
demuxer_stream_provider_ = demuxer_stream_provider;
DCHECK(demuxer_stream_provider_);
::media::DemuxerStream* audio_stream =
demuxer_stream_provider_->GetStream(::media::DemuxerStream::AUDIO);
::media::DemuxerStream* video_stream =
demuxer_stream_provider_->GetStream(::media::DemuxerStream::VIDEO);
if (audio_stream && !video_stream) {
// If the CMA backend does not support the audio codec and there is no
// video stream, use the default renderer with software decoding.
// Currently this applies to FLAC and Opus only due to difficulties in
// correctly generating the codec string (eg for mp4a variants).
::media::AudioCodec codec = audio_stream->audio_decoder_config().codec();
bool flac_supported = (MediaCodecSupportShlib::IsSupported("flac") !=
MediaCodecSupportShlib::kNotSupported);
bool opus_supported = (MediaCodecSupportShlib::IsSupported("opus") !=
MediaCodecSupportShlib::kNotSupported);
if ((codec == ::media::kCodecFLAC && !flac_supported) ||
(codec == ::media::kCodecOpus && !opus_supported)) {
cma_renderer_.reset();
}
}
if (cma_renderer_) {
// If the CMA renderer was not reset above, then we will use it; the
// default renderer is not needed.
default_renderer_.reset();
}
return GetRenderer()->Initialize(
demuxer_stream_provider, init_cb, statistics_cb, buffering_state_cb,
ended_cb, error_cb, waiting_for_decryption_key_cb);
}
::media::Renderer* SwitchingMediaRenderer::GetRenderer() const {
DCHECK(default_renderer_ || cma_renderer_);
if (cma_renderer_)
return cma_renderer_.get();
DCHECK(default_renderer_);
return default_renderer_.get();
}
void SwitchingMediaRenderer::SetCdm(
::media::CdmContext* cdm_context,
const ::media::CdmAttachedCB& cdm_attached_cb) {
GetRenderer()->SetCdm(cdm_context, cdm_attached_cb);
}
void SwitchingMediaRenderer::Flush(const base::Closure& flush_cb) {
GetRenderer()->Flush(flush_cb);
}
void SwitchingMediaRenderer::StartPlayingFrom(base::TimeDelta time) {
GetRenderer()->StartPlayingFrom(time);
}
void SwitchingMediaRenderer::SetPlaybackRate(double playback_rate) {
GetRenderer()->SetPlaybackRate(playback_rate);
}
void SwitchingMediaRenderer::SetVolume(float volume) {
GetRenderer()->SetVolume(volume);
}
base::TimeDelta SwitchingMediaRenderer::GetMediaTime() {
return GetRenderer()->GetMediaTime();
}
bool SwitchingMediaRenderer::HasAudio() {
return GetRenderer()->HasAudio();
}
bool SwitchingMediaRenderer::HasVideo() {
return GetRenderer()->HasVideo();
}
} // namespace media
} // namespace chromecast
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,706 |
{"url":"https:\/\/jarrettbillingsley.github.io\/teaching\/classes\/cs0445\/slides\/09.html","text":"## Announcements\n\n\u2022 There\u2019s a quiz today!\n\n## Recap\n\n\u2022 Selection sort\n\u2022 Select min from unsorted part\n\u2022 Swap with first unsorted item\n\u2022 Repeat until whole array is sorted\n\u2022 Bubble sort\n\u2022 Find inversions and swap\n\u2022 Repeat until no more inversions are found\n\u2022 Insertion sort\n\u2022 Take first unsorted item\n\u2022 Insert into sorted part in correct order, sliding things over if needed\n\u2022 Repeat until whole array is sorted\n\u2022 Last time we couldn\u2019t get any better than $O(n^2)$ worst case for sorting\n\u2022 Selection and bubble sort were tied\n\u2022 Insertion sort could be $O(n)$ best case, but was still $O(n^2)$ worst case\n\u2022 Let\u2019s try to do better!\n\n## Divide-and-conquer sorting\n\n\u2022 A divide-and-conquer algorithm\u2026\n\u2022 Divides a problem into subproblems which are a multiplicative fraction of the original size\n\u2022 Combines the results of the subproblems into a solution for the larger problem\n\u2022 They\u2019re typically written recursively, as it\u2019s the most natural way to express this behavior\n\u2022 How could we divide an array? What\u2019s the most obvious solution?\n\u2022 Chop it in half!\n\u2022 And then sort each half, somehow?\n\u2022 And then combining those halves together\u2026\n\u2022 That\u2019s a bit less obvious\n\u2022 It depends on how we chopped the array in half\n\u2022 Let\u2019s try doing the subproblems with a simple sort, like selection sort\n\u2022 If we start with $\\{ 7, 5, 2, 1, 0, 3, 6, 4 \\}$\n\u2022 We divide to get $\\{ 7, 5, 2, 1 \\}$ and $\\{ 0, 3, 6, 4 \\}$\n\u2022 We sort each half to get $\\{ 1, 2, 5, 7 \\}$ and $\\{ 0, 3, 4, 6 \\}$\n\u2022 And now we have to combine the two halves\u2026\n\u2022 Think of it like cars merging onto a highway\n\u2022 Except the cars have numbers on them\n\u2022 And whoever has the smaller number goes first\n\u2022 So this \u201cmerging\u201d procedure looks like this:\n1. Look at the two values at the beginnings of the arrays.\n2. Remove the smaller one and put it at the end of a new \u201csorted\u201d array.\n3. Repeat until both arrays are empty.\n\n\u2022 Did we make anything better?\n\u2022 How long does this merging procedure take?\n\u2022 There\u2019s just a single loop with two constant time operations inside\u2026\n\u2022 So it\u2019s $O(n)$\n\u2022 How long did sorting each sub-array take?\n\u2022 We used selection sort, which is $O(n^2)$\n\u2022 So our procedure took $O(2n^2 + n) = O(n^2)$ time\n\u2022 GOD\n\u2022 DAMN IT\n\u2022 WHY AREN\u2019T THINGS GETTING BETTER??!?\n\n## Merge Sort\n\n\u2022 We just did most of merge sort!\n\u2022 Here\u2019s merge sort:\n1. If the array length is 0 or 1, it\u2019s sorted.\n2. Else:\n\u2022 split the array into two halves\n\u2022 recursively merge sort each half\n\u2022 merge the two halves back together using the procedure we just talked about\n\u2022 Wait, but how does this make things any better???\n\u2022 Well\u2026 it\u2019s a little tricky, but\u2026\n\u2022 Let\u2019s use a recursion tree diagram to see\n\u2022 mergesort({ 7, 5, 2, 1, 0, 3, 6, 4 }) recursively calls\u2026\n\u2022 mergesort({ 7, 5, 2, 1 }) which recursively calls\u2026\n\u2022 mergesort({ 7, 5 }) which recursively calls\u2026\n\u2022 mergesort({ 7 }) which is sorted.\n\u2022 mergesort({ 5 }) which is sorted.\n\u2022 mergesort({ 2, 1 }) which recursively calls\u2026\n\u2022 mergesort({ 2 }) which is sorted.\n\u2022 mergesort({ 1 }) which is sorted.\n\u2022 mergesort({ 0, 3, 6, 4 }) which recursively calls\u2026\n\u2022 mergesort({ 0, 3 }) which recursively calls\u2026\n\u2022 mergesort({ 0 }) which is sorted.\n\u2022 mergesort({ 3 }) which is sorted.\n\u2022 mergesort({ 6, 4 }) which recursively calls\u2026\n\u2022 mergesort({ 6 }) which is sorted.\n\u2022 mergesort({ 4 }) which is sorted.\n\u2022 When analyzing a \u201cbranching tree\u201d structure, it\u2019s best to look at it \u201cby levels.\u201d\n\u2022 So the first \u201clevel\u201d has 1 recursive call\n\u2022 The second \u201clevel\u201d has 2 recursive calls\n\u2022 The third has 4, then the fourth has 8\u2026\n\u2022 At each level, how many comparisons are done?\n\u2022 Comparisons are only done during merging, so the base cases can be ignored.\n\u2022 At the top level, we have to merge two n\/2-sized arrays\n\u2022 What would be the worst possible case there?\n\u2022 n comparisons - first array A, then B, then A, then B\u2026 all the way down\n\u2022 At the next level, we have to merge two n\/4-sized arrays, twice\n\u2022 so it\u2019s 2n\/4, twice\u2026 so again, n comparisons\n\u2022 At the third level, we have to merge four n\/4-sized arrays, 4 times\n\u2022 again, it\u2019s n comparisons!\n\u2022 But here\u2019s the kicker: how many levels are there?\n\u2022 There are $\\log(n)$ levels.\n\u2022 So it\u2019s not $O(n^2)$ anymore.\n\u2022 It\u2019s $O(n \\log n)$! \ud83c\udfba\ud83c\udf89\ud83c\udf8a\n\u2022 We did it!\n\u2022 We broke the $O(n^2)$ barrier\n\u2022 What does $O(n \\log n)$ look like?\n\u2022 Well, it grows faster than linear\u2026\n\u2022 \u2026but not as fast as quadratic.\n\u2022 We also call this linearithmic (it\u2019s a fun portmanteau)\n\u2022 It comes up a lot in sorting and tree algorithms so it\u2019s a useful term\n\u2022 For $n=100$, linear is $100$, linearithmic is $200$, quadratic is $10{,}000$\n\u2022 For $n=1{,}000$, linear is $1{,}000$, linearithmic is $3{,}000$, quadratic is $1{,}000{,}000$\n\u2022 Now why did this work while the earlier example didn\u2019t do better than $O(n^2)$?\n\u2022 Cause before, we didn\u2019t keep splitting the problem up.\n\u2022 We just made the problem size $\\frac{n}{2}$.\n\u2022 It\u2019s the recursive splitting that\u2019s the secret.\n\u2022 However\u2026\n\u2022 One of the big downsides of mergesort is that we need $O(n)$ additional space.\n\u2022 Or in English, to sort an array of n items, we need to allocate a second array of n items.\n\u2022 So if you don\u2019t have memory to spare\u2026\n\n## Quick Sort\n\n\u2022 Despite its name, it\u2019s not really any \u201cquicker\u201d than mergesort\u2026\n\u2022 Here\u2019s quicksort:\n1. If the array length is 0 or 1, it\u2019s sorted.\n2. Else:\n\u2022 pick a value from the array. This is the \u201cpivot\u201d.\n\u2022 partition the array into two halves: everything less than the pivot and everything greater than the pivot.\n\u2022 Now we know where the pivot goes, so put the pivot there.\n\u2022 recursively quicksort each half of the array.\n\u2022 It feels almost like binary search, but backwards!\n\u2022 We could be lazy and just allocate new arrays for the partitions.\n\u2022 But unlike mergesort, quicksort can be performed without using any extra space!\n\u2022 Here\u2019s the partitioning algorithm:\n1. look at the last value in the array (at length - 1). that is your \u201cpivot.\u201d\n\n2. have two \u201cfingers\u201d, one at each end of the array (at 0 and length - 2). then in a loop:\n1. move the left one right until you find something >= pivot (or you cross the right).\n2. move the right one left until you find something < pivot (or you cross the left).\n3. if they cross (left finger > right finger), then break.\n\u2022 the > is super important here. If you use >=, it doesn\u2019t work!\n4. swap the values at the left and right fingers, and move them inwards by 1.\n\n3. swap the pivot (at length - 1) with the thing at the \u201cleft\u201d finger (value larger than pivot):\n\n\u2022 Now everything to the left of the pivot is less than it,\n\u2022 and everything to its right is greater than or equal to it.\n\u2022 hey, this feels like binary search again!\n\u2022 and to sort, we just recursively partition the left and right sides.\n\u2022 To analyze it, let\u2019s consider the best and worst cases.\n\u2022 How efficient the partitioning is really has to do with one important decision:\n\u2022 What pivot value do we use?\n\u2022 Above, we used a simple method: use the last value in the array.\n\u2022 But what if the last value in the array happens to be the biggest?\n\u2022 Then how many things would be to the left?\n\u2022 All of them!\n\u2022 And how many to the right?\n\u2022 None of them!\n\u2022 And we\u2019d repeat the process on the left side\u2026\n\u2022 and maybe its last value is also the biggest\u2026\n\u2022 and so on and so on\u2026\n\u2022 So the worst case is an array that is already sorted.\n\u2022 So each time, we have to look at n values, but each recursion we\u2019re only making it smaller by 1.\n\u2022 If we look at the recursion tree, it\u2019s not much of a tree at all.\n\u2022 More of a recursion linked list\u2026\n\u2022 Since there are n levels, and n steps on each level, it has a worst case performance of $O(n^2)$.\n\u2022 Unlike mergesort, which is always $O(n \\log n)$.\n\u2022 What is the best case? That is, when will we get the same number of values on either side of the pivot?\n\u2022 When the pivot is the median.\n\u2022 That\u2019s \u201cthe value with equal numbers of values on either side.\u201d\n\u2022 If we keep picking the median, then the array keeps getting split evenly\u2026\n\u2022 And just like mergesort, we end up with $\\log n$ levels, so $O(n \\log n)$.\n\u2022 Picking the pivot is an important part of implementing quicksort.\n\u2022 If you always pick the first, the last, the nth\u2026\n\u2022 Then there will always be pathological cases (arrays that take $O(n^2)$ time).\n\u2022 You could try picking a random pivot\n\u2022 But then you can\u2019t really predict the performance.\n\u2022 Maybe you get lucky every time! Maybe you get unlucky every time!\n\u2022 We can get a sample of values and pick the median from those\n\u2022 A common technique is median-of-three\n\u2022 You pick the first, last, and middle items, and whichever is in the middle of the other two becomes your pivot\n\u2022 It greatly reduces the chances of getting a pathological case, but\u2026\n\u2022 It is possible to have them\n\u2022 Consider an array where every item is the same!\n\u2022 Or we can sample the whole array and get the real median\n\u2022 It can be found in $O(n)$ time, so that we always get the \u201cbest\u201d split\n\u2022 And therefore it\u2019s $O(n \\log n)$ in all cases!\n\u2022 But in practice\u2026 it ends up not being worth it.\n\n## Stability\n\n\u2022 One important property of sorting algorithms is stability.\n\u2022 If you have two equal items in the input array\u2026\n\u2022 Let\u2019s call them i and j, where i comes before j in the input array\n\u2022 In a stable sort, it will keep i and j in the same order as in the input array.\n\u2022 An unstable sort might swap them so that j comes before i in the output array.\n\u2022 Why on earth is this important??\n\u2022 They\u2019re equal, right? Who cares?\n\u2022 Well it doesn\u2019t matter much for numbers. But for other things\u2026\n\u2022 Consider a spreadsheet of users.\n\u2022 Each has a first (given) name, last (family) name, user ID, email, and department\n\u2022 Let\u2019s say I want to sort them so that they are sorted by department, and then within each department, they are sorted by last name.\n\n\u2022 We work backwards: first sort by last name. (This does not have to be done stably.)\n\n\u2022 Then, stably sort on the department.\n\u2022 Notice that the two people in CS will not swap places. Their names will remain in alphabetical order. (Same with the Business people.)\n\n\u2022 Do we need stability?\n\u2022 I\u2019m kinda talking out of my butt here, but\n\u2022 Honestly? For the above problem? I\u2019d just use a comparator (compareTo() method) that compares both the name and the department at the same time, rather than doing 2 sorts.\n\u2022 That would work even if we used an unstable sort.\n\u2022 I\u2019m sure I\u2019m wrong about something.\n\u2022 I\u2019m like 80% sure.\n\n## The Best of Both Worlds\n\n\u2022 Remember what the graphs of log n, n, and n^2 look like?\n\u2022 What do you notice about the values of these functions near n = 0?\n\u2022 They\u2019re all about the same\u2026 and in fact, n^2 is smaller in some cases!\n\u2022 Quicksort and mergesort work great for large values of n\n\u2022 But their performance for small arrays is not really significantly better than the simple sorts\n\u2022 And can be worse in some cases!\n\u2022 So instead of using one sorting algorithm\u2026\n\u2022 We can use two.\n\u2022 We modify the recursive versions of quicksort or mergesort by adding this condition:\n\u2022 If the array size < k, perform an insertion sort.\n\u2022 (or selection sort, or bubble sort, whatever you want)\n\u2022 Else, proceed as normal.\n\u2022 k is some arbitrary constant.\n\u2022 It decides when we \u201cswitch over\u201d from the divide-and-conquer sort to the simpler sort.\n\u2022 How we pick k is\u2026 kind of throwing stuff at the wall and seeing what sticks.\n\u2022 It\u2019s usually on the order of magnitude of 10 to 100.\n\u2022 Yep, even for 100 items, an $O(n^2)$ sort is more than fast enough in most cases!\n\n\u2022 One last sort, and we\u2019re not gonna go too deeply into it cause it\u2019s kind of scary to analyze\n\u2022 But I want you to have a peek of \u201canother kind\u201d of sort.\n\u2022 All the algorithms we talked about so far are comparison-based algorithms.\n\u2022 It has been proven that these kinds of algorithms can never do better than $O(n \\log n)$ in all cases.\n\u2022 We saw cases where they could be as fast as $O(n)$ in the best case!\n\u2022 But not in all cases.\n\u2022 However\u2026\n\u2022 If we know something about the data to be sorted\u2026\n\u2022 We can take advantage of that information to do things in a faster way.\n\u2022 Radix Sort is a non-comparison sort.\n\u2022 You have probably done a radix sort yourself without knowing it.\n\u2022 If I were to give you 300 cards with a bunch of different words on them, all shuffled\u2026\n\u2022 You might go \u201cwell, this is too much to sort at once.\u201d\n\u2022 \u201cI\u2019m going to put all the \u2018a\u2019 words in one pile, and all the \u2018b\u2019 words in another, and\u2026\u201c\n\u2022 This is the idea behind radix sort.\n\u2022 If you have data that can be represented as some sort of string\u2026\n\u2022 Where each position in the string can have a small number of possible values\u2026\n\u2022 Then you can do a radix sort.\n\u2022 Fortunately, many common cases fit this criteria!\n\u2022 But this is how we get the better performance: we lose generality.\n\u2022 We can\u2019t apply this sort to ALL kinds of data.\n\u2022 There are two important variables:\n\u2022 The maximum length (k) of the values to be sorted\n\u2022 For integers, that would be how many digits the biggest number has.\n\u2022 The number of possibilities (d) for each position in the value.\n\u2022 For integers, each place can be one of 10 digits (0 through 9).\n\u2022 Here\u2019s the idea:\n\u2022 Create d buckets.\n\u2022 These are probably just arrays.\n\u2022 For i = 0 to k:\n\u2022 For j = 0 to n:\n1. Look at the digit at position i.\n2. Place it in the matching bucket.\n\u2022 Now all the values have been sorted into buckets.\n\u2022 Next, take the values out of the buckets, in order, and put them back into the original array.\n\u2022 It doesn\u2019t seem obvious that this works, but\u2026\n\u2022 If we\u2019re sorting 3 digit numbers, the first pass makes sure the rightmost digits are in order.\n\u2022 Then the second pass makes sure the middle digits are in order, and the rightmost ones will maintain their order. (It\u2019s a stable sort!)\n\u2022 Then the last pass makes sure the first digits are in order, and the numbers are sorted!\n\u2022 If we look at the loops, it\u2019s like\u2026 $O(kn)$.\n\u2022 It\u2019s not technically linear.\n\u2022 But for most practical cases, k is usually pretty small.\n\u2022 So it\u2019s almost like a constant\u2026 meaning we can sort in $O(n)$ in all cases.","date":"2019-01-17 06:54:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 39, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.44458234310150146, \"perplexity\": 1481.5722794231574}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-04\/segments\/1547583658844.27\/warc\/CC-MAIN-20190117062012-20190117084012-00532.warc.gz\"}"} | null | null |
Q: Confusing NODE_MODULE error I am using the greenworks SDK. I don't understand this error message. I can't tell if it's telling me to recompile the greenworks-osx64.node or change my system node settings.
ELECTRON_ASAR.js:173 Uncaught Error: The module '/Users/quantum/ele/electron-quick-start/lib/greenworks-osx64.node'
was compiled against a different Node.js version using
NODE_MODULE_VERSION 57. This version of Node.js requires
NODE_MODULE_VERSION 53. Please try re-compiling or re-installing
the module (for instance, using `npm rebuild` or`npm install`).
at process.module.(anonymous function) [as dlopen] (ELECTRON_ASAR.js:173:20)
at Object.Module._extensions..node (module.js:598:18)
at Object.module.(anonymous function) [as .node] (ELECTRON_ASAR.js:173:20)
at Module.load (module.js:488:32)
at tryModuleLoad (module.js:447:12)
at Function.Module._load (module.js:439:3)
at Module.require (module.js:498:17)
at require (internal/module.js:20:19)
at Object.<anonymous> (/Users/quantum/ele/electron-quick-start/greenworks.js:12:18)
at Object.<anonymous> (/Users/quantum/ele/electron-quick-start/greenworks.js:133:3)
I am using electron ~1.6.2 and node -v v8.6.0
I missed there were other releases further down the page. I installed the one for electron 1.6.1. Now my code runs this:
var greenworks = require('./greenworks'); var f = greenworks.initAPI(); document.write(f); console.log(f);
"false" logged into the console.
A:
The native Node modules are supported by Electron, but since Electron is very likely to use a different V8 version from the Node binary installed in your system, you have to manually specify the location of Electron's headers when building native modules.
Source: here
| {
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Clanci Jo
Each playlist I create is intentional in their tracklist, and are constantly being expanded upon. Some collections are inspired by various genres and subgenres, while others aim to convey a certain mood and feature songs from a myriad of backgrounds.
To listen to whole playlists on Spotify, click the icon in the upper right corner of each playlist.
tuyo [a summer playlist]
tuyo (yours) is my international playlist, featuring artists from across the world that all convey a sense of place, a soulful connection to the ground that they walk upon.
Features: Buena Vista Social Club, Djelimady Tounkara, Rising Appalachia, Lura +++
remmuS [a summer playlist]
I started this playlist in 2014 after I had finished my Freshman year of undergrad and have been adding summer tunes to it since. It should be listened to in order, but can be shuffled if you really want to.
Features: Sugar Ray, Grouplove, Dirty Heads, Grateful Dead, The Cars, Lafa Taylor +++
S/H/I\V/A\ [a summer playlist]
Named for the Hindu god Shiva, this playlist is all reggae.
Features: Toots & the Maytals, Michael Franti, Bigga Hatian, Slightly Stoopid, most of the Marley family +++
This constantly changing playlist is a breezeway for indecisive songs that require multiple listens before being added to a permanent playlist, updated weekly.
Currently features: Tokimonsta, Manatee Commune, Erykah Badu, Phony Ppl, G Eazy +++
Nothing But a Gangster Party
Golden age rap from the 90's & early 2000's.
Features: 2Pac, Nas, Juvenile, R. Kelly, Jay Z +++
Gutterfly
Rap like butterflies trapped in the gutter.
Features: Taylor Bennet, Kooley High, The Grouch, Sampa the Great, Jay Electronica +++
Marmalade Fade
These tracks should remind you of porch swings, dancing in the living room with your love, and forgetting your umbrella on a rainy day.
Features: Tegan & Sara, The Beatles, Neutral Milk Hotel, The Mamas & The Papas, Beck +++
This playlist title is taken from Alina Baraz's song Fantasy. The song also served as an inspiration for this series. There is no one genre that goes with Private Island, but it has a definite chill vibe and is a reflection of my sound.
Features: Snakehips, Doja Cat, Kanye West, Chet Faker, Tory Lanez +++
Private Island Vol. II
Volume I's older sibling.
Features: Blackbear, FKJ, Swell, Kultur, Blended Babies +++
Bad News, Class, a Piano, and Some Brass
Piano and brass go together like chips and salsa. Ella Fitzgerald is the centerpiece for this collection…like me, she is from Newport News, VA, or "Bad News," hence the name of the playlist.
Features: Nat King Cole, the Ink Spots, Etta James, Billie Holiday, Frank Sinatra +++
Her ***** Like Me (Her Heart Like Fck It)
Urban/Experimental Rap, Hip-Hop, and R&B. The title comes from Chance the Rapper's song, Lost, which features Noname Gypsy (ironically, the song isn't on Spotify).
Features: Donnie Trumpet, Hiatus Kaiyote, Blu, Mick Jenkins, Matthias, ProbCause +++
M0t1vVt10n
A playlist for when you need to get motivated.
Features: Bryson Tiller, Future, J. Cole, Kodak Black, Travis Scott +++ | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,115 |
\section{Introduction}
The contractive (or unitary) perturbations $U+K$ of a unitary operator $U$ on a Hilbert space $H$ by finite rank $d<\infty$ operators $K$ with fixed range are parametrized by the $(d\times d)$ contractive (resp.~unitary) matrices $\Gamma$. Namely, if $\Ran K\subset {\mathfrak{R}}$, where ${\mathfrak{R}}\subset H$, $\dim {\mathfrak{R}}=d$ is fixed, and $\mathbf{B}:{\mathbb C}^d\to {\mathfrak{R}}$ is a fixed unitary operator (which we call the coordinate operator), then $K$ is represented as $K= \mathbf{B}(\Gamma-\mathbf{I}\ci{{\mathbb C}^d}) \mathbf{B}^*U$ where $\Gamma$ is a contraction (resp.~a unitary operator) on ${\mathbb C}^d$. Therefore, all such perturbations with $\Ran K\subset {\mathfrak{R}}$ are represented as $T\ci{\Gamma}= U+ \mathbf{B}(\Gamma-\mathbf{I}\ci{{\mathbb C}^d}) \mathbf{B}^*U$, where $\Gamma $ runs over all $(d\times d)$ contractive (resp.~unitary) matrices.
Recall that $T$ being a \emph{contraction} (contractive) means that $\|T\|\le 1$.
Focusing on the non-trivial part of the perturbation, we can assume that $\Ran \mathbf{B} = {\mathfrak{R}}$ is a star-cyclic subspace for $U$, i.e.~$H = \overline{\spa}\{U^k {\mathfrak{R}}, (U^*)^k{\mathfrak{R}}: k\in{\mathbb Z}_+\}.$ Below we will show that star-cyclicity together with the assumption that $\Gamma$ is a pure contraction ensures that the operator $T\ci{\Gamma}$ is what is called a completely non-unitary contraction, meaning that $T\ci \Gamma$ does not have a non-trivial unitary part. The model theory informs us that such $T\ci{\Gamma}$ is unitarily equivalent to its functional model $\mathcal{M}_{\theta}$, $\theta=\theta\ci\Gamma$, that is, the compression of the shift operator on the model space ${\mathcal K}_{\theta}$ with the characteristic function $\theta=\theta\ci\Gamma$ of $T\ci{\Gamma}$.
In this paper we investigate
the so-called Clark operator, i.e.~a unitary operator $\Phi$ that intertwines the contraction $T\ci{\Gamma}$ (in the spectral representation of the unperturbed operator $U$) with its model: $\mathcal{M}_\theta\Phi = \Phi T\ci{\Gamma}$, $\theta=\theta\ci\Gamma$. The case of rank one perturbations ($d=1$) was treated by D.~Clark when $\theta$ is inner \cite{Clark}, and later by D.~Sarason under the assumption that $\theta$ is an extreme point of the unit ball of $H^\infty$, \cite{SAR}. For finite rank perturbations with inner characteristic matrix-valued functions $\theta$, V.~Kapustin and A.~Poltoratski \cite{KP06} studied boundary convergence of functions in the model space ${\mathcal K}_\theta$. The setting of inner characteristic function corresponds to the operators $U$ that have purely singular spectrum (no a.c.~component), see e.g.~\cite{DL2013}.
In \cite{LT15} we completely described the general case of rank one perturbations (when the measure can have absolutely continuous part, or equivalently, the characteristic function is not not necessarily inner).
In the present paper we extend the results from \cite{LT15} to finite rank perturbations with general matrix-valued characteristic functions. We first find a universal representation of the adjoint Clark operator, which features a special case of a matrix-valued Cauchy integral operator. By universal we mean that our formula is valid in any transcription of the functional model.
This representation is a pretty straightforward, albeit more algebraically involved, generalization of the corresponding result from \cite{LT15}; it might look like an ``abstract nonsense'', since it is proved under the assumption that we picked a model operator that ``agrees'' with the Clark model (more precisely that the corresponding coordinate/parametrizing operators agree).
However, by careful investigation of the construction of the functional model, using the coordinate free Nikolski--Vasyunin model we were able to present a formula giving the parametrizing operators for the model that agree with given coordinate operators for a general contraction $T$, see Lemma \ref{l:C-C*}.
Moreover, for the Sz.-Nagy--Foia\c s transcription of the model we get explicit formulas for the parametrizing operators in terms of the characteristic function, see Lemma \ref{l:C_N-F}; similar formulas can be obtained for other transcriptions of the model.
We also compute the characteristic function of the perturbed operator $T\ci \Gamma$; the formula involves the Cauchy integral of the matrix-valued measure.
For the Sz.-Nagy--Foia\c s transcription of the model we give a more concrete representation of the adjoint Clark operator in terms of vector-valued Cauchy transform, see Theorem \ref{t-repr3}. This representation looks more natural when one considers spectral representations of the non-perturbed operator $U$ defined with the help of matrix-valued measures, see Theorem \ref{t:repr04}.
\subsection{Plan of the paper}
In Section \ref{s-prelims} we set the stage by introducing finite rank perturbations and studying some their basic properties. In particular, we discuss the concept of a star-cyclic subspace and find a measure-theoretic characterization for it.
Main result of Section \ref{s-adjClark} is the universal representation formula for the adjoint Clark operator, see Theorem \ref{t-repr}. In this section we also introduce the notion of agreement of the coordinate/parametrizing operators and make some preliminary observations about such an agreement.
Section \ref{s-ModAgree} is devoted to the detailed investigation of the agreement of the coordinate/parametrizing operators. Careful analysis of the construction of the model from the coordinate free point of view of Nikolski--Vasyunin allows us to get for a general contraction $T$ formulas for the parametrizing operators for the model that agree with the coordinate operators, see Lemma \ref{l:C-C*}.
Explicit formulas (in terms of the characteristic function) are presented for the case of Sz.-Nagy--Foia\c s transcription, see Lemma \ref{l:C_N-F}.
The characteristic function $\theta\ci\Gamma$ of the perturbed operator $T\ci \Gamma$ is the topic of Sections \ref{s-charfunc} and \ref{s:PropCharFunct}. Theorem \ref{t-theta} gives a formula for $\theta\ci\Gamma$ in terms of a Cauchy integral of a matrix-valued measure. In Section \ref{s:PropCharFunct} we show that, similarly to the rank one case, the characteristic functions $\theta\ci\Gamma$ and $\theta\ci{{\mathbf{0}}}$ are related via a special linear fractional transformation. Relations between defect functions $\Delta\ci{\mathbf{0}}$ and $\Delta\ci\Gamma$ are also described.
Section \ref{s-Explanations} contains a brief heuristic overview of what subtle techniques are to come in Sections \ref{s-SIO} and \ref{ss-PhiStarSNF}.
In Section \ref{s-SIO} we present results about regularizations of the Cauchy transform, and about uniform boundedness of such generalizations, that we need to get the representation formulas in Section \ref{ss-PhiStarSNF}.
In Section \ref{ss-PhiStarSNF} we give a formula for the adjoint Clark operator in the Sz.-Nagy--Foia\c s transcription of the model. As in the scalar case the adjoint Clark operator is given by the sum of two terms: one is in essence a vector-valued Cauchy transform (postmultiplied by a matrix-valued function
, and the second one is just a multiplication operator by a matrix-valued function, see Theorem \ref{t-repr3}. In the case of inner characteristic function (purely singular spectral measure of $U$) the second term disappears, and the adjoint Clark operator is given by what can be considered a matrix-valued analogue of the scalar \emph{normalized Cauchy transform}, see Section \ref{ss:GCT}.
Section \ref{s:directClark} is devoted to a description of the Clark operator $\Phi$, see Theorem \ref{t:direct Clark}.
\section{Preliminaries}\label{s-prelims}
Consider the family of rank $d$ perturbations $U+K$ of a unitary operator $U$ on a separable Hilbert space $H$. If we fix a subspace ${\mathfrak{R}}\subset H$, $\dim {\mathfrak{R}}=d$ such that $\Ran K\subset {\mathfrak{R}}$, then all unitary perturbations of $U+K$ of $U$ can be parametrized as
\begin{align}
\label{pert-01}
T= U + ({X}-\mathbf{I}\ci {\mathfrak{R}}) P\ci {\mathfrak{R}} U,
\end{align}
where ${X}$ runs over all possible unitary operators in ${\mathfrak{R}}$.
It is more convenient to factorize the representation of ${X}$ through the fixed space ${\mathfrak D}:={\mathbb C}^d$ by picking an isometric operator ${\mathbf{B}}:\varphi
\to H$, $\Ran \mathbf{B} = {\mathfrak{R}}$.
Then any ${X}$ in \eqref{pert-01} can be represented as ${X}=\mathbf{B} \Gamma \mathbf{B}^*$ where $\Gamma: {\mathfrak D}\to {\mathfrak D}$ (i.e. $\Gamma$ is a $(d\times d)$ matrix). The perturbed operator $T=T\ci\Gamma$ can be rewritten as
\begin{align}
\label{pert-02}
T= U+ \mathbf{B}(\Gamma-\mathbf{I}\ci{{\mathfrak D}}) \mathbf{B}^*U.
\end{align}
If we decompose the space $H$ treated as the domain as $H= U^*{\mathfrak{R}}\oplus (U^*{\mathfrak{R}})^\perp$, and the same space treated as the target space as $H={\mathfrak{R}}\oplus {\mathfrak{R}}^\perp$, then the operator $T$ can be represented with respect to this decomposition as
\begin{align}
\label{BlDec-01}
T= \left( \begin{array}{cc} \mathbf{B}\Gamma \mathbf{B}^*U & 0\\ 0 & T_1 \end{array} \right) ,
\end{align}
where block $T_1$ is unitary.
From the above decomposition we can immediately see that if $\Gamma$ is a contraction then $T$ is a contraction (and if $\Gamma$ is unitary then $T$ is unitary).
In this formula we slightly abuse notation, since formally the operator $\mathbf{B}\Gamma \mathbf{B}^*U$ is defined on the whole space $H$. However, this operator clearly annihilates $ (U^*{\mathfrak{R}})^\perp$, and its range belongs to ${\mathfrak{R}}$, so we can restrict its domain and target space to $U^*{\mathfrak{R}}$ and ${\mathfrak{R}}$ respectively. So when such operators appear in the block decomposition we will assume that its domain and target space are restricted.
In this paper we assume that the isometry $\mathbf{B}$ is fixed and that all the perturbations are parametrized by the $(d\times d)$ matrix $\Gamma$.
\subsection{Spectral representation of \texorpdfstring{$U$}{U}}
By the Spectral Theorem the operator $U$ is unitarily equivalent to the multiplication $M_\xi$ by the independent variable $\xi$ in the von Neumann direct integral
\begin{align}
\label{DirInt}
\mathcal{H}= \int_\mathbb{T}^\oplus E(\xi) {\mathrm{d}} \mu(\xi),
\end{align}
where $\mu$ is a finite Borel measure on $\mathbb{T}$ (without loss of generality we can assume that $\mu$ is a probability measure, $\mu(\mathbb{T}) =1$).
Let us recall the construction of the direct integral; we present not the most general one, but one that is sufficient for our purposes. Let $E$ be a separable Hilbert space with an orthonormal basis $\{e_n\}_{n=1}^\infty$, and let $N:\mathbb{T}\to {\mathbb N}\cup\{\infty\}$ be a measurable function (the so-called \emph{dimension function}). Define
\[
E(\xi) = \cspa\{ e_n\in E:1\le n \le N(\xi)\}.
\]
Then the direct integral $\mathcal{H}$ is the subspace of the $E$-valued space $L^2(\mu;E)=L^2(\mathbb{T}, \mu;E)$ consisting of the functions $f$ such that $f(\xi) \in E(\xi)$ for $\mu$-a.e.~$\xi$.
Note, that the dimension function $N$ and the spectral type $[\mu]$ of $\mu$ (i.e.~the collection of all measures that are mutually absolutely continuous with $\mu$) are spectral invariants of $U$, meaning that they define operator $U$ up to unitary equivalence.
So, without loss of generality, we assume that $U$ is the multiplication $M_\xi$ by the independent variable $\xi$ in the direct integral \eqref{DirInt}.
An important particular case is the case when $U$ is star-cyclic, meaning that there exists a vector $h\in H$ such that $\overline\spa\{U^n h: n\in{\mathbb Z}\}=H$. In this case $N(\xi)\equiv 1$, and the operator $U$ is unitary equivalent to the multiplication operator $M_\xi$ in the scalar space $L^2(\mu)=L^2(\mathbb{T},\mu)$.
In the representation of $U$ in the direct integral it is convenient to give a ``matrix'' representation of the isometry $\mathbf{B}$. Namely, for $k=1, 2, \ldots, d$ define functions $b_k\in\mathcal{H} \subset L^2(\mu;E)$ by $b_k:= \mathbf{B} e_k $; here $\{e_k\}_{k=1}^d$ is the standard orthonormal basis in ${\mathbb C}^d$.
In this notation the operator $\mathbf{B}$, if we follow the standard rules of the linear algebra is the multiplication by a row $B$ of vector-valued functions,
\[
B(\xi) = (b_1(\xi), b_2(\xi), \ldots, b_d(\xi)).
\]
If we represent $b_k(\xi)$ in the standard basis in $E$ that we used to construct the direct integral \eqref{DirInt}, then $\mathbf{B}$ is just the multiplication by the matrix-valued function of size $(\dim E)\times d$.
\subsection{Star-cyclic subspaces and completely non-unitary contractions} \label{ss-cyccnu}
\begin{defn}\label{d-cyclic}
A subspace ${\mathfrak{R}}$ is said to be \emph{star-cyclic} for an operator $T$ on $H$, if
\[
H = \overline{\spa}\{T^k {\mathfrak{R}}, (T^*)^k{\mathfrak{R}}: k\in{\mathbb Z}_+\}.
\]
\end{defn}
For a perturbation (not necessarily unitary) $T=T\ci\Gamma$ of the unitary operator $U$ given by \eqref{pert-02}
the subspace
\begin{align}
\label{SpanOrbit-01}
\mathcal{E}=\cspa\{ U^k {\mathfrak{R}}, (U^*)^k{\mathfrak{R}}: k\in{\mathbb Z}_+\} = \cspa\{ U^k{\mathfrak{R}}: k\in{\mathbb Z}\}
\end{align}
is a reducing subspace for both $U$ and $T\ci\Gamma$ (i.e.~$\mathcal{E}$ and $\mathcal{E}^\perp$ are invariant for both $U$ and $T\ci\Gamma$).
Since $T\ci \Gamma \bigm|_{\mathcal{E}^\perp}=U \bigm|_{\mathcal{E}^\perp}$, the perturbation does not influence the action of $T\ci \Gamma$ on $\mathcal{E}^\perp$, so nothing interesting for perturbation theory happens on $\mathcal{E}^\perp$; all action happens on $\mathcal{E}$. Therefore, we can restrict our attention to $T\ci \Gamma \bigm|_{\mathcal{E}}$, i.e.~assume without loss of generality that $\mathfrak{R} = \Ran \mathbf{B}$ is a star-cyclic subspace for $U$.
We note that if ${\mathfrak{R}}$ is a star-cyclic subspace for $U$ and $\Gamma$ is unitary, then ${\mathfrak{R}}$ is also a star-cyclic subspace for all perturbed unitary operators given by \eqref{pert-02}.
\begin{lem}
\label{l:*cycl}
Let ${\mathfrak{R}}=\Ran \mathbf{B}$ be a star-cyclic subspace for $U$ and let $\Gamma$ be unitary. Then ${\mathfrak{R}}$ is also a star-cyclic subspace for all perturbed unitary operators $U\ci\Gamma=T\ci{\Gamma}$ given by \eqref{pert-02}.
\end{lem}
We postpone for a moment a proof of this well-known fact.
\begin{defn}
\label{d:cnu} A contraction $T$ in a Hilbert space $H$ is called \emph{completely non-unitary} (c.n.u.~for short) if there is no non-zero reducing subspace on which $T$ acts unitarily.
\end{defn}
Recall that a contraction is called \emph{strict} if $\|Tx\|<\|x\|$ for all $x\ne\mathbf{0}$.
\begin{lem}\label{l-cnu}
If ${\mathfrak{R}}=\Ran \mathbf{B}$ is a star-cyclic subspace for $U$ and $\Gamma$ is a strict contraction, then $T$ defined by \eqref{pert-02} is a c.n.u.~contraction.
\end{lem}
\begin{proof}
Since $\Gamma$ is a strict contraction,
we get that ${\mathbf{B}}\Gamma {\mathbf{B}}^*U|\ci{U^*{\mathfrak{R}}}$ is also a strict contraction. Therefore \eqref{BlDec-01} implies that
\begin{align*}
\|Tx\| = \|x\| \quad&\Longleftrightarrow\quad x\perp U^{-1} {\mathfrak{R}}\\
\|T^*x\| = \|x\| \quad&\Longleftrightarrow\quad x\perp {\mathfrak{R}}.
\end{align*}
Moreover, we can see from \eqref{BlDec-01} that if $x\perp U^{-1}{\mathfrak{R}}$ then $Tx= Uf$ and if $x\perp {\mathfrak{R}}$ then $T^*x = U^{-1}x$.
Consider a reducing subspace $G$ for $T$ such that $T|\ci{G}$ is unitary. Then the above observations imply $G\perp {\mathfrak{R}}$ and $G\perp U^{-1}{\mathfrak{R}}$, and that
for any $x\in G$
\begin{align*}
T^n x = U^n x \qquad\text{as well as}\qquad
\left(T^*\right)^n x = U^{-n}x.
\end{align*}
Since $G$ is a reducing subspace for $T$ it follows that $U^k x\in G$ for all integers $k$. But this implies that $U^nx\perp {\mathfrak{R}}$, or equivalently $x\perp U^n {\mathfrak{R}}$ for all $n\in{\mathbb Z}$. But ${\mathfrak{R}}$ is a star-cyclic subspace for $U$, so we get a contradiction.
\end{proof}
\begin{proof}[Proof of Lemma \ref{l:*cycl}]
Assume now that for unitary $\Gamma$, the subspace $\Ran\mathbf{B}$ is not a star-cyclic subspace for $U\ci\Gamma= T\ci\Gamma$ (but is a star-cyclic subspace for $U$). Consider the perturbation
$T_\mathbf{0}$
\[
T\ci\mathbf{0} = U + \mathbf{B}(\mathbf{0}-\mathbf{I}\ci{\mathfrak D})\mathbf{B}^*U.
\]
We will show that
\begin{align}
\label{ChPert-01}
T\ci\mathbf{0} = U\ci \Gamma + \mathbf{B}(\mathbf{0}- \mathbf{I}\ci{\mathfrak D})\mathbf{B}^*U\ci\Gamma
\end{align}
By Lemma \ref{l-cnu} the operator $T\ci\mathbf{0}$ is a c.n.u.~contraction.
But, as we discussed in the beginning of this subsection, if $\Ran \mathbf{B}$ is not star-cyclic for $U$, then for $\mathcal{E}$ defined by \eqref{SpanOrbit-01} the subspace $\mathcal{E}^\perp$ is a reducing subspace for $T\ci\Gamma$ (with any $\Gamma$) on which $T\ci\Gamma$ acts unitarily.
Since by \eqref{ChPert-01} the operator $T_\mathbf{0}$ is a perturbation of form \eqref{pert-02} of the unitary operator $T\ci\Gamma$, we conclude that the operator $T\ci\mathbf{0}$ has a non-trivial unitary part, and arrive to a contradiction.
To prove \eqref{ChPert-01} we notice that
\begin{align}
\label{ChPert-02}
T\ci\mathbf{0}= U-\mathbf{B}\bB^* U = U\ci\Gamma - \mathbf{B}\Gamma \mathbf{B}^*U.
\end{align}
Direct computations show that
\begin{align*}
U\ci\Gamma U^*\mathbf{B} = UU^*\mathbf{B} + \mathbf{B}(\Gamma-\mathbf{I}\ci{\mathfrak D}) \mathbf{B}^*UU^*\mathbf{B} = \mathbf{B} + \mathbf{B} (\Gamma-\mathbf{I}\ci{\mathfrak D}) =\mathbf{B}\Gamma.
\end{align*}
Taking the adjoint of this identity we get that $\mathbf{B}^*UU^*\ci\Gamma = \mathbf{B}^*\Gamma^*$, and so $\Gamma \mathbf{B}^*U = \mathbf{B}^*U\ci\Gamma$. Substituting $\mathbf{B}^*U\ci\Gamma$ instead of $\Gamma\mathbf{B}^*U$ in \eqref{ChPert-02} we get \eqref{ChPert-01}.
\end{proof}
\subsection{Characterization of star-cyclic subspaces}
Recall that for an isometry $\mathbf{B}:\mathcal{D} \to \mathcal{H}$ (where $\mathcal{H}$ is the direct integral \eqref{DirInt}) we denoted by $b_k\in\mathcal{H}$ the ``columns'' of $\mathbf{B}$,
\[
b_k=\mathbf{B} e_k,
\]
where $e_1, e_2, \ldots, e_d$ is the standard basis in ${\mathbb C}^d$.
\begin{lem}
\label{l:cycl-02} Let $U$ be the multiplication $M_\xi$ by the independent variable $\xi$ in the direct integral $\mathcal{H}$ given by \eqref{DirInt}, and let $\mathbf{B}:{\mathbb C}^d\to \mathcal{H}$ be as above. The space $\Ran \mathbf{B}=\spa\{b_k:1\le k\le d\}$ is star-cyclic for $U$ if and only if $\cspa\{b_k(\xi) : 1\le k \le d\} = E(\xi)$ for $\mu$-a.e.~$\xi$.
\end{lem}
\begin{proof
First assume that $\Ran \mathbf{B}$ is not a star-cyclic subspace for $U$. Then there exists $f\in \mathcal{H}\subset L^2(\mu;E)$, $f\ne 0$ $\mu$-a.e., such that
\begin{align*}
U^l f\perp b_k
\qquad\text{for all }l\in {\mathbb Z}, \text{ and } k=1,\hdots,d,
\end{align*}
or, equivalently
\begin{align*}
\int_\mathbb{T} \Bigl( f(\xi) , {b_k(\xi)}\Bigr)\ci E \xi^l {\mathrm{d}} \mu(\xi) = 0 \qquad\text{for all }l\in {\mathbb Z}, \text{ and } k=1,\hdots,d.
\end{align*}
But that means for all $k=1, 2, \ldots, d$ we have
\begin{align*}
\Bigl( f(\xi), b_k(\xi) \Bigr)\ci E= 0 \qquad \mu\text{-a.e.},
\end{align*}
so on some set of positive $\mu$ measure (where $f(\xi)\ne \mathbf{0}$) we have
\begin{align}
\label{Span-ne-E}
\cspa\{b_k(\xi): 1\le k\le d\}\subsetneqq E(\xi).
\end{align}
Vice versa, assume that \eqref{Span-ne-E} holds on some Borel subset $A\subset\mathbb{T}$ with $\mu(A)>0$.
For $n=1, 2, \ldots, \infty$ define sets
$
A_n:=\{\xi\in A: \dim E(\xi)=n\}
$.
Then $\mu(A_n)>0$ for some $n$. Fix this $n$ and denote the corresponding space $E(\xi)$, $\xi\in A_n$ by $E_n$.
We know that $\cspa\{b_k(\xi): 1\le k\le d\}\subsetneqq E_n$ on $A_n$, so there exists $e\in E_n$ such that
\[
e\notin \cspa\{b_k(\xi): 1\le k\le d\}
\]
on a set of positive measure in $A_n$.
Trivially, if $f\in \cspa\{U^k \Ran\mathbf{B}: k\in{\mathbb Z}\}$ then
\[
f(\xi)\in \cspa\{b_k(\xi): 1\le k\le d\}\qquad \mu\text{-a.e.},
\]
and therefore $f=\mathbf{1}\ci{A_n}e$ is not in $\cspa\{U^k \Ran\mathbf{B}: k\in{\mathbb Z}\}$.
\end{proof}
\subsection{The case of star-cyclic \texorpdfstring{$U$}{U}}
If $U$ is star-cyclic (i.e.~it has a one-dimensional star-cyclic subspace/vector), $U$ is unitarily equivalent to the multiplication operator $M_\xi$ in the scalar space $L^2(\mu)$; of course the scalar space $L^2(\mu)$ is a particular case of the direct integral, where all spaces $E(\xi) $ are one-dimensional.
In our general vector-valued case, Lemma \ref{l:cycl-02} says that $\Ran \mathbf{B}$ is star-cyclic for $U$ if and only if there is no measurable set $A$, $\mu(A)>0$, on which all the functions $b_k$ vanish.
So, we know that $U$ has a star-cyclic vector. Here we ask the question:
\begin{center}
\emph{Does operator $U$ have a star-cyclic vector that belongs to a prescribed (finite-dimensional) star-cyclic subspace?}
\end{center}
The following lemma answers ``yes" to that question. Moreover, it implies that if $\Ran \mathbf{B}$ is star-cyclic for $U=M_\xi$ on the scalar-valued space $L^2(\mu)$, then almost all vectors $b\in \Ran \mathbf{B}$ are star-cyclic for $U$. As the result is measure-theoretic in nature, we formulate it in a general context.
\begin{lem}
\label{l:cycl-03}
Consider a $\sigma$-finite scalar-valued measure $\tau$ on a measure space $\mathcal{X}$. Let $b_1, b_2, \ldots, b_d\in L^2(\tau)$ be such that
\begin{align*}
\sum_{k=1}^d |b_k| \ne 0 \qquad\tau\text{-a.e.}
\end{align*}
Then for almost all (with respect to the Lebesgue measure) $\alpha=(\alpha_1, \alpha_2, \ldots, \alpha_d)\in{\mathbb C}^d$ we have
\begin{align*}
\sum_{k=1}^d \alpha_k b_k \ne 0 \qquad\tau\text{-a.e.~on }\mathcal{X}.
\end{align*}
\end{lem}
\begin{rem*}
The above lemma also holds for almost all $\alpha\in{\mathbb R}^d$.
\end{rem*}
\begin{proof}[Proof of Lemma \ref{l:cycl-03}]
Consider first the case $\tau(\mathcal{X})<\infty$.
We proceed by induction in $d$. Clearly, if $|b_1|\neq 0$ $\tau$-a.e.~on $\mathcal{X}$, then $\alpha b_1\neq 0$ $\tau$-a.e.~on $\mathcal{X}$ for all $\alpha\in {\mathbb C}\setminus\{0\}$.
Now assume the statement of the Lemma for $d=n$ for some $n\in {\mathbb N}$. Deleting a set of $\tau$-measure $0$, we can assume that $\sum_{k=1}^{n+1} |b_k| \ne 0$ on $\mathcal{X}$.
Let $\mathcal{Y}:=\{x\in\mathcal{X}: \sum_{k=1}^n |b_k(x)|>0\}$.
By the induction assumption for almost all $\alpha'=(\alpha_1, \alpha_2, \ldots, \alpha_n)$
\begin{align*}
b(\alpha',x) : =\sum_{k=1}^n \alpha_k b_k(x) \ne 0 \qquad {on\ } \mathcal{Y}.
\end{align*}
Fix $\alpha'=(\alpha_1, \alpha_2, \ldots, \alpha_n)$ such that $b(\alpha', x) \ne 0$ on $\mathcal{Y}$. We will show that for any such fixed $\alpha'$ the measure
\begin{align}
\label{CountMany-alpha}
\tau\left(\left\{x\in\mathcal{X}: \sum_{k=1}^{n+1} \alpha_k b_k(x) =0\right\} \right)>0
\end{align}
only for countably many values of $\alpha_{n+1}$.
To show that define for $\beta=\alpha_{n+1} \in {\mathbb C}$ the set
\[
\mathcal{X}_{\beta}:=\left\{x\in \mathcal{X}:b(\alpha',x)+\beta b_{n+1}(x)=0\right\}.
\]
Let $\widetilde\beta \in {\mathbb C}\setminus\{0\}$, $\widetilde\beta\ne \beta$. We claim that the sets $X_\beta$ and $X_{\widetilde\beta}$ are disjoint.
Indeed, the assumption that $\sum_{k=1}^{n+1} |b_k|>0$ implies that $b_{n+1}\ne 0$ on $\mathcal{X}\setminus\mathcal{Y}$, so $\mathcal{X}_\beta, \mathcal{X}_{\widetilde\beta}\in\mathcal{Y}$. Moreover, solving for $b_{n+1}$ we get that if $\beta\ne0$, then
\begin{align*}
\mathcal{X}_\beta= \{x\in\mathcal{Y}: b_{n+1}(x) = -b(\alpha', x)/\beta\},
\end{align*}
and similarly for $\mathcal{X}_{\widetilde\beta}$. Since $b(\alpha',x)\ne0$ on $\mathcal{Y}$, we get that
\begin{align*}
b(\alpha', x)/\beta \ne b(\alpha', x)/\widetilde\beta\qquad \forall x\in\mathcal{Y},
\end{align*}
so if $\beta\ne0$, then $\mathcal{X}_\beta$ and $\mathcal{X}_{\widetilde\beta} $ are disjoint as preimages of disjoint sets (points).
If $\beta=0$, then $\mathcal{X}_0=\mathcal{X}\setminus\mathcal{Y}$, so the sets $\mathcal{X}_{\widetilde\beta}$ and $\mathcal{X}_0$ are disjoint.
The set $\mathcal{X}$ has finite measure, and $\mathcal{X}$ is the union of disjoint sets $\mathcal{X}_\beta$, $\beta\in {\mathbb C}$. So, only countably many sets $\mathcal{X}_\beta$ can satisfy $\tau(\mathcal{X}_\beta)>0$.
We have proved the lemma for $\tau(\mathcal X)<\infty$.
The rest can be obtained by Tonelli's theorem. Namely, define
\[
\mathcal{A}:= \left\{(x,\alpha): x\in\mathcal{X}, \alpha\in{\mathbb C}^{n+1} , \sum_{k=1}^{n+1}\alpha_k b_k(x) =0\right\}
\]
and let $F=\mathbf{1}\ci\mathcal{A}$. From the Tonelli Theorem we can see that
\begin{align}
\label{Int1_A}
\int\mathbf{1}\ci\mathcal{A} (x,\alpha) {\mathrm{d}} m(\alpha) {\mathrm{d}}\tau(x)>0
\end{align}
if and only if for the set of $\alpha\in{\mathbb C}^{n+1}$ of positive Lebesgue measure
\[
\tau\left( \left\{x\in\mathcal{X}: \sum_{k=1}^{n+1}\alpha_k b_k(x) =0 \right\}\right) >0 .
\]
It follows from \eqref{CountMany-alpha} that for almost all $\alpha'=(\alpha_1, \alpha_2, \ldots, \alpha_n)\in{\mathbb C}^n$
\[
\int\mathbf{1}\ci\mathcal{A} (x,\alpha', \alpha_{n+1}) {\mathrm{d}} m(\alpha_{n+1}) {\mathrm{d}}\tau(x) = 0,
\]
so, by Tonelli, the integral in \eqref{Int1_A} equals $0$.
\end{proof}
\section{Abstract formula for the adjoint Clark operator}\label{s-adjClark}
In this section we introduce necessary known facts about functional models and then give a general abstract formula for the adjoint Clark operator. To do this we need a new notion of coordinate/parametrizing operators for the model and their agreement: the abstract representation formula (Theorem \ref{t-repr}) holds under the assumption that the coordinate operators $\mathbf{C}$ and $\mathbf{C}_*$ agree with the Clark model.
Later in Section \ref{s-ModAgree} we construct the coordinate operators that agree with the Clark, and in Section \ref{s-charfunc} we compute the characteristic function, so the abstract Theorem \ref{t-repr} will give us concrete, albeit complicated formulas.
\subsection{Functional model
}
\begin{defn}
\label{d:defects}
Recall that for a contraction $T$ its \emph{defect operators} $D\ci T$ and $D\ci{T^*}$ are defined as
\begin{align*}
D\ci{T} := (\mathbf{I} -T^*T)^{1/2}, \qquad D\ci{T^*}:= (\mathbf{I}-TT^*)^{1/2}.
\end{align*}
The \emph{defect spaces} ${\mathfrak D}\ci T$ and ${\mathfrak D}\ci{T^*}$ are defined as
\begin{align*}
{\mathfrak D}\ci{T}:=\clos\Ran D\ci T, \qquad {\mathfrak D}\ci{T^*}:=\clos\Ran D\ci{T^*}.
\end{align*}
\end{defn}
The characteristic function is an (explicitly computed from the contraction $T$) op\-erator-valued function $\theta\in H^\infty({\mathfrak D}\!\!\to\! {\mathfrak D}_*)$, where ${\mathfrak D}$ and ${\mathfrak D}_*$ are Hilbert spaces of appropriate dimensions,
\[
\dim {\mathfrak D}=\dim {\mathfrak D}\ci{T}, \qquad \dim {\mathfrak D}_*=\dim {\mathfrak D}\ci{T^*}\,.
\]
Using the characteristic function $\theta$ one can then construct the so-called \emph{model space} ${\mathcal K}_\theta$, which is a subspace of a weighted $L^2$ space $L^2(\mathbb{T}, W; {\mathfrak D}_*\oplus{\mathfrak D})= L^2(W; {\mathfrak D}_*\oplus{\mathfrak D})$ with an operator-valued weight $W$. The model operator $\mathcal{M}_\theta:{\mathcal K}_\theta \to {\mathcal K}_\theta$ is then defined as the \emph{compression} of the multiplication $M_z$ by the independent variable $z$,
\begin{align*}
\mathcal{M}_\theta f = P\ci{{\mathcal K}_\theta} M_z f, \qquad f\in{\mathcal K}_\theta;
\end{align*}
here $M_z f(z) =zf(z)$.
Let as remind the reader, that the norm in the weighted space $L^2(\mathbb{T},W;H)$ with an operator weight $W$ is given by
\[
\|f\|\ci{L^2(W;H)}^2 = \int_\mathbb{T} (W(z) f(z), f(z) )\ci H {\mathrm{d}} m(z) ;
\]
in the case $\dim H=\infty$ there are some technical details,
but in the finite-dimensional case considered in this paper everything is pretty straightforward.
The best known example of a model is the Sz.-Nagy--Foia\c{s} (transcription of a) model, \cite{SzNF2010}. The Sz.-Nagy--Foia\c{s} model space ${\mathcal K}_\theta$ is a subspace of a non-weighted space $L^2({\mathfrak D}_*\oplus{\mathfrak D})$ (the weight $W\equiv\mathbf{I}$),
given by
\begin{align*
{\mathcal K}_\theta
:=
\begin{pmatrix}H^2({\mathfrak D}_*)\\\clos\Delta L^2({\mathfrak D})\end{pmatrix}
\ominus \begin{pmatrix}\theta\\\Delta\end{pmatrix} H^2({\mathfrak D}),
\end{align*}
where
\[
\Delta(z) := ({\mathbf{I}}_\mathfrak{D}-\theta^\ast(z)\theta(z))^{1/2}
\quad
\text{and}\quad
\begin{pmatrix}\theta\\\Delta\end{pmatrix} H^2({\mathfrak D})=
\left\{ \left(\begin{array}{c}\theta f \\ \Delta f\end{array}\right) : f\in H^2({\mathfrak D}) \right\}.
\]
In literature, the case when the vector-valued characteristic function $\theta$ is \emph{inner} (i.e.~its boundary values are isometries for a.e.~$z\in\mathbb{T}$) is often considered. Then $\Delta(z) = {\mathbf{0}}$ on $\mathbb{T}$, so in that case the second component of ${\mathcal K}_\theta$ collapses completely and the Sz.-Nagy--Foia\c{s} model space reduces to the familiar space
\[
{\mathcal K}_\theta
=
H^2({\mathfrak D}_*)\ominus \theta H^2({\mathfrak D}).
\]
Also, in the literature, cf \cite{SzNF2010}, the characteristic function is defined up to multiplication by constant unitary factors from the right and from the left. Namely, two functions $\theta\in H^\infty({\mathfrak D}\to{\mathfrak D}_*)$ and $\widetilde\theta \in H^\infty(\widetilde{\mathfrak D}\to\widetilde{\mathfrak D}_*)$ are equivalent if there exist unitary operators $U:{\mathfrak D}\to\widetilde{\mathfrak D}$ and $U_*:{\mathfrak D}_*\to\widetilde{\mathfrak D}_*$ such that $\widetilde\theta = U_*\theta U^*$.
It is a well-known fact, cf \cite{SzNF2010}, that two c.n.u.~contractions are unitarily equivalent if and only if their characteristic functions are equivalent as described above. So, usually in the literature the characteristic function was understood as the corresponding equivalence class, or an arbitrary representative in this class. However, in this paper, to get correct formulas it is essential to track which representative is chosen.
\subsection{Coordinate operators, parameterizing operators,
and their agreement
}
\label{s:agree-01}
Let $T:H\to H$ be a contraction, and let ${\mathfrak D}$, ${\mathfrak D}_*$ be Hilbert spaces, $\dim {\mathfrak D}= \dim{\mathfrak D}\ci T$, $\dim {\mathfrak D}_* = \dim {\mathfrak D}\ci{T^*}$. Unitary operators $V:{\mathfrak D}\ci{T}\to {\mathfrak D}$ and $V_*:{\mathfrak D}\ci{T^*}\to {\mathfrak D}_*$ will be called \emph{coordinate operators} for the corresponding defect spaces; the reason for that name is that often spaces ${\mathfrak D}$ and ${\mathfrak D}_*$ are spaces with a fixed orthonormal basis (and one can introduce coordinates there), so the operators introduce coordinates on the defect spaces.
The inverse operators $V^*:{\mathfrak D}\to {\mathfrak D}\ci{T}$ and $V_*^*:{\mathfrak D}_*\to {\mathfrak D}\ci{T^*}$ will be called \emph{parameterizing} operators. For a contraction $T$ we will use symbols $V$ and $V_*$ for the coordinate operators, but for its model $\mathcal{M}_\theta$ the parametrizing operators will be used, and we reserve letters $\mathbf{C}$ and $\mathbf{C}_*$ for these operators.
Let $T$ be a c.n.u.~contraction with characteristic function $\theta\in H^\infty({\mathfrak D}\!\!\to\! {\mathfrak D}_*)$, and let $\mathcal{M}_\theta: {\mathcal K}_\theta\to {\mathcal K}_\theta$ be its model. Let also $V:{\mathfrak D}\ci{T}\to {\mathfrak D}$ and $V_*:{\mathfrak D}\ci{T^*}\to {\mathfrak D}_*$ be coordinate operators for the defect spaces of $T$, and
$\mathbf{C}:{\mathfrak D}\ci{\mathcal{M}_\theta}\to {\mathfrak D}$ and $\mathbf{C}_*:{\mathfrak D}\ci{\mathcal{M}_\theta^*}\to {\mathfrak D}_*$ be the parameterizing operators for the defect spaces of $\mathcal{M}_\theta$ (this simply means that all 4 operators are unitary).
We say that the operators $V$, $V_*$ agree with operators $\mathbf{C}$, $\mathbf{C}_*$ if there exists a unitary operator $\Phi:{\mathcal K}_\theta\to H$ intertwining $T$ and $\mathcal{M}_\theta$,
\[
T\Phi = \Phi \mathcal{M}_\theta,
\]
and such that
\begin{align}
\label{agree-01}
\mathbf{C}^* = V\Phi\Bigm|_{{\mathfrak D}\ci{\mathcal{M}_\theta}}, \qquad \mathbf{C}_*^* = V_*\Phi\Bigm|_{{\mathfrak D}\ci{\mathcal{M}_\theta^*}}\,.
\end{align}
The above identities simply mean that the diagrams below are commutative.
\begin{center}
\begin{tikzpicture}[>=angle 90]
\matrix(a)[matrix of math nodes,
row sep=2.5em, column sep=2.5em,
text height=1.5ex, text depth=0.25ex]
{D\ci{T}&&{\mathfrak D}&{\mathfrak D}_*&&D\ci{T^*}\\
&&&&&\\
&{\mathfrak D}\ci{\mathcal{M}_\theta}&&&{\mathfrak D}\ci{\mathcal{M}_\theta^*}&\\};
\path[->](a-1-1) edge node[auto] {$V$} (a-1-3);
\path[->](a-1-4) edge node[auto] {$ V_*^*$} (a-1-6);
\path[->](a-3-2) edge node[auto] {$\Phi$} (a-1-1);
\path[<-](a-1-3) edge node[auto] {$\mathbf{C}$} (a-3-2);
\path[<-](a-1-6) edge node[auto] {$\Phi$} (a-3-5);
\path[->](a-3-5) edge node[auto] {$\mathbf{C}_*$} (a-1-4);
\end{tikzpicture}
\end{center}
In this paper, when convenient, we always extend an operator between subspaces to the operator between the whole spaces, by extending it by $0$ on the orthogonal complement of the domain;
slightly abusing notation we will use the same symbol for both operators. Thus a unitary operator between subspaces $E$ and $F$ can be treated as a partial isometry with initial space $E$ and final space $F$, and vice versa. With this agreement \eqref{agree-01} can be rewritten as
\begin{align*}
\mathbf{C}^* = V\Phi, \qquad \mathbf{C}_*^* = V_*\Phi .
\end{align*}
\subsection{Clark operator} Consider a contraction $T$ given by \eqref{pert-02} with $\Gamma$ being a strict contraction. We also assume that $\Ran \mathbf{B}$ is a star-cyclic subspace for $U$, so $T$ is a c.n.u.~contraction, see Lemma \ref{l-cnu}.
We assume that $U$ is given in its spectral representation, so $U$ is the multiplication operator $M_\xi$ in the direct integral $\mathcal{H}$.
A Clark operator $\Phi:{\mathcal K}_\theta\to \mathcal{H}$ is a unitary operator, intertwining this special contraction $T$ and its model $\mathcal{M}_\theta$, $\Phi\mathcal{M}_\theta = T\Phi$, or equivalently
\begin{align}
\label{e-intertwinePhi*}
\Phi^* T
=
\mathcal{M}_\theta \Phi^*.
\end{align}
We name it so after D.~Clark, who in \cite{Clark} described it for rank one perturbations of unitary operators with purely singular spectrum.
We want to describe the operator $\Phi$ (more precisely, its adjoint $\Phi^*$) in our situation.
In our case, $\dim {\mathfrak D}\ci{T} = \dim {\mathfrak D}\ci{T^*}=d$, and it will be convenient for us to consider models with ${\mathfrak D}={\mathfrak D}_*={\mathbb C}^d$.
As it was discussed above, it can be easily seen from the representation \eqref{BlDec-01} that the operators $U^*\mathbf{B}:{\mathfrak D}={\mathbb C}^d\to {\mathfrak D}\ci{T}$ and $\mathbf{B}:{\mathfrak D}={\mathbb C}^d\to {\mathfrak D}\ci{T^*}$ are unitary operators canonically (for our setup) identifying ${\mathfrak D}$ with the corresponding defect spaces, i.e.~the canonical parameterizing operators for these spaces. The corresponding coordinate operators are given by $V=\mathbf{B}^* U$, $V_*=\mathbf{B}^*$.
We say that parametrizing operators $\mathbf{C}:{\mathfrak D}\to {\mathfrak D}\ci{\mathcal{M}_\theta}$, $\mathbf{C}_* : {\mathfrak D}\to {\mathfrak D}\ci{\mathcal{M}_\theta^*}$ \emph{agree} with the Clark model, if the above coordinate operators $V=\mathbf{B}^* U$, $V_*=\mathbf{B}^*$ agree with the parametrizing operators $\mathbf{C}$, $\mathbf{C}_*$ in the sense of Subsection \ref{s:agree-01}. In other words, they agree
if there exists a Clark operator $\Phi$ such that the following diagram commutes.
\begin{equation}
\label{d:agree-02}
\begin{tikzpicture}[>=angle 90, baseline={([yshift=-.5ex]current bounding box.center)}]
\matrix(a)[matrix of math nodes,
row sep=2.5em, column sep=2.5em,
text height=1.5ex, text depth=0.25ex]
{{\mathfrak D}\ci{T}&&{\mathfrak D} = {\mathbb C}^d&&{\mathfrak D}\ci{T^*}\\
&&&&\\
&{\mathfrak D}\ci{\mathcal{M}_\theta}&&{\mathfrak D}\ci{\mathcal{M}_\theta^*}&\\};
\path[->](a-1-1) edge node[auto] {$\mathbf{B}^* U$} (a-1-3);
\path[->](a-1-3) edge node[auto] {$ \mathbf{B}$} (a-1-5);
\path[<-](a-3-2) edge node[auto] {$\Phi^*$} (a-1-1);
\path[->](a-1-3) edge node[auto] {$\mathbf{C}$} (a-3-2);
\path[->](a-1-5) edge node[auto] {$\Phi^*$} (a-3-4);
\path[<-](a-3-4) edge node[auto] {$\mathbf{C}_*$} (a-1-3);
\end{tikzpicture}
\end{equation}
Note, that in this diagram one can travel in both directions: to change the direction one just needs to take the adjoint of the corresponding operator.
Slightly abusing notation, we use $\mathbf{C}$ to also denote the extension of $\mathbf{C}$ to the model space ${\mathcal K}\ci\theta$ by the zero operator, and similarly for $\mathbf{C}_*$.
Note that agreement of $\mathbf{C}$ and $\mathbf{C}_*$ with the Clark model can be rewritten as
\begin{align}\label{e-comm}
\Phi^* (\mathbf{B}^* U)^*= \mathbf{C}, \qqua
\Phi^*\mathbf{B} = \mathbf{C}_*.
\end{align}
And by taking restrictions (where necessary) we find
\begin{align}\label{e-INTER}
\mathcal{M}_\theta \mathbf{C} = \mathbf{C}_*\Gamma
\qquad
\text{and}
\qquad
\mathcal{M}_\theta^* \mathbf{C}_* = \mathbf{C}\Gamma^*.
\end{align}
We express the action of the model operator and its adjoint in an auxiliary result. The result holds in any transcription of the model.
We will need the following simple fact.
\begin{lem}
\label{l:Tdefect}
For a contraction $T$
\begin{align*}
T {\mathfrak D}\ci T \subset {\mathfrak D}\ci{T^*}, \qquad T^* {\mathfrak D}\ci{T^*} \subset {\mathfrak D}\ci{T} \,.
\end{align*}
\end{lem}
\begin{proof} Since $D\ci T $ is a strict contraction on ${\mathfrak D}\ci T$ we get that
\begin{align}
\nota
\|T x\|=\|x\|\quad &\iff \quad x\perp {\mathfrak D}\ci{T}, \\
\intertext{and similarly, since $T^*$ is a strict contraction on ${\mathfrak D}\ci{T^*}$,}
\label{IsomT-02}
\|T^* x\|=\|x\|\quad &\iff \quad x\perp {\mathfrak D}\ci{T^*} \,.
\end{align}
Thus the operator $T$ is an isometry on ${\mathfrak D}\ci{T}^\perp$, so the polarization identity implies that $T^*Tx=x$ for all $x\in {\mathfrak D}\ci{T}^\perp$. Together with \eqref{IsomT-02} this implies that $T({\mathfrak D}\ci{T}^\perp) \subset {\mathfrak D}\ci{T^*}^\perp$, which is equivalent to the inclusion $T^* {\mathfrak D}\ci{T^*} \subset {\mathfrak D}\ci{T}$\,.
Replacing $T$ by $T^*$ we get $T {\mathfrak D}\ci{T} \subset {\mathfrak D}\ci{T^*}$.
\end{proof}
\begin{lem}\label{l-model}
Let $T$ be as defined in \eqref{pert-02} with $\Gamma$ being a strict contraction. Assume also that
$\Ran \mathbf{B}$ is star-cyclic (so $T$ is completely non-unitary, see Lemma \ref{l-cnu}).
Let $\theta\in H^\infty({\mathfrak D}\!\!\to\! {\mathfrak D}_*)$, ${\mathfrak D}={\mathfrak D}_*={\mathbb C}^d$, be the characteristic function of $T$, and let $\mathcal{M}_\theta:{\mathcal K}_\theta\to {\mathcal K}_\theta$ be a model operator. Let $\mathbf{C}:{\mathfrak D}\to {\mathfrak D}\ci{\mathcal{M}_\theta}$ and $\mathbf{C}_* :{\mathfrak D}\to {\mathfrak D}\ci{\mathcal{M}_\theta^*}$ be the parametrizing unitary operators, that agree with a Clark model.
Then
\[
\mathcal{M}_\theta = M_z +(\mathbf{C}_* \Gamma - M_z \mathbf{C}) \mathbf{C}^*
\qquad\text{and}\qquad
\mathcal{M}_\theta^* = M_{\bar z}+ (\mathbf{C} \Gamma^* - M_{\bar{z}} \mathbf{C}_*)\mathbf{C}_*^* .
\]
\end{lem}
\begin{proof}
Since operator $\mathcal{M}_\theta$ acts on ${\mathcal K}_\theta\ominus{\mathfrak D}\ci{\mathcal{M}_\theta}$ as the multiplication operator $M_z$, we can trivially write
\[
\mathcal{M}_\theta = M_z (\mathbf{I} - P\ci{{\mathfrak D}\ci{\mathcal{M}_\theta}}) + \mathcal{M}_\theta P\ci{{\mathfrak D}\ci{\mathcal{M}_\theta}} .
\]
Recalling that $\mathbf{C}:{\mathfrak D}\to{\mathcal K}_\theta$ is an isometry with range ${\mathfrak D}\ci{\mathcal{M}_\theta}$, we can see that $P\ci{{\mathfrak D}\ci{\mathcal{M}_\theta}} = \mathbf{C} \mathbf{C}^* $, so
\begin{align}
\label{MzP-MzCC}
M_z (\mathbf{I} - P\ci{{\mathfrak D}\ci{\mathcal{M}_\theta}})= M_z (\mathbf{I} - \mathbf{C} \mathbf{C}^*) .
\end{align}
Using the identity $P\ci{{\mathfrak D}\ci{\mathcal{M}_\theta}} = \mathbf{C} \mathbf{C}^* $ and the first equation of \eqref{e-INTER} we get
\[
\mathcal{M}_\theta P\ci{{\mathfrak D}\ci{\mathcal{M}_\theta}} = \mathcal{M}_\theta \mathbf{C}\bC^*
=
\mathbf{C}_*\Gamma\mathbf{C}^*,
\]
which together with \eqref{MzP-MzCC} gives us the desired formula for $\mathcal{M}_\theta$.
To get the formula for $\mathcal{M}_\theta^*$ we represent it as
\[
\mathcal{M}_\theta^* = M_{\overline z} (\mathbf{I} - P\ci{{\mathfrak D}\ci{\mathcal{M}_\theta^*}}) + \mathcal{M}_\theta^* P\ci{{\mathfrak D}\ci{\mathcal{M}_\theta^*}}.
\]
Using the identities
\[
P\ci{{\mathfrak D}\ci{\mathcal{M}_\theta^*}} = \mathbf{C}_*\mathbf{C}_*^* , \qquad \mathcal{M}_\theta^* P\ci{{\mathfrak D}\ci{\mathcal{M}_\theta^*}} = \mathbf{C} \Gamma^* \mathbf{C}_*^*
\]
(the first holds because ${\mathfrak D}\ci{\mathcal{M}_\theta^*}$ is the range of the isometry $\mathbf{C}_*$, and the second one follows from the second equation in \eqref{e-INTER}), we get the desired formula.
\end{proof}
\subsection{Representation Theorem}
For a (general) model operator $\mathcal{M}_\theta$, $\theta\in H^\infty({\mathfrak D}\to {\mathfrak D}_*)$,
the parametrizing operators $\mathbf{C}:{\mathfrak D}\to {\mathfrak D}\ci{\mathcal{M}_\theta}$, $\mathbf{C}_* : {\mathfrak D}_*\to {\mathfrak D}\ci{\mathcal{M}_\theta^*}$ give rise to (uniquely defined) operator-valued functions $C$ and $C_*$, where $C(\xi):{\mathfrak D}\to {\mathfrak D}\oplus {\mathfrak D}_*$, $C_*(\xi):{\mathfrak D}_*\to {\mathfrak D}\oplus {\mathfrak D}_*$ and
\begin{align}\label{d-Cz}
(\mathbf{C} e)(z)
& =
C(z) e
&&\text{for all}
\quad e\in {\mathfrak D},
\\
\label{d-C*z}
(\mathbf{C}_* e_*)(z)
&=
C_*(z) e_*
&&\text{for all}
\quad e_*\in {\mathfrak D}_*
.
\end{align}
If we fix orthonormal bases in ${\mathfrak D}$ and ${\mathfrak D}_*$, then the $k$th column of the matrix of $C(\xi)$ is defined as $(\mathbf{C}_*e_k)(\xi)$, where $e_k$ it the $k$th vector in the orthonormal basis in ${\mathfrak D}$, and similarly for $C_*$.
If $\mathcal{M}_\theta$ is a model for a contraction $T=T\ci\Gamma$ with $\Gamma$ being a strict contraction on ${\mathfrak D}={\mathbb C}^d$, we can see from \eqref{BlDec-01} that $\dim {\mathfrak D}\ci T = \dim{\mathfrak D}\ci{T^*}=d$, so we can always pick a characteristic function $\theta\in H^\infty({\mathfrak D}\to{\mathfrak D}_*)$ (i.e.~with ${\mathfrak D}_*={\mathfrak D}={\mathbb C}^d$).
The following formula for the adjoint $\Phi^*$ of the Clark operator $\Phi$ generalizes the ``universal" representation theorem \cite[Theorem 3.1]{LT15} to higher rank perturbations.
\begin{theo}[Representation Theorem]\label{t-repr}
Let $T$ be as defined in \eqref{pert-02} with $\Gamma$ being a strict contraction and $U=M_\xi$ in $\mathcal{H}\subset L^2(\mu;E)$.
Let $\theta =\theta\ci{T}$ be a characteristic function of $T$, and let ${\mathcal K}_\theta$ and $\mathcal{M}_\theta$ be the corresponding model space and model operator.
Let $\mathbf{C}:{\mathfrak D}\to {\mathfrak D}\ci{\mathcal{M}_\theta}$ and $\mathbf{C}_* :{\mathfrak D}\to {\mathfrak D}\ci{\mathcal{M}_\theta^*}$ be the parameterizing unitary operators%
\footnote{Note that here we set ${\mathfrak D}_*={\mathfrak D}$, which is possible because the dimensions of the defect spaces are equal.}
that agree with Clark model, i.e.~such that \eqref{e-comm} is satisfied for some Clark operator $\Phi$. And let $C(z)$ and $C_*(z)$ be given by \eqref{d-Cz} and \eqref{d-C*z}, respectively.
Then the action of the adjoint Clark operator $\Phi^*$ is given by
\begin{align}
\label{e-repr}
\bigl(\Phi^* hb\bigr)(z)
=
h(z)C_*(z){\mathbf{B}}^*b
+
(C_*(z)-z C(z))
\int\ci\mathbb{T}
\frac{h(\xi) - h(z)}{1-z\bar\xi} B^*(\xi)b(\xi)
{\mathrm{d}}\mu(\xi),
\end{align}
for any $b\in\Ran \mathbf{B}$ and for all $h\in C^1(\mathbb{T})$;
here
\[
B^*(\xi) = \left( \begin{array}{c} b_1(\xi)^* \\ b_2(\xi)^* \\ \vdots \\ b_d(\xi)^* \end{array}\right)
\]
and
$
\mathbf{B}^* b
=
\int\ci\mathbb{T} B^*(\xi) b(\xi) d\mu(\xi),
$ as explained more thoroughly in the proof below.
\end{theo}
\begin{rem*}
The above theorem looks like an abstract nonsense, because right now it is not clear how to find the parametrizing operators $\mathbf{C}$ and $\mathbf{C}_*$ that agree with the Clark model. However, Theorem \ref{t-theta} below gives an explicit formula for the characteristic function $\theta$ (one of the representative in the equivalence class), and
Lemma \ref{l:C_N-F} gives an explicit formulas for $\mathbf{C}$ and $\mathbf{C}^*$ in the Sz.-Nagy--Foia\c s transcription, that agree with Clark model for our $\theta$.
\end{rem*}
When $d=1$ this formula agrees with the special case of the representation formula derived in \cite{LT15}. While some of the ideas of the following proof were originally developed there, the current extension to rank $d$ perturbations requires several new ideas and a more abstract way of thinking.
\begin{proof}[Proof of Theorem \ref{t-repr}]
Recall that $U= M_\xi$,
so $T = M_\xi+ {\mathbf{B}}(\Gamma - {\mathbf{I}}\ci{{\mathbb C}^d}){\mathbf{B}}^* M_\xi$. The intertwining relation $\Phi^* T=\mathcal{M}_\theta\Phi^*$ then can be rewritten as
\begin{align}\label{equation}
\Phi^* M_\xi
+
\Phi^*{\mathbf{B}}(\Gamma - {\mathbf{I}}\ci{{\mathbb C}^d}){\mathbf{B}}^* U
=
\Phi^* T
=
\mathcal{M}_\theta\Phi^*
=
[M_z +(\mathbf{C}_* \Gamma - M_z \mathbf{C}) \mathbf{C}^*]\Phi^*;
\end{align}
here we used Lemma \ref{l-model} to express the model operator in the right hand side of \eqref{equation}.
By the commutation relations in equation \eqref{e-comm}, the term $\Phi^*{\mathbf{B}}\Gamma{\mathbf{B}}^* U$ on the left hand side of \eqref{equation} cancels with the term $\mathbf{C}_* \Gamma \mathbf{C}^*\Phi^*$ on the right hand side of \eqref{equation}. Then \eqref{equation} can be rewritten as
\begin{align}
\notag
\Phi^* M_\xi
&=
M_z \Phi^*
+\Phi^*{\mathbf{B}} {\mathbf{I}}\ci{{\mathbb C}^d}{\mathbf{B}}^* U - M_z \mathbf{C} \mathbf{C}^*\Phi^*
\\ \label{e-PhiUn-01}
&=
M_z \Phi^*
+(\mathbf{C}_*-M_z\mathbf{C}){\mathbf{B}}^* M_\xi ;
\end{align}
the last identity holds because, by \eqref{e-comm}, we have $\Phi^*\mathbf{B} =\mathbf{C}_*$ and $\mathbf{C}^*\Phi^* = {\mathbf{B}}^* U =\mathbf{B}^* M_\xi $.
Right multiplying \eqref{e-PhiUn-01} by $M_\xi$ and using \eqref{e-PhiUn-01} we get
\begin{align*}
\Phi^* M_\xi^2
&=
M_z \Phi^*M_\xi+(\mathbf{C}_*- M_z \mathbf{C}){\mathbf{B}}^*M_\xi^2
\\&=
M_z^2 \Phi^*+M_z (\mathbf{C}_*- M_z \mathbf{C}){\mathbf{B}}^*M_\xi+ (\mathbf{C}_*- M_z \mathbf{C}){\mathbf{B}}^*M_\xi^2 .
\end{align*}
Right multiplying the above equation by $M_\xi$ and using \eqref{e-PhiUn-01} again we get the identity
\begin{align}\label{e-PhiUn}
\Phi^* M_\xi^n
=
M^n_z\Phi^*
+
\sum_{k=1}^n M_z^{k-1} (\mathbf{C}_* - M_z \mathbf{C})
{\mathbf{B}}^* M_\xi^{n-k+1},
\end{align}
with $n=3$. Right multiplying by $M_\xi$ and applying \eqref{e-PhiUn-01} we get by induction that \eqref{e-PhiUn} holds for all $n\ge 0$.
(The case $n=0$ trivially reads $\Phi^*=\Phi^*$, and equation \eqref{e-PhiUn-01} is precisely the case $n=1$.)
We now apply \eqref{e-PhiUn} to some $b\in\Ran \mathbf{B}$. By commutative diagram \eqref{d:agree-02} we get that $\Phi^* b =\mathbf{C}_*\mathbf{B}^* b$, i.e.~$(\Phi^* b)(z) = C_*(z) \mathbf{B}^*b$. Using this identity we get
\begin{align}\label{e-application}
\bigl( \Phi^* M_\xi^n b \bigr)(z)
& =
z^n ( \Phi^* b)(z)
+
\sum_{k=1}^n
z^{k-1}(C_*(z)- z C(z) ) {\mathbf{B}}^* M_\xi^{n-k+1} b
\\ \notag
& = z^n C_*(z)( \mathbf{B}^* b)(z)
+
(C_*(z)- z C(z) ) \sum_{k=1}^n
z^{k-1} {\mathbf{B}}^* M_\xi^{n-k+1} b .
\end{align}
To continue, we recall that $\mathbf{B}:{\mathbb C}^d\to L^2(\mu;E)$ acts as multiplication by matrix $B(\xi) = (b_1(\xi), b_2(\xi), \ldots, b_d(\xi))$, so its adjoint $\mathbf{B}^*:\mathcal{H}\subset L^2(\mu;E)\to {\mathbb C}^d$ is given by
\[
\mathbf{B}^* f
=
\int\ci\mathbb{T} B^*(\xi) f(\xi) d\mu(\xi)
\qquad\text{for }f\in \mathcal{H},
\]
where the integral can be expanded as
\[
\int\ci\mathbb{T} B^*(\xi) f(\xi) d\mu(\xi)
=
\begin{pmatrix}
\int\ci\mathbb{T} b_1(\xi)^* f(\xi) d\mu(\xi)\\
\int\ci\mathbb{T} b_2(\xi)^* f(\xi) d\mu(\xi)\\
\vdots\\
\int\ci\mathbb{T} b_d(\xi)^* f(\xi) d\mu(\xi)\\
\end{pmatrix}.
\]
Using the sum of geometric progression formula we evaluate the sum in \eqref{e-application} to
\begin{align}
\notag
\sum_{k=1}^n
z^{k-1} {\mathbf{B}}^* M_\xi^{n-k+1} b
&=
\sum_{k=1}^n
z^{k-1} \int\ci\mathbb{T} \xi^{n-k+1} B^*(\xi) b(\xi)
d\mu(\xi)
\\
\notag&=
\int\ci\mathbb{T} \sum_{k=1}^n
z^{k-1} \xi^{n-k+1} B^*(\xi) b(\xi)
d\mu(\xi)
\\
\label{e-geometric}&=
\int\ci\mathbb{T}
\frac{\xi^n - z^n}{1-z\bar\xi}
B^*(\xi) b(\xi)
d\mu(\xi) .
\end{align}
Thus, we have proved \eqref{e-repr} for monomials $h(\xi)= \xi^n$, $n\ge 0$. And by linearity of $\Phi^*$ the representation \eqref{e-repr} holds for (analytic) polynomials $h$ in $\xi$.
The argument leading to determine the action of $\Phi^*$ on polynomials $h$ in $\bar \xi$ is similar. But we found that the devil is in the details and therefore decided to include much of the argument.
First observe that the intertwining relation \eqref{e-intertwinePhi*} is equivalent to $\mathcal{M}_\theta^*\Phi^*=\Phi^* T^*$. Recalling $T^* = U^* + U^* \mathbf{B} (\Gamma^*- {\mathbf{I}}\ci{{\mathbb C}^d}) \mathbf{B}^*$ and the resolution of the adjoint model operator $\mathcal{M}_\theta^*$ (see second statement of Lemma \ref{l-model}), we obtain
\[
M_{\bar z}\Phi^*+ (\mathbf{C} \Gamma^* - \bar{z} \mathbf{C}_*)\mathbf{C}_*^* \Phi^*=
\mathcal{M}_\theta^*\Phi^*
=\Phi^* T^*
=
\Phi^* U^* - \Phi^* U^* \mathbf{B} (\Gamma^*- {\mathbf{I}}\ci{{\mathbb C}^d}) \mathbf{B}^*.
\]
The terms involving $\Gamma^*$ on the left hand side and the right hand side cancel by the commutation relations in equation \eqref{e-comm} (actually by their adjoints). Now, rearrangement and another application of the adjoints of the commutation relations in equation \eqref{e-comm} yields
\begin{align}
\notag
\Phi^* M_{\bar\xi}
&=
\Phi^* U^*
=
M_{\bar z}\Phi^*
+
\Phi^* U^* \mathbf{B} {\mathbf{I}}\ci{{\mathbb C}^d} \mathbf{B}^* - \bar{z} \mathbf{C}_*\mathbf{C}_*^* \Phi^*
=
M_{\bar z}\Phi^*
+
(\mathbf{C} - M_{\bar{z}}\mathbf{C}_* ) \mathbf{B}^*\\
\label{e-mxi}
&=
M_{\bar z}\Phi^*
+
M_{\bar{z}}(M_{{z}}\mathbf{C} - \mathbf{C}_* ) \mathbf{B}^*.
\end{align}
In analogy to the above, we right multiply \eqref{e-mxi} by $M_{\bar \xi}$ and apply \eqref{e-mxi} twice to obtain
\[
\Phi^* M_{\bar\xi}^2
=
M_{\bar z}^2\Phi^*
+
\sum_{k=1}^2
M_{\bar z}^{k} (M_z \mathbf{C}-\mathbf{C}_*)
{\mathbf{B}}^*M_{\bar \xi}^{2-k}
.
\]
Inductively, we conclude
\[
\Phi^* M_{\bar\xi}^n
=
M^n_{\bar z}\Phi^*
-
\sum_{k=1}^n
M_{\bar z}^{k} (\mathbf{C}_*-M_{z} \mathbf{C}) {\mathbf{B}}^*M_{\bar\xi}^{n-k},
\]
which \emph{differs in the exponents and in the sign from its counterpart expression} in equation \eqref{e-PhiUn}.
Through an application of this identity to $b$ and by the commutative diagram \eqref{d:agree-02}, we see
\begin{align*
\bigl( \Phi^* M_{\bar \xi}^n b \bigr)(z)
& =
\bar z^n ( \Phi^* b)(z)
-
\sum_{k=1}^n
\bar z^{k}(C_*(z)- z C(z) ) {\mathbf{B}}^* M_{\bar\xi}^{n-k} b
\\ \notag
& = \bar z^n C_*(z)( \mathbf{B}^* b)(z)
-
(C_*(z)- z C(z) ) \sum_{k=1}^n
\bar z^{k} {\mathbf{B}}^* M_{\bar\xi}^{n-k} b .
\end{align*}
As in equation \eqref{e-geometric}, but here with the geometric progression
\[
-\sum_{k=1}^n (\bar z)^k (\bar \xi)^{n-k} = \frac{(\bar \xi)^n - (\bar z)^n}{1-\bar \xi z},
\]
we can see equation \eqref{e-repr} for monomials $\bar \xi^n$, $n\in {\mathbb N}$.
And by linearity of $\Phi^*$, we obtain the \emph{same formula} \eqref{e-repr} for functions $h$ that are polynomials in $\bar\xi$.
We have proved \eqref{e-repr} for trigonometric polynomials $f$. The theorem now follows by a standard approximation argument, developed in \cite{LT09}. The application of this argument to the current situation is a slight extension of the one used in \cite{LT15}.
Fix $f\in C^1(\mathbb{T})$ and let $\{p_k\}$ be a sequence of trigonometric polynomials with uniform on $\mathbb{T}$ approximations $p_k\rightrightarrows f$ and $p_k'\rightrightarrows f'$.
In particular, we have $|p_k'|$ is bounded (with bound independent of $k$) and $p_k\to f$ as well as $p_kb\to fb$ in $L^2(\mu;E)$. Since $\Phi^*$ is a unitary operator, it is bounded and therefore we have convergence on the left hand side
$\Phi^* p_k b \to \Phi^* f b$
in ${\mathcal K}_\theta$.
To investigate convergence on the right hand side, first recall that the model space is a subspace of the weighted space $L^2(W;{\mathfrak D}_*\oplus {\mathfrak D})$.
So convergence of the first term on the right hand side happens, since $p_k\rightrightarrows f$ and the operator norm $\|\mathbf{C}_* \mathbf{B}^*\| = 1$ implies $p_kC_*(z) \mathbf{B}^*b=p_k\mathbf{C}_* \mathbf{B}^*b\to f\mathbf{C}_* \mathbf{B}^*b=fC_*(z) \mathbf{B}^*b$ in ${\mathcal K}_\theta$.
Lastly, to see convergence of the second term on the right hand side, consider auxiliary functions $f_k:=f-p_k$. We have $f_k\rightrightarrows 0$ and $f_k'\rightrightarrows 0$. Let $I_{\xi, z}\subset \mathbb{T}$ denote the shortest arc connecting $\xi $ and $z$. Then by the intermediate value theorem
\begin{align*}
|f_k(\xi )- f_k(z)|
\le \|f_k'\|_\infty |I_{\xi,z}|
\qquad\text{for all } \xi,z\in\mathbb{T}.
\end{align*}
In virtue of the geometric estimate $|I_{\xi,z}|\le\frac\pi2 |\xi-z|$, we obtain
\[
\left| \frac{f_k(\xi)-f_k(z)}{1-\overline \xi z} \right| \le \frac\pi2 \|f_k'\|_\infty \to 0 \qquad \text{as } k\to \infty.
\]
And since $\mathbf{B}^*$ is bounded as a partial isometry, we conclude the componentwise uniform convergenc
\begin{align*
\int\frac{p_k(\xi)-p_k(z)}{1-\overline \xi z}B^*(\xi) b(\xi) \, d\mu(\xi)
\quad
\rightrightarrows
\quad
\int\frac{f(\xi)-f(z)}{1-\overline\xi z}B^*(\xi) b(\xi) \, d\mu(\xi)
\qquad
z\in \mathbb{T}.
\end{align*}
By Lemma \ref{l:bound-C} below the functions $W^{1/2}C$ and $W^{1/2}C_*$ are bounded, and so is the function $W^{1/2}C_1$, $C_1(z) := C_*(z) - z C(z)$. That means the multiplication operator $f\mapsto C_1 f$ is a bounded operator $L^2({\mathfrak D})\to L^2(W;{\mathfrak D}_*\oplus {\mathfrak D})$ (recall that in our case ${\mathfrak D}={\mathfrak D}_*$ and we use ${\mathfrak D}_*$ here only for the consistency with the general model notation).
The uniform convergence implies the convergence in $L^2({\mathfrak D})$, so the boundedness of the multiplication by $C_1$ implies the convergence in norm in the second term in the right hand side of \eqref{e-repr} (in the norm of $L^2(W;{\mathfrak D}_*\oplus{\mathfrak D})$).
\end{proof}
\section{Model and agreement of operators}\label{s-ModAgree}
We want to explain how to get operators $\mathbf{C}$ and $\mathbf{C}_*$ that agree with each other.
To do that we need to understand in more detail how the model is constructed, and what operator gives the unitary equivalence of the function and its model.
Everything starts with the notion of unitary dilation. Recall that for a contraction $T$ in a Hilbert space $H$ its unitary dilation is a unitary operator $\mathcal{U}$ on a bigger space $\mathcal{H}$, $H\subset \mathcal{H}$ such that for all $n\ge 0$
\begin{align}
\label{dila-01}
T^n = P\ci H \mathcal{U}^n \bigm|_{H}.
\end{align}
Taking the adjoint of this identity we immediately get that
\begin{align}
\label{dila-02}
(T^*)^n = P\ci H \mathcal{U}^{-n} \bigm|_{H}.
\end{align}
A dilation is called \emph{minimal} if it is impossible to replace $\mathcal{U}$ by its restriction to a reducing subspace and still have the identities \eqref{dila-01} and \eqref{dila-02}.
The structure of minimal unitary dilations is well known.
\begin{theo}[{\cite[Theorem 1.4]{Nik-Vas_model_MSRI_1998}} and {\cite[Theorem 1.1.16]{Nik-book-v2}}]
\label{t:MUDil}
Let $\mathcal{U}:\mathcal{H}\to\mathcal{H}$ be a minimal unitary dilation of a contraction $T$. Then $\mathcal{H}$ can be decomposed as $\mathcal{H} = G_*\oplus H \oplus G$, and with respect to this decomposition $\mathcal{U}$ can be represented as
\begin{align}
\label{MUDil}
\mathcal{U} = \left(
\begin{array}{ccc}
\mathcal{E}_*^* & 0 & 0 \\
D\ci{T^*}V_*^* & T & 0 \\
-V T^* V_*^* & V D\ci{T} & \mathcal{E}
\end{array}
\right)
\end{align}
where $\mathcal{E}:G\to G$ and $\mathcal{E}_*:G_*\to G_*$ are pure isometries, $V$ is a partial isometry with initial space ${\mathfrak D}_{T}$ and the final space $\ker \mathcal{E}^*$ and $V_*$ is a partial isometry with initial space ${\mathfrak D}\ci{T^*}$ and final space $\ker \mathcal{E}_*^*$.
Moreover, any minimal unitary dilation of\, $T$ can be obtained this way. Namely if we pick auxiliary Hilbert spaces $G$ and $G_*$ and isometries $\mathcal{E}$ and $\mathcal{E}_*$ there with $\dim \ker \mathcal{E}^* = \dim{\mathfrak D}\ci{T}$, $\dim\ker \mathcal{E}_*^* = \dim{\mathfrak D}\ci{T*}$ and then pick arbitrary partial isometries $V$ and $V_*$ with initial and final spaces as above, then \eqref{MUDil} will give us a minimal unitary dilation of $T$.
\end{theo}
The construction of the model then goes as follows. We take auxiliary Hilbert spaces ${\mathfrak D}$ and ${\mathfrak D}_*$, $\dim {\mathfrak D} = \dim{\mathfrak D}\ci{T}$, $\dim {\mathfrak D}_* = \dim{\mathfrak D}\ci{T*}$, and construct operators $\mathcal{E}$ and $\mathcal{E}_*$ such that $\ker \mathcal{E}^* ={\mathfrak D}$, $\ker\mathcal{E}_*^* ={\mathfrak D}_*$. We can do that by putting $G=\ell^2({\mathfrak D})=\ell^2({\mathbb Z}_+;{\mathfrak D})$, and defining $\mathcal{E} (x_0, x_1, x_2, \ldots ) = (0, x_0, x_1, x_2, \ldots)$, $x_k\in {\mathfrak D}$, and similarly for $\mathcal{E}_*$.
Picking arbitrary partial isometries $V$ and $V_*$ with initial and final spaces as in the above Theorem \ref{MUDil} we get a minimal unitary dilation $U$ of $T$ given by \eqref{MUDil}.
\begin{rem*}
Above, we were speaking a bit informally, by identifying $x\in {\mathfrak D}$ with the sequence $(x, 0, 0, 0, \ldots) \in \ell^2({\mathfrak D})$, and $x_*\in {\mathfrak D}_*$ with $(x_*, 0, 0, 0, \ldots) \in \ell^2({\mathfrak D})$.
To be absolutely formal, we need to define canonical embeddings $\mathbf{e}: {\mathfrak D}\to G=\ell^2({\mathfrak D})$, $\mathbf{e}_*:{\mathfrak D}_*\to G_*=\ell^2({\mathfrak D}_*)$ with
\begin{align}
\label{eq:e}
\mathbf{e}(x) &:=(x, 0, 0, 0, \ldots), \qquad x\in {\mathfrak D}, \\
\label{eq:e_*}
\mathbf{e}_*(x_*) &:=(x_*, 0, 0, 0, \ldots), \qquad x\in {\mathfrak D}_* .
\end{align}
Then, picking arbitrary unitary operators $V:{\mathfrak D}\ci{T}\to {\mathfrak D}$, $V_*:{\mathfrak D}\ci{T^*}\to {\mathfrak D}_*$, we rewrite \eqref{MUDil} to define the corresponding unitary dilation as
\begin{align}
\label{MUDil-01}
\mathcal{U} = \left(
\begin{array}{ccc}
\mathcal{E}_*^* & 0 & 0 \\
D\ci{T^*}V_*^* \mathbf{e}_*^* & T & 0 \\
-\mathbf{e} V T^* V_*^*\mathbf{e}_*^* & \mathbf{e} V D\ci{T} & \mathcal{E}
\end{array}
\right) \,.
\end{align}
The reason for being so formal is that if $\dim {\mathfrak D}\ci{T}=\dim{\mathfrak D}\ci{T^*}$it is often convenient to put ${\mathfrak D}={\mathfrak D}_*$, but we definitely want to be able to distinguish between the cases when ${\mathfrak D}$ is identified with $\ker \mathcal{E}$ and when with $\ker\mathcal{E}_*$.
\end{rem*}
\medskip
We then define \emph{functional embeddings} $\pi:L^2({\mathfrak D})\to \mathcal{H}$ and $\pi_*: L^2({\mathfrak D}_*)\to \mathcal{H}$ by
\begin{align*}
\pi \left(\sum_{k\in{\mathbb Z}} z^k e_k \right) & = \sum_{k\in{\mathbb Z}} \mathcal{U}^k \mathbf{e} (e_k), \qquad e_k\in {\mathfrak D}, \\
\pi_* \left(\sum_{k\in{\mathbb Z}} z^k e_{k} \right) & = \sum_{k\in{\mathbb Z}} \mathcal{U}^{k+1} \mathbf{e}_*(e_{k}), \qquad e_{k}\in {\mathfrak D}_* .
\end{align*}
We refer the reader to \cite[Section 1.6]{Nik-Vas_model_MSRI_1998} or to \cite[Section 1.2]{Nik-book-v2} for the details. Note that there ${\mathfrak D}$ and ${\mathfrak D}_*$ were abstract spaces, $\dim {\mathfrak D} = \dim\ker\mathcal{E}^*$ and $\dim {\mathfrak D}_*=\dim\ker\mathcal{E}_*^*$, and the unitary operators $v:{\mathfrak D}\to\ker \mathcal{E}^*$, $v_* :{\mathfrak D}_* \to \ker \mathcal{E}_*^*$ used in the formulas there are just the canonical embeddings $\mathbf{e}$ and $\mathbf{e}_*$ in our case.
Note that $\pi$ and $\pi_*$ are isometries.
Note also that for $k\ge0$
\begin{align*}
\mathcal{U}^k \mathbf{e}(e) &= \mathcal{E}^k e, &&e\in {\mathfrak D},\\
\mathcal{U}^{-k} \mathbf{e}_*(e_*) & = \mathcal{E}_*^k e_*, && e_*\in {\mathfrak D}_*,
\end{align*}
so
\begin{align*}
\pi (H^2({\mathfrak D})) = G, \qquad \pi_* (H^2_-({\mathfrak D}_*)) = G_* .
\end{align*}
The characteristic function is then defined as follows. We consider
the operator $\boldsymbol{\theta}=\pi_*^*\pi :L^2({\mathfrak D})\to L^2({\mathfrak D}_*)$. It is easy to check that $M_z\boldsymbol{\theta} = \boldsymbol{\theta} M_z$, so the $\boldsymbol{\theta}$ is a multiplication by a function $\theta\in L^\infty({\mathfrak D}\!\!\to\! {\mathfrak D}_*)$. It is not hard to check that $\boldsymbol{\theta} $ is a contraction, so $\|\theta\|_\infty\le 1$. Since
\begin{align*}
\pi(H^2({\mathfrak D})) = G \perp G_* = \pi_*(H^2_-({\mathfrak D}_*)),
\end{align*}
we can conclude that $\theta\in H^\infty({\mathfrak D}\!\!\to\! {\mathfrak D}_*)$.
The characteristic function $\theta=\theta\ci T$ can be explicitly computed, see \cite[Theorem 1.2.10]{Nik-book-v2},
\begin{align}
\label{CharFunction-01}
\theta_T(z) = V_*\left( -T + z D_{T^*}\left(\mathbf{I}\ci\mathcal{H} - z T^*\right)^{-1} D_T \right) V^* \Bigm|_{{\mathfrak D}}, \qquad z\in {\mathbb D}.
\end{align}
Note that the particular representation of $\theta$ depends on the coordinate operators $V$ and $V_*$ identifying defect spaces ${\mathfrak D}\ci{T}$ and ${\mathfrak D}\ci{T^*}$ with the abstract spaces ${\mathfrak D}$ and ${\mathfrak D}_*$.
To construct a model (more precisely its particular transcription) we need to construct a unitary map $\Psi$ between the space $\mathcal{H}$ of the minimal unitary dilation $\mathcal{U}$ and its spectral representation.
Namely, we represent $\mathcal{U}$ as a multiplication operator in some subspace $\widetilde{\mathcal K}=\widetilde{\mathcal K}_\theta$ of $L^2({\mathfrak D}_*\oplus {\mathfrak D})$ or its weighted version.
We need to construct a unitary operator $\Psi:\mathcal{H}\to \widetilde {\mathcal K}$ intertwining $\mathcal{U}$ and $M_z$ on $\widetilde{\mathcal K}$, i.e. such that
\begin{align}
\label{eq:Psi-01}
\Psi \mathcal{U} = M_z \Psi .
\end{align}
Note that if $T$ is a completely non-unitary contraction, then $\pi(L^2({\mathfrak D})) + \pi_* (L^2({\mathfrak D}_*))$ is dense in $\mathcal{H}$.
So, for $\Psi$ to be unitary it is necessary and sufficient that $\Psi^*$ acts isometrically on $\pi (L^2({\mathfrak D}))$ and on $\pi_*( L^2({\mathfrak D}_*))$, and that for all $f\in L^2({\mathfrak D})$, $g\in L^2({\mathfrak D}_*)$
\begin{align}
\label{pi^pi*}
(\Psi^* \pi f , \Psi^*\pi g)\ci{\widetilde{\mathcal K}} = (\pi f, \pi_* g)\ci{\mathcal{H}} =(\boldsymbol{\theta} f, g)\ci{L^2({\mathfrak D}_*)} ;
\end{align}
the last equality here is just the definition of $\boldsymbol{\theta}$.
Of course, we need $\Psi^*$ to be onto, but that can be easily accomplished by restricting the target space $\widetilde{\mathcal K}$ to $\Ran \Psi^*$.
Summing up, we have:
\[
\begin{array}{ccccccc}
\mathcal{H}&=&G&\oplus& H &\oplus&G_*\vspace{.4cm}\\\vspace{.4cm}
\,\,\,\phantom{\hat{\varphi}}\Bigg\downarrow{\Psi^*}
&&\quad\,\,\phantom{\hat{\varphi}}\Bigg\downarrow{\Psi^*|\ci{G}}
&&\quad\quad\,\phantom{\hat{\varphi}}\Bigg\downarrow{\Psi^*|\ci{H}}
&&\,\,\quad\phantom{\hat{\varphi}}\Bigg\downarrow{\Psi^*|\ci{G_*}}\\
\widetilde{\mathcal K}&=&\mathcal{G}&\oplus&{\mathcal K}_\theta&\oplus&\mathcal{G}_*
\end{array}
\]
\subsection{Pavlov transcription}
Probably the easiest way to construct the model is to take $\widetilde {\mathcal K}$ to be the weighted space $L^2({\mathfrak D}_*\oplus{\mathfrak D} , W)$ where the weight $W$ is picked to make the simplest operator $\Psi^*$ to an isometry, and is given by
\begin{align}
\label{PavlovWeight}
W (z)= \left(\begin{array}{cc}\mathbf{I}\ci{{\mathfrak D}_*} & \theta (z)\\ \theta(z)^* & \mathbf{I}\ci{{\mathfrak D}} \end{array}\right).
\end{align}
Now operator $\Psi^*$ is defined on $\pi(L^2({\mathfrak D}))$ and on $\pi_* (L^2({\mathfrak D}_*))$ as
\begin{align}
\nota
\Psi^*\Bigl(\sum_{k\in{\mathbb Z}}\mathcal{U}^k \mathbf{e}(e_k) \Bigr) & = \sum_{k\in{\mathbb Z}} z^k
\left(\begin{array}{c}0 \\varepsilon_k \end{array}\right), \qquad e_k\in{\mathfrak D}, \\
\label{Psi*Pavlov-02}
\Psi^*\Bigl(\sum_{k\in{\mathbb Z}}\mathcal{U}^k \mathbf{e}_*(e_{k}) \Bigr) & = \sum_{k\in{\mathbb Z}} z^{k-1}
\left(\begin{array}{c}e_k \\ 0\end{array}\right) , \qquad e_{k}\in{\mathfrak D}_*\,,
\end{align}
or equivalently
\begin{align*}
\Psi^* (\pi f ) & =
\left(\begin{array}{c}0 \\ f \end{array}\right), \qquad f\in L^2({\mathfrak D}), \\
\Psi^*(\pi_* f) & =
\left(\begin{array}{c} f \\ 0\end{array}\right) , \qquad f \in L^2({\mathfrak D}_*)\,,
\end{align*}
The incoming and outgoing spaces $\mathcal{G}_*= \Psi^* G_*$, $\mathcal{G}= \Psi^* G$ are given by
\begin{align*}
\mathcal{G}_* := \clos\ci{\widetilde{\mathcal K}}\left\{\left(\begin{array}{c} f \\ 0 \end{array} \right): f\in H^2_-({\mathfrak D}_*) \right\}, \qquad
\mathcal{G} := \clos\ci{\widetilde{\mathcal K}}\left\{\left(\begin{array}{c} 0 \\ f \end{array} \right): f\in H^2({\mathfrak D}) \right\},
\end{align*}
and the model space ${\mathcal K}={\mathcal K}_\theta$ is defined as
\[
{\mathcal K}_\theta= \widetilde{\mathcal K} \ominus (\mathcal{G}_*\oplus\mathcal{G}).
\]
\subsection{Sz.-Nagy--\texorpdfstring{Foia\c{s}}{Foias} transcription}
This transcription appears when one tries to make the operator $\Psi^*$ to act into a non-weighted space $L^2({\mathfrak D}_*\oplus{\mathfrak D})$. We make the action of the operator $\Psi^*$ on $\pi_*(L^2({\mathfrak D}_*))$ as simple as possible,
\begin{align}
\label{Psi*N-F-01}
\Psi^*\Bigl(\sum_{k\in{\mathbb Z}}\mathcal{U}^k \mathbf{e}_*(e_{k}) \Bigr) & = \sum_{k\in{\mathbb Z}} z^{k-1}
\left(\begin{array}{c}e_k \\ 0\end{array}\right), \qquad e_{k}\in{\mathfrak D}_*
\end{align}
(this is exactly as in \eqref{Psi*Pavlov-02}). Action of $\Psi^*$ on $\pi(L^2({\mathfrak D}))$ is defined as
\begin{align}
\label{Psi*N-F-02}
\Psi^*\Bigl(\sum_{k\in{\mathbb Z}}\mathcal{U}^k \mathbf{e}(e_{k}) \Bigr) & = \sum_{k\in{\mathbb Z}} z^{k}
\left(\begin{array}{c}\theta e_k \\ \Delta e_k\end{array}\right), \qquad e_{k}\in{\mathfrak D}\,,
\end{align}
where $\Delta(z) = (\mathbf{I} - \theta(z)^*\theta(z))^{1/2}$. The equations \eqref{Psi*N-F-01} and \eqref{Psi*N-F-02} can clearly be rewritten as
\begin{align}
\label{Psi*N-F-03}
\Psi^* (\pi f ) & =
\left(\begin{array}{c}\theta f \\ \Delta f \end{array}\right), \qquad f\in L^2({\mathfrak D}), \\
\label{Psi*N-F-04}
\Psi^*(\pi_* f) & =
\left(\begin{array}{c} f \\ 0\end{array}\right) , \qquad f \in L^2({\mathfrak D}_*)\, .
\end{align}
Note, that $\theta$ in the top entry in \eqref{Psi*N-F-02} and \eqref{Psi*N-F-03} is necessary to get
\eqref{pi^pi*}; after \eqref{Psi*N-F-01} (equivalently \eqref{Psi*N-F-04}) is chosen, one does not have any choice here. The term $\Delta$ in the bottom entry of \eqref{Psi*N-F-02} and \eqref{Psi*N-F-03} is there to make $\Psi^*$ act isometrically on $\pi(L^2({\mathfrak D}))$. There is some freedom here; one can left multiply $\Delta$ by any operator-valued function $\phi$ such that $\phi(z)$ acts isometrically on $\Ran \Delta(z)$. However, picking just $\Delta$ is the canonical choice for the Sz.-Nagy--Foia\c{s} transcription, and we will follow it.
The incoming and outgoing spaces are given by
\begin{align*}
\mathcal{G}_* := \left(\begin{array}{c} H^2_-({\mathfrak D}_*) \\ 0\end{array}\right), \qquad
\mathcal{G} := \left(\begin{array}{c}\theta \\ \Delta\end{array}\right) H^2({\mathfrak D})
.
\end{align*}
The model space is given by
\begin{align}
\label{N-F-K_theta}
{\mathcal K}_\theta :=
\left(\begin{array}{c} L^2({\mathfrak D}_*) \\ \clos \Delta L^2({\mathfrak D}) \end{array}\right) \ominus
(\mathcal{G}_*\oplus\mathcal{G})
=
\left(\begin{array}{c} H^2({\mathfrak D}_*) \\ \clos \Delta L^2({\mathfrak D}) \end{array}\right) \ominus
\left(\begin{array}{c}\theta \\ \Delta\end{array}\right) H^2({\mathfrak D}) .
\end{align}
\begin{rem*}
While the orthogonal projection from
\[
\left(\begin{array}{c} L^2({\mathfrak D}_*) \\ \clos \Delta L^2({\mathfrak D}) \end{array}\right)
\quad\text{to}\quad
\left(\begin{array}{c} L^2({\mathfrak D}_*) \\ \clos \Delta L^2({\mathfrak D}) \end{array}\right)\ominus \mathcal{G}_*
\]
is rather simple, the one from
\[
\left(\begin{array}{c} L^2({\mathfrak D}_*) \\ \clos \Delta L^2({\mathfrak D}) \end{array}\right)
\quad\text{to}\quad
\left(\begin{array}{c} L^2({\mathfrak D}_*) \\ \clos \Delta L^2({\mathfrak D}) \end{array}\right)\ominus \mathcal{G}
\]
involves the range of a Toeplitz operator.
\end{rem*}
\subsection{De Branges--Rovnyak transcription} This transcription looks most complicated, but its advantage is that both coordinates are analytic functions. To describe this transcription, we use the auxiliary weight $W = W(z)$ as in the Pavlov transcription, see \eqref{PavlovWeight}.
The model space is the subspace of $L^2({\mathfrak D}_*\oplus {\mathfrak D}, W^{[-1]})$, where for a self-adjoint operator $A$ the symbol $A^{[-1]}$ denotes its Moore--Penrose (pseudo)inverse, i.e.~$A^{[-1]}=0$ on $\Ker A$ and $A^{[-1]}$ is the left inverse of $A$ on $(\Ker A)^\perp$.
The operator $\Psi^*: \mathcal{H} \to L^2({\mathfrak D}_*\oplus {\mathfrak D}, W^{[-1]})$ is defined by
\begin{align*}
\Psi^* (\pi f ) & = W \left(\begin{array}{c} 0 \\ f \end{array}\right) =
\left(\begin{array}{c} \theta f\\ f\end{array}\right), \qquad f\in L^2({\mathfrak D}), \\
\Psi^*(\pi_* f) & = W \left(\begin{array}{c} f \\ 0\end{array}\right) =
\left(\begin{array}{c} f \\ \theta^* f\end{array}\right) , \qquad f \in L^2({\mathfrak D}_*)\, .
\end{align*}
The incoming and outgoing spaces are
\begin{align*}
\mathcal{G}_*:= \left(\begin{array}{c} \mathbf{I} \\ \theta^* \end{array}\right) H^2({\mathfrak D}_*), \qquad
\mathcal{G}:= \left(\begin{array}{c} \theta \\ \mathbf{I} \end{array}\right) H^2({\mathfrak D}),
\end{align*}
and the model space is defined as
\begin{align*}
{\mathcal K}_\theta := \left\{ \left(\begin{array}{c} f \\ g \end{array}\right) : f\in H^2({\mathfrak D}_*),\ g\in H^2_-({\mathfrak D}), \ g-\theta^* f \in \Delta L^2({\mathfrak D}) \right\},
\end{align*}
see \cite[Section 3.7]{Nik-Vas_model_MSRI_1998} for the details (there is a typo in \cite[Section 3.7]{Nik-Vas_model_MSRI_1998}, in the definition of ${\mathcal K}_\theta$ on p.~251 it should be $f\in H^2(E_*)$, $g\in H^2(E)$) .
\subsection{Parametrizing operators for the model, agreeing with coordinate operators}
The parametrizing operators that agree with the coordinate operators $V$ and $V_*$ are described in the following lemma, which holds for any transcription of the model.
Let $T$ be a c.n.u.~contraction, and let $V:{\mathfrak D}\ci{T} \to {\mathfrak D}$ and $V_*:{\mathfrak D}\ci{T^*} \to {\mathfrak D}_*$ be coordinate operators for the defect spaces of $T$. Let $\theta=\theta\ci{T} = \theta\ci{T, V, V_*}\in H^\infty({\mathfrak D}\!\!\to\! {\mathfrak D}_*)$ be the characteristic function of $T$, defined by \eqref{CharFunction-01}, and let
$\mathcal{M}_\theta$ be the corresponding model operator (in any transcription).
Recall that $\Psi $ is a unitary operator intertwining the minimal unitary dilation $\mathcal{U}$ of $T$ and the multiplication operator $M_z$ in the corresponding function space, see \eqref{eq:Psi-01}. The operator $\Psi$ determines transcription of the model, so for any particular transcription it is known.
Define
\begin{align}
\label{eq:tilde e}
\widetilde \mathbf{e} := \Psi^*\mathbf{e}, \qquad \widetilde\mathbf{e}_* := \Psi^* \mathbf{e}_*,
\end{align}
where the embedding $\mathbf{e}$ and $\mathbf{e}_*$ are defined by \eqref{eq:e}, \eqref{eq:e_*}.
\begin{lem}
\label{l:C-C*}
Under the above assumptions the parametrizing operators $\mathbf{C}_*:{\mathfrak D}_*\to{\mathfrak D}\ci{\mathcal{M}_\theta^*}$ and $\mathbf{C}:{\mathfrak D}\to{\mathfrak D}\ci{\mathcal{M}_\theta}$ given by
\begin{align}
\label{C*_CoFree}
\mathbf{C}_* e_* &= \left(D\ci{\mathcal{M}_\theta^*}\bigm|_{{\mathfrak D}_{\mathcal{M}_\theta^*}}\right)^{-1} P\ci{{\mathcal K}_\theta} M_z \widetilde\mathbf{e}_* (e_*) , \qquad e_*\in {\mathfrak D}_*,
\\
\label{C_CoFree}
\mathbf{C} e &= \left(D\ci{\mathcal{M}_\theta}\bigm|_{{\mathfrak D}_{\mathcal{M}_\theta}}\right)^{-1} P\ci{{\mathcal K}_\theta} M_{\bar z} \widetilde\mathbf{e} (e) , \qquad e\in{\mathfrak D},
\end{align}
agree with the coordinate operators $V$ and $V_*$.
\end{lem}
\begin{rem*}It follows from the equation \eqref{MUDil-02} below that $P\ci{{\mathcal K}_\theta} M_z \widetilde\mathbf{e}_* (e_*) \in \Ran D\ci{\mathcal{M}_\theta^*}$ as well as $P\ci{{\mathcal K}_\theta} M_{\bar z} \widetilde\mathbf{e} (e) \in \Ran D\ci{\mathcal{M}_\theta}$, so everything in \eqref{C*_CoFree}, \eqref{C_CoFree} is well defined.
\end{rem*}
\begin{proof}[Proof of Lemma \ref{l:C-C*}]
Right and left multiplying \eqref{MUDil-01} by $\Psi$ and $\Psi^*$ respectively, we get
\begin{align}
\label{MUDil-02}
\Psi^* \mathcal{U} \Psi = \left(
\begin{array}{ccc}
\widetilde\mathcal{E}_*^* & 0 & 0 \\
D\ci{\mathcal{M}_\theta^*}\mathbf{C}_* \widetilde\mathbf{e}_*^* & \mathcal{M}_\theta & 0 \\
-\widetilde\mathbf{e} \mathbf{C}^* \mathcal{M}_\theta^* \mathbf{C}_*\widetilde\mathbf{e}_*^* & \widetilde\mathbf{e} \mathbf{C}^* D\ci{\mathcal{M}_\theta} & \widetilde\mathcal{E}
\end{array}
\right) \,,
\end{align}
where $\widetilde \mathcal{E}= \Psi^* \mathcal{E}\Psi$, $\widetilde \mathcal{E}_*= \Psi \mathcal{E}_*\Psi$, $\mathbf{C}^* = V\Psi$, $\mathbf{C}_*^* = V_*\Psi$, $\widetilde \mathbf{e}= \Psi^*\mathbf{e}$, $\widetilde \mathbf{e}_*=\Psi^*\mathbf{e}_*$.
The operators $\widetilde\mathbf{e}$ and $\widetilde\mathbf{e}_*$ are the canonical embeddings of ${\mathfrak D}$ and ${\mathfrak D}_*$ into $\mathcal{G}$ and $\mathcal{G}_*$ that agree with the canonical embeddings $\mathbf{e}$ and $\mathbf{e}_*$. The operators $\mathbf{C}$ and $\mathbf{C}_*$ are the parameterizing operators for the defect spaces of the model operator $\mathcal{M}_\theta$ that agree with the coordinate operators $V$ and $V_*$ for the defect spaces of the operator $T$.
In any particular transcription of the model, the operator $\Psi^*\mathcal{U}\Psi$ is known (it is just the multiplication by $z$ in an appropriate function space), so we get from the decomposition \eqref{MUDil-02}
\begin{align*}
D\ci{\mathcal{M}_\theta^*}\mathbf{C}_* \widetilde\mathbf{e}_*^* = P\ci{{\mathcal K}_\theta} M_z \Bigm|_{\mathcal{G}_*} , \qquad
D\ci{\mathcal{M}_\theta}\mathbf{C} \widetilde\mathbf{e}^* = P\ci{{\mathcal K}_\theta} M_{\bar z} \Bigm|_{\mathcal{G}_*} .
\end{align*}
Right and left multiplying the first identity by $\mathbf{e}_*$ and $\Bigl(D\ci{\mathcal{M}_\theta^*}\bigm|_{{\mathfrak D}_{\mathcal{M}_\theta^*}}\Bigr)^{-1}$ respectively, we get \eqref{C*_CoFree}. Similarly, to get \eqref{C_CoFree} we just right and left multiply the second identity by $\mathbf{e}$ and $\left(D\ci{\mathcal{M}_\theta}\bigm|_{{\mathfrak D}_{\mathcal{M}_\theta}}\right)^{-1}$.
\end{proof}
Applying the above Lemma \ref{l:C-C*} to a particular transcription of the model, we can get more concrete formulas for $\mathbf{C}$, $\mathbf{C}_*$ just in terms of characteristic function $\theta$.
For example, the following lemma gives formulas for $\mathbf{C}$ and $\mathbf{C}_*$ in the Sz.-Nagy--Foia\c{s} transcription.
\begin{lem}\label{l:C_N-F}
Let $T$ be a c.n.u.~contraction, and let $\mathcal{M}_\theta $ be its model in Sz.-Nagy--Foia\c{s} transcription, with the characteristic function $\theta=\theta\ci{T, V, V_*}$, $\theta\in H^\infty({\mathfrak D}\!\!\to\!{\mathfrak D}_*)$.
Then the maps $\mathbf{C}_*:{\mathfrak D}_*\to{\mathfrak D}\ci{\mathcal{M}_\theta^*}$ and $\mathbf{C}:{\mathfrak D}\to{\mathfrak D}\ci{\mathcal{M}_\theta}$ given by
\begin{align}\label{C*_N-F}
\mathbf{C}_* e_* &=
\kf{ {\mathbf{I}}-{\theta} (z){\theta}^\ast(0)}{ -\Delta (z){\theta}^\ast(0) } \left( {\mathbf{I}}- {\theta}(0){\theta}^\ast(0) \right)^{-1/2} e_*,& e_*&\in {\mathfrak D}_*,
\\
\label{C_N-F}
\mathbf{C} e & =
\kf{ z^{-1} \left({\theta}(z)- {\theta}(0)\right)}{ z^{-1} \Delta (z)}\left( {\mathbf{I}}- {\theta}^\ast(0){\theta}(0) \right)^{-1/2} e,&e&\in {\mathfrak D},
\end{align}
agree with the coordinate operators $V$ and $V_*$.
\end{lem}
\begin{proof}
To prove \eqref{C*_N-F} we will use \eqref{C*_CoFree}. It follows from \eqref{Psi*N-F-01} that
\[
\widetilde\mathbf{e}_*(e_*)=z^{-1} \left(\begin{array}{c} e_* \\mathbb{O}\end{array}\right),
\]
so by \eqref{C*_CoFree}
\begin{align}
\label{C*_N-F-02}
\mathbf{C}_* e_* = (\mathbf{I} - \mathcal{M}_\theta\mathcal{M}_\theta^*)\bigm|_{{\mathfrak D}_{\mathcal{M}_\theta^*}}^{-1/2} P\ci{{\mathcal K}_\theta} \left(\begin{array}{c} e_* \\mathbb{O}\end{array}\right),\qquad e_*\in{\mathfrak D}_*.
\end{align}
It is not hard to show that
\begin{align}
\label{P_theta_N-F}
P\ci{{\mathcal K}_\theta} \left(\begin{array}{c} e_* \\mathbb{O}\end{array}\right) = \left(\begin{array}{c} \mathbf{I} - \theta\theta(0)^* \\ -\Delta\theta(0)^*\end{array}\right) e_*\,.
\end{align}
One also can compute
\begin{align}
\label{comm-defects-00}
(\mathbf{I}-\mathcal{M}_\theta\mathcal{M}_\theta^*) \left(\begin{array}{c} f \\ g\end{array}\right) = \left(\begin{array}{c} \mathbf{I} - \theta\theta(0)^* \\ -\Delta\theta(0)^*\end{array}\right) f(0), \qquad \left(\begin{array}{c} f \\ g\end{array}\right)\in{\mathcal K}_\theta.
\end{align}
Combining the above identities we get that
\begin{align}
\label{comm-defects-01}
(\mathbf{I}-\mathcal{M}_\theta\mathcal{M}_\theta^*) P\ci{{\mathcal K}_\theta} \left(\begin{array}{c} e_* \\mathbb{O}\end{array}\right)
=\left(\begin{array}{c} \mathbf{I} - \theta\theta(0)^* \\ -\Delta\theta(0)^*\end{array}\right)
(e_*- \theta(0)\theta^*(0) e_*).
\end{align}
As we discussed above just after \eqref{C_CoFree}, $P\ci{{\mathcal K}_\theta} \left(\begin{array}{c} e_* \\mathbb{O}\end{array}\right)\in \Ran D\ci{\mathcal{M}_\theta^*}$, so in \eqref{comm-defects-01} we can replace
$(\mathbf{I}-\mathcal{M}_\theta\mathcal{M}_\theta^*)$ by its restriction onto ${\mathfrak D}\ci{\mathcal{M}_\theta^*}$.
Applying $(\mathbf{I}-\mathcal{M}_\theta\mathcal{M}_\theta^*)\bigm|_{{\mathfrak D}\ci{\mathcal{M}_\theta^*}}$ to \eqref{comm-defects-01} (with $(\mathbf{I}-\mathcal{M}_\theta\mathcal{M}_\theta^*)$ replaced by its restriction onto ${\mathfrak D}\ci{\mathcal{M}_\theta}$) and using \eqref{comm-defects-00} we get
\begin{align*}
\bigl( (\mathbf{I}-\mathcal{M}_\theta\mathcal{M}_\theta^*)\bigm|_{{\mathfrak D}_{\mathcal{M}_\theta^*}} \bigr)^2 P\ci{{\mathcal K}_\theta} \left(\begin{array}{c} e_* \\mathbb{O}\end{array}\right)
=\left(\begin{array}{c} \mathbf{I} - \theta\theta(0)^* \\ -\Delta\theta(0)^*\end{array}\right)
\bigl(\mathbf{I}\ci{{\mathfrak D}_*}- \theta(0)\theta^*(0) \bigr)^2 e_*
\end{align*}
Applying $(\mathbf{I}-\mathcal{M}_\theta\mathcal{M}_\theta^*)\bigm|_{{\mathfrak D}\ci{\mathcal{M}_\theta^*}}$ to the above identity, and using again \eqref{comm-defects-00}, we get by induction that
\begin{align}
\label{comm-defects-02}
\varphi\bigl( (\mathbf{I}-\mathcal{M}_\theta\mathcal{M}_\theta^*)\bigm|_{{\mathfrak D}_{\mathcal{M}_\theta^*}} \bigr) P\ci{{\mathcal K}_\theta} \left(\begin{array}{c} e_* \\mathbb{O}\end{array}\right)
=\left(\begin{array}{c} \mathbf{I} - \theta\theta(0)^* \\ -\Delta\theta(0)^*\end{array}\right)
\varphi\bigl(\mathbf{I}\ci{{\mathfrak D}_*}- \theta(0)\theta^*(0) \bigr) e_*
\end{align}
for any monomial $\varphi$, $\varphi(x) =x^n$, $n\ge0$ (the case $n=0$ is just the identity \eqref{P_theta_N-F}).
Linearity implies that \eqref{comm-defects-02} holds for any polynomial $\varphi$.
Using standard approximation reasoning we get that $\varphi$ in \eqref{comm-defects-02} can be any measurable function. In particular, we can take $\varphi(x) =x^{-1/2}$, which together with \eqref{C*_N-F-02} gives us \eqref{C*_N-F}.
To prove \eqref{C_N-F} we proceed similarly. Equation \eqref{Psi*N-F-02} implies that
\begin{align*}
\widetilde\mathbf{e}(e) = \left(\begin{array}{c}\theta\\ \Delta \end{array}\right) e,
\end{align*}
so by \eqref{C_CoFree}
\begin{align}
\label{C_N-F-02}
\mathbf{C} e = \bigl((\mathbf{I} - \mathcal{M}\ci\theta^*\mathcal{M}\ci\theta)\bigm|_{{\mathfrak D}_{\mathcal{M}_\theta}}\bigr)^{-1/2} P\ci{{\mathcal K}_\theta} M_{\bar z} \left(\begin{array}{c} \theta \\ \Delta \end{array}\right) e,\qquad e\in{\mathfrak D}.
\end{align}
One can see that
\begin{align*}
P\ci{{\mathcal K}_\theta} M_{\bar z} \left(\begin{array}{c} \theta \\ \Delta \end{array}\right) e =
M_{\bar z} \left(\begin{array}{c} \theta -\theta(0) \\ \Delta \end{array}\right) e,
\end{align*}
so
\begin{align*}
\mathcal{M}_\theta P\ci{{\mathcal K}_\theta} M_{\bar z} \left(\begin{array}{c} \theta \\ \Delta \end{array}\right) e
=
P\ci{{\mathcal K}_\theta} \left(\begin{array}{c} \theta -\theta(0) \\ \Delta \end{array}\right) e
=
-P\ci{{\mathcal K}_\theta} \left(\begin{array}{c} \theta(0) \\ 0 \end{array}\right) e.
\end{align*}
Combining this with \eqref{P_theta_N-F}, we get
\begin{align*}
\mathcal{M}_\theta P\ci{{\mathcal K}_\theta} M_{\bar z} \left(\begin{array}{c} \theta \\ \Delta \end{array}\right) e =
\left(\begin{array}{c} \theta\theta(0)^* -\mathbf{I} \\ \Delta\theta(0)^*\end{array}\right) \theta(0) e .
\end{align*}
Using the fact that
\begin{align*}
\mathcal{M}_\theta^* \left(\begin{array}{c} f \\ g \end{array}\right) = M_{\bar z}\left(\begin{array}{c} f - f(0)\\ g \end{array}\right),
\end{align*}
we arrive at
\begin{align*}
\mathcal{M}_\theta^*\mathcal{M}_\theta P\ci{{\mathcal K}_\theta} M_{\bar z} \left(\begin{array}{c} \theta \\ \Delta \end{array}\right) e = M_{\bar z} \left(\begin{array}{c} \theta -\theta(0)\\ \Delta \end{array}\right) \theta(0)^*\theta(0) e,
\end{align*}
so
\begin{align*}
(\mathbf{I} - \mathcal{M}_\theta^*\mathcal{M}_\theta) P\ci{{\mathcal K}_\theta} M_{\bar z} \left(\begin{array}{c} \theta \\ \Delta \end{array}\right)e = M_{\bar z} \left(\begin{array}{c} \theta -\theta(0)\\ \Delta \end{array}\right)(\mathbf{I} - \theta(0)^*\theta(0) )e.
\end{align*}
Using the same reasoning as in the above proof of \eqref{C*_N-F} we get that
\begin{align}
\label{comm-defects-03}
\varphi\bigl( (\mathbf{I} - \mathcal{M}_\theta^*\mathcal{M}_\theta)\bigm|_{{\mathfrak D}_{\mathcal{M}_\theta}} \bigr)
P\ci{{\mathcal K}_\theta} & M_{\bar z} \left(\begin{array}{c} \theta \\ \Delta \end{array}\right)e
\\
\notag
&= M_{\bar z} \left(\begin{array}{c} \theta -\theta(0)\\ \Delta \end{array}\right)\varphi\bigl( \mathbf{I} - \theta(0)^*\theta(0) \bigr)e,
\end{align}
first with $\varphi$ being a polynomial, and then any measurable function.
Using \eqref{comm-defects-03} with $\varphi(x)=x^{-1/2}$ and taking \eqref{C_N-F-02} into account, we get \eqref{C_N-F}.
\end{proof}
\subsection{An auxiliary lemma}
\label{s:bound-C}
We already used, and we will also need later the following simple Lemma.
\begin{lem}
\label{l:bound-C}
Let $\mathcal{M}=\mathcal{M}_\theta$ be model operator on a model space ${\mathcal K}_\theta\subset L^2(W;{\mathfrak D}_*\oplus {\mathfrak D})$, and let $\mathbf{C}: {\mathfrak D}_*\to {\mathfrak D}\ci{\mathcal{M}_\theta}$, $\mathbf{C}_*: {\mathfrak D}\to {\mathfrak D}\ci{\mathcal{M}_\theta^*}$ be bounded operators.
If $C$ and $C_*$ are the operator-valued functions, defined by
\begin{align*}
C(z) e & = \mathbf{C} e (z), &&z \in \mathbb{T}, \ e\in {\mathfrak D} , \\
C_*(z) e_* & = \mathbf{C}_* e_* (z), &&z \in \mathbb{T}, \ e_*\in {\mathfrak D}_*.
\end{align*}
then the functions $W^{1/2} C$ and $W^{1/2}C^*$ are bounded,
\begin{align*}
\| W^{1/2} C \|\ci{L^\infty} = \|\mathbf{C}\|, \qquad \| W^{1/2} C_* \|\ci{L^\infty} = \|\mathbf{C}_*\| .
\end{align*}
\end{lem}
\begin{proof}
It is well-known and is not hard to show, that if $T$ is a contraction and $\mathcal{U} $ is its unitary dilation, then then the subspaces $\mathcal{U}^n {\mathfrak D}\ci T$, $n\in{\mathbb Z}$ (where recall ${\mathfrak D}\ci T$ is the defect space of $T$) are mutually orthogonal, and similarly for subspaces $\mathcal{U}^n{\mathfrak D}\ci{T^*}$, $n\in{\mathbb Z}$.
Therefore, the subspaces $z^n {\mathfrak D}\ci{\mathcal{M}}$, $n\in{\mathbb Z}$ are mutually orthogonal in $L^2(W; {\mathfrak D}_*\oplus {\mathfrak D})$. and the same holds for the subspaces $z^n {\mathfrak D}\ci{\mathcal{M}^*}$, $n\in{\mathbb Z}$.
The subspaces $z^n {\mathfrak D}\subset L^2(\mathbb{T};{\mathfrak D})$ are mutually orthogonal, and since
\[
C(z) \sum_{n\in{\mathbb Z}} z^n \hat f(n) = \sum_{n\in{\mathbb Z}} z^n \mathbf{C} f_n , \qquad \hat f(n)\in {\mathfrak D},
\]
we conclude that the operator $f\mapsto Cf$ is a bounded operator acting $L^2({\mathfrak D})\to L^2(W;{\mathfrak D}_*\oplus{\mathfrak D})$, and its norm is exactly $\|\mathbf{C}\|$.
But that means the multiplication operator $f\mapsto W^{1/2}f$ between the non-weighted spaces $L^2({\mathfrak D})\to L^2({\mathfrak D}_*\oplus{\mathfrak D})$ is bounded with the same norm, which immediately implies that $\| W^{1/2} C \|\ci{L^\infty} = \|\mathbf{C}\|$.
The proof for $C_*$ follows similarly.
\end{proof}
\section{Characteristic function}\label{s-charfunc}
In this section we derive formulas for the (matrix-valued)
characteristic function $\theta\ci\Gamma$, see Theorem \ref{t-theta} below.
\subsection{An inverse of a perturbation}
We begin with an auxiliary result.
\begin{lem}
\label{l:pert-01}
Let $D$ be an operator in an auxiliary Hilbert space ${\mathfrak{R}}$ and let $B, C:{\mathfrak{R}}\to \mathcal{H}$. Then $\mathbf{I}\ci\mathcal{H} - CDB^*$ is invertible if and only if $\mathbf{I}\ci {\mathfrak{R}} - DB^*C$ is invertible, and if and only if $\mathbf{I}\ci {\mathfrak{R}} -B^*CD$ is invertible.
Moreover, in this case
\begin{align}
\label{InvPert-01}
(\mathbf{I}\ci\mathcal{H} - CDB^*)^{-1} & = \mathbf{I}\ci \mathcal{H} + C (\mathbf{I}\ci {\mathfrak{R}} - DB^*C)^{-1} DB^*\\ \notag
& = \mathbf{I}\ci \mathcal{H} + C D (\mathbf{I}\ci {\mathfrak{R}} - B^*CD)^{-1} B^*.
\end{align}
\end{lem}
We will apply this lemma for $D:{\mathbb C}^d\to {\mathbb C}^d$, so in this case the inversion of $\mathbf{I}\ci\mathcal{H} - CDB$ is reduced to inverting $(d\times d)$ matrix.
This lemma can be obtained from the Woodbury inversion formula \cite{Wood}, although formally in \cite{Wood} only the matrix case was treated.
\begin{proof}[Proof of Lemma \ref{l:pert-01}]
First let us note that it is sufficient to prove lemma with $D=\mathbf{I}\ci {\mathfrak{R}}$, because $D$ can be incorporated either into $C$ or into $B^*$.
One could guess the formula by writing the power series expansion of $\mathbf{I}\ci\mathcal{H} - CDB^*$, and we can get the result for the case when the series converges. This method can be made rigorous for finite rank perturbations by considering the family $(\mathbf{I}\ci\mathcal{H} -\lambda CDB^*)^{-1}$, $\lambda\in{\mathbb C}$ and using analytic continuation.
However, the simplest way to prove the formula is just by performing multiplication,
\begin{align*}
(\mathbf{I}\ci\mathcal{H} - CB^*) \Bigl(\mathbf{I}\ci\mathcal{H} & + C(\mathbf{I}\ci {\mathfrak{R}} -B^*C)^{-1}B^*\Bigr)\\
& = \mathbf{I}\ci\mathcal{H} -CB^* + C(\mathbf{I}\ci {\mathfrak{R}} -B^*C)^{-1}B^* -CB^*C(\mathbf{I}\ci {\mathfrak{R}} -B^*C)^{-1}B^* \\
&= \mathbf{I}\ci\mathcal{H} + C\Bigl( -\mathbf{I}\ci {\mathfrak{R}} (\mathbf{I}\ci {\mathfrak{R}} - B^*C) +\mathbf{I}\ci {\mathfrak{R}} - B^*C \Bigr) (\mathbf{I}\ci {\mathfrak{R}} - B^*C)^{-1}B^* \\
&=\mathbf{I}\ci\mathcal{H}.
\end{align*}
Thus, when $\mathbf{I}\ci {\mathfrak{R}} -B^*C$ is invertible, the operator $\mathbf{I}\ci\mathcal{H} + C(\mathbf{I}\ci {\mathfrak{R}} -B^*C)^{-1}B^*$ is the right inverse of $\mathbf{I}\ci\mathcal{H} - CB^*$. To prove that it is also a right inverse we even do not need to perform the multiplication: we can just take the adjoint of the above identity and then interchange $B$ and $C$.
So, the invertibility of $\mathbf{I}\ci {\mathfrak{R}} -B^*C$ implies the invertibility of $\mathbf{I}\ci\mathcal{H} - CB^*$ and the formula for the inverse. To prove the ``if and only if'' statement we just need to change the roles of $\mathcal{H}$ and ${\mathfrak{R}}$ and express, using the just proved formula, the inverse of $\mathbf{I}\ci {\mathfrak{R}} -B^*C$ in terms of $(\mathbf{I}\ci\mathcal{H} - CB^*)^{-1}$.
\end{proof}
\subsection{Computation of the characteristic function}
We turn to computing the characteristic function of $T= U+ \mathbf{B}(\Gamma-\mathbf{I}\ci{{\mathbb C}^d}) \mathbf{B}^*U$, $\|\Gamma\|<1$, where $U$ is the multiplication operator $M_\xi$ in $L^2(\mu;E)$.
We will use formula \eqref{CharFunction-01} with $V= {\mathbf{B}}^*U$, $V_*={\mathbf{B}}^*$, ${\mathfrak D}={\mathfrak D}_*={\mathbb C}^d$.
Let us first calculate for $|z|<1$:
\begin{align*}
(\mathbf{I}\ci\mathcal{H} - z T^*)^{-1} &= \left[ (\mathbf{I}\ci\mathcal{H} - zU^*) \left(\mathbf{I}\ci\mathcal{H} - z (\mathbf{I}\ci\mathcal{H} - zU^*)^{-1} U^*{\mathbf{B}}(\Gamma^* -\mathbf{I}\ci {{\mathbb C}^d}){\mathbf{B}}^* \right)\right]^{-1} \\
& = \left[\mathbf{I}\ci\mathcal{H} - z (\mathbf{I}\ci\mathcal{H} - zU^*)^{-1} U^*{\mathbf{B}}(\Gamma^* -\mathbf{I}\ci {{\mathbb C}^d}){\mathbf{B}}^* \right]^{-1} (\mathbf{I}\ci\mathcal{H} - zU^*)^{-1} \\
&=: X(z) (\mathbf{I}\ci\mathcal{H} - zU^*)^{-1}.
\end{align*}
To compute the inverse $X(z)$ we use Lemma \ref{l:pert-01} with $z (\mathbf{I}\ci\mathcal{H} - zU^*)^{-1} U^*{\mathbf{B}}$ instead of $C$, $\Gamma^* -\mathbf{I}\ci {{\mathbb C}^d}$ instead of $D$ and $\mathbf{B}$ instead of $B$. Together with the first identity in \eqref{InvPert-01} we get
\begin{align}
\label{X_1}
X(z) =
\mathbf{I}\ci \mathcal{H} + z(\mathbf{I}\ci\mathcal{H} -z U^*)^{-1} U^*{\mathbf{B}} \Bigl(\mathbf{I}\ci {{\mathbb C}^d} - z
D {\mathbf{B}}^*(\mathbf{I}\ci \mathcal{H} - z U^*)^{-1}U^*{\mathbf{B}}\Bigr)^{-1} D
{\mathbf{B}}^*,
\end{align}
where $D=\Gamma^*-\mathbf{I}\ci{{\mathbb C}^d}$.
Now, let us express $z{\mathbf{B}}^*(\mathbf{I}\ci \mathcal{H} - z U^*)^{-1}U^*{\mathbf{B}}$ as a Cauchy integral of some matrix-valued measure. Recall that $U$ is a multiplication by the independent variable $\xi$ in $\mathcal{H}\subset L^2(\mu;E)$. Recall that $b_1, b_2, \ldots , b_d\in \mathcal{H}$ denote the ``columns'' of ${\mathbf{B}}$ (i.e.~$b_k = {\mathbf{B}}e_k$, where $e_1, e_2, \ldots, e_d$ is the standard basis in ${\mathbb C}^d$), and $B(\xi)= (b_1(\xi), b_2(\xi),\ldots, b_d(\xi))$ is the matrix with columns $b_k(\xi)$. Then
\begin{align*}
b_j^* (\mathbf{I}\ci{{\mathbb C}^d} - zU^*)^{-1} U^* b_k = \int_\mathbb{T} \frac{\overline\xi}{1-z\overline\xi}\, {b_j(\xi)^*} b_k(\xi) {\mathrm{d}}\mu(\xi),
\end{align*}
so
\begin{align}
\label{MatrCauchy-01}
z {\mathbf{B}}^*(\mathbf{I}\ci \mathcal{H} - z U^*)^{-1}U^*{\mathbf{B}} = \int_\mathbb{T} \frac{z\overline\xi}{1-z\overline\xi} \,M(\xi){\mathrm{d}}\mu(\xi) =: \mathcal{C}_1[M\mu](z)=: F_1(z).
\end{align}
where $M $ is the matrix-valued function $M(\xi)=B(\xi)^*B(\xi)$, or equivalently $M_{j,k}(\xi) = {b_j(\xi)^*} b_k(\xi)$, $1\le j, k \le d$.
Using \eqref{MatrCauchy-01} and denoting $D:= \Gamma^* -\mathbf{I}\ci{{\mathbb C}^d}$ we get from the above calculations that
\begin{align*}
(\mathbf{I}\ci\mathcal{H} - z T^*)^{-1} = \, & (\mathbf{I}\ci\mathcal{H} - zU^*)^{-1} \\
& + z (\mathbf{I}\ci\mathcal{H} - zU^*)^{-1} U^*{\mathbf{B}} \Bigl(\mathbf{I}\ci{{\mathbb C}^d} - D F_1(z) \Bigr)^{-1} D{\mathbf{B}}^* (\mathbf{I}\ci\mathcal{H} - zU^*)^{-1} .
\end{align*}
Applying formula \eqref{CharFunction-01}, with $V= {\mathbf{B}}^*U$, $V_*={\mathbf{B}}_*$, ${\mathfrak D}={\mathfrak D}_*={\mathbb C}^d$, we see that the characteristic function is an analytic function $\theta=\theta\ci T$, whose values are bounded linear operators acting on ${\mathfrak D}$, defined by the
formula
\begin{align}
\label{CharFunction}
\theta_T(z) = {\mathbf{B}}^*\left( -T + z D_{T^*}\left(\mathbf{I}\ci\mathcal{H} - z T^*\right)^{-1} D_T \right) U^*{\mathbf{B}} \Bigm|_{{\mathfrak D}}, \qquad z\in {\mathbb D}.
\end{align}
We can see from \eqref{BlDec-01} that the defect operators $D\ci T$ and $D\ci{T^*}$ are given by
\begin{align*}
D\ci T= U^*{\mathbf{B}} D\ci\Gamma {\mathbf{B}}^*U, \qquad D\ci{T^*} = {\mathbf{B}} D\ci{\Gamma^*} {\mathbf{B}}^*.
\end{align*}
We can also see from \eqref{BlDec-01} that the term $-T$ in \eqref{CharFunction} contributes $-\Gamma$ to the matrix
$\theta\ci T$. The rest can be obtained from the above representation formula for $(\mathbf{I}\ci\mathcal{H} -zT^*)^{-1}$.
Thus, recalling the definition \eqref{MatrCauchy-01} of $\mathcal{C}_1M\mu$
we get, denoting $F_1(z):= (\mathcal{C}_1M\mu)(z)$, that
\begin{align*}
\theta\ci T (z) & = -\Gamma + D\ci{\Gamma^*}\Biggl[ F_1(z) + F_1(z) \Bigl( \mathbf{I}\ci{{\mathfrak D}} -(\Gamma^*-\mathbf{I}\ci{{\mathfrak D}})F_1(z)\Bigr)^{-1} (\Gamma^*-\mathbf{I}\ci{{\mathfrak D}}) F_1(z) \Biggr] D\ci\Gamma
\\
& = -\Gamma + D\ci{\Gamma^*} F_1(z) \Bigl( \mathbf{I}\ci{{\mathfrak D}} -(\Gamma^*-\mathbf{I}\ci{{\mathfrak D}})F_1(z)\Bigr)^{-1} D\ci\Gamma.
\end{align*}
In the above computation to compute $X(z)$ we can use the second formula in \eqref{InvPert-01}. We get instead of \eqref{X_1} an alternative representation
\begin{align*}
X(z) =
\mathbf{I}\ci \mathcal{H} + z(\mathbf{I}\ci\mathcal{H} -z U^*)^{-1} U^*{\mathbf{B}} D \Bigl(\mathbf{I}\ci{{\mathfrak D}} - z
{\mathbf{B}}^*(\mathbf{I}\ci \mathcal{H} - z U^*)^{-1}U^*{\mathbf{B}}D\Bigr)^{-1}
{\mathbf{B}}^*.
\end{align*}
Repeating the same computations as above we get another formula for $\theta\ci T$,
\begin{align*}
\theta\ci T (z) = -\Gamma + D\ci{\Gamma^*} \Bigl( \mathbf{I}\ci{{\mathfrak D}} -F_1(z)(\Gamma^*-\mathbf{I}\ci{{\mathfrak D}})\Bigr)^{-1} F_1(z) D\ci\Gamma.
\end{align*}
To summarize we have proved two representations of the
characteristic operator-valued function.
\begin{theo}
\label{t-theta}
Let $T=T_\Gamma$ be the operator given in \eqref{BlDec-01}, with $\Gamma$ being a strict contraction. Then the characteristic function $\theta\ci{T}=\theta\ci{T_\Gamma}\in H^\infty({\mathfrak D}\,\!\!\to\!{\mathfrak D}_*)$, with coordinate operators $V= {\mathbf{B}}^*U$, $V_*={\mathbf{B}}^*$ (and with ${\mathfrak D}={\mathfrak D}_*={\mathbb C}^d$) is given by
\begin{align*}
\theta\ci{T_\Gamma} (z) &= -\Gamma + D\ci{\Gamma^*} F_1(z) \Bigl( \mathbf{I}\ci{{\mathfrak D}} -(\Gamma^*-\mathbf{I}\ci{{\mathfrak D}})F_1(z)\Bigr)^{-1} D\ci\Gamma
\\
&= -\Gamma + D\ci{\Gamma^*} \Bigl( \mathbf{I}\ci{{\mathfrak D}} -F_1(z)(\Gamma^*-\mathbf{I}\ci{{\mathfrak D}})\Bigr)^{-1} F_1(z) D\ci\Gamma,
\end{align*}
where $F_1(z)$ is the matrix-valued function given by \eqref{MatrCauchy-01}.
\end{theo}
In these formulas, the inverse is taken of a $(d\times d)$ matrix-valued function, which is much simpler than computing the inverse in \eqref{CharFunction}.
\subsection
Characteristic function and the Cauchy integrals of matrix-valued measures}
For a (possibly complex-valued) measure $\tau$ on $\mathbb{T}$ and $z\notin\mathbb{T}$ define the following Cauchy type transforms $\mathcal{C}$, $\mathcal{C}_1$ and $\mathcal{C}_2$
\begin{align*}
\mathcal{C} \tau (z) := \int_\mathbb{T} \frac{d\tau(\xi)}{1-\overline\xi z}, \qquad \mathcal{C}_1 \tau ( z) := \int_\mathbb{T} \frac{\overline\xi z d\tau(\xi)}{1-\overline\xi z}, \qquad \mathcal{C}_2\tau( z):= \int_\mathbb{T} \frac{1+ \overline\xi z }{1-\overline\xi z} d\tau(\xi).
\end{align*}
Performing the Cauchy transforms component-wise we can define them for matrix-valued measures as well.
Thus $F_1$ from the above Theorem \ref{t-theta} is given by $F_1 =\mathcal{C}_1[M\mu]$, where $M(\xi) = B^*(\xi) B(\xi)$. We would like to give the representation of $\theta\ci{T_\Gamma}$ in terms of function $F_2:= \mathcal{C}_2 [M\mu]$.
Slightly abusing notation we will write $\theta_\Gamma$ instead of
$\theta\ci{T_\Gamma}$.
\begin{cor}
\label{c:theta_0}
For $\theta\ci{\mathbf{0}} :=\theta\ci{T_{\mathbf{0}}}$ we have
\begin{align}
\label{theta_0-01}
\theta\ci{\mathbf{0}}(z) & = F_1(z) (\mathbf{I} + F_1(z))^{-1} = (\mathbf{I} + F_1(z))^{-1}F_1(z) \\
\label{theta_0-02}
& = (F_2(z)-\mathbf{I}) (F_2(z)+\mathbf{I})^{-1} = (F_2(z)+\mathbf{I})^{-1} (F_2(z)-\mathbf{I}) .
\end{align}
\end{cor}
\begin{proof}
The identity \eqref{theta_0-01} is a direct application of Theorem \ref{t-theta}. The identity \eqref{theta_0-02} follows immediately from the trivial relation
\[
F_2(z) = \int_\mathbb{T} M{\mathrm{d}} \mu + 2F_1(z) = \mathbf{I}\ci{\mathfrak D} + 2 F_1(z);
\]
the equality $\int_\mathbb{T} M{\mathrm{d}} \mu =\mathbf{I}\ci{\mathfrak D}=\mathbf{I}\ci{{\mathbb C}^d}$ is just a re-statement of the fact that the functions $b_1, b_2, \ldots, b_d$ form an orthonormal basis in $\mathcal{H}$.
\end{proof}
\section{Relations between characteristic functions \texorpdfstring{$\theta\ci\Gamma$}{theta<sub>Gamma} }
\label{s:PropCharFunct}
\subsection{Characteristic functions and linear fractional transformations}\label{ss:LFT}
When $d=1$, it is known that the characteristic functions are related by a linear fractional transformation
\[
\theta_\gamma(z) = \frac{\theta_0(z)-\gamma}{1-\overline\gamma \theta_0(z)}\,,
\]
see \cite[Equation (2.9)]{LT15}.
It turns out that a similar formula holds for finite rank perturbations.
\begin{theo}
\label{t-LFT}
Let $T$ be the operator given in \eqref{BlDec-01}, with $\Gamma$ being a strict contraction. Then the characteristic functions $\theta\ci{\Gamma}:=\theta\ci{T_\Gamma}$ and $\theta\ci{\mathbf{0}} = \theta\ci{T_{\mathbf{0}}} $ are related via linear fractional transformation
\[
\theta\ci{\Gamma}
=
D\ci{\Gamma^*}^{-1}( \theta\ci{\mathbf{0}}-\Gamma) ({\mathbf{I}}\ci{{\mathfrak D}} - \Gamma^* \theta\ci{\mathbf{0}})^{-1} D\ci{\Gamma}
=
D\ci{\Gamma^*}({\mathbf{I}}\ci{{\mathfrak D}} - \theta\ci{\mathbf{0}}\Gamma^* )^{-1} ( \theta\ci{\mathbf{0}}-\Gamma) D\ci{\Gamma}^{-1}
.
\]
\end{theo}
\begin{rem*}
At first sight, this formula looks like a formula in \cite[p.~234]{Nik-Vas_model_MSRI_1998}. However, their result expresses the characteristic function in terms of a linear fractional transformation in $T$; whereas, here we have a linear fractional transformation in $\Gamma$.
\end{rem*}
\begin{theo}
\label{t:LFT-02}
Under assumptions of the above Theorem \ref{t-LFT}
\[
\theta_{\mathbf{0}} = D\ci{\Gamma^*} (\mathbf{I} + \theta\ci\Gamma \Gamma^*)^{-1} (\theta\ci\Gamma + \Gamma) D\ci\Gamma^{-1}
= D\ci{\Gamma^*}^{-1} (\theta\ci\Gamma + \Gamma) (\mathbf{I} + \Gamma^* \theta\ci\Gamma)^{-1} D\ci\Gamma.
\]
\end{theo}
To prove Theorem \ref{t-LFT} we start with the following simpler statement.
\begin{prop}\label{p-LFT}
The matrix-valued characteristic functions $\theta\ci{\Gamma}$ and $\theta\ci{\mathbf{0}}$ are related via
\[
\theta\ci{\Gamma}
=
-\Gamma + D\ci{\Gamma^*} \theta\ci{\mathbf{0}} \left({\mathbf{I}}\ci{{\mathfrak D}} - \Gamma^* \theta\ci{\mathbf{0}}\right)^{-1} D\ci{\Gamma}
=
-\Gamma + D\ci{\Gamma^*} \left({\mathbf{I}}\ci{{\mathfrak D}} - \theta\ci{\mathbf{0}}\Gamma^*\right)^{-1} \theta\ci{\mathbf{0}} D\ci{\Gamma}
.
\]
\end{prop}
\begin{proof}
Solving \eqref{theta_0-01} for $F_1$ we get that
\[
F_1(z) = \theta\ci{\mathbf{0}}(z)[\mathbf{I}-\theta\ci{\mathbf{0}}(z)]^{-1}.
\]
Substituting this expression into the formula for the characteristic function from Theorem \ref{t-theta}, we see that
\begin{align}
\label{theta_Gamma-03}
\theta\ci\Gamma = -\Gamma + D\ci{\Gamma^*} \theta\ci{\mathbf{0}}[\mathbf{I}\ci{{\mathfrak D}}- \theta\ci{\mathbf{0}}]^{-1} \Bigl\{ \mathbf{I}\ci{{\mathfrak D}} -(\Gamma^*-\mathbf{I}\ci{{\mathfrak D}}) \theta\ci{\mathbf{0}}[\mathbf{I}\ci{{\mathfrak D}}- \theta\ci{\mathbf{0}}]^{-1}\Bigr\}^{-1} D\ci\Gamma.
\end{align}
We manipulate the term inside the curly brackets
\begin{align*}
\mathbf{I}\ci{{\mathfrak D}} -(\Gamma^*-\mathbf{I}\ci{{\mathfrak D}}) \theta\ci{\mathbf{0}}
[\mathbf{I}\ci{{\mathfrak D}}- \theta\ci{\mathbf{0}}]^{-1}
&=
\left(
\mathbf{I}\ci{{\mathfrak D}} - \theta\ci{\mathbf{0}} -(\Gamma^*-\mathbf{I}\ci{{\mathfrak D}}) \theta\ci{\mathbf{0}}
\right)
[\mathbf{I}\ci{{\mathfrak D}}- \theta\ci{\mathbf{0}}]^{-1}
\\
&=
\left(
\mathbf{I}\ci{{\mathfrak D}} -\Gamma^* \theta\ci{\mathbf{0}}
\right)
[\mathbf{I}\ci{{\mathfrak D}}- \theta\ci{\mathbf{0}}]^{-1},
\end{align*}
so that
\[
\Bigl\{ \mathbf{I}\ci{{\mathfrak D}} -(\Gamma^*-\mathbf{I}\ci{{\mathfrak D}}) \theta\ci{\mathbf{0}}[\mathbf{I}\ci{{\mathfrak D}}- \theta\ci{\mathbf{0}}]^{-1}\Bigr\}^{-1}
=
[\mathbf{I}\ci{{\mathfrak D}}- \theta\ci{\mathbf{0}}]
\left(
\mathbf{I}\ci{{\mathfrak D}} -\Gamma^* \theta\ci{\mathbf{0}}
\right)
^{-1}.
\]
Substituting this back into \eqref{theta_Gamma-03}, we get the first equation the first equation in the proposition.
The second equation is obtained similarly.
\end{proof}
\begin{lem}
\label{l:G-D_G}
For $\|\Gamma\|<1$ we have for all $\alpha\in{\mathbb R}$
\begin{align}
\label{G-D_G-01}
D\ci{\Gamma^*}^{\alpha}\Gamma & = \Gamma D\ci{\Gamma}^{\alpha} \,,\\
\label{G-D_G-02}
D\ci{\Gamma}^{\alpha}\Gamma^* & = \Gamma^* D\ci{\Gamma^*}^{\alpha}\,,
\end{align}
where, recall $D\ci \Gamma := (\mathbf{I} - \Gamma^*\Gamma)^{1/2}$, $D\ci{\Gamma^*} := (\mathbf{I} - \Gamma\Gamma^*)^{1/2}$ are the defect operators.
\end{lem}
\begin{proof}
Let us prove \eqref{G-D_G-01}. It is trivially true for $\alpha=2$, and by induction we get that it is true for $\alpha= 2n$, $n\in{\mathbb N}$. Since $\|\Gamma\|<1$, the spectrum of $D\ci\Gamma$ lies in the interval $[ a, 1]$, $a=(1-\|\Gamma\|^2)^{1/2}>0$.
Approximating $\varphi(x)= x^\alpha$ uniformly on $[a,1]$ by polynomials of $x^2$ we get \eqref{G-D_G-01}.
Applying \eqref{G-D_G-01} to $\Gamma^*$ we get \eqref{G-D_G-02}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{t-LFT}]
From \eqref{G-D_G-01} we get that $D\ci{\Gamma^*}^{-1}\Gamma D\ci{\Gamma}^{-1} = D\ci{\Gamma^*}^{-2}\Gamma$, so
\begin{align*}
\theta\ci{\Gamma}
&=
-\Gamma + D\ci{\Gamma^*} \theta\ci{\mathbf{0}} \left({\mathbf{I}}\ci{{\mathfrak D}} - \Gamma^* \theta\ci{\mathbf{0}}\right)^{-1} D\ci{\Gamma}
\\&
=
D\ci{\Gamma^*}\left[-D\ci{\Gamma^*}^{-2}\Gamma + \theta\ci{\mathbf{0}} \left({\mathbf{I}}\ci{{\mathfrak D}} - \Gamma^* \theta\ci{\mathbf{0}}\right)^{-1} \right]
D\ci{\Gamma}
\\&
=
D\ci{\Gamma^*}^{-1}\left[-\Gamma + D\ci{\Gamma^*}^{2} \theta\ci{\mathbf{0}} \left({\mathbf{I}}\ci{{\mathfrak D}} - \Gamma^* \theta\ci{\mathbf{0}}\right)^{-1} \right]
D\ci{\Gamma}
\\&
=
D\ci{\Gamma^*}^{-1}\left[-\Gamma ({\mathbf{I}}\ci{{\mathfrak D}} - \Gamma^* \theta\ci{\mathbf{0}})+(\mathbf{I} -\Gamma\Gamma^*) \theta\ci{\mathbf{0}} \right]
\left({\mathbf{I}}\ci{{\mathfrak D}} - \Gamma^* \theta\ci{\mathbf{0}}\right)^{-1}
D\ci{\Gamma}
\\&
=
D\ci{\Gamma^*}^{-1}\left[-\Gamma + \theta\ci{\mathbf{0}} \right]
\left({\mathbf{I}}\ci{{\mathfrak D}} - \Gamma^* \theta\ci{\mathbf{0}}\right)^{-1}
D\ci{\Gamma},
\end{align*}
which is exactly the first identity.
The second identity is obtained similarly, using the formula $D\ci{\Gamma^*}^{-1}\Gamma D\ci{\Gamma}^{-1} = \Gamma D\ci{\Gamma}^{-2}$ and taking the factor $\left({\mathbf{I}}\ci{{\mathfrak D}} - \Gamma^* \theta\ci{\mathbf{0}}\right)^{-1}$ out of brackets on the left.
\end{proof}
\begin{proof}[Proof of Theorem \ref{t:LFT-02}]
Right multiplying the first identity in Theorem \ref{t-LFT} by $D\ci\Gamma^{-1} (\mathbf{I}-\Gamma^* \theta\ci{\mathbf{0}})$ we get
\[
\theta\ci\Gamma D\ci\Gamma^{-1} - \theta\ci\Gamma D\ci\Gamma^{-1} \Gamma^* \theta\ci{\mathbf{0}} = D\ci{\Gamma^*}^{-1} \theta\ci{\mathbf{0}} - D\ci{\Gamma^*}^{-1} \Gamma.
\]
Using identities $D\ci{\Gamma^*}^{-1}\Gamma =\Gamma D\ci\Gamma^{-1}$ and $D\ci\Gamma^{-1} \Gamma^* = \Gamma^* D\ci{\Gamma^*}^{-1}$, see Lemma \ref{l:G-D_G}, we rewrite the above equality as
\[
\theta\ci\Gamma D\ci\Gamma^{-1} + \Gamma D\ci\Gamma^{-1} =
\theta\ci\Gamma \Gamma^* D\ci{\Gamma^*}^{-1} \theta\ci{\mathbf{0}} + D\ci{\Gamma^*}^{-1} \theta\ci{\mathbf{0}}.
\]
Right multiplying both sides by $D\ci{\Gamma^*}(\theta\ci\Gamma \Gamma^* + \mathbf{I})^{-1} $ we get the first equality in the theorem.
The second one is proved similarly.
\end{proof}
\subsection{The defect functions \texorpdfstring{$\Delta\ci\Gamma$}{Delta<sub>Gamma} and relations between them}\label{s-Delta}
Recall that every strict contraction $\Gamma$ yields a characteristic matrix-valued function $\theta\ci\Gamma$ through association with the c.n.u.~contraction $U\ci\Gamma$. The definition of the Sz.-Nagy--Foia\c s model space (see e.g.~formula \eqref{N-F-K_theta}) reveals immediately that the defect functions $\Delta\ci\Gamma= ({\mathbf{I}} - \theta^*\ci\Gamma \theta\ci\Gamma)^{1/2}$ are central objects in model theory. We express defect function $\Delta\ci\Gamma$ in terms of $\Delta\ci{\mathbf{0}}$ (and $\Gamma$ and $\theta\ci{\mathbf{0}}$).
\begin{theo}\label{t-Delta}
The defect functions of $\theta\ci\Gamma$ and $\theta\ci{\mathbf{0}}$ are related by
\[
\Delta\ci\Gamma^2
=
D\ci\Gamma (I-\theta\ci{\mathbf{0}}^*\Gamma)^{-1} \Delta\ci{\mathbf{0}}^2(I-\Gamma^* \theta\ci{\mathbf{0}})^{-1}
D\ci\Gamma .
\]
\end{theo}
\begin{proof}
By Theorem \ref{t-LFT}
\begin{align*}
\theta\ci{\Gamma}
=
D\ci{\Gamma^*}^{-1}( \theta\ci{\mathbf{0}}-\Gamma) ({\mathbf{I}}\ci{{\mathfrak D}} - \Gamma^* \theta\ci{\mathbf{0}})^{-1} D\ci{\Gamma},
\end{align*}
so $\theta\ci\Gamma^*\theta\ci\Gamma = A^*B A$, where
\begin{align*}
A=(\mathbf{I} -\Gamma^* \theta\ci{\mathbf{0}}) D\ci\Gamma, \qquad B = (\theta\ci{\mathbf{0}}^*-\Gamma^*)
D\ci{\Gamma*}^{-2} (\theta\ci{\mathbf{0}}-\Gamma) .
\end{align*}
Then
$
\Delta\ci\Gamma = \mathbf{I} - \theta\ci\Gamma^* \theta\ci\Gamma = A^*XA$,
where
\begin{align*}
X &=(A^*)^{-1} A^{-1} - B
= (\mathbf{I} -\theta\ci{\mathbf{0}}^*\Gamma ) D\ci\Gamma^{-2} (\mathbf{I} -\Gamma^* \theta\ci{\mathbf{0}}) -
(\theta\ci{\mathbf{0}}^*-\Gamma^*) D\ci{\Gamma*}^{-2} (\theta\ci{\mathbf{0}}-\Gamma) \\
& =
D\ci\Gamma^{-2} - \theta\ci{\mathbf{0}}^*\Gamma D\ci\Gamma^{-2} - D\ci\Gamma^{-2}\Gamma^* \theta\ci{\mathbf{0}} +
\theta\ci{\mathbf{0}}^*\Gamma D\ci\Gamma^{-2} \Gamma^* \theta\ci{\mathbf{0}}
\\ &
\qquad\qquad\qquad -\theta\ci{\mathbf{0}}^* D\ci{\Gamma^*}^{-2} \theta\ci{\mathbf{0}} + \Gamma^* D\ci{\Gamma^*}^{-2} \theta\ci{\mathbf{0}} +
\theta\ci{\mathbf{0}}^* D\ci{\Gamma^*}^{-2} \Gamma -\Gamma^* D\ci{\Gamma^*}^{-2} \Gamma
\end{align*}
It follows from Lemma \ref{l:G-D_G} that $ D\ci\Gamma^{-2} \Gamma^* = \Gamma^*D\ci{\Gamma^*}^{-2}$ and that $\Gamma^* D\ci\Gamma^{-2} = D\ci{\Gamma^*}^{-2} \Gamma$, so in the above identity we have cancellation of non-symmetric terms,
\begin{align*}
- \theta\ci{\mathbf{0}}^*\Gamma D\ci\Gamma^{-2} - D\ci\Gamma^{-2}\Gamma^* \theta\ci{\mathbf{0}}
+ \Gamma^* D\ci{\Gamma^*}^{-2} \theta\ci{\mathbf{0}} +
\theta\ci{\mathbf{0}}^* D\ci{\Gamma^*}^{-2} \Gamma = 0.
\end{align*}
Therefore
\begin{align*}
X & = D\ci\Gamma^{-2} + \theta\ci{\mathbf{0}}^*\Gamma D\ci\Gamma^{-2} \Gamma^* \theta\ci{\mathbf{0}}
-\theta\ci{\mathbf{0}}^* D\ci{\Gamma^*}^{-2} \theta\ci{\mathbf{0}} -\Gamma^* D\ci{\Gamma^*}^{-2} \Gamma
\\
& = D\ci\Gamma^{-2} + \theta\ci{\mathbf{0}}^* D\ci{\Gamma^*}^{-2} \Gamma\Gamma^* \theta\ci{\mathbf{0}}
-\theta\ci{\mathbf{0}}^* D\ci{\Gamma^*}^{-2} \theta\ci{\mathbf{0}} - D\ci{\Gamma}^{-2} \Gamma^* \Gamma
\\ &
= D\ci\Gamma^{-2} ( \mathbf{I} - \Gamma^*\Gamma) + \theta\ci{\mathbf{0}}^* D\ci\Gamma^{-2}( \Gamma^*\Gamma -\mathbf{I}) \theta\ci{\mathbf{0}} =\mathbf{I} - \theta\ci{\mathbf{0}}^*\theta\ci{\mathbf{0}} =\Delta\ci{\mathbf{0}} .
\end{align*}
Thus we get that $\Delta\ci\Gamma = A^* \Delta\ci{\mathbf{0}} A$, which is exactly the conclusion of the theorem.
\end{proof}
\subsection{Multiplicity of the absolutely continuous spectrum}
It is well-known that the Sz.-Nagy--Foia\c s model space reduces to the familiar one-story setting with $\mathcal{K}_\theta = H^2({\mathfrak D}_*)\ominus \theta H^2({\mathfrak D})$ when $\theta$ is inner. Indeed, for inner $\theta$ the non-tangential boundary values of the defect $\Delta(\xi) = ({\mathbf{I}} - \theta^*(\xi) \theta(\xi))^{1/2}= 0$ Lebesgue a.e.~$\xi\in \mathbb{T}$. So, the second component of the Sz.-Nagy--Foia\c s model space collapses completely.
Here we provide a finer result that reveals the matrix-valued weight function and the multiplicity of $U$'s absolutely continuous part.
Before we formulate the statement, we recall some terminology. First, we Lebesgue decompose the (scalar) measure $d\mu = d\mu\ti{ac}+d\mu\ti{sing}$. The absolutely continuous part of $U$ is unitarily equivalent to the multiplication by the independent variable $\xi$ on the von Neumann direct integral $\mathcal{H}\ti{ac}= \int_\mathbb{T}^\oplus E(\xi) {\mathrm{d}} \mu\ti{ac}(\xi).$ Note that the dimension of $E(\xi)$ is the multiplicity function of the spectrum.
Let $w$ denote the density of the absolutely continuous part of $\mu$, i.e.~${\mathrm{d}}\mu\ti{ac}(\xi) = w(\xi){\mathrm{d}} m(\xi)$. Then the matrix-valued function $\xi\mapsto B^*(\xi)B(\xi)w(\xi)$ is the absolutely continuous part of the matrix-valued measure
$
B^*B\mu
$.
\begin{theo}\label{t-AC}
The defect function $\Delta\ci{\mathbf{0}}$ of $\theta\ci{\mathbf{0}}$ and the absolutely continuous part $B^*Bw$ of the matrix-valued measure $B^*B\mu$ are related by
\begin{align}
\label{e-NOT}
({\mathbf{I}} - \theta\ci{\mathbf{0}}^*(\xi))B^*(\xi)B(\xi)w(\xi)
({\mathbf{I}} - \theta\ci{\mathbf{0}}(\xi))=
(\Delta\ci{\mathbf{0}}(\xi))^2
\end{align}
for Lebesgue a.e.~$\xi\in \mathbb{T}$.
The function $\mathbf{I}-\theta\ci{\mathbf{0}}$ is invertible a.e.~on $\mathbb{T}$, so
the multiplicity of the absolutely continuous part of $\mu$ is given by
\begin{align}\label{e-3}
\dim E(\xi)
=
\rank ({\mathbf{I}} - \theta\ci{\mathbf{0}}^*(\xi)\theta\ci{\mathbf{0}}(\xi))
=
\rank\bigtriangleup\ci{\mathbf{0}}(\xi),
\end{align}
of course, with respect to Lebesgue a.e.~$\xi\in \mathbb{T}$.
\end{theo}
Combining \eqref{e-3} with Theorem \ref{t-Delta} we obtain:
\begin{cor}
For Lebesgue a.e.~$\xi\in\mathbb{T}$ we have
$\dim E(\xi)
=
\rank\bigtriangleup\ci\Gamma(\xi)
$ for all strict contractions $\Gamma$.
\end{cor}
Another immediate consequence is the following:
\begin{cor}
Operator $U$ has no absolutely continuous part on a Borel set $B\subset \mathbb{T}$ if and only if $\theta\ci{\mathbf{0}}(\xi)$ (or, equivalently, $\theta\ci\Gamma(\xi)$ for all strict contractions $\Gamma$) is unitary for Lebesgue almost every $\xi\in B$.
\end{cor}
This corollary is closely related to the main result of \cite[Theorem 3.1]{DL2013}. Interestingly, it appears that the proof (in \cite{DL2013}) of that result cannot be refined to yield our current result (Theorem \ref{t-AC}).
\begin{cor}\label{c-TFAE}
In particular, we confirm that the following are equivalent:
\begin{enumerate}
\item $U$ is purely singular,
\item $\theta\ci\Gamma(\xi)$ is inner for one (equivalently any) strict contraction $\Gamma$,
\item $\Delta\ci\Gamma \equiv {\mathbf{0}}$ for one (equivalently any) strict contraction $\Gamma$,
\item the second story of the Sz.-Nagy--Foia\c s model space collapses (and we are dealing with the model space ${\mathcal K}_{\theta\ci\Gamma} = H^2({\mathbb C}^d) \ominus \theta\ci\Gamma H^2({\mathbb C}^d)$ for one (equivalently any) strict contraction $\Gamma$).
\end{enumerate}
\end{cor}
\begin{proof}[Proof of Theorem \ref{t-AC}]
Take $\Gamma\equiv{\mathbf{0}}$. Solving \eqref{theta_0-02} for $F_2$ we see
\[
F_2(z) = [{\mathbf{I}} + \theta\ci{\mathbf{0}}(z)][{\mathbf{I}} - \theta\ci{\mathbf{0}}(z)]^{-1}
.
\]
Let $\mathcal{P}(B^*B\mu)$ denote the Poisson extension of the matrix-valued measure $B^*B\mu$ to the unit disc ${\mathbb D}$. Since $F_2= \mathcal{C}_2 B^*B\mu$, we can see that $\mathcal{P}(B^*B\mu)= \re F_2$ on ${\mathbb D}$, so
\[
\mathcal{P}(B^*B\mu)=\re F_2
=
\re[({\mathbf{I}}+\theta\ci{\mathbf{0}})({\mathbf{I}}-\theta\ci{\mathbf{0}})^{-1}].
\]
Standard computations yield
\begin{align*}
\mathcal{P}(B^*B\mu)
&=
\re[({\mathbf{I}}+\theta\ci{\mathbf{0}})({\mathbf{I}}-\theta\ci{\mathbf{0}})^{-1}]
=
\frac{1}{2} [({\mathbf{I}}+\theta\ci{\mathbf{0}})({\mathbf{I}}-\theta\ci{\mathbf{0}})^{-1}+({\mathbf{I}}-\theta\ci{\mathbf{0}}^*)^{-1}({\mathbf{I}}+\theta\ci{\mathbf{0}}^*)]\\
&=
\frac{1}{2} ({\mathbf{I}}-\theta\ci{\mathbf{0}}^*)^{-1}\left[({\mathbf{I}}-\theta\ci{\mathbf{0}}^*)({\mathbf{I}}+\theta\ci{\mathbf{0}})+({\mathbf{I}}+\theta\ci{\mathbf{0}}^*)({\mathbf{I}}-\theta\ci{\mathbf{0}})\right]({\mathbf{I}}-\theta\ci{\mathbf{0}})^{-1}\\
&=
\frac{1}{2} ({\mathbf{I}}-\theta\ci{\mathbf{0}}^*)^{-1}[{\mathbf{I}} - \theta\ci{\mathbf{0}}^*\theta\ci{\mathbf{0}}]({\mathbf{I}}-\theta\ci{\mathbf{0}})^{-1}
=
({\mathbf{I}}-\theta\ci{\mathbf{0}}^*)^{-1}\re [{\mathbf{I}} - \theta\ci{\mathbf{0}}^*\theta\ci{\mathbf{0}}]({\mathbf{I}}-\theta\ci{\mathbf{0}})^{-1}\\
&=({\mathbf{I}}-\theta\ci{\mathbf{0}}^*)^{-1}[{\mathbf{I}} - \theta\ci{\mathbf{0}}^*\theta\ci{\mathbf{0}}]({\mathbf{I}}-\theta\ci{\mathbf{0}})^{-1}
\end{align*}
on ${\mathbb D}$. Note that for any characteristic function $\theta$ and $z\in{\mathbb D}$ the matrix $\theta(z)$ is a strict contraction, so in our case $\mathbf{I} -\theta\ci{\mathbf{0}}$ is invertible on ${\mathbb D}$, and all computations are justified.
We can rewrite the above identity as
\begin{align*}
(\mathbf{I}-\theta\ci{\mathbf{0}})^*\mathcal{P}(B^*B\mu) (\mathbf{I} -\theta\ci{\mathbf{0}}) = {\mathbf{I}} - \theta\ci{\mathbf{0}}^*\theta\ci{\mathbf{0}},
\end{align*}
and taking the non-tangential boundary values we get \eqref{e-NOT}. Here we used the Fatou Lemma (see e.g.~\cite[Theorem 3.11.7]{NikERI}) which says that for a complex measure $\tau$ the non-tangential boundary values of its Poisson extension $\mathcal{P}\tau$ coincide a.e.~with the density of the absolutely continuous part of $\tau$; applying this lemma entrywise we get what we need in the left hand side.
To see that the boundary values of $\mathbf{I}-\theta\ci{\mathbf{0}}$ are invertible a.e.~on $\mathbb{T}$ we notice that $z\mapsto \det (\mathbf{I}- \theta\ci{\mathbf{0}}(z)) $ is a bounded analytic function on ${\mathbb D}$, so its boundary values are non-zero a.e.~on $\mathbb{T}$.
\end{proof}
\section{What is wrong with the universal representation formula and what to do about it?}\label{s-Explanations}
There are several things that are not completely satisfactory with the universal representation formula given by Theorem \ref{t-repr}.
First of all, it is defined only on functions of form $hb$, where $h\in C^1$ is a scalar function and $b\in\Ran\mathbf{B}$. Of course, one can than define it on a dense set, for example on the dense set of linear combinations $f=\sum_k h_k, b_k$, where $b_k$ are columns of the matrix $B$, $b_k=\mathbf{B} e_k$, and $h_k\in C^1(\mathbb{T})$. But the use of functions $b$ (or $b_k$) in the representation is a bit bothersome, especially taking into account that the representation $f=\sum_k h_k b_k$ is not always unique. So, it would be a good idea to get rid of the function $b$.
The second thing is that while the representation formula looks like a singular integral operator (Cauchy transform), it is not represented as a classical singular integral operator, so it is not especially clear if the (well developed) theory of such operators apply in our case. So, we would like to represent the operator in more classical way.
Denoting
$C_1(z):=C_*(z)- z C(z)$ and using the formal Cauchy-type expression
\[
(T^{B^* \mu} f )(z)
=
\int\ci{\mathbb{T}}
\frac{1}{1-z\bar\xi}\, B^*(\xi) f(\xi) d\mu(\xi),
\]
we can, performing formal algebraic manipulations, rewrite \eqref{e-repr} as
\begin{align}\label{e-reprSzNFExplanations}
(\Phi^* h b)(z)
=
C_1(z)
(T^{B^* \mu} h b)(z)
+h(z)
[C_*(z){\mathbf{B}}^*b
-C_1(z)
(T^{B^* \mu} b)(z)]
,\quad z\in \mathbb{T}.
\end{align}
So, is it possible to turn these formal manipulations into meaningful mathematics? And the answer is ``yes'': the formula \eqref{e-reprSzNFExplanations} gives the representation of $\Phi^*$ if one interprets $T^{B^*\mu}f$ as the boundary values of the Cauchy Transform $\mathcal{C}[B^*f\mu](z)$, $z\notin \mathbb{T}$, see the definition in the next section.
In the next section (Section \ref{s-SIO}) we present necessary facts about (vector-valued) Cauchy transform and its regularization, that will allow us to interpret and justify the formal expression \eqref{e-reprSzNFExplanations}. We will complete this justification in Section \ref{ss-PhiStarSNF}, see \eqref{form-repr-01}. This representation is a universal one, meaning that it works in any transcription of the model, but still involves the function $b\in \Ran\mathbf{B}$.
The function $b$ is kind of eliminated Proposition \ref{p:repr2} below, and as it is usually happens in the theory of singular integral operators, the operator $\Phi^*$ splits into the singular integral part (weighted boundary values of the Cauchy transform) and the multiplication part. The function $b$ becomes hidden in the multiplication part, and at the first glance it is not clear why this part is well defined.
Thus the representation given by Proposition \ref{p:repr2} is still not completely satisfactory (the price one pays for the universality), but it is a step to obtain a nice representations for a fixed transcription of a model. Thus we were able to obtain a precise and unambiguous representation of $\Phi^*$ in the Sz.-Nagy--Foia\c{s} transcription, see Theorem \ref{t-repr3} which is the main result of Section \ref{ss-PhiStarSNF}.
\section{Singular integral operators}\label{s-SIO}
\subsection{Cauchy type integrals}
For a finite (signed or even complex-valued) measure $\nu$ on $\mathbb{T}$ its Cauchy Transform $\mathcal{C}\nu$ is defined as
\begin{align*}
\mathcal{C}\nu(z) =\mathcal{C}[\nu](z) = \int_\mathbb{T}\frac{{\mathrm{d}}\nu(\xi)}{1-\bar\xi z} \,, \qquad z\in {\mathbb C}\setminus \mathbb{T}.
\end{align*}
It is a classical fact that $\mathcal{C}\nu(z) $ has non-tangential boundary values as $z\to z_0\in\mathbb{T}$ from the inside and from the outside of the disc ${\mathbb D}$. So, given a finite positive Borel measure $\mu$ one can define operators $T_\pm^\mu$ from $L^1(\mu; E)$ to the space of measurable functions on $\mathbb{T}$ as the non-tangential boundary values from inside and outside of the unit disc ${\mathbb D}$,
\[
(T_{+}^\mu f)(z_0)
=
\text{n.t.-}\lim_{\substack{z\to z_0\\z\in{\mathbb D}}}
\mathcal{C} [f\mu](z)\,,
\qquad
\qquad
(T_{-}^\mu f)(z_0)
=
\text{n.t.-}\lim_{\substack{z\to z_0\\ z\notin \overline{{\mathbb D}}}}
\mathcal{C} [f\mu](z)
\,.
\]
One can also define the regularized operators $T^\mu_r$, $r\in (0,\infty)\setminus \{1\}$, and the restriction of $\mathcal{C} [f\mu]$ to the circle of radius $r$,
\begin{align*}
T^\mu_r f(z) = \mathcal{C} [f \mu ](rz).
\end{align*}
Everything can be extended to the case of vector and matrix valued measures; there are some technical details that should be taken care of in the infinite dimensional case, but in our case everything is finite dimensional ($\dim E\le d<\infty$), so the generalization is pretty straightforward.
So, given a (finite, positive) scalar measure $\mu$ and a matrix-valued function $B^*$ (with entries in $L^2(\mu)$) and vector-valued function $f\in L^2(\mu; E)$ we can define $T^{B^*\mu}_\pm f$ and $T_r^{B^*\mu} f$ as the non-tangential boundary values and the restriction to the circle of radius $r$ respectively of the Cauchy transform $\mathcal{C}[ B^*f\mu](z)$. Modulo slight abuse of notation this notation agrees with the accepted notation for the scalar case.
In what follows the function $B^*$ will be the function $B^*$ from Theorem \ref{t-repr}.
\subsection{Uniform boundedness of the boundary Cauchy operator and its regularization}
For a finite Borel measure $\nu$ on $\mathbb{T}$ and $n\in {\mathbb Z}$ define
\begin{align*}
P_n \nu (z) = \left\{
\begin{array}{ll} \sum_{k=0}^n \hat \nu (k) z^k \qquad & n\ge 0, \\
\sum_{k=n}^{-1} \hat \nu (k) z^k \qquad & n< 0 ; \end{array} \right.
\end{align*}
here $\hat\nu(k)$ is the Fourier coefficient of $\nu$, $\hat\nu(k) = \int_\mathbb{T} \xi^{-k} {\mathrm{d}}\nu(\xi)$.
Recall that $C_1(z) := C_*(z) - zC(z)$ where $C_*$ and $C$ are from Theorem \ref{t-repr}.
Recall that if $W$ is a matrix-valued weight (i.e.~a function whose values $W(\xi)$ are positive semidefinite operators on a finite-dimensional space $H$), then the norm in the weighted space $L^2(W;H)$ is defined as
\[
\|f\|\ci{L^2(W; H)}^2 = \int_\mathbb{T} (W(\xi) f(\xi) , f(\xi))\ci H {\mathrm{d}} m(\xi).
\]
We are working with the model space ${\mathcal K}_\theta$ which is a subspace of a weighted space $L^2(W;{\mathfrak D}_*\oplus{\mathfrak D})$ (the weight could be trivial, $W\equiv \mathbf{I}$, as in the case of Sz.-Nagy--Foia\c{s} model).
Define $\widetilde C_1:= W^{1/2} C_1$.
The function $\widetilde C_1^*\widetilde C_1$ is a matrix-valued weight, whose values are operators on ${\mathfrak D}_*\oplus {\mathfrak D}$, so we can define the weighted space $L^2(\widetilde C_1^*\widetilde C_1) = L^2(\widetilde C_1^*\widetilde C_1;{\mathfrak D}_*\oplus{\mathfrak D})$. Note that
\[
\|f\|\ci{L^2(\widetilde C_1^*\widetilde C_1)} = \|\widetilde C_1 f\|\ci{L^2({\mathfrak D}_*\oplus{\mathfrak D})} = \| C_1 f\|\ci{L^2(W;{\mathfrak D}_*\oplus{\mathfrak D})} .
\]
\begin{lem}
\label{l:BdPn}
The operators $P_n^{B^*\mu}:\mathcal{H}\subset L^2(\mu;E) \to L^2(\widetilde C_1^*\widetilde C_1;{\mathfrak D}_*\oplus{\mathfrak D})$ defined by
\begin{align*}
P_n^{B^*\mu} f := P_n (B^*f\mu), \qquad n\in {\mathbb Z}
\end{align*}
are uniformly in $n$ bounded with norm at most $2$, i.e.
\begin{align*}
\| \widetilde C_1P_n({B^*\mu} f)\|\ci{ L^2({\mathfrak D}_*\oplus{\mathfrak D})} \le 2\|f\|\ci{L^2(\mu;E)}.
\end{align*}
\end{lem}
\begin{proof}
The columns $b_k$ of $B$ are in $\mathcal{H}\subset L^2(\mu;E)$, so $B^*f\mu \in L^1(\mu;{\mathfrak D})$, and therefore operators $P_n^{B^*\mu}$ are bounded operators $\mathcal{H}\to L^2({\mathfrak D})$. It follows from Lemma \ref{l:bound-C} that $\| \widetilde C_1\|_\infty \le 2$, so operator $f\mapsto \widetilde C_1 P_n^{B^*\mu} f$ are bounded operators $\mathcal{H}\to L^2({\mathfrak D}_*\oplus{\mathfrak D})$ (notice that we do not claim the uniform in $n$ bounds here).
Therefore, it is sufficient to check the uniform boundedness on a dense set.
Take $f=hb$ where $b\in \Ran\mathbf{B}$ and $h\in C^1(\mathbb{T})$ is scalar-valued. Then for $n\in{\mathbb Z}$ we have by Theorem \ref{t-repr}
\begin{align*}
\Phi^* f & - z^{n} \Phi^*(\bar\xi^{n} f ) \\
& = C_1(z)\int_\mathbb{T} \frac{h(\xi) - h(z)}{1-\overline \xi z} B^*b{\mathrm{d}}\mu(\xi) -
z^nC_1(z)\int_\mathbb{T} \frac{\bar\xi^{n}h(\xi) - \bar z^{n}h(z)}{1-\overline \xi z} B^*b{\mathrm{d}}\mu(\xi)
\\
& = C_1(z)\int_\mathbb{T} \frac{ 1- (\bar\xi z)^{n} }{1-\overline \xi z} B^*hb{\mathrm{d}}\mu(\xi)
\end{align*}
Expressing $\frac{ 1- (\bar\xi z)^{n} }{1-\overline \xi z}$ as a sum of geometric series we get that for $f=hb$, $h\in{\mathbb C}^1(\mathbb{T})$
\begin{align*}
\Phi^* f - z^{n} \Phi^*(\bar\xi^{n} f ) =
\left\{ \begin{array}{ll} C_1 P_{n-1}(B^*f\mu), \qquad & n\ge 1, \\ -C_1 P_n(B^*f\mu) , & n<0. \end{array}\right.
\end{align*}
By linearity the above identity holds for a dense set of linear combinations $f= \sum_k h_k b_k$, $h_k\in C^1(\mathbb{T})$. The operators $\Phi^*:\mathcal{H} \to {\mathcal K}_{\theta}\subset L^2(W; {\mathfrak D}_*\oplus {\mathfrak D})$ are bounded (unitary) operators, so the desired estimate holds on the above dense set.
\end{proof}
For a measure $\nu$ on $\mathbb{T}$ let $T_r\nu$ be the restriction of the Cauchy transform of $\nu$ to the circle of radius $r\ne 1$,
\[
T_r\nu(z) = \int_\mathbb{T} \frac{{\mathrm{d}}\nu(\xi)}{1- r\bar \xi z}, \qquad z\in \mathbb{T}.
\]
Define operators $T_r^{B^*\mu} $ on $L^2(\mu;E)$ as
\[
T_r^{B^*\mu} f = T_r(B^*f\mu).
\]
The lemma below is an immediate corollary of the above Lemma \ref{l:BdPn}.
\begin{lem}
\label{l:BdT_r}
The operators $T_r^{B^*\mu}:\mathcal{H}\subset L^2(\mu; E) \to L^2(\widetilde C_1^*\widetilde C_1;{\mathfrak D}_*\oplus{\mathfrak D})$ are uniformly in $r$ bounded with norm at most $2$, i.e.
\begin{align*}
\| \widetilde C_1 T_r^{B^*\mu} f\|\ci{L^2({\mathfrak D}_*\oplus{\mathfrak D})} \le 2 \|f\|\ci{L^2(\mu;E)}
\end{align*}
\end{lem}
\begin{proof}
The result follows immediately from Lemma \ref{l:BdPn}, since the operators $T_r^{B^*\mu}$ can be represented as averages of operators $P_n^{B^*\mu}$,
\[
P_r^{B^*\mu} = \left\{
\begin{array}{ll} \displaystyle\sum_{n=0}^\infty (r^n - r^{n+1}) P_n^{B^*\mu} , \qquad & 0<r<1, \\
\displaystyle\sum_{n=1}^\infty (r^{-n} - r^{-n-1}) P_{-n}^{B^*\mu} , \qquad & r>1. \end{array}\right.
\]
\end{proof}
Using uniform boundedness of the operators $\widetilde C_1 T_r^{ B^* \mu}$ (Lemma \ref{l:BdT_r}) and existence of non-tangential boundary values $T_\pm^{B^*\mu}f$ we can get the convergence of operators $\widetilde C_1 T_r^{ B^* \mu}$ in the weak operator topology.
\begin{prop}\label{d-TPlusMinus}
The operators $\widetilde C_1 T_\pm^{ B^* \mu}:\mathcal{H}\subset L^2(\mu;E)\to L^2(W;{\mathfrak D}_*\oplus{\mathfrak D})$ are bounded and
\begin{align*}
C_1 T_\pm^{B^* \mu}
=
\text{\rm w.o.t.-}\lim_{r\to 1^\mp} C_1 T_r^{ B^* \mu}.
\end{align*}
\end{prop}
\begin{proof}
We want to show that for any $f\in\mathcal{H}\subset L^2(\mu;E)$
\[
C_1 T_\pm^{B^*\mu} f = \text{w-}\lim_{r\to 1^{\mp}} C_1 T_r^{B^*\mu} f,
\]
where the limit is in the weak topology of $L^2(W;{\mathfrak D}_*\oplus{\mathfrak D})$. This is equivalent to
\[
\widetilde C_1 T_\pm^{B^*\mu} f = \text{w-}\lim_{r\to 1^{\mp}} \widetilde C_1 T_r^{B^*\mu} f,
\]
with the limit being in the weak topology of $L^2({\mathfrak D}_*\oplus{\mathfrak D})$.
Let us prove this identity for $\widetilde C_1T^{B^*\mu}_+f$. Assume that for some $f\in L^2(\mu;E)$
\[
\widetilde C_1 T_+^{B^*\mu} f \ne \text{w-}\lim_{r\to 1^{-}} \widetilde C_1 T_r^{B^*\mu} f.
\]
Then for some $h\in L^2({\mathfrak D}_*\oplus{\mathfrak D})$
\begin{align}
\label{w-ne}
\Bigl(\widetilde C_1 T_r^{B^*\mu} f, h\Bigr)\ci{L^2({\mathfrak D}_*\oplus{\mathfrak D})} \nrightarrow \Bigl(\widetilde C_1 T_+^{B^*\mu} f, h\Bigr)\ci{L^2({\mathfrak D}_*\oplus{\mathfrak D})} \qquad\text{as } r\to 1^-,
\end{align}
so there exists a sequence $r_k\nearrow 1$ such that
\[
\lim_{k\to \infty} \Bigl(\widetilde C_1 T_{r_k}^{B^*\mu} f, h\Bigr)\ci{L^2({\mathfrak D}_*\oplus{\mathfrak D})} \ne \Bigl(\widetilde C_1 T_+^{B^*\mu} f, h\Bigr)\ci{L^2({\mathfrak D}_*\oplus{\mathfrak D})};
\]
note that taking a subsequence we can assume without loss of generality that the limit in the left hand side exists.
Taking a subsequence again, we can assume without loss of generality that \linebreak $\widetilde C_1 T_{r_k}^{B^*\mu} f \to g$ the weak topology, and \eqref{w-ne} implies that $g\ne \widetilde C_1 T_+^{B^*\mu} f$.
The existence of non-tangential boundary values and the definition of $T^{B^*\mu}_+$ implies that $\widetilde C_1 T_{r_k}^{B^*\mu} f \to \widetilde C_1 T_+^{B^*\mu} f$ a.e.~on $\mathbb{T}$. But as \cite[Lemma 3.3]{LT09} asserts, if $f_n\to f$ a.e.~and $f_n\to g$ in the weak topology of $L^2$, then $f=g$, so we arrived at a contradiction.
Note, that in \cite[Lemma 3.3]{LT09} everything was stated for scalar functions, but applying this scalar lemma componentwise we immediately get the same result for $L^2(\mu; E)$ with values in a separable Hilbert space.
\end{proof}
\section{Adjoint Clark operator in Sz.-Nagy--\texorpdfstring{Foia\c s}{Foias} transcription}\label{ss-PhiStarSNF}
The main result of this section is Theorem \ref{t-repr3} below, giving a formula for the adjoint Clark operator $\Phi^*$.
Denote by $F$ the Cauchy transform of the matrix-valued measure $B^*B\mu$,
\begin{align}
\label{F-02}
F(z) =\mathcal{C}[B^*B\mu](z) = \int_\mathbb{T} \frac{1}{1-z\overline\xi} B^*(\xi) B(\xi) {\mathrm{d}}\mu(\xi), \qquad z\in{\mathbb D},
\end{align}
and let us use the same symbol for its non-tangential boundary values, which exist a.e. on $\mathbb{T}$. Using the operator $T_+^{B^*\mu}$ introduced in the previous section, we give the following formula for $\Phi^*$.
\begin{theo}\label{t-repr3}
The adjoint Clark operator in Sz.-Nagy--Foia\c s transcription reduces to
\begin{align}
\label{Phi*-03}
\Phi^*f
=
\kf{0}{\Psi_2} f
+
\kf{({\mathbf{I}}+\theta\ci\Gamma \Gamma^*)D\ci{\Gamma^*}^{-1}F^{-1}}{\Delta\ci\Gamma D\ci{\Gamma}^{-1}
( \Gamma^* - {\mathbf{I}})}
T_+^{B^* \mu} f, \qquad f\in\mathcal{H},
\end{align}
with $\Psi_2(z)= \widetilde \Psi_2(z) R(z)$, where
\begin{align}
\label{Psi_2-alt}
\widetilde \Psi_2(z) & = \Delta\ci\Gamma D\ci{\Gamma}^{-1}
(\Gamma^*
+
({\mathbf{I}} - \Gamma^*) F(z))
\\ \notag
& = \Delta\ci\Gamma D\ci{\Gamma}^{-1}(\mathbf{I} -\Gamma^* \theta\ci{\mathbf{0}}(z)) F(z) \qquad \text{a.e.~on }\mathbb{T},
\end{align}
and $R$ is a measurable right inverse for the matrix-valued function $B$.
\end{theo}
\begin{rem*}
When $d=1$, this result reduces to \cite[Equation (4.5)]{LT15}.
\end{rem*}
\begin{rem}
\label{r:Psi_well-defined}
As one should expect, the matrix-valued function $\Psi_2$ does not depend on the choice of the right inverse $R$. To prove this it is sufficient to show that $\ker B(z) \subset\ker \widetilde\Psi_2(z)$ a.e., which follows from the proposition below.
\end{rem}
\begin{prop}
\label{p:Psi^*Psi}
For $\widetilde\Psi_2$ defined above in \eqref{wtPsi_2} and $w$ being the density of $\mu\ti{ac} $ we have
\begin{align}
\label{Psi^*Psi}
\widetilde\Psi_2(\xi)^*\widetilde\Psi_2(\xi)= F(\xi)^* \Delta\ci{\mathbf{0}}(\xi)^2 F(\xi) &= B(\xi)^*B(\xi) w(\xi)\qquad && \mu\ti{ac}\text{-a.e.,}
\intertext{and so}
\label{Psi^*Psi-01}
\Psi_2(\xi)^*\Psi_2(\xi) & = w(\xi) \mathbf{I}\ci{E(\xi)} && \mu\ti{ac}\text{-a.e.}
\end{align}
\end{prop}
\begin{proof
Since $\Psi_2 =\widetilde \Psi_2 R$, \eqref{Psi^*Psi-01} follows immediately from \eqref{Psi^*Psi}.
To prove \eqref{Psi^*Psi},
consider first the case $\Gamma = {\mathbf{0}}$. In this case $\widetilde\Psi = \Delta\ci{\mathbf{0}} F$, so
\begin{align}
\notag
\widetilde \Psi^*_2 \widetilde\Psi_2 & = F^* \Delta\ci{\mathbf{0}}^2 F = (\mathbf{I} - \theta\ci{\mathbf{0}}^*)^{-1} \Delta\ci{\mathbf{0}}^2 (\mathbf{I} - \theta\ci{\mathbf{0}})^{-1} &&
\\
\label{Psi^*Psi-02}
& = B^*Bw . && \text{by \eqref{e-NOT} }
\end{align}
Consider now the case of general $\Gamma$. We get
\begin{align*}
\widetilde \Psi_2^* \widetilde\Psi_2 & = F^*(\mathbf{I} - \theta\ci{\mathbf{0}}^*\Gamma) D\ci{\Gamma}^{-1}\Delta\ci\Gamma^2 D\ci{\Gamma}^{-1}(\mathbf{I} -\Gamma^* \theta\ci{\mathbf{0}}) F &&
\\
& = F^* \Delta\ci{\mathbf{0}}^2 F && \text{by Theorem \ref{t-Delta}}
\\
& = B^*Bw && \text{by \eqref{Psi^*Psi-02}. }
\end{align*}
\end{proof}
\subsection{A preliminary formula}
\label{s:Phi^*-prelim}
We start proving Theorem \ref{t-repr3} by first proving this preliminary result, that holds for any transcription of the model. Below the matrix-valued functions $C_*$ and $C$ are from Theorem \ref{t-repr},
and $C_1(z):= C_*(z)-zC(z)$.
\begin{prop}\label{p:repr2}
The adjoint Clark operator represented for $f\in \mathcal{H}\subset L^2(\mu;E)$ by
\begin{align}
\label{Phi^*NF-02}
(\Phi^* f)(z) =
C_1(z)
(T_\pm^{B^* \mu} f)(z)
+\Psi_\pm(z) f(z)
,\quad z\in \mathbb{T},
\end{align}
where the matrix-functions $\Psi_\pm$, $\Psi_\pm(z): E(z) \to {\mathbb C}^{2d}={\mathfrak D}_*\oplus{\mathfrak D}$ are defined via the identities
\begin{align}
\label{Psi}
\Psi_\pm(z)b(z) :=C_*(z){\mathbf{B}}^*b-C_1(z)(T_\pm^{B^* \mu} b)(z), \qquad b\in \Ran \mathbf{B};
\end{align}
here two choices of sign (the same sign for all terms) gives two different representation formulas.
\end{prop}
\begin{rem*}
When $d=1$ and $b\equiv 1$ this alternative representation formula reduces to a formula that occurs in the proof of \cite[Theorem 4.7]{LT15}.
\end{rem*}
\begin{rem*}
It is clear that relations \eqref{Psi} with $b=b_k$, $k=1,2,\ldots, d$, completely defines
the matrix-valued function $\Psi$. However, it is not immediately clear why such function $\Psi$ exists; the existence of $\Psi$ will be shown in the proof.
Recalling the definition \eqref{F-02} of the function $F$,
we can see that $\Psi (z) b_k(z)$ can be given as the (non-tangential) boundary values of the vector-valued function
\begin{align}
\label{Psi-01}
C_*(z) e_k - C_1(z) F(z) e_k, \qquad z\in{\mathbb D},
\end{align}
where $e_1, e_2, \ldots, e_d$ is the standard orthonormal basis in ${\mathbb C}^d$.
\end{rem*}
\begin{proof}[Proof of Proposition \ref{p:repr2}]
Let us first show the result for functions of the form $f=hb\in L^2(\mu; E)$, where $b\in \Ran\mathbf{B}$ and $h$ is a scalar function. We want to show that
\begin{align}\label{t-reprL2}
(\Phi^* hb)(z)
=
C_1(z)
(T_\pm^{B^* \mu} hb )(z)
+h(z)\psi_b^\pm(z)
,\quad z\in \mathbb{T},
\end{align}
where
\[
\psi_b^\pm(z)
:=
C_*(z){\mathbf{B}}^*b
-C_1(z)
(T_\pm^{B^* \mu} b)(z).
\]
First note that \eqref{e-repr} implies that for $b\in \Ran \mathbf{B}$
\begin{align*}
\Phi^* b (z) = C_*(z) \mathbf{B}^* b.
\end{align*}
Observe that for (scalar) $h\in C^1$ we have uniform on $z\in\mathbb{T}$ convergence as $r\to 1^\mp$:
\begin{align}
\label{uniform_conv-01}
\int\ci{\mathbb{T}}
\frac{h(\xi)-h(z)}{1-rz\bar\xi}B^*(\xi)b(\xi)
d\mu(\xi)
&\rightrightarrows
\int\ci{\mathbb{T}}
\frac{h(\xi)-h(z)}{1-z\bar\xi}B^*(\xi)b(\xi)
d\mu(\xi).
\end{align}
Multiplying both sides by $C_1(z)$ we get in
the left hand side exactly $C_1(z) (T_r^{B^* \mu} hb)(z) - h(z) C_1(z) (T_r^{B^* \mu} b)(z)$,
and in the right hand side the part with the integral in the representation \eqref{e-repr}.
Recall that the model space ${\mathcal K}\ci{\theta_\Gamma}$ is a subspace of a weighted space $L^2(W, {\mathfrak D}_*\oplus{\mathfrak D})$.
Uniform convergence in \eqref{uniform_conv-01} implies the convergence in $L^2({\mathfrak D}_*\oplus{\mathfrak D})$, and by Lemma \ref{l:bound-C} the multiplication by $C_*$ and $C_1$ are bounded operators $L^2({\mathfrak D})\to L^2(W;{\mathfrak D}_*\oplus{\mathfrak D})$. Thus (because $h$ is bounded)
\[
h C_*\mathbf{B}^* b + C_1 T_r^{B^* \mu} hb - h C_1T_r^{B^* \mu} b \to \Phi^* hb
\]
as $r\to 1^\mp$ in the norm of $L^2(W;{\mathfrak D}_*\oplus{\mathfrak D})$. By Proposition \ref{d-TPlusMinus} the operators $C_1 T_r^{B^*\mu} \to C_1 T_\pm^{B^*\mu}$ in weak operator topology as $r\to 1^\mp$, so
\begin{align}
\label{form-repr-01}
\Phi^* hb = C_1 T_\pm^{B^* \mu} hb + h C_*\mathbf{B}^* b - h C_1T_\pm^{B^* \mu} b ,
\end{align}
which immediately implies \eqref{t-reprL2}. Thus, \eqref{t-reprL2} is proved for $h\in C^1(\mathbb{T})$.
To get \eqref{form-repr-01}, and so \eqref{t-reprL2} for for general $h$ such that $hb\in L^2(\mu;E)$ (recall that $b\in\Ran \mathbf{B}$) we use the standard approximation argument: the operators $\Phi^*, C_1 T_\pm^{B^*\mu}:\mathcal{H}\to
L^2(W;{\mathfrak D}_*\oplus {\mathfrak D})$ are bounded, and therefore for a fixed $b\in\Ran\mathbf{B}$ the operators $hb \mapsto h\psi_b^\pm$ (which are defined initially on a submanifold of $\mathcal{H}$ consisting of functions of form $hb$, $h\in C^1(\mathbb{T})$) are bounded (as a difference of two bounded operators). Approximating in $L^2(\mu;E)$ the function $hb$ by functions $h_n b$, $h_n\in C^1(\mathbb{T})$ we get \eqref{form-repr-01} and \eqref{t-reprL2} for general $h$.
Let us now proof existence of $\Psi$.
Consider the (bounded) linear operator $\Phi^* - C_1 T^{B^*\mu}$. We know that for $f=hb\in L^2(\mu;E)$ with $b\in\Ran \mathbf{B}$ and scalar $h$
\[
(\Phi^* - C_1 T_\pm^{B^*\mu})hb = h\psi_b^\pm,
\]
so on functions $f=hb$ the operators $\Phi^* - C_1 T_\pm^{B^*\mu}$ intertwine the multiplication operators $M_\xi$ and $M_z$. Since linear combinations of functions $h_kb_k$ are dense in $\mathcal{H}$, we conclude that the operators $\Phi^* - C_1 T_\pm^{B^*\mu}$ intertwine $M_\xi$ and $M_z$ on all $\mathcal{H}$, and so these operators are the multiplications by some matrix functions $\Psi_\pm$.
Using \eqref{form-repr-01} with $h=1$ we can see that
\[
\Psi_\pm b = \Phi^* b -C_1 T^{B^*\mu}_\pm b = \mathbf{C}_* B^* b - C_1 T^{B^*\mu}_\pm b,
\]
so $\Psi_\pm$ are defined exactly as stated in the proposition.
\end{proof}
\subsection{Some calculations}\label{ss-repr3}
Let us start with writing more detailed formulas for the matrix functions $C_*$ and $C_1$ from Proposition \ref{p:repr2}.
\begin{lem}\label{l-C1simple}
We have
\begin{align*
C_*(z)
&=
\kf{ {\mathbf{I}}+ {\theta}\ci\Gamma (z)\Gamma^\ast}{ \Delta\ci\Gamma (z)\Gamma^\ast}
D\ci{\Gamma^*}^{-1},
\qquad
C_1(z)
&=
\kf{\mathbf{I}}{{\mathbf{0}}}D\ci{\Gamma^*}^{-1}({\mathbf{I}} - \Gamma)+\kf{\theta\ci\Gamma(z)}{\Delta\ci\Gamma(z)}D\ci{\Gamma}^{-1}(\Gamma^*-{\mathbf{I}}).
\end{align*}
\end{lem}
\begin{proof}
The formula for $C_*(z)$ is just \eqref{C*_N-F} and the identity $\theta\ci\Gamma(0)=-\Gamma$.
Similarly, equation \eqref{C_N-F} gives us
\[
C(z)
=
\kf{z^{-1}(\theta\ci\Gamma (z)+\Gamma)}{z^{-1}\Delta\ci\Gamma(z)}D\ci{\Gamma}^{-1}.
\]
Substituting these expressions into $C_1(z) = C_*(z) - z C(z)$ and applying the commutation relations from Lemma \ref{l:G-D_G} we see
\begin{align*}
C_1(z)
&
=
\kf{D\ci{\Gamma^*}^{-1}+ \theta\ci\Gamma \Gamma^* D\ci{\Gamma^*}^{-1}-\theta\ci\Gamma D\ci{\Gamma}^{-1}-\Gamma D\ci{\Gamma}^{-1}}{\Delta\ci\Gamma \Gamma^*D\ci{\Gamma^*}^{-1}-\Delta\ci\Gamma D\ci{\Gamma}^{-1}}
\\
& =
\kf{D\ci{\Gamma^*}^{-1}+ \theta\ci\Gamma D\ci{\Gamma}^{-1}\Gamma^*-\theta\ci\Gamma D\ci{\Gamma}^{-1}- D\ci{\Gamma^*}^{-1}\Gamma}{\Delta\ci\Gamma D\ci{\Gamma}^{-1}\Gamma^*-\Delta\ci\Gamma D\ci{\Gamma}^{-1}}
\\&
=
\kf{D\ci{\Gamma^*}^{-1}({\mathbf{I}}-\Gamma)+ \theta\ci\Gamma D\ci{\Gamma}^{-1}(\Gamma^*-{\mathbf{I}})}{\Delta\ci\Gamma D\ci{\Gamma}^{-1}(\Gamma^*-{\mathbf{I}})}
\\ & =
\kf{\mathbf{I}}{{\mathbf{0}}}D\ci{\Gamma^*}^{-1}({\mathbf{I}}-\Gamma)
+
\kf{\theta\ci\Gamma }{\Delta\ci\Gamma} D\ci{\Gamma}^{-1}(\Gamma^*-{\mathbf{I}})
,
\end{align*}
and the second statement in the lemma is verified.
\end{proof}
Recall that $F(z)$, $z\in{\mathbb D}$ is the matrix-valued Cauchy transform of the measure $B^*B\mu$, see \eqref{F-02}, and that for $z\in\mathbb{T}$ the symbol $F(z)$ denotes the non-tangential boundary values of $F$. We need the following simple relations between $F$ and $\theta\ci{\mathbf{0}}$.
\begin{lem}
\label{l:theta-F}
For all $z\in{\mathbb D}$ and a.e.~on $\mathbb{T}$
\begin{align*}
F(z)= ( \mathbf{I} - \theta\ci{\mathbf{0}}(z) )^{-1};
\end{align*}
note that for all $z\in{\mathbb D}$ the matrix $\theta\ci{\mathbf{0}}(z)$ is a strict contraction, so $\mathbf{I} -\theta\ci{\mathbf{0}}(z) $ is invertible.
\end{lem}
\begin{proof}
Recall that the function $F_1$ was defined by $F_1(z)=\mathcal{C}_1[B^*B\mu](z)$. Since $F(z)=\mathbf{I}+ F_1(z)$, we get from \eqref{theta_0-01} that
\[
\theta\ci{\mathbf{0}} (z) = F_1(z) (\mathbf{I} + F_1(z))^{-1} = \bigl( F(z) - \mathbf{I} \bigr) F(z)^{-1}.
\]
Solving for $F$ we get the conclusion of the lemma.
\end{proof}
\subsection{Proof of Theorem \ref{t-repr3}}
Let us first prove the second identity in \eqref{Psi_2-alt}. Using the identity $F=(\mathbf{I} - \theta\ci{\mathbf{0}})^{-1}$ we compute
\begin{align*}
\Gamma^* + (\mathbf{I} -\Gamma^* ) F = (\Gamma^* (\mathbf{I} -\theta\ci{\mathbf{0}} ) + \mathbf{I} -\Gamma^*) F = (\mathbf{I} - \Gamma^* \theta\ci{\mathbf{0}})F,
\end{align*}
which is exactly what we need.
Let us now prove that $\Psi$ from Proposition \ref{p:repr2} if given by $\Psi =\kf{{\mathbf{0}}}{\Psi_2}$ with $\Psi_2$ defined above in Theorem \ref{t-repr3}. Since $R(z) b_k(z)= e_k$, it is sufficient to show that $\Psi =\kf{{\mathbf{0}}}{\Psi_2}$ and that
\begin{align}
\label{Psi_2}
\Psi_2(z) b_k(z) =\Delta\ci\Gamma D\ci{\Gamma}^{-1}
(\Gamma^*
+
({\mathbf{I}} - \Gamma^*)
F(z))e_k, \qquad k=1, 2, \ldots, d.
\end{align}
Using the formulas for $C_*$ and $C_1$ provided in Lemma \ref{l-C1simple} we get from \eqref{Psi-01}
\begin{align*}
\Psi(z) b_k(z) & = C_*(z) e_k - C_1(z) F(z) e_k
\\
&=
\kf{
({\mathbf{I}}+{\theta}\ci\Gamma \Gamma^\ast)D\ci{\Gamma^*}^{-1}
-
[D\ci{\Gamma^*}^{-1}({\mathbf{I}}- \Gamma)+ {\theta}\ci\Gamma D\ci{\Gamma}^{-1} (\Gamma^\ast-{\mathbf{I}})]
F
}
{
\Delta\ci\Gamma \Gamma^* D\ci{\Gamma^*}^{-1}
-\Delta\ci\Gamma D\ci{\Gamma}^{-1}
( \Gamma^*-{\mathbf{I}})
F
}
e_k
.
\end{align*}
Note that it is clear from the representation \eqref{Phi^*NF-02} that the top entry of $\Psi$ should disappear, i.e.~that
\begin{align}
\label{Psi_1=0}
({\mathbf{I}}+{\theta}\ci\Gamma \Gamma^\ast)D\ci{\Gamma^*}^{-1} = [D\ci{\Gamma^*}^{-1}({\mathbf{I}}- \Gamma)+ {\theta}\ci\Gamma D\ci{\Gamma}^{-1} (\Gamma^\ast-{\mathbf{I}})] F.
\end{align}
Indeed, by the definition of ${\mathcal K}\ci\theta$ in the Sz.-Nagy--Foia\c{s} transcription the top entry of $\Phi^* f$ belongs to $H^2({\mathfrak D}_*)$. One can see from Lemma \ref{l-C1simple}, for example, that the top entry of $C_1$ belongs to matrix-valued $H^\infty$, so the top entry of $C_1 T_+^{B^*\mu} f$ is also in $H^2({\mathfrak D}_*)$. Therefore the top entry of $\Psi f$ must be in $H^2({\mathfrak D}_*)$ for all $f$. But that is impossible, because $f$ can be any function in $L^2(\mu;E)$.
For a reader that is not comfortable with such ``soft'' reasoning, we present a ``hard'' computational proof of \eqref{Psi_1=0}. This computation also helps to assure the reader that the previous computations were correct.
To do the computation, consider the term in the square brackets in the right hand side of \eqref{Psi_1=0}.
Using the commutation relations from Lemma \ref{l:G-D_G} in the second equality, we get
\begin{align*}
D\ci{\Gamma^*}^{-1}({\mathbf{I}}- \Gamma) + {\theta}\ci\Gamma D\ci{\Gamma}^{-1}( \Gamma^\ast-{\mathbf{I}})
&=
D\ci{\Gamma^*}^{-1}+ \theta D\ci{\Gamma}^{-1}\Gamma^*-\theta D\ci{\Gamma}^{-1}- D\ci{\Gamma^*}^{-1}\Gamma
\\
&=
D\ci{\Gamma^*}^{-1}+ \theta\Gamma^* D\ci{\Gamma^*}^{-1}-\theta D\ci{\Gamma}^{-1}-\Gamma D\ci{\Gamma}^{-1}
\\
&=
({\mathbf{I}}+\theta\ci\Gamma \Gamma^*)D\ci{\Gamma^*}^{-1}
\{\mathbf{I} - D\ci{\Gamma^*}({\mathbf{I}}+\theta\ci\Gamma \Gamma^*)^{-1}(\theta\ci\Gamma+\Gamma)D\ci{\Gamma}^{-1}\}
\\
& = ({\mathbf{I}}+\theta\ci\Gamma \Gamma^*)D\ci{\Gamma^*}^{-1} \{\mathbf{I}-\theta\ci{\mathbf{0}}\};
\end{align*}
the last equality holds by Theorem \ref{t:LFT-02}.
By Lemma \ref{l:theta-F} we have ${\mathbf{I}} - \theta\ci{\mathbf{0}} = F^{-1}$,
so we have for the term in the square brackets
\[
[D\ci{\Gamma^*}^{-1}({\mathbf{I}}- \Gamma)+ {\theta}\ci\Gamma D\ci{\Gamma}^{-1} (\Gamma^\ast-{\mathbf{I}})] =
({\mathbf{I}}+\theta\ci\Gamma \Gamma^*)D\ci{\Gamma^*}^{-1} F^{-1},
\]
which proves \eqref{Psi_1=0}.
To deal with the bottom entry of $\Psi$ we use the commutation relations from Lemma \ref{l:G-D_G},
\begin{align*}
\Delta\ci\Gamma \Gamma^* D\ci{\Gamma^*}^{-1} -\Delta\ci\Gamma D\ci{\Gamma}^{-1} ( \Gamma^*-{\mathbf{I}}) F & =
\Delta\ci\Gamma D\ci{\Gamma}^{-1} \Gamma^* - \Delta\ci\Gamma D\ci{\Gamma}^{-1} \Gamma^* F + \Delta\ci\Gamma D\ci{\Gamma}^{-1} F
\\
& = \Delta\ci\Gamma D\ci{\Gamma}^{-1} \left( \Gamma^* + (\mathbf{I} - \Gamma^*) F \right) ,
\end{align*}
which gives the desired formula \eqref{Psi_2} for $\Psi_2$.
Finally, let us deal with the second term in the right had side of \eqref{Phi*-03}. We know from Proposition \ref{p:repr2} that the term in front of $T^{B^*\mu}_+ f$ is given by $C_1$. From Lemma \ref{l-C1simple} we get
\begin{align*}
C_1 =
\kf{
D\ci{\Gamma^*}^{-1}({\mathbf{I}}- \Gamma)+ {\theta}\ci\Gamma D\ci{\Gamma}^{-1} (\Gamma^\ast-{\mathbf{I}})
}
{
\Delta\ci\Gamma D\ci\Gamma^{-1} (\Gamma^*-\mathbf{I})
}.
\end{align*}
But the top entry of $C_1$ here is the expression in brackets in the right hand side of \eqref{Psi_1=0}, so it is equal to $({\mathbf{I}}+{\theta}\ci\Gamma \Gamma^\ast)D\ci{\Gamma^*}^{-1}F^{-1}$. Therefore
\[
C_1 =
\kf{
({\mathbf{I}}+{\theta}\ci\Gamma \Gamma^\ast)D\ci{\Gamma^*}^{-1}F^{-1}
}
{
\Delta\ci\Gamma D\ci\Gamma^{-1} (\Gamma^*-\mathbf{I})
},
\]
which is exactly what we have in \eqref{Phi*-03}.
\ \hfill \qed
\subsection{Representation of \texorpdfstring{$\Phi^*$}{Phi*} using matrix-valued measures}
The above Theorem \ref{t-repr3} is more transparent if we represent the direct integral $\mathcal{H}$ as the weighted $L^2$ space with a matrix-valued measure.
Namely, consider the weighted space $L^2(B^*B\mu)$
\[
\|f\|\ci{L^2(B^*B\mu)}^2 = \int_\mathbb{T} \bigl(B(\xi)^*B(\xi) f(\xi), f(\xi) \bigr)\ci{{\mathbb C}^d} {\mathrm{d}}\mu(\xi) =\int_\mathbb{T} \| B(\xi) f(\xi)\|^2\ci{{\mathbb C}^d} {\mathrm{d}}\mu(\xi)
\]
(of course one needs to take the quotient space over the set of function with norm $0$).
Then for all scalar functions $\varphi_k$ we have
\[
\biggl\|\sum_{k=1}^d \varphi_k e_k \biggr\|_{L^2(B^*B\mu)} = \biggl\|\sum_{k=1}^d \varphi_k b_k \biggr\|_{L^2} ;
\]
recall that $e_1, e_2, \ldots, e_d$ is the standard basis in ${\mathbb C}^d$ and $b_k(\xi) = B(\xi)e_k$.
Then the map $\mathcal{U}$
\[
\mathcal{U} \biggl( \sum_{k=1}^d \varphi_k e_k \biggr) = \sum_{k=1}^d \varphi_k b_k, \qquad \text{or, equivalently }\quad \mathcal{U} f = Bf,
\]
defines a unitary operator from $L^2(B^*B\mu)$ to $\mathcal{H}$.
The inverse operator $\mathcal{U}^*$ is given by $\mathcal{U}^* f (\xi) = R(\xi) f(\xi)$, where, recall, $R$ is a measurable pointwise right inverse of $B$, $B(\xi) R(\xi) = \mathbf{I}\ci{E(\xi)}$ $\mu$-a.e.
We denote by $\widetilde\Phi := \mathcal{U}^* \Phi$, so $\widetilde\Phi^* = \Phi^* \mathcal{U}$, and by $T_+^{B^*B\mu} f$ the non-tangential boundary values of the Cauchy integral $\mathcal{C}[B^*Bf\mu](z)$, $z\in{\mathbb D}$. Substituting $f=Bg$ into \eqref{Phi*-03} we can restate Theorem \ref{t-repr3} as follows.
\begin{theo}
\label{t:repr04}
The adjoint Clark operator $\widetilde\Phi^*:L^2(B^*B\mu) \to {\mathcal K}_\theta$ in Sz.-Nagy--Foia\c s transcription is given by
\begin{align}
\label{Phi*-04}
\widetilde\Phi^*g
=
\kf{0}{\widetilde\Psi_2} g
+
\kf{({\mathbf{I}}+\theta\ci\Gamma \Gamma^*)D\ci{\Gamma^*}^{-1}F^{-1}}{\Delta\ci\Gamma D\ci{\Gamma}^{-1}
( \Gamma^* - {\mathbf{I}})}
T_+^{ B^*B \mu} g, \qquad g\in L^2(B^*B\mu),
\end{align}
where the matrix-valued function $\widetilde\Psi_2(z)$ is defined as
\begin{align}
\label{wtPsi_2}
\widetilde\Psi_2(z) =\Delta\ci\Gamma D\ci{\Gamma}^{-1}
(\Gamma^*
+
({\mathbf{I}} - \Gamma^*) F(z)) .
\end{align}
\end{theo}
\subsection{A generalization of the normalized Cauchy transform}
\label{ss:GCT}
Consider the case when the unitary operator $U$ has purely singular spectrum. By virtue of Corollary \ref{c-TFAE}, the second component of the Sz.-Nagy--Foia\c s model space collapses, i.e.~${\mathcal K}_{\theta\ci\Gamma} = H^2({\mathbb C}^d) \ominus \theta\ci\Gamma H^2({\mathbb C}^d)$ for all strict contractions $\Gamma$.
The representation formula \eqref{Phi*-03} then reduces to a generalization of the well-studied normalized Cauchy transform.
\begin{cor}\label{c-KapustinPolt}
If $\theta = \theta\ci{\mathbf{0}}$ is inner, then
\[
(\Phi^* f)(z)
=
({\mathbf{I}}-\theta(z)) (T_+^{B^* \mu}f)(z)
=
(F(z))^{-1} (T_+^{B^* \mu}f)(z)
\]
for $z\in {\mathbb D}, f\in L^2(\mu;E)$.
\end{cor}
The first equation was also obtained in \cite[Theorem 1]{KP06}.
Here we used $\Gamma = {\mathbf{0}}$ only for simplicity. With the linear fractional relation \eqref{t:LFT-02}, it is not hard to write the result in terms of $\theta\ci\Gamma$ for any strict contraction $\Gamma$.
\begin{proof}
Theorem \ref{t-repr3} for inner $\theta$ and $\Gamma={\mathbf{0}}$ immediately reduces to the first statement.
The equality of the second expression follows immediately from Lemma \ref{l:theta-F}.
\end{proof}
\section{The Clark operator}
\label{s:directClark}
Let $f\in\mathcal{H}\subset L^2(\mu; E)$ and let
\begin{align}
\label{Phi^*f=h}
\Phi^* f = h = \left(\begin{array}{c} h_1\\ h_2\end{array}\right) \in{\mathcal K}\ci\theta .
\end{align}
From the representation \eqref{Phi*-04} we get, subtracting from the second component the first component multiplied by an appropriate matrix-valued function, that
\begin{align*}
\Psi_2 f = h_2 -\Delta\ci\Gamma D\ci\Gamma^{-1}(\Gamma^*-\mathbf{I}) F D\ci{\Gamma^*} (\mathbf{I}+\theta\ci\Gamma \Gamma^*)^{-1} h_1.
\end{align*}
Right multiplying this identity by $\Psi_2^*$, and using Proposition \ref{p:Psi^*Psi} and formulas for $\Psi_2$, $\widetilde\Psi_2$ from Theorem \ref{t-repr3}, we get an expression for the density of the absolutely continuous part of $\mu\ti{ac}$. Namely, we find that a.e.~(with respect to Lebesgue measure on $\mathbb{T}$)
\begin{align}
\label{g_{ac}-01}
w f & = R^*F^*(\mathbf{I} - \theta\ci{\mathbf{0}}^*\Gamma) D\ci{\Gamma}^{-1}\Delta\ci\Gamma h_2
\\ \notag
& \qquad -
R^*F^*(\mathbf{I} - \theta\ci{\mathbf{0}}^*\Gamma) D\ci{\Gamma}^{-1}\Delta\ci\Gamma^2 D\ci\Gamma^{-1}(\Gamma^*-\mathbf{I}) F D\ci{\Gamma^*} (\mathbf{I}+\theta\ci\Gamma \Gamma^*)^{-1} h_1
\\ \notag
& = R^*F^*(\mathbf{I} - \theta\ci{\mathbf{0}}^*\Gamma) D\ci{\Gamma}^{-1}\Delta\ci\Gamma h_2
\\ \notag
& \qquad - R^*F^* \Delta\ci{\mathbf{0}}^2 (\mathbf{I} - \Gamma^*\theta\ci{\mathbf{0}})^{-1} (\Gamma^*-\mathbf{I}) F D\ci{\Gamma^*} (\mathbf{I}+\theta\ci\Gamma \Gamma^*)^{-1} h_1 .
\end{align}
In the case $\Gamma={\mathbf{0}}$ the above equation simplifies:
\begin{align}
\label{g_{ac}-02}
w f & = R^*F^*\Delta\ci{\mathbf{0}} h_2 + R^*F^* \Delta\ci{\mathbf{0}}^2 F h_1
\\ \notag
& = R^*F\Delta\ci{\mathbf{0}} h_2 + wB h_1;
\end{align}
in the second equality we use \eqref{Psi^*Psi}.
The above formulas \eqref{g_{ac}-01}, \eqref{g_{ac}-02} determine the absolutely continuous part of $f$.
The singular part of $f$ was in essence computed in \cite{KP06}. Formally it was computed there only for inner functions $\theta$, but using the ideas and results from \cite{KP06} it is easy to get the general case from our Theorem \ref{t-repr3}.
For the convenience of the reader, we give a self-contained presentation.
\begin{lem}
\label{l:polt}
Let $f\in L^2(\mathbb{T}, \mu; {\mathbb C}^d)$. Then $\mu\ti s$-a.e.~the nontagential boundary values of $\mathcal{C} [f\mu](z)/\mathcal{C}[\mu](z)$, $z\in{\mathbb D}$ exist and equal $f(\xi)$, $\xi\in\mathbb{T}$.
\end{lem}
This lemma was proved in \cite{KP06} even for a more general case of $f\in L^2(\mu;E)$, where $E$ is a separable Hilbert space. Note that our case $E={\mathbb C}^d$ follows trivially by applying the corresponding scalar result ($E={\mathbb C}$) proved in \cite{NONTAN} to entries of the vector $f$.
Applying the above Lemma to the representation giving by the first coordinate of \eqref{Phi*-03} from Theorem \ref{t-repr3} we get that for $f$ and $h$ related by \eqref{Phi^*f=h} we have
\begin{align*}
B^*f = \frac1{\mathcal{C}[\mu]} F D\ci{\Gamma^*} (\mathbf{I}+\theta\ci \Gamma \Gamma^*)^{-1} h_1 \qquad \mu\ti s\text{-a.e.}
\end{align*}
Left multiplying this identity by $R^*$ we get that
\begin{align}
\label{f_sing}
\Phi h = f = \frac1{\mathcal{C}[\mu]} R^* F D\ci{\Gamma^*} (\mathbf{I}+\theta\ci \Gamma \Gamma^*)^{-1} h_1 \qquad \mu\ti s\text{-a.e.}
\end{align}
Summarizing, we get the following theorem, describing the direct Clark operator $\Phi$.
\begin{theo}
\label{t:direct Clark}
If $\Phi^* f = h$ as in \eqref{Phi^*f=h}, so $f=\Phi h$, then the absolutely continuous part of $f$ is given by \eqref{g_{ac}-01} and the singular part of $f$ is given by \eqref{f_sing}.
\end{theo}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
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\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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| {
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} | 5,715 |
Friends of South Asia is the name of several organizations in North America:
An activist group:
Friends of South Asia (FOSA), a primarily Indo-Pakistani peace group in the San Francisco Bay Area
A museum members' group:
Friends of South Asia, a special interest group at the Royal Ontario Museum in Toronto
Groups in support of South Asian Studies departments:
Friends of South Asia at the University of Chicago
Friends of the South Asian Studies Council at Yale | {
"redpajama_set_name": "RedPajamaWikipedia"
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Ludwig Glauert (5 de mayo de 1879 – 1 de febrero de 1963) fue un paleontólogo, herpetólogo y curador de museos brito-australiano. Es conocido por su obra sobre fósiles mamíferos del Pleistoceno, y como curador de museo jugando un papel importante en las ciencias naturales de Australia Occidental.
Glauert era nacido en Ecclesall, Sheffield, Inglaterra. Su padre Johann Ernst Louis Henry Glauert, comerciante y fabricante de cubiertos, y madre Amanda Watkinson. Se educó en Sheffield, en el Sheffield Royal Grammar School, en el Firth University College, y en la Technical School, estudiando geología, convirtiéndose, en 1900, en miembro de la Sociedad Geológica de Londres.
En 1908, con su esposa migraron a Perth, Australia Occidental, donde se unió al Servicio Geológico, como paleontólogo, trabajando en la organización de las colecciones del Museo de Australia Occidental.
En 1910, se convirtió en parte del personal permanente del museo; y, en 1914 fue ascendido a Curador de geología y etnología. De 1909 a 1915, llevó a cabo trabajo de campo en cuevas del Margaret River, Australia Occidental, hallando fósiles de varias especies de extintos Monotremata y marsupiales en calizas del Pleistoceno.
Fue miembro del Western Australian Naturalists Club; y publicó regularmente en el West Australian Naturalist como también en las columnas de'The Naturalist' del Western Mail
Glauert falleció en Perth.
Honores
nombrado MBE en los Honores del Año Nuevo 1960.
Eponimia
Varanus glauerti, una especie de Australian lagarto monitor se nombró en su honor.
Galardones
1948 Medalla Australiana de Historia Natural
Referencias
Paleontólogos de Australia
Herpetólogos de Australia
Miembros de la Orden del Imperio Británico
Medalla Australiana de Historia Natural
Personas de Australia Occidental
Nacidos en Sheffield
Fallecidos en Perth | {
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Gayelle Television (Gayelle) is a privately owned television station, serving Trinidad and Tobago. The television station broadcasts on UHF channel 23 to the city of Port of Spain and has recently started broadcasting to Central Trinidad on UHF channel 27. It is available on the Flow Trinidad cable system as channel 7 and also on cable systems in Tobago and Grenada. The station offers 100% local and Caribbean programming, and much of its programming consists of live talk-shows. The station's studio is located at 13 Southern Main Road, Curepe, Trinidad and Tobago.
News
Gayelle TV carried a weeknight news-hour at 7pm, anchored by Paolo Kernahan and a thirty-minute equivalent on weekends, anchored by Adonis Ballah. The station also carries hourly news updates on weekdays from 6am to 6pm. Gayelle News is known for its focus of community features such as We The People and Community Connection.On March 31, 2009, the station closed its news department and laid off 16 members of staff due to the global economic downturn. As of April 1, 2009, the station entered into an agreement with CNC3 Television to simulcast the CNC3 7pm newscast.
Network Slogans
At last we own television! (16 February 2004 – 19 January 2010, currently used on website)
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References
External links
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Television stations in Trinidad and Tobago | {
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\section{Introduction}
\label{sec:Introduction}
Majorana zero modes (MZMs) are particle-hole symmetric zero energy bound states predicted to appear on edges and topological defects in a certain class of superconductors (SCs)~\cite{Kitaev2001,Alicea2012}. Besides characterizing this novel state of matter, topological superconductivity, MZMs are also expected to exhibit non-Abelian statistics, which in turn is believed to be a promising route towards fault-tolerant topological quantum computation~\cite{Nayak2008}.
Much of the activity in current research on MZMs is to propose and design hybrid superconducting structures and experimental setups where the modes are expected to appear through suitable tuning of various parameters. Among these setups, one that has attracted particular attention is a one-dimensional (1D) semiconducting nanowire with strong spin-orbit coupling lying in proximity with an $s$-wave SC~\cite{Oreg2010,Lutchyn2010}. When such a wire is immersed in an external magnetic field above a certain critical field strength, the system is expected to enter a topological superconducting regime with MZMs exponentially localized at the two wire endpoints.
The zero-energy and particle-hole symmetry properties of MZMs are further predicted to result in a robust quantized tunneling conductance of $2e^2/h$ at zero voltage bias due to perfect Andreev reflection\cite{Flensberg2010,Fidkowski2012,Diez2012,Setiawan2015}
and also a $4\pi$-periodic Josephson current due to the conservation of fermion parity\cite{Kitaev2001,Kwon2004,Fu2009}. Deeply connected to this $4\pi$-Josephson effect is the appearance of an odd number of zero energy crossings in the Andreev spectrum when the junction phase difference is tuned. While the existence of an odd number crossings is guaranteed from the underlying particle-hole symmetry\cite{Lutchyn2010}, their specific locations in the spectrum are non-universal and will generally depend on microscopic details.
In this paper, we consider Josephson junctions of the aforementioned nanowires with an additional chiral symmetry and show that a single crossing can be viewed as a phase transition between two topologically distinct regimes. Hence, both its existence and location in the spectrum are universal and can be obtained from topological arguments. We further exploit this observation to analyze junctions in which the Andreev spectrum is strongly affected by the system geometry. Our key observation is that, in the presence of chiral symmetry, the directional nature of Rashba spin-orbit coupling (RSOC) is inherited by the induced $p$-wave pairing in the topological regime. Consequently, two proximity induced wires connected with an offset angle can therefore exhibit an effective phase difference, despite the lack of external phase bias. Hence, such a configuration realizes a so-called $\phi_0$-junction, which has recently been analyzed in the context of topological superconductivity\cite{Dolcini2015,Nesterov2016,Huang2017}.
We also show that by introducing local curvature $\kappa(w)$ into a single nanowire parametrized by the coordinate $w$, there exists an explicit relation between the superconducting current density and the curvature
\begin{equation}
\label{eq:IntroEquation}
\langle J(w) \rangle = \frac{\rho(w)}{2m_\text{eff}} \kappa (w),
\end{equation}
where $\rho(w)$ is the local charge density along the wire and $m_\text{eff}$ is the effective electron mass in the topological regime. Similar ideas, that curvature may induce currents (and accompanying magnetic fields) in chiral superconductors have recently been proposed in 2D\cite{Kvorning2017} and we argue that the effects presented in this paper are of similar origin.
The remainder of this paper is organized as follows. We introduce our model of a 1D topological SC in Sec.~\ref{sec:Model} and discuss its symmetries. In Sec.~\ref{sec:GeoJosephson} ,we consider this model in the setting of Josephson junctions and show how the junction geometry affects the Andreev spectrum. These results are confirmed numerically in Sec.~\ref{sec:Numerics}. In Sec.~\ref{sec:Curvature}, we use the previous analysis to derive a proportionality relation between the curvature and current density for a single curved wire. We end with a summary and a discussion in Sec.~\ref{sec:Summary}. Throughout the paper we use units where $e=\hbar=1$ and we also assume zero temperature.
\section{Model of a 1D topological superconductor}
\label{sec:Model}
\subsection{Hamiltonian}
\label{sec:Hamiltonian}
Our starting point is a single straight 1D semi-conducting nanowire lying in the $x$-$y$ plane. The wire has a RSOC with strength $\alpha_R$ and lies in proximity with an $s$-wave SC with order parameter $\Delta$. The wire is also immersed in an external magnetic field $\mathbf{B}$ which defines the Zeeman field $\mathbf{h}\equiv \frac{1}{2}g\mu_B \mathbf{B}$, with $g$ the effective g-factor in the wire and $\mu_B$ being the Bohr magneton.
Assuming the wire to be thin, so that only a single channel is occupied, we model this system with a BdG Hamiltonian acting on basis spinors
$\psi(w) = (u^{}_\uparrow(w),u^{}_\downarrow(w),v^{}_\uparrow(w),v^{}_\downarrow(w))^T$, where $w$ is the coordinate along the wire, $u$ and $v$ represents electron and hole components and $\uparrow,\downarrow$ refers to the spin-projection along the $z$-axis. The Hamiltonian reads
\begin{subequations}
\label{eq:FullNanowire}
\begin{align}
\label{eq:NanoWire}
&\mathcal{H}_\text{BdG}(p_w) = \begin{pmatrix}
h(p_w) & h_\Delta \\
h^\dagger_\Delta & -h^T(-p_w)
\end{pmatrix},\\
& h(p_w) = \frac{p_w^2}{2m^*}-\mu -\alpha_R p_w (\sigma_y\cos\varphi -\sigma_x\sin\varphi) + \mathbf{h} \cdot \boldsymbol{\sigma}, \label{eq:normalHam} \\
& h_\Delta = |\Delta|e^{i\phi_s}(i\sigma_y).
\end{align}
\end{subequations}
In these expressions, $\mathbf{p}_w=p_w(\cos \varphi,\sin \varphi)$ is the planar momentum operator parametrized by $\varphi$, the angle between the wire and the positive $x$-axis. The effective electron mass is denoted by $m^*$ and $\mu$ is the chemical potential. The RSOC is assumed to arise from some electrical field pointing in the $\mathbf{\hat{z}}$ direction so that it favours spin-alignment along some vector lying in the $x$-$y$ plane. This direction will be referred to as the RSOC direction and can in principle depend on $w$. We decompose the proximity induced SC order parameter as $|\Delta|e^{i\phi_s}$ with the phase parameter inherited directly from the underlying $s$-wave SC. The set of Pauli-matrices $\boldsymbol{\sigma}=\{\sigma_x,\sigma_y,\sigma_z\}$ act in spin space, and for later convenience we also define similarly the set of particle-hole Pauli matrices $\boldsymbol{\tau}=\{\tau_x,\tau_y,\tau_z\}$. The accompanying $2\times2$ unit matrices are denoted $\sigma_0$ and $\tau_0$ respectively.
It has been shown\cite{Oreg2010,Lutchyn2010,Halperin2012} that the Hamiltonian~\eqref{eq:FullNanowire} can be mapped onto a spinless $p$-wave SC model with a topological regime hosting edge MZMs \cite{Kitaev2001}.
This topological regime occurs when two conditions on $\mathbf{h}$ are met\cite{Osca2014,Rex2014}:
\begin{enumerate}[(i)]
\item The full field must satisfy $|\mathbf{h}|>h_c \equiv \sqrt{|\Delta|^2+\mu^2}$. Furthermore, if the Hamiltonian~\eqref{eq:FullNanowire} is put on a lattice (with unit lattice constant) which limits the spectrum, there is an additional upper critical field $\tilde{h}_c\equiv\sqrt{|\Delta|^2+(\mu-4t)^2}$, where $t\equiv\frac{1}{2m^*}$ is the hopping parameter. In that case, the full field must also satisfy $|\mathbf{h}|<\tilde{h}_c$.
\item The projection of the Zeeman field onto the direction of the RSOC vector, $\mathbf{h}_\alpha$, must obey $|\mathbf{h}_\alpha| < |\Delta|$, otherwise the energy gap closes, and the system becomes metallic.
\end{enumerate}
\subsection{Symmetries}
\label{sec:Symmetries}
The BdG Hamiltonian \eqref{eq:FullNanowire} belongs generally to symmetry class $\mathcal{D}$\cite{Schnyder2008,Ryu2010,Kitaev2009} with a single antiunitary symmetry, the particle-hole symmetry $\mathcal{P}\mathcal{H}_\text{BdG}(p_w)\mathcal{P}^{-1} = -\mathcal{H}_\text{BdG}(-p_w)$, $\mathcal{P}^2=+1$. In our basis $\mathcal{P}=\tau_x \mathcal{K}$, with $\mathcal{K}$ being the complex conjugation operator.
However, with perpendicular or parallel orientations of the Zeeman field; $\mathbf{h}=h\mathbf{\hat{z}}$ or $\mathbf{h}=h \left(\cos\varphi\mathbf{\hat{x}}+ \sin \varphi\mathbf{\hat{y}}\right)$ respectively, the model has an additional unitary chiral symmetry $\mathcal{C}\mathcal{H}_\text{BdG}(p_w)\mathcal{C}^\dag = -\mathcal{H}_\text{BdG}(p_w)$ and the system belongs to symmetry class $\mathcal{BDI}$ characterized by a topological winding number invariant~\cite{Schnyder2008,Ryu2010,Kitaev2009,Diez2012,Tewari2012,VolovikBook}. In a translationally invariant (with compact Brillouin zone) and gapped system, this invariant can be computed as\cite{Tewari2012}
\begin{equation}
\label{eq:windingnumber}
\nu \equiv \frac{1}{\pi i} \int_0^\pi d\theta(p_w) \in \mathbbm{Z},
\end{equation}
which cannot change as long as the energy gap is maintained. Here, $\theta(p_w)$ is the phase of $\text{det}A(p_w)$ and $A(p_w)$ is the off-diagonal block of the BdG Hamiltonian in the chiral basis
\begin{equation}
\mathcal{H}^\mathcal{C}_{\text{BdG}}(p_w) = \begin{pmatrix}
0 & A(p_w) \\
A^T(-p_w) & 0
\end{pmatrix}.
\end{equation}
The particular model \eqref{eq:FullNanowire}, realizes only $\nu \in \{0,-1,+1\}$, where $\nu=0$ is the trivial regime and $\nu = \pm 1$ are topological regimes with edge MZMs. Crucially, in these regimes $\nu$ typically assigns wires with SC phases that are even or odd multiples of $\pi$ with opposite winding numbers. Consequently, the chiral wires has a notion of directionality in the pairing. This feature is particularly visible in Andreev reflection, where low energy electron-hole and hole-electron scattering processes are phase shifted by $\pi$ since they experience the SC gap with different signs due to their relative direction\cite{Tanaka1995,Beenakker2012} (see also Ref.~\onlinecite{Spanslatt2017} for a recent discussion).
\section{Geometrical Josephson junctions}
\label{sec:GeoJosephson}
It follows that if two chiral 1D topological SCs with a phase difference of $\pi$ (mod $2\pi$) are connected in a Josephson junction, the junction must host an interface where the gap closes, since $\nu$ necessarily changes\cite{Spanslatt2015}. On this interface, a number of zero energy bound states, related to the difference in $\nu$, appear. The Majorana nature of these states follows from particle-hole symmetry. Note also that in class $\mathcal{BDI}$, both the existence and the location of the crossing is topologically protected, and it will therefore not change in the presence of symmetry respecting perturbations and is also independent on whether the junction is in the short or long junction limit.
We therefore consider the model~\eqref{eq:FullNanowire} in the context of Josephson junctions for two different configurations. First, the Zeeman field is taken perpendicular to the junction, and secondly in the same plane as the junction.
\subsection{Junctions with a perpendicular Zeeman field}
\label{sec:PerpJunctions}
\begin{figure}[t]
\captionsetup[subfigure]{position=top,justification=raggedright}
\subfloat[][]{
\includegraphics[width=0.96\columnwidth]{SetupA.pdf}
\label{fig:SetupA}}
\\
\subfloat[][]{
\includegraphics[width=0.96\columnwidth]{SetupB.pdf}
\label{fig:SetupB}}
\label{fig:Setups}
\caption{(a) Schematics of a ``hinge'' type of Josephson junction with phase difference $\Delta \phi_s$, where the two wires are offset by the angle $\varphi$. The Zeeman field $\mathbf{h}$ is either long the $z$-direction or in the $x$-$y$ plane parametrized by the angle $\phi_B$. (b) Proximitized and curved wire in perpendicular Zeeman field. The wire is parametrized by a local Frenet frame $\mathbf{\hat{t}}(w)$-$\mathbf{\hat{n}}(w)$ defining the local curvature as $\partial_w\mathbf{\hat{n}}(w)=-\kappa(w)\mathbf{\hat{t}}(w)$. In the topological regime, the curvature induces an effective phase gradient $\kappa(w) = \partial_w\phi_p(w)$.}
\end{figure}
We consider a ``hinge'' type of junction where wire $1$ lies along the $x$-axis: $\mathbf{\hat{w}}_1=(1,0)$ of which the end point is coupled to wire $2$ lying along the direction $\mathbf{\hat{w}}_2=(\cos\varphi,\sin\varphi,0)$, see Fig.~\ref{fig:SetupA}. In addition, the wires lie in proximity to two separate $s$-wave SCs with externally controllable phases $\phi_{s1}$ and $\phi_{s2}$ respectively. The Zeeman field is taken as $\mathbf{h}=h\mathbf{\hat{z}}$, and we also choose a phase convention such that $\phi_{s1}=0$ and $\phi_{s2}\equiv\Delta \phi_s$: the $s$-wave phase difference between the wires.
Considering first wire $1$ in isolation, the chiral symmetry operator is given by $\mathcal{C}=\tau_x$. For wire $2$ in isolation, the unitary transformation $U=\text{diag}(e^{i \varphi},1,e^{-i \varphi},1)$ transfers the angle $\varphi$ to become a contribution to $\Delta\phi_s$. To satisfy the same chiral symmetry condition as wire 1 (that is using $\mathcal{C}=\tau_x$ which is required for consistency if the wires are to be coupled), we find that the second wire must fulfill the condition $\Delta\phi_{s}+\varphi = n\pi$, where $n \in \mathbbm{Z}$. Therefore, for $n$ odd, we then see by using Eq.~\eqref{eq:windingnumber} that the two wires have opposite winding numbers at this instance. Hence, if the wires are connected, there must exist a gap closing, a crossing, in the Andreev spectrum precisely at
\begin{equation}
\label{eq:crossingcondition}
\Delta \phi_s = \pi-\varphi \quad \text{mod}\;2\pi.
\end{equation}
It then follows that by varying $\varphi$, the whole spectrum can be shifted because of the topological protection and $2\pi$-periodicity of the crossing.
To verify this reasoning further we now review the mapping onto the effective low energy $p$-wave superconductor. To this end, we construct the following eigen-spinors of the normal state Hamiltonian \eqref{eq:normalHam} as
\begin{subequations}
\label{eq:electronspinors}
\begin{align}
& \ket{u_+(p_w)} = e^{-i\pi/4}\begin{pmatrix}
e^{-i\varphi}\cos\frac{\theta_p}{2} \\
-i\sin\frac{\theta_p}{2}
\end{pmatrix}, \\
&\ket{u_-(p_w)} = e^{-i\pi/4}\begin{pmatrix}
-e^{-i\varphi}\sin\frac{\theta_p}{2} \\
-i\cos\frac{\theta_p}{2}
\end{pmatrix},
\end{align}
\end{subequations}
where $\tan\theta_{p}=|\alpha p_w/h|$. By particle-hole symmetry, the corresponding hole spinors read
\begin{subequations}
\label{eq:holespinors}
\begin{align}
& \ket{v_+(p_w)} = e^{i\pi/4}\begin{pmatrix}
-e^{i\varphi}\cos\frac{\theta_p}{2} \\
i\sin\frac{\theta_p}{2}
\end{pmatrix}, \\
&\ket{v_-(p_w)} = e^{i\pi/4}\begin{pmatrix}
e^{i\varphi}\sin\frac{\theta_p}{2} \\
i\cos\frac{\theta_p}{2}
\end{pmatrix}.
\end{align}
\end{subequations}
The electron and hole energies are $\pm E_\pm(p_w) = \frac{p_w^2}{2m^*}-\mu\pm\sqrt{\alpha_R^2p_w^2+h^2}$ respectively. In the strong magnetic field regime, $h\gg \alpha_R$, the spins are almost completely polarized within each band, and if $\mu$ lies in the gap between the two electron bands, one can project the full Hamiltonian~\eqref{eq:NanoWire} onto the bands $\ket{u_-(p_w)}$ and $\ket{v_-(p_w)}$. The result is an effective spin-less $p$-wave Hamiltonian\cite{Halperin2012,Alicea2011}
\begin{equation}
\label{eq:effectivepwave}
\mathcal{H}_p = \begin{pmatrix}
\frac{p_w^2}{2m_\text{eff}}-\mu_\text{eff} & p_w |\Delta_p| e^{i \phi_p} \\
p_w |\Delta_p| e^{-i \phi_p} & -\frac{p_w^2}{2m_\text{eff}}+\mu_\text{eff}
\end{pmatrix},
\end{equation}
where the effective parameters are $\mu_\text{eff}=\mu+h$, $\frac{1}{m_\text{eff}}=\frac{1}{m^*}(1-m^*\alpha_R^2/h)$, $|\Delta_p|= |\alpha_R \Delta/h|$. Most interestingly, in this basis, the effective $p$-wave phase is given by
\begin{equation}
\label{eq:phases}
\phi_p = \phi_s + \varphi.
\end{equation}
For a single uniform wire, the extra geometrical phase contribution $\varphi$ is not important since it can be removed adjusting the overall phase factors on the spinors in \eqref{eq:electronspinors} and \eqref{eq:holespinors}. However, in the Josephson junction setup outlined above, the effective phase difference reads
\begin{equation}
\label{eq:phasediff}
\Delta \phi_p \equiv \phi_{p2}-\phi_{p1}= \Delta \phi_s + \varphi.
\end{equation}
In the effective model limit, there will be gap closing in the Andreev spectrum whenever $\Delta \phi_p=\pi$ (mod $2\pi$)\cite{Kwon2004,Spanslatt2015}, which by Eq.~\eqref{eq:phasediff} again leads to the condition~\eqref{eq:crossingcondition}.
In the short junction limit, where there are only two Andreev bound states, the spectrum and the zero temperature DC-current for a 1D $p$-wave Josephson junction were calculated in Ref.~\onlinecite{Kwon2004} (assuming fermion parity conservation). They read
\begin{subequations}
\label{eq:ABS}
\begin{align}
\epsilon_{\pm} &= \pm |\Delta_p|\sqrt{D}\cos(\frac{\Delta\phi_p}{2}), \\
I &=\sqrt{D}|\Delta_p| \sin(\frac{\Delta\phi_p}{2}),
\end{align}
\end{subequations}
where $D$ is the junction transparency for a single channel.
By simply inserting the phase relation in Eq.~\eqref{eq:phasediff} into Eq.~\eqref{eq:ABS} we obtain a geometrically dependent spectrum and current
\begin{subequations}
\label{eq:ABSGeo}
\begin{align}
\label{eq:ABSGeo1}
\epsilon_{\varphi,\pm} &= \pm |\Delta_p|\sqrt{D}\cos(\frac{\Delta\phi_s + \varphi}{2}), \\
I_\varphi &= \sqrt{D}|\Delta_p|\sin(\frac{\Delta\phi_s + \varphi}{2}).
\end{align}
\end{subequations}
In particular, there is a finite Josephson current, even if the externally controlled $\Delta \phi_s=0$. We denote this anomalous current as geometrical and define
\begin{equation}
\label{eq:GeoCurrent}
I_{Geo} \equiv I_{\varphi = 0}=\sqrt{D}|\Delta_p|\sin(\frac{\varphi}{2}) ,
\end{equation}
which is one of the main results of this paper. $I_{Geo}$ is maximal for anti-parallel wires $\varphi = \pi$, which is equivalent to the wires having opposite signs of the Rashba coupling $\alpha_R$. The junction then realizes a topological $\pi$-junction~\cite{Ojanen2013,Klinovaja2015}. In essence, our proposal of the geometrical Josephson junction is a generalization of such junctions which are based on chiral symmetry.
\subsection{Junctions with a planar Zeeman field}
\label{sec:PlanarJunctions}
Next, we discuss the same setup as in the previous section, but with a planar Zeeman field $\mathbf{h}=h \left(\cos\phi_B\mathbf{\hat{x}}+ \sin \phi_B\mathbf{\hat{y}}\right)$, parametrized by the angle $\phi_B$.
As follows from the discussion in Sec.~\ref{sec:Symmetries}, the chiral symmetry will be broken for all values of $\Delta \phi_s$ unless the Zeeman field lies in parallel with both wires. If there is such parallel alignment, the crossing necessarily occurs at $\Delta \phi_s = \pi$ (mod $2\pi$)\cite{Lutchyn2010} and both the crossing and its location is topologically protected.
Accordingly, a finite $\varphi$ for wire $2$ will also break the chiral symmetry for all $\Delta \phi_s$, and we can not use winding number arguments for determining the location of the crossing in the Andreev spectrum. Still, the presence of a single crossing in the Andreev spectrum is protected by particle-hole symmetry\cite{Lutchyn2010}, but its location is non-universal.
Indeed, for both wires along the $x$-axis ($\varphi=0$) and with a small $h_y=h\sin \phi_B$ , it was reported that location of the crossing occurs when $\Delta \phi_s = \pi+2\arcsin(h_y/|\Delta|)$ in the short junction limit~\cite{Dolcini2015,Nesterov2016,Huang2017}, due to a Fermi momentum mismatch mechanism. However, for longer junctions, the crossing location changes, indicating the non-universality (see Sec.~\ref{sec:Numerics}).
When $\mathbf{h}=h\mathbf{\hat{x}}$ and $\varphi$ is finite, we have found numerically (see Sec.~\ref{sec:Numerics}) that the crossing occurs when $\Delta\phi_s = \pi-\arcsin(h\sin\varphi/|\Delta|)$ in the short junction limit, but again, for longer junctions this is not the case. Nevertheless, the geometry of the junction affects the Andreev spectrum, but from a similar mismatch mechanism, rather than by changes in the condition for a topological phase transition as in Sec.~\ref{sec:PerpJunctions}. We leave the investigation of the combination of both an arbitrary planar Zeeman field and a finite ``hinge'' angle $\varphi$ to the future.
We do note however that due to point (ii) in Sec.~\ref{sec:Hamiltonian}, there is an upper bound on the Zeeman field component in the RSOC direction which restricts the range of possible $\varphi$ for the junction to be topological. This restriction is determined by the following expressions
\begin{subequations}
\label{eq:anglereq}
\begin{align}
\label{eq:CritangleA}
\sin(\phi_{B,c}) &= \pm \frac{|\Delta|}{h}, \\
\label{eq:CritangleB}
\sin(\phi_B-\varphi_c) &= \pm \frac{|\Delta|}{h}.
\end{align}
\end{subequations}
In Eq.~\eqref{eq:CritangleA} $\phi_{B,c}$ is the maximally allowed angle for condition (ii) to be fulfilled for wire $1$. Given that $\phi_B<\phi_{B,c}$, Eq.~\eqref{eq:CritangleB} then limits the field projection on the $\varphi$-dependent RSOC vector in wire $2$ and thereby determines $\varphi_c$, the maximally allowed offset angle.
\section{Numerical results}
\label{sec:Numerics}
\begin{figure}[t]
\captionsetup[subfigure]{position=top,justification=raggedright}
\subfloat[Subfigure 1 list of figures text][]{
\includegraphics[width=0.48\columnwidth]{3pi4SpectrumA.pdf}
\label{fig:Spectrum1}}
\subfloat[Subfigure 2 list of figures text][]{
\includegraphics[width=0.48\columnwidth]{3pi4SpectrumB.pdf}
\label{fig:Spectrum2}}
\label{fig:Spectrum12}
\caption{(a) Andreev (black) and bulk (gray) spectrum of a short, $L_N=0$, geometrical Josephson junction as a function of the phase difference $\Delta \phi_s$. Flat midgap states, corresponding to outer edge Majorana zero modes states have been removed for clarity. The parameters are $L_S=120$, $t=1.0$, $|\Delta|=1.0$, $\mu=0.0$, $h=3.0$ and $\alpha_R=0.5$. (b) Same as (a) but with $L_N=10$.}
\end{figure}
\begin{figure}[t]
\captionsetup[subfigure]{position=top,justification=raggedright}
\subfloat[Subfigure 1 list of figures text][]{
\includegraphics[width=\columnwidth]{Spec1.pdf}
\label{fig:SpectrumA}}
\\
\subfloat[Subfigure 2 list of figures text][]{
\includegraphics[width=\columnwidth]{Spec2.pdf}
\label{fig:SpectrumB}}
\\
\subfloat[Subfigure 2 list of figures text][]{
\includegraphics[width=\columnwidth]{Spec3.pdf}
\label{fig:SpectrumC}}
\label{fig:SpectrumABC}
\caption{Numerically calculated locations of zero energy crossings, $\Delta \phi_s^*$, in the $s$-wave phase difference $\Delta \phi_s$ for various configurations of the topological Josephson junction. In (a) $\mathbf{h}=h\mathbf{\hat{z}}$ and the gray solid line denotes $\Delta \phi_s^*=\pi-\varphi$, (b) $\mathbf{h}=h \left(\cos\phi_B\mathbf{\hat{x}}+ \sin \phi_B\mathbf{\hat{y}}\right)$, $\varphi=0$, and the gray line is $\Delta \phi_s^*=\pi+2\arcsin(h\sin(\phi_B)/|\Delta|)$, (c) $\mathbf{h}=h\mathbf{\hat{x}}$ and the gray line is $\Delta \phi_s^*=\pi-\arcsin(h\sin(\varphi)/|\Delta|)$. For all plots: $L_S=120$, $t=1.0$, $|\Delta|=1.0$, $\mu=0.0$, $h=3.0$ and $\alpha_R=0.5$. The normal segment lengths are $L_N=0$ (circles), $L_N=10$ (squares) and $L_N=40$ (stars).}
\end{figure}
To verify our analytical calculations, we next numerically compute the low energy spectrum of the various Josephson junction setups. To this end, we use a discretized version of~\eqref{eq:FullNanowire} (on a lattice with unit lattice constant) given by
\begin{align}
\label{eq:Tightbinding}
\mathcal{H} &= -t\sum_{j,\sigma} c^{\dag}_{j\sigma} c^{}_{j+1\sigma}+h.c.-(\mu-2t)\sum_{j,\sigma} c^{\dag}_{j\sigma} c^{}_{j\sigma}\notag\\
& + \sum_{j,\sigma, \sigma'} c^{\dag}_{j\sigma} \left(\mathbf{h} \cdot \boldsymbol{\sigma}\right)_{\sigma \sigma'} c^{}_{j\sigma'}\notag\\
&-\frac{i\alpha_R}{2}\sum_{j,\sigma, \sigma'} c^{\dag}_{j\sigma}\left(\sigma_y\cos\varphi_{j}-\sigma_x\sin\varphi_{j}\right)_{\sigma \sigma'} c^{}_{j+1\sigma'}-h.c. \notag \\
&+\sum_{j} \Delta_j c^{}_{j\uparrow}c^{}_{j\downarrow}+h.c.,
\end{align}
where $t\equiv 1/2m^*$ is the hopping parameter and $\varphi_{j}$ is the angle between site $j$ and the positive $x$ axis. We model the SC-normal-metal-SC Josephson junction setup by taking
\begin{equation}
\Delta_j =\begin{cases}
|\Delta| &\text{for } 1\leq j\leq L_S \\
~0 &\text{for } L_S < j \leq L_S+L_N \\
|\Delta|e^{i \Delta \phi_s} &\text{for } L_S+L_N<j\leq 2L_S+L_N,
\end{cases}
\end{equation}
where $L_S$ denotes the number of sites in each of the SC regions, and $L_N$ that of the normal region, see Fig.~\ref{fig:SetupA}. Throughout this section, we choose $L_S=120$, $t=1.0$, $|\Delta|=1.0$, $\mu=0.0$, $h\equiv|\mathbf{h}|=3.0$ and $\alpha_R=0.5$.
We first model the setup described in Sec.~\ref{sec:PerpJunctions} and accordingly we take $\mathbf{h}=h\mathbf{\hat{z}}$, wire 1 along the direction $\mathbf{\hat{w}}_1=(1,0)$ and wire 2 lies along $\mathbf{\hat{w}}_2=(\cos\varphi,\sin\varphi)$, see Fig.~\ref{fig:SetupA}. For concreteness, we also choose $\varphi=\pi/4$. The resulting spectra for $L_N=0$ and $L_N=10$ are depicted in Fig.~\ref{fig:Spectrum1} and~\ref{fig:Spectrum2} respectively. We note that the crossing in both plots occurs at $\Delta \phi_s = 3\pi/4$ in agreement with Eq.~\eqref{eq:crossingcondition}. Furthermore, the Andreev spectrum in Fig.~\ref{fig:Spectrum1} matches Eq.~\eqref{eq:ABSGeo1} very well with $D\approx 0.85$.
We next generalize this calculation to general angles $\varphi$ and for a few different junction lengths $L_N$. The result is shown in Fig.~\ref{fig:SpectrumA}. The location of the crossings, $\Delta \phi_s^*$, agree very well with Eq.~\eqref{eq:crossingcondition}, represented by the gray solid line, and does not change when increasing $L_N$. We attribute this feature to the topological nature of the crossing location.
We repeat the previous calculation for the planar setup of Sec.~\ref{sec:PlanarJunctions}. We first take $\mathbf{h}=h \left(\cos\phi_B\mathbf{\hat{x}}+ \sin \phi_B\mathbf{\hat{y}}\right)$ and $\varphi=0$. The results are shown in Fig.~\ref{fig:SpectrumB} and are in agreement (gray solid line) with Refs.~\onlinecite{Dolcini2015,Nesterov2016,Huang2017} in the short junction limit $L_N=0$. For longer junctions, the crossing locations deviate significantly from this line, and indicates that the crossing locations are not protected by topology. We also note that in this limit, the crossings changes from below to above $\pi$ and vice versa, indicated by the discontinuities in Fig.~\ref{fig:SpectrumB} and Fig.~\ref{fig:SpectrumC}.
Finally, we choose $\mathbf{h}=h\mathbf{\hat{x}}$ and vary $\varphi$. The results are presented Fig.~\ref{fig:SpectrumC}. In the short junction limit, we find the approximate relation $\Delta\phi_s^* = \pi-\arcsin(h\sin\varphi/|\Delta|)$ which is violated for longer junctions. In all figures, we associate the slight deviations from the dashed lines to finite size effects. We have also checked that weak scalar disorder does not qualitatively change our results.
We conclude that our numerical calculations support the concept of geometrical Josephson junctions in Sec.~\ref{sec:GeoJosephson}.
\section{From curvature to phase gradient to current}
\label{sec:Curvature}
Given the result of mapping from Eq.~\eqref{eq:FullNanowire} onto the low energy theory~\eqref{eq:effectivepwave}, we anticipate that the induced phase shift in Eq.~\eqref{eq:phases} should influence a wire where $\varphi$ varies continuously, which is the situation for a curved wire, see Fig.~\ref{fig:SetupB}. We therefore move on to generalise the mapping onto the effective theory to a smoothly curved wire in the $x$-$y$ plane. The symmetrized and Hermitian normal state Hamiltonian in real space reads\cite{Ortix2015a,Ortix2015b,Ying2017}
\begin{equation}
\label{eq:curvedRashba}
h_\text{curved}(w) = -\frac{\partial^2_w}{2m^*}-\mu -\frac{\alpha_R}{2}\lbrace-i\partial_w , \sigma(w) \rbrace + h \sigma_z,
\end{equation}
where $\lbrace\cdot,\cdot\rbrace$ denote the anti-commutator and $\sigma(w)$ is a local Pauli matrix co-moving along with the wire. Explicitly, $\sigma(w)=\boldsymbol{\sigma}\cdot \mathbf{\hat{n}}(w)$, where $\mathbf{\hat{n}}(w)$ is the local unit vector normal to the wire direction. The Frenet-Serret equations then define the local curvature, $\kappa(w)$, of the wire through $\partial_w\mathbf{\hat{n}}(w)=-\kappa(w)\mathbf{\hat{t}}(w)$ with $\mathbf{\hat{t}}(w)$ being the local unit tangent vector along the wire. Consequently, the RSOC favours spin-alignment in a direction determined by the local curvature of the wire according to
\begin{equation}
\label{eq:curvature}
\varphi(w) = \int^w_0\;dw' \kappa(w').
\end{equation}
By again following the steps of the mapping onto the topological superconductor, we obtain
\begin{equation}
\label{eq:curvedpwave}
\mathcal{H}_p = \begin{pmatrix}
-\frac{\partial_w^2}{2m_\text{eff}}-\mu_\text{eff} & \frac{1}{2} \lbrace-i\partial_w , |\Delta_p| e^{i \phi_p(w)} \rbrace\\
\frac{1}{2}\lbrace-i\partial_w , |\Delta_p| e^{-i \phi_p(w)} \rbrace & \frac{\partial_w^2}{2m_\text{eff}}+\mu_\text{eff}
\end{pmatrix},
\end{equation}
and the local curvature in Eq.~\eqref{eq:curvature} enters the local $p$-wave pairing phase through
\begin{equation}
\label{eq:curvaturephase}
\phi_p(w) = \phi_s +\int^w_0\;dw' \kappa(w').
\end{equation}
Hence, the presence of curvature in the wire is equivalent to a phase gradient of the effective order parameter.
By using the unitary transformation $U=\exp(-i \phi_p(w)\tau_z/2)$, the phase $\phi_p(w)$ can be removed from the pairing term, and the transformed Hamiltonian $\mathcal{H}_p \mapsto U \mathcal{H}_p U^\dag$ becomes
\begin{equation}
\label{eq:curvedpwavetransformed}
\mathcal{H}_p = \begin{pmatrix}
\frac{\left(-i\partial_w + \partial_w \phi_p/2\right)^2}{2m_\text{eff}}-\mu_\text{eff} & -|\Delta_p|i\partial_w , \\
-|\Delta_p|i\partial_w & -\frac{\left(-i\partial_w - \partial_w \phi_p/2\right)^2}{2m_\text{eff}}+\mu_\text{eff}
\end{pmatrix},
\end{equation}
where it has been assumed that $|\Delta_p|$ is spatially constant. This expression makes manifest that the local curvature $\kappa(w) = \partial_w\phi_p(w)$ enters the Hamiltonian just as an electromagnetic vector potential, similar to the 2D geo-Josephson effect\cite{Kvorning2017}.
The charge current density operator $J(w)$ can be derived by first employing the usual minimal substitution $-i\partial_w \rightarrow -i\partial_w \pm A_w$ where the signs are different for the electron and hole components. Note however, that to ensure invariance under electromagnetic gauge transformations
\begin{subequations}
\begin{align}
\label{eq:Gaugetransformation}
A_w &\rightarrow A_w' = A_w+\partial_w \chi(w) \\
\phi_p(w) &\rightarrow \phi_p'(w) = \phi_p(w)-2\chi(w),
\end{align}
\end{subequations}
the minimal substitution rule does not apply to the pairing terms. We then obtain the current density operator from $\langle J(w) \rangle \equiv \frac{\delta H_p^A}{\delta A(w)}$, where $H_p^A=\int dw \Psi^\dag(w) \mathcal{H}_p(w) \Psi(w)$ is the full $p$-wave Hamiltonian, as
\begin{equation}
\label{eq:currentoperator}
J(w) = \frac{1}{2m_\text{eff}}\left(-i \overset{\rightarrow}{\partial}_w + i\overset{\leftarrow}{\partial}_w \right)\tau_0 + \frac{\left(\partial_w \phi_p(w) -2A(w) \right)}{2 m_\text{eff}}\tau_z.
\end{equation}
Assuming further that the the system is in its zero temperature ground state with no single particle excitations, we omit the first term in \eqref{eq:currentoperator} and obtain the supercurrent contribution
\begin{equation}
\label{eq:current3}
\langle J(w)\rangle = \frac{\rho(w)}{2m_\text{eff}} (\partial_w \phi_p(w)-2A(w)),
\end{equation}
where $\rho(w) = |u(w)|^2-|v(w)|^2$ is the local charge density along the wire in the ground state. Finally, setting $A(w)=0$ and using Eq.~\eqref{eq:curvaturephase}, the geometrically induced supercurrent density reads
\begin{equation}
\label{eq:current2}
\langle J(w)\rangle = \frac{\rho(w)}{2m_\text{eff}} \kappa(w),
\end{equation}
which is Eq.~\eqref{eq:IntroEquation} in Sec.~\ref{sec:Introduction}. This effect is similar to that described in Ref.~\onlinecite{Klinovaja2015}, where the Rashba electrical field direction is changed constinously along a straight wire and thereby generates a phase gradient and a supercurrent in the topological regime. Here instead, the Rashba electrical field is fixed, but the wire direction changes continously, making the connection between current and curvature manifest and also relates the effect to a higher dimensional analogue~\cite{Kvorning2017}.
\section{Summary and Discussion}
\label{sec:Summary}
In this paper we considered Josephson junctions composed of 1D topological superconducting wires. We showed that it is possible to generate a contribution to the Josephson current which originates from a geometric offset angle between two wires. In particular, this contribution results in a geometrically induced anomalous Josephson current which can be traced to the directional nature of Rashba spin-orbit coupling inherited by the effective $p$-wave pairing in the topological regime. We also showed that this directional dependence manifests itself by inducing a phase gradient in the presence of curvature. Since a superconductor responds to a phase gradient by generating a supercurrent, we could derive an explicit current-curvature relationship for chiral 1D $p$-wave SCs.
Regarding the experimental feasibility, observing a curvature induced supercurrent through the curving of a planar nanowire is perhaps not within current technological range. In addition, setups with proximity coated nanowires and a perpendicular magnetic field configuration has the drawback of low critical magnetic fields, which complicates the transition into the topological regime~\cite{Higginbotham2015}. A more reasonable proposal would instead be to use etched narrow channels in a two-dimensional electron gas with strong spin-orbit coupling\cite{Hell2017}. Such a system, immersed in a thin semiconducting layer sandwiched between a conventional $s$-wave superconductor and a ferromagnetic insulator could realize the model \eqref{eq:FullNanowire} with $\mathbf{h}=h\mathbf{\hat{z}}$~\cite{Sato2010,Black-Schaffer2011}.
Going beyond the single channel description of the wires, it has been shown\cite{Tewari2012,Tewari2012b,Diez2012} that the chiral symmetry is still approximately present as long as the wire width is smaller than the spin-orbit length $l_{so}=(m^*\alpha_R)^{-1}$.
As a final remark, we note that the geometrical supercurrent \eqref{eq:IntroEquation} should be thought of as \textit{extrinsically} induced in the sense that it originates from curvature in the embedding 2D space, since no intrinsic geometry can be defined on a 1D line. That the 1D confined electrons still experience this extrinsic curvature is because of the Rashba effect, through which the electron spin introduces the notion of direction. This feature stands in contrast to the recently proposed geo-Meissner and geo-Josephson effects for 2D chiral $p$-wave superconductors\cite{Kvorning2017}, where the geometrical contribution to the Josephson current stems from the \textit{intrinsic} 2D curvature. Nonetheless, both effects arise because of a directional or chiral nature of the $p$-wave pairing through the electron spin degree of freedom.
\section*{ACKNOWLEDGMENTS}
The author is very grateful to Eddy Ardonne, Thors Hans Hansson, Iman Mahyaeh and Axel Gagge for numerous helpful discussions and input. Thomas Kvorning, Jens Bardarsson, Ady Stern and Jorge Cayao are also acknowledged for useful comments.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,963 |
A study conducted by a Canadian institution dispels several myths about piracy in third-world countries.
We all know that pirating games is bad, right? That downloading someone's work without paying for it is tantamount to stealing? Of course, you do, you're a good person. But there is a persuasive argument made by some that piracy is sometimes the only way to play games, whether because there is no legal avenues to purchase them or because the legal marketplaces that do exist are too highly priced to be feasible. A three-year study conducted by the Social Science Research Council and funded by Canada's International Development Research Centre seems to state that these two points are the major contributor to piracy in developing economies. The study focused on six nations (South Africa, Russia, Brazil, Mexico, Bolivia, and India) and its findings were printed in a book called Media Piracy in Emerging Economies edited by Joe Karaganis.
Among the major points in the book is the assertion that organized crime has nearly nothing to do with piracy in these nations because it's impossible for even "well-organized" criminals to compete with the low, low price of "free." Also, there are over 300 anti-piracy education programs identified in these nations but the study concludes that these have zero affect on consumer's behavior.
The cause of piracy in these nations, the study purports, is that the price of media is far too high for local populations to afford it. "Relative to local incomes in Brazil, Russia, or South Africa, the retail price of a CD, DVD, or copy of [Microsoft] Office is five to ten times higher than in the US or Europe," the study's website said.
The study also believes that the industries involved (such as the music, movie or game businesses) have been concentrating on legal solutions to the problem of piracy by demanding laws prohibiting it. But with judges and law enforcers already taxed to help the well-being of a nation by reducing violent crime, the study found that anti-piracy laws were deemed less important. "After a decade of ramped up enforcement, the authors can find no impact on the overall supply of pirated goods."
For the gaming industry, the economies of developing nations cannot support the cost of owning expensive game consoles and buying $60 games. So it only makes sense for knockoffs and pirated material to be rampant; if the choice is between not playing and playing, the gamer will always choose to play, no matter if he's pirating the game or not. If a game publisher wants to market games legally to these developing nations, then it might be necessary to reduce the prices to something that the people can afford.
Now, that might not be realistic for AAA titles hot off the presses, but what's wrong with suppliers selling 1 or 2 year old titles to developing countries at a steep discount? I think the worry that people might then try to import games from India back into Western markets is a little crazy, honestly.
In any case, I'm glad that Media Piracy in Emerging Economies is attempting to discuss these issues so that there is an independent source of facts other than the information released by media companies. 'Cause, you know, they kind of want to think about this stuff the way that they do. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,937 |
What is a Podcast Feed?
A laptop with a webcam built in, which can be used to make video podcasts.
Written By: Mary McMahon
Edited By: O. Wallace
A podcast feed is a document on a server which updates when new podcasts in a series are uploaded. People can subscribe to the feed to arrange to have new podcasts delivered to their computers through a program which handles podcasts, or through a feed reader which allows them to click to listen. Podcast feeds are used to allow people to keep up with a podcast they enjoy without having to visit the site where the podcast is hosted to check for new episodes.
Podcasts are video or audio presentations which can vary in length considerably. This format for content delivery is used by a wide range of websites, and can include things like brief commentary on the news, talk shows, game shows, and so forth. Once the podcast is generated, it is placed on a server, and people can access it at any time. This contrasts with live streaming broadcasts, which are only available at set times.
Setting up a feed is usually strongly recommended to people who run a podcast, because it ensures that people will be able to access their content. With a podcast feed, when new episodes are uploaded, the feed updates. When people open their feed readers or podcast management programs, the new episodes are downloaded. Listeners and viewers never need to worry about missing an episode with a podcast feed delivery to their computers.
Typically, links to the podcast feed are located on the website which hosts the feed. In addition, many podcast management programs allow people to search a database of podcasts, and to subscribe from there. Podcast designers make sure to embed information in their feed which will make their podcasts show up on a search. For example, two people running "The Dave and Shelly" show on automotive repair would make sure that the files in their podcast feed were clearly labeled and tagged with keywords so that people searching for "Dave and Shelly," "car repair," "automotive repair," and so forth could find their podcast when they searched through a third party client.
It is easy to manage subscriptions of podcast feeds. People can usually subscribe and unsubscribe with a single click, and they can sort their subscriptions. Someone who listens to a lot of podcasts might break them up by category so that new material got delivered into different folders, for example. It is also possible to use devices like personal media players which handle subscriptions, allowing people to connect the devices to get the latest updates downloaded directly onto a device they will use to listen while taking a train, biking, or engaging in other activities where a portable podcast player might be desired.
What is a Podcast RSS?
What is a Podcast Directory?
What is a Podcast Blog?
What is Podcast Software?
What is a Video Podcast?
What is a Podcasting Aggregator? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,966 |
package de.littlerolf.sav.data;
public class HistoryItem {
public int[] values;
public int[] index = {-1};
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,528 |
#ifndef SRC_TRACE_PROCESSOR_IMPORTERS_PROTO_PROFILE_PACKET_UTILS_H_
#define SRC_TRACE_PROCESSOR_IMPORTERS_PROTO_PROFILE_PACKET_UTILS_H_
#include "perfetto/ext/base/optional.h"
#include "perfetto/ext/base/string_view.h"
#include "src/trace_processor/importers/proto/packet_sequence_state.h"
#include "src/trace_processor/importers/proto/stack_profile_tracker.h"
#include "protos/perfetto/trace/interned_data/interned_data.pbzero.h"
#include "protos/perfetto/trace/profiling/profile_common.pbzero.h"
#include "protos/perfetto/trace/profiling/profile_packet.pbzero.h"
namespace perfetto {
namespace trace_processor {
class ProfilePacketUtils {
public:
static SequenceStackProfileTracker::SourceMapping MakeSourceMapping(
const protos::pbzero::Mapping::Decoder& entry) {
SequenceStackProfileTracker::SourceMapping src_mapping{};
src_mapping.build_id = entry.build_id();
src_mapping.exact_offset = entry.exact_offset();
src_mapping.start_offset = entry.start_offset();
src_mapping.start = entry.start();
src_mapping.end = entry.end();
src_mapping.load_bias = entry.load_bias();
for (auto path_string_id_it = entry.path_string_ids(); path_string_id_it;
++path_string_id_it) {
src_mapping.name_ids.emplace_back(*path_string_id_it);
}
return src_mapping;
}
static SequenceStackProfileTracker::SourceFrame MakeSourceFrame(
const protos::pbzero::Frame::Decoder& entry) {
SequenceStackProfileTracker::SourceFrame src_frame;
src_frame.name_id = entry.function_name_id();
src_frame.mapping_id = entry.mapping_id();
src_frame.rel_pc = entry.rel_pc();
return src_frame;
}
static SequenceStackProfileTracker::SourceCallstack MakeSourceCallstack(
const protos::pbzero::Callstack::Decoder& entry) {
SequenceStackProfileTracker::SourceCallstack src_callstack;
for (auto frame_it = entry.frame_ids(); frame_it; ++frame_it)
src_callstack.emplace_back(*frame_it);
return src_callstack;
}
static const char* StringifyCpuMode(
protos::pbzero::Profiling::CpuMode cpu_mode) {
using protos::pbzero::Profiling;
switch (cpu_mode) {
case Profiling::MODE_UNKNOWN:
return "unknown";
case Profiling::MODE_KERNEL:
return "kernel";
case Profiling::MODE_USER:
return "user";
case Profiling::MODE_HYPERVISOR:
return "hypervisor";
case Profiling::MODE_GUEST_KERNEL:
return "guest_kernel";
case Profiling::MODE_GUEST_USER:
return "guest_user";
}
return "unknown"; // switch should be complete, but gcc needs a hint
}
static const char* StringifyStackUnwindError(
protos::pbzero::Profiling::StackUnwindError unwind_error) {
using protos::pbzero::Profiling;
switch (unwind_error) {
case Profiling::UNWIND_ERROR_UNKNOWN:
return "unknown";
case Profiling::UNWIND_ERROR_NONE:
return "none"; // should never see this serialized by traced_perf, the
// field should be unset instead
case Profiling::UNWIND_ERROR_MEMORY_INVALID:
return "memory_invalid";
case Profiling::UNWIND_ERROR_UNWIND_INFO:
return "unwind_info";
case Profiling::UNWIND_ERROR_UNSUPPORTED:
return "unsupported";
case Profiling::UNWIND_ERROR_INVALID_MAP:
return "invalid_map";
case Profiling::UNWIND_ERROR_MAX_FRAMES_EXCEEDED:
return "max_frames_exceeded";
case Profiling::UNWIND_ERROR_REPEATED_FRAME:
return "repeated_frame";
case Profiling::UNWIND_ERROR_INVALID_ELF:
return "invalid_elf";
case Profiling::UNWIND_ERROR_SYSTEM_CALL:
return "system_call";
case Profiling::UNWIND_ERROR_THREAD_TIMEOUT:
return "thread_timeout";
case Profiling::UNWIND_ERROR_THREAD_DOES_NOT_EXIST:
return "thread_does_not_exist";
case Profiling::UNWIND_ERROR_BAD_ARCH:
return "bad_arch";
case Profiling::UNWIND_ERROR_MAPS_PARSE:
return "maps_parse";
case Profiling::UNWIND_ERROR_INVALID_PARAMETER:
return "invalid_parameter";
case Profiling::UNWIND_ERROR_PTRACE_CALL:
return "ptrace_call";
}
return "unknown"; // switch should be complete, but gcc needs a hint
}
};
class ProfilePacketInternLookup
: public SequenceStackProfileTracker::InternLookup {
public:
explicit ProfilePacketInternLookup(PacketSequenceStateGeneration* seq_state)
: seq_state_(seq_state) {}
~ProfilePacketInternLookup() override;
base::Optional<base::StringView> GetString(
SequenceStackProfileTracker::SourceStringId iid,
SequenceStackProfileTracker::InternedStringType type) const override {
protos::pbzero::InternedString::Decoder* decoder = nullptr;
switch (type) {
case SequenceStackProfileTracker::InternedStringType::kBuildId:
decoder = seq_state_->LookupInternedMessage<
protos::pbzero::InternedData::kBuildIdsFieldNumber,
protos::pbzero::InternedString>(iid);
break;
case SequenceStackProfileTracker::InternedStringType::kFunctionName:
decoder = seq_state_->LookupInternedMessage<
protos::pbzero::InternedData::kFunctionNamesFieldNumber,
protos::pbzero::InternedString>(iid);
break;
case SequenceStackProfileTracker::InternedStringType::kMappingPath:
decoder = seq_state_->LookupInternedMessage<
protos::pbzero::InternedData::kMappingPathsFieldNumber,
protos::pbzero::InternedString>(iid);
break;
}
if (!decoder)
return base::nullopt;
return base::StringView(reinterpret_cast<const char*>(decoder->str().data),
decoder->str().size);
}
base::Optional<SequenceStackProfileTracker::SourceMapping> GetMapping(
SequenceStackProfileTracker::SourceMappingId iid) const override {
auto* decoder = seq_state_->LookupInternedMessage<
protos::pbzero::InternedData::kMappingsFieldNumber,
protos::pbzero::Mapping>(iid);
if (!decoder)
return base::nullopt;
return ProfilePacketUtils::MakeSourceMapping(*decoder);
}
base::Optional<SequenceStackProfileTracker::SourceFrame> GetFrame(
SequenceStackProfileTracker::SourceFrameId iid) const override {
auto* decoder = seq_state_->LookupInternedMessage<
protos::pbzero::InternedData::kFramesFieldNumber,
protos::pbzero::Frame>(iid);
if (!decoder)
return base::nullopt;
return ProfilePacketUtils::MakeSourceFrame(*decoder);
}
base::Optional<SequenceStackProfileTracker::SourceCallstack> GetCallstack(
SequenceStackProfileTracker::SourceCallstackId iid) const override {
auto* interned_message_view = seq_state_->GetInternedMessageView(
protos::pbzero::InternedData::kCallstacksFieldNumber, iid);
if (!interned_message_view)
return base::nullopt;
protos::pbzero::Callstack::Decoder decoder(
interned_message_view->message().data(),
interned_message_view->message().length());
return ProfilePacketUtils::MakeSourceCallstack(std::move(decoder));
}
private:
PacketSequenceStateGeneration* seq_state_;
};
} // namespace trace_processor
} // namespace perfetto
#endif // SRC_TRACE_PROCESSOR_IMPORTERS_PROTO_PROFILE_PACKET_UTILS_H_
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,097 |
package azkaban.jobtype;
import static azkaban.test.Utils.initServiceProvider;
import static org.junit.Assert.assertEquals;
import static org.junit.Assert.assertFalse;
import static org.junit.Assert.assertNotNull;
import static org.junit.Assert.assertNull;
import static org.junit.Assert.assertTrue;
import azkaban.flow.CommonJobProperties;
import azkaban.jobExecutor.Job;
import azkaban.utils.Props;
import com.google.common.io.Resources;
import java.io.File;
import java.io.IOException;
import java.net.URL;
import java.util.Optional;
import java.util.Set;
import org.apache.commons.io.FileUtils;
import org.apache.log4j.Logger;
import org.junit.After;
import org.junit.Before;
import org.junit.Rule;
import org.junit.Test;
import org.junit.rules.TemporaryFolder;
/**
* Test the flow run, especially with embedded flows. Files are in unit/plugins/jobtypes
*/
public class JobTypeManagerTest {
public final static String TEST_PLUGIN_DIR = "jobtypes_test";
private final Logger logger = Logger.getLogger(JobTypeManagerTest.class);
@Rule
public TemporaryFolder temp = new TemporaryFolder();
private String testPluginDirPath;
private JobTypeManager manager;
public JobTypeManagerTest() {
}
@Before
public void setUp() throws Exception {
// TODO: reallocf Remove initServiceProvider when ProcessJob fully guiced
initServiceProvider();
final File jobTypeDir = this.temp.newFolder(TEST_PLUGIN_DIR);
this.testPluginDirPath = jobTypeDir.getCanonicalPath();
final URL resourceUrl = Resources.getResource("plugins/jobtypes");
assertNotNull(resourceUrl);
FileUtils.copyDirectory(new File(resourceUrl.toURI()), jobTypeDir);
this.manager = new JobTypeManager(this.testPluginDirPath, null,
this.getClass().getClassLoader());
}
@After
public void tearDown() throws IOException {
this.temp.delete();
}
/**
* Tests that the common and common private properties are loaded correctly
*/
@Test
public void testCommonPluginProps() throws Exception {
final JobTypePluginSet pluginSet = this.manager.getJobTypePluginSet();
final Props props = pluginSet.getCommonPluginJobProps();
System.out.println(props.toString());
assertEquals("commonprop1", props.getString("commonprop1"));
assertEquals("commonprop2", props.getString("commonprop2"));
assertEquals("commonprop3", props.getString("commonprop3"));
final Props priv = pluginSet.getCommonPluginLoadProps();
assertEquals("commonprivate1", priv.getString("commonprivate1"));
assertEquals("commonprivate2", priv.getString("commonprivate2"));
assertEquals("commonprivate3", priv.getString("commonprivate3"));
assertEquals("azkaban.jobtype.FakeJavaJob2",
priv.getString("default.proxyusers.jobtype.classes"));
}
/**
* Tests that the proper classes were loaded and that the common and the load properties are
* properly loaded.
*/
@Test
public void testLoadedClasses() throws Exception {
final JobTypePluginSet pluginSet = this.manager.getJobTypePluginSet();
final Props props = pluginSet.getCommonPluginJobProps();
System.out.println(props.toString());
assertEquals("commonprop1", props.getString("commonprop1"));
assertEquals("commonprop2", props.getString("commonprop2"));
assertEquals("commonprop3", props.getString("commonprop3"));
assertNull(props.get("commonprivate1"));
final Props priv = pluginSet.getCommonPluginLoadProps();
assertEquals("commonprivate1", priv.getString("commonprivate1"));
assertEquals("commonprivate2", priv.getString("commonprivate2"));
assertEquals("commonprivate3", priv.getString("commonprivate3"));
// Testing the anothertestjobtype
assertEquals("azkaban.jobtype.FakeJavaJob",
pluginSet.getPluginClassName("anothertestjob"));
final Props ajobProps = pluginSet.getPluginJobProps("anothertestjob");
final Props aloadProps = pluginSet.getPluginLoaderProps("anothertestjob");
// Loader props
assertEquals("lib/*", aloadProps.get("jobtype.classpath"));
assertEquals("azkaban.jobtype.FakeJavaJob",
aloadProps.get("jobtype.class"));
assertEquals("commonprivate1", aloadProps.get("commonprivate1"));
assertEquals("commonprivate2", aloadProps.get("commonprivate2"));
assertEquals("commonprivate3", aloadProps.get("commonprivate3"));
// Job props
assertEquals("commonprop1", ajobProps.get("commonprop1"));
assertEquals("commonprop2", ajobProps.get("commonprop2"));
assertEquals("commonprop3", ajobProps.get("commonprop3"));
assertNull(ajobProps.get("commonprivate1"));
assertEquals("azkaban.jobtype.FakeJavaJob2",
pluginSet.getPluginClassName("testjob"));
final Props tjobProps = pluginSet.getPluginJobProps("testjob");
final Props tloadProps = pluginSet.getPluginLoaderProps("testjob");
// Loader props
assertNull(tloadProps.get("jobtype.classpath"));
assertEquals("azkaban.jobtype.FakeJavaJob2",
tloadProps.get("jobtype.class"));
assertEquals("commonprivate1", tloadProps.get("commonprivate1"));
assertEquals("commonprivate2", tloadProps.get("commonprivate2"));
assertEquals("private3", tloadProps.get("commonprivate3"));
assertEquals("0", tloadProps.get("testprivate"));
// Job props
assertEquals("commonprop1", tjobProps.get("commonprop1"));
assertEquals("commonprop2", tjobProps.get("commonprop2"));
assertEquals("1", tjobProps.get("pluginprops1"));
assertEquals("2", tjobProps.get("pluginprops2"));
assertEquals("3", tjobProps.get("pluginprops3"));
assertEquals("pluginprops", tjobProps.get("commonprop3"));
// Testing that the private properties aren't shared with the public ones
assertNull(tjobProps.get("commonprivate1"));
assertNull(tjobProps.get("testprivate"));
}
/**
* Test building classes
*/
@Test
public void testBuildClass() throws Exception {
final Props jobProps = new Props();
jobProps.put("type", "anothertestjob");
jobProps.put("test", "test1");
jobProps.put("pluginprops3", "4");
final Job job = this.manager.buildJobExecutor("anothertestjob", jobProps, this.logger);
assertTrue(job instanceof FakeJavaJob);
final FakeJavaJob fjj = (FakeJavaJob) job;
final Props props = fjj.getJobProps();
assertEquals("test1", props.get("test"));
assertNull(props.get("pluginprops1"));
assertEquals("4", props.get("pluginprops3"));
assertEquals("commonprop1", props.get("commonprop1"));
assertEquals("commonprop2", props.get("commonprop2"));
assertEquals("commonprop3", props.get("commonprop3"));
assertNull(props.get("commonprivate1"));
assertNull(props.get(CommonJobProperties.TARGET_CLUSTER_CLASSPATH));
assertNull(props.get(CommonJobProperties.TARGET_CLUSTER_NATIVE_LIB));
}
/**
* Test building classes 2
*/
@Test
public void testBuildClass2() throws Exception {
final Props jobProps = new Props();
jobProps.put("type", "testjob");
jobProps.put("test", "test1");
jobProps.put("pluginprops3", "4");
final Job job = this.manager.buildJobExecutor("testjob", jobProps, this.logger);
assertTrue(job instanceof FakeJavaJob2);
final FakeJavaJob2 fjj = (FakeJavaJob2) job;
final Props props = fjj.getJobProps();
assertEquals("test1", props.get("test"));
assertEquals("1", props.get("pluginprops1"));
assertEquals("2", props.get("pluginprops2"));
assertEquals("4", props.get("pluginprops3")); // Overridden value
assertEquals("commonprop1", props.get("commonprop1"));
assertEquals("commonprop2", props.get("commonprop2"));
assertEquals("pluginprops", props.get("commonprop3"));
assertNull(props.get("commonprivate1"));
assertNull(props.get(CommonJobProperties.TARGET_CLUSTER_CLASSPATH));
assertNull(props.get(CommonJobProperties.TARGET_CLUSTER_NATIVE_LIB));
}
/**
* Configure a {@link JobPropsProcessor} for a jobtype plugin and verify the JobPropsProcessor
* are invoked correctly for jobs of that type.
*/
@Test
public void testJobPropsProcessor() throws Exception {
final Props jobProps = new Props();
jobProps.put("type", "testjobwithpropsprocessor");
final Job job = this.manager.buildJobExecutor("testjobwithpropsprocessor", jobProps, this.logger);
assertTrue(job instanceof FakeJavaJob);
final FakeJavaJob fjj = (FakeJavaJob) job;
final Props props = fjj.getJobProps();
assertEquals("commonprop1", props.get("commonprop1"));
assertEquals("commonprop2", props.get("commonprop2"));
assertEquals("commonprop3", props.get("commonprop3"));
assertNull(props.get("commonprivate1"));
assertEquals(TestJobPropsProcessor.INJECTED_ADDITION_PROP,
props.get(TestJobPropsProcessor.INJECTED_ADDITION_PROP));
}
/**
* Test out reloading properties
*/
@Test
public void testResetPlugins() throws Exception {
// Add a plugins file to the anothertestjob folder
final File anothertestfolder = new File(this.testPluginDirPath + "/anothertestjob");
final Props pluginProps = new Props();
pluginProps.put("test1", "1");
pluginProps.put("test2", "2");
pluginProps.put("pluginprops3", "4");
pluginProps
.storeFlattened(new File(anothertestfolder, "plugin.properties"));
// clone the testjob folder
final File testFolder = new File(this.testPluginDirPath + "/testjob");
FileUtils.copyDirectory(testFolder, new File(this.testPluginDirPath
+ "/newtestjob"));
// change the common properties
final Props commonPlugin =
new Props(null, this.testPluginDirPath + "/common.properties");
commonPlugin.put("commonprop1", "1");
commonPlugin.put("newcommonprop1", "2");
commonPlugin.removeLocal("commonprop2");
commonPlugin
.storeFlattened(new File(this.testPluginDirPath + "/common.properties"));
// change the common properties
final Props commonPrivate =
new Props(null, this.testPluginDirPath + "/commonprivate.properties");
commonPrivate.put("commonprivate1", "1");
commonPrivate.put("newcommonprivate1", "2");
commonPrivate.removeLocal("commonprivate2");
commonPrivate.storeFlattened(new File(this.testPluginDirPath
+ "/commonprivate.properties"));
// change testjob private property
final Props loadProps =
new Props(null, this.testPluginDirPath + "/testjob/private.properties");
loadProps.put("privatetest", "test");
// Reload the plugins here!!
this.manager.loadPlugins();
// Checkout common props
final JobTypePluginSet pluginSet = this.manager.getJobTypePluginSet();
final Props commonProps = pluginSet.getCommonPluginJobProps();
assertEquals("1", commonProps.get("commonprop1"));
assertEquals("commonprop3", commonProps.get("commonprop3"));
assertEquals("2", commonProps.get("newcommonprop1"));
assertNull(commonProps.get("commonprop2"));
// Checkout common private
final Props commonPrivateProps = pluginSet.getCommonPluginLoadProps();
assertEquals("1", commonPrivateProps.get("commonprivate1"));
assertEquals("commonprivate3", commonPrivateProps.get("commonprivate3"));
assertEquals("2", commonPrivateProps.get("newcommonprivate1"));
assertNull(commonPrivateProps.get("commonprivate2"));
// Verify anothertestjob changes
assertEquals("azkaban.jobtype.FakeJavaJob",
pluginSet.getPluginClassName("anothertestjob"));
final Props ajobProps = pluginSet.getPluginJobProps("anothertestjob");
assertEquals("1", ajobProps.get("test1"));
assertEquals("2", ajobProps.get("test2"));
assertEquals("4", ajobProps.get("pluginprops3"));
assertEquals("commonprop3", ajobProps.get("commonprop3"));
final Props aloadProps = pluginSet.getPluginLoaderProps("anothertestjob");
assertEquals("1", aloadProps.get("commonprivate1"));
assertNull(aloadProps.get("commonprivate2"));
assertEquals("commonprivate3", aloadProps.get("commonprivate3"));
// Verify testjob changes
assertEquals("azkaban.jobtype.FakeJavaJob2",
pluginSet.getPluginClassName("testjob"));
final Props tjobProps = pluginSet.getPluginJobProps("testjob");
assertEquals("1", tjobProps.get("commonprop1"));
assertEquals("2", tjobProps.get("newcommonprop1"));
assertEquals("1", tjobProps.get("pluginprops1"));
assertEquals("2", tjobProps.get("pluginprops2"));
assertEquals("3", tjobProps.get("pluginprops3"));
assertEquals("pluginprops", tjobProps.get("commonprop3"));
assertNull(tjobProps.get("commonprop2"));
final Props tloadProps = pluginSet.getPluginLoaderProps("testjob");
assertNull(tloadProps.get("jobtype.classpath"));
assertEquals("azkaban.jobtype.FakeJavaJob2",
tloadProps.get("jobtype.class"));
assertEquals("1", tloadProps.get("commonprivate1"));
assertNull(tloadProps.get("commonprivate2"));
assertEquals("private3", tloadProps.get("commonprivate3"));
// Verify newtestjob
assertEquals("azkaban.jobtype.FakeJavaJob2",
pluginSet.getPluginClassName("newtestjob"));
final Props ntjobProps = pluginSet.getPluginJobProps("newtestjob");
final Props ntloadProps = pluginSet.getPluginLoaderProps("newtestjob");
// Loader props
assertNull(ntloadProps.get("jobtype.classpath"));
assertEquals("azkaban.jobtype.FakeJavaJob2",
ntloadProps.get("jobtype.class"));
assertEquals("1", ntloadProps.get("commonprivate1"));
assertNull(ntloadProps.get("commonprivate2"));
assertEquals("private3", ntloadProps.get("commonprivate3"));
assertEquals("0", ntloadProps.get("testprivate"));
// Job props
assertEquals("1", ntjobProps.get("commonprop1"));
assertNull(ntjobProps.get("commonprop2"));
assertEquals("1", ntjobProps.get("pluginprops1"));
assertEquals("2", ntjobProps.get("pluginprops2"));
assertEquals("3", ntjobProps.get("pluginprops3"));
assertEquals("pluginprops", ntjobProps.get("commonprop3"));
}
@Test
public void testDefaultProxyUsers() throws Exception {
final JobTypePluginSet pluginSet = this.manager.getJobTypePluginSet();
// Verify the allowed jobType classes for defaultProxyUser feature
Set<String> defaultProxyUsersJobTypeClasses = pluginSet.getDefaultProxyUsersJobTypeClasses();
assertEquals(1, defaultProxyUsersJobTypeClasses.size());
assertTrue(defaultProxyUsersJobTypeClasses.contains("azkaban.jobtype.FakeJavaJob2"));
// Verify defaultProxyUser
Optional<String> proxyUser = pluginSet.getDefaultProxyUser("testjob");
assertEquals("azkabanUser1", proxyUser.get());
// JobType class is not allowed for defaultProxyUser
proxyUser = pluginSet.getDefaultProxyUser("anothertestjob");
assertFalse(proxyUser.isPresent());
// Plugin itself doesn't exist even if it is part of default-proxy-user mapping
proxyUser = pluginSet.getDefaultProxyUser("notestjob");
assertFalse(proxyUser.isPresent());
// defaultProxyUser is part of the filter list
proxyUser = pluginSet.getDefaultProxyUser("testjobwithpropsprocessor");
assertFalse(proxyUser.isPresent());
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,175 |
Manuscript/Mixed Material Elizabeth Cady Stanton Papers: Speeches and Writings, 1848-1902; Speeches; [circa 1876], "The Subjection of Woman"
Download: Text ( all pages ) JPEG (460x389px) JPEG (920x779px) JPEG (1840x1559px) JPEG (3681x3119px) JPEG2000 (872.7 KB) TIFF (11.0 MB)
Elizabeth Cady Stanton Papers: Speeches and Writings, 1848-1902; Speeches; [circa 1876], "The Subjection of Woman"
Stanton, Elizabeth Cady. Elizabeth Cady Stanton Papers: Speeches and Writings, 1848 to 1902; Speeches; , "The Subjection of Woman" . 1876. Manuscript/Mixed Material. https://www.loc.gov/item/mss412100089/.
Stanton, E. C. (1876) Elizabeth Cady Stanton Papers: Speeches and Writings, 1848 to 1902; Speeches; , "The Subjection of Woman" . [Manuscript/Mixed Material] Retrieved from the Library of Congress, https://www.loc.gov/item/mss412100089/.
Stanton, Elizabeth Cady. Elizabeth Cady Stanton Papers: Speeches and Writings, 1848 to 1902; Speeches; , "The Subjection of Woman" . 1876. Manuscript/Mixed Material. Retrieved from the Library of Congress, <www.loc.gov/item/mss412100089/>. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,131 |
Q: Building a Email sending application in C# I got the assignment to unify and simplyfy the companies Email sendouts from their site, with the possibility to edit the emails them selves. So Im scetching on a C# Window Form application with WYSIWYG-editor to manage all the emails. The email is stored in SQL-DB
But im in dire need of some tips and pointers on the logic of some of the sendouts.
Some of the sendouts is action triggerd from signups on the site etc. But some sendouts is intervall based, like search-match-email-notification and other reminder-emails wich is sent out in intervals from 5 min to every midnight.
My dilema is this:
How do you best handle interval based sendouts? Can you implement some kind of deamon or service the checks stored procedurs at given intervals and trigger sendouts if there is any hits? I would prefer if the Application could handle both the managing of the email content and the schedueled sendouts, (the 5 min checks and every midnight)
Or is there any other smarter way to tackle the interval based sendouts?
Thankfull for tips and pointers on how to tackle this
A: For the sign-up emails you can just add code to send the email to whatever is being executed when users are signing up.
For the scheduled emails there are a couple of ways that you could handle it.
If you think just managing it all in the database is better, and are on SQL 2005 or above, there is built-in Database Mail functionality. http://msdn.microsoft.com/en-us/library/ms175887.aspx
Recurring emails can be sent by scheduled jobs that check a queue of emails that you maintain.
If you want to handle the emails in your code you can use System.Net.Mail. The scheduling can be handled by a Windows Service or a recurring task in the OS.http://support.microsoft.com/kb/226795
A: Google for "C# windows service" - a service is the Windows equivalent of a daemon, the ideal way to write a program that runs in the background, starts up automatically on a server, and doesn't require the user to log in before it starts.
Make the service Thread.Sleep for a few seconds at a time and check whether it is time to shutdown, or whether to poll the database for emails to send.
http://www.systemnetmail.com/
A: Try a scheduling engine like http://quartznet.sourceforge.net/
A: Some sort of queue would be ideal for you. you would place your request in the queue and it would get generated. Trigger can place the request in queue as well as your UI. And ofcourse I expect you are doing that through the code (smtp)
A: This is a very broad question but if you are using SQL Server you have alot of email options built in. Also Queues and scheduler built in.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,963 |
Garry Kasparov: A History of Profesional Chess
4/8/2002 – At the end of February Yasser Seirawan presented a detailed proposal for sorting out the chaos that current afflicts top-level chess. Soon after that Garry Kasparov replied directly to Seirawan's proposal. Michael ("Mig") Greengard, who had spent the weekend interviewing the world's number one player in New York, also got a full chronology of the GMA and the PCA from Kasparov. He now passes ít on to the chess public in an extensive keynote article here.
A Recent History of Professional Chess with Garry Kasparov
Using the past to help the present "History not used is nothing, for all intellectual life is action, like practical life, and if you don't use the stuff—well, it might as well be dead." — A. J. Toynbee
It is no coincidence that the political upheavals in the chess world in the past 16 years coincide with the domination of the chess board by Garry Kasparov. His style of play -- dynamic, aggressive, mercurial -- also reflects his temperament in the realm of business and political negotiations, where these attributes are often more a handicap than an advantage. And while you can easily lose count of Kasparov's tournament triumphs, his victories in the political arena have been few and fleeting.
An old business proverb says that pioneers are the ones who get filled full of arrows and Kasparov has more than his share of scars to prove it. Time and again he has led the charge to change the chess world and he has often met with Apache war cries from those he was trying to help. A list of his ventures includes a trade union for players, professional sponsorship, a world championship cycle and match outside of FIDE, and an international tournament grand prix (twice). That several of these undertakings were in part destroyed by the same Kasparov energy that built them does not diminish the level of dedication and passion they represent.
Critics have said that these efforts stemmed from self-interest on the part of Kasparov, and he would be the last person to say he has only been motivated by altruism. If Kasparov was looking out for himself while trying to create meaningful changes in the chess world, that is not a crime. If in some cases he had the most to gain, he almost always had the most to lose, and it is to his credit that he continued the fight regardless. His so-called war against FIDE is almost older than the current FIDE champion and he continues to assert that a counterbalance to the official federation is essential. Now that dissatisfaction with FIDE is again on the rise thanks to their recent experiments with the world championship, drug testing, and short time controls, it is the perfect time to look back and see how exactly we arrived at such a bleak situation.
Kasparov's lengthy discourse was more a history lesson than a press release or plan of attack. He told the tale of his efforts from the birth of the Grandmaster Association (GMA) in his Dubai hotel room in 1986 to the collapse of the Professional Chess Association in 1996. Several times he emphasized that this was only his take on things, his memories, and that he welcomed contributions and corrections from others. More than wanting to set the record straight, Kasparov wants to use the past to help the present. Many of the problems faced by the GMA and the PCA are still relevant issues today.
American GM Yasser Seirawan, author of the "Fresh Start" proposal that has stirred up so much activity lately, was a co-founder of the GMA and continues to be an important activist on the chess scene. He was kind enough to help with this article, both in supplying his memories and archives of his now defunct magazine, "Inside Chess" and by providing valuable perspective to many of Kasparov's recollections.
Kasparov summarized his stance eloquently after talking for over an hour, and his conclusion serves as a perfect introduction.
"Now, no matter how upset some people are about the past, or what negative memories they have, what I want is for people to read these stories and see that we are not starting from scratch. "Fresh Start" is a good figure of speech but there is a history here and we have to learn from it. I'm quite happy for others to give their suggestions based on our experience, which is the most valuable things we have. It would be foolish to ignore the past."
Allow me an "Amen." From a call to unite in 2002 we look back to the controversial Dubai Olympiad at the end of 1986. Several teams had boycotted because the Arab nation would not permit the Israeli team to participate, a sad situation that certainly made many wonder if FIDE was in its collective right mind. Several other recent events already had tensions high. One was the way FIDE President Florencio Campomanes had aborted the 1984-85 Karpov-Kasparov world championship marathon match. Another was the collapse of the Lucena/Keene ticket that had tried to take power from Campomanes in 1986. There had been so much politicking and double-dealing by various federations that, according to Kasparov, "we all began to think that it was useless to try and find a solution with FIDE unless we had our own trade union to protect our professional livelihood and the game itself."
Seirawan points out that if the Kasparov-backed Lucena bid for the FIDE presidency had succeeded, it is unlikely that the GMA would have been founded. When Campomanes maintained his grasp the formation of a players union, Plan B, as it were, was essential.
Apart from discussing the topics of the day, Kasparov added, "By the way, there were still many people who remembered the vote in 1975 when the small federations decided the fate of Bobby Fischer. It was just not appropriate that these small federations had the same vote as the USSR and the other chess powers and that these votes could decide everything in the lives of professional chessplayers, including the rules of the world championship match."
We, the Grandmasters of the world…
The Grandmaster Association's primordial constitution was scribbled on a napkin in Kasparov's hotel room at the Dubai Hilton. The small group of players was joined by businessman Bessel Kok, then the CEO of the SWIFT corporation. Conversation centered around protecting the players' interests at a time when FIDE under Campomanes had reached a then-record level of corruption and confusion. Kasparov's approach to FIDE about the creation of the GMA was less than auspicious.
"During the Olympiad I had a rest day (Karpov played board one against Georgiev) and I went to the FIDE General Assembly to present our case. There was this FIDE delegate, from Syria I think, who was shouting at me, "What is this, Grandmaster Association? Next an IM association! They have to obey the rules!" It's unheard of in any professional sport, this pawn yelling at the world champion. And there were even other delegates who backed him up.
Eventually it ended up with FIDE giving its permission, they all thought that it would be a failure. Everybody expected there would be fighting amongst the players, that there would be a lack of money, etc. But that was a mistake."
Those assumptions were not wholly incorrect since a level of inevitable infighting in the GMA existed from the beginning. But the players were more dedicated than FIDE believed and it survived the first crucial months. The first Board was Jan Timman (Netherlands), Ljubomir Ljubojevic (Yugoslavia), Lajos Portisch (Hungary), John Nunn (England), Yasser Seirawan (USA), Anatoly Karpov, and Kasparov, both USSR. The key player, however, was from Brussels and was no Grandmaster.
The involvement of Bessel Kok was the difference between the GMA being a serious and professional organization and being just a loose alliance of players. He and his then-wife Pierette, a lawyer, put together the necessary legal apparatus and raised money for the nascent organization. Kok, who has recently returned to the chess world with big events in his new base of Prague, provided a crucial guiding hand as well as a deep pocket. The GMA office in Brussels was run by Angela Day, and Seirawan calls her the glue that kept things together. Among many other things, she produced the GMA newsletter and arranged agendas and meetings. The assembled board of players knew a lot about chess, but had a lot to learn about what they wanted and how to get it.
"We didn't have a clear plan, we were running in the dark. We had no experience. We had ideas about the GMA, the world championship, but there was nothing fixed. We were pioneers. There were conflicts within the GMA as well. There was the traditional view, primarily from Karpov and Portisch, who wanted to stick with FIDE as an affiliated organization. Were we to be an affiliated group or an independent organization? That was a key moment. Eventually, as you can guess, the vote was for an independent trade union. The majority of the board decided to be independent and to work with FIDE, but not be a part of it."
The office in Brussels was the focal point. Dutch GM Jan Timman, not present in Dubai because his team was boycotting the event, was the vice-president and a key figure. Lubomir Kavalek, a Czech émigré to the US, was later brought in as technical director and fundraiser. He was responsible for organizing the remarkable World Cup tournament series that became the permanent landmark of the GMA. Seirawan highlights the importance of Anatoly Karpov's participation. "Karpov was not only a former World Champion he was the second strongest player in the world. His support for the GMA was vital."
Both Kasparov and Seirawan speak fondly of the early days of the GMA. It was a time of big ideas and big changes. Items on the table included a player retirement fund, sponsorship for more tournaments, game copyrights, and, of course, the world chess championship. The subject of time controls had yet to rear its ugly head.
In the earliest days the board was everything, but this changed in 1988. Kasparov states: "We knew the board couldn't run it forever so we needed to build the membership and have assemblies. The first one was in April, 1988, after the first Grand Prix in Brussels. That was a great system, by the way, classical chess (no debates at that time, it was the only game we knew). We wanted to build the professional world on the cornerstones of the past.
Yes, we made mistakes, perhaps we could have been more progressive. But it was important to preserve these elements of the past and to do it on our own, without FIDE's dictate."
Taking action on active chess
Speaking of classical chess (a good example of a retronym), it was around this time that the first threats to the heritage of how the game was played were beginning to surface. Interestingly similar concepts would come from both FIDE and Kasparov, but with very different intent. FIDE was proposing a rapid (30/min. per player) championship tournament that would, and this is the key, also be good for rating points and international "active" titles. Many players were horrified and the GMA mobilized to prevent the worst. Meanwhile, Kasparov was playing in what could be called the first chess spectacular of the modern era.
"The first big speed match was played in February, 1987, between me and Nigel Short. We played in the famous Hippodrome discotheque in London at 25 minutes per side, on a fancy stage in tuxedos. [They played at g/25 instead of g/30 so the games would fit into the one-hour television slot! This control continues to be standard today. –Mig] I won 4-2 with no draws, it was a great event. I remember showing the tape to my colleagues in Brussels. There was an opinion in the room that it was a form of prostitution! There were strong opinions back then that rapid chess was a threat. In my view, then and now, rapid chess is an excellent tool to promote the game. Now they are talking about getting television coverage, commercial sponsorship, etc. We discussed this 15 years ago! And we reached a consensus on how to use rapid chess, within limits. We realized it was great for promotion.
Now we come to the first GMA general assembly. We had over 100 members already. (At its peak we had probably 95% of all the GMs in the world.) I would say we had 100+ represented in Brussels. We had a proxy system, and thanks to Bessel we had a professional organization that allowed the entire thing to function.
The big topic was that FIDE had announced a rapid tournament, a rapid chess world championship. Not just promotional, but one in which they could give titles, including the GM title. That was Campomanes' idea. They were already frightened of the GMA and wanted to dilute our influence. So the key debate of this assembly was how to deal with this threat. You hear everyone talking about time controls now and we have already lived through this. We had the same fears back then.
It was a tough situation because some of the players thought it was okay, and they wanted to play. Many others thought it was clearly a threat, a danger, and we should boycott it completely. There was a strong resolution by GMA member Valery Salov, then a big enemy of FIDE![In recent years Salov has become a zealous FIDE supporter. – Mig] He proposed the outright condemnation of anyone who played in that tournament. I remember Lajos Portisch nearly crying, leaving the stage, saying he was resigning because no one could condemn him for his professional activities. It was a very hard moment, an emotional crisis.
In the end we passed a compromise solution that we wouldn't denounce anyone who played there, but that we condemned giving titles, that we did not recognize the validity of a world champion of rapid chess or GM of rapid chess."
Yasser Seirawan has helped greatly by digging through both his own memories and his archives of Inside Chess, the American chess magazine that he published for over a decade. Both sources concur that it was Kasparov, not Salov, who introduced the motion of "moral condemnation" at the GMA. (Salov's proposal for a boycott of the FIDE event had come earlier and he sided with Kasparov on these issues.) While naming names is really not the point – or at least should not be the point – of this article, it does serve the purpose of highlighting Kasparov's strong anti-FIDE stance at an early stage. It wasn't rapid chess that was the problem, it was FIDE's rapid chess and its use for titles and ratings.
In the end FIDE did go on to create a parallel rapid chess rating system and to drop the idea of titles based on rapid play. (They called it "active chess" back then, prompting many jokes about a potential "passive vs. active" break in the chess world. As Kasparov jokes, "Did that mean I played passive chess?") Of note is that this rapid world championship, which took place at the end of 1988 in Mazatlán, Mexico, was a factor in another GMA crisis. In Seirawan's "Fresh Start" article he mentions that he resigned from the GMA board when Kasparov asked. The story behind Seirawan's resignation at the beginning of September, 1988, is a disputed one and beyond the scope of this article, but the basic storyline is as follows.
Kasparov states that after the GMA assembly rejected FIDE's plans for the rapid event they found out that Seirawan was an active supporter of the Mazatlán event. "Bessel asked Yasser to resign on behalf of the board, and he did. Everybody is responsible for his or her actions, and things were not so simple as to be summarized in one sentence. Everyone has their own views but if the views are contradictory to those of the organization, and someone does something alien to the interests of the organization, he has to resign. Perhaps that incident caused some bad blood, and for years after that Yasser's views of everything I did were negative."
That last sentence is not uncharacteristic of Kasparov when he really gets going. Although by his own admission things were quite complicated at the time, there is a tendency for memory to reduce everything to a binary formula. If you weren't for him, you were against him, and not just what he stood for and what he wanted to do, but against him personally. There is no doubt that personal friendship and animosity have both played large parts in Kasparov's successes and failures throughout his career, but as he says himself there were also splits on the issues.
Seirawan's interest in active chess was inspired by nothing less than the Kasparov-Short speed match and he set out to use rapid chess to as a promotional tool. To this end he supported the Mazatlán event and two years later helped create the FIDE-I.C.E. rapid chess rating list, for which he engaged in a membership drive using his magazine, Inside Chess, as a vehicle. Seirawan makes clear in an e-mail that he wanted to set up a separate active chess rating list, not allow the dilution of the main list with rapid games. Obviously, working on a joint project with FIDE was not going to put Seirawan in Kasparov's good graces. This was a precursor to the more dramatic breakdown to come.
Seirawan is adamant that his resignation stemmed from the publication of long insider cover story on the GMA in Inside Chess, and that while Bessel Kok served as the messenger, it was Kasparov's request he was delivering when he asked Seirawan to resign from the board. The problem with the article wasn't any "anti-Kasparov" content as much as its mere existence. According to Seirawan, Kasparov (among others) was annoyed that Seirawan had included in his article things from a closed GMA board meeting. Seirawan admits having made a mistake in that regard, but he points out that Inside Chess subsequently published several articles favorable to Kasparov. Seirawan says that things did not go sour between them until six years later in Moscow when they were on opposite sides of the big 1994 Olympiad and FIDE election brawl. He also points out that Mazatlán was won by Karpov, who continued as a GMA board member.
Seirawan was active within the GMA despite his resignation from the board, where he was replaced by another American GM, Maxim Dlugy. He says he held no grudge against Kasparov and was even a little relieved because of the extensive travel required to attend GMA functions from his home in Seattle. In Seirawan's own words on his resignation: "Furthermore, I often called, faxed and helped Angela and Lubosh both with a myriad number of trivial matters, contacts, player names, journalists, letters, even though I was off the board. Later, I was asked to return, I politely declined. You see, I actually preferred to be off the board. The GMA had been established and would do just fine without me! Even after resigning I always had good discussions with Garry for the next several years! Skelleftea, Barcelona '89, Moscow '90 and even in Murcia in '90 we were on good terms. So, probably for the wrong reasons, Garry did us both a favor!"
Kasparov says he doesn't remember the article being a serious enough issue to lead to a fracture in the board, although Seirawan adds that Kavalek was also quite displeased with a board member spilling the beans as a journalist. However, we should again strive to avoid getting bogged down in minutiae when we are really trying to establish what of the past can be of use to the present. In general these discussions slip rapidly into "who said what to whom when and where," which really doesn't help us much here in 2002.
The issues that would dominate the GMA until its collapse were whether it should develop its own commercial activities and whether it should work closely with FIDE or be independent from it. Meanwhile, the series of World Cup tournaments were perhaps the greatest series of events in chess history. They were huge round robins, dwarfing current elite events like Wijk aan Zee, that brought the world's best players together again and again around the world. The prizes were substantial, the conditions were good, and there were qualifiers that gave up-and-comers a chance to join the fun. It was simply too good to last! Kasparov explains what led up to the next crisis.
"The GMA was doing extremely well. We had the successful World Cup cycle, major qualification tournaments in Belgrade, Moscow, Palma de Mallorca, and everybody enjoyed it. The GMA was rising but at the same time there was a growing crisis.
The GMA was a trade union and being such we had to deal with the commercial aspects of things. We didn't have a structure for this. If you want to build a professional organization you need a commissioner, Yasser is right. You need a commercial department. Every time I tried to build the commercial structure of the GMA I would lose the vote in the board. We needed people organizing events, finding sponsors, etc., it couldn't be just us. We had the support of the players, we could have dictated terms to FIDE, but we needed events.
Dlugy supported me while he was there, and then [Alexander] Beliavsky, so the votes were always two to five! I understood Bessel's reservations and those of the other players who supported him. It would have meant a shift in the power structure. The money people would have had more power, but I didn't care. We needed someone to raise money and run events, we could still have controlled the rules.
There was a view among the players that we had to make a deal with FIDE. Now we come to Murcia, but Murcia had a pre-history. One of the biggest misconceptions is that the choice in Murcia was between war with FIDE promoted by Garry or peace with FIDE. That was not the case. The choice was between what kind of deal we would have with FIDE."
It might be hard to imagine for those new to the scene, but back then the world championship was far and away the biggest event going and controlling it meant controlling the chess world. Almost all of FIDE's operating funds came from its cut of world championship matches and there was a tremendous amount of back-room dealing when it came time to receive the bids. (Yes, different sites actually competed to host the championship, with bids running over four million dollars. Nowadays it's like trying to give away a landmine.)
Kasparov was due to defend his title in 1990 and the GMA had grown powerful enough to largely take control of the decision process. According to Kasparov, some FIDE officials told him that at one point in 1989 Campomanes was just about ready to give it up and close the doors on FIDE. The players were raising the money and talking with the organizers themselves, particularly Kasparov working with Ted Field, an American multimillionaire well known for his passion for chess. It was one of his entertainment companies, Interscope, that sponsored the New York leg of the 1990 world championship match.
Some GMA board members thought that things were going too far, while Kasparov thought it was the perfect chance to break FIDE's stranglehold on world chess. This division came to a head when Bessel Kok and Jan Timman were sent to negotiate an agreement with FIDE regarding the world championship.
"My view is that they negotiated a very bad document. Not because they had bad intentions, but because it would take things all the way back to 1987 and make the GMA an affiliate of FIDE. GMA would be subordinate to FIDE in the decision making process. I was adamant, I vehemently opposed it. I could not accept that we would move backwards and waste three years of our lives.
And then Murcia came, and I think Murcia was the tragedy of the GMA. It was a lose-lose situation. Bessel said it best, it was like a plane with two engines. If you remove one it crashes, and he was right. I didn't push really hard in Murcia. I think I could have won that vote. But if I had won, so what? I had a match with Karpov coming up and frankly I didn't know what to do."
Seirawan agrees that this was the key moment that caused the collapse of the GMA. He states that there was a great deal of confusion over what exactly was contained in the agreement. "The GMA membership were told by the GMA Chairman, Bessel Kok, that the contract was ideal and that it would place the GMA on sound financial footing, whereas the GMA President, Garry Kasparov, complained that the agreement would place the GMA in a subordinate role to FIDE. Who to believe? How to vote?"
To the Hustings!
The exact sequence of events is hard to nail down, particularly so many years later. It makes one wish Kasparov and Timman had gotten together and written a book on the history of the GMA, as Kasparov says he once suggested. Kasparov was having trouble with the Soviet chess federation and also wanted a full vote of the GMA membership to ratify agreements instead of having the board decide things. This led Kasparov to reject an agreement that granted favorable conditions to the GMA because it fell short of Kasparov's desire for the GMA to be both independent and in control. Kok and Timman both resigned when Kasparov wouldn't agree, only to come back after making a few changes in a Barcelona meeting.
Things broke down again and finally there would be a vote in the assembly to decide. "Yes" to sign the agreement, "no" to hold out for more (or, more correctly, less). Kasparov viewed it as between dependence or independence for the GMA and accuses his opposition of turning the referendum into "for Garry or against Garry."
"We were calling the shots and we had to take control of the world championship. And we could have done it. We had all the GMs behind us. We needed to go forward with commercial sponsorship. FIDE was irrelevant, we could have gone forward without them, build a new world! Let FIDE do what they want, we were in control. Why the hell go back? Fresh start, fresh means! Everything was in place, a unique situation. From late 1989 to early 1990 we could have done anything we wanted."
There was quite a bit of campaigning and both sides accused the other of not playing fair. Kasparov brought in a consultant to speak to the members about the need for commercialization. Those in favor invested considerable effort in convincing the many members from the newly opening Eastern Europe that a further break with FIDE would create dangerous instability. Kasparov wanted more control for the GMA and the players, the opposition said he wanted control for himself.
Things had already reached the point of no return. Kasparov's strong words above cast doubt on Seirawan's assertion that Kasparov "failed to understand that his colleagues were well and truly split." Perhaps no one understood what was about to happen but Kasparov knew what he wanted. According to Seirawan, the Soviet players supported Kasparov and the Western players mostly took Bessel's side, with few, if any, of the voters actually having laid eyes on the agreement itself.
Kasparov states: "Eventually it ended up 62-65 and I bet 80% of the voters didn't understand what was at stake." This might be a bit high, but since everything was conducted in English and many members didn't even speak English, it is fair to say that the "what" was less important than the "who" for many in attendance.
"There was a parallel election of the board members. There were, I think, 128 votes for the board. I got 125 votes. After that vote I announced I would suspend my membership on the board, and some people went bananas and they still tell these stories now. But it was a clear-cut situation. They were saying that I was the best fundraiser and important for moving the organization so they wanted me in charge. I mean, in five of the biggest GMA tournaments I had raised probably 90% of the money. But now, by a narrow vote, I would have to support their policies. How can you remain the president of an organization if your view was just defeated? I said I would go ahead and play my match with Karpov and then perhaps come back.
But we all knew it was the beginning of the end. The GMA was strong, even dominant, but after Murcia it just lost its cohesiveness. Such a close vote fragmented the organization and it lost much of its power."
Seirawan adds, "The lesson here is that future chess unions shouldn't rely upon mere majorities for such major controversial actions." The logic of this is powerful. Had a typical two-thirds majority been needed, the issue could have been sent to a committee until it was better understood or until changes could be made.
Kasparov and Seirawan both call the GMA period a golden age and both refer to Murcia as a missed opportunity. Kasparov saw it as a chance to relegate FIDE to a minor role in the affairs of chessplayers. Seirawan wishes that the dramatic vote had never taken place echoes Kasparov's "lose-lose" description when he writes, "Had Kasparov won that vote, it would also have torn the GMA apart."
The five tournaments Kasparov refers to are the three giant GMA pre-qualifiers plus the Moscow qualifier and the Murcia rapid event. The bulk of the sponsorship for the GMA World Cup events was brought in by the redoubtable Lubosh Kavalek. He did most of his work on a commission basis, something that became a source of internal friction at one point.
The schism heard round the world
After Kasparov resigned from the board the GMA gradually collapsed. There was too much bad blood, too little unity. If the assembly had been able to unite on either side of the FIDE proposal (or even postponed the divisive decision) it would have been much better than the down-the-middle split that occurred. The World Cup cycle shut down and FIDE was back in charge. Kasparov was exhausted after his 1990 match with Karpov and the next few years passed relatively quietly. The GMA leadership passed to Timman and then to England's Nigel Short, who would soon be the first person in eight years other than Karpov to challenge Kasparov for the title. The confluence of these factors led Kasparov to make what he has called the worst blunder of his career.
"My frustration with the situation eventually led to a big mistake in 1993. But I have to tell you that what happened in 1993 was also dictated by what happened in the past. We had the usual crisis with the world championship. Campomanes was playing one against the other, with Galicia, Manchester…, I don't remember all the bids. Manchester was the obvious choice.
You can blame me for what happened but we can't forget that Nigel Short called me on the phone and said "Garry, do you want to play outside of FIDE?" I mean, Short, who is now a big supporter of FIDE and kissing up to Ilyumzhinov, he made this offer. Nigel did it for the money, but at that time I thought "great, now with Nigel we can rally the support of the Western players." We could have momentum. That was a huge miscalculation. I thought we could revive the GMA, which had pretty much collapsed by then, and Nigel was its last president. I was thinking that Nigel represented the anti-Kasparov group in Murcia and now he was making this offer. It was now me and Nigel, not Karpov, and we could rebuild things and get support in Britain. Of course this was a horrible blunder. Nigel had no support behind him at all, it was completely his personal desire.
In the cold light of morning I can tell you we could have made more money in Manchester, and it was the best for everybody to make a deal with Campomanes at the time. He had already agreed to give up some power and we could have done things quietly, played the match under FIDE and dealt with rebuilding the GMA later, after the match.
But things were moving quickly and [English GM and writer Raymond] Keene, who saw that he had much to gain from a split, revealed the story in, I think, the Telegraph, and this put me in an awkward position and he, and others, pushed Nigel to the extreme. I still had a chance to tell Nigel to forget it, but I had already given my word, and I stood by it. That turned out to be a giant mistake. We had no support in the world of chess. Everywhere it was 'chess championship hijacked.'"
Two World Champions and the rise and fall of the PCA
Nowadays it is fashionable to look at that moment in 1993 as the chess world's lapsarian instant. Short and Kasparov created the Professional Chess Association on the fly and left FIDE hanging in order to play their match in London. FIDE reacted with equal destructiveness, removing the two renegades from the rating list and staging their own world championship between Timman and Karpov, who had both been defeated by Short in the candidates matches. The breach grew with incredible speed. But Kasparov is not ready to let those with short memories say that the years of the PCA were a complete waste.
"Yes, I made a mistake. A mistake that cost me strength as well as money. My results in 1994 and 95 were not up to my standards, for example. But I'm confused because everyone is talking about television and sponsorship and professionalism these days, and it was all there.
The PCA was not a big organization but it had a commissioner, Bob Rice, and a few people who helped the PCA operate. It had virtually no money and so no administrative core. We spent all the money on the prizes, to impress the players, which was probably a mistake looking back. We needed to strengthen the organization but we gave all the money to the players.
It had commercial sponsorship, the only time in the history of chess that we had the sponsorship of a blue-chip company, Intel. There was a two year contract. It had never happened before and hasn't happened since. There was television coverage in limited fashion. The PCA Grand Prix, the speed chess events, were on ESPN and EuroSport.
Okay, it wasn't huge, it was quite small, but it was unique. It was something that even the GMA had failed to do. The irony is that both parts of the solution were there, but at different times. The GMA had no commercial solution and the PCA had no trade union support. We needed both. That was the tragedy.
One of the incidents worth mentioning is that in 1995 we introduced a code of ethics, under pressure from Intel. We needed to protect the sponsors and organizers. It was 'anti-Kamsky,' nobody tried to hide that. He had made some statements that irritated Intel and he was playing Anand for the right to face me in the 1995 New York world championship, and that was a potential disaster.
Then in Linares that year the players, led by Karpov and others, signed a petition to protect what they called their "human rights" against this code of ethics. This petition was not missed by Intel and it did not make them very happy. A few years later FIDE introduced a draconian code of ethics and I didn't see any letter, any protest, about that. So these players have to bear some responsibility for their actions, for us losing momentum.
You could dislike what I did, you could call me the hijacker of the world championship, but at the end of the day I brought commercial sponsorship. We struggled but we ran some great events, the Grand Prix was unprecedented. Kramnik, Anand, Ivanchuk, they made a lot of money and they thrived thanks to these events. And there was not a word of support from them. There was no support from the elite but there were plenty of complaints and attacks and Intel saw this."
There were indeed many vocal critics of the PCA at the time. Several players considered it a rogue organization and refused outright to play in the PCA championship cycle. But the majority benefited greatly from Intel's money and the existence of two world championship cycles. Of the top players, only Karpov, Salov, and Seirawan refused to participate in any PCA event. Many players even played successfully in both cycles at the same time, with Kamsky and Anand facing each other in both the FIDE and PCA cycles. (Anand won in the PCA to play Kasparov, Kamsky won in the FIDE match to face Karpov. Both lost the title match.)
Kasparov and FIDE?!
Kasparov again surprised the chess world when he made a rapprochement with Campomanes to save the 1994 FIDE Olympiad and bring it to Moscow. There Kasparov tried to engineer some sort of unification even if it meant making a deal with his former worst enemy. But as often happened, when one side had a change of heart the other side took it as a sign of weakness and slammed the door.
"I tried desperately to close the gap. In 1994 I went as far as trying to make a deal with Campomanes in Moscow. I was talking about reunification and they were adamant, "No unification, Kasparov wants to come back, no way!" Who did this? The Western federations. They tried on legal issues to block Campomanes. And in 1994 we saved the Olympiad. In only 55 days Andrei Makarov and I organized the Olympiad in Moscow when the choice was that or no Olympiad at all. And everyone heaped garbage on us, complaining about the conditions, criticizing constantly.
Ironically, this Olympiad brought in [current FIDE president Kirsan] Ilyumzhinov; he made his first appearance at a FIDE congress. So I have a share of that responsibility, I admit. Maybe I should receive a finder's fee commission of all the prize money he has paid out to other chessplayers over the years!
In the FIDE general assembly of 1995 in Paris it all came down to not granting me 12-12 draw odds in a unification match. It was the Western federations again, and this anti-Kasparov sentiment. I had to be "punished" for 1993. I insisted that if I played against Karpov I deserved draw odds because I had already played him, but if it was Kamsky then I could compromise. But they insisted that they could not discriminate against "their champion" and things broke down. And many people were quite happy to see this, to keep me outside and prevent unification. In Moscow and then later in Paris those that opposed me torpedoed reunification."
The 1994 FIDE election in Moscow could have been held in Florida. It saw every parliamentary trick in the book, both dirty and clean, as well as a few tricks that weren't even in the book. The ticket of Kouatly and Karpov met resistance by Makarov and Campomanes, now supported by none other than Garry Kasparov! Some Western players reported being shocked by the strong-arm tactics and this as much as anything ruined Kasparov's hopes for a compromise with FIDE and a potential reunification match. Seirawan gives it as the moment at which he and Kasparov ceased being on the best of terms, at least for a time. It can only be good that these two prime movers have now come together for the best cause.
Intel goes and a legend is born
By 1996 Kasparov and Karpov had won their respective matches and Ilyumzhinov had taken over FIDE to begin his plan to remake the chess world in his image. At the same time, the PCA ran into a brick wall when Intel declined to renew their sponsorship of the Grand Prix. The conventional wisdom now is that Intel pulled the plug when Kasparov played Deep Blue under the auspices of IBM, an Intel competitor. You can see the frustration in Kasparov's face when he hears this story yet again.
"November, 1995 is when chess really hit its low. This was a crucial moment and it is important to clear this up. Everyone simply repeats the fairy tale that Intel pulled its sponsorship of the PCA because I played Deep Blue. Every player and journalist just repeats this. At the end of November, 1995, I was in London in the office of Rod Alexander [whose sports promotion company, SBI, had Intel Europe as a client], and we got a call from Intel Europe.
Intel Europe, in Germany, they backed our idea, but they reported to the Intel board. And the board rejected the sponsorship proposal. We wanted two more years, and they supported us in Europe, but the board rejected it. That was at the end of November, 1995, and I nearly died when I heard the news.
Why? They didn't give their reasons, but the Germans told us, unofficially, that there had been bad reports. That chess was struggling, having an endless internal war, and that the PCA had failed to build up an internal administrative structure. Yes, everything was true. That is why I don't want these Grandmasters hiding in the corners. I made mistakes, fine, but the fact that Intel stopped their sponsorship is due to the lack of support and unity in the chess world at that time and everyone was responsible. Those who wrote the letters, complained, and blocked unification have their share of responsibility. Intel did not want to be associated with it anymore. I raised the PCA, I protected it, I fought as hard as I could to keep Intel and I failed.
Three weeks later I got the letter from ACM [the Association for Computing Machinery], before Christmas. It was three weeks after the Intel call and you can ask David Levy, or other people from there; I could track down the names. These were two separate events; Intel's decision was made earlier. I don't have the exact date of their decision, but when I was in Paris in November and played the final PCA Grand Prix match, and I talked with Campomanes, we still expected Intel to come back. So it was probably at the end of November.
The Deep Blue match was organized very quickly, there was no (as some suspect) conspiracy about how it was organized. There were no IBM representatives anywhere around the match at that time. It was organized by ACM and they didn't expect any public, journalists, or heavy interest in the match. The first game was the surprise, with the huge interest shown by the world in the match. It was a huge surprise for IBM and the organizers. But IBM was not even involved. It was ACM and it was all organized very quickly around Christmas time.
I wish I had all these letters on hand and if it's important, and someone insists, and tells me I am lying, then I can start collecting all the data and all the dates. But I want them to stop, Yasser and everyone else. I want them to stop telling everyone "Oh of course, Garry went with IBM and played Deep Blue and Intel dropped the PCA sponsorship." It's simply not true."
Seirawan says he never heard anything about Intel abandoning the PCA prior to the first Deep Blue match, and he is certainly not alone. The chain of announcements that are public knowledge give credence to the "traditional" story that Intel did not pull the plug until after the Deep Blue match. Seirawan recollects that the Intel representative at the 1995 Kasparov-Anand match was "all smiles" and committed to doing it again. Then, after the Deep Blue match was announced, rumors began that said Intel might withdraw, and this was only confirmed publicly after the Deep Blue match.
Since Kasparov's London phone call refutes the conventional wisdom that was so harmful to him at the time ("Kasparov sold out Intel and the players to line his pockets with IBM cash" was the refrain) we are left wondering why Kasparov has waited so long to clear things up. As Seirawan writes, "After Deep Blue, it was confirmed, no Intel. What else could I think? Intel was upset was my only conclusion. I didn't know that they had definitely pulled out beforehand. I'm quite happy to stand corrected. Had Garry at any time written me a letter to correct the false impression that I was under, I would have published it immediately!"
Considering Kasparov's relationship with Seirawan and the rest of the chess press at the time it is no surprise that he wasn't writing many letters. But his secretiveness definitely did not help his reputation and the IBM/Intel story was rapidly accepted, however spurious it may have been. As Brian Friel wrote in the play "Dancing at Lughnasa," "What fascinates me about history is that it owes nothing to fact. In that memory, atmosphere is more real than incident and everything is simultaneously actual and illusory."
Kasparov did try to jump into bed with IBM after the match, but was given the cold shoulder. He tried to get a combined investment from them to support the Grand Prix and other PCA activities along with the Deep Blue rematch. It was a last-ditch effort to save the PCA and had it succeeded it would have put more money into the pockets of chess professionals. (The latest twist is FIDE's new Grand Prix, which kicked off in Dubai this week. A knock-out series of tournaments at rapid time controls, it has everything in common with the PCA Grand Prix, except it is funded by Kirsan Ilyumzhinov instead of an Intel.)
Friends of Kasparov sponsored a Grand Prix event in Moscow in 1996 and then the Credit-Suisse Masters tournament was transformed into another Grand Prix event after Kasparov convinced organizer William Wirth. "And then that was the end."
Kasparov finished by highlighting the various parallels that are appearing today.
"Look at what we were discussing in 1986, how FIDE was trying to replace classical chess with rapid chess. Now it is happening again and we need to reach a consensus and take action. I'm not calling for a boycott, we need to provide alternatives.
From 1986 to 2000 I tried to create alternatives to FIDE. To create an alternative force to balance the power, to raise sponsorship, to protect the players. So I failed in the end, but I didn't fail in a vacuum. Many professional players did not support me, others attacked me directly. When they complain how FIDE is calling all the shots now they have to take their share of responsibility for the current situation. If you destroy the alternatives what do you expect to happen?
The need for alternatives is greater now than ever before. I see the potential for positive changes. There is a lot of frustration out there, you can see it in all this activity. But unlike in the 1980s there is no unity in the chess world. Today there are diverse interests and it will be hard to reach a consensus. Frankly, I'm not terribly optimistic. It will be hard to bring all the parties to the table. It seems they really don't care.
Not to self-promote but at least I've always cared, always tried, and I'm still ready to make compromises. I hope I'm not alone in this. If Yasser succeeds in bringing everyone to the table, if Bessel can play the role he played 15 years ago, then I will be the happiest person. I wish them well and I am ready to support these efforts."
Seirawan concludes, "Garry is to be commended for his article and more importantly for his undertakings. He has worked extremely hard trying to raise the level of awareness and done his best to vitalize the sport. His efforts have been extraordinary. While I have pointed out two areas of different views [His resignation and the events in Murcia. –Mig], my admiration for what he achieved with the GMA have never dimmed. The GMA's were "golden years" for chess players and if a future for professional players exists, a key will be to create a union to protect their self-interest. If they can avoid the mistakes made by the GMA, and yours truly, the rewards will be great. Hopefully, Garry's article and this contribution will help them to identify pitfalls and avoid repeating our mistakes. My final parting word is to not forget that we live in a Human Comedy. Things happen, good and bad. Face them with a sense of humor not foreboding and all will be overcome."
It might not be an entirely uplifting tale, but it gives us room for optimism. Kasparov has grown weary of people saying that things would be better if he just kept his mouth shut. For many years the chess world has enjoyed alternately supporting and attacking Garry Kasparov, letting him be the leader and the lightning rod. Now he is still willing to lend a hand, but it is clear that like the rest of us he is waiting for someone else to pick up the torch that has burned him a few too many times.
Of the current candidates for torch-bearer, Vladimir Kramnik has been quiet, insisting against all evidence that the Dortmund qualifier will unite something other than his bank account and a nice check. FIDE has not made a public comment on Seirawan's unification proposal but the whispering winds say that Prague may bring a few surprises. Bessel Kok has organized a players' workshop this month and all the top players will be there. We can only hope that if history does repeat itself, we will get the happy ending this time.
Seirawan's original proposal
Kasparov's direct reply
Bessel Kok's press release
Smbat Lputian's letter to Seirawan (with Seirawan's reply)
GM Larry Evans on Seirawan's proposal | {
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Nun's recovery recognized as 70th official miraculous healing at Lourdes
In this 2010 file photo, pilgrims visit the grotto where Mary appeared in Lourdes, France. (CNS photo/Jose Navarro, EPA)
By Cindy Wooden • Catholic News Service • Posted February 12, 2018
ROME (CNS) — As the Catholic Church celebrated the feast of Our Lady of Lourdes, a French bishop announced the 70th officially recognized miraculous cure of a pilgrim to the Lourdes grotto where Mary appeared 160 years ago.
Bishop Jacques Benoit-Gonnin of Beauvais formally declared Feb. 11 "the prodigious, miraculous character" of the healing of Sister Bernadette Moriau, a French member of the Franciscan Oblates of the Sacred Heart of Jesus, who had been partially paralyzed for more than 20 years despite repeated surgeries to relieve pressure on the nerve roots of her lower back.
In November 2016, the International Medical Committee of Lourdes confirmed the nun's "unexplained healing, in the current state of scientific knowledge." But it is up to the bishop, not the physicians, to declare a healing miraculous.
Lourdes, close to the Pyrenees in southern France, attracts millions of visitors each year and has been a place of pilgrimage since St. Bernadette Soubirous reported the first of 18 visions of the Virgin Mary while gathering firewood in February 1858.
To be declared miraculous, cures must be "found complete and lasting," involving a "serious illness which is incurable," and must involve a sudden "indisputable change from a precise medical diagnosis of a known illness to a situation of restored health."
Sister Moriau, now 78, made her pilgrimage to Lourdes in 2008, the 150th anniversary of the apparitions. She had experienced lower back pain, the first symptom of her disease, in 1966 at the age of 27. Four surgeries did not stop the progressive worsening of her neurological deficits.
"This pilgrimage was for me a source of grace," she said in a statement posted on the website of the Diocese of Beauvais. In the cave where St. Bernadette reported seeing Mary, "I felt the mysterious presence of Mary and little Bernadette."
She said she went to confession and received the anointing of the sick during the pilgrimage. "In no case did I ask for healing, but only for the conversion of heart and the strength to continue my journey as an invalid."
A few days after returning to her convent, she said she felt unusually relaxed and she experienced warmth throughout her body. Sister Moriau said an inner voice asked her to remove the rigid corset that helped hold her erect, the splint that kept her foot straight and the neurostimulator she used for pain control. She began walking unaided and without pain.
Before her case went to the International Medical Committee of Lourdes, she underwent batteries of tests and examinations, which were studied by committees of the Lourdes Medical Bureau in 2009, 2013 and 2016.
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} | 1,758 |
The Character You Likely Forgot Forest Whitaker Played In Bloodsport
Johnny Nunez/Getty
By Mark D. McKee/April 15, 2022 1:19 pm EST
We can thank the 1980s for giving us Forest Whitaker and some of the biggest stars of the last generation. With movies catapulting actors like Whitaker, Tom Cruise, Arnold Schwarzenegger, Sigourney Weaver, Michael Keaton, Bruce Willis, and Tom Hanks, the decade set us up for the 90s. One of the more influential movies to land in the 80s was "Bloodsport."
The 1988 in-your-face martial arts flick built on the martial arts tournament trope essentially invented by Bruce Lee and "The Karate Kid" before it by upping the brutality. "Bloodsport" follows American Frank Dux as he travels to Hong Kong to compete in a last-man-standing tournament called The Kumite. The event is by invitation only, and the winner is crowned the greatest fighter in the world.
Along with contributing to establishing the martial arts genre and continuing it into the next decade, "Bloodsport" introduced the world to none other than the "Muscles from Brussels," Jean Claude Van Damm. The movie would catapult Van Damm to be one of the premier action stars in the world over the next decade. On top of that, it would showcase the talents of a future superstar in the background, Forest Whitaker.
He chased Van Damm around Hong Kong
Cannon International/YouTube
One of the aspects of "Bloodsport" that made the film so compelling was that Frank Dux (Van Damm) wasn't supposed to be there. Not only is he a westerner, who are rarely invited to the tournament, but he went AWOL from the Army to compete in the tournament to bring honor to his shidoshi, Tanaka. The Army, however, wasn't going to take his absence lying down.
Colonel Cook (Ken Boyle) enlists the services of Helmer and Rawlins (Norman Burton and Forest Whitaker) to track Dux down and bring him back to his duty station to face the consequences. What follows is a chase all over Hong Kong for the two (let's call them bounty hunters). What begins as a comical chase scene through the streets ending in a fall off a boat culminates in the enlistment of the Hong Kong police. During a confrontation on the last day of the tournament, the two finally relent and allow Dux to compete (clearly, they had no choice as he already alluded them and bested the police). The final scene sees them all get on a plane together to head home after Dux won the championship, as he promised in the beginning.
While it was a small part, Whitaker's Rawlins went a long way to giving the film more depth and showcased more of what Forest Whitaker had to offer, assisting in jumpstarting a career still going strong today. | {
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PRS for Music Foundation Helps Unsigned Artists Tap Into Vinyl Revival
The initiative sees vinyl pressing funds available for new acts.
The vinyl revival may only add up to a small part of the overall recorded music market, but for new and unsigned acts the chance to have their music released in the form of a 7" or 12" record retains a special appeal that goes beyond simply sales.
From today that process gets a little bit easier for a select number of U.K. acts, thanks to a new and novel funding scheme from PRS for Music Foundation, the independent charitable arm of U.K. collecting society PRS for Music.
U.K. Government Launches Competition Review Into PRS for Music
Launched in association with Brighton-based music specialist packaging company Modo, the "Flash Funding" scheme is open to British artists across all genres and offers five emerging acts the opportunity to get 500 12" vinyl pressings worth over £1,500 ($2,250) free of charge.
As well as physical vinyl product, the chosen artists will receive bespoke design support from the Modo creative team, as well as release planning, promotion and sales strategy support. Acts wanting to apply have until midnight GMT on Wednesday, Dec. 9 to do so at the PRS for Music Foundation website with the first wave of successful applicants due to be announced 14 December.
Included among the panel of judges is John Kennedy (Radio X), Julie Weir (Visible Noise), Nigel Adams (Full Time Hobby), Pip Newby (Play It Again Sam) and Sade Lawson (Relentless Records).
"Flash Funding for vinyl is a brilliant idea as it helps give bands something to sell to people at gigs after they've been blown away by an amazing set," commented Radio X's John Kennedy.
UK Collection Society PRS Brings In $1 Billion
His words were echoed by Vanessa Reed, PRS for Music Foundation's executive director, who called the initiative "an exciting new way for us to offer and deliver targeted support to music creators quickly and directly."
Since its inception in 2000, PRS for Music Foundation has given more than £22 million ($33 million) to over 5,000 new music initiatives by awarding grants and leading partnership programs that support music sector development, ranging from composer residencies and commissions to live showcases in the U.K. and overseas. Each year it receives £1.5 million ($2.3 million) from PRS for Music to back the creation and promotion of new British talent. | {
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import {Component, EventEmitter, Input, Output} from "@angular/core";
import {MatchesConstants} from "../../matches.constant.service";
import {AbstractControl, FormBuilder, FormGroup, Validators} from "@angular/forms";
import "rxjs/add/operator/startWith";
import "rxjs/add/operator/map";
import {MatchesDataStoreService} from "../../matches-data-store";
import {Subject} from "rxjs/Subject";
import {MatchesService} from "../../../../common/services/matches.service";
/*/!**
* Created by HudaZulifqar on 6/27/2017.
*!/*/
@Component({
selector: 'submit-score-totals',
templateUrl: 'totals.html',
//styleUrls: ['../../../../theme/components/baCheckbox/baCheckbox.scss'],
styleUrls: ['../submitScore.scss'],
})
export class SubmitScoreTotalsComponent {
@Output() notify: EventEmitter<any> = new EventEmitter<any>();
@Output() notify_Extras: EventEmitter<any> = new EventEmitter<any>();
private ngUnsubscribe: Subject<void> = new Subject<void>()
@Input() innings: string;
@Input() isFirstInnings: boolean = true;
isSubmitted: boolean = false;
public totalsResStatus: string;
public totalsForm: FormGroup;
public name: AbstractControl;
public game_id: AbstractControl;
public innings_id: AbstractControl;
public team: AbstractControl;
public wickets: AbstractControl;
public overs: AbstractControl;
public total: AbstractControl;
constructor(fb: FormBuilder, private matchesService: MatchesService, private matchesConstants: MatchesConstants,
private matchesDataStoreService: MatchesDataStoreService) {
this.totalsForm = fb.group({
'name': ['', Validators.compose([Validators.required, Validators.minLength(1), Validators.maxLength(2)]), Validators.pattern('^(0|[1-9][0-9]*)')],
'game_id': ['', Validators.compose([Validators.required, Validators.minLength(1), Validators.maxLength(2), Validators.pattern('^(0|[1-9][0-9]*)')])],
'team': ['', Validators.compose([Validators.required, Validators.minLength(1), Validators.maxLength(2), Validators.pattern('^(0|[1-9][0-9]*)')])],
'innings_id': ['', Validators.compose([Validators.required, Validators.minLength(1), Validators.maxLength(2), Validators.pattern('^(0|[1-9][0-9]*)')])],
'wickets': ['', Validators.compose([Validators.required, Validators.minLength(1), Validators.maxLength(2), Validators.pattern('^(0|[1-9][0-9]*)')])],
'overs': ['', Validators.compose([Validators.required, Validators.minLength(1), Validators.maxLength(4)])],
'total': ['', Validators.compose([Validators.required, Validators.minLength(1), Validators.maxLength(3), Validators.pattern('^(0|[1-9][0-9]*)')])],
});
this.name = this.totalsForm.controls['name'];
this.game_id = this.totalsForm.controls['game_id'];
this.innings_id = this.totalsForm.controls['innings_id'];
this.team = this.totalsForm.controls['team'];
this.wickets = this.totalsForm.controls['wickets'];
this.overs = this.totalsForm.controls['overs'];
this.total = this.totalsForm.controls['total'];
}
batting_poistion = this.matchesConstants.getBattingPositions();
onClick() {
console.log('this.totalsForm.value from total: ', this.totalsForm.value)
this.notify.emit(this.totalsForm.value);
}
getMatchData() {
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,092 |
Johan Jakob Jakobsen (* 15. April 1937 in Namsos, Nord-Trøndelag; † 30. Juni 2018) war ein norwegischer Politiker der Senterpartiet (Sp), der unter anderem 28 Jahre lang Abgeordneter des Storting, mehrfach Minister sowie zwölf Jahre Vorsitzender der Senterpartiet war.
Leben
Berufliche Tätigkeiten und Kommunalpolitiker
Jakobsen, Sohn eines Provinzagronomen, trat nach dem Abitur 1956 in die Offiziersanwärterschule in Bergen ein und begann dann 1959 eine Ausbildung an der Landwirtschaftsschule in Jønsberg, ehe er von 1959 bis 1962 ein Studium an der Norwegischen Landwirtschaftshochschule in Ås absolvierte. Im Anschluss war er von 1962 bis 1972 Abteilungsleiter bei der Landwirtschaftskooperative Felleskjøpet in Namsos und bei A/L Namdal Kornsilo. Zugleich betrieb er von 1963 bis 1993 einen eigenen Bauernhof in Namsos und erwarb darüber hinaus 1971 ein Examen an der Handelsschule.
Mitte der 1960er Jahre begann er seine politische Laufbahn in der Kommunalpolitik und war von 1967 bis 1972 Mitglied des Gemeinderates von Namsos sowie zugleich zwischen 1971 und 1973 Mitglied des Rates des Fylke (Fylkesting) von Nord-Trøndelag. Darüber hinaus war er von 1967 bis 1969 Vizevorsitzender des Senterungdommen, des Jugendverbandes der Zentrumspartei, sowie Mitglied der Kommissionen zur Ausarbeitung der Parteiprogramme für die Wahlen 1969 und 1973.
Storting-Abgeordneter, Fraktions- und Parteivorsitzender
Nachdem er von 1969 bis 1973 stellvertretendes Mitglied des Storting sowie zwischen 1972 und 1973 Politischer Sekretär im Verkehrsministerium war, wurde er 1973 erstmals für die Senterpartiet als Abgeordneter in das Storting gewählt und vertrat in diesem bis 2001 28 Jahre lang den Wahlkreis Nord-Trøndelag. Zu Beginn seiner Parlamentszugehörigkeit war er von Oktober 1973 bis September 1977 Sekretär des Odelsting, der damals noch bestehenden ersten Kammer des Parlaments.
Im Oktober 1977 wurde Jakobsen, der Vorsitzender der Kommission zur Ausarbeitung des Parteiprogramms für die Wahl 1977 war, als Nachfolger von Erland Steenberg erstmals Vorsitzender der Fraktion der Sp im Storting, und hatte diese Funktion bis zu seiner Ablösung durch Johan Buttedahl im Juni 1983 inne. Danach war er Mitglied des Fraktionsvorstands seiner Partei.
1979 wurde er zugleich als Nachfolger von Gunnar Stålsett Vorsitzender der Senterpartiet und übte diese Funktion zwölf Jahre lang bis 1991 aus.
Im Laufe seiner Abgeordnetentätigkeit war er Mitglied der Storting-Ausschüsse für Landwirtschaft (Landbrukskomité), für Auswärtige und Verfassungsangelegenheiten (Utenriks- og konstitusjonskomité), für Wahlen (Valgkomité), für Verteidigung (Forsvarskomité) sowie für Arbeitsplanung (Arbeidsordningskomité). Darüber hinaus war er zwischen Januar 1978 und September 2001 auch Delegierter beim Nordischen Rat sowie zuletzt von November 1997 bis September 2001 Mitglied der Delegation des Storting für Beziehungen zum Europäischen Parlament.
Minister und Funktionen in der Privatwirtschaft
Am 8. Juni 1983 wurde er von Ministerpräsident Kåre Willoch im Rahmen einer Kabinettsumbildung zum Verkehrsminister in dessen Regierung berufen und gehörte dieser bis zum Ende von Willochs Amtszeit am 9. Mai 1986 an. Während dieser Zeit war er auch Mitglied des Verteidigungsrates.
Im April 1991 folgte er Anne Enger Lahnstein im Amt des Vorsitzenden der Sp-Fraktion im Storting und hatte diese Funktion nunmehr bis zu seiner Ablösung durch Odd Roger Enoksen im März 2000 inne. Anne Enger Lahnstein wurde stattdessen seine Nachfolgerin als Parteivorsitzende.
Am 16. Oktober 1989 wurde er von Ministerpräsident Jan P. Syse zum Minister für Kommunales und Arbeit in dessen Regierung berufen, einer aus Høyre, Kristelig Folkeparti und Senterpartiet gebildeten Minderheitsregierung. In dieser war er nach einer Neuausrichtung der Ressorts zuletzt vom 1. Januar 1990 bis zum Ende von Syses Amtszeit am 3. November 1990 Kommunalminister.
Nach seinem Ausscheiden aus dem Parlament 2001 übernahm Jakobsen zahlreiche Positionen in Unternehmen der Privatwirtschaft, aber auch in gesellschaftlichen und sozialen Institutionen. Zum einen war er seit 2001 Vorstandsmitglied der Stiftung für schwedisch-norwegische Zusammenarbeit (Svensk-norske Samarbeidsfond) sowie der Norwegischen Veterinärhochschule und war zwischen 2000 und 2007 Vorstandsmitglied von Falstadsenteret, des norwegischen Zentrums für Menschenrechte in Ekne sowie von 2001 bis 2002 Berater der Phytochem Holding ASA. Ferner war er von 2004 bis 2007 Vorstandsmitglied des Regionalmuseums von Namdalen und ab 2006 Berater des Norwegischen Instituts für strategische Studien (Norsk institutt for strategiske studier).
Veröffentlichungen
Mot Strømmen, Gyldendal, Oslo 2000
Muntre minner fra ting og torg, Gyldendal, Oslo 2001
Makten og æren : en biografi om Nils Trældal, Gyldendal, Oslo 2005
Weblinks
Biografie in Store norske leksikon (Onlineversion)
Eintrag auf der Homepage des Storting
Verkehrsminister (Norwegen)
Kommunalminister (Norwegen)
Storting-Abgeordneter (Nord-Trøndelag)
Fylkesting-Abgeordneter (Nord-Trøndelag)
Parteivorsitzender der Senterpartiet
Landwirt (Norwegen)
Norweger
Geboren 1937
Gestorben 2018
Mann | {
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Mary Ella is a fourth-year medical student at ACOM. She completed her undergraduate degree in Biology at Point Loma Nazarene University in San Diego, CA. During her undergraduate studies, she volunteered as an EMT at a clinic in Tijuana, Mexico monthly–an experience that sparked her passion to pursue a career in global health. In addition to spending time in Tijuana, she has traveled to many other countries to learn about their healthcare systems including Guatemala and Argentina most recently. She has a particular interest in refugee medicine, women's health, and sustainability in global health practices.
Outside of her medical studies, Mary Ella writes for her blog, Mod Med (www.modmedblog.com). Through her blog, she intends to empower people to work towards their dreams as she writes about becoming a doctor in this day and age. She also writes about controversial issues in medicine and global health as well as maintaining a balance while having a career. Mary Ella ultimately wants to pursue a career in obstetrics & gynecology. She hopes to work with people around the globe to collectively improve health care practices, increase access to medicine worldwide, and advocate for global women's health issues. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,772 |
\section{Introduction}
\label{section1}
The aim of this article is to study the local time for a certain class of Gaussian processes. Since the works of S. Berman \cite{7} the existence and properties of the local time are studied for a wide class of Gaussian processes and fields \cite{17}. In the case of Brownian motion the local time can be investigated using the parabolic equations and potential theory due to the independence of increments and self-similarity. For general Gaussian processes S. Berman proposed the notion of local nondeterminism which in some sense means almost independency of increments on small intervals. Under some technical assumptions this property leads to the existence and regularity of local time with respect to both spatial and time variables. Different authors proposed the version of local nondeterminism property for Gaussian processes and fields and proved not only existence of the local time but investigated some its asymptotic properties such as the law of iterated logarithm or small ball probabilities \cite{18}. Nevertheless the local nondeterminism property is hard to check for an arbitrary Gaussian process. Simple sufficient conditions were given for processes with stationary increments or for self-similar processes \cite{19}. In the case of planar Gaussian process the situation is much worse. Namely, for Brownian motion on the plane the existence of multiple self-intersection points is well-known \cite{20}. The corresponding local time of multiple self-intersection needs to be properly renormalized. Such renormalization was done by S. Varadhan \cite{11}, E.B. Dynkin \cite{12}, J. Rosen \cite{21}. Later J.-F. Le Gall \cite{14} obtained asymptotic expansion for the area of Brownian sausage which contains renormalized self-intersection local times as the coefficients. As all these results are essentially based on the structure of Brownian motion, used here technic can not be expanded on other Gaussian processes. But the question of such generalization is quiet interesting in view of constructing the random polymer models using not only Markov processes (in some cases there are no reasons for molecule to differ the starting point and the end-point). All mentioned reasons lead to the attempt to find such class of Gaussian processes for which some version of local nondeterminism holds and results related to existence and renormalization of the local time and self--intersection local times can be achieved. Such class of processes was introduced by A.A. Dorogovtsev in \cite{1} in connection with the anticipating stochastic integration. The original definition is the following.
\begin{defn}[\cite{1}]
\label{defn1}
A centered Gaussian process $x(t),\ t\in[0; 1]$ is said to be an integrator if there exists the constant $c>0$ such that for an arbitrary partition $0=t_0<t_1<\ldots<t_n=1$ and real numbers $a_0, \ldots, a_{n-1} $
\begin{equation}
\label{eq1}
E\Big(\sum^{n-1}_{k=0}a_k(x(t_{k+1})-x(t_k))\Big)^2\leq c\sum^{n-1}_{k=0}a^2_k\Delta t_k.
\end{equation}
\end{defn}
The inequality \eqref{eq1} allows to integrate functions from $L_2([0; 1])$ with respect to $x.$ This naturally leads to the definition of Skorokhod type stochastic integral with respect to $x.$ In \cite{1} the corresponding stochastic calculus including It\^o formula for $x$ was considered. The following statement describes the structure of integrators.
\begin{prop} [\cite{1}]
\label{prop2}
The centered Gaussian process $x(t),\ t\in[0; 1]$ is an integrator iff there exist a Gaussian white noise $\xi$ \cite{3, 4} in $L_2([0; 1])$ and a continuous linear operator $A$ in the same space such that
\begin{equation}
\label{eq2}
x(t)=(A\1_{[0; 1]}, \xi),\ t\in[0; 1].
\end{equation}
\end{prop}
In the article we use the language of white noise analysis \cite{3,4,15,16}.
Note that if $A$ equals identity, then $x$ in the expression \eqref{eq2} is a Wiener process. For continuously invertible operator $A$ one can expect that $x$ will inherit some properties of the Wiener process. For example, it will be proved in the section 2 that if $A$ is continuously invertible, then $x$ has a local time at any point $u\in\mbR.$ Such local time can be obtained as the occupation density. Also it will be checked that this density is a continuous function in spatial and time variables. In the section 3 we will prove a continuous dependence of local times of integrators on generating them operators.
The main method of our investigations is based on the studying of functional properties of the Hilbert-valued functions. In particular, we obtain some estimations for the Gram determinant constructed by the increments of such function. These estimations allow to investigate the conditional moments of the Brownian self-intersection local time in dimensions one and two when the end point (the value $w(1)$) tends to infinity. We establish the speed of decreasing of the mentioned moments. The question of the conditional behavior of the self-intersection local times is inspired by the studying properties of continuous polymer models [22-24].
The real polymers can not have self-intersections due to the excluded volume effect \cite{22}. But the energy of interaction between the monomers from different places in polymer molecule influences on its form. Flory proposed the evaluation of the size of polymer based on the counting of interaction energy \cite{22}. The Brownian path can be viewed as an ideal Gaussian model of polymer \cite{22}. Applying the Flory method to it one have to substitute the energy of interaction by the self-intersection local time. So the question of the dependence the self-intersection local time on the size of Brownian path is natural. We present the corresponding estimations in the section 4.
Some necessary facts from geometry of Hilbert-valued functions are proved in the appendix of the paper.
\section{Existence of local time for Gaussian integrators}
\label{section2}
Let us recall approaches of defining the local time for the one-dimensional Wiener process $w(t),\ t\in[0; 1].$ Put
$$
f_\ve(y)=\frac{1}{\sqrt{2\pi\ve}}e^{-\frac{y^2}{2\ve}},\ y\in\mbR,\ \ve>0.
$$
\begin{defn}
\label{defn3}
For any $t\in[0; 1]$ and $u\in\mbR$
$$
\int^t_0\delta_u(w(s))ds:=L_2\mbox{-}\lim_{\ve\to0}\int^t_0f_\ve(w(s)-u)ds
$$
is said to be a local time of the Wiener process at the point $u$ up to time $t.$
\end{defn}
Consider an occupation measure of $w$ up to time $t$ defined by the formula
$$
\mu_t(D)=\int^t_0\1_D(w(s))ds,\ D\in B(\mbR)
$$
($B(\mbR)$ is the Borel $\sigma$-field on $\mbR$). $\mu_t(D)$ equals the Lebesgue measure of the time which the trajectory of the Wiener process spends in the set $D$ up to time $t.$ Levy in \cite{5} proved that for almost all trajectories of $w$ and any $t\in[0; 1]$ the measure $\mu_t$ has a density, i.e. there exists the random function $l(u, t),\ u\in\mbR$ such that a.s. for any $t\in[0; 1]$ and $D\in B(\mbR)$
$$
\mu_t(D)=\int_Dl(u, t)du.
$$
Trotter in \cite{6} proved that the density of an occupation measure of the Wiener process is continuous in $u$ and $t.$ Useful consequence of joint continuity is the following occupation density formula. For every continuous function $\vf$ on $\mbR$ with compact support
\begin{equation}
\label{eq4}
\int^t_0\vf(w(s))ds=\int_{\mbR}\vf(u)l(u, t)du.
\end{equation}
It follows from \eqref{eq4} that
$$
\int^t_0\delta_u(w(s))ds=\lim_{\ve\to0}\int^t_0f_\ve(w(s)-u)ds=
$$
$$
=\lim_{\ve\to0}\int_{\mbR}f_\ve(v-u)l(v, t)dv=l(u, t).
$$
Therefore, the value of the density of occupation measure $l(u, t)$ is the local time of the Wiener process at $u$ up to time $t.$ In this section we will establish the same properties of the local time for Gaussian integrators.
To prove the existence of the local time for the Gaussian integrator $x$ with the representation \eqref{eq2} we need the notion of local nondeterminism for Gaussian processes introduced by Berman in \cite{7}. Let $\{y(t),\ t\in J\}$ be $\mbR$-valued zero mean Gaussian process on an open interval $J.$ Suppose that there exists $d>0$ such that
1) $E(y(t)-y(s))^2>0,$ for all
$
s, t\in J: 0\leq|t-s|\leq d;
$
2) $Ey^2(t)>0$ for all $t\in J.$
For $m\geq2, \ t_1, \ldots, t_m\in J, \ t_1<t_2<\ldots< t_m $ put
$$
V_m=\frac{\Var(y(t_m)-y(t_{m-1})/y(t_1), \ldots, y(t_{m-1}))}
{\Var(y(t_m)-y(t_{m-1}))}
$$
which is the ratio of the conditional and the unconditional variance.
\begin{defn} [\cite{7}]
\label{defn4}
A Gaussian process $y$ is said to be locally nondetermined on $J$ if for every $m\geq2$
$$
\lim_{c\to0}\inf_{t_m-t_1\leq c}V_m>0.
$$
The following statement was proved in \cite{7} and demonstrates that the local nondeterminism property can be used as one of the sufficient conditions for the existence and smoothness of the local time for general Gaussian process.
\end{defn}
\begin{thm} [\rm \cite{7}]
\label{thm5}
Let $y(t),\ t\in[0; T]$ be a centered Gaussian process satisfying the following three conditions
1) $y(0)=0;$
2) $y$ is locally nondetermined on $(0; T);$
3) there exist positive real numbers $\gamma, \delta$ and $a$ continuous even function $b(t)$ such that $b(0)=0,\ b(t)>0,\ t\in(0; T],$
$$
\lim_{h\to0}h^{-\gamma}\int^h_0(b(t))^{-1-2\delta}dt=0
$$
and $E(y(t)-y(s))^2\geq b^2(t-s)$ for all $s, t\in[0; T].$ Then there exists a version $l(u, t),\ u\in\mbR,\ t\in[0; T]$ of the local time of the process $y$ which is jointly continuous in $(u, t)$ and which satisfies a H\"older condition in $t$ uniformly in $u,$ i.e. for every $\gamma_1<\gamma,$ i.e. there exist positive and finite random variables $\eta$ and $\eta_1$ such that
$$
\sup_u|l(u, t+h)-l(u, t)|\leq \eta_1|h|^{\gamma_1}
$$
for all $s, t, t+h\in[0; T]$ and all $|h|<\eta.$
\end{thm}
To discuss the existence of the local time for Gaussian integrator $x$ we need reformulation of the notion of local nondeterminism. Denote by $G(e_1, \ldots, e_n)$ the Gram determinant constructed by vectors $e_1, \ldots, e_n.$ Let $g\in C([0; 1], L_2([0; 1])),$\
$
\Delta g(t_i)=g(t_{i+1})-g(t_i),\ i=\ov{1, m-1}
$
($C([0; 1], L_2(0; 1))$ is the space of all continuous functions from $[0; 1]$ into $L_2([0; 1])$).
\begin{lem}
\label{lem6}
The Gaussian process $y(t)=(g(t), \xi),$ where $\xi$ is a white noise in $L_2([0; 1])$ is locally nondetermined on $J$ iff for every $m\geq2$
$$
\lim_{c\to0}\inf_{t_m-t_1\leq c}
\frac{G(g(t_1), \Delta g(t_1), \ldots, \Delta g(t_{m-1}))}
{\|g(t_1)\|^2\|\Delta g(t_1)\|^2\ldots\|\Delta g(t_{m-1})\|^2}>0.
$$
\end{lem}
\begin{proof}
This is a consequence of definition $V_m$ and relation
$$
V_2\cdot\ldots\cdot V_m=\frac
{\det\cov(x(t_i), x(t_j))^m_{ij=1}}
{\Var x(t_1)\Var(x(t_2)-x(t_1))\cdot\ldots\cdot\Var(x(t_m)-x(t_{m-1}))}=
$$
$$
\frac{G(g(t_1), \Delta g(t_1), \ldots, \Delta g(t_{m-1}))}
{\|g(t_1)\|^2\|\Delta g(t_1)\|^2\ldots\|\Delta g(t_{m-1})\|^2}.
$$
\end{proof}
By using the lemma \ref{lem6} one can establish the following statement.
\begin{thm}[\rm \cite{8}]
\label{thm8}
Suppose that the operator $A$ in the representation \eqref{eq2} of $x$ is continuously invertible. Then there exists a version $l(u, t),\ u\in\mbR,\ t\in[0; 1]$ of the local time of $x$ which is jointly continuous in $(u, t)$ and which satisfies a H\"older condition in $t$ uniformly in $u,$ i.e. for every $\gamma<\frac{1}{2}$ there exist positive and finite random variables $\eta$ and $\eta_1$ such that
$$
\sup_u|l(u, t+h)-l(u, t)|\leq\eta_1|h|^\gamma
$$
for all $s, t, t+h\in[0; 1]$ and all $|h|<\eta.$
\end{thm}
The theorem \ref{thm8} was proved in the article of the second author \cite{8}. Here we briefly recall the main steps of the proof, which is based on the following key property of the Gram determinant.
\begin{lem}
\label{lem8}
Suppose that $A$ is a continuously invertible operator in the Hilbert space $H.$ Then for all $k\geq1$ there exists a positive constant $c(k)$ which depends on $k$ such that for any $e_1, \ldots, e_k\in H$ the following relation holds
$$
G(Ae_1, \ldots, Ae_k)\geq c(k)G(e_1, \ldots, e_k).
$$
\end{lem}
The lemma \ref{lem8} is proved in the appendix (Lemma A.1).
\begin{proof}[Proof of the theorem \ref{thm8}.]
To prove the theorem let us check that $x$ satisfies conditions 1)--3) of the theorem \ref{thm5}. It is obvious that $x(0)=0.$ The lemma \ref{lem6} and the lemma \ref{lem8} imply that $x$ is locally nondetermined. Let us check that $x$ satisfies the condition 3) of the theorem \ref{thm5}. Let $b(t)=c\sqrt{t},\ c>0.$ Pick $\delta<\frac{1}{2}$ and then $\gamma$ such that $\gamma<\frac{1}{2}-\delta.$ One can see that
$$
\lim_{h\to0}h^{-\gamma}\int^h_0t^{-\frac{1}{2}-\delta}dt=\frac{2}{1-2\delta}\lim_{h\to0}h^{\frac{1}{2}-\delta-\gamma}=0.
$$
\end{proof}
\section{On continuous dependence of local times of integrators on generating them operators}
\label{section4}
Suppose that $A_n, A$ are continuously invertible operators in $L_2([0; 1])$ which generate Gaussian integrators $x_n,\ x,$ i.e.
$$
x_n(t)=(A_n\1_{[0; t]}, \xi), \ x(t)=(A\1_{[0; t]}, \xi), \ t\in[0; 1].
$$
We proved in the section 2 that there exist the random variables
$$
l_n(u):=l_n(u, 1)=\int^1_0\delta_u(x_n(t))dt,
$$
$$
l(u):=l(u, 1)=\int^1_0\delta_u(x(t))dt, \ u\in\mbR.
$$
The following statement shows that if the sequence of operators $A_n$ converges strongly to an operator $A,$ then the sequence of local times of integrators $x_n$ converges in mean square to the local time of $x.$
\begin{thm}
\label{thm11}
Suppose that $A_n, A$ are continuously invertible operators in $L_2([0; 1])$ such that
1) for any $y\in L_2([0; 1])$
$$
\|A_ny-Ay\|\to0, \ n\to\infty;
$$
2) $\sup_{n\geq1}\|A^{-1}_n\|<\infty.$
Then
$$
E\int_{\mbR}(l_n(u)-l(u))^2du\to0, \ n\to\infty.
$$
\end{thm}
\begin{proof}
To prove the theorem it suffices to check that
$$
E\int_{\mbR}l^2_n(u)du\to E\int_{\mbR}l^2(u)du,\ n\to\infty,
$$
$$
E\int_{\mbR}l_n(u)l(u)du\to E\int_{\mbR}l^2(u)du,\ n\to\infty.
$$
It follows from the theorem B.1 that
$$
E\int_{\mbR}l_n(u)l_n(u)du=E\int^1_0\int^1_0\delta_0(x_n(t)-x_n(s))dsdt=
$$
$$
=\lim_{\ve\to0}E\int^1_0\int^1_0f_\ve(x_n(t)-x_n(s))dsdt=
$$
$$
=\frac{2}{\sqrt{2\pi}}\int_{\Delta_2}
\frac{dsdt}{\|A_n\1_{[s; t]}\|}.
$$
It follows from the invertibility of operators $A_n$ and the condition 2) that
$$
\frac{1}{\|A_n\1_{[s; t]}\|}\leq\sup_{n\geq1}\|A^{-1}_n\|\frac{1}{\sqrt{t-s}}.
$$
The Lebesgue dominated convergence theorem implies that
$$
E\int_{\mbR}l^2_n(u)du\to E\int_{\mbR}l^2(u)du, \ n\to\infty.
$$
Let us check that
$$
E\int_{\mbR}l_n(u)l(u)du\to E\int_{\mbR}l^2(u)du, \ n\to\infty.
$$
Again by using the theorem B.1 one can write
$$
E\int_{\mbR}l_n(u)l(u)du= E\int_0^1\int^1_0\delta_0(x_n(t)-x(s))dsdt=
$$
$$
=\frac{2}{\sqrt{2\pi}}\int_{\Delta_2}\frac{dsdt}{\|A_n\1_{[0; t]}-A\1_{[0; s]}\|}=
$$
$$
=\frac{2}{\sqrt{2\pi}}\int_{\Delta_2}\frac{dsdt}{\|A_n(\1_{[0; t]}-A^{-1}_nA\1_{[0; s]})\|}\leq
$$
$$
\leq\frac{2}{\sqrt{2\pi}}\sup_{n\geq1}\|A^{-1}_n\|
\int_{\Delta_2}\frac{dsdt}{\|\1_{[0; t]}-\vk_n(s)\|},
$$
where $\vk_n(s)=A^{-1}_nA\1_{[0; s]}.$ It follows from the lemma A.4 (see appendix A) that the sequence $\Big\{\frac{1}{\|\1_{[0 ; t]}-\vk_n(s)\|}\Big\}_{n\geq1}$ is uniformly integrable. Consequently,
$$
E\int_{\mbR}l_n(u)l(u)du\to E\int_{\mbR}l^2(u)du, \ n\to\infty.
$$
\end{proof}
\section{Conditional moments of Brownian self-intersection local time}
\label{section2'}
In this part of the article we will discuss the relationships between the norm of the end-point of Brownian path and its self-intersection local time. As it was mentioned in the introduction, such relation reflects the fact that the real polymers have biggest Flory number then ideal due to the excluded volume effect. Here we will study the conditional distribution of the self-intersection local time for Brownian motion under condition that its end-point tends to infinity. Begin with the one-dimensional Brownian motion $w.$ As it was disscused for example in \cite{25}, the self-intersection local time for $w$ exists. Denote it by
$$
T_2=\int_{\Delta_2}\delta_0(w(t_2)-w(t_1))dt_1dt_2,\ \Delta_2=\{0\leq t_1\leq t_2\leq 1\}.
$$
The following statement holds.
\noindent
{\bf Theorem 10}. {\it For any $p>0$ and $\beta\in(0; 1)$
$$
E(T^p_2/w(1)=a)=O(|a|^{-\beta}), \ a\to\infty.
$$
}
\begin{proof}
It is enough to consider $p$ integer. Then
$$
E(T^p_2/w(1)=a)=
$$
$$
=E\int_{\Delta^p_2}\prod^p_{i=1}\delta_0(\eta(t^i_2)-\eta(t^i_1))d\vec{t},
$$
where $\eta(t)=w(t)-tw(1)+at, \ t\in[0; 1].$ In terms of white noise $\xi=\dot{w}$ the process $\eta$ has a representation
$$
\eta(t)=(Qg_0(t), \xi)+at.
$$
Here $g_0(t)=\1_{[0; t]}$ and $Q$ is a projection onto orthogonal complement to $g_0(1)=\1_{[0; 1]}.$ To estimate the conditional expectation for $T^p_2$ let us use the following lemma from Appendix (Lemma A.5).
\noindent
{\bf Lemma 11}. {\it Let the elements $e_1, \ldots, e_n$ of $L_2([0; 1])$ and a projection $Q$ are such that $Qe_1, \ldots, Qe_n$ are linearly independent. Suppose that the elements $f, g$ satisfy relationships \newline $\forall \ i=1, \ldots, n:$
$$
(f, e_i)=(g, Qe_i).
$$
Then
$$
\|P_1f\|\leq \|P_2g\|,
$$
where $P_1$ and $P_2$ are the orthogonal projections on the linear span of $e_1, \ldots, e_n$ and $Qe_1, \ldots, Qe_n$ respectively.
}
To apply this lemma for our situation denote by $\Gamma^Q_{\vec{t}}$ and $P^Q_{\vec{t}}$ the Gram determinant for $Qe_1, \ldots, Qe_p$ and the projection on its linear span, where $e_i=\1_{[t^i_1; t^i_2]}, i=1, \ldots, p,$ and $Q$ is a projection onto $\1^\perp_{[0; 1]}.$ Then
$$
E\int_{\Delta^p_2}\prod^p_{i=1}\delta_0(\eta(t^i_2)-\eta(t^i_1))d\vec{t}=
$$
$$
=\int_{\Delta^p_2}
\frac{
e^{
-\frac{1}{2}\|P^Q_{\vec{t}}h_{\vec{t}}\|^2a^2
}
}
{\Gamma^Q_{\vec{t}}}d\vec{t}.
$$
Here $h_{\vec{t}}$ is taken in a such way that \newline $\forall \ i=1, \ldots, p:$
$$
(h_{\vec{t}}, Qe_i)=t^i_2-t^i_1.
$$
It follows from the previous lemma, that
$$
\|P^Q_{\vec{t}}h_{\vec{t}}\|\geq \|P_{\vec{t}}\1_{[0; 1]}\|,
$$
where $P_{\vec{t}}$ is a projection onto the linear span of $\1_{[t^1_1; t^1_2]}, \ldots, \1_{[t^p_1; t^p_2]}.$ Consequently, for arbitrary $k=1, \ldots, p$
$$
e^{-\frac{1}{2}\|P^Q_{\vec{t}}h_{\vec{t}}\|^2a^2}\leq e^{-\frac{1}{2}(t^k_2-t^k_1)a^2}.
$$
To find the estimation for $\Gamma^Q_{\vec{t}}$ note that
$$
\Gamma^Q_{\vec{t}}=\Gamma(\1_{[0; 1]}, \1_{[t^1_1; t^1_2]}, \ldots, \1_{[t^p_1; t^p_2]}).
$$
Let us use the following lemma (Lemma A.6).
\noindent
{\bf Lemma 12}. {\it
Let $\Delta_0=\O,$ and $\Delta_1, \ldots, \Delta_n$ be the subsets of $[0; 1].$ Then
$$
\Gamma(\1_{\Delta_1}, \ldots, \1_{\Delta_n})\geq\prod^n_{k=1}|\Delta_k\setminus\mathop{\cup}\limits^{k-1}_{j=1}\Delta_j|.
$$
}
As a consequence of this lemma one can obtain the following estimation for the Gram determinant
$$
\Gamma^Q_{\vec{t}}\geq\prod^N_{j=1}|\wt{\Delta}_j|,
$$
where $\wt{\Delta}_j, j=1, \ldots, N$ are the intervals from the partition of $[0; 1]$ by the end-points of the intervals $[t^1_k, t^2_k], k=1, \ldots, p.$
Now using the previous estimation for $\|P^Q_{\vec{t}}h_{\vec{t}}\|$ one can get that
$$
E\int_{\Delta^p_2}\prod^p_{i=1}\delta_0(\eta(t^i_2)-\eta(t^i_1))d\vec{t}\leq
$$
$$
\leq(2p)!\frac{1}{\sqrt{2\pi}^p}\int_{\Delta_{2p}}
\frac
{e^{-\frac{1}{2}a^2(t_{2p}-t_{2p-1})}}
{(\prod^{2p}_{j=0}(t_{j+1}-t_j))^{\frac{1}{2}}}d\vec{t},
$$
where $t_0=1$ and $t_{2p+1}=1.$ Consider the integral with respect to the last variable $t_{2p}$
$$
\int^1_{2p-1}\frac
{e^{-\frac{1}{2}a^2(t_{2p}-t_{2p-1})}}
{\sqrt{(t_{2p}-t_{2p-1})(1-t_{2p})}}dt_{2p}.
$$
Using expression $\delta=1-t_{2p-1}$ and changing variable one can rewrite the last integral as
$$
\int^\delta_0\frac{e^{-\frac{1}{2}a^2s}}
{\sqrt{s(\delta-s)}}ds=
\int^1_0\frac{e^{-\frac{1}{2}a^2\delta s'}}
{\sqrt{s'(1-s')}}ds'.
$$
Using H\"{o}lder inequality one can get for $\alpha\in(1; 2)$
$$
\int^1_0\frac{e^{-\frac{1}{2}a^2\delta s'}}
{\sqrt{s'(1-s')}}ds'\leq
c_\alpha\Big(
\int^1_0e^{-\frac{1}{2}a^2\delta\frac{\alpha}{\alpha-1} s'ds'}
\Big)^{\frac{\alpha}{\alpha-1}}\leq
$$
$$
\leq\wt{c}_\alpha
\frac{1}{a^{2-\frac{2}{\alpha}}}
\frac{1}{\delta^{1-\frac{1}{\alpha}}},
$$
where $c_\alpha$ and $\wt{c}_\alpha$ are the positive constants which depend on $\alpha.$ Finally, for any $\alpha\in(1; 2)$
$$
E\int_{\Delta^p_2}\prod^p_{i=1}\delta_0(\eta(t^i_2)-\eta(t^i_1))d\vec{t}\leq
$$
$$
\leq\wt{\wt{c}}_\alpha
\int_{\Delta_{2p-1}}
\prod^{2p-2}_{j=0}
\frac{1}{\sqrt{t_{j+1}-t_j}}\cdot
\frac{1}{(1-t_{2p-1})^{1-\frac{1}{\alpha}+\frac{1}{2}}}d\vec{t}
\cdot a^{-2+\frac{2}{\alpha}}.
$$
\end{proof}
For planar Wiener process $w$ on the interval $[0; 1]$ consider trajectories with $w(1)=a.$ One can expect that if $\|a\|$ is large, then the trajectory of $w$ has a small number of self-intersections. The conditional distribution of the Wiener process under the condition $w(1)=a$ coincides with the distribution of the Brownian bridge
$$
y_a(t)=w(t)-tw(1)+at, \ t\in[0; 1].
$$
Let us investigate the dependence of the self-intersection local time of the process $y_a(t),\ t\in[0; 1]$ on $\|a\|.$ Denote by
$$
T_2(a, \alpha)=\int_{\Delta_2(a, \alpha)}
\delta_0(y_a(t_2)-y_a(t_1))dt_1dt_2,
$$
where
$$
\Delta_2(a, \alpha)=\{(t_1, t_2): \ 0\leq t_1\leq 1-\|a\|^{-\alpha},\ t_1+\|a\|^{-\alpha}\leq t_2\leq1\}.
$$
The self-intersection local time
$$
\int_{\Delta_2(a,\alpha)}\delta_{0}(w(t_2)-w(t_1))dt_1dt_2
$$
exists (see \cite{25}). As before, one can check that
$$
T_{2}(a,\alpha)=E\left(\int_{\Delta_2(a,\alpha)}\delta_{0}(w(t_2)-w(t_1))dt_1dt_2/w(1)=a\right).
$$
Let us prove that the following statement holds.
\noindent
{\bf Theorem 13}. {\it
For
1) $\alpha=2:$
$$
\lim_{\|a\|\to+\infty}ET_2(a, \alpha)=\frac{1}{2\pi}\int^{+\infty}_1\frac{1}{y}e^{-\frac{y}{2}}dy;
$$
2) $\alpha<2:$
$$
\lim_{\|a\|\to+\infty}ET_2(a, \alpha)=0;
$$
3) $\alpha>2:$
$$
\lim_{\|a\|\to\infty}ET_2(a, \alpha)=+\infty.
$$
}
\begin{proof}
Let $\Delta t_1=t_2-t_1.$
\begin{equation}
\label{eq1_9}
ET_2(a, \alpha)=
\int_{\Delta_2(a, \alpha)}
\frac{1}{2\pi\Delta t_1(1-\Delta t_1)}e^{\frac{1}{2}\frac{\Delta t_1\|a\|^2}{1-\Delta t_1}}d\vec{t}.
\end{equation}
Let $t_1=s_1, \ \|a\|^2\Delta t_1=s_2.$ Then \eqref{eq1_9} equals
$$
\int^{1-\|a\|^{-\alpha}}_0\int^{(1-s_1)\|a\|^2}_{\|a\|^{-\alpha+2}}
\frac{\|a\|^2}{2\pi s_2(\|a\|^2-s_2)}
e^{-\frac{1}{2}\frac{\|a\|^2s_2}{\|a\|^2-s_2}}ds_2ds_1=
$$
$$
=\int^{\|a\|^2}_{\|a\|^{-\alpha+2}}
\int^{1-\frac{s_2}{\|a\|^2}}_{0}
\frac{\|a\|^2}{2\pi s_2(\|a\|^2-s_2)}
e^{-\frac{1}{2}\frac{\|a\|^2s_2}{\|a\|^2-s_2}}ds_1ds_2=
$$
$$
=
\int^{\|a\|^2}_{\|a\|^{-\alpha+2}}
\Big(1-\frac{s_2}{\|a\|^2}\Big)
\frac{\|a\|^2}{2\pi s_2(\|a\|^2-s_2)}
e^{-\frac{1}{2}\frac{\|a\|^2s_2}{\|a\|^2-s_2}}ds_2=
$$
\begin{equation}
\label{eq2_9}
=
\frac{1}{2\pi}
\int^{\|a\|^2}_{\|a\|^{-\alpha+2}}
\frac{1}{s_2}
e^{-\frac{1}{2}\frac{\|a\|^2s_2}{\|a\|^2-s_2}}ds_2.
\end{equation}
Put $\frac{\|a\|^2s_2}{\|a\|^2-s_2}=y.$ Then \eqref{eq2_9} equals
\begin{equation}
\label{eq3_9}
\frac{1}{2\pi}
\int^{+\infty}_{\frac{\|a\|^{2-\alpha}}{1-\|a\|^{-\alpha}}}
\frac{\|a\|^2}{y(\|a\|^2+y)}e^{-\frac{y}{2}}dy.
\end{equation}
Note that for $\alpha=2$
$$
\frac{1}{2\pi}\int^{+\infty}_{\frac{1}{1-\|a\|^{-2}}}
\frac{\|a\|^2}{y(\|a\|^2+y)}e^{-\frac{y}{2}}dy\to
\frac{1}{2\pi}\int^{+\infty}_1\frac{1}{y}e^{-\frac{y}{2}}dy, \ \|a\|\to+\infty.
$$
One can see that for $\alpha<2$
$$
\frac{1}{2\pi}\int^{+\infty}_{\frac{\|a\|^{2-\alpha}}{1-\|a\|^{-2}}}
\frac{\|a\|^2}{y(\|a\|^2+y)}e^{-\frac{y}{2}}dy\leq
$$
\begin{equation}
\label{eq4_9}
\leq\frac{1}{2\pi}
\frac{(1-\|a\|^{-\alpha})^2}{\|a\|^{2-\alpha}}
e^{-\frac{\|a\|^{2-\alpha}}{1-\|a\|^{-\alpha}}}.
\end{equation}
The estimate \eqref{eq4_9} implies that for $\alpha<2$
$$
\lim_{\|a\|\to+\infty}
\frac{1}{2\pi}
\int^{+\infty}_{\frac{\|a\|^{2-\alpha}}{1-\|a\|^{-\alpha}}}
\frac{\|a\|^2}{y(\|a\|^2+y)}
e^{-\frac{y}{2}}dy=0.
$$
On the other hand for $\alpha>2$
$$
\frac{1}{2\pi}
\int^{+\infty}_{\frac{\|a\|^{2-\alpha}}{1-\|a\|^{-\alpha}}}
\frac{\|a\|^2}{y(\|a\|^2+y)}
e^{-\frac{y}{2}}dy\geq
$$
$$
\geq
\frac{1}{2\pi}
\int^{\frac{1}{m}}_{\frac{\|a\|^{2-\alpha}}{1-\|a\|^{-\alpha}}}
\frac{\|a\|^2}{y(\|a\|^2+y)}
e^{-y}dy\geq
$$
\begin{equation}
\label{eq5_9}
\geq
\frac{1}{2\pi}
\frac{m\|a\|^2}{\|a\|^2+\frac{1}{m}}
\Big(-e^{\frac{1}{m}}+e^{-\frac{\|a\|^{2-\alpha}}{1-\|a\|^{-\alpha}}}\Big).
\end{equation}
It follows from \eqref{eq5_9} that for $\alpha>2$
$$
\lim_{\|a\|\to+\infty}
\frac{1}{2\pi}
\int^{+\infty}_{\frac{\|a\|^{2-\alpha}}{1-\|a\|^{-\alpha}}}
\frac{\|a\|^2}{y(\|a\|^2+y)}
e^{-\frac{y}{2}}dy=+\infty.
$$
\end{proof}
\section* { Appendix A. {\large On some geometry of Hilbert-valued functions}}
In this appendix we collect some useful estimates for Gramian matrix and Gram determinant which describe the changing of geometry of Hilbert-valued functions under the action of a linear continuous operator. Also for $ 0\leq\alpha<1$ we check that
$$
\sup_{y\in L_2([0;1])}\int^1_0\frac{dt}{\|\1_{[0; t]}-y\|^{1+\alpha}}.
$$
Let $B(e_1, \ldots, e_n)$ be the Gramian matrix constructed from the vectors $e_1,\ldots, e_n$ in the Hilbert space $H.$
\noindent
{\bf Lemma A.1.} {\it Suppose that $A$ is a continuously invertible operator in the Hilbert space $H.$ Then for all $k\geq1$ there exists a positive constant $c(k)$ which depends on $k$ and $A$ such that for any $e_1, \ldots, e_k\in H$ the following relation holds
$$
G(Ae_1, \ldots, Ae_k)\geq c(k)G(e_1, \ldots, e_k).
$$
}
\begin{proof}
To prove the lemma it suffices to check that
$$
\inf G\left(\frac{Af_1}{\|Af_1\|}, \ldots,\frac{Af_k}{\|Af_k\|} \right)>0,
$$
where infimum is taking over all orthonormal systems $(f_1, \ldots, f_k).$ Using the Gram--Schmidt orthogonalization procedure build the orthogonal system $q_1, \ldots, q_k$ from $\frac{Af_1}{\|Af_1\|}, \ldots,\frac{Af_k}{\|Af_k\|}.$
Here
$$
q_j=\frac{Af_j}{\|Af_j\|}-\sum^{j-1}_{i=1}a_{ij}\frac{Af_i}{\|Af_i\|}
$$
with some $a_{ij}.$ Let us prove that
$$
\inf_{(f_1, \ldots, f_k)} G\left(\frac{Af_1}{\|Af_1\|}, \ldots,\frac{Af_k}{\|Af_k\|} \right)=
$$
$$
=\inf_{(f_1, \ldots, f_k)}\prod^k_{i=1}\|q_i\|^2>0.
$$
If it is not so, then there exists the sequence $\{f^n_1, \ldots, f^n_k\}_{n\geq1}$ and $j=\ov{1, k}$ such that $\|q^n_j\|\to0,\ n\to\infty.$ The invertibility of the operator $A$ implies that
$$
\left\|\frac{f^n_j}{\|Af^n)j\|}-\sum^{j-1}_{i=1}a^n_{ij}\frac{f^n_i}{\|Af^n_i\|}
\right\|\to0, n\to\infty.
$$
But
$$
\left\|\frac{f^n_j}{\|Af^n_j\|}-\sum^{j-1}_{i=1}a^n_{ij}\frac{f^n_i}{\|Af^n_i\|}
\right\|\geq\frac{1}{\|Af^n_j\|}>0.
$$
\end{proof}
\noindent
{\bf Lemma A.2.}
{\it
Suppose that $A$ is a continuously invertible operator in the Hilbert space $H.$ Then for any $e_0=0,\ e_1, \ldots, e_n \in H$ such that $e_{i+1}-e_i\perp e_{j+1}-e_j,\ i,j=\ov{1, n-1},\ i\ne j$ there exists a positive constant $c$ such that for all $\vec{u}\in\mbR^n$ with $u_0=0$ the following relation holds
$$
(B^{-1}(Ae_1, \ldots, Ae_n)\vec{u}, \vec{u})\geq c\sum^{n-1}_{i=0}
\frac{(u_{i+1}-u_i)^2}{\|e_{i+1}-e_i\|^2}.
$$
}
\begin{proof}
It was proved in \cite{10} that in the case $\vec{u}=((h_0, Ae_1), \ldots, (h_0, Ae_n)), $ \ $h_0\in H$ the following relation holds
$$
(B^{-1}(Ae_1, \ldots, Ae_n)\vec{u}, \vec{u})=\|P_{Ae_1\ldots Ae_n}h_0\|^2,
$$
where $P_{e_1\ldots e_n}$ is a projection on $LS\{e_1, \ldots, e_n\}$ (linear span generated by elements $e_1, \ldots, e_n$). Note that
$$
((h_0, Ae_1), \ldots, (h_0, Ae_n))=((A^*h_0, e_1), \ldots, (A^*h_0, e_n)).
$$
Since $(A^*h_0, e_1)=u_1, \ldots, (A^*h_0, e_n)=u_n,$ then
$$
A^*h_0=\sum^{n-1}_{i=0}\frac{e_{i+1}-e_i}{\|e_{i+1}-e_i\|^2}(u_{i+1}-u_i)+r,
\eqno(A.1)
$$
where $r\perp e_i,\ i=\ov{0, n}.$
Consequently,
$$
h_0=\sum^{n-1}_{i=0}{A^*}^{-1}\Big(\frac{e_{i+1}-e_i}{\|e_{i+1}-e_i\|^2}(u_{i+1}-u_i)\Big)+{A^*}^{-1}r.
$$
Let us remark that continuous invertibility of the operator $A$ implies the existence of ${A^*}^{-1}.$ It follows from (A.1) that
$$
(B^{-1}(Ae_1, \ldots, Ae_n)\vec{u}, \vec{u})=
$$
$$
=\Big\|{A^*}^{-1}\Big(\sum^{n-1}_{i=0}
\frac{(e_{i+1}-e_i)(u_{i+1}-u_i)}{\|e_{i+1}-e_i\|^2}+r\Big)\Big\|^2
$$
$$
\geq c\sum^{n-1}_{i=0}
\frac{(u_{i+1}-u_i)^2}{\|e_{i+1}-e_i\|^2}+c\|r\|^2\geq
$$
$$
\geq
c\sum^{n-1}_{i=0}
\frac{(u_{i+1}-u_i)^2}{\|e_{i+1}-e_i\|^2}.
$$
\end{proof}
The following statement describes the direct application of lemma A.1 and lemma A.2.
For $s_1, \ldots, s_n\in \Delta_n, u_1, \ldots, u_n\in\mbR$ let $p_{s_1\ldots s_n}(u_1, \ldots, u_n)$ be the density of Gaussian vector $(x(s_1), \ldots, x(s_n))$ in $\mbR^n.$ Here $x$ is Gaussian integrator with the representation \eqref{eq2}.
\noindent
{\bf Lemma A.3} \cite{8}. {\it Suppose that $A$ in the representation \eqref{eq2} is continuously invertible. Then there exist positive constants $c_1(n), c_2$ such that the following relation holds
$$
p_{s_1\ldots s_n}(u_1, \ldots, u_n)\leq
\frac{c_1(n)}
{\sqrt{s_1(s_2-s_1)\ldots(s_n-s_{n-1})}}e^{-\frac{1}{2}c_2}\sum^{n-1}_{j=0}\frac{(u_{j+1}-u_j)^2}{s_{j+1}-s_j}.
$$
}
For a proof see \cite{8}.
\noindent
{\bf Lemma A.4.}
{\it
For any $0\leq\alpha<1$
$$
\sup_{y\in L_2([0;1])}\int^1_0\frac{1}{\|\1_{[0; t]}-y\|^{1+\alpha}}dt<+\infty.
$$
}
\begin{proof}
Put $g_0(t):=\1_{[0; t]}.$ Note that
$$
\int^1_0\frac{1}{\|g_0(t)-y\|^{1+\alpha}}dt=\int^{+\infty}_0\lambda\{t: \|g_0(t)-y\|^{-(1+\alpha)}\geq z\}dz,
$$
where $\lambda$ is the Lebesgue measure on $
\mbR.$ Then to prove the statement of the lemma it suffices to check that for $b>0$
$$
\sup_{y\in L_2([0;1])}\int^{+\infty}_b\lambda\{t: \|g_0(t)-y\|^{-(1+\alpha)}\geq z\}dz<+\infty.
$$
For any $g_0(t_0), g_0(t_1)$ from the closed ball $\ov{B}\Big(y, \frac{1}{z^{\frac{1}{1+\alpha}}}\Big)$ the following relation holds
$$
|t_0-t_1|=\|g_0(t_0)-g_0(t_1)\|^2{\leq}\frac{4}{z^{\frac{2}{1+\alpha}}}.\eqno(A.2)
$$
(A.2) implies that
$$
\Big\{t: \|g_0(t)-y\|\leq\frac{1}{z^{\frac{1}{1+\alpha}}}\Big\}\subset\Big[t_0-\frac{4}{z^{\frac{2}{1+\alpha}}}; t_0+\frac{4}{z^{\frac{2}{1+\alpha}}}\Big],
$$
for some $t_0$ such that
$$
\|g_0(t)-y\|\leq\frac{1}{z^{\frac{1}{1+\alpha}}}.\eqno(A.3)
$$
It follows from (A.3) that for $0\leq\alpha<1$
$$
\int^\infty_b
\lambda\{t:
\|g_0(t)-y\|\leq\frac{1}{z^{\frac{1}{1+\alpha}}}
\}dz\leq 4\int^{+\infty}_b\frac{dz}{z^{\frac{2}{1+\alpha}}}<+\infty.
$$
\end{proof}
\noindent
{\bf Lemma A.5}. {\it Let the elements $e_1, \ldots, e_n$ of $L_2([0; 1])$ and a projection $Q$ are such that $Qe_1, \ldots, Qe_n$ are linearly independent. Suppose that the elements $f, g$ satisfy relationships \newline $\forall \ i=1, \ldots, n:$
$$
(f, e_i)=(g, Qe_i).
$$
Then
$$
\|P_1f\|\leq \|P_2g\|,
$$
where $P_1$ and $P_2$ are the orthogonal projections on the linear span of $e_1, \ldots, e_n$ and $Qe_1, \ldots, Qe_n$ respectively.
}
\begin{proof}
Note, that \newline $\forall \ i=1, \ldots, n$
$$
(g, Qe_i)=(P_2g, Qe_i)=(QP_2g, e_i)=(P_1f, e_i).
$$
Consequently,
$$
P_1f=P_1QP_2g=P_1P_2g.
$$
Hence
$$
\|P_1f\|\leq\|P_2g\|.
$$
\end{proof}
\noindent
{\bf Lemma A.6}. {\it
Let $\Delta_0=\O,$ and $\Delta_1, \ldots, \Delta_n$ be the subsets of $[0; 1].$ Then
$$
\Gamma(\1_{\Delta_1}, \ldots, \1_{\Delta_n})\geq\prod^n_{k=1}|\Delta_k\setminus\mathop{\cup}\limits^{k-1}_{j=1}\Delta_j|.
$$
}
\begin{proof}
Since
$$
\Gamma(\1_{\Delta_1}, \ldots, \1_{\Delta_n})=|\Delta_1|\prod^n_{k=2}\|h_k\|^2,
$$
where $h_k$ is the orthogonal component of $\1_{\Delta_k}$ with respect to the linear span of $\1_{\Delta_1}, \ldots, \1_{\Delta_{k-1}},$ then it is enough to prove that for $k=2, \ldots, n$
$$
\|h_k\|^2\geq|\Delta_k\setminus\mathop{\cup}\limits^{k-1}_{j=1}\Delta_j|.
$$
For the set $\Delta$ such that $|\Delta|>0$ denote by $P_\Delta$ the orthogonal projection into $\1_\Delta.$ Then
$$
\|P_{\Delta_j}\1_{\Delta_k}\|^2=\frac{|\Delta_k\cap\Delta|^2}{|\Delta|}\leq\big|\Delta_k\cap\Delta\big|.
$$
Consider the representation
$$
\cup^{k-1}_{j=1}\Delta_j=\cup^l_{i=1}H_i,
$$
where $|H_i|>0$ and disjoint, all $H_i$ belong to the algebra generated by $\{\Delta_j\}$ and every $\Delta_j$ can be obtained as the union of the certain $H_i.$ Then the linear span of $\1_{\Delta_1, \ldots, \1_{\Delta_{k-1}}}$ is a subset of the linear span of $\1_{H_1}, \ldots, \1_{H_l}.$ Hence
$$
\|h_k\|^2\geq|\Delta_k|-\sum^l_{i=1}|\Delta_k\cap H_i|=|\Delta_k\setminus\mathop{\cup}\limits^{k-1}_{j=1}\Delta_j|.
$$
\end{proof}
\section*{Appendix B. {\large On some relations between generalized func\-tio\-nals from white noise}}
Consider linearly independent elements $f_1, \ldots, f_n\in L_2([0; 1]).$ Here we investigate conditions on elements $r_j\in L_2([0; 1]),\ j=\ov{1, n-1}$ that allow to establish the following relation
$$
\int_{\mbR}\prod^n_{k=1}\delta_0((f_k, \xi)-u)du=\prod^{n-1}_{j=1}\delta_{0}((r_j, \xi)),
\eqno{\rm(B.1)}
$$
which is understood as equality of the generalized functionals from white noise \cite{15} and will be checked using Fourier--Wiener transform.
The following statement discribes one of the possible choices for $r_j,\ j=\ov{1, n-1}.$
\noindent
{\bf Theorem B.1.} {\it
Let $f_1, \ldots, f_n$ be linearly independent elements in $L_2([0; 1]).$ Then
$$
\int_{\mbR}\prod^n_{k=1}\delta_0((f_k, \xi)-u)du=\prod^{n-1}_{k=1}\delta_0((f_{k+1}-f_k, \xi)).
\eqno{\rm (B.2)}
$$
}
\begin{proof}
To prove the statement let us calculate the Fourier--Wiener transform of the left-hand side and the right-hand side of the equality (B.2). Denote by $\cT(\alpha)(h)$ the Fourier--Wiener transform of random variable $\alpha.$ One can check that
$$
\cT\Big(\prod^{n-1}_{j=1}\delta_0((r_j, \xi))\Big)(h)=
$$
$$
=\frac{1}{(2\pi)^{\frac{n-1}{2}}\sqrt{G(r_1,\ldots, r_{n-1})}}
e^{-\frac{1}{2}\|P_{r_1\ldots r_{n-1}}h\|^2}
\eqno{\rm (B.3)}
$$
(see \cite{1}). Let us find the Fourier--Wiener transform of $\int_{\mbR}\prod^n_{k=1}\delta_0((f_k, \xi)-u)du$
$$
\cT\Big(\int_{\mbR}\prod^{n}_{k=1}\delta_0((f_k, \xi)-u)du\Big)(h)=
$$
$$
=\int_{\mbR}\frac{1}{(2\pi)^{\frac{n}{2}}\sqrt{G(f_1,\ldots, f_{n})}}
e^{-\frac{1}{2}(B^{-1}(f_1,\ldots, f_n)(u\vec{e}-\vec{a}), u\vec{e}-\vec{a})}du,
\eqno{\rm(B.4)}
$$
where $\vec{e}=\begin{pmatrix}1\\
\vdots\\
1\end{pmatrix}, $ $\vec{a}=\begin{pmatrix}(f_1, h)\\
\vdots\\
(f_n, h)
\end{pmatrix}.$ By integrating (B.4) over $u$ one can get
$$
\frac{1}
{(2\pi)^{\frac{n-1}{2}}
\sqrt{G(f_1, \ldots, f_n)}\sqrt{(B^{-1}(f_1,\ldots, f_n)e, e)}}\cdot
$$
$$
\cdot \exp\Big\{-\frac{1}{2}\Big((B^{-1}(f_1,\ldots, f_n)a,a)-
\frac{(B^{-1}(f_1,\ldots, f_n)a,e)^2}
{(B^{-1}(f_1,\ldots, f_n)e, e)}\Big)\Big\}.
\eqno{\rm(B.5)}
$$
It is not difficult to check that
$$
(B^{-1}(f_1,\ldots, f_n)a,a)=\|P_{f_1\ldots f_n}h\|^2.
$$
Consider the function $f\in LS\{f_1, \ldots, f_n\}$ such that $(f, f_k)=1,\ k=\ov{1, n}.$
Then
$$
(B^{-1}_{f_1\ldots f_n}\vec{e}, \vec{e})=\|P_{f_1\ldots f_n}f\|^2=\|f\|^2
$$
$$
(B^{-1}_{f_1\ldots f_n}\vec{a}, \vec{e})=(P_{f_1\ldots f_n} h, f).
$$
Therefore, (B.5) equals
$$
\frac{1}
{(2\pi)^{\frac{n-1}{2}}\sqrt{
G(f_1, \ldots, f_n)}\|f\|}
e^{-\frac{1}{2}(\|P_{f_1\ldots f_n}h\|^2-\|P_fP_{f_1\ldots f_n}h\|^2)}.
$$
Denote by $f\overset{\perp}{=}\{v\in LS\{f_1, \ldots, f_n\}: (v, f)=0\}.$ Then
$$
\cT\Big(\int_{\mbR}\prod^{n}_{k=1}\delta_0((f_k, \xi)-u)du\Big)(h)=
$$
$$
=\frac{1}
{(2\pi)^{\frac{n-1}{2}}\sqrt{
G(f_1, \ldots, f_n)}\|f\|}
e^{-\frac{1}{2}\|P_{f^\perp}h\|^2}.
\eqno{\rm(B.6)}
$$
By comparing (B.3) and (B.6) we obtain the following conditions on elements $r_k,\ k=\ov{1, n-1}$
1) $LS\{r_1, \ldots, r_{n-1}\}=f^\perp;$
2) $G(r_1, \ldots, r_{n-1})=G(f_1, \ldots, f_n)\|f\|^2.$
Let us check that $r_j:=f_{j+1}-f_j$ satisfy conditions 1), 2). Really, put $M=LS\{f_2-f_1, \ldots, f_n-f_{n-1}\}.$ Then $f\perp M.$ Denote by $r$ the distance from $f_1$ to $M.$ One can see that
$$
G(f_1,\ldots, f_n)=G(f_1, f_2-f_1, \ldots, f_n-f_{n-1})=r^2G(f_2-f_1, \ldots, f_n-f_{n-1}).
$$
Since
$$
(f_1, \frac{f}{\|f\|})=\|f_1\|\cos\alpha=r,
$$
then $r=\frac{1}{\|f\|}.$ Consequently,
$$
\|f\|^2G(f_1, \ldots, f_{n-1})=G(f_2-f_1, \ldots, f_n-f_{n-1}).
$$
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,045 |
{"url":"https:\/\/mathematica.stackexchange.com\/questions\/182595\/density-map-for-complex-and-imaginary-parts-of-eigenvalues-on-one-graph","text":"# Density map for complex and imaginary parts of eigenvalues on one graph\n\nI have three eigenvalues of a particular 3x3 matrix. Namely:\n\neigen1 = 1\/3*(p + sig - u2\/(2^(8\/3)*sig) - 3*I*\u03c9);\neigen2 = 1\/3*(p - 1\/2*(sig - u2\/(2^(8\/3)*sig)) + (I*Sqrt[3])\/2*(sig + u2\/(2^(8\/3)*sig)) - 3*I*\u03c9);\neigen3 = 1\/3*(p - 1\/2*(sig - u2\/(2^(8\/3)*sig)) - (I*Sqrt[3])\/2*(sig + u2\/(2^(8\/3)*sig)) - 3*I*\u03c9);\n\n\nWhere\n\np = (\u0393 + \u03ba1 + \u03ba2)\/2;\nu1 = 36 g1^2 (-2 p + 3 \u03ba2) + (36 g2^2 + (2 p - 3 \u03ba1) (2 p - 3 \u03ba2)) (4 p - 3 (\u03ba1 + \u03ba2));\nu2 = (2^(2\/3)*(12 g1^2 + 12 g2^2 - 4 p^2 + 6 p (\u03ba1 + \u03ba2) - 3 (\u03ba1^2 + \u03ba1 \u03ba2 + \u03ba2^2)));\nsig = ((u1 + Sqrt[u1^2 + u2^3])\/16)^(1\/3);\n\n\nNow, I intend to make a density plot for all three eigenvalues on the same graph as a function of g1 and g2 with fixed parameters (\u0393,\u03ba1,\u03ba2). However, the eigenvalues may or may not be complex (depending on the selection of parameters). My goal is to be able to illustrate a territory that demarcates the real and imaginary part of the eigenvalues.\n\nI start by doing just for the first eigenvalue (eigen1)\n\nDensityPlot[{Evaluate@eigen1 \/. {\u0393 -> 0.01, \u03ba1 -> 1, \u03ba2 -> 5, \u03c9 -> 0}}, {g1, 0, 10}, {g2, 0, 10}, PlotRange -> All, PlotLegends -> Automatic]\n\n\nand I was given something like this\n\nWhereas I am unable to show anything for the density plot for eigen2 with respect to g1, g2 and the same parameters\n\nDensityPlot[{Evaluate@eigen2 \/. {\u0393 -> 0.01, \u03ba1 -> 1, \u03ba2 -> 5, \u03c9 -> 0}}, {g1, 0, 10}, {g2, 0, 10}, PlotRange -> All, PlotLegends -> Automatic]\n\n\nAnd finally, doing the same thing for eigen3 respectively gives\n\nDensityPlot[{Evaluate@eigen3 \/. {\u0393 -> 0.01, \u03ba1 -> 1, \u03ba2 -> 5, \u03c9 -> 0}}, {g1, 0, 10}, {g2, 0, 10}, PlotRange -> All, PlotLegends -> Automatic]\n\n\nNotice that there are white spaces that permeates throughout all three plots and I'm not sure what's going on. My hunch is that the white spaces might correspond to complex components of the eigenvalues and hence DensityPlot was unable to show it.\n\nEdit: Collecting the Real and Imaginary parts of the eigenvalues using Re[] and Im[]seems to do the trick in eliminating the white spaces.\n\nDensityPlot[{Evaluate@Re[eigen2 \/. {\u0393 -> 0.01, \u03ba1 -> 1, \u03ba2 -> 5, \u03c9 -> 0}]}, {g1, 0, 10}, {g2, 0, 10}, PlotRange -> All, PlotLegends -> Automatic]\n\n\nDensityPlot[{Evaluate@Im[eigen2 \/. {\u0393 -> 0.01, \u03ba1 -> 1, \u03ba2 -> 5, \u03c9 -> 0}]}, {g1, 0, 10}, {g2, 0, 10}, PlotRange -> All, PlotLegends -> Automatic]\n\n\nAnd so forth (for eigen1 and eigen3). But it doesn't seem like the white spaces in the bad plots correspond to the Imaginary parts and I can't diagnose what the exact problem is here. Nonetheless, my issue now is that I wish to combine the Real and Imaginary parts of the eigenvalues in the same density plot for all 3 eigenvalues. But I'm struggling to combine just the Real and Imaginary parts of the same eigenvalue onto the same plot. Doing\n\nDensityPlot[{Evaluate@Re[eigen2 \/. {\u0393 -> 0.01, \u03ba1 -> 1, \u03ba2 -> 5, \u03c9 -> 0}],Evaluate@Im[eigen2 \/. {\u0393 -> 0.01, \u03ba1 -> 1, \u03ba2 -> 5, \u03c9 -> 0}}, {g1, 0, 10}, {g2, 0, 10}, PlotRange -> All, PlotLegends -> Automatic]\n\n\nWill only show the Imaginary part of eigen2. Recall that the idea of doing this is to show the parameter space of the eigenvalues demarcated by real parts and imaginary parts of the eigenvalue on the same density plot. What should be my approach here?\n\n\u2022 Why combine together the real and imaginary part, when they can be reproduced separately and placed side by side? For example, {DensityPlot[{Evaluate@ Re[eigen2] \/. {\\[CapitalGamma] -> 0.01, \\[Kappa]1 -> 1, \\[Kappa]2 -> 5, \\[Omega] -> 0}}, {g1, 0, 10}, {g2, 0, 10}, PlotRange -> All, PlotLegends -> Automatic], DensityPlot[{Evaluate@ Im[eigen2] \/. {\\[CapitalGamma] -> 0.01, \\[Kappa]1 -> 1, \\[Kappa]2 -> 5, \\[Omega] -> 0}}, {g1, 0, 10}, {g2, 0, 10}, PlotRange -> All, PlotLegends -> Automatic]} \u2013\u00a0Alex Trounev Sep 26 '18 at 2:23","date":"2019-04-22 16:53:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.18587443232536316, \"perplexity\": 2324.451875953817}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-18\/segments\/1555578558125.45\/warc\/CC-MAIN-20190422155337-20190422181337-00148.warc.gz\"}"} | null | null |
The 1954 Asian Games (), officially known as the Second Asian Games – Manila 1954 was a multi-sport event held in Manila, Philippines, from May 1 to 9, 1954. A total of 970 athletes from 19 Asian National Olympic Committees (NOCs) competed in 76 events from eight sports. The number of participating NOCs and athletes were larger than the previous Asian Games held in New Delhi in 1951. This edition of the games has a different twist where it did not implement a medal tally system to determine the overall champion but a pointing system. The pointing system is a complex system where each athlete were given points according to their achievement like position in athletics or in swimming. In the end the pointing system showed to be worthless as it simply ranked the nations the same way in the medal tally system. The pointing system was not implemented in future games ever since. Jorge B. Vargas was the head of the Philippine Amateur Athletic Federation (In 1976, was renamed as Philippine Olympic Committee) and the Manila Asian Games Organizing Committee. With the second-place finish of the Philippines, only around 9,000 spectators attended the closing ceremony at the Rizal Memorial Stadium. The events were broadcast on radio live at DZRH and DZAQ-TV ABS-3 on delayed telecast.
Opening ceremony
The Games were formally opened by President Ramon Magsaysay on May 1, 1954, at 16:02 local time. Around 20,000 spectators filled the Rizal Memorial Stadium in Malate, Manila, for the opening ceremony. As requested by the IOC, the torch relay and lighting of the cauldron were excluded from the Opening Ceremony to preserve the tradition of the Olympic Games. The torch ceremony were returned at the 1958 Asian Games. The host however gave a solution by giving a special citation to the last athlete to enter the parade. The Philippines, as host, was the last country to enter the stadium. The flag bearer for the Philippines squad was Andres Franco, who won a gold medal in the 1951 Asian Games in high jump event, the sole gold medal of any Filipino in the athletics events of the previous Asian Games.
Sports
The 1954 Asian Games featured eight sports divided into 10 events, aquatics included three events namely diving, swimming and water polo. This version of the Asian Games comprised more sports and events than the last one, as six sports and seven events were in the calendar of 1951 Asian Games. Three sports—boxing, shooting and wrestling—made their debut, while cycling was dropped out.
Participating nations
National Olympic Committees (NOCs) are named and arranged according to their official IOC country codes and designations at the time.
Calendar
In the following calendar for the 1954 Asian Games, each blue box represents an event competition, such as a qualification round, on that day. The yellow boxes represent days during which medal-awarding finals for a sport were held. The numeral indicates the number of event finals for each sport held that day. On the left, the calendar lists each sport with events held during the Games, and at the right, how many gold medals were won in that sport. There is a key at the top of the calendar to aid the reader.
Medal table
Japan led the medal table, athletes from Japan won most medals, including most gold, silver and bronze. Host nation, Philippines finished second with 45 total medals (including 14 gold).
The top ten ranked NOCs at these Games are listed below. The host nation, Philippines, is highlighted.
References
Asian Games by year
Asian Games, 1954
Asian Games
Sports in Manila
Asian games
Multi-sport events in the Philippines
Asian Games
20th century in Manila
May 1954 sports events in Asia | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,703 |
DENNE PLAKAT ER EN BESTILLINGSVARE MED CA 14 DAGES LEVERINGSTID / PLEASE NOTE THIS POSTER IS MADE-TO-ORDER. UP TO TWO WEEKS PRODUCTIONS TIME, YOU WILL BE NOTIFIED WHEN WE SHIP THE POSTER.
Special edition Oslo. Plakaten af den smukke norske hovedstad i maskuline toner fra petroleum til karry. Måler 50x70cm. Plakaten er printet på lækkert 200gr højkvalitetspapir i let elfenbensfarvet nuance.
Special edition art print of Oslo. The beautiful poster of Oslo features the Norwegian capitals neighbourhoods. This poster is a limited edition multicoloured print. The colours vary in Kortkartellets signature colours petrol blue, charcoal grey, strong mint etc. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,771 |
Q: How to use debezium with views, instead of tables with sql server connect I have been trying to find a way to use views from an sql server, to be used in kafka source connect with debezium, but the cdc doesn't allow that.
I also used indexed views, but cdc still doesnt recognize it and asks for a Table.
Is there a way around this, where I can monitor real-time changes
from multiple views using debezium in particular?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,459 |
Wallis- och Futunaöarnas officiella flagga är Frankrikes flagga eftersom öarna är ett franskt territorium.
Wallis- och Futunaöarnas inofficiella flagga är röd och har ett rött andreaskors i en vit ruta. Flaggan finns också i en alternativ variant med ett mantuanskt kors istället för andreaskors.
I officiella ärenden används den franska flaggan.
Se även
Wallis- och Futunaöarna
Nationsflaggor i Oceanien
Kultur i Wallis- och Futunaöarna
Franska flaggor | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,876 |
package org.sistemafinanciero.entity.dto;
import java.io.Serializable;
import java.math.BigInteger;
import javax.persistence.Column;
import javax.persistence.Entity;
import javax.persistence.Id;
import javax.persistence.NamedQueries;
import javax.persistence.NamedQuery;
import javax.persistence.Table;
import javax.xml.bind.annotation.XmlAccessType;
import javax.xml.bind.annotation.XmlAccessorType;
import javax.xml.bind.annotation.XmlElement;
import javax.xml.bind.annotation.XmlRootElement;
import org.hibernate.annotations.Subselect;
/*@Subselect(value = "SELECT c.ID_CAJA as idCaja, "
+ "c.denominacion, "
+ "c.abreviatura, "
+ "c.estado, "
+ "c.abierto, "
+ "c.ESTADO_MOVIMIENTO as estadoMovimiento, "
+ "listagg(b.denominacion || ',') WITHIN GROUP (ORDER BY c.ID_CAJA) AS bovedas, "
+ "a.id_agencia as idAgencia "
+ "FROM Caja c "
+ "INNER JOIN BOVEDA_CAJA bc ON C.id_caja = bc.id_caja "
+ "INNER JOIN BOVEDA b ON B.id_boveda = bc.id_boveda "
+ "INNER JOIN AGENCIA a ON a.id_agencia = b.id_agencia "
+ "GROUP BY c.ID_CAJA, c.denominacion, c.abreviatura, c.estado, c.abierto, c.ESTADO_MOVIMIENTO, a.id_agencia")*/
@NamedQueries({ @NamedQuery(name = CajaView.findByIdAgencia, query = "SELECT c FROM CajaView c WHERE c.idAgencia = :idAgencia") })
@Entity
@Table(name = "CAJA_VIEW", schema = "C##BDSISTEMAFINANCIERO")
@XmlRootElement(name = "cajaView")
@XmlAccessorType(XmlAccessType.NONE)
public class CajaView implements Serializable {
/**
*
*/
private static final long serialVersionUID = 1L;
public final static String findByIdAgencia = "CajaView.findByIdAgencia";
private BigInteger idCaja;
private String denominacion;
private String abreviatura;
private int estado;
private int abierto;
private int estadoMovimiento;
private String bovedas;
private String saldos;
private BigInteger idAgencia;
public CajaView() {
// TODO Auto-generated constructor stub
}
@XmlElement(name = "id")
@Id
@Column(name = "ID_CAJA", unique = true, nullable = false, precision = 22, scale = 0)
public BigInteger getIdCaja() {
return idCaja;
}
public void setIdCaja(BigInteger idCaja) {
this.idCaja = idCaja;
}
@XmlElement(name = "denominacion")
@Column(name = "DENOMINACION", nullable = false, length = 100, columnDefinition = "nvarchar2")
public String getDenominacion() {
return denominacion;
}
public void setDenominacion(String denominacion) {
this.denominacion = denominacion;
}
@XmlElement(name = "abreviatura")
@Column(name = "ABREVIATURA", nullable = false, length = 100, columnDefinition = "nvarchar2")
public String getAbreviatura() {
return abreviatura;
}
public void setAbreviatura(String abreviatura) {
this.abreviatura = abreviatura;
}
@XmlElement(name = "estado")
@Column(name = "ESTADO", nullable = false, precision = 22, scale = 0)
public boolean getEstado() {
return estado == 1;
}
public void setEstado(boolean estado) {
this.estado = (estado ? 1 : 0);
}
@XmlElement(name = "abierto")
@Column(name = "ABIERTO", nullable = false, precision = 22, scale = 0)
public boolean getAbierto() {
return abierto == 1;
}
public void setAbierto(boolean abierto) {
this.abierto = (abierto ? 1 : 0);
}
@XmlElement(name = "estadoMovimiento")
@Column(name = "ESTADO_MOVIMIENTO", nullable = false, precision = 22, scale = 0)
public boolean getEstadoMovimiento() {
return estadoMovimiento == 1;
}
public void setEstadoMovimiento(boolean estadomovimiento) {
this.estadoMovimiento = (estadomovimiento ? 1 : 0);
}
@XmlElement(name = "bovedas")
@Column(name = "BOVEDAS", nullable = false, length = 100)
public String getBovedas() {
return bovedas;
}
public void setBovedas(String bovedas) {
this.bovedas = bovedas;
}
@XmlElement(name = "idAgencia")
@Column(name = "ID_AGENCIA", precision = 22, scale = 0)
public BigInteger getIdAgencia() {
return idAgencia;
}
public void setIdAgencia(BigInteger idAgencia) {
this.idAgencia = idAgencia;
}
@XmlElement(name = "saldos")
@Column(name = "SALDOS", nullable = false, length = 100)
public String getSaldos() {
return saldos;
}
public void setSaldos(String saldos) {
this.saldos = saldos;
}
@Override
public int hashCode() {
final int prime = 31;
int result = 1;
result = prime * result + ((idCaja == null) ? 0 : idCaja.hashCode());
return result;
}
@Override
public boolean equals(Object obj) {
if (this == obj)
return true;
if (obj == null)
return false;
if (getClass() != obj.getClass())
return false;
CajaView other = (CajaView) obj;
if (idCaja == null) {
if (other.idCaja != null)
return false;
} else if (!idCaja.equals(other.idCaja))
return false;
return true;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,580 |
Follow @IanDunt
https://twitter.com/PattyJenks/status/1329211121790664704
We obvs can't go near cinemas at the moment, but fwiw, we felt very safe in cinemas before lockdown: you're well away from anyone, people are wearing masks and the audience is not talking.
I struggled with first WW film and really dislike DC's films in general. But fuck me if going to see a good Wonder Woman film isn't exactly what I need right now.
Not saying it will be good. But I'll definitely be going in there willing it to be good.
Keep Current with Ian Dunt
More from @IanDunt
@IanDunt
It's happening tonight. The Doritos Lasagna will be challenged by a new experiment: the tamale pie.
This was some of the best shit I ever cooked. Looked like baby sick, obviously, but then Latin food so often does. Tasted like you were being touched by God.
I think what I find so irritating about the fleets (urgh) is that this site, for all its faults, is predominantly about what's going on in someone's head, not what they look like.
Now, it turns out that what's going on in many people's heads is absolutely horrific and I wish I'd never encountered it, but that's another matter.
I use Instagram as a kind of Facebook-without-words - a locked account just following people I know in real life, with a few exceptions. But I'm aware that much of Instagram features people doing their very best to look beautiful and charmed.
What the fuck is this Instagram stories shit in my Twitter
What the fuck have they done. This is a war crime.
This can fuck the fuck off and fuck off again while it's doing it.
Bit of news - we've got a few changes going on at Politics.co.uk towers.
I'm leaving my position as editor and turning into…. editor at large.
In effect, we're splitting my role, so that I keep the writing and let go of the other editorial responsibilities, which have been occupying a lot of my time.
Biden's victory showed what is possible if liberals and the left work together against nationalism - my latest for the @TheNewEuropean theneweuropean.co.uk/brexit-news/it…
We can of course go back to usual pattern of mutual loathing and frustration. Or people can take the fight to the real enemy.
That involves at least one concession on each side. The Labour left should accept the EHRC antisemitism report in its entirely. It has to commit to anti-racism. Some never will - and they should fuck off. The majority of the left does and will.
PMQs in five minutes parliamentlive.tv/Event/Index/a2…
Wouldn't want you to have developed a sense of hope or spiritual reassurance. We'll have none of that here.
Johnson says Windrush was a "scandal" and the best is being done to "collectively to make amends". This is false, of course. If it were true, they would not be creating the conditions for precisely the same situation to arise with European citizens.
Read all threads by @IanDunt | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,165 |
{"url":"https:\/\/crypto.stackexchange.com\/questions\/56282\/the-counter-mode-of-encryption-ctr-aes","text":"# the counter mode of encryption CTR - AES [closed]\n\nTo attack an encryption algorithm, you ask the encryption oracle to encrypt a polynomial number of messages of your choice and observe the outputs. Then, you give the oracle two messages m0 and m1. The oracle will choose one of the messages uniformly at random, encrypts it, and return the ciphertext c to you. The encryption algorithm is said to be broken if you can determine to which one of the two messages the ciphertext corresponds. Assuming the counter is 48-bit long and the used block cipher is AES (recall that AES operates on 128-bit blocks) answer the following questions:\n\na. Assume the counter is always zero (i.e., the counter does not change from block to block). Can you come up with a successful attack against this mode of encryption?\n\nb. Assume now the nonce is fixed (i.e., the nonce does not change from message to message). Can you come up with a successful attack against this mode of encryption?\n\nc. What is the maximum length of the messages that can be encrypted using this mode?\n\nd. In the counter mode of encryption, the nonce cannot be used again unless a new block cipher key is chosen. What is the maximum number of messages that can be encrypted using the same key?\n\n\u2022 Posting hw questions here is a waste of your time, our time, and your lecturer's time. \u2013\u00a0redplum Mar 10 '18 at 0:48\n\u2022 I see your homework\/assignment, but I can't detect your question. Therefore, it's unclear what you are asking. What research have you done? What have you tried? Where exactly did you get stuck solving this? Please edit your question accordingly. I'll be happy to reopen it once you do. \u2013\u00a0e-sushi Mar 10 '18 at 1:23\n\u2022 I copied the whole question. \u2013\u00a0AFB Mar 10 '18 at 11:52\n\u2022 @AFB \"This is not a homework solving service. What have you tried? What do you not understand about these questions?\" \u2013\u00a0cypherfox Mar 10 '18 at 12:10\n\u2022 @AFB You indeed copied the whole question\/assignment\/homework (nothing really new there), but you still fail to describe your own question related to it. This results in the question still being off-topic as it\u2019s still unclear what you are asking (and not the person who wrote the question\/assignment\/homework you are quoting). Again \u2013 What research have you done? What have you tried? Where exactly did you get stuck solving this? Please edit your question accordingly to pull this on-topic. One thing is clear: no one will do your work and solve the quoted assignment\/homework for you. \u2013\u00a0e-sushi Mar 10 '18 at 12:14\n\nYour questions were directly copied form a text book. This is not a homework solving service. The following hints should help you understand the questions. What have you tried? What do you not understand about these questions?\n\na. Assume the counter is always zero (i.e., the counter does not change from block to block). Can you come up with a successful attack against this mode of encryption?\n\nCTR with a fixed counter is worse than ECB. Penguins anyone?\n\nb. Assume now the nonce is fixed (i.e., the nonce does not change from message to message). Can you come up with a successful attack against this mode of encryption?\n\nKey-nonce reuse is critical. Especially with a stream cipher like AES-CTR. What happens when you use a one-time pad twice?\n\nc. What is the maximum length of the messages that can be encrypted using this mode?\n\nHow large is a single block?\n\nHow many blocks can a block cipher in counter mode produce?\n\nd. In the counter mode of encryption, the nonce cannot be used again unless a new block cipher key is chosen. What is the maximum number of messages that can be encrypted using the same key?\n\nHow many different nonces are there?\n\n\u2022 \u2018CTR\u2019 with a fixed counter is not ~ECB\u2014it's much worse than that. $\\operatorname{AES-ECB}_k(a \\mathbin\\Vert b) = \\operatorname{AES}_k(a) \\mathbin\\Vert \\operatorname{AES}_k(b)$, but $\\operatorname{AES-CTRLOLWUPS}_k(a \\mathbin\\Vert b) = (a \\oplus p) \\mathbin\\Vert (b \\oplus p)$, where $p = \\operatorname{AES}_k(0)$. \u2013\u00a0Squeamish Ossifrage Mar 9 '18 at 20:32\n\u2022 @SqueamishOssifrage Er yes of course. \u2013\u00a0cypherfox Mar 9 '18 at 22:43\n\u2022 I think c is the maximum message size of 2^32 blocks but I am not sure \u2013\u00a0AFB Mar 10 '18 at 6:43\n\u2022 about d .. the number of messages encrypted under a single key is small. but I can not specify a number??? \u2013\u00a0AFB Mar 10 '18 at 6:44\n\u2022 Please revise your question with respect to e-sushi's comment on the question. \u2013\u00a0cypherfox Mar 10 '18 at 7:05","date":"2019-12-12 19:54:20","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4208601117134094, \"perplexity\": 784.921102418433}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575540545146.75\/warc\/CC-MAIN-20191212181310-20191212205310-00214.warc.gz\"}"} | null | null |
Q: average proportions over time Let us say I have a number of events per week over some time periods. During each week a proportion of these events are important (important_events_per_week/number_of_events_per_week). The number_of_events_per_week also vary over time. How does one calculate the average important number of events per week over time please? I would think it is too naive to simply calculate the proportion per week and than take the arithmetic mean, so I thought I ask proper mathematicians? Should one use a weighted average to account for the varying number of events? Thanks!
A: Maybe the number $X$ of events per week has $X \sim \mathsf{Pois}(\lambda = 10),$ so that on average there are ten events a week. If $1/5$ of these events are "important," then the number $Y$ of important events per week has
$Y \sim \mathsf{Pois}(\lambda_I = \lambda/5 = 2),$ so that there are two important events in a week on average.
Then $P(Y = k) = e^{-2}2^k/k!$, for $k = 0, 1, 2, \dots.$
In particular $P(Y = 5) = 0.0361$ and $P(Y \le 3) = 0.8571,$ (rounded to four places) can be
found from the PDF formula above or in R, as follows:
dpois(5, 2)
[1] 0.03608941
exp(-2)*2^5/factorial(5)
[1] 0.03608941
sum(dpois(0:3, 2))
[1] 0.8571235
ppois(3, 2)
[1] 0.8571235
In R a Poisson PDF is denoted dpois and a Poisson CDF
is denoted ppois, each with appropriate parameters.
If you don't know the average number of important events per week, you might look at numbers of
important events over the last 52 weeks.
y
[1] 2 3 3 2 3 2 0 2 0 4
[11] 0 1 2 1 3 1 2 3 2 1
[21] 2 2 3 1 4 0 1 4 2 3
[31] 3 0 2 1 1 1 1 3 1 1
[41] 1 3 2 1 3 1 0 2 2 1
[51] 2 2
summary(y)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.000 1.000 2.000 1.788 3.000 4.000
So a reasonable estimate of $\lambda_i$ as $\hat \lambda_I = 1.788.$
Here is one way to find a 95% confidence interval (CI) $(1.46, 2.19)$ for $\lambda_I,$ based on the total number of important events in a year:
sum(y)
[1] 93
CI.52 = 93 + 2 + qnorm(c(.025,.975))*sqrt(93+1)
CI.52/52
[1] 1.461489 2.192357
That's $(93+2 \pm 1.96\sqrt{93+1})/52.$
Note: My 'data' above were sampled from
$\mathsf{Pois}(2)$ as shown below. In a real
application you wouldn't know the exact true
value of $\lambda_I.$
set.seed(315)
y = rpois(52, 2)
A: If you want the average proportion of the number of interesting events to the number of total events over some period of multiple weeks,
one simple method is to add up the total number of events during that period
(which you can do because you have the total number of events in each week of the period),
then add up the total number of interesting events during that period
(which you can do because you have the total number of interesting events in each week of the period).
Finally, divide the total number of interesting events by the total number of events.
This is how many such averages are computed in real life.
For example, to find the average speed for a trip of $20$ miles, you don't take the speed on each mile and do some kind of fancy weighted average of those $20$ speeds to find the average speed, you just take the total trip time and divide by the total distance (which is $20$ miles).
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A Nuclear Plant Braces for Impact With Hurricane Florence
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In 2014, a storm sent 50,000 gallons of water flooding into the St. Lucie nuclear power plant, on the Florida coast, damaging critical emergency equipment. Flooding is a major concern for nuclear plants sitting in the path of Hurricane Florence.
Alamy
On March 11, 2011, a one-two, earthquake-tsunami punch knocked out the safety systems at the Fukushima Dai-ichi nuclear power plant, triggering an explosion of hydrogen gas and meltdowns in three of its six reactors—the world's worst nuclear disaster since Chernobyl. Fukushima's facility was built with 1960s technology, designed at a time when engineers underestimated plant vulnerabilities during natural disasters. In the US, 20 plants with similar designs are currently operating.
One of them is slated for a head-on collision with Hurricane Florence.
Duke Energy Corp's dual-reactor, 1,870-megawatt Brunswick plant sits four miles inland from Cape Fear, a pointy headland jutting out into the Atlantic Ocean just south of the city of Wilmington, North Carolina. Brunswick has survived decades of run-ins with hurricanes, but Florence could be its biggest test yet. The plant perches near the banks of the Cape Fear River, which drains 9,000 square miles of the state's most densely populated regions. Like Hurricane Harvey in 2017, Florence is predicted to stall out for days, pounding the Carolinas with unrelenting amounts of water, leading to life-threatening storm surges and catastrophic flooding. NOAA's National Hurricane Center is projecting 110 mile-per-hour winds, waves as high as 13 feet, and in some places, up to 40 inches of rain.
https://twitter.com/NHC_Atlantic/status/1040262760607703040
Officials at Brunswick say the plant is bracing for the impending destruction. "We're monitoring the meteorological conditions, and if we have certainty that the winds onsite will reach 73 miles per hour, then we'll begin an orderly shutdown of the units," said Karen Williams, a spokeswoman for Duke Energy, reached by phone Wednesday afternoon.
The company also brought in workers ahead of the storm's landfall who will stay through its duration, sleeping on cots and blow-up mattresses, so that the facility has enough staff to handle multiple shifts. In the last few days they've been doing walk-throughs of the plant, inspecting diesel-powered backup generators and installing waterproof steel barriers on nine doors that house important safety equipment.
These precautions are relatively new for Brunswick. They're part of a sweep of changes nuclear plants around the US have adopted post-Fukushima.
Following the accident in Japan, a task force of senior Nuclear Regulatory Commission staff used the lessons from that disaster to draft new rules for the US. When the earthquake's tremors hit Fukushima, knocking out the electrical grid, the plant's emergency diesel generators kicked in as expected to provide emergency power. It was the wave of water that hit 40 minutes later that damaged that backup equipment, plunging the plant into total blackout. Without power, operators lost the ability to pump water into the reactors, exposing the cores, and leading to the explosive meltdown. From this, the NRC's big initiative to make US nuclear plants better prepared for such extreme events included the particular goal of making them less vulnerable to flooding.
"Every plant in the country was required to re-examine potential flooding hazards from any source—be it storm surge, intense rainfall, river flooding—with up-to-date models," says Scott Burnell, a public affairs officer for the NRC. The Commission then compared the results of those reports to the plants' flood protection features.
Duke predicted a maximum storm surge of 7 feet at the plant's safety-related buildings. But the plant was originally designed to cope with only 3.6 feet of expected surge, according to the NRC's 2017 summary assessment of Duke's hazard reevaluation report, which has not been made public.
In a letter earlier this year, the NRC reminded Duke that the plant's current design falls short of the reevaluated flood risks. According to Burnell, Duke has since submitted an assessment of how it will cope—including the use of those steel door reinforcements—which the NRC is still evaluating. "The review is not complete but there's nothing in there to this point that causes us any concern," says Burnell.
Duke's Williams echoed the sentiment, saying that the company doesn't expect any flooding damage at Brunswick, which sits 20 feet above sea level. "Our plant is designed to handle any kind of natural event, including a hurricane," she said.
Storms can be unpredictable, however. Dave Lochbaum, who directs a nuclear safety watchdog group at the Union of Concerned Scientists, has spent a lifetime studying nuclear failures. Brunswick troubles him because in 2012, Duke found hundreds of missing or damaged flood protections at the plant, such as cracked seals and corroded pipes. According to the group, none of the NRC's subsequent reports have mentioned repairs. "Hopefully they've been fixed," says Lochbaum. "But we've not been able to confirm that with the available documentation."
He credits Brunswick for following through on the NRC's post-Fukushima orders to install additional equipment—pumps, generators, hoses, cables, battery-powered sensors—to maintain safe levels of cooling in the event the plant loses its connection to the grid and use of its emergency diesel generators. But Lochbaum points out that history proves such preparation might not be enough.
In its 2012 post-Fukushima review, Florida Power & Light told the NRC that flood protections at its St. Lucie plant on South Hutchinson Island were adequate, despite failing to discover six electrical conduits with missing seals in one of the emergency core cooling systems. Two years later, a freak storm inundated Florida's central coast with record rainfall, flooding one of the plant's reactors with 50,000 gallons of stormwater. The deluge submerged core cooling pumps, rendering them useless. Had the reactor faltered during the storm, the plant would not have been able to maintain a safe and stable status beyond 24 hours, according to an NRC notice of violation issued to FPL after the incident.
Something similarly freakish happened at Entergy's Arkansas Nuclear One plant in March 2013. Workers were transporting a 525-ton generator during a maintenance outage when the rigging collapsed, sending it crashing through the floor, rupturing a fire main. Emergency systems began pumping water into the facility, causing flooding and damage to electrical components shared by both reactors.
"I'm not projecting that Florence is going to cause the next St. Lucie, or Arkansas," says Lochbaum. But those incidents serve as a reminder that nuclear plants are vulnerable to extreme events, like superstorms. "The only two times we've been challenged by floods since Fukushima we've come up short-handed," he says. "Both those plants thought they were ready, until they weren't."
Duke is also preparing five other nuclear plants in the projected impact area of the 400-mile-wide hurricane. The good news is that local residents have had ample warning. More than 1.5 million residents across North and South Carolina have been ordered to evacuate their homes before the eye of the storm makes landfall later today.
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The Simple Idea Behind Einstein's Greatest Discoveries | {
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\section{Introduction}
\label{sec:introduction}
Software development generally begins with the specification of the functional and non-functional requirements of the software. Both the functional and non-functional requirements are listed in the requirements specification of software. The inter-dependency of the functional and non-functional requirements makes both critical for software quality. The non-functional requirements are specified as quality attributes (QAs), such as performance, security, and reliability. The functional requirements model the functional correctness of a software, while the non-functional requirements capture the degree to which a software achieves the intended functionality \cite{chung2012non}.
QAs have been termed as reusable architectural building blocks used in developing an application architecture \cite{kim2009quality}. However, in order to embed the QAs in software design, one implements various software tactics. For instance, \emph{authentication} tactic is incorporated into a software design to attain \emph{security}; \emph{queues} are incorporated to attain better \emph{performance}, and so on. The design choices made by software developers among various architectural patterns and their tactics determine the software quality.
Novice software architects often lack the knowledge and expertise of using various software tactics to embed different software patterns in an architecture. Further, traditional teaching methods do not provide much opportunity to teach the design essentials and their effective implementation to the novice software architects \cite{capilla2020teaching,pinto2017training}. Thus, it becomes challenging to make adequate design decisions in selecting the appropriate design alternatives while developing a software. However, while performing a software architecture evaluation (SoftArchEval), it becomes essential to validate if a particular software design fulfills the required QAs and meet the necessary quality standards.
\subsection{Motivation}
Broadly, the prime objective of our work is to develop a framework that:
\begin{enumerate}
\item Given an input architectural design image, determines and lists the QAs it meets or lacks, and
\item Recommends the relevant software tactics to achieve the lacking QAs.
\end{enumerate}
For instance, the pipe and filter architectural pattern distributes a task's processing into various independent sub-tasks and is thus primarily used to attain performance, availability, and reliability. Timely detection of software quality by analyzing its architectural design can lower the overall defect-fixing cost \cite{kapur2020defect}. Some of the critical questions in this context could be:
\begin{enumerate}
\item What are the significant QAs and tactics associated with architectural design?
\item How to evaluate the architectural designs using QAs?
\item What are the necessary architectural tactics to achieve various QAs?
\item Is there any automated method to detect the standard architectural patterns present in an architectural design?
\item Given an input architectural design image, is it possible to detect the QAs it captures or lacks in and the necessary tactics that can be implemented to achieve the lacking QAs?
\end{enumerate}
This paper addresses these questions via an in-depth examination of QAs associated with various architectural design patterns and the necessary tactics to implement them.
\subsection{Broad Idea of our work}
\begin{figure}
\centerline{\includegraphics[scale=0.8]{figures/broadIdeaTactics.png}}
\caption{Broad idea of the proposed approach}
\vspace{-0.56cm}
\label{fig:basic-idea}
\end{figure}
Fig. \ref{fig:basic-idea} illustrates the central idea underlying our approach:
\begin{enumerate}
\item Every software has an architectural design, which comprises one or more patterns.
\item Similar software designs can be identified using the existing Image Similarity detection techniques.
\item Every architectural pattern may incorporates several software tactics. For instance, the \emph{pipe-and-filter pattern} incorporates \emph{fault detection,} \emph{recovery,} and \emph{performance} tactics by various design concepts, such as ping (or echo), active redundancy, shadowing, spare, and so on.
\item Successful execution of software tactics leads to the fulfillment of QAs. For instance, the successful execution of recovery and performance tactics leads to the \emph{availability,} \emph{reliability,} and \emph{maintainability} of QAs.
\item Software quality is defined as the degree to which a system, component, or process meets the specified requirements \cite{ieee1990ieee}.
\item Software QAs metrics can be used to measure software quality.
\end{enumerate}
\pagebreak
\section{Related Work}
\label{sec:related-work}
\subsection{Broad Categorization of SoftArchEval methods}
\label{sec:existing-soft-eval-methods}
A broad categorization of the existing evaluation methods \cite{abowd1997recommended,dobrica2002survey} is as follows:
\begin{enumerate}
\item \emph{Questioning methods} comprise asking qualitative questions on an architecture derived from the software quality aspect. These can be used to evaluate the quality of any given architecture and are classified as follows:
\begin{enumerate}
\item \emph{Scenario: } It is a sequence of steps involving the use or modification of the system. It provides a means to characterize how well a particular architecture responds to the demands placed on it by those scenarios.
\item \emph{Questionnaire:} It is a list of general and relatively open questions applicable to all architectures. The questions generally cover various topics, such as the architectural generation method or the architectural description details.
\item \emph{Checklist:} It is a detailed list of questions developed after evaluating a domain-specific set of systems.
\end{enumerate}
\item \emph{Measuring methods} comprise quantitative measurements made on the architecture by addressing specific QAs, and have the following salient types:
\begin{enumerate}
\item \emph{Experience-based methods:} These are based on the experiential knowledge of the experts \cite{bosch2000design}.
\item \emph{Metrics-based methods:} Metrics are termed as the quantitative interpretations of some observable measurements on architectural elements, such as the fan-in/ fan-out of various components. The metrics-based evaluations focus on the metrics values extracted from various architectural components while considering the assumptions involved in such metrics \cite{selby1995interconnectivity}.
\item \emph{Mathematical modeling-based methods:} These comprise mathematical proofs and methods for evaluating the operational QAs, such as the performance and reliability of architectural components \cite{reussner2003reliability}.
\item \emph{Simulations, prototypes, and experiment-based methods:} Simulations, prototypes, and experimental results are often a part of the software development process and play an important role in answering various questions during software reviews. For instance, simulation results can be used to validate specific assertions. However, this tends to be an expensive approach if a prototype is specifically developed to perform the evaluation \cite{bengtsson1998scenario}.
\end{enumerate}
\end{enumerate}
\subsection{Generic SoftArchEval works}
The \emph{scenario-based Architectural Analysis method (SAAM)} \cite{kazman1994saam} was introduced in 1993 to describe and analyze the software architecture based on various QAs. It was stated that software architectural analysis could help detect software defects in the early phases of software development, reducing the overall cost. With the emphasis on different QAs, different SAAM versions were developed. For instance, SAAM majorly focused on modifiability, while \emph{SAAM-founded on Complex Scenarios (SAAMCS)} \cite{lassing1999software} which was an extension of SAAM emphasized on flexibility. Similarly, \emph{Extending-SAAM by Integration in the domain (ESAAMI)} \cite{molter1999integrating} is an improved version of SAAM, which combines analytical and reuse concepts and integrates the SAAM in the domain-specific and reuse-based development process, and \emph{Software Architectural Analysis Method for Evolution and Re-usability (SAAMER)} \cite{lung1997approach} is an extension of SAAM emphasizing \emph{evolution} and \emph{reusability}.
\emph{The Architectural Trade-off Analysis Method (ATAM)}\cite{kazman1998architecture} provides a framework for evaluating software architecture concerning multiple QAs, particularly modifiability, performance, availability, and security. ATAM introduces the notion of tradeoff among multiple QAs, given a software architectural description. ATAM requires a software architectural description based on Kruchten's \enquote{$4+1$} views \cite{kruchten19954+} and requires several views, viz., a dynamic view, a system view, and a source view.
\emph{Quality-Driven Architecture Derivation and Improvement (QuaDAI)} \cite{gonzalez2013defining} is a metrics-based method used for derivation, evaluation, and improvement of software product architectures obtained in Software Product Line (SPL) development processes. QuaDAI performs the evaluations based on a) SPL viewpoints: functional, variability, quality, transformation, and b) and a process consisting of a set of activities conducted by model transformations to allow the automatic derivation, evaluation, and improvement of a product architecture from the SPL architecture.
\subsection{Training novice software architects}
Some recent works have felt the need of devising better methods for training software engineers to improve their decision making process \cite{pinto2019training}. Automated solutions for software design and development can help improve the decision making process of novice software architects or undergrad students \cite{kapur2019towards,7332519,kapur2020defect,kapur2021using}. However, the existing studies have developed solutions to facilitate the collaborative decision making process and study the effect on cognitive and modelling tasks \cite{capilla2020teaching}. In some of the existing studies proposing software solutions for training students, it has been reported to result in a positive boost of student's confidence and the improvement of their technical skills\cite{pinto2017training}.
\subsection{Recovering Architectural information from design diagrams}
The Image Extractor for Architectural Views (IMEAV) \cite{maggiori2014towards} is proposed to extract the architectural views from design images. However IMEAV is only applicable to Unified Modeling Language (UML) design diagrams, and cannot be used to detect architectural patterns in other type of design images. Similarly, most of the other existing studies are focused on the detection of images based on photographic scenes (based on image depths) \cite{jin2021image}, Synthetic Aperture Radar (SAR) images (based on Intensities) \cite{ye2017robust}, or on high-speed tracking by detection \cite{bochinski2017high}. However, the applications of these works and the different image types considered differ considerably from architectural design diagrams. However, it has been validated by some of the existing studies that SIFT when tuned effectively, acts as an effective image detection technique \cite{jin2021image}.
\textbf{Limitations of the existing works:}
\begin{enumerate}
\item Most of the existing SoftArchEval works consider a subset of QAs for software evaluations.
\item The complexity in the existing SoftArchEval methods limit their use. Also, there is an overhead of learning the evaluation methods and training the novice software architect to use them.
\item Almost all existing methods require inputs from stakeholders, software architects, and various experts. The dependency comes with an additional cost and limits the scope of knowledge to the involved participants' experience.
\item Most of the image datasets are based on photographic scenes or non-architectural design images.
\item To the best of our knowledge, none of the existing works recommend the necessary software tactics to achieve various QAs. Knowledge of the essential software tactics used for specific QAs is unexplored in the existing works.
\end{enumerate}
To overcome the limitations listed above, we propose a machine-learning-based assisting framework for software architectural evaluation that leverages the knowledge present in architectural images and the relationship between various architectural patterns and QAs to perform the evaluations. Using the relationship between various software tactics and QAs, our framework also recommends the necessary tactics to achieve specific QAs. We have not come across any work exploring such research direction for architectural evaluation to the best of our knowledge.
\begin{figure}
\centerline{\includegraphics[scale=0.75]{figures/SoftwareEvalSystem.png}}
\caption{An overview of the proposed method}
\vspace{-0.57cm}
\label{fig:framework}
\end{figure}
\pagebreak
\section{Proposed Methodology}
\label{sec:proposed-method}
Fig. \ref{fig:framework} illustrates the essential stages involved in implementing our approach:
\begin{enumerate}
\item \textbf{Dataset development:} \label{step:image-dataset} We developed an Architectural Design Patterns dataset, named as ArchPatterns dataset, comprising 2,035 architectural design images from fourteen different architectural patterns viz., broker, layered, event-bus, pipe-and-filter, repository, microkernel, microservices, model-view controller, peer-to-peer, presentation abstraction controller, client-server, space-based, representational state transfer (REST), and publisher-subscriber. We considered more than 100 design images in each pattern category.
The images were collected from online sources such as official blogs and technical write-ups with ground truth. After collecting these images, we manually filtered them to remove the images not projecting the relevant design patterns or not of the desired quality. Ground truth for the remaining images is manually annotated for the training. Images are of three channels with RGB color coding of different resolutions and sizes.
We consider the QAs and sub-QAs listed by ISO/ IEC 25010 standard as a reference point for our study. The necessary tactics required to achieve these QAs were derived by reviewing some standard software architectural reference books \cite{bass2003software,bachmann2003deriving,scott2009realizing,kalra2018towards,bachmann2003deriving,kim2009quality,li2020understanding,osses2018exploration,bi2018architecture,mark2015software,trowbridge2004integration,fielding2000architectural,bachmann2007modifiability} and the information is stored in the form of structured tables available at \url{https://doi.org/10.6084/m9.figshare.14623005}.
Our ArchPatterns dataset is publicly shared at \url{https://doi.org/10.6084/m9.figshare.14156408}.
\item \textbf{Image Preprocessing and Enhancement:} To improve the quality of our images and remove noise, we preprocessed all the ArchPatterns dataset images and scaled them to same size. We considered the images of only a considerable resolution ($>=350*350$ pixels) to have an effective image detection and matching.
\item \textbf{Feature Extraction:} \label{step:feature-extraction} We extract the image features, software components, their interconnections, and component labels present in the ArchPatterns dataset. To speed-up the image matching (or lookup) task during the evaluation stage, we also store the similarity of an image when compared to all the rest present in the dataset. We store these image features as a structured dataset, named as \emph{ImageFeatures} dataset.
We used SIFT \cite{lowe1999object}, a scale-invariant interest point detector with corner properties at different scales for feature extraction. SIFT also devises a suitable feature descriptor to uniquely represent the interest points.
\item \textbf{Image matching and Similarity detection:} \label{step:image-matching} Every image has multiple and different numbers of interest points. Interest point descriptors across different architectural images are subjected to achieve a mutual similarity, and then the count of matching interest points is used as a similarity measure between the two architectural images. For instance, if the images A and B having $N_a$ and $N_b$ number of SIFT interest points have $N_{ab}$ number of highly correlated descriptors in common, then the match score is computed as:
\begin{equation}
\label{eq:1}
Score = 1 - \frac{N_{ab}}{min(N_a , N_b)}
\end{equation}
Each image in the database is matched with all other database images to obtain a dis-similarity score during testing. The images belonging to the same architectural pattern are expected to have a low score, whereas those belonging to different pattern classes have a high score value. The matching score of every pair of images is obtained, and the score list is sorted in increasing order of dis-similarity. Further, for a given query architectural image, the most similar architectural image in the database is determined, and the corresponding label is assigned to the query image. The suitability of SIFT lies in the fact that the method builds a scale-space pyramid and only chooses prominent feature points that appear in all scales. By this, it achieves scale-invariance, which is a much-needed property for the architecture images. The same is evident from the result of achieving a higher correct recognition rate (CRR), which is defined as the percentage of images, out of total images, for which the recognition is correct at Rank-1 retrieval. Suppose out of $n$ test images, $x$ images are found to be true matches at Rank-1, then:
\begin{equation}
\label{eq:CRR}
CRR= (x/n)\times 100
\end{equation}
\item \textbf{Software Architecture Design Evaluation: } \label{step:eval} When a software architect starts evaluating an architectural design image using our framework, the steps involved are:
\begin{enumerate}
\item \emph{Feature extraction} of the image as described in Step \ref{step:feature-extraction}.
\item Using the extracted features, the \emph{image matching and similarity detection} is performed to determine the top-similar match from the \emph{ImageFeatures} dataset. This step provides the necessary information about the most-likely architectural pattern prominent in the considered design image.
\item \emph{Architectural Evaluation:} The software architect can then conclude about the strengths and weaknesses of the design (in terms of QAs) using our knowledge-based provided in the form of tables (discussed in the Section \ref{sec:proposed-method}). The software architect can then work in the direction of improving the design by implementing the necessary tactics as listed by the tables.
\end{enumerate}
\end{enumerate}
\section{Experimental System}
The essential objective of our work lies in improving the decision making process of software architects by improving the understanding behind software architectural patterns, QAs, and tactics. We achieve our goal by providing a framework for supporting the architectural design decision making process where:
\begin{itemize}
\item The framework's \emph{image detection and matching module} helps the novice software architects in determining the software architectural patterns in a design, and
\item The framework's \emph{knowledge-base} present in the form of tables provides the necessary information of the software tactics required to embed various QAs in the design.
\end{itemize}
The research questions that guide this study are:
\begin{enumerate}
\item What is our framework's highest CRR in identifying the software architectural design patterns?
\item How does our framework impact the decision making process of the software architects?
\end{enumerate}
\vspace*{-1cm}
\begin{figure}[hbtp]
\centering
\includegraphics[scale=0.4]{figures/chart.pdf}
\vspace*{-0.5cm}
\caption{Effect of the number of classes on the CRR}
\vspace*{-0.5cm}
\label{fig:result_matching}
\end{figure}
\subsection{Experimental setting and results analysis for image matching}
Architectural images in our database are scaled to the size $100\times 100$. We followed the one-vs-rest matching strategy where each image in the database is matched with all the remaining images to obtain the matching score. Since we have 2,035 images in the database, we obtained 4133088 scores that contain 337958 genuine and 3795130 imposter matching scores. Our framework achieves a Correct Recognition Rate (CRR) of 17.51\% in identifying the architectural patterns when all the classes have been considered. However, CRR keeps increasing when fewer classes are considered and 75.52\% when architectural images of any two classes are compared. The highest CRR obtained with our framework is 98.71\%. The change in CRR value with a change in the number of classes is annotated and depicted using the graph shown in the Fig. \ref{fig:result_matching}. Here, the classes represent the architectural patterns (discussed in Section \ref{sec:proposed-method}) considered in this work.
\subsection{Experimental setting for the experiments with software architects}
As part of a post-graduate course on Software and Data Engineering, we identified 64 suitable participants for this study such that 30 participants were selected who were having IT industry exposure of more than one year and rest of the 34 participants were not having any industry exposure or less than one year experience. All the participants were taught IEC/ISO 25010 Software Quality Models and Design Patterns through online class lectures. To conduct the study, we divided each of this group in equal halves and formed controlled and experimental groups as shown in Table \ref{tab:groups}.
During the experiment, all the participants were provided the access to course lectures and teaching material on ISO 25010 and Design Patterns. However, for the experimental group, an additional access to the image-processing tool interface and the knowledge base was also provided. This was done to study the impact on decision making process due to our framework's support.
\vspace*{-1cm}
\begin{table*}
\caption{Groups and participants}
\label{tab:groups}
\begin{center}
\resizebox{0.7\textwidth}{!}{
\begin{tabular}{c|c|c}
\toprule
\multirow{2}{*}{\textbf{\shortstack{Participant Type\\ or Group}}}&\multicolumn{2}{c}{\textbf{Type of Experimental Setting}}\\
\cline{2-3}
& \textbf{Experimental}& \textbf{Controlled}\\
\midrule
\textbf{Without Industry Experience} & 17 & 17\\ \hline
\textbf{With Industry Experience} & 15 & 15\\
\bottomrule
\end{tabular}}
\end{center}
\end{table*}
\vspace*{-1cm}
\subsection{Evaluation Metrics}
The proposed system provides QA analysis for a given architectural diagram of a software system. To evaluate the proposed framework's effectiveness, we conducted experiments by sharing the generated QA analysis with experimental group participants and compared the performance with control group participants. As part of the experiment, we ask the following questions:
\begin{enumerate}
\item \emph{Accuracy:} How accurately experimental group participants performed in terms of QA analysis's correctness compared to the control group participants?
\item \emph{Time Performance:} Is there any improvement in the time taken to analyze a given software architectural diagram by the experimental group participants compared to control group participants?
\item \emph{Explainability:} How much appropriate reasoning/ explanation was provided for the experimental group's analysis compared to the control group participants?
\item \emph{Robustness: } How many different architectural patterns can be processed more accurately with the help of the proposed system by the experimental group compared to the control group? We selected architectural diagrams from 14 categories of architectural patterns. Our framework is extendable and can consider more categories given the corresponding data for architectural diagrams and descriptions.
\end{enumerate}
\begin{figure}
\centering
\includegraphics[width=1.1\columnwidth]{figures/PerformanceBroad}
\caption{Performance Metrics}
\label{fig:perf-metrics}
\end{figure}
\subsection{Procedure}
As part of our experiments, we distributed various architectural diagrams to the controlled group and the experimental group. Each of the participants was given 28 architectural diagrams, two from each of the 14 categories. We asked them to identify the significant architectural components, patterns, and overall system properties (quality attributes and tactics) perceived by them after analyzing the architectural diagrams without any additional documents. The participants were free to use any other information source available to them with appropriate citations. We manually evaluated all the assignment responses and scored them based on the following parameters for each architectural diagram to measure the accuracy and explainability:
\begin{enumerate}
\item The number of components and their characteristics correctly identified.
\item The number of connections and their characteristics correctly identified.
\item The number of design patterns correctly identified with justification.
\item The number of quality attributes correctly identified with justification.
\item The number of architectural tactics correctly identified with justification.
\end{enumerate}
We also asked each participant to share the time spent on the evaluation of each architectural image to measure the time performance.
\subsection{Results and Discussion}
Each of the box plot graphs shown in Fig. \ref{fig:perf-metrics} represents the distribution of actual values at a minimum value, 25 percentile, 75 percentile, and maximum values, respectively. The Groups of participants are annotated as:
\begin{itemize}
\item Controlled group with Industrial Experience: Cont\_With\_Ind
\item Controlled group without Industrial Experience: Cont\_WithO\_Ind
\item Experimental group with Industrial Experience: Exp\_With\_Ind
\item Experimental group without Industrial Experience:
Exp\_WithO\_Ind
\end{itemize}
\subsubsection{Accuracy and Explainability}
The responses of each candidate and inference outcome of the automated model were evaluated by teaching assistants manually for their accuracy.
\textbf{Salient Observations:}
\begin{enumerate}
\item The Controlled group without Industrial Experience, i.e., the group without Industrial Experience and without the help of our framework performs the worst, while the Experimental Group With Industrial Experience performs the best achieving an highest accuracy of 94.37\%.
\item As the accuracy of Exp\_With\_Ind group is more than the accuracy of Exp\_WithO\_Ind, and a similar trend is observed in case of controlled groups, it can be concluded that the accuracy of the group improves with the help of our framework.
\item The improvement in accuracy as observed in the category of the candidates Without Industrial experience, viz., Cont\_WithO\_Ind to Exp\_WithO\_Ind is huge (48.9\% on average). Therefore, our framework contributes significantly in improving the design decision process of the novice software architects.
\item The accuracy improvement in case of candidates with prior industrial experience, viz., Cont\_With\_Ind to Exp\_With\_Ind is 8.42\%, and thus is beneficial in improving the decision process of industrial experts as well.
\end{enumerate}
\textbf{Inference:} It is evident that the usage of our proposed model to evaluate the architectural designs reduces the variation in the results and, at the same time, increases the overall correctness of the responses. The explainability score indicates that the experimental group participants are performing better than the controlled group participants.
\subsubsection{Time Performance}
The results depicted in the time boxplots of the figure validate that the experimental group performed far better than the controlled group and took lesser time in evaluation the architectural designs (47.8\% improvement on average). Further, the use of the proposed approach helped speed up the evaluation process (37.82\% for novice architects and 62.2\% for experienced architects). Hence, it shows that using the proposed approach will help novice and experienced candidates to evaluate the architectural designs in lesser time as compared to when performed withour our framework's support. Also, the use of our framework leads to results with limited variation in the responses in terms of time, efforts, and accuracy.
\textbf{Critical Reasoning:} The time improvement is more in case of experienced architects, which might imply that using the proposed framework experts were able to evaluate the design diagrams in a considerable lesser time. The reduced variation the results leads to an increase in the certainty in effort and time estimation for the architectural evaluation tasks, and help in better resource and cost planning.
\subsubsection{Robustness}
We included the robustness parameter to avoid bias due to candidates' prior experience in the control and experimental groups. Candidates might be good at evaluating the architectural patterns that they have worked with in the past. This parameter analyzed whether a candidate could answer with higher accuracy across the different categories of architectural design images. To measure this parameter, we aggregated each candidate's score for each category and calculated the percentage of categories for which they have scored more than 80\%. As shown in the robustness boxplots, the experimental group shows a better hold on robustness than the controlled group (158.6\% average improvement). The largest improvement was observed from the Cont\_WithO\_Ind group to Exp\_WithO\_Ind of 1953.85\%, with the respective average robustness scores as 0.65 vs. 13.35.
\textbf{Inference:} A significant improvement in robustness scores for the novice architects signifies that the framework helps them to understand and evaluate the design diagrams from different architectural patterns effectively. It also in a way validates that the framework is not biased to support the decision making process of any specific architectural pattern type.
\section{Threats to validity}
Several threats affect the validity of this work. Firstly, the scope of our work is limited only to the considered set of architectural patterns. Our framework cannot detect the patterns not considered currently and thus does not comment about their QAs or tactics. However, this can be overcome by extending our study for the left-out architectural patterns by appending the dataset with appropriate images.
Secondly, the images have been acquired from the web, which calls for many challenges, such as size or resolution variation, illumination variation, blurriness, noise, and color in the images, affecting the overall process; because feature extraction is primarily dependent on the quality of the images. Therefore, manual filtering for removing the images not projecting the considered patterns and low quality is performed.
Further, the pre-processing of acquired images is done in order to enhance them for better feature extraction. Another threat could be the use of OCR in the mapping procedure. The images can have labels written in various font styles and sizes, or it could be the case that some images may contain handwritten labels. Many independent variables were used in these models, which could affect the accuracy of predictions. However, we used step-wise techniques to identify the optimum number of variables for the models and tested multi-col-linearity models. These risks were partially mitigated using a classic N-cross-fold validation to evaluate the models and demonstrated that they were generalizable.
Lastly, the experiments conducted with the controlled group and experimental group may have some limitations. We asked candidates to follow the guidelines as much as possible to ensure that equivalence among them in terms of resources available to them. Bigger group size should have been better to reduce noisy and biased responses from the results. We had to discard the three submissions from both the groups due to the lack of enough data for estimating time performance and robustness.
\section{Conclusion and Future Work}
An automated framework to evaluate a software architecture with image processing, and inference from the QA knowledge-base is quite helpful. It reduces the variation among responses and increases the accuracy across the multiple categories of architectural patterns. Further, the effectiveness of architecture image matching could be improved by exploring graph-based techniques on a larger image dataset that can connect vision components and determine combinatorial similarity. Possible improvements using OCR methods could be explored for better handling of design artifacts and handwritten annotations. These improvements will help increase the explainability of our framework and improve the overall model's robustness. Developing the framework in the form of a full-fledged automated application for community use is a part of our future work. We are currently working on the system prototype and are planning to add more architectural knowledge artifacts to increase the effectiveness of this approach and provide more advanced recommendations.
\bibliographystyle{splncs04}
| {
"redpajama_set_name": "RedPajamaArXiv"
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La Divisione Nazionale A, conocida con anterioridad como Serie A Dilettanti, Serie B d'Eccellenza y Serie B1, fue la tercera división del baloncesto en Italia hasta 2013, tras la Serie A y la Legadue. Estaba organizada por la desaparecida Lega Nazionale Pallacanestro. Actualmente la tercera división italiana es la Serie B.
Estaba formada por 24 equipos en lugar de los 32 de los que constaba hasta 2011. La competición se desarrollaba entre los meses de septiembre a mayo.
Formato de competición
El campeonato lo formaban 24 equipos, divididos en dos conferencias, y éstas a su vez en dos divisiones. Cada equipo se enfrentaba a doble vuelta con el resto de equipos de su conferencia, y a un único partido con los equipos de la otra, componiendo 34 jornadas de competición.
Accedían a los play-offs 12 equipos, los dos primeros de cada división y los cuatro (dos por conferencia) con más puntos acumulados. El descenso se lo jugaban el último clasificado de cada división y otros cuatro equipos, los dos peores de cada conferencia.
Palmarés
Enlaces externos
web oficial de la Serie A Dilettanti
Serie A Dilettanti en eurobasket.com
3
Ligas de baloncesto desaparecidas | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,279 |
A unique collection of Olympic symbols. The original symbol was designed for table tennis, with the concept of it's Victorian origins behind the design using the shape of the penny-farthing. The full set of pictograms depicting each sport, continue the original style. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,191 |
Дэвид Мартин Скотт Стил, барон Стил Эйквудский (; 31 марта 1938, Керколди) — британский политический деятель. C 1976 по 1988 год занимал пост председателя Либеральной партии. Также он являлся членом Палаты лордов в Великобритании. С 1999 по 2003 год он занимал пост спикера Шотландского парламента.
Биография
Дэвид Стил родился 31 марта 1938 года в Шотландии в городе Керколди. Его отец, также носивший имя Дэвид Стил, был служителем Шотландской церкви. Дэвид Стил получил своё образование в Школе Принца Уэльского, а также в колледже Джорджа Вашингтона в Эдинбурге. Ещё во время обучения в колледже Дэвид Стил стал председателем университетского либерального клуба. В 1964 году, в возрасте 25 лет он стал членом парламента. Являясь членом парламента, Дэвид Стил голосовал за принятие в 1967 году закона о легализации абортов в Великобритании.
В 1976 году Дэвид Стилл занял пост председателя Либеральной партии. На парламентских выборах 1983 года, руководимая им Либеральная партия вместе с Социал-демократической партией, выступила в составе либерально-социал-демократического альянса. В 1988 году по предложению Стила Либеральная партия объединилась с Социал-демократической партией в Партию либеральных демократов. Дэвид Стил являлся временным председателем Партии либеральных демократов, до тех пор пока не был избран лидером партии Пэдди Эшдаун. В 1990 году Дэвид Стил получил наследственное дворянское звание и Орден Британской империи. В 1994 году Дэвид Стил принял предложение участвовать в выборах в Европейский парламент. Хотя он не был избран, избирательная кампания прошла очень успешно. В этом же году он стал президентом Либерального интернационала — международной организации, объединяющей либеральные партии. В 1999 году Дэвид Стил был избран спикером парламента Шотландии.
Примечания
Ссылки
David Steel, Liberal Party (UK) Leader
Политики Шотландии
Члены Либеральной партии Великобритании
Члены Партии либеральных демократов Великобритании
Члены парламента Соединённого Королевства (1964—1966)
Члены парламента Соединённого Королевства (1966—1970)
Члены парламента Соединённого Королевства (1970—1974)
Члены парламента Соединённого королевства (1974)
Члены парламента Соединённого Королевства (1974—1979)
Члены парламента Соединённого Королевства (1979—1983)
Члены парламента Соединённого Королевства (1983—1987)
Члены парламента Соединённого Королевства (1987—1992)
Члены парламента Соединённого Королевства (1992—1997)
Пожизненные пэры | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,505 |
Criminalisation of sex work normalises violence, review finds
Sex workers three times more likely to experience violence from client where trade is criminalised, data shows
Badges promoting the decriminalisation of sex work from the Scarlet Alliance, the organisation for sex workers in Australia. Photograph: Michael Wickham/The Guardian
Sarah Boseley Health editor
Tue 11 Dec 2018 14.00 EST
Last modified on Tue 11 Dec 2018 15.50 EST
Sex workers are more likely to suffer poor health, violence and abuse in countries where their trade is criminalised, a major review has found.
The review, by researchers from the London School of Hygiene and Tropical Medicine, found that sex workers suffering repressive policing – including arrest, imprisonment and extortion by officers – were three times more likely to experience sexual or physical violence from a client and were twice as likely to have HIV or another sexually transmitted infection as those who lived in countries where sex work was tolerated.
Sex workers who fear that they, or their clients, may be picked up by the police are more likely to engage in risky encounters, unable to take the time to talk to a client before getting into a car or negotiate terms in advance, the researchers found.
Their health and safety were at risk not only in countries where sex work was criminalised, but also in Canada, which has introduced the "Nordic model" pioneered by Sweden, under which the client can be arrested for a criminal offence, but not the sex worker.
Published in the journal PLOS Medicine, the paper by Lucy Platt, associate professor in public health epidemiology, and Pippa Grenfell, assistant professor of public health sociology, is a review of data from 33 countries. They included comments from sex workers who took part in some of the studies.
Canada's 'anti-prostitution law' raises fears for sex workers' safety
Canada passed a law in 2014 to make it illegal to pay for sex, but some sex workers say that has made their lives more risky.
"They couldn't have designed a law better to make it less safe," said one sex worker. "It's like you have to hide out, you can't talk to a guy, and there's no discussion about what you're willing to do and for how much. The negotiation has to take place afterwards, which is always so much scarier. It's designed to set it up to be dangerous. I don't think it was the original intention, but that's what it does."
Another woman working on the streets in Canada said she was no longer able to talk through the car window to ensure they felt safe. "Because of being so cold and being harassed, I got into a car where I normally wouldn't have. The guy didn't look at my face right away. And I just hopped in cause I was cold and tired of standing out there. And you know, he put something to my throat. And I had to do it for nothing."
France, Iceland, Northern Ireland, Norway, the Republic of Ireland and Sweden also criminalise the client. Guatemala, Mexico, Turkey and the US state of Nevada have regulated sex work, which allows better conditions for some, but worse for the many who operate outside the regulated arrangements.
Criminalise the sex buyers, not the prostitutes | Catherine Bennett
A man in the UK said the ideal situation was working from shared premises, where everybody had companionship and greater security. But, although buying and selling sexual services is legal, that can fall foul of the law. "Because of the legal situation, you have to be very, very careful. Because obviously it's running a brothel, which has … really dangerous consequences these days," he said.
New Zealand is the only country to have decriminalised sex work, in 2003, although it is not legal for migrants. Sex workers said they were more able to refuse clients and insist on condom use, while relationships with police were better. "We always have police coming up and down the street every night," said one woman. "We'd even have them coming over to make sure that we were all right and making sure … that we've got minders and that they were taking registration plates and the identity of the clients. So … it changed the whole street, it's changed everything."
Grenfell said: "It is clear from our review that criminalisation of sex work normalises violence and reinforces gender, racial, economic and other inequalities. Decriminalisation of sex work is urgently needed, but other areas must also be addressed.
"Wider political action is required to tackle the inequalities, stigma and exclusion that sex workers face, not only within criminal justice systems but also in health, domestic violence, housing, welfare, employment, education and immigration sectors." | {
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Q: Test Successful with 'Cannot start PhantomJS' Error I'm get the strangest error with NodeJS v0.10.11 and karma tests via grunt.
Initially, the tests would not run as PhantomJS did not start. After updating NodeJS, I get this :
[2013-06-19 03:16:12.743] [DEBUG] config - autoWatch set to false, because of singleRun
INFO [karma]: Karma server started at http://localhost:8080/
INFO [launcher]: Starting browser PhantomJS
INFO [PhantomJS 1.9 (Linux)]: Connected on socket id KVDuQs5niBzzc8KQSnFm
PhantomJS 1.9 (Linux): Executed 1 of 1 SUCCESS (0.327 secs / 0.041 secs)
ERROR [launcher]: Cannot start PhantomJS
Why is there an error after the tests have run successfully ?
UPDATE: Adding code
Gruntfile karma task:
karma: {
unit: {
configFile: 'karma.conf.js',
singleRun: true
},
e2e: {
configFile: 'karma-e2e.conf.js',
singleRun: true
}
}
karma.conf.js
// Karma configuration
// base path, that will be used to resolve files and exclude
basePath = '';
// list of files / patterns to load in the browser
// removed the following because of warning 'test/mock/**/*.js',
files = [
JASMINE,
JASMINE_ADAPTER,
'app/components/angular/angular.js',
'app/components/angular-mocks/angular-mocks.js',
'app/scripts/*.js',
'app/scripts/**/*.js',
'test/spec/**/*.js'
];
// list of files to exclude
exclude = [];
// test results reporter to use
// possible values: dots || progress || growl
reporters = ['progress'];
// web server port
port = 8080;
// cli runner port
runnerPort = 9100;
// enable / disable colors in the output (reporters and logs)
colors = true;
// level of logging
// possible values: LOG_DISABLE || LOG_ERROR || LOG_WARN || LOG_INFO || LOG_DEBUG
logLevel = LOG_INFO;
// enable / disable watching file and executing tests whenever any file changes
autoWatch = false;
// Start these browsers, currently available:
// - Chrome
// - ChromeCanary
// - Firefox
// - Opera
// - Safari (only Mac)
// - PhantomJS
// - IE (only Windows)
browsers = ['PhantomJS'];
// If browser does not capture in given timeout [ms], kill it
captureTimeout = 5000;
// Continuous Integration mode
// if true, it capture browsers, run tests and exit
singleRun = false;
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Q: Is there any extension for Google Chrome that would let me use the find function (CTRL+F) over multiple tabs? Say I had two tabs open and I wanted to search for the term "my term", I would like to view the results from both tabs. I've looked at the chrome extension website ( https://chrome.google.com/webstore/search/find ) and I've searched on Googlebut couldn't find such an extension. It seems like an obvious extension so I assume it must be impossible due to Chrome's design (each tab being a separate process causing the problem maybe?).
Two questions:
1. Is there already an extension for finding text over multiple tabs?
2. Is it technically possible to build such an extension given Chrome's design?
A: There is a Chrome extension named "Search Plus"
It supports search over all tabs.
The UI also allows to search in only some selected tabs, instead of all, which can be quite useful.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,337 |
\section{Introduction}
Given $s\in(0,\infty)$, the Riesz $s$--energy of a set on points $\omega_{n} = \{ x_{1}, \ldots ,x_{n} \}$ on a subset $X\subseteq\mathbb{R}^{m}$ is
\begin{equation}\label{eq:Riesz}
\E{s}(\omega_{n}) = \displaystyle\sum_{i \neq j} \frac{1}{\| x_{i} - x_{j} \|^{s}}.
\end{equation}
This energy has a physical interpretation for some particular values of $s$, i.e. for $s=1$ the Riesz energy is the Coulomb potential and for $s=d-2$ ($d\geq 3$) is the Newtonian potential. In the special case $s=0$ the energy is defined by
\[
\E{0}(\omega_{n})=\left. \frac{d}{ds} \right|_{s=0}\E{s}(\omega_n) = \sum_{i \neq j} \log\| x_{i} - x_{j} \|^{-1}
\]
and is related to the transfinite diameter and the capacity of the set by classical potential theory, see for example \cite{doohovskoy2011foundations}.
The minimal value of this energy and its asymptotic behavior have been extensively studied, most remarkably in the case that $X=\mathbb{S}^{d}\subseteq\mathbb{R}^{d+1}$ is the $d$--dimensional unit sphere.
In \cite{10.2307/117605} it was proved that for $d > 2$ and $0<s<d$ there exist constants $c>C> 0$ (depending only on $d$ and $s$) such that
\begin{equation}\label{eq:bounds}
-cn^{1 + \frac{s}{d}}
\leq
\min_{\omega_n}\left(\E{s}(\omega_{n})\right) - V_{s}(\mathbb{S}^{d})n^{2}
\leq
-Cn^{1 + \frac{s}{d}},
\end{equation}
where $V_{s}(\mathbb{S}^{d})$ is the continuous s-energy for the normalized Lebesgue measure,
\begin{equation}
V_{s}(\mathbb{S}^{d})
=
\frac{1}{Vol(\mathbb{S}^{d})^{2}} \displaystyle\int_{p,q \in \mathbb{S}^{d}} \frac{1}{\|p-q\|^{s}}dpdq
=
2^{d-s-1} \frac{\Gamma \left( \frac{d+1}{2} \right) \Gamma \left( \frac{d-s}{2} \right)}{\sqrt{\pi} \Gamma \left( d- \frac{s}{2} \right)}.
\end{equation}
Finding the precise value of the constants in \eqref{eq:bounds} is an important open problem and has been addressed by several authors, see \cites{ BHS2012b, Sandi, LB2015, MR1306011} for some very precise conjectures and \cite{Brauchart2015293} or \cite{BHSlibro} for surveys. One can post the problem as follows
\begin{problem}
For $s\in(0,d)$, let $C_{s,d,n}$ be defined by
\[
0<C_{s,d,n}=\frac{ V_{s}(\mathbb{S}^{d})n^{2} -\min_{\omega_n}\left(\E{s}(\omega_{n})\right)}{n^{1+\frac{s}{d}}}.
\]
Find asymptotic values for $C_{s,d,n}$ as $n\to\infty$. In particular, prove if the limit exists.
\end{problem}
A sometimes successful strategy for the upper bound in the constant $C_{s,d,n}$ is to take collections of random points in $\S{d}$ and then compute the expected value of the energy (which is of course greater than or equal to the minimum possible value). Simply taking $n$ points with the uniform distribution in $\S{d}$ already gives the correct term $V_{s}(\mathbb{S}^{d})n^{2}$, and other distributions with nice separation properties have proved successful in bounding the constant $C_{s,d,n}$.
We are thus interested in computationally feasible random procedures to generate points in sets which exhibit local repulsion. One natural choice is using {\em determinantal point processes} which have these two properties (see \cite{Hough_zerosof} for theoretical properties and \cite{PhysRevE.79.041108} for an implementation). A brief summary of the fundamental properties of determinantal point processes is given in Section \ref{sec:DPP}.
In a recent paper \cite{EJP3733} a determinantal point process named the {\em spherical ensemble} is used to produce low--energy random configurations on $\S{2}$. This process was previously studied by Krishnapur \cite{krishnapur2009} who proved a remarkable fact: the spherical ensemble is equivalent to taking eigenvalues of $A^{-1}B$ (where $A,B$ have Gaussian entries) and sending them to the sphere through the stereographic projection.
In \cite{BMOC2015energy} a different determinantal point process rooted on the use of spherical harmonics is described, producing low--energy random configurations in $\S{d}$ for some infinite sequence of values of $n$. In particular, it is proved in that paper that
\begin{equation}\label{eq:BMOCliminf}
\limsup_{n\to\infty} C_{s,d,n}\geq 2^{s-s/d}V_s(\S{d})
\frac{ d\,\Gamma\left( 1+\frac{d}{2} \right) \Gamma\left( \frac{1+s}{2}
\right)\Gamma\left(d-\frac{s}{2}\right) }
{ \sqrt{\pi} \Gamma\left( 1+\frac{s}{2} \right) \Gamma\left( 1+\frac{s+d}{2}
\right)\left(d!\right)^{1-\frac{s}{d}}},\quad 0<s<d.
\end{equation}
If $d-1<s<d$, $\limsup$ can be changed to $\liminf$ in \eqref{eq:BMOCliminf} (see \cite[Cor. 2]{BMOC2015energy}).
The bound in \eqref{eq:BMOCliminf} is the best known to the date for general $d$ (although more precise bounds exist for particular values of $d$ including $d=1,2$, see \cites{BLMS:BLMS0621, LB2015}). In particular, for $s=2$ and odd dimensions the formula in \eqref{eq:BMOCliminf} reads
\begin{equation}\label{eq:BMOCliminf2}
\limsup_{n\to\infty} C_{2,2d+1,n}\geq \frac{2^{1-\frac{2}{2d+1}}\left({(2d+1)!}\right)^\frac{2}{2d+1}}{(2d-1)(2d+3)}\stackrel{d\to\infty}{\to}\frac{2}{e^2}.
\end{equation}
The determinantal point process in \cite{BMOC2015energy} is called the {\em harmonic ensemble} and it is shown to be the optimal one (at least for $s=2$) among a certain class of determinantal point processes obtained from subspaces of functions with real values defined in $\S{d}$.
However, the bound in \cite{BMOC2015energy} for the case $d=2$ is worse than that of \cite{EJP3733}, which is not surprising since the spherical ensemble uses complex functions and is thus of a different nature.
An alternative natural interpretation of Krishnapur's result is to consider eigenvalues $(\alpha,\beta)\in\mathbb P(\mathbb{C}^2)$ of the generalized eigenvalue problem $\det(\beta A-\alpha B)=0$ and to identify $\mathbb P(\mathbb{C}^2)$ with the Riemann sphere. An homotety then generates the points in the unit sphere $\S{2}$. This remark suggests that the spherical ensemble can be seen as a natural point process in the complex projective space, and a search for an extension to higher dimensions is in order. In this paper we extend this process in a very natural manner to $\mathbb{P}(\mathbb{C}^{d+1})$ for any $d\geq2$. We will propose the name {\em projective ensemble}.
In order to show the separation properties of the projective ensemble we will define a (probably non--determinantal) point process in odd--dimensional spheres, which will allow us to compare our results to those of \cite{BMOC2015energy}. This point process is as follows: first, choose a number $r$ of points in $\mathbb{P}(\mathbb{C}^{d+1})$ coming from the projective ensemble. Then, consider $k$ equally spaced unit norm affine representatives of each of the projective points. We allow these points to be rotated by a randomly chosen phase. As a result, we get $rk$ points in the odd--dimensional sphere $\mathbb{S}(\mathbb{C}^{d+1})\equiv\mathbb{S}^{2d+1}$.
We study the expected $2$--energy of such a point process. Our first main result can be succinctly written as follows.
\begin{thm}\label{th:main}
With the notations above,
\begin{equation}\label{eq:nuestracota}
\begin{split}
&\limsup_{n\to\infty} C_{2,2d+1,n}
\geq
\frac{3^{1 - \frac{2}{2d+1}} (2d-1)^{1 - \frac{2}{2d+1}} (2d+1) \Gamma\left( d- \frac{1}{2} \right)^{2 - \frac{2}{2d+1}}}{2^{4 - \frac{2}{2d+1}} (d!)^{2 - \frac{4}{2d+1}}}\stackrel{d\to\infty}{\to}\frac{3}{4e}.
\end{split}
\end{equation}
\end{thm}
The bound in Theorem \ref{th:main} is larger than that of \eqref{eq:BMOCliminf2}, which shows that random configurations of points coming from this point process are, at least from the point of view of the $2$--energy, better distributed than those coming from the harmonic ensemble. See Figure \ref{fig:comparacion} for a graphical comparison of both bounds.
\begin{figure}[htp]
\centering
\includegraphics[width=0.9\textwidth]{comparison.jpg}
\caption{Comparison of the values of the constants in \eqref{eq:nuestracota} (blue solid line) and \eqref{eq:BMOCliminf2} (red dashed line).} \label{fig:comparacion}
\end{figure}
Since the point process we have defined in $\mathbb{S}^{2d+1}$ starts by choosing points in $\mathbb{P}(\mathbb{C}^{d+1})$ coming from the projective ensemble, Theorem \ref{th:main} gives us arguments to think that the projective ensemble produces quite well distributed points in $\mathbb{P}(\mathbb{C}^{d+1})$ (for $d=1$ this property is quantitatively described in \cite{EJP3733}). There are several ways to measure how well distributed a collection of points is in $\mathbb{P}(\mathbb{C}^{d+1})$. For example, one can study the natural analogues of Riesz's energy as in Theorem \ref{Prop_proy} below. A very natural measure is given by the value of {\em Green's energy} of \cite{Juan}: let $G:\mathbb{P}(\mathbb{C}^{d+1})\times\mathbb{P}(\mathbb{C}^{d+1})\to[0,\infty]$ be the Green function of $\mathbb{P}(\mathbb{C}^{d+1})$, that is, $G(x,\cdot)$ is zero--mean for all $x$, $G$ is symmetric and $\Delta_yG(x,y)=\delta_x(y)-vol(\mathbb{P}(\mathbb{C}^{d+1}))^{-1}$, with $\delta_x$ the Dirac's delta function, in the distributional sense. The Green energy of a collection of $r$ points $\omega_r=(x_1,\ldots,x_r)\in\mathbb{P}(\mathbb{C}^{d+1})^r$ is defined as
\[
\E{G}(\omega_r)=\sum_{i\neq j}G(x_i,x_j).
\]
Minimizers of Green's energy are assymptotically well--distributed (see \cite[Main Theorem]{Juan}). Our second main result will follow from the computation of the expected value of Green's energy for the projective ensemble.
\begin{thm}\label{th:main2}
Let $d\geq 2$. Then,
\begin{equation}\label{eq:final}
\liminf_{r\to\infty}\frac{\min_{\omega_r}\left(\E{G}^{\mathbb{P}}(\omega_{r})\right)}{r^{2 - \frac{1}{d}}}\leq-\frac{ (d!)^{1 - \frac{1}{d}}}{4\pi^{d} (d-1)}.
\end{equation}
\end{thm}
Theorem \ref{th:main2} gives a criterium to decide how well--distributed a collection of projective points is: compute their Green's energy and compare to that of \eqref{eq:final}.
\section{Determinantal point processes}\label{sec:DPP}
\subsection{Basic notions}
In this section we follow \cite{Hough_zerosof}.
\begin{defn}\label{def:pp}
Let $\Lambda$ be a locally compact, polish topological space with a Radon measure $\mu$. A \textit{simple point process} $\mathfrak{X}$ of $n$ points in $\Lambda$ is a random variable taking values in the space of $n$ point subsets of $\Lambda$.
\end{defn}
There are some subtle issues in the general definition of point processes, see \cite[Section 1.2]{Hough_zerosof}. For our purposes we will only use simple point processes with a fixed, finite number of points.
\noindent For some point processes there exist \textit{joint intensities} satisfying the following definition.
\begin{defn}
Let $\Lambda,\mathfrak{X}$ be as in Definition \ref{def:pp}. The joint intensities are functions (if any exist) $\rho_{k}:\Lambda^k\rightarrow[0,\infty)$, $k\geq1$ such that for any family of mutually disjoint subsets $D_1,\ldots,D_k$ of $\Lambda$ we have
\[
\Esp{x\sim\mathfrak{X}}{\left(\prod_{i=1}^k \sharp(x\cap D_i)\right)}=\int_{\prod D_i}\rho_k(x_1,\ldots,x_k)\,d\mu(x_1,\ldots,x_k).
\]
Here, $\mathrm{E}$ denotes expectation and by $x\sim\mathfrak{X}$ we mean that $x$ is a subset of $\Lambda$ with $n$ elements, obtained from the point process $\mathfrak{X}$.
\end{defn}
From \cite[Formula (1.2.2)]{Hough_zerosof}, for any measurable function $\phi: \Lambda^{k} \longrightarrow [0, \infty)$ the following equality holds.
\begin{multline}
\label{eq_1}
\Esp{x\sim\mathfrak{X}}{\sum_{i_{1} \ldots i_{k}\text{ distinct}} \phi(x_{i_{1}} ,\ldots , x_{i_{k}})} \\ =\int_{y_{1},\ldots ,y_{k} \in \Lambda} \phi(y_{1},\ldots ,y_{k}) \rho_{k}(y_{1},\ldots ,y_{k})\,d\mu(y_1,\ldots,y_k).
\end{multline}
\noindent Sometimes these intensity joint functions can be written as $\rho_{k}(x_{1},\ldots ,x_{k}) = det(K(x_{i}, x_{j})_{i,j = 1,\ldots ,k})$ for some function $K:\Lambda\times\Lambda\to\mathbb{C}$.
In this case, we say that $\mathfrak{X}$ is a \textit{determinantal point process}.
A particularly amenable collection of such processes is obtained from $n$--dimensional subspaces of the Hilbert space $\mathrm{L}^2(\Lambda,\mathbb{C})$ (i.e. the set of square--integrable complex functions in $\Lambda$).
Recall that the reproducing kernel of $H$ is the unique continuous, skew--symmetric, positive--definite function $K_H:\Lambda\times\Lambda\to\mathbb{C}$ such that
\[
f(x)=\langle f,K_H(\cdot,x)\rangle=\int_{y\in\Lambda}f(y)K_H(x,y)\,dy,\quad x\in\Lambda,f\in H.
\]
Given any orthonormal basis $\varphi_{1},\ldots ,\varphi_{n}$ of $H$, we have
\begin{equation}\label{eq:kernel}
K_H(x,y)=\sum_{i=1}^n\varphi_i(x)\overline{\varphi_i(y)}.
\end{equation}
Such a kernel $K_H$ is usually called a {\em projection kernel} of trace $n$.
\begin{prop}\label{prop:MS}
Let $\Lambda$ be as in Definition \ref{def:pp} and let $H \subset \text{\rm L}^{2}(\Lambda,\mathbb{C})$ have dimension $n$. Then there exists a point process $\mathfrak{X}_H$ in $\Lambda$ of $n$ points with associated join intensity functions
\begin{equation*}
\rho_{k}(x_{1},\ldots ,x_{k}) = det(K_H(x_{i}, x_{j})_{i,j = 1,\ldots ,k}).
\end{equation*}
In particular for any measurable function $f:\Lambda \times \Lambda \longrightarrow [0,\infty)$ we have
\begin{multline}
\label{th_1}
\Esp{x \sim \mathfrak{X}_H}{
\displaystyle\sum_{i \neq j} f(x_{i}, x_{j})}
\\
= \int_{p,q \in \Lambda}
\left( K_H(p,p)K_H(q,q) - |K_H(p,q)|^{2} \right) f(p,q)\,d\mu(p,q).
\end{multline}
We will call $\mathfrak{X}_H$ a projection determinantal point process with kernel $K_H$.
\end{prop}
\begin{dem}
This proposition is a direct consequence of the Macchi--Soshnikov Theorem, see \cite{Macchi, Soshni} or \cite[Theorem 4.5.5]{Hough_zerosof}. $\square$
\end{dem}
\begin{obs}\label{obs:constant}
We note that in the hypotheses of Proposition \ref{prop:MS}, from \eqref{eq_1} with $\phi\equiv1$ we also have
\begin{equation*}
n=\Esp{x\sim\mathfrak{X}_H}{n}
= \int_{p\in\Lambda} K_H(p,p)\,d\mu(p),
\end{equation*}
\noindent In particular, if $K_H(p,p)$ is constant then we must have $K_H(p,p)=n/Vol(\Lambda)$.
\end{obs}
\subsection{Transformation under diffeomorphisms}
We now describe the push--forward of a projection determinantal point process. We are most interested in the case that the spaces are Riemannian manifolds (which are locally compact, Polish and measurable spaces).
\begin{prop}
\label{prop_1}
Let $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$ be two Riemannian manifolds and let $\phi: M_{1} \longrightarrow \mathrm{M}_{2}$ be a $\boldsymbol{\mathcal{C}^{1}}$ diffeomorphism. Let $H \subset \mathrm{L}^{2}(\mathrm{M}_{1},\mathbb{C})$ be an $n$--dimensional subspace. Then, the set
\begin{align*}
H_{*} =& \left \{ f: \mathrm{M}_{2} \longrightarrow \mathbb{C}: \sqrt{|\text{\rm Jac}(\phi)(x)|}( f \circ \phi)(x) \in H \right \} \\
=&\left \{ g \circ \phi^{-1}(\cdot)\sqrt{|\text{\rm Jac}(\phi^{-1})(\cdot)|}: g \in H \right \}
\end{align*}
is an $n$--dimensional subspace of $\mathrm{L}^2(\mathrm{M}_2,\mathbb{C})$. Its associated determinantal point process $\mathfrak{X}_{H_*}$ has kernel
\begin{equation}
\begin{split}
K_{H_*}(a,b) &= \frac{K_H(\phi^{-1}(a), \phi^{-1}(b))}{\sqrt{|\text{\rm Jac}(\phi)(\phi^{-1}(a)) \text{\rm Jac}(\phi)(\phi^{-1}(b))|}} \\
& = K_H(\phi^{-1}(a), \phi^{-1}(b))\sqrt{|\text{\rm Jac}(\phi^{-1})(a) \text{\rm Jac}(\phi^{-1})(b)|}.
\end{split}
\end{equation}
(We are denoting by $\mathrm{Jac}$ the Jacobian determinant).
\end{prop}
This proposition is a direct consequence of the change of variables formula, see Section \ref{subsec:proof1} for a short proof.
\section{The projective ensemble}\label{sec:iran}
Consider the standard Fubini--Study metric in the complex projective space of complex dimension $d$, denoted by $\mathbb{P}(\mathbb{C}^{d+1})$. The distance between two points $p,q\in\mathbb{P}(\mathbb{C}^{d+1})$ is given by:
\begin{equation*}
\sin d_{\mathbb{P}(\mathbb{C}^{d+1})}(p,q)
=
\sqrt{1 - \frac{| \left\langle p,q \right\rangle |^{2}}{|| q ||^{2} || p ||^{2}}}
=
\sqrt{1 - \left| \left\langle \frac{p}{||p||},\frac{q}{||q||}\right\rangle \right|^{2}}.
\end{equation*}
\begin{defn}
\label{def_2}
\noindent Let $L \geq 0$ and consider the set of the following functions defined in $\mathbb{C}^{d}$:
\begin{equation}
\label{eq_3}
I_{d,L} = \left\lbrace \sqrt{C_{\alpha_{1},\ldots ,\alpha_{d}}^{L}} \frac{z_{1}^{\alpha_{1}}\ldots z_{d}^{\alpha_{d}}}{(1+\|z\|^{2})^{\frac{d+L+1}{2}}} \right\rbrace_{\alpha_{1}+\ldots +\alpha_{d} \leq L}
\end{equation}
\noindent where $\alpha_{1},\ldots ,\alpha_{d}$ are non--negative integers and
\begin{equation*}
C_{\alpha_{1},\ldots ,\alpha_{d}}^{L}
=
\frac{1}{{\pi}^{d}} \frac{(d+L)!}{\alpha_{1}!\ldots \alpha_{d}! (L-(\alpha_{1}+\ldots +\alpha_{d}))!}.
\end{equation*}
\noindent Let $\mathcal{H}_{d,L}=Span(I_{d,L})\subset\text{\rm L}^{2}(\mathbb{C}^{d},\mathbb{C})$ which is a subspace of complex dimension $r = {d+L \choose d}$.
The collection $I_{d,L}$ given in \eqref{eq_3} is an orthonormal basis of $\mathcal{H}_{d,L}$ (for the usual Lebesgue measure in $\mathbb{C}^{d}$) and the reproducing kernel $K: \mathbb{C}^{d} \times \mathbb{C}^{d} \longrightarrow \mathbb{C}$ is given by:
\begin{equation*}
K(z,w) = \frac{r d!}{{\pi}^{d}} \frac{(1 + \left\langle z,w \right\rangle)^{L}}{(1+\|z\|^{2})^{\frac{d+L+1}{2}} (1+\|w\|^{2})^{\frac{d+L+1}{2}}}.
\end{equation*}
From Proposition \ref{prop:MS}, there is an associated determinantal point process of $r$ points in $\mathbb{C}^d$ that we denote by $\mathfrak{X}^{(r,d)}$.
\end{defn}
\begin{lem}\label{propdef_1}
Let $d\geq1$ and let $r$ be of the form $r={d+L \choose d}$. Then, the pushforward $\mathfrak{X}_*^{(r,d)}$ of $\mathfrak{X}^{(r,d)}$ under the mapping
\[
\begin{matrix}
\psi_d:&\mathbb{C}^d&\to&\mathbb{P}(\mathbb{C}^{d+1})\\
&z&\mapsto&(1,z)
\end{matrix}
\]
is a determinantal point process in $\mathbb{P}(\mathbb{C}^{d+1})$ whose associated kernel satisfies
\begin{equation*}
\left| K_*^{(r,d)}(p,q) \right|
=
\frac{r d!}{\pi^{d}}
\left| \left\langle \frac{p}{||p||}, \frac{q}{||q||} \right\rangle \right|^L.
\end{equation*}
We call this process the projective ensemble.
\end{lem}
\noindent See Section \ref{propLemma32} for a proof of Lemma \ref{propdef_1}.
The spherical ensemble described in \cite{krishnapur2009,EJP3733} is just the case $d=1$ of the projective ensemble identifying $\mathbb{P}(\mathbb{C}^{2})$ with the Riemmann sphere and translating the process to the unit sphere.
The next result computes the expected value of a Riesz--like energy for the projective ensemble.
\begin{thm}
\label{Prop_proy}
Let $L\geq1$. For $r=\binom{d+L}{d}$ and $\omega_{r}=(x_1,\ldots,x_r)\in\mathbb{P}(\mathbb{C}^{d+1})^r$ let
\begin{equation*}
\mathcal{E}_{s}^{\mathbb{P}}(\omega_{r})
=
\sum_{i \neq j} \frac{1}{\sin \left( d_{\mathbb{P}(\mathbb{C}^{d+1})} (x_{i}, x_{j}) \right)^{s}}.
\end{equation*}
Then, for $0 < s < 2d$,
\begin{equation*}
\begin{split}
\Esp{\mathfrak{X}_{*}^{(r)}}{\mathcal{E}_{s}^{\mathbb{P}}(\omega_{r}) }
& =
\frac{d}{d - \frac{s}{2}}r^{2} - r^{2}dB \left( d - \frac{s}{2}, L + 1 \right)\\&= \frac{d}{d - \frac{s}{2}}r^{2}-r^{1+\frac{s}{2d}}\frac{d \Gamma\left(d - \frac{s}{2}\right)}{(d!)^{1-\frac{s}{2d}}}+o\left(r^{1+\frac{s}{2d}}\right).
\end{split}
\end{equation*}
Note that $d/(d-s/2)$ is precisely the continuous $s$--energy for the uniform measure in $\mathbb{P}(\mathbb{C}^{d+1})$.
\end{thm}
\begin{cor}\label{cor:logaritmica}
Let $L\geq1$. For $r=\binom{d+L}{d}$ and $\omega_{r}=(x_1,\ldots,x_r)\in\mathbb{P}(\mathbb{C}^{d+1})^r$ let
\begin{equation*}
\mathcal{E}_{0}^{\mathbb{P}}(\omega_{r})
=
\sum_{i \neq j} \log\frac{1}{\sin \left( d_{\mathbb{P}(\mathbb{C}^{d+1})} (x_{i}, x_{j}) \right)}.
\end{equation*}
Then,
\begin{equation*}
\begin{split}
\Esp{\mathfrak{X}_{*}^{(r)}}{\mathcal{E}_{0}^{\mathbb{P}}(\omega_{r}) }
& =
\frac{r^2}{2d}+\frac{r^2d}{2}B(d,L+1)\sum_{j=0}^L\frac{1}{d+j}\\&= \frac{r^2}{2d}+\frac{r\log r}{2d}+o\left(r\log r\right).
\end{split}
\end{equation*}
\end{cor}
Theorem \ref{Prop_proy} and Corollary \ref{cor:logaritmica} are proved in sections \ref{sec:proofofProp_proy} and \ref{sec:prooflog}.
\section{A new point process in odd-dimensional spheres}\label{section4}
We now describe a point process of $n$ points, for certain values of $n$, in $\S{2d+1}$ in the following manner.
\begin{defn}\label{def:pps2dmas1}
Given integers $d,\,k,\,L\geq0$, let $r=\binom{d+L}{d}$ and $n=k\,r$. We define the following point process of $n$ points in $\S{2d+1}$. First, let
\[
x_1,\ldots,x_r\in\mathbb{P}(\mathbb{C}^{d+1})
\]
be chosen from the projective ensemble $\mathfrak{X}_*^{(r,d)}$. Choose, for each $x_i$, one affine representative (which we denote by the same letter). Then, let $\theta_1,\ldots,\theta_r\in[0,2\pi)$ be chosen uniformly and independently and define
\begin{equation}\label{eq:yes}
y_{i}^{j}
=
e^{\mathbf{i}\left( \theta_{i} + \frac{2\pi j}{k} \right)} x_{i},\quad 1\leq i\leq r,\;0\leq j\leq k-1.
\end{equation}
We denote this point process by $\mathfrak{X}_{\S{2d+1}}^{(k,L)}$.
\end{defn}
Note that the way to generate a collection of $n$ points coming from $\mathfrak{X}_{\S{2d+1}}^{(k,L)}$ amounts to taking $r$ points from the projective ensemble and taking, for each of these points, $k$ affine unit norm representatives, uniformly spaced in the great circle corresponding to each point, with a random phase.
The following statement shows that the expected $2$--energy of points generated from the point process of Definition \ref{def:pps2dmas1} can be computed with high precision. It will be proved in Section \ref{sec:proofUju1}.
\begin{prop}\label{Uju1}
\begin{multline}
\label{eq3bis}
\Esp{\mathfrak{X}_{\S{2d+1}}^{(k,L)}}{\E{2}(y_{1}^{0},...,y_{1}^{k-1},y_{2}^{0},...,y_{r}^{k-1})}\\ =\frac{d}{2d-1} (kr)^{2}+\frac{rk^{3}}{12}-\frac{d\Gamma\left(d-\frac12\right)}{2(d!)^{1-\frac1{2d}}} k^2r^{1+\frac{1}{2d}}+o\left( k^2r^{1+\frac{1}{2d}}\right).
\end{multline}
\end{prop}
Following the same ideas one can also compute the expected $s$-energy for $n$ points coming from the point process $\mathfrak{X}_{\S{2d+1}}^{(k,L)}$ for other even integer values $s \in 2\mathbb{N}$, and a bound can be found for other values of $s>0$. The computations, though, are quite involved.
Proposition \ref{Uju1} describes how different choices of $L$ (i.e. of $r$) and $k$ produce different values of the expected $2$--energy of the associated $n=rk$ points. An optimization argument is in order: for given $n$, which is the optimal choice of $r$ and $k$? Since we know from \eqref{eq:bounds} that the second order term in the assymptotics is $\sim n^{1+2/(2d+1)}=(rk)^{1+2/(2d+1)}$, it is easy to conclude that the optimal values of $r$ and $k$ satisfy:
\[
k\sim r^{\frac{1}{2d}}.
\]
The following corollary follows inmediately from Proposition \ref{Uju1}.
\begin{cor}\label{cor:casi}
If we choose $k=A r^{\frac{1}{2d}}$ for some $A\in\mathbb{R}$ making that quantity a positive integer, then:
\begin{multline}
\label{eq3bis2}
\Esp{\mathfrak{X}_{\S{2d+1}}^{(k,L)}}{\E{2}(y_{1}^{0},...,y_{1}^{k-1},y_{2}^{0},...,y_{r}^{k-1})}\\ = \frac{d}{2d-1} n^{2}
+
\left(
\frac{A^{2 - \frac{2}{2d+1}} }{12}
-
\frac{d \Gamma\left( d - \frac{1}{2} \right) A^{1 - \frac{2}{2d+1}} }{2 \left(d!\right)^{1 - \frac{1}{2d}} }
\right)
n^{1+\frac{2}{2d+1}}
+
o\left(n^{1+\frac{2}{2d+1}}\right).
\end{multline}
\end{cor}
The proof of our first main theorem will follow easily from Corollary \ref{cor:casi}.
\section{Proof of the main results}
\subsection{Proof of Proposition \ref{prop_1}}\label{subsec:proof1}
We first prove that $H_*\subseteq\mathrm{L}^2(\mathrm{M}_2,\mathbb{C})$. Indeed, for $f\in H_*$ we have
\[
\int_{y\in \mathrm{M}_2}|f|^2\,dy=\int_{y\in \mathrm{M}_2}|g\circ\phi^{-1}(y)|^2|\mathrm{Jac}(\phi^{-1})(y)|\,dy,
\]
for some $g\in H$. Since it is in one--to--one correspondence with $H$, the dimension of $H_*$ is also $n$.
Now, by the change of variables formula this last equals the squared $\text{L}^2$ norm of $g$ which is finite since $H\subseteq\mathrm{L}^2(\mathrm{M}_1,\mathbb{C})$.
\noindent We now prove the formula for $K_{H*}$. Let $\varphi_1,\ldots,\varphi_n$ be an orthonormal basis of $H$. Then, $\varphi_{i,*}=\varphi_i \circ \phi^{-1}(\cdot)\sqrt{|\text{\rm Jac}(\phi^{-1})(\cdot)|}$, $1\leq i\leq n$, are elements in $H_*$ and using the change of variables formula we have:
\begin{align*}
\int_{y\in \mathrm{M}_2} \varphi_{i,*}(y)\overline{\varphi_{j,*}(y)}\,dy=&\int_{y\in \mathrm{M}_2} \varphi_i \circ \phi^{-1}(y)\,\overline{\varphi_j \circ \phi^{-1}(y)}\,|\text{\rm Jac}(\phi^{-1})(y)|\,dy\\
=&\int_{x\in\mathrm{M}_1}\varphi_i(x)\overline{\varphi_j(y)}\,dx=\delta_{ij},
\end{align*}
where we use the Kronecker delta notation. Hence, $\{\varphi_{i,*}\}$ form an orthonormal basis and
\begin{align*}
K_{H_*}(a,b)=&\sum_{i=1}^n\varphi_{i,*}(a)\overline{\varphi_{i,*}(b)}\\
=&\sum_{i=1}^n\varphi_i \circ \phi^{-1}(a)\overline{\varphi_i \circ \phi^{-1}(b)}\sqrt{|\text{\rm Jac}(\phi^{-1})(a)\,\text{\rm Jac}(\phi^{-1})(b)|}\\
=&K_H(\phi^{-1}(a), \phi^{-1}(b))\sqrt{|\text{\rm Jac}(\varphi^{-1}(a))\, \text{\rm Jac}(\phi^{-1}(b))|}.
\end{align*}
The other formula for $K_{H_*}$ follows from this last one, using that
\[
\mathrm{Jac}(\phi)(\phi^{-1}(a))=\mathrm{Jac}(\phi^{-1})(a)^{-1}.
\]
$\square$
\subsection{Proof of Lemma \ref{propdef_1}}\label{propLemma32}
From Proposition \ref{prop_1}, $\mathfrak{X}_*^{(r,d)}$ has reproducing kernel
\begin{equation*}
K_*^{(r,d)}(p,q)
=
\frac{K(\psi_d^{-1}(p), \psi_d^{-1}(q))}{\sqrt{|\text{\rm Jac}(\psi_d)(\psi_d^{-1}(p)) \text{\rm Jac}(\psi_d)(\psi_d^{-1}(q))|}}.
\end{equation*}
The Jacobian of $\psi_d$ is:
\begin{equation}\label{eq:jacobianpsid}
\text{\rm Jac}(\psi_d)(z)
=
\left( \frac{1}{1 + ||z||^{2}} \right)^{d+1}.
\end{equation}
We thus have (denoting $p=(z,1)$ and $q=(w,1)$):
\begin{align*}
\left| K_*^{(r,d)}(p,q) \right|
=&
\frac{
\frac{r d!}{\pi^{d}}
\frac{
\left| 1 + \left\langle \psi^{-1}(p), \psi^{-1}(q) \right\rangle \right|^{L}
}{
\left( 1 + ||\psi^{-1}(p)||^{2} \right)^{\frac{L+d+1}{2}}
\left( 1 + ||\psi^{-1}(q)||^{2} \right)^{\frac{L+d+1}{2}}
}
}{
\left( \frac{1}{1 + ||\psi^{-1}(p)||^{2}} \right)^{\frac{d+1}{2}}
\left( \frac{1}{1 + ||\psi^{-1}(q)||^{2}} \right)^{\frac{d+1}{2}}
}\\
=&
\frac{r d!}{\pi^{d}} \frac{\left| 1 + \left\langle z,w \right\rangle \right|^{L}}{(1 + \|z\|^{2})^{\frac{L}{2}} (1 + \|w\|^{2})^{\frac{L}{2}}}\\
=&\frac{r d!}{\pi^{d}} \frac{\left|\left\langle p,q \right\rangle\right|^{L}}{\|p\|^{L} \|q\|^{L}},
\end{align*}
and the lemma follows.
$\square$
\subsection{Proof of Theorem \ref{Prop_proy}}\label{sec:proofofProp_proy}
Let $J$ be the quantity we want to compute. Following Proposition \ref{prop:MS} we have that
\begin{equation*}
\begin{split}
J=& \Esp{\mathfrak{X}_{*}^{(r)}}{ \sum_{i \neq j} \frac{1}{\sin \left( d_{\mathbb{P}(\mathbb{C}^{d+1})} (x_{i}, x_{j}) \right)^{s}} } \\&
= \int_{\mathbb{P}(\mathbb{C}^{d+1}) \times \mathbb{P}(\mathbb{C}^{d+1})} \frac{K(p,p)^{2} - |K(p,q)|^{2}}{\sin \left( d_{\mathbb{P}(\mathbb{C}^{d+1})} (p, q) \right)^{s}} dp dq
\\
& = \frac{r^{2} d!^{2} }{\pi^{2d}}\int_{\mathbb{P}(\mathbb{C}^{d+1}) \times \mathbb{P}(\mathbb{C}^{d+1})} \frac{1 - \left| \left\langle p,q \right\rangle \right|^{2L} }{\left(1 - \left| \left\langle p,q \right\rangle \right|^{2}\right)^{\frac{s}{2}}} dp dq,
\end{split}
\end{equation*}
where we choose unit norm representatives $p,q$. Since the integrand only depends on the distance between $p$ and $q$ and $\mathbb{P}(\mathbb{C}^{d+1})$ is a homogeneous space, we can fix $p=e_1=(1,0,\ldots,0)$ to get:
\[
J=\frac{r^{2} d! }{\pi^{d}} \int_{\mathbb{P}(\mathbb{C}^{d+1})} \frac{ 1 - \left| \left\langle e_{1},q \right\rangle \right|^{2L} }{\left(1 - \left| \left\langle e_{1},q \right\rangle \right|^{2}\right)^{\frac{s}{2}}} dq,
\]
where we have used that the volume of $\mathbb{P}(\mathbb{C}^{d+1})$ is equal to ${\pi^{d}}/{d! }$. In order to compute this integral, we use the change of variables theorem with the map $\psi_d$ whose Jacobian is given in \eqref{eq:jacobianpsid}, getting:
\begin{equation*}
\begin{split}
J& = \frac{r^{2} d! }{\pi^{d}} \int_{\mathbb{C}^{d}} \frac{ 1 - \left| \left\langle e_{1},\frac{(1,z)}{\sqrt{1 + ||z||^{2}}} \right\rangle \right|^{2L} }{\left(1 - \left| \left\langle e_{1},\frac{(1,z)}{\sqrt{1 + ||z||^{2}}} \right\rangle \right|^{2}\right)^{\frac{s}{2}}} \frac{1}{(1 + ||z||^{2})^{d+1}} dz
\\
& = \frac{r^{2} d! }{\pi^{d}} \int_{\mathbb{C}^d} \frac{ 1 - \left( \frac{1}{1 + ||z||^{2}} \right)^{L} }{\left(1 - \frac{1}{1 + ||z||^{2}}\right)^{\frac{s}{2}}} \left(\frac{1}{1 + ||z||^{2}}\right)^{d+1} dz.
\\
\end{split}
\end{equation*}
Integrating in polar coordinates,
\begin{equation*}
\begin{split}
J&= \frac{r^{2} d! }{\pi^{d}} \frac{2\pi^{d}}{(d-1)!} \int_{0}^{\infty} \frac{ 1 - \left( \frac{1}{1 + t^{2}} \right)^{L} }{\left(1 - \frac{1}{1 + t^{2}}\right)^{\frac{s}{2}}} \left(\frac{1}{1 + t^{2}}\right)^{d+1} t^{2d-1} dt
\\
& = 2r^{2}d \left[ \int_{0}^{\infty} \frac{ t^{2d-1-s} }{(1 + t^{2})^{d + 1 - \frac{s}{2}}} dt - \int_{0}^{\infty} \frac{ t^{2d-1-s} }{(1 + t^{2})^{d + 1 - \frac{s}{2} + L}} dt \right]
\\
& = 2r^{2}d \left[ \frac{B \left( d - \frac{s}{2}, 1 \right)}{2} - \frac{B \left( d - \frac{s}{2}, L + 1 \right)}{2} \right]
\\
& = \frac{d}{d - \frac{s}{2}}r^{2} - r^{2}dB \left( d - \frac{s}{2}, L + 1 \right),
\end{split}
\end{equation*}
as claimed. For the assymptotics, note that for $L\to\infty$ (equiv. $r\to\infty$)
\[
B \left( d - \frac{s}{2}, L + 1 \right)=\frac{\Gamma\left(d - \frac{s}{2}\right)\Gamma(L+1)}{\Gamma\left(d - \frac{s}{2}+ L + 1 \right)}\sim\Gamma\left(d - \frac{s}{2}\right)L^{\frac{s}{2}-d},\quad r=\binom{L+d}{d}\sim\frac{L^d}{d!},
\]
and hence
\[
B \left( d - \frac{s}{2}, L + 1 \right)\sim\Gamma\left(d - \frac{s}{2}\right)(d!r)^{\frac{s}{2d}-1}.
\]
The assymptotic expansion claimed in the theorem follows.
$\square$
\subsection{Proof of Corollary \ref{cor:logaritmica} }\label{sec:prooflog}
Note that $\mathcal{E}_0(\omega_r)= \left. \frac{d}{ds}\right|_{s=0}\mathcal{E}_s(\omega_r)$. In particular, interchanging the order of expected value and derivative (it is an exercise to check that this change is justified), from Theorem \ref{Prop_proy} we have
\[
\Esp{\mathfrak{X}_{*}^{(r)}}{\mathcal{E}_{0}^{\mathbb{P}}(\omega_{r}) } = \left. \frac{d}{ds} \right|_{s=0}\left(\frac{d}{d - \frac{s}{2}}r^{2} - r^{2}dB \left( d - \frac{s}{2}, L + 1 \right)\right).
\]
The proof of the corollary is now a straightforward computation of that derivative and it is left to the reader. It is helpful to recall the derivative of Euler's Beta function in terms of the digamma function $\psi_0$ for $m \in \mathbb{N}$:
\begin{align*}
\frac{d}{dt}B(t,m)=&\frac{d}{dt}\frac{\Gamma(t)\Gamma(m)}{\Gamma(t+m)}\\=&\frac{\Gamma'(t)\Gamma(m)\Gamma(t+m)-\Gamma(t)\Gamma(m)\Gamma'(t+m)}{\Gamma(t+m)^2}\\=&\frac{\psi_0(t)\Gamma(t)\Gamma(m)-\Gamma(t)\Gamma(m)\psi_0(t+m)}{\Gamma(t+m)}\\=&B(t,m)(\psi_0(t)-\psi_0(t+m))\\=&-B(t,m)\sum_{j=0}^{m-1}\frac{1}{t+j}.
\end{align*}
\subsection{Proof of Proposition \ref{Uju1}}\label{sec:proofUju1}
We will use the following equality, valid for $y\in(-1,1)$:
\begin{equation}\label{eq:integrallibro}
\int_{0}^{2\pi} \frac{d\theta}{1 - y \cos(\theta)}=\frac{2\pi}{\sqrt{1-y^2}}.
\end{equation}
See for example \cite[3.792--1]{zwillinger2014table} from which the equality above easily follows.
We have to compute
\begin{equation}\label{eq2}
\begin{split}
& \frac{1}{(2\pi)^{r}} \int_{\theta_{1},...,\theta_{r} \in [0, 2\pi]} \Esp{\mathfrak{X}_{*}^{(r)}}{\E{2}(y_{1}^{0},...,y_{1}^{k-1},y_{2}^{0},...,y_{r}^{k-1})} d(\theta_{1},...,\theta_{r})
\\
& = \frac{1}{(2\pi)^{r}} \int_{\theta_{1},...,\theta_{r} \in [0, 2\pi]} \Esp{\mathfrak{X}_{*}^{(r)}}{ \sum_{i_{1} \neq i_{2}\text{ or } j_{1} \neq j_{2}} \frac{1}{\left|\left| y_{i_{1}}^{j_{1}} - y_{i_{2}}^{j_{2}} \right|\right|^ {2}} } d(\theta_{1},...,\theta_{r})
\\
&=J_1+J_2,
\end{split}
\end{equation}
where
\begin{align*}
J_1=& \frac{1}{(2\pi)^{r}} \int_{\theta_{1},...,\theta_{r} \in [0, 2\pi]} \Esp{\mathfrak{X}_{*}^{(r)}}{ \sum_{i=1}^{r} \sum_{j_{1} \neq j_{2}} \frac{1}{\left|\left| y_{i}^{j_{1}} - y_{i}^{j_{2}} \right|\right|^ {2}}}\,d(\theta_{1},...,\theta_{r}) ,
\\
J_2=& \frac{1}{(2\pi)^{r}} \int_{\theta_{1},...,\theta_{r} \in [0, 2\pi]}\Esp{\mathfrak{X}_{*}^{(r)}}{ \sum_{j_{1},j_2=0}^{k-1} \sum_{i_{1} \neq i_{2}} \frac{1}{\left|\left| y_{i_{1}}^{j_{1}} - y_{i_{2}}^{j_{2}} \right|\right|^ {2}} }\, d(\theta_{1},...,\theta_{r}).
\end{align*}
From \eqref{eq:yes} we have:
\begin{align*}
J_1=& \frac{1}{2\pi}\sum_{i=1}^{r}\int_{\theta\in [0, 2\pi]} \Esp{\mathfrak{X}_{*}^{(r)}}{ \sum_{j_{1} \neq j_{2}} \frac{1}{\left|\left| e^{\mathbf{i}\left( \theta + \frac{2\pi j_1}{k} \right)} x_i - e^{\mathbf{i}\left( \theta + \frac{2\pi j_2}{k} \right)} x_i \right|\right|^ {2}}}\,d\theta.
\end{align*}
Now, the integral does not depend on $\theta$ nor in the (unit norm) vector $x_i\in\mathbb{C}^{n+1}$, so we actually have that
\[
\frac{J_1}{r}=\sum_{j_{1} \neq j_{2}} \frac{1}{\left|e^{\mathbf{i} \frac{2\pi j_1}{k} } - e^{\mathbf{i} \frac{2\pi j_2}{k}} \right|^ {2}},
\]
is the $2$--energy of the $k$ roots of unity. This quantity has been studied with much more detail than we need in \cite[Theorem 1.1]{BLMS:BLMS0621}. In particular, we know that it is of the form $k^3/12+o(k)$. We thus conclude:
\begin{equation}\label{eq:J1}
J_1=\frac{rk^3}{12}+o(rk).
\end{equation}
We now compute $J_2$. Interchanging the order of integration we have:
\begin{equation*}
\begin{split}
J_2
& = \Esp{\mathfrak{X}_{*}^{(r)}}{ \sum_{j_{1},j_2=0}^{k-1} \sum_{i_{1} \neq i_{2}} \frac{1}{4\pi^{2}}\int_{0}^{2\pi} \int_{0}^{2\pi} \frac{d\theta_{i_{1}} d\theta_{i_{2}} }{\left|\left| e^{\mathbf{i}\left( \theta_{i_{1}} + \frac{2\pi j_{1}}{k} \right)} x_{i_{1}} - e^{\mathbf{i}\left( \theta_{i_{2}} + \frac{2\pi j_{2}}{k} \right)} x_{i_{2}} \right|\right|^ {2}} },
\end{split}
\end{equation*}
where we can choose whatever unit norm representatives we wish of $x_{i_1}$ and $x_{i_2}$. In order to compute the inner integral, for any fixed $i_i,i_2$ we assume that our choice satisfies $\langle x_{i_1},x_{i_2}\rangle\in[0,1]$ (i.e. it is real and non--negative), which readily implies
\begin{equation}\label{eq:real}
\sin d_{\mathbb{P}(\mathbb{C}^{d+1})}(x_{i_1},x_{i_2})
=
\sqrt{1 -\left\langle x_{i_1},x_{i_2}\right\rangle ^{2}}.
\end{equation}
A simple computation using the invariance of the integral under rotations yields:
\begin{multline*}
\frac{1}{4\pi^{2}}\int_{0}^{2\pi} \int_{0}^{2\pi} \frac{d\theta_{i_{1}} d\theta_{i_{2}} }{\left|\left| e^{\mathbf{i}\left( \theta_{i_{1}} + \frac{2\pi j_{1}}{k} \right)} x_{i_{1}} - e^{\mathbf{i}\left( \theta_{i_{2}} + \frac{2\pi j_{2}}{k} \right)} x_{i_{2}} \right|\right|^ {2}}\\=\frac{1}{2\pi}\int_0^{2\pi}\frac{d\theta}{2-2\left\langle x_{i_1},x_{i_2}\right\rangle\cos\theta}\stackrel{\eqref{eq:integrallibro}}{=}\frac{1}{2\sqrt{1-\left\langle x_{i_1},x_{i_2}\right\rangle^2}}\stackrel{\eqref{eq:real}}{=}\frac{1}{2\sin d_{\mathbb{P}(\mathbb{C}^{d+1})}(x_{i_1},x_{i_2})},
\end{multline*}
and this last value is independent of $j_1,j_2$. We thus have:
\[
J_2= \frac{k^2}{2}\Esp{\mathfrak{X}_{*}^{(r)}}{ \sum_{i_{1} \neq i_{2}} \frac{1}{\sin d_{\mathbb{P}(\mathbb{C}^{d+1})}(x_{i_1},x_{i_2})}}
\]
This last expected value has been computed in Theorem \ref{Prop_proy}, which yields:
\begin{equation}\label{eq:J2}
J_2=
\frac{d}{2d-1} (kr)^{2}-\frac{d\Gamma\left(d-\frac12\right)}{2(d!)^{1-\frac1{2d}}} k^2r^{1+\frac{1}{2d}}+o\left( k^2r^{1+\frac{1}{2d}}\right).
\end{equation}
Proposition \ref{Uju1} follows from \eqref{eq2}, \eqref{eq:J1} and \eqref{eq:J2}.
$\square$
\subsection{Proof of Theorem \ref{th:main}}
Fix $d\geq1$ and let
\[
f(A)=\frac{A^{2 - \frac{2}{2d+1}} }{12}
-\frac{d \Gamma\left( d - \frac{1}{2} \right) A^{1 - \frac{2}{2d+1}} }{2 \left(d!\right)^{1 - \frac{1}{2d}} }
\]
be the coefficient of $n^{1+\frac{2}{2d+1}}$ in \eqref{eq3bis2}. The function $f(A)$ has a strict global minimum at
\[
A_d=\frac{3 \Gamma\left( d - \frac{1}{2} \right) (2d-1)}{2 \left(d!\right)^{1 - \frac{1}{2d}} }.
\]
Indeed,
\[
f(A_d)=-\frac{3^{1 - \frac{2}{2d+1}} (2d-1)^{1 - \frac{2}{2d+1}} (2d+1) \Gamma\left( d- \frac{1}{2} \right)^{2 - \frac{2}{2d+1}}}{2^{4 - \frac{2}{2d+1}} (d!)^{2 - \frac{4}{2d+1}}},
\]
gives the bound for the $\limsup$ given in Theorem \ref{th:main}. We cannot just let $k=A_dr^{\frac{1}{2d}}$ in Corollary \ref{cor:casi} since it might happen that $k\not\in\mathbb{Z}$, but we will easily go over this problem. Let $L\geq1$ be any positive integer, let $r=\binom{d+L}{d}$ and let $A$ be the unique number in the interval
\[
\big[A_d,A_d+r^{-\frac{1}{2d}}\big)
\]
such that $k=Ar^{\frac{1}{2d}}\in\mathbb{Z}$. Finally, let $n=n_L=rk$, which depends uniquely on $d$ and $L$, and which satisfies $n_L\to\infty$ as $L\to\infty$. For any $\epsilon>0$ we then have:
\begin{align*}
\limsup_{L\to\infty}\frac{ V_{2}(\mathbb{S}^{2d+1})n_L^{2}-\min_{\omega_{n_L}}\left(\E{2}(\omega_{n_L})\right) }{n_L^{1+\frac{2}{2d+1}}}
\geq -f(A)\geq -f(A_d)-\epsilon,
\end{align*}
the first inequality from Corollary \ref{cor:casi} and the second inequality due to $r\to\infty$ as $L\to\infty$, which implies for some constant $C>0$:
\[
|f(A)-f(A_d)| \leq Cr^{-\frac{1}{2d}}\to0,\quad L\to\infty.
\]
We have thus proved
\[
\limsup_{L\to\infty}\frac{ V_{2}(\mathbb{S}^{2d+1})n_L^{2}-\min_{\omega_{n_L}}\left(\E{2}(\omega_{n_L})\right) }{n_L^{1+\frac{2}{2d+1}}}
\geq -f(A_d),
\]
which finishes the proof of our Theorem \ref{th:main}.
\subsection{Proof of Theorem \ref{th:main2}}
From \cite{Juan}, the Green function of $\mathbb{P}(\mathbb{C}^{d+1})$ is given $G(x,y)=\phi(r)$ where $r=d_{\mathbb{P}(\mathbb{C}^{d+1})}(x,y)$ and
\[
\phi'(r)=-\frac{1}{Vol(\mathbb{P}(\mathbb{C}^{d+1}))}\frac{\int_r^{\pi/2}\sin^{2d-1}t\cos t\,dt}{\sin^{2d-1}r\cos r}=-\frac{1}{2dVol(\mathbb{P}(\mathbb{C}^{d+1}))}\frac{1-\sin^{2d}r}{\sin^{2d-1}r\cos r}.
\]
Integrating the formula above (see for example \cite[2.517--1]{zwillinger2014table} we have:
\begin{equation*}
\begin{split}
\phi(r)
=
& \frac{1}{2dVol(\mathbb{P}(\mathbb{C}^{d+1}))}
\left[
\frac{1}{2} \displaystyle\sum_{k=1}^{d-1} \frac{1}{(d-k) \left( \sin r \right)^{2d-2k}}
-
\log \left( \sin r \right)
\right] +C.
\end{split}
\end{equation*}
In order to compute the constant we need to impose that the average of $G(x,\cdot)$ equals $0$ for all (i.e. for some) $x\in\mathbb{P}(\mathbb{C}^{d+1})$. Let $x=(1,0)$ and change variables using $\psi_d$ from Lemma \ref{propdef_1} whose Jacobian is given in \eqref{eq:jacobianpsid} to compute:
\[
C=- \frac{1}{2dVol(\mathbb{P}(\mathbb{C}^{d+1}))^2}
\left[
\frac{1}{2} \displaystyle\sum_{k=1}^{d-1} \int_{z\in\mathbb{C}^d}\frac{(1+\|z\|^2)^{-k-1}}{(d-k) \|z\|^{2d-2k}}
\,dz
-
\frac{1}{2}\int_{z\in\mathbb{C}^d}\frac{\log \left( \frac{\|z\|^2}{1+\|z\|^2} \right)}{(1+\|z\|^2)^{d+1}}\,dz
\right].
\]
Integrating in polar coordinates,
\begin{align*}
C=&
\frac{1}{2Vol(\mathbb{P}(\mathbb{C}^{d+1}))}\left(\int_{0}^\infty\frac{t^{2d-1}\log \left( \frac{t^2}{1+t^2} \right)}{(1+t^2)^{d+1}}\,dt- \displaystyle\sum_{k=1}^{d-1} \int_{0}^\infty\frac{t^{2k-1}}{(d-k)(1+t^2)^{k+1}} \,dt\right)
\\=&
-\frac{d!}{4\pi^d}\left(\frac{1}{d^{2}}
+
\displaystyle\sum_{k=1}^{d-1} \frac{1}{k(d-k)}\right)\\=&
-\frac{(d-1)!}{4\pi^d}\left(\frac{1}{d}+2\sum_{k=1}^{d-1}\frac1k\right).
\end{align*}
(for the computation of the integrals, use the change of variables $s=t^2/(1+t^2)$ and \cite[4.272--15]{zwillinger2014table}, for example).
We thus conclude for $r=d_{\mathbb{P}(\mathbb{C}^{d+1})}(x,y)$ :
\begin{equation*}
\begin{split}
G(x,y)
=
& \frac{(d-1)!}{2\pi^{d}}
\left[
\left(
\frac{1}{2} \displaystyle\sum_{k=1}^{d-1} \frac{1}{(d-k) \left( \sin r \right)^{2d-2k}}
\right)
-
\log \left( \sin r\right)
\right] \\
&
-\frac{(d-1)!}{4\pi^d}\left(\frac{1}{d}+2\sum_{k=1}^{d-1}\frac1k\right).
\end{split}
\end{equation*}
\noindent Following the definitions of Theorem \ref{Prop_proy} and Corollary \ref{cor:logaritmica}, the expected value of Green energy may be expressed as
\begin{equation*}
\begin{split}
& \Esp{\mathfrak{X}_{*}^{(r)}}{\E{G}^{\mathbb{P}}(\omega_{r})}
=
\overbrace{\frac{(d-1)!}{2\pi^{d}}
\left[
\left(
\frac{1}{2} \displaystyle\sum_{k=1}^{d-1} \frac{1}{d-k}
\Esp{\mathfrak{X}_{*}^{(r)}}{\mathcal{E}_{2d-2k}^{\mathbb{P}}(\omega) }
\right)
+
\Esp{\mathfrak{X}_{*}^{(r)}}{\mathcal{E}_{0}^{\mathbb{P}}(\omega) }
\right]}^{A} \\
&-\frac{r(r-1)(d-1)!}{4\pi^d}\left(\frac{1}{d}+2\sum_{k=1}^{d-1}\frac1k\right). \\
\end{split}
\end{equation*}
Each of the expected values in the last expression has been computed in Theorem \ref{Prop_proy} and Corollary \ref{cor:logaritmica}, producing:
\begin{equation*}
\begin{split}
&
A
=
\frac{(d-1)!}{4\pi^{d}}
\left[
\left(
\displaystyle\sum_{k=1}^{d-1} \frac{1}{d-k}
\left(
\frac{d}{k}r^{2}-r^{2-\frac{k}{d}}\frac{d \Gamma\left(k\right)}{(d!)^{\frac{k}{d}}}
\right)
\right)
+
\frac{r^2}{d} + \frac{r\log r}{d}
\right]
+o\left(r^{2-\frac{1}{d}}\right)\\
& =
r^{2} \frac{d!}{4\pi^{d}}
\left(
\sum_{k=1}^{d-1} \frac{1}{k(d-k)} + \frac{1}{d^2}
\right)
-
\frac{(d!)^{1- \frac{1}{d}}}{4\pi^{d}(d-1)}
r^{2 - \frac{1}{d}}
+o\left(r^{2-\frac{1}{d}}\right)
\\
& =
r^{2} \frac{(d-1)!}{4\pi^{d}}
\left(
\frac{1}{d}+2\sum_{k=1}^{d-1} \frac{1}{k}
\right)
-
\frac{(d!)^{1- \frac{1}{d}}}{4\pi^{d}(d-1)}
r^{2 - \frac{1}{d}}
+o\left(r^{2-\frac{1}{d}}\right)
\end{split}
\end{equation*}
\noindent We thus have:
\begin{equation*}
\begin{split}
\Esp{\mathfrak{X}_{*}^{(r)}}{\E{G}^{\mathbb{P}}(\omega_{r})}
& =
-
\frac{(d!)^{1- \frac{1}{d}}}{4\pi^{d}(d-1)}
r^{2 - \frac{1}{d}}
+o\left(r^{2-\frac{1}{d}}\right).
\end{split}
\end{equation*}
Since this last equation holds for an infinite sequence of numbers (those of the form $r=\binom{d+L}{d}$, Theorem \ref{th:main2} follows.
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URL = {http://dx.doi.org/10.1214/EJP.v20-3733},
}
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Translation edited and with a preface by Daniel Zwillinger and
Victor Moll,
Revised from the seventh edition},
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ISBN = {978-0-8218-4373-4},
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DOI = {10.1090/ulect/051},
URL = {http://dx.doi.org/10.1090/ulect/051},
}
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\bib{10.2307/117605}{article}{
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URL = {http://dx.doi.org/10.1090/S0002-9947-98-02119-9},
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\bib{Macchi}{article}{
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\bib{Sandi}{article}{
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\bib{PhysRevE.79.041108}{article}{
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url = {http://link.aps.org/doi/10.1103/PhysRevE.79.041108},
}
\bib{Soshni}{article}{
AUTHOR = {Soshnikov, A.},
TITLE = {Determinantal random point fields},
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Obshchestvo. Uspekhi Matematicheskikh Nauk},
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NUMBER = {5--335},
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}
\end{biblist}
\end{bibdiv}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,851 |
Makita Canada Inc. Search: GO FRANÇAIS
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Providing Goods and Services to Persons with Disabilities
1. Corporate Philosophy
The management philosophy of Makita Corporation and Makita Canada Inc. is to exist in harmony with society through observance of laws and regulations; constant demonstration of ethical behavior; value and respect for our customers; consistent and proactive management; and encouragement of each individual to perform to his or her highest level.
2. Our commitment
In fulfilling our mission, Makita is committed to providing its goods and services in a way that respects the dignity and independence of persons with disabilities. We are also committed to giving persons with disabilities the same opportunity to access our goods and services and allowing them to benefit from the same services, in the same place and in a similar way as other customers.
3. Providing goods and services to persons with disabilities
Makita is committed to excellence in serving all customers including persons with disabilities and we will carry out our functions and responsibilities in the following areas:
a. Communication
We will communicate with persons with disabilities in ways that take into account their disability.
We will train associates who communicate with customers on how to interact and communicate with people with various types of disabilities.
b. Telephone services
We are committed to providing fully accessible telephone service to our customers. We will train associates to communicate with customers over the telephone in clear and plain language and to speak clearly and slowly.
We will offer to communicate with customers in person, by email or social media if telephone communication is not suitable to their communication needs or is not available, and we encourage our customers to share other means that may be of assistance to them.
c. Assistive devices
We are committed to serving persons with disabilities who use assistive devices to obtain, use or benefit from our goods and services. We will ensure that our associates are trained and familiar with various assistive devices that may be used by customers with disabilities while accessing our goods or services.
We will also ensure that associates receive training on how to use any assistive devices that may be available on our premises for customers.
d. Billing
We are committed to providing accessible invoices to all of our customers. For this reason, invoices will be provided upon request in the following formats: PDF, large print, or email. We will answer any questions customers may have about the content of the invoice in person, by telephone or e-mail.
4. Use of service animals and support persons
We are committed to welcoming persons with disabilities who are accompanied by a service animal on the parts of our premises that are open to the public and other third parties. We will also ensure that all associates, volunteers and others dealing with the public are properly trained in how to interact with persons with disabilities who are accompanied by a service animal.
We are committed to welcoming persons with disabilities who are accompanied by a support person. Any person with a disability who is accompanied by a support person will be allowed to enter Makita's premises with his or her support person. At no time will a person with a disability who is accompanied by a support person be prevented from having access to his or her support person while on our premises.
5. Notice of temporary disruption
Makita will provide customers with notice in the event of a planned or unexpected disruption in the facilities or services usually used by persons with disabilities. This notice will include information about the reason for the disruption, the anticipated duration, and a description of alternative facilities or services, if available.
The notice will be placed at all public entrances and service counters on our premises.
6. Training for associates
Makita will provide training to all associates, volunteers and others that provide service to customers in Ontario, deal with the public or other third parties on their behalf, and all those who are involved in the development and approvals of customer service policies, practices and procedures.
This training will be provided as part of the training program for new associates and includes:
The purposes of the Accessibility for Ontarians with Disabilities Act, 2005 and the requirements of the customer service standard
How to interact and communicate with people with various types of disabilities
How to interact with persons with disabilities who use an assistive device or require the assistance of a service animal or a support person
How to use any assistive device that Makita may offer to persons with disabilities
What to do if a person with a disability is having difficulty in accessing Makita's goods and services
Makita's policies, practices and procedures relating to the customer service standard.
Associates that provide service in the province of Ontario will be trained on policies, practices and procedures that affect the way goods and services are provided to persons with disabilities. Associates will also be trained on an ongoing basis when changes are made to these policies, practices and procedures.
7. Feedback process
The ultimate goal of Makita is to meet and surpass customer expectations while serving customers with disabilities. Comments on our services regarding how well those expectations are being met are welcome and appreciated.
Feedback regarding the way Makita provides goods and services to persons with disabilities can be made by formal feedback process, email, social media or in person. All feedback will be directed to the Manager, Corporate Administration. Customers can expect to hear back within 10 business days.
8. Modifications to this or other policies
We are committed to developing customer service policies that respect and promote the dignity and independence of persons with disabilities. Therefore, no changes will be made to this policy before considering the impact on persons with disabilities.
Any policy of Makita that does not respect and promote the dignity and independence of persons with disabilities will be modified or removed.
AODA – Accessible Employment Policy
The provision of accessible employment services for persons with disabilities.
3. Recruitment, Assessment and Selection
Makita Canada Inc. will notify employees and the public about the availability of accommodation for job applicants who have disabilities. Applicants will be informed that these accommodations are available, upon request, for the interview process and for other candidate selection methods. Where an accommodation is requested, Makita Canada Inc. will consult with the applicant and provide or arrange for suitable accommodation.
Successful applicants will be made aware of Makita Canada Inc.'s policies and supports for accommodating people with disabilities.
4. Accessible Formats and Communication Supports for Employees
Makita Canada Inc. will ensure that employees are aware of our policies for employees with disabilities and any changes to these policies as they occur.
If an employee with a disability requests it, Makita Canada Inc. will provide or arrange for the provision of accessible formats and communication supports for the following:
a. Information needed in order to perform his/her job; and
b. Information that is generally available to all employees in the workplace.
Makita Canada Inc. will consult with the employee making the request to determine the best way to provide the accessible format or communication support.
5. Workplace Emergency Response Information
Where required, Makita Canada Inc. will create individual workplace emergency response information for employees with disabilities. This information will take into account the unique challenges created by the individual's disability and the physical nature of the workplace and will be created in consultation with the employee.
This information will be reviewed when:
The employee moves to a different physical location in the organization;
The employee's overall accommodation needs or plans are reviewed; and/or
Makita Canada Inc. reviews general emergency response policies.
6. Documented Individual Accommodation Plans
Makita Canada Inc. will ensure that our website and all web content published after January 1, 2012, conform to the Web Content Accessibility Guidelines (WCAG) 2.0 and will refer to the schedule set out in the IASR for specific compliance deadlines.
Makita Canada Inc. will also develop and have in place written processes for documenting individual accommodation plans for employees with disabilities. The process for the development of these accommodation plans should include specific elements, including:
The ways in which the employee can participate in the development of the plan;
The means by which the employee is assessed on an individual basis;
The ways that an employer can request an evaluation by an outside medical expert, or other experts (at the employer's expense) to determine if accommodation can be achieved, or how it can be achieved;
The steps taken to protect the privacy of the employee's personal information;
The frequency with which the individual accommodation plan should be reviewed or updated and how it should be done;
The way in which the reasons for the denial of an individual accommodation plan will be provided to the employee; and
The means of providing the accommodation plan in an accessible format, based on the employee's accessibility needs.
The individual accommodation will also:
Include information regarding accessible formats and communication supports upon request;
Where needed, include individualized workplace emergency response information; and
Outline all other accommodation provided.
7. Performance Management and Career Development and Advancement
Makita Canada Inc. will consider the accessibility needs of employees with disabilities when implementing performance management processes, or when offering career development or advancement opportunities.
Individual accommodation plans will be consulted, as required.
8. Return to Work
Makita Canada Inc. will develop and implement return to work processes for employees who are absent from work due to a disability and require disability-related accommodation(s) in order to return to work.
The return to work process will outline the steps Makita Canada Inc. will take to facilitate the employee's return to work and shall use documented individual accommodation plans (as described in section 28 of the regulation).
9. Redeployment
The accessibility needs of employees with disabilities will be taken into account in the event of redeployment.
10. Review
This policy will be reviewed regularly to ensure that it is reflective of Makita Canada Inc.'s current practices as well as legislative requirements.
AODA – Multi-Year Accessibility Plan
Outlines the plan to improve opportunities for people with disabilities in accordance with the requirements communicated under the Integrated Accessibility Standards, Ontario Regulation 191/11.
Accessibility Requirement: Accessible Website and web content Complianace Deadline:
Current Barriers: Complete website is not currently at WCAG 2.0
Plan to Meet Requirements: April 2020
Potential Future Barriers:
Responsible Authority: Marketing Department
This document was created on October 2019 and must be reviewed and updated by March 2020.
AODA – Training Policy
Providing training for employees on IASR (Integrated Accessibility Standards Regulations [Ontario Regulation 191/11]) and the Ontario Human Rights Code as it pertains to individuals with disabilities.
3. Training Requirements
Makita Canada Inc. will provide training for its employees and regarding the IASR and the Ontario Human Rights Code as they pertain to individuals with disabilities. Training will also be provided to individuals who are responsible for developing Makita Canada Inc.'s policies, and all other persons who provide goods, services or facilities on behalf of Makita Canada Inc.
Training will be provided as soon as is reasonably practicable. Training will be provided on an ongoing basis to new employees and as changes to Makita Canada Inc.'s accessibility policies occur.
4. Records
Makita Canada Inc. will maintain records on the training provided, when it was provided and the number of employees that were trained.
Questions about policies
These policies exists to achieve service excellence to customers with disabilities. If anyone has a question about the policy, or if the purpose of a policy is not understood, please contact:
Manager - Corporate Administration (CorporateAdmin@makita.ca)
Makita Canada Inc.
1950 Forbes Street
Whitby, ON L1N 7B7
P: (905) 571-2200 ext.1410
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Copyright © 2017 Makita Canada Inc. All Rights Reserved - 1950 Forbes Street, Whitby, ON. L1N 7B7, 1 800 263-3734 | {
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} | 4,299 |
<?php
class La_Controller_Action_Helper_Download extends Zend_Controller_Action_Helper_Abstract
{
public function direct($filepath, $filename = null)
{
if (file_exists($filepath)) {
$controller = $this->getActionController();
$controller->getHelper('layout')->disableLayout();
$controller->getHelper('viewRenderer')->setNoRender();
if (!$filename) {
$filename = basename($filepath);
}
$etag = md5_file($filepath);
$size = filesize($filepath);
header('Content-Type: application/octet-stream');
header('Content-Disposition: attachment; filename=' . $filename);
header('Content-Transfer-Encoding: binary');
header('Expires: 0');
header('Etag: ' . $etag);
header('Cache-Control: must-revalidate');
header('Pragma: public');
header('Content-Length: ' . $size);
ob_clean();
flush();
readfile($filepath);
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,487 |
Q: Serialize any type of list to XML? I have 4 entity class . A,B,C,D.It has different properties like ID,Address and even complex type.I want to Write a common Method where i can pass any list and it will convert to XML. Let say
public string GetAnyListtoXML(Any type of list)
{
string myXML=string.Empty;
return myXML;
}
A: This method allow you to serialize whatever you want.
Your entity class should have parameterless constructor.
This link may be helpful to control serialization: http://msdn.microsoft.com/en-us/library/2baksw0z%28v=vs.100%29.aspx
public string ObjectToXml<T>(T obj)
{
var stream = new StringWriter();
string xmlDoc = string.Empty;
try
{
var xmlSerializer = new XmlSerializer(typeof (T));
xmlSerializer.Serialize(stream, obj);
xmlDoc = stream.GetStringBuilder().ToString();
}
catch (Exception ex)
{
Console.WriteLine("Błąd pliku xml: " + ex);
}
finally
{
stream.Close();
}
return xmlDoc;
}
public static T XmlToObject<T>(string xmlDoc)
{
var stream = new MemoryStream();
byte[] xmlObject = Encoding.Unicode.GetBytes(xmlDoc);
stream.Write(xmlObject, 0, xmlObject.Length);
stream.Position = 0;
T message;
var ss = new XmlSerializer(typeof (T));
try
{
message = (T) ss.Deserialize(stream);
}
catch (Exception)
{
message = default(T);
}
finally
{
stream.Close();
}
return message;
}
If you want to this method take only list, you can use it:
public string ObjectToXml<T>(List<T> obj)
{
var stream = new StringWriter();
string xmlDoc = string.Empty;
try
{
var xmlSerializer = new XmlSerializer(typeof (List<T>));
xmlSerializer.Serialize(stream, obj);
xmlDoc = stream.GetStringBuilder().ToString();
}
catch (Exception ex)
{
Console.WriteLine("Błąd pliku xml: " + ex);
}
finally
{
stream.Close();
}
return xmlDoc;
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,129 |
READY TO GO...
DARREN KNOWLES' TASH
Well, the new season is almost here and we're ready to see if our beloved Pools can make it out of the National League at the fifth time of asking.
If we thought this was bad then maybe we should to look at Wrexham's situation as a comparison. Our National League counterparts fell out of the FL in 2008 and haven't made it back since. So there's definitely some hope out there.
As I said in my returning column last month, this is Dave Challinor's first real season in charge of Pools as he came midway through the last one. The squad has again changed with DC taking numerous trialists on pre-season to see if they are good enough to be part of his reshaped squad. He's been busy with no less than 10 signings made in the summer (at the time of writing) which goes to show he knew where the squad needed strengthening. Now it's time for this new look squad to make the town proud again and by gosh it's needed.
"The big question is: can we do it this season?"
One player that some will hope can make an impact is Luke Williams if DC offers him a new deal. Luke, it has to be said, has never had the chance to shine due to one injury after another. On his day he can be a world beater, and could be the difference-maker. If he can remain fit, it's like having a new signing on your books. Fingers crossed Luke, you could be our secret weapon in the months ahead.
The big question is: can we do it this season? Putting my realistic head on, I don't think we will win the title at all. But if the team can stay solid at home and pick up points here and there away then a play-off spot is definitely not beyond them. If I have any concerns, then I think the squad depth is a bit on the lightweight side but finances have probably had an impact on this. DC is going to have juggle what he has at his disposal whilst wheeling and dealing in the loan markets. Unfortunately with the gap between the bigger and smaller clubs widening all the time, this is how it is.
So whilst being exiled here in Northern Ireland, I've been also following the fortunes of two former Poolies south of the border, those being Scott Fenwick and Connor Simpson who are at Cork City. Unfortunately at the time of writing, things aren't going well for City as they're bottom of the table. Both strikers haven't found the net either. There's though a third Poolie connection at Turners Cross, City's assistant manager is Joe Gamble who made 61 appearances in Pools colours between January 2010 and July 2011. I'll be keeping you updated on their performances every month...
Well that's it for now. Strap yourselves in Poolies and get ready for another bumpy ride of emotions. Here's hoping that come May 2021, there is going to be cause for celebration, but Pools being Pools... it certainly won't be easy! Pools never do things the simple way, do they? | {
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} | 9,598 |
\section{Introduction}
\label{Intro}
The unified model of active galactic nuclei (AGN; \citealt{Antonucci1993}) states that most of the observational
differences between type-1 and type-2 Seyfert-like galaxies arise from an orientation effect. According to this theory,
the disappearance of ultraviolet (UV) and optical broad emission features at type-2 inclinations can be explained
by the presence of an obscuring, circumnuclear material along the equatorial plane of the AGN (the so-called dusty torus)
that hides both the central engine and the photoionized broad line regions (BLRs; the low ionization line LIL BLR and
the highly ionized line HIL BLR; \citealt{Rowan1977,Osterbrock1978}). A type-2 viewing angle can then be defined as a
line of sight towards the central source that intercepts the equatorial dust, while type-1 inclination allows a direct view
of the central engine. The observational lack of type-1 AGN with edge-on host galaxy \citep{Keel1980,Lawrence1982} suggests that
dust along Seyfert~1 galaxy discs may obscure the HIL and LIL BLR and make the AGN appear like Seyfert~2s \citep{Maiolino1995}.
The number count of Seyfert~1 objects is thus expected to be small, even if the fraction of type-1 against type-2 AGN in the
nearby Universe still needs to be properly determined. Estimating the orientation of a large sample of AGN is necessary to
verify the assumptions of the unified model, and check whether all the differences between Seyfert~1 and Seyfert~2 objects
can be explained by inclination or if morphological differences must also be taken into account.
In this regard, polarization has proven to be a solid tool to investigate the inner structure of AGN. The spectropolarimetric
measurements of NGC~1068 by \citet{Miller1983} helped to identify electron and dust scattering as the main
mechanisms producing a continuum polarization in radio-quiet AGN. Going further, the extensive high-resolution, high
signal-to-noise spectropolarimetric observation of the same AGN by \citet{Antonucci1985} revealed the presence of
highly polarized, broad, symmetric Balmer and permitted Fe~{\sc ii} lines. The polarization spectrum was found to be closely similar
to typical Seyfert~1 galaxies, supporting the idea that Seyfert~2 AGN are hiding Seyfert~1 core behind the dusty torus. This
discovery was a key argument in favour of a unified model of AGN.
Spectropolarimetry is thus a powerful method to probe the validity of any AGN model, as the computed fluxes shall match both
observational intensity, polarization percentage and polarization angle, reducing the number of free parameters/degeneracies
\citep{Kartje1995,Young2000,Goosmann2007,Marin2012a}. In order to model a peculiar source, the observer's viewing angle $i$ has
to be set (e.g. $i \sim$ 70$^\circ$ for NGC~1068; \citealt{Honig2007,Raban2009}) to explore the resulting polarization
\citep{Goosmann2011,Marin2012c}. The impact of the system's orientation on to the net polarization can lead to significantly
different results, especially when the observer's line of sight matches the half-opening angle of the obscuring region
\citep{Marin2012a}. To be consistent with observation, an investigation of the model over a broad range of inclination
must be undertaken. However, any comparison between the observed polarization and the theoretical orientation of individual AGN
is hampered by the lack of a data base that combines inclination and polarization.
It is the aim of this paper to provide the first spectropolarimetric compendium of Seyfert-like galaxies, gathering
observed continuum polarizations from literature correlated with estimated inclinations of individual AGN. In Sect.~\ref{Comp}, I
investigate the different observational and numerical techniques used to estimate the inclination of 53 objects, and match
the sources with their archival UV/optical polarization measurement, whenever it is feasible. To illustrate the significance
of this catalogue when comparing models to observations, in Sect.~\ref{Analysis}, I pick four different, competitive AGN models
from the literature and analyse them in the framework of this compendium. In Sect.~\ref{Discussion}, I review the
successes and potential improvements of AGN modelling, explore the problematic, high polarization levels of peculiar type-1
objects and discuss possible bias on the estimation of inclination. Finally, conclusions are drawn in Sect.~\ref{Conclusion}.
\section{The compendium}
\label{Comp}
While polarimetric measurement of AGN are difficult to obtain due to intrinsically low polarization degrees in type-1 objects
\citep{Berriman1989,Berriman1990,Smith2002a} and flux dilution by unpolarized starlight in type-2s \citep{Antonucci2002a}, it is
more straightforward to measure polarization than to estimate the inclination of the system, as orientation is not easily derived directly
from observations. Hence, the following sections present different techniques to infer the orientation of AGN (with potential caveats
discussed in Sect.~\ref{Discussion:Bias}), the selection criteria used to select/remove estimated inclinations, corrections that have
to be made to most of the Seyfert-2 AGN and the final compilation of data. For the remaining of this paper, the terminology
``inclination of the system'' will refer to the nuclear, not the host, inclination.
\subsection{Estimation of the system's inclination}
\label{Comp:Incl}
{\sc method A}. Based on the tight correlation found by \citet{Gebhardt2000}, \citet{Ferrrarese2000} and \citet{Merritt2001} between the mass of
the central supermassive black hole and the bulge velocity dispersion in nearby galaxies, \citet{Wu2001} and \citet{Zhang2002} developed
a method to derive the inclination angle of nearby Seyfert~1 AGN. By assuming a Keplerian motion of the LIL BLR and a similar mass/velocity
dispersion between type-1 AGN and regular galaxies, they estimated the orientation angles $i$ for a variety of Seyfert~1 objects, with associated
errors calculated from the uncertainties of both the black hole mass (obtained by reverberation mapping techniques; \citealt{Blandford1982})
and the measured velocity dispersion.
{\sc method B}. The inclination estimation of the Seyfert~1 galaxy ESO~323-G077 comes from the optical spectropolarimetric measurement
achieved by \citet{Schmid2003}, who detected very high levels of linear polarization (up to 7.5~\% at 3600~\AA). Within the framework of
the unified model, those levels are inconsistent with the polarization degrees produced by an object seen in the polar orientation
\citep{Marin2012a}. \citet{Schmid2003} argued that the system must be partially hidden by the dusty torus and tilted by $\sim$~45$^\circ$
with respect to the observer's line of sight to produce such a high polarization degree. The same method was previously applied to
Fairall~51 (continuum polarization 4.12~\% $\pm$ 0.03~\%) by \citet{Schmid2001}, who also derived an inclination of $\sim$~45$^\circ$.
{\sc method C}. An increasing amount of X-ray bright, type-1 AGN shows an asymmetrically blurred emission feature at 6.4~keV, associated with
iron fluorescence in near-neutral material \citep{Reeves2006}. Interestingly, the line broadening caused by Doppler effects and gravitational
plus transverse redshifts can be used to numerically probe the inclination of the system \citep{Fabian1989}. In \citet{Nandra1997},
this characteristic line profile is equally fitted within a Schwarzschild or a Kerr metric (even if recent modelling seems to favour maximally
rotating black holes in the centre of type-1 AGN; \citealt{Bambi2011,Bambi2013}), giving a mean Seyfert~1 inclination of 30$^\circ$.
{\sc method D}. Constraints on the inclination of NGC~1097 are derived by \citet{Storchi1997}, who applied an eccentric accretion ring model
to the observed broad, double-peaked Balmer emission lines. Between 1991 and 1996, the double-peaked H$\alpha$ line of NGC~1097 evolved from a
red-peak dominance \citep{Storchi1993} to a nearly symmetrical profile \citep{Storchi1995} and up to a blue-peak dominance \citep{Storchi1997}.
This line profile evolution can be explained by a refinement of the precessing, planar, elliptical accretion-ring model developed by
\citet{Storchi1995} and \citet{Eracleous1995}, to fit the data using an eccentric accretion disc inclined by 34$^\circ$.
{\sc method E}. \citet{Hicks2008} measured the two-dimensional distribution and kinematics of the molecular, ionized, and highly ionized gas in
the inner regions of a sample of radio-quiet, type-1 AGN using high spatial resolution, near-infrared (IR) spectroscopy. Based on a model developed
by \citet{Macchetto1997}, they assumed that a gravitational well, created by the combined action of a central supermassive black hole and
a distant stellar population, is driving the circular motion of a coplanar thin disc, reproducing the observed emission line gas kinematics.
Exploring four free parameters (disc inclination, position angle of its major axis, black hole mass and mass-to-light ratio), \citet{Hicks2008}
statistically estimated the inclination of NGC~3227, NGC~4151 and NGC~7469 using a Bootstrap Gaussian fit \citep{Efron1979}.
{\sc method F}. To determine the inclination of a sample of nearby AGN, \citet{Fischer2013} explored the three-dimensional geometry and kinematics of
the narrow-line regions (NLRs) of AGN, observing both type-1 and type-2 objects. Resolved by [O~{\sc iii}] imaging and long-slit spectroscopy, most of
the AGN show a bi-conical structure which can be morphologically and kinematically constrained. However, to extract information about the AGN
orientation, a kinematic model must be generated. Using uniform, hollow, bi-conical models with sharp edges, \citet{Fischer2013} were able to
statistically derive a set of morphological parameters (including orientation) for 17 objects out of 53.
{\sc method G}. Relatively bright and situated in the nearby Universe, NGC~1068 is an archetypal Seyfert~2 galaxy, observed during the last fifty years.
Taking advantage of past near and mid-IR photometric and interferometric observations \citep{Jaffe2004,Wittkowski2004}, \citet{Honig2007}
applied a three-dimensional radiative transfer code to a clumpy, dusty structure in order to reproduce the observed spectral energy distribution (SED).
Among new constraints on the bolometric luminosity and the IR optical depth of the torus, \citet{Honig2007} estimated the overall inclination of
NGC~1068 to be close to 70$^\circ$.
{\sc method H}. \citet{Borguet2010} examined the generation of C~{\sc iv} line profiles in broad absorption line (BAL) quasars using a two-component wind model.
By modelling a structure based on stellar wind laws and composed of axisymmetric, polar and equatorial outflows filled with 2-level atoms, they
succeeded to reproduce a large set of BAL profiles and concluded that the viewing angle to the wind is generally large. Unfortunately, degeneracies
in line profile fitting do not allow stronger constraints.
{\sc method I}. Finally, \citet{Wills1992} investigated their own polarimetric and photometric observations of the type-2 quasar IRAS~13349+2438
in the context of an axisymmetric distribution of scatterers to explain the alignment of polarization with the major axis of the host galaxy.
Using a model of a dusty disc parallel to the plane of the galaxy, similar to a usual dusty torus, surrounding the continuum source and the LIL BLR,
they showed that both the observed polarization in the continuum and in the broad H$\alpha$ line could be reproduced if the inclination of the observer
is about 52$^\circ$ with respect to the symmetry axis of their model.
\subsection{Selection criteria}
\label{Comp:Tables}
Once inclinations are obtained, I match them with UV/optical spectropolarimetric measurements, whenever it was possible. The methods presented in
Sect.~\ref{Comp:Incl} derive about 100 AGN orientations but only 53 of them have published continuum polarization measurements. Moreover, not all
of the estimated inclinations are unique and a selection has to be done whenever two methods, or more, give different estimations for the
same AGN. Tab.~\ref{Table:Reject} lists discrepancies of duplicate inclinations. Reasons for the choice of a given inclination are discussed below.
\begin{table*}
\centering
{
\begin{tabular}{|c|c|c|c|}
\hline
{\bf Object} & {\bf Type} & {\bf $i_{\rm sel}$ ($^\circ$)} & {\bf $i_{\rm rej}$ ($^\circ$)} \\
\hline
3C~120 & 1 & 22.0$^{+9.3}_{-7.7}$ \citep{Wu2001} & 88$^{+2}_{-1}$ \citep{Nandra1997} \\
Fairall~9 & 1 & 35.0 \citep{Zhang2002} & 89$^{+1}_{-49}$ \citep{Nandra1997} \\
IC~4329A & 1 & 10$^{+13.0}_{-10.0}$ \citep{Nandra1997} & 5.0 \citep{Zhang2002} \\
Mrk~279 & 1 & 35.0 \citep{Fischer2013} & 13.0 \citep{Zhang2002} \\
Mrk~509 & 1 & 19.0 \citep{Zhang2002} & 89$^{+1}_{-89}$ \citep{Nandra1997} \\
NGC~3227 & 1 & 14.2 $\pm$ 2.5 \citep{Hicks2008} & 15.0 \citep{Fischer2013} \\
~ & ~ & ~ & 21$^{+7}_{-21}$ \citep{Nandra1997}\\
~ & ~ & ~ & 37.5$^{+17.3}_{-25.4}$ \citep{Wu2001}\\
NGC~3516 & 1 & 26$^{+3}_{-4}$ \citep{Nandra1997} & 38.3 $\pm$ 7.6 \citep{Wu2001} \\
NGC~3783 & 1 & 15.0 \citep{Fischer2013} & 40$^{+12}_{-40}$ \citep{Nandra1997}\\
~ & ~ & ~ & 38.0 \citep{Zhang2002} \\
NGC~4051 & 1 & 19.6$^{+10.4}_{-6.6}$ \citep{Wu2001} & 25$^{+12}_{-4}$ \citep{Nandra1997}\\
~ & ~ & ~ & 10.0 \citep{Fischer2013} \\
NGC~4151 & 1 & 9$^{+18}_{-9}$ \citep{Nandra1997} & 19.8 $\pm$ 2.9 \citep{Hicks2008}\\
~ & ~ & ~ & 45 \citep{Fischer2013}\\
~ & ~ & ~ & 60$^{+30}_{-30.6}$ \citep{Wu2001} \\
NGC~5548 & 1 & 47.3$^{+7.6}_{-6.9}$ \citep{Wu2001} & 10$^{+80}_{-10}$ \citep{Nandra1997} \\
NGC~7469 & 1 & 15.0 $\pm$ 1.8 \citep{Hicks2008} & 20$^{+70}_{-20}$ \citep{Nandra1997}\\
~ & ~ & ~ & 13.0 \citep{Zhang2002} \\
NGC~1068 & 2 & 70.0 \citep{Honig2007} & 85.0 \citep{Fischer2013} \\
\hline
\end{tabular}
}
\caption{Selected ($i_{\rm sel}$) and rejected ($i_{\rm rej}$) nuclear inclinations according to the
selection criteria presented in Sect.~\ref{Comp:Incl}.}
\label{Table:Reject}
\end{table*}
~\
{\sc 3c~120}. The inclination of 3C~120 found by \citet{Nandra1997}, $i$~=~88${^\circ}^{+2}_{-1}$, is rejected in favour of the estimation by
\citet{Wu2001}, who found $i$~=~21${^\circ}^{+9.3}_{-7.7}$, a result in a better agreement with the type-1 classification of 3C~120\footnote{3C~120
is a type-1, broad-line, X-ray bright radio galaxy showing an episodic superluminal jet outflow \citep{Marscher2002}. 3C~120 is sometimes
included in radio-quiet AGN surveys as its X-ray spectrum shows a strong relativistic iron K$\alpha$ emission \citep{Nandra1997}.}.
{\sc fairall~9}. Due to the huge error bars derived by \citet{Nandra1997} on the orientation of Fairall~9 (89${^\circ}^{+1}_{-49}$), covering the full
permitted range of inclination for a type-2 object plus a fraction of the permitted range of type 1s, the viewing angle computed by
\citet{Zhang2002} is favored.
{\sc ic~4329a}. The inclination derived by \citet{Zhang2002} in the case of IC~4329A (5.0$^\circ$) is compatible within the error bars of the
estimation computed by \citet{Nandra1997}. The latest (10${^\circ}^{+13.0}_{-10.0}$) is thus selected in order to concur with the two values.
{\sc mrk~279}. Estimated inclinations of Mrk~279 are rather different between \citet{Zhang2002}, $i$~=~13.0$^\circ$, and \citet{Fischer2013},
$i$~=~35.0$^\circ$, especially since they do not have overlapping error bars. The inclination derived by \citet{Fischer2013} is favoured as
\citet{Zhang2002} were not able to recover the measured stellar velocity dispersion of Mrk~279 and had to estimate it from the [O {\sc iii}] emission
line, introducing another potential bias in their final AGN orientation.
{\sc mrk~509}. The inclination estimated by \citet{Nandra1997} covers the whole range of inclination possible for an AGN (89${^\circ}^{+1}_{-89}$)
and therefore does not make much sense. The inclination of Mrk~509 by \citet{Zhang2002} is thus selected.
{\sc ngc~3227}. There are four different estimations for the viewing angle of NGC~3227. Two have very large error bars \citep{Nandra1997,Wu2001}
overlapping the two others estimates by \citet{Hicks2008} and \citet{Fischer2013}. Since the two later inclinations are very similar but from
totally different estimation methods, they are likely to be representative of the real inclination of NGC~3227. Hence I favour the one of
\citet{Hicks2008}, 14.2$^\circ$ $\pm$ 2.5$^\circ$, which agrees with the value found by \citet{Fischer2013},
15.0$^\circ$.
{\sc ngc~3516}. The values estimated by \citet{Nandra1997}, $i$~=~26${^\circ}^{+3}_{-4}$, and \citet{Wu2001}, $i$~=~38.3$^\circ$ $\pm$ 7.6$^\circ$,
are not overlapping but still very close to each other. However, provided that \citet{Wu2001} had to artificially estimate the errors on the
black hole mass for NGC~3516 while \citet{Nandra1997} derived it from their simulation, the estimation made by \citet{Nandra1997} is used.
{\sc ngc~3783}. Similarly to the case of Mrk~279, estimations of the orientation of NGC~3783 are rather different between \citet{Zhang2002} and
\citet{Fischer2013} and, for the same reasons, the value derived by \citet{Fischer2013} is selected. The inclination recovered by \citet{Nandra1997}
for NGC~3783, covering the whole possible inclination range for type-1 objects, is discarded.
{\sc ngc~4051}. Both estimations made by \citet{Nandra1997}, $i$~=~25${^\circ}^{+12}_{-4}$, and \citet{Wu2001}, $i$~=~19.6${^\circ}^{+10.4}_{-6.6}$,
are compatible and their error bars nearly fully overlap. The viewing angle taken from \citet{Wu2001} having slightly higher error bars, this value
is chosen to fully concur with the estimations from \citet{Nandra1997} and to be representative of the value derived by \citet{Fischer2013}, 10.0$^\circ$.
{\sc ngc~4151}. The inclination angle of NGC~4151 derived by \citet{Wu2001} is not considered due to its huge error bars that cover two thirds of
the possible AGN inclinations (60${^\circ}^{+30}_{-30.6}$). The overlapping values, from different methods, found by \citet{Nandra1997}, 9${^\circ}^{+18}_{-9}$,
and \citet{Hicks2008}, 19.8$^\circ$ $\pm$ 2.9$^\circ$, exclude the fourth one derived by \citet{Fischer2013}, 45.0$^\circ$. Finally, the estimation made
by \citet{Nandra1997} is preferred as it fully covers the potential inclination calculated by \citet{Hicks2008}.
{\sc ngc~5548}. Similar to the case of Mrk~509, the inclination of NGC~5548 evaluated by \citet{Nandra1997}, $i$~=~10${^\circ}^{+80}_{-10}$,
is rejected in favour of the one derived by \citet{Wu2001}, $i$~=~47.3${^\circ}^{+7.6}_{-6.9}$.
{\sc ngc~7469}. The viewing angle derived by \citet{Zhang2002} is consistent with the one derived by \citet{Hicks2008}, within the error
bars of the later. The inclination of NGC~7469 estimated by \citet{Nandra1997} is not worth considering due to its huge error bars
(20${^\circ}^{+70}_{-20}$).
{\sc ngc~1068}. Finally, the estimations of \citet{Fischer2013} and \citet{Honig2007} are different and do not overlap. However, the inclination
derived by \citet{Honig2007} is supported by the three-dimensional structure of the nuclear region of NGC~1068 reconstructed by
\citet{Kishimoto1999} using a different set of observations. As \citet{Kishimoto1999} derived a similar, $\sim$~70$^\circ$, inclination, the
estimation of \citet{Honig2007} is thus favoured.
\subsection{Revisited polarization of Seyfert~2s}
\label{Comp:Seyfert2}
Similarly to the inclination estimates in the previous section, I have found for a very few number of objects several continuum polarization
measurements achieved by different authors. The level of polarization and the polarization position angle (measured from north through east) were
coherent between the different observing campaigns, regardless of the epoch. I have thus favoured polarimetric measurement from Seyfert atlases
\citep{Martin1983,Brindle1990,Kay1994,Ogle1999,Smith2002a}.
However, not all of these measurements are reliable. Most, if not all, type-2 AGN are dominated by relatively large, unpolarized starlight fluxes
\citep{Antonucci2002b}. Removing the contribution from old stellar populations drastically increases the resulting continuum polarization but previous
polarimetric type-2 atlases still recorded low, 1 -- 3~\%, polarization degrees \citep{Martin1983,Kay1994,Smith2002b}. Such behaviour is in disagreement
with the unified model, where radiation escapes from the inner parts of the obscuring equatorial torus by perpendicular scattering into the polar outflows,
carrying a large amount of polarization \citep{Antonucci1993}. High polarization percentages are thus expected, but not observed, for type-2 objects
\citep{Miller1990}. \citet{Miller1990} reported that even after starlight subtraction, the remaining continuum is dominated by another component
responsible for dilution of the polarized flux, now identified as originating from starburst regions. To estimate the continuum polarization of a
given object after corrections for the interstellar polarization and dilution by host galaxy starlight, \citet{Tran1995a,Tran1995b,Tran1995c} proposed
to measure the equivalent width (EW) of the broad Balmer lines in both intensity and polarized flux spectra, since polarized flux spectra have the advantage
to suppress low polarization emission from starlight and narrow emission lines. If no additional unpolarized, or very little polarized, continuum
superimposes on the polarization originating from the scattered light alone, the intrinsic polarization in the line and the adjacent continuum should
be equal. This is the case for NGC~1068, where the broad lines have the same polarization as the continuum \citep{Antonucci1994}. Unfortunately, nearly
all the remaining measurements of type-2 AGN polarization are likely to be biased downward.
In the following I revisit and, when necessary and possible, revise the estimated continuum polarization of type-2 objects to be included in the compendium.
The best way to estimate the continuum polarization due only to scattered light is to divide the polarized flux by the total flux across the broad emission lines,
as suggested by \citet{Tran1995c} and \citet{Antonucci2002a}. Only the broad lines polarization is a reliable indicator of the polarization of the scattered
component. However, by definition, broad lines are not detected in the total flux spectra of type-2s. To overcome this problem, I compare the typical
EW of type-1 polarized, broad, Balmer emission lines H$\alpha$~$\lambda$6563 (EW~$\approx$~400~\AA; \citealt{Smith2002a}) and
H$\beta$~$\lambda$4861 (EW~$\approx$~80~\AA; \citealt{Young1997}) to the EW of the polarized H$\alpha$~$\lambda$6563 and H$\beta$~$\lambda$4861 lines of
the compendium-selected Seyfert 2s. This is a first-order approximation, but it is justified by the fact that the broad lines have the same EW in polarized flux in type-1
and type-2 AGN \citep{Antonucci2002b}. If the ratio is equal to unity, the continuum polarization is correct; if larger than unity, the continuum must be
revised by the same factor. In the case where no spectra are available to estimate the EW of broad emission lines in polarized flux, the continuum
polarization reported by previous Seyfert atlases will be used as a lower limit \citep{Goodrich1994}.
~\
I revisit the polarization of the Circinus galaxy measured by \citet{Alexander2000}, 1.9~\%, using the only broad emission line detected
in its polarized flux spectrum, namely H$\alpha$~$\lambda$6563 (H$\beta$~$\lambda$4861 being only marginally detected, $\sim$~2$\sigma$). The EW,
estimated using a Lorentzian profile \citep{Kollatschny2012}, is about 34~\AA. From then, the ratio between type-1 H$\alpha$ EW ($\approx$~400~\AA)
and EW$_{\rm H\alpha~Circinus}$ is equal to 11.8 and the resulting continuum polarization is equal to 22.4~\%. Interestingly, \citet{Oliva1998}
have derived a similar result (25~\%) using a simple, numerical model applied to their own spectropolarimetric measurement of the Circinus galaxy.
Mrk~3 has been observed by \citet{Tran1995a}, who found after correction for interstellar polarization and dilution by starlight a continuum
polarization of 7.0~\%. Using H$\alpha$~$\lambda$6563 and H$\beta$~$\lambda$4861 as diagnostic lines from \citet{Tran1995b}, the EW of H$\alpha$
is estimated to be $\sim$~360~\AA~and EW H$\beta$ $\sim$~65~\AA. The ratio are 1.11 and 1.23, respectively. The scattered light is thus expected
to have an intrinsic polarization of 7.77 -- 8.61~\%.
The small bump around 6563~\AA~in the percentage of polarization spectrum of NGC~1667 was attributed to the red wing of the H$\alpha$ +
[N {\sc ii}] profile by \citet{Barth1999}, who consequently stated that no polarized H$\alpha$ emission have been detected. However, the polarization
spectra are dominated by high noise levels, probably hiding the appearance of the broad wings of the line. The EW of the polarized H$\alpha$~$\lambda$6563
line is consequently very similar to the EW measured in the total flux spectrum, i.e. 14.3~\AA. The resulting ratio, 28.0, can be used to account for
an upper limit of the intrinsic continuum polarization, which is then set between 0.35~\% \citep{Barth1999} and 9.8~\%.
Both H$\alpha$~$\lambda$6563 and H$\beta$~$\lambda$4861 polarized broad lines can be detected in the polarized flux spectrum of NGC~4507
\citep{Moran2000}. The estimated EW are 165~\AA~(H$\alpha$) and 30~\AA~(H$\beta$), raising the continuum polarization from its initial
value (6.1~\%; \citealt{Moran2000}) to 14.8 -- 16.3~\%.
NGC~5506 has been observed by \citet{Kay1994}, who found a continuum polarization of 2.60~\% $\pm$ 0.41~\%. From the spectropolarimetric measurements
of a sample of nearby Compton-thin ($N_{\rm H}~<$~10$^{23}$~cm$^{-2}$) Seyfert 2s, \citet{Lumsden2004} found no evidence for a broad H$\alpha$~$\lambda$6563
line in NGC~5506. They argued that the cause of non-detection of the broad lines in the polarized spectrum of this AGN can be due to an extended
obscuring region rather than non-existence. I estimate an upper limit on the EW for the polarized H$\alpha$~$\lambda$6563 line ($\sim$~10~\AA)
looking at the total flux spectrum and set the corrected polarization degree between 2.6~\% and 100~\%.
\citet{Tran1995b} measured the continuum polarization of NGC~7674 to be 3.8~\% after removing the starlight contribution. From their
spectra, I estimate the EW of the H$\alpha$~$\lambda$6563 and H$\beta$~$\lambda$4861 lines to be 208~\AA~and 40~\AA, respectively.
The ratio are 1.72 and 2.0, increasing the intrinsic polarization of the scattered light to 6.54 -- 7.6~\%.
NGC~1068 has proven to be remarkable in a sense that the polarization degree of its broad emission lines is similar to that of the continuum.
This peculiar feature is also shared by the broad H$\alpha$ line/continuum polarization in IRAS~13349+2438 and no revisions are necessary for
these two objects. The high level of polarization detected in Mrk~78 by \citet{Kay1994}, 21.0~\% $\pm$ 9.0~\%, is significantly different from
the rest of the sample and might face the same physical conditions, if not suffering from an upward bias (since polarization is a positive-definite
quantity). The absence of spectra make it impossible to verify this assumption.
The polarization spectra of the seven BAL quasars extracted from \citet{Ogle1999}, 0019+0107, 0145+0416, 0226--1024, 0842+3431,
1235+1453, 1333+2840 and 1413+1143, are too noisy to measure the EW of isolated broad emission lines in polarized flux. The few broad emission
lines detected in total flux are either merged with some other lines (L$\alpha$ + N~V; Al~III + C~III]), truncated or poorly resolved.
Most of them are not even detected in polarized flux. The reported polarization continuum will then be used as lower limits.
Unfortunately, there are no detailed polarization spectra published so far for Mrk~34, Mrk~573, Mrk~1066 and NGC~5643.
~\
Over the 20 type-2 AGN, 3 of them did not need any correction, 6 were revised and 11 can only be used as basic lower limits for the intrinsic polarization
of the scattered light. Type-1 AGN, once corrected for starlight contribution, do not suffer from an additional dilution component and are thus directly
exploitable. The detailed lists of the AGN sampled, with inclination matched to continuum polarization, are given in Tab.~\ref{Table:Type1} and
Tab.~\ref{Table:Type2}, for type-1 and type-2 objects respectively.
\begin{table*}
\centering
{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline {\bf Object} & {\bf Waveband (\AA)} & {\bf Pol. degree (\%)} & {\bf Pol. angle ($^\circ$)} & {\bf Ref.} & {\bf Inclination ($^\circ$)} & {\bf Ref.} & {\bf Method}\\
\hline 3C~120 & 3800 -- 5600 & 0.92 $\pm$ 0.25 & 103.5 $\pm$ 7.9 & Mar83 & 22.0$^{+9.3}_{-7.7}$ & Wu01 & A\\
Akn~120 & 3800 -- 5600 & 0.65 $\pm$ 0.13 & 78.6 $\pm$ 5.7 & Mar83 & 42.0 & Zha02 & A\\
Akn~564 & 6000 -- 7500 & 0.52 $\pm$ 0.02 & 87.0 $\pm$ 1.3 & Smi02 & 26.0 & Zha02 & A\\
ESO~323-G077 & 3600 & 7.5 & 84 & Sch03 & 45.0 & Sch03 & B\\
Fairall~9 & 3800 -- 5600 & 0.4 $\pm$ 0.11 & 2.4 $\pm$ 7.6 & Mar83 & 35.0 & Zha02 & A\\
Fairall~51 & 4700 -- 7200 & 4.12 $\pm$ 0.03 & 141.2 $\pm$ 0.2 & Smi02 & 45.0 & Sch01 & B\\
IC~4329A & 5000 -- 5800 & 5.80 $\pm$ 0.26 & 42.0 $\pm$ 1.0 & Bri90 & 10$^{+13.0}_{-10.0}$ & Nan97 & C\\
PG~1211+143 & 4700 -- 7200 & 0.27 $\pm$ 0.04 & 137.7 $\pm$ 4.5 & Smi02 & 31.0 & Zha02 & A\\
MCG-6-30-15 & 5000 -- 5800 & 4.06 $\pm$ 0.45 & 120.0 $\pm$ 3.0 & Bri90 & 34.0$^{+5.0}_{-6.0}$ & Nan97 & C\\
Mrk~79 & 3800 -- 5600 & 0.34 $\pm$ 0.19 & 0.4 $\pm$ 16.2 & Mar83 & 58.0 & Zha02 & A\\
Mrk~110 & 3200 -- 8600 & 0.17 $\pm$ 0.08 & 18.0 $\pm$ 15.0 & Ber90 & 37.4$^{+9.2}_{-9.5}$ & Wu01 & A\\
Mrk~279 & 6000 -- 7500 & 0.48 $\pm$ 0.04 & 58.9 $\pm$ 2.4 & Smi02 & 35.0 & Fis13 & F\\
Mrk~335 & 3800 -- 5600 & 0.48 $\pm$ 0.11 & 107.6 $\pm$ 6.9 & Mar83 & 20.0 & Zha02 & A\\
Mrk~478 & 3800 -- 5600 & 0.46 $\pm$ 0.15 & 44.9 $\pm$ 9.5 & Mar83 & 25.0 & Zha02 & A\\
Mrk~486 & 3800 -- 5600 & 3.40 $\pm$ 0.14 & 136.8 $\pm$ 1.2 & Mar83 & 16.0 & Zha02 & A\\
Mrk~509 & 3800 -- 5600 & 1.09 $\pm$ 0.15 & 146.5 $\pm$ 4.0 & Mar83 & 19.0 & Zha02 & A\\
Mrk~590 & 3800 -- 5600 & 0.32 $\pm$ 0.30 & 105.9 $\pm$ 26.6 & Mar83 & 17.8$^{+6.1}_{-5.9}$ & Wu01 & A\\
Mrk~705 & 4700 -- 7200 & 0.46 $\pm$ 0.07 & 49.3 $\pm$ 6.5 & Smi02 & 16.0 & Zha02 & A\\
Mrk~707 & 3800 -- 5600 & 0.20 $\pm$ 0.24 & 140.9 $\pm$ 52.0 & Mar83 & 15.0 & Zha02 & A\\
Mrk~766 & 4500 -- 7100 & 3.10 $\pm$ 0.80 & 90.0 & Bat11 & 36.0$^{+8.0}_{-7.0}$ & Nan97 & C\\
Mrk~841 & 4500 -- 7500 & 1.00 $\pm$ 0.03 & 103.4 $\pm$ 1.0 & Smi02 & 26.0$^{+8.0}_{-5.0}$ & Nan97 & C\\
Mrk~896 & 3800 -- 5600 & 0.55 $\pm$ 0.13 & 1.9 $\pm$ 7.1 & Mar83 & 15.0 & Zha02 & A\\
Mrk~1239 & 3800 -- 5600 & 4.09 $\pm$ 0.14 & 136.0 $\pm$ 1.0 & Mar83 & 7.0 & Zha02 & A\\
NGC~1097 & 5100 -- 6100 & 0.26 $\pm$ 0.02 & 178 $\pm$ 2.0 & Bar99 & 34.0 & Sto97 & D\\
NGC~1365 & 5000 -- 5900 & 0.91 $\pm$ 0.18 & 157 $\pm$ 6.0 & Bri90 & 57.5 $\pm$ 2.5 & Risa13 & C\\
NGC~3227 & 5000 & 1.3 $\pm$ 0.1 & 133 $\pm$ 3.0 & Sch85 & 14.2 $\pm$ 2.5 & Hic08 & E\\
NGC~3516 & 4500 -- 7500 & 0.15 $\pm$ 0.04 & 30.1 $\pm$ 8.0 & Smi02 & 26$^{+3}_{-4}$ & Nan97 & C\\
NGC~3783 & 4500 -- 7500 & 0.52 $\pm$ 0.02 & 135.5 $\pm$ 1.0 & Smi02 & 15.0 & Fis13 & F\\
NGC~4051 & 4500 -- 7500 & 0.55 $\pm$ 0.04 & 82.8 $\pm$ 1.8 & Smi02 & 19.6$^{+10.4}_{-6.6}$ & Wu01 & A\\
NGC~4151 & 4600 -- 7400 & 0.26 $\pm$ 0.08 & 62.8 $\pm$ 8.4 & Mar83 & 9$^{+18}_{-9}$ & Nan97 & C\\
NGC~4593 & 6000 -- 7600 & 0.14 $\pm$ 0.05 & 109.5 $\pm$ 10.8 & Smi02 & 21.6 $\pm$ 10.5 & Wu01 & A\\
NGC~5548 & 6000 -- 7500 & 0.69 $\pm$ 0.01 & 33.2 $\pm$ 0.5 & Smi02 & 47.3$^{+7.6}_{-6.9}$ & Wu01 & A\\
NGC~7469 & 6000 -- 7500 & 0.18 $\pm$ 0.01 & 76.8 $\pm$ 1.7 & Smi02 & 15.0 $\pm$ 1.8 & Hic08 & E\\
\hline
\end{tabular}
}
\caption{Recorded average continuum polarization states and inclinations of 33 type-1 AGN.
The first reference column is related to polarization measurements, the second to estimations
of the orientation. Methods used to determine the inclination of the system are described in Sect.~\ref{Comp:Incl}.
Legend: Mar83 - \citet{Martin1983}; Sch85 - \citet{Schmidt1985}; Ber90 - \citet{Berriman1990};
Bri90 - \citet{Brindle1990}; Nan97 - \citet{Nandra1997}; Sto97 - \citet{Storchi1997};
Bar99 - \citet{Barth1999}; Sch01 - \citet{Schmid2001}; Wu01 - \citet{Wu2001}; Smi02 - \citet{Smith2002a};
Zha02 - \citet{Zhang2002}; Sch03 - \citet{Schmid2003}; Hic08 - \citet{Hicks2008};
Bat11 - \citet{Batcheldor2011}; Fis13 - \citet{Fischer2013} and Ris13 - \citet{Risaliti2013}.}
\label{Table:Type1}
\end{table*}
\begin{table*}
\centering
{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline {\bf Object} & {\bf Waveband (\AA)} & {\bf Pol. degree (\%)} & {\bf Pol. angle ($^\circ$)} & {\bf Ref.} & {\bf Inclination ($^\circ$)} & {\bf Ref.} & {\bf Method}\\
\hline 0019+0107 & 4000 -- 8600 & $>$ 0.98 & 35.0 $\pm$ 0.5 & Ogl99 & 90.0 & Bor10 & H\\
0145+0416 & 1960 -- 2260 & $>$ 2.14 & 126.0 $\pm$ 1.0 & Ogl99 & 80.0 & Bor10 & H\\
0226-1024 & 4000 -- 8600 & $>$ 1.81 & 167.1 $\pm$ 0.2 & Ogl99 & 87.0 & Bor10 & H\\
0842+3431 & 4000 -- 8600 & $>$ 0.51 & 27.1 $\pm$ 0.6 & Ogl99 & 78.0 & Bor10 & H\\
1235+1453 & 1600 -- 1840 & $>$ 0.75 & 175.0 $\pm$ 12.0 & Ogl99 & 76.0 & Bor10 & H\\
1333+2840 & 4000 -- 8600 & $>$ 4.67 & 161.5 $\pm$ 0.1 & Ogl99 & 80.0 & Bor10 & H\\
1413+1143 & 4000 -- 8600 & $>$ 1.52 & 55.7 $\pm$ 0.9 & Ogl99 & 88.0 & Bor10 & H\\
Circinus & 5650 -- 6800 & 22.4 -- 25.0 & 45.0 & Ale00 & 65.0 & Fis13 & F\\
IRAS~13349+2438 & 3200 -- 8320 & 23 -- 35 & 124.0 $\pm$ 5.0 & Wil92 & 52.0 & Wil92 & I\\
Mrk~3 & 5000 & 7.77 -- 8.61 & 167.0 & Tra95 & 85.0 & Fis13 & F\\
Mrk~34 & 3200 -- 6200 & $>$ 3.92 & 53.0 $\pm$ 4.5 & Kay94 & 65.0 & Fis13 & F\\
Mrk~78 & 3200 -- 6200 & 21.0 $\pm$ 9.0 & 75.3 $\pm$ 11.2 & Kay94 & 60.0 & Fis13 & F\\
Mrk~573 & 3200 -- 6200 & $>$ 5.56 & 48.0 $\pm$ 2.0 & Kay94 & 60.0 & Fis13 & F\\
Mrk~1066 & 3200 -- 6200 & $>$ 1.99 & 135.1 $\pm$ 2.6 & Kay94 & 80.0 & Fis13 & F\\
NGC~1068 & 3500 -- 5200 & 16.0 $\pm$ 2.0 & 95.0 & Mill83 & 70.0 & Hon07 & G\\
NGC~1667 & 5100 -- 6100 & 0.35 -- 9.8 & 94.0 $\pm$ 1.0 & Bar99 & 72.0 & Fis13 & F\\
NGC~4507 & 5400 -- 5600 & 14.8 -- 16.3 & 37.0 $\pm$ 2.0 & Mor00 & 47.0 & Fis13 & F\\
NGC~5506 & 3200 -- 6200 & $>$ 2.6 & 72.8 $\pm$ 4.5 & Kay94 & 80.0 & Fis13 & F\\
NGC~5643 & 5000 -- 5900 & $>$ 0.75 & 57.0 $\pm$ 9.0 & Bri90 & 65.0 & Fis13 & F\\
NGC~7674 & 3200 -- 6200 & 6.54 -- 7.6 & 31.0 & Tra95 & 60.0 & Fis13 & F\\
\hline
\end{tabular}
}
\caption{Recorded average continuum polarization states and inclinations of 20 type-2 AGN.
The first reference column is related to polarization measurements, the second to estimations
of the orientation. Methods used to determine the inclination of the system are described in Sect.~\ref{Comp:Incl}.
Legend: Mill83 - \citet{Miller1983}; Bri90 - \citet{Brindle1990}; Wil92 - \citet{Wills1992};
Kay94 - \citet{Kay1994}; Tra95 - \citet{Tran1995a}; Bar99 - \citet{Barth1999}; Ogl99 - \citet{Ogle1999};
Ale00 - \citet{Alexander2000}; Mor00 - \citet{Moran2000}; Hon07 - \citet{Honig2007};
Bor10 - \citet{Borguet2010} and Fis13 - \citet{Fischer2013}.}
\label{Table:Type2}
\end{table*}
\subsection{Inclination versus polarization}
\label{Comp:Survey}
The resulting compendium, comparing the polarization percentage $P$ versus the inclination $i$ for 53 objects, is presented in Fig.~\ref{Fig:Survey}.
Type-1 AGN are shown in red, type-2 AGN in violet.
\begin{figure}
\centering
\includegraphics[trim = 10mm 0mm 0mm 0mm, clip, width=9.2cm]{P_vs_incl.pdf}
\caption{The polarization degree $P$ is plotted versus the
AGN inclination $i$. Type-1 Seyfert-like galaxies are
shown in red, type-2 objects in violet.}
\label{Fig:Survey}
\end{figure}
According to the inclination estimations, type-1 AGN cover a range of orientation from $i$~=~0$^\circ$ (pole-on) to $i$~=~45$^\circ$ -- 60$^\circ$,
which is in agreement with the estimations of the overall half opening angle of the system, $\theta$~$>$~58$^\circ$, made by \citet{Osterbrock1993}
and \citet{Ho1995}. Type-1 objects exhibit low polarization degrees ($\le$~1~\%), except for seven unusually, highly polarized sources (ESO~323-G077,
Fairall~51, IC~4329A, MCG-6-30-15, Mrk~486, Mrk~766 and mrk~1239). Those high levels of polarization are similar to the detection of polarization degrees
up to 4~\% in Mrk~231 \citep{Gallagher2005}, or in the Warm Infrared Ultraluminous AGN survey done by \citet{Hines1994}, but they still need to be
explained (see Sect.~\ref{Discussion:Type1}). However, most of the type-1 sources collected here follow the empirical ascertainment, started with
the observational surveys of Seyfert~1 AGN realized by \citet{Berriman1989}, \citet{Berriman1990} and \citet{Smith2002a}: type-1 AGN predominantly show low levels
of polarization, associated with polarization position angles roughly parallel\footnote{Exceptions in the compendium are NGC~5548, Fairall~51, Mrk~486
\citep{Smith2002a}, ESO~323-G077, NGC~3227, and probably Mrk~766 and NGC~4593 \citep{Batcheldor2011}, which exhibit polarization position angles
perpendicular to their radio axis.} to the radio axis of the system (when radio axis measurements were available). It is particularly interesting
to note that, among the seven polar scattering dominated AGN (with ``perpendicular'' polarization) identified, five exhibit a polarization degree
higher than 1~\%. If a perpendicular polarization position angle can be explained by an increase of the opacity of the polar outflows (through which
the observer's line of sight is passing), it is more difficult to explain such high $P$ at inclinations below 45$^\circ$ \citep{Marin2012a}.
In the case of type-2 AGN, objects show perpendicular polarization position angles associated with higher polarization percentages, not uncommonly
$>$~7~\% (after first-order correction for the intrinsic continuum polarization). As electron-induced $P$ depends on the cosine square of the scattering
angle, Seyfert galaxies seen edge-on are expected to be intrinsically more polarized \citep{Miller1983,Brindle1990,Kay1994}. Radiation scatters
perpendicularly on to the ionized, polar outflows detected in type-2 AGN, leading to higher polarization percentages than at polar inclinations,
where the combination of forward scattering and dilution by the unobscured nucleus diminishes the net polarization. A clear estimation of the average
polarization percentage of Seyfert 2 objects remains problematic since most of the quoted values are lower limits. However, the trend is that $P$ seems
to increase with inclination, with a possible decrease at extreme type-2 inclinations that has to be properly observed by future, rigorous measurements
of the scattered light alone. The overall inclination of Seyfert~2 AGN lies between 60$^\circ$ and 90$^\circ$ (edge-on).
No clear polarization break is detected at the transition angles (45$^\circ$ -- 60$^\circ$) between type-1 and type-2 objects; $P$ increases
continuously from pole-on to edge-on view. Note that the range of orientation that separates type-1 and type-2 Seyfert galaxies is not artificially
enhanced by the selection criteria (see Sect.~\ref{Comp:Tables}), as none of the rejected inclinations cover the 45$^\circ$ -- 60$^\circ$
range. However, the reader is advised to note that possible bias in the estimation of $i$ may shift some objects to different inclination values.
ESO~323-G077 (type-1), NGC~1365 (type-1), IRAS~13349+2438 (type-2) and the Circinus galaxy (ESO~97-G13, type-2) are in the intermediate zone
between the two classifications. While the type-2 classification of the Circinus galaxy is undisputed, the cases of ESO~323-G077, NGC~1365 and
IRAS~13349+2438 are more ambiguous. Spectropolarimetric observations achieved by \citet{Schmid2003} showed that, at least, a fraction of the inner
region of ESO~323-G077 must be hidden behind the torus horizon, classifying this AGN as a borderline Seyfert~1 galaxy. NGC~1365 is another intriguing,
borderline object, showing rapid transition between Compton-thin and Compton-thick regimes due to X-ray eclipses \citep{Risaliti2005}. Such behaviour
leads to a difficult classification of NGC~1365, either type-1 \citep{Schulz1994}, type-1.5 \citep{Veron1980}, type-1.8 \citep{Alloin1981,Risaliti2005}
or even type-2 \citep{Chun1982,Rush1993}. The case of IRAS~13349+2438 is somewhat similar to the previous object. The H$\beta$ line width measured
by \citet{Lee2013} favours a type-2 classification, an argument in contradiction with \citet{Brandt1996} who found that IRAS~13349+2438 shares many
properties with narrow-line Seyfert 1s, though they noted that this object must have a peculiar geometry. Such statement is supported by the
spectropolarimetric observations realized by \citet{Wills1992}, who have shown that the observer's line of sight is probably intercepting part
of the equatorial dusty material, classifying IRAS~13349+2438 as a borderline type-2 AGN. Thus, from a polarimetric point of view and if its
estimated inclination is correct, the Circinus galaxy should be considered as a borderline Seyfert~2 object.
\section{Polarization predictions from theoretical models}
\label{Analysis}
A direct application of this polarization/inclination study concerns radiative transfer in numerical AGN models. While more recent simulations tend to
complexify in terms of morphology, composition and kinematic, testing the relevance of a model against observations is a significant consistency check.
In the following section, I run Monte Carlo simulations on four different AGN models from the literature and compare the polarimetric results to the
compendium.
Simulations of emission, multiple scattering and radiative coupling in complex AGN environments were achieved using {\sc stokes}
\citep{Goosmann2007,Marin2012a}, a public\footnote{http://www.stokes-program.info/} Monte Carlo code including scattering-induced polarization.
The results presented hereafter are representative of each different model, characterized by a unique set of parameters. The resulting wavelength-independent
polarization percentage is integrated over 2000 to 8000~\AA. For consistency, the same input spectrum is used for all the models, namely an isotropic
source emitting an unpolarized spectrum with a power-law SED $F_{\rm *}~\propto~\nu^{-\alpha}$ ($\alpha$~=~1). Finally, an important condition on the models
investigated below is that, at least, they reproduce the expected polarization dichotomy (i.e. parallel polarization position angles for type-1 inclinations
and perpendicular polarization position angles for type-2s).
\subsection{Three-component AGN}
\label{Analysis:3Comp}
According to the axisymmetric unified model \citep{Antonucci1993}, the central supermassive black hole and its accretion disc, that radiates
most of its bolometric luminosity in the UV/optical bands, are obscured by an equatorial, dusty torus. A common hypothesis is that the
funnel of the torus collimates ejection winds in the form of a bi-conical polar outflow. Past spectropolarimetric models of AGN,
composed by a central irradiating source, a dusty torus and a bi-conical, electron-filled wind, showed that only perpendicular polarization
(with respect to the symmetry axis of the torus) can emerge \citep{Kartje1995,Marin2012a}. To introduce the production of parallel polarization
in polar viewing angles, a third, equatorial region lying between the torus and the source has been proposed \citep{Antonucci1984,Young2000}.
This highly ionized, geometrically thin disc can be associated with the accretion flow between the torus and the BLR
\citep{Young2000,Goosmann2007} and is necessary to reproduce the observed polarization dichotomy \citep{Goosmann2007,Marin2012a}.
\begin{figure}
\centering
\includegraphics[trim = 0mm 125mm 120mm 120mm, clip, width=8cm]{Schema.pdf}
\includegraphics[trim = 10mm 0mm 0mm 8mm, clip, width=9.2cm]{Marin_3model.pdf}
\caption{Top: schematic view of the three-component model. The dusty torus
is shown in dark brown.
Bottom: the resulting polarization (black line) of a three-component
model (see \citealt{Marin2012a}) is plotted against observations.
The dashed section corresponds to parallel polarization, the solid line
to perpendicular polarization.}
\label{Fig:3Comp}
\end{figure}
Following the parametrization from \citet{Marin2012a}, the central, unpolarized source is surrounded by an equatorial, scattering flared disc
with a half-opening angle of 20$^\circ$ with respect to the equatorial plane and a Thomson optical depth in the $V$ band of $\tau_{disc}$~=~1.
Along the same plane, an optically thick ($\tau_{torus}~\gg$~1), dusty torus, filled with a standard ``Milky Way'' dust mixture \citep{Mathis1977}
prevents radiation to escape along the equator. An hourglass-shaped, electron-filled region ($\tau_{wind}$~=~0.3) accounts for the polar
ejection flow. The torus and the collimated ionized wind sustain the same half-opening angle, 60$^\circ$ with respect to the symmetry axis of the
model, see Fig.~\ref{Fig:3Comp} (top). The reader may refer to \citet{Marin2012a} for further details about the model.
The wavelength-integrated polarization spectrum of the three-component model is shown in Fig.~\ref{Fig:3Comp} (bottom). From 0$^\circ$ to 60$^\circ$
(where the transition between type-1 and type-2 classification occurs), the three-component model successfully reproduces the average
polarization level expected from type-1 AGN (i.e. $P~\le$~1~\%) as well as a parallel polarization position angle. $P$ rises from pole-on
view to intermediate inclination without exceeding 1~\%. When the observer's line of sight crosses the torus height, $P$ decreases. This is
due to the competition between parallel polarization produced by the equatorial, scattering disc and perpendicular polarization originating from the
torus/polar regions, cancelling each other. Such behaviour is expected in any axisymmetric model and $P$ may decrease down to zero, depending
on the global morphology of the system. Once the equatorial, electron-filled disc disappears behind the torus horizon, $P$ strongly increases,
up to 30~\%, as radiation becomes dominated by perpendicular, Thomson scattering inside the polar outflows. The resulting polarization is not
high enough in the 45$^\circ$ -- 65$^\circ$ range and becomes too strong at large inclinations to fit the majority of type-2 objects.
\subsection{Four-component AGN}
\label{Analysis:4Comp}
A three-component model produces too much polarization at extreme type-2 inclinations. A natural way to decrease the amount of $P$ is to add an
absorbing medium to the previous model. We know from observations that, beyond the dust sublimation radius, the ionized outflows merge continuously
with the dusty environment of the host galaxy, forming the so-called (low-density) NLR \citep{Capetti1996,Capetti1999}. The next step is then to
investigate, in the framework of this compendium, the polarization signature of a four-component model that includes a dust-filled, low opacity
($\tau_{NLR}$~=~0.3) NLR \citep{Marin2012b}. I used the same parametrization as in Sect.~\ref{Analysis:3Comp}, with the NLR bi-cone sustaining
the same half-opening angle as the ionized outflows (Fig.~\ref{Fig:4Comp}, top).
\begin{figure}
\centering
\includegraphics[trim = 80mm 120mm 30mm 120mm, clip, width=8cm]{Schema.pdf}
\includegraphics[trim = 10mm 0mm 0mm 8mm, clip, width=9.2cm]{Marin_4model.pdf}
\caption{Top: schematic view of the four-component model. The model is the same as the
three-component one with the addition of dusty NLR (shown in light brown).
Bottom: the resulting polarization (black line) of a four-component
model (see \citealt{Marin2012b}) is plotted against observations.
The dashed section corresponds to parallel polarization, the solid line
to perpendicular polarization.}
\label{Fig:4Comp}
\end{figure}
The addition of NLR into the modelling of an AGN does not strongly impact the overall polarization signature. It slightly decreases the net polarization in
type-1 viewing angles (see Fig.~\ref{Fig:4Comp}, bottom) but does not alter its polarization position angle due to the large opening angle of the torus.
However, the possibility to reach polarization degrees of few percents (such as for IC~4329A, MCG-6-30-15 or Mrk~766) becomes less likely due to absorption.
The transition between parallel and perpendicular polarization occurs at the same inclination but the net polarization is lower, due to absorption. Finally,
in comparison with observations, $P$ is found to be still too small between 45$^\circ < i <$ 65$^\circ$ ($P~\sim$~1 -- 10~\%).
\subsection{Fragmented media}
\label{Analysis:Uniform:Frag}
Optical and UV observations of the NLR of NGC~1068 \citep{Evans1991,Capetti1995a,Capetti1997,Packham1997} revealed the presence of many
knots of different luminosity in the outflowing gas that can be attributed to inhomogeneities of the medium. Similar results are found for other sources
(e.g. Mrk~3; \citealt{Capetti1995b}), strengthening the idea that AGN outflows may not be a continuous flow \citep{Dai2008}. Due to the torus
compactness, there is less direct evidence for a clumpy torus and most of the suppositions about the fragmented nature of the circumnuclear matter
comes from numerical simulations \citep{Pier1992,Pier1993,Nenkova2002}.
I now investigate a model in which the ionization cones are fragmented, while maintaining a compact dusty torus and an equatorial,
scattering disc responsible for the production of parallel polarization. The equatorial disc and the circumnuclear dusty region retain the same
morphological and composition parameters as in Sect.~\ref{Analysis:3Comp} and Sect.~\ref{Analysis:4Comp}. The fragmented outflows now consist of
2000 electron-filled spheres of constant density ($\tau_{spheres}$~=~0.3; \citealt{Ogle2003}) and radius (filling factor $\sim$~5~\%). The
filling factor is evaluated by summing up the volume of the clumps and dividing the total by the volume of the same unfragmented bi-conical NLR.
A schematic view of the model is presented in Fig.~\ref{Fig:Frag} (top).
\begin{figure}
\centering
\includegraphics[trim = 80mm 180mm 30mm 65mm, clip, width=8cm]{Schema.pdf}
\includegraphics[trim = 10mm 0mm 0mm 8mm, clip, width=9.2cm]{Marin_Frag.pdf}
\caption{Top: schematic view of the clumpy model. The torus and the
equatorial disc are the same as in Fig.~\ref{Fig:3Comp}.
Bottom: the resulting polarization (black line) of a
clumpy model is plotted against observations.
The dashed section corresponds to parallel polarization, the solid line
to perpendicular polarization.}
\label{Fig:Frag}
\end{figure}
Similarly to the three and four-component models, a clumpy AGN (Fig.~\ref{Fig:Frag}, bottom) reproduces both the low polarization levels and the
expected polarization position angle in type-1 orientations. However, due to multiple scattering on the outflow's clumps increasing the polarization
degree, and gaps along the type-1 line of sights that allow a direct view of the electron disc, the net polarization percentage is higher, up to 2~\%
for intermediate inclinations. This level is still not sufficient to reproduce the observed polarization of highly polarized type-1 objects, but
a fragmented medium enables higher polarization degrees for type-1 modelling. Fragmentation also impacts the inclination at which perpendicular polarization
starts to dominate the production of parallel polarization, but the resulting transition inclination is not consistent with the observed polarization
position angle of type-2 AGN. At type-2 inclinations, $P$ is much lower ($P~<$~3~\%) than in previous modellings due to the enhanced escape probability
from the outflows. A fragmented model can match the lower limit on polarization of a large fraction of type-2 AGN but fails to reproduce the high
continuum polarization of NGC~1068, Mrk~78, IRAS~13349+2438 or the Circinus galaxy. However, the cloudlet distribution is probably different for each
individual object and should be adapted case by case, i.e. by increasing/decreasing the filling factor. By increasing the filling factor of the
clumpy outflows, the model will start to behave like the AGN model presented in Sect.~\ref{Analysis:3Comp}, strengthening the production of
perpendicular polarization at type-2 viewing angles and matching higher polarization percentages.
\subsection{A structure for quasars}
\label{Analysis:Elvis}
The hydrostatic equilibrium hypothesis, postulated for the equatorial, toroidal region, is slowly evolving. Based on the pioneering work done by
\citet{Blandford1982}, a hydrodynamical scenario is now considered as an alternative to the usual dusty torus, involving clumps of dusty matter
embedded in a hydromagnetic disc-born wind (see \citealt{Elitzur2006}, and references therein). \citet{Elvis2000} took advantage of this scenario to
build a model which attempts to explain the broad and narrow absorption line regions, as well as the broad emission line region, of type-1 quasars.
In its phenomenologically derived structure, a flow of warm, highly ionized matter (WHIM) arises from an accretion disc in a narrow range of
radii, bent outward and driven into a radial direction by radiation pressure. The model of \citet{Elvis2000} was recently explored by
\citet{Marin2013a,Marin2013b,Marin2013c}, who proposed a number of adjustments to match observed polarization data in the UV and optical bands.
To explore the consistency of the model described by \citet{Elvis2000} and modified by \citet{Marin2013a}, I plotted in Fig.~\ref{Fig:Elvis} (bottom)
the adjusted model proposed by \citet{Marin2013a}. The WHIM bending angle is set to 45$^\circ$ and its collimation angle to 3$^\circ$. It arises
at a distance $r$~=~0.0032~pc from the central source and extends up to 0.032~pc. The Thomson optical depth at the outflow's base and inside the
conical, outflowing direction are, respectively, set to $\tau_{base}$~=~0.02 and $\tau_{flow}$~=~2. A failed wind, composed of cold dust, is
self-shielded from the continuum source by the WHIM. The dusty outflow sustains a half-opening angle of 51$^\circ$ with respect to the symmetry
axis of the system, and a collimation angle of 3$^\circ$. Its opacity along the equator is set to $\tau_{dust}$~=~4. Refer to \citet{Marin2013a}
for details about this choice of parameters.
\begin{figure}
\centering
\includegraphics[trim = 0mm 180mm 120mm 70mm, clip, width=8cm]{Schema.pdf}
\includegraphics[trim = 10mm 0mm 0mm 8mm, clip, width=9.2cm]{Elvis_model.pdf}
\caption{Top: schematic view of the structure for quasar as
proposed by \citet{Elvis2000} and modified by \citet{Marin2013a}.
The WHIM appears in white, the failed dusty wind in brown.
Bottom: the resulting polarization (black line) of the
disc-born wind model is plotted against observations.
The dashed section corresponds to parallel polarization, the solid line
to perpendicular polarization.}
\label{Fig:Elvis}
\end{figure}
From Fig.~\ref{Fig:Elvis} (bottom), it can be seen that the continuum polarization arising at type-1 inclinations follow the same trend as the three
previous models: $P$ reaches a maximum value of 1~\% and cannot account for the highly polarized type-1 objects of the compendium. A local diminution
of $P$ appears at $i$~=~42$^\circ$, when the observer's line of sight crosses the outflowing material. The polarization position angle then switches
from parallel to perpendicular, with respect to the projected symmetry axis of the system. While the transition between parallel and perpendicular
polarization occurs at a smaller $i$ in comparison with previous modelling, it is still coherent with the polarization position angle measurements of
NGC~5548 and ESO~323-G077, which exhibit perpendicular polarization at $i$~=~47.3${^\circ}^{+7.6}_{-6.9}$ and $i$~=~45$^\circ$, respectively.
Beyond 54$^\circ$, when the observer's line of sight no longer passes through the radial outflows, $P$ reaches 10~\% -- 11~\% then slowly decreases
because of the overwhelming impact of dust absorption along the equatorial direction. The polarization predicted by the line-driven wind model
fits nearly all the observational, highly polarized type-2 objects in the 45$^\circ$ -- 65$^\circ$ range and can account for objects with
lower $P$ at extreme inclinations. It is noteworthy that while a disc-born wind is by far the closest model to observations, one must
be cautious as type-2 polarizations have first-order corrections and estimated inclinations are subject to potential biases.
\section{Discussion}
\label{Discussion}
\subsection{AGN modelling within the compendium}
\label{Discussion:Models}
The polarization-versus-inclination study presented in this paper allows a test of the relevance of four different AGN models from the literature.
All of them successfully reproduce both the observed polarization dichotomy and the average polarization percentage of type-1 AGN, but strongly differ
in the 45$^\circ$ -- 90$^\circ$ inclination range. It is then easier to discriminate between several AGN models at intermediate inclinations.
Models composed of uniform, homogeneous reprocessing regions (an equatorial scattering disc, a dusty torus and a pair of collimated cones, with the
possible addition of dusty NLR) tend to create high perpendicular polarization degrees for type-2 AGN, while the polarization produced by a model
with fragmented polar outflows do not extend farther than 3~\% in the same orientation range. Moreover, in the case of the model with clumpy ionization
cones, the transition between parallel and perpendicular polarization happens at typical type-2 inclinations, which is in disagreement with observations.
A refinement of the clumpy model is necessary. A deeper analysis of fragmented media is ongoing, targeting equatorial scattering discs, LIL and HIL BLR,
tori, ionization cones and NLR. Preliminary results show that the polarization degree at type-2 inclinations can rise up to few tens of percent for dense
cloudlet distributions. The transition between parallel and perpendicular polarization is correlated with the half-opening angle of the torus, and a
fragmented circumnuclear region with a half-opening angle of 45$^\circ$ can produce a switch between parallel and perpendicular polarization position
angle at $\sim$~60$^\circ$. The exploration of the parameter space of the models (optical depth, filling factor, covering factor ...) will be considered.
The model of \citet{Elvis2000} is undoubtedly the closest to observation, as it produces both high and low polarization degrees at type-2 viewing angle,
strengthening the hypothesis that at least some undermined fraction of AGN components are wind-like structures.
\subsection{Highly polarized type-1 AGN}
\label{Discussion:Type1}
Even by varying the parameters, none of the model can reach the high polarization levels of the inventoried type-1s ESO~323-G077 (7.5~\%),
Fairall~51 (4.12~\% $\pm$ 0.03~\%), IC~4329A (5.80~\%~$\pm$~0.26~\%), MCG-6-30-15 (4.06~\%~$\pm$~0.45~\%), Mrk~486 (3.40~\%~$\pm$~0.14~\%),
Mrk~766 (3.10~\%~$\pm$~0.80~\%), Mrk~1239 (4.09~\%~$\pm$~0.14~\%) or Mrk~231 ($\sim$~4~\%; \citealt{Gallagher2005}). Results presented in
Sect.~\ref{Comp:Survey} show that their inclination ranges from 0$^\circ$ to 45$^\circ$, indicating that scattering at large angles between
the photon source, the polar winds and the observer is unlikely to be responsible for all the atypically high continuum polarization observed.
As the optical thickness of AGN polar outflows is estimated to be relatively small ($\tau~\ll$~1), the major contribution to scattering-induced
type-1 polarization comes from the equatorial scattering disc. \citet{Goosmann2007} showed that the resulting polarization percentage from an
equatorial, electron-filled disc is fairly low, independently of its half opening angle and Thomson opacity. A major challenge to numerical models
of AGN is to increase the net polarization degree at type-1 viewing angles for isolated cases. It is even more challenging taking into account that
the polarization position angle of five out of the seven objects (ESO~323-G077, Fairall~51, Mrk~486, Mrk~766 and NGC~3227) is found to be
perpendicular to their radio axis, similarly to Mrk~231 \citep{Smith2004}.
There are several potential ways to strengthen the net polarization of the models. One can consider perpendicular scattering between the source, a
reprocessing region located along the equatorial plane, and the observer. A promising target could be the HIL and LIL BLR. Considering the constraints on
the HIL and LIL BLR structure derived by \citet{Kollatschny2013}, using kinematic measurements of emission lines in four nearby AGN, one can estimate the
half-opening angle of the emission line region. According to \citet{Kollatschny2013}, the HILs (i.e. emitted close to the photoionizing source)
originate from a medium with half-opening angle 3.8$^\circ$ -- 26$^\circ$ from the equatorial plane, while the LILs are created in a
structure with a half-opening angle 11$^\circ$ -- 60$^\circ$. Further tests must be achieved to explore the polarization position angle and the amount
of parallel polarization that HIL and LIL BLR can generate but it is unlikely that scattering within an axisymmetrical model can reach up to
few percents at type-1 inclinations.
As stated by \citet{Gaskell2010,Gaskell2011}, breaking the symmetric pattern of irradiation can help to understand the velocity dependence of broad
emission line variability detected in many AGN. Strong off-axis flares could then explain the observed, extremely asymmetric, Balmer lines with broad peak
redshifted or blueshifted by thousands of km$\cdot$s$^{-1}$ \citep{Smith2002a}, and produce higher polarization degrees even at polar orientation. It is
important to test the off-axis flare model as, if correct, the azimuthal phase of the continuum source would play a critical role in the measurement of
the supermassive black hole mass. \citet{Goosmann2013} have recently started the investigation of the off-axis irradiation theory by looking at the velocity
dependence of the polarization of the broad emission lines. Preliminary results, compared to spectropolarimetric data for type-1 AGN from the literature,
indicate that both the degree and position angle of polarization should be affected by asymmetrical emission. The net polarization percentage of the optical
continuum is also slightly stronger. Further modelling will be achieved to explore how far optical and UV continuum polarization can be strengthened by
temporary off-axis irradiation.
Finally, highly polarized AGN are not uncommon in type-1 radio-loud objects, with $P$ up to 45.5~\%~$\pm$~0.9~\% \citep{Mead1990}. The net polarization
is quite variable from the radio to the optical band, often on short time-scales, pointing towards well-ordered magnetic fields surrounding a spatially
small emitting region. While most of the observed polarization of radio-quiet AGN is undoubtedly originating from scattering off small particles
\citep{Stockman1979,Antonucci1984,Antonucci1993}, the high, optical polarization of blazing quasars is thought to be associated with Doppler-boosted
synchrotron emission from relativistic jets pointing towards us. The polarization position angle of blazar cores is usually perpendicular to the jet axis
while a parallel component is detected for emerging superluminal knots \citep{Darcangelo2009}. If electron and dust reprocessing appear to be unable to
reach $P$~$\ge$~3~\% along poloidal directions, the correct interpretation might lie somewhere in the middle. The presence of a sub-parsec, aborted jet
has yet to be proven but could explain the time variability and spectra of Narrow Line Seyfert~1 galaxies \citep{Ghisellini2004} and potentially create
perpendicular polarization degree up to a few percent, while the lines and continuum polarization would be still mainly produced by reprocessing. If an
undetermined fraction of the total polarization of highly polarized type-1 AGN indeed originates from synchrotron emission, the net polarization could be
expected to vary, but with a much smaller amplitude than for synchrotron-dominated, radio-loud objects. Long term monitoring of radio-quiet, highly-polarized
type-1 AGN would then help to evaluate the fraction of polarization arising from reprocessing and from synchrotron emission.
\subsection{Potential caveats on the determination of inclination}
\label{Discussion:Bias}
Estimations made by \citet{Wu2001} and \citet{Zhang2002} are primarily based on the assumptions that Seyfert~1 and normal galaxies follow the same black
hole mass\footnote{To rectify black hole masses obtained from reverberation mapping, a correction factor $f$~=~5 was used by \citet{Ho1999}.} - bulge velocity
dispersion correlation, and that the LIL BLR are in pure Keplerian rotation, coplanar to the system inclination. While detected, the motion and the morphology
of the LIL BLR remain uncertain \citep{Peterson2006}. There are no strong constraints from the emission line profiles as a wide variety of kinematic
models are able to reproduce the non-Gaussian profiles detected in AGN \citep{Bon2009}. The technique of velocity-resolved reverberation mapping
\citep{Gaskell1988} is a step forward and tends to rule out any significant outflow from the AGN, while detecting a slight inflow \citep{Gaskell2007}
and fast Keplerian motion.
Fitting the distorted red wing of the Fe K$\alpha$ fluorescent line in X-ray bright, type-1 AGN can lead to possible bias. The procedure used by
\citet{Nandra1997} assumes that the asymmetrical broadening of the iron line is caused by Doppler and general relativistic effects close to the central
black hole; however, a competitive mechanism was proposed by \citet{Inoue2003,Miller2008,Miller2009} and \citet{Miller2013}. In this scenario,
line broadening occurs at larger distances from the accretion disc, where a distribution of cold, absorbing gas blocks a fraction of the initial
continuum. Transmitted and scattered radiation through the cloudlet environment finally carves out the distorted red wing. If distant absorption
dominates relativistic effects, the estimated inclinations might then be questionable\footnote{In this context, a future X-ray polarimetric mission
would be a solid tool to identify the preponderant mechanism responsible for line distortion \citep{Marin2012d,Marin2013d,Marin2013e}.}.
The method developed by \citet{Fischer2013}, based on the work achieved by \citet{Crenshaw2000} and \citet{Das2005,Das2006}, relies on the nature of
the NLR kinematics to determine the orientation of the system. One of their fundamental hypothesis is consistent with the unified model: AGN are
axisymmetrical objects. In this picture, the NLR structure sustains the same symmetry axis as the dusty torus, which is coplanar with the accretion
disc. Thus, determining the inclination of the NLR is equivalent to determining the orientation of the whole system. However, a recent IR interferometric
campaign carried out by \citet{Raban2009} found that the extended outflows of NGC~1068 are likely to be inclined by 18$^\circ$ with respect to
the obscuring torus axis. If this trend is confirmed, and observed for other Seyfert-like galaxies, the overall AGN picture will become more complex.
Finally, the two-component model produced by \citet{Borguet2010} shows degeneracies between the various parameter combinations. It disallows the
characterization of the outflow geometry in quasars showing broad absorption features, and weakens the constraints brought on the inclination of
the system. The viewing angles of the outflow derived by the authors are quite large, a conclusion shared by \citet{Schmidt1999} and \citet{Ogle1999},
based on optical polarization surveys of BAL quasars, but contested by several other authors \citep{Barvainis1997,Punsly2010}.
In particular, \citet{Punsly2010} showed that two-thirds of the BAL quasars they observed using H$\beta$ line width as a diagnostic are well
represented by objects with gas flowing along polar directions.
\section{Conclusions}
\label{Conclusion}
The first match of 53 AGN inclinations with their intrinsic continuum polarization originating from electron and dust scattering is presented
in this paper. Different techniques to retrieve the nuclear orientation of type-1 and type-2 Seyfert galaxies were presented and discussed,
highlighting their potential caveats. The continuum polarization of several Seyfert-2s was corrected using broad H$\alpha$ and H$\beta$ line
polarization as a reliable indicator of the true polarization of the scattered light, and lower limits were put for the remaining AGN whose
polarization spectra were either noise-saturated or unpublished.
~\
The resulting compendium\footnote{The compendium will be regularly updated and available upon email request.} is in agreement with past
observational/theoretical literature, and warrants additional conclusions and remarks the following.
\begin{itemize}
\item Seyfert 1 AGN are associated with low polarization degrees, $P~\le$~1~\%, and predominantly characterized by a polarization position
angle parallel to the projected radio axis of the system. The inclination of type-1 objects ranges from 0$^\circ$ to 60$^\circ$.
\item Seven type-1s have been identified as polar scattering dominated AGN, i.e. showing a perpendicular polarization position angle.
Among them, five have an atypical continuum polarization higher than 1~\%, mostly associated with 10$^\circ$ -- 45$^\circ$ inclinations.
As scattering-induced polarization is unlikely to produce such high polarization degrees at type-1 orientation, a more elaborate scenario
must be considered.
\item After correction, Seyfert 2 AGN show polarization degrees higher than 7~\% and perpendicular polarization position angle.
Unfortunately, most of the objects have only lower limits. The inclination of type-2 objects is ranging from 47$^\circ$ to 90$^\circ$.
\item The transition between type-1 and type-2 AGN occurs between 45$^\circ$ -- 60$^\circ$. This range of inclination is likely to include
AGN classified as borderline objects, where the observer's line of sight crosses the horizon of the equatorial dusty medium. Four
objects lie in this range, three of them (ESO~323-G077, NGC~1365 and IRAS~13349+2438) being already considered as borderline Seyfert
galaxies from spectroscopic observations. If the estimated inclination of the fourth object (the Circinus galaxy) is correct, it
should be considered as another borderline AGN.
\item The usual axisymmetric AGN models have difficulties to reproduce the trend of polarization with inclination. Fragmenting
the reprocessing regions is helpful to cover a wide range of continuum polarization but a disc-born wind model is found to be
already quite close from observations.A fine tuning of the line-driven disc wind could easily match a substantial fraction of the
reported measurements.
\end{itemize}
~\
Problems determining the inclination of AGN must be taken into consideration but, despite potential caveats, the associated continuum
polarization lies within the margins of past empirical results, consolidating the basis of the compendium. It is then important for
future models, as a consistency check, to reproduce the average continuum polarization, the polarization dichotomy and the transition
between type-1 and type-2 classification (45$^\circ < i <$ 60$^\circ$). By improving the quality of the methods to determine the inclination
of AGN and properly removing the contribution of both stellar and starburst light in future polarimetric measurement of type-2 objects,
it will possible to bring very strong constraints on the morphology, composition and kinematics of AGN. To achieve this goal, a new
UV/optical spectropolarimetric atlas of Seyfert 2s is necessary.
\section*{Acknowledgements}
The author is grateful to the referee Ski Antonucci for his useful and constructive comments on the manuscript. I also acknowledge
the Academy of Sciences of the Czech Republic for its hospitality, and the French grant ANR-11-JS56-013-01 of the project POLIOPTIX
and the COST Action MP1104 for financial support.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,048 |
The Hashish Connection: Iconic Music & Pop Culture Inspired by Hash
by The Hash Corporation | Jul 22, 2021 | Blog
Has your frontal lobe been stimulated lately? If the question sounds vaguely arousing, we'll ask it another way: Have you enjoyed quality hash lately? Because some of your favourite musicians, artists and filmmakers certainly have.
Not that you need to know this, but when you indulge in hashish, billions of neurons in your frontal lobes receive a cue to get up and dance. By stimulating those lobes, Hashish elevates your emotions, memory, attention, and other cognitive functions, creating a sensory cascade that can become the catalyst for many cool feelings, including creativity.
Ah yes, creativity on hash. Like when you composed the perfect sequel to Bohemian Rhapsody, or had the inspiration for the Great Canadian Novel, or discovered the formula for time-travel. Obviously, not every hash-induced epiphany is a homerun.
But some are. In fact, hash-fuelled creativity has helped inspire some of the world's greatest music, art, literature and film. Hashish helped these artists imagine and complete works that have become part of pop-culture history. Through the centuries, musicians, filmmakers, writers, and artists have used hashish to reach higher levels of creativity. Some even made it the subject of their work.
Here are some famous pop-culture artists and moments that owe at least some of their notoriety to hashish. Would the art and the artists have been as good without the buzz? Maybe, maybe not. But it's fair to say the final products we know today wouldn't look or sound quite the same without it.
Charlie Sheen, Platoon (1996)
Platoon (1986): You have to go back a long way to see Charlie Sheen get stoned for the first time. Winner of the Academy Award for Best Picture, this film was an unflinching depiction of Vietnam through the eyes of a young private, Sheen, who was sent to fight a foreign enemy but ended up in the internal war Americans were fighting against themselves. Amidst the ceaseless carnage, Sheen is invited into his sergeant's tent that's thick with authentic eastern hash. Willem Dafoe places a rifle barrel into Sheen's mouth and shotguns smoke through it. For a moment the world makes sense and a warm glow of humanity appears amidst the ugliness of war.
Brad Davis, Midnight Express (1978)
Midnight Express (1978): Spoiler alert – don't smuggle hashish out of Turkey. The film adapts the real-life story of Billy Hayes' imprisonment in a Turkish prison after being caught at the airport with packets of hash strapped to his waist. Life quickly goes from bad to worse for Billy, as he's sentenced to four years in prison, and then resentenced to 30 years. Hash is featured in the pivotal scene at the airport and in prison, treated as a valued commodity by prisoners and guards, and a forbidden evil by the government. The film won the Academy Award for Best Adapted Screenplay and was accused of destroying Turkey's tourism industry because of its negative depiction of Turks.
Oliver Stone: Stone was the screenwriter for both Platoon and Midnight Express, and was the Director of Platoon. He won a Best-Director Oscar for Platoon and Best-Adapted Screenplay Oscar for Midnight Express. He became an advocate for hash and cannabis while serving in Vietnam, a lifetime appreciation that was carried over into many of his films.
"When I was in Vietnam, [hash] made the difference between being human and being a beast. There were a lot of guys who were drinking and doing a lot of the killing that was so unnecessary and raping. The guys who did [hash] were much more conscious of the value of life." – Oliver Stone
Cast of Hair (1968)
Hair: In 1968, a raggedy upstart musical blasted onto the Broadway stage. Its rock-based score gave audiences a decibel-pounding alternative to the stately orchestras playing in other theatres. Hair not only looked and sounded different than any other play, its message was a revolutionary F-YOU! to America's establishment. Amidst the graphic tunes about sex, racism, and the awfulness of Vietnam, there was a little ditty called "Hashish". While Hello Dolly and Funny Girl were playing down the street, audiences at Hair got to hum along to lyrics like, "Hashish, Cocaine, Marijuana, Opium, LSD… shoe polish, cough syrup, peyote…".
Written by two hardcore hippies, Gerome Ragni and James Rado, Hair came straight out of the hash culture of the 1960's, transforming the theatre stage and inspiring the next generations of raw, gritty productions, including Rent.
Some of rock's giants found inspiration and sanctuary in hashish. Bob Dylan used hash as a way to stretch his already prodigious lyrical genius. According to Dylan, "…hash and pot, those things aren't drugs. They just bend your mind a little. I think everyone's mind should be bent once in a while."Who knows – without hashish, Like A Rolling Stone might've been a 3-minute song with just two verses. Aside from his own mystical connection to the substance, we know that Dylan was responsible for introducing hash to another notable group of musicians.
The Beatles met Dylan through a mutual friend in 1964. The five musical icons spent an evening hanging together out in a New York suite where Dylan gave the lads their first hit of hash. Dylan had assumed they'd already had some experience. "What about your song? The one about getting high?"
"Which song?", John asked.
"You know…" he sung, "and when I touch you I get high, I get high…"
"Those aren't the words," John replied sheepishly. "The words are, 'I can't hide, I can't hide, I can't hide…'"The Beatles' menu of substances famously increased from pot to hash to harder chemicals, influencing the sounds and writing on their records from the mid '60s on. Their famous stay in an Indian ashram placed them at ground-zero for some of the planet's highest-quality hash. Its influence is heard in the evolution of their music that began around the time Revolver was released, as the trippy effects of hash began wafting into tunes like Got To Get You Into My Life, Yellow Submarine, and Tomorrow Never Knows. Hashish remained a source of creativity for The Beatles, especially for George Harrison, whose love for the substance culminated with police raiding his home in 1969 and finding large blocks of hashish.
Essaouira, Morocco
Many famous musicians from the 1960s were drawn to Essaouira, a surfing city on the coast of Morocco. The country is one of the world's largest producers of hashish, an attraction for tourists looking to rediscover the spiritual hippie vibe from the 1960s. A trip to Essaouira puts them in good company. Paul Simon, The Rolling Stones, Frank Zappa, Jefferson Airplane and Cat Stevens spent time in the port city. Its most famous legendary visitor was Jimi Hendrix, who spent 11 well-documented days in Essaouira, enjoying its shores, culture, and resin. And of course, Crosby, Stills & Nash immortalized another Morroccan city in Marrakesh Express.
Les Misérables, Victor Hugo
The Club des Hashischins was a group of French intellectuals who dedicated themselves to the experimentation and enjoyment of hashish. Lasting from 1844 to 1849, the Club held monthly seances where they consumed the resin in various forms, often mixed into a green paste with honey, pistachios and fat. Some of France's most renowned writers and poets were among the members, including Victor Hugo, Alexandre Dumas, Charles Baudelaire, Gérard de Nerval, and Honoré de Balzac.
Imagine a world without such classic works as Les Misérables. It's possible that without the creative encouragement of hash, Hugo's magnum opus may never have been written, or would've been considerably different.
So next time you're listening to a Beatles tune, settling in with a classic book, or getting lost in an intense film, consider that somewhere in its creative genealogy a very specific plant resin might have played an inspirational role.
The world is due for the next wave of great art. Better grab that doob and get to work!
Sign up for more info about HASHCO…
The Hash Corporation Announces Ontario Cannabis Store Product Listings 28/10/2021
The Hash Corporation's Premium Craft-Style Products Are Now Available to Ontario Cannabis Retailers Through the OCS 21/10/2021
The Hash Corporation Completes First Commercial Batch of Rosin in Collaboration with Black Rose 16/09/2021
The Hash Corporation Implements Custom-Made System for Making Bubble Hash Jointly Designed with Maratek 14/09/2021
© The Hash Corporation.
We use cookies to optimize our website and services. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 367 |
ACCEPTED
#### According to
Index Fungorum
#### Published in
null
#### Original name
Pholiota pseudoerebia A. Pearson
### Remarks
null | {
"redpajama_set_name": "RedPajamaGithub"
} | 8,946 |
class ToolbarActionsBarBubbleDelegate {
public:
enum CloseAction {
ACKNOWLEDGED,
DISMISSED,
};
virtual void OnToolbarActionsBarBubbleClosed(CloseAction action) = 0;
};
#endif // CHROME_BROWSER_UI_TOOLBAR_TOOLBAR_ACTIONS_BAR_BUBBLE_DELEGATE_H_
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,147 |
using namespace Halley;
CopyFileImporter::CopyFileImporter(ImportAssetType importType, AssetType outputType)
: importType(importType)
, outputType(outputType)
{
}
void CopyFileImporter::import(const ImportingAsset& asset, IAssetCollector& collector)
{
collector.output(asset.assetId, outputType, asset.inputFiles[0].data, asset.inputFiles[0].metadata);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,223 |
\section{Introduction} \label{sec-1}
In this paper, we study the following variant of the multiple zeta value,
\[ \sum_{0<m_1<\cdots <m_r\atop m_i\equiv i\bmod 2}
\frac1{m_1^{k_1}m_2^{k_2}\cdots m_r^{k_r}},\]
which was introduced in \cite[Section 5]{KT2} in connection with a `level 2'
generalization of the zeta function studied by Arakawa and the first named
author in \cite{AK1999}. We regard this value as a level 2 multiple zeta value
because of the congruence condition in the summation and of the (easily proved) fact that
this value can be written as a linear combination of alternating multiple zeta values (also referred to as
Euler sums or colored multiple zeta values)
\begin{equation}\label{eulersum} \sum_{0<m_1<\cdots <m_r}
\frac{(\pm1)^{m_1}(\pm1)^{m_2}\cdots(\pm1)^{m_r}}{m_1^{k_1}m_2^{k_2}\cdots m_r^{k_r}}.
\end{equation}
It turned out that the value with the normalizing factor $2^r$,
\begin{equation}\label{MTV}
T(k_1,k_2,\ldots,k_r):=2^r\!\!\!\sum_{0<m_1<\cdots <m_r\atop m_i\equiv i\bmod 2}
\frac1{m_1^{k_1}m_2^{k_2}\cdots m_r^{k_r}},
\end{equation} was more natural and convenient, and we often refer this value as the `multiple $T$-value' (MTV).
This is in contrast to Hoffman's multiple $t$-value defined by
\begin{equation}\label{hoft} t(k_1,\ldots,k_r)=\sum_{0<m_1<\cdots <m_r\atop \forall m_i\,:\text{ odd}}
\frac1{m_1^{k_1}m_2^{k_2}\cdots m_r^{k_r}},
\end{equation}
which was introduced and studied in his recent paper \cite{Hoffman16} as another variant of
multiple zeta values of level 2. In the next section and in \S\ref{relTt}, we discuss in more detail
the comparison between $T$- and $t$-values.
In the case of depth $r=2$, Tasaka and the first named author studied in \cite{KanekoTasaka}
both versions in connection to modular forms of level 2, and gave some results generalizing the previous
work by Gangle-Kaneko-Zagier \cite{GKZ}. We do not pursue any modular aspects in this paper.
In the following sections, we show several properties of MTVs such as an integral expression,
the duality relation, certain sum formulas, the parity result, and the generating series of
`height one' MTVs, all similar to those properties for classical multiple zeta values (MZVs)
\begin{equation}\label{mzv}
\zeta(k_1,\ldots,k_r)=\sum_{0<m_1<\cdots <m_r}\frac {1}{m_1^{k_1}m_2^{k_2}\cdots m_r^{k_r}},
\end{equation}
and give some conjectures concerning the space spanned by the multiple $T$-values
and a speculation on a basis of multiple zeta values in terms of Hoffman's $t$-values.
\section{The space of multiple $T$-values}\label{sec-2}
As in the study of multiple zeta values, let us introduce the $\mathbb{Q}$-vector space
\[ \mathcal{T}^\sh=\sum_{k=0}^\infty\,\mathcal{T}^\sh_k\] spanned by all MTVs, where
\[ \mathcal{T}^\sh_0=\mathbb{Q},\quad \mathcal{T}^\sh_1=\{0\},\quad
\mathcal{T}^\sh_{k}=\sum_{1\leq r \leq k-1 \atop {{k_1,\ldots,k_{r-1}\geq 1,\ k_r\geq 2} \atop k_1+\cdots+k_r=k}}
\mathbb{Q}\cdot T(k_1,\ldots,k_r)\quad (k\geq 2).
\]
The space $\mathcal{T}^\sh$ becomes a $\mathbb{Q}$-algebra, the product of two MTVs being described by the
{\it shuffle product}. This is clear from the following integral expression of MTVs,
which is exactly parallel to that of multiple zeta values.
For a given tuple of numbers $\varepsilon_i \in \{0,1\}$ ($1\le i\le k$) with $\varepsilon_1=1$
and $\varepsilon_k=0$, set
\begin{equation*}
I(\varepsilon_1,\ldots,\varepsilon_k)=\mathop{\int\cdots\int}\limits_{0<t_1<\cdots <t_k<1}\varOmega_{\varepsilon_1}(t_1)\cdots \varOmega_{\varepsilon_k}(t_k),
\end{equation*}
where
\[\varOmega_0(t)=\frac{dt}{t},\quad \varOmega_1(t)=\frac{2\,dt}{1-t^2}.\]
Recall that an index set ${\bf k}=(k_1,k_2,\ldots,k_r)\in\mathbb{N}^r$ is {\it admissible} if $k_r\ge2$.
This ensures the convergence of the series \eqref{MTV} (as well as \eqref{hoft} and \eqref{mzv}).
\begin{theorem}\label{Th-3-1}\ For any admissible index set $(k_1,k_{2},\ldots,k_{r})$, we have
\begin{equation}
T(k_1,k_{2},\ldots,k_{r})=I(1,\underbrace{0,\ldots,0}_{k_1-1},1,\underbrace{0,\ldots,0}_{k_{2}-1},\cdots,1,\underbrace{0,\ldots,0}_{k_r-1}). \label{integ-exp}
\end{equation}
\end{theorem}
This can be seen by expanding $1/(1-t^2)$ into geometric series and integrate from left to right,
just as in the standard iterated integral expression of the multiple zeta value \eqref{mzv},
which is given in exactly the same form with $\varOmega_1(t)$ replaced by $dt/(1-t)$. We should
remark that this integral expression~\eqref{integ-exp} is essentially given in \cite{Sasaki2012},
although one needs some change of variables to obtain the current form.
From this integral expression, we immediately see that the
{\em same} shuffle product rule holds for MTVs as for MZVs, an example being
\[ T(2)^2=4T(1,3)+2T(2,2)\quad \text{and}\quad \zeta(2)^2=4\zeta(1,3)+2\zeta(2,2). \]
Another immediate consequence of the integral expression is the duality. We state and prove this in the
next section, but remark here that the formula is again exactly the same as the duality formula of
ordinary multiple zeta values. \\
Returning to the space $\mathcal{T}^\sh$, the first question would be the dimension $d_k^T$
over $\mathbb{Q}$ of each subspace $\mathcal{T}^\sh_k$ of weight $k$ elements.
We have conducted numerical experiments with Pari-GP,
and obtained the following conjectural table.
\vspace{7pt}
\begin{center}
\begin{tabular}{|c|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline
$k$ & 0&1&2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13& 14& 15\\
\hline &&\\[-13pt]\hline
$d_k^T$ & 1& 0& 1& 1& 2& 2& 4& 5& 9& 10& 19& 23& 42& 49& 91& 110\\
\hline
\end{tabular}
\end{center}
\vspace{7pt}
Interestingly enough, the Fibonacci-like relation $d_k^T=d_{k-1}^T+d_{k-2}^T$ can be read off
from the table for {\it even} $k$, but no immediate pattern is recognizable for general $k$.
Recall that the conjectural dimension of the space of
alternating MZVs of weight $k$ (spanned by the numbers \eqref{eulersum} with all
possible signs and $k_i$'s with $k_1+\cdots+k_r=k$, with $r$ varying) is given by the Fibonacci number
$F_k$ with $F_0=F_1=1$ and $F_k=F_{k-1}+F_{k-2}$.
We also note that Hoffman in \cite{Hoffman16} conjectures that the dimension
$d_k^t$ of the space spanned by his $t$-values \eqref{hoft} of weight $k$ is given
by the Fibonacci number $F_{k-1}$ (for $k\ge2$).
\vspace{7pt}
\begin{center}
\begin{tabular}{|c|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline
$k$ & 0&1&2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13& 14& 15\\
\hline &&\\[-13pt]\hline
$d_k^t$ & 1& 0& 1& 2& 3& 5& 8& 13& 21& 34& 55& 89& 144& 233& 377& 610\\
\hline
$F_k$ & 1& 1 & 2& 3& 5& 8& 13& 21& 34& 55& 89& 144& 233& 377& 610 & 987\\
\hline
\end{tabular}
\end{center}
\vspace{7pt}
In \S\ref{relTt}, we present some speculations on relations between multiple $T$-values,
Hoffman's $t$-values, and multiple zeta values.
\section{Several identities among $T$-values}\label{sec-5}
In this section, we describe some formulas we have obtained so far.
\subsection{Duality} Denote by ${\bf k}^\dagger$ the usual dual index of an admissible index set ${\bf k}$.
We assume the reader is familiar with the precise definition of the dual index and omit it here.
See for instance the textbook of Zhao \cite{Zh}.
\begin{theorem}\label{Th-3-1}\ For any admissible index set ${\bf k}$, we have
\begin{equation}\label{duality}
T({\bf k}^\dagger)=T({\bf k}).
\end{equation}
\end{theorem}
\begin{proof}
The involution $t\to (1-t)/(1+t)$, which interchanges the differential forms
$\varOmega_0(t)$ and $\varOmega_1(t)$ and sends the interval $(0,1)$ to itself (with opposite orientation),
plays the role for the involution $t\to 1-t$ in the case of MZVs. That is to say, the change of variables
\[ s_i=\frac{1-t_{k-i+1}}{1+t_{k-i+1}}\ \ (1\le i\le k) \]
in \eqref{integ-exp} immediately gives
\[ I(\varepsilon_1,\ldots,\varepsilon_k)=I(1-\varepsilon_k,\ldots,1-\varepsilon_1), \]
which is the required duality.
\end{proof}
\subsection{Sum formulas}
For multiple zeta values, the classical sum formula is widely known and its variants
are enormous (see \cite[Chapter~5]{Zh} for some of them). For our $T$-values, we only have certain formulas in depths 2 and 3. In depth~2, we obtain an analogue of the weighted sum formula of Ohno and Zudilin \cite{OZ},
but in depth~3, we only obtain a formula which looks incomplete to be called as a sum formula.
\begin{theorem}\label{Th-sum1} \
For $k\in \mathbb{Z}_{\geq 3}$, we have
\begin{equation}
\sum_{j=2}^{k-1} 2^{j-1}\,T(k-j,j)=(k-1)T(k). \label{SF-2}
\end{equation}
\end{theorem}
\begin{theorem}\label{Th-sum2}\ For $k\in \mathbb{Z}_{\geq 4}$,
\begin{equation} \sum_{a+b+c=k \atop a, b\ge 1, c\ge 2}T(a,b,c)+\sum_{j=2}^{k-2}T(1,k-1-j,j)=\frac{2}{3}T(2)T(k-2). \label{SF-3}
\end{equation}
\end{theorem}
We give proofs of these two theorems in the next section, and also present a conjectural (weighted)
sum formula in depth 3.
\subsection{Parity result}
The so-called \textit{parity result}, proved in the case of MZVs in \cite{IKZ,Tsu2004},
also holds for MTVs.
\begin{theorem}\label{Th-5-2}\ Let ${\bf k}=(k_1,\ldots,k_{r})$ be an admissible index and assume
its depth $r$ and weight $k_1+\cdots+k_{r}$ are of different parity.
Then $T ({\bf k})$ can be expressed as a $\mathbb{Q}$-linear combination of multiple $T$-values
of lower depths and products of multiple $T$-values with sum of depths not exceeding $r$.
\end{theorem}
This was essentially proved in a previous paper \cite{Tsu2007} of the second named author.
Actually, what we have shown there was a reduction of $T$-values having depth and weight of opposite parity
into a mixture of $T$-values and a certain multiple $L$-values with the character of conductor~$4$ of lower depth.
But by checking carefully the proof of \cite[Theorem~1]{Tsu2007},
we see that Theorem~\ref{Th-5-2} is in principle already proved there. We plan to write a detailed proof
separately in \cite{Tsu-parity}.
\begin{example}\label{Exam-5-3} For the case of depth $2$, we obtain the following formulas.
For $p\ge1$ and $q\ge2$ with $p+q$ odd, we have
\begin{align}
(-1)^q\,T(p,q) & =\binom{p+q-1}{q}T(p+q)\label{eq-5-1}\\
& \quad -\sum_{\mu=1\atop \mu\equiv q\bmod2}^{q-2}\binom{p+\mu-1}{\mu}
\frac{1}{2^{q-\mu}-1}T(p+\mu)T(q-\mu)\notag\\
& \quad -\sum_{\mu=0\atop \mu\equiv p\bmod2}^{p-2}
\binom{q+\mu-1}{\mu}T(p-\mu)T(q+\mu). \notag
\end{align}
We discuss a bit of a special case in depth 3 in \S\ref{relTt}.
\end{example}
\subsection{Height one $T$-values}
It is well-known (\cite{Ao, Dr}, see also \cite{Ohno-Zagier}) that the generating function of the `height one' multiple zeta values
is given in terms of the gamma function:
\[ 1-\sum_{m,n=1}^\infty \zeta(\underbrace{1,\ldots,1}_{n-1},m+1)X^m Y^n
=\frac{\Gamma(1-X)\Gamma(1-Y)}{\Gamma(1-X-Y)}.\]
We can give the following $T$-version of this formula.
\begin{theorem}\label{Th-2-2} We have the generating series identity
\[ 1-\!\!\sum_{m,n=1}^\infty T(\underbrace{1,\ldots,1}_{n-1},m+1)X^m Y^n=
\frac{2\,\Gamma(1-X)\Gamma(1-Y)}{\Gamma(1-X-Y)}F(1-X,1-Y;1-X-Y;-1), \]
where $F(a,b;c;z)$ is the Gauss hypergeometric function and we assume $|X|<1, -1<Y<0$.
\end{theorem}
\begin{proof}
From the integral expression \eqref{integ-exp}, we have
\begin{align*}
T(\underbrace{1,\ldots,1}_{n-1},m+1)&=\mathop{\int\cdots\int}\limits_{0<t_1<\cdots<t_n<u_1<\cdots<u_m<1}
\frac{2dt_1}{1-t_1^2}\,\cdots\,\frac{2t_n}{1-t_n^2}\,\frac{du_1}{u_1}\,\cdots\,\frac{du_m}{u_m} \\
&=\int_{0<t_n<1} \frac1{(n-1)!}\left(\int_0^{t_n} \frac2{1-t^2}\,dt\right)^{n-1}
\frac1{m!}\left(\int_{t_n}^1\frac1u\,du\right)^m\,\frac{2dt_n}{1-t_n^2}\\
& =\frac{1}{(n-1)!\,m!}\int_0^{1}\left\{\log\left( \frac{1+t_n}{1-t_n}\right)\right\}^{n-1}
\left\{\log\left( \frac{1}{t_n}\right)\right\}^{m}\frac{2dt_n}{1-t_n^2}.
\end{align*}
Hence we have
\begin{align*}
& \sum_{m,n=1}^\infty T(\underbrace{1,\ldots,1}_{n-1},m+1)X^m Y^{n-1}\\
& \quad = \int_0^1 \left( \frac{1+t}{1-t}\right)^Y(t^{-X}-1)\frac{2dt}{1-t^2}\\
& \quad = 2\int_0^1 t^{-X}(1-t)^{-Y-1}(1+t)^{Y-1}dt -\int_0^1 \left( \frac{1+t}{1-t}\right)^Y\frac{2dt}{1-t^2}.
\end{align*}
Denote the two integrals on the last line by $I_1$ and $I_2$ respectively. It follows from
the Euler integral
$$F(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_0^1 t^{a-1}(1-t)^{c-a-1}(1-zt)^{-b}dt\quad (0<\Re a <\Re c)$$
that
$$I_1=\frac{\Gamma(1-X)\Gamma(-Y)}{\Gamma(1-X-Y)}F(1-X,1-Y;1-X-Y;-1).$$
As for $I_2$, setting $w=\log\bigl((1+t)/(1-t)\bigr)$,
we have
$$I_2=\int_0^\infty e^{Yw}dw=-\frac{1}{Y}\quad (\text{if}\ Y<0).$$
Thus, multiplying $-Y$ and using $(-Y)\Gamma(-Y)=\Gamma(1-Y)$, we obtain the desired formula.
\end{proof}
\section{Proofs of the sum formulas}
In this section, we prove Theorems~\ref{Th-sum1} and \ref{Th-sum2}, and give a conjectural sum formula
for depth 3.
\subsection{Proof of Theorem~\ref{Th-sum1}}
We use two formulas of the function
\begin{align}
& \psi(k_1,\ldots,k_r;s) =\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}\frac{{A}(k_1,\ldots,k_{r};\tanh (t/2))}{\sinh(t)}\,dt\label{ee-6-1}
\end{align}
which was studied in our previous paper \cite{KT2}. Here, ${A}(k_1,\ldots,k_r;z)$ is
given by
\[
{A}(k_1,\ldots,k_r;z)=2^r\sum_{0<m_1<\cdots <m_r\atop m_i\equiv i\bmod 2}
\frac{z^{m_r}}{m_1^{k_1}\cdots m_{r}^{k_{r}}}.
\]
(In \cite{KT2}, $2^{-r}{A}(k_1,\ldots,k_r;z)$ is denoted by ${\rm Ath}(k_1,\ldots,k_r;z)$.)
The formulas we need are special cases of \cite[Theorems~5.3 and 5.5]{KT2}, which read
(by letting $k=2$, $r\to k-2$ and $m=0$)
\begin{align}
& \psi(\underbrace{1,\ldots,1}_{k-3},2;s) \label{psi}\\
&\quad =-\sum_{j=2}^{k-1}
\binom{s+j-2}{j-1}\,T(k-j,j-1+s)-T(k-1,s)+T(k-1)T(s) \notag
\end{align}
and
\begin{equation}\label{psivalue}
\psi(\underbrace{1,\ldots,1}_{k-3},2;1)=T(1,k-1).
\end{equation}
We also use the fact that the function $\psi(\underbrace{1,\ldots,1}_{k-3},2;s)$ is holomorphic everywhere.
Since each function $T(k-j,j-1+s)$ in the sum on the right of \eqref{psi} is holomorphic at $s=1$,
the remaining sum $-T(k-1,s)+T(k-1)T(s)$ should be holomorphic at $s=1$ (each of
$T(k-1,s)$ and $T(k-1)T(s)$ has a pole of order 1 at $s=1$). To evaluate
the value of $-T(k-1,s)+T(k-1)T(s)$ at $s=1$, we compute the `stuffle
product'
\begin{align}\label{zetaeo}
&\frac12\,T(k-1)\cdot 2^{-s}\zeta(s) \\
&=\sum_{m=1\atop m:\text{odd}}\frac1{m^{k-1}}\,\sum_{n=2\atop n:\text{even}}\frac1{n^s}
=\sum_{0<m<n\atop m:\text{odd,}\,n:\text{even}}\frac1{m^{k-1}n^s}+
\sum_{0<n<m\atop n:\text{even,}\,m:\text{odd}}\frac1{n^s m^{k-1}}\notag \\
&=\frac14\,T(k-1,s)+\zeta^{eo}(s,k-1), \notag
\end{align}
($\zeta^{eo}(s,k-1)$ is the last sum in \eqref{zetaeo})
from which we have
\begin{align*}
&-T(k-1,s)+T(k-1)T(s) \\
&\quad = 4\zeta^{eo}(s,k-1)- 2T(k-1)\cdot 2^{-s}\zeta(s)+T(k-1)T(s) \\
&\quad = 4\zeta^{eo}(s,k-1)- 2T(k-1)\cdot 2^{-s}\zeta(s)+T(k-1)\cdot 2(1-2^{-s})\zeta(s)\\
&\quad = 4\zeta^{eo}(s,k-1)+T(k-1)\cdot 2(1-2^{1-s})\zeta(s).
\end{align*}
We then see that $\zeta^{eo}(s,k-1)$ is finite at $s=1$ and so is
\[ (1-2^{1-s})\zeta(s)=1-\frac1{2^s}+\frac1{3^s}-\frac1{4^s}+\cdots \]
whose value at $s=1$ is $\log2$. Hence we have
\[ \lim_{s\to 1} \left(-T(k-1,s)+T(k-1)T(s)\right)=4\zeta^{eo}(1,k-1)+
(2\log2)T(k-1). \]
To compute the value $\zeta^{eo}(1,k-1)$, we consider the specific alternating MZV
\begin{equation}
\sigma_a(1,k-1)=\sum_{1\leq m<n}\frac{(-1)^{m-1}}{m n^{k-1}}=\left(1-2^{-k+1}\right)\zeta(1,k-1)
-2{\zeta}^{eo}(1,k-1).\label{eq-4-1}
\end{equation}
We use the formula by Borwein et al \cite[\S 4]{Borwein95} (we are using the notation there)
\begin{align*}
\sigma_a(1,k-1)&=(2\log 2)(1-2^{-k+1})\zeta(k-1)-\frac{k-1}2\zeta(k)\\
&+\frac12\sum_{j=2}^{k-2}(1-2^{1-j})(1-2^{j-k+1})\zeta(j)\zeta(k-j)
\end{align*}
and by Euler
\begin{equation}\label{Euler}
\zeta(1,k-1)=\frac{k-1}2\zeta(k)-\frac12\sum_{j=2}^{k-2} \zeta(j)\zeta(k-j)
\end{equation}
to conclude
\begin{align*}
4{\zeta}^{eo}(1,k-1)&=2(1-2^{-k+1})\zeta(1,k-1)-2\sigma_a(1,k-1)\\
&=(2-2^{-k+1})(k-1)\zeta(k)-(4\log 2)(1-2^{-k+1})\zeta(k-1)\\
& \ -\sum_{j=2}^{k-2}\left\{(1-2^{-k+1})+(1-2^{1-j})(1-2^{j-k+1})\right\}\zeta(j)\zeta(k-j)\\
& =(k-1)T(k)-(2\log 2)T(k-1)-\frac{1}{2}\sum_{j=2}^{k-2}T(j)T(k-j).
\end{align*}
We have used $2(1-2^{-m})\zeta(m)=T(m)$ and
\[(1-2^{-k+1})+(1-2^{1-j})(1-2^{j-k+1})=2(1-2^{-j})(1-2^{-k+j}).\]
We therefore have
\[
\lim_{s\to 1} \left(-T(k-1,s)+T(k-1)T(s)\right)=(k-1)T(k)
-\frac{1}{2}\sum_{j=2}^{k-2}T(j)T(k-j), \]
and by letting $s\to1$ in \eqref{psi} together with \eqref{psivalue} we obtain
\begin{equation}
\sum_{j=2}^{k-1} T(k-j,j)+T(1,k-1) =(k-1)T(k)-\frac{1}{2}\sum_{j=2}^{k-2} T(j)T(k-j). \label{interm}
\end{equation}
Now, recall the shuffle product expansion of $T(j)T(k-j)$ has the same form as that of
$\zeta(j)\zeta(k-j)$ given in {\em e.g.} \cite[p.~72, (3)]{GKZ}, which is
$$T(j)T(k-j)=\sum_{\nu=2}^{k-1}\left\{\binom{\nu-1}{j-1}+\binom{\nu-1}{k-j-1}\right\}T(k-\nu,\nu).$$
Summing up, we obtain
\begin{align}\label{sumprodT}
\frac12\sum_{j=2}^{k-2}T(j)T(k-j)&=\frac12\sum_{\nu=2}^{k-1}\left(\sum_{j=2}^{k-2}\left\{\binom{\nu-1}{j-1}
+\binom{\nu-1}{k-j-1}\right\}\right)T(k-\nu,\nu) \\
&=\sum_{\nu=2}^{k-1}\left(\sum_{j=2}^{k-2}\binom{\nu-1}{j-1}\right)T(k-\nu,\nu)\notag\\
&=\sum_{\nu=2}^{k-2}(2^{\nu-1}-1)T(k-\nu,\nu)+(2^{k-2}-2)T(1,k-1).\notag
\end{align}
Here, we have used
\begin{equation*}
\sum_{j=2}^{k-2}\binom{\nu-1}{j-1}=
\begin{cases}
2^{\nu-1}-1 & (\nu\leq k-2),\\
2^{k-2}-2 & (\nu=k-1).
\end{cases}
\end{equation*}
Combining \eqref{interm} and \eqref{sumprodT}, we obtain Theorem~\ref{Th-sum1}.
\begin{remark}\label{Remark-4-7}\
The weighted sum formula for the double zeta values is
\[ \sum_{j=2}^{k-1} 2^{j-1}\,\zeta(k-j,j)=\frac{k+1}{2}\zeta(k). \]
This can also be proved in the same manner as in the above last step by starting with Euler's \eqref{Euler}
and expressing $\zeta(j)\zeta(k-j)$ as a sum of double zeta values by the shuffle
product, and by using the ordinary sum formula.
\end{remark}
\subsection{Triple $T$-values}\label{sec-5-2}
The method of proof here is different from that in the previous subsection and uses
partial fraction decompositions.
We start with a lemma.
\begin{lemma}\label{Lemma-4-8}\ For $q\in \mathbb{N}$, it holds
\begin{align}
&\sum_{l=-\infty}^\infty \sum_{m,n= 0}^\infty \frac{1}{(2l+1)(2m+1)(2n+1)^q(2l+2m+2n+3)}=0. \label{eq-4-2}
\end{align}
\end{lemma}
\begin{proof}
It is well-known that
$$\sum_{l=0}^\infty \frac{\sin((2l+1)x)}{2l+1}=\frac{\pi}{4},$$
which is uniformly convergent for $0<x<\pi$
(see \cite[\S 2.2]{Titch}). Setting $x=\pi/2+\theta$, we have
\begin{equation*}
\lim_{L\to \infty}\sum_{l=-L}^L \frac{(-1)^l e^{(2l+1)i\theta}}{2l+1}=\frac{\pi}{2}\quad \left(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\right),
\end{equation*}
where $i=\sqrt{-1}$. For simplicity we write the limit of the left-hand side as $\sum_{l=-\infty}^{\infty}$.
Hence, for $q\in \mathbb{N}$ and $z\in (-1,1)$, we have
\begin{align*}
0=&\left(\sum_{l=-\infty}^\infty \frac{(-1)^l e^{(2l+1)i\theta}}{2l+1}-\frac{\pi}{2}\right)\sum_{m,n=0}^\infty \frac{(-z)^{m+n}e^{(2m+2n+2)i\theta}}{(2m+1)(2n+1)^q}\\
&=\sum_{l=-\infty}^\infty \frac{(-1)^l}{2l+1}\sum_{m,n= 0}^\infty \frac{(-z)^{m+n}e^{(2l+2m+2n+3)i\theta}}{(2m+1)(2n+1)^q}-\frac{\pi}{2}\sum_{m,n=0}^\infty \frac{(-z)^{m+n}e^{(2m+2n+2)i\theta}}{(2m+1)(2n+1)^q}
\end{align*}
for $-\pi/2<\theta<\pi/2$.
Integrating the both sides from $-t$ to $t$ $(-\pi/2<t<\pi/2)$, we obtain
\begin{align*}
0=&2 \sum_{l=-\infty}^\infty \frac{(-1)^l}{2l+1}\sum_{m,n= 0}^\infty \frac{(-z)^{m+n}\sin((2l+2m+2n+3)t)}{(2m+1)(2n+1)^q(2l+2m+2n+3)}\\
& \quad -\pi\sum_{m,n=0}^\infty \frac{(-z)^{m+n}\sin((2m+2n+2)t)}{(2m+1)(2n+1)^q(2m+2n+2)}.
\end{align*}
We can easily see that the right-hand side is absolutely and uniformly convergent for $t\in [-\pi/2,\pi/2]$ and $z\in [-1,1]$. Hence, letting $t\to \pi/2$ and $z\to 1$, we obtain \eqref{eq-4-2}.
\end{proof}
We can write \eqref{eq-4-2} as
\begin{align}
&\sum_{l,m,n\geq 0} \frac{1}{(2l+1)(2m+1)(2n+1)^q(2l+2m+2n+3)}\label{eq-4-3} \\
& \quad -\sum_{l,m,n\geq 0} \frac{1}{(2l+1)(2m+1)(2n+1)^q(-2l+2m+2n+1)}=0 .\notag
\end{align}
Denote the sums on the left-hand side by $I_1$ and $I_2$ respectively, so that $I_1=I_2$ holds.
We shall write down each $I_1$ and $I_2$ in terms of $T$-values, by using the following partial fraction decomposition formulas.
\begin{lemma}\label{Lemma-4-9}\ For $q\in \mathbb{N}$,
\begin{align}
\frac{1}{xy^q}&=\sum_{j=0}^{q-1}\frac{1}{y^{q-j}(x+y)^{j+1}}+\frac{1}{x(x+y)^q}, \label{eq-4-4}\\
\frac{1}{xyz^q}&=\sum_{j=0}^{q-1}\bigg\{\sum_{\nu=0}^{q-j-1}\frac{1}{z^{q-j-\nu}(x+z)^{\nu+1}(x+y+z)^{j+1}}+\frac{1}{x(x+z)^{q-j}(x+y+z)^{j+1}}\label{eq-4-5}\\
& \qquad +\sum_{\nu=0}^{q-j-1}\frac{1}{z^{q-j-\nu}(y+z)^{\nu+1}(x+y+z)^{j+1}}+\frac{1}{y(y+z)^{q-j}(x+y+z)^{j+1}}\bigg\} \notag\\
& \quad +\frac{1}{x(x+y)(x+y+z)^q}+\frac{1}{y(x+y)(x+y+z)^q}. \notag
\end{align}
\end{lemma}
\begin{proof}
Equation \eqref{eq-4-4} immediately follows from the factorization
\[ \frac1{y^q} -\frac1{(x+y)^q}=\left(\frac1y-\frac1{x+y}\right)
\sum_{j=0}^{q-1}\frac1{y^{q-1-j}(x+y)^j}=\frac{x}{y(x+y)}\sum_{j=0}^{q-1}\frac1{y^{q-1-j}(x+y)^j}. \]
Replacing $y$ by $z$ and $x$ by $x+y$ in \eqref{eq-4-4} and then multiplying $(x+y)/xy=1/x+1/y$, we have
$$
\frac{1}{xyz^q}=\sum_{j=0}^{q-1}\left(\frac{1}{xz^{q-j}}+\frac{1}{yz^{q-j}}\right)\frac{1}{(x+y+z)^{j+1}}+\frac{1}{xy(x+y+z)^q}.
$$
Applying \eqref{eq-4-4} to $1/xz^{q-j}$ and $1/yz^{q-j}$ and writing $1/xy$ as $1/x(x+y)+1/y(x+y)$
in the last term, we obtain \eqref{eq-4-5}.
\end{proof}
\noindent {\it Proof of Theorem~\ref{Th-sum2}.}\quad
Using \eqref{eq-4-5} with $x=2l+1, y=2m+1, z=2n+1$, we readily have
(note $2l+2m+2n+3=x+y+z$)
\begin{align}
I_1&=\frac14\sum_{j=0}^{q-1}\left\{\sum_{\nu=0}^{q-1-j}T(q-j-\nu,\nu+1,j+2)+
T(1,q-j,j+2)\right\}+\frac14T(1,1,q+1)\label{I1}\\
&=\frac14\sum_{a+b+c=q+3 \atop a, b\ge 1, c\ge 2}T(a,b,c)+\frac14\sum_{j=2}^{q+1}T(1,q+2-j,j)
+\frac14T(1,1,q+1).\notag
\end{align}
As for $I_2$, set $d=n-l$ or $e=l-n$ according as $l<n$ or $l\geq n$. Then
\begin{align}
I_2=&\sum_{d\geq 1 \atop l,m\geq 0} \frac{1}{(2l+1)(2m+1)(2d+2l+1)^q(2d+2m+1)}\label{I2}\\
& +\sum_{e,m,n\geq 0} \frac{1}{(2e+2n+1)(2m+1)(2n+1)^q(-2e+2m+1)}.\notag
\end{align}
The first sum on the right is equal to
\begin{align*}
&\sum_{d\geq 1 \atop l\geq 0} \frac{1}{(2l+1)(2d+2l+1)^q}\,\frac1{(2d)}\sum_{m=0}^\infty \left(\frac{1}{2m+1}-\frac{1}{2m+2d+1}\right)\\
&=\sum_{d\geq 1, l\geq 0 \atop 0\le m\le d-1} \frac{1}{(2l+1)(2d+2l+1)^q(2d)(2m+1)}\\
& =\sum_{l,m,k\geq 0} \frac{1}{(2l+1)(2m+1)(2m+2k+2)(2l+2m+2k+3)^q}\quad (d=m+k+1)\\
& = \sum_{l,m,k\geq 0} \frac{1}{(2l+1)(2m+1)(2l+2m+2k+3)^{q+1}}\\
& \qquad +\sum_{l,m,k\geq 0} \frac{1}{(2m+1)(2m+2k+2)(2l+2m+2k+3)^{q+1}}\\
&=\sum_{l,m,k\geq 0} \frac{1}{(2l+1)(2l+2m+2)(2l+2m+2k+3)^{q+1}}\\
&\qquad+\sum_{l,m,k\geq 0} \frac{1}{(2m+1)(2l+2m+2)(2l+2m+2k+3)^{q+1}}\\
& \qquad\qquad +\sum_{l,m,k\geq 0} \frac{1}{(2m+1)(2m+2k+2)(2l+2m+2k+3)^{q+1}}\\
&=\frac38\,T(1,1,q+1).
\end{align*}
The second sum in \eqref{I2} is, by setting $f=m-e$ or $g=e-m$ according as
$e\leq m$ or $e>m$, transformed into
\begin{align*}
&\sum_{e,f,n\geq 0} \frac{1}{(2e+2n+1)(2e+2f+1)(2n+1)^q(2f+1)}\\
& +\sum_{g\geq 1 \atop m,n\geq 0} \frac{1}{(2g+2m+2n+1)(2m+1)(2n+1)^q(-2g+1)}.
\end{align*}
The second sum of this is equal to $-I_1$ as seen by setting $g=l+1$. We write the first sum,
first by separating the terms with $e=0$ and $e>0$, as
\begin{align*}
&\sum_{f,n\geq 0} \frac{1}{(2f+1)^2(2n+1)^{q+1}}
+\sum_{e\geq 1, f,n\geq 0} \frac{1}{(2e+2n+1)(2n+1)^q(2e+2f+1)(2f+1)}\\
&=\frac14 T(2)T(q+1) +\sum_{e\geq 1 \atop n\geq 0} \frac{1}{(2e+2n+1)(2n+1)^q}\,\frac1{(2e)}
\sum_{f=0}^\infty \left(\frac{1}{2f+1}-\frac{1}{2f+2e+1}\right)\\
&=\frac14 T(2)T(q+1) +\sum_{e\geq 1, n\geq 0 \atop 0\le f\le e-1}
\frac{1}{(2e+2n+1)(2n+1)^q(2e)(2f+1)}\\
& =\frac14 T(2)T(q+1)
+\sum_{f,l,n\geq 0}\frac{1}{(2f+1)(2f+2l+2)(2n+1)^q(2f+2l+2n+3)}
\end{align*}
($e=f+l+1$). Using \eqref{eq-4-4} repeatedly, we have
\begin{align*}
& \sum_{f,l,n\geq 0}\frac{1}{(2f+1)(2f+2l+2)(2n+1)^q(2f+2l+2n+3)}\\
& = \sum_{f,l,n\geq 0}\bigg\{ \sum_{j=0}^{q-1}\frac{1}{(2f+1)(2n+1)^{q-j}(2f+2l+2n+3)^{j+2}}\\
&\qquad +\frac{1}{(2f+1)(2f+2l+2)(2f+2l+2n+3)^{q+1}}\bigg\}\\
& = \sum_{f,l,n\geq 0}\sum_{j=0}^{q-1}\bigg\{ \sum_{\nu=0}^{q-j-1}\frac{1}{(2n+1)^{q-j-\nu}(2f+2n+2)^{\nu+1}
(2f+2l+2n+3)^{j+2}}\\
&\qquad\qquad +\frac1{(2f+1)(2f+2n+2)^{q-j}(2f+2l+2n+3)^{j+2}}\biggr\}\\
&\qquad +\frac18 T(1,1,q+1)\\
&=\frac18\sum_{j=0}^{q-1}\left\{\sum_{\nu=0}^{q-1-j}T(q-j-\nu,\nu+1,j+2)
+T(1,q-j,j+2)\right\} + \frac18T(1,1,q+1)\\
&=\frac18\sum_{a+b+c=q+3 \atop a, b\ge 1, c\ge 2}T(a,b,c)+\frac18\sum_{j=2}^{q+1}T(1,q+2-j,j)
+ \frac18T(1,1,q+1).
\end{align*}
We therefore have
\begin{align*}
I_2&=\frac38\,T(1,1,q+1)-I_1+ \frac14 T(2)T(q+1)\\
&+\frac18\sum_{a+b+c=q+3 \atop a, b\ge 1, c\ge 2}T(a,b,c)+\frac18\sum_{j=2}^{q+1}T(1,q+2-j,j)
+ \frac18T(1,1,q+1).
\end{align*}
Combining this and \eqref{I1} together with $I_1=I_2$ and setting $q+3=k$ gives the theorem. \hfill \qed
\begin{example}\label{Ex-3-3}
The case $k=5$ of Theorem~\ref{Th-sum2} is
\begin{align}
& 2T(1,1,3)+2T(1,2,2)+T(2,1,2)=\frac{2}{3}T(2)T(3). \label{SFwt5}
\end{align}
This is not quite parallel to the case of ordinary MZVs, where the identity
\[ 2\zeta(1,1,3)+2\zeta(1,2,2)+\zeta(2,1,2)=2\zeta(2)\zeta(3)-\frac{5}{2}\zeta(5) \]
holds. It is unlikely that the right-hand side is a multiple of $\zeta(2)\zeta(3)$.
\end{example}
\begin{remark}\label{Rem-4-11}\ We may prove \eqref{interm} by a similar argument
starting from
\begin{align*}
0=&\left(\sum_{l=-\infty}^\infty \frac{(-1)^l e^{(2l+1)i\theta}}{2l+1}-\frac{\pi}{2}\right)\sum_{m=0}^\infty \frac{(-z)^{m}e^{(2m+1)i\theta}}{(2m+1)^q}.
\end{align*}
Hence Theorem~\ref{Th-sum2} can be regarded as a triple version of \eqref{interm}.
\end{remark}
We end this section by proposing the following conjecture as an analogue of
Machide's formula \cite[Corollary~4.1]{M}.
\begin{conj}\label{machide} For $k\ge4$, we have
\[ \sum_{a+b+c=k \atop a, b\ge 1, c\ge 2}2^b(3^{c-1}-1)T(a,b,c)=\frac23(k-1)(k-2)T(k). \]
\end{conj}
\section{Relations among multiple $T$-, $t$-, and zeta values}\label{relTt}
If we denote by $\mathcal{T}^*$ the $\mathbb{Q}$-vector space spanned by all Hoffman's multiple
$t$-values, then, as can be directly seen from the definition \eqref{hoft}, the space
$\mathcal{T}^*$ also becomes a $\mathbb{Q}$-algebra by the {\it stuffle} (or {\it harmonic}) product,
an example being $t(2)^2=2t(2,2)+t(4)$. Hence, we have two $\mathbb{Q}$-subalgebas $\mathcal{T}^\sh$ and
$\mathcal{T}^*$ of the algebra of alternating multiple zeta values, one being closed under the shuffle product
and the other under the stuffle product. There are both shuffle and stuffle product structures
on the whole space of alternating multiple zeta values.
It seems that the sum $\mathcal{T}^\sh+\mathcal{T}^*$ does not exhaust all alternating MZVs, and that the
seemingly smaller space $\mathcal{T}^\sh$
is not contained in $\mathcal{T}^*$, as the following table (numerically computed, only up to
weight 8) suggests.
\vspace{7pt}
\begin{center}
\begin{tabular}{|c|r|r|r|r|r|r|r|r|r|} \hline
$k$ & 0&1&2& 3& 4& 5& 6& 7& 8\\
\hline &&\\[-13pt]\hline
$\dim(\mathcal{T}^\sh_k+\mathcal{T}^*_k)$ & 1& 0& 1& 2& 4& 5& 9& 14& 24\\
\hline
$\dim(\mathcal{T}^\sh_k\cap\mathcal{T}^*_k)$& 1& 0 & 1& 1& 1& 2& 3& 4& 6\\
\hline
\end{tabular}
\end{center}
\vspace{7pt}
Let $\mathcal{Z}$ be the space of usual multiple zeta values. The well-known conjectural dimension
(Zagier~\cite{Z}) of the subspace $\mathcal{Z}_k$ of weight $k$
is given by the sequence $d_k$ which satisfies $d_k=d_{k-2}+d_{k-3}$ with $d_0=1, d_1=0, d_2=1$.
\vspace{7pt}
\begin{center}
\begin{tabular}{|c|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline
$k$ & 0&1&2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13& 14& 15\\
\hline &&\\[-13pt]\hline
$d_k$ & 1& 0& 1& 1& 1& 2& 2& 3& 4& 5& 7& 9& 12& 16& 21& 28\\
\hline
\end{tabular}
\end{center}
\vspace{7pt}
It appears that $d_k^T\ge d_k$ holds for all $k$ ($F_{k-1}\ge d_k$ is certainly true, where $F_{k-1}$
is conjectured to be equal to $d_k^t$).
Moreover, we conjecture (also based on our numerical experiments) that the
space $\mathcal{T}^\sh$ as well as $\mathcal{T}^*$ contains the space~$\mathcal{Z}$.
\begin{conj}
Both $\mathcal{T}^\sh$ and $\mathcal{T}^*$ contain $\mathcal{Z}$ as a $\mathbb{Q}$-subalgebra.
\end{conj}
The intersection $\mathcal{T}^\sh\cap\mathcal{T}^*$ seems strictly larger than $\mathcal{Z}$, as the tables above
suggest. If this conjecture is true, then both $\mathcal{T}^\sh$ and $\mathcal{T}^*$ are modules over
$\mathcal{Z}$. What are the structures of these modules?
Specific elements show an interesting pattern.
By definition, the single $T$-values $T(k)$ and $t$-values $t(k)$ are multiples of $\zeta(k)$ and
hence contained in $\mathcal{Z}$. And by our parity result (Theorem~\ref{Th-5-2}), every double $T$-value
of odd weight is also contained in $\mathcal{Z}$.
For higher depths, we conjecture the following. Since we have the duality for $T$-values,
we may restrict ourselves to the $T$-values with depth smaller than or equal to $\text{weight}/2$.
\begin{conj}\label{conj5-2}
1) For even weights, other than the single $T$-value $T(k)$, only $T(p,q,r)$ with $p,r:\text{odd}\ge3$ and $q:\text{even}$ (and their duals) are in $\mathcal{Z}$.
2) If the weight is odd, other than the single and the double $T$-values,
only $T(p,1,r)$ with $p,r:\text{even}$ (and their duals) are in $\mathcal{Z}$.
\end{conj}
Recall that, from our parity result, the triple $T$-value $T(p,q,r)$ of even weight
can be written in terms of single and the double $T$-values. From an explicit formula for such an expression
(see \cite{Tsu-parity} for the detail), we surmise that the following is true.
\begin{conj}\label{conj5-3} For $m\ge1, p\ge1, q\ge2$ with $p+q+m$ even, we have
\[ \sum_{i+j=m\atop i,j\ge0} \binom{p+i-1}{i}\binom{q+j-1}{j}T(p+i,q+j)\,\in\,\mathcal{Z}.\]
\end{conj}
For instance, the case $m=1$ predicts $qT(p,q+1)+pT(p+1,q)\in\mathcal{Z}$.
\begin{remark} Denoting the sum in the conjecture above by $s(p,q,m)$,
the form of the parity reduction for $T(2p+1,2q,2r+1)$ is
\begin{align*}
&T(2p+1,2q,2r+1)\\
&=-\sum_{j=0}^{p-1}T(2p-2j)s(2q-1,2r+1,2j+1)
-\sum_{j=0}^{r-1} T(2r-2j)s(2q,2j+2,2p)\\
&\quad +\text{sum of products of single $T(n)$'s}.
\end{align*}
\end{remark}
\vspace{7pt}
As for $t$-values, we experimentally observe that any $t(k_1,\ldots,k_r)$ with $\forall k_i\ge2$ is in $\mathcal{Z}$.
Among those, we may choose the following elements as linear and algebraic bases of $\mathcal{Z}$.
\begin{conj}
1) A linear basis of the space $\mathcal{Z}_k$ of multiple zeta values of weight $k$ is given by
\[ \{t(2)^n t(k_1,\ldots,k_r)\,\mid\, n,r\ge0, \forall k_i:\text{odd} \ge3,\, 2n+k_1+\cdots+k_r=k \} \]
2) An algebra basis of $\mathcal{Z}$ is given by $t(2)$ and $t(k_1,\ldots,k_r)$ with $\forall k_i:\text{odd} \ge3$
and the sequence $(k_1,\ldots,k_r)$ being Lyndon.
\end{conj}
With the usual order by magnitude, a sequence $(k_1,\ldots,k_r)$ is Lyndon if any right
subsequence $(k_i,\ldots,k_r)$ ($i\ge2$) is greater than $(k_1,\ldots,k_r)$ in lexicographical order.
\begin{remark}
Quite recently, T.~Murakami proved our observation $t(k_1,\ldots,k_r)\in\mathcal{Z}$ if $\forall k_i\ge2$,
by using the motivic method employed in \cite{Gla}. Also he could prove Conjecture~\ref{conj5-3}
or Conjecture~\ref{conj5-2} except the `only' parts.
\end{remark}
\section{Description of the space $\mathcal{T}^\sh_k$ for low weights}
Obviously $\mathcal{T}^\sh_2=\mathbb{Q}\cdot T(2)$ is one dimensional, and by the duality \eqref{duality} the space
\[ \mathcal{T}^\sh_3=\mathbb{Q}\cdot T(3)+\mathbb{Q}\cdot T(1,2) =\mathbb{Q}\cdot T(3)\] is also one dimensional.
At weight 4, we have $T(1,1,2)=T(4)$ by the duality and $T(2,2)=\frac12 T(4)-2T(1,3)$ by
the sum formula \eqref{SF-2}, and thus we see that
\[ \mathcal{T}^\sh_4=\mathbb{Q}\cdot T(4)+\mathbb{Q}\cdot T(1,3). \]
According to our conjecture (see the table in \S2), this would give a basis of $\mathcal{T}^\sh_4$.
We conjecture that the space $\mathcal{T}^\sh_5$ of weight 5 is also two dimensional. By the duality,
we see that $\mathcal{T}^\sh_5$ is spanned by $T(5)$ and elements of depth 2. We have two independent
relations
\begin{align*}
& 4T(1,4)+2T(2,3)+T(3,2)=2T(5), \\
& 2 T(1, 4) + 2T(2,3) + T(3, 2)=4T(1,4)+2T(2,3)+\frac23 T(3,2)
\end{align*}
coming from \eqref{SF-2} and \eqref{SFwt5} (as for the latter, we used the duality on the left-hand side
and the shuffle product on the right), and from these we obtain
\[ T(3,2)=6T(1,4)\quad\text{and}\quad T(2, 3) =T(5)- 5T(1, 4). \]
Hence we conclude
\[ \mathcal{T}^\sh_5=\mathbb{Q}\cdot T(5)+\mathbb{Q}\cdot T(1,4). \]
Already at weight 6, known identities appear not to be enough to reduce the dimension to the
conjectural 4. Using Theorems~\ref{Th-3-1} through \ref{Th-5-2} and relations obtained
by applying the shuffle product to lower weight relations, we may deduce
\begin{align*}
T(1, 2, 3) &= -\frac{25}{12} T(6) + 12 T(1, 5) + 6 T(2, 4) + 2 T(3, 3) - 2 T(1, 1, 4), \\
T(1, 3, 2) &= \frac{55}{12} T(6) - 24 T(1, 5) - 12 T(2, 4) - 4 T(3, 3) - T(1, 1, 4), \\
T(2, 1, 3) &= \frac{55}{12} T(6) - 24 T(1, 5) - 12 T(2, 4) - 4 T(3, 3) - T(1, 1, 4), \\
T(2, 2, 2) &= -\frac{35}4 T(6) + 48 T(1, 5) + 24 T(2, 4) + 8 T(3, 3) + 6 T(1, 1, 4), \\
T(3, 1, 2) &= \frac56 T(6) - T(1, 1, 4), \\
T(4, 2) &= \frac52 T(6) - 8 T(1, 5) - 4 T(2, 4) - 2 T(3, 3).
\end{align*}
One missing relation would be supplied by Conjecture~\ref{conj5-3}, which predicts for instance
\begin{align*} 3T(2,4)+2T(3,3)&=-\frac{15}7 T(6)+\frac{10}7 T(3)^2\\
&=-\frac{15}7 T(6) + \frac{120}7 T(1, 5)+ \frac{60}7 T(2, 4) + \frac{20}7 T(3, 3).
\end{align*}
(Note that the space of multiple zeta values of weight 6 is spanned by $\zeta(6)=\frac{32}{63} T(6)$ and
$\zeta(3)^2=\frac{16}{49}T(3)^2$.) From this we could conclude
\[ \mathcal{T}^\sh_6=\mathbb{Q}\cdot T(6)+\mathbb{Q}\cdot T(1,5)+\mathbb{Q}\cdot T(2,4)+\mathbb{Q}\cdot T(1,1,4). \]
In a similar vein, we may deduce by using proven relations that the space $\mathcal{T}^\sh_7$ is at most 6 dimensional,
and by assuming Conjecture~\ref{machide}, we may reduce the dimension to the
conjectural 5.\\
Since it becomes more and more tedious to write down the parity reduction
explicitly as the depth gets larger, we have not checked if all relations obtained and
conjectured in this paper are enough to give the conjectural upper bound of the
dimension of $\mathcal{T}^\sh_k$ for $k$ greater than 7. Are there any other nicer families of
relations among MTVs, and what is the complete set of linear relations?
\
{\bf Acknowledgements.}\
{This work was supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (S) 16H06336 (Kaneko) and (C) 18K03218 (Tsumura).}
\
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,199 |
CRT monitor acting like an LCD !?
Re: CRT monitor acting like an LCD !?
What would cause this kind of fault !?
Proposed ban on LCD and Plasma screens ?
On Sun, 11 Dec 2005 23:59:38 +1100, "(Just) Allan"
grill. All Sony Trinitron CRT monitors have these 'lines'.
the width of the screen?
may be seen against a white background.
Weird CRT monitor acting like an LCD !
to point these out to people that have never noticed them before.
On Sun, 11 Dec 2005 14:16:01 GMT, "Anthony Fremont"
They are all like this, have been for many years and they've sold plenty.
On Sun, 11 Dec 2005 23:55:31 GMT, "Mitchell McCreath"
over time because of corrosion/contamination.
See if there is another one about a third from the bottom.
I think there wil be one.
"support for shadow mask". This is how Trinitrons ara made.
You're kidding? Mine has that. I thought I inherited a piece of junk.
it is supposed to have faint lines across the screen?
Look up Trinitron on the Internet. This will explain why you have the lines.
A heart monitor for next to nothing!
I could have killed her right there and then!
a dark line and not a gap?
it which was running at the same vertical sync rate.
If it's a used monitor it could be burn in.
lies within - hint "shadow line".
Monitor dies after HDMI to Receiver + DVI experiment, what is HDMI+ethernet cable ???
it sound like they nailed it.
» Bass Link HV DC? | {
"redpajama_set_name": "RedPajamaC4"
} | 7,831 |
{"url":"https:\/\/infoscience.epfl.ch\/record\/158749","text":"## Low-rank Matrix Approximation Using Point-wise Operators\n\nExtracting low dimensional structure from high dimensional data arises in many applications such as machine learning, statistical pattern recognition, wireless sensor networks, and data compression. If the data is restricted to a lower dimensional subspace, then simple algorithms using linear projections can find the subspace and consequently estimate its dimensionality. However, if the data lies on a low dimensional but nonlinear space (e.g., manifolds), then its structure may be highly nonlinear and hence linear methods are doomed to fail. In this paper we introduce a new technique for dimensionality reduction based on point-wise operators. More precisely, let $\\mathbf{A}_{n\\times n}$ be a matrix of rank $k\\ll n$ and assume that the matrix $\\mathbf{B}_{n\\times n}$ is generated by taking the elements of $\\mathbf{A}$ to some real power $p$. In this paper we show that based on the values of the data matrix $\\mathbf{B}$, one can estimate the value $p$ and therefore, the underlying low-rank matrix $\\mathbf{A}$; i.e., we are reducing the dimensionality of $\\mathbf{B}$ by using point-wise operators. Moreover, the estimation algorithm does not need to know the rank of $\\mathbf{A}$.We also provide bounds on the quality of the approximation and validate the stability of the proposed algorithm with simulations in noisy environments.\n\nPublished in:\nIEEE Transactions on Information Theory\nYear:\n2012\nPublisher:\nInstitute of Electrical and Electronics Engineers\nISSN:\n0018-9448\nKeywords:\nLaboratories:","date":"2018-02-25 23:34:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7732205986976624, \"perplexity\": 151.27005190874942}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-09\/segments\/1518891817523.0\/warc\/CC-MAIN-20180225225657-20180226005657-00529.warc.gz\"}"} | null | null |
package repositories.auth
import common.slick.SchemaInitializer
import javax.inject.Inject
import model.auth.facebook.FacebookAuthTable
import model.auth.facebook.FacebookAuthTable.FacebookAuthTable
import slick.lifted.TableQuery
import scala.concurrent.ExecutionContext
class FacebookAuthSchemaInitializer @Inject()(implicit val executionContext: ExecutionContext)
extends SchemaInitializer[FacebookAuthTable] {
override val name: String = FacebookAuthTable.name
override val table = TableQuery[FacebookAuthTable]
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 106 |
Home • Encephalitozoon romaleae SJ-2008
Gene Ontology PFAM Domains KEGG KOG CAZymes Peptidases Transporters Transcription Factors
Encephalitozoon romaleae SJ-2008
Please confirm that you want to SAVE all your changes for 'Encephalitozoon romaleae SJ-2008'.
Please confirm that you want to DISCARD all your changes for 'Encephalitozoon romaleae SJ-2008'.
Phylogenetic tree of Microsporidia including E. romaleae.
The genome sequence and gene predictions of Encephalitozoon romaleae were not determined by the JGI, but were downloaded from NCBI and have been published (Jean-François Pombert, Mohammed Selmanb, Fabien Burki et al., 2012). Please note that this copy of the genome is not maintained by the author and is therefore not automatically updated.
Microsporidia are highly derived relatives of fungi that are obligate intracellular parasites of virtually all animal lineages and which lead to a number of economically and medically important diseases, particularly in sericulture and apiculture. To date, more than 1,200 microsporidian species have been described, and at least 13 of these species infect humans; many are opportunistic pathogens found in immune-compromised patients. The group is distinguished by a number of cellular characteristics, including the presence of a specialized host-invasion apparatus (the polar tube), an unconventional Golgi apparatus, and highly reduced mitochondria called "mitosomes".
The smallest (nonorganellar) nuclear genomes currently known are those of microsporidian species in the genus Encephalitozoon, making them a model for extreme reductive forces in nuclear genome evolution. Complete genome sequences from two species were found to encode about 2,000 genes making up a reduced set of sometimes simplified molecular and biochemical pathways. Their high degree of host dependence also is reflected in the relatively large number of transporters encoded in these genomes (e.g., ATP transporters), which allow them to acquire essential energy and nutrients from their hosts. Some of these transporters are thought to have originated by horizontal gene transfer (HGT), possibly from coexisting bacterial pathogens, and the recent finding of an animal-derived gene in both Encephalitozoon romaleae and Encephalitozoon hellem also raised the intriguing possibility that microsporidia can acquire genes from their hosts.
Pombert JF, Selman M, Burki F, Bardell FT, Farinelli L, Solter LF, Whitman DW, Weiss LM, Corradi N, Keeling PJ
Gain and loss of multiple functionally related, horizontally transferred genes in the reduced genomes of two microsporidian parasites.
Proc Natl Acad Sci U S A. 2012 Jul 31;109(31):12638-43. doi: 10.1073/pnas.1205020109 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,993 |
\section{Introduction}
\setcounter{equation}{0}
The integrable many-body systems discovered by Calogero and Sutherland around 40 years ago
still enjoy extraordinary popularity due to the wealth of their physical applications and
connections to mathematics, which are described in the surveys \cite{Nekr, Per,Banff,Suth}.
In correspondence to the many variants of these systems (associated with different
interaction potentials,
root systems, relativistic deformations, inclusion of two types
of particles, and so on) there exist also
several approaches to studying them.
The systems based on trigonometric/hyperbolic interaction potentials
are usually called Sutherland type and here we study a particular case of such systems
in the classical Hamiltonian reduction framework, reviewed for example in \cite{Per}.
A Sutherland type integrable system describing the interaction
of $m$ `positively charged' and
$(n-m)$ `negatively charged' particles was first introduced by Calogero
\cite{Cal} by means
of shifting the positions of $m$ out of the $n$ particles by $\ri \frac{\pi}{2}$.
This trick converts the repulsive interaction potential $\sinh^{-2}(q_j-q_k)$ into the attractive
potential $-\cosh^{-2}(q_j-q_k)$ between the particles of opposite charge (indexed say by
$1\leq j \leq m < k \leq n$).
Then Olshanetsky and Rogov \cite{OR} derived
the Calogero-Sutherland model by Hamiltonian reduction of free
motion on an affine symmetric space.
The dynamics of the model and its relativistic deformation was
analyzed in detail by Ruijsenaars \cite{RIMS94}, motivated mainly by the relation of this problem
to the interaction of sine-Gordon solitons and anti-solitons.
In a little noticed paper Hashizume \cite{Ha} generalized the Olshanetsky-Rogov derivation and
thereby proved the integrability of a family of hyperbolic Sutherland models associated
to the so-called root systems with signature.
In the case of the $BC(n)$ root system, his model involves two types of particles
moving on the half-line with the interaction governed by two independent coupling parameters.
It is well-known that integrable $BC(n)$ Sutherland models involve in general three arbitrary
couplings corresponding to the three different root lengths.
However, it has been explained only rather recently, by Pusztai and one of us \cite{FP1},
how the three couplings arise in the setting of Hamiltonian reduction.
In the present paper, we generalize the result of \cite{FP1} and derive the following
$BC(n)$ type Sutherland Hamiltonian,
\bea
&& H= \frac{1}{2} \sum_{j=1}^n p_j^2
- \sum_{1\leq j\leq m<k \leq n} (\frac{\kappa^2}{\cosh^2(q_j - q_k)} +
\frac{\kappa^2}{\cosh^2(q_j + q_k)} )
\nonumber\\
&&\phantom{X}
+ \sum_{1\leq j < k \leq m} ( \frac{\kappa^2 }{\sinh^2(q_j - q_k)} +
\frac{\kappa^2 }{\sinh^2(q_j + q_k)} )
+ \sum_{m<j< k\leq n} (\frac{\kappa^2}{\sinh^2(q_j - q_k)} + \frac{\kappa^2}{\sinh^2(q_j + q_k)} )
\nonumber\\
&&\phantom{XXX}
+ \frac{1}{2}\sum_{j=1}^n \frac{(x - y)^2}{\sinh^2(2 q_j)}
+ \frac{1}{2}\sum_{j=1}^m \frac{xy}{\sinh^2(q_j)} -
\frac{1}{2}\sum_{j=m+1}^n \frac{ xy }{\cosh^2( q_j)},
\label{I1}\eea
where $\kappa>0$, $x$ and $y$ are real coupling constants.
If $(x^2 - y^2)\neq 0$, then energy conservation
ensures that the corresponding dynamics can be consistently restricted
to the domain where
\be
q_1>q_2>...>q_m>0\quad\hbox{and}\quad
\quad
q_{m+1}> q_{m+2} >...> q_n>0.
\ee
Supposing also that $xy>0$,
the Hamiltonian (\ref{I1}) describes attractive-repulsive
interactions between $m$ `positively charged' and $(n-m)$ `negatively charged'
particles influenced also by their mirror images and a positive charge
fixed at the origin.
We shall derive
the model (\ref{I1}) by reduction of the free geodesic motion
on the group $G= SU(n,n)$.
Our reduction relies on a symmetry group of the form
$G_+ \times G^+$, where $G_+<G$ is a maximal compact subgroup and $G^+<G$ is
the (non-compact)
fixed point subgroup of a $G$-involution that commutes with the
Cartan involution fixing $G_+$.
Hashizume \cite{Ha} reduced the geodesic motion on affine symmetric spaces
such as $G/G^+$, which itself is the reduction of the free motion on $G$
associated with the zero value of
the moment map of the $G^+$ symmetry.
He obtained the model (\ref{I1}) with two independent
couplings, while we obtain it with three
arbitrary couplings since we use non-trivial
one-point coadjoint orbits of $G^+$ having a free parameter
(corresponding to $y$ in (\ref{I1})) to define our reduction.
In \cite{FP1} (see also \cite{Obl,FP2}) the symmetry
group $G_+ \times G_+$ was used
in an analogous manner to describe the $m=0$ case.
To be more precise regarding the comparison with Ref.~\cite{Ha}, note that
the model (\ref{I1}) with $y=0$ was obtained in \cite{Ha}
by using $G=U(n,n)$ and the model with a certain non-linear relation between
the three couplings was obtained by using $G=U(n+1,n)$.
The possible alternative reduction treatments
of the model (\ref{I1}) are briefly discussed also in the concluding section.
Our derivation implies the Liouville integrability of the model (\ref{I1})
in the general case, and it also gives rise to a simple linear-algebraic
algorithm for constructing the solutions.
It could be interesting to analyze the dynamics of the model
in the future by utilizing this algorithm, and to possibly relate
it to special solutions
in a field theory on the half-line.
Further comments on open problems are offered at the end of the paper.
\section{Group theoretic preliminaries}
\setcounter{equation}{0}
We here fix our notations and recall some group theoretic results
that will be needed later.
To begin, we choose some integers
\be
1\leq m < n
\label{2.}\ee
and define the matrices
\be
Q_{n,n}:=\left[
\begin{array}{cc}
0 & \1_n\\
\1_n & 0
\end{array}\right]\in gl(2n,\bC),
\quad
I_{m}:=\operatorname{diag}(\1_m,-\1_{n-m})
\in gl(n,\mathbb{C}),
\label{2.2}\ee
where $\1_n$ denotes the $n\times n$ unit matrix. We also introduce
\be
D_m := \operatorname{diag}(I_{m}, I_{m}) =
\operatorname{diag}(\1_m, -\1_{n-m}, \1_{m}, -\1_{n-m})
\in gl(2n,\bC).
\label{2.3}\ee
We adopt the convention in which the group $G:= SU(n,n)$ and its Lie algebra $\cG:= su(n,n)$
are given by
\be
SU(n,n) = \{ g \in SL(2n,\bC)\,\vert\, g^\dagger Q_{n,n} g = Q_{n,n}\}
\label{2.4}\ee
and
\be
su(n,n) = \{ V \in sl(2n,\bC)\,\vert\, V^\dagger Q_{n,n} + Q_{n,n} V =0\}.
\label{2.5}\ee
In the obvious $n\times n$ block notation the elements $V\in su(n,n)$ have the form
\be
V=
\left[
\begin{array}{cc}
X & Y\\
Z & -X^\dagger
\end{array}\right],
\quad
Y^\dagger = -Y, \quad Z^\dagger = -Z,
\quad
\Im(\oper{tr}(X))=0.
\label{2.6}\ee
We consider the commuting involutions of $SU(n,n)$ provided by the Cartan involution $\Theta$
and the involution $\Gamma$:
\be
\Theta(g):= (g^\dagger)^{-1},
\qquad
\Gamma(g):= D_m \Theta(g) D_m,
\qquad
\forall g\in G.
\label{2.7}\ee
The fixed point subgroup of $\Theta$ is the maximal compact subgroup $G_+ <G$
and the (non-compact) fixed point subgroup
of $\Gamma$ is
denoted by $G^+$.
Let $\theta$ and $\gamma$ be the corresponding involutions of $\cG = su(n,n)$.
Using the $n\times n$ block notation,
the Lie algebra $\cG_+$ of $G_+$ reads
\be
\cG_+=\left\{
\left[
\begin{array}{cc}
X & Y\\
Y & X
\end{array}\right]:
\,\,
X^\dagger = -X, \,\, Y^\dagger = -Y, \,\, \oper{tr}(X)=0\right\},
\label{2.8}\ee
and is isomorphic to $s(u(n) \oplus u(n))$ according to
\be
s(u(n) \oplus u(n))\ni \left[
\begin{array}{cc}
\alpha & 0\\
0 & \beta
\end{array}\right] \mapsto \psi(\alpha,\beta):=
\frac{1}{2}\left[
\begin{array}{cc}
\alpha+\beta & \alpha-\beta\\
\alpha-\beta & \alpha+\beta
\end{array}\right]\in \cG_+.
\label{2.9}\ee
Correspondingly, the group $G_+$ is isomorphic to $S(U(n) \times U(n))$
via the formula
\be
S(U(n) \times U(n))\ni \left[
\begin{array}{cc}
a & 0\\
0 & b
\end{array}\right] \mapsto
g(a,b):=
\frac{1}{2}\left[
\begin{array}{cc}
a+b & a-b\\
a-b & a+b
\end{array}\right]\in G_+,
\label{2.9+}\ee
which can be written also as
\be
g(a,b) = K (\oper{diag}(a,b)) K^{-1}
\quad
\hbox{with}\quad
K:= \frac{1}{\sqrt{2}} \left[
\begin{array}{cc}
\1_n & \1_n\\
\1_n & -\1_n
\end{array}\right].
\label{C}\ee
The Lie algebra $\cG^+$ of $G^+$ is furnished by
\be
\cG^+= \left\{
\left[
\begin{array}{cc}
X & Y\\
I_{m}Y I_{m} & I_{m}X I_{m}
\end{array}\right]:
\,\,
X^\dagger = -I_{m}X I_{m}, \,\, Y^\dagger = -Y, \,\, \oper{tr}(X)=0\right\},
\label{2.10}\ee
and is isomorphic to $s(u(m,n-m) \oplus u(m,n-m))$ via the map
\be
s(u(m,n-m) \oplus u(m,n-m))\ni \left[
\begin{array}{cc}
\alpha & 0\\
0 & \beta
\end{array}\right] \mapsto \chi(\alpha,\beta):=
\frac{1}{2}\left[
\begin{array}{cc}
\alpha+\beta & (\alpha-\beta)I_m\\
I_m(\alpha-\beta) & I_m(\alpha+\beta)I_m
\end{array}\right]\in \cG^+.
\label{E1}\ee
In the above formula $u(m,n-m)$ is realized as the Lie algebra of
the $n \times n$ matrices satisfying the relation
\be
\alpha^\dagger I_m + I_m \alpha = 0
\label{E2}\ee
and it holds that
\be
\chi(\alpha,\beta)
= {\tilde K} (\oper{diag}(\alpha,\beta)) {\tilde K}^{-1}
\quad\hbox{with}\quad \tilde K:= \frac{1}{\sqrt{2}} \left[
\begin{array}{cc}
\1_n & \1_n\\
I_m & - I_m
\end{array}\right].
\ee
Similarly, $G^+$ is isomorphic to
$S( U(m,n-m) \times U(m,n-m))$ by means of conjugation by $\tilde K$.
The eigensubspaces $\cG_-$ of $\theta$ and $\cG^-$ of $\gamma$
having eigenvalue $-1$ can be displayed as
\be
\cG_- =\left\{
\left[
\begin{array}{cc}
X & Y\\
-Y & -X
\end{array}\right]:
\,\,
X^\dagger = X, \,\, Y^\dagger = -Y\right\}
\label{2.11}\ee
and
\be
\cG^-=
\left\{\left[
\begin{array}{cc}
X & Y\\
-I_mYI_m & -I_m X I_m
\end{array}\right]:
\,\,
X^\dagger = I_m X I_m, \,\, Y^\dagger = -Y\right\}.
\label{2.12}\ee
Introducing $\cG_s^r:= \cG_s \cap \cG^r$ for any signs $s,r \in \{\pm\}$, we can
decompose $\cG$ as the direct sum of disjoint subspaces,
\be
\cG = \cG_-^- \oplus \cG_-^+ \oplus \cG_+^- \oplus \cG_+^+,
\label{2.13}\ee
which are pairwise perpendicular to each other
with respect to the invariant scalar product on $\cG$ defined by
\be
\langle V, W \rangle := \frac{1}{2} \tr (V W),
\qquad
\forall V, W \in \cG.
\label{2.14}\ee
We have
\be
\cG_+^+=\left\{
\left[
\begin{array}{cc}
X & Y\\
Y & X
\end{array}\right]:
\,\,
X = I_m X I_m=-X^\dagger, \,\, Y=I_mY I_m=-Y^\dagger, \,\, \oper{tr}(X)=0\right\},
\label{G++}\ee
and the appropriate restriction of the map (\ref{2.9}) gives rise to an isomorphism
\be
\cG_+^+\simeq s((u(m)\oplus u(n-m)) \oplus (u(m)\oplus u(n-m)) ).
\ee
We shall also use the explicit form of the other subspaces:
\bea
&&\cG_+^-=\left\{
\left[
\begin{array}{cc}
X & Y\\
Y & X
\end{array}\right]:
\,\,
X = -I_m X I_m=-X^\dagger, \,\, Y=-I_mY I_m=-Y^\dagger \right\},
\nonumber\\
&&\cG_-^+=\left\{
\left[
\begin{array}{cc}
X & Y\\
-Y & -X
\end{array}\right]:
\,\,
X = -I_m X I_m=X^\dagger, \,\, Y=-I_mY I_m=-Y^\dagger \right\},\nonumber\\
&&\cG_-^-=\left\{
\left[
\begin{array}{cc}
X & Y\\
-Y & -X
\end{array}\right]:
\,\,
X = I_m X I_m=X^\dagger, \,\, Y=I_mY I_m=-Y^\dagger \right\}.
\label{2.15+}\eea
Next, for our purpose we choose a maximal Abelian subspace $\cA$ of $\cG^-_-$,
i.e., an Abelian subalgebra of $\cG$ which lies in $\cG_-^-$ and
is not properly contained in any Abelian subalgebra of the same kind.
It is known \cite{KAH} that
any two choices are equivalent by the conjugation action of $G_+^+ $ on $\cG_-^-$,
and concretely we choose
\be
\cA:=
\left\{ q:=\left[
\begin{array}{cc}
\mathbf{q} & 0\\
0 & -\mathbf{q}
\end{array}\right]:
\,\,
\mathbf{q}=\oper{diag}(q_1,..., q_n),\,\, q_k\in \bR\right\}.
\label{2.15}\ee
One can verify that
the centralizer of $\cA$ in $\cG$ is given by the direct sum
\be
\cC = \cA \oplus \cM,
\quad
\cM=\left\{ d:=\ri \left[
\begin{array}{cc}
\mathbf{d} & 0\\
0 & \mathbf{d}
\end{array}\right]:
\,\,
\mathbf{d}=\oper{diag}(d_1,..., d_n),\,\, d_k\in \bR,\,\, \tr(d) =0\right\}< \cG_+^+.
\label{2.16}\ee
Denote by $A$ and $M$ the connected subgroups of $G$ corresponding to the Abelian
subalgebras $\cA$ and $\cM$, respectively.
In fact, $M$ is precisely the subgroup of $G_+^+$ whose elements $g$ satisfy
\be
g q g^{-1} = q,
\qquad
\forall q\in \cA.
\label{2.17}\ee
Furthermore, we call an element $q\in \cA$ \emph{regular} if the elements $g\in G_+^+$ satisfying
the relation $g q g^{-1}=q$ all belong to $M$.
It is not difficult to check that $q\in \cA$ is regular in this sense if and only
if the following conditions hold:
\be
q_i \neq 0 \quad i=1,..., n,
\quad (q_j - q_k) (q_j + q_k) \neq 0
\quad 1\leq j < k\leq m
\quad\hbox{and}
\quad m<j < k\leq n.
\label{qreg}\ee
Choose a connected component $\cA_c$ of the open set formed by regular elements
of $\cA$, and denote the closure of
this `open Weyl chamber' by $\bar\cA_c$.
According to general results \cite{KAH}, every element $g\in G$ can be decomposed in the form
\be
g = g_+ e^q g^+
\quad\hbox{with}\quad q\in \bar\cA_c,\, g_+\in G_+,\, g^+ \in G^+.
\label{2.18}\ee
The constituent $q$ that enters this decomposition is unique, and if $q$ is regular then the
ambiguity of the pair $(g_+, g^+)$ is exhausted by the replacement
\be
(g_+, g^+) \to (g_+ \mu, \mu^{-1} g^+)
\qquad
\forall \mu \in M.
\label{2.19}\ee
In the the generalized Cartan decomposition (\ref{2.18}) the
open Weyl chamber $\cA_c$ can be taken to consist of the elements $q$
in (\ref{2.15}) that are subject to the condition
\be
q_1 > q_2 > ... > q_m >0
\quad\hbox{and}\quad
q_{m+1} > q_{m+2} > ...> q_n >0.
\label{2.20}\ee
Both $\cG_+$ and $\cG^+$ possess one-dimensional centres.
The centre of $\cG_+$ is generated by
\be
C^l:= \ri Q_{n,n} =\ri \left[\begin{array}{cc}
0 & \1_n\\
\1_n & 0
\end{array}\right]
\label{2.21}\ee
and the centre of $\cG^+$ is spanned by
\be
C^r:= \ri \left[\begin{array}{cc}
0 & I_m\\
I_m & 0
\end{array}\right].
\label{2.22}\ee
These elements enjoy the property
\be
C^\lambda \in \cM^\perp \cap \cG^+_+
\quad\hbox{for}\quad \lambda =l, r.
\label{2.23}\ee
The decomposition (\ref{2.18}) and the property
(\ref{2.23}) will be important for us in Section 3.
By means of the invariant scalar product (\ref{2.14}),
we can regard $\cG$, $\cG_+$ and $\cG^+$ as their own dual spaces, respectively.
This then also identifies the respective coadjoint actions with the
adjoint actions.
In the next section we shall utilize particular coadjoint orbits of $G_+$.
To describe them,
for any non-zero column vector $u\in \bC^n$ define
the matrices
\be
X(u) := \ri (u u^\dagger - \frac{u^\dagger u}{n} \1_n)
\quad\hbox{and}\quad
\xi(u):=
\frac{1}{2}
\left[
\begin{array}{cc}
X(u) & X(u)\\
X(u) & X(u)
\end{array}\right].
\label{2.24}\ee
Fixing arbitrary real constants $\kappa>0$ and $x\neq 0$,
it is easy to see (cf.~(\ref{2.9})-(\ref{C})) that the set
\be
\cO_{\kappa,x} :=
\{x C^l +\xi(u) \vert\, u\in \bC^n, \, u^\dagger u = 2 \kappa n \}
\label{2.25}\ee
is
a coadjoint orbit of $G_+$ of minimal non-zero dimension.
The action of $g(a,b) \in G_+$ on $\cO_{\kappa,x}$ takes the form
\be
g(a,b) (x C^l +\xi(u)) g(a,b)^{-1} = (xC^l + \xi(au)).
\ee
Since $\xi(u)$ determines $u$ up to an overall $U(1)$ phase,
the orbit $\cO_{\kappa,x}$ can be identified with the
the complex projective space $\bC P_{n-1}$.
We remark that, for any real constants $x$ and $y$, $x C^l $ and $y C^r$
represent one-point coadjoint orbits of $G_+$ and $G^+$, respectively.
\section{Hamiltonian reduction}
\setcounter{equation}{0}
We shall reduce the free geodesic motion
on the group $G=SU(n,n)$ formulated as a Hamiltonian system on
the cotangent bundle $T^* G$.
We find it convenient to analyze the reduction
by using the so-called shifting trick of symplectic geometry, which amounts to extending the
phase space by a coadjoint orbit before reduction \cite{OrtRat}.
Specifically, trivializing $T^* G$ by right-translations and identifying
$\cG^*$ with $\cG$ by means of the invariant scalar product,
we consider the phase space
\be
P:= T^* G \times \cO_{\kappa, x} \simeq (G\times \cG)\times \cO_{\kappa, x} \equiv
\{ (g, J, \zeta)\, \vert\, g\in G, \, J\in \cG,\, \zeta\in
\cO_{\kappa, x} \}.
\label{3.1}\ee
The symplectic form on $P$ is given by
\be
\Omega = \Omega_{T^*G} + \Omega_{\cO_{\kappa, x}},
\label{Om}\ee
where $\Omega_{T^*G}$ can be written explicitly as
\be
\Omega_{T^*G} = d \langle J, dg g^{-1} \rangle
\label{3.2}\ee
while the explicit form of
the Kirillov-Kostant-Souriau symplectic form $\Omega_{\cO_{\kappa, x}}$ of the coadjoint
orbit $\cO_{\kappa, x}$ (\ref{2.25}) will not be needed.
The phase space $P$ carries the commuting family of Hamiltonians
provided by
\be
H_k(g,J,\zeta):= \frac{1}{4k} \tr(J^{2k}),
\qquad
k=1,2,..., n,
\label{Hk}\ee
the first member of which is responsible for the geodesic motion.
These Hamiltonians are explicitly integrable;
the flow of $H_k$ with initial value $(g_0, J_0, \zeta_0)$ is readily verified to
be
\be
(g(t), J(t), \zeta(t)) = (e^{t V_{k}} g_0, J_0, \zeta_0)
\quad\hbox{with}\quad
V_{k}:= J_0^{2k-1} - \frac{1}{2n} \tr (J_0^{2k-1}) \1_{2n}.
\label{freesol}\ee
Note that $H_k$ is real since $J^{2k}$ satisfies
$(J^{2k})^\dagger = Q_{n,n} J^{2k} (Q_{n,n})^{-1}$, and $V_{k}$ in (\ref{freesol}) belongs to
$\cG=su(n,n)$.
We introduce an action of the group $G_+ \times G^+$ on $P$ by sending
the pair $(\eta, h)\in G_+ \times G^+$ to the symplectomorphism $\Psi_{\eta, h}$ of $P$
operating as follows:
\be
\Psi_{\eta, h}(g,J,\zeta):= (\eta g h^{-1}, \eta J \eta^{-1}, \eta\zeta \eta^{-1}).
\label{3.5}\ee
The Hamiltonians $H_{k}$ (\ref{Hk}) are invariant under this
group action, which is generated by the equivariant moment map
\be
\Phi = (\Phi_+, \Phi^+): P \to (\cG_+, \cG^+),
\label{3.6}\ee
\be
\Phi_+(g, J, \zeta) = \pi_+(J) + \zeta,
\qquad
\Phi^+(g, J, \zeta)= - \pi^+(g^{-1} J g),
\label{3.7}\ee
where the projection $\pi_+:\cG \to \cG_+$ is
given by means of the decomposition $\cG = \cG_+ \oplus \cG_-$ and
$\pi^+: \cG \to \cG^+$ by $\cG= \cG^+ \oplus \cG^-$.
We are interested in the reduction defined by imposing the moment map constraint
\be
\Phi=\nu
\quad
\hbox{with}\quad
\nu := (0, -y C^r),
\label{3.8}\ee
where $y\neq 0$ is a real constant and we refer to (\ref{3.6}).
The action of the symmetry group $G_+ \times G^+$ preserves
the `constraint surface'
\be
P_c:= \Phi^{-1}(\nu)\subset P.
\label{Pc} \ee
We require that the constants $x$ and $y$ verify
\be
(x^2 - y^2) \neq 0.
\label{xy}\ee
Then the corresponding space of orbits,
\be
P_\red:= P_c / (G_+ \times G^+),
\label{3.10}\ee
will turn out to be a smooth manifold.
According to the general theory \cite{OrtRat},
$P_\red$ inherits the symplectic form $\Omega_\red$ and the
reduced Hamiltonians $H_{k}^\red$
defined by the formulas
\be
\pi^* \Omega_\red = \Omega\vert_{P_c},
\qquad
\pi^* H_{k}^\red = H_{k}\vert_{P_c},
\label{3.11}\ee
where $\pi: P_c \to P_\red$ is the natural projection and $\Omega\vert_{P_c}$ is
the restriction of $\Omega$ (\ref{Om}) on $P_c \subset P$.
\medskip
\noindent
{\bf Remark 3.1.}
In this technical remark we explain why the space of orbits (\ref{3.10}) is a smooth manifold.
First, we note that the action (\ref{3.5}) of
$G_+ \times G^+$ is \emph{proper}.
By definition \cite{OrtRat},
this means that for any sequences
$(\eta_n, h_n)$ in $G_+ \times G^+$ and
$(g_n,J_n,\zeta_n)$ in $P$ (with $n\in \bN)$ for which
$(g_n,J_n,\zeta_n)$ and $\Psi_{(\eta_n, h_n)}(g_n,J_n,\zeta_n)$ are both convergent, there
exists a convergent subsequence of the sequence $(\eta_n, h_n)$.
To show this, choose a convergent subsequence $\eta_{n_i}$ of the sequence
$\eta_n$ in $G_+$.
This is always possible since $G_+$ is compact.
Then, by considering the convergent sequences $\eta_{n_i} g_{n_i} (h_{n_i})^{-1}$ and
$g_{n_i}$ one can immediately conclude that $h_{n_i}$ must be a convergent sequence in $G^+$, which
proves the claim.
To continue, notice from (\ref{3.5}) that the effectively acting symmetry group is
the factor group
$(G_+ \times G^+)/ (\bZ_n)_\diag$, where
$(\bZ_n)_\diag$ is formed by the pairs $ (z \1_{2n}, z \1_{2n}) \in G_+ \times G^+$ with $z$ running
over the $n^{\mathrm{th}}$ roots of unity.
We shall demonstrate in the proof of Theorem 3.4 that
the action of $(G_+ \times G^+)/ (\bZ_n)_\diag$ on $P_c$ is a \emph{free} action.
Moreover, $P_c$ is
a closed, embedded submanifold of $P$, as it follows from the definition (\ref{Pc}) of $P_c$
and from the locally free character of the $(G_+\times G^+)$-action on it.
Since we have a free and proper action on the manifold $P_c$, the general theory \cite{OrtRat}
guarantees that $P_\red\simeq P_c/ ((G_+ \times G^+)/ (\bZ_n)_\diag)$
is a smooth symplectic manifold.
This is manifest
by the model of $P_\red$ constructed below.
\medskip
Our goal in what follows is to exhibit a global cross section (a global `gauge slice')
of the orbits of $G_+ \times G^+$ in $P_c$, which will yield a concrete model
of the reduced Hamiltonian systems $(P_\red, \Omega_\red, H_{k}^\red)$.
We first present the following lemma, whose proof will also show
how to construct a convenient global gauge slice.
\medskip
\noindent
{\bf Lemma 3.2.}
\emph{The element $e^q$, $q\in \cA$ in (\ref{2.15}), and $u\in \bC^n$
enter a triple
$(e^q,J, x C^l + \xi(u))\in P_c$ (\ref{Pc}) if and only if
$\vert u_j\vert^2 = 2 \kappa$ for all $j=1,..., n$
and $q$ is regular in the sense of
Eq.~(\ref{qreg}).}
\begin{proof}
Let us inspect the moment map constraint for an element of $P$ of the form
\be
(e^q, J, x C^l + \xi(u))
\quad\hbox{with some}\quad
q\in \cA.
\label{3.12}\ee
Denoting the projections associated to the decomposition (\ref{2.13}) as $\pi^r_s$
and decomposing $J$ as
\be
J= J_+^+ + J_+^- + J^+_- + J_-^-,
\label{3.13}\ee
we can spell out the moment map constraint as the conditions
\be
J_+^+ = - x C^l - \pi_+^+ (\xi(u)),
\qquad
J_+^- = - \pi_+^-(\xi(u)),
\label{3.14}\ee
and
\be
\pi^+(e^{-\ad_q}(J))\equiv(\cosh\ad_q)(J_+^+ + J_-^+) - (\sinh \ad_q)(J_+^- + J_-^-) = y C^r.
\label{3.15}\ee
Since $C^r \in \cG_+^+$, the $\pi^+_-$ projection of equation (\ref{3.15}) says that
\be
(\cosh\ad_q)( J_-^+) - (\sinh \ad_q)(J_+^-) = 0,
\label{3.16}\ee
and its $\pi^+_+$ projection requires that
\be
(\cosh\ad_q)(J_+^+ ) - (\sinh \ad_q)(J_-^-) = y C^r.
\label{3.17}\ee
We here used that $\cosh\ad_q$ maps $\cG_s^r$ to $\cG_s^r$ and
$\sinh\ad_q$ maps $\cG_s^r$ to $\cG_{-s}^{-r}$ (with $-s= \mp$ for $s=\pm$).
By substituting $J_+^+$ from (\ref{3.14}) into (\ref{3.17}) and then
taking the scalar product of both sides of equation (\ref{3.17}) with an
arbitrary $T\in \cM$ (\ref{2.16}),
we obtain the requirement
\be
\langle T, \xi(u) \rangle =0
\qquad
\forall T\in \cM,
\label{3.18}\ee
where we also took into account that $C^l$ and $C^r$ belong to
$\cM^\perp$ (\ref{2.23}).
By using the form of $\cM$ (\ref{2.16}) and that of $\xi(u)$ (\ref{2.24}),
we can rewrite (\ref{3.18}) as
the condition
\be
\vert u_j \vert^2 = 2\kappa,
\qquad
\forall j=1,..., n.
\label{3.19}\ee
If (\ref{3.19}) holds, then we can apply the action of the subgroup
$M_\diag$ of $G_+ \times G^+$,
\be
M_\diag:= \{(\mu, \mu) \in G_+ \times G^+ \,\vert\, \mu \in M\}
\label{3.20}\ee
to replace (without changing $q$) the element in (\ref{3.12})
by an element of the form
\be
(e^q, J, xC^l+ \xi(u^\kappa))
\quad\hbox{with the vector}\quad
u^\kappa_j := \sqrt{2\kappa},
\qquad
j=1,..., n.
\label{3.21}\ee
We further inspect the moment map constraint for the element (\ref{3.21}).
First looking at
the block-diagonal components of (\ref{3.17}), we see that
the matrix elements
${(J_-^-)}_{k,k}$
are arbitrary real numbers
for all $k=1,..., n$, and that we must have
\be
-\ri \kappa \cosh (q_j - q_k) - {(J_-^-)}_{j,k} \sinh(q_j - q_k) =0
\quad
\hbox{for}\quad
1\leq j < k \leq m
\quad\hbox{and}\quad
m<j<k \leq n.
\label{3.22}\ee
The last equation can be solved for ${(J_-^-)}_{j,k}$ if and only if $(q_j - q_k)\neq 0$ for the
pertinent indices.
Next,
the block off-diagonal components of (\ref{3.17}) can be spelled out as the conditions
\be
-\ri \kappa \cosh (q_j + q_k) - {(J_-^-)}_{j,n+k} \sinh(q_j + q_k) =0
\quad
\hbox{for}\quad
1\leq j < k \leq m
\quad\hbox{and}\quad
m<j<k \leq n,
\label{3.23}\ee
and
\be
- \ri x \cosh(2q_k) - {(J_-^-)}_{k,n+k} \sinh(2q_k) = y(C^r)_{k,n+k}
\quad \hbox{for}\quad k=1,..., n.
\label{3.24}\ee
Equation (\ref{3.23}) can be solved for ${(J_-^-)}_{j,n+k}$ if and only if
$(q_j + q_k) \neq 0$ for the relevant indices.
Taking into account the assumption $(x^2 - y^2)\neq 0$ (\ref{xy}) and the formula
(\ref{2.22}) of $C^r$,
equation (\ref{3.24}) can be solved for ${(J_-^-)}_{k,n+k}$ if and only if $q_k\neq 0$ for all $k$.
We have seen that equation (\ref{3.17}) admits a solution if and only if $u$ satisfies
(\ref{3.19}) and $q$ is regular (\ref{qreg}). The proof is finished by noting
that the remaining equation (\ref{3.16}) can always be solved
for $J_-^+$ if $J_+^- =- \pi_+^-(\xi(u))$ is given, since
$\cosh \ad_q$ yields an invertible map on $\cG_-^+$.
\end{proof}
\medskip
\noindent
{\bf Definition 3.3.}
\emph{
Suppose that $\kappa>0$ and $x,y$ satisfy (\ref{xy}).
For any $q\in \cA_c$ and $p\in \cA$ define the function $J(q,p)$ by the formula
\be
J(q,p) := - xC^l - \xi(u^\kappa) + L(q,p),
\label{Jqp}\ee
where $(u^\kappa)_j=\sqrt{2\kappa}$ ($j=1,..., n$) and the matrix elements of $L(q,p)= \pi_-(J(q,p))$ are
the following.
Firstly, if $1\leq j < k \leq m$ or $m<j<k \leq n$, then
\be
L_{j,k} = -L_{k,j}=-L_{j+n, k+n}=L_{k+n,j+n} = -\ri \kappa \coth(q_j-q_k),
\ee
\be
L_{j,k+n} = L_{k,j+n}=-L_{j+n, k}=-L_{k+n,j} = -\ri \kappa \coth(q_j+q_k).
\ee
Secondly, if $1\leq j \leq m$ and $m<k \leq n$, then
\be
L_{j,k} = -L_{k,j}=-L_{j+n, k+n}=L_{k+n,j+n} = -\ri \kappa \tanh(q_j-q_k),
\ee
\be
L_{j,k+n} = L_{k,j+n}=-L_{j+n, k}=-L_{k+n,j} = -\ri \kappa \tanh(q_j+q_k).
\ee
Finally, for any $1\leq j\leq m$, $m<k\leq n$, and $1\leq l \leq n$, we have
\be
L_{j,j+n} = - L_{j+n, j}=-\frac{\ri y}{\sinh (2q_j)} - \ri x \coth(2q_j),
\ee
\be
L_{k,k+n}=- L_{k+n,k} = \frac{\ri y}{\sinh (2q_k)} - \ri x \coth(2q_k).
\ee
\be
L_{l,l} =- L_{l+n,l+n}= p_l.
\ee
}
\medskip
\noindent
{\bf Theorem 3.4.} \emph{By using the above definition of $J(q,p)$, consider the set
\be
S= \{ (e^q, J(q,p), xC^l + \xi(u^\kappa))\,\vert\, q\in \cA_c,\,p\in \cA\,\}.
\label{S}\ee
The submanifold $S\subset P$ lies in the constraint surface $P_c$ (\ref{Pc}) and intersects
every orbit of $G_+ \times G^+$ in $P_c$ precisely in one point.
The pull-back $\Omega_S$ of the symplectic form $\Omega$ (\ref{Om}) on $S$ is given by
\be
\Omega_S =\sum_{k=1}^n d p_k \wedge d q_k.
\label{OmS}\ee
Thus the symplectic manifold $(S, \Omega_S)$ provides a model
of the reduced phase space $(P_\red, \Omega_\red)$ (\ref{3.10}), which can be identified
with the cotangent bundle
$T^* \cA_c$.}
\begin{proof}
We know that every $g\in G$ can be decomposed according to (\ref{2.18}),
and Lemma 3.2 implies that every gauge orbit (i.e.~$G_+\times G^+$ orbit) in $P_c$
admits a representative of the form
\be
(e^q, J, xC^l + \xi(u^\kappa))
\quad
\hbox{with}\quad
q\in \cA_c,
\label{3.37}\ee
where $\cA_c$ is an open Weyl chamber (for example the one defined in (\ref{2.20})).
Following the proof of Lemma 3.2, it is easy to check that $J$ in (\ref{3.37})
can be written as $J= J(q,p)$ in (\ref{Jqp}) with some $p\in \cA$. Indeed, the formula
(\ref{Jqp}) was obtained by directly solving
the constraint equations listed in the proof of Lemma 3.2.
To check that $S$ intersects every gauge orbit only once, suppose that we have
\be
(\eta e^q h^{-1},\eta J(q,p) \eta^{-1}, x C^l + \eta \xi(u^\kappa) \eta^{-1}) =
(e^{q'}, J(q', p'),xC^l+ \xi(u^\kappa)),
\quad
(\eta, h) \in G_+ \times G^+,
\label{3.38}\ee
for two triples in $S$.
The uniqueness property of the decomposition (\ref{2.18})
entails that
$e^q= e^{q'}$, which is equivalent to $q=q'$, and $(\eta,h) = (\mu,\mu)$ for some
$\mu \in M$. Then it follows from the second component of the equality in (\ref{3.38}) that $p=p'$ holds,
i.e., the two representatives of the orbit coincide.
Incidentally, the equality $\mu \xi(u^\kappa) \mu^{-1} = \xi(u^\kappa)$
implies that $\mu\in M$ must belong to the centre of $M$, which
is isomorphic to the group $\bZ_n$ and equals the centre of $G$.
The corresponding subgroup $(\bZ_n)_\diag < M_\diag$ acts trivially on $P$, and hence
we can also conclude that the factor group $(G_+ \times G^+)/ (\bZ_n)_\diag$ acts
freely on the constraint surface $P_c$.
We can compute the pull-back of $\Omega$ (\ref{Om}) on the submanifold
$S\subset P$, which gives the formula (\ref{OmS}). Since we have seen that
$S$ is a global cross section of the gauge orbits in $P_c$, it follows that
$(S,\Omega_S)$ represents a model of the reduced phase space $(P_\red,\Omega_\red)$.
Referring to the identification $\cA \simeq \cA^*$ defined by the scalar product
(\ref{2.14}) of $\cG$,
$(S,\Omega_S)$ is symplectomorphic to the
cotangent bundle $T^*\cA_c \simeq \cA_c \times \cA^*$ equipped with the Darboux symplectic form.
\end{proof}
Let us recall that
a Hamiltonian given by a smooth function on a $2n$ dimensional symplectic manifold
is called
\emph{Liouville integrable}
if it is contained in a family of $n$ functionally independent, globally smooth functions
on the phase space
whose mutual Poisson brackets vanish and their Hamiltonian flows
are complete.
Now the following result is an immediate consequence of the Hamiltonian reduction.
\medskip
\noindent
{\bf Corollary 3.5.}
\emph{A family of functionally independent Hamiltonians that are in involution with respect to the
canonical Darboux Poisson structure on $T^* \cA_c$ is provided by
\be
H_{k}^\red = \frac{1}{4k} \tr (J(q,p)^{2k}),
\quad
k=1,..., n.
\label{3.39}\ee
The generalized Sutherland
Hamiltonian $H(q,p)$ (\ref{I1}) is Liouville integrable, since it obeys
\be
H(q,p) =\frac{1}{4} \tr( J(q,p)^2)=H_1^\red(q,p).
\label{3.40}\ee}
\medskip
\begin{proof}
The reduced Hamiltonians (\ref{3.39}) are in involution with respect to the canonical
Poisson structure
derived from $\Omega_S$ (\ref{OmS}) since the original Hamiltonians $H_{k}$ (\ref{Hk}) are in involution
with respect to the Poisson structure on $(P, \Omega)$.
By using Definition 3.3, the identity (\ref{3.40}) is a matter of direct verification.
At generic points of the phase space, the Hamiltonians (\ref{3.39}) are independent,
since they start with independent
`leading terms' given by respective homogeneous polynomials in $p_1,..., p_n$.
The reduction guarantees that the corresponding Hamiltonian flows are complete, and
thus $H_{k}^\red$ (and in particular $H=H_1^\red$) is Liouville integrable.
\end{proof}
Finally, let us describe how the flows of the reduced Hamiltonians $H_k^\red$ can be
constructed from the `free flows' given in (\ref{freesol}).
Take an arbitrary initial value $(q(0), p(0))$.
As a consequence of the Hamiltonian reduction, the corresponding
solution $(q(t), p(t))$ of Hamilton's equation for $H_{k}^\red$
can be read off from the equality
\bea
&&\left(e^{q(t)}, J(q(t),p(t)),xC^l+ \xi(u^\kappa)\right) = \qquad
\label{sol}\\
&& \quad =\left(\eta(t) e^{t V_{k}} e^{q(0)} h(t)^{-1},
\eta(t) J(q(0), p(0)) \eta(t)^{-1},
\eta(t)(xC^l+ \xi(u^\kappa)\eta(t)^{-1}\right),
\nonumber\eea
where
\be
V_{k}= J(q(0), p(0))^{2k-1} - \frac{1}{2n} \tr (J(q(0),p(0))^{2k-1}) \1_{2n}
\ee
and
$(\eta(t), h(t))\in G_+ \times G^+$ is determined by the condition that
the left-hand-side of (\ref{sol}) must belong to the gauge slice $S$ (\ref{S}).
Thus, finding the solution requires the determination of the generalized Cartan decomposition
\be
e^{t V_{k}} e^{q(0)} = \eta(t)^{-1} e^{q(t)} h(t),
\qquad
(\eta(t), q(t), h(t))\in G_+ \times \cA_c \times G^+,
\label{factor1}\ee
made unique by the initial condition $\eta(0)=h(0)=\1_{2n}\in G$ and continuity in $t$
together with
the auxiliary condition
\be
\eta(t) \xi(u^\kappa) \eta(t)^{-1} = \xi(u^\kappa).
\ee
Then $p(t)$ obeys
\be
p(t)= \pi_-^-( \eta(t) J(q(0), p(0)) \eta(t)^{-1}) = \pi_-^-(\eta(t)L(q(0), p(0))\eta(t)^{-1}).
\ee
If one is interested only in $q(t)$, then a simpler solution algorithm is also available.
For this, notice that
the evaluation of the expression $g \Gamma(g^{-1})$ (with $\Gamma$ defined by (\ref{2.7})) for both sides of
the equality in (\ref{factor1}) leads
to the relation
\be
e^{2 q(t)} D_m = \eta(t) e^{t V_k} e^{2q(0)} D_m
e^{t V_{k}^\dagger} \eta(t)^{-1}.
\label{factor2}\ee
This means that the entries of the diagonal matrix $e^{2 q(t)} D_m$ are the eigenvalues
of the Hermitian matrix
\be
e^{t V_{k}} e^{2q(0)} D_m
e^{t V_{k}^\dagger}.
\ee
It follows from (\ref{factor2}) and the forms of $D_m$ (\ref{2.3}) and the Weyl alcove (\ref{2.20}) that
the eigenvalues of the above Hermitian matrix are all different, and
therefore \emph{finding
$q(t)$ boils down to an ordinary diagonalization problem.}
The above algorithm could be particularly useful to analyze
the generalized Sutherland dynamics, which arises as the $k=1$ special case of the reduced systems
(\ref{3.39}).
The formula (\ref{I1}) entails that in this case $p(t)= \dot{q}(t)$.
If desired,
one could also use the above derivation to obtain a Lax representation for the
equations of motion, taking $J(q,p)$ in (\ref{Jqp}),
or alternatively $L(q,p)$, as the Lax matrix.
However, the Lax pair would not give anything substantial to our knowledge about the
generalized Sutherland model, whose
main features follow from its realization as a reduction of the free geodesic motion on $SU(n,n)$.
\section{Conclusion}
\setcounter{equation}{0}
In this paper we applied the Hamiltonian reduction approach
to a particular many-body system.
Our derivation of the generalized Sutherland model (\ref{I1}) by reduction
of the free geodesic motion on the group $SU(n,n)$
proves the integrability of the model in the new case of three independent coupling
constants, and provides a simple algorithm for constructing the solutions.
This potentially paves the way for future work to analyze the scattering
characteristics of the model
along the lines of the papers \cite{RIMS94,PG}.
The investigation of the quantum mechanics of the model (for example by
quantum Hamiltonian reduction) is also a challenging problem.
Another interesting problem is to find duality properties for the generalized Sutherland model,
which would extend the action-angle dualities of the integrable many-body systems
studied by Ruijsenaars (see e.g.~the review \cite{Banff}).
This problem exists in general for the Sutherland models with two types of particles,
whose duality properties are not even known in the $A_n$ case.
For the description of dualities in the reduction approach, see also \cite{FA,FK}
and references therein.
Recently \cite{Bog}
new integrable random matrix models have been constructed in association
with certain integrable many-body systems of Calogero-Sutherland type.
It could be feasible to extend this correspondence between
random matrix models and integrable-many body systems to other cases, possibly including
generalized Sutherland models with two types of particles.
We end with a remark on the Lax matrices that can be associated to the model (\ref{I1}).
Namely, we
note that our usage of $SU(n,n)$ as the starting point leads to a $2n \times 2n$
Lax matrix, but it should be also possible to derive a $(2n+1) \times (2n+1)$ Lax matrix
for the same model,
with 3 independent couplings, by reduction of the free motion
on $SU(n+1,n)$ (cf.~Ref.~\cite{FP1}).
In the case of equal couplings ($x = 2 y = 2 \sqrt{2}\kappa$ in (\ref{I1})), it is this latter Lax matrix
that one may expect to obtain directly as well from the standard Lax matrix of the
original Sutherland
model of $2n+1$ particles by applying imaginary shifts and restriction to `mirror symmetric'
configurations.
Although in the reduction approach the role of the Lax matrices is somewhat secondary, they
are central in other approaches \cite{Banff,Suth}. For this reason, we plan to describe
the alternative Lax matrices of different size and their relationship elsewhere.
\bigskip
\medskip
\noindent{\bf Acknowledgements.}
This work was supported in part
by the Hungarian
Scientific Research Fund (OTKA) under the grant K 77400.
We thank J. Balog for useful discussions, and
thank the anonymous referee for pointing out an interesting question about alternative Lax matrices.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,412 |
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If re‑evaluation does not reveal a diagnosis, you CANNOT rule out smallpox. Patient's risk of smallpox is HIGH.
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The Response team from the local and/or state public health department will advise on management and specimen collection.
If variola testing rules out smallpox, continue other diagnostic testing. Patient's risk of smallpox is low.
If variola is positive, then smallpox is confirmed. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,237 |
/*****************************************************************************/
/**
*
* @file xintc_l.c
*
* This file contains low-level driver functions that can be used to access the
* device. The user should refer to the hardware device specification for more
* details of the device operation.
*
* <pre>
* MODIFICATION HISTORY:
*
* Ver Who Date Changes
* ----- ---- -------- -------------------------------------------------------
* 1.00b jhl 04/24/02 First release
* 1.00c rpm 10/17/03 New release. Support the static vector table created
* in the xintc_g.c configuration table.
* 1.00c rpm 04/09/04 Added conditional compilation around the old handler
* XIntc_LowLevelInterruptHandler(). This handler will only
* be include/compiled if XPAR_INTC_SINGLE_DEVICE_ID is
* defined.
* 1.10c mta 03/21/07 Updated to new coding style
* 1.10c ecm 07/09/07 Read the ISR after the Acknowledge in the interrupt
* handler to support architectures with posted write bus
* access issues.
* 2.00a ktn 10/20/09 Updated to use HAL Processor APIs and _m is removed
* from all the macro definitions.
* 2.04a bss 01/13/12 Removed the unused Register variable for warnings.
* 2.05a bss 08/18/12 Added XIntc_RegisterFastHandler API to register fast
* interrupt handlers using base address.
* 2.06a bss 01/28/13 To support Cascade mode:
* Modified XIntc_DeviceInterruptHandler,
* XIntc_SetIntrSvcOption,XIntc_RegisterHandler and
* XIntc_RegisterFastHandler APIs.
* Added XIntc_CascadeHandler API.
* 2.07a bss 10/18/13 Modified XIntc_DeviceInterruptHandler to support
* nested interrupts.
*
* </pre>
*
******************************************************************************/
/***************************** Include Files *********************************/
#include "xparameters.h"
#include "xil_types.h"
#include "xil_assert.h"
#include "xintc.h"
#include "xintc_i.h"
/************************** Constant Definitions *****************************/
/**************************** Type Definitions *******************************/
/***************** Macros (Inline Functions) Definitions *********************/
/************************** Function Prototypes ******************************/
static XIntc_Config *LookupConfigByBaseAddress(u32 BaseAddress);
#if XPAR_INTC_0_INTC_TYPE != XIN_INTC_NOCASCADE
static void XIntc_CascadeHandler(void *DeviceId);
#endif
/************************** Variable Definitions *****************************/
/*****************************************************************************/
/**
*
* This is the interrupt handler for the driver interface provided in this file
* when there can be no argument passed to the handler. In this case, we just
* use the globally defined device ID for the interrupt controller. This function
* is provided mostly for backward compatibility. The user should use
* XIntc_DeviceInterruptHandler() if possible.
*
* This function does not support multiple interrupt controller instances to be
* handled.
*
* The user must connect this function to the interrupt system such that it is
* called whenever the devices which are connected to it cause an interrupt.
*
* @return None.
*
* @note
*
* The constant XPAR_INTC_SINGLE_DEVICE_ID must be defined for this handler
* to be included in the driver compilation.
*
******************************************************************************/
#ifdef XPAR_INTC_SINGLE_DEVICE_ID
void XIntc_LowLevelInterruptHandler(void)
{
/*
* A level of indirection here because the interrupt handler used with
* the driver interface given in this file needs to remain void - no
* arguments. So we need the globally defined device ID of THE
* interrupt controller.
*/
XIntc_DeviceInterruptHandler((void *) XPAR_INTC_SINGLE_DEVICE_ID);
}
#endif
/*****************************************************************************/
/**
*
* This function is the primary interrupt handler for the driver. It must be
* connected to the interrupt source such that is called when an interrupt of
* the interrupt controller is active. It will resolve which interrupts are
* active and enabled and call the appropriate interrupt handler. It uses
* the AckBeforeService flag in the configuration data to determine when to
* acknowledge the interrupt. Highest priority interrupts are serviced first.
* This function assumes that an interrupt vector table has been previously
* initialized.It does not verify that entries in the table are valid before
* calling an interrupt handler. In Cascade mode this function calls
* XIntc_CascadeHandler to handle interrupts of Master and Slave controllers.
* This functions also handles interrupts nesting by saving and restoring link
* register of Microblaze and Interrupt Level register of interrupt controller
* properly.
* @param DeviceId is the zero-based device ID defined in xparameters.h
* of the interrupting interrupt controller. It is used as a direct
* index into the configuration data, which contains the vector
* table for the interrupt controller. Note that even though the
* argument is a void pointer, the value is not a pointer but the
* actual device ID. The void pointer type is necessary to meet
* the XInterruptHandler typedef for interrupt handlers.
*
* @return None.
*
* @note For nested interrupts, this function saves microblaze r14
* register on entry and restores on exit. This is required since
* compiler does not support nesting. This function enables
* Microblaze interrupts after blocking further interrupts
* from the current interrupt number and interrupts below current
* interrupt proirity by writing to Interrupt Level Register of
* INTC on entry. On exit, it disables microblaze interrupts and
* restores ILR register default value(0xFFFFFFFF)back. It is
* recommended to increase STACK_SIZE in linker script for nested
* interrupts.
*
******************************************************************************/
void XIntc_DeviceInterruptHandler(void *DeviceId)
{
u32 IntrStatus;
u32 IntrMask = 1;
int IntrNumber;
XIntc_Config *CfgPtr;
u32 Imr;
/* Get the configuration data using the device ID */
CfgPtr = &XIntc_ConfigTable[(u32)DeviceId];
#if XPAR_INTC_0_INTC_TYPE != XIN_INTC_NOCASCADE
if (CfgPtr->IntcType != XIN_INTC_NOCASCADE) {
XIntc_CascadeHandler(DeviceId);
}
else
#endif
{ /* This extra brace is required for compilation in Cascade Mode */
#if XPAR_XINTC_HAS_ILR == TRUE
#ifdef __MICROBLAZE__
volatile u32 R14_register;
/* Save r14 register */
R14_register = mfgpr(r14);
#endif
volatile u32 ILR_reg;
/* Save ILR register */
ILR_reg = Xil_In32(CfgPtr->BaseAddress + XIN_ILR_OFFSET);
#endif
/* Get the interrupts that are waiting to be serviced */
IntrStatus = XIntc_GetIntrStatus(CfgPtr->BaseAddress);
/* Mask the Fast Interrupts */
if (CfgPtr->FastIntr == TRUE) {
Imr = XIntc_In32(CfgPtr->BaseAddress + XIN_IMR_OFFSET);
IntrStatus &= ~Imr;
}
/* Service each interrupt that is active and enabled by
* checking each bit in the register from LSB to MSB which
* corresponds to an interrupt input signal
*/
for (IntrNumber = 0; IntrNumber < CfgPtr->NumberofIntrs;
IntrNumber++) {
if (IntrStatus & 1) {
XIntc_VectorTableEntry *TablePtr;
#if XPAR_XINTC_HAS_ILR == TRUE
/* Write to ILR the current interrupt
* number
*/
Xil_Out32(CfgPtr->BaseAddress +
XIN_ILR_OFFSET, IntrNumber);
/* Read back ILR to ensure the value
* has been updated and it is safe to
* enable interrupts
*/
Xil_In32(CfgPtr->BaseAddress +
XIN_ILR_OFFSET);
/* Enable interrupts */
Xil_ExceptionEnable();
#endif
/* If the interrupt has been setup to
* acknowledge it before servicing the
* interrupt, then ack it */
if (CfgPtr->AckBeforeService & IntrMask) {
XIntc_AckIntr(CfgPtr->BaseAddress,
IntrMask);
}
/* The interrupt is active and enabled, call
* the interrupt handler that was setup with
* the specified parameter
*/
TablePtr = &(CfgPtr->HandlerTable[IntrNumber]);
TablePtr->Handler(TablePtr->CallBackRef);
/* If the interrupt has been setup to
* acknowledge it after it has been serviced
* then ack it
*/
if ((CfgPtr->AckBeforeService &
IntrMask) == 0) {
XIntc_AckIntr(CfgPtr->BaseAddress,
IntrMask);
}
#if XPAR_XINTC_HAS_ILR == TRUE
/* Disable interrupts */
Xil_ExceptionDisable();
/* Restore ILR */
Xil_Out32(CfgPtr->BaseAddress + XIN_ILR_OFFSET,
ILR_reg);
#endif
/*
* Read the ISR again to handle architectures
* with posted write bus access issues.
*/
XIntc_GetIntrStatus(CfgPtr->BaseAddress);
/*
* If only the highest priority interrupt is to
* be serviced, exit loop and return after
* servicing
* the interrupt
*/
if (CfgPtr->Options == XIN_SVC_SGL_ISR_OPTION) {
#if XPAR_XINTC_HAS_ILR == TRUE
#ifdef __MICROBLAZE__
/* Restore r14 */
mtgpr(r14, R14_register);
#endif
#endif
return;
}
}
/* Move to the next interrupt to check */
IntrMask <<= 1;
IntrStatus >>= 1;
/* If there are no other bits set indicating that all
* interrupts have been serviced, then exit the loop
*/
if (IntrStatus == 0) {
break;
}
}
#if XPAR_XINTC_HAS_ILR == TRUE
#ifdef __MICROBLAZE__
/* Restore r14 */
mtgpr(r14, R14_register);
#endif
#endif
}
}
/*****************************************************************************/
/**
*
* Set the interrupt service option, which can configure the driver so that it
* services only a single interrupt at a time when an interrupt occurs, or
* services all pending interrupts when an interrupt occurs. The default
* behavior when using the driver interface given in xintc.h file is to service
* only a single interrupt, whereas the default behavior when using the driver
* interface given in this file is to service all outstanding interrupts when an
* interrupt occurs. In Cascade mode same Option is set to Slave controllers.
*
* @param BaseAddress is the unique identifier for a device.
* @param Option is XIN_SVC_SGL_ISR_OPTION if you want only a single
* interrupt serviced when an interrupt occurs, or
* XIN_SVC_ALL_ISRS_OPTION if you want all pending interrupts
* serviced when an interrupt occurs.
*
* @return None.
*
* @note
*
* Note that this function has no effect if the input base address is invalid.
*
******************************************************************************/
void XIntc_SetIntrSvcOption(u32 BaseAddress, int Option)
{
XIntc_Config *CfgPtr;
CfgPtr = LookupConfigByBaseAddress(BaseAddress);
if (CfgPtr != NULL) {
CfgPtr->Options = Option;
/* If Cascade mode set the option for all Slaves */
if (CfgPtr->IntcType != XIN_INTC_NOCASCADE) {
int Index;
for (Index = 1; Index <= XPAR_XINTC_NUM_INSTANCES - 1;
Index++) {
CfgPtr = XIntc_LookupConfig(Index);
CfgPtr->Options = Option;
}
}
}
}
/*****************************************************************************/
/**
*
* Register a handler function for a specific interrupt ID. The vector table
* of the interrupt controller is updated, overwriting any previous handler.
* The handler function will be called when an interrupt occurs for the given
* interrupt ID.
*
* This function can also be used to remove a handler from the vector table
* by passing in the XIntc_DefaultHandler() as the handler and NULL as the
* callback reference.
* In Cascade mode Interrupt Id is used to set Handler for corresponding Slave
* Controller
*
* @param BaseAddress is the base address of the interrupt controller
* whose vector table will be modified.
* @param InterruptId is the interrupt ID to be associated with the input
* handler.
* @param Handler is the function pointer that will be added to
* the vector table for the given interrupt ID.
* @param CallBackRef is the argument that will be passed to the new
* handler function when it is called. This is user-specific.
*
* @return None.
*
* @note
*
* Note that this function has no effect if the input base address is invalid.
*
******************************************************************************/
void XIntc_RegisterHandler(u32 BaseAddress, int InterruptId,
XInterruptHandler Handler, void *CallBackRef)
{
XIntc_Config *CfgPtr;
CfgPtr = LookupConfigByBaseAddress(BaseAddress);
if (CfgPtr != NULL) {
if (InterruptId > 31) {
CfgPtr = XIntc_LookupConfig(InterruptId/32);
CfgPtr->HandlerTable[InterruptId%32].Handler = Handler;
CfgPtr->HandlerTable[InterruptId%32].CallBackRef =
CallBackRef;
}
else {
CfgPtr->HandlerTable[InterruptId].Handler = Handler;
CfgPtr->HandlerTable[InterruptId].CallBackRef =
CallBackRef;
}
}
}
/*****************************************************************************/
/**
*
* Looks up the device configuration based on the base address of the device.
* A table contains the configuration info for each device in the system.
*
* @param BaseAddress is the unique identifier for a device.
*
* @return
*
* A pointer to the configuration structure for the specified device, or
* NULL if the device was not found.
*
* @note None.
*
******************************************************************************/
static XIntc_Config *LookupConfigByBaseAddress(u32 BaseAddress)
{
XIntc_Config *CfgPtr = NULL;
int Index;
for (Index = 0; Index < XPAR_XINTC_NUM_INSTANCES; Index++) {
if (XIntc_ConfigTable[Index].BaseAddress == BaseAddress) {
CfgPtr = &XIntc_ConfigTable[Index];
break;
}
}
return CfgPtr;
}
/*****************************************************************************/
/**
*
* Register a fast handler function for a specific interrupt ID. The handler
* function will be called when an interrupt occurs for the given interrupt ID.
* In Cascade mode Interrupt Id is used to set Handler for corresponding Slave
* Controller
*
* @param BaseAddress is the base address of the interrupt controller
* whose vector table will be modified.
* @param InterruptId is the interrupt ID to be associated with the input
* handler.
* @param FastHandler is the function pointer that will be called when
* interrupt occurs
*
* @return None.
*
* @note
*
* Note that this function has no effect if the input base address is invalid.
*
******************************************************************************/
void XIntc_RegisterFastHandler(u32 BaseAddress, u8 Id,
XFastInterruptHandler FastHandler)
{
u32 CurrentIER;
u32 Mask;
u32 Imr;
XIntc_Config *CfgPtr;
if (Id > 31) {
/* Enable user required Id in Slave controller */
CfgPtr = XIntc_LookupConfig(Id/32);
/* Get the Enabled Interrupts */
CurrentIER = XIntc_In32(CfgPtr->BaseAddress + XIN_IER_OFFSET);
/* Convert from integer id to bit mask */
Mask = XIntc_BitPosMask[(Id%32)];
/* Disable the Interrupt if it was enabled before calling
* this function
*/
if (CurrentIER & Mask) {
XIntc_Out32(CfgPtr->BaseAddress + XIN_IER_OFFSET,
(CurrentIER & ~Mask));
}
XIntc_Out32(CfgPtr->BaseAddress + XIN_IVAR_OFFSET +
((Id%32) * 4), (u32) FastHandler);
/* Slave controllers in Cascade Mode should have all as Fast
* interrupts or Normal interrupts, mixed interrupts are not
* supported
*/
XIntc_Out32(CfgPtr->BaseAddress + XIN_IMR_OFFSET, 0xFFFFFFFF);
/* Enable the Interrupt if it was enabled before calling this
* function
*/
if (CurrentIER & Mask) {
XIntc_Out32(CfgPtr->BaseAddress + XIN_IER_OFFSET,
(CurrentIER | Mask));
}
}
else {
CurrentIER = XIntc_In32(BaseAddress + XIN_IER_OFFSET);
/* Convert from integer id to bit mask */
Mask = XIntc_BitPosMask[Id];
if (CurrentIER & Mask) {
/* Disable Interrupt if it was enabled */
CurrentIER = XIntc_In32(BaseAddress + XIN_IER_OFFSET);
XIntc_Out32(BaseAddress + XIN_IER_OFFSET,
(CurrentIER & ~Mask));
}
XIntc_Out32(BaseAddress + XIN_IVAR_OFFSET + (Id * 4),
(u32) FastHandler);
Imr = XIntc_In32(BaseAddress + XIN_IMR_OFFSET);
XIntc_Out32(BaseAddress + XIN_IMR_OFFSET, Imr | Mask);
/* Enable Interrupt if it was enabled before calling
* this function
*/
if (CurrentIER & Mask) {
CurrentIER = XIntc_In32(BaseAddress + XIN_IER_OFFSET);
XIntc_Out32(BaseAddress + XIN_IER_OFFSET,
(CurrentIER | Mask));
}
}
}
#if XPAR_INTC_0_INTC_TYPE != XIN_INTC_NOCASCADE
/*****************************************************************************/
/**
*
* This function is called by primary interrupt handler for the driver to handle
* all Controllers in Cascade mode.It will resolve which interrupts are active
* and enabled and call the appropriate interrupt handler. It uses the
* AckBeforeService flag in the configuration data to determine when to
* acknowledge the interrupt. Highest priority interrupts are serviced first.
* This function assumes that an interrupt vector table has been previously
* initialized. It does not verify that entries in the table are valid before
* calling an interrupt handler.This function calls itself recursively to handle
* all interrupt controllers.
*
* @param DeviceId is the zero-based device ID defined in xparameters.h
* of the interrupting interrupt controller. It is used as a direct
* index into the configuration data, which contains the vector
* table for the interrupt controller.
*
* @return None.
*
* @note
*
******************************************************************************/
static void XIntc_CascadeHandler(void *DeviceId)
{
u32 IntrStatus;
u32 IntrMask = 1;
int IntrNumber;
u32 Imr;
XIntc_Config *CfgPtr;
static int Id = 0;
/* Get the configuration data using the device ID */
CfgPtr = &XIntc_ConfigTable[(u32)DeviceId];
/* Get the interrupts that are waiting to be serviced */
IntrStatus = XIntc_GetIntrStatus(CfgPtr->BaseAddress);
/* Mask the Fast Interrupts */
if (CfgPtr->FastIntr == TRUE) {
Imr = XIntc_In32(CfgPtr->BaseAddress + XIN_IMR_OFFSET);
IntrStatus &= ~Imr;
}
/* Service each interrupt that is active and enabled by
* checking each bit in the register from LSB to MSB which
* corresponds to an interrupt input signal
*/
for (IntrNumber = 0; IntrNumber < CfgPtr->NumberofIntrs; IntrNumber++) {
if (IntrStatus & 1) {
XIntc_VectorTableEntry *TablePtr;
/* In Cascade mode call this function recursively
* for interrupt id 31 and until interrupts of last
* instance/controller are handled
*/
if ((IntrNumber == 31) &&
(CfgPtr->IntcType != XIN_INTC_LAST) &&
(CfgPtr->IntcType != XIN_INTC_NOCASCADE)) {
XIntc_CascadeHandler((void *)++Id);
Id--;
}
/* If the interrupt has been setup to
* acknowledge it before servicing the
* interrupt, then ack it */
if (CfgPtr->AckBeforeService & IntrMask) {
XIntc_AckIntr(CfgPtr->BaseAddress, IntrMask);
}
/* Handler of 31 interrupt Id has to be called only
* for Last controller in cascade Mode
*/
if (!((IntrNumber == 31) &&
(CfgPtr->IntcType != XIN_INTC_LAST) &&
(CfgPtr->IntcType != XIN_INTC_NOCASCADE))) {
/* The interrupt is active and enabled, call
* the interrupt handler that was setup with
* the specified parameter
*/
TablePtr = &(CfgPtr->HandlerTable[IntrNumber]);
TablePtr->Handler(TablePtr->CallBackRef);
}
/* If the interrupt has been setup to acknowledge it
* after it has been serviced then ack it
*/
if ((CfgPtr->AckBeforeService & IntrMask) == 0) {
XIntc_AckIntr(CfgPtr->BaseAddress, IntrMask);
}
/*
* Read the ISR again to handle architectures with
* posted write bus access issues.
*/
XIntc_GetIntrStatus(CfgPtr->BaseAddress);
/*
* If only the highest priority interrupt is to be
* serviced, exit loop and return after servicing
* the interrupt
*/
if (CfgPtr->Options == XIN_SVC_SGL_ISR_OPTION) {
return;
}
}
/* Move to the next interrupt to check */
IntrMask <<= 1;
IntrStatus >>= 1;
/* If there are no other bits set indicating that all interrupts
* have been serviced, then exit the loop
*/
if (IntrStatus == 0) {
break;
}
}
}
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,527 |
Andrew Saul Levin (born August 10, 1960) is an American attorney and politician who served as the U.S. representative for from 2019 to 2023. A member of the Democratic Party, Levin was elected to the House in 2018, succeeding his retiring father, Sander Levin. He is the nephew of Carl Levin, a former U.S. senator.
Early life and education
Levin was born on August 10, 1960, to parents Sander Levin and Vicki Schlafer. Sander was elected to the United States House of Representatives in 1982. Andy grew up with two sisters, Jennifer and Madeleine, and a brother, Matthew.
Levin graduated from Williams College with a bachelor's degree. He earned a master's degree in Asian languages and culture from the University of Michigan and a Juris Doctor from Harvard Law School.
Early career
Levin was a staff attorney for the U.S. Commission on the Future of Worker-Management Relations in 1994 and worked as a trade union organizer and director. He ran as a Democrat for the 13th district seat in the Michigan State Senate in 2006. He lost to Republican John Pappageorge by 0.6% of the vote. After the election, he directed Voice@Work, a program seeking to expand trade union membership.
In 2007, Governor Jennifer Granholm appointed Levin deputy director in the Michigan Department of Energy, Labor, and Economic Growth (DELEG). He oversaw the "No Worker Left Behind" program, which provided job training to unemployed workers. In 2009, Granholm named him chief workforce officer. In 2010, Granholm named him acting director of DELEG, a role he served in until the end of her administration in 2011. He founded the clean energy firm Levin Energy Partners LLC and serves as president of Lean & Green Michigan.
U.S. House of Representatives
Elections
2018
Levin ran to succeed his father in the U.S. House of Representatives in . He defeated former State Representative Ellen Lipton and attorney Martin Brook in the primary election with 52.5% of the vote. Levin defeated Republican businesswoman Candius Stearns in the general election.
2020
Levin ran for a second term in 2020. He defeated Republican Charles Langworthy and several minor candidates, with 57.8% of the vote.
2022
In the 2022 Democratic primary, Levin lost to fellow incumbent Democrat Haley Stevens. As a result of redistricting, Michigan lost a seat in the House of Representatives, resulting in Stevens' and Levin's districts being combined, though the resulting district contained more of Stevens' original voters.
A Zionist and former synagogue leader known for his critical views of hard-line Israeli policies, Levin was opposed by the American Israel Public Affairs Committee (AIPAC), which provided $4 million for a negative publicity campaign against his candidacy. Levin has said, "AIPAC can't stand the idea that I am the clearest, strongest Jewish voice in Congress standing for a simple proposition: that there is no way to have a secure, democratic homeland for the Jewish people unless we achieve the political and human rights of the Palestinian people."
Tenure
In November 2020, The New York Times reported rumors that Levin was considered a possible candidate for Secretary of Labor in the Biden administration; Mayor of Boston Marty Walsh was ultimately named to the post in 2021.
Committee assignments
Committee on Education and Labor (Vice Chair)
Subcommittee on Health, Employment, Labor, and Pensions
Subcommittee on Higher Education and Workforce Investment
Committee on Foreign Affairs
Subcommittee on Asia, the Pacific and Nonproliferation
Subcommittee on the Western Hemisphere, Civilian Security and Trade
Caucus memberships
Congressional Progressive Caucus (Deputy Whip)
Medicare for All Caucus
House Pro-Choice Caucus
Electoral history
Personal life
Levin and his wife Mary (née Freeman) have four children, and live in Bloomfield Township. Levin is Jewish.
See also
2022 United States House of Representatives elections in Michigan
List of Jewish members of the United States Congress
References
External links
|-
1960 births
21st-century American Jews
21st-century American politicians
American Jews from Michigan
Democratic Party members of the United States House of Representatives from Michigan
Harvard Law School alumni
Jewish members of the United States House of Representatives
Jewish American people in Michigan politics
Levin family
Living people
Michigan lawyers
University of Michigan alumni
Williams College alumni | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 500 |
{"url":"https:\/\/support.bioconductor.org\/p\/87353\/","text":"Question: Ability to use a particular package via Docker\n0\n3.1 years ago by\njordan.xanthopoulos0 wrote:\n\nHi, I just started using docker and I what I want to know is if I can actually download with a command a particular package, with its dependencies,inside the Dockerfile, instead of dl'ing the whole category, cause it makes my image too big for no reason.\n\nWhat I am doing at the moment is just adding: FROM bioconductor\/release_flow in the top of the docker file.\n\nThe problem is, I only need the flowcore library, which is part of the flow release. I tried running this command, from somewhere else:\n\nRUN Rscript -e 'source(\"http:\/\/bioconductor.org\/biocLite.R\");library(BiocInstaller); biocLite(\"flowCore\", dep = TRUE)'\n\nbut I got some errors, since some dependencies failed to install, rendering the image useless.\n\nflowcore docker dockerfile \u2022 701 views\nwritten 3.1 years ago by jordan.xanthopoulos0\n\nPlease post the sessionInfo and error message.\n\nhttp:\/\/imgur.com\/a\/qiUHx\n\nThese are the final lines. Note that the only steps I have are, adding the rocker\/shiny, adding some CRAN libraries (successfully) and then running this line to get my bioconductor package.\n\nCan you also show the errors during \u00a0XML and ncdfFlow installations? Without these information, my best guess is some C libraries are missing \u00a0(e.g. libxml2 and hdf5)\n\nhttp:\/\/imgur.com\/a\/7DOJM\n\nI managed only to catch the XML. Can you understand what the problem may be? I can see the libxml2 is indeed missing. How can I add this via the dockerfile?\n\nYou need to install libxml2 library.\n\nI fixed the XML problem. There are more problems to be solved though. The HDF5 package is no longer on CRAN. What am I supposed to do?\n\nhttp:\/\/imgur.com\/a\/aHFcM\n\nAs I said, you need c library for hdf5. The same for GL.\n\nHey, thank you for the replies. I've come up with more questions.\n\nRUN Rscript -e 'source(\"http:\/\/bioconductor.org\/biocLite.R\");library(BiocInstaller); biocLite(\"flowCore\", dep = TRUE)'\n\nwhat i have added to the rocker dockerfile are these C libraries\nlibxml2-dev \\\nlibxml2 \\\nmesa-common-dev \\\nlibglu1-mesa-dev \\\nfreeglut3-dev \\\nhdf5-tools \\\nlibhdf5-openmpi-dev \\\nlibhdf5-serial-dev\nneeded for various packages from CRAN or bioconductor. Still, I get a warning about the rgl http:\/\/imgur.com\/a\/QQ7Bx (no errors though) and when I try to run a simple container i get this\n\"standard_init_linux.go:175: exec user process caused \"exec format error\". I am guessing this has to do something with the rgl error I posted.\n\nlibhdf5-openmpi-dev and\u00a0hdf5-tools are not necessary.\n\nI don't know how to resolve the dock container issue through. Maybe you should install the flowCore directly to your linux instead of through docker.\n\nyeah, the thing is I want to have a ready-to-go image of my R app for anyone, not just locally. Thanks for the replies. I am closing the thread","date":"2019-10-17 06:16:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.27508240938186646, \"perplexity\": 3490.68526036184}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986672723.50\/warc\/CC-MAIN-20191017045957-20191017073457-00295.warc.gz\"}"} | null | null |
'use strict';
import express from 'express';
import { getBoardsAndOrganizations } from './homeController';
const router = express.Router();
router.route('/').get(getBoardsAndOrganizations);
export default router; | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,552 |
Le canal du Grand Morin est un canal de Seine-et-Marne, long de , reliant le Grand Morin (Saint-Germain-sur-Morin) au canal de Chalifert à (Esbly), Il compte une écluse.
Communes traversées
Esbly ~ Montry ~ Saint-Germain-sur-Morin ~ Couilly-Pont-aux-Dames.
Notes et références
Cours d'eau en Seine-et-Marne
Système hydrologique de la Marne | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,306 |
Speaking Highlights on "The Hill" at Prairie View A&M University + Photos
Speaking at Prairie View A&M University on April 12
The month of April kicked off strong with two major speaking engagements. I was invited to be a panelist at the Civil Liberties & Public Policy 30th Anniversary Conference hosted at Hampshire College in Amherst, Massachusetts. This was my second time speaking in this city. As some of you may remember, the first time was back in September 2010 when I was a part of the national and international Community Resistance Tour coordinated by my good friend Jordan Flaherty.
The panel I was on took place April 9th with the theme "Media Voices Beyond the Mainstream." The discussion surrounded the power, growth, establishment, influence and future of grassroots media entities. It was a great hour and a half discussion. I still have a ton of the audio to upload from it, but if you would like to hear my 8:43 opening words just press "play" below. Thank you to the organizers for being so gracious to invite me, fly me out there, house me in a nice hotel and give me the opportunity to network with some great people from across the country.
I returned to Houston on April 11th and the next day I was set to deliver a message that would be very dear to me. Of course, I count every speaking moment as a blessing and don't take any for granted. However, this one was very special because I was returning to the school that helped to shape me into the young man I am today. The school where I honed many of my organizing skills. The school where I was able to test that which I was learning in the ministry class at Muhammad Mosque No. 45. The school where I got my first speeding ticket. That school is Prairie View A&M University.
Isis McGraw, who works in the PVAMU Special Programs and Cultural Series, invited me to be the final speaker in their second annual Students Participating In Transcendent Knowledge (S.P.I.T. Knowledge) Lecture Series. They invite several speakers per semester as guests and I was blessed to close out the 2010-2011 year. I was preceded by CNBC Contributor Keith Boykins, Domestic Violence survivor Carolyn Thomas, anti-racism activist Tim Wise, and Dr. Maya Rockeymoore.
This was my first time speaking on "The Hill" since 2001, when I led a protest against former President George W. Bush, who was the keynote speaker for the commencement ceremony. Long story short, that almost got me kicked out of school!
When we arrived at the new student memorial center I was in awe of how nice it was especially since my tuition fees contributed towards its development. Yea, we protested that too back then. The event took place inside the auditorium, and before I spoke I was interviewed in the holding room by Panther Newspaper writer Shei'Tia Benson. (Click here to read her story)
I was then brought out to be seated and was given a surprise that honestly was a tear jerker. Members of the dynamic Charles Gilpin Players performance group (pictured below) practiced for weeks to do a skit in honor of me. They performed many skits of some of my stories, blogs and motivational quotes! OMG! I was not prepared for that and I humbly thank them.
I was introduced by Trenton Johnson, who is a part of the schools PALs (Panther Advisor Leaders). That was the same week as the annual PALs week-long of activities and they graciously added my talk to their calendar. Thanks Trenton (pictured below) for the Twitter promotions and great introduction.
It had been a long time since I was nervous about a talk, but once I started I felt like was at home talking to family because PVAMU was my home for nearly five years. Within 45 minutes I touched on various organizing experiences I had as a student, the future of America as Minister Farrakhan has warned, finding purpose for one's existence, several news stories I've written included the recent Mississippi hanging of Frederick Jermaine Carter, the true purpose of education, self-identity, love, and the sexual assault of women on campus. I talked about the impact that Mayor Frank Jackson and the late Dr. Imari Obadele had on my development. I closed out my talk by inviting them to take the "Red Pill and the Blue Pill" like the movie The Matrix--meaning I shared with them some of the contents of the books "The Secret Relationships Between Blacks & Jews" Vols. 1 & 2 by the Nation of Islam Research Department. It sparked a lot of interest.
I then fielded many questions from the audience that covered a vast amount of topics. Plus I entertained more questions from a line of students at the end of the evening. What a joyous experience, and I thank the Honorable Minister Louis Farrakhan for this and all opportunities to strive to represent him as a student in his class.
I'm sure you're wondering if anyone recorded it, right? Well I was so focused on being sure I delivered the message properly that I did not think to ask about any audio/visual setup. However, I did press record on my handheld digital recorder just before I mounted the rostrum, but somehow it did not record one second of my talk. Urgh! Oh well, it happens.
I was blessed to receive feedback from Ms. McGraw (pictured below) who said that the feedback from students showed that they were most pleased with how I was able to relate to them and also many of them said I should have spoken longer. Interesting. Thank you to Ryan C. Versey for all of these wonderful photos.
All I have to say is: P-V-U!
Praise be to Allah!
(You're welcome to follow Brother Jesse's boring tweets on Twitter @BrotherJesse or his sometimes funny posts on Facebook. Follow at your own risk!) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,798 |
For Fullerton PD motor officer, it's all about old school and honoring a beloved role model
Fullerton PD Motor Officer Kyle Baas talks about his grandfather John McAllister, former Los Angeles PD Deputy Chief, while holding his 3-month-old son John William Baas. Photo by Steven Georges/Behind the Badge OC
By Greg Hardesty
Although he's only 27, when it comes to his uniform, Fullerton PD motor officer Kyle J. Baas prefers the old-school look.
Instead of wearing modern Kevlar-blend motor breeches, he steps into triple-thick wool duty pants — which are hotter and don't breathe nearly as well as the modern synthetic material.
And his shiny black riding boots rise to his knees — a style that has become less common with motor officers who prefer shorter boots that are more comfortable to wear an entire shift.
Baas' uniform choice is about much more than personal style.
Holding his 3-month-old son and waiting for his wife, Jeannette, a records and dispatcher supervisor for the Covina PD, to return home, Baas gestures to a black-and-white photo on the dining room table.
John McAllister, shown here as a motor officer for the LAPD in the early 1950s, is Baas' grandfather. Photo courtesy of Baas family
In the grainy picture from the early 1950s, a handsome, square-jawed cop wearing shades is standing next to his 1949 Harley-Davidson police motorcycle.
The bike has headlights and a red light, and not much else — not even a windshield.
Baas' eyes well up when he talks about the man by the bike.
He's John A. McAllister, who ascended the ranks of the Los Angeles Police Department with his friend and colleague Daryl Gates, the late legendary LAPD chief of police.
McAllister was deputy chief and former chief of staff when he retired from the LAPD in June 1979 after a stellar 30-year run.
A former motor officer who maintained a passion for Motors throughout his career, McAllister also is Baas' grandfather – a man of great distinction and integrity who was a close father figure to Baas.
Baas bounces his son, John, on his knee.
"When my grandfather passed away on Jan. 1, 2004, when I was 15 — that's when I really decided I wanted to follow in his footsteps," says Baas, who apologizes for getting emotional.
Fullerton PD Motor Officer Kyle Baas says law enforcement runs in the family. His great uncle, grandfather and aunt were also were motorcycle cops.
The 6-foot-4, 285-pound Baas may be a giant of a man, but a big part of him still is a boy in awe of his grandfather.
John Alexander McAllister was born Sept. 12, 1923 in San Francisco to Lieutenant Commander Harvey Ross, MD, and Edith McAllister.
John grew up with a brother, Harvey Ross McAllister Jr. — who also would make a big name for himself at the LAPD — in Taft, a small town in the foothills of the San Joaquin Valley.
After serving as a P-51 fighter pilot in the U.S. Army Air Corp and fighting in the Pacific Theater, John McAllister returned home, finished college and decided he wanted to be a cop.
Baas, who can instantly recall key dates and facts about his grandfather's life and career at the LAPD, says Gates never forgave McAllister when McAllister graduated at the top of his police academy class in December 1949 — one slot ahead of Gates, chief of police of the LAPD from 1978 to 1992.
Gates was a close friend of the McAllisters and Baas, who grew up in Chino Hills, recalls getting to know Gates at family functions.
And he fondly recalls boating with his grandfather during visits to McAllister's cabin at Green Valley Lake, near Big Bear Lake in the San Bernardino Mountains.
Baas inherited several of his grandfather's police memorabilia when McAllister died in 2004 at age 80 of amyotrophic lateral sclerosis (ALS), or Lou Gehrig's Disease.
Baas' other role model at the LAPD — Harvey Ross McAllister Jr., John McAllister's brother — was a close friend of Officer Ian J. Campbell, the 31-year-old victim of the notorious Onion Field execution-style killing on March 9, 1963.
John McAllister was close friends with legendary LAPD Police Chief Daryl Gates. McAllister was deputy chief when he retired from the LAPD in 1979. Photo courtesy of Baas family
Harvey McAllister was the lone bagpipe player at Campbell's funeral — the first time the LAPD had bagpipes played at a funeral for an officer.
Since then, bagpipes have been played at the funeral of every LAPD officer killed in the line of duty — and McAllister's bagpipes are on display at the Los Angeles Police Museum.
Baas has a third relative on his mother's side in law enforcement who helped inspire him to become a cop: his aunt Julie McAllister, a retired detective sergeant with the LAPD.
After working at a pizza joint and grocery story after high school, Baas took the advice of officers he had met through his relatives and became a cadet at the Fullerton PD, in 2007.
Baas recalls meeting the chief of police at the time, Pat McKinley.
"You're John's grandson?" McKinley asked Baas.
Turns out McAllister was McKinley's supervisor when the former Fullerton PD chief was an LAPD officer.
"He's one of the best chiefs I've ever seen," McKinley told Baas.
Baas went on to graduate from the Fullerton College Police Academy in May 2009.
Because there were no immediate positions available at the FPD, Baas worked for 2 ½ years as a police officer for Sierra Madre before the FPD made good on its promise to hire him.
He joined the FPD in October 2011.
Baas first worked patrol and started training for his dream job — motor officer — in June 2014.
Fullerton PD Motor Officer Kyle Baas on his Honda ST1300 police motorcycle. "It's the best-kept secret of the police department," Baas says. "I get paid to ride a motorcycle every day."
Photo by Steven Georges/Behind the Badge OC
He went to motor school in July 2014 and after graduating started patrolling Fullerton on his Honda ST1300 — a motorcycle equipped with a windshield and much, much more.
"It's the best-kept secret of the police department," Baas said. "I get paid to ride a motorcycle every day."
A couple of years ago, Baas started playing bagpipes in honor of his great uncle and now plays them at official Fullerton PD functions.
Baas hopes to be a motor officer for several years.
An early shot of John McAllister, an LAPD cop from 1949-1979. He died in 2004.
He looks lovingly at his son.
"I would love for him to follow in my footsteps," says Baas, who named his son after his grandfather.
He adds: "For him to carry on our family history of nearly 70 years in law enforcement would be very special.
"I will make sure he always knows what a great leader my grandfather was — and still is to our family today.
"I will teach him the importance of integrity, honesty and pride — just like I learned from my grandfather.
"And if he does become a police officer," Baas says, "I hope he will hold those traditions true to his heart."
Baas named his 3-month-old son after his grandfather and says he hopes his son, John, follows in his footsteps and becomes a cop, too.
Posted with Permission by the Author
Categories: NAMOA Blog
| Tags: fullerton, Greg Hardesty, history, motor cop, story
← Don Bailey – The Gray Ghost
OR Training Officer Hurt in Collision → | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,189 |
const {
PIXEL_SHIFT_WRAP,
PIXEL_CONFIG,
PIXEL_SHOW,
PIXEL_SET_STRIP,
PIXEL_SHIFT,
MAX_STRIPS,
I2C_DEFAULT,
COLOR_ORDER,
GAMMA_DEFAULT,
FIRMATA_7BIT_MASK
} = require('../constants');
const { Pixel } = require('../pixel');
const { create_gamma_table } = require('../utils');
const IC2Backpack = {
initialize: {
value(opts, strips) {
const MAX_PIXELS = 500; // based on # bytes available in firmata
const strip_length = opts.length || 6; // just an arbitrary val
const strip_definition = opts.strips || new Array();
const color_order = opts.color_order || COLOR_ORDER.GRB; // default GRB
const gamma = opts.gamma || GAMMA_DEFAULT; // Changing to 2.8 in v0.10
// set up the gamma table
const gtable = create_gamma_table(256, gamma, this.dep_warning.gamma);
const io = opts.firmata || opts.board.io;
if (!opts.address) {
opts.address = I2C_DEFAULT;
}
if (io == undefined) {
const err = new Error('An IO object is required to I2C controller');
err.name = 'NoIOError';
throw err;
}
// work out the map of strips and pixels.
if (typeof(strip_definition[0]) == 'undefined') {
// there is nothing specified so it's probably a single strip
// using the length and colour type.
strip_definition.push( {
color_order,
length: strip_length
});
} else if (parseInt(strip_definition[0], 10) != NaN) {
// we have the array of pin lengths but do we have the colour
for (let i = 0; i< strip_definition.length; i++) {
const len = strip_definition[i];
strip_definition[i] = {
color_order,
length: len
};
}
}
// put in check if it's gone over.
if (strip_definition.length > MAX_STRIPS) {
const err = new RangeError('Maximum number of strips ' + MAX_STRIPS + ' exceeded');
this.emit('error', err);
}
let total_length = 0;
strip_definition.forEach(function(data) {
total_length += data.length;
});
// put in check if there are too many pixels.
if (total_length > MAX_PIXELS) {
const err = new RangeError('Maximum number of pixels ' + MAX_PIXELS + ' exceeded');
this.emit('error', err);
}
const pixel_list = [];
for (let i=0; i < total_length; i++) {
pixel_list.push(new Pixel({
addr: i,
io,
controller: 'I2CBACKPACK',
i2c_address: opts.address,
strip: this
}, strips) );
}
strips.set(this, {
pixels: pixel_list,
io,
i2c_address: opts.address,
gtable,
gamma
});
this.strips_internal = strips;
// now send the config message with length and data point.
const data = [];
data.push(PIXEL_CONFIG);
strip_definition.forEach(function(strip) {
data.push( (strip.color_order << 5) | strip.pin);
data.push( strip.length & FIRMATA_7BIT_MASK);
data.push( (strip.length >> 7) & FIRMATA_7BIT_MASK);
});
// send the I2C config message.
io.i2cConfig(opts);
process.nextTick(function() {
try {
io.i2cWrite(opts.address, data);
} catch (e) {
if (e instanceof Error && e.name == 'EIO') {
this.emit('np_i2c_write_error', data);
}
}
process.nextTick(function() {
this.emit('ready', null)
}.bind(this) );
}.bind(this) );
}
},
show: {
value() {
const strip = this.strips_internal.get(this);
try {
strip.io.i2cWrite(strip.i2c_address, [PIXEL_SHOW]);
} catch (e) {
if (e instanceof Error && e.name == 'EIO') {
this.emit('np_i2c_write_error', 'PIXEL_SHOW');
}
}
}
},
strip_color: {
value(color) {
const strip = this.strips_internal.get(this);
const data = [];
data[0] = PIXEL_SET_STRIP;
data[1] = color & FIRMATA_7BIT_MASK;
data[2] = (color >> 7) & FIRMATA_7BIT_MASK;
data[3] = (color >> 14) & FIRMATA_7BIT_MASK;
data[4] = (color >> 21) & FIRMATA_7BIT_MASK;
try {
strip.io.i2cWrite(strip.i2c_address, data);
} catch (e) {
if (e instanceof Error && e.name == 'EIO') {
this.emit('np_i2c_write_error', data);
}
}
}
},
_shift: {
value(amt, direction, wrap) {
// shifts the strip in the appropriate direction.
//
const wrap_val = wrap ? PIXEL_SHIFT_WRAP : 0;
const strip = this.strips_internal.get(this);
const data = [];
data[0] = PIXEL_SHIFT;
data[1] = (amt | direction | wrap_val) & FIRMATA_7BIT_MASK;
try {
strip.io.i2cWrite(strip.i2c_address, data);
} catch (e) {
if (e instanceof Error && e.name == 'EIO') {
this.emit('np_i2c_write_error', data);
}
}
}
}
}
module.exports = {
IC2Backpack
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,732 |
\section{Introduction}
Bipolar outflows are an ubiquitous phenomena associated with star formation.
The leading theories on the origin of protostellar outflows indicate that they are generated through
the interaction of ionized disk material and the magnetic fields from the forming star and/or disk
\citep[and references therein]{li2014}.
Protostellar winds entrain the host cloud's molecular gas thereby producing molecular outflows \citep[e.g.][]{arce2007}.
There is substantial observational evidence showing
that molecular outflows from low-mass young stars tend to have relatively well-collimated lobes early
in their development and then evolve to have wider opening angle flows at later stages \citep{rich2000,arce2006,sea2008}.
However, there is no consensus on the detailed physics that produce this evolutionary trend in molecular outflows.
It has been proposed that jet axis wandering (precession) could produce wider cavities as the protostar
evolves \citep{mass1993}. However, in most sources the opening angle of the precession cone is smaller
than the observed wide-angle outflow cavity \citep[e.g.,][]{rei2000}. A second possibility is that protostellar
winds have both a collimated and a wide-angle component, and the observed molecular outflow is
predominantly driven by one of the two components, depending on the age of the protostar
\citep[e.g.,][]{sha2006,rom2009}. In this scenario the
opening angle and axis direction of the wind arising from the
protostar/disk system remain approximately constant, but the wind increasingly
disperses more of the circumstellar gas, thereby creating wider outflow cavities, as the protostar evolves and
the circumstellar envelope decreases in density \citep[e.g.,][]{off2011}.
Barnard 5 - IRS 1 (hereafter, B5-IRS1) is a young stellar object located at the eastern end of the Perseus molecular cloud complex
at a distance of about 240 pc \citep{hiro2011}. This object is embedded in a dense core with a filamentary structure \citep{pin2011}.
B5-IRS1 has a bolometric luminosity of $\sim$ 5 L$_\odot$ \citep{bei1984,eva2009}
and drives a giant bipolar flow associated with two clusters
of HH objects: HH 366 E to the northeast and HH 366 W to the southwest \citep{bally1996, yu1999}.
A molecular outflow with a bipolar wide-angle morphology and a southwest-northeast orientation
is found at the base of these HH objects \citep{full1991,lan1996,vel1998}.
Both the blueshifted lobe (northeast of the source) and the redshifted lobe (southwest of the source)
show a wide-angle cone-like structure with a projected opening angle of about 90$^\circ$ \citep{lan1996}.
Near infrared and optical images shown by \citet{yu1999} reveal
H$_2$ and H$\alpha$ emission within 20$''$ of B5-IRS1,
arising from a jet that bisects the limb-brightened CO cones.
\citet{yu1999} suggested that the presence of both axial jet-knots and a wide-angle cavity implies that the
central source may simultaneously power both a jet and a wide-angle wind.
In this paper, we present new line and continuum Submillimeter Array\footnote{The Submillimeter Array (SMA)
is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and
Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica.} (SMA) observations of the young
star B5-IRS1. These observations reveal further structure in the bipolar outflow associated with this young stellar object
and show that it has a peculiar morphology.
\section[]{Observations}
The observations were made with the SMA, and were collected on 2007 September 13,
when the array was in its subcompact configuration. The 21 independent baselines in this
configuration ranged in projected length from 5 to 35 k$\lambda$. The phase reference center
for the observations was set to $\alpha_{J2000.0}$ = \dechms{03}{47}{41}{60}, $\delta_{J2000.0}$ =
\decdms{32}{51}{43}{5}. Two frequency bands, centered at 230.53799 GHz (Upper Sideband) and
220.53799 GHz (Lower Sideband) were observed simultaneously. The primary beam of the SMA at
230 GHz has a FWHM of about $50''$, and the continuum emission arising from B5-IRS1
falls well within it. The line emission, on the other hand, extends beyond the FWHM of the primary beam,
so we corrected the images for the primary beam attenuation.
The SMA digital correlator was configured to have 24 spectral windows (``chunks'') of 104 MHz and
128 channels each. This provided a spectral resolution of 0.812 MHz ($\sim$ 1 km s$^{-1}$)
per channel. Observations of Uranus provided the absolute scale for the flux density calibration.
The gain calibrators were the quasars 3C 111 and 3C 84, while 3C 454.3 was used for bandpass calibration.
The uncertainty in the flux scale is estimated to be between 15 and 20$\%$, based on the SMA monitoring of
quasars.
The data were calibrated using the IDL superset MIR, originally developed for the Owens Valley
Radio Observatory \citep[OVRO,][]{Scovilleetal1993} and adapted for the SMA.\footnote{The
MIR-IDL cookbook by C. Qi can be found at http://cfa-www.harvard.edu/$\sim$cqi/mircook.html.}
The calibrated data were imaged and analyzed in the standard manner using the {\emph MIRIAD} \citep{sau1995} and
{\emph KARMA} \citep{goo96} softwares\footnote{The calibrated data can be obtained from:
http://www.cfa.harvard.edu/rtdc/sciImages/070913\_125512\_b5irs1.html}.
A 1300 $\micron$ continuum image was obtained by averaging line-free
channels in the lower sideband with a bandwidth of about 2 GHz.
For the line emission, the continuum was also removed.
For the continuum emission, we set the {\emph ROBUST} parameter of the task {\emph INVERT} to 0 to obtain
an optimal compromise between resolution and sensitivity,
while for the line emission we set this to -2 in order to obtain a higher angular
resolution. The resulting r.m.s.\ noise for the
continuum image was about 2 mJy beam$^{-1}$ at an angular resolution of $7\rlap.{''}74$
$\times$ $4\rlap.{''}87$ with a P.A. = $66.3^\circ$. The r.m.s.\ noise in each channel of the
spectral line data was about 80 mJy beam$^{-1}$ at an angular resolution of
$7\rlap.{''}51$ $\times$ $4\rlap.{''}18$ with a P.A. = $63.2^\circ$.
\begin{figure}
\begin{center}\bigskip
\includegraphics[scale=0.4]{f3.eps}
\caption{
Sum spectra of the three lines detected towards B5-IRS1:
$^{12}$CO(2-1) (blue), $^{13}$CO(2-1) (green), and
SO(6$_5$-5$_4$) (red).
The sum spectra were obtained from an area delimited by a
$45"\times45"$ square for $^{12}$CO(2-1), and a
$20"\times20"$ square for $^{13}$CO(2-1)
and SO(6$_5$-5$_4$), all centered at the position of the source.
The vertical dashed line represents the location of the
systemic velocity of the cloud, V$_{LSR}$ $\sim$ 10.0 km s$^{-1}$.
The spectra were obtained with
the task {\emph VELPLOT} of {\emph MIRIAD}.
}
\label{fig1}
\end{center}
\end{figure}
\section[]{Results and Discussion}
\subsection[]{Continuum}
In Figure 1, we show the 1300 $\micron$ continuum image resulting from these millimeter SMA observations.
We only detect a single source that is associated with B5-IRS 1 at the position
of $\alpha_{J2000.0}$ = \dechms{03}{47}{41}{5}, $\delta_{J2000.0}$ =
\decdms{32}{51}{43}{8}, with a positional error of less than 1\rlap.{"}0.
From a Gaussian fit to the continuum emission we obtain that
the flux density and peak intensity values of this compact source at this wavelength
are 54$\pm$8 mJy and 31$\pm$5 mJy beam$^{-1}$, respectively.
The Gaussian fit was also used to determine that the deconvolved size for this continuum source is
$5\rlap.{''}5$ $\pm$ $0\rlap.{''}5$ $\times$ $4\rlap.{''}0$ $\pm$ $0\rlap.{''}6$ with a P.A. = $-$28$^\circ$ $\pm$ 10$^\circ$.
Therefore, at the distance of the Perseus molecular cloud complex
the size of the continuum source is about 1200 AU.
Following \citet{hil1983} and assuming optically thin isothermal dust emission, a gas-to-dust ratio of
100, a dust temperature of 30 K, a dust mass opacity $\kappa_{1300 \micron}$
= 1.1 cm$^2$ g$^{-1}$ \citep{oss1994},
and that this object is located at 240 pc, we estimate
the total mass associated with the dust continuum emission to be 0.02 M$_\odot$, this in a good agreement
with the value obtained by \citet{bra2011}. However, we remark that the mass estimate presented in \citet{bra2011}
was obtained using a restricted UV-range trying to avoid the envelope contribution.
The millimeter continuum emission is probably tracing the envelope and the circumstellar disk surrounding B5-IRS 1.
Furthermore, we remark that the detected continuum emission could be partially optically thick, and therefore the reported mass estimate is a lower limit.
\begin{figure}
\begin{center}\bigskip
\includegraphics[scale=0.4]{f4.eps}
\caption{
Moment zero map of the SO line emission (grey scale) shown in combination with the blue- and red-shifted $^{13}$CO
emission (blue and red contours, respectively).
For SO(6$_5$-5$_4$) the velocity integration range is from $+$7 to $+$12 km s$^{-1}$.
For $^{13}$CO(2-1) the velocity integration range for the blueshifted lobe is from $+$6.5 to $+$8.6
km s$^{-1}$, while for the redshifted lobe is from $+$11.8 to $+$16.0 km s$^{-1}$.
The blue and red contours range from 20\% to 90\%
of their respective peak emission, in steps of 10\%. The peak of the blue and red emission are
6.4 and 2.5 Jy beam$^{-1}$ km s$^{-1}$, respectively.
The synthesized beam of the SO and the $^{13}$CO lines is shown in the lower
left corner. The grey-scale bar on the right pertain to the SO line emission map.}
\label{fig4}
\end{center}
\end{figure}
\subsection[]{Spectral Lines}
In Figure 2, we show the lines that were detected in the two sidebands and that are discussed throughout
this article. We simultaneously detected three lines, the $^{12}$CO(2-1) at a rest frequency of about 230.5 GHz, $^{13}$CO(2-1)
at a rest frequency of about 220.5 GHz, and the SO(6$_5$-5$_4$) line at a rest frequency of about 219.9 GHz. The exact values
for the rest frequency of these lines can be found in the database for astronomical spectroscopy: {\emph http://splatalogue.net}.
The sum spectra of $^{13}$CO(2-1) and SO(6$_5$-5$_4$) are narrower and fainter compared to the $^{12}$CO(2-1) spectrum.
The former have a FWHM between 3-5 km s$^{-1}$, while the latter has a FWHM of 20 km s$^{-1}$.
The sum spectra of $^{12}$CO and $^{13}$CO show wings at high velocities that are attributed
to the presence of molecular outflows. This is confirmed by our integrated intensity maps showing a clear bipolar
structure, see Figures 1 and 3.
\begin{figure*}
\begin{center}\bigskip
\includegraphics[scale=0.35]{f5.eps}
\caption{
Position velocity diagram (PVD) along the axis of the outflow that is shown in Figure 1.
The PVD was made at a position angle of 67.1$^\circ$. The black contours range from 6\% to 90\%
of the peak emission, in steps of 4\%. The peak of the line emission is 6 Jy beam$^{-1}$.
The grey contours represent the negative
emission arising mainly from velocities close to the cloud's systemic velocity
and have the same value as the black contours, but negative.
The curves mark the different structures found in our PVD. The circles mark
the highest velocity features in the outflow. Velocity is shown with respect to the Local Standard of Rest (LSR). }
\label{fig3}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}\bigskip
\includegraphics[scale=0.4]{f6.eps}
\caption{
Left panel: PVD perpendicular to the outflow axis in the redshifted lobe. The PVD was made at a position angle of 157.1$^\circ$.
The contour levels and the peak emission are the same as Figure 4.
Right panel: The same as in the left panel, but for the blueshifted lobe.
The contour levels and the peak emission are the same as Figure 4.
The blue and red dots represent the position and velocities of the high-velocity bullets shown in Figure 1{\it b}. The dashed curves mark
the different structures found in the PVDs. Both PVDs were made approximately 15$''$ away from the continuum source in both sides of the outflow.}
\label{fig3}
\end{center}
\end{figure*}
In Figure 1a we show the velocity-integrated intensity, or zero-moment map,
of $^{12}$CO(2-1) for both the blue and red outflow lobes.
The velocity range of integration for the blueshifted lobe is from $-$50.9 to $+$6.2 km s$^{-1}$,
while for the redshifted lobe is from $+$14.7 to $+$70.8 km s$^{-1}$.
With our SMA observations we detect a substantially larger spread of velocities
because the SMA data have significantly better sensitivity than previous studies \citep[see ][]{vel1998}.
The left panel in Figure 1 shows the wide-angle bipolar outflow that has been reported
in previous high-angular resolution studies of this source \citep{lan1996,vel1998}.
In addition, our higher sensitivity and fidelity maps reveal new features in the outflow,
especially at high velocities, that are shown in Figure 1b.
In Figure 1b, we split the range of integration into three different velocity ranges
in both the blueshifted and redshifted lobes. The velocity ranges were selected such that each velocity
range reveals distinct features that appear to be similar (symmetrical) in both lobes of the molecular outflow.
These features are clearly separated in velocity (and space) and do not appear be a single continuous structure in the outflow.
At low velocities the wide-angle bipolar outflow reported by previous studies is clearly seen.
Our map reveals that the low-velocity blueshifted lobe has a parabolical morphology,
while the redshifted lobe is a conical (or triangular) shape. The vertices of the cones
have a projected opening angle of about 90$^\circ$ for the blue lobe and
about 80$^\circ$ for the red lobe. At intermediate velocities the outflow
shows parabolic lobes that have narrower opening angles than the
low-velocity outflow gas; we measured an opening angle of 50$^\circ$ for the red and 60$^\circ$ for the blue lobe.
At high velocities we detect very compact structures (or bullets) in the
blue and red lobes that are equidistant from the position of the source.
Including the maps of all three different velocity ranges, for both lobes, together in the
same figure the outflow shows a ``spider-like'' structure (see Figure 1b).
In Figure 1b, we also included the positions of the strongest
knots from the jet mapped in H$_2$ and H$\alpha$ by \citet{yu1999}.
There is a good correspondence between the molecular and optical knots
with a small offset of about 5$''$ (or $\sim 1200$ AU) between corresponding peaks, with both
molecular bullets being farther away from the source than the optical/IR knots.
One possible explanation to this offset is that the knots and the molecular
bullets are produced by the same mass ejection episode, and that the difference
in position is caused by the fact that the two different datasets were taken approximately
a decade apart. A tangential velocity of approximately 570 km s$^{-1}$ would cause the
observed positional offset between the optical knot and the
molecular bullet. Such very high velocities have been observed in several
Herbig-Haro objects \citep[e.g. HH 111; HH 39 ][]{jon1982, sch1984, har2001}.
We note that both the blueshifted and redshifted bullets are
almost symmetrical in position (with respect to the source) which suggests
that they were both ejected in the same mass ejection event.
We tentatively
suggest that the optical and molecular bullets are different manifestations of the same mass ejection episode.
However, more optical/IR and
millimeter data are needed to confirm this.
Assuming local thermodynamic equilibrium (LTE), and that the $^{12}$CO(2-1) molecular emission
is optically thin, we estimate the outflow mass using the following equation:
$$
\frac{M(H_2)}{M_\odot}=6.3 \times 10^{-20} m(H_2) X_\frac{H_2}{CO} \left (\frac{c^2 d^2}{2k\nu^2} \right ) \frac{ \exp \left ( \frac{5.5}{T_{ex}} \right ) \int
S_{\nu} dv \Delta \Omega}{ \left (1 - \exp \left[\frac{-11.0}{T_{ex}} \right ] \right ) },
$$
where all units are in cgs, m(H$_2$) is mass of the molecular hydrogen with a value of 3.34$\times$10$^{-24} gr$, X $_\frac{H_2}{CO}$
is the fractional abundance between the carbon monoxide and the molecular hydrogen (10$^4$), $c$ is the speed of light with a value of 3$\times$10$^{10}$ cm s$^{-1}$, $k$ is the Boltzmann constant with a value of 1.38$\times$10$^{-16}$
erg K$^{-1}$, $\nu$ is the rest frequency of the CO line in Hz, $d$ is distance (240 pc), a parsec is equivalent to 3.08 $\times$10$^{18}$ cm,
$S_{\nu}$ is the flux density of the CO (Jansky), a Jansky is equivalent to 1.0 $\times$10$^{-23}$ erg s$^{-1}$ cm$^{-2}$ Hz$^{-1}$, $dv$ is the velocity range in cm s$^{-1}$,
$ \Delta \Omega$ is the solid angle of the source in steradians,
and T$_{ex}$ is excitation temperature taken to be 50 K. Using only emission above 4-$\sigma$ in every velocity channel map, we
estimate a mass for the outflow powered by B5-IRS 1 to be 7$\times$10$^{-3}$ M$_\odot$.
This value is consistent with the mass of other molecular outflows powered
by young low-mass protostars, see \citet{wu2004}. The mass estimated here is only a lower limit because the CO emission is likely to be optically thick
\citep[see, e.g.,][]{dun2014}.
We also estimate a kinematical energy ($E_k=\sum_{i} \frac{1}{2} \cdot m \cdot v_i^2$) of 2 $\times$ 10$^{44}$ ergs,
and an outflow momentum ($p=\sum_{i} m \cdot v_i$) of 0.5 M$_\odot$ km s$^{-1}$. From Figure 1, we estimate a dynamical age approximately of 700 years, which results
in a mechanical luminosity ($L=\frac{E_k}{t}$, where $E_k$ is the kinematical energy and $t$ is the dynamical age) of about 2 L$_\odot$.
The dynamical age is estimated assuming a size for the molecular outflow of 30$''$ and a velocity of 50 km s$^{-1}$.
These estimates do not take into consideration the inclination of the outflow with respect to the line-of-sight
(which is thought to be high for this source, see Yu et al. 1999).
In Figure 3, we show the integrated intensity map of the SO(6$_5$-5$_4$) and the $^{13}$CO(2-1) emission.
The SO line emission reveals a single compact
object associated with the continuum source in the position of B5-IRS 1, and does not show any clear velocity gradient in our data.
On the other hand, the $^{13}$CO(2-1) emission reveals cone-like lobes in the inner region of the bipolar outflow.
\subsection[]{ Shells from a spider-like outflow}
In Figure 4 we present the Positional-Velocity Diagram (PVD) of the $^{12}$CO(2-1) emission
along the outflow axis (P.A. = 67.1$^\circ$).
The PVD shows two structures at different velocity ranges. The main structure in the middle of the diagram,
at low to intermediate velocities (within outflow velocities of 30
km s$^{-1}$), shows a rotated ``{\it X}" morphology.
Comparing the molecular outflow properties predicted by different models presented in \citet{arce2007} with
the velocity structure (from the PVD) and the cone-like morphology in both lobes of the B5-IRS 1 outflow (in Figure 1{\it a})
we propose that this outflow is mostly driven by a wide-angle wind, as concluded by \citet[][]{lan1996}.
There is also evidence,
from the optical and infrared observations by \citet{yu1999} and our SMA data ({\it i.e.},
the high-velocity bullets shown in Figure 1{\it b}), that B5-IRS 1 powers a very collimated (jet-like) wind.
The molecular outflow jet component is also observed in our PVD as a structure at high outflow velocities
distinct from the central rotated ``{\it X}" structure in Figure 4. This is most evident in the redshifted lobe
where faint emission, at offsets more negative than $-10\arcsec$ and at LSR velocities greater than $+$ 20 km s$^{-1}$
(and up to about $+$ 60 km s$^{-1}$), shows outflow velocity increasing with offset from the source,
as expected in a jet-driven molecular outflow \citep[see, {\it e.g.},][]{lee2000, lee2001}.
The PVD in Figure 4 shows that the molecular bullets in both lobes reach outflow velocities of about 50 km s$^{-1}$.
In Figure 5 we present position-velocity diagrams of both outflow lobes, along cuts
perpendicular to the outflow axis (P.A. = 157.1$^\circ$).
These PVDs present different structures compared to those shown in Figure 4.
At low outflow velocities ($\lesssim 10$~km s$^{-1}$)
there are two bright curved structures that have similar velocity ranges in
both outflow lobes, and are related to the low-velocity and very wide-angle structures seen in Figure 1{\it b}.
In addition, there are two (faint) ring-like structures in both sides of the flow at intermediate velocities (at
outflow velocities of about 10 to 45~km s$^{-1}$). Such structures, in position-velocity space,
are reminiscent of radially expanding shells or bubbles
\citep[see for example][]{zap2011, arce2011}, but are also similar to the elliptical structures
expected in the PVD of a jet with a low inclination with respect to the plane of the sky
\citep[see Figure 26 of][]{lee2000}.
As discussed above, the integrated intensity map at intermediate velocities show
narrow parabolic structures, very different from the circular (or semi-circular)
structures associated with radially expanding bubbles. Furthermore, the observed structure
is somewhat similar to the structures seen in
in young (Class 0) jet-powered molecular outflows like L1448 \citep{bachi1995}, L1157 \citep{gueth1996} and
HH 211 \citep{gueth1999, pal2006, lee2007}. In these young sources the molecular outflow cavities have a moderate opening angle
(similar to the intermediate-velocity structures we detect in B5-IRS1)
that are presumably formed by the passage of jet bow-shocks, consistent with the
bow shock-driven molecular outflow model of \citet{raga1993}.
It therefore seems more likely that the observed
intermediate-velocity features are caused by propagating bow shocks in a jet, or highly collimated wind.
The high-velocity bullets
are located at the tips of the intermediate-velocity elliptical structure in the
position-velocity diagrams shown in Figure 5 (and at the far ends of the PVD shown in Figure 4).
Hence, it seems possible that these high-velocity features are tracing the tip (or head) of the bow-shock responsible for
the intermediate-velocity cavities.
\subsection[]{Nature of the morphology of the B5-IRS1 outflow}
One of the most striking characteristics of the B5-IRS1 outflow is the different structures observed at
different ranges of velocities. As discussed above, the high-velocity bullets and intermediate velocity structure are driven by a
collimated (jet-like) wind, while the low-velocity lobes are consistent with being formed by
a wide-angle wind.
An increasing number of outflows have been observed that show two components
(a wide-angle structure and a collimated feature). They include L1551 \citep{ito2000},
HH 46/47 \citep{vel2007,arce2013},
Cepheus HW2 \citep{tor2011},
and Source I \citep{mat2010, zap2012}.
However, to our knowledge, none of these outflows show a spider-like structure like the one we
observe in B5-IRS 1.
These ``dual component'' molecular outflows are generally explained by assuming that the underlying
protostellar wind that entrains the surrounding ambient gas has a wide-angle morphology with a
narrow component (with a much higher outflow momentum rate) along the wind axis
as in the X-wind models of \citet{sha2006} and
disc-magnetosphere boundary outflow launching models of \citet{rom2009}.
The spider-like morphology of the B5-IRS 1 molecular outflow may be explained with
these kinds of models. One hypothesis for the observed evolutionary trend in outflow
opening angle is that the observed molecular outflow (produced by the interaction between
the protostellar wind and the surrounding ambient gas) is mostly driven by one of the two
different components, depending on the evolutionary stage of the protostar and the density
distribution of the circumstellar material. During the early deeply embedded stage of protostellar evolution
(Class 0) only the high-momentum component along the outflow axis is able to puncture through
the dense circumstellar envelope, producing a collimated (jet-like) outflow.
At later stages ({\it i.e.}, Class I and II), after the envelope looses mass through outflow
entrainment and infall onto the protostar, the wider component will be
able to entrain the remaining circumstellar material at
larger angles away from the outflow axis. If such picture is correct,
we should then expect a transitional phase when there is little or no molecular gas along the outflow axis, yet
enough molecular gas at intermediate angles ({\it i.e.}, between the outflow axis and the edge of the wide-angle wind)
that would result in a molecular outflow with a
component that is approximately midway
between the collimated and wide-angle components, and a faint (or no) on-axis component.
We suggest this scenario explains the spider-like structure
seen in B5-IRS1, which in fact may be considered a young
Class I source (based on its spectral energy distribution, e.g., Arce \& Sargent 2006) and hence
likely to be in an evolutionary phase close to the transition between Class 0 and Class I (see also Yu et al.~1999).
We argue that in B5-IRS1 the molecular outflow material is detected along (and within 30$^\circ$~of) the outflow axis (as shown by the high-velocity bullet and the intermediate-velocity structure) because there is still enough molecular material in the cavity for it to be entrained by the collimated (jet) component of the wind
(as expected in a young source). However, as seen in Figure 1 (where the integrated intensity of the outflowing CO emission is dominated by the very wide angle structure) most of the entrainment is taking place at larger angles from the axis where the wide-angle wind is currently interacting with the
denser parts of the surrounding envelope.
\section*{Summary}
We observed in the millimeter regime the dust and molecular gas surrounding the young stellar object B5-IRS 1
using the Submillimeter Array. Our conclusions are as follow:
\begin{itemize}
\item The millimeter dust continuum emission reported here is tracing the
envelope and circumstellar disk surrounding B5-IRS 1.
\item Our $^{12}$CO(2-1) observations resolve the bipolar northeast-southwest outflow
associated with B5-IRS 1 and find that its morphology is reminiscent of a ``spider", where
three velocity components with different morphologies are present in each lobe.
\item In addition to detecting the previously observed wide-angle cone-like lobes, our observations
for the first time reveal
the presence of intermediate-velocity, parabolic shells emerging very close
to the young stellar object as well as high-velocity compact molecular bullets
which we argue are associated with the optical/IR jet in this source.
These high-velocity features reach outflow (radial) velocities of about 50 km s$^{-1}$.
\item We interpret the peculiar spider-like morphology as a result of the molecular
material being entrained by a wind in which the momentum
has an angular dependence (i.e., larger towards the outflow axis). We believe
the peculiar outflow morphology is evident in this source because it is in a transitional evolutionary
phase, at a stage that is slightly older than the phase in which the outflow is completely dominated
by the on-axis (collimated) part of the wind, but slightly younger than the stage in which
the outflow entrainment is fully dominated by the wide-angle component of the wind.
\item We report the detection $^{13}$CO(2-1) and SO(6$_5$-5$_4$) emission,
which arises from the outflow and the vicinity of B5-IRS1.
\end{itemize}
We conclude that the protostar B5-IRS 1 is a great laboratory to study the process of
outflow formation and evolution. Further observations at different wavelengths and at higher
angular and velocity resolution are needed to better understand its kinematics as well its launching
and entrainment mechanisms.
\section*{Acknowledgments}
L.A.Z. acknowledge the financial support from DGAPA, UNAM, and CONACyT, M\'exico.
H.G.A. acknowledges support from his NSF CAREER award AST-0845619. A.P. is supported
by the Spanish MICINN grant AYA2011-30228-C03-02 (co-funded with FEDER funds),
and by the AGAUR grant 2009SGR1172 (Catalonia).
| {
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We are so happy to have the Tammy Fender collection back in our shop. We love this luxurious, healing line of natural skincare products. In honour of our reconnection, we'd love to introduce you to the beautiful lady behind the line, Tammy Fender.
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Irritants severely impact sensitive skin, including the potential toxins found in some skincare products (petrochemicals, parabens and synthetic compounds), as well as environmental stressors. Skin is an eliminatory organ, and when it rejects what's put onto it, the results are redness and irritation, and, in some cases, skin conditions can develop. All of my formulas are chemical-free. But some remedies, including Spontaneous Recovery Crème, which incorporates calming lavender, helichrysum, and matricaria, and Roman Chamomile Tonic, work especially well to soothe stressed skin, while reducing the signs of irritation.
4/ Can you tell us something surprising about yourself that most people likely don't know?
I find handmade clothes, art, and jewelry irresistible. I love finding things made by local artisans when I travel, and I am always especially fond of the handcrafted and heart-full gifts made by the children at my daughter's Waldorf school.
Check out Tammy Fender Products. | {
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Niels Bohr Bohr Model
1 Molecule Of Co2 The use of a molecular ruthenium catalyst with selected triphos ligands. Moreover, the catalytic combination of recycled diols with carbon dioxide allows the sustainable synthesis of linear and. In respiration, most of the energy in the original glucose molecule is: a) stored in molecules of ADP b) stored in molecules of ATP c) stored in
Bohr's atomic model was utterly revolutionary when it was presented. as a "probability cloud" around the nucleus of the atom. Plataforma SINC. "Bohr's quantum theory revised." ScienceDaily.
Bohr model of the atom was proposed by Neil Bohr in 1915. It came into existence with the modification of Rutherford's model of an atom. Rutherford's model.
Niels Bohr completely transformed our view of the atom and of the world. Realizing that classical physics fails catastrophically when things are atom-sized or smaller, he remodeled the atom so electrons occupied 'allowed' orbits around the nucleus while all other orbits were forbidden.
Over the course of five weeks, the groups visited Germany, Switzerland and Denmark where they not only learned, but experienced first-hand the sights and sounds out of which modern physics grew.
Niels Bohr's father was Christian Bohr and his mother was Ellen Adler. Christian Bohr was awarded a doctorate in physiology from the University of Copenhagen in 1880 and in 1881 he became a Privatdozent at the university.
Niels Bohr (1885 – 1962) Niels Henrik David Bohr was born in Copenhagen in 1885 to Christian Bohr, a professor of physiology at the University of Copenhagen and a Nobel Prize winner, and Ellen Adler Bohr, who came from a wealthy Sephardic Jewish family prominent in Danish banking and parliamentary circles.
The Bohr model of the hydrogen atom explains the connection between the quantization of photons and the quantized emission from atoms. Bohr described the hydrogen atom in terms of an electron moving in a circular orbit about a nucleus. He postulated that the electron was restricted to certain orbits characterized by discrete energies.
Fearing arrest by the Germans, Niels fled to Britain during World War II and then to New Mexico to work on the Manhattan Project – a research and development program by the US and UK – that produced.
Niels Bohr was a physicist who worked on the Manhattan Project during World War II to develop the atomic bomb. Bohr was a renowned physicist who developed the Bohr model of atoms and the principle of.
Oct 31, 2012. Bohr studied the planetary model of an atom with his mentor, Ernest Rutherford. He noticed that there were some problems with Rutherford's.
Niels Bohr (7 October 1885 – 18 November 1962) was a Danish physicist who helped discover quantum physics, the structure of the atom, and the atomic bomb.Bohr was awarded the Nobel Prize in 1922 for, in short, discovering the quantization of atomic energy levels. He worked also in a project of physicists known as Manhattan Project.He married Margrethe Nørlund in 1912.
and this year marks 100 years since the model was first proposed by Danish physicist Niels Bohr. The idea that matter comprises indivisible units dates back to Indian and Greek philosophers. But it.
Between 1905 and 1945, scientific luminaries like Niels Bohr, Albert Einstein, Werner Heisenberg and Max Planck forever changed the modern world of physics. The Bohr model of the atom, Einstein's.
Interesting Facts Rachel Carson Home Latest News Articles, Celebrity Profiles, Wiki, Facts. Latest News Articles, Celebrity Profiles, Wiki, Facts. Are you interested in the unpopular details you probably didn't know about some of your favorite celebrities? We let you in on all the scoops including the lesser-known details about their family, dating relationships, careers. Random B. F. Skinner Facts
Summary of Flame Testing and Bohr's Quantum Model of the Atom. Who? Niels Bohr; Copenhagen, Denmark; October 7, 1885- November 18, 1962.
Dec 4, 2015. In this installment in the history of atom theory, physics professor (and my dad) Dean Zollman explains how Niels Bohr built on the work of.
Niels Bohr was one of the most influential scientists of the 20th century. He was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, and received the Nobel Prize for Physics in 1922.
Despite winning the Nobel physics prize and being more than a match for Einstein, Niels Bohr is far from being the household name. creating the atomic model that became the basis for our.
Niels Bohr Collected Works. Niels Bohr on the 500 Danish Kroner banknote. Encyclopaedia Britannica article on Niels Bohr; Niels Bohr Archive; Nobel Foundation: Niels Bohr; Annotated bibliography for Niels Bohr from the Alsos Digital Library for Nuclear Issues; Quantum Chemistry I Lecture – Bohr's Model of the Atom
In 1913 Niels Bohr came to work in the laboratory of Ernest Rutherford. Rutherford, who had a few years earlier, discovered the planetary model of the atom asked Bohr to work on it because there were some problems with the model: According to the physics of the time, Rutherford's planetary atom should have an extremely short lifetime.
Niels Bohr's model of hydrogen depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the.
Niels Bohr won the Nobel Prize in Physics in 1922 for his then-radical model of the structure of the atom. Bohr proposed an atomic structure, known as the Bohr model, that placed the atom's neutral.
Niels Bohr and the Quantum Atom is the first book that focuses in detail on the birth and development of Bohr's atomic theory and gives a comprehensive picture.
Mar 21, 2016 · Niels Bohr applied Max Planck's quantum theory to the Rutherford model to come up with his famous Bohr model of the atom. The structure of the Bohr model was similar to that of a solar system with electrons orbiting the positively charged atomic nucleus in fixed orbits.
Jun 04, 2019 · Learn the in and out of Bohr's Atomic Model JEE Chemistry basic concepts, tricks to solve JEE problems and questions entailed in JEE Main and JEE Advanced 2020 preparation.Pahul sir enlightens you.
Syllabus For B.sc 1st Year Zoology Promotion of Science Education (Pose) Scholarship , Haryana Haryana State Council for Science and Technology For the first year students of B.Sc. & M.Sc. in basic science subjects viz Physics, Zoology 195-229 Ancillary Courses 1. Biology 231-232 2. Chemistry 232 3. Computer Science 233 4. The credits of each of the three main subjects shall
Danish physicist and Nobel Laureate Niels Henrik David Bohr's atomic model, popularly called the Bohr's Atomic Model, is search engine major Google's new doodle. The doodle marks Bohr's 127th birth.
Km Matol Botanical International Cdc Fungi Media Morphology What Is Math Studies Equivalent To K Social Science Center Enhance Pre K-8th grade classroom science with activities focused on physics, biology, chemistry, earth science, and more! All programs support Next Generation Science Standards. Professional Development. Infuse your practice with innovative ideas and strategies to engage students in science. The FTCE
Clearly I am not a hipster, because I love the Bohr model, and will staunchly defend its use — at least in popular physics books for general audiences, and introductory courses for undergraduates.
I Niels Henrik David Bohr (Template:IPA-da; 7 Octubre 1885 – 18 Noviembre 1962) metung yang physicist ibat Denmark a minambag king pamitátag ning pangabalangkas ning atomo ampo king pamanintindi king pangabalangkas ning atomo ampo king quantum mechanics, at para kaniti, tinggap ne ing Nobel Prize in Physics anyang 1922. Ayalkus (developed) ne ing modelu ning atomo nung nu.
Niels Bohr Biographical N iels Henrik David Bohr was born in Copenhagen on October 7, 1885, as the son of Christian Bohr, Professor of Physiology at Copenhagen University, and his wife Ellen, née Adler. Niels, together with his younger brother Harald (the future Professor in Mathematics), grew up in an atmosphere most favourable to the development of his genius – his father was an eminent.
In 1927, when Albert Einstein began his famous series of battles at the Solvay Conference in Brussels with Danish physicist Niels. Bohr arrived there for a research stay lasting several months.
Niels Bohr. Model of the Atom (Niels Bohr) In 1913 one of Rutherford's students, Niels Bohr, proposed a model for the hydrogen atom that was consistent with Rutherford's model and yet also explained the spectrum of the hydrogen atom. The Bohr model was based on the following assumptions. 1. The electron in a hydrogen atom travels around the nucleus in a circular orbit.
Niels Henrik David Bohr (Danish: [nels ˈb̥oɐ̯ˀ]; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. Bohr was also a philosopher and a promoter of scientific research. Bohr developed the Bohr model of the atom, in which he proposed.
Image caption Bohr's model of atomic structure was at odds with established. The letters will be excerpted in a forthcoming book by Prof Heilbron and Finn Aaserud of the Niels Bohr Archive, titled.
One of the more interesting personalities of the early 20th century was Niels Bohr. Bohr was quite athletic. And he came up with the idea of the "liquid drop" model of the nucleus. Leading up to.
Nov 15, 2017 · Bohr model was proposed by Niels Bohr in 1915. Bohr model is considered as a modification of Rutherford model. The main difference between Rutherford and Bohr model is that Rutherford model does not explain the energy levels in an atom whereas Bohr model explains the energy levels in an atom.
Feb 8, 2017. Niels Bohr's atomic model was utterly revolutionary when it was presented in 1913. Although it is still taught in schools, it became obsolete.
Aged 18, Niels Bohr started studying philosophy and mathematics. and became part of the group of eminent scientists who studied the structure of the atom. Bohr published his model of atomic.
They are all descendants of the revolution in atomic theory catalysed by Danish physicist Niels Bohr 100 years ago. in concentric orbits around a positively charged nucleus. In Bohr's model,
Here's an animation of the Bohr model of the hydrogen atom:* Hydrogen. Niels Bohr in about 1922 (1885-1962), Founding Father of quantum.
Niels Bohr's expertise was crucial to the Allies. Within two years the pair had published their "collective model" of nuclear structure. It combined the two existing theories, noting that, as.
The Bohr model of atomic structure was developed by Danish physicist and Nobel laureate Niels Bohr (1885–1962). Published in 1913, Bohr's model improved the classical atomic models of physicists J. J. Thomson and Ernest Rutherford by incorporating quantum theory. While working on his doctoral dissertation at Copenhagen University, Bohr studied physicist Max Planck's quantum theory of.
Worksheet Of Science For Class 1 11 Feb 2019. In this article, we have mentioned the usefull worksheet for CBSE Class 1 EVS. you can download CBSE Class 1 EVS Worksheet in PDF for. students get a parade of substitutes who might give them worksheets or worse — spend time sitting in an auditorium without. 1 Molecule Of Co2 The use
Precipitation Science For Kids Weather Wiz Kids is a fun and safe website for kids about all the weather info they need to know. It contains tools for weather education, including weather games, activities, experiments, photos, a glossary and educational teaching materials for the classroom. Precipitation definition is – the quality or state of being precipitate : hastiness. Kids
Born in 1885 to Christian Bohr – a professor of physiology – and Ellen Adler Bohr in Copenhagen, Denmark, Niels Bohr made numerous contributions. Bohr's atomic orbit model (Credit: S. Egts) After.
The most important properties of atomic and molecular structure may be exemplified using a simplified picture of an atom that is called the Bohr Model.This model was proposed by Niels Bohr in 1915; it is not completely correct, but it has many features that are approximately correct and it is sufficient for much of our discussion.
We shall start with the semi-classical model, as suggested in 1913 by Niels Bohr, and called: The Bohr model of the atom. According to this model, every atom is.
Because of his work, Bohr was awarded various accolades and awards. Niels Bohr discovered his atomic theory in 1913. He initially named this the Energy Level Model, as the theory proposed the.
Mar 11, 2014 · The model of the atom made by Neil Bohr depicts a positively charged nucleus surrounded by a negatively charged ring of electrons that travel in circular orbits. It was a large advancement in the field because Bohr's model described, for the first time, that an electron must absorb or omit energy to move between orbits.
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L'hélicon est un instrument de musique à vent en cuivre, de la famille des tubas.
Souvent confondu avec le soubassophone à cause de la similitude de sa forme globale (le musicien porte l'instrument autour de son buste), il se distingue de celui-ci par une absence de pavillon ou un très petit pavillon positionné dans le prolongement de l'enroulement du tube (ou corps de l'instrument) et une très forte conicité (le diamètre du tube à son extrémité est comparable au diamètre du pavillon du soubassophone, mais il ne présente pas la modification de l'angle de conicité qui délimite habituellement les pavillons). Il présente l'avantage d'avoir moins de prise au vent et un meilleur équilibre global que le soubassophone. Cette particularité lui vaut d'avoir été choisi par les musiques défilant à cheval, à l'instar de la Garde républicaine.
Il existe des hélicons en si, mi, et en fa.
Très répandu en Europe centrale et de l'est, c'est la contrebasse à vent de prédilection des fanfares gitanes de Macédoine.
Il a été popularisé par la chanson de Boby Lapointe L'hélicon.
Voir aussi
Soubassophone ou Sousaphone
Tuba, Tuba wagnérien
L'hélicon dans la culture populaire
L'Hélicon, chanson, Boby Lapointe (1962).
L'hélicon, livre disque pour enfant "Weepers Circus à la récré", par Weepers Circus et Juliette (2009).
Hélicon, chanson, Cliché (2014).
Liens externes
Helicon | {
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The next closest planet to Sol is Two and little is known about it.
The third closest planet to Sol is Three, and is naturally habitable to homo sapiens without any need for terraforming. The climate and geography are very similar to Earth That Was, but sufficient history has been lost such that whether Three actually is Earth That Was or simply a similar planet is unknown.
Three has one moon which is called Luna. Luna is tidally locked to Three, and has two scientific-oriented bases which are limited to scientists of various fields, their families, and support staff. Unlike other planets and moons in the system, there is no tourism industry, permanent population, or government on Luna.
The two bases, one on each side of the moon, one facing Three and one facing away, are connected by a high-speed, equatorial train.
The seventh planet in the Cups system is a large gas giant that is uninhabitable, known as Seven, but has many moons, most of which have a high enough gravity to comfortably support human life.
Nanna and Vali are the only two moons with permanent populations.
Nanna is the largest moon of Seven also has the largest permanent human colony of all the moons of Seven, and has the fourth smallest orbit. It has several dome colonies around it's surface and spins on its axis once every 46 earth hours. Since Nanna is so far from Sol, it is Seven that provides the most light and occupies the greatest percentage of the sky. Visitors to Nanna and Vidar who are unnaccostomed to Seven hanging over them often find themselves unconsciously hunched over in response to Seven hanging in the sky.
Vidar is a pleasure moon, with domes dedicated to various forms of entertainment. It also has a deep mine system that fully penetrates the solid core of the moon and features special zero gravity entertainment. There are actually two parallel inner core spheres, one for tourists and one exclusively for permanent residents of Nanna.
Ull is the second smallest moon of seven with the most elliptical and inclined orbit. At it's furthest from Seven, it swings slightly outside Sif's orbit (without overlapping). It houses a scientific base dedicated to early detection of meteors, and tracking and indexing asteroids.
Vali is the second most distant moon from Seven and caters to interstellar corporations as a headquarters and tax haven. It's entire economy is funded by banking and legal services.
Sif is a captured asteroid that is unusually rich in rare ores. In terms of asteroids, it is unusually large, but in terms of moons, it is rather small. | {
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\section{Introduction}
In basis-set quantum chemistry, we divide the challenge of solving the non-relativistic electronic Schr\"odinger equation into
two parts: the treatment of $n$-electron correlations, and the saturation of the one-particle basis.
Recent years have seen significant advances in both these areas. In the first case, methods
such as general order coupled cluster (CC) \cite{kallay}, the density matrix renormalization group (DMRG) \cite{chan2011}, and
initiator full configuration interaction quantum Monte Carlo (\mbox{{\it i}-FCIQMC}) \cite{cleland,booth1} have been developed to
achieve an efficient treatment of arbitrary $n$-electron correlations in modestly sized molecules. In the second case, explicit correlation (F12) techniques\cite{Ten-no2004}
augment the one-particle basis with geminal functions that represent the electron-electron cusp.
Taken together, these advances provide the potential to converge to near-exact solutions of the non-relativistic electronic Schr\"odinger equation
at the basis set limit.
In this report, we describe the efficient combination of explicit correlation, via the canonical transcorrelation approach \cite{yanai2012canonical},
with DMRG and with \mbox{{\it i}-FCIQMC}, and apply these combinations to determine the ground and excited state
electronic structure of the beryllium dimer {to very high} accuracy.
\section{Methods}
The essence of explicit correlation (henceforth referred to as F12 theory) is to use a geminal correlation
factor, $f(r_{12})=-\frac{1}{\gamma}\exp(-\gamma r_{12})$ to augment the doubles manifold of the virtual space \cite{Kong2011,tennoreview}.
The geminal can be thought of as including some excitations into a formally infinite basis of virtuals. Labelling the
infinite virtual basis by $\alpha, \beta, \gamma, \ldots$, the geminal doubles excitation operator is written as
\begin{align}
T^{F_{12}} = \sum_{ij\alpha \beta} G^{ij}_{\alpha \beta} E_{ij}^{\alpha\beta} \label{eq:geminal_excite}
\end{align}
where $G^{ij}_{\alpha\beta}$ are the geminal doubles amplitudes.
A significant practical advance was the realization that the geminal
amplitudes are fixed to linear order by the electron-electron cusp condition \cite{Kutzelnigg1985,kut_klopp,Ten-no2004},
\begin{align}
G^{ij}_{\alpha\beta}= \frac{3}{8}\langle \alpha\beta|{Q}_{12} f(r_{12})|ij\rangle + \frac{1}{8} \langle \alpha\beta|{Q}_{12}f(r_{12})|ji\rangle
\end{align}
where ${Q}_{12}$ is a projector, defined in terms of projectors ${O}$ and ${V}$ into
the occupied and virtual space of the standard orbital basis,
\begin{align}
{Q}_{12} = (1-{O}_1)(1-{O}_2) - {V}_1 {V}_2
\end{align}
that ensures that excitations of the geminal factor are orthogonal to those of the standard orbital space \cite{sinanoglu}.
Combining F12 methodology with the DMRG and \mbox{{\it i}-FCIQMC} methods involves practical hurdles not
present in prior combinations of F12 theory with other correlation methods.
For example, explicitly correlated coupled cluster theory formally starts from
a non-Hermitian effective Hamiltonian, obtained by similarity transforming with the geminal excitation operator $\exp(T^{F_{12}})$ \cite{torheyden}
but the DMRG is most conveniently implemented with a Hermitian effective Hamiltonian. Similarly, the
universal perturbative correction of Torheyden and Valeev, which has been used with \mbox{{\it i}-FCIQMC} \cite{booth,torheyden2} does not introduce non-Hermiticity, but
requires
the one- and two-particle reduced density matrices which can be expensive to compute precisely in Monte Carlo methods.
These practical complications are removed within the recently introduced canonical transcorrelation
form of F12 theory of Yanai and Shiozaki \cite{yanai2012canonical}. In this method,
a Hermitian effective Hamiltonian is obtained from
an {\it anti-hermitian} geminal doubles excitation operator, ${A}^{F_{12}}$
\begin{align}
{A}^{F_{12}} = \frac{1}{2} ({T}^{F_{12}} - {T}^{{F_{12}\dag}})
\end{align}
The canonical transcorrelated Hamiltonian is formally defined as
\begin{align}
\bar{H} = \exp({-{A}^{F^{12}}}) H \exp({{A}^{F_{12}}}) \label{eq:ct_h}
\end{align}
The fully transformed Hamiltonian involves operators
of high particle rank. To ameliorate the complexity, Yanai and Shiozaki invoke the same
commutator approximations used in the canonical transformation theory \cite{Neuscamman2010, yanaict1, yanaict2}, and further simplify the
quadratic commutator term by replacing the
Hamiltonian with a generalized Fock operator ${f}$,
\begin{align}
\bar{H}^{F_{12}} = H + [H, A^{F_{12}}]_{1,2} + \frac{1}{2}[[f, A^{F_{12}}], A^{F_{12}}]_{1,2} \label{eq:ct_f12}
\end{align}
an approximation which is valid through second-order in perturbation theory. The subscript $1,2$ denotes
that only one- and two-particle rank operators and density matrices are kept in the Mukherjee-Kutzelnigg normal-ordered form \cite{Mukherjee1997,kutzelnigg97}.
In our calculations here, normal-ordering is carried out with respect to the Hartree-Fock reference, thus no density
cumulants \cite{Mazziottiijqc,kutzelniggcumulant,Mazziotticpl,MazziottiReview} appear. The only error arises from the neglect of the three-particle normal-ordered operator\cite{EricQuadraticCT} generated by the $A^{F_{12}}$ excitations. As the orbital basis increases, $A^{F_{12}}$ tends to zero and the three-particle error also goes to zero, very
different behaviour from density cumulant theories where full three-particle quantity reconstruction is performed \cite{EricQuadraticCT,Mazziottijcp}. In this sense,
the three-particle error in this theory is part of the basis set error.
The practical advantages of the canonical transcorrelation formulation are that no correlated density matrices are required
and the effective Hamiltonian is Hermitian and of two-particle form. It may thus be combined readily
with {\it any correlation treatment}.
Beyond the practical advantages, the canonical transcorrelation formulation uses a
``perturb then diagonalize'' approach, rather than the ``diagonalize then perturb'' approach of the a posteriori F12 treatment of Valeev previously combined with \mbox{{\it i}-FCIQMC}. This allows
the geminal factors to automatically relax the parameters of the subsequent correlation treatment. This approach is similar in spirit to the similarity transformed F12 method of Ten-no which has been used with the PMC-SD method of Ohtsuka \cite{ohtsuka}, but there the excitation operator is not anti-hermitian and alternative approximations are used in the simplification of the resulting equations. The use of the effective Hamiltonian (\ref{eq:ct_f12}) was denoted by Yanai and Shiozaki by the prefix
F12-, thus in their nomenclature, the combinations with DMRG and \mbox{{\it i}-FCIQMC} in this work would be \mbox{F12-DMRG} and \mbox{F12-{\it i}-FCIQMC} respectively. However,
as all our DMRG and \mbox{{\it i}-FCIQMC} calculations use this effective Hamiltonian here, we will usually omit the F12 prefix and simply refer to DMRG and \mbox{{\it i}-FCIQMC}.
We now briefly introduce the DMRG and \mbox{{\it i}-FCIQMC} correlation methods used in this work. The DMRG is
a variational ansatz based on a matrix-product representation of the FCI amplitudes.
Expanding the FCI wavefunction as
\begin{align}
|\Psi\rangle = \sum_{\{n\}} C^{n_1 n_2 \ldots n_k}|n_1 n_2 \ldots n_k\rangle
\end{align}
where $n_i$ is the occupancy of orbital $i$ in the occupancy vector representation of the $n$-particle determinant $|n_1 n_2 \ldots n_k\rangle$,
$\sum_i n_i = n$, the one-site DMRG wavefunction approximates the FCI coefficient $C^{n_1 n_2 \ldots n_k}$
as the vector, matrix, \ldots, matrix, vector product
\begin{align}
C^{n_1 n_2 \ldots n_k} = \sum_{\{i \}} A^{n_1}_{i_1} A^{n_2}_{i_1 i_2} \ldots A^{n_k}_{i_{k-1}} \label{eq:mps}
\end{align}
For each occupancy $n_i$, the dimension of the corresponding matrix(vector) is $M\times M$($M$). $M$ is usually referred to
as the number of renormalised states. The energy is determined by minimizing
$\langle \Psi | H |\Psi\rangle / \langle \Psi|\Psi\rangle$ with respect to the matrix and vector coefficients in Eq. (\ref{eq:mps}) \cite{Chan2002,Hachmann2006,chan2011,marti11}.
As $M$ is increased, the DMRG energy converges towards the FCI limit. In practical DMRG calculations, the minimization is
carried out with a slightly more flexible wavefunction form, where two $A^{n_r}$, $A^{n_{r}+1}$ matrices on adjacent orbitals are fused into a single larger composite (two-site) matrix, $A^{n_rn_{r+1}}$.
This introduces a larger variational space than in Eq. (\ref{eq:mps}), which improves the numerical convergence.
A measure of the error in a DMRG calculation is provided by the ``discarded'' weight, which is the (squared) difference in overlap between the two-site wavefunction
and its best one-site approximation: this difference vanishes as $M\to \infty$. The discarded weight usually exhibits
a linear relationship with the energy, and thus provides a convenient way to extrapolate the energy to the exact FCI result \cite{legeza,Chan2002}.
The FCIQMC algorithm has been recently introduced by Alavi and co-workers
\cite{Booth2013,booth1,cleland,semistochastic}.
FCIQMC is a projector Monte Carlo method
wherein the stochastic walk is done in determinant space~\cite{BlaSug-PRD-83,trivedi},
but instead of imposing a fixed node approximation~\cite{Haaf1995} it uses computational
power and cancellation algorithms to control the fermion sign problem.
The exact ground state wavefunction is obtained by repeatedly applying a ``projector" to an initial state,
\begin{align}
|\Psi\rangle = \lim_{n \to \infty} \left(\hat{\mathbf{1}} + \tau (E\hat{\mathbf{1}}-\hat{H})\right)^n |\Phi\rangle
\end{align}
If $|\Psi\rangle$ is expanded in an orthogonal basis of $N_s$ determinants, $|\Psi\rangle= \sum_{i=1}^{N_s} c_i |\mathbf{n}_i\rangle$
the expansion coefficients evolve according to
\begin{align}
c_i(t+1) = \left(1 + \tau(E-H_{ii})\right) c_i(t) - \tau \sum_{j\neq i}^{N_s} H_{ij} c_j \label{eq:pop_dynamics}
\end{align}
where $t$ labels the iterations, and $\tau$ is a time step, the maximum value of which is constrained
by the inverse of the spectral range of the Hamiltonian.
Since the number of basis states, $N_s$ is too large to permit storing all the coefficients, $c_j$, a stochastic
approach is used wherein $N_w$ ``walkers" ($N_w \ll N_s$) sample the wavefunction.
Although the distribution of walkers among the states at any time step $t$ is a crude approximation to the
wavefunction, the infinite time average yields the ground state wavefunction exactly.
The term $1+\tau(E-H_{ii})$ in Eq.~(\ref{eq:pop_dynamics}) leads to an increase or decrease in the weight of the walker on determinant $i$ while $-\tau H_{ij}$ causes transitions of
walkers from determinant $j$ to determinant $i$. If walkers land on the same determinant, their weights are combined.
However, because contributions to a given determinant can be of either sign for most systems,
a fermionic sign problem results, where the signal becomes exponentially small compared to the noise\cite{foulkes}.
As demonstrated by Alavi and coworkers, however, when cancellations are employed, for {\it sufficiently large} $N_w$, the walk undergoes a transition into a regime where the sign problem is controlled\cite{booth1}.
For the sufficiently large $N_w$ such that this cancellation is effective, FCIQMC is exact within a statistical error of order $\sim (N_w N_t)^{-1/2}$. However, the cost of this brute-force approach prevents application to
realistic problems. A significant advance was the introduction of the initiator approximation (\mbox{{\it i}-FCIQMC}) \cite{booth1,cleland,cleland11}. In the initiator approximation, only walkers
beyond a certain {\it initiator threshold} $n_{\rm init}$ are allowed to generate walkers on the unsampled determinants. The result is that low-weight determinants whose sign may not be sufficiently accurate, propagate according to a dynamically truncated hamiltonian, defined by the space of instantaneously occupied determinants. This concentrates the stochastic walk within a subspace of the
full Hilbert space allowing for more effective cancellation, at the cost of introducing an initiator error
that may be either positive or negative. However, as the total number of walkers $N_w$ is increased
(for fixed $n_{\rm init}$) the {\it i}-FCIQMC energy converges to the FCI limit.
Additional large efficiency gains can be made by carrying out some of the walk
non-stochastically and by using a multi-determinantal trial wave function when computing the energy estimator,
giving rise to semistochastic quantum Monte Carlo \cite{semistochastic}.
This is not used in the results presented here, but, future studies will investigate the gain in efficiency
and the possible reduction in initiator bias from doing so.
\section{Results and Discussion}
We now describe the application of the DMRG and {{\it i}-FCIQMC} methods
to the beryllium dimer. The beryllium dimer has been of long-standing interest
to theory and experiment. (See Refs. \cite{Patkowski1, Merritt} for an
overview of earlier theoretical and experimental work). Simple molecular orbital arguments
would say that the molecule is unbound, however, Be$_2$ can in fact be observed in the gas phase. The observed bond is
significantly stronger than that of other van der Waal's closed shell diatomics such as He$_2$ and Ne$_2$ \cite{Patkowski1,Patkowski2}.
The unusual bonding arises from electron correlation effects that are enhanced by the near $sp$ degeneracy of the Be atom.
This near-degeneracy, coupled with the need for very large basis sets to describe the long bond-lengths,
presents a challenge for modern electronic structure methods, while the weak bond makes accurate experimental measurement challenging.
The lack of accurate theoretical data has also hindered the intepretation of experiment, as the
a priori assumed functional form of the potential energy curve biases the extraction of parameters from the spectral lines.
Thus, for many years, there had been significant disagreement between theory and experiment.
The earliest experimental estimate of the well-depth ($D_e$) was 790$\pm$30~cm$^{-1}$ (Ref. \onlinecite{Bondybey, Bondybey2}), but this used a Morse potential
in the fitting that has the wrong shape at large distances, where
van der Waal's forces dominate.
Theoretical calculations generally yielded much deeper wells. Composite
coupled cluster/full-configuration interaction schemes that sum over core/valence (CV), complete basis set (CBS), high-order
correlation effects, and relativistic corrections, gave $D_e$ as 944$\pm 25$~cm$^{-1}$ (Ref. \onlinecite{Martin}),
938$\pm$15~cm$^{-1}$ (Ref. \onlinecite{Patkowski1})
and 935$\pm$10~cm$^{-1}$ (Ref. \onlinecite{koput}). We believe the latter calculation to be the most accurate to date.
Variants of multireference configuration
interaction gave similar, but slightly shallower wells: 903$\pm$8~cm$^{-1}$
(Ref. \onlinecite{Gdanitz1999}, $r_{12}$-MR-ACPF with relativistic corrections),
912~cm$^{-1}$ (Ref. \onlinecite{Schmidt2010}, MRCI with CV, CBS, and relativistic corrections), and 923~cm$^{-1}$ (Ref. \onlinecite{koput}, MRCI+Q, no error bar).
Only recently, remeasurements by Merritt {et al.}\cite{Merritt}, together with an improved fitting of the experimental spectrum,
yielded an experimentally derived $D_e$ consistent with theory: 929.7$\pm$2.0~cm$^{-1}$, which lies within the error bars of the calculations.
A further refit of Merritt {et al.}'s measurements to a ``fine-tuned'' version of the potential of Ref. \onlinecite{Patkowski2} gave a slightly modified
well-depth of $D_e=$934.6 cm$^{-1}$, presumably with similar error bars to Ref. \onlinecite{Merritt}. This can be regarded as the most
accurate ``experimental'' estimate of $D_e$ to date.
With the recent resolution of the disagreement between theory and experiment, bonding in the beryllium dimer
can now be considered to be satisfactorily understood, at least
from a computational perspective. Nonetheless, the theoretical efforts so far have required
careful composite schemes to separately saturate basis set effects, high-order correlation, and core contributions.
While such additive schemes perform quite well,
{the need to assume additivity between large contributions is theoretically unsatisfactory
and can potentially introduce some uncertainty into the final predicted result}.
For example, the all-electron FCI calculation
in Ref. \onlinecite{Patkowski1} could only be carried out in an aug-cc-pVDZ basis, and gave a well-depth of only 181~cm$^{-1}$, while
the CCSD(T) calculations in the largest aug-cc-pV7Z basis \cite{koput} gave a well-depth of only 696~cm$^{-1}$.
Thus, in reaching the value of $D_e \approx$ 935~cm$^{-1}$ a large degree of transferability amongst incremental
contributions was assumed. The only non-composite method, the $r_{12}$-MR-ACPF calculation of Gdanitz \cite{Gdanitz1999} gave
a non-relativistic $D_e=898$~cm$^{-1}$, which remains quite far from the best experimental or theoretical results.
\begin{table*}
\caption{\label{tab:dmrgm} Energy in $E_h$ and discarded weights of the DMRG calculation with the canonical transcorrelated Hamiltonian and cc-pCVQZ-F12 basis set for
the Be$_2$ dimer at a bond length of 2.45 \AA. (l.) and (q.) denote the results of linear and quadratic extrapolations.
}
\begin{tabular}{lcc}
\hline
\hline
M&Energy&Discarded weight\\
\hline
500 &1.57$\times 10^{-7}$ & -29.338592\\
1000& 2.28$\times 10^{-8}$& -29.338647\\
1500& 5.81$\times 10^{-9}$& -29.338655\\
2000& 1.56$\times 10^{-9}$& -29.338657\\
$\infty$(l.)&-&-29.338657\\
$\infty$(q.)&-&-29.338658\\
\hline
\hline
\end{tabular}
\end{table*}
We can now carry out a direct calculation, with saturated large basis sets and explicit correlation as well as a full account of the $n$-electron correlations,
using the canonically transcorrelated DMRG and \mbox{{\it i}-FCIQMC} methods, thus eliminating the need for
composite approaches.
We have computed several points along the ground-state
$1^1\Sigma^+_g$ Be$_2$ potential energy curve using a series of
cc-pCVnZ-F12 basis sets \cite{basis} with $n$=D, T, Q (henceforth referred to as DZ, TZ, and QZ, for short) and cc-pCVnZ-F12\_OPTRI basis \cite{basis} sets with $n$=D, T, Q respectively for the resolution of the identity (RI) basis sets.
These basis sets contain 68, 124, and 192 basis functions respectively, with
up to $g$ functions in the QZ basis, and the RI basis sets contain 164, 190 and 188 basis functions respectively.
The DMRG calculations
were carried out using the
\textsc{Block} code \cite{sharma2012spin}.
This DMRG implementation incorporates two symmetries not commonly found
in other implementations: spin-adaptation (SU(2))
and $D_{\infty h}$ symmetries. Spin-adapted DMRG implementations for
quantum chemistry
were described by Wouters et al. \cite{wouters}
and our group \cite{sharma2012spin},
based on earlier work by McCulloch \cite{mcculloch3}.
Compared to non-spin-adapted DMRG with only $S_z$ symmetry, we find that calculations with $M$ spin-adapted states correspond
in accuracy to approximately $2M$ renormalized non-spin-adapted states in the calculation \cite{sharma2012spin}. Our implementation of $D_{\infty h}$
symmetry resembles that for spin-symmetry, where the Wigner-Eckart theorem is used to simplify the evaluation of matrix elements as well as to reduce
storage. We find that $D_{\infty h}$ symmetry brings an additional factor of 2 in the effective $M$ over the use of only $D_{2h}$ symmetry. Consequently, with
both spin and $D_{\infty h}$ adaptation, our reported energies here with $M$ renormalized states are { roughly comparable
in accuracy to similar calculations with $4M$ renormalized states in a conventional DMRG code with only $S_z$ and $D_{2h}$ symmetries}. {Our calculation at the bond length of $2.45\AA$ took a wall clock time of 150 hours running in parallel on 72 Intel Xeon E5-2670 cores, totalling 10,800 core hours.}
\begin{figure*}
\begin{center}
\includegraphics[width=2.5in]{extrap.eps}
\end{center}
\caption{Convergence of the DMRG energy (E+29.0) in $E_h$ as a function of the discarded weight
and renormalized states $M$ with the canonical transcorrelated Hamiltonian and cc-pCVQZ-F12 basis set. \label{fig:dmrgm}}
\end{figure*}
The \mbox{{\it i}-FCIQMC} calculations were carried
out using the \textsc{Neci} code \cite{booth1,booth2011,Booth-neci}. These calculations used the Abelian rotational subgroup of $D_{\infty h}$, as described in Ref. \onlinecite{booth2011}. This symmetrized determinant space is smaller than that for the $D_{2h}$ group, especially for large angular momentum basis sets, but is larger than
that for the full $D_{\infty h}$ group by less than a factor of 2 because it retains only one-dimensional irreducible representations.
The F12 integrals and transcorrelated Hamiltonian were generated
using the \textsc{Orz} code, using the F12 exponent $\gamma=1.0$~$a_0^{-1}$.
The well-depth was calculated from the energy at $r=2.45$~\AA.
All 8 electrons were correlated, thus the largest calculation formally involved more than $3\times 10^{15}$ determinants.
In the DMRG calculations, we also computed the lowest 4 excited states in the $\Sigma$ class of irreps ($2^1\Sigma^+_g$, $1^1\Sigma^+_u$, $1^1\Sigma^-_g$, $1^1\Sigma^-_u$)
at the ground-state equilibrium geometry of $r=2.45\AA$.
For comparison, we also present results of CCSD(T), CCSD(T)-F12, and F12-CCSD(T) \cite{adler2007,knizia,yanai2012canonical} calculations for the ground-state curve,
and MRCI-F12 \cite{shiozaki2011}, MRCI, and EOM-CCSD calculations for the excited states. These computations
were performed using the \textsc{Molpro} package\cite{werner2012}; the
F12-CCSD(T) calculations used the MRCC program with the
transcorrelated Hamiltonian as input\cite{mrcc}.
\begin{table*}
\caption{\label{tab:ccsdtf12bench}Binding energies in units of m$E_h$ from CCSD(T)/aug-cc-pCVnZ ($n$=4, 5, and 6), F12-CCSD(T)/cc-pCVQZ-F12 and CCSD(T)-F12b/cc-pCVQZ-F12,
as a function of bond length $r$. All the binding energies are counterpoise corrected. 2 different values of the complete basis set limit of the CCSD(T) method are
calculated by extrapolating the correlation energies of the Be$_2$ dimer and the Be atom (no extrapolation of the HF energy was performed) using Eqs.\ref{extrap1},\ref{extrap3}.}
\begin{tabular}{ccccccccc}
\hline
\hline
&\multicolumn{3}{c}{CCSD(T)}&&\multicolumn{2}{c}{CCSD(T)/CBS}&CCSD(T)-F12b&F12-CCSD(T)\\
\cline{2-4}\cline{6-7}
$r$/\AA&QZ&5Z&6Z&& $(1)$ &$(2)$ &QZ&QZ\\
\hline
2.20& 0.46& 0.73& 0.86&& 0.97& 1.05& 0.88& 0.86\\
2.40& 2.73& 2.94& 3.04&& 3.12& 3.18& 3.07& 3.05\\
2.45& 2.83& 3.03& 3.12&& 3.20& 3.25& 3.15& 3.14\\
2.50& 2.83& 3.02& 3.11&& 3.18& 3.23& 3.14& 3.12\\
3.00& 1.45& 1.55& 1.60&& 1.64& 1.67& 1.60& 1.61\\
5.00& 0.34& 0.35& 0.36&& 0.36& 0.36& 0.35& 0.35\\
\hline
\hline
\end{tabular}
\end{table*}
\begin{table*}
\caption{\label{tab:be2data} Be$_2$ binding energies in units of m$E_h$ as a function of bond-distance
using various methods. The atomic Be energy is \mbox{-14.666740}$E_h$ (DZ-DMRG),
-14.666691$E_h$ (TZ-DMRG), -14.667207$E_h$ (QZ-DMRG). DMRG binding energies for the three basis sets cc-pCVnZ-F12 , where $n$=2, 3, and 4, are tabulated and a fourth column gives our best estimate with error bars (see text for more details). Two sets of {\it i}-FCIQMC calculations are performed, the results in the columns marked QZ(50) and QZ(200) are calculations with 50 million and 200 million walkers respectively. The statistical
error of {\it i}-FCIQMC is denoted in brackets. The difference between the DMRG and {\it i}-FCIQMC numbers is a measure of initiator
error, see text. Merritt, Patkowski denote
experimentally derived fits from Refs. \cite{Merritt,Patkowski2}.}
\begin{tabular}{cccccccccccc}
\hline
\hline
&\multicolumn{4}{c} {DMRG}&&\multicolumn{2}{c}{{\it i}-FCIQMC}&&\multicolumn{2}{c}{experiment}&\\
\cline{2-5}
\cline{7-8}
\cline{10-11}
r/\AA& DZ & TZ & QZ & CBS/BSSE/rel.&& QZ (50) &QZ(200) && Merritt & Patkowski \\
\hline
2.20 & 0.68 & 1.76 & 2.11 & 2.23(0.08)&& 2.06 (0.02)&&& 2.41 & 2.21 \\
2.30 & 2.33 & 3.41 & -- & -- &&--& &&3.67 & 3.71 \\
2.40 & 3.02 & 3.99 & 4.20 & 4.26(0.05) && 4.22 (0.02)&&& 4.17 & 4.22 \\
2.45 & 3.13 & 4.09 & 4.24 & 4.30(0.04)&& 4.27 (0.05)&4.21(0.04)&& 4.24 & 4.26 \\
2.50 & 3.13 & 4.00 & 4.18 & 4.24(0.04) && 4.32 (0.03)&4.11(0.04)&& 4.20 & 4.19 \\
2.60 & 2.95 & 3.70 & -- & -- && -- &&& 3.89 & 3.86 \\
2.70 & 2.64 & 3.27 & -- &--&& -- &&& 3.44 & 3.42 \\
3.00 & 1.72 & 2.12 & 2.23 &2.26(0.02) && 2.32 (0.05)&&& 2.18 & 2.22 \\
5.00 & 0.31 & 0.37 & 0.39 &0.39(0.01) && 0.55 (0.03)&&& 0.39 & 0.40 \\
\hline
\hline
\end{tabular}
\end{table*}
\begin{table*}
\caption{\label{tab:des} A comparison of $D_e$ cm$^{-1}$ from this work and
from the literature. Here BSSE, CBS and rel. respectively indicate that corrections have been made for basis set superposition error, basis set incompleteness error and relativistic effects. }
\begin{tabular}{cc}
\hline
\hline
Method & \\
\hline
CCSD(T)-F12b/BSSE & 699.3 \\
DMRG & 931.2 \\
{\it i}-FCIQMC & 924(9) \\
DMRG/CBS/BSSE/rel.& 944(10) \\
\hline
Author & \\
\hline
Merritt(E/T)\cite{Merritt} & 929.7(2) \\
Patkowski(E/T)\cite{Patkowski2} & 934.6 \\
Patkowski(T)\cite{Patkowski1} & 938.0(15) \\
Schmidt(T)\cite{Schmidt2010} & 915.5 \\
Koput(T)\cite{koput} & 935.1(10) \\
\hline
\hline
\end{tabular}
\end{table*}
Tables \ref{tab:be2data} and \ref{tab:des} present our accumulated data for the DMRG and \mbox{{\it i}-FCIQMC} ground-state Be$_2$ calculations,
as well as selected computed and reference data for the well-depths. All DMRG energies correspond to $M=2000$ (see below) while all \mbox{{\it i}-FCIQMC} calculations
were carried out with $n_{\rm init}=3$ and $N_w=5 \times 10^7$ (see below).
Figure \ref{fig:dmrgm} shows
the convergence of the DMRG energy as a function of the discarded weight and $M$ for the QZ basis at $r=2.45\AA$; {energies as a function of $M$ are given in Table \ref{tab:dmrgm}. We note that the DMRG energies presented in Table~\ref{tab:dmrgm} and Figure~\ref{fig:dmrgm}
were obtained by first carrying out standard DMRG calculations up to $M$=2500, and then {\it backtracking} (by decreasing $M$ in subsequent sweeps)
down to $M$=500 in steps of 500, to obtain the tabulated energies at $M$=500, 1000, 1500, 2000. This ensures that the energy at each $M$ is well converged
and free from any initialization bias, leading to more accurate extrapolation.
We calculate the DMRG extrapolated energy by fitting to
linear and quadratic functions of the discarded weight. Due to the high cost of calculation, insufficient
sweeps were performed at $M$=2500 to attain full convergence, hence the DMRG energies at $M$=2500 were not themselves used in the extrapolation.
The maximum difference between the linear and quadratic extrapolations is 6 $\mu E_h$, and we use this as an upper estimate of the remaining error
in the DMRG energy. Examining Fig. \ref{fig:dmrgm}, we find that the DMRG energy converges extremely rapidly with $M$: even by $M=1000$, the total DMRG energy in the QZ
basis appears within 10~$\mu E_h$ (2 cm$^{-1}$) of the extrapolated
$M=\infty$ result!
The \mbox{{\it i}-FCIQMC} energies contain two sources of error: statistical error (due to the finite simulation time), and initiator error (due to the
finite walker population). The statistical errors are listed in the Table~\ref{tab:be2data} and are
on the order of 20-50~$\mu E_h$. The remaining discrepancy between the \mbox{{\it i}-FCIQMC}
energies and the DMRG energies is due to initiator error. Note
that the initiator error can be of either sign. Because of the small energy scales
of this system, the initiator error is significant at some bond-lengths. For
example, at $r=2.5\AA$, the initiator error with $N_w=5\times 10^7$ is 0.14~m$ E_h$, or about $5 \sigma$,
causing the \mbox{{\it i}-FCIQMC} curve to have an unphysical shape (the energy
at $2.50\AA$ is below that at the equilibrium distance $r=2.45\AA$). The initiator error can be removed by carrying out simulations with larger number of walkers. At $r=2.45\AA$ and $r=2.50\AA$ we recomputed the \mbox{{\it i}-FCIQMC} using $N_w=2\times 10^8$ walkers. These \mbox{{\it i}-FCIQMC} are now in better agreement with the
converged DMRG energies and restore the physical shape of the potential. However, such calculations were 3-4 times more expensive than the corresponding
DMRG calculations.}
We now discuss the possible remaining sources of error and non-optimality in our
calculations.
These include basis set superposition error (BSSE), relativistic effects, non-optimality of the F12 $\gamma$ exponent, geometry effects, errors associated with the F12 approximations in the canonical transcorrelation approach and basis set incompleteness error. BSSE error can be estimated from the
counterpoise correction \cite{boys}. We find the counterpoise contribution to
the F12-DMRG well-depth to be -11~$\mu E_h$ (-2.4 cm$^{-1}$) at the QZ level.
Our relativistic correction using the CCSD(T)/aug-cc-pCVQZ method with the second-order Douglas-Kroll-Hess (DKH) one-electron Hamiltonian is
-4.2 cm$^{-1}$, which is in good agreement with previous studies\cite{Patkowski1,Gdanitz1999}.
We have checked the optimality of the F12 exponent and the bond-length effects
through CCSD(T)-F12 calculations \cite{adler2007,knizia}. At the QZ level, $\gamma=0.8-1.2$ yielded the same CCSD(T)-F12 $D_e$=3.2~m$E_h$ to within 2~$\mu E_h$ (0.4 cm$^{-1}$) and thus we conclude that
our exponent of $\gamma=1.0$ is near-optimal. The difference in energy between the CCSD(T)-F12/QZ equilibrium bond-length energy (at 2.46\AA),
and the energy at our assumed $r_e=2.45\AA$ is only 3~$\mu E_h$ (0.6 cm$^{-1}$).
{The F12 canonical transcorrelation approach contains two kinds of error. The first is the auxiliary basis integral approximations used
to compute the F12 integrals, and the second is the neglect of normal-ordered three-particle operators in the canonical transcorrelated Hamiltonian
as described above. (We recall that in this work all three-particle cumulants are zero in our definition of $\bar{H}^{F_{12}}$, since we normal order
with respect to a Hartree-Fock reference).
Both the above errors are non-variational, which can be seen from the DMRG atomic energies as we increase the basis cardinal number; these are
-14.66674~$E_h$ (DZ), -14.66669~$E_h$ (TZ), -14.66721~$E_h$ (QZ). For comparison, the best variational calculation for the beryllium atom that we are aware of, using exponentially correlated Gaussian expansions, is -14.66736 $E_h$\cite{beatom}. However, both errors also go identically to zero as the orbital basis is increased, because
the F12 factor (and the $A^{F_{12}}$ amplitude) is only used to represent the correlation not captured within the basis set.
To obtain more insight into the error from the F12 canonical transcorrelated Hamiltonian, we have computed in Table \ref{tab:ccsdtf12bench} the
F12-CCSD(T)/cc-pCVQZ-F12 binding energies (i.e. CCSD(T) using the canonical transcorrelated Hamiltonian) using the MRCC
program of K\'{a}llay \cite{mrcc}, and the conventional CCSD(T)-F12b/cc-pCVQZ-F12 binding energies using the \textsc{Molpro} program package \cite{werner2012}. (As pointed out by Knizia et al.\cite{ccsdf12b}, the CCSD(T)-F12b variant is to be preferred with the large basis sets used here).
We observe that the CCSD(T)-F12b/cc-pCVQZ-F12 and F12-CCSD(T)/cc-pCVQZ-F12 binding energies agree very well (to within 5 cm$^{-1}$ along the entire binding curve).
Perfect agreement between the methods is not expected as they correspond to different
F12 theories, but these results show that the neglect of three-particle operators in the canonical transcorrelated Hamiltonian produces a description
with no significant differences
from a standard F12 approach.
To extrapolate the remaining F12 and basis set errors to zero, we carry out a further basis-set completeness (CBS) study.
In Table \ref{tab:ccsdtf12bench}, we give the CCSD(T)/aug-cc-pCVnZ binding energies for $n$=4, 5, 6. Following Koput\cite{koput}
we use the following two basis extrapolation formulae to provide error bars on the complete basis result:
\begin{align}
E_n &= E_{\infty} + a \exp\left( -b(n-2)\right) \label{extrap1}\\
E_n &= E_{\infty} + a /(n+0.5)^b \label{extrap3}
\end{align}
From Table \ref{tab:ccsdtf12bench} we observe that the F12-CCSD(T)/cc-pCVQZ-F12 binding energies correspond closely
to those of CCSD(T)/aug-cc-pCV6Z. Using Koput's prescription, we obtain the extrapolated energy as the average of Eqs.~(\ref{extrap1}), (\ref{extrap3}).
At the equilibrium bond length we obtain a basis set limit correction to the DMRG calculation of 87~$\mu E_h$ (19 cm$^{-1}$) and an uncertainty of 43~$\mu E_h$ (10 cm$^{-1}$). (We estimate the uncertainty as half
the extrapolation correction). Thus, the basis-set error remains the largest source of uncertainty in our calculations.
Compared to the experimentally derived well-depths, we find that our directly calculated DMRG (and {\it i}-FCIQMC) well-depths, 931.2 cm$^{-1}$ (924$\pm$ 9 cm$^{-1}$),
are in excellent agreement with the ``experimental'' $D_e$ of 929.7~cm$^{-1}$ (Merritt et al \cite{Merritt})
and 934.6~cm$^{-1}$ (Patkowski et al \cite{Patkowski2}) (Table \ref{tab:des}).
Including the estimated CBS correction (19 cm$^{-1}$), the
counterpoise correction (-2 cm$^{-1}$), and the relativistic correction
(-4~cm$^{-1}$), yields a corrected well-depth of 944~cm$^{-1}$ (DMRG) {with an error estimate of 10 cm$^{-1}$}, which is slightly larger,
but still in good agreement with the experimental well-depths. Thus, corrected or otherwise, our calculations compare favorably to the very best experimentally derived well-depths to date.
Compared to CCSD(T)-F12, we find that quadruples and higher correlations contribute 25\% of the binding energy,
indicating significant correlation effects in the ground-state.
The largest absolute discrepancy between our calculations and the experimentally derived curve appears at the shorter
bond-length of $r=2.20\AA$, where we find the energy (CBS/BSSE/rel. corrected) to be 2.07~m$E_h$ above the equilibrium point as compared to 1.83~m$E_h$ and 2.05~m$E_h$ respectively, in the experimental numbers of Merritt et al. \cite{Merritt} and Patkowski et al. \cite{Patkowski2}. Given the close agreement between
our computations and experiment at all other points on the curve (the agreement between
the corrected DMRG curve with Patkowski's curve is better than 0.05~m$E_h$ at all points) the
discrepancy with Merritt's experimental number is quite large. When measured as a multiple of the theoretical uncertainty, we also find that the largest errors are at $r=2.20\AA$ (2.4$\sigma$) and at $r=3.00\AA$ (4.0$\sigma$). We note that
the inadequacies of Merritt's fit at {\it longer} distances have already been discussed in Ref. \cite{Patkowski2}. Our results further suggest that there are
inaccuracies in Merritt's experimental fit at shorter distances as well.
\begin{table*}
\caption{\label{tab:exc}Low-lying $\Sigma$ excited state energies (in eV) of Be$_2$ calculated using (F12-)DMRG and the cc-pCVTZ-F12 and cc-pCVQZ-F12 basis sets. The complete
basis set limit and error estimate of the (F12-)DMRG is also given (see text for more details). Excited state energies from the MRCI-F12 and MRCI+Q-F12 methods using the cc-pCVQZ-F12 basis, and the EOM-CCSD method using the cc-pCV5Z basis are also shown.}
\begin{tabular}{ccccccc}
\hline
\hline
\text{State} & \text{DMRG/TZ} &\text{DMRG/QZ}& \text{DMRG/CBS}&\text{MRCI-F12} & \text{MRCI+Q-F12} &\text{EOM-CCSD}\\
\hline
$2^1\Sigma^+_g$ &3.61& 3.59 &3.57(0.02)&3.60&3.54&3.97\\
$1^1\Sigma^+_u$ & 3.58&3.56 &3.55(0.01)&3.70&3.55&3.48\\
$1^1\Sigma^-_g$ & 7.69&7.66 &7.64(0.03)&8.27&8.13&7.33\\
$1^1\Sigma^-_u$ & 4.81&4.78 &4.77(0.02)&4.80&4.75&5.96\\
\hline
\hline
\end{tabular}
\end{table*}
\begin{figure*}
\begin{center}
\subfigure[$2^1\Sigma^+_g$]{\includegraphics[width=2.5in]{extrap_irrep1.eps}}
\subfigure[$1^1\Sigma^+_u$]{\includegraphics[width=2.5in]{extrap_irrep2.eps}}\\
\subfigure[$1^1\Sigma^-_g$]{\includegraphics[width=2.5in]{extrap_irrep3.eps}}
\subfigure[$1^1\Sigma^-_u$]{\includegraphics[width=2.5in]{extrap_irrep4.eps}}
\end{center}
\caption{Convergence of the DMRG energies (E+29.0) in $E_h$ for the four excited states as a function of the discarded weight
and renormalized states $M$ with the canonical transcorrelated Hamiltonian and cc-pCVQZ-F12 basis set. \label{fig:dmrg_excited}}
\end{figure*}
\begin{table*}
\caption{\label{tab:dmrg_excited} Energy in $E_h$ and discarded weights of the DMRG with transcorrelated cc-pCVQZ-F12 basis set on the Be$_2$ dimer at a bond length of 2.45 \AA.
(l.) and (q.) denote linear and quadratic extrapolations.
}
\begin{tabular}{lcc ccc ccc ccc}
\hline
\hline
&\multicolumn{2}{c}{$2^1\Sigma^+_g$}&&\multicolumn{2}{c}{$1^1\Sigma^+_u$}&&\multicolumn{2}{c}{$1^1\Sigma^-_g$}&&\multicolumn{2}{c}{$1^1\Sigma^-_u$}\\
\cline{2-3}
\cline{5-6}
\cline{8-9}
\cline{11-12}
M&Discarded weight&Energy&&Discarded weight&Energy&&Discarded weight&Energy&&Discarded weight&Energy\\
\hline
500 &7.40$\times 10^{-7}$& -29.206566&&2.43$\times 10^{-7}$& -29.207784&&1.83$\times 10^{-6}$& -29.056883 &&6.18$\times 10^{-7}$& -29.162740\\
1000 &1.27$\times 10^{-7}$& -29.206794&&3.40$\times 10^{-8}$& -29.207877&&1.01$\times 10^{-7}$& -29.057249&&1.02$\times 10^{-7}$& -29.162905\\
1500 &4.71$\times 10^{-8}$& -29.206827&&9.48$\times 10^{-9}$& -29.207890&&2.92$\times 10^{-8}$& -29.057277&&2.87$\times 10^{-8}$& -29.162931\\
2000 &2.18$\times 10^{-8}$& -29.206836&&2.67$\times 10^{-9}$& -29.207894&&9.74$\times 10^{-9}$& -29.057282&&1.11$\times 10^{-8}$& -29.162939\\
$\infty$(l.)& & -29.206844&& & -29.207894&& & -29.057280&& & -29.162940\\
$\infty$(q.)& & -29.206845&& & -29.207895&& & -29.057287&& & -29.162943\\
\hline
\hline
\end{tabular}
\end{table*}
We now turn to the excited state DMRG calculations. While accurate ground-state energies can
be obtained through composite techniques, this is much more
difficult for excited states, due to significantly larger correlation effects. Near exact excited states, however,
can be accessed through a state-averaged DMRG calculation \cite{Dorando2007}. Combined
with the saturated basis set treatment here, the DMRG excitation energies now allow us to present
very accurate excitation energies for large basis sets, against which other methods
may be compared.
The (F12-)DMRG excitation energies, with comparison MRCI-F12, MRCI+Q-F12, and EOM-CCSD
energies, are shown in Table \ref{tab:exc}.
The active space used in the MRCI calculations was a 4 electron, 8 orbital complete active space.
The convergence of the DMRG excitation energies with $M$ is shown in Table \ref{tab:dmrg_excited} and is plotted in Figure \ref{fig:dmrg_excited}.
These show that the DMRG energies are converged to within 10 $\mu E_h$ of the formal exact result, and are thus negligible on the eV scale (on the order of tenths of meV's). The basis set errors for
the excitation energies are larger than for the ground-state, because we use the F12 canonical transcorrelated Hamiltonian derived
for the ground $1\Sigma_g^+$ state to compute all the excitation energies, thus the $A^{F_{12}}$ correlation
factor is biased towards the ground-state. To estimate the complete basis set limit of the excitation energy we use Eq.~(\ref{extrap5}) (derived from a fit to CCSD-F12b energies across a large data set\cite{hill-extrap}).
\begin{align}
E_n &= E_{\infty} + a/n^{4.6} \label{extrap5}
\end{align}
Since excited state complete basis set extrapolation is less well studied,
we estimate the uncertainty conservatively as twice the difference between the estimated complete basis value and the QZ value. As for the ground-states, the basis set error
remains the largest uncertainty in the calculations, but even
with our conservative estimate ranges only from 0.01 to 0.03 eV.
Overall, MRCI+Q-F12 gives the best agreement with DMRG, with errors of less than 0.05 eV for 3 out of the 4 states. The effect of the Q size-consistency correction
is significant, contributing as much as 0.15 eV to the excitation energy. The EOM-CCSD excitation energy errors are large for all states,
which is unsurprising given the multireference nature of the ground-state. However, what is most surprising is that
for the $1\Sigma_g^-$ state, the error of the MRCI+Q excitation energy is as large as 0.4 eV! This indicates extremely
strong correlation effects in this state. The $1\Sigma_g^-$ state of the beryllium dimer is thus a good benchmark state for the development of excited state methods.
To summarize, in this work we have used explicit correlation via the canonical transcorrelation approach, in conjunction with
the density matrix renormalization group and initiator full configuration interaction quantum Monte Carlo methods, to compute the binding curve of the
beryllium dimer without the
use of composite methods. Our calculations correlate all 8 electrons in basis sets with an orbital basis set of up to 192 basis functions (cc-pCVQZ-F12).
Our direct DMRG calculations produce a well-depth of {$D_e$=931.2 cm$^{-1}$} which
agrees very well with the best experimental and theoretical estimates.
The remaining basis set effects, BSSE, and relativistic effects, contribute to
a final well-depth of {$D_e$=944 $\pm 10$ cm$^{-1}$}. We find a significant discrepancy between our computed binding energies and the experimentally derived
energies of Merritt et al. at shorter bond-lengths ($r=2.20\AA$) that suggest inaccuracies in the experimental fits. Finally, using DMRG, we have also computed
the excited states at the equilibrium geometry to unprecedented accuracy,
highlighting surprisingly strong
correlation in the excited states. Overall, we have demonstrated that, by combining explicit correlation with
the DMRG or {\it i}-FCIQMC methods, it is now possible to directly solve the non-relativistic Schr\"odinger equation without significant
basis set or correlation error for small molecules.
\textbf{Acknowledgements}\\
This work was supported by National Science Foundation (NSF) through Grant No.
NSF-CHE-1265277. TY was supported in part by the Core Research for Grant-in-Aid for Scientific Research (C) (Grant No. 21550027) from Ministry of Education, Culture, Sports, Science and Technology-Japan (MEXT). CJU would like to acknowledge the NSF grant NSF-CHE-1112097 and the Department of Energy (DOE) grant DE-SC0006650.
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\section{Introduction}
\label{sec:intro}
Scientific data is often complex in nature and difficult to visualise. As a result,
analytic tools have become increasingly prominent in scientific visualisation,
and in particular topological analysis. While earlier work dealt primarily with
scalar data~\cite{2003-Chiang-Simplification, 2004-Carr-simplification, 2012-Tierny-tvcg},
multivariate topological analysis in the form of the Reeb
Space~\cite{2004-Saeki, 2008-edels-reebspace}
has started to become feasible using a quantised approximation called the Joint
Contour Net (JCN)~\cite{2013-Carr-TVCG}.
Prior experience in scalar and vector topology shows that simplification of
topological structures is required, as real data sets are often noisy and
complex. Although most of the work required is practical and algorithmic in
nature, mathematical formalisms are also needed, in this case based on fiber
analysis, in the same way that Reeb Graphs and contour trees rely on Morse theory.
This paper therefore:
\begin{enumerate}\itemsep=.8pt
\item Clarifies relationships between the Reeb Space of a multivariate map $f$,
the Jacobi Set of $f$, and fiber topology,
\item Introduces the \emph{Jacobi Structure} in the Reeb Space that decomposes the
Reeb Space into \emph{regular} and \emph{singular} components equivalent to edges and vertices
in the Reeb Graph, then reduces it further to a \textit{Reeb Skeleton},
\item Proves that Reeb Spaces for topologically simple domains have simple structures
with properties analogous to properties of the contour tree, allowing
\emph{lip-pruning} based simplification,
\item Introduces the \emph{range measure} and other geometric measures
for a total ordering of regular components of the Reeb Space,
\item Describes an algorithm that extracts the Jacobi Structure from the Joint Contour Net
using a \textit{Multi-dimensional Reeb Graph} (MDRG) and computes the Reeb Skeleton,
and
\item Simplifies the Reeb Skeleton and the corresponding Reeb Space
computing the range and other geometric measures using the Joint Contour Net.
\end{enumerate}
To clarify the relationships between the newly introduced
data-structures in the current paper, note
that the JCN is an approximation of the Reeb Space. We compute
a MDRG from the JCN. The critical nodes of
the MDRG form the Jacobi Structure of the JCN. The Jacobi Structure then
separates the JCN into regular and singular components. The dual graph
of such components gives a Reeb Skeleton which is used in the
multivariate topology simplification.
As a result, much of this paper addresses the theoretical machinery for simplification of the
Reeb Space and its approximation, the Joint Contour Net. ~\secref{PreviousWork} reviews
relevant background material on simplification, followed by a more
detailed review of the fiber topology, Jacobi Set and Reeb Space in~\secref{Background}. \secref{Theory} provides
theoretical analysis and results needed for the lip-simplification of
the Reeb Space. For simple domains, the Reeb Space can have detachable
(lip) components: this is used in \secref{Simplification} to generalise leaf-pruning simplification from the
contour tree to the Reeb Space. Once this has been done, we introduce a range persistence and
other geometric measures to govern the simplification process.
In \secref{SimplifyingJCN}, we give an algorithm for simplifying the Joint Contour
Net (an approximation of the Reeb Space). We start by building a hierarchical structure called the Multi-Dimensional Reeb
Graph (MDRG) that captures the Jacobi Structure of the Joint Contour Net,
and then show how to reduce the JCN to a Reeb Skeleton - a graph with properties similar to a contour tree.
In \secref{Implementation}, we illustrate these reductions first with
analytic data where the correct solution is known \emph{a priori},
then for a real data from the nuclear physics. As part of this, we provide performance figures and other implementation details in
\secref{Implementation}, then draw conclusions and lay out a road map
for further work in \secref{Conclusions}.
\section{Previous Work}
\label{sec:PreviousWork}
Topology-based simplification aims to reduce the topological
complexity of the underlying data. There are different ways to
measure such topological complexity depending on the nature of the
underlying data. Here we mention some well-known approaches from the
literature for measuring the topological complexity and their
simplification procedure.
\subsection*{Scalar Field Simplification.}
The topological complexity of the scalar field data is measured in
terms of the number of critical points and their connectivities -
captured by its Reeb Graph or contour tree. Another way to capture
the topological complexity of the scalar field is by computing the Morse-Smale complex of the corresponding
gradient field. Therefore, the topological simplification
in this case is driven by reducing the number of critical points via
simplification of the Reeb Graph/ contour tree or the Morse-Smale complex.
Carr et al. \cite{2004-Carr-simplification} describe a method for associating local geometric measures such
as the surface area and the contained volume of contours with the contour
tree and then simplifying the contour tree by suppressing the minor
\emph{topological features} of the data. Note that a feature is any
prominent or distinctive part or quality that characterises the data
and topological features captures the topological phenomena of the underlying data. Wood et al. \cite{2004-Wood-excessTopo} give a
Reeb Graph based simplification strategy for removing the excess
topology created by unwanted handles in an isosurface using a
measure for computing the handle-size in the isosurface and associating them
with the loops of the Reeb Graph. Gyulassy et al. \cite{2006-Gyulassy} describe a
technique for simplifying a three-dimensional scalar field by
repeatedly removing pair of critical points from the Morse-Smale
complex of its gradient field, by repeated application of a critical-point
simplification operation. Mathematically, the simplification of
``lips'' proposed in this paper is a direct generalization of this
idea (for scalar fields) to multi-fields.
Luo et al \cite{2009-Luo-Jacobi} describe a method for computing and simplifying gradients and critical points of a function from a
point cloud. Tierny et al. \cite{2012-Tierny-tvcg} present a
combinatorial algorithm for simplifying the topology of a scalar field on a surface
by approximating with a simpler scalar field having a subset of
critical points of the given field, while guaranteeing a small error distance
between the fields.
The topological complexity of a point cloud
data can be measured by its homology. For a point cloud data in
$\mathbb{R}^3$ this is expressed by the topological invariants, such as the Betti
numbers corresponding to a simplicial complex of the point cloud
- denoted by $\beta_0$ (number of connected components), $\beta_1$ (number of
tunnels or 1-dimensional holes) and
$\beta_2$ (number of voids or 2-dimensional holes). The $i$-th Betti number represents the
rank of the $i$-th homology group ($i=0, 1, 2$). Edelsbrunner et al. \cite{2002-Edels-Persist} introduce the
idea of persistence homology for the topological simplification of a
point cloud by reducing the Betti numbers using a filtration
technique. Cohen-Steiner et al. \cite{2007-Cohen-Steiner} extend the persistence diagram
for scalar functions on topological spaces and analyze its
stability.
\subsection*{Mesh Simplification.} Mesh-simplification is well-known
in the computational geometry and graphics community. Topological complexity of a mesh can be
determined by its genus. Guskov et al. \cite{2001-Guskov-topologicalnoise} remove
the unnecessary topological noise from
meshes of laser scanner data by reducing their genera.
Nooruddin et al. \cite{2003-Nooruddin-tvcg} give a voxel-based simplification and repair method of polygonal models
using a volumetric morphological operation. Ni et al. \cite{2004-Ni-ATOG}
generate a fair Morse function for extracting the topological
structure of a surface mesh by user-controlled number and
configuration of critical points. Hoppe et al. \cite{1993-Hoppe-mesh-opti} describe
a energy-minimization technique for generating an optimal mesh by
reducing the number of vertices from a given mesh. Also Hoppe et al. \cite{1996-Hoppe-progressive-mesh, 1997-Hoppe-progressive-mesh} give a new progressive mesh
representation, a new scheme for storing and transmitting arbitrary
triangle meshes, and their simplification technique. Chiang et al. \cite{2003-Chiang-Simplification}
describe a technique of progressive simplification of tetrahedral
meshes preserving isosurface topologies. Their method works in two
stages - first they segment the volume data into
topological-equivalence regions and in the second step they simplify
each topological-equivalence region independently by edge collapsing, preserving the
iso-surface topologies. There are many cost-driven methods of mesh-simplification
(in the literature) which attempt to measure only the cost of each individual edge collapse
and the entire simplification process is considered as a sequence of
steps of increasing cost
\cite{1998-Dey-topologypreserving, 1998-LindstromT,
1999-Lindstrom-Memless, 1997-GarlandH}.
\subsection*{Vector Field Simplification.} Topology based
methods for vector field simplification are based on the
idea of \textit{singularity pair cancellation} to reduce the number
of singularities and thus the topological complexity. This method
iteratively eliminates suitable pairs of singularities with opposite
Poincar\'e-Hopf indices so that total sum of the indices remain
invariant to keep the global structure of the field the same.
This idea has been exploited in
\cite{2001-Tricoche-vec-simpli, 2006-Zhang-vec, 2011-Reininghaus-Visweek}.
There are also non-topology based methods for vector-field
simplification which are mainly based on
smoothing operations. Smoothing operations reduce vector and tensor-field
complexity and remove large percentage of
singularities. Polthier et al. \cite{2003-Polthier} apply Laplacian
smoothing on the potential of a
vector-field. Tong et al. \cite{2003-Tong} decompose a vector field into three
components: curl free, divergence free and harmonic. Each component
is smoothed individually and results are summed to obtain simplified
vector field.
\subsection*{Multi-Field Simplification.}
To the best of our knowledge, until now there is no prior work
on topology-based simplification of general multi-field data.
All those techniques, cited so far, for simplifying scalar fields,
meshes and vector fields are not directly applicable in case of
multi-fields, mainly because the computation of the equivalent tools
such as, Jacobi Set \cite{2004-edels-localglobal}, Reeb Space \cite{2008-edels-reebspace} are
not well-developed.
A generalization of the persistence homology is proven to be difficult
for the multi-fields \cite{2007-carlsson-persist-multi}.
However, few attempts have been made for
simplifying the Jacobi Sets in restrictive cases.
Snyder et al. \cite{2004-snyder} give two metrics for
measuring persistence of the Jacobi Sets. Bremer et al. \cite{2007-Bremer} describe a
method for noise removal from the Jacobi Sets of time varying data.
Suthambhara et al. \cite{2009-Nataraj-Jacobi} give a technique for the
Jacobi Set simplification of bivariate fields based on simplification
of the Reeb Graphs of their comparison measures.
Huettenberger et al. propose multi-field
simplification method using Pareto sets \cite{2013-Huettenberger-pareto,
2014-Huettenberger-tvcg}. However, these methods lack
mathematical justification for simplifying the corresponding input multi-fields and
work mostly for bivariate data.
In a similar context, Bhatia et al. \cite{2013-Bhatia} provide a simplification
method by generalising the critical point cancellation of scalar
functions to the Jacobi Sets in two dimensional domains.
However, current research shows that the Jacobi Sets are unable to capture
the actual topological changes of multi-fields,
instead one should consider their Reeb Spaces,
introduced in \cite{2008-edels-reebspace}.
Recently, Multi-Dimensional Reeb Graphs \cite{2014-EuroVis-short} and
Layered Reeb Graphs \cite{2014-Strodthoff} have been introduced from
two different perspectives to extend the Reeb Graph for multi-fields.
In the current paper, we use the recently introduced Jacobi
Structure \cite{2014-EuroVis-short} to separate the Reeb Space into regular and singular components.
Thus we obtain a dual Reeb Skeleton corresponding to the Reeb Space.
Our simplification strategy is based on simplifying this Reeb Skeleton
by associating different measures with the nodes of the
Reeb Skeleton.
\section{Necessary Background}
\label{sec:Background}
\begin{figure*}[th!]
\begin{center}
\subfloat[\label{fig:1a}]{\includegraphics[height=5.5cm]{./Fig/3d-2d-sphere-height.eps}}
\subfloat[\label{fig:1b}]{\includegraphics[height=5.5cm]{./Fig/distance-ht-domain-separate.eps}}\qquad \qquad
\subfloat[\label{fig:1c}]{\includegraphics[height=5.5cm]{./Fig/tangle-height-labeled.eps}}
\subfloat[\label{fig:1d}]{\includegraphics[width=4.9cm]{./Fig/RS-sphere-height.eps}}\qquad \qquad
\subfloat[\label{fig:1e}]{\includegraphics[width=5.1cm]{./Fig/HemisphereHeight2.eps}}\qquad \qquad
\subfloat[\label{fig:1f}]{\includegraphics[width=5.1cm]{./Fig/tangle-reebspace0-labeled.eps}}
\end{center}
\vspace*{-2ex}
\caption{ (a) A stable bivariate field $(f_1, f_2)\equiv(x^2 + y^2 + z^2, \, z)$
from $\mathbb{R}^3$ to $\mathbb{R}^2$ that is visualized using the transparent isosurfaces of the
first component field; black curves are the fiber-components of the
bivariate field; the red line represents the Jacobi Set;
(d) The Reeb Space corresponding to (a) that is comprising one sheet (in pink) and the Jacobi
Structure (red parabolic curve);
(b) The Jacobi Set (consists of the red
lines, top face and bottom face of the box) of the bivariate field $(f_1, f_2)\equiv(x^2 + y^2 + z^2, \, z)$ in the
box $[-1,\,1]\times [-1,\,1] \times[0,\,1]$; singular fibers passing through
the boundary tangent points form a cylindrical surface that
separates the domain into five components, denoted
as A, B, C, D and E;
(e) The Reeb Space of the multi-field
corresponding to (b) that is comprising five sheets (in grey) and
the Jacobi Structure (red lines); the regular
components of the Reeb Space are marked to match the corresponding
components in the domain; components of the Jacobi Set in the domain and
their corresponding projections in the Reeb Space are denoted by
numbers;
(c) An unstable bivariate field $(f_1, f_2)\equiv (x^4 + y^4 + z^4 - 5(x^2 + y^2 +
z^2) + 10, \, z)$ from $\mathbb{R}^3$ to $\mathbb{R}^2$ that is visualized using the transparent isosurfaces of the
first component field; black curves are the fiber-components of the
bivariate field; the Jacobi Set consists of 9 red lines;
(f) The Reeb Space corresponding to (c) that is comprising six sheets
and the Jacobi Structure (6 red lines).}
\label{fig:jacobi-reeb}
\end{figure*}
Over the last two decades, scalar topology has been used to support scientific data analysis and visualization,
in particular through the use of the Reeb Graph and its
specialisation, the contour tree \cite{1997-Kreveld-CT,
2000-Carr-CT, 2001-Hilaga, 2004-edels-timevarying, 2007-Pascucci}.
The subject of multi-field topology in data analysis is
rather new. In this section we briefly describe the multi-field
topological analysis and existing tools for capturing them, viz. the
Jacobi Set and the Reeb Space.
\begin{table}[h!]
\begin{center}
\caption{Important Notations}
\begin{tabular}{ll}
\hline
\textbf{Notation} & \textbf{Name} \\
\hline
$\mathbb{W}_f$ & Reeb Space\\
$\mathbb{K}_f$ & Reeb Skeleton\\
$\mathbb{J}_f$ & Jacobi Set\\
$\Bd{\mathbb{J}}_f$ & Boundary Jacobi Set\\
$\Int{\mathbb{J}}_f$ & Interior Jacobi Set \\
$\mathfrak{J}_f$ & Jacobi Structure\\
$\JCN(f, m_Q)$ & Joint Contour Net with quantization
level $m_Q$\\
$\mathbb{R}_{f_i}$ & Reeb Graph\\
$\mathbb{M}_f$& Multi-dimensional Reeb Graph\\
\hline
\end{tabular}
\label{tab:notations}
\end{center}
\end{table}
\subsection*{Multi-Field Analysis.}
A multi-field on a $d$-manifold $\mathbb{X}\, (\subseteq\mathbb{R}^d)$ with $r$ component scalar fields
$f_i:\mathbb{X}\rightarrow \mathbb{R}$ ($i=1,\,\ldots, r$) is a \textit{map} $f=(f_1,\,f_2,\,\ldots,\,f_r): \mathbb{X}\rightarrow
\mathbb{R}^r$. Table~\ref{tab:notations} shows the notations used to denote
various structures corresponding to a multi-field $f$ in the current paper.
In differential topology, $f$ is considered to be a \textit{smooth map} when all its partial derivatives of any order
are continuous. A point $\mathbf{x}\in \mathbb{X}$ is called a \textit{singular point} (or
\textit{critical point}) of $f$ if the
rank of its differential map $df_{\mathbf{x}}$ is strictly less than $\min\{d,
r\}$ where $df_{\mathbf{x}}$ is the $r \times d$ matrix whose rows are the
gradients of $f_1$ to $f_r$ at $\mathbf{x}$. And the corresponding
value $f(\mathbf{x})=\mathbf{c}=(c_1,\, c_2,\, \ldots,\, c_r)$ in $\mathbb{R}^r$ is a \textit{singular
value}. Otherwise if the rank of the differential map $df_{\mathbf{x}}$ is $\min\{d,
r\}$ then $\mathbf{x}$ is called a \textit{regular point} and a point $\mathbf{y}\in \mathbb{R}^r$
is a \textit{regular value} if $f^{-1}(\mathbf{y})$ does not contain
a singular point.
The inverse image of the map $f$ corresponding to a
value $\mathbf{c}\in \mathbb{R}^r$, $f^{-1}(\mathbf{c})$ is called a \textit{fiber} and each
connected component of the fiber is called a \emph{fiber-component}
\cite{Saeki2014, 2004-Saeki}. In particular, for a scalar field these are known as
the \emph{level set} and the \emph{contour}, respectively.
The inverse image of a singular value is called a \textit{singular fiber} and the inverse image
of a regular value is called a \textit{regular fiber}. If a
fiber-component passes through a singular point, it is called a \emph{singular
fiber-component}. Otherwise, it is known as a \emph{regular
fiber-component}. Note that a singular fiber may contain a regular fiber-component.
A continuous map is said to be \emph{proper} if the pre-image
of a compact set is always compact and it is said to be \emph{stable} if its topological properties remain unchanged
by small perturbations \cite{Levine1985}.
Let $f : \mathbb{X} \subset \mathbb{R}^3 \to \mathbb{R}^2$ be a proper smooth map. Then, it is stable if and only if it satisfies the following local and global conditions.
Around each singular
point $\mathbf{s}$, $f$ is locally described as either (i) $(u,\,
x^2+y^2)$: $\mathbf{s}$ is a definite fold point, or (ii) $(u,\, x^2-y^2)$: $\mathbf{s}$ is an indefinite fold point, or (iii) $(u,\,
y^2+ux-x^3/3)$: $\mathbf{s}$ is a cusp point, for some local coordinates $(u,\, x,\, y)$ around $\mathbf{s}$
and an appropriate set of local coordinates around $f(\mathbf{s})$ in the range $\mathbb{R}^2$.
Moreover, no cusp point is a double point of $f$ restricted to the set
of singular points and $f$ restricted to the set of all definite and
indefinite fold points is an immersion with normal crossings.
Thus for a proper stable map, a singular fiber-component passing through a definite fold
point or a cusp point contains exactly one such point, while
a singular fiber-component passing through an indefinite fold point may pass through one or two indefinite fold
points. Otherwise, the map is called an \emph{unstable map}.
We note, characterizing the stability of the maps on compact 3-manifold domains with boundary
needs additional types of singularities which is discussed in \cite{Saeki2015}.
\figref{1a} is an example of a map from $\mathbb{R}^3$ to $\mathbb{R}^2$ where $f=(x^2+y^2+z^2,\,z)$.
All its singular points are definite fold points, so this is an example of a stable map. \figref{1c}
is an example of a map from $\mathbb{R}^3$ to $\mathbb{R}^2$ where $f=(x^4 + y^4 + z^4 - 5(x^2 + y^2 +
z^2) + 10, \, z)$. It has singular fiber-components which pass
through four indefinite fold points (on corresponding four 1-manifold components
numbered as 5, 6, 7 and 8 in \figref{1c}) and so is an example of an unstable map.
From the pre-image theorem \cite{1974-gp-dt}, generically a regular fiber
$f^{-1}(\mathbf{c})$ is a $(d-r)$-manifold for the regular value $\mathbf{c}=(c_1,\, c_2,\, \ldots,\, c_r)$.
We note for $d<r$, $f^{-1}(\mathbf{c})$ is an empty set or a discrete set of points.
A fiber $f^{-1}(\mathbf{c})$ can be considered as the intersection of
the fibers of the component scalar fields
$f_1^{-1}(c_1),\, f_2^{-1}(c_2),\ldots ,\, f_r^{-1}(c_r)$ and
a connected component of this intersection is a
fiber-component. Alternatively, fiber-components of
$(f_1,\,f_2,\,\ldots,\,f_r)$ can be considered as the contours of a
component field $f_i$, restricted to the fiber-components of the
remaining component fields. This is a \textit{key observation}, we use
in building our Multi-Dimensional Reeb Graph data-structure.
\subsection*{Jacobi Set.} The compact $d$-manifold domain $\mathbb{X} \, (\subseteq\mathbb{R}^d)$ of the map $f$ can be expressed as
$\mathbb{X}=\Int{\mathbb{X}}\cup \partial\mathbb{X}$ where
$\Int{\mathbb{X}}$ denotes the interior (the set of interior points) of
the domain and $\partial\mathbb{X}$ denotes the boundary (the set of boundary
points) of $\mathbb{X}$. In case the domain $\mathbb{X}$ is without boundary, $\partial
\mathbb{X}=\emptyset$ and $\mathbb{X}=\Int{\mathbb{X}}$.
Now the \emph{interior Jacobi Set} of the map
$f:\mathbb{X}\rightarrow\mathbb{R}^r$
is denoted by $\Int{\mathbb{J}}_f$ and is defined by the set
$\Int{\mathbb{J}}_f := \left\{ \mathbf{x} \in \Int{\mathbb{X}} \mid \text{ rank }
d\Int{f}_x<\min\{d, r\} \right\}$ \cite{2004-edels-jacobi} where
$\Int{f}$ is the restriction of $f$ to $\Int{\mathbb{X}}$, i.e., $\Int{f}:= f|_{\Int{\mathbb{X}}}:{\Int{\mathbb{X}}}\rightarrow \mathbb{R}^r$.
In other words, $\Int{\mathbb{J}}_f$ is the set of singular points of the map
$f$ interior to the domain $\mathbb{X}$. Similarly, the \emph{boundary Jacobi
Set} of the map $f$ is denoted by $\Bd{\mathbb{J}}_f$ and is defined as the set of singular points of the
restriction of $f$ to the boundary $\partial\mathbb{X}$, i.e. $f_\partial:=f|_{\partial \mathbb{X}}: \partial \mathbb{X}\rightarrow \mathbb{R}^r$.
Finally, by the \emph{Jacobi Set} of the map $f: \mathbb{X}\rightarrow\mathbb{R}^r$ we mean
the union of the interior and the boundary Jacobi Set of the map $f$,
and is denoted by $\mathbb{J}_f$, i.e. $\mathbb{J}_f=\Int{\mathbb{J}}_f\cup \Bd{\mathbb{J}}_f$.
Now the boundary of a domain may come with \emph{corners}, e.g. a
3-dimensional cube has corners
of two types: 12 edge corners, and 8 vertex corners (as in \figref{1b}). Let $\mathbb{X}$ be a compact $3$-dimensional manifold
with corners and $f : \mathbb{X} \to \mathbb{R}^2$ be a smooth
map. A point $q \in \partial \mathbb{X}$ is a (boundary)
regular point if $f$ restricted to $\partial \mathbb{X}$
is a local homeomorphism around $q$, where
$\partial \mathbb{X}$ stands for the boundary of $\mathbb{X}$ which
includes all the boundary points and corner points.
Otherwise, $q$ is a (boundary) singular point.
For example, if we take a point in a vertical edge in \figref{1b},
then in its (2-dimensional) neighborhood on
the boundary, there are always a pair of points
that are mapped to the same point.
Thus, it is never injective, and hence is never a
local homeomorphism. Therefore, the point
is a (boundary) singular point.
Alternatively, the Jacobi Set is the set of critical
points of one component field (say $f_i$) of $f$ restricted to the intersection of the level
sets of the remaining component fields. Edelsbrunner et al. \cite{2004-edels-jacobi} studied properties of the Jacobi Set for $r$
Morse functions. They proved the Jacobi Set is symmetric with respect to its component
fields. They also showed, generically, the Jacobi Set of two Morse functions is a smoothly embedded
$1$-manifold where the gradients of the functions become
parallel. However, in general Jacobi Sets
are not sub-manifolds of the domain of the multi-field $f$,
and are the disjoint union of sub-manifolds
of the domain \cite{2004-edels-jacobi}.
The red lines in \twofigref{1a}{1c}
illustrate the Jacobi Sets of multi-fields on domains without boundary.
\subsection*{Reeb Space.} As with the Reeb Graph of a scalar field,
the Reeb Space parametrizes the fiber-components of a
multi-field and its topology is described by the
standard quotient space topology.
We note the fiber-components of a continuous map
$f:\mathbb{X}\rightarrow \mathbb{R}^r$ ($\mathbb{X}\subseteq
\mathbb{R}^d$) partition the domain $\mathbb{X}$ into a set
of equivalence classes, denoted by $\mathbb{W}_f:=\mathbb{X}/\sim$,
where two points $a,\,b\in \mathbb{X}$ are equivalent or $a\sim b$ if
$f(a)=f(b)$ and $a,\, b$ belong to the same fiber-component of $f^{-1}(f(a))$ and $f^{-1}(f(b))$.
Now the canonical projection map
$q_f:\mathbb{X}\rightarrow \mathbb{X}/\sim$ that maps each element of $\mathbb{X}$
to its equivalence class defines the standard quotient topology where
open sets are defined to be those sets of equivalence classes with an
open pre-image, under map $q_f$. The Reeb Space of $f$ is the quotient
space $\mathbb{W}_f$ together with this quotient topology.
The decomposition of $f$ as the composition of $q_f$ and $\bar{f}$,
where $\bar{f}:\mathbb{W}_f\rightarrow\mathbb{R}^r$ is such that $f=\bar{f}\circ q_f$. This is called the Stein factorisation of $f$.
The following commutative diagram describes this relationship between the maps.
\begin{center}
\begin{tikzcd}[column sep=normal]
\mathbb{X} \arrow{dr}[swap]{q_f}\arrow{rr}{f} & & \mathbb{R}^r\\
& \mathbb{W}_f \arrow{ur}[swap]{\bar{f}} &
\end{tikzcd}
\end{center}
\noindent
Now to construct a fiber $f^{-1}(a)$, instead of going directly from
$\mathbb{R}^r$ to $\mathbb{X}$ one can compute the pre-image under $\bar{f}$ of
$a$. Each fiber consists of a number of components, one for each point in $\bar{f}^{-1}(a)$.
Generically, the Reeb Space is a Hausdorff space, i.e., any two distinct points of
$\mathbb{W}_f$ have disjoint neighbourhoods.
Moreover, when $r\leq d$ the Reeb Space corresponding to the multi-field
$f$ consists of a collection of $r$-manifolds glued together in
complicated ways \cite{2008-edels-reebspace}.
\threefigref{1d}{1e}{1f} show three examples of the Reeb Spaces
corresponding to a stable bivariate
field in $\mathbb{R}^3$, an unstable bivariate field on a closed 3-dimensional interval
and an unstable bivariate-field in $\mathbb{R}^3$,
respectively. We indicate the dark (red) lines in the Reeb Spaces as the Jacobi Structures which are introduced in the next section.
Note that the structures of
the Reeb Spaces as in \threefigref{1d}{1e}{1f} are obtained by analyzing the evolution of the
fiber-components of the corresponding bivariate fields. For example, if we consider evolution
of the fiber-components of the map in \figref{1b},
they start at the definite fold points on the line numbered as 1.
Then these fiber-components start growing and meet at the boundary
Jacobi Set points (on the lines numbered as 2, 3, 4, 5).
Then each of them splits into four fiber-components which continue to shrink
and die at the corner singular points (on the lines numbered as 6, 7,
8, 9). This evolution phenomenon is captured in its Reeb Space (e).
\section{Theoretical Results}
\label{sec:Theory}
In this section we exploit the underlying structure of the Reeb Space
to decompose it into a set of simple manifold-like components (namely
regular and singular components) and capture
their connectivities by a dual skeleton graph (namely Reeb Skeleton).
Then, since in real applications most of the data come with simple
domains (such as, a cube or a box), we study properties of the Reeb Space and its representative skeleton graph
for such topologically simple data-domains. More precisely, to prove our
theoretical results in this section we consider a stable bivariate
field $f=(f_1,f_2):\mathbb{X}\rightarrow\mathbb{R}^2$ where $\mathbb{X}\; (\subseteq\mathbb{R}^3)$ is a
three-dimensional bounded, closed interval.
However, most of the results
are straight-forward to generalize for multi-fields of
higher dimensions. Thus the domain $\mathbb{X}$, we consider, is a compact domain with
boundary and is simply-connected and in this case
the Reeb Space is path-connected.
\subsection{Path Connectedness}
For a continuous map $f: \mathbb{X}\subseteq \mathbb{R}^3\rightarrow
\mathbb{R}^2$ the Reeb Space is a quotient space of the
fiber-components and is path-connected (or $0$-connected). That is, any two points $p_0$ and $p_1$ of the
Reeb Space can be connected by a path $\gamma: [0,1]\rightarrow \mathbb{W}_f$
so that $\gamma(0)=p_0$ and $\gamma(1)=p_1$. In other wards, we say $0$-connectivity is
preserved by the quotient map $q_f: \mathbb{X} \rightarrow \mathbb{W}_f$. This can also be stated by
saying that $0$-th homotopy group of the Reeb Space $\pi_0(\mathbb{W}_f)$ remains trivial.
Next we prove the following important property of the Reeb Space.
\begin{lemma}
\label{lem:path}
Let $f:\mathbb{X}\subseteq \mathbb{R}^3\rightarrow \mathbb{R}^2$ be a continuous, generic map on a $3$-dimensional interval $\mathbb{X}$ and
$\mathbb{W}_f$ be the corresponding Reeb Space. Let $P$ be a
continuous path between any two points on the Reeb Space. Then
if $\mathbb{W}_f \setminus P$ is path-connected, then so
is $\mathbb{X}\setminus q_f^{-1}(P)$.
\end{lemma}
\noindent
\begin{proof}
Consider any two points $p_0,\,p_1\in \mathbb{W}_f \setminus
P$. Since $\mathbb{W}_f \setminus P$ is path-connected, $\exists$ a path
$\gamma: [0, 1]\rightarrow \mathbb{W}_f \setminus P$ with $\gamma(0)=p_0$ and
$\gamma(1)=p_1$. Using conditions like
the genericity of $f$, $\gamma(t)$ lifts to $\mathbb{X} \setminus q_f^{-1}(P)$,
i.e., there exists a path $\tilde{\gamma}$ in
$\mathbb{X}$ such that $q_f \circ \tilde{\gamma}(t) = \gamma(t)$ (using the
\emph{path-lifting} property \cite{2002-Hatcher}).
Now $\tilde{\gamma}$ is a path between any point of $q_f^{-1}(p_0)$ to any point
of $q_f^{-1}(p_1)$ in $\mathbb{X}\setminus q_f^{-1}(P)$. Therefore, $\mathbb{X}\setminus q_f^{-1}(P)$ must be path-connected.
\end{proof}
\noindent
Thus Lemma~\ref{lem:path} implies if there
exists a path $P$ in the Reeb Space whose preimage $q_f^{-1}(P)$ separates
the domain then $P$ must also separate the Reeb Space. This is a
useful property in detaching unimportant components from the Reeb Space.
\subsection{Jacobi Structure}
\label{subsec:Jacobi}
As noted in Section \ref{sec:Background}, the Jacobi Set of a function is not the same as the set of singular fibers, as each
point in the Jacobi Set is merely a representative of a singular fiber. Moreover, the structure of
the Reeb Space is actually given by a projection of the Jacobi Set or
the singular fibers. For example, in
\twofigref{1c}{1f}, the Jacobi Set consists of 9 parallel lines in the domain, but
they correspond to 6 1-manifold structures in the Reeb Space.
Note that if the input multi-field domain is with boundary, there are
additional edges (corresponding to the boundary
Jacobi Set) needed to describe the Reeb Space.
We therefore introduce the \emph{Jacobi Structure}: the
manifold structure of the Reeb Space corresponding to the Jacobi Set
in the domain.
\begin{dfn}
The \textbf{Jacobi Structure} of a Reeb Space $\mathbb{W}_f$ corresponding to a
multi-field $f:\mathbb{X}\subseteq \mathbb{R}^3\rightarrow \mathbb{R}^2$ is denoted
by $\mathfrak{J}_f$ and is defined by $\mathfrak{J}_f:=q_f(\mathbb{J}_f)$, i.e., the projection of the Jacobi Set
$\mathbb{J}_f$ to the Reeb Space by the quotient map $q_f:\mathbb{X}\rightarrow\mathbb{W}_f$.
\end{dfn}
\noindent
Note that according to our definition the Jacobi Set $\mathbb{J}_f$ consists of both the interior and the
boundary Jacobi Set, i.e., $\mathbb{J}_f=\Int{\mathbb{J}}_f\cup \Bd{\mathbb{J}}_f$.
Thus each point of the Jacobi Structure corresponds to a singular fiber-component
in the domain $\mathbb{X}$ of $f$, and vice-versa.
To understand the underlying structure of the Reeb Space
one needs to understand both the topology of the singular fibers
and the corresponding local configurations of the Jacobi Structure. Classification of singular
fibers and their local configurations in the quotient space have been studied
for stable maps from $\mathbb{R}^3$ to $\mathbb{R}^2$ and $\mathbb{R}^4$ to $\mathbb{R}^3$~\cite{2004-Saeki}.
Figure \ref{fig:jacobi} illustrates examples of a regular and few
singular fiber-components, and their local structures in the Reeb Space
\cite{Kushner-1984, Levine1985} for a stable map $f:\mathbb{X}\subseteq
\mathbb{R}^3\rightarrow \mathbb{R}^2$.
For a more complete classification of singular fibers for maps on
3-manifolds (with boundary) to plane and for local configurations of the Reeb spaces we refer to \cite{Saeki2015, SaekiCobordism2015}.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=.47\textwidth]{./Fig/rs-singularities.eps}
\end{center}
\caption{Regular and singular fiber-components and corresponding local configurations in
the Reeb Space. The Jacobi Structures are in red lines in the Reeb
Space \cite{Saeki2015}.}
\label{fig:jacobi}
\end{figure}
For a generic map $f:\mathbb{X}\rightarrow\mathbb{R}^2$, $\mathbb{W}_f$ is a two-dimensional
polyhedron and Jacobi structure embedded in the Reeb Space consists of
1-dimensional components which are at the boundary of the
two-dimensional sheets in $\mathbb{W}_f$.
Now a 1-manifold component of the Jacobi Structure can be classified into three types
based on the transition of number of regular fiber-components if one
passes across the component \cite{Saeki2014}:
\begin{enumerate}
\item \emph{Birth-Death} or \emph{Boundary component} - where a fiber-component takes
birth or dies (Figure~\ref{fig:jacobi} (b)),
\item \emph{Merge-Split component} or \emph{Bifurcation locus} - where
two (or more) fiber-components merge together or one component splits into two (or more) (Figure~\ref{fig:jacobi}(c))
and
\item \emph{Neutral component} - where there is no
change in the number of fiber-components if one passes through such
components (Figure~\ref{fig:jacobi}(g)), but here, the topology of the regular fiber-component changes from a circle to an arc (or vice versa).
\end{enumerate}
A connected component of the Jacobi structure may also consist of a composition of these
three types, e.g. in Figure~\ref{fig:jacobi}(d) the Jacobi structure
component consists of a boundary and a merge-split
component connected at a discrete \emph{cusp point}. In
Figure~\ref{fig:jacobi}(e) four merge-split components are connected
at a \emph{double point} on the Jacobi Structure.
\twofigref{1b}{1e} respectively show an example of 8 1-manifold components
of the boundary Jacobi Set (red lines in the boundary of the domain) and their
corresponding projection in the Reeb Space as 5 1-manifold parts of
the Jacobi Structure. From this example, it is clear
that a boundary Jacobi Set component may not be the boundary
component of the Jacobi Structure in the Reeb Space or vice-versa.
In Section~\ref{sec:SimplifyingJCN} we propose an algorithm for computing the Jacobi Structure
by constructing a Multi-Dimensional Reeb Graph corresponding to a multi-field.
\subsection{Regular and Singular Components}
As the number of dimensions increases, the projections of the singular fibers
develop more internal structure in the Reeb Space.
Consider the Reeb Graph of a scalar function: in this, the projection
images of the critical points are single points (0-manifolds)
separating edges (1-manifolds). Similarly, for the bivariate fields
shown in \threefigref{1d}{1e}{1f}, the projections of the
singular fibers are arranged in a Reeb Space along 1-manifold curves which separate 2-manifold sheets.
This induces a natural stratification or partition of the Reeb Space
into disjoint subspaces (or strata).
To describe a stratification of the Reeb Space and the corresponding
domain of the multi-field we first classify the fiber-components of the generic map $f: \mathbb{X}\subseteq
\mathbb{R}^3\rightarrow\mathbb{R}^2$ according to their complexity or codimension of the subspace where they lie \cite{Saeki2014}.
Given the Stein factorization $f=\bar{f}\circ q_f$, fiber-components of $f$ can be classified into three classes.
\begin{enumerate}
\item $\mathcal{C}^0=\{q_f^{-1}(s): s\in \mathbb{W}_f \text{ and } q_f^{-1}(s) \text{ does
not contain any }$ $\text{singular point of } f\}$. Fiber-components
of this class are the regular fiber-components and their $q_f$-images
form codimension 0 subspaces in $\mathbb{W}_f$, denoted as $\mathbb{W}_f^0$.
\item $\mathcal{C}^1=\{q_f^{-1}(s): s\in \mathbb{W}_f \text{ and } q_f^{-1}(s)
\text{ contains exactly one de-}$ $\text{finite or indefinite fold point}\}.$
Singular fiber-components of this class are moderately complex and
their $q_f$-images form codimension 1 subspaces in $\mathbb{W}_f$, denoted as $\mathbb{W}_f^1$.
\item $\mathcal{C}^2=\{q_f^{-1}(s): s\in \mathbb{W}_f \text{ and } q_f^{-1}(s)
\text{ contains a cusp point}$ $\text{or two indefinite fold points}\}.$
Singular fiber-components of this class
are the most complex and their $q_f$-images form codimension 2
subspaces in $\mathbb{W}_f$, denoted as $\mathbb{W}_f^2$.
\end{enumerate}
\noindent
Complexity of a fiber-component increases as the codimension of the
corresponding subspace in the Reeb Space increases.
Note that $q_f$-images of the fiber-components in $\mathcal{C}^1$ and $\mathcal{C}^2$ form the Jacobi
Structure $\mathfrak{J}_f$ of the Reeb Space, i.e., $\mathfrak{J}_f=\mathbb{W}_f^1\cup \mathbb{W}_f^2$. Topologically, regular fiber-components are
either a circle or an arc \cite{Saeki2015}. For stable maps $f: \mathbb{X}\subseteq
\mathbb{R}^3\rightarrow\mathbb{R}^2$, topologically there are $7$ different types of singular
fibers in $\mathcal{C}^1$ and $21$ different types of singular
fibers in $\mathcal{C}^2$ \cite{Saeki2015}.
Two regular points $a,\, b \in \mathbb{W}_f^0$ are \emph{topologically
equivalent} in the Reeb Space $\mathbb{W}_f$ or $a\sim_\rho b$ if there exists a path
between $a$ and $b$ without intersecting the Jacobi Structure $\mathfrak{J}_f$.
It is not difficult to check that `$\sim_\rho$' is an equivalence relation.
Therefore, the equivalence relation `$\sim_\rho$' partitions the regular points
of $\mathbb{W}_f$ into a set of equivalence classes.
Now we prove that each such equivalence class is a 2-dimensional sheet.
\begin{lemma}[\textbf{Partition}]
\label{lem:partition}
The Jacobi structure $\mathfrak{J}_f$ of a Reeb space $\mathbb{W}_f$ corresponding to a smooth
stable map $f:\mathbb{X}\subseteq \mathbb{R}^3\rightarrow \mathbb{R}^2$ separates the Reeb Space into a set
of $2$-manifold components.
\end{lemma}
\noindent
\begin{proof}
Let $D$ be a small disk in the range consisting of
regular values (i.e., $D$ does not intersect $f(\mathbb{J}_f)$).
Then, by Ehresmann's fibration theorem,
$f$ restricted to $ f^{-1}(D)$ is equivalent to
the projection $D \times F \rightarrow D$, where
$F$ is a 1-dimensional compact manifold.
So, this means that $q_f(f^{-1}(D))$ can
be identified with a disjoint union
of some copies of $D$, where the number of copies
is the same as the number of connected components
of $F$. Even when $D$ intersects with $f(\mathbb{J}_f)$,
if we restrict $f$ to the components
of the inverse image $f^{-1}(D)$ that do not
intersect $\mathbb{J}_f$, then the same consequence holds.
So, the regular sheets of $\mathbb{W}_f$ are locally
homeomorphic to $D$, and hence is a 2-manifold.
\end{proof}
\noindent
Thus we have the following definition of regular components.
\begin{dfn}
A path-connected component of $\mathbb{W}_f\setminus\mathfrak{J}_f$ or $\mathbb{W}_f^0$ is called a regular component.
\end{dfn}
\noindent
Generically, the $0$-dimensional strata are in the boundary of
the $1$-dimensional strata in the $\mathfrak{J}_f$.
Therefore, an equivalence relation on the set of points in $\mathbb{W}_f^1$ can
be defined, similarly, where two points of $\mathbb{W}_f^1$ are equivalent if there exists a continuous path between
them without crossing the $0$-dimensional strata in $\mathfrak{J}_f$ and each such equivalence
class will be considered as a 1-singular component.
\begin{dfn}
A path-connected component of $\mathfrak{J}_f\setminus \mathbb{W}_f^2$ or $\mathbb{W}_f^1$ is called a 1-singular component.
\end{dfn}
\noindent
Note that a 1-singular component in $\mathbb{W}_f$ may be an arc or a circle. An arc
1-singular component will also be called as an \emph{edge}.
\begin{dfn}
Each component of $\mathbb{W}_f^2$ is called a 0-singular component.
\end{dfn}
\noindent
To extract a skeleton graph from the Reeb Space we need adjacency of
these regular and 1-singular components which are defined as follows.
\begin{dfn}
\begin{enumerate}
\item A circle 1-singular component is self-adjacent (adjacent to itself).
\item If two end points of an arc 1-singular component coincide, then the
1-singular component is self-adjacent.
\item Two distinct 1-singular components $S_1,\,S_2$ are
adjacent if $\exists$ a 0-singular component $\alpha_0$
such that $S_1~\cup~ S_2~\cup~\alpha_0$ form a connected space.
\end{enumerate}
\end{dfn}
\begin{dfn}
A 1-singular component $S_i$ is adjacent to a
regular component $R_j$ if $S_i\cup R_j$ forms a connected space.
\end{dfn}
Next we define a connectivity graph of regular and singular components
based on their adjacency.
\begin{figure}[t!]
\begin{center}
\fbox{\includegraphics[width=.47\textwidth]{./Fig/RS.eps}}
\end{center}
\vspace*{-2ex}
\caption{Reeb Skeletons: (a) corresponding to the Reeb Space in \figref{1d}, (b)
corresponding to the Reeb Space in \figref{1e}, (c) corresponding to the
Reeb Space in \figref{1f}.}
\label{fig:reeb-skeleton}
\end{figure}
\subsection{Reeb Skeleton}
\label{sec:ReebSkeleton}
Once the Reeb space $\mathbb{W}_f$ is split into $2$-manifold regular components
and $1$-or-lower manifold singular components, it is possible to
perform a further reduction from the Reeb Space. To do so, we
represent both these regular and 1-singular components as points (or nodes), and add edges representing their
adjacency: in short, we can build the dual graph of these components of the Reeb Space. This has
the merit of further reducing the Reeb Space from a $2$-dimensional structure to a fundamentally
$1$-dimensional structure which is easier to represent, to reason about and to visualise. We refer to this
as the \emph{Reeb Skeleton} and formally define as follows.
\begin{dfn}
\label{dfn:reeb-skeleton}
Let $R_1, R_2, \ldots, R_m$ be the regular components and $S_1, S_2,
\ldots, S_n$ be the 1-singular components of $\mathbb{W}_f$. Then
the Reeb Skeleton of $f$, denoted by $\mathbb{K}_f$, is the adjacency graph which consists of
(i) nodes $n_{R_i}$ and $n_{S_j}$ ($i=1, 2, \ldots, m$ and $j=1, 2,
\ldots, n$) corresponding to each of the regular and 1-singular
components, and (ii) edges $e(S_j, S_{j'})$ and $e(R_i, S_j)$ that are
defined as follows:
\begin{enumerate}
\item If $S_j$ is self-adjacent, then $e(S_j, S_j)=1$. In other words, $n_{S_j}$ has
a self-loop.
\item If $S_j$ is self-adjacent and $S_j$ is adjacent with a regular component $R_i$, then
$e(R_i, S_j)=2$. In other words, $n_{S_j}$ is connected with $n_{R_i}$ by two edges.
\item If $S_j$ and $S_{j'}$ are two distinct \textbf{non-boundary}
1-singular components, then
\begin{align*}
e(S_j, S_{j'})=\left\{
\begin{array}{ccc}
1, & \text{ if } S_j,\, S_{j'} \text{ are adjacent}\\
0, & \text{ otherwise.}
\end{array}\right.
\end{align*}
\item For any regular component $R_i$ and any 1-singular component $S_j$
\begin{align*}
e(R_i, S_j)=\left\{
\begin{array}{ccc}
1, & \text{ if } R_i,\, S_j \text{ are adjacent}\\
0, & \text{ otherwise.}
\end{array}\right.
\end{align*}
\end{enumerate}
\end{dfn}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=.47\textwidth]{./Fig/reeb-skeleton-loop.eps}
\end{center}
\vspace*{-2ex}
\caption{(a) Reeb Space with self-adjacent 1-singular component (b)
Corresponding Reeb Skeleton.}
\label{fig:reeb-skeleton2}
\end{figure}
\noindent
The regular and 1-singular components of
the Reeb Space are represented as the \emph{regular} and \emph{singular nodes},
respectively, in the Reeb Skeleton. \figref{reeb-skeleton} shows some
examples of Reeb Skeletons corresponding to the Reeb Spaces in \figref{jacobi-reeb}.
\figref{reeb-skeleton2} illustrates an example of the Reeb Skeleton
with a self-adjacent singular node.
Note that although the Reeb Skeleton gives a simple abstraction of 0-connectivity in
the Reeb Space, it loses information of higher-dimensional
connectivities, like higher dimensional holes (tunnels, voids) in the
Reeb Space. But on the other hand, the Reeb Skeleton is extremely useful for
extracting any ``fork''-like structure (corresponding to a merge-split
feature) in the Reeb Space. And we will see later
by a little simplification we can extract the most prominent merge-split
feature in the Reeb Skeleton and so in the Reeb Space.
Therefore, next we study properties of the Reeb Skeleton to simplify it
further.
\subsection{Simple Domains}
\label{sec:SimpleDomains}
We know from scalar fields that topologically simple domains have a useful property: the Reeb Graph is guaranteed
to be a tree - i.e. the contour tree. This not only enables more efficient computation, but also provides
straightforward mechanisms for feature extraction, simplification and
visualisation. Ideally, in multi-fields, the Reeb Space
would also be contractible to a point. But we show this is not true, in general.
In topology, simple domains are characterised by
\emph{simply-connected space}. A topological space is simply-connected
if it is path-connected and every \textit{loop} in that space can be
continuously shrunk to a point without leaving the space.
In terms of homotopy theory this means a simply-connected space is
without any ``handle-shaped hole'' (as in \figref{reeb-tunnel}) or it has trivial fundamental group.
For example, a sphere (that has a hollow center) is a simply-connected space
whereas a torus (that has a handle-shaped hole) is not. Even a simpler
topological space is known as \emph{contractible space} which
is homotopically equivalent to a point. Note that a contractible space is
simply-connected, but the converse is not true. For example, a sphere
is simply-connected as every loop on it can be contracted to a point on
it, although the sphere is not a contractible space because of the center
hole in it. In the following lemma, we prove that the Reeb
Space corresponding to a map defined on a simply-connected domain is
simply-connected, but later we show it may not be contractible.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=8cm]{./Fig/reeb-space-tunnel.eps}
\caption{Example of Reeb Space with a tunnel.}
\label{fig:reeb-tunnel}
\end{center}
\end{figure}
\begin{lemma}[\textbf{Simply-Connected}]
\label{lem:contractible}
The Reeb Space of a generic continuous map $f: \mathbb{X} \subseteq\mathbb{R}^3\rightarrow \mathbb{R}^2$ is simply-connected.
\end{lemma}
\noindent
\begin{proof} We consider any loop
in the Reeb space $\mathbb{W}_f$. Then, it lifts to an arc
in $\mathbb{X}$. But, every fiber of $q_f$ is connected,
and therefore, it lifts to a loop.
As $\mathbb{X}$ is simply-connected, this lifted loop
is null-homotopic. Therefore, its $q_f$-image
is also null-homotopic from the continuity of $q_f$. This means that $\mathbb{W}_f$
is simply-connected.
\end{proof}
\noindent
Therefore, if $f$ is good enough (for example,
triangulable or piecewise linear), then the Reeb space
is simply-connected. This implies that the 1st homology of the Reeb
Space also vanishes (or is the trivial group),
and therefore the Reeb space does not have a tunnel or $1$-dimensional
hole (i.e., a hole inside a circle $S^1$, e.g. \figref{reeb-tunnel}). Thus we
have the following theorem.
\begin{theorem}
\label{thm:tunnel}
The Reeb Space of a generic map $f: \mathbb{X}
\subseteq\mathbb{R}^3\rightarrow \mathbb{R}^2$ does not contain any
tunnel or $1$-dimensional hole.
\end{theorem}
On the other hand, for void or $2$-dimensional hole (i.e., hole inside a sphere $S^2$),
this is no longer true. We can construct a (piecewise
linear) map $f : \mathbb{X} \rightarrow \mathbb{R}^2$ whose Reeb space does have
a $2$-dimensional hole. For example, consider the Hopf
fibration $\mathbb{S}^3~\rightarrow~\mathbb{S}^2$ and its composition with
a standard projection $\mathbb{S}^2~\rightarrow~\mathbb{R}^2$. The resulting map
$\mathbb{S}^3~\rightarrow~\mathbb{R}^2$ is not generic, but perturbing
it slightly along its Jacobi set, we can obtain
a generic map $\mathbb{S}^3~\rightarrow~\mathbb{R}^2$, whose Reeb space is
the union of a $2$-sphere and an annulus attached
along the equator (and one boundary component of
the annulus). Then, by extracting a 3-ball
in the preimage of a two disk in the interior
of the annulus part, we get the desired map $\mathbb{X}~\rightarrow~\mathbb{R}^2$.
The Reeb space is the same space; the union of
$\mathbb{S}^2$ and an annulus (Figure~\ref{fig:hopf}).
Over each blue point lies a point (definite fold)
and it corresponds to a birth-death.
Over each red point lies a fiber as in Figure~\ref{fig:jacobi}(c) (with an
indefinite fold) and the splitting of a circle fiber
occurs. Over each green point lies a circle touching the
boundary of the domain cube $\mathbb{X}$. Thus,
over each point in the shaded disk bounded
by the green circle lies an interval.
Note this disk is a subset of the
annulus part. Therefore, a Reeb Space of a multi-field
on a contractible domain may not be contractible and
simplification of such space may not be simple as in
the scalar case.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=6cm]{./Fig/hopf1.eps}
\caption{Reeb Space with a void.}
\label{fig:hopf}
\end{center}
\end{figure}
According to Theorem~\ref{thm:tunnel},
we can conclude that each regular component of $\mathbb{W}_f$ is
planar; i.e., each regular component is a disk possibly with
holes. For example, torus with holes (or a 1-dimensional hole as in Figure~\ref{fig:reeb-tunnel}) never appears!
This is essential in applying our simplification
rules for the Reeb Skeleton as will be discussed in
Section~\ref{sec:rules} (Figure~\ref{fig:rules-simplification}).
Next we focus on finding a criterion for detachability of such regular
components from the Reeb Space for simplifying the
corresponding the multi-field.
\subsection{Detachability}
\label{subsec:detach}
In the case of scalar field in a simply-connected domain,
the Reeb Space (Graph) is a contour tree and there always exists a leaf edge that can be
detached in a mathematically correct way,
unless the contour tree consists only of one edge.
We find similar criteria for defining
\emph{detachable} regular components in the Reeb Space.
We say that it is possible to detach a regular component from a Reeb Space
to obtain a simplified Reeb Space
if the multi-field corresponding to
the initial Reeb Space could be simplified
to the multi-field corresponding to the modified one,
and then the regular component is said to be detachable from the Reeb Space.
Mathematically, any map could be simplified to a simpler map
in the following sense. Since $\mathbb{R}^2$ is contractible,
any two stable maps $f_0$ and $f_1 : \mathbb{X} \rightarrow \mathbb{R}^2$ are homotopic.
So, using singularity theory, we can show that
$f_0$ and $f_1$ are connected by a generic 1-parameter
family of maps. So, if we take an arbitrary stable
map as $f_0$ and a very simple map as $f_1$, then $f_0$
is simplified to $f_1$ after the generic 1-parameter
family. Such a 1-parameter family passes through
finitely many bifurcation parameters, and such
bifurcations can be classified \cite{Mata-Lorenzo-1989}.
Such transitions of the Reeb Spaces for generic smooth maps on a
closed 3-dimensional manifold into $\mathbb{R}^2$ have been studied in \cite{Mata-Lorenzo-1989},
although for maps on a 3-dimensional
manifold with boundary these results need further extension. In the current paper, we consider only
a simple type of singularities and show that corresponding regular
component is detachable from the Reeb Space.
These components are known as \emph{lips}
and are defined as follows.
\begin{dfn}
\label{dfn:lips}
A lip is a regular component that is attached to the other
sheets of the Reeb Space exactly along one edge or an arc 1-singular component,
and it should not contain any vertex on the boundary,
except for the two cuspidal points (Figure~\ref{fig:lips-dfn}(a)).
\end{dfn}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.47\textwidth]{./Fig/lips.eps}
\caption{Lip simplification: (a) Reeb Space with a lip, (b) Simplified
Reeb Space.}
\label{fig:lips-dfn}
\end{center}
\end{figure}
\noindent
Next we prove the following lemma to show that the underlying map
corresponding to a lip can be simplified.
\begin{lemma}
\label{lem:detachability}
For a generic bivariate field $f: \mathbb{X}\subseteq \mathbb{R}^3\rightarrow\mathbb{R}^2$, a ``lip'' can always be detached safely.
\end{lemma}
\noindent
\begin{proof}
If we have a lip in the Reeb Space, there are
three possibilities as shown in Figure~\ref{fig:lips}.
That is, we may consider the map near the inverse image of
the lip as a 1-parameter family of functions on a piece of surface.
Let $S_0$ be a ``piece of surface'' (a cylinder or a square as in Figure~\ref{fig:lips}),
and $f_t : S_0 \rightarrow \mathbb{R},\, t \in \mathbb{I}$,
be the 1-parameter family of height functions as in Figure~\ref{fig:lips}.
Then, the original map $f$ is equivalent to the map $(x, t) \rightarrow
(f_t(x), t)$, $x \in S_0,\, t \in \mathbb{I}$, around the inverse image of a neighborhood of the lip by $q_f$.
The Figure~\ref{fig:lips} presents the three such families of functions.
As we can see easily, these can be eliminated continuously,
by just shrinking the ``time interval'' for which a pair
of critical points appear.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=8.3cm]{./Fig/lip1.eps}
\caption{The behavior of the stable map
near the pre-image of a neighborhood of the lip. The red lines indicate the boundary of the domain.}
\label{fig:lips}
\end{center}
\end{figure}
\end{proof}
\noindent
Thus, we see that lips are detachable and they can be simplified as in
Figure~\ref{fig:lips-dfn}. Therefore, we get our simplification rule for
detaching the lip components
as follows.\\
\noindent
\textbf{Simplification Rule:}
\emph{
Let $R_i$ be a detachable lip component of the Reeb
Space $\mathbb{W}_f$. Then we simplify the Reeb Space by (i) deleting $R_i$ with
its adjacent boundary 1-singular component and
(ii) converting the attached arc 1-singular component (merge-split)
and two 0-singular components (cusp vertices) as regular.}
Next we discuss the Reeb Space (Skeleton) simplification based on the
rule developed in this section.
\section{Reeb Space Simplification and Measures}
\label{sec:Simplification}
In the real multi-field data because of noise
very often there are ``lip''-like components which occlude the
original feature captured by the Reeb Space.
Therefore it is important to simplify such components to
understand the topology of the underlying data.
Given that it is possible to detach such ``lip''-like regular components from the Reeb Space, we follow a similar strategy
to that used for the contour tree~\cite{CSv10}. There, a leaf edge was chosen for pruning
and removed from the tree. If as a result a saddle point became regular (i.e. 1-manifold), it too
was removed, simplifying the graph further. By tracking which leaves, saddles and edges are
removed, the branch decomposition~\cite{PCS04} then gave a natural simplification
hierarchy for any ordering of leaves.
In any Reeb space where \lemref{detachability} applies, we can use the same strategy, building
a simplification hierarchy in the process. \textcolor{black}{To do so, we simply choose a detachable
component and remove it from the Reeb Space as described in the
simplification rule of Section~\ref{subsec:detach}.}
We illustrate this process in \figref{demo-simplification}, where we
progressively remove detachable regular
components from the Reeb Space, reducing the Jacobi Structure accordingly as much as desired. As
in leaf-pruning of contour trees, ``lip''-simplification reduces the number of regular components in the Reeb Space by one each
time, and also remove components of the Jacobi Structure, guaranteeing that the number of steps
required is linear in the number of regular components of the Reeb Space. Moreover, the editing operations to update
the Reeb Space, Jacobi Structure and Reeb Skeleton are constant at
every step, \textcolor{black}{making the
simplification effectively linear (in the number of regular components) once the order of reduction is known.}
Therefore, we study different measures to associate with the regular
components (nodes) of the Reeb Space (Skeleton).
\subsection{Range Measure}
\label{sec:MeasurePersistence}
In simplifying the contour tree, Reeb Graph and Morse-Smale Complex, simplification can be defined
by cancelling pairs of critical points according to an ordering given by a \emph{filtration} - i.e.
a sequence by which simplices are added to a complex. For any given filtration, a unique ordering
exists, and the persistence of a feature is defined by the distance in the filtration between the
critical points defining the feature.
For scalar data, however, the order in the filtration is dictated by the isovalues associated with
each vertex of the simplex, with the result that persistence can also be formalised as the isovalue
difference between the critical points that cancel each other. In multi-fields, the persistence of
a feature gives rise to tuples rather than a single value~\cite{Carlsson-MultiVariatePersistence},
which does not naturally give rise to a total ordering of the features.
This is however, not the only way to define a simplification ordering.
Carr et al. \cite{CSv10}
showed that pruning leaves individually could be ordered by geometric properties such as area,
volume etc. of the features defined by the contour tree. In this model, persistence is the vertical
height of a feature corresponding to a branch of the contour tree, and removing leaves can be done
with simple queue-based processing.
Recently, Duffy et al. \cite{DCM13} demonstrated that many properties of isosurfaces in scalar
and multi-fields relate to geometric measure theory. In this model, statistical and geometric
properties of a function are measured by integration over the range.
Following a similar approach we introduce a \emph{range measure} for
computing area of the regular components using the induced measure
from the range to the Reeb Space. Note that, in general a regular component of a Reeb Space
is projected to the range with multiplicities: i.e., this map is an
immersion, but may not be injective.
Consider for example the Reeb spaces shown in \figref{jacobi-reeb} for bivariate volumetric
maps.
Mathematically, range measure of a regular component in the Reeb space $\mathbb{W}_f$ is
defined as the area of the 2-dimensional sheets with respect to the measure induced from the usual
area measure of the range Euclidean space.
The range measure of each regular component in the Reeb space is a fixed scalar value. Thus,
there is a unique induced ordering for simplification. If two
components have identical range measure, some form of perturbation will be required to guarantee a
strict ordering.
\subsection{Geometric Measures}
\label{sec:geomMeasures}
Similarly, it is also possible to compute geometric properties of the regular components, either
in the domain, in the range, or in some combination of the two, using
geometric measure theory.
As with the contour tree~\cite{CSv10}, obvious properties of interest include the measure of the
region's boundary in the domain (contour length in 2D, isosurface surface area in 3D), the measure
of the region in the domain (area in 2D, volume in 3D), the measure of the function over the region
(a generalisation of the volume in 2D, hypervolume in 3D), and so forth. \textcolor{black}{However, as in that work,
rules will be needed in each case for combination of measure with parents in the simplification
hierarchy based on the theory in Section~\ref{subsec:detach}.}
\subsection{Summary of Theoretical Contributions}
\label{sec:mathSummary}
We have now completed the theoretical groundwork for practical
simplification algorithm of Reeb Spaces. In particular our theoretical
results could be summarised as follows.
\begin{enumerate}\itemsep1pt
\item The Reeb Space consists of regular components corresponding to regions in the domain
of the function, and singular components describing their relationships.
\item The Jacobi Set in the domain does not capture all of the structure of the singular
components in the Reeb Space, and the Jacobi Structure is needed to do so.
\item The Jacobi Structure of the Reeb Space can be used to
further collapse the Reeb Space into the Reeb
Skeleton.
\item Multifields with topologically simple domains can be
simplified using a variation on the leaf-pruning used for
contour trees.
\item A Reeb Space measure and other geometric measures are introduced to guide
the Reeb Space simplification process.
\end{enumerate}
We now turn to the practical and algorithmic part of this paper: how to simplify
the Joint Contour Net, an approximation of the Reeb Space.
\section{Algorithm: Simplifying the Joint Contour Net}
\label{sec:SimplifyingJCN}
In this section, first we introduce the Joint Contour Net, a graph
data-structure that approximates the Reeb Space.
As described in \cite{2013-Carr-TVCG}, the Joint Contour Net is a
quantized approximation of the Reeb Space. Therefore, to avoid having
duplicate terminology we will use the same terminology for JCN
as what we have developed for the Reeb Space, namely, Jacobi Structure, Regular
Component, Singular Components, Reeb Skeleton etc.
\subsection{Joint Contour Net}
\label{sec:jcn}
The Joint Contour Net (JCN) \cite{2013-Carr-TVCG, 2012-Duke-VisWeek} approximates the
Reeb Space $\mathbb{W}_f$ of a multi-field $f=(f_1,\,f_2,\,\ldots,\,f_r):\mathbb{X}\subset\mathbb{R}^d\rightarrow \mathbb{R}^r$
in a $d$-dimensional interval $\mathbb{X}$. Let
$\tilde{f}=(\tilde{f}_1,\,\tilde{f}_2,\,\ldots,\,\tilde{f}_r):
M\rightarrow \mathbb{R}^r$ be a piecewise-linear (PL) approximation of $f$ corresponding to a
mesh $M$ of $\mathbb{X}$. The idea of computing the JCN is based on
quantization of the fiber-components of $\tilde{f}$. The JCN $\tilde{f}$ with a
quantization level (or level of resolution) $m_Q$ is denoted as
$\JCN(\tilde{f}, m_Q)$, where $m_Q$ refers to how fine the rectangular mesh
for the range is.
A \textit{quantized level set} of $\tilde{f}_i$ at an isovalue $h \in
\mathbb{Z}/m_Q$ is denoted by $Q\tilde{f}_i^{-1}(h)$ and is
defined as:
$ Q\tilde{f}_i^{-1}(h):=\big\{x\in M : (\frac{1}{m_Q}) \operatorname{round}(m_Q\tilde{f}_i(x))=h\}$.
A connected component of the quantized level set in the mesh is called
a \textit{quantized contour} or a \textit{contour slab}. The part of the
contour slab in a single cell of the mesh is called a \textit{contour
fragment}.
Now the first step of the JCN algorithm constructs all the
contour fragments corresponding to a quantization of each component
field. In the second step, the \textit{joint contour fragments} are computed by
computing the intersections of these contour fragments for the component
fields in a cell. The third step is to construct an adjacency graph of these
joint contour fragments where a node in the graph corresponds to a joint
contour fragment and there is an edge between two nodes if the corresponding
joint contour fragments are adjacent. Finally, the JCN is obtained by
collapsing the neighbouring redundant nodes with identical isovalues.
Thus, each node in the JCN corresponds to a \textit{joint contour
slab} (or \emph{quantized fiber-component}) and an edge represents the
adjacency between two quantized fiber-components (with quantization level $m_Q$) of $\tilde{f}$.
Note that one can build a multi-resolution JCN by increasing or
decreasing the quantization level using a scaling factor for the
ranges of the component fields. An example of a small JCN is given in \figref{mdrg}, but
we refer the interested reader to \cite{2013-Carr-TVCG} for details.
The following lemma shows that in the limiting case, when the
quantization level increases and the domain-mesh becomes more
refined, then the JCN converges to the corresponding Reeb Space.
\begin{lemma}[\textbf{Convergence}]
\label{lem:convergence}
Let $f : \mathbb{X} \subset \mathbb{R}^d \to \mathbb{R}^r$, $d \geq r$, be a tiangulable continuous
multi-field with the Reeb space $\mathbb{W}_f$.
Choose an increasing sequence of quantization levels $\{m_Q^{(n)}\}$ for $f$
such that $m_Q^{(n)}$ is an integer multiple of $m_Q^{(n-1)}$ for each $n$
and $\displaystyle\lim_{n \to \infty}m_Q^{(n)} = \infty$.
Furthermore, let $\{M_n\}$ be sequence of sufficiently fine meshes
of $\mathbb{X}$ such that $M_n$ is a refinement of $M_{n-1}$ for each $n$ and
$\displaystyle\lim_{n\to \infty} d(M_n) = 0$, where $d(M_n)$ stands for
the maximum of the diameters of the cells of $M_n$.
Finally, let $f^{(n)} : M_n \to \mathbb{R}^r$ be the PL map associated with $f$
corresponding to the mesh $M_n$.
Then the sequence $\left\{\JCN(f^{(n)}, m_Q^{(n)})\right\}$
converges to $W_f$.
\end{lemma}
\noindent
\begin{proof}
Hiratuka et al. \cite{Hiratuka2013} show
for a PL map $f : A \to B$ of a compact polyhedron $A$ into
another polyhedron $B$, if we subdivide the range polyhedron $B$
appropriately, then $A$ is subdivided accordingly and the quotient map
$q_f : A \to \mathbb{W}_f$ to the Reeb Space $\mathbb{W}_f$ is triangulable with respect
to the triangulations. In the proof, it is also shown that the inverse
image by $q_f$ of a small regular neighborhood of a vertex $v$ in
$\mathbb{W}_f$ is always a regular neighborhood of $(q_f)^{-1}(v)$ in $A$. This
implies that if the quantization level is high enough, then the
quantized fiber-component is actually a regular neighborhood of the
central fiber-component. Consequently, we have a natural embedding
$\rho_0: JCN_0^{(n)} \to \mathbb{W}_f$, where $JCN_0^{(n)}$ is the set of
vertices of the Joint Contour Net $\JCN(f^{(n)},m_Q^{(n)})$ for
sufficiently large $n$. (For each quantized
fiber-component, associate the central fiber-component.) Furthermore,
as is shown in \cite{Hiratuka2013}, this embedding preserves the
adjacencies. This implies that the embedding $\rho_0$ extends to
an embedding $\rho : \JCN(f^{(n)},m_Q^{(n)}) \to \mathbb{W}_f$. Hence, the required result holds,
since the triangulations of $\JCN(f^{(n)},m_Q^{(n)})$ and $\mathbb{W}_f$
becomes finer and finer as $n$ increases.
If $f$ itself is not a PL map, then we can consider its triangulation
$g : A \to B$ and obtain the required result for the triangulation. As
the domain $\mathbb{X}$ is compact, this implies the same consequence for the
original map $f$ as well. This completes the proof.
\end{proof}
Next we see that the simplification will have four stages: 1. extraction of the
Jacobi Structure from the JCN, 2. computing regular and singular
components for construction of the Reeb Skeleton,
3. computation of measures for each regular node in the Reeb Skeleton,
and 4. simplification by pruning nodes corresponding to the ``lip'' components. In practice, the first stage is the most difficult - identifying the regular
components, and this requires an intermediate data-structure, which we introduce now.
\subsection{Multi-Dimensional Reeb Graphs}
\label{sec:MDRG}
The first step in detecting and analysing the Jacobi Structure is to identify the
nodes in the JCN that capture changes in the topology - i.e. the quantized
representatives of the Jacobi Structure. To do so, we exploit a simple property
of the JCN - that the slabs can be arranged hierarchically, with the levels of the
hierarchy corresponding to the individual fields. At the highest level of the
hierarchy, the slabs are only defined by field $f_1$, and are therefore equivalent
to interval volumes: as such, we can compute the Reeb Graph for field
$f_1$ (see Figure \ref{fig:mdrg}~(middle)).
\begin{algorithm}
\caption{{\sc CreateReebGraph}$(G, f_i)$}
\label{alg:rg}
{\bf Input:} A subgraph $G$ of $JCN$ and a chosen field $f_i$\\
{\bf Output:} The Reeb Graph $RG$ with respect to field $f_i$
\begin{algorithmic}[1]
\State Create Union-Find Structure $UF$ for field $f_i$.
\State For each adjacent $g_1,g_2 \in G$ with $f_i(g_1) = f_i(g_2)$, UFAdd($g_1,g_2)$
\For{ each component $C_l$ in UF}
\State Create a node $n_{C_l}$ in $RG$
\State Map graph node-id(s) and field-values from $G$ to $n_{C_l}$
\EndFor
\State Order nodes ${n_{C_1},\ldots,n_{C_n}}$ according to $f_i$ field values.
\For{edge $e_1e_2$ in $G$}
\If {$e_1, e_2 \in$ components $C_j \neq C_k$ and $f_i(e_1) \neq f_i(e_2)$}
\State Add edge $e(n_{C_j}, n_{C_k})$ in $RG$ if not already present
\EndIf
\EndFor
\State \Return{$RG$}
\end{algorithmic}
\end{algorithm}
Each slab (i.e. interval volume) of $f_1$ can be broken up into smaller slabs with respect to
field $f_2$ in a similar way (which form a subgraph $G$ in the JCN), and
the Reeb graph for these slabs computed similarly, as
shown in \algoref{rg}. Proceeding recursively, we then compute a hierarchy of Reeb graphs,
each of which represents the internal topology of a slab of the parent Reeb Graph
with respect to the child's field. We call this hierarchy the \emph{Multi-Dimensional
Reeb Graph} or MDRG and denote this as $\mathbb{M}_f$.
Computing the MDRG is straightforward once the full JCN has been extracted: we
start with the JCN and compute the Reeb Graph for property $f_1$ by performing
union-find processing over the nodes of the JCN. This breaks the JCN into subgraphs
corresponding to slabs in the Reeb Graph of property $f_1$. The MDRG for each
subgraph is then computed recursively, and stored in the node of the parent Reeb
Graph to which its slab corresponds. In the process, the slabs get separated out into
smaller and smaller components.
\begin{algorithm}
\caption{{\sc MultiDimensionalReebGraph}$(G, f_i,\ldots, f_r)$}
\label{alg:mdrg}
{\bf Input:} Graph $G$, fields $f_i, \ldots, f_r$
{\bf Output:} MDRG $M$
\begin{algorithmic}[1]
\If {$i \leq r$}
\State Let $R = $ CreateReebGraph($G, f_i$)
\State Store $R$ as root node of $M$
\For {Each slab $s$ of $R$}
\State Extract subgraph $G_s$ of nodes of $G$ belonging to $s$ in $R$
\State Compute $M_s = $ \small{MultiDimensionalReebGraph($G_s, f_{i+1}, \allowbreak \ldots, f_r$)}
\State Store $M_s$ at node $s$ of $R$
\EndFor
\State \Return $M$
\Else
\State \Return $M = \emptyset$
\EndIf
\end{algorithmic}
\end{algorithm}
We state this as an algorithm in \algoref{mdrg} and illustrate with a bivariate
field in \figref{mdrg}. This algorithm is stated recursively for simplicity, but can also
be implemented with queue processing for speed. Moreover, the division of subgraphs
at each level into slabs can be performed more efficiently by exploiting the connectivity
already encoded in the JCN.
The principal value of the MDRG is that every node of the JCN in the Jacobi Structure is
guaranteed to be a critical node of the finest-resolution Reeb
Graphs (denoted as the critical nodes of the MDRG). This immediately gives a method of computing the Jacobi Structure once the MDRG
is known \cite{2014-EuroVis-short}.
\subsection{Jacobi Structure Extraction}
\label{sec:JacobiStructureExtraction}
Since every node belonging to the Jacobi Structure is guaranteed to appear as a critical node
of the lowest level of an MDRG, the initial stage in Jacobi Structure extraction is simply to
mark these nodes. Unmarked nodes are then guaranteed to be regular, and can be collected
into regular components. Once this has been done, any remaining nodes that are adjacent
to each other and to the same set of regular components are identified, as these form a
1-singular component between the regular components.
The first stage of this can be seen in \figref{mdrg}, where the critical nodes of the lowest
level of the MDRG together mark all of the Jacobi Structure nodes in the JCN (in colour).
\subsection{Reeb Skeleton Construction}
\label{sec:reeb-skel}
\begin{figure}[t!]
\begin{tabular}{|c|c|}
\hline
\subfloat[\label{fig:6a}]{\includegraphics[height=3.5cm]{./Fig/reebSkel1/3d-2d-sphere-parab2.eps}} & \subfloat[\label{fig:6b}]{\includegraphics[height=3.5cm]{./Fig/reebSkel1/3d-2d-sphere-parab-JCN.eps}}\\
\hline
\subfloat[\label{fig:6c}]{\includegraphics[height=3.5cm]{./Fig/reebSkel1/3d-2d-sphere-parab-regular.eps}}& \subfloat[\label{fig:6d}]{\includegraphics[height=3.5cm]{./Fig/reebSkel1/3d-2d-sphere-parab-RS.eps}}\\
\hline
\end{tabular}
\caption{(a) Bivariate Field: $(x^2+y^2-z$, $x^2+y^2+z^2)$ in a box
$[-5,\,5]^3$, the `red' components are the Jacobi Set, (b) the Joint
Contour Net with the Jacobi Structure (in red), (c) Regular
Components, (d) the Reeb Skeleton.}
\end{figure}
\label{fig:reeb-skeleton-demo}
Once we have extracted the Jacobi Structure for the JCN, it is straightforward to compute the
corresponding Reeb Skeleton by creating a single node for each regular or 1-singular component,
and connecting them using the adjacency of components in the Jacobi
Structure. In \figref{reeb-skeleton-demo} we see an example of the Reeb
Skeleton construction for a volumetric bivariate field. We note, in
this case, the Reeb Skeleton has no
detachable lip-like regular node whereas the Reeb Skeleton
\figref{demo-simplification} (d) has such detachable nodes.
Now in the simplification algorithm, the order in which detachable
Reeb Skeleton nodes are removed is determined by the metrics associated with those
nodes. Computation of such metrics are described next.
\begin{algorithm}
\caption{\sc{ SimplifyReebSpace}}
\label{alg:simplify-jcn}
{\textbf{Input:}} JCN $\JCN$\\
{\textbf{Output:}} Reeb Skeleton $\mathbb{K}_f$
\begin{algorithmic}[1]
\State Build MDRG $\mathbb{M}_f$ and Jacobi Structure $\mathfrak{J}_f$ from JCN $\JCN$.
\State Partition $\JCN$ into disjoint regular components $C = \{R_1, \ldots, R_m\}$ by deleting $\mathfrak{J}_f$ from $\JCN$.
\State Partition $\mathfrak{J}_f$ into disjoint 1-singular components $\{S_1, \ldots, S_n\}$ based on adjacency to regular components in $C$.
\State Use adjacencies of $\{R_1, \ldots, R_m, S_1, \ldots, S_n\}$ to
construct $\mathbb{K}_f$ (following defintion~\ref{dfn:reeb-skeleton}).
\State Push ``detachable'' nodes of $\mathbb{K}_f$ on priority queue $PQ$ with priority determined by geometric measures.
\While {$PQ$ not empty (or priority is below a threshold value)}
\State Pop node $r$ from queue
\State Prune $r$ from $\mathbb{K}_f$
\EndWhile
\State \Return {Simplified Reeb Skeleton $\mathbb{K}_f$.}
\end{algorithmic}
\end{algorithm}
\subsection{Computing Simplification Metrics}
\label{sec:metric}
Our simplification algorithm
can use any desired measure of importance for components of the Reeb
space, including but not limited to
\begin{itemize}
\item \textbf{Range measure.} As described in \subsecref{MeasurePersistence}, we can measure
the size of the regular components by the induced measure of the
range. This is easy to approximate - in this case, by the number
of unique JCN slabs (i.e. pixels in the range) that map to a given
regular component
\item \textbf{Surface area.} A regular component of the Reeb space is separated from
other regular components by one or more singular components in the Jacobi Structure. Since
the regular components correspond to features and the singular components to boundaries
between features, we can associate the area of the bounding surface with the regular component
for the purpose of simplification. For the JCN, we can approximate this with the
surface area of the fragments adjacent to the bounding region
\item \textbf{Volume.} Similarly, we can measure volume in the domain for each feature represented
by a regular component, and approximate it by summing the volume of fragments mapping
to a given regular component
\item \textbf{Other measures.} As shown by Duffy~et~al.~\cite{DCM13} and Carr~et~al.~\cite{CSv10},
almost any geometric or other property of features can be used for simplification provided that it is
correctly approximated and suitable rules for composition during
simplification are established.
\end{itemize}
\subsection{Simplifying the Reeb Skeleton}
\label{sec:rules}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=5.3cm]{./Fig/rule1.eps}
\caption{Simplification rule for detaching a lip-like node (regular
node B in (a)) from the Reeb skeleton.}
\label{fig:rules-simplification}
\end{center}
\end{figure}
The lip-simplification rule of the Reeb Space, as described in
Section~\ref{subsec:detach}, can be translated similarly in the
corresponding Reeb Skeleton. We note, according to Theorem 4.4,
each regular component of our Reeb Space
is always a disk possibly with holes. Therefore, the degree 2 regular
node B as in a Reeb Skeleton \figref{rules-simplification}(a) is always a
lip-like detachable node. \figref{rules-simplification}(b) shows the simplified Reeb
Skeleton after pruning the lip-like node and attached singular nodes.
Finally, we give simplification strategies of the
Reeb Skeleton, based on geometric and range measures of the
components. The Reeb skeleton simplification method simplifies the Reeb skeleton
and the corresponding Reeb space given a threshold (between 0 and 1) adapting the previous approaches
in the literature \cite{2004-Carr-simplification, 2012-Tierny-tvcg}. The threshold represents a
``scale'', under which detachable regular nodes of the Reeb skeleton
are considered as unimportant (noise).
The threshold is expressed as a fraction of the range of the metric used. It can vary from 0 (no simplification) to 1
(maximal simplification).
\figref{demo-simplification} demonstrates the simplification
of components from the Reeb Space, of an unstable bivariate
volumetric data. We use range measure for ordering the components.
We note, regular nodes 4 and 2 in \figref{demo-simplification} are not strictly the lips according to our
definition of lips. However, a perturbation can be applied first
to convert such components to lips and then lip
simplifications can be applied. In \figref{demo-simplification} we apply our lip-simplification rule
directly at the regular nodes 4 and 2, sequentially (node with
smaller measure is pruned first).
\begin{figure}[t]
\begin{center}
\includegraphics[width=.47\textwidth]{./Fig/demo-simplification2.eps}
\caption{(\textbf{Simplification Demo}) (a) Original JCN/Reeb Space of bivariate field (Paraboloid, Height)
(b) Jacobi Structure, (c) Regular components (d) Reeb Skeleton (`blue'
corresponds to regular components and `red'
corresponds to adjacent 1-singular components) (e)
Simplified JCN
(f)-(g) Simplified Reeb Skeleton using range measure.}
\label{fig:demo-simplification}
\end{center}
\end{figure}
\section{Implementation and Application}
\label{sec:Implementation}
We implement our Reeb Space or JCN simplification algorithm using the Visualization Toolkit (VTK)
\cite{vtk}. Details of the JCN implementation can be found in \cite{2013-Carr-TVCG,
2012-Duke-VisWeek}. Our Reeb Space simplification implementation takes the JCN (a vtkGraph
structure) as input and builds four filters: (1) The first filter computes the
Jacobi Structure by implementing the Multi-dimensional Reeb graph
algorithm, (2) The second filter builds the Reeb Skeleton structure by
partitioning the JCN, (3) The third filter implements persistence and geometric measures and (4) The fourth filter implements the
simplification rules of the Reeb Skeleton.
We use a vtkTree structure to store the Multi-Dimensional Reeb Graph (MDRG):
Reeb graphs at each level of MDRG are stored in a vtkReebGraph structure. For capturing
the Reeb skeleton we use the vtkGraph structure.
\begin{table}[h!]
\begin{centering}
\caption{Data Statistics}
\scalebox{0.65}{
\begin{tabular}{lccccccc}
\hline
datasets & spatial-dimensions & slab widths & no. of nodes (JCN) & no. of edges (JCN)\\
\hline
(Circle, Line) &(29, 29, 1) &(1, 1) &500 &1057 \\
(Paraboloid, Height) &(40, 40, 40) &(1, 1) &1260 &2383 \\
(Sphere, Height) &(40, 40, 40) &(1, 1) &1308 &2428 \\
(Paraboloid, Sphere) &(40, 40, 40) &(1, 1) &6554 &12795 \\
(Cubic, Height) &(40, 40, 40) &(1, 1) &3149 &5928 \\
\end{tabular}
}
\label{tab:perf-mdrg}
\end{centering}
\end{table}
A force-directed graph-layout from the OGDF - an Open Graph Drawing
Framework \cite{ogdf} strategy has been used for the graph
visualization as shown in the demonstrations and outputs.
We run our implementation on different synthetic and simulated data sets for
testing the performance. In \tabref{perf-mdrg} the synthetic data sets are
labelled by the combination of scalar fields used: Circle: $x^2+y^2$, Line: $y$,
Sphere: $x^2+y^2+z^2$, Paraboloid: $x^2+y^2-z$, Height: $z$ and Cubic: $(y^3-xy+z^2, \,
x)$. Circle and Line are in the 2D-box $[-5,\, 5]^2$ and other fields are
considered in the 3D-box $[-5,\, 5]^3$.
\begin{table}[h!]
\begin{centering}
\caption{Performance results for Simplification}
\scalebox{0.7}{
\begin{tabular}{lccccccc}
\hline
Data & Spatial & Slab &Jacobi & Reeb & \\
& Dimensions & Widths &Structure & Skeleton & Simplification \\
\hline
(Circle, Line) &(29, 29, 1) &(1, 1) &0.06s & 0.242s &0.00s\\
(Paraboloid, Height) &(40, 40, 40) &(1, 1) &0.10s &1.97s &0.00s\\
(Paraboloid, Sphere) &(40, 40, 40) &(1, 1) &0.80s &33.02s &0.00s\\
Nucleon &(40, 40, 66) &(8,2) & 0.45s& 48.91s & 0.45s\\
\end{tabular}
}
\label{tab:perf-simpl}
\end{centering}
\end{table}
\paragraph{Performance results} Table~\ref{tab:perf-simpl} shows the performance
results of the JCN and MDRG algorithms for some simulated
data. All timings were performed on a 3.06 GHz 6-Core Intel Xeon with 64GB memory, running
OSX 10.8.5, and using VTK 5.10.1.
The number of nodes in the MDRG is actually the number of Reeb graphs
computed by the MDRG Algorithm~\ref{alg:mdrg}. From the table it is clear that
performance of the MDRG algorithm is quite impressive for these
simulated data. The complexity of the CreateReebGraph on a graph with $n$ nodes is $O(n+p\log n)$
which is the complexity of a sequence of $p$ UF operations (here,
$p\leq n$) \cite{1975-tarjan}.
\subsection*{Nuclear Scission Data}
\begin{figure*}
\centering
\includegraphics[width= 0.71\textwidth]{./Fig/physics.eps} \\
\caption{ This figure shows the simplification process of the
JCN corresponding to a ``nuclear scission'' data set used in \cite{2012-Duke-VisWeek}.
The process includes: (a) The original JCN graph; (b) The Reeb
Skeleton; (c) A modified representation of the Reeb Skeleton
by changing the colour of the degree 2 singular nodes as
``blue'', since they capture only minor topological information, (d)
Simplified Reeb Skeleton after applying lip-pruning using
the range measure for ordering the nodes (e) Resultant Reeb
Skeleton is represented as the ``Y''-fork;
(f) Geometry corresponding to the ``scission'' point. }
\label{fig:nuclear}
\end{figure*}
In a previous application paper~\cite{2012-Duke-VisWeek}, the JCN was
applied to nuclear data set (time-varying bivariate field of proton and neutron densities) and used to visualise the scission points in
high-dimensional parameter spaces. Here the scission refers to the
point where a single plutonium nucleus breaks into two fragments.
However, this was based on visual analysis, and was complicated by a
number of artefacts such as the recurring chains of star-like 'motifs' within
the JCN. Moreover, the eight corners of the domain boundary induced
eight corresponding small sets of features in the JCN.
Here we apply our simplification algorithm to
overcome these artefacts and preserve the principal topological
feature. Note that apparently there are two red nodes that are adjacent to
5 blue nodes in the Reeb Skeleton \figref{nuclear}(b). This never
happens if the given multi-field is stable, so this is an example of
an unstable bivariate field in 3-dimensional interval. We demonstrate the process of simplification for one of these
scission data sets in \figref{nuclear}. Note that the final simplification
results a simple Y-fork where two regular nodes correspond to the separated
nuclei, while the third represents the exterior.
\section{Conclusions and Discussions}
\label{sec:Conclusions}
In this paper, we provide a rigorous mathematical and computational foundation of
multivariate simplification based on lip-pruning from its Reeb Space
and this generalises approaches that are effective for scalar fields.
We note, lip-simplification can be applied only when there is a
lip-component in the Reeb Space and might not always be possible, but
nevertheless, it is quite effective when applied to real data sets,
which usually contain a lof of noise (as demonstrated in
\figref{nuclear}). The Jacobi Structure
that characterises the Reeb Space is richer than the Jacobi
Set and decomposes the Reeb Space. This is proved to be a useful
property for the simplification procedure.
In addition, we have shown how to extract a reliable
approximate Jacobi Structure and Reeb Skeleton from the JCN that can be simplified to
improve the use of the JCN for multivariate analysis, and illustrated this with analytical
datasets and a real-world data. However, there are few open issues which
need to be addressed in future research.
\begin{itemize}\itemsep1pt
\item \textbf{False lips:} Currently using our lip simplification
approach we are not able to distinguish or simplify the false-lips,
similar as in \figref{falselip}. This is because our Reeb Skeleton cannot compute
the multiplicity of the adjacency between a regular and a 1-singular
node which will be important for detecting such a false lip in the Reeb Space. So, in our current simplification we assume no false lip
appears and we can simplify only the ``genuine'' lips as defined in Definition~\ref{dfn:lips}.
\begin{figure}[h!]
\centering
\includegraphics[width= 0.22\textwidth]{./Fig/false-lip1.eps} \\
\caption{False lip}
\label{fig:falselip}
\end{figure}
\item \textbf{Discontinuity in components of Jacobi Structure:}
Computed 1-singular components of
the Jacobi Structure in the JCN may be discontinuous because of the degeneracy, as all degenerate singular points may not be captured by the critical nodes of
MDRG. Moreover, a choice of the quantization level may also
result in the discontinuous Jacobi Structure in the JCN.
\item \textbf{Further Structures in the Reeb Skeleton:} In the current
implementation of the Reeb Skeleton we have considered only the
regular and the 1-singular components and their adjacency graph. But, there
is further hierarchy possible in
the Jacobi Structure. For more than bivariate case,
singular components can be decomposed into lower
dimensional manifolds (strata) and can be represented in the Reeb Skeleton,
hierarchically. However, detecting such lower dimensional strata needs
further theoretical analysis and an algorithm for detecting them.
\end{itemize}
\noindent
Apart from these issues, in the future, we intend to work on further simplification and acceleration of these techniques,
and on alternate methods for Reeb Space computation and / or approximation. We also expect
to examine more data sets from multiple domains, now that we have solved more of the main theoretical
issues.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 461 |
A. Petrignani, P. U. Andersson, J. B. C. Pettersson, R. D. Thomas, F. Hellberg, A. Ehlerding, M. Larsson and W. J. van der Zande
Dissociative recombination of the weakly bound NO-dimer cation: cross sections and three-body dynamics
J. Chem. Phys. , Volume 123 - Issue Article number: 194306 p. 1- 11
Dissociative recombination (DR) of the dimer ion (NO)2+ has been studied at the heavy-ion storage ring CRYRING at the Manne Siegbahn Laboratory, Stockholm. The experiments were aimed at determining details on the strongly enhanced thermal rate coefficient for the dimer, interpreting the dissociation dynamics of the dimer ion, and studying the degree of similarity to the behavior in the monomer. The DR rate reveals that the very large efficiency of the dimer rate with respect to the monomer is limited to electron energies below 0.2 eV. The fragmentation products reveal that the breakup into the three-body channel NO+O+N dominates with a probability of 0.69±0.02. The second most important channel yields NO+NO fragments with a probability of 0.23±0.03. Furthermore, the dominant three-body breakup yields electronic and vibrational ground-state products, NO(?=0)+N(4S)+O(3P), in about 45% of the cases. The internal product-state distribution of the NO fragment shows a similarity with the product-state distribution as predicted by the Franck-Condon overlap between a NO moiety of the dimer ion and a free NO. The dissociation dynamics seem to be independent of the NO internal energy. Finally, the dissociation dynamics reveal a correlation between the kinetic energy of the NO fragment and the degree of conservation of linear momentum between the O and N product atoms. The observations support a mechanism in which the recoil takes place along one of the NO bonds in the dimer.
Petrignani, A, Andersson, P. U, Pettersson, J. B. C, Thomas, R. D, Hellberg, F, Ehlerding, A, … van der Zande, W. J. (2005). Dissociative recombination of the weakly bound NO-dimer cation: cross sections and three-body dynamics. J. Chem. Phys., 123(Article number: 194306), 1–11. doi:10.1063/1.2116927 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,664 |
import os.path
import random
import string
import sys
import pytest
import salt.config
import salt.states.boto_vpc as boto_vpc
import salt.utils.botomod as botomod
from salt.utils.versions import LooseVersion
from tests.support.mixins import LoaderModuleMockMixin
from tests.support.mock import patch
from tests.support.runtests import RUNTIME_VARS
from tests.support.unit import TestCase, skipIf
# pylint: disable=import-error,unused-import
from tests.unit.modules.test_boto_vpc import BotoVpcTestCaseMixin
try:
import boto
boto.ENDPOINTS_PATH = os.path.join(
RUNTIME_VARS.TESTS_DIR, "unit/files/endpoints.json"
)
import boto3
from boto.exception import BotoServerError
HAS_BOTO = True
except ImportError:
HAS_BOTO = False
try:
from moto import mock_ec2_deprecated # pylint: disable=no-name-in-module
HAS_MOTO = True
except ImportError:
HAS_MOTO = False
def mock_ec2_deprecated(self):
"""
if the mock_ec2_deprecated function is not available due to import failure
this replaces the decorated function with stub_function.
Allows boto_vpc unit tests to use the @mock_ec2_deprecated decorator
without a "NameError: name 'mock_ec2_deprecated' is not defined" error.
"""
def stub_function(self):
pass
return stub_function
# pylint: enable=import-error,unused-import
# the boto_vpc module relies on the connect_to_region() method
# which was added in boto 2.8.0
# https://github.com/boto/boto/commit/33ac26b416fbb48a60602542b4ce15dcc7029f12
required_boto_version = "2.8.0"
region = "us-east-1"
access_key = "GKTADJGHEIQSXMKKRBJ08H"
secret_key = "askdjghsdfjkghWupUjasdflkdfklgjsdfjajkghs"
conn_parameters = {
"region": region,
"key": access_key,
"keyid": secret_key,
"profile": {},
}
cidr_block = "10.0.0.0/24"
subnet_id = "subnet-123456"
dhcp_options_parameters = {
"domain_name": "example.com",
"domain_name_servers": ["1.2.3.4"],
"ntp_servers": ["5.6.7.8"],
"netbios_name_servers": ["10.0.0.1"],
"netbios_node_type": 2,
}
network_acl_entry_parameters = ("fake", 100, -1, "allow", cidr_block)
dhcp_options_parameters.update(conn_parameters)
def _has_required_boto():
"""
Returns True/False boolean depending on if Boto is installed and correct
version.
"""
if not HAS_BOTO:
return False
elif LooseVersion(boto.__version__) < LooseVersion(required_boto_version):
return False
else:
return True
class BotoVpcStateTestCaseBase(TestCase, LoaderModuleMockMixin):
def setup_loader_modules(self):
ctx = {}
utils = salt.loader.utils(
self.opts,
whitelist=["boto", "boto3", "args", "systemd", "path", "platform", "reg"],
context=ctx,
)
serializers = salt.loader.serializers(self.opts)
self.funcs = salt.loader.minion_mods(
self.opts, context=ctx, utils=utils, whitelist=["boto_vpc", "config"]
)
self.salt_states = salt.loader.states(
opts=self.opts,
functions=self.funcs,
utils=utils,
whitelist=["boto_vpc"],
serializers=serializers,
)
return {
boto_vpc: {
"__opts__": self.opts,
"__salt__": self.funcs,
"__utils__": utils,
"__states__": self.salt_states,
"__serializers__": serializers,
},
botomod: {},
}
@classmethod
def setUpClass(cls):
cls.opts = salt.config.DEFAULT_MINION_OPTS.copy()
cls.opts["grains"] = salt.loader.grains(cls.opts)
@classmethod
def tearDownClass(cls):
del cls.opts
def setUp(self):
self.addCleanup(delattr, self, "funcs")
self.addCleanup(delattr, self, "salt_states")
# connections keep getting cached from prior tests, can't find the
# correct context object to clear it. So randomize the cache key, to prevent any
# cache hits
conn_parameters["key"] = "".join(
random.choice(string.ascii_lowercase + string.digits) for _ in range(50)
)
@skipIf(HAS_BOTO is False, "The boto module must be installed.")
@skipIf(HAS_MOTO is False, "The moto module must be installed.")
@skipIf(
_has_required_boto() is False,
"The boto module must be greater than or equal to version {}".format(
required_boto_version
),
)
class BotoVpcTestCase(BotoVpcStateTestCaseBase, BotoVpcTestCaseMixin):
"""
TestCase for salt.states.boto_vpc state.module
"""
@skipIf(
sys.version_info > (3, 6),
"Disabled for 3.7+ pending https://github.com/spulec/moto/issues/1706.",
)
@mock_ec2_deprecated
@pytest.mark.slow_test
def test_present_when_vpc_does_not_exist(self):
"""
Tests present on a VPC that does not exist.
"""
with patch.dict(botomod.__salt__, self.funcs):
vpc_present_result = self.salt_states["boto_vpc.present"](
"test", cidr_block
)
self.assertTrue(vpc_present_result["result"])
self.assertEqual(
vpc_present_result["changes"]["new"]["vpc"]["state"], "available"
)
@skipIf(
sys.version_info > (3, 6),
"Disabled for 3.7+ pending https://github.com/spulec/moto/issues/1706.",
)
@mock_ec2_deprecated
def test_present_when_vpc_exists(self):
vpc = self._create_vpc(name="test")
vpc_present_result = self.salt_states["boto_vpc.present"]("test", cidr_block)
self.assertTrue(vpc_present_result["result"])
self.assertEqual(vpc_present_result["changes"], {})
@mock_ec2_deprecated
@skipIf(True, "Disabled pending https://github.com/spulec/moto/issues/493")
def test_present_with_failure(self):
with patch(
"moto.ec2.models.VPCBackend.create_vpc",
side_effect=BotoServerError(400, "Mocked error"),
):
vpc_present_result = self.salt_states["boto_vpc.present"](
"test", cidr_block
)
self.assertFalse(vpc_present_result["result"])
self.assertTrue("Mocked error" in vpc_present_result["comment"])
@skipIf(
sys.version_info > (3, 6),
"Disabled for 3.7+ pending https://github.com/spulec/moto/issues/1706.",
)
@mock_ec2_deprecated
@pytest.mark.slow_test
def test_absent_when_vpc_does_not_exist(self):
"""
Tests absent on a VPC that does not exist.
"""
with patch.dict(botomod.__salt__, self.funcs):
vpc_absent_result = self.salt_states["boto_vpc.absent"]("test")
self.assertTrue(vpc_absent_result["result"])
self.assertEqual(vpc_absent_result["changes"], {})
@skipIf(
sys.version_info > (3, 6),
"Disabled for 3.7+ pending https://github.com/spulec/moto/issues/1706.",
)
@mock_ec2_deprecated
@pytest.mark.slow_test
def test_absent_when_vpc_exists(self):
vpc = self._create_vpc(name="test")
with patch.dict(botomod.__salt__, self.funcs):
vpc_absent_result = self.salt_states["boto_vpc.absent"]("test")
self.assertTrue(vpc_absent_result["result"])
self.assertEqual(vpc_absent_result["changes"]["new"]["vpc"], None)
@mock_ec2_deprecated
@skipIf(True, "Disabled pending https://github.com/spulec/moto/issues/493")
def test_absent_with_failure(self):
vpc = self._create_vpc(name="test")
with patch(
"moto.ec2.models.VPCBackend.delete_vpc",
side_effect=BotoServerError(400, "Mocked error"),
):
vpc_absent_result = self.salt_states["boto_vpc.absent"]("test")
self.assertFalse(vpc_absent_result["result"])
self.assertTrue("Mocked error" in vpc_absent_result["comment"])
class BotoVpcResourceTestCaseMixin(BotoVpcTestCaseMixin):
resource_type = None
backend_create = None
backend_delete = None
extra_kwargs = {}
def _create_resource(self, vpc_id=None, name=None):
_create = getattr(self, "_create_" + self.resource_type)
_create(vpc_id=vpc_id, name=name, **self.extra_kwargs)
@skipIf(
sys.version_info > (3, 6),
"Disabled for 3.7+ pending https://github.com/spulec/moto/issues/1706.",
)
@mock_ec2_deprecated
@pytest.mark.slow_test
def test_present_when_resource_does_not_exist(self):
"""
Tests present on a resource that does not exist.
"""
vpc = self._create_vpc(name="test")
with patch.dict(botomod.__salt__, self.funcs):
resource_present_result = self.salt_states[
"boto_vpc.{}_present".format(self.resource_type)
](name="test", vpc_name="test", **self.extra_kwargs)
self.assertTrue(resource_present_result["result"])
exists = self.funcs["boto_vpc.resource_exists"](self.resource_type, "test").get(
"exists"
)
self.assertTrue(exists)
@skipIf(
sys.version_info > (3, 6),
"Disabled for 3.7+ pending https://github.com/spulec/moto/issues/1706.",
)
@mock_ec2_deprecated
@pytest.mark.slow_test
def test_present_when_resource_exists(self):
vpc = self._create_vpc(name="test")
self._create_resource(vpc_id=vpc.id, name="test")
with patch.dict(botomod.__salt__, self.funcs):
resource_present_result = self.salt_states[
"boto_vpc.{}_present".format(self.resource_type)
](name="test", vpc_name="test", **self.extra_kwargs)
self.assertTrue(resource_present_result["result"])
self.assertEqual(resource_present_result["changes"], {})
@mock_ec2_deprecated
@skipIf(True, "Disabled pending https://github.com/spulec/moto/issues/493")
def test_present_with_failure(self):
vpc = self._create_vpc(name="test")
with patch(
"moto.ec2.models.{}".format(self.backend_create),
side_effect=BotoServerError(400, "Mocked error"),
):
resource_present_result = self.salt_states[
"boto_vpc.{}_present".format(self.resource_type)
](name="test", vpc_name="test", **self.extra_kwargs)
self.assertFalse(resource_present_result["result"])
self.assertTrue("Mocked error" in resource_present_result["comment"])
@skipIf(
sys.version_info > (3, 6),
"Disabled for 3.7+ pending https://github.com/spulec/moto/issues/1706.",
)
@mock_ec2_deprecated
@pytest.mark.slow_test
def test_absent_when_resource_does_not_exist(self):
"""
Tests absent on a resource that does not exist.
"""
with patch.dict(botomod.__salt__, self.funcs):
resource_absent_result = self.salt_states[
"boto_vpc.{}_absent".format(self.resource_type)
]("test")
self.assertTrue(resource_absent_result["result"])
self.assertEqual(resource_absent_result["changes"], {})
@skipIf(
sys.version_info > (3, 6),
"Disabled for 3.7+ pending https://github.com/spulec/moto/issues/1706.",
)
@mock_ec2_deprecated
@pytest.mark.slow_test
def test_absent_when_resource_exists(self):
vpc = self._create_vpc(name="test")
self._create_resource(vpc_id=vpc.id, name="test")
with patch.dict(botomod.__salt__, self.funcs):
resource_absent_result = self.salt_states[
"boto_vpc.{}_absent".format(self.resource_type)
]("test")
self.assertTrue(resource_absent_result["result"])
self.assertEqual(
resource_absent_result["changes"]["new"][self.resource_type], None
)
exists = self.funcs["boto_vpc.resource_exists"](self.resource_type, "test").get(
"exists"
)
self.assertFalse(exists)
@mock_ec2_deprecated
@skipIf(True, "Disabled pending https://github.com/spulec/moto/issues/493")
def test_absent_with_failure(self):
vpc = self._create_vpc(name="test")
self._create_resource(vpc_id=vpc.id, name="test")
with patch(
"moto.ec2.models.{}".format(self.backend_delete),
side_effect=BotoServerError(400, "Mocked error"),
):
resource_absent_result = self.salt_states[
"boto_vpc.{}_absent".format(self.resource_type)
]("test")
self.assertFalse(resource_absent_result["result"])
self.assertTrue("Mocked error" in resource_absent_result["comment"])
@skipIf(HAS_BOTO is False, "The boto module must be installed.")
@skipIf(HAS_MOTO is False, "The moto module must be installed.")
@skipIf(
_has_required_boto() is False,
"The boto module must be greater than or equal to version {}".format(
required_boto_version
),
)
class BotoVpcSubnetsTestCase(BotoVpcStateTestCaseBase, BotoVpcResourceTestCaseMixin):
resource_type = "subnet"
backend_create = "SubnetBackend.create_subnet"
backend_delete = "SubnetBackend.delete_subnet"
extra_kwargs = {"cidr_block": cidr_block}
@skipIf(HAS_BOTO is False, "The boto module must be installed.")
@skipIf(HAS_MOTO is False, "The moto module must be installed.")
@skipIf(
_has_required_boto() is False,
"The boto module must be greater than or equal to version {}".format(
required_boto_version
),
)
class BotoVpcInternetGatewayTestCase(
BotoVpcStateTestCaseBase, BotoVpcResourceTestCaseMixin
):
resource_type = "internet_gateway"
backend_create = "InternetGatewayBackend.create_internet_gateway"
backend_delete = "InternetGatewayBackend.delete_internet_gateway"
@skipIf(
True,
"Disabled for Python 3 due to upstream bugs: "
"https://github.com/spulec/moto/issues/548 and "
"https://github.com/gabrielfalcao/HTTPretty/issues/325",
)
@skipIf(HAS_BOTO is False, "The boto module must be installed.")
@skipIf(HAS_MOTO is False, "The moto module must be installed.")
@skipIf(
_has_required_boto() is False,
"The boto module must be greater than or equal to version {}".format(
required_boto_version
),
)
class BotoVpcRouteTableTestCase(BotoVpcStateTestCaseBase, BotoVpcResourceTestCaseMixin):
resource_type = "route_table"
backend_create = "RouteTableBackend.create_route_table"
backend_delete = "RouteTableBackend.delete_route_table"
@mock_ec2_deprecated
def test_present_with_subnets(self):
vpc = self._create_vpc(name="test")
subnet1 = self._create_subnet(
vpc_id=vpc.id, cidr_block="10.0.0.0/25", name="test1"
)
subnet2 = self._create_subnet(
vpc_id=vpc.id, cidr_block="10.0.0.128/25", name="test2"
)
route_table_present_result = self.salt_states["boto_vpc.route_table_present"](
name="test",
vpc_name="test",
subnet_names=["test1"],
subnet_ids=[subnet2.id],
)
associations = route_table_present_result["changes"]["new"][
"subnets_associations"
]
assoc_subnets = [x["subnet_id"] for x in associations]
self.assertEqual(set(assoc_subnets), {subnet1.id, subnet2.id})
route_table_present_result = self.salt_states["boto_vpc.route_table_present"](
name="test", vpc_name="test", subnet_ids=[subnet2.id]
)
changes = route_table_present_result["changes"]
old_subnets = [x["subnet_id"] for x in changes["old"]["subnets_associations"]]
self.assertEqual(set(assoc_subnets), set(old_subnets))
new_subnets = changes["new"]["subnets_associations"]
self.assertEqual(new_subnets[0]["subnet_id"], subnet2.id)
@mock_ec2_deprecated
def test_present_with_routes(self):
vpc = self._create_vpc(name="test")
igw = self._create_internet_gateway(name="test", vpc_id=vpc.id)
with patch.dict(botomod.__salt__, self.funcs):
route_table_present_result = self.salt_states[
"boto_vpc.route_table_present"
](
name="test",
vpc_name="test",
routes=[
{"destination_cidr_block": "0.0.0.0/0", "gateway_id": igw.id},
{"destination_cidr_block": "10.0.0.0/24", "gateway_id": "local"},
],
)
routes = [
x["gateway_id"]
for x in route_table_present_result["changes"]["new"]["routes"]
]
self.assertEqual(set(routes), {"local", igw.id})
route_table_present_result = self.salt_states["boto_vpc.route_table_present"](
name="test",
vpc_name="test",
routes=[{"destination_cidr_block": "10.0.0.0/24", "gateway_id": "local"}],
)
changes = route_table_present_result["changes"]
old_routes = [x["gateway_id"] for x in changes["old"]["routes"]]
self.assertEqual(set(routes), set(old_routes))
self.assertEqual(changes["new"]["routes"][0]["gateway_id"], "local")
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,808 |
Q: Lambda function is not recognizing LexBot I have created simple Lambda function (simple nodejs application) and trying to integrate with Lex bot. have followed the aws doc and created the execution role. configured the lex bot with proper alias and associated the version as well.
this is the doc being followed
when I am testing the lambda function execution it throws following error:
{
"errorType": "BadRequestException",
"errorMessage": "INVALID_REQUEST - Invalid bot name or alias",
"trace": [
"BadRequestException: INVALID_REQUEST - Invalid bot name or alias",
" at Object.extractError (/var/task/node_modules/aws-sdk/lib/protocol/json.js:52:27)",
" at Request.extractError (/var/task/node_modules/aws-sdk/lib/protocol/rest_json.js:55:8)",
" at Request.callListeners (/var/task/node_modules/aws-sdk/lib/sequential_executor.js:106:20)",
" at Request.emit (/var/task/node_modules/aws-sdk/lib/sequential_executor.js:78:10)",
" at Request.emit (/var/task/node_modules/aws-sdk/lib/request.js:688:14)",
" at Request.transition (/var/task/node_modules/aws-sdk/lib/request.js:22:10)",
" at AcceptorStateMachine.runTo (/var/task/node_modules/aws-sdk/lib/state_machine.js:14:12)",
" at /var/task/node_modules/aws-sdk/lib/state_machine.js:26:10",
" at Request.<anonymous> (/var/task/node_modules/aws-sdk/lib/request.js:38:9)",
" at Request.<anonymous> (/var/task/node_modules/aws-sdk/lib/request.js:690:12)"
]
}
bot name and alias is valid. i checked that. still it throws error.
how to debug this?
A: I am facing the same thing, aws lex-models get-bots returns an empty list although bots are present.
aws lexv2-runtime recognize-text --bot-id <botId> --bot-alias-id TSTALIASID --locale-id 'en_GB' --session-id 'test_sessio1n' --text 'my input test'
The above gives a proper response.
I am guessing this is an issue with Lex v1 vs v2.
A: In the Lex Bot make sure you create a separate alias from the default one that is present and try it. Also, ensure to provide the right alias ID and the bot ID in your code.
A: I encountered the same error when trying to use the @aws_sdk/client-lex-runtime-service client's PutText command (which uses the botName and botAlias) with a Lex V2 chatbot, which apparently isn't compatible. The fix was to use the newer @aws-sdk/client-lex-runtime-v2 client's RecognizeText command (which uses the botAliasId, botId, localeId and sessionId).
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,129 |
\section{Introduction}
We are interested in studying non-Newtonian phenomena in general;
in particular, we have studied the shapes of free surfaces
in complex fluids\cite{soto2006}.
\section{Experimental Conditions}
We use a selctive withdrawal device to produce cusps in the free surface of a non-Newtonian liquid. By varying the flow rate through the tube, a variety of shapes are observed. The liqui used in this study is an associative polymer (HASE 1.5\%).
\section{Videos}
Our video contributions can be found at:
\begin{itemize}
\item \href{http://hdl.handle.net/1813/14110}{Video 1, mpeg2, full
resolution}
\item \href{http://hdl.handle.net/1813/14110}{Video 2, mpeg1, low
resolution}
\end{itemize}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,686 |
PN's Voice 15, 14-10-2014
PN's Voice 15, 14.10.2014
S. Korea Open to Talking to N. Korea About Lifting Sanctions and Resuming Geumgang Tours
The South Korean Unification Minister Ryu Kihl-Jae spoke last week of the possibility for the two Koreas to discuss raising economic sanctions against the North, and the resumption of tours to Mount Geumgang during the scheduled senior-level inter-Korean dialogue. Ryu, who was speaking at a parliamentary audit of the unification ministry, said that "the May 24 measures and Mount Geumgang tours can be discussed if the high-level contact is held. All can be on the dialogue table." Ryu added that it would be "important for the two Koreas to overcome the issues through dialogue".
The May 24th sanctions have been in place since the Cheonan, a South Korean vessel, was sunk by, what was believed to be, a North Korean torpedo attack in 2010. Pyongyang has repeatedly denied responsibility for the incident and thus has demanded the lifting of these sanctions, and deliberately has kept inter-Korean exchanges to a bare minimum until the sanctions are lifted. While tours to Mount Geumgang, a source of considerable income for the North, have been on ice since a South Korean tourist was shot dead by a North Korean solider there in 2008.
Opposition lawmakers have been calling for the lifting of the sanctions to facilitate an expansion in inter-Korean exchanges that could lead to the North opening up; many experts believe such actions would be key step towards peace on the Korean Peninsula. Calls for the lifting of sanctions have become stronger after a top-level North Korean delegation visited South Korea for the closing ceremony of the Incheon Asian Games. This surprise visit has ignited hopes for a mood of reconciliation on the Korean peninsula.
However, Minister Ryu was keen to play down excessive expectations, pointing out that the recent North Korean visit wouldn't change Seoul's position or policy on inter-Korean issues, including the May 24th sanctions and Mount Geumgang tours.
NK Admits to Labor Camps
Last week marked the first time North Korea has publically admitted the existence of its labor camps. The admission appears to be a response to the UN's highly critical human rights report on North Korea that was released earlier this year.
Choi Myong-Nam, a North Korean foreign ministry official in charge of UN affairs and human rights, said at a briefing with reporters that North Korea had no prison camps and, in practice, "no prison, or things like that". However, he did say that, "we do have reform through labor detention camps — not, detention centers — where people are improved through their mentality and look on their wrongdoings." These so-called 're-education labor camps' are for petty offenders and some political prisoners, however most political prisoners are held in a harsher system of political prison camps.
A group of diplomats who were with Choi, told reporters that a "top North Korean official" had visited the headquarters of the EU and expressed interest in starting dialogue on human rights; talks are expected to take place sometime next year. Greg Scarlatoiu, executive director of the Washington-based Committee for Human Rights in North Korea, sums up the promise of this recent move by Pyongyang: "while the North Korean human rights record remains abysmal, it is very important that senior North Korean officials are now speaking about human rights."
N. Korean Gulags Estimated to be Twice the Size of Seoul
Sticking with the subject of North Korean labor camps: A South Korean lawmaker claimed last week that satellite imagery shows that the total area of North Korea's concentration camps for political prisoners amounts to an area double the size of Seoul. Rep. Yoon Sang-Hyun of the ruling Saenuri Party said "based on material from research institutes at home and abroad and the analysis of satellite imagery, the total area of North Korea's five prison camps was found to be 1,247.9 square kilometers…this amounts to twice the area of Seoul, which is 605.2 square kilometers."
North Korea is estimated to hold as many as 80,000-120,000 prisoners across its system of five concentration camps, according to a report published this year by Seoul-based Korea Institute for National Unification. The size of the infamously gruesome Yodok prison camp in the eastern South Hamgyong Province of the communist country is particularly noticeable as it occupies some 40 percent of the entire Yodok area at 551.6 square kilometers.
NK-SK Exchange Fire
Last week saw the two Koreas exchange fire in two separate incidents, despite the raised optimism for improved inter-Korean relations caused just days earlier by a top level North Korean delegation's visit to South Korea. First, on Tuesday North and South Korean warships exchanged warning shots after a North Korean ship briefly violated the disputed western sea boundary. A South Korean official stated that the North Korean ship in South Korean waters for about 10 minutes before retreating. He said that a South Korean navy ship first broadcast a warning, then fired warning shots; the North Korean ship replied with shots of its own, the South Korean ship fired again leading the North Korean ship to retreat. The South Korean official stated that there haven't been any reports of injuries or damage on either side.
Pyongyang has long disputed this boundary as it was drawn without the North's consent by the American-led UN command at the end of the Korean War. As the line cuts off North Korea from rich fishing waters, North Korean navy ships and fishing boats frequently violate the sea boundary and therefore exchanges around the West Seas boundary are not uncommon. Indeed this area has been the scene of several deadly maritime skirmishes between the Koreas in recent years.
However, the second skirmish of the week, which took place on Friday, was a much rarer occurrence as the two Korea's exchanged fire over their land border. The incident was sparked by South Korean activists' release of balloons filled with leaflets condemning North Korean leader Kim Jong-Un. Pyongyang had given pre-warning through harsh rhetoric that it would respond strongly if the balloon release went ahead.
South Korean media reported that the North fired to try and down the balloons, and South Korea responded as some of the shots landed on the Southern side of the border. As with the incident by the maritime border, there have been no reports of casualties.
North Korean leader Kim Jong-Un Returns to the Public Gaze
The North Korean state run Korean Central News Agency reported that leader Kim Jong-Un "gave field guidance" at a newly-built scientists' residential district yesterday, this was Kim's first public appearance in nearly 6 weeks. Kim's prolonged absence had been partially explained by a North Korean official publically admitting that the leader was unwell, although no specific details were given. Nonetheless, rumors suggesting that Kim was laying low amid a struggle for power or even that he had been toppled in a coup were plentiful.
North Korean newspaper Rodong Sinmun carried several images of Kim Jong-Un's visit. In these images it's noticeable that Kim is now using a walking stick which further compounds the belief that he has been struggling against ill health.
During Kim's lengthy public his absence, Kim missed two high-profile public events - the 10 October anniversary of the establishment of the Korean Worker's Party and the 9 September Foundation Day of the North Korean State. Experts say these are two days in the political calendar when the leader would be expected to make an appearance. Several experts have suggested that yesterday's public appearance was a response to the growing international media attention that Kim's absence had caused.
For more information about Peace Network, visit our website at: peacekorea.org/zbxe/?mid=Eng_main
Seoul, Mapo-gu, Mangwon-dong 423-2 (World Cup Gil 25, 55), 5th floor, Peace Network Tel: +82 2 733 3509
Mit freundlicher Erlaubnis von Peace Network.
2019: Renewing and Reframing Hiroshima
2014: Japan's Collective Self-Defense | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,889 |
Almost 200 new coronavirus cases reported in South Carolina, DHEC says
One Upstate death among the 9 reported in state
>> WE BEGIN WITH CORONAVIRUS HEADLINES. MORE THAN 5 MILLION PEOPLE WORLDWIDE INFECTED, ACCORDING TO JOHN'S AND BASED ON GOVERNMENT DATA. JOHN HOPKINS REPORTS MORE THAN 328,000 HAVE DIED, INCLUDING 93,000 PEOPLE IN THE UNITED STATES. THE TRANSPORTATION SECURITY ADMINISTRATION OUT NEW RULES. THEY INCLUDE SOCIAL DISTANCING, IMPLEMENTATION, AND FACE COVERING PRACTICES. THEY ALSO WANT PASSENGERS TO HOLD BOARDING PASSES INSTEAD OF GIVING THEM TO OFFICERS. MACY'S SAID IT EXPECTS TO LOSE $1.1 BILLION BETWEEN FEBRUARY AND MAY. THEY WERE FORCED TO CLOSE THEIR DOORS ON MARCH 18, CAUSING SALES TO PLUNGE AS MUCH AS 45%. >> WERE WAITING FOR THE DEPARTMENT OF HEALTH TO RELEASE TODAY'S CUT CASE NUMBERS. AS OF YESTERDAY, SOUTH CAROLINA HEALTH OFFICIALS REPORTED 407 DEATHS AND 9000 POSITIVE CASES.
(Above are the coronavirus headlines from WYFF News 4 at 5.)The South Carolina Department of Health and Environmental Control announced Thursday 199 new cases of COVID-19 and nine additional deaths.This brings the total number of people confirmed to have COVID-19 in South Carolina to 9,379 and those who have died to 416.Six of the nine deaths occurred in elderly individuals from Clarendon (1), Darlington (1), Horry (1), Kershaw (1), Lee (1) and Spartanburg (1) counties. Three of the nine deaths occurred in middle-aged individuals from Dillon (1), Florence (1), and Lee (1) counties.The number of new cases by county are listed below.Aiken (1), Allendale (1), Anderson (5), Bamberg (2), Beaufort (2), Berkeley (4), Charleston (7), Chesterfield (3), Clarendon (4), Colleton (2), Darlington (8), Dillon (3), Dorchester (2), Edgefield (1), Fairfield (13), Florence (13), Greenville (27), Greenwood (4), Horry (13), Kershaw (6), Lancaster (8), Lee (3), Lexington (5), Marion (1), Marlboro (8), Newberry (2), Orangeburg (2), Pickens (5), Richland (13), Saluda (5), Spartanburg (6), Sumter (7), Williamsburg (6), York (7)55 Mobile Testing Clinics Scheduled StatewideAs part of our ongoing efforts to increase testing in underserved and rural communities across the state, DHEC is working with community partners to set up mobile testing clinics that bring testing to these communities. Currently, there are 55 mobile testing events scheduled through June 26 with new testing events added regularly. Find a mobile testing clinic event near you at www.scdhec.gov/covid19testing.More than 140 Permanent Testing Sites Across the StateIn addition to the mobile testing events, there are currently 145 permanent testing locations at health care facilities throughout the state. Find a location near you – including address, hours of operation an additional details – at www.scdhec.gov/covid19testing.DHEC Trains National Guard EMTs to Conduct Nasal Specimen CollectionToday, staff from DHEC's Bureau of EMS conducted training for COVID-19 specimen collection at McEntire Joint National Guard Base in Richland County. All 25 of the military detachment's EMTs received training in collecting nasal and oral swabs from individuals for COVID-19 testing. Testing in South CarolinaAs of May 20, a total of 148,901 total tests by both DHEC's Public Health Laboratory and private labs have been conducted in the state. DHEC's Public Health Laboratory is operating extended hours and is testing specimens seven days a week. The Public Health Laboratory's current time frame for providing results to health care providers is 24-48 hours.Hospital Bed OccupancyAs of this morning, 3,142 inpatient hospital beds are available and 7,199 are in use, which is a 69.62% statewide hospital bed utilization rate. Of the 7,199 inpatient beds currently used, 438 are occupied by patients who have either tested positive or are under investigation for COVID-19.*As new information is provided to the department, some changes in cases may occur. Cases are reported based on the person's county of residence, as it is provided to the department. DHEC's COVID-19 map will adjust to reflect any reclassified cases.Additional coronavirus resources: Tracking COVID-19 curve of cases, deaths in the Carolinas, Georgia Latest update on coronavirus cases, latest headlines in Carolinas, Georgia COVID-19 maps of Carolinas, Georgia: Latest coronavirus cases by county Sign up for WYFF News 4 coronavirus daily newsletter
COLUMBIA, S.C. —
(Above are the coronavirus headlines from WYFF News 4 at 5.)
The South Carolina Department of Health and Environmental Control announced Thursday 199 new cases of COVID-19 and nine additional deaths.
This brings the total number of people confirmed to have COVID-19 in South Carolina to 9,379 and those who have died to 416.
Six of the nine deaths occurred in elderly individuals from Clarendon (1), Darlington (1), Horry (1), Kershaw (1), Lee (1) and Spartanburg (1) counties. Three of the nine deaths occurred in middle-aged individuals from Dillon (1), Florence (1), and Lee (1) counties.
The number of new cases by county are listed below.
Aiken (1), Allendale (1), Anderson (5), Bamberg (2), Beaufort (2), Berkeley (4), Charleston (7), Chesterfield (3), Clarendon (4), Colleton (2), Darlington (8), Dillon (3), Dorchester (2), Edgefield (1), Fairfield (13), Florence (13), Greenville (27), Greenwood (4), Horry (13), Kershaw (6), Lancaster (8), Lee (3), Lexington (5), Marion (1), Marlboro (8), Newberry (2), Orangeburg (2), Pickens (5), Richland (13), Saluda (5), Spartanburg (6), Sumter (7), Williamsburg (6), York (7)
55 Mobile Testing Clinics Scheduled Statewide
As part of our ongoing efforts to increase testing in underserved and rural communities across the state, DHEC is working with community partners to set up mobile testing clinics that bring testing to these communities. Currently, there are 55 mobile testing events scheduled through June 26 with new testing events added regularly. Find a mobile testing clinic event near you at www.scdhec.gov/covid19testing.
More than 140 Permanent Testing Sites Across the State
In addition to the mobile testing events, there are currently 145 permanent testing locations at health care facilities throughout the state. Find a location near you – including address, hours of operation an additional details – at www.scdhec.gov/covid19testing.
DHEC Trains National Guard EMTs to Conduct Nasal Specimen Collection
Today, staff from DHEC's Bureau of EMS conducted training for COVID-19 specimen collection at McEntire Joint National Guard Base in Richland County. All 25 of the military detachment's EMTs received training in collecting nasal and oral swabs from individuals for COVID-19 testing.
Testing in South Carolina
As of May 20, a total of 148,901 total tests by both DHEC's Public Health Laboratory and private labs have been conducted in the state. DHEC's Public Health Laboratory is operating extended hours and is testing specimens seven days a week. The Public Health Laboratory's current time frame for providing results to health care providers is 24-48 hours.
Hospital Bed Occupancy
As of this morning, 3,142 inpatient hospital beds are available and 7,199 are in use, which is a 69.62% statewide hospital bed utilization rate. Of the 7,199 inpatient beds currently used, 438 are occupied by patients who have either tested positive or are under investigation for COVID-19.
*As new information is provided to the department, some changes in cases may occur. Cases are reported based on the person's county of residence, as it is provided to the department. DHEC's COVID-19 map will adjust to reflect any reclassified cases.
Additional coronavirus resources:
Tracking COVID-19 curve of cases, deaths in the Carolinas, Georgia
Latest update on coronavirus cases, latest headlines in Carolinas, Georgia
COVID-19 maps of Carolinas, Georgia: Latest coronavirus cases by county
Sign up for WYFF News 4 coronavirus daily newsletter | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,808 |
\section{Introduction}
The puzzling nature of the hidden order (HO) phase of URu$_2$Si$_2$\ is still not understood. The central interest of this enigma is that it reflects the duality between the local and itinerant characters of the $5f$ electrons. These different facets often play a major role in the field of strongly correlated electronic systems. In spite of more than two decades of intense search\cite{Palstra:1985}, there is still no direct access to the order parameter (OP) as it occurs for the sublattice magnetization of the high pressure antiferromagnetic (AF) ground state with the wave-vector \textbf{Q$_{AF}$}=(0,0,1)\cite{Amitsuka:1999,Bourdarot:2005} which appears above $P_x\simeq0.5$ GPa via a first order transition switching from HO to AF phases\cite{Motoyama:2003,Amitsuka:2007,Hassinger:2008}. However, recently it was noticed that an unambiguous signature of the HO phase is the sharp resonance at $E_0\simeq1.8$ meV for the commensurate wave-vector \textbf{Q$_0$}=(1,0,0) (equivalent wave-vector to \textbf{Q$_{AF}$}) as this resonance mode collapses through $P_x$ while the other resonance at $E_1\simeq4.1$ meV for the incommensurate wave-vector \textbf{Q$_1$}=(1.4,0,0) persists through $P_x$\cite{villaume:2008}. Furthermore, above $P_x$, a magnetic field leads to the ''resurrection'' of the resonance at $E_0$, when the HO phase is restored \cite{aoki:2009}. As the strong inelastic signal at \textbf{Q$_0$} is replaced above $P_x$ by a large elastic signal, fingerprint of the AF ground state with \textbf{Q$_{AF}$}=(0,0,1), it was proposed that, in both HO and AF phases, a lattice doubling along the \textbf{c} axis occurs at the transition from paramagnetic (PM) to either HO or AF ground states\cite{aoki:2009}. The nice feedback is that the change in the class of tetragonal symmetry via development of a new Brillouin zone generates a drastic decrease in the carrier number as pointed out by a large number of theories\cite{Yamagami:2000,Harima:2009,Elgazzar:2009} and experiments\cite{Hassinger:2008a}. Furthermore, as the HO-AF line touches the PM-HO and PM-AF lines at a critical pressure $P_c\simeq 1.4$ GPa and critical temperature $T_c\simeq19.5$ K, a supplementary symmetry breaking must occur between the HO and AF phases. Two recent theoretical proposals for the OP of the HO phase were a hexadecapole - from Dynamical Mean Field Theory (DMFT) calculations\cite{Haule:2009}; or a $O_{xy}$ type antiferroquadrupole - from group theory analysis \cite{Harima::2010}. For both models, the pressure-switch from HO to AF will add a supplementary time reversal breaking in the AF phase. For these models as well as for recent band structure calculations \cite{Elgazzar:2009}, partial gapping of the Fermi surface may happen at $T_0=17.8$ K with a characteristic gap $\Delta_G$. It was even proposed in the last model, that in the HO phase the gapping is produced by a spontaneous symmetry breaking occurring through collective AF moment excitations.
This article presents a careful revisit of the inelastic neutron response at \textbf{Q$_0$} \cite{Broholm:1991,Mason:1995,Wiebe:2007} using recent progress in polarized inelastic neutron configuration of the spectrometer IN22 and in the performance of the cold-neutron three-axis spectrometer IN12, both installed at the high flux reactor of the Institute Laue Langevin (ILL). Thus, this new generation of experiments provide a careful basis on the temperature and pressure evolution of $E_0$. They give a new insight on previous neutron studies thanks to our recent proof\cite{villaume:2008} that the resonance at $E_0$ is up to now the major signature of the HO phase. Comparison will be made with the case of charge ordering observed in the skutterudite PrRu$_4$P$_{12}$ as well as with PrFe$_4$P$_{12}$ where a sequence of HO-AF phases are observed \cite{JPSJ77SA:2008}.
\section{Experimental set up}
High quality single crystals of URu$_2$Si$_2$\ were grown by the Czochralski method in tetra-arc furnace. The details are described elsewhere \cite{aoki:2010}. The sample already used for inelastic neutron scattering in the superconducting state \cite{Hassinger:2010}, was installed in an ILL-type orange cryostat with $\bf{c}$ axis oriented vertically on the thermal triple-axis IN22 in its polarized configuration and on the cold triple-axis IN12 in its standard configuration (both CEA-CRG spectrometer at ILL).
The IN22 experiment was performed at 1.5 K, with two fixed final energies : 14.7 meV ($k_f$=2.662 \AA$^{-1}$) and 30.3 meV ($k_f$=3.84 \AA$^{-1}$). The beam was polarized by a Heusler monochromator vertically focusing and analyzed in energy and polarization by a Heusler analyzer vertically (fixed) and horizontally focusing. The flipping ratio was around 17 and the energy resolutions were 0.95 meV and 2.4 meV for both energies, respectively. No collimation was installed. The background was optimized by an optical calculation of the dimension openings of the slits placed before and after the sample. The measurements were performed in the non-spin-flip channel which provides all the necessary information. The inelastic scans were performed with $\bf{Q}$ parallel to the $\bf{a}$ axis. When the polarization is along the $\bf{a}$ axis, the intensity measured (I$^a_{NSF}$) corresponds to the nuclear and background contributions; when the polarization is along the $\bf{b}$ or $\bf{c}$ axis, the intensities measured (I$^b_{NSF}$) and (I$^c_{NSF}$) correspond to the same contributions plus the (imaginary part of the) susceptibility along the $\bf{b}$ or $\bf{c}$ directions. In the following, the data shown are the subtraction of inelastic scans performed in the non-spin-flip channel with polarization along the $\bf{b}$ or $\bf{c}$ axis, with an inelastic scan performed in the same conditions with polarization along the $\bf{a}$ axis. The subtractions correspond to the imaginary part of the dynamical susceptibilities with for the $\bf{b}$ axis $\chi''_y\propto (I^b_{NSF}-I^a_{NSF})$ the transverse susceptibility and for the $\bf{c}$ axis $\chi''_z\propto (I^c_{NSF}-I^a_{NSF})$ the longitudinal susceptibility.
The IN12 experiment was performed with a fixed final energy $E_f=4.7$ meV ($k_f$=1.5 \AA$^{-1}$) that gives a good compromise between intensity of the excitation and energy resolution $l=0.11$ meV. The incident neutrons were selected by a (0,0,2) graphite vertically focusing monochromator with vertically focusing and analyzed in energy by a (0,0,2) graphite analyzer with horizontal focusing. No collimation was installed. The temperature was measured by a calibrated carbon thermometer and it was checked before each scan that the temperature was stable. As for IN22, the background was optimized by an optical calculation of the dimension openings of the slits placed before and after the sample. The raw scans were corrected for the electronic background (29 counts per hour) and for the $\lambda/2$ contamination of the monitor.
\section{Description of the model used for fitting the inelastic spectrum}
In a neutron scattering experiment, the neutron intensity $I(q,\omega)$ in the detector is proportional to the convolution of the scattering function $S(q,\omega)$ with the instrumental resolution function. $S(q,\omega)$ is related to the imaginary part of the dynamical spin susceptibility $\chi''(q,\omega)$ ($\chi(q,\omega)= \chi'(q,\omega)+\imath\chi''(q,\omega)$) via the fluctuation-dissipation theorem : $S(q,\omega)=n(\omega,T)\chi''(q,\omega)$ where $n(\omega,T)= 1/(1-e^{-\hbar\omega/{k_BT}})$ is the detailed balance factor.
Below $T_{0}$, the excitation is well-defined with an asymmetrical shape related to the finite extent of the resolution function that integrates the dispersion of the excitation in the vicinity of the nominal $q$ wave-vector at which the spectrometer is set-up. To analyze the data, firstly a harmonic oscillator function is taken for $\chi"(q,\omega$) ; this corresponds to the difference of two normalized Lorentzian functions multiplied by $\frac{\chi_o\,\Omega^2_0}{\omega_0\gamma}$, where $\chi_0=\chi(q,\omega=0)$ is the static susceptibility, $\Omega_0$ is the oscillator frequency, $\gamma$ its damping, and $\omega_0$ is given by the equation $\Omega_0=\sqrt{\omega^2_0+(\gamma/2)^2}$. Secondly, a simplified convolution with the resolution function is made : to this aim the resolution function is approximated by a 4D parallelepiped (instead of an ellipsoid) and the $q$ dispersion is taken as linear in all directions. This linear dispersion simplified the calculation and gives a better description of the dispersion than a usual quadratic law in the case of URu$_2$Si$_2$. This description is called the \bm{$\gamma$} \textbf{model} when $\gamma \gg l$ and the \bm{$l$} \textbf{model} when $l \gg \gamma$, $l$ being the energy resolution. It leads to a simple analytic expression for $I(q,\omega)$ at the minimums of the dispersion ($q_0$):
\begin{equation}
I(q_0,\omega)=n(\omega,T)\left(F(q_0,\omega,)-F(q_0,-\omega)\right)\\
\label{m3}
\end{equation}
with
\begin{eqnarray}
\begin{split}
F(q_0,\omega)&=L\,K(q_0,\omega)\left(\sum_{i=x,y,z}\sqrt{1+(2\alpha_i\,l/\omega_0(q_0))^2}\right.\nonumber\\
&\left.e^{-\left(4\ln{2}\left(\alpha_i\frac{\omega-\omega_0(q_0)}{\omega_0(q_0)}\right)^2\right)} \right)\\
K(q_0,\omega)&=\frac{1}{2}\left(1+\frac{2}{\pi}arctan\left(\frac{2\,(\omega-\omega_0(q_0))}{\beta}\right)\right)\nonumber\\
\end{split}
\end{eqnarray}
\\
\noindent
where $\beta=\gamma(q_0)$ for the \bm{$\gamma$} \textbf{model} and $\beta=l$ for the \bm{$l$} \textbf{model}
$\alpha_i=\alpha_{i0}(\omega_0/\Omega_0)^2$, where $\alpha_{i0}$ is the ratio between the slope of the dispersion and the $q$-width of the resolution function. The L parameter, which depends of the magnetic form factor, the incident and final energies ($k_i$ and $k_f$) is supposed to stay constant at first approximation.
Finally the susceptibility $\chi_{0}$ for $T<T_0$ is determined using the integrated intensity by applying the formula:
\begin{equation}
\begin{split}
I_{\Omega_0}=\int^\infty_0\chi''(q_0,\omega)d\omega=\\
\frac{\chi_0 \Omega^2_0(q_0)}{\omega_0(q_0)}\arctan\left(\frac{\omega_0(q_0)}{\gamma(q_0)/2}\right)
\end{split}
\label{m4}
\end{equation}
Above $T_{0}$, the signal is much broader than the resolution and no convolution is needed (at least on IN12). For $\chi"(q,\omega)$, we use the magnetic quasi-elastic model, which corresponds to a Lorentzian function of susceptibility $\chi_{L0}$ and the full-width at half-maximum $\Gamma_L$, multiplied by $\omega$:
\begin{equation}
\chi''(q_0,\omega)=\frac{\chi_{L0}\,\omega\,\Gamma_L}{\omega^2+\Gamma^2_L}
\label{lor}
\end{equation}
Between \bm{$\gamma$} \textbf{model} and the quasi-elastic model, the widths and the susceptibilities are linked by the relations: $\Gamma_L=\gamma/2$ and $\chi_0=\chi_{L0}$.
\section{Results}
\subsection{Inelastic Polarized neutron scattering}
At first this experiment was to unambiguously determine the polarization of the resonances at \textbf{Q$_0$} = (1,0,0) and \textbf{Q$_1$} = (1.4,0,0) and to determine the nature of the signal occurring at much higher energy than the resonant modes. The origin of such a signal is often referred to as coming from multi-phonons. However, it may come from magnetic process as was hinted by C. Broholm \cite{Broholm:1991}.
Figure \ref{fig5} shows the longitudinal and transverse magnetic response at \textbf{Q$_0$} measured on IN22 with a final energy $E_f=14.7$ meV. No transverse magnetic response is detected for a range of energy transfer going from -3 meV to 27 meV. The longitudinal magnetic response shows two contributions: the well-known and well-defined resonance with a gap value $E_0$ fitted to 1.86(5) meV, where $E_0$ is the harmonic oscillator energy $\Omega_{0}$ (the gap value $E_0$ measured with the cold triple-axis spectrometer IN12 is around 1.70(5) meV) and a broad magnetic contribution. This broad magnetic contribution, which looks like to a magnetic continuum persists at least up to 27 meV as seen in the inset of Figure \ref{fig5} for measurements performed with a final energy of 30.3 meV (27\ meV was the maximum energy transfer we could reach in this configuration). In this paper the intensity of this continuum is described by a Lorentzian function (eq.\ref{lor}) of width $\Gamma_{c}$. An extra elastic signal (at $\omega=0$) corresponding to the small antiferromagnetic moment is detected. This well known signal is currently believed to be a parasitic contribution due to the survival of AF droplets generated near defects \cite{Matsuda:2001,Amato:2004}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=80mm]{F5.eps}
\end{center}
\caption{(color online) Transverse (open circles) and longitudinal (filled circles) magnetic response of URu$_2$Si$_2$\ at \textbf{Q$_0$} and $T=1.5$ K with a final energy $E_f=14.7$ meV. The full and dashed curves correspond to a \bm{$l$}\textbf{-model} of eq.(\ref{m3}) and quasi-elastic function for the magnetic continuum, respectively. The vertical black arrow indicates the gap position of the resonance (1.86(5) meV) for the \bm{$l$}\textbf{-model}. The inset shows the same magnetic response but with a final energy $E_f=30.3$ meV.}
\label{fig5}
\end{figure}
Figure \ref{fig6} shows the longitudinal and transverse magnetic response at \textbf{Q$_1$} measured with a final energy $E_f=14.7$ meV. As previously at \textbf{Q$_0$}, no transverse magnetic response is detected in the range of 0 meV to 27 meV. Again the longitudinal response shows two inelastic magnetic contributions: the well-known and well-defined resonance with a gap $E_1$ of 4.06(6) meV, and the broad continuum of linewidth $\Gamma_c$. Note that the gap $E_1=4.06(6)$ meV is slightly lower than the usual gap value $E_1\simeq4.5$ meV found in previous experiment \cite{Bourdarot:2003}. Let us point out that it is the first time that the resolution and the dispersion are taken into account to analyse this excitation.
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{F6.eps}
\end{center}
\caption{Transverse (open circles) and longitudinal (filled circles) response of URu$_2$Si$_2$\ at \textbf{Q$_1$} and $T=1.5$ K. The full and dashed curves correspond to a \bm{$l$}\textbf{-model} of eq.(\ref{m3}) and quasi-elastic function for the magnetic continuum, respectively. The vertical black arrow indicates the gap position of the resonance (4.06(6) meV) for the \bm{$l$}\textbf{-model}.}
\label{fig6}
\end{figure}
Figure \ref{fig8} shows the longitudinal and transverse magnetic response for a \textbf{Q}-scan performed with an energy transfer of 15 meV. A constant signal corresponding to the continuum is measured from \textbf{Q}=(1,0,0) to \textbf{Q}=(1.7,0,0), then the magnetic signal decreases approaching to nuclear zone center \textbf{Q}=(2,0,0). The vanishing of the continuum has to be verified by new \textbf{Q}-scans at different energy transfer in the futur.
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{F8.eps}
\end{center}
\caption{Transverse (open circles) and longitudinal (filled circles) response of URu$_2$Si$_2$\ at $T=1.5$ K for a \textbf{Q}-scan (Q$_x$,0,0) at 15 meV. The hatched area corresponds to the magnetic continuum.}
\label{fig8}
\end{figure}
To summarize, the inelastic polarized neutron scattering experiments performed at \textbf{Q$_0$} and \textbf{Q$_1$} confirm without any ambiguity that, at low temperature (below $T_0=17.8$ K), the magnetic response is exclusively longitudinal. We also evidence a broad magnetic continuum that may be fitted at least for the two main \textbf{Q} positions (\textbf{Q$_0$} and \textbf{Q$_1$}) by exactly the same quasi-elastic function with a half-width $\Gamma_c=7.8$ meV. This magnetic contribution was never taken into account in the previous studies of temperature dependence of the gaps at \textbf{Q$_0$} and at \textbf{Q$_1$}. This motivates us to reinvestigate the temperature dependence of the excitation at \textbf{Q$_0$}.
\subsection{Low energy study at \textbf{Q$_0$}}
The aim of this study is a precise determination of the temperature dependence of the magnetic resonance at the position \textbf{Q$_0$}. This experiment was already performed by T.E. Mason \cite{Mason:1995b} but without a large precision and more important without taking into account the magnetic continuum described by a quasi-elastic function and revealed by our recent inelastic polarized neutron scattering. From their study they concluded that the gap follows a singlet ground state model but moreover that the evolution of the susceptibility $\chi(\mathbf{Q}_0,\omega=0)$ shows a large enhancement at $T_0$. We will discuss these points below.
Figures \ref{fig11} and \ref{fig12} show some representative inelastic spectrums obtained at \textbf{Q$_0$} just below and above the transition temperature $T_0=17.8$ K respectively. At $T$=1.5 K, the signal measured at energies above 5-6 meV is derived from the continuum previously detected by inelastic polarized neutron scattering. As the half-width $\Gamma_c=7.8$ meV of the quasi-elastic function which fit the continuum is already known at low temperature, and assuming that for temperatures lower than $\Gamma_c/k_B \approx 88$ K, this width does not change, only the amplitude can depend on temperature. A good fit is obtained by taking a constant amplitude for all temperatures: the continuum being, at $T=27.1$ K, the unique contribution (as seen in Fig. \ref{fig12}).
For $T<T_0$ (Fig. \ref{fig11}), in parallel to the continuum, the well-defined resonance is detected as for $T=1.5$ K. The inelastic spectrums show clearly that the width $\gamma_0$ and the gap $E_0$ change substantially with temperature only close to $T_0$. The gap $E_0$ and the width $\gamma_0$ determined using the \bm{$\gamma$} \textbf{model} are shown in Fig. \ref{fig10} and \ref{fig14}. As expected from the spectrum, the gap $E_0$ decreases only close to $T_0$ but more abruptly as the width $\gamma_0$ increases. At $T_0$, $E_0\approx\gamma_0/2$, that confirms that we enter into an over-damped regime for temperatures larger than $T_0$.
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{F11.eps}
\end{center}
\caption{Energy scans measured at \textbf{Q$_0$} for temperatures below $T_0=17.8$ K. Fits are performed with the \bm{$\gamma$} \textbf{model} plus the continuum (quasi-elastic with $\Gamma_c=7.8$ meV).}
\label{fig11}
\end{figure}
For $T>T_0$ (Fig. \ref{fig12}), the magnetic response is over-damped, and the spectrum is treated with a quasi-elastic magnetic function of width $\Gamma_L$ added to the continuum ($\Gamma_c=7.8$ meV). As expected, $\Gamma_L=\gamma_0/2$ at $T_0$, which validates the \bm{$\gamma$} \textbf{model} and indicates that the gap $E_0$ drops to zero in the paramagnetic state (above $T_0$). The half-width of this excitation is plotted versus temperature in Figure \ref{fig14} (open circles). $\Gamma_L$ increases rapidly (maybe linearly) with temperature. It was not possible to follow this signal for temperatures approaching $T=27.5$ K: above this temperature only the large tail of the continuum can be detected.
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{F12.eps}
\end{center}
\caption{Energy scans measured at \textbf{Q$_0$} for temperatures just around the transition temperature $T_0=17.8$ K (and at 1.5 K for reference). Fits are described in the text. At $T=27. 1$ K (triangles), the scan is only fit by the continuum (quasi-elastic signal with $\Gamma_c=7.8$ meV).}
\label{fig12}
\end{figure}
As the temperature evolution of the signal is not usual, the Figure \ref{fig20} gives the variation of $\chi''(\mathbf{Q}_0,\omega)/\omega$ as a function of $\omega$ at different temperatures. Thus, this plot shows that this signal saturates at low temperature and the abrupt drop of E$_0$ when approaching $T_0$. Clearly the integration of $\chi''(\mathbf{Q}_0,\omega)/\omega$ increases on cooling below $T_0$ as discussed latter as a consequence of Fermi Surface reconstruction.
Figure \ref{fig15} shows the magnetic susceptibility at $Q_0$, $\chi(\mathbf{Q}_0,\omega=0)$. The susceptibility for $T<T_0$ is determined using eq. (\ref{m4}) (filled circles), then $\chi_{L0}$ is a fitted parameter of the quasi-elastic expression for $T>T_0$ (open circles). Of course, in addition, the susceptibility coming from the magnetic continuum (with $\Gamma_c=7.8$ meV) has to be added (constant low contribution in Fig \ref{fig15}). Without any scaling-factor, the susceptibilities from the \bm{$\gamma$} \textbf{model} and the second magnetic quasi-elastic contribution ($\Gamma_L$) are equal at $T_0$. The total susceptibility versus temperature shows a saturation at low temperature, then decreases but with a bump around $T_0$. The susceptibility stays almost constant above $T = 30$ K. In contrast to Mason's analysis no marked divergence of $\chi(\mathbf{Q}_0,\omega=0)$ is observed at $T_0$. Furthermore, in our data below 16 K, we found that $\chi(\mathbf{Q}_0,\omega=0)$ increases on cooling before saturating at low temperature.
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{F10.eps}
\end{center}
\caption{Temperature dependence of the gap $E_0$. The curves are guide for the eyes.}
\label{fig10}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{F14.eps}
\end{center}
\caption{Temperature dependence of the half-width $\gamma_0$ of the resonance at \textbf{$Q_0$} below $T_0$ (filled circle) and of the quasi-elastic $\Gamma_L$ above $T_0$ (open circle). The curve below $T_0$ corresponds to a fit with a Korringa model.}
\label{fig14}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{F20.eps}
\end{center}
\caption{Imaginary part of the dynamical spin susceptibility without the magnetic continuum divided by energy $(\chi''(\mathbf{Q}_0,\omega)/\omega$) for different temperatures. The curves are deduced from the $\gamma$ model corrected by the thermal factor and divided by $\omega$. The integration of these curves gives $\chi(\textbf{Q$_0$},\omega=0)$.}
\label{fig20}
\end{figure}
The integration $\int^{6.3 meV}_0\chi''(\mathbf{Q}_0,\omega)d\omega$ ($\simeq I_{E_0}$) of the magnetic resonance at \textbf{$Q_0$} without the continuum contribution ($\Gamma_c$) is plotted in Fig. \ref{fig13}. The filled circles correspond to the contribution with the oscillator model and the open circles correspond to the integration of the quasi-elastic contribution of width $\Gamma_L$. This integration, below $T_0$, seems to mimic the temperature variation of an OP vanishing at $T_0$, while above $T_0$ it behaves like the contribution of a critical regime. Furthermore, the temperature variation of $I_{E_0}(T)$ is well described below $T_0$ by BCS formula used for the temperature variation of the superconducting gap \cite{Bardeen:1957}.
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{F15.eps}
\end{center}
\caption{Temperature dependence of the static susceptibility $\chi(\textbf{Q}_0,\omega=0)$. In filled circles are data coming from the \bm{$\gamma$} \textbf{model}, the open circles from the quasi-elastic model $\Gamma_L$, and the hatched area coming from the magnetic continuum ($\Gamma_c=7.8$ meV).}
\label{fig15}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{F13.eps}
\end{center}
\caption{Temperature dependence of the integrated imaginary part of the dynamical spin susceptibility. In filled circles are data coming from the \bm{$\gamma$} \textbf{model}, the open circles from the quasi-elastic model. The intensity coming from the magnetic continuum ($\Gamma_c=7.8$ meV) was not taken into account. The line is the BCS-type gap below $T_0$, and a guide for the eyes above $T_0$.}
\label{fig13}
\end{figure}
\section {Discussion}
The occurrence of a magnetic continuum ($\Gamma_c = 7.8$ meV) for energies $\omega$ far higher than the energy $k_BT_0$ of the hidden order phase is a general phenomenon: large energy transfer leads to excite energy states not only from the ground state.
As the study is focused on the hidden order phase, we have concentrated our studies on the frequency response at \textbf{Q$_0$}. Below $T_0$, the resonance at $E_0$ represents a collective mode with a half-width $\gamma_0/2$ much smaller than the gap energy $E_0$: the collective mode is long lifetime as shown by the large ratio $\frac{E_0}{\gamma_0/2}\approx35$. As shown on Fig. \ref{fig10}, in decreasing temperature $E_0$ very rapidly reaches its final value $\approx1.7$ meV. On crossing $T_0$, the resonance collapses and the magnetic response appears quasi-elastic with a half-width $\Gamma_L$ becoming rapidly larger than the resonance energy $E_0$ at $T = 0$ K.
The susceptibility $\chi($\textbf{Q$_0$},$\,\omega$=0) reported in Fig. \ref{fig15} is calculated for $T<T_0$ using eq. (\ref{m4}). Due to the weakness of the resonance signal close to $T_0$ the uncertainty in the determination of $\chi($\textbf{Q$_0$},$\,\omega$=0) is large. However it is obvious that no divergence of $\chi($\textbf{Q$_0$},$\,\omega$=0) occurs at $T_0$ as it is observed at the onset of an antiferromagnetic ordering. In heavy fermion compounds close to AF-PM critical point, as in Ce$_{1-x}$La$_x$Ru$_2$Si$_2$ series, a sharp maximum of $\chi(\textbf{Q$_0$},\omega=0$) occurs at $T_N$ (for x$<$0.075) which is accompanied on cooling , below $T_N$ by a decrease of $\chi(\textbf{Q$_0$},\omega=0)$ \cite{Knafo:2009}. Here on cooling below $T_0$, by contrast, $\chi(\textbf{Q$_0$},\omega=0)$ increases at it is observed for critical concentration x$_c=0.075$ in Ce$_{0.925}$La$_{0.075}$Ru$_2$Si$_2$. However, the major difference is that for URu$_2$Si$_2$\ the saturation regime is achieved very steeply in temperature as the key energy is the gap energy $\Delta_G$ (see Figure \ref{fig10}). Also a drastic difference between URu$_2$Si$_2$\ and CeRu$_2$Si$_2$ cases is that the resonance at E$_0$ collapses suddenly above P$_x$ at the benefit of the establishment of a large sublattice magnetization. The occurrence of a weak maximum of $\chi(\textbf{Q$_0$},\omega=0)$ at $T_0$ may be consequence of the Fermi surface reconstruction with the particularity that the U ions through $T_0$ will go from an intermediate valence behavior above $T_0$ to a quasi-tetravalent dressing below $T_0$ \cite{Hassinger:2008a}. This image, based on thermodynamical considerations, is supported by the recent tunneling microscope experiments which appear during the revision of our paper \cite{aynajan:2010,schmidt:2010}. Thus this observation confirms the lack of antiferromagnetism at \textbf{Q$_0$} in the hidden order phase. Thanks to our previous measurements\cite{villaume:2008}, we have specified the emergence of the resonance at \textbf{Q$_0$}=(1,0,0) as the signature of the hidden order (see also \cite{Ohkawa:1999,Miyake:2010,Miyake:2010b}), we interpret the temperature dependence $I_{E_0}(T)$ plotted in Fig. \ref{fig13} as the temperature dependence of the OP\cite{villaume:2008}.
It was suggested by P.M. Oppeneer according the model developed in reference \cite{Elgazzar:2009,Oppeneer:2009} that $I_{E_0}(T)$ may be related with the hidden order parameter via an even function of the magnetization amplitude which will not vanish in time even in the hidden order phase as the sublattice magnetization amplitude for an AF.
As the quasi-elastic contribution with the characteristic energy half-width $\Gamma_L$ above $T_0$ seems to vanish above $T\approx30$ K, a simple picture would be that only the continuum with $\Gamma_c=7.8$ meV persists above $T=30$ K. According to the Kondo impurity model which predicts \cite{1}
$\Gamma\gamma=750$ mJ mol$^{-1}$K$^{-2}$, the linear term of the specific heat of $\gamma$ can be approximated to $\approx 96$ mJ mol$^{-1}$K$^{-2}$ with the continuum $\Gamma_c$. This agrees with the observed magnitude of the linear term of the specific heat \cite{vanDijk:1997b}. Of course an open problem is the modification of this continuum at low energy when crossing $T_0$ since transport measurements, as well as NMR and specific heat measurements, indicate clearly a Fermi surface reconstruction with a carrier drop by a factor 3 to 10 \cite{Schoenes:1987,Kasahara:2007,Behnia:2005,Bel:2004,Kohara:1986,Kohara:1987,Maple:1986}, which favors localized magnetism.
A key point is the rapid temperature evolution of the energy gap $E_0$ which is much faster than the temperature evolution of the BCS-type gap. The emerging image is that, at $T_0$, the change in lattice symmetry associated with the paramagnetic to hidden order phase transition (from $body-centered\ tetragonal$ to $simple\ tetragonal$) induced presumably by a multipolar ordering leads to large gapping of the Fermi surface with characteristic gap energy $\Delta_G$ much larger than $E_0$. The origin of $E_0$ may come from crystal-field splitting with dispersion coming from dipolar and quadrupolar interactions (see for example \cite{JPSJ77SA:2008,Kuramoto:2009,Balatsky:2009}). Opening a gap $\Delta_G$ at $T_0$ leads to a drop of the carrier number which allows the observation and development of the resonance: the resonance at \textbf{Q$_0$} is over-damped far above $T_0$ become a well-defined excitation below $T_0$. Of course the change of carrier number acts as well on the over-damped mode observed above $T_0$ for \textbf{Q}$_1$=(1.4,0,0), allowing again the appearance of a sharp resonance below $T_0$ at $E_1\approx4$ meV \cite{Broholm:1991,Wiebe:2007}.
Since $E_1$ is quite comparable to $\Gamma_c$ below $T_0$, this explains why the inelastic contribution at \textbf{Q$_1$} is still observed in the over-damped regime, above $T_0$. It was suggested that the hidden order phase may be an incommensurate antiferromagnet at \textbf{Q}$_1$ \cite{Balatsky:2009}. However, inelastic magnetic response at \textbf{Q}$_1$ does not give evidence of a crossing through a phase transition: no divergence of the static susceptibility $\chi(\mathbf{Q}_1,\omega=0)$ has been reported \cite{Broholm:1991}. Even the shallow maxima of $\chi(\mathbf{Q}_1,\omega=0)$ at $T_0$ may be an artifact of the fitting, or it can also be, as discussed for $\chi(\textbf{Q$_0$},\omega=0)$ a consequence of the Fermi Surface reconstruction.
As recently proposed in two different approaches, DMFT \cite{Haule:2009} and group theory analysis \cite{Harima::2010}, the promising explanation is that the hidden order phase would be a multipolar phase: a hexadecapolar order in DMFT studies, a quadrupolar order in group theory. In this last scenario, the hidden order phase is still hidden as it corresponds to a second order phase transition from the space group $I4/mmm$ (No.139) to the space group $P4_2/mnm$ (No.136) with no lattice distortion and invariance of the Ru-site at the crossover transition from hidden order to antiferromagnetic phase. Switching from the hidden order to the antiferromagnetic phase will preserve the $P4_2/mnm$ (No.136) symmetry of the lattice but will add of course the time-reversal symmetry operator. Let us emphazise that even if the resonance at \textbf{Q}$_0$=(1,0,0) is not a direct proof of the hidden order parameter, there is no doubt that it is a key signature which supports strongly the change from $I4/mmm$ (No.139) to $P4_2/mnm$ (No.136) symmetry at the paramagnetic-hidden order border. If no quadrupolar signature will be detected, a possibility is that the HO phase of URu$_2$Si$_2$\ may be regarded as an electronic spin Peierls transition with only tiny displacement of the atoms ($\delta d/d <10^{-6}$) \cite{fomin}.
Furthermore the collection of previous data with fine pressure tuning \cite{Bourdarot:2004b,Bourdarot:2005,villaume:2008} and of recent data at a fixed pressure $P$ between $P_x$ and $P_c$ with supplementary magnetic field $H$ scans\cite{aoki:2009}, allow to extract the $P$ dependence of $E_0$ and $E_1$ and to compare with the $P$ dependence of the gap $\Delta_G$ derived from resistivity measurements (figure \ref{fig16}) \cite{Hassinger:2010,Hassinger:2010b,Jeffries:2007}. This gap $\Delta_G$ is directly related to the partial gap opening at the Fermi surface which occurs at $T_0$.
Using a simple formula $\rho = \rho_0+A\,T^2 + B\,e^{-\Delta_G/T}$ and not the currently used $\rho = \rho_0+A\,T^2 + B\,T/\Delta_G\,(1+2\,T/\Delta_G)\,e^{-\Delta_G/T}$, which is suitable only if the scattering process is due to spin waves, $\Delta_G$ is quite close to $E_1$ by comparison to $E_0$ and comparable to $\Gamma_c$. In the vicinity of $T_0$, as pointed out for example by the entropy drop, the contribution of the $E_1$ resonance may play a major role. However on cooling, the \textbf{Q}$_1$ role is defeated by the \textbf{Q}$_0$ wave-vector response which occurs at lower energy than the $E_1$ one.
Under pressure, as shown in Fig.\ref{fig16}, $E_0(P)$ decreases from 1.8 meV at $P=0$ to 0.8 meV just below $P_x$ and collapses above $P_x$ as the response at \textbf{Q}$_0$ is dominated by the onset of a large static sublattice magnetization. On the contrary, for the wave-vector \textbf{Q}$_1$=(1.4,0,0), the resonance at $E_1$ persists through $P_x$; $E_1$ increases smoothly under pressure and exhibits a jump at $P_x$. As for the gap $E_0$, the resonance at \textbf{Q}$_1$ becomes sharp below $T_0$, where the number of carriers drops, but broadens, with a width comparable to $\Gamma_c$ in the over-damped regime. The thermal dependence of the width $\gamma_0$ can be fitted using the Korringa model \cite{Iwasa:2005}: $\gamma_0 \sim \gamma_{0,T=0K} + a\ n(T)^2\ k_BT$ where $n(T)=n_0\,e^{-\Delta(T)/k_BT}$ is the number of carriers which reduced when the gap $\Delta(T)$ opens at the Fermi surface. A gap (temperature independent) $\Delta=7.7$ meV is found below $T_0$ surprisingly very close to our derivation of $\Delta_G$ (70 K) or to the gap value deduced from specific heat measurements ($\Delta_{cp}=73$ K)\cite{vanDijk:1997}.
\begin{figure}
\begin{center}
\includegraphics[width=72mm]{F16.eps}
\end{center}
\caption{Behavior of the resonance energies $E_0$ and $E_1$ versus pressure, above and below $P_x\simeq5$ kbar and of the the gap $\Delta_G$ reflecting the gap opening below $T_0$.}
\label{fig16}
\end{figure}
Thus our data give sound basis for further theoretical developments. They confirm the dual character of the phase transition with $\Delta_G$ directly linked to the itinerant nature of the $5f$ electrons and $E_0$ and $E_1$ collective modes associated to the local character of the $5f$ electrons.
The description of quantities such as $I_{E_0}$ and presumably $I_{E_1}$ as well as the temperature variation of the specific heat below $T_0$ \cite{Balatsky:2009}, by BCS-type formula, may reflect the feedback between the local and itinerant properties. Of course, the possibility that the strong resonance ($E_1$) at \textbf{Q}$_1$ is a mark of an incommensurate spin density wave cannot completely be ruled out. Our support for the choice of \textbf{Q}$_{AF}$ as the wave-vector of the HO phase are; its occurrence only in the HO phase \cite{villaume:2008}, the quasi-invariance of the frequencies detected in the de Haas-van Alphen effect\cite{Nakashima:2003} and Shubnikov-de Haas effect through $P_x$ \cite{Hassinger:2010b} that indicate no change of the wave-vector between HO and AF phases, the lower value of $E_0$ by comparison to $E_1$ with furthermore a field convergence of $E_0$ towards $E_1$ in high magnetic field when the intersite dipolar and quadrupolar interaction are smeared out on entering in the paramagnetic polarized phase where the magnetic response will be $q$ independent \cite{Bourdarot:2003,Levallois:2009}.
Let us compare the results on URu$_2$Si$_2$\ with
two Pr skutterudite systems PrRu$_4$P$_{12}$ and PrFe$_4$P$_{12}$ where strong feedbacks occur between Fermi sea and multipole dynamics (see references in \cite{JPSJ77SA:2008}). The interest in the last reference is that the U ions in their tetravalent configuration will have two electrons in the $5f$ shell as do Pr ions in their trivalent configuration in the $4f$ shell.
These are systems where a strong feedback may occur between band structure, charge and multipolar ordering \cite{JPSJ77SA:2008}. In PrRu$_4$P$_{12}$, it is now well established that at low pressure a charge order phase transition at $T_0=63$ K occurs through a switch from $body\ centered\ cubic$ to $simple\ cubic$ lattice with clear evidence of a formation of two sublattice leading here to an unambiguous detection of $Ru-ion$ displacement \cite{Lee:2001,Iwasa:2005}. The strong similarity of PrRu$_4$P$_{12}$ with URu$_2$Si$_2$\ in inelastic neutron scattering experiment is the smearing of the inelastic response above $T_0$ and the appearance of a sharp feature below $T_0$ with nuclear Bragg displacement following BCS-type dependence leading to the claim that the crystal-field level variation through $T_0$ is coupled to the carrier change itself \cite{Iwasa:2005}(as for URu$_2$Si$_2$, the temperature variation of integrated intensity of $E_0$). Another interesting case is PrFe$_4$P$_{12}$ where for $P_x=2$ GPa, the system switches from HO semi-metallic phase to AF insulator phase \cite{Sato:2000}. NMR experiments on P sites \cite{Kikuchi:2007} have recently lead to the conclusion that the HO phase has a scalar OP\cite{Sakai:2007,Kiss:2008}. The difference between URu$_2$Si$_2$\ and the last two skutterudite system appears that for the first case in the paramagnetic regime the system is clearly in a mixed valence state for the U ions (valence v $\sim$ 3.5)\cite{Hassinger:2008a,Barzykin:1995}. As discussed for systems like TmSe \cite{Derr:2006}, it is the crossing to a well ordered phase at $T_0$, which makes that the uranium centers look to be tetravalent and leads consequently to the idea that a ThRu$_2$Si$_2$ description for the electronic bands may be a good starting point \cite{Haule:2009,Harima::2010}.
\section {Conclusion}
The present work leads to a precise study of the resonance $E_0$ at \textbf{Q$_0$} which is up to now the main signature of the OP of the HO phase. The key results are; the control of the temperature dependence of the resonance energy $E_0$ by the partial gap opening at the Fermi surface ($\Delta_G$), the temperature like BCS dependence of the integrated inelastic intensity of the resonance. It was suggested that this variation may reflect the temperature evolution of the order parameter. Clearly, the itinerant and local character of the $5f$ electrons must be treated in equal footing. These new data will certainly push to theoretical developments.
\section{Acknowledgements}
We thank H. Harima, K. Miyake, G. Knebel, L. Malone, M. Zhitomirsky, V Mineev, J.P. Sanchez and J.P. Brison for useful and fruitful comments.
This work is supported by the Agence Nationale de la Recherche through the ANR contracts Delice, Sinus, and Cormat.
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\section{Encrypted domains and the \texttt{referer}}
\label{appendix:a}
Visits to web pages in encrypted domains are not visible through passive measurements of the network. Moreover, RFC~7231 mandates that ``a user agent MUST NOT send a Referer header field in an unsecured HTTP request if the referring page was received with a secure protocol''. Therefore, \texttt{referer} fields on HTTPS to HTTP transitions should not be visible in the network.
However, RFC~7231 is not strictly followed by popular domains that promote content. Content promoters have incentives to pass on the \texttt{referer} to domains they promote, including non-HTTPS destinations. For illustration, we quote Facebook when it switches to HTTPS in 2013:\footnote{\url{https://www.facebook.com/notes/facebook-engineering/secure-browsing-by-defa\%20ult/10151590414803920}}
\begin{quote}
Browsers traditionally omit the referrer when navigating from https to http. When you click on an external link, we'd like the destination site to know you came from Facebook while not learning your user id or other sensitive information. On most browsers, we redirect through an http page to accomplish this, but on Google Chrome we can skip this redirect by using their meta referrer feature.
\end{quote}
Thus, many different strategies have been adopted over the years when moving from HTTPS to HTTP. Facebook statement explains that modern web browsers include a mechanism to allow HTTPS domains to declare whether the \texttt{referer} field should be passed on or not (the HTML~5 \texttt{meta referrer} tag). By the time of writing, large content promoters rely on the mechanism to pass on some \texttt{referer} information, such as Facebook, Twitter, Bing and Yahoo.
The HTML~5 \texttt{meta referrer} tag is however rather recent. Before it, each content promoter used to implement its own solution to pass on the \texttt{referer} to non-HTTPS domains. In Facebook statement, for instance, we see that for browsers not supporting the \texttt{meta referrer} tag Facebook first redirects users to a Facebook domain that is still running on HTTP. In this first redirection, the original \texttt{referer} is lost as specified by RFC~7231, but users are still in Facebook systems. Then, users are again redirected to the final destination domain. The final destination receives as \texttt{referer} the HTTP domain under control of Facebook.
As we have seen, the majority of domains was not under HTTPS during the data capture. Popular content promoters were the exception, since they were the early-adopters of full HTTPS support. Since content promoters implement the above techniques to pass on the \texttt{referer}, transitions from the content promoters to HTTP domains are indeed considered. Instead, HTTPS to HTTPS transitions are not visible -- \mbox{e.g.,}\xspace internal navigation on HTTPS domains (\mbox{e.g.,}\xspace Google$\rightarrow$Google) or transition between two HTTPS domains (\mbox{e.g.,}\xspace Google$\rightarrow$Facebook). Clearly, as more and more domains move to HTTPS-only, more transitions become invisible.
\section{Impact of device on browsing habits}
\label{sec:ContentConsumption}
We now focus on the analysis of the clickstream graphs to study the long-term evolution of browsing habits. To exemplify the kind of data used in the remaining sections, we show in Figure~\ref{fig:clickstreams} two cases from an arbitrary household of browser clickstream graphs during one day of navigation. Figure~\ref{fig:left} refers to a graph in which the user was using a PC to browse the web, while Figure~\ref{fig:right} is related to sessions on a smartphone browser. Bigger colored nodes are SEs and OSNs (red for SEs, blue for OSNs). For instance, Figure~\ref{fig:left} shows four independent components: The bigger starts from web searches from Google; The second one are all pages originated from Facebook; The third and forth components are instead visits not originated by any SE or OSN, and could be possibly due to bookmarks or links received via, e.g., email. These examples already give the intuition about peculiar topologies and the diversity among them. We will quantify such aspects in aggregated and statistical ways in the following section taking into account all clickstream graphs in the dataset. We will refer again to Figure~\ref{fig:clickstreams} just to illustrate the introduced metrics.
Our aim is to highlight properties of differences in distributions that are of statistical significance. For this, we run a two Kolmogorov-Smirnov test with the null hypothesis that the two empirical distributions come from the same distribution. The null hypothesis is rejected with a level of significance of 5\%. As a counter-proof, we extract two samples from the same empirical distribution and see if the test does not reject the null hypothesis.
Given the huge amount of data we are using for our analysis (31 million of user actions per month on average), we observe statistically significant differences even when comparing two empirical distributions that have only small observable differences on the plots. In a nutshell, all statistical tests show significant differences and no statistically significant difference is observed when sampling from the same distribution.
\begin{figure}[t!]
\subfloat[PC]{
\centering
\includegraphics[width=0.50\textwidth]{fig/PC23.pdf}
\label{fig:left}
}\hfill
\centering
\subfloat[Smartphone]{
\centering
\includegraphics[width=0.45\textwidth]{fig/Mobile14.pdf}
\label{fig:right}
}
\caption{Browser clickstream samples for a PC and a smartphone browser in the same household in an arbitrary day. SEs and OSNs (and their neighbors) are marked in red and blue, respectively.
}
\label{fig:clickstreams}
\end{figure}
For simplicity, from now on we report results from \mbox{\emph{PoP~1}}\xspace only, since the other probes show analogous results.
\subsection{Device usage}
We investigate the evolution in popularity of device categories (PCs, tablets and smartphones). We compute, for each device category, its share in terms of number of active browsers, \mbox{i.e.,}\xspace browsers that generate at least one user-action, and its share in terms of user-actions.
Figure~\ref{fig:UA_share_action_a} shows the fraction of active browsers over the last three years. Observe the stunning increase of smartphone browsers, from $26\%$ to $55\%$ of the total active browsers, with PCs that are now less than $40\%$ of the active browsers.
However, contrast this with Figure~\ref{fig:UA_share_action_b}, which shows the fraction of user-actions per device category.
Although we see an even larger relative growth in the fraction of user-actions coming from smartphones -- \mbox{e.g.,}\xspace from $7\%$ to $27\%$ -- in absolute terms PCs are still creating more user-actions in 2016. For tablets, the increase is more limited, but more visible than in Figure~\ref{fig:UA_share_action_a} -- from $5\%$ to $9\%$.
Considering the median number of browsers per household, this number has increased from 4 in July 2013 to 7 in June 2016. Smartphones had the largest increase from 1 to 4 (from 1.5 to 6.4 considering the average). PC category remains constant with a median moving between 3 and 4 throughout the years. Tablets are not widespread and the majority of households does not see any browser of this category (with its mean increasing from 0.4 to 1.2). Remember that when an application is updated, we see it as two distinct browsers.
Concluding, we see an increasing number of custom \texttt{user-agent} strings used by different apps in mobile applications, which, recall, we identify as distinct browsers; however, we see a more limited usage of each browser in mobiles as compared to PCs. We next investigate this latter effect in more details.
\begin{figure}[!t]
\centering
\subfloat[Fraction of active browsers per device category] {
\label{fig:UA_share_action_a}
\includegraphics[width=0.45\textwidth]{fig/UserAgents.pdf}
}
\subfloat[Fraction of user-actions per device category] {
\label{fig:UA_share_action_b}
\includegraphics[width=0.45\textwidth]{fig/actions.pdf}
}
\caption{Smartphones present a higher number of browsers, but PCs still dominate the number of user-actions. Browsing from smartphones has increased almost 4 times from 2013.}
\label{fig:UA_share_action}
\end{figure}
\subsection{Browsing sessions}
\subsubsection{Think-time}
Consider the time between two consecutive user-actions, commonly referred as {\it think-time}. Figure~\ref{fig:interarrival} reports Empirical Cumulative Distribution Functions (ECDFs) for smartphones and PCs, comparing July 2013 with June 2016. Tablets are left out to improve visualization.
We observe that in all cases, more than $60\%$ of user-actions are separated by less than 1 minute. The long tail exceeds one day, and peaks are present at typical automatic refreshing time of popular web pages. Think-time is shorter on PCs than on smartphones, suggesting more interactive browsing sessions on PCs.
Interestingly, think-time slightly increased from 2013 to 2016 (median increased of about 10\% for both PCs and smartphones). We hypothesize that this small increase in think-time could be due to the lack of visibility that the increase in the use of encryption causes to us. In a nutshell, the shift toward HTTPS reduces our visibility on pages and links which may cause the shift in the think-time we observe. To verify this hypotesis, we perform the experiment described in Section \ref{sec:impact}. We start from 2013 trace, and remove about 45\% of services as if they migrated to HTTPS. We then recompute the think-time, and obtain a CDF that indeed is very similar to the actual one obtained using 2016 data. We do not report results for the sake of brevity.
\begin{figure}[!t]
\centering
\includegraphics[width=0.45\textwidth]{fig/inter_years_device.pdf}
\caption{ECDF of think-time for browsers.
Think-time is shorter on PCs than on smartphones.}
\label{fig:interarrival}
\end{figure}
\subsubsection{Session time and activity}
We next consider {\it browsing sessions}, \mbox{i.e.,}\xspace the grouped and consecutive generation of user-actions by the same browser. While defining a browsing session is complicated~\cite{Catledge1995,bianco2009web,fomitchev2010google}, we consider a think-time larger than 30~minutes as an indication of the session end. This is a conservative threshold (see Figure~\ref{fig:interarrival}), and it is often seen in previous works (\mbox{e.g.,}\xspace \cite{Catledge1995}), and in applications like Google Analytics.\footnote{\url{https://support.google.com/analytics/answer/2731565?hl=en}}
Figure~\ref{fig:session_a} shows the distribution of session durations and Figure~\ref{fig:session_b} that of user-actions per session. A session with just 1 action is considered of duration 0\,s. Observe that PC sessions last longer and contain more user-actions than smartphone ones.
The median number of user-actions per browsing session is small: half of the smartphone (resp. PC) sessions consist in less than 5 (resp. 9) web pages per session, which do not last more than 2~min (resp. 8~min) in 2016. On the other hand, some few heavy sessions are present: as the tails show, some sessions contain hundreds of web pages and last many hours.
From July 2013 to June 2016 both session duration and number of user-actions per session decreased.
To investigate this change, we run again the experiment of Section \ref{sec:impact}: from the 2013 trace, we keep removing websites to the point in which 45\% of the user actions would be missing (due to adoption of HTTPS).
We want to observe if this shift would be compatible with the observed changes in actual data of 2016.
To better highlight this, Figure~\ref{fig:duration2} shows the number of user actions per session, detailing the trend when 15, 30 and 45 \% of services migrated to HTTPS. We consider the PC scenario. As it can be clearly seen, the more services move to HTTPS, the more the curve moves closer and closer to the actual measurements observed in 2016. Notice indeed the tail of the distributions that becomes practically identical.
Whereas this is a not a proof of causality, it provides strong evidences that in this case the differences in three years are likely artefacts due to the shift to HTTPS.
\begin{figure}[!t]
\centering
\subfloat[Browsing session duration] {
\label{fig:session_a}
\includegraphics[width=0.45\textwidth]{fig/duration_years_device.pdf}
}
\subfloat[Number of user-actions per browsing session] {
\label{fig:session_b}
\includegraphics[width=0.45\textwidth]{fig/actions_years_device.pdf}
}
\caption{Browsing session characteristics. Sessions on smartphones last shorter with fewer web pages than on PCs. }
\label{fig:session}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[width=0.45\textwidth]{./fig/EVOactions_years_deviceFINALEfilterhttps}
\caption{Simulated impact of the HTTPS migration on the number of user-actions per session for increasing percentage of domains that migrated to
HTTPS.}
\label{fig:duration2}
\end{figure}
\subsubsection{Inter-session time}
To complete the analysis, Figure~\ref{fig:idle} shows the distribution of idle time between sessions. Results are clearly affected by the periodicity of human life: notice jumps at 24 hours, 48 hours, etc. Also in this case, idle time is shorter on PCs than on smartphones, with median values 3 and 7 hours, respectively. We remind that we are considering web page browsing, which is different from other typical usages of mobile terminals, e.g., for instant messaging. Interestingly, smartphone idle time decreased from 2013 to 2016, meaning that the frequency of their usage for browsing the web at home is increasing. The number of times a smartphone browser is used has increased, as well as the number of smartphone browsers.
Again, simulating the migration of services to HTTPS provides further evidences to this claim: HTTPS migration should have caused an increase in the metric. Therefore, the simulation not only reinforces that there has been a decrease in idle time from 2013 to 2016, but also provides an indication about the bias (around 25\%) introduced by the HTTPS migration in the 2016 figures.
In a nutshell, people seem to have short sessions on smartphone and tablet browsers, in which they visit a handful of web pages. Nevertheless, the browsing intensity is increasing on smartphones (as well as the number of used apps), but PCs are still the preferred means for long, concentrated sessions from home.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{fig/idle_years_device.pdf}
\caption{ECDF of session idle time. PC browsers access the web more frequently than smartphone ones, but frequency of use is increasing for the latter.}
\label{fig:idle}
\end{figure}
\subsection{Content consumption}
Next, we quantify the consumption of content per browser and per device category. Figure~\ref{fig:number_nodes} shows the ECDF of the number of web pages visited per day per browser, considering all days in June 2016. PC, tablet, and smartphone browsers are depicted in separate lines. As expected, the number of web pages visited on smartphone browsers is significantly lower than the number of web pages visited on PCs, with tablets in between. This reinforces the observation that smartphone browsers are not used for long web page browsing at home.
The total daily number of visited web pages for each browser is in general small: on average, browsers visit 27 distinct web pages per day on PCs, 15 on tablets, and 10 on smartphones. Only few browsers consume more than 300 web pages in total. Comparing the number of user-actions (not shown in the figure) to the number of \emph{unique} visited web pages, we see that each web page is visited $1.5$ times on average.\footnote{Results are overestimated since parameters are removed from URLs.}
Each browser is seen online only 3.8 days per month (mean values).
Recall that households have a median of $7$ browsers per month (see previous subsections).
\begin{figure}[!t]
\centering
\includegraphics[width=0.45\textwidth]{fig/PagesDistribution_day.pdf}
\caption{Number of visited web pages per browser per day. PC browsers visit 3 times more web pages per day than smartphone ones.}
\label{fig:number_nodes}
\end{figure}
\begin{table}[!tb]
\centering
\small
\caption{Details per popular browser application of the average number of visited web pages per day.}
\label{tab:pop}
\begin{tabular}{|c||c|c||c|c||c|c|}
\cline{1-7}
\multirow{2}{*}{Browser type}
& \multicolumn{2}{c||}{PC} & \multicolumn{2}{c||}{Smartphone} & \multicolumn{2}{c|}{Tablet} \\
\cline{2-7}
& avg & pop.\% & avg & pop.\% & avg & pop.\% \\
\cline{1-7}
Chrome & 30.9 & 46\% & 9.8 & 39\% & 19.3 & 42\% \\
Safari & 23.6 & 6\% & 12.4 & 17\% & 11.8 & 30\% \\
Internet Explorer & 16.5 & 20\% & - & - & - & - \\
Firefox & 30.4 & 20\% & - & - & - & - \\
\cline{1-7}
Others & 22.4 & 8\% & 8.4 & 44\% & 13.8 & 28\% \\
\cline{1-7}
\end{tabular}
\end{table}
To give more details, Table~\ref{tab:pop} reports the average number of visited web pages per day, considering popular browser types for each category (PCs, smartphones and tablets). On PCs, the popular browsers account for 92\% of the observed browsers, with Chrome being the most popular browser type. On smartphones and tablets, instead, there are many more browser types, so that the `others' class accounts for a significant amount of user-actions. Considering user behavior on PCs, we observe that Firefox and Chrome users visit more than 30 web pages per day, while Internet Explorer users are much less active, visiting only 16.5 web pages. On smartphones, Safari users are more active than Chrome users.
To determine how content consumption is evolving, Figure~\ref{fig:PDF_pages} shows the evolution of the mean number of daily visited web pages and domains for each browser in the dataset.
Results confirm that Internet browsing habits are in general light: the median number of domains visited in each day is pretty low and constant over time.
We observe a decreasing trend for both smartphones and for PCs. This trend is a consequence of the increasing usage of HTTPS (see~Figure~\ref{fig:https}).
In fact, the figure reports the period in which popular domains started deploying HTTPS (see~Figure~\ref{fig:timeline_protocols}). Google and Facebook had already started their migration before July 2013, but there is still a visible reduction in the series for web pages on PCs, likely connected to the final steps of Google Search migration until September 2013. The migration of some popular domains, such as the Wikipedia, caused a minor, but visible, reduction in numbers for PCs too. The trend is accelerated in the final months of the capture, when other domains starting deploying HTTPS.
To corroborate this claim, we run again the experiment of Section \ref{sec:impact}. If we remove domains that moved to HTTPS in June 2016 from the data of July 2013, the number of daily visited web pages for PCs (smartphones) lowers to 37 (10, respectively). The actual numbers for June 2016 are slightly higher of those numbers, suggesting that the number of web pages visited daily actually did not decrease.
In summary, we conclude that each smartphone browser is used for browsing few web pages at home, while PC browsers are used for more time and to visit more web pages. The number of different domains visited over time is typically small and rather constant.
\begin{figure}[!t]
\centering
\subfloat[PC] {
\includegraphics[width=0.4\textwidth]{fig/PagesDomainMean_day_PC.pdf}
}
\subfloat[Smartphone] {
\includegraphics[width=0.4\textwidth]{fig/PagesDomainMean_day_Mobile.pdf}
}
\caption{Mean number of web pages and domains visited per day.}
\label{fig:PDF_pages}
\end{figure}
\section{User-action classifier design}
\label{sec:validation}
Our classification problem consists in identifying a URL as either user-action or automatic-action. In the past, this problem has been faced by designing ad-hoc heuristics driven by domain knowledge, e.g., by rebuilding the web page structure~\cite{ihm_towards_2011,xie_resurf_2013}, or manually building blacklists and simple tests~\cite{houidi_gold_2014}. We here revisit the problem and introduce a machine learning methodology. It is given a labeled dataset where the classes of observations are known. Observations are characterized by {\it features}, \mbox{i.e.,}\xspace explanatory variables that describe observations. The classifier uses the knowledge of the class to build a {\it model} that, from features, allows it to separate objects into classes. In the following we summarize our approach that we presented earlier in a workshop~\cite{VD16}.
\subsection{Feature extraction}
\label{sec:proposed}
Instead of a priori selecting features that we believe might be useful for classification, we follow the best practice of machine learning and extract a large number of possibly generic features. We let the classifier build the model and automatically choose the most valuable features for the goal.
Table~\ref{tab.features} lists the features extracted from traffic traces. We consider 17 features that can be coarsely grouped into four non-independent categories: (i)~based on referring relations among URLs; (ii)~based on timestamps; (iii)~describing properties of objects; and (iv)~describing properties of URLs. Features are sorted by their {Information Gain} (IG), a notion that we discuss later.
Some are inspired from prior works.
Some are boolean or categorical, i.e., they can take a limited number of labels, while others are counters or real-valued.
\begin{table}[!tb]
\centering
\caption{Features and IG with respect to the {user-actions}.}
\label{tab.features}
{
\small
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Feature & \texttt{referer} & Time & Object & URL & Type & IG\cr
\hline
Number of Children~\cite{houidi_gold_2014}\cite{xie_resurf_2013}\cite{ihm_towards_2011} & x & & & & Count & 0.2706 \cr
Content Type~\cite{houidi_gold_2014}\cite{xie_resurf_2013}\cite{ihm_towards_2011} & & & x & & Cat. & 0.0287 \cr
$\Delta_t$ -- Previous Request & & x & & & Real &0.0140 \cr
HTTP Status Code~\cite{xie_resurf_2013} & & & x & & Cat. & 0.0061 \cr
URL length & & & & x & Count &0.0060 \cr
$\Delta_t$ -- Sibling & x & x & & & Real & 0.0048 \cr
Ads in URL & & & & x & Bool & 0.0040 \cr
$\Delta_t$ -- Parent~\cite{xie_resurf_2013}\cite{ihm_towards_2011} & x & x & & & Real & 0.0036 \cr
Content Length~\cite{xie_resurf_2013} & & & x & & Count & 0.0027 \cr
Parent Status Code & x & & x & & Cat. &0.0016 \cr
Has \texttt{referer}? & x & & & & Bool &0.0014 \cr
Max $\Delta_t$ -- Child & x & x & & & Real & 0.0010 \cr
Parent Content Type & x & & x & & Cat. & 0.0007 \cr
Ads in \texttt{referer} & x & & & x & Bool & 0.0005 \cr
Max Length -- Child & x & & x & & Count & 0.0005 \cr
Min $\Delta_t$ -- Child & x & x & & & Real &0.0003 \cr
Parent Content Length & x & & x & & Count& 0.0002 \cr
\hline
\end{tabular}
}
\end{table}
Features are extracted from HTTP logs. Given a URL, we calculate the time interval ($\Delta_t$) from the previous request from the same browser. If the request has a \texttt{referer}, we call the URL in the \texttt{referer} its \emph{parent}. We also determine whether a URL in a request has \emph{children}, i.e., subsequent requests that have this particular URL in the \texttt{referer} field. Based on parent-child relations, we extract the number of children, the time interval between the request and its eventual parent, and the time interval between the request and its last \emph{sibling}, i.e., previous request sharing the same parent. If the request has children, we compute the minimum and maximum time to see a child.
We consider features in server responses, such as the Status Code, Content Type and Content Length. We also augment the feature set with statistics of the request of the parent (if it exists), e.g., the Content Length, Content Type and Status Code of the parent request. Finally, we include features that describe the URL strings. We count the number of characters in the URL and we check if the URL (or the \texttt{referer}) is included in a blacklist of terms associated with advertisement domains.
\subsection{Classifier choice}
Given the heterogeneity of features and their diverse nature, the choice of which classification model to adopt requires ingenuity. For instance, algorithms based on notion of distance such as Support Vector Machines or nearest neighbor methods are particularly sensitive to the presence of boolean and categorical features. Similarly, the presence of dependencies between the features challenges regression based classifiers. Generally, when there are complex dependencies among features, decision trees and neural networks offer the best performance~\cite{Mitchell_ML}. As such, we consider:
\noindent {\bf Decision Tree (DT):} It is a tree-like classifier for making sequential decisions on features~\cite{cart84}. Internal nodes represent tests, branches are the outcomes of tests, and leaves represent classes. We use \texttt{J48} -- an open source implementation of the \texttt{C4.5} decision tree training model.
\noindent{\bf Random Forest (RF)~\cite{Br2001}:} It improves and generalizes decision trees. It constructs a multitude of decision trees at training time using subsets of features, outputting the class that is the mode among those trees. RF is more robust to over-fitting.
\noindent {\bf Multi-Layer Perceptron Neural Network (NN):} It is a feedforward neural network that maps features into classes~\cite{Ha94}. It consists of multiple layers of nodes in a directed graph, where each node is a processing element with a nonlinear activation function. Training is performed with the backpropagation algorithm.
We use the implementations offered by Weka in our experiments.\footnote{http://www.cs.waikato.ac.nz/ml/weka/}
\subsection{Performance metrics and methodology}
The classification performance measures the ability to correctly return the class of an object.
Performance is typically summarized using $Accuracy$, i.e., the fraction of objects from any class that are correctly classified.
Accuracy is often misleading, especially when object classes are unbalanced, i.e., a naive classifier returning always the most popular class would achieve a high accuracy. In such cases, per-class performance metrics must be considered.
Therefore, given we are interested in user-action classification, we also evaluate performance metrics related to this class:
(i) \mbox{$Precision$}: the fraction of requests correctly classified as user-action (the number of true positives among the requests that the classifier selected as user-actions);
(ii) \mbox{$Recall$}: the fraction of user-actions that the classifier captures (number of detected user-actions over the total number of user-actions);
and (iii) $F-Measure$: the harmonic mean of Precision and Recall.
We use the standard stratified 10-fold cross-validation to measure model performance and select the best classification setup, \mbox{i.e.,}\xspace the best tuning parameters.
\section{User-action classifier performance}
\label{sec:clasPerf}
We now provide a performance evaluation of the user-action classifier on ground truth traces. Instead of only building such traces in a controlled environment as done in previous works, we also rely on real traces from actual end-users to build the ground truth.
\subsection{Annotated dataset for training and testing}
Training a classifier and testing its performance require data in which the ground truth is known, i.e., requests are annotated with class labels. In our scenario, we need HTTP logs in which requests are annotated as user- or automatic-actions.
To obtain the ground truth of user-actions, we rely on volunteers. Specifically, we collect the visited web pages from volunteers' PCs by extracting their browsing history from three major browsers: Safari, Chrome and Firefox. Referring to Figure~\ref{fig:classificationProblem}, a volunteer's browsing history exposes the timeline of user-actions. It includes (i)~timestamps of web page visits; (ii)~the requested URLs; and (iii)~codes describing web page transitions -- \mbox{e.g.,}\xspace whether the visit resulted in a redirection to another web page. In total, we observed more than 12\,000 visits to more than 2\,000 websites in 3 months of browsing activity of 10 volunteers.
During the same period, we instrumented our campus network to passively collect the raw HTTP logs of these volunteers by observing the traffic flowing in and out of our university.
We use Tstat~\cite{finamore_experiences_2011}, a passive monitoring tool to perform the collection. The tool exposes information from both the TCP and HTTP headers, including (i) TCP flow-ID, (ii)~timestamps; (iii)~requested URLs; (iv)~user agent; (v)~\texttt{referer}; (vi)~content type; (vii)~content length; and (viii)~status code. Referring to Figure~\ref{fig:classificationProblem}, Tstat exposes the timeline of \emph{all} HTTP requests observed in the network. All in all, we record more than 0.6 million HTTP requests related to the volunteers.
We next match entries in HTTP logs with entries extracted from browsing histories to label user-actions. The matching of entries however requires care. We primarily use the URL and timestamps as keys, but web page redirections may create issues. For instance, shortened URLs or web page redirections are logged in different ways by various browsers. We decide to label as user-action the last request in a redirection chain that is present in both HTTP logs and browsing histories. At the end of this process, we mark about 2\% of all HTTP requests as actual user-actions.
\subsection{Feature relevance}
The central idea when doing feature selection is that the data may contain irrelevant or redundant features.
To check which features are relevant to separate user- from automatic- actions, we compute the Information Gain, also known as the mutual information. It quantifies the reduction in entropy caused by partitioning the dataset according to the values of the specific feature. In Table~\ref{tab.features}, we rank features: the higher the information gain, the higher is the information about the user-action class that the specific feature carries in isolation.
We see that the Number of Children is by far the feature with the highest IG. Content Type is well-ranked as well. These results confirm the intuition of prior work~\cite{houidi_gold_2014,xie_resurf_2013,ihm_towards_2011} that \texttt{referer} relations and the analysis of Content Types strongly help in user-actions detection. Next to these features, we find the time interval ($\Delta_t$) between consecutive requests of a single browser, and some other features that are independent of \texttt{referer}, such as HTTP Status Code and Size of URL.
\subsection{Classification performance}
We evaluate the performance of the different classification models. Table~\ref{tab.models} reports the classification accuracy, and F-Measure, precision and recall for the user-action class in the 10-fold cross validation experiments. These results are obtained using all features listed in Table~\ref{tab.features}. Experiments with the best-ranked features result in minor performance variations, which we do not report for brevity (see~\cite{VD16} for more details). The table also lists results for the heuristic proposed by Ben-Houidi \textit{et al}.~\cite{houidi_gold_2014}, which was manually crafted using domain knowledge.
\begin{table} [!t]
\centering
\caption{Performance of the classifiers. F-measure, precision and recall for the user-action class.}
\subfloat[Cross-validation with PC volunteers' training set.] {
\small
\begin{tabular}{|x{2.8cm}|x{1.5cm}||x{1.5cm}|x{1.5cm}|x{1.5cm}|x{0cm}}
\cline{1-5}
Model & Accuracy & F-measure & Precision & Recall & \\[0.75ex]
\cline{1-5}
Decision Tree & 0.996 & 0.906 & 0.905 & 0.907 & \\[0.75ex]
Random Forest & 0.996 & 0.912 & 0.891 & 0.943 & \\[0.75ex]
Neural Network & 0.994 & 0.888 & 0.860 & 0.917 & \\[0.75ex]
Manual Heuristic~\cite{houidi_gold_2014} & 0.988 & 0.784 & 0.711 & 0.870 & \\[0.75ex]
\cline{1-5}
\end{tabular}
\label{tab.models}
} \\
\subfloat[Decision tree tested on smartphone traces for different apps.] {
\small
\begin{tabular}{|x{2.8cm}|x{1.5cm}||x{1.5cm}|x{1.5cm}|x{1.5cm}|x{0cm}}
\cline{1-5}
Testing Dataset & Accuracy & F-measure & Precision & Recall & \\[0.75ex]
\cline{1-5}
Chrome & 0.999 & 0.935 & 0.935 & 0.935 & \\[0.75ex]
UC browser & 0.996 & 0.863 & 0.923 & 0.810 & \\[0.75ex]
Facebook & 0.998 & 0.931 & 0.902 & 0.961 & \\[0.75ex]
Instagram & 0.998 & 0.930 & 0.890 & 0.973 & \\[0.75ex]
\cline{1-5}
\end{tabular}
\label{tab.models.test}
}
\end{table}
We see in Table~\ref{tab.models} that the accuracy is higher than $99\%$ for the three classification models. Recalling that we have about 98\% of automatic-actions, i.e., classes are strongly unbalanced, a naive classifier that always returns ``automatic'' would have about $98\%$ accuracy. As such, we focus on performance for the user-action class. All three machine learning classifiers deliver very good performance. Neural network has the lowest F-Measure among the three. The \texttt{J48} decision tree F-Measure equals to $90.6\%$, while the Random Forest performs marginally better. Interestingly, all machine learning alternatives have much better performance than the manual heuristic (last line in the table) thanks to our larger set of features, and the higher variability in actual user browsing habits that we used to train our models. Recall that the manual heuristic we compare to has been trained and tested on a smaller ground truth trace built in a controlled environment.
Given a decision tree is simpler than a random forest, but with similar performance, we decide to use the former from now on. An appealing characteristic of decision trees is the easy interpretation of the built model. Manually inspecting the tree, we see that indeed features with high IG (\mbox{e.g.,}\xspace number of children) are among the top features of the tree.\footnote{Classifier code can be downloaded from \url{http://bigdata.polito.it/clickstream}.}
\subsection{Training set size}
To get more insights on performance, we run experiments varying the training set. On each round, we consider an increasing number of volunteers in the training set. We then assess performance (i)~using 10-fold cross validation on the same training data; (ii)~validating the model on the remaining volunteers' traces that were not included in the training set, i.e., an independent test set.
The results are on Figure~\ref{fig:varying_users}, where F-Measure for the user-action class is depicted. Considering the cross validation estimates, the F-Measure reaches a plateau when two volunteers are considered. That is, the classifier is able to model the browsing habits of the volunteers included in the training set. More importantly, the validation with independent users shows consistent results, reaching more than 90\% of F-measure when seven or more volunteers are in the training set. In a nutshell, the behavior of the independent volunteers has been learned from other users. These figures provide additional evidence that the produced model is robust and generic.
\begin{figure}[]
\centering
\includegraphics[width=0.43\textwidth]{./fig/Fmeasure.pdf}
\caption{Effects of varying the number of volunteers for training.}
\label{fig:varying_users}
\end{figure}
\subsection{Testing on smartphone traffic}
Training and testing have been done so far considering annotated traces from browsers running on PCs. It is not clear whether the classifier would perform well for smartphone or tablet users too. In particolar we are interested in validating the methodology for mobile browsers, i.e., apps that can be used to navigate through multiple web pages and domains. To answer this question, we collect synthetic traces of visits to popular HTTP websites, taken from top-100 Alexa websites\footnote{https://www.alexa.com/topsites} and by randomly following two links inside each of them.
We performed the experiment by re-opening the URLs with different apps in an Android smartphone, while connected via WiFi to the campus network instrumented with Tstat. We choose four different browsers: Chrome, UC browser, Instagram and Facebook. Instagram and Facebook are apps primarly thought to exploit their internal services, but that allows as well to follow links to external web pages in a in-app browser.
The traces have been collected by visiting these web pages manually, waiting for each web page to be fully-loaded before visiting the next web page. Even if this behavior cannot be considered totally natural as for our PC volunteers, the dataset has some ingredients of real user interactions. Moreover, we do not filter out background traffic of the smartphone.
At the instrumented network, $\approx {5\,000}$-${8\,000}$ HTTP requests have been recorded, depending on the browser, which we classify using the previously described decision tree.
The results are in Table~\ref{tab.models.test}.
Performances are inline with previous experiments. , \mbox{i.e.,}\xspace precision and F-Measure close to $90\%$. There are very few false positives and even less false negative. Only UC browser shows a lightly smaller value of recall with respect to the others. Indeed, this browser performs compression of the web pages and some requests are dropped. Therefore, they cannot be recognized as user-actions.
The validation that we performed on different mobile browsers suggests that our machine learning classifier, although trained on PC datasets, adapts well also to traffic towards other devices.
\section{The clickstream graph}
\label{sec:clickstream}
We now characterize properties of the clickstream graphs looking at how people explore the web over time. For each day, we extract and analyze the clickstream graph for each browser (on average, more than 7\,500 graphs per day), and then compute statistics by aggregating all graphs in each month.
\subsection{Paths characteristics}
First, we gauge how extensive and deep browser explorations are. To reach this goal, we compute how many consecutive and related (by the \texttt{referer} relationship) user-actions, forming a \textit{path}, browsers visit. More precisely, we extracted the longest among the finite directed shortest paths between all vertices in each graph. Such a path is simply called \textit{longest path} further in the paper. Its length gives a hint on how far the user goes from its navigation starting point over a one-day navigation period.
As seen in the examples of Figure~\ref{fig:clickstreams}, the clickstream graphs are not random graphs, apparently following a preferential attachment structure, with some hubs with a large number of connected web pages and some relatively long branches that form the longest paths we are studying.
Figure~\ref{fig:page_path} shows the evolution of median and average number of user-actions in the longest paths for PC and smartphones. Paths are quite short, and longer on PCs than on smartphones. Interestingly, path length is stable throughout the years, even if the number of web pages is reduced (see PCs in Figure~\ref{fig:number_nodes}).
The migration to HTTPS has little impact here: the longest paths do not appear to be exclusively through encrypted domains, even if their popularity (in terms of user-actions) is quite high.
\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{fig/length_path_domain_day.pdf}
\caption{Daily number of visited web pages in the longest path of a clickstream. Depth of the graphs is very limited.}
\label{fig:page_path}
\end{figure}
If we count the number of unique domains in each path, we obtain that, on average, only $1.8$ domains are present in the longest path. This suggests that people tend to perform deep navigation in web pages from the same domain, rather than moving among multiple consecutive domains.
One could expect this to be a consequence of the fact that most paths are very short. To check whether long paths differ from short paths, we extract path characteristics conditioned to the number of user-actions in the path, i.e., for paths with less than 10 user-actions, from 10 to 100 user-actions, or with more than 100 user-actions. Figure~\ref{fig:domain_per_url} summarizes with whiskers box-plots the distributions of the number of domains in the paths and it is cut at 6 domains for ease of visualization. Outliers are recognized with the classical Tukey rule on quartile (1.5 IQR method) and shown with diamond markers. In our case, only 3.8\% of all the paths are outliers, with the remaining 96.2\% of samples that have a number of domains per path that is not greater than 3, with the median number (starred mark) of domains that is always 1 or 2.
Therefore the number of domains in paths is always extremely limited, and this number does not increase for longer paths. Unexpectedly, all paths with more than 100 visited web pages are within 3 domains too. We investigated this behavior and found that long paths are related to peculiar domains, such as comics or galleries with hundreds of images.
In conclusion, user paths among web pages are rather short, and rarely users move through many consecutive domains when navigating.
\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{fig/boxplot_path_domain.pdf}
\caption{Box-plots of the number of domains in the paths, per path length. Users stay in very few domains, even when navigating through hundreds of web pages.}
\label{fig:domain_per_url}
\end{figure}
\subsection{Connected components}
We now study the structure of the overall clickstream graph to check whether the several paths forming a clickstream are independent or connected. To this end, we consider the size of the biggest \emph{Weakly Connected Components} (WCC). A WCC is any maximum sub-graph such that there exists a path between any pair of vertices, considering {\it undirected} edges. In a large WCC, the vertices are connected by edges, i.e., web pages are visited by following hyperlinks. Recalling the examples of Figure~\ref{fig:clickstreams}, in those cases we have 4 and 3 WCCs, respectively for the PC and for the smartphone browser, with the biggest WCCs accounting for the majority of the visited web pages.
As usual, we report the evolution over time of the measurements. Figure~\ref{fig:wcc} shows the median of the ratio between the {biggest WCC size} and the {entire graph size}, i.e., the relative extension of the WCC.\footnote{Our WCC is a lower bound estimation of the actual WCC: visits to web pages served over HTTPS are invisible to us; the user-action classifier could miss some user-actions, and some \texttt{referer} fields may be missing in requests. These artifacts correspond to missing edges, thus shrinking the largest WCC.} Interestingly, minor changes are observed over the three years, showing minor modification in browsing habits.
Observe that the biggest WCC covers more than $50\%$ of the entire graph, surpassing $70\%$ for smartphones. By manually inspecting the biggest WCCs, we notice that they usually include SEs and/or OSNs, which act as hubs connecting many user-actions. Recalling that paths are usually short this suggests that SEs and OSNs act as starting web pages used by people to reach other content. We will better investigate this in the next section.
\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{fig/WCCrelativesize.pdf}
\caption{Size of the WCC over time (median over browsers). WCCs typically include SE and OSN. On smartphones, the WCC is bigger than on PCs. This is likely due to the higher usage of SEs and OSNs to reach content.}
\label{fig:wcc}
\end{figure}
We now focus on those vertices which are not connected to the biggest WCC, but form other (small) WCCs. These typically do not contain SEs or OSNs.
A new WCC can indeed be created when there is no \texttt{referer} in a user-action, i.e., when the user directly visits a web page from a bookmark, or by directly entering the address in the browser bar.
These would be web pages users are familiar with, and they reach without following hyperlinks.
Notice how the number of vertices that are not part of the biggest WCC is higher for PCs than for smartphones. We conjecture that this can be explained by the different usability of devices: PC browsers facilitate the direct access to content (via both bookmarks, and by auto-completing URLs manually entered). On smartphones, users tend to get support of SEs and hyperlinks to reach the desired content, which results in a more extended WCC.
\section{Conclusions}\label{sec:future}
Clickstreams offer invaluable information to understand how people explore the web. This paper provides a thorough and longitudinal characterization of clickstreams from passive measurements.
Key challenges for such a study are the identification of URLs effectively requested by users in HTTP traces, and the availability of data summarizing users' activity. We proposed a machine learning approach to identify user-actions in HTTP traces, which we showed to perform well in different scenarios. We then applied it to a large dataset describing three years of activity in thousands of households. We offer the anonymized data to the community in the hope to foster further studies on browsing behaviors. To the best of our knowledge, this is one of the largest datasets available to the study of clickstreams.
Our characterization answers two research questions. First, we quantified several aspects of browsing behaviors and how clickstreams have evolved for 3 years. Second, we uncovered properties of clickstream graphs according to the different types of client devices. We observed interesting trends, such as the typical short paths followed by people while navigating the web and the fast increasing trend in browsing from mobile devices. We observed the different roles of search engines and social networks in promoting content in PCs and smartphones. Finally, we also highlighted how the deployment of HTTPS impacts the study of clickstreams from network traffic. Our results, while sometimes confirming intuitions, precisely quantify the various aspects of clickstreams, with implications for targeted on-line advertisement, content recommendation, the study of on-line privacy, among others.
We envision some promising directions for future work. First, whereas HTTPS prevents the full reconstruction of clickstreams, contacted domains are still visible in the network. We plan to investigate whether machine learning approaches are able to reconstruct clickstreams -- at a domain level -- from HTTPS traffic. Such an approach would provide a coarse view of paths followed by users while surfing the HTTPS web. Second, this study has focused in structural properties of clickstreams without considering the type of content people visit. Characterizing clickstreams according to properties of the visited web pages is a natural continuation.
\section{Content Discovery}\label{sec:content_discovery}
In this section we investigate in more details how users reach content on the web, and which are the domains that act as content promoters. We use the out-degree of a domain\footnote{The out-degree of a domain is the number of hyperlinks leaving web pages from this domain to web pages of another one.} as a metric to pinpoint the promoters: the higher this number is, the more different domains are reached from it. We consider all clickstream graphs in June of each year, and, for each domain, we count the total number of unique domains it appears as a parent at least once.
Table~\ref{tab:prom} shows the evolution of the top promoters in terms of the fraction of different domains they promoted. The great majority are SEs and OSNs. Both facilitate the user to look for content, and are the typical way to start browsing activities and to discover new domains. Google has a dominant position, with more than 6 times more domains promoted than Facebook which comes second. Its leading position has been constant over years even if in slight decrease, with 55\% of domains that have been reached at least one time from \url{google.it} domains in June 2016. We observe little changes in the ranking, with Facebook, Bing and Twitter becoming more popular means to reach content.
\begin{table}[!tb]
\centering
\small
\caption{Ranking of content promoters popularity with the percentage of domains visited from them.}
\label{tab:prom}
\begin{tabular}{|l|c|c|c|c|}
\hline
Domain & 2013 & 2014 & 2015 & 2016 \\
\hline
google.it & 1 (58.3\%) & 1 (56.1\%)& 1 (54.5\%)& 1 (54.5\%) \\
facebook.com & 3 (6.2\%)& 3 (6.9\%)& 2 (8.1\%)& 2 (9.1\%)\\
bing.com & 5 (2.0\%)& 5 (2.3\%) & 4 (2.8\%) & 3 (4.0\%) \\
yahoo.com & 4 (3.1\%)& 4 (2.6\%) & 5 (2.4\%) & 4 (2.0\%)\\
google.com &2 (6.5\%)& 2 (6.9\%)& $3$ (3.0\%) & $5$ (1.7\%) \\
twitter.com & 13 ($0.5$\%)& 9 ($0.5$\%) & 8 ($0.6$\%)& 8 (0.6\%)\\
\hline
\end{tabular}
\end{table}
\begin{figure}[]
\centering
\includegraphics[width=0.46\textwidth]{fig/SE_OSN_distribution.pdf}
\caption{Fraction of distinct web pages per clickstream graph that are directly connected to or reachable from SEs and OSNs. Paths starting from SEs include a sequence of multiple web page visits, while OSNs typically promote a single web page.}
\label{fig:reach_search_engines}
\end{figure}
\begin{figure}[]
\centering
\subfloat[PC]{
\label{fig:PC_OSN_a}
\centering
\includegraphics[width=0.45\textwidth]{fig/distanceGoogleFBmean_20_PC.pdf}
}
\subfloat[Smartphone]{
\label{fig:PC_OSN_b}
\centering
\includegraphics[width=0.45\textwidth]{fig/distanceGoogleFBmean_20_mobile.pdf}
}
\caption{Average number of web pages directly connected to and reachable from SEs and OSNs.
SEs promote more content than OSNs. On smartphones, OSNs have become as important as SEs to directly reach content.}
\label{fig:PC_OSN}
\end{figure}
We now focus on individual clickstream graph to better understand (i) how the usage of SEs and OSNs varies across users and devices; and (ii) if users keep visiting other web pages after their first visit from a promoter. We manually build a list of the top-50 SEs and OSNs that promote content. To verify to what extent users explore the web after leaving any of these domains, we define the concept of \textit{reachability}: we say that a destination web page is \textit{reachable} if there exists a direct path in the clickstream graph from the promoter web page to the destination web page. A web page is {\it directly connected} if it is a child of the promoter. We consider June 2016, and browsers that visited at least 20 web pages.
Given a browser clickstream graph, consider the fraction of visited web pages that are directly connected to (or reachable from) an SE or an OSN.
Figure~\ref{fig:reach_search_engines} shows the ECDFs of this fraction across all graphs.
For instance, consider the SE curves. 10\% of graphs have a fraction of 0\%, meaning that 10\% of browsers have no web page connected to any SE.
The median fraction of web pages per graph which are directly connected to a SE is approximately 17\%, i.e., 17\% of web pages in a browser are found thanks to the direct support of a SE.
Compare now the fraction of web pages reachable and directly connected to SEs. Figure~\ref{fig:reach_search_engines} shows that there are many more web pages that are reachable from search engines (median of $42\%$) than web pages directly connected to them. This indicates that users keep browsing the web after leaving a SE, and keep visiting web pages after the initial one.
OSNs, on the other hand, exhibit a very different pattern. First, the fraction of web pages to which they generate visits is much lower, with about $65\%$ of graphs that contains no web pages visited from any OSNs. Second, the \textit{Direct} and \textit{Reachable} curves are very close to each other, suggesting that OSN users visit some external web page (direct visit), but then do not continue visiting any other web page (indirect visit). Overall, $61\%$ of the web pages connected to an OSN have no other child.
This shows some ``addictiveness'' of OSN users: they click on external links, but then get back to the OSN web page, without continuing exploring the web.
At last, we track the evolution over time of the average direct and reachable fraction of web pages from SEs and OSNs. Figs.~\ref{fig:PC_OSN_a} and \ref{fig:PC_OSN_b} report results for PCs and for smartphones, respectively. PC browsers show little changes, with SEs being their preferred means to start their browsing paths and to discover content. OSNs are less often used to discover web pages, and very rarely to reach web pages not directly promoted on the OSN itself (cf. Figure~\ref{fig:reach_search_engines}).
Instead, results for smartphones show a clear evolution over time: OSNs are much more important to discover content than for PC browsers. Even more interestingly, we observe a decreasing usage of SEs to look for content. Notice indeed that the fraction of web pages reached from OSNs is now comparable to the one reached from SEs.
\section{Introduction}
\label{sec:intro}
Since its introduction, the web has become the preferred means to access online information. Technology has evolved, from simple static web pages to dynamic applications that let users search for content, buy goods, spend time on social networks, i.e., to ``browse'' the web. Understanding how people interact with the web has been always a fascinating problem~\cite{bucklin_click_2009,kammenhuber_web_2006,broder_graph_2000,meusel_graph_2014} for a variety of purposes like improving search engines~\cite{bai_discovering_2011}, comparing rankings in the web~\cite{craswell_random_2007,joachims_optimizing_2002}, recommending content to people~\cite{Mele:2013:WUM:2433396.2433493}, or increasing privacy and security~\cite{wang2013you}.
Browsing activities have been typically modeled using graphs -- or clickstream graphs as they are typically called -- where vertices are the visited web pages, and edges are the followed hyperlinks. They capture the paths that users take when navigating through websites. The evolution of the web, obviously, changes how users interact with it. We today witness the predominance of a handful of popular online services~\cite{gehlen_uncovering_2012} and the rise of mobile devices. How are these factors changing the way we browse the web?
In this paper we are interested in answering the following two questions:
\begin{itemize}
\item How are the clickstream graphs affected by the web evolution over the past years?
\item What are the differences between clickstream graphs from different browsing devices (\mbox{e.g.,}\xspace PCs and smartphones)?
\end{itemize}
We provide a longitudinal characterization of the clickstream graphs. Fundamental to answer these questions is the availability of data. Previous studies are either outdated~\cite{huberman_strong_1998}, or focused on small sets of volunteers~\cite{tossell_characterizing_2012,sellen_how_2002,weinreich_not_2008}, on user interactions with search engines~\cite{DuarteTorres:2014:ASB:2600093.2555595} or proxy logs~\cite{adar_large_2008}. Only few studies have used passive network traces to study browsing behavior~\cite{kammenhuber_web_2006,xie_resurf_2013}. In this paper, we leverage a three-year long anonymized dataset of passive measurements from an ISP network, thus offering a privileged observation point. The probes monitored 25\,000 \emph{households} and observed more than 64 billion HTTP requests. From this dataset, we extract the subset of requests related to explicitly visited web pages (user-actions) from the mass of objects automatically fetched by browsers (automatic-actions). This is a complicated task, since modern web pages are complex and include many HTML files, JavaScript, and multimedia objects. We tackle it by adopting a novel approach based on machine learning algorithms.
Given the requested web pages and the hyperlinks followed by people, we build \emph{clickstream graphs for each browser in a household}. In total, we have 5.5 million graphs corresponding to over 1 billion visited web pages. We then exploit this dataset to investigate browsing habits, providing the evolution over three years, and carefully characterizing differences in usage according to device types. In summary, this paper makes three contributions:
\begin{itemize}
\item We propose a new approach for the identification of web pages explicitly visited by users in HTTP logs collected by passive network monitors. Our approach generalizes ad-hoc designed heuristics~\cite{ihm_towards_2011,xie_resurf_2013,houidi_gold_2014}, automatically learning the patterns that characterize explicit visits, with detection precision and recall over 90\%.
\item We present a characterization of clickstreams that differs from previous efforts~\cite{xie_resurf_2013,kumar_characterization_2010} for (i)~covering a large population during a long consecutive period of time, and (ii)~accounting for different device types used to browser the web at home.
\item We contribute to the community a three-year long dataset of anonymized clickstreams, covering thousands of households in Europe. To the best of our knowledge, this is one of the largest datasets that includes clickstream graphs from regular internet users, browsing with multiple devices.
\end{itemize}
We focus on global patterns, highlighting when there are general trends in surfing habits, rather than unexpected but rare events. Our analysis confirms and precisely quantifies many intuitions about the way people navigate the web, besides leading to a number of interesting findings:
\begin{itemize}
\item Search Engines (SEs) and Online Social Networks (OSNs) are among the preferred means to discover content. As of 2016, 54\% of web domains were visited starting from Google, and 9\% (6\% in 2013) starting from Facebook. SEs are starting point of longer and deeper navigation, while content promoted by OSNs typically generates visits to a single or very few web pages. Interestingly, OSNs are much more important to discover content on smartphones than on PCs, a result previously not highlighted.
\item Web page complexity has continuously increased from 2013 to 2016, with URLs intentionally visited by users going from 2\% to 1.5\% of the total number of URLs requested by browsers.
\item The number of devices and applications used to browse the web at home has increased significantly, with smartphones and tablets accounting for 29\% and 9\% of the visited web pages in 2016, respectively. Users are interacting more frequently with the web from their smartphones at home than in the past~\cite{xie_resurf_2013}. However, in a session on a mobile app on average only $5$ web pages are visited, in a time span of only 2 minutes.
\item When considering the number of visited web pages, we observe that 50\% of the clickstream graphs include less than 27 web pages per day for PCs (8 for smartphones), belonging to less than 9 domains (4 for smartphones). Considering consecutive visited web pages, \mbox{i.e.,}\xspace a path, we observe that people stay in very few domains, even when
navigating through hundreds of web pages. These numbers have mostly remained constant over the years, despite changes in devices and applications used to browse the web.
\item Encryption has gained momentum in the web with many popular domains migrating to HTTPS. We clearly see the impact of HTTPS on properties of the clickstream graphs. Still, in June 2016, only around 13\% of the domains are served (partly or totally) in HTTPS, and 85\% of the encrypted traffic is related to the top-20 content providers, like Google and Facebook. Through data-driven simulations, we also provide estimations for the impact of HTTPS migration on the clickstream graphs.
\end{itemize}
Findings and the contributed dataset have several implications to the Internet actors. For example, they can (i)~help advertisers to make informed decisions on whether to target ads campaigns on mobile or PC users; (ii)~help network operators to understand interests of users and recommend products and services; (iii)~help researchers to investigate privacy aspects related to properties of clickstreams learned from traffic; and, more generally, (iv)~help the research community to study the place of web technologies in people's life.
\subsection{Scope and Limitations}
The scope of our study is obviously limited by the coverage and characteristics of the dataset.
First, the evaluated dataset is limited to the non-encrypted part of the web. It however covers a particularly interesting period, in which the usage of HTTPS has grown from negligible to noticeable percentages. We analyze from different points of view the impact of encryption on our dataset. Despite the growth on HTTPS usage, the majority of the domains were still not encrypted by the end of the data capture in 2016. Moreover, transitions {\it from} popular encrypted domains {\it to} the unencrypted ones are still visible in the analysis. This happens because early adopters of full HTTPS deployments are large content promoters (\mbox{e.g.,}\xspace Google and Facebook) that still inform non-encrypted domains the origin of visits. As such, we have no information about actions performed \emph{inside} these encrypted domains, but we see the transition when users eventually leave them towards unencrypted domains. An encrypted domain appears as a single vertex in a clickstream graph, connected to all vertices representing domains visited from it.
Second, no information to identify people exists in the dataset. Section~\ref{sec:datasets} will show that households (\mbox{i.e.,}\xspace home internet installations) are identified by anonymized keys, and browsers by \texttt{user agent} strings. Analyses are thus performed in a \emph{per-browser} level -- \mbox{i.e.,}\xspace each \texttt{user agent} string observed in a household. Naturally, people use several browsers to explore the web, and several persons are aggregated in a household. Privacy requirements however limit any different granularity.
Third, the evaluated dataset includes only a regional sample of households in Europe. Users in other regions may have diverse browsing habits that result in different clickstreams. Equally, mobile devices have been monitored only while connected to home WiFi networks. As such, our quantification of browsing on mobile terminals is actually a lower-bound, since visits while connected to other technologies are not captured.
Last, as in any large-scale analysis of real-world measurements, many preparation steps have proven essential to clean up spurious data and reduce biases on results. For instance, we have observed non-standard implementations of HTTP protocols by some browsers that prevent the reconstruction of the clickstreams in certain situations. Equally, we were faced with many challenges to reconstruct clickstreams on mobile terminals, given the diverse ways that mobile apps operate. We will elaborate further about these technical aspects in Section~\ref{sec:datasets} and on Appendix~\ref{appendix:a}.
\subsection{Reading Map}
After defining the problem, introducing the terminology used throughout the paper (Section~\ref{sec:probTerm}) and discussing related works (Section~\ref{sec:related}), we describe (Section~\ref{sec:validation}) and assess performance (Section~\ref{sec:clasPerf}) of the classifier used to extract visited web pages from raw HTTP logs.
We then apply it to a longitudinal dataset (described in Section~\ref{sec:datasets}) and characterize the browsing habits (Section~\ref{sec:ContentConsumption}), clickstreams (Section~\ref{sec:clickstream}) and content promoters (Section~\ref{sec:content_discovery}). We then conclude the paper (Section~\ref{sec:future}) and present some final technical details about the extraction of clickstreams from HTTPS to HTTP transitions (Appendix~\ref{appendix:a}).
\section{Dataset and traffic characteristics}
\label{sec:datasets}
\subsection{ISP traces}
We evaluate a long-term dataset of HTTP logs captured in a European ISP network to study how users interact with the web and how such interactions evolve over time. Three probes running Tstat have been installed in Points of Presence (PoPs) in different cities, where they observe about 25\,000 households overall. Each household is assigned and uniquely identified by a static IP address. Users connect to the Internet via DSL or FTTH, using a single access gateway offering Ethernet and WiFi home network.\footnote{The sets of households may have slowly changed over 3 years. We however consider always daily and monthly statistics which are marginally affected by such changes.}
Tstat was used to capture HTTP logs, and it saves flow-level statistics simultaneously to the collection of HTTP logs. These statistics include the number of exchanged bytes/packets, flow start/end timestamps, and, for each HTTP request/response pair, timestamp, server hostname, client IP address, URL, \texttt{referer}, user agent string, content type, content length, and status code.\footnote{To reduce the privacy risks, Tstat anonymizes IP addresses and removes parameters from URLs in both \texttt{GET} and in the \texttt{referer} fields.} We rely on the \emph{Universal Device Detection} library\footnote{https://github.com/piwik/device-detector} to parse user agent strings and infer the type of devices (\mbox{e.g.,}\xspace PC, tablet, smartphone, etc.) and the application used. The library operates by matching the user agent strings against a collection of regular expressions describing the different devices.
Tstat also implements DPI mechanisms to identify application layer protocols, such as HTTP and HTTPS. Moreover, Tstat records the server Fully Qualified Domain Name (FQDN) the client resolved via previous DNS queries, using its DN-Hunter plugin~\cite{bermudez_dns_2012}. This mechanism allows us to know which FQDN the client contacted when accessing a given server IP address, and track the usage of HTTP/HTTPS per domain.
We evaluate data collected during 3 years from July 2013 until June 2016. Table~\ref{tab:datasets} summarizes the dataset. In total, Tstat logged information about more than 64 billions of HTTP requests, from which 1.1 billion user-actions are identified. Note that the probes have had some outages during the course of the data collection -- the exact number of days in which each probe was active is shown in Table~\ref{tab:datasets}. We will not show results for the analysis affected by partial outages.
\begin{table}[!tb]
\centering
\small
\caption{Summary of the ISP traces.}
\label{tab:datasets}
\begin{tabular}{|x{0.9cm}|x{2.0cm}|x{2.2cm}|x{2.0cm}|x{1.0cm}|x{0cm}}
\cline{1-5}
Name & Households & HTTP Requests & User-actions & Days & \\[0.75ex]
\cline{1-5}
\mbox{\emph{PoP~1}}\xspace & $\approx 10\,000$ & 28.8~billions & 477~millions & 1\,068 & \\[0.75ex]
\mbox{\emph{PoP~2}}\xspace & $\approx 13\,000$ & 30.3~billions & 494~millions & 752 & \\[0.75ex]
\mbox{\emph{PoP~3}}\xspace & $\approx 2\,000$ & 5.3~billions & 79~millions & 600 & \\[0.75ex]
\cline{1-5}
Total & $\approx 25\,000$ & 64.4~billions & 1.1~billions & -- & \\[0.75ex]
\cline{1-5}
\end{tabular}
\end{table}
The dataset captures how users of this ISP interact with the web. Users in different parts of the world will certainly access other domains and services. Thus, some of the results we will present next, such as about top domains promoting content, are certainly specific to this dataset. However, it covers tens of thousands of households and appear representative of the monitored country. For instance, no significant differences are observed among probes.
In the remaining of this section we provide an overview of the dataset and discuss possible limitations, issues and steps we adopt to avoid biases in the analyses.
\subsection{Impact of HTTPS} \label{sec:impact}
Our study is limited to the non-encrypted part of the web. A recent work~\cite{felt2017measuring} reports that HTTPS was responsible for around 45\% of the user-actions by the last month of our data capture. Authors rely on direct instrumentation of Chrome and Firefox. Usage of HTTPS is similar across terminals, but with lower figures on smartphones than on PCs. For example, 38\% (47\%) of the user-actions are over HTTPS for Chrome on Android (Windows) on 4th June 2016. A steady increasing trend on the deployment of HTTPS is observed, but numbers only cover the last year of the ISP traces.
We rely on flow-level statistics saved by Tstat to gauge the effect of HTTPS during the complete duration of the evaluated traces. We quantify how many domains were running over HTTPS and their traffic characteristics (in terms of downloaded bytes). Naturally, we cannot see the exact number of user-actions on HTTPS, but only conjecture how trends reported by~\cite{felt2017measuring} have evolved during the data capture.
Figure~\ref{fig:https} shows the share of domains in \mbox{\emph{PoP~1}}\xspace relying on HTTP only, \mbox{i.e.,}\xspace completely without encryption. The figure also reports the share of bytes on HTTP. Other probes are omitted since they lead to similar results.
\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{fig/https.pdf}
\caption{Incidence of HTTP in the dataset. Fraction of domains in HTTP only, and fraction of their byte volumes. Only around 13\% of the domains adopted HTTPS (partly or totally) in June 2016.
}
\label{fig:https}
\end{figure}
The vast majority of domains relied only on HTTP in 2013 (left-most point in the figure). About $96\%$ of the domains were exclusively running over HTTP, with another $3\%$ using both HTTP and HTTPS. Yet, the overall number of domains using exclusively HTTP remains high -- \mbox{e.g.,}\xspace $87\%$ of the domains, with $6\%$ relying on both protocols in 2016. In terms of downloaded bytes, HTTP was responsible for more than $90\%$ of the traffic in 2013, and the percentage of was reduced to around $55\%$ in 2016 -- \mbox{i.e.,}\xspace $45\%$ of the download traffic was encrypted by the end of our capture.
This discrepancy between domains on HTTP and the HTTP traffic, both in terms of bytes seen in the traces and user-actions reported by~\cite{felt2017measuring}, can be explained by the fact that early-adopters of HTTPS-only deployments are among the most popular domains in the Internet. In fact, observe in the figure the increasing trend on the deployment of HTTPS in 2014, which is related to the migration of YouTube to HTTPS.
To further understand which user-actions are missing due to HTTPS, we have studied the domains that were popular in July 2013. For domains presenting a sharp decrease on popularity (more than 60\% reduction on the number of user-actions), we have manually investigated whether they have switched to HTTPS or not. Figure~\ref{fig:timeline_protocols} reports the timeline of some of the relevant migrations. The traces capture the final periods of Facebook and Google Search migration to HTTPS by default. Other popular domains, such as those from Yahoo and LinkedIn, migrated during 2014. CloudFlare started offering universal HTTPS support for its customers towards the end of 2014. Wikipedia switched to HTTPS in 2015. The migration trend accelerated in the last months of the capture with several (less-popular) domains switching to HTTPS. By the end of the capture, 24 out of the top-100 domains in 2013 have switched to HTTPS.
\begin{figure*}[!t]
\scriptsize
\startchronology[startyear=2013,stopyear=2017,dates=false,color=gray,arrowheight=8px,arrowwidth=15px]
\chronoevent[date=false,markdepth=70pt,textwidth=1.5cm]{01/09/2013}{Google Search completely HTTPS}
\chronoevent[date=false,markdepth=10pt,textwidth=1.5cm]{01/01/2014}{Yahoo}
\chronoevent[date=false,markdepth=50pt,textwidth=1.5cm]{01/03/2014}{YouTube starts the HTTPS switch}
\chronoevent[date=false,markdepth=35pt,textwidth=1.5cm]{15/07/2013}{Facebook finishes the HTTPS switch}
\chronoevent[date=false,markdepth=25pt,textwidth=1.5cm]{01/04/2014}{LinkedIn}
\chronoevent[date=false,markdepth=15pt,textwidth=1.5cm]{01/09/2014}{CloudFlare starts offering Universal SSL}
\chronoevent[date=false,markdepth=20pt,textwidth=1.5cm]{01/06/2015}{Wikipedia}
\chronoevent[date=false,markdepth=35pt,textwidth=1.5cm]{01/12/2015}{TripAdvisor, Twoo and others}
\chronoevent[date=false,markdepth=15pt,textwidth=1.5cm]{01/11/2015}{Ask.fm}
\chronoevent[date=false,textwidth=1.5cm]{01/07/2013}{Capture start (07/2013)}
\chronoevent[date=false,textwidth=1.5cm]{01/07/2016}{Capture end (06/2016)}
\stopchronology
\caption{Timeline showing when popular domains running on HTTP in 2013 migrated to HTTPS.}
\label{fig:timeline_protocols}
\end{figure*}
Further checking the HTTPS traffic, we found that around $85\%$ of the HTTPS bytes in 2016 come from the top-20 domains. Google and Facebook alone account for around $65\%$ of the HTTPS traffic. Relevant for our analysis, a large number of HTTPS domains still pass on the \texttt{referer} information when users transition from the HTTPS domains to any other HTTP domain. This happens because content promoters (\mbox{e.g.,}\xspace Google, Facebook, Yahoo, Twitter etc) have interest in informing others the origin of visits. Technical details are discussed in Appendix~\ref{appendix:a}. Thus, whereas we miss user-actions \emph{inside} HTTPS domains of content promoters, such as Google and Facebook, we still see the information about users' origin when they leave these services. For instance, Google appears as a single vertex in the graph, with edges linking it to web pages visited after users leave its services.
To observe what the migration of services to HTTPS may cause, we set up an experiment, based on trace-driven simulations, to estimate the impact of this. We took all graphs from one full month in 2013 (July), when there was still only a minor portion of HTTPS traffic, and recomputed metrics after removing user actions of an increasing number of websites from the data.~\footnote{Note that pinpointing \emph{all} websites that migrated to HTTPS is not possible. In our simulation, we assume the most popular services are among the first to migrate (which is close to what is observed in reality).} We then incrementally recalculated the metrics until the percentage of missing user actions have reached around 45\% of the total, which is the estimated percentage of missing user actions by the end of our capture in 2016. Notice that this procedure is different from removing 45\% user actions at random, since user actions related to a single site have temporal correlations. Our methodology instead mimics the migration of full popular sites to HTTPS.
The simulation provides more evidences to support the conjectures in the paper. Detailed results will be described in the next sections, alongside the real data.
\subsection{Caveats for the detection of user-actions and clickstreams}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.44\textwidth]{fig/user_actions.pdf}
\caption{Effects of the increasing complexity of web pages on the percentage of user-actions among HTTP requests.}
\label{fig:click_percentage}
\end{center}
\end{figure}
\subsubsection{Effect of web page complexity evolution}
We investigate how the complexity of web pages has changed during the data collection. In particular, given our machine learning approach to detect user-actions, we are interested in checking whether key features have varied significantly throughout the years. Major changes in features would affect the performance of the classification models, which have been trained with data collected from volunteers simultaneously to the last months of the data capture at the ISP network.
We observe that web pages have become more complex in recent years. This observation is inline with previous works~\cite{butkiewicz_characterizing_2014}, which report an increasing trend in the number of objects needed to render web pages. As an example, the median number of children for user-actions is increased by about $40\%$ from 2013 to 2016. As a consequence, the percentage of user-actions among all HTTP requests is decreasing. Figure~\ref{fig:click_percentage} illustrates this effect by depicting the overall percentage of user-actions over time. Observe how similar the trends are across the datasets. The percentage of HTTP requests corresponding to user-actions has decreased from close to $2\%$ in 2013 to less than $1.4\%$ in 2016.
Even if web pages are becoming more complex, the impact on features relevant to our classifier is limited. As an example, the overall mean number of children per HTTP request is more or less constant between 0.71 and 0.75 in three years, and the vast majority of HTTP requests has no children at all. We remind that the number of children is one key feature used by the decision tree to identify user-actions. Similar observation holds for other features used by the trained decision tree.
\subsubsection{\texttt{referer} artifacts}
\begin{figure}[!t]
\centering
\includegraphics[width=0.45\textwidth]{fig/referer.pdf}
\caption{Fraction of missing \texttt{referer} for PC browsers and for the anomalous Android Samsung browser in Tablets.}
\label{fig:ref_missing}
\end{figure}
The analysis of clickstreams depends on the \texttt{referer} field. Previous work~\cite{schneider_understanding_2009} has reported artifacts related to lack of \texttt{referer} in HTTP requests. Artifacts related to missing \texttt{referer} are expected to be caused by bugged browser implementations or middle-boxes. The latter is not present in our scenario. We then study the number of HTTP requests that miss \texttt{referer} per browser.
Figure~\ref{fig:ref_missing} illustrates the percentages of missing \texttt{referer} when aggregating all browsers running on PCs (red points). Percentages are between $10\%$ and $15\%$ in the three datasets. This behavior is consistent if we check different PC browsers in isolation, as well as most browsers running on smartphones and tablets. This small percentage of missing \texttt{referer} is expected. It is caused by normal browsing activity, such as when users request the first web page after the browser is loaded or when web pages are loaded starting from bookmarks.
However, a completely different picture emerges for few browsers. Figure~\ref{fig:ref_missing} shows that the percentage of missing \texttt{referer} for a specific Android browser running on Samsung Tablets is much higher than in other cases (green points). This behavior is restricted to particular versions of the browser. We see that the percentage of missing \texttt{referer} was close to $60\%$ in 2013 and has continuously grown as more users updated to the versions that skip the \texttt{referer} in HTTP requests. We discard such abnormal browsers in the remaining analyses to avoid biases.
\subsubsection{Clickstreams on mobile terminals}
Some data preparation and filtering is needed to study clickstreams on mobile terminals. Several apps are simply ordinary browsers that behave like PC browsers -- \mbox{e.g.,}\xspace Chrome, Firefox, Safari, Samsung Browser etc. They allow users to move between web pages and domains and pass on the \texttt{referer} information on each transition. Similarly, many apps include their own browsers and allow users to navigate to other web pages and domains without leaving the app. This category includes Facebook, Instagram, Flipboard, Messenger, Gmail among others. Here again, the \texttt{referer} information is passed on normally. Each of these apps sends out a customized user agent string and, as such, they are treated as independent browsers given our definitions (see Table~\ref{tab.nom}).
However, several apps constrain users to few operations, and rely on other browsers (\mbox{e.g.,}\xspace Chrome) or third-party apps (\mbox{e.g.,}\xspace Google Maps) to handle external links. These inter-app transitions are built based on different APIs, and the behavior is not standard across apps and mobile operating systems. As an outcome, the \texttt{referer} information, as observed in the network, when switching between apps is not reliable -- sometimes the \texttt{referer} is an arbitrary string instead of a URL, and often the \texttt{referer} is simply not present.
We have manually evaluated all popular browsers seen in the traces, and ignored browsers that do not allow users to navigate through different pages and domains.
\section{Problem and Terminology}
\label{sec:probTerm}
The key terminology used throughout the paper is summarized in Table~\ref{tab.nom}.
\begin{table}[!h]
\centering
\caption{Summary of the key terminology used throughout the paper.}
\label{tab.nom}
{
\small
\begin{tabular}{|x{2cm}|y{10cm}|x{0cm}}
\cline{1-2}
User-action & The initial HTTP request triggered by a user interaction with a browser. Informally, a user-action is a click or a visit to a web page. & \\[0.75ex]
\cline{1-2}
Web page & A URL without parameters (\mbox{e.g.,}\xspace \texttt{http://www.example.com/index.html}). Many user-actions thus can be related to the same web page. & \\[0.75ex]
\cline{1-2}
Domain & The pair formed by the second-level domain and the top-level domain of a URL (\mbox{e.g.,}\xspace \texttt{example.com}). Informally, a domains is a website. & \\[0.75ex]
\cline{1-2}
Clickstream & An ordered list of user-actions from a browser that can be represented through a graph.& \\[0.75ex]
\cline{1-2}
Browser & An application, identified by its user agent string, used in a \textit{household} to navigate through multiple web pages and domains. & \\[0.75ex]
\cline{1-2}
\end{tabular}
}
\end{table}
We are interested in studying how people surf the web, based on the analysis of raw HTTP logs. One challenge is to extract \emph{user-actions} from such noisy logs. Indeed, rendering a web page is a rather complex process that requires the browser to download HTML files, JavaScript, multimedia objects and dynamically generated content.
All these objects are retrieved by the browser by means of independent HTTP requests. We define the initial HTTP request triggered by a user interaction with the browser as a \emph{user-action}, and as \emph{automatic-actions} all the remaining HTTP requests fired to render web pages.
User-actions thus correspond to \emph{web pages} explicitly visited by a user (\mbox{e.g.,}\xspace \texttt{http://www.example.com/index.html}), and the two terms will be interchangeable in this paper. For each web page, we know its second-level domain (\mbox{i.e.,}\xspace \texttt{example}) and top-level domain (\mbox{i.e.,}\xspace \texttt{com}), which combination (\mbox{e.g.,}\xspace \texttt{example.com}) we will simply call a \emph{domain}.
Figure~\ref{fig:classificationProblem} illustrates these definitions. It depicts the timeline of a user surfing the web. The user visits five web pages, whose corresponding user-actions are marked by tall red arrows. Following each user-action, the browser fires automatic-actions to fetch objects, which we mark with short blue arrows.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{./fig/Drawing1.pdf}
\caption{Example of a client browser activity and its relation to our terminology.}
\label{fig:classificationProblem}
\end{figure}
We aim at characterizing browsing habits and check how they are evolving. We call the \emph{clickstream} the list of user-actions. The clickstream is typically modeled as a directed graph, where web pages constitute the vertices, and edges represent the movement of a user through web pages, i.e., when the user clicks on a hyperlink.
The right-hand side of Figure~\ref{fig:classificationProblem} illustrates the clickstream extracted from the navigation example. Two {\it components} are present: user-actions 2--3--4 are reached following hyperlinks starting from user-action 1, while user-action~5 is a web page reached independently.
We assume a monitoring infrastructure exposes HTTP logs. Examples of such infrastructure are web proxies and network probes that extract information from packets crossing a measurement point. HTTP logs contain records about HTTP requests. Each record includes (i) the time of the request; (ii) the requested URL; (iii) the HTTP \texttt{referer}, i.e., the HTTP header that identifies the URL of the web page that linked to the resource being requested; and (iv) the user agent, i.e., an identification of the browser application sending the HTTP requests.
HTTP does not specify any mechanism for signaling if a request is a user-action or not. As such, HTTP URLs are indistinctly mixed in HTTP logs. Thus, the first problem we target is the identification of user-actions in the stream HTTP requests. URLs identified as user-actions become the clickstream vertices. If a \texttt{referer} is present, it represents a directed edge {\it from} the URL in the \texttt{referer} {\it to} the user-action URL.
As mentioned above, the data we use in our analysis do not provide any user identifier for obvious privacy reasons. We will rely on anonymized IP addresses which, in our case, uniquely identify a \textit{household}, from which several users using different applications and devices may connect. We define the \textit{browser clickstream} as the user-actions performed by a particular {\it browser}\footnote{We use the term browser to refer to any specific application that uses HTTP to fetch resources from the web, e.g., traditional web browsers, mobile apps, etc.} in a household, and thus characterized by the pair of anonymized IP address and user agent string. Obviously, a browser clickstream is not equivalent to the entire activity of a physical person since the same person could use different browsers.
At last, while the user agent string identifies a specific application and version, it also exposes the operating system and type of device being used. We here coarsely group browser clickstreams into three classes of browser devices, namely PCs (including desktops and laptops), tablets, and smartphones.
\section{Related work }
\label{sec:related}
\subsection{Identification of user-actions}
The user-action detection problem we address is similar to web page view identification, a part of the data cleaning process in web usage mining~\cite{Sri00}. Web usage mining is historically the task of extracting knowledge from HTTP server-side log files. As such, this task was traditionally tailored on a per website basis. For instance, web page view identification from web server logs leverages the a priori known structure of the website, and is performed often by discarding a manually constructed list of embedded object extensions from the logs~\cite{S15,SU13}.
This approach does not work in our setting because we aggregate logs from a variety of heterogeneous web servers with different website structures, naming conventions, Ad networks and CDN providers. It is thus not feasible to manually construct a list of extensions to discard.
Previous works have introduced different methods for identifying user-actions from network-based HTTP traces. \mbox{StreamStructure}~\cite{ihm_towards_2011} exploits the \texttt{referer} field in HTTP requests and the widespread deployment of the \emph{Google Analytics beacon} to reconstruct web page structures from HTTP traces and identify user-actions. The authors of~\cite{xie_resurf_2013} follow a similar approach, exploiting the \texttt{referer} field to group requests into HTTP streams. A series of manual rules are used to decide whether a request is the consequence of a user-action or not. Precision and recall above 90\% are claimed on synthetic traces. Finally, the authors of~\cite{houidi_gold_2014} present a heuristic to identify user-actions that operates only with the HTTP requests. The proposed heuristic is shown to scale well in high-speed networks, claiming 66\%--80\% precision and 91\%--97\% recall, depending on parameter choices.
In our previous work~\cite{VD16}, we have introduced a machine learning approach to detect user-actions. In contrast to previous efforts, our approach is fully automatic, does not require manually tuned parameters and is validated with traces from real users.
The present work extends~\cite{VD16} by validating the methodology on mobile browsers and by using a long-term dataset of HTTP logs collected in an operational network.
\subsection{Web characterization and clickstream analyses}
In the last decade, several works focus on the behavior of end-users -- \mbox{e.g.,}\xspace to determine how often web pages are revisited and how users discover and arrive to web pages. The authors of~\cite{weinreich_not_2008, obendorf_web_2007} characterize web usage exploiting few volunteers' browsing histories. They find that navigation based on search engines and multi-tabbing are changing the way users interact with browsers -- \mbox{e.g.,}\xspace direct web page visits based on bookmarks and backtracking are becoming less popular. A similar study is presented in~\cite{sellen_how_2002}, based on device instrumentation and a small user population, and in~\cite{adar_large_2008}, based on proxy logs.
The closest to our work is~\cite{kumar_characterization_2010}, which leveraged the Yahoo toolbar to analyze web page views from a large population of heterogeneous users over a period of one week. Some of our findings confirm theirs (e.g., deep browsing after leaving search engines) while others present different figures compared to what they discovered (e.g., we observe a higher weight of social networks in referral share and less no-referer traffic). However, with the very fast evolution of the networking technologies and the web, the question whether these results still hold nowadays is raised. Our study answers it on various aspects.
With the emergence of connected mobile devices, the works focusing on mobile users' behavior have multiplied.
The authors of \cite{cui_how_2008} propose a taxonomy of usage of the Internet for mobile users, where data are extracted thanks to contextual inquiries with volunteers. They retrieve three already known categories (\textit{information seeking}, \textit{communication} and \textit{transaction}) and identify a new one: \textit{personal space extension}, anticipating the wide usage of cloud storage systems. The authors of \cite{bohmer_falling_2011} focus on the usage of mobile applications, showing that sessions are generally very short (less than one minute on average).
The authors of \cite{gerpott_empirical_2014} perform a survey on mobile Internet behavior, concluding that the approach to measure the usage (passive/active, objective/subjective, etc.) could heavily impact the results.
A recent work~\cite{ren_analyzing_2017} analyses the Wi-Fi access logs of a city shopping mall, showing that the user revisit periodically the same web content.
Considering HTTPS usage on mobile devices, the authors of~\cite{finamore_mind_2017} study HTTP/HTTPS deployment in mobile networks, finding that HTTPS is mainly used by large internet services. Our work confirm many of these trends. However, we not only present aggregated statistics about protocol usage, but also extract clickstreams from HTTP traffic.
The comparison between mobile and PC Internet usage gave rise to a lot of studies. For example, the authors of~\cite{papapanagiotou_smartphones_2012} present a comparison of objects retrieved from PCs and smartphones, and implications for caching, but without distinguishing user-actions. The authors of~\cite{tossell_characterizing_2012} compare smartphone and PC navigation, concluding that web page revisits are rare in smartphones, while bookmarks are more widely used on smartphones than on PCs.
The authors of~\cite{song_exploring_2013} study the differences in searching behavior of mobile, tablet and PC users. They show that most clicked websites depends on the used device, suggesting that these differences should be taken into account to design specific ranking systems.
On a more sociological side, the authors of~\cite{pearce_digital_2013} note that the computers can increase the use of \textit{capital enhancing} activities and to palliate the unavailability of Internet access using mobile devices is not enough to restrain the digital divide. The authors of~\cite{oulasvirta_habits_2012} note that mobile usage induces a \textit{checking habit} for smartphone, consisting of quick and repetitive content inspection.
Considering clickstream analyses, the work presented in~\cite{huberman_strong_1998} studies the number of web pages that a user visits within a single website, while the one in~\cite{kammenhuber_web_2006} analyses the relations between queries on a search engine and followed paths. On another side, the authors of~\cite{tikhonov_what_2015} define navigation profiles considering data exported by a browser toolbar in Russia, showing that the navigation path leading users to web pages characterizes properties of the destination web page. Finally, the authors of~\cite{olmedilla_mobile_2010} focus primarily on the categories of websites that mobile users access.
In contrast to our work, all these previous works rely on somehow limited vantage points. Here we capture HTTP traces from a large ISP to extract clickstreams, considering different devices and users while connected at residential networks. Moreover, thanks to the three years of data capture, we provide a comprehensive analysis on how browsing behavior is evolving over time. Equally importantly, we study more aspects like the graph structure of clickstreams and the comparison between OSNs and SEs.
The authors of~\cite{xie_resurf_2013} present analyses of clickstreams from passive traces collected for two months. Our work reappraisals some results in~\cite{xie_resurf_2013} -- \mbox{e.g.,}\xspace we confirm that clickstreams are small and restricted to few domains. However, the duration and richness of our data shed light on novel aspects of browsing habits. For instance, in contrast to the previous work, we show that browsing from mobile devices is getting more and more frequent, suggesting that it will soon surpass PC browsing even for users connected at home networks.
The web graph can be obtained thanks to active crawlers~\cite{broder_graph_2000,meusel_graph_2014,bai_discovering_2011}. Our work is orthogonal to those efforts, since they miss how users interact with the web. Contrasting the full web graph with the portion effectively visited by users is planned for future work. Our work can be of interest for the authors of~\cite{wang_unsupervised_2016}, who introduce a system to cluster clickstreams, aiming at mining knowledge from them. Our dataset is a unique source for such analyses, since it covers a large user population for a long period.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,181 |
Q: How to make python code read a certain line of text in a .txt file? So i have been fiddling around with python and league of legends. And i found out you can make notes in game. So i thought about making python code read a line of text from the note i made in game, like "Lux no flash", but it doesn't seems to be able to read it at all, it only works when i do it manually with the exact same code. Here is my code:
import os
import time
def main():
os.chdir('C:\\Riot Games\\League of Legends\\RADS\\solutions\\lol_game_client_sln\\releases\\0.0.1.237')
f=open("MyNotes.txt", "r")
if f.mode == 'r':
lines=f.readlines()
text = lines[4]
time.sleep(0.1)
if text == 'Lux no flash':
print('Done')
else:
print('Something went wrong')
f.close()
if __name__== "__main__":
main()
The output is "something went wrong", but when i do it manually it says "done". I feel like python cant read league code. Maybe you guys know how to do this... This is the .txt file im trying to access:
##################################################
2018-09-13_18-57-33_
##################################################
Lux no flash
A: I'm Just taking a file on an assumption basis:
# cat MyNotes.txt
there is Lux no flash in line
there is Something went wrong
There is nothing lux no flash
this is all test
So, just looking for the word 'Lux no flash' you are searching into your file, we can simply do as below.. but its case sensitive.
It's always best practice to use with open() method to read a file.
import os
import time
def main():
with open("MyNotes.txt") as f:
for line in f.readlines():
if 'Lux no flash' in line:
print('Done')
else:
print('Something went wrong')
if __name__== "__main__":
main()
Output result will be :
Done
Something went wrong
Something went wrong
Something went wrong
Even tried using the lux.txt , it works as expected with my code.
import os
import time
def main():
with open("lux.txt") as f:
for line in f.readlines():
#line = line.strip() # use can use the strip() or strip("\n")
#line = line.strip("\n") # if you see white spaces in the file
if 'Lux no flash' in line:
print('Done')
else:
pass
if __name__== "__main__":
main()
Resulted outout is:
# test.py
Done
A: Using lux.txt:
##################################################
2018-09-13_18-57-33_
##################################################
Lux no flash
Code:
content = []
with open('lux.txt', 'r') as f:
for line in f:
content.append(line.strip('\n'))
for i in content:
if 'Lux no flash' == i:
print("Done")
else:
pass
Better @pygo
with open('lux.txt', 'r') as f:
content = f.read()
if 'Lux no flash' in content:
print("Done")
else:
print("No else, this works :)")
Output:
(xenial)vash@localhost:~/python/stack_overflow$ python3.7 lux.py
Done
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,264 |
Basking in new light and energy.
Being open to the beauty of Earth and the passionate embrace of Love while still recognizing in your heart the pain, the grief, and the problems of the world. You are unafraid of this paradox.
Finding in the outside world or creating within your home a place to retreat to for prayer, meditation, or dream incubation.
Creating something beautiful by weaving with paper or yarn or twigs, and as you are weaving concentrate on your desires and dreams. Then their energy will be woven into the completed piece.
"Feathering your nest" in some special way, perhaps by cleaning up a space and bringing in a new inspiring element: art, flowers, curtains, lace.
POSTURE: Use Pile 3 / People cards for a question such as: What attitudes and behaviors will be most helpful to undertake this practice and walk with path? CHILD OF EARTH invites you to follow your path and practices with attitudes of ….
Learning from the physical reality before you (the peeking bulbs, the food on your plate, the animals in your life).
Seeing the mysteries of the universe in the common and mundane things of your world.
Tuning into your senses as teachers.
OVERALL MUSINGS: As a set, this week's cards invite us to make space in our life for what brings us joy – and then commune deeply with this joy-brining gift in our lives. We will have to do this amidst the daily demands of our lives and the raucous happenings in the wider world, but our time spent with joy can fill us with the energy needed to engage with these challenges.
Most people find the most joy in the simple things in life – being with loved ones, good food, a walk in nature, time spent fiddling with a creative project – so joy is generally waiting close by waiting for us to turn to its radiant presence.
What keeps us from doing so? Perhaps the most insidious distraction of our time is the constant pull of technology and social media. Our devices are ringing and dinging all day. The technology is not really the problem but rather our response to it. We are letting our ability to be constantly plugged in numb us and distract us rather than aid us, but it does not have to be this way.
This morning I read a NY Times piece on Digital Minimalism to support Deep Work. Just the words inspired me. The practices suggested are not difficult – and don't require you to give up all technology or social media use – but just require a shift to be intentional and proactive rather than distracted and responsive in our use. Making a shift such as this can open up space in our life for what really matters; it can make a space for joy.
If the work and words here at Soul Path Sanctuary speak to you, I hope it can be part of your intentional and proactive use of social media. I send an e-news letter once a moonth right before the new lunar phase begins and here on the blog offer weekly posts like these to support your spiritual practice. Be sure to be on my list to get the e-news. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,219 |
\section{Introduction}\label{sec1}
Coagulation-fragmentation equations are mean-field models describing the growth of clusters changing their sizes under the combined effects of (binary) merging and breakup. Denoting the size distribution function of the particles with mass $x>0$ at time $t>0$ by $f=f(t,x)\ge 0$, the coagulation equation with multiple fragmentation reads
\begin{eqnarray}
\partial_t f(t,x) & = & \frac{1}{2} \int_0^x K(x-y,y) f(t,x-y) f(t,y)\ dy - \int_0^\infty K(x,y) f(t,x) f(t,y)\ dy \nonumber \\
& & -\ a(x) f(t,x) + \int_x^\infty a(y) b(x|y) f(t,y)\ dy\ , \label{i1}
\end{eqnarray}
for $(t,x)\in (0,\infty)^2$. In \eqref{i1}, $K$ denotes the coagulation kernel which is a non-negative and symmetric function $K(x,y)=K(y,x)\ge 0$ of $(x,y)$ accounting for the likelihood of two particles with respective masses $x$ and $y$ to merge. Next, $a(x)\ge 0$ denotes the overall rate of breakup of particles of size $x>0$ and $b(x|y)$ is the daughter distribution function describing the probability that the fragmentation of a particle with mass $y$ produces a particle with mass $x\in (0,y)$. Conservation of matter during fragmentation events requires
\begin{equation}
\int_0^y x b(x|y)\ dx = y\ , \qquad y\in (0,\infty)\ , \label{i2}
\end{equation}
and since the total mass of particles is preserved during coagulation events, the total mass is expected to be constant throughout time evolution, that is,
\begin{equation}
\int_0^\infty x f(t,x)\ dx = \int_0^\infty x f(0,x)\ dx\ , \quad t>0\ . \label{i100}
\end{equation}
The total mass conservation \eqref{i100} is actually a key issue in the analysis of coagulation-fragmentation equations as it might be infringed during time evolution. More precisely, in the absence of fragmentation ($a\equiv 0$), it is by now known that the coagulation kernels $K$ split into two classes according to their growth for large values of $x$ and $y$: either
\begin{equation}
K(x,y)\le \kappa (2+x+y)\ , \quad (x,y)\in (0,\infty)^2\ , \label{i101}
\end{equation}
for some $\kappa>0$ and solutions to the coagulation equation are expected to be \textit{mass-conserving} and satisfies \eqref{i100}. Or
\begin{equation}
K(x,y)\ge \kappa (xy)^{\lambda/2}\ , \quad (x,y)\in (0,\infty)^2\ , \label{i102}
\end{equation}
for some $\lambda>1$ and $\kappa>0$ and conservation of mass breaks down in finite time, a phenomenon commonly referred to as gelation. From a physical viewpoint, it corresponds to a runaway growth of the dynamics, leading to the formation of a particle with infinite mass in finite time and resulting in a loss of matter as \eqref{i1} only accounts for the evolution of particles with finite mass. In other words, there is a finite time $T_{gel}$, the so-called \textit{gelation time}, such that
$$
\int_0^\infty x f(t,x) = \int_0^\infty x f(0,x)\ dx\ , \;\; t\in [0,T_{gel}) \;\;\text{ and }\;\; \int_0^\infty x f(t,x) < \int_0^\infty x f(0,x)\ dx\ , \;\; t>T_{gel}\ .
$$
Alternatively,
\begin{equation*}
T_{gel} := \inf{\left\{ t>0\ :\ \int_0^\infty x f(t,x)\ dx < \int_0^\infty x f(0,x)\ dx \right\}}\ .
\end{equation*}
The fact that gelation occurs for coagulation kernels satisfying \eqref{i102} was conjectured at the beginning of the eighties and supported by a few examples of explicit or closed-form solutions to \eqref{i1} \cite{LT81, Le83}. That it indeed takes place for such coagulation kernels and arbitrary initial data was shown twenty years later in \cite{EMP02}. Since fragmentation reduces the sizes of the clusters, it is rather expected to counteract gelation and it has indeed been established that strong fragmentation prevents the occurrence of gelation \cite{dC95} but does not impede the occurrence of gelation if it is too weak \cite{ELMP03, EMP02, Lt00, VZ89}. More specifically, for the following typical choices of coagulation kernel~$K$ and overall breakup rate~$a$
\begin{eqnarray}
K(x,y) & = & \kappa \left( x^\alpha y^\beta + x^\beta y^\alpha \right)\ , \quad (x,y) \in (0,\infty)\times (0,\infty)\ , \label{i3} \\
a(x) & = & k x^\gamma\ , \quad x\in (0,\infty)\ , \nonumber
\end{eqnarray}
with $\alpha \le \beta \le 1$, $\lambda:= \alpha+\beta>1$, $\gamma>0$, $\kappa>0$, and $k>0$, it is conjectured in \cite{VZ89, Pi12} that mass-conserving solutions exist when $\gamma>\lambda-1$ while gelation occurs when $\gamma<\lambda-1$, the critical case $\gamma=\lambda-1$ being more involved and possibly depending as well on the properties of the daughter distribution function $b$. Except for the critical case $\gamma=\lambda-1$, mathematical proofs of these conjectures are provided in \cite{dC95, ELMP03, EMP02} for some specific classes of daughter distribution functions $b$.
\medskip
More generally, due to the possible breakdown of mass conservation, the analysis of the existence of solutions to \eqref{i1} follows two directions since the pioneering works by Melzak \cite{Me57}, McLeod \cite{ML62a,ML62b,ML64}, White \cite{Wh80}, and Spouge \cite{Sp84}. On the one hand, several works have been devoted to the construction of mass-conserving solutions to \eqref{i1} for coagulation kernels satisfying \eqref{i101} under various assumptions on the breakup rate $a$ and the daughter distribution function $b$, see \cite{BC90, BL11, BL12, DS96b, ELMP03, La04, Lt02, MLM97, Me57, Wh80} and the references therein. On the other hand, weak solutions which need not satisfy \eqref{i100} have been constructed for large classes of coagulation kernels $K$, breakup rate $a$, and daughter distribution function $b$, see \cite{Ca06, EW01, ELMP03, EMRR05, GKW11, GLW12, Lt00, No99, St89}. Existence of mass-conserving solutions to \eqref{i1} for coagulation kernels satisfying \eqref{i102} with strong fragmentation has been established in \cite{dC95, ELMP03}. In addition, uniqueness of mass-conserving solutions has been investigated in \cite{dC95, GW11, No99, St90b}. A survey of earlier results and additional references may be found in \cite{Du94b, LM04}. Let us mention here that a common feature of the above mentioned works is that the total number of particles $n(y)$ resulting from the breakup of a cluster of size $y>0$ defined by
\begin{equation}
n(y) := \int_0^y b(x|y)\ dx\ , \label{i104}
\end{equation}
is assumed to be bounded or even constant and thus does not include the case where the breakup of particles could produce infinitely many particles. This possibility shows up in the class of breakup rates $a$ and daughter distribution functions $b$ satisfying \eqref{i2} which are derived in \cite{MZ87} and given by
\begin{equation}
a(x) = k x^\gamma\ , \quad b(x|y) = (\omega+2) \frac{x^\omega}{y^{\omega+1}}\ , \quad 0<x<y\ , \label{i4}
\end{equation}
with $k>0$, $\gamma\in\mathbb{R}$, and $\omega\in (-2,0]$. Clearly, the total number of particles $n(y)$ resulting from the breakup of a particle of size $y>0$ defined in \eqref{i104} is finite if $\omega\in (-1,0]$ and infinite if $\omega\in (-2,-1]$, the latter being then excluded from the already available studies.
\medskip
The purpose of this note is to contribute to the analysis of the critical case when the homogeneity indices $\lambda = \alpha+\beta$ and $\gamma$ of $K$ in \eqref{i3} and $a$ in \eqref{i4} are related by $\gamma=\lambda-1$ in the particular case
\begin{equation}
\alpha=\beta=1\ , \quad \kappa = \frac{1}{2} \ , \quad \gamma=1\ , \quad \omega=-1\ , \quad k>0 \ . \label{i5}
\end{equation}
Note that the choice of $\omega$ in \eqref{i5} corresponds to the production of an infinite number of particles during each fragmentation event. With this choice of parameters, equation~\eqref{i1} reads
\begin{eqnarray}
\partial_t f(t,x) & = & \frac{1}{2} \int_0^x (x-y)y f(t,x-y) f(t,y)\ dy - \int_0^\infty xy f(t,x) f(t,y)\ dy \nonumber \\
& & -\ k x f(t,x) + k \int_x^\infty \frac{y}{x} f(t,y)\ dy\ . \label{i6}
\end{eqnarray}
At this point we realize that, introducing $\nu(t,x) := x f(t,x)$ and multiplying \eqref{i6} by $x$, an alternative formulation of \eqref{i6} reads
\begin{eqnarray}
\partial_t \nu(t,x) & = & \frac{x}{2} \int_0^x \nu(t,x-y) \nu(t,y)\ dy - x \int_0^\infty \nu(t,x) \nu(t,y)\ dy \nonumber \\
& & -\ k x \nu(t,x) + k \int_x^\infty \nu(t,y)\ dy\ . \label{i7}
\end{eqnarray}
One advantage of this formulation is that it cancels the possible singularity of $f(t,x)$ as $x\to 0$. More precisely, while $\nu(t)$ is expected to be integrable due to the boundedness of the total mass, it is unclear whether $f(t)$ is integrable for positive times, even if the initial total number of particles is finite. Indeed, recall that, for the fragmentation rate under consideration, an infinite number of particles is produced during each fragmentation event. Another advantage of this formulation and the choice of the rate coefficients $K$, $a$, $b$ is that taking the Laplace transform of \eqref{i6} leads us to the following partial differential equation
\begin{equation}
\partial_t L(t,s) = (L(t,0)+k-L(t,s))\ \partial_s L(t,s) + k\ \frac{L(t,0)-L(t,s)}{s}\ , \quad t>0\ ,\ s>0\ , \label{i8}
\end{equation}
for the Laplace transform
$$
L(t,s) := \int_0^\infty \nu(t,x) e^{-sx}\ dx = \int_0^\infty x f(t,x) e^{-sx}\ dx\ , \quad (t,s)\in (0,\infty)^2\ ,
$$
of $\nu$. Note that $L(t,0)$ is the total mass at time $t$ and does not depend on time in the absence of gelation. Our aim is then to study the behavior of the solutions to \eqref{i8} and thereby obtain some information on the behavior of solutions to \eqref{i7} (and thus also on \eqref{i6}). To this end, it turns out that it is more appropriate to work with \textit{measure-valued} solutions to \eqref{i7}. Let $\mathfrak{M}^+$ be the space of non-negative bounded measures on $(0,\infty)$ and fix an initial condition $\nu^{in}$ satisfying
\begin{equation}
\nu^{in} \in \mathfrak{M}^+\ , \quad \int_0^\infty \nu^{in}(dx) =1\ . \label{i9}
\end{equation}
We may actually assume without loss of generality that $\nu^{in}$ is a probability measure after a suitable rescaling. Weak solutions to \eqref{i7} are then defined as follows:
\begin{definition}\label{defws}
Given an initial condition $\nu^{in}$ satisfying \eqref{i9}, a weak solution to \eqref{i7} with initial condition $\nu^{in}$ is a weakly continuous map $\nu: [0,\infty)\to \mathfrak{M}^+$ such that
\begin{eqnarray}
\int_0^\infty \vartheta(x) \nu(t,dx) & = & \int_0^\infty \vartheta(x) \nu^{in}(dx) + \int_0^t \int_0^\infty \int_0^\infty x [\vartheta(x+y) - \vartheta(x)] \nu(\tau,dx) \nu(\tau,dy)\ d\tau \nonumber \\
& & +\ k \int_0^t \int_0^\infty \left[ \int_0^x \vartheta(y)\ dy - x \vartheta(x) \right]\ \nu(\tau,dx)\ d\tau \label{i10}
\end{eqnarray}
for all $t>0$ and $\vartheta\in C([0,\infty))$ with compact support.
\end{definition}
We next denote the subset of non-negative bounded measures on $(0,\infty)$ with finite first moment by $\mathfrak{M}_1^+$. We now state the main result:
\begin{theorem}\label{thint0}
Let $\nu^{in}$ be an initial condition satisfying \eqref{i9}. There is a weak solution $\nu$ to \eqref{i7} in the sense of Definition~\ref{defws} which is mass-conserving for all times, that is,
\begin{equation*}
\int_0^\infty \nu(t,dx) = 1 = \int_0^\infty \nu^{in}(dx)\ , \quad t>0\ .
\end{equation*}
Moreover, $\nu(t)\in \mathfrak{M}^+_1$ for all $t>0$ and
\begin{equation}
\int_0^\infty x \nu(t,dx) \sim \frac{e^{1/k}-1}{t} \;\;\text{ as }\;\; t\to \infty\ . \label{i12}
\end{equation}
Furthermore, introducing
$$
M(t,x) := \int_0^x \nu(t,dy)\ , \quad (t,x)\in (0,\infty)\times (0,\infty)\ ,
$$
there is a probability measure $\nu_\star\in\mathfrak{M}^+$ such that
\begin{equation}
\lim_{t\to\infty} M\left( t , \frac{x}{t} \right) = M_\star(x) := \int_0^x \nu_\star(dy) \;\;\text{ for all }\;\; x\in (0,\infty)\ . \label{i13}
\end{equation}
\end{theorem}
More information is actually available on $\nu_\star$. Indeed, it follows from Proposition~\ref{prltb0} that the Laplace transform $L_\star$ of $\nu_\star$ is given by
\begin{equation}
L_\star(s) := 1 + s - k W\left( \frac{s}{k} e^{(1+s)/k} \right)\ , \quad s\ge 0\ , \label{i14}
\end{equation}
where $W$ is the so-called Lambert $W$-function, that is, the inverse function of $z\mapsto z e^z$ in $(0,\infty)$.
Recalling that the gelation time is finite for all solutions to the coagulation equation \eqref{i1} in the absence of fragmentation ($k=0$), we deduce from Theorem~\ref{thint0} that adding fragmentation prevents the occurrence of gelation whatever the value of $k>0$. The value of the parameter $k$ thus plays only a minor role in that direction but it comes into play in the large time behavior as the self-similar profile \eqref{i14} depends explicitly on $k$. That a self-similar behavior for large times is plausible for rate coefficients $K$ and $a$ given by \eqref{i3} and \eqref{i4} and satisfying $\gamma=\alpha+\beta-1$ is expected from the scaling analysis performed in \cite{Pi12,VZ89}, and we show in Theorem~\ref{thint0} that this is indeed the case for the particular choice \eqref{i5}. Moreover, the convergence to zero of the first moment of $\nu$ (which corresponds to the second moment of $f$) as $t\to \infty$ gives a positive answer to a conjecture in \cite{VZ89}, providing in addition an optimal rate of convergence to zero.
\medskip
As already mentioned, the proof of Theorem~\ref{thint0} relies on the study of the Laplace transform $L$ of solutions $\nu$ to \eqref{i7} which are related to solutions $f$ of \eqref{i6} by $\nu(t,x)=x f(t,x)$. A similar technique has already been used for the coagulation equation with multiplicative kernel $K(x,y)=xy$ and without fragmentation \cite{MP04, NZ11, vRS01}. However, the fragmentation term complicates the analysis as it adds a singular reaction term in the first-order hyperbolic equation \eqref{i8} solved by the Laplace transform $L$. This additional difficulty is met again later on in the proof when the method of characteristics is used. Indeed, in contrast to the case without fragmentation, it is no longer a single ordinary differential equation which shows up in the study of characteristics but a nonlinear system of two ordinary differential equations with a singularity. To be more precise, the strategy to show that \eqref{i8} has a global solution $L$ satisfying $L(t,0)=1$ for all times $t\ge 0$ is the following: we employ the method of characteristics to establish that the equation
$$
\partial_t \tilde{L}(t,s) = (1+k-\tilde{L}(t,s))\ \partial_s \tilde{L}(t,s) + k\ \frac{1-\tilde{L}(t,s)}{s}\ , \quad t>0\ ,\ s>0\ ,
$$
(which is nothing but \eqref{i8} where we have replaced $L(t,0)$ by $1$) has a global solution $\tilde{L}$ satisfying $\tilde{L}(t,0)=1$ for $t\ge 0$. Setting $L=\tilde{L}$ obviously gives a global solution $L$ to \eqref{i8} satisfying $L(t,0)=1$ for $t\ge 0$. As already pointed out, the method of characteristics requires a detailed analysis of a nonlinear system of two ordinary differential equations with a singularity which turns out to be rather involved and is performed in Section~\ref{sec31}. As a consequence of this analysis we obtain the existence of a global solution $L$ to \eqref{i8} satisfying $L(t,0)=1$ for $t\ge 0$ in Section~\ref{sec32}. The connection with \eqref{i7} is then made in Section~\ref{sec4} where we show that the just obtained solution $L$ to \eqref{i8} is completely monotone for all positive times and thus the Laplace transform of a probability measure. Several auxiliary results on completely monotone functions are needed for this step. The existence of a mass-conserving solution to \eqref{i7} in the sense of Definition~\ref{defws} results from the outcome of Sections~\ref{sec3} and~\ref{sec4}. Additional information can be retrieved from the detailed study of $L$ performed in Sections~\ref{sec3} and~\ref{sec4}. This allows us to identify the large time behavior of $L$ in Section~\ref{sec5} as well as the behavior of $\partial_s L(t,s)$ as $s\to 0$ in Section~\ref{sec6}, thereby providing the large time limits \eqref{i12} and \eqref{i13} stated in Theorem~\ref{thint0}.
\section{Alternative representation}\label{sec2}
Let $\nu$ be a weak solution to \eqref{i7} in the sense of Definition~\ref{defws}. Introducing its Laplace transform
\begin{equation*}
L(t,s) := \int_0^\infty e^{-sx} \nu(t,dx)\ , \quad (t,s)\in [0,\infty)\times (0,\infty)\ ,
\end{equation*}
and observing that $x\mapsto e^{-sx}$ is bounded and continuous for $s>0$, we infer from \eqref{i10} that $L$ solves
\begin{eqnarray}
\partial_t L(t,s) & = & (L(t,0)+k-L(t,s))\ \partial_s L(t,s) + k\ \frac{L(t,0)-L(t,s)}{s}\ , \quad t>0\ ,\ s>0\ , \label{rep3} \\
L(0,s) & = & L_0(s)\ , \quad s>0\ , \label{rep4}
\end{eqnarray}
where
\begin{equation}
L_0(s) := \int_0^\infty e^{-sx} \nu^{in}(dx) \ , \quad s>0\ . \label{rep5}
\end{equation}
Introducing the characteristic equation
\begin{equation}
\frac{dS}{dt}(t) = L(t,S(t)) - L(t,0) - k\ , \label{rep6}
\end{equation}
we infer from \eqref{rep3} that
\begin{equation}
\frac{d}{dt} L(t,S(t)) = k\ \frac{L(t,0)-L(t,S(t))}{S(t)}\ . \label{rep7}
\end{equation}
Consequently, $t\mapsto (S(t),L(t,S(t))$ solves the differential system \eqref{rep6}-\eqref{rep7} which has a singularity when $S(t)$ vanishes and is not closed as it features the yet unknown time dependent function $L(t,0)$. Nevertheless, as long as the total mass is conserved, it follows from \eqref{i7} that $L(t,0)=1$ and the differential system \eqref{rep6}-\eqref{rep7} is closed and can in principle be solved.
\section{Well-posedness}\label{sec3}
We first list some useful properties of the Laplace transform $L_0$ of $\nu^{in}$ defined in \eqref{rep5}. Owing to \eqref{i7}, $L_0\in C([0,\infty))\cap C^\infty((0,\infty))$ and satisfies
\begin{subequations}
\begin{eqnarray}
& & L_0'(s) = - \int_0^\infty x e^{-sx} \nu^{in}(dx) \;\text{ and }\; 0 < L_0(s) < 1\ , \quad s>0\ , \label{wp0a} \\
& & s L_0'(s) + 1 - L_0(s) = \int_0^\infty \left( e^{sx} - 1- sx \right) e^{-sx} \nu^{in}(dx) > 0\ , \quad s>0\ , \label{wp0b}
\end{eqnarray}
\end{subequations}
the second statement being a consequence of the elementary inequality $e^{sx}\ge 1+sx$ for $x>0$ and $s>0$. For further use, we also define
\begin{equation}
L_1(s) := \frac{L_0(s)-1}{s}<0 \;\;\text{ with }\;\; L_1'(s) = \frac{s L_0'(s) + 1 - L_0(s)}{s^2}>0\ , \quad s>0\ , \label{wp0c}
\end{equation}
the positivity properties of $-L_1$ and $L_1'$ being a consequence of \eqref{wp0a} and \eqref{wp0b}.
\subsection{An auxiliary differential system}\label{sec31}
According to the previous discussion, we focus in this section on the following initial value problem: given $s>0$,
\begin{eqnarray}
\frac{d\Sigma}{dt}(t) & = & \ell(t) - 1 - k\ , \label{wp1} \\
\frac{d\ell}{dt}(t) & = & k\ \frac{1-\ell(t)}{\Sigma(t)}\ , \label{wp2} \\
(\Sigma,\ell)(0) & = & (s,L_0(s))\ . \label{wp3}
\end{eqnarray}
We infer from the Cauchy-Lipschitz theorem that there is a unique maximal solution $(\Sigma,\ell)(\cdot,s)\in C([0,T(s));\mathbb{R }^2)$ to \eqref{wp1}-\eqref{wp3} such that
\begin{equation}
\Sigma(t,s)>0\ , \quad t\in [0,T(s))\ . \label{wp4}
\end{equation}
In addition,
\begin{equation}
T(s)<\infty \iff \lim_{t\to T(s)} \Sigma(t,s)=0 \;\text{ or }\; \lim_{t\to T(s)} \left( \Sigma(t,s) + |\ell(t,s)| \right) = \infty\ . \label{wp5}
\end{equation}
Since $L_0(s)\in (0,1)$ by \eqref{wp0a}, a first consequence of \eqref{wp2} and the comparison principle is that $\ell(t,s)<1$ for $t\in [0,T(s))$. This fact and \eqref{wp4} ensure that the right hand side of \eqref{wp2} is then positive, hence
\begin{equation}
\partial_t \ell(t,s)>0 \;\;\text{ and }\;\; 0 \le \ell(t,s) < 1\ , \quad t\in [0,T(s))\ . \label{wp6a}
\end{equation}
We then deduce from \eqref{wp1}, \eqref{wp6a}, and the positivity of $k$ that
\begin{equation}
\partial_t \Sigma(t,s) < -k < 0 \;\text{ and }\; \Sigma(t,s)\le s\ , \quad t\in [0,T(s))\ . \label{wp6b}
\end{equation}
Two interesting consequences can be drawn from the estimates \eqref{wp6a} and \eqref{wp6b}: they clearly exclude the occurrence of finite time blowup and imply that $\Sigma(\cdot,s)$ vanishes at a finite time. Recalling \eqref{wp5}, we conclude that
$$
T(s)< \infty \;\text{ and }\; \lim_{t\to T(s)} \Sigma(t,s)=0\ .
$$
In fact, $\Sigma(\cdot,s)$, $\ell(\cdot,s)$ and $T(s)$ can be computed explicitly as we show now. Owing to \eqref{wp4} and \eqref{wp6a}, the function $\ln{\Sigma} - \ln{(1-\ell)} + (\ell/k)$ is well-defined on $[0,T(s))$ and we infer from \eqref{wp1} and \eqref{wp2} that
$$
\frac{d}{dt} \left( \ln{\Sigma} - \ln{(1-\ell)} + \frac{\ell}{k} \right)(t,s) = 0\ , \quad t\in [0,T(s)\ .
$$
Therefore
\begin{subequations}
\begin{equation}
\Sigma(t,s) = s\ \frac{1-\ell(t,s)}{1-L_0(s)}\ e^{(L_0(s)-\ell(t,s))/k}\ , \quad t\in [0,T(s))\ , \label{wp7a}
\end{equation}
or, alternatively,
\begin{equation}
\frac{1-\ell(t,s)}{\Sigma(t,s)} = \frac{1-L_0(s)}{s} e^{(\ell(t,s)-L_0(s))/k}\ , \quad t\in [0,T(s))\ . \label{wp7b}
\end{equation} \label{wp7}
\end{subequations}
Inserting \eqref{wp7b} in \eqref{wp2} gives
\begin{equation}
\frac{d\ell}{dt}(t,s) = k\ \frac{1-L_0(s)}{s}\ e^{-L_0(s)/k} e^{\ell(t,s)/k}\ , \quad t\in [0,T(s))\ ,
\label{wp8b}
\end{equation}
whence, after integration,
\begin{equation}
\ell(t,s) = L_0(s) - k \ln{(1+tL_1(s))}\ , \quad t\in [0,T(s))\ , \label{wp8}
\end{equation}
the function $L_1$ being defined in \eqref{wp0c}. We next combine \eqref{wp7a} and \eqref{wp8} to obtain
\begin{equation}
\Sigma(t,s) = -\frac{1+t L_1(s)}{L_1(s)}\ \left[ 1 - L_0(s) + k\ \ln{(1+tL_1(s))} \right]\ , \quad t\in [0,T(s))\ . \label{wp9}
\end{equation}
The first term of the right hand side of \eqref{wp9} vanishes when $t=-1/L_1(s)=s/(1-L_0(s))$ while the second term is a decreasing function of time (recall that $L_0(s)<1$) and ranges in $(-\infty,1-L_0(s)]$ when $t$ ranges in $[0,-1/L_1(s))$. It thus vanishes only once in $[0,-1/L_1(s))$ and since we have already excluded finite time blowup, we conclude that $T(s)$ is the unique zero in $[0,-1/L_1(s))$ of the second term of the right hand side of \eqref{wp9}, that is, $T(s)$ solves
\begin{subequations}
\begin{equation}
1 - L_0(s) + k\ \ln{(1+ T(s) L_1(s))} = 0\ , \label{wp10a}
\end{equation}
or, alternatively,
\begin{equation}
T(s) = \frac{s}{1-L_0(s)}\ \left( 1 - e^{(L_0(s)-1)/k} \right) < \frac{s}{1-L_0(s)}\ . \label{wp10b}
\end{equation}
\end{subequations}
The last bound implies in particular that
\begin{equation}
s(1+t L_1(s)) = s + (L_0(s)-1) t > s e^{(L_0(s)-1)/k} > 0\ , \quad t\in [0,T(s))\ . \label{wp10c}
\end{equation}
Recalling that the differential system \eqref{wp3}-\eqref{wp4} features a singularity as $t\to T(s)$, let us investigate further the behavior of $(\Sigma,\ell)(t,s)$ as $t\to T(s)$.
\begin{lemma}\label{lewp1b}
The functions $\Sigma(\cdot,s)$ and $\ell(\cdot,s)$ both belong to $C^1([0,T(s)])$ with
\begin{align*}
\Sigma(T(s),s) = 0\ , & \quad \ell(T(s),s) = 1\ , \\
\partial_t \Sigma(T(s),s) = - k\ , & \quad \partial_t \ell(T(s),s) = - k L_1(s) e^{(L_0(s)-1)/k}\ .
\end{align*}
\end{lemma}
\begin{proof}
The behavior of $\Sigma(t,s)$ and $\ell(t,s)$ as $t\to T(s)$ is a straightforward consequence of \eqref{wp8}, \eqref{wp9}, and \eqref{wp10a}. Next, it readily follows from \eqref{wp1} that
$$
\lim_{t\to T(s)} \partial_t \Sigma(t,s) = -k \ .
$$
Consequently, $\Sigma(\cdot,s)\in C^1([0,T(s)])$ and $\ell(\cdot, s)$ shares the same regularity thanks to a similar argument relying on \eqref{wp8b}.
\end{proof}
We now study the behavior of $T(s)$ as a function of $s>0$ and gather the outcome of the analysis in the next lemma.
\begin{lemma}\label{lewp2}
The function $T$ is an increasing $C^\infty$-smooth diffeomorphism from $(0,\infty)$ onto $(0,\infty)$ and enjoys the following properties:
\begin{subequations}
\begin{eqnarray}
& & \lim_{s\to 0} T(s) = 0 \;\text{ and }\; T(s) \sim \frac{s}{k} \;\text{ as } s \to 0\ , \label{wp11a} \\
& & \lim_{s\to \infty} T(s) = \infty \;\text{ and }\; T(s) \sim (1- e^{-1/k}) s \;\text{ as } s \to \infty\ . \label{wp11b}
\end{eqnarray}
\end{subequations}
\end{lemma}
\begin{proof}
The smoothness of $T$ readily follows from that of $L_0$ and \eqref{wp0a} by \eqref{wp10b} while \eqref{wp11a} and \eqref{wp11b} are consequences of \eqref{wp10b} and the properties of $L_0$. To establish the monotonicity of $T$, we differentiate \eqref{wp10a} with respect to $s$ and find
$$
-L_0'(s) + k\ \frac{1 + L_1'(s) T(s) + L_1(s) T'(s)}{1 + L_1(s) T(s)} = 0\ .
$$
Since $1+L_1(s) T(s) = e^{(L_0(s)-1)/k}$ by \eqref{wp10a}, we obtain
$$
- L_0'(s) e^{(L_0(s)-1)/k} + k \left( 1+ L_1'(s) T(s) + L_1(s) T'(s) \right) = 0\ ,
$$
and then
$$
- k L_1(s) T'(s) = - L_0'(s) e^{(L_0(s)-1)/k} + k ( 1+ L_1'(s) T(s))\ .
$$
Since $-L_0'$ and $L_1'$ are both positive by \eqref{wp0a} and \eqref{wp0c}, we conclude that
\begin{equation*}
T'(s)>0\ , \quad s>0\ ,
\end{equation*}
which, together with \eqref{wp11a} and \eqref{wp11b}, implies that $T$ is an increasing $C^\infty$-smooth diffeomorphism from $(0,\infty)$ onto $(0,\infty)$ and completes the proof.
\end{proof}
Denoting the inverse of $T$ by $T^{-1}$, we deduce from \eqref{wp10a} that
\begin{equation}
1 = L_0(T^{-1}(t)) - k \ln{(1+t L_1(T^{-1}(t))}\ , \quad t>0\ . \label{wp10d}
\end{equation}
We next turn to the properties of $\Sigma$ with respect to the variable $s$ and establish the following monotonicity result:
\begin{lemma}\label{lewp3}
Let $t>0$. The function $\Sigma(t,\cdot)$ is an increasing $C^\infty$-smooth diffeomorphism from $(T^{-1}(t),\infty)$ onto $(0,\infty)$ and satisfies
\begin{equation}
\partial_s \Sigma (t,s)> 0 \;\;\text{ for }\;\; s\in (T^{-1}(t),\infty) \;\;\text{ and }\;\; \Sigma(t,s)\sim s \;\;\text{ as }\;\; s\to \infty\ . \label{wp14}
\end{equation}
\end{lemma}
\begin{proof}
Let $s>0$ and $t\in (0,T(s))$. We differentiate \eqref{wp9} with respect to $s$ and obtain
\begin{align*}
\partial_s \Sigma(t,s) = & - \frac{1+t L_1(s)}{L_1(s)} \left[ -L_0'(s) + kt\ \frac{L_1'(s)}{1+t L_1(s)} \right] \\
+ & \frac{L_1'(s)}{L_1(s)^2}\ \left( 1 - L_0(s) + k \ln{ (1+t L_1(s))} \right) \\
= & \frac{L_0'(s)}{L_1(s)} + t L_0'(s) - kt \frac{L_1'(s)}{L_1(s)} - \frac{s L_1'(s)}{L_1(s)} + k\ \frac{L_1'(s)}{L_1(s)^2} \ln{(1+t L_1(s))} \ ,
\end{align*}
\begin{equation}
\partial_s \Sigma(t,s) = 1 + t L_0'(s) + k\ \frac{L_1'(s)}{L_1(s)^2} \ \left[ \ln{(1+t L_1(s))} - t L_1(s) \right]\ . \label{wp13}
\end{equation}
Differentiating \eqref{wp13} with respect to $t$ gives
\begin{align*}
\partial_t \partial_s \Sigma(t,s) = & L_0'(s) + k\ \frac{L_1'(s)}{L_1(s)^2}\ \left[ \frac{L_1(s)}{1 + t L_1(s)} - L_1(s) \right] \\
= & L_0'(s) - kt\ \frac{L_1'(s)}{1 + t L_1(s))}< 0
\end{align*}
for $t\in [0,T(s))$, the negativity of the right hand side of the above inequality being a consequence of \eqref{wp0a}, \eqref{wp0c}, and \eqref{wp10c}. Therefore, for all $t\in [0,T(s))$,
\begin{equation}
\partial_s \Sigma(t,s) > \tau(s) := \lim_{t\to T(s)} \partial_s \Sigma(t,s)\ , \label{wp13b}
\end{equation}
where
$$
\tau(s) = 1 + T(s) L_0'(s) + k\ \frac{L_1'(s)}{L_1(s)^2} \ \left[ \ln{(1+T(s) L_1(s))} - L_1(s) T(s) \right]\ .
$$
Since
$$
\ln{(1+T(s) L_1(s))} = \frac{L_0(s)-1}{k} = \frac{s L_1(s)}{k}
$$
by \eqref{wp10a}, we realize that
\begin{align*}
\tau(s) = & 1 + T(s) L_0'(s) + k\ \frac{L_1'(s)}{L_1(s)} \ \left[ \frac{s}{k} - T(s) \right] \\
= & L_0'(s)\ \left( T(s) - \frac{s}{1-L_0(s)} \right) - k\ \frac{L_1'(s)}{L_1(s)} \ T(s) > 0\ ,
\end{align*}
due to \eqref{wp0a}, \eqref{wp0c}, and \eqref{wp10b}. Recalling \eqref{wp13b}, we have shown that $\partial_s \Sigma(t,s)>0$ for $t\in (0,T(s))$ and $s>0$ or equivalently for $s\in (T^{-1}(t),\infty)$ and $t>0$. Consequently, for each $t>0$, $\Sigma(t,\cdot)$ is an increasing $C^\infty$-smooth diffeomorphism from $(T^{-1}(t),\infty)$ onto its range which is nothing but $(0,\infty)$ since $\Sigma(t,s)\sim s$ as $s\to\infty$ by \eqref{wp9} and $\Sigma(t,s)\to 0$ as $s\to T^{-1}(t)$ by Lemma~\ref{lewp1b}. The proof of Lemma~\ref{lewp3} is then complete.
\end{proof}
For $t>0$, we denote the inverse of $\Sigma(t,\cdot)$ by $\zeta(t,\cdot)$ and observe that it is a $C^\infty$-smooth increasing function from $(0,\infty)$ onto $(T^{-1}(t),\infty)$. Since
\begin{equation}
\Sigma(t,\zeta(t,s))=s \;\text{ for }\; (t,s)\in (0,\infty)^2\ , \label{wp100}
\end{equation}
we infer from \eqref{wp14}, \eqref{wp100}, and the implicit function theorem that $\zeta\in C^\infty((0,\infty)^2)$ with
\begin{equation}
\partial_s \zeta(t,s) = \frac{1}{\partial_s \Sigma(t,\zeta(t,s))} \;\text{ and }\; \partial_t \zeta(t,s) = - \frac{\partial_t \Sigma(t,\zeta(t,s))}{\partial_s \Sigma(t,\zeta(t,s))}\ , \quad (t,s)\in (0,\infty)^2\ . \label{wp101}
\end{equation}
We end up this section with the differentiability of $\Sigma(t,\cdot)$ and $\ell(t,\cdot)$ at $s=T^{-1}(t)$.
\begin{lemma}\label{lewp4}
Let $t>0$. Both $\Sigma(t,\cdot)$ and $\ell(t,\cdot)$ belong to $C^1\left( [T^{-1}(t),\infty) \right)$.
\end{lemma}
\begin{proof}
Since $T^{-1}(t)>0$ by Lemma~\ref{lewp2} and $L_0, L_1\in C^1((0,\infty))$, we infer from \eqref{wp13} that $\partial_s \Sigma(t,s)$ has a limit as $s\to T^{-1}(t)$ and thus that $\Sigma(t,\cdot)\in C^1([T^{-1}(t),\infty)$. Similarly, by \eqref{wp8},
$$
\partial_s \ell(t,s) = L_0'(s) - kt\ \frac{L_1'(s)}{1+t L_1(s)}\ ,
$$
which has a limit as $s\to T^{-1}(t)$ as $L_0$ and $L_1$ belong to $C^1((0,\infty))$.
\end{proof}
\subsection{Existence of a solution to \eqref{rep3}-\eqref{rep4}}\label{sec32}
After this preparation, we are in a position to show the existence of a solution to \eqref{rep3}-\eqref{rep4}. As expected from the analysis performed in Section~\ref{sec2}, we set
\begin{equation}
L(t,s) := \ell(t,\zeta(t,s))\ , \quad (t,s)\in (0,\infty)^2\ , \label{wp102}
\end{equation}
and aim at showing that $L$ solves \eqref{rep3}-\eqref{rep4} as well as identifying $L(t,0)$ for all $t>0$.
On the one hand, the properties of $\zeta$, \eqref{wp8}, \eqref{wp10d}, and \eqref{wp102} entail that, for $t>0$,
\begin{align*}
L(t,0) = &\ \lim_{s\to 0} L(t,s) = \lim_{s\to 0} \ell(t,\zeta(t,s)) = \lim_{s\to T^{-1}(t)} \ell(t,s) \\
= &\ L_0(T^{-1}(t)) - k \ln{(1+t L_1(T^{-1}(t)))} =1\ ,
\end{align*}
\begin{equation}
L(t,0) = 1\ , \qquad t>0\ . \label{wp103}
\end{equation}
On the other hand, it follows from \eqref{wp1}, \eqref{wp2}, \eqref{wp101}, and \eqref{wp102} that, for $(t,s)\in (0,\infty)^2$,
\begin{align*}
\partial_t L(t,s) = & k\ \frac{1 - L(t,s)}{s} + \partial_s \ell(t,\zeta(t,s))\ \partial_t\zeta(t,s) \\
= & k\ \frac{1 - L(t,s)}{s} - \partial_s \ell(t,\zeta(t,s))\ \frac{\partial_t\Sigma(t,\zeta(t,s))}{\partial_s\Sigma(t,\zeta(t,s))} \\
= & k\ \frac{1 - L(t,s)}{s} - \partial_s L(t,s)\ (L(t,s)-k-1)\ ,
\end{align*}
whence, thanks to \eqref{wp103},
\begin{equation*}
\partial_t L(t,s) = \partial_s L(t,s)\ (L(t,0)+k-L(t,s)) + k\ \frac{L(t,0) - L(t,s)}{s}\ , \quad (t,s)\in (0,\infty)^2\ .
\end{equation*}
Finally, the properties of $\Sigma$, $\zeta$, and \eqref{wp3} imply that, for $s>0$,
$$
\lim_{t\to 0} L(t,s) = \lim_{t\to 0} \ell(t,\zeta(t,s)) = \ell(0,s) = L_0(s)\ .
$$
We have thus shown that the function $L$ defined in \eqref{wp102} solves \eqref{rep3}-\eqref{rep4} and enjoys the additional property \eqref{wp103}. To obtain a solution to the coagulation-fragmentation equation \eqref{i6} it remains to show that, for all $t>0$, $L(t,\cdot)$ is the Laplace transform of a non-negative bounded measure or alternatively that it is completely monotone. This will be the aim of Section~\ref{sec4}.
\section{Complete monotonicity}\label{sec4}
\begin{proposition}\label{prcm0}
For all $t>0$, the function $L(t,\cdot)$ defined in \eqref{wp102} is completely monotone.
\end{proposition}
We first recall two important criteria guaranteeing complete monotonicity, see \cite[Chapter~XIII.4, Criterion~1 \& Criterion~2]{Fe71} for instance.
\begin{lemma}\label{lecm1}
\begin{enumerate}
\item If $\varphi$ and $\psi$ are completely monotone functions then their product $\varphi \psi$ is also a completely monotone function.
\item If $\varphi$ is a completely monotone function and $\psi$ is a non-negative function with a completely monotone derivative, then $\varphi \circ \psi$ is a completely monotone function.
\end{enumerate}
\end{lemma}
In particular, the complete monotonicity of $r\mapsto -\ln{r}$ and Lemma~\ref{lecm1}~(2) have the following consequence.
\begin{lemma}\label{lecm2}
Let $I$ be an open interval of $\RR$ and $g: I \mapsto (0,1)$ be a $C^\infty$-smooth function such that $g'$ is completely monotone. Then $-\ln{g}$ is completely monotone.
\end{lemma}
Let $t>0$. Recalling that $L(t,\cdot)$ is given by
$$
L(t,s) = \ell(t,\zeta(t,s))\ , \qquad s>0\ ,
$$
the proof of its complete monotonicity is performed in two steps. More precisely, we prove that $\ell(t,\cdot)$ and the derivative of $\zeta(t,\cdot)$ are completely monotone. The complete monotonicity of $L(t,\cdot)$ is then a consequence of Lemma~\ref{lecm1}~(2) and the non-negativity of $\zeta(t,\cdot)$. To prove the complete monotonicity of $\ell(t,\cdot)$ we need the following result:
\begin{lemma}\label{lecm3}
Let $g: (0,\infty) \mapsto (0,1)$ be a completely monotone function satisfying $g(0)=1$. Setting $G(x) := (g(x)-1)/x$ for $x\in (0,\infty)$, its derivative $G'$ is completely monotone.
\end{lemma}
\begin{proof}
We first note that
\begin{equation}
G'(x) = \frac{x g'(x) - g(x) + 1}{x^2} = \frac{g'(x) - G(x)}{x}\,, \qquad x\in (0,\infty)\,. \label{cm1}
\end{equation}
\noindent\textbf{Step~1:} We first prove by induction that, for $n\ge 0$,
\begin{equation}
G^{(n+1)}(x) = \frac{g^{(n+1)}(x) - (n+1) G^{(n)}(x)}{x}\,, \qquad x\in (0,\infty)\,, \label{cm2}
\end{equation}
with the obvious notation $G^{(0)}=G$. Indeed, \eqref{cm2} is clearly true for $n=0$ by \eqref{cm1}. Assume next that \eqref{cm2} is true for some $n\ge 0$. Differentiating the corresponding identity gives, for $x>0$,
\begin{equation*}
G^{(n+2)}(x) = \frac{g^{(n+2)}(x) - (n+1) G^{(n+1)}(x)}{x} - \frac{g^{(n+1)}(x) - (n+1) G^{(n)}(x)}{x^2}
\end{equation*}
We next use \eqref{cm2} for $n$ and find
\begin{equation*}
G^{(n+2)}(x) = \frac{g^{(n+2)}(x) - (n+1) G^{(n+1)}(x)}{x} - \frac{G^{(n+1)}(x)}{x}\,,
\end{equation*}
whence \eqref{cm2} for $n+1$.
\noindent\textbf{Step~2:} We next prove by induction that, for $n\ge 1$,
\begin{equation}
G^{(n)}(x) = \frac{n!}{x^{n+1}}\ \left[ (-1)^n\ (g(x)-1) + \sum_{j=1}^n (-1)^{n-j}\ \frac{x^j}{j!}\ g^{(j)}(x) \right]\,, \quad x\in (0,\infty)\,. \label{cm3}
\end{equation}
Indeed, the identity \eqref{cm3} is true for $n=1$ by \eqref{cm1}. Assume next that $G^{(n)}$ is given by \eqref{cm3} for some $n\ge 1$. It then follows from \eqref{cm2} and \eqref{cm3} that, for $x>0$,
\begin{align*}
G^{(n+1)}(x) = & \frac{g^{(n+1)}(x)}{x} - \frac{(n+1)!}{x^{n+2}}\ \left[ (-1)^n\ (g(x)-1) + \sum_{j=1}^n (-1)^{n-j}\ \frac{x^j}{j!}\ g^{(j)}(x) \right] \\
= & \frac{(n+1)!}{x^{n+2}}\ \left[ (-1)^{n+1}\ (g(x)-1) + \sum_{j=1}^n (-1)^{n+1-j}\ \frac{x^j}{j!}\ g^{(j)}(x) + \frac{x^{n+1}}{(n+1)!}\ g^{(n+1)}(x) \right]\,.
\end{align*}
Thus, $G^{(n+1)}$ is also given by \eqref{cm3}.
\noindent\textbf{Step~3:} Define $h_n(x) := x^{n+1}\ G^{(n)}(x) / n!$ for $n\ge 0$ and $x>0$. Thanks to \eqref{cm3},
$$
h_n(x) = (-1)^n\ (g(x)-1) + \sum_{j=1}^n (-1)^{n-j}\ \frac{x^j}{j!}\ g^{(j)}(x)\,, \quad x\in (0,\infty)\,.
$$
Since $g(0)=1$, we have $h_n(0)=0$ and, for $x>0$,
\begin{align*}
h_n'(x) = & (-1)^n\ g'(x) + \sum_{j=1}^n (-1)^{n-j}\ \frac{x^j}{j!}\ g^{(j+1)}(x) + \sum_{j=1}^n (-1)^{n-j}\ \frac{x^{j-1}}{(j-1)!}\ g^{(j)}(x) \\
= & \sum_{j=1}^{n+1} (-1)^{n+1-j}\ \frac{x^{j-1}}{(j-1)!}\ g^{(j)}(x) + \sum_{j=1}^n (-1)^{n-j}\ \frac{x^{j-1}}{(j-1)!}\ g^{(j)}(x) \\
= & \frac{x^n}{n!}\ g^{(n+1)}(x)\,.
\end{align*}
Since $g$ is completely monotone, the previous identity implies that $(-1)^{n+1}\ h_n' \ge 0$. Recalling that $h_n(0)=0$, we have thus shown that $(-1)^{n+1}\ h_n\ge 0$ which in turn gives $(-1)^{n+1}\ G^{(n)}\ge 0$ and the complete monotonicity of $G'$.
\end{proof}
\begin{lemma}\label{lecm4}
For each $t>0$, $\ell(t,\cdot)$ is completely monotone in $(T^{-1}(t),\infty)$.
\end{lemma}
\begin{proof}
Fix $t>0$ and recall that $\ell(t,\cdot)$ is given by
$$
\ell(t,s) = L_0(s) - k \ln{( 1 + t L_1(s))}\ , \quad s\in (T^{-1}(t),1)\ ,
$$
with $L_1(s) := (L_0(s)-1)/s$ for $s>0$. Since $L_0$ is completely monotone with $L_0(0)=1$, we infer from Lemma~\ref{lecm3} that the derivative of the function $L_1$ is completely monotone in $(0,\infty)$. Then, so is the derivative of $1+tL_1$ and $s\mapsto 1+tL_1(s)$ ranges in $(0,1)$ when $s\in (T^{-1}(t),\infty)$ according to \eqref{wp10c}. We are thus in a position to apply Lemma~\ref{lecm2} and conclude that $s\mapsto - \ln{( 1 + tL_1(s))}$ is completely monotone in $(T^{-1}(t),\infty)$. Since $k>0$ and $L_0$ is completely monotone in $(0,\infty)$, we conclude that $\ell(t,\cdot)$ is completely monotone in $(T^{-1}(t),\infty)$.
\end{proof}
We next turn to $\zeta(t,\cdot)$ and first establish the following auxiliary result.
\begin{lemma}\label{lecm5}
Let $g$ be a non-negative function in $C^\infty(0,\infty)$ such that $g'$ is completely monotone and $g'<1$. Then the function $(\mathop{\rm id}\nolimits-g)^{-1}$ has a completely monotone derivative.
\end{lemma}
\begin{proof}
For the sake of completeness, we give a sketch of the proof which is actually outlined in \cite[p.~1209]{MP04}. We set $\sigma_\infty := (\mathop{\rm id}\nolimits - g)^{-1}$ and define by induction a sequence of functions $(\sigma_n)_{n\ge 0}$ as follows:
\begin{equation}
\sigma_0(s) := s \;\;\text{ and }\;\; \sigma_{n+1}(s) := s + g(\sigma_n(s))\,, \qquad s>0\,, \quad n\ge 0\,. \label{cm40}
\end{equation}
On the one hand, thanks to the properties of $g$ (and in particular the bounds $0\le g'<1$), the function $\sigma_n$ is well-defined and non-negative for all $n\ge 0$ and belongs to $C^\infty(0,\infty)$. In addition, it satisfies
\begin{equation}
\sigma_0(s) = s \le \sigma_n(s) \le \sigma_{n+1}(s) \le \sigma_\infty(s)\,, \qquad s>0\,, \quad n\ge 0\,. \label{cm4}
\end{equation}
Now, fix $s_1>0$ and $s_2>s_1$. Owing to the property $g'<1$, there is $\delta\in (0,1)$ such that $g'(s)\le \delta$ for each $s\in [s_1,\sigma_\infty(s_2)]$. Since $\sigma_n(s)\in [s_1,\sigma_\infty(s_2)]$ for $n\ge 0$ and $s\in [s_1,s_2]$ by \eqref{cm4}, we obtain
$$
0 \le \sigma_\infty(s) - \sigma_{n+1}(s) = g(\sigma_\infty(s)) - g(\sigma_{n}(s)) \le \delta\ \left( \sigma_\infty(s) - \sigma_{n}(s)\right)\,.
$$
This estimate readily implies that
\begin{equation}
(\sigma_n)_{n\ge 0} \;\text{ converges uniformly towards }\; \sigma_\infty \;\text{ on compact subsets of }\; (0,\infty)\,.
\label{cm5}
\end{equation}
On the other hand, it follows from \eqref{cm40} by induction that \begin{equation}
\sigma_n \;\text{ is completely monotone for every }\; n\ge 0\,.
\label{cm6}
\end{equation}
Indeed, $\sigma_0=\mathop{\rm id}\nolimits$ is clearly completely monotone and, if $\sigma_n$ is assumed to be completely monotone, the complete monotonicity of $\sigma_{n+1}$ follows from \eqref{cm40} and that of $\sigma_n$ and $g'$ with the help of \cite[Chapter~XIII.4, Criterion~2]{Fe71}.
The assertion of Lemma~\ref{lecm5} then follows from \eqref{cm5} and \eqref{cm6} by \cite[Chapter~XIII.1, Theorem~2 \& Chapter~XIII.4, Theorem~1]{Fe71}.
\end{proof}
\begin{lemma}\label{lecm6}
For each $t>0$, $\partial_s \zeta(t,\cdot)$ is completely monotone in $(0,\infty)$.
\end{lemma}
\begin{proof}
Fix $t>0$. Introducing
$$
\Phi(s) := s - \Sigma(t,s) = (1-L_0(s)) t + k\ \frac{1+t L_1(s)}{L_1(s)} \ln{(1+tL_1(s))}\ , \quad s\in (T^{-1}(t),\infty)\ ,
$$
the formula \eqref{wp9} also reads $\Sigma(t,s) = s -\Phi(s)$ for $s\in (T^{-1}(t),\infty)$ from which we deduce that
\begin{equation}
s = \zeta(t,s) - \Phi(\zeta(t,s))\ , \quad s>0\ . \label{wp16}
\end{equation}
Therefore, $\zeta(t,\cdot)$ satisfies a functional identity of the form required to apply Lemma~\ref{lecm5}. To go on, we have to show that $\Phi'$ is completely monotone. To this end, we compute $\Phi'$ and find
$$
\Phi'(s) = - t L_0'(s) + k\ \frac{L_1'(s)}{L_1(s)^2}\ \left[ t L_1(s) - \ln{(1+t L_1(s))} \right]\ , \quad \quad s\in (T^{-1}(t),\infty)\ .
$$
Given $\theta\in [0,t]$, the monotonicity of $T^{-1}$ ensures that $T^{-1}(\theta)\le T^{-1}(t)$ so that $1+\theta L_1>0$ in $(T^{-1}(t),\infty)$ by \eqref{wp10c}. In addition, $1+\theta L_1$ has a completely monotone derivative since $L_1'$ is completely monotone by Lemma~\ref{lecm3} which, together with the complete monotonicity of $z\mapsto 1/z$ and Lemma~\ref{cm1}~(2) entails the complete monotonicity of $1/(1+\theta L_1)$ in $(T^{-1}(t),\infty)$ for all $\theta\in [0,t]$. Observing that
$$
\frac{t L_1(s) - \ln{(1+t L_1(s))}}{L_1(s)^2} = \int_0^t \frac{\theta}{1+\theta L_1(s)}\ d\theta\ , \quad s\in (T^{-1}(t),\infty)\ ,
$$
we infer from \cite[Theorem~4]{MS01} that
$$
\frac{t L_1 - \ln{(1+t L_1)}}{L_1^2} \;\text{ is completely monotone in }\; (T^{-1}(t),\infty)\ .
$$
Using now Lemma~\ref{cm1}~(1) along with the positivity of $k$ and the complete monotonicity of $L_1'$ and $-L_0'$ entails that $\Phi'$ is completely monotone in $(T^{-1}(t),\infty)$. Furthermore, recalling the definition of $\Phi$, there holds $\Phi'=1 - \partial_s \Sigma(t,\cdot)<1$ by \eqref{wp14}.
Summarizing, we have shown that $\Phi\in C^\infty((T^{-1}(t),\infty))$ has a completely monotone derivative $\Phi'$ satisfying $\Phi'<1$ while $\zeta(t,\cdot)$ solves \eqref{wp16}. Lemma~\ref{lecm5} then ensures that $\partial_s \zeta(t,\cdot)$ is completely monotone in $(0,\infty)$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prcm0}]
Fix $t>0$. The complete monotonicity of $L(t,\cdot)$ is a straightforward consequence of the non-negativity of $\zeta(t,\cdot)$, Lemma~\ref{cm1}~(2), Lemma~\ref{lecm4}, and Lemma~\ref{lecm6}.
\end{proof}
\medskip
Now, for each $t>0$, $L(t,\cdot)$ is completely monotone in $(0,\infty)$ with $L(t,0)=1$ by Proposition~\ref{prcm0} and \eqref{wp103} so that it is the Laplace transform of a probability measure $\nu(t)\in \mathfrak{M}^+$ by \cite[Chapter~XIII.4, Theorem~1]{Fe71}. In addition, it follows from the time continuity of $L$ and \cite[Chapter~XIII.1, Theorem~2]{Fe71} that the map $\nu: [0,\infty)\to \mathfrak{M}^+$ is weakly continuous. We finally argue as in \cite[Section~2]{MP04} to show that $\nu$ satisfies \eqref{i10} for all $C^1$-smooth functions $\vartheta$ with compact support. An additional approximation argument allows us to extend the validity of \eqref{i10} to all continuous functions $\vartheta$ with compact support and complete the proof of the first statement of Theorem~\ref{thint0}.
\section{Large time behavior}\label{sec5}
We now aim at investigating the behavior of $L(t,ts)$ as $t\to \infty$ for any given $s>0$. More specifically, we prove the following convergence result.
\begin{proposition}\label{prltb0}
For all $s>0$, there holds
\begin{equation}
\lim_{t\to \infty} L(t,ts) = 1 + s - k W\left( \frac{s}{k} e^{(1+s)/k} \right)\ , \label{ltb0}
\end{equation}
where we recall that $W$ is the Lambert $W$-function, that is, the inverse function of $z\mapsto z e^z$ in $(0,\infty)$.
\end{proposition}
\begin{proof}
We fix $s>0$ and set
\begin{equation}
\eta(t) := \zeta(t,ts) \;\;\text{ and }\;\; \mu(t) := - t L_1(\eta(t))\ , \quad t>0\ . \label{ltb1}
\end{equation}
Since $\eta(t) = \zeta(t,ts) \ge T^{-1}(t)$, Lemma~\ref{lewp2} ensures that
\begin{equation}
\lim_{t\to\infty} \eta(t) = \infty\ . \label{ltb2}
\end{equation}
Next, for $\sigma>T^{-1}(t)$, we infer from the properties of $L_0$ and \eqref{wp10b} that
$$
1 > 1 + t L_1(\sigma) > 1 + L_1(\sigma) T(\sigma) = e^{(L_0(\sigma)-1)/k}\ .
$$
Taking $\sigma = \eta(t)$ in the above estimate gives
\begin{equation}
1 > 1 - \mu(t) > e^{(L_0(\eta(t))-1)/k} > e^{-1/k}\ , \quad t>0\ . \label{ltb3}
\end{equation}
Now, we infer from \eqref{wp9} (with $\eta(t)=\zeta(t,s)$ instead of $s$) that
\begin{align*}
ts = & \frac{\eta(t) (1+t L_1(\eta(t)))}{1-L_0(\eta(t))} \left[ 1 - L_0(\eta(t)) + k \ln{(1-\mu(t))} \right] \\
s = & \frac{1-\mu(t)}{\mu(t)} \left[ 1 - L_0(\eta(t)) + k \ln{(1-\mu(t))} \right] \\
\mu(t) s = & (1-\mu(t))\ (1 - L_0(\eta(t))) + k (1-\mu(t)) \ln{(1-\mu(t))}\ ,
\end{align*}
whence
\begin{equation}
s + (1-\mu(t))\ (1 - L_0(\eta(t))) = h(1-\mu(t))\ , \quad t>0\ , \label{ltb4}
\end{equation}
where
\begin{equation}
h(z) := (1+s) z + k z \ln{z}\ , \quad z\in (e^{-1/k},1)\ . \label{ltb5}
\end{equation}
On the one hand, it follows from \eqref{ltb2}, \eqref{ltb3}, and \eqref{ltb4} that
\begin{equation}
\lim_{t\to\infty} h(1-\mu(t)) = s\ . \label{ltb6}
\end{equation}
On the other hand, $h$ is increasing on $(e^{-1/k},1)$ with $h(e^{-1/k}) = s e^{-1/k} < s < s+1 = h(1)$, so that $h$ is a one-to-one function from $(e^{-1/k},1)$ onto $(s e^{-1/k},s+1)$. Introducing its inverse $h^{-1}$, we deduce from \eqref{ltb6} that
\begin{equation}
\lim_{t\to\infty} \mu(t) = 1 - h^{-1}(s)\ . \label{ltb7}
\end{equation}
Recalling \eqref{ltb1} and \eqref{ltb2}, it follows from \eqref{ltb7} that
\begin{equation}
\lim_{t\to\infty} \frac{\zeta(t,ts)}{t} = \frac{1}{1 - h^{-1}(s)}\ . \label{ltb8}
\end{equation}
Finally, since
$$
L(t,ts) = \ell(t,\zeta(t,ts)) = L_0(\zeta(t,ts)) - k \ln{(1-\mu(t))}
$$
by \eqref{wp8}, \eqref{wp102}, and \eqref{ltb1}, we infer from \eqref{ltb7}, \eqref{ltb8}, and the properties of $L_0$ that
\begin{equation}
\lim_{t\to\infty} L(t,ts) = -k \ln{h^{-1}(s)}\ . \label{ltb9}
\end{equation}
We are left with expressing the right hand side of \eqref{ltb9} with the help of the Lambert $W$-function. To this end, we note that $h^{-1}(s)$ solves
$$
h^{-1}(s)\ \left[ (1+s) + k \ln{h^{-1}(s)} \right] = s
$$
by \eqref{ltb5} or, equivalently,
$$
W^{-1}\left( \ln{\left( e^{(1+s)/k} h^{-1}(s) \right)} \right) = \frac{s}{k}\ e^{(1+s)/k}\ .
$$
Therefore,
$$
e^{(1+s)/k} h^{-1}(s) = \exp{\left\{ W\left( \frac{s}{k}\ e^{(1+s)/k} \right) \right\}}\ ,
$$
and applying $-k\ln{}$ to both sides of the above identity leads us to \eqref{ltb0} thanks to \eqref{ltb9}.
\end{proof}
\medskip
We finally use \cite[Chapter~XIII.1, Theorem~2]{Fe71} to express the outcome of Proposition~\ref{prltb0} in terms of $\nu$ and obtain the last statement of Theorem~\ref{thint0}.
\section{Second moment estimate}\label{sec6}
We establish in this section an interesting smoothing property of \eqref{i6}, namely that the second moment of the solution becomes instantaneously finite for positive times, even it is initially infinite. We also investigate its large time behavior and answer by the positive a conjecture of Vigil \& Ziff \cite{VZ89}.
\begin{proposition}\label{prsme0}
For $t>0$, there holds
\begin{equation}
\partial_s L(t,0) = \frac{L_1(T^{-1}(t))}{1+ t L_1(T^{-1}(t))} \;\;\text{ and }\;\; \partial_s L(t,0) \mathop{\sim}_{t\to\infty} \frac{1-e^{1/k}}{t}\ . \label{sme0a}
\end{equation}
\end{proposition}
Note that the positivity of $T^{-1}(t)$ for $t>0$ guaranteed by Lemma~\ref{lewp2} ensures that $L_1(T^{-1}(t))$ is finite whatever the value $t>0$. Recalling that $L_1(s)=(L_0(s)-1)/s$, we realize that $L_1(T^{-1}(t))$ however blows down as $t\to 0$ if $L_0'(0)=-\infty$, that is, $f^{in}$ has an infinite second moment.
\begin{proof}
Fix $t>0$ and set $\theta:=T^{-1}(t)$. Recalling that $L(t,s)=\ell(t,\zeta(t,s))$ by \eqref{wp102}, it follows from \eqref{wp101} that
$$
\partial_s L(t,s) = \frac{\partial_s \ell(t,\zeta(t,s))}{\partial_s \Sigma(t,\zeta(t,s))}\ , \quad s>0\ ,
$$
while \eqref{wp8} and \eqref{wp13} give
\begin{align*}
\partial_s \ell(t,\sigma) = & L_0'(\sigma) - \frac{kt L_1'(\sigma)}{1+t L_1(\sigma)}\ , \quad \sigma>\theta\ , \\
\partial_s \Sigma(t,\sigma) = & 1 + t L_0'(\sigma) + k\ \frac{L_1'(\sigma)}{L_1(\sigma)^2} \ \left[ \ln{(1+t L_1(\sigma))} - t L_1(\sigma) \right]\ , \quad \sigma>\theta\ .
\end{align*}
Owing to Lemma~\ref{lewp4}, we may take $\sigma=\theta$ in the previous identities and use \eqref{wp10d} to obtain
$$
\partial_s \ell(t,\theta) = L_0'(\theta) - \frac{kt L_1'(\theta)}{1+t L_1(\theta)} = \frac{(1+t L_1(\theta)) L_0'(\theta) - kt L_1'(\theta)}{1+t L_1(\theta)}\ ,
$$
and
\begin{align*}
\partial_s \Sigma(t,\theta) = & 1 + t L_0'(\theta) + \frac{L_1'(\theta)}{L_1(\theta)^2} \ \left( \theta L_1(\theta)) - kt L_1(\theta) \right) \\
= & 1 + t L_0'(\theta) +\frac{\theta L_0'(\theta) + 1 - L_0(\theta)}{L_0(\theta)-1} - kt \frac{L_1'(\theta)}{L_1(\theta)} \\
= & \left( t + \frac{1}{L_1(\theta)} \right) L_0'(\theta) - kt \frac{L_1'(\theta)}{L_1(\theta)} \\
= & \frac{(1+t L_1(\theta)) L_0'(\theta) - kt L_1'(\theta)}{L_1(\theta)} \ .
\end{align*}
Combining the above formulas for $\partial_s L(t,s)$, $\partial_s \ell(t,\theta)$, and $\partial_s \Sigma(t,\theta)$, and recalling that $\zeta(t,0)=\theta$ give the first statement in \eqref{sme0a} and imply in particular that $\partial_s L(t,0)$ is finite.
To prove the second statement in \eqref{sme0a}, we use the first one to obtain
$$
\frac{1}{|\partial_s L(t,0)|} = - t - \frac{1}{L_1(T^{-1}(t))} = T^{-1}(t) -t + \frac{T(t)^{-1} L_0(T^{-1}(t))}{1 - L_0(T^{-1}(t))}\ .
$$
Since $T^{-1}(t) \sim t/(1-e^{-1/k})$ as $t\to \infty$ by \eqref{wp11b}, it follows from the properties of $L_0$ and the above identity that
$$
\frac{1}{|\partial_s L(t,0)|} \sim \left( \frac{1}{1-e^{-1/k}} - 1 \right) t \;\;\text{ as }\;\; t\to \infty\ ,
$$
and the proof of \eqref{sme0a} is complete.
\end{proof}
\medskip
Since
$$
\partial_s L(t,0) = - \int_0^\infty y \nu(t,dy)\ , \quad t>0\ ,
$$
the second statement \eqref{i12} of Theorem~\ref{thint0} readily follows from Proposition~\ref{prsme0}.
\section*{Acknowledgments}
Part of this work was done while PhL enjoyed the hospitality and support of the Department of Mathematical and Statistical Sciences of the University of Alberta, Edmonton, Canada and the African Institute for Mathematical Sciences, Muizenberg, South Africa.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,749 |
Q: Is there any way to know whether a reader is associated with a command object or not I have a layer where function openconnection closeconnection ... are written .
I have a big business logic file which has many data reader has used.
My problem is this that the data-readers has not been closed.
So the number of connection at in my app pool keep increasing.
Is there any wany to find is there any reader is associated with the commnad and if then in the closeconnection function I can check command object and close the reader.
Since we are using ref-cusor of oracle to read the data. And as far I know this type of cursor is not managed by oracle.
It is very much difficult for me to check all the bll.
Kindly suggest.
Edit1:
I have also some confusion about dispose if I dispose the command object weather or not the reader dispose will be called. Or If I dispose the Connection object the reader will be disposed.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,516 |
{"url":"https:\/\/www.learn-mlms.com\/01-module-1.html","text":"# Chapter 1 Introduction\n\n## 1.1 Overview\n\nThese materials focus on conceptual foundations of multilevel models (MLMs), specifiying them, and interpreting the results. Topics include multilevel data and approaches to dependence, specifying and interpreting fixed and random effects, model estimation, centering, repeated measures and longitudinal models, assumptions testing, and effect sizes in MLMs.\n\n## 1.2 Goals\n\nThese materials are intended for students and instructors.\n\nBy the end of this course, students will be able to:\n\n1. Estimate variance components and interpret the intraclass correlation coefficient;\n2. Decide if and when a multilevel model is needed;\n3. Specify and build multilevel models with covariates at level 1 and 2 with both cross-sectional and repeated measures designs;\n4. Interpret regression coefficients and variance components from multilevel models;\n5. Assess the assumptions of multilevel models;\n6. Calculate effect sizes for multilevel models.\n\n## 1.3 Prerequisites\n\nReaders should be comfortable with multiple linear regression, including building regression models, interpreting regression output, and testing for and interpreting regression coefficients including interactions. The first module reviews multiple regression and can be used to gauge your preparedness for continuing. For those wishing to brush up their regression skills before working through these materials, we recommend UCLA\u2019s Statistical Methods and Data Analytics resources and online seminars: https:\/\/stats.oarc.ucla.edu\/other\/mult-pkg\/seminars\/\n\nThe worked examples will be conducted using lme4 in R. The lme4 documentation provides details of the workings of lme4, for interested readers.","date":"2022-11-26 18:21:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.35592982172966003, \"perplexity\": 4182.723159269443}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446708046.99\/warc\/CC-MAIN-20221126180719-20221126210719-00500.warc.gz\"}"} | null | null |
\section{Introduction}
It has been well known for more than two decades that the Wilson
renormalization group (RG) offers practical tools as well as profound
insights for investigation of non-perturbative phenomena in field
theories. The continuum versions of the Wilson RG equation are called
the exact renormalization group (ERG) equations, which are given in
the form of non-linear functional differential equations.
There have been proposed several formulations of the ERG, which are
found to be mutually equivalent. These ERG equations give the change
of the so-called Wilsonian effective actions\cite{WH,WK,P} ~ or 1PI
cutoff effective actions\cite{Wet1993,Bonini} ~ under scale variation
leaving the low energy physics unaltered. The Wilsonian effective
action may be regarded as a point in the infinite dimensional space of
theories, or the space of coupling constants, and the ERG generates
flows of the coupling constants in this space.
However, in practical use, it is inevitable to approximate such an
infinite dimensional theory space by a much smaller subspace in order
to solve the ERG equations. Needless to say, the non-perturbative
nature of the ERG should be maintained in this approximation. The
method generally applied is the so-called derivative expansion, which
expands the interactions in powers of the derivatives and truncates
the series at a certain order.\cite{Wetderi,Wet1,Morrisderi,Ball} ~ With
this approximation the full equation is reduced to coupled partial
differential equations. Recently, the ERG in the derivative expansion
approximation for scalar field theories has been extensively studied
at the order of the derivative squared and has been found to offer
fairly good non-perturbative results even
quantitatively.\cite{Berges,Morris1,Morris2,Morris3} ~ The lowest
order of the derivative expansion, neglecting all corrections to
derivative operators, is called the ``local potential
approximation'' (LPA).\cite{NCS,HH} ~ Although the wave function
renormalization and,
therefore, the anomalous dimension is ignored in the LPA, the leading
exponent $\nu$ is known to be estimated rather well by using the ERG.
However, the number of the couplings to be incorporated, or the number
of the beta functions, is still infinite in the derivative
expansion. If we consider application to more complicated systems
it would be favorable to approximate it further by a finite number of
couplings, as long as this is sufficiently effective.
This method is advantageous not only because it simplifies the
analysis but also because the effective couplings of physical interest
are treated directly. Such an approximation is naively performed by
truncated expansion of the Wilsonian effective action, in turn, in
powers of the fields. Actually it is found to work well practically
without loss of the non-perturbative nature as long as we adopt a
special expansion scheme, as is discussed later.
One of the advantageous points of the ERG is certainly that it is able
to allow for the systematic improvement of the approximations, as
mentioned above. However, the improvement totally relies on the
convergence of the results in non-perturbative analysis. We may
approve the results obtained by the ERG only after confirming their
sufficient convergence, since there is no small parameter which
controls the approximation. It should be noted here that the commonly
used expansion methods, e.g., the perturbation theory, $1/N$
expansion and $\varepsilon$-expansion etc., generally produce asymptotic
series at best, in contrast to the convergence property exhibited by
the ERG.\cite{AMSSTeffective}
The main subject of this paper is the convergence of the expansion
scheme in terms of the fields. It has been claimed that this
convergence is rather poor,\cite{MOP} ~ or that the results cease to
converge.\cite{Morristrunc} ~ If such behavior appears commonly, it
would be a fatal defect of the ERG approach.\footnote{
The convergence of the derivative expansion in the
non-perturbative calculation remains difficult to see due to
complication in the higher orders. Morris examined this problem
perturbatively at two loops and found that the ERG with certain
cutoff profiles indeed displays convergence.\cite{Morris1}}
However, it has been realized that the expansion around the potential
minimum drastically improves this convergence
property.\cite{Wetderi,tetwet,Wet1,A} ~
Indeed, it is good news for the ERG approach that we may obtain
good convergence by adopting the appropriate truncation scheme.
However, it has not yet been investigated in detail how effective this
scheme is generally, nor has it been determined the origin of this
improvement.
In this paper we discuss convergence properties in different
expansions schemes by examining $Z_2$ symmetric scalar theories in
three dimensions. As the physical quantities, the critical exponents
and also the infrared effective potentials, or the renormalized
trajectories, are compared among the different schemes. It is found
that the convergence property is significantly improved in the new
truncation scheme. We will also discuss the physical reason of the
improvement by studying $O(N)$ symmetric scalar theories in the large
$N$ limit. From this observation it is speculated that good
convergence depends on how accurately the relevant operator is covered
within the truncated subspace.
\section{Exact renormalization group equations}
First let us briefly review the formulations of the ERG with which
we will analyze the scalar theories. The ERG equation widely
studied recently is given by considering renormalization of the
so-called ``cutoff effective action'',
$\Gamma_{\mbox{eff}}[\phi]$.\cite{Wet1993,Bonini} ~ The cutoff
effective action is the 1PI part of the Wilsonian effective
action, namely, the generating functional of the connected and amputated
cutoff Green functions. Therefore the ERG equation may be obtained by
the Legendre transformation of the Polchinski equation for the
Wilsonian effective action.
In this formulation the cutoff is performed by introducing a proper
smooth function in terms of the momentum $q$ and the cutoff scale
$\Lambda$; $C(q,\Lambda)$, into the partition function as
\begin{equation}
\exp W[J] = \int {\cal D}\phi \exp
\left\{
-\frac{1}{2} \phi \cdot C^{-1} \cdot \phi -S[\phi] + J \cdot \phi
\right\},
\label{partitionfunction}
\end{equation}
where dot $(\cdot)$ denotes matrix contraction in momentum space.
From Eq.~(\ref{partitionfunction}) we may obtain the variation of $W$
with respect to the cutoff as \cite{P}
\begin{equation}
\frac{\partial W[J]}{\partial \Lambda} =
-\frac{1}{2}
\left\{
\frac{\delta W[J]}{\delta J}\cdot
\frac{\partial C^{-1}}{\partial \Lambda}\cdot
\frac{\delta W[J]}{\delta J}
+\mbox{tr}
\left(
\frac{\partial C^{-1}}{\partial \Lambda}\cdot
\frac{\delta^2 W[J]}{\delta J \delta J}
\right)
\right\}.
\end{equation}
The ERG for the cutoff effective action $\Gamma_{\mbox{eff}}$ is
defined by the Legendre transformation:
\begin{equation}
\Gamma_{\mbox{eff}}[\phi] + \frac{1}{2}\phi \cdot C^{-1} \cdot \phi
= -W[J]+J\cdot \phi.
\end{equation}
By taking the canonical scaling under the shift of the cutoff scale
into consideration, the ERG equation for $\Gamma_{\mbox{eff}}$
in $D$ dimensions may
be written down as \cite{Morrisderi}
\begin{equation}
\left(
\frac{\partial}{\partial t}
+d_{\phi}\phi\cdot\frac{\delta}{\delta\phi}+\Delta_{\partial} -D
\right)
\Gamma_{\mbox{eff}}[\phi]
=
\frac{1}{2} \mbox{tr}
\left\{
\frac{\partial C^{-1}}{\partial t} \cdot
\left(
C^{-1}+\frac{\delta^2 \Gamma_{\mbox{eff}}[\phi]}{\delta \phi \delta \phi}
\right) ^{-1}
\right\} ,
\label{legendrefloweq}
\end{equation}
where $t =\ln ( \Lambda_0/\Lambda)$, and $d_{\phi}$ is the scaling
dimension of the scalar field which is given by $(D-2+\eta)/2$
with the anomalous dimension $\eta$.
The operator $\Delta_{\partial}$ counts
the number of the derivatives, which is given explicitly by
\begin{equation}
\Delta_{\partial}
\equiv D + \int \frac{d^D q}{(2\pi)^D}
\phi(q)q_{\mu}\frac{\partial}{\partial q_{\mu}}
\frac{\delta}{\delta\phi(q)}.
\end{equation}
Thus the ERG equation is defined depending on the cutoff functions.
The physical quantities such as the exponents are found to be
independent of the cutoff scheme, as is expected.
However, this does not hold once some approximations have been
performed to these equations.\footnote{
In practice, as far as the scalar theories are concerned, this cutoff
scheme dependence is found to be rather weak, and the exponent changes
smoothly under variation of the cutoff
profiles.\cite{Ball,AMSSTcutoff}}
In this paper we are going to examine the extreme cases, i.e. schemes
with a very smooth cutoff and with the sharp cutoff limit.
We adopt the cutoff function
\begin{equation}
C(q,\Lambda)=\frac{1}{q^2}\frac{\theta(q,\Lambda)}{1-\theta(q,\Lambda)},
\hspace{10mm}
\theta(q,\Lambda)=\frac{1}{1+(\Lambda^2/q^2)^2} \hspace{3mm}
\label{cutofffunction}
\end{equation}
as the smooth one in the practical calculations,
{\it a la} Ref.~\cite{Morrisderi}.
The sharp cutoff version of the ERG will be discussed later.
In the derivative expansion,
the full ERG equation(\ref{legendrefloweq}) is reduced to a
set of partial differential equations by substituting the effective
action $\Gamma_{\mbox{eff}}$ of the form
\begin{equation}
\Gamma_{\mbox{eff}}[\phi]=\int d^D x \left\{
V(\phi;t)+\frac{1}{2}K(\phi;t)(\partial_{\mu}\phi)^2 + \cdots~~~~
\right\},
\end{equation}
where $V(\phi;t)$ and $K(\phi;t)$ are cutoff-dependent functions. In the
second order of the expansion we may take the variation of the first two
terms only, ${\partial V}/{\partial t}$ and ${\partial K}/{\partial
t}$.
For the cutoff function given by (\ref{cutofffunction}),
these ERG equations in three dimensions are found to be
\begin{eqnarray}
\frac{\partial V}{\partial t}
&=&-\frac{1}{2}(1+\eta)\phi V'+3V
-\frac{1-\eta/4}{\sqrt{K}\sqrt{V''+2\sqrt{K}}} \hspace{3mm} ,
\label{dVdt} \\
\frac{\partial K}{\partial t}
&=&-\frac{1}{2}(1+\eta)\phi K'-\eta K
+(1- \frac{\eta}{4})(\frac{1}{48}
\frac{24KK''-19(K')^2}{K^{3/2}(V''+2\sqrt{K})^{3/2}} \nonumber \\
& &-\frac{1}{48} \frac{58V'''K' \sqrt{K} +57(K')^2 +(V''')^2 K}
{K(V''+2 \sqrt{K})^{5/2}} \nonumber \\
& &+\frac{5}{12} \frac{(V''')^2 K+2V'''K' \sqrt{K} + (K')^2}
{\sqrt{K} (V''+2 \sqrt{K})^{7/2}}
) \hspace{1mm} ,
\label{dKdt}
\end{eqnarray}
where the prime denotes differentiation
with respect to $\phi$. \cite{Morrisderi} ~ The anomalous dimension
$\eta(t)$ is determined by imposing
the renormalization condition for the wave function, $K(\phi=0;t)=1$.
In the LPA we may solve only Eq.~(\ref{dVdt})
with respect to the effective potential $V(\phi;t)$
by reducing $K=1$ and, therefore, $\eta=0$.
Actually these partial differential equations have been solved and
found to give the exponents with fairly
good accuracy.\cite{Morris1,Morris2,Morris3,Morristrunc} ~ In the next
section these equations are examined by expansion in powers of the
fields.
In the case of the sharp cutoff scheme, we examine the
Wegner-Houghton (WH) equation\cite{WH} instead of
Eq.~(\ref{legendrefloweq}) with the sharp cutoff limit.
The WH equation is formulated so that the fields are integrated
gradually from the modes with higher momentum by introducing the sharp
cutoff into the path integral measure. Indeed, the ERG equation for the
1PI effective action as well as the Polchinski equation\cite{P} turns
out to be equivalent to the WH equation in the sharp cutoff
limit.\cite{MorrisIJMP,AMSSTcutoff} ~
However, the ERG equation is known to exhibit non-analytic dependence
on the momentum in this limit. Therefore we examine the sharp cutoff
scheme only in the LPA. The WH equation in the LPA is given simply by
\begin{equation}
\frac{\partial V}{\partial t}=
3V-\frac{1}{2}\phi V'+\frac{1}{4\pi^2} \ln(1+V'') \hspace{3mm} ,
\label{LPAWH}
\end{equation}
in three dimensions.\cite{NCS,HH}
These partial differential equations, of course, may be solved
directly. However, this would not be practical for more
complicated systems. Indeed, it turns out to be much more economical
to solve them by reducing them to coupled ordinary differential
equations. Besides, if we are interested in the effective
coupling constants of the theories, e.g. masses,
self-interactions, gauge couplings and so on, it is
natural to solve the ERG equations for these coupling constants
by expanding the effective action into a sum of operators.
Note that Equations (\ref{legendrefloweq}) and (\ref{LPAWH}) given
in the two different cutoff schemes do not produce the same results
for physical quantities even if both are analyzed in the LPA.
Rather, in this paper we are interested in the convergence properties
of the solutions of the equations obtained by truncation of their
power expansion in each case. In the next section the convergence
behavior is explicitly examined.
\section{Truncation in the comoving frame}
First let us examine the ERG equation (\ref{LPAWH})
for the $Z_2$ symmetric scalar theory. If we approximate the
effective potential $V(\phi ;t)$ by a finite order power series
in terms of the $Z_2$ invariant variable $\rho=\phi^2/2$ (Scheme I),
\begin{equation}
V(\rho; t)=\sum _{n=1}^{M} \frac{a_n(t)}{n!} \rho ^n,
\end{equation}
then we obtain $M$ ordinary differential equations for the running
couplings $a_n$.\footnote{
The $M$ coupled beta functions lead us to $M$ distinct fixed point
solutions, all of which but two (the trivial fixed point and the
so-called Wilson-Fisher fixed point) should be fake in this
approximation.
However, we may easily identify the true fixed point among these
solutions by looking at their stability against the truncation.}
We may suppose naively that the results obtained with these truncated
equations converge to the solutions of the partial differential
equation (10) as the order of the truncation $M$ is increased.
However, this is not the case.
In Fig.~1 the truncation dependence of the leading exponent $\nu$ is
shown. It is seen that the solutions cease to converge beyond a
certain order and finally to oscillate with 4-fold periodicity around
the expected value from the partial differential equation (10).
Actually, the non-trivial fixed point itself is also found to oscillate
similarly in the truncated approximation.
Morris pointed out in Ref.~\cite{Morristrunc}
that this oscillatory behavior is related to the singularity of the
fixed point solution of Eq.~(10) in the complex plane.
The singularities $\rho_*=|\rho_*|e^{i \theta_*}$ closest to the
origin are located at $(|\rho_{*}|,\theta_*)=(0.123,\pm 0.514 \pi)$.
The existence of these singularities implies that the coefficients of
the fixed point solution expanded in powers around the origin appear
with 4-fold oscillation and that the convergence radius of the
expansion series is given by 0.123.
From these observations Morris has explained the behavior of the
truncated solutions in Scheme I and that the leading exponent cannot
converge to a definite value with precision beyond an error of 0.008.
\begin{figure}[htb]
\epsfxsize=0.85\textwidth
\centerline{\epsffile{exposharp2ver1.eps}}
\vspace{3mm}
\centerline{
\parbox{140mm}{
\footnotesize
Fig.~1~~The truncation dependence of the leading exponent evaluated
using various truncation schemes. The solutions not displayed
($M=3$ of Scheme III (0.08), $M=3,4$ of Scheme III (0.1) and
$M=3,4,5$ of Scheme III (0.12))
lie outside the range on the vertical axis.
The values in the parentheses in the legend correspond to the
expansion point $\rho_0$ in Scheme III.
The term ``minimum'' means the minimum of the fixed point potential
in each truncation.
}
}
\end{figure}
\vspace{3mm}
This undesirable behavior, however, is drastically improved if we
expand the potential around its minimum. The effective potential may
be approximated in turn by (Scheme II)
\begin{equation}
V(\rho; t)=\sum _{n=2}^{M} \frac{b_n(t)}{n!} (\rho - \rho _{0}(t))^n,
\end{equation}
where $\rho_0$ is the potential minimum. Therefore here the term
linear in $\rho-\rho_0$ is absent.
Indeed, as is seen in Fig.~1 (see also Fig.~4), the results for the
exponent calculated in this scheme converge very rapidly to
0.689459056$\pm$2, which is consistent with the results obtained by
analysis of the partial differential equation (10), 0.689459056.
(The latter will be reported elsewhere.)
Thus we may say that the expansion scheme II is a quite effective
method in obtaining an accurate answer in a fairly small-dimensional
subspace. The fixed point potential obtained with this
method (Scheme II) is also shown in Fig.~2 (right).
It is rather surprising that the potential of the fixed point solution
can be obtained within a certain range of the field variable $\rho$
quite well with such a simple analysis.
\begin{figure}[htb]
\parbox{77mm}{
\epsfxsize=0.45\textwidth
\begin{center}
\leavevmode
\epsfbox{fppotori2.eps}
\end{center}
}
\parbox{77mm}{
\epsfxsize=0.45\textwidth
\begin{center}
\leavevmode
\epsfbox{fppotcom.eps}
\end{center}
}
\end{figure}
\vspace{-8mm}
\begin{center}
\parbox{140mm}{
\footnotesize
Fig.~2~~The truncation dependences of the fixed point potential
evaluated using Schemes I and II are shown to the left and right in
the figure, respectively. The integers in the figures denote the
orders of the truncation.
}
\end{center}
\vspace{5mm}
However, the series of the solution obtained in truncated approximation
of Scheme II does not converge perfectly. We examined the leading
exponent $\nu$ in this analysis up to the 60th order of truncation.
The logarithmic plot of the obtained coefficients $\rho _0$ and $b_n$
of the fixed point solution and the leading exponent against the order
of truncation is shown up to 60th in Figs.~3 and 4, respectively.
It is seen that the truncation dependence does not disappear
completely, and the results display oscillatory behavior with 3-fold
approximate periodicity, as is shown in Figs.~3 and 4.
These behavior can be clearly understood by considering the
singularities of the untruncated fixed point
solution.\cite{Morrisprivate}
Since the minimum of the fixed point potential is at
$\rho_0 =0.0471$, we have $|\rho_* - \rho_0|=0.134$ and
${\rm arg}(\rho_* - \rho_0)=0.64 \pi$.
This angle is close to $2\pi/3$, which tells us that
the truncated solution in Scheme II oscillates with 3-fold periodicity.
Also, the expansion should have a finite convergence radius of 0.134.
Therefore the boundary of the convergence radius appears around
$\rho =0.181$. This is indeed seen in Fig.~2 (right).
Although Scheme II extends the convergence radius slightly,
the convergence of the leading exponent is much improved up to the
error $10^{-9}$, as is seen in Fig.~4 (left).\footnote{
It is noted that in the analysis of Eq.~(10) the minimum expansion
in terms of the variable $\phi$ becomes worse than Scheme I
due to the smaller convergence radius.
}
\begin{figure}[htb]
\parbox{77mm}{
\epsfxsize=0.45\textwidth
\begin{center}
\leavevmode
\epsfbox{coefflarge.eps}
\end{center}
}
\parbox{77mm}{
\epsfxsize=0.48\textwidth
\vspace{2mm}
\begin{center}
\leavevmode
\epsfbox{rho02.eps}
\end{center}
}
\end{figure}
\vspace{-8mm}
\begin{center}
\parbox{140mm}{
\footnotesize
Fig.~3~~The large order behavior of the coefficients $b_n$
and the potential minimum $\rho_0$ evaluated using Scheme II.
The vertical axis of the left figure denotes
$\ln (|b_i-\langle b_i \rangle |10^{11}+1)
{\rm sign}(b_i - \langle b_i \rangle)$.
}
\end{center}
\begin{figure}[htb]
\parbox{77mm}{
\epsfxsize=0.45\textwidth
\begin{center}
\leavevmode
\epsfbox{expolarge.eps}
\end{center}
}
\parbox{77mm}{
\epsfxsize=0.45\textwidth
\begin{center}
\leavevmode
\epsfbox{scheme3expo1.eps}
\end{center}
}
\end{figure}
\vspace{-8mm}
\begin{center}
\parbox{140mm}{
\footnotesize
Fig.~4~~The large order behavior of the leading exponent evaluated
using Scheme II and III. The vertical axis of the left figure denotes
$\ln( |\nu - \langle \nu \rangle|10^{10}+1)
{\rm sign}(\nu - \langle \nu \rangle)$.
}
\end{center}
\vspace{5mm}
Such truncation dependence in the two schemes is seen in the ERG
equations with smooth cutoff, (\ref{dVdt}) and (\ref{dKdt}), as well.
In Fig.~5 the leading exponent obtained by Scheme I and Scheme II in
the LPA is shown. We find that Scheme II is again clearly
superior to Scheme I, while the oscillation, even in Scheme I,
is significantly attenuated in the smooth cutoff scheme.
The value to which the leading exponent converges is
0.660.\footnote{
The partial ERG equations given by (\ref{dVdt}) and (\ref{dKdt})
are examined in great detail in Ref.~\cite{Morris1,Morris2}.
Our results are consistent with those of that analysis.}
In the second order of the derivative expansion we examined the
truncation dependence of the leading exponent and also of the anomalous
dimension in Scheme II.
\footnote{The exponents could not even be evaluated in Scheme I.}
In this analysis the function $K(\rho;t)$ is also expanded around
$\rho_0$ and is truncated at the same order as $V(\rho;t)$.
The results are shown in Figs.~5 and 6.
The values so obtained are $\nu=0.617476$ and $\eta=0.05425$.
The world standard values are $\nu=0.6310$ and $\eta=0.0375$ from the
$\varepsilon$-expansion.\cite{ZJ} ~ It is worth while to mention
that the leading exponent indeed approaches the world standard
value when going up to the next-to-leading order of the derivative
expansion.
\begin{figure}[htb]
\parbox{77mm}{
\epsfxsize=0.45\textwidth
\begin{center}
\leavevmode
\epsfbox{exposmooth.eps}
\end{center}
\begin{center}
\parbox{63mm}{\footnotesize
Fig.~5~~The leading exponent evaluated using Scheme II}
\end{center}
}
\parbox{77mm}{
\epsfxsize=0.45\textwidth
\begin{center}
\leavevmode
\epsfbox{anomalous.eps}
\end{center}
\begin{center}
\parbox{65mm}{\footnotesize
Fig.~6~~The anomalous dimension evaluated using Scheme II}
\end{center}
}
\end{figure}
In general $d/dt$ in the ERG defines a vector field of RG flow on the
theory space and is given in the coordinate system $\{g_i\}$ by
\begin{equation}
\frac{d}{dt}=\beta_i(g)\frac{\partial}{\partial g_i},
\end{equation}
where we have introduced the generalized beta functions
$\beta_i(g)=dg_i/dt$. Correspondingly, the variation of the effective
action $S$ may be written as
\begin{equation}
\frac{dS}{dt}=\beta_i(g)\xi_i(g),
\end{equation}
where $\xi_i(g)=\partial S/\partial g_i$ are the base vectors. In
Scheme II the base vectors are dependent on the position in the theory
space, while they are fixed in Scheme I. We refer to these
coordinate systems as the ``comoving frame'' and ``fixed frame'',
respectively.\cite{AMSSTeffective}
Note that Schemes I and II with same maximum powers $M$ employ the
same subspace of the polynomials. We project the flow in the full
theory space on the subspace and evaluate the beta functions for the
projected flow. The projection depends on the choice of the
coordinates, or on the manner of expansion. In fact, the beta
functions in the truncation Schemes I and II are evaluated for
different projections. This causes deviation of the results between
the two schemes discussed here.
From the preceding argument it may be expected that more accurate
convergence is achieved if we expand at a point with a larger
convergence radius. In order to see this, we examine another type of
truncation, similar to Scheme I, but with the expansion point shifted
from the origin by some fixed values $\rho _0$ (Scheme III):
\begin{equation}
V(\rho; t)=\sum _{n=1}^{M} \frac{c_n(t)}{n!} (\rho - \rho _{0})^n.
\end{equation}
This scheme is an example of the fixed frame. The leading exponent
obtained in this approximation is also shown in Fig.~1 and in Fig.~4
(right). Here we employ Eq.~(\ref{LPAWH}). It is reasonable that the
value will converge to a definite value with high precision if the
expansion point $\rho_0$ is far from singularities. Indeed, we can
obtain a convergent value of leading exponent with precision on the
order of $10^{-16}$, 0.68945905616213484$\pm$1, by choosing
$\rho_0=0.08$. However, in order to obtain a highly accurate value, we
need a very large order of truncation.
One should employ a truncation method which provides a result with
sufficient convergence and precision even with a small order of
truncation. As is seen in the next section, in the large $N$ limit,
Scheme II is found to give us the exact solution. Apart from this
extreme case of large $N$ theory, however, it is significant for the
practical analysis of complicated systems to be converging with small
order of truncation, because it is in general difficult to evaluate at
the large order of truncation. It is seen from Fig.~1 that we should
set the value of $\rho_0$ to the minimum of the fixed point potential
if we demand better convergence at small order of truncation.
If the highly accurate precision is not required, or if it is
difficult to analyze with a large order of truncation, the truncated
approximation of Scheme II is sufficiently workable owing to its
simplicity. The fact that Scheme III with $\rho_0$ set to the minimum
of the fixed point potential also gives good convergence in the small
subspace implies that the improvement of the approximation originates
in the choice of the base vectors $\xi_i$ around the fixed point.
This seems reasonable, since the exponent is determined solely by the
structure of the ERG equation around the non-trivial fixed point.
Needless to say, the truncated approximation of the comoving frame
is much more advantageous in practical analysis than that of the fixed
frame, since the position of the minimum of the fixed point potential
cannot be known {\it a priori}.
\section{Large $N$ limit}
If we extend the observation examined in the previous section to
$O(N)$ symmetric scalar theories in three dimensions, then the
approximation in Scheme II is found to show stronger convergence as
$N$ increases, while the exponents obtained in Scheme I become more
fluctuating.\cite{WH,MOP,Changpr,RTWlargen,Wet1,A,Lit,Comeon,Morlargen,Morris3}
~Moreover, the truncation dependence turns out to
disappear completely in the large $N$ limit, as discussed in
Ref.~\cite{AMSSTeffective}. Therefore we discuss the physical
reason behind this remarkable improvement of convergence by
considering $O(N)$ models in the large $N$ limit.
The LPA WH equation in $D$ dimensions is reduced in the large $N$
limit to
\begin{equation}
\frac{\partial V}{\partial t} =
DV+(2-D)\rho V_{\rho}+ \frac{A_D}{2}\ln(1+V_{\rho}),
\label{largeNLPAWH}
\end{equation}
where $\rho$ denotes $\Sigma_{a=1}^N(\phi^a)^2/2$, $V_{\rho}$ denotes
the differentiation of $V$ with respect to $\rho$ and $A_D$ is the
surface of the $D$-dimensional unit sphere divided by $(2\pi)^D$.
Here we have rescaled $\rho$ and $V$ properly by $N$ before taking the
limit. It is known that Eq.~(\ref{largeNLPAWH}) gives the exact
effective potential in the large $N$ limit. If we expand
Eq.~(\ref{largeNLPAWH}) in Scheme II, the resulting differential
equations,
\begin{eqnarray}
& & \frac{d\rho_0}{dt}\equiv \beta_1
=(D-2)\rho_0-\frac{A_D}{2}, \nonumber \\
& & \frac{db_2}{dt}\equiv \beta_2
=(4-D)b_2-\frac{A_D}{2}b_2^2, \\
& & \frac{db_3}{dt}\equiv \beta_3
=(6-2D)b_3-\frac{A_D}{2}(3b_2b_3-2b_2^3),
~~~ {\rm etc.}, \nonumber
\end{eqnarray}
are found to be analytically soluble order by order.
This means that the entire flow diagram in the theory space can be
exactly determined within any finite dimensional subspaces.
We refer to such a special truncation scheme as the ``perfect
coordinates''.\cite{AMSSTeffective} Generally, it would be difficult
to find out such coordinates that enable us to solve the ERG equations
exactly. However, Scheme II turns out to be an example of perfect
coordinates in the large $N$ limit. Such a remarkable simplification
does not occur in Scheme I.
Due to the perfectness of the coordinates employed, the exact values
for the exponents can be obtained. The exponents are given by the
eigenvalues of the matrix $\Omega_{ij}=\partial \beta_i/\partial b_j$
evaluated at the non-trivial fixed point:
\begin{equation}
\Omega=\left(
\begin{array}{cccc}
D-2 & 0 & 0 & \cdots \\
0 & D-4 & 0 & \cdots \\
0 & \frac{24}{A_D} \frac{(4-D)^2}{6-D} & D-6 & \cdots \\
\cdots & \cdots & \cdots & \cdots
\end{array}
\right).
\end{equation}
Thus the exponents are exactly determined from the eigenvalues,
or the diagonal elements of this matrix, and $\nu=1/(D-2)$.
Also the corresponding eigenvectors of this matrix give us the
so-called relevant and the irrelevant operators.
The most characteristic feature of the ERG equations in Scheme II
is that the relevant coupling precisely coincides with $\rho_0$ and is
not influenced by the truncation. We can say that this is the direct
reason why the leading exponent is calculated in a
truncation-independent way.
Thus the relevant operator corresponding to the coupling $\rho_0$
has been found to be given by $V_{\rho}^*$. In addition to the
relevant operator, all eigen-operators can be derived exactly from
Eq.~(\ref{largeNLPAWH}) as follows. Suppose $V^*(\rho)$ is the
non-trivial fixed point solution of the ERG equation and consider an
infinitesimal deviation from this; $V(\rho;t)=V^*(\rho)+\delta
V(\rho;t)$. Then we obtain the eigenvalue equation with respect to
$\delta V$ as
\begin{equation}
D\delta V - 2\frac{V^*_{\rho}}{V^*_{\rho\rho}}\delta V_{\rho}
=\lambda \delta V.
\end{equation}
By solving this equation, all of the eigenvectors are found to be
given by
\begin{equation}
\delta V(\rho;t) \propto (V^*_{\rho}(\rho))^{\frac{D-\lambda}{2}}.
\end{equation}
If we demand the analyticity of $\delta V$, then the eigenvalues
are determined to be $\lambda=D-2,D-4,D-6,\cdots$, as expected.
Moreover, it turns out to be possible to reformulate the ERG so that
the effective potential $V(\rho;t)$ is expanded into a power series of
the (ir)relevant operators $(V_{\rho})^n$. For this purpose, let us
first introduce two auxiliary fields $\chi$ and $\eta$ to the theory,
\begin{equation}
Z= \int {\cal D}\phi^a {\cal D}\chi {\cal D}\eta
\exp \left\{
-\int d^D x \left[ \frac{1}{2}(\partial _{\mu}\phi^a)^2
+ \chi\left( {1\over 2}\left(\phi^a \right)^2 - N \eta \right)
+N V(\eta)
\right]\right\}.
\end{equation}
In the large $N$ limit, the path integral of these auxiliary fields is
evaluated by the saddle point method. Then the effective potential
$V({\bar \eta};t)$ is given by solving the coupled equations
\begin{eqnarray}
V(\bar{\eta};t) &=&
\frac{1}{2} {\rm trln} (- \Box + \chi)
+ \chi (\bar{\eta} - \eta) + V(\eta) ,\\
\eta &=&
\bar{\eta} + \frac{1}{2} {\rm tr} \frac{1}{- \Box + \chi},\\
\chi &=&
\frac{\partial V(\eta)}{\partial \eta},
\end{eqnarray}
where $\bar{\eta}$ denotes $\bar{\phi}^2/2N$.
Then we change the variable $\bar{\eta}$ to $\bar{\chi}$
through the Legendre transformation:
\begin{eqnarray}
\bar{\chi} &=& \frac{\partial V(\bar{\eta};t)}{\partial \bar{\eta}}, \\
U(\bar{\chi};t) &=& - \bar{\chi} \bar{\eta} + V(\bar{\eta};t).
\end{eqnarray}
Therefore the WH equation for $U(\bar{\chi};t)$ is simply given by
\begin{equation}
\frac{\partial U}{\partial t}
=DU-2\bar{\chi} U_{\bar{\chi}}+\frac{A_D}{2}\ln (1+\bar{\chi}).
\end{equation}
It is readily seen that this equation is indeed identical to
Eq.~(\ref{largeNLPAWH}) owing to the saddle point equation.
This form of the ERG equation, in turn, is exactly solved by expanding $U$
into an ordinary Taylor series of $\bar{\chi}$ as
\begin{equation}
U(\bar{\chi};t)=
a_0(t)+a_1(t)\bar{\chi} + \frac{1}{2}a_2(t)\bar{\chi}^2 +\cdots,
\end{equation}
where $a_1$ is just the potential minimum parametrized previously by
$\rho_0$. The relevant operator is found to be $\bar{\chi}$ itself,
which has dimension $D-2$ at the fixed point. The irrelevant operators
are also simply given by $\bar{\chi}^2, \bar{\chi}^3, \cdots$.
Thus we can reformulate the large $N$ model in terms of the purely
(ir)relevant operators by introducing a new variable, which is a
composite operator of the original scalar fields. On the renormalized
trajectory, we may ignore these irrelevant operators. Once they are
eliminated, the theory turns out to be identical to the non-linear
$\sigma$ model.
What do these relations found in the large $N$ limit imply for the
finite $N$ cases? It would be natural to suppose from the above
observation that good convergence in Scheme II for a finite $N$ may be
explained similarly. Actually, if we compare the forms of the
eigenvectors of the matrix $\Omega$ in Schemes I and II, then we will
see a clear difference. That is, the eigenvectors are approximated
well by the first several components in Scheme II, while this is not
the case in Scheme I. Thus we may deduce that Scheme II is able to
capture the relevant operator in the small dimensional subspace and,
therefore, to make the truncation dependence diminish rapidly.
In practice, it is not hard to extend the formulation of ERG so as to
incorporate the auxiliary field to finite $N$ cases. The results of
numerical analyses of such ERG equations will be reported elsewhere.
To summarize, the physical reason for the good convergence in the
comoving frame is speculated to be that the leading operator
defined in this scheme covers the relevant operator quite well.
\section{Infrared effective potentials}
It is significant to observe the truncation dependence of not only the
exponents but also other physical quantities in various approximation
schemes. We here discuss the infrared effective potentials for the
scalar theory by using the LPA WH equation (\ref{LPAWH}).
The infrared effective potentials enable us to calculate physical
quantities such as the effective mass and the effective couplings at a
low energy scale. The low energy physics is completely described by
the one dimensional renormalized trajectory of the relevant operator
extending from the non-trivial fixed point on the critical surface.
The renormalized trajectory is divided into two parts in the symmetric
phase and in the symmetry broken phase. We evaluate the infrared
effective potentials by tracing the running coupling constants on the
renormalized trajectory. As the cutoff is lowered, the minimum of the
effective potential in the symmetric phase shrinks, while it grows in
the broken phase.
The renormalized trajectories obtained using different truncation
approximations do not coincide. We should be careful when examining
the renormalized trajectory. We need to impose a common appropriate
renormalization condition to compare the infrared effective potentials
evaluated in the various approximation schemes.
Actually, we evaluate the infrared potentials by employing the point
on the renormalized trajectory satisfying the following
renormalization condition; the gradient at the origin of the potential
is equal to $0.1$ in the symmetric phase and the field value at the
minimum of the potential is equal to $0.1$ in the broken phase.
The effective potentials obtained in Schemes I and II are shown in
Figs.~7 and 8, respectively. Note that the absolute height of the
potential is adjusted so as to vanish at the origin, since it is not
taken into account correctly in the ERG equation.\footnote{
In the analysis in terms of Scheme II, the minimum of the potential
moves from the positive region ($\rho >0$) to the negative region
($\rho <0$) in the symmetric phase. Therefore one should switch the
evaluation of the potential using Scheme II to Scheme I at the point
where the minimum of the potential passes the origin $\rho =0$.
}
\begin{figure}[htb]
\parbox{77mm}{
\epsfxsize=0.45\textwidth
\begin{center}
\leavevmode
\epsfbox{infrapotsymori1.eps}
\end{center}
}
\parbox{77mm}{
\epsfxsize=0.45\textwidth
\begin{center}
\leavevmode
\epsfbox{infrapotbroori1.eps}
\end{center}
}
\end{figure}
\vspace{-8mm}
\begin{center}
\parbox{140mm}{
\footnotesize
Fig.~7~~The truncation dependence of the infrared effective potential
in the symmetric and the symmetry broken phases evaluated using
Scheme I
}
\end{center}
\begin{figure}[htb]
\parbox{77mm}{
\epsfxsize=0.45\textwidth
\begin{center}
\leavevmode
\epsfbox{infrapotsymmv1.eps}
\end{center}
}
\parbox{77mm}{
\epsfxsize=0.45\textwidth
\begin{center}
\leavevmode
\epsfbox{infrapotbromv1.eps}
\end{center}
}
\end{figure}
\vspace{-8mm}
\begin{center}
\parbox{140mm}{
\footnotesize
Fig.~8~~The truncation dependence of the infrared effective potential
in the symmetric and the symmetry broken phases evaluated using
Scheme II
}
\end{center}
\vspace{5mm}
As a result, the truncated approximation in the comoving frame leads
to good convergence for the effective potential in both phases as well
as for the exponent. It is remarkable property of the comoving frame
that it remains so effective after truncation, even away from
criticality, and moreover, that this occurs irrespective of the
phases. This result would imply that the relevant operator ruling the
renormalized trajectory can be approximated well enough in the small
dimensional subspace truncated in the comoving frame.
\section{Summary and discussion}
We considered the convergence properties of physical quantities
evaluated using the ERG in various truncated approximation schemes in
operator expansions. The $Z_2$ symmetric scalar theory in three
dimensions was numerically analyzed, and the approximated solutions
for the exponents and the infrared effective potentials were compared
in the various schemes. In particular we focused on studies of the
difference between the truncation schemes in the expansion at the
field origin (Scheme I) and at the minimum of the effective potential
(Scheme II).
It was found that Scheme II displays a remarkably strong convergence
property with respect to the order of truncation as far as the
quantities we examined are concerned; the leading exponent, the
anomalous dimension and the infrared effective potentials. Although it
is seen that the exponent obtained in Scheme II also ceases to
converge eventually at a certain large order, the width of fluctuation
is very small, and we can obtain the value with great accuracy.
Indeed we may examine the partial differential equations derived in
the derivative expansion scheme directly for such a simple model.
However, such analyses would become difficult for more complicated
systems, e.g., triviality mass bound for the Higgs particle in the
standard model, non-perturbative analysis of dynamical chiral symmetry
breaking of strongly coupled
fermions, etc.\cite{nager,chl,chl-massb,scgt96} ~
Therefore we would like to stress here that the operator expansion
scheme is desired if it gives converging value effectively enough.
Actually, Scheme II is found to satisfy such a practical demand for
scalar theories. It will be necessary to examine the presence of such
a good approximation scheme for the non-perturbative analysis of
various models. Naturally, it would be desirable to seek general
methods to offer us such effective schemes. Such problems in the ERG
approach deserve further study.
We also discussed the physical reason for this rapid convergence in
the comoving frame by considering $O(N)$ symmetric scalar theories in
the large $N$ limit. It was shown that the exponents are derived
exactly by operator expansion in the comoving frame. Not only the
exponents but also each RG flow of the coupling is exactly derived in
every finite order of truncation. Moreover, all of the (ir)relevant
operators at the non-trivial fixed point have been given exactly and
are found to be highly complicated composite operators in terms of the
original scalar fields. We found that the coupling $\rho_0$, the
potential minimum, defined in the comoving frame corresponds to the
exact relevant operator. If such structure remains in the finite $N$
cases in an approximate sense, it could be regarded as the physical
reason for the good convergence of Scheme II. Indeed, the relevant
operator is found to be described well within the subspace of the
first several operators in Scheme II.
The ERG equation in terms of the composite operators has been
proposed. This is equivalent to the ERG equation for scalar theories
in the large $N$ limit. The reformulation is achieved by introducing a
composite field to the original theory. Interestingly, this operator
turns out to be the exact ``relevant'' operator after evolution to
infrared. Other irrelevant operators are also simply given by the
products of this composite. Namely, the naive polynomial expansion
leads to the perfect coordinates in turn. Thus this offers an example
in which the good expansion scheme in the operators is revealed
through a proper change of field variables. A numerical study of the
ERG in terms of the composite operators in finite $N$ cases will be
reported elsewhere. Indeed, such a variable change incorporating
composite operators has been found to be significant in the RG
analysis of the dynamical chiral symmetry breaking.\cite{scgt96}
\vspace{8mm}
\noindent
{\Large\bf Acknowledgements}
\vspace{4mm}
\noindent
We would like to thank T. R. Morris for valuable discussions at
YKIS '97. Collaboration in the early stages with K. Sakakibara and
Y. Yoshida is also acknowledged. K-I. A. and H. T. are supported in
part by Grants-in-Aid for Scientific Research (\#08240216 and
\#08640361) from the Ministry of Education, Science and Culture.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,479 |
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,750 |
Мария Беатриче Витория Джузепина Савойска (, * 6 декември 1792, Торино, † 15 септември 1840, Кастело дел Катайо, Баталя Терме, Падуа) е принцеса от Савойската династия и чрез женитба от 1814 г. до смъртта си херцогиня на Модена.
Биография
Тя е най-голямото дете на Виктор-Емануил I (1759 – 1824), крал на Сардиния-Пиемонт (1802 – 1821), и съпругата му Мария Тереза Австрийска-Есте (1773 – 1832), дъщеря на Фердинанд Карл Австрийски (1754 – 1806) и Мария Беатриче д'Есте (1750 – 1829), принцеса от Модена. Нейният баща е четвъртият син на императрица Мария Тереза и Франц I Стефан. Тя е сестра на Мария Анна, омъжена през 1831 г. за австрийския кайзер Фердинанд I.
Мария Беатриче се омъжва на 20 юни 1812 г. в Каляри, с разрешение на папа Пий VII, за нейния вуйчо по майчина линия Франц IV фон Австрия-Есте (1779 – 1846), ерцхерцог на Австрия, от 14 юли 1814 г. херцог на Модена и Реджо.
Мария Беатриче е потомка на Стюартите. Затова след смъртта на баща ѝ (1824) Якобитите я смятат за последничка на британския трон като Мари II, кралица на Англия, Шотландия, Ирландия и Франция (10 януари 1824 – 15 септември 1840).
Тя умира на 15 септември 1840 г. от сърце на 47 години. Погребана е в църквата "Чиеза ди Сан Винченцо" в Модена. Нейната позиция като наследничка на Дом Стюарт отива на нейния син Франц.
Деца
Мария Беатриче и Франц IV имат децата:
Мария Терезия (1817 – 1886), ∞ Анри д'Артоа (1820 – 1883), граф де Шамбор, последният мъж от френските Бурбони
Франц V (1819 – 1875), последният херцог на Модена
Фердинанд (1821 – 1849), ерцхерцог на Австрия
Мария Беатрикс (1824 – 1906) ∞ инфант Хуан Карлос дьо Бурбон Испански (1822 – 1887), граф Монтисон
Източници
Elena Bianchini Braglia: Maria Beatrice Vittoria. Rivoluzione e Risorgimento tra Estensi e Savoia. Edizioni TEI, Modena 2004.
Cesare Galvani: Brevi cenni biografici intorno l'altezza reale di Maria Beatrice Vittoria principessa di Savoja arciduchessa d'Austria. Camera, Modena 1850.
Maria Beatrix, Brigitte Hamann: Die Habsburger (1988), S. 311.
Constantin von Wurzbach: Habsburg, Maria Beatrix (kön. Prinzessin von Sardinien). Biographisches Lexikon des Kaiserthums Oesterreich. Band 7, Verlag L. C. Zamarski, Wien 1861, S. 42.
Habsburg 5, genealogy.euweb.cz
Genealogy of the House of Savoy
Външни препратки
Mary III and II (The Jacobite Heritage)
Херцози на Модена
Савойска династия
Дом Австрия-Есте
История на Англия
Жени
Италианска аристокрация
Италианки | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 474 |
\section{Introduction}
In electric power systems, correcting the mismatch between demand and supply is crucial for reliable operation \cite{Kundur}. This is usually achieved through regulation services. Recently developed coordination and control schemes allow distributed energy resources (DERs), such as air conditioners and electric water heaters, to provide regulation services such as frequency regulation using automatic generation control (AGC) \cite{Almassalkhi:2018IMA,Callaway:2011wq,energies,optres}. In such a coordinating scheme, an aggregate of regulating resources tracks an AGC signal, resulting in the frequency being maintained at the required value. Moreover, due to the increasing penetration of renewable sources of energy like solar/wind, there is added variability (due to the fluctuating nature of the sources) and uncertainty (due to lack of accurate forecasts) in the demand \cite{balancing}. When the \textcolor{black}{generation from variable renewables} is under-predicted, more generation is scheduled than necessary, leading to increased costs. When it is over-predicted, less generation is scheduled, requiring more expensive quick-start power generators and/or load shedding. Hence, there is a need for balancing authorities like the Pennsylvania-New Jersey-Maryland interconnection (PJM) ~\cite{pjmmanual} that coordinate these resources effectively to maintain the balance between demand and supply.
By considering the characteristics of the specific DER coordination scheme, controllers can be designed to guarantee high tracking performance of the regulation resources and, thus, ensure optimal and reliable operation. As a case in point, the authors are involved in a flexible load coordination scheme called Packetized Energy Management (PEM) \cite{Almassalkhi:2018IMA,almassalkhi2017packetized}, which is a demand dispatch scheme where loads individually request and can be granted uninterruptible access to the grid for a pre-specified time interval called the packet length. Such a bottom-up coordination method centers on preserving the end-user quality of service (QoS). However, this results in a down ramp-limited response because the loads only consume power from the grid and do not transition from on to off until their pre-specified interval is completed. By taking into account the down ramp-limited nature of PEM, a predictive control scheme can be designed, which can precompensate the AGC signal, using knowledge of its future values, to improve the tracking performance of PEM. \textcolor{black}{The tracking performance of up-ramp limited thermal generators can be likewise improved in this manner.} To maximize the performance of PEM and other such demand dispatch schemes, however, requires modeling and forecasting of the AGC signal.
Controllers for frequency regulating units can be more effectively designed if there is knowledge of the statistical properties of the specific AGC signal being tracked. Of commonly available AGC regulation signals, the Reg-D signal, provided by the regulatory authority, PJM, which is part of the Eastern Interconnection in the United States, is an ``energy-neutral'' regulation AGC signal, typically dispatched every two seconds. Compared to the Reg-A, which is another, slower, PJM regulation signal that is sent to traditional resources and meant to recover larger, longer fluctuations in system conditions, the Reg-D is a fast, dynamic signal that is sent to dynamic resources. Its hourly average tends toward zero (i.e., it is \textit{energy-neutral}), but it requires resources to respond rapidly~\cite{pjmmanual}. Reg-D is normalized between $-1$ and $1$, with $-1$ and $1$ representing minimum and maximum power capacity (MW) bid into the frequency regulation market by an aggregator, respectively. To the best knowledge of the authors, there is currently no work in the literature that provides a detailed statistical analysis of AGC signals, such as PJM Reg-D, intending to derive accurate models and forecasts, which are essential for effective controller design. \textcolor{black}{The paper \cite{agcmod} provides brief analyses on a specific AGC signal from the Bonneville Power Administration (unlike on the commonly available AGC signal PJM Reg-D attempted here), specifically regarding its statistical distribution and change in energy content across hours. However, unlike this paper, \cite{agcmod} does not conduct other important statistical analyses on the AGC signal or provide a statistical model. The paper \cite{agcmod} also provides ARMA forecasting models to predict the \textit{hourly energy content} of the AGC signal (unlike the AGC signal itself that is predicted here), and describes a method to predict the state-of-charge of an energy storage resource based on forecasts of the AGC hourly energy content. However, it does not provide simulations regarding the effectiveness of the forecasting model on the practical application (of predicting the state-of-charge).}
This paper fills the above gap. Specifically, we investigate the statistical properties of a widely known AGC signal, PJM Reg-D, intending to develop two models: a statistical model that fairly accurately captures the second moments of the Reg-D signal and its saturated nature, and a time-series-based forecasting model. A statistical model for the AGC signal enables the model-based design of controllers by providing accurate representations of its variability and/or saturated nature \cite{qlc,chen1998multivariate}, whereas a time series based forecasting model can be used to predict the future AGC signal (either its value or direction) and make decisions on allocating resources effectively based on that prediction, including designing model predictive controllers. For example, when there are steep ramp-ups followed immediately by ramp-downs in the AGC signal, \textcolor{black}{tracking them optimally can maximize performance score, which increases revenue under pay-for-performance schemes.} If it can be predicted that the AGC signal will ramp up and down quickly in the future, then resources can be optimally utilized.
The analyses in this paper are conducted on a year-long historical data of the signal from July 2018 to June 2019 with a 2~s resolution obtained from PJM \cite{dataminer}.
First, the statistical distribution of Reg-D is investigated.
Second, the variability and stationarity of Reg-D are investigated. The mean-variance of AGC across minutes, hours, days, and months are computed, as well as the running mean and variance.
Third, the statistics of saturation of Reg-D (i.e., values $-1$ and $1$), known as ``pegging", are evaluated. It is found that the amount of pegging is directly related to the variance of Reg-D. Fourth, the power spectral density of Reg-D is computed for different months.
Fifth, using the information on mean, variance, stationarity, and bandwidth of Reg-D, a stochastic model is constructed that consists of zero-mean stationary white Gaussian noise passed through a coloring filter of appropriate bandwidth, and the output scaled by the standard deviation of Reg-D and saturated between $-1$ and $1$. It is found that this statistical model fairly accurately captures the second moments of Reg-D (with $<3.5$\% error) and its saturated nature (with $<2.5$\% error).
Sixth, using the autocorrelation and partial autocorrelation functions of Reg-D,
an autoregressive moving average (ARMA) model for predicting Reg-D is designed. It is found that an AR(3) model can provide a directionally salient prediction, i.e., the slope of Reg-D and the ARMA forecast are highly correlated (correlation coefficient $>0.5$) for up to 30 s.
Finally, to further improve forecasts, the cross-correlation of AGC with frequency is evaluated using the additional historical data obtained from phasor measurement units (PMUs) in PJM's territory. Specifically, a vector autoregressive moving average (VARMA) model is constructed using both Reg-D and frequency data. It is found that a VAR(3) model can provide a significant improvement in the forecasts (by about 3.5\% over a horizon of 1 min) compared to the corresponding AR(3) model. To illustrate the effectiveness of the forecasts in a practical application, the VAR(3) forecasts are also employed in a model predictive controller (MPC) described in \cite{brahma2021optimal}. \textcolor{black}{The MPC designed in \cite{brahma2021optimal} uses AR(3) predictions of the power output of a DER-coordination scheme, Packetized Energy Management (PEM), to predict its down-ramp-limited nature and pre-compensate the AGC input to improve its tracking performance, compared to the case with no precompensator.} In this paper, it is shown that the tracking performance of this down ramp-limited demand dispatch scheme with MPC can be improved by around 1\% compared to AR(3) forecasts.
The original contributions of this paper are, thus, as follows:
\begin{itemize}
\item Statistical analysis is systematically conducted on the AGC regulation signal, PJM Reg-D.
\item Using the results of the analysis, a linear stochastic model of AGC is derived that is driven by stationary white noise,
\item Time series-based forecasting models are developed using ARMA and VARMA models that are effective in predicting the future values of the AGC signal.
\item The VARMA-based forecasting model is applied to a model predictive controller for the Packetized Energy Management scheme from \cite{brahma2021optimal} to indicate the effectiveness of the forecasts and the resulting improvement in tracking performance compared to ARMA forecasts.
\end{itemize}
\section{Statistical Modeling of AGC signal}
\begin{figure}[t]
\centering
\subfloat[Full data]{\includegraphics[width=0.75\columnwidth]{Figures/pdfregdfull.png}\label{fig:regdfull}}
\hfil
\subfloat[Excluding values near -1 or 1]{\includegraphics[width=0.75\columnwidth]{Figures/pdfregdzoom.png}\label{fig:regdzoom}}
\caption{Probability Density of Reg-D}
\label{fig:regdprob}
\end{figure}
This section aims to derive a stochastic model of an AGC signal, specifically the PJM Reg-D, driven by a random noise process. To motivate the appropriate form of the statistical model, statistical analyses are first conducted on the Reg-D signal. \textcolor{black}{Specific analyses on the variability of Reg-D across different time scales were conducted in \cite{brahma2021optimal}, the details of which have been omitted here.}
\subsection{Statistical Distribution}\label{ssec:dist}
First, to obtain an idea of the distribution of the values of Reg-D, its histogram was plotted (using the probability density normalization) on the data for the entire year from July 2018 to June 2019. It can be seen from Fig. \ref{fig:regdfull} that the signal is mostly saturated at $1$ or $-1$. However, on zooming near the value of zero (Fig. \ref{fig:regdzoom}), it can be seen that the distribution of Reg-D can be approximated to be a zero-mean truncated Gaussian \cite{gubner2006probability}.
\subsection{Wide-sense Stationarity and Ergodicity}\label{ssec:wss}
\begin{figure}
\centering
\subfloat[Sample Mean]{\includegraphics[width=0.6\columnwidth]{Figures/SampleMean1month.png}\label{fig:sampmean}}
\hfil
\subfloat[Sample Variance]{\includegraphics[width=0.6\columnwidth]{Figures/SampleVar1month.png}\label{fig:sampvar}}
\caption{Sample Mean and Variance of PJM Reg-D}
\label{fig:sampmeanvar}
\end{figure}
\begin{figure}
\centering
\subfloat[Settling Time of Sample Mean]{\includegraphics[width=0.9\columnwidth]{Figures/SampleMeanST12months.png}\label{fig:sampmst}}
\hfil
\subfloat[Settling Time of Sample Variance]{\includegraphics[width=0.9\columnwidth]{Figures/SampleVarST12months.png}\label{fig:sampvst}}
\caption{Settling Time of Sample Mean and Variance of PJM Reg-D}
\label{regdmeanvarst}
\end{figure}
Next, to determine whether a statistical model for Reg-D can be driven by a \textit{stationary} random process and whether the model will be valid across multiple time ranges, the stationarity of PJM Reg-D was investigated. Specifically, the sample mean and variance of Reg-D are plotted in Figs. \ref{fig:sampmean} and \ref{fig:sampvar}. It can be seen that after about 1 hour, the sample mean settles to a value close to 0, while the sample variance settles after about 25 hours to a value close to 0.42. To quantify this systematically, the settling times (i.e., the time when the signal is within 2\% of its final value) of the sample mean and sample variance of Reg-D are evaluated at different months of the year. The results are shown in Figs. \ref{fig:sampmst} and \ref{fig:sampvst}. \textcolor{black}{It was found that the settling time of the sample mean is, on average, less} than two hours, except for April 2019 (during April 2019, it was observed that there were more peg-up events than peg-down events). \textcolor{black}{Since the mean is close to zero, it indicates that Reg-D is \textit{energy-neutral}. This reduces the likelihood that an electric storage resource would have insufficient energy to respond to Reg-D, thereby reducing its potential
compensation and ability to provide regulation in a future interval \cite{pjmmanual}. The above results indicate that Reg-D is fairly wide-sense stationary (WSS) and ergodic in the mean and variance, which provides confidence that a WSS statistical model can be considered for it that would be valid across multiple time ranges.}
\subsection{Pegging Amount}\label{ssec:pegam}
\begin{figure}[t]
\centering
\subfloat[Minutely]{\includegraphics[width=0.62\columnwidth]{Figures/TotalVarPegMinute.png}\label{fig:pegmin}}\\
\subfloat[Hourly]{\includegraphics[width=0.62\columnwidth]{Figures/TotalVarPegHour.png}\label{fig:peghour}}
\hfil
\subfloat[Daily]{\includegraphics[width=0.7\columnwidth]{Figures/TotalVarPegDay.png}\label{fig:pegday}}\\
\subfloat[Monthly]{\includegraphics[width=0.8\columnwidth]{Figures/TotalVarPegMonth.png}\label{fig:pegmonth}}
\caption{Pegging Amount of PJM Reg-D. For details on the mean variance, please see \cite{brahma2021optimal}.}
\label{regdpeg}
\end{figure}
The Reg-D signal is often saturated between $-1$ and $1$, referred to as ``pegging", which can be due to unexpected and sudden changes in generation or load, unexpected large interchange swings, generation lagging or not following economic dispatch, frequency excursion outside PJM or load forecast error \cite{martini}. The actual AGC signal is usually a scaled and biased version of the Reg-D signal. In this and the following subsections, an analysis of the amount and duration of pegging in the AGC signal are analyzed. This is important since any controller that is specifically designed for non-saturated signals will not perform well when the signal is saturated. In such a case, it will have to be adapted to handle saturation.
The amount of pegging in the AGC signal is evaluated by finding the percentage of the samples that are saturated over the total number of samples considered in the particular group, i.e., the minute, hour, day, or month. The results are shown in Fig. \ref{regdpeg}. \textcolor{black}{It can be seen that the amount of pegging in the AGC signal is directly related to the variability of Reg-D.}
\subsection{Pegging Duration}\label{ssec:pegdur}
\begin{figure}[t]
\centering
\subfloat[Hourly]{\includegraphics[width=0.7\columnwidth]{Figures/PegDurHour.png}\label{fig:pdurhour}}
\hfil
\subfloat[Daily]{\includegraphics[width=0.7\columnwidth]{Figures/PegDurDay.png}\label{fig:pdurday}}\\
\subfloat[Monthly]{\includegraphics[width=0.8\columnwidth]{Figures/PegDurMonth.png}\label{fig:pdurmonth}}
\caption{Pegging Duration of PJM Reg-D}
\label{pegdur}
\end{figure}
Usually, the majority of the pegging events are isolated and of short duration, which may not detrimentally affect the performance of a controller, as compared to long-duration pegging events. To obtain a sense of when and for how long the pegging takes place, apart from the amount of pegging, which was considered above, the continuous pegging duration of the signal was evaluated at different time scales. The results are shown in Fig. \ref{pegdur}. From all the subfigures, it can be seen that the maximum pegging duration can be up to 40 minutes, while the average pegging duration is generally around one minute. Moreover, there is no variability among the different hours, days, and months in the average and the 95th percentile of the pegging duration. However, the maximum pegging duration is seen to be the highest during the start of the week, on Monday, and lowest at the end of the week, on Friday, while the maximum monthly pegging duration is the highest around September and the lowest around February. There is no general trend observed in the maximum hourly pegging duration.
\subsection{Power Spectral Density}\label{ssec:psd}
\begin{figure}[t]
\centering
\subfloat[Spectrum]{\includegraphics[width=0.7\columnwidth]{Figures/Spectrum1month.png}\label{fig:spec}}
\hfil
\subfloat[Bandwidth]{\includegraphics[width=1.05\columnwidth]{Figures/Bandwidth.png}\label{fig:bandwidth}}
\caption{PJM Reg-D Statistics}
\label{forecast}
\end{figure}
Next, the power spectral density (PSD) of Reg-D is analyzed, which provides information about the bandwidth and structure of the filter to be used in the statistical model. The PSD of Reg-D is shown in Fig. \ref{fig:spec}, obtained using the Welch estimate, involving Hanning windows with 50\% overlap \cite{welch}. It can be seen that the signal has a lowpass nature, possibly a slight bandpass nature. The peak of the PSD is at around 0.7 mHz. The 3-dB bandwidth, i.e., the frequency of Reg-D at 3 dB less than the peak PSD of Reg-D, is around 12 mHz. Fig. \ref{fig:bandwidth} shows the peak and bandwidth frequencies across different months of the year 2018-19. It can be seen that both the peak and bandwidth frequencies are relatively constant across the year, which provides evidence that a filter model representing Reg-D can be used across a wide range of periods.
\subsection{Statistical Modeling of AGC}
\begin{figure}
\centering
\begin{tikzpicture}[>=latex,scale=0.85,every node/.style={transform shape}, every text node part/.style={align=center}]
\node (wr) at (0,0){$w_r(t)$};
\node[rectangle,draw] (rf) [right=0.5 of wr,label={above:$\lVert {F_{\Omega}\left(s\right)\rVert}_2=1$}]{$F_{\Omega}\left(s\right)$};
\node[rectangle,draw] (sr) [right=0.5 of rf]{$\sigma_r$};
\node[rectangle,draw] (sl) [right=0.5 of sr]{
\begin{tikzpicture}
\draw (0,0)--(0.3,0)--(0.7,0.8)--(1,0.8);
\end{tikzpicture}
};
\draw [->] (wr)--(rf);
\draw [->] (rf)--(sr);
\draw [->] (sr)--(sl);
\draw [->] (sl)--($(sl.east)+(0.4,0)$);
\node (wr) at ($(sl.east)+(0.8,0)$){$r(t)$};
\end{tikzpicture}
\caption{Statistical Model of PJM Reg-D}
\label{fig:agcmod}
\end{figure}
\begin{figure}[t]
\centering
\subfloat[Time Series]{\includegraphics[width=0.6\columnwidth]{Figures/regdmod.png}\label{fig:tmregd}}\hfil
\subfloat[\% Error in Standard Deviation]{\includegraphics[width=0.6\columnwidth]{Figures/stderr.png}\label{fig:stderr}}\\
\subfloat[Absolute Error in Mean]{\includegraphics[width=0.6\columnwidth]{Figures/mnerr.png}\label{fig:mnerr}}\\
\subfloat[\% Error in Pegging]{\includegraphics[width=0.6\columnwidth]{Figures/pegerr.png}\label{fig:pegerr}}
\caption{Effectiveness of Statistical Model}
\label{modagc}
\end{figure}
\subsubsection{Model} Using the above information, a statistical model of AGC can be derived. The proposed structure of the model is shown in Fig. \ref{fig:agcmod}. It consists of a standard stationary Gaussian white noise process $w_r(t)$ passed through a coloring low pass filter, for example a Butterworth filter, $F_{\Omega}\left(s\right)$ with unit $\mathcal{H}_2$-norm, the output scaled by a constant, $\sigma_r$, and finally the result is saturated between -1 and 1. The rationale for choosing the above elements is explained as follows. Since Reg-D is found to be fairly wide-sense stationary (Section \ref{ssec:wss}), and the values of Reg-D are found to fairly obey a Gaussian distribution for values close to zero (Section \ref{ssec:dist}), the model is driven by a standard Gaussian stationary white noise. The low pass filter is chosen since the PSD of Reg-D (Section \ref{ssec:psd}) informs that it has a low pass nature (which is close in shape to a Butterworth filter in the manner of its roll-off at high frequencies). The bandwidth of the filter can be obtained from the plot of the PSD of Reg-D. The $\mathcal{H}_2$-norm ensures that the variability of the output of the filter is unity, and that it can be scaled to a desired value of variance through the gain, $\sigma_r$, that would be representative of the AGC signal. Finally, the saturation models the pegged nature of Reg-D (Section \ref{ssec:pegam}).
\subsubsection{Validation}
To validate the statistical model of Reg-D, a 3rd order Butterworth filter, with the transfer function,
\begin{equation*}
F_{\Omega}\left(s\right)=\frac{\sqrt{\frac{3}{\omega_{n}}}}{\left(\frac{s}{\omega_{n}}\right)^{3}+2\left(\frac{s}{\omega_{n}}\right)^{2}+2\left(\frac{s}{\omega_{n}}\right)+1}
\end{equation*}
was chosen to filter the standard stationary white Gaussian noise, where $\Omega_n$ is the bandwidth, taken to be around 5 mHz. A third-order filter is chosen as opposed to, \textcolor{black}{for example}, a first-order filter, since a higher-order filter \textcolor{black}{was found to }result in a smoother signal \textcolor{black}{that is} similar to Reg-D, \textcolor{black}{resulting in a higher fidelity of the statistical model}. Moreover, the filter bandwidth of 5 mHz, which is close to the 3-dB bandwidth obtained from the power spectral density plot, was found to provide a similar rate of fluctuations as in Reg-D (Fig. \ref{fig:tmregd}). The constant, $\sigma_r$, was taken to be 1.25 times the standard deviation of Reg-D. The factor 1.25 was chosen since the AGC signal is saturated and saturation accounts for about 25\% of the signal, as can be seen from Fig. \ref{regdpeg}. Ten 100,000 sample snippets of Reg-D were considered for validation (about 55.5 h each, by which time Reg-D attains fairly wide-sense stationarity, as seen from Fig. \ref{fig:sampmean}-\ref{fig:sampvar}), and the error between the model and the AGC signals in the standard deviation and mean are reported in Figs. \ref{fig:stderr} and \ref{fig:mnerr} for all the validation sets. It can be seen that the error in standard deviation is less than 3.5\% and that in mean is also very small - less than 0.05 in absolute value. Moreover, the error in pegging amount is also small, less than 2.5\%. This indicates that the derived statistical model is accurate both in determining the second moments of the AGC signal, Reg-D, and also its pegged nature.
\section{Forecasting of AGC Signal}
In this section, a forecasting model for an AGC signal is developed. First, an autoregressive moving average (ARMA) model is developed and its effectiveness evaluated. \textcolor{black}{While in \cite{brahma2021optimal}, an ARMA model was briefly described, its effectiveness across different lead times and robustness to coefficients were not explored. A contribution of this section is to provide those analyses and also to use them as a baseline to show improvements with multivariate vector autoregressive moving average (VARMA) forecasts.} Second, using historical frequency data, the cross-correlation between historical AGC and the power grid frequency is evaluated, and a VARMA model is developed. It is shown that the VARMA model can improve prediction performance compared to an ARMA model.
\subsection{ARMA Modeling}
\subsubsection{Model}
To forecast an AGC signal, the following ARMA model can be used \cite{makridakis1997arma}:
\begin{align}
r[k]={}&\mu+\phi_{1}r[k-1]+\ldots+\phi_{p}r\left[k-g\right]\nonumber\\
&+a\left[k\right]-\theta_{1}a\left[k-1\right]-\ldots-\theta_{q}a\left[k-h\right]
\end{align}
where $\phi_{i}$ are the autoregressive components, $\theta_i$ the moving average components and $\mu$ is the main level of the process, and $a[k]$ is a stationary zero-mean random Gaussian innovation. $(g,h)$ determines the order of the ARMA model. Since the Reg-D signal is saturated between -1 and 1, the output $r[k]$ of the ARMA model is also saturated between -1 and 1.
The process of determining the order $(g,h)$ of the ARMA model using its autocorrelation and partial correlation functions has been described in Section IIIC of \cite{brahma2021optimal} and is omitted here.
\subsubsection{Forecast Accuracy}\label{ssec:forarmaacc}
The effectiveness of an AR(3) forecast is shown in Fig. \ref{arm:forcast} for a snapshot of the Reg-D signal in February 2019. Fig. \ref{fig:nosat} shows the forecast if there was no saturation in the ARMA model output. It can be seen that the forecast is outside the acceptable range of -1 and 1, and is hence not valid and leads to a high forecast error. However, if the ARMA model is saturated, the forecast during saturation (Fig. \ref{fig:sat}) is exact compared to the Reg-D signal. Hence, using a saturated ARMA model increases accuracy during pegging by a large amount. From both the figures, however, it can be seen that the forecast is within 95\% confidence of the mean prediction, indicating that it is fairly accurate.
\begin{figure}[t]
\centering
\subfloat[Unsaturated ARMA model]{\includegraphics[width=0.6\columnwidth]{Figures/forecastpegnosat.png}\label{fig:nosat}}
\hfil
\subfloat[Saturated ARMA model]{\includegraphics[width=0.6\columnwidth]{Figures/forecastpegsat.png}\label{fig:sat}}
\caption{ARMA forecast}
\label{arm:forcast}
\end{figure}
The effectiveness of the ARMA model depends highly on the lead time of the forecast. To quantify this, the ARMA model was subjected to varying lead times, and the accuracy of the forecasts was evaluated for twelve one-hour Reg-D signals from all the months of the year, \textcolor{black}{sampled at 2-s intervals. That is, the ARMA model was tested on $12\times 60\times 30=21600$ samples of the Reg-D signal. } The mean results are shown in Fig. \ref{fig:armaforacc}, where TE stands for the total mean absolute error (MAE) between the output of the ARMA model and that of the Reg-D signal, SE stands \textcolor{black}{for the MAE considering only the samples when Reg-D is saturated at -1 or 1, USE stands for MAE considering only the samples when Reg-D is not saturated. SLE stands for slope error, which is the MAE between the slopes of the Reg-D signal and that of the output of the ARMA model, with the slope computed over a specified lead time.} It can be seen that all the errors increase as the lead time is increased. However, the error when the signal is saturated (SE), while lower than other errors, increases sharply with the increase in lead time, while the error between the slopes (SLE) increases at a slower rate when the lead time increases. Hence, when only the direction is required from the AGC signal, the ARMA model can be effective even for a relatively large lead time. From these results, it is found that in any case, the error in predicting the value of Reg-D is less than 15\%, for up to 30 s. \textcolor{black}{Moreover, the ARMA model was tested against common neural network (NN) architectures: dense feedforward NN, recurrent NN (RNN), long short term memory (LSTM), and gated recurrent unit (GRU), each with two layers of 32 units each, and the results were found to be within 1\% of their predictions (Fig. \ref{fig:nn}), indicating that it is a useful and systematic approach to forecasting compared to black box-based data-driven NN models.}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\columnwidth]{Figures/maeplot2layers.png}
\caption{Comparision of AR(3) model to common NN models}
\label{fig:nn}
\end{figure}
\begin{figure}[t]
\centering
\subfloat[ARMA Forecast accuracy]{\includegraphics[width=0.7\columnwidth]{Figures/ErrorsVsLag.png}\label{fig:armaforacc}}
\hfil
\subfloat[Correlation of Slopes of AGC]{\includegraphics[width=0.7\columnwidth]{Figures/CorrVsLag.png} \label{fig:armacorr}}
\caption{Effectiveness of ARMA forecast as a function of lead time}
\end{figure}
\subsubsection{Correlation of slopes vs. lead time}
Occasionally, it is necessary to only predict the slope of the AGC signal and not its exact value. Such a case may arise when we want to make a decision based on only the future \textit{direction} of the AGC signal. For example, if the AGC signal is predicted to be lower in value in the future, we can use that information to pre-compensate the AGC signal such that the distributed resources are properly utilized. Hence, a study was conducted to find how much the slope of the ARMA model and that of the original AGC signal are correlated.
The effect of lead times on the correlation between the slopes of the output of the ARMA model and Reg-D is shown in Fig. \ref{fig:armacorr} for the same one-hour AGC signals considered above. It can be seen that the mean correlation decreases mostly linearly as the lead time increases to about 30-time steps or 1 min. Hence, the AR(3) model can predict the slope of Reg-D fairly accurately within less than half a minute.
\subsubsection{Detection of Slopes}
\textcolor{black}{The correlation between slopes of AGC as described above can be used to \textit{classify} the AGC signal into three classes: ``Up", ``Down", and ``Flat", based on whether it is moving up, going down, or remaining flat respectively.} To define what is meant by \textit{flat}, a certain threshold of slope may be accepted, so that if the slope is outside or larger than that threshold, the AGC signal will be either determined to move up or go down in the future. However, if the slope of the AGC signal remains within that threshold band, it will be deemed to remain flat. Of course, the accuracy of the classification would depend on the value of the threshold. Hence, a study was conducted that describes the effect of the threshold on the classification by using confusion matrices \cite{forbes1995classification}. The threshold is defined as a certain percentage of the range of slopes of the AGC signals considered.
\begin{figure}[t]
\centering
\subfloat[10\% of Slope Range]{\includegraphics[width=0.6\columnwidth]{Figures/ConfusionMatrix10.png}\label{fig:confmat10}}
\hfil
\subfloat[20\% of Slope Range]{\includegraphics[width=0.6\columnwidth]{Figures/ConfusionMatrix20.png}\label{fig:confmat20}}
\caption{Confusion Matrix for Detection of Slope}
\label{confmat}
\end{figure}
The results of one such classification on Reg-D signal are shown in Fig \ref{confmat}. It can be seen that when the threshold is low (10\% of range of slope, as in Fig. \ref{fig:confmat10}), then more cases are wrongly identified as flat, up, or down, but when the threshold is high (20\% of range of slope, as in Fig. \ref{fig:confmat20}), there are fewer errors in classification, although there are more cases which are considered flat. The choice of the proper threshold will be determined by the particular application and the tolerance allowed for the regulating resources.
\subsubsection{Sensitivity of the ARMA model to training set}
To determine if the coefficients of the AR(3) model are sensitive to the training set used for training that ARMA model, different training sets were considered, and the ARMA model was re-fitted on those training sets. Specifically, twelve training sets of Reg-D were considered from July 2018 to June 2019, one for each month of the year. The results are shown in Fig. \ref{fig:armarob}. It can be seen that the three autoregressive coefficients of the AR(3) are relatively flat with respect to the training sets. \textcolor{black}{This provides confidence that the ARMA model is fairly insensitive to the training sets, and thus, to the specific portion of the AGC used for training.}
\subsection{VARMA Modeling}
The ARMA model developed in the previous subsection can be further improved if we have other time-series information. Typically, the power system grid frequency data is available along with the AGC signal. Since the AGC signal is generated to control the frequency, there is a high correlation between the frequency and the AGC signal. This correlation can be utilized to form a \textit{multivariate} time series forecasting model to improve the forecasts that would be otherwise generated by an ARMA model \cite{reinsel2003elements}.
\subsubsection{Cross Correlation of Reg-D with measured grid frequency}
Grid frequency data were available for one day, 20th June 2019, at a sample time of 100 ms from The University of Tennessee Knoxville. The data was collected on the same grid as Reg-D was used. Since the frequency data were sampled every 100 milliseconds while the AGC signal is sampled every 2 seconds, to enable the highest utilization of information, the frequency data needs to be filtered appropriately.
A first-order lag filter was chosen for filtering the frequency data. To select the time constant optimally, the cross-correlation between frequency and Reg-D was evaluated, after the frequency was filtered with a first-order transfer function, $1/(\tau s+1)$, with the specified time constant, $\tau$. The result is shown in Fig. \ref{fig:tauagc}. It can be seen that a time constant of 10 min leads to the highest magnitude of cross-correlation between the (filtered) frequency and the AGC signal. Hence, this filtered frequency is used to fit the VARMA model described below. \textcolor{black}{Note that the cross-correlation is negative as expected since when the frequency is \textit{low}, there is not enough generation, thus requiring a \textit{high} value for AGC to balance demand and supply.}
\begin{figure}[t]
\centering
\subfloat[Sensitivity of ARMA coefficients]{\includegraphics[width=0.8\columnwidth]{Figures/RobustnessFit.png}\label{fig:armarob}}
\hfil
\subfloat[Filtering Frequency]{\includegraphics[width=0.7\columnwidth]{Figures/tauagc.png}\label{fig:tauagc}}
\caption{Robustness of ARMA model and Filtering Frequency for VARMA model}
\label{agcfreq}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1.05\columnwidth]{Figures/varmaerrlag.png}
\caption{Effectiveness of VAR(3) model}
\label{fig:var3}
\end{figure}
\subsubsection{Model}
To be consistent with the ARMA model considered earlier, the VARMA model considered here is a VAR($g$) model \cite{lutkepohl2006forecasting}, given by:
\begin{equation*}
\mathbf{r}[k] =\boldsymbol{\mu}+\Phi_{1}\mathbf{r}[k-1]+\Phi_{2}\mathbf{r}[k-2]
+...+\Phi_{g}\mathbf{r}[k-g]+\mathbf{a}\left[k\right]
\end{equation*}
where, $g$ is the number of lags, e.g., 3 as considered for the ARMA model, $\mathbf{r}[k]$ is the multivariate time series (Reg-D and frequency in this case), $\Phi_{i}$ are autoregressive coefficient matrices, and $\mathbf{a}[k]$ is the multivariate Gaussian innovation with covariance matrix $\Sigma$ and $\boldsymbol{\mu}$ is the mean level vector of the process. The coefficients can be obtained using maximum likelihood estimation, for example, using \texttt{estimate} command of MATLAB after creating a VARMA model using \texttt{varm}.
\subsubsection{Effectiveness of VARMA forecast}
To investigate the effectiveness of the VAR(3) model, experiments were conducted on a four-hour Reg-D signal and a corresponding length of the frequency signal on 20th June 2019, to find the total error (TE), saturated error (SE), and unsaturated error (USE), as defined in Section \ref{ssec:forarmaacc}, as a function of the lead times. It can be seen from Fig. \ref{fig:var3} that VAR(3) provides 3.5\% less TE, 5\% less SE, and about 2\% less USE than the corresponding AR(3) model. This shows that VAR(3) is more accurate in predicting the AGC signal compared to the corresponding AR(3) model.
\subsubsection{Application of VARMA model to Model Predictive Control Framework}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{Figures/PEMschematic.png}
\caption{\textcolor{black}{Closed-loop feedback system for PEM with the reference power $P_{\rm{ref}}$ provided by the grid or market operator and the aggregate net-load $P_{\rm{dem}}$ measured by the coordinator.}}
\label{fig:pem_schem}
\end{figure}
\begin{figure}[!t]
\centering
\begin{tikzpicture}[>=latex,scale=1.1, transform shape]
\node[draw] (mpc) at (0,0){MPC};
\node[draw] (pem) [right= of mpc]{PEM};
\node[draw] (pred) [left= of mpc]{Predictor};
\draw[->,thick] (pred)--node[above]{$R[k]$}(mpc);
\draw[->,thick] (mpc)--node[above]{$u[k]$}(pem);
\draw[->,thick] ($(pred)-(1.5,0)$)--node[above]{$r[k]$}(pred);
\draw[->,thick] (pem)--node[above]{$y[k]$}++(1.5,0)--++(0,-1)-|(mpc);
\draw[->,thick] ($(pem)+(1.5,0)$)--++(0.5,0);
\end{tikzpicture}
\caption{\textcolor{black}{MPC-based Precompensator}}
\label{fig:mpc}
\end{figure}
\textcolor{black}{The VARMA forecasting model can be applied in a predictive control setting, where it can provide valuable forecasts of the future AGC signal to generate optimal control decisions for the regulating grid resources, especially when their outputs are down/up-ramp limited. For example, in \cite{brahma2021optimal}}, a model predictive controller (MPC) was designed by the authors to ensure optimal tracking performance of the Packetized Energy Management (PEM) scheme, and its performance was tested using ARMA forecasts. \textcolor{black}{PEM (Fig. \ref{fig:pem_schem}) is a demand dispatch scheme that can be used to provide ancillary services such as frequency regulation. In PEM, DERs stochastically request access for power from a DER coordinator, which then grants or rejects them uninterruptible access to the grid for a specified period called a packet length. Details on PEM can be found in \cite{Almassalkhi:2018IMA}. A characteristic of PEM is that once packet requests are accepted by the coordinator, it locks devices ON for the duration of their packet length. This causes the aggregate response of DERs to become down ramp-limited, and consequently low tracking performance while tracking down ramps in the AGC signal. The MPC design (Fig. \ref{fig:mpc}) in \cite{brahma2021optimal} overcomes this issue and improves the tracking performance of PEM while ensuring less device switching. The objective of this subsection is to investigate whether the tracking performance of PEM with MPC can be improved using VARMA forecasts compared to ARMA forecasts presented in \cite{brahma2021optimal} (which the reader is encouraged to refer to for background and context).}
To investigate the effect of improved forecasts from the VARMA model on the tracking performance of PEM, simulations were conducted on PEM, equipped with the MPC but this time with VAR(3) forecasts using \textit{both} Reg-D and frequency data. Four representative 1-h datasets of Reg-D and frequency on 20th June 2019 were chosen (specifically, 6-7 AM, 12-1 PM, 6-7 PM, and 12-1 AM EST) for the simulations, and the MPC horizon was varied. The average relative mean absolute tracking errors (RMAE) are shown in Fig. \ref{fig:varm}. It can be seen that RMAE with VAR(3) forecast is smaller by about 0.7\%. With a horizon of 10 min, it can be seen that while the ARMA forecast performs worse than with no MPC (horizon tending to 0), the VAR(3) forecast improves and also results in RMAE lesser by 0.8\% than the corresponding AR(3) forecast. This indicates that utilizing additional information from the frequency data results in improved forecasts even at a high prediction horizon, which then leads to improved tracking with the MPC.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\columnwidth]{Figures/varmf.png}
\caption{Effectiveness of MPC with VARMA forecast}
\label{fig:varm}
\end{figure}
\section{Conclusion}
In this paper, a statistical model and a time series-based forecasting model are developed for the modeling and forecasting of AGC signals \textcolor{black}{to provide a useful starting point for designing model-based controllers for frequency regulating units.} By conducting a statistical analysis on a widely used AGC regulation signal, PJM Reg-D, including its variability, power spectrum, and saturation, a stochastic model, driven by stationary white noise, is derived that is shown to fairly accurately model the Reg-D signal and capture its second moments and saturated nature. By conducting studies on the autocorrelation and partial autocorrelation functions, an ARMA model is derived that fairly accurately forecasts the Reg-D signal within less than half a minute, both directionally and in predicting its value. Further, by incorporating information from the power grid frequency data, it can be seen that a VARMA model can further improve the forecasts obtained using AGC data alone. The VARMA forecasts have been used in an MPC framework to improve the tracking performance of a DER coordination scheme compared to ARMA forecasts.
\textcolor{black}{Future work includes investigation of better forecasts and models, including machine learning-based approaches and architectures, studies on different lead times and orders of ARMA and VARMA models, correlation with power load data, testing different probability distributions to model AGC signals from other ISOs, and extending analyses to incorporating more frequency data and locations.}
\section*{Acknowledgment}
\textcolor{black}{The authors would like to acknowledge the support of the U.S. Department of Energy through its Advanced Research Projects Agency-Energy (ARPA-E) award: DE-AR0000694. The authors would like to thank Dr. Weikang Wang and Prof. Yilu Liu at the University of Tennessee Knoxville (FNET) for providing us with PMU data, as well as Danielle Croop and Anthony Giacomi at PJM for helpful discussions on PJM Performance Scoring.}
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,077 |
Heavy fighting erupts during CAR referendum voting
By Al Jazeera
At least two people have been killed and 20 others wounded when heavy fighting broke out in a Muslim enclave of Central African Republic's capital Bangui as people voted in a constitutional referendum.
Medical charity Medecins Sans Frontieres said on Sunday that it treated 12 people for gunshot wounds.
Gunfire and the explosion of rocket-propelled grenades were heard in the PK5 neighbourhood, witnesses told the Reuters news agency.
Residents of the isolated enclave, which was visited by Pope Francis during a trip to Bangui last month, marched to the headquarters of the country's UN peacekeeping mission, MINUSCA, to complain they were unable to cast their ballots.
The mission responded by sending soldiers to protect voters and poll workers in PK5.
The new constitution – seen as crucial to restoring stability in the restive country – is set to replace a transitional charter that currently governs the former French colony.
About one-quarter of CAR's 4.7 million population has been displaced since March 2013, when Muslim Seleka rebels overthrew Christian president Francois Bozize.
Fighting has been ongoing between armed Christian groups and Muslim rebels since then despite the deployment of the 11,000-troop strong MINUSCA last year.
Voting got off to a late start in many neighbourhoods in the sprawling, riverside capital, but queues formed in front of polling stations across the city.
"We received all the necessary materials for voting yesterday, and it was all monitored by MINUSCA agents," said a member of the National Elections Authority (ANE), Fabien Kodou.
"This morning, we started work very early. We were missing certain things, including the electoral list. It's quite disorganized and there's nothing we could do about it. But we're trying our best and at least we had a good start."
Following months of relative calm in Bangui, clashes in late September sparked a fresh wave of inter-religious fighting that has killed more than 100 people in and around PK5, according to rights campaigner Human Rights Watch.
If approved, the new charter will bar members of current interim government from standing in December 27 legislative and presidential elections.
It will also limit the presidency to two terms and has provision for a senate – which will be the main legislative body.
The landlocked nation is currently led by interim President Catherine Samba-Panza, a Christian, and Prime Minister Mahamat Kamoun, a Muslim.
Source:: Al Jazeera
The post Heavy fighting erupts during CAR referendum voting appeared first on African Media Agency.
Source:: http://amediaagency.com/heavy-fighting-erupts-during-car-referendum-voting/ | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,749 |
For our struggle is not against flesh and blood, but against the rulers, against the powers, against the world forces of this darkness, against the spiritual forces of wickedness in the heavenly places.
"Our struggle is not against flesh and blood," which is to say, against human beings who move about in the same way that we do, but "against the world forces of this darkness, against the spiritual forces of wickedness," that is, against the organized, powerful, closely-knit kingdom of the prince of darkness.
The "we" in whose name the apostle speaks are the brothers in Ephesus, the church to which he directs his word.
He had just admonished them to "be strong in the Lord and in the strength of His might" that they might be able to withstand "the schemes of the devil." In our text, he reminds them of the power that opposes them. He wishes to convince them that they not only need to seek an ally in this struggle, but also need to be outfitted with armor that protects head and breast, keeps their feet from stumbling, and enables them to "extinguish all the flaming arrows of the evil one."
It is self-evident that all of this is applicable to the contemporary church.
Why did we hesitate to lay this word at the foundation of our meditation?
Not because the church no longer needs the admonition.
Not because anyone among us would be inclined to cast into doubt what the apostle here expresses.
But because he sees something that lies outside our field of vision, and with an eye to what he saw, recommends something for which we feel no immediate need.
After all, it does not say, "the prince of darkness struggles against you"; that may be true even if we do not know it, even if we do not believe it. What it does say is, "our struggle is… against the world forces of this darkness, against the spiritual forces of wickedness," and this is something that does not take place outside of our consciousness.
For if someone were to tell you that you struggle against something, in such and such a manner, you should be the first to know that.
But we have here to do with a power that keeps itself hidden, a danger to which our attention needs to be directed.
You see, we lived and live in days of struggle. Struggle with all sorts of origins, all manner of forms, struggle in ecclesiastical, societal, and political terrain, struggle for existence, for the majority, for power and influence. But in our eyes it was and always has been a struggle against flesh and blood, against notions, orientations, temporal powers and trends, in a word, against everything that proceeds from man and is limited to man.
Many Christians seem to have noticed nothing of a power of darkness, even less of a kingdom of darkness.
When the apostle Paul arrived in Ephesus for the first time, and found a few disciples there, he asked them, "Did you receive the Holy Spirit when you believed?", to which they responded, "No, we have not even heard whether there is a Holy Spirit" (Acts 19:2).
Should we take this to mean that, up until then, the Holy Spirit had acted outside of them, that they were not brought to faith through His power? They certainly later learned to understand that this was not the case. It only means that to that point He had operated outside of their conscious experience.
The same thing is true here of the work of the devil.
Many of whom we consider to be believers would answer the question, "do you know that we struggle against rulers of this age, powers of darkness, spiritual forces of evil in the heavenly places?", may not have responded in precisely the same way as the Ephesians with regard to the Holy Spirit, for they have all heard that "the devil prowls around like a roaring lion, seeking someone to devour" (1 Peter 5:8); but nevertheless, they still have to recognize what amounts to the same thing: "in our own lives, and in the midst of our struggle, we have noticed nothing of the working of those spiritual forces of evil."
There are all sorts of reasons for this, which I do not need to go into now. But the most obvious is the shallowness of our spiritual life.
We are like children playing in the surf or running around on the sand, without any idea of what lies hidden in the depths of the sea. Not even when the storm hits and the waters are stirred up; not even when the waves break against the shore.
For this reason, it is so necessary to make clear to ourselves what the expression "the kingdom of darkness" signifies, and what the admonition of the apostle wishes to impart with respect to that kingdom.
How can we properly struggle against a power the existence of which we do not suspect, and against which we are powerless with our ordinary weapons?
We usually speak of Satan as if this was the inclusive name of all hellish power. But according to Scripture, Satan is the prince of the kingdom of darkness. Often the head of state is named when we wish to speak of the entire people. Satan, as far as Scripture makes known to us, appeared only twice in this world. The first time was in Paradise before the first Adam, the second time in the wilderness before the second Adam. But the apostle describes for us the kingdom of darkness in the words of my text: "the rulers of this world, the darkness of this age," the entire hierarchy of the Evil One. He speaks here of an ascent of power and indicates the contiguous organization of that kingdom, of which the Evil One is the head.
Our Catechism says that a child of God has three main enemies, the devil, the world, and the flesh.
We know this.
And we understand, even if not entirely from our own experience, what it is from which God preserves us! What are the temptations and combats of the devil; what are the fiery darts which must strike the soul if the shield of faith did not ward them off. But now that we, in response to our text, speak of the kingdom of darkness, we have in mind no specific experience of spiritual life indicated by the word "temptation," but that which in the history of the world is seen of the action of the powers of destruction, even where we had not discovered it before.
I will show you:
1. the two powers which rule history, namely, the power of God and the power of Satan;
2. the resources and aids available to the prince of darkness;
3. what of this struggle can be seen in our days; but most importantly
4. what, with this in mind, we have to hope and to fear, to do and to learn.
I speak thus over this struggle, which the apostle observes, the struggle not against flesh and blood but against rulers, powers, the world forces of this world, the darkness of this age, against the spiritual forces of evil in the heavenly places.
This world is a stage displaying the tremendous struggle between two powers, God and Satan, a struggle which survives the ages and which will not end as long as the kingdoms of this world have not been subjected to the Christ, who must reign from sea to sea and from the rivers to the ends of the earth.
The start of this struggle lies outside our field of vision. The conflict between God and Satan was already fought and settled before man appeared on earth. The prince of darkness has already "been cast out of heaven." But here below he has managed to get a foothold. It is the continuation of the struggle to which we have to draw attention.
It is foretold in the promise in Paradise: "I will put enmity between you and the woman, and between your seed and her seed; he shall bruise you on the head, and you shall bruise him on the heel" [Genesis 3:15]. We owe that battle to God's grace, which prevented us from descending into the depths and lamentation of misery into which Satan has brought his followers. That word of promise is a brief summary of the entire history of the world.
If you ask whether that battle was not already settled at Golgotha, when Christ bruised the head of Satan, I answer no, certainly not! There the king of the Kingdom of God, Jesus Christ, bought Himself a people with His own blood, a people who will be willing in the day of God's power (Psalm 110:3).
There He fought the good fight and gained the victory. There, in the words of the hymn we just sang, He "received the Kingdom of God as reward for His struggle."[1] There He in principle accepted the rulership, so that He might soon say: "All power is given to Me in heaven and on earth." He proceeded from there to receive a Kingdom in the place where the word is spoken: "Sit at My right hand Until I make Your enemies a footstool for Your feet" (Psalm 110:1).
Or do Da Costa's beautiful words:
"Our wounds
Are healed
Our sins
Are repealed
And the snake's head crushed"
contain no truth?
Doubtless, if we only pay attention to the principle and overlook what the apostle Paul calls "what is lacking in Christ's afflictions" (Colossians 1:24).
"It is finished!" But the end is not yet!
There is a difference between what we see in faith, and what we perceive in reality.
For faith, time and distance do not exist. It sees the things that are not as though they were. It boasts where unbelief would have every right to complain. In the promise, it receives the fulfillment; in the bud, the blossom and the fruit.
In hope we have been saved, Romans 8:24. For us, faith here still exists in the form of hope, the Christian's hope. But the Word will soon be fulfilled: "The God of peace will soon crush Satan under your feet" (Romans 16:20).
That was spoken to the church, and through the church to you and me.
Not only is there a revelation of God which culminated in the ascension of Christ, but there is also a revelation of Satan. The Lord God manifested Himself from out of the mystery of His being, but there is also a mystery of iniquity already being wrought (2 Thess. 2: 7), and this satanic revelation shall be realized when the apostasy of Christendom allows him to bring about the man of sin, in whom we will be able to worship man, genius, the man who will be girded with all the gifts and powers that art and science impart and who will possess everything that makes one great and mighty, as may be expected of a satanic genius (2 Thess. 2:3). Lying signs and miracles still have to be done in the midst of a society that had been taught that miracles were impossible (2 Thess. 2:9).
But before that, the nations' boundaries have to be erased.
What path will this take? God is busy teaching us.
For, as has become apparent in the battle being fought in our city,[2] there is a struggle which cannot be confined to a city or a country, but extends over the whole world. When all the world will be a republic, and this republic established on new foundations, when violence is recognized as the redeemer of the oppressed, then the moment will have come, the fulness of time, in which he can appear, who takes the reins of government and reigns with a lordship that demands not only tribute but also enslaves souls – as long as God permits!
Holy Scripture again and again shows us the evil one in the background of things, the power that sets itself up against God and His anointed one.
When David counts the people (1 Chron. 21:1), when Job has so much to suffer (Job 2:1ff.), when Israel returns from Babylon and Joshua the high priest is lacking his robe, so that the letter of the law condemns him (Zechariah 3), when Christ is led into the wilderness to be tempted of the devil (Matt. 4), when Judas betrays his Savior (Luke 22:3), when Peter is sifted like wheat (Luke 22:31), when the demon-possessed fight against their Redeemer (Matt. 8:29).
But Scripture goes even farther and speaks of the kingdom of Satan as a kingdom established in opposition to the Kingdom of God.
So, we must in the second place indicate the aids and the resources Satan possesses. It is precisely this aspect of the matter that we have so little awareness of. Our text speaks of "rulers, powers, and forces" and hereby gives us a sign that there are things in heaven and earth of which man, as Hamlet says, has no idea.
There is an invisible world that is not so far away. In that world, a struggle is under way, apart from man, which influences the course of events here below.
We cannot and may not say more about this. Except:
If what happened with the servant of Elisha in Dothan were to happen to us, we would also see the powers that surround us, and then we would see what he saw and – praise God! – we would also see that those who are with us are more than those who are against us.
"The angel of the Lord encamps around those who fear Him, and rescues them" (Psalm 34:7).
To penetrate into this mystery would only serve to satisfy curiosity.
Scripture is very sober in this.
Even so, it is not entirely silent.
In the Revelation of John, a scene on earth always follows after a scene in heaven has been sketched, so that it is not difficult to discover the connection that turns out to exist between the realm there and here.
Something similar is shown to us in the vision of Daniel by the river Tigris (ch. 10). There was opposition to be overcome in the spirit realm before the blessing for which the prophet had begged for his people could descend (ch. 9:3ff).
All this, however, lies outside the field of vision in which the text places us. Here is talk of a battle on this earth fought by the church of God against the principalities and the powers that the kingdom of Satan is attempting to establish here on earth.
The struggle would not be so dangerous, however, if the Evil One did not possess resources and aids in this world.
We color our map of missions black where Christ's church has not been established and the gospel has not penetrated, and in so doing we say that the kingdom of darkness exercises unlimited rule there.
There is certainly a truth in this.
But Scripture takes a different point of view and teaches us a more humbling truth. If Peter wants nothing to do with Jesus' suffering and death, and exclaims, "Lord! That will by no means happen to you!", Jesus responds to him: "Get behind me, Satan! "
Because he is thinking not of the things of God, but of the things of man (Matthew 16:22ff.).
And if the church of Corinth was to conceive of discipline in such a way that it no longer would accept the apostate even if he were to repent, even if the Lord were to grant him forgiveness, then Paul comes and says, know that you are an instrument of Satan! (2 Cor. 2:11).
When Paul speaks of the battle "against principalities and powers" he actually means a battle waged against the persecutors, a battle also against brethren, against Demas who loved the present world, against Peter, who let himself be coaxed by the Jewish-minded in a way that leads to great sadness. The struggle to which the apostle Paul points would not be so frightful and would not pose so much danger if we only faced the Evil One. The kingdom of Satan could then be demarcated with a certain color, and the boundaries of that kingdom would certainly not be crossed. In that case the soldiers of King Jesus could be identified by a band around the cap or a sign on the arm.
Unfortunately, Eve's experience is also ours. She called her first-born Cain and clearly indicated both in the name (weapon, gain) and in the cry, "I have acquired a man from the Lord" (Gen. 4:1), that in him she thought she saw the promised seed of the woman.
But in the name of her second son Abel (vanity, disappointment) she proclaimed no less loudly that she had already discovered the seed of the serpent in the firstborn of her generation.
It is indeed deeply disturbing to discover the influence of the Evil One most often where one least expects it.
In these days we have become acquainted with a power hitherto underestimated, the power of an organization so entirely obedient that the individuals are joined into one body, and that body animated by a spirit, a purpose, a plan, a will. This many-headed unity, however, was brought about by the voluntary accession of members individually, and thus distinguishes itself from that other organization of which the apostle speaks in our text.
No earthly power has such aids as Satan.
Where sin dwells, where error reigns, self-seeking, worldly motives, where even when we have the best intentions we do not submit to God's Word alone in the choice of means, there Satan discovers a point of contact, an aid, an ally, an instrument which he can use when the time comes.
He takes possession of Judas, for he belongs to him; but he sifts Peter as wheat and discovers, in that which in him proceeds apart from grace, an opened door and a weak spot.
While the Savior exercises His prophetic, His redeeming activity in the circle of souls seeking salvation, so that the crowds flock to Him impressed by His Word and work, he has at his disposal none other than the mother of the Lord, who with His brothers, the spirit of whom evidently animates her this time, seeks a means of causing Him to cease His work (Matthew 12:47).
There is a reverent man at Bethel, an old prophet, who lends himself to the unenviable work of defeating the witness of God standing up to the altar of Jeroboam in the center of the apostasy (1 Kings 13).
And if there be no sin and no error enabling him to enslave a man, there will be some one-sidedness in that man that can be put to effective use with any policy. There will be some truth which, when misapplied, performs the same operation as the lie it replaces. The first lie was, in a sense, a truth (Gen. 3:5).
These things are not entirely unknown to us, although the existing division into ecclesiastical and political parties as well as into social groups gives too much reason to forget them.
And because of this, be it said in passing, faction shows itself to be from the evil one. It is a master deception of Satan by which he leads us to call good what comes from one particular side, while neglecting what in it must be reckoned to the account of the prince of darkness. No, Johan Huss had a more correct view when, at the stake, a woman of the commonality came with exemplary zeal with a branch, adding to the stack of wood where she saw a heretic not worthy to behold the light: her enthusiasm caused him to exclaim "Sancta simplicitas!" Holy simplicity.
For the apostle, what we have here indicated as separate facts stand, according to our text, in the service of a plan, something of which one occasionally discerns but which serves a mighty purpose as part of an interlocking wheelwork, i.e., is made subservient to the establishment and expansion of the kingdom of darkness.
As children we shuddered when we were told of the Jesuits who penetrated everywhere, with and without their cassock, who from the living room, the study, the council room, the office, everywhere and in all kinds of ways, would take hold of the strings held in the hand of a man, the general of the order, who could lead the movement that was to paralyze the power of truth in the religious sphere and every other.
And one might well shudder, for what can be seen in the Counter-reformation is only a faint picture of what the Apostle Paul perceived when he spoke of the battle against "the spiritual forces of evil in the heavenly places."
God is these days graphically teaching us the meaning of such an organization.
We see an imperium in imperio being formed.
We see a power in the hands of the masses, but in fact of a few, soon perhaps of a single one, which will gradually spread over the city, the region, the country, the world, a power from which soon no one will be able to withdraw, and which decides the fate of all, a power that will only gradually emerge as the initial outcome of the ideas which found their first initial expression in the upheaval of more than a century ago.
Seeing, we must be blind if we limited our attention to Jesuitism or socialism, let alone the present strike, and did not discover in the background of things the power which holds the threads of the whole fabric in hand and competes for the rulership of the world.
Our struggle is not against concepts and opinions but against a goal to which they are made serviceable.
Finally, let me say to you what it is we have to hope or fear from this, what we have to do or to learn.
Without forgetting that I am speaking here officially, from the pulpit, to the congregation, I must then briefly mention the circumstances of the times, because here they both ask for and provide the necessary explanation.
The struggle between workers and employers, which involves the public and of which many have been and will become victims, does not concern the interests of individuals or groups, but of the entire citizenry.
It is now a question of authority.
Of authority and freedom.
Not today or yesterday but almost a century ago, the power which we call the Revolution tore apart the nation and, along with the nation, the body of the state and of the church.
This took place under the motto, "liberty, equality, fraternity."
This was done under the pretense that faith is a purely private affair.
We reap what we sow!
We are only at the beginning of the things which are coming.
But already now, anyone can know that freedom means coercion, intolerable coercion in the name of the common good, exercised by a few; that equality signifies a revolution, whereby that which lies below comes to the surface; that brotherhood is hard to find when not grounded in self-interest.
And one can also clearly see that authority has shifted. Instead of the authority with which God has vested the government, the authority of the Committees takes its place, as in the French Revolution; a power that is emerging, from the people, from the majority: an authority before which the government will eventually also have to yield, for the simple reason that even the army eventually ceases to support that which may not profess any faith, and without faith can neither rule nor find obedience.
Be that as it may.
In the light of God's Word, we see deeper and further. It is the prince of darkness who is establishing his kingdom, and is to this end that he makes use, under a variety of names, of the error, the one-sidedness, the sloth, the self-interest, the often well-meaning deviations from the only rule by which government and subject should walk.
On every battlefield, in every struggle, there is great activity. There are troop movements here, there, everywhere, baffling the uninitiated. But the general whose eye is on everything knows very well that what matters, after all is said and done, is the key position.
I spoke of a plan that Satan is busy executing, and say with Paul that "we are not ignorant of his schemes" [2 Cor. 2:11], not because it does not take so much discernment to see through that plan, but because God has made known His plan and established His kingdom. The antithesis is made known by the thesis.
And what is the thesis?
In this world where sin dwells and selfishness reigns, God has established two powers to expound the truth, to teach the fear of God, to maintain justice, and to restrain the license of men.
While He has entrusted no earthly power to the church, He has given her a mouth with which to confess and a hand to hold high the torch of the truth.
Satan has used all manner of people and means and still uses them to silence that mouth and to rob this power of influence, as it is this day.
Every error, every deficiency hands him a means to achieve his goal. The outposts he establishes do not have bad intentions and in most cases have no idea of the system they are serving.
With many it is the same as with the 200 men whom Absalom carried away in his rebellion. They went in their simplicity, for they knew nothing of the affair (2 Sam. 15:11).
What he cannot use directly for his purpose, he occupies with the care of the poor, with the ministry of mercy, with some form of education and so on, good things, useful things all, but applied so that the one thing that is needful remains invisible; applied as a surrogate.
To the government, God gave the vocation to rule in accordance with His will, in His name, as His viceroy.
And Satan has made use of all sorts of things in order to get even the Christian people to understand that the government can and may not have any fixed rule by which it is to rule; that it is much better if sovereignty is transferred to various spheres in the life of the nation, so that we no longer have a nation at all, but all manner of spheres with their own resources and aids in the struggle for existence.
We see before our eyes what this results in. Society must be established on new foundations, which these days are clearly being exposed.
God's Word has been or will be fulfilled.
The two witnesses are put to death, for the court is given to the Gentiles. And their dead bodies lie in the streets of the great city, "where also our Lord was crucified" (Rev. 11:2, 7–10).
What do we have to fear, you ask? The movement, which seemed confined to Amsterdam, must sooner or later expand all over the world; it ends up in the world republic. And that Republic is what it already is in the very essence of the matter, the single-headed government girded with the Satanic power, turning, when the nations find not what they seek, against God and His Anointed One, even against all who confess Him.
What do we hope for?
What God promised when He comes to judge.
To do and to learn.
To open our eyes for what history has to tell us, and our eyes for what God has said in His Word.
Furthermore, to organize, not in the sense in which interested, like-minded people have been doing since time immemorial.
Every organization is weak when confronted with the close-knit power of self-interest.
God has organized. He anointed His Son as King over Zion.
He is the king of justice.
The prince of peace.
But His kingdom must be established in heart and home, in society and state.
Established in this manner, that we ever more subject ourselves to Him. "Because He is your King, bow before Him!"
In that case, what is important to you is His honor.
You then ask, in accordance with His Word.
You will then look differently at life and at persons.
David refused to have Shimei killed, because God had said to him: curse David (2 Sam. 16:10).
You will go very easy on another if you know that when you act against him most zealously, perhaps Satan has already laid his hand on you.
But it will also let you know that you and your work are not so isolated as you had suspected.
And if you ask, How shall we be able to stand against this power that both presses on us and resides within us, then I can tell you that Satan needs double permission, first God's, then our own.
When we pray, Lord, keep us from evil, and seek to learn the wiles of the Evil One from God's Word, then we need not fear.
The last word in the history of the world will be: the kingdoms of this earth have become of the kingdoms of our Lord and of His Christ!
Delivered Sunday morning, February 1st, 1903, in the Westerkerk, Amsterdam, by Dr. P. J. Hoedemaker.
Recorded during the sermon and checked by the speaker.
[1] Hymn 166 of the Oude Hervormde Bundel [Old Reformed Bundle].
[2] A reference to the railroad strike of 1903. A good description of this event can be found here; a Marxist interpretation can be found here. The sermon was delivered in the midst of this event, before it had been settled, forcibly, by the government led by Abraham Kuyper. | {
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