text stringlengths 12 4.76M | timestamp stringlengths 26 26 | url stringlengths 32 32 |
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If you are going to need a room in Arts and Sciences to show DVDs during your lectures, try using S302 and S218. They have fixed DVD players in each room. For more information contact me, Cliff Johnson at ext: 2386 or email me at: cjohnson@ltu.edu.
The LCD ceiling mounted projectors at LTU are protected with
both cable locks and sonic alarms. If you hear a projector alarm going
off please call campus security first.In the event of a theft they will apprehend the thief or reset a false
alarm.
You should have received an email message indicating your
classroom LCD projectors were cleaned by eLearning Services, Classroom
Technologies.Please email Classroom Technologies
at elearning@ltu.edu and we will schedule
to inspect and clean your department portable LCD projectors Jan 3rd
through Jan 7th 2011. | 2023-11-03T01:26:58.646260 | https://example.com/article/8071 |
mon multiple of 55 and 40350.
443850
What is the common denominator of -133/984 and -47/11152?
33456
What is the least common multiple of 4095 and 38205?
3476655
Find the common denominator of 79/8256 and 151/78.
107328
What is the least common multiple of 404 and 1818?
3636
Calculate the common denominator of 127/6 and -19/17375.
104250
Calculate the common denominator of 64/225 and -59/27.
675
What is the smallest common multiple of 24696 and 1568?
98784
What is the lowest common multiple of 129918 and 9?
389754
What is the least common multiple of 150744 and 59384?
1959672
Calculate the common denominator of 24/473 and 151/327660.
3604260
What is the lowest common multiple of 270 and 9280?
250560
What is the common denominator of -83/15386 and -89/835240?
5846680
Find the common denominator of 97/4 and 31/228370.
456740
Calculate the smallest common multiple of 17043 and 273.
119301
What is the smallest common multiple of 3 and 171353?
514059
Find the common denominator of -27/26512 and -71/145816.
291632
Find the common denominator of -18/7 and 6/101.
707
Calculate the common denominator of 7/4734 and 61/21303.
42606
What is the common denominator of -1/493800 and 113/29628?
1481400
Find the common denominator of 85/61784 and -145/92676.
185352
What is the common denominator of 91/4455 and 11/567?
31185
Find the common denominator of 43/84885 and -29/6.
169770
What is the least common multiple of 102597 and 330?
1025970
Calculate the least common multiple of 247465 and 3.
742395
Find the common denominator of 27/57920 and 27/9050.
289600
Calculate the common denominator of -91/5976 and 79/9296.
83664
Calculate the common denominator of 4/9313 and 93/92.
856796
What is the common denominator of -99/28724 and 32/50267?
201068
Calculate the smallest common multiple of 2133 and 27018.
81054
Calculate the common denominator of -87/2254 and 1/77.
24794
What is the common denominator of -4/1670415 and 133/1336332?
6681660
Calculate the smallest common multiple of 292950 and 113925.
2050650
Calculate the lowest common multiple of 1558 and 164.
3116
Calculate the common denominator of 74/842247 and -51/1310162.
11791458
What is the least common multiple of 168 and 42?
168
What is the lowest common multiple of 50400 and 525?
50400
Calculate the least common multiple of 4986 and 1662.
4986
Find the common denominator of 83/430 and 11/281349.
2813490
Calculate the common denominator of 131/864 and -91/157968.
947808
Calculate the common denominator of -181/758430 and -89/1590.
758430
Calculate the common denominator of 41/51 and 62/57.
969
What is the least common multiple of 425696 and 76?
8088224
What is the common denominator of -87/3560 and -27/554915?
4439320
What is the common denominator of -133/12330 and -71/10275?
61650
What is the common denominator of -46/39463 and -48/14003?
434093
What is the smallest common multiple of 10 and 70930?
70930
Find the common denominator of 91/1704 and -43/17622.
5004648
Find the common denominator of 29/5353 and -30/18887.
1001011
What is the smallest common multiple of 1698414 and 10?
8492070
What is the common denominator of -31/1863 and 145/828?
7452
Find the common denominator of -53/108 and 29/546.
9828
What is the common denominator of -19/42 and -155/16884?
16884
Calculate the common denominator of -53/4 and 109/163280.
163280
Calculate the lowest common multiple of 352 and 184.
8096
Find the common denominator of 61/4 and 44/426143.
1704572
Calculate the lowest common multiple of 447567 and 12.
1790268
Find the common denominator of 73/8640 and 56/15.
8640
What is the smallest common multiple of 713250 and 450?
713250
Calculate the smallest common multiple of 19260 and 430.
828180
What is the common denominator of 125/21996 and -101/1326?
373932
What is the lowest common multiple of 880 and 55?
880
What is the lowest common multiple of 77334 and 8?
309336
What is the least common multiple of 1092544 and 5?
5462720
Find the common denominator of -99/64984 and -69/80.
649840
Calculate the common denominator of -46/65 and -6/35789.
178945
Find the common denominator of -79/2 and -41/2912974.
2912974
Calculate the common denominator of -119/12 and 67/43296.
43296
What is the lowest common multiple of 12 and 58882?
353292
Calculate the least common multiple of 390816 and 3186.
1172448
What is the common denominator of 37/407100 and 58/9?
1221300
Calculate the smallest common multiple of 255420 and 119196.
1787940
Calculate the common denominator of -43/212 and 89/196.
10388
What is the common denominator of -59/7749 and 2/6027?
54243
What is the common denominator of 13/33 and 59/77865?
856515
What is the smallest common multiple of 62 and 9674?
299894
Calculate the lowest common multiple of 111874 and 11.
1230614
Calculate the smallest common multiple of 1171 and 1.
1171
Calculate the smallest common multiple of 568 and 20874.
83496
What is the common denominator of 17/98568 and -121/370?
492840
Calculate the common denominator of -58/284043 and 91/12.
1136172
Calculate the lowest common multiple of 4570 and 80.
36560
Calculate the smallest common multiple of 589225 and 200.
4713800
Calculate the least common multiple of 90 and 16335.
32670
What is the lowest common multiple of 21650 and 6928?
173200
Find the common denominator of 9/2 and -13/78330.
78330
Find the common denominator of 97/67856 and 89/8482.
67856
Calculate the lowest common multiple of 21 and 263790.
1846530
What is the lowest common multiple of 1775 and 239375?
16995625
What is the lowest common multiple of 78974 and 122?
4817414
Find the common denominator of -48/232339 and -37/12087.
2091051
What is the smallest common multiple of 20 and 64412?
322060
Calculate the common denominator of -71/100590 and -145/120708.
603540
Calculate the smallest common multiple of 36337 and 5012.
145348
Calculate the common denominator of 51/6754 and -55/60172.
661892
What is the common denominator of 11/244770 and 169/330?
2692470
Calculate the common denominator of -23/1892223 and -71/1892223.
1892223
Calculate the common denominator of 19/598782 and -77/20.
5987820
Find the common denominator of 52/225 and -9/266900.
2402100
Calculate the smallest common multiple of 25 and 8240.
41200
Calculate the common denominator of -93/376 and -79/564.
1128
Find the common denominator of 54/1645 and 17/3290.
3290
Calculate the smallest common multiple of 8976 and 17760.
3321120
Find the common denominator of -91/405 and -89/50.
4050
Calculate the least common multiple of 2 and 47092.
47092
Calculate the smallest common multiple of 23304 and 58260.
116520
Calculate the common denominator of -55/418746 and -57/5528.
1674984
Calculate the lowest common multiple of 38 and 4019.
152722
What is the lowest common multiple of 22 and 17560?
193160
Find the common denominator of -161/7650 and -11/150.
7650
Find the common denominator of 86/753075 and 117/669400.
6024600
Find the common denominator of -121/1390 and 43/2910.
404490
Calculate the common denominator of -121/504660 and 77/1429870.
8579220
What is the lowest common multiple of 36136 and 48?
216816
Calculate the common denominator of 16/22485 and 123/20.
89940
Find the common denominator of 139/564 and 127/3864.
181608
Calculate the lowest common multiple of 190 and 88755.
3372690
What is the smallest common multiple of 204258 and 15579?
1838322
Calculate the common denominator of 80/51 and 43/51300.
872100
Calculate the least common multiple of 2955 and 47871.
239355
What is the smallest common multiple of 1137810 and 1896350?
5689050
Calculate the common denominator of -37/60631 and -3/4.
242524
What is the least common multiple of 92274 and 3?
92274
What is the least common multiple of 36485 and 124049?
620245
Find the common denominator of -129/227920 and -39/284900.
1139600
What is the common denominator of -59/15456 and 83/952?
262752
Find the common denominator of -23/546 and -107/66066.
66066
Find the common denominator of 67/18 and 73/14616.
14616
Calculate the least common multiple of 510 and 51483.
8752110
Calculate the lowest common multiple of 20208 and 18.
60624
Calculate the least common multiple of 228340 and 163100.
1141700
Find the common denominator of -62/1974427 and 57/182.
3948854
What | 2023-11-26T01:26:58.646260 | https://example.com/article/7276 |
<?xml version="1.0" encoding="UTF-8"?>
<ui version="4.0">
<class>DatabaseOpenWidget</class>
<widget class="QWidget" name="DatabaseOpenWidget">
<property name="geometry">
<rect>
<x>0</x>
<y>0</y>
<width>596</width>
<height>250</height>
</rect>
</property>
<layout class="QVBoxLayout" name="verticalLayout" stretch="1,0,0,1,0,0,3">
<property name="spacing">
<number>8</number>
</property>
<item>
<spacer name="verticalSpacer_2">
<property name="orientation">
<enum>Qt::Vertical</enum>
</property>
<property name="sizeHint" stdset="0">
<size>
<width>20</width>
<height>40</height>
</size>
</property>
</spacer>
</item>
<item>
<widget class="QLabel" name="labelHeadline">
<property name="text">
<string>Enter master key</string>
</property>
</widget>
</item>
<item>
<widget class="QLabel" name="labelFilename"/>
</item>
<item>
<spacer name="verticalSpacer_3">
<property name="orientation">
<enum>Qt::Vertical</enum>
</property>
<property name="sizeHint" stdset="0">
<size>
<width>20</width>
<height>40</height>
</size>
</property>
</spacer>
</item>
<item>
<layout class="QGridLayout" name="gridLayout">
<property name="verticalSpacing">
<number>8</number>
</property>
<item row="1" column="0">
<widget class="QCheckBox" name="checkKeyFile">
<property name="text">
<string>Key File:</string>
</property>
</widget>
</item>
<item row="0" column="0">
<widget class="QCheckBox" name="checkPassword">
<property name="text">
<string>Password:</string>
</property>
</widget>
</item>
<item row="0" column="1">
<layout class="QHBoxLayout" name="horizontalLayout">
<item>
<widget class="PasswordEdit" name="editPassword">
<property name="echoMode">
<enum>QLineEdit::Password</enum>
</property>
</widget>
</item>
<item>
<widget class="QToolButton" name="buttonTogglePassword">
<property name="checkable">
<bool>true</bool>
</property>
</widget>
</item>
</layout>
</item>
<item row="1" column="1">
<layout class="QHBoxLayout" name="horizontalLayout_2">
<item>
<widget class="QComboBox" name="comboKeyFile">
<property name="sizePolicy">
<sizepolicy hsizetype="Expanding" vsizetype="Fixed">
<horstretch>0</horstretch>
<verstretch>0</verstretch>
</sizepolicy>
</property>
<property name="editable">
<bool>true</bool>
</property>
</widget>
</item>
<item>
<widget class="QPushButton" name="buttonBrowseFile">
<property name="text">
<string>Browse</string>
</property>
</widget>
</item>
</layout>
</item>
</layout>
</item>
<item>
<widget class="QDialogButtonBox" name="buttonBox">
<property name="orientation">
<enum>Qt::Horizontal</enum>
</property>
<property name="standardButtons">
<set>QDialogButtonBox::Cancel|QDialogButtonBox::Ok</set>
</property>
</widget>
</item>
<item>
<spacer name="verticalSpacer">
<property name="orientation">
<enum>Qt::Vertical</enum>
</property>
<property name="sizeHint" stdset="0">
<size>
<width>20</width>
<height>40</height>
</size>
</property>
</spacer>
</item>
</layout>
</widget>
<customwidgets>
<customwidget>
<class>PasswordEdit</class>
<extends>QLineEdit</extends>
<header>gui/PasswordEdit.h</header>
</customwidget>
</customwidgets>
<tabstops>
<tabstop>checkPassword</tabstop>
<tabstop>editPassword</tabstop>
<tabstop>buttonTogglePassword</tabstop>
<tabstop>checkKeyFile</tabstop>
<tabstop>comboKeyFile</tabstop>
<tabstop>buttonBrowseFile</tabstop>
<tabstop>buttonBox</tabstop>
</tabstops>
<resources/>
<connections/>
</ui>
| 2024-03-20T01:26:58.646260 | https://example.com/article/7600 |
Followup to: An Alien God, The Wonder of Evolution, Evolutions Are Stupid
Yesterday, I wrote:
Humans can do things that evolutions probably can't do period over the expected lifetime of the universe. As the eminent biologist Cynthia Kenyon once put it at a dinner I had the honor of attending, "One grad student can do things in an hour that evolution could not do in a billion years." According to biologists' best current knowledge, evolutions have invented a fully rotating wheel on a grand total of three occasions.
But then, natural selection has not been running for a mere million years. It's been running for 3.85 billion years. That's enough to do something natural selection "could not do in a billion years" three times. Surely the cumulative power of natural selection is beyond human intelligence?
Not necessarily. There's a limit on how much complexity an evolution can support against the degenerative pressure of copying errors.
(Warning: A simulation I wrote to verify the following arguments did not return the expected results. See addendum and comments.)
(Addendum 2: This discussion has now been summarized in the Less Wrong Wiki. I recommend reading that instead.)
The vast majority of mutations are either neutral or detrimental; here we are focusing on detrimental mutations. At equilibrium, the rate at which a detrimental mutation is introduced by copying errors, will equal the rate at which it is eliminated by selection.
A copying error introduces a single instantiation of the mutated gene. A death eliminates a single instantiation of the mutated gene. (We'll ignore the possibility that it's a homozygote, etc; a failure to mate also works, etc.) If the mutation is severely detrimental, it will be eliminated very quickly - the embryo might just fail to develop. But if the mutation only leads to a 0.01% probability of dying, it might spread to 10,000 people before one of them died. On average, one detrimental mutation leads to one death; the weaker the selection pressure against it, the more likely it is to spread. Again, at equilibrium, copying errors will introduce mutations at the same rate that selection eliminates them. One mutation, one death.
This means that you need the same amount of selection pressure to keep a gene intact, whether it's a relatively important gene or a relatively unimportant one. The more genes are around, the more selection pressure required. Under too much selection pressure - too many children eliminated in each generation - a species will die out.
We can quantify selection pressure as follows: Suppose that 2 parents give birth to an average of 16 children. On average all but 2 children must either die or fail to reproduce. Otherwise the species population very quickly goes to zero or infinity. From 16 possibilities, all but 2 are eliminated - we can call this 3 bits of selection pressure. Not bits like bytes on a hard drive, but mathematician's bits, information-theoretical bits; one bit is the ability to eliminate half the possibilities. This is the speed limit on evolution.
Among mammals, it's safe to say that the selection pressure per generation is on the rough order of 1 bit. Yes, many mammals give birth to more than 4 children, but neither does selection perfectly eliminate all but the most fit organisms. The speed limit on evolution is an upper bound, not an average.
This 1 bit per generation has to be divided up among all the genetic variants being selected on, for the whole population. It's not 1 bit per organism per generation, it's 1 bit per gene pool per generation. Suppose there's some amazingly beneficial mutation making the rounds, so that organisms with the mutation have 50% more offspring. And suppose there's another less beneficial mutation, that only contributes 1% to fitness. Very often, an organism that lacks the 1% mutation, but has the 50% mutation, will outreproduce another who has the 1% mutation but not the 50% mutation.
There are limiting forces on variance; going from 10 to 20 children is harder than going from 1 to 2 children. There's only so much selection to go around, and beneficial mutations compete to be promoted by it (metaphorically speaking). There's an upper bound, a speed limit to evolution: If Nature kills off a grand total of half the children, then the gene pool of the next generation can acquire a grand total of 1 bit of information.
I am informed that this speed limit holds even with semi-isolated breeding subpopulations, sexual reproduction, chromosomal linkages, and other complications.
Let's repeat that. It's worth repeating. A mammalian gene pool can acquire at most 1 bit of information per generation.
Among mammals, the rate of DNA copying errors is roughly 10^-8 per base per generation. Copy a hundred million DNA bases, and on average, one will copy incorrectly. One mutation, one death; each non-junk base of DNA soaks up the same amount of selection pressure to counter the degenerative pressure of copying errors. It's a truism among biologists that most selection pressure goes toward maintaining existing genetic information, rather than promoting new mutations.
Natural selection probably hit its complexity bound no more than a hundred million generations after multicellular organisms got started. Since then, over the last 600 million years, evolutions have substituted new complexity for lost complexity, rather than accumulating adaptations. Anyone who doubts this should read George Williams's classic "Adaptation and Natural Selection", which treats the point at much greater length.
In material terms, a Homo sapiens genome contains roughly 3 billion bases. We can see, however, that mammalian selection pressures aren't going to support 3 billion bases of useful information. This was realized on purely mathematical grounds before "junk DNA" was discovered, before the Genome Project announced that humans probably had only 20-25,000 protein-coding genes. Yes, there's genetic information that doesn't code for proteins - all sorts of regulatory regions and such. But it is an excellent bet that nearly all the DNA which appears to be junk, really is junk. Because, roughly speaking, an evolution isn't going to support more than 10^8 meaningful bases with 1 bit of selection pressure and a 10^-8 error rate.
Each base is 2 bits. A byte is 8 bits. So the meaningful DNA specifying a human must fit into at most 25 megabytes.
(Pause.)
Yes. Really.
And the Human Genome Project gave the final confirmation. 25,000 genes plus regulatory regions will fit in 100,000,000 bases with lots of room to spare.
Amazing, isn't it?
Addendum: genetics.py, a simple Python program that simulates mutation and selection in a sexually reproducing population, is failing to match the result described above. Sexual recombination is random, each pair of parents have 4 children, and the top half of the population is selected each time. Wei Dai rewrote the program in C++ and reports that the supportable amount of genetic information increases as the inverse square of the mutation rate(?!) which if generally true would make it possible for the entire human genome to be meaningful.
In the above post, George Williams's arguments date back to 1966, and the result that the human genome contains <25,000 protein-coding regions comes from the Genome Project. The argument that 2 parents having 16 children with 2 surviving implies a speed limit of 3 bits per generation was found here, and I understand that it dates back to Kimura's work in the 1950s. However, the attempt to calculate a specific bound of 25 megabytes was my own.
It's possible that the simulation contains a bug, or that I used unrealistic assumptions. If the entire human genome of 3 billion DNA bases could be meaningful, it's not clear why it would contain <25,000 genes. Empirically, an average of O(1) bits of genetic information per generation seems to square well with observed evolutionary times; we don't actually see species gaining thousands of bits per generation. There is also no reason to believe that a dog has greater morphological or biochemical complexity than a dinosaur. In short, only the math I tried to calculate myself should be regarded as having failed, not the beliefs that are wider currency in evolutionary biology. But until I understand what's going on, I would suggest citing only George Williams's arguments and the Genome Project result, not the specific mathematical calculation shown above. | 2023-11-17T01:26:58.646260 | https://example.com/article/4841 |
Q:
Disable an asp.net dynamic button click event during postback and enable it afterwards
I am creating a button dynamically in my code and attaching a click event to it. However I have to prevent people to click it while there is a process going on. So when it is clicked once, it should be disabled and when the process ends it should be enabled. How can I do that?
Thanks.
A:
onclick="this.enabled=false" add this from your code behind to your control
btnAdd.Attributes.Add("onclick", "this.enabled=false;");
This link explains in detail http://encosia.com/2007/04/17/disable-a-button-control-during-postback/
| 2023-09-04T01:26:58.646260 | https://example.com/article/3878 |
Morning Jolt: Monday, July 28
The philosophies of Mike Tomlin and Casey Hampton clashed yesterday, prompting the Steelers coach to prohibit his Pro Bowl nose tackle from joining his teammates on the training camp practice field until he gets in shape. Tomlin placed Hampton on the physically unable to perform (PUP) list after watching him labor through five of a required eight 100-yard jogs that were part of the players' conditioning test their first day at Saint Vincent College in Latrobe. "He wasn't able to finish the test," Tomlin said. "He's overweight and he's not conditioned enough to participate at this point." Tomlin said when he determines Hampton is in shape, he'll take him off the PUP list and allow him to practice. "I could be in better shape," said Hampton, who the Steelers officially list on their roster at 325 pounds. "But my thing is the only way you can get into football shape is to play football. You can do all the running you want, know what I mean?" (Pittsburgh Post-Gazette) Comment
The Yankees' acquisition of Bobby Abreu went so well two years ago they are trying to recreate it now with Jarrod Washburn. The ploy works like this: The Yanks locate a player who fills a need, a very expensive player whose contract does not expire until after the following season. The player's current team badly wants to excise the contract. There is just one team, however, with the financial heft to absorb the money. So the Yanks say they will assume every penny, but in exchange will give up marginal prospects, at best. The Yanks have told Seattle they will take on the $13 million-plus left on Washburn through next year, but to do that the Mariners will get no better than a Grade C-type prospect. For now, Seattle is refusing that offer. A game of chicken is ongoing, and all indications are the Yanks will see if the Mariners blink at the deadline as the Phillies did in 2006. (New York Post)Comment
It has taken five years for the aftershocks of the ACC's raid on the Big East to subside. Now the league that commissioner Mike Tranghese rebuilt seems rock solid. So what's next for the league? Expansion. The Post has learned that the Big East has explored deals with Army and Navy in football. The concept is for each of the service academies to play four Big East opponents each season on a rotating basis. A ninth member, even limited partners such as Army and Navy (both have a significant TV following) would give league members eight league games. That would allow the flexibility of four non-conference games so teams could pursue non-conference rivals such as Pittsburgh and Penn State. Army and Navy initially rejected the idea of a Big East association but because both are independents, that could change. (New York Post)Comment
Must-See Photo
Supporters of Manchester United and Portsmouth FC watch side by side their friendly soccer match in Abuja, Nigeria, on Sunday. Manchester United won the match 2-1. (AP Photo/George Osodi)
Must-See Video
Little Leaguer flashes some leather.
Game To Watch
Cubs at Brewers, 8:05 p.m. ET -- The NL Central's top two teams open a four-game series.
1984 -- The Summer Olympics begin in Los Angeles. 1991 -- Montreal's Dennis Martinez pitches a perfect game against the Dodgers in Los Angeles. 1994 -- Baseball players announce a strike date of Aug. 12, 1994. 1994 -- Texas' Kenny Rogers pitches a perfect game against the California Angels at the The Ballpark in Arlington.
More More Sports
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Get expert analysis, unrivaled access, and the award-winning storytelling only SI can provide - from Peter King, Tom Verducci, Lee Jenkins, Seth Davis, and more - delivered straight to you, along with up-to-the-minute news and live scores. | 2024-02-07T01:26:58.646260 | https://example.com/article/5502 |
The facility integrates server, storage, networking and management resources, and will provide the infrastructure required for cloud computing services, application modernisation and data centre transformation. The data centre will also come equipped with a HP Carbon Emissions Management Service, an assessment service that helps organisations calculate energy consumption and greenhouse gas emissions emanating from the use of IT.
Opening the centre, Communications Minister Stephen Conroy said infrastructure "such as this Next Generation Data Centre exemplifies the type of forward-looking investment activity that the NBN is encouraging in Australia. It is a tangible demonstration of how the government’s investment in the NBN is driving corporate investment in the Australian ICT sector.”
The multi-million dollar investment will be an employment stimulant, immediately creating “hundreds” of new local jobs in the construction and associated industries, the minister said
The centre is expected to be operational by the end of the year.
HP is the latest in a run of new data centre operators banking on the benefits of the NBN. A new data centre was launched by Tier 5 and Dell in Adelaide, South Australia in December which is attracting government and educational facility customers.
A fleet of data centres are poised to launch this year from freshly ASX listed company NextDC, a data hosting company founded by former Pipe Networks chief Bevan Slattery. The company is slated to launch a flagship data centre in Brisbane in March, Melbourne in November and another in Sydney at an as yet unspecified date. ® | 2024-05-15T01:26:58.646260 | https://example.com/article/3065 |
Q:
Displaying image taken by camera API android
I have been following this tutorial: AndroidHive - working with Camera API to try and enable taking photos in my app. However, I'm getting an error once I press my "take photo button".
LogCat:
09-16 11:04:00.539 19561-19561/au.gov.nsw.shellharbour.saferroadsshellharbour D/Dob_in_a_Hoon_photos﹕ Failed to create Dob_in_a_Hoon_photos directory
09-16 11:04:00.539 19561-19561/au.gov.nsw.shellharbour.saferroadsshellharbour D/AndroidRuntime﹕ Shutting down VM
09-16 11:04:00.539 19561-19561/au.gov.nsw.shellharbour.saferroadsshellharbour W/dalvikvm﹕ threadid=1: thread exiting with uncaught exception (group=0x40e21438)
09-16 11:04:00.559 19561-19561/au.gov.nsw.shellharbour.saferroadsshellharbour E/AndroidRuntime﹕ FATAL EXCEPTION: main
java.lang.NullPointerException: file
at android.net.Uri.fromFile(Uri.java:441)
at au.gov.nsw.shellharbour.saferroadsshellharbour.dob_in_a_hoon.getOutputMediaFileUri(dob_in_a_hoon.java:97)
at au.gov.nsw.shellharbour.saferroadsshellharbour.dob_in_a_hoon.captureImage(dob_in_a_hoon.java:89)
at au.gov.nsw.shellharbour.saferroadsshellharbour.dob_in_a_hoon.access$000(dob_in_a_hoon.java:28)
at au.gov.nsw.shellharbour.saferroadsshellharbour.dob_in_a_hoon$1.onClick(dob_in_a_hoon.java:60)
at android.view.View.performClick(View.java:4191)
at android.view.View$PerformClick.run(View.java:17229)
at android.os.Handler.handleCallback(Handler.java:615)
at android.os.Handler.dispatchMessage(Handler.java:92)
at android.os.Looper.loop(Looper.java:137)
at android.app.ActivityThread.main(ActivityThread.java:4963)
at java.lang.reflect.Method.invokeNative(Native Method)
at java.lang.reflect.Method.invoke(Method.java:511)
at com.android.internal.os.ZygoteInit$MethodAndArgsCaller.run(ZygoteInit.java:1038)
at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:805)
at dalvik.system.NativeStart.main(Native Method)
09-16 11:04:10.529 19561-19561/au.gov.nsw.shellharbour.saferroadsshellharbour I/Process﹕ Sending signal. PID: 19561 SIG: 9
Here is my .java file:
public class dob_in_a_hoon extends ActionBarActivity {
private static final int CAMERA_CAPTURE_IMAGE_REQUEST_CODE = 100;
public static final int MEDIA_TYPE_IMAGE = 1;
private static final String IMAGE_DIRECTORY_NAME = "Dob_in_a_Hoon_photos";
private Uri fileUri;
private ImageView Hoon_Image;
private Button button_take_photo;
private String driver_spinner_array[];
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_dob_in_a_hoon);
driver_spinner_array = new String[2];
driver_spinner_array[0] = "Yes";
driver_spinner_array[1] = "No";
Spinner Driver_spinner = (Spinner) findViewById(R.id.driver_selector);
ArrayAdapter adapter = new ArrayAdapter(this, android.R.layout.simple_dropdown_item_1line, driver_spinner_array);
Driver_spinner.setAdapter(adapter);
Hoon_Image = (ImageView) findViewById(R.id.CapturedImage);
button_take_photo = (Button)findViewById(R.id.btn_take_photo);
button_take_photo.setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View v) {
captureImage();
}
});
}
@Override
public boolean onCreateOptionsMenu(Menu menu) {
// Inflate the menu; this adds items to the action bar if it is present.
getMenuInflater().inflate(R.menu.dob_in_a_hoon, menu);
return true;
}
@Override
public boolean onOptionsItemSelected(MenuItem item) {
// Handle action bar item clicks here. The action bar will
// automatically handle clicks on the Home/Up button, so long
// as you specify a parent activity in AndroidManifest.xml.
int id = item.getItemId();
if (id == R.id.action_settings) {
return true;
}
return super.onOptionsItemSelected(item);
}
private void captureImage(){
Intent intent = new Intent(MediaStore.ACTION_IMAGE_CAPTURE);
fileUri = getOutputMediaFileUri(MEDIA_TYPE_IMAGE);
intent.putExtra(MediaStore.EXTRA_OUTPUT, fileUri);
startActivityForResult(intent, CAMERA_CAPTURE_IMAGE_REQUEST_CODE);
}
public Uri getOutputMediaFileUri(int type){
return Uri.fromFile(getOutputMediaFile(type));
}
private static File getOutputMediaFile(int type){
File mediaStorageDir = new File(Environment.getExternalStoragePublicDirectory(Environment.DIRECTORY_PICTURES),IMAGE_DIRECTORY_NAME);
if (!mediaStorageDir.exists()){
if (!mediaStorageDir.mkdirs()){
Log.d(IMAGE_DIRECTORY_NAME, "Failed to create " + IMAGE_DIRECTORY_NAME + " directory");
return null;
}
}
String timeStamp = new SimpleDateFormat("yyyyMMdd_HHmmss", Locale.getDefault()).format(new Date());
File mediaFile;
if (type==MEDIA_TYPE_IMAGE){
mediaFile = new File(mediaStorageDir.getPath()+File.separator+"IMG_"+timeStamp+".jpg");
}else {
return null;
}
return mediaFile;
}
@Override
protected void onActivityResult(int requestCode, int resultCode, Intent data){
if (requestCode == CAMERA_CAPTURE_IMAGE_REQUEST_CODE){
if (resultCode==RESULT_OK){
previewCapturedImage();
}else if (resultCode == RESULT_CANCELED){
Toast.makeText(getApplicationContext(),"User Cancelled image Capture", Toast.LENGTH_SHORT).show();
}else {
Toast.makeText(getApplicationContext(),"Failed to capture image", Toast.LENGTH_SHORT).show();
}
}
}
private void previewCapturedImage(){
try{
Hoon_Image.setVisibility(View.VISIBLE);
BitmapFactory.Options options = new BitmapFactory.Options();
options.inSampleSize = 8;
final Bitmap bitmap = BitmapFactory.decodeFile(fileUri.getPath(),options);
Hoon_Image.setImageBitmap(bitmap);
}catch (NullPointerException e){
e.printStackTrace();
}
}
@Override
protected void onSaveInstanceState(Bundle outState){
super .onSaveInstanceState(outState);
outState.putParcelable("file_uri", fileUri);
}
@Override
protected void onRestoreInstanceState(Bundle savedInstanceState){
super.onRestoreInstanceState(savedInstanceState);
fileUri = savedInstanceState.getParcelable("file_uri");
}
}
Can anybody see where I am going wrong in my code, and what I have to do to fix it?
A:
I created a sample snippet and tested out the mkdirs as shown in your code, it causes similar exception if i removed the permissions
<uses-permission android:name="android.permission.WRITE_EXTERNAL_STORAGE"/>
<uses-permission android:name="android.permission.READ_EXTERNAL_STORAGE"/>
Can you please check your manifest to make sure you have these defined.
If above is not the problem, another (unlikely) source of error could be your SD card is running out of space or some permissions restrictions. In this case, i would try the app on android emulator or another device.
| 2024-04-18T01:26:58.646260 | https://example.com/article/1810 |
The origin and characteristics of a pig kidney cell strain, LLC-PK.
A stable epithelial-like pig kidney cell strain has been established. This strain has been carried through more than 300 serial passages, has remained free of microbial and viral contaminants, and has retained a near diploid number of chromosomes. Attempts to produce tumors with these cells in immunosuppressed laboratory animals have been uniformly negative. The cells have grown rapidly in monolayer cultures with a split ratio of 1 to 15 at weekly intervals, but have failed to proliferate in suspension cultures. A subline adapted to growth on serum-free medium 199 has been carried through 145 passages on this medium. Several unusual morphologic features have been observed in these cultures including three-dimensional "domelike" structures. These cells have been found susceptible to some viruses and have been especially useful for viruses of domestic animals. LLC-PK1 cells have produced significant levels of plasminogen activator. | 2024-04-08T01:26:58.646260 | https://example.com/article/9357 |
First Unofficial Picture of Benedict Cumberbatch in Star Trek Sequel
Trek Movie today posted two unofficial location shots from the Los Angeles set of JJ Abrams’ Star Trek sequel. The pictures feature what has been described as a a fight scene on a space barge. The images feature Zachary Quinto as Spock, Zoe Saldana as Uhura and Benedict Cumberbatch in the role that has caused so much speculation ever since his involvement with the project was first announced.
Although Abrams and Cumberbatch were both coy on the potentially villainous nature of the role, the pictures below indicate that Spock is clearly not happy, with Cumberbatch’s character appearing to be on the receiving end of a Vulcan neck pinch. | 2024-03-22T01:26:58.646260 | https://example.com/article/4670 |
Background {#Sec1}
==========
Hereditary spastic paraplegia (HSP), also called familial spastic paraparesis or Stru ¨mpell-Lorrain disease, is a group of neurodegenerative and inherited heterogeneous neurological disorders characterized by a length-dependent distal axonal degeneration of the corticospinal tracts \[[@CR1]\]. The progressive spasticity and pyramidal signs of the lower limbs are the prominent features of HSP, which can be well explained by the fact that the innervated function of the longest fibers toward to the lower extremity is prone to be affected \[[@CR2]\].
According to whether accompanied by additional neurological or psychiatric symptoms such as ataxia, mental and cognitive changes, extrapyramidal signs, visual dysfunction or epilepsy, or with extra neurological signs, the diseases can be categorized into either pure HSP (pHSP) or complicated HSP (cHSP) \[[@CR3], [@CR4]\]. The onset age of HSP exhibits a wide range from childhood to over 70 years old depending on the underlying genetic defect, even in the family with a same mutation \[[@CR5]\]. Therefore, it is hard to explain the interaction between the genotype and the phenotype of HSP.
Based on the distinguishably inherited trait, HSP can be classified into as autosomal dominant, autosomal recessive, X-linked, mitochondrial, or de novo \[[@CR6]\]. Meanwhile, autosomal dominant HSP is the most common mode that accounts for approximately 70% of all HSP patients \[[@CR7]\]. All modes of inheritance are associated with multiple genes or loci. Until now, there have been at least 76 spastic paraplegia associated with loci and more than 59 corresponding spastic paraplegia genes (SPG) have been identified \[[@CR8]\]. The *SPG4/ SPAST* gene comprising 17 exons, identified as the 90-kb genomic region on chromosome 2 (2p22.3, \[[@CR9]\], has been reported to be the most frequent cause of HSP and accounts for approximately 40% of pure autosomal dominant HSP and 10% of sporadic cases \[[@CR10], [@CR11]\]. Over 500 mutations have been identified in the *SPAST* gene. Generally, *SPAST* gene mutations have a tendency to cause pure HSP \[[@CR12]\] and are more common in males than females \[[@CR5]\]. Here, we report a novel *SPAST* gene mutation site (c.1710_1712delGAA) that presented in a Chinese family with HSP, significantly enriching the mutation spectrum of HSP gene.
Methods {#Sec2}
=======
Subjects {#Sec3}
--------
In this study, we recruited 15 subjects (female: male ratio is about 1:1; age range: 3--63 years) in total from a Chinese family with HSP. This family was enrolled in our study on the basis of the following criteria: (1) Based on Harding's criteria \[[@CR4]\], the proband, a 63-years-old female, was diagnosed with HSP; (2) The family of the proband had at least four affected relatives with HSP; (3) The family was potentially informative for designing a study to investigate the genetic mutations. Total of 15 individuals were performed with neurologic examination, four of which had the same clinical manifestations and the rest were all asymptomatic. The study was conducted according to the Declaration of Helsinki and was authorized by the Ethics Committee of Peking University First Hospital. The related written informed consents for publication of details and images were obtained from all the participants and the legal guardian of the patient aged 11 years in our study.
DNA extraction {#Sec4}
--------------
Whole genomic DNA was extracted from peripheral blood of the 15 family members using DNA Isolation Kit (Bioteke, AU1802) as previously described \[[@CR13]\]. Concentrations of each DNA sample were measured on a Qubit fluorometer (Invitrogen, Q33216) using Qubit dsDNA HS Assay Kit (Invitrogen, Q32851). Meanwhile, 1% agarose gel electrophoresis was performed for quality control of each DNA sample.
Libraries preparation and amplification {#Sec5}
---------------------------------------
DNA libraries were established with KAPA Library Preparation Kit (Kapa Biosystems, KR0453) following the manufacturer's instructions, which mainly contains three major procedures: end-repair of fragmented DNA, A-tailing, adapter ligation and amplification \[[@CR14]\]. Purifications between procedures were achieved using Agencourt AMPure XP beads. After the ligation reaction with beads, 50 μl ligation was totally resuspended in 45 μl PEG/NaCl SPRI® Solution and then incubated at 37 °C for 2 min. Subsequently, the captured beads via a magnet were incubated until the liquid was clear. The beads were washed for three times using 200 μl 80% ethanol after discarding the clear supernatant and then was dried at room temperature (RT). Eventually, the beads captured on a magnet were thoroughly resuspended in 25 μl water and incubated for 2 min at RT until the liquid was clear to be proceed with library amplification.
Libraries amplification was fulfilled by polymerase chain reaction (PCR) under the following 25 μl reaction system: 12.5 μl 2× KAPA HiFi HotStart ReadyMix, 1 μl 5 μM each primer, 10 μl captured library beads suspension and 1.5 μl water. PCR amplification program was set up: 98 °C 2 min; 98 °C 30 s; 65 °C 30 s; 72 °C 30 s, 13 cycles; and a final step at 72 °C for 4 min. Subsequently, repeat the steps of washing and resuspending the beads as described above. The amplified libraries that were prepared for array capture were assessed with Qubit dsDNA HS Assay kit (Invitrogen, Q32851).
Array capture and sequencing {#Sec6}
----------------------------
Array capture was performed via the Agilent SureSelectXT2 Target Enrichment System as previously described \[[@CR14]\]. Briefly, array hybridization was captured by mixing the pooled libraries with a buffer solution and oligo-blockers, which was incubated for 24 h at 65 °C. The hybridized library molecules were performed with Dynabeads® MyOne™ Streptavidin T1 (Invitrogen, \#65601). The captured library was amplified as following: 21 μl 2× KAPA HiFi HotStart ReadyMix, 1 μl 5 μM primer, 20 μl captured library beads suspension. PCR amplification program was 98 °C 2 min; 98 °C 30 s; 65 °C 30 s; 72 °C 30 s, 13 cycles and a final step at 72 °C for 4 min. Purifications between procedures were conducted using Agencourt AMPure XP beads and the libraries were evaluated with Qubit dsDNA HS Assay kit (Invitrogen, Q32851). Finally, DNA libraries of the proband were analyzed by whole exome sequencing (WES). WES was carried out on the HiSeq2500 platform as paired-end 200-bp reads. Illumina Sequence Control Software (SCS) was used to evaluate the sequencing data, thus removing adapter sequences in the raw data and discarding low-quality sequencing reads. Conventional Sanger sequencing of the *SPAST* gene was further performed in 15 individuals from the Chinese family.
In-silico predictions {#Sec7}
---------------------
Effects of the novel mutation on SPAST tertiary structure were predicted by RaptorX prediction tool \[[@CR15]\]. Additionally, PROVEAN (Protein Variation Effect Analyzer) \[[@CR16]\], a new algorithm, is also adopted to predict whether the mutation has an functional impact on the SPAST protein sequence variations.
Results {#Sec8}
=======
Clinical characteristics {#Sec9}
------------------------
The proband (II:2) (Fig. [1](#Fig1){ref-type="fig"}a), a 63-years-old female, presented to the outpatient clinic in our hospital due to her progressive difficulty walking caused by moderate spasticity of the lower limbs for 24 years. She has felt unknown gait disorder as early as in 1992. For the last 11 years, this gait disorder has gotten worse, especially in cold weather. And she was confined to a wheelchair at the age of 52. In the last year, the proband suffered from the frequently urged to urinating and bowel functions. Physical examination showed that she had brisk deep tendon reflexes in all four limbs, simultaneously accompanied with obvious corticospinal tract signs (Babinski^'^s signs was positive), and decreased sense of pain, light touch and vibration in the lower limbs characterized with stocking pattern-distributed sensory loss. Muscle strength of the upper limbs was normal, while both the extensors and flexors in the lower limbs were 3/5. The results obtained from routine laboratory tests, electromyography, cranial and cervical MRIs did not reveal any obvious pathognomonic alteration. For her previous history, the proband had ever received some treatment on rheumatoid arthritis because of the pain in both hips and knees 26 years ago, but the uncomfortable symptom was not getting better. 9 years ago, a traumatic injury on her back further aggravated her discomfort though the cranial and cervical MRIs were both normal at that time. Fig. 1Pedigree of the investigated HSP family harbouring a novel SPAST gene mutation. **a**: The black arrow indicates the proband II:6. Squares indicate male, circles indicate females. Individuals affected with HSP are represented by black filled, while Healthy members are indicated by empty symbols. Slashes indicate already dead. **b**: Detection of the mutation of SPAST gene in a Chinese family. Sequence analysis revealed a newly identified in-frame deletion mutation in a heterozygous form in four affected individuals (II:2,III:1,III:9 and IV:9) within the family. The exon16 consists three nucleotides deletion(c.1710_1712delGAA). III:5 is the representative wild type sequences of the investigated healthy family members
With regard to the proband's family history (Fig. [1](#Fig1){ref-type="fig"}a), her parents were deceased, but her father had similar symptoms. In addition, she had three brothers and two sisters. One of her brothers who had similar symptoms has passed away and the rest were all asymptomatic. Details about her symptomatic father and brother are not clear. Physical examination gave the below results. III:9 aged 32 years had an abnormal spine physiological curvature. The shoulders of IV:9 aged 11 years were not equal. The (III:1) aged 46 years, III:9 and IV:9 showed brisk deep tendon reflexes in all four limbs and positive Babinski signs. All the three symptomatic patients all could not run and squat since young age. Despite their motor symptoms, the proband's nephew(III:1) and the third daughter (III:9) have frequent urge to urinate and to have bowel functions at the age of 40 years and 27 years.
The proband's the third daughter (III:9) and grandson(IV:9) suffer from other disease except HSP. The III:9 patient who was diagnosed with pulmonary hypertension has been suffering from chest tightness and shortness of breath for 3 years, losing the ability to work. Similarly, the echocardiography of IV:9 aged 11 years showed mild reflux at mitral valve and tricuspid valve.
Clinical features of the four affected individuals in the family have been summarized in Table [1](#Tab1){ref-type="table"}, and their clinical commonalities and personalities are exhibited respectively. The four symptomatic patients have the different degrees of disability. The disability score was evaluated according to a four-point scale (1: normal, 2: able to walk but not run, 3: need the help of a walking aid or support, 4: walk on wheelchair) \[[@CR17]\]. In addition, the onset age ranges from 3 to 30 years old, although they have the same mutation in the exon 16. Table 1Clinical features of the affected individuals within the familyIndividual IDII:6III:1III:9IV:9SexFMFMAge at onset (years)aearly 30searly 10searly 10s3Age at examination (years)63463210Disease duration (years)\> 33\> 36\> 227Disability score b4222Lower limb hyperreflexia++++Lower limb spasticity++++Lower limb pyramidal weakness--------Babinski sign++++Upper limb hyperreflexia++++Upper limb spasticity--------Sphincter disturbances+++--Scoliosis----+--Pes cavus+----+Sensory deficits+------Mental retardation--------concomitant diseases----Pulmonary hypertensionReflux at mitral and tricuspid valve+ indicates the presence of a feature, − indicates the absence of a feature, respectivelya: Age at onset was calculated approximately when appeared to have difficulty in walking firstb: Disability stages: 1: normal, 2: able to walk alone but not run, 3: need the help of a walking aid or support, 4: wheelchair user
Genetic findings and prediction results of protein structure and function by different methods {#Sec10}
----------------------------------------------------------------------------------------------
The Exome Sequencing analysis of the proband exhibited a novel disease-associated mutation in exon 16 of the already known disease-associated *SPAST* gene, and the in-frame deletion was identified in the three affected family members (Fig. [1](#Fig1){ref-type="fig"}b II:2, III:1,III:9,IV:9). It is an in-frame deletion mutation in the heterozygous state: the GAA nucleotides deletion at codon 1710--1712 position and the circled nucleotide represents the codon 1710 position, where the mutation starts. (Fig. [1](#Fig1){ref-type="fig"}b II:2, III:1, III:9,IV:9). According to Human Gene Mutation Database (HGMDpro), the pathogenic mutation site c.1710_1712delGAA has not been reported until now. Therefore, it is a novel mutation. And there are no mutations were identified when analysis of other genes associated with HSP was performed: *PLP1, L1CAM, SPG11, SPG7, ATL1* and so on. While the rest asymptomatic family members had no mutations at this site (Fig. [1](#Fig1){ref-type="fig"}b III:5).
As highlighted in Fig. [1](#Fig1){ref-type="fig"}b, the results of RaptorX prediction showed that this new-found mutation we reported resulted in the synthesis of misfolded protein (Fig. [2](#Fig2){ref-type="fig"}b) in comparison to native one (Fig. [2](#Fig2){ref-type="fig"}a). We performed a protein sequence alignment across species showing the area of this in-frame amino acid deletion and the surrounding residues (https://www.uniprot.org/align/A20200502216DA2B77BFBD2E6699CA9B6D1C41EB2087CC0O). The result revealed that the spastin protein sequence across species is highly conserved at the position 570 of the protein (red frame) (Fig. [2](#Fig2){ref-type="fig"}c). Therefore, the lysine deficiency at the position 570 of the protein has a significant impact on the function. Additionally, the result of PROVEAN demonstrated that the mutation site c.1710_1712delGAA has an functional impact on the SPAST protein sequence variations. Given a list of genomic coordinates and variants (232,372,308,AGAA,A), the amino acid change(p.K570del) can be quickly determined and PROVEAN score is computed to be − 11.55, which is significantly lower than the score threshold (cutoff = − 2.5). The deletion variant was predicted as deleterious (Table [2](#Tab2){ref-type="table"}). According to American College of Medical Genetics and Genomics (ACMG) criteria \[[@CR18]\], we score this variant as likely pathogenic PM1, PM2, PM4, PP3. Fig. 2Tertiary structure alteration prediction of SPAST by RaptorX tool. **a**. The tertiary structure of native protein. **b**. Tertiary structure of p.K570del affected protein. The novel mutation we reported resulted in the synthesis of misfolded protein. **c**. The spastin protein sequence alignment across species showing the area of this in-frame amino acid deletion (red frame) and the surrounding residuesTable 2The genome variants results are represented as PROVEAN scores and predictions
Discussion {#Sec11}
==========
According to the family history, accurate description of the clinical phenotype as well as cerebral and spinal MRI, we diagnosed the proband with HSP. Analyses of the Exome Sequencing revealed a novel disease-associated in-frame deletion in the *SPAST* gene. This mutation consists of three nucleotides deletions (c.1710_1712delGAA) within the exon 16. The onset age of patients in this family is highly variable, which is accordance with the previous study \[[@CR19]\]. However, the onset age of affected individuals in our study is obviously earlier than previous studies \[[@CR20]\], which promotes us to consider the existence of the new mutation site c.1710_1712delGAA. Additionally, affected individuals in our study presented clinical features of the pHSP, which is consistent with Orlacchio A' s study \[[@CR21]\].
The *SPAST* gene has 17 coding exons and encodes the protein spastin, a member of the AAA ATPase protein family. The protein spastin contains two main structural domains: the microtubule interacting and trafficking (MIT) domain in the N- terminus and the catalytic AAA domain at the C-terminus, in which the former mainly regulates microtubule organization and the later focus on ATPase activity associated with various cellular activities \[[@CR22]\]. The two main domains are both essential to accomplish the main known function of spastin: microtubule (MT) severing \[[@CR23]\]. More than 200 different mutations located in sites within the AAA region have been identified in patients with HSP-*SPG4* \[[@CR24]\]. Therefore, it is believed that some mutated spastins may result in insufficient microtubule-severing activity by dominant-negative fashion. Additionally, another prevalent hypothesis is neurotoxicity of mutant spastin proteins. The *SPAST* gene presents two translation initiation codons, which allows to synthesis two spastin isoforms: a full-length isoform called M1(616 amino acid) and a slightly shorter isoform called M87(530 amino acid) that lacks the first 87 amino acid \[[@CR25]\]. Studies on rodents show that M87 is more abundant in various tissues, whereas M1 is only appreciably detected in brain and spinal cord \[[@CR26]\]. Besides that, axonal transport and neurite growth are not affected by the mutated M87 \[[@CR27]\]. However, Mutant spastin proteins can form defective heterohexamers with wild-type (WT) spastin, and simultaneously produce toxic effect when presented as the tissue-specific M1 isoform \[[@CR28], [@CR29]\]. Even though M87 likely harbors the same AAA mutations as the M1 isoform, it is somehow degraded more effectively than mutated M1 in a dominant-negative scenario \[[@CR30]\], thus possessing a lower toxicity.
Conclusion {#Sec12}
==========
In Our study, a novel mutation in *SPAST* gene was found in a Chinese family with multiple affected family members, which significantly enrich the mutation spectrum of HSP. This novel mutation has been inherited at least four generations according to the related investigation, further emphasizing the closed interaction between the phenotypic and genetic heterogeneity of HSP. However, some shortcomings are still existed. For example, we haven't performed functional laboratory studies about the novel mutation due to the unavailable patient cell lines and the related verification on which one molecular mechanisms above of *SPAST* gene mutants in our study cannot be realized. Haploinsufficiency is the prevalent mechanism at present. Using the haploinsufficiency model of HSP-SPG4 with a 50% decrease in active spastin levels has shown a lower MT severing \[[@CR31], [@CR32]\]. Another hypothesis is mainly that mutated spastin can form defective heterohexamers with wild-type (WT) spastin, and exert a dominant-negative effect. However, there is still considerable debate about the latter one hypothesis. It is unclear whether the mutant M87 can effectively impair the enzymatic activity of WT spastin \[[@CR33]\]. Thus, further study should ascertain the role of causative genes to help better understand the relationship between genotypes and phenotypes.
HSP
: Hereditary spastic paraplegia
SPG
: Spastic paraplegia genes
CT
: Computed tomography
MRI
: Magnetic resonance imaging
SCS
: Sequence Control Software
NGS
: Next Generation Sequence
HGMDpro
: Human Gene Mutation Database
MIT
: Microtubule interacting and trafficking
MT
: Microtubule
WT
: Wild- type
ER
: Reticulum membrane
CK2
: Casein kinase 2
FAT
: Fast axonal transport
PROVEAN
: Protein Variation Effect Analyzer
ACMG
: American College of Medical Genetics and Genomics
PCR
: Polymerase chain reaction
RT
: Room temperature
**Publisher's Note**
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We would like to thank all members of the family participating in the study for agreeing to publish their available clinical data in medical journals.
YN H and WW Y designed the work and analyzed the clinical data. HQ J and JW D performed the neurologic examination and interpreted the clinical data related to the HSP. D N collected the material of these family members and analyzed the genetic data. WW Y was a major contributor to writing the manuscript. All authors contributed toward drafting and critically revising the paper, gave final approval of the version to be published.
No grant was provided for this analysis and this study.
The datasets generated and/or analyzed during the current study are not publicly available in order to protect participant confidentiality.
The study was conducted in accordance with the Declaration of Helsinki and the Ethics Committee of Peking University First Hospital (PKUFH-2019-181). Written informed consents were obtained from all of the participants of this family in the study. The written informed consent of the boy aged 11 years in our study was obtained from his parents.
Written informed consent was obtained from all of the participants in the study to publish the clinical details and images on the internet. The written informed consent to publish of the boy aged 11 years in our study was obtained from his parents.
The authors have no conflict of interests.
| 2024-02-22T01:26:58.646260 | https://example.com/article/1511 |
Q:
взять данные из другого формы через user control (winforms)
Добрый день. Есть одна форма и user control. В форме есть tabcontrol с tabPages. а user control надо записывать. Я с помощью класса TabControl хотел попробовать выдаёт ошибка. Можно ли вызывать элементы формы из usercontrol? Теперь значение Id берётся, но при заказе Id обнуляется.
Вот код:
UserControl.cs
public int index;
public void InsertIndex(int index)
{
this.index = index;
}
public void InsertTempOutcomes()
{
string query = "INSERT INTO TempOutcomes(RoomId,UserId,NameGoods,PriceGoods,AmountGoods,DataStart,DataFinish) VALUES((SELECT Name FROM Rooms WHERE Id=@roomId),(SELECT Name FROM Users WHERE Id=@userId),(SELECT Name FROM Store WHERE Id=@id),(SELECT Price FROM Store WHERE Id=@id),@amount,@dataStart,@dataFinish)";
int goodsId = comboBoxGoods.SelectedIndex + 1;
try
{
command = new SQLiteCommand(query,connection);
command.Parameters.AddWithValue("@amount",textBoxAmount.Text);
command.Parameters.AddWithValue("@dataStart",dateTimePicker.Value);
command.Parameters.AddWithValue("@id", goodsId);
command.Parameters.AddWithValue("@dataFinish", dateTimePicker.Value.AddSeconds(seconds));
command.Parameters.AddWithValue("@roomId",index);
command.ExecuteNonQuery();
}
catch(Exception ex) { MessageBox.Show(ex.Message); }
finally { connection.Close(); }
}
Form.cs
UserContrElems user = new UserContrElems();
string querySelect = "SELECT Name FROM Rooms WHERE Id=@id";
int index = tabControl1.SelectedIndex + 1;
user.InsertIndex(index);
A:
Решил свою проблему. Я создавал новый User Control. Мне надо было использовать уже созданный user control.
int index = tabControl1.SelectedIndex + 1;
switch(index)
{
case 1: userContrElems1.Insert(index); break;
case 2: userContrElems2.Insert(index); break;
case 3: userContrElems3.Insert(index); break;
case 4: userContrElems4.Insert(index); break;
case 5: userContrElems5.Insert(index); break;
case 6: userContrElems6.Insert(index); break;
}
| 2024-05-19T01:26:58.646260 | https://example.com/article/9861 |
Q:
Is it possible to combine two integers in such a way that you can always take them apart later?
Given two integers $n$ and $m$ (assuming for both $0 < n < 1000000$) is there some function $f$ so that if I know $f(n, m) = x$ I can determine $n$ and $m$, given $x$? Order is important, so $f(n, m) \not= f(m,n)$ (unless $n=m$).
A:
$f(n,m) = 2^n 3^m$.
Alternatively, use the bijection between $\mathbb N \times \mathbb N$ and $\mathbb N$ which is given by $$f(n,m) = \frac{(n+m)(n+m+1)}{2} + m$$
A:
Since others have answered, here is another idea - less easy to write down explicitly as a mathematical function, but easy to describe and easy to implement if you have a machine which can handle strings. Send the digits of the first number to odd positions and the digits of the other to even positions so (31, 5681) would go to 50,608,311.
A:
If there are no bounds on your integers, use the Cantor pairing function. It is pleasantly easy to compute, as are its two "inverses."
For the case where your integers are bounded by say $10^6$, you can simply concatenate the decimal expansions, padding with initial $0$'s as appropriate. Or do something similar with binary expansions. Dirt cheap to combine and uncombine, an easy string manipulation even when we allow bounds much larger than $10^6$.
| 2024-07-23T01:26:58.646260 | https://example.com/article/3154 |
---
title: Config Options for javascript-flowtyped
sidebar_label: javascript-flowtyped
---
| Option | Description | Values | Default |
| ------ | ----------- | ------ | ------- |
|allowUnicodeIdentifiers|boolean, toggles whether unicode identifiers are allowed in names or not, default is false| |false|
|disallowAdditionalPropertiesIfNotPresent|Specify the behavior when the 'additionalProperties' keyword is not present in the OAS document. If false: the 'additionalProperties' implementation is compliant with the OAS and JSON schema specifications. If true: when the 'additionalProperties' keyword is not present in a schema, the value of 'additionalProperties' is set to false, i.e. no additional properties are allowed. Note: this mode is not compliant with the JSON schema specification. This is the original openapi-generator behavior.This setting is currently ignored for OAS 2.0 documents: 1) When the 'additionalProperties' keyword is not present in a 2.0 schema, additional properties are NOT allowed. 2) Boolean values of the 'additionalProperties' keyword are ignored. It's as if additional properties are NOT allowed.Note: the root cause are issues #1369 and #1371, which must be resolved in the swagger-parser project.|<dl><dt>**false**</dt><dd>The 'additionalProperties' implementation is compliant with the OAS and JSON schema specifications.</dd><dt>**true**</dt><dd>when the 'additionalProperties' keyword is not present in a schema, the value of 'additionalProperties' is automatically set to false, i.e. no additional properties are allowed. Note: this mode is not compliant with the JSON schema specification. This is the original openapi-generator behavior.</dd></dl>|true|
|ensureUniqueParams|Whether to ensure parameter names are unique in an operation (rename parameters that are not).| |true|
|enumNameSuffix|Suffix that will be appended to all enum names.| |Enum|
|enumPropertyNaming|Naming convention for enum properties: 'camelCase', 'PascalCase', 'snake_case', 'UPPERCASE', and 'original'| |PascalCase|
|legacyDiscriminatorBehavior|This flag is used by OpenAPITools codegen to influence the processing of the discriminator attribute in OpenAPI documents. This flag has no impact if the OAS document does not use the discriminator attribute. The default value of this flag is set in each language-specific code generator (e.g. Python, Java, go...)using the method toModelName. Note to developers supporting a language generator in OpenAPITools; to fully support the discriminator attribute as defined in the OAS specification 3.x, language generators should set this flag to true by default; however this requires updating the mustache templates to generate a language-specific discriminator lookup function that iterates over {{#mappedModels}} and does not iterate over {{children}}, {{#anyOf}}, or {{#oneOf}}.|<dl><dt>**true**</dt><dd>The mapping in the discriminator includes descendent schemas that allOf inherit from self and the discriminator mapping schemas in the OAS document.</dd><dt>**false**</dt><dd>The mapping in the discriminator includes any descendent schemas that allOf inherit from self, any oneOf schemas, any anyOf schemas, any x-discriminator-values, and the discriminator mapping schemas in the OAS document AND Codegen validates that oneOf and anyOf schemas contain the required discriminator and throws an error if the discriminator is missing.</dd></dl>|true|
|modelPropertyNaming|Naming convention for the property: 'camelCase', 'PascalCase', 'snake_case' and 'original', which keeps the original name. Only change it if you provide your own run-time code for (de-)serialization of models| |original|
|npmName|The name under which you want to publish generated npm package. Required to generate a full package| |null|
|npmRepository|Use this property to set an url your private npmRepo in the package.json| |null|
|npmVersion|The version of your npm package. If not provided, using the version from the OpenAPI specification file.| |1.0.0|
|nullSafeAdditionalProps|Set to make additional properties types declare that their indexer may return undefined| |false|
|prependFormOrBodyParameters|Add form or body parameters to the beginning of the parameter list.| |false|
|snapshot|When setting this property to true, the version will be suffixed with -SNAPSHOT.yyyyMMddHHmm| |false|
|sortModelPropertiesByRequiredFlag|Sort model properties to place required parameters before optional parameters.| |true|
|sortParamsByRequiredFlag|Sort method arguments to place required parameters before optional parameters.| |true|
|supportsES6|Generate code that conforms to ES6.| |false|
## IMPORT MAPPING
| Type/Alias | Imports |
| ---------- | ------- |
## INSTANTIATION TYPES
| Type/Alias | Instantiated By |
| ---------- | --------------- |
|array|Array|
|list|Array|
|map|Object|
## LANGUAGE PRIMITIVES
<ul class="column-ul">
<li>Array</li>
<li>Blob</li>
<li>Date</li>
<li>File</li>
<li>Object</li>
<li>boolean</li>
<li>number</li>
<li>string</li>
</ul>
## RESERVED WORDS
<ul class="column-ul">
<li>abstract</li>
<li>arguments</li>
<li>array</li>
<li>boolean</li>
<li>break</li>
<li>byte</li>
<li>case</li>
<li>catch</li>
<li>char</li>
<li>class</li>
<li>const</li>
<li>continue</li>
<li>date</li>
<li>debugger</li>
<li>default</li>
<li>delete</li>
<li>do</li>
<li>double</li>
<li>else</li>
<li>enum</li>
<li>eval</li>
<li>export</li>
<li>extends</li>
<li>false</li>
<li>final</li>
<li>finally</li>
<li>float</li>
<li>for</li>
<li>formparams</li>
<li>function</li>
<li>goto</li>
<li>hasownproperty</li>
<li>headerparams</li>
<li>if</li>
<li>implements</li>
<li>import</li>
<li>in</li>
<li>infinity</li>
<li>instanceof</li>
<li>int</li>
<li>interface</li>
<li>isfinite</li>
<li>isnan</li>
<li>isprototypeof</li>
<li>let</li>
<li>long</li>
<li>math</li>
<li>nan</li>
<li>native</li>
<li>new</li>
<li>null</li>
<li>number</li>
<li>object</li>
<li>package</li>
<li>private</li>
<li>protected</li>
<li>prototype</li>
<li>public</li>
<li>queryparameters</li>
<li>requestoptions</li>
<li>return</li>
<li>short</li>
<li>static</li>
<li>string</li>
<li>super</li>
<li>switch</li>
<li>synchronized</li>
<li>this</li>
<li>throw</li>
<li>throws</li>
<li>tostring</li>
<li>transient</li>
<li>true</li>
<li>try</li>
<li>typeof</li>
<li>undefined</li>
<li>useformdata</li>
<li>valueof</li>
<li>var</li>
<li>varlocaldeferred</li>
<li>varlocalpath</li>
<li>void</li>
<li>volatile</li>
<li>while</li>
<li>with</li>
<li>yield</li>
</ul>
## FEATURE SET
### Client Modification Feature
| Name | Supported | Defined By |
| ---- | --------- | ---------- |
|BasePath|✓|ToolingExtension
|Authorizations|✗|ToolingExtension
|UserAgent|✗|ToolingExtension
|MockServer|✗|ToolingExtension
### Data Type Feature
| Name | Supported | Defined By |
| ---- | --------- | ---------- |
|Custom|✗|OAS2,OAS3
|Int32|✓|OAS2,OAS3
|Int64|✓|OAS2,OAS3
|Float|✓|OAS2,OAS3
|Double|✓|OAS2,OAS3
|Decimal|✓|ToolingExtension
|String|✓|OAS2,OAS3
|Byte|✓|OAS2,OAS3
|Binary|✓|OAS2,OAS3
|Boolean|✓|OAS2,OAS3
|Date|✓|OAS2,OAS3
|DateTime|✓|OAS2,OAS3
|Password|✓|OAS2,OAS3
|File|✓|OAS2
|Array|✓|OAS2,OAS3
|Maps|✓|ToolingExtension
|CollectionFormat|✓|OAS2
|CollectionFormatMulti|✓|OAS2
|Enum|✓|OAS2,OAS3
|ArrayOfEnum|✓|ToolingExtension
|ArrayOfModel|✓|ToolingExtension
|ArrayOfCollectionOfPrimitives|✓|ToolingExtension
|ArrayOfCollectionOfModel|✓|ToolingExtension
|ArrayOfCollectionOfEnum|✓|ToolingExtension
|MapOfEnum|✓|ToolingExtension
|MapOfModel|✓|ToolingExtension
|MapOfCollectionOfPrimitives|✓|ToolingExtension
|MapOfCollectionOfModel|✓|ToolingExtension
|MapOfCollectionOfEnum|✓|ToolingExtension
### Documentation Feature
| Name | Supported | Defined By |
| ---- | --------- | ---------- |
|Readme|✓|ToolingExtension
|Model|✓|ToolingExtension
|Api|✓|ToolingExtension
### Global Feature
| Name | Supported | Defined By |
| ---- | --------- | ---------- |
|Host|✓|OAS2,OAS3
|BasePath|✓|OAS2,OAS3
|Info|✓|OAS2,OAS3
|Schemes|✗|OAS2,OAS3
|PartialSchemes|✓|OAS2,OAS3
|Consumes|✓|OAS2
|Produces|✓|OAS2
|ExternalDocumentation|✓|OAS2,OAS3
|Examples|✓|OAS2,OAS3
|XMLStructureDefinitions|✗|OAS2,OAS3
|MultiServer|✗|OAS3
|ParameterizedServer|✗|OAS3
|ParameterStyling|✗|OAS3
|Callbacks|✗|OAS3
|LinkObjects|✗|OAS3
### Parameter Feature
| Name | Supported | Defined By |
| ---- | --------- | ---------- |
|Path|✓|OAS2,OAS3
|Query|✓|OAS2,OAS3
|Header|✓|OAS2,OAS3
|Body|✓|OAS2
|FormUnencoded|✓|OAS2
|FormMultipart|✓|OAS2
|Cookie|✓|OAS3
### Schema Support Feature
| Name | Supported | Defined By |
| ---- | --------- | ---------- |
|Simple|✓|OAS2,OAS3
|Composite|✓|OAS2,OAS3
|Polymorphism|✓|OAS2,OAS3
|Union|✗|OAS3
### Security Feature
| Name | Supported | Defined By |
| ---- | --------- | ---------- |
|BasicAuth|✗|OAS2,OAS3
|ApiKey|✗|OAS2,OAS3
|OpenIDConnect|✗|OAS3
|BearerToken|✗|OAS3
|OAuth2_Implicit|✗|OAS2,OAS3
|OAuth2_Password|✗|OAS2,OAS3
|OAuth2_ClientCredentials|✗|OAS2,OAS3
|OAuth2_AuthorizationCode|✗|OAS2,OAS3
### Wire Format Feature
| Name | Supported | Defined By |
| ---- | --------- | ---------- |
|JSON|✓|OAS2,OAS3
|XML|✓|OAS2,OAS3
|PROTOBUF|✗|ToolingExtension
|Custom|✗|OAS2,OAS3
| 2024-02-09T01:26:58.646260 | https://example.com/article/9358 |
Featured Research
from universities, journals, and other organizations
'New' human adenovirus may not make for good vaccines, after all
Date:
August 11, 2010
Source:
The Wistar Institute
Summary:
In a new study of four adenovirus vectors, researchers show that a reportedly rare human adenovirus, AdHu26, is not so rare, after all, and would not be optimal as a vaccine carrier. As previous research has shown, a viral vector may be ineffective if the virus it is based on is common. The study supports the use of chimpanzee adenoviruses as vaccine vectors, since humans have little exposure to these viruses.
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In recent years, scientists have studied the possibility of using engineered human adenoviruses as vaccines against diseases such as HIV, tuberculosis, and malaria. In this approach, adenoviruses, which commonly cause respiratory-tract infections, are rendered relatively harmless before they are used as vectors to deliver genes from pathogens, which in turn stimulate the body to generate a protective immune response.
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In a new study of four adenovirus vectors, researchers from The Wistar Institute show that a reportedly rare human adenovirus, called AdHu26, is not so rare, after all, and would thus be unlikely to be optimal as a vaccine carrier for mass vaccination. As previous research has shown, a viral vector may be ineffective if the virus it is based on is common in a given population. According to the Wistar scientists, their study also supports the use of chimpanzee adenoviruses as vaccine vectors, since humans have little exposure to these viruses. Their findings were published online, ahead of print, in the Journal of Virology.
"Despite previous reports to the contrary, we find that AdHu26 commonly infects people, particularly those in Sub-Saharan Africa, the very people for whom the need for novel vaccine strategies is most dire," said senior author Hildegund C. J. Ertl, M.D., Wistar professor and director of The Wistar Institute's Vaccine Center. "HIV, malaria, and other infectious diseases take a tremendous toll in the developing world, especially in Sub-Saharan Africa, and a vaccine platform that could be used in those regions could save the lives of millions."
Scientists believe that prior immunity to human adenoviruses is what led, in part, to the failure in 2007 of the STEP trial, a large vaccine trial in the US and other countries that used an adenovirus vector as the basis for an HIV vaccine.
In the current study, Ertl and her colleagues analyzed blood samples collected from people at seven sites around the world, including Thailand, the United States, and five sub-Saharan African nations. They tested the samples to see if they contained neutralizing antibodies and responsive immune cells when exposed to AdHu26 and AdHu5, the virus used in the STEP trial. Surprisingly, neutralizing antibodies to AdHu26 were very prevalent in blood.
According to Ertl, adenoviruses are still good vaccine vectors, just not necessarily human adenoviruses. In addition to testing AdHu5 and AdHu26, the Wistar scientists also tested two adenoviruses that originated in chimpanzees, called AdC6 and AdC7. As expected, neutralizing antibodies were far less likely to be detected in human samples. Mouse studies of all four vectors demonstrated that that were similar in their ability to generate cellular immune responses.
"This study also confirms our current line of research that suggests engineered chimpanzee adenovirus vectors could be superior to related, native human adenoviruses," Ertl said. "Both human and chimpanzee adenoviruses function in similar ways, but the simple benefit is that humans are rarely exposed to adenoviruses of chimpanzee origin."
The Ertl laboratory is currently developing an HIV vaccine utilizing chimpanzee adenoviruses.
Along with senior author Ertl, co-authors from the Ertl laboratory include senior staff scientists Zhi Quan Xiang, M.D., and Xiang Zhou, M.D.; staff scientist Dongming Zhou, M.D., Ph.D.; research technician Yan Li; research assistant Ang Bian; and visiting scientists Heng Chen and Raj Kurupati, Ph.D. Co-authors also include Michael Betts, Ph.D.; Natalie Hutnick; Sally Yuan, Ph.D.; and Clive Gray, Ph.D.; of the University of Pennsylvania School of Medicine's Department of Microbiology; Jennifer Serwanga, Ph.D., of the National Institute for Communicable Diseases in South Africa; and Betty Auma, Ph.D.; and Pontiano Kaleebu, Ph.D.; of the Uganda Research Unit on AIDS in Uganda.
Funding for this study was provided through grants from the National Institutes of Health.
The Wistar Institute. "'New' human adenovirus may not make for good vaccines, after all." ScienceDaily. ScienceDaily, 11 August 2010. <www.sciencedaily.com/releases/2010/08/100811130019.htm>.
The Wistar Institute. (2010, August 11). 'New' human adenovirus may not make for good vaccines, after all. ScienceDaily. Retrieved March 31, 2015 from www.sciencedaily.com/releases/2010/08/100811130019.htm
The Wistar Institute. "'New' human adenovirus may not make for good vaccines, after all." ScienceDaily. www.sciencedaily.com/releases/2010/08/100811130019.htm (accessed March 31, 2015).
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ScienceDaily features breaking news and videos about the latest discoveries in health, technology, the environment, and more -- from major news services and leading universities, scientific journals, and research organizations. | 2024-06-23T01:26:58.646260 | https://example.com/article/9637 |
FANTASY FOOTBALL IN-SEASON FEATURES
IDP Interactive - Week 15 Steve Gallo December 13, 2008
Welcome to our week 15 edition of IDP INTERACTIVE. In our IDP Forum you will find a thread entitled "IDP INTERACTIVE" pinned to the top where you can submit your questions for consideration. If you question is not chosen to be answered in this column every effort will still be made to answer your question in the IDP Forum.
Let’s take a look at a “do I bench my stud question” forum member Eagles2010 has this week.
Eagles2010 wrote,
“I need some help with which LB to start week 15.
Beason vs. Denver.....do they pass even more than ever now??
Julian Peterson at St. Louis......seems to have the Rams number
Bradi James vs. Giants.... Have a good feeling about James this week
Stephen Cooper at Kansas City.... Don't have a feel for this play at all
Scoring is 1 point for tackles and pass deflections, 2.25 points for every half sack, 5 point Int's, 3 point fumble recoveries and forced fumbles, and 4.5 point for a safety. We do not score assisted tackles.
With the little research I did muster into what type of numbers MLB's ( 4-3 ) score verse Denver, what I found wasn't very promising. I think they may even get worse if Denver isnt running the ball with any success. Peterson had a good game verse the Rams earlier this season and Bradie James has been a hot hand lately. Any good insight into which LB is the best play this week? I'm leaning toward's Bradie James, but I've learned from experience that sitting your studs like Beason can hurt sometimes and I do not want to make that mistake as I did with Morrison last week. Thanks”
Always start your studs?
Yes, it is that time of year when fantasy owners find themselves second guessing their second guesses.
The, “Always start your studs” saying is so simple, straight forward and to the point but we know that it’s not always right. Jon Beason is a stud, that there is no denying. His IDP TOP* for the year stands at .180. To put that in a bit of perspective, Patrick Willis who is the #1 ranked LB in most scoring systems has an IDP TOP of .183. The problem is that no matter how much of a stud a player is there are games where they just don’t produce up to their normal levels. The last thing anyone wants to do at this time of year is to start someone that leaves you high and dry. Never has “what have you done for me lately” meant more than it does during fantasy playoffs.
Missing cut.
First let’s get the guys out of the way that aren’t going to make the final cut this week.
Julian Peterson
Over the last 5 weeks Peterson has been pretty inconsistent and he sure didn’t take advantage of a choice matchup last week. The last thing that you want this week is another 3 tackle effort and that is what Peterson has done in 3 of the past 4 weeks. He did put up a nice score versus the Rams earlier this year but most of that scoring occurred on just one play. Yep, he recorded a tackle, sack, forced fumble & fumble recovery on just on snap. Want me to make it look even scarier? That play happened on the Rams third offensive snap of the game. Without that one play, Peterson would have been looking at a 1 tackle, 1 assist day.
Stephen Cooper
Cooper came on like gang busters after returning from his four game league imposed suspension but he has cooled off of late. Over his last 5 weeks Cooper has recorded 4 or fewer tackles 3 times and to make matters worse they have come in the last 3 weeks. Sure he posted a nice score in week 14 but that was due to his 2 interceptions. I highly doubt you would be excited about plugging Cooper in and seeing him put up between 3-4 tackles.
And then there were two.
Wait, did you think I actually ruled Jon Beason out? The final decision in my opinion comes down to between Beason and Bradie James. To see who gets the final call we are going to take a look at some numbers. I think it is safe to say that if you knew you could get the type of production you normally get out of Beason that you would take it in a heartbeat. So let’s go one better, let’s shoot for Patrick Willis type production. After all we are talking about the playoffs.
The Matchups
Jon Beason vs Broncos
Season to date the Broncos have given up 82 combined tackles to MLB/ILBs. Those tackles have come on 622 total tackle opportunities, yielding an IDP TOP of .132. That isn’t exactly an enticing IDP TOP. However, when we look at the past 5 weeks we see that the Broncos have been a better matchup for MLB/ILBs. Over those 5 weeks the Broncos have given up 39 total tackles on 242 tackle opportunities for an IDP TOP of .161. Now that is a bit more palatable, still not studly but much better (+22% increase better). Seems like Beason might not be such a bad option after all. The Broncos should provide between 40-50 tackle opportunities which would put Beason between 6-8 tackles.
Bradie James vs Giants
James is facing a Giants team that has given up138 tackles so far this season on 658 tackle opportunities to MLB/ILBs. That would give opposing MLB/ILBs and IDP TOP of .210 versus the Giants this year.
To be fair we should look at the last 5 weeks just like we did for Beason. Over the last 5 weeks the Giants have given up 56 tackles on 254 tackle opportunities to opposing MLB/ILBs. That is an average of over 11 total tackles/game and an even more impressive .220 IDP TOP. Now, I know that is giving you a warm and fuzzy feeling but don’t forget to take a look at what James did in week 9 versus the Giants. All he did back in week 9 was log 9 tackles, 3 assist & 1 sack on 50 tackle opportunities for a mind blowing IDP TOP of .240. Now there is a bit of risk involved since the Giants have announced that Brandon Jacobs will be inactive for this game but I wouldn’t expect the Giants to deviate much from their game plan. Both Derrick Ward and Ahmad Bradshaw should carry the load without Jacobs just fine but even if the Giants struggle offensively look no further than last week’s game versus Philadelphia to see that the Giants are in a giving mood when it comes to MLB/ILBs this year.
And the winner is…
Eagle2010 is the winner that’s who as long as he starts Bradie James that is. I know, after seeing the above numbers announcing the winner was pretty anticlimactic. But truthfully you couldn’t ask for a better matchup in a playoff week. It is like the perfect storm. Ride the wave, enjoy it and hopefully a playoff win too.
*IDP TOP
The basic definition of IDP TOP (Tackle Opportunity Production) is that it measures the performance of a player based on the number of tackle opportunities that they have. The higher the IDP TOP the better. Tackle opportunity is currently being defined as the number of rushing attempts plus the number of completions that a defense faces in a game. Below is an example to help better understand IDP TOP.
Example:
NY Giants Offense has: 26 rushes and 19 completions for a total of 45 tackle opportunities (TOs).
New England Patriots LB Tedy Bruschi recorded 5 solo tackles and 3 assists. TOP for Bruschi on solo tackles is .111 (5 divided by 45)TOP for Bruschi on assists is .067 (3 divided by 45)TOP for Bruschi on combined tackles is .178 (8 divided by 45)
Green Bay Packers Offense has: 14 rushes and 19 completions for a total of 33 TOs.
New York Giants DB Gibril Wilson recorded 5 solo tackles and 3 assists. TOP for Wilson on solo tackles is .152 (5 divided by 33)TOP for Wilson on assists is .091 (3 divided by 33)TOP for Wilson on combined tackles is .242 (8 divided by 33)
In the above example what TOP helps to illustrate is that just looking at the recorded tackles doesn’t tell the entire story. Both players recorded 5 solo tackles and 3 assists yet Gibril Wilson was 36% to 37% more productive then Bruschi was. If Bruschi had achieved the same TOP as Wilson he would have recorded 7 solo tackles and 4 assists. It might not seem like much but it surely can make a difference.
I hope that everyone has enjoyed our first installment with IDP Interactive and I look forward to seeing more questions in the coming weeks. If you have any comments or recommendations on what you would like to see in IDP Interactive you can email me at idpsteve@gmail.com | 2023-11-29T01:26:58.646260 | https://example.com/article/6030 |
With the integrated circuit (IC) fabrication technology developing to a stage where the device feature sizes reach deep-submicron dimensions, all circuit MOS devices are employed as lightly-doped drain (LDD) structures, and the silicide process has been widely used in the diffusion layers of such MOS devices. Meanwhile, in order to reduce the series resistance due to diffusion in gate polysilicon, synthesis of polycrystalline compounds is also employed. These fabrication process improvements can result in significant increases in IC operation speed and integration, along with the MOS device gate oxide layer being increasingly thinned due to the scaling down of the IC component. However, the improvements also lead to a significant disadvantage. Deep submicron ICs are more vulnerable to electrostatic discharge (ESD) strikes which can cause failure of the circuits, leading to lower reliability of products in which such ICs are used.
ESD is an event that transfers an amount of charge from one object (e.g., the human body) to another (e.g., a chip). The existing anti-ESD requirements with respect to ICs all mainly concern the protection of static electricity from human body, and the human-body model (HBM) has been established which is the earliest and one of the most commonly-used ESD models.
HBM simulates a discharge from an electrostatically charged person to an IC chip pin when the person touches the pin with the hand. Therefore, an ESD event often occurs within the IC's input and output units, and internal supply-to-ground paths as well. This event may cause a very large current flowing through the IC chip in a very short period of time. In fact, ESD events account for 35% or higher of the causes of chip failure.
ESD protection circuit is designed to prevent a working circuit from acting as an ESD path and thus being damaged by guaranteeing that, for any pin of the circuit, there is an appropriate low-impedance bypass for guiding the current caused by an ESD event occurring at the pin to the power line, which is then discharged through an ESD current path established by another pin.
Referring to FIG. 1, which is a schematic circuit diagram of a first ESD clamp of the prior art, including an RC trigger circuit formed of a resistor RI and a capacitor C1, a plurality of inverters 10 constituted by pMOS and nMOS transistors, and a discharge transistor 20. The discharge transistor 20 is realized as an nMOS transistor. As an ESD pulse is typically a high-voltage, high-frequency signal, the discharge transistor 20 is required to be a bulky device with a large footprint, which is conflictive with the high integration trend in this industry. In addition, the bulky discharge transistor 20 tends to be associated with a significant leakage current Ioff which may affect the proper operation of the circuit.
Referring to FIG. 2, which is a schematic circuit diagram of a second ESD clamp of the prior art. The ESD clamp depicted in FIG. 2 is a silicon-controlled rectifier (SCR) in which N+ and P+ regions are formed in an Nwell substrate, and P+ and N+ regions in a Pwell substrate. Additionally, two transistors T1 and T2 are also formed in the substrates. In this case, ESD protection triggering requires breakdown of the p-n junctions, which leads to low efficiency and low sensitivity. In addition, there is a possibility of latch-up occurring in the SCR, which poses an additional risk. | 2024-06-03T01:26:58.646260 | https://example.com/article/1634 |
Tag: Greatest basketball players
The basketball is a sport that is loved by a lot of people around the world today. The history of the NBA is indeed a rich one, dating back nearly sixty-nine years. During this time, there have been notable players who have trilled and inspirited many. Their names have been engraved in the history book. Many of them were able to take their team to victory.
In this article, we are going to be looking at some of the best basketball players of all times.
Hakeem Olajuwon
This big man is regarded by many as one of the most skilled players in the history of the game. Hakeem dominance and resilience draw the attention of a lot of sports enthusiast. He humiliated many defenders with his legendary move (Dream Snake).
The fact that Hakeem is not only an attacking player is what makes him truly remarkable. He is one of the best defenders in the game of basketball. He bagged the title- most defensive player of the year in 1994.
Shaquille O’Neal
This beast of a player was one of the most dominant players in the game of basketball. He used his massive strength to brutalize defenders. He led the Lakers to three NBA Championship. He was able to bag a good number of trophies in his long career.
Tim Duncan
What brought this player to the spotlight was his consistency. There is not much variation in his statistics. This big man was able to bag four NBA Championship in his entire. His unblemished record was tainted when his team lost to Miami Heat.
LeBron James
James is one of the highest paid athletes in the world. Within just ten seasons he has become one of the greatest players in the sport he loved. He has been the MVP a good number of times. His statistics say a lot about how skilled he has become.
Micheal Jordan
This legend is known by many as the greatest basketball player in history. Back in the days, he was able to make anything possible. When he was with the Chicago blue, he dominated both sides of the ball. 1988 was the year he won the most valuable and defensive player. Nobody has currently been able to match the performance of this legend.
These men become legends because they were consistent and worked very hard. Down to this day, Micheal Jordan remains the greatest basketball player. | 2024-06-22T01:26:58.646260 | https://example.com/article/9214 |
Mercury and Selenium Balance in Endangered Saimaa Ringed Seal Depend on Age and Sex.
The endangered Saimaa ringed seal (Pusa hispida saimensis) is exposed to relatively high concentrations of mercury (Hg) in freshwaters poor in selenium (Se), a known antagonist of Hg. The impact of age and sex on the bioaccumulation of Hg and Se was studied by analyzing liver, muscle, and hair samples from seals of different age groups. Adult females were found to accumulate significantly more Hg in the liver (with ca. 60% as HgSe), and less Hg in the muscles compared to adult males, which may be explained by accelerated metabolism during gestation and lactation. In adult seals, molar Se:Hg ratios in the muscles fall below one, which is considered a threshold for the emergence of adverse effects. As a result, Saimaa ringed seals may be at risk of developing health and reproductive problems. According to mass balance calculations, the pups are exposed to considerable amounts (μg/d) of mercury during gestation, although lactation is their main exposure route. In lanugo pups, Hg concentrates in the hair, and molting serves as a main detoxification route. For other age groups, demethylation followed by the formation of HgSe is the main detoxification route, and the demethylation capability develops in pups by the time of weaning. | 2024-03-28T01:26:58.646260 | https://example.com/article/6670 |
SGAC Indonesia
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Selamat datang di SGAC Indonesia!
This page describes the SGAC Indonesia. You'll find information regarding space activities of Indonesian Youth on this page specifically. This SGAC page basically is intended to inspire the Indonesian youth to be more enthusiastic and interested in space activities. Moreover, generally it is to share, strengthen and broaden knowledge, to build a network and to voice the vision relating hot space topics of the Indonesian golden age.Fundamentally, Indonesian space sector has begun since 1963, marked by establishing the National Council for Aeronautic and Space (DEPANRI), and then continued by creating National Institute of Aeronautics and Space (LAPAN), in the same year. The Indonesian space sector was strengthened by the successful launching of Satellite Palapa A1, the first Indonesian communication satellite making Indonesia to be as the first emerging country in the world which operates its own domestic satellite system, and then sent other series into outer space. Those satellites programs also encouraged the Indonesian human spaceflight program in 1985 by selecting Dr. Pratiwi Sudarmono to fly on STS-61H, the planned shuttle mission for the deployment of commercial communication satellites.Although the human spaceflight program finally was cancelled due to the unfortunate disaster of the Challenger shuttle mission, both things have been inspiring Indonesian people, mainly young generation to learn, know and to take part in the space sector. Students, young professional and young researchers turn on some space activities and forums, and also attend several conferences/seminars in order not only to share, strengthen and broaden knowledge but also to build a networking and to voice their vision relating hot space topics. Studying in universities about space-related subjects is also another thing that those groups do to reach their passion. Some facilities that support research and development in the space field are under construction. One of them is National Observatory that will be located in Kupang, Indonesia.
All of activities in fact are important assets for developing Indonesian space sector these days and in the future generally.
Two Indonesian NPOCs actively involve both in space awareness activities and space commercialization because they share same vision: to inspire and improve the technological education skills for the younger generation, to bring us to the advancement of space. Space bussiness development is a potential strength that will improve the economy and welfare of the nation of Indonesia. On several occasions, NPOC introduce SGAC to young generation. One of them on the Jakarta Model United Nations event held at Atmajaya University, Jakarta, 25 July 2015. Therefore, feel free to connect with us!
Young Space Activities Overview in Indonesia
Space in Indonesia has become a trend among young Indonesia. This is marked by the beginning of the rampant activities that carried out activities and involved the younger generation. The Indonesian rocket competition, better known as KOMURINDO, was conducted in 2007, followed by other competitions such as the Atmospheric Balloon Competition, Water Rocket Competition for the kindergarten children, and Space Debate organized by LAPAN.The young generation of Indonesia is also actively involved in several international events, either in seminars/conferences such as IAC, APRSAF, APSCO, ITU or in international competitions. Many awards are achieved by Indonesia's young generation in the competition such as becoming the 2nd Winner of APSCO Cansat Competition held in Ulaanbatar, Mongolia, 22-23 September 2016, 2nd and 3rd Winner of APRSAF-Water Rocket Competition at Los Banos, Philipine, 12 -13 November 2016. In addition, one research by IT Del's high school students namely 'Micro-Aerobic Fermentation In Space With Micro Gravity' successfully escaped on the ISS project. Their research result, was launched into space on March 23, 2016 , Using Atlas 5 rocket from Cape Canaveral, Florida, USA.
Ms. Rumiris is presenting about Space Generation Advisory Council #JMUN2015
In the event of natural phenomena of total solar eclipse, dated March 9, 2016, Indonesia became the only country whose mainland most crossed the path of total solar eclipse. It was used to organize activities such as teaching children to make eyeglasses or sun binoculars, educating local students about a total solar eclipse, solar observation with LAPAN researchers, and the Amateur Astronomy Communities in 9 of total solar eclipse location which involved many young people.
Menembus Langit Campaign is also a parameter of interest of Indonesia's young generation. On this expedition, the unmanned aerial vehicle (UAV) Ai-X1 is flown by a weather balloon into the stratosphere and will return to the starting point automatically. The Ai-X1 UAV is equipped with a meteorological sensor to collect atmospheric data. Meteorological data and aviation research reports will be processed and will be distributed to Universities in Indonesia. Moreover, as a national space agency, LAPAN also attracted several universities to develop satellite research and development, such as ITB, IPB, Surya University, Telkom University.
Regarding the space business activities in Indonesia, only few have started to pursue the space business domains. In fact, following Working Group 2 in Asia Pacific Space Generation, the idea of space business incubator in Indonesia has never been setup. However, a space-based startup has been accepted to provide the Grant Programme of Technology Business Incubation 2015 of Ministry of Research and Technology (RISTEK-DIKTI) as the implementation of triple helix concept in Indonesia. The two-year program has been started to support young entrepreneurs not only financially but also train in business knowledge. After finishing the program, the startups are prepared to ready enter the market.
Country-Specific Events in 2017
August 21-24: Komurindo and Kombat, at LAPAN Facilities in Pameungpeuk - The implementation of the Indonesian Rocket and Payload Rocket Competition (Komurindo) and the Atmospheric Balloon Payload Competition (Kombat) is the annual national competition for rocket and rocket-level EDF rocket designs organized since 2009.
September 27-29: International Seminar Aerospace and Science Technology 2017 / V, North Sumatra - The international seminar is the continuation of ISAST series, which has been held by LAPAN since 2013, about the results of research and development the aerospace science and technology conducted in LAPAN and its partners in Indonesia.
Water Rocket Competition: Water rocket competition is a Prestigious competition level in Jakarta and its neighborhood areas, for Junior High School students aged 13-16 years. There is also a water rocket workshop. This activity is intended for students, for science experiments enthusiast of Primary-Senior High School students, for instance Astronomy field.
October: The Space Science Festival, for kindergarten pupils to high school students, is organized by LAPAN. This event will be held in LAPAN Bandung in October 2017.
Star Party is routinely performed by some Indonesian amateur astronomy societies. This activity will be conducted for children, to recognize and observe stars at night by using star binoculars. | 2024-06-07T01:26:58.646260 | https://example.com/article/6387 |
using Bridge;
namespace System.Collections.Generic
{
[External]
[Namespace("Bridge")]
public interface IDictionary<TKey, TValue> : IEnumerable<KeyValuePair<TKey, TValue>>, IBridgeClass
{
TValue this[TKey key]
{
get;
set;
}
ICollection<TKey> Keys
{
get;
}
ICollection<TValue> Values
{
get;
}
int Count
{
get;
}
bool ContainsKey(TKey key);
void Add(TKey key, TValue value);
bool Remove(TKey key);
bool TryGetValue(TKey key, out TValue value);
}
}
| 2023-09-08T01:26:58.646260 | https://example.com/article/2157 |
Website Exposes Loads of Personal Information
How many people know you on a first and last name basis? Thanks to spokeo.com, anyone can gain access to your personal information and you don't even have to accept a friend request for them to see it.
"You have no privacy anyway. Get over it!" Words of Scott, McNealy, former CEO of Sun Microsystems on today's information and web technology era.
Sites like Spokeo.com back up the notion finding a way to make a buck off your public information by compiling it all in one place.. All someone searching needs is a name.
Matthew Kissner, a FSU student says, "All of this information I knew was online somewhere but this is the first I've seen it all in the same place."
And nearly everyone's "digital identity" is there- whether they know it or not. "That's pretty crazy. Family tree? hum"
Personal information doesn't float away into "Internet Space". Sites, like Spokeo gather it from phone books, social networks, marketing surveys, business websites, even real estate listings. "I think it was just a matter of time before somebody put it all together. I'm okay with it as long as it's not too intrusive." says Carey.
Faye Jones with FSU's College of Law says, "Certain business professionals, psychiatrist, public officials and others who feel that having all this information makes them more vulnerable to identity theft, possibly opens them up to ethical considerations."
Where is the line drawn?
Jones pointed out- Spokeo partnered with a protection agency called Reputation Defender in an attempt to sell a consumer protection for the information Spokeo has already tapped into.
You do have to subscribe to Spokeo to get more and more detailed information and some who say they were okay with the basic version say being able to pay for deeper personal information, concerns them.
The Commerce Department task force did call for the creation of a "Privacy Bill of Rights" for Internet users in December. But for now people will have to go in and manually remove themselves from the site.
Online Public Information File
Viewers with disabilities can get assistance accessing this station's FCC Public Inspection File by contacting the station with the information listed below. Questions or concerns relating to the accessibility of the FCC's online public file system should be directed to the FCC at 888-225-5322, 888-835-5322 (TTY), or fccinfo@fcc.gov. | 2023-08-15T01:26:58.646260 | https://example.com/article/8335 |
Bitcoin Whitepaper Limited Edition Poster » Gadget Flow
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Celebrate your crypto wealth with the Bitcoin Whitepaper Limited Edition Poster. The Bitcoin Whitepaper Poster pays homage to the founding father of cryptocurrencies, Satoshi Nakamoto, and his revolutionary whitepaper. The limited edition poster features Satoshi’s ‘Bitcoin: A Peer-to-Peer Electronic Cash System’ whitepaper from October 2008. Printed in Amsterdam, the Bitcoin Whitepaper Poster is a high-quality print on thick and durable matte paper. In addition, the poster size is 70 x 100 centimeters, making it suitable for any space. You can pay for the poster through traditional payment methods such as PayPal or credit cards. However, cryptocurrency enthusiasts will appreciate that they can also pay with BTC, ETH, and LTC. With free worldwide shipping, the Bitcoin Whitepaper Paper also makes an excellent gift for friends who are into crypto.
Make checking your phone one less thing to worry about with the Bose QuietComfort 35 Wireless Headphones II. These headphones feature the excellent noise-canceling technology of Bose with the addition of built-in Google Assistant and Amazon Alexa. Now you only.. | 2024-05-27T01:26:58.646260 | https://example.com/article/5568 |
Q:
Adding unsupported Framework to project
I'm trying to add new library to my project. This library contains code to deal with bar code scanning and uses CoreVideoFramework which was introduced in iOS 4.0. Currently I had my iOS deployment target set to 3.1.2 to support those devices. But now when I try to run new version of application on iPhone with 3.1.2 it crashes at very beginning saying it can't find CoreVideoFramework. On 4.2 simulator everything work fine.
I'm wondering if it is possible to add CoreVideoFramework to project without crashes on 3.x devices. I'm thinking about a situation when on 3.x BarCodeScanning is turned off and on 4.x it is enabled.
Simply I'm trying to avoid requesting from user to have iOS 4.x. I want my app to still support 3.x.
A:
Yes, that's possible
In your target settings go to "Link with libraries" section and for CoreVideo.framework set link option to "Optional" (by default it is "Required").
| 2023-09-03T01:26:58.646260 | https://example.com/article/6823 |
Megyn Kelly on NBC. Charles Sykes/Invision/AP
When Megyn Kelly's Sunday-night talk show debuted earlier this year, it was panned by critics, protested by high-profile public figures, and ignored by many viewers.
The pattern seems to be repeating itself for her new daytime talk show.
In its first week on the air, NBC's new 9 a.m. ET talk show, "Megyn Kelly Today," has been blasted by critics, drawn middling ratings, and inspired a series of negative press reports about her banter with celebrity guests.
Largely known for mercilessly grilling high-profile guests and at-times inflammatory segments about race, Kelly declared on her first show that she was "done" with politics and would focus on broader lifestyle topics and more positive-focused stories. Asked, for example, whether she would take a knee with NFL players protesting police brutality and racial injustice, Kelly offered platitudes about the right for everyone to the First Amendment, saying, "Go USA, that's my feeling in watching it."
Critics were unmoved. Both CNN and USA Today dubbed the premiere "awkward," The Washington Post called it a "morning-show Bride of Frankenstein," and Time said she "fails to connect" with her audience.
The new NBC host also bungled a series of softball interviews with celebrities, asking cringe-inducing questions that ended up going viral.
In her first major interview, Kelly sat with the cast of NBC's "Will & Grace," the resurrected early-2000s sitcom that starred a fictional gay lawyer.
Kelly brought a superfan onstage during the interview, asking the fan whether it was "true you became a lawyer, and you became gay, because of Will."
The question provoked an immediate wave of criticism, even from the cast: The star Debra Messing said she regretted going on the show and was "dismayed" by Kelly's comments, a remark the New York Post's Page Six said got Messing in trouble privately with NBC.
An NBC insider who was there said Kelly was "obviously making a joke, that was clear to the 'superfan.'"
"It became even clearer when she spent much of the interview asking the show’s creators how proud they were of the positive social impact of Will & Grace and the progress it brought about for gay rights," the insider said.
But the host again appeared to alienate a guest just two days later when she asked actress Jane Fonda about her history of plastic surgery.
"We really want to talk about that now?" Fonda asked.
"One of the things people think about when they look at you is how amazing you look," Kelly replied.
"Well, thanks. Good attitude, good posture, take care of myself," Fonda said. "But let me tell you why I love this movie that we did, 'Our Souls at Night,' rather than plastic surgery."
The clip quickly went viral: The Washington Post cited the moment as an example that Kelly was "striking out with her guests," while USA Today said Fonda "shut down" Kelly.
It's unclear whether viewers are on board with the NBC host's rebranding.
Kelly's show had 2.9 million viewers overall on Monday and 917,000 in advertisers' coveted demographic of 25- to 54-year-olds, a number down from the same time slot last year (NBC noted that it would be difficult for any network to match that day, which featured the first debate between former Secretary of State Hillary Clinton and now President Donald Trump).
That was higher overall and in the key demo than the show's main competitor at 9 a.m., "LIVE with Kelly & Ryan."
But that number has appeared to decline steadily over the week.
NBC didn't immediately respond to a request for comment. | 2024-05-11T01:26:58.646260 | https://example.com/article/9326 |
[Clinical manifestation of Kaposi sarcoma in otorhinolaryngology head and neck surgery].
To improve the knowledge of Kaposi sarcoma and the relationship between Kaposi sarcoma and human immunodeficiency virus (HIV) infection, and to improve the ability to diagnose and treat Kaposi sarcoma and acquired immune deficiency syndrome (AIDS). Symptoms, signs and results of 121 patients encountered in the department of otorhinolaryngology head and neck surgery in Tanzania, who was diagnosed as Kaposi sarcoma actually with HIV infection and AIDS, were retrospectively analyzed in this study. There were 46 males and 75 females with age ranged from 5 to 65 years, medium 30 year. The mucous membranes and skin lesions was the most commonly seen clinical manifestation in 121 cases, these lesions appeared as raised blotches or lumps that might be purple, brown, or red, early stages typical lesions began as flat or slightly raised colored spots. Among the cases reported here, 25 patients (20.66%) showed progressive nose blockage and nose bleeding and the purple-red new-grows were found in the nose of these patients. Fifteen patients (12.40%) had flat or slightly raised colored spots in their mucous membrane of mouth (palate or tongue), and in other 7 patients, purple small lumps were found in the gums of the patients. There were same lesions in their pharynx in 9 cases. In 10 patients (8.26%), Kaposi sarcoma was found in tonsil looked like tonsillitis with enlarged tonsils by two to three degree. Twelve patients (9.92%) had masses in the neck with no pain. Thirty-five patients (28.92%)had lesions of purple black nodules, including 10 patients who had the same lesions with ulcer formation in the nodules. All patients had been followed-up for at least two-years. Eighty-five patients passed away in one year, survival rate of one year was 21.48% (26/121), only 12 patients survived from the disease over two years, two years' survival rate was 9.92% (12/121). Kaposi sarcoma is the characteristic disease for AIDS, mainly found on the membranes and skin. These lesions appears as raised blotches or lumps that may be purple, brown, or red, early stages typical lesions begin as flat or slightly raised colored spots. Patients who had kaposi sarcoma often died in a short-time without treatment. | 2024-07-22T01:26:58.646260 | https://example.com/article/5674 |
17 comments:
Oh, I always love when the magazines put out their big fall issues. Fun. Makes me want to sketch the models--they always look great. I thought of you at Trader Joe's yesterday when I saw their macarons. They're "OK."..for Trader Joe's. (They're no Laduree, mind you, and they're certainly no M'lle Gillott's, either, for that matter.)
OMG! I am losing it - our TWO Trader Joes have opened and I have shopped and shopped and totally forgot about the macs! I deserve to have to wear that cape with the platform shoes!!!Stick with PH for le rentree - at least he seems to have it right!
What a lot of trouble macarons are to make! Just the idea of getting each one to be the correct size puts me off. No wonder they cost! Australia is in the grip of the macaron craze too. French Fantasies in South Yarra, Melbourne make great ones; you can also buy good ones with your coffee in the National Gallery of Victoria in Melbourne.Those crazy yellow shoes are made for stomping along not for walking in. Gwendoline, Australia | 2024-01-02T01:26:58.646260 | https://example.com/article/2806 |
Influence of phospholipid on bile salt binding to calcium hydroxyapatite and on the poisoning of nascent hydroxyapatite crystals.
Glycine-conjugated, dihydroxy bile salts inhibit calcium hydroxyapatite (HAP) formation by binding to and poisoning nascent crystal embryos. Their taurine-conjugated counterparts bind less well to hydroxyapatite and do not inhibit its formation; but more hydrophobic, synthetic analogs of the taurine conjugated bile salts are inhibitors of hydroxyapatite formation. Because hydrophobicity is an important determinant of the ability of bile salts to inhibit hydroxyapatite crystal growth, experiments were performed to study the effect of the physiologically important mixed micelles of bile salt and phospholipid. Taurodeoxycholate/phosphatidylcholine (10:1) mixed micelles bound to HAP at lower total lipid concentrations than did pure taurodeoxycholate. At low total lipid concentrations, phosphatidylcholine (PC) binding appeared to predominate, suggesting that PC had a higher affinity than did taurodeoxycholate (TDC) for the HAP surface. Although glycodeoxycholate (3 mM) significantly (> 95%) inhibited hydroxyapatite precipitation, higher concentrations of taurodeoxycholate, either alone or mixed with phosphatidylcholine, did not affect hydroxyapatite formation. These results suggest that biliary phospholipids do not modulate the ability of bile salts to inhibit hydroxyapatite crystal growth. | 2024-05-16T01:26:58.646260 | https://example.com/article/9253 |
Electrocardiogram reconstruction from high resolution voltage optical mapping.
Electrocardiogram recordings during opucal mapping experiments in heart tissue are commonly used tu monitor the health of the preparation and to obtain dominant frequencies during arrhythmic and defibrillatory studies. However the use of ECG reconstructed from optical mapping is seldom used and to date it has not been strictly validated. In this manuscript we present the first detailed validation and comparison of Optical Mapping ECG, or OM-ECG, with standard ECG recordings by calculating the electrostatic potential in space as a function of the voltage measured optically and describe the different approximations that can be used to obtain unipolar or bipolar ECG recordings. We found that in small/medium hearts, such as rabbits, leads that are aligned apex to base only require activation recording from one surface (anterior or posterior) for the OM-ECG to match the ECG while leads aligned left to right may require both an anterior and posterior optical mapping recording. The discrepancy between leads is due to symmetries in the ventricular activations. In the case of ischemic hearts where activations even-out more, the match between the OM-ECG and standard ECG may require only one surface recording for both left-right and base-apex leads. We believe that this methodology has two main and direct applications in the study of cardiac dynamics. The first is during studies of defibrillation where information after the shock may be crucial in the development of new strategies, OM-ECGs do not suffer the current artifacts of standard ECGs during shocks and can be calculated during the entire activation. We present examples in rabbit ventricles where even low amplitude pacing artifacts are captured by the ECG but do not appear in the OM-ECG. The second use of this technique is for reconstructions of intramural dynamics in larger hearts where differences between the ECG and OM-ECG obtained from anterior and posterior recordings can be used to derive the intramural activation. | 2024-07-28T01:26:58.646260 | https://example.com/article/2181 |
Controlled doping of carbon nanotubes with metallocenes for application in hybrid carbon nanotube/Si solar cells.
There is considerable interest in the controlled p-type and n-type doping of carbon nanotubes (CNT) for use in a range of important electronics applications, including the development of hybrid CNT/silicon (Si) photovoltaic devices. Here, we demonstrate that easy to handle metallocenes and related complexes can be used to both p-type and n-type dope single-walled carbon nanotube (SWNT) thin films, using a simple spin coating process. We report n-SWNT/p-Si photovoltaic devices that are >450 times more efficient than the best solar cells of this type currently reported and show that the performance of both our n-SWNT/p-Si and p-SWNT/n-Si devices is related to the doping level of the SWNT. Furthermore, we establish that the electronic structure of the metallocene or related molecule can be correlated to the doping level of the SWNT, which may provide the foundation for controlled doping of SWNT thin films in the future. | 2023-11-24T01:26:58.646260 | https://example.com/article/4059 |
Immunoblot using recombinant antigens derived from different genospecies of Borrelia burgdorferi sensu lato.
Immunodominant proteins are variable in molecular and antigenic structure among different genospecies of Borrelia burgdorferi sensu lato. We have recently developed an immunoblot using five recombinant antigens: the chromosomal-encoded B. burgdorferi proteins p100, the flagellin and an internal flagellin fragment thereof, and the plasmid-encoded outersurface proteins A (OspA) and C (OspC). In the present study the same antigens (derived from strain PKo, genospecies B. afzelii) were compared with the homologous recombinant proteins from strain B31 (genospecies B. burgdorferi sensu stricto) and with OspA, OspC and the internal flagellin fragment from strain PBi (genospecies B. garinii). Patients with neuroborreliosis (n = 28) and patients with acrodermatitis chronica atrophicans (n = 20) were investigated in the IgG immunoblot; the IgM immunoblot was performed only in patients with neuroborreliosis. There was a small increase in the detection rate of OspA-specific IgG or IgM antibodies using the different variants of recombinant OspA; however, OspA remained an insensitive antigen for antibody detection in Lyme borreliosis. The same was true to OspC-specific IgG antibodies. The sensitivity of OspC, which is the immunodominant antigen for IgM antibody detection, could not be increased using recombinant antigens derived from different strains. However, some sera which were negative in the recombinant immunoblot reacted with OspC in the conventional immunoblot using B. burgdorferi whole cell lysate as antigen. The most unexpected finding was the high degree of immunological heterogeneity of the internal flagellin fragments: IgG antibodies were detected in 18 of 48 patients using B31 fragments, in 25 of 48 using PKo fragments, in 23 of 48 using PBi fragments versus 33 of 48 when the three recombinant proteins were combined. PKo-derived fragments were more sensitive for antibody detection in patients with acrodermatitis chronica atrophicans, B31- and PBi-derived fragments for antibody detection in patients with neuroborreliosis. This is in agreement with the fact that isolates from patients with neuroborreliosis are predominantly belonging to the genospecies B. burgdorferi sensu stricto and B. garinii. For detection of IgM antibodies in sera from patients with neuroborreliosis, recombinant internal fragments derived from strains B31 and PBi were more sensitive than the PKo-derived fragment. The best discrimination between neuroborreliosis sera and control sera was achieved when the IgM blot was performed using recombinant internal flagellin fragments derived from strains PKo and PBi and OspC derived from B31 or PKo. | 2024-05-15T01:26:58.646260 | https://example.com/article/9240 |
Q:
Android Webview tel: link ERR_UNKNOWN_URL_SCHEME
I want to use tel: links in my website under webview, but when i click it i got this error message: ERR_UNKNOWN_URL_SCHEME.
I dont know what is the problem, everything is working fine, except this special link.
Here is my MainActivity.java:
import android.content.Intent;
import android.net.Uri;
import android.support.v7.app.AppCompatActivity;
import android.os.Bundle;
import android.view.KeyEvent;
import android.view.Window;
import android.view.WindowManager;
import android.webkit.URLUtil;
import android.webkit.WebSettings;
import android.webkit.WebView;
import android.webkit.WebViewClient;
public class MainActivity extends AppCompatActivity {
private WebView webView;
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
this.requestWindowFeature(Window.FEATURE_NO_TITLE);
this.getWindow().setFlags(WindowManager.LayoutParams.FLAG_FULLSCREEN, WindowManager.LayoutParams.FLAG_FULLSCREEN);
setContentView(R.layout.activity_main);
webView = (WebView) findViewById(R.id.webView);
WebSettings webSettings = webView.getSettings();
webSettings.setJavaScriptEnabled(true);
webSettings.setRenderPriority(WebSettings.RenderPriority.HIGH);
webView.loadUrl("http://mywebsite.com");
webView.setWebViewClient(new WebViewClient());
}
public boolean shouldOverrideUrlLoading(WebView view, String url) {
if( URLUtil.isNetworkUrl(url) ) {
return false;
}
Intent intent = new Intent(Intent.ACTION_VIEW, Uri.parse(url));
startActivity( intent );
return true;
}
@Override
public boolean onKeyDown(int keyCode, KeyEvent event) {
if (event.getAction() == KeyEvent.ACTION_DOWN) {
switch (keyCode) {
case KeyEvent.KEYCODE_BACK:
if (webView.canGoBack()) {
webView.goBack();
} else {
finish();
}
case KeyEvent.KEYCODE_MENU:
webView.loadUrl("javascript:open_menu()");
return true;
}
}
return super.onKeyDown(keyCode, event);
}
}
A:
Finally i solved the problem!
Here is the good code:
import android.content.Intent;
import android.net.Uri;
import android.support.v7.app.AppCompatActivity;
import android.os.Bundle;
import android.view.KeyEvent;
import android.view.Window;
import android.view.WindowManager;
import android.webkit.URLUtil;
import android.webkit.WebSettings;
import android.webkit.WebView;
import android.webkit.WebViewClient;
public class MainActivity extends AppCompatActivity {
private WebView webView;
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
this.requestWindowFeature(Window.FEATURE_NO_TITLE);
this.getWindow().setFlags(WindowManager.LayoutParams.FLAG_FULLSCREEN, WindowManager.LayoutParams.FLAG_FULLSCREEN);
setContentView(R.layout.activity_main);
webView = (WebView) findViewById(R.id.webView);
WebSettings webSettings = webView.getSettings();
webSettings.setJavaScriptEnabled(true);
webSettings.setRenderPriority(WebSettings.RenderPriority.HIGH);
webView.loadUrl("http://www.mywebpage.com");
webView.setWebViewClient(new WebViewClient() {
@Override
public boolean shouldOverrideUrlLoading(WebView view, String url) {
if( URLUtil.isNetworkUrl(url) ) {
return false;
}
Intent intent = new Intent(Intent.ACTION_VIEW, Uri.parse(url));
startActivity( intent );
return true;
}
});
}
@Override
public boolean onKeyDown(int keyCode, KeyEvent event) {
if (event.getAction() == KeyEvent.ACTION_DOWN) {
switch (keyCode) {
case KeyEvent.KEYCODE_BACK:
if (webView.canGoBack()) {
webView.goBack();
} else {
finish();
}
case KeyEvent.KEYCODE_MENU:
webView.loadUrl("javascript:open_menu()");
return true;
}
}
return super.onKeyDown(keyCode, event);
}
}
| 2024-07-11T01:26:58.646260 | https://example.com/article/6379 |
Nanban-ji
Nanban-ji (南蛮寺, also pronounced Nanbandera) is a name applied to spaces or structures used by Christian missionaries and Japanese Christian converts in the early History of the Catholic Church in Japan.
Whether converted from existing temples or built for purpose as churches and centers for Christian education, buildings known as Nanban-ji (temple of/for the southern barbarians) were present in Kyōto, Nagasaki, Hirado, Azuchi, Osaka, Kanazawa, Sunpu, and Edō. Using the term Deus for God, the temples were also called Daiusu-ji だいうす寺 and Daiusu-dō だいうす堂.
Structures known as Nanban-ji were destroyed from Toyotomi Hideyoshi's 1588 edict against Christians in Japan, with some fragments of construction remaining and eventually being deposited in museum. There are also depictions in contemporary art, and in the narratives of missionaries such as Luís Fróis.
See also
Christianity in Japan
Nanban art, Japanese art of the sixteenth and seventeenth centuries influenced by contact with the Nanban
Nanban trade, trade between Japan and Western countries from 1543 to 1614
Category:History of Christianity in Japan
Category:History of Catholicism in Asia
Category:Catholic Church in Japan
ja:南蛮寺
de:Namban-ji | 2024-01-29T01:26:58.646260 | https://example.com/article/3452 |
Dallas Cowboys Disney Minnie Mouse Hugger with Blanket
Dallas Cowboys Disney Minnie Mouse Hugger with Blanket
The Official Shop of the Dallas Cowboys brings you the cute and cuddly Dallas Cowboys Disney Minnie Mouse Hugger. This combo includes a cuddly, stuffed Minnie Mouse character pillow in a Dallas Cowboys dress that is 14 inches tall, PLUS a removable plush fleece Dallas Cowboys blanket that measures 40 by 50 inches. The plush fleece throw blanket features the Dallas Cowboys team name and logo, as well as Minnie in a cheer outfit cheering for your team!
Set comes with one stuffed animal pillow and one fleece blanket
Minnie is 14" tall and wears a pink dress with Cowboys logo and wordmark screen printed on front | 2024-02-29T01:26:58.646260 | https://example.com/article/6112 |
Q:
Python detect changing (global) variable
To make a somewhat long explanation rather simple for someone who's not fully into my project as me; I'm trying to find a way to detect a global variables that change in Python 2.7.
I'm trying to send updated to another device who registers them.
To reduce traffic and CPU load, instead of opting for a periodic update message, I was thinking of only sending an update message when a variable changes, and I might be tired right now but I don't know how I can detect a variable that changes.
Is there a library or something I can take advantage of?
Thank you!
A:
You could use the Pub/Sub feature of Redis if this kind of behaviour is typical in your codebase. https://redis.io/topics/pubsub. Every time your variable changes, you publish this event on a channel.
For example let's call the channel variableUpdates.
The devices that depend on your variable value subscribe to the channel variableUpdates.
Every time when your variable changes you publish this event on the channel variableUpdates.
When this happens, your listeners get notified of this event, read the new variable value and use it in their own context.
| 2023-09-19T01:26:58.646260 | https://example.com/article/8691 |
Q:
Disable logging completly SP2013
I have followed every guide there is, and also even tried to disable logging through this PowerShell command.
Set-SPUsageService -LoggingEnabled $false
Set-SPLogLevel -TraceSeverity None -EventSeverity None
But still my logs folder is spammed with log files.
C:\Program Files\Common Files\microsoft shared\Web Server Extensions\15\LOGS
https://technet.microsoft.com/en-us/library/ee663480.aspx
How can this be? Do I have to restart some kind of service in order for it to work?
A:
I just tried in my farm the below steps and its worked as expcted.
Set-SPLogLevel -TraceSeverity None
Set-SPLogLevel -EventSeverity None
Now all categories showing the none. and i checked the ULS logs, nothing after that.
Another way to stop it, their is one Timer services which is responsible for this called sharepoint tracing services, try to stop on all servers.
To Stop net stop SPTrace
To start net start SPTrace
| 2023-10-04T01:26:58.646260 | https://example.com/article/4268 |
NOT RECOMMENDED FOR PUBLICATION
File Name: 18a0110n.06
No. 17-5301
UNITED STATES COURT OF APPEALS FILED
FOR THE SIXTH CIRCUIT Mar 02, 2018
DEBORAH S. HUNT, Clerk
UNITED STATES OF AMERICA, )
)
Plaintiff-Appellee, )
) ON APPEAL FROM THE
v. ) UNITED STATES DISTRICT
) COURT FOR THE EASTERN
BROC KALON WHITFIELD, ) DISTRICT OF KENTUCKY
)
Defendant-Appellant. ) OPINION
)
)
BEFORE: GILMAN, ROGERS, and STRANCH, Circuit Judges.
JANE B. STRANCH, Circuit Judge. Broc Kalon Whitfield pled guilty to a single
count of distribution of cocaine base, in violation of 21 U.S.C. § 841(a)(1). The district court
sentenced Whitfield as a career offender under the United States Sentencing Guidelines (USSG)
§ 4B1.1(b) based on two prior controlled substances convictions and imposed a sentence of 198
months of imprisonment followed by an eight-year term of supervised release. Whitfield appeals
his sentence, arguing that his designation as a career offender violates the Fourteenth
Amendment’s due process and equal protection provisions. We AFFIRM.
I. BACKGROUND
Between April 21 and June 24, 2016, confidential informants acting on behalf of the
Kentucky State Police purchased a total of 85.822 grams of crack cocaine from Whitfield. In
total, six controlled buys occurred, and the largest single transaction involved just over an ounce
No. 17-5301
United States v. Whitfield
of crack cocaine.1 As a result of these transactions, Whitfield was indicted on September 1,
2016, and charged with two counts of distributing crack cocaine, in violation of 21 U.S.C.
§ 841(a)(1). Prior to sentencing, Whitfield submitted a pro se motion requesting the district
court “to consider a 1 to 1 crack to powder ratio.” In addition to Whitfield’s pro se motion,
defense counsel prepared a sentencing memorandum reiterating Whitfield’s arguments regarding
the crack-to-powder cocaine disparities and arguing that the relatively small quantities of drugs
involved in both the instant case and Whitfield’s prior offenses merited a downward variance.
At the sentencing hearing, Whitfield personally addressed the district court at length,
requesting the court to “waiver[ ] against applying the 4B1 career criminal enhancement or
depart downward for the category VI, final offense level 34.” Whitfield emphasized the
relatively small quantities of drugs involved in his offenses and implored the court to determine
that these circumstances warranted a downward departure. Defense counsel reiterated that the
drug transactions Whitfield had conducted involved small quantities and added that Whitfield
had a stable work history and sold drugs to pay for legal assistance in his battle to obtain
custody of his daughter and remove her from a dangerous situation.
The district court concluded that Whitfield was a career offender based on two prior
felony controlled substance offenses. In May 2011, Whitfield was found guilty of trafficking an
unspecified controlled substance in the second degree, in violation of Kentucky Revised Statutes
(Ky. Rev. Stat.) § 218A.1413, and sentenced to thirty months in prison. In January 2014,
Whitfield was sentenced to five years of imprisonment for trafficking cocaine in the first degree,
in violation of Ky. Rev. Stat. § 218A.1412. Id. Based on a total offense level of 34 and a
criminal history category of VI, Whitfield’s advisory Guidelines range was 262 to 327 months of
1
The transactions included: 3.428 grams sold on April 16, 2016; 2.211 grams sold on April 22, 2016; 3.304 grams
sold on May 4, 2016; 3.87 grams sold on May 9, 2016; 6.424 grams sold on June 9, 2016; 31.831 grams sold on
June 22, 2016; and 34.754 grams sold on June 24, 2016.
-2-
No. 17-5301
United States v. Whitfield
imprisonment. The district court applied a downward variance and imposed a sentence of 198
months of imprisonment. Whitfield filed this timely appeal of his sentence.
II. ANALYSIS
A. Standard of Review
Whitfield raises constitutional due process and equal protection challenges to his
sentence. “While constitutional challenges are typically reviewed de novo, when the argument
was not raised at the district court[,] ‘Sixth Circuit precedent requires application of the plain
error standard.’” United States v. Dedman, 527 F.3d 577, 591 (6th Cir. 2008) (quoting United
States v. Barton, 455 F.3d 649, 652 (6th Cir. 2006)). Whitfield argues that by raising a pro se
general objection to his career offender designation, his constitutional challenges to his sentence
were adequately preserved, warranting de novo review. Even under the liberal pleading
standards afforded pro se litigants, Whitfield’s constitutional challenges to his sentence and
designation as a career offender were not sufficiently raised before the district court. See United
States v. Houston, 792 F.3d 663, 666–67 (6th Cir. 2015) (requiring adequate specificity to
objections from pro se litigants). Therefore, we review Whitfield’s argument that his sentence
violates constitutional due process and equal protection requirements for plain error.
Under plain-error review, “the burden is on the defendant to show (1) [an] error that
(2) was plain, (3) affected defendant’s substantial rights, and (4) seriously affected the fairness,
integrity, or public reputation of the judicial proceedings.” United States v. Ushery, 785 F.3d
210, 218 (6th Cir. 2015) (citing United States v. McCreary-Redd, 475 F.3d 718, 721 (6th Cir.
2007)).
-3-
No. 17-5301
United States v. Whitfield
B. Constitutional Challenges
The gravamen of Whitfield’s argument is that because his prior conviction for trafficking
an unspecified amount of an unspecified substance would not constitute a § 4B1.1(b) qualifying
conviction in certain other states, his designation as a career offender violates the Constitution’s
equal protection and due process provisions. Whitfield reasons that had the identical trafficking
activity that led to his 2011 Kentucky conviction occurred across the state line in Ohio, the Ohio
drug offender would not have been convicted of a career criminal predicate offense. This is so
because in some states, such as Ohio, trafficking of smaller amounts of cocaine does not
constitute a felony offense carrying a term of imprisonment in excess of one year.2 Therefore, a
similarly situated Ohio defendant would not qualify as a career criminal, whereas Whitfield does,
despite engaging in identical conduct.
At the outset, we must clarify that Whitfield’s argument invoking the Fourteenth
Amendment’s due process and equal protection guarantees is instead properly grounded in the
Fifth Amendment, which is “applicable to the federal government.” United States v. Baker, 197
F.3d 211, 215 n.1 (6th Cir. 1999). Although the Fifth Amendment “does not explicitly guarantee
equal protection of the laws[,] . . . the United States Supreme Court has found that the Due
Process Clause of the Fifth Amendment encompasses an equal protection guarantee.” Id. (citing
Bolling v. Sharpe, 347 U.S. 497, 499 (1954)).
We have previously addressed Whitfield’s argument, albeit in an unpublished opinion.
See United States v. Smith, 681 F. App’x 483 (6th Cir. 2017). Smith held that “[t]he fact that
different states punish the possession of a certain amount of a controlled substance differently,
thus making the same conduct a predicate for a career-criminal enhancement for some
2
In Ohio, trafficking less than ten grams of cocaine is a fourth degree felony. Ohio Rev. Code § 2925.03(4)(c).
Pursuant to Ohio Rev. Code § 2929.13, non-violent first offenders convicted of fourth degree felonies shall be
sentenced to community corrections rather than imprisonment.
-4-
No. 17-5301
United States v. Whitfield
defendants but not for others, does not give rise to a constitutional challenge to the Guidelines.”
Id. at 490 (citing United States v. Kubosh, 63 F.3d 404, 407 (5th Cir. 1995), vacated on other
grounds).
The reasoning in Smith is persuasive. Under our federal system, “the States possess
primary authority for defining and enforcing the criminal law.” United States v. Lopez, 514 U.S.
549, 561 n.3 (1995). In crafting the federal sentencing Guidelines and substantive federal
criminal laws, Congress was well aware of the significant variations that existed in state criminal
law. See Kubosh, 63 F.3d at 407 (“Congress was well aware that different states classify similar
crimes differently. Congress’ deference to the states in this matter is not irrational.”).
Whitfield has also failed to articulate how the decision of Congress to rely on the
definitions of criminal conduct in the various states to determine career offender designations
implicates a suspect class or a fundamental right. We thus examine Whitfield’s equal protection
argument under rational-basis review. Baker, 197 F.3d at 216. This level of review “is highly
deferential to Congress’s judgment in enacting a particular statute. To survive rational basis
review, a statute need only be rationally related to a legitimate governmental interest.” Id.
Reliance on the states’ definitions of criminal conduct is deeply rooted in our federal system, in
which “[t]he States possess primary authority for defining and enforcing the criminal law.”
Brecht v. Abrahamson, 507 U.S. 619, 635 (1993) (internal quotation marks and citations
omitted). Congressional deference to state definitions of criminal conduct, moreover, promotes
comity and principles of federalism. Whitfield has failed to carry his burden of demonstrating
that Congress’s deference to state definitions of criminal conduct bears no rational relationship to
a legitimate government interest. His due process and equal protection arguments are therefore
unavailing.
-5-
No. 17-5301
United States v. Whitfield
C. Vagueness
Whitfield’s contention that his case raises an issue of “absolute vagueness” fares no
better. Whitfield argues that because one of his predicate offense convictions involves a
conviction that on its face neither specifies the quantity nor the controlled substance that was
trafficked, “one cannot really discern whether the offense involved a banned, controlled
substance in any jurisdiction, but for the fact that the only information in the notice of intent to
enhance and the [Presentence Investigative Report (PSR)] simply says so.”
The appropriate time to object to the factual basis of a PSR is before the district court,
prior to sentencing, which Whitfield failed to do. A “[d]efendant’s failure to raise any sort of
challenge in the proceedings below operates as an admission as to the drug types and quantities
set forth in the [PSR], and thereby provides the requisite factual basis to sustain” a defendant’s
conviction. United States v. Stafford, 258 F.3d 465, 476 (6th Cir. 2001). Under Federal Rule of
Criminal Procedure 32, the district court “may accept any undisputed portion of the presentence
report as a finding of fact.” Fed. R. Crim. P. 32(i)(3)(A). The district court’s obligation to make
factual findings under the preponderance of the evidence standard is triggered only when the
content of the PSR is disputed. Fed. R. Crim. P. 32(i)(3)(B); United States v. White, 492 F.3d
380, 415 (6th Cir. 2007) (“As a threshold matter, the defendant must actively raise the dispute
during the sentencing hearing before the district court’s duty to find facts arises.”). Because
Whitfield did not object to the PSR below, this obligation was not triggered, and the court
reasonably relied on the PSR.
Whitfield urges that his case is similar to United States v. Hernandez, 145 F.3d 1433
(11th Cir. 1998), in which the statute of conviction encompassed both purchase and sale of a
controlled substance, only the latter of which constitutes a controlled substance offense under the
-6-
No. 17-5301
United States v. Whitfield
Guidelines. The Eleventh Circuit reversed the district court because the latter utilized the arrest
affidavits rather than conviction documents such as the plea agreement and plea transcripts to
evaluate the defendant’s conduct. Id. at 1440. Hernandez is distinguishable from this case
because Whitfield’s PSR unambiguously makes references to the transcript of the sentencing
proceedings to determine that Whitfield pled guilty to trafficking a Schedule II substance. In
short, the district court did not err in determining that Whitfield’s prior crimes constituted career
offender predicate offenses.
Whitfield was a small-time drug dealer. The largest single transaction underlying his
conviction was barely more than an ounce of crack cocaine. We cannot conclude, however, that
Whitfield’s criminal record did not qualify him as a career offender as a matter of law.
III. CONCLUSION
For the reasons stated above, we AFFIRM.
-7-
| 2024-06-16T01:26:58.646260 | https://example.com/article/9246 |
WASHINGTON — U.S. nuclear power regulators are scaling back tougher safety rules proposed in the wake of Japan’s Fukushima nuclear disaster, saying additional measures imposed after the 2011 incident are sufficient.
In a 3-2 decision split along partisan lines, the Republican-majority Nuclear Regulatory Commission on Thursday voted to strike language from a proposed new regulation that would have required nuclear plants to plan for earthquakes, floods and other disasters of an intensity unimagined before the Japanese catastrophe, which was triggered by a tsunami. | 2023-10-12T01:26:58.646260 | https://example.com/article/7362 |
Friday, October 17, 2008
List Scalability
List scalability is a hot topic in the WSS/MOSS arena, and for good reason. Pretty much everything in SharePoint is stored in a list. Knowing if you're likely to hit a wall with scalability is a proactive step that can save you a tonne of time (and help you decide if a List is an appropriate container in the first place).
Rules of Thumb
When it comes to putting a tonne of data into a SharePoint list I would consider the following guidelines:
No more than 2000 items per view/container. That could be 1900 items and 100 folders but either way, no more than 2000 per view. A folder is an example of a container, so is the "root" level of a list.
No more than 5 million items per list. This is the max, it's only really do-able by nesting many folders together which in turn hold more items.
No more than 2000 lists per sub site (SPWeb).
Methods of Access
It's also worth noting that depending on how you access data from a list you can get dramatically different performance (especially for large data sets). The following white paper from Microsoft (which was described as a must read by Joel Oleson) compares the performance of getting data out of lists with the following techniques:
SPList with For/Each
SPList with SPQuery
SPList with DataTable
List Web service
Search
PortalSiteMapProvider (yes you can query a list with a portal site map provider...who would have known).
For those simply to busy/lazy to give the article a read, I'll try say that in general getting data from the PortalSiteMapProvider is worth looking into (since it's cached) if the data isn't going to be changing that often.
Search and SPList with SPQuery were also strong performers when it came to getting data from lists with 100k+ items.
What's more important is that SPList with For/Each and SPList with DataTable (both below) just don't scale well for large lists.
//Doesn't scale even if you use a data viewforeach (DataRow in items){//do something clever}
You can probably see what the above codes have in common, they both get EVERYTHING out of the list which is a big no, no. What scales a lot better is retrieving a subset of what you're after from the view with an SPQuery, or other like minded code.
Know The Limits
SharePoint will a LOT of work for you, but there's a balance that needs to be observed. Knowing when the platform is at its limits is key to differentiating what the platform can and cannot do for a project.
Hopefully the above code will help you get a little more out of your lists. If you're more interested in general scalability (how many lists to a site, how many sites to a site collection etc...) there's a great TechNet article here on the topic.
1 comment:
Hi. Check out my blog series about the same topic @ http://blog.dynatrace.com/category/net/sharepoint-net/Here I actually show what is going on the database layer which explains why some of the approaches do not scale
About Me
Tyler Holmes is a Solutions Architect working in Portland, Oregon. He lives mostly in the MS tech stack and is currently treading the waters of Communication/Collaboration and Business Intelligence with off the shelf/open source technologies. | 2024-05-07T01:26:58.646260 | https://example.com/article/9937 |
Why Nepal will not suffer from the Covid-19 Pandemic like the west?
Home » Blog » Why Nepal will not suffer from the Covid-19 Pandemic like the west?
Why Nepal will not suffer from the Covid-19 Pandemic like the west?
Nepal is a tiny country compared to its giant neighbors regarding any matters. There are very less cases of COVID-19 in Nepal. Till now, 21st of April as I am writing this blog post the underlying reasons why Nepal likely to suffer less from the global pandemic COVID-19, we got 31 cases so far. Those cases are also imported cases from outsiders and they found very less transmission even to the nearest of the person who found positive for COVID-19. There are certain factors which are playing vital role which helps for less spreading of the Corona. The patients with the cases we got at the beginning are already discharged from the hospital. We have lot of traditional practices that saves us and our practices are different than the westerners. The major reasons are listed below.
Traditional habits of eating with hands
Many Nepalese households outside the big cities people don’t have spoon and forks and that is very common for families not having that. Even if they have spoon and forks they don’t use them. They eat with their hands all the time. Before eating they have a culture of washing their hands and even face. Traditionally many families even they were changing the dress to go to kitchens and eat. This culture put us on cleaning our hands frequently. The cooking is done with the firewood; the ash is used for cleaning if they don’t get soap. Even the red clay is utilized for all cleansing purpose. After defecation also they use only water with soap for cleaning and they clean their hands again too.
Pre-exposure to unhygienic condition
Majority of the Nepalese even who are living in big cities right now are exposed to unhygienic condition back in the village because these people lived and grown up in the village between and among the domestic animals. They have been played in the mud, swamps and dust made them more resistant towards the any microorganism. We handled cow dungs, buffalo dungs and other type of animal excreta and used them as manure in our farmland. As handling all these thing made us stronger against any type of external organism.
Spices that boost our immune system
The basic spice in the Nepalese kitchen is the cumin, coriander, turmeric, garlic, ginger and onions. Besides these basic spices many Nepalese kitchen have fenugreek, black pepper, cardamom, cloves, chilies and many others. All these spices help us to boost our immune system. With boosted immune system we can handle the microorganism quiet well. Lots of households still use traditional medication for simple illnesses like flues. If somebody suffers from cold and flu they make a drink out of turmeric, ginger, lemon and honey, which is also an immune system booster.
Culture of cooking fresh foods every meal
We have to cook foods for every meal. We don’t use frozen foods that much. Even the meat we use fresh. In western societies they have a culture once they cook, cook for a week and at the time of eating they just warm them but we do cook for every meal, a fresh food. We buy fresh vegetables every day and we consume very less meat. The research showed the virus originated from see food market in Wuhan.
Coming summer with high temperature
They claim the virus will not survive in relatively high temperature. As we are in northern hemisphere and temperature rises every day it is less likely that virus spread fast.
Pre-exposure to deadly disease
We have suffered from Malaria, Chicken pox, Small pox, Tuberculosis, Leprosy, Dengue and many other deadly diseases in the past. With pre-exposures to these deadly diseases made us stronger and help us handle well with COVID-19.
BCG Vaccine, the game changer in the scenario
BCG vaccine is the mandatory vaccine for all children and they get vaccinated soon after their birth. Researches proved that who got this vaccine would suffer less from COVID-19. As almost everybody got this vaccine in Nepal this would help to cope against the virus quiet well.
Lock Down already
Nepal is already in lock down for more than four weeks. They probably are going to continue as a preventive measure. As everybody knows there are no infrastructures in the hospitals and if lot of people flood to the hospital there will be severe crisis so public are obeying the lock down order by government. Public are now already aware of social distancing and self-isolation and its importance to mitigate the hazards of COVID-19.
No mass transport and nightlife
Unlike western countries Nepal does not have any means of mass transport like metro, rails in the city that will lower down the spread rate drastically. The transport here is a motorbike and small microbus so spreading is limited to insignificant population. We have no night clubs and discos besides many Nepalese people cannot afford to eat outside so that helps to keep the number much lower and spread very slow.
Prompt healthcare and environmental factors
To avoid pandemic, whoever suspected are kept in self-isolation until the test is done. If the result of the test shows negative with Corona than they can go for home quarantine. Hospitals are prepared to handle limited number of people but till now they are quiet prompt. As most of the Nepalese live in solitary houses with enough exposure to sunshine and even most of the offices not furnished with air conditioning they are quiet prone to Pandemic.
As the tourism and travel trade stopped totally we are waiting to welcome to our valuable clients in this mysterious and beautiful land. Hope this will be over soon and people start moving again. The people from the land of Buddha and Everest eagerly waiting to see the world will go normal emotionally, financially and socially. You will be greatly appreciated if you share this to people who are interested to visit this marvelous country after pandemic is over. | 2024-01-16T01:26:58.646260 | https://example.com/article/1253 |
Novel Genes/Genetic Defects in IRDs Our WES analyses have led to identification of two novel IRD genes providing insights into retinal pathways and function. (1) We identified 7 different variants in IDH3A in 4 unrelated families as a cause of Retinitis Pigmentosa (PMID 28412069) (2) We also reported mutation in REEP6, a rod-specific gene in patient with RP and showed in mouse model that REEP6 mediates trafficking of a subset of Clathrin-coated vesicle critical for rod photoreceptor function and survival (PMID 28369466). Additionally, we collaborated with Sam Jocobson to characterize structure, function and disease progression in patients with autosomal recessive RP caused by EYS mutation (PMID 28704921) Genetics of AMD Age-related macular degeneration (AMD) is a leading cause of blindness in the developed world. While many AMD susceptibility variants have been identified, their influence on AMD progression has not been elucidated. We were involved in several different projects, as follows. Using data from two large clinical trials, Age-Related Eye Disease Study (AREDS) and AREDS2, we evaluated the effects of 34 known risk variants on disease progression. We show that for prediction of AMD progression, addition of GRS to the demographic/environmental risk factors considerably improved the prediction performance. Our model for predicting the disease progression risk demonstrated satisfactory performance in both cohorts, and we recommend its use with baseline AMD severity scores plus baseline age, education level, and smoking status, either with or without GRS (PMID 28341650). We are now extending this analysis to genome-wide bivariate time-to-event test for AMD progression with 9 million variants on 2,721 AREDS participants (Yan et al., submitted) We also participated in studying the genetic pleiotrophy between AMD with other complex diseases. We demonstrate a substantial overlap of the genetics of several complex diseases/traits with AMD and provide statistically significant evidence for an additional 20 loci associated with AMD. This highlights the possibility that so far unrelated pathologies may have disease pathways in common (PMID 28347358). To discover additional rare variants and characterize the GWAS locus further, we performed whole genome sequencing of 2,394 cases and 2,393 controls and the analysis is in progress. We have looked into the contribution of rare variants in GWAS loci of sub-types of AMD (Pietraszkiewicz et al., submitted). We have also participated in a deep phenotype association study in AREDS2 reveals specific phenotype association with genetic variants involved in AMD. This shows the association of the SNP at the ARMS2/HTRA1 locus with subretinal/sub-RPE hemorrhage and poorer visual acuity and of SNPs at CFH locus with drusen area may provide new insights in pathophysiological pathways underlying different stages of AMD (Freekje et al., submitted) We also generated WES data in 19 large multigenerational AMD families to ascertain high-penetrance causative allele(s). To enhance the power of analysis, we are collaborating with other groups to extract useful information. Functional Genomic Analysis of AMD Current variants/loci can explain 50-60% of AMD heritability, with ARMS2 and CFH accounting for bulk of the effect. Our collaborative studies suggest increased retinal mitochondrial DNA damage in patients with CFH risk alleles (Ferrington et al. 2016). However, causal variants and underlying mechanisms remain largely unknown for majority of the loci. Thus, we are now focusing on dissection of relative contributions of variants, genes and pathways to AMD pathology. A Reference Transcriptome of Adult Human Retina We have generated a comprehensive reference transcriptome of the normal human retina by RNA-seq analysis of 105 healthy donor samples (from Dr. D Ferrington) We have characterized both annotated coding regions as well as un-annotated transcripts using de novo assembly. This dataset would be valuable for the vision community while designing interventions for retinal diseases. Gene and Pathways Associated with AMD Progression To elucidate genetic underpinnings of progression of AMD phenotypes, AREDS1 data is being examined in collaborative studies. However, lack of longitudinal clinical information in most datasets makes it harder to replicate our findings. Thus, we are taking advantage of transcriptomic studies of 390 donor AMD retinas (197 early, 127 intermediate, 66 advanced). Differential expression, pathway and co-expression analysis has revealed the core transcriptome signature and pathways dysregulated in AMD, providing several novel candidates and insights into the AMD pathobiology. A comprehensive resource of eQTL in retina interpretation of the GWAS findings, and application in risk evaluation and therapeutic interventions remains one of the major challenges for researchers. A large proportion GWAS signals reside in the non-coding region and thus underlying genes/variants are not conspicuous and are most likely to mediate their effect through regulation. To understand this regulation, we used the DNA from 523 donor subjects and genotyped 603,583 markers using UM HUNT Biobank v1.0 chip then obtained the inferred genotypes for 6,554,241 variants using 1000G reference panel through imputation. After adjusting for LD structure, this resulted in 41,364 cis and 8,165 trans eQTL, several of which are in AMD GWAS loci and can help elucidate the causal variants. | 2023-09-07T01:26:58.646260 | https://example.com/article/6329 |
Story highlights Donald Trump leads Marco Rubio by 8 percentage points in Florida, a new poll shows
Florida holds its critical winner-take-all primary on March 15
(CNN) Donald Trump is leading Marco Rubio in the Florida senator's home state by 8 percentage points, a Monmouth University poll out Monday shows.
Trump has the support of 38% of Florida's likely GOP primary voters, compared to 30% who back Rubio, 17% for Texas Sen. Ted Cruz and 10% for Ohio Gov. John Kasich.
The snapshot of Florida comes before the state's crucial 99-delegate, winner-take-all primary set for March 15.
It's a must-win for Rubio, who has fallen behind Trump and Cruz in the delegate race after taking only two of the first 20 Republican contests.
Read More | 2023-09-25T01:26:58.646260 | https://example.com/article/1703 |
President Trump Donald John TrumpUS reimposes UN sanctions on Iran amid increasing tensions Jeff Flake: Republicans 'should hold the same position' on SCOTUS vacancy as 2016 Trump supporters chant 'Fill that seat' at North Carolina rally MORE floated the idea of arming teachers in an effort to prevent future school shootings during a listening session on school shootings at the White House on Wednesday.
"If you had a teacher who was adept at firearms, they could very well end the attack very quickly, and the good thing about a suggestion like that — and we're going to be looking at it very strongly, and I think a lot of people are going to be opposed to it. I think a lot of people are going to like it. But the good thing is you're going to have a lot of [armed] people with that," the president said.
He said the "coach," presumably meaning athletic director Chris Hixon, who was killed during the Florida shooting last week, "saved a lot of lives" but could have saved more if he had a gun.
"He wouldn't have had to run, he would have shot, and that would have been the end of it," he said, adding that he only supported concealed carry for people "adept" with guns.
"And there are many of them," he added.
He also criticized gun-free zones around schools.
"A gun-free zone to a maniac, because they're all cowards, a gun-free zone is 'let's go in and attack,' " he said. "I really believe if these cowards knew that the school was well-guarded from the standpoint of pretty much having professionals with great training, I think they wouldn't go into the schools to start with, it would pretty much solve your problem."
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Trump then surveyed the room, asking how attendees felt about the idea.
"So does anybody like that idea here, does anybody like it?" he said. "Do people feel strongly against it, anybody? Anybody? Strongly against it? We can understand both sides. Certainly, it's controversial, but we'll study that along with many other ideas."
Some attendees voiced their opposition to the idea.
"Schoolteachers have more than enough responsibilities right now than to have to have the awesome responsibility of lethal force to take a life," said Mark Barden, whose son was killed in the 2012 Sandy Hook Elementary School shooting. Some reacted with applause.
"Nobody wants to see a shoot-out in school. And a deranged sociopath on his way to commit an act of murder in a school with the outcome, knowing the outcome is going to be suicide, is not going to care if there is somebody there with a gun," he continued.
The listening session was comprised of survivors and loved ones of victims from last week's shooting at Florida's Marjory Stoneman Douglas High School, the Connecticut Sandy Hook Elementary School shooting and the Colorado Columbine High School shooting.
Vice President Pence and Education Secretary Betsy DeVos Elizabeth (Betsy) Dee DeVosNEA president says Azar and DeVos should resign over school reopening guidance The Hill's 12:30 Report - Presented by Facebook - You might want to download TikTok now Former DeVos chief of staff joins anti-Trump group MORE also attended the session.
Last week's shooting in Parkland, Fla., left 17 people dead and numerous others injured.
The shooting has reignited the U.S. debate on gun control.
Students across the country staged walkouts on Wednesday to demand action on gun laws from lawmakers, while survivors of the shooting visited Florida's capital in Tallahassee with hopes of lobbying state lawmakers to take action on gun control.
Updated at 5:55 p.m. | 2024-05-25T01:26:58.646260 | https://example.com/article/5155 |
Q:
Python string capture
I have the below sample paragraph:
Some paragraph contents
email address: 1234532@aol.com
seq_id : 1234567
The seq_id line may contain the below possible patterns. There will always be : with prefix of
seq, SEQ, seq_id, SEQ_ID, etc., and I just need the digits after the :
which is 1234567, but not the email id which may have the same or different digits.
seq id:1234567
seq_id : 1234567
seq_iD : 1234567
seq_iD:1234567
seq_ID: 1234567
So far I tried to store this as a list and was checking if a word is a digit but that's returning true for email Id data as well.
Sometimes I get id:1234567 as a word which makes it not get detected as a digit.
Is there any better way to get this done?
A:
You could do this with a regular expression.
import re
s = "some string or data input, in this case your paragraph"
re.match('seq(?:[_\s]id\s?):\s?(\d+)', s, flags=re.IGNORECASE).group(1)
This ignores case, then looks for optinal spacing just about everywhere and puts all the digits into a group which is returned by group(1)
You can test the regex and see it work interactively:
Update to handle missing spaces per request:
For example: Seq Id:1234567 doesn't work with the regular expression above.
Change the regex to: seq(?:[_?\s?]id\s?):\s?(\d+)
| 2024-04-29T01:26:58.646260 | https://example.com/article/7815 |
Braised Pork Ribs with Soya Sauce and Rock Sugar
Ran out of ideas of what to cook the other day. Yes, I too have those days when I try my hardest to WILL food into existence just by a blink of an eye. Had some pork ribs in my trusted freezer but no recipes found in my head.
Decided to take a chance. Thankfully it turned out pretty good and I might just whip it up again sometime.
Marinade pork ribs in half the soy sauces for a few hours in the fridge.
Heat up a non-stick saucepan and drizzle in olive oil.
Fry garlic cloves for 1 minute. Add spices and fry another minute.
Put in the ribs and stir till the ribs are slightly brown, lower heat, then add in the remaining soya sauces, rock sugar and water. Stir until sugar dissolved, simmer for 20 minutes on low heat uncovered and stirring from time to time. Â This is to let some liquid evaporate thus creating a thicker gravy.
Cover with a lid and simmer for another 40 – 50Â minutes. Checking frequently to avoid gravy from totally drying up and burning the contents. If gravy is rapidly drying up, add a little more hot water and stir.
When almost ready, put in eggs and stir until well coated with gravy.
Ribs are ready when thoroughly cooked through, meat is tender, ribs well coated with thick gravy. | 2023-12-15T01:26:58.646260 | https://example.com/article/9233 |
Q:
PyQt5 - Position a widget at top right corner of a QTextEdit
I'm developping a text editor with pyqt5 and I want to implement a Find Box which sticks to the top right corner of my textarea just like this:
image
textarea = QTextEdit()
layout = QHBoxLayout() # tried hbox, vbox and grid
find_box = QLineEdit()
empty = QTabWidget()
layout.addWidget(empty)
layout.addWidget(empty)
layout.addWidget(find_box)
textarea.setLayout(layout)
So with this code, I managed to have my Find Box stick to the left of my texarea, even when window gets resized. But somehow the y position of my textarea's layout starts from the middle:
image
An awful solution is to set the textarea as my Find Box parent, use move(x, y) to set the Find Box's position but I'll have to catch whenever my window or my textarea get resized and use move() again to set a new position.
So why my QTextEdit's layout starts from the middle? And is there anyway to avoid this?
A:
I got it working by using a gridlayout and setting stretch factors to rows and columns.
from PyQt5.QtWidgets import *
import sys
class Wind(QWidget):
def __init__(self):
super().__init__()
self.setupUI()
def setupUI(self):
self.setGeometry(300,300, 300,500)
self.show()
text_area= QTextEdit()
find_box = QLineEdit()
# this layout is not of interest
layout = QVBoxLayout(self)
layout.addWidget(text_area)
# set a grid layout put stuff on the text area
self.setLayout(layout)
text_layout= QGridLayout()
# put find box in the top right cell (in a 2 by 2 grid)
text_layout.addWidget(find_box, 0, 1)
# set stretch factors to 2nd row and 1st column so they push the find box to the top right
text_layout.setColumnStretch(0, 1)
text_layout.setRowStretch(1, 1)
text_area.setLayout(text_layout)
def main():
app= QApplication(sys.argv)
w = Wind()
exit(app.exec_())
if __name__ == '__main__':
main()
| 2023-11-15T01:26:58.646260 | https://example.com/article/4388 |
Sarah Horwich
THRIVE = Less Waste + More Money
THRIVE saves my family money and time, it is as simple as that. THRIVE allows me to use just the amount of whatever I need for a recipe, whether it's meat or veggies or fruit. No more throwing out limp celery because I needed only one stalk. No more last minute runs to the grocery to get kid-friendly foods. No more cooking, cooling, then chopping chicken. No more splatter clean ups from frying burger or sausage.
The kids love to snack on the yogurt bites and any of the fruits and I don't have to worry about rotten fruit or spoiled dairy from my once-a-week trip to the grocery. In our household we try not to eat a lot of processed foods and strive to eat whole and clean. THRIVE make that easier and less stressful as well. | 2023-08-25T01:26:58.646260 | https://example.com/article/3466 |
Crater Historic District
The Crater Historic District encompasses National Park Service structures within Haleakala National Park. The buildings include utility structures, employee housing, administration facilities and visitor services facilities. Most were built by the Civilian Conservation Corps to standard Park Service designs in the 1930s. A few World War II era buildings survive from U.S. Army construction, and are included in the historic district. The Crater Historic District was added to the National Register of Historic Places on November 1, 1974.
The Park Service adapted its preferred National Park Service rustic style to the Hawaiian Islands, avoiding the heavy log construction characteristic of the western continental United States parks in favor of a frame-construction interpretation for most buildings. The House of the Sun Visitor Center stands as the closest example of the mainland style of rubble construction with heavy framing. Designed by Park Service architect Merel Sager, it is also one of the few buildings that were not built with CCC labor.
References
Category:Historic districts on the National Register of Historic Places in Hawaii
Category:Geography of Maui County, Hawaii
Category:Haleakalā National Park
Category:Civilian Conservation Corps
Category:Rustic architecture in Hawaii
Category:History of Maui
Category:Protected areas established in 1974
Category:1974 establishments in Hawaii
Category:National Register of Historic Places in Maui County, Hawaii | 2023-09-17T01:26:58.646260 | https://example.com/article/7528 |
Q:
Could you please solve this problem?
15 persons are arranged in a row. Find the number of ways of selecting 6 persons so that no two persons sit next to each other.
I think we have to select 6 persons from either the group of 8 people sitting 1st, 3rd, 5th.....15th or the group of 7 people sitting 2nd, 4th...14th.
I tried to solve this way but I'm not getting the required answer. Please some one help me.
A:
If you are selecting $6$ persons,
$9$ persons are left behind, and the selections must have been made from the $10$ gaps (see diagram)
$-P-P-P-P-P-P-P-P-P-$
thus $\binom{10}{6}$ ways
| 2023-10-06T01:26:58.646260 | https://example.com/article/6732 |
Dagen McDowell
In an extraordinary moment on Fox News yesterday, host Dagen McDowell openly blurted out what Trump supporters secretly worry about when assessing a potential Robert Mueller interview with president Trump. What if he lies on record? "How in the world could [Trump] ever cooperate ...read more | 2024-01-07T01:26:58.646260 | https://example.com/article/8263 |
Don Shula Autographed Miami Dolphins Mini Helmet (17-0)
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Don Shula Autographed Miami Dolphins Mini Helmet (17-0)
Don Shula signed Miami Dolphins Mini Helmet. Includes inscription "17-0 Perfect Season". Comes with a COA from Dave & Adam's Card World. Also is accompanied by a signing ticket from the Collector's Showcase of America show that it was signed at. Please note the decal on the signed side of the helmet has started to peal slightly. | 2023-08-21T01:26:58.646260 | https://example.com/article/5725 |
Speckled lentiginous naevus: which of the two disorders do you mean?
Speckled lentiginous naevus (synonym: naevus spilus) no longer represents one clinical entity, but rather, two different disorders can be distinguished. Naevus spilus maculosus is consistently found in phacomatosis spilorosea, whereas naevus spilus papulosus represents a hallmark of phacomatosis pigmentokeratotica. The macular type is characterized by dark speckles that are completely flat and rather evenly distributed on a light brown background, resembling a polka-dot pattern. In contrast, naevus spilus papulosus is defined by dark papules that are of different sizes and rather unevenly distributed, reminiscent of a star map. Histopathologically, the dark spots of naevus spilus maculosus show a 'jentigo' pattern and several nests of melanocytes involving the dermoepidermal junction at the tips of the papillae, whereas most of the dark speckles of naevus spilus papulosus are found to be dermal or compound melanocytic naevi. The propensity to develop Spitz naevi appears to be the same in both types of speckled lentiginous naevus, whereas development of malignant melanoma has been reported far more commonly in naevus spilus maculosus. | 2024-03-20T01:26:58.646260 | https://example.com/article/5062 |
Introduction
============
*Clostridium difficile* (*C. difficile*) is increasingly being recognized as a major cause of gastrointestinal infections worldwide, with 70%--80% of *C. difficile* infections (CDIs) occurring in adults aged 65 and older.[@b1-cia-12-1799]--[@b3-cia-12-1799] The inciting agent *C. difficile* is a ubiquitous anaerobic, spore-forming, Gram-positive bacterium. The elderly are especially vulnerable to CDI.[@b4-cia-12-1799] Indeed, reducing the incidence of CDI in this population is crucial because of the significant morbidity, mortality, and financial cost associated with this infection.[@b5-cia-12-1799] There are a number of therapeutic agents in development and currently being utilized for CDI, including antibiotics, probiotics, fecal transplantation therapy, antibody-based immunotherapy, and vaccines.[@b6-cia-12-1799]--[@b9-cia-12-1799] In this article, we review the epidemiology of CDI, discuss risk factors, and outline current and emerging therapeutic options as it pertains to the geriatric population.
Pathogenesis and epidemiology
=============================
The pathogenesis of CDI lies in the dysregulation of the normal indigenous gastrointestinal microbiota typically secondary to systemic antimicrobial use.[@b10-cia-12-1799],[@b11-cia-12-1799] The histopathologic hallmark of CDI is damage to the mucosal epithelial cell lining with generation of an acute, neutrophil-predominant inflammatory response and the formation of pseudomembranes.[@b10-cia-12-1799],[@b12-cia-12-1799] Damage to the epithelium is caused by *C. difficile* virulence factors, the glucosyltransferase toxin A (TcdA) and toxin B (TcdB). The clinical manifestations of CDI range from mild diarrhea to life-threatening conditions such as pseudomembranous colitis and toxic megacolon. It should be noted, however, that *C. difficile* burden varies dramatically by geographic region, between institutions, and even between units of the same hospital.[@b12-cia-12-1799],[@b13-cia-12-1799]
Over the last few decades there has been a dramatic rise in CDI incidence. Rates of CDI tripled in the USA and Canada.[@b1-cia-12-1799],[@b14-cia-12-1799] Of great concern is the fact that severe and fatal CDI predominantly affects elderly, nursing home patients, and those with poor functional status.[@b1-cia-12-1799],[@b15-cia-12-1799] A 2015 report from the Center for Disease Control and Prevention noted that one out of every three CDIs occurs in patients 65 years or older and two out of every three health care--associated CDIs occur in patients 65 years or older.[@b16-cia-12-1799] Indeed, CDI hospitalization rates were approximately fourfold for adults 65--84 years old and tenfold for adults ≥85 years old compared to adults 45--64 years old utilizing data from the Healthcare Cost and Utilization Project.[@b17-cia-12-1799] Another study found that US rates of hospital discharges with CDI increased from \~5 per 1,000 discharges in 2,000 to greater than 10 per 1,000 discharges in 2008; increases were especially prominent among those ≥65 years of age ([Figure 1](#f1-cia-12-1799){ref-type="fig"}).[@b3-cia-12-1799]
According to national mortality data records, *C. difficile*-related deaths in the USA rose from 5.7 deaths per million in 1999 to 23.7 in 2004 with a median age of death reported as 82 years.[@b18-cia-12-1799] The substantial increase in CDI incidence has been primarily attributed to the emergence of a more virulent strain categorized as North American pulsed-field 1/PCR-ribotype 027 (NAP1/BI/027). NAP1/BI/027 virulence is characterized by increasing fluoroquinolone resistance, production of binary toxin, increased toxin production, and higher sporulation rates.[@b1-cia-12-1799],[@b12-cia-12-1799]
The aging host: risk factors for CDI
====================================
Several prospective and retrospective trials have looked into risk factors, including advanced age, as being contributors to the development and severity of CDI. The three main factors are exposure to systemic antimicrobial therapy for other infections, exposure to *C. difficile* spores, and the host immune response ([Table 1](#t1-cia-12-1799){ref-type="table"}).[@b1-cia-12-1799],[@b2-cia-12-1799],[@b10-cia-12-1799],[@b19-cia-12-1799]--[@b23-cia-12-1799]
The risk of CDI is the highest during systemic antimicrobial therapy and in the first month after cessation of antimicrobial therapy thereafter.[@b1-cia-12-1799] Antimicrobials that pose the greatest risk of CDI are clindamycin, cephalosporins, and fluoroquinolones, and to a lesser frequency macrolides and sulfonamides. A meta-analysis identified that fluoroquinolone use and age over 65 years were associated with a higher risk of CDI because of the NAP1/BI/027 strain.[@b23-cia-12-1799],[@b24-cia-12-1799] Studies also suggest probable association between proton-pump inhibitor (PPI) use and incident and recurrent CDI. In a 15-month prospective Canadian cohort study, Loo et al found that older age, use of antibiotics, and use of PPI were significantly associated with health care--associated CDI. Specifically, the authors found that for each additional year of age \>18 years, the risk of health care--acquired CDI increased by 2% (odds ratio \[OR\] 1.02; 95% CI 1.00--1.04).[@b25-cia-12-1799] Among the many risk factors for CDI, the most readily modifiable is antimicrobial utilization. In the USA, 25%--75% of antibiotic prescriptions for long-term care residents have been found to be inappropriate.[@b26-cia-12-1799] Undeniably, reducing antimicrobial use also reduces CDI rates. For example, an effort to improve antimicrobial utilization and stewardship at a Veterans Affairs (VA) long-term care facility (LTCF) resulted in an infectious disease consult service achieving a 30% reduction in antimicrobial use, which correlated with a significant decrease in the rate of positive *C. difficile* tests.[@b27-cia-12-1799] Advanced age and receipt of non-CDI anti-microbials during or after CDI treatment were significantly associated with CDI recurrence.[@b28-cia-12-1799],[@b29-cia-12-1799] The validated results of a prediction tool by Hu et al[@b29-cia-12-1799] consistently predicted CDI recurrence in patients with three clinical factors: age \>65 years, severe or fulminant underlying illness (assessed by Horn Index), and additional antimicrobial use after initial CDI treatment. Median age of patients in the cohort was 69 years.[@b29-cia-12-1799]
Host factors are also important CDI risks, with advanced age, immunosuppression, prior hospitalization, and severity of underlying illness contributing to an increased risk. Aging alters important physiologic barriers to infection, ranging from changes in genitourinary physiology that impairs bladder function to decreased gastrointestinal microbial diversity.[@b30-cia-12-1799] In addition, the complex changes in the immune system related to advancing age, collectively called immunosenescence, play a key role in increased susceptibility in the elderly. Immunosenescence has been associated with a decrease in T-cell and B-cell counts as well as a decline in cell function.[@b31-cia-12-1799]--[@b33-cia-12-1799] This age-related pathophysiology enhances morbidity and mortality risk as it limits the ability of older adults to respond to microbes. Indeed, older adults have been shown to exhibit an increase in incidence of infections compared to their younger counterparts.[@b30-cia-12-1799]
In addition, decreased functional status is increasingly being recognized as an important and independent risk factor for poor outcomes among older adults, further enhancing the risk and severity of infections.[@b15-cia-12-1799],[@b34-cia-12-1799] Utilizing an assessment of activities of daily living prior to hospitalization and at onset of CDI, Rao et al identified impaired functional status as an independent risk factor for severe CDI in patients 50 years and older.[@b15-cia-12-1799]
Therapeutic agents
==================
The management of CDI involves three basic principles: 1) supportive care with fluid and electrolyte replacement, 2) discontinuation of the precipitating antimicrobials when appropriate, and 3) the initiation of effective anti-*Clostridium difficile* therapy. The drugs available in the USA for the treatment of CDI are listed in [Table 2](#t2-cia-12-1799){ref-type="table"}.
The goals of successful treatment are the elimination of symptoms and the prevention of recurrent CDI. Currently, CDI treatment regimens depend on severity of CDI and if the presentation is an index or recurrent episode ([Table 3](#t3-cia-12-1799){ref-type="table"}).[@b19-cia-12-1799]--[@b21-cia-12-1799] While certainly a consideration for severe CDI, treatment recommendations are not currently stratified by patient age.
Metronidazole and vancomycin
============================
Early studies suggested that oral metronidazole and oral vancomycin had equivalent efficacy, with similar tolerability.[@b35-cia-12-1799] Newer data suggest higher treatment failure rates when metronidazole is used in severe or complicated CDI.[@b36-cia-12-1799]--[@b38-cia-12-1799] In the first randomized controlled trial (RCT) comparing vancomycin to metronidazole for the treatment of CDI, vancomycin therapy was superior to metronidazole therapy overall, but this treatment benefit was limited to patients with severe disease. Approximately half of the study participants (N=150) were older than 60 years (47%). While age was not evaluated in subgroup analysis, patient characteristics that were statistically more common in the metronidazole treatment failure group were a low albumin level, admission to the intensive care unit, and the presence of pseudomembranous colitis on endoscopic examination.[@b37-cia-12-1799] In response to metronidazole's lower drug cost, vancomycin efficacy data, and a theoretical risk of promoting vancomycin-resistant enterococci, major guidelines consider oral metronidazole as the primary agent for only mild-to-moderate CDI.[@b19-cia-12-1799]--[@b21-cia-12-1799],[@b37-cia-12-1799] Of note, vancomycin is also inexpensive if the intravenous form of the drug is formulated for oral administration.
Tolevamer is a toxin-binding polymer that neutralizes the effects of *C. difficile* toxins A and B in vitro. Despite encouraging early-phase results, tolevamer failed to meet its primary endpoint of noninferiority to vancomycin in Phase III clinical trials.[@b39-cia-12-1799] In these Phase III trials comparing tolevamer with vancomycin and metronidazole, the investigators found that while tolevamer was inferior to both metronidazole and vancomycin, metronidazole was inferior to vancomycin (clinical success rates of 44.2%, 72.7%, and 81.1%, respectively). These differences were more pronounced in severe CDI (clinical success rates of 66.3% for metronidazole and 78.5% for vancomycin). Due to the randomization of patients to each tolevamer, metronidazole, and vancomycin treatment arm, this study actually represented the largest randomized study comparing metronidazole (n=278) to vancomycin (n=259) for the treatment of CDI. In post hoc analysis, age ≤65 years compared to age \>65 years was not shown to influence clinical success.[@b39-cia-12-1799] Despite the tolevamer study providing no evidence for an impact of age on treatment success, advancing age has been shown in numerous studies to influence treatment outcomes. For example, a systematic review of 39 articles from 2001 to 2010 by Vardakas et al allowed an assessment of the impact of age on treatment failures.[@b36-cia-12-1799] The median age was greater than 65 years in 22 studies and 65 years and younger in 15 other studies. In age-specific analysis, more total treatment failures were reported in studies with older patients (median age \>65 years) compared to younger patients (24.7% vs 19.6%; *p*=0.005). Total CDI recurrences were also higher in studies with older patients than in studies with younger patients (23.4% vs 19.4%; *p*=0.003). Treatment failure with metronidazole in studies with older patients was 27.4% and that of younger patients was 17.6% (*p*\<0.001). The corresponding recurrence was 33.9% in older patients and 17.9% in younger patients (*p*\<0.001). No age-related difference was observed in treatment failure and recurrence with vancomycin, suggesting that metronidazole may be associated with poorer outcomes in the elderly population.[@b36-cia-12-1799]
Seventy patients were identified in a retrospective chart review (January--December 2006) to examine the clinical course of CDI in the patients 80 years and older (mean age: 84.0±4.1; range 80--94). The aim of this study was to characterize CDI in the "oldest" old population. Majority of patients received antibiotics (81.4%) and PPI (58.5%) during the 30 days prior to CDI presentation. Twelve patients (17.1%) died within 90 days of initial presentation, with one death directly attributable to CDI. Overall, treatment failure occurred in 18 (25.7%) patients and correlated with leukocytosis on presentation. While the small number of patients on vancomycin precluded a comparison of efficacy between metronidazole and vancomycin, the authors concluded that initial CDI therapy with vancomycin may be appropriate for elderly patients, especially those with elevated white blood cell counts.[@b40-cia-12-1799]
Mounting evidence, therefore, suggests that in older adults with CDI, recurrence and treatment failure with metronidazole may be higher, so it may be reasonable to initiate therapy with vancomycin in all older adults with CDI.
Fidaxomicin
===========
Fidaxomicin, US Food and Drug Administration (FDA) approved in May 2011 for CDI, is a bactericidal macrolide that inhibits nucleic acid synthesis by impairing bacterial RNA polymerase activity.[@b41-cia-12-1799] Fidaxomicin has a narrower spectrum of antimicrobial activity than metronidazole or vancomycin, thus limiting disruption to the normal gastrointestinal flora.[@b42-cia-12-1799] In addition, fidaxomicin has a prolonged post-antibiotic effect (\~10 hours) allowing for twice-daily dosing.[@b43-cia-12-1799]
The in vitro effect of fidaxomicin and its metabolite, OpT-1118, on *C. difficile* growth and sporulation dynamics was compared to vancomycin, metronidazole, and rifaximin. In comparison to the three comparator drugs, fidaxomicin and OpT-118 effectively inhibited *C. difficile* sporulation.[@b43-cia-12-1799] More recently, Housman et al sought to compare the number of *C. difficile* vegetative cells and spores in stool among patients receiving fidaxomicin or vancomycin as treatment for their first CDI episode. Thirty-four patients were enrolled, majority of them elderly: mean ages of the fidaxomicin (n=18) and vancomycin groups (n=16) were 69 years (±15 years) and 66 years (±15 years), respectively. Vancomycin and fidaxomicin therapy both resulted in rapid decreases in vegetative *C. difficile* counts throughout therapy; however, more patients receiving fidaxomicin achieved at least a 2 log~10~ colony-forming units/g reduction in spores at the 2-week follow-up visit (*p*=0.02).[@b44-cia-12-1799]
Several clinical trials, with an adequate representation of elderly patients, have been conducted to compare the efficacy and safety of fidaxomicin in CDI treatment. In the two Phase III noninferiority RCTs, fidaxomicin was compared with vancomycin in the treatment of new-onset or first recurrence of CDI with a 28-day follow-up period.[@b45-cia-12-1799]--[@b47-cia-12-1799] A total of 1,164 participants were evaluated in the pooled dataset. The mean reported age was 63 years and 61 years in the Louie study and Cornely study, respectively.[@b45-cia-12-1799],[@b46-cia-12-1799] Fidaxomicin was proven to be noninferior to vancomycin for CDI treatment and more effective than vancomycin in reducing the rate of recurrence. These findings were not influenced by age stratification (age \<65 vs ≥65 years) in subgroup analysis. It should be noted that fidaxomicin was not associated with fewer recurrences among patients infected with the NAP1/BI/027 strain versus those infected with other *C. difficile* strains, possibly due to the small numbers of NAP1/BI/027 strain-infected patients. With regard to adverse events, fidaxomicin was well tolerated with a similar safety profile compared to oral vancomycin.[@b47-cia-12-1799]
Utilizing combined data from the two RCTs conducted for fidaxomicin drug approval, Louie et al examined the effects of age (characterized in decades: ≤40, 41--50, 51--60, 61--70, 71--80, and \>80 years) and study drug on CDI outcomes.[@b48-cia-12-1799] They reported a statistically significant linear effect of age on CDI outcomes, specifically a 17% lower clinical cure, 17% greater recurrence, and 13% lower sustained clinical response by advancing decade than in those younger than 40 years (*p*\<0.01 each). Vancomycin and fidaxomicin were comparably effective in attaining clinical cure in all age strata; however, for participants who achieved clinical cure, fidaxomicin-treated participants were half as likely to have had a recurrence as participants treated with vancomycin (OR =0.46; 95% CI 0.32--0.67; *p*\<0.001). Consequently, the authors suggest that fidaxomicin be considered an alternative to vancomycin for treatment of CDI, particularly in elderly adults, who have a higher likelihood of developing recurrent disease.[@b48-cia-12-1799]
While fidaxomicin has a favorable safety and twice-a-day dosing profile, its current high drug acquisition cost poses a significant barrier to adoption in clinical practice. However, given its advantage in reducing the risk of recurrent CDI, targeting its use to populations at highest risk of relapse, including elderly patients, may prove to be cost-effective.[@b19-cia-12-1799]--[@b23-cia-12-1799],[@b49-cia-12-1799],[@b50-cia-12-1799]
Oral vancomycin and fidaxomicin are poorly absorbed; thus, systemic adverse effects are minimal. In addition, oral vancomycin and fidaxomicin do not require dose adjustment in the elderly or in patients with hepatic or renal dysfunction. On the other hand, the oral formulation of metronidazole is systemically absorbed but achieves effective concentrations in the colon after secretion back into the lumen.[@b19-cia-12-1799],[@b51-cia-12-1799] Intravenous and oral metronidazole have frequently been reported to cause diarrhea, nausea, gastrointestinal discomfort, and dysgeusia. Severe adverse effects of metronidazole include seizures, encephalopathy, and peripheral neuropathy. Metronidazole is also implicated in several drug interactions including an increased risk of bleeding with concomitant warfarin, a commonly utilized anticoagulant in the elderly. Fortunately, metronidazole dose adjustment is not required in the elderly.[@b52-cia-12-1799]
Bezlotoxumab
============
Although metronidazole, vancomycin, and fidaxomicin are effective in the treatment of CDI, they each disrupt the indigenous gastrointestinal microbiota to varying degrees. This presents a considerable challenge in the risk reduction of recurrent CDI episodes. Because the pathogenesis of CDI is closely linked to the dysregulation of the gastrointestinal microbiota and host immune response, the development of immunotherapy is a rational therapeutic strategy and an area of increased interest. The severity and range of the symptoms of CDI are caused by the two *C. difficile* virulence factors, TcdA and TcdB. The magnitude of antibody response to these *C. difficile* virulence toxins is inversely correlated with the relative risk of developing recurrent disease. Indeed, studies have identified low endogenous anti-TcdA and -TcdB antibody levels as a risk factor for CDI recurrence.[@b53-cia-12-1799]
Bezlotoxumab, approved in October 2016 by the FDA, is a human monoclonal antibody that binds to and neutralizes TcdB. This therapeutic strategy represents a recent advance in antibody-based immunotherapy for managing CDI. Bezlotoxumab binds to the combined repetitive oligopeptide domains of TcdB, and, through x-ray crystallography, has been shown to prevent binding of TcdB to mammalian cells.[@b54-cia-12-1799],[@b55-cia-12-1799] In addition to inciting a release of proinflammatory factors such as interleukin 8, TcdA and TcdB disrupt gastrointestinal epithelial cell tight junction resulting in acute diarrhea.[@b54-cia-12-1799]--[@b57-cia-12-1799] The postulated mechanism of action of bezlotoxumab is direct toxin neutralization, thereby preventing the deleterious toxin effects and leading to restoration of a healthy microbiota.[@b58-cia-12-1799],[@b59-cia-12-1799]
Bezlotoxumab is indicated in patients who are receiving standard-of-care anti-*C. difficile* treatment and are at a high risk for CDI recurrence.[@b60-cia-12-1799] The median age of participants was 66 years in the pivotal Phase III trials. CDI recurrence occurred in 16.5% of the bezlotoxumab group compared to 26.6% (*p*\<0.0001) in the placebo group. Sustained cure (defined as initial clinical cure of the baseline episode of CDI and no recurrent infection through the 12-week follow-up period) was 64% with bezlotoxumab compared to 54% with placebo. Across prespecified groups who were at high risk for recurrent CDI, the rates of recurrent infection were lower with receipt of bezlotoxumab. In particular, among patients 65 years or older, bezlotoxumab was associated with a CDI recurrence rate that was 51% lower than that associated with placebo.[@b59-cia-12-1799] While bezlotoxumab was found to protect against CDI morbidity, like all medications, potential adverse events exist. Heart failure was more commonly reported in patients who received bezlotoxumab compared to placebo (12.7% vs 4.8%, respectively), prompting the FDA to require a warning label in the bezlotoxumab package insert.[@b59-cia-12-1799],[@b60-cia-12-1799]
In addition, the impact of systemic concomitant antibiotics on the efficacy of bezlotoxumab is necessary to add valuation to this new therapy. Interestingly, actoxumab, developed in tandem with bezlotoxumab, is another human monoclonal antibody that neutralizes toxin A. However, actoxumab alone did not decrease *C. difficile* recurrence and had a worse adverse event profile.[@b59-cia-12-1799] Antibodies are poised to become an essential therapeutic strategy in the management of CDI and bezlotoxumab represents a significant advancement. However, like most first-in-class agents, concerns over real-world effectiveness and drug cost remain.
Fecal microbiota transplant
===========================
Relapse of CDI occurs in 10%--25% of patients treated with metronidazole or vancomycin. Furthermore, multiple relapses in the same individual are common.[@b28-cia-12-1799],[@b37-cia-12-1799] In recognition of the importance of restoring balance to the disrupted gastrointestinal flora, major guidelines have addressed the role of fecal microbiota transplant (FMT) but differ in their recommendations given the limited evidence at time of respective publications.[@b19-cia-12-1799]--[@b21-cia-12-1799] For example, the 2010 Infectious Diseases Society of America/Society for Healthcare Epidemiology of America guidelines recognized FMT as a promising emerging therapy but due to a lack of randomized controlled trials were unable to evaluate its efficacy and safety.[@b19-cia-12-1799] On the other hand, for multiple recurrent CDIs unresponsive to repeated antibiotic treatment, European Society of Clinical Microbiology and Infection strongly recommends the use of FMT in combination with oral antibiotic treatment.[@b21-cia-12-1799] The American College of Gastroenterology offered a more reserved recommendation, "if there is a third recurrence after a pulsed vancomycin regimen, FMT should be considered (Conditional recommendation, moderate-quality evidence)".[@b20-cia-12-1799]
Since the major guidelines were published, interest in FMT has grown rapidly. A review of the literature reveals that FMT is gaining acceptance as an effective therapy for recurrent CDI.[@b61-cia-12-1799] Cumulative experience from case series and controlled trials shows that FMT is effective (80%--90%) when used to treat relapsing CDI.[@b62-cia-12-1799]--[@b65-cia-12-1799] For example, a recent systemic review evaluated data from two RCTs, 28 case-series studies, and five case reports. The study subjects were predominantly elderly, and symptom resolution was seen in 85% of cases.[@b65-cia-12-1799]
To better understand the impact of FMT on CDI in the elderly, Burke et al identified 115 patients from 10 pooled case studies, ranging in age from 60 to 101 years (mean age: 77 years). Durable remission of CDI was achieved in 103 (89.6%) patients over a follow-up period of 2 months to 5 years (mean 5.9 months). Cure rate in the older population (89.6%) was not significantly different from that of the 52 younger individuals (80.8%) in the included studies (*p*=0.26). Although most achieved bacteriological cure without complication, one patient died of peritonitis that may have resulted from nasogastric tube perforation during fecal transplantation.[@b66-cia-12-1799] In the subgroup analysis of a more recent meta-analysis, long-term outcomes of FMT for CDI were compared between older individuals (≥65 years old) and younger individuals (\<65 years old). The primary cure rate (resolution of diarrhea without recurrence within 90 days of FMT) was higher in younger individuals compared to older individuals (99.4% vs 87.0%; *p*=0.0003). Among younger groups, the overall recurrence rate post-FMT was 4.6% compared to 9.3% for older individuals. The authors concluded that while FMT is likely a highly effective and robust therapy for recurrent CDI in adults, old age (≥65 years) should be considered as a risk factor for early CDI recurrence post-FMT therapy.[@b67-cia-12-1799]
Identification of a healthy stool donor is an essential initial step to successful FMT. Because the indigenous gastrointestinal microbiota undergoes age-related changes, the selection of healthy FMT donors from among the elderly population may prove a challenge. In practice, younger donors tend to donate stool samples for their older relatives while older donors commonly donate specimens for their spouses. Guidelines do not suggest an upper limit of age to exclude donors for the purpose of FMT.[@b68-cia-12-1799] To address the lack of data regarding the effect of donor age on fecal microbiota and its clinical efficacy in patients with recurrent CDI, Anand et al utilized stool sample rRNA sequencing and demonstrated that while there was a decrease in the abundance of phylum Actinobacteria in donors above 60 years of age compared to the younger donor group (\<60 years), there was no significant difference in the alpha diversity between the two donor groups.[@b69-cia-12-1799] Additional larger studies in both age and ethnic diverse populations are required to corroborate these findings.
Despite the growing support for FMT, clinicians and patients need to be cognizant of the inevitable risk of communicable disease transmission.[@b70-cia-12-1799] Another important consideration is the route of administration. A number of delivery modalities have been described for FMT: nasogastric or nasojejunal tube, colonoscopy, and enemas. Recently, the development of oral FMT capsules has garnered interest. The safety and rate of diarrhea resolution following administration of oral capsulized frozen FMT was evaluated in a feasibility study with 20 patients (median age 64.5 years; interquartile range 53.5--78.3). Resolution of diarrhea was achieved in 14 patients (70%; 95% CI 47%--85%) after a single-capsule-based regimen and in 90% of patients after non-responders were retreated. Age was not associated with CDI relapse.[@b71-cia-12-1799] Having a variety of delivery modalities, especially oral FMT capsules, may benefit the elderly population because of ease of administration and the avoidance of procedure-associated risk with invasive administration modalities such as colonoscopy.[@b70-cia-12-1799]--[@b72-cia-12-1799]
Probiotics
==========
Probiotics, a nutritional supplement, contain either a single culture or a mixed culture of live microorganisms such as *Lactobacillus* and *Bifidobacterium* strains and the yeast *Saccharomyces boulardii*.[@b73-cia-12-1799],[@b74-cia-12-1799] They represent another therapeutic strategy targeting the restoration of microbiota flora.
Evidence around the probiotic effect has been mixed.[@b75-cia-12-1799],[@b76-cia-12-1799] For example, the largest placebo-controlled randomized trial conducted in 2,941 inpatients aged 65 years or older that received probiotics (multistrain preparation of *Lactobacillus* and *Bifidobacterium*) failed to demonstrate a reduction in antibiotic-associated diarrhea or *C. difficile* rates.[@b75-cia-12-1799] On the other hand, a recent meta-analysis, incorporating the aforementioned trial in addition to 25 other studies, did show a significantly lower risk of developing CDI in the probiotics group compared to the control group (relative risk \[RR\] =0.395; 95% CI 0.294--0.531; *p*\<0.001).[@b77-cia-12-1799] Subgroup analysis identified that *Lactobacillus*, *Saccharomyces*, or a mixture of probiotics was beneficial in reducing the risk of developing CDI. Probiotics were beneficial for both adults (RR =0.405; 95% CI 0.294--0.556; *p*\<0.001) and children (RR =0.341; 95% CI 0.153--0.759; *p*=0.008).[@b77-cia-12-1799]
Though there are numerous studies and several systematic reviews evaluating the use of probiotics, the wide variety of probiotic strains, dosages, and durations of therapy makes it difficult to interpret. Overall, there is moderate-quality evidence supporting a protective effect of probiotics in preventing CDI in patients taking antibiotics.[@b10-cia-12-1799],[@b75-cia-12-1799],[@b77-cia-12-1799] In addition, the use of probiotics has been controversial because of the rare case reports of fungemia in both immunocompromised and immunocompetent patients.[@b75-cia-12-1799],[@b78-cia-12-1799] High-quality studies, utilizing standardized regimens, are certainly required in diverse populations including the elderly.
Combination antibiotics
=======================
There is a paucity of data on the efficacy of combination therapy in the management of CDI. Njoku et al sought to shed light on the impact of combination therapy versus monotherapy for CDI in a recent single-center study.[@b79-cia-12-1799] Median age was 59 years and 63 years in the monotherapy and combination therapy groups, respectively (*p*=0.08). Approximately 9% of patients were admitted from a nursing or LTCF. Overall, 177 of 248 patients (71.4%) achieved clinical cure. There were no differences in time to return of daily bowel movements to ≤2/day, clinical cure, length of stay, recurrence, or mortality, and while not clinically significant the combination therapy group had longer duration of therapy than the monotherapy group (15 vs 14 days; *p*=0.009).[@b79-cia-12-1799]
In a systematic review comparing metronidazole mono-therapy with vancomycin monotherapy and combination therapy in CDI patients, no statistically significant difference was observed between monotherapy and combination therapy. The rate of adverse drug events was lower for monotherapy than that for combination therapy (OR =0.30; 95% CI 0.17--0.51; *p*\<0.0001).[@b80-cia-12-1799]
Miscellaneous agents
====================
Besides the therapies discussed earlier, other therapeutic agents have been utilized for the treatment of CDI including rifaximin, nitazoxanide, and tigecycline. Most of the evidence for these agents comes from case reports and their utility in the elderly is largely unknown.
Nitazoxanide
============
Nitazoxanide, a nitrothiazolide, is FDA approved for the treatment of cryptosporidiosis and giardiasis, and is routinely employed in the management of parasitic intestinal infections through inhibition of anaerobic metabolism. However, for the treatment of CDI, there appears to be limited evidence.
In a noninferior, RCT, nitazoxanide was shown to be at least as effective as metronidazole in the treatment of *C. difficile* colitis. This study was conducted across seven VA medical centers with a predominance of elderly male patients.[@b81-cia-12-1799] Subsequently, a similarly designed study by the same investigators was designed to compare vancomycin and nitazoxanide therapy.[@b82-cia-12-1799] Among those who completed therapy, sustained response rates were 78% for the vancomycin group and 89% for the nitazoxanide group. Forty percent of patients were categorized as severe CDI and mean age was 59.6 years and 65.7 years in the nitazoxanide and vancomycin groups, respectively (*p*=0.19). The small sample (N=49) precluded any noninferiority analysis; nonetheless, the results suggest that nitazoxanide may be as effective as vancomycin.[@b82-cia-12-1799]
Tigecycline
===========
A derivative of minocycline, tigecycline has broad-spectrum activity against Gram-positive and Gram-negative organisms and anaerobic bacteria such as *Bacteroides fragilis*.[@b83-cia-12-1799] Several case reports have reported the use of intravenous tigecycline as salvage therapy for severe refractory cases of CDI with varying outcomes. A limited number of these reports involved elderly adults.[@b84-cia-12-1799],[@b85-cia-12-1799]
In one such case series, Herpers et al present four patients with severe refractory CDI who were successfully treated with tigecycline. Three patients had previously failed standard CDI therapy while one patient was treated with tigecycline upon CDI onset. The pertinent demographics of the patients are as follows: 59-year-old male, 36-year-old female, 36-year-old male, and an 82-year-old female.[@b84-cia-12-1799]
A single-center retrospective study by Thomas et al compared the outcomes of patients who received standard-of-care therapy with tigecycline (n=18) versus standard-of-care therapy without tigecycline (n=26) for severe CDI.[@b86-cia-12-1799] Median age of patients in the tigecycline group was 55 years and 63 years in the non-tigecycline group. No difference in treatment outcomes including overall survival, colectomy rates, and relapse rates were observed between the two groups.[@b86-cia-12-1799]
Rifaximin
=========
Rifaximin is a nonabsorbable derivative of rifamycin. It is primarily used in the management of irritable bowel syndrome, hepatic encephalopathy, and traveler's diarrhea. Rifaximin shows potent activity against *C. difficile*, and clinical anecdotes have reported use as an adjunctive antibiotic for the treatment of recurrent and refractory CDI. For the treatment of mild-to-moderate CDI, clinical success with rifaximin (57%) was similar to vancomycin (64%) therapy but failed to achieve the goal of noninferiority in a RCT.[@b87-cia-12-1799]
Johnson et al reported the clinical courses of the six CDI recurrent patients treated with rifaximin (post-vancomycin treatment).[@b88-cia-12-1799] The six patients were 88, 33, 78, 85, 81, and 66 years old, with a mean age of 72 years. Four of the six patients (67%) had no further diarrhea episodes, but two patients relapsed shortly after or during the rifaximin treatment. Of note, the two patients classified as treatment failure were elderly (88 and 85 years old), and one of these two patients had a *C. difficile* isolate minimum inhibitory concentration of \>256 μg/mL to rifampin.[@b88-cia-12-1799]
Cost-effectiveness
==================
In addition to contributing to patient morbidity and mortality, CDI exerts a substantial financial toll on health systems, with a total US economic burden thought to exceed \$1 billion per year.[@b89-cia-12-1799] As a result, hospitals and third-party payers are increasingly relying on the economic analysis of available and emerging therapeutic agents in their formulary decision-making. The varied purchasing, pricing, and insurance reimbursement structures utilized in different countries limit extrapolation of these analyses.
For example, analysis from a Scottish public health care provider perspective showed that compared to vancomycin, fidaxomicin is cost-effective in either patients with severe CDI or a first CDI recurrence.[@b50-cia-12-1799] In the USA, Konijeti et al compared four treatment strategies (metronidazole, vancomycin, fidaxomicin, and FMT via colonoscopy) for first-line treatment of recurrent CDI in a hypothetical cohort of patients with a median age of 65 years.[@b90-cia-12-1799] Initial treatment with FMT via colonoscopy was the most cost-effective strategy for recurrent CDI at cure rates greater than 88.4%. In clinical setting where FMT via colonoscopy is not available or cure rates are lower than threshold, oral vancomycin was more cost-effective.[@b90-cia-12-1799] Similarly, FMT by colonoscopy (or enema, if colonoscopy is unavailable) was concluded to be cost-effective for treating recurrent CDI in Canada. The modeled patient in this particular study was a 70-year-old community-dweller.[@b91-cia-12-1799]
With regard to fidaxomicin in particular, cost-effectiveness analysis has been mixed given the varied methodological approaches. For example, utilizing a number-needed-to-treat of 7.1 for sustained clinical response from the two pivotal fidaxomicin trials, an epidemiologic study estimated that at \$280 US dollars, fidaxomicin represents value for money in the treatment of CDAD.[@b92-cia-12-1799] On the other hand, Bartsch et al utilized a decision analytic simulation model to demonstrate that using fidaxomicin as a first-line treatment for CDI is not cost-effective when NAP1/BI/027 accounts for \~50% of infecting strains. In fact, a course of fidaxomicin would need to cost ≤\$150 to be cost-effective in the treatment of all CDI cases. The authors suggest that treatment with fidaxomicin based on strain may be a reasonable approach.[@b93-cia-12-1799]
Conclusion
==========
Our understanding of CDI continues to evolve but it is apparent that advanced age is a major risk factor and one that results in substantial morbidity and mortality. Appropriate CDI prevention and management strategies involve antimicrobial and non-antimicrobial complimentary approaches. Metronidazole remains the initial treatment for mild-to-moderate CDI in majority of patients; however, evidence suggests that vancomycin or fidaxomicin may be considered as first-line options in the elderly. Certainly, there is no one-size-fits-all approach. For each elderly patient, therapeutic decisions should be guided by several factors, including the severity of the primary infection, underlying comorbidities, the severity of CDI, and the patient's end-of-life wishes.
An updated *C. difficile* management guideline by Infectious Diseases Society of America/Society for Healthcare Epidemiology of America is anticipated in 2017 and will likely provide evidence-based recommendations on current and emerging treatment options, including FMT and bezlotoxumab, especially in populations at greatest risk of relapse. A concerted effort from national and state public health agencies, health care providers, and antimicrobial stewardship teams is required to decrease the burden of CDI in our aging population. Finally, the limited studies on CDI management among the elderly, especially LTCF residents, warrant further research to identify poor prognostic indicators and to validate interventions that may improve outcomes among this vulnerable population.
**Disclosure**
The authors report no conflicts of interest in this work.
![Incidence of nosocomial *Clostridium difficile* infection.\
**Notes:** The overall incidence of nosocomial *C. difficile* infection is shown by year (blue), as is the incidence according to patient age (black). From *N Engl J Med*, Leffler DA, Lamont JT, *Clostridium difficile* infection, 372(16):1539--1548. Copyright © (2015) Massachusetts Medical Society. Reprinted with permission from Massachusetts Medical Society.[@b3-cia-12-1799]](cia-12-1799Fig1){#f1-cia-12-1799}
######
Risk factors associated with CDI development and recurrence
--------------------------------------------------------------------------------------------------------------------------------
Risk factors
--------------------------- ----------------------------------------------------------------------------------------------------
Pharmacotherapy Number and days of systemic concomitant antibiotic use\
High-risk antibiotic (clindamycin, fluoroquinolones, second generation cephalosoprins and higher)\
Proton-pump inhibitors and histamine type 2 blockers
Past health care exposure Prior hospitalization\
Duration of hospitalization\
Long-term care residency
Host immunity Lack of antibody response to *Clostridium difficile* toxin\
Severity of underlying illness\
Comorbidities
Increasing age \>65 years and older\
Per-year increment over 18 years
CDI experience Previous CDI infection
--------------------------------------------------------------------------------------------------------------------------------
**Abbreviation:** CDI, *Clostridium difficile* infection.
######
Recommended medical therapy for *Clostridium difficile* infection
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Disease severity Therapeutic agent If significant risk of recurrence
--------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------
Mild to moderate Metronidazole 500 mg by mouth, three times daily, for 10--14 days\ Vancomycin 125 mg by mouth, four times daily, for 10--14 days\
If intolerant to metronidazole: vancomycin 125 mg by mouth, four times daily, for 10--14 days Fidaxomicin, 200 mg by mouth, twice daily, for 10 days
Severe Vancomycin 125 mg by mouth, four times daily, for 10--14 days Fidaxomicin 200 mg by mouth, twice daily, for 10 days
Severe, complicated Vancomycin 125 mg or 500 mg\* by mouth, four times daily and/or vancomycin 500 mg per rectum four times daily\* and metronidazole 500 mg intravenously every 8 hours\ Fidaxomicin 200 mg by mouth, twice daily, for 10 days
Surgical consultation/management
Recurrent First recurrence: repeat same regimen used for initial episode\ Fidaxomicin 200 mg by mouth, twice daily, for 10 days
Second recurrence: pulsed or tapered oral vancomycin regimen\
Third recurrence: vancomycin plus fecal microbiota transplant
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
**Notes:** Based on IDSA/SHEA, ACG, and ESCMID guideline recommendations. \*If ileus, toxic colon, or significant abdominal distension.
**Abbreviations:** IDSA, Infectious Diseases Society of America; SHEA, Society for Healthcare Epidemiology of America; ACG, American College of Gastroenterology; ESCMID, European Society of Clinical Microbiology and Infection.
######
Current treatment options available in the USA
Drug name Class Dose and frequency
------------------ ------------------------ ---------------------------------------------------------
Metronidazole Nitroimidazole 500 mg by mouth or IV three times daily
Vancomycin Glycopeptide 125--500 mg by mouth four times daily
Fidaxomicin Macrolide 200 mg by mouth twice a day
Nitazoxanide Nitrothiazolide 500 mg by mouth twice a day
Tigecycline Tetracycline 100 mg IV loading dose followed by 50 mg IV twice daily
Rifaximin Rifamycin 200--400 mg by mouth twice or three times daily
Bezlotoxumab Monoclonal antibody Single dose of 10 mg/kg intravenously
Fecal microbiota -- Various formulations and regimens
Probiotics Nutritional supplement Various formulations and regimens
**Abbreviation:** IV, intravenous.
| 2023-12-02T01:26:58.646260 | https://example.com/article/3839 |
With or without MARTHA KARUA, we will win like TRUMP in 2017 - RAILA & ODM brag
Friday November 11, 2016 – On Thursday, Narc Kenya
chairperson, Martha Karua, caught the opposition leaders unaware when she
dropped her presidential ambitions and endorsed President Uhuru Kenyatta for a
second term in office in 2017.
Karua, who has been nicknamed the Iron Lady by the
opposition, said she will support Uhuru in 2017 and also vie for the Kirinyaga
gubernatorial seat in 2017. | 2024-02-16T01:26:58.646260 | https://example.com/article/1253 |
Atomkraft med øget sikkerhed og forbedret økonomi kan blive en eksportvare til Asien.
Atomkraft behøver ikke være gigantiske anlæg med rygende reaktorer og frygt for ulykker.
Står det til danske Seaborg Technologies, ligger svaret på fremtidens energikilde i en nyudviklet reaktor, som både er væsentligt mindre og har en bedre brændselsudnyttelse end de traditionelle atomkraftanlæg.
Den danske iværksættervirksomhed er så langt med udviklingen af reaktoren, at man har landet en kæmpe aftale med et stort sydkoreansk energiselskab, hvis navn endnu ikke er offentligt. Det skriver Finans.dk.
Kan transporteres i lille container
Aftalen er et partnerskab, hvor Seaborg Technologies og sydkoreanerne vil bygge 7.500 eksemplarer af den danskudviklede reaktor i Sydøstasien frem mod 2040.
Den nyudviklede reaktor adskiller sig ved at være mindre end kernekraftværker, som vi kender dem. Den kan transporteres i en container på syv til otte meter og vil kunne fremstilles i serieproduktion, hvilket letter omkostningerne.
- Det er et meget stort skridt på vejen mod næste generation af atomkraft. Sydkoreanerne vil sørge for al infrastruktur til at kunne bygge reaktorerne, og så skal vi bare koncentrere os om at færdigudvikle selve reaktoren, siger Troels Schönfeldt, medstifter og administrerende direktør i Seaborg Technologies.
Det er ikke bare i Sydøstasien, den danske reaktor har fået bevågenhed.
Da det amerikanske tech-institut MIT for nylig kårede en liste med 35 navne over de mest lovende og nytænkende opfindere, figurerede Seaborg Technologies norske medgrundlægger Eirik Eide Pettersen på listen på grund af sit arbejde med at udvikle atomkraft til en verden, som skal belave sig på at klare sig uden fossile brændstoffer.
A-kraft med dansk accent er billig
Atomkraft med dansk accent er en slags miniatureversion af et klassisk reaktoranlæg. En af Seaborgs reaktorer kan levere energi til en by på 50.000 indbyggere, inklusiv al industri.
Men hvad er det egentlig for en størrelse atomreaktor, Seaborg har udviklet?
Det har Bent Lauritzen, som er afdelingschef ved Center for Nukleare Teknologier på DTU, et bud på.
- Det, der er mest interessant ved små, modulære reaktorer er, at de kan bygges industrielt og i serieproduktion. Det bliver billigere, men det bliver også muligt at placere dem på steder, hvor el-nettet ikke er så udviklet. Den vil også være nemmere både at transportere og udskifte, siger Bent Lauritzen til TV 2.
En anden nyskabelse er, at den danske model kører på en nyudviklet blanding af uran, smeltet salt og thorium.
- Denne type reaktor har en langt større brændselsudnyttelse end tidligere. Hvis reaktoren tillige kan køre med den høje udnyttelse af brændstoffet, får du mere energi ud af den samme mængde brændsel. Og dermed vil man alt andet lige kunne deponere mindre affald, siger Bent Lauritzen.
Kan redde klimaet
Ifølge Seaborg Technologies betyder den danske a-kraft model, at lande i Asien hurtigere vil kunne gøre sig uafhængige af kul.
- Vores teknologi vil kunne forsyne Sydøstasien med energi, som er en tredjedel billigere end kulkraft. Men det vigtigste er, at atomkraften kan redde klimaet, siger Troels Schönfeldt ifølge Finans.dk.
Man taler ikke atomkraft uden også at tale sikkerhed. I en ideel verden er atomkraft en mere ren energikilde end fossile brændstoffer og dermed også en del af løsningen på udledningen af drivhusgasser. Men er reaktoren fra Seaborg også et fremskridt rent sikkerhedsmæssigt?
Det er ifølge Bent Lauritzen endnu for tidligt at udtale sig om sikkerhed på lang sigt.
- Den har potentialet til at være det, men det ved vi af gode grunde ikke, før reaktoren er testet over et længere tidsspand. Det er det, vi lærer af. Man kan sige, at den i hvert fald er lige så sikker som eksisterende reaktorer, siger den danske professor. Han peger på, at sikkerheden ved atomkraft er strammet op og mere betryggende end de billeder, mange har fået på nethinden efter at have fået genkaldt historien om ulykken på det russiske kernekraftværk Tjernobyl, som HBO har lavet en gruopvækkende tv-serie over.
- Det er vigtigt at understrege, at sikkerhed er en anden end for 40 år siden. For det første, fordi man ikke ville bygge en reaktortype som Tjernobyl i den vestlige verden, og i forhold til de reaktorer, vi kender i dag, er der et meget større erfaringsgrundlag - både i forhold til sikkerhed og teknologi. | 2023-10-15T01:26:58.646260 | https://example.com/article/4080 |
Vegan Cashew “Cheese” Spread
As a way to ring in 2016, I asked my followers on Social Media to pick one food item for me to take out of my diet for the month of January. I received so many great options like cheese, sugar, meat, and Richard (literally, two different people wrote Richard . . . not really sure what to do with that information, but I’ll roll with it). I put all of their suggestions into a drawing for one lucky winner (or loser, depending how you look at it): DAIRY!
I knew this was going to be a tough cookie, because, well, dairy encapsulates so many of my favorites: cheese, chocolate, cheese, ice cream, cheese, and did I mention cheese? This may come as a surprise to some of you, but I really like cheese. Nay, I really love cheese. There was a moment in my life when I had to be physically removed from my station on a sandwich making team, because being in charge of the cheese meant that I ate every other piece that slice that came my way. Here you go sandwich, one for you, one for me, one for you, one for me . . . I’m not even exaggerating, I have witnesses. All of that to say, I was a little bit nervy about giving up dairy (aka cheese) for an entire month. I said I was going to do it, though, so by golly I’m going to do it!
I’ve heard about this alleged “Cashew Cheese” from some of my vegan friends, and I’ve seen it listed as ingredients at some of my favorite restaurants, so I figured I’d try to look through a few recipes to see if I could come up with my own version of this Cheese-postor (that’s cheese-impostor, in case it doesn’t come across so soundly in written form). So, here’s what I did, and you can too!
Here’s what you’ll need:
1 cup cashews (soaked for at least one hour & drained)
1/4 cup filtered water
1/2 cup nutritional yeast
1/4 cup lemon juice
1 tablespoon Apple Cider Vinegar
1 Tablespoon Dijon Mustard
3 Garlic Cloves
Himalayan Sea Salt
Cracked Pepper
Here’s what you do:
Throw everything in a blender
Blend (supes complicated, right?)
Just in case you want my two cents, here you go: it.is.exquisite! Seriously, I loved every single tasty bite I got, and the family and friends who got to it before I licked the bowl also report thoroughly enjoying this scrumptious treat. However, and this is a big however, I’m not sure why someone would call this “cheese” exactly. Please don’t misunderstand, it is SUPER DELICIOUS, just not in a cheesy kinda way. I can see how it might be used in lieu of a cheese spread or dip, but my feta favoring palette can’t seem to get on board with calling this cheese just yet.
Am I being too harsh? Are my cheese standards just a little too high? You tell me! Comment below with your thoughts on this Vegan “Cheese” Spread. Are there other recipes or variations I should try? | 2024-02-28T01:26:58.646260 | https://example.com/article/8073 |
NBC Fires Lauer Over Sexual Misconduct Allegation
The reckoning over sexual harassment in the workplace claimed another leading television personality Wednesday when NBC fired its leading morning news anchor, Matt Lauer, over a sexual harassment allegation.
“On Monday night, we received a detailed complaint from a colleague about inappropriate sexual behavior in the workplace by Matt Lauer,” Andrew Lack, the NBC News president, said in a memo to the staff.
He said the allegation against Lauer “represented, after serious review, a clear violation of our company’s standards. As a result, we’ve decided to terminate his employment.”
“While it is the first complaint about his behavior in the over 20 years he’s been at NBC News, we were also presented with reason to believe this may not have been an isolated incident,” Lack said.
Ari Wilkenfeld, a civil rights lawyer with the firm Wilkenfeld, Herendeen & Atkinson in Washington, said he represented the woman who made the complaint to NBC but declined to publicly identify her. In a statement provided to The New York Times, he said: “My client and I met with representatives from NBC’s Human Resources and Legal Departments at 6 p.m. on Monday for an interview that lasted several hours. Our impression at this point is that NBC acted quickly, as all companies should, when confronted with credible allegations of sexual misconduct in the workplace.
“While I am encouraged by NBC’s response to date, I am in awe of the courage my client showed to be the first to raise a complaint and to do so without making any demands other than the company do the right thing.”
The Times met with the woman Monday afternoon, but she said she was not ready to come forward and tell her story publicly.
Lauer’s co-host, Savannah Guthrie, announced the news on “Today” on Wednesday morning. Appearing on the verge of tears, Guthrie said, “All we can say is we are heartbroken; I’m heartbroken.”
She described Lauer as “a dear, dear friend,” and said she was “heartbroken for the brave colleague who came forward to tell her story.”
Calling Lauer’s dismissal part of a national reckoning, she continued, “How do you reconcile your love for someone with the revelation that they have behaved badly?” | 2023-12-29T01:26:58.646260 | https://example.com/article/2084 |
Q:
Branching Rule for alternating groups
Let $A_n$ be the alternating group of degree $n$. What is the branching rule for the subgroup $A_{n-1}\subset A_n$, i.e., the structure of the restriction of ordinary irreducible representations of $A_n$ to $A_{n-1}$? Are there some nice books or references which provide detailed answer to this question?
A:
This is answered in Theorem 4 of my paper Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups, with Geetha Thangavelu, which is published in Algebras and Representation Theory, and is also available on the arXiv. See also Ruff, O.: Weight theory for alternating groups. Algebra Colloq. 15(03), 391–404 (2008).
The alternating groups have two types of representations, those corresponding to mutually conjugate non-self-conjugate pairs $(\lambda,\lambda')$ which are simply the restriction of the irreducible representation $V_\lambda$ or $V_{\lambda'}$ of $S_n$ to $A_n$, and two corresponding to each self-conjugate partition $\lambda$, denoted $V_\lambda^\pm$, which are the two irreducible summands of the irreducible representation $V_\lambda$ of $S_n$, when resticted to $A_n$.
If $\lambda$ is a partition of $n$ and $\mu$ is a partition of $n-1$, write $\mu\in \lambda^-$ if the representation $V_\lambda$ of $S_n$ contains the representation $V_\mu$ of $S_{n-1}$ upon restriction, then we have:
If $\lambda$ and $\mu$ are non-self-conjugate then the representation $V_\mu$ of $A_{n-1}$ is contained in the restriction of $V_\lambda$ from $A_n$ to $A_{n-1}$ if either $\mu\in \lambda^-$, or $\mu'\in \lambda^-$.
If $\lambda$ is non-self-conjugate and $\mu\in \lambda^-$ is self-conjugate, then $V_\mu^\pm$ are both contained in the restriction of $V_\lambda$ from $A_n$ to $A_{n-1}$.
If $\lambda$ is self-conjugate and $\mu\in \lambda^-$ is non-self-conjugate, then $V_\mu$ is contained in the restriction of both $V_\lambda^\pm$ from $A_n$ to $A_{n-1}$.
Finally, if $\lambda$ and $\mu\in \lambda^-$ are both self-conjugate, then $V_\mu^+$ is contained in $V_\lambda^+$ and $V_\mu^-$ is contained in $V_\lambda^-$. This result is based on a careful choice of sign in defining the representations $V_\lambda^\pm$ (deciding which gets the $+$ sign, and which gets the $-$ minus sing among the irreducible $A_n$ representations contained in a self-conjugate representation of $S_n$).
See the figure below. Please write to me if you would like an e-print of the the published version of the article.
| 2024-05-28T01:26:58.646260 | https://example.com/article/7933 |
1. Field of the Invention
This invention relates in general to earth boring drill bits, and in particular to a cutter mounting for a large hole earth boring drill bit.
2. Description of the Prior Art
One phase of the earth boring industry is concerned with drilling large diameter holes for mining. Normally several cutters are mounted on frames which are mounted to a large cutter support plate. The cutter support plate is connected to a drill string and rotated with each cutter rotating on its individual frame. Drilling is performed by pulling upward through a small diameter pilot hole, or by blind shaft drilling without using the pilot hole.
The cutters are carried by bearings and mounted to individual frames that are welded to the cutter support plate. Various devices including pins and bolts are used to mount the bearing carrier to the frame. Bolts require high torques and still occasionally become unscrewed because of the high vibratory forces. If a pin is used in a clevis-type arrangement, the two holes of the frame must be aligned with the hole in the bearing carrier within very small tolerances to minimize any play between them. This type of clevis is expensive to construct and a certain amount of play always remains. | 2024-05-17T01:26:58.646260 | https://example.com/article/7129 |
Bloomberg: How China Is Buying Its Way Into Europe
For more than a decade, Chinese political and corporate leaders have been scouring the globe with seemingly bottomless wallets in hand. From Asia to Africa, the U.S. and Latin America, the results are hard to ignore as China has asserted itself as an emerging world power. Less well known is China’s diffuse but expanding footprint in Europe.
snip
We analyzed data for 678 completed or pending deals in 30 countries since 2008 for which financial terms were released, and found that Chinese state-backed and private companies have been involved in deals worth at least $255 billion across the European continent. Approximately 360 companies have been taken over, from Italian tire maker Pirelli & C. SpA to Irish aircraft leasing company Avolon Holdings Ltd., while Chinese entities also partially or wholly own at least four airports, six seaports, wind farms in at least nine countries and 13 professional soccer teams.
Importantly, the available figures underestimate the true size and scope of China’s ambitions in Europe. They notably exclude 355 mergers, investments and joint ventures—the primary types of deals examined here—for which terms were not disclosed. Bloomberg estimates or reporting on a dozen of the higher-profile deals among this group suggest an additional total value of $13.3 billion. Also not included: greenfield developments or stock-market operations totaling at least $40 billion, as compiled by researchers at the American Enterprise Institute and the European Council on Foreign Relations, plus a $9 billion stake in Mercedes-Benz parent company Daimler AG by Zhejiang Geely Holding Group Co. chairman Li Shufu reported by Bloomberg.
snip
Looking ahead, Chinese companies have expressed interest in a slew of European deals that haven’t been officially announced yet, based on Bloomberg data and reporting, as well as a recent ECFR report. These include building nuclear reactors in Romania and Bulgaria, buying a Croatian container terminal and building a Swedish port, taking over Czech carmaker Skoda Transportation AS and an Ireland-based oil and gas producer, investing in French ski-lift firm Compagnie des Alpes and a German electricity grid operator and providing financing for a bridge in Croatia and a Budapest-Belgrade rail link.
1. Was in Italy recently and there is a town that is outside of Florence
Where there is a large Chinese population who make clothes. The clothes carry the label
“Made in Italy”. The clothes are inexpensive $5-$15 and the shop owners are Chinese. I brought some really nice items with Italian styling. But it is something to hear a Chinese person speak Italian. All over Venice, Rome and Florence I saw these stores $5, 10 and 15 clothes.
2. Long range planning
the US is folding in on itself while China reaches out with purchases and trade agreements. They will overtake the US economically at some point. tRump was the worst thing to happen - he sees a tiny issue that he believes is a negotiaion while China sees the big picture.
3. Right. And Americans respond to our national decline by voting for Trump.
This response is very dangerous because it will lead to a worse and worse and worse economy here.
We are flying, rudderless, into the unknown.
We should be organizing ourselves and planning how to deal with the challenges of the future.
Trump is a total waste of time. That's the worst thing about his presidency.
Here we are talking about his constant changing staff and the icebergs of his scandals that are emerging on our horizon and we are not dealing with the reality we face at all.
Europeans were talking about the changing reality in terms of the economy and planning to deal with it back in the 1970s and early 1980s. I was there. In contrast, we are not talking about what is really going on and how to deal with it. | 2023-12-27T01:26:58.646260 | https://example.com/article/9275 |
/**
* Provides default sources, sinks and sanitizers for reasoning about
* unvalidated URL redirection problems on the server side, as well as
* extension points for adding your own.
*/
import javascript
import RemoteFlowSources
private import UrlConcatenation
module ServerSideUrlRedirect {
/**
* A data flow source for unvalidated URL redirect vulnerabilities.
*/
abstract class Source extends DataFlow::Node { }
/**
* A data flow sink for unvalidated URL redirect vulnerabilities.
*/
abstract class Sink extends DataFlow::Node { }
/**
* A sanitizer for unvalidated URL redirect vulnerabilities.
*/
abstract class Sanitizer extends DataFlow::Node { }
/** A source of third-party user input, considered as a flow source for URL redirects. */
class ThirdPartyRequestInputAccessAsSource extends Source {
ThirdPartyRequestInputAccessAsSource() {
this.(HTTP::RequestInputAccess).isThirdPartyControllable()
}
}
/**
* An HTTP redirect, considered as a sink for `Configuration`.
*/
class RedirectSink extends Sink, DataFlow::ValueNode {
RedirectSink() { astNode = any(HTTP::RedirectInvocation redir).getUrlArgument() }
}
/**
* A definition of the HTTP "Location" header, considered as a sink for
* `Configuration`.
*/
class LocationHeaderSink extends Sink, DataFlow::ValueNode {
LocationHeaderSink() {
any(HTTP::ExplicitHeaderDefinition def).definesExplicitly("location", astNode)
}
}
/**
* A call to a function called `isLocalUrl` or similar, which is
* considered to sanitize a variable for purposes of URL redirection.
*/
class LocalUrlSanitizingGuard extends TaintTracking::SanitizerGuardNode, DataFlow::CallNode {
LocalUrlSanitizingGuard() { this.getCalleeName().regexpMatch("(?i)(is_?)?local_?url") }
override predicate sanitizes(boolean outcome, Expr e) {
// `isLocalUrl(e)` sanitizes `e` if it evaluates to `true`
getAnArgument().asExpr() = e and
outcome = true
}
}
/**
* A URL attribute for a React Native `WebView`.
*/
class WebViewUrlSink extends Sink {
WebViewUrlSink() {
// `url` or `source.uri` properties of React Native `WebView`
exists(ReactNative::WebViewElement webView, DataFlow::SourceNode source, string prop |
source = webView and prop = "url"
or
source = webView.getAPropertyWrite("source").getRhs().getALocalSource() and prop = "uri"
|
this = source.getAPropertyWrite(prop).getRhs()
)
}
}
}
| 2024-01-03T01:26:58.646260 | https://example.com/article/7812 |
Tokyo
JAPAN'S government on Tuesday pledged to modestly boost the amount of energy coming from renewable sources to around a quarter in a new plan that also keeps nuclear power central to the country's policy.
The plan aims to have 22-24 per cent of Japan's energy needs met by renewable sources including wind and solar by 2030, a figure critics describe as unambitious based on current levels of around 15 per cent.
Japan's own foreign minister Taro Kono earlier this year called the goal "significantly low" and described the country's commitment to renewables as "lamentable".
The European Union this month agreed to raise its renewable energy target to 32 per cent by 2030.
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Japan's policy also envisions nuclear providing more than 20 per cent of the country's energy needs by 2030, reflecting the government's ongoing commitment to the sector despite deep public concern after the 2011 Fukushima disaster.
The government has reduced Japan's reliance on the sector, but defends nuclear as an emissions-free energy source that will help the country meet its climate change commitments.
Critics though say the government has done too little to push renewable energy as a viable option.
Japan currently generates around 90 per cent of its energy from fossil fuels, and the plan calls for that figure to drop to just over half, with energy efficiency policies to cut demand.
Reliance on fossil fuels like coal increased in Japan after the Fukushima disaster, as public anger over the accident pushed all of the country's nuclear reactors offline temporarily. Six reactors are currently operating, and utilities face public opposition to activating more despite political support for the nuclear industry.
Japan's Tepco, which operated the Fukushima plant, signalled last week that it was ready to resume work on the construction of a new nuclear plant in the country's north.
"While we have strong obligations resulting from the Fukushima accident, we believe that it is our duty to ensure sufficient electricity supplies to avoid cuts," Tepco chief Tomoaki Kobayakawa said on Friday.
The government's plan also includes a pledge to reduce the country's 47-tonne stockpile of pluto-nium, which is large enough to produce 6,000 atomic bombs, though it is mostly stored overseas.
Japan has sought to generate energy from the material, but decades of research has not produced an effective and commercially viable method, leading to international criticism of Tokyo for continuing to produce and possess plutonium. AFP | 2024-04-19T01:26:58.646260 | https://example.com/article/8348 |
Has it ever felt odd to you that for Survivor: Fiji, the season started with 19 castaways? An odd number of players had never been done before and it forced the show to use Exile Island before a player had even been voted off. If it seems like it wouldn’t be the best idea in the world, it’s because it was never intentional from the producers’ perspective.
Some may already be aware of this but Fiji did actually have a 20th player intended to appear on the season. Her name was Mellisa McNulty and she had gone through the same casting process as the other 19. She was on location in Fiji and ready to go into the game… until she decided she simply could not go through with it and quit.
By the time Mellisa made her decision, it was too late to bring anybody else in. She had shot her pre-game footage and taken the promotional pictures. From the footage we got to see in TV Guide’s Fiji preview, Mellisa was an attractive single woman who was, per her words, going to use her looks to her advantage. Unfortunately I have not been able to uncover any of this footage online. The gist of it though, was that she was going to try to be Cook Islands Parvati 2.0. She was officially part of the Survivor: Fiji cast and there was no going back.
So why did she suddenly back out at the last possible minute? As she told TV Guide back in the day, Mellisa suffered a series of panic attacks that would not go away.
My#1 trigger is being in a situation where I can’t get out. I get a claustrophobia kind of panic. I had multiple panic attacks in one day, which I’ve never experienced before. I tried to pull myself together, but….
To finish the sentence, what she is saying is that she simply couldn’t get it together. She talked to on-site psychologists and they deemed her ready to go out and play. Producers left it up to Mellisa to decide whether she wanted to go out or not. Unfortunately for Mellisa, panic attacks had been an ongoing trauma in her life and this time, she just couldn’t shake them. As she told the Star Scoop, she believes the cause of the attacks to be the confined sequester the contestants are placed under before the game begins and not Survivor itself.
I don’t think I’ve ever cried so hard in my life, I was so scared because it does feel like you don’t have control of your body. They needed to call the producers to let them know what was going on, and one of the producers [came] in, and they basically asked me are you in, are you out, and I said, I don’t think I’m in any state to complete this. I don’t know why I can’t get a hold of myself; I was very distraught. They didn’t push me either way to stay or to go. They really wanted what was best for me and I absolutely appreciate that. They did not make me feel bad about it, they didn’t pressure me to just stay. Rumors have it that the game started the very next morning, but I have no way of knowing that. I flew home the next day. I don’t know when they started. If I would have known it was starting the next morning, maybe I would have been able to make it. I think it was being in a confined space, not speaking to anybody for a long period of time, reading every single book that I brought, my iPod was out of batteries. For somebody who suffers from panic, and the need to get out, and you know you can’t get out, just, far more than I could handle.
You might have guessed it given her background in modeling and living in Los Angeles, Mellisa was not an applicant. She was recruited through sheer chance at a bar.
I met a girl out at a bar in Hollywood, a mutual friend had known her. I don’t know how we got into such a deep conversation, but we got into a very deep conversation and at the end of the night, she gave me her business card and said, would you ever consider trying out for Survivor I think you would be awesome. And I said, I’m not so sure about that, and I really didn’t think about it. And I went home that night, and I actually went to Blockbuster to try and find the latest Survivor. I’d seen Survivor before, but I really wanted to get a feel for exactly what the game was. I went to three different stores before I found it, and I went home and I watched it that night by myself. Personally at that point in time in my life, I don’t know why she gave me her business card, but I felt like this is something I needed to do. I’ve been through many challenges in my life, been through rough situations, and I said, if I had an opportunity for 40 days, by choice to not eat and do these things, I was suffering from the panic attacks at the time…and I said what better way could I get over these things then put myself in the worst possible scenario I could, and that’s just the kind of person I am.
While I wouldn’t bet any money on it, mostly because I’m broke, I have a strong suspicion that the woman in question is Lynn Spillman. She was also the one who met Jonny Fairplay at a gas station and has done most of the recruiting for Survivor since the beginning of time.
As she says herself, production was really good with Mellisa and handled her situation well. Still, for the audience, the damage had been done and some already had knowledge that she had been slotted to appear on the season. As Jeff Probst said, the production team did not think they would need an alternate for the season and were thus left with an uneven number of people.
We didn’t bring an alternate with us. It’s a little weird. We had an alternate with us in Cook Islands because we had somebody we weren’t sure on, weren’t sure about. So we brought somebody. We had an alternate for Fiji, but when we got out there, Mellisa quit five hours before we started. There was no way to get anybody there. Had she quit three days before, we would have probably brought an alternate out. We really don’t typically bring anyone on location that’s not going to be on the show because you’re getting their hopes up and it’s pretty emotional on them. We did do it in Cook Islands, we didn’t do it on Fiji and maybe we should have.
In a way, Mellisa made a huge impact on the show, at least from behind the curtains. Since that situation, I know that production has started taking alternates on location to pretty much every season in order to avoid having something like this happen again. In fact, Jay Starrett of Millennials vs Gen X has stated that he had been an alternate for Survivor: Kaoh Rong before getting the chance to appear on his own season.
Going by the same TV Guide interview, Mellisa states that she still planned on watching Fiji and actually felt like she might be ready to try Survivor again should production have wanted her back.
I don’t know if they’d even consider having me back, but I would absolutely try Survivor again next year. I’m a stronger person. There are a lot of people laughing at me right now, and I’d like to have the opportunity to tell them I’m not this “weak prissy model” that everybody is calling me on the Internet. I’m a lot stronger now.
So my question in all of this: why she was never brought back for Game Changers? She was “technically” part of a previous Survivor cast. Besides, if they can call Sierra Dawn Thomas or Brad Culpepper Game Changers, why not Mellisa McNulty? At least she had an impact on how the show handles alternates. What was Brad’s impact? The implementation of the Culpepper math strategy?
All in all, Mellisa has been largely forgotten by the fan base. Many people in the internet community remember her simply for being the only person to ever quit the show without actually being on it. Otherwise, she goes by as a blip in the Survivor timeline mostly forgotten. Hopefully I was able to teach a few of you something new about the show and if not, at least help you remember a quirky part of the show’s history. | 2024-02-07T01:26:58.646260 | https://example.com/article/5794 |
Steeped in mystery, the operation spans at least 36 countries with an estimated cost of close to $750,000 for trademark filings alone. Due to the breadth of this ongoing activity, and with high-profile brands such as BMW, Western Digital and even US President Donald Trump currently challenging some of his marks, every rights holder should take notice.
Over the past couple of weeks, World Trademark Review – along with assistance from trademark watching platform CompuMark – has built up an extensive data set in an attempt to measure the scope of Gleissner’s global portfolio, all of which is based on publically available information. After collating it into a single document, The Gleissner Files (pdf with 69 pages), it reveals more than 1,100 registered company names, over 2,500 trademark applications and 5,300 domain names – the scope of which has been described by one expert as more “organised”, “professional” and “sophisticated” than anything they’ve seen before.
Gleissner now has 756 trademark applications at the USPTO. The most common legal representative for Gleissner-related filing at the firm is Jonathan Grant Morton, a former USPTO patent examiner who is listed on LinkedIn as a partner at Morton & Associates and also general counsel at Gleissner’s companies Bigfoot Entertainment and Fashion One.
The Gleissner Files: most used trademark jurisdictions
United States – 756 trademark applications
United Kingdom – 652
Portugal – 394
Benelux – 380
Canada – 89
Mexico – 67
European Union – 55
Turkey – 27
Philippines – 21
France – 17
With over 1,000 applications now registered (and a further 1,314 that could reach registration), Gleissner already has a sizeable trademark portfolio. A costly one too; in total, according to our estimate, the applications alone would have cost just under $750,000.
World Trademark Review in their latest analysis, they found at least 13 marks currently in opposition proceedings at the EUIPO, including the mark PEPPER being opposed by BMW, and distribution giant Connect Group opposing the mark CONNECT. In the UK, the mark TRUMP TV is currently being opposed by DTTM Operations LLC, the New York-based entity used to protect the intellectual property of US president Donald Trump. Furthermore, a number of registered marks are being sought for cancellation, including the UK mark PURPLE.COM which Western Digital is seeking to cancel. Apple has also been a previous foe in proceedings.
Over 82% of the trademark applications use one of the 1,161 UK company names that Gleissner has personally registered in the past two years – with the names spread out across trademark applications to make tracking nearly impossible.
World Trademark Review has recorded at least 5,418 domains owned by the entrepreneur (data accessed at ViewDNS.info), with 267 identifying Gleissner as the owner and the rest his domain subsidiary NextEngine Ventures. At the time of going to press, more than 3,000 of the domains are not currently active. A majority of the domains that are active are used in two ways. The first is as a holding page promoting Gleissner’s Fashion One TV channel. The second use is, bafflingly, over 1,500 domains redirecting internet users to a video entitled “Counting Numbers for Kids with Lego Figures”, hosted on the YouTube channel Kiddo TV. The video is currently at 99,000 views, with most comments referencing the unintended redirect that led users to the page.
Some of the domains that Gleissner owns has, in the past, suggested a pattern of typosquatting (eg, ‘frienjdster.com’, ‘hotmaol.com’, ‘hotrmail.com’) and there are current examples of unusual use, such as the copycat website ‘tmview.com’ we wrote about in August. There’s also ‘ui.com’, which appears to be a website for a marketing company. However, the site closely mimics that of Texas-based marketing agency TM, including copying its work and staff pages (there’s no indication the two are related; we’ve contacted TM to confirm). Furthermore, in the past 18 months, Gleissner has lost numerous domains through UDRP actions, and hundreds of the trademark applications are for one-word terms followed by ‘.com’. Meanwhile, some of the marks – many of which are now registered – appear to match well-known brand names, including BAIDU, IPHONE, ITUNES, and TESLA, and due to most of Gleissner’s trademarks being for a single word, it is possible that clashes exist with other third party product or service names.
About Konstantinos Zournas
I studied Computer Engineering and Computer Science in London, UK and now live in Athens, Greece. I love domains and building websites. I am online since 1995, learned about HTML in 1996 and about domains in 2002. I started publishing the OnlineDomain.com blog in 2012.
6 comments
Hello Konstantinos,
Monopolies, blocking Small Business Online Growth and the resulting atmosphere are setting the stage for the greatest small business Online growth spurt in history. The most efficient way to harness this growth is by leveraging your investment dollars into .COM Equimoddity Platform Assets. JAS 10/6/17 Former (Rockefeller I.B.E.C. Marketing Intelligence Analyst/Strategist) (Licensed C.B.O.E. Commodity Hedge Strategist.)
Google/Alphabets Renegade Cartel ambitions and its current losing battle with the E.U. Anti-Trust Ruling cannot stop Google/Alphabets ambitions achieved through Traffic controlled Digital Manipulation.Any ongoing algorithm changes they offer will surely foil any policing authorities Radar. The mere design of any algorithm is to accomplish dominant control of Traffic patterns to some recipients advantage, while at the same instant, forcing competitors to an extreme disadvantage, all the while effectively excluding others from the free market place. This is how Monopolies price fix and bury their competitors access to free and open markets.
Googles incessant gaming of algorithms is easily cameflauging unscrupulous skewing, in the favor of high paying Incumbent Monopolistic giants.This smothers the free open market places access to compete effectively. Squelching New Small Business’s right to compete. Making Google/Alphabet the ring leader in standing in the way of Economic Expansion.
The Google/Alphabets Traffic platform consists of a vast all encompassing Network of Manipulated Algorithm controlled Digital code operating under the radar of detection. Google/Alphabets Traffic platforms function is to effectively intercept All its incoming traffic and reroute traffic to its Googleopoly’s competitive advantage. This devious Enigma plot has become even more treacherous for competitors by Google/Alphabets introduction of New Gtlds they have added to their anti-Competitive Cartels Arsenals effectiveness in the hijacking of the internet’s traffic, giving algorithm concealed preferences to their new Tlds.
SEO, SEM(Search Engine Manipulation), is a cesspool of Digital Thieves employed to steel businesses lifesblood traffic through the SEM Platform. Its the largest assemblage of Bad Actors ever assembled in history. All supported and coddled by the (Googleopoly SEM culture). Are you willing to expose your Online business to this SEM Culture of dishonor, set in an atmosphere of criminal acts ? Get your valuable Brand out of the S.E.M. Platform before your Brands Reputation and Traffic are decimated.
Bottom Line: Google/Alphabets glaring anti-competitive business platforms strangle hold, is ominously standing in the way of our nations Economic Prosperity. Ultimately, standing in the way of our national security interests striving for Small business expansion. Small Business Expansion has been languishing in the lower 1/3 tranche of historical efficiency. The worlds competing economies that wake to these realities and reinstate free economy laws and regulations enforcement, to curb these fundamental barriers to progress, will be the free Worlds Global leaders.Will the U.S. awake soon enough? Someone will take the baton if not. JAS 9/29/17 Former (Rockefeller I.B.E.C. Marketing Intelligence Analyst/Strategist) (Licensed C.B.O.E. Commodity Hedge Strategist.) | 2024-03-24T01:26:58.646260 | https://example.com/article/2193 |
Employee surveys
Employee surveys are tools used by organizational leadership to gain feedback on and measure employee engagement, employee morale, and performance. Usually answered anonymously, surveys are also used to gain a holistic picture of employees' feelings on such areas as working conditions, supervisory impact, and motivation that regular channels of communication may not. Surveys are considered effective in this regard provided they are well-designed, effectively administered, have validity, and evoke changes and improvements.
History
The first employee surveys, commonly known as employee-attitude surveys, surfaced in industrial companies in the 1920s. Between 1944 and 1947, the National Industrial Advisory Board saw a 250% jump in companies that chose to conduct an attitude survey (within a 3,500 company group). The increased awareness in measurement tools regarding employees’ attitudes is attributed to research and observation conducted during World War II, which sought to measure morale and replicate high-morale environments. The United States Army Research Branch, for example, conducted Soldier Surveys, which recorded the opinions of more than half a million soldiers on topics ranging from food quality to confidence in leadership. Examples of early survey methods include printed questionnaires, directive interviews, and unguided interviews.
Reasons for use
Present day employee surveys are used by an estimated 50 to 75% of companies to evaluate and progress organizational health as it pertains to personnel. This may include a focus on topics such as employee engagement, workplace culture, return on human capital (ROHC), and commitment.
United States federal agencies are required by law to conduct an annual survey. The Office of Personnel Management states that employee influence is a primary reason for conducting surveys, stating, “This is your opportunity to influence change in your agency. Your participation is voluntary and your responses are confidential.”
Methodology
Organizations conduct their own surveys, contract with a survey provider, or use a combination of both. Main-line survey providers have traditionally used similar survey question types and survey length over the course of years and throughout various industries. Comparison databases provide standard ranges on which certain factors can be placed, as well as correlations between coexisting factors (allowing for emphasis on the factor with highest correlation to a desired outcome). In contrast, the advent of survey software, particularly online programs, has given organizational leadership tools to design and conduct their own surveys. In this case, the conducting leadership are responsible for tabulating and assessing the data.
Questions
A key component of employee surveys is the styling of questions. Variables in question design include:
number and sequence
length and wording
closed or open answer
factual or attitudinal
Questions that are vague, use technical jargon, are relevant to only a segment of survey-takers, or use phrasing that is interpreted differently across audiences sabotage survey effectiveness. Multiple choice answers, likewise, are a concern when there are missing plausible choices, or when choices are too wordy or too numerous.
References
Category:Workplace
Category:Employee relations | 2023-09-06T01:26:58.646260 | https://example.com/article/3148 |
Solvent promoted reversible cyclometalation in a tethered NHC iridium complex.
Reaction of [Ir(COD)(py-I(t)Bu)](+) (py-I(t)Bu = 3-tert-butyl-1-picolylimidazol-2-ylidene) with acetonitrile results in reversible intramolecular C-H bond activation of the NHC ligand and formation of [Ir(η(2):η(1)-C8H13)(py-I(t)Bu')(NCMe)](+). Coordinated COD acts as an internal hydride acceptor and acetonitrile coordination offsets the otherwise unfavourable thermodynamics of the process. | 2024-03-07T01:26:58.646260 | https://example.com/article/4771 |
1. Field of the Invention
The invention relates in general to a sense amplifier, and more particularly to a sense amplifier with a shortened voltage sensing time.
2. Description of the Related Art
In the modern age having the technology changing with each passing day, a non-volatile memory, such as a flash, is widely used in various electronic products. Conventionally, when recorded data stored in a memory cell of the flash is to be read, the memory cell in a flash array is biased through a column decoder and a row decoder so that the memory cell generates a sensing current. Thereafter, a sense amplifier detects the sensing current to judge data values stored in the memory cell.
As the technology is gradually developed, the requirement of the access speed for the flash in the market is getting higher and higher. Therefore, it is one of the important subjects in the industry to develop the technique capable of effectively increasing the data sensing speed of the sense amplifier. | 2023-12-18T01:26:58.646260 | https://example.com/article/9653 |
WASHINGTON — The United States has obtained intelligence that the son and potential successor of al Qaeda leader Osama bin Laden, Hamza bin Laden, is dead, according to three U.S. officials.
The officials would not provide details of where or when Hamza bin Laden died or if the U.S. played a role in his death. It is unclear if the U.S. has confirmed his death.
Asked by reporters on Wednesday whether the U.S. had intelligence that Hamza is dead, President Donald Trump said, "I don't want to comment on that."
Hamza bin Laden's last known public statement was released by al Qaeda's media arm in 2018. In that message, he threatened Saudi Arabia and called on the people of the Arabian peninsula to revolt.
Hamza bin Laden is believed to have been born around 1989. His father moved to Afghanistan in 1996 and declared war against the U.S. Hamza went with him and appeared in al Qaeda propaganda videos. As leader of al Qaeda, Osama bin Laden oversaw operations against Western targets that culminated in the Sept. 11, 2001, attacks on New York's World Trade Center and on the Pentagon.
__________________
Quote:
Death is a farcical pile of bullshit.
I refuse to participate.
The Oatmeal
Quote:
...it could be raining pu$$y and troops will complain and blame the leadership for not providing an equal ration of a$$holes
Billy L-Bach
Quote:
In Special Forces we had a saying: "Work hard in silence, let your success do the talking." | 2024-03-22T01:26:58.646260 | https://example.com/article/9378 |
Q:
GCM update 7.5 to 8.3.0 fatal exception
I tried to update the GCM services (Google cloud messages) libraries (from 7.5 to 8.3.0) in my Gradle project.
But now, with this new version, i'm unable to launch my previous activity which was working perfectly before.
The code which handle the following error is :
Intent in = new Intent(this, MyGcmListenerService.class);
startService(in);
MyGcmListenerService.java :
public class MyGcmListenerService extends GcmListenerService
{
private static final String TAG = "MyGcmListenerService";
@Override
public void onMessageReceived(String from, Bundle data)
{
Log.w(TAG, "onMessageReceived");
}
The returned error :
FATAL EXCEPTION: AsyncTask #1
java.lang.NullPointerException: Attempt to invoke virtual method 'int java.lang.String.hashCode()' on a null object reference
E/AndroidRuntime: at com.google.android.gms.gcm.GcmListenerService.zzo(Unknown Source)
E/AndroidRuntime: at com.google.android.gms.gcm.GcmListenerService.zza(Unknown Source)
E/AndroidRuntime: at com.google.android.gms.gcm.GcmListenerService$1.run(Unknown Source)
E/AndroidRuntime: at java.util.concurrent.ThreadPoolExecutor.runWorker(ThreadPoolExecutor.java:1112)
E/AndroidRuntime: at java.util.concurrent.ThreadPoolExecutor$Worker.run(ThreadPoolExecutor.java:587)
E/AndroidRuntime: at java.lang.Thread.run(Thread.java:818)
Gradle files :
dependencies {
compile fileTree(dir: 'libs', include: ['*.jar'])
compile 'com.android.support:appcompat-v7:21.0.3'
compile 'com.google.android.gms:play-services-gcm:8.3.0'
}
If I rollback to the previous GCM version (7.5) it's working back as expected.
Do you know what changed ? I checked the changelogs but I'm unable to found any informations about it.
https://developers.google.com/android/guides/releases
Thank you for your help
A:
Try looking at the decompiled class file for GcmListenerService using your IDE. When I do that with Android Studio for version 8.3.0, the code that appears to be throwing the exception is attempting to get the ACTION from the intent that invoked the service. Because you are invoking the service with an explicit intent, the ACTION is null.
I'm wondering why you are invoking your subclass of GcmListenerService explicitly? The normal GCM processing for message receipt is for the message to be delivered to GcmReceiver, which then passes it to the app's instance of GcmListenerService for processing. You should not be invoking your listener service explicitly, GcmReceiver does that. Take a look at the sample project.
| 2024-01-29T01:26:58.646260 | https://example.com/article/7446 |
Jesus speaks to us in the gospel today about how we need to live in the world. The rules are simple – we must live by love. Living by love is living by the way of God. The choice is clear – we can live by the world’s ways or we can live by God’s ways. We hear in the first reading that we can choose to be burned or we can choose the living waters. We hear over and over again in Scripture that God’s ways are not our ways. But where does that leave us? We are part of this world and it is not always a pretty one. I don’t have to tell you of the problems and atrocities of this world. We read about it and hear it every day in the news. It can be overwhelming.
But then Jesus calls. He calls us to be something other than the hatred, violence and oppression of this world. He calls us to react to this world only in love and to be a shining beacon of light for the rest of the world. Jesus tells us that all our actions, all our thoughts and words, all our relationships must be lived out of love. Hate is not part of the equation. Any thought, word or action that does not demonstrate love is not part of God’s plan. Jesus asks us to live a different life from the ways of the world; a life that if lived according to God’s ways can transform us and the world around us.
How can we accomplish this? How can we reveal to others the light and love of God? Lawrence reminds us every month to observe “Kindness Week” and share a little extra kindness to our brothers and sisters. It is a wonderful way of shining light in this world and giving love. But kindness is more than a gesture, it is also an attitude. In all our encounters even when we disagree, we need to approach one another with love and mutual respect. That is kindness and that is essential to following Jesus’ teaching. Everyone we meet should be able to see the light of Jesus in us and be allowed to share in that love.
That can be easier said than done! When we encounter hatred and negativity in the world, it can consume us and we forget Jesus’ message of love. That is where our faith comes in. We turn to our God of love in prayer, communion, worship, fellowship so that no matter what the world throws at us, we remain rooted in this love and can withstand any opposition.
We are not separate from this world and it is a beautiful world. God has created us to be in this world and to be part of His creation to experience and enjoy it fully. It is our reaction to this creation that matters. In our Advent Liturgy we used the analogy of the great significance of a single drop of water in a pool and how it transforms the water around it.
“A drop of water, insignificant in isolation, when dropped in a pool the ripples move out into infinity. The water is changed by one drop of water. A drop is falling into the pool of our spirit. The water begins to ripple … Creator God, as we consider the magnitude of a drop of water to touch and change the pool, may we feel the power of Your touch in our lives, the potential for our lives to touch others. All creation is Your handiwork; Your touch gives life to all that is. You who shape and form us by the breath of Your Spirit and the touch of Your Grace. You alone are able to call forth from the depths of our being the beauty of Your Spirit in us. May we surrender more deeply to Your loving touch. May You fashion and form Your heart’s desire in us.”
As we speak this prayer in our hearts and contemplate the effects of a single drop of water, may each and every one of us become co-creators in the transformation of the world by revealing God’s love. Amen. | 2024-01-07T01:26:58.646260 | https://example.com/article/9746 |
Swifty Sharpe Motorized Knife Sharpener in Pakistan
New Swifty Sharpe Motorized Knife Sharpener Incredible Wireless, Motorized Blade Sharpener! Quickly is a wireless electric knife sharpener to give you sharpening precision for all your knives and tools? Swifty Sharpe Motorized Knife Sharpener in Pakistan has a professional grade sapphire stone, which will sharpen at the perfect angle and will make any sharp razor blade in just a few seconds. The original Swifty Sharp knife sharpener and fishing tray under manual to collect metal joinery. Restores the edge of the razor on any blade in just seconds with precision sharpening power at a fraction of the price. The built in manual holds the blade at the perfect angle. Professional grade, high speed sharpening knife stones.
Swifty Sharpe Motorized Knife Sharpener
Features:
– Sharpening and edge blade edge
– Restores the edge of the razor on any blade in just a few seconds
– Precision – Sharpen power
– The wireless knife sharpener features a compact design for easy storage
– Includes safety slot for different functions: all kinds of knives, precision tools, scissors and household tools
When you call, don't forget to mention that you found this ad on OLX.com.pk
I do not wish to be contacted by telemarketers or representatives of any other website.
OLX.com.pk is a free local classifieds site. Sell anything from used cars to mobiles, furniture, laptops, clothing and more. Submit ads for free and without creating an account. If you want to buy something - here you will find interesting items, cheaper than in the store. Start buying and selling in the most easy way on OLX.com.pk. | 2023-10-13T01:26:58.646260 | https://example.com/article/3300 |
BANDAI NAMCO Studios are providing a variety of products ranging from console games to smartphone games. Utilizing the unique know-how and technology of our group, we have created many hit titles. On this page, we introduce those titles. | 2024-07-15T01:26:58.646260 | https://example.com/article/9597 |
Over the past several days, I have seen at least two dozen friends and acquaintances on Facebook and Twitter post a link to this web comic from The Oatmeal entitled “How To Suck at Your Religion.” This comic, written by Oatmeal founder Matthew Inman, was re-posted several times by friends of mine who are atheists and agnostics and who frequently pepper my news feed with thoughtful critiques of religion, so I was eager to see what Inman had to say. While he obviously paints with some broad brushstrokes––I mean, it is a comic strip after all––I am in agreement with nearly all of the basic premises. I too would like to be a part of a religious community that is not in the business of hindering the advancement of scientific knowledge, giving people weird anxieties about their sexuality, or using the name of God to “kill, hurt, hinder, or condemn.” These are caricatures which the Church has undoubtedly brought upon itself through centuries of irresponsibly using its power. But in the final lines of the comic, he says
However, does your religion inspire you to help people? Does it make you happier? Does it help you cope with the fact that you are a bag of meat sitting on a rock in outer space that someday you will DIE and you are completely powerless, helpless, and insignificant in the wake of this beautiful cosmic shitstorm we call existence? Does it help with that? Yes? Excellent! Carry on with your religion! Just keep it to your fucking self.
In addition to having a laughably narrow view of religion (1)––as some sort of existentially comforting teddy bear to which people cling in blind defiance of life’s obvious meaninglessness––I feel that Inman makes the crucial error to which so many contemporary secularists fall prey when attempting to offer a “way forward” after their critique. They are fine with allowing the narcissistic fantasy of religion to perpetuate itself, as long as its practitioners are willing to recognize it as such. This is the logic of liberal toleration: you are free to live in any number of insane or impractical ways, as long as you keep your hands to yourself. But in addition to its patron saints like John Locke and Adam Smith, Western liberalism has a few religious skeletons in its closet.
I, for one, am glad that folks like William Wilberforce, Martin Luther King Jr., and Dorothy Day were unable (and unwilling) to keep their religion to themselves, and I would assume that the improvements they offered to the liberal establishment in the West are ones that Inman and others would like to maintain. Here, I feel that the secular left might fall into the same rhetorical trap that plagues so many Tea Partiers in the United States. Just as the solution to bad government is not no government, the solution to bad religion is not no religion, but better religion (2). So for all of the injustice and hatred perpetrated in the name of God(s) around the world, religions must prayerfully and humbly repent. But the solution is not to naively abandon religion, it is to remain faithful the prophetic vision of justice, peace, and love that only religion can offer.
_______________________________________________________________________________
(1) The sort of blatant “scientism”––unwillingness to recognize that scientific discourse can be just as irrational and ideologically driven as any other––in an argument such as this one has been so well diagnosed in the half-century since Thomas Kuhn’s The Structure of Scientific Revolutions that I don’t feel the need expound on it here. If you’re interested in the topic, I’d highly recommend Terry Eagleton’s masterful arguments against the new atheists in Reason, Faith, and Revolution.
(2) Parenthetically, I would like to suggest that it is precisely the type of individualism encouraged by Inman and others that breeds this “bad religion” in the first place. | 2023-08-26T01:26:58.646260 | https://example.com/article/1347 |
Syria war: Government attacks IS enclave in south-west Published duration 11 July 2018 Related Topics Syrian civil war
image copyright Reuters image caption Syrian government forces have retaken much of Deraa province in the past three weeks
Syrian government and Russian forces are reportedly attacking an enclave held by the jihadist group Islamic State (IS) in south-western Syria.
Activists and a monitoring group said aircraft were bombing the Yarmouk Basin area, which borders Jordan and the Israeli-occupied Golan Heights.
The militants are said to have counter-attacked, targeting nearby villages.
The fighting comes after the government recaptured most of the surrounding province of Deraa from rebel factions.
Rebel commanders agreed on Friday to surrender their heavy weapons and begin handing over towns as part of a Russian-brokered agreement.
In return, the Russian military is believed to have guaranteed the safe return of the 320,000 civilians who fled their homes after the government's offensive began on 19 June, as well as the evacuation to rebel-held parts of north-western Syria for people who wish to leave.
An IS-affiliated group, the Khalid Ibn al-Walid Army, has controlled the south-western corner of Deraa province since 2014, when jihadists overran vast swathes of Syria and neighbouring Iraq and proclaimed the establishment of a "caliphate".
IS was not covered by last week's ceasefire deal and on Wednesday its positions in the Yarmouk Basin was subjected to air strikes and artillery fire.
The Syrian Observatory for Human Rights, a UK-based monitoring group, said Russian warplanes had targeted the town of Saham al-Golan early on Wednesday and that government helicopters had also dropped barrel bombs on the area.
In retaliation, IS militants attacked Hait, a rebel-held town that recently agreed to return to surrender, it added.
The pro-opposition Horan Free Media group reported that IS artillery fire killed four children and a woman in Hait.
Syrian state media meanwhile reported that troops were advancing towards Tal al-Ashari, Jallain and Zaizoun, rebel-held villages in Deraa's western countryside that have agreed to surrender.
On Tuesday, IS claimed it had carried out a suicide bombing in Zeizoun.
An IS statement said the attack had targeted a gathering of Russian and Syrian troops, killing more than 35 of them. But the Syrian Observatory said the attack put the death toll at 14 and said they were soldiers and rebel fighters "who recently reconciled" with the government.
Thousands of civilians have reportedly fled the Yarmouk Basin in anticipation of a government ground assault and headed towards the frontier with the occupied Golan Heights, which Israel captured from Syria during the 1967 Middle East war.
image copyright Reuters image caption Displaced Syrians have sought refuge along the frontier with the Israeli-occupied Golan Heights
Up to 190,000 people displaced by the assault on rebel-held areas are also gathered near the armistice line, according to the United Nations. Many do not have any shelter, leaving them exposed to harsh weather conditions, such as dusty desert winds and high temperatures.
The Syrian army's advance towards the occupied Golan Heights has also alarmed Israeli officials, who believe it may attempt to deploy soldiers along the frontier in defiance of a 1974 Separation of Forces Agreement that created a buffer zone patrolled by UN peacekeepers.
On Wednesday, the Israeli military said it had launched a Patriot missile at a drone launched from Syria, setting off air defence sirens in Israeli communities.
Israeli Prime Minister Benjamin Netanyahu meanwhile flew to Moscow to discuss "Syria, Iran and Israel's security needs" with Russian President Vladimir Putin.
Iran, Israel's arch-enemy, has deployed hundreds of troops to Syria, ostensibly as advisers to the government. Thousands of Shia militiamen armed, trained and financed by Iran have also been battling rebels alongside the Syrian army. | 2023-10-04T01:26:58.646260 | https://example.com/article/3684 |
I know that the meaty, chewy texture is something a lot of people love about beef ribs, but I have never been too fond of them personally. I find them too tough and the meat gets stuck in my teeth and I just don’t end up enjoying the actual flavour. But, I recently discovered a way I do like ribs. And it starts with the cut. We asked our butcher for ribs and I talked to him about my concern that we would end up with tough short ribs so he said let me worry about that and cut the ribs really, really thin so they absorb the marinade and also cook super fast, compared to slow cooking thicker cuts. He did this by cutting them the opposite way you normally would, right through the bone! Brilliant! We made 3 different recipes with the ribs. Root Beer Marinated Sweet Ribs, Korean BBQ Pan Fried Ribs, and Spicy/Tangy Masala Ribs. The latter were my personal favourite. Try them out, they will melt in your mouth! Mmm MmmMmm!
Ramadan Specials!
The holy month of Ramadan will be starting soon and for all those who will be fasting in these very, very long summer days, it's not always easy to work, prepare food for everyone for Iftar and to be out in the heat. We're very fortunate to have ample food, our health, the capability, and immense blessings to be able to fast comfortably but some times a little help can go a long way right? So let us do the work for you and you can spend your free time with family and on various modes of self realization and growth that is an essential part of this special month. Look at our list of specials for this month and see what sounds appealing! Call us and we'll talk more. The question really is, what will you order first?
Who wants to drink some Algae? No, it's not a dare - if we can down smoothies made of all sorts of funky veggies and use Algae face masks, why not learn more about what other benefits it has to offer? Check out what Nida has for us today!
W.P
I had promised some more detoxes or smoothies and here we are - Today I bring you a smoothie with the single ingredient that is 1,000 more powerful than fruits and veggies. You might be saying what the wha..?! If you're hoping its chocolate, sadly its not (believe me I would be a champion in that category if it were haha!) Its Klamath Blue-green algae. I'm going to give you some detail on the benefits and its source, since its not a commonly heard of ingredient. I also want to stress on the fact that LESS is MORE here.
It comes from the Klamath Lake - home to a particular type of blue green algae called Aphanizomenon flos-aquae, which is similar to spirulina and chlorella, but just a wee bit different.
Aphanizomenon flos-aquae is a type of cyano-bacteria that biologically is quite similar to spirulina. It’s actually toxic in most places where it’s grown, but Klamath Lake is remarkably pure and free of toxins so the blue-green algae there can be ingested by humans.
It is the ultimate brain food and a powerful PINEAL GLAND ACTIVATOR! It contains more protein, B12 and chlorophyll than any other food on the planet. It is rich in neuropeptide precursors which are vital for neurotransmission which allows brain neurons (neurotransmitters) to communicate with each other in learning and memory.
“The greatest value of A.F.A. is not only its nutrient concentration, but its effect on the nervous system, specifically the pituitary, pineal, and hypothalamus. People taking A.F.A. have reported an overall increase in mental alertness, mental stamina, short and long term memory, problem solving, creativity, dream recall, a greater sense of well being and centeredness.” – Dr Gabriel Cousens, health guru
Not only is Klamath Blue Green Algae a powerful brain food for the pineal,pituitary and hypothalamus, it is the most nutrient dense food on the planet! In addition to its high potency, the body only has to use a small amount of energy to metabolize it. This makes it easy for the body to absorb and assimilate its nutrients, which it is full of. Klamath Blue Green Algae is full of vitamins, minerals, trace minerals, chlorophyll, simple carbohydrates, fatty acids, neuropeptide precursors, lipids, glyco-proteins and enzymes.
Summer is finally in full swing! How many barbecues have you been to already? I know our calendar is full of outdoor events the next couple of weeks. Check out my post on Zardozi for colourful, healthy, raw, gluten free ideas on what to take to your next picnic in the park or a barbecue. If you're still stuck on ideas for what to make for an outdoor event, call us- that's where we can help sort you out. We will handle all the work for you and you just sit back and relax.
We have been working with Muneeba from Cakes by Muneeba for the past one year. We have done several events together and are also her clients. One of our clients introduced us and we have been hooked since then. So we thought it was about time we introduce her to all of you! Her cakes and cookies are not only gorgeous, they're extremely tasty too. She is extremely passionate about what she does and always goes above and beyond our expectations. That is one of the biggest reasons she has a clientele which has built up through word of mouth. My heart breaks every time I see someone cutting a cake made by her. Whenever I see such elaborate cakes, I wish I could preserve them forever just as they are. That is what cameras are for, so take a look at some of her work. I had requested Muneeba to send us a short story about her work along with some pictures. Enjoy! We know you will want to reach out to her for your next big event.
A.K
My passion for baking, decorating cakes and making desserts goes back to my childhood days. As a family, we are obsessed with sweets and desserts. My father in particular has an intense appetite for sweets and my mom loved to create new recipes. Growing up, I remember watching and admiring her. It was wonderful seeing her prepare mouthwatering desserts and gourmet cakes routinely. I aspire to be in her place one day. As a kid, whenever my mom called me to the kitchen for some assistance, I was always delighted to help her. Whenever she would let me gave her a hand I helped her and these were my happiest moments. I have to admit, I just didn't pick up the profession of making cakes and desserts randomly. It was always in my genes and I worked towards making it into a career for myself.
While my interest grows with time, I didn't start professionally until recently. Friends and families, who had tasted my products, motivated me greatly to advance my work to the next level. To polish my skills further, I took a refresher course and then started offering my products to clients. While my portfolio contains a variety of desserts, baking and decorating cakes is my core competency. I offer customized cakes from a wide range of sizes, designs, themes, fillings and decorations. Other products that have attracted customers and gained popularity include fruit tarts, customized cookies and cupcakes. I love to share ideas and immensely value any input that comes from my clients in fulfilling their orders. I feel honored to receive some very positive and encouraging reviews from my clients and they only make me strive for betterment. It has been a pleasure working with Kitchen Cultures on all their events and personal celebrations as well. Now I hope to bake for you too!
You can reach me via www.cakesbymuneeba.ca and my work can also be viewed on my Facebook page “cakes by muneeba”. I sincerely look forward to filling your life with sweetness as well as joy from my cakes. | 2023-08-09T01:26:58.646260 | https://example.com/article/3507 |
h = 0.239826 + d. What is h rounded to 5 dps?
-0.00017
Let b be -2*(-1)/(-16) - (-48833208)/(-576). What is b rounded to the nearest one thousand?
-85000
Let a = 0.11 - 5.53. Let r = a + 0.12. Round r to 0 dps.
-5
Let v = 274.181 - 272.6. What is v rounded to the nearest integer?
2
Suppose -2*f - 1034 = -7*f + b, -1039 = -5*f - 4*b. Round f to the nearest 100.
200
Let c = 6.4590394 - 3.759042. Let m = -2.7 + c. Round m to 6 decimal places.
-0.000003
Let o = 198 - 209.98. Let a = o - -11.98000493. Round a to seven dps.
0.0000049
Let d = 31.0292 - 32.74. Let m = d - -1.74. What is m rounded to 2 dps?
0.03
Let q = 25425.85 + -25424.130193. Let h = q - 1.72. What is h rounded to 5 decimal places?
-0.00019
Let b = 1.053 - 34.053. Let s = 163006.199955 - 162973.2. Let f = b + s. Round f to five decimal places.
-0.00005
Let g = -67.5 + 67.50000381. What is g rounded to seven decimal places?
0.0000038
Let z(f) = -f**2 - 12*f - 10. Let m be z(-10). Suppose 0 = -m*r - 6*r. Round r to seven dps.
0
Suppose -5*f = -3*o + 11 + 6, -o + 5*f = 11. Suppose 0 = 3*b - 62 + o. What is b rounded to the nearest 100?
0
Let u(f) = -f**3 + 8*f**2 + 11*f - 10. Let d = -8 + 17. Let c be u(d). Let y(o) = -5*o + 5. Let n be y(c). Round n to the nearest 10.
-40
Let a = 285.1 - 308. Round a to the nearest integer.
-23
Let y(k) = -46749*k**2 + 12*k + 13. Let l(w) = 140246*w**2 - 35*w - 38. Let g(a) = 4*l(a) + 11*y(a). Let j be g(-13). What is j rounded to the nearest 1000000?
8000000
Let j = 9.4 + -9.242. Let b = 0.712 + j. Let w = 10.67 - b. Round w to the nearest integer.
10
Let c = 34 + -59. Let o = -20 - c. Let y = -4.999972 + o. What is y rounded to 5 dps?
0.00003
Let p = 195359.543914822 - -1.508185178. Let k = p - 195410.05210058. Let z = k - -49. Round z to seven dps.
-0.0000006
Let w = 0.2684 + 44.6936. Let h = 9.7 - -35.3. Let o = h - w. What is o rounded to 2 dps?
0.04
Let n(a) = -a**3 + 19*a**2 - 46*a - 13. Let d be n(16). Let q(u) = u**3 - 19*u**2 - 26*u - 10. Let w be q(d). What is w rounded to the nearest 100?
-500
Let o = -70.06 + 12.51. Let l = o + 62. Let u = l - 2.26. Round u to 1 decimal place.
2.2
Let l = 387 - 365.7. Let i = l - -2.7. Let h = -23.997 + i. What is h rounded to two decimal places?
0
Let c be 1 - -2 - 4 - -3. Suppose 2692 = c*u - 89308. What is u rounded to the nearest one thousand?
46000
Let j = -1.06 - -1.06000396. Round j to six dps.
0.000004
Let h = -42.9 - -45. Let u = -8 - -7.8. Let f = u - h. Round f to the nearest integer.
-2
Let w = -3.192 + -0.048. Let b = 0.36 - w. Round b to 0 dps.
4
Let h = -140240.023 + 140211.0229995. Let w = 29 + h. What is w rounded to seven dps?
-0.0000005
Let b = -31 + 19. Let h(j) = -j**2 - 12*j - 2. Let r be h(b). Let n be 148004 + (-2 - r/(-1)). What is n rounded to the nearest ten thousand?
150000
Let o = -252.00895 - -252. What is o rounded to three decimal places?
-0.009
Let i(t) = 4*t - 31. Let h be i(7). Let y(s) = 0 + 1332818*s - 3 + 467181*s. Let q be y(h). What is q rounded to the nearest 1000000?
-5000000
Let p = -127.63 + -8.07. Round p to the nearest ten.
-140
Let o = -4.3 - -4.299796. What is o rounded to 5 dps?
-0.0002
Let n = -236992 - -236803.971. Let f = 0.029 + n. Let u = f - -188.0194. What is u rounded to three dps?
0.019
Let u be ((-8989602)/(-2))/(3/(-16)*-4). Let z = -9106932 - u. What is z rounded to the nearest 1000000?
-15000000
Let q = -4 - -7. Let n = q - 2.5. Let f = -0.4999904 + n. What is f rounded to six dps?
0.00001
Suppose f = -4*a + 8120, 32431 = 5*f + 3*a - 8237. Suppose 0 = 4*s + f + 744. Round s to the nearest 100.
-2200
Let m(n) be the first derivative of -32*n**3/3 + 7*n**2/2 - 11*n + 11. Let y be m(-6). Let i = -535 - y. What is i rounded to the nearest one hundred?
700
Let k = -14564 - -14642.9953. Let o = k + -79. Round o to three decimal places.
-0.005
Let o = -15080.0217 + 15092. Let l = 11.92 - o. Round l to two dps.
-0.06
Let t = 47.2 - 0.2. Let p = 48.9 - t. Let w = p + -1.90055. What is w rounded to 4 dps?
-0.0006
Suppose 4*q + 2*b + 193 = -3*b, 3*q + 147 = -3*b. Let v = 124 + q. Suppose -15 = 3*i + 3*t + v, 2*t = -i - 28. Round i to the nearest one hundred.
0
Let v(w) = 81839957*w + 43. Let n be v(1). Round n to the nearest 1000000.
82000000
Let y = 1.03 - -43.97. Let x = y - 44.9999966. What is x rounded to 6 decimal places?
0.000003
Let h = 0.668 + 0.002. Let l = 275 - 275.612. Let f = h + l. Round f to 2 dps.
0.06
Suppose 4*f + 2*t = -0*t + 380850, 0 = -3*f + 3*t + 285633. Suppose 44788 = 2*i - f. Round i to the nearest one hundred thousand.
100000
Suppose 0 = 17*p - 8*p - 10*p. Suppose -16*i + 14*i + 2064000 = p. Round i to the nearest 100000.
1000000
Let p = -14933 - -42542. Suppose -p = -3*m + 4*d - d, -m - 5*d = -9227. Suppose m = -2*j - 1793. Round j to the nearest one thousand.
-6000
Let o = 170.65 + -167. What is o rounded to 1 dp?
3.7
Let p(q) = 18*q**3 - 10*q**2 + 12*q - 20. Suppose 4*m - 41 = v, -5*m = 2*v - 36 + 1. Let f be p(m). Round f to the nearest one thousand.
12000
Let l = -0.858 + 0.858000879. Round l to 7 dps.
0.0000009
Let b = -9.165 + 14.5. Round b to 1 decimal place.
5.3
Suppose 4 = -s + 3*s. Let a be (-44)/12 - s/6. Let r = 13 + a. Round r to the nearest 10.
10
Let x = -0.163 + 0.162998199. What is x rounded to seven dps?
-0.0000018
Let a = 1715 - 1672.84. Let j = a + -45. Let f = -2.840173 - j. What is f rounded to five decimal places?
-0.00017
Let n = -1411.0617 + 1411. Round n to two dps.
-0.06
Let y(v) = v**2 - v - 7. Let r be y(-4). Suppose -3*k + r - 1 = 0. Suppose -s + 487996 = k*c, -c + 4*s - 2*s + 122008 = 0. Round c to the nearest ten thousand.
120000
Suppose n + 3*z = -22, -3*z - 32 = 3*n - n. Let u(p) = p**2 + 8*p - 12. Let l be u(n). Suppose -g + 154000 = -l*g. Round g to the nearest 10000.
-20000
Let o = -33 + 26. Let w = o - -7.0000002. Round w to 6 dps.
0
Let i be 768000/((-20)/25*(-3)/78). Suppose 0 = -8*p + 2*p - i. What is p rounded to the nearest 100000?
-4200000
Let t = 0.3 - 0.342. Let z = -13394.041943 + 13394. Let o = t - z. Round o to five dps.
-0.00006
Let q = 593256854 + -593257466.000706. Let m = q + 612. Round m to four dps.
-0.0007
Let k = -15307426.0758 + 15285952.5. Let t = -21430.6 - k. Let o = t + -43. Round o to 3 decimal places.
-0.024
Let v = 1.6 - 6.1. Let x = v - -4.49999911. What is x rounded to seven dps?
-0.0000009
Let t = -1395.5353 + 1396. What is t rounded to 1 decimal place?
0.5
Let m = -455984.99915 + 455886. Let t = m - -99. Round t to 4 dps.
0.0009
Let g = 469 + -468.999999327. What is g rounded to seven dps?
0.0000007
Let b = -41 + 24. Let g = b + 18.7. Let a = g - 1. What is a rounded to the nearest integer?
1
Let m be ((-23362500)/49)/(1 + (-452)/448). What is m rounded to the nearest one million?
53000000
Suppose -3*t = -t - 18. Let z = -10 + t. Let l be -41997 - 1 - z - 3. Round l to the nearest ten thousand.
-40000
Let z = 5230.008 - 5248. Let p = z + 18. Round p to two dps.
0.01
Let p = 117 + -117.015. Let r = 10.3 + -4.515. Let k = p - r. Round k to the nearest integer.
-6
Suppose -4*s = 2*v - 4*v - 10, 4*s - 4*v = 0. Suppose 2*r + 3*h = 6*r - 439, 0 = s*r - 2*h - 554. Round r to the nearest 10.
110
Suppose 5*g + 11 = 2*y, 0*g = 3*y - g - 10. Suppose -4*k - 5*v = -535985, 2 = -y*v - 7. Round k to the nearest 10000.
130000
Let p = 991362 - 1693368. Let x be 6/45 - p/(-45). Round x to the nearest one thousand.
-16000
Let p = -1022 + 1021.9923. What is p rounded to three decimal places?
-0.008
Let l = 0.02787 - 0.0375. Round l to 3 dps.
-0.01
Let l = 241 + -240.9999572. Round l to 6 dps.
0.000043
Let k(z) = -89*z - 4. Let n be k(-4). Suppose -3*m + 2*h - n = 0, 2*h = 4*m + 243 + 229. Round m to the nearest 10.
-120
Let x = -173 + 156.7. Let b = -31.9 - x. Round b to 0 dps.
-16
Let s = 0.0544 + -0.054399298. Round s to seven dps.
0.0000007
Let l(d) = 22 + 6*d - 30 - 1978*d - 1364*d. Let t be l(-3). Round t to the nearest 10000.
10000
Let w = -129218.36777 + 129219. Let p = w - 0.11753. Let o = -0.52 + p. Round o to three dps.
-0.005
Let t = -473.08131041 + 886.0813122. Let n = 413 - t. Round n to 6 dps.
-0.000002
Let f = 0.25 - 5.4. Let z = f + 0.15. Let o = z + 4.99999972. Round o to 7 decimal places.
-0.0000003
Let z be 9702/8 + (-1)/(-4). Let h(x) = 739*x + 74. Let i | 2024-01-26T01:26:58.646260 | https://example.com/article/4302 |
Oftentimes, public health overlaps with a variety of other domains and competencies in the fight to improve health. What makes the field unique is its focus on creating systems that contribute to positive health outcomes.
Leah Pope is a Health Educator at East Liberty Family Health Care Center, a Federally Qualified Health Care Center dedicated to serving Pittsburgh’s, and its surrounding area’s, underserved, underinsured, and uninsured population. At her site, she is organizing patient education groups and assisting patients in transitions of care to reduce hospital admissions.
NHC member Nicole Paul reflects on how theoretical public health care discussions as an undergrad compare to her experiences serving vulnerable populations at the Pittsburgh Mercy Family Health Center.
Abby Smith is a Maternal Child Health Coordinator at the UPMC Shadyside Family Health Center. She provides health education and case management for moms with risk factors that increase their chances of preterm birth or low birth weight in future pregnancies. | 2024-03-10T01:26:58.646260 | https://example.com/article/2978 |
/*
* See the NOTICE file distributed with this work for additional
* information regarding copyright ownership.
*
* This is free software; you can redistribute it and/or modify it
* under the terms of the GNU Lesser General Public License as
* published by the Free Software Foundation; either version 2.1 of
* the License, or (at your option) any later version.
*
* This software is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public
* License along with this software; if not, write to the Free
* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA
* 02110-1301 USA, or see the FSF site: http://www.fsf.org.
*/
package org.xwiki.mail.internal.factory.template;
import java.util.Arrays;
import java.util.Collections;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import javax.inject.Named;
import javax.mail.MessagingException;
import org.junit.jupiter.api.BeforeEach;
import org.junit.jupiter.api.Test;
import org.xwiki.bridge.DocumentAccessBridge;
import org.xwiki.mail.MimeBodyPartFactory;
import org.xwiki.model.reference.DocumentReference;
import org.xwiki.test.junit5.mockito.ComponentTest;
import org.xwiki.test.junit5.mockito.InjectMockComponents;
import org.xwiki.test.junit5.mockito.MockComponent;
import com.xpn.xwiki.api.Attachment;
import com.xpn.xwiki.doc.XWikiAttachment;
import com.xpn.xwiki.doc.XWikiDocument;
import static org.junit.jupiter.api.Assertions.assertEquals;
import static org.junit.jupiter.api.Assertions.assertThrows;
import static org.mockito.Mockito.mock;
import static org.mockito.Mockito.verify;
import static org.mockito.Mockito.verifyNoMoreInteractions;
import static org.mockito.Mockito.when;
/**
* Unit tests for {@link org.xwiki.mail.internal.factory.template.TemplateMimeBodyPartFactory}.
*
* @version $Id: 72f0feb6b5ee020827b41a1ff5e024d3120fee32 $
* @since 6.1RC1
*/
@ComponentTest
public class TemplateMimeBodyPartFactoryTest
{
private DocumentReference documentReference = new DocumentReference("wiki", "space", "page");
@InjectMockComponents
private TemplateMimeBodyPartFactory templateMimeBodyPartFactory;
@MockComponent
private MailTemplateManager templateManager;
@MockComponent
@Named("text/html")
private MimeBodyPartFactory<String> htmlMimeBodyPartFactory;
@MockComponent
private DocumentAccessBridge dab;
@MockComponent
private AttachmentConverter attachmentConverter;
@BeforeEach
void setUp() throws Exception
{
when(this.templateManager.evaluate(this.documentReference, "text", new HashMap<>(), null))
.thenReturn("Hello John Doe, john@doe.com");
when(this.templateManager.evaluate(this.documentReference, "html", new HashMap<>(), null))
.thenReturn("Hello <b>John Doe</b> <br />john@doe.com");
}
@Test
void createWithoutAttachment() throws Exception
{
this.templateMimeBodyPartFactory.create(this.documentReference,
Collections.singletonMap("velocityVariables", new HashMap<String, String>()));
verify(this.htmlMimeBodyPartFactory).create("Hello <b>John Doe</b> <br />john@doe.com",
Collections.singletonMap("alternate", "Hello John Doe, john@doe.com"));
}
@Test
void createWithAttachment() throws Exception
{
Attachment attachment = mock(Attachment.class);
List<Attachment> attachments = Collections.singletonList(attachment);
Map<String, Object> bodyPartParameters = new HashMap<>();
bodyPartParameters.put("velocityVariables", new HashMap<String, String>());
bodyPartParameters.put("attachments", attachments);
this.templateMimeBodyPartFactory.create(this.documentReference, bodyPartParameters);
Map<String, Object> htmlParameters = new HashMap<>();
htmlParameters.put("alternate", "Hello John Doe, john@doe.com");
htmlParameters.put("attachments", attachments);
verify(this.htmlMimeBodyPartFactory).create("Hello <b>John Doe</b> <br />john@doe.com", htmlParameters);
}
@Test
void createWithAttachmentAndTemplateAttachments() throws Exception
{
Attachment attachment1 = mock(Attachment.class, "attachment1");
Map<String, Object> bodyPartParameters = new HashMap<>();
bodyPartParameters.put("velocityVariables", new HashMap<String, String>());
bodyPartParameters.put("attachments", Collections.singletonList(attachment1));
bodyPartParameters.put("includeTemplateAttachments", true);
// Mock the retrieval and conversion of attachments from the Template document
XWikiDocument xwikiDocument = mock(XWikiDocument.class);
when(this.dab.getDocumentInstance(this.documentReference)).thenReturn(xwikiDocument);
XWikiAttachment xwikiAttachment = mock(XWikiAttachment.class);
when(xwikiDocument.getAttachmentList()).thenReturn(Collections.singletonList(xwikiAttachment));
Attachment attachment2 = mock(Attachment.class, "attachment2");
when(this.attachmentConverter.convert(Collections.singletonList(xwikiAttachment))).thenReturn(
Collections.singletonList(attachment2));
this.templateMimeBodyPartFactory.create(this.documentReference, bodyPartParameters);
Map<String, Object> htmlParameters = new HashMap<>();
htmlParameters.put("alternate", "Hello John Doe, john@doe.com");
htmlParameters.put("attachments", Arrays.asList(attachment1, attachment2));
verify(this.htmlMimeBodyPartFactory).create("Hello <b>John Doe</b> <br />john@doe.com", htmlParameters);
}
@Test
void createWithAttachmentAndTemplateAttachmentsWhenError() throws Exception
{
Attachment attachment1 = mock(Attachment.class, "attachment1");
Map<String, Object> bodyPartParameters = new HashMap<>();
bodyPartParameters.put("velocityVariables", new HashMap<String, String>());
bodyPartParameters.put("attachments", Collections.singletonList(attachment1));
bodyPartParameters.put("includeTemplateAttachments", true);
// Mock the retrieval and conversion of attachments from the Template document
when(this.dab.getDocumentInstance(this.documentReference)).thenThrow(new Exception("error"));
Throwable exception = assertThrows(MessagingException.class, () -> {
this.templateMimeBodyPartFactory.create(this.documentReference, bodyPartParameters);
});
assertEquals("Failed to include attachments from the Mail Template [wiki:space.page]", exception.getMessage());
verifyNoMoreInteractions(this.htmlMimeBodyPartFactory);
}
} | 2024-07-01T01:26:58.646260 | https://example.com/article/1033 |
15.09.2003
+ 3.05.2004
Central hadron production {#central-hadron-production .unnumbered}
=========================
in crossing of dedicated hadronic beams [^1] {#in-crossing-of-dedicated-hadronic-beams .unnumbered}
=============================================
[**Peter Minkowski**]{}\
Institute for Theoretical Physics, University of Bern,\
CH-3012 Bern, Switzerland
[**Abstract\
**]{}
The physics potential {#the-physics-potential .unnumbered}
---------------------
of an in depth experimental investigation {#of-an-in-depth-experimental-investigation .unnumbered}
-----------------------------------------
Introduction
============
I shall begin a historical survey, quoting a recent article [@KaKhoMaRy] entitled
“Central exclusive diffractive production as a spin–parity analyser:\
from hadrons to Higgs” , written by four authors : A.B. Kaidalov, V.A. Khoze, A.D. Martin and M.G. Ryskin.
’Pour fixer les idées’ , let me reproduce the first figure of the above paper
1.cm
-1.5cm
In figure \[fig1\] the reaction of central type
$$\label{eq:1}
\begin{array}{l}
\left \lbrace
\begin{array}{c}
\hspace*{0.3cm} H_{\ 1} \ ( \ p_{\ 1} \ ; \ q.\# 1 \ )
\vspace*{0.3cm} \\
+
\ H_{\ 2} \ ( \ p_{\ 2} \ ; \ q.\# 2 \ )
\end{array}
\ \right \rbrace
\ \rightarrow
\left \lbrace
\begin{array}{c}
\hspace*{0.3cm} H_{\ 3} \ ( \ p_{\ 3} \ ; \ q.\# 3 \ )
\vspace*{0.3cm} \\
+
\ H_{\ 4} \ ( \ p_{\ 4} \ ; \ q.\# 4 \ )
\vspace*{0.3cm} \\
+
\ \left \lbrack
\ h_{\ c} \ ( \ p_{\ c} \ ; \ q.\# c \ )
\ + \ X_{\ c}
\ \right \rbrack
\end{array}
\right \rbrace
\end{array}$$
is represented with the following identifications
1\) initial and tagged final hadron pairs inducing central production
$$\label{eq:2}
\begin{array}{l}
H_{\ 1 \ , \ 2} \ :
\ \begin{array}{ll}
\mbox{initial hadron pair} & \hspace*{1.2cm} \mbox{with}
\end{array}
\ \left \lbrace
\begin{array}{cr}
\mbox{momenta} & p_{\ 1 \ , \ 2}
\vspace*{0.3cm} \\
\mbox{and $q. \#$} & q_{\ 1 \ , \ 2}
\end{array}
\right \rbrace
\vspace*{0.3cm} \\
H_{\ 3 \ ,\ 4} \ :
\ \begin{array}{ll}
\begin{array}{l}
\ \mbox{$3 \ (\leftarrow \ 1)$ and $4 \ (\leftarrow \ 2)$}
\vspace*{0.3cm} \\
\mbox{associated hadron pair}
\end{array}
& \mbox{with}
\end{array}
\ \left \lbrace
\begin{array}{cr}
\mbox{momenta} & p_{\ 3 \ , \ 4}
\vspace*{0.3cm} \\
\mbox{and $q. \#$} & q_{\ 3 \ , \ 4}
\end{array}
\right \rbrace
\end{array}$$
2\) centrally produced (hadronic) system $h_{\ c}$ [*conditioned*]{} by $h_{\ c} \ | \ X_{\ c}$
$$\label{eq:3}
\begin{array}{l}
\begin{array}{lll ll}
h_{\ c} & : &
\begin{array}{l}
\mbox{centrally produced}
\vspace*{0.3cm} \\
\mbox{system of interest}
\end{array}
& \mbox{with}
& \hspace*{-0.1cm} \left \lbrace
\begin{array}{cl}
\mbox{momentum} & p_{\ c}
\vspace*{0.3cm} \\
\mbox{mass} & M_{\ c} = \sqrt{p_{\ c}^{\ 2}}
\vspace*{0.3cm} \\
\mbox{and $q. \#$} & q_{\ c}
\end{array}
\right \rbrace
\vspace*{0.3cm} \\
X_{\ c} & : &
\mbox{{\it specified conditions}}
& &
\left \lbrace
\begin{array}{cl}
\mbox{optimized to isolate} \ h_{\ c}
\vspace*{0.3cm} \\
\mbox{from background}
\end{array}
\right \rbrace
\end{array}
\end{array}$$
As is illustrated by the range of topics discussed in ref. [@KaKhoMaRy] , the general issue of central production is not restricted to strong interactions limited as far as quark- and antiquark flavors are concerned to the three light ones u,d and s, denoted $ QCD_{\ 3}$ hereafter.
[*This is our main focus here.*]{}
Rather at sufficiently high c.m. energy strong and electrweak synthesis of the central system ’$h_{\ c}$’ well includes the following processes, becoming dominantly reducible to fusion of virtual gauge boson pairs formed out of the sequence gluon (g) , photon ($\gamma$) , W , Z. We list only the combinations where $h_{\ c} \ = \ Q \overline{Q}$ $g \ g \ ( \ g \ \gamma \ , \ \gamma \gamma \ )
\ \rightarrow \ Q \ \overline{Q}$ , $g \ \gamma \ \rightarrow \ Q \ \overline{Q}$ for heavy flavors $Q \ = \ c \ , \ b \ , \ t$ and the top quark induced hadronic production of Higgs boson(s), where $h_{\ c} \ = h^{\ JP}$ $g \ g \ \rightarrow \ t \overline{t} \ \rightarrow \ h^{\ JP}$.
i\) hadronic production of (single) heavy quark-antiquark pairs, both bound and open
$$\label{eq:4}
\begin{array}{l}
h_{\ c} \ ( \ QCD_{\ 6} \ ) \ =
\left \lbrace
\begin{array}{lll ll}
c \overline{c} & : &
J/ \Psi, \chi \ , \cdots & , & D \overline{D} \ , \cdots
\vspace*{0.3cm} \\
b \overline{b} & : &
Y \ , \chi_{\ b} \ , \cdots & , & B \overline{B} \ , \cdots
\vspace*{0.3cm} \\
t \overline{t} & &
\end{array}
\right \rbrace
\vspace*{0.3cm} \\
g \ g \ \rightarrow \ Q \overline{Q}
\end{array}$$
ii\) hadronic production of (single) Higgs bosons $h^{\ PC}$
$$\label{eq:5}
\begin{array}{l}
h_{\ c} \ ( \ QCD_{\ 6} \ , \ {\cal{Y}}_{\ h^{\ PC} \ t \overline{t}} \ ) \ =
\left \lbrace
\begin{array}{l}
h^{\ ++} \ , \ h^{\ -+}
\end{array}
\right \rbrace
\vspace*{0.3cm} \\
g \ g \ \rightarrow \ t \overline{t} \ \rightarrow \ h^{\ JP}
\end{array}$$
In eq. (\[eq:5\]) ${\cal{Y}}_{\ h^{\ PC} \ t \overline{t}}$ denotes the Yukawa coupling between the Higgs boson(s) and the top quark.
The association of central production with [*perturbatively preconceived*]{} gauge boson fusion is not fortuitous. It goes back to seminal work on multiparticle production mainly of electrons and positrons in QED, by Landau, Lifschitz, Pomeranchuk and others. I only wish to cite a selcted subset for historical accuracy [@LandauPom] .
The perturbative approach to QED governed high energy elastic scattering amplitudes for initial particle pairs $e^{-} e^{\pm}$ , $e^{-} p$ , $e^{-} \gamma$ , $\gamma p$ and $\gamma \gamma$ was pioneered by Cheng and Wu [@ChengWu] . The proton can be replaced by a nucleus (A, Z), where the nuclear charge $Q \ = \ Z e$ serves to represent ’strong’ coupling, for large Z.
Theoretical expectations for primary gluonic binary (gb) Regge trajectories
===========================================================================
We follow the identification of the gluonic binary states lowest in mass discussed in ref. [@PMWO].
1.cm
-1.5cm
It shall be clear, that here we follow a combination of [*hypotheses and theoretical expectations*]{}. We will comment on alternatives below.
We begin with the spectrosopic classification of gluonic binaries [@HFPM], which, apart from the confined nature of binary gluons, is identical to the classification of photon binaries [@Land], [@Yang].
It will prove useful to obtain a ’Richtwert’ for the inverse slope of the rho-Regge trajectory at 0 momentum transfer, since it is this quantity which sets the unit of mass square with respect to which hadronic resonances, gb and others are to be placed in the simplified harmonic [*and zero width-*]{} approximation.
In this objective we define the quantity $m_{\ \varrho}^{\ 2} \ ( \ 0 \ = \ t \ )$, which represents the $\varrho$ mass square as seen in the limit of spacelike momentum transfer $t \ \rightarrow \ 0$ through the electromagnetic form factor of charged pions $F_{\ \gamma}^{\ \pi^{\ +}} \ ( \ t \ )$
$$\label{eq:6}
\begin{array}{l}
m_{\ \varrho}^{\ - 2} \ ( \ 0 \ ) \ =
\ \left .
d \ F_{\ \gamma}^{\ \pi^{\ +}} \ ( \ t \ ) \ / \ ( \ dt \ )
\ \right |_{\ t \ = \ 0}
\ = \ \frac{1}{6} \ \left \langle \ r^{\ 2}
\ \right \rangle^{\ \pi^{\ +}}_{\ \gamma}
\end{array}$$
In eq. (\[eq:6\]) $\left \langle \ r^{\ 2} \ \right \rangle^{\ \pi^{\ +}}_{\ \gamma}$ denotes the e.m. mean square charge radius of charged pions.
This quantity is presently beeing investigated by Caprini, Colangelo and\
Leutwyler [@CaCoLeu], from where I quote the preliminary result
$$\label{eq:7}
\begin{array}{l}
\left \langle \ r^{\ 2}
\ \right \rangle^{\ \pi^{\ +}}_{\ \gamma}
\ =
\ \left \lbrace
\begin{array}{l}
0.4332 \ \pm \ 0.005 \ (stat.) \ \pm
\ 0.0004 \ (syst.) \ \pm \ 0.0004 \ (P)
\vspace*{0.3cm} \\
\ \rightarrow \ 0.4332 \ \pm \ 1.3 \ \% \ \mbox{fm}^{\ 2}
\end{array}
\right .
\end{array}$$
Converting to GeV units we obtain
$$\label{eq:8}
\begin{array}{lll}
m_{\ \varrho}^{\ 2} \ ( \ 0 \ ) & = &
0.5393 \ \pm \ 1.3 \ \% \ \mbox{GeV}^{\ 2}
\vspace*{0.3cm} \\
m_{\ \varrho}^{\ 2} \ ( \ 0 \ ) \ - \ m_{\ \pi^{0}}^{\ 2}
& = &
0.5211 \ \pm \ 1.3 \ \% \ \mbox{GeV}^{\ 2}
\vspace*{0.3cm} \\
& = &
\left ( \ 721.9 \ \pm \ 0.65 \ \% \ \mbox{MeV} \ \right )^{\ 2}
\end{array}$$
The quantity $m_{\ \varrho} \ ( \ 0 \ )$ in eq. (\[eq:8\]) deviates substantially from the resonance parameters of the rho, whether obtained from the pole position in the complex energy plane or other parametrizations of physical cross sections. For comparison I quote a recent determination by the Kloe collaboration [@Kloe]
$$\label{eq:8a}
\begin{array}{l}
m_{\ \varrho} \ = \ 775.9 \ \pm \ 0.5 \ \pm \ 0.3 \ \mbox{MeV}
\vspace*{0.3cm} \\
\Gamma_{\ \varrho} \ = \ 143.9 \ \pm \ 1.3 \ \pm \ 1.1 \ \mbox{MeV}
\end{array}$$
The relation to the inverse Regge slope parameter $ ( \ \alpha^{'} \ )^{\ -1}$ is
$$\label{eq:9}
\begin{array}{l}
\begin{array}{lll}
( \ \alpha^{'} \ )^{\ -1}
& = &
2 \ ( \ m_{\ \varrho}^{\ 2} \ ( \ 0 \ ) \ - \ m_{\ \pi^{0}}^{\ 2} \ )
\vspace*{0.3cm} \\
& = &
1.0422 \ \pm \ 1.3 \ \% \ \mbox{GeV}^{\ 2}
\vspace*{0.3cm} \\
& = & ( \ 1.0209 \ \pm \ 0.65 \ \% \ \mbox{GeV} \ )^{\ 2}
\end{array}
\end{array}$$
We remark here, that the relation in eq. (\[eq:9\]) is [*not*]{} a rigorous one. We can compare with the direct $t \ > \ 0$ spectroscopic masses along the $\Lambda$ baryon trajectory, assumed unperturbed
$$\label{eq:10}
\begin{array}{l}
\frac{1}{2} \left ( \ m^{2} \ ( \ \Lambda^{\ 5/2 +} \ ) \ -
m^{2} \ ( \ \Lambda^{\ 1/2 +} \ ) \ \right )
\ = \ ( \ \alpha^{'}_{\ \Lambda} \ )^{\ -1}
= 1.034 \ \pm \ 0.010 \ \mbox{GeV}^{\ 2}
\vspace*{0.3cm} \\
\frac{1}{4} \left ( \ m^{2} \ ( \ \Lambda^{\ 9/2 +} ) \ -
m^{2} \ ( \ \Lambda^{\ 1/2 +} )
\ \right ) \ =
\ ( \ \alpha^{'}_{\ \Lambda} \ )^{\ -1}
\ =
\ 1.069 \ \pm \ 0.024 \ \mbox{GeV}^{\ 2}
\end{array}$$
Quantum numbers of binary gluonic mesons
========================================
The binary gluon system is only singled out in the present discussion, because it is expected to contain gluonic meson resonances lower in mass, than ternary or more complex multi-gluonic mesons.
Let us first consider the finite dimensional (nonunitary) representations of $SL2C \ \times \ SL2C$ restricted and unrestricted to the covering group of the real Lorentz group. Details are presented in appendices A.1 and A.2 .
To this end we associate with a bosonic resonance a free, massive state or collection of spin states. Let the total spin be J. The spinor wave functions are obtained by direct products of full and chiral spin 1/2 spinors, and the four-momentumi p, neglecting here the width of the associated resonance.
$$\label{eq:11}
\begin{array}{l}
t_{\ \alpha_{1} \alpha_{ 2} \cdots \alpha_{N}}
\ ( \ p \ ; \ \left \lbrace spin \right \rbrace \ ) \ e^{\ - i p x}
\ =
\ \left \langle \ \Omega \ \right |
\ \phi_{\ \alpha_{1} \alpha_{ 2} \cdots \alpha_{N}} \ ( \ x \ )
\ \left | \ p \ ; \ \left \lbrace spin \right \rbrace
\ \right \rangle
\vspace*{0.3cm} \\
N \ = \ 2 \ J
\hspace*{0.3cm} , \hspace*{0.3cm}
p^{\ 2} \ = \ M^{\ 2}
\hspace*{0.3cm} , \hspace*{0.3cm}
p^{\ 0} \ = \ E \ \ge \ M
\vspace*{0.3cm} \\
t_{\ \alpha_{1} \alpha_{ 2} \cdots \alpha_{N}}
\ = \ t_{\ \underline{\alpha}}
\hspace*{0.3cm} : \hspace*{0.3cm}
\begin{array}{l}
\mbox{totally symmetric under}
\vspace*{0.3cm} \\
\mbox{permutations of the indices}
\end{array}
\ \alpha_{1} \ \cdots \ \alpha_{N}
\vspace*{0.3cm} \\
\alpha_{j} \ = \ 1,2
\hspace*{0.3cm} , \hspace*{0.3cm}
j \ = \ 1 \ \cdots \ N
\vspace*{0.2cm} \\
\hline
\vspace*{-0.2cm} \\
\widetilde{t}^{\ \dot{\gamma}_{1} \dot{\gamma}_{ 2} \cdots
\dot{\gamma}_{N}}
\ ( \ p \ ; \ \left \lbrace spin \right \rbrace \ ) \ e^{\ - i p x}
\ =
\ \left \langle \ \Omega \ \right |
\ \psi^{\ \dot{\gamma}_{1} \dot{\gamma}_{ 2} \cdots
\dot{\gamma}_{N}} \ ( \ x \ )
\ \left | \ p \ ; \ \left \lbrace spin \right \rbrace
\ \right \rangle
\vspace*{0.3cm} \\
\widetilde{t}^{\ \dot{\gamma}_{1} \dot{\gamma}_{ 2} \cdots
\dot{\gamma}_{N}}
\ = \ \widetilde{t}^{\ \underline{\dot{\gamma}}}
\hspace*{0.3cm} : \hspace*{0.3cm}
\begin{array}{l}
\mbox{totally symmetric under}
\vspace*{0.3cm} \\
\mbox{permutations of the indices}
\end{array}
\ \dot{\gamma}_{1} \ \cdots \ \dot{\gamma}_{N}
\vspace*{0.3cm} \\
\dot{\gamma}_{j} \ = \ 1,2
\hspace*{0.3cm} , \hspace*{0.3cm}
j \ = \ 1 \ \cdots \ N
\end{array}$$
In eq. (\[eq:11\]) $\left \lbrace spin \right \rbrace$ denotes the spin state, to be specified in a general frame of motion, and $( \ \phi_{\ \underline{\alpha}} \ ,
\ \psi^{\ \underline{\dot{\gamma}}} \ )$ a pair of free fields, ( right chiral , left chiral ) , associated with the particle in question.
The transformation rules of the spinor wave functions $( \ t_{\ \underline{\alpha}}
\ , \ \widetilde{t}^{\ \underline{\dot{\gamma}}} \ )$ are
$$\label{eq:12}
\begin{array}{l}
\left \lbrace spin \right \rbrace \ \rightarrow \ s
\hspace*{0.3cm} , \hspace*{0.3cm}
\# \ s \ = \ N \ + \ 1
\vspace*{0.3cm} \\
t_{\ \underline{\alpha}}
\ ( \ \Lambda p \ ; \ s \ )
\ =
\ S_{\ \underline{\alpha}}^{\hspace*{0.3cm} \underline{\beta}} \ ( \ a \ )
\ t_{\ \underline{\beta}}
\ ( \ p \ ; \ s^{'} \ )
\ D_{\ s \ s^{'} }
\vspace*{0.3cm} \\
\widetilde{t}^{\ \underline{\dot{\gamma}}}
\ ( \ \Lambda p \ ; \ s \ )
\ =
\ \widetilde{S}^{\ \underline{\dot{\gamma}}}_{\hspace*{0.4cm}
\underline{\dot{\delta}}} \ ( \ b \ )
\ \widetilde{t}^{\ \underline{\dot{\delta}}}
\ ( \ p \ ; \ s^{'} \ )
\ D_{\ s \ s^{'} }
\vspace*{0.3cm} \\
D_{\ s \ s^{'} } \ = \ D_{\ s \ s^{'} }^{\ J} \ ( \ \Lambda \ , \ p \ )
\hspace*{0.3cm} ; \hspace*{0.3cm}
b \ = \overline{a}
\end{array}$$
The sought representations of the Lorentz group are obtained as symmetric products of the spin 1/2 chiral spinors. They are presented in appendix A.1 .
There is a [*small*]{} step from binary photon to binary gluon compounds, even though their classification with respect to quantum numbers $J^{\ PC}$ is identical.
To see this let me first discuss the $SU3_{\ c}$ gauge invariant binary gauge boson operator
$$\label{eq:101}
\begin{array}{l}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ =
\vspace*{0.3cm} \\
\hspace*{0.5cm}
F_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack}
\ ( \ x_{\ 1} \ ; \ A \ )
\ U \ ( \ x_{\ 1} \ , \ A \ ; \ x_{\ 2} \ , \ B \ )
\ F_{\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 2} \ ; \ B \ )
\vspace*{0.3cm} \\
A \ , \ B \ , \cdots \ = \ 1, \cdots , 8
\end{array}$$
Adjoint representation indices referring to the color gauge group are denoted by A , B in eq. (\[eq:101\]). Summation over repeated such indices is implied.
$F_{\ \left \lbrack \ \mu \ \nu \ \right \rbrack} \ ( \ x \ ; \ A \ )$ denote the color octet of field strengths.
The quantity $ U \ ( \ x \ , \ A \ ; \ y \ , \ B \ )$ in eq. (\[eq:101\]) denotes the octet string operator, i. e. the path ordered exponential over a straight line path ${\cal{C}}$ from y to x.
$$\label{eq:102}
\begin{array}{l}
U \ ( \ x \ , \ A \ ; \ y \ , \ B \ )
\ =
P \ \exp
\ \left (
\ \left .
{\displaystyle{\int}}_{\ y}^{\ x}
\ \right |_{\ {\cal{C}}}
\ d \ z^{\ \mu}
\ \frac{1}{i} \ V_{\ \mu} \ ( \ z \ , \ D \ ) \ {\cal{F}}_{\ D}
\ \right )_{\ A \ B}
\vspace*{0.3cm} \\
\left ( \ {\cal{F}}_{\ D} \ \right )_{\ A B} \ = \ i \ f_{\ A \ D \ B}
\end{array}$$
In eq. (\[eq:102\]) $f_{\ A \ D \ B}$ denotes the structure constants of $SU3_{\ c}$ and ${\cal{F}}_{\ D}$ the generators of its adjoint representation. $ V_{\ \mu} \ ( \ z \ , \ D \ )$ denote the octet of field potentials.
Properties pertaining to the octet string operators, field strengths and their potentials are collected in appendix A.3 .
The extensive discussion of Stokes relations in appendix A.3 serves here to present as clear an argument as possible, why the octet string operator\
$\left . U \ ( \ x \ , \ A \ ; \ y \ , \ B \ ) \ \right |_{\ {\cal{C}}}$ taken over a straight line path ${\cal{C}}$ attached to two field strength operators $F_{\ \left \lbrack \ \mu_{k} \ \nu_{k} \ \right \rbrack}
\ ( \ x_{\ k} \ ; \ A_{\ k} \ ) \ ; \ k \ = 1,2$ at the ends of the string – as displayed in eqs. (\[eq:101\]) and (\[eq:102\] – form a configuration similar to an $H_{\ 2}$ molecule, representing an energetically favored gluonic meson, i.e. a hadronic resonance susceptible of identification specifically in central production.
The band structure of the $H_{\ 2}$ molecule would then translate into the possible quantum numbers of the associated binary gluonic mesons, [*so defined*]{}, in appropriately adapted analogy.
From a purely theoretical point of view it has to be stressed, that this remains at the present stage [*a hypothesis*]{}, subject to further tests, extending the existing analyses in refs. [@PMWO] and [@HFPM] as well as related and/or alternative points of view, to be substancified below.
To illustrate the molecular aspect I reproduce in figure \[fig10\] the gauge boson action density in a lattice calculcation of a nucleon [@Schier]
-0.0cm
-0.5cm
-0.5cm
The bilinear operator in eq. (\[eq:101\]) satisfies Bose symmetry
$$\label{eq:103}
\begin{array}{l}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ =
B_{\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack}
\ ( \ x_{\ 2} \ , \ x_{\ 1} \ )
\ \rightarrow
\vspace*{0.3cm} \\
\widehat{C}^{\ -1}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ \widehat{C}
\ =
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\end{array}$$
In eq. (\[eq:103\]) $\widehat{C}$ denotes the charge conjugation operator.
We shall consider matrix elements of the form
$$\label{eq:104}
\begin{array}{l}
\left \langle
\ \emptyset
\ \right |
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ \left |
\ gb \ ( \ J^{\ P\ C} \ ) \ ; \ p \ ,
\ \left \lbrace spin \right \rbrace
\ \right \rangle \ \rightarrow
\vspace*{0.3cm} \\
\exp^{\ - i p X}
\ \widetilde{t}_{\ \underline{.}} \ ( \ z \ , \ p \ , \ J^{\ P\ C} \ ; \ . \ )
\hspace*{0.3cm} \mbox{with} \hspace*{0.3cm}
\vspace*{0.3cm} \\
\left |
\ \emptyset
\ \right \rangle \ : \ \mbox{ground state}
\hspace*{0.2cm} , \hspace*{0.2cm}
X \ = \ \frac{1}{2}
\ \left ( \ x_{\ 1} \ + \ x_{\ 2} \ \right )
\hspace*{0.2cm} , \hspace*{0.2cm}
z \ = \ \ \left ( \ x_{\ 1} \ - \ x_{\ 2} \ \right )
\vspace*{0.3cm} \\
J^{\ P \ C} \ : \ \mbox{total spin , parity , C-parity}
\hspace*{0.2cm} ; \hspace*{0.2cm}
p \ : \ \mbox{c.m. four momentum}
\vspace*{0.3cm} \\
\underline{.} \ : \ \mbox{spinor representation for}
\ \left \lbrack \ \mu_{\ 1} \ \nu_{\ 1} \ \right \rbrack \ ,
\ \left \lbrack \ \mu_{\ 2} \ \nu_{\ 2} \ \right \rbrack
\vspace*{0.3cm} \\
. \ : \ \mbox{spin state}
\end{array}$$
In eq. (\[eq:104\]) $J^{\ P \ C}$ , p and $. \ = \ \left \lbrace spin \right \rbrace$ refer to properties of the gluonic meson in question (gb) , whereas $\underline{.} \ \leftrightarrow
\ \left \lbrack \ \mu_{\ 1} \ \nu_{\ 1} \ \right \rbrack \ ,
\ \left \lbrack \ \mu_{\ 2} \ \nu_{\ 2} \ \right \rbrack$ and z refer to variables initrinsic to the operator $B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )$ .
The four momentum p is introduced, as if $gb \ ( \ J^{\ P\ C} \ ) \ ; \ p \ , \ \left \lbrace spin \right \rbrace$ would correspond to a [*stable*]{} particle. This is at best approximately justified in the zero width approximation, which we shall [*not*]{} a priori assume to be valid.
Nevertheless gb-s will manifest themselves as poles in complex momentum planes, corresponding to analytic continuation of strong interaction scattering amplitudes. The latter refer to stable particles, like pions, kaons and baryons, in the limit where both electromagnetic and weak interactions are neglected.
Hence, ignoring the above complication for the time beeing, the mass of $gb \ ( \ J^{\ P \ C} \ )$ is defined through p
$$\label{eq:105}
\begin{array}{l}
m^{\ 2} \ = \ p_{\ \mu} \ p^{\ \mu}
\hspace*{0.2cm} ; \hspace*{0.2cm}
E_{\ p} \ = \ \sqrt{\ p^{\ 2} \ + \ m^{\ 2}}
\vspace*{0.3cm} \\
m \ = \ m \ \left ( \ gb \ ( \ J^{\ P \ C} \ ) \ \right )
\end{array}$$
As a consequence of eq. (\[eq:103\]) we have for [*binary*]{} gluonic mesons $C \ = \ +$ throughout.
$$\label{eq:106}
\begin{array}{l}
gb \ ( \ J^{\ P \ C} \ ) \ \rightarrow
gb \ ( \ J^{\ P \ +} \ )
\end{array}$$
The relativistic spin $\underline{.}$ , processed as outlined in appendix A.1 , combines the same way as in the nonrelativistic case to a total spin $S_{\ 12 }$
$$\label{eq:107}
\begin{array}{l}
S_{\ 12} \ = \ S_{\ 12}^{\ +} \ + \ S_{\ 12}^{\ -}
\vspace*{0.3cm} \\
\left (
\ S_{\ 12}^{\ +} \ = \ 0 \ , \ 2
\hspace*{0.2cm} \leftrightarrow \hspace*{0.2cm}
P \ = \ +
\ \right )
\hspace*{0.2cm} ; \hspace*{0.2cm}
\left (
\ S_{\ 12}^{\ -} \ = \ 1
\hspace*{0.2cm} \leftrightarrow \hspace*{0.2cm}
P \ = \ -
\ \right )
\end{array}$$
The total spin states $S_{\ 12}$ in eq. (\[eq:107\]) are subject to transversity conditions, to which we will turn below. But independtly thereof the spectrum of $gb \ ( \ J^{\ P \ +} \ )$ at this stage splits into three
$$\label{eq:108}
\begin{array}{l}
gb \ ( \ J^{\ P \ +} \ )
\ \begin{array}{lll ll}
\nearrow & gb \ ( \ J^{\ + \ +} \ ) & , & S_{\ 12}^{\ +} \ = \ 2
& \hspace*{0.3cm} : \hspace*{0.3cm} II^{\ +}
\vspace*{0.3cm} \\
\rightarrow & gb \ ( \ J^{\ + \ +} \ ) & , & S_{\ 12}^{\ +} \ = \ 0
& \hspace*{0.3cm} : \hspace*{0.3cm} I^{\ +}
\vspace*{0.3cm} \\
\searrow & gb \ ( \ J^{\ - \ +} \ ) & , & S_{\ 12}^{\ -} \ = \ 1
& \hspace*{0.3cm} : \hspace*{0.3cm} I^{\ -}
\end{array}
\end{array}$$
The three spectral types shall be denoted as in eq. (\[eq:108\]) : $II^{\ +}$ , $I^{\ +}$ and $I^{\ -}$ , where the superfix stands for parity.
To clarify the spin structure we discuss the spectral classes $I^{\ \pm}$ first. To this end we label the amplitudes $\widetilde{t}_{\ \underline{.}}
\ ( \ z \ , \ p \ , \ J^{\ P\ C} \ ; \ . \ )$ in eq. (\[eq:104\])
$$\label{eq:122}
\begin{array}{l}
\widetilde{t}_{\ \underline{.}}
\ ( \ z \ , \ p \ , \ J^{\ P \ +} \ ; \ . \ )
\ \rightarrow
\ \widetilde{t}_{ \
\underline{.} \ ; \ S_{\ 12}^{\ \pm}}
\ ( \ z \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
\widetilde{t}_{\ \underline{.} \ ; \ S_{\ 12}^{\ \pm}}
\ ( \ z \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )
\begin{array}{ll}
\nearrow &
\widetilde{t}_{\ \underline{.} \ ; \ II^{\ +}}
\ ( \ z \ , \ p \ , \ J^{\ + \ +} \ ; \ . \ )
\vspace*{0.3cm} \\
\rightarrow &
\widetilde{t}_{\ \underline{.} \ , \ I^{\ +}}
\ ( \ z \ , \ p \ , \ J^{\ + \ +} \ ; \ . \ )
\vspace*{0.3cm} \\
\searrow &
\widetilde{t}_{\ \underline{.} \ ; \ I^{\ -}}
\ ( \ z \ , \ p \ , \ J^{\ - \ +} \ ; \ . \ )
\end{array}
\vspace*{0.3cm} \\
\underline{.} \ ; S_{\ 12}^{\ \pm} \ \rightarrow \ \underline{.}
\ \rightarrow
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ ,
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack
\end{array}$$
The two spectral classes $I^{\ \pm}$ exhibit the relativistic factorization patterns
$$\label{eq:123}
\begin{array}{l}
\widetilde{t}_{\ \underline{.} \ ; \ I^{\ \pm}}
\ ( \ z \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )
\ =
\ \left (
\begin{array}{l}
\ \left (
\ K^{\ \pm}
\ \right )_{\
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack}
\ \times
\vspace*{0.3cm} \\
\hspace*{0.7cm} \times
\ \widetilde{t}_{\ I^{\ \pm}}
\ ( \ z \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )
\end{array}
\ \right )
\vspace*{0.5cm} \\
\ \left (
\ K^{\ +}
\ \right )_{\
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack}
\ =
\ g_{\ \mu_{\ 1} \ \mu_{\ 2}}
\ g_{\ \nu_{\ 1} \ \nu_{\ 2}}
\ -
\ g_{\ \mu_{\ 1} \ \nu_{\ 2}}
\ g_{\ \mu_{\ 2} \ \nu_{\ 1}}
\vspace*{0.5cm} \\
\ \left (
\ K^{\ -}
\ \right )_{\
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack}
\ =
\ \varepsilon_{\ \mu_{\ 1} \ \mu_{\ 2} \ \nu_{\ 1} \ \nu_{\ 2}}
\end{array}$$
In eq. (\[eq:123\]) $K^{\ \pm}$ denote the two Lorentz invariant tensors with parity $\pm$ respectively and $g_{\ \mu \nu}$ the Lorentz metric tensor.
The tensors $K^{\ \pm}$ form projection operations on the octet string operators
$$B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )$$
introduced in eq. (\[eq:101\]) , described in appendix A.4 .
The projections yield
$$\label{eq:124}
\begin{array}{l}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ =
\left (
\hspace*{-0.1cm}
\begin{array}{r}
\left (
\ K^{\ +}
\ \right )_{\
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack}
\ B^{\ (+)}
\vspace*{0.3cm} \\
+
\ \left (
\ K^{\ -}
\ \right )_{\
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack}
\ B^{\ (-)}
\vspace*{0.3cm} \\
+
\ B^{\ '}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\end{array}
\right )
\vspace*{0.2cm}
\end{array}$$
where the quantities $B^{\ (\pm)}$ are derived in appendix A.4 . They are of the form given in eq. (\[eq:1014\]) reproduced below
$$\label{eq:125}
\begin{array}{l}
B^{\ (+)} \ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ =
\vspace*{0.3cm} \\
\hspace*{0.6cm} =
\ \frac{1}{12}
\ F_{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 1} \ ; \ A \ )
\ U \ ( \ x_{\ 1} \ , \ A \ ; \ x_{\ 2} \ , \ B \ )
\ F^{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 2} \ ; \ B \ )
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
B^{\ (-)} \ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ =
\vspace*{0.3cm} \\
\hspace*{0.6cm} =
\ - \ \frac{1}{12}
\ F_{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 1} \ ; \ A \ )
\ U \ ( \ x_{\ 1} \ , \ A \ ; \ x_{\ 2} \ , \ B \ )
\ \widetilde{F}^{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 2} \ ; \ B \ )
\vspace*{0.3cm} \\
\hspace*{0.6cm} =
\ - \ \frac{1}{12}
\ \widetilde{F}_{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 1} \ ; \ A \ )
\ U \ ( \ x_{\ 1} \ , \ A \ ; \ x_{\ 2} \ , \ B \ )
\ F^{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 2} \ ; \ B \ )
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
\widetilde{F}_{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 2} \ ; \ B \ )
\ =
\ \frac{1}{2}
\ \varepsilon_{ \ \alpha \ \beta \ \gamma \ \delta}
\ F^{\ \left \lbrack \ \gamma \ \delta \ \right \rbrack}
\ ( \ x_{\ 2} \ ; \ B \ )
\vspace*{0.3cm} \\
\mbox{and} \hspace*{0.3cm}
( \ x_{\ 2} \ ; \ B \ ) \ \leftrightarrow
\ ( \ x_{\ 1} \ ; \ A \ )
\end{array}$$
Returning to the (spin-) reduced amplitudes $\widetilde{t}_{\ I^{\ \pm}}
\ ( \ z \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )$ introduced in eq. (\[eq:123\]) we obtain using the notation of eq. (\[eq:104\])
$$\label{eq:126}
\begin{array}{l}
\left \langle
\ \emptyset
\ \right |
\ B^{\ (\pm)}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ \left |
\ gb_{\ I^{\ \pm}} \ ( \ J^{\ \pm \ +} \ ) \ ; \ p \ ,
\ \left \lbrace spin \right \rbrace
\ \right \rangle \ =
\vspace*{0.3cm} \\
\hspace*{0.3cm} =
\ \exp^{\ - i p X}
\ \widetilde{t}_{\ I^{\ \pm}} \ ( \ z \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )
\end{array}$$
with $B^{\ (\pm)}$ given in eq. (\[eq:125\]) . In eq. (\[eq:126\]) the suffix $I^{\ \pm}$ of the states $gb_{\ I^{\ \pm}}$ indicates, that these are restricted to the spectral types denoted $I^{\ \pm}$ in eq. (\[eq:108\]) .
In the local limit of $z \ \rightarrow \ 0$ , i.e. shrinking the extension of the adjoint string to zero length, we recognize in $B^{\ (\pm)}$ two local operators shaping the dynamics of QCD. We ignore here for clarity all short distance singularities, in this limit.
$$\label{eq:127}
\begin{array}{l}
z \ \rightarrow \ 0
\hspace*{0.3cm} : \hspace*{0.3cm}
\begin{array}{l lr}
B^{\ (+)} & \rightarrow &
\frac{1}{12}
\ F_{\ \left \lbrack \ \mu \ \nu \ \right \rbrack}
\ ( \ X \ ; \ A \ )
\ F^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack}
\ ( \ X \ ; \ A \ )
\vspace*{0.3cm} \\
B^{\ (-)} & \rightarrow &
- \ \frac{1}{12}
\ F_{\ \left \lbrack \ \mu \ \nu \ \right \rbrack}
\ ( \ X \ ; \ A \ )
\ \widetilde{F}^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack}
\ ( \ X \ ; \ A \ )
\end{array}
\ \rightarrow
\vspace*{0.4cm} \\
\hline
\vspace*{-0.3cm} \\
\begin{array}{l|l}
\begin{array}{rll}
3 \ \left . B^{\ (+)} \ \right |_{\ 0}
& = &
{\cal{L}}^{\ (+)} \ ( \ X \ )
\vspace*{0.3cm} \\
- \ 3 \ \left . B^{\ (-)} \ \right |_{\ 0}
& = &
{\cal{L}}^{\ (-)} \ ( \ X \ )
\end{array}
&
\begin{array}{rll}
{\cal{L}}^{\ (+)} \ ( \ X \ )
& = &
g^{\ 2} \ s \ ( \ X \ )
\vspace*{0.3cm} \\
{\cal{L}}^{\ (-)} \ ( \ X \ )
& = &
8 \ \pi^{\ 2} \ ch_{\ 2} \ ( \ X \ )
\end{array}
\end{array}
\vspace*{0.4cm} \\
\hline
\vspace*{-0.3cm} \\
\begin{array}{rll}
{\cal{L}}^{\ (+)} \ ( \ X \ )
& = &
\ \frac{1}{4}
\ F_{\ \left \lbrack \ \mu \ \nu \ \right \rbrack}
\ ( \ X \ ; \ A \ )
\ F^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack}
\ ( \ X \ ; \ A \ )
\vspace*{0.3cm} \\
{\cal{L}}^{\ (-)} \ ( \ X \ )
& = &
\frac{1}{4}
\ F_{\ \left \lbrack \ \mu \ \nu \ \right \rbrack}
\ ( \ X \ ; \ A \ )
\ \widetilde{F}^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack}
\ ( \ X \ ; \ A \ )
\end{array}
\end{array}$$
In eq. (\[eq:127\]) $s$ denotes the action density pertaining to gauge bosons and g the (strong) coupling constant, while $ch_{\ 2}$ represents the density of the second Chern character.
We return to the wave functions $\widetilde{t}_{\ I^{\ \pm}}
\ ( \ z \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )$ defined in eqs. (\[eq:123\]) and (\[eq:126\]) pertaining to the gb spectral types $I^{\ \pm}$ in eq. (\[eq:126\]) . As a consequence of eq. (\[eq:103\]) they satisfy the Bose symmetry relation
$$\label{eq:128}
\begin{array}{l}
\widetilde{t}_{\ I^{\ \pm}}
\ ( \ z \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )
\ =
\widetilde{t}_{\ I^{\ \pm}}
\ ( \ - \ z \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )
\end{array}$$
We meet a problem of interpretation of the bilinear wave functions $\widetilde{t}_{\ I^{\ \pm}}$ and the symmetry in eq. (\[eq:128\]) , known (also) from the study of $q \overline{q}$ and 3 q composite systems [@PMBar].
This is recognized, decomposing the Lorentz four vector z into parallel and transverse components relative to the c.m. momentum p
$$\label{eq:129}
\begin{array}{l}
z \ = \ z_{\ p} \ + \ \eta \ p \ / \ m^{\ 2}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\eta \ = \ z \ p
\hspace*{0.3cm} \rightarrow \hspace*{0.3cm}
z_{\ p} \ p \ = \ 0
\end{array}$$
In the c.m. system the scalar product $\eta$ in eq. (\[eq:129\]) becomes relative time, which is not a genuine degree of freedom of the dynamical system in question.
$$\label{eq:130}
\begin{array}{l}
\mbox{c.m. :}
\hspace*{0.3cm}
p \ \rightarrow
\ p_{\ c.m.} \ ( \ m \ , \ \vec{0} \ )
\vspace*{0.3cm} \\
\eta \ \rightarrow
\ m \ z_{\ 0} \ = \ m \ t_{\ rel}
\end{array}$$
Let me illustrate what is addressed here, considering the decay $\varrho^{\ 0} \ \rightarrow \ 2 \ \pi$ .
First we shall assume pions to be absolutely stable. Then the decay
$$\varrho^{\ 0} \ \rightarrow \ 2 \ \pi^{\ 0}$$
is forbidden by Bose symmetry. Next we take into account, that $\pi^{\ 0}$ s decay (mainly) into two photons, over the width of $\pi^{\ 0}$. The latter is according to the PDG [@PDG]
$$\label{eq:131}
\begin{array}{l}
\tau_{\ \pi^{\ 0}} \ =
\ \left (
\ 8.4 \ \pm \ 0.6
\ \right ) \ 10^{\ -17} \ \mbox{sec}
\hspace*{0.3cm} \leftrightarrow \hspace*{0.3cm}
\Gamma_{\ \pi^{\ 0}} \ =
\ \left (
\ 7.84 \ \pm \ 0.53
\ \right ) \ \mbox{eV}
\end{array}$$
To be specific we consider the reaction
$$\label{eq:132}
\begin{array}{l}
e^+ \ e^- \ \rightarrow \ \varrho^{\ 0} \ \rightarrow \ 4 \ \gamma
\end{array}$$
and ask the question, whether it can proceed, when the two pairs of photons, $\gamma_{\ 1} \ \gamma_{\ 2}$ and $\gamma_{\ 3} \ \gamma_{\ 4}$ say, form each a $\pi^{\ 0}$ , with invariant masses $m_{\ 12}$ and $m_{\ 34}$ differing by a well defined fraction of $\Gamma_{\ \pi^{\ 0}}$
$$\label{eq:133}
\begin{array}{l}
f_{ +} \ \Gamma_{\ \pi^{\ 0}}
\ \ge
\left |
\ m_{\ 12} \ - \ m_{\ 34}
\ \right |
\ \ge
\ f_{ -} \ \Gamma_{\ \pi^{\ 0}}
\vspace*{0.3cm} \\
\mbox{with}
\hspace*{0.4cm}
\ f_{+} \ = \ 1
\hspace*{0.3cm} , \hspace*{0.3cm}
\ f_{-} \ = \ 0.1
\hspace*{0.4cm}
\mbox{say}
\end{array}$$
The answer relevant here is, that there is a [*multitude*]{} of equivalent irreducible wave functions out of the [*family*]{} defined in eqs. (\[eq:123\]) and (\[eq:126\])
$$\label{eq:134}
\begin{array}{l}
\widetilde{t}_{\ I^{\ \pm}}
\ ( \ z \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )
\ =
\widetilde{t}_{\ I^{\ \pm}}
\ ( \ z_{\ p} \ , \ p \ , \ \eta \ , \ J^{\ \pm \ +} \ ; \ . \ )
\end{array}$$
distinguished by the [*parameter*]{} $\eta$ as defined in eqs. (\[eq:129\]) and (\[eq:130\]).
Thus we choose the representative with $\eta \ = \ 0$
$$\label{eq:135}
\begin{array}{l}
t_{\ I^{\ \pm}}
\ ( \ z_{\ p} \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )
\ =
\widetilde{t}_{\ I^{\ \pm}}
\ ( \ z_{\ p} \ , \ p \ , \ \eta \ = \ 0 \ , \ J^{\ \pm \ +} \ ; \ . \ )
\end{array}$$
The above procedure illustrates the [*difference*]{} between decay amplitudes of resonances into two photons and their selection rules, derived in refs. [@Land] and [@Yang], and the wave functions of binary gluonic mesons.
The irreducible wave functions $t_{\ I^{\ \pm}}$ can readily be discussed in the rest system of the momentum p, where
$$\label{eq:136}
\begin{array}{l}
\mbox{c.m. :}
\hspace*{0.3cm}
\left (
\ \eta \ = \ 0 \ , \ z_{\ p}
\ \right )
\ \rightarrow
\ z_{\ p} \ =
\ \left (
\ 0 \ , \ \vec{z}
\ \right )
\ \rightarrow
\ \vec{z}
\vspace*{0.3cm} \\
t_{\ I^{\ \pm}}
\ ( \ z_{\ p} \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )
\ \rightarrow
t_{\ I^{\ \pm}}
\ ( \ \vec{z} \ , \ J^{\ \pm \ +} \ ; \ . \ )
\end{array}$$
The procedure outlined above implies, that the octet string bilinear operators
$$\label{eq:137}
\begin{array}{l}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\end{array}$$
defined in eq. (\[eq:101\]) are to be evaluated for spacelike relative positions\
$z \ = \ x_{\ 1} \ - \ x_{\ 2}$ only.
Now eq. (\[eq:128\]) takes the form
$$\label{eq:138}
\begin{array}{l}
t_{\ I^{\ \pm}}
\ ( \ \vec{z} \ , \ J^{\ \pm \ +} \ ; \ . \ )
\ =
\ t_{\ I^{\ \pm}}
\ ( \ - \ \vec{z} \ , \ J^{\ \pm \ +} \ ; \ . \ )
\end{array}$$
The consequence for the angular momentum composition of the spectral types $I^{\ \pm}$ is indeed identical to the situation of decay into two photons [@Land] , [@Yang]
$$\label{eq:139}
\begin{array}{l}
t_{\ I^{\ \pm}}
\ ( \ \vec{z} \ , \ J^{\ \pm \ +} \ ; \ . \ )
\ \rightarrow
\ t_{\ I^{\ \pm}}
\ ( \ \vec{z} \ , \ J^{\ \pm \ +} \ ; \ M \ )
\vspace*{0.3cm} \\
t_{\ I^{\ \pm}}
\ ( \ \vec{z} \ , \ J^{\ \pm \ +} \ ; \ M \ )
\ = \ R^{\ J}_{\ I^{\ \pm}} \ ( \ r \ ) \ Y^{\ J}_{\ M} \ ( \ \vec{e} \ )
\hspace*{0.3cm} ; \hspace*{0.3cm}
J \ = \ \mbox{even}
\vspace*{0.3cm} \\
r \ =
\ \left | \ \vec{z} \ \right |
\hspace*{0.3cm} , \hspace*{0.3cm}
\vec{e} \ = \ \vec{z} \ / \ r
\vspace*{0.4cm} \\
\hline
\vspace*{-0.3cm} \\
\begin{array}{lll}
I^{\ +} & : &
J^{\ P \ C} \ = \ 0^{\ ++} \ , \ 2^{\ ++} \ , \ 4^{\ ++} \ \cdots
\vspace*{0.3cm} \\
I^{\ -} & : &
J^{\ P \ C} \ = \ 0^{\ -+} \ , \ 2^{\ -+} \ , \ 4^{\ -+} \ \cdots
\end{array}
\end{array}$$
In eq. (\[eq:139\]) $Y^{\ J}_{\ M}$ denote the orbital spherical harmonics with angular momentum J , while $\left \lbrace \ R^{\ J}_{\ I^{\ \pm}} \ \right \rbrace$ stand for a [*family*]{} of radial wave functions. Neither nature nor extension of this family, nor its ordering in mass can be deduced from the spectral type.
For the sake of absolute clarity let me emphasize that the family of wave functions (of all types pertinent to binary gluonic mesons) can be empty, since no first principle proof to the contrary exists.
[*Quantum numbers of binary gluonic mesons continued*]{}
[*The $II^{\ +}$ spectral type*]{}
We turn to the remaining spectral type denoted $II^{\ +}$ in eq. (\[eq:108\]) .
The properties of this spectral series, represented by the quantities\
$B^{\ '}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}$ defined in eq. (\[eq:124\]) are derived in appendix A.5 .
As shown there the wave functions of the gb spectral type $II^{\ +}$ are uniquely associated with the classical energy momentum bilinear pertaining to gauge bosons.
We retain here the characteristic composition of the wave function associated bilinears $B^{\ '}$ in eqs. (\[eq:1072\]) and (\[eq:1073\]) in the summary remarks of appendix A.5:
$$\label{eq:140}
\begin{array}{l}
\hspace*{-0.2cm}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\ \left \lbrace
\begin{array}{c}
\ \frac{1}{2}
\left (
\begin{array}{l}
g_{\ \mu_{\ 1} \ \mu_{\ 2}} \ \varrho_{\ \nu_{\ 1} \ \nu_{\ 2}}
- \ g_{\ \nu_{\ 1} \ \mu_{\ 2}} \ \varrho_{\ \mu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.3cm} \\
- \ g_{\ \mu_{\ 1} \ \nu_{\ 2}} \ \varrho_{\ \nu_{\ 1} \ \mu_{\ 2}}
\ + \ g_{\ \nu_{\ 1} \ \nu_{\ 2}} \ \varrho_{\ \mu_{\ 1} \ \mu_{\ 2}}
\end{array}
\right )
\vspace*{0.3cm} \\
+
\ K^{\ +}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ B^{\ (+)}
\vspace*{0.3cm} \\
+
\ K^{\ -}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ B^{\ (-)}
\vspace*{0.3cm} \\
\end{array}
\right \rbrace
\vspace*{0.4cm} \\
\hline
\vspace*{-0.3cm} \\
\mbox{with :}
\ g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.5cm} \\
- \ \varrho^{\ \mu \ \nu} \ =
\ - \ R^{\ \mu \ \nu} \ +
\ \frac{1}{4}
\ g^{\ \mu \ \nu} \ R \ =
\ \vartheta_{\ cl}^{\ \mu \ \nu}
\vspace*{0.5cm} \\
R \ =
\ g^{\ \nu_{\ 1} \ \nu_{\ 2}}
\ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\ =
\ 12 \ B^{\ (+)}
\end{array}$$
The bilinears $B^{\ '}$ are thus given by
$$\label{eq:141}
\begin{array}{l}
B^{\ '}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\ \frac{1}{2}
\left (
\begin{array}{l}
g_{\ \mu_{\ 1} \ \mu_{\ 2}} \ \varrho_{\ \nu_{\ 1} \ \nu_{\ 2}}
- \ g_{\ \nu_{\ 1} \ \mu_{\ 2}} \ \varrho_{\ \mu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.3cm} \\
- \ g_{\ \mu_{\ 1} \ \nu_{\ 2}} \ \varrho_{\ \nu_{\ 1} \ \mu_{\ 2}}
\ + \ g_{\ \nu_{\ 1} \ \nu_{\ 2}} \ \varrho_{\ \mu_{\ 1} \ \mu_{\ 2}}
\end{array}
\right )
\vspace*{0.3cm} \\
B^{\ '} \ \leftrightarrow \ \left \lbrace \ II^{\ +} \ \right \rbrace
\ \longleftrightarrow \ \vartheta_{\ cl}^{\ \mu \ \nu}
\end{array}$$
The three spectral types are given in eq. (\[eq:142\]) below, completing the types $I^{\ \pm}$ in eq. (\[eq:139\])
$$\label{eq:142}
\begin{array}{l}
t_{\ I^{\ \pm}}
\ ( \ \vec{z} \ , \ J^{\ \pm \ +} \ ; \ . \ )
\ \rightarrow
\ t_{\ I^{\ \pm}}
\ ( \ \vec{z} \ , \ J^{\ \pm \ +} \ ; \ M \ )
\vspace*{0.3cm} \\
t_{\ I^{\ \pm}}
\ ( \ \vec{z} \ , \ J^{\ \pm \ +} \ ; \ M \ )
\ = \ R^{\ J}_{\ I^{\ \pm}} \ ( \ r \ ) \ Y^{\ J}_{\ M} \ ( \ \vec{e} \ )
\hspace*{0.3cm} ; \hspace*{0.3cm}
J \ = \ \mbox{even}
\vspace*{0.3cm} \\
r \ =
\ \left | \ \vec{z} \ \right |
\hspace*{0.3cm} , \hspace*{0.3cm}
\vec{e} \ = \ \vec{z} \ / \ r
\vspace*{0.4cm} \\
\hline
\vspace*{-0.3cm} \\
\begin{array}{lll}
I^{\ +} & : &
J^{\ P \ C} \ = \ 0^{\ ++} \ , \ 2^{\ ++} \ , \ 4^{\ ++} \ \cdots
\vspace*{0.3cm} \\
I^{\ -} & : &
J^{\ P \ C} \ = \ 0^{\ -+} \ , \ 2^{\ -+} \ , \ 4^{\ -+} \ \cdots
\end{array}
\vspace*{0.4cm} \\
\hline
\hline
\vspace*{-0.3cm} \\
t_{\ II^{\ +}}
\ ( \ \vec{z} \ , \ J^{\ \pm \ +} \ ; \ . \ )
\ \rightarrow
\ t_{\ II^{\ +}}
\ ( \ \vec{z} \ , \ J^{\ II \ +} \ ; \ M \ , \ \vec{E}_{\ \pm} \ )
\vspace*{0.4cm} \\
t_{\ II^{\ +}}
\ ( \ \vec{z} \ , \ J^{\ II \ +} \ ; \ M \ , \ \vec{E}_{\ \pm} \ )
\ =
\vspace*{0.4cm} \\
\hspace*{0.4cm} =
\ R^{\ J}_{\ II^{\ +}} \ ( \ r \ , \ \vec{E}_{\ \pm} \ )
\ D^{\ J}_{\ M \ \pm 2} \ ( \ \vec{e} \ , \ \vec{E}_{\ \pm} \ )
\vspace*{0.4cm} \\
\begin{array}{lll}
II^{\ +} & : &
J^{\ P \ C} \ = \ 2^{\ ++} \ , \ 3^{\ ++} \ , \ 4^{\ ++} \ , \ 5^{\ ++}
\ \cdots
\end{array}
\vspace*{0.3cm} \\
\hline
\end{array}$$
In eq. (\[eq:142\]) the chromoelectric field strengths $\vec{E}_{\ \pm}$ are retained in the arguments of the wave functions.
The functions $D^{\ J}_{\ M \ \sigma} \ ( \ \vec{e} \ , \ \vec{E}_{\ \pm} \ )$ , with $\sigma \ = \ \pm 2$ , denote the eigenfunctions of a (symmetric) top, with the full orientation involving three Euler angles provided by the correlation between the two chromoelectric field strengths $\vec{E}_{\ \pm}$ of the adjoint string, discussed in appendix A.5 .
Let us end here the theoretical discussion of binary gluonic modes associated with the octet gauge boson string. Theoretical expectations of spectral characteristics of states representing the spectral types $I^{\ \pm}$ and $II^{\ +}$ shall be addressed in the next section.
Spectral patterns of gb - facts and fancy
=========================================
[**a) Lattice QCD calculations**]{}
The most promising and widely accepted framework to derive spectral patterns of hadrons, including gluonic mesons, is lattice gauge theory and therein the restriction to gauge boson degrees of freedom only. I shall quote several papers instead of a review : [@Ruefe] , [@Teper] , [@Michael] and [@Kuni] .
I shall discuss the above papers one by one. In ref. [@Ruefe] a careful and dedicated study is devoted to the determination of the mass of $gb \ ( \ 0^{\ ++} \ )$ , the lowest lying gluonic meson in pure Yang-Mills theory based on $SU3_{\ c}$ , and also $gb \ ( \ 2^{++} \ )$ , with the results
$$\label{eq:143}
\begin{array}{l}
m \ \left ( \ gb \ ( \ 0^{++} \ ) \ \right )
\ = \ 1627 \ \pm \ 83 \ \mbox{MeV}
\ \rightarrow
\ m^{\ 2} \ = \ 2.65 \ \pm \ 0.27 \ \mbox{GeV}^{\ 2}
\vspace*{0.3cm} \\
m \ \left ( \ gb \ ( \ 2^{++} \ ) \ \right )
\ = \ 2354 \ \pm \ 95 \ \mbox{MeV}
\ \rightarrow
\ m^{\ 2} \ = \ 5.54 \ \pm \ 0.6 \ \mbox{GeV}^{\ 2}
\end{array}$$
The main result refers to $gb \ ( \ 0^{\ ++} \ )$ and is in very good agreement with all lattice gauge theory calculations, concerning the same state.
I compare the above results with the assignment made here in figure \[fig2\]
$$\label{eq:144}
\begin{array}{l}
m^{\ 2} \ \left ( \ gb \ ( \ 0^{++} \ ) \ \right )
\ =
\ 1.04 \ \mbox{GeV}^{\ 2}
\vspace*{0.3cm} \\
m^{\ 2} \ \left ( \ gb \ ( \ 2^{++} \ ) \ \right )
\ =
\ 3.13 \ \mbox{GeV}^{\ 2}
\end{array}$$
While it is difficult to associate an error with the tentative pattern represented in figure \[fig2\] and eq. (\[eq:144\]) , to which I will return below, the essentially smaller mass square scale, by factors of $\sim \ 2.5$ and $\sim \ 1.8$ for $gb \ ( \ 0^{\ ++} \ )$ and $gb \ ( \ 2^{\ ++} \ )$ respectively, is indeed a [*basic*]{} controversy, seemingly disproving the mass square range considered in eq. (\[eq:144\]) .
In ref. [@Teper] an attempt is made to align gb resonances on the Pomeron trajectory, as done here in figure \[fig2\], but with very different assignments : the slope of the Pomeron trajectory is assumed to be
$$\label{eq:145}
\begin{array}{l}
\cite{Teper} \ :
\ \alpha^{\ '}_{\ P} \ = \ \ \alpha^{\ '}_{\ g \ b}
\ \sim \ 0.22 \ \pm \ 0.4 \ \mbox{GeV}^{\ -2}
\vspace*{0.3cm} \\
\mbox{here} \ :
\ \ \alpha^{\ '}_{\ g \ b} \ = \ \frac{1}{2} \ \alpha^{\ '}
\ = \ 0.5211 \ \pm \ 1.3 \ \% \ \mbox{GeV}^{\ -2}
\end{array}$$
Again a factor of two opens up, with respect to the value of $\alpha^{\ '}_{\ gb}$, between ref. [@Teper] and our present discussion, where indeed the relation $\alpha^{\ '}_{\ gb} \ = \ \frac{1}{2} \ \alpha^{\ '}$ , also discussed below, can be in doubt.
As a consequence of the calculations in ref. [@Teper] the deduced mass square value for $gb \ ( \ 2^{\ ++} \ )$ , which is supposed to lye on the Pomeron trajectory, becomes
$$\label{eq:146}
\begin{array}{l}
\cite{Teper} \ :
\ m^{\ 2} \ \left ( \ gb \ ( \ 2^{++} \ ) \ \right )
\ = \ 4.4 \ \pm \ 1.2 \ \mbox{GeV}^{\ 2}
\vspace*{0.3cm} \\
\mbox{here} \ :
\ m^{\ 2} \ \left ( \ gb \ ( \ 2^{++} \ ) \ \right )
\ =
\ 3.13 \ \mbox{GeV}^{\ 2}
\vspace*{0.3cm} \\
\hspace*{1.0cm}
\left (
\ m^{\ 2} \ \left ( \ gb \ ( \ 3^{++} \ ) \ \right )
\ = \ 4.17 \ \mbox{GeV}^{\ 2}
\ \right )
\end{array}$$
Comparing the mass square values of ref. [@Teper] in eq. (\[eq:146\]) with the one of ref. [@Ruefe] in eq. (\[eq:143\]) we see (marginal) agreement.
In ref. [@Michael] lattice calculations are presented to determin the masses of hybrid mesons, composed of at least one gluon bound with a (nonstrange) $q \overline{q}$ pair, and exhibiting $q \overline{q}$ exotic quantum numbers, such as
$$\begin{array}{l}
J^{\ P C} \ = \ 0^{\ --} \ , \ 0^{\ +-}
\ , \ 1^{\ -+} \ , \ 2^{\ +-} \ \cdots
\end{array}$$
The lightest hybrid states with nonstrange quarks is found with characteristics
$$\label{eq:147}
\begin{array}{l}
\cite{Michael} \ :
\ J^{\ PC} \ = \ 1^{\ -+}
\hspace*{0.3cm} ; \hspace*{0.3cm}
m_{\ hyb} \ = \ 1.9 \ \pm \ 0.2 \ \mbox{GeV}
\vspace*{0.3cm} \\
\hspace*{1.0cm} \rightarrow
\ m^{\ 2}_{\ hyb} \ = \ 3.6 \ \pm \ 0.6 \ \mbox{GeV}^{\ 2}
\end{array}$$
Also in lattice calculations of hybrid meson masses agreement between different groups is very satisfactory. The above is not directly related to the discussion of binary gluonic mesons, but the result in eq. (\[eq:147\]) is apparently contradicted by the experimental finding of (at least) two exotic mesons with $J^{\ PC} \ = \ 1^{\ -+}$ quantum numbers in p wave decay to $\eta \ \pi$ and $\eta^{\ '} \ \pi$ [@Chung] .
These resonances carry the name $\pi_{\ 1} \ ( \ 1400 \ )$ and $\pi_{\ 1} \ ( \ 1600 \ )$ , where the mass in MeV is the argument.
The two resonances in question were attributed the following characteristics [@Chung] ( beyond $J^{\ PC} \ = \ 1^{\ -+}$ )
$$\label{eq:148}
\begin{array}{l}
\begin{array}{lll}
\pi_{\ 1} \ ( \ 1400 \ )
& : &
m \ = \ 1370 \ \pm \ 16 \ ^{\ + \ 50}_{\ - \ 30} \ \mbox{MeV}
\vspace*{0.3cm} \\
& &
\Gamma \ = \hspace*{0.4cm} 385 \ \pm \ 40 \ ^{\ + \ 65}_{\ - \ 105} \ \mbox{MeV}
\vspace*{0.3cm} \\
\pi_{\ 1} \ ( \ 1600 \ )
& : & m \ = \ 1597 \ \pm \ 10 \ ^{\ + \ 45}_{\ - \ 10} \ \mbox{MeV}
\vspace*{0.3cm} \\
& &
\Gamma \ = \hspace*{0.4cm} 340 \ \pm \ 40 \ ^{\ + \ 50}_{\ - \ 50} \ \mbox{MeV}
\end{array}
\end{array}$$
In the first paper in ref [@Chung] the authors remark, that the exotic quantum numbers violate $SU3_{\ fl}$ symmetry, in the decay $\pi_{\ 1} \ ( \ 1400 \ ) \ \rightarrow \ \pi \ \eta_{\ 8}$ , assigning pure flavor octet quantum numbers to $\eta$ , unless it is not a hybrid meson but rather composed of two quarks and two antiquarks. There is a $\sim \ 20^{\ \circ}$ singlet octet mixing between $\eta$ and $\eta^{\ '}$ , which, given the mass of $\pi_{\ 1} \ ( \ 1400 \ )$ , i.e. below decay threshald for $\pi \ \eta^{\ '}$ ( modulo the width ) becomes essential, even though we would then expect a reduction of the width by $\sim$ a factor of 5.
Alternatively, it can not be excluded, that $\pi \ ( \ 1400 \ )$ is in a quark flavor configuration corresponding to $q \overline{q} \ q \overline{q}$ and thus is not a hybrid meson in the first place. This discussion, even if at the side of the issue of gluonic mesons, gives a taste of the [*interpretation-*]{} difficulties, facing the recognition of gb-s.
But even if we assume that precisely $\pi \ ( \ 1600 \ )$ is a genuine hybrid meson, and further that the result given in ref. [@Michael] can be made to agree with a mass value of 1600 MeV, it is difficult to conceive that $gb \ ( \ 0^{\ ++} \ )$ would have a mass in excess of 1600 MeV as indicated in the value given in ref. [@Ruefe] . To be fair to all lattice calculations, let me stress, that the mass values of gluonic mesons refer to the (unrealistic) case of no quark flavors (or all quark flavors very heavy), and that a considerable shift in mass of e.g. $gb \ ( \ 0^{\ ++} \ )$ can be the result of the light quark flavors, unaccounted for in [@Ruefe] and all comparable calculations.
In ref. [@Kuni] the calculations focus on the question of low mass scalar mesons, not gb-s. This issue is a prerequisite for the successful identification of $gb \ ( \ 0^{\ ++} \ )$, lowest in mass and thus was examined as to the structure of the scalar $q \overline{q}$ nonet, lowest in mass, in ref. [@PMWO], where this nonet was [*assumed*]{} to be identifyable.
In ref. [@Kuni] the local, composite field called $\sigma$ was investigated on the lattice
$$\label{eq:149}
\begin{array}{l}
\sigma \ ( \ x \ ) \ = \ \frac{1}{\sqrt{2}}
\ \left (
\ \overline{u}_{\ c} \ ( \ x \ ) \ u_{\ c} \ ( \ x \ )
\ + \ \overline{d}_{\ c} \ ( \ x \ ) \ d_{\ c} \ ( \ x \ )
\ \right )
\end{array}$$
where the suffix c denotes triplet color.
Irrespective of the pattern of the full nonet it is valid to consider the two point function of two $\sigma$ fields, on the lattice, and to deduce the mass of the lowest scalar resonance, coupling to the $\sigma$ field.
The authors of ref. [@Kuni] declare their calculation preliminary, so it is not yet possible to evaluate the error of their mass determination. Nevertheless they indicate the following result
$$\label{eq:150}
\begin{array}{l}
\cite{Kuni} \ :
\ m_{\ \pi} \ < \ m_{\ \sigma} \ < \ m_{\ \varrho}
\ \sim \ 776 \ \mbox{MeV}
\vspace*{0.3cm} \\
\cite{PMWO} \hspace*{0.3cm} :
\ \sigma \ ( \ q \overline{q} \ ) \ \rightarrow
\ f_{\ 0} \ ( \ 980 \ ) \ ;
\ m_{\ f_{\ 0}} \ \sim \ 980 \ \mbox{MeV}
\end{array}$$
We continue the discussion of the scalar $q \overline{q}$ nonet, beyond lattice calculations only, in the next subsection.
[**b) $\pi \ \pi \ -$ and related ps-ps scattering and scalars**]{}
In this contex let me start with quoting a recent paper devoted to $\pi \ \pi$ elastic scattering in the framework of chiral perturbation theory, and the Roy equation for full control of analyticity, unitarity and crossing relations [@CLG].
In ref. [@CLG] in a dedicated chapter “Poles on the second sheet” , op. cit., the following pole parameters are quoted for the s wave $I \ = \ 0 \ , \ \pi \pi$ partial wave amplitude
$$\label{eq:151}
\begin{array}{l}
\begin{array}{lll}
\cite{CLG}
& : &
\sqrt{s} \ =
\ \left (
\ 430 \ \pm \ 30 \ - \ i \ ( \ 295 \ \pm \ 20 \ )
\ \right ) \ \mbox{MeV}
\ \rightarrow
\vspace*{0.3cm} \\
& &
\hspace*{0.2cm}
s \ =
\ \left (
\ 0.098 \ \pm 0.037 \ - \ i \ ( \ 0.254 \ \pm \ 0.032 \ )
\ \right ) \ \mbox{GeV}^{\ 2}
\vspace*{0.3cm} \\
& &
s_{\ thr} \ = \ 4 \ m_{\ \pi}^{\ 2} \ = \ 0.078 \ \mbox{GeV}^{\ 2}
\end{array}
\end{array}$$
The result in eq. (\[eq:151\]) is indeed of highest interest.
Within the quoted errors the resonance parameters are compatible with a [*threshold*]{} resonance, when considered in the complex s plane. For the properties of Jost functions in this and in general cases I refer to Res Jost’s original work [@Jfunc] .
The deeper question related to the existence ( or nonexistence ) of the threshold resonance, as derived in ref. [@CLG] is, whether there exists a symmetry, which would enforce the stability of the resonance position, in particular in the chiral limit, i.e. of
$$\label{eq:152}
\begin{array}{l}
s_{\ R} \ = \ \Re \ s \ \sim \ s_{\ thr} \ \rightarrow \ 0
\end{array}$$
The role of the threshold resonance is then apparently that of a dilaton zero mode, arising from spontaneous breaking of dilatation invariance. It is the trace anomaly, which prevents the dilatation symmetry to be broken [*exclusively*]{} spontaneously.
The relations with respect to the (Lorentz) invariant amplitude are
$$\label{eq:153}
\begin{array}{l}
T_{\ 0} \ ( \ s \ ) \ =
\ \frac{1}{2} \ {\displaystyle{\int}}_{\ -1}^{\ +1} \ d \ z
\ T \ ( \ s \ , \ z \ )
\hspace*{0.2cm} ; \hspace*{0.2cm}
f_{\ 0} \ ( \ s \ ) \ = \ \frac{1}{8 \ \pi \ \sqrt{s}} \ T_{\ 0} \ ( \ s \ )
\vspace*{0.3cm} \\
f_{\ 0} \ ( \ s \ ) \ = \ \frac{1}{q} \ t \ ( \ q \ )
\hspace*{0.2cm} ; \hspace*{0.2cm}
t \ ( \ q \ ) \ = \ \left ( \ S \ ( \ q \ ) \ - \ 1 \ \right ) \ /
\ ( \ 2 \ i \ )
\vspace*{0.3cm} \\
t \ ( \ q \ ) \ = \ \frac{1}{16 \ \pi}
\ \sqrt{\ 1 \ - \ s_{\ thr} \ / \ s \ } \ T_{\ 0} \ ( \ s \ )
\vspace*{0.3cm} \\
\sigma_{\ el} \ ( \ s \ ) \ = \ 4 \ \pi \ | \ f_{\ 0} \ ( \ s \ ) \ |^{\ 2}
\hspace*{0.2cm} \mbox{for} \hspace*{0.2cm}
s \ \mbox{real} \ \ge \ s_{\ thr}
\end{array}$$
In eq. (\[eq:153\]) q denotes the c.m. momentum.
So the function $S \ ( \ q \ )$ is of the form, assuming indeed a threshold resonance with $s_{\ R} \ =
\ \left . \Re \ s \ \right |_{\ R} \ = \ s_{\ thr}$ and $\Im \ s_{\ R} \ = \ - \ \gamma_{\ R}$
$$\label{eq:154}
\begin{array}{l}
S \ ( \ q \ ) \ =
\ \begin{array}{c}
s_{\ R} \ + \ i \ \gamma_{\ R} \ - \ s
\vspace*{0.3cm} \\
\hline
\vspace*{-0.4cm} \\
s_{\ R} \ - \ i \ \gamma_{\ R} \ - \ s
\end{array}
\hspace*{0.3cm} S_{\ 1} \ ( \ q \ )
\end{array}$$
The most interesting situation arises if [*first*]{} we assume that $S_{\ 1}$ tends to 1 at threshold
$$\label{eq:155}
\begin{array}{l}
S_{\ 1} \ ( \ q \ \rightarrow \ 0 \ ) \ \rightarrow \ 1
\vspace*{0.3cm} \\
\rightarrow \ \ f_{\ 0} \ \sim \ a \ \rightarrow \ i \ / \ q
\end{array}$$
The behaviour displayed in eq. (\[eq:155\]) is obviously at variance with the restrictions imposed by (approximate) chiral symmetry, but this is not the interesting part, due to a threshold resonance. Rather it is the intrinsic interdependence of the [*remaining*]{} contribution $S_{\ 1} \ ( \ q \ \sim \ 0 \ )$ near threshold with the threshold phase of $90^{\ \circ}$ of the threshold resonance, which is most striking.
The latter must be moved either backward or forward by another $90^{\ \circ}$ at threshold in order to achieve a finite scattering length.
It is this interdependence, which is unlikely not to move the threshold resonance even very far from its initial threshold position.
A measure for the width of the deduced threshold resonance is the ratio
$$\label{eq:156}
\begin{array}{l}
\gamma_{\ R} \ / \ s_{\ R} \ \sim \ 2.5
\ \leftrightarrow
\ \Gamma \ = \ 590 \ \pm \ 40 \ \mbox{MeV}
\end{array}$$
as obtained in ref. [@CLG] . While we do not pursue the above discussion further here, it is necessary to retain, that the value of the mass derived in ref. [@CLG]: $m_{\ R} \ = \ 430 \ \pm \ 30 \ \mbox{MeV}$ , especially when the width is just ignored, leads to an increased uncertainty concerning the very possibility of recognizing the mass and mixing pattern of scalar mesons, in the sense of spectroscopy.
[*Alternative discussions of scalar resonances*]{}
Besides the new derivation of the $I \ = \ 0$ s-wave $\pi \pi$ scattering amplitude in [@CLG], the phase shifts in this channel are by now fairly well established from threshold to a c.m. energy of $\sim \ 1400$ MeV [@Lesniak], but only as far as resolution of phase ambiguities is concerned.
The $I \ = \ 0$ s-wave amplitude from the second reference in [@Lesniak] is reproduced below
-2.8cm
-1.5cm
-2.9cm
It becomes clear from the errrors both in the phase shift ( figure \[fig11\] a ) as well as in the inelasticity ( figure \[fig11\] b ) that the details are, despite a remarkable effort in analysis, rather uncertain in the range of c.m. energies $600 \ \mbox{MeV} \ \le \ \sqrt{s} \ \le \ 1600
\ \mbox{MeV}$.
[**The red dragon and “$\sigma$” in $\pi \pi \ ; \ I \ = \ 0$ s wave**]{}
The discussion of the partial wave amplitude, corresponding to the projection on $I \ = \ 0$ and on the s wave, denoted $t \ ( \ q \ )$ in eq. \[eq:153\]
$$\label{eq:157}
\begin{array}{l}
t \ ( \ q \ ) \ = \ \left ( \ S \ ( \ q \ ) \ - \ 1 \ \right ) \ /
\ ( \ 2 \ i \ )
\end{array}$$
has been the subject of many recent papers, to which we turn now. But we first show the result of combining elastic and quasi elastic pseudoscalar meson scattering, corresponding to the same quantum numbers, performed in ref. [@PMWO]. For a detailed discussion I refer back to ref. [@PMWO].
-0.2cm
-0.2cm
-1.0cm
The absolute values $\left | \ t \ ( \ q \ ) \ \right |^{\ 2}$ (with only relative normalization) for $\pi \pi \ \rightarrow \ \pi \pi \ , \ K \overline{K} \ , \ \eta \eta$ are shown in figure \[fig12\] together with the full shape of the red dragon, amputating the negative interference due to $f_{\ 0} \ ( \ 980 \ )$ and $f_{\ 0} \ ( \ 1500 \ )$.
Several comments are necessary here :
i\) “data”
The compilation of figures \[fig12\] a - c makes it appear as if actual data is displayed. This is by no means the case, rather between the real data from the reactions
$$\begin{array}{l}
\pi \ {\cal{N}} \ \rightarrow
\ \left \lbrace
\ \begin{array}{l}
\pi^{\ 0} \ \pi^{\ 0} \ {\cal{N}} \ ( \ \Delta \ )
\vspace*{0.3cm} \\
K_{\ s} \ K_{\ s} \ {\cal{N}} \ ( \ \Delta \ )
\vspace*{0.3cm} \\
\eta \ \eta \ {\cal{N}} \ ( \ \Delta \ )
\end{array}
\right .
\end{array}$$
and the displayed absolute values there is a [*series*]{} of analysis steps. The latter make it difficult to assess the overall errors.
ii\) the second interference minimum due to $f_{\ 0} \ ( \ 1500 \ )$
The pattern showing two interfering narrow states : $f_{\ 0} \ ( \ 980 \ )$ and $f_{\ 0} \ ( \ 1500 \ )$ by todays notation, has been inferred from the peripheral $\pi \ {\cal{N}}$ reactions listed above.
The latter resonance has clearly been observed in $p \overline{p}$ annihilation at rest by the Crystal Barrel collaboration at the Lear facility of CERN [@Amsler] , adding a new element with high statistics and precision of analysis.
iii\) the red dragon proper
The unfolding of the interference due to $f_{\ 0} \ ( \ 980 \ )$ and $f_{\ 0} \ ( \ 1500 \ )$ reveals a broad structure, the red dragon proper, as sketched in fig. \[fig12\] d.
The c.m. energy over which this structure is extended comprises the range $400 \ \mbox{MeV} \ \le \ \sqrt{s} \ \le \ 1600 \ \mbox{MeV}$ . Within all Breit-Wigner like strong interaction resonances, there does not exist a comparably wide one. This establishes the singular feature of the $\pi \pi$ s wave scattering amplitude in this range, and also considerably below 400 MeV, i.e. down to the two pion threshold, as well as above 1600 MeV.
The combined experimental and theoretical evaluation of data, which led to the clear picture represented by the red dragon in figure \[fig12\] is [*not*]{} subject to the remaining large inherent errors of details of the respective scattering amplitudes. This contrasts with all attempts : [@PMWO] , [@CLG] and those discussed below, where further [*interpretation*]{} of details of the red dragon are undertaken.
[**$\sigma \ ( \ \sim \ 500 \ )$ and/or $\kappa \ ( \ \sim \ 750 \ )$ scalar mesons**]{}
The claims of the existence of an isospin singlet, nonstrange scalar state $\sigma$ in a mass region clearly below $f_{\ 0} \ ( \ 980 \ )$ are numerous besides ref. [@CLG]. Another light scalar state, $\kappa$ with isospin 1/2, well below $K^{\ *}_{\ 0} \ ( \ 1430 \ )$ has also received much attention. These claims have been recently repeated on various grounds. We cite two reviews compiled within the PDG [@PDG] : on scalar mesons [@spanier] and on non $q \overline{q}$ candidates [@amslerrev] .
A new window has opened up in the study of the decay of charmed [@aitala] , [@bediaga] and b flavored mesons [@garmash] , [@babar] .
What is emerging from c- and b-flavored meson decays is the clear fact, that in three pseudoscalar meson ( $\pi$ and $K$ ) decays two out of the three pseudoscalars are produced amply in their relative s wave. This is quite in line with analogous decays from $p \overline{p}$ and hence the analysis in terms of two body amplitudes, the third pseudoscalar beeing treated as ’kinematical spectator, modulo constraints from Bose statistics’, was performed in all reactions in a similar way.
A few decays are listed below for definiteness
$$\label{eq:158}
\begin{array}{lll ll}
\begin{array}{l}
D^{\ +}
\vspace*{0.3cm} \\
D^{\ +}_{\ s}
\end{array}
& \rightarrow &
\hspace*{0.3cm} \pi^{\ -} \ \pi^{\ +} \ \pi^{\ +}
\hspace*{0.3cm} & ; & \hspace*{0.3cm}
\cite{aitala} \ , \ \cite{bediaga}
\vspace*{0.3cm} \\
B^{\ +}
& \rightarrow &
\ \begin{array}{l}
\pi^{\ -} \ K^{\ +} \ \pi^{\ +}
\vspace*{0.3cm} \\
K^{\ -} \ K^{\ +} \ K^{\ +}
\end{array}
\hspace*{0.3cm} & ; & \hspace*{0.3cm}
\cite{garmash} \ , \ \cite{babar}
\vspace*{0.3cm} \\
p \ \overline{p}
& \rightarrow &
\hspace*{0.3cm} \pi^{\ -} \ \pi^{\ +} \ \eta
\hspace*{0.3cm} & ; & \hspace*{0.3cm}
\cite{amslerrev2}
\end{array}$$
The present results from the study of the above decays do favour the derived existence of a ’$\sigma$’ isoscalar state, called $f_{\ 0} \ ( \ 600 \ )$ by the PDG [@PDG] as well as indications of an isospin 1/2 state called ’$\kappa$’, with masses $\sim \ 500$ MeV for $\sigma$ and $\sim \ 750$ MeV for $\kappa$ respectively. The determination of the widths is rather uncertain, but follows the widths of peaks in the projected Dalitz plot distributions of the order of 200-400 MeV.
It is fair to say, that as welcome as these new channels are, the present stage of analysis has not led to a clear picture of scalar meson states.
[**c) Central production experiments**]{}
The first experiment searching for gluonic mesons in central production was performed at the ISR at CERN [@akesson] , at $\sqrt{s} \ = \ 63$ GeV.
I reproduce here the invariant mass distribution of $\pi^{\ +} \ \pi^{\ -}$ pairs as observed in ref. [@akesson]
+0.2cm
-0.2cm
-1.0cm
Even though figure \[fig14\] represents the (absolute) square of an amplitude and figure \[fig12\] the square of [*another*]{} amplitude, the similarity and shape of the red dragon is clearly visible. This similarity does not need any further analysis.
The more recent experiment WA102 and its predecessor WA76 are using a fixed target configuration and thus the c.m. energies studied are lower $\sqrt{s} \ \le \ 29$ GeV [@WA102] .
Despite dedicated studies [@Close], no clear understanding of central production [*and*]{} spectroscopic information encoded in ps-ps scattering amplitudes ( section b) of this chapter ) nor any convincing evidence for the mass region from lattice QCD calculations ( section a) of this chapter ) for the gluonic binary $gb ( \ 0^{\ ++} \ )$ is emerging.
Rather a choice of apparent possibilities is offered, where in order not to offend any individuals I follow the PDG [@PDG]
$$\label{eq:159}
\begin{array}{l}
f_{\ 0} \ : \ 600 \ , \ 980 \ , \ 1370 \ , \ 1500 \ , \ 1710
\ \cdots \ \mbox{MeV}
\end{array}$$
The present [*controversial*]{} situation does - in my opinion - reflect human shortcomings more than intrinsic difficulty to understand the strong interaction dynamics underlying gluonic binaries as well as $q \ \overline{q}$ scalar mesons.
Conclusion
===========
In view of the previous sentence and in summary of the present outline, I think that a dedicated experiment of central production, at the highest achievable c.m. energies as well as with an optimally adapted detector is scientifically worth while.
Appendix
========
Spinor wave functions, spin states and transformation rules
-----------------------------------------------------------
We present the spin 1/2 chiral building blocks below, as they determine the general spin transformation rules defined in eq. (\[eq:12\]) .
$$\label{eq:14}
\begin{array}{l}
S_{\ \underline{\alpha}}^{\hspace*{0.3cm} \underline{\beta}} \ ( \ a \ )
\ = \ \left \lbrace
\ S_{\ \alpha_{1}}^{\hspace*{0.3cm} \beta_{1}} \ ( \ a \ )
\ \times \ \cdots \ \times
\ S_{\ \alpha_{N}}^{\hspace*{0.3cm} \beta_{N}} \ ( \ a \ )
\ \right \rbrace_{\ symm}
\end{array}$$
The irreducible blocks $S_{\ \alpha_{j}}^{\hspace*{0.3cm} \beta_{j}} \ ( \ a \ )$ in eq. (\[eq:14\]) correspond to spin 1/2
$$\label{eq:15}
\begin{array}{l}
a \ = \ \left ( \ a_{\ 0} \ , \ a_{\ 1} \ , \ a_{\ 2} \ , \ a_{\ 3}
\ \right ) \ = \ a_{\ \mu}
\hspace*{0.3cm} : \hspace*{0.3cm}
\begin{array}{l}
\mbox{complex four-vector}
\vspace*{0.1cm} \\
\mbox{{\it not} a Lorentz vector}
\end{array}
\vspace*{0.3cm} \\
S_{\ \alpha}^{\hspace*{0.3cm} \beta} \ ( \ a \ ) \ =
\ S_{\ 1}^{\ 1} \ ( \ a \ )
\ =
\ \left (
\ a_{\ 0} \ \Sigma_{\ 0} \ + \ \frac{1}{i} \ \vec{a}
\ \vec{\Sigma}
\ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\vspace*{0.3cm} \\
S_{\ 1}^{\ 1} \ ( \ a \ ) \ =
\ \left (
\begin{array}{rr}
a_{\ 0} \ - \ i \ a_{\ 3} \hspace*{0.2cm} & \hspace*{0.2cm}
- \ a_{\ 2} \ - \ i \ a_{\ 1}
\vspace*{0.3cm} \\
a_{\ 2} \ - \ i \ a_{\ 1} \hspace*{0.2cm} & \hspace*{0.2cm}
a_{\ 0} \ + \ i \ a_{\ 3}
\end{array}
\ \right )
\vspace*{0.3cm} \\
a^{\ 2} \ = \ a_{\ 0}^{\ 2} \ + \ \vec{a}^{\ 2} \ = \ Det \ S_{\ 1}^{\ 1}
\ = \ 1
\end{array}$$
The quadratic constraint restricts $S_{\ 1}^{\ 1}$ as defined in eq. \[eq:15\] to be unimodular (i.e. to have Det = 1). Rotations ( by half angles in bosonic terms ) correspond to $a_{\ \mu}$ real. This is parametrizing the sphere ( over the real numbers ) : $S_{\ 3} \ \equiv \ SU2$. Lorentz boosts ( by hyperbolic half angles in bosonic terms ) correspond to $a_{\ 0}$ real, $\vec{a}$ pure imaginary. This is parametrizing the double hyperboloid ( over the real numbers ) : $( \ \Re \ a_{\ 0} \ )^{\ 2} \ - \ ( \ \Im \ \vec{a} \ )^{\ 2} \ = \ 1$.
The matrices $\Sigma_{\ 0} \ , \ \Sigma_{\ k} \ ; \ k = \ 1,2,3$ are the Pauli matrices, as arising in the right chiral representation of the full $\gamma \ -$ matrix algebra.
$$\label{eq:16}
\begin{array}{l}
\left ( \ \sigma_{\ \mu \ \nu} \ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\ \leftrightarrow
\ P_{\ R} \ \frac{i}{2}
\ \left \lbrack \ \gamma_{\ \mu} \ , \ \gamma_{\ \nu} \ \right \rbrack
\ P_{\ R}
\hspace*{0.3cm} ; \hspace*{0.3cm}
P_{\ R} \ = \ \frac{1}{2} \ ( \ 1 \ + \ \gamma_{\ 5 \ R} \ )
\vspace*{0.3cm} \\
\gamma_{\ 5 \ R} \ = \ \frac{1}{i}
\ \gamma_{\ 0} \ \gamma_{\ 1} \ \gamma_{\ 2} \ \gamma_{\ 3}
\vspace*{0.3cm} \\
\left ( \ \sigma_{\ \mu \ \nu} \ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\ =
\ \begin{array}{ll}
\ \left (
\begin{array}{l}
- \ i \ \Sigma_{\ k}
\vspace*{0.3cm} \\
\varepsilon_{\ m n r} \ \Sigma_{\ r}
\end{array}
\ \right )_{ \alpha}^{\hspace*{0.3cm} \beta}
\vspace*{1.0cm}
&
\vspace*{-0.5cm}
\begin{array}{l}
\mbox{for} \ \mu \ = \ 0 \ , \ \nu \ = \ k \ = \ 1,2,3
\vspace*{0.3cm} \\
\mbox{for} \ \mu \ = \ m \ , \ \nu \ = \ n \ ;
\vspace*{-0.2cm} \\
\hspace*{2.7cm}
m,n,r \ = \ 1,2,3
\end{array}
\end{array}
\end{array}$$
The right chiral quantities $\left ( \ \sigma_{\ \mu \ \nu}
\ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}$ in eq. (\[eq:16\]) satisfy the duality relation
$$\label{eq:17}
\begin{array}{l}
\left ( \ \sigma_{\ \mu \ \nu}
\ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\ \rightarrow \ \sigma_{\ \mu \ \nu}^{\ R}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\sigma_{\ \mu \ \nu}^{\ R} \ = \ - \ i
\ \frac{1}{2} \ \varepsilon_{\ \mu \nu \varrho \tau}
\ \sigma^{\ \varrho \ \tau \ R}
\end{array}$$
[*Half angles (6) , rotational and hyperbolic - a) to the right*]{}
An infinitesimal Lorentz transformation is covered by the spin 1/2 half angles $\omega^{\ \mu \nu}$ , defined below, multiplying the (right chiral) base transformations $\sigma_{\ \mu \ \nu}^{\ R}$
$$\label{eq:18}
\begin{array}{l}
\omega^{\ \mu \nu} \ = \ \frac{1}{2} \ \Omega^{\ \mu \nu}
\vspace*{0.3cm} \\
\omega^{\ \mu \nu} \ =
\ \left (
\begin{array}{ccc c}
0 & \varepsilon_{\ 1} & \varepsilon_{\ 2} & \varepsilon_{\ 3}
\vspace*{0.1cm} \\
- \varepsilon_{\ 1} & 0 & \Theta_{\ 3} & - \Theta_{\ 2}
\vspace*{0.1cm} \\
- \varepsilon_{\ 2} & - \Theta_{\ 3} & 0 & \Theta_{\ 1}
\vspace*{0.1cm} \\
- \varepsilon_{\ 3} & \Theta_{\ 2} & - \Theta_{\ 1} & 0
\end{array}
\ \right )
\ =
\ \omega^{\ \mu \nu} \ ( \ \vec{\Theta} \ , \ \vec{\varepsilon} \ )
\end{array}$$
Projecting $\omega$ onto $\sigma^{\ R}$ we obtain
$$\label{eq:19}
\begin{array}{l}
\left ( \ \omega^{\ R} \ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\hspace*{0.3cm} \rightarrow \hspace*{0.3cm}
\omega_{\ \alpha}^{\hspace*{0.3cm} \beta}
\vspace*{0.3cm} \\
\omega_{\ \alpha}^{\hspace*{0.3cm} \beta}
\ = \ \frac{1}{2}
\ \omega^{\ \mu \nu}
\ \left ( \ \sigma_{\ \mu \ \nu}
\ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\ =
\ i
\ \left (
\ \left \lbrace
\ \vec{\Theta} \ - i \ \vec{\varepsilon}
\ \right \rbrace
\ \frac{1}{i} \ \vec{\Sigma}
\ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\vspace*{0.3cm} \\
\rightarrow
\ \vec{\omega} \ \equiv \ \vec{\omega}^{\ R}
\ = \ \vec{\Theta} \ - \ i \ \vec{\varepsilon}
\end{array}$$
$S_{\ 1}^{\ 1}$ in eq. (\[eq:15\]) then represents the exponential of $\left ( \ \omega \ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}$ (multiplied with $\frac{1}{i}$)
$$\label{eq:20}
\begin{array}{l}
S_{\ \alpha}^{\hspace*{0.3cm} \beta} \ ( \ a \ ) \ =
\ \exp
\ \left (
\ \frac{1}{i} \ \omega
\ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\ =
\ \exp
\ \left (
\ \frac{1}{i} \ \vec{\omega} \ \vec{\Sigma}
\ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\end{array}$$
Leaving out the (right chiral) spinor indices eq. (\[eq:20\]) becomes
$$\label{eq:21}
\begin{array}{l}
S \ ( \ a \ ) \ =
\ \cos \ ( \ \vec{\omega} \ \vec{\Sigma} \ )
\ - \ i \ \sin \ ( \ \vec{\omega} \ \vec{\Sigma} \ )
\end{array}$$
Thus we introduce the orthogonal complex invariant of $\vec{\omega}$
$$\label{eq:22}
\begin{array}{l}
Z \ ( \ \omega \ ) \ =
\ z^{\ 2} \ ( \ \omega \ ) \ =
\ \vec{\omega}^{\ 2} \ =
\ \left \lbrace
\ \vec{\Theta}^{\ 2} \ - \ \vec{\varepsilon}^{\ 2}
\ \right \rbrace
\ - \ i
\ \left \lbrace
\ 2 \ \vec{\Theta} \ \vec{\varepsilon}
\ \right \rbrace
\vspace*{0.3cm} \\
\rightarrow
\hspace*{0.3cm}
S \ ( \ a \ ) \ =
\ \cos \ ( \ z \ ) \ \Sigma_{\ 0}
\ - \ i \ \left \lbrack
\ \sin \ ( \ z \ ) \ / \ z
\ \right \rbrack
\ \vec{\omega} \ \vec{\Sigma}
\vspace*{0.3cm} \\
\hspace*{0.3cm}
a \ = \ a \ ( \ \omega \ )
\hspace*{0.3cm} ; \hspace*{0.3cm}
a_{\ 0} \ = \ \cos \ ( \ z \ )
\hspace*{0.3cm} ; \hspace*{0.3cm}
\vec{a} \ = \ \left \lbrack
\ \sin \ ( \ z \ ) \ / \ z
\ \right \rbrack \ \vec{\omega}
\end{array}$$
The square root ambiguity of $z \ ( \ \omega \ ) \ = \ \pm \ \sqrt{Z \ ( \ \omega \ )}$ does not affect the functional relation $a \ = \ a \ ( \ \omega \ )$, as becomes clear from eq. (\[eq:22\]).
[*From right chiral to left chiral spinors*]{}
The right chiral base representations of $SL2C_{\ R}$ are by construction not parity invariant, nor are the matrices $S \ ( \ a \ ) \ \equiv \ {\cal{A}}$ over the real numbers.
Se we shall transform the defining equations (\[eq:14\]-\[eq:16\]) to the left chiral side
$$\label{eq:23}
\begin{array}{l}
\hspace*{0.6cm}
S_{\ \underline{\alpha}}^{\hspace*{0.3cm} \underline{\beta}} \ ( \ a \ )
\ = \ \left \lbrace
\ S_{\ \alpha_{1}}^{\hspace*{0.3cm} \beta_{1}} \ ( \ a \ )
\ \times \ \cdots \ \times
\ S_{\ \alpha_{N}}^{\hspace*{0.3cm} \beta_{N}} \ ( \ a \ )
\ \right \rbrace_{\ symm}
\ \rightarrow
\vspace*{0.3cm} \\
\rightarrow
\ \widetilde{S}^{\ \underline{\dot{\gamma}}}_{\hspace*{0.3cm}
\underline{\dot{\delta}}} \ ( \ b \ )
\ = \ \left \lbrace
\ \widetilde{S}^{\ \dot{\gamma}_{1}}_{\hspace*{0.3cm} \dot{\delta}_{1}}
\ ( \ b \ )
\ \times \ \cdots \ \times
\ \widetilde{S}^{\ \dot{\gamma}_{N}}_{\hspace*{0.3cm} \dot{\delta}_{N}}
\ ( \ b \ )
\ \right \rbrace_{\ symm}
\end{array}$$
$$\label{eq:24}
\begin{array}{l}
b \ = \ \left ( \ b_{\ 0} \ , \ b_{\ 1} \ , \ b_{\ 2} \ , \ b_{\ 3}
\ \right ) \ = \ b_{\ \mu}
\hspace*{0.3cm} : \hspace*{0.3cm}
\begin{array}{l}
\mbox{complex four-vector}
\vspace*{0.1cm} \\
\mbox{{\it not} a Lorentz vector}
\end{array}
\vspace*{0.3cm} \\
\widetilde{S}^{\ \dot{\gamma}}_{\hspace*{0.3cm} \dot{\delta}} \ ( \ b \ ) \ =
\ \widetilde{S}_{\ 2}^{\ 2} \ ( \ b \ )
\ =
\ \left (
\ b_{\ 0} \ \widetilde{\Sigma}_{\ 0} \ + \ \frac{1}{i} \ \vec{b}
\ \vec{\widetilde{\Sigma}}
\ \right )^{\ \dot{\gamma}}_{\hspace*{0.3cm} \dot{\delta}}
\vspace*{0.3cm} \\
\widetilde{S}_{\ 2}^{\ 2} \ ( \ b \ ) \ =
\ \left (
\begin{array}{rr}
b_{\ 0} \ - \ i \ b_{\ 3} \hspace*{0.2cm} & \hspace*{0.2cm}
- \ b_{\ 2} \ - \ i \ b_{\ 1}
\vspace*{0.3cm} \\
b_{\ 2} \ - \ i \ b_{\ 1} \hspace*{0.2cm} & \hspace*{0.2cm}
b_{\ 0} \ + \ i \ b_{\ 3}
\end{array}
\ \right )
\vspace*{0.3cm} \\
b^{\ 2} \ = \ b_{\ 0}^{\ 2} \ + \ \vec{b}^{\ 2} \ =
\ Det \ \widetilde{S}_{\ 2}^{\ 2}
\ = \ 1
\end{array}$$
The transformation from ${\cal{A}}$ to ${\cal{B}}$ corresponds to the substitution
$$\label{eq:25}
\begin{array}{l}
{\cal{A}} \ \rightarrow
\ \left ( \ {\cal{A}}^{\ \dagger} \ \right )^{\ -1}
\ = \ {\cal{B}}
\end{array}$$
The substitution in eq. (\[eq:25\]) makes use of the four base representations of SL2C, best represented in the associated quadrangle
$$\label{eq:26}
\begin{array}{l}
\begin{array}{ccc}
{\cal{A}} & \longleftrightarrow &
\left ( \ {\cal{A}}^{\ T} \ \right )^{\ -1}
\vspace*{0.3cm} \\
\updownarrow \ \equiv \ c.c. & &
\updownarrow \ \equiv \ c.c.
\vspace*{0.3cm} \\
\overline{\cal{A}} & \longleftrightarrow &
\left ( \ {\cal{A}}^{\ \dagger} \ \right )^{\ -1}
\end{array}
\end{array}$$
In the quadrangle in eq. (\[eq:26\]) the up-down operation means complex conjugation of each matrix element, forming the involutory chains
$$\begin{array}{c}
{\cal{A}} \ \rightarrow \ \overline{{\cal{A}}} \ \rightarrow \ {\cal{A}}
\vspace*{0.3cm} \\
\mbox{and}
\vspace*{0.3cm} \\
\left ( \ {\cal{A}}^{\ T} \ \right )^{\ -1} \ \rightarrow
\ \left ( \ {\cal{A}}^{\ \dagger} \ \right )^{\ -1}
\ \rightarrow \ \left ( \ {\cal{A}}^{\ T} \ \right )^{\ -1}
\end{array}$$
whereas the left-right operation associates the symplectic dual, forming the equally involutary chains
$$\begin{array}{c}
{\cal{A}} \ \rightarrow \ \left ( \ {\cal{A}}^{\ T} \ \right )^{\ -1}
\ \rightarrow \ {\cal{A}}
\vspace*{0.3cm} \\
\mbox{and}
\vspace*{0.3cm} \\
\overline{{\cal{A}}} \ \rightarrow
\ \left ( \ {\cal{A}}^{\ \dagger} \ \right )^{\ -1}
\ \rightarrow \ \overline{{\cal{A}}}
\end{array}$$
Thus both up-down and left-right transformations along the quadrangle in eq. (\[eq:26\]) are commutative as well as involutory.
Yet the left-right transformation associates equivalent representations, contrary to the up- down one, which associates inequivalent representations, of which we have chosen the two residing in the upper left and lower right corners of the triangle in eq. (\[eq:26\].
The symplectic equivalence is realized in the right chiral basis by
$$\label{eq:27}
\begin{array}{l}
\left ( \ {\cal{A}}^{\ T} \ \right )^{\ -1}
\ =
\ s \ {\cal{A}} \ s^{\ -1}
\hspace*{0.3cm} ; \hspace*{0.3cm}
s \ = \ ( \ \pm \ ) \ i \ \sigma_{\ 2} \ =
\ ( \ \pm \ ) \ \left (
\begin{array}{rl}
0 & 1
\vspace*{0.2cm} \\
- 1 & 0
\end{array}
\right )
\end{array}$$
The base Pauli matrices go into each other under the substitution in eq. (\[eq:25\])
$$\label{eq:28}
\begin{array}{l}
\Sigma_{\ \mu} \ =
\ \Sigma_{\ \mu}^{\ \dagger}
\ =
\ \left ( \ \Sigma_{\ \mu} \ \right )^{\ -1}
\ =
\ \left ( \ \Sigma_{\ \mu}^{\ \dagger} \ \right )^{\ -1}
\vspace*{0.2cm} \\
\rightarrow
\ \left (
\ \widetilde{\Sigma}_{\ 0} \ ,
\ \vec{\widetilde{\Sigma}}
\ \right )
\ =
\ \left (
\ \Sigma_{\ 0} \ ,
\ \vec{\Sigma}
\ \right )
\end{array}$$
Hence we have
$$\label{eq:29}
\begin{array}{l}
{\cal{A}} \ \rightarrow
\ \left ( \ {\cal{A}}^{\ \dagger} \ \right )^{\ -1}
\ = \ {\cal{B}} \ ( \ {\cal{A}} \ )
\vspace*{0.3cm} \\
b \ = \ \overline{a}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\forall \ \mbox{components}
\end{array}$$
Thus the quadrangle in eq. (\[eq:26\]) leads to the right- and left-chiral reality restricted form of $SL2C_{\ R} \ \times \ SL2C_{\ L}$
$$\label{eq:30}
\begin{array}{l}
\left \lbrace spin \right \rbrace \ \rightarrow \ s
\hspace*{0.3cm} , \hspace*{0.3cm}
\# \ s \ = \ N \ + \ 1
\hspace*{0.3cm} ; \hspace*{0.3cm}
D_{\ s \ s^{'} } \ = \ D_{\ s \ s^{'} }^{\ J} \ ( \ \Lambda \ , \ p \ )
\vspace*{0.5cm} \\
\begin{array}{lll lll l}
R & : & t_{\ \underline{\alpha}}
\ ( \ \Lambda p \ ; \ s \ )
& = &
S_{\ \underline{\alpha}}^{\hspace*{0.3cm} \underline{\beta}} \ ( \ a \ )
& \hspace*{0.2cm} t_{\ \underline{\beta}}
\ ( \ p \ ; \ s^{'} \ )
& \hspace*{0.2cm} D_{\ s \ s^{'} }
\vspace*{0.3cm} \\
L & : & \widetilde{t}^{\ \underline{\dot{\gamma}}}
\ ( \ \Lambda p \ ; \ s \ )
& = &
\widetilde{S}^{\ \underline{\dot{\gamma}}}_{\hspace*{0.4cm}
\underline{\dot{\delta}}} \ ( \ b \ )
& \hspace*{0.2cm} \widetilde{t}^{\ \underline{\dot{\delta}}}
\ ( \ p \ ; \ s^{'} \ )
& \hspace*{0.2cm} D_{\ s \ s^{'} }
\end{array}
\end{array}$$
While we proceed in steps, let me quote Res Jost [@RJ] , illustrating the L-R chiral aspects.
The decomposition in eq. (\[eq:14\]) expands (doubles) into
$$\label{eq:31}
\begin{array}{l}
\begin{array}{lll}
R & : &
S_{\ \underline{\alpha}}^{\hspace*{0.3cm} \underline{\beta}} \ ( \ a \ )
\ = \ \left \lbrace
\ S_{\ \alpha_{1}}^{\hspace*{0.3cm} \beta_{1}} \ ( \ a \ )
\ \times \ \cdots \ \times
\ S_{\ \alpha_{N}}^{\hspace*{0.3cm} \beta_{N}} \ ( \ a \ )
\ \right \rbrace_{\ symm}
\vspace*{0.3cm} \\
L & : &
\widetilde{S}^{\ \underline{\dot{\gamma}}}_{\hspace*{0.4cm}
\underline{\dot{\delta}}} \ ( \ b \ )
\ = \ \left \lbrace
\ \widetilde{S}^{\ \dot{\gamma_{1}}}_{\hspace*{0.4cm} \dot{\delta_{1}}}
\ ( \ b \ )
\ \times \ \cdots \ \times
\ \widetilde{S}^{\ \dot{\gamma_{N}}}_{\hspace*{0.4cm} \dot{\delta_{N}}}
\ ( \ b \ )
\ \right \rbrace_{\ symm}
\end{array}
\end{array}$$
and then reduces to the R-L spin 1/2 building blocks
$$\label{eq:32}
\begin{array}{l}
\begin{array}{lll ll ll}
R & : &
S_{\ \alpha}^{\hspace*{0.3cm} \beta} \ ( \ a \ )
& = &
S_{\ 1}^{\ 1} \ ( \ a \ ) \ \equiv \ {\cal{A}} \ ( \ a \ )
& = &
\left (
\ a_{\ 0} \ \Sigma_{\ 0} \ + \ \frac{1}{i} \ \vec{a}
\ \vec{\Sigma}
\ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\vspace*{0.3cm} \\
L & : &
\widetilde{S}^{\ \dot{\gamma}}_{\hspace*{0.3cm} \dot{\delta}} \ ( \ b \ )
& = &
\widetilde{S}_{\ 2}^{\ 2} \ ( \ b \ ) \ \equiv \ {\cal{B}} \ ( \ b \ )
& = &
\left (
\ b_{\ 0} \ \widetilde{\Sigma}_{\ 0} \ + \ \frac{1}{i} \ \vec{b}
\ \vec{\widetilde{\Sigma}}
\ \right )^{\ \dot{\gamma}}_{\hspace*{0.3cm} \dot{\delta}}
\end{array}
\vspace*{0.3cm} \\
\widetilde{\Sigma}_{\ \mu} \ = \ \Sigma_{\ \mu}
\end{array}$$
The reality condition corresponds to a diagonal in the quadrangle in eq. (\[eq:26\])
$$\label{eq:33}
\begin{array}{l}
\begin{array}{rcr c}
{\cal{A}} \ ( \ a \ ) & \longleftrightarrow &
{\cal{B}} \ ( \ b \ ) &
\vspace*{-0.1cm} \\
\searrow & & \searrow &
\vspace*{-0.0cm} \\
& {\cal{A}} & = \hspace*{0.4cm} &
\left ( \ {\cal{B}}^{\ \dagger} \ \right )^{\ -1}
\vspace*{-0.1cm} \\
& \downarrow & & \downarrow
\vspace*{-0.0cm} \\
& a & = \hspace*{0.4cm} & \overline{b}
\end{array}
\end{array}$$
The so constrained pair
$$\label{eq:34}
\begin{array}{l}
\left (
\ {\cal{A}} \ ( \ a \ )
\ , \ {\cal{B}} \ ( \ \overline{a} \ )
\ \right )
\ \equiv \ spin \ ( \ 1 \ , \ 3 \ ; \ \Re \ )
\ \simeq \ SL2C
\end{array}$$
defines the (self covered) group $spin \ ( \ 1 \ , \ 3 \ ; \ \Re \ )$ : $\Re$ indicates that the spin group is over the [*real*]{} numbers, whereas $1 \ , \ 3$ denote the signature of the derived metric, i.e. 1 time and 3 space (real) dimensions.
[*Half angles (6) , rotational and hyperbolic - b) to the left*]{}
The left-chiral representation of $spin \ ( \ 1 \ , \ 3 \ ; \ \Re \ )$ complements the right-chiral one defined in eq. (\[eq:16\])
$$\label{eq:35}
\begin{array}{l}
\left ( \ \sigma_{\ \mu \ \nu} \ \right )^{\ \dot{\gamma}}_{\hspace*{0.4cm}
\dot{\delta}}
\ \leftrightarrow
\ P_{\ L} \ \frac{i}{2}
\ \left \lbrack \ \gamma_{\ \mu} \ , \ \gamma_{\ \nu} \ \right \rbrack
\ P_{\ L}
\hspace*{0.3cm} ; \hspace*{0.3cm}
P_{\ L} \ = \ \frac{1}{2} \ ( \ 1 \ - \ \gamma_{\ 5 \ R} \ )
\vspace*{0.3cm} \\
\gamma_{\ 5 \ R} \ = \ \frac{1}{i}
\ \gamma_{\ 0} \ \gamma_{\ 1} \ \gamma_{\ 2} \ \gamma_{\ 3}
\vspace*{0.3cm} \\
\left ( \ \sigma_{\ \mu \ \nu} \ \right )^{\ \dot{\gamma}}_{\hspace*{0.4cm}
\dot{\delta}}
\ =
\ \begin{array}{ll}
\ \left (
\begin{array}{l}
i \ \Sigma_{\ k}
\vspace*{0.3cm} \\
\varepsilon_{\ m n r} \ \Sigma_{\ r}
\end{array}
\ \right )^{ \dot{\gamma}}_{\hspace*{0.4cm} \dot{\delta}}
\vspace*{1.0cm}
&
\vspace*{-0.5cm}
\begin{array}{l}
\mbox{for} \ \mu \ = \ 0 \ , \ \nu \ = \ k \ = \ 1,2,3
\vspace*{0.3cm} \\
\mbox{for} \ \mu \ = \ m \ , \ \nu \ = \ n \ ;
\vspace*{-0.2cm} \\
\hspace*{2.7cm}
m,n,r \ = \ 1,2,3
\end{array}
\end{array}
\end{array}$$
In principle we should have distinguished the left chiral matrices $\widetilde{\Sigma}_{\ \mu}$ characterizing the left chiral SL2C basis in eq. (\[eq:35\]) but we have chosen (without loss of generality) to identify $\widetilde{\Sigma}_{\ \mu} \ = \ \Sigma_{\ \mu}$ as specified in eq. (\[eq:32\]).
The left chiral variant of eq. (\[eq:17\]) becomes
$$\label{eq:36}
\begin{array}{l}
\left ( \ \sigma_{\ \mu \ \nu}
\ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\ \rightarrow \ \sigma_{\ \mu \ \nu}^{\ R}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\sigma_{\ \mu \ \nu}^{\ R} \ = \ - \ i
\ \frac{1}{2} \ \varepsilon_{\ \mu \nu \varrho \tau}
\ \sigma^{\ \varrho \ \tau \ R}
\vspace*{0.3cm} \\
\rightarrow
\ \left ( \ \sigma_{\ \mu \ \nu} \ \right )^{\ \dot{\gamma}}_{\hspace*{0.4cm}
\dot{\delta}}
\ \rightarrow \ \sigma_{\ \mu \ \nu}^{\ L}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\sigma_{\ \mu \ \nu}^{\ L} \ = \ + \ i
\ \frac{1}{2} \ \varepsilon_{\ \mu \nu \varrho \tau}
\ \sigma^{\ \varrho \ \tau \ L}
\end{array}$$
Eq. (\[eq:19\]) when reflected to the left takes on the form
$$\label{eq:37}
\begin{array}{l}
\left ( \ \omega^{\ L} \ \right )^{\ \dot{\gamma}}_{\hspace*{0.4cm}
\dot{\delta}}
\hspace*{0.3cm} \rightarrow \hspace*{0.3cm}
\widetilde{\omega}^{\ \dot{\gamma}}_{\hspace*{0.4cm} \dot{\delta}}
\vspace*{0.3cm} \\
\widetilde{\omega}^{\ \dot{\gamma}}_{\hspace*{0.4cm} \dot{\delta}}
\ = \ \frac{1}{2}
\ \omega^{\ \mu \nu}
\ \left ( \ \sigma_{\ \mu \ \nu}
\ \right )^{\ \dot{\gamma}}_{\hspace*{0.4cm} \dot{\delta}}
\ =
\ i
\ \left (
\ \left \lbrace
\ \vec{\Theta} \ + i \ \vec{\varepsilon}
\ \right \rbrace
\ \frac{1}{i} \ \vec{\Sigma}
\ \right )^{\ \dot{\gamma}}_{\hspace*{0.4cm} \dot{\delta}}
\vspace*{0.3cm} \\
\rightarrow
\ \vec{\widetilde{\omega}} \ \equiv \ \vec{\omega}^{\ L}
\ = \ \vec{\Theta} \ + \ i \ \vec{\varepsilon}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\vec{\omega} \ \equiv \ \vec{\omega}^{\ R}
\ = \ \vec{\Theta} \ - \ i \ \vec{\varepsilon}
\end{array}$$
$\widetilde{S}_{\ 2}^{\ 2}$ in eq. (\[eq:32\] is along with the right counterpart in eq. (\[eq:20\])
$$\label{eq:38}
\begin{array}{l}
S_{\ \alpha}^{\hspace*{0.3cm} \beta} \ ( \ a \ ) \ =
\ \exp
\ \left (
\ \frac{1}{i} \ \omega
\ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\ =
\ \exp
\ \left (
\ \frac{1}{i} \ \vec{\omega} \ \vec{\Sigma}
\ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\vspace*{0.3cm} \\
\rightarrow
\ \widetilde{S}^{\ \dot{\gamma}}_{\hspace*{0.3cm} \dot{\delta}} \ ( \ b \ )
\ = \ \exp
\ \left (
\ \frac{1}{i} \ \widetilde{\omega}
\ \right )^{\ \dot{\gamma}}_{\hspace*{0.4cm} \dot{\delta}}
\ =
\ \exp
\ \left (
\ \frac{1}{i} \ \vec{\widetilde{\omega}} \ \vec{\Sigma}
\ \right )^{\ \dot{\gamma}}_{\hspace*{0.4cm} \dot{\delta}}
\end{array}$$
Eq. (\[eq:21\]) extends to
$$\label{eq:39}
\begin{array}{l}
S \ ( \ a \ ) \ =
\ \cos \ ( \ \vec{\omega} \ \vec{\Sigma} \ )
\ - \ i \ \sin \ ( \ \vec{\omega} \ \vec{\Sigma} \ )
\vspace*{0.3cm} \\
\rightarrow
\ \widetilde{S} \ ( \ b \ )
\ = \ S \ ( \ b \ )
\ = \ \exp
\ \left (
\ \frac{1}{i} \ \widetilde{\omega}
\ \right )
\ = \ \cos \ ( \ \vec{\widetilde{\omega}} \ \vec{\Sigma} \ )
\ - \ i \ \sin \ ( \ \vec{\widetilde{\omega}} \ \vec{\Sigma} \ )
\vspace*{0.3cm} \\
\hspace*{0.5cm}
\vec{\widetilde{\omega}} \ =
\ \overline{\vec{\omega}}
\hspace*{0.3cm} ; \hspace*{0.3cm}
b \ = \ \overline{a}
\vspace*{0.3cm} \\
\hspace*{0.5cm}
S \ ( \ a \ ) \ = \ a_{\ 0} \ \Sigma_{\ 0} \ + \ \frac{1}{i} \ \vec{a}
\ \vec{\Sigma}
\hspace*{0.3cm} ; \hspace*{0.3cm}
S \ ( \ b \ ) \ = \ b_{\ 0} \ \Sigma_{\ 0} \ + \ \frac{1}{i} \ \vec{b}
\ \vec{\Sigma}
\end{array}$$
Eq. (\[eq:22\]) completes to
$$\label{eq:40}
\begin{array}{l}
Z \ ( \ \omega \ ) \ =
\ z^{\ 2} \ ( \ \omega \ ) \ =
\ \vec{\omega}^{\ 2} \ =
\ \left \lbrace
\ \vec{\Theta}^{\ 2} \ - \ \vec{\varepsilon}^{\ 2}
\ \right \rbrace
\ - \ i
\ \left \lbrace
\ 2 \ \vec{\Theta} \ \vec{\varepsilon}
\ \right \rbrace
\vspace*{0.3cm} \\
\rightarrow
\hspace*{0.3cm}
S \ ( \ a \ ) \ =
\ \cos \ ( \ z \ ) \ \Sigma_{\ 0}
\ - \ i \ \left \lbrack
\ \sin \ ( \ z \ ) \ / \ z
\ \right \rbrack
\ \vec{\omega} \ \vec{\Sigma}
\vspace*{0.3cm} \\
\hspace*{0.3cm}
a \ = \ a \ ( \ \omega \ )
\hspace*{0.3cm} ; \hspace*{0.3cm}
a_{\ 0} \ = \ \cos \ ( \ z \ )
\hspace*{0.3cm} ; \hspace*{0.3cm}
\vec{a} \ = \ \left \lbrack
\ \sin \ ( \ z \ ) \ / \ z
\ \right \rbrack \ \vec{\omega}
\vspace*{0.5cm} \\
\rightarrow
\hspace*{0.3cm}
Z \ ( \ \widetilde{\omega} \ ) \ =
\ z^{\ 2} \ ( \ \widetilde{\omega} \ ) \ =
\ \vec{\widetilde{\omega}}^{\ 2} \ =
\ \left \lbrace
\ \vec{\Theta}^{\ 2} \ - \ \vec{\varepsilon}^{\ 2}
\ \right \rbrace
\ + \ i
\ \left \lbrace
\ 2 \ \vec{\Theta} \ \vec{\varepsilon}
\ \right \rbrace
\vspace*{0.3cm} \\
Z \ ( \ \widetilde{\omega} \ ) \ =
\ z^{\ 2} \ ( \ \widetilde{\omega} \ ) \ =
\ \overline{Z} \ ( \ \omega \ )
\ =
\ \overline{z^{\ 2}} \ ( \ \omega \ )
\vspace*{0.3cm} \\
S \ ( \ b \ ) \ =
\ \cos \ ( \ \overline{z} \ ) \ \Sigma_{\ 0}
\ - \ i \ \left \lbrack
\ \sin \ ( \ \overline{z} \ ) \ / \ \overline{z}
\ \right \rbrack
\ \overline{\vec{\omega}} \ \vec{\Sigma}
\vspace*{0.3cm} \\
b \ = \ a \ ( \ \widetilde{\omega} \ ) \ = \ \overline{a \ ( \ \omega \ )}
\end{array}$$
[**Realization of (half) angles through an antisymmetric pair of vectors**]{}
The complex three vectors defining the half angles $\omega^{\ \mu \nu}$ in eq. (\[eq:18\])
$$\label{eq:41}
\begin{array}{l}
\vec{\omega}^{\ R}
\ = \ \vec{\Theta} \ - \ i \ \vec{\varepsilon}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\vec{\omega}^{\ L}
\ = \ \vec{\Theta} \ + \ i \ \vec{\varepsilon}
\end{array}$$
can be realized as antisymmetric combinations of two real Lorentz vectors $x^{\ \mu} \ , \ y^{\ \nu}$ . [*This is however a restricted realization.*]{}\
Here Lorentz vector does not distinguish between vector and axial vector. In fact we shall think of x as a genuine Lorentz four vector and of y as an axial vector.
$$\label{eq:42}
\begin{array}{l}
\omega_{\ \mu \nu} \ ( \ [ \ x \ , \ y \ ] \ )
\ =
\ \varepsilon_{ \ \mu \nu \sigma \tau}
\ x^{\ \sigma} \ y^{\ \mu}
\vspace*{0.3cm} \\
\omega_{\ 0 k} \ = \ \left ( \ \vec{x} \ \wedge \ \vec{y} \ \right )^{\ k}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\omega_{\ m n} \ =
\ \varepsilon_{ \ m n r }
\ \left ( \ x^{\ 0} \ y^{\ r} \ - \ x^{\ r} \ y^{\ 0} \ \right )
\vspace*{0.3cm} \\
\vec{\varepsilon} \ = \ - \ \vec{x} \ \wedge \ \vec{y}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\vec{\Theta} \ = \ x^{\ 0} \ \vec{y} \ - \ y^{\ 0} \ \vec{x}
\hspace*{0.3cm} : \hspace*{0.3cm}
\vec{\Theta} \ \vec{\varepsilon} \ = \ 0
\vspace*{0.5cm} \\
\begin{array}{ll}
\rightarrow &
\vec{\omega}^{\ R} \ = \ x^{\ 0} \ \vec{y} \ - \ y^{\ 0} \ \vec{x}
\ + \ i \ \left ( \ \vec{x} \ \wedge \ \vec{y} \ \right )
\vspace*{0.3cm} \\
& \vec{\omega}^{\ L} \ = \ x^{\ 0} \ \vec{y} \ - \ y^{\ 0} \ \vec{x}
\ - \ i \ \left ( \ \vec{x} \ \wedge \ \vec{y} \ \right )
\end{array}
\end{array}$$
The invariants $Z \ ( \ \omega \ ) \ , \ Z \ ( \ \widetilde{\omega} \ )$ in eqs. (\[eq:22\]) and (\[eq:40\]) are purely real
$$\label{eq:43}
\begin{array}{l}
Z \ ( \ \omega \ ) \ =
\ \left ( \ \vec{\omega}^{\ R} \ \right )^{\ 2}
\ =
\ \left ( \ x \ y \ \right )^{\ 2}
\ -
\ x^{\ 2} \ y^{\ 2}
\ =
\ \left ( \ \vec{\omega}^{\ L} \ \right )^{\ 2}
\ =
\ Z \ ( \ \widetilde{\omega} \ )
\end{array}$$
In eq. (\[eq:43\]) we used the timelike Lorentz scalar product $x^{\ 2} \ = \ ( \ x^{\ 0} \ )^{\ 2} \ - \ \vec{x}^{\ 2}$.
The realization given in eqs. (\[eq:42\]) and (\[eq:43\]) is useful when $x^{\ \mu}$ is proportional to a four-velocity, i.e. $x^{\ 0} \ \ge \ \lambda \ > \ 0 \ , \ x^{\ 2} \ = \ \lambda^{\ 2}$ and y describes a spin direction, chosen in such a way, that $x y \ = \ 0$, and $y^{\ 2} \ = \ - \ 1$.
Note on the complex Lorentz group and associated operations
-----------------------------------------------------------
We recall the reality constrained covering of the Lorentz group $spin \ ( \ 1 \ , \ 3 \ ; \ \Re \ )$ defined in eq. (\[eq:34\])
$$\label{eq:44}
\begin{array}{l}
\left (
\ {\cal{A}} \ ( \ a \ )
\ , \ {\cal{B}} \ ( \ \overline{a} \ )
\ \right )
\ \equiv \ spin \ ( \ 1 \ , \ 3 \ ; \ \Re \ )
\ \simeq \ SL2C
\end{array}$$
I list the operations on amplitudes or fields, which demand an extension of spin representations to [*covering of*]{} the complex Lorentz group. This latter extension is denoted by $\stackrel{C}{\rightarrow}$ , defined in eq. (\[eq:45\]) below
$$\label{eq:45}
\begin{array}{l}
\begin{array}{ccc cc}
spin \ ( \ 1 \ , \ 3 \ ; \ \Re \ )
& \stackrel{C}{\rightarrow} &
spin \ ( \ 1 \ , \ 3 \ ; \ C \ )
& \simeq & SL2C \ \times \ SL2C
\vspace*{0.3cm} \\
\left (
\ {\cal{A}} \ ( \ a \ )
\ , \ {\cal{B}} \ ( \ \overline{a} \ )
\ \right )
& \stackrel{C}{\rightarrow} &
\left (
\ {\cal{A}} \ ( \ a \ )
\ , \ {\cal{B}} \ ( \ b \ )
\ \right )
& ; &
a \ , \ b \ \mbox{unrestricted}
\end{array}
\end{array}$$
In the list below we number and specify the operation in the first and second columns, the operand in the third, inducing the parallel operation $\stackrel{C}{\rightarrow}$
$$\label{eq:46}
\begin{array}{l}
\begin{array}{|c|c|c|c|}
\hline
& \mbox{operation} & \mbox{operand} & \stackrel{C}{\rightarrow}
\\ \hline
1 & \mbox{crossing}
&
\mbox{scattering amplitudes}
& \surd \\
\hline
2 & \mbox{extension to complex momenta} &
\mbox{scattering amplitudes}
& \surd \\
\hline
3 & \mbox{extension to Euclidean space} &
\mbox{local fields}
& \surd \\
\hline
\end{array}
\end{array}$$
Operations 1 - 3 in eq. (\[eq:46\]) are [*not*]{} independent of each other. A profound consequence is the symmetry under the antiunitary CPT transformation [@RJ] for local field theories.
Field strengths, potentials and adjoint string operators
--------------------------------------------------------
Potentials and field strengths have been introduced in eqs. (\[eq:101\]) and (\[eq:102\]). We shall specify their local gauge transformation properties below. For simplicity we shall only discuss the octet or adjoint representation of $SU3_{\ c}$ .
The Lie algebra generators of the octet representation $\left ( \ {\cal{F}}_{\ D} \ \right )_{\ A B} \ = \ i \ f_{\ A \ D \ B}$ in eq. (\[eq:102\]) lead to the finite (local) transformations
$$\label{eq:47}
\begin{array}{l}
\Omega_{ \ A \ B} \ ( \ x \ )
\ =
\ \left (
\ \exp
\ \frac{1}{i} \ \omega_{\ D} \ ( \ x \ ) \ {\cal{F}}_{\ D}
\ \right )_{\ A \ B}
\vspace*{0.3cm} \\
\left \lbrack
\ {\cal{F}}_{\ A} \ , \ {\cal{F}}_{\ B}
\ \right \rbrack \ = \ i \ f_{\ A \ B \ C}
\ {\cal{F}}_{\ C}
\end{array}$$
The $SU3_{\ c}$ [*angles*]{} $\omega_{\ D} \ ( \ x \ )$ shall not be confused with the Euler half angles $\omega^{\ \mu \nu}$ in eq. (\[eq:21\]), while the group analogy is obvious. $\omega_{\ D} \ ( \ x \ )$ shall be chosen varying over space time $x$ , restricted by differentiability requirements.
Let $X \ ( \ x \ , \ A \ )$ be a classical field transforming under the local octet transformations $\Omega$
$$\label{eq:48}
\begin{array}{l}
X^{\ \Omega} \ ( \ x \ , \ A \ )
\ =
\ \Omega_{ \ A \ B} \ ( \ x \ )
\ X \ ( \ x \ , \ B \ )
\vspace*{0.3cm} \\
\mbox{in short :}
\hspace*{0.3cm}
X^{\ \Omega} \ ( \ x \ )
\ =
\ \Omega \ ( \ x \ )
\ X \ ( \ x \ )
\ \rightarrow
\ X^{\ \Omega} \ = \ \Omega \ X
\end{array}$$
The extension of the local [*adjoint*]{} transformations in eq. (\[eq:48\]) to other representations of $SU3_{\ c}$ is straightforward. $\Omega$ are real, orthogonal $8 \ \times \ 8$ matrices with determinant 1.
Here we treat gauge potentials and field strengths as classical fields ( test fields in the sence of distributions ). The potentials $ V_{\ \mu} \ ( \ x \ , \ D \ )$ are defined through the (octet) covariant derivatives acting on $X$
$$\label{eq:49}
\begin{array}{l}
\left ( \ D_{\ \mu} \ \right )_{\ A \ B}
\ = \ \partial_{\ \mu} \ \delta_{\ A \ B} \ +
\ \left ( \ {\cal{W}}_{\ \mu} \ \right )_{\ A \ B}
\hspace*{0.3cm} ; \hspace*{0.3cm} \partial_{\ \mu}
\ = \ \partial \ / \ \partial \ x^{\ \mu}
\vspace*{0.3cm} \\
\left ( \ {\cal{W}}_{\ \mu} \ \right )_{\ A \ B}
\ = \ i \ V_{\ \mu} \ ( \ x \ , \ D \ )
\ \left ( \ {\cal{F}}_{\ D} \ \right )_{\ A \ B}
\ =
\ V_{\ \mu} \ ( \ x \ , \ D \ ) \ f_{\ D \ A \ B}
\vspace*{0.3cm} \\
\mbox{in short :}
\hspace*{0.3cm}
{\cal{W}}_{\ \mu} \ = \ i \ V_{\ \mu \ D}
\ {\cal{F}}_{\ D}
\hspace*{0.3cm} ; \hspace*{0.3cm}
D_{\ \mu} \ = \ \partial_{\ \mu} \ + \ {\cal{W}}_{\ \mu}
\end{array}$$
In eq. (\[eq:49\]) the quantities $ V_{\ \mu} \ ( \ x \ , \ D \ )$ , $\ \left ( \ {\cal{W}}_{\ \mu} \ \right )_{\ A \ B} \ ( \ x \ )$ are real.
[*Parallel transport*]{}
We turn to the parallel transport operators, defined in eq. (\[eq:102\]) repeated below
$$\label{eq:50}
\begin{array}{l}
U \ ( \ x \ , \ A \ ; \ y \ , \ B \ )
\ =
P \ \exp
\ \left (
\ \left .
{\displaystyle{\int}}_{\ y}^{\ x}
\ \right |_{\ {\cal{C}}}
\ d \ z^{\ \mu}
\ \frac{1}{i} \ V_{\ \mu} \ ( \ z \ , \ D \ ) \ {\cal{F}}_{\ D}
\ \right )_{\ A \ B}
\vspace*{0.3cm} \\
\hspace*{0.8cm} =
\ P \ \exp
\ \left (
\ \left .
\ -
\ {\displaystyle{\int}}_{\ y}^{\ x}
\ \right |_{\ {\cal{C}}}
\ d \ z^{\ \mu}
\ {\cal{W}}_{\ \mu} \ ( \ z \ , \ D \ ) \ {\cal{F}}_{\ D}
\ \right )_{\ A \ B}
\vspace*{0.3cm} \\
\left ( \ {\cal{F}}_{\ D} \ \right )_{\ A B} \ = \ i \ f_{\ A \ D \ B}
\hspace*{0.3cm} ; \hspace*{0.3cm}
{\cal{W}}_{\ \mu} \ ( \ z \ , \ D \ ) \ =
\ i \ V_{\ \mu} \ ( \ z \ , \ D \ )
\vspace*{0.5cm} \\
{\cal{W}}_{\ \mu \ ; \ A \ B} \ ( \ z \ ) \ =
\ {\cal{W}}_{\ \mu} \ ( \ z \ , \ D \ )
\ \left ( \ {\cal{F}}_{\ D} \ \right )_{\ A B}
\vspace*{0.5cm} \\
\mbox{in short :}
\ U \ ( \ x \ ; \ y \ )
\ =
\ P \ \exp
\ \left (
\ \left .
\ -
\ {\displaystyle{\int}}_{\ y}^{\ x}
\ \right |_{\ {\cal{C}}}
\ d \ z^{\ \mu}
\ {\cal{W}}_{\ \mu}
\ \right )
\end{array}$$
For classical field configurations $\left . U \ ( \ x \ ; \ y \ ) \ \right |_{\ {\cal{C}}}$ is the operation of parallel transport of a tangent (octet) vector, e.g. $X \ ( \ y \ ) \ \left \lbrace
\ \rightarrow \ X \ ( \ y \ , \ B \ ) \ \right \rbrace$ , at the point $y$ along the curve ${\cal{C}}$ to $x$ .
$$\label{eq:51}
\begin{array}{l}
\begin{array}{ccc cc}
X \ ( \ y \ , \ B \ )
& \stackrel{{\cal{C}}}{\longrightarrow} &
X_{\ \parallel} \ ( \ x \ , \ A \ )
& = &
\ U \ ( \ x \ , \ A \ ; \ y \ , \ B \ )
\ X \ ( \ y \ , \ B \ )
\vspace*{0.5cm} \\
X \ ( \ y \ )
& \stackrel{{\cal{C}}}{\longrightarrow} &
\ X_{\ \parallel} \ ( \ x \ )
& = &
\ U \ ( \ x \ ; \ y \ ) \ X \ ( \ y \ )
\vspace*{0.3cm} \\
y
& \stackrel{{\cal{C}}}{\longrightarrow} &
x & &
\end{array}
\vspace*{0.3cm} \\
U \ ( \ x \ ; \ y \ ) \ =
\ \left . U \ ( \ x \ ; \ y \ ) \ \right |_{\ {\cal{C}}}
\end{array}$$
If $X \ ( \ x \ )$ is itself an octet field defined at all $x$ , then $\left . X_{\ \parallel} \ ( \ x \ )
\ \right |_{\ x \ \stackrel{\leftarrow}{\cal{C}} \ y}$ has to be distinguished from the given value $X \ ( \ x \ )$ .
$\left . U \ ( \ x \ ; \ y \ ) \ \right |_{\ {\cal{C}}}$ defined in eqs. (\[eq:50\]) and (\[eq:51\]) follows from the parallel transport differential equation, using a parameter representation of the curve ${\cal{C}}$
$$\label{eq:52}
\begin{array}{l}
{\cal{C}} \ :
\ \left \lbrace
\ 1 \ \ge \ \tau \ \ge \ 0
\ \left |
\ \begin{array}{l}
z \ = \ z \ ( \ \tau \ )
\vspace*{0.3cm} \\
z \ ( \ 0 \ ) \ = \ y
\vspace*{0.3cm} \\
z \ ( \ 1 \ ) \ = \ x
\end{array}
\right .
\ \right \rbrace
\vspace*{0.3cm} \\
v \ ( \ \tau \ ) \ = \dot{z} \ ( \ \tau \ )
\ = \ \left ( \ d \ / \ d \ \tau \ \right ) \ z \ ( \ \tau \ )
\end{array}$$
Lets follow the $\tau$ development of the family of parallel transports from y along ${\cal{C}}$ to the point $z \ ( \ \tau \ )$, as the latter moves from $y$ to $x$
$$\label{eq:53}
\begin{array}{l}
U \ ( \ \tau \ ) \ =
\ \left .
\ U \ ( \ z \ ( \ \tau \ ) \ ; \ y \ ) \ \right |_{\ {\cal{C}}}
\vspace*{0.3cm} \\
{\cal{W}}_{\ \mu} \ ( \ \tau \ )
\ =
\ {\cal{W}}_{\ \mu} \ ( \ z \ ( \ \tau \ ) \ )
\ \rightarrow
\vspace*{0.3cm} \\
\left ( \ d \ / \ d \ \tau \ \right )
\ U \ ( \ \tau \ )
\ =
\ - \ v^{\ \mu} \ ( \ \tau \ )
\ {\cal{W}}_{\ \mu} \ ( \ \tau \ )
\ U \ ( \ \tau \ )
\vspace*{0.5cm} \\
U \ ( \ 0 \ ) \ = \ \P
\hspace*{0.3cm} \leftrightarrow \hspace*{0.3cm}
\ U \ ( \ y \ , \ A \ ; \ y \ , \ B \ ) \ = \ \delta_{\ A \ B}
\end{array}$$
The parallel transport equation (\[eq:53\]) is subjected to the initial conditions defined in its last line.
It can be integrated by successive iterations
$$\label{eq:54}
\begin{array}{l}
U \ ( \ \tau \ ) \ =
\ \sum_{\ n=0}^{\ \infty}
\ {\displaystyle{\int}}_{\ 0}^{\ \tau} \ d \tau_{\ 1}
\ {\displaystyle{\int}}_{\ 0}^{\ \tau_{\ 1}} \ d \tau_{\ 2}
\cdots
\ {\displaystyle{\int}}_{\ 0}^{\ \tau_{\ n-1}} \ d \tau_{\ n}
\ \times
\vspace*{0.3cm} \\
\hspace*{4.0cm} \times
\ w \ ( \ \tau_{\ 1} \ )
\ w \ ( \ \tau_{\ 2} \ )
\cdots
\ w \ ( \ \tau_{\ n} \ )
\vspace*{0.3cm} \\
w \ ( \ \tau \ ) \ =
\ w_{\ A \ B} \ ( \ \tau \ ) \ =
\ - \ v^{\ \mu} \ ( \ \tau \ )
\ {\cal{W}}_{\ \mu \ ; \ A \ B} \ ( \ z \ ( \ \tau \ ) \ )
\vspace*{0.3cm} \\
\tau
\ \ge \ \tau_{\ 1}
\ \ge \ \tau_{\ 2}
\ \cdots
\hspace*{0.3cm} ; \hspace*{0.3cm}
U \ ( \ 1 \ ) \ =
\ \left . U \ ( \ x \ ; \ y \ ) \ \right |_{\ {\cal{C}}}
\vspace*{0.5cm} \\
\hspace*{0.5cm} \rightarrow
\left . U \ ( \ x \ A \ ; \ y \ B \ ) \ \right |_{\ {\cal{C}}}
\ =
\vspace*{0.3cm} \\
\hspace*{1.5cm} =
\ P \ \exp
\ \left (
\ \left .
\ -
\ {\displaystyle{\int}}_{\ y}^{\ x}
\ \right |_{\ {\cal{C}}}
\ d \ z^{\ \mu}
\ {\cal{W}}_{\ \mu} \ ( \ z \ , \ D \ ) \ {\cal{F}}_{\ D}
\ \right )_{\ A \ B}
\end{array}$$
The path ordering in eq. (\[eq:50\]) reflects the path ordered sequence $\tau \ \ge \ \tau_{\ 1} \ \ge \ \tau_{\ 2} \ \cdots$ in the multiple integrals in eq. (\[eq:54\]) , thus established.
[*Parallel transport and gauge transformations*]{}
We go back to eqs. (\[eq:48\]) and (\[eq:49\]) , implying the action of a local gauge transformation on the connection ${\cal{W}}_{\ \mu} \ ( \ x \ , \ D \ ) \ {\cal{F}}_{\ D}$
$$\label{eq:55}
\begin{array}{l}
D_{\ \mu} \ = \ \partial_{\ \mu} \ + \ {\cal{W}}_{\ \mu}
\hspace*{0.3cm} ; \hspace*{0.3cm}
D_{\ \mu}^{\ \Omega} \ = \ \partial_{\ \mu} \ + \ {\cal{W}}^{\ \Omega}_{\ \mu}
\vspace*{0.3cm} \\
X^{\ \Omega} \ ( \ x \ )
\ = \ \Omega \ ( \ x \ )
\ X \ ( \ x \ ) \ \rightarrow
\ D_{\ \mu}^{\ \Omega} \ X^{\ \Omega} \ =
\Omega \ D_{\ \mu} \ X
\end{array}$$
The local gauge transformation $\Omega$ thus induces the transformation law for the connection
$$\label{eq:56}
\begin{array}{l}
{\cal{W}}^{\ \Omega}_{\ \mu}
\ = \ \Omega \ \partial_{\ \mu} \ \Omega^{\ -1}
\ +
\ \Omega
\ {\cal{W}}_{\ \mu} \ \Omega^{\ -1}
\end{array}$$
The parallel transport of tangent vectors $X^{ \Omega} \ ( y )$ along the curve ${\cal{C}}$ with connection ${\cal{W}}^{\ \Omega}_{\ \mu}$ should be equivalent to the same operation on tangent vectors $X \ ( \ y \ )$ with ${\cal{W}}$ modulo the transformation induced on the tangent vectors. This implies using the relations in eq. (\[eq:51\])
$$\label{eq:57}
\begin{array}{l}
X^{\ \Omega} \ ( \ x \ ) \ = \ \Omega \ ( \ x \ ) \ X \ ( \ x \ )
\hspace*{0.3cm} \leftrightarrow \hspace*{0.3cm}
X^{\ \Omega} \ ( \ y \ ) \ = \ \Omega \ ( \ y \ ) \ X \ ( \ y \ )
\ \rightarrow
\vspace*{0.3cm} \\
\begin{array}{ccc cc}
X^{\ \Omega} \ ( \ y \ )
& \stackrel{{\cal{C}}}{\longrightarrow} &
\ X_{\ \parallel}^{\ \Omega} \ ( \ x \ )
& = &
\ U^{\ \Omega} \ ( \ x \ ; \ y \ ) \ X^{\ \Omega} \ ( \ y \ )
\vspace*{0.3cm} \\
X \ ( \ y \ )
& \stackrel{{\cal{C}}}{\longrightarrow} &
\ X_{\ \parallel} \ ( \ x \ )
& = &
\ U \ ( \ x \ ; \ y \ ) \ X \ ( \ y \ )
\vspace*{0.3cm} \\
y
& \stackrel{{\cal{C}}}{\longrightarrow} &
x & &
\end{array}
\end{array}$$
Thus we expect the relations
$$\label{eq:58}
\begin{array}{l}
X_{\ \parallel}^{\ \Omega} \ ( \ x \ )
\ = \ \Omega \ ( \ x \ ) \ X_{\ \parallel} \ ( \ x \ )
\hspace*{0.3cm} \leftrightarrow \hspace*{0.3cm}
X^{\ \Omega} \ ( \ y \ )
\ = \ \Omega \ ( \ y \ ) \ X \ ( \ y \ )
\ \rightarrow
\vspace*{0.3cm} \\
\Omega \ ( \ x \ )
\ U \ ( \ x \ ; \ y \ )
\ X \ ( \ y \ )
\ = \ U^{\ \Omega} \ ( \ x \ ; \ y \ )
\ \Omega \ ( \ y \ )
\ X \ ( \ y \ )
\ \forall \ X \ ( \ y \ )
\ \rightarrow
\vspace*{0.5cm} \\
\hspace*{2.0cm}
U^{\ \Omega} \ ( \ x \ ; \ y \ )
\ =
\ \Omega \ ( \ x \ )
\ U \ ( \ x \ ; \ y \ )
\ \Omega^{\ -1} \ ( \ y \ )
\end{array}$$
We want to verify the relation inferred in eq. (\[eq:58\]). To this end we form the two , a priori different, matric valued functions of $\tau$ along ${\cal{C}}$
$$\label{eq:59}
\begin{array}{l}
U_{\ 1} \ ( \ \tau \ ) \ = \ U^{\ \Omega} \ ( \ \tau \ )
\hspace*{0.3cm} \leftrightarrow \hspace*{0.3cm}
U_{\ 2} \ ( \ \tau \ ) \ = \ \Omega \ ( \ z_{\ \tau} \ )
\ U \ ( \ \tau \ ) \ \Omega^{\ -1} \ ( \ y \ )
\vspace*{0.3cm} \\
z_{\ \tau} \ = \ z \ ( \ \tau \ )
\end{array}$$
From eq. (\[eq:53\]) we infer
$$\label{eq:60}
\begin{array}{l}
\partial_{\ \tau}
\ U_{\ 1} \ ( \ \tau \ )
\ =
\ - \ v^{\ \mu} \ ( \ \tau \ )
\ {\cal{W}}^{\ \Omega}_{\ \mu} \ ( \ \tau \ )
\ U_{\ 1} \ ( \ \tau \ )
\vspace*{0.3cm} \\
\partial_{\ \tau}
\ U_{\ 2} \ ( \ \tau \ )
\ =
\left \lbrack
\begin{array}{l}
\left (
\ \partial_{\ \tau} \ \Omega \ ( \ z_{\ \tau} \ )
\ \right )
\ \Omega^{\ -1} \ ( \ z_{\ \tau} \ ) \ -
\vspace*{0.3cm} \\
- \ v^{\ \mu} \ ( \ \tau \ ) \ \Omega \ ( \ z_{\ \tau} \ )
\ {\cal{W}}_{\ \mu} \ ( \ \tau \ )
\ \Omega^{\ -1} \ ( \ z_{\ \tau} \ )
\end{array}
\right \rbrack
U_{\ 2} \ ( \ \tau \ )
\end{array}$$
The expression in the first line of the bracket in eq. (\[eq:60\]) transforms into
$$\label{eq:61}
\begin{array}{l}
\left (
\ \partial_{\ \tau} \ \Omega \ ( \ z_{\ \tau} \ )
\ \right )
\ \Omega^{\ -1} \ ( \ z_{\ \tau} \ ) \ =
\ -
\ v^{\ \mu} \ ( \ \tau \ )
\ \Omega \ ( \ z_{\ \tau} \ ) \ \partial_{\ z \ \mu}
\ \Omega^{\ -1} \ ( \ z_{\ \tau} \ )
\end{array}$$
Thus the differential equation for $U_{\ 2} \ ( \ \tau \ $ in eq. (\[eq:60\]) takes the form
$$\label{eq:62}
\begin{array}{l}
\partial_{\ \tau}
\ U_{\ 2} \ ( \ \tau \ )
=
\ - \ v^{\ \mu} \ ( \ \tau \ )
\left \lbrack
\begin{array}{c}
\ \Omega \ ( \ z_{\ \tau} \ )
\ \partial_{\ z \ \mu}
\ \Omega^{\ -1} \ ( \ z_{\ \tau} \ )
\vspace*{0.5cm} \\
+ \ \Omega \ ( \ z_{\ \tau} \ )
\ {\cal{W}}_{\ \mu} \ ( \ \tau \ )
\ \Omega^{\ -1} \ ( \ z_{\ \tau} \ )
\end{array}
\right \rbrack
U_{\ 2} \ ( \ \tau \ )
\vspace*{0.3cm} \\
\hspace*{2.3cm}
=
\ - \ v^{\ \mu} \ ( \ \tau \ )
\ \left \lbrack
\hspace*{2.2cm}
\ {\cal{W}}^{\ \Omega}_{\ \mu} \ ( \ \tau \ )
\hspace*{2.2cm}
\ \right \rbrack
\ U_{\ 2} \ ( \ \tau \ )
\vspace*{0.4cm} \\
\hline
\vspace*{-0.2cm} \\
{\cal{W}}^{\ \Omega}_{\ \mu} \ ( \ z \ )
\ = \ \Omega \ ( \ z \ ) \ \partial_{\ z \ \mu} \ \Omega^{\ -1} \ ( \ z \ )
\ +
\ \Omega \ ( \ z \ )
\ {\cal{W}}_{\ \mu} \ ( \ z \ )
\ \Omega^{\ -1} \ ( \ z \ )
\end{array}$$
Comparing eqs. (\[eq:60\]) and (\[eq:62\]) we see that $U_{\ 1}$ and $U_{\ 2}$ fulfill the same differential equation, as a consequence of the gauge tranformation law of the connection ${\cal{W}}$ . They also have the same initial value
$$\label{eq:63}
\begin{array}{l}
U_{\ 1} \ ( \ 0 \ ) \ = \ U_{\ 2} \ ( 0 \ ) \ = \ \P
\ \rightarrow
\ U_{\ 1} \ ( \ \tau \ ) \ = \ U_{\ 2} \ ( \tau \ )
\vspace*{0.3cm} \\
\hspace*{0.3cm} \rightarrow \hspace*{0.3cm}
U^{\ \Omega} \ ( \ x \ ; \ y \ )
\ =
\ \Omega \ ( \ x \ )
\ U \ ( \ x \ ; \ y \ )
\ \Omega^{\ -1} \ ( \ y \ )
\hspace*{0.3cm} \mbox{qed}
\end{array}$$
[*On the nonabelian Stokes relation*]{}
For our purpose here, to describe the degrees of freedom of [*binary*]{} gluonic mesons, the set of parallel transport matrices ( matrix valued bilocal field operators ) as displayed in eq. (\[eq:54\])
$$\label{eq:64}
\begin{array}{l}
\left . U \ ( \ x \ A \ ; \ y \ B \ ) \ \right |_{\ {\cal{C}}}
\ =
\ \left (
\left . U \ ( \ x \ ; \ y \ ) \ \right |_{\ {\cal{C}}}
\ \right )_{\ A \ B}
\vspace*{0.3cm} \\
\hspace*{1.5cm} =
P \ \exp
\ \left (
\ \left .
\ -
\ {\displaystyle{\int}}_{\ y}^{\ x}
\ \right |_{\ {\cal{C}}}
\ d \ z^{\ \mu}
\ {\cal{W}}_{\ \mu} \ ( \ z \ , \ D \ ) \ {\cal{F}}_{\ D}
\ \right )_{\ A \ B}
\vspace*{0.5cm} \\
U^{\ \Omega} \ ( \ x \ ; \ y \ )
\ =
\ \Omega \ ( \ x \ )
\ U \ ( \ x \ ; \ y \ )
\ \Omega^{\ -1} \ ( \ y \ )
\end{array}$$
along [*straight line*]{} pathes ${\cal{C}}$ restricting general ones, as defined in eq. (\[eq:52\]), are sufficient.
$$\label{eq:65}
\begin{array}{l}
\stackrel{{\cal{C}}}{\longleftarrow} \ :
\ \left \lbrace
\ 1 \ \ge \ \tau \ \ge \ 0
\ \left |
\ \begin{array}{l}
z \ = \ z \ ( \ \tau \ ) \ = \ y \ + \ \tau \ ( \ x \ - \ y \ )
\vspace*{0.3cm} \\
z \ ( \ 1 \ ) \ = \ x
\hspace*{0.3cm} \longleftarrow \hspace*{0.3cm}
z \ ( \ 0 \ ) \ = \ y
\end{array}
\right .
\ \right \rbrace
\vspace*{0.3cm} \\
v \ ( \ \tau \ ) \ = \dot{z} \ ( \ \tau \ )
\ = \ z \ = \ x \ - \ y
\end{array}$$
Parallel transport beeing generated by the connection 1-form
$$\label{eq:66}
\begin{array}{l}
\left . \mbox{{\Large (}}
\ {\cal{W}}^{\ (1)}
\ \equiv
d \ z^{\ \mu}
\ {\cal{W}}_{\ \mu} \ ( \ z \ , \ D \ ) \ {\cal{F}}_{\ D}
\ \right . \mbox{{\Large )}}_{\ A \ B}
\ \rightarrow
\vspace*{0.3cm} \\
U \ ( \ x \ ; \ y \ )
\ =
P \ \exp
\ \left (
\ \left .
\ -
\ {\displaystyle{\int}}_{\ y}^{\ x}
\ \right |_{\ {\cal{C}}}
\ {\cal{W}}^{\ (1)}
\ \right )
\hspace*{0.3cm} ; \hspace*{0.3cm}
P \ \equiv \ P^{\ (1)}
\end{array}$$
the matrix valued 1-forms naturally acquire the line ordering, appropriate for one dimensional integrals.
[*Yet*]{} connection 1-forms and their path $P^{\ (1)}$ ordered integrals do not exhaust the range of r-forms and their r dimensional $P^{\ (r)}$ ordered integrals, associated with nonabelian degrees of freedom.
Next in line are the curvature 2-form and its [*surface*]{} $P^{\ (2)}$ ordered integral.
We follow the covariant derivative path with the octet field $X \ ( \ x \ )$ introduced in eqs. (\[eq:47\]) - (\[eq:49\])
$$\label{eq:67}
\begin{array}{l}
D_{\ \mu} \ X \ ( \ x \ ) \ =
\ \left (
\ \partial_{\ \mu} \ +
\ {\cal{W}}_{\ \mu}
\ \right )
\ X \ ( \ x \ )
\vspace*{0.3cm} \\
\left ( \ D_{\ \mu} \ D_{\ \nu}
\ - \ D_{\ \nu} \ D_{\ \mu}
\ \right )
\ X \ ( \ x \ )
\ =
\ {\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack}
\ X \ ( \ x \ )
\end{array}$$
In eq. (\[eq:67\]) ${\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack}$ denotes the (antisymmetric Yang-Mills) curvature tensor, i.e. the field strengths
$$\label{eq:68}
\begin{array}{l}
{\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack}
\ =
\ \partial_{\ \mu} \ {\cal{W}}_{\ \nu}
\ -
\ \partial_{\ \nu} \ {\cal{W}}_{\ \mu}
\ +
\ \left \lbrack
\ {\cal{W}}_{\ \mu} \ , \ {\cal{W}}_{\ \nu}
\ \right \rbrack
\vspace*{0.3cm} \\
{\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack} \ ( \ x \ )
\ =
\ {\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack}
\ ( \ x \ , \ D \ ) \ {\cal{F}}_{\ D} \ =
\vspace*{0.3cm} \\
\hspace*{0.8cm} =
\ \left \lbrack
\begin{array}{l}
\ \left . \mbox{{\Large (}}
\ \partial_{\ \mu} \ {\cal{W}}_{\ \nu} \ ( \ x \ , \ D \ )
\ - \ \partial_{\ \nu} \ {\cal{W}}_{\ \mu} \ ( \ x \ , \ D \ )
\ \mbox{{\Large )}} \right .
\ {\cal{F}}_{\ D}
\vspace*{0.3cm} \\
\ +
\ {\cal{W}}_{\ \mu} \ ( \ x \ , \ A \ )
\ {\cal{W}}_{\ \nu} \ ( \ x \ , \ B \ )
\ \left \lbrack
\ {\cal{F}}_{\ A} \ , \ {\cal{F}}_{\ B}
\ \right \rbrack
\end{array}
\right \rbrack
\vspace*{0.3cm} \\
{\cal{W}}_{\ \mu} \ ( \ z \ , \ D \ ) \ =
\ i \ V_{\ \mu} \ ( \ z \ , \ D \ )
\hspace*{0.3cm} ; \hspace*{0.3cm}
\left \lbrack
\ {\cal{F}}_{\ A} \ , \ {\cal{F}}_{\ B}
\ \right \rbrack \ = \ i \ f_{\ A \ B \ C}
\ {\cal{F}}_{\ C}
\end{array}$$
In eq. (\[eq:68\]) we have included the relations in eqs. (\[eq:47\]) and (\[eq:50\]).
The form of the curvature tensor ${\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack}$ in eq. (\[eq:68\]) becomes
$$\label{eq:69}
\begin{array}{l}
{\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack}
\ ( \ x \ , \ D \ ) \ =
\ \frac{1}{i} \ F_{\ \left \lbrack \ \mu \nu \ \right \rbrack}
\ ( \ x \ , \ D \ ) \ =
\vspace*{0.3cm} \\
\hspace*{0.8cm} =
\ \left \lbrack
\begin{array}{l}
\ \partial_{\ \mu} \ {\cal{W}}_{\ \nu} \ ( \ x \ , \ D \ )
\ - \ \partial_{\ \nu} \ {\cal{W}}_{\ \mu} \ ( \ x \ , \ D \ )
\vspace*{0.3cm} \\
\ + \ i
\ {\cal{W}}_{\ \mu} \ ( \ x \ , \ A \ )
\ {\cal{W}}_{\ \nu} \ ( \ x \ , \ B \ )
\ f_{\ A \ B \ D}
\end{array}
\right \rbrack
\vspace*{0.5cm} \\
F_{\ \left \lbrack \ \mu \nu \ \right \rbrack}
\ ( \ x \ , \ D \ ) \ =
\vspace*{0.3cm} \\
\hspace*{0.8cm} =
\ \left \lbrack
\begin{array}{l}
\ \partial_{\ \nu} \ V_{\ \mu} \ ( \ x \ , \ D \ )
\ - \ \partial_{\ \mu} \ V_{\ \nu} \ ( \ x \ , \ D \ )
\vspace*{0.3cm} \\
\ -
\ V_{\ \nu} \ ( \ x \ , \ A \ )
\ V_{\ \mu} \ ( \ x \ , \ B \ )
\ f_{\ A \ B \ D}
\end{array}
\right \rbrack
\end{array}$$
We recast the quantities ${\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack}$ in eqs. (\[eq:68\]) and (\[eq:69\]) into their Lie algebra valued form
$$\label{eq:70}
\begin{array}{l}
\begin{array}{lll}
{\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack
\ \left \lbrack \ A B \ \right \rbrack} \ ( \ x \ )
& = &
\ \left (
\ {\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack}
\ ( \ x \ , \ D \ ) \ {\cal{F}}_{\ D}
\ \right )_{\ A \ B}
\vspace*{0.3cm} \\
& = &
\ \left (
\ F_{\ \left \lbrack \ \mu \nu \ \right \rbrack}
\ ( \ x \ , \ D \ ) \ L_{\ D}
\ \right )_{\ A \ B}
\end{array}
\vspace*{0.3cm} \\
\left ( \ L_{\ D} \ \right )_{\ A \ B}
\ = \ \frac{1}{i}
\ \left ( \ {\cal{F}}_{\ D} \ \right )_{\ A \ B}
\ = \ f_{\ A \ D \ B}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\left \lbrack
\ L_{\ R} \ , \ L_{\ S}
\ \right \rbrack
\ = \ f_{\ R \ S \ T} \ L_{\ T}
\end{array}$$
We also cast eq. (\[eq:66\]) into the $L_{\ D}$ form
$$\label{eq:71}
\begin{array}{l}
\left . \mbox{{\Large (}}
\ {\cal{W}}^{\ (1)}
\ \equiv
d \ z^{\ \mu}
\ {\cal{W}}_{\ \mu} \ ( \ z \ , \ D \ ) \ {\cal{F}}_{\ D}
\ \right . \mbox{{\Large )}}_{\ A \ B}
\ \rightarrow
\vspace*{0.3cm} \\
{\cal{W}}_{\ \mu} \ ( \ z \ , \ D \ ) \ {\cal{F}}_{\ D}
\ = \ i \ {\cal{W}}_{\ \mu} \ ( \ z \ , \ D \ ) \ L_{\ D}
\ = \ - \ V_{\ \mu} \ ( \ z \ , \ D \ ) \ L_{\ D}
\ \rightarrow
\vspace*{0.3cm} \\
{\cal{W}}_{\ \mu \ \left \lbrack \ A B \ \right \rbrack}
\ ( \ x \ ) \ = \ - \ V_{\ \mu} \ ( \ x \ , \ D \ )
\ \left (
\ L_{\ D}
\ \right )_{\ A \ B}
\end{array}$$
Comparing the connection and curvature representations in eqs. (\[eq:70\]) and (\[eq:71\]) we learn that the local quantities ${\cal{W}}_{\ \mu \ \left \lbrack \ A B \ \right \rbrack} \ ( \ x \ )$ and ${\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack
\ \left \lbrack \ A B \ \right \rbrack} \ ( \ x \ )$, as well as the components $- \ V_{\ \mu} \ ( \ x \ , \ D \ )$ and $F_{\ \left \lbrack \ \mu \nu \ \right \rbrack}
\ ( \ x \ , \ D \ )$ are real. This is usus in the mathematical literature.
Local gauge transformations as defined for the connection in eqs. (\[eq:56\]) and (\[eq:62\]) are naturally extended to the curvature
$$\label{eq:72}
\begin{array}{l}
\left \lbrace
\begin{array}{l}
{\cal{W}}_{\ \mu \ \left \lbrack \ A B \ \right \rbrack}
\vspace*{0.3cm} \\
{\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack
\ \left \lbrack \ A B \ \right \rbrack}
\end{array}
\ \right \rbrace
\ ( \ x \ )
\ \rightarrow
\ \left \lbrace
\begin{array}{l}
{\cal{W}}_{\ \mu }
\vspace*{0.3cm} \\
{\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack}
\end{array}
\ \right \rbrace
\ ( \ x \ )
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
{\cal{W}}_{\ \mu }^{\ \Omega} \ ( \ x \ )
\ = \ \Omega \ ( \ x \ ) \ \partial_{\ \mu} \ \Omega^{\ -1} \ ( \ x \ )
\ +
\ \Omega \ ( \ x \ )
\ {\cal{W}}_{\ \mu} \ ( \ x \ )
\ \Omega^{\ -1} \ ( \ x \ )
\vspace*{0.3cm} \\
{\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack}^{\ \Omega}
\ ( \ x \ )
\ =
\ \Omega \ ( \ x \ )
\ {\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack} \ ( \ x \ )
\ \Omega^{\ -1} \ ( \ x \ )
\end{array}$$
[*Lie cohomology and de Rham cohomology*]{}
With connection and curvature we associate the Lie algebra valued one and two forms, as defined in eqs. (\[eq:66\]) - (\[eq:71\])
$$\label{eq:73}
\begin{array}{l}
\left (
\ \begin{array}{l}
{\cal{W}}^{\ (1)}
\ \equiv
\ d \ x^{\ \mu} \ {\cal{W}}_{\ \mu}
\vspace*{0.3cm} \\
{\cal{W}}^{\ (2)}
\ \equiv
\ \frac{1}{2} \ d \ x^{\ \mu} \ \wedge \ d \ x^{\ \nu}
\ {\cal{W}}_{\ \left \lbrack \ \mu \nu \ \right \rbrack}
\end{array}
\ \right )
\ ( \ x \ , \ \left \lbrack \ A B \ \right \rbrack \ )
\vspace*{0.3cm} \\
{\cal{W}}^{\ (2)} \ = \ d \ {\cal{W}}^{\ (1)}
\ + \ {\cal{W}}^{\ (1)} \ \circ \ {\cal{W}}^{\ (1)}
\ \equiv \ D \ {\cal{W}}^{\ (1)}
\end{array}$$
In eq. (\[eq:73\]) the symbol $\circ$ denotes normal matrix multiplication [**to be distinguished**]{} from the Lie product denoted below by $\odot$.
It is the antisymmetric nature of the wedge product $d \ x^{\ \mu} \ \wedge \ d \ x^{\ \nu}$ which renders the $\circ$ product equivalent to a Lie algebra product $\odot$
$$\label{eq:74}
\begin{array}{l}
{\cal{W}}^{\ (1)} \ \circ \ {\cal{W}}^{\ (1)}
\ = \ \frac{1}{2}
\ {\cal{W}}^{\ (1)} \ \odot \ {\cal{W}}^{\ (1)}
\end{array}$$
We shall verify eq. (\[eq:74\]) by components
$$\label{eq:75}
\begin{array}{l}
{\cal{W}}^{\ (1)} \ \circ \ {\cal{W}}^{\ (1)}
\ ( \ x \ , \ \left \lbrack \ A B \ \right \rbrack \ )
\ = \ d \ x^{\ \mu} \ \wedge \ d \ x^{\ \nu}
\ {\cal{W}}_{\ \mu \ \left \lbrack \ A A' \ \right \rbrack}
\ {\cal{W}}_{\ \nu \ \left \lbrack \ A' B \ \right \rbrack}
\ \rightarrow
\vspace*{0.3cm} \\
\hspace*{0.8cm} =
\ \frac{1}{2} \ d \ x^{\ \mu} \ \wedge \ d \ x^{\ \nu}
\ \left \lbrack
\begin{array}{l}
{\cal{W}}_{\ \mu \ \left \lbrack \ A A' \ \right \rbrack}
\ {\cal{W}}_{\ \nu \ \left \lbrack \ A' B \ \right \rbrack}
\ -
\vspace*{0.3cm} \\
- \ {\cal{W}}_{\ \nu \ \left \lbrack \ A \ A' \ \right \rbrack}
\ {\cal{W}}_{\ \mu \ \left \lbrack \ A' B \ \right \rbrack}
\end{array}
\right \rbrack
\vspace*{0.3cm} \\
\hspace*{0.8cm} =
\ \frac{1}{2} \ d \ x^{\ \mu} \ \wedge \ d \ x^{\ \nu}
\ \left \lbrack
\begin{array}{l}
{\cal{W}}_{\ \mu} \ \odot
\ {\cal{W}}_{\ \nu}
\end{array}
\right \rbrack_{\ \left \lbrack \ A \ B \ \right \rbrack}
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
{\cal{W}}_{\ \mu} \ \odot
\ {\cal{W}}_{\ \nu}
\ =
\ \left \lbrack
\ {\cal{W}}_{\ \mu} \ , \ {\cal{W}}_{\ \nu}
\ \right \rbrack
\ \equiv
\ {\cal{W}}_{\ \mu} \ \circ \ {\cal{W}}_{\ \nu}
\ - \ {\cal{W}}_{\ \nu} \ \circ \ {\cal{W}}_{\ \mu}
\end{array}$$
Eq. (\[eq:73\]) yields the first relation in the [*adaptive*]{} Lie cohomology chain, generated by the the sequence of operations $D \ \rightarrow \ D^{\ '} \ \neq \ D$
$$\label{eq:76}
\begin{array}{l}
{\cal{W}}^{\ (2)} \ =
\ D \ {\cal{W}}^{\ (1)}
\hspace*{0.3cm} \rightarrow \hspace*{0.3cm}
{\cal{W}}^{\ (3)} \ = \ D^{\ '} \ {\cal{W}}^{\ (2)} \ = \ 0
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
\begin{array}{lll}
D & : &
{\cal{W}}^{\ (2)} \ = \ d \ {\cal{W}}^{\ (1)}
\ + \ \frac{1}{2} \ {\cal{W}}^{\ (1)} \ \odot \ {\cal{W}}^{\ (1)}
\vspace*{0.3cm} \\
D^{\ '} & : &
{\cal{W}}^{\ (3)} \ =
\ d \ {\cal{W}}^{\ (2)}
\ + \ {\cal{W}}^{\ (1)} \ \odot \ {\cal{W}}^{\ (2)}
\ = \ 0
\end{array}
\end{array}$$
The termination of the [*adaptive*]{} $D \ \rightarrow \ D^{\ '}$ sequence follows from the antisymmetry of the wedge product [*and*]{} the Jacobi identity of cyclic double commutators
$$\label{eq:77}
\begin{array}{l}
{\cal{W}}^{\ (3)} \ =
\ \left \lbrace
\begin{array}{l}
d \
\ \left (
\ ( \ {\cal{W}}^{\ (1)} \ \circ \ )^{\ 2}
\ \right )
\ + \ {\cal{W}}^{\ (1)} \ \odot
\ d \ {\cal{W}}^{\ (1)}
\vspace*{0.3cm} \\
+
\ {\cal{W}}^{\ (1)} \ \odot \
\ ( \ {\cal{W}}^{\ (1)} \ \circ \ )^{\ 2}
\end{array}
\ \right \rbrace
\vspace*{0.3cm} \\
( \ {\cal{W}}^{\ (1)} \ \circ \ )^{\ n}
\ = \ {\cal{W}}^{\ (1)} \ \circ
\ \left (
\ ( \ {\cal{W}}^{\ (1)} \ \circ \ )^{\ n - 1}
\ \right )
\ , \ \cdots
\end{array}$$
Expressing the $\odot$ product in eq. (\[eq:77\]) in $\circ$ products it follows
$$\label{eq:78}
\begin{array}{l}
{\cal{W}}^{\ (3)} \ =
\ \left \lbrace
\begin{array}{l}
d \
\ \left (
\ ( \ {\cal{W}}^{\ (1)} \ \circ \ )^{\ 2}
\ \right )
\vspace*{0.3cm} \\
+ \ {\cal{W}}^{\ (1)} \ \circ
\ \left ( \ d \ {\cal{W}}^{\ (1)} \ \right )
\ - \ \left ( \ d \ {\cal{W}}^{\ (1)} \ \right )
\ \circ
\ {\cal{W}}^{\ (1)}
\vspace*{0.3cm} \\
+
\ {\cal{W}}^{\ (1)} \ \circ
\ \left ( \ ( \ {\cal{W}}^{\ (1)} \ \circ \ )^{\ 2} \ \right )
\ -
\ \left ( \ ( \ {\cal{W}}^{\ (1)} \ \circ \ )^{\ 2} \ \right )
\ \circ
\ {\cal{W}}^{\ (1)}
\end{array}
\ \right \rbrace
\end{array}$$
The contribution cubic in ${\cal{W}}^{\ (1)}$ vanishes on the ground of the associative product $\circ$, while the first three cancel due to the identity
$$\label{eq:79}
\begin{array}{l}
d \
\ \left (
\ ( \ {\cal{W}}^{\ (1)} \ \circ \ )^{\ 2}
\ \right )
\ = \ \left ( \ d \ {\cal{W}}^{\ (1)} \ \right )
\ \circ \ {\cal{W}}^{\ (1)}
\ -
\ {\cal{W}}^{\ (1)}
\ \circ
\ \left ( \ d \ {\cal{W}}^{\ (1)} \ \right )
\end{array}$$
[*Loops of parallel transports, local holonomy groups*]{}
In the inverse of the differential Lie cohomology chain the parallel transport matrices
$$\label{eq:80}
\begin{array}{l}
\ U \ ( \ x \ , \ y \ ; \ \stackrel{{\cal{C}}}{\longrightarrow} \ )
\ =
\ P \ \exp
\ \left (
\ \left .
\ -
\ {\displaystyle{\int}}_{\ y}^{\ x}
\ \right |_{\ {\cal{C}}}
\ d \ z^{\ \mu}
\ {\cal{W}}_{\ \mu}
\ \right )
\end{array}$$
defined in eqs. (\[eq:50\]) - (\[eq:51\]) can be combined to form a closed curve starting and ending at $y$.
$$\label{eq:81}
\begin{array}{l}
U \ \left (
\ y \ , \ y \ ; \ {\cal{CL}}
\hspace*{0.5cm} \begin{array}{c}
\longleftarrow
\vspace*{-0.42cm} \\
\hspace*{-0.36cm} \swarrow \hspace*{0.5cm} \nearrow
\vspace*{-0.43cm} \\
\hspace*{-0.6cm} \longrightarrow
\end{array}
\right )_{\ A \ B}
\ \rightarrow
\ U \ ( \ y \ , \ y \ ; \ {\cal{CL}} \ )
\end{array}$$
The quantities $U \ ( \ x \ , \ y \ ; \ \stackrel{{\cal{C}}}{\longrightarrow} \ )$ , called adjoint strings here, are rarely used in lattice discretized Yang-Mills theory. The associated fundamental strings, projected on the fundamental representation of the local gauge group ( the triplet strings for $SU3_{\ c}$ ) are the dynamical [*link*]{} variables therein [@latt].
The quantities $U \ ( \ y \ , \ y \ ; \ {\cal{CL}} \ )$ , defined in eq. (\[eq:81\]) we shall call closed adjoint (octet) strings. Their counterparts, projected on the fundamental (triplet) representation, assigned to a minimal closed lattice loop, a plaquette, are used to generate the lattice action.
Closed loop matrices or operators are widely studied in their own right. For the fundamental representation they are called Wilson loops ( $W \ ( \ {\cal{C}} \ )$ )\
within Yang-Mills theories [@alvarez].
We continue to focus on open and closed adjoint strings here. Nevertheless it is tacitly assumed, that the configurations obey the regularity requirements of extensions of these strings to [*all*]{} representations of the gauge group. This framework is called the universal bundle in the mathematical literature.
The gauge transformation properties of open and closed (adjoint) strings in eqs. (\[eq:80\]) and (\[eq:81\]) follow from eqs. (\[eq:62\]) and (\[eq:63\])
$$\label{eq:82}
\begin{array}{l}
U^{\ \Omega} \ ( \ x \ , \ y \ ; \ \stackrel{{\cal{C}}}{\longrightarrow} \ )
\ =
\ \Omega \ ( \ x \ )
\ U \ ( \ x \ , \ y \ ; \ \stackrel{{\cal{C}}}{\longrightarrow} \ )
\ \Omega^{\ -1} \ ( \ y \ )
\ \rightarrow
\vspace*{0.3cm} \\
U^{\ \Omega} \ ( \ y \ , \ y \ ; \ {\cal{CL}} \ )
\ =
\ \Omega \ ( \ y \ )
\ U \ ( \ y \ , \ y \ ; \ {\cal{CL}} \ )
\ \Omega^{\ -1} \ ( \ y \ )
\end{array}$$
The closed curve $ {\cal{CL}}$ is still punctuated at its beginning and ending. Yet the gauge transformation act [*locally*]{} at this point. However the simply connected closed loop can be repeatedly transcurred, leading to the multiple positve as well as negative powers, all transforming the same way under gauge transformations
$$\label{eq:83}
\begin{array}{l}
U^{\ n} \ ( \ y \ , \ y \ ; \ {\cal{CL}} \ )
\ =
\ U \ ( \ y \ , \ y \ ; \ {\cal{CL}}^{\ (n)} \ )
\hspace*{0.3cm} , \hspace*{0.3cm} \ n \ = \ 0 \ , \ \pm 1 \ , \ \cdots
\vspace*{0.3cm} \\
U^{\ \Omega} \ ( \ y \ , \ y \ ; \ {\cal{CL}}^{\ (n)} \ )
\ =
\ \Omega \ ( \ y \ )
\ U \ ( \ y \ , \ y \ ; \ {\cal{CL}}^{\ (n)} \ )
\ \Omega^{\ -1} \ ( \ y \ )
\end{array}$$
The closed curve ${\cal{CL}}^{\ (n)}$ shall represent the n-fold transcurred simple curev ${\cal{CL}}$, whereby negative powers mean to reverse the orientation, from clockwise to anticlockwise say.
Gauge invariant quantities are thus all (adjoint) traces
$$\label{eq:84}
\begin{array}{l}
W_{\ (n)} \ ( \ {\cal{CL}} \ ) \ = \ tr
\left \lbrack
\ U^{\ n} \ ( \ y \ , \ y \ ; \ {\cal{CL}} \ )
\right \rbrack
\ =
\ \sum_{ \left \lbrace \lambda \ \right \rbrace} \ \lambda^{\ n}
\ \left (
\ U \ ( \ y \ , \ y \ ; \ {\cal{CL}} \ )
\ \right )
\end{array}$$
In eq. (\[eq:84\]) $\lambda$ runs over all the eigenvalues of the (real orthogonal) matrix $U \ ( \ y \ , \ y \ ; \ {\cal{CL}} \ )$ .
The quantities $W_{\ (n)} \ ( \ {\cal{CL}} \ )$ in eq. (\[eq:84\]) do depend on the shape of the simply laced curve $ {\cal{CL}}$ , but they are the same for all points along $ {\cal{CL}}$, when adopted as alternative starting and ending points.
They represent the adjoint characters of the (self covering) Lie group, dependent only on the angles of the Cartan subalgebra. Thus they depend, for a simple gauge group with rank r ( r = 2 for $SU3_{\ c}$ ) on the r Cartan subalgebra angles, characterising any of the representatives $U \ ( \ y^{\ '} \ , \ y^{\ '} \ ; \ {\cal{CL}} \ )$ with $y^{\ '}$ anywhere on the curve ${\cal{CL}}$. The characteristic coefficients are determined from the roots of the Lie algebra and, through its universal extension to all representations, from its r fundamental weights. For $SU3_{\ c}$ these are the weights of the $3$ and $\overline{3}$ fundamental representations.
For $SU3_{\ c}$ let the two Cartan algebra angles be $\phi \ \leftrightarrow \ I_{\ 3}$ and $\psi \ \leftrightarrow \ Y \ / \ ( \ 2 \sqrt{3} \ )$, using standard weight normalization, where $I_{\ 3}$ and Y denote isospin and hypercharge respectively.
The $3$ and $\overline{3}$ Cartan matrices shall be $u$ and $\overline{u}$ respectively, $u \ = \ u \ ( \ \phi \ , \ \psi \ )$ . The two fundamental characters thus become
$$\label{eq:85}
\begin{array}{l}
\chi \ = \ \chi \ ( \ u \ ) \ \rightarrow \ \chi \ ( \ \phi \ , \ \psi \ )
\hspace*{0.3cm} \mbox{and} \hspace*{0.3cm}
\overline{\chi}
\vspace*{0.3cm} \\
\chi \ = \ \sum_{\ k = 1}^{\ 3}
\ \exp \ \left ( \ \frac{1}{i} \ \kappa_{\ k} \ \right )
\hspace*{0.3cm} ; \hspace*{0.3cm}
\kappa_{\ 3} \ = \ - \ \kappa_{\ 1} \ - \ \kappa_{\ 2}
\vspace*{0.3cm} \\
\kappa_{\ 1} \ = \ \frac{1}{2} \ \phi \ + \ \frac{1}{2 \ \sqrt{3}} \ \psi
\hspace*{0.3cm} , \hspace*{0.3cm}
\kappa_{\ 2} \ = \ - \ \frac{1}{2} \ \phi \ + \ \frac{1}{2 \ \sqrt{3}} \ \psi
\end{array}$$
Then for a reducible direct product representation $D_{\ red \ ; \ M \ , \ N}$ of M copies of $u$ and N copies of $\overline{u}$, the character is multiplicative
$$\label{eq:86}
\begin{array}{l}
\chi_{\ red \ ; \ M \ , \ N} \ = \ \chi^{\ M} \ \overline{\chi}^{\ N}
\ =
\ \overline{\chi}_{\ red \ ; \ N \ , \ M}
\end{array}$$
From $D_{\ red \ ; \ M \ , \ N}$ the characters of all irreducible representations of the gauge group can be derived. We only give the lowest charcters for the i\
$3$ , $\overline{3}$ , $6$ , $\overline{6}$ , $10$ , $\overline{10}$ and adjoint ( $8$ ) representations of $SU3_{\ c} \ \rightarrow \ SU3_{\ .}$ , with the association
$$\label{eq:87}
\begin{array}{l}
\begin{array}{lll clll}
3
& = &
D_{\ ird \ ; \ 1 \ , \ 0}
& cc &
\overline{3}
& = &
D_{\ ird \ ; \ 0 \ , \ 1}
\vspace*{0.3cm} \\
6
& = &
D_{\ ird \ ; \ 2 \ , \ 0}
& cc &
\overline{6}
& = &
D_{\ ird \ ; \ 0 \ , \ 2}
\vspace*{0.3cm} \\
10
& = &
D_{\ ird \ ; \ 3 \ , \ 0}
& cc &
\overline{10}
& = &
D_{\ ird \ ; \ 0 \ , \ 3}
\vspace*{0.3cm} \\
8
& = &
D_{\ ird \ ; \ 1 \ , \ 1}
& \mbox{real} &
& &
\end{array}
\vspace*{0.0cm} \\
\begin{array}{lll clll}
\chi^{\ ird}_{\ 1 \ , \ 0}
& = & \chi
& cc &
\chi^{\ ird}_{\ 0 \ , \ 1}
& = &
\overline{\chi}
\vspace*{0.3cm} \\
\chi^{\ ird}_{\ 2 \ , \ 0}
& = &
\chi^{\ 2} \ - \ \overline{\chi}
& cc &
\chi^{\ ird}_{\ 0 \ , \ 2}
& = &
\overline{\chi}^{\ 2} \ - \ \chi
\vspace*{0.3cm} \\
\chi^{\ ird}_{\ 3 \ , \ 0}
& = &
\left (
\begin{array}{c}
\chi^{\ 3}
\ - \ 2 \ | \ \chi \ |^{\ 2}
\vspace*{0.3cm} \\
+ \ 1
\end{array}
\right )
& cc &
\chi^{\ ird}_{\ 0 \ , \ 3}
& = &
\left (
\begin{array}{c}
\overline{\chi}^{\ 3}
\ - \ 2 \ | \ \chi \ |^{\ 2}
\vspace*{0.3cm} \\
+ \ 1
\end{array}
\right )
\vspace*{0.3cm} \\
\chi^{\ ird}_{\ 1 \ , \ 1}
& = &
| \ \chi \ |^{\ 2} \ - \ 1
& \mbox{real} &
& &
\end{array}
\vspace*{0.5cm} \\
\left (
\ \chi \ , \ \overline{\chi}
\ \right )
\ =
\ \left (
\ \chi \ , \ \overline{\chi}
\ \right ) \ ( \ \phi \ , \ \psi \ )
\end{array}$$
In our case the quantities $U \ ( \ y \ , \ y \ ; \ {\cal{CL}} \ )$ in eq. (\[eq:81\]) are in the adjoint representation, i.e. in $D_{\ ird \ ; \ 1 \ , \ 1}$ . Hence all equivalent representatives $U^{\ n} \ ( \ y^{\ '} \ , \ y^{\ '} \ ; \ {\cal{CL}} \ )$ with $y^{\ '}$ anywhere on ${\cal{CL}}$ are characterized by the two Cartan subalgebra angles
$$\label{eq:88}
\begin{array}{l}
U^{\ n} \ ( \ y^{\ '} \ , \ y^{\ '} \ ; \ {\cal{CL}} \ ) \ \rightarrow
\ \left ( \ n \ \phi \ , \ n \ \psi \ \right )
\ \left \lbrace
\ {\cal{CL}}
\ \right \rbrace
\end{array}$$
The invariant quantities $W_{\ (n)} \ ( \ {\cal{CL}} \ )$ in eq. (\[eq:84\]) are thus given by
$$\label{eq:89}
\begin{array}{l}
W_{\ (n)} \ ( \ {\cal{CL}} \ )
\ = \ | \ \chi_{\ n}
\ \left \lbrace
\ {\cal{CL}}
\ \right \rbrace
\ |^{\ 2} \ - \ 1
\vspace*{0.3cm} \\
\chi_{\ n}
\ \left \lbrace
\ {\cal{CL}}
\ \right \rbrace
\ = \ \chi
\ \left ( \ n \ \phi \ , \ n \ \psi \ \right )
\ \left \lbrace
\ {\cal{CL}}
\ \right \rbrace
\vspace*{0.3cm} \\
n \ = \ 0 \ , \ \pm \ 1 \ \cdots
\hspace*{0.3cm} ; \hspace*{0.3cm}
W_{\ (0)} \ ( \ {\cal{CL}} \ ) \ = \ 8
\end{array}$$
Because the fundamental $SU3_{\ .}$ matrices $u$ ( and $\overline{u}$ ) are three dimensional, with determinant 1 , only two out of the infinite $n$ sequence of [*fundamental*]{} characters $\chi_{\ n} \ \left \lbrace\ {\cal{CL}} \ \right \rbrace$ : ( $n \ = \ 1 \ , \ 2 \ \mod{3}$ ) are independent of each other.
The fundamental characters of any element u of the 3 representation of $SU3_{\ .}$ obey the [*elementary*]{} generating identity, expressing the fundamental polynomial $P_{\ 3} \ ( \ \mu \ ; \ u \ ) \ = \ Det \ ( \ \P - \ \mu \ u \ )$ in terms of fundamental characters
$$\label{eq:90}
\begin{array}{l}
P_{\ 3} \ ( \ \mu \ ; \ u \ ) \ = \ Det \ ( \ \P - \ \mu \ u \ )
\ = \ 1 \ - \ p_{\ 1} \ \mu \ + \ p_{\ 2} \ \mu^{\ 2} \ - \ \mu^{\ 3}
\vspace*{0.3cm} \\
E \ ( \ \mu \ ; \ u \ )
\ = \ \exp
\ \left (
\ - \ \sum_{\ n = 1}^{\ \infty} \ \mu^{\ n} \ \chi_{\ n} \ / \ n
\ \right )
\vspace*{0.3cm} \\
E \ ( \ \mu \ ; \ u \ )
\ - \ P_{\ 3} \ ( \ \mu \ ; \ u \ ) \ = \ 0
\hspace*{1.0cm}
\forall \ \mu
\vspace*{0.3cm} \\
\hline
\vspace*{-0.4cm} \\
p_{\ k} \ = \ p_{\ k} \ ( \ u \ )
\ ; \ k \ = \ 1 \ , \ 2
\hspace*{0.3cm} \leftrightarrow \hspace*{0.3cm}
\chi_{\ n} \ = \ tr \ u^{\ n} \ = \ \chi_{\ n} \ ( \ u \ )
\vspace*{0.3cm} \\
\Pi_{\ k} \ = \ p_{\ k} \ ( \ u \ = \ \P \ )
\vspace*{0.5cm} \\
p_{\ 1} \ = \ \chi_{\ 1}
\hspace*{0.3cm} , \hspace*{0.3cm}
p_{\ 2} \ = \ \frac{1}{2}
\ ( \ \chi_{\ 1}^{\ 2} \ - \ \chi_{\ 2} \ )
\vspace*{0.3cm} \\
\chi_{\ - n} \ = \overline{\chi}_{\ n} \ ,
\ \chi_{\ 0} \ = \ 3
\hspace*{0.3cm} ; \hspace*{0.3cm}
\Xi_{\ n} \ = \ \chi_{\ n} \ ( \ u \ = \ \P \ ) \ = \ 3
\hspace*{0.3cm} \forall \ n
\vspace*{0.3cm} \\
\Xi_{\ n} \ = \ \Xi
\hspace*{0.3cm} ; \hspace*{0.3cm}
\Pi_{\ k} \ = \ \Xi
\end{array}$$
The first of the reducing identities is
$$\label{eq:91}
\begin{array}{l}
P_{\ 3} \ ( \ \mu \ = \ u \ ; \ u \ ) \ = \ 0
\ \rightarrow
\ \chi_{\ 3}
\ =
\ \chi_{\ 0} \ - \ p_{\ 1} \ \chi_{\ 1}
\ + \ p_{\ 2} \ \chi_{\ 2}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\ \cdots
\vspace*{0.3cm} \\
\rightarrow
\ \chi_{\ 3}
\ = \ \chi_{\ 0} \ - \ \chi_{\ 1}^{\ 2}
\ + \ \frac{1}{2}
\ ( \ \chi_{\ 1}^{\ 2} \ \chi_{\ 2} \ - \ \chi_{\ 2}^{\ 2} \ )
\end{array}$$
Substituting $\chi^{\ ird}_{\ M \ , \ N} \ ( \ \chi \ , \ \overline{\chi} \ )$ the values for $u \ = \ \P$ we obtain the dimension of the associated irreducible representation ( eq. (\[eq:87\]) )
$$\label{eq:92}
\begin{array}{l}
dim \ ( \ D_{\ ird \ ; \ M \ , \ N} \ )
\ =
\ \chi^{\ ird}_{\ M \ , \ N} \ ( \ \Xi \ , \ \Xi \ )
\hspace*{0.3cm} ; \hspace*{0.3cm}
\Xi \ = \ 3
\end{array}$$
All this notwithstanding, the adjoint matrices $U \ ( \ y \ , \ y \ ; \ {\cal{CL}} \ )$ represent the holonomy group at the point $y$ mapping through ${\cal{CL}}$ parallel transport all adjoint tangent space into itself.
The entire range of adjoint matrices forms the group $SU3_{\ .} \ / \ Z_{\ 3}$. The mapping
$$\label{eq:93}
\begin{array}{l}
\begin{array}[t]{c}
u
\vspace*{0.3cm} \\
\in \ D_{\ ird \ ; \ 1 , \ 0}
\end{array}
\ \longrightarrow
\begin{array}[t]{c}
U
\vspace*{0.3cm} \\
\in \ D_{\ ird \ ; \ 1 , \ 1}
\end{array}
\vspace*{0.3cm} \\
( \ u \ z_{\ 0} \ , \ u \ z_{\ 1} \ , \ u \ z_{\ 2} \ )
\ \rightarrow \ U
\end{array}$$
is covering the adjoint representation $D_{\ ird \ ; \ 1 , \ 1}$ three times. In eq. (\[eq:93\]) $z_{\ s} \ , \ s \ = \ 0,1,2$ denote the elements forming the center $Z_{\ 3}$ of $SU3_{\ .}$ .
But there is no loss of information in considering only $U \ ( \ y \ , \ y \ ; \ {\cal{CL}} \ )$ , assuming the analytic extension of the underlying Lie group to be implementable in the classical field configurations, which resolves the above threefold covering through the analytic extension inherent to the Lie algebra leading from $SU3_{\ .} \ / \ Z_{\ 3} \ \rightarrow \ SL3C \ \rightarrow \ SU3_{\ .}$ . This is in accordance with the universal fibre bundle structure.
At the end of this appendix we shall go back to $U \ ( \ y \ , \ y \ ; \ {\cal{CL}} \ )$ as defined in eq. (\[eq:81\]) and state the nonabeliean Stokes relation [@Nachtmann] :
$$\label{eq:94}
\begin{array}{l}
U \ \left (
\ y \ , \ y \ ; \ {\cal{CL}}
\hspace*{0.5cm} \begin{array}{c}
\longleftarrow
\vspace*{-0.42cm} \\
\hspace*{-0.36cm} \swarrow \hspace*{0.5cm} \nearrow
\vspace*{-0.43cm} \\
\hspace*{-0.6cm} \longrightarrow
\end{array}
\right )_{\ A \ B}
\ =
\vspace*{0.3cm} \\
\hspace*{0.3cm} =
\ \left (
\ U \ ( \ y \ , \ x \ ; \ \stackrel{{\cal{C}}}{\longrightarrow} \ )
\ \right )_{\ A \ G}
\ \times
\vspace*{0.3cm} \\
\hspace*{1.7cm} \times
\ \left (
\ P_{\ 2}^{\ \Omega} \ \exp
\ \left (
\ -
\ {\displaystyle{\int}}_{\ S_{\ x}} \ {\cal{W}}^{\ (2)}
\ \right )
\ \right )_{\ G \ H}
\ \times
\vspace*{0.3cm} \\
\hspace*{5.2cm} \times
\ \left (
\ U \ ( \ x \ , \ y \ ; \ \stackrel{{\cal{C}}}{\longrightarrow} \ )
\ \right )_{\ H \ B}
\vspace*{0.3cm} \\
\ \left (
\ U_{\ (2)} \ ( \ x \ ; \ P_{\ 2}^{\ \Omega}
\ | \ {\cal{CL}} \ = \ \partial \ S \ )
\ \right )_{\ G \ H}
\hspace*{0.4cm} \rightarrow
\ U_{\ (2)}^{\ \Omega} \ ( \ x \ ; \ {\cal{CL}} \ = \ \partial \ S \ )
\vspace*{0.3cm} \\
\hspace*{2.5cm} =
\ \left (
\ P_{\ 2}^{\ \Omega} \ \exp
\ \left (
\ -
\ {\displaystyle{\int}}_{\ S_{\ x}} \ {\cal{W}}^{\ (2)}
\ \right )
\ \right )_{\ G \ H}
\end{array}$$
In eq. (\[eq:94\]) $U_{\ (2)} \ ( \ x \ ; \ P_{\ 2}^{\ \Omega}
\ | \ {\cal{CL}} \ = \ \partial \ S \ )
\ \rightarrow \ U_{\ (2)}^{\ \Omega}
\ ( \ x \ ; \ {\cal{CL}} \ = \ \partial \ S \ ) $ denotes the Stokes surface integral proper, punctuated at an internal point $x$ and oriented in a coil like wiring fashion, denoted by $P_{\ 2}^{\ \Omega}$ .
The $P_{\ 2}^{\ \Omega}$ ordering for four coils and two wiring layers is shown in figure \[fig3\] below.
0.5cm
-0.5cm
-1.5cm
The ordering $P_{\ 2}^{\ \Omega}$ for the segmnents at fixed distance from the base point $x$ converges to a flagpole path, shown in the lower right corner of figure \[fig3\] .
This path , denoted $\natural$ , starts and ends at the base point $x$ and turns around the plaquette at the point $z$ on the surface ${\cal{S}}$ .
Its contribution inside the ordering $P_{\ 2}^{\ \Omega}$ is
$$\label{eq:95}
\begin{array}{l}
\natural \ =
\ U ( \ x \ , \ z \ ; \ \stackrel{{\cal{C}}}{\longrightarrow} \ )
\ P_{\ 2}^{\ \Omega \ | \ z} \ \exp
\ \left (
\ -
\ {\displaystyle{\int}}_{\ S_{\ x}} \ {\cal{W}}^{\ (2)}
\ \right )
\ U \ ( \ z \ , \ x \ ; \ \stackrel{{\cal{C}}}{\longrightarrow} \ )
\end{array}$$
The superscript $\Omega$ characterizing the surface ordering $P_{\ 2}^{\ \Omega}$ is chosen to associate a local gauge transformation with the surface ${\cal{S}}$ . This follows from the similarity transformation induced on the flagpole path $\natural$ as defined in eq. (\[eq:95\]).
To make this explicit we rename the parallel transport matrix $U \ ( \ z \ , \ x \ ; \ \stackrel{{\cal{C}}}{\longrightarrow} \ )$ associated with the fixed base point $x$ and the point $z$ varying over the entire surface ${\cal{S}}$
$$\label{eq:96}
\begin{array}{l}
U \ ( \ z \ , \ x \ ; \ \stackrel{{\cal{C}}}{\longrightarrow} \ )
\ \longrightarrow \ \omega_{\ x} \ ( \ z \ )
\ \rightarrow
\vspace*{0.3cm} \\
\natural \ =
\ \left (
\ \omega_{\ x} \ ( \ z \ )
\ \right )^{\ -1}
\ P_{\ 2}^{\ \Omega \ | \ z} \ \exp
\ \left (
\ -
\ {\displaystyle{\int}}_{\ S_{\ x}} \ {\cal{W}}^{\ (2)}
\ \right )
\ \omega_{\ x} \ ( \ z \ )
\vspace*{0.5cm} \\
\omega_{\ x} \ ( \ z \ )
\ =
\left .
\omega_{\ x} \ ( \ z \ )
\ \right |_{\ {\cal{C}}}
\end{array}$$
The last line in eq. (\[eq:96\]) shall make it explicit, that the parallel transport matrices $\omega_{\ x} \ ( \ z \ )$ are not local functions of the surface point z. Rather they depend on the path, one each from $x$ to $z$ .
The [*family*]{} of similarity transformations $\left \lbrace \ \omega_{\ x} \ ( \ z \ ) \ \right \rbrace$ induced on the [*local*]{} field strength differential ${\cal{W}}^{\ (2)} \ = \ {\cal{W}}^{\ (2)} \ ( \ z \ )$ reflects the nested structure of the weaving pattern defining $P_{\ 2}^{\ \Omega}$ as a whole [^2] .
Looking at the structure of the similarity transformations forming the [*nonlocal*]{} structure $\natural$ in eq. (\[eq:96\]) the question arises, whether there exists a local gauge transformation $\widehat{\Omega}$ - [*adapted*]{} to ${\cal{S}}$ - which would render the gauge transformed set $\left \lbrace \ \omega_{\ x}^{\ \widehat{\Omega}}
\ ( \ z \ ) \ \right \rbrace$ trivial
$$\label{eq:97}
\begin{array}{l}
\left .
\omega_{\ x}^{\ \widehat{\Omega}} \ =
\ \widehat{\Omega} \ ( \ z \ )
\ \omega_{\ x}
\ \left (
\ \widehat{\Omega} \ ( \ x \ )
\ \right )^{\ -1}
\ \right |_{\ {\cal{C}}}
\ = \ \P
\hspace*{0.3cm} , \hspace*{0.3cm} \forall
\ z \ \in \ {\cal{S}}
\vspace*{0.3cm} \\
\widehat{\Omega} \ \rightarrow
\ \mbox{Riemann normal gauge}
\end{array}$$
Indeed the gauge transformation with the requirements in eq. (\[eq:97\]) exists and can be found [*together*]{} with a coordinate transformation of local coordinates on ${\cal{S}}$ and the original contour ${\cal{CL}}$ such that ${\cal{S}}$ becomes the inner part of a bounding circle. The latter forms in the new coordinates the closed contour ${\cal{CL}}$ and the family of curves from the base point $x$ to $z$ becomes the family of straight [*radial*]{} lines. The point $y$ punctuating the contour ${\cal{CL}}$ then can be mapped on the south pole of the bounding circle ( to be definite ) .
The transformed variables are well known in the analogous situation, where gauge transformations refer to coordinate transformations, i.e. the tangent space (universal) spin bundle. The respective coordinates are called Riemann normal coordinates. The gauge equivalent we shall call the Riemann normal gauge as indicated in eq. (\[eq:97\]) .
The Riemann normal gauge is also known as radial gauge, at least in the case of an abelian gauge group.
It is precisely in the Riemann normal gauge where the $P_{\ 2}^{\ \Omega} \ \rightarrow
\ P_{\ 2}^{\ \widehat{\Omega}}$ ordering becomes ’normal’ . Transforming to the Riemann normal gauge $R.n.g. \ ( \ {\cal{S}} \ )$ we have
$$\label{eq:98}
\begin{array}{l}
R.n.g. \ ( \ {\cal{S}} \ ) \ :
\vspace*{0.3cm} \\
\begin{array}{ccl}
\left .
\omega_{\ x} \ ( \ z \ )
\ \right |_{\ {\cal{C}}}
& \rightarrow &
\left .
\ \omega_{\ x}^{\ \widehat{\Omega}} \ ( \ z \ )
\ \right |_{\ {\cal{C}}}
\ \equiv \ \P
\vspace*{0.3cm} \\
P_{\ 2}^{\ \Omega}
& \rightarrow &
P_{\ 2}^{\ \widehat{\Omega}}
\vspace*{0.3cm} \\
{\cal{W}}^{\ (2)} \ ( \ z \ )
& \rightarrow &
{\cal{W}}^{\ (2) \ \widehat{\Omega}} \ ( \ z \ )
\ =
\ \widehat{\Omega} \ ( \ z \ )
\ {\cal{W}}^{\ (2)} \ ( \ z \ )
\ \left (
\ \widehat{\Omega} \ ( \ z \ )
\ \right )^{\ -1}
\end{array}
\end{array}$$
Using the transformad quantities on the right hand side of eq. (\[eq:98\]) we undo first the flagpole sequence $\natural$ in eq. (\[eq:96\])
$$\label{eq:99}
\begin{array}{l}
\natural \ =
\ \left (
\ \omega_{\ x} \ ( \ z \ )
\ \right )^{\ -1}
\ P_{\ 2}^{\ \Omega \ | \ z} \ \exp
\ \left (
\ -
\ {\displaystyle{\int}}_{\ S_{\ x}} \ {\cal{W}}^{\ (2)}
\ \right )
\ \omega_{\ x} \ ( \ z \ )
\ \rightarrow \ \natural^{\ \widehat{\Omega}}
\vspace*{0.3cm} \\
\natural^{\ \widehat{\Omega}}
\ =
\ P_{\ 2}^{\ \widehat{\Omega} \ | \ z} \ \exp
\ \left (
\ -
\ {\displaystyle{\int}}_{\ S_{\ x}} \ {\cal{W}}^{\ (2) \ \widehat{\Omega}}
\ \right )
\end{array}$$
Next eq. (\[eq:94\]) becomes
$$\label{eq:100}
\begin{array}{l}
U^{\ \widehat{\Omega}} \ \left (
\ y \ , \ y \ ; \ {\cal{CL}}
\hspace*{0.5cm} \begin{array}{c}
\longleftarrow
\vspace*{-0.42cm} \\
\hspace*{-0.36cm} \swarrow \hspace*{0.5cm} \nearrow
\vspace*{-0.43cm} \\
\hspace*{-0.6cm} \longrightarrow
\end{array}
\right )_{\ A \ B}
\ =
\vspace*{0.3cm} \\
\hspace*{0.3cm} =
\ \left (
\ P_{\ 2}^{\ \widehat{\Omega}} \ \exp
\ \left (
\ -
\ {\displaystyle{\int}}_{\ S_{\ x}} \ {\cal{W}}^{\ (2) \ \widehat{\Omega}}
\ \right )
\ \right )_{\ A \ B}
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
\ \left (
\ U_{\ (2)} \ ( \ x \ ; \ P_{\ 2}^{\ \widehat{\Omega}}
\ | \ {\cal{CL}} \ = \ \partial \ S \ )
\ \right )_{\ A \ B}
\hspace*{0.4cm} \rightarrow
\ U_{\ (2)}^{\ \widehat{\Omega}}
\ ( \ x \ ; \ {\cal{CL}} \ = \ \partial \ S \ )
\vspace*{0.3cm} \\
\hspace*{2.5cm} =
\ \left (
\ P_{\ 2}^{\ \widehat{\Omega}} \ \exp
\ \left (
\ -
\ {\displaystyle{\int}}_{\ S_{\ x}} \ {\cal{W}}^{\ (2) \ \widehat{\Omega}}
\ \right )
\ \right )_{\ A \ B}
\end{array}$$
At this stage, although implicit in the original definition of the general $P_{\ 2}^{\ \Omega}$ ordering, it remains to indicate the (or a) simplified ordering in the Riemann normal gauge. This is shown in figure \[fig4\] .
0.0cm
-0.2cm
-3.6cm
In the Riemann normal gauge the repeated intermediate returns to the base point $x$ are no more necessary [^3] .
An interesting shortcut is shown in an actual spiderweb in figure \[fig4a\].
0.0cm
-0.2cm
Several remarks conclude this discussion :
i\) back to the original gauge
In eq. (\[eq:100\]) we have to transform back from Riemann normal gauge on the surface ${\cal{S}}_{\ x \ | \ y}$ to the original gauge
$$\label{eq:1001}
\begin{array}{l}
U \ \left (
\ y \ , \ y \ ; \ {\cal{CL}}
\hspace*{0.5cm} \begin{array}{c}
\longleftarrow
\vspace*{-0.42cm} \\
\hspace*{-0.36cm} \swarrow \hspace*{0.5cm} \nearrow
\vspace*{-0.43cm} \\
\hspace*{-0.6cm} \longrightarrow
\end{array}
\right )_{\ A \ B}
\ =
\vspace*{0.3cm} \\
\hspace*{0.3cm} =
\ \left (
\ \left (
\ \widehat{\Omega} \ ( \ y \ )
\ \right )^{\ -1}
\ U^{\ \widehat{\Omega}} \ \left (
\ y \ , \ y \ ; \ {\cal{CL}}
\hspace*{0.5cm} \begin{array}{c}
\longleftarrow
\vspace*{-0.42cm} \\
\hspace*{-0.36cm} \swarrow \hspace*{0.5cm} \nearrow
\vspace*{-0.43cm} \\
\hspace*{-0.6cm} \longrightarrow
\end{array}
\ \right )
\ \widehat{\Omega} \ ( \ y \ )
\ \right )_{\ A \ B}
\vspace*{0.1cm}
\end{array}$$
ii\) cut the edges of the contour ${\cal{CL}}$
In order to transform the map of the contour ${\cal{CL}}$ continuously into a circle, the edges marked in the corresponding symbol in eqs. (\[eq:81\]), (\[eq:94\]), (\[eq:100\]) and (\[eq:1001\]) need to be cut
$$\label{eq:1002}
\begin{array}{l}
\begin{array}{c}
\longleftarrow
\vspace*{-0.42cm} \\
\hspace*{-0.36cm} \swarrow \hspace*{0.5cm} \nearrow
\vspace*{-0.43cm} \\
\hspace*{-0.6cm} \longrightarrow
\end{array}
\ \rightarrow
\ \hspace*{0.05cm}
\bigcirc \vspace*{0.1cm} \hspace*{-0.14cm} {\scriptstyle{\wedge}}
\end{array}$$
iii\) the full collection of surfaces and Riemann normal gauges
As indicated in point i) the meaning of Stokes relations summarized in eqs. (\[eq:100\]) and (\[eq:1001\]) is to consider [*all*]{} surfaces with boundary ${\cal{CL}} \ ( \ y \ )$, the latter punctuated at the point $y$, the former with base point $x$, inheriting the point $y$, [*and*]{} the associated Riemann normal gauges $\widehat{\Omega} \ ( \ x \ )$ .
The collection of surfaces and associated Riemann normal gauges shall be denoted
$$\label{eq:1003}
\begin{array}{l}
\left \lbrace
\ {\cal{S}}_{\ x \ | \ y} \ ; \ \widehat{\Omega} \ ( \ x \ )
\ \right \rbrace
\end{array}$$
iv\) the Stokes relations proper
Stokes relations in eq. (\[eq:100\]) in Riemann normal gauges take the form
$$\label{eq:1004}
\begin{array}{l}
U^{\ \widehat{\Omega}_{\ x}} \ \left (
\ y \ , \ y \ ; \ {\cal{CL}} \ ( \ y \ )
\hspace*{0.2cm}
\begin{array}{l}
\hspace*{0.05cm}
\bigcirc \vspace*{0.1cm} \hspace*{-0.14cm} {\scriptstyle{\wedge}}
\end{array}
\ \right )_{\ A \ B}
\ =
\vspace*{0.3cm} \\
\hspace*{0.3cm} =
\ \left (
\ P_{\ 2}^{\ \widehat{\Omega}_{\ x}} \ \exp
\ \left (
\ -
\ {\displaystyle{\int}}_{\ S_{\ x \ | \ y}}
\ {\cal{W}}^{\ (2) \ \widehat{\Omega}_{\ x}}
\ \right )
\ \right )_{\ A \ B}
\vspace*{0.3cm} \\
\hspace*{0.3cm} \forall \hspace*{0.3cm}
\left \lbrace
\ {\cal{S}}_{\ x \ | \ y} \ ; \ \widehat{\Omega}_{\ x}
\ \right \rbrace
\end{array}$$
The true form of Stokes relations returns to a general [*common*]{} gauge, combining eqs. (\[eq:1001\]) and (\[eq:1004\])
$$\label{eq:1005}
\begin{array}{l}
U \ \left (
\ y \ , \ y \ ; \ {\cal{CL}} \ ( \ y \ )
\hspace*{0.2cm}
\begin{array}{l}
\hspace*{0.05cm}
\bigcirc \vspace*{0.1cm} \hspace*{-0.14cm} {\scriptstyle{\wedge}}
\end{array}
\ \right )_{\ A \ B}
\ =
\vspace*{0.3cm} \\
\hspace*{0.3cm} =
\ \left (
\begin{array}{l}
\ \left (
\ \widehat{\Omega}_{\ x} \ ( \ y \ )
\ \right )^{\ -1}
\ \times
\vspace*{0.3cm} \\
\hspace*{1.5cm} \times
\ P_{\ 2}^{\ \widehat{\Omega}_{\ x}} \ \exp
\ \left (
\ -
\ {\displaystyle{\int}}_{\ S_{\ x \ | \ y}}
\ {\cal{W}}^{\ (2) \ \widehat{\Omega}_{\ x}}
\ \right )
\ \times
\vspace*{0.3cm} \\
\hspace*{3.5cm} \times
\ \widehat{\Omega}_{\ x} \ ( \ y \ )
\end{array}
\ \right )_{\ A \ B}
\vspace*{0.7cm} \\
\hspace*{0.3cm} \forall \hspace*{0.3cm}
\left \lbrace
\ {\cal{S}}_{\ x \ | \ y} \ ; \ \widehat{\Omega}_{\ x}
\ \right \rbrace
\end{array}$$
The closed contour ${\cal{CL}} \ ( \ y \ )$ integral $U \ \left (
\ y \ , \ y \ ; \ {\cal{CL}} \ ( \ y \ )
\hspace*{0.2cm}
\begin{array}{l}
\hspace*{0.05cm}
\bigcirc \vspace*{0.1cm} \hspace*{-0.14cm} {\scriptstyle{\wedge}}
\end{array}
\ \right )$ on the left hand side of eq. (\[eq:1005\]) is dependent on the point $y$, where the contour begins and ends, but [*not*]{} on any surface and associated Riemann normal gauge forming the collection $\left \lbrace
\ {\cal{S}}_{\ x \ | \ y} \ ; \ \widehat{\Omega}_{\ x}
\ \right \rbrace$ .
v\) the surface integral proper in Riemann normal gauge
The main ingredient in the Stokes relations in eq. (\[eq:1005\]) is\
– selecting a surface and a Riemann normal gauge\
out of the collection $\left \lbrace
\ {\cal{S}}_{\ x \ | \ y} \ ; \ \widehat{\Omega}_{\ x}
\ \right \rbrace$ – the surface integral proper as summarized in eq. (\[eq:1004\])
$$\label{eq:1006}
\begin{array}{l}
U^{\ \widehat{\Omega}_{\ x}} \ \left (
\ y \ , \ y \ ; \ {\cal{CL}} \ ( \ y \ )
\hspace*{0.2cm}
\begin{array}{l}
\hspace*{0.05cm}
\bigcirc \vspace*{0.1cm} \hspace*{-0.14cm} {\scriptstyle{\wedge}}
\end{array}
\ \right )_{\ A \ B}
\ =
\vspace*{0.3cm} \\
\hspace*{0.3cm} =
\ \left (
\ P_{\ 2}^{\ \widehat{\Omega}_{\ x}} \ \exp
\ \left (
\ -
\ {\displaystyle{\int}}_{\ S_{\ x \ | \ y}}
\ {\cal{W}}^{\ (2) \ \widehat{\Omega}_{\ x}}
\ \right )
\ \right )_{\ A \ B}
\end{array}$$
The surface integral on the right hand side of eq. (\[eq:1006\]) – [*in Riemann normal gauge*]{} – involves the ordering, denoted $P_{\ 2}^{\ \widehat{\Omega}_{\ x}}$ , of products of [*local*]{} surface differentials ${\cal{W}}^{\ (2) \ \widehat{\Omega}_{\ x}}$ . By this local property the surface ’integral’ is indeed an integral.
In any gauge other than a Riemannian normal one, the corresponding differentials are [*not*]{} local functions of the plaquette differentials, rather they depend on the entire set of flagpole paths, described in figure \[fig3\] and eqs. (\[eq:95\]) and (\[eq:96\]).
vi\) the ordering of surface elements in Riemann normal gauge
The path ordering $P_{\ 2}^{\ \widehat{\Omega}_{\ x}}$ of the – matrix valued – surface elements is very special to Riemann normal gauges.
It derives from two steps, starting in a general (original) gauge.
They are described in the text following eq.(\[eq:94\]) and in figures \[fig3\] and \[fig4\].
An appropriate name for $P_{\ 2}^{\ \widehat{\Omega}_{\ x}}$ is ’spider-web ordering’ illustrated in figure \[fig4a\] , [@spiderfot] .
Spin projection operations on adjoint string operators
------------------------------------------------------
The adjoint string operators forming binary gluonic mesons are interoduced in eq. (\[eq:101\]) , repeated below
$$\label{eq:1007}
\begin{array}{l}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ =
\vspace*{0.3cm} \\
\hspace*{0.5cm}
F_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack}
\ ( \ x_{\ 1} \ ; \ A \ )
\ U \ ( \ x_{\ 1} \ , \ A \ ; \ x_{\ 2} \ , \ B \ )
\ F_{\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 2} \ ; \ B \ )
\vspace*{0.3cm} \\
A \ , \ B \ , \cdots \ = \ 1, \cdots , 8
\end{array}$$
The Lorentz invariant tensors $K^{\ \pm}$ are introduced in eq. (\[eq:123\]) , repeated below
$$\label{eq:1008}
\begin{array}{l}
\widetilde{t}_{\ \underline{.} \ ; \ I^{\ \pm}}
\ ( \ z \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )
\ =
\ \left (
\begin{array}{l}
\ \left (
\ K^{\ \pm}
\ \right )_{\
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack}
\ \times
\vspace*{0.3cm} \\
\hspace*{0.7cm} \times
\ \widetilde{t}_{\ I^{\ \pm}}
\ ( \ z \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )
\end{array}
\ \right )
\vspace*{0.5cm} \\
\ \left (
\ K^{\ +}
\ \right )_{\
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack}
\ =
\ g_{\ \mu_{\ 1} \ \mu_{\ 2}}
\ g_{\ \nu_{\ 1} \ \nu_{\ 2}}
\ -
\ g_{\ \mu_{\ 1} \ \nu_{\ 2}}
\ g_{\ \mu_{\ 2} \ \nu_{\ 1}}
\vspace*{0.5cm} \\
\ \left (
\ K^{\ -}
\ \right )_{\
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack}
\ =
\ \varepsilon_{\ \mu_{\ 1} \ \mu_{\ 2} \ \nu_{\ 1} \ \nu_{\ 2}}
\end{array}$$
We perform the associated projections
$$\label{eq:1009}
\begin{array}{l}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\ \left (
\ \begin{array}{r}
K^{\ +}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ B^{\ (+)} \ +
\vspace*{0.3cm} \\
+ \ K^{\ -}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ B^{\ (-)} \ +
\vspace*{0.3cm} \\
+ \ B^{\ '}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\end{array}
\ \right )
\end{array}$$
The decomposition according to eq. (\[eq:1009\]) is the same for the Riemann curvature tensor, where $B^{\ (+)}$ relates to the curvature scalar , $B^{\ '}$ to the Ricci and Weyl tensors and $B^{\ (-)} \ = \ 0$, unlike here.
So we form the metric (Ricci-) contraction
$$\label{eq:1010}
\begin{array}{l}
g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.3cm} \\
R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\ =
\ \left (
\ \begin{array}{l}
3 \ g_{\ \nu_{\ 1} \ \nu_{\ 2}} \ B^{\ (+)} \ +
\vspace*{0.3cm} \\
+ \ g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ B^{\ '}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\end{array}
\ \right )
\vspace*{0.3cm} \\
R_{\ \nu_{\ 1} \ \nu_{\ 2}} \ =
\ \varrho_{\ \nu_{\ 1} \ \nu_{\ 2}} \ + \ \frac{1}{4}
\ g_{\ \nu_{\ 1} \ \nu_{\ 2}} \ R
\hspace*{0.3cm} ; \hspace*{0.3cm}
R \ =
\ g^{\ \nu_{\ 1} \ \nu_{\ 2}} \ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.3cm} \\
\varrho_{\ \nu_{\ 1} \ \nu_{\ 2}} \ =
\ R_{\ \nu_{\ 1} \ \nu_{\ 2}} \ -
\ \frac{1}{4}
\ g_{\ \nu_{\ 1} \ \nu_{\ 2}} \ R
\end{array}$$
It follows from eq. (\[eq:1010\])
$$\label{eq:1011}
\begin{array}{l}
\varrho_{\ \nu_{\ 1} \ \nu_{\ 2}} \ =
\ g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ B^{\ '}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ \rightarrow
\vspace*{0.3cm} \\
g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ g^{\ \nu_{\ 1} \ \nu_{\ 2}}
\ B^{\ '}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ 0
\hspace*{0.3cm} \rightarrow \hspace*{0.3cm}
R \ = \ 12 \ B^{\ (+)}
\end{array}$$
The quantity $B^{\ '}$ with the trace condition in eq. (\[eq:1011\]) forms the irreducible [*relativistic*]{} spin two part $S_{\ 12}^{\ +} \ = \ 2$ as defined in eqs. (\[eq:108\]) and (\[eq:122\]) in the main text. Here we concentrate on the projection on $B^{\ (\pm)}$ .
From eq. (\[eq:1011\]) we obtain
$$\label{eq:1012}
\begin{array}{l}
B^{\ (+)} \ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ =
\vspace*{0.3cm} \\
\hspace*{0.6cm} =
\ \frac{1}{12}
\ F_{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 1} \ ; \ A \ )
\ U \ ( \ x_{\ 1} \ , \ A \ ; \ x_{\ 2} \ , \ B \ )
\ F^{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 2} \ ; \ B \ )
\end{array}$$
The projection on $B^{\ (-)}$ proceeds in a similar way
$$\label{eq:1013}
\begin{array}{l}
\varepsilon^{\ \mu_{\ 1} \ \mu_{\ 2} \ \nu_{\ 1} \ \nu_{\ 2}}
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\ - \ 24 \ B^{\ (-)}
\ +
\vspace*{0.3cm} \\
\hspace*{4.5cm} +
\ \varepsilon^{\ \mu_{\ 1} \ \mu_{\ 2} \ \nu_{\ 1} \ \nu_{\ 2}}
\ B^{\ '}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ \rightarrow
\vspace*{0.3cm} \\
\varepsilon^{\ \mu_{\ 1} \ \mu_{\ 2} \ \nu_{\ 1} \ \nu_{\ 2}}
\ B^{\ '}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ 0
\end{array}$$
The structure pf $B^{\ (-)}$ follows similarly as for $B^{\ (+)}$ in eq. (\[eq:1012\]) . We thus give both expressions together below
$$\label{eq:1014}
\begin{array}{l}
B^{\ (+)} \ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ =
\vspace*{0.3cm} \\
\hspace*{0.6cm} =
\ \frac{1}{12}
\ F_{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 1} \ ; \ A \ )
\ U \ ( \ x_{\ 1} \ , \ A \ ; \ x_{\ 2} \ , \ B \ )
\ F^{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 2} \ ; \ B \ )
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
B^{\ (-)} \ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ =
\vspace*{0.3cm} \\
\hspace*{0.6cm} =
\ - \ \frac{1}{12}
\ F_{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 1} \ ; \ A \ )
\ U \ ( \ x_{\ 1} \ , \ A \ ; \ x_{\ 2} \ , \ B \ )
\ \widetilde{F}^{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 2} \ ; \ B \ )
\vspace*{0.3cm} \\
\hspace*{0.6cm} =
\ - \ \frac{1}{12}
\ \widetilde{F}_{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 1} \ ; \ A \ )
\ U \ ( \ x_{\ 1} \ , \ A \ ; \ x_{\ 2} \ , \ B \ )
\ F^{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 2} \ ; \ B \ )
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
\widetilde{F}_{\ \left \lbrack \ \alpha \ \beta \ \right \rbrack}
\ ( \ x_{\ 2} \ ; \ B \ )
\ =
\ \frac{1}{2}
\ \varepsilon_{ \ \alpha \ \beta \ \gamma \ \delta}
\ F^{\ \left \lbrack \ \gamma \ \delta \ \right \rbrack}
\ ( \ x_{\ 2} \ ; \ B \ )
\vspace*{0.3cm} \\
\mbox{and} \hspace*{0.3cm}
( \ x_{\ 2} \ ; \ B \ ) \ \leftrightarrow
\ ( \ x_{\ 1} \ ; \ A \ )
\end{array}$$
Spin projection operations on adjoint string operators - extended
-----------------------------------------------------------------
We continue the projection operations carried out in appendix A.4 in order to extend them to the remaining gb spectral series of type $II^{\ +}$ . This is related to the quantities $B^{\ '}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}$ defined in eq. (\[eq:124\]) and refined in appendix A.4 ( eq. (\[eq:1009\]) ) .
To that end we perform the [*full*]{} decomposition of the tensorial structure of the octet string operators interoduced in eq. (\[eq:101\]) and rewritten in eq. (\[eq:1007\])
$$\label{eq:1015}
\begin{array}{l}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ \rightarrow
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\end{array}$$
which is analogous to that of the Riemann curvature tensor, without the metric constraints of the latter.
The Ricci contraction introduced in eq. (\[eq:1010\]) yields the follwoing structure
$$\label{eq:1016}
\begin{array}{l}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ w_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
+ \Delta
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\vspace*{0.3cm} \\
\Delta
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\begin{array}{l}
\ \frac{1}{2}
\left (
\begin{array}{l}
g_{\ \mu_{\ 1} \ \mu_{\ 2}} \ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
- \ g_{\ \nu_{\ 1} \ \mu_{\ 2}} \ R_{\ \mu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.3cm} \\
- \ g_{\ \mu_{\ 1} \ \nu_{\ 2}} \ R_{\ \nu_{\ 1} \ \mu_{\ 2}}
\ + \ g_{\ \nu_{\ 1} \ \nu_{\ 2}} \ R_{\ \mu_{\ 1} \ \mu_{\ 2}}
\end{array}
\right )
\vspace*{0.3cm} \\
-
\ \frac{1}{6}
\ K^{\ +}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ R
\end{array}
\vspace*{0.4cm} \\
\hline
\vspace*{-0.3cm} \\
\mbox{with :}
\hspace*{0.3cm}
g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ w_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ 0
\vspace*{0.5cm} \\
g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\ g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ \Delta
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.5cm} \\
R \ =
\ g^{\ \nu_{\ 1} \ \nu_{\ 2}}
\ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\ =
\ 12 \ B^{\ (+)}
\end{array}$$
Before proceeding lets express the Ricci bilinear in terms of the base octet string operators $B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}$
$$\label{eq:1017}
\begin{array}{l}
R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\ =
\vspace*{0.3cm} \\
\hspace*{0.3cm} =
\ - \ F_{\ \nu_{\ 1} \ \alpha}
\ ( \ x_{\ 1} \ ; \ A \ )
\ U \ ( \ x_{\ 1} \ , \ A \ ; \ x_{\ 2} \ , \ B \ )
\ F^{\ \alpha}_{\hspace*{0.3cm} \nu_{\ 2}}
\ ( \ x_{\ 2} \ ; \ B \ )
\end{array}$$
In order to simplify notation we shall suppress the position arguments and use chromoelectric and -magnetic fields for the field strength tensor.
$$\label{eq:1018}
\begin{array}{l}
- \ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\ =
\ \left (
\begin{array}{cc}
\vec{E}^{\ A} \ \vec{E}^{\ D}
& \vec{S}^{\ k \ A \ D}
\vspace*{0.5cm} \\
\vec{S}^{\ i \ A \ D} &
\begin{array}{ll}
- \ \vec{E}^{\ i \ A} \ \vec{E}^{\ k \ D}
- \ \vec{B}^{\ i \ A} \ \vec{B}^{\ k \ D}
\vspace*{0.3cm} \\
\ + \ \delta_{\ i k} \ \vec{B}^{\ A} \ \vec{B}^{\ D}
\end{array}
\end{array}
\ \right )
\ U_{\ A \ D}
\vspace*{0.5cm} \\
R \ = \ 2
\ \left (
\ \vec{B}^{\ A} \ \vec{B}^{\ D}
\ - \ \vec{E}^{\ A} \ \vec{E}^{\ D}
\ \right )
\ U_{\ A \ D}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\vec{S}^{\ A \ D} \ = \ \vec{E}^{\ A} \ \wedge \ \vec{B}^{\ D}
\vspace*{0.5cm} \\
\vec{E}^{\ i \ A}
\ = \ F^{\ 0 \ i \ A}
\hspace*{0.3cm} , \hspace*{0.3cm}
\vec{B}^{\ i \ A} \ = \ \frac{1}{2} \ \varepsilon_{\ ikl}
\ F^{\ k \ l \ A}
\end{array}$$
In eq. (\[eq:1018\]) we recognize the Maxwell energy momentum like (bilinear) expression, where $\vec{S}^{\ A \ D}$ shall be called the bilinear Poynting vector.
Next we substitute the traceless part of the Ricci bilinear ( eq. (\[eq:1010\]) ) in eq. (\[eq:1016\])
$$\label{eq:1019}
\begin{array}{l}
- \ \varrho^{\ \mu \ \nu} \ =
\ - \ R^{\ \mu \ \nu} \ +
\ \frac{1}{4}
\ g^{\ \mu \ \nu} \ R \ =
\ \vartheta_{\ cl}^{\ \mu \ \nu} \ =
\vspace*{0.5cm} \\
\hspace*{0.4cm} =
\ \left (
\begin{array}{cc}
\frac{1}{2}
\ \left (
\begin{array}{l}
\vec{E}^{\ A} \ \vec{E}^{\ D}
\vspace*{0.3cm} \\
\hspace*{0.3cm} + \ \vec{B}^{\ A} \ \vec{B}^{\ D}
\end{array}
\ \right )
& - \ \vec{S}^{\ k \ A \ D}
\vspace*{0.5cm} \\
- \ \vec{S}^{\ i \ A \ D} &
\begin{array}{ll}
- \ \vec{E}^{\ i \ A} \ \vec{E}^{\ k \ D}
- \ \vec{B}^{\ i \ A} \ \vec{B}^{\ k \ D}
\vspace*{0.3cm} \\
\ + \ \frac{1}{2} \ \delta_{\ i k}
\ \left (
\begin{array}{l}
\vec{E}^{\ A} \ \vec{E}^{\ D}
\vspace*{0.3cm} \\
\hspace*{0.3cm} + \ \vec{B}^{\ A} \ \vec{B}^{\ D}
\end{array}
\ \right )
\end{array}
\end{array}
\ \right )
\ U_{\ A \ D}
\end{array}$$
In eq. (\[eq:1019\]) we recognize the bilinear with the structure of the classical (traceless) Maxwell energy momentum tensor of nonabelian gauge field strengths.
Eq. (\[eq:1016\]) becomes decomposed into positive parity irreducible parts
$$\label{eq:1020}
\begin{array}{l}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ w_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
+ \Delta
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\vspace*{0.3cm} \\
\Delta
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\begin{array}{l}
\ \frac{1}{2}
\left (
\begin{array}{l}
g_{\ \mu_{\ 1} \ \mu_{\ 2}} \ \varrho_{\ \nu_{\ 1} \ \nu_{\ 2}}
- \ g_{\ \nu_{\ 1} \ \mu_{\ 2}} \ \varrho_{\ \mu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.3cm} \\
- \ g_{\ \mu_{\ 1} \ \nu_{\ 2}} \ \varrho_{\ \nu_{\ 1} \ \mu_{\ 2}}
\ + \ g_{\ \nu_{\ 1} \ \nu_{\ 2}} \ \varrho_{\ \mu_{\ 1} \ \mu_{\ 2}}
\end{array}
\right )
\vspace*{0.3cm} \\
+
\ \frac{1}{12}
\ K^{\ +}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ R
\end{array}
\vspace*{0.4cm} \\
\hline
\vspace*{-0.3cm} \\
\mbox{with :}
\hspace*{0.3cm}
g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ w_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ 0
\vspace*{0.5cm} \\
g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\ g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ \Delta
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.5cm} \\
R \ =
\ g^{\ \nu_{\ 1} \ \nu_{\ 2}}
\ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\ =
\ 12 \ B^{\ (+)}
\end{array}$$
As a side remark to the (Lorentz-) tensorial reduction of the bilinear quantities $ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}$ it is necessary to include the spatio-temporal [*nonlocal*]{} parallel transport matrices pertaining to a general connection and metric $\Gamma^{\hspace*{0.3cm} \mu}_{\ \sigma \hspace*{0.3cm} \nu}$ and $g_{\ \mu \nu}$ . This is necessary to render $ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}$ a true [*nonlocal*]{} Lorentz-tensor. We do not do this here. In globally flat Minkowski space coordinates this parallel transport is trivial.
The sequence of projections on first Lorentz spin ($\underline{.}$) and second on rotational spin ($.$) , needs two steps , rearranging the structure of $- \ \varrho^{\ \mu \nu}$ in eq. (\[eq:1019\])
$$\label{eq:1021}
\begin{array}{l}
- \ \varrho^{\ \mu \ \nu} \ =
\ \vartheta_{\ cl}^{\ \mu \ \nu} \ =
\vspace*{0.5cm} \\
\hspace*{0.1cm} =
\left (
\begin{array}{cc}
\frac{1}{2}
\ \left (
\begin{array}{l}
\vec{E}^{\ A} \ \vec{E}^{\ D}
\vspace*{0.3cm} \\
\hspace*{0.3cm} + \ \vec{B}^{\ A} \ \vec{B}^{\ D}
\end{array}
\ \right )
& - \ \vec{S}^{\ k \ A \ D}
\vspace*{0.5cm} \\
- \ \vec{S}^{\ i \ A \ D} &
\begin{array}{ll}
\left (
\begin{array}{l}
- \ \vec{E}^{\ i \ A} \ \vec{E}^{\ k \ D}
- \ \vec{B}^{\ i \ A} \ \vec{B}^{\ k \ D}
\vspace*{0.3cm} \\
+ \ \frac{1}{3} \ \delta_{\ i k}
\ \left (
\begin{array}{l}
\vec{E}^{\ A} \ \vec{E}^{\ D}
\vspace*{0.3cm} \\
\hspace*{0.3cm} + \ \vec{B}^{\ A} \ \vec{B}^{\ D}
\end{array}
\ \right )
\end{array}
\ \right )
\vspace*{0.6cm} \\
+ \ \frac{1}{6} \ \delta_{\ i k}
\ \left (
\begin{array}{l}
\vec{E}^{\ A} \ \vec{E}^{\ D}
\vspace*{0.3cm} \\
\hspace*{0.3cm} + \ \vec{B}^{\ A} \ \vec{B}^{\ D}
\end{array}
\ \right )
\end{array}
\end{array}
\right )
\ \times
\vspace*{0.3cm} \\
\hspace*{5.6cm} \times \ U_{\ A \ D}
\end{array}$$
The tensor structure of $\vartheta_{\ cl}^{\ \mu \ \nu}$ in eq. (\[eq:1021\]) follows the hydrodynamic nomenclature
$$\label{eq:1022}
\begin{array}{l}
- \ \varrho^{\ \mu \ \nu} \ =
\ \vartheta_{\ cl}^{\ \mu \ \nu} \ =
\ \left (
\begin{array}{cc}
\varrho_{\ e}
& - \ \vec{S}^{\ k}
\vspace*{0.5cm} \\
- \ \vec{S}^{\ i} &
\left (
\ \begin{array}{cc}
\pi_{\ i \ k}
\vspace*{0.3cm} \\
+ \ \delta_{\ i k} \ p
\end{array}
\ \right )
\end{array}
\right )
\vspace*{0.3cm} \\
\varrho_{\ e} \ = \ 3 \ p \ =
\ \frac{1}{2}
\ \left (
\ \vec{E}^{\ A} \ \vec{E}^{\ D}
\ + \ \vec{B}^{\ A} \ \vec{B}^{\ D}
\ \right ) \ U_{\ A \ D}
\vspace*{0.3cm} \\
\pi_{\ i \ k} \ = \ 2 \ p \ \delta_{\ i \ k}
\ -
\ \left (
\ \vec{E}^{\ i \ A} \ \vec{E}^{\ k \ D}
\ + \ \vec{B}^{\ i \ A} \ \vec{B}^{\ k \ D}
\ \right )
\ U_{\ A \ D}
\vspace*{0.3cm} \\
\sum_{\ i} \ \pi_{\ i \ i} \ = \ 0
\end{array}$$
with the identifications given in eq. (\[eq:1022\]) .
The chain of irreducible components of $\Delta \ B_{\ \underline{.}}$ is shown in eq. (\[eq:1023\]) below
$$\label{eq:1023}
\begin{array}{l}
\begin{array}{|c|cc cc|}
\hline
& & & & \vspace*{-0.3cm} \\
\mbox{step} & \mbox{name} & \mbox{\# comp.} & \mbox{L.-spin} & \mbox{R.-spin} \\
& & & & \vspace*{-0.3cm} \\
\hline
& & & & \vspace*{-0.3cm} \\
1 & \Delta \ B_{\ \underline{.}} & 10 & \mbox{mixed} & \mbox{mixed} \\
2 & B^{\ (+)} & 1 & 1 & 1 \\
2 & \varrho^{\ \underline{.}} & 9 & D^{\ 1 \ , \ \overline{1}}
& \mbox{mixed} \\
3 & \varrho_{\ e} & 1 & - & 1 \\
3 & \vec{S} & 3 & - & D^{\ 1} \\
3 & \pi_{\ .} & 5 & - & D^{\ 2} \vspace*{-0.3cm} \\
& & & & \\
\hline
\end{array}
\end{array}$$
It is the last term in eq. (\[eq:1023\]) $\pi_{\ i \ k}$ as displayed in eq. (\[eq:1022\]) which characterizes the $S_{\ 12}^{\ +} \ = \ 2$ spectral series of binary gluonic mesons. The R-spin 2 tensor $\pi_{\ i \ k}$ is related to the corresponding components of the Weyl bilinear $w_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}$ introduced in eq. (\[eq:1016\]), which represents the traceless part of the bilinear Riemann like tensor $B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}$ , to which we turn next.
[*Weyl bilinear and circular polarization basis for gauge field strengths*]{}
We recall the right and left chiral spin matrices defined in appendix A.1 ( eqs. (\[eq:16\]) and (\[eq:17\]) ) reproduced below, first for the right circular part. Here the notion [*right (left) circular*]{} refers to a fixed spin axis and [*not*]{} to the individual momenta of the two gauge bosons at the end of the octet string, in question. The spin axis is common to both and an axial vector.
$$\label{eq:1024}
\begin{array}{l}
\left ( \ \sigma_{\ \mu \ \nu} \ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\ \leftrightarrow
\ P_{\ R} \ \frac{i}{2}
\ \left \lbrack \ \gamma_{\ \mu} \ , \ \gamma_{\ \nu} \ \right \rbrack
\ P_{\ R}
\hspace*{0.3cm} ; \hspace*{0.3cm}
P_{\ R} \ = \ \frac{1}{2} \ ( \ 1 \ + \ \gamma_{\ 5 \ R} \ )
\vspace*{0.3cm} \\
\gamma_{\ 5 \ R} \ = \ \frac{1}{i}
\ \gamma_{\ 0} \ \gamma_{\ 1} \ \gamma_{\ 2} \ \gamma_{\ 3}
\vspace*{0.3cm} \\
\left ( \ \sigma_{\ \mu \ \nu} \ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\ =
\ \begin{array}{ll}
\ \left (
\begin{array}{l}
- \ i \ \Sigma_{\ k}
\vspace*{0.3cm} \\
\varepsilon_{\ m n r} \ \Sigma_{\ r}
\end{array}
\ \right )_{ \alpha}^{\hspace*{0.3cm} \beta}
\vspace*{1.0cm}
&
\vspace*{-0.5cm}
\begin{array}{l}
\mbox{for} \ \mu \ = \ 0 \ , \ \nu \ = \ k \ = \ 1,2,3
\vspace*{0.3cm} \\
\mbox{for} \ \mu \ = \ m \ , \ \nu \ = \ n \ ;
\vspace*{-0.2cm} \\
\hspace*{2.7cm}
m,n,r \ = \ 1,2,3
\end{array}
\end{array}
\vspace*{0.3cm} \\
\left ( \ \sigma_{\ \mu \ \nu}
\ \right )_{\ \alpha}^{\hspace*{0.3cm} \beta}
\ \rightarrow \ \sigma_{\ \mu \ \nu}^{\ R}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\sigma_{\ \mu \ \nu}^{\ R} \ = \ - \ i
\ \frac{1}{2} \ \varepsilon_{\ \mu \nu \varrho \tau}
\ \sigma^{\ \varrho \ \tau \ R}
\end{array}$$
The right circular spinor basis in eq. (\[eq:1024\]) yields the projection on the gauge field strengths
$$\label{eq:1025}
\begin{array}{l}
\frac{1}{2}
\ \sigma_{\ \mu \ \nu}^{\ R}
\ F^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack}
\ ( \ x \ ; \ A \ )
\ = \ \Sigma_{\ r} \ \vec{C}^{ \ r \ A} \ ( \ x \ )
\vspace*{0.3cm} \\
\vec{C}^{ \ r \ A} \ ( \ x \ )
\ =
\ \left (
\ \vec{B} \ - \ i \ \vec{E}
\ \right )^{\ r \ A} \ ( \ x \ )
\hspace*{0.3cm} \rightarrow \hspace*{0.3cm}
\vec{C}^{ \ r \ A}
\vspace*{0.3cm} \\
r \ = \ 1,2,3
\end{array}$$
The right circular quantities $\vec{C}^{ \ A}$ in eq. (\[eq:1025\]) are complex combinations of the hermitian field strengths $F^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}$ in the adjoint representation of $SU3_{\ c}$ .
We note the right circular identity, following from eq. (\[eq:1024\])
$$\label{eq:1026}
\begin{array}{l}
\frac{1}{2}
\ \sigma_{\ \mu \ \nu}^{\ R}
\ F^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\ \equiv
\ \frac{1}{2}
\ \sigma_{\ \mu \ \nu}^{\ R}
\ \left ( \ F^{\ R}
\ \right )^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\vspace*{0.3cm} \\
\ \left ( \ F^{\ R}
\ \right )^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\ =
\ \frac{1}{2}
\ \left (
\ F^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\ - \ i
\ \widetilde{F}^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\ \right )
\vspace*{0.3cm} \\
\widetilde{F}_{\ \left \lbrack \ \mu \ \nu \ \right \rbrack}^{\ A}
\ =
\ \frac{1}{2}
\ \varepsilon_{\ \mu \nu \sigma \tau}
\ F^{\ \left \lbrack \ \sigma \ \tau \ \right \rbrack \ A}
\end{array}$$
When the space-time component $r$ in $\vec{C}^{\ r \ A}$ is explicitely denoted, the vector symbol of $\vec{C}$ shall be omitted for simplicity.
Now we recall the left chiral spinor matrices defined in eqs. (\[eq:35\]) and (\[eq:36\]) in appendix A.1
$$\label{eq:1027}
\begin{array}{l}
\left ( \ \sigma_{\ \mu \ \nu} \ \right )^{\ \dot{\gamma}}_{\hspace*{0.4cm}
\dot{\delta}}
\ \leftrightarrow
\ P_{\ L} \ \frac{i}{2}
\ \left \lbrack \ \gamma_{\ \mu} \ , \ \gamma_{\ \nu} \ \right \rbrack
\ P_{\ L}
\hspace*{0.3cm} ; \hspace*{0.3cm}
P_{\ L} \ = \ \frac{1}{2} \ ( \ 1 \ - \ \gamma_{\ 5 \ R} \ )
\vspace*{0.3cm} \\
\gamma_{\ 5 \ R} \ = \ \frac{1}{i}
\ \gamma_{\ 0} \ \gamma_{\ 1} \ \gamma_{\ 2} \ \gamma_{\ 3}
\vspace*{0.3cm} \\
\left ( \ \sigma_{\ \mu \ \nu} \ \right )^{\ \dot{\gamma}}_{\hspace*{0.4cm}
\dot{\delta}}
\ =
\ \begin{array}{ll}
\ \left (
\begin{array}{l}
i \ \Sigma_{\ k}
\vspace*{0.3cm} \\
\varepsilon_{\ m n r} \ \Sigma_{\ r}
\end{array}
\ \right )^{ \dot{\gamma}}_{\hspace*{0.4cm} \dot{\delta}}
\vspace*{1.0cm}
&
\vspace*{-0.5cm}
\begin{array}{l}
\mbox{for} \ \mu \ = \ 0 \ , \ \nu \ = \ k \ = \ 1,2,3
\vspace*{0.3cm} \\
\mbox{for} \ \mu \ = \ m \ , \ \nu \ = \ n \ ;
\vspace*{-0.2cm} \\
\hspace*{2.7cm}
m,n,r \ = \ 1,2,3
\end{array}
\end{array}
\vspace*{0.3cm} \\
\left ( \ \sigma_{\ \mu \ \nu} \ \right )^{\ \dot{\gamma}}_{\hspace*{0.4cm}
\dot{\delta}}
\ \rightarrow \ \sigma_{\ \mu \ \nu}^{\ L}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\sigma_{\ \mu \ \nu}^{\ L} \ = \ + \ i
\ \frac{1}{2} \ \varepsilon_{\ \mu \nu \varrho \tau}
\ \sigma^{\ \varrho \ \tau \ L}
\end{array}$$
Correspondingly to eq. (\[eq:1024\]) , the left circular spinor basis in eq. (\[eq:1027\]) yields the projection on the left circular gauge field strengths, completing the right circular one in eq. (\[eq:1025\])
$$\label{eq:1028}
\begin{array}{l}
\frac{1}{2}
\ \sigma_{\ \mu \ \nu}^{\ L}
\ F^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack}
\ ( \ x \ ; \ A \ )
\ = \ \Sigma_{\ r} \ \vec{G}^{ \ r \ A} \ ( \ x \ )
\vspace*{0.3cm} \\
\vec{G}^{ \ r \ A} \ ( \ x \ )
\ =
\ \left (
\ \vec{B} \ + \ i \ \vec{E}
\ \right )^{\ r \ A} \ ( \ x \ )
\hspace*{0.3cm} \rightarrow \hspace*{0.3cm}
\vec{G}^{ \ r \ A}
\vspace*{0.3cm} \\
r \ = \ 1,2,3
\end{array}$$
Analogous to the right circular identity in eq. (\[eq:1026\]) is the left circular one, displayed together in eq. (\[eq:1029\]) below
$$\label{eq:1029}
\begin{array}{l}
\frac{1}{2}
\ \sigma_{\ \mu \ \nu}^{\ R}
\ F^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\ \equiv
\ \frac{1}{2}
\ \sigma_{\ \mu \ \nu}^{\ R}
\ \left ( \ F^{\ R}
\ \right )^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\vspace*{0.3cm} \\
\ \left ( \ F^{\ R}
\ \right )^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\ =
\ \frac{1}{2}
\ \left (
\ F^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\ - \ i
\ \widetilde{F}^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\ \right )
\vspace*{0.3cm} \\
\widetilde{F}_{\ \left \lbrack \ \mu \ \nu \ \right \rbrack}^{\ A}
\ =
\ \frac{1}{2}
\ \varepsilon_{\ \mu \nu \sigma \tau}
\ F^{\ \left \lbrack \ \sigma \ \tau \ \right \rbrack \ A}
\vspace*{0.5cm} \\
\ \sigma_{\ \mu \ \nu}^{\ L}
\ F^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\ \equiv
\ \frac{1}{2}
\ \sigma_{\ \mu \ \nu}^{\ L}
\ \left ( \ F^{\ L}
\ \right )^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\vspace*{0.3cm} \\
\ \left ( \ F^{\ L}
\ \right )^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\ =
\ \frac{1}{2}
\ \left (
\ F^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\ + \ i
\ \widetilde{F}^{\ \left \lbrack \ \mu \ \nu \ \right \rbrack \ A}
\ \right )
\end{array}$$
As long as we remain within the real Lorentz group, as discussed in appendix A.1, the three vector quantities $\vec{C}^{\ A}$ and $\vec{G}^{\ A}$ are relative hermitian conjugates of each other. They transform according to the $D^{\ 1 \ , \ 0}$ and $D^{\ 0 \ , \ \dot{1}}$ representations of the $spin \ ( \ 1 \ , \ 3 \ ; \ \Re \ ) \ \simeq \ SL2C$ group, as defined in eqs. (\[eq:34\]) and (\[eq:35\]) in appendix A.2 .
$$\label{eq:1030}
\begin{array}{l}
\vec{C}^{\ A} \ =
\ \left (
\ \vec{G}^{\ A}
\ \right )^{\ *}
\ \rightarrow
\vspace*{0.3cm} \\
D^{\ 1 \ , \ 0}
\hspace*{0.2cm} : \hspace*{0.2cm}
C^{\ r \ A} \ \rightarrow \ R_{\ rs} \ C^{\ s \ A}
\hspace*{0.4cm} ; \hspace*{0.4cm}
D^{\ 0 \ , \ \dot{1}}
\hspace*{0.2cm} : \hspace*{0.2cm}
G^{\ r \ A} \ \rightarrow \ L_{\ rs} \ G^{\ s \ A}
\vspace*{0.3cm} \\
R_{\ rs} \ = \ R_{\ rs} \ ( \ {\cal{A}} \ )
\hspace*{0.3cm} , \hspace*{0.3cm}
L_{\ rs} \ = \ L_{\ rs} \ ( \ {\cal{B}} \ )
\vspace*{0.3cm} \\
R_{\ rs} \ = \ \overline{L}_{\ rs}
\hspace*{0.3cm} \mbox{within} \hspace*{0.3cm}
spin \ ( \ 1 \ , \ 3 \ ; \ \Re \ )
\end{array}$$
The three by three matrices $R_{\ rs}$ and $L_{\ rs}$ are complex orthogonal with determinant 1, forming the group $SO3C \ \simeq \ SL2C \ / \ Z_{\ 2}$ , where $Z_{\ 2}$ denotes the center of $SL2C$ .
We are now ready to decompose the Weyl bilinear $w_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}$ in eq. (\[eq:1016\]), into its irreducible parts.
$$\label{eq:1031}
\begin{array}{l}
w_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\ \left (
\begin{array}{r}
\left (
\ P^{\ RR} \ \left ( \ \vec{C} \ \otimes \ \vec{C} \ \right )
\ \right )_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\vspace*{0.3cm} \\
+
\ \left (
\ P^{\ LL} \ \left ( \ \vec{G} \ \otimes \ \vec{G} \ \right )
\ \right )_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\vspace*{0.3cm} \\
+
\ B^{\ (-)}
\ \left (
\ K^{\ -}
\ \right )_{\
\ \left \lbrack
\ \mu_{ 1} \ \nu_{ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{ 2} \ \nu_{ 2}
\ \right \rbrack}
\end{array}
\ \right )
\vspace*{0.4cm} \\
\left (
\ K^{\ -}
\ \right )_{\
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack}
\ =
\ \varepsilon_{\ \mu_{\ 1} \ \mu_{\ 2} \ \nu_{\ 1} \ \nu_{\ 2}}
\end{array}$$
In eq. (\[eq:1031\]) the quantities $B^{\ (-)}$ and $\left ( \ K^{\ -} \ \right )$ are defined in eqs. (\[eq:123\]) - (\[eq:125\]) .
The projections denoted $P^{\ RR} \ \left ( \ \vec{C} \ \otimes \ \vec{C} \ \right )$ and $P^{\ LL} \ \left ( \ \vec{G} \ \otimes \ \vec{G} \ \right )$ , operate on the doubly right- and left circular, direct product combinations indicated as respective arguments in eq. (\[eq:1031\]) . They are of the form, omitting the explicit dependence on the Lorentz indices $\left \lbrack \ \mu_{\ 1} \ \nu_{\ 1} \ \right \rbrack \ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2} \ \right \rbrack$ for simplicity
$$\label{eq:1032}
\begin{array}{l}
P^{\ RR} \ \left ( \ \vec{C} \ \otimes \ \vec{C} \ \right )
\ = \ \pi^{\ (2)}_{\ r \ s}
\ \left (
\ C^{\ r \ A} \ C^{\ s \ D} \ -
\ \frac{1}{3} \ \delta^{\ r s} \ \vec{C}^{\ A} \ \vec{C}^{\ D}
\ \right )
\ U_{\ A \ D}
\vspace*{0.3cm} \\
P^{\ LL} \ \left ( \ \vec{G} \ \otimes \ \vec{G} \ \right )
\ = \ \pi^{\ (2)}_{\ r \ s}
\ \left (
\ G^{\ r \ A} \ G^{\ s \ D} \ -
\ \frac{1}{3} \ \delta^{\ r s} \ \vec{G}^{\ A} \ \vec{G}^{\ D}
\ \right )
\ U_{\ A \ D}
\end{array}$$
We thus introduce the abbreviations following eq. (\[eq:1032\])
$$\label{eq:1033}
\begin{array}{l}
P^{\ RR} \ \left ( \ \vec{C} \ \otimes \ \vec{C} \ \right )
\ \rightarrow
\ w^{\ RR}_{\ \underline{.}}
\vspace*{0.3cm} \\
P^{\ LL} \ \left ( \ \vec{G} \ \otimes \ \vec{G} \ \right )
\ \rightarrow
\ w^{\ LL}_{\ \underline{.}}
\vspace*{0.3cm} \\
w^{\ RR}_{\ \underline{.}}
\ =
\ \left (
\ C^{\ r \ A} \ C^{\ s \ D} \ -
\ \frac{1}{3} \ \delta^{\ r s} \ \vec{C}^{\ A} \ \vec{C}^{\ D}
\ \right )
\ U_{\ A \ D}
\vspace*{0.3cm} \\
w^{\ LL}_{\ \underline{.}}
\ =
\ \left (
\ G^{\ r \ A} \ G^{\ s \ D} \ -
\ \frac{1}{3} \ \delta^{\ r s} \ \vec{G}^{\ A} \ \vec{G}^{\ D}
\ \right )
\ U_{\ A \ D}
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
w^{\ LL}_{\ \underline{.}}
\ =
\ \left (
\ w^{\ RR}_{\ \underline{.}}
\ \right )^{\ *}
\end{array}$$
The bilinears $w^{\ RR}_{\ \underline{.}}$ and $w^{\ LL}_{\ \underline{.}}$ defined in eq. (\[eq:1033\]) transform according to the complex representations $D^{\ 2 \ , \ 0}$ and $D^{\ 0 \ , \ \dot{2}}$ of $spin \ ( \ 1 \ , \ 3 \ ; \ \Re \ ) \ \simeq \ SL2C$ respectively. These two representations are complex conjugate to each other.
As in the case of the Lorentz tensor $\varrho^{\ \mu \nu}$ in eqs. (\[eq:1019\]) , (\[eq:1021\]) and (\[eq:1022\]) the space time indices $r \ s$ for the quantities $w^{\ LL}_{\ \underline{.}}$ and $w^{\ RR}_{\ \underline{.}}$ in eq. (\[eq:1033\]) are understood to be symmetrized.
This completes the decomposition of the Weyl bilinear. We compare the structure of the irreducible components $w_{\ \underline{.}}$ with that of $\Delta \ B_{\ \underline{.}}$ , as shown in eq. (\[eq:1023\]) reproduced below
$$\label{eq:1034}
\begin{array}{l}
\begin{array}{|c|cc cc|}
\hline
& & & & \vspace*{-0.3cm} \\
\mbox{step} & \mbox{name} & \mbox{\# comp.} & \mbox{L.-spin} & \mbox{R.-spin} \\
& & & & \vspace*{-0.3cm} \\
\hline
& & & & \vspace*{-0.3cm} \\
1 & \Delta \ B_{\ \underline{.}} & 10 & \mbox{mixed} & \mbox{mixed} \\
2 & B^{\ (+)} & 1 & 1 & 1 \\
2 & \varrho^{\ \underline{.}} & 9 & D^{\ 1 \ , \ \overline{1}}
& \mbox{mixed} \\
3 & \varrho_{\ e} & 1 & - & 1 \\
3 & \vec{S} & 3 & - & D^{\ 1} \\
3 & \pi_{\ .} & 5 & - & D^{\ 2} \vspace*{-0.3cm} \\
& & & & \\
\hline
\end{array}
\end{array}$$
The corresponding structure of the Weyl bilinears is displayed in eq. (\[eq:1035\])
$$\label{eq:1035}
\begin{array}{l}
\begin{array}{|c|cc cc|}
\hline
& & & & \vspace*{-0.3cm} \\
\mbox{step} & \mbox{name} & \mbox{\# comp.} & \mbox{L.-spin} & \mbox{R.-spin} \\
& & & & \vspace*{-0.3cm} \\
\hline
& & & & \vspace*{-0.3cm} \\
1 & w_{\ \underline{.}} & 11 & \mbox{mixed} & \mbox{mixed} \\
2 & B^{\ (-)} & 1 & 1 & 1 \\
2 & w^{\ RR}_{\ \underline{.}} & 5 & D^{\ 2 \ , \ 0}
& D^{\ 2} \\
2 & w^{\ LL}_{\ \underline{.}} & \overline{5} & D^{\ 0 \ , \ \dot{2}}
& D^{\ 2} \vspace*{-0.3cm} \\
& & & & \\
\hline
\end{array}
\end{array}$$
Comparing the counting in the two tables ( eqs. (\[eq:1034\]) and (\[eq:1035\]) we should keep in mind that the colums labeled $\# \ comp.$ are based on counting independent [*hermitian operators*]{} among the bilinears $B_{\ \underline{.}}$ .
In this respect we verify the correctness of the counting : the Riemann tensor like bilinears have $6 \ \times \ 7 \ / \ 2 \ = \ 21$ hermitian components, which combine into 10 for the Ricci tensor like quantities further decomposed according to eq. (\[eq:1034\]) and 11 for the Weyl tensor like in eq. (\[eq:1035\]).
For the Ricci tensor the decomposition into $B^{\ (+)}$ corresponding to the curvature scalar and the traceless part, called $\varrho^{\ \underline{.}}$ here, is straightforward.
For the Weyl tensor the splitting into 10 + 1 hermitian components, corresponding to the pseudoscalar $B^{\ (-)}$ and the right- and left circular bilinears $w^{\ RR}_{\ \underline{.}}$ and $w^{\ LL}_{\ \underline{.}}$ , with together 10 [*hermitian*]{} components is also quite clear.
What appears impossible, is to find a common contribution to the so defined irreducibles : $\varrho^{\ \underline{.}}$ with 9 hermitian components on the one hand and $w^{\ RR}_{\ \underline{.}}$ and $w^{\ LL}_{\ \underline{.}}$ with 10 on the other. It follows from the discussion below, that this is indeed impossible.
In this connection we have to remember, that we are considering matrix elements of the form defined in eq. (\[eq:104\])
$$\label{eq:1036}
\begin{array}{l}
\left \langle
\ \emptyset
\ \right |
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ \left |
\ gb \ ( \ J^{\ P\ C} \ ) \ ; \ p \ ,
\ \left \lbrace spin \right \rbrace
\ \right \rangle \ \rightarrow
\vspace*{0.3cm} \\
\exp^{\ - i p X}
\ \widetilde{t}_{\ \underline{.}} \ ( \ z \ , \ p \ , \ J^{\ P\ C} \ ; \ . \ )
\end{array}$$
where hermition bilinears induce [*complex*]{} amplitudes.
Next we focus on the continuity equation for the classical energy momentum tensor pertaining to the field strengths, extended to the nonlocal situation, conditioned by the c.m. four momentum p. This follows the relations in eqs. (\[eq:1021\]) and (\[eq:1022\])
$$\label{eq:1037}
\begin{array}{l}
- \ \varrho^{\ \mu \ \nu} \ =
\ \vartheta_{\ cl}^{\ \mu \ \nu} \ ( \ X \ ; \ z \ )
\hspace*{0.3cm}
\rightarrow
\hspace*{0.3cm}
\partial_{\ X \ \mu}
\ \vartheta_{\ cl}^{\ \mu \ \nu} \ ( \ X \ ; \ z \ )
\ = \ 0
\end{array}$$
Eq. (\[eq:1037\]) is valid for classical field configurations and follows from the analogous classical treatment of the Stokes relation discussed in appendix A.3. It is not straightforward for quantized local gauge fields. In the latter case the identical relation is not sufficiently established and deserves further study. Nevertheless we use it here for consistency. Thus it follows that the quantities $\varrho_{\ e}$ and $\vec{S}$ defined in eq. (\[eq:1022\]) do not contribute to the amplitudes associated with the classical energy momentum tensor $\vartheta_{\ cl}^{\ \mu \ \nu} \ ( \ X \ ; \ z \ )$ .
Thus we associate each bilinear irreducible to the (family of) wave functions, following the notation introduced in eq. (\[eq:104\]) and repeated in eq. (\[eq:1036\]) . Hereby the complete family of wave functions is accordingly projected
$$\label{eq:1038}
\begin{array}{l}
\begin{array}{lll ll}
\varrho^{\ \underline{.}}
& \leftrightarrow &
\ \widetilde{t}_{\ \underline{.}}
\ ( \ \left \lbrace \ \varrho \ \right \rbrace
\ ; \ z \ , \ p \ , \ J^{\ P\ C} \ ; \ . \ )
& \rightarrow &
\widetilde{t}
\ ( \ \left \lbrace \ \varrho \ \right \rbrace \ )
\vspace*{0.3cm} \\
w_{\ \underline{.}}
& \leftrightarrow &
\widetilde{t}_{\ \underline{.}}
\ ( \ \left \lbrace \ w \ \right \rbrace
\ ; \ z \ , \ p \ , \ J^{\ P\ C} \ ; \ . \ )
& \rightarrow &
\widetilde{t}
\ ( \ \left \lbrace \ w \ \right \rbrace \ )
\vspace*{0.3cm} \\
w^{\ RR}_{\ \underline{.}}
& \leftrightarrow &
\widetilde{t}_{\ \underline{.}}
\ ( \ \left \lbrace \ w^{\ RR} \ \right \rbrace
\ ; \ z \ , \ p \ , \ J^{\ P\ C} \ ; \ . \ )
& \rightarrow &
\widetilde{t}
\ ( \ \left \lbrace \ w^{\ RR} \ \right \rbrace \ )
\vspace*{0.3cm} \\
w^{\ LL}_{\ \underline{.}}
& \leftrightarrow &
\widetilde{t}_{\ \underline{.}}
\ ( \ \left \lbrace \ w^{\ LL} \ \right \rbrace
\ ; \ z \ , \ p \ , \ J^{\ P\ C} \ ; \ . \ )
& \rightarrow &
\widetilde{t}
\ ( \ \left \lbrace \ w^{\ LL} \ \right \rbrace \ )
\end{array}
\end{array}$$
The bilinears $B^{\ (\pm)}$ are already fully characterized. They do not contribute to the wave functions of the $II^{\ +}$ , i.e. $S_{\ 12}^{\ +} \ = \ 2$ spectral type, and thus we will not discuss them any further here.
The tables in eqs. (\[eq:1034\]) and (\[eq:1035\]) are thus reduced and adapted to the wave functions $\widetilde{t}$ defined in eq. (\[eq:1038\])
$$\label{eq:1039}
\begin{array}{l}
\begin{array}{|c|cc cc|}
\hline
& & & & \vspace*{-0.3cm} \\
\mbox{step} & \mbox{name} & \mbox{\# comp.} & \mbox{L.-spin} & \mbox{R.-spin} \\
& & & & \vspace*{-0.3cm} \\
\hline
& & & & \vspace*{-0.3cm} \\
2 &
\widetilde{t} \ ( \ \left \lbrace \ \varrho \ \right \rbrace \ )
& 9 & D^{\ 1 \ , \ \overline{1}} & D^{\ 2} \\
3 & \widetilde{t} \ ( \ \left \lbrace \ \varrho_{\ e} \ \right \rbrace \ )
& 0 & - & - \\
3 & \widetilde{t} \ ( \ \left \lbrace \ \vec{S} \ \right \rbrace \ )
& 0 & - & - \\
3 & \widetilde{t} \ ( \ \left \lbrace \ \pi \ \right \rbrace \ )
& 5 & - & D^{\ 2} \vspace*{-0.3cm} \\
& & & & \vspace*{-0.3cm} \\
\hline \vspace*{-0.4cm} \\
2 & \widetilde{t} \ ( \ \left \lbrace \ w^{\ RR} \ \right \rbrace \ )
& 5 & D^{\ 2 \ , \ 0} & D^{\ 2} \\
2 & \widetilde{t} \ ( \ \left \lbrace \ w^{\ LL} \ \right \rbrace \ )
& 5 & D^{\ 0 \ , \ 2} & D^{\ 2} \vspace*{-0.3cm} \\
& & & & \\
\hline
\end{array}
\end{array}$$
The wave functions denoted $\widetilde{t} \ ( \ \left \lbrace \ \varrho_{\ e} \ \right \rbrace \ )$ and $\widetilde{t} \ ( \ \left \lbrace \ \vec{S} \ \right \rbrace \ )$ in eq. (\[eq:1039\]) vanish, as a consequence of the continuity equation in eq. (\[eq:1037\]) .
Thus the table in eq. (\[eq:1039\]) reduces to
$$\label{eq:1040}
\begin{array}{l}
\begin{array}{|c|cc cc|}
\hline
& & & & \vspace*{-0.3cm} \\
\mbox{step} & \mbox{name} & \mbox{\# comp.} & \mbox{L.-spin} & \mbox{R.-spin} \\
& & & & \vspace*{-0.3cm} \\
\hline
& & & & \vspace*{-0.3cm} \\
2 &
\widetilde{t} \ ( \ \left \lbrace \ \varrho \ \right \rbrace \ )
& 9 & D^{\ 1 \ , \ \overline{1}} & D^{\ 2} \\
3 & \widetilde{t} \ ( \ \left \lbrace \ \pi \ \right \rbrace \ )
& 5 & - & D^{\ 2} \vspace*{-0.3cm} \\
& & & & \vspace*{-0.3cm} \\
\hline \vspace*{-0.4cm} \\
2 & \widetilde{t} \ ( \ \left \lbrace \ w^{\ RR} \ \right \rbrace \ )
& 5 & D^{\ 2 \ , \ 0} & D^{\ 2} \\
2 & \widetilde{t} \ ( \ \left \lbrace \ w^{\ LL} \ \right \rbrace \ )
& 5 & D^{\ 0 \ , \ 2} & D^{\ 2} \vspace*{-0.3cm} \\
& & & & \\
\hline
\end{array}
\end{array}$$
In the tables ( eqs. (\[eq:1039\]) and (\[eq:1040\]) ) the column labelled $\mbox{\# comp.}$ refers to wave function components over the complex numbers.
The entries and properties displayed in eq. (\[eq:1040\]) look more coherent than in the tables in eqs. (\[eq:1034\]) and (\[eq:1035\]) , but the puzzle of 5 versus 10 components for $\widetilde{t} \ ( \ \left \lbrace \ \varrho \ \right \rbrace \ )$ compared to $\widetilde{t} \ ( \ \left \lbrace \ w^{\ RR} \ \right \rbrace \ )$ and $\widetilde{t} \ ( \ \left \lbrace \ w^{\ LL} \ \right \rbrace \ )$ remains.
To understand this difference we compare the structure of the bilinears associated with $\widetilde{t} \ ( \ \left \lbrace \ \pi \ \right \rbrace \ )$ ( eq. (\[eq:1022\]) ) with the one pertaining to $\widetilde{t} \ ( \ \left \lbrace \ w^{\ RR} \ \right \rbrace \ )$ and $\widetilde{t} \ ( \ \left \lbrace \ w^{\ LL} \ \right \rbrace \ )$ ( (\[eq:1032\]) ) . To this end we use the relations in eqs. (\[eq:1025\]) defining the quantities $\vec{C}^{\ A}$ and (\[eq:1028\]) for $\vec{G}^{\ A}$ respectively.
$$\label{eq:1041}
\begin{array}{l}
- \ \varrho^{\ \mu \ \nu} \ =
\ \vartheta_{\ cl}^{\ \mu \ \nu} \ \rightarrow
\vspace*{0.3cm} \\
\pi_{\ i \ k} \ =
\ \left (
\begin{array}{l}
\ \frac{1}{3} \ \delta_{\ i \ k}
\ \left (
\ \vec{E}^{\ A} \ \vec{E}^{\ D}
\ + \ \vec{B}^{\ A} \ \vec{B}^{\ D}
\ \right )
\vspace*{0.3cm} \\
\ -
\ \left (
\ \vec{E}^{\ i \ A} \ \vec{E}^{\ k \ D}
\ + \ \vec{B}^{\ i \ A} \ \vec{B}^{\ k \ D}
\ \right )
\end{array}
\ \right )
\ U_{\ A \ D}
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
P^{\ RR} \ \left ( \ \vec{C} \ \otimes \ \vec{C} \ \right )
\ \rightarrow
\vspace*{0.3cm} \\
\left (
\ \pi^{\ R \ R}
\ \right )^{\ r \ s}
\ =
\ \left (
\ C^{\ r \ A} \ C^{\ s \ D} \ -
\ \frac{1}{3} \ \delta^{\ r s} \ \vec{C}^{\ A} \ \vec{C}^{\ D}
\ \right )
\ U_{\ A \ D}
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
P^{\ LL} \ \left ( \ \vec{G} \ \otimes \ \vec{G} \ \right )
\ \rightarrow
\vspace*{0.3cm} \\
\left (
\ \pi^{\ L \ L}
\ \right )^{\ r \ s}
\ =
\ \left (
\ G^{\ r \ A} \ G^{\ s \ D} \ -
\ \frac{1}{3} \ \delta^{\ r s} \ \vec{G}^{\ A} \ \vec{G}^{\ D}
\ \right )
\ U_{\ A \ D}
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
\vec{C}^{ \ r \ A} \ ( \ x \ )
\ =
\ \left (
\ \vec{B} \ - \ i \ \vec{E}
\ \right )^{\ r \ A} \ ( \ x \ )
\vspace*{0.3cm} \\
\vec{G}^{ \ r \ A} \ ( \ x \ )
\ =
\ \left (
\ \vec{B} \ + \ i \ \vec{E}
\ \right )^{\ r \ A} \ ( \ x \ )
\vspace*{0.3cm} \\
\sum_{\ i} \ \pi_{\ i \ i} \ = \ 0
\hspace*{0.3cm} ; \hspace*{0.3cm}
\sum_{\ r}
\ \left (
\ \pi^{\ R \ R}
\ \right )^{\ r \ r}
\ =
\ \sum_{\ r}
\left (
\ \pi^{\ L \ L}
\ \right )^{\ r \ r}
\ = \ 0
\end{array}$$
In eq. (\[eq:1041\]) the 21 complex components of the octet string bilinear wave functions are reduced to 3 complex, traceless and symmetric $3 \ \times \ 3$ matrices, denoted $\pi$ , $\pi^{\ R \ R}$ and $\pi^{\ L \ L}$ respectively. There is at this stage an essential ingredient missing. The property distinguishing the above matrices is the spatial ’Dreibein’ nature of chromoelectric- , chromomagnetic and orientation axis vectors [@MF] .
In order to realize the ’Dreibein’ property, we reduce the three quantities $\pi$ , $\pi^{\ R \ R}$ and $\pi^{\ L \ L}$ in eq. (\[eq:41\]) to their common (chromo-) electric and magnetic components. This leaves $\pi$ unchanged
$$\label{eq:1042}
\begin{array}{l}
\pi^{\ r \ s} \ =
\ \left (
\begin{array}{l}
\ \frac{1}{3} \ \delta^{\ r \ s}
\ \left (
\ \vec{E}^{\ A} \ \vec{E}^{\ D}
\ + \ \vec{B}^{\ A} \ \vec{B}^{\ D}
\ \right )
\vspace*{0.3cm} \\
\ -
\ \left (
\ \vec{E}^{\ r \ A} \ \vec{E}^{\ s \ D}
\ + \ \vec{B}^{\ r \ A} \ \vec{B}^{\ s \ D}
\ \right )
\end{array}
\ \right )
\ U_{\ A \ D}
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
\left (
\ \pi^{\ R \ R}
\ \right )^{\ r \ s}
\ = \ a^{\ r \ s} \ - \ i \ b^{\ r \ s}
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
\left (
\ \pi^{\ L \ L}
\ \right )^{\ r \ s}
\ = \ a^{\ r \ s} \ + \ i \ b^{\ r \ s}
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
a^{\ r \ s} \ =
\ \left (
\begin{array}{l}
\ \frac{1}{3} \ \delta^{\ r \ s}
\ \left (
\ - \ \vec{E}^{\ A} \ \vec{E}^{\ D}
\ + \ \vec{B}^{\ A} \ \vec{B}^{\ D}
\ \right )
\vspace*{0.3cm} \\
\ -
\ \left (
\ - \ \vec{E}^{\ r \ A} \ \vec{E}^{\ s \ D}
\ + \ \vec{B}^{\ r \ A} \ \vec{B}^{\ s \ D}
\ \right )
\end{array}
\ \right )
\ U_{\ A \ D}
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
b^{\ r \ s} \ =
\ \left (
\begin{array}{l}
\ \frac{1}{3} \ \delta^{\ r \ s}
\ \left (
\ \vec{E}^{\ A} \ \vec{B}^{\ D}
\ + \ \vec{B}^{\ A} \ \vec{E}^{\ D}
\ \right )
\vspace*{0.3cm} \\
\ -
\ \left (
\ \vec{E}^{\ r \ A} \ \vec{B}^{\ s \ D}
\ + \ \vec{B}^{\ r \ A} \ \vec{E}^{\ s \ D}
\ \right )
\end{array}
\ \right )
\ U_{\ A \ D}
\end{array}$$
The adjoint representation indices $A \ , \ D$ and the position difference $z$ make it necessary to symmetrize the above expressions with respect to the indices $r \ , s$ taking into account the dependence on the relative (Lorentz-) coordinate $z$ , not explicitely shown in eq. (\[eq:1042\]) .
In order to retain the relevant degrees of freedom we streamline the displayed indices and kinematic variables to the electromagnetic case. But in no way is this implying, that the nonabelian character of the underlying variables is sacrificed, to the contrary.
With this in mind we introduce the abbreviating notation
$$\label{eq:1043}
\begin{array}{l}
\left (
\ \vec{E}^{\ A} \ , \ \vec{B}^{\ A}
\ \right )
\ \rightarrow
\ \left (
\ \vec{E} \ , \ \vec{B}
\ \right ) \ ( \hspace*{0.3cm} \zeta \hspace*{0.5cm} )
\ \rightarrow
\ \left (
\ \vec{E} \ , \ \vec{B}
\ \right )_{\ +}
\vspace*{0.3cm} \\
\left (
\ \vec{E}^{\ D} \ , \ \vec{B}^{\ D}
\ \right )
\ \rightarrow
\ \left (
\ \vec{E} \ , \ \vec{B}
\ \right ) \ ( \ - \ \zeta \ )
\ \rightarrow
\ \left (
\ \vec{E} \ , \ \vec{B}
\ \right )_{\ -}
\vspace*{0.3cm} \\
U_{\ A \ D} \ \rightarrow \ .
\hspace*{0.3cm} ; \hspace*{0.3cm}
\zeta \ = \ z \ / \ 2
\end{array}$$
The trace parts proportional to $\delta^{\ r \ s}$ of the matrix $\pi$ in eq. (\[eq:1042\]) can be neglected. This follows from eq. (\[eq:1037\]) .
Thus eq. (\[eq:1042\]) takes the form using the notation introduced in eq. (\[eq:1043\])
$$\label{eq:1044}
\begin{array}{l}
\begin{array}{rlr}
- \ \pi^{\ r \ s}
& \sim &
\vec{E}_{\ +}^{\ r } \ \vec{E}_{\ -}^{\ s}
\ + \ \vec{B}_{\ +}^{\ r} \ \vec{B}_{\ -}^{\ s}
\vspace*{0.3cm} \\
a^{\ r \ s}
& \sim &
\vec{E}_{\ +}^{\ r} \ \vec{E}_{\ -}^{\ s}
\ - \ \vec{B}_{\ +}^{\ r} \ \vec{B}_{\ -}^{\ s}
\vspace*{0.3cm} \\
- \ b^{\ r \ s}
& \sim &
\vec{E}_{\ +}^{\ r} \ \vec{B}_{\ -}^{\ s}
\ + \ \vec{B}_{\ +}^{\ r} \ \vec{E}_{\ -}^{\ s}
\end{array}
\ + \ ( \ r \ \leftrightarrow \ s \ )
\vspace*{0.3cm} \\
\mbox{modulo trace parts}
\end{array}$$
Here we need the variable $\vec{e} \ = \ \vec{z} \ / \ r$ introduced in eq. (\[eq:139\]) in order to orient chromoelectric and -magnetic fields, in a radial gauge, where the parallel transport matrix $U_{\ A \ D} \ \rightarrow \ \delta_{\ A \ D}$ .
The chromomagnetic fields then become related to the -electric ones
$$\label{eq:1045}
\begin{array}{l}
\vec{B}_{\ +} \ = \ \vec{e} \ \wedge \ \vec{E}_{\ +}
\hspace*{0.3cm} ; \hspace*{0.3cm}
\vec{B}_{\ -} \ = \ - \ \vec{e} \ \wedge \ \vec{E}_{\ -}
\vspace*{0.3cm} \\
\vec{e} \ \vec{E}_{\ \pm}
\ = \ \vec{e} \ \vec{B}_{\ \pm} \ = \ 0
\end{array}$$
Eq. (\[eq:1045\]) establishes the nonabelian ’Dreibein’ form.
As the wave functions associated with the field strengths in eqs. (\[eq:1045\]) and (\[eq:1045\]) are complex, we can choose the $\vec{e}$ associated helicity basis. Choosing the $z$ axis along $\vec{e}$ eq. (\[eq:1045\]) takes the component form
$$\label{eq:1046}
\begin{array}{l}
\vec{E}_{\ \pm} \ =
\ \left ( \ E_{\ x \ \pm} \ , \ E_{\ y \ \pm} \ , \ 0 \ \right )
\vspace*{0.3cm} \\
\vec{B}_{\ +} \ =
\ \left ( \ - \ E_{\ y \ +} \ , \ E_{\ x \ +} \ , \ 0 \ \right )
\hspace*{0.3cm} ; \hspace*{0.3cm}
\vec{B}_{\ -} \ =
\ \left ( \ E_{\ y \ -} \ , \ - \ E_{\ x \ -} \ , \ 0 \ \right )
\ \rightarrow
\vspace*{0.3cm} \\
\left ( \ {\cal{E}} \ , \ {\cal{B}} \ \right )^{\ R}_{\ +} \ =
\left (
\ E \ , \ B
\ \right )_{\ x \ +} \ - \ i
\left (
\ E \ , \ B
\ \right )_{\ y \ +}
\vspace*{0.3cm} \\
\left ( \ {\cal{E}} \ , \ {\cal{B}} \ \right )^{\ L}_{\ +} \ =
\left (
\ E \ , \ B
\ \right )_{\ x \ +} \ + \ i
\left (
\ E \ , \ B
\ \right )_{\ y \ +}
\vspace*{0.3cm} \\
\left ( \ {\cal{E}} \ , \ {\cal{B}} \ \right )^{\ R}_{\ -} \ =
\left (
\ E \ , \ B
\ \right )_{\ x \ -} \ + \ i
\left (
\ E \ , \ B
\ \right )_{\ y \ -}
\vspace*{0.3cm} \\
\left ( \ {\cal{E}} \ , \ {\cal{B}} \ \right )^{\ L}_{\ -} \ =
\left (
\ E \ , \ B
\ \right )_{\ x \ -} \ - \ i
\left (
\ E \ , \ B
\ \right )_{\ y \ -}
\end{array}$$
The (complex) components $\left ( \ {\cal{E}} \ , \ {\cal{B}} \ \right )^{\ R \ (L)}_{\ \pm}$ in eq. (\[eq:1046\]) [^4] define the sought helicity basis, where eq. (\[eq:1046\]) takes the form
$$\label{eq:1047}
\begin{array}{l}
{\cal{B}}^{\ R}_{\ \pm} \ =
\ \left ( \ - \ i \ \right )
\ {\cal{E}}^{\ R}_{\ \pm}
\hspace*{0.3cm} ; \hspace*{0.3cm}
{\cal{B}}^{\ L}_{\ \pm} \ =
\ \left ( \ i \ \right )
\ {\cal{E}}^{\ L}_{\ \pm}
\end{array}$$
The spin component associated with the operation
$$\label{eq:1048}
\begin{array}{l}
\widehat{S}_{\ \vec{e}} \ = \ i \ \vec{e} \ \wedge \ .
\end{array}$$
which represents an infinitesimal rotation around the $\vec{e} \ -$ axis (i.e. its derivative with respect to the rotation angle), takes the eigenvalues implied by the $R \ , \ L$ components in eq. (\[eq:1047\])
$$\label{eq:1049}
\begin{array}{l}
\widehat{S}_{\ \vec{e}}
\ \left (
\ {\cal{E}}^{\ R}_{\ +}
\ , \ {\cal{B}}^{\ R}_{\ +}
\ \right ) \ = \ + \ 1
\hspace*{0.3cm} , \hspace*{0.3cm}
\widehat{S}_{\ \vec{e}}
\ \left (
\ {\cal{E}}^{\ L}_{\ -}
\ , \ {\cal{B}}^{\ L}_{\ -}
\ \right ) \ = \ + \ 1
\vspace*{0.3cm} \\
\widehat{S}_{\ \vec{e}}
\ \left (
\ {\cal{E}}^{\ L}_{\ +}
\ , \ {\cal{B}}^{\ L}_{\ +}
\ \right ) \ = \ - \ 1
\hspace*{0.3cm} , \hspace*{0.3cm}
\widehat{S}_{\ \vec{e}}
\ \left (
\ {\cal{E}}^{\ R}_{\ -}
\ , \ {\cal{B}}^{\ R}_{\ -}
\ \right ) \ = \ - \ 1
\end{array}$$
Hence the direct product components $R_{\ +} \ R_{\ -}$ , $R_{\ +} \ L_{\ -}$ , $L_{\ +} \ R_{\ -}$ and $L_{\ +} \ L_{\ -}$ describe four $S_{\ \vec{e} \ 12}$ spin states
$$\label{eq:1050}
\begin{array}{l}
\begin{array}{clr}
\Delta \ \mbox{helicity components} & & S_{\ \vec{e} \ 12}
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
\ \left (
\ {\cal{E}}^{\ R}_{\ +}
\ , \ {\cal{B}}^{\ R}_{\ +}
\ \right )
\ \otimes
\ \left (
\ {\cal{E}}^{\ R}_{\ -}
\ , \ {\cal{B}}^{\ R}_{\ -}
\ \right )
& & 0
\vspace*{0.3cm} \\
\ \left (
\ {\cal{E}}^{\ L}_{\ +}
\ , \ {\cal{B}}^{\ L}_{\ +}
\ \right )
\ \otimes
\ \left (
\ {\cal{E}}^{\ L}_{\ -}
\ , \ {\cal{B}}^{\ L}_{\ -}
\ \right )
& & 0
\vspace*{0.3cm} \\
\ \left (
\ {\cal{E}}^{\ R}_{\ +}
\ , \ {\cal{B}}^{\ R}_{\ +}
\ \right )
\ \otimes
\ \left (
\ {\cal{E}}^{\ L}_{\ -}
\ , \ {\cal{B}}^{\ L}_{\ -}
\ \right )
& & 2
\vspace*{0.3cm} \\
\ \left (
\ {\cal{E}}^{\ L}_{\ +}
\ , \ {\cal{B}}^{\ L}_{\ +}
\ \right )
\ \otimes
\ \left (
\ {\cal{E}}^{\ R}_{\ -}
\ , \ {\cal{B}}^{\ R}_{\ -}
\ \right )
& & - \ 2
\end{array}
\end{array}$$
For clarity let us associate a pair of complex, transverse three vectors $\left ( \vec{v} , \vec{w} \right )$ [^5] with the two sides of the octet string denoted by $+$ and $-$
$$\label{eq:1051}
\begin{array}{l}
\begin{array}{lll}
\vec{v}
\hspace*{0.3cm} \leftrightarrow \hspace*{0.3cm}
\ \left (
\ \vec{E}
\ , \ \vec{B}
\ \right )_{\ +}
& , &
\vec{w}
\hspace*{0.3cm} \leftrightarrow \hspace*{0.3cm}
\ \left (
\ \vec{E}
\ , \ \vec{B}
\ \right )_{\ -}
\vspace*{0.3cm} \\
\vec{e} \ \vec{v} \ = \ 0
& , &
\vec{e} \ \vec{w} \ = \ 0
\vspace*{0.3cm} \\
v^{\ R} \ = \ v_{\ x} \ - \ i \ v_{\ y}
& , & w^{\ R} \ = \ w_{\ x} \ + \ i \ w_{\ y}
\vspace*{0.3cm} \\
v^{\ L} \ = \ v_{\ x} \ + \ i \ v_{\ y}
& , & w^{\ L} \ = \ w_{\ x} \ - \ i \ w_{\ y}
\end{array}
\end{array}$$
Using the components $R \ , \ L$ as defined in eqs. (\[eq:1046\]) and (\[eq:1051\]) the (complex orthogonal) scalar product takes the form
$$\label{eq:1052}
\begin{array}{l}
u \ . \ v \ = \ u_{\ x} \ v_{\ x} \ + \ \ u_{\ y} \ v_{\ y}
\ = \frac{1}{2}
\ \left (
\ u^{\ R} \ w^{\ R} \ + \ u^{\ L} \ w^{\ L}
\ \right )
\end{array}$$
We adapt the tensor structure of the symmetric matrices $\pi \ , \ a \ , b$ in eq. (\[eq:1044\]) to the $R \ R$ , $R \ L$ , $L \ R$ and $L \ L$ basis defined in eq. (\[eq:1046\])
$$\label{eq:1053}
\begin{array}{l}
\pi^{\ \widetilde{r} \ \widetilde{s}} \ \sim
\ \left (
\begin{array}{ll}
\pi^{\ RR} & \pi^{\ RL}
\vspace*{0.3cm} \\
\pi^{\ LR} & \pi^{\ LL}
\end{array}
\ \right ) \ +
\ ( \ r \ \leftrightarrow \ s \ )
\hspace*{0.3cm} ; \hspace*{0.3cm}
\hspace*{0.3cm} \mbox{and} \hspace*{0.3cm}
\pi \ \rightarrow \ a \ , \ b
\vspace*{0.3cm} \\
\mbox{modulo trace parts}
\end{array}$$
It follows from eqs. (\[eq:1052\]) and (\[eq:1053\]) that the trace part remains valid in the specific $\widetilde{r} \ \widetilde{s}$ assignment chosen. However symmetrization with respect to the indices $r \ s$ is not equivalent to symmetrization with respect to $\widetilde{r} \ \widetilde{s}$
$$\label{eq:1054}
\begin{array}{l}
\left (
\ r \ \leftrightarrow \ s
\ \right )
\ \equiv
\ \left (
\begin{array}{c}
\pi^{\ RR} \ \leftrightarrow \ \pi^{\ LL}
\vspace*{0.3cm} \\
\pi^{\ RL} \ \rightarrow \ \pi^{\ RL}
\hspace*{0.3cm} \mbox{and} \hspace*{0.3cm}
\pi^{\ LR} \ \rightarrow \ \pi^{\ LR}
\end{array}
\ \right )
\vspace*{0.3cm} \\
\mbox{modulo trace parts}
\hspace*{0.3cm} \mbox{and} \hspace*{0.3cm}
\pi \ \rightarrow \ a \ , \ b
\end{array}$$
Eq. (\[eq:1054\]) implies for the symmetrized matrices $\pi \ , \ a \ , \ b$
$$\label{eq:1055}
\begin{array}{l}
\pi^{\ \widetilde{r} \ \widetilde{s}} \ \sim
\ \left (
\begin{array}{cc}
\frac{1}{2} \ \left (
\ \pi^{\ RR} \ + \ \pi^{\ LL}
\ \right ) & \pi^{\ RL}
\vspace*{0.3cm} \\
\pi^{\ LR} &
\frac{1}{2} \ \left (
\ \pi^{\ RR} \ + \ \pi^{\ LL}
\ \right )
\end{array}
\ \right )
\vspace*{0.3cm} \\
\mbox{modulo trace parts}
\hspace*{0.3cm} \mbox{and} \hspace*{0.3cm}
\pi \ \rightarrow \ a \ , \ b
\end{array}$$
Now taking the traceless parts simply removes the $R \ R$ and $L \ L$ components
$$\label{eq:1056}
\begin{array}{l}
\pi^{\ \widetilde{r} \ \widetilde{s}} \ \sim
\ \left (
\begin{array}{cc}
0 & \pi^{\ RL}
\vspace*{0.3cm} \\
\pi^{\ LR} & 0
\end{array}
\ \right )
\hspace*{0.3cm} \mbox{and} \hspace*{0.3cm}
\pi \ \rightarrow \ a \ , \ b
\end{array}$$
Now we cast $\pi \ , \ a \ , \ b$ in eq. (\[eq:1044\]) into the $\widetilde{r} \ \widetilde{s}$ basis
$$\label{eq:1057}
\begin{array}{l}
\begin{array}{rll rrll}
- \pi^{\ R \ L}
& \sim &
{\cal{E}}_{\ +}^{\ R } \ {\cal{E}}_{\ -}^{\ L}
\ + \ {\cal{B}}_{\ +}^{\ R} \ {\cal{B}}_{\ -}^{\ L}
& , &
- \pi^{\ L \ R}
& \sim &
\ {\cal{E}}_{\ +}^{\ L } \ {\cal{E}}_{\ -}^{\ R}
\ + \ {\cal{B}}_{\ +}^{\ L} \ {\cal{B}}_{\ -}^{\ R}
\vspace*{0.3cm} \\
a^{\ R \ L}
& \sim &
\ {\cal{E}}_{\ +}^{\ R} \ {\cal{E}}_{\ -}^{\ L}
\ - \ {\cal{B}}_{\ +}^{\ R} \ {\cal{B}}_{\ -}^{\ L}
& , &
a^{\ L \ R}
& \sim &
{\cal{E}}_{\ +}^{\ L} \ {\cal{E}}_{\ -}^{\ R}
\ - \ {\cal{B}}_{\ +}^{\ L} \ {\cal{B}}_{\ -}^{\ R}
\vspace*{0.3cm} \\
- b^{\ R \ L}
& \sim &
{\cal{E}}_{\ +}^{\ R} \ {\cal{B}}_{\ -}^{\ L}
\ + \ {\cal{B}}_{\ +}^{\ R} \ {\cal{E}}_{\ -}^{\ L}
& , &
- b^{\ L \ R}
& \sim &
\ {\cal{E}}_{\ +}^{\ L} \ {\cal{B}}_{\ -}^{\ R}
\ + \ {\cal{B}}_{\ +}^{\ L} \ {\cal{E}}_{\ -}^{\ R}
\end{array}
\vspace*{0.2cm}
\end{array}$$
We substitute the relations in eq. (\[eq:1047\]) in eq. (\[eq:1057\])
$$\label{eq:1058}
\begin{array}{l}
{\cal{B}}^{\ R}_{\ \pm} \ =
\ \left ( \ - \ i \ \right )
\ {\cal{E}}^{\ R}_{\ \pm}
\hspace*{0.3cm} ; \hspace*{0.3cm}
{\cal{B}}^{\ L}_{\ \pm} \ =
\ \left ( \ i \ \right )
\ {\cal{E}}^{\ L}_{\ \pm}
\ \rightarrow
\end{array}$$
with the result that [*all contributions from the spin 2 Weyl bilinear vanish*]{} , as a consequence of the ’Dreibein’ conditions
$$\label{eq:1059}
\begin{array}{l}
- \ \pi^{\ R \ L}
\ \sim
\ 2 \ {\cal{E}}_{\ +}^{\ R } \ {\cal{E}}_{\ -}^{\ L}
\hspace*{0.3cm} , \hspace*{0.3cm}
- \ \pi^{\ L \ R}
\ \sim
\ 2 \ {\cal{E}}_{\ +}^{\ L } \ {\cal{E}}_{\ -}^{\ R}
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
\left (
\ a^{\ R \ L} \ , \ a^{\ L \ R}
\ \right )
\ \sim \ 0
\hspace*{0.3cm} , \hspace*{0.3cm}
\left (
\ b^{\ R \ L} \ , \ b^{\ L \ R}
\ \right )
\ \sim \ 0
\end{array}$$
Comparing with the helicity structure in eq. (\[eq:1050\]) we find
$$\label{eq:1060}
\begin{array}{l}
\begin{array}{clr}
\Delta \ \mbox{helicity components} & & S_{\ \vec{e} \ 12}
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
- \ \frac{1}{2} \ \pi^{\ R \ L}
\ =
\ {\cal{E}}_{\ +}^{\ R } \ {\cal{E}}_{\ -}^{\ L}
& & 2
\vspace*{0.3cm} \\
- \ \frac{1}{2} \ \pi^{\ L \ R}
\ =
\ {\cal{E}}_{\ +}^{\ L } \ {\cal{E}}_{\ -}^{\ R}
& & - \ 2
\vspace*{0.3cm} \\
a \ = \ b \ = \ 0 & & 0
\end{array}
\end{array}$$
[*Summary remarks on the construction of the $II^{\ +}$ spectral gb series*]{}
i\) Decomposition of adjoint string bilinears and their associated wave functions
We reproduce here the structure of the adjoint string bilinears introduced in eq. (\[eq:101\])
$$\label{eq:1061}
\begin{array}{l}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ =
\vspace*{0.3cm} \\
\hspace*{0.5cm}
F_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack}
\ ( \ x_{\ 1} \ ; \ A \ )
\ U \ ( \ x_{\ 1} \ , \ A \ ; \ x_{\ 2} \ , \ B \ )
\ F_{\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 2} \ ; \ B \ )
\vspace*{0.3cm} \\
A \ , \ B \ , \cdots \ = \ 1, \cdots , 8
\end{array}$$
The bilinear quantities in eq. (\[eq:1061\]) yield the gb wave functions in the three spectral series $I^{\ +}$ , $I^{\ -}$ and $II^{\ +}$ introduced in eqs. (\[eq:108\]) - (\[eq:123\]) summarized in eq. (\[eq:1062\]) below
$$\label{eq:1062}
\begin{array}{l}
\widetilde{t}_{\ \underline{.}}
\ ( \ z \ , \ p \ , \ J^{\ P \ +} \ ; \ . \ )
\ \rightarrow
\ \widetilde{t}_{ \
\underline{.} \ ; \ S_{\ 12}^{\ \pm}}
\ ( \ z \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )
\vspace*{0.3cm} \\
\hline
\vspace*{-0.2cm} \\
\widetilde{t}_{\ \underline{.} \ ; \ S_{\ 12}^{\ \pm}}
\ ( \ z \ , \ p \ , \ J^{\ \pm \ +} \ ; \ . \ )
\begin{array}{ll}
\nearrow &
\widetilde{t}_{\ \underline{.} \ ; \ II^{\ +}}
\ ( \ z \ , \ p \ , \ J^{\ + \ +} \ ; \ . \ )
\vspace*{0.3cm} \\
\rightarrow &
\widetilde{t}_{\ \underline{.} \ , \ I^{\ +}}
\ ( \ z \ , \ p \ , \ J^{\ + \ +} \ ; \ . \ )
\vspace*{0.3cm} \\
\searrow &
\widetilde{t}_{\ \underline{.} \ ; \ I^{\ -}}
\ ( \ z \ , \ p \ , \ J^{\ - \ +} \ ; \ . \ )
\end{array}
\vspace*{0.3cm} \\
\underline{.} \ ; S_{\ 12}^{\ \pm} \ \rightarrow \ \underline{.}
\ \rightarrow
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ ,
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack
\end{array}$$
The decomposition of the adjoint string bilinears is introduced in eq. (\[eq:124\]) repeated below
$$\label{eq:1063}
\begin{array}{l}
\hspace*{-1.0cm}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ ( \ x_{\ 1} \ , \ x_{\ 2} \ )
\ =
\left (
\hspace*{-0.1cm}
\begin{array}{r}
\left (
\ K^{\ +}
\ \right )_{\
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack}
\ B^{\ (+)}
\vspace*{0.3cm} \\
+
\ \left (
\ K^{\ -}
\ \right )_{\
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack}
\ B^{\ (-)}
\vspace*{0.3cm} \\
+
\ B^{\ '}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\end{array}
\right )
\vspace*{0.5cm} \\
\ \left (
\ K^{\ +}
\ \right )_{\
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack}
\ =
\ g_{\ \mu_{\ 1} \ \mu_{\ 2}}
\ g_{\ \nu_{\ 1} \ \nu_{\ 2}}
\ -
\ g_{\ \mu_{\ 1} \ \nu_{\ 2}}
\ g_{\ \mu_{\ 2} \ \nu_{\ 1}}
\vspace*{0.5cm} \\
\ \left (
\ K^{\ -}
\ \right )_{\
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack}
\ =
\ \varepsilon_{\ \mu_{\ 1} \ \mu_{\ 2} \ \nu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.2cm}
\end{array}$$
In eq. (\[eq:1063\]) $K^{\ \pm}$ and the associated bilinears and their induced amplitudes denoted $B{\ (\pm})$ project on the spectral types $I^{\ \pm}$ according to eq. (\[eq:1062\]) , whereas $B^{\ '}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}$ refers to the projection on the $II^{\ +}$ spectral type. iThe unique projection of $B^{\ '}$ on the traceless part of the Ricci bilinear identical to the classical energy momentum tensor pertaining to gauge bosons is derived in this appendix (A.5) .
The general decomposition of $B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}$ in eq. (\[eq:1061\]) is introduced in eq. (\[eq:1016\]) reproduced below
$$\label{eq:1064}
\begin{array}{l}
\hspace*{-1.0cm}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ w_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
+ \Delta
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\vspace*{0.3cm} \\
\hspace*{-1.0cm}
\Delta
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\begin{array}{l}
\ \frac{1}{2}
\left (
\begin{array}{l}
g_{\ \mu_{\ 1} \ \mu_{\ 2}} \ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
- \ g_{\ \nu_{\ 1} \ \mu_{\ 2}} \ R_{\ \mu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.3cm} \\
- \ g_{\ \mu_{\ 1} \ \nu_{\ 2}} \ R_{\ \nu_{\ 1} \ \mu_{\ 2}}
\ + \ g_{\ \nu_{\ 1} \ \nu_{\ 2}} \ R_{\ \mu_{\ 1} \ \mu_{\ 2}}
\end{array}
\right )
\vspace*{0.3cm} \\
-
\ \frac{1}{6}
\ K^{\ +}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ R
\end{array}
\vspace*{0.4cm} \\
\hline
\vspace*{-0.3cm} \\
\hspace*{-1.0cm}
\mbox{with :}
\hspace*{0.3cm}
g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ w_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ 0
\vspace*{0.5cm} \\
\hspace*{-1.0cm}
g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\ g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ \Delta
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.5cm} \\
\hspace*{-1.0cm}
R \ =
\ g^{\ \nu_{\ 1} \ \nu_{\ 2}}
\ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\ =
\ 12 \ B^{\ (+)}
\end{array}$$
In eq. (\[eq:1064\]) we have denoted the still reducible parts in the following way
$$\label{eq:1065}
\begin{array}{l}
\begin{array}{lll}
w_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
& : &
\mbox{Weyl bilinear}
\vspace*{0.3cm} \\
R_{\ \mu \ \nu}
& : &
\mbox{Ricci bilinear}
\vspace*{0.3cm} \\
R
& : &
\mbox{Riemann scalar bilinear}
\end{array}
\end{array}$$
The decomposition in eq. (\[eq:1064\]) into positive parity irreducible parts, leaving the Weyl bilinear reducible, is introduced in eq. (\[eq:1020\]) repeated below
$$\label{eq:1066}
\begin{array}{l}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ w_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
+ \Delta
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\vspace*{0.3cm} \\
\hspace*{-1.0cm}
\Delta
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\begin{array}{l}
\ \frac{1}{2}
\left (
\begin{array}{l}
g_{\ \mu_{\ 1} \ \mu_{\ 2}} \ \varrho_{\ \nu_{\ 1} \ \nu_{\ 2}}
- \ g_{\ \nu_{\ 1} \ \mu_{\ 2}} \ \varrho_{\ \mu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.3cm} \\
- \ g_{\ \mu_{\ 1} \ \nu_{\ 2}} \ \varrho_{\ \nu_{\ 1} \ \mu_{\ 2}}
\ + \ g_{\ \nu_{\ 1} \ \nu_{\ 2}} \ \varrho_{\ \mu_{\ 1} \ \mu_{\ 2}}
\end{array}
\right )
\vspace*{0.3cm} \\
+
\ \frac{1}{12}
\ K^{\ +}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ R
\end{array}
\vspace*{0.4cm} \\
\hline
\vspace*{-0.3cm} \\
\mbox{with :}
\hspace*{0.3cm}
g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ w_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ 0
\vspace*{0.5cm} \\
\hspace*{-1.0cm}
g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\ g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ \Delta
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ = \ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.5cm} \\
R \ =
\ g^{\ \nu_{\ 1} \ \nu_{\ 2}}
\ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\ =
\ 12 \ B^{\ (+)}
\end{array}$$
In eq. (\[eq:1066\]) the irreducible positive parity parts introduce the traceless part of the Ricci bilinear, i.e. the classical traceless energy momentum bilinear pertaining to gauge bosons. Thus the notations in eq. (\[eq:1065\]) are extended, as shown in eq. (\[eq:1019\])
$$\label{eq:1067}
\begin{array}{l}
\begin{array}{lll}
w_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
& : &
\mbox{Weyl bilinear}
\vspace*{0.3cm} \\
R_{\ \mu \ \nu}
& : &
\mbox{Ricci bilinear}
\vspace*{0.3cm} \\
R
& : &
\mbox{Riemann scalar bilinear}
\vspace*{0.3cm} \\
\begin{array}{l}
- \ \varrho^{\ \mu \ \nu} \ =
\ - \ R^{\ \mu \ \nu} \ +
\ \frac{1}{4}
\ g^{\ \mu \ \nu} \ R
\vspace*{0.3cm} \\
\equiv
\ \vartheta_{\ cl}^{\ \mu \ \nu}
\end{array}
& : &
\begin{array}{l}
\mbox{energy momentum}
\vspace*{0.0cm} \\
\mbox{bilinear}
\end{array}
\end{array}
\end{array}$$
The structure of the energy momentum bilinear introduced in eq. (\[eq:1019\]) ( and eqs. \[eq:1066\] - \[eq:1067\] ) with respect to chromoelectric and -magnetic field strengths is reproduced in eq. (\[eq:1068\]) below
$$\label{eq:1068}
\begin{array}{l}
- \ \varrho^{\ \mu \ \nu} \ =
\ - \ R^{\ \mu \ \nu} \ +
\ \frac{1}{4}
\ g^{\ \mu \ \nu} \ R \ =
\ \vartheta_{\ cl}^{\ \mu \ \nu} \ =
\vspace*{0.5cm} \\
\hspace*{-1.0cm} =
\ \left (
\begin{array}{cc}
\frac{1}{2}
\ \left (
\begin{array}{l}
\vec{E}^{\ A} \ \vec{E}^{\ D}
\vspace*{0.3cm} \\
\hspace*{0.3cm} + \ \vec{B}^{\ A} \ \vec{B}^{\ D}
\end{array}
\ \right )
& - \ \vec{S}^{\ k \ A \ D}
\vspace*{0.5cm} \\
- \ \vec{S}^{\ i \ A \ D} &
\begin{array}{ll}
- \ \vec{E}^{\ i \ A} \ \vec{E}^{\ k \ D}
- \ \vec{B}^{\ i \ A} \ \vec{B}^{\ k \ D}
\vspace*{0.3cm} \\
\ + \ \frac{1}{2} \ \delta_{\ i k}
\ \left (
\begin{array}{l}
\vec{E}^{\ A} \ \vec{E}^{\ D}
\vspace*{0.3cm} \\
\hspace*{0.3cm} + \ \vec{B}^{\ A} \ \vec{B}^{\ D}
\end{array}
\ \right )
\end{array}
\end{array}
\ \right )
\ U_{\ A \ D}
\end{array}$$
The Weyl bilinear is decomposed into irreducible parts according to eq. (\[eq:1031\]) repeated in eq. (\[eq:1069\]) below
$$\label{eq:1069}
\begin{array}{l}
w_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\ \left (
\begin{array}{r}
\left (
\ P^{\ RR} \ \left ( \ \vec{C} \ \otimes \ \vec{C} \ \right )
\ \right )_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\vspace*{0.3cm} \\
+
\ \left (
\ P^{\ LL} \ \left ( \ \vec{G} \ \otimes \ \vec{G} \ \right )
\ \right )_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\vspace*{0.3cm} \\
+
\ B^{\ (-)}
\ \left (
\ K^{\ -}
\ \right )_{\
\ \left \lbrack
\ \mu_{ 1} \ \nu_{ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{ 2} \ \nu_{ 2}
\ \right \rbrack}
\end{array}
\ \right )
\vspace*{0.4cm} \\
\left (
\ K^{\ -}
\ \right )_{\
\ \left \lbrack
\ \mu_{\ 1} \ \nu_{\ 1}
\ \right \rbrack
\ \left \lbrack
\ \mu_{\ 2} \ \nu_{\ 2}
\ \right \rbrack}
\ =
\ \varepsilon_{\ \mu_{\ 1} \ \mu_{\ 2} \ \nu_{\ 1} \ \nu_{\ 2}}
\end{array}$$
The irreducible parts $\varrho$ , $P^{\ RR}$ and $P^{\ LL}$ representing the spin 2 parts of the energy momentum bilinear ( $\varrho$ ) and the Weyl bilinear ($P^{\ RR}$ , $P^{\ LL}$) are identified with the wave functions of the $II^{\ +}$ spectral series in eq. (\[eq:1038\]) repeated in eq. (\[eq:1070\]) below
$$\label{eq:1070}
\begin{array}{l}
\begin{array}{lll ll}
\varrho^{\ \underline{.}}
& \leftrightarrow &
\ \widetilde{t}_{\ \underline{.}}
\ ( \ \left \lbrace \ \varrho \ \right \rbrace
\ ; \ z \ , \ p \ , \ J^{\ P\ C} \ ; \ . \ )
& \rightarrow &
\widetilde{t}
\ ( \ \left \lbrace \ \varrho \ \right \rbrace \ )
\vspace*{0.3cm} \\
w_{\ \underline{.}}
& \leftrightarrow &
\widetilde{t}_{\ \underline{.}}
\ ( \ \left \lbrace \ w \ \right \rbrace
\ ; \ z \ , \ p \ , \ J^{\ P\ C} \ ; \ . \ )
& \rightarrow &
\widetilde{t}
\ ( \ \left \lbrace \ w \ \right \rbrace \ )
\vspace*{0.3cm} \\
w^{\ RR}_{\ \underline{.}}
& \leftrightarrow &
\widetilde{t}_{\ \underline{.}}
\ ( \ \left \lbrace \ w^{\ RR} \ \right \rbrace
\ ; \ z \ , \ p \ , \ J^{\ P\ C} \ ; \ . \ )
& \rightarrow &
\widetilde{t}
\ ( \ \left \lbrace \ w^{\ RR} \ \right \rbrace \ )
\vspace*{0.3cm} \\
w^{\ LL}_{\ \underline{.}}
& \leftrightarrow &
\widetilde{t}_{\ \underline{.}}
\ ( \ \left \lbrace \ w^{\ LL} \ \right \rbrace
\ ; \ z \ , \ p \ , \ J^{\ P\ C} \ ; \ . \ )
& \rightarrow &
\widetilde{t}
\ ( \ \left \lbrace \ w^{\ LL} \ \right \rbrace \ )
\end{array}
\end{array}$$
The [*new*]{} result, worked out in this appendix (A.5) , shows, that the wave functions pertaining to $P^{\ RR}$ and $P^{\ LL}$ , i.e. to the spin 2 irreducible parts of the Weyl bilinear vanish.
In this sense we identify here the bilinear with its gb wave functions, keeping in mind that the full spin 2 Weyl bilinear operator does [*not*]{} vanish identically. This implies for the wave functions defined in eq. (\[eq:1070\])
$$\label{eq:1071}
\begin{array}{l}
\widetilde{t}
\ ( \ \left \lbrace \ w^{\ RR} \ \right \rbrace \ )
\ =
\ \widetilde{t}
\ ( \ \left \lbrace \ w^{\ LL} \ \right \rbrace \ )
\ = \ 0
\end{array}$$
Eq. (\[eq:1071\]) is the main result of this appendix (A.5) .
With the above identification the full decomposition of the adjoint string bilinear in eq. (\[eq:1066\]) becomes
$$\label{eq:1072}
\begin{array}{l}
\hspace*{-1.2cm}
B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\ \left \lbrace
\begin{array}{c}
\ \frac{1}{2}
\left (
\begin{array}{l}
g_{\ \mu_{\ 1} \ \mu_{\ 2}} \ \varrho_{\ \nu_{\ 1} \ \nu_{\ 2}}
- \ g_{\ \nu_{\ 1} \ \mu_{\ 2}} \ \varrho_{\ \mu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.3cm} \\
- \ g_{\ \mu_{\ 1} \ \nu_{\ 2}} \ \varrho_{\ \nu_{\ 1} \ \mu_{\ 2}}
\ + \ g_{\ \nu_{\ 1} \ \nu_{\ 2}} \ \varrho_{\ \mu_{\ 1} \ \mu_{\ 2}}
\end{array}
\right )
\vspace*{0.3cm} \\
+
\ K^{\ +}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ B^{\ (+)}
\vspace*{0.3cm} \\
+
\ K^{\ -}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ B^{\ (-)}
\vspace*{0.3cm} \\
\end{array}
\right \rbrace
\vspace*{0.4cm} \\
\hline
\vspace*{-0.3cm} \\
\mbox{with :}
\ g^{\ \mu_{\ 1} \ \mu_{\ 2}}
\ B_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.5cm} \\
- \ \varrho^{\ \mu \ \nu} \ =
\ - \ R^{\ \mu \ \nu} \ +
\ \frac{1}{4}
\ g^{\ \mu \ \nu} \ R \ =
\ \vartheta_{\ cl}^{\ \mu \ \nu}
\vspace*{0.5cm} \\
R \ =
\ g^{\ \nu_{\ 1} \ \nu_{\ 2}}
\ R_{\ \nu_{\ 1} \ \nu_{\ 2}}
\ =
\ 12 \ B^{\ (+)}
\end{array}$$
Comparing the form of the adjoint string components in eq. (\[eq:1072\]) with eq. (\[eq:1063\]) we find
$$\label{eq:1073}
\begin{array}{l}
\hspace*{-0.6cm}
B^{\ '}_{\ \left \lbrack \ \mu_{1} \ \nu_{1} \ \right \rbrack
\ , \ \ \left \lbrack \ \mu_{2} \ \nu_{2} \ \right \rbrack}
\ =
\ \frac{1}{2}
\left (
\begin{array}{l}
g_{\ \mu_{\ 1} \ \mu_{\ 2}} \ \varrho_{\ \nu_{\ 1} \ \nu_{\ 2}}
- \ g_{\ \nu_{\ 1} \ \mu_{\ 2}} \ \varrho_{\ \mu_{\ 1} \ \nu_{\ 2}}
\vspace*{0.3cm} \\
- \ g_{\ \mu_{\ 1} \ \nu_{\ 2}} \ \varrho_{\ \nu_{\ 1} \ \mu_{\ 2}}
\ + \ g_{\ \nu_{\ 1} \ \nu_{\ 2}} \ \varrho_{\ \mu_{\ 1} \ \mu_{\ 2}}
\end{array}
\right )
\vspace*{0.3cm} \\
B^{\ '} \ \leftrightarrow \ \left \lbrace \ II^{\ +} \ \right \rbrace
\ \longleftrightarrow \ \vartheta_{\ cl}^{\ \mu \ \nu}
\end{array}$$
ii\) The parallel to abelian gauge fields
The relations of the nonabelian adjoint string variables, contained in eqs. (\[eq:1072\]) and (\[eq:1072\]) , with the three spectral types, denoted $I^{\ +}$ , $I^{\ -}$ and $II^{\ +}$ here, equally apply to the corresponding string variables pertaining to electromagnetic fields [@Land] , [@Yang] . This has not been explicitely done [@HFPM] , since the structure of the two isolated decay photons yields a considerable simplification.
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and references cited therein.
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and references cited therein.
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and references cited therein.
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\ \pi^{\ -} \ \pi^{\ +} \ \pi^{\ +}$ Dalitz plots, Phys.Lett.B407 (1997) 79,
and Phys.Lett.B351 (1995) 591.
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K. Abe et al., Belle Collaboration, Study of charmless B decays to three kaon final states, Contributed to 31st International Conference on High Energy Physics (ICHEP 2002), Amsterdam, The Netherlands, 24-31 Jul 2002, hep-ex/0208030.
B. Aubert et al., BABAR Collaboration, Measurements of the branching fractions of charged B decays to $K^{\ \pm} \ \pi^{\ \mp} \ \pi^{\ \pm}$ final states, hep-ex/0308065,
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P. C. Bloom for the BABAR Collaboration, Measurements of rare B decays at BABAR, eConf C020805 (2002) TTH06, hep-ex/0302030,
B. Aubert et al., BABAR Collaboration, Measurements of the branching fractions of charmless three body charged B decays, hep-ex/0206004.
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F. Close , Light hadron spectroscopy and experiment, Int.J.Mod.Phys.A17 (2002) 3239, hep-ph/0110081,
F. Close and A. Kirk, Scalar glueball $q \ \overline{q}$ mixing above 1 GeV and implications for lattice QCD, Eur.Phys.J.C21 (2001) 531, hep-ph/0103173,
F. Close, Glueballs: a central mystery, Acta Phys.Polon.B31 (2000) 2557, hep-ph/0006288,
F.Close, A. Kirk and G. Schuler, Dynamics of glueball and $q \ \overline{q}$ production in the central region of p p collisions, Phys.Lett.B477 (2000) 13, hep-ph/0001158,
and references cited therein.
R. Jost, The general theory of quantized fields, American Mathematical Society, Providence R.I., 1965,
R. Jost, CTP-Invarianz der Streumatrix und interpolierende Felder, Helv. Phys. Acta 36 (1963) 77,
R. Jost, Einiges über die Lorentzgruppe und das einäugige Sehen (engl. transl. : Some things on the Lorentz group and monocular vision) , Verein Schweizerischer Mathematik- und Physiklehrer, Bulletin Nr. 2, 1966.
for a review see M. Creutz, The early days of lattice gauge theory, hep-lat/0306024,
J. Greensite, The confinement problem in lattice gauge theory, hep-lat/0301023,
M. Luescher, S.Sint, R. Sommer and P. Weisz, Chiral symmetry and O(a) improvement in lattice QCD, Nucl.Phys. B478 (1996) 365 , hep-lat/9605038.
for a review see E. Alvarez, Loops versus strings, gr-qc/0307090,
A. M. Polyakov and V. S. Rychkov, Gauge fields-strings duality and the loop equation, Nucl.Phys.B581 (2000) 116, hep-th/0002106,
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O. Nachtmann, Perturbative and nonperturbative aspects of QCD, H. Latal and W. Schweiger edts. Springer Verlag, Berlin, Heidelberg 1997, hep-ph/9609365.
Spider, The Encyclopedia Americana, Int. edition, Vol. 25, p. 494, Americana Corporation 1976.
Lui Bernard, www.onlinekunst.de/ bernard/04.html .
M. Fierz, Die unitären Darstellungen der homogenen Lorentzgruppe, in Preludes in theoretical physics, in honor of V. F. Weisskopf, A. De-Shalit, H. Feshbach and L. van Hove ets., North Holland, Amsterdam 1966, p.1.
[^1]: This material served as base for the contributions of the author to the 9th Adriatic Meeting on Particle Physics and the Universe, Dubrovnik, Croatia, 4 - 14 September 2003, and to the 310th Heraeus Seminar, Quarks in Hadrons and Nuclei II, September 15 - 20, 2003, Rothenfels Castle, Oberwoelz, Austria.
[^2]: Nachtmann [@Nachtmann] compares the repetitious return of the weaving pattern to the base point $x$ with the way a spider weaves its [*fan-type anchoring part*]{} of the net.
[^3]: This also is the spiders path, in the second stage : the scaffolding spiral [@encyc] , [@spiderfot] .
[^4]: The helicity basis components ${\cal{B}}$ shall not be confused with the SL2C matrix ${\cal{B}}$ defined in appendix A.1 .
[^5]: The auxiliary vector $\vec{w}$ introduced here shall not be confused with the Weyl bilinears $w_{\ \underline{.}}^{\ RR}$ and $w_{\ \underline{.}}^{\ RR}$ in eqs. (\[eq:1031\]) and (\[eq:1033\]) .
| 2024-07-24T01:26:58.646260 | https://example.com/article/4248 |
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| 2023-09-03T01:26:58.646260 | https://example.com/article/5351 |
{
"homepage": "https://www.xmind.net/",
"description": "XMind is the most professional and popular mind mapping tool.",
"version": "3.7.8",
"license": {
"identifier": "Shareware",
"url": "https://www.xmind.net/terms/"
},
"url": "https://www.xmind.net/xmind/downloads/xmind-8-update8-windows.zip",
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"extract_dir": "XMind 8 Update 8",
"persist": "workspace",
"shortcuts": [
[
"XMind.exe",
"XMind"
]
]
}
| 2024-02-12T01:26:58.646260 | https://example.com/article/1446 |
He added: "2 were charged (by others) & punished. The other has since died."
Now 40, he also referred to Weinstein's alleged actions and said: "I understand the unwarranted shame, powerlessness and inability to blow the whistle. There's a power dynamic that feels impossible to overcome." | 2023-08-13T01:26:58.646260 | https://example.com/article/6366 |
Captive Heart: Hearts In Peril (Episode 22)
Difficult? Had Helen actually imagined preparing for the arrival of Bukola’s family would be only difficult? She had been wrong. It was a nightmare.
For some reason, Bukola had decided it was time to step up and run the house, preparing for her role as madam-to-be. Naturally, she had thrown Helen’s precise organization of that chore into total chaos. The house staff was confused and annoyed at the conflicting instructions Bukola kept giving them and Helen’s job became a lot harder. It had not been easy to keep everything running efficiently, with Bukola underfoot. The woman couldn’t cook and had no idea what detergent was, but she thought she could tell everyone what to do.
It had really been a nightmare.
Helen sighed and rubbed the back of her neck, happy to be out of the house, and in the front passenger seat of Lionel’s jeep, heading to the airport. She had volunteered to help pick up Bukola’s family when they arrived at the airport. In truth, she had just been looking for a reason to get away from Bukola before she choked her out of frustration.
“I’ve never seen anyone dive into a car that fast.” Joe-B, Lionel’s new bodyguard said with a loud laugh. He had offered to drive Helen to the airport, and she was a bit comforted by his hulking presence. It would be additional security for the Deindes as well. “What was chasing you?”
Helen grimaced. Joe-B was referring to how eagerly she had left the Thomas mansion. “I had to get out of that house, abeg. I left so fast, I forgot my phone. I’ll have to commandeer yours, Joe-B.”
“The phone is yours to command, pretty lady. Now I understand, still it’s not like waiting in the airport for a flight to arrive is fun, so I don’t see how this is a better alternative.” Joe-B quipped. He smoothly rounded a corner, his black blazer straining at the seams as his big hands turned the steering wheel. “Or were you just interested in spending time with me?”
Helen chuckled and hit his muscular arm playfully. The bodyguard was always flirting with her, but she knew he was just being playful.
“You couldn’t handle me, Joe-B.” she teased. “I know you could probably lift me with one finger, but e no mean o! I’m small, but mighty.”
The bodyguard shook his head in mock-sorrow. “You’re such a violent woman, Helen. I wasn’t asking for a wrestling match abeg. Women have very sharp teeth, they love to bite…”
Helen’s laugh was cut short by the rest of his comment.
“…and besides I’m not your type.”
She stared at him curiously. “My type? How would you know my type, Joe-B?”
The man grinned, eyes watching the rear-view mirror and the busy road as he drove carefully. “I’ve seen the way you look at Mr Thomas, my friend. I may not have sharp teeth, but I have sharp eyes. Part of my job requirement.”
Helen forced out a strangled laugh, surprised.
Joe-B noticed her expression and laughed. “Don’t worry your small head there. Your secret is safe with me. I keep secrets very well; also another job requirement. If you know how many big men i have helped to sneak out so they can see their girlfriends eh…”
Helen let Joe-B chatter on, thinking. Was her fondness for her boss so evident? She would have to be more discreet. And what Joe-B said about keeping secrets struck her as true. Even though Bukola had told her to say and do nothing about Bernard’s and Kelechi Dike’s kidnap plot, she suddenly felt an urge to tell Joe-B everything. Surely, she could convince him to keep it from Lionel but it would help him be more alert to any danger to the man he was hired to protect.
Helen bit her lip. It was the smart thing to do.
She turned to the bodyguard, “Joe-B, I have to tell you something…”
“Not the time, dear.” Joe-B said, his eyes flicking to the rear-view mirror again. “I don’t mean to alarm you, but I think you should hunch down in your seat a bit. Two jeeps have been following us for a while now.”
Helen’s heart jumped in her chest, and she twisted around, looking through the back windshield. “What? Where..?”
“Please, Helen.” Joe-B cut in firmly, using his free hand to turn her around. “Just do as I said. It may be nothing, and its daytime, so nothing may happen. But just to be on the safe side…”
She hunched down in her seat. “Oh God! Oh God! Maybe they think Lionel is in here. Maybe they have finally come to kidnap him.”
Joe-B shot her a look of concern. “Kidnap?”
“Yes! That’s what I wanted to tell you. I overheard plans to kidnap Lionel… Mr Thomas. This must be the kidnappers following us.”
“Damn. Then daylight won’t matter.” Joe-B threw another quick glance in the rear-view mirror, and then pressed down on the accelerator. The vehicle sped up smoothly, its purr increasing. “It would have been really nice to know about this before now, Helen.” He ground out between clenched jaws.
Helen felt fear fill her. He was right. She should have told him before now. She risked peeping over the edge of her seat. She could see two jeeps that had picked up speed and were getting closer to them.
She ducked back into her seat, shivering with fear, and looked at Joe-B. He was concentrating on driving very fast and carefully, but she could see the concern in his eyes. | 2024-07-25T01:26:58.646260 | https://example.com/article/6327 |
Impactoftheinternet.Com : Nobody Does It Like Patio
Sunbrella Hammock
By Quentin Bousquet. Hammocks. Published at Friday, November 03rd, 2017 - 09:52:00 AM.
One more thing you should know is that rope and fabric hammocks can have spreader bars at the ends or be stave-less (traditional hammock without bars). You might think that spreader bars add comfort, but in fact it is the other way around - hammocks without the bars are more comfortable, because they wrap around your body better. If for appearance reasons you prefer hammocks with spreader bars, go for hammocks with 3 hanging points instead of just 2 - these have much less chance of tipping over.
Mayan hammocks are extremely comfortable and provide good ventilation, but don't leave any prints on your skin. They are also very light-weight, so string hammocks can be used for camping or taken to the beach. There is one disadvantage though - it is easy to damage a string hammock. Any sharp object can easily cut the string, so they are not as durable as rope or fabric hammocks.
Attractiveness. Within a few years, most cotton hammocks loose much of their original beauty due to inevitable, continual exposure to water and humidity. Also, typically, their colors fade in the sun.
Texsport Wilderness Hammock with Mosquito Netting: This is a multipurpose hammock that can convert into a tent very easily. The hammock bottom is made of cotton canvas. The hammock has mesh to protect yourself from mosquitoes and other insects and bugs. With this hammock you can rest assured of your safety and comfort. The hammock is quite spacious and measures 82" x 28" x 18". The capacity of the hammock is 250 pounds. It comes with a nylon storage bag and weighs around 3 pounds. The price of this hammock / tent is only $35. | 2024-04-06T01:26:58.646260 | https://example.com/article/7011 |
Gender and family composition related to discharge destination and length of hospital stay after acute stroke.
Informal care by family members still plays an important role in home care after acute stroke. This study determined the clinical and demographic factors, such as family structure, that predict discharge to home and length of hospital stay (LOS) after acute stroke hospitalization. We reviewed the sex, age, family structure before stroke, type of stroke, size of the lesion, activities of daily living (ADL) function at discharge, discharge destination, and LOS of stroke patients (114 cerebral infarctions and 44 intracerebral hemorrhages) admitted to a neurosurgical hospital. Patients with cerebral infarction were older than those with intracerebral hemorrhage (median 75 vs 66 years). Ninety-eight were discharged to home (62%). In the logistic regression analysis, low ADL function, medium or large infarction, and intracerebral hemorrhage (vs lacunar infarction) were significantly associated with discharge to a destination other than home. Of the patients discharged home, low ADL function was strongly associated with LOS in the multiple regression analysis. In addition, living with a spouse only had the opposite effect on LOS in men and women (p=0.050 and 0.071, respectively). LOS tended to be shorter for men with a wife, but longer for women with a husband. The structure and gender roles in a stroke patient's household may need further attention for the efficient use of hospital resources. | 2024-06-27T01:26:58.646260 | https://example.com/article/3616 |
Swatara Creek
Swatara Creek (nicknamed the Swattie) is a tributary of the Susquehanna River in east central Pennsylvania in the United States. It rises in the Appalachian Mountains in central Schuylkill County and passes through northwest Lebanon Valley before draining into the Susquehanna at Middletown.
The name Swatara is said to derive from a Susquehannock word, Swahadowry or Schaha-dawa, which means 'where we feed on eels'.
Drinking Water
Further, three water companies — Suez Water Pennsylvania, American Water, and the Lebanon Water Authority draw drinking water for hundreds of thousands of residents of the Swatara Watershed.
Geography
Swatara Creek rises in the Appalachian Mountains in central Schuylkill County, north of the Sharp Mountain ridge, approximately west of Minersville. It then flows southwest in a winding course, passing south of Tremont, then cutting south through Second Mountain ridge. It passes through Swatara State Park then turns south to pass through Swatara Gap in the Blue Mountain ridge northwest of Lebanon. After emerging from the ridge it flows southwest, north of Hershey, past Hummelstown, and joins the Susquehanna at Middletown. It receives Quittapahilla Creek from the east north of Palmyra.
History
The creek was a significant transportation route in the colonial period of North America up through the late 19th century. A canal linking the Susquehanna and Delaware valleys in southeastern Pennsylvania was first proposed in 1690 by William Penn, the founder of the Pennsylvania Colony. Nearly a century passed before a route for the canal was surveyed by David Rittenhouse and William Smith between 1762 and 1770, the first canal ever surveyed in the U.S. Spurred by the 1791 discovery of anthracite in the upper Susquehanna Valley, the Pennsylvania General Assembly chartered two companies to undertake the project: the Schuylkill and Susquehanna Navigation Company and the Delaware and Schuylkill Navigation Company. At the time of the initial construction in 1792, Philadelphia was involved in an intense rivalry with Baltimore for the supremacy as a shipping port. The canal was backed by Philadelphia businessmen as a means to divert commercial traffic from following the Susquehanna downriver to the Chesapeake Bay, its more natural destination. Although the Schuylkill and Susquehanna Navigation Company project failed for lack of funds, the project was restarted and ultimately completed by its successor company, the Union canal in 1828.
From west to east, the route of the Schuylkill and Susquehanna Navigation Company canal in 1792 was to follow Swatara Creek upstream from Middletown to Quittapahilla Creek, which it then followed upstream past Lebanon and Myerstown to its headwaters. It then crossed overland to Clarks Run at the headwaters of Tulpehocken Creek, following Tulpehocken Creek downstream to Reading on the Schuylkill River. It was to follow the Schuylkill downriver to the Delaware River at Philadelphia. The route of the Union Canal followed the same route up the Swatara Creek and continue up the creek to Union Water Works. The canal then went up Clark's run to the summit and thence by a 729-foot tunnel over to Lebanon. The upper course above Union Water Works into the mountains provided the route of a feeder to the main canal, as well as providing a route to ship anthracite from the mountains to Philadelphia.
On September 8, 2011, The creek reached a record height of 26.8 feet near Hershey, following devastating rains from Tropical Storm Lee and remnants of Hurricane Irene, the highest since measurements began in 1975. Farther upstream at the Harpers Tavern gauge, 24.60' was recorded, making it the worst flooding since 1889. The flooding caused thousands of people to be evacuated from their homes throughout Central Pennsylvania, and at least one death.
Several covered bridges once crossed the Swatara Creek, including the Fiddler's Elbow Covered Bridge, near Hummelstown and Clifton Covered Bridge near Middletown, both destroyed by Hurricane Agnes in 1972. The Sand Beach Covered Bridge burned down in 1966 by arson.
Recreation
Today, the Swatara Creek is part of a national and statewide water trail system, providing outdoor recreation and a wellness activity for families canoeing and kayaking a 60-mile segment that connects to the Susquehanna River and Captain John Smith Water Trails.
Tributaries
Iron Run
Beaver Creek (Swatara Creek)
Spring Creek (Swatara Creek)
Manada Creek
Quittapahilla Creek
Brandywine Creek (Quittapahilla Creek)
Bow Creek (Swatara Creek)
Indiantown Run
Little Swatara Creek
Lower Little Swatara Creek
Upper Little Swatara Creek
Good Spring Creek
Ships
Two ships in the United States Navy have been named USS Swatara after the creek:
The first USS Swatara (1865) was a wooden, screw sloop, launched in 1865 and dismantled in 1872 to become the second ship of this name.
The second USS Swatara (1873) was a screw sloop, launched in 1873 and decommissioned in 1891.
See also
List of rivers of Pennsylvania
References
External links
Manada Conservancy - Swatara Greenway Stewardship Program
U.S. Geological Survey: PA stream gaging stations
Swatara Creek Water Trail
Canal Museum: Union Canal
Swatara Creek Greenway
Northern Swatara Creek Watershed Association
Fred Yenerall Collection - Includes Fiddler's Elbow, Clifton and Sand Beach Covered Bridge
Category:Rivers of Pennsylvania
Category:Tributaries of Swatara Creek
Category:Rivers of Dauphin County, Pennsylvania
Category:Rivers of Schuylkill County, Pennsylvania
Category:Rivers of Lebanon County, Pennsylvania | 2024-04-27T01:26:58.646260 | https://example.com/article/9552 |
package de.robv.android.xposed;
import android.os.Environment;
import java.io.ByteArrayOutputStream;
import java.io.File;
import java.io.FileInputStream;
import java.io.FileOutputStream;
import java.io.IOException;
import java.io.OutputStream;
import java.security.DigestException;
import java.security.MessageDigest;
import java.security.NoSuchAlgorithmException;
import java.util.zip.Adler32;
import static de.robv.android.xposed.XposedHelpers.inputStreamToByteArray;
/**
* Helper class which can create a very simple .dex file, containing only a class definition
* with a super class (no methods, fields, ...).
*/
/*package*/ class DexCreator {
public static File DALVIK_CACHE = new File(Environment.getDataDirectory(), "dalvik-cache");
/** Returns the default dex file name for the class. */
public static File getDefaultFile(String childClz) {
return new File(DALVIK_CACHE, "xposed_" + childClz.substring(childClz.lastIndexOf('.') + 1) + ".dex");
}
/**
* Creates (or returns) the path to a dex file which defines the superclass of {@clz} as extending
* {@code realSuperClz}, which by itself must extend {@code topClz}.
*/
public static File ensure(String clz, Class<?> realSuperClz, Class<?> topClz) throws IOException {
if (!topClz.isAssignableFrom(realSuperClz)) {
throw new ClassCastException("Cannot initialize " + clz + " because " + realSuperClz + " does not extend " + topClz);
}
try {
return ensure("xposed.dummy." + clz + "SuperClass", realSuperClz);
} catch (IOException e) {
throw new IOException("Failed to create a superclass for " + clz, e);
}
}
/** Like {@link #ensure(File, String, String)}, just for the default dex file name. */
public static File ensure(String childClz, Class<?> superClz) throws IOException {
return ensure(getDefaultFile(childClz), childClz, superClz.getName());
}
/**
* Makes sure that the given file is a simple dex file containing the given classes.
* Creates the file if that's not the case.
*/
public static File ensure(File file, String childClz, String superClz) throws IOException {
// First check if a valid file exists.
try {
byte[] dex = inputStreamToByteArray(new FileInputStream(file));
if (matches(dex, childClz, superClz)) {
return file;
} else {
file.delete();
}
} catch (IOException e) {
file.delete();
}
// If not, create a new dex file.
byte[] dex = create(childClz, superClz);
FileOutputStream fos = new FileOutputStream(file);
fos.write(dex);
fos.close();
return file;
}
/**
* Checks whether the Dex file fits to the class names.
* Assumes that the file has been created with this class.
*/
public static boolean matches(byte[] dex, String childClz, String superClz) throws IOException {
boolean childFirst = childClz.compareTo(superClz) < 0;
byte[] childBytes = stringToBytes("L" + childClz.replace('.', '/') + ";");
byte[] superBytes = stringToBytes("L" + superClz.replace('.', '/') + ";");
int pos = 0xa0;
if (pos + childBytes.length + superBytes.length >= dex.length) {
return false;
}
for (byte b : childFirst ? childBytes : superBytes) {
if (dex[pos++] != b) {
return false;
}
}
for (byte b : childFirst ? superBytes: childBytes) {
if (dex[pos++] != b) {
return false;
}
}
return true;
}
/** Creates the byte array for the dex file. */
public static byte[] create(String childClz, String superClz) throws IOException {
boolean childFirst = childClz.compareTo(superClz) < 0;
byte[] childBytes = stringToBytes("L" + childClz.replace('.', '/') + ";");
byte[] superBytes = stringToBytes("L" + superClz.replace('.', '/') + ";");
int stringsSize = childBytes.length + superBytes.length;
int padding = -stringsSize & 3;
stringsSize += padding;
ByteArrayOutputStream out = new ByteArrayOutputStream();
// header
out.write("dex\n035\0".getBytes()); // magic
out.write(new byte[24]); // placeholder for checksum and signature
writeInt(out, 0xfc + stringsSize); // file size
writeInt(out, 0x70); // header size
writeInt(out, 0x12345678); // endian constant
writeInt(out, 0); // link size
writeInt(out, 0); // link offset
writeInt(out, 0xa4 + stringsSize); // map offset
writeInt(out, 2); // strings count
writeInt(out, 0x70); // strings offset
writeInt(out, 2); // types count
writeInt(out, 0x78); // types offset
writeInt(out, 0); // prototypes count
writeInt(out, 0); // prototypes offset
writeInt(out, 0); // fields count
writeInt(out, 0); // fields offset
writeInt(out, 0); // methods count
writeInt(out, 0); // methods offset
writeInt(out, 1); // classes count
writeInt(out, 0x80); // classes offset
writeInt(out, 0x5c + stringsSize); // data size
writeInt(out, 0xa0); // data offset
// string map
writeInt(out, 0xa0);
writeInt(out, 0xa0 + (childFirst ? childBytes.length : superBytes.length));
// types
writeInt(out, 0); // first type = first string
writeInt(out, 1); // second type = second string
// class definitions
writeInt(out, childFirst ? 0 : 1); // class to define = child type
writeInt(out, 1); // access flags = public
writeInt(out, childFirst ? 1 : 0); // super class = super type
writeInt(out, 0); // no interface
writeInt(out, -1); // no source file
writeInt(out, 0); // no annotations
writeInt(out, 0); // no class data
writeInt(out, 0); // no static values
// string data
out.write(childFirst ? childBytes : superBytes);
out.write(childFirst ? superBytes : childBytes);
out.write(new byte[padding]);
// annotations
writeInt(out, 0); // no items
// map
writeInt(out, 7); // items count
writeMapItem(out, 0, 1, 0); // header
writeMapItem(out, 1, 2, 0x70); // strings
writeMapItem(out, 2, 2, 0x78); // types
writeMapItem(out, 6, 1, 0x80); // classes
writeMapItem(out, 0x2002, 2, 0xa0); // string data
writeMapItem(out, 0x1003, 1, 0xa0 + stringsSize); // annotations
writeMapItem(out, 0x1000, 1, 0xa4 + stringsSize); // map list
byte[] buf = out.toByteArray();
updateSignature(buf);
updateChecksum(buf);
return buf;
}
private static void updateSignature(byte[] dex) {
// Update SHA-1 signature
try {
MessageDigest md = MessageDigest.getInstance("SHA-1");
md.update(dex, 32, dex.length - 32);
md.digest(dex, 12, 20);
} catch (NoSuchAlgorithmException | DigestException e) {
throw new RuntimeException(e);
}
}
private static void updateChecksum(byte[] dex) {
// Update Adler32 checksum
Adler32 a32 = new Adler32();
a32.update(dex, 12, dex.length - 12);
int chksum = (int) a32.getValue();
dex[8] = (byte) (chksum & 0xff);
dex[9] = (byte) (chksum >> 8 & 0xff);
dex[10] = (byte) (chksum >> 16 & 0xff);
dex[11] = (byte) (chksum >> 24 & 0xff);
}
private static void writeUleb128(OutputStream out, int value) throws IOException {
while (value > 0x7f) {
out.write((value & 0x7f) | 0x80);
value >>>= 7;
}
out.write(value);
}
private static void writeInt(OutputStream out, int value) throws IOException {
out.write(value);
out.write(value >> 8);
out.write(value >> 16);
out.write(value >> 24);
}
private static void writeMapItem(OutputStream out, int type, int count, int offset) throws IOException {
writeInt(out, type);
writeInt(out, count);
writeInt(out, offset);
}
private static byte[] stringToBytes(String s) throws IOException {
ByteArrayOutputStream bytes = new ByteArrayOutputStream();
writeUleb128(bytes, s.length());
// This isn't MUTF-8, but should be OK.
bytes.write(s.getBytes("UTF-8"));
bytes.write(0);
return bytes.toByteArray();
}
private DexCreator() {}
}
| 2024-07-31T01:26:58.646260 | https://example.com/article/4237 |
/*
Copyright (C) 1997,1998,1999,2000,2001 Franz Josef Och
mkcls - a program for making word classes .
This program is free software; you can redistribute it and/or
modify it under the terms of the GNU General Public License
as published by the Free Software Foundation; either version 2
of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307,
USA.
*/
#ifndef MY_ASSERT_DEFINED
#define MY_ASSERT_DEFINED
void myerror(int line,const char *file,const char *expression);
void imyerror(int line,const char *file,const char *expression);
#define iassert(expression) do {if (!(expression)) {imyerror(__LINE__,__FILE__,#expression);}} while (0)
#define massert(expr) do {} while(0)
#define vassert(expr) do {} while(0)
#include <assert.h>
#endif
| 2024-04-10T01:26:58.646260 | https://example.com/article/2573 |
When Ademola Lookman joined RB Leipzig from Everton for £22.5m last week, the transfer fees spent on four former Charlton youth teammates hit £50m. Aston Villa signed England Under-21 defender Ezri Konsa from Brentford for £12m in July, following Karlan Grant’s move to Huddersfield in January, Lookman’s £11m move to Everton in 2017, and Liverpool’s £3.5m deal for Joe Gomez in 2015. How does a club that has spent the last three seasons in the third tier produce so much talent – including four youngsters who all played together in their Under-18s five years ago?
I had the privilege of working with Charlton’s academy for seven years. In that time, they signed 72 apprentices (known in football as scholars). Twenty-nine of them (40%) are still playing professionally in the Premier League, Football League or top European leagues. Given that less than 15% of scholars become professionals, that is quite an achievement.
After earning promotion from League One through the play-offs, Lee Bowyer’s team will cross paths with several academy graduates in the Championship this season: Sheffield Wednesday defender Morgan Fox, West Brom’s goalscoring centre-back Semi Ajayi and new Stoke midfielder Jordan Cousins are among six current professionals from the crop of 11 scholars who completed their apprenticeship at the club in 2012.
Despite having their belts tightened considerably under unpopular chairman Roland Duchâtelet – the redevelopment of their training ground in New Eltham was brought to a halt when they were relegated to the third tier, leaving them with two new 4G pitches alongside rapidly deteriorating facilities – Charlton have continued to fund a Category 2 academy.
They have to fish alongside five Category 1 academies in London (Arsenal, Chelsea, Fulham, Tottenham and West Ham) as well as local Category 2 rivals Millwall and Crystal Palace, yet still bring through players who make the grade. There are a few reasons for their success. While Duchâtelet has employed 11 first-team managers in five years, the club’s academy has remained relatively stable.
Academy manager Steve Avory, a former teacher and England Schoolboys manager, has been full-time at the club for 18 years; and his avuncular colleague Joe Francis has been educating Charlton’s young players for even longer. Francis – in his own words “a failed footballer” having been released by Charlton at 18 – provides the daily enthusiasm, paternal guidance, realism, support, encouragement and, vitally, discipline, that teenage players need to make the most of their ability.
Francis and Avory have developed a culture in which young players are encouraged to pursue other interests. Francis is as proud of Alex Willis – who is making waves on a scholarship at Northern Kentucky University – as he is Lookman, Grant, Gomez, Konsa, or George Lapslie, the fifth current professional from that youth team, who is now playing in Charlton’s midfield and completing a coaching degree at Anglia Ruskin University.
The Under-18 and Under-23 players attend two 90-minute personal development workshops every week. “A young player’s journey to becoming a professional is always a rollercoaster,” says Francis. “There are peaks and troughs. They are stretched and tested in their training but we want to do that outside of the white lines. We felt we can build in some kind of personal growth workshops that will benefit their performances both in training and their day-to-day lives.”
Ademola Lookman leaves Everton with potential unfulfilled there | Andy Hunter Read more
These players come from all sorts of backgrounds. Lapslie grew up in Brentwood and joined the club as a nine-year-old; Grant was brought up by his Scottish mother a few minutes’ walk from The Valley; Konsa, who is fluent in the French and Portuguese spoken by his family, is from Silvertown; and Gomez grew up in Catford, the son of a Gambian father and an English mother who taught art in London’s prisons. Lookman, a quiet wizard from Camberwell, was an outlier: the academic son of a Nigerian solicitor, he impressed Avory so much while playing for his school that he was offered a scholarship almost immediately. He rose through the ranks at Charlton quickly, making his debut at 18 and being snapped up by Everton a year later for £11m.
Charlton are patient. They give players with long-term injuries years rather than months to recover and they give chances to players who are unwanted at smaller clubs. Toby Stevenson and Reeco Fairchild-Hackett were released by Conference clubs Leyton Orient and Dagenham & Redbridge, respectively, when they were 18. They were given trials at Charlton, earned contracts and worked their way into the first-team.
The Addicks are proving there are diamonds to be found everywhere if you look closely and treat them kindly.
Fantasy football
Pre-season is an annual reminder of how reserve football used to be. Star names, youth players and forgotten senior pros are all teammates for one strange month. Arsenal’s tour of the US is a classic example. Unai Emery gave plenty of minutes to their young attacking talents Bukayo Saka and Eddie Nketiah, and also took Tyreece John-Jules, Robbie Burton, James Olayinka, Dominic Thompson and keeper Matt Macey. Saka, in particular, did his prospects no harm whatsoever, by all accounts.
Facebook Twitter Pinterest Bukayo Saka, Eddie Nketiah, Tyreece John-Jules and James Olayinka pose before Arsenal’s game against Real Madrid in the International Champions Cup. Photograph: David Price/Arsenal FC via Getty Images
Remember me?
Danny Drinkwater may have played his last game for Chelsea. The 29-year-old, who joined the club for £35m in 2017, played the first half at Reading last Sunday. It was Drinkwater’s fourth appearance in pre-season but Frank Lampard has not given him more than 45 minutes and is likely to sell the midfielder.
It’s is incredible to think that Drinkwater, a Premier League-winner and owner of three England caps, has not played for Chelsea since coming on as a substitute in the Community Shield last summer and has only made 12 league appearances for the club.
Facebook Twitter Pinterest The rare sight of Danny Drinkwater in a Chelsea shirt against Reading in pre-season. Photograph: Joe Toth/BPI/Rex/Shutterstock
Next man up
Sometimes a player just needs the right stage. Preston fans frustrated by the deliberating displays of the talented but lackadaisical on-loan forward Lukas Nmecha last season will have been amazed to see him leading Manchester City’s frontline in pre-season.
Manchester United boss Ole Gunnar Solskjær has given roles of responsibility to Angel Gomes and Mason Greenwood in Asia, while Tottenham took several Under-23 regulars to the US, with Cyprus midfielder Anthony Georgiou and 20-year-old centre-back Japhet Tanganga both given game time. Keep an eye out for striker Troy Parrott: the Dubliner is just 17 but Mauricio Pochettino trusted him to start against Juventus.
Facebook Twitter Pinterest Troy Parrott in action for Tottenham against Juventus in the International Champions Cup. Photograph: Pakawich Damrongkiattisak/Getty Images
This week in 2003
With most Premier League clubs now heading to Asia or the US for lucrative pre-season tours, it seems rather quaint that Swedish club Bodens BK used to host a top-flight club near the Arctic circle each July. Lakeside campers would be greeted by Premier League stars on the wide open Bjorknasvallen stadium next door.
“Tottenham Hotspurs” (as they were billed on the posters), Charlton, Southampton, Aston Villa and Leeds all visited. Tourists flocked in from the campsite, with crowds hitting 5,000. Once a top-flight club, Bodens now attract just a few hundred fans as they struggle to stay in the third tier at their smart new stadium on the edge of the small Baltic town, where it never goes completely dark in summer.
Follow Gavin Willacy and Playing in the Shadows on Twitter | 2024-07-02T01:26:58.646260 | https://example.com/article/6338 |
PORTLAND, Ore., Sept. 27, 2011 /PRNewswire-USNewswire/ -- Earth Advantage Institute, a nonprofit green building resource that has certified more than 11,000 homes nationally, invites Expressions of Interest (EOI) to participate in an innovative Eco Neighborhoods pilot program that will rate and certify the livability and accomplishments of existing neighborhoods. The request for EOIs is intended to determine whether there is sufficient market interest in such a certification program. EOIs will be accepted until November 30, 2011.
Eco Neighborhoods ConceptNeighborhoods are critical elements of community vitality and prosperity, and essential contributors to social, economic, and environmental well-being. A wide variety of neighborhoods across the country are working to create exceptionally livable residential and commercial districts, including:
Certifying the accomplishments of these existing neighborhoods can strengthen local participation, reinforce values, and improve outcomes. To achieve these results, the Eco Neighborhoods concept is distinguished in two respects. First, it will focus on existing, fully developed neighborhoods that are at least five years old and have a demonstrated record of accomplishments. Second, the program's rating and certification will be grounded in the principles of sustainability, but, importantly, will "go beyond green" to encompass a broader set of social, economic, and cultural accomplishments.
For example, Eco Neighborhoods may have deployed some of the following measures:
For purposes of this request, "existing neighborhoods" are buildings or groups of buildings that have been constructed and occupied for at least five years, including, but not limited to, residential areas, condominiums, public housing projects, business districts, office/industrial parks, shopping centers, resorts, institutional campuses, and military housing areas. There is no minimum or maximum number of buildings or acres required for an existing neighborhood.
Who May be InterestedEarth Advantage encourages a broad range of organizations to submit EOIs. Potential responders may include, but are not limited to, neighborhood and homeowner associations; public housing tenant associations; business improvement districts and transportation management organizations; community development corporations; and owners/managers of resorts, shopping centers, office/industrial parks, institutional campuses, and military installation housing areas.
Earth Advantage InstituteEarth Advantage Institute is a national nonprofit organization that works with the building industry to implement sustainable building practices. Its mission is to advance green building science and create an immediate, practical and cost-effective path to sustainability and reduction of carbon in the built environment. The organization achieves its objectives through an innovative range of programs for certification of high performance homes, remodels, sustainable communities, and commercial spaces. More information is available at www.earthadvantage.org. | 2024-05-29T01:26:58.646260 | https://example.com/article/4078 |
local smb = require "smb"
local vulns = require "vulns"
local stdnse = require "stdnse"
local string = require "string"
description = [[
Checks if the target machine is running the Double Pulsar SMB backdoor.
Based on the python detection script by Luke Jennings of Countercept.
https://github.com/countercept/doublepulsar-detection-script
]]
---
-- @usage nmap -p 445 <target> --script=smb-double-pulsar-backdoor
--
-- @see smb-vuln-ms17-010.nse
--
-- @output
-- | smb-double-pulsar-backdoor:
-- | VULNERABLE:
-- | Double Pulsar SMB Backdoor
-- | State: VULNERABLE
-- | Risk factor: HIGH CVSSv2: 10.0 (HIGH) (AV:N/AC:L/Au:N/C:C/I:C/A:C)
-- | The Double Pulsar SMB backdoor was detected running on the remote machine.
-- |
-- | Disclosure date: 2017-04-14
-- | References:
-- | https://isc.sans.edu/forums/diary/Detecting+SMB+Covert+Channel+Double+Pulsar/22312/
-- | https://github.com/countercept/doublepulsar-detection-script
-- |_ https://steemit.com/shadowbrokers/@theshadowbrokers/lost-in-translation
author = "Andrew Orr"
license = "Same as Nmap--See https://nmap.org/book/man-legal.html"
categories = {"vuln", "safe", "malware"}
hostrule = function(host)
return smb.get_port(host) ~= nil
end
-- stolen from smb.lua as timeout needs to be modified to get a response
local function send_transaction2(smbstate, sub_command, function_parameters, function_data, overrides)
overrides = overrides or {}
local header1, header2, header3, header4, command, status, flags, flags2, pid_high, signature, unused, pid, mid
local header, parameters, data
local parameter_offset = 0
local parameter_size = 0
local data_offset = 0
local data_size = 0
local total_word_count, total_data_count, reserved1, parameter_count, parameter_displacement, data_count, data_displacement, setup_count, reserved2
local response = {}
-- Header is 0x20 bytes long (not counting NetBIOS header).
header = smb.smb_encode_header(smbstate, 0x32, overrides) -- 0x32 = SMB_COM_TRANSACTION2
if(function_parameters) then
parameter_offset = 0x44
parameter_size = #function_parameters
data_offset = #function_parameters + 33 + 32
end
-- Parameters are 0x20 bytes long.
parameters = string.pack("<I2 I2 I2 I2 B B I2 I4 I2 I2 I2 I2 I2 B B I2",
parameter_size, -- Total parameter count.
data_size, -- Total data count.
0x000a, -- Max parameter count.
0x3984, -- Max data count.
0x00, -- Max setup count.
0x00, -- Reserved.
0x0000, -- Flags (0x0000 = 2-way transaction, don't disconnect TIDs).
10803622, -- Timeout
0x0000, -- Reserved.
parameter_size, -- Parameter bytes.
parameter_offset, -- Parameter offset.
data_size, -- Data bytes.
data_offset, -- Data offset.
0x01, -- Setup Count
0x00, -- Reserved
sub_command -- Sub command
)
local data = "\0\0\0" .. (function_parameters or '')
.. (function_data or '')
-- Send the transaction request
stdnse.debug2("SMB: Sending SMB_COM_TRANSACTION2")
local result, err = smb.smb_send(smbstate, header, parameters, data, overrides)
if(result == false) then
return false, err
end
return true
end
action = function(host,port)
local double_pulsar = {
title = "Double Pulsar SMB Backdoor",
-- IDS = {CVE = 'CVE-2010-2550'},
risk_factor = "HIGH",
scores = {
CVSSv2 = "10.0 (HIGH) (AV:N/AC:L/Au:N/C:C/I:C/A:C)",
},
description = [[
The Double Pulsar SMB backdoor was detected running on the remote machine.
]],
references = {
'https://github.com/countercept/doublepulsar-detection-script',
'https://isc.sans.edu/forums/diary/Detecting+SMB+Covert+Channel+Double+Pulsar/22312/',
'https://steemit.com/shadowbrokers/@theshadowbrokers/lost-in-translation'
},
dates = {
disclosure = {year = '2017', month = '04', day = '14'},
},
exploit_results = {},
}
local report = vulns.Report:new(SCRIPT_NAME, host, port)
double_pulsar.state = vulns.STATE.NOT_VULN
local share = "IPC$"
local status, smbstate = smb.start_ex(host, true, true, share, nil, nil, nil)
if not status then
stdnse.debug1("Could not connect to IPC$ share over SMB.")
else
-- the multiplex ID needs to be 65
smbstate["mid"] = 65;
-- 12 (not 11, not 13) nulls
local param = ("\0"):rep(12)
-- 0x000e is SESSION_SETUP
local status, result = send_transaction2(smbstate, 0xe, param)
if not status then
stdnse.debug1("Error: ", result)
else
local status, header, parameters, data = smb.smb_read(smbstate)
local multiplex_id = string.unpack("<I2", header, 1 + string.packsize("BBBBB I4 B I2 I2 i8 I2 I2 I2 I2"))
if (multiplex_id == 81) then
double_pulsar.state = vulns.STATE.VULN
else
stdnse.debug1("Machine is not vulnerable")
end
end
end
return report:make_output(double_pulsar)
end
| 2024-04-06T01:26:58.646260 | https://example.com/article/5413 |
Re: Pacific Rim
Re: Pacific Rim
Originally Posted by Hardkore
A total bad ass video Hardkore!! It really shows how much you can make with CGI!! I'm so waiting for the Blu-ray release, so that I can watch it at home again! One of this years best movies hands down!
Joel Chiodo, the Legendary Pictures executive in charge of marketing Pacific Rim, has left the company.
Legendary also has let go of Christopher Erb, who had been brought over in June from EA Sports to head up the company's brand strategy.
Sources say Chiodo, who had been an executive VP of branding and social marketing, left on his own accord to join Paramount Pictures. The studio confirmed the move.
Legendary has been shaking up its marketing ranks over the past few weeks. The company recently hired Emily Castel as its chief marketing officer (earlier in the year, Legendary bought the company she founded, Five33).
Erb, who had been an executive VP brand marketing, left EA Sports to oversee Legendary's social media and consumer marketing campaigns for such upcoming films as Godzilla (which Warner Bros. will release May 16) and Seventh Son (which Universal will release Feb. 6, 2015).
Both Chiodo and Erb reported to Legendary president and chief creative officer Jon Jashni.
Legendary, the finance and production company behind The Dark Knight films and The Hangover trilogy, made a number of significant moves in 2013, leaving its longtime partner Warner Bros. to team up with Universal Pictures, where it will co-finance a chunk of the studio's slate.
Released July 12, the $200 million Pacific Rim was initially dismissed as a high-priced bomb. But the film made a huge showing internationally, particularly in China, and wound up with a $408 million worldwide haul.
Re: Pacific Rim 2
Goldberg:I don’t mean to come out with so much skepticism, I’m just curious about Godzilla and Dracula, these beloved properties. Also you had an original with Pacific Rim, I really enjoyed that film a lot and it did incredibly well overseas, and I was just curious what’s the status of the sequel?
TULL:
We love being in business with Guillermo and frankly that movie, if you look it up, did I think more business than the first X-Men, did more than Batman Begins, our first movie, did more than Superman Returns, The Fast and the Furious, Star Trek- so for a movie that was an original property that we made up it’s done really well. It did north of 400 million dollars globally and both the home video sales and the merchandise have way over-indexed, so it seems like fans really loved the world. So we’re going to sit down with Guillermo and as long as we think it’s authentic and there’s something to say, we’re certainly open to it.
Re: Pacific Rim vs Godzilla
This is taken from the text that the link leads to and is the reason why I liked Pacific Rim a hell of a lot more than Godzilla.
But in Godzilla, the main point of the story is to be rescued. All our efforts to fight the MUTO are doomed to fail, and it's only when Godzilla randomly decides to fight them off that we're saved. In fact, there's a scene where one of the scientists essentially says that our only hope is for Godzilla to save the world while we run away. Like Ford with his son, Godzilla becomes our parent figure, protecting us from destruction. This is a perfect emotional touch in a movie that works in part by infantilizing its audience.
Godzilla may have succeeded because it treated its audiences more like kids, leading them gradually into an unfamiliar world — and guiding them with a parental hand, instead of forcing them to imagine growing up and fighting for themselves.
"It's only when Godzilla randomly decides to fight them off that we're saved". Randomly? Really?
"Succeeded because it treated its audiences more like kids". Kids? Is this guy serious?
Instead of randomly as this guy writes, it's more like Godzilla, being a mutated lizard, suddenly got the intelligence of a human to realize that if it don't kill the MUTOS, they will breed and grow in numbers and eventually take over the world and kill him as well. So it comes to the conclusion that if it kills the MUTOS and survives, humans will most likely let it live instead of killing it, cause he's the hero of the day and the humans new best friend, even if it happens to be one giant lizard.
And this would make us feel more like kids? It doesn't make us feel more like kids. The movie treats it's audiences like they were idiots! Not kids. Idiots!
At least in Pacific Rim, the Kaiju's wanna take over the world an kill everything in it, though they are not very intelligent when they send these monsters to take over the world. They send their weakest ones first and then send stronger and stronger as the humans keeps killing them. Why not just send the best you got right away and if it gets killed, then you'll know to stay the hell away from that place?
It's like if you wanted to take over a town. Why first send a man with a pistol if you have a M1 Abrams battle tank in your arsenal?
In Godzilla the movie makers just assume that we would accept that Godzilla would be able to make intelligent decisions and help humans by being their new best friend, instead of being an animal with animal instincts and attack humans if it's attack himself and stay in hiding rest of the time and try to avoid humans all together. Well, you know what they say about assumption right? Assumption is the mother of all fuckups...
Re: Pacific Rim vs Godzilla
Originally Posted by frolunda71
Instead of randomly as this guy writes, it's more like Godzilla, being a mutated lizard, suddenly got the intelligence of a human to realize that if it don't kill the MUTOS, they will breed and grow in numbers and eventually take over the world and kill him as well.
You don't understand animal instincts very much do you? The MUTOs would have deprived Godzilla of his natural food source, and it's established quite clearly that in the past, the Godzilla species naturally fought against the MUTOs all the time. They were natural enemies. Godzilla didn't show up "just to save the humans", he was totally ignorant of humanity. He didn't give a damn about the humans. He appeared because he heard the MUTOs' mating calls. Did you even watch the movie?
In Godzilla the movie makers just assume that we would accept that Godzilla would be able to make intelligent decisions and help humans by being their new best friend, instead of being an animal with animal instincts and attack humans if it's attack himself and stay in hiding rest of the time and try to avoid humans all together.
Godzilla wasn't trying to "help humans by being their new best friend" though. He was seeking out his natural enemy, acting on as legitimate an animal instinct as any other. He had no reason to attack humans. Consider how long he's been alive. Humanity has only existed for a very small fraction of his lifespan. He wouldn't have any real reason to attack us because we're like ants to him. He avoided things like the aircraft carrier because they were large objects in his path, not because he was trying to be friendly with the humans on board. It was just an obstacle to him.
Maybe you should see the movie again, and this time pay attention. And don't get cute and repeat that back to me. I've seen it three times. I'm right.
Re: Pacific Rim
Godzilla didn’t live up to my expectations.
Okay there are good things about the movie that I’m not taking away. This Godzilla was better looking than the one in the first Godzilla US film. But let’s not confuse that with originality. Originality wise, the design of this Godzilla was just copied from the Japanese Godzilla. So hey it’s not as if they’ve put a lot of effort in the design of the creature. In fact the first US film Godzilla design was more original since it was really designed from scratch and not copied from the Japanese design. Unfortunately originality does not always translate to box office hit. And the first US Godzilla film tanked.
Effects wise, okay I’ll give it to them. This Godzilla film had good effects. Story wasn’t bad. But it lacked the intensity action-wise. Overall Pacific Rim really set the benchmark high and leaves this Godzilla film half-dead.
And after watching X-Men DOFP I totally forgot everything about Godzilla. X-Men just kinda took over. Because it was just so good. Want to be impressed? Watch X-Men DOFP. I totally recommend it.
Re: Pacific Rim vs Godzilla
Originally Posted by Horror Sober
You don't understand animal instincts very much do you? The MUTOs would have deprived Godzilla of his natural food source, and it's established quite clearly that in the past, the Godzilla species naturally fought against the MUTOs all the time. They were natural enemies. Godzilla didn't show up "just to save the humans", he was totally ignorant of humanity. He didn't give a damn about the humans. He appeared because he heard the MUTOs' mating calls. Did you even watch the movie?
Godzilla wasn't trying to "help humans by being their new best friend" though. He was seeking out his natural enemy, acting on as legitimate an animal instinct as any other. He had no reason to attack humans. Consider how long he's been alive. Humanity has only existed for a very small fraction of his lifespan. He wouldn't have any real reason to attack us because we're like ants to him. He avoided things like the aircraft carrier because they were large objects in his path, not because he was trying to be friendly with the humans on board. It was just an obstacle to him.
Maybe you should see the movie again, and this time pay attention. And don't get cute and repeat that back to me. I've seen it three times. I'm right.
"The MUTOs would have deprived Godzilla of his natural food source" and that would be nuclear energy right? since that's what the MUTOs ate in the movie. So, how come hasn't Godzilla attacked any nuclear power plants if he feeds on nuclear power? Maybe cause wasn't he living in a cave so deep that it got it's energy from the Earths core? So he showed up only cause he heard the MUTOs calling each other and knew that if they would breed and multiply, they would kill him, so he had to kill them first. And how did the humans react to Godzilla? They let him fight the MUTOs and just let him swim away at the end. Yeah right. The US military would have used everything in it's arsenal all from Tomahawk cruise missiles to MOAB (Massive Ordnance Air Blast) bombs to napalm to blow Godzilla to pieces till the military would have no bombs left. And that is the main reason why the movie is so stupid. The humans don't even try to kill it and therefore I see it in the way that the humans think of Godzilla as their new best friend, only cause he killed the MUTOs and they let him live only because of that.
And don't worry, I'm not gonna spend any money to see it again. Tonight I'm gonna do what Razor wrote and see X-Men: Days of Future Past and after that Godzilla will be forgotten all together.
Re: Pacific Rim vs Godzilla
Originally Posted by frolunda71
"The MUTOs would have deprived Godzilla of his natural food source" and that would be nuclear energy right? since that's what the MUTOs ate in the movie. So, how come hasn't Godzilla attacked any nuclear power plants if he feeds on nuclear power? Maybe cause wasn't he living in a cave so deep that it got it's energy from the Earths core? So he showed up only cause he heard the MUTOs calling each other and knew that if they would breed and multiply, they would kill him, so he had to kill them first. And how did the humans react to Godzilla? They let him fight the MUTOs and just let him swim away at the end. Yeah right. The US military would have used everything in it's arsenal all from Tomahawk cruise missiles to MOAB (Massive Ordnance Air Blast) bombs to napalm to blow Godzilla to pieces till the military would have no bombs left. And that is the main reason why the movie is so stupid. The humans don't even try to kill it and therefore I see it in the way that the humans think of Godzilla as their new best friend, only cause he killed the MUTOs and they let him live only because of that.
And don't worry, I'm not gonna spend any money to see it again. Tonight I'm gonna do what Razor wrote and see X-Men: Days of Future Past and after that Godzilla will be forgotten all together.
The humans didn't even try to kill it? They dropped a nuke on it in '54 and that didn't do it, so they knew they'd have to try and kill it with an even bigger nuke.
Which they did. Except the MUTOs screwed up their plan. Try watching it again, and this time don't roll your eyes and scoff every five seconds. That's a good way to miss important scenes.
Enjoy "Wolverine saves the world again" while Bryan Singer is still getting work.
“I’m working very, very hard with Zak Penn,” he told BuzzFeed. “We’ve been working for a few months now in secret. We found a way to twist it around. Travis Beacham [co-writer of the first film, now working on Fox’s Hieroglyph] was involved in the storyline and now I’m writing with Zak because Travis has become a TV mogul.”
While the filmmaker stressed that Legendary Pictures has not formally agreed to put the script into production, he said that he and Penn are working on it as if it will be made one day.
“I don’t have the money, but I’m proceeding like it is happening,” he said with a laugh, adding that he hopes to tackle the sequel after producing his next project.
Del Toro also revealed that the script, as it stands now, includes the characters Raleigh Becket (Charlie Hunnam) and Mako Mori (Rinko Kikuchi) — and the story will follow the events of the first movie.
“Some people were wondering if we were going to do the prequel. I was never interested in doing that first wave of invasion,” del Toro said. “I’m going for very new, very crazy ideas on the second one, which are very different from the first one — but you will get really great spectacle.”
While Pacific Rim was a very modest hit in the U.S., grossing $101.8 million domestically, it was a sensation overseas, pulling in $309.2 million internationally (including $111.9 million in China alone). Legendary Pictures Chief Thomas Tull recently told the website I Am Rogue that his production company would only make Pacific Rim 2 “if we can crack the story [and] we all think it’s great,” and if del Toro is on board to direct it.
Re: Pacific Rim
Hope they get the green light to do the second one. I really liked the first one cause it was so totally different compared to Transformers in the way that the action was slow and literally heavy and not fast cut like Transformers where you barely see what happens with all the transformations and shootings and split second cutting all happening at the same time. 2017 might be the year when we see Pacific Rim 2 if they get the script written in the near future and the movie gets the green light from the movie company. Just gotta keep my thumbs up and hope that they'll make the movie.
Re: Pacific Rim
Originally Posted by frolunda71
Hope they get the green light to do the second one. I really liked the first one cause it was so totally different compared to Transformers in the way that the action was slow and literally heavy and not fast cut like Transformers where you barely see what happens with all the transformations and shootings and split second cutting all happening at the same time.
Well, let's balance things out then. At least most of the battles in "Transformers" occur during daylight and you can actually see the creatures on the screen. "Pacific Rim" on the other hand is night, night, night + water. It's like they were too cheap to show a big battle in broad daylight. So thanks, but no thanks.
I enjoyed 'Pacific Rim", but I have no intention of seeing it again. "Transformers" wins in my book. By a mile.
I like Del Toro, I think he's a good fantasy director, but when it comes to pure action cinema Bay is heads and tails above him. It's not even a contest.
"You know why the departures and the arrivals at LAX are on separate levels? So the 30,000 heartbreakers that come here each month don't notice the 30,000 that are leaving with their hearts broken."
Re: Pacific Rim
Originally Posted by Bayhem
Well, let's balance things out then. At least most of the battles in "Transformers" occur during daylight and you can actually see the creatures on the screen. "Pacific Rim" on the other hand is night, night, night + water. It's like they were too cheap to show a big battle in broad daylight. So thanks, but no thanks.
I enjoyed 'Pacific Rim", but I have no intention of seeing it again. "Transformers" wins in my book. By a mile.
I like Del Toro, I think he's a good fantasy director, but when it comes to pure action cinema Bay is heads and tails above him. It's not even a contest.
Hey Bayhem, have you seen his other films like Cronos, Devil's Backbone, Mimic (better in it's blu-ray Director's Cut) and Pan's Labyrinth to stuff he produced like The Orphanage? great stuff.
Re: Pacific Rim
Hey Bayhem, have you seen his other films like Cronos, Devil's Backbone, Mimic (better in it's blu-ray Director's Cut) and Pan's Labyrinth to stuff he produced like The Orphanage? great stuff.
I've seen them all and I've enjoyed them all. "Mimic" is pretty underrated and "The Orphanage" is actually one of my favorite horror films. I would also add "Don't Be Afraid of the Dark" (the new version with Katie Holmes). Del Toro and Troy Nixey did a great job with that movie. At least in my view.
"You know why the departures and the arrivals at LAX are on separate levels? So the 30,000 heartbreakers that come here each month don't notice the 30,000 that are leaving with their hearts broken."
Re: Pacific Rim
Originally Posted by Bayhem
Well, let's balance things out then. At least most of the battles in "Transformers" occur during daylight and you can actually see the creatures on the screen. "Pacific Rim" on the other hand is night, night, night + water. It's like they were too cheap to show a big battle in broad daylight. So thanks, but no thanks.
I enjoyed 'Pacific Rim", but I have no intention of seeing it again. "Transformers" wins in my book. By a mile.
I like Del Toro, I think he's a good fantasy director, but when it comes to pure action cinema Bay is heads and tails above him. It's not even a contest.
I like both movies a lot, but what Bay has done with Transformers is that the action has gotten faster and faster in every movie. In the first one you really could see all the transformations and the action was great and not so fast and that's why it's still my favorite one. Then in DOTM it was like three times faster and the Optimus jetpack scene is one of those scenes were all happens so fast that you sit and just watch and think shit that was cool! and then afterwards you're like what the hell happened there? In five seconds he flies down a street, lands, loses his jetpack, kills like seven Decepticons of which you have no idea who they were and then the action just goes on. Only Bay can pull off those scenes and make the movie still look cool. But as I said, I liked Pacific Rim a lot cause when I saw it at the movies, you could feel the weight in every punch in your stomach when the robots fought the monsters. The sound was so loud that it was just awesome. Then that the fights happened at night I had no problem with cause everything still looked great on the screen. I watched it at home on Blu-ray a while ago on my 46" TV and it still looked just as good as it did at the movies. I still have to wait a month for AOE and I hope that it's gonna be the best one yet and not all the action scenes are gonna be fast and speedy like the one where Optimus rides to town on Grimlock and jumps and kills like three-four Decepticons in three seconds.
Re: Pacific Rim
Originally Posted by frolunda71
I like both movies a lot, but what Bay has done with Transformers is that the action has gotten faster and faster in every movie. In the first one you really could see all the transformations and the action was great and not so fast and that's why it's still my favorite one.
I see what you mean, but we have to look at the big picture here. Bay's action style was always fast and aggressive, ever since "Bad Boys 1". And I don't really see anything different in his "Transformers" franchise. Sometimes he shoots an action scene with 10 different cameras and, as you would expect, during editing he takes advantage of all that footage. He's not doing it to cover some flaws, it's just how he likes to present his action sequences. Fast. Personally I think "Bad Boys 2" is his most aggressive (action-wise) and fast-paced movie to date, but that's not a problem for me. I've been following the guy's career for more than 15 years, I know exactly what to expect, and I have no problem with his fast cuts and kinetic camera work. I actually appreciate his approach.
And not to trash the late Tony Scott (I love his work), but his last few movies are 10 times faster (action/editing/camera-wise) than Bay's movies. Watch Scott's "Unstoppable" and then watch Bay's "Dark of the Moon" and you'll see what I mean.
Speaking for myself, I actually prefer the fast action-style of Bay, Scott, Greengrass and Dominic Sena. Yes, there are a lot of people who don't like the shaky-cam and the fast cuts, but you know what - I love it all. It just works for me.
There's place for all types of filmmakers and I think we can agree that Bay has earned his right to shoot his action the way he wants to shoot it. Thankfully he's not influenced by his detractors and he sticks to his guns. For 20 years now.
By the way, Del Toro's "Blade 2" is also known for its fast action sequences. Some scenes actually rival the crazy action in "Bad Boys 2". So it seems that Del Toro also has a soft spot for "fast action".
P.S. With all of this I'm not trying to change your views on the matter. It's just my observation and opinion.
"You know why the departures and the arrivals at LAX are on separate levels? So the 30,000 heartbreakers that come here each month don't notice the 30,000 that are leaving with their hearts broken."
Re: Pacific Rim
Originally Posted by frolunda71
I like both movies a lot, but what Bay has done with Transformers is that the action has gotten faster and faster in every movie. In the first one you really could see all the transformations and the action was great and not so fast and that's why it's still my favorite one. Then in DOTM it was like three times faster and the Optimus jetpack scene is one of those scenes were all happens so fast that you sit and just watch and think shit that was cool! and then afterwards you're like what the hell happened there? In five seconds he flies down a street, lands, loses his jetpack, kills like seven Decepticons of which you have no idea who they were and then the action just goes on. Only Bay can pull off those scenes and make the movie still look cool. But as I said, I liked Pacific Rim a lot cause when I saw it at the movies, you could feel the weight in every punch in your stomach when the robots fought the monsters. The sound was so loud that it was just awesome. Then that the fights happened at night I had no problem with cause everything still looked great on the screen. I watched it at home on Blu-ray a while ago on my 46" TV and it still looked just as good as it did at the movies. I still have to wait a month for AOE and I hope that it's gonna be the best one yet and not all the action scenes are gonna be fast and speedy like the one where Optimus rides to town on Grimlock and jumps and kills like three-four Decepticons in three seconds.
I really don't understand this kind of criticism about action scenes of bay's films. You can't barely see what's happening? I surely can see them! Yeah, his movies are fast and aggressive. But today is not 1990s, but 2010s. His style is so familiar. And as film critic Scott Foundas & Director Joseph Kahn said, Bay's action sequences are constructed in more classical way than today's many chaotic movies like Bourne series or found footage/shaky cam movies or whatever.
And I think Transformers 1 has most unrecognizable & aggressive action scenes in the series. Sequels, especially third one because of 3d, have longer shot length and use many slow motions in action scenes. So those films convey information and energy and emotion in actions more clearly.
Re: Pacific Rim
I'm not criticizing Bay as director. I have ALL his movies on Blu-ray and dvd. I wouldn't have bought them if I didn't like his movies.
In my case, the whole Pacific Rim vs Transformers was about the speed in the action scenes. What I meant by saying that I liked the slow and heavy speed in Pacific Rim cause it was a total opposite to the fast paced speed in Transformers and in that way the movie isn't a Transformers rip off or a wanna be movie. I've heard and read that people didn't wanna see Pacific Rim cause it was just another Transformers movie. Then there are scenes in Transformers that are really fast and a lot happens in a couple of seconds and you really have to look hard to notice everything that happens, but at the same time those scenes are almost every time the highlight scenes in the movies cause they give you that what I call "Did you just see that shit?!! That was awesome!" feeling and you are more amazed of what you just saw than thinking of what just happened there.
So as I said before I like both movies a lot and hope that AOE will be the best Transformers so far and that the movie company will give the green light to Pacific Rim 2 as well. | 2023-10-04T01:26:58.646260 | https://example.com/article/5073 |
The federal bank and thrift regulatory agencies announce today they are reopening the comment period on proposed guidance on correspondent concentration risks, originally published on September 25. The comment period will be open until November 27, 2009, an additional 30 days after the original comment period closed.
A copy of the proposed guidance, along with the notice to reopen the comment period, are attached. | 2024-05-23T01:26:58.646260 | https://example.com/article/7313 |
And in preparation of watching the new season, it’s time to publish Thuy’s take on last season’s finale.
***************************
I know that Khiem hated this episode the most. As a writer though, I totally relished in that jaw-dropping, shocker twist. I know it seems totally implausible that they’d wind up together, especially after I watched the attempted “threesome” scene again.
As an audience, I did feel cheated in that I didn’t see the emotional turn in Jessa. But I guess that IS the magic of Hollywood to create that dramatic opener.
Drama is the theme of Jessa’s life. I don’t agree with Evan’s mom in that Jessa chose the successful guy to maintain her bohemian lifestyle. What I do agree with is that it all started with her talk with the “mom”.
Jessa creates the drama in her life as a way of compensating for the void in her life. The excitement is replacing the lack of happiness she feels. She had always jumped from guy to guy – being unpredictable, feeling the thrill of the conquest, and experiencing the extreme highs and lows.
However, she never really slowed down to create a true emotional connection with another human being. What she felt in the past was akin to being under the influence of drugs. She never felt a real, natural high. She never felt real love because she was manipulating everything.
The talk with the “mom” gave her insight into herself and she finally saw the light. For her to finally experience happiness, she let go of having power over someone for the first time. She let go of her judgments. She let go of allowing drama in her life. She probably thought it was a grown up thing to finally commit to something and a safe bet to be with the successful guy.
But with her motivations half-misguided, we can only foreshadow that she may have just created more drama for herself.
Look into your life. See where your actions are motivated from. Are you just creating excitement to fill emptiness in your life?
Everything does seem flipped in this episode – from Shoshanna and Ray’s quirky exchange to Marnie’s hot kiss with the little fat boy.
I do get why Shoshanna asked:
Are you punking me?
Ray wasn’t just forward. What he gave her was a backhanded compliment when he told her she was the strangest girl he ever met.
Should she have felt flattered?
I have never had a guy say that to me before… but I do wonder how it works on a girl. It does put a girl on her toes. It’s not a typical compliment like “You have beautiful eyes” or “You’re so sexy” (which at times are actually pretty generic and boring, when said too often or ingenuinely).
Please don’t misunderstand me though. I would never tell guys to stop giving those compliments altogether. When you are in a relationship, girls love to hear that you still feel they are attractive. However, when you just meet them for the first time, compliments given too fast or too early just sound like cheap pick-up lines.
What is great about Ray is that he is not putting Shoshanna on a pedestal. He’s pointing out what could be seen as a flaw but he’s actually completely mesmerized by it! With that intention, his backhanded compliment takes a different meaning… which is what women love: to know that someone recognizes what is truly unique and real about her.
I had a guy once give me a compliment about my freckles. Had I been red- headed, that would have been a typical compliment… but I’m Asian! Asians usually get: “You have such nice, smooth, porcelain skin.”
That was the first time anyone had ever mentioned my freckles. I was so flattered. It showed me he really paid attention to the little details, to who I was… and that he noticed in me something that others hadn’t.
Marnie taking a soft spot for the little fat boy seemed like a man’s fantasy that Hollywood has portrayed far too often. When is it going to be time for the hot dude to fall for the little fat girl?!!!
Protest aside, unlike Khiem and Evan, I can see it happening just for the same reason we sometimes see a hot chick walking down the street arm in arm with a geeky guy while wondering what does she see in him. As unbelievable as it may sound, some attractive women like Marnie go for less attractive guys because:
They are afraid hot guys will cheat on them. Hot guys get constant attention and distraction from other hot girls. It gives the girlfriend a lot of reasons to be leery, jealous, and insecure.
To a certain degree (I say this because even I have a limit to how much of a blind eye I’ll give to a man’s looks), women put less weight on sheer physical beauty than men do and more weight on the emotional connection.
The only caveat to this idyllic fantasy is that Marnie once again has found a boy who might put her on a pedestal… just like Charlie!
Now, the biggest flip in this episode is Hannah and Adam. Typically, it’s the girl who pressures the guy to move in and the guy gets cold feet about it. Here, the roles are reversed.
I’ve had similar experiences on both spectrums, strangely with the same guy. After about five months of dating, my then boyfriend said that we should start thinking about moving in the direction of marriage. I was surprised! We hadn’t even said our “I love you’s” yet!
I gave him some of the same excuses that Hannah did, like I had so many bills to pay off and I had to focus on my career. Although I thought his talk of marriage was premature, I did admire how much faith he had in our relationship at that moment.
Adam had that pure faith but Hannah did not believe in herself. She was afraid that she was going to screw things up. When I gave my boyfriend the career excuse, he gave me the same response as Adam did:
That’s why we have each other to support.
I didn’t buy it. I grew up underprivileged. I didn’t want to start a marriage in financial strife and to have my children lacking what they need. In retrospect though, he did have a point. Wouldn’t you want someone to be there for you as your cheerleader through your struggles? If they have stuck with you through the challenging times in your life, then wasn’t the relationship more likely to last in the long run?
If you wait for things to line up perfectly before you take that plunge, it may never happen.
I realize now that it is sometimes prudent to take things slow and not move a relationship too fast. Yet you have to ask yourself what is holding you back?
Procrastination is rooted in fear and Hannah was paralyzed with fear. She didn’t say it out loud but I’m sure she was jumping ahead into the thought of what living together could lead to – responsibility, marriage, and family – all the things that she wasn’t ready to have, all the things that meant she was going to be accountable to someone.
You don’t necessarily have to think that far ahead when you make the decision to move in with someone. (I know I’m going to get girls hating on me for saying that.) Years ago, I went to Belgium to visit my great uncle. His son just started dating a girl. She moved in to live with him and his family just a couple weeks after they had started dating! It sounds totally radical to us Americans but it turns out it is commonplace to do in Europe. It’s their way of figuring out if they are compatible with each other.
What a novel idea?! Why spend such an expanse of time meeting just a couple times a week over a period of a couple years when you can just cut through the chase and figure it out in a couple months? Making that next step in a relationship may be a scary thing but you will waste less time in figuring out if he or she is the right one for you.
Speaking of the right guy, in every podcast Khiem and Evan complain about how there are no strong men in this show.
Admittedly, I would like to see a male version of Jessa. Ray is probably the closest character to Jessa but he doesn’t draw in the women like how Jessa does men. The only time that I can foresee a strong man appearing in the show is if he is a guest character who plays the “straight shooting man” that calls out on these girls or that briefly acts as a romantic mentor to one of them.
Otherwise, how can there be a strong man in the show when there are no strong women? Each one of the women on the show is so flawed. They’re still figuring themselves out. None of them have a clear direction of where they are going or who they are. The fact of the matter is you attract people who are in the same place as you are in your life.
… I can’t wait for next season to see what other guys the Girls will attract!
So you’ve done all this “work.” You’ve gone from total loser to complete stud.
You walk slowly in the bar, you scan the room, a girl is checking you out. You half-smile back at her but you know not to eagerly walk up to her yet. She’s actually not the only one who’s been glancing at you ever since you stepped in. You have the choice. Hmmm… who are you going to pick tonight? There, that one! She’s sitting in the back corner talking to her friends. She’s pretty. Cute face, boobilicious body, curvacious hips. She notices you as you start walking to her. She pretends not to see you but you can tell she’s getting nervous as she realizes the prospect of meeting you.
You start the conversation with something you can’t even remember. Within minutes, all her friends love you. Her entire social circle is mesmerized by you. Without knowing why or how, she senses the sexual tension rising between you and her. She’s feeling a little bit flustered inside, maybe a little warm. With calm confidence, you pull her to the side. She smiles or laughs at the drop of your every word. She touches your arm and brushes herself against you. She tells you things she haven’t shared with many guys before. She really likes you. By the end of the night, you get her phone number. Maybe you’ve kissed her. Maybe you’ve made out with her. Maybe you’ve even slept with her.
Pick-up is easy for you now. If you wanted, the same scenario could play over and over again every night. Women call you non-stop to hang out with you. You have more dates than you even know what to do with. Now what?
You actually have a few girlfriends, some more serious than others. Some are actually so in love with you that they want a committed relationship from you. However, you don’t know if you can honor that commitment. Now what?
Let me tell you this. Don’t be afraid to go to the next level.
I see so many Pick-Up Artists working so hard at staying in the position to “have the choice in women.” After suffering so many years of not having any options, they are now constantly looking for new, better, hotter women. It’s like they collect them. They always need “one more.” I know… having someone new is fun. It’s like having a new toy every day. If that’s what you enjoy and want, keep playing the field. I respect that decision and you should stop reading now.
But if you are done sowing your royal oats, if you have now decided to find yourself a “real” girlfriend/wife or if you are not finding fulfillment in meeting more women, then ask yourself: what’s the point of having a choice in women if you don’t exercise that choice? How long are you wanting to stay unattached for? Do you want to be that creepy 65 year old man who’s still chasing after 3-4 women?
It’s time to look hard at what you have already. I am sure that some of the women you are seeing today are worth pursuing a real relationship with. Why aren’t you?
In pick-up, you learn to conquer your own fears and insecurities. Be that confident man, they say. Don’t let fear take over you when it comes to talking to a woman. Just approach and so… you’ve learned to overcome that anxiety. Well… now that you have the choice in women, what are you afraid of?
My roommate used to tell me:
In the end, you only need one woman to be happy.
I have a friend who was dating a wonderful girl. He met her only one month after taking his bootcamp with theApproach. He came to a dilemma that many Pick-Up Artists would eventually have to face. After a few months of dating, he sensed that she would soon want a committed relationship with him. She was hot, she was caring, she was fun, she was sexy and for a while, she did tolerate his uncommitted ways. Yes, she liked him a lot and so did he. But what was he to do: go exclusive with her or keep seeing other women?
At this moment, a lot of Pick-Up Artists would have chosen to break-up with the girl to maintain their position of choice. They prefer to break it off rather than become more attached to her. They don’t want to miss out on meeting and seeing other girls. Maybe they are afraid of liking that ONE girl too much. Maybe they don’t want to hurt her because she’s falling too in love with them and they don’t want that to get out of hand. Whatever.
My friend decided to give exclusivity a shot. It’s not like he couldn’t break-up with her later if things didn’t work out. When he made that decision, he wasn’t thinking marriage either but now, I can say he’s happier than ever. You should have seen the big smile he had at his wedding. He married her earlier this month. If he didn’t take the time to connect to her on a deeper level, he wouldn’t have realized how much she meant to him. When you are done with the thrills of “just sex,” only a committed relationship can allow you to grow on a deeper personal level. We sometimes forget to notice the very gem that is in front of us. If you have a quality woman in your life who is worth savouring, don’t let her slip you by.
So when the time comes, don’t fall for the fallacy of wanting more. Recall why you became a Pick-Up Artist in the first place. Wasn’t it to have the ability to eventually find that special someone? Look at Hugh Hefner. At 80 year old, even he is getting married again.
I am not asking you to settle down but if a girl feels right, stop looking around. Allow yourself to grow with her. Try things out.
Another friend of mine who has always been very successful with women reminded me of something I’ve told him in my college years. I didn’t remember it but it was something that have stayed on his mind for years:
On a deep emotional level, when you sleep with someone, you give away a little piece of your soul. Don’t spread your soul too thin. | 2024-07-19T01:26:58.646260 | https://example.com/article/4372 |
1. Field of the Invention
Our invention is an electric switch that selects data by moving contacts along tracks of electrodes such as a rotary type switch having a switch rotor mounted to or with respect to a printed circuit board.
2. Prior Art
Many data switches in this field are known. One example, disclosed in the Japanese Published Examined Utility Model Application No. 53-49308 and shown in FIG. 3, is rotated to a data selection position, and selects the data at that position when a read button (not shown) is pressed. This switch has a contact base 55 with several contacts 55a-55g. The contacts 55a-55g are arranged to correspond to concentric annular electrodes 52a-52g printed on a printed circuit board 51. Each of the electrodes 52a-52g has terminals 53 that can be touched by one of the contacts 55a-55g. The terminals 53 for each electrode 52a-52g are connected by narrower leads 54 that the contacts 55a-55g cannot touch. The contacts 55a-55g slide in unison over the electrodes 52a-55g when the switch is rotated. The contacts 55a-55g are electrically interconnected.
In FIG. 3, the contacts 55a-55g rest on the printed circuit board 51 at the hatched squares. In this example, four contacts 55a, 55c, 55e and 55f touch four electrodes 52a, 52c, 52e and 52f, respectively, thus conducting signal current. Three contacts 55b, 55d and 55g do not touch any of electrodes 52a-52g, and thus conduct no signal current. As a result, the switch, at this position, would read "0110101", where the innermost electrode 52a represents the least significant bit.
In this arrangement, the electrodes 52a-52g are so narrow that attempts to miniaturize the switch leave little margin for error. The contacts 55a-55g have to be made so small that the manufacturing process becomes very complex and expensive, while the strength of the contacts 55a-55g declines allowing greater deterioration during use. | 2024-05-03T01:26:58.646260 | https://example.com/article/3470 |
This invention relates to a method and apparatus for encoding/decoding image data. It is particularly applicable to the encoding/decoding of images that can be separated into their constituent parts as may be used in composite picture systems used in law enforcement, artistic creations, recreation and education.
It is known in the art to create images on the basis of components that are assembled to form a complete image. For example, a common technique for synthesizing single images of faces involves horizontally dividing the image of a face into bands for different features of the face such as hair, eyes, nose, mouth, and chin, respectively. Paper strips containing exemplary features are then be combined to form a composite drawing of a face. Yet another example involves a program element running on a computing platform which allows a user to select individual components and combining them on a pre-selecled face. In a typical interaction, the user first selects the shape of the face then eyes, nose, mouth and other components and combines them to form a facial image. Many variations on this theme can be used as described in Kakiyama et al. U.S. Pat. No. 5,600,767, Yoshino et al. U.S. Pat. No. 5,644,690, Sato et al. U.S. Pat. No. 5,537,662 and Belfer et al. U.S. Pat. No. 5,649,06 whose contents are hereby incorporated by reference. For example, the Sato et al. Patent, entitled Electronic Montage composing apparatus, describes a system for creating a montage image of a face using a plurality of basic parts stored in a library.
In constructing an image, pictorial entities are selected from a library of pictorial entities as assembled into images. These images may then be stored on a computer readable medium commonly referred to as a database or repository. Often, the storage of an image requires significant amounts of memory, often necessitating large repositories. For example, a composite picture system used in a police department often requires maintaining records of thousands of Individuals. The images are typically stored in files in some graphical format such as a xe2x80x9cbitmapxe2x80x9d, xe2x80x9cgifxe2x80x9d or xe2x80x9cjpegxe2x80x9d are other format. Although such encoding schemes provide a compressed representation of the image, the memory required for storing the image remains significant. In addition, compression methods of the type described above generally degrade the quality of the image. The size and quality of images is also particularly significant when the images are transmitted from one site to another via a digital link. For example, a given police station may transmit a composite picture to another police station in order to share information about a given suspect.
Thus, there exists a need in the Industry to refine the process of encoding images such as to reduce the memory requirements for storage and the bandwidth required for the transmission of the image.
The invention provides a novel method and an apparatus for encoding images.
For the purpose of the specification, the expression xe2x80x9cbasic elementsxe2x80x9d is used to describe a part of a specific image. In the preferred embodiment, a basic element is comprised of a pictorial entity conditioned by a set of image qualifiers. Examples of pictorial entities in a facial image are noses, eyes, mouths and eyebrows. In the preferred embodiment, pictorial entities are grouped into classes. For example, in composite picture system, all nose pictorial entities are grouped into the xe2x80x9cNOSExe2x80x9d class and all the eye pictorial entities are grouped in the xe2x80x9cEYExe2x80x9d class. Each class of pictorial entities is associated to a set of image qualifiers that are used to condition the pictorial entities in the associated class. The image qualifiers may include position qualifiers, zoom qualifiers, color qualifier and the likes.
For the purpose of this specification, the basic elements used in the special case of a facial image are referred to as xe2x80x9cbasic morphological elementsxe2x80x9d.
For the purpose of this specification, the word xe2x80x9csymbolxe2x80x9d is used to designate a representation of an object, image, qualifier or the likes. In a specific example, a symbol may be an index mapped to a memory location storing data elements such as a pictorial entity or image qualifier.
According to a broad aspect, the invention provides, a computer readable storage medium comprising a program element suitable for use on a computer having a memory. The program element is operative to create a first input to receive a set of element codes. The element codes characterized a portion of an image and included at least one symbol. A given symbol is a representation of a certain characteristic of the portion of the element code. A given symbol can acquire a set of possible values indicative of variations of the certain characteristic with which it is associated. The program element is also operative to create a second input to receive code factors associated to respective symbols of the set of element codes. A given code factor is assigned a value that exceeds the highest value that the symbol with which it is associated can acquire. The program element is operative to process the set of element codes to derive an image code. The image code is a compressed digital representation of the image, and is derived at least in part on the basis of the plurality of code factors. The image code can then be released as the output.
In a preferred embodiment, the image code is a number in a given base. Preferably, a large base is used in order to obtain a reduced number of characters in the image code.
In a preferred embodiment of the invention, the encoding method and apparatus is integrated into a picture system. The picture system creates images on the basis of images of basic individual parts, herein referred to as basic elements. In the preferred embodiment, the picture system includes a library of pictorial entities and qualifiers, an image builder unit, an encoding unit, a decoding unit and a factor table.
Each basic element in an image is assigned a unique identifier, herein referred to as element code. The element code contains information data elements, herein referred to as symbols. In a specific embodiment, the element code for each basic element includes a symbol that characterizes the pictorial element. In a preferred embodiment, the element code includes a plurality if symbols. In a specific example two (2) symbols are used namely an pictorial entity symbol and a position qualifier symbol. The element code may contain additional symbols without detracting from the spirit of the invention. For example symbols representative of other image qualifiers may be used such as color, zoom and other image effects may be used. An image is constructed by a set of basic elements. The basic elements present in a given facial image are said be xe2x80x9cactivexe2x80x9d in the given image. The set of active elements is stored in a data structure suitable for that purpose. In a specific example this data structure s an image data table. The image data table stores for each class a record, each record containing a set of fields, each field describing the active pictorial entity and qualifiers.
The number variations in each of the symbols for each of the classes is stored in a table, herein referred to as a code factor table. The code factor table provides information about the number of possible variations in an image. For each class, the code factor table stores a record, each record containing a set of fields, each field describing a maximum factor. The maximum factor in the code factor table is the largest identifier used for the given factor. Each symbol in the image data table is mapped to a factor in the code factor table.
According to another broad aspect, the invention provides an apparatus for encoding an image, the image comprising a set of basic elements, each basic element of the set of basic elements being associated to an element code. An encoding unit receiving as input the code factors and the element codes. The encoding unit processes the set of element codes to derive an image code, the image code being a compressed digital representation of the image derived at least in part on the basis of said plurality of code factors. The encoding unit then outputs the image code.
According to another broad aspect, the invention provides a method for encoding an image, the image comprising a set of basic elements, each basic element of the set of basic elements being associated to an element code. A processing step receives as input the code factors and the element codes to derive an image code. The image code is a compressed digital representation of the image derived at least in part on the basis of said plurality of code factors. The image code is the released.
In a preferred embodiment, the image code can be used to reproduce the image described by the image code. Image data may be obtained by combining the code factors and the image code with a decoding device. The image data is obtained by applying the inverse operations in the reverse order than those applied in the encoding process to the image code.
The image code allows each image to be described with a very small number of characters permitting the rapid transmission of the image over a data transmission medium. The receiving device has a decoding unit that is capable of extracting data information from the image code.
According to another broad aspect, the invention provides a method, apparatus and computer readable medium for decoding an image, the image comprising a set of basic elements, each basic element of the set of basic elements being associated to an element code. A processing step receives as input the code factors and the image code to derive the element codes. The element codes are then released. | 2024-07-25T01:26:58.646260 | https://example.com/article/6130 |
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abstract: |
Let $k$ be a field, $G$ be an abelian group and $r\in {\mathbb N}.$ Let $L$ be an infinite dimensional $k$-vector space. For any $m\in {\operatorname{End}}_k(L)$ we denote by $r(m)\in [0,\infty ]$ the rank of $m$. We define by $R(G,r,k)\in [0,\infty ]$ the minimal $R$ such that for any map $A:G \to {\operatorname{End}}_k(L)$ with $r(A(g'+g'')-A(g')-A(g''))\leq r$, $g',g''\in G$ there exists a homomorphism $\chi :G\to {\operatorname{End}}_k(L)$ such that $r(A(g)-\chi (g))\leq R(G, r, k)$ for all $g\in G$.
We show the finiteness of $R(G,r,k)$ for the case when $k$ is a finite field, $G=V$ is a $k$-vector space $V$ of countable dimension. We actually prove a generalization of this result.
In addition we introduce a notion of [*Approximate Cohomology*]{} groups $H^k_{\mathcal F}(V,M)$ (which is a purely algebraic analogue of the notion of ${\epsilon}$-representation ([@ep])) and interperate our result as a computation of the group $H^1_{\mathcal F}(V,M)$ for some $V$-modules $M$.
address:
- 'Einstein Institute of Mathematics, Edmond J. Safra Campus, Givaat Ram The Hebrew University of Jerusalem, Jerusalem, 91904, Israel'
- 'Einstein Institute of Mathematics, Edmond J. Safra Campus, Givaat Ram The Hebrew University of Jerusalem, Jerusalem, 91904, Israel'
author:
- David Kazhdan
- Tamar Ziegler
title: Approximate cohomology
---
[^1]
Introduction
============
Let $k$ be a field, $G$ be an abelian group and $r\in {\mathbb N}.$ Let $L$ be an infinite dimensional $k$-vector space, $End(L)=End_k(L)$ and $M\subset {\operatorname{End}}(L)$ be the subspace of operators of finite rank. For any $m\in {\operatorname{End}}(L)$ we denote by $r(m)\in [0,\infty ]$ the rank of $m$. We define by $R(G,r,k)\in [0,\infty ]$(correspondingly $R^f(G,r,k))$ the minimal number $R$ such that for any map $A:G\to {\operatorname{End}}(L)$(correspondingly a map $A:G\to M)$, $g\to m_g$ with $r(A(g'+g'')-A(g')-A(g''))\leq r$, $g',g''\in G$ there exists a homomorphism $\chi :G\to {\operatorname{End}}(L)$ such that $r(A(g)-\chi (g))\leq R(G,r,k)$ for all $g\in G$.
It is easy to see that in the case when $G={{\mathbb Z}}$ and $\operatorname{char} (k)\neq 2$ we have $R({{\mathbb Z}},1, {\mathbb F})\leq 2$. We sketch the proof: one studies the rank $\le 1$ operators $r_{m,n}=A(m+n)-A(m)-A(n)$. Since $r_{0,0}$ of rank $\le 1$, we replace $r_{m,n}$ by $r_{n,m}-r_{0,0}$ and can then assume that $r_{0,0}=0$. Under this condition one must show that $r_{n,m}$ is a coboundary of rank one operators. The operators $r_{n,m}$ satisfy the equation $r_{a,-a}=r_{a+c, -a}+r_{a,c}$. From this one can deduce that either there is a subspace of codimension $1$ in the kernel of all three operators, or a subspace of dimension $1$ containing the image of all three. One shows inductively that this property holds for all operators $r_{m,n}$. Unfortunately we don’t know whether $R({{\mathbb Z}},2,{{\mathbb C}})<\infty$.
In this paper we first show that $R^f(V,r,k)<\infty$ in the case when $k={\mathbb F}_p$ and $G$ is a $k$- vector space $V$ of countable dimension and then show that $R(V,r,k)=R^f(V,r,k)$.
Actually we prove the analogous bound in a more general case when $M$ is replaced by the space of tensors $L^1\otimes L^2\otimes ...\otimes L^n$. To simplify the exposition we assume that $L_i=L^\vee$ and that $Im(A)$ is contained in the subset $Sym^d(L^\vee)\subset {L^\vee}^{\otimes n}$ of symmetric tensors. In other words we consider map $A:V\to M^d$ where $M^d$ is isomorphic to the space of homogeneous polynomials of degree $d$ on $L$. We denote by $N^d\subset M^d$ the subspace of multilinear polynomials.
Now some formal definitions.
Let $M$ be an abelian group. A [*filtration*]{} ${\mathcal F}$ on $M$ is an increasing sequence of subsets $M_i, 1\leq i\leq \infty$ of $M$ such that for each $i,j$ there exists $c(i,j)$ such that $M_i+M_j\subset M_{c(i,j)}$ and $M=\cup _iM_i$. Two filtrations $M_i,M_i'$ on $M$ are equivalent if there exist functions $a,b: \mathbb Z _+\to \mathbb Z _+$ such that $M_i\subset M_{a(i)}'$ and $M'_i\subset M_{b(i)}$.
\[Algebraic rank filtration\] Let $k$ be a field. Fix $d\geq 2$ and consider the $k$-vector space $ M=M^d$ of homogeneous polynomials $P$ of degree $d$ in variables $x_j$, $j\geq 1$. For a non-zero homogeneous polynomial $P$ on a $k$-vector space $W$, $P\in k[W^\vee ]$ of degree $d\geq 2$ we define the [*rank*]{} $r(P)$ of $P$ as $r$ ,where $r$ is the minimal number $r$ such that it is possible to write $P$ in the form $$P = \sum ^ r_{i=1} l_iR_i,$$ where $l_i, R_i\in \bar k[W^\vee ]$ are homogeneous polynomials of positive degrees (in [@schmidt] this is called the $h$-invariant). We denote by ${\mathcal A}_d$ the filtration on $M$ such that $M_n$ is the subset of polynomials $P$ with $r(P) < n$. For the $k$-space $\mathcal P^d$ of non-homogeneous polynomials of degree $\le d$ define the rank similarly, and denote by ${\mathcal B}_d$ the corresponding filtration.\
Let $V$ be a countable vector space over $k$. We say that a linear map $P:V\to M^d$ is of [*finite rank*]{} if we can write $P$ as a sum $\sum _{k=1}^{d-1}P_k$ where each $P_k$ is a finite sum $P_k=\sum _jQ_jR_j$ where $R_j\in M^k$ and $Q_j$ is a linear map from $V$ to $M^{d-k}$. Denote ${\operatorname{Hom}}_f(V,M^d)$ the subspace of ${\operatorname{Hom}}(V,M^d) $ of finite rank maps.
Now we can formulate our main result.
\[Main\] For any finite field $k={\mathbb F}_p ,d<p-1, r\geq 0$ there exists $R=R(r,k,d)$ such that for any map $$A:V\to M=M^d, \quad r(A(v'+v'')-A(v')-A(v''))\leq r, \quad v',v''\in V$$ there exists a homomorphism $\chi_A : V\to M$ such that $r(A(v)-\chi_A (v))\leq R$. Moreover the homomorphism $\chi _A$ is unique up to an addition of a homomorphism of a finite rank.
$a)$ Is there a bound on $R$ independent of $k$ ? Moreover, does there exist $c(d)$ such that $R(V,r,k,d)\leq c(d)r$?
$b)$ Could we drop the condition $d<p-1$ if $Im(A)\subset N^d$?
Theorem \[Main\] does not hold for $p = d= 2$, see [@tao-example] for a function from ${\mathbb F}_2^n$ to the space of quadratic forms over ${\mathbb F}_2$ such that $f(u+v)-f(u)-f(v)$ is of rank $\le 3$ but for $n$ sufficiently large $f$ does not differ from a linear function by a function taking values in bounded rank quadratics. In the low characteristic case the same proof shows that obstructions come from [*non-classical polynomials*]{} see Remark \[nonclassical\]. In the case when $G$ is a finite cyclic group, $k$ any field, $d=2$, one can show that $C(1,k)\le 2$.
We can reformulate Theorem \[Main\] as an example of a computation of [*approximate cohomology* ]{} groups.
1. Let $M$ be an abelian group, and let ${\mathcal F}=\{M_i\}$ be a filtration on $M$.
2. Let $G$ be a discrete group acting on $M$ preserving the subsets $M_i$. A cochain $r:G^n\to M$ is an [*approximate $n$-cocycle*]{} if $Im
(\partial r)\subset M_i$ for some $i\in {{\mathbb Z}}_+$. It is clear that the set $Z^n_{\mathcal F}$ of approximate $n$-cocycles is a subgroup of the group $C^n$ of $n$-chains which depends only on the equivalence class of a filtration ${\mathcal F}$.
3. A cochain $r:G^n\to M$ is an [*approximate $n$-coboundary*]{} if there exists an $n-1$-cochain $t\in C^{n-1}$ such that $Im (r-\partial t)\subset M_i$ for some $ i\in {{\mathbb Z}}_+$. It is clear that the set $B^n_{\mathcal F}$ of approximate $n$-coboundaries is a subgroup of $ \tilde Z ^n_{\mathcal F}$.
4. We define $H^n_{\mathcal F}=Z^n_{\mathcal F}/B^n_{\mathcal F}$.
5. Since any cocycle is an approximate cocycle and any coboundary is an approximate coboundary we have a morphism $a^n_{\mathcal F}:H^n(G,M)\to H^n_{\mathcal F}(G,M)$.
In this paper we consider the case when the group $V$ acts trivially on $M$. So the group $Z^1(V,M)$ of $1$-cocycles coincides with the group ${\operatorname{Hom}}(V,M)$ of linear maps, the subgroup of coboundaries $B^1(V,M)\subset Z^1(V,M)$ is equal to $\{ 0\}$ and therefore $H^1(V,M)={\operatorname{Hom}}(V,M)$. In this case we can reformulate the Theorem \[Main\] in terms of a computation of the map $a^1_{\mathcal F}$.
\[trivial-action-V\] Let $k$ be a prime finite field of characteristic $p$, $V$ be a countable vector space over $k$ acting trivially on $(M, {\mathcal F})=(M^d,{\mathcal A}_d)$ and assume that $p>d+1$. Then the map $a^1: H^1(V,M)={\operatorname{Hom}}(V,M)\to \tilde H_{\mathcal F}^1(V,M)$ is surjective, and ${\operatorname{Ker}}(a^1) = {\operatorname{Hom}}_f(V,M_d)$.
How to describe $H_{\mathcal F}^n(V,M)$ for $n>1$ ?
\[translation-action-V\] Let $k$ be a prime finite field of characteristic $p$, and let $V$ be a countable vector space over $k$. Consider the filtration $(\mathcal P^d, {\mathcal B}_d)$ with $W=V$ and with $V$-acting by translations $(v .P)(x) = P(x+v)$ and assume that $p>d+1$. Then the map $a^1_{\mathcal B}$ is surjective.
Let $P: V \to \mathcal P^d$ where $\mathcal P^d$ is the space of polynomial of degree $\le d$. . We assume $$\partial P(v,v')(x)= P(v+v')(x)-P(v)(x+v')-P(v')(x)$$ is of rank $\le i$ for any $v,v' \in V$. Let $Q(v)$ be the homogeneous degree $d$ term of $P(v)$. Then since $P(v)(x+v')-P(v)(x)$ is of degree $<d$ we have $$Q(v+v')(x)-Q(v)(x)-Q(v')(x)$$ is of rank $\le i+1$.
The proof of Theorem \[Main\] uses the inverse theorem for the Gowers norms [@btz; @tz-inverse]. One can use also prove the reverse implication modifying the arguments in [@sam], and thus an independent proof of Theorem \[Main\] could lead to a new proof of the inverse conjecture for the Gowers norms.
Proof of Theorem \[Main\]
=========================
For a function $f$ on a finite set $X$ we define $${{\mathbb E}}_{x \in X}f(x) = \frac{1}{|X|}\sum_{x \in X}f(x).$$ We use $X \ll_L Y$ to denote the estimate $|X| \le C(L) |Y|$, where the constant $C$ depends only on $L$. We fix a prime finite field $k$ of order $p$ and degree $d<p-1$ and suppress the dependence of all bounds on $k,d$. We also fix a non-trivial additive character $\psi$ on $k$.\
\
For a function $f:G \to H$ a function between abelian groups we denote $\Delta_h f(x) = f(x+h) - f(x)$. if $f:G_1 \times G_2 \to H$ then for $g_1 \in G_1$ we write $\Delta_{g_1} f$ shorthand for $\Delta_{(g_1,0)}f$.\
\
Let $V$ be a finite vector space over $k$. Let $F: V \to k$. The $m$-th [*Gowers norm*]{} of $F$ is defined by $$\|\psi(F)\|_{U_m}^{2^m} = {{\mathbb E}}_{v, v_1, \ldots, v_m \in V} \psi(\Delta_{v_m} \ldots \Delta_{v_1} F(v)).$$ These were introduced by Gowers in [@gowers], and were shown to be norms for $m>1$.\
\
For a homogeneous polynomial $P$ on $V$ of degree $d$ we define $$\label{multilinear}
\tilde P(x_1, \ldots, x_d) = \Delta_{x_d}\ldots \Delta_{x_1}P(x).$$ This is a multilinear homogeneous form in $x_1, \ldots, x_d \in V$ such that $$P(x) = \frac{1}{d!}\tilde P(x,\ldots, x).$$
\[main-finite\] There exists a function $R(L)$ such that for any finite dimensional $k$-vector space $V$ and a map $P : V \to M= M_d$ such that $r(P(v+v') -P(v) -P(v')) \le L$ for $v,v' \in V$ there exists a linear map $Q : V \to M$ such that $r(P( v) - Q(v))\le R(L)$.
The proof is based on an argument of [@gt-equivalence], [@lovett]. Our aim is to show that if $\partial P(v,v')(x)$ is of rank $\le L$ for all $v,v' \in V$ then there exists a homogeneous polynomial $Q(v)(x)$ of degree $\le d$ such that $\partial P(v,v')=0$ and $P(v)-Q(v) \ll_{L} 1$.
Let $P:V^d \to k$ be a multilinear homogeneous polynomial of degree $d \ge 2$ and rank $L$. Then ${{\mathbb E}}_{\bar x \in V} \psi(P(x))\ge C_{L,d}$ for some positive constant $C_{L,d}$ depending only on $L,d$.
We prove this by induction on $d$. For quadratics: $P(x_1,x_2)= \sum_{i=1}^L l^1_i(x_1)l^2_i(x_2)$. If $x_1 \in \bigcap_i {\operatorname{Ker}}(l_i^1)$ then $ \psi(P({\bar x})) \equiv 1$, thus on a subspace $W$ of codimension at most $L$ we have $ \psi(P({\bar x})) \equiv 1$, so that $$\sum_{x_1 \in W} {{\mathbb E}}_{x_2 \in V }\psi(P({\bar x}))=|W|.$$ If $x_1$ is outside $W$ then the inner sum is nonnegative.\
\
Suppose now that $d>2$. Let $P(x_1,\ldots, x_d)$ be multilinear homogeneous polynomial of degree $d$ and rank $L$. Write $$P(x_1, \ldots, x_d) = \sum_{j \le d} \sum_{i\le L_j} l_i^j(x_j)Q_i^j(x_1, \ldots, \hat x_j, \ldots, x_d) + \sum_{k\le M} T_k({\bar x})R_k({\bar x})$$ with ${\bar x} = (x_1, \ldots, x_d)$, $l_i^j$ are linear and $T_k({\bar x})R_k({\bar x})$ is homogenous multilinear in $x_1, \ldots x_d$ such that the degrees of $T_k({\bar x})$ and $R_k({\bar x})$ are $\ge 2$, and $\sum_j L_j+M=L$.\
\
If for all $j$ we have $L_j=0$, then $P({\bar x})= \sum_{k\le L} T_k({\bar x})R_k({\bar x})$ with the degree of $T_k({\bar x}),R_k({\bar x}) \ge 2$. For $x_1 \in V$ write $P_{x_1}(x_2, \ldots, x_d) = P(x_1,x_2, \ldots, x_d)$. Then $P_{x_1}$ is of rank $L$ and degree $d-1$ for all $x_1$ and we obtain the claim by induction.\
\
Otherwise there is a $j$, such that $L_j>0$, without loss of generality $j=1$. Let $W = \bigcap_{i} {\operatorname{Ker}}(l_i^1)$, then $W$ is of codimension at most $L_1$. For $x_1 \in W$ let $P_{x_1}(x_2, \ldots, x_d) = P(x_1,x_2, \ldots, x_d)$. Consider the sum $${{\mathbb E}}_{x_2, \ldots, x_d \in V} \psi(P_{x_1}(x_2, \ldots, x_d)).$$ For $x_1 \in W$ we have $P_{x_1}$ is of rank $\le L-L_1$ and homogeneous of degree $d-1$. By the induction hypothesis the above sum is $\ge C_{L, d-1}$, so that $$\sum_{x_1 \in W} {{\mathbb E}}_{x_2, \ldots, x_d \in V} \psi(P_{x_1}(x_2, \ldots, x_d)) \ge C_{L, d-1} |W|.$$ For any $x_1 \notin W$, $P_{x_1}$ is of degree $d-1$, and of rank $<\infty$ thus by the induction hypothesis, $${{\mathbb E}}_{x_2, \ldots, x_d \in V} \psi(P_{x_1}(x_2, \ldots, x_d)) \ge 0.$$ Thus $${{\mathbb E}}_{x_1 \in V} {{\mathbb E}}_{x_2, \ldots, x_d \in V} \psi(P_{x_1}(x_2, \ldots, x_d)) \ge C_{L, d-1} |W|/|V|,$$ and we obtain the claim.
\
Let $P:V\to M_d$ be a map such that for all $u,v$: $$rk(P(v)+P(u) - P(v+u))<L.$$ We define a function on $V\times W^d$ by $$f(v,x_1, \ldots, x_d) = \psi(\tilde P(v)(x_1, \ldots, x_d)) = \psi(\tilde P(v)({\bar x})).$$
$\|f\|_{U^{d+2}} \ge c_{L}$.
We expand $$\Delta_{(v_{d+2},{\bar h_{d+2}})} \ldots \Delta_{(v_1,{\bar h_1})}\tilde P(v)({\bar x}) = \sum_{k=0}^{d+2} \Delta_{{\bar h_{d+2}}} \ldots \Delta_{{\bar h_{k+1}}}( \Delta_{v_k} \ldots \Delta_{v_1}(\tilde P(v)))({\bar x}+{\bar h_1 + \ldots + h_k})$$ with $v_i \in V$ and ${\bar h_i} \in V^d$. Since $P_v$ is of degree $d$ the above is equal $$\sum_{k=2}^{d+2} \Delta_{{\bar h_{d+2}}} \ldots \Delta_{{\bar h_{k+1}}}( \Delta_{v_k} \ldots \Delta_{v_1} \tilde P(v))({\bar x}+{\bar h_1 + \ldots +\bar h_k})$$ Since $rk(P(v)+P(u) - P(v+u))<L$, for any $v_1+u_1=v_2+u_2$ we have $$rk(P(v_1)+P(u_1) - P(v_2)-P(u_2) )< 2L.$$ that for $k\ge 2$ we have $\Delta_{v_k} \ldots \Delta_{v_1}\tilde P(v)$ is of rank $\ll_L 1$. For fixed $v, v_1, \ldots, v_k$, the above polynomial can be expresses as a multilinear homogeneous polynomial of degree $d$ in $y_1, \ldots, y_d$ with $$y_j= (x_j,h_1^j, \ldots, h_{d+2}^j).$$ which is of rank that is bounded in terms of $L, d$. Now apply previous lemma.
By the inverse theorem for the Gowers norm [@btz; @tz-inverse] there is a polynomial $Q$ on $V^{d+1}$ of degree $d+1$ on with $Q:V \times V^d \to k$ s.t. such that $$|{{\mathbb E}}_{v,x_1, \ldots, x_d }\psi(\tilde P(v)(x_1, \ldots, x_d)-Q(v, x_1, \ldots,x_d)) | \gg_L 1.$$ By an application of the triangle and Cauchy-Schwarz inequalities we obtain $$\begin{aligned}
&|{{\mathbb E}}_{v,x_1, \ldots, x_d }\psi(\tilde P(v)(x_1, \ldots, x_d)-Q(v, x_1, \ldots,x_d)) |^2 \\
& \le \left|{{\mathbb E}}_{v,x_1, \ldots,x_{d-1}}\left| {{\mathbb E}}_{x_d }\psi(\tilde P(v)(x_1, \ldots, x_d)-Q(v, x_1, \ldots,x_d))\right| \right|^2\\
&\le {{\mathbb E}}_{v,x_1, \ldots,x_{d-1}}\left| {{\mathbb E}}_{x_d }\psi(\tilde P(v)(x_1, \ldots, x_d)-Q(v, x_1, \ldots,x_d)) \right|^2\\
&= {{\mathbb E}}_{v,x_1, \ldots,x_{d-1}} {{\mathbb E}}_{x_d,x'_d }\psi(\tilde P(v)(x_1, \ldots, x_d+x'_d) -\tilde P(v)(x_1, \ldots, x_d)-Q(v, x_1, \ldots,x_d+x_d')+Q(v, x_1, \ldots,x_d)) \\
&= {{\mathbb E}}_{v,x_1, \ldots,x_{d-1}} {{\mathbb E}}_{x_d,x'_d }\psi(\tilde P(v)(x_1, \ldots, x_{d-1},x'_d)-\Delta_{x_d'}Q(v, x_1, \ldots,x_d)).
\end{aligned}$$ Where the last equality follows from the fact that $\tilde P$ is homogeneous multilinear form and thus $$\tilde P(v)(x_1, \ldots, x_d+x'_d) -\tilde P(v)(x_1, \ldots, x_d)= \tilde P(v)(x_1, \ldots, x_{d-1},x'_d).$$ Applying Cauchy-Schwarz $d-1$ more times we obtain $${{\mathbb E}}_{v,x_1, x_1', \ldots, x_d,x'_d }\psi(\tilde P_v(x'_1, \ldots, x'_d)-\Delta_{x_1'}\ldots \Delta_{x_d'}Q(v, x_1, \ldots,x_d)) \gg_{L} 1$$ One more application of Cauchy-Schwarz gives $${{\mathbb E}}_{v,v',x_1, x_1', \ldots x_d,x'_d }\psi((\tilde P(v+v')-\tilde P(v))(x'_1, \ldots, x'_d)-\Delta_{v'}\Delta_{x_1'}\ldots \Delta_{x_d'}Q(v, x_1, \ldots,x_d)) \gg_{L} 1.$$ Since $Q$ is a polynomial of degree $d+1$, $\Delta_{v'}\Delta_{x_1'}\ldots \Delta_{x_d'}Q$ is independent of $v,x_1, \ldots, x_d$ so we obtain $$|{{\mathbb E}}_{v,v',x'_1, \ldots, x'_d }\psi((\tilde P(v+v')-\tilde P(v))(x'_1, \ldots, x'_d)-\tilde Q(v', x'_1, \ldots,x'_d)) | \gg_{L} 1$$ with $\tilde Q$ a multilinear homogeneous form in $v', x'_1, \ldots,x'_d$. Denote by $\tilde Q(v)$ the function on $W^d$ given by $\tilde Q(v, \bar x)=Q(v,\bar x)$. Then $${{\mathbb E}}_{v,v'}|{{\mathbb E}}_{x'_1, \ldots, x'_d }\psi((\tilde P(v+v')-\tilde P(v)- \tilde Q(v'))(x'_1, \ldots, x'_d) | \gg_{L} 1.$$ By [@gt-polynomial] (Proposition 6.1) and [@BL] (Lemma 4.17) it follows that for at least $\gg_{L} |V|^2$ values of $v,v'$ we have $ P(v+v')-P(v)- Q'(v')$ is of rank $\ll_{L} 1$, where $Q'(v)(x) = \tilde Q(v)(x, \ldots, x)/d!$ where $Q'(v)$ is linear in $v$. Recall now that $P(v+v')-P(v) - P(v')$ is of rank $\le L$, so that we get a set $E$ of size $\gg_{L} |V|$ of $v$ for which $$P(v) = Q'(v) + R(v)$$ with $R(v)$ of rank $\le L$. Since $E \gg_{L} |V|$, by the Bogolyubov lemma (see e.g. [@wolf])) $2E$ contains a subspace $E'$ of codimension $K \ll_{L} 1$ in $V$. For $v \in E'$, define $$P'(v)=Q'(v).$$ Let $P':V\to M_d$ be any extension of $P'$ linear in $v\in V$. Then $P'(v) - P(v)$ is of rank $\ll_{L} 1$, and $P'(v)$ is a cocycle.
\[nonclassical\] In the case where $p\le d+2$, by the inverse theorem for the Gowers norms over finite fields the polynomial $Q$ in the above argument on $V \times V^d$ would be replaced by a [*nonclassical polynomial*]{} see [@tz-low], and the same argument would give that the approximate cohomology obstructions lie in the nonclassical degree $d$ polynomials - these are functions $P: V \to {\mathbb T}$ satisfying $\Delta_{h_{d+1}} \ldots \Delta_{h_1}P \equiv 0$.
[*Proof of Theorem \[Main\]*]{}. Let $k={\mathbb F}_q$, and let $V$ be an countable vector space over $k$. Denote $V_n=k^n$, then $V=\cup V_n$. Let $M\subset k[x_1,...x_n,...]$ be the subspace of homogeneous polynomials of degree $d$. For any $l\geq 1$ we denote by $N_l\subset M$ the subset of polynomials of in $x_1,...,x_l$ and denote by $p_l:M\to N_l$ the projection defined by $x_i\to 0$ for $ i> l$. Observe that $p_n$ does not increase the rank. By Proposition \[main-finite\] there is a constant $C$ depending only on $L,d$ (and $k$) such that for any $n$ there exists a linear map $\phi _n:V_n\to M$ such that rank $(P(v)-\phi _n (v))\le C$, for $v\in V_n$.
We now show that the existence of such linear maps $\phi _n$ implies the existence of a linear map $\psi :V\to M$ such that $rank (R(v)-\psi (v))\leq C$, for $v\in V$.
\[compatible\] Let $X_n$, $ n\geq 0$ be finite not empty sets and $f_n:X_{n+1}\to X_n$ be maps. Then one can find $x_n\in X_n$ such that $f_n(x_{n+1})=x_n$.
This result is standard, but for the convenience of a reader we provide a proof. The claim is obviously true if the maps $f_n$ are surjective. For any $m>n$ we define the subset $X_{m,n}\subset X_n$ as the image of $$f_n\circ \ldots \circ f_{m-1}:X_m\to X_n$$ It is clear that for a fixed $n$ we have $$X_n\supset X_{n+1,n}\supset \ldots \supset X_{m,n}\supset \ldots$$ \
We define $Y_n$ as the intersection $\cap _{m>n}X_{m,n}$. Since the set $X_n$ is finite, the sets $X_{m,n}$ stabilize as m grows and hence $Y_n$ is not empty. Let ${\tilde}f_n$ be the restriction of $f_n$ on $Y_{n+1}$. Now the maps ${\tilde}f_n:Y_{n+1}\to Y_n$ are surjective, thus the lemma follows.
\[lift\] Let $P:V\to M$ be a map such that for any $n$ there exists a linear map $\phi _n:V_n\to M$ such that rank $(P(v)-\phi _n (v))\leq C$, $v\in V_n$. Then there exists a linear map $\psi :V\to M$ such that $rank (P(v)-\psi (v))\leq C$, for all $v\in V$.
Let $l(n)$ be such that $P(V_n)\subset N_{l(n)}$ and $\psi _n=p_{l(n)}\circ \phi _n$. Since $p_n$ does not increase the rank, and since $(p_{l(n)}\circ P)(V_n)= R(V_n)$ we have $$(\star _n) \qquad rank(P(v)-\psi _n (v))\leq C, \quad v\in V_n.$$ We apply Lemma \[compatible\] to the case when $X_n$ is the set of linear maps $\psi _n:V_n\to N_{l(n)}$ satisfying $(\star _n)$ and $f_n$ are the restriction from $V_{n+1}$ onto $V_n$ we find the existence of linear maps $\psi _n:V_n\to N_{l(n)}$ satisfying the condition $(\star _n)$ and such that the restriction of $\psi _{n+1}$ onto $V_n$ is equal to $\psi _n.$ The system $\{ \psi _n\}$ defines a linear map $\psi :V\to M$.
We now prove the result stated in the abstract by proving the equality $R(V,r,k)=R^f(V,r,k)$. Let $\Gamma$ be an abelian group, $\Gamma =\cup \Gamma _n$ where $\Gamma_n$ are finitely generated groups. Let $k$ be a finite field, and let $V,W$ be $k$-vector spaces with bases $v_j, w_j$. For $n\geq 1$ let $V_n, W_n$ be the spans of $v_j,w_j,1\leq j\leq n$. We denote by $i_n:V_n\to V_{n+1}$ the natural imbedding and by $\beta _n:W_{n+1}\to W_n$ the natural projection. Denote ${\operatorname{Hom}}^f(V,W)$ the finite rank homomorphisms from $V \to W$.
Suppose there exists $C=C(c)$ such that for any map $a^f:\Gamma \to {\operatorname{Hom}}^f(V,W)$ such that $$r(a^f({\gamma}'+{\gamma}'')-a^f({\gamma}')-a^f({\gamma}''))\leq c$$ there exists a homomorphism $\chi^f:{\Gamma}\to {\operatorname{Hom}}^f(V,W)$ such that $r(a^f({\gamma})-\chi ^f({\gamma}))\leq C$. Then for any map $a:{\Gamma}\to {\operatorname{Hom}}(V,W)$ such that $$r(a({\gamma}'+{\gamma}'')-a({\gamma}')-a({\gamma}''))\leq c$$ there exists a homomorphism $\chi :\Gamma \to {\operatorname{Hom}}(V,W)$ such that $r(a({\gamma})-\chi ({\gamma}))\leq C$.
We will use the following fact that is an immediate consequence of König’s lemma : Let $X$ be a locally finite tree, $x\in X$. If for any $N$ there exists a branch starting at $x$ of length $N$ then there exists an infinite branch starting at $x$.\
\
Let $a:\Gamma \to {\operatorname{Hom}}(V,W)$ be a map such that $$r(a({\gamma}'+{\gamma}'')-a({\gamma}')-a({\gamma}''))\leq c$$ We define $F_n={\operatorname{Hom}}(V_n,W_n)$. Let $Y_n={\operatorname{Hom}}(\Gamma _n, F_n)$ and $q_n :Y_{n+1}\to Y_n$ be given by $$q_n (\chi _{n+1})=\beta _n\circ \chi '_{n+1}\circ i_n$$ where $\chi '_{n+1}$ is the restriction of $\chi _{n+1}$ on ${\Gamma}_n$.\
\
We denote by $X_n\subset Y_n$ the subset of homomorphisms $\chi _n$ of $\Gamma _n$ such that $$r(\beta _n\circ a({\gamma})\circ i_n -\beta _n\circ \chi _n({\gamma})\circ i_n )\leq C.$$ Let $X$ be the disjoint union of $X_n$ and we connect $\chi _n\in X_n$ with $\chi _{n+1}\in X_{n+1}$ if $\chi _n=q_n(\chi _{n+1})$.\
\
By the assumption $X_n$ are finite not empty sets and for any $n$ there exists a branch from $X_0$ to $X_n$ (any $\chi _n\in X_n$ defines such a branch). Now the Lemma \[compatible\] implies the existence a character $\chi :\Gamma \to {\operatorname{Hom}}(V,W)$ such that $r(a({\gamma})-\chi ({\gamma}))\leq C$.
To conclude the proof of Theorem \[Main\] we calculate the kennel of the map $a^1$:
\[kernel\] The kernel of $a^1$ consists of maps $P$ of finite rank.
Suppose $V, W$ are of dimension $n_1, n_2$ respectively. All the bounds below are independent of $n_1, n_2$. Suppose $P:V \to M_d$ is a linear map with $r(P) \le L$. Let $\tilde P$ be the multilinear version of $P$ as in . Let $f(v,\bar x) = \psi(\tilde P(v)(\bar x))$. Now $\tilde P(v)(\bar x)$ is a multilinear polynomial on $V\times W^d$ of degree $d+1$.\
\
For any fixed $v$ we have $$\mathbb E_{\bar x \in W^d} \psi(\tilde P(v)(\bar x)) \ge C(L),$$ and thus $$\mathbb E_{v\in V} \mathbb E_{\bar x \in W^d} \psi(\tilde P(v)(\bar x)) \ge C(L).$$ It follows that $\tilde P(v)(\bar x)$ is of bounded rank $\ll_L 1$ and thus of the form $$\tilde P(v)(\bar x) = \sum_{j=1}^K \tilde Q_j(v,\bar x) \tilde R_j(v, \bar x)$$ with $\tilde Q_j, \tilde R_j$ of degree $\ge 1$ ,for any fixed $v$ also $\tilde Q_j, \tilde R_j$ are of degree $\ge 1$, and $K \ll_L 1$. For any fixed $x$, $\tilde P(V)$ is linear and thus either $\tilde Q_j$ or $\tilde R_j$ are constant as a function of $v$. Recall that $P(v)(x) = \frac{1}{d!}\tilde P(v)(x,\ldots, x)$, and let $Q_j(v,x) = \frac{1}{d!}\tilde Q_j(v,x,\ldots, x)$, similarly $R_j$. Let $J$ be the set of $j$ in the sum $P=\sum_j Q_jR_j$ such that $Q_j$ is linear in $v$ and does not depend on $x$. Let $V'=\bigcap _j {\operatorname{Ker}}Q_j$. The restriction of $P$ to $V'$ has finite (that is by a constant which does not depend on $n_1,n_2$ ) rank. Since $codim (V')\leq |J|$ we see that $P$ has rank that is bounded by a constant which does not depend on $n_1$ and $n_2$.\
\
Now let $V, W$ be infinite. Let $V_n, l(n), p_{l(n)}$ be as in Lemma \[lift\]. Len $ P_n(V_n)= (p_{l(n)}\circ P)(V_n)$. Now apply Lemma \[compatible\] for $X_n$ the collection of finite rank maps from $V_n \to N_{l(n)}$, and $f_n$ the restriction as before. This finishes a proof of Theorem \[Main\].
[99]{} Bhowmick A.,Lovett S. [*Bias vs structure of polynomials in large fields, and applications in effective algebraic geometry and coding theory*]{}. Bergelson, V., Tao, T., Ziegler, T. [*An inverse theorem for the uniformity seminorms associated with the action of ${\mathbb F}_p^{\infty}$.*]{} Geom. Funct. Anal. 19 (2010), no. 6, 1539-1596. Gowers, T. [*A new proof of Szemerédi’s theorem*]{}. Geom. Funct. Anal. 11, (2001) 465Ð588. Green, B., Tao, T. [*The distribution of polynomials over finite fields, with applications to the Gowers norms.*]{} Contrib. Discrete Math. 4 (2009), no. 2, 1-36. Green, B., Tao, T. [*An equivalence between inverse sumset theorems and inverse conjectures for the $U_3$ norm.*]{} Math. Proc. Cambridge Philos. Soc. 149 (2010), no. 1, 1-19. Kazhdan, D. On ${\epsilon}$-representations. Israel J. Math. 43 (1982), no. 4, 315-323. Lovett, S. [*Equivalence of polynomial conjectures in additive combinatorics.* ]{} Combinatorica 32 (2012), no. 5, 607 -618. Samorodnitsky, A. [*Low-degree tests at large distances*]{}, STOC Ã07. Tao, T. [*https://terrytao.wordpress.com/2008/11/09/a-counterexample-to-a-strong-polynomial-freiman-ruzsa-conjecture/*]{} Schmidt, W.M. [*Bounds for exponential sums*]{} Acta Arith. 44 (1984), 281-297. Tao, T., Ziegler, T. [*The inverse conjecture for the Gowers norm over finite fields via the correspondence principle.* ]{} Anal. PDE 3 (2010), no. 1, 1-20. Tao, T., Ziegler, T. [*The inverse conjecture for the Gowers norm over finite fields in low characteristic.*]{} Ann. Comb. 16 (2012), no. 1, 121-188. Wolf, J. [*Finite field models in arithmetic combinatorics ten years on*]{}. Finite Fields Appl. 32 (2015), 233-274.
[^1]: The second author is supported by ERC grant ErgComNum 682150
| 2024-07-25T01:26:58.646260 | https://example.com/article/3206 |
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abstract: |
Stochastic discount factor (SDF) processes in dynamic economies admit a permanent-transitory decomposition in which the permanent component characterizes pricing over long investment horizons. This paper introduces econometric methods to extract the permanent and transitory components of the SDF process. We show how to estimate the solution to the Perron-Frobenius eigenfunction problem of [@HS2009] using data on the Markov state and the SDF process. Estimating directly the eigenvalue and eigenfunction allows one to (1) construct empirically the time series of the permanent and transitory components of the SDF process and (2) estimate the yield and the change of measure which characterize pricing over long investment horizons. The methodology is nonparametric, i.e., it does not impose any tight parametric restrictions on the dynamics of the state variables and the SDF process. We derive the large-sample properties of the estimators and illustrate favorable performance in simulations. The methodology is applied to study an economy where the representative agent is endowed with recursive preferences, allowing for general (nonlinear) consumption and earnings growth dynamics.
**Keywords:** Nonparametric estimation, sieve estimation, stochastic discount factor, permanent-transitory decomposition, nonparametric value function estimation.
**JEL codes:** C13, C14, C58.
author:
- '[ Timothy M. Christensen]{}[^1]'
date: 'First version: May 2, 2013. Revised: . '
title: 'Nonparametric Stochastic Discount Factor Decomposition[^2] '
---
Introduction
============
In dynamic asset pricing models, stochastic discount factors (SDFs) are stochastic processes that assign prices to claims to future payoffs over different investment horizons. [@AJ], [@HS2009] and [@Hansen2012] show that SDF processes may be decomposed into permanent and transitory components. The [*permanent component*]{} is a martingale that induces an alternative probability measure which is used to characterize pricing over long investment horizons. The [*transitory component*]{} is related to the return on a discount bond of (asymptotically) long maturity. [@AJ] and the subsequent literature on bounds has found that SDFs must have nontrivial permanent and transitory components in order to explain a number of salient features of historical returns data. [@QL] show that the permanent-transitory decomposition obtains even in very general semimartingale environments, suggesting that the decomposition is a fundamental feature of arbitrage-free asset pricing models.
This paper introduces econometric methods to extract the permanent and transitory components of the SDF process. Specifically, we show how to estimate the solution to the Perron-Frobenius eigenfunction problem introduced by [@HS2009] from a time series of data on state variables and the SDF process. By estimating directly the eigenvalue and eigenfunction, one can reconstruct empirically the time series of the permanent and transitory components and investigate their properties (e.g. the size of the components, their correlation, etc). The methodology also allows one to estimate both the yield and the change of measure which characterizes pricing over long investment horizons. This approach is fundamentally different from existing empirical methods for studying the permanent-transitory decomposition, which produce bounds on various moments of the permanent and transitory components as functions of asset returns [@AJ; @BakshiChabiYo; @BC-YG; @BC-YG:rec].[^3]
The pathbreaking work of [@HS2009] shows that the SDF decomposition may be obtained analytically in Markovian environments by solving a Perron-Frobenius eigenfunction problem. The permanent and transitory components are formed from the SDF, the eigenvalue, and its eigenfunction. The eigenvalue is a long-run discount factor that determines the average yield on long-horizon payoffs. The eigenfunction captures dependence of the price of long-horizon payoffs on the state. The probability measure that characterizes pricing over long investment horizons may be expressed in terms of the eigenfunction and another eigenfunction that is obtained from a time-reversed Perron-Frobenius problem. See [@HS2012; @HS2014], [@BackusChernovZin], [@BHS], and [@QL; @QL:or] for related theoretical developments. The methodology introduced in this paper complements the existing theoretical literature by providing an [*empirical*]{} framework for estimating the Perron-Frobenius eigenvalue and eigenfunctions from a time series of data on the Markov state and the SDF process. Empirical versions of the permanent and transitory components can then be recovered from the estimated eigenvalue and eigenfunction.
The methodology is [*nonparametric*]{}, i.e., it does not place any tight parametric restrictions on the law of motion of state variables or the joint distribution of the state variables and the SDF. This approach is coherent with the existing literature on bounds on the permanent and transitory components, which are derived without placing any parametric restrictions on the joint distribution of the SDF, its permanent and transitory components, and asset returns. This approach is also in line with conventional moment-based estimators for asset pricing models, such as GMM [@Hansen1982] and its various extensions. Examples include conditional moment based estimation methodology of [@AiChen2003] which has been applied to estimate asset pricing models featuring habits [@ChenLudvigson] and recursive preferences [@ChenFavilukisLudvigson] and the extended method of moments methodology of [@GGR2011] which is particularly relevant for derivative pricing.
In structural macro-finance models, SDF processes (and their permanent and transitory components) are determined by both the preferences of economic agents and the dynamics of state variables. Several works have shown that standard preference and state specifications can struggle to explain salient features of historical returns data. [@BackusChernovZin] find that certain specifications appear unable to generate a SDF whose permanent component is large enough to explain historical return premia without also generating unrealistically large spreads between long- and short-term yields. [@BakshiChabiYo] find that historical returns data support positive covariance between the permanent and transitory components, but that this positive association cannot be replicated by workhorse models such as the long-run risks model [@BansalYaron]. The role of dynamics can be subtle, especially with recursive preferences that feature forward-looking components. The nonparametric methodology introduced in this paper may be used together with parametric methods to better understand the roles of dynamics and preferences in building models whose permanent and transitory components have empirically realistic properties.
Of course, if state dynamics are treated nonparametrically then certain forward-looking components, such as the continuation value function under [@EpsteinZin1989] recursive preferences, are not available analytically. We therefore introduce nonparametric estimators of the continuation value function in models with [@EpsteinZin1989] recursive preferences with elasticity of intertemporal substitution (EIS) equal to unity.[^4]$^,$[^5] This class of preferences is used in prominent empirical work, such as [@HansenHeatonLi], and may also be interpreted as risk-sensitive preferences as formulated by [@HansenSargent1995] (see [@Tallarini2000]). We reinterpret the fixed-point problem solved by the value function as a [*nonlinear*]{} Perron-Frobenius problem. In so doing, we draw connections with the literature on nonlinear Perron-Frobenius theory following [@SolowSamuelson].
The methodology is applied to study an environment similar to that in [@HansenHeatonLi]. We assume a representative agent with [@EpsteinZin1989] preferences with unit elasticity of intertemporal substitution. However, instead of modeling consumption and earnings using a homoskedastic Gaussian VAR as in [@HansenHeatonLi], we model consumption growth and earnings growth as a general (nonlinear) Markov process. We recover the time series of the SDF process and its permanent and transitory components without assuming any parametric law of motion for the state. The permanent component is large enough to explain historical returns on equities relative to long-term bonds, strongly countercyclical, and highly correlated with the SDF. We also show that the permanent component induces a probability measure that tilts the historical distribution of consumption and earnings growth towards regions of low earnings and consumption growth and away from regions of high consumption growth. To understand better the role of dynamics, we estimate the permanent and transitory components corresponding to different calibrations of preference parameters. We find that the permanent and transitory components can be positively correlated for high (but not unreasonable) values of risk aversion. In contrast, we find that parametric linear-Gaussian and linear models with stochastic volatility fitted to the same data have permanent and transitory components that are negatively associated. These findings suggest that nonlinear dynamics may have a useful role to play in explaining the long end of the term structure.
Although the paper is presented in the context of SDF decomposition, the methodology can be applied to study more general processes such as the valuation and stochastic growth processes in [@HansenHeatonLi], [@HS2009], and [@Hansen2012].
The sieve approach that we use for estimation builds on earlier work on nonparametric estimation of Markov diffusions by [@CHS2000] and [@Gobetetal]. This approach approximates the infinite-dimensional eigenfunction problem by a low-dimensional matrix eigenvector problem whose solution is trivial to compute.[^6] This approach also sidesteps nonparametric estimation of the transition density of the state.
The main theoretical contributions of the paper may be summarized as follows. First, we study formally the large-sample properties of the estimators of the Perron-Frobenius eigenvalue and eigenfunctions. We show that the estimators are consistent, establish convergence rates for the function estimators, and establish asymptotic normality of the eigenvalue estimator and estimators of other functionals. These large-sample properties are established in a manner that is sufficiently general that it can allow for the SDF process to be either of a known functional form or containing components that are first estimated from data (such as preference parameters and continuation value functions). Although the analysis is confined to models in which the state vector is observable, the main theoretical result applies equally to models in which components of the state are latent. Second, semiparametric efficiency bounds for the eigenvalue and related functionals are also derived for the case in which the SDF is of a known functional form and the estimators are shown to attain their bounds. Third, we establish consistency and convergence rates for sieve estimators of the continuation value function for a class of models with recursive preferences.
The derivation of the large sample properties of the eigenfunction/value estimators is nonstandard as the eigenfunction/value are defined implicitly by an unknown, nonselfadjoint operator. The literature on nonparametric eigenfunction estimation to date has focused almost exclusively on the selfadjoint case (see [@CHS2000] and [@Gobetetal] for sieve estimation and [@DFR], [@DFG], and [@CFR2007] for a kernel approach). The exception is [@LintonLewbelSrisuma2011] and [@EHLLS] who establish asymptotic normality of kernel-based eigenfunction and eigenvalue estimators for nonparametric Euler equation models.[^7] The derivation of convergence rates for the time-reversed eigenfunction, semiparametric efficiency bounds, and asymptotic normality of the eigenvalue estimator (and estimators of related functionals) are all new.
The remainder of the paper is as follows. Section \[s:setup\] briefly reviews the theoretical framework in [@HS2009] and related literature and discusses both the scope of the analysis and identification issues. Section \[s:est\] introduces the estimators of the eigenvalue, eigenfunctions, and related functionals and establishes their large-sample properties. Nonparametric continuation value function estimation is studied in Section \[s:recursive\]. Section \[s:mc\] presents a simulation exercise which illustrates favorable finite-sample performance of the estimators. Section \[s:emp\] presents the empirical application and Section \[s:conc\] concludes. An appendix contains additional results on estimation, identification, and all proofs.
Setup {#s:setup}
=====
Theoretical framework
---------------------
This subsection summarizes the theoretical framework in [@AJ], [@HS2009] (HS hereafter), [@Hansen2012], and [@BHS] (BHS hereafter). We work in discrete time with $T$ denoting the set of non-negative integers.
In arbitrage-free environments, there is a positive [*stochastic discount factor*]{} process $M = \{M_t : t \in T\}$ that satisfies: $$\label{e:ee}
{\mathbb{E}} \Big[ \frac{M_{t+\tau}}{M_t} R_{t,t+\tau} \Big| {\mathcal{I}}_t \Big] = 1$$ where $R_{t,t+\tau}$ is the (gross) return on a traded asset over the period from $t$ to $t+\tau$, ${\mathcal{I}}_t$ denotes the information available to all investors at date $t$, and ${\mathbb{E}}[\,\cdot\,]$ denotes expectation with respect to investors’ beliefs (see, e.g., [@HansenRenault]). Throughout this paper, we impose rational expectations by assuming that investors’ beliefs agree with the data-generating probability measure.
[@AJ] introduce the *permanent-transitory decomposition*: $$\begin{aligned}
\label{e:decomp}
\frac{M_{t+\tau}}{M_t} & = \frac{M_{t+\tau}^P}{M_t^P} \frac{M_{t+\tau}^T}{M_t^T}\,.\end{aligned}$$ The permanent component $M^P_{t+\tau}/M^P_t$ is a martingale: ${\mathbb{E}}[M_{t+\tau}^P/M_t^P | {\mathcal{I}}_t] = 1$ (almost surely). HS show that the martingale induces an alternative probability measure which is used to characterize pricing over long investment horizons. The transitory component $M_{t+\tau}^T/M_t^T$ is the reciprocal of the return to holding a discount bond of (asymptotically) long maturity from date $t$ to date $t + \tau$. [@AJ] provide conditions under which the permanent and transitory components exist. [@QL] show that the decomposition obtains in very general semimartingale environments.
To formally introduce the framework in HS and BHS, consider a probability space $(\Omega,{\mathscr{F}}, {\mathbb{P}})$ on which there is a time homogeneous, strictly stationary and ergodic Markov process $X = \{X_t : t \in T\}$ taking values in ${\mathcal{X}} \subseteq {\mathbb{R}}^d$. Let ${\mathcal{F}} = \{ {\mathcal{F}}_t : t \in T\} \subseteq {\mathscr{F}}$ be the filtration generated by the histories of $X$. When we consider payoffs that depend only on future values of the state and allow trading at intermediate dates, we may assume the SDF process is a [*positive multiplicative functional*]{} of $X$. That is, $M_t$ is adapted to ${\mathcal{F}}_t$, $M_t > 0$ for each $t \in T$ (almost surely) and: $$M_{t + \tau} (\omega) = M_t (\omega) M_\tau (\theta_t (\omega))$$ where $\theta_t : \Omega \to \Omega$ is the time-shift operator given by $X_\tau(\theta_t(\omega)) = X_{t+\tau}(\omega)$ for each $\tau,t \in T$.[^8] Define the operators $\{{\mathbb{M}}_\tau : \tau \in T\}$ which assign date-$t$ prices to date-$(t+\tau)$ payoffs $\psi(X_{t+\tau})$ by: $$\label{e:mtau:def}
{\mathbb{M}}_\tau \psi(x) = {\mathbb{E}} \Big[ \frac{M_{t+\tau}}{M_t} \psi(X_{t+\tau}) \Big| X_t = x \Big]\,.$$ It follows from the time-homogeneous Markov structure of $X$ and the multiplicative functional property of $M$ that ${\mathbb{M}}_\tau = {\mathbb{M}}^\tau$ for each $\tau \in T$, where: $$\begin{aligned}
{\mathbb{M}} \psi(x) & = {\mathbb{E}} \Big[ m(X_t,X_{t+1}) \psi(X_{t+\tau}) \Big| X_t = x \Big] \label{e:m:def} \\
\frac{M_{t+1}}{M_t} & = m(X_t,X_{t+1}) \label{e:sdf:m}\end{aligned}$$ for some positive function $m$. For convenience, we occasionally refer to $m$ as the SDF.
HS introduce and study the Perron-Frobenius eigenfunction problem: $$\label{e:pev}
{\mathbb{M}} \phi(x) = \rho \phi(x)$$ where the eigenvalue $\rho$ is a positive scalar and the eigenfunction $\phi$ is positive.[^9] We are also interested in the time-reversed Perron-Frobenius problem: $$\label{e:pev:star}
{\mathbb{M}}^* \phi^*(x) = \rho \phi^*(x)$$ where ${\mathbb{M}}^* \psi(x) = {\mathbb{E}}[ m(X_t,X_{t+1}) \psi(X_t) | X_{t+1} = x]$, $\rho$ is the eigenvalue from (\[e:pev\]), and the eigenfunction $\phi^*$ is positive. The permanent and transitory components are constructed using $\rho$ and $\phi$. The function $\phi^*$ will play a role in characterizing the long-run pricing approximation and the asymptotic variance of estimators of $\rho$. Clearly $\rho$, $\phi$ and $\phi^*$ are determined by both the form of the SDF process and the dynamics of the state process. We will assume that the function $m$ is either pre-specified by the researcher or has been estimated from data on asset returns. However, we will not restrict the dynamics of $X$ to be of any parametric form.
Given $\rho$ and $\phi$ which solve the Perron-Frobenius problem (\[e:pev\]), we may define: $$\begin{aligned}
\label{e:pctc}
\frac{M_{t+\tau}^P}{M_t^P} & = \rho^{-\tau} \frac{M_{t+\tau}}{M_t} \frac{\phi(X_{t+\tau})}{\phi(X_t)} & \frac{M_{t+\tau}^T}{M_t^T} & = \rho^\tau \frac{\phi(X_t)}{\phi(X_{t+\tau})}\,.\end{aligned}$$ It follows from (\[e:pev\]) that ${\mathbb{E}}[M_{t+\tau}^P/M_t^P | {\mathcal{F}}_t] = 1$ (almost surely) for each $\tau,t \in T$. HS show that there may exist multiple solutions to (\[e:pev\]), but only one solution leads to processes $M^P$ and $M^T$ that may be interpreted correctly as permanent and transitory components. Such a solution has a martingale term that induces a change of measure under which $X$ is *stochastically stable*; see Condition 4.1 in BHS for sufficient conditions. Loosely speaking, stochastic stability requires that conditional expectations under the distorted probability measure converge (as the horizon increases) to an unconditional expectation ${\widetilde{{\mathbb{E}}}}[\,\cdot\,]$. The expectation ${\widetilde{{\mathbb{E}}}}[\,\cdot\,]$ will typically be different from the expectation ${{\mathbb{E}}}[\,\cdot\,]$ associated with the stationary distribution of $X$. Under stochastic stability, the one-factor representation: $$\label{e:lrr}
\lim_{\tau \to \infty} \rho^{-\tau} {\mathbb{M}}_\tau \psi(x) = {\widetilde{{\mathbb{E}}}} \left[ \frac{\psi(X_t)}{\phi(X_t)} \right] \phi(x)$$ holds for each $\psi$ for which $ {\widetilde{{\mathbb{E}}}} [ {\psi(X_t)}/{\phi(X_t)} ]$ is finite.[^10] When a result like (\[e:lrr\]) holds, we may interpret $M_{t+\tau}^P/M_t^P$ and $M_{t+\tau}^T/M_t^T$ from (\[e:pctc\]) as the permanent and transitory components. We may also interpret $ -\log \rho$ as the *long-run yield*. Further, the result shows that $\phi$ captures state dependence of long-horizon asset prices.
The theoretical framework of HS may be used to characterize properties of the permanent and transitory components [analytically]{} by solving the Perron-Frobenius eigenfunction problem. Below, we describe an [empirical]{} framework to estimate the eigenvalue and eigenfunctions from time series data on $X$ and the SDF process.
Scope of the analysis
---------------------
The Markov state vector $X_t$ is assumed throughout to be fully observable to the econometrician. However, we do not constrain the transition law of $X$ to be of any parametric form.[^11] Appendix \[ax:filter\] discusses potential extensions of our methodology to models in which $X_t$ or a subvector of $X_t$ is unobserved by the econometrician. The main theoretical results on consistency and convergence rates (Theorem \[t:rate\] and \[t:fpest\]) apply equally to such cases.
We assume the SDF function $m$ is either observable or known up to some parameter which is first estimated from data on $X$ (and possibly asset returns).
#### Case 1: SDF is observable
Here the functional form of $m$ is known ex ante. For example, consider the CCAPM with time discount parameter $\beta$ and risk aversion parameter $\gamma$ both pre-specified by the researcher. Here we would simply take $m(X_t,X_{t+1}) = \beta G_{t+1}^{-\gamma}$ provided consumption growth $G_{t+1}$ is of the form $G_{t+1} = G(X_t,X_{t+1})$ for some function $G$. Other structural examples include models with external habits and models with durables with pre-specified preference parameters.
#### Case 2: SDF is estimated
Here the $m$ is of the form $m(X_t,X_{t+1};\alpha_0)$ where the functional form of $m$ is known up to a parameter $\alpha_0$. Here $\alpha_0$ could be of several forms:
- A finite-dimensional vector of preference parameters in structural models (e.g. [@HansenSingleton1982] and [@HHY]) or risk-premium parameters in reduced-form models (e.g. [@GGR2011]).
- A vector of parameters $\theta_0$ together with a function $h_0$, so $\alpha_0 = (\theta_0,h_0)$. One example is models with [@EpsteinZin1989] recursive preferences, where the continuation value function is not known when the transition law of the Markov state is modeled nonparametrically (see [@ChenFavilukisLudvigson] and the application in Section \[s:emp\]). For such models, $\theta_0$ would consist of discount, risk aversion, and intertemporal substitution parameters and $h_0$ would be the continuation value function. Another example is [@ChenLudvigson] in which $\theta_0$ consists of time discount and homogeneity parameters and $h_0$ is a nonparametric internal or external habit formation component.
- We could also take $\alpha_0$ to be $m$ itself, in which case $\hat \alpha$ would be a nonparametric estimate of the SDF. Prominent examples include [@BansalViswanathan], [@AitSahaliaLo], and [@RosenbergEngle].
In this second case, we consider a two-step approach to SDF decomposition. In the first step we estimate $\alpha_0$ from a time series of data on the state, and possibly also contemporaneous data on asset returns. In the second step we plug the first-stage estimator $\hat \alpha$ into the nonparametric procedure to recover $\rho$, $\phi$, $\phi^*$, and related quantities.
Identification {#s:id}
--------------
In this section we present some sufficient conditions that ensure there is a unique solution to the Perron-Frobenius problems (\[e:pev\]) and (\[e:pev:star\]). The conditions also ensure that a long-run approximation of the form (\[e:lrr\]) holds. Therefore, the resulting $M^P$ and $M^T$ constructed from $\rho$ and $\phi$ as in (\[e:pctc\]) may be interpreted correctly as the permanent and transitory components.
HS and BHS established very general identification, existence and long-run approximation results that draw upon Markov process theory. The operator-theoretic conditions that we use are more restrictive than the conditions in HS and BHS but they are convenient for deriving the large-sample theory that follows. Specifically, the conditions ensure certain continuity properties of $\rho$, $\phi$ and $\phi^*$ with respect to perturbations of the operator ${\mathbb{M}}$. Our results are also derived for the specific parameter (function) space that is relevant for estimation, whereas the results in HS and BHS apply to a larger class of functions. Connections between our conditions and the conditions in HS and BHS are discussed in detail in Appendix \[ax:id\], which also treats separately the issues of identification, existence, and long-run approximation. For estimation, all that we require is for the conclusions of Proposition \[p:id\] below hold. Therefore, the following conditions could be replaced by other sets of sufficient conditions.
Let $L^2$ denote the space of all measurable $\psi : {\mathcal{X}} \to {\mathbb{R}}$ such that $\int \psi^2 \, {\mathrm{d}} Q < \infty$, where $Q$ denotes the stationary distribution of $X$. Our parameter space for $\phi$ is the set of all positive functions in $L^2$. Let $\|\cdot\|$ and $\langle \cdot,\cdot \rangle$ denote the $L^2$ norm and inner product. We say that ${\mathbb{M}}$ is bounded if $\|{\mathbb{M}}\| := \sup\{\|{\mathbb{M}} \psi\| : \psi \in L^2 , \|\psi\| = 1 \} < \infty$ and compact if ${\mathbb{M}}$ maps bounded sets into relatively compact sets. Finally, let $Q \otimes Q$ denote the product measure on ${\mathcal{X}}^2$.
\[a:id:0\] Let ${\mathbb{M}}_\tau$ in (\[e:mtau:def\]) and ${\mathbb{M}}$ in (\[e:m:def\]) satisfy the following:
1. ${\mathbb{M}}$ is a bounded linear operator of the form: $${\mathbb{M}} \psi(x_t) = \int {\mathcal{K}}_m (x_t,x_{t+1}) \psi (x_{t+1}) \, {\mathrm{d}} Q(x_{t+1})$$ for some (measurable) ${\mathcal{K}}_{m}: {\mathcal{X}}^2 \to {\mathbb{R}}$ that is positive ($Q \otimes Q$-almost everywhere)
2. ${\mathbb{M}}_\tau$ is compact for some $\tau \in T$.
[**Discussion of assumptions:**]{} Part (a) are mild boundedness and positivity conditions. If the unconditional density $f(x_t)$ and the transition density $f(x_{t+1}|x_t)$ of $X$ exist, then ${\mathcal{K}}_m$ is of the form: $${\mathcal{K}}_m (x_t,x_{t+1}) = m(x_t,x_{t+1}) \frac{f(x_{t+1}|x_t)}{f(x_{t+1})}\,.$$ In this case, the positivity condition will hold provided $m$ and the densities are positive (almost everywhere). Part (b) is weaker than requiring ${\mathbb{M}}$ itself to be compact.
To introduce the identification result, let $\sigma({\mathbb{M}}) \subset {\mathbb{C}} $ denote the spectrum of ${\mathbb{M}}$ (see, e.g., Chapter VII in [@DunfordSchwartz]). We say that $\rho$ is *simple* if it has a unique eigenfunction (up to scale) and *isolated* if there exists a neighborhood $N$ of $\rho$ such that $\sigma({\mathbb{M}}) \cap N = \{\rho\}$. As $\phi$ and $\phi^*$ are defined up to scale, we say that $\phi$ and $\phi^*$ are unique if they are unique up to scale, i.e. if $\zeta \in L^2$ is a positive eigenfunction of ${\mathbb{M}}$ then $\zeta = s \phi$ (almost everywhere) for some $s > 0$.
\[p:id\] Let Assumption \[a:id:0\] hold. Then:
1. There exists positive functions $\phi,\phi^* \in L^2$ and a positive scalar $\rho$ such that $(\rho,\phi)$ solves (\[e:pev\]) and $(\rho,\phi^*)$ solves (\[e:pev:star\]).
2. The functions $\phi$ and $\phi^*$ are the unique positive solutions (in $L^2$) to (\[e:pev\]) and (\[e:pev:star\]).
3. The eigenvalue $\rho$ is simple and isolated and it is the largest eigenvalue of ${\mathbb{M}}$.
4. The representation (\[e:lrr\]) holds for all $\psi \in L^2$ with ${\widetilde{{\mathbb{E}}}}$ given by: $${\widetilde{{\mathbb{E}}}} \left[ \frac{\psi(X_t)}{\phi(X_t)} \right] = {\mathbb{E}}\left[ \psi (X_t) \phi^*(X_t) \right]$$ under the scale normalization ${\mathbb{E}}[\phi(X_t)\phi^*(X_t)] = 1$.
Parts (a) and (b) are existence and identification results. Part (c) guarantees that $\rho$ is isolated and simple, which is used extensively in the derivation of the large sample theory. Finally, part (d) says that $\rho$ and $\phi$ are the relevant eigenvalue-eigenfunction pair for constructing the permanent and transitory components and links the expectation ${\widetilde{{\mathbb{E}}}}$ to $\phi^*$. In particular, if ${\widetilde{Q}}$ denotes the expectation under which ${\widetilde{{\mathbb{E}}}}$ is calculated, then the change of measure between ${\widetilde{Q}}$ and $Q$ is: $$\label{e:rnderiv}
\frac{{\mathrm{d}} {\widetilde{Q}}(x)}{{\mathrm{d}} Q(x)} = \phi(x)\phi^*(x)$$ under the scale normalization ${\mathbb{E}}[\phi(X_t)\phi^*(X_t)] = 1$. Therefore, estimating $\phi$ and $\phi^*$ directly allows one to estimate this change of measure.
The above identification result is based on an extension of the classical Krein-Rutman theorem in the mathematics literature. Recently, similar operator-theoretic results have been applied to study identification in nonparametric Euler equation models (see [@EscancianoHoderlein], [@LintonLewbelSrisuma2011], and [@EHLLS]) and semiparametric Euler equation models featuring habits [@Chenetal2012]. Time-reversed Perron-Frobenius problems and long-run approximation do not feature in these other works, whereas they are important from our perspective. Identification under weaker, but related, operator-theoretic conditions is studied in [@Christensen-idpev].
Estimation {#s:est}
==========
This section introduces the estimators of the Perron-Frobenius eigenvalue $\rho$ and eigenfunctions $\phi$ and $\phi^*$ and presents the large-sample properties of the estimators.
Sieve estimation
----------------
We use a sieve approach in which the infinite-dimensional eigenfunction problem is approximated by a low-dimensional matrix eigenvector problem whose solution is easily estimated. This methodology is an empirical counterpart to projection methods in numerical analysis (see, e.g., Chapter 11 in [@Judd]). Let $b_{k1},\ldots,b_{kk} \in L^2$ be a dictionary of linearly independent basis functions (polynomials, splines, wavelets, Fourier basis, etc) and let $B_k \subset L^2$ denote the linear subspace spanned by $b_{k1},\ldots,b_{kk}$. The sieve dimension $k < \infty$ is a smoothing parameter chosen by the econometrician and should increase with the sample size. Let $\Pi_k : L^2 \to B_k$ denote the orthogonal projection onto $B_k$. Consider the projected eigenfunction problem: $$\label{e:symprob}
(\Pi_k \mathbb M) \phi_k = \rho_k \phi_k$$ where $\rho_k$ is the largest real eigenvalue of $\Pi_k \mathbb M$ and $\phi_k : {\mathcal{X}} \to {\mathbb{R}}$ is its eigenfunction. Under regularity conditions, $\phi_k$ is unique (up to scale) for all $k$ large enough (see Lemma \[lem:exist\]).
The solution $\phi_k$ to (\[e:symprob\]) must belong to the space $B_k$. Therefore, $\phi_k(x) = b^k(x)' c_k$ for a vector $c_k \in {\mathbb{R}}^k$, where $b^k(x) = (b_{k1}(x),\ldots,b_{kk}(x))'$. We may rewrite the eigenvalue problem in display (\[e:symprob\]) as:[^12] $${\mathbf{G}}^{-1}_k {\mathbf{M}}_k^{\phantom {1}} c_k = \rho_k c_k$$ where the $k \times k$ matrices ${\mathbf{G}}_k$ and ${\mathbf{M}}_k$ are given by: $$\begin{aligned}
{\mathbf{G}}_k &= & {\mathbb{E}}[b^k(X_t) b^k(X_t)'] \label{e:gmat} \\
{\mathbf{M}}_k &=& {\mathbb{E}}[b^k(X_t) m(X_t,X_{t+1}) b^k(X_{t+1})'] \label{e:mmat}\end{aligned}$$ and where $\rho_k$ is the largest real eigenvalue of ${\mathbf{G}}_k^{-1} {\mathbf{M}}_k^{\phantom{1}}$ and $c_k$ is its eigenvector. We refer to $\phi_k(x) = b^k(x)'c_k$ as the *approximate solution* for $\phi$. The approximate solution for $\phi^*$ is $\phi_k^*(x) = b^k(x)'c_k^{*}$ where $c_k^*$ is the eigenvector of ${\mathbf{G}}^{-1}_k {\mathbf{M}}_k^{\prime}$ corresponding to $\rho_k$. Together, $(\rho_k,c_k^{\phantom *},c_k^*)$ solve the generalized eigenvector problem:$$\label{e:gev}
\begin{array}{rcl}
{\mathbf{M}}_k c_k & = & \rho_k {\mathbf{G}}_k c_k \\
c_k^{*\prime} {\mathbf{M}}_k & = & \rho_k c_k^{*\prime} {\mathbf{G}}_k
\end{array}$$ where $\rho_k$ is the largest real generalized eigenvalue of the pair $({\mathbf{M}}_k,{\mathbf{G}}_k)$. We suppress dependence of ${\mathbf{M}}_k$ and ${\mathbf{G}}_k$ on $k$ hereafter to simplify notation.
To estimate $\rho$, $\phi$ and $\phi^*$, we solve the sample analogue of (\[e:gev\]), namely: $$\label{e:est}
\begin{array}{rcl}
{\widehat{{\mathbf{M}}}} \hat c & = & \hat \rho {\widehat{{\mathbf{G}}}} \hat c \\
\hat c^{* \prime} {\widehat{{\mathbf{M}}}} & = & \hat \rho \hat c^{* \prime} {\widehat{{\mathbf{G}}}}
\end{array}$$ where ${\widehat{{\mathbf{M}}}}$ and ${\widehat{{\mathbf{G}}}}$ are described below and where $\hat \rho$ is the maximum real generalized eigenvalue of the matrix pair $({\widehat{{\mathbf{M}}}},{\widehat{{\mathbf{G}}}})$.[^13] The estimators of $\phi$ and $\phi^*$ are: $$\begin{aligned}
\hat \phi(x) & = b^k(x)' \hat c \\
\hat \phi^*(x) & = b^k(x)' \hat c^*\,.\end{aligned}$$ Under the regularity conditions below, the eigenvalue $\hat \rho$ and its right- and left-eigenvectors $\hat c$ and $\hat c^*$ will be unique with probability approaching one (see Lemma \[lem:exist:hat\]).[^14]
Given a time series of data $\{X_0,X_1,\ldots,X_n\}$, a natural estimator for ${\mathbf{G}}$ is: $$\label{e:ghat}
{\widehat{{\mathbf{G}}}} = \frac{1}{n} \sum_{t=0}^{n-1} b^k(X_t)b^k(X_t)'\,.$$ We consider two possibilities for estimating ${\mathbf{M}}$.
#### Case 1: SDF is observable
First, consider the case in which the function $m(X_t,X_{t+1})$ is specified by the researcher. In this case our estimator of ${\mathbf{M}}$ is $$\label{e:mhat1}
{\widehat{{\mathbf{M}}}} = \frac{1}{n} \sum_{t=0}^{n-1} b^k(X_t)m(X_t,X_{t+1})b^k(X_{t+1})'\,.$$
#### Case 2: SDF is estimated
Now suppose that the SDF is of the form $m(X_t,X_{t+1};\alpha_0)$ where the functional form of $m$ is known up to the parameter $\alpha_0$ which is to be estimated first from the data on $X$ and possibly also asset returns. Let $\hat \alpha$ denote this first-stage estimator. In this case, we take: $$\label{e:mhat2}
{\widehat{{\mathbf{M}}}} = \frac{1}{n} \sum_{t=0}^{n-1} b^k(X_t) m(X_t,X_{t+1};\hat \alpha)b^k(X_{t+1})'\,.$$
### Other functionals
Recall that the [*long-run yield*]{} is $y = -\log \rho$. We may estimate $y$ using: $$\label{e-rhohat}
\hat y = - \log \hat \rho \,.$$ We may also estimate the size of the permanent component. The [*entropy*]{} of the permanent component, namely $L := \log {\mathbb{E}}[M^P_{t+1}/M^P_t] - {\mathbb{E}}[\log(M^P_{t+1}/M^P_t)]$, is known to be bounded from below by the expected excess return of any traded asset relative to a discount bond of asymptotically long maturity [@AJ Proposition 2]. Previous empirical work has estimated this bound from data on equity returns and (proxies for) holding period returns for long-maturity discount bonds (see, e.g., [@AJ] and [@BakshiChabiYo]). In contrast, here we can estimate directly the entropy and do so without data on returns on long-maturity discount bonds (though we require data on $X$ and must take a stand on the SDF process). In Markovian environments, the entropy has the simple form $L = \log \rho - {\mathbb{E}}[ \log m(X_t,X_{t+1})]$ (see [@Hansen2012] and [@BackusChernovZin]). Given $\hat \rho$, a natural estimator of $L$ is: $$\label{e:Lhat1}
\hat L = \displaystyle \log \hat \rho - \frac{1}{n} \sum_{t=0}^{n-1} \log m(X_t,X_{t+1})$$ in Case 1; in Case 2 we replace $m(X_t,X_{t+1})$ by $m(X_t,X_{t+1};\hat \alpha)$ in (\[e:Lhat1\]). The size of the permanent component may also be measured by other types of statistical discrepancies besides entropy (e.g. Cressie-Read divergences) which may be computed from the time series of the permanent component recovered empirically using $\hat \rho$ and $\hat \phi$. We confine our attention to entropy because the theoretical literature has typically used entropy to measure the size of SDFs and their permanent components over different horizons (see, e.g., [@Hansen2012] and [@BackusChernovZin]) and for sake of comparison with the empirical literature on bounds.
Consistency and convergence rates
---------------------------------
Here we establish consistency of the estimators and derive the convergence rates of the eigenfunction estimators under mild regularity conditions.
\[a:id\] ${\mathbb{M}}$ is bounded and the conclusions of Proposition \[p:id\] hold.
\[a:bias\] $\|\Pi_k {\mathbb{M}} - {\mathbb{M}}\| =o(1)$.
Let ${\mathbf{G}}^{-1/2}$ denote the inverse of the positive definite square root of ${\mathbf{G}}$ and ${\mathbf{I}}$ denote the $k \times k$ identity matrix. Define the “orthogonalized” matrices ${\mathbf{M}}^o = {\mathbf{G}}^{-1/2} {\mathbf{M}} {\mathbf{G}}^{-1/2}$, ${\widehat{{\mathbf{G}}}}^o = {\mathbf{G}}^{-1/2} {\widehat{{\mathbf{G}}}} {\mathbf{G}}^{-1/2}$, and ${\widehat{{\mathbf{M}}}}^o = {\mathbf{G}}^{-1/2} {\widehat{{\mathbf{M}}}} {\mathbf{G}}^{-1/2}$. Let $\|\cdot\|$ also denote the Euclidean norm when applied to vectors and the operator norm (largest singular value) when applied to matrices.
\[a:var\] $\|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| = o_p(1)$ and $\|{\widehat{{\mathbf{M}}}}^o - {{\mathbf{M}}}^o\| = o_p(1)$.
[**Discussion of assumptions:**]{} Assumption \[a:bias\] requires that the space $B_k$ be chosen such that it approximates well the range of ${\mathbb{M}}$ (as $k \to \infty$). Similar assumptions are made in the literature on projection methods. Assumption \[a:bias\] requires that ${\mathbb{M}}$ is compact, as has been assumed previously in the literature on sieve estimation of eigenfunctions (see, e.g., [@Gobetetal]).[^15] Assumption \[a:var\] ensures that the sampling error in estimating ${{\mathbf{G}}}^{-1} {\mathbf{M}}$ vanishes asymptotically. This condition implicitly restricts the maximum rate at which $k$ can grow with $n$, which will be determined by both the type of sieve and the weak dependence properties of the data. See Appendix \[ax:est:mat\] for sufficient conditions. Note that ${\widehat{{\mathbf{G}}}}^o$ and ${\widehat{{\mathbf{M}}}}^o$ are a proof device and do not need to be calculated in practice.
Before presenting the main result on convergence rates, we first we introduce the sequence of positive constants $\delta_k^{\phantom *}$, $\delta_k^*$, $\eta_{n,k}^{\phantom *}$, and $\eta_{n,k}^*$ that bound the approximation bias ($\delta_k$ and $\delta_k^*$) and sampling error ($\eta_{n,k}$ and $\eta_{n,k}^*$). As eigenfunctions are only normalized up to scale, in what follows we impose the normalizations $\|\phi\| = 1$ and $\|\phi^*\| = 1$. Define: $$\label{e:deltas}
\delta_k^{\phantom *} = \| \Pi_k \phi - \phi\| \quad \mbox{and} \quad \delta_k^* = \| \Pi_k \phi^* - \phi^*\|\,.$$ The quantities $\delta_k^{\phantom *}$ and $\delta_k^*$ measure the bias incurred by approximating $\phi$ and $\phi^*$ by elements of $B_k$.[^16] Let $\tilde c_k = {\mathbf{G}}^{1/2} c_k$ and $\tilde c_k^* = {\mathbf{G}}^{1/2} c_k^*$ and normalize $c_k$ and $c_k^*$ so that $\|\tilde c_k\| = 1$ and $\|\tilde c_k^*\| = 1$ (these normalizations are equivalent to setting $\|\phi_k\| = 1$ and $\|\phi_k^*\| = 1$). Under Assumption \[a:var\], we may choose positive sequences $\eta_{n,k}$ and $\eta_{n,k}^*$ which are both $o(1)$, so that: $$\label{e:etas}
\|(({\widehat{{\mathbf{G}}}}^o)^{-1}{\widehat{{\mathbf{M}}}}^o - {{\mathbf{M}}}^o) \tilde c_k^{\phantom *}\| = O_p(\eta_{n,k}^{\phantom *}) \quad \mbox{and} \quad \|(({\widehat{{\mathbf{G}}}}^o)^{-1}{\widehat{{\mathbf{M}}}}^{o\prime} - {{\mathbf{M}}}^{o\prime}) \tilde c_k^*\| = O_p(\eta_{n,k}^{ *})\,.$$ (see Lemma \[lem:matcgce\] in the Appendix for further details). The terms $\eta_{n,k}^{\phantom *}$ and $\eta_{n,k}^*$ will typically be increasing in $k$ and decreasing in $n$: with more data we can estimate the entries of ${\mathbf{G}}$ and ${\mathbf{M}}$ more precisely, but with larger $k$ there are more parameters to estimate.
\[t:rate\] Let Assumptions \[a:id\]–\[a:var\] hold. Then:
1. $|\hat \rho - \rho| = O_p( \delta_k + \eta_{n,k})$
2. $\|\hat \phi - \phi\| = O_p( \delta_k + \eta_{n,k})$
3. $\|\hat \phi^* - \phi^*\| = O_p( \delta_k^* + \eta_{n,k}^*)$
where $\delta_k^{\phantom *}$ and $\delta_k^*$ are defined in (\[e:deltas\]) and $\eta_{n,k}^{\phantom *}$ and $\eta_{n,k}^*$ are defined in (\[e:etas\]). The convergence rates for $\hat \phi$ and $\hat \phi^*$ should be understood to hold under the scale normalizations $\|\phi\| = 1$, $\|\hat \phi\| = 1$, $\| \phi^* \| = 1$ and $\|\hat \phi^* \| = 1$ and sign normalizations $\langle \phi,\hat \phi \rangle \geq 0$ and $\langle \phi^*, \hat \phi^* \rangle \geq 0$.
\[rmk:gen\] Theorem \[t:rate\] holds for $\hat \rho$, $\hat \phi$ and $\hat \phi^*$ calculated from any estimators ${\widehat{{\mathbf{G}}}}$ and ${\widehat{{\mathbf{M}}}}$ that satisfy Assumption \[a:var\] with ${\widehat{{\mathbf{G}}}}$ positive definite and symmetric (almost surely).
Theorem \[t:rate\] is sufficiently general that it applies to models with latent state vectors without modification: all that is required is that one can construct estimators of ${\mathbf{G}}$ and ${\mathbf{M}}$ that satisfy Assumption \[a:var\]. Appendix \[ax:filter\] describes two approaches for extending the methodology to models with latent variables.
The relation (\[e:gev\]) may also be used to numerically compute $\rho$, $\phi$, and $\phi^*$ in models for which analytical solutions are unavailable. For such models, the matrices ${\mathbf{M}}$ and ${\mathbf{G}}$ may be computed directly (e.g. via simulation or numerical integration). The approximate solutions $\rho_k$, $\phi_k$ and $\phi_k^*$ for $\rho$, $\phi$ and $\phi^*$ can be recovered by solving (\[e:gev\]). Lemma \[lem:bias\] gives the convergence rates $|\rho_k - \rho| = O(\delta_k)$, $\|\phi_k - \phi\| = O(\delta_k^{\phantom *})$, and $\|\phi_k^* - \phi^*\| = O(\delta_k^*)$.
Theorem \[t:rate\] displays the usual bias-variance tradeoff encountered in nonparametric estimation. The bias terms $\delta_k^{\phantom *}$ and $\delta_k^*$ will be decreasing in $k$ (since $\phi$ and $\phi^*$ are approximated over increasingly rich subspaces as $k$ increases). On the other hand, the variance terms $\eta_{n,k}^{\phantom *}$ and $\eta_{n,k}^*$ will typically be increasing in $k$ (larger matrices) and decreasing in $n$ (more data). Choosing $k$ to balance the bias and variance terms will yield the best convergence rate.
To investigate the theoretical properties of the estimators, we derive the convergence rate of $\hat \phi$ in Case 1, where ${\widehat{{\mathbf{G}}}}$ and ${\widehat{{\mathbf{M}}}}$ are as in (\[e:ghat\]) and (\[e:mhat1\]) under standard conditions from the statistics literature on optimal convergence rates. Although the following conditions are not particularly appropriate in an asset pricing context, the result is informative about the convergence properties of $\hat \phi$ relative to conventional nonparametric estimators.
\[c:rate\] Let Assumption \[a:id\] and the following conditions hold: (i) ${\mathcal{X}} \subset {\mathbb{R}}^d$ is compact, rectangular and has nonempty interior; (ii) the stationary distribution $Q$ of $X$ has a continuous and positive density; (iii) ${\mathbb{M}}$ is a bounded operator from $L^2$ into a Hölder class of functions $\Lambda^{p_0} ({\mathcal{X}})$ of smoothness $p_0 > 0$ [@Stone1982 p. 1043]; (iv) $\phi \in \Lambda^p({\mathcal{X}})$ with $p \geq \min\{p_0,d/2\}$; (v) ${\mathbb{E}}[|m(X_0,X_1)|^s] < \infty$ for some $s > 2$; (vi) $B_k$ is a spanned by (a tensor product of) polynomial splines of degree $\nu \geq p$ with uniformly bounded mesh ratio [@Schumaker2007 Chapter 12] with $k^{(2s+2)/s} = o(n)$; and (vii) $\{X_t\}$ is exponentially rho-mixing. Then: Assumptions \[a:bias\] and \[a:var\] hold, $\delta_k = O(k^{-p/d})$, and $\eta_{n,k} = O(k^{(s+2)/2s}/\sqrt n)$. Choosing $k \asymp n^{\frac{sd}{2sp+(2+s)d}}$ yields: $$\|\hat \phi - \phi\| = O_p(n^{-\frac{sp}{2sp+(2+s)d}}) \,.$$
When $m$ is bounded (i.e. $s = \infty$), the convergence rate obtained in Corollary \[c:rate\] is $n^{-p/(2p+d)}$, which is the optimal $L^2$ convergence rate for nonparametric regression estimators with i.i.d. data when the unknown regression function belongs to $\Lambda^p(\mathcal X)$ (see, e.g., [@Stone1982]).
Asymptotic normality
--------------------
In this section we establish the asymptotic normality of $\hat \rho$, which is also useful in establishing asymptotic normality of the estimator of the long-run yield and entropy of the permanent component (see Appendix \[ax:inf\]). The semiparametric efficiency bound in Case 1 is also derived and $\hat \rho$ is shown to be efficient.
### Asymptotic normality in Case 1
To establish asymptotic normality of $\hat \rho$, we derive the representation: $$\label{e:ale:1}
\sqrt n (\hat \rho - \rho) = \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \psi_\rho(X_t,X_{t+1}) + o_p(1)$$ where the influence function $\psi_\rho$ is given by: $$\label{e:inf:def}
\psi_\rho(x_0,x_1) = \phi^*(x_0) m(x_0,x_1) \phi(x_1) - \rho \phi^*(x_0) \phi(x_0)$$ with $\phi$ and $\phi^*$ normalized so that $\|\phi\| = 1$ and $\langle \phi, \phi^* \rangle = 1$. The process $\{\psi_\rho(X_t,X_{t+1}) : t \in T\}$ is a martingale difference sequence (relative to the filtration $\{{\mathcal{F}}_t : t \in T\}$). The asymptotic distribution of $\hat \rho$ then follows from (\[e:ale:1\]) by a central limit theorem for martingale differences. To formalize this argument, define: $$\psi_{k,\rho}(x_0,x_1) = \phi_k^*(x_0) m(x_0,x_1) \phi_k^{\phantom{*}}(x_1) - \rho_k \phi^*_k(x_0) \phi_k^{\phantom{*}}(x_0)$$ with $\phi_k^{\phantom *}$ and $\phi_k^*$ normalized so that $\|\phi_k\| = 1$ and $\langle \phi_k^{\phantom *}, \phi_k^* \rangle = 1$, and: $$\Delta_{\psi,n,k} = \frac{1}{n} \sum_{t=0}^{n-1} \big( \psi_{\rho,k}(X_t,X_{t+1}) - \psi_\rho(X_t,X_{t+1}) \big)\,.$$
\[a:asydist\] Let the following hold:
1. $\delta_k = o(n^{-1/2})$
2. $ \|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| = o_p(n^{-1/4})$ and $\|{\widehat{{\mathbf{M}}}}^o - {{\mathbf{M}}}^o\| = o_p(n^{-1/4})$
3. $\Delta_{\psi,n,k} = o_p(n^{-1/2})$
4. ${\mathbb{E}}[(\phi^*(X_t)m(X_t,X_{t+1})\phi(X_{t+1}))^2] < \infty$.
[**Discussion of assumptions:**]{} Assumption \[a:asydist\](a) is an undersmoothing condition which ensures that the approximation bias $\rho - \rho_k$ does not distort the asymptotic distribution of $\hat \rho$. Assumption \[a:asydist\](b)(c) ensures the remaining terms in (\[e:ale:1\]) are $o_p(1)$; sufficient conditions for Assumption \[a:asydist\](b) are presented in Appendix \[ax:est:mat\]. Assumption \[a:asydist\](c) requires that higher-order approximation error be asymptotically negligible. This condition is mild: the summands in $\Delta_{\psi,n,k}$ have expectation zero, and $\phi_k^{\phantom *}$, $\phi_k^*$ and $\rho_k$ are converging to $\phi$, $\phi^*$, and $\rho$ by Lemma \[lem:bias\]. Finally, Assumption \[a:asydist\](d) ensures the asymptotic variance of $\hat \rho$ is finite.
The following result establishes asymptotic normality of $\hat \rho$ in Case 1. Appendix \[ax:inf\] contains further results on inference on functionals of $\phi$ and $\rho$, such as the entropy of the permanent component. Define $V_\rho = {\mathbb{E}}[\psi_\rho(X_0,X_1)^2]$.
\[t:asydist:1\] Let Assumptions \[a:id\]–\[a:asydist\] hold. Then: the asymptotic linear expansion (\[e:ale:1\]) holds and $\sqrt n (\hat \rho - \rho) \to_d N(0,V_\rho)$.
The following corollary lists some sufficient conditions for Assumptions \[a:var\]–\[a:asydist\]. Define $\xi_k = \sup_{x} \|{\mathbf{G}}^{-1/2} b^k(x)\|$ and let $\|\cdot\|_p$ denote the $L^p$ norm $\|\psi\|_p = (\int |\psi|^p \, {\mathrm{d}} Q)^{1/p}$ (so $\|\cdot\|_2 \equiv \|\cdot\|$).
\[c:inf:1\] Let Assumptions \[a:id\] and \[a:bias\] hold, and let: (i) $\{X_t\}$ be exponentially beta-mixing (see Appendix \[ax:est:mat\]); (ii) $\xi_k = O(k^\lambda)$ for some $\lambda$, (iii) $\delta_k^{\phantom *} = O(k^{-\omega})$ and $\delta_k^* = O(k^{-\omega})$ for some $\omega > 0$, (iv) ${\mathbb{E}}[|m(X_t,X_{t+1})|^s] < \infty$, $\|\phi\|_{\frac{2s}{s-2}} < \infty$ and $\| \phi^*\|_{\frac{2s}{s-2}} < \infty$ for some $s > 2$, (v) $\|\phi - \phi_k\|_{\frac{2s}{s-2}} = O(1)$, and (vi) $k$ increase with $n$ such that $\sqrt n k^{-\omega} = o(1)$ and $k^{2\lambda(s+1)/s} (\log n)^2 / \sqrt n = o(1)$. Then: (\[e:ale:1\]) holds and $\sqrt n (\hat \rho - \rho) \to_d N(0,V_\rho)$.
We conclude by deriving the semiparametric efficiency bounds for Case 1. To derive the efficiency bound we require a further technical condition (see Appendix \[ax:inf\]).
\[t:eff\] Let Assumptions \[a:id\]–\[a:asydist\] and \[a:eff\] hold. Then: the semiparametric efficiency bound for $\rho$ is $V_\rho$ and $\hat \rho$ is semiparametrically efficient.
Theorem \[t:eff\] provides further theoretical justification for using sieve methods to nonparametrically estimate $\rho$, $\phi$, and related quantities. In Appendix \[ax:inf\] we derive efficiency bounds for $ y$ and $ L$ and show that $\hat y$ and $\hat L$ attain their bounds.
### Asymptotic normality in Case 2
For Case 2, we obtain the following expansion (under regularity conditions): $$\label{e:ale:2}
\sqrt n (\hat \rho - \rho) = \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \Big( \psi_\rho(X_t,X_{t+1}) + \psi_{\alpha,k}(X_t,X_{t+1}) \Big) + o_p(1)$$ with $\psi_\rho$ from display (\[e:inf:def\]) with $m(x_0,x_1) = m(x_0,x_1;\alpha_0)$ and: $$\label{e:inf:alpha:def}
\psi_{\alpha,k}(x_0,x_1) = \phi^*_k(x_0) \big( m(x_0,x_1;\hat \alpha) - m(x_0,x_1;\alpha_0) ) \phi_k^{\phantom *}(x_1) \,.$$ The expansion (\[e:ale:2\]) indicates that the asymptotic distribution of $\hat \rho$ and related functionals will depend on the properties of the first stage estimator $\hat \alpha$.
We first suppose that $\alpha_0$ is a finite-dimensional parameter and the plug-in estimator $\hat \alpha$ is root-$n$ consistent and asymptotically normally distributed. The following regularity conditions are deliberately general so as to allow for $\hat \alpha$ to be any conventional parametric estimator of $\alpha_0$. To simplify notation, let $\psi_{\rho,t} = \psi_\rho(X_t,X_{t+1})$.
\[a:parametric\] Let the following hold:
1. $\sqrt n (\hat \alpha - \alpha_0) = \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \psi_{\alpha,t} + o_p(1)$ for some ${\mathbb{R}}^{{d_\alpha}}$-valued random process $\{\psi_{\alpha,t} : t \in T\}$
2. $\{(\psi_{\rho,t}^{\phantom \prime},\psi_{\alpha,t}')' : t \in T\}$ satisfies a CLT, i.e.: $$\frac{1}{\sqrt n } \sum_{t=0}^{n-1} \bigg( \begin{array}{c} \psi_{\rho,t} \\ \psi_{\alpha,t} \end{array} \bigg) \to_d N(0,V_{[{\mathrm{2a}}]})$$ where the $({d_\alpha}+1) \times ({d_\alpha}+1)$ matrix $V_{[{\mathrm{2a}}]}$ is positive definite
3. There exists a neighborhood $N$ of $\alpha_0$ upon which $m(x_0,x_1,\alpha)$ is continuously differentiable in $\alpha$ for each $(x_0,x_1) \in {\mathcal{X}}^2$ with: $${\mathbb{E}} \bigg[ \sup_{\alpha \in N} \Big\| \frac{\partial m(X_t,X_{t+1};\alpha)}{\partial \alpha} \Big\|^s \bigg] < \infty$$ for some $s > 2$
4. ${\mathbb{E}}[(\phi(X_t)\phi^*(X_t))^{s/(s-1)}] < \infty$ and $\delta_k^* = O(n^{-\omega})$ for some $\omega > 1/s$.
Let $h_{[{\mathrm{2a}}]} = (1\,,\, {\mathbb{E}}[\phi^*(X_t) \phi(X_{t+1})\frac{\partial m(X_t,X_{t+1};\alpha_0)}{\partial \alpha'}])'$ and define $V_\rho^{[{\mathrm{2a}}]} = h_{[{\mathrm{2a}}]}'V_{[{\mathrm{2a}}]}^{\phantom \prime} h_{[{\mathrm{2a}}]}^{\phantom \prime}$.
\[t:asydist:2a\] Let Assumptions \[a:id\]–\[a:parametric\] hold. Then: $\sqrt n (\hat \rho - \rho) \to_d N(0,V_\rho^{[{\mathrm{2a}}]})$.
We now suppose that $\alpha_0$ is an infinite-dimensional parameter. The parameter space is ${\mathcal{A}} \subseteq {\mathbb{A}}$ (a Banach space) equipped with some norm $\|\cdot\|_{{\mathcal{A}}}$. This includes the case in which (1) $\alpha$ is a function, i.e. $\alpha = h$ with ${\mathbb{H}}$ a function space, and (2) $\alpha$ consists of both finite-dimensional and function parts, i.e. $\alpha = (\theta,h)$ with ${\mathbb{A}} = \Theta \times {\mathbb{H}}$ with $\Theta \subseteq {\mathbb{R}}^{\dim (\theta)} $. For example, under recursive preferences the vector $\theta$ could consist of discount, risk-aversion and EIS parameters and $h$ could be the continuation value function.
Inference in this case involves the (nonlinear) functional $\ell : {\mathcal{A}} \to {\mathbb{R}}$, given by: $$\ell(\alpha) = {\mathbb{E}}[\phi^*(X_t) \phi(X_{t+1}) m(X_t,X_{t+1};\alpha)]\,.$$ We focus on the case in which the functional $\ell(\alpha_0)$ is regular (i.e. root-$n$ estimable). We say that the functional $\ell : {\mathcal{A}} \to {\mathbb{R}}$ is *Gateaux differentiable* at $\alpha_0$ in the direction $[\alpha - \alpha_0]$ if $\lim_{\tau \to 0^+} (\ell(\alpha_0 + \tau[\alpha - \alpha_0]) - \ell(\alpha_0))/\tau$ exists for every fixed $\alpha \in {\mathcal{A}}$. If so, we denote the derivative by $\dot \ell_{\alpha_0}[\alpha - \alpha_0]$. Let $g_\alpha(x_t,x_{t+1}) =\phi^*(x_t) \phi(x_{t+1}) (m(x_t,x_{t+1};\alpha) - m(x_t,x_{t+1};\alpha_0))$ and let ${\mathcal{G}} = \{ g_\alpha : \alpha \in {\mathcal{A}}\}$. Let ${\mathcal{Z}}_n$ denote the centered empirical process on ${\mathcal{G}}$. We say that ${\mathcal{G}}$ is a *Donsker class* if $\sum_{t \in {\mathbb{Z}}} {\mathrm{Cov}}(g(X_0,X_1),g(X_t,X_{t+1}))$ is absolutely convergent over ${\mathcal{G}}$ to a non-negative quadratic form $\Gamma(g,g)$ and there exists a sequence of Gaussian processes ${\mathcal{Z}}^{(n)}$ indexed by ${\mathcal{G}}$ with covariance function $\Gamma$ and a.s. uniformly continuous sample paths such that $\sup_{g \in G} | {\mathcal{Z}}_n (g) - {\mathcal{Z}}^{(n)}(g)| \to_p 0$ as $n \to \infty$ (see [@DoukhanMassartRio]).
\[a:nonpara\] Let the following hold:
1. $\ell$ is Gateaux differentiable at $\alpha_0$ and $| \ell(\alpha) - \ell(\alpha_0) - \dot \ell_{\alpha_0} [\alpha - \alpha_0]| = O(\|\alpha - \alpha_0\|^2_{{\mathcal{A}}})$ as $\|\alpha - \alpha_0\|_{{\mathcal{A}}} \to 0$
2. $\sqrt n \dot \ell_{\alpha_0}[\hat \alpha - \alpha_0] = \frac{1}{\sqrt n } \sum_{t=0}^{n-1} \psi_{\ell,t} + o_p(1)$ for some ${\mathbb{R}}$-valued random process $\{ \psi_{\ell,t} : t \in T\}$, $\|\hat \alpha - \alpha_0\|_{{\mathcal{A}}} = o_p(n^{-1/4})$, and $\|g_{\hat \alpha}\|_{2+\epsilon} = o_p(1)$ for some $\epsilon > 0$
3. $\{(\psi_{\rho,t},\psi_{\ell,t})' : t \in T\}$ satisfies a CLT, i.e.: $$\frac{1}{\sqrt n } \sum_{t=0}^{n-1} \bigg( \begin{array}{c} \psi_{\rho,t} \\ \psi_{\ell,t} \end{array} \bigg) \to_d N(0,V_{[{\mathrm{2b}}]})$$ where the $2 \times 2$ matrix $V_{[{\mathrm{2b}}]}$ is positive definite
4. ${\mathcal{G}}$ is a Donsker class and $\{X_t : t \in T\}$ is exponentially beta-mixing
5. ${\mathbb{E}}[ \sup_{\alpha \in {\mathcal{A}}} m(X_t,X_{t+1};\alpha)^s] < \infty$ and $\|\phi_k - \phi\|_{\frac{2s}{s-2}} = O(1)$ for some $s > 2$, $\|\phi\|_{s_1} < \infty$ and $\|\phi^*\|_{s_1} < \infty$ for some $s_1 > \frac{2s}{s-2}$, and $\delta_k^* = o(n^{-1/2})$.
Parts (a) and (b) are standard conditions for inference in nonlinear semiparametric models (see, e.g., Theorem 4.3 in [@Chen2007]). Part (c) is a mild CLT condition. The covariance function $\Gamma$ is well defined under parts (d) and (e). Sufficient conditions for the class ${\mathcal{G}}$ to be Donsker are well known (see, e.g., [@DoukhanMassartRio]).
For the following theorem, let $h_{[{\mathrm{2b}}]} = (1,1)'$ and define $V_\rho^{[{\mathrm{2b}}]} = h_{[{\mathrm{2b}}]}'V_{[{\mathrm{2b}}]}^{\phantom \prime} h_{[{\mathrm{2b}}]}^{\phantom \prime}$. Sufficient conditions for Assumption \[a:var\] are presented in Appendix \[ax:est:mat\] for this case.
\[t:asydist:2b\] Let Assumptions \[a:id\]–\[a:asydist\] and \[a:nonpara\] hold. Then: $\sqrt n (\hat \rho - \rho) \to_d N(0,V_\rho^{[{\mathrm{2b}}]})$.
For case $2$, the efficiency bounds will depend on the assumptions made about the model from which $\alpha$ was estimated (say, whether it is an unconditional or conditional moment restriction model). We therefore do not derive efficiency bounds for this case.
Value functions and nonlinear eigenfunctions {#s:recursive}
============================================
This section describe how to nonparametrically estimate the value function and SDF in a class of recursive preference models by solving a *nonlinear* Perron-Frobenius eigenfunction problem. We focus on [@EpsteinZin1989] recursive preferences with unit elasticity of intertemporal substitution (EIS). This class of preferences may also be interpreted as risk-sensitive preferences as formulated by [@HansenSargent1995] (see [@Tallarini2000]). After describing the setup, we present some regularity conditions for local identification. The local identification argument is constructive. We then introduce the estimators and derive their large-sample properties.
Setup {#setup}
-----
Under Epstein-Zin preferences, the date-$t$ utility of the representative agent is defined via the recursion: $$V_t = \left\{ (1-\beta) C_t^{1-\theta} + \beta {\mathbb{E}}[V_{t+1}^{1-\gamma}|\mathcal F_t]^{\frac{1-\theta}{1-\gamma}} \right\}^{\frac{1}{1-\theta}}$$ where $C_t$ is date-$t$ consumption, $1/\theta$ is the EIS, $\beta \in (0,1)$ is the time discount parameter, and $\gamma > 1$ is the relative risk aversion parameter. We maintain our assumption of a Markov state process $X$. Let consumption growth, namely $G_{t+1} = C_{t+1}/C_t$, be a measurable function of $(X_t,X_{t+1})$. [@HansenHeatonLi] show that the scaled continuation value $V_t/C_t$ may be written as $V(X_t) $ where $V : {\mathcal{X}} \to {\mathbb{R}}_+$ solves the fixed point equation: $$\label{e:ezrecur}
V(X_t) = \bigg\{ (1-\beta) + \beta {\mathbb{E}}\Big[ \left(V(X_{t+1})G_{t+1}\right)^{1-\gamma} \Big|X_t \Big]^{\frac{1-\theta}{1-\gamma}} \bigg\}^{\frac{1}{1-\theta}} \,.$$ With unit EIS (i.e. $\theta = 1$) the fixed point equation (\[e:ezrecur\]) reduces to: $$\label{e:fp:prelim}
v(X_t) =\frac{\beta}{1-\gamma} \log {\mathbb{E}} \Big[e^{ (1-\gamma) (v(X_{t+1}) +\log G_{t+1} ) } \Big|X_t \Big]$$ where $v= \log V$. The SDF is of the form: $$\begin{aligned}
\frac{M_{t+1}}{M_t} & = \beta G_{t+1}^{-1} \frac{(C_{t+1} e^{ v(X_{t+1})})^{1-\gamma}}{{\mathbb{E}}[(C_{t+1} e^{ v(X_{t+1})})^{1-\gamma}|X_t]}\,.\end{aligned}$$ The dynamics of $X$ determine the value function $v$ and the conditional expectation in the denominator of the SDF. The value function and conditional expectation are unknown when the dynamics of $X$ are treated nonparametrically.
The following reformulation is convenient for estimation. Setting $h(x) = \exp(\frac{1-\gamma}{\beta} v(x))$ and rearranging (\[e:fp:prelim\]) yields: $$h(X_t) = {\mathbb{E}} \Big[ G_{t+1}^{1-\gamma} \big( h(X_{t+1}) \big)^\beta\Big|X_t \Big] \,.$$ This fixed-point equation may be written more compactly as ${\mathbb{T}} h = h
$, where: $$\begin{aligned}
{\mathbb{T}} \psi (x) & = {\mathbb{E}}\Big[ G_{t+1}^{1-\gamma} \big|\psi(X_{t+1}) \big|^\beta \Big| X_t = x \Big] \end{aligned}$$ (as we seek a positive solution, taking an absolute value inside the conditional expectation does not change the fixed point). We may then rewrite the SDF as: $$\begin{aligned}
\label{e:efn:nl:0}
\frac{M_{t+1}}{M_t} & = \beta G_{t+1}^{-\gamma} \frac{(h(X_{t+1}))^\beta}{{\mathbb{T}} h(X_t)}\,.\end{aligned}$$ Dividing ${\mathbb{T}} h = h$ by $\|h\|$ and using homogeneity allows us to reformulate the fixed-point problem as a nonlinear Perron-Frobenius problem: $$\label{e:efn:nl}
{\mathbb{T}} \chi (x) = \lambda \chi (x)$$ where $\chi(x) = h(x)/\|h\|$ is a positive eigenfunction of ${\mathbb{T}}$ and $\lambda = \|h\|^{1-\beta}$ is its eigenvalue. Since ${\mathbb{T}}$ is nonlinear, in this section we normalize the eigenfunction $\chi$ to have unit norm.[^17] Combining equations (\[e:efn:nl:0\]) and (\[e:efn:nl\]) yields: $$\begin{aligned}
\label{e:rec:sdf}
\frac{M_{t+1}}{M_t}
& = \frac{\beta}{\lambda} G_{t+1}^{-\gamma} \frac{ \big(\chi(X_{t+1}) \big)^\beta}{ \chi(X_t) } \,.\end{aligned}$$ We show how to estimate $\chi$ and $\lambda$ from time series data on $X$. The estimates $\hat \chi$ and $\hat \lambda$ can be plugged into (\[e:rec:sdf\]) to obtain empirically a time series of the SDF process without assuming a parametric law of motion for $X$.
Local identification
--------------------
In this section we provide sufficient conditions for local identification of $h$. We establish the results for the parameter (function) space $L^2$ because it is convenient for sieve estimation. One cannot establish (global) identification using contraction mapping arguments because ${\mathbb{T}}$ is not a contraction on $L^2$.[^18] Some of the regularity conditions we require for estimation are sufficient for ${\mathbb{T}}$ to satisfy a local ergodicity property. This, in turn, is sufficient for local identification. It may be possible to strengthen the conditions to achieve global identification.
To describe the local ergodicity property, first choose some (nonzero) function $\psi \in L^2$ and set $\chi_1(\psi) = \psi$. Then calculate: $${\chi_{n+1}(\psi)} = \frac{{\mathbb{T}} \chi_n(\psi) }{\| {\mathbb{T}} \chi_n(\psi)\|}$$ for $n \geq 1$. Proposition \[p:nl\] below shows that the sequence $\chi_n(\psi)$ converges to $\chi$ for any starting value $\psi$ in a suitably defined region ($h$ and $\lambda$ can then be obtained by setting $\lambda = \| {\mathbb{T}} \chi\|$ and $h = \lambda^{\frac{1}{1-\beta}}\chi$). This is similar to various “stability” results in the literature on balanced growth following [@SolowSamuelson].[^19] There, ${\mathbb{T}} : {\mathbb{R}}^K \to {\mathbb{R}}^K$ is a homogeneous of degree one input-output system, $\chi_n \in {\mathbb{R}}^K$ lists the proportions of commodities in the economy in period $n$, and ${\mathbb{T}} \chi_n$ is normalized by its $\ell^1$ norm so that $\chi_{n+1} := {\mathbb{T}} \chi_n/\|{\mathbb{T}} \chi_n\|_{\ell^1}$ lists the proportions in period $n+1$. “Stability” concerns convergence of the sequence $\chi_n$ to a positive eigenvector $\chi$ of ${\mathbb{T}}$ (representing balanced growth proportions).
Write ${\mathbb{T}} = {\mathbb{G}} {\mathbb{F}}$ where ${\mathbb{F}} \psi(x) = |\psi(x)|^\beta$ and: $${\mathbb{G}} \psi(x) = {\mathbb{E}}\Big[ G_{t+1}^{1-\gamma} \psi(X_{t+1}) \Big| X_t = x \Big] \,.$$ As is known from the theory of Hammerstein operators (see Chapter 5 of [@KZPS]), ${\mathbb{T}}$ is a bounded (respectively, compact) operator on $L^2$ whenever ${\mathbb{G}}$ is bounded (compact) on $L^2$. We say that ${\mathbb{G}}$ is [*positive*]{} if ${\mathbb{G}} \psi$ is positive for any non-negative $\psi \in L^2$ that is not identically zero. Positivity of ${\mathbb{G}}$ ensures that the iterative procedure is well defined and that any non-zero fixed point of ${\mathbb{T}}$ is strictly positive (almost everywhere). We say that ${\mathbb{T}}$ is *Fréchet differentiable* at $h$ if there exists a bounded linear operator ${\mathbb{D}}_h : L^2 \to L^2$ such that: $$\| {\mathbb{T}}(h + \psi) - {\mathbb{T}} h - {\mathbb{D}}_h \psi\| = o( \|\psi\| )\,.$$ If it exists, the Fréchet derivative ${\mathbb{D}}_h$ of ${\mathbb{T}}$ is given by: $${\mathbb{D}}_h \psi(x) = {\mathbb{E}} \Big[ \beta G_{t+1}^{1-\gamma} h(X_t)^{\beta-1} \psi(X_{t+1}) \Big| X_t = x \Big] \,.$$ Let $r({\mathbb{D}}_h):= \sup\{ |\lambda | : \lambda \in \sigma({\mathbb{D}}_h)\}$ denote the spectral radius of ${\mathbb{D}}_h$
\[p:nl\] Let ${\mathbb{G}}$ be positive and bounded and let ${\mathbb{T}}$ be Fréchet differentiable at $h$ with $r({\mathbb{D}}_h) < 1$. Then: there exists finite positive constants $C,b$ with $b < 1$ and a neighborhood $N$ of $\chi$ such that: $$\| \chi_{n+1}(\psi) - \chi\| \leq C b^n$$ for any initial point $\psi$ in the cone $\{ a N : a \in {\mathbb{R}}, a \neq 0\}$.
We say that $\chi$ is [*locally identified*]{} if there exists a neighborhood $N$ of $\chi$ such that $\chi$ is the unique eigenfunction of ${\mathbb{T}}$ belonging to $N \cap S_1$ where $S_1$ denotes the unit sphere in $L^2$ (recall we normalize eigenfunctions of ${\mathbb{T}}$ to have unit norm). Similarly, we say that $h$ is locally identified if it is the unique fixed point of ${\mathbb{T}}$ belonging to some neighborhood $N'$ of $h$. To see why local identification follows from Proposition \[p:nl\], suppose $\bar \chi$ is a positive eigenfunction of ${\mathbb{T}}$ belonging to $N \cap S_1$. Proposition \[p:nl\] implies that $\|\bar \chi - \chi\| \leq C b^n$ for each $n$, hence $\bar \chi = \chi$. Local identification of $h$ follows similarly.
\[c:local-id\] $h$ and $\chi$ are locally identified under the conditions of Proposition \[p:nl\].
Local identification of $\chi$ and positive homogeneity of ${\mathbb{T}}$ imply that $h$ is the unique fixed point of ${\mathbb{T}}$ in the cone $\{ a (N \cap S_1) : a \in {\mathbb{R}} , a \neq 0\}$, which is stronger than local identification of $h$.
Existence and (global) identification of value functions in models with recursive preferences has been studied previously (see [@MarinacciMontrucchio], [@HS2012], and references therein). The most closely related work to ours is [@HS2012], who study existence and uniqueness of value functions for Markovian environments in $L^1$ spaces (whose cones of non-negative functions also have empty interior). [@HS2012] provide conditions under which a fixed point may exist when the EIS is equal to unity but do not establish its uniqueness. Their existence conditions are based, in part, on existence of a positive eigenfunction of the operator ${\mathbb{G}}$.
There is also a connection between Corollary \[c:local-id\] and the literature on local identification of nonlinear, nonparametric econometric models. We can write ${\mathbb{T}} h = h$ as the conditional moment restriction: $${\mathbb{E}}\Big[ G_{t+1}^{1-\gamma} \big|h(X_{t+1}) \big|^\beta - h(X_t) \Big| X_t \Big] = 0$$ (almost surely). The conditions of Proposition \[p:nl\] ensure that the above moment restriction is Fréchet differentiable at $h$ with derivative ${\mathbb{D}}_h - I$. The condition $r({\mathbb{D}}_h) < 1$ implies that ${\mathbb{D}}_h - I$ is invertible on $L^2$. The conditions in Proposition \[p:nl\] are therefore similar to the differentiability and rank conditions that [@Chenetal2012] use to study local identification in nonlinear conditional moment restriction models.
Estimation {#estimation}
----------
We again use a sieve approach to reduce the infinite-dimensional problem to a low-dimensional (nonlinear) eigenvector problem. Consider the projected fixed-point problem: $$(\Pi_k \mathbb T) h_k = h_k$$ where $\Pi_k : L^2 \to B_k$ is the orthogonal projection onto the sieve space defined in Section \[s:est\]. Under the assumptions below, the approximate solution $h_k$ is well defined for all $k$ sufficiently large. As $h_k \in B_k$, we may write $h_k = b^k(x)' v_k$ for some vector $v_k \in {\mathbb{R}}^k$ which solves: $$\label{e:fp:vec}
{\mathbf{G}}_k^{-1} {\mathbf{T}}_k^{\phantom {-1}}\!\!\! v_k = v_k$$ where ${\mathbf{T}}_k v = {\mathbb{E}}[ b^k(X_{t}) G_{t+1}^{1-\gamma} |b^k(X_{t+1})'v |^\beta ]$. To simplify notation we drop dependence of ${\mathbf{G}}_k$ and ${\mathbf{T}}_k$ on $k$ hereafter. For estimation, we solve a sample analogue of (\[e:fp:vec\]), namely: $$\label{e:fp:sample}
{\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{T}}}} \hat v = \hat v$$ where ${\widehat{{\mathbf{G}}}}$ is defined in display (\[e:ghat\]) and ${\widehat{{\mathbf{T}}}} : {\mathbb{R}}^k \to {\mathbb{R}}^k$ is given by: $${\widehat{{\mathbf{T}}}}v = \frac{1}{n} \sum_{t=0}^{n-1} b^k(X_{t}) G_{t+1}^{1-\gamma} |b^k(X_{t+1})'v |^\beta \,.$$ Under the regularity conditions below, a solution $\hat v$ on a neighborhood of $v_k$ necessarily exists wpa1 (see Lemma \[lem:fphat:exist\]). The estimators of $h$, $\chi$ and $\lambda$ are: $$\begin{aligned}
\label{e:fpest}
\hat h(x) & = b^k(x)' \hat v &
\hat \chi(x) & = \frac{b^k(x)' \hat v}{(\hat v' {\widehat{{\mathbf{G}}}} \hat v)^{1/2}} &
\hat \lambda & = (\hat v' {\widehat{{\mathbf{G}}}} \hat v)^{\frac{1-\beta}{2}} \,.\end{aligned}$$ The estimators $\hat \chi$ and $\hat \lambda$ can then be plugged into display (\[e:rec:sdf\]) to obtain the SDF consistent with preference parameters $\beta$, $\gamma$ and the observed law of motion of the state.
We say that ${\mathbb{T}}$ is *continuously Fréchet differentiable* at $h$ if ${\mathbb{T}}$ is differentiable on a neighborhood of $h$ and $\|{\mathbb{D}}_g - {\mathbb{D}}_{h} \| \to 0$ as $\|g - h\| \to 0$.
\[a:fp:exist\] Let the following hold:
1. ${\mathbb{T}}$ has a unique positive fixed point $h \in L^2$
2. ${\mathbb{G}}$ is compact and positive
3. ${\mathbb{T}}$ is continuously Fréchet differentiable at $h$ with $r({\mathbb{D}}_h) < 1$.
\[a:fp:bias\] Let the following hold:
1. $\|\Pi_k {\mathbb{D}}_h - {\mathbb{D}}_h\| = o(1)$
2. $B_k$ is dense in the range of ${\mathbb{T}}$ as $k \to \infty$.
Let ${\widehat{{\mathbf{G}}}}^o$ be as in Assumption \[a:var\]. Let ${\mathbf{T}}^o v = {\mathbf{G}}^{-1/2} {\mathbf{T}} ({\mathbf{G}}^{-1/2} v)$ and ${\widehat{{\mathbf{T}}}}^o v = {\mathbf{G}}^{-1/2} {\widehat{{\mathbf{T}}}} ({\mathbf{G}}^{-1/2} v)$.
\[a:fp:var\] $\| {\widehat{{\mathbf{G}}}}^o - {\mathbf{I}} \| = o_p(1)$ and $\sup_{v \in {\mathbb{R}}^k : \| v\| \leq c} \| {\widehat{{\mathbf{T}}}}^ov - {\mathbf{T}}^o v\| = o_p(1)$ for each $c > 0$.
[**Discussion of assumptions:**]{} Assumption \[a:fp:exist\] imposes some mild structure on ${\mathbb{T}}$ which ensures that fixed points of ${\mathbb{T}}$ are continuous under perturbations. Assumption \[a:fp:bias\](a) is analogous to Assumption \[a:bias\]. Assumptions \[a:fp:exist\] and \[a:fp:bias\] are standard in the literature on solving nonlinear equations with projection methods. Finally, Assumption \[a:fp:var\] is similar to Assumption \[a:var\] and restricts the rate at which the sieve dimension $k$ can grow with $n$. Sufficient conditions for Assumption \[a:fp:var\] are presented in Appendix \[ax:est:mat:fp\].
Let $\tau_k = \|\Pi_k h - h\|$ denote the bias in approximating $h$ by an element of the sieve space. Assumption \[a:fp:bias\](b) implies that $\tau_k = o(1)$. To control the sampling error, fix any small $\varepsilon > 0$. By Assumption \[a:fp:var\] we may choose a sequence of positive constants $\nu_{n,k}$ with $\nu_{n,k} = o(1)$ such that: $$\label{e:nudef}
\sup_{v \in {\mathbb{R}}^k : \| v- {\mathbf{G}}^{1/2} v_k\| \leq \varepsilon} \| ({\widehat{{\mathbf{G}}}}^o)^{-1} {\widehat{{\mathbf{T}}}}^ov - {\mathbf{T}}^o v\| = O_p(\nu_{n,k})\,.$$
\[t:fpest\] Let Assumptions \[a:fp:exist\]–\[a:fp:var\] hold. Then:
1. $|\hat \lambda - \lambda| = O_p( \tau_k + \nu_{n,k})$
2. $\|\hat \chi - \chi\| = O_p( \tau_k + \nu_{n,k})$
3. $\|\hat h - h\| = O_p( \tau_k + \nu_{n,k})$.
The convergence rates obtained in Theorem \[t:fpest\] again exhibit a bias-variance tradeoff. The bias terms $\tau_k$ are decreasing in $k$, whereas the variance term $\nu_{n,k}$ is typically increasing in $k$ but decreasing in $n$. Choosing $k$ to balance the terms will lead to the best convergence rate.
For implementation, we recommend the following iterative scheme based on Proposition \[p:nl\]. Set $z_1 = {\widehat{{\mathbf{G}}}}^{-1} (\frac{1}{n} \sum_{t=0}^{n-1} b^k(X_t))$, then calculate: $$\begin{aligned}
y_{k} & = \frac{z_k}{(z_k' {\widehat{{\mathbf{G}}}} z_k^{\phantom \prime})^{1/2}} &
z_{k+1} & = {\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{T}}}} y_k\end{aligned}$$ for $k \geq 1$. If the sequence $\{(y_k,z_k) : k \geq 1\}$ converges to $(\hat y,\hat z)$ (say), we then set: $$\begin{aligned}
\hat h(x) & = \hat \lambda^{\frac{1}{1-\beta}}b^k(x)' \hat y &
\hat \chi(x) & = b^k(x)' \hat y &
\hat \lambda = (\hat z' {\widehat{{\mathbf{G}}}} \hat z^{\phantom \prime})^{1/2}\,.\end{aligned}$$ This procedure converged in the simulations and empirical application and was more efficient than numerical solution of the sample fixed-point problem (\[e:fp:sample\]).
Simulation evidence {#s:mc}
===================
The following Monte Carlo experiment illustrates the performance of the estimators in consumption-based models with power utility and recursive preferences. The state variable is log consumption growth, denoted $g_t$, which evolves as a Gaussian AR(1) process: $$g_{t+1} - \mu = \kappa (g_t - \mu) + \sigma e_{t+1}\,, \quad e_t \sim \mbox{ i.i.d. N$(0,1)$.}$$ The parameters for the simulation are $\mu = 0.005$, $\kappa = 0.6$, and $\sigma = 0.01$. The data are constructed to be somewhat representative of quarterly growth in U.S. real per capita consumption of nondurables and services (for which $\kappa \approx 0.3$ and $\sigma \approx 0.005$). However, we make the consumption growth process twice as persistent to produce more nonlinear eigenfunctions and twice as volatile to produce a more challenging estimation problem.
We consider a power utility design in which $m(X_t,X_{t+1}) = \beta G_{t+1}^{-\gamma}$ and a design with recursive preferences with unit EIS, whose SDF is presented in display (\[e:rec:sdf\]). For both designs we set $\beta = 0.994$ and $\gamma = 15$. The parameterization $\beta = 0.994$ and $\gamma = 10$ is typically used in calibrations of long-run risks models ([@BansalYaron; @CDJL]); here we take $\gamma = 15$ to produce greater nonlinearity in the eigenfunctions. For each design we generate 50000 samples of length 400, 800, 1600, and 3200. Results reported in this section use either a Hermite polynomial or cubic B-spline basis of dimension $k=8$. The Hermite polynomials are normalized by the mean and standard deviation of the data to be approximately orthonormal and the knots of the B-splines are placed evenly at the quantiles of the data. The results were reasonably insensitive both to the choice of sieve and to the dimension of the sieve space.
We estimate $\phi$, $\phi^*$, $\rho$, $y$, and $L$ for both designs and we also estimate $\chi$ and $\lambda$ for the recursive preference design. To implement the estimators $\hat \rho$, $\hat \phi$, and $\hat \phi^*$, we use the estimator ${\widehat{{\mathbf{G}}}}$ in (\[e:ghat\]) for both preference specifications. For power utility we use the estimator ${\widehat{{\mathbf{M}}}}$ in (\[e:mhat1\]). For recursive preferences we first estimate $(\lambda,\chi)$ using the method described in the previous section, then construct the estimator ${\widehat{{\mathbf{M}}}}$ as in display (\[e:mhat2\]), using the plug-in SDF: $$m(X_t,X_{t+1};\hat \lambda,\hat \chi) = \frac{\beta}{\hat \lambda} G_{t+1}^{-\gamma} \frac{ \big(\hat \chi(X_{t+1}) \big)^\beta}{ \hat \chi(X_t) }$$ based on first-stage estimators $(\hat \lambda,\hat \chi)$ of $(\lambda,\chi)$. We normalize $\hat \phi$ so that $\frac{1}{n} \sum_{t=0}^{n-1} \hat \phi(X_t)^2 = 1$ and $\hat \phi^*$ so that $\frac{1}{n} \sum_{t=0}^{n-1} \hat \phi(X_t)\hat\phi^*(X_t) = 1$. We also normalize $\hat \chi$ so that $\frac{1}{n} \sum_{t=0}^{n-1} \hat \chi(X_t)^2 = 1$.
The bias and RMSE of the estimators are presented in Tables \[tab:mc1\] and \[tab:mc2\].[^20] Table \[tab:mc1\] shows that $\phi$, $\phi^*$ and $\chi$ may be estimated with small bias and RMSE using a reasonably low-dimensional sieve for both Hermite polynomials and cubic B-splines. Table \[tab:mc2\] presents similar results for $\hat \rho$, $\hat y$, $\hat L$ and $\hat \lambda$. The RMSEs for $\hat \phi$ and $\hat \rho$ under recursive preferences are typically smaller than the RMSEs for $\hat \phi$ and $\hat \rho$ under power utility, even though with recursive preferences the continuation value must be first estimated nonparametrically. In contrast, the RMSE for $\hat \phi^*$ is larger under recursive preferences, which is likely due to the fact that $\phi^*$ is much more curved for that design (as evident from comparing the vertical scales Figures \[fig:mc:sfig2\] and \[fig:mc:sfig4\]). The results in Table \[tab:mc1\] also show that $\chi$ may be estimated with a reasonably small degree of bias and RMSE in moderate samples.
Figures \[fig:mc:sfig1\]–\[fig:mc:sfig5:bs\] also present (pointwise) confidence intervals for $\phi$, $\phi^*$ and $\chi$ computed across simulations of different sample sizes. For each figure, the true function lies approximately in the center of the pointwise confidence intervals, and the widths of the intervals shrink noticeably as the sample size $n$ increases. The results for Hermite polynomials and cubic B-splines are similar. However, the intervals are somewhat narrower at their extremities for cubic B-splines than for Hermite polynomials.
-- ------ ------------- --------------- ------------- --------------- -------------
$n$ $\hat \phi$ $\hat \phi^*$ $\hat \phi$ $\hat \phi^*$ $\hat \chi$
400 0.0144 0.0129 0.0027 0.0247 0.0119
800 0.0115 0.0129 0.0020 0.0187 0.0090
1600 0.0084 0.0104 0.0016 0.0128 0.0062
3200 0.0058 0.0077 0.0014 0.0095 0.0040
400 0.1136 0.1683 0.0458 0.4068 0.1034
800 0.0872 0.1060 0.0413 0.3513 0.0760
1600 0.0681 0.0837 0.0361 0.1763 0.0577
3200 0.0552 0.0677 0.0317 0.1591 0.0455
400 0.0144 0.0141 0.0009 0.0241 0.0116
800 0.0113 0.0132 0.0011 0.0190 0.0086
1600 0.0078 0.0101 0.0010 0.0145 0.0057
3200 0.0049 0.0068 0.0009 0.0128 0.0034
400 0.1106 0.1334 0.0283 0.3479 0.0988
800 0.0851 0.1043 0.0270 0.3151 0.0734
1600 0.0650 0.0814 0.0235 0.2747 0.0547
3200 0.0500 0.0627 0.0222 0.1702 0.0414
-- ------ ------------- --------------- ------------- --------------- -------------
-- ------ ------------- ---------- ---------- ------------- ---------- ---------- ----------------
$n$ $\hat \rho$ $\hat y$ $\hat L$ $\hat \rho$ $\hat y$ $\hat L$ $\hat \lambda$
400 0.0035 -0.0029 0.0006 0.0010 -0.0008 0.0004 0.0040
800 0.0027 -0.0024 0.0000 0.0011 -0.0010 0.0001 0.0022
1600 0.0020 -0.0018 -0.0006 0.0010 -0.0008 -0.0005 0.0014
3200 0.0014 -0.0013 -0.0011 0.0010 -0.0009 -0.0008 0.0009
400 0.0358 0.0338 0.0261 0.0216 0.0179 0.0323 0.1005
800 0.0264 0.0251 0.0198 0.0217 0.0172 0.0252 0.0318
1600 0.0204 0.0192 0.0155 0.0190 0.0151 0.0202 0.0179
3200 0.0159 0.0149 0.0124 0.0192 0.0155 0.0184 0.0123
400 0.0036 -0.0030 0.0006 0.0010 -0.0009 0.0003 0.0028
800 0.0027 -0.0024 0.0000 0.0011 -0.0011 0.0001 0.0019
1600 0.0019 -0.0017 -0.0007 0.0010 -0.0009 -0.0005 0.0013
3200 0.0012 -0.0011 -0.0013 0.0007 -0.0006 -0.0011 0.0008
400 0.0345 0.0330 0.0251 0.0154 0.0130 0.0264 0.0348
800 0.0254 0.0244 0.0190 0.0155 0.0133 0.0217 0.0209
1600 0.0190 0.0182 0.0145 0.0163 0.0136 0.0185 0.0153
3200 0.0142 0.0135 0.0110 0.0148 0.0123 0.0150 0.0110
-- ------ ------------- ---------- ---------- ------------- ---------- ---------- ----------------
[.5]{} ![$\hat \chi$ for recursive preferences[]{data-label="fig:mc:sfig5"}](phi_cc_hp.pdf "fig:"){width="\linewidth"}
[.5]{} ![$\hat \chi$ for recursive preferences[]{data-label="fig:mc:sfig5"}](phistar_cc_hp.pdf "fig:"){width="\linewidth"}
\
[.5]{} ![$\hat \chi$ for recursive preferences[]{data-label="fig:mc:sfig5"}](phi_ez_hp.pdf "fig:"){width="\linewidth"}
[.5]{} ![$\hat \chi$ for recursive preferences[]{data-label="fig:mc:sfig5"}](phistar_ez_hp.pdf "fig:"){width="\linewidth"}
\
[.5]{} ![$\hat \chi$ for recursive preferences[]{data-label="fig:mc:sfig5"}](chi_ez_hp.pdf "fig:"){width="\linewidth"}
\[fig:mc\]
[.5]{} ![$\hat \chi$ for recursive preferences[]{data-label="fig:mc:sfig5:bs"}](phi_cc_bs.pdf "fig:"){width="\linewidth"}
[.5]{} ![$\hat \chi$ for recursive preferences[]{data-label="fig:mc:sfig5:bs"}](phistar_cc_bs.pdf "fig:"){width="\linewidth"}
\
[.5]{} ![$\hat \chi$ for recursive preferences[]{data-label="fig:mc:sfig5:bs"}](phi_ez_bs.pdf "fig:"){width="\linewidth"}
[.5]{} ![$\hat \chi$ for recursive preferences[]{data-label="fig:mc:sfig5:bs"}](phistar_ez_bs.pdf "fig:"){width="\linewidth"}
\
[.5]{} ![$\hat \chi$ for recursive preferences[]{data-label="fig:mc:sfig5:bs"}](chi_ez_bs.pdf "fig:"){width="\linewidth"}
\[fig:mc:bs\]
Empirical application {#s:emp}
=====================
In this section we apply the nonparametric SDF decomposition methodology to an economy similar to that in [@HansenHeatonLi]. We assume a representative agent with [@EpsteinZin1989] recursive preferences with EIS equal to unity.[^21] We work with a two-dimensional state process in consumption and earnings growth. Our analysis may be summarized as follows. First, under recursive preferences with discount and risk aversion preferences estimated from asset returns data (around $\beta = 0.98$ and $\gamma = 25$), we show that this bivariate specification is able to generate a permanent component which implies a long-run equity premium (i.e. return on assets relative to long-term bonds) of around 2.1% per quarter which is in the ballpark of empirically reasonable estimates. Second, we document the business cycle properties of the permanent and transitory components. Third, we describe the wedge required to tilt the distribution of the state to that which is relevant for long-run pricing. Finally, we show that, unlike the linear-Gaussian case, allowing for flexible treatment of the state process can lead to different behavior of long-run yields and different signs of correlation between the permanent and transitory components for different preference parameters. This suggests that nonlinearities in dynamics can be important in explaining the long end of the yield curve.
All data are quarterly and span the period 1947:Q1 to 2016:Q1 (277 observations). Data on consumption, dividends, inflation, and population are sourced from the National Income and Product Accounts (NIPA) tables. Real per capita consumption and dividend growth series are formed by taking seasonally adjusted consumption of nondurables and services (Table 2.3.5, lines 8 plus 13) and dividends (Table 1.12, line 16), deflating by the personal consumption implicit price deflator (Table 2.3.4, line 1), then converting to per capita growth rates using population data (Table 2.1, line 40). The resulting state variable is $X_t = (g_t,d_t)$ where $g_t$ and $d_t$ are real per capita consumption and dividend growth in quarter $t$, respectively.[^22]
We also use a time series of seven asset returns spanning 1947:Q1 to 2016:Q1. The first six are the returns on the six value-weighted portfolios sorted on size and book-to-market values, sourced from Kenneth French’s website. The final asset return is that of the 90-day Treasury bill. All asset returns series are converted to real returns using the implicit price deflator for personal consumption expenditures.
We estimate the preference parameters $(\beta,\gamma)$ and the pair $(\lambda,\chi)$ using the data on $X_t$ and the time series of seven asset returns. This falls into the setup of “Case 2” with $\alpha = (\beta,\gamma,\lambda,\chi)$. We estimate the parameters $(\beta,\gamma)$ using a series conditional moment estimation procedure [@AiChen2003]. This methodology was used recently in a similar context by [-@ChenFavilukisLudvigson].[^23] For each $(\beta,\gamma)$, we estimate the solution to the nonlinear eigenfunction problem, namely $(\hat \lambda_{(\beta,\gamma)},\hat \chi_{(\beta,\gamma)})$, using the procedure introduced in Section \[s:recursive\]. Here we make explicit the dependence of $(\lambda,\chi)$ on $\beta $ and $\gamma$, since different preference parameters will correspond to different continuation value functions. We then form the approximate SDF for $(\beta,\gamma)$ as: $$m(X_t,X_{t+1};(\beta,\gamma,\hat \lambda_{(\beta,\gamma)},\hat \chi_{(\beta,\gamma)})) = \frac{\beta}{\hat \lambda_{(\beta,\gamma)}} G_{t+1}^{-\gamma} \frac{ \big(\hat \chi_{(\beta,\gamma)}(X_{t+1}) \big)^\beta}{ \hat \chi_{(\beta,\gamma)}(X_t) }\,.$$ Let ${\mathbf{R}}_{t+1}$ denote a vector of (gross) asset returns from time $t$ to $t+1$ and ${\mathbf{1}}$ and ${\mathbf{0}}$ denote conformable vectors of ones and zeros. As the Euler equation ${\mathbb{E}}[m(X_t,X_{t+1}) {\mathbf{R}}_{t+1} - {\mathbf{1}} | X_t] = {\mathbf{0}}$ holds conditionally, we instrument the generalized residuals, namely: $$m(X_t,X_{t+1};(\beta,\gamma,\hat \lambda_{(\beta,\gamma)},\hat \chi_{(\beta,\gamma)})) {\mathbf{R}}_{t+1} - {\mathbf{1}}\,,$$ by functions of $X_t$ to form a criterion function which exploits the conditional nature of the Euler equation. We instrument the generalized residuals with the vector $b^k(X_t)$ used in estimation of the eigenvalues/functions. This leads to the criterion function: $$\begin{aligned}
L_n(\beta,\gamma) = &\, {\mathrm{trace}} \Bigg\{ \left( \frac{1}{n} \sum_{t=1}^n \Big( m \big(X_t,X_{t+1};(\beta,\gamma,\hat \lambda_{(\beta,\gamma)},\hat \chi_{(\beta,\gamma)})\big) {\mathbf{R}}_{t+1} - {\mathbf{1}} \Big)b^k(X_t) ' \right) \\
& \quad \quad {\widehat{{\mathbf{G}}}}^{-} \left( \frac{1}{n} \sum_{t=1}^n b^k(X_t) \Big( m \big(X_t,X_{t+1};(\beta,\gamma,\hat \lambda_{(\beta,\gamma)},\hat \chi_{(\beta,\gamma)})\big) {\mathbf{R}}_{t+1} - {\mathbf{1}} \Big)' \right) \Bigg\}\,.\end{aligned}$$ We minimize $L_n(\beta,\gamma)$ to obtain $(\hat \beta,\hat \gamma)$ and we set $\hat \alpha = (\hat \beta,\hat \gamma,\hat \lambda_{(\hat \beta,\hat \gamma)},\hat \chi_{(\hat \beta,\hat \gamma)})$. We then estimate $\rho$, $\phi$, $\phi^*$ and related quantities using the estimator ${\widehat{{\mathbf{M}}}}$ in display (\[e:mhat2\]) for this choice of $\hat \alpha$.
To implement the procedure, we use the same basis functions for estimation of $(\rho$, $\phi$, $\phi^*)$ and $(\lambda,\chi)$. To construct the basis, we form fifth-order univariate Hermite polynomial bases for each of the $g_t$ and $d_t$ series. We then construct a tensor product basis from the univariate bases but we discard any tensor-product polynomials whose total degree is order six or higher. The resulting sparse basis has dimension 15 (a tensor product basis would have dimension 25) and it is no less efficient in terms of approximation error than a tensor-product basis.[^24] We sometimes compare with the univariate state process $X_t = g_t$ for which we use an eighth-order Hermite polynomial basis. We instrument the generalized residuals with a lower-dimensional vector of basis functions to form the criterion function $L_n$ (dimension 6 in the univariate case and 10 in the bivariate case) to that the number of moment conditions is not unreasonably large. Similar results are obtained with different dimensional bases.
$X_t = (g_t,d_t)$ $X_t = g_t$ $X_t = (g_t,d_t)$
------------------------ ----------------------------------------- ----------------------------------------- -------------------------------------- -------------------------------------- --------------------------------------
\[-12pt\] $\hat \rho$ $\underset{(0.9701 , 0.9866)}{ 0.9779}$ $\underset{(0.9686, 0.9859)}{ 0.9784}$ $\underset{(0.9851,0.9872)}{0.9859}$ $\underset{(0.9850,0.9881)}{0.9861}$ $\underset{(0.9842,0.9913)}{0.9860}$
$\hat y $ $\underset{(0.0134 , 0.0303)}{ 0.0223}$ $\underset{(0.0141, 0.0318)}{ 0.0218}$ $\underset{(0.0129,0.0150)}{0.0142}$ $\underset{(0.0120,0.0151)}{0.0140}$ $\underset{(0.0087,0.0159)}{0.0141}$
$\hat L $ $\underset{(0.0000 , 0.0410)}{ 0.0208}$ $\underset{(0.0000, 0.0432)}{ 0.0219}$ $\underset{(0.0090,0.0185)}{0.0128}$ $\underset{(0.0146,0.0295)}{0.0203}$ $\underset{(0.0198,0.0435)}{0.0297}$
\[-12pt\] $\hat \beta$ $\underset{(0.9747 , 0.9892)}{ 0.9818}$ $\underset{(0.9734 , 0.9885)}{ 0.9819}$ 0.99 0.99 0.99
$\hat \gamma$ $\underset{(1.0000 ,46.2299)}{25.4383}$ $\underset{(1.0000 ,50.0228)}{27.7928}$ 20 25 30
$\hat \lambda$ $\underset{(0.8114 , 0.9891)}{ 0.8967}$ $\underset{(0.7960 , 0.9955)}{ 0.8862}$ $\underset{(0.9008,0.9324)}{0.9154}$ $\underset{(0.8789,0.9205)}{0.8983}$ $\underset{(0.8579,0.9111)}{0.8834}$
Table \[tab:emp\] presents the estimates and 90% confidence intervals.[^25] There are several notable aspects. First, both state specifications are able to generate a permanent component whose entropy is consistent with a return premium of around 2.1% per quarter relative to the long bond, which is in the ballpark of empirically reasonable estimates. Second, the estimated long-run yield of around 2.2% per quarter is too large, which is explained by the low value of $\hat \beta$. Third, the estimated entropy of the SDF itself is $0.0206$ for the bivariate specification and $0.0212$ for the univariate specification.[^26] Therefore, the estimated horizon dependence (the difference between the entropy of the permanent component and the entropy of the SDF; see [@BackusChernovZin]) is about 0.02% and 0.07% per quarter, respectively. These estimates are well inside the bound of $\pm$0.1% per month that [@BackusChernovZin] argue is required to match the spread in average yields between short- and long-term nominal bonds.[^27] Finally, the estimates of $\gamma$ are quite imprecise, in agreement with previous studies (see, e.g., [@ChenFavilukisLudvigson] who use a similar data set and a different, but related, estimation technique). The confidence intervals for $\rho$, $y$ and $L$ in the left panel of Table \[tab:emp\] are quite wide because they reflect, in large part, the uncertainty in estimating $\beta$ and $\gamma$. Experimentation with different sieve dimensions resulted in estimates of $\gamma$ between $20$ and $30$, which is in agreement with previous studies.[^28] The right panel of Table \[tab:emp\] presents estimates of $\rho$, $y$ and $L$ fixing $\beta = 0.99$ and $\gamma = 20$, $25$, and $30$ ($\chi$ and $\lambda$ are still estimated nonparametrically). It is clear that the resulting confidence intervals are much narrower once the uncertainty in estimating $\beta$ and especially $\gamma$ is shut down. For example, with $\gamma = 25$ the estimate of $L$ is very similar to that obtained for $\hat \gamma$, but the confidence interval shrinks from $(0.000 , 0.041)$ to $(0.015,0.030)$. Confidence intervals for $\rho$ and $y$ are also considerably narrower.
We now turn to analyzing the time-series properties of the permanent and transitory components. The upper panel of Figure \[fig:ts\] presents time-series plots of the SDF under recursive preferences obtained at the parameter estimates $\hat \alpha = (\hat \beta,\hat \gamma,\hat \lambda_{(\hat \beta,\hat \gamma)},\hat \chi_{(\hat \beta,\hat \gamma)})$ for the bivariate state specification. The upper panel also plots the time series of the permanent and transitory components, which are constructed as: $$\begin{aligned}
\frac{\hat M_{t+1}^P}{\hat M_t^P} & = \hat \rho^{-1} m(X_t,X_{t+1};\hat \alpha) \frac{\hat \phi(X_{t+1})}{\hat \phi(X_t)} & \frac{\hat M_{t+1}^T}{\hat M_t^T} & = \hat \rho \frac{\hat \phi(X_t)}{\hat \phi(X_{t+1})}\,.\end{aligned}$$ As can be seen, the great majority of the movement in the SDF is attributable to movements in the permanent component. Both evolve closely over time and exhibit strong counter-cyclicality. The transitory component for recursive preferences is small, coherent with the literature on bounds which finds that the transitory component should be substantially smaller than the permanent component. The correlation of the permanent component series $\hat M_{t+1}^P/\hat M_t^P$ and GDP growth is approximately $-0.39$ whereas the correlation of the transitory component series $\hat M_{t+1}^T/\hat M_t^T$ and $g_{t+1}$ is approximately $0.05$. The lower panel of Figure \[fig:ts\] displays the time series of the SDF and permanent and transitory components obtained under power utility using the same $(\hat \beta,\hat \gamma)$. This panel shows that the permanent component, which is similar to that obtained under recursive preferences, is much more volatile than the SDF series. The large difference between the SDF and permanent component series under power utility is due to a very volatile transitory component, which implies a counterfactually large spread in average yields between short- and long-term bonds ([@BackusChernovZin]).
{width="0.9\linewidth"}
To understand further the long-run pricing implications of the estimates of $\rho$, $\phi$ and $\phi^*$, Figures \[fig:emp:phi\_1\]–\[fig:emp:phistar\_2\] plot the estimated $\phi$ and $\phi^*$ under recursive preferences for both the bivariate and univariate state specifications. It is clear that both estimates of $\phi$ are reasonably flat, which explains the small variation in the transitory component in Figure \[fig:ts\]. However, the estimated $\phi^*$ has a pronounced downward slope in $g$. The bivariate specification shows that this is only part of the story, as the estimated $\phi^*$ is also downward-sloping in $d$, especially when consumption growth is relatively low. What is the economic significance of these estimates? Proposition \[p:id\] shows that $\phi\phi^*$ is the Radon-Nikodym derivative of the measure corresponding to ${\widetilde{{\mathbb{E}}}}$, say ${\widetilde{Q}}$, with respect to $Q$. Figures \[fig:emp:rn\_1\]–\[fig:emp:rn\_2\] plot the estimated change of measure for the two state specifications. As the estimate of $\phi$ is relatively flat, the estimated change of measure is characterized largely by $\hat \phi^*$. Therefore, in the bivariate case, the distribution ${\widetilde{Q}}$ assigns relatively more mass to regions of the state space in which there is low dividend and consumption growth than the stationary distribution $Q$, and relatively less mass to regions with high consumption growth.
[.5]{} ![Contour plot of $\hat \phi(x) \hat \phi^*(x)$ for $X_t = (g_t,d_t)$[]{data-label="fig:emp:rn_2"}](phi_emp_1.pdf "fig:"){width="\linewidth"}
[.5]{} ![Contour plot of $\hat \phi(x) \hat \phi^*(x)$ for $X_t = (g_t,d_t)$[]{data-label="fig:emp:rn_2"}](phi_emp_2.pdf "fig:"){width="\linewidth"}
\
[.5]{} ![Contour plot of $\hat \phi(x) \hat \phi^*(x)$ for $X_t = (g_t,d_t)$[]{data-label="fig:emp:rn_2"}](phistar_emp_1.pdf "fig:"){width="\linewidth"}
[.5]{} ![Contour plot of $\hat \phi(x) \hat \phi^*(x)$ for $X_t = (g_t,d_t)$[]{data-label="fig:emp:rn_2"}](phistar_emp_2.pdf "fig:"){width="\linewidth"}
\
[.5]{} ![Contour plot of $\hat \phi(x) \hat \phi^*(x)$ for $X_t = (g_t,d_t)$[]{data-label="fig:emp:rn_2"}](rn_emp_1.pdf "fig:"){width="\linewidth"}
[.5]{} ![Contour plot of $\hat \phi(x) \hat \phi^*(x)$ for $X_t = (g_t,d_t)$[]{data-label="fig:emp:rn_2"}](rn_emp_2.pdf "fig:"){width="\linewidth"}
Finally, we turn to the role of nonlinearities and non-Gaussianity in explaining certain features of the long-end of the term structure. Figure \[fig:emp:yield\] presents nonparametric estimates of (a) the (quarterly) long-run yield and (b) the covariance between the logarithm of the permanent and transitory components, namely $\hat m_t^P = \log (\hat M^P_{t+1}/\hat M^P_t)$ and $\hat m^T_t = \log (\hat M^T_{t+1}/\hat M^T_t)$, recovered from the data on $X_t = (g_t,d_t)$ with $\beta = 0.994$ and $\gamma$ increased from $\gamma = 0$ to $\gamma = 40$. The nonparametric estimates are presented alongside estimates for two parametric specifications of the state process. The first specification assumes $X_t = (g_t,d_t)$ evolves as a Gaussian VAR(1) with constant variance. The second specification is a Gaussian AR(1) for log consumption growth with stochastic volatility: $$g_{t+1} - \mu = \kappa ( g_{t} - \mu ) + \sqrt{v_t} e_{t+1}\,, \quad e_t \sim \mbox{ i.i.d. N$(0,1)$}$$ where $\{v_t\}$ is a first-order autoregressive gamma process (i.e. a discrete-time versions of the Feller square-root process; see [@GourierouxJasiak2006]). The state vector for the second specification is $X_t = (g_t,v_t)$. We refer to this second specification as SV-AR(1). The long-run yield and the covariance Cov($ m^P_{t+1}$,$ m^T_{t+1}$) were obtained analytically as functions of $\beta$, $\gamma$, and the estimates of the VAR(1) and SV-AR(1) parameters.[^29]
[.5]{} ![Cov($\hat m^P_{t+1}$,$\hat m^T_{t+1}$)[]{data-label="fig:emp:cov"}](emp_yld_sv.pdf "fig:"){width="\linewidth"}
[.5]{} ![Cov($\hat m^P_{t+1}$,$\hat m^T_{t+1}$)[]{data-label="fig:emp:cov"}](emp_cov_sv.pdf "fig:"){width="\linewidth"}
Figure \[fig:emp:yld\] shows that the nonparametric estimates of the long-run yield are non-monotontic, whereas the parametric estimates are monotonically decreasing. This non-monotonicity is not apparent in the nonparametric estimates using $X_t = g_t$. It is also clear that the nonparametric estimates of the long-run yield are much larger for larger values of $\gamma$ than the corresponding results from either of the two parametric models.
Figure \[fig:emp:cov\] displays the covariance between the log of the nonparametric estimates of the permanent and transitory components for different values of $\gamma$ against those obtained for the two parametric state processes. The covariance of the nonparametric estimates of the permanent and transitory components is negative for low to moderate values of $\gamma$, but becomes positive for larger values of $\gamma$.[^30] In contrast, the parametric estimates are negative and decreasing in $\gamma$. A recent literature has emphasized the role of positive dependence between the permanent and transitory components in explaining excess returns of long-term bonds [@BakshiChabiYo; @BC-YG; @BC-YG:rec]. Positive dependence also features in models in which the term structure of risk prices is downward sloping (see, e.g., the example presented in section 7.2 in [@BorovickaHansen]). However, positive dependence is known to be difficult to generate via conventional preference specifications in workhorse models with exponentially-affine dynamics, such as the long-run risks model of [@BansalYaron]. Although the covariance is poorly estimated for large values of $\gamma$, this finding at least suggests that nonlinearities in state dynamics may have a role to play in explaining salient features of the long end of the yield curve.
Conclusion {#s:conc}
==========
This paper introduces econometric methods to extract the permanent and transitory components of the SDF process in the long-run factorization of [@AJ], [@HS2009], and [@Hansen2012]. We show how to estimate the solution to the Perron-Frobenius eigenfunction problem directly from a time series of data on state variables and the SDF process. By estimating directly the Perron-Frobenius eigenvalue and eigenfunction, we can (1) reconstruct the time series of the permanent and transitory components and investigate their properties (e.g. the size of the components, their correlation, etc), and (2) estimate both the yield and the change of measure which characterizes pricing over long investment horizons. This represents a useful contribution relative to existing empirical work which has established bounds on various moments of the permanent and transitory components as functions of asset returns, but has not extracted these components directly from data. The methodology is nonparametric in that it does not impose tight parametric restrictions on the dynamics of the state. We view this approach as being complementary to parametric methods and useful to help better understand the roles of dynamics and preferences in structural macro-finance models.
The main technical contributions of this paper are to introduce nonparametric sieve estimators of the eigenvalue and eigenfunctions and establish consistency, convergence rates and asymptotic normality of the estimators, and some efficiency properties. We also introduce nonparametric estimators of the continuation value function under [@EpsteinZin1989] recursive preferences with unit EIS and study their large-sample properties.
The econometric methodology may be extended and applied in several different ways. First, the methodology can be applied to study more general multiplicative functional processes such as the valuation and stochastic growth processes in [@HansenHeatonLi], [@HS2009], and [@Hansen2012]. Second, the methodology can be applied to models with latent state variables. The main theoretical results (Theorems \[t:rate\] and \[t:fpest\]) are sufficiently general that they apply equally to such cases. Finally, our analysis was conducted within the context of structural models in which the SDF process was linked tightly to preferences. A further extension would be to apply the methodology to SDF processes which are extracted flexibly from panels of asset returns data. To this end, this methodology could be applied in conjunction with information projection-based SDF estimation procedures developed recently by [@SchneiderTrojani] and [@GJT].
[ ]{}
Supplementary Appendix for
Nonparametric Stochastic Discount Factor Decomposition
This appendix contains material to support the paper “Nonparametric Stochastic Discount Factor Decomposition”. Appendix \[ax:est\] presents further results on estimation of the eigenvalue and eigenfunctions used for SDF decomposition and on the estimation of the continuation value function under recursive preferences. Appendix \[ax:inf\] contains additional results on inference. Appendix \[ax:id\] further details on the relation between the identification and existence conditions in Section \[s:id\] and the identification and existence conditions in [@HS2009] and [@BHS]. Appendix \[ax:filter\] discusses extension of the SDF decomposition methodology to models with latent state variables. Proofs of all results are presented in Appendix \[ax:proofs\].
Additional results on estimation {#ax:est}
================================
In this section we present some supplementary lemmas from which Theorems \[t:rate\] and \[t:fpest\] follow immediately. We also present some sufficient conditions for Assumption \[a:var\] and \[a:fp:var\].
Bias and variance calculations for Theorem \[t:rate\] {#ax:est:cgce}
-----------------------------------------------------
When we refer to an eigenfunction/eigenvector as being unique, we mean unique up to sign and scale normalization. The results below for $\rho$ and $\phi$ are derived using arguments from [@Gobetetal] (note that [@Gobetetal] study a selfadjoint operator whereas here the operator is nonselfadjoint). Our results for $\phi^*$ are new.
The first result shows that the approximate solutions $\rho_k$, $\phi_k^{\phantom *}$ and $\phi_k^*$ from the eigenvalue problem (\[e:gev\]) are well defined and unique for all $k$ sufficiently large. It should be understood throughout that $c_k^{\phantom *}$, $c_k^*$, $\hat c$ and $\hat c^*$ are all real-valued eigenvectors.
\[lem:exist\] Let Assumptions \[a:id\] and \[a:bias\] hold. Then there exists $K \in {\mathbb{N}}$ such that for all $k \geq K$, the maximum eigenvalue $\rho_k$ of the eigenvector problem (\[e:gev\]) is real and simple, and hence $({\mathbf{M}}, {\mathbf{G}})$ has unique right- and left-eigenvectors $c_k^{\phantom *}$ and $c_k^*$ corresponding to $\rho_k$.
\[lem:bias\] Let Assumptions \[a:id\] and \[a:bias\] hold. Then:
1. $|\rho_k - \rho| = O(\delta_k^{\phantom *})$
2. $\|\phi_k - \phi\| = O(\delta_k^{\phantom *})$
3. $\|\phi_k^* - \phi^*\| = O(\delta_k^*)$
where $\delta_k^{\phantom *}$ and $\delta_k^*$ are defined in display (\[e:deltas\]). The rates should be understood to hold under the scale normalizations $\|\phi\| = 1$, $\| \phi_k\| = 1$, $\| \phi^*_{\phantom k} \| = 1$, and $\| \phi^*_k \| = 1$ and the sign normalizations $\langle \phi_k,\phi \rangle \geq 0$ and $\langle \phi_k^*,\phi^* \rangle \geq 0$.
The following result shows that the solutions $\hat \rho$, $\hat c^{\phantom *}$ and $\hat c^*$ to the sample eigenvalue problem (\[e:est\]) are well defined and unique with probability approaching one (wpa1).
\[lem:exist:hat\] Let Assumptions \[a:id\], \[a:bias\], and \[a:var\] hold. Then wpa1, the maximum eigenvalue $\hat \rho$ of the generalized eigenvector problem (\[e:est\]) is real and simple, and hence $({\widehat{{\mathbf{M}}}},{\widehat{ {\mathbf{G}}}})$ has unique right- and left-eigenvectors $\hat c$ and $\hat c^*$ corresponding to $\hat \rho$.
\[lem:var\] Let Assumptions \[a:id\], \[a:bias\], and \[a:var\] hold. Then:
1. $|\hat \rho - \rho_k| = O_p(\eta_{n,k}^{\phantom *})$
2. $\|\hat \phi - \phi_k\| = O_p(\eta_{n,k}^{\phantom *})$
3. $\|\hat \phi^* - \phi_k^*\| = O_p(\eta_{n,k}^{*})$
where $\eta_{n,k}^{\phantom *}$ and $\eta_{n,k}^*$ are defined in display (\[e:etas\]). The rates should be understood to hold under the scale normalizations $\|\hat \phi\| = 1$, $\| \phi_k\| = 1$, $\| \hat \phi^*_{\phantom k} \| = 1$ and $\| \phi^*_k \| = 1$ and the sign normalizations $\langle \hat \phi,\phi_k\rangle \geq 0$ and $\langle \hat \phi^* ,\phi^*_k \rangle \geq 0$.
Bias and variance calculations for Theorem \[t:fpest\] {#ax:est:fp}
------------------------------------------------------
The following two Lemmas are known in the literature on solution of nonlinear equations by projection methods (see, e.g., Chapter 19 of [@Kras]). The first result shows that the approximate solution $h_k$ is well defined for all $k$ sufficiently large and converges to the fixed point $h$ as $k \to \infty$.
\[lem:fp:exist\] Let Assumptions \[a:fp:exist\] and \[a:fp:bias\] hold. Then there exists $K \in {\mathbb{N}}$ and $\varepsilon > 0$ such that for all $k \geq K$ the projected problem $(\Pi_k \mathbb T) h_k = h_k$ has a unique solution $h_k$ in the ball $\|h - h_k\| \leq \varepsilon$. Moreover, $\|h - h_k\| \to 0$ as $k \to \infty$.
Let $\chi_k = h_k /\|h_k\|$ and $\lambda_k = \| \Pi_k {\mathbb{T}} \chi_k\|$.
\[lem:bias:fp\] Let Assumptions \[a:fp:exist\] and \[a:fp:bias\] hold. Then $\lambda_k$, $\chi_k$, and $h_k$ are each well defined for all $k$ sufficiently large, and:
1. $|\lambda_k - \lambda| = O( \tau_k )$
2. $\|\chi_k - \chi\| = O( \tau_k )$
3. $\|h_k - h\| = O( \tau_k )$.
We now show that the sample fixed-point problem has a solution wpa1, and that the solution is a consistent estimator of $h$. The following two results are new.
\[lem:fphat:exist\] Let Assumptions \[a:fp:exist\]–\[a:fp:var\] hold. Then wpa1, there exists a fixed point $\hat v$ of ${\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{T}}}}$ on a neighborhood of $v_k$ and the estimator $\hat h$ from (\[e:fpest\]) satisfies $\|\hat h - h\| = o_p(1)$.
The following result bounds the sampling error. Notice that the “bias term” $\tau_k$ shows up due to the nonlinearity of ${\mathbb{T}}$. In contrast, the bias term $\delta_k$ does not appear in the corresponding “variance terms” for the linear eigenvalue problem (see Lemma \[lem:var\]).
\[lem:var:fp\] Let Assumptions \[a:fp:exist\]–\[a:fp:var\] hold. Then wpa1 there exists a fixed point $\hat v$ of ${\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{T}}}}$ such that the estimators $\hat \lambda$, $\hat \chi$, and $\hat h$ from (\[e:fpest\]) satisfy:
1. $|\hat \lambda - \lambda_k| = O_p( \nu_{n,k} ) + o_p(\tau_k) $
2. $\|\hat \chi - \chi_k \| = O_p( \nu_{n,k} ) + o_p(\tau_k)$
3. $\|\hat h - h_k\| = O_p( \nu_{n,k} ) + o_p(\tau_k)$.
Sufficient conditions for Assumption \[a:var\] {#ax:est:mat}
----------------------------------------------
We derive the results assuming that the state process $\{X_t\}$ is either beta-mixing (absolutely regular) or rho-mixing. We use these dependence concepts because a variety of models for macroeconomic and financial time series exhibit these types weak dependence. Examples include copula-based models [@ChenWuYi2009; @Beare2010] and discretely sampled diffusion processes [@ChenHansenCarrasco].
The beta-mixing coefficient between two $\sigma $-algebras ${\mathcal{A}}$ and ${\mathcal{B}}$ is: $$2\beta ({\mathcal{A}},{\mathcal{B}})=\sup \sum_{(i,j)\in I\times J}|{\mathbb{P}}(A_{i}\cap B_{j})-{\mathbb{P}}(A_{i}){\mathbb{P}}(B_{j})|$$ with the supremum taken over all ${\mathcal{A}}$-measurable finite partitions $\{A_{i}\}_{i\in I}$ and ${\mathcal{B}}$-measurable finite partitions $\{B_{j}\}_{j\in J}$. The beta-mixing coefficients of $\{X_{t} : t \in T\}$ are defined as: $$\beta_q =\sup_{t}\beta (\sigma (\ldots ,X_{t-1},X_{t}),\sigma (X_{t+q},X_{t+q+1},\ldots ))\,.$$ We say that $\{X_t : t \in T \}$ is *exponentially beta-mixing* if $\beta_q \leq ce^{-\gamma q}$ for some $\gamma >0$ and $c\geq 0$. The rho-mixing coefficients of $\{X_t : t \in T\}$ are defined as: $$\rho_q = \sup_{ \psi \in L^2 : {\mathbb{E}}[\psi] = 0, \|\psi\| = 1} {\mathbb{E}}\big[ {\mathbb{E}}[\psi(X_{t+q})|X_t]^2\big]^{1/2}\,.$$ We say that $\{X_t : t \in T\}$ is *exponentially rho-mixing* if $\rho_q \leq e^{-\gamma q}$ for some $\gamma > 0$.
### Sufficient conditions in Case 1
The following two lemmas derive convergence rates for the estimators ${\widehat{{\mathbf{G}}}}$ in (\[e:ghat\]) and ${\widehat{{\mathbf{M}}}}$ in (\[e:mhat1\]). These results may be used to verify Assumptions \[a:var\] and \[a:asydist\](b) and bound the terms $\eta_{n,k}$ and $\eta_{n,k}^*$ in display (\[e:etas\]).
Lemma \[lem:beta:1\] uses an exponential inequality for sums of weakly-dependent random matrices derived in [@ChenChristensen-reg]. Recall that $\xi_k = \sup_x \| {\mathbf{G}}^{-1/2} b^k(x)\|$.
\[lem:beta:1\] Let the following hold:
1. $\{X_t\}$ is strictly stationary and exponentially beta-mixing
2. ${\mathbb{E}}[|m(X_t,X_{t+1})|^r] < \infty$ for some $2 < r \leq \infty$
3. $\xi_k^{1+2/r} (\log n)/\sqrt n = o(1)$.
Then: Assumption \[a:var\] holds, $$\begin{aligned}
\|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| & = O_p( \xi_k (\log n)/\sqrt n) &
\|{\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o \| & = O_p ( \xi_k^{1+2/r} (\log n)/\sqrt n ) \end{aligned}$$ and we may take $\eta_{n,k}^{\phantom *} = \eta_{n,k}^* = \xi_k^{1+2/r} (\log n)/\sqrt n$.
The following Lemma extends arguments from [@Gobetetal].
\[lem:rho:1\] Let the following hold:
1. $\{X_t\}$ is strictly stationary and exponentially rho-mixing
2. ${\mathbb{E}}[|m(X_t,X_{t+1})|^r] < \infty$ for some $2 < r \leq \infty$
3. $\xi_k^{1+2/r} \sqrt{k/n} = o(1)$.
Then: Assumption \[a:var\] holds, $$\begin{aligned}
\|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| & = O_p( \xi_k \sqrt{k/n}) &
\|{\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o \| & = O_p ( \xi_k^{1+2/r} \sqrt{k/n} ) \end{aligned}$$ and we may take $\eta_{n,k}^{\phantom *} = \eta_{n,k}^* = O(\xi_k^{1+2/r} /\sqrt n)$.
### Sufficient conditions in Case 2 with parametric first-stage
The following lemma derives convergence rates for the estimators ${\widehat{{\mathbf{G}}}}$ in (\[e:ghat\]) and ${\widehat{{\mathbf{M}}}}$ in (\[e:mhat2\]) when $\alpha_0 \in {\mathcal{A}} \subseteq {\mathbb{R}}^{{d_\alpha}}$ is a finite-dimensional parameter. If the first-stage estimator $\hat \alpha$ converges at a root-$n$ rate (which corresponds to taking $\beta = 1/2$ in the following lemma), then the same convergence rates for $\hat \rho$, $\hat \phi$ and $\hat \phi^*$ are obtained as for Case 1. For brevity, in the following two results we consider only the case in which the state process is beta-mixing.
\[lem:beta:2\] Let the conditions of Lemma \[lem:beta:1\] hold for $m(x_0,x_1) = m(x_0,x_1;\alpha_0)$, and let:
1. $\|\hat \alpha - \alpha_0\| = O_p(n^{-\beta})$ for some $\beta > 0$
2. $m(x_0,x_1;\alpha)$ be continuously differentiable in $\alpha$ on a neighborhood $N$ of $\alpha_0$ for all $(x_0,x_1) \in {\mathcal{X}}^2$ with: $${\mathbb{E}} \Bigg[ \sup_{\alpha \in N} \left\| \frac{\partial m(X_t,X_{t+1};\alpha)}{\partial \alpha} \right\|^2 \Bigg] < \infty$$
3. $\xi_k^{1+2/r} (\log n)/\sqrt n + \xi_k/ n^{\beta} = o(1)$.
Then: Assumption \[a:var\] holds, $$\begin{aligned}
\|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| & = O_p( \xi_k (\log n)/\sqrt n ) &
\|{\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o \| & = O_p ( \xi_k^{1+2/r} (\log n)/\sqrt n + \xi_k/ n^{\beta} ) \end{aligned}$$ and we may take $\eta_{n,k}^{\phantom *} = \eta_{n,k}^* = \xi_k^{1+2/r} (\log n)/\sqrt n + \xi_k/ n^{\beta}$.
### Sufficient conditions in Case 2 with semi/nonparametric first-stage
We now derive convergence rates for the estimators ${\widehat{{\mathbf{G}}}}$ in (\[e:ghat\]) and ${\widehat{{\mathbf{M}}}}$ in (\[e:mhat2\]) for the semi/nonparametric case in which $\alpha_0 \in {\mathcal{A}} \subseteq {\mathbb{A}}$ is an infinite-dimensional parameter. The parameter space is ${\mathcal{A}} \subseteq {\mathbb{A}}$ (a Banach space) equipped with some norm $\|\cdot\|_{{\mathcal{A}}}$. This includes the case in which $\alpha$ is a function, i.e. $\alpha = h$ with ${\mathbb{H}}$ a function space, and in which $\alpha$ consists of both finite-dimensional and function parts, i.e. $\alpha = (\theta,h)$ with ${\mathbb{A}} = \Theta \times {\mathbb{H}}$ where $\Theta \subseteq {\mathbb{R}}^{\dim (\theta)}$. For each $\alpha \in {\mathcal{A}}$ we define ${\mathbb{M}}^{(\alpha)}$ as the operator ${\mathbb{M}}^{(\alpha)}\psi(x) = {\mathbb{E}}[m(X_t,X_{t+1};\alpha) \psi(X_{t+1}) | X_t = x]$ with the understanding that ${\mathbb{M}}^{(\alpha_0)} = {\mathbb{M}}$. Let ${\mathcal{M}} = \{ m(x_0,x_1;\alpha) - m(x_0,x_1;\alpha_0) : \alpha \in {\mathcal{A}}\}$. We say that ${\mathcal{M}}$ has an envelope function $E$ if there exists some measurable $E : {\mathcal{X}}^2 \to [1,\infty)$ such that $|m(x_0,x_1)| \leq E(x_0,x_1)$ for every $(x_0,x_1)$ and $m \in {\mathcal{M}}$. Let ${\mathcal{M}}^* = \{ m /E : m \in {\mathcal{M}}\}$. The functions in ${\mathcal{M}}^*$ are clearly bounded by $\pm 1$. Let $N_{[\,\,]}(u,{\mathcal{M}}^*,\|\cdot\|_{p})$ denote the entropy with bracketing of ${\mathcal{M}}^*$ with respect to the norm $\|\cdot\|_{p}$. Finally, let $\ell^*(\alpha) = \|{\mathbb{M}}^{(\alpha)} - {\mathbb{M}}\|$ and observe that $\ell^*(\alpha_0) = 0$.
\[lem:beta:3\] Let the conditions of Lemma \[lem:beta:1\] hold for $m(x_0,x_1) = m(x_0,x_1;\alpha_0)$, and let:
1. ${\mathcal{M}}$ have envelope function $E$ with $\|E\|_{4s} < \infty$ for some $s > 1$
2. $\log N_{[\,\,]}(u,{\mathcal{M}}^*,\|\cdot\|_{\frac{4vs}{2s-v}}) \leq {\mathrm{const}} \times u^{-2\zeta}$ for some $\zeta \in (0,1)$ and $v \in (0,2s)$
3. $\ell^*(\alpha)$ be Gateaux differentiable at $\alpha_0$ with $| \ell^*(\alpha) - \ell^*(\alpha_0) - \dot \ell^*_{\alpha_0} [\alpha - \alpha_0]| = O(\|\alpha - \alpha_0\|^2_{{\mathcal{A}}})$ as $\|\alpha - \alpha_0\|_{{\mathcal{A}}} \to 0$
4. $\|\hat \alpha - \alpha_0\|_{{\mathcal{A}}} = o_p(n^{-1/4})$ and $\sqrt n \dot \ell_{\alpha_0}^*[\hat \alpha - \alpha_0] = O_p(1)$
5. $\xi_k^{1+2/r} (\log n)/\sqrt n + \xi_k^2 \sqrt{k/n} = o(1)$.
Then: Assumption \[a:var\] holds, $$\begin{aligned}
\|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| & = O_p( \xi_k (\log n)/\sqrt n ) &
\|{\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o \| & = O_p \bigg( \frac{\xi_k^{1+2/r} (\log n)}{\sqrt n} + \frac{ \xi_k^2 \sqrt k }{\sqrt n} \bigg) \end{aligned}$$ and we may take $\eta_{n,k}^{\phantom *} = \eta_{n,k}^* = \xi_k^{1+2/r} (\log n)/\sqrt n + \xi_k^2 \sqrt{k/n}$.
Sufficient conditions for Assumption \[a:fp:var\] {#ax:est:mat:fp}
-------------------------------------------------
The following conditions are sufficient for Assumption \[a:fp:var\] and for bounding the term $\nu_{n,k}$ in display (\[e:nudef\]). Let $(a \vee b) = \max\{a,b\}$.
\[lem:beta:4\] Let the following hold:
1. $\{X_t\}$ is strictly stationary and exponentially beta-mixing
2. ${\mathbb{E}}[(G_{t+1}^{1-\gamma})^{2s}] < \infty$ for some $s > 1$
3. $\xi_k ( (\log n) \vee \xi_k^{\beta} \sqrt k)/\sqrt{n} = o(1)$.
Then: Assumption \[a:fp:var\] holds, $$\begin{aligned}
\|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| & = O_p( \xi_k (\log n)/\sqrt n ) &
\sup_{v \in {\mathbb{R}}^k : \| v\| \leq c} \| {\widehat{{\mathbf{T}}}}^ov - {\mathbf{T}}^o v\| & = O_p \bigg( \frac{\xi_k^{1+\beta}\sqrt{k}}{\sqrt n} \bigg) \end{aligned}$$ and we may take $\nu_{n,k} = \xi_k^{1+\beta} \sqrt{k/n}$.
Additional results on inference {#ax:inf}
===============================
Asymptotic normality of long-run entropy estimators
---------------------------------------------------
Here we consider the asymptotic distribution the estimator $\hat L$ of the entropy of the permanent component of the SDF, namely $L = \log \rho - {\mathbb{E}}[\log m(X_t,X_{t+1})]$.
In Case 1, the estimator of the long-run entropy is: $$\hat L = \log \hat \rho - \frac{1}{n} \sum_{t=0}^{n-1} \log m(X_t,X_{t+1})\,.$$ Recall that $\psi_{\rho,t} = \psi_\rho(X_t,X_{t+1})$ where the influence function $\psi_\rho$ is defined in (\[e:inf:def\]). Define: $$\psi_{lm}(x_t,x_{t+1}) = \log m(x_t,x_{t+1}) - {\mathbb{E}}[\log m(X_t,X_{t+1})]$$ set $\psi_{lm,t} = \psi_{lm}(X_t,X_{t+1})$.
\[p:asydist:L:1\] Let the assumptions of Theorem \[t:asydist:1\] hold and let $\{(\psi_{\rho,t},\psi_{lm,t})' : t \in T\}$ satisfy a CLT, i.e.: $$\frac{1}{\sqrt n} \sum_{t=0}^{n-1} \bigg( \begin{array}{c}
\psi_{\rho,t} \\
\psi_{lm,t} \end{array} \bigg)
\to_d N(0,W)$$ where the $2 \times 2$ matrix $W$ is positive definite. Then: $$\sqrt n (\hat L - L) \to_d N(0,V_L)$$ where $V_L = (\rho^{-1}\,,-1) W (\rho^{-1}\,,-1)'$.
In the preceding proposition, $V_L$ will be the long-run variance: $$V_L = \sum_{t \in {\mathbb{Z}}} {\mathrm{Cov}}(\psi_L(X_0,X_1),\psi_L(X_t,X_{t+1}))$$ where $\psi_L(X_t,X_{t+1}) = \rho^{-1} \psi_\rho(X_t,X_{t+1}) - \psi_{lm}(X_t,X_{t+1})$. Theorem \[t:eff:main\] below shows that $V_L$ is the semiparametric efficiency bound for $L$.
In Case 2, the estimator of the long-run entropy is: $$\hat L = \log \hat \rho - \frac{1}{n} \sum_{t=0}^{n-1} \log m(X_t,X_{t+1},\hat \alpha)\,.$$ As with asymptotic normality of $\hat \rho$, the asymptotic distribution of $\hat L$ will depend on the manner in which $\hat \alpha$ was estimated. For brevity, we just consider the parametric case studied in Theorem \[t:asydist:2a\]. Let $\psi_{lm}$ and $\psi_{lm,t}$ be as previously defined with $m(x_t,x_{t+1}) = m(x_t,x_{t+1},\alpha_0)$. Recall $\psi_{\alpha,t}$ and neighborhood $N$ from Assumption \[a:parametric\] and define: $$\hbar_{[{\mathrm{2a}}]} = \left( \rho^{-1} \; , \; - {\mathbb{E}} \left[\frac{1}{m(X_t,X_{t+1},\alpha)} \frac{\partial m(X_t,X_{t+1},\alpha)}{\partial \alpha'} \right] \; , \; -1 \right)'\,.$$
\[p:asydist:L:2a\] Let the assumptions of Theorem \[t:asydist:2a\] hold. Also let (i) there exist a neighborhood $N_1$ of $\alpha_0$ upon which the function $\log m(x_0,x_1,\alpha)$ is continuously differentiable in $\alpha$ for all $(x_0,x_1) \in {\mathcal{X}}^2$ with: $${\mathbb{E}} \bigg[ \sup_{\alpha \in N_1} \left\| \frac{1}{m(x_0,x_1,\alpha)} \frac{\partial m(x_0,x_1,\alpha) }{\partial \alpha} \right\| \bigg] < \infty$$ and (ii) $\{ (\psi_{\rho,t}^{\phantom \prime},\psi_{\alpha,t}',\psi_{lm,t}^{\phantom \prime})' : t \in T\}$ satisfies a CLT, i.e.: $$\frac{1}{\sqrt n } \sum_{t=0}^{n-1} \left( \begin{array}{c} \psi_{\rho,t} \\ \psi_{\alpha,t} \\ \psi_{lm,t} \end{array} \right) \to_d N(0,W_{[{\mathrm{2a}}]})$$ where the $({d_\alpha}+2) \times ({d_\alpha}+2)$ matrix $W_{[{\mathrm{2a}}]}$ is positive definite. Then: $$\sqrt n (\hat L - L) \to_d N(0,V_L^{[{\mathrm{2a}}]})$$ where $V_L^{[{\mathrm{2a}}]} = \hbar_{[{\mathrm{2a}}]}'W_{[{\mathrm{2a}}]}^{\phantom \prime} \hbar_{[{\mathrm{2a}}]}^{\phantom \prime}$.
Semiparametric efficiency bounds in Case 1
------------------------------------------
To establish efficiency, we first obtain the efficient influence function for the parameter $\rho$ by showing that $\rho$ is differentiable relative to an appropriately chosen tangent set (see Chapter 25 of [@vdV] for details). We then show that the efficient influence function for $\rho$ is precisely the influence function $\psi_\rho$ in expressions (\[e:ale:1\])–(\[e:inf:def\]).
Let $P_n(x,A) = \Pr( X_{t+n} \in A | X_t = x)$ denote the $n$-step transition probability of $X$ for any Borel set $A$. We say that $\{X_t\}_{t \in {\mathbb{Z}}}$ is *uniformly ergodic* if: $$\lim_{n \to \infty} \sup_{x \in {\mathcal{X}}} \| P_n(x,\cdot) - Q\|_{TV} =0$$ where $\|\cdot\|_{TV}$ denotes total variation norm and $Q$ denotes the stationary distribution of $X$.
\[a:eff\] $\{X_t\}_{t \in {\mathbb{Z}}}$ is uniformly ergodic.
Equivalent conditions for Assumption \[a:eff\], such as Doeblin’s condition, are well known (see, e.g., Theorem 16.0.2 of [@MeynTweedie2009]). Assumption \[a:eff\] also implies that $\{X_t\}_{t \in {\mathbb{Z}}}$ is exponentially phi-mixing, and hence also exponentially beta-mixing and exponentially rho-mixing [@IL pp. 365–366]. This assumption is used to construct the tangent set in Markov process models; see [@BickelKwon2001] and references therein.
\[t:eff:main\] (i) Let Assumption \[a:id\], \[a:asydist\](d), and \[a:eff\] hold and let $h : {\mathbb{R}} \to {\mathbb{R}}$ be continuously differentiable at $\rho$ with $h'(\rho) \neq 0$. Then: the efficiency bound for $h(\rho)$ is $h'(\rho)^2V_\rho$.\
(ii) If, in addition, ${\mathbb{E}}[(\log m(X_t,X_{t+1}))^2] < \infty$, then: the efficiency bound for $L$ is $V_L$.
Sieve perturbation expansion {#ax:est:inf}
----------------------------
The following result shows that $\hat \rho - \rho_k$ behaves as a linear functional of ${\widehat{{\mathbf{M}}}} - \rho_k {\widehat{{\mathbf{G}}}}$. This is used to derive the asymptotic distribution of $\hat \rho$ in Theorem \[t:asydist:1\]. It follows from Assumption \[a:var\] that we can choose sequences of positive constants $\eta_{n,k,1}$ and $\eta_{n,k,2}$ such that: $$\|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| = O_p(\eta_{n,k,1}) \quad \mbox{and} \quad \|{\widehat{{\mathbf{M}}}}^o - {{\mathbf{M}}}^o\| = O_p(\eta_{n,k,2})$$ with $\eta_{n,k,1} = o(1)$ and $\eta_{n,k,2} = o(1)$ as $n,k \to \infty$.
\[lem:expansion\] Let Assumptions \[a:id\], \[a:bias\] and \[a:var\] hold. Then: $$\hat \rho - \rho_k = c_k^{*\prime} ( {\widehat{{\mathbf{M}}}} - \rho_k {\widehat{{\mathbf{G}}}} ) c_k + O_p( \eta_{n,k,1} \times ( \eta_{n,k,1} \vee \eta_{n,k,2}) ) \,.$$ with $c_k$ and $c_k^*$ normalized so that $\|{\mathbf{G}} c_k\| = 1$ and $c_k^{* \prime } {\mathbf{G}} c_k^{\phantom *} = 1$.
In particular, if $\|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| = o_p(n^{-1/4})$ and $\|{\widehat{{\mathbf{M}}}}^o - {{\mathbf{M}}}^o\| = o_p(n^{-1/4})$ then: $$\sqrt n (\hat \rho - \rho_k) = \sqrt n c_k^{*\prime} ( {\widehat{{\mathbf{M}}}} - \rho_k {\widehat{{\mathbf{G}}}} ) c_k + o_p(1)\,.$$
Additional results on identification {#ax:id}
====================================
In this appendix we discuss separately existence and identification, and compare the conditions in the present paper with the stochastic stability conditions in [@HS2009] (HS hereafter) and [@BHS] (BHS hereafter).
Identification {#identification}
--------------
\[a:id:1\] Let the following hold:
1. ${\mathbb{M}}$ is bounded
2. There exists positive functions $\phi,\phi^* \in L^2$ and a positive scalar $\rho$ such that $(\rho,\phi)$ solves (\[e:pev\]) and $(\rho,\phi^*)$ solves (\[e:pev:star\])
3. ${\mathbb{M}} \psi$ is positive for each non-negative $\psi \in L^2$ that is not identically zero.
**Discussion of Assumptions:** Parts (a) and (c) are implied by the boundedness of ${\mathbb{M}}$ and positivity of ${\mathcal{K}}_m$ in Assumption \[a:id:0\](a). Proposition \[p:a:exist\] below shows that the conditions in Assumption \[a:id:0\] are sufficient for existence (part (b)). No compactness condition is assumed.
\[p:a:id\] Let Assumption \[a:id:1\] hold. Then: the functions $\phi$ and $\phi^*$ are the unique solutions (in $L^2$) to (\[e:pev\]) and (\[e:pev:star\]), respectively.
We now compare the identification results with those in HS and BHS. Some of HS’s conditions related to the generator of the semigroup of conditional expectation operators ${\widetilde{{\mathbb{E}}}}[\cdot|X_t = x]$ under the change of conditional probability induced by $M_t^P$, namely: $$\label{e:twist}
{\widetilde{{\mathbb{E}}}} [\psi(X_{t+\tau})|X_t = x] := {\mathbb{E}} \bigg[ \frac{M_{t+ \tau}^P}{M_t^P} \psi(X_{t+\tau}) \bigg| X_t = x\bigg] \,.$$ In discrete-time environments, both multiplicative functionals and semigroups are indexed by non-negative integers. Therefore, the “generator” in discrete-time is just the single-period distorted conditional expectation operator $\psi \mapsto {\widetilde{{\mathbb{E}}}}[\psi(X_{t+1})|X_t = x]$.
The following are discrete-time versions of Assumptions 6.1, 7.1, 7.2, 7.3, and 7.4 in HS.
\[c:HS\]
1. $\{ M^P_t : t \in T\}$ is a positive multiplicative functional
2. There exists a probability measure $\hat \varsigma$ such that $$\int {\widetilde{{\mathbb{E}}}}[\psi(X_{t+1})|X_t = x]\,{\mathrm{d}} \hat \varsigma(x) = \int \psi(x)\,{\mathrm{d}} \hat \varsigma(x)$$ for all bounded measurable $\psi : \mathcal X \to {\mathbb{R}}$
3. For any $\Lambda \in \mathscr X$ with $\hat \varsigma(\Lambda) > 0$, $${\widetilde{{\mathbb{E}}}}\left[ \left. \sum_{t=1}^\infty {1\!\mathrm{l}}\{X_t \in \Lambda\} \right| X_0 = x \right] > 0$$ for all $x \in \mathcal X$
4. For any $\Lambda \in \mathscr X$ with $\hat \varsigma(\Lambda) > 0$, $${\widetilde{{\mathbb{P}}}} \left( \left. \sum_{t=1}^\infty {1\!\mathrm{l}}\{X_t \in \Lambda\} = \infty \right| X_0 = x \right) = 1$$ for all $x \in \mathcal X$, where $${\widetilde{{\mathbb{P}}}} (\{X_s\}_{s=0}^t \in A | X_0 = x) = \int {{\mathbb{E}}}[(M_t^P/M_0^P) {1\!\mathrm{l}}\{\{X_s\}_{s=0}^t \in A\} |X_0 = x]\,{\mathrm{d}} \hat \varsigma(x)$$ for each $A \in \mathcal F_t$.
Condition \[c:HS\](a) is satisfied by construction of $M^P$ in (\[e:pctc\]). For Condition \[c:HS\](b), let $\phi$ and $\phi^*$ be as in Assumption \[a:id:1\](b) and normalize $\phi^*$ such that ${\mathbb{E}}[\phi(X_t)\phi^*(X_t)] = 1$. Under this normalization we can define a probability measure $\hat \varsigma$ by $\hat \varsigma(A) = {\mathbb{E}}[\phi(X_t) \phi^*(X_t) {1\!\mathrm{l}}\{X_t \in A\}]$ for all $A \in \mathscr X$.[^31] Recall that $Q$ is the stationary distribution of $X$. We then have: $$\begin{aligned}
& \int {\widetilde{{\mathbb{E}}}}[\psi(X_{t+1})|X_t = x]\,{\mathrm{d}} \hat \varsigma(x) \\
& = \int {\mathbb{E}} \left[ \left. \rho^{-1} m(X_t,X_{t+1}) \frac{\phi(X_{t+1})}{\phi(X_t)} \psi(X_{t+1}) \right| X_t = x \right] \phi(x) \phi^*(x)\,{\mathrm{d}} Q(x) \\
& = \rho^{-1} {\mathbb{E}} \left[ \phi^*(X_t) ({\mathbb{M}}(\phi \psi) ( X_t)) \right] \\
& = \rho^{-1} {\mathbb{E}} \left[ (({\mathbb{M}}^* \phi^*)(X_{t+1})) \phi(X_{t+1}) \psi(X_{t+1}) \right] \\
& = {\mathbb{E}}[\phi^*(X_{t+1}) \phi(X_{t+1}) \psi(X_{t+1})] = \int \psi(x) \,{\mathrm{d}} \hat \varsigma(x)\,.\end{aligned}$$ Therefore, Condition \[c:HS\](b) is satisfied. A similar derivation is reported for continuous-time semigroups in an preliminary 2005 draft of HS with $Q$ replaced by an arbitrary measure.
For Condition \[c:HS\](c), note that $\hat \varsigma(\Lambda) > 0$ implies $Q(\Lambda ) > 0$ under our construction of $\hat \varsigma$. Therefore, $\hat \varsigma(\Lambda) > 0$ implies $\phi(x) {1\!\mathrm{l}}\{x \in \Lambda\}$ is positive on a set of positive $Q$ measure. Moreover, by definition of ${\widetilde{{\mathbb{E}}}}$ we have: $$\begin{aligned}
{\widetilde{{\mathbb{E}}}}\left[ \left. \sum_{t=1}^\infty {1\!\mathrm{l}}\{X_t \in \Lambda\} \right| X_0 = x \right] & = & \frac{1}{\phi(x)} \sum_{t=1}^\infty \rho^{-t} \mathbb M_t (\phi(\cdot) {1\!\mathrm{l}}\{ \cdot \in \Lambda\})(x) \\
& \geq & \frac{1}{\phi(x)} \sum_{t=1}^\infty \lambda^{-t} \mathbb M_t(\phi(\cdot) {1\!\mathrm{l}}\{ \cdot \in \Lambda\})(x) \end{aligned}$$ for any $\lambda \geq r({\mathbb{M}})$ where $r({\mathbb{M}})$ denotes the spectral radius of ${\mathbb{M}}$. Assumption \[a:id:1\](c) implies ${\mathbb{M}}$ is irreducible and, by definition of irreducibility, $\sum_{t=1}^\infty \lambda^{-t} \mathbb M_t(\phi(\cdot) {1\!\mathrm{l}}\{ \cdot \in \Lambda\})(x) > 0$ (almost everywhere) holds for $\lambda > r({\mathbb{M}})$. Therefore, Assumption \[a:id:1\](c) implies Condition \[c:HS\](c), up to the “almost everywhere” qualification.
Part (d) is a Harris recurrence condition which does not translate clearly in terms of the operator $\mathbb M$. When combined with existence of an invariant measure and irreducibility (Condition \[c:HS\](b) and (c), respectively), it ensures both uniqueness of $\hat \varsigma$ as the invariant measure for the distorted expectations as well as $\phi$-ergodicity, i.e., $$\label{e:HScgce}
\lim_{\tau \to \infty} \sup_{0 \leq \psi \leq \phi} \left| {\widetilde{{\mathbb{E}}}}\left[\left. \frac{\psi(X_{t+\tau})}{\phi(X_{t+\tau})}\right|X_t = x\right] - \int \frac{\psi(x)}{\phi(x)}\,{\mathrm{d}} \hat \varsigma(x) \right| = 0$$ (almost everywhere) where the supremum is taken over all measurable $\psi$ such that $0 \leq \psi \leq \phi$ [@MeynTweedie2009 Proposition 14.0.1]. Result (\[e:HScgce\]) is a discrete-time version of Proposition 7.1 in HS, which they use to establish identification of $\phi$. Assumption \[a:id:1\] alone is not enough to obtain a convergence result like (\[e:HScgce\]). On the other hand, the conditions in the present paper assume existence of $\phi^*$ whereas no positive eigenfunction of the adjoint of ${\mathbb{M}}$ is guaranteed under the conditions in HS.[^32] This suggests the Harris recurrence condition is of a very different nature from Assumption \[a:id:1\].
BHS assume that $X$ is ergodic under the ${\widetilde{{\mathbb{P}}}}$ probability measure, for which Conditions \[c:HS\](b)–(d) are sufficient. Also notice that Condition \[c:HS\](a) is satisfied by construction in BHS.
The identification results in HS and the proof of proposition 3.3 in BHS shows that uniqueness is established in the space of functions $\psi$ for which ${\widetilde{{\mathbb{E}}}}[\psi(X_t)/\phi(X_t)]$ is finite, where ${\widetilde{{\mathbb{E}}}}$ denotes expectation under the stationary distribution corresponding to (\[e:twist\]). Under Assumption \[a:id:1\], their result establishes identification in the space of functions $\psi$ for which $${\widetilde{{\mathbb{E}}}}[\psi(X_t)/\phi(X_t)] = {\mathbb{E}}[ \psi(X_t) \phi^*(X_t)]$$ is finite. The right-hand side is finite for all $\psi \in L^2$ (by Cauchy-Schwarz). So in this sense the identification result in HS and BHS applies to a larger class of functions than our result.
Existence
---------
We obtain the following existence result by replacing Assumption \[a:id:1\](b)(c) by the slightly stronger quasi-compactness and positivity conditions in Assumption \[a:id:0\]. The following result is essentially Theorems 6 and 7 of [@Sasser].[^33] Say that ${\mathbb{M}}$ is *quasi-compact* if ${\mathbb{M}}$ is bounded and there exists $\tau \in T$ and a bounded linear operator ${\mathbb{V}}$ such that ${\mathbb{M}}_\tau - {\mathbb{V}}$ is compact and $r({\mathbb{V}}) < r({\mathbb{M}})^\tau$. Quasi-compactness of ${\mathbb{M}}$ is implied by Assumption \[a:id:0\].
\[p:a:exist\] Let Assumption \[a:id:0\](a) hold and let ${\mathbb{M}}$ be quasi-compact. Then:
1. There exists positive functions $\phi,\phi^* \in L^2$ and a positive scalar $\rho$ such that $(\rho,\phi)$ solves (\[e:pev\]) and $(\rho,\phi^*)$ solves (\[e:pev:star\]).
2. The functions $\phi$ and $\phi^*$ are the unique solutions (in $L^2$) to (\[e:pev\]) and (\[e:pev:star\]), respectively.
3. The eigenvalue $\rho$ is simple and isolated and it is the largest eigenvalue of ${\mathbb{M}}$.
A similar existence result to part (a) was presented in a 2005 preliminary version of HS. For that result, HS assumed that $r({\mathbb{M}})$ was positive and that the (continuous-time) semigroup of operators had an element which was compact. The further properties of $\rho$ that we establish in part (c) of Proposition \[p:a:exist\] are essential to our derivation of the large-sample theory. A similar proposition was derived under different conditions in [@Christensen-idpev].
HS establish existence of $\phi$ in possibly non-stationary, continuous-time environments by appealing to the theory of ergodic Markov processes. Equivalent conditions for discrete-time environments are now presented and compared with our identification conditions. As with the identification conditions, we use analogues of generators and resolvents for discrete-time semigroups where appropriate.
\[c:HS:ex\]
1. There exists a function $V : \mathcal X \to {\mathbb{R}}$ with $V \geq 1$ and a finite constant $\underline a> 0 $ such that ${\mathbb{M}} V(x) \leq \underline a V(x)$ for all $x \in \mathcal X$
2. There exists a measure $\nu$ on $(\mathcal X, \mathscr X)$ such that ${\mathbb{J}}{1\!\mathrm{l}}\{ \cdot \in \Lambda\}(x) > 0$ for any $\Lambda \in \mathscr X$ with $\nu(\Lambda) > 0$, where ${\mathbb{J}}$ is given by $${\mathbb{J}} \psi(x) = \sum_{t=0}^\infty a^{-(t+1)} \frac{{\mathbb{M}}_t (V \psi)(x)}{V(x)}$$ for $a > \underline a$
3. The operators ${\mathbb{J}}$ and ${\mathbb{K}}$ are bounded, where ${\mathbb{K}}$ is given by $${\mathbb{K}} \psi(x) = \sum_{t=0}^\infty \lambda^{-t}(({\mathbb{J}} - s \otimes \nu)^t \psi)(x)$$ where $s: \mathcal X \to {\mathbb{R}}_+$ is such that $\int s \, {\mathrm{d}} \nu > 0$ and ${\mathbb{J}} \psi(x) \geq s(x) \int \psi\,{\mathrm{d}} \nu$ for all $\psi \geq 0$ ($s$ exists by part (b)), $(s \otimes \nu) \psi(x) := s(x) \int \psi \,\mathrm d \nu$, and $\lambda\in \sigma({\mathbb{J}})$.
HS show that ${\mathbb{K}} s$ is a positive eigenfunction of ${\mathbb{M}}$ under the preceding conditions (see their Lemma D.3). Condition \[c:HS:ex\](b) is satisfied under Assumption \[a:id:0\] with $\nu = Q$ whenever $a > r({\mathbb{M}})$. To see this, take $\Lambda \in \mathscr X$ with $Q(\Lambda) > 0$ and observe that: $$\sum_{t=1}^\infty a^{-t} {\mathbb{M}}_t (V(\cdot) {1\!\mathrm{l}}\{ \cdot \in \Lambda\})(x) \geq \sum_{t=1}^\infty a^{-t} {\mathbb{M}}_t {1\!\mathrm{l}}\{ \cdot \in \Lambda\} > 0$$ (almost everywhere) where the first inequality is by positivity and the second is by irreducibility. It follows that ${\mathbb{J}} {1\!\mathrm{l}}\{ \cdot \in \Lambda\}(x) > 0$ (almost everywhere). This verifies part (b), up to the “almost everywhere” qualification.
On the other hand, Conditions \[c:HS:ex\](a)(c) seem quite different from the conditions of Proposition \[p:a:exist\]. For instance, Assumption \[a:id:0\] does not presume existence of the function $V$ but imposes a quasi-compactness condition. HS do not restrict the function space for ${\mathbb{M}}$ ex ante so there is no notion of a bounded or power-compact operator on the space to which $\phi$ belongs. The requirement that ${\mathbb{K}}$ be bounded (or the sufficient conditions for this provided in HS) do not seem to translate clearly in terms of the operator ${\mathbb{M}}$.
Long-run pricing
----------------
We now present a version of the long-run pricing approximation of HS that holds under our existence and identification conditions. We impose the normalizations ${\mathbb{E}}[\phi(X_t)^2] = 1$ and ${\mathbb{E}}[\phi(X_t)\phi^*(X_t)] = 1$ and define the operator $(\phi \otimes \phi^*) : L^2 \to L^2$ by: $$(\phi \otimes \phi^*) \psi(x) = \phi(x) \int \phi^* \psi\,{\mathrm{d}} Q\,.$$
\[p:lr\] Let Assumption \[a:id:0\] hold. Then: there exists $c > 0$ such that: $$\|\rho^{-\tau}{\mathbb{M}}_\tau - (\phi \otimes \phi^*)\| = O( e^{-c\tau})$$ as $\tau \to \infty$.
Proposition \[p:lr\] is similar to Proposition 7.4 in HS. Proposition \[p:lr\] establishes convergence of $\rho^{-\tau} {\mathbb{M}}_\tau$ to $(\phi \otimes \phi^*)$, with the approximation error vanishing exponentially in the payoff horizon $n$. A similar proposition (without the rate of convergence) was reported in a 2005 draft of HS. There, HS assumed directly that the distorted conditional expectations converged to an unconditional expectation characterized by $\phi$, $\phi^*$, and an arbitrary measure. Proposition \[p:lr\] shows that in stationary environments the unconditional expectation is characterized by $\phi$, $\phi^*$ and $Q$.
More generally, Section 7 of HS establishes long-run approximations of the form: $$\label{e-meas-HS}
\rho^{-\tau} {\mathbb{M}}_\tau \psi(x) \to {\widetilde{{\mathbb{E}}}}[ \psi(X_t)/\phi(X_t)] \phi(x)$$ as $\tau \to \infty$, where ${\widetilde{{\mathbb{E}}}}$ is defined in terms of some probability measure, say ${\widetilde{Q}}$. By Proposition \[p:lr\], we obtain: $$\label{e-meas-stat}
\rho^{-\tau} {\mathbb{M}}_\tau \psi(x) \to {\mathbb{E}}[ \psi(X_t) \phi^*(X_t)] \phi(x) \,.$$ Comparing (\[e-meas-HS\]) and (\[e-meas-stat\]), we see that the probability measure ${\widetilde{Q}}$ is given by: $$\frac{{\mathrm{d}} {\widetilde{Q}}(x)}{{\mathrm{d}} Q(x)} = \phi(x) \phi^*(x)\,.$$ Therefore, in environments for which our identification conditions hold, the expectation ${\widetilde{{\mathbb{E}}}}[ \psi(X_t)/\phi(X_t)]$ is characterized by the stationary distribution of $X$ and the positive eigenfunctions of ${\mathbb{M}}$ and ${\mathbb{M}}^*$.
Estimation with latent states {#ax:filter}
=============================
In this appendix we describe two approaches that may be used to extend the methodology presented in this paper to models with latent state variables. Theorem \[t:rate\] will continue to apply provided that one can construct estimators of ${\mathbf{G}}$ and ${\mathbf{M}}$ that satisfy Assumption \[a:var\]. Formal verification of Assumption \[a:var\] for the following approaches requires a nontrivial extension of the statistical theory in Appendix \[ax:est:mat\], which we defer to future research.
A first approach to dealing with models with a latent volatility state variable is via *high-frequency proxies.* Suppose that the state vector may be partitioned as $X_t = (Y_t, V_t)$ where $Y_t$ is observable and $V_t$ is spot volatility or integrated volatility. Here we could use an estimator of $V_t$, say $\hat V_t$, based on high-frequency data.[^34] The estimators ${\widehat{{\mathbf{G}}}}$ and ${\widehat{{\mathbf{M}}}}$ may be formed as described in Section \[s:est\] but using $\hat X_t = (Y_t,\hat V_t)$ in place of $X_t$.
A second approach is via *filtering and smoothing.* Fully nonparametric models with latent variables are not, in general, well identified. Nevertheless, with sufficient structure (e.g. by specifying a semiparametric state-space representation for the unobservable components in terms of an auxiliary vector of observables) it may be feasible to use a filter and smoother to construct an estimate of the distribution of the latent state variables given the time series of observables. Suppose that the state vector $X_t$ may be partitioned as $X_t = (Y_t,\zeta_t)$ where $Y_t$ is observable and $\zeta_t$ is latent. To simplify presentation, we assume that the joint density of $(Y_{t+1},\zeta_{t+1})$ given $(Y_t,\zeta_t)$ factorizes as: $$\label{e:latent:y}
f(Y_{t+1},\zeta_{t+1}|Y_t,\zeta_t) = f(Y_{t+1}|Y_t) f(\zeta_{t+1}|\zeta_t)$$ where we assume that $f(Y_{t+1}|Y_t)$ is unknown and $f(\zeta_{t+1}|\zeta_t)$ is known up to a parameter $\theta_1$. We further assume that the econometrician observes a time series of data on an auxiliary vector $Z_t$ and that the joint density for $(Z_{t+1},\zeta_{t+1})$ given $(Z_{t},\zeta_{t})$ factorizes as: $$\label{e:latent:z}
f(Z_{t+1},\zeta_{t+1}|Z_t,\zeta_t) = f(Z_{t+1}|Z_t,\zeta_t) f(\zeta_{t+1}|\zeta_t)$$ with $f(Z_{t+1}|Z_t,\zeta_t)$ known up to some parameter $\theta_2$. Note that (\[e:latent:z\]) describes a state-space representation in which $f(\zeta_{t+1}|\zeta_t)$ is the state equation and $f(Z_{t+1}|Z_t,\zeta_t)$ is the observation equation. Finally, the sequence $(Y_t,\zeta_t,Z_t)$ is assumed to be (jointly) strictly stationary and ergodic. The econometrician observes the vector $X_t^{obs} = (Y_t,Z_t)$ at dates $t=0,1,\ldots,n$.
To estimate ${\mathbf{G}}$ we could proceed as follows:
1. Calculate the maximum likelihood estimate $\hat \theta$ of $\theta = (\theta_1,\theta_2)$ using the time series $Z_0,Z_1,\ldots,Z_n$.
2. 3. Partition $Z_0,Z_1,\ldots,Z_n$ into $(N+1)$ blocks of length $q := \frac{n+1}{N+1}$. That is, for each $j = 0 ,\ldots,N$, let $\vec Z_{[j]} = (Z_{jq},\ldots,Z_{(j+1)q-1})$. Fix a prior distribution $\pi$ for each $\zeta_{jq}$, $j = 0,\ldots,N$, such as the unconditional distribution for $\zeta_{jq}$ implied by $f(\zeta_{t+1}|\zeta_t;\hat \theta)$. For each $j = 0,\ldots,N$ and for each $t = jq,\ldots,(j+1)q-1$ we compute the posterior density $f(\zeta_t | \vec Z_{[j]};\hat \theta,\pi)$ of $\zeta_{t}$ given $\vec Z_{[j]}$, $\hat \theta$ and $\pi$.
4. 5. Let $b^{k_1}_1(\cdot)$ be a $k_1$-vector of basis functions for $Y$ and let $b^{k_2}_2(\cdot)$ be a $k_2$-vector of basis functions for $\zeta$. We form basis functions for $x = (y,\zeta)$ by taking the tensor product $b^k((y,\zeta)) = b_1^{k_1}(y) \otimes b_2^{k_2}(\zeta)$ with $k = k_1 \times k_2$. The estimator of ${\mathbf{G}}$ is: $$\begin{aligned}
{\widehat{{\mathbf{G}}}} & = \frac{1}{N+1} \sum_{j=0}^N {\widehat{{\mathbf{G}}}}_{[j]} \\
{\widehat{{\mathbf{G}}}}_{[j]} & = \frac{1}{q} \sum_{t=jq}^{(j+1)q-1} \left( \big( b_1^{k_1}(Y_t)b_1^{k_1}(Y_t)' \big) \otimes {{\mathbb{E}}}_{\hat \theta,\pi} \big[ b_2^{k_2}(\zeta_t)b_2^{k_2}(\zeta_t)' \big|\vec Z_{[j]} \big] \right) \end{aligned}$$
where ${{\mathbb{E}}}_{\hat \theta,\pi} \big[ \cdot \big|\vec Z_{[j]} \big]$ denotes expectation under the posterior $f(\zeta_{t} | \vec Z_{[j]}; \hat \theta,\pi)$.
To gain some intuition for this estimator, suppose that the true $\theta_0$ is known. Then: $$\frac{1}{q} \sum_{t=jq}^{(j+1)q-1} {{\mathbb{E}}}_{\theta_0,\pi} \big[ b_2^{k_2}(\zeta_t)b_2^{k_2}(\zeta_t)' \big|\vec Z_{[j]} \big] \approx
\frac{1}{q} \sum_{t=jq}^{(j+1)q-1} {{\mathbb{E}}} \big[ b_2^{k_2}(\zeta_t)b_2^{k_2}(\zeta_t)' \big|\vec Z_{[j]} \big]$$ where “$\approx$” is because, as the block length $q$ increases, the posterior distribution will typically become independent of the prior $\pi$ and the posterior expectation should approach the true conditional expectation. Then for each $t = jN,\ldots,(j+1)N-1$ we have: $$\begin{aligned}
\frac{1}{q} \sum_{t=jq}^{(j+1)q-1} \big( b_1^{k_1}(Y_t)b_1^{k_1}(Y_t)' \big) \otimes {\mathbb{E}} \big[ b_2^{k_2}(\zeta_t)b_2^{k_2}(\zeta_t)' \big|\vec Z_{[j]} \big]
& = \frac{1}{q} \sum_{t=jq}^{(j+1)q-1} {\mathbb{E}}\big[ b^k(X_t) b^k(X_t)' \big| \vec X^{obs}_{[j]} \big]\end{aligned}$$ where $\vec X^{obs}_{[j]} = (\vec X_{jq}^{obs},\ldots,\vec X_{(j+1)q-1}^{obs})$. But notice that the term on the right-hand side is a $k \times k$ random matrix which is a function of $\vec X^{obs}_{[j]}$ and whose expected value is ${\mathbf{G}}$ (by iterated expectations). Therefore, the estimator ${\widehat{{\mathbf{G}}}}$ is approximately the average of $N + 1$ such random matrices. As the sequence $\vec X_{[0]}^{obs},\ldots,\vec X_{[N]}^{obs}$ is strictly stationary and ergodic, its average ought to approach ${\mathbf{G}}$ as $N$ increases.[^35] When $\theta$ is unknown but the MLE $\hat \theta$ is consistent, the same intuition goes through except it would also have to be established that: $${{\mathbb{E}}}_{\hat \theta,\pi} \big[ b_2^{k_2}(\zeta_t)b_2^{k_2}(\zeta_t)' \big|\vec Z_{[j]} \big] - {{\mathbb{E}}}_{\theta_0,\pi} \big[ b_2^{k_2}(\zeta_t)b_2^{k_2}(\zeta_t)' \big|\vec Z_{[j]} \big]$$ becomes negligible (in an appropriate sense) as $n,q,k \to \infty$.
To estimate ${\mathbf{M}}$ we proceed similarly:
1. With $\hat \theta$ and $\pi$ as before, for each $j = 0,\ldots,N$ and each $t = jq,\ldots,(j+1)q-2$ we compute the posterior density $f(\zeta_t,\zeta_{t+1} | \vec Z_{[j]};\hat \theta,\pi)$ of $(\zeta_{t},\zeta_{t+1})$ given $\vec Z_{[j]}$, $\hat \theta$ and $\pi$.
2. 3. When the function $m$ is known, the estimator of ${\mathbf{M}}$ is: $$\begin{aligned}
{\widehat{{\mathbf{M}}}} & = \frac{1}{N+1} \sum_{j=0}^N {\widehat{{\mathbf{M}}}}_{[j]} \\
{\widehat{{\mathbf{M}}}}_{[j]} & = \frac{1}{q-1} \sum_{t=jq}^{(j+1)q-2} \bigg( \big( b_1^{k_1}(Y_t)b_1^{k_1}(Y_{t+1})' \big) \\
& \quad \quad \quad \quad \quad \otimes \bigg( \int b_2^{k_2}(\zeta_t)b_2^{k_2}(\zeta_{t+1})'m(Y_t,\zeta_t,Y_{t+1},\zeta_{t+1} ) f(\zeta_t;\zeta_{t+1}| \vec Z_{[j]},\hat \theta,\pi) {\mathrm{d}} \zeta_{t+1} {\mathrm{d}} \zeta_t \bigg) \bigg) \end{aligned}$$ When $m$ contains an estimated component $\hat \alpha$ we replace $m(Y_t,\zeta_t,Y_{t+1},\zeta_{t+1} )$ in the above display by $m(Y_t,\zeta_t,Y_{t+1},\zeta_{t+1};\hat \alpha)$.
Proofs {#ax:proofs}
======
For any vector $v \in {\mathbb{R}}^k$, define: $$\|v\|_{{\mathbf{G}}}^2 = v' {\mathbf{G}}_k v$$ or equivalently $\|v\|_{{\mathbf{G}}} = \|{\mathbf{G}}_k^{1/2} v\|$. For any matrix ${\mathbf{A}} \in {\mathbb{R}}^{k \times k}$ we define: $$\|{\mathbf{A}}\|_{{\mathbf{G}}} = \sup\{ \|{\mathbf{A}} v\|_{{\mathbf{G}}} : v \in {\mathbb{R}}^k, \|v\|_{{\mathbf{G}}} = 1\}\,.$$ We also define the inner product weighted by ${\mathbf{G}}_k$, namely $\langle u,v \rangle_{{\mathbf{G}}} = u' {\mathbf{G}}_k v$. The inner product $\langle \cdot,\cdot \rangle_{{\mathbf{G}}}$ and its norm $\|\cdot\|_{{\mathbf{G}}}$ are germane for studying convergence of the matrix estimators, as $({\mathbb{R}}^k,\langle \cdot,\cdot\rangle_{{\mathbf{G}}})$ is isometrically isomorphic to $(B_k,\langle \cdot,\cdot \rangle)$.
Proofs of main results
----------------------
Immediate from Propositions \[p:a:id\], \[p:a:exist\], and \[p:lr\].
Immediate from Lemmas \[lem:bias\] and \[lem:var\].
First note that: $$\begin{aligned}
\sqrt n ( \hat \rho - \rho) & = \sqrt n ( \hat \rho - \rho_k) + \sqrt n ( \rho_k - \rho) \notag \\
& = \sqrt n ( \hat \rho - \rho_k) + o(1) \notag \\
& = \sqrt n c_k^{*\prime} ( {\widehat{{\mathbf{M}}}} - \rho_k {\widehat{{\mathbf{G}}}}) c_k + o_p(1) \label{a:asydist:pf:1}\end{aligned}$$ where the second line is by Assumption \[a:asydist\](a) and the third line is by Lemma \[lem:expansion\] and Assumption \[a:asydist\](b) (under the normalizations $\|{\mathbf{G}} c_k\| = 1$ and $c_k^{* \prime } {\mathbf{G}} c_k^{\phantom *} = 1$). By identity, we may write the first term on the right-hand side of display (\[a:asydist:pf:1\]) as: $$\begin{aligned}
\sqrt n c_k^{*\prime} ( {\widehat{{\mathbf{M}}}} - \rho_k {\widehat{{\mathbf{G}}}}) c_k
& = \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \psi_{\rho,k}(X_t,X_{t+1}) + o_p(1) \notag \\
& = \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \psi_{\rho}(X_t,X_{t+1}) + o_p(1) \label{a:asydist:pf:2}\end{aligned}$$ where the second line is by Assumption \[a:asydist\](c).
The result follows by substituting (\[a:asydist:pf:2\]) into (\[a:asydist:pf:1\]) and applying a CLT for stationary and ergodic martingale differences (e.g. [@Billingsley1961]).
We verify the conditions of Theorem \[t:asydist:1\]. Since the data are exponentially beta-mixing, $\xi_k = O(k^\lambda)$, and $m$ has finite $s$th moment, Lemma \[lem:beta:1\] implies that $\|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| = O_p( k^\lambda (\log n)/\sqrt{n}) $ and $\|{\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o \| = O_p ( k^{\lambda(s+2)/s} (\log n) /\sqrt{n} ) $. Both terms are $o_p(n^{-1/4})$ under condition (vi), verifying Assumptions \[a:var\] and \[a:asydist\](b). Part (a) of Assumption \[a:asydist\] is satisfied by conditions (iii) and (vi) and part (d) is satisfied by condition (iv). Finally, for part (c), by stationarity and the triangle inequality we have: $$\begin{aligned}
{\mathbb{E}}[ |\Delta_{\psi,n,k}| ] & \leq \sqrt n {\mathbb{E}}[ | \psi_{\rho,k}(X_t,X_{t+1}) - \psi_\rho(X_t,X_{t+1}) |] \notag \\
& \leq \sqrt n \Big( | \rho_k - \rho| \times {\mathbb{E}}[ \phi ^* (X_t) \phi(X_t)] + \rho_k \times {\mathbb{E}}\big[ | \phi^*(X_t)\phi(X_t) - \phi^*_k(X_t) \phi_k^{\phantom *}(X_t)|\big] \Big) \notag \\
& \quad + \sqrt n \Big( {\mathbb{E}}\big[ m(X_t,X_{t+1}) | \phi^*(X_t)\phi(X_{t+1}) - \phi^*_k(X_t) \phi_k^{\phantom *}(X_{t+1})|\big] \Big)\,. \label{e:corrnorm:0}\end{aligned}$$ For the leading term, Lemma \[lem:bias\](a) and the conditions on $\delta_k$ and $k$ yield: $$\begin{aligned}
\sqrt n \times | \rho_k - \rho| \times {\mathbb{E}}[ \phi ^* (X_t) \phi(X_t)] & = \sqrt n \times O(\delta_k) \times 1 \notag \\
& = \sqrt n \times k^{-\omega} \times 1 = o(1) \label{e:corrnorm:1} \,.\end{aligned}$$ For the second term, the Cauchy-Schwarz inequality, Lemma \[lem:bias\](b)(c) and the conditions on $\delta_k^{\phantom *}$, $\delta_k^*$ and $k$ yield: $$\begin{aligned}
\sqrt n \times \rho_k \times {\mathbb{E}} \big[ | \phi^*(X_t)\phi(X_t) - \phi^*_k(X_t) \phi_k^{\phantom *}(X_t)| \big] & \leq \sqrt n \times \rho_k \times \Big( \| \phi^* - \phi_k^*\| \|\phi\| + \|\phi_k^*\| \| \phi - \phi_k\| \Big) \notag \\
& = \sqrt n \times O(1) \times O( \delta_k^* + \delta_k^{\phantom *} ) \notag \\
& = \sqrt n \times O(1) \times O( k^{-\omega } ) = o(1) \label{e:corrnorm:2}\,.\end{aligned}$$ We split the final term in (\[e:corrnorm:0\]) in two, to obtain: $$\begin{aligned}
& \sqrt n \Big( {\mathbb{E}}[ m(X_t,X_{t+1}) | \phi^*(X_t)\phi(X_{t+1}) - \phi^*_k(X_t) \phi_k^{\phantom *}(X_{t+1})|] \Big) \\
& \quad \quad \leq \sqrt n \Big( {\mathbb{E}}[ m(X_t,X_{t+1}) | \phi(X_{t+1}) - \phi_k(X_{t+1})| \phi^*(X_{t+1}) ] \\
& \quad \quad \quad + {\mathbb{E}}[ m(X_t,X_{t+1}) | \phi^*(X_t) - \phi_k^*(X_t)| |\phi_k(X_{t+1})| ]\Big) \,.\end{aligned}$$ Using the Cauchy-Schwarz inequality and Lemma \[lem:bias\](b)(c) as above, we may deduce that these terms are $O( \sqrt n \times \delta_k)$ and $O(\sqrt n \times \delta_k^*)$ provided that ${\mathbb{E}}[m(X_t,X_{t+1})^2 \phi^*(X_{t+1})^2 ] < \infty$ (which holds under the moment conditions on $\phi^*$ and $m$) and ${\mathbb{E}}[m(X_t,X_{t+1})^2 \phi_k(X_{t+1})^2 ] < \infty$. This latter condition holds under the moment conditions on $\phi$ and $m$ because: $$\begin{aligned}
& {\mathbb{E}}[m(X_t,X_{t+1})^2 \phi_k(X_{t+1})^2 ] \\
& \leq 2{\mathbb{E}}[m(X_t,X_{t+1})^2 \phi(X_{t+1})^2 ] + 2{\mathbb{E}}[m(X_t,X_{t+1})^2 (\phi(X_{t+1}) - \phi_k(X_{t+1}))^2 ] \\
& \leq 2{\mathbb{E}}[m(X_t,X_{t+1})^2 \phi(X_{t+1})^2 ] + 2{\mathbb{E}}[m(X_t,X_{t+1})^s]^{2/s} \times \|\phi - \phi_k\|_{\frac{2s}{s-2}} ^2\end{aligned}$$ and $\|\phi - \phi_k\|_{\frac{2s}{s-2}} = O(1)$. Therefore: $$\sqrt n \Big( {\mathbb{E}}\big[ m(X_t,X_{t+1}) | \phi^*(X_t)\phi(X_{t+1}) - \phi^*_k(X_t) \phi_k^{\phantom *}(X_{t+1})|\big] \Big) = O( \sqrt n \times k^{-\omega} ) = o(1) \label{e:corrnorm:3}$$ under the conditions on $k$. Substituting (\[e:corrnorm:1\]), (\[e:corrnorm:2\]), and (\[e:corrnorm:3\]), into (\[e:corrnorm:0\]) and using Markov’s inequality yields $\Delta_{\psi,n,k} = o_p(n^{-1/2})$, as required.
This is a consequence of Theorem \[t:eff:main\] in Appendix \[ax:inf\].
As in the proof of Theorem \[t:asydist:1\], we have: $$\begin{aligned}
\sqrt n ( \hat \rho - \rho)
& = \frac{1}{\sqrt n } \sum_{t=0}^{n-1} \Big( \phi_k^*(X_t)\phi_k^{\phantom *}(X_{t+1}) m(X_t,X_{t+1},\hat \alpha) - \rho_k \phi_k^*(X_t)\phi_k^{\phantom *}(X_t) \Big) + o_p(1)\,.\end{aligned}$$ By adding and subtracting terms and using Assumption \[a:asydist\](c), we obtain: $$\begin{aligned}
\sqrt n (\hat \rho - \rho) & = \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \psi_\rho(X_t,X_{t+1}) \notag \\
& \quad + \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \phi_k^*(X_t)\phi_k^{\phantom *}(X_{t+1})\Big( m(X_t,X_{t+1},\hat \alpha) - m(X_t,X_{t+1}, \alpha_0) \Big) + o_p(1) \label{e:2a:pf1} \,.\end{aligned}$$ Let $\phi_{k,t} = \phi_k(X_t)$, $\phi_t = \phi(X_t)$, $\phi^*_{k,t} = \phi^*_k(X_t)$, $\phi^*_t = \phi^*(X_t)$, and $m_t(\alpha) = m(X_t,X_{t+1};\alpha)$. We decompose the second term on the right-hand side of (\[e:2a:pf1\]) as: $$\begin{aligned}
\frac{1}{\sqrt n} \sum_{t=0}^{n-1} \phi_{k,t}^*\phi_{k,t+1}^{\phantom *} \big( m_t(\hat \alpha) - m_t( \alpha_0) \big) \notag
& = \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \phi^*_t \phi_{t+1}^{\phantom *}\frac{\partial m_t(\alpha_0)}{\partial \alpha'} (\hat \alpha - \alpha_0) \\
& \quad + \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \phi_t^*\phi_{t+1}^{\phantom *}\Big( m_t(\hat \alpha) - m_t(\alpha_0) - \frac{\partial m_t(\alpha_0)}{\partial \alpha'} (\hat \alpha - \alpha_0) \Big) \notag \\
& \quad + \frac{1}{\sqrt n} \sum_{t=0}^{n-1} ( \phi_{k,t}^*\phi_{k,t+1}^{\phantom *} - \phi_t^*\phi_{t+1}^{\phantom *} ) ( m_t(\hat \alpha) - m_t(\alpha_0) ) \notag \\
& =: \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \phi^*_t \phi_{t+1}^{\phantom *}\frac{\partial m_t(\alpha_0)}{\partial \alpha'} (\hat \alpha - \alpha_0) + T_{1,n} + T_{2,n}\label{e:2a:pf2} \,.\end{aligned}$$ For term $T_{1,n}$, we know that $\hat \alpha \in N$ wpa1. Whenever $\hat \alpha \in N$ we may take a mean value expansion to obtain: $$T_{1,n} = \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \phi_t^*\phi_{t+1}^{\phantom *}\Big( \frac{\partial m_t(\tilde \alpha)}{\partial \alpha'} - \frac{\partial m_t(\alpha_0)}{\partial \alpha'}\Big) (\hat \alpha - \alpha_0)$$ wpa1, where $\tilde \alpha$ is in the segment between $\hat \alpha$ and $\alpha_0$. Since $\hat \alpha - \alpha_0 = O_p(n^{-1/2})$, it suffices to show that: $$\label{e:2a:pf2:1}
\frac{1}{ n} \sum_{t=0}^{n-1} \phi_t^*\phi_{t+1}^{\phantom *}\Big( \frac{\partial m_t(\tilde \alpha)}{\partial \alpha} - \frac{\partial m_t(\alpha_0)}{\partial \alpha}\Big) = o_p(1)\,.$$ Let $N_1 \subseteq N$ be a compact neighborhood of $\alpha_0$. Assumption \[a:parametric\](c)(d) implies that: $$\label{e:moments}
{\mathbb{E}}\left[ \sup_{\alpha \in N} \Big\|\phi^*_t \phi^{\phantom *}_{t+1} \frac{\partial m_t(\alpha)}{\partial \alpha} \Big\| \right] < \infty$$ and so by dominated convergence we may deduce that the ${\mathbb{R}}^{d_\alpha}$-valued function: $$\alpha \mapsto {\mathbb{E}}\left[ \phi^*_t \phi^{\phantom *}_{t+1} \Big( \frac{\partial m_t(\alpha)}{\partial \alpha} - \frac{\partial m_t(\alpha_0)}{\partial \alpha} \Big) \right]$$ is continuous at $\alpha_0$. It is also straightforward to show that: $$\sup_{\alpha \in N_1} \left| \frac{1}{ n} \sum_{t=0}^{n-1} \phi_t^*\phi_{t+1}^{\phantom *}\Big( \frac{\partial m_t( \alpha)}{\partial \alpha} - \frac{\partial m_t(\alpha_0)}{\partial \alpha}\Big) - {\mathbb{E}}\left[ \phi^*_t \phi^{\phantom *}_{t+1} \Big( \frac{\partial m_t(\alpha)}{\partial \alpha} - \frac{\partial m_t(\alpha_0)}{\partial \alpha} \Big) \right] \right| = o_p(1)$$ by (\[e:moments\]), the ergodic theorem, and compactness of $N_1$. Therefore, $T_{1,n} = o_p(1)$.
For $T_{2,n}$, using the fact that $\Pr(A) \leq \Pr(A \cap B) + \Pr(B^c)$ and root-$n$ consistency of $\hat \alpha$, it follows that for any $\varepsilon >0$ we have: $$\begin{aligned}
& \Pr \left( \max_{0 \leq t \leq n-1} \left|m_t(\hat \alpha) - m_t(\alpha_0) \right| > \varepsilon \right) \\
& \leq \Pr \left( \max_{0 \leq t \leq n-1} \left\{ \left| \frac{\partial m_t(\tilde \alpha)}{\partial \alpha'} (\hat \alpha - \alpha_0) \right| > \varepsilon \right\} \cap \left\{ \hat \alpha \in N, \| \hat \alpha - \alpha_0 \| \leq \sqrt{(\log n)/n} \right\} \right) + o(1) \\
& \leq \Pr \bigg( \bigcup_{t=0}^{n-1} \left\{ \sup_{\alpha \in N} \left\| \frac{\partial m_t( \alpha)}{\partial \alpha'} \right\| \sqrt{(\log n)/n} > \varepsilon \right\} \bigg) + o(1) \\
& \leq \sum_{t=0}^{n-1} \Pr \bigg( \sup_{\alpha \in N} \left\| \frac{\partial m_t( \alpha)}{\partial \alpha'} \right\| \sqrt{(\log n)/n} > \varepsilon \bigg) + o(1) \\
& \leq \frac{n}{\varepsilon^s} \left( \log n/n \right)^{s/2} {\mathbb{E}}\left[ \sup_{\alpha \in N} \Big\| \frac{\partial m_t( \alpha)}{\partial \alpha'} \Big\|^s {1\!\mathrm{l}}\left\{ \sup_{\alpha \in N} \left\| \frac{\partial m_t( \alpha)}{\partial \alpha'} \right\| \sqrt{(\log n)/n} > \varepsilon \right \} \right]\end{aligned}$$ where the second-last line is by the union bound and Cauchy-Schwarz, and the final line is by Markov’s inequality and Assumption \[a:parametric\](c) and where the $o(1)$ term is independent of $\varepsilon$. Therefore $\max_{0 \leq t \leq n-1} \left|m_t(\hat \alpha) - m_t(\alpha_0) \right| = o_p( n^{-(s-2)/(2s)} \log n)$ and hence: $$\begin{aligned}
T_{2,n} & = o_p( n^{-(s-2)/(2s)} \log n) \times n^{1/2} \times \frac{1}{ n} \sum_{t=0}^{n-1} | \phi_{k,t}^*\phi_{k,t+1}^{\phantom *} - \phi_t^*\phi_{t+1}^{\phantom *} | \\
& \leq o_p( n^{1/s} \log n) \times O_p(1) \times \left( \|\phi_k^* - \phi^*\| + \|\phi_k - \phi\| \right) \\
& = o_p\big( n^{1/s} \log n \times(\delta_k^{\phantom *} + \delta_k^*)\big) \,.\end{aligned}$$ where the second line is by Markov’s inequality the Cauchy-Schwarz inequality and the third is by Lemma \[lem:bias\](b)(c). It follows by Assumptions \[a:asydist\](a) and \[a:parametric\](d) that $T_{2,n} = o_p(1)$.
Since $T_{1,n} $ and $T_{2,n}$ in display (\[e:2a:pf2\]) are both $o_p(1)$, we have: $$\begin{aligned}
\frac{1}{\sqrt n} \sum_{t=0}^{n-1} \phi_{k,t}^*\phi_{k,t+1}^{\phantom *} \big( m_t(\hat \alpha) - m_t( \alpha_0) \big) & = \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \phi^*_t \phi_{t+1}^{\phantom *}\frac{\partial m_t(\alpha_0)}{\partial \alpha'} (\hat \alpha - \alpha_0) + o_p(1) \\
& = \bigg( \frac{1}{ n} \sum_{t=0}^{n-1} \phi^*_t \phi_{t+1}^{\phantom *}\frac{\partial m_t(\alpha_0)}{\partial \alpha'} \bigg) \sqrt n (\hat \alpha - \alpha_0) + o_p(1) \\
& = {\mathbb{E}} \left[ \phi^*(X_t) \phi(X_{t+1})\frac{\partial m(X_t,X_{t+1};\alpha_0)}{\partial \alpha'} \right] \sqrt n (\hat \alpha - \alpha_0) + o_p(1)\end{aligned}$$ where the final line is by the ergodic theorem.
Substituting into (\[e:2a:pf1\]) and using Assumption \[a:parametric\](a): $$\sqrt n (\hat \rho - \rho) = \frac{1}{\sqrt n } \sum_{t=0}^{n-1} h_{[{\mathrm{2a}}]}' \bigg( \begin{array}{c} \psi_{\rho,t} \\ \psi_{\alpha,t} \end{array} \bigg) + o_p(1)$$ and the result follows by Assumption \[a:parametric\](b).
By arguments to the proof of Theorem \[t:asydist:1\] and \[t:asydist:2a\] and using Assumption \[a:nonpara\](a)(b), we may deduce: $$\begin{aligned}
\sqrt n (\hat \rho - \rho) & = \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \psi_{\rho,t} + \phi_{k,t}^*\phi_{k,t+1}^{\phantom *} (m_t(\hat \alpha) - m_t(\alpha_0)) + o_p(1) \\
& = \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \psi_{\rho,t} + \sqrt n \dot \ell_{\alpha_0}[\hat \alpha - \alpha_0] + T_{1,n} + T_{2,n} + o_p(1)\end{aligned}$$ where: $$\begin{aligned}
T_{1,n} & = \frac{1}{\sqrt n} \sum_{t=0}^{n-1} (\phi_{k,t}^*\phi_{k,t+1}^{\phantom *} - \phi_t^* \phi_{t+1}^{\phantom *}) (m_t(\hat \alpha) - m_t(\alpha_0)) \\
T_{2,n} & = \frac{1}{\sqrt n} \sum_{t=0}^{n-1} \Big( \phi_t^* \phi_{t+1}^{\phantom *} (m_t(\hat \alpha) - m_t(\alpha_0)) - (\ell(\hat \alpha) - \ell(\alpha_0)) \Big) \,.\end{aligned}$$ The result will follow by Assumption \[a:nonpara\](b)(c) provided we can show that the terms $T_{1,n}$ and $T_{2,n}$ are both $o_p(1)$.
Similar arguments to the proof of Corollary \[c:inf:1\] imply that $T_{1,n} = o_p(1)$ under Assumption \[a:nonpara\](e). For term $T_{2,n}$, notice that: $$T_{2,n} = {\mathcal{Z}}_n(g_{\hat \alpha})$$ where ${\mathcal{Z}}_n$ denotes the centered empirical process on ${\mathcal{G}}$. Let $\|\cdot\|_{2,\beta}$ denote the norm defined on p. 400 of [@DoukhanMassartRio]. Display (a) in Lemma 2 of [@DoukhanMassartRio] shows that, under exponential beta-mixing (condition (d)), the norm $\|\cdot\|_{2,\beta}$ is dominated by the $L^p$ norm for any $p > 2$. It follows from condition (e) that $\sup_{g \in {\mathcal{G}}}\|g\|_{2 + \epsilon} < \infty$ for some $\epsilon > 0$. Therefore, ${\mathcal{G}}$ is uniformly bounded under $\|\cdot\|_{2,\beta}$ and $\Gamma$ is well defined on ${\mathcal{G}}$ since $\Gamma(g,g)\leq 4\|g\|_{2,\beta}^2$ [@DoukhanMassartRio p. 401]. It follows by condition (b) that $\Gamma(g_{\hat \alpha},g_{\hat \alpha}) = o_p(1)$. Appropriately modifying the arguments of Lemma 19.24 in [@vdV] (i.e. replacing the $L^2$ norm by the norm induced by $\Gamma$ which is the appropriate norm for the weakly dependent case) give ${\mathcal{Z}}_n(g_{\hat \alpha}) \to_p 0$, as required.
Take $C$, $b$, and $N$ as in Lemma \[lem:fpiter\]. Fix $\varepsilon > 0$ and consider $B_{\varepsilon} (\chi) = \{ \psi \in L^2 : \| \psi - \chi\| < \varepsilon\}$. As $\{ \|h\| \psi : \psi \in B_{\varepsilon}(\chi)\} = B_{\varepsilon \|h\|}(h)$, choose $\varepsilon$ sufficiently small that $B_{\varepsilon \|h\|}(h) \subseteq N$. ($B_{\varepsilon} (\chi)$ is the neighborhood in the statement of the proposition.)
Take any $\psi \in \{ a f : f \in B_{\varepsilon}(\chi), a \in {\mathbb{R}} \setminus \{0\}\}$. For any such $\psi$ we can write $\psi = (a /\|h\|) (\|h\| f)$ where $\|h\|f \in B_{\varepsilon \|h\|}(h) \subseteq N$. Write $f^* = \|h\| f$. By homogeneity of ${\mathbb{T}}$: $$\chi_{n+1}(\psi) = \frac{{\mathbb{T}}^n (\chi_1( \psi))}{\|{\mathbb{T}}^n (\chi_1( \psi))\|} = \frac{{\mathbb{T}}^n (\chi_1(f^*))}{\|{\mathbb{T}}^n (\chi_1(f^*))\|} = \chi_{n+1}(f^*)$$ for each $n \geq 1$, where $f^* \in B_{\varepsilon \|h\|} \subseteq N$ (note positivity of ${\mathbb{G}}$ ensures that $\| {\mathbb{T}}^n f^* \| > 0$ for each $n$ and each $f^* \in N$). It follows from Lemma \[lem:fpiter\] that: $$\| \chi_{n+1}(\psi) - \chi\| = \| \chi_{n+1}(f^*) - \chi\| = \left\| \frac{{\mathbb{T}}^n (f^*)}{\| {\mathbb{T}}^n (f^*)\|} - \frac{h}{\|h\|} \right\| \leq \frac{2}{\|h\|} \| {\mathbb{T}}^n (f^*) - h\| \leq \frac{2}{\|h\|} C b^n$$ as required.
The next Lemma is loosely based on Lemma 6.10 in [@AGN].
\[lem:fpiter\] Let the conditions of Proposition \[p:nl\] hold. Then: there exists finite positive constants $C,b$ with $b < 1$ and a neighborhood $N$ of $h$ such that: $$\| {\mathbb{T}}^n \psi - h\| \leq C b^n$$ for all $\psi \in N$.
Fix some constant $a$ such that $r({\mathbb{D}}_h) < a < 1$. By the Gelfand formula, there exists $m \in {\mathbb{N}}$ such that $\|{\mathbb{D}}_h^m\| < a^m$. Fréchet differentiability of ${\mathbb{T}}$ at $h$ together with the chain rule for Fréchet derivatives implies that: $$\|{\mathbb{T}}^m \psi - {\mathbb{T}}^m h - {\mathbb{D}}_h^m (\psi-h)\| = o(\|\psi - h\|)$$ hence: $$\| {\mathbb{T}}^m \psi - h\| \leq \| {\mathbb{D}}_h^m \| \| \psi - h\| + o(\| \psi - h\|) < (a^m+o(1)) \times \| \psi - h\|\,.$$ We may therefore choose $\epsilon > 0$ such that $\| {\mathbb{T}}^m \psi - h\| \leq a^m \| \psi - h\|$ for all $\psi \in B_\epsilon(h)$ where $B_\epsilon(h) = \{ \psi \in L^2 : \| \psi - h\| < \epsilon\}$. Then for any $\psi \in B_\epsilon(h)$ and any $k \in {\mathbb{N}}$ we have: $$\label{lem:fpiter:1}
\| {\mathbb{T}}^{km} \psi - h\| \leq a^{km} \| \psi - h\|\,.$$ It is straightforward to show via induction that boundedness of ${\mathbb{G}}$ and homogeneity of degree $\beta$ of ${\mathbb{T}}$ together imply: $$\label{lem:fpiter:2}
\| {\mathbb{T}} \psi_1 - {\mathbb{T}} \psi_2 \| \leq ( 1 + \| {\mathbb{G}}\|)^{\frac{1}{1-\beta}} \| \psi_1 - \psi_2\|^{\beta^n}$$ for any $\psi_1,\psi_2 \in L^2$.
Take any $n \geq m$ and let $k = \lfloor n/m \rfloor $. By (\[lem:fpiter:1\]) and (\[lem:fpiter:2\]) we have: $$\begin{aligned}
\| {\mathbb{T}}^n \psi - h\| & = \| {\mathbb{T}}^{(n-km)} {\mathbb{T}}^{km} \psi - {\mathbb{T}}^{(n-km)} h\| \\
& \leq (1+\|{\mathbb{G}}\|)^{\frac{1}{1-\beta}} \|{\mathbb{T}}^{km} \psi - h\|^{\beta^{(n-km)}} \\
& \leq (1+\|{\mathbb{G}}\|)^{\frac{1}{1-\beta}}\epsilon^{\beta^{(n-km)}}(a^{ km}) ^{\beta^{(n-km)}}\end{aligned}$$ for any $\psi \in B_\epsilon(h)$. The result follows for suitable choice of $b$ and $C$.
The result for $\chi$ is stated in the text. For $h$, let $C$, $b$, and $N$ be as in Lemma \[lem:fpiter\]. Suppose $h'$ is a fixed point of ${\mathbb{T}}$ belonging to $N$. Then by Lemma \[lem:fpiter\]: $$\| h' - h\| = \| {\mathbb{T}}^n h' - h\| \leq C b^n \to 0$$ hence $h' = h$.
Immediate from Lemmas \[lem:fp:exist\]–\[lem:var:fp\].
Proofs for Appendix \[ax:est:cgce\]
-----------------------------------
Step 1: We prove that there exists $K \in {\mathbb{N}}$ such that the maximum eigenvalue $\rho_k$ of the operator $\Pi_k {\mathbb{M}} : L^2 \to L^2$ is real and simple whenever $k \geq K$.
First note that $\rho$ is a simple isolated eigenvalue of ${\mathbb{M}}$ under Assumption \[a:id\]. Therefore, there exists an $\epsilon > 0$ such that $|\lambda - \rho| > 2\epsilon$ for all $\lambda \in \sigma({\mathbb{M}})$. Let $\Gamma$ denote a positively oriented circle in ${\mathbb{C}}$ centered at $\rho$ with radius $\epsilon$. Let ${\mathcal{R}}({\mathbb{M}},z) = ({\mathbb{M}} - zI)^{-1}$ denote the resolvent of ${\mathbb{M}}$ evaluated at $z \in {\mathbb{C}} \setminus \sigma({\mathbb{M}})$, where $I$ is the identity operator. Note that the number $C_{{\mathcal{R}}}$, given by: $$\label{e:crdef}
C_{{\mathcal{R}}} = \sup_{z \in \Gamma}\|{\mathcal{R}}({\mathbb{M}},z)\| < \infty$$ because ${\mathcal{R}}({\mathbb{M}},z)$ is a holomorphic function on $\Gamma$ and $\Gamma$ is compact.
By Assumption \[a:bias\], there exists $K \in {\mathbb{N}}$ such that $\|\Pi_k {\mathbb{M}} - {\mathbb{M}}\| < C_{{\mathcal{R}}}^{-1}$ for all $k \geq K$. Therefore, for all $k \geq K$ the inequality: $$\|\Pi_k {\mathbb{M}} - {\mathbb{M}}\| \sup_{z \in \Gamma}\|{\mathcal{R}}({\mathbb{M}},z)\| \leq C_{{\mathcal{R}}} \|\Pi_k {\mathbb{M}} - {\mathbb{M}}\| < 1$$ holds. It follows by Theorem IV.3.18 on p. 214 of [@Kato] that whenever $k \geq K$: (i) the operator $\Pi_k {\mathbb{M}}$ has precisely one eigenvalue $\rho_k$ inside $\Gamma$ and $\rho_k$ is simple; (ii) $\Gamma \subset ({\mathbb{C}} \setminus \sigma(\Pi_k {\mathbb{M}}))$; and (iii) $\sigma(\Pi_k {\mathbb{M}})\setminus \{\rho\}$ lies on the exterior of $\Gamma$. Note that $\rho_k$ must be real whenever $k \geq K$ because complex eigenvalues come in conjugate pairs. Thus, if $\rho_k$ were complex-valued then its conjugate would also be in $\Gamma$, which would contradict the fact that $\rho_k$ is the unique eigenvalue of $\Pi_k {\mathbb{M}}$ on the interior of $\Gamma$.
Step 2: Any nonzero eigenvalue of $\Pi_k {\mathbb{M}}$ is also a nonzero eigenvalue of $({\mathbf{M}},{\mathbf{G}})$ with the same multiplicity. So by Step 1 we know that the largest eigenvalue $\rho_k$ of $({\mathbf{M}},{\mathbf{G}})$ is positive and simple whenever $k \geq K$.
Recall that $\phi^*_k(x) = b^k(x)'c_k^*$ where $c_k^*$ solves the left-eigenvector problem in (\[e:gev\]). Here, $\phi_k^*$ is the eigenfunction corresponding to $\rho_k$ of the adjoint of $\Pi_k {\mathbb{M}}$ with respect to the space $B_k$. We now introduce the term $\phi_k^+$, which is the eigenfunction corresponding to $\rho_k$ of the adjoint of $\Pi_k {\mathbb{M}}$ with respect to the space $L^2$ (it follows from Lemma \[lem:exist\] that $\phi_k^*$ and $\phi_k^+$ are uniquely defined (up to scale) for all $k$ sufficiently large under Assumptions \[a:id\] and \[a:bias\]). That is: $$\begin{aligned}
{\mathbb{E}}[\phi_k^+(X) \Pi_k {\mathbb{M}} \psi(X)] & = \rho_k {\mathbb{E}}[\phi_k^+(X) \psi(X)] \\
{\mathbb{E}}[\phi_k^*(X) \Pi_k {\mathbb{M}} \psi_k(X)] & = \rho_k {\mathbb{E}}[\phi_k^*(X) \psi_k(X)]\end{aligned}$$ for all $\psi \in L^2$ and $\psi_k \in B_k$. Notice that $\phi_k^+$ does not necessarily belong to $B_k$ whereas $\phi_k^*$ does belong to $B_k$. However, it follows from the preceding display that $\Pi_k^{\phantom *} \phi^+_k = \phi^*_k$. The proof of Lemma \[lem:bias\] shows that $\phi_k^*$ and $\phi_k^+$ converge to $\phi^*$ at the same rate.
In this proof, all adjoints are taken with respect to the space $L^2$.
Step 1: Proof of part (b). We use similar arguments to the proof of Proposition 4.2 of [@Gobetetal]. Take $k \geq K$ (where $K$ is from Lemma \[lem:exist\]) and let $P_k$ denote the spectral projection of $\Pi_k {\mathbb{M}}$ corresponding to $\rho_k$. By Lemma 6.4 on p. 279 of [@Chatelin], we have: $$\phi - P_k \phi = \left( \frac{-1}{2 \pi {\mathrm{i}}} \int_{\Gamma} \frac{{\mathcal{R}}(\Pi_k {\mathbb{M}},z)}{\rho - z}\,{\mathrm{d}} z \right) (\Pi_k {\mathbb{M}} - {\mathbb{M}})\phi$$ where $\Gamma$ is defined in the proof of Lemma \[lem:exist\]. It follows that: $$\begin{aligned}
\|\phi - P_k \phi\| & \leq & \frac{1}{2 \pi} \left\| \left( \int_{\Gamma} \frac{{\mathcal{R}}(\Pi_k {\mathbb{M}},z)}{\rho - z}\,{\mathrm{d}} z \right)(\Pi_k {\mathbb{M}} - {\mathbb{M}})\phi \right\| \notag \\
& \leq & \frac{1}{2 \pi} (2 \pi \epsilon) \frac{\sup_{z \in \Gamma} \|{\mathcal{R}}(\Pi_k {\mathbb{M}},z)\|}{\epsilon} \|(\Pi_k {\mathbb{M}} - {\mathbb{M}})\phi\| \notag \\
& \leq & \left( \textstyle \sup_{z \in \Gamma} \|{\mathcal{R}}(\Pi_k {\mathbb{M}},z)\|\right) \|(\Pi_k {\mathbb{M}} - {\mathbb{M}})\phi\|\,. \label{e:projbd}\end{aligned}$$ Moreover, for each $z \in \Gamma$ we have: $$\label{e:resbd}
\|{\mathcal{R}}(\Pi_k {\mathbb{M}},z)\| \leq \frac{\|{\mathcal{R}}({\mathbb{M}},z)\|}{1 - \|\Pi_k {\mathbb{M}} - {\mathbb{M}} \| C_{{\mathcal{R}}}} \leq \frac{C_{{\mathcal{R}}}}{1 - \|\Pi_k {\mathbb{M}} - {\mathbb{M}} \| C_{{\mathcal{R}}}} = O(1)$$ where the first inequality is by Theorem IV.3.17 on p. 214 of [@Kato] and the second is by display (\[e:crdef\]). This inequality holds uniformly for $z \in \Gamma$. Substituting (\[e:resbd\]) into (\[e:projbd\]): $$\label{e:resbd:00}
\|\phi - P_k \phi\| = O (1) \times \|(\Pi_k {\mathbb{M}} - {\mathbb{M}})\phi\| = O (1) \times \rho \times \|\Pi_k \phi -\phi\| = O(\delta_k)$$ where the final equality is by definition of $\delta_k$ in display (\[e:deltas\]).
Since $\rho_k$ is simple, the spectral projection $P_k$ is of the form:[^36] $$P_k = \frac{1}{\langle \phi_k^{\phantom +},\phi_k^+ \rangle}(\phi_k^{\phantom +}\! \otimes \phi_k^+)$$ under the normalizations $\|\phi_k\| = 1$ and $\|\phi_k^+\| = 1$. By the proof of Proposition 4.2 of [@Gobetetal], under the sign normalization $\langle \phi,\phi_k \rangle \geq 0$ we have: $$\|\phi - \phi_k\|^2 \leq 2 \|\phi - (\phi_k \otimes \phi_k)\phi\|^2
\leq 2 \left\|\phi - \bigg( \phi_k \otimes \frac{\phi_k^+}{\langle \phi_k^{\phantom +},\phi_k^+ \rangle}\bigg) \phi\right\|^2 = 2 \|\phi - P_k \phi\|^2 \label{e:oblique}$$ whence $\|\phi - \phi_k\| = O(\delta_k)$ by (\[e:resbd:00\]), proving (b).
Step 2: Proof of part (a). Here we use similar arguments to the proof of Corollary 4.3 of [@Gobetetal]. By the triangle inequality: $$\begin{aligned}
|\rho_k - \rho| & = & \left| \|\Pi_k {\mathbb{M}} \phi_k\| - \|{\mathbb{M}} \phi\| \right| \\
& \leq & \|\Pi_k {\mathbb{M}} \phi_k - {\mathbb{M}} \phi\| \\
& \leq & \|\Pi_k {\mathbb{M}} \phi_k - \Pi_k {\mathbb{M}} \phi \| + \| \Pi_k {\mathbb{M}} \phi - {\mathbb{M}} \phi\| \\
& \leq & \|\Pi_k {\mathbb{M}} \| \|\phi_k - \phi \| + \rho \| \Pi_k \phi - \phi\| = O(\delta_k)\end{aligned}$$ because $\|\phi_k - \phi\| = O(\delta_k)$ by part (b), $\|\Pi_k \phi - \phi\| = \delta_k$ by definition of $\delta_k$, and $\|\Pi_k {\mathbb{M}} \| = O(1)$ because ${\mathbb{M}}$ is bounded and $\Pi_k$ is a (weak) contraction on $L^2$.
Step 3: Proof that $\|\phi_k^+ - \phi^*\| = O(\delta_k^*)$ under the normalizations $\| \phi^*_{\phantom k} \| = 1$ and $\| \phi^+_k \| = 1$.
First observe that $\|{\mathcal{R}}({\mathbb{M}}^*,z)\| = \|{\mathcal{R}}({\mathbb{M}},\bar z)\|$ holds for all $z \in \Gamma$ (where $\bar z$ denotes the conjugate of $z$) because ${\mathcal{R}}({\mathbb{M}}^*,z) = {\mathcal{R}}({\mathbb{M}},\bar z)^*$ [@Kato Theorem 6.22, p. 184]. Similarly, $\|{\mathcal{R}}((\Pi_k {\mathbb{M}})^*,z)\| = \|{\mathcal{R}}(\Pi_k {\mathbb{M}},\bar z)\|$ holds for all $z \in \Gamma$ whenever $k \geq K$. By similar arguments to the proof of part (b): $$\begin{aligned}
\|\phi^* - P_k^* \phi^* \| = O(1) \times \|((\Pi_k {\mathbb{M}})^* - {\mathbb{M}}^*) \phi^*\| = O(1) \times \rho \times \|\Pi_k \phi^* - \phi^*\| = O(\delta_k^*) \label{e:projbdstar}\end{aligned}$$ where the final equality is by definition of $\delta_k^*$ (see display (\[e:deltas\])).
Now we use the fact that $P_k^*$ is of the form: $$P_k^* = \frac{1}{\langle \phi_k^{\phantom +},\phi_k^+ \rangle}(\phi_k^+ \otimes \phi_k^{\phantom +}\!)$$ under the normalizations $\|\phi_k\| = 1$ and $\|\phi_k^+\| = 1$. By similar arguments to the proof of part (b), we have: $$\left\| \phi^*- \phi_k^+ \right\|^2
\leq 2 \| \phi^* - ( \phi^+_k \otimes \phi^+_k) \phi^* \|^2
\leq 2 \left\| \phi^* - \bigg( \phi_k^+ \otimes \frac{\phi_k}{\langle \phi_k^{\phantom +},\phi_k^+ \rangle} \bigg) \phi^* \right\|^2
= 2 \|\phi^* - P_k^* \phi^* \|^2$$ under the sign normalization $\langle \phi^*, \phi_k^+ \rangle \geq 0$. It follows by (\[e:projbdstar\]) that $\|\phi^* - \phi_k^+\| = O(\delta_k^*)$.
Step 4: Proof that $\|\phi_k^* - \phi^*\| = O(\delta_k^*)$. Recall that $\phi_k^* = \Pi_k \phi_k^+$. Then by the triangle inequality and the fact that $\Pi_k$ is a weak contraction, we have: $$\begin{aligned}
\|\phi^* - \phi_k^*\| = \|\phi^* - \Pi_k \phi^+_k\|
& \leq \|\phi^* - \Pi_k \phi^* \| + \|\Pi_k \phi^* - \Pi_k \phi^+_k\| \\
& \leq \|\phi^* - \Pi_k \phi^* \| + \| \phi^* - \phi^+_k\|
= O(\delta_k^*) + O(\delta_k^*) \end{aligned}$$ where the final equality is by definition of $\delta_k^*$ (see display (\[e:deltas\])) and Step 3.
The following lemma collects some useful bounds on the orthogonalized estimators.
\[lem:matcgce\] (a) If ${\widehat{{\mathbf{G}}}}$ is invertible then: $$({\widehat{{\mathbf{G}}}}^o)^{-1} {\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o = {\widehat{{\mathbf{M}}}}^o - {\widehat{{\mathbf{G}}}}^o {\mathbf{M}}^o + ({\widehat{{\mathbf{G}}}}^o)^{-1} \Big(({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}})^2{{\mathbf{M}}}^o + ({\mathbf{I}} - {\widehat{{\mathbf{G}}}}^o)({\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o ) \Big)\,.$$ (b) In particular, if $\|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| \leq \frac{1}{2}$ we obtain: $$\|({\widehat{{\mathbf{G}}}}^o)^{-1} {\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o \| \leq \|{\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o\| + 2 \|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\|\times (\|{\mathbf{M}}^o\| + \|{\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o\| )\,.$$
If ${\widehat{{\mathbf{G}}}}$ is invertible we have: $$\begin{aligned}
({\widehat{{\mathbf{G}}}}^o)^{-1} {\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o & = ( {\mathbf{I}} - ({\widehat{{\mathbf{G}}}}^o)^{-1}({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}})) {\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o \notag \\
& = {\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o - ({\widehat{{\mathbf{G}}}}^o)^{-1}({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}) {\mathbf{M}}^o - ({\widehat{{\mathbf{G}}}}^o)^{-1}({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}) ({\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o)\,. \end{aligned}$$ Part (b) follows by the triangle inequality, noting that $\|({\widehat{{\mathbf{G}}}}^o)^{-1}\| \leq 2$ whenever $\|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| \leq \frac{1}{2}$. Substituting $({\widehat{{\mathbf{G}}}}^o)^{-1} = ( {\mathbf{I}} - ({\widehat{{\mathbf{G}}}}^o)^{-1}({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}})) $ into the preceding display yields: $$\begin{aligned}
& ({\widehat{{\mathbf{G}}}}^o)^{-1} {\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o \\
& = {\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o - ( {\mathbf{I}} - ({\widehat{{\mathbf{G}}}}^o)^{-1}({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}))({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}) {\mathbf{M}}^o - ({\widehat{{\mathbf{G}}}}^o)^{-1}({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}) ({\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o) \\
& = {\widehat{{\mathbf{M}}}}^o - {\widehat{{\mathbf{G}}}}^o {\mathbf{M}}^o + ({\widehat{{\mathbf{G}}}}^o)^{-1}({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}})^2 {\mathbf{M}}^o - ({\widehat{{\mathbf{G}}}}^o)^{-1}({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}) ({\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o) \,. \end{aligned}$$ as required.
Step 1: Let $\Pi_k {\mathbb{M}}|_{B_k}$ denote the restriction of $\Pi_k {\mathbb{M}}$ to $B_k$ (i.e. $\Pi_k {\mathbb{M}}|_{B_k}$ is a linear operator on the space $B_k$ rather than the full space $L^2$). We show that: $$\|{\mathcal{R}}(\Pi_k {\mathbb{M}}|_{B_k},z)\| \leq \|{\mathcal{R}}(\Pi_k {\mathbb{M}},z)\|$$ holds for all $z \in {\mathbb{C}} \setminus (\sigma(\Pi_k {\mathbb{M}}) \cup \sigma(\Pi_k {\mathbb{M}}|_{B_k}))$.
Fix any such $z$. Then for any $\psi_k \in B_k$ we have ${\mathcal{R}}(\Pi_k {\mathbb{M}}|_{B_k},z) \psi_k = \zeta_k$ where $\zeta_k = \zeta_k(\psi_k) \in B_k$ is given by $\psi_k = (\Pi_k {\mathbb{M}} - z I)\zeta_k$. Similarly, for any $\psi \in L^2$ we have ${\mathcal{R}}(\Pi_k {\mathbb{M}},z) \psi = \zeta$ where $\zeta = \zeta(\psi) \in L^2$ is given by $ \psi = (\Pi_k {\mathbb{M}} - z I) \zeta$. Therefore, for any $\psi_k \in B_k$ we must have $\zeta_k(\psi_k) = \zeta(\psi_k)$, i.e., ${\mathcal{R}}(\Pi_k {\mathbb{M}}|_{B_k},z) \psi_k = {\mathcal{R}}(\Pi_k {\mathbb{M}},z) \psi_k$ holds for all $\psi_k \in B_k$. Therefore: $$\begin{aligned}
\|{\mathcal{R}}(\Pi_k {\mathbb{M}}|_{B_k},z)\| & =\sup\{ \|{\mathcal{R}}(\Pi_k {\mathbb{M}}|_{B_k},z) \psi_k\| : \psi_k \in B_k , \|\psi_k\| = 1\}\\
& =\sup\{ \|{\mathcal{R}}(\Pi_k {\mathbb{M}},z) \psi_k\| : \psi_k \in B_k , \|\psi_k\| = 1\}\\
& \leq\sup\{ \|{\mathcal{R}}(\Pi_k {\mathbb{M}},z) \psi\| : \psi \in L^2 , \|\psi\| = 1\} = \|{\mathcal{R}}(\Pi_k {\mathbb{M}},z) \| \,.\end{aligned}$$
Step 2: We now show that $({\widehat{{\mathbf{M}}}},{\widehat{{\mathbf{G}}}})$ has a unique eigenvalue $\hat \rho$ inside $\Gamma$ wpa1, where $\Gamma$ is from the proof of Lemma \[lem:exist\]. As the nonzero eigenvalues of $\Pi_k {\mathbb{M}}$, $\Pi_k {\mathbb{M}}|_{B_k}$, and $({\mathbf{M}},{\mathbf{G}})$ are the same, it follows from the proof of Lemma \[lem:exist\] that for all $k \geq K$ the curve $\Gamma$ separates $\rho_k$ from $\sigma(\Pi_k {\mathbb{M}}|_{B_k})\setminus \{\rho_k\}$ and $\rho_k$ from $\sigma({\mathbf{G}}^{-1} {\mathbf{M}})\setminus \{\rho_k\}$.
Taking $k \geq K$ from Lemma \[lem:exist\], by Step 1 and display (\[e:resbd\]), we have: $$\label{e:resbd:proj}
\sup_{z \in \Gamma} \|{\mathcal{R}}(\Pi_k {\mathbb{M}}|_{B_k},z)\| \leq \sup_{z \in \Gamma} \|{\mathcal{R}}(\Pi_k {\mathbb{M}},z)\| = O(1)\,.$$ Recall that ${\mathbf{G}}^{-1} {\mathbf{M}}^{\phantom 1}$ is isomorphic to $\Pi_k {\mathbb{M}} |_{B_k}$ on $({\mathbb{R}}^k,\langle \cdot,\cdot\rangle_{{\mathbf{G}}})$. Let ${\mathcal{R}}({\mathbf{G}}^{-1} {\mathbf{M}},z)$ denote the resolvent of ${\mathbf{G}}^{-1} {\mathbf{M}}$ on the space $({\mathbb{R}}^k,\langle \cdot,\cdot\rangle_{{\mathbf{G}}})$. By isometry and display (\[e:resbd:proj\]): $$\label{e:resbd:k}
\sup_{z \in \Gamma} \|{\mathcal{R}}({\mathbf{G}}^{-1} {\mathbf{M}},z)\|_{{\mathbf{G}}} = \sup_{z \in \Gamma} \|{\mathcal{R}}(\Pi_k {\mathbb{M}}|_{B_k},z)\| = O(1) \,.$$ By Lemma \[lem:matcgce\](b), Assumption \[a:var\], and boundedness of ${\mathbb{M}}$, we have: $$\| {\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} - {\mathbf{G}}^{-1} {\mathbf{M}} \|_{{\mathbf{G}}} = \| ({\widehat{{\mathbf{G}}}}^o)^{-1} {\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o \| = o_p(1)\,.$$ Hence: $$\label{e:kato}
\|{\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{M}}}} - {{\mathbf{G}}}^{-1}{{\mathbf{M}}}\|_{{\mathbf{G}}} \times \sup_{z \in \Gamma} \|{\mathcal{R}}({\mathbf{G}}^{-1} {\mathbf{M}},z)\|_{{\mathbf{G}}} < 1$$ holds wpa1.
The proof of Lemma \[lem:exist\] shows that ${\mathbf{G}}^{-1} {\mathbf{M}}$ has a unique eigenvalue $\rho_k$ inside $\Gamma$ whenever $k \geq K$ and that $\rho_k$ is a simple eigenvalue of ${\mathbf{G}}^{-1} {\mathbf{M}}$. Therefore, whenever (\[e:kato\]) holds we have that: (i) ${\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{M}}}}$ has precisely one simple eigenvalue $\hat \rho$ inside $\Gamma$; (ii) $\Gamma \subset ({\mathbb{C}} \setminus \sigma({\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{M}}}}))$; and (iii) $\sigma({\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{M}}}})\setminus \{\hat \rho\}$ lies on the exterior of $\Gamma$ (see Theorem IV.3.18 on p. 214 of [@Kato]). Again, $\hat \rho$ must be real and simple whenever (\[e:kato\]) holds (because complex eigenvalues come in conjugate pairs) hence the corresponding left- and right-eigenvectors $\hat c^*$ and $\hat c$ are unique.
Take $k \geq K$ from Lemma \[lem:exist\] and work on the sequence of events upon which the solutions $\hat \rho$, $\hat c$ and $\hat c^*$ to (\[e:est\]) are unique (this holds wpa1 by Lemma \[lem:exist:hat\]).
Step 1: Proof of part (b). Let ${\widehat{{\mathbf{P}}}}_k$ denote the spectral projection of ${\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{M}}}}$ corresponding to the eigenvalue ${\widehat{\rho}}$. By similar arguments to the proof of Lemma \[lem:bias\](b), $$c_k - {\widehat{{\mathbf{P}}}}_k c_k \leq \left( \frac{-1}{2 \pi {\mathrm{i}}} \int_{\Gamma} \frac{{\mathcal{R}}({\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{M}}}},z)}{\rho_k - z}\,{\mathrm{d}} z \right) ({\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{M}}}} - {{\mathbf{G}}}^{-1}{{\mathbf{M}}})c_k$$ and so: $$\label{e:hatbd}
\| c_k - {\widehat{{\mathbf{P}}}}_k c_k \|_{{\mathbf{G}}} \leq \frac{\epsilon}{\inf_{z \in \Gamma}|z - \rho_k|} \times \textstyle \sup_{z \in \Gamma} \|{\mathcal{R}}({\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{M}}}},z)\|_{{\mathbf{G}}} \times \|({\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{M}}}} - {{\mathbf{G}}}^{-1}{{\mathbf{M}}})c_k\|_{{\mathbf{G}}}\,.$$ Note that $\inf_{z \in \Gamma}|z - \rho_k| \to \epsilon$ because $\Gamma$ is centered at $\rho$ and $|\rho - \rho_k| = o(1)$ by Lemma \[lem:bias\]. Further, whenever (\[e:kato\]) holds, for each $z \in \Gamma$ we have: $$\begin{aligned}
\|{\mathcal{R}}({\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{M}}}},z)\|_{{\mathbf{G}}}
& \leq & \frac{\sup_{z \in \Gamma}\|{\mathcal{R}}({{\mathbf{G}}}^{-1}{{\mathbf{M}}},z)\|_{{\mathbf{G}}}}{1 - \|{\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{M}}}} - {{\mathbf{G}}}^{-1}{{\mathbf{M}}}\|_{{\mathbf{G}}} \sup_{z \in \Gamma} \|{\mathcal{R}}({{\mathbf{G}}}^{-1}{{\mathbf{M}}},z)\|_{{\mathbf{G}}}} \label{e:resbd2}\end{aligned}$$ which is $O_p(1)$ by Assumption \[a:var\] and display (\[e:resbd:k\]). Substituting into (\[e:hatbd\]) and using the definition of $\eta_{n,k}$ (cf. display (\[e:etas\])) yields: $$\begin{aligned}
\| c_k - {\widehat{{\mathbf{P}}}}_k c_k \|_{{\mathbf{G}}} & \leq O_p(1) \times \|({\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{M}}}} - {{\mathbf{G}}}^{-1}{{\mathbf{M}}})c_k\|_{{\mathbf{G}}} \notag \\
& = O_p(1) \times \|(({\widehat{{\mathbf{G}}}}^o)^{-1}{\widehat{{\mathbf{M}}}}^o - {{\mathbf{M}}}^o)\tilde c_k\| = O_p (\eta_{n,k}) \label{e:ckhatbd}\,.\end{aligned}$$ Let $u \otimes_{{\mathbf{G}}} v = u (v' {\mathbf{G}})$. Observe that ${\widehat{{\mathbf{P}}}}_k$ is given by: $$\label{e:hat:pk:def}
{\widehat{{\mathbf{P}}}}_k = \frac{1}{\langle \hat c , \hat c^* \rangle_{{\mathbf{G}}}} (\hat c \otimes_{{\mathbf{G}}} \hat c^*)$$ under the normalizations $\|\hat c\|_{{\mathbf{G}}} = 1$ and $ \|\hat c^*\|_{{\mathbf{G}}} = 1$. By similar arguments to (\[e:oblique\]), we obtain: $$\begin{aligned}
\|\hat \phi - \phi_k\|^2 = \|\hat c - c_k\|^2_{{\mathbf{G}}}
& \leq \|c_k - (\hat c \otimes_{{\mathbf{G}}} \hat c) c_k \|_{{\mathbf{G}}}^2 \notag \\
& \leq \left\|c_k - \left( \hat c \otimes_{{\mathbf{G}}} \frac{\hat c^*}{\langle \hat c,\hat c^* \rangle_{{\mathbf{G}}}} \right) c_k \right\|^2_{{\mathbf{G}}}
= \|c_k - {\widehat{{\mathbf{P}}}}_k c_k\|^2_{{\mathbf{G}}} \label{e:ckineq}\end{aligned}$$ under the sign normalization $\langle \hat c ,c_k \rangle_{{\mathbf{G}}} \geq 0$. It follows that $\|\hat \phi - \phi_k\| = O_p(\eta_{n,k})$ by (\[e:ckhatbd\]).
Step 2: proof of part (a). Similar arguments to the proof of Lemma \[lem:bias\](a) yield: $$\begin{aligned}
|\hat \rho - \rho_k| & = & \left| \|{\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} \hat c\|_{{\mathbf{G}}} - \|{\mathbf{G}}^{-1} {\mathbf{M}} c_k\|_{{\mathbf{G}}} \right| \notag \\
& \leq & \|{\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} \hat c - {\mathbf{G}}^{-1} {\mathbf{M}} c_k \|_{{\mathbf{G}}} \notag \\
& \leq & \|{\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}}\|_{{\mathbf{G}}} \| \hat c - c_k \|_{{\mathbf{G}}} + \|(({\widehat{{\mathbf{G}}}}^o)^{-1} {\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o) \tilde c_k\|_{{\mathbf{G}}} \,.\label{e:rhatineq}\end{aligned}$$ The second term on the right-hand side of (\[e:rhatineq\]) is $O_p(\eta_{n,k})$ (cf. display (\[e:etas\])). For the first term, Assumption \[a:var\] implies that $\|{\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} \|_{{\mathbf{G}}} = \|{{\mathbf{G}}}^{-1} {{\mathbf{M}}} \|_{{\mathbf{G}}} + o_p(1)$. But $\|{{\mathbf{G}}}^{-1} {{\mathbf{M}}} \|_{{\mathbf{G}}} = \|\Pi_k {\mathbb{M}}|_{B_k}\| \leq \|{\mathbb{M}}\| < \infty$. Therefore, $\|{\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}}\|_{{\mathbf{G}}} = O_p(1)$ and the result follows by (\[e:ckineq\]).
Step 3: Proof of part (c). Identical arguments to the proof of part (b) yield: $$\label{e:projbdstarhat}
\|c_k^* - {\widehat{{\mathbf{P}}}}_k^* c_k\|_{{\mathbf{G}}} \leq O_p(1) \times \|({\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{M}}}}' - {{\mathbf{G}}}^{-1}{{\mathbf{M}}}')c_k^*\|_{{\mathbf{G}}} = O_p(\eta_{n,k}^*)$$ where: $${\widehat{{\mathbf{P}}}}_k^* = \frac{1}{\langle \hat c , \hat c^* \rangle_{{\mathbf{G}}}} (\hat c^* \otimes_{{\mathbf{G}}} \hat c)\,.$$ and $\eta_{n,k}^*$ is from display (\[e:etas\]). Therefore, under the sign normalization $\langle \hat c^*,c_k^* \rangle_{{\mathbf{G}}}$ we have: $$\begin{aligned}
\| \hat \phi^* - \phi_k^* \|^2 = \| \hat c^* - c_k^* \|^2_{{\mathbf{G}}} & \leq 2 \| c_k^* - (\hat c^* \otimes_{{\mathbf{G}}} \hat c^*) c_k^* \|_{{\mathbf{G}}}^2 \\
& \leq 2 \left\| c_k^* - \left( \hat c^* \otimes_{{\mathbf{G}}} \frac{\hat c}{\langle \hat c , \hat c^* \rangle_{{\mathbf{G}}}} \right) c_k^* \right\|_{{\mathbf{G}}}^2 = 2 \| c_k^* - {\widehat{{\mathbf{P}}}}_k^* c_k^*\|^2_{{\mathbf{G}}}
\end{aligned}$$ whence $\| \hat \phi^* - \phi_k^* \| = O_p(\eta_{n,k}^*)$ by (\[e:projbdstarhat\]).
Proofs for Appendix \[ax:est:fp\]
---------------------------------
We verify the conditions of Theorem 19.1 in [@Kras]. We verify the conditions noting that, in our notation, $\Omega$ is a neighborhood of $h$, $P_n = \Pi_k$, $T = {\mathbb{T}}$, and $T_n = \Pi_k {\mathbb{T}}|_{B_k}$ (the restriction of $\Pi_k {\mathbb{T}}$ to $B_k$).
${\mathbb{T}}$ is continuously Fréchet differentiable on a neighborhood $N$ of $h$ by Assumption \[a:fp:exist\](c). As a consequence, $\Pi_k {\mathbb{T}}|_{B_k}$ is continuously Fréchet differentiable on $N \cap B_k$. Assumption \[a:fp:exist\](b)(c) implies that $I - {\mathbb{D}}_h$ is continuously invertible in $L^2$. Also notice that $\|\Pi_k h - h\| \to 0$ by Assumption \[a:fp:bias\](b) (this verifies condition (19.8) in [@Kras]). Moreover: $$\begin{aligned}
\|\Pi_k {\mathbb{T}} (\Pi_k h) - {\mathbb{T}} h\| & \leq \|\Pi_k {\mathbb{T}} (\Pi_k h) - \Pi_k {\mathbb{T}} h\| + \|\Pi_k {\mathbb{T}} h - {\mathbb{T}} h\| \\
& \leq \|{\mathbb{T}} (\Pi_k h) - {\mathbb{T}} h\| + \|\Pi_k h - h\| \to 0 \end{aligned}$$ by continuity of ${\mathbb{T}}$. Moreover, $$\begin{aligned}
\|\Pi_k {\mathbb{D}}_{\Pi_k h} - {\mathbb{D}}_h\| & \leq \|\Pi_k {\mathbb{D}}_{\Pi_k h} - \Pi_k {\mathbb{D}}_h\| + \|\Pi_k {\mathbb{D}}_{h} - {\mathbb{D}}_h\| \\
& \leq \| {\mathbb{D}}_{\Pi_k h} - {\mathbb{D}}_h\| + \|\Pi_k {\mathbb{D}}_{h} - {\mathbb{D}}_h\| \to 0 \end{aligned}$$ by continuous Fréchet differentiability of ${\mathbb{T}}$ and Assumption \[a:fp:bias\](a). This verifies conditions (19.9) in [@Kras], and their condition (19.10) is trivially satisfied. Fix any $\epsilon > 0$. By continuous Fréchet differentiability, there exists $\delta_\epsilon > 0$ such that $\| {\mathbb{D}}_g - {\mathbb{D}}_h \| \leq \epsilon/2$ whenever $\|g - h\| \leq 2\delta_\epsilon$. We may also choose $K_\epsilon \in {\mathbb{N}}$ such that $\|h - \Pi_k h\| \leq \delta_\epsilon$ and $\|{\mathbb{D}}_{\Pi_k h} - {\mathbb{D}}_h\| \leq \epsilon /2$ for all $k \geq K_\epsilon$. Whenever $k \geq K_\epsilon$ and $\|g - \Pi_k h\| \leq \delta_\epsilon$ we have: $$\|g - h\| \leq \|g - \Pi_k h\| + \| \Pi_k h - h\| \leq 2 \delta_\epsilon$$ and hence: $$\| {\mathbb{D}}_g - {\mathbb{D}}_{\Pi_k h} \| \leq \| {\mathbb{D}}_g - {\mathbb{D}}_h \| + \| {\mathbb{D}}_{\Pi_k h} - {\mathbb{D}}_h \| \leq \epsilon$$ as required. Theorem 19.1 in [@Kras] then ensures (1) existence of $h_k$ and uniqueness as a fixed point of $\Pi_k {\mathbb{T}}$ on a neighborhood of $h$ for all sufficiently large $k$ and (2) the upper bound: $$\|h_k - h\| \leq \|h - \Pi_k h \| + {\mathrm{const}} \times \|\Pi_k {\mathbb{T}} h - \Pi_k {\mathbb{T}} (\Pi_k h)\|$$ (see equations (19.12)–(19.13) on p. 295). It follows by continuity of $\Pi_k$ and ${\mathbb{T}}$ and Assumption \[a:fp:bias\](b) that $\|h_k - h\| = o(1)$.
We first prove part (c) by standard arguments (see p. 310 in [@Kras]). Take $k \geq K$ from Lemma \[lem:fp:exist\]. We then have: $$\begin{aligned}
(I - \Pi_k {\mathbb{D}}_h )(h - h_k) & = (I - \Pi_k) h - \Pi_k \big( {\mathbb{T}} h_k - {\mathbb{T}} h - {\mathbb{D}}_h (h_k - h) \big) \notag \\
\| (I - \Pi_k {\mathbb{D}}_h )(h - h_k)\| & \leq \|h - \Pi_k h\| + \| {\mathbb{T}} h_k - {\mathbb{T}} h - {\mathbb{D}}_h (h_k - h)\| \,. \label{e:bias:fp:0}\end{aligned}$$ Fréchet differentiability of ${\mathbb{T}}$ at $h$ (Assumption \[a:fp:exist\](c)) implies: $$\begin{aligned}
\label{e:bias:fp:1}
\| {\mathbb{T}} h_k - {\mathbb{T}} h - {\mathbb{D}}_h (h_k - h)\| = o(1) \times \|h_k - h\|\,.\end{aligned}$$ As $r({\mathbb{D}}_h) < 1$ and $\|{\mathbb{D}}_h - \Pi_k {\mathbb{D}}_h\| = o(1)$ (Assumptions \[a:fp:exist\](c) and \[a:fp:bias\](a)), a similar argument to Lemma \[lem:exist\] ensures that there exists $\epsilon > 0$ such that $r(\Pi_k {\mathbb{D}}_h) \leq 1-\epsilon$ for all $k$ sufficiently large. Therefore $I - \Pi_k {\mathbb{D}}_h$ is invertible and $$\begin{aligned}
\label{e:bias:fp:2}
\|(I - \Pi_k {\mathbb{D}}) \psi\| \geq C \|\psi\| \quad \mbox{for all $\psi \in L^2$}\end{aligned}$$ holds for all $k$ sufficiently large. Substituting (\[e:bias:fp:1\]) and (\[e:bias:fp:2\]) into (\[e:bias:fp:0\]) yields: $$(1- o(1)) \|h - h_k\| \leq {\mathrm{const}} \times \|h - \Pi_k h\|$$ hence $\|h - h_k\| = O(\tau_k)$. Part (b) then follows from the inequality: $$\left\| \frac{h}{\|h\|} - \frac{h_k}{\|h_k\|} \right\| \leq \frac{2}{\|h\|} \|h - h_k\| \,.$$ Finally, part (a) follows from the fact that $\big| \|h\| - \|h_k\| \big| = O(\tau_k)$ and continuous differentiability of $x \mapsto x^{1-\beta}$ at each $x > 0$.
The next lemma presents some bounds on the estimators which are used in the proof of Lemmas \[lem:fphat:exist\] and \[lem:var:fp\].
\[lem:fp:matcgce\] (a) Let Assumption \[a:fp:var\] hold. Then: $$\sup_{v \in {\mathbb{R}}^k : \| v\|_{{\mathbf{G}}} \leq c} \| {\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{T}}}}v - {\mathbf{G}}^{-1} {\mathbf{T}} v\|_{{\mathbf{G}}} = o_p(1)$$ (b) Moreover: $$\sup_{v \in {\mathbb{R}}^k : \| v - v_k\|_{{\mathbf{G}}} \leq \varepsilon} \|{\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{T}}}} v - {\mathbf{G}}^{-1} {\mathbf{T}} v \|_{{\mathbf{G}}} = O_p(\nu_{n,k})$$ where $\nu_{n,k}$ is from display (\[e:nudef\]).
By definition of ${\widehat{{\mathbf{G}}}}^o$, ${\widehat{{\mathbf{T}}}}^o$, and ${\mathbf{T}}^o$, we have $$\sup_{v \in {\mathbb{R}}^k : \| v\|_{{\mathbf{G}}} \leq c} \| {\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{T}}}}v - {\mathbf{G}}^{-1} {\mathbf{T}} v\|_{{\mathbf{G}}} = \sup_{v \in {\mathbb{R}}^k : \| v\| \leq c} \| ({\widehat{{\mathbf{G}}}}^o)^{-1}{\widehat{{\mathbf{T}}}}^ov - {\mathbf{T}}^o v\|\,.$$ As $\|({\widehat{{\mathbf{G}}}}^o)^{-1}\| \leq 2$ holds wpa1 (by the first part of Assumption \[a:fp:var\]), we have: $$\begin{aligned}
({\widehat{{\mathbf{G}}}}^o)^{-1} {\widehat{{\mathbf{T}}}}^o - {\mathbf{T}}^o & = ( {\mathbf{I}} - ({\widehat{{\mathbf{G}}}}^o)^{-1}({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}})) {\widehat{{\mathbf{T}}}}^o - {\mathbf{T}}^o \notag \\
& = {\widehat{{\mathbf{T}}}}^ov - {\mathbf{T}}^ov - ({\widehat{{\mathbf{G}}}}^o)^{-1}({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}) {\mathbf{T}}^o - ({\widehat{{\mathbf{G}}}}^o)^{-1}({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}) ({\widehat{{\mathbf{T}}}}^o - {\mathbf{T}}^o) \label{e:gtineq}\end{aligned}$$ wpa1, hence: $$\begin{aligned}
\sup_{v \in {\mathbb{R}}^k : \| v\|_{{\mathbf{G}}} \leq c} \| {\widehat{{\mathbf{G}}}}^{-1}{\widehat{{\mathbf{T}}}}v - {\mathbf{G}}^{-1} {\mathbf{T}} v\|_{{\mathbf{G}}} & \leq \sup_{v \in {\mathbb{R}}^k : \| v\| \leq c} \|{\widehat{{\mathbf{T}}}}^ov - {\mathbf{T}}^ov\| + 2 \| {\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| \times \sup_{v \in {\mathbb{R}}^k : \|v\| \leq c} \| {\mathbf{T}}^o v\| \\
& \quad \quad + 2 \| {\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| \times \sup_{v \in {\mathbb{R}}^k : \| v\| \leq c} \|{\widehat{{\mathbf{T}}}}^ov - {\mathbf{T}}^ov\|\end{aligned}$$ wpa1. The result follows by Assumption \[a:fp:var\].
Part (b) follows similarly by definition of ${\widehat{{\mathbf{G}}}}^o$, ${\widehat{{\mathbf{T}}}}^o$, ${\mathbf{T}}^o$, and $\nu_{n,k}$ in display (\[e:nudef\]).
The following proof uses certain properties of fixed point indices and the rotation of compact vector fields. We refer the reader to Section 14 of [@Kras] for details.
Let $\varepsilon$ be as in Lemma \[lem:fp:exist\]. Define $\Gamma = \{ \psi \in L^2 : \|\psi - h\| \leq \varepsilon\}$. Let $\Gamma_k = \Gamma \cap B_k$ and $\partial \Gamma_k$ denote the boundary of $\Gamma_k$ (in $B_k$), $\boldsymbol \Gamma_k = \{ v \in {\mathbb{R}}^k : v'b^k(x) \in \Gamma_k\}$, and $\partial \boldsymbol \Gamma_k = \{ v \in {\mathbb{R}}^k : v'b^k(x) \in \partial \Gamma_k\}$.
Lemma \[lem:fp:exist\] implies that $\Pi_k {\mathbb{T}}$ has a unique fixed point $h_k$ in $\Gamma\setminus \partial \Gamma$ and no fixed point on $\partial \Gamma$ for all $k \geq K$. It also follows from the proof of Lemma \[lem:fp:exist\] that $I - \Pi_k {\mathbb{D}}_{h_k}$ is invertible for all $k \geq K$ (increasing $K$ if necessary). Let $\gamma(I - \Pi_k {\mathbb{T}};\partial \Gamma)$ denote the rotation of $I - \Pi_k {\mathbb{T}}$ on $\partial \Gamma$. Then for all $k \geq K$ we have $|\gamma(I - \Pi_k {\mathbb{T}};\partial \Gamma)|=1$ and hence $|\gamma(I - \Pi_k {\mathbb{T}}|_{B_k};\partial \Gamma_k)|=1$ by Propositions (3)–(5) on pp. 299-300 of [@Kras].
We now show that the inequality: $$\label{e:fpindex:ineq}
\sup_{v \in {\mathbb{R}}^k : v'b^k(x) \in \partial \Gamma_k} \|{\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{T}}}} - {{\mathbf{G}}}^{-1} {{\mathbf{T}}}\|_{{\mathbf{G}}} < \inf_{ \psi \in \partial \Gamma_k} \| \psi - \Pi_k {\mathbb{T}} \psi\|$$ holds wpa1. The left-hand side is $o_p(1)$ by Lemma \[lem:fp:matcgce\] and Assumption \[a:fp:var\]. We claim that $\liminf_{k \to \infty} \inf_{ \psi \in \partial B_k} \| \psi - \Pi_k {\mathbb{T}} \psi\| > 0$. Suppose the claim is false. Then there exists a subsequence $\{\psi_{k_l} : l \geq 1\}$ with $\psi_{k_l} \in \partial \Gamma_{k_l}$ such that $\psi_{k_l} - \Pi_{k_l} {\mathbb{T}} \psi_{k_l} \to 0$. Since $\{\psi_{k_l} : l \geq 1\}$ is bounded and ${\mathbb{T}}$ is compact, there exists a convergent subsequence $\{\psi_{k_{l_j}} : j \geq 1\}$. Let $\psi^* = \lim_{j \to \infty} {\mathbb{T}} \psi_{k_{l_j}}$. Then: $$\begin{aligned}
\| \psi_{k_{l_j}} - \psi^*\| & \leq \| \psi_{k_{l_j}} - \Pi_{k_{l_j}} {\mathbb{T}} \psi_{k_{l_j}} \| + \| \Pi_{k_{l_j}} {\mathbb{T}} \psi_{k_{l_j}} -\Pi_{k_{l_j}} \psi^*\| + \| \Pi_{k_{l_j}} \psi^* - \psi^*\| \to 0\end{aligned}$$ as $j \to \infty$, where the first term vanishes by definition of $\psi_{k_l}$, the second vanishes by definition of $\psi_{k_{l_j}}$, and the third vanishes because $B_k$ is dense in the range of ${\mathbb{T}}$. It follows that $\psi^* \in \partial \Gamma$. Finally, by continuity of ${\mathbb{T}}$ and definition of $\psi^*$: $$\| {\mathbb{T}} \psi^* - \psi^*\| \leq \| {\mathbb{T}} \psi^* - {\mathbb{T}} \psi_{k_{l_j}}\| + \| {\mathbb{T}} \psi_{k_{l_j}} - \psi^*\| \to 0$$ as $j \to \infty$, hence $\psi^* \in \partial \Gamma$ is a fixed point of ${\mathbb{T}}$. But this contradicts the fact that $h$ is the unique fixed point of ${\mathbb{T}}$ in $\Gamma$. This proves the claim.
It follows by Proposition (2) on p. 299 of [@Kras] that whenever inequality (\[e:fpindex:ineq\]) holds, we have: $$\gamma( {\mathbf{I}} - {\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{T}}}} ; \partial \boldsymbol \Gamma_k) = \gamma( {\mathbf{I}} - {{\mathbf{G}}}^{-1} {{\mathbf{T}}} ; \partial \boldsymbol \Gamma_k) \,.$$ By isomorphism: $$\gamma( {\mathbf{I}} - {{\mathbf{G}}}^{-1} {{\mathbf{T}}} ; \partial \boldsymbol \Gamma_k) = \gamma(I - \Pi_k {\mathbb{T}}|_{B_k};\partial \Gamma_k)$$ hence $|\gamma( {\mathbf{I}} - {\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{T}}}} ; \partial \boldsymbol \Gamma_k)| = 1$ holds wpa1. Therefore, ${\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{T}}}}$ has at least one fixed point $\hat v \in \boldsymbol \Gamma_k $ wpa1. Clearly, for $\hat h(x) = \hat v'b^k(x)$ we must have $\|\hat h - h\| \leq \varepsilon$. Repeating the argument for any $\varepsilon' < \varepsilon$ implies that $\|\hat h - h \| \leq \varepsilon'$ wpa1. Therefore $\|\hat h - h\| = o_p(1)$.
The Fréchet derivative of $\Pi_k {\mathbb{T}}|_{B_k}$ at $h$ is $\Pi_k {\mathbb{D}}_h |_{B_k}$. This may be represented on $({\mathbb{R}}^k,\langle \cdot,\cdot\rangle_{{\mathbf{G}}})$ by ${\mathbf{G}}^{-1} {\mathbf{D}}_h$ where ${\mathbf{D}}_h = {\mathbb{E}}[b^k(X_t) \beta G_{t+1}^{1-\gamma} h(X_t)^{\beta-1} b^k(X_{t+1})']$.
We first prove part (c). By Lemma \[lem:fphat:exist\], $\hat h$ is well defined (wpa1) and $\|\hat h - h_k\| = o_p(1)$. Recall that $\hat h (x)= b^k(x) \hat v$ and $h_k(x) = b^k(x) v_k$. Then wpa1, we have: $$\begin{aligned}
({\mathbf{I}} - {\mathbf{G}}^{-1} {\mathbf{D}}_{h_k})(v_k - \hat v) & = {\mathbf{G}}^{-1} {\mathbf{T}} \hat v - {\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{T}}}} \hat v - \big( {\mathbf{G}}^{-1} {\mathbf{T}} \hat v - {\mathbf{G}}^{-1} {\mathbf{T}} v_k - {\mathbf{G}}^{-1} {\mathbf{D}}_{h_k} (\hat v - v_k) \big) \end{aligned}$$ so by Lemma \[lem:fp:matcgce\](b) and the triangle inequality: $$\begin{aligned}
\label{e:var:fp:0}
\| ({\mathbf{I}} - {\mathbf{G}}^{-1} {\mathbf{D}}_{h_k})(v_k - \hat v)\|_{{\mathbf{G}}}
& \leq O_p(\nu_{n,k}) + \| {\mathbf{G}}^{-1} {\mathbf{T}} \hat v - {\mathbf{G}}^{-1} {\mathbf{T}} v_k - {\mathbf{G}}^{-1} {\mathbf{D}}_{h_k} (\hat v - v_k) \|_{{\mathbf{G}}}\,.\end{aligned}$$ Notice that: $$\begin{aligned}
\| ({\mathbf{I}} - {\mathbf{G}}^{-1} {\mathbf{D}}_{h_k})(v_k - \hat v)\|_{{\mathbf{G}}} & = \| (I - \Pi_k {\mathbb{D}}_{h_k})(h_k - \hat h)\| \\
& \geq \| (I - {\mathbb{D}}_{h_k})(h_k - \hat h)\| - \| ({\mathbb{D}}_{h_k} - \Pi_k {\mathbb{D}}_{h_k})(h_k - \hat h)\| \\
& = \| (I - {\mathbb{D}}_{h})(h_k - \hat h)\| - o(1) \times \|h_k - \hat h\| \,.\end{aligned}$$ Moreover, $I - {\mathbb{D}}_h$ is continuously invertible by Assumption \[a:fp:exist\](b). Therefore: $$\label{e:var:fp:1}
\| ({\mathbf{I}} - {\mathbf{G}}^{-1} {\mathbf{D}}_{h_k})(v_k - \hat v)\|_{{\mathbf{G}}} \geq {\mathrm{const}} \times \|h_k - \hat h\|$$ holds for all $k$ sufficiently large. Also notice that: $$\begin{aligned}
& \| {\mathbf{G}}^{-1} {\mathbf{T}} \hat v - {\mathbf{G}}^{-1} {\mathbf{T}} v_k - {\mathbf{G}}^{-1} {\mathbf{D}}_{h_k} (\hat v - v_k) \|_{{\mathbf{G}}} \notag \\
& = \| \Pi_k {\mathbb{T}} \hat h - \Pi_k {\mathbb{T}} h_k - \Pi_k {\mathbb{D}}_{h_k}(\hat h - h_k)\| \notag \\
& \leq \| {\mathbb{T}} \hat h - {\mathbb{T}} h - {\mathbb{D}}_h(\hat h - h) + ( {\mathbb{D}}_h - {\mathbb{D}}_{h_k})(\hat h - h_k) - ({\mathbb{T}} h_k - {\mathbb{T}} h - {\mathbb{D}}_h( h_k - h)) \| \notag \\
& \leq \| {\mathbb{T}} \hat h - {\mathbb{T}} h - {\mathbb{D}}_h(\hat h - h) \| + \| ( {\mathbb{D}}_h - {\mathbb{D}}_{h_k})(\hat h - h_k)\| +\|{\mathbb{T}} h_k - {\mathbb{T}} h - {\mathbb{D}}_h( h_k - h)\| \notag \\
& = o_p(1) \times ( \| \hat h - h_k\| + \|h_k - h\|) + o(1) \times \| \hat h - h_k\| + o(1) \times \|h - h_k\| \label{e:var:fp:2}\end{aligned}$$ where the first inequality is because $\Pi_k$ is a (weak) contraction on $L^2$, the second is by the triangle inequality, and the final line is by Frechet differentiability of ${\mathbb{T}}$ at $h$ (first and third terms) and continuity of $g \mapsto {\mathbb{D}}_g$ at $h$ (second term).
Substituting (\[e:var:fp:1\]) and (\[e:var:fp:2\]) into (\[e:var:fp:0\]) and rearranging, we obtain: $$(1-o_p(1)) \| h_k - \hat h\| \leq O_p(\nu_{n,k}) + o_p(\tau_k)$$ as required. Parts (a) and (b) follow by similar arguments to the proof of Lemma \[lem:bias:fp\].
Proofs for Appendix \[ax:est:mat\]
----------------------------------
\[lem:G:beta\] Let the conditions of Lemma \[lem:beta:1\] hold. Then:
1. $\|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| = O_p ( \xi_k (\log n)/\sqrt n )$
2. $\|({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}})\tilde c_k\| = O_p ( \xi_k (\log n)/\sqrt n )$ and similarly for $\tilde c_k^*$.
Part (a) is just Lemma 2.2 of [@ChenChristensen-reg]; part (b) follows directly by definition of the spectral norm.
\[lem:M:beta\] Let the conditions of Lemma \[lem:beta:1\] hold. Then:
1. $\|{\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o\| = O_p ( \xi_k^{(r+2)/r} (\log n)/\sqrt n )$
2. $\|({\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o)\tilde c_k\| = O_p ( \xi_k ^{(r+2)/r} (\log n)/\sqrt n )$ and $\|\tilde c_k^{*\prime }({\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o)\| = O_p ( \xi_k ^{(r+2)/r} (\log n)/\sqrt n )$.
For part (a) we use a truncation argument. Let $\{T_n : n \geq 1\}$ be a sequence of positive constants to be defined subsequently, and write: $${\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o = \sum_{t=0}^{n-1} \Xi_{1,t,n} + \sum_{t=0}^{n-1} \Xi_{2,t,n}$$ where, with $\tilde b^k = {\mathbf{G}}^{-1/2} b^k$ and ${1\!\mathrm{l}}\{ \}$ denoting the indicator function, we have: $$\begin{aligned}
\Xi_{1,t,n} & = & n^{-1} \tilde b^k(X_t) m(X_t,X_{t+1}) \tilde b^k(X_{t+1})' {1\!\mathrm{l}}\{ \|\tilde b^k(X_t) m(X_t,X_{t+1}) \tilde b^k(X_{t+1})'\| \leq T_n\} \\
& & - {\mathbb{E}}[n^{-1} \tilde b^k(X_t) m(X_t,X_{t+1}) \tilde b^k(X_{t+1})' {1\!\mathrm{l}}\{ \|\tilde b^k(X_t) m(X_t,X_{t+1}) \tilde b^k(X_{t+1})'\| \leq T_n\} ] \\
\Xi_{2,t,n} & = & n^{-1} \tilde b^k(X_t) m(X_t,X_{t+1}) \tilde b^k(X_{t+1})' {1\!\mathrm{l}}\{ \|\tilde b^k(X_t) m(X_t,X_{t+1}) \tilde b^k(X_{t+1})'\| > T_n\} \\
& & - {\mathbb{E}}[n^{-1} \tilde b^k(X_t) m(X_t,X_{t+1}) \tilde b^k(X_{t+1})' {1\!\mathrm{l}}\{ \|\tilde b^k(X_t) m(X_t,X_{t+1}) \tilde b^k(X_{t+1})'\| > T_n\} ] \,.\end{aligned}$$ Note ${\mathbb{E}}[\Xi_{1,t,n}] = 0$ and $\|\Xi_{1,t,n}\| \leq 2 n^{-1} T_n$ by construction. Let $S^{k-1} = \{u \in {\mathbb{R}}^k : \|u\| = 1\}$. For any $u,v \in S^{k-1}$ and any $0 \leq t,s \leq n-1$, we have: $$\begin{aligned}
|u'{\mathbb{E}}[ \Xi_{1,t,n}^{\phantom \prime} \Xi_{1,s,n}' ]v|
& \leq n^{-2} \xi_k^2 {\mathbb{E}}[|u'\tilde b^k(X_t) m(X_t,X_{t+1}) m(X_s,X_{s+1}) \tilde b^k(X_s)'v|] \\
& \leq n^{-2} \xi_k^2 {\mathbb{E}}[|m(X_t,X_{t+1})|^r]^{1/r} {\mathbb{E}}[|m(X_s,X_{s+1})|^r]^{1/r} \\
& \quad \quad \times {\mathbb{E}}[|(u'\tilde b^k(X_t))|^q]^{1/q} {\mathbb{E}}[|(v'\tilde b^k(X_s))|^q]^{1/q} \end{aligned}$$ by Hölder’s inequality with $q$ chosen such that $1 = \frac{2}{r} + \frac{2}{q}$. Using the fact that $\| u\| = 1$ and $E[(\tilde b^k(X_0)'u)^2] = \|u\|^2 = 1$ (and similarly for $v$), we obtain: $${\mathbb{E}}[|(u'\tilde b^k(X_t))|^q]^{1/q} \leq ( \xi_k^{q-2} {\mathbb{E}}[(u'\tilde b^k(X_t))^2])^{1/q} = \xi_k^{1-2/q}$$ and hence: $$|u'{\mathbb{E}}[ \Xi_{1,t,n}^{\phantom \prime} \Xi_{1,s,n}' ]v| \leq \frac{\xi_k^{2+4/r} }{n^2} {\mathbb{E}}[|m(X_0,X_1)|^r]^{2/r}\,.$$ As this bound holds uniformly for $u,v \in S^{k-1}$, it follows from the variational characterization of the operator norm that: $$\|{\mathbb{E}}[ \Xi_{1,t,n}^{\phantom \prime} \Xi_{1,s,n}' ]\| = \sup_{u,v \in S^{k-1}}|u'{\mathbb{E}}[ \Xi_{1,t,n}^{\phantom \prime} \Xi_{1,s,n}' ]v| = O(\xi_k^{2+4/r}/n^2)\,.$$ This bound holds uniformly for $0 \leq t,s \leq n-1$, and also holds for $\|{\mathbb{E}}[ \Xi_{1,t,n}^{\prime} \Xi_{1,s,n}^{\phantom \prime} ]\|$. Applying Corollary 4.2 of [@ChenChristensen-reg] yields: $$\bigg\| \sum_{t=0}^{n-1} \Xi_{1,t,n} \bigg\| = O_p ( \xi_k^{1+2/r}(\log n)/\sqrt n )$$ provided $T_n (\log n)/n = o(\xi_k^{1+2/r}/\sqrt n)$.
Now consider the remaining term. If $r = \infty$ then we can set $\Xi_{2,t,n} \equiv 0$ by taking $T_n = C\xi_k^2$ for sufficiently large $C$. Now suppose $2 < r < \infty$. By the triangle and Jensen inequalities: $$\begin{aligned}
& {\mathbb{E}} \bigg[ \bigg\| \sum_{t=0}^{n-1} \Xi_{2,t,n} \bigg\| \bigg] \\
& \leq \frac{2}{n} \sum_{t=0}^{n-1} {\mathbb{E}}[ \| \tilde b^k(X_t) m(X_t,X_{t+1}) \tilde b^k(X_{t+1})' \|{1\!\mathrm{l}}\{ \|\tilde b^k(X_t) m(X_t,X_{t+1}) \tilde b^k(X_{t+1})'\| > T_n\} ] \\
& \leq \frac{2}{n T_n^{r-1}} \sum_{t=0}^{n-1} {\mathbb{E}}[ \| \tilde b^k(X_t) m(X_t,X_{t+1}) \tilde b^k(X_{t+1})' \|^r {1\!\mathrm{l}}\{ \|\tilde b^k(X_t) m(X_t,X_{t+1}) \tilde b^k(X_{t+1})'\| > T_n\} ] \\
& \leq \frac{2\xi_k^{2r}}{T_n^{r-1}} {\mathbb{E}}[|m(X_0,X_1)|^r]\,.\end{aligned}$$ So by Markov’s inequality: $$\bigg\| \sum_{t=0}^{n-1} \Xi_{2,t,n} \bigg\| = O_p ( \xi_k^{2r}/T_n^{r-1} )\,.$$ We choose $T_n$ so that: $$\frac{\xi_k^{2r}}{T_n^{r-1}} = \frac{\xi_k^{1+2/r}(\log n)}{\sqrt n}$$ so that $\| \sum_{t=0}^{n-1} \Xi_{2,t,n}\| = O_p ( \xi_k^{1+2/r}(\log n)/\sqrt n )$. Solving the above display for $T_n$, we obtain $T_n = \xi_k^{2+2/r}(\xi_k (\log n)/\sqrt n)^{-1/(r-1)}$. The condition $T_n (\log n)/n = o(\xi_k^{1+2/r}/\sqrt n)$ is, with this choice of $T_n$, equivalent to $(\xi_k (\log n)/\sqrt n )^{(r-2)/(r-1)} = o(1)$. This condition is therefore implied by the conditions $\xi_k (\log n)/\sqrt n = o(1)$ and $r > 2$.
Part (b) follows from part (a) by definition of the operator norm.
Follows from Lemmas \[lem:matcgce\](a), \[lem:G:beta\], and \[lem:M:beta\].
\[lem:G:rho\] Let the conditions of Lemma \[lem:rho:1\] hold. Then:
1. $\|{\widehat{{\mathbf{G}}}}^o - {\mathbf{I}}\| = O_p ( \xi_k \sqrt{ k/n} )$
2. $\|({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}})\tilde c_k\| = O_p ( \xi_k /\sqrt n )$ and $\|\tilde c_k^{* \prime}({\widehat{{\mathbf{G}}}}^o - I)\| = O_p ( \xi_k /\sqrt n )$.
Parts (a) and (b) follow by similar arguments to the proof of Lemmas 4.8 and 4.12 of [@Gobetetal].
\[lem:M:rho\] Let the conditions of Lemma \[lem:rho:1\] hold. Then:
1. $\|{\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o\| = O_p ( \xi_k^{(r+2)/r}\sqrt {k/n} )$
2. $\|({\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o)\tilde c_k\| = O_p ( \xi_k^{(r+2)/r}/\sqrt n )$ and $\|\tilde c_k^{* \prime}({\widehat{{\mathbf{M}}}}^{o} - {\mathbf{M}}^{o})\| = O_p ( \xi_k^{(r+2)/r}/\sqrt n )$.
We use similar arguments to Lemmas 4.8 and 4.9 of [@Gobetetal]. For part (b), by the covariance inequality for exponentially rho-mixing processes, there is a finite positive constant $C$ (depending on the rho-mixing coefficients) such that: $${\mathbb{E}}[ \|({\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o)\tilde c_k\|^2] \leq \frac{C}{n} \sum_{l=1}^k {\mathbb{E}} \left[ \tilde b_{kl}(X_t)^2 m(X_t,X_{t+1})^2 (\tilde b^k(X_{t+1})'\tilde c_k)^2 \right]\,.$$ It follows by definition of $\xi_k$, Cauchy-Schwarz, and Hölder’s inequality that: $$\begin{aligned}
{\mathbb{E}}[ \|({\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o)\tilde c_k\|^2] & \leq \frac{C \xi_k^2}{n} {\mathbb{E}}[|m(X_0,X_1)|^r]^{2/r} {\mathbb{E}}[ (\tilde b^k(X_0)' \tilde c_k)^{2r/(r-2)}]^{(r-2)/r} \notag \\
& \leq \frac{C \xi_k^{(2r+4)/r}}{n} {\mathbb{E}}[|m(X_0,X_1)|^r]^{2/r} \label{e-mhatvec}\end{aligned}$$ where the final line is because $\|\tilde c_k\| = 1$ and $E[(\tilde b^k(X_0)'\tilde c_k)^2] = \|\tilde c_k\|^2 = 1$. This proves $\|({\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o)\tilde c_k\| = O_p ( \xi_k^{(r+2)/r}/\sqrt n )$. The result for $\tilde c^*_k$ follows similarly.
For part (a), let $u_1,\ldots,u_k$ be an orthonormal basis for ${\mathbb{R}}^k$. Using the fact that the Frobenius norm $\|\cdot\|_F$ dominates the $L^2$ norm, we have: $$\begin{aligned}
{\mathbb{E}}[ \|{\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o\|^2] & \leq & {\mathbb{E}}[ \|{\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o\|^2_F] \\
& = & \sum_{l=1}^k {\mathbb{E}}[ \|({\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o)u_l\|^2] \\
& \leq & \frac{Ck \xi_k^{(2r+4)/r}}{n} {\mathbb{E}}[|m(X_0,X_1)|^r]^{2/r} \end{aligned}$$ by similar arguments to (\[e-mhatvec\]). The result follows by Chebyshev’s inequality.
Follows from Lemmas \[lem:matcgce\](a), \[lem:G:rho\], and \[lem:M:rho\].
The convergence rate of ${\widehat{{\mathbf{G}}}}$ is as in Lemma \[lem:beta:1\]. First write: $$\begin{aligned}
{\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o
& = \left( \frac{1}{n} \sum_{t=0}^{n-1} \tilde b^k(X_t) \Big( m(X_t,X_{t+1};\hat \alpha) - m(X_t,X_{t+1};\alpha_0) \Big) \tilde b^{k}(X_{t+1}) \right) \\
& \quad + \left( \frac{1}{n} \sum_{t=0}^{n-1} \tilde b^k(X_t) m(X_t,X_{t+1};\alpha_0) \tilde b^{k}(X_{t+1}) - {\mathbf{M}}^o \right)
=: {\widehat{\Delta}}_{1,k} + {\widehat{\Delta}}_{2,k}\,.\end{aligned}$$ Lemma \[lem:M:beta\](a) yields ${\widehat{\Delta}}_{2,k} = O_p ( \xi_k^{(r+2)/r} (\log n)/\sqrt n )$. For the convergence rate for the remaining term, condition (a) implies that $\hat \alpha \in N$ wpa1. A mean-value expansion (using conditions (a) and (b)) then yields: $$\begin{aligned}
\left\| {\widehat{\Delta}}_{1,k} \right\|
& = \left\| \frac{1}{n} \sum_{t=0}^{n-1} \tilde b^k(X_t) \tilde b^{k}(X_{t+1})' \left( \frac{\partial m(X_t,X_{t+1};\tilde \alpha)}{\partial \alpha'} (\hat \alpha - \alpha_0) \right) \right\| \end{aligned}$$ wpa1, for $\tilde \alpha$ in the segment between $\hat \alpha$ and $\alpha_0$. Let $S^{k-1} = \{u \in {\mathbb{R}}^k : \|u\| = 1\}$. Therefore, wpa1 we have: $$\begin{aligned}
\left\| {\widehat{\Delta}}_{1,k} \right\| & = \sup_{u,v \in S^{k-1}} \left| \frac{1}{n} \sum_{t=0}^{n-1} (u'\tilde b^k(X_t) ) (v' \tilde b^{k}(X_{t+1}) \left( \frac{\partial m(X_t,X_{t+1};\tilde \alpha)}{\partial \alpha'} (\hat \alpha - \alpha_0) \right) \right| \\
& \leq \xi_k \times \left( \sup_{u \in S^{k-1}} \frac{1}{n} \sum_{t=0}^{n-1} |u'\tilde b^k(X_t) | \bar m(X_t,X_{t+1}) \right) \times \|\hat \alpha - \alpha_0 \| \\
& \leq \xi_k \times \left( \sup_{u \in S^{k-1}} u' {\widehat{{\mathbf{G}}}}^o u \right)^{1/2} \times \left( \frac{1}{n} \sum_{t=0}^{n-1} \sup_{\alpha \in N} \left\| \frac{\partial m(X_t,X_{t+1};\alpha)}{\partial \alpha} \right\|^2 \right)^{1/2} \times \|\hat \alpha - \alpha_0 \| \end{aligned}$$ where the first line is because $\|{\mathbf{A}}\| = \sup_{u,v \in S^{k-1}} |u'{\mathbf{A}} v|$ and the second and third lines are by several applications of the Cauchy-Schwarz inequality. Finally, notice that $\sup_{u \in S^{k-1}} u' {\widehat{{\mathbf{G}}}}^o u = \| {\widehat{{\mathbf{G}}}}^o \| = 1 + o_p(1)$ by Lemma \[lem:G:beta\](a) and condition (c), and: $$\frac{1}{n} \sum_{t=0}^{n-1} \sup_{\alpha \in N} \left\| \frac{\partial m(X_t,X_{t+1};\alpha)}{\partial \alpha} \right\|^2 = O_p(1)$$ by the ergodic theorem and condition (b). The bounds for $\eta_{n,k}$ and $\eta_{n,k}^*$ follow by Lemma \[lem:matcgce\](a).
The convergence rate for ${\widehat{{\mathbf{G}}}}$ is from Lemma \[lem:beta:1\]. To establish the rate for ${\widehat{{\mathbf{M}}}}$, it suffices to bound: $${\widehat{\Delta}}_{k} := \frac{1}{n} \sum_{t=0}^{n-1} \tilde b^k(X_t) \Big( m(X_t,X_{t+1};\hat \alpha) - m(X_t,X_{t+1};\alpha_0) \Big) \tilde b^{k}(X_{t+1}) \,.$$ Let $h_\alpha(x_0,x_1) = m(x_0,x_1; \alpha) - m(x_0,x_1;\alpha_0)$ and let $\{T_n : n \geq 1\}$ be a sequence of positive constants to be defined. Define the functions: $$\begin{aligned}
h_\alpha^{trunc}(x_0,x_1) & = h_\alpha(x_0,x_1) {1\!\mathrm{l}}\{ \| \tilde b^k(x_0) \tilde b^k(x_1)' E(x_0,x_1)\| \leq T_n\} \\
h_\alpha^{tail}(x_0,x_1) & = h_\alpha(x_0,x_1) {1\!\mathrm{l}}\{ \| \tilde b^k(x_0) \tilde b^k(x_1)' E(x_0,x_1)\| > T_n\} \,.\end{aligned}$$ Then by the triangle inequality: $$\begin{aligned}
\| {\widehat{\Delta}}_k\| \leq & \,\sup_{\alpha \in {\mathcal{A}}} \left\| \frac{1}{n} \sum_{t=0}^{n-1} \tilde b^k(X_t) h^{trunc}_\alpha(X_t,X_{t+1}) \tilde b^{k}(X_{t+1}) - {\mathbb{E}}[ \tilde b^k(X_t) h^{trunc}_\alpha(X_t,X_{t+1}) \tilde b^{k}(X_{t+1}) ] \right\| \\
& + \sup_{\alpha \in {\mathcal{A}}} \left\| \frac{1}{n} \sum_{t=0}^{n-1} \tilde b^k(X_t) h^{tail}_\alpha(X_t,X_{t+1}) \tilde b^{k}(X_{t+1})\right\| \\
& + \sup_{\alpha \in {\mathcal{A}}} \left\| {\mathbb{E}}[ \tilde b^k(X_t) h^{tail}_\alpha(X_t,X_{t+1}) \tilde b^{k}(X_{t+1}) ] \right\| + \left\| {\mathbb{E}}[ \tilde b^k(X_t) h_{\hat \alpha}(X_t,X_{t+1}) \tilde b^{k}(X_{t+1})] \right\| \\
=: &\, {\widehat{T}}_1 + {\widehat{T}}_2 + {\widehat{T}}_3 + {\widehat{T}}_4 \,.\end{aligned}$$ To control ${\widehat{T}}_1$, consider the class of functions: $${\mathcal{H}}_{n,k} = \{ (c_0'\tilde b^k(x_0))(c_1'\tilde b^k(x_1))h^{trunc}_\alpha(x_0,x_1) : c_0,c_1 \in S^{k-1}, \alpha \in {\mathcal{A}}\}$$ where $S^{k-1}$ is the unit sphere in ${\mathbb{R}}^k$. Each function in ${\mathcal{H}}_{n,k}$ is bounded by $T_n$. Moreover: $${\widehat{T}}_1 \leq \frac{1}{\sqrt n } \times \sup_{h \in {\mathcal{H}}_{n,k}} | {\mathcal{Z}}_n(h)|$$ where ${\mathcal{Z}}_n$ is the centered empirical process on ${\mathcal{H}}_{n,k}$. By Theorem 2 of [@DoukhanMassartRio]: $$\label{e:dmr:bd}
{\mathbb{E}}[{\textstyle \sup_{h \in {\mathcal{H}}_{n,k}} | {\mathcal{Z}}_n(h)|}] = O \left( \varphi(\sigma_{n,k}) + \frac{T_n q \varphi^2(\sigma_{n,k})}{\sigma_{n,k}^2 \sqrt n} + \sqrt n T_n \beta_q \right)$$ where $q$ is a positive integer, $\sigma_{n,k} \geq \sup_{h \in {\mathcal{H}}_{n,k}} \|h\|_{2,\beta}$ for the norm $\|\cdot\|_{2,\beta}$ is defined on p. 400 of [@DoukhanMassartRio], and: $$\varphi(\sigma) = \int_0^\sigma \sqrt{\log N_{[\,\,]}(u,{\mathcal{H}}_{n,k},\|\cdot\|_{2,\beta})} \, {\mathrm{d}} u \,.$$ Exponential $\beta$-mixing and display (a) in Lemma 2 of [@DoukhanMassartRio] (with $\phi(x) = x^{v}$) imply: $$\label{e:beta:norm}
\|\cdot\|_{2,\beta} \leq C \|\cdot\|_{2v} \quad \mbox{on $L^{2v}$}$$ for some constant $C < \infty$ that depends only on the $\beta$-mixing coefficients. Therefore: $$\sup_{h \in {\mathcal{H}}_{n,k}} \|h\|_{2,\beta} \leq C \sup_{h \in {\mathcal{H}}_{n,k}} \|h\|_{2v} \leq C \xi_k^2 \|E\|_{2v}$$ where $\|E\|_{2v}$ is finite by conditions (a) and (b). Take $\sigma_{n,k} = C \xi_k^2 \|E\|_{2v}$.
Define ${\mathcal{H}}_{n,k}^* = \{ h/(\xi_k^2 E) : h \in {\mathcal{H}}_{n,k}\}$ and ${\mathcal{H}}_{n}^* = \{ h^{trunc}_\alpha/E : \alpha \in {\mathcal{A}}\}$. Each function in ${\mathcal{H}}_{n,k}^*$ is of the form: $$\frac{c_0'\tilde b^k(x_0)}{\xi_k} \times \frac{c_1'\tilde b^k(x_1)}{\xi_k} \times\frac{h^{trunc}_\alpha(x_0,x_1)}{E}$$ For $c_0,c_1 \in S^{k-1}$ we have: $$\begin{aligned}
|c_0' \tilde b^k(x)/\xi_k - c_1' \tilde b^k(x)/\xi_k| & \leq \|\tilde b^k(x)\|/\xi_k \times \|c_0 - c_1\|\end{aligned}$$ where: $$\big\| \|\tilde b^k(x)\|/\xi_k \big\|_{q} \leq (k/\xi_k^2)^{1/q}$$ for any $q > 2$. Let ${\mathcal{B}}_k^* = \{ (c'\tilde b^k)/\xi^k : c \in S^{k-1}\}$. By Theorem 2.7.11 of [@vdVW], we have: $$N_{[\,\,]}(u,{\mathcal{B}}_k^*,\|\cdot\|_{q}) \leq N\big (u/(2(k/\xi_k^2)^{1/q}),S^{k-1},\|\cdot\| \big) \leq \bigg( \frac{4 (k/\xi_k^2)^{1/q} }{u} + 1\bigg)^k$$ since the $\varepsilon$-covering number for the unit ball in ${\mathbb{R}}^k$ is bounded above by $(2/\varepsilon + 1)^k$. It follows by Lemma 9.25(ii) in [@Kosorok] that: $$\begin{aligned}
N_{[\,\,]}(3u,{\mathcal{H}}_{n,k}^*,\|\cdot\|_{q})
& \leq \bigg( \frac{4 (k/\xi_k^2)^{1/q} }{u} + 1\bigg)^{2k} N_{[\,\,]}(u,{\mathcal{H}}_n^*,\|\cdot\|_{q})\,. \label{e:bracket:euclid}\end{aligned}$$ Let $[f_l,f_u]$ be a $\varepsilon$-bracket for ${\mathcal{H}}_{n,k}^*$ under $\|\cdot\|_{\frac{4vs}{2s-v}}$. Then $[ \xi_k^2 E f_l, \xi_k^2 E f_u]$ is a $(\xi_k^2 \| E \|_{4s}\varepsilon)$-bracket for the $\|\cdot\|_{2v}$ norm for ${\mathcal{H}}_{n,k}$, because: $$\| \xi_k^2 E (f_u - f_l)\|_{2v} = \xi_k^2 \| E (f_u - f_l)\|_{2v} \leq \xi_k^2 \| E \|_{4s} \| f_u - f_l\|_{\frac{4vs}{2s-v}}$$ by Hölder’s inequality. Taking $q = \frac{4vs}{2s-v}$ in (\[e:bracket:euclid\]) and using the fact that truncation of ${\mathcal{M}}^*$ doesn’t increase its bracketing entropy, we obtain: $$\begin{aligned}
N_{[\,\,]} ( u ,{\mathcal{H}}_{n,k}, \|\cdot\|_{2v})
& \leq N_{[\,\,]} \Big( \frac{u}{\xi_k^2 \| E \|_{4s}} ,{\mathcal{H}}_{n,k}^*, \|\cdot\|_{\frac{4vs}{2s-v}} \Big) \notag \\
& \leq \Big( \frac{12 \|E\|_{4s} \xi_k^{2} (k/\xi_k^2)^{\frac{2s-v}{4vs}}}{u} + 1\Big)^{2k} N_{[\,\,]}\Big( \frac{u}{3 \xi_k^2 \|E\|_{4s}},{\mathcal{M}}^*,\|\cdot\|_{\frac{4vs}{2s-v}}\Big) \,. \label{e:entropy:Hnk}\end{aligned}$$ Finally, by displays (\[e:beta:norm\]) and (\[e:entropy:Hnk\]) and condition (b): $$\begin{aligned}
\varphi(\sigma) & = \int_0^\sigma \sqrt{\log N_{[\,\,]}(u,{\mathcal{H}}_{n,k},\|\cdot\|_{2,\beta})} \, {\mathrm{d}} u \\
& \leq \int_0^\sigma \sqrt{\log N_{[\,\,]}(u/C,{\mathcal{H}}_{n,k},\|\cdot\|_{2v})} \, {\mathrm{d}} u \\
& \leq \int_0^\sigma \sqrt{2 k \log \Big(1+ 12 C \|E\|_{4v} \xi_k^{2} (k/\xi_k^2)^{\frac{2s-v}{4vs}} /u \Big)} \, {\mathrm{d}} u + {\mathrm{const}}^{1/2} (3C \xi_k^2 \|E\|_{4s})^\zeta \frac{\sigma^{1-\zeta}}{1-\zeta} \\
& \leq 12\sqrt{2k} C \|E\|_{4s} \xi_k^{2} (k/\xi_k^2)^{\frac{2s-v}{4vs}} \int_0^{\sigma/(12C \|E\|_{4s} \xi_k^{2} (k/\xi_k^2)^{\frac{2s-v}{4vs}} )} \sqrt{ \log (1 + 1/u )} \, {\mathrm{d}} u \\
& \quad + {\mathrm{const}}^{1/2} (3 \xi_k^2 \|E\|_{4s})^\zeta \frac{\sigma^{1-\zeta}}{1-\zeta} \\
& \leq 24 \sqrt{2k} C \|E\|_{4s} \xi_k^{2} (k/\xi_k^2)^{\frac{2s-v}{4vs}} \bigg( \frac{\sigma}{12C \|E\|_{4s} \xi_k^{2} (k/\xi_k^2)^{\frac{2s-v}{4vs}} } \vee 1 \bigg) \\
& \quad + {\mathrm{const}}^{1/2} (3 \xi_k^2 \|E\|_{4s})^\zeta \frac{\sigma^{1-\zeta}}{1-\zeta} \,.\end{aligned}$$ Using the fact that $k \leq \xi_k^2$ and $\sigma_{n,k} = C \xi_k^2 \|E\|_{2v}$, we obtain $\varphi(\sigma_{n,k}) = O ( \xi_k^2 \sqrt k )$. Substituting into (\[e:dmr:bd\]) and using Markov’s inequality: $${\widehat{T}}_1 = O_p \bigg( \frac{ \xi_k^2 \sqrt k }{\sqrt n} + \frac{T_n q k }{ n} + T_n \beta_q \bigg) \,.$$
Similar arguments to the proof of Lemma \[lem:M:beta\] yield ${\widehat{T}}_2 = O_p ( \xi_k^{8s}/T_n^{4s-1}) $ and ${\widehat{T}}_3 = O ( \xi_k^{8s}/T_n^{4s-1})$. We choose $T_n$ so that: $$\frac{\xi_k^{8s}}{T_n^{4s-1}} = \frac{ \xi_k^2 \sqrt k }{\sqrt n}$$ which makes $T_n \to \infty$ (since $k/n = o(1)$). We will also choose $q = C_0 \log n $ for sufficiently large $C_0$, so that $T_n \beta_q = o(\xi_k^2 \sqrt {k/ n})$. This also ensures that the term $T_n q k/n = O (\xi_k^2 \sqrt {k/n})$. Therefore, ${\widehat{T}}_1$, ${\widehat{T}}_2$, and ${\widehat{T}}_3$ are all $O_p(\xi_k^2 \sqrt{k/n})$.
For the remaining term, by conditions (c) and (d) of the lemma we may deduce: $$\begin{aligned}
{\widehat{T}}_4 \leq \ell^*(\hat \alpha)
& = \frac{1}{\sqrt n } \sqrt n \dot \ell_{\alpha_0}^*[\hat \alpha - \alpha_0] + O( \|\hat \alpha - \alpha_0\|_{{\mathcal{A}}}^2 ) \\
& = O_p(n^{-1/2}) + O( \|\hat \alpha - \alpha_0\|_{{\mathcal{A}}}^2 ) = O_p(n^{-1/2}) \,.\end{aligned}$$ Combining the above bounds for ${\widehat{T}}_1,\ldots,{\widehat{T}}_4$ yields ${\widehat{\Delta}}_k = O_p( \xi_k^2 \sqrt {k/ n} )$. The bounds for $\eta_{n,k}$ and $\eta_{n,k}^*$ follow by Lemma \[lem:matcgce\](a).
Proofs for Appendix \[ax:est:mat:fp\]
-------------------------------------
The convergence rate for ${\widehat{{\mathbf{G}}}}^o$ is from Lemma 2.2 of [@ChenChristensen-reg]. For ${\widehat{{\mathbf{T}}}}^o$, let $\{T_n : n \geq 1\}$ be a sequence of positive constants to be defined. Also define: $$\begin{aligned}
G^{trunc}_{t+1} & = G_{t+1}^{1-\gamma} {1\!\mathrm{l}}\{ \| \tilde b^k(x_t) \| \|\tilde b^k(x_1)\|^\beta |G_{t+1}^{1-\gamma}| \leq T_n\} \\
G^{tail}_{t+1} & = G_{t+1}^{1-\gamma} {1\!\mathrm{l}}\{ \| \tilde b^k(x_t) \| \|\tilde b^k(x_1)\|^\beta |G_{t+1}^{1-\gamma}| > T_n\} \,.\end{aligned}$$ We then have: $$\begin{aligned}
\sup_{v : \| v\| \leq c} \| {\widehat{{\mathbf{T}}}}^ov - {\mathbf{T}}^o v\|
& \leq \sup_{v : \| v\| \leq c} \left\| \frac{1}{n} \sum_{t=0}^{n-1} \tilde b^k(X_t) G^{trunc}_{t+1} |\tilde b^{k}(X_{t+1})'v|^\beta - {\mathbb{E}}[ \tilde b^k(X_t) G^{trunc}_{t+1} |\tilde b^{k}(X_{t+1})'v|^\beta ] \right\| \\
& \quad + \sup_{v : \| v\| \leq c} \left\| \frac{1}{n} \sum_{t=0}^{n-1} \tilde b^k(X_t) G^{tail}_{t+1} |\tilde b^{k}(X_{t+1})'v|^\beta \right\| \\
& \quad + \sup_{v : \| v\| \leq c} \left\| {\mathbb{E}}[ \tilde b^k(X_t) G^{tail}_{t+1} |\tilde b^{k}(X_{t+1})'v|^\beta ] \right\| \quad =: \quad {\widehat{T}}_1 + {\widehat{T}}_2 + {\widehat{T}}_3 \,.\end{aligned}$$ Let $\tilde b_i^k$ denote the $i$th element of $\tilde b^k$. Also define $${\mathcal{H}}_{n,k} = \{ w' \tilde b^k_i(x_0) G_1^{trunc} |\tilde b^k(x_1)'v|^\beta : v \in {\mathbb{R}}^k, \|v\| \leq c, w \in S^{k-1} \}\,.$$ Observe that ${\widehat{T}}_1 \leq n^{-1/2} \times \sup_{h \in {\mathcal{H}}_{n,k}} | {\mathcal{Z}}_n(h) |$ where ${\mathcal{Z}}_n$ is the centered empirical process on ${\mathcal{H}}_{n,k}$. By construction, each $h \in {\mathcal{H}}_{n,k}$ is uniformly bounded by $c^\beta T_n$. Theorem 2 of [@DoukhanMassartRio] gives the bound: $$\label{e:dmr:bd:fp}
{\mathbb{E}}[{\textstyle \sup_{h \in {\mathcal{H}}_{n,k}} | {\mathcal{Z}}_n(h)|}] = O \left( \varphi(\sigma_{n,k}) + \frac{c^\beta T_n q \varphi^2(\sigma_{n,k})}{\sigma_{n,k}^2 \sqrt n} + \sqrt nc^\beta T_n \beta_q \right)$$ where $q$ is a positive integer, $\sigma_{n,k} \geq \sup_{h \in {\mathcal{H}}_{n,k}} \|h\|_{2,\beta}$ where the norm $\|\cdot\|_{2,\beta}$ is defined on p. 400 of [@DoukhanMassartRio], and: $$\varphi(\sigma) = \int_0^\sigma \sqrt{\log N_{[\,\,]}(u,{\mathcal{H}}_{n,k},\|\cdot\|_{2,\beta})} \, {\mathrm{d}} u \,.$$ Exponential $\beta$-mixing and display (a) in Lemma 2 of [@DoukhanMassartRio] (with $\phi(x) = x^s$), imply: $$\label{e:beta:norm:fp}
\|\cdot\|_{2,\beta} \leq C \|\cdot\|_{2s} \quad \mbox{on $L^{2s}$}$$ for some constant $C<\infty$ that depends only on the $\beta$-mixing coefficients. Therefore: $$\sup_{h \in {\mathcal{H}}_{n,k}} \|h\|_{2,\beta} \leq C \sup_{h \in {\mathcal{H}}_{n,k}} \|h\|_{2s} \leq C c^\beta \|G^{1-\gamma}\|_{2s} \xi_k^{1+\beta}$$ where $\|G^{1-\gamma}\|_{2s}$ is finite by condition (b). Set $\sigma_{n,k} = C c^\beta \|G^{1-\gamma}\|_{2s} \xi_k^{1+\beta}$.
To calculate the bracketing entropy integral, first fix $q > 2$ and let $w_1,\ldots,w_{N_1}$ be a $\varepsilon$-cover for $S^{k-1}$ and $v_1,\ldots,v_{N_2}$ be a $\varepsilon^{1/\beta}$-cover for $\{ v \in {\mathbb{R}}^k : \|v\| \leq c\}$. For any $ w \in S^{k-1}$ and $v \in \{ v \in {\mathbb{R}}^k : \|v\| \leq c\}$ there exist $v_i \in \{v_1,\ldots,v_{N_1}\}$ and $w_j \in \{w_1,\ldots,w_{N_2}\}$ such that: $$\begin{aligned}
& w_j' \tilde b^k_i(x_0) G_1^{trunc} |\tilde b^k(x_1)'v_i|^\beta - 2 \varepsilon \Big( (1+c^\beta)\| \tilde b^k(x_0) \| \| \tilde b^k(x_1)\|^\beta | G_1^{trunc}| \Big) \\
& \leq w' \tilde b^k_i(x_0) G_1^{trunc} |\tilde b^k(x_1)'v|^\beta \\
& \leq w_j' \tilde b^k_i(x_0) G_1^{trunc} |\tilde b^k(x_1)'v_i|^\beta + 2 \varepsilon \Big( (1+c^\beta)\| \tilde b^k(x_0) \| \| \tilde b^k(x_1)\|^\beta | G_1^{trunc}| \Big) \end{aligned}$$ where: $$\left\| 4 \varepsilon \Big( (1+c^\beta)\| \tilde b^k(x_0) \| \| \tilde b^k(x_1)\|^\beta | G_1^{trunc}| \Big)\right\|_{2s} \leq 4 \varepsilon (1+c^\beta) \xi_k^{1+\beta} \| G^{1-\gamma}\|_{2s} =: K \varepsilon \xi_k^{1+\beta}\,.$$ Therefore: $$N_{[\,\,]}\big(u ,{\mathcal{H}}_{n,k}, \|\,\cdot\,\|_{2s} \big) \leq \Big(\frac{2K\xi_k^{1+\beta} }{u} + 1\Big)^k \Big(\frac{2(cK\xi_k^{1+\beta})^{1/\beta} }{u^{1/\beta}} + 1\Big)^k$$ since the $\varepsilon$-covering number for the unit ball in ${\mathbb{R}}^k$ is bounded above by $(2/\varepsilon + 1)^k$.
It follows by (\[e:beta:norm:fp\]) and the above display that: $$\begin{aligned}
\varphi(\sigma) & = \int_0^\sigma \sqrt{\log N_{[\,\,]}(u,{\mathcal{H}}_{n,k},\|\cdot\|_{2,\beta})} \, {\mathrm{d}} u \\
& \leq \int_0^\sigma \sqrt{\log N_{[\,\,]}(u/C,{\mathcal{H}}_{n,k},\|\cdot\|_{2s})} \, {\mathrm{d}} u \\
& \leq \int_0^\sigma \sqrt{k \log \big(1+ 2 C K \xi_k^{1+\beta}/u \big)} \, {\mathrm{d}} u + \int_0^\sigma \sqrt{k \log \big(1+ 2 (c CK \xi_k^{1+\beta}/u)^{1/\beta} \big)} \, {\mathrm{d}} u \\
& \leq 4 C K \sqrt k \xi_k^{1+\beta} \bigg( \frac{\sigma}{4CK\xi_k^{1+\beta}} \vee 1 \bigg) \\
& \quad \quad + 2^\beta c C K \sqrt k \xi_k^{1+\beta} \int_0^{\sigma/(2^\beta c C K \xi_k^{1+\beta})} \sqrt{\log(1+u^{-1/\beta})} \, {\mathrm{d}} u \\
& \leq 4 C K \sqrt k \xi_k^{1+\beta} \bigg( \frac{\sigma}{4CK\xi_k^{1+\beta}} \vee 1 \bigg) + 2^\beta c C K \sqrt k \xi_k^{1+\beta} \frac{\big( \sigma/(2^\beta c C K \xi_k^{1+\beta}) \big)^{1-\frac{1}{2\beta}}}{1-\frac{1}{2\beta}} \,.\end{aligned}$$ where the final line is because $\log(1+u^{-1/\beta}) \leq u^{-1/\beta}$. Since $\sigma_{n,k} = C c^\beta \|G^{1-\gamma}\|_{2s} \xi_k^{1+\beta}$, we obtain $\varphi(\sigma_{n,k}) = O(\xi_k^{1+\beta} \sqrt k)$. Substituting into (\[e:dmr:bd\]) and using Markov’s inequality: $${\widehat{T}}_1 = O_p \bigg( \frac{ \xi_k^{1+\beta} \sqrt k }{\sqrt n} + \frac{T_n q k }{ n} + T_n \beta_q \bigg) \,.$$
By similar arguments to the proof of Lemma \[lem:M:beta\], we have ${\widehat{T}}_2 = O_p ( \xi_k^{(1+\beta)2s}/T_n^{2s-1}) $ and ${\widehat{T}}_3 = O ( \xi_k^{(1+\beta)2s}/T_n^{2s-1})$. We choose $T_n$ so that: $$\frac{\xi_k^{(1+\beta)2s}}{T_n^{2s-1}} = \frac{ \xi_k^{1+\beta} \sqrt k }{\sqrt n}$$ which makes $T_n \to \infty$ (since $k/n = o(1)$). We will also choose $q = C_0 \log n $ for sufficiently large $C_0$, so that $T_n \beta_q = o(\xi_k^{1+\beta} \sqrt {k/ n})$. This also ensures that the term $T_n q k/n = O (\xi_k^{1+\beta} \sqrt {k/n})$. Therefore, ${\widehat{T}}_1$, ${\widehat{T}}_2$, and ${\widehat{T}}_3$ are all $O_p(\xi_k^{1+\beta} \sqrt{k/n})$.
The expression for $\nu_{n,k}$ now follows from display (\[e:gtineq\]) and the rates for ${\widehat{{\mathbf{G}}}}^o$ and ${\widehat{{\mathbf{T}}}}^o$.
Proofs for Appendix \[ax:inf\]
------------------------------
Step 1: Normalize $c_k$ and $c_k^*$ so that $\|c_k\|_{{\mathbf{G}}} = \| c_k^*\|_{{\mathbf{G}}} = 1$. We first show that: $$\hat \rho - \rho_k = \frac{1}{\langle c_k^{\phantom *} , c^*_k \rangle_{{\mathbf{G}}}} c_k^{*\prime} {\mathbf{G}} ({\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} - {\mathbf{G}}^{-1} {\mathbf{M}}) c_k + O_p( ( \eta_{n,k} \vee \eta_{n,k,1} \vee \eta_{n,k,2})) \times O_p(\eta_{n,k})$$ Notice that dividing by $\langle c_k^{\phantom *} , c^*_k \rangle_{{\mathbf{G}}}$ establishes the result under the normalization $c_k^{* \prime} {\mathbf{G}} c_k = 1$ in the statement of the lemma.
Recall the definitions of ${\widehat{{\mathbf{P}}}}_k$ in display (\[e:hat:pk:def\]). Also define: $${\mathbf{P}}_k = \frac{1}{\langle c_k^{\phantom *} , c^*_k \rangle_{{\mathbf{G}}}} ( c_k^{\phantom *} \otimes_{{\mathbf{G}}} c^*_k)$$ Whenever $k \geq K$ (from Lemma \[lem:exist\]) and $\hat \rho$ is positive and simple (which it is wpa1 by Lemma \[lem:exist:hat\]) then $ {{\mathbf{P}}}_k$ and ${\widehat{{\mathbf{P}}}}_k$ are well defined and we have: $$\begin{aligned}
\hat \rho - \rho_k
& = & {\mathrm{trace}({ {\widehat{{\mathbf{P}}}}_k {\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} - {\mathbf{P}}_k {\mathbf{G}}^{-1} {\mathbf{M}} })} \\
& = & {\mathrm{trace}({ {\mathbf{P}}_k ({\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} -{\mathbf{G}}^{-1} {\mathbf{M}} ) + ({\widehat{{\mathbf{P}}}}_k - {\mathbf{P}}_k){\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} })} \\
& = & \frac{1}{\langle c_k^{\phantom *} , c^*_k \rangle_{{\mathbf{G}}}} c_k^{*\prime} {\mathbf{G}} ({\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} - {\mathbf{G}}^{-1} {\mathbf{M}}) c_k + {\mathrm{trace}({({\widehat{{\mathbf{P}}}}_k - {\mathbf{P}}_k){\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} })}\end{aligned}$$ by linearity of trace. Observe that: $$\begin{aligned}
& {\mathrm{trace}({({\widehat{{\mathbf{P}}}}_k - {\mathbf{P}}_k){\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} })} \notag \\
& =\hat \rho - \frac{1}{\langle c_k^{\phantom *} , c^*_k \rangle_{{\mathbf{G}}}}c_k^{*\prime} {\mathbf{G}} {\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} {\widehat{{\mathbf{P}}}}_k c_k + \frac{1}{\langle c_k^{\phantom *} , c^*_k \rangle_{{\mathbf{G}}}} c_k^{*\prime} {\mathbf{G}} {\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} ({\widehat{{\mathbf{P}}}}_k c_k - c_k ) \notag \\
& = \frac{1}{\langle c_k^{\phantom *} , c^*_k \rangle_{{\mathbf{G}}}} \Big(\hat \rho c_k^{*\prime} {\mathbf{G}} (c_k - {\widehat{{\mathbf{P}}}}_k c_k) + c_k^{*\prime} {\mathbf{G}} {\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} ({\widehat{{\mathbf{P}}}}_k c_k - c_k ) \Big) \notag \\
& = \frac{1}{\langle c_k^{\phantom *} , c^*_k \rangle_{{\mathbf{G}}}} \Big( (\hat \rho-\rho_k) c_k^{*\prime} {\mathbf{G}} (c_k - {\widehat{{\mathbf{P}}}}_k c_k) + \rho_k c_k^{*\prime} {\mathbf{G}} (c_k - {\widehat{{\mathbf{P}}}}_k c_k) + c_k^{*\prime} {\mathbf{G}} {\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} ({\widehat{{\mathbf{P}}}}_k c_k - c_k ) \Big) \notag \\
& = \frac{1}{\langle c_k^{\phantom *} , c^*_k \rangle_{{\mathbf{G}}}} \Big( (\hat \rho-\rho_k) c_k^{*\prime} {\mathbf{G}} (c_k - {\widehat{{\mathbf{P}}}}_k c_k) + c_k^{*\prime} {\mathbf{M}} (c_k - {\widehat{{\mathbf{P}}}}_k c_k) + c_k^{*\prime} {\mathbf{G}} {\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} ({\widehat{{\mathbf{P}}}}_k c_k - c_k ) \Big) \notag \\
& = \frac{1}{\langle c_k^{\phantom *} , c^*_k \rangle_{{\mathbf{G}}}} \Big( (\hat \rho-\rho_k) c_k^{*\prime} {\mathbf{G}} (c_k - {\widehat{{\mathbf{P}}}}_k c_k) + c_k^{*\prime} {\mathbf{G}} ({\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} - {\mathbf{G}}^{-1} {\mathbf{M}})({\widehat{{\mathbf{P}}}}_k c_k - c_k ) \Big) \label{e:pgbd0} \,.\end{aligned}$$ But we have: $$\label{e:pgbd1}
| c_k^{*\prime} {\mathbf{G}} (c_k - {\widehat{{\mathbf{P}}}}_k c_k) | \leq \| \tilde c_k^*\| \| c_k - {\widehat{{\mathbf{P}}}}_k c_k\|_{{\mathbf{G}}} = O_p( \eta_{n,k})$$ by Cauchy-Schwarz, display (\[e:ckhatbd\]), and the normalization $\| \tilde c_k^*\| = 1$.
Moreover: $$\begin{aligned}
| c_k^{*\prime} {\mathbf{G}} ({\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} - {\mathbf{G}}^{-1} {\mathbf{M}})({\widehat{{\mathbf{P}}}}_k c_k - c_k ) | \notag
& = | \tilde c_k^{*\prime} (({\widehat{{\mathbf{G}}}}^o)^{-1} {\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o) {\mathbf{G}}^{1/2}({\widehat{{\mathbf{P}}}}_k c_k - c_k ) | \\
& \leq \| \tilde c_k^*\| \| ({\widehat{{\mathbf{G}}}}^o)^{-1} {\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o\| \| c_k - {\widehat{{\mathbf{P}}}}_k c_k\|_{{\mathbf{G}}} \notag \\
& = O_p( ( \eta_{n,k,1} \vee \eta_{n,k,2})) \times O_p(\eta_{n,k}) \label{e:pgbd2}\end{aligned}$$ by the normalization $\| \tilde c_k^*\| = 1$, Lemma \[lem:matcgce\](b), and display (\[e:ckhatbd\]).
It follows from (\[e:pgbd0\]), (\[e:pgbd1\]), (\[e:pgbd2\]), and Lemma \[lem:var\](a) that: $$\left| {\mathrm{trace}({({\widehat{{\mathbf{P}}}}_k - {\mathbf{P}}_k){\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} })} \right| \leq \left| \frac{1}{\langle c_k^{\phantom *} , c^*_k \rangle_{{\mathbf{G}}}} \right| \times O_p( ( \eta_{n,k} \vee \eta_{n,k,1} \vee \eta_{n,k,2})) \times O_p(\eta_{n,k})\,.$$ But notice that: $$\left| \frac{1}{\langle c_k^{\phantom *} , c^*_k \rangle_{{\mathbf{G}}}} \right| = \| {\mathbf{P}}_k c_k^* \|_{{\mathbf{G}}} \leq \| {\mathbf{P}}_k \|_{{\mathbf{G}}} \leq \left\| \frac{-1}{2 \pi {\mathrm{i}}} \int_\Gamma \frac{{\mathcal{R}}({\mathbf{G}}^{-1} {\mathbf{M}},z)}{\rho_k - z }\, {\mathrm{d}}z \right\|_{{\mathbf{G}}}$$ whenever $k \geq K$ [@Kato expression (6.19) on p. 178] where $\Gamma$ is as in the proof of Lemma \[lem:exist\]. Finally, by display (\[e:resbd:k\]) we have: $$\left| \frac{1}{\langle c_k^{\phantom *} , c^*_k \rangle_{{\mathbf{G}}}} \right| \leq O(1) \times \sup_{z \in \Gamma} \|{\mathcal{R}}({\mathbf{G}}^{-1} {\mathbf{M}},z)\|_{{\mathbf{G}}} = O(1)\,.$$ This completes the proof of step 1.
Step 2: Here we show that $$c_k^{*\prime} {\mathbf{G}} ({\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} - {\mathbf{G}}^{-1} {\mathbf{M}}) c_k = c_k^{*\prime} ( {\widehat{{\mathbf{M}}}} - \rho_k {\widehat{{\mathbf{G}}}} ) c_k + O_p( \eta_{n,k,1} \times (\eta_{n,k,1} \vee \eta_{n,k,2}))$$ By Lemma \[lem:matcgce\](a) we have: $$\begin{aligned}
& c_k^{*\prime} {\mathbf{G}} ({\widehat{{\mathbf{G}}}}^{-1} {\widehat{{\mathbf{M}}}} - {\mathbf{G}}^{-1} {\mathbf{M}}) c_k \\
& = \tilde c_k^{*\prime} (({\widehat{{\mathbf{G}}}}^o)^{-1} {\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o) \tilde c_k \\
& = \tilde c_k^{*\prime} ( {\widehat{{\mathbf{M}}}}^o - {\widehat{{\mathbf{G}}}}^o {\mathbf{M}}^o +({\widehat{{\mathbf{G}}}}^o)^{-1} ({\mathbf{I}} - {\widehat{{\mathbf{G}}}}^o)({\widehat{{\mathbf{M}}}}^o - {\mathbf{M}}^o ) + ({\widehat{{\mathbf{G}}}}^o)^{-1}({\widehat{{\mathbf{G}}}}^o - {\mathbf{I}})^2 {{\mathbf{M}}}^o) \tilde c_k \\
& = \tilde c_k^{*\prime} ( {\widehat{{\mathbf{M}}}}^o - {\widehat{{\mathbf{G}}}}^o {\mathbf{M}}^o ) \tilde c_k + O_p( \eta_{n,k,1} \times (\eta_{n,k,1} \vee \eta_{n,k,2}))\\
& = c_k^{*\prime} ( {\widehat{{\mathbf{M}}}} - \rho_k {\widehat{{\mathbf{G}}}} ) c_k + O_p( \eta_{n,k,1} \times (\eta_{n,k,1} \vee \eta_{n,k,2}))\,.\end{aligned}$$ Which completes the proof of step 2. The result follows by combining steps 1 and 2 and using $\eta_{n,k} \leq (\eta_{n,k,1} \vee \eta_{n,k,2})$ (which follows from Lemma \[lem:matcgce\](b) and definition of $\eta_{n,k}$).
First note that: $$\begin{aligned}
\sqrt n (\hat L - L) & = \sqrt n \left( \log \hat \rho - \log \rho - \frac{1}{n} \sum_{t=0}^{n-1} \log m(X_t,X_{t+1}) + {\mathbb{E}}[\log m(X_t,X_{t+1})] \right) \\
& = \frac{1}{\sqrt n} \sum_{t=0}^{n-1} ( \rho^{-1} \psi_{\rho,t} - \psi_{lm,t} ) + o_p(1)\end{aligned}$$ where the second line is by display (\[e:ale:1\]) and a delta-method type argument. The result now follows from the joint convergence in the statement of the proposition.
Similar arguments to the proof of Proposition \[p:asydist:L:1\] yield: $$\sqrt n (\hat L - L) = \frac{1}{\sqrt n} \sum_{t=1}^n \left( \rho^{-1} \psi_{\rho,t} - \log m(X_t,X_{t+1};\hat \alpha) + \log m(X_t,X_{t+1};\alpha_0) - \psi_{lm,t} \right) + o_p(1)\,.$$ By continuous differentiability of $\log m(X_t,X_{t+1};\hat \alpha)$ on a neighborhood of $\alpha_0$ and the dominance condition in the statement of the proposition, we may deduce: $$\frac{1}{\sqrt n } \sum_{t=0}^{n-1} \Big( \log m(X_t,X_{t+1};\hat \alpha) - \log m(X_t,X_{t+1};\alpha_0) \Big) = D_{\alpha,lm} \sqrt n (\hat \alpha - \alpha_0) + o_p(1)$$ where $$D_{\alpha,lm} = {\mathbb{E}} \left[\frac{1}{m(X_t,X_{t+1},\alpha)} \frac{\partial m(X_t,X_{t+1},\alpha)}{\partial \alpha'} \right] \,.$$ Substituting into the expansion for $\hat L$ and using Assumption \[a:parametric\](a) yields: $$\sqrt n (\hat L - L) = \frac{1}{\sqrt n} \sum_{t=1}^n \left( \rho^{-1} \psi_{\rho,t} - D_{\alpha,lm} \psi_{\alpha,t} - \psi_{lm,t} \right) + o_p(1)\,.$$ The result follows by the joint CLT assumed in the statement of the proposition.
We prove part (i) first. Let $Q_2$ denote the stationary distribution of $(X_t,X_{t+1})$. Under Assumption \[a:eff\] and the maintained assumption of strict stationarity of $\{X_t\}_{t \in {\mathbb{Z}}}$, the tangent space is ${\mathcal{H}}_0 = \{ h(X_t,X_{t+1}) : {\mathbb{E}}[h(X_t,X_{t+1})^2] < \infty$ and ${\mathbb{E}}[ h(X_t,X_{t+1})|X_t = x] = 0$ almost surely$\}$ endowed with the $L^2(Q_2)$ norm (see pp. 878–880 of [@BickelKwon2001] which is for a real-valued Markov process, but trivially extends to vector-valued Markov processes).
Take any bounded $h \in {\mathcal{H}}_0$ and consider the one-dimensional parametric model which we identify with the collection of transition probabilities $\{ P_1^{\tau,h} : |\tau| \leq 1\}$ where each transition probability $P_1^{\tau,h}$ is dominated by $P_1$ and is given by: $$\frac{{\mathrm{d}} P_1^{\tau,h}(x_{t+1}|x_t)}{{\mathrm{d}} P_1(x_{t+1}|x_t)} = e^{\tau h(x_t,x_{t+1})-A(\tau,x_t)}$$ where: $$A(\tau,x_t) = \log\left( \int e^{\tau h(x_t,x_{t+1})} P_1({\mathrm{d}} x_{t+1}|x_t) \right) \,.$$ For each $\tau$ we define the linear operator ${\mathbb{M}}^{(\tau,h)}$ on $L^2$ by: $${\mathbb{M}}^{(\tau,h)} \psi(x_t) = \int m(x_t,x_{t+1}) \psi(x_{t+1}) P_1^{\tau,h}({\mathrm{d}} x_{t+1}|x_t) \,.$$ Observe that: $$\begin{aligned}
\label{e:eff:pf1}
({\mathbb{M}}^{(\tau,h)}-{\mathbb{M}})\psi(x_t) = \int m(x_t,x_{t+1}) \psi(x_{t+1}) \left( e^{\tau h(x_t,x_{t+1})-A(\tau,x_t)}-1 \right)P_1 ({\mathrm{d}} x_{t+1}|x_t)\,.\end{aligned}$$ which is a bounded linear operator on $L^2$ (since ${\mathbb{M}}$ is a bounded linear operator on $L^2$ and $h$ is bounded). By Taylor’s theorem and boundedness of $h$ we can deduce that $A(\tau,x_t) = O(\tau^2)$ and therefore: $$\label{e:eff:pf2}
e^{\tau h(x_t,x_{t+1})-A(\tau,x_t)}-1 = \tau h(x_t,x_{t+1}) + O(\tau^2)$$ where the $O(\tau^2)$ term is uniform in $(x_t,x_{t+1})$. It then follows by this and boundedness of $h$ that $\|{\mathbb{M}}^{(\tau,h)}-{\mathbb{M}}\| = O(\tau)$. Similar arguments to the proof of Lemma \[lem:exist\] imply that there exists $\epsilon > 0$ and $\bar \tau > 0$ such that the largest eigenvalue $\rho_{(\tau,h)}$ of ${\mathbb{M}}^{(\tau,h)}$ is simple and lies in the interval $(\rho - \epsilon,\rho + \epsilon)$ for each $\tau < \bar \tau$.
It follows from a perturbation expansion of $\rho_{(\tau,h)}$ about $\tau = 0$ (see, for example, equation (3.6) on p. 89 of [@Kato]) that:[^37] $$\begin{aligned}
\rho_{(\tau,h)} - \rho & = \langle ({\mathbb{M}}^{(\tau,h)}-{\mathbb{M}}) \phi, \phi^* \rangle + O(\tau^2) \notag \\
& = \tau {\mathbb{E}}[ m(X_t,X_{t+1}) h(X_t,X_{t+1}) \phi(X_{t+1}) \phi^* (X_t)] + O(\tau^2) \notag \\
& = \tau \int m(x_t,x_{t+1})\phi(x_{t+1})\phi^*(x_t) h(x_t,x_{t+1}){\mathrm{d}}Q_2(x_t,x_{t+1}) + O(\tau^2) \label{e:eff:pf3}\end{aligned}$$ under the normalization $\langle \phi,\phi^*\rangle = 1$, where the second line is by (\[e:eff:pf1\]) and (\[e:eff:pf2\]). Expression (\[e:eff:pf3\]) shows that the derivative of $\rho_{(\tau,h)}$ at $\tau = 0$ is $\tilde \psi_\rho = m(x_t,x_{t+1})\phi(x_{t+1})\phi^*(x_t)$. Since bounded functions $h$ are dense in ${\mathcal{H}}_0$, we have that $\rho$ is differentiable relative to the tangent set ${\mathcal{H}}_0$ with derivative $\tilde \psi_\rho$. The efficient influence function is the projection of $\tilde \psi_\rho$ onto ${\mathcal{H}}_0$, namely: $$\tilde \psi_\rho(x_t,x_{t+1}) - {\mathbb{E}}[\tilde \psi_\rho(X_t,X_{t+1}) |X_t = x_t] = \psi_\rho(x_t,x_{t+1})$$ because ${\mathbb{E}}[\tilde \psi_\rho(X_t,X_{t+1}) |X_t = x_t] = \phi^*(x_{t}) {\mathbb{M}} \phi(x_t) = \rho \phi(x_t) \phi^*(x_t)$. It follows that $V_\rho = {\mathbb{E}}[\psi_\rho(X_t,X_{t+1})^2]$ is the efficiency bound for $\rho$.
We now prove part (ii). By linearity, the efficient influence function of $L$ is: $$\psi_L = \rho^{-1} \psi_\rho - \psi_{\log m}$$ where $\psi_{\log m}$ is the efficient influence function for ${\mathbb{E}}[\log m(X_t,X_{t+1})]$. To simplify notation, let $l(x_t,x_{t+1}) = \log m(x_t,x_{t+1})$. It is well known that: $$\begin{aligned}
\psi_{\log m}(x_0,x_1) = l(x_0,x_1) + \sum_{t=0}^\infty \Big( {\mathbb{E}}[l(X_{t+1},X_{t+2})|X_1 = x_1] - {\mathbb{E}}[l(X_t,X_{t+1})|X_0 = x_0] \Big) \label{e-phim}\end{aligned}$$ (see, e.g., p. 879 of [@BickelKwon2001]). It may be verified using the telescoping property of the sum (\[e-phim\]) that $V_L = {\mathbb{E}}[ \psi_L(X_0,X_1)^2]$, as required.
Proofs for Appendix \[ax:id\]
-----------------------------
We first show that any positive eigenfunction of ${\mathbb{M}}$ must have eigenvalue $\rho$. Suppose that there is some positive $\psi \in L^2$ and scalar $\lambda$ such that ${\mathbb{M}} \psi(x) = \lambda \psi(x)$. Then we obtain: $$\lambda \langle \phi^*, \psi \rangle = \langle \phi^*, {\mathbb{M}} \psi \rangle = \langle {\mathbb{M}}^* \phi^*, \psi \rangle = \rho \langle \phi^*, \psi \rangle$$ with $\langle \phi^*, \psi \rangle > 0$ because $\phi^*$ and $\psi$ are positive, hence $\lambda = \rho$. A similar argument shows that any positive eigenfunction of ${\mathbb{M}}^*$ must correspond to the eigenvalue $\rho$.
It remains to show that $\phi$ and $\phi^*$ are the unique eigenfunctions (in $L^2$) of ${\mathbb{M}}$ and ${\mathbb{M}}^*$ with eigenvalue $\rho$. We do this in the following three steps. Let $F = \{ \psi \in L^2 : {\mathbb{M}} \psi = \rho \psi\}$. We first show that if $\psi \in F$ then the function $|\psi|$ given by $|\psi|(x) = |\psi(x)|$ also is in $F$. In the second step we show that $\psi \in F$ implies $\psi = |\psi|$ or $\psi = - |\psi|$. Finally, in the third step we show that $F = \{ s \phi : s \in {\mathbb{R}}\}$.
For the first step, first observe that $F \neq \{0\}$ because $\phi \in F$ by Assumption \[a:id:1\](b). Then by Assumption \[a:id:1\](c), for any $\psi \in F$ we have ${\mathbb{M}} |\psi| \geq |{\mathbb{M}} \psi| = \rho |\psi|$ and so ${\mathbb{M}} |\psi| - \rho |\psi| \geq 0$ (almost everywhere). On the other hand, $$\langle \phi^*,{\mathbb{M}} |\psi| - \rho |\psi|\rangle = \langle {\mathbb{M}}^* \phi^*,|\psi| \rangle - \rho \langle \phi^*,|\psi| \rangle = 0$$ which implies that ${\mathbb{M}} |\psi| = \rho |\psi|$ and hence $|\psi| \in F$.
For the second step, take any $\psi \in F$ that is not identically zero. Suppose that $\psi = |\psi|$ on a set of positive $Q$ measure (otherwise we can take $- \psi$ in place of $\psi$). We will prove by contradiction that this implies $|\psi| = \psi$. Assume not, i.e. $|\psi| \neq \psi$ on a set of positive $Q$ measure. Then $|\psi| - \psi \geq 0$ (almost everywhere) and $|\psi| - \psi \neq 0$. But by step 1 we also have that ${\mathbb{M}}(|\psi| - \psi) = \rho(|\psi| - \psi)$. Then for any $\lambda > r({\mathbb{M}})$ we have $$\frac{(\rho/\lambda)}{1-(\rho/\lambda)}(|\xi| - \xi) = \sum_{n\geq 1} \left(\frac{\rho}{\lambda}\right)^n (|\xi| - \xi) = \sum_{n \geq 1}\lambda^{-n} {\mathbb{M}}^n (|\xi| - \xi) > 0$$ (almost everywhere) by Assumption \[a:id:1\](c). Therefore, $|\psi| > \psi $ (almost everywhere). This contradicts the fact that $\psi = |\psi|$ on a set of positive $Q$ measure. A similar proof shows that if $-\psi = |\psi|$ holds on a set of positive $Q$ measure then $-\psi = |\psi|$.
For the third step we use an argument based on the Archimedean axiom (see, e.g., p. 66 of [@Schaefer1974]). Take any positive $\psi \in F$ and define the sets $S_+ = \{s \in {\mathbb{R}} : \psi \geq s \phi\}$ and $S_- = \{s \in {\mathbb{R}} : \zeta \leq s \phi\}$ (where the inequalities are understood to hold almost everywhere). It is easy to see that $S_+$ and $S_-$ are convex and closed. We also have $(-\infty,0] \subseteq S_+$ so $S_+$ is nonempty. Suppose $S_-$ is empty. Then $\psi > s \phi$ on a set of positive measure for all $s \in (0,\infty)$. By step 2 we therefore have $\psi > s \phi$ (almost everywhere). But then because $L^2$ is a lattice we must have $\|\psi\| \geq s \|\phi\|$ for all $s \in (0,\infty)$ which is impossible because $\psi \in L^2$. Therefore $S_-$ is nonempty. Finally, we show that ${\mathbb{R}} = S_+ \cup S_-$. Take any $s \in {\mathbb{R}}$. Clearly $\psi - s \phi \in F$. By Claim 2 we know that either: $\psi - s \phi \geq 0$ (almost everywhere) which implies $s \in S_+$ or $\psi - s \phi \leq 0$ (almost everywhere) which implies $s \in S_-$. Therefore ${\mathbb{R}} = S_+ \cup S_-$. The Archimedean axiom implies that the intersection $S_+ \cap S_-$ must be nonempty. Therefore $S_+ \cap S_- = \{s^*\}$ (the intersection must be a singleton else $\psi = s \phi$ and $\psi = s'\phi$ with $s \neq s'$) and so $\psi = s^* \phi$ (almost everywhere). This completes the proof of the third step.
A similar argument implies that $\phi^*$ is the unique positive eigenfunction of ${\mathbb{M}}^*$.
Assumption \[a:id:0\](a) implies that $r({\mathbb{M}}) > 0$ (see Proposition IV.9.8 and Theorem V.6.5 of [@Schaefer1974]). The result now follows by Theorems 6 and 7 of [@Sasser] with $\rho = r({\mathbb{M}})$. That $\rho$ is isolated follows from the discussion on p. 1030 of [@Sasser].
Consider the operator ${\overline{{\mathbb{M}}}} = \rho^{-1} {\mathbb{M}}$ with $\rho = r({\mathbb{M}})$. Proposition \[p:a:exist\] implies that $\{1\} = \{ \lambda \in \sigma({\overline{{\mathbb{M}}}}) : |\lambda| = 1\}$. Further, since ${\mathbb{M}}$ is power compact it has discrete spectrum [@DunfordSchwartz Theorem 6, p. 579]. We therefore have $\sup\{ |\lambda| : \lambda \in \sigma({\overline{{\mathbb{M}}}}) , \lambda \neq 1\} < 1$ and hence ${\overline{{\mathbb{M}}}} = (\phi \otimes \phi^*) + {\mathbb{V}}$ where $r({\mathbb{V}}) < 1$ and ${\overline{{\mathbb{M}}}}$, $(\phi \otimes \phi^*)$ and ${\mathbb{V}}$ commute (see, e.g., p. 331 of [@Schaefer1974] or pp. 1034-1035 of [@Sasser]). Since these operators commute, a simple inductive argument yields: $${\mathbb{V}}^\tau = (\overline{{\mathbb{M}}} - (\phi \otimes \phi^*))^\tau
= \overline{{\mathbb{M}}}^\tau - (\phi \otimes \phi^*) = \rho^{-\tau} {\mathbb{M}}_\tau - (\phi \otimes \phi^*)$$ for each $\tau \in T$. By the Gelfand formula, there exists $\epsilon > 0$ such that: $$\label{gelfand}
\lim_{\tau \to \infty}\|{\mathbb{V}}^\tau \|^{1/\tau} = r({\mathbb{V}}) \leq 1-\epsilon$$ Let $\{\tau_k : k \geq 1\} \subseteq T$ be the maximal subset of $T$ for which $\|{\mathbb{V}}^{\tau_k}\| > 0$. If this subsequence is finite then the proof is complete. If this subsequence is infinite, then by expression (\[gelfand\]), $$\limsup_{\tau_k \to \infty} \frac{\log \|{\mathbb{V}}^{\tau_k}\|}{\tau_k} < 0\,.$$ Therefore, there exists a finite positive constant $c$ such that for all $\tau_k$ large enough, we have: $$\log \|{\mathbb{V}}^{\tau_k}\| \leq -c \tau_k$$ and hence: $$\| \rho^{-\tau_k} {\mathbb{M}}_{\tau_k} - (\phi \otimes \phi^*) \| \leq e^{-c \tau_k}$$ as required.
[^1]: Department of Economics, New York University, 19 W. 4th Street, 6th floor, New York, NY 10012, USA. E-mail address: `timothy.christensen@nyu.edu`
[^2]: This paper is based on my job market paper, which was titled “Estimating the Long-Run Implications of Dynamic Asset Pricing Models” and dated November 8, 2013. I am very grateful to my advisors Xiaohong Chen and Peter Phillips for their advice and support. I would like to thank the co-editor Lars Peter Hansen and three anonymous referees for insightful and helpful comments. I have benefited from feedback from Caio Almeida (discussant), Jarda Borovička, Tim Cogley, Jean-Pierre Florens (discussant), Nour Meddahi, Keith O’Hara, Eric Renault, Guillaume Roussellet, Tom Sargent and participants of seminars at Chicago Booth, Columbia, Cornell, Duke, Indiana, Monash, Montréal, Northwestern, NYU, Penn, Princeton, Rutgers, Sydney, UCL, UNSW, Vanderbilt, Wisconsin, and several conferences. All errors are my own.
[^3]: Recently, [@QLN] used a complementary parametric approach to recover the permanent component in a dynamic term structure model. Elements of our methodology are still applicable to such parametric approaches when the dynamics are sufficiently nonlinear that the positive eigenfunction and eigenvalue is not available analytically.
[^4]: The continuation value function is only known analytically when the conditional moment generating function of the Markov state is exponentially affine and the EIS $=1$.
[^5]: [@ChenFavilukisLudvigson] introduced estimators for models with EIS $\neq 1$. Our approach is different from theirs for a number of reasons discussed in Section \[s:emp\]. Further, we study formally the large-sample properties of our estimators whereas [@ChenFavilukisLudvigson] do not.
[^6]: Kernel-based estimators may also be represented as a matrix eigenvector problem. However, with kernels the resulting problem is of dimension $n$ ($n$ is the sample size) whereas with sieves the dimension is $k \ll n$.
[^7]: [@EHLLS] do not study estimation of time-reversed eigenfunctions or semiparametric efficiency of their estimators. The large-sample theory for kernel estimators of nonparametric Euler equations in [@EHLLS] uses different arguments from those for sieve estimators in this paper. Moreover, they assume a short panel of cross-sectional consumption data and returns data which they assume is i.i.d. whereas we study a long time series of data of the Markov state process and the SDF process.
[^8]: That is, $M_t(\omega)$ is a function of $X_0(\omega),\ldots,X_t(\omega)$ and $M_\tau(\theta_t(\omega))$ is a function of $X_t(\omega),\ldots,X_{t+\tau}(\omega)$. The methodology applies to models in which the increments of the process $M$ are functionals of both $X$ and an additional noise process as in BHS. We maintain this simpler presentation for notational convenience.
[^9]: We say a function is positive (non-negative) if it is positive (non-negative) almost everywhere with respect to the stationary distribution of $X$.
[^10]: See, e.g., Theorem 6.3 in [@Hansen2012] and Proposition 7.1 in HS.
[^11]: [@GGR2011] also presume the existence of a strictly stationary and ergodic, time-homogeneous Markov state process that is observable to the econometrician but do not constrain its transition law to be of any parametric form.
[^12]: We assume throughout that ${\mathbf{G}}_k$ is nonsingular.
[^13]: In [Matlab]{}, the estimators $\hat \rho$, $\hat c$, $\hat c^*$ can be computed using the command `[C,D,Cstar]=eig(Mhat,Ghat)` where `Mhat` and `Ghat` are the estimators ${\widehat{{\mathbf{M}}}}$ and ${\widehat{{\mathbf{G}}}}$. Then $\hat \rho$ is the maximum diagonal entry of `D`, $\hat c$ is the column of `C` corresponding to $\hat \rho$, and $\hat c^{*}$ is the column of `Cstar` corresponding to $\hat \rho$. Simultaneous computation of $\hat c$ and $\hat c^*$ is also possible in [R]{} using, for example, the function `qz.dggev` in the `QZ` package.
[^14]: When there is no real positive eigenvalue that solves (\[e:est\]) or when $\hat c$ and $\hat c^*$ are not unique, then we can simply take $\rho = 1$ and set $\hat \phi(x) = \hat \phi^*(x) = 1$ for all $x$ without altering the convergence rates or limiting distribution of the estimators. This was not an issue in simulations or the empirical application.
[^15]: An operator is compact if and only if it is the limit of a norm-convergent sequence of operators with finite-dimensional range [@DunfordSchwartz p. 515]. Each $\Pi_k {\mathbb{M}}$ has range $B_k$ where $\dim(B_k) = k < \infty$. If ${\mathbb{M}}$ is not compact but ${\mathbb{M}}_{\tau}$ is compact for some $\tau \geq 2$, then one can apply the estimators to ${\mathbb{M}}_{\tau}$ in place of ${\mathbb{M}}$ and estimate the solution $(\rho^\tau,\phi)$ to ${\mathbb{M}}_\tau \phi = \rho^\tau \phi$ and similarly for $\phi^*$. Consistency and convergence rates of $\rho^\tau$, $\phi$ and $\phi^*$ would then follow directly from Theorem \[t:rate\].
[^16]: Bounds for $\delta_k^{\phantom *}$ and $\delta_k^*$ are available under standard smoothness assumptions (see [@Chen2007]).
[^17]: If the eigenfunction is rescaled then the corresponding eigenvalue will be different due to nonlinearity of ${\mathbb{T}}$. For example, $c \chi$ is a positive eigenfunction of ${\mathbb{T}}$ with eigenvalue $c^{\beta-1} \lambda$ for any $c > 0$.
[^18]: Suppose that ${\mathbb{T}}$ has a positive fixed point $h \in L^2$. The function $\bar h \equiv 0$ is also a fixed point. Therefore, ${\mathbb{T}}$ is not a contraction on $L^2$ (else the Banach contraction mapping theorem would yield a unique fixed point).
[^19]: Much of the literature on nonlinear Perron-Frobenius theory has focused on the finite-dimensional case. The literature on infinite-dimensional Perron-Frobenius problems has typically worked in spaces for which cone of non-negative functions has nonempty interior (see [@Krause] for a recent overview). The non-negative cone in the $L^2$ space we study has empty interior. If ${\mathcal{X}}$ is bounded then these previous results may be used to derive (global) identification conditions in the space $C({\mathcal{X}})$. However, bounded support seems inappropriate for common choices of state variable, such as consumption growth and dividend growth.
[^20]: To calculate the root mean square error (RMSE) for $\hat \phi$, $\hat \phi^*$, and $\hat \chi$, for each replication we calculate the $L^2$ distance between the estimators and their population counterparts, then take the average over the MC replications. To calculate the bias we take the average of the estimators across the MC replications to produce $\bar \phi(x)$, $\bar \phi^*(x)$, and $\bar \chi(x)$ (say), then compute the $L^2$ distance between $\bar \phi$, $\bar \phi^*$ and $\bar \chi$ and the true $\phi$, $\phi^*$ and $\chi$. The use of the “bias” here is not to be confused with the bias term in the convergence rate calculations. There “bias” measures how close $\phi_k$, $\phi_k^*$, and $\rho_k$ are to $\phi$, $\phi^*$, and $\rho$. Here “bias” of an estimator refers to the distance between the parameter and the average of its estimates across the MC replications. Similar calculations are performed for $\hat \rho$, $\hat y$, $\hat L$, and $\hat \lambda$.
[^21]: Recursive preferences with EIS equal to unity may be interpreted as those of an agent with a preference for robustness; see [@Tallarini2000].
[^22]: Qualitatively and quantitatively similar results are obtained replacing $d_t$ with real per capita growth in corporate earnings (using after-tax profits from line 15 of Table 1.12) and with real per capita growth in a four-quarter geometric moving average of dividends, as in [@HansenHeatonLi].
[^23]: The differences between our methodology and [@ChenFavilukisLudvigson] are as follows. First, we focus on the case in which the EIS $=1$ whereas [@ChenFavilukisLudvigson] have EIS $\neq 1$ and treat the EIS as a free parameter. Second, we exploit the eigenfunction representation of the specific recursion we study. Third, we “profile out” continuation value function estimation by solving for $(\lambda,\chi)$ separately from estimating the preference parameters. Therefore, so our criterion function depends only on $(\beta,\gamma)$. In contrast, [@ChenFavilukisLudvigson] estimate structural parameters and the continuation value function jointly, resulting in a higher-dimensional optimization problem. Fourth, the continuation value is a function of the Markov state in our analysis whereas the continuation value function in [@ChenFavilukisLudvigson] depends on contemporaneous consumption growth and the lagged continuation value. It is beyond the scope of the paper to compare the two procedures.
[^24]: This construction is related to a complete polynomial basis, but it is built from tensor products Hermite polynomials instead of tensor products of regular polynomials. See, e.g., pp. 239–240 of [@Judd] for a discussion of complete polynomial bases and their approximation properties.
[^25]: We resample the data 1000 times using the stationary bootstrap with an expected block length of six quarters. In the left panel we re-estimate $\beta$, $\gamma$, $\lambda$, $\chi$, $\rho$, $y$, and $L$ for each bootstrap replication. We discard the fraction of replications in which the estimator of $(\beta,\gamma)$ failed to converge. In the right panel we fix $\beta$ and $\gamma$ and re-estimate $\lambda$, $\chi$, $\rho$, $y$, and $L$ for each bootstrap replication.
[^26]: The entropy of the SDF is estimated using $\log (\frac{1}{n} \sum_{t=0}^{n-1} m(X_t,X_{t+1};\hat \alpha)) - \frac{1}{n} \sum_{t=0}^{n-1} \log (m(X_t,X_{t+1};\hat \alpha))$.
[^27]: [@ChernovMueller] provide evidence that yield spreads are smaller on real bonds.
[^28]: For example, [@ChenFavilukisLudvigson] obtain $\hat \gamma \approx 60$ using aggregate consumption data and $\hat \gamma \approx 20$ using stockholder consumption data. Further, with stockholder data their estimated EIS is not significantly different from zero. This suggests that our estimates of $\gamma$ and maintained assumption of a unit EIS are empirically plausible.
[^29]: The VAR(1) parameters are estimated by maximum likelihood. The parameters $\mu$, $\kappa$ and the three autoregressive gamma process parameters for the SV-AR(1) model are estimated via indirect inference [@GMR] using a GARCH(1,1) auxiliary model. Analytic solutions for the stochastic volatility specification are available by following arguments in Appendix H of [@BackusChernovZin].
[^30]: Similar results are obtained for the permanent and transitory components themselves (not their logarithms) as well as for robust measures of association such as Kendall’s tau and Spearman’s rho.
[^31]: Proposition \[p:lr\] below shows that this probability measure is precisely the measure used to define the unconditional expectation ${\widetilde{{\mathbb{E}}}}$ in the long-run approximation (\[e:lrr\]).
[^32]: Indeed, for non-stationary environments it is not even clear how to restrict the class of functions appropriately to define an adjoint (for instance, HS do not appear to restrict $\phi$ to belong to a Banach space).
[^33]: I thank an anonymous referee for bringing Theorems 6 and 7 of [@Sasser] to my attention. Theorems 6 and 7 of [@Sasser] replace Assumption \[a:id:0\](a) in Proposition \[p:a:exist\] by the condition that ${\mathbb{M}}$ is *quasi-positive*, i.e. for each non-negative $\psi$ and $\psi^*$ in $L^2$ that are not identically zero there exists $\tau \in T$ such that $\langle \psi^*, {\mathbb{M}}_\tau \psi \rangle > 0$. Notice that quasi-compactness also requires that $r({\mathbb{M}}) > 0$. Assumption \[a:id:0\](a) is sufficient for these two conditions (i.e. quasi-positivity and $r({\mathbb{M}}) > 0$). The condition $r({\mathbb{M}}) > 0$ together with power-compactness of ${\mathbb{M}}$ (Assumption \[a:id:0\](b)) is sufficient for quasi-compactness.
[^34]: See, e.g., [@FosterNelson] and [@LiXu] for estimating spot volatility and [@ABDL] for estimating integrated volatility based on high-frequency data.
[^35]: This may be established formally using, for example, Corollary 4.2 in [@ChenChristensen-reg].
[^36]: See p. 113 of [@Chatelin]. Note that [@Chatelin] defines the spectral projection as $(\phi_k^{\phantom +}\! \otimes \phi_k^+)$ under the normalizations $\|\phi_k\| = 1$ and $\langle \phi_k^{\phantom +}\!, \phi_k^+ \rangle = 1$. Here we are imposing the normalizations $\|\phi_k\| = 1$ and $\|\phi_k^+\| = 1$ and and so we scale the definition of the projection accordingly.
[^37]: The result on p. 89 of [@Kato] is presented for the finite-dimensional case but it also applies in the infinite-dimensional case, as made clear in Section VII.1.5 of [@Kato].
| 2023-12-12T01:26:58.646260 | https://example.com/article/2564 |
The Matrix Revolutions is a 2003 American science fiction film and the third and final installment of The Matrix trilogy. The film is a sequel to The Matrix Reloaded. The film was written and directed by the Wachowski brothers and released simultaneously in sixty countries on November 5, 2003. Despite the fact that it is the final film in the series, the Matrix storyline continued in The Matrix Online.
The film was the first live-action film to be released in both regular and IMAX movie theaters at the same time. The Wachowskis were present in Tokyo at the opening of the movie, as were stars Jada Pinkett Smith and Keanu Reeves.
Contents
Plot
The film begins where The Matrix Reloaded ended. Bane and Neo are both unconscious, but Neo shows neural patterns suggesting he is in the Matrix. Morpheus decides to start a search for Neo within the Matrix. Neo finds himself trapped in a subway station; a transition zone between the Matrix and the machine world. At this station, Neo meets a 'family' of programs, including a girl named Sati, whose father tells Neo that the station is controlled by a program called The Trainman, an exile loyal only to the Merovingian who exerts complete control over the subway. When Neo tries to board the train with the family, the Trainman refuses to let him aboard.
Seraph contacts Morpheus on behalf of the Oracle. The Oracle, with a changed appearance, informs Morpheus and Trinity of Neo's confinement. Seraph, Morpheus, and Trinity pursue the Trainman to secure Neo's release. The trio enters Club Hel to confront the Merovingian to release Neo. The Merovingian demands "the eyes of the Oracle" in exchange for Neo's release. Trinity loses her patience and provokes a Mexican standoff, forcing the Merovingian to release Neo.
Troubled by visions of the Machine City, Neo decides to visit the Oracle. She informs him that as the One, he has developed a wireless connection with the Source. All of Neo's abilities - inside and outside the Matrix - exist because of this connection. She characterizes Smith as Neo's exact "opposite" and "negative", who threatens to destroy the Matrix and eventually the Machine City. She states that "everything that has a beginning has an end", and that the war is about to end "one way or another." After Neo leaves the Oracle, a large group of "Smiths," after assimilating Sati and Seraph, arrive to assimilate the unresisting Oracle, gaining her powers of precognition.
In the real world, the remaining crew of the Nebuchadnezzar and the Hammer encounter Niobe's deactivated ship, the Logos, and its crew. They successfully reactivate the ship and begin to interrogate the now awakened Bane, who claims he has no memory of the events of the earlier battle. As the ship captains plan to return to defend Zion, Neo announces that he needs a ship to travel to the Machine City. The captain of the Hammer refuses to allow it, citing his rights as ship's captain. However, Niobe provides him the Logos, rebuking the captain of the Hammer when he attempts to prevent her from exercising the rights he just extolled.
Niobe pilots the Hammer through a series of service tunnels, which are nearly impossible to navigate in order to avoid the Sentinel army. Shortly after departure, they discover that Bane has murdered a crew member and has hidden aboard the Logos. They also come to realize that it was Bane who fired the EMP which disabled the human fleet after it had engaged the Sentinel army. Despite this discovery, they are unable to warn Trinity and Neo. Before the Logos can depart, Bane ambushes Trinity and takes her hostage. Neo fights Bane, who reveals that he is Smith. During the struggle, Bane blinds Neo by cauterizing his eyes with a severed power cable. In spite of his blindness, Neo can see the glowing form of Smith and kills him. Trinity pilots them towards the Machine City.
In Zion, the defenders deploy infantry armed with rocket launchers and Armored Personnel Units. The docks are invaded by a massive horde of Sentinels and two giant drilling machines. Outnumbered and overwhelmed by the attackers, the APUs are unable to hold the Dock, and their leader Captain Mifune is fatally wounded. With his last breath, Mifune tells the Kid, who has been rearming his APU, to open the gate for the Hammer. Encouraged by Mifune, Kid pilots the APU and opens the gate with the help of Link's wife Zee who kills a Sentinel and saves him. The Sentinels are on the verge of overwhelming the remaining humans when the Hammer, with more Sentinels in pursuit, arrives at Zion and sets off its EMP, disabling the Sentinels, but also the remaining defenses. Although they have bought a temporary reprieve, the humans are forced to retreat to the temple's entrance and wait for the next attack, preparing for what they believe will be their last stand.
Nearing the Machine City, Neo and Trinity are attacked by the city's huge defense machines. Neo uses his power to destroy the incoming bombs, but Sentinels overwhelm the ship. To evade them, Trinity flies the Logos into an electrical storm cloud. Her actions disable the Sentinels, but also disable the ship's engines. Above the cloud layer, Trinity sees a glimpse of sunlight and blue sky for the first time. The ship then free-falls directly toward the Machine City, and despite Trinity's attempts to ignite the engines in time, the ship crashes. The impact of the collision fatally wounds Trinity, and she dies in Neo's embrace.
Neo enters into the Machine City to strike a bargain with the Machines, personified by the Deus Ex Machina. Neo warns that Smith is beyond the Machines' control and will soon assault the Source. He offers to help stop Smith in exchange for a ceasefire with Zion: the Machines agree, causing all the Sentinels attacking Zion to stand down and await orders. The Machines provide a connection for Neo to enter the Matrix and confront Smith. In the Matrix, which is now wholly populated by Smith's copies, the clone with the Oracle's powers steps forth, claiming that he has already foreseen his own victory.
The Smith clones stand by and watch while Neo and the primary Smith fight on the streets, through buildings, and in the sky, until they finally brawl in a flooded crater. Neo is outmatched by Smith but refuses to give up. A frustrated Smith continues to attack, but when he suddenly repeats the Oracle's words, "Everything that has a beginning has an end," Neo baits Smith into assimilating him. As the Machines sense the assimilation, Neo's body spasms as a surge of energy enters his body. The Neo-Smith and the rest of the clones then explode, restoring The Matrix and its citizens to normal. In the crater full of water, we now see the Oracle lying on the ground. The waiting Sentinels withdraw from Zion. Neo, having sacrificed himself to save both the Machines and humans, is unplugged from the Matrix and his body is carried away by the Machines.
The Matrix "reboots", repairing the damage caused by Neo and Smith's battle, and without its usual green tint. The cat from the first movie is seen in the same fashion as in the first film and is picked up by Sati. The Architect and the Oracle meet, and agree to unplug all humans who want to be freed, and that peace will last "as long as it can." Sati, who has created a colorful dawn sky in memory of Neo, asks the Oracle if they will ever see him again to which she replies that she believes they will. Seraph asks the Oracle if she knew all along that this would happen, and she replies that she didn't know, but she did believe.
Actress Gloria Foster, who played the Oracle in the first film, died before the completion of her filming for the third and was replaced by actress Mary Alice. Her changed appearance is addressed in the film's plot, and the directors state they had coincidentally explored such a change early in the script's development.
Sound design
Sound editing on The Matrix trilogy was completed by Danetracks in West Hollywood, CA.
Soundtrack
Main article: The Matrix Revolutions: Music from the Motion Picture
In contrast to the movie's predecessors, very few "source" tracks are used in the movie. Aside from Don Davis' score, again collaborating with Juno Reactor, only one external track (by Pale 3) is used.
Although Davis rarely focuses on strong melodies, familiar leitmotifs from earlier in the series reappear. For example, Neo and Trinity's love theme—which briefly surfaces in the two preceding movies—is finally fully expanded into "Trinity Definitely"; the theme from the Zion docks in Reloaded returns as "Men in Metal", and the energetic drumming from the Reloaded teahouse fight between Neo and Seraph opens "Tetsujin", as Seraph, Trinity and Morpheus fight off Club Hel's three doormen.
The climactic battle theme, named "Neodämmerung" (in reference to Wagner's Götterdämmerung), features a choir singing extracts (shlokas) from the Upanishads. The chorus can be roughly translated from Sanskrit as follows: "lead us from untruth to truth, lead us from darkness to light, lead us from death to immortality, peace peace peace" [1]. The extracts were brought to Davis by the Wachowski brothers when he informed them that it would be wasteful for such a large choir to be singing simple "ooh"s and "aah"s (according to the DVD commentary, Davis felt that the dramatic impact of the piece would be lost if the choir was to sing 'This is the one, see what he can do' in plain English). These extracts return in the denouement of the movie, and in Navras, the track that plays over the closing credits (which may be considered a loose remix of "Neodämmerung").
Reception
The budget of the movie is an estimated US$150 million, grossing over $139 million in North America and approximately $427 million worldwide,[2] roughly half of The Matrix Reloaded box-office total. The Matrix Revolutions was released on DVD and VHS on April 6, 2004. The film grossed $116 million in DVD sales.
The film received mixed reviews from critics. The Matrix Revolutions received a score of 37% on Rotten Tomatoes, a movie review aggregation site (with a score of 14% when filtered to include only Top critics).[3] The film's average critic score on Metacritic is 48/100.[4]
The Matrix Revolutions grossed $83.8 million within its first five days of being released in North America.[5] The film's opening was not as commercially successful as the opening of its predecessor in the series. Some have attributed this to a more subdued marketing campaign[citation needed] compared to that of The Matrix Reloaded, which was considered a summer blockbuster event.
The film was criticized for being anticlimactic.[6][7] Additionally, some critics regard the movie as less philosophically ambiguous than its predecessor, The Matrix Reloaded.[8][9] Critics had difficulty finding closure pertaining to events from The Matrix Reloaded, and were generally dissatisfied.[10][11] The film's earnings dropped 66% during the second week after its release to theaters.[5]
The Matrix: Music from the Motion Picture ·The Matrix: Original Motion Picture Score ·The Matrix Reloaded: The Album ·The Matrix Reloaded: Complete 2 CD Score ·The Matrix Revolutions: Music from the Motion Picture ·The Matrix Revolutions: Complete 2 CD Score ·The Animatrix: The Album ·Enter the Matrix: Original Soundtrack from the Videogame ·The Matrix: Path of Neo - Music from the Video Game | 2024-07-30T01:26:58.646260 | https://example.com/article/9800 |
using MvcTemplate.Components.Extensions;
using MvcTemplate.Components.Security;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Web.Mvc;
using System.Xml.Linq;
namespace MvcTemplate.Components.Mvc
{
public class MvcSiteMapProvider : IMvcSiteMapProvider
{
private IEnumerable<MvcSiteMapNode> AllNodes { get; }
private IEnumerable<MvcSiteMapNode> NodeTree { get; }
public MvcSiteMapProvider(String path, IMvcSiteMapParser parser)
{
XElement siteMap = XElement.Load(path);
NodeTree = parser.GetNodeTree(siteMap);
AllNodes = ToList(NodeTree);
}
public IEnumerable<MvcSiteMapNode> GetSiteMap(ViewContext context)
{
Int32? account = context.HttpContext.User.Id();
String area = context.RouteData.Values["area"] as String;
String action = context.RouteData.Values["action"] as String;
String controller = context.RouteData.Values["controller"] as String;
IEnumerable<MvcSiteMapNode> nodes = CopyAndSetState(NodeTree, area, controller, action);
return GetAuthorizedNodes(account, nodes);
}
public IEnumerable<MvcSiteMapNode> GetBreadcrumb(ViewContext context)
{
String area = context.RouteData.Values["area"] as String;
String action = context.RouteData.Values["action"] as String;
String controller = context.RouteData.Values["controller"] as String;
MvcSiteMapNode current = AllNodes.SingleOrDefault(node =>
String.Equals(node.Area, area, StringComparison.OrdinalIgnoreCase) &&
String.Equals(node.Action, action, StringComparison.OrdinalIgnoreCase) &&
String.Equals(node.Controller, controller, StringComparison.OrdinalIgnoreCase));
List<MvcSiteMapNode> breadcrumb = new List<MvcSiteMapNode>();
while (current != null)
{
breadcrumb.Insert(0, new MvcSiteMapNode
{
IconClass = current.IconClass,
Controller = current.Controller,
Action = current.Action,
Area = current.Area
});
current = current.Parent;
}
return breadcrumb;
}
private IEnumerable<MvcSiteMapNode> CopyAndSetState(IEnumerable<MvcSiteMapNode> nodes, String area, String controller, String action)
{
List<MvcSiteMapNode> copies = new List<MvcSiteMapNode>();
foreach (MvcSiteMapNode node in nodes)
{
MvcSiteMapNode copy = new MvcSiteMapNode();
copy.IconClass = node.IconClass;
copy.IsMenu = node.IsMenu;
copy.Controller = node.Controller;
copy.Action = node.Action;
copy.Area = node.Area;
copy.Children = CopyAndSetState(node.Children, area, controller, action);
copy.HasActiveChildren = copy.Children.Any(child => child.IsActive || child.HasActiveChildren);
copy.IsActive =
copy.Children.Any(child => child.IsActive && !child.IsMenu) ||
(
String.Equals(node.Area, area, StringComparison.OrdinalIgnoreCase) &&
String.Equals(node.Action, action, StringComparison.OrdinalIgnoreCase) &&
String.Equals(node.Controller, controller, StringComparison.OrdinalIgnoreCase)
);
copies.Add(copy);
}
return copies;
}
private IEnumerable<MvcSiteMapNode> GetAuthorizedNodes(Int32? accountId, IEnumerable<MvcSiteMapNode> nodes)
{
List<MvcSiteMapNode> authorized = new List<MvcSiteMapNode>();
foreach (MvcSiteMapNode node in nodes)
{
node.Children = GetAuthorizedNodes(accountId, node.Children);
if (node.IsMenu && IsAuthorizedToView(accountId, node.Area, node.Controller, node.Action) && !IsEmpty(node))
authorized.Add(node);
else
authorized.AddRange(node.Children);
}
return authorized;
}
private Boolean IsAuthorizedToView(Int32? accountId, String area, String controller, String action)
{
return action == null || Authorization.Provider?.IsAuthorizedFor(accountId, area, controller, action) != false;
}
private IEnumerable<MvcSiteMapNode> ToList(IEnumerable<MvcSiteMapNode> nodes)
{
List<MvcSiteMapNode> list = new List<MvcSiteMapNode>();
foreach (MvcSiteMapNode node in nodes)
{
list.Add(node);
list.AddRange(ToList(node.Children));
}
return list;
}
private Boolean IsEmpty(MvcSiteMapNode node)
{
return node.Action == null && !node.Children.Any();
}
}
}
| 2024-03-28T01:26:58.646260 | https://example.com/article/4314 |
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