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Q:
How to check if an arbitrary number of matlab.unittest.constraints is satisfied by a cell array?
I have a cell array of matlab.unittest.constraints and a cell array of values. I'd like to see if the values match the constraints (respectively). Of course, I can just use a for cycle, something like the following code:
satisfied = zeros(1,argLength);
for i=1:argLength
satisfied(i) = satisfiedBy(cons{i}, val{i});
end;
answer = all(satisfied);
but knowing MATLAB, there must be a way to condense all that into a single line, I just don't know it. I compare the lengths of the arrays beforehand and return false if they're not equal.
A:
Here is a possible CELLFUN statement:
satisfied = cellfun(@satisfiedBy, cons, val);
Make sure satisfiedBy returns only single numeric/logical value.
| 2023-11-15T01:27:17.760571 | https://example.com/article/1859 |
### `tf.contrib.layers.parse_feature_columns_from_sequence_examples(serialized, context_feature_columns, sequence_feature_columns, name=None, example_name=None)` {#parse_feature_columns_from_sequence_examples}
Parses tf.SequenceExamples to extract tensors for given `FeatureColumn`s.
##### Args:
* <b>`serialized`</b>: A scalar (0-D Tensor) of type string, a single serialized
`SequenceExample` proto.
* <b>`context_feature_columns`</b>: An iterable containing the feature columns for
context features. All items should be instances of classes derived from
`_FeatureColumn`. Can be `None`.
* <b>`sequence_feature_columns`</b>: An iterable containing the feature columns for
sequence features. All items should be instances of classes derived from
`_FeatureColumn`. Can be `None`.
* <b>`name`</b>: A name for this operation (optional).
* <b>`example_name`</b>: A scalar (0-D Tensor) of type string (optional), the names of
the serialized proto.
##### Returns:
A tuple consisting of:
* <b>`context_features`</b>: a dict mapping `FeatureColumns` from
`context_feature_columns` to their parsed `Tensors`/`SparseTensor`s.
* <b>`sequence_features`</b>: a dict mapping `FeatureColumns` from
`sequence_feature_columns` to their parsed `Tensors`/`SparseTensor`s.
| 2024-04-01T01:27:17.760571 | https://example.com/article/8683 |
Instructions
Whisk flour, cornmeal, sugar, baking powder, baking soda and salt together in a large bowl. In a medium bowl, stir together yogurt, oil, syrup, vanilla and eggs. Using a rubber spatula, fold wet ingredients into dry ingredients until just mixed. Cover bowl and refrigerate overnight.
Preheat oven to 375F. Grease two standard 12-cup muffin tins.
Remove batter from refrigerator and fold in blackberries. Fill 16 muffin cups just to the top and bake 10 to 15 minutes. Rotate tins and bake an additional 10 to 12 minutes, until puffed and slightly golden and a toothpick inserted into the center of a muffin comes out clean.
Allow muffins to cool in their tins 5 minutes before inverting onto a cooling rack.
*Note: If using frozen berries, stir them in frozen just before baking so the batter doesn’t become too streaky. For even baking, fill the empty muffin cups halfway with water.
Recipe by Paula Disbrowe. | 2023-08-05T01:27:17.760571 | https://example.com/article/5038 |
Signature
Dex Entry: "Alolan Raticate gained the Dark type and used it to beat up local Alakazam and steal their mustaches. This explains both the clean-shaven Alolan Alakazam, and the luxuriousness of Alolan Raticate's 'stache" | 2023-08-29T01:27:17.760571 | https://example.com/article/2210 |
/*
* Licensed to the Apache Software Foundation (ASF) under one
* or more contributor license agreements. See the NOTICE file
* distributed with this work for additional information
* regarding copyright ownership. The ASF licenses this file
* to you under the Apache License, Version 2.0 (the
* "License"); you may not use this file except in compliance
* with the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing,
* software distributed under the License is distributed on an
* "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
* KIND, either express or implied. See the License for the
* specific language governing permissions and limitations
* under the License.
*/
package org.apache.druid.query.movingaverage;
import org.apache.druid.data.input.Row;
import org.apache.druid.java.util.common.guava.YieldingAccumulator;
import java.util.ArrayList;
import java.util.List;
/**
* Groups all the rows for a specific period together.
* Rows of each period are placed in a single {@link RowBucket} (timed through the dateTime field).
* (Assumpltion: Input arrives sorted by timestamp).
*/
public class BucketingAccumulator extends YieldingAccumulator<RowBucket, Row>
{
/* (non-Javadoc)
* @see YieldingAccumulator#accumulate(java.lang.Object, java.lang.Object)
*/
@Override
public RowBucket accumulate(RowBucket accumulated, Row in)
{
List<Row> rows;
if (accumulated == null) {
// first row, initializing
rows = new ArrayList<>();
accumulated = new RowBucket(in.getTimestamp(), rows);
} else if (accumulated.getNextBucket() != null) {
accumulated = accumulated.getNextBucket();
}
if (!accumulated.getDateTime().equals(in.getTimestamp())) {
// day change detected
rows = new ArrayList<>();
rows.add(in);
RowBucket nextBucket = new RowBucket(in.getTimestamp(), rows);
accumulated.setNextBucket(nextBucket);
yield();
} else {
// still on the same day
rows = accumulated.getRows();
rows.add(in);
}
return accumulated;
}
}
| 2023-08-14T01:27:17.760571 | https://example.com/article/2546 |
[The spectroscopic study of the iodine beta-cyclodextrin complexes].
In this paper, the iodine beta-cyclodextrin complexes prepared in different techniques were investigated by Fourier transformation infrared spectrum(FTIR), FT-Raman, Ultra-violet absorption spectrum in qualitative and quantitative analysis. Results showed that the iodine existed in dissociated, bundle and polymerized forms in the complexes. Contras to the method of alcohol craft, the sample prepared in water craft method included a higher capacity of iodine, but it was not stable enough to keep a constant high iodine concentration. This analysis method was simple and convenient, it is useful for quality control of the iodine beta-cyclodextrin complexes raw material medicine. | 2024-04-22T01:27:17.760571 | https://example.com/article/3949 |
WASHINGTON, D.C. -- "Explosion Kills Former Prime Minister." "Suicide Terrorist Kills Five at Nightclub." "Car Bomber Kills 125 Police Recruits." "Iraqi Judge Assassinated." These recent headlines describe bloody events in Lebanon, Israel and Iraq, where IEDs, or "body bombs," have killed and maimed hundreds. Though true, these reports have apparently distracted many in the so-called mainstream media from a discomfiting reality: Freedom is on the march down the "Arab street."
Ever since U.S. troops first went to Afghanistan in October 2001, our supposedly more experienced "betters" in Europe and the "prudent potentates of the press" have said that U.S. military action against an Islamic nation would cause the "Arab street" to rise up and crush us. This theme was widely replayed in the build-up for Operation Iraqi Freedom -- and has been reiterated many times in the aftermath of Saddam Hussein's capture.
Since his Inaugural Address, President Bush has been repeatedly castigated for his "naivety" on one hand and for his "aggressive arrogance" on the other -- because he boldly tells those who suffer tyranny that "the United States will not ignore your oppression or excuse your oppressors. When you stand for your liberty, we will stand with you."
Yet, despite the carping critics -- and the carnage caused by those who would rather die than see freedom flourish -- any objective observer has to conclude that George W. Bush is right. "The call of freedom" does indeed come "to every mind and every soul." Freedom is indeed on the march -- even down the "Arab street."
It was evident last October in Afghanistan, in the ballots cast by Palestinians in early January and again in late January on the ink-stained fingers of Iraqi men and women, raised in proud defiance against murderous thugs who would return them to brutal bondage.
Whether the America-haters and Bush-bashers want to acknowledge it or not, the "call of freedom" is now being heard in places where American "influence" has long been deemed by the "experts" to be minimal, at best:
-- In December, hundreds of thousands of Ukrainians peacefully protested to force a new election when a rigged vote installed Vladimir Putin's handpicked presidential candidate. Today, reformist Viktor Yushchenko governs in Kiev. The Bush administration needed to do little more than lend its voice to the calls for a free and fair election.
-- Last week, in long-suffering Syrian-occupied Lebanon, 25,000 unarmed Christian and Muslim civilians, protesting the assassination of former Prime Minister Rafik Hariri, forced the resignation of Syria's puppet government in Beirut. In the aftermath, the new Iraqi government -- and even the French -- have joined our call for the Syrians to withdraw their forces from Lebanon and deport the residue of Saddam's regime hiding there.
Though they have yet to fully comply, the Syrians have arrested and turned over the former dictator's half-brother, Sabawi Ibrahim al Hassan. And to ensure that those in Damascus who support terror don't get the idea that this is sufficient, President Bush has since told them to "get your troops and your secret services out of Lebanon, and give democracy a chance."
-- In Cairo, Hosni Mubarak, never known to be a friend of liberty or democratic institutions, has announced that opposition candidates will be allowed to run for office in the upcoming Egyptian elections. Mubarak has been the only presidential "candidate" since taking power in 1981. While questions remain about who will be "allowed" to run, a taste of liberty in a democratic election may ignite the "fire of freedom" among the "people of the Nile."
-- And now, even the royal family in Riyadh, Saudi Arabia, seems to be getting the message. Though the recent Saudi "municipal elections" were more show than substance -- the elected councilors wield little power, the ruling House of Saud appoints as many councilors as were elected and only men could vote -- the taste of democracy has intensified the call on the "Arab street" for real elections.
Last week, the kingdom's Foreign Minister Prince Saud al-Faisal, speaking the heretofore unthinkable, said that in future elections women "may" have the right to vote. Unfortunately, he then added, "We know we want to reform, we know we want to modernize, but for God's sake leave us alone."
And therein lies the first problem: The prince doesn't get it. It's not just Bush's promise, "When you stand for your liberty we will stand with you," at work in Saudi Arabia -- it really is a quest for freedom that is sweeping down his "Arab streets," right past minarets preaching repression and hatred for all things "Western."
But Saud al-Faisal isn't alone in misunderstanding what freedom really means -- and from whence it springs. Last week, when President Bush confronted Vladimir Putin about Russia's freedom of the press, Putin shot back with: "We didn't criticize you when you fired those reporters at CBS."
Thus the second problem: Saud al-Faisal and Putin apparently believe that holding an election is enough. It's not. As we have learned from the "election" of Hugo Chavez in Venezuela, there is much more to freedom than casting a ballot. Liberty also means a free press; freedom to worship-- or not; the rule of law where justice is tempered with mercy; freedom from fear -- of government, criminals or outsiders -- and the freedom to come and go, to speak politically, to work and create wealth.
All of this -- and more -- is what freedom is about. Elections are not the end of the process, just the beginning. That's what's wrong with the argument being waged by some in Congress to start withdrawing American forces from Iraq now that there has been an election. Whether it's the "Arab street," or elsewhere, liberty doesn't march to the beat of a cadence -- it arrives to the sound of many drummers, and impatience is never the friend of freedom. | 2023-12-31T01:27:17.760571 | https://example.com/article/5505 |
Prevalence of psychopathology in children of bipolar parents.
The purpose of this study was to determine psychopathology in a large sample of adolescent children (N = 140) of parents (N = 86) with bipolar disorders, using multiple sources of information (child, parent, teacher). The main aims of the study were: 1) to compare parent, teacher and self reported problem behavior ratings of children of bipolar parents with those of children from the general population, and 2) to determine the prevalence of DSM-IV diagnoses in offspring of bipolar parents, and 3) to examine the possible association between gender of the bipolar parent and psychopathology among the offspring. | 2024-07-11T01:27:17.760571 | https://example.com/article/5269 |
Mandi Burewala railway station
Mandi Burewala railway station (, ) is located in Pakistan.
See also
List of railway stations in Pakistan
Pakistan Railways
References
External links
Official Web Site of Pakistan Railways
Category:Railway stations in Vehari District
Category:Railway stations on Lodhran–Raiwind Line | 2023-12-08T01:27:17.760571 | https://example.com/article/8717 |
Question & Answer Service Providers
Click on a Service Provider or Reference below for more information.
What is the Women, Infants, and Children (WIC) Program?
The purpose of the Special Supplemental Nutrition Program for Women, Infants, and Children - better known as the WIC Program - is to help safeguard the health of low-income women, infants, & children up to age 5 who are at nutritional risk by providing nutritious foods to supplement diets, information on healthy eating, and referrals to health care. | 2023-09-13T01:27:17.760571 | https://example.com/article/6930 |
"Okay, ladies, listen up." "Since it's "Focus on Agility" month, for the rest of the class," "I want you all to work on your cartwheels, splits and aerials." "Brittany, that's a perfect split." "Thanks, Ms. Morris." "Funny how all the drills for "Focus on Agility" month are the same ones you'd do if you were trying to sneak a cheerleader practice into regular gym class." "Yes, and I don't intend to stand for it." "They can have my squat-thrusts when they pry them from my cold, dead hands." "What?" "I don't know." "See something you like or just browsing?" "I'm thinking about getting those running shoes." "Uh-huh." "What color were they?" "Just kick the bleachers out from underneath me, okay?" "Uh-oh, busted." "Ladies." "Why aren't you exercising with the rest of the class?" "We are exercising." "We're exercising our right to abstain from cheerleader practice." "That's not cheerleading, it's agility." "Jane Lane, you're just like your sisters, aren't you?" "We share certain chromosome pairs." "Beyond that, I'm not supposed to say." "You know what I mean." "Can't be part of a group." "Always have to be different." "Your sister Penny never wanted to participate, either." "I taught her a thing or two about the American competitive spirit." "You sure did." "That's why she's spent the last ten years out of the country." "I know what kind of upbringing she's had." "What's your excuse?" "I'm just plain no good?" "Life's just one big smartass joke, isn't it?" "Well, I'm here to prepare you for the real world." "I'll see you both after school to make up this class." "Can somebody help me, please?" "Yo, how come you look so down, Mack Daddy?" "Don't call me that." "I think I screwed up that ethics test today." "How'd you do?" "Pretty good." "Wait a minute." "You weren't even there today." "You cut class on the day of a midterm?" "I was excused." "I didn't have to take the test." "Coach talked to the teacher, and I got by." "You mean you got "a bye."" "Yeah, that's it." "Coach says I gotta maintain a "C" average to stay on the team, so he took care of it or something." "Hey, that's not fair." "I'm on the team and I had to study." "I'm the Q.B., Mack Daddy." "I've told you a million times, don't call me that!" "When are you going to get it through your thick skull already?" "If that's how you feel all you had to do was say so, bro." "So, bro... what's wrong, bro?" "Did you hear that?" "Whoever said life was fair?" "I don't know, but I'll bet he was a quarterback." "You know, I had this weird anxiety dream where we had this pop quiz in math that I was totally unprepared for." "Were you in your underwear?" "No." "That's how I figured out it wasn't a dream." "Uh-oh." "Not a great score on the quiz?" "My straight "C" average in math is now in serious danger of becoming a "D."" "So I'm off to study hall." "To finish your nap?" "You know me too well." "Yo, bro, what's wrong?" "I'm here for you, bro." "Hey, bro, wait up!" "Gee, Daria, you're home late from school today." "Quinn's always home this late." "Yeah, but that's because she's usually with some... oh, my gosh, Daria, you met a guy!" "Actually, I'm late because I had to make up gym class." "Oh, you don't have to pretend with old Dad." "Go on, tell me, who's the lucky fella?" "Gee, Daria, I hope he's not pressuring you to, uh, do... anything." "All right, you figured me out." "He's a senior and he carves words into his arm." "Together we're hoping to buck the odds against teen marriages." "That's a good one, Daria." "I'm kidding." "Of course you are, kiddo." "I knew that." "Oh." "If someone named Knuckles calls, it's for me." "Daria, just getting home?" "I had to make up gym." "Really?" "Why?" "The teacher was running a cheerleading practice during class, so I sat out." "I'm not going to jump up and down like an idiot to prove to the world what I don't believe, which is that jocks are great." "Good for you, honey." "The only way it will stop being a man's world is if we get the boys out there in their short shorts cheering, too." "What I mean is, this double standard has got to end, right, Jake?" "Equal pay for equal work." "Is that right?" "Daria, I don't approve of refusing to participate in class, but no daughter of mine is going to buy into the mindset that a woman's purpose in life is pleasing men." "Like my new shoes?" "They pinch my toes and give me blisters but they make my legs look hot so I'm going to buy another pair." "So Helen actually supported you on this gym class thing?" "Her vestigial sense of right and wrong was acting up again." "What does your mother think about it?" "I don't know." "When she gets back from the painted desert I'll ask her if she had a vision." "You know, I'd really like to show that Ms. Morris that the Lanes are no bunch of ordinary deadbeats." "Of course not." "You're deadbeats with style." "Exactly, and speaking of style: fashion fiends at high noon." "Can you imagine joining an actual sport?" "You'd get all sweaty and your makeup would smudge." "You're supposed to date jocks, not be one." "Oh, and I read in Waif that running makes your feet spread to a bigger shoe size, which would look really, really, really bad." "How are your feet in those new shoes, Quinn?" "They're killing me." "Oh... that's too bad." "What kind of loser would sign up for the track team?" "You girls don't think I'm a loser, do you?" "Are you asking if you can hang out with us?" "Nope." "I prefer women with a slightly more enlightened attitude toward fitness." "Oh, yeah?" "I guess you can be born in the '80s and still stuck in the '50s." "It's hard to believe there's anyone alive who still thinks being athletic isn't ladylike." "Women can excel at any sport they put their minds to." "I definitely agree with that." "Me, too." "I'd like to sign up for the track team." "What?" "Miss Lane, what are you doing here?" "Trying out for track team." "You won't participate in regular gym class, but you want to join an extracurricular sport?" "I like to run." "Okay." "Let's see if you're any good at it." "I'd particularly like to run up and down your spine wearing track spikes." "She isn't your favorite teacher?" "We have a score to settle." "I'm Evan, by the way." "Jane." "So, you think you're ready for track?" "Yeah." "I figure if the people suck I can always wear headphones." "I'm way ahead of you." "Until we get out there." "You walking home?" "Um, not today." "Practice." "So you made the track team?" "Yeah." "This is the part where you say," ""Hey, way to go, congratulations."" "Hey, way to go." "Congratulations...?" "Congratulations." "Now, don't get mushy on me." "Next, on Sick, Sad World." "Jane." "On a school team." "That's pathetic." "Hey, if she's the pathetic one, why am I talking to myself?" "And more importantly, why am I waiting for a reply?" "Quinn, it is so great of you to invite us over to watch the Miss Continents of the Earth Pageant." "Oh, Sandi, it's so great of you to come over." "Quinn, your cousin or whatever sure spends a lot of time at your house." "Yes, well, um, she has nowhere else to go, what with her parents being in jail and all." "Look, it's starting." "Shouldn't you all be running around in teddies and giving each other makeovers by now?" "Mustard is not her color." "Oh, the no-salt, no-oil, fat-free popcorn is ready." "Popcorn?" "Stop, my mouth is watering." "Oh, yeah, I know I like to wear heels with my bathing suit." "And I never go swimming without my lip liner." "Quinn, I need to ask you something." "All right, so Tuesday night I wasn't washing cars for the United Way, but Mom, if you'd seen his boyish smile you'd understand." "Actually, I wanted to ask you to spend some time with your sister." "Why don't you just lock me in the attic till I'm 25?" "I can't sacrifice any more for her, Mother, I just can't." "Quinn, I'm worried." "She's alone too much and I think she's talking to herself." "Just give her an afternoon." "She'd do it for you." "Should I agree to perform this sensitive and difficult task, and I'm not saying I will, what's in it for me?" "I am disappointed that it's come to this, but... you can take my credit card to the mall." " Platinum?" " Green." " Gold." " Done." "Someone's knocking." "Come in." "Hi, Daria, how's it going?" "Okay." "Out with it." "I was thinking that you and I could... you know, do something together." "All right, who are you, and where are you hiding Quinn?" " What?" "Can't a sister spend time with her..." " Cousin?" "All right, Mom is forcing me to hang out with you." "Forcing?" "I've got her gold card." "Why didn't you say so?" "Give me the first cash advance and you can come with me to the library." "The library?" "If anyone saw me there, I'd die." "Okay, then, how about the mall?" "You know, where your friends hang out." "Um, hi." "Will you read to me?" "I really like stories." "You don't have to tell me a story." "Just talk." "So Jack and Jill went up the hill or whatever to the sample sale, where Jill found the perfect ensemble and Jack gave it to her for the one-month anniversary of their first date." "Hey, Quinn." "I just came by to pick up my little brother from story hour." "I didn't know you were a reader." "Hi, Calvin." "I'm not actually..." "That's really cool." "I mean, you're popular; you don't have to volunteer." "Oh, well, you know, I get back more than I give." "I mean, the children appreciate it so." "Uh, librarian... could you return these magazines for me?" "Thanks a bunch, dear." "So, where are you guys headed now?" ""Thin Thighs For Your Man."" "But I don't like men with thin thighs." "Land that big sneaker endorsement yet?" "Nah, they wouldn't meet my price." "Today's our first track meet." "I wouldn't mind if you were there." "Oh." "Well, you know how I hate to cancel my line-dancing lesson... but okay." "Cool." "The question is, am I supporting my friend or her surrender to the system?" "Do you mind?" "This is a private conversation." "Hey, Daria." "Trent?" "What are you doing here?" "I came to support Janey." "They're power-scrubbing my room again." "Anyway, this running stuff means lot to her." "What she's doing takes guts." "All that practicing, day in, day out." "Yeah." "Hey, Jane's good." "She's great!" "Go, go, kick butt." "And you missed a lot of really good TV while you were at those practices." "The other day, they had this special, When Animals Hold a Grudge." "Jane, you were amazing." "First place?" "Yeah, congratulations." "Thanks, I just pretend I'm running away from a pep rally." "Yo, bro!" "Gotta go." "So anyway, this one parakeet was really ticked off at its owner for not changing its water..." "Hey, teammate." "Did you see this girl run like the wind?" "Have you ever heard her break wind?" "Daria!" "That's pretty funny." "See you at practice, speedy." "So anyway, the parakeet waits until its owner..." "Jeez, Daria, what the hell was that about?" "What?" "Hey, hey, hey, it's the track star." "You're gonna like being a jock, man." "When you're a winner, everybody wants to be your friend." "Not like those boring friends you used to have who liked you even when you lost." "You mean the ones who made incredibly humiliating jokes at my expense?" "Oh, come on." "You come on, Daria." "So, I tried something different." "I went out on a limb and participated in something and it worked out." "Why do you have to be so pissy about it?" "Uh, babe, this is really awkward, right?" "Yeah, babe." "It's creeping me out!" "Oh, Kevvy, come on." "Nurse Brittany gonna make you feel all better." "All right, so I thought I was making a joke, but I was really humiliating you in front of this guy you like..." "Appreciate." "This guy you appreciate, which was stupid and insensitive." "So I'm an idiot and I'm sorry." "Okay?" "Not so fast." "How do I know this is a sincere apology and not a cheap ploy to get back on my good side?" "Why does there have to be a difference?" "I accept." "Thank you." "Pizza after school?" "Wish I could." "Practice." "I guess we'd better join the rest of the cheerleaders before we get detention again." "Don't worry about it." "What do you mean?" "How are the legs, Jane?" "Okay." "Maybe a little sore after the meet." "Sore?" "You better rest up for tomorrow." "Here, go watch TV in my office." "I don't know." "I-I don't really want to be all by myself..." "Ms. Morgendorffer, you join her." "Yes, ma'am." "Anything for the team, ma'am." "So that's your archenemy." "You think it's hypocritical of me to accept favors from a woman I despise?" "Does she have cable?" "Let's find out." "Oh, they're eating dinner." "Should I sit down and join them or fake a headache?" "I'm not feeling too well." "I think I'll eat later." "Oh, boy, cuckoo girl is at it again." "I'm really worried." "She's been acting this way for days." "It's that guy, Knuckles!" "I tell you, he's messing with her head!" "Quinn, I thought you were going to spend some time with your sister." "Oh, yeah, here." "What's this?" "Well, that one is my time sheet showing the hours and locations where I baby-sat Daria." "And that other one is a receipt for my new pleather jacket." "Thanks." "$300?" "!" "Dad, it's very difficult to keep her amused." "Oh, and when the cash advance shows up on the statement?" "Don't blame me, that's Daria's." "How'd the math test go?" "I passed." "You hope." "I know." "Well, aren't we confident." "If I had as much trouble wi..." "You got a bye on the test, didn't you?" "I probably would have gotten a "C" anyway." "I can't believe it!" "Hey, you watched TV with me instead of going to gym." "So?" "So, that was one perk of being on the team." "The math test was another." "You don't see any difference?" "Yeah, the math test was a hell of a lot better." "Give me a break." "What's up with your friend?" "Oh, I rubbed her self-righteousness the wrong way." "I don't know why you hang out with her anyway." "What do you mean?" "Well, you're sort of a sports star now." "It's not so cool to be hanging around with... you know, losers." "Yeah, you're right." "I've been spending too much time with losers." "I'm glad you... see it my way." "Admit it, she had a point." "Who am I to criticize her when I was getting spillover perks?" "God, I wish she'd just quit that damn team." "It's corrupting me." "I'll get it." "Yo, it's me." "Act natural." "You're not hurt." "What?" "Look, I've been thinking and... you're right." "It was really screwed up of me to take a bye on that math test." "I should have gone in there and failed it on my own." "They gave you the chance to cut a corner." "I can't blame you for taking it." "This all happened because I was trying to prove to Ms. Morris that I'm not a deadbeat." "And that Evan guy factors in exactly how?" "That jerk?" "I can't believe you thought he was cute." "Yeah." "Aren't I a dope?" "Hey, don't you have practice now?" "Yeah." "I quit." "What?" "I quit." "I'm afraid I might come out of high school knowing as much as Kevin." "You came out of kindergarten knowing more than Kevin." "Remind me of that when he gets his football scholarship." "I'll bring the spikes back tomorrow." "Honest." "Jane, I wish you would consider coming back to the team." "Thanks, but I stand by my decision." "Jane, I wish you would consider coming back to the team." "Am I missing something?" "The team needs you, and you need the team... if you don't want to be here taking math again this summer." "Gee, that almost sounds like blackmail." "Fortunately, I can pull up my math grade on my own." "Then I'll flunk you." "Why don't I go to Ms. Li and expose this grade-changing arrangement?" "She already knows." "Okay, then, back off or I'll tell the P.T.A." "They know, too." "Congress?" "You're beaten, Lane." "How about if I call the three local TV stations and tell each one that the other two are running the story?" "Damn." "I really don't get you." "No." "You don't." "What happened to your come-and-get-it heels?" "Ha-ha, Daria." "I decided I shouldn't wear shoes that kill me just because they make my legs look hot." "Why, Quinn, how mature." "I mean, my legs look hot no matter what I wear." "They'd probably even look good in those things you have on." "I didn't think things could get any more pathetic, but obviously I was wrong." "Come on, ladies." "You've got a lot of work to make up." "Tell me that at least I have my integrity." "Integrity is a funny word." "Well, then, at least tell me I'm marginally less corrupt than the jocks." "You refused to participate in a crooked system where good grades are exchanged for athletic performance." "But you didn't try to reform the system, either." "For fear of complete teenage exile." "Right." "So the system continues, you haven't redeemed yourself, and we're ostracized anyway." "Come on, now, stop trying to paint a rosy picture." "You know what?" "What?" "They really are preparing us for the real world." "Daria!" "Jane!" "Come on!" "It's time for splits!" "Réponses au blindtest :" "Jane running Jimmie's Chicken Shack" " Droppin' Anchor" "Jane sees Evan Spice Girls" " Too Much" "Quinn shows off her new shoes Super 8" " King Of The World" "Jane running Jimmie's Chicken Shack" " Droppin' Anchor" "Jane trying out for track The Getaway People" " She Gave Me Love" "Miss Continents of the Earth Pageant Daft Punk" " Revolution 909 at the library Garbage" " Push It at Pizza King Montell Jordan" " Let's Ride" "Daria walking around in room Beastie Boys" " So Whatcha' Want track team pressures Jane to re-join No Doubt" " The Simple Kind of Life" "Quinn back in her regular shoes Beastie Boys" " Jimmy James" "Daria in gym class Ben Harper" " Faded closing credits Clara Thomas" " Talkin' To Myself" | 2024-05-25T01:27:17.760571 | https://example.com/article/9803 |
Yang She
Yang She (楊涉), courtesy name Wenchuan (文川), was an official of the Chinese dynasty Tang Dynasty and Tang's successor dynasty Later Liang, serving as a chancellor during the reigns of Tang's final emperor Emperor Ai and Later Liang's both commonly recognized emperors, Emperor Taizu (Zhu Quanzhong) and Emperor Taizu's son Zhu Zhen.
During Tang Dynasty
Background and early career
It is not known when Yang She was born. His family was originally from Tong Prefecture (同州, in modern Weinan, Shaanxi), and claimed ancestry from the Sui Dynasty general Yang Su. His grandfather Yang Yizhi (楊遺直) served as a personnel officer at Hao Prefecture (濠州, in modern Chuzhou, Anhui). At one point, Yang Yizhi became a teaching scholar at Su Prefecture (蘇州, in modern Suzhou, Jiangsu), and therefore settled there, making it the childhood home of Yang She's father Yang Yan (楊嚴), who would eventually serve as the deputy minister of defense and the acting director of finances by the time of his death in 878. (Yang Yan's older brother (Yang She's uncle) Yang Shou had been earlier a chancellor during the reign of then-reigning Emperor Xizong, who was eventually removed and executed in exile for corruption; Yang Yan's career had been thus itself derailed by demotions due to taint by his relationship with Yang Shou, but had been resumed by that point.) Yang She had at least two younger brothers, Yang Zhu (楊注) and Yang Dong (楊洞).
Yang She himself passed the imperial examinations in the Jinshi (進士) class in 875. During the subsequent reign of Emperor Xizong's brother and successor Emperor Zhaozong, he successively served as Libu Langzhong (吏部郎中), a supervisory official at the ministry of civil service affairs (吏部, Libu); the deputy minister of rites (禮部侍郎, Libu Shilang); and deputy minister of justice (刑部侍郎, Xingbu Shilang). In 897, he became the deputy minister of civil service affairs (吏部侍郎, Libu Shilang). In early 904, he became Shangshu Zuo Cheng (尚書左丞), one of the secretaries general of the executive bureau of government (尚書省, Shangshu Sheng). When Emperor Zhaozong was forced by the major warlord Zhu Quanzhong the military governor (Jiedushi) of Xuanwu Circuit (宣武, headquartered in modern Kaifeng, Henan) to move the capital from Chang'an to Luoyang later that year, Yang accompanied the emperor to Luoyang. He was distressed by the physical hold that Zhu had over the emperor and the imperial government, but did not dare to leave governmental service. After the emperor's arrival in Luoyang, he was made the minister of civil service affairs (吏部尚書, Libu Shangshu).
Chancellorship
Later in the year, Zhu Quanzhong had Emperor Zhaozong assassinated and replaced with his son Emperor Ai. In spring 905, Zhu further removed the chancellors Pei Shu, Dugu Sun, and Cui Yuan, believing the accusations of their fellow chancellor Liu Can that they would not be obedient to Zhu. In their place, Zhang Wenwei and Yang She were named chancellors, with the designations Tong Zhongshu Menxia Pingzhangshi (同中書門下平章事). It was said that Yang, who was peaceful and humble in disposition, wept with his family members upon hearing that he was commissioned a chancellor, and stated to his son Yang Ningshi (楊凝式) that this commission would surely bring disaster on Yang Ningshi. At that time, his younger brother Yang Zhu was an imperial scholar, and chose to resign to avoid appearance of impropriety.
In 907, Emperor Ai was preparing to yield the throne to Zhu. As part of the ceremonial passage of the throne, Yang, acting as Shizhong (侍中), the head of the examination bureau of government (門下省, Menxia Sheng), was put in delivering the main imperial seal to Zhu. (Yang Ningshi, pointing out that ceremonial, Yang would be terminating Tang Dynasty, suggested that Yang She resign and let someone else carry out the duty; Yang She, fearful that this would bring disaster on the whole family, did not do so.) Yang She, as well as Zhang (who was to announce the passage of the throne to Zhu) and Xue Yiju (who was to deliver the secondary imperial seals), went in a grand procession from Luoyang to Daliang (i.e., the capital of Xuanwu Circuit), where Zhu was at the time, for the ceremony. Zhu accepted the throne, ending Tang and starting a new Later Liang (as its Emperor Taizu).
During Later Liang
During Emperor Taizu's reign
Yang She continued as a chancellor of the new Later Liang until 908, when he was removed from the chancellor position and made You Pushe (右僕射), one of the heads of the executive bureau. Later in the year, however, when he was Zuo Pushe (左僕射) (also head of the executive bureau, but considered higher than You Pushe), he was again made chancellor with the designation Tong Zhongshu Menxia Pingzhangshi, when one of the chancellors who replaced him, Zhang Ce, retired. He was again removed from his chancellor position in 909 and kept only the position of Zuo Pushe. He apparently was also in charge of the imperial examinations, but was said to have had few accomplishments otherwise.
During Zhu Zhen's reign
At some point, Yang She must have returned to chancellorship, for he was described by the History of the Five Dynasties (which did not contain a biography for him) as a chancellor as of 915, by which point Emperor Taizu had died and his son Zhu Zhen was emperor. That year—when Yang, in addition to being chancellor, was said to be You Pushe, Menxia Shilang (門下侍郎, the deputy head of the examination bureau), and the director of salt and iron monopoly, was removed from his chancellor post (after he had requested to resign due to illness) and made only Zuo Pushe. In 916, he further retired from the Zuo Pushe post and made a senior advisor to the Crown Prince—an entirely honorary post as there was no crown prince at the time. That was the last historical reference to Yang, and it is not known when he died.
Notes and references
Old Book of Tang, vol. 177.
New Book of Tang, vol. 184.
New History of the Five Dynasties, vol. 35.
Zizhi Tongjian, vols. 265, 266, 267.
Category:9th-century births
Category:10th-century deaths
Category:Chancellors under Emperor Ai of Tang
Category:Later Liang (Five Dynasties) chancellors
Category:Later Liang (Five Dynasties) people born during Tang | 2024-02-25T01:27:17.760571 | https://example.com/article/7057 |
---
abstract: 'For a random vector $X$ in ${{\mathbb R}}^n$, we obtain bounds on the size of a sample, for which the empirical $p$-th moments of linear functionals are close to the exact ones uniformly on a convex body $K \subset {{\mathbb R}}^n$. We prove an estimate for a general random vector and apply it to several problems arising in geometric functional analysis. In particular, we find a short Lewis type decomposition for any finite dimensional subspace of $L_p$. We also prove that for an isotropic $\log$-concave random vector, we only need $\lfloor n^{p/2} \log n \rfloor$ sample points so that the empirical $p$-th moments of the linear functionals are almost isometrically the same as the exact ones. We obtain a concentration estimate for the empirical moments. The main ingredient of the proof is the construction of an appropriate majorizing measure to bound a certain Gaussian process.'
author:
- 'Olivier Guédon, Mark Rudelson [^1]'
title: '$L_p$ moments of random vectors via majorizing measures'
---
Introduction
============
In many problems of geometric functional analysis it is necessary to approximate a given random vector by an empirical sample. More precisely, given a random vector $X \in {{\mathbb R}}^n$, we want to find the smallest number $m$ such that the properties of $X$ can be recovered from the empirical measure $1/m \sum_{j=1}^m
\delta_{X_j}$, constructed with independent copies $X_1, \ldots,
X_m$ of the vector $X$. In particular, for $p \ge 2$ and for $y
\in {{\mathbb R}}^n$, we want to approximate the moments ${{\mathbb E}}|{\langle}X,y {\rangle}|^p$ by the empirical averages $1/m \sum_{j=1}^m |{\langle}X_j,y {\rangle}|^p$ with high probability. Moreover, we require this approximation to be uniform over $y$ belonging to some convex symmetric set in ${{\mathbb R}}^n$. A problem of this type was considered in [@BouLinMil]. Formulated in analytic language, it asks about finding the smallest $m$ and a set of points $x_1, \ldots x_m \in
X$ such that for any function $f$ from an $n$-dimensional function space $F \subset L_1(X,\mu)$, $$(1-{\varepsilon})\|f\|_1
\le \frac{1}{m} \sum_{j=1}^m |f(x_j)|
\le (1-{\varepsilon})\|f\|_1.$$ Another example of such problems originates in Computer Science. The probabilistic algorithm for estimating the volume of an $n$-dimensional convex body, constructed by Kannan, Lovász, and Simonovits [@KLS] required to bring the body to a nearly isotropic position as a preliminary step. To this end, one has to sample $m$ random points $x_1, \ldots, x_m$ in the body $L$ so that the empirical isotropy tensor will be close to the exact one, namely $$\label{inertia}
{\left \| \frac{1}{m} \sum_{j=1}^m x_j \otimes x_j
- \frac{1}{\text{vol}(L)} \int_L x \otimes x \, dx \right \|}
< {\varepsilon}.$$ This problem was attacked with different probabilistic techniques. The original estimate of [@KLS] was significantly improved by Bourgain [@Bou]. Using the decoupling method he proved that $m=C({\varepsilon})n \log^3 n$ vectors $x_1, \ldots x_m$ uniformly distributed in the body $L$ satisfy with high probability. This estimate was farther improved to $(Cn/{\varepsilon}^2)
\cdot \log^2 (Cn/{\varepsilon}^2)$ in [@Rud2], [@Rud3]. The proof in [@Rud2] used majorizing measures, while the later proof in [@Rud3] was based on the non-commutative Khinchine inequality.
These problems were put into a general framework by Giannopoulos and Milman [@GiaMil], who related them to the concentration properties of a random vector. Let ${\alpha}>0$ and let $\nu$ be a probability measure on $(X,\Omega)$. For a function $f: X \to {{\mathbb R}}$ define the $\psi_{{\alpha}}$-norm by $${\left \| f \right \|}_{\psi_{\alpha}}
= \inf \{ {\lambda}>0 \mid \int_X \exp (|f|/{\lambda})^{{\alpha}} d\nu \le 2 \}.$$ Chebychev’s inequality shows that the functions with bounded $\psi_{\alpha}$-norm are strongly concentrated, namely $\nu\{x \mid
|f(x)|> {\lambda}t\} \le C \exp(-t^{\alpha})$. Let $\mu$ be a Borel measure in ${{\mathbb R}}^n$. It is called isotropic if $$\int_{{{\mathbb R}}^n} x \otimes x \, d \mu(x) = {\rm Id},$$ where ${\rm Id}$ is the identity operator in ${{\mathbb R}}^n$. Note that this normalization is consistent with the one used in [@KLS; @Rud2; @Rud3]. The normalization used in [@MilPaj; @GiaMil] differs from it by the multiplicative coefficient $L_{\mu}^2$, where $L_{\mu}$ is the [*isotropic constant*]{} of $\mu$ (see [@MilPaj]).
The paper [@GiaMil] considers isotropic measures which satisfy the $\psi_{\alpha}$-condition for scalar products: $${\left \| {\langle {\cdot} , {y} \rangle} \right \|}_{\psi_{\alpha}} \le C$$ for all $y \in S^{n-1}$. Here and below $C,c, \ldots$ denote absolute constants, whose value may change at each occurrence.
Note that by Borell’s lemma, any log-concave measure in ${{\mathbb R}}^n$ satisfies the $\psi_1$-condition [@MilSch], [@MilPaj]. Let $p \ge 1$ and let $\mu$ be an isotropic log-concave measure satisfying the $\psi_{\alpha}$ condition for scalar products with some ${\alpha}\in [1,2]$. The central result of [@GiaMil] provides an estimate for the minimal size of a set of independent random vectors $X_1, \ldots, X_m$ distributed according to the measure $\mu$ such that the empirical $p$-moments satisfy the inequality $$\label{GM result}
\Gamma_1(p)
\le \left ( \frac{1}{m} \sum_{j=1}^m |{\langle {X} , {y} \rangle}|^p
\right )^{1/p}
\le \Gamma_2(p),
\quad
\forall y \in S^{n-1}.$$ The $\psi_{\alpha}$-condition implies that the $L_p(\mu)$ and $L_2(\mu)$-norms of the function $f_y(x)={\langle {x} , {y} \rangle}$ are equivalent. Thus the inequality means that the empirical $p$-moment of $f_y$ is equivalent to the real $p$-moment up to a constant coefficient.
In the present paper we use a different approach to this problem based on the majorizing measure technique developed by Talagrand [@Tal:genericchaining]. This approach lead to breakthrough results in various problems in probabilistic combinatorics and analysis (see [@Tal:genericchaining] and references therein). In a similar context the majorizing measures were applied in [@Rud4] to select small almost orthogonal submatrices of an orthogonal matrix, and in [@Rud2] to prove the estimate with small $m$. To state the results we have to introduce some notation. Let $({{\mathbb R}}^n, \langle \cdot, \cdot \rangle)$ be a Euclidean space, and let $|\cdot|_2$ be the associated Euclidean norm. For a symmetric convex body $K$ in ${{\mathbb R}}^n$, we denote by $\| \cdot
\|_K$ the norm, whose unit ball is $K$, and by $K^o = \{y\in {{\mathbb R}}^n
\mid \forall x \in K, \langle x,y \rangle \le 1 \}$ the polar of $K$. We assume that the body $K$ has the modulus of convexity of power type $q \ge 2$ (see Section \[generalrandomvector\] for the definition). Classical examples of convex bodies satisfying this property are unit balls of finite dimensional subspaces of $L_q$ [@Clarkson] or of non-commutative $L_q$-spaces (like Schatten trace class matrices [@Tomczak]). We denote by $D$ the radius of the symmetric convex set $K$ i.e. the smallest $D$ such that $K \subset D B_{2}^n$. For every $1 \le
q \le +\infty$, we define $q^*$ to be the conjugate of $q$, i.e. $1/q + 1/q^* = 1$.
Given a random vector $X$ in ${{\mathbb R}}^n$, let $X_1, \ldots, X_m$ be $m$ independent copies of $X$. Let $K \subset {{\mathbb R}}^n$ be a convex symmetric body. Denote by $$V_{p}(K) = \sup_{y \in K} \bigg| \frac{1}{m} \sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p \ - \ {{\mathbb E}}|{\langle}X, y {\rangle}|^p \bigg|$$ the maximal deviation of the empirical $p$-moment of $X$ from the exact one. We would like to bound $V_p(K)$ under minimal assumptions on the body $K$ and random vector $X$. This will allow us to choose the size of the sample $m$ for which this deviation is small with high probability. Although the resulting statement is pretty technical, it is applicable to a wide range of problems arising in geometric functional analysis. We discuss some examples in Sections \[subspace\], \[seclogconcave\].
To bound such random process, we must have some control of the random variable $\max_{1 \le j \le m} |X_j|_2$. To this end we introduce the parameter ${\kappa}_{p,m}(X)$, which plays a key role below $${\kappa}_{p,m}(X) = \bigg( {{\mathbb E}}\max_{1 \le j \le m} |X_j|_2^p
\bigg)^{1/p}.$$ We prove the following estimate for $V_p(K)$.
\[thrandomintro\] Let $K \subset ({{\mathbb R}}^n, \langle \cdot, \cdot \rangle)$ be a symmetric convex body of radius $D$. Assume that $K$ has modulus of convexity of power type $q$ for some $q \ge 2$. Let $p \ge q$ and let $q^*$ be the conjugate of $q$.\
Let $X$ be a random vector in ${{\mathbb R}}^n$, and let $X_1, \ldots, X_m$ be independent copies of $X$. Assume that $$C_{p,\lambda} \frac{(\log m)^{2/q^*}}{ m} {\displaystyle}(D \cdot {\kappa}_{p,m}(X))^{p}
\le
{\delta}^2 \cdot \sup_{y \in K} {{\mathbb E}}|{\langle}X, y {\rangle}|^p$$ for some ${\delta}<1$. Then $${{\mathbb E}}V_p(K) \le 2 {\delta}\cdot \sup_{y \in K} {{\mathbb E}}|{\langle}X, y {\rangle}|^p.$$
The constant $C_{p,\lambda}$ in Theorem 1 depends on $p$ and on the parameter $\lambda$ in the definition of the modulus of convexity of power type $q$ (see Section 2.1 for the definition).
Note that minimal assumptions on the vector $X$ are enough to guarantee that ${{\mathbb E}}V_p(K)$ becomes small for large $m$. Indeed, assume that the variable $|X|_2$ possesses a finite moment of order $p + {\varepsilon}$ for some positive ${\varepsilon}$. Then $${\kappa}_{p,m}(X) \le \left({{\mathbb E}}\sum_{j=1}^m |X_j|_2^{p+{\varepsilon}}\right)^{1/p+{\varepsilon}}
\le m^{1/p+{\varepsilon}} ({{\mathbb E}}|X|_2^{p+{\varepsilon}})^{1/p+{\varepsilon}},$$ so the quantity $$\frac{(\log m)^{2/q^*}}{ m} \cdot {\kappa}_{p,m}^{p}(X)$$ tends to $0$ when $m$ goes to $\infty$. Moreover, in most cases, ${\kappa}_{p,m}(X)$ may be bounded by a simpler quantity: $$\label{simplerMs}
{\kappa}_{p,m}(X) \le \left( {{\mathbb E}}\sum_{j=1}^m |X_j|_2^{p}Ê\right)^{1/p}
\le e \left( {{\mathbb E}}|X|^s \right )^{1/s} =: e M_s,$$ where $s=\max(p, \log m)$. Theorem \[thrandomintro\] improves the results of [@GiaMil] in two ways. First, it contains an almost isometric approximation of the $L_p$-moments of the random vector by empirical samples (see Theorem \[logconcaveintro\] below). Second, the assumption on the norm of a random vector $X$ used in Theorem \[thrandomintro\] is weaker than the $\psi_{\alpha}$-assumption on the scalar products, appearing in [@GiaMil]. This allows to handle the situations, where the $\psi_{\alpha}$-estimate does not hold (see e.g. approximate Lewis decompositions, discussed in Section \[subspace\]).
While Theorem \[thrandomintro\] combined with Chebychev’s inequality provides a bound for $V_p(K)$, which holds with high probability, it is often useful to have this probability exponentially close to $1$. Using a measure concentration result of Talagrand ([@LT] Theorem 6.21), we obtain such probability estimate in Theorem \[concentration\].
We apply Theorem \[thrandomintro\] to isotropic $\log$-concave random vectors. This class includes many naturally arising types of random vectors, in particular a vector uniformly distributed in an isotropic convex body (see Section \[seclogconcave\] for exact definitions). The empirical moments of $\log$-concave vectors have been extensively studied in the last years [@KLS], [@Bou], [@Rud3], [@GiaMil], [@GHT]. We will prove the following
\[logconcaveintro\] For any ${\varepsilon}\in (0,1)$ and $p \ge 2$ there exists $n_0({\varepsilon}, p)$ such that for any $n \ge n_0({\varepsilon}, p)$, the following holds: let $X$ be a log-concave isotropic random vector in ${{\mathbb R}}^n$, let $X_1, \ldots, X_m$ be independent copies of $X$, if $$m = \lfloor C_p \ {\varepsilon}^{-2} n^{p/2} \log n \rfloor$$ then for any $t > {\varepsilon}$, with probabiblity greater than $1 - C \exp (- \left( t / C_p' {\varepsilon}\right)^{1/p})$, for any $y \in {{\mathbb R}}^n$, $$(1 - t) {{\mathbb E}}| {\langle}X, y {\rangle}|^p \le
\frac{1}{m} \sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p \le (1 + t) {{\mathbb E}}| {\langle}X, y {\rangle}|^p.$$ The constants $C_p$ and $C_p'$ are positive real numbers depending only on $p$.
Theorem \[logconcaveintro\] provides an almost isometric approximation of the exact moments, instead of the isomorphic estimates of [@GiaMil], and achieves it with fewer sample vectors. In the case $p=2$, it also improves the estimate of [@Rud3], and extends to the general setting the estimate obtained by Giannopoulos, Hartzoulaki and Tsolomitis [@GHT] for a random vector uniformly distributed in a 1-unconditional isotropic convex body.
The rest of the paper is organized as follows. In Section \[generalrandomvector\] we formulate and prove the main results for abstract random vectors. The key step of the proof of Theorem \[thrandomintro\] is the estimate of the Gaussian random process $$Z_y = \sum_{j=1}^m g_j | \langle X_j,y \rangle |^p,$$ where $g_j$ are independent standard Gaussian random variables ${\cal N}(0,1)$. To obtain such estimate we construct an appropriate majorizing measure and apply the Majorizing measure theorem of Talagrand [@Tal:genericchaining]. In Sections \[subspace\] and \[seclogconcave\], we provide applications of Theorem \[thrandomintro\]. Since we require only the existence of high order moments of the norm of $X$ we can apply Theorem \[thrandomintro\] to the measures supported by the contact points of a convex body, like in [@Rud1], [@Rud2], as well as to finding a short Lewis-type decomposition, as described in Section \[subspace\]. In Section \[seclogconcave\], we study in detail the case of $\log$-concave random vectors $X$. In the last part of this paper, we extend the results obtained in [@GiaMil] for a uniform distribution on a discrete cube to a general random vector $X$, which satisfies a $\psi_2$ estimate for the scalar products ${\langle {X} , {y} \rangle}, \ y \in {{\mathbb R}}^n$.
Maximal deviation of the empirical $p$-moment {#generalrandomvector}
=============================================
Statement of the results
------------------------
Let $K \subset {{\mathbb R}}^n$ be a convex symmetric body. The modulus of convexity of $K$ is defined for any ${\varepsilon}\in (0,2)$ by $$\delta_K ({\varepsilon}) = \inf \left \{1 - {\left \| \frac{x+y}{2} \right \|}_K, \|x\|_K = 1,
\|y\|_K = 1, \|x-y\|_K > {\varepsilon}\right \}.$$ We say that $K$ has modulus of convexity of power type $q \ge 2$ if $\delta_K({\varepsilon}) \ge c {\varepsilon}^q$ for every ${\varepsilon}\in (0,2)$. It is known (see e.g., [@Pisier], Proposition 2.4 or [@Figiel]) that this property is equivalent to the fact that the inequality $${\left \| \frac{x+y}{2} \right \|}_K^q + {\lambda}^{-q} {\left \| \frac{x-y}{2} \right \|}_K^q \le
\frac{1}{2} (\|x\|_K^q + \|y\|_K^q).$$ holds for all $x,y \in {{\mathbb R}}^n$. Here ${\lambda}>0$ is a constant depending only on $c$ and $q$. Referring to this inequality below, we shall say that $K$ has modulus of convexity of power type $q$ with constant ${\lambda}$.
Our main result is the following theorem, which implies Theorem \[thrandomintro\] from the Introduction.
\[thrandom\] Let $K \subset {{\mathbb R}}^n$ be a symmetric convex body of radius $D$. Assume that $K$ has modulus of convexity of power type $q$ with constant ${\lambda}$ for some $q \ge
2$, and let $q^*$ be the conjugate of $q$.\
Let $X$ be a random vector in ${{\mathbb R}}^n$ and let $X_1, \ldots, X_m$ be independent copies of $X$. For $p \ge q$ set $$A = C^p \, {\lambda}^p \frac{(\log m)^{1/q^*}}{\sqrt m} {\displaystyle}(D
{\kappa}_{p,m}(X))^{p/2} \hbox{ {\it and} } B= \sup_{y \in K} {{\mathbb E}}|{\langle}X, y {\rangle}|^p.$$ Then $${{\mathbb E}}\sup_{y \in K} \bigg| \frac{1}{m} \sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p
- \quad {{\mathbb E}}|{\langle}X, y {\rangle}|^p \bigg|
\le
A^2 + A \sqrt{B}.$$
The assumption of Theorem \[thrandomintro\] reads $A^2 \le {\delta}^2
\cdot B$, hence $A^2+A \sqrt{B} \le 2 {\delta}B$. Thus, Theorem \[thrandomintro\] follows immediately from Theorem \[thrandom\].
[**Remark.**]{} In fact we shall prove a slightly better inequality. Define $${\kappa}_{p,m}'(X, K) = \bigg( {{\mathbb E}}\max_{1 \le j \le m} |X_j|_2^2
\
\max_{1 \le j \le m} \|X_j\|_{K^o}^{p-2}
\bigg)^{1/p},$$ then Theorem \[thrandom\] holds, if the quantity $(D
{\kappa}_{p,m}(X))^{p/2} $ is replaced by $D {\kappa}_{p,m}'(X,K)^{p/2}$. Since $K \subset D B_2^n$, it is clear that $${\kappa}_{p,m}'(X, K)^{p/2}
\le
D^{p/2 -1}
{\kappa}_{p,m}(X)^{p/2}.$$
The proof of this Theorem is based on the following lemma.
\[majorizing\] Let $K \subset {{\mathbb R}}^n$ be a symmetric convex body of radius $D$. Assume that $K$ has modulus of convexity of power type $q$ with constant ${\lambda}$ for some $q \ge 2, {\lambda}>0$. Let $q^*$ be the conjugate of $q$.\
Then for every $p \ge q$, and every deterministic vectors $X_{1}, \ldots, X_{m}$ in ${{\mathbb R}}^n,$ $$\begin{array}{c}
{\displaystyle}{{\mathbb E}}\sup_{y \in K}
\bigg|
\sum_{j=1}^m {\varepsilon}_{j} |{\langle}X_j, y {\rangle}|^p
\bigg|
\le
\\
{\displaystyle}C^p \, {\lambda}^p \ (\log m)^{1/q^*} \, D \,
\max_{1 \le j \le m} |X_{j}|_{2} \,
\sup_{y \in K} \left(\sum_{j=1}^m |{\langle}X_j, y
{\rangle}|^{2(p-1)} \right)^{1/2}
\end{array}$$ where expectation is taken over the Bernoulli random variables $({\varepsilon}_{j})_{1 \le j \le m}$.
The proof of the Lemma uses a specific construction of a majorizing measure. It will be presented in part \[construction\].\
[**Proof of Theorem \[thrandom\].**]{} The proof is based on a standard symmetrization argument. We denote by $X'_{1}, \ldots,
X'_{m}$ independent copies of $X_1, \ldots, X_m$. Let $({\varepsilon}_{j})_{j=1}^m$ be independent symmetric Bernoulli random variables, which are independent of all others. Then the expectation of $$V_p(K) = \sup_{y \in K} \bigg| \frac{1}{m} \sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p
- \quad {{\mathbb E}}|{\langle}X, y {\rangle}|^p \bigg|$$ can be estimated as follows: $$\begin{array}{rcl}
m \ {{\mathbb E}}V_p(K) & = &
{\displaystyle}{{\mathbb E}}\sup_{y \in K} \bigg|
\sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p
\ - \ m \, {{\mathbb E}}|{\langle}X, y {\rangle}|^p
\bigg|
\\
& = &
{\displaystyle}{{\mathbb E}}\sup_{y \in K}
\bigg|
\sum_{j=1}^m
\big(|{\langle}X_j, y {\rangle}|^p - {{\mathbb E}}|{\langle}X'_j, y {\rangle}|^p \big)
\bigg|
\\
& \le &
{\displaystyle}{{\mathbb E}}_X {{\mathbb E}}_{X'} \sup_{y \in K}
\bigg|
\sum_{j=1}^m
\big( |{\langle}X_j, y {\rangle}|^p - |{\langle}X_j', y {\rangle}|^p \big)
\bigg|
\\
& = &
{\displaystyle}{{\mathbb E}}_X {{\mathbb E}}_{X'} {{\mathbb E}}_{{\varepsilon}}
\sup_{y \in K}
\bigg|
\sum_{j=1}^m
{\varepsilon}_j \big( |{\langle}X_j, y {\rangle}|^p - |{\langle}X_j', y {\rangle}|^p \big)
\bigg|
\\
& \le &
{\displaystyle}2 {{\mathbb E}}_X {{\mathbb E}}_{{\varepsilon}} \sup_{y \in K}
\bigg|
\sum_{j=1}^m {\varepsilon}_j |{\langle}X_j, y {\rangle}|^p
\bigg|.
\end{array}$$ Therefore, Lemma \[majorizing\] implies $${{\mathbb E}}V_p(K) \le C^p \, {\lambda}^p D \frac{(\log m)^{1/q^*}}{\sqrt m}
{{\mathbb E}}_{X} \max_{1 \le j \le m} |X_{j}|_{2} \sup_{y \in K} \bigg(
\frac{1}{m}\sum_{j=1}^m |{\langle}X_j, y {\rangle}|^{2(p-1)} \bigg)^{1/2}.$$ Since $p \ge 2$, it is easy to see that $$\begin{array}{rl}
{\displaystyle}{{\mathbb E}}_{X} \max_{1 \le j \le m} |X_{j}|_{2}
\sup_{y \in K}
\bigg( \frac{1}{m}\sum_{j=1}^m |{\langle}X_j, y {\rangle}|^{2(p-1)} \bigg)^{1/2}
& \le
\\
{\displaystyle}{{\mathbb E}}_{X} \max_{1 \le j \le m} |X_{j}|_{2}
\max_{1 \le j \le m} \|X_{j}\|_{K^o}^{p/2 -1}
\sup_{y \in K}
\bigg( \frac{1}{m}\sum_{j=1}^m |{\langle}X_j, y {\rangle}|^{p} \bigg)^{1/2}
& \le
\\
{\displaystyle}{\kappa}_{p,m}'(X, K)^{p/2}
\bigg( {{\mathbb E}}_{X} \sup_{y \in K}
\frac{1}{m}\sum_{j=1}^m |{\langle}X_j, y {\rangle}|^{p} \bigg)^{1/2}
& \le
\\
{\displaystyle}{\kappa}_{p,m}'(X, K)^{p/2}
\bigg( {{\mathbb E}}V_p(K) + \sup_{y \in K} {{\mathbb E}}|{\langle}X, y {\rangle}|^p \bigg)^{1/2}.
\end{array}$$ We get that ${{\mathbb E}}V_p(K) \le A' ( {{\mathbb E}}V_p(K) + B)^{1/2}$ where $$A' = C^p\, {\lambda}^p D \frac{(\log m)^{1/q^*}}{\sqrt m} {\kappa}_{p,m}'(X,
K)^{p/2} \hbox{ and } B = \sup_{y \in K} {{\mathbb E}}|{\langle}X, y {\rangle}|^p$$ which proves the announced result. $\hfill \Box$ We present now a deviation inequality for the positive random variable $V_p(K)$ under the assumption that $|X|_2$ satisfies some $\psi_{\alpha}$ estimate. Mendelson and Pajor [@MenPa] studied the same deviation inequality in the case $p=2$ and $K = B_2^n$ using a symmetrization argument. Our approach is based on a concentration result of Talagrand (Theorem 6.21 in [@LT]).
\[concentration\] With the same notation as in Theorem \[thrandom\], let $V_p(K)$ be the random variable $$V_p(K) = \sup_{y \in K} \bigg| \frac{1}{m} \sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p
- \quad {{\mathbb E}}|{\langle}X, y {\rangle}|^p \bigg|.$$ Assume that $\| |X|_2 \|_{\psi_{\alpha}} < \infty$ for some $0<
\alpha \le p$. Then there exists a positive constant $c_{\alpha,p}$ depending only on $\alpha$ and $p$ such that $$\forall t > 0, \ {{\mathbb P}}(V_p(K) \ge t) \le 2 \exp\left( - ( t /
Q)^{\alpha / p} \right)$$ where $$Q = c_{\alpha,p} \left( {{\mathbb E}}V_p(K) + \frac{(\log m)^{p/\alpha}}{m} D^p \| |X|_2 \|_{\psi_{\alpha}}^p \right).$$
[**Remark.**]{} Observe that in the typical case, $Q$ is of the order ${{\mathbb E}}V_p(K)$ for which we may use Theorem \[thrandom\]. By Lemma \[psia\] (see below), $${\kappa}_{p,m} (X) \le C (p \, \log m)^{1/{\alpha}} \| |X|_2 \|_{\psi_{\alpha}}$$ therefore, using Theorem \[thrandom\], $$Q \le C_{{\alpha},p} (2 A_1^2 + A_1 \sqrt B)$$ where $$A_1 = {\lambda}^p D^{p/2} \frac{(\log m)^{1/q^* + p/2 {\alpha}}}{\sqrt m} \| |X|_2 \|_{\psi_{\alpha}}
\hbox{ and }
B = \sup_{y \in K} {{\mathbb E}}|{\langle}X,y{\rangle}|^p.$$ For the proof of this theorem, we need an elementary lemma.
\[psia\] Let $\delta>0$ and let $Z_1, \ldots, Z_m$ be independent copies of a random variable $Z$. Then $$\| \, \max_{j=1, \ldots, m} |Z_j| \ \|_{\psi_{\delta}}
\le C \log^{1/{\delta}} m \cdot \| Z \|_{\psi_{{\delta}}}.$$
[**Proof.**]{} Note that for any random variable $Y$ the inequality ${\left \| Y \right \|}_{\psi_{\delta}} \le A$ is equivalent to $${\left \| Y \right \|}_r \le C A r^{1/{\delta}}$$ for all $r >1$. Assume that $r< \log m$. Then $$\begin{array}{rcl}
{\displaystyle}\| \, \max_{j=1 \ldots m} |Z_j| \ \|_r
&
\le & {\displaystyle}\| ( \sum_{j=1}^m |Z_j|^{\log m} )^{1/\log m} \|_r
\le
\left (
{{\mathbb E}}( \sum_{j=1}^m |Z_j|^{\log m})^{r/\log m}
\right )^{1/r}
\\
&\le & {\displaystyle}\left ( \sum_{j=1}^m {{\mathbb E}}|Z_j|^{\log m}
\right )^{1/\log m}
\le C \log^{1/{\delta}} m \cdot {\left \| Z \right \|}_{\psi_{\delta}}.
\end{array}$$ If $r> \log m$, then using $\max_{j=1, \ldots, m} a_j \le
(\sum_{j=1}^m a_j^r)^{1/r}$, we get $$\| \, \max_{j=1, \ldots, m} |Z_j| \ \|_r
\le ( \sum_{j=1}^m {{\mathbb E}}|Z_j|^r )^{1/r}
\le m^{1/r} {\left \| Z \right \|}_r
\le C r^{1/{\delta}} \cdot {\left \| Z \right \|}_{\psi_{\delta}}.$$ These two inequalities imply the Lemma. $\hfill \Box$
[**Proof of Theorem \[concentration\].**]{} To any vector $x \in {{\mathbb R}}^n$ we associate the function $f_x$ defined on $K$ by $$\begin{array}{rcl}
f_x : K & \to & {{\mathbb R}}\\
y & \mapsto & \frac{1}{m} \left( |{\langle}x, y {\rangle}|^p - {{\mathbb E}}|{\langle}X,y{\rangle}|^p \right).
\end{array}$$ Let $f_X$ be the random vector of $L_{\infty}(K)$ associated to $X$. Now we apply Theorem 6.21 of Ledoux–Talagrand [@LT] to ${\left \|
\sum_{j=1}^m f_{X_j} \right \|}$ where the $X_j$’s are independent copies of $X$. By definition, $$\| \sum_{j=1}^m f_{X_j} \|_{L_{\infty}(K)} = V_p(K),$$ and $$\| f_{X} \|_{L_{\infty}(K)}
\le
\frac{1}{m} \left( \sup_{y\in K} |{\langle}X,y {\rangle}|^p +
\sup_{y\in K} {{\mathbb E}}|{\langle}X,y {\rangle}|^p \right)
\le
\frac{D^p}{m} \left( |X|_2^p + {{\mathbb E}}|X|_2^p \right).$$ Theorem 6.21 of Ledoux-Talagrand [@LT] states that if $\alpha / p \le 1$, there exists a constant $c_{\alpha,p}$ depending only on $\alpha/p$ such that $$\| V_p(K) \|_{\psi_{\alpha/p}}
\le
c_{\alpha,p} \left(
{{\mathbb E}}V_p(K) + \| \max_{1 \le j \le m} \|f_{X_j}\|_{L_{\infty}(K)} \, \|_{\psi_{\alpha/p}}
\right).$$ Moreover, $$\begin{aligned}
\| \max_{1 \le j \le m} \|f_{X_j}\|_{L_{\infty}(K)} \,
\|_{\psi_{\alpha/p}}
&\le \frac{2 D^p}{m} \ \| \max_{1 \le j \le m}
|X_j|_2^p \, \|_{\psi_{\alpha/p}}
\\
&=
\frac{2 D^p}{m} \ \| \max_{1 \le j \le m} |X_j|_2 \,
\|_{\psi_{\alpha}}^p.\end{aligned}$$ Lemma \[psia\] implies $$\label{replace}
\| \max_{1 \le j \le m} |X_j|_2 \, \|_{\psi_{\alpha}}
\le
C (\log m)^{1/\alpha} \| \ |X|_2 \|_{\psi_{\alpha}}.$$ This proves that $$\| V_p(K) \|_{\psi_{\alpha/p}}
\le
c_{\alpha,p} \left(
{{\mathbb E}}V_p(K) + \frac{(\log m)^{p/\alpha}}{m} \, D^p \, \| \ |X|_2 \|_{\psi_{\alpha}}^p \right).$$ The deviation inequality follows from the Chebychev inequality. $\hfill \Box$
Construction of majorizing measures {#construction}
-----------------------------------
Let us recall the assumptions of Lemma \[majorizing\]. The ambient space is ${{\mathbb R}}^n$ equipped with a Euclidean structure and we denote by $|\cdot|_2$ the norm associated. The symmetric convex body $K$ has a modulus of convexity of power type $q \ge 2$ with a constant ${\lambda}$, which means that $$\begin{aligned}
\label{Clark}
\forall x,y \in {{\mathbb R}}^n, {\left \| \frac{x+y}{2} \right \|}_K^q + {\lambda}^{-q}
{\left \| \frac{x-y}{2} \right \|}_K^q \le \frac{1}{2} (\|x\|_K^q + \|y\|_K^q).\end{aligned}$$ and satisfies also the inclusion $K \subset D B_2^n$, which means that $$\forall x \in {{\mathbb R}}^n, |x|_2 \le D \|x\|_K.$$ Let $p \ge q \ge 2$, and $X_1, \ldots, X_m$ be $m$ fixed vectors in ${{\mathbb R}}^n$. We define the random process $V_y$ for all $y \in
{{\mathbb R}}^n$ by $$V_y = \sum_{j=1}^m {\varepsilon}_j | \langle X_j,y \rangle |^p,$$ where ${\varepsilon}_j$ are independent symmetric Bernoulli random variables. It is well known that this process satisfies a sub-Gaussian tail estimate: $\forall y, \overline y \in {{\mathbb R}}^n$, $\forall t > 0$, $$P(|V_y - V_{\overline y}| \ge t)
\le
2
\exp\left(- \frac{c t^2}{\tilde{d}^2(y,\overline y)} \right)$$ where $$\tilde{d}^2(y,\overline y) =
\sum_{j=1}^m \left(
| \langle X_j,y \rangle |^p
- | \langle X_j, \overline y \rangle |^p
\right)^2.$$ Instead of working with this function which is not a metric, it will be preferable to consider the following quasi-metric $$d^2(y,\overline y) =
\sum_{j=1}^m | \langle X_j,y-\overline y \rangle |^{2}
\left(
| \langle X_j,y \rangle |^{2(p-1)}
+ | \langle X_j, \overline y \rangle |^{2(p-1)}
\right).$$ The following propositions state inequalities that we will need to prove Lemma \[majorizing\]. Proposition \[prop:geometry\] gives some information concerning the geometry of the balls associated to the metric $d$ and Proposition \[prop:d,sup,eucl\] explains relation between metric $d$, new Euclidean norm and the following norm defined by $$\|x\|_{\infty} = \max_{1 \le j \le m} |{\langle}X_j,x{\rangle}|.$$ We denote by ${{\cal B}}_{\rho} (x)$ the ball of center $x$ with radius $\rho$ for the quasi-metric $d$.
\[prop:geometry\] For all $y, \overline y \in K$ $$\begin{aligned}
\label{ineg1}
\tilde{d}(y, \overline y) \le p \, d(y,\overline y),
\\
\label{ineg2}
d(y,\overline y)
\le \sqrt 2 \|y-\overline y\|_{\infty} \
\sup_{y \in K}
\left( \sum_{j=1}^m
| \langle X_j,y \rangle |^{2(p-1)}
\right)^{1/2},
\\
\label{ineg3}
\|y - \overline y\|_{\infty} \le D \max_{1 \le j \le m} |X_j|_2
\ \|y-\overline y\|_K.\end{aligned}$$ Moreover, the quasi-metric $d$ satisfies the generalized triangle inequality, and for any point $x$, the ball ${{\cal B}}_{\rho} (x)$ is a convex set: for all $u_1, \ldots, u_N \in {{\mathbb R}}^n$ and all $x, y,
z \in {{\mathbb R}}^n$, $$\begin{aligned}
\label{ineg:triangle}
d(u_1, u_N) \le 2p \sum_{i=1}^{N-1} d(u_i, u_{i+1})
\hbox{ and }
d^2(x, \frac{y+z}{2}) \le \frac{1}{2} \big(d^2(x,y) + d^2(x,z) \big).\end{aligned}$$
To prove it, we will need the following basic inequalities on real numbers.
\[lem:dist\] For every $x, y \in {{\mathbb R}}^+$ and $p\ge 2$, we have $$\begin{aligned}
\label{ineg:classic}
|x^p - y^p| \le p | x-y | \sqrt{x^{2p-2} + y^{2p-2}}\end{aligned}$$ Moreover, if $f(s,t) = |s-t| \sqrt{|s|^{2p-2} + |t|^{2p-2}}$ then for all $r_1, \ldots, r_N \in {{\mathbb R}}$ $$f(r_1, r_N) \le 2p \sum_{i=1}^{N-1} f(r_i, r_{i+1})$$ and for all $r,$ $s$, $t \in {{\mathbb R}}$, $$f(r,(s+t)/2)^2 \le \big(f(r,s)^2 + f(r,t)^2\big)/2$$
[**Proof.**]{} The first inequality is straightforward. To prove the second one, consider two cases. When $r_1 r_N \ge 0$, since $$|r_1-r_N| \sqrt{|r_1|^{2p-2} + |r_N|^{2p-2}}
\le \sqrt 2 \Big ||r_1|^p - |r_N|^p \Big |,$$ the conclusion follows from the triangle inequality and inequality (\[ineg:classic\]). When $r_1 r_N \le 0$, we can assume without loss of generality that $r_1 \ge 0$ and $r_N \le 0$. Then $$\begin{aligned}
f(r_1, r_N)
&= (r_1+|r_N|) \sqrt{r_1^{2p-2} + |r_N|^{2p-2}}
\le
(r_1+|r_N|)(r_1^{p-1} + |r_N|^{p-1}) \\
&\le 2(r_1^p + |r_N|^{p}).\end{aligned}$$ Let $m<N$ be a number such that $r_m \ge 0$ and $r_{m+1} \le 0$. Then $$r_1^p + |r_N|^{p}
\le \sum_{i=1}^{m-1} \Big | |r_i|^p-|r_{i+1}|^p \Big |
+r_{m}^p + |r_{m+1}|^{p}
+ \sum_{i=m+1}^{N-1} \Big | |r_i|^p-|r_{i+1}|^p \Big |.$$ Combining the previous inequalities with (\[ineg:classic\]), we get $$\begin{aligned}
f(r_1, r_N)
& \le 2\sum_{i=1}^{m-1} \Big | |r_i|^p-|r_{i+1}|^p \Big |
+2(r_{m}^p + |r_{m+1}|^{p})
+ 2\sum_{i=m+1}^{N-1} \Big | |r_i|^p-|r_{i+1}|^p \Big |
\\
& \le 2p \sum_{i=1}^{m-1} f(r_i, r_{i+1})
+ 2(r_{m} - r_{m+1})(r_{m}^{p-1} + |r_{m+1}|^{p-1}) \\
& + 2p \sum_{i=m+1}^{N-1} f(r_i, r_{i+1})
\\
& \le
2p \sum_{i=1}^{m-1} f(r_i, r_{i+1})
+ 2 \sqrt 2 f(r_{m}, r_{m+1})
+ 2p \sum_{i=m+1}^{N-1} f(r_i, r_{i+1})\end{aligned}$$ which proves the announced result. The last inequality follows from the fact that for $p \ge 2,$ the function $v \mapsto
(1-v)^2(1+v^{2p-2})$ is convex on ${{\mathbb R}}$, which can be checked by computing the second derivative. $\hfill \Box$
[**Proof of Proposition \[prop:geometry\].**]{} Inequalities (\[ineg1\]) and (\[ineg:triangle\]) clearly follow from the three inequalities proved in Lemma \[lem:dist\]. Inequalities (\[ineg2\]) and (\[ineg3\]) follow from simple observations about $d$ and the fact that $K \subset D B_2^n$. $\hfill \Box$
\[prop:d,sup,eucl\] Let $
{\displaystyle}M = \sup_{y \in K}\sum_{j=1}^m
| \langle X_j,y \rangle |^{2(p-1)}.
$ For a fixed $u \in K$, we define the Euclidean norm $|\cdot|_{{\cal E}_u}$ associated to $u$ by $$|z|_{{\cal E}_u}^2 =
\sum_{\ell = 1}^m
| \langle X_{\ell}, z \rangle |^2
| \langle X_{\ell}, u \rangle |^{2(p-1)},
\ \forall z \in {{\mathbb R}}^n.$$ Then the following inequality holds for all $z$, $\overline z \in {{\mathbb R}}^n$: $$d^2(z, \overline z) \le
2 \cdot 4^{p-1} \big( |z-\overline z|_{{\cal E}_u}^2 +
M \ \|z-\overline z\|_{\infty}^2
(\|z-u\|_K^{2p-2}
+\|\overline z -u \|_K^{2p-2} )\big).$$
[**Proof.**]{} By homogeneity of the statement, we can assume that $$M = \sup_{y \in K}\sum_{j=1}^m
| \langle X_j,y \rangle |^{2(p-1)}=1.$$ For any $z \in {{\mathbb R}}^n$, let $L_z = \{\ell \in \{1, \ldots, m\}\,
\big | \ |{\langle}X_{\ell}, z {\rangle}| \ge 2 |{\langle}X_{\ell}, u {\rangle}|
\}$. Then by convexity of the function $t \mapsto t^{2p-2}$, we have $$\begin{array}{rcl}
{\displaystyle}\sum_{\ell \in L_z}
| \langle X_{\ell}, z \rangle |^{2(p-1)}
& \le &
{\displaystyle}2^{2p-3} \sum_{\ell \in L_z}
| \langle X_{\ell}, z - u \rangle |^{2(p-1)}
+
2^{2p-3} \sum_{\ell \in L_z}
| \langle X_{\ell}, u \rangle |^{2(p-1)}
\\
& \le & {\displaystyle}2^{2p-3} \sum_{\ell \in L_z} | \langle X_{\ell}, z -
u \rangle |^{2(p-1)} + \frac{1}{2} \sum_{\ell \in L_z} | \langle
X_{\ell}, z \rangle |^{2(p-1)},
\end{array}$$ which proves (since $M=1$) that for any $z \in {{\mathbb R}}^n$, $$\sum_{\ell \in L_z}
| \langle X_{\ell}, z \rangle |^{2(p-1)}
\le 4^{p-1} \|z - u\|_K^{2p-2}.$$ Hence, for any $z, \overline z \in {{\mathbb R}}^n$, $$\begin{aligned}
\sum_{\ell \in L_z} | \langle X_{\ell}, z - \overline z
\rangle |^2 | \langle X_{\ell}, z \rangle |^{2(p-1)}
& \le {\displaystyle}\|z - \overline z\|_{\infty}^2
\sum_{\ell \in L_z}| \langle X_{\ell}, z \rangle |^{2(p-1)}
\\
& \le
4^{p-1} \|z - u\|_K^{2p-2} \, \|z- \overline z
\|_{\infty}^2.\end{aligned}$$ For any $l \notin L_z$ we have $|{\langle {X_l} , {z} \rangle}| \le 2
|{\langle {X_l} , {u} \rangle}|$, so $$\sum_{\ell \notin L_z} | \langle X_{\ell}, z -
\overline z \rangle |^2 | \langle X_{\ell}, z \rangle |^{2(p-1)}
\le 4^{p-1} \sum_{\ell = 1}^m | \langle X_{\ell}, z - \overline z
\rangle |^2 | \langle X_{\ell}, u \rangle |^{2(p-1)}.$$ The same inequalities hold if we exchange the roles of $z$ and $\overline z$. To compute $d^2(z_i, z_j)$, we split the sum in four parts and apply the inequalities above: $$\begin{array}{l}
d^2(z, \overline z) =
{\displaystyle}\sum_{\ell = 1}^m
| \langle X_{\ell}, z - \overline z \rangle |^2
| \langle X_{\ell}, z \rangle |^{2(p-1)}
+
| \langle X_{\ell}, z - \overline z \rangle |^2
| \langle X_{\ell}, \overline z \rangle |^{2(p-1)}
\\
=
{\displaystyle}\sum_{\ell \in L_z}
| \langle X_{\ell}, z - \overline z \rangle |^2
| \langle X_{\ell}, z \rangle |^{2(p-1)}
+ \sum_{\ell \notin L_z}
| \langle X_{\ell}, z - \overline z \rangle |^2
| \langle X_{\ell}, z \rangle |^{2(p-1)}
\\
+ {\displaystyle}\sum_{\ell \in L_{\overline z}}
| \langle X_{\ell}, z - \overline z \rangle |^2
| \langle X_{\ell}, \overline z \rangle |^{2(p-1)}
+ \sum_{\ell \notin L_{\overline z}}
| \langle X_{\ell}, z - \overline z \rangle |^2
| \langle X_{\ell}, \overline z \rangle |^{2(p-1)}
\\
\le {\displaystyle}2 \cdot 4^{p-1} \left(|z - \overline z |_{{\cal E}_u}^2 +
\|z- \overline z \|_{\infty}^2 (\|z - u\|_K^{2p-2} + \|z -
u\|_K^{2p-2}) \right). \hfill \Box
\end{array}$$ [**Proof of Lemma \[majorizing\].**]{} By inequality (\[ineg1\]), we may treat $V_y$ as a sub-Gaussian process with the quasi-metric $p \cdot d$. By homogeneity of the statement, we can assume that $$\sup_{y \in K} \sum_{j=1}^m | \langle X_j,y \rangle |^{2(p-1)} =
1.$$ Denote $Q = \max_{1\le j \le m} |X_j|_2$. We want to show that $$\begin{aligned}
\label{sup}
{{\mathbb E}}\sup_{y \in K} |V_y| \le C^p \, {\lambda}^p \ Q \ (\log m)^{1/q^*} \
D.\end{aligned}$$ By Proposition \[prop:geometry\], the diameter of the set $K$ with respect to the metric $d$ is bounded by $2\sqrt 2 Q D$. Let $r$ be a fixed number chosen such that $r = c p^2$ for a large universal constant $c$ and $k_0$ be the largest integer such that $r^{-k_0} \ge 2\sqrt 2 QD$.
The proof of inequality $(\ref{sup})$ is based on the majorizing measure theory of Talagrand [@Tal:genericchaining]. The following theorem is a combination of Proposition 2.3, Theorem 4.1 and Proposition 4.5 of [@Tal:genericchaining]. Note that assuming that $r \ge 2$, one can set $K(r)=C$ in Proposition 2.3, and $K(2,1,r)=C$ in Proposition 4.5.
[*Let $r \ge 2$. Let $\phi_k : K \to {{\mathbb R}}^+$ for $k \ge k_0$ be a family of maps satisfying the following assumption: there exists $A>0$ such that for any point $x \in K$, for any $k \ge
k_0$ and any $N \in {{\mathbb N}}$\
$
(H)
\left\{
\begin{array}{l}
for \ any \ points \ x_1, \ldots, x_N \in {{\cal B}}_{r^{-k}}(x)
\ with \ d(x_i, x_j) \ge r^{-k-1}, i \ne j
\\
we \ have \ {\displaystyle}\max_{i=1, \ldots, N} \phi_{k+2}(x_i)
\ge \phi_k(x) + \, \frac{1}{A} \ r^{-k} \sqrt{\log N}.
\end{array}
\right.
$\
Then for any fixed $y_0 \in K$, $${{\mathbb E}}\sup_{y \in K} |V_y - V_{y_0}|
\le c \ A \cdot \sup_{k\ge k_0, x\in K} \phi_k(x).$$* ]{}
To obtain the conclusion of Lemma \[majorizing\], set $y_0 =
0$.\
To complete the proof , we have to define the functionals $\phi_k
: K \to {{\mathbb R}}^+$. Let $k_1$ be the smallest integer such that $r^{-k_1} \le QD / \sqrt n$. For $k \ge k_1 +1$, set
$$\phi_k(x) =
1 + \frac{1}{2 \log r} + \frac{\sqrt n}{Q \, D \, (\log m)^{1/q^*}}
\sum_{l=k_1}^k r^{-l}
\sqrt{\log(1 + 4 Q D r^{l})}.$$ Note that in this range of $k$ the functionals $\phi_k$ do not depend on $x$. We shall show that with this choice of $\phi_k$, the condition $(H)$ follows from the classical volumetric estimate of the covering numbers.
For $k_0 \le k \le k_1$, the functionals $\phi_k$ are defined by $$\phi_k(x) =
\min\{ \|y\|_K^q, y \in {{\cal B}}_{4pr^{-k}}(x) \} +
\frac{k-k_0}{\log m}.$$ Since $q \ge 2$ then $1\le q^* \le 2$ and $(\log m)^{1/q^*} \ge \sqrt{\log m}$. It is easy to see using definitions of $k_0$ and $k_1$ that $$\sup_{x \in K, k\ge k_0} \phi_k(x) \le c.$$ We shall prove that our functionals satisfy condition $(H)$ for $$A = (C {\lambda})^p \ Q \ D \ (\log m)^{1/q^*}$$ where $C$ is a large numerical constant. That will conclude the proof of Lemma \[majorizing\] with a new constant $C$. $\hfill \Box$\
[**Proof of condition $(H)$.**]{} Let $N \in {{\mathbb N}}$, $x \in K$, $x_1, \ldots, x_N \in {{\cal B}}_{r^{-k}}(x)$ with $d(x_i, x_j) \ge
r^{-k-1}$. We have to prove that $$\max_{i=1, \ldots, N} \phi_{k+2} (x_i) - \phi_k(x)
\ge \frac{r^{-k} \sqrt{\log N}}{ (C \lambda)^p \ Q \ D \ (\log
m)^{1/q^*}}.$$
For $k \ge k_1-1$, we always have $$\phi_{k+2}(x_i) - \phi_k(x)
\ge
\frac{\sqrt {n \log(1 + 4 Q D r^{k+2})}}
{Q D (\log m)^{1/q^*}} \ r^{-k-2}.$$ Since the points $x_1, \ldots, x_N$ are well separated in the metric $d$, they are also well separated in the norm $\|\cdot\|_K$. Indeed, by $(\ref{ineg2})$ and $(\ref{ineg3})$, we have $$\|x_i - x_j\|_K \ge r^{-k-1} / Q D \sqrt 2.$$ By the classical volumetric estimate, the maximal cardinality of a $t$-net in a convex symmetric body $K \subset {{\mathbb R}}^n$ with respect to $\|\cdot
\|_K$ does not exceed $
\left ( 1+ 2/t \right )^n.
$ Therefore, $$\sqrt{\log N} \le \sqrt{n \log(1 + 2 \sqrt{2} QD r^{k+1})},$$ which proves the desired inequality.
The case $k_0 \le k \le k_1-2$ is much more difficult. Our proof uses estimates of the covering numbers, in particular, the dual Sudakov inequality [@PajTom]. Recall that the covering number $N(W,\|\cdot \|_X,t)$ is the minimal cardinality of $\|\cdot\|_X$-balls of radius $t$ needed to cover the $W$.
For $j = 1, \ldots, N$ denote by $z_j \in K$ the points which satisfy $ \|z_j\|_K^q = \min \{ \|y\|_K^q, y \in
{{\cal B}}_{4pr^{-k-2}}(x_j) \} $. Denote by $u \in K$ a point such that $
\|u\|_K^q = \min \{ \|y\|_K^q, y \in {{\cal B}}_{4pr^{-k}}(x) \}. $ Set $$\theta = \max_{j} \|z_j\|_K^q - \|u\|_K^q.$$ Then we have $ \max_{j} \phi_{k+2} (x_j) - \phi_k(x) = \theta +
\frac{2}{\log m} $. We shall prove that $$\begin{aligned}
\label{wanted}
\theta + \frac{2}{\log m}
\ge
r^{-k} \sqrt{\log N} / A.\end{aligned}$$ Since $d(x_i, x_j) \ge r^{-k-1}$, $z_l \in {{\cal B}}_{4pr^{-k-2}}(x_l)$, and $d$ satisfies a generalized triangle inequality, the points $(z_j)_{1\le j \le N}$ remain well separated. Indeed, $$r^{-k-1}
\le
d(x_i, x_j) \le 2p (d(x_i, z_i) + d(z_i, z_j) + d(z_j, x_j))
\le
2p d(z_i, z_j) + 16 p^2 r^{-k-2}$$ and since $r=cp^2$, we have $$d(z_i, z_j) \ge r^{-k-1}/cp$$ for all $i \ne j$. Recall that $r=cp^2$. Using again the generalized triangle inequality, we get that $$d(x,z_j) \le 2p(d(x, x_j) + d(x_j, z_j)) \le 2p (r^{-k} + 4p
r^{-k-2}) \le 4p r^{-k}.$$ It means that $z_j \in {{\cal B}}_{4pr^{-k}}(x)$, $u \in
{{\cal B}}_{4pr^{-k}}(x)$, and the convexity of the balls for the quasi-metric $d$ proved in Proposition \[prop:geometry\] implies $(u+z_j)/2 \in {{\cal B}}_{4pr^{-k}}(x)$. Since $K$ has modulus of convexity of power type $q$, inequality $(\ref{Clark})$ holds. By the definition of $u$, we get that for all $j=1, \ldots, N$ $${\lambda}^{-q} {\left \| \frac{z_j - u}{2} \right \|}_K^q \le \frac{1}{2} \left(
\|z_j\|_K^q + \|u\|_K^q \right) - {\left \| \frac{z_j + u}{2} \right \|}_K^q \le
\frac{\|z_j\|_K^q - \|u\|_K^q}{2} \le \frac{\theta}{2}.$$ This proves that $\forall j=1, \ldots, N, \|z_j - u \|_K \le 2 {\lambda}\theta^{1/q}. $ Let $\delta>0$. Consider the set $$U = u + 2 {\lambda}\theta^{1/q} K$$ which contains all the $z_j$’s and let $S$ be the maximal number of points in $U$ that are $2 \delta$ separated in $\|\cdot\|_{\infty}$. Then $U$ is covered by $S$ subsets of diameter smaller than $2\delta$ in $\|\cdot\|_{\infty}$ metric, and so $ S \le N(U, \|\cdot\|_{\infty}, 2 \delta). $ Set $$\delta = \tilde{c}^p {\lambda}^{1-p} r^{-k} \theta^{1/q - 1}$$ where the constant $\tilde{c}$ will be chosen later. Since $U = u
+ 2 {\lambda}\theta^{1/q} K$ and $K \subset D B_2^n$,the dual Sudakov inequality [@PajTom] implies $$\sqrt{\log S} \le \sqrt{\log N(B_2^n, \|\cdot\|_{\infty}, \delta /
D {\lambda}\theta^{1/q})} \le c \ D \ {\lambda}\ \theta^{1/q} \ {{\mathbb E}}\|G\|_{\infty} \, / \, \delta.$$ Here $G$ denotes a standard Gaussian vector in ${{\mathbb R}}^n$. It is well known that $${{\mathbb E}}\|G\|_{\infty} = {{\mathbb E}}\max_{j=1, \ldots, m} |{\langle}X_j, G {\rangle}|
\le c \ Q \ \sqrt{\log m}.$$ We consider now two cases.
First, assume that $S \ge \sqrt N$. Then by previous estimate and the definition of $\delta$, we get $$\sqrt{\log N}
\le c \ Q \ {\lambda}\ D \ \sqrt{\log m} \ \theta^{1/q} \, / \, \delta
\le \theta \ \tilde{c}^p
\ r^k \ Q \ {\lambda}^p \ D \ \sqrt{\log m}$$ which easily proves $(\ref{wanted})$ (since $q^* \le 2$).
The second case is when $S\le \sqrt N$. Since $U$ is covered by $S$ balls of diameter smaller than $2\delta$ in $\|\cdot\|_{\infty}$, there exists a subset $J$ of $\{1, \ldots,
N\}$ with $\# J \ge \sqrt N$ such that $$\forall i,j \in J, \|z_i - z_j\|_{\infty} \le 2 \delta.$$ By Proposition \[prop:d,sup,eucl\] applied to the Euclidean norm defined by $$|y|_{{\cal E}_u}^2 = \sum_{\ell = 1}^m
| \langle X_{\ell}, y \rangle |^2
| \langle X_{\ell}, u \rangle |^{2(p-1)},$$ we get that $$d^2(z_i, z_j)
\le
2 \cdot 4^{p-1} \left(|z_i - z_j|_{{\cal E}_u}^2 + 4^p {\lambda}^{2p -2} \theta^{(2p-2)/q}
\delta^2 \right).$$ Since $\theta \le 1$ and $q \le p$, the definition of $\delta$ implies $$4^{2p} {\lambda}^{2p -2} \theta^{(2p-2)/q} \, \delta^2
\le (4\tilde{c})^{2p} r^{-2k} \theta^{2(p/q -1)}
\le (4\tilde{c})^{2p} r^{-2k}.$$ Recall that $d(z_i, z_j) \ge r^{-k-1} / cp$ and $r = c p^2$. Hence, $$r^{-2k} / cp^6 \le d(z_i, z_j)^2
\le 2 \cdot 4^{p-1} |z_i - z_j|_{{\cal E}_u}^2
+ 2 (4\tilde{c})^{2p} r^{-2k}.$$ Choosing $\tilde{c}$ small enough, we get that for all $i, j \in
J$, $$|z_i - z_j|_{{\cal E}_u} \ge r^{-k-1} c^p.$$ Since $K \subset D B_2^n$, we have the following estimate for the covering numbers: $$\begin{array}{rcl}
\# J \le N(U, |\cdot|_{{\cal E}_u}, c^p r^{-k-1})
& = &
N(K, |\cdot|_{{\cal E}_u}, c^p r^{-k-1} / 2 {\lambda}\theta^{1/q})
\\
& \le &
N(B_2^n, |\cdot|_{{\cal E}_u}, c^p r^{-k-1}/ 2 {\lambda}\theta^{1/q} D).
\end{array}$$ Recall that $G$ denotes a standard Gaussian vector in ${{\mathbb R}}^n$. By the dual Sudakov inequality [@PajTom], we have $$\begin{array}{rcl}
{\displaystyle}\sqrt{\log N(B_2^n, |\cdot|_{{\cal E}_u},
\frac{c^p r^{-k-1}}{2 {\lambda}D \theta^{1/q}})}
& \le &
C^p \ r^{k+1} \ \theta^{1/q} \ {\lambda}\ D \
{{\mathbb E}}|G|_{{\cal E}_u}
\\
& \le & {\displaystyle}C^p \ r^{k+1} \ \theta^{1/q} \ {\lambda}\ D \left( {{\mathbb E}}|G|_{{\cal E}_u}^2\right)^{1/2}.
\end{array}$$ Since for all $y \in K$, $\sum_{j=1}^m | \langle X_j,y \rangle
|^{2(p-1)} \le 1$, we obtain $${{\mathbb E}}|G|_{{\cal E}_u}^2 = \sum_{\ell = 1}^m |X_{\ell}|_2^2
| \langle X_{\ell}, u \rangle |^{2(p-1)} \le Q^2.$$ Since $\# J \ge \sqrt N$, we have $ \sqrt{\log N} \le C^p r^{k+1}
{\lambda}D Q \theta^{1/q} $ with a universal constant $C$. Moreover, by Young’s inequality $$\theta^{1/q} \le (\log m)^{1/q^*}
\left( \theta / q + 1 / (q^* \log m) \right)$$ and since $q^* \le 2 \le q$ and ${\lambda}\ge 1$, we get $$\sqrt{\log N}
\le
(C {\lambda})^p \ r^{k+1} \ D \ Q \
(\log m)^{1/q^*}
\left( \theta + \frac{2}{\log m} \right).$$ This completes the proof of $(\ref{wanted})$ and the proof of condition $(H)$ for the functionals $\phi_k$. $\hfill \Box$
Approximate Lewis decomposition {#subspace}
===============================
It is well known that if $E$ is an $n$-dimensional subspace of $L_p$, then $E$ is $(1+{\varepsilon})$-isomorphic to an $n$-dimensional subspace of $\ell_p^N$ with $N$ depending on $n$, $p$ and ${\varepsilon}$. Lewis [@Lewis] proved that any linear subspace $E$ of $\ell_p^N$ possesses a special decomposition of the identity. More precisely, there exists a Euclidean structure on $E$ with the scalar product $\langle \cdot, \cdot
\rangle$, vectors $y_{1}, \ldots, y_{N} \in E$ and scalars $c_{1},
\ldots, c_{N} > 0$ such that $$\left \{
\begin{array}{l}
\forall i, {\langle {y_i} , {y_i} \rangle} = 1 ,
\\
\| x \|_{E} = {\displaystyle}\bigg( \sum_{i=1}^{N} c_{i} |\langle x, y_{i} \rangle|^p
\bigg)^{1/p} , \forall x \in E,
\\
{\rm Id}_{E} = {\displaystyle}\sum_{i=1}^{N} c_{i} y_{i} \otimes y_{i}.
\end{array}
\right.$$
Denote by $(H, | \cdot |_{H})$ the linear space $E$ equipped with this Euclidean structure. Recall that $p^*$ denotes the conjugate of $p$. In the following Theorem, we prove that both spaces $E$ and $H$ can be $(1+ {\varepsilon})$-embedded in $\ell_p^m$ and $\ell_2^m$ respectively via the same linear operator $T: {{\mathbb R}}^N \to {{\mathbb R}}^m$, whenever $m$ is of the order of ${\varepsilon}^{-2} n^{p/2} \log^{2/p^*} (n / {\varepsilon}^{4/p})$. This extends a classical result of Bourgain, Lindenstrauss and Milman [@BouLinMil] (and [@LT] for a better dependance on ${\varepsilon}$) and some results in [@Rud1] concerning the number of contact points of a convex body needed to approximate the identity decomposition.
\[subLp\] Let $E$ be an $n$-dimensional subspace of $L_p$ for some $p \ge
2$. Then for every ${\varepsilon}> 0$ there exists a Euclidean structure $H= (E,{\langle}\cdot, \cdot {\rangle})$ on $E$ and $m$ points $x_1, \ldots, x_m$ in $E$ with $$m \le \frac{C^p}{{\varepsilon}^{2}} \ n^{p/2} \ \log^{2/p^*} \left (
\frac{n}{{\varepsilon}^{4/p}} \right )
\le \frac{C^p}{{\varepsilon}^{2}} \ n^{p/2} \log^2 \left (
\frac{n}{{\varepsilon}^{4/p}} \right )$$ such that $ \forall j, | x_{j} |_{H} = 1 $ and for all $y \in E$, $$\left \{
\begin{array}{l}
(1 - {\varepsilon}) \| y \|_{E}
\le
{\displaystyle}\bigg( \frac{n}{m} \sum_{j=1}^{m} |\langle y, x_{j} \rangle|^p
\bigg)^{1/p}
\le (1 + {\varepsilon}) \| y \|_{E}
\\
(1 - {\varepsilon}) | y |_{H}
\le
{\displaystyle}\bigg( \frac{n}{m} \sum_{j=1}^{m} |\langle y, x_{j} \rangle|^2
\bigg)^{1/2}
\le (1 + {\varepsilon}) | y |_{H}.
\end{array}
\right.$$
[**Proof.**]{} Let $X$ be the random vector taking values $y_{i}$ with probability $c_{i} / n$. Then for all $y \in E$, $${{\mathbb E}}|\langle X, y \rangle|^p = \| y \|_{E}^p / n
\quad
\hbox{ and }
\quad
{{\mathbb E}}|\langle X, y \rangle|^2 = |y|_{H}^{2} / n
\quad
\hbox{ and }
\quad
|X|_{H} = 1.$$
We will apply Theorem \[thrandomintro\] twice: first time for the unit ball of $E$, and then for the unit ball of $H$.
Since $E$ is a subspace of $L_p$, by Clarkson’s inequality [@Clarkson], $B_E$ has modulus of convexity of power type $p$ with constant ${\lambda}= 1$. From Lewis decomposition, we get $\| y \|_{E} \le |y|_{H} \le n^{\frac{1}{2} - \frac{1}{p}}
\| y \|_{E}$ which means that for $ D = n^{\frac{1}{2} -
\frac{1}{p}}$, $$B_H \subset B_E \subset
D B_H.$$ Let $X_{1}, \ldots, X_{m}$ be independent copies of $X$, then $$\sup_{y \in B_E}
{{\mathbb E}}|{\langle}X, y {\rangle}|^p = 1 / n
\quad
\hbox{ and }
\quad
{\kappa}_{p,m}(X) =
\bigg( {{\mathbb E}}\max_{1 \le j \le m} |X_j|_H^p
\bigg)^{1/p}
= 1.$$ Applying Theorem \[thrandomintro\] with ${\delta}={\varepsilon}$, we get that if $m \ge C^p n^{p/2} (\log m)^{2/p^*} / {\varepsilon}^{2}$, then $${{\mathbb E}}\sup_{y \in B_E} \bigg| \frac{n}{m} \sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p
- \| y \|_{E}^p \bigg| \le
{\varepsilon}.$$
Now, we apply Theorem \[thrandomintro\] for $K=B_H$ which clearly has modulus of convexity of power type 2 (i.e. satisfies inequality (\[Clark\]) for $q=2$). In that case, $D = 1$, and $$\sup_{|y|_H \le 1}
{{\mathbb E}}|{\langle}X, y {\rangle}|^2 = 1 / n
\quad
\hbox{ and }
\quad
{\kappa}_{2,m}(X) =
\bigg( {{\mathbb E}}\max_{1 \le j \le m} |X_j|_2^2
\bigg)^{1/2}
=1.$$ Applying Theorem \[thrandomintro\] for $q=p=2$ and ${\delta}={\varepsilon}$, we get that if $m \ge C^2 n \log m / {\varepsilon}^2$, $${{\mathbb E}}\sup_{|y|_{H} \le 1} \bigg| \frac{n}{m} \sum_{j=1}^m |{\langle}X_j, y
{\rangle}|^2
- |y|_{H}^{2} \bigg| \le {\varepsilon}.$$ Choosing the smallest integer $m$ such that, for a new constant $\tilde C$, $$m \ge \frac{\tilde{C}^p}{{\varepsilon}^{2}} \ n^{p/2} \
(\log n / {\varepsilon}^{4/p})^{2/p^*}$$ we get by Chebychev’s inequality that there exist $m$ vectors $x_{1}, \ldots, x_{m}$ of Euclidean norm 1 such that for all $y \in E,$ $$\bigg| \frac{n}{m} \sum_{j=1}^m |{\langle}x_j, y {\rangle}|^p
- \| y \|_{E}^p \bigg| \le
{\varepsilon}\|y\|_E^p$$ and $$\bigg| \frac{n}{m} \sum_{j=1}^m |{\langle}x_j, y
{\rangle}|^2
- |y|_{H}^{2} \bigg| \le
{\varepsilon}|y|_{H}^{2}$$ which gives the desired result. $\Box$
Isotropic log-concave vectors in ${{\mathbb R}}^n$ {#seclogconcave}
==================================================
We investigate the case of $X$ being an isotropic log-concave vector in ${{\mathbb R}}^n$ (or also a vector uniformly distributed in an isotropic convex body). Let us recall some definitions and classical facts about $\log$-concave measures. A probability measure $\mu$ on ${{\mathbb R}}^n$ is said to be $\log$-concave if for every compact sets $A, B$, and every $ {\lambda}\in [0,1]$, $\mu({\lambda}A +
(1-{\lambda}) B) \ge \mu(A)^{{\lambda}} \mu(B)^{1-{\lambda}}$. There is always a Euclidean structure ${\langle}\cdot, \cdot {\rangle}$ on ${{\mathbb R}}^n$ for which this measure is isotropic, i.e. for every $y\in {{\mathbb R}}^n$, $${{\mathbb E}}{\langle}X,y {\rangle}^2 =
\int_{{{\mathbb R}}^n} {\langle}x,y {\rangle}^2 d\mu(x) = | y |_2^2.$$ A particular case of a $\log$-concave probability measure is the normalized uniform (Lebesgue) measure on a convex body. Borell’s inequality [@Borell] (see also [@MilSch; @MilPaj]) implies that the linear functionals $x \mapsto {\langle}x,y {\rangle}$ satisfy Khintchine type inequalities with respect to $\log$-concave probability measures. Namely, if $p
\ge 2$, then for every $y \in {{\mathbb R}}^n$, $$\begin{aligned}
\label{KKlog}
\left( {{\mathbb E}}{\langle}X,y {\rangle}^2 \right)^{1/2} \le \left( {{\mathbb E}}|{\langle}X,y
{\rangle}|^p \right)^{1/p} \le C p \left( {{\mathbb E}}{\langle}X,y {\rangle}^2
\right)^{1/2},\end{aligned}$$ or in other words $$\| {\langle}\cdot, y {\rangle}\|_{\psi_1} \le C \left( {{\mathbb E}}{\langle}X,y {\rangle}^2 \right)^{1/2}.$$ We have stated in $(\ref{simplerMs})$ that it is easy to deduce some information about the parameter ${\kappa}_{p,m}(X)$ from the behavior of the moment $M_s$ of order $s= \max(p, \log m)$ of the Euclidean norm of the random vector $X$. These moments were studied for a random vector uniformly distributed in an isotropic 1-unconditional convex body in [@BN], and for a vector uniformly distributed in the unit ball of a Schatten trace class in [@GP], where it was proved that when $s \le c \sqrt n$, $M_s$ is of the same order as $M_2$ (up to constant not depending on $s$). Very recently, Paouris [@Paouris05] proved that the same statement is valid for any $\log$-concave isotropic random vector in ${{\mathbb R}}^n$. We state precisely his result.
\[[@Paouris05]\][There exist constants $c,C > 0$ such that for any $\log$-concave isotropic random vector $X$ in ${{\mathbb R}}^n$, for any $p \le c \, \sqrt
n$, $$\left( {{\mathbb E}}|X|_2^p \right)^{1/p} \le C \left( {{\mathbb E}}|X|_2^2 \right)^{1/2}.$$ ]{}
From this sharp estimate, we will deduce the following
\[max\] Let $X$ be an isotropic log-concave random vector in ${{\mathbb R}}^n$ and let $(X_{j})_{1 \le j \le m}$ be independent copies of $X$. If $m
\le e^{c \, \sqrt n}$, then for any $p \ge 2$ $${\kappa}_{p,m}(X) =
\bigg({{\mathbb E}}\max_{1 \le j \le m} |X_{j}|_{2}^p\bigg)^{1/p}
\le
\, \left\{\begin{array}{l}
C \sqrt n \hbox{ if } p \le \log m\\
C \ p \ \sqrt n \hbox{ if } p \ge \log m
\end{array}
\right.$$
[**Proof.**]{} Since $X$ is isotropic, and for every $y \in {{\mathbb R}}^n$, ${{\mathbb E}}{\langle}X, y {\rangle}^2 = |y|_2^2$, we get ${{\mathbb E}}|X|_{2}^2 = n$. By Borell’s inequality [@Borell], $ \forall q \ge 2$, $({{\mathbb E}}|X|_2^q)^{1/q} \le C q \sqrt n$. Therefore if $p \ge \log m$, $$\bigg({{\mathbb E}}\max_{1 \le j \le m} |X_{j}|_{2}^p\bigg)^{1/p}
\le
\bigg({{\mathbb E}}\sum_{1 \le j \le m} |X_{j}|_{2}^p\bigg)^{1/p}
\le C p m^{1/p} \sqrt n \le C p \sqrt n.$$ If $p \le \log m$, by $(\ref{simplerMs})$ $$\bigg({{\mathbb E}}\max_{1 \le j \le m} |X_{j}|_{2}^p\bigg)^{1/p}
\le
e \left( {{\mathbb E}}|X|_2^{\log m} \right)^{1/ \log m}.$$ Since $m \le e^{c \, \sqrt n}$, $\log m \le c \, \sqrt n$, the Theorem of Paouris implies $$\left( {{\mathbb E}}|X|_2^{\log m} \right)^{1/ \log m} \le C \sqrt n,$$ which concludes the proof of the Lemma. $\Box$
\[psi1maxlog\] Let $X$ be an isotropic log-concave random vector in ${{\mathbb R}}^n$, and let $(X_{j})_{1 \le j \le m}$ be independent copies of $X$. Then for every $m \le e^{c \, \sqrt n}$ $$\| \, \max_{1 \le j \le m} |X_j|_2 \ \|_{\psi_1} \le C \sqrt{n}.$$
[**Proof.**]{} By Lemma \[max\], we know that $$\forall r \ge 2, \,
\left( {{\mathbb E}}\max_{1 \le j \le m} |X_j|_2^r \right)^{1/r} \le C r \sqrt{n}$$ which proves the claimed estimate for the $\psi_1$-norm. $\hfill \Box$
[**Remark.**]{} Recall that for a random isotropic log-concave vector, Borell’s inequality implies that $$\| \ |X|_2 \ \|_{\psi_1} \le C \sqrt n.$$ Therefore, a direct application of Lemma \[psia\] is not enough to obtain the desired estimate. We are now able to give a proof of Theorem \[logconcaveintro\]. It is based on the estimates of ${\kappa}_{p,m}(X)$ proved above. [**Proof of Theorem \[logconcaveintro\].**]{} Let ${\varepsilon}\in (0,1)$ and $p \ge 2$ and set $n_0({\varepsilon}, p) = c_p +
{\varepsilon}^{-4/p}$ where $c_p$ depends only on $p$. For any $n \ge
n_0({\varepsilon}, p)$, for any log-concave isotropic random vector $X$ in ${{\mathbb R}}^n$, set $$V_p = \sup_{y \in B_2^n} \bigg| \frac{1}{m} \sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p
\ - \ {{\mathbb E}}|{\langle}X, y {\rangle}|^p \bigg|$$ where $X_1, \ldots, X_m$ are independent copies of $X$. Assume that $p \le \log m$ and $m \le e^{c \, \sqrt n}$ then by Lemma \[max\], we know that $${\kappa}_{p,m}(X)^p \le c_1^p n^{p/2}.$$ We shall use Theorem \[thrandomintro\] with $K = B_2^n$ which is uniformly convex of power type 2 with constant 1 and for which $D=1$. By (\[KKlog\]), $$1 \le \sup_{y \in B_2^n} {{\mathbb E}}| {\langle}X, y
{\rangle}|^p \le p^p,$$ therefore Theorem \[thrandomintro\] implies that for every $\delta \in (0,1)$, satisfying $ C^p n^{p/2} (\log m) \le \delta^{2} m$, we have $${{\mathbb E}}\sup_{y \in B_2^n} \bigg| \frac{1}{m} \sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p
\ - \ {{\mathbb E}}|{\langle}X, y {\rangle}|^p \bigg| \le 2 \delta p^{p}.$$ By taking $\delta$ such that $2 \delta p^{p} = {\varepsilon}$, we deduce that $$\hbox{if }
m \ge C_p' \ {\varepsilon}^{-2} n^{p/2} \log (n {\varepsilon}^{-4/p})
\hbox{ then }
{{\mathbb E}}V_p \le {\varepsilon}.$$ Since $n \ge n_0({\varepsilon},p)$, it is easy to see that if $$m = \lfloor C_p \ {\varepsilon}^{-2} n^{p/2} \log n \rfloor,$$ then $m \ge C_p' \ {\varepsilon}^{-2} n^{p/2} \log (n {\varepsilon}^{-4/p})$, $m \le e^{c \sqrt n}$ and $p \le \log m$ which allows us to use the estimate ${{\mathbb E}}V_p \le {\varepsilon}$. To get a deviation inequality for $V_p$, we will apply a result similar to Theorem \[concentration\]. We know by Corollary \[psi1maxlog\] that $$\| \, \max_{1 \le j \le m} |X|_2 \ \|_{\psi_1} \le C \sqrt{n}.$$ Following the proof of Theorem \[concentration\] and replacing inequality $(\ref{replace})$ by the previous estimate, we easily see that $$\| V_p \|_{\psi_{1/p}}
\le
C_p \left(
{{\mathbb E}}V_p + \frac{2 n^{p/2}}{m}
\right).$$ Since $ \lfloor C_p \ {\varepsilon}^{-2} n^{p/2} \ \log n \rfloor = m$ then $${{\mathbb E}}V_p \le {\varepsilon}\hbox{ and } 2 n ^{p/2}/m \le {\varepsilon}$$ and we deduce from the Chebychev inequality that for any $t > 0$, $${{\mathbb P}}( V_p \ge t ) \le C \exp (- \left( t / C_p' {\varepsilon}\right)^{1/p}).$$ Therefore, for any $t \ge {\varepsilon}$, with probability greater than $1 - C \exp (- \left( t / C_p' {\varepsilon}\right)^{1/p})$, $V_p \le t$ which means that $$\forall y \in {{\mathbb R}}^n,
\bigg| \frac{1}{m} \sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p
\ - \ {{\mathbb E}}|{\langle}X, y {\rangle}|^p \bigg|
\le t |y|_2^p.$$ Since $|y|_2 = ({{\mathbb E}}{\langle}X, y {\rangle}^2)^{1/2} \le ({{\mathbb E}}|{\langle}X, y {\rangle}|^p)^{1/p}$, we get the claimed result of Theorem \[logconcaveintro\]. $\Box$ [**Remark.**]{} Since by Borell’s inequality (\[KKlog\]), for any $ y \in {{\mathbb R}}^n$, $$|y|_2 = \left( {{\mathbb E}}{\langle}X,y {\rangle}^2 \right)^{1/2}
\le
\left( {{\mathbb E}}|{\langle}X,y {\rangle}|^p \right)^{1/p}
\le
C p \left( {{\mathbb E}}{\langle}X,y {\rangle}^2 \right)^{1/2} = C p |y|_2,$$ it is clear that Theorem \[logconcaveintro\] improves the results of Giannopoulos and Milman [@GiaMil].
When the linear functionals associated to the random vector $X$ satisfy a $\psi_2$ condition
============================================================================================
Let start this section considering the case when $X$ is a Gaussian vector in ${{\mathbb R}}^n$. Let $X_j, \ j= 1, \ldots, m,$ be independent copies of $X$. For $t \in {{\mathbb R}}^m$ denote by $X_{t,y}$ the Gaussian random variable $$X_{t,y} = \sum_{j=1}^m t_{j} {\langle}X_j, y {\rangle}.$$ Observe that if $p^*$ denotes the conjugate of $p$, then $$\sup_{t \in B_{p^*}^m} X_{t,y} =
\bigg(\sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p \bigg)^{1/p}.$$ Let $Z$ and $Y$ be Gaussian vectors in ${{\mathbb R}}^m$ and ${{\mathbb R}}^n$ respectively. Using Gordon’s inequalities [@Gordon], it is easy to show that whenever ${{\mathbb E}}|Z|_p \ge {\varepsilon}^{-1} {{\mathbb E}}|Y|_2$ (i.e. for a universal constant $c$, $m \ge c^p p^{p/2} {\varepsilon}^{-p}
n^{p/2}$) $$\begin{array}{rcl}
{{\mathbb E}}|Z|_p - {{\mathbb E}}|Y|_2 \le
{\displaystyle}{{\mathbb E}}\inf_{y \in S^{n-1}}
\bigg(\sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p \bigg)^{1/p}
& \le &
\\
\le {\displaystyle}{{\mathbb E}}\sup_{y \in S^{n-1}} \bigg(\sum_{j=1}^m |{\langle}X_j, y
{\rangle}|^p \bigg)^{1/p} & \le & {{\mathbb E}}|Z|_p + {{\mathbb E}}|Y|_2,
\end{array}$$ where $({{\mathbb E}}|Z|_p + {{\mathbb E}}|Y|_2) / ({{\mathbb E}}|Z|_p - {{\mathbb E}}|Y|_2) \le
(1+{\varepsilon})/(1-{\varepsilon})$. It is therefore possible to get (with high probability with respect to the dimension $n$, see [@Gordon2]) a family of $m$ random vectors $X_1, \ldots, X_m$ such that for every $y \in {{\mathbb R}}^n$, $$A \ |y|_2 \le \bigg(\frac{1}{m} \sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p
\bigg)^{1/p} \le A \ \frac{1 + {\varepsilon}}{1 - {\varepsilon}} \ |y|_2.$$ This argument significantly improves the bound on $m$ in Theorem \[logconcaveintro\] for Gaussian random vectors.
In this part we will be interested in isomorphic moment estimates (instead of almost isometric as in Theorem \[logconcaveintro\]). We will be able to extend the estimate for the Gaussian random vector to random vector $X$ satisfying the $\psi_2$ condition for linear functionals $y \mapsto {\langle}X, y {\rangle}$ with the same dependance on $m$.
Recall that a random variable $Z$ satisfies the $\psi_2$ condition if and only if for any ${\lambda}\in {{\mathbb R}}$ $$\label{randpsi2}
{{\mathbb E}}\exp ({\lambda}Z) \le 2 \exp( c {\lambda}^2 \cdot {\left \| Z \right \|}_2^2).$$ We prove the following
\[logconcavepsi2\] Let $X$ be an isotropic random vector in ${{\mathbb R}}^n$ such that all functionals $y \mapsto {\langle}X, y {\rangle}$ satisfy the $\psi_2$ condition. Let $X_1, \ldots, X_m$ be independent copies of $X$. Then for every $p \ge 2$ and every $m \ge n^{p/2}$ $${{\mathbb E}}\sup_{y \in B_2^n} \bigg( \frac{1}{m} \sum_{j=1}^m |{\langle}X_j, y
{\rangle}|^p \bigg)^{1/p} \le c \, \sqrt p.$$
Note that the results of Part 3 of [@GiaMil] follow immediately from Theorem \[logconcavepsi2\], since the random vector with independent $\pm 1$ coordinates satisfies the $\psi_2$ condition for scalar products.
[**Proof.**]{} Since $X$ is isotropic, $${\left \| {\langle {X} , {y} \rangle} \right \|}_{\psi_2} \le c {\left \| {\langle {X} , {y} \rangle} \right \|}_{2}
= c |y|.$$ Hence, for any ${\lambda}\in {{\mathbb R}}$ $${{\mathbb E}}\exp {\lambda}{\langle}X,y {\rangle}\le 2 e^{c {\lambda}^2 |y|_2^{2}}.$$ Writing $$\Delta = X_{t,y} - X_{t',y'} = \sum_{j=1}^m \left( (t_j - t'_j)
{\langle}X_j,y {\rangle}+ t'_j {\langle}X_j, y - y' {\rangle}\right),$$ it is easy to find a new constant $c \ge 1$ such that for every $t, t' \in B_{p^*}^m,$ $y, y' \in B_2^n$ and every ${\lambda}\in {{\mathbb R}}^{+}$, $${{\mathbb E}}\exp( {\lambda}\Delta ) \le 2 e^{c {\lambda}^2 (|t-t'|_{2}^{2} +
|y-y'|_{2}^{2})}.$$
This means that ${\left \| \Delta \right \|}_{\psi_2} \le c (|t-t'|_{2}^{2} +
|y-y'|_{2}^{2})^{1/2}$, and so $X_{t,y}$ is a sub-Gaussian random process with respect to the distance $$d\big( (t,y) ; (t',y') \big)
= \big(|t-t'|_{2}^{2} + |y-y'|_{2}^{2} \big)^{1/2}.$$ Let $G_{t,y} = \langle Z, t \rangle + \langle Y, y \rangle$, where $Z \in {{\mathbb R}}^m$ and $Y \in {{\mathbb R}}^n$ are two independent Gaussian vectors. Then $$\big( {{\mathbb E}}|G_{t,y} - G_{t', y'}|^{2} \big)^{1/2}
=
d\big( (t,y) ; (t',y') \big)$$ The natural metric for the random process $X_{t,y}$ is bounded by the metric of the process $G_{t,y}$. The Majorizing Measure theorem of Talagrand [@Tal:genericchaining] implies that $${{\mathbb E}}\sup_{(t,y) \in V} X_{t,y}
\le C \sup_{(t,y) \in V} G_{t,y}$$ for any compact set $V \subset {{\mathbb R}}^m \times {{\mathbb R}}^n$. Therefore, $$\begin{array}{l}
{\displaystyle}{{\mathbb E}}\sup_{y \in B_2^n} \bigg(\frac{1}{m} \sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p \bigg)^{1/p}
=
\frac{1}{m^{1/p}} {{\mathbb E}}\sup_{t \in B_{p^*}^m} \sup_{y \in B_2^n}
\sum_{j=1}^m t_{j} {\langle}X_j, y {\rangle}\\
\le
{\displaystyle}\frac{C}{m^{1/p}}
{{\mathbb E}}\sup_{t \in B_{p^*}^m} \sup_{y \in B_2^n}
G_{t,y} = \frac{C}{m^{1/p}} \big( {{\mathbb E}}|Z|_p + {{\mathbb E}}|Y|_2 \big)
\\
\le
{\displaystyle}C \big( \sqrt p + \frac{\sqrt n}{m^{1/p}} \big).
\end{array}$$ This proves that if $ m \ge n^{p/2}$, then $${{\mathbb E}}\sup_{y \in B_2^n} \bigg(\frac{1}{m} \sum_{j=1}^m |{\langle}X_j, y
{\rangle}|^p \bigg)^{1/p} \le c \sqrt p,$$ as claimed. $\Box$
[**Remark.**]{} Let $X$ be an isotropic random vector in ${{\mathbb R}}^n$ satisfying the $\psi_2$ estimate for the scalar products. It is not difficult to see, using Corollary 2.7 in [@GiaMil], that if $m \ge C n$, then with probability greater than $3/4$ $$c_2 \, |y|_2 \le \bigg( \frac{1}{m} \sum_{j=1}^m |{\langle}X_j, y
{\rangle}|^p \bigg)^{1/p}$$ for every $y \in {{\mathbb R}}^n$. Therefore, using Theorem \[logconcavepsi2\], it is easy to deduce that if $m \ge
n^{p/2}$, then with probability greater than $1/2$ $$\forall y \in {{\mathbb R}}^n \quad
c_2 \ |y|_2
\le \bigg( \frac{1}{m} \sum_{j=1}^m |{\langle}X_j, y {\rangle}|^p
\bigg)^{1/p}
\le c_1 \, \sqrt p \, |y|_2$$ with universal constants $c_1, c_2\ge 1$. This generalizes results of [@GiaMil] and gives an isomorphic version of the result of Klartag and Mendelson [@KM] valid for every $p\ge 2.$
A part of this work was done when the first author was visiting University of Missouri (Columbia). We wish to thank this institution for its hospitality.
[Zz 99]{} S. Alesker, $\psi\sb 2$-estimate for the Euclidean norm on a convex body in isotropic position, [*Geometric aspects of functional analysis (Israel, 1992–1994)*]{}, 1–4, Birkhäuser, Basel, 1995.
S. Bobkov, S. Nazarov, On convex bodies and log-concave probability measures with unconditional basis, [*Geometric aspects of functional analysis*]{}, 53–69, Lecture Notes in Math., 1807, Springer, Berlin, 2003.
C. Borell, Complements of Lyapunov’s inequality, [*Math. Ann.*]{} [**205**]{} (1973), 323–331.
J. Bourgain, Random points in isotropic convex sets, [*Convex geometric analysis (Berkeley, CA, 1996)*]{}, 53–58, Cambridge Univ. Press, Cambridge, 1999
J. Bourgain, J. Lindenstrauss and V. Milman, Approximation of zonoids by zonotopes, [*Acta Math.*]{} [**162**]{} (1989), no. 1-2, 73–141.
J. A. Clarkson, Uniformly convex spaces, [*Trans. Amer. Math. Soc.*]{} [**40**]{} (1936), no. 3, 396–414
T. Figiel, An example of infinite dimensional reflexive Banach space non-isomorphic to its Cartesian square, [*Studia Math.*]{} [**42**]{} (1972), 295–306.
A. A. Giannopoulos, M. Hartzoulaki and A. Tsolomitis, Random points in isotropic unconditional convex bodies, [*J. Lond. Math. Soc.*]{} 72 (2005), 779-798.
A. A. Giannopoulos and V. D. Milman, Concentration property on probability spaces, [*Adv. Math.*]{}, [**156**]{} (2000), no. 1, 77–106.
Y. Gordon, Gaussian processes and almost spherical sections of convex bodies, [*Ann. Probab.*]{} [**16**]{} (1988), no. 1, 180–188.
Y. Gordon, On Milman’s inequality and random subspaces which escape through a mesh in $ R\sp n$ [*Geometric aspects of functional analysis (1986/87)*]{}, 84–106, Lecture Notes in Math., 1317, Springer, Berlin, 1988
O. Guédon, G. Paouris, Concentration of mass on the Schatten classes, to appear in [*Annales de l’Institut Henri Poincaré, Probabilités et Statistiques*]{}.
R. Kannan, L. Lovász, M. Simonovits, Random walks and an $O\sp *(n\sp 5)$ volume algorithm for convex bodies, [*Random Structures Algorithms*]{} [**11**]{} (1997), no. 1, 1–50.
B. Klartag, S. Mendelson, Empirical processes and random projections, to appear in [*J. Funct. Anal.*]{}
M. Ledoux and M. Talagrand, [*Probability in Banach spaces*]{}, Springer, Berlin, 1991.
D. R. Lewis, Finite dimensional subspaces of $L\sb{p}$, [*Studia Math.*]{} [**63**]{} (1978), no. 2, 207–212.
L. Lovász and M. Simonovits, Random walks in a convex body and an improved volume algorithm, [*Random Structures Algorithms*]{} [**4**]{} (1993), no. 4, 359–412.
S. Mendelson and A. Pajor, On singular values of matrices with independent rows, to appear in [*Bernoulli*]{}.
V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $n$-dimensional space. Geometric aspects of functional analysis (1987–88), 64–104, Lecture Notes in Math., 1376, Springer, Berlin, 1989.
V. D. Milman and G. Schechtman, [*Asymptotic theory of finite-dimensional normed spaces*]{}, Springer, Berlin, 1986
G. Paouris, Concentration of mass on symmetric convex bodies, to appear in [*Geom. Funct. Anal.*]{}
A. Pajor and N. Tomczak-Jaegermann, Subspaces of small codimension of finite-dimensional Banach spaces, [*Proc. Amer. Math. Soc.*]{} [**97**]{} (1986), no. 4, 637–642.
G. Pisier, Martingales with values in uniformly convex spaces, [*Israel J. Math.*]{} [**20**]{} (1975), no. 3-4, 326–350.
M. Rudelson, Contact points of convex bodies, [*Israel J. Math.*]{} [**101**]{} (1997), 93–124.
M. Rudelson, Random vectors in isotropipc position, MSRI preprint.
M. Rudelson, Random vectors in isotropipc position, [*J. Funct. Anal.*]{} [**164**]{} (1999), no. 1, 60–72.
M. Rudelson, Almost orthogonal submatrices of an orthogonal matrix, [*Israel J. of Math.*]{} [**111**]{} (1999), 143-155.
M. Talagrand, Majorizing measures: the generic chaining, [*Ann. Probab.*]{} [**24**]{} (1996), no. 3, 1049–1103.
N. Tomczak-Jaegermann, The moduli of smoothness and convexity and the Rademacher averages of trace classes $S\sb{p} (1\leq p<\infty )$ [*Studia Math.*]{} [**50**]{} (1974), 163–182.
Olivier Guédon:\
Université Paris 6, Institut de Mathématiques de Jussieu, Projet Analyse Fonctionnelle, 4, place Jussieu, 75005 Paris, France;\
e-mail:
Mark Rudelson:\
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA;\
e-mail:
[^1]: Research was supported in part by NSF grant DMS-024380.
| 2023-08-01T01:27:17.760571 | https://example.com/article/1543 |
MAM
Jahan Desh ke Rakshak wahan tak DishTV; special offer launched
17 Jan, 2017 - 06:13 PM IST | By indiantelevision.com Team
MUMBAI: There is no greater love, than the love for one’s country. As India celebrated the 69th Army Day on 15 January, DishTV paid tribute to the valour, indomitable spirit and determination of the brave soldiers of the Indian Armed Forces.
DishTV network reaches far-flung areas where the jawans of the Indian Army are posted. Highlighting this strength, DishTV has launched a campaign under the aegis ‘Jahan Desh ke Rakshak wahan tak DishTV.’
Saluting the spirit of The Indian Armed Forces – DishTV has also announced special offers. The offers that are extended on the occasion of Army Day, give distinct benefits to the jawans of the Defence and Paramilitary forces and their families, to avail discounts on DishTV connections under the ‘Jai Jawan’ scheme. Under this scheme, on new acquisition of SD or HD connection, a Jawan can enjoy a host of special offers on the new set top box as well as on monthly recharge amount. The extra relocation charges have also been waived off for our brave sentinels.
DishTV India CEO Arun Kapoor expressed, “It is truly an honour to salute our superheroes and brave hearts on Army Day. As an acknowledgement of their devotion to the nation, we have launched special offers for our Jawans in the Indian armed forces and their families. This is our commitment to recognising the armed forces’ selfless services and sacrifices.”
As a tribute to the Armed forces altruistic services, the brand has extended services to the nation’s protectors by introducing a relocation plan specifically designed for armed force personnel and families who constantly on the move and are deputed to numerous locations. DishTV will waive off the relocation charges when the officers and Jawans shift to a new location. With these bundle of offers, they can take their DishTV connection everywhere they go at pocket-friendly prices and convenience.
As an extension to the brand’s tribute to the army’s altruistic service to the nation, DishTV has also come up with a relocation plan specifically designed for armed forces personnel, who are constantly on the move and are posted to different locations. DishTV will waive off the relocation charges whenever the person shifts to a different area. Along with the bundle of benefits offered to the Jawans, they also take their DishTV connection everywhere they go at pocket-friendly prices and convenience. | 2023-09-26T01:27:17.760571 | https://example.com/article/8724 |
click to enlarge Photo via Solar City
By the year 2050, Orlando commissioners want all the electricity used in the city to come from renewable sources of energy like solar and wind power.In a unanimous vote Tuesday, the Orlando City Council agreed to adopt this goal , joining San Diego, Salt Lake City and 37 other cities across the U.S. that have adopted a 100 percent clean-energy target. Orlando is the largest city in Florida committing to this goal so far, according to the Sierra Club , with St. Petersburg and Sarasota right behind. Aside from combating climate change and pollution, the city argues the move toward renewable energy increases economic opportunities in Central Florida by creating local jobs in the industry. Orlando Mayor Buddy Dyer didn't call out the climate change aversion of President Donald Trump or the Republican-led Florida Legislature by name, but did say city mayors had to lead the fight against rising seas and increasing temperatures."This administration has decided not to honor our commitment to the Paris climate accord, but a lot of mayors around the country have picked up the reins to say if we're not doing it at the federal level, it's incumbent that we lead at the local level," Dyer says. "More than 50 percent of the world's population now lives in cities, so we have to be the ones that are leading on the important issues that are of consequence for not just this year, but for decades and even centuries to come."Chris Castro, director of sustainability for the city, says over the last decade, Orlando has been trying to move the needle through its Green Works Orlando initiative to become one of the most sustainable cities in the Southeast. The city has already committed to reducing 90 percent of its air pollution and greenhouse gas emissions by 2040 in accord with the Paris Climate Agreement. To achieve this, Orlando has already set a goal of powering 100 percent of municipal operations using renewable energy by 2030. Last year, the U.S. Department of Energy and the Solar Foundation declared Orlando a "SolSmart City" for its leadership on expanding clean energy sources."The power from the sun is cheaper to produce electricity than the power from fossil fuels, including coal and even natural gas," Castro says. "What we want to do is maintain the affordability of our electricity rates. A lot of people think that just by going solar, it's going to be more expensive, and that is not the case. We're actually going to be able to levelize our cost of power over decades, and we'll be able to maintain the affordability and the reliability of our power here in the City of Orlando."The council is also particularly interested in the potential employment opportunities created by having this commitment to clean energy. Castro says last year in Florida, solar jobs grew 10 times faster than the overall state economy, adding 1,700 new jobs. Commissioner Sam Ings proposed moving the citywide goal of 100 percent renewable energy to 2035, and Castro said that target could be updated as more technology comes along.The resolution was applauded by members of the First 50 Coalition, a broad alliance pushing for local sustainability issues that includes the League of Women Voters of Orange County, the Sierra Club and FL SUN."I see this vote as historic and a first step toward what we can do in leadership on the national stage," says Sara Isaac, director of partnerships for the League. "I think that you are sending a signal across the nation of the kind of city Orlando wants to be."In a statement, Phil Compton from the Sierra Club’s Ready for 100 Campaign in Florida also praised the decision by Orlando commissioners."All across our state and our nation, cities are committing to a future powered by 100 percent clean and renewable energy for all," Compton says. "Today, Orlando joins this growing movement of cities that are ready for 100 percent clean, renewable energy." | 2023-10-15T01:27:17.760571 | https://example.com/article/2981 |
All relevant data are within the manuscript and its Supporting Information files.
Introduction {#sec001}
============
Mitochondria are double membrane-bound organelles that generate 90% of cellular ATP \[[@pone.0213116.ref001]\]. In most cells, mitochondria form extensive and dynamic networks, undergoing continuous cycles of fission and fusion. This creates a highly adaptable and efficient energy transfer system to rapidly deliver ATP to where it is most needed \[[@pone.0213116.ref002]\]. In addition, fission plays a central role in the sequestration and selective degradation of defective mitochondria by mitophagy \[[@pone.0213116.ref003], [@pone.0213116.ref004]\]. Mitochondrial fission and fusion are both tightly regulated processes that are largely orchestrated by GTPases. Dynamin and dynamin-related protein (Drp) mediate fission \[[@pone.0213116.ref005], [@pone.0213116.ref006]\] whereas fusion of the mitochondrial outer and inner membranes is driven by the GTPases mitofusins (Mfn) 1 and 2 and Opa1, respectively \[[@pone.0213116.ref007]\].
Dynamin-related protein 1 (Drp1) is a predominantly cytosolic protein with only \~3% bound to mitochondria under basal conditions \[[@pone.0213116.ref008]\]. Nonetheless, in cells lacking functional Drp1 the equilibrium between fission and fusion is perturbed, leading to highly elongated and interconnected mitochondria, largely localised in perinuclear clusters \[[@pone.0213116.ref009]\]. During cell stress, rates of fission and fragmentation increase, causing the release of pro-apoptotic cytochrome *c* from mitochondria, a process that can be delayed by mutation or deletion of Drp1 \[[@pone.0213116.ref010]\].
Because Drp1 lacks the membrane targeting PH-domain present in conventional dynamins, it requires membrane-bound adaptor/receptor proteins to recruit it to the mitochondrial outer membrane (MOM) \[[@pone.0213116.ref011]\]. Four mitochondrial Drp1 receptors have been identified; Fis1, MiD49, MiD51 and Mff \[[@pone.0213116.ref012]\]. Of these, Fis1 is dispensable for mammalian mitochondrial fission \[[@pone.0213116.ref013]\]. The MiD proteins are specific to higher eukaryotes and although they can each recruit Drp1 to mitochondrial fission sites \[[@pone.0213116.ref014], [@pone.0213116.ref015]\] it remains unclear if MiD proteins facilitate fusion or inhibit fission \[[@pone.0213116.ref016]\]. Mff facilitates the majority of Drp1 recruitment and is the best characterised Drp1 receptor. It is a \~35kDa protein with a single C-terminal transmembrane domain and interacts with Drp1 via its N-terminus \[[@pone.0213116.ref017]\]. Like Drp1-null cells, Mff-knockout cells have grossly elongated mitochondria under basal conditions, and attenuated fragmentation and apoptosis following stress \[[@pone.0213116.ref018]\].
Parkin is a ubiquitin ligase that is inactive in the cytosol but is recruited to damaged/depolarised mitochondria where it is activated by the MOM protein PTEN-induced protein kinase 1 (PINK1). PINK1 is basally maintained at very low levels by rapid proteolytic degradation soon after mitochondrial import \[[@pone.0213116.ref019], [@pone.0213116.ref020]\]. However, loss of membrane potential in damaged or defective mitochondria inhibits PINK1-proteolysis, resulting in its accumulation on the outer membrane, where it phosphorylates mitochondrial ubiquitin at Serine 65 and triggers mitophagy via a multi-step process \[[@pone.0213116.ref021], [@pone.0213116.ref022]\].
Briefly, PINK1-phosphorylated ubiquitin (pUb) binds to and alters the conformation of Parkin. This makes Serine 65 within the Ubiquitin-like domain (UbL) of Parkin accessible for PINK1-mediated phosphorylation, which initiates a cascade of subsequent conformational changes exposing the catalytic site of Parkin \[[@pone.0213116.ref022]--[@pone.0213116.ref024]\]. In a positive-feedback loop, Parkin ubiquitinates mitochondrial proteins, providing further substrates for PINK1-mediated phosphorylation, which then recruit more Parkin \[[@pone.0213116.ref025], [@pone.0213116.ref026]\]. For example, mitophagy induced by the mitochondrial proton gradient uncoupler carbonyl cyanide m-chlorophenyl hydrazine (CCCP) is largely dependent on Parkin-mediated, non-selective ubiquitination of mitochondrial proteins with K48- and K63-linked ubiquitin chains \[[@pone.0213116.ref027], [@pone.0213116.ref028]\]. Mitochondrial depolarisation leads to PINK1 accumulation on the surface of mitochondria that recruits Parkin to indiscriminately tag MOM proteins with K48- linked ubiquitin chains, marking them for excision and proteasomal degradation \[[@pone.0213116.ref027], [@pone.0213116.ref029]\]. The remaining portion of the mitochondrion is then tagged with K63-linked ubiquitin that recruits phagosomal adaptors including p62 \[[@pone.0213116.ref030]\] resulting in the engulfment of the organelle into an autophagosome prior to lysosomal fusion and degradation \[[@pone.0213116.ref031], [@pone.0213116.ref032]\]. Thus, this elegant quality control mechanism identifies damaged mitochondria and targets proteins for degradation.
Moreover, in cells lacking functional PINK1 and/or Parkin, mitochondria undergo fragmentation due to excessive Drp1-mediated fission \[[@pone.0213116.ref033]--[@pone.0213116.ref035]\]. However, the roles of Parkin in non-stressed mitochondria have not been extensively investigated. Here we show that, independent of stress-induced mitophagy, Mff is ubiquitinated by Parkin and at least one other E3 ligase under basal conditions. Our data indicate that Parkin-mediated ubiquitination triggers lysosomal degradation of Mff, suggesting a role for Parkin in homeostatic maintenance of Mff levels and mitochondrial integrity.
Materials and methods {#sec002}
=====================
Molecular biology {#sec003}
-----------------
21bp short hairpin (shRNA) constructs used in Figs [1](#pone.0213116.g001){ref-type="fig"}--[4](#pone.0213116.g004){ref-type="fig"}: targeting human shParkin: `5’- ACCAGCATCTTCCAGCTCAAG-3’`
non-specific shControl: `5’- AACGTACGCGGAATACTTCGA-3’`
were cloned under a H1 promoter into a modified pSUPER vector co-expressing mCherry driven by a PGK promoter. Alternative Parkin shRNAs ([S1 Fig](#pone.0213116.s001){ref-type="supplementary-material"}) were cloned in the same way, with target sequences:
`5’- GCTTAGACTGTTTCCACTTAT-3’` (Parkin (Berger))
`5’-AACTCCAGCCATGGTTTCCCA-3’` (Parkin (other)).
Parkin (Berger) target sequence was previously published \[[@pone.0213116.ref036]\]. Other Parkin shRNA target sequences were designed as part of this study. PINK1 knockdown was performed using MISSION esiRNA human PINK1 (EHU057101, Sigma Aldrich). Mff knockdown ([S2 Fig](#pone.0213116.s002){ref-type="supplementary-material"}) was performed using siRNA with the target sequence `5’- CCAUUGAAGGAACGUCAGA-3’` (Eurofins genomics). Firefly Luciferase siRNA was used as a negative control (MISSION esiRNA Firefly Luciferase, EHUFLUC, Sigma Aldrich).
{#pone.0213116.g001}
{#pone.0213116.g002}
{#pone.0213116.g003}
{ref-type="fig"}, knockdown of Parkin reduces total Mff ubiquitination. b) Samples from (a), showing no co-immunoprecipitation of K48-linked ubiquitin with CFP-Mff. c) Samples from (a), showing co-immunoprecipitation of a single K63-linked ubiquitinated CFP-Mff species. d) HEK293T cells were transfected with untagged Parkin. Prior to lysis 48 hours post- transfection, cells were treated with DMSO (control), bortezomib (BTZ, proteasomal inhibitor, 1μM) or bafilomycin (Baf, lysosomal inhibitor, 100nM) for 6 hours. Lysates were then Western blotted for Mff and other mitochondrial membrane proteins. e) Quantitative analysis of (d), data presented as mean ± SEM. Analysed using ordinary two-way ANOVA with Dunnett's correction for multiple comparisons. N = 3. \* p \< 0.05, \*\*\* p \< 0.001.](pone.0213116.g004){#pone.0213116.g004}
The open reading frame of human Mff (isoform I, accession number: Q9GZY8) was cloned into pECFP between 5' KpnI and 3' BamHI restriction sites. CFP-Mff expression was driven by a CMV promoter. The open reading frame of human Parkin (full length, accession number: O60260) was cloned into pcDNA3.1(+) between 5' HindIII and 3' BamHI restriction sites. Parkin expression was driven by a CMV promoter. CFP-Mff K151R, K302R and 2KR, Parkin S65A and S65D were generated by site-directed mutagenesis.
HEK293T cell culture and transfection {#sec004}
-------------------------------------
Human Embryonic Kidney (HEK293T) cells were obtained from The European Collection of Cell Cultures (ECACC). Cultures were maintained at 37°C in a humidified cell culture incubator, supplied with 5% CO2, in Dulbecco's Modified Eagle's Medium (Lonza) supplemented with 10% (v/v) Foetal Bovine Serum (Sigma) and 2mM L-Glutamine (Gibco). For transfection, cells were plated on dishes pre-coated with 0.1mg/mL poly-L-lysine to promote adhesion. Lipofectamine 2000 transfection reagent (Invitrogen) was used according to manufacturer's protocol. Cells were lysed 48 hours (protein over-expression) or 72 hours (protein knockdown) post-transfection.
Protein biochemistry {#sec005}
--------------------
For immunoblotting, cells were lysed in sample buffer (1x concentrate) containing 2% SDS (w/v), 5% glycerol (v/v), 62.5mM Tris-HCl pH6.8 and 5% (v/v) 2-β-mercaptoethanol. Lysates were heated to 95°C for 10 minutes prior to gel electrophoresis. For immunoprecipitation, cells were lysed in lysis buffer containing 20mM Tris pH7.4, 137mM NaCl, 2mM sodium pyrophosphate, 2mM EDTA, 1% (v/v) Triton X-100, 0.1% (w/v) SDS, 25mM β-glycerophosphate, 10% glycerol (v/v), 1x cOmplete protease inhibitor cocktail (Roche) and 20mM N-Ethylmaleimide (NEM, Sigma). Lysates were incubated with GFP-Trap agarose beads (ChromoTek) at 4°C for 90 minutes with gentle agitation. Beads were pelleted, washed 3 times with wash buffer (lysis buffer without protease inhibitor cocktail or NEM) and unbound material aspirated. 2x concentrate sample buffer was used to elute immunoprecipitated proteins from the beads. Samples were heated to 95°C for 10 minutes prior to gel electrophoresis.
Denaturing SDS-PAGE was performed on 10--15% (v/v) poly-acrylamide gels. Western blotted PDVF membranes were blocked in 5% (w/v) non-fat milk powder or 4% (w/v) Bovine Serum Albumin (BSA, Sigma) in PBS-T. Primary antibodies used were: Parkin (mouse monoclonal, 1:1000, Santa Cruz sc-32282), Mff (mouse monoclonal, 1:1000, Santa Cruz sc-398731), PINK1 (rabbit monoclonal, 1:1000, Cell Signaling D83G 6946), ubiquitin (mouse monoclonal, 1:1000, P4D1 3936S), Mfn2 (rabbit monoclonal, 1:1000, Cell Signaling, D2D10 9482S), Mid49 (rabbit polyclonal, 1:1000, ProteinTech, I64I3-I-AP), VDAC (rabbit polyclonal, 1:1000, Santa Cruz, FL-283 sc-98708), Fis1 (rabbit monoclonal, 1:1500, ProteinTech, 10956-1-AP), LC3 (rabbit polyclonal, 1:1000, Cell Signaling, 2775S), GFP (rat monoclonal, 1:10,000, Chromotek 3H9), β-actin (mouse monoclonal, 1:10,000, Sigma-Aldrich A5441). For protein detection, membranes were incubated with HRP-conjugated secondary antibodies (1:10,000; Sigma-Aldrich) and visualised by enhanced chemiluminescence. Protein bands were quantified by densitometry using ImageJ (NIH). Mff- and Parkin-antibody specificity was validated by endogenous protein knockdown ([S1](#pone.0213116.s001){ref-type="supplementary-material"} and [S2](#pone.0213116.s002){ref-type="supplementary-material"} Figs).
Results {#sec006}
=======
Knockdown of PINK1 or Parkin increase levels of Mff {#sec007}
---------------------------------------------------
We first investigated the effects of Parkin knockdown on Mff by transfecting HEK293T cells with plasmids encoding an shRNA sequence targeted to the human Parkin transcript (shParkin) or a control shRNA sequence. shParkin reduced Parkin levels to 6% of control levels 72 hours after transfection ([Fig 1A and 1C](#pone.0213116.g001){ref-type="fig"}). Consistent with Parkin-mediated degradation of Mff there was a corresponding increase in levels of Mff ([Fig 1A and 1B](#pone.0213116.g001){ref-type="fig"}). To exclude the possibility of the Mff increase being due off-target effects of the Parkin shRNA, two further Parkin shRNAs were generated and tested in HEK293T cells, with the same effect on Mff levels ([S1 Fig](#pone.0213116.s001){ref-type="supplementary-material"}).
To determine if the effect of Parkin on Mff levels was PINK1-dependent we next knocked down PINK1 in HEK293T cells \[[@pone.0213116.ref001]--[@pone.0213116.ref003]\]. 72 hours post-transfection, PINK1 was significantly knocked down to 30% of control values ([Fig 1D and 1F](#pone.0213116.g001){ref-type="fig"}). As expected, loss of PINK1 also significantly increased total levels of Mff ([Fig 1E](#pone.0213116.g001){ref-type="fig"}). Moreover, total levels of Parkin were unaffected by knockdown of PINK1, indicating that it is Parkin activity, rather than expression, that causes the increase in Mff ([Fig 1G](#pone.0213116.g001){ref-type="fig"}).
Knockdown of Parkin reduces steady-state ubiquitination of Mff {#sec008}
--------------------------------------------------------------
Parkin has been reported to ubiquitinate Mff at K251 in HEK293 cells over-expressing HA-tagged Parkin and treated with CCCP to induce mitochondrial depolarisation \[[@pone.0213116.ref030]\]. Our data suggest that the PINK1/Parkin pathway may also control Mff levels in the absence of global or applied cell stress. To determine if endogenous Parkin regulates Mff expression under basal, non-stressed conditions, HEK293T cells were co-transfected with CFP-tagged Mff and shRNA targeting Parkin or a scrambled control. 72 hours after transfection, cells were lysed and CFP-Mff retained on GFP-Trap agarose beads. Steady-state levels of Mff ubiquitination were significantly decreased by Parkin knockdown (Figs [2A](#pone.0213116.g002){ref-type="fig"} and [1B](#pone.0213116.g001){ref-type="fig"}). These data demonstrate a role for Parkin in Mff ubiquitination that is not dependent on global mitochondrial stress.
Parkin ubiquitinates Mff at K151 {#sec009}
--------------------------------
A recent study has reported that Mff is phosphorylated by AMPK at conserved sites S155 and S172 (highlighted in [S2 Fig](#pone.0213116.s002){ref-type="supplementary-material"}) \[[@pone.0213116.ref037]\]. We hypothesised that, given its conservation and density of modifiable residues, this region could be a 'hotspot' for Mff post-translational modifications. We therefore selected K151 as an alternative possible target for ubiquitination. To specifically test if Parkin ubiquitinates Mff at K151 under basal conditions, HEK293T cells were co-transfected with CFP-Mff WT or a mutant in which lysine 151 had been replaced with a non-ubiquitinatable arginine (K151R), and either Parkin-targeting or control shRNA. 72 hours post-transfection, cells were lysed and GFP-Trap agarose beads used to precipitate CFP-Mff. Once again, knockdown of Parkin reduced levels of WT Mff ubiquitination ([Fig 2C and 2D](#pone.0213116.g002){ref-type="fig"}). However, Parkin knockdown had no significant effect on ubiquitination of CFP-Mff K151R, indicating that under these conditions Parkin preferentially ubiquitinates Mff at K151 ([Fig 2C and 2D](#pone.0213116.g002){ref-type="fig"}). Interestingly, while, CFP-Mff K151R had significantly reduced ubiquitination compared to WT, some ubiquitin modification was still present.
The Mff isoform used in the study by Gao et al \[[@pone.0213116.ref030]\] was isoform II (291 amino acids) which is a truncated protein, lacking a large portion of the N-terminus as well as a central region. Here we used Mff isoform I, which is the full-length protein comprising 342 amino acids. Accounting for these differences in the length of the isoforms, K251 in Isoform II corresponds to K302 in Isoform I ([S2 Fig](#pone.0213116.s002){ref-type="supplementary-material"}). We therefore generated a CFP-Mff mutant in which lysine 302 was replaced with arginine (K302R). HEK293T cells were co-transfected with CFP-Mff WT or K302R. 48 hours post-transfection, cells were lysed and GFP-Trap agarose beads used to precipitate CFP-Mff. Mutation of Mff K302 to arginine (K302R) reduced, but did not abolish, ubiquitination of Mff isoform I ([Fig 2E and 2F](#pone.0213116.g002){ref-type="fig"}). Furthermore, replacing both K151 and K302 with arginines (K151R and K302R; 2KR) also fails to abolish Mff ubiquitination ([Fig 2E and 2F](#pone.0213116.g002){ref-type="fig"}). These results indicate that Mff isoform I has several sites of ubiquitination under non-stressed conditions which include K151 and K302, as well as at least one other lysine.
Parkin mediates PINK1-dependent turnover of Mff {#sec010}
-----------------------------------------------
Our results show that Parkin knockdown significantly decreases Mff ubiquitination and increases Mff levels. We next directly tested the role of Parkin in Mff degradation. HEK293T cells were transfected with Parkin-targeting shRNA or a scrambled shRNA control. Prior to lysis 72 hours post-transfection, cells were treated with the protein translation inhibitor cycloheximide (CHX) for up to 24 hours. In cells expressing control shRNA \~90% of Mff was degraded within 24 hours, with a half-life of \~5 hours. In the Parkin knockdown cells, however, the rate of degradation was dramatically slower, increasing the half-life of Mff to \~24 hours ([Fig 3A and 3B](#pone.0213116.g003){ref-type="fig"}). Taken together, these data indicate that Parkin-mediated ubiquitination plays a role in physiological Mff degradation under non-stressed conditions.
Parkin ligase activity requires phosphorylation at S65 by PINK1 \[[@pone.0213116.ref022]\]. To further investigate the roles of Parkin and PINK1 in Mff turnover, we generated phospho-null (S65A) and phospho- mimetic (S65D) untagged Parkin S65 mutants. We reasoned that Parkin S65A would be unable to efficiently translocate to mitochondria or catalyse ubiquitin-transfer due to its PINK1-insensitivity, whereas Parkin S65D would be constitutively active.
HEK293T cells were transfected with WT, S65A or S65D Parkin. Prior to lysis 48 hours post- transfection, cells were treated with CHX or DMSO (vehicle control) for 6 hours. Samples were then Western blotted for Mff ([Fig 3C and 3D](#pone.0213116.g003){ref-type="fig"}). Levels of over-expressed Parkin WT, S65A and S65D relative to endogenous Parkin under the same conditions are shown in [S3 Fig](#pone.0213116.s003){ref-type="supplementary-material"}. Consistent with a PINK1-dependent role for Parkin in Mff turnover, expression of Parkin WT or S65D resulted in significant loss of Mff during 6 hours of inhibited protein translation, whereas Parkin S65A had no effect on Mff levels compared to control. These data demonstrate that PINK1-mediated activation via phosphorylation of Parkin at S65 is required for its activity in Mff turnover.
Parkin mediates specific lysosomal degradation of Mff {#sec011}
-----------------------------------------------------
To establish whether constitutive Parkin-mediated degradation of Mff is mediated via the lysosome or the proteasome, we used GFP-Trap immunoprecipitation to pull down CFP-Mff from cells expressing control or Parkin-targeting shRNA. These samples were then probed for total ubiquitin, K48-linked ubiquitin and K63-linked ubiquitin ([Fig 4A](#pone.0213116.g004){ref-type="fig"}, [4B and 4C](#pone.0213116.g004){ref-type="fig"} respectively). Indicative of Mff not being a proteasome substrate, no K48-linked ubiquitin was detected on CFP-Mff ([Fig 4B](#pone.0213116.g004){ref-type="fig"}). However, a single K63-linked ubiquitin-reactive species was present ([Fig 4C](#pone.0213116.g004){ref-type="fig"}), which was reduced in the absence of Parkin, suggesting a role for the lysosome in Parkin-mediated degradation of Mff.
We next investigated the mechanism of Parkin-mediated degradation of Mff by transfecting HEK293T cells with untagged WT Parkin and treating with the proteasomal inhibitor bortezomib \[[@pone.0213116.ref038]\] or bafilomycin, which inhibits fusion of the autophagosome and lysosome \[[@pone.0213116.ref039]\] for 6 hours. Over-expression of Parkin WT compared to non-transfected samples is shown in [S3B Fig](#pone.0213116.s003){ref-type="supplementary-material"}. The efficacy of proteasomal inhibition by 1μM bortezomib was demonstrated by the accumulation of high molecular weight ubiquitin conjugates and a reduction of free ubiquitin compared to DMSO or bafilomycin treatment ([S3 Fig](#pone.0213116.s003){ref-type="supplementary-material"}). The efficacy of autophagic inhibition by 100nM bafilomycin was confirmed by an increase in LC3-ii/i ratio ([S3 Fig](#pone.0213116.s003){ref-type="supplementary-material"}). Consistent with Parkin mediating lysosomal degradation of Mff, bafilomycin significantly increased Mff levels, whereas bortezomib treatment had no effect ([Fig 4D and 4E](#pone.0213116.g004){ref-type="fig"}). Interestingly, other MOM proteins tested (Mfn2, Mid49, VDAC) were not affected by bafilomycin, while Fis1 was increased by both proteasomal and lysosomal inhibition ([Fig 4D and 4E](#pone.0213116.g004){ref-type="fig"}). VDAC and Mfn2 have both been reported to be ubiquitinated by Parkin during induced mitophagy, yet neither were increased by inhibition of mitophagy under the basal conditions of this experiment \[[@pone.0213116.ref032], [@pone.0213116.ref040]\]. These data therefore suggest that under conditions of basal mitophagy, the Parkin-mediated turnover of Mff far exceeds that of other MOM proteins, which could be indicative of an additional selective Parkin-mediated degradative pathway.
Discussion {#sec012}
==========
Our data show that under basal conditions endogenous Parkin ubiquitinates Mff at K151. For this ubiquitination and subsequent Mff degradation Parkin needs to be activated by PINK1-dependent phosphorylation at S65. This Parkin-mediated ubiquitination of Mff coincides with Parkin-mediated Mff degradation, suggesting that Mff turnover is regulated by a Parkin-dependent, ubiquitin-mediated pathway. Mff is not a substrate of K48-linked ubiquitination but is a substrate of K63-linked ubiquitination. Furthermore, inhibition of the lysosome, but not the proteasome, rescues Mff from Parkin-mediated degradation. These data support a model in which Parkin-mediated degradation of Mff occurs via K63-linked ubiquitination and the lysosome. Interestingly, this activity appears to be in addition to mitophagy, in which depolarised mitochondria recruit Parkin to indiscriminately ubiquitinate MOM proteins prior to their degradation. Of the five MOM proteins assayed in Parkin-overexpressing cells, only Mff and, to a lesser extent, Fis1 were significantly rescued from degradation by inhibition of the lysosome (Fis1 was also rescued by proteasomal inhibition). Mfn2, Mid49 and VDAC were not significantly changed, despite Mfn2 and VDAC being known targets of CCCP-induced, Parkin-mediated mitophagy \[[@pone.0213116.ref032], [@pone.0213116.ref040], [@pone.0213116.ref041]\]. These data may indicate that mitophagy is not solely responsible for the changes we observe. Thus, we propose that Parkin may have a selective effect on the turnover of Mff, in addition to its role in mitophagy-dependent Mff degradation. The degradation of Mff by Parkin under basal conditions, together with its inactivity toward other known substrates under the same conditions, suggest that Parkin-mediated degradation of Mff is a regulatory mechanism independent of stress-dependent mechanisms. Moreover, since PINK1 is maintained at low levels in the MOM of healthy mitochondria, we propose that this mechanism plays a critical background role in maintaining mitochondrial integrity in the absence of induced stress.
Intriguingly, our data demonstrate that K302 is not the sole site of ubiquitination of Mff isoform I, and we identified an additional ubiquitination site at K151. However, even in mutants in which both K151 and K302 were ablated, residual Mff ubiquitination remained, indicating the presence of at least one other ubiquitination site. Moreover, our observation that ubiquitination of CFP-Mff K151R is unaffected by Parkin knockdown strongly suggests that the sole, or at least predominant, site of Parkin-mediated ubiquitination under non-stressed conditions is K151.
Our data, combined with previous reports of Mff ubiquitination and phosphorylation \[[@pone.0213116.ref030], [@pone.0213116.ref037]\] indicate Mff is subject to multiple post-translational modifications and suggest that the region around K151 could be an important regulatory 'hotspot'. Given its role as the primary receptor for Drp1 in mitochondrial fission, further work will be needed to elucidate how these modifications, and the interplay between them, regulate Mff abundance and activity in health and disease.
Supporting information {#sec013}
======================
###### Validation of Parkin antibody and knockdown constructs.
a\) Parkin antibody and shRNA are specific. HEK293T cells were transfected with control (scrambled sequence) shRNA or shRNA targeting human Parkin. Cells were lysed 72 hours post-transfection and lysates used for Western blotting with anti-Parkin antibody (Santa Cruz sc-32282). A single band of around 55kDa was detected (predicted MW: 52kDa), which was abolished by Parkin shRNA.
b\) Effect of Parkin knockdown on Mff is specific. HEK293T cells were transfected with control (scrambled sequence) shRNA or one of 3 shRNAs targeting human Parkin. Cells were lysed 72 hours post-transfection and lysates used for Western blotting with anti-Parkin antibody (Santa Cruz sc-32282) and anti-Mff antibody (Santa Cruz sc-398731). shRNA construct target sequences: Parkin (blue) `5’-ACCAGCATCTTCCAGCTCAAG-3’`, Parkin-Berger (purple) `5’-GCTTAGACTGTTTCCACTTAT-3’`, Parkin-other (red) `5’-AACTCCAGCCATGGTTTCCCA-3’`. Parkin (Berger) shRNA target sequence taken from \[[@pone.0213116.ref004]\].
c\) Partial sequence alignments of human Parkin (Uniprot: O60260) and shRNAs, coloured as in a) and b).
(TIF)
######
Click here for additional data file.
###### Mff isoform alignment.
a\) Mff antibody is specific. HEK293T cells were transfected with control (Firefly luciferase) siRNA or human Mff siRNA (`5’-CCAUUGAAGGAACGUCAGATT-3’`, Eurofins genomics). Cells were lysed 72 hours post-transfection and lysates used for Western blotting with anti-Mff antibody (Santa Cruz sc-398731). No bands were detected in Mff knockdown cells.
b\) Alignment of all five isoforms of human Mff. Isoform I (green, 342 amino acids) is the longest and was used to generate CFP-Mff constructs. Isoform II (orange, 291 amino acids) was used in the study by Gao et al. Residue numbers are given according to their position in Isoform I. S155 and S172 are present in all isoforms of Mff (red). K151 (blue) and K302 (purple) are also present in all isoforms. Alignment produced using ClustalOmega. Uniprot identifiers are as shown (Q9GZY8).
(TIF)
######
Click here for additional data file.
###### Parkin over-expression and further controls for Figs [3](#pone.0213116.g003){ref-type="fig"} and [4](#pone.0213116.g004){ref-type="fig"}.
a\) Samples from [Fig 3C](#pone.0213116.g003){ref-type="fig"} (over-expressing Parkin WT, S65A or S65D, in the presence of CHX) alongside non-transfected HEK293T cells in the presence of CHX. Lysates probed for Parkin, Mff and β-actin.
b\) Samples from [Fig 4D](#pone.0213116.g004){ref-type="fig"} (over-expressing Parkin WT, in the presence of DMSO, BTZ or Baf) alongside non-transfected HEK293T cells in the presence of DMSO. Lysates probed for Parkin, ubiquitin, LC3 and β-actin.
c\) Quantitative analysis of (c), data presented as mean ± SEM. Analysed using ordinary one-way ANOVA with Tukey's correction for multiple comparisons with a pooled variance. N = 3. \* p \< 0.05.
(TIF)
######
Click here for additional data file.
[^1]: **Competing Interests:**The authors have declared that no competing interests exist.
| 2024-01-15T01:27:17.760571 | https://example.com/article/9498 |
1. Introduction: Technology and society {#section1-0963662510392485}
=======================================
Technological advances are continuing to be part of the trajectory of evolving civilization. The quest for knowledge and scientific enquiry has driven humanity to explore developments in science and apply them to human requirements and needs. Technology has been defined as "a technological process, invention or method," or "the application of knowledge for practical ends" or "the sum of ways in which social groups provide themselves with the material objects of their civilization" ([@bibr38-0963662510392485]). An example of the last is the definition of technology provided by [@bibr29-0963662510392485] where technology is defined as a "social practice that embodies the capacity of societies to transform themselves by creating and manipulating not only physical objects, but also symbols and cultural forms." Considerable debate exists over the definition of technology and different approaches to technology ([@bibr28-0963662510392485]; [@bibr30-0963662510392485]; [@bibr32-0963662510392485]; [@bibr57-0963662510392485]). Social science studies of technology have included perspectives drawn from a large number of disciplines, including, for example, sociology, political science and economics ([@bibr23-0963662510392485]; [@bibr31-0963662510392485]; [@bibr34-0963662510392485]; [@bibr56-0963662510392485]). In the research presented here, the focus of the review is confined to social psychological approaches to understanding societal responses to technology.
Evident from the definition by [@bibr29-0963662510392485] is that sequentially evolving technologies are not isolated from the society in which they are embedded, but are integral to the social environment. Increased societal dependency on technologies necessitates the examination of "society--technology" interactions. In this context, it is important to note that on one hand a new technology may bring about radical changes in society, while on the other hand the fate of that technology rests with the society in which it is being applied. A negative societal response may be caused by the fact that, while many technologies deliver benefits to society, they may also introduce new risks ([@bibr17-0963662510392485]). As a consequence, such developments are often shaped by public controversies and concerns ([@bibr21-0963662510392485]).
Public rejection of technologies has frequently resulted in negative consequences for the commercialization of technologies. In particular, unpredicted events and accidents affecting the public have acted as a signal which has resulted in fear and reluctance to adopt certain technologies, and resulted in consumer rejection of the products of these technologies. Perhaps as a consequence, much of the research focused on understanding societal acceptance of technologies has been directed towards risk perception. As a case in point, the Three Mile Island accident sparked controversy and public negativity towards nuclear technology ([@bibr7-0963662510392485]; [@bibr16-0963662510392485]; [@bibr55-0963662510392485]). Another example is the market introduction of the first generation of genetically modified (GM) food crops, which led to polarized GM food debate internationally ([@bibr10-0963662510392485]; [@bibr19-0963662510392485]). The intensive societal discussion that followed was detrimental for the adoption and commercialization of GM crops and food products at least in some regions of the world ([@bibr1-0963662510392485]; [@bibr4-0963662510392485]; [@bibr14-0963662510392485]; [@bibr24-0963662510392485]; [@bibr54-0963662510392485]). Occurrence of such events and controversies over the use of technology, emphasize the importance of public acceptance in strategic development, application and commercialization of technologies.
Resistance to technologies and factors influencing public acceptance of technologies have generated wide interest in academia, particularly in the arena of social and behavioural research ([@bibr48-0963662510392485]). A lot of research has been conducted on risk and (more recently) benefit perceptions and public attitudes as these are believed to be the major factors influencing public acceptance ([@bibr2-0963662510392485]; [@bibr3-0963662510392485]; [@bibr8-0963662510392485]; [@bibr15-0963662510392485]; [@bibr25-0963662510392485]; [@bibr36-0963662510392485]; [@bibr37-0963662510392485]; [@bibr39-0963662510392485]; [@bibr44-0963662510392485]; [@bibr49-0963662510392485]; [@bibr50-0963662510392485]; [@bibr53-0963662510392485]). Research in psychology focused on how individuals define risks and understanding the key factors influencing such processes ([@bibr41-0963662510392485]), and although originally most emphasis was on cognitive processes (e.g. [@bibr22-0963662510392485]), psychometric study of attitudes towards technological risks and benefits has explored the emotional basis of risk judgements ([@bibr13-0963662510392485]).
More recently the emotional approach of risk perception has become more dominant with the proposition of a theoretical framework that describes the importance of "affect-heuristic" in guiding risk perceptions and risk-related behaviour ([@bibr12-0963662510392485]; [@bibr51-0963662510392485], [@bibr52-0963662510392485]), and the "risk as feeling" perspective suggests that intuitions experienced at the moment of decision-making can play a vital role in the choice an individual eventually makes ([@bibr27-0963662510392485]). All of these studies imply that people's attitudes towards technological risks and benefits are influenced by risk dimensions that have little to do with the possible consequences of the technology. An individual can evaluate a risk cognitively and react to it emotionally. Pesticides, while considered to be the technology driving the "Green Revolution," and contributing to international improvement in food security, are primarily associated with consumer negativity linked to "negative affect," or emotional responses, rather than systematic cognitive evaluation of the issues, although these are also a topic of societal discourse ([@bibr2-0963662510392485]). Thus showing that cognitive evaluation and emotional response do not necessarily align. Although these two reactions are interrelated, they have different determinants. Exploring these determinants in detail can facilitate our understanding of the socio-psychological process affecting public acceptance of technology ([@bibr33-0963662510392485]).
The aim of this paper is to present an overview of the socio-psychological determinants of relevance to understanding public acceptance of technologies by analyzing the literature in social psychology and risk perception.
The main research question of the study is to identify which socio-psychological determinants of public acceptance of technology have been studied in the social science literature in the field of social psychology and risk perception. To do so the following sub-questions were addressed.
1. What potential socio-psychological determinants which influence public acceptance of technologies have been researched?
2. Are some socio-psychological determinants more relevant to specific technologies?
3. How have the socio-psychological determinants addressed in research of public acceptance of technology developed and changed over time?
4. Are there regional differences in determinants of public acceptance of technologies which have been researched?
2. Methods {#section2-0963662510392485}
==========
The database {#section3-0963662510392485}
------------
A search was conducted using the Scopus (electronic) database in order to identify papers that included information about the determinants of public acceptance of technology. First, a scoping search was conducted to gain information about technologies that have been controversial or have raised discussion about their use. The second, main, search was conducted in order to identify papers focused on these technologies. The time scale for the search was between 1977 and 2008 (one paper of 2009 appeared online as a prepublication) and the last search was done on 12 November 2008. The search was limited to peer-reviewed articles and review papers and the subject area was confined to social science and psychology. Duplicate articles, opinion papers, and articles which did not include relevant data were excluded from the main analysis. A total of 292^[1](#fn1-0963662510392485){ref-type="fn"}^ research papers were selected for the main analysis which included papers that reanalyzed data using a new analytical approach (*N* = 108). However, these 108 papers were excluded in the regional analysis as country of data collection could not be identified for these papers, therefore leaving 184 papers to be included in the regional analysis. The title, authors, abstract, keywords and bibliographical data of the articles were stored in Endnote. Although Scopus covers over 15,000 journals, a limitation of selecting publications from the Scopus database is that only articles cited in this database, and keywords assigned to the papers by their authors have been included in the review.
Selection of technologies {#section4-0963662510392485}
-------------------------
The initial scoping exercise was done to quickly scan papers for selecting the technologies in the analysis. Search terms were developed to identify articles that focused on technology and societal controversy. Ten technologies were prominent (although not necessarily evenly distributed in occurrence with times). These were *nuclear technology*, *information and communication technology* (ICT) (including computers and the internet), *mobile phones*, *chemicals used in agriculture* (pesticides and insecticides), *genetic modification*, *genomics*, *cloning*, *hydrogen technology*, *radio frequency identification technology* (RFID) and *nanotechnology*.
After the preliminary scanning, a literature search was conducted to collate papers addressing specific issues with regard to risk perception and its determinants for the selected technologies. The keywords used with each of the technologies were: technologies (as listed above) AND "scare OR fear" AND "controversy" AND "risk perception" AND "consumer acceptance OR consumer response OR consumer acceptability" AND "societal response OR societal acceptance OR societal concern OR social acceptability." In total 292 papers ([Table 1](#table1-0963662510392485){ref-type="table"}) were found to be relevant, i.e. investigating determinants of social acceptance of technology.
######
Distribution of articles and frequency of determinants over technologies.
Technology No. of articles (out of 292) Frequency of determinants (out of 558) Ratio (frequency of determinants/ no. of articles)
---------------------- ------------------------------ ---------------------------------------- ----------------------------------------------------
Genetic modification 104 210 2.02
Nuclear power 49 99 2.02
ICT 45 93 2.07
Pesticides 30 50 1.67
Nanotechnology 16 30 1.87
Cloning 11 21 1.91
Mobile phones 11 20 1.82
Hydrogen power 7 11 1.57
Genomics 13 14 1.08
RFID 6 10 1.67
Coding {#section5-0963662510392485}
------
The year of publication, research question, methodology, and the results were extracted from research articles. The factors influencing public acceptance were recorded from the research articles. These factors were coded into 31 different determinants of technology acceptance using thematic analysis. These were: *Impact (general, positive and negative)*; *Expert versus lay knowledge*; *Affect (general, negative and positive)*; *Impact health (positive and negative)*; *Impact environment (positive and negative)*; *Heuristics*; *Values (general and positive)*; *Perceived risk*; *Perceived benefit*; *Perceived cost*; *Risk management*; *Risk assessment*; *Attitudes (general, positive and negative)*; *Ethics and values*; *Role of societal actors*; *Trust and culpability*; *Concern*; *Citizen knowledge*; *Individual differences*; *Communication*; *Costs*; and *Technology characteristics*.
Countries where data were collected were also coded for all the articles. In total, 39 countries were identified (including research that compared data from consumers in different countries or cultural contexts). These countries were then categorized into seven regions: North-West Europe (UK, Germany, The Netherlands, Switzerland, Belgium, Sweden, Ireland, Norway, Austria, Finland, France, Poland and Denmark); Southern Europe (Romania, Turkey, Italy, Portugal and Spain); North America (USA and Canada); Latin America (Trinidad, Mexico and Argentina); Asia (Singapore, Korea, Hong Kong, Vietnam, Nepal, Bangladesh, Philippines, India, China, Kazakhstan, Malaysia, Japan and Taiwan); Oceania (Australia and New Zealand); and Africa.
Data analysis {#section6-0963662510392485}
-------------
The content of the papers was analyzed on distribution of coded scores across year and region. In addition, correspondence analysis was used to investigate the relationship and trends across different determinants and technologies, to depict the results in categories on a few dimensions ([@bibr18-0963662510392485]; [@bibr20-0963662510392485]). To avoid determinants with very low frequencies distorting the analysis, the determinants that appeared only once were merged into super-ordinate categories (the role of societal actors positive and negative were merged into role of societal actors; risk management complete and incomplete were merged into risk management; and risk assessment complete and incomplete were merged into risk assessment).
3. Results {#section7-0963662510392485}
==========
Determinants influencing public acceptance of technologies {#section8-0963662510392485}
----------------------------------------------------------
Thirty-one potential determinants which emerged from the coding scheme were found to influence public acceptance of new technologies. More than one determinant was found to influence public acceptance in most of the articles. In terms of the technology that was the focus of the research, the most frequently investigated technology was Genetic Modification (*N* = 104). On average an article includes between 1 and 2 determinants ([Table 1](#table1-0963662510392485){ref-type="table"}).
Of the 31 determinants, 6 determinants accounted for about 60% of all determinants mentioned across the sample. Of these, *perceived risk* was found to be the most frequently investigated determinant, and was reported 86 times. *Trust* was used 63 times; *perceived benefit* 51 times; *knowledge* 50 times; *individual differences* 44 times; and *attitudes* 42 times. Other influential determinants were *negative affect* coded 27 times; *technology characteristics* and *role of societal actors* each coded 22 times. In the sample determinants like *negative impact general*, *positive impact general* and *positive attitude* were coded 12 times each and *ethics* and *cost* were coded 11 times. *Communication*, *negative health impact*, *negative environment impact* and *values* were found to be coded about 10 times. Less researched determinants were *expert versus lay knowledge*, *heuristics*, *perceived cost*, *risk management*, *negative attitude*, *general affect*, *concern* and *positive affect* (coded about 6 times each). Determinants that were coded the least number of times (1--2 times) were *positive environment impact*, *risk assessment*, *general impact*, *positive health impact* and *positive value*.
Correspondence analysis between technologies and determinants showed that certain determinants were associated more with specific technologies (χ^2^ = 332.64, *p* = .006; [Figure 1](#fig1-0963662510392485){ref-type="fig"}). To classify these groups, hierarchical cluster analysis was applied to determine which technologies and determinants are associated more closely with each other. The four clusters identified in the cluster analysis comprised the technologies and the associated determinants. Clusters one and two came out as very clear clusters each including one technology, and one or more determinant. Cluster one showed the association of pesticides with the seven determinants *positive impact (health and environment)*, *negative impact (health and environment)*, *positive value*, *communication* and *cost*. The second cluster suggested that *concern* is associated with mobile phones. In cluster three, genomics and cloning appeared together with two determinants: *ethics* and *expert versus lay knowledge*. While these two determinants were associated strongly with cloning, they were weakly associated with genomics. In contrast to the first three clusters where a clear picture emerges for one single or two related technologies, the fourth cluster consisted of 6 technologies and 17 associated determinants. In this cluster nuclear technology and RFID were closely associated with *values*, *role of societal actors*, *impact general (positive and negative)*, *risk management*, *perceived risk*, *attitude general*, *perceived cost* and *affect (general and negative)*. In the same cluster ICT, nanotechnology, hydrogen power and genetic modification exhibited close association with *attitude (positive and negative)*, *technology characteristics*, *individual differences*, *trust*, *perceived benefits* and *knowledge*. While most of the determinants were found in the four clusters, some determinants were not found to have strong association with any of the technologies. These were: *heuristics*, *impact general*, *risk assessment* and *positive affect*. *Heuristics* and *impact* *general* were related to each other but they did not associate strongly with any of the technologies specifically.
{#fig1-0963662510392485}
Temporal trends in research on public acceptance of the technologies {#section9-0963662510392485}
--------------------------------------------------------------------
An increase in the number of studies and determinants dealing with public acceptance of technologies occurred over time ([Figure 2](#fig2-0963662510392485){ref-type="fig"}). A linear regression confirms an increase in publication over the years (*F* (1, 26) = 52.22, *p* \< .01, *R*^2^ = .66). Earlier publications focused on nuclear technology (first paper in 1977) and pesticides (first paper in 1988). In 1994 publications on genetic modification started appearing and the topic continues to attract scholarly attention, making it the most extensively researched upon technology. Research articles on hydrogen power, cloning, genomics and RFID were sporadic. Most recent in these technologies is nanotechnology, with papers being published in 2006, 2007 and 2008.
{#fig2-0963662510392485}
Over time, the number of determinants that have been investigated has increased ([Figure 3](#fig3-0963662510392485){ref-type="fig"}), implying that research directed towards understanding public acceptance of technologies is becoming increasingly sophisticated. From [Figure 3](#fig3-0963662510392485){ref-type="fig"} we can see that the models used to predict public acceptance are getting more complex, with a wide coverage of determinants influencing technology acceptance. "Classical" determinants, for example *risk perception*, *benefit perception*, *trust*, *knowledge*, *attitude*, *negative impact* and *individual differences* continue to be included in research designs. In addition new determinants (such as *heuristics*, *concern*, *risk assessment*, *positive impact* and *positive value*) have been the topic of more recent research. In terms of risk and benefit perception, *perceived risk* was cited more often than *perceived benefit*, showing researcher prioritization of risk perception over and above benefit perception as an important determinant of consumer acceptance.
{#fig3-0963662510392485}
Regional trends in research on public acceptance {#section10-0963662510392485}
------------------------------------------------
Regional trends in research on various determinants were examined for 184 research articles that included information that enabled the identification of country of collection ([Table 2](#table2-0963662510392485){ref-type="table"}). Looking at the frequency of determinants investigated in different regions across the world, research originating in North-West Europe investigated the greatest number of determinants (44%). This was followed by research originating in North America (30%). Fewer determinants were investigated in research studies originating in Asia and Southern Europe, the least were reported for research originating in Latin America and Oceania. This sharp decline in frequency of determinants is, in part, attributable to fewer publications addressing few technologies in these regions. Of the 184 research articles, 19 articles were comparative, as data were collected in different countries or regions. However, these articles were again dominated by data originating in North-West Europe and North America.
######
Regional distribution of articles and determinants on public acceptance.
Region No. of technologies covered No. of determinants
------------------- ----------------------------- ---------------------
North-West Europe 10 29
North America 9 27
Asia 5 16
Southern Europe 5 18
Latin America 1 8
Africa 2 8
Oceania 3 9
4. Discussion and conclusion {#section11-0963662510392485}
============================
Public acceptance of technologies continues to be a focus of scholarly attention, as demonstrated by the steady rise in the number of publications and determinants investigated that are found to impact the acceptance. Regional trends show that most of the research has been carried out in North America and North-West Europe. While this may be in part, because the search was limited to the English language, it is nevertheless clear from this that most of this type of research is concentrated in the developed world. More research is needed in developing countries and with developing economies, to present a more comprehensive picture of societal response to new technologies.
Of the ten controversial technologies studied, research was most frequently focused on nuclear power, genetic modification and ICT compared to genomics, cloning, mobile phones and hydrogen power. The study of public acceptance of nanotechnology and RFID has only recently been initiated, in line with the recency of technological advances, and it is therefore too early to judge whether research into these technologies will provide major contributions to the technology acceptance literature. The publication trend for different technologies can be partly related to the year of their introduction or commercialization. Discussion about the consequences of using nuclear technology has been a topic of public debate since World War 2. Application of synthetic pesticides in agriculture drew public criticism in 1962 with the publication of *Silent Spring* by Rachel Carson, inspiring widespread public concerns associated with pesticide use and environmental pollution ([@bibr26-0963662510392485]; [@bibr35-0963662510392485]). The consequences of using genetic modification escalated the already existing public debate on the use of new technologies in 1994 with commercialization of genetically modified food crops and products. Ever since its introduction, the technology has been exposed to media attention and societal debate about its merits or otherwise ([@bibr5-0963662510392485], [@bibr6-0963662510392485]). Research focused on the application of cloning technology started appearing around 1997 when the first cloned higher animal "Dolly" (a sheep) was developed ([@bibr5-0963662510392485]). An important point is that research into public acceptance of new technologies has tended to occur post-commercialization, when public concerns have begun to arise. In the case of nanotechnology, the discussion has been initiated at about the same time, in response to concerns from developers about public negativity to its application. This indicates a shift in focus on public acceptance of technologies, from *post hoc* studies to a more proactive effort to identify public opinions and values prior to commercialization. The extent to which such information will be used to shape science strategy and the specific application of nanotechnology remains to be evaluated.
Social science analysis focused on public acceptance of specific technologies is typically conducted after the technology has been introduced and commercialized, and subsequently has been associated with societal disquiet or negativity. Thus, in the past, it would appear social science research funding has been allocated to those technologies that have become societally controversial. In order to better understand the process of technology acceptance in society, research into non-controversial technologies might be applied to identify what factors drive societal acceptance (assuming that comparative analysis can be applied across technological areas that are inherently associated with different levels of impact or political controversy). It would also resolve whether the inherently "dramatic" qualities of some technologies also drive researcher interest, which in turn drives funding cycles and societal discourse about the technologies in question.
It is recognized that the socio-political context in which technologies are embedded also shapes public debate and acceptance of these same technologies. Further discussion of this is beyond the scope of the current research paper, as defined by the original research question. The question of why some technologies become societally controversial, whereas others do not, is worthy of further research.
The sophistication of the socio-psychological factors used to assess attitudes has also increased with time, reflecting theoretical advances in this area. *Perceived risk*, *perceived benefit*, *trust and culpability*, *knowledge*, *individual differences* and *attitude* are traditionally the most often reported or cited determinants, and these remain dominant. Temporal analysis of the data indicates that the postulated models explaining public acceptance have increased in complexity, by adding, rather than replacing determinants. Determinants that were found to influence public acceptance of one technology contribute in shaping the acceptance for other technologies. The analysis has demonstrated that the number of social psychological determinants investigated in the context of technology acceptance has increased, perhaps reflecting theoretical advances in understanding public responses to technologies. Some determinants (for example, *positive affect*, *risk assessment* and *heuristics*) have been less frequently studied. A systematic critique of the relative predictive capacity of these different determinants is not currently available. A first step in the development of such a systematic review would be the simultaneous analysis of all potential determinants in a single study, or (possibly) through application of formal meta-analysis if appropriate data are available. This is a topic worthy of future investigation.
The temporal analysis has also confirmed that research that has focused on the individual as a "non-rational" actor has increased. This research suggests considerable support for the socio-psychological determinants of acceptance of technology underpinning lay opinion, as well as providing an explanation as to why these might differ from expert views. Further investigation into the disparity between lay and expert opinions of technology may systematically contrast the extent to which the different determinants predict technology acceptance in each group.
Certain determinants are seen to have more impact on public acceptance of specific technologies. Pesticides were mainly associated with health and the environment, cloning and genomics with ethics; while a large group of technologies was associated with most of the remaining determinants. This association between certain types of technologies and determinants to some extent can help us to understand and predict the factors that will set the stage for discussion of new and emerging technologies.
In this paper a specific class of technologies has been investigated, that is, those technologies that have been enabling a myriad of applications with the potential to change society, as the impact of these applications ripples through society. These technologies can be called "transformative," as they have the power to transform society by introducing completely new social phenomena. Previous transformative technologies (agricultural technology, printing, aircraft and vaccinations) have had lasting effects on human values, power structures and ideas and acted as potential drivers of socio-economic, political and institutional change ([@bibr9-0963662510392485]; [@bibr11-0963662510392485]; [@bibr46-0963662510392485]). The emergence of the technologies reviewed in this paper not only fuelled the engines of economy and growth, but also raised critical issues of political and military influence (e.g. nuclear power), international competition (e.g. GMO), environmental crisis (e.g. pesticides), ethical debates in relation to the discussions on the ethical acceptability of human control over and manipulation of nature (cloning and genomics), social changes resulting from expectations of being connected 24 hours a day, seven days a week (mobile phones) and protection of privacy (RFID, ICT).
Nanotechnology has the potential to become a transformative technology. It is among the recent emerging technologies which have been the focus of attention on the part of stakeholders, opinion leaders and media discussion. On one hand it presents unmatched opportunities to develop new products and services, and may result in longevity, public health benefits, and more sustainable production, but on the other raises concern, fear and anxiety among the public ([@bibr43-0963662510392485]; [@bibr45-0963662510392485]; [@bibr47-0963662510392485]). Understanding the socio-psychological factors would allow contextualization of its development and implementation, and potentially facilitate allocation of resources in areas of application relevant to the wider needs of society ([@bibr40-0963662510392485]; [@bibr42-0963662510392485]).
Future research needs to explore the interrelationships between determinants, particularly those which have emerged as being influential in recent years, such as the relationship between perceived risk and benefit, but also identify the knowledge gaps and explore other psychological factors that have recently started appearing in the literature such as heuristics and affective responses.
Upon email request to the authors, a list of all 292 articles is available.
**Nidhi Gupta** is a PhD scholar at the Marketing and Consumer Behaviour Group, Wageningen University, the Netherlands. She has a Master's degree in Natural Resources from TERI University, India. Her PhD research focuses on the societal response to nanotechnology and the determinants of consumer acceptance of nanotechnology.
**Arnout R.H. Fischer** is assistant professor with the Marketing and Consumer Behaviour Group at Wageningen University, the Netherlands. He aims to understand public behaviour through social psychological approaches. He has published on consumer behaviour in relation to food safety and new food technologies in academic books and peer-reviewed journals.
**Lynn J. Frewer** is professor of food safety and consumer behaviour in the Marketing and Consumer Behaviour Group in the Social Sciences Department at Wageningen University, the Netherlands. Previously Lynn was Head of Consumer Science at the Institute of Food Research at Norwich in the UK. Lynn has a background in psychology and has published extensively in the area of risk perception and communication.
| 2023-09-16T01:27:17.760571 | https://example.com/article/3595 |
CCAAT/enhancer-binding proteins (C/EBP) beta and delta activate osteocalcin gene transcription and synergize with Runx2 at the C/EBP element to regulate bone-specific expression.
CCAAT/enhancer-binding proteins (C/EBP) are critical determinants for cellular differentiation and cell type-specific gene expression. Their functional roles in osteoblast development have not been determined. We addressed a key component of the mechanisms by which C/EBP factors regulate transcription of a tissue-specific gene during osteoblast differentiation. Expression of both C/EBPbeta and C/EBPdelta increases from the growth to maturation developmental stages and, like the bone-specific osteocalcin (OC) gene, is also stimulated 3-6-fold by vitamin D(3), a regulator of osteoblast differentiation. We characterized a C/EBP enhancer element in the proximal promoter of the rat osteocalcin gene, which resides in close proximity to a Runx2 (Cbfa1) element, essential for tissue-specific activation. We find that C/EBP and Runx2 factors interact together in a synergistic manner to enhance OC transcription (35-40-fold) in cell culture systems. We show by mutational analysis that this synergism is mediated through the C/EBP-responsive element in the OC promoter and by a direct interaction between Runx2 and C/EBPbeta. Furthermore, we have mapped a domain in Runx2 necessary for this interaction by immunoprecipitation. A Runx2 mutant lacking this interaction domain does not exhibit functional synergism. We conclude that, in addition to Runx2 DNA binding functions, Runx2 can also form a protein complex at C/EBP sites to regulate transcription. Taken together, our findings indicate that C/EBP is a principal transactivator of the OC gene and the synergism with Runx2 suggests that a combinatorial interaction of these factors is a principal mechanism for regulating tissue-specific expression during osteoblast differentiation. | 2024-04-18T01:27:17.760571 | https://example.com/article/7406 |
Beech Mushrooms Recipe
In my Asian market, there are always abundant choices when it comes to mushrooms—shiitake, enoki, button, portobello, buna shimeji, chicken drumstick/king oyster mushrooms, etc. I love mushrooms as they are one of the healthiest foods to eat, plus they are low in calories and are often organically grown.
This week, both white and brown beech mushrooms (buna shimeji mushrooms) are on sale. I got a couple of boxes of them but don’t really have a good recipe to prepare them. I then think of Japanese recipes as mushroom is a mainstay in Japanese cuisine and I had personally savored delicately flavored mushroom dishes during my trips to Tokyo. After flipping through my Japanese cookbook, I decided to cook the mushrooms with some dashi, soy sauce, sake, mirin, and butter.
The end result was simple, pleasing, and delicious. I really enjoyed the dish. Now, do you guys love mushrooms? And which is your favorite?
For me, it’s a tossup between shiitake and enoki! I love the silky and earthy qualities of shiitake, but the sight of cute, delicate enoki always delights me. Never had beech mushrooms, must try them soon.
This dish looks deilicious. I love mushrooms,how ever our selection here in middle Georgia is a bit limited. Its even difficult to find canned Straw mushrooms. If you are very lucky, you will happen across some shitake that are a bit long in the tooth, very sad indeed. I’m afraid we are still at the mercy of the commercial growers, white button mushrooms. But as always you give us hope with your beautiful dishes. On a brighter note, we have been serching the Pecan orchards for the elusive “Blond Pecan Truffle”, native to middle and south Georgia. So far all we have found are past their prime. :-(
The Korean market near my house sometimes has these for only $.99 per little carton! I will try your recipe it sounds good. I sauteed two packs with a little olive oil and butter mixed, with thin sliced red onion and garlic. After a few minutes I added a little red wine (I didnt have white!) and a little ume vinegar and a touch of soy sauce. I sprinkled with a little freshly dried thyme and black pepper…they were SOO good. | 2023-12-21T01:27:17.760571 | https://example.com/article/2479 |
Saturday, June 30, 2012
Originale philosophia
"Another Greek term whose ancient meaning is retained is philosophia. The Greek Fathers had defined the monk's life as 'philosophy according to Christ' and 'the only true philosophy', or even simply as 'philosophy'. This term, meaning the practical discernment of the value of things and of the vanity of the world which must be renounced, is applied to those whose whole existence manifests this renunciation. In the same way, in the monastic Middle Ages as well as in antiquity, philosophia designates, not a theory or a way of knowing, but a lived wisdom, a way of living according to reason. There are, in effect, two ways of living according to reason. Either one lives according to worldly wisdom, as taught by the pagan philosophers, and that is the philosophia saecularis or mundialis, or one lives according to Christian wisdom which is not of this world but already of the world to come, and this is the philosophia caelestis or spiritualis or divina. The philosopher par excellence, and philosophy itself, is Christ: ipsa philosophia Christus. He was the Wisdom itself of God incarnate; and the Virgin Mary, in whom was accomplished the mystery of the Incarnation, is called 'the philosophy of Christians'. They must learn from her: philosophari in Maria. Those who had heralded the advent of the Lord Jesus or who have transmitted His message are the philosophers the Christians obey; they speak of philosophia Pauli, and of 'David the philosopher'.
"Now, this integral Christianity, this way of life entirely consecrated to God, this conuersatio caelestis, is indeed realized in monastic life. That is why the lawgivers and the models of monasticism are considered masters of philosophy. The cloisters are schools of philosophy, 'gymnasia' where the 'philosophy of St. Benedict' is learned. St. Bernard is praised because he formed the monks of Clairvaux in the 'disciplines of celestial philosophy', and Adam of Perseigne declares he is committed to the 'Cistercian philosophy'. To lead a monastic life is simply 'to philosophize'. Du Cange gives no other medieval equivalent for the word philosophari than: monachum agere. The verb philosophari is applied to cenobites living in monasteries as well as to the solitary in his hermit's cell. In monastic literature well into the twelfth century, the expression christiana philosophia when employed without commentary or explanation very often stands for monastic life itself."
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About Me
An obscurantist who takes pleasure in holding the unpopular view, dynamiter of the Train of Progress, and vehement defender of Dead White Men.
Also a guitarist: listen to demos at ReverbNation.com/JohnnieOrson | 2024-01-12T01:27:17.760571 | https://example.com/article/6708 |
a girl who loves her dog, traveling, food, and writing.
ROME, ITALY.
After spending a few days in Venice, I took a train to Rome! I have to say, I had THE BEST TIME in Roma! It was so HOT, and I did so much walking- but so worth it.
Where do I even begin? So, as I said before- Italy was my SOLO-TRIP! I stayed at the yellow. It had a bar right next door, so it’s a great way to meet people! I highly recommend staying there if you want to have a good time. The bar has different theme nights, good drinks + food.
There were a couple of things I just HAD to check off my list:
The Colosseum
Trevi Fountain
Spanish Steps
If you know me, I’m a HUGE Audrey Hepburn fan, and Roman Holiday is one of my favorite movies. Sitting on the Spanish Steps was so surreal. It was beyond beautiful. I wish I got a picture of myself sitting on the steps– but truthfully, I didn’t trust giving anyone my iPhone! That was my only lifeline (I used google maps to get everywhere! & that was my only way to stay connected with friends + family). So, when I go back to Rome, I’ll for sure get more pictures with myself in them. (i’m assuming i’ll be there with someone! ha!) | 2023-11-12T01:27:17.760571 | https://example.com/article/4063 |
Q:
Generating and checking random numbers in JavaScript
I have some problems with JavaScript.
So far, I have this code:
<!DOCTYPE html>
<html>
<body>
<p>This program generates random numbers from 100 to 999.</p>
<button onclick="maingen()">Start</button> <-- Here's the FUNCTION OF FUNCTIONS..It should generate 3 digits, but only generates 1 :(
<p id="numbers"></p>
<script>
function generate1() {
var a = Math.floor((Math.random() * 9) + 1);
document.getElementById("numbers").innerHTML = a; <--generating the first digit (from 1 to 9)
}
function generate2() {
var b = Math.floor((Math.random() * 9) + 0);
document.getElementById("numbers").innerHTML = b; <--generating the second digit (from 0 to 9)
}
function generate3() {
var c = Math.floor((Math.random() * 9) + 0);
document.getElementById("numbers").innerHTML = c; <--generating the third digit (from 0 to 9)
}
function maingen(){
generate1();
generate2();
generate3();
}
</script>
</body>
</html>
And it doesn't work like I intended to. It should generate a random number from 100 to 999.
(I am generating separate digits because later I will need to check if there are same digits in that number (for example 222)).
So what did I do wrong? Any kind of help would be nice. Thank you.
A:
You could do it using the += in each function or you could simplify it a little more.
function generate1() {
return Math.floor((Math.random() * 9) + 1);
}
function generate2() {
return Math.floor((Math.random() * 9) + 0);
}
function generate3() {
return Math.floor((Math.random() * 9) + 0);
}
function maingen() {
var a = generate1();
var b = generate2();
var c = generate3();
document.getElementById("numbers").innerHTML = a + '' + b + '' + c;
}
You can also do it like this
function generate_rand() {
return Math.floor((Math.random() * 9));
}
function maingen() {
var a = generate_rand() + 1;
var b = generate_rand();
var c = generate_rand();
document.getElementById("numbers").innerHTML = a + '' + b + '' + c;
}
| 2023-10-12T01:27:17.760571 | https://example.com/article/5685 |
56 Ill. App.3d 806 (1978)
372 N.E.2d 679
LYNN THOMAS, Plaintiff-Appellee,
v.
KENNETH R. THOMAS, Defendant-Appellant.
No. 77-82.
Illinois Appellate Court Second District.
Opinion filed January 23, 1978.
*807 Albert S. Salvi, of Lake Zurich, for appellant.
H. Joseph Gitlin and Victor P. Lundemo, both of Woodstock, for appellee.
Judgment affirmed.
Mr. JUSTICE WOODWARD delivered the opinion of the court:
Plaintiff, Lynn Thomas, and defendant, Kenneth Thomas, were divorced in 1971, and plaintiff was given custody of their minor son. In 1974 she obtained an order allowing her and the child to move to Georgia. While there, she initially lived with her parents, but later obtained an apartment for herself and her son. A number of relevant events occurred thereafter: she allowed a male friend to live in her apartment; she remarried and has been divorced again; she pleaded guilty to a charge of shoplifting; she was charged with theft, although the charge was dismissed; and after being robbed, she purchased a gun, although the purchase was a violation of probation. Defendant obtained information about these incidents and now seeks to change the custody of the minor son to him. The trial court refused to interview the child, denied the requested modification and ordered defendant to pay plaintiff's attorney's fees totaling $870.
1, 2 We are first called on to consider whether the denial of custody modification was against the manifest weight of the evidence. Defendant here has the burden of proof of showing a change of circumstances materially affecting the welfare of the minor child. (Nye v. Nye (1952), 411 Ill. 408, 105 N.E.2d 300.) We view Nye as controlling because there is involved a modification of an original decree of custody; we have already distinguished this situation from one in which there has been no previous grant of custody of children by court order. (See Strand v. Strand (1976), 41 Ill. App.3d 651, 355 N.E.2d 47.) The record shows that plaintiff provided some explanation on each alleged impropriety which was raised by defendant. Because the weight to be given testimony and the determination of what constitutes the best interest of the child are properly matters for the trier of fact, absent an abuse by the trial court, we will not substitute our judgment for that of the finder of fact. (Wells v. Wells (1976), 36 Ill. App.3d 488, 344 N.E.2d 37.) Our review of the testimony given below does not show such an abuse, nor is the decision of *808 the trial court palpably erroneous. We accordingly hold that the decision of the trial court was not contrary to the manifest weight of the evidence.
3 We next consider whether the trial court abused its discretion by not interviewing the minor in question. Defendant requested that the court make an in camera examination of the minor. The court properly noted that such an examination of a minor, particularly one of tender years, is discretionary and noted further that if an interview were to be conducted, the court would defer it. We distinguish this case from Crownover v. Crownover (1975), 33 Ill. App.3d 327, 337 N.E.2d 56. In that case the trial court rejected outright the proposed testimony of an 11-year-old, and the Third District held that such an outright rejection was improper. In the cause before us, the court indicated that it would first consider whether to interview the 7-year-old. It is manifest that because the court did not interview the child, it had considered the matter and decided that there was no purpose to interview the child in this particular situation. Again, in light of the better position of the trial court and the discretion allowed it, we hold that there was no error in refusing to conduct the requested interview.
4 We turn finally to the question of whether the order to pay plaintiff's attorney's fees was proper. Defendant argues that the record shows that no hearing was held on the plaintiff's need to have defendant pay her attorney's fees. The judgment order does note that the court was advised in the premises. The rule of construction in determining the meaning of a judgment or decree is that one must examine the situation as it existed at the rendition of the judgment. (Waldron v. Waldron (1973), 13 Ill. App.3d 964, 301 N.E.2d 167.) We note that the record shows that at the time of the proceedings below, September 15, 1976, plaintiff had not been employed since the end of July of 1975. It is our view that this testimony provided the court with sufficient information to conclude that in this case plaintiff was unable to pay her own attorney's fees.
We accordingly find no error and we affirm the decision of the trial court on all points.
Affirmed.
SEIDENFELD, P.J., and NASH, J., concur.
| 2024-04-09T01:27:17.760571 | https://example.com/article/1455 |
When Gen. Michael Flynn marched into the White House Briefing Room to declare that "we are officially putting Iran on notice," he drew a red line for President Trump. In tweeting the threat, Trump agreed.
His credibility is now on the line.
And what triggered this virtual ultimatum?
Iran-backed Houthi rebels, said Flynn, attacked a Saudi warship, and Tehran tested a missile, undermining "security, prosperity and stability throughout the Middle East," placing "American lives at risk."
But how so?
TRENDING: Americans against unconstitutional mask mandates
The Saudis have been bombing the Houthi rebels and ravaging their country, Yemen, for two years. Are the Saudis entitled to immunity from retaliation in wars they start?
Where is the evidence Iran had a role in the Red Sea attack on the Saudi ship? And why would President Trump make this war his war?
As for the Iranian missile test, a 2015 U.N. resolution "called upon" Iran not to test nuclear-capable missiles. It did not forbid Iran from testing conventional missiles, which Tehran insists this was.
Is the United States making new demands on Iran not written into the nuclear treaty or international law – to provoke a confrontation?
Did Flynn coordinate with our allies about this warning of possible military action against Iran? Is NATO obligated to join any action we might take?
Or are we going to carry out any retaliation alone, as our NATO allies observe, while the Israelis, Gulf Arabs, Saudis and the Beltway War Party, which wishes to be rid of Trump, cheer him on?
Bibi Netanyahu hailed Flynn's statement, calling Iran's missile test a flagrant violation of the U.N. resolution and declaring, "Iranian aggression must not go unanswered." By whom, besides us?
The Saudi king spoke with Trump Sunday. Did he persuade the president to get America more engaged against Iran?
Senate Foreign Relations Committee Chairman Bob Corker is among those delighted with the White House warning:
"No longer will Iran be given a pass for its repeated ballistic missile violations, continued support of terrorism, human rights abuses and other hostile activities that threaten international peace and security."
The problem with making a threat public – Iran is "on notice" – is that it makes it almost impossible for Iran, or Trump, to back away.
Tehran seems almost obliged to defy it, especially the demand that it cease testing conventional missiles for its own defense.
This U.S. threat will surely strengthen those Iranians opposed to the nuclear deal and who wish to see its architects, President Hassan Rouhani and Foreign Minister Mohammad Javad Zarif, thrown out in this year's elections.
If Rex Tillerson is not to become a wartime secretary of state like Colin Powell or Dean Rusk, he is going to have to speak to the Iranians, not with defiant declarations, but in a diplomatic dialogue.
Tillerson, of course, is on record as saying the Chinese should be blocked from visiting the half-dozen fortified islets they have built on rocks and reefs in the South China Sea.
A prediction: The Chinese will not be departing from their islands, and the Iranians will defy the U.S. threat against testing their missiles.
Wednesday's White House statement makes a collision with Iran almost unavoidable, and a war with Iran quite possible.
Why did Trump and Flynn feel the need to do this now?
Like the reporting you see here? Sign up for free news alerts from WND.com, America's independent news network.
There is an awful lot already on the foreign-policy plate of the new president after only two weeks, as pro-Russian rebels in Ukraine are firing artillery again, and North Korea's nuclear missile threat, which, unlike Iran's, is real, has yet to be addressed.
High among the reasons that many supported Trump was his understanding that George W. Bush blundered horribly in launching an unprovoked and unnecessary war on Iraq.
Along with the 15-year war in Afghanistan and our wars in Libya, Syria and Yemen, our 21st-century U.S. Mideast wars have cost us trillions of dollars and thousands of dead. And they have produced a harvest of hatred of America that was exploited by al-Qaida and ISIS to recruit jihadists to murder and massacre Westerners.
Osama's bin Laden's greatest achievement was not to bring down the twin towers and kill 3,000 Americans, but to goad America into plunging headlong into the Middle East, a reckless and ruinous adventure that ended her post-Cold War global primacy.
Unlike the other candidates, Trump seemed to recognize this.
It was thought he would disengage us from these wars, not rattle a saber at an Iran that is three times the size of Iraq and has as its primary weapons supplier and partner Vladimir Putin's Russia.
When Barack Obama drew his red line against Bashar Assad's use of chemical weapons in Syria's civil war, and Assad appeared to cross it, Obama discovered that his countrymen wanted no part of the war that his military action might bring on.
President Obama backed down – in humiliation.
Neither the Ayatollah Khamenei nor Trump appears to be in a mood to back away, especially now that the president has made the threat public. | 2024-01-17T01:27:17.760571 | https://example.com/article/3012 |
Broward County Deputy Scot Peterson is refuting claims that he’s a coward, saying he did not believe gunfire was happening inside the building during the Parkland, Florida school shooting.
In a response sent out by his lawyer, Peterson fights back against the narrative that he “failed to meet the standards of police officers.”
#BREAKING: Broward County Deputy Scot Peterson releases a statement through his attorney, saying: "the allegations that Mr. Peterson was a coward and that his performance, under the circumstances, failed to meet the standards of police officers are patently untrue." pic.twitter.com/f5wZFMVy4f — Ryan Saavedra 🇺🇸 (@RealSaavedra) February 26, 2018
The statement, released by Fort Lauderdale attorney Joseph DiRuzzo, says, “Allegations that Mr. Peterson was a coward and that his performance, under the circumstances, failed to meet the standards of police officers are patently untrue.”
“Mr. Peterson is confident that his actions on that day were appropriate under the circumstances and that the video (together with the eye-witness testimony of those on the scene) will exonerate him of any sub-par performance,” the letter continues.
Pointing out Sheriff Israel’s hypocrisy when it comes to “jumping to conclusions,” the statement reads:
We note that Sheriff Scott Israel’s accusations are wholly inconsistent with the position he has taken in response to Representative Bill Hager’s criticisms. It is our understanding that Sheriff Israel acknowledged that the investigation remains on-going and that “[i]nvestigations will not be rushed or asked to jump to conclusions.” But this is exactly what Sheriff Israel did, he jumped to a conclusion regarding Mr. Peterson’s performance on February 14th even though Israel claims that “[i]t is more important for us to wait and let the investigators get it right[.]” We question why this statement would not also apply to Mr. Peterson?
The response is referring to Broward Sheriff Israel’s revelation that Peterson waited outside of the school for four minutes while the shooter gunned down students and staff.
In addition to Peterson’s declaration, a medical emergency responder on the scene says he was told to stand down, not enter the building and that law enforcement didn’t follow mass casualty event protocols.
The Emergency Election Sale is now live! Get 30% to 60% off our most popular products today! | 2024-01-17T01:27:17.760571 | https://example.com/article/3739 |
Q:
casting out nines: division
the division in the casting out nines is not clear to me. In the given example he got a 8 and a 4, and later he got an 8 and a 3. Shouldn't it be an 8 and a 4 (or 3) in both?
A:
The example seems to be:
$$275462 \div 877 = 314 \quad \text{ r. } \quad 84 $$ which which can be checked by casting out nines and verifying $$\quad 8 \quad \quad \equiv \quad 4 \quad\times \quad 8 \: \quad + \: \quad 3$$ (modulo 9).
In fact what is being checked is the equivalent multiplication statement
$$275462 = 877 \times 314 \quad + \quad 84 $$
which could also be written as
$$(30606 \times 9 + 8) = (97 \times 9 + 4)\times(34\times 9 +8) +(9 \times 9 + 3)$$
and (modulo 9) all the $\times 9$ terms are equivalent to 0.
| 2024-02-15T01:27:17.760571 | https://example.com/article/7355 |
Q:
boost::msm usage: how to set properties of target states
States in my fsm has different properties. I define a event which will invoke a transition to a target state. I want to set properties of target by event data. My choices are:
1. set properties in the action or guard: but it is still in the source state. I fill not good if i set the target state properties at the time
2. set properties in the target entry: but the value of properties have been calculated in guard on receiving the event so i should recalculated again :(
Any one can give me more ideas. Thanks !
A:
I'm not sure your situation but I think that you have two different issue. One is where is the best place to set the target state's property. The other is how to avoid re-calculation.
Let's consider the former issue. As you mentioned, there are two candidates. They are a transition action and a target state's entry action. I think both are OK. If you use the target state's entry action, you need to care about event type. If you write the handler as follows, all events are caught by the handler.
template <class Event, class Fsm>
void on_entry(Event const& e, Fsm&) {
std::cout << "State2::on_entry()" << std::endl;
std::cout << "You can also set the property here." << std::endl;
property = e.data;
}
You can specify the event as follows:
template <class Fsm>
void on_entry(Event1 const& e, Fsm&) {
std::cout << "State2::on_entry()" << std::endl;
std::cout << "You can also set the property here." << std::endl;
property = e.data;
}
To solve the latter issue, how to avoid re-calculation, you need to declare the member variable in the event as mutable as follows:
struct Event1 {
mutable int data; // mutable is required because Event is passed as const&.
};
Because Boost.MSM passes the Event parameter as a const reference.
Here is a code that describes how to set the target state property without re-calculation:
#include <iostream>
#include <boost/msm/back/state_machine.hpp>
#include <boost/msm/front/state_machine_def.hpp>
#include <boost/msm/front/functor_row.hpp>
#include <boost/static_assert.hpp>
namespace msm = boost::msm;
namespace msmf = boost::msm::front;
namespace mpl = boost::mpl;
// StateMachine [Sm]
//
// (initial)
// |
// V
// State1 --Event1[Guard1]/Action1--> State2
//
// ----- Events
struct Event1 {
mutable int data; // mutable is required because Event is passed as const&.
};
struct Sm_:msmf::state_machine_def<Sm_>
{
// States
struct State1:msmf::state<>
{
template <class Event,class Fsm>
void on_entry(Event const&, Fsm&) const {
std::cout << "State1::on_entry()" << std::endl;
}
template <class Event,class Fsm>
void on_exit(Event const&, Fsm&) const {
std::cout << "State1::on_exit()" << std::endl;
}
};
struct State2:msmf::state<>
{
template <class Event,class Fsm>
void on_entry(Event const& e, Fsm&) {
std::cout << "State2::on_entry()" << std::endl;
std::cout << "You can also set the property here." << std::endl;
property = e.data;
}
template <class Event,class Fsm>
void on_exit(Event const&, Fsm&) const {
std::cout << "State2::on_exit()" << std::endl;
}
int property;
};
// Set initial state
typedef State1 initial_state;
// Guards
struct Guard1 {
template <class Event, class Fsm, class Source, class Target>
bool operator()(Event const& e, Fsm&, Source&, Target&) const {
e.data *= 2; // calc
std::cout << "In Guard1, e.data *= 2. e.data=" << e.data << std::endl;
return true;
}
};
// Actions
struct Action1 {
template <class Event, class Fsm, class Source, class Target>
void operator()(Event const& e, Fsm&, Source&, Target& t) const {
std::cout << "In Action1, set target state property based on event data that is calculated in Guard1." << std::endl;
t.property = e.data;
}
};
// Transition table
struct transition_table:mpl::vector<
// Start Event Next Action Guard
msmf::Row < State1, Event1, State2, Action1, Guard1 >
> {};
};
// Pick a back-end
typedef msm::back::state_machine<Sm_> Sm;
void test()
{
Sm sm;
sm.start();
std::cout << "> Send Event1()" << std::endl;
Event1 e1;
e1.data = 12;
sm.process_event(e1);
}
int main()
{
test();
}
You can compile, run, and modify the code using Wandbox, online compiler.
http://melpon.org/wandbox/permlink/L1HAjIiLU091906u
| 2023-11-07T01:27:17.760571 | https://example.com/article/2905 |
Share
Heeding diners' demands for more than a menu centered on sandwiches over dinner, Abbot Kinney's new produce-commanded Feed Body & Soul is vastly expanding its evening offerings with a new selection of entrees. The healthy-leaning restaurant, which adheres to recipes shunning excess salt, sugar, and oil, is instituting a few other changes since opening back in January to spice things up. Quite literally.
Matthew Dickson is now making the house-made hot sauces you see above to accompany every table. The chef prepares both his own red sauce and tomatillo, along with a diminutive dish of salt and pepper for those wholike us on our first visitmight have found too many of the dishes in need of a little more punch.
The new menu, meanwhile, offers guests more traditional dinner options by building on Dickson's daily changing rotisserie specials (which recently included a serving of bison) in entrees like chili and hemp seed-crusted albacore, vegetable and grain bowls, and healthy spreads of salads and small plates. You can see Feed Body & Soul's full new dinner menu below.
SMALL PLATES
Farmers Market Pickled Vegetables GF 7
In-House Fermented Red Beet, Cabbage and Cauliflower GF 7
A Selection of Three Farmers Market Pickled Vegetables GF 7
Farmers Market Spreads GF 7
A Selection of Three Farmers Market Spreads GF 10
All of our spreads are served with our whole wheat
and sea salt crackers (GF upon request) | 2023-09-04T01:27:17.760571 | https://example.com/article/3314 |
Q:
Accessing files from a multiple datastores on a VSAN in VMware
I have a VMware 6.7 environment running as a single datacenter, however with two clusters (say A-cluster and a B-cluster), each with 3 esxi hosts. A VSAN is configured on these clusters and has two VSAN datastrores (A-datastore and B-datastore) for A-cluster and B-cluster respectively.
Now, I have a iso file in the A-datastore and I'm trying to deploy a VM in B-cluster and in the process of deploying, I would like to mount the iso file present in the A-datastore. When I try to attach the CD/DVD and choose "Datastore ISO File" option, I only get to browse the B-datastore, but not the A-datastore.
Is this even possible?
A:
The Virtual SAN datastores are ONLY presented to the hosts that are members of a particular cluster. But you can implement Content Library and store there all templates/iso/ etc.
Here is a detailed guide, how to create it - https://www.starwindsoftware.com/blog/working-with-content-libraries-in-vsphere-6-5
| 2024-01-24T01:27:17.760571 | https://example.com/article/2802 |
Deprecated: mysql_connect(): The mysql extension is deprecated and will be removed in the future: use mysqli or PDO instead in /home/clearvie/public_html/system/database/mysql.php on line 6Warning: session_start(): Cannot send session cookie - headers already sent by (output started at /home/clearvie/public_html/system/database/mysql.php:6) in /home/clearvie/public_html/system/library/session.php on line 11Warning: session_start(): Cannot send session cache limiter - headers already sent (output started at /home/clearvie/public_html/system/database/mysql.php:6) in /home/clearvie/public_html/system/library/session.php on line 11Warning: Cannot modify header information - headers already sent by (output started at /home/clearvie/public_html/system/database/mysql.php:6) in /home/clearvie/public_html/index.php on line 181Warning: Cannot modify header information - headers already sent by (output started at /home/clearvie/public_html/system/database/mysql.php:6) in /home/clearvie/public_html/system/library/currency.php on line 45Onderhoud | 2023-08-25T01:27:17.760571 | https://example.com/article/9300 |
---
abstract: 'In this paper, we study relations between automorphism groups of cubic fourfolds and Kuznetsov components. Firstly, we characterize automorphism groups of cubic fourfolds as subgroups of autoequivalence groups of Kuznetsov components using Bridgeland stability conditions. Secondly, we compare automorphism groups of cubic fourfolds with automorphism groups of their associated K3 surfaces. Thirdly, we note that the existence of a non-trivial symplectic automorphism on a cubic fourfold is related to the existence of associated K3 surfaces.'
address: 'Interdisciplinary Theoretical and Mathematical Sciences Program, RIKEN, 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan'
author:
- Genki Ouchi
title: Automorphism groups of cubic fourfolds and K3 categories
---
Introduction
============
Background and results
----------------------
In this paper, we study relations between symmetries of cubic fourfolds and symmetries of K3 surfaces. Finite symmetries of K3 surfaces are related to sporadic finite simple groups as the Mathieu groups $M_{23}, M_{24}$ and the Conway groups $\mathrm{Co}_0, \mathrm{Co}_1$ in both mathematics and physics. Mukai [@Muk] proved that finite groups of symplectic automorphisms of K3 surfaces are certain subgroups of the Mathieu group $M_{23}$. In the context of physics, Eguchi, Ooguri and Tachikawa [@EOT] found Mathieu moonshine phenomena for the elliptic genera of K3 surfaces. This is the mysterious relation between the elliptic genera of K3 surfaces and the Mathieu group $M_{24}$. After [@EOT], Gaberdiel, Hohenegger and Volpato [@GHV] studied symmetries of K3 sigma models. From the mathematical point of view, Gaberdiel, Hohenegger and Volpato [@GHV] compared symmetries of the Mukai lattice with the symmetries of the Leech lattice. Huybrechts [@Huy] interpreted the symmetries of the Mukai lattice in [@GHV] as the symmetries of derived categories of K3 surfaces and stability conditions. We recall the precise statement of Huybrechts’s theorem in [@Huy]. For a K3 surface $S$, let ${\mathrm{Stab}}^*(S)$ be the distinguished connected component of the space of stability conditions on the derived category $D^b(S)$ of coherent sheaves on $S$ as in [@Bri08]. For a K3 surface $S$ and a stability condition $\sigma \in {\mathrm{Stab}}^*(S)$, denote the group of symplectic autoequivalences of $D^b(S)$ fixing $\sigma$ by ${\mathrm{Aut}}_{\mathrm{s}}(D^b(S),\sigma)$ (Definition \[stabilizer K3\]). The Conway group $\mathrm{Co}_0$ is the automorphism group of the Leech lattice $N$. The Conway group $\mathrm{Co}_1=\mathrm{Co}_0/\{\pm1\}$ is known as one of sporadic finite simple groups. Huybrechts [@Huy] proved the analogue of Mukai’s theorem in [@Muk].
\[Conway\] Let $G$ be a group. The following are equivalent.
- There is a K3 surface $S$ and a stability condition $\sigma \in {\mathrm{Stab}}^*(S)$ such that $G$ can be embedded into ${\mathrm{Aut}}_{\mathrm{s}}(D^b(S),\sigma)$.
- The group $G$ can be embedded into the Conway group $\mathrm{Co}_0=\mathrm{O}(N)$ such that $\mathrm{rk}(N^G) \geq 4$.
Gaberdiel, Hohenegger and Volpato [@GHV] computed the automorphism group of the Gepner model $(1)^6$. The automorphism group of the Gepner model $(1)^6$ is same as the symplectic automorphism group of the Fermat cubic fourfold. Cheng, Ferrari, Harrison and Paquette [@CFHP] studied Landau-Ginzburg models related to homogeneous polynomials of degree three with six variables. In [@CFHP], symplectic automorphism groups of cubic fourfolds appeared as the symmetries of K3 sigma models. Recently, Laza and Zheng [@Laza] classified symplectic automorphism groups of cubic fourfolds. In particular, symplectic automorphism groups of cubic fourfolds are certain subgroups of the Conway group $\mathrm{Co}_0$. The first main result of our paper is that the interpretation of automorphism groups of cubic fourfolds as certain subgroups of autoequivalence groups of Kuznetsov components of cubic fourfolds (cf. Theorem \[Conway\]). Kuznetsov [@Kuz] studied semi-orthogonal decompositions of derived categories of cubic fourfolds. For a cubic fourfold $X$, there is a semi-orthogonal decomposition $$D^b(X)=\langle \D_X, \OO_X, \OO_X(1), \OO_X(2) \rangle.$$ The admissible subcategory $\D_X$ of the derived category $D^b(X)$ of $X$ is called the Kuznetsov component of $X$. The kuznetsov component $\D_X$ of $X$ is a two dimensional Calabi-Yau category [@Kuz]. By Orlov’s theorem [@Orl], the Kuznetsov component of a cubic fourfold $X$ is equivalent to the category of graded matrix factorizations of the defining polynomial of $X$. Kuznetsov components of cubic fourfolds share various properties with derived categories of K3 surfaces. Addington and Thomas [@AT] introduced weight two Hodge structures on Mukai lattices for Kuznetsov components of cubic fourfolds. Then we have the notion of symplectic autoequivalences of Kuznetsov components of cubic fourfolds. Bayer, Lahoz, Macrì and Stellari [@BLMS] constructed stability conditions on Kuznetsov components of cubic fourfolds. We give the statement of the second result in our paper. Let $X$ be a cubic fourfold. Denote the stability condition on $\D_X$ constructed in [@BLMS] by $\sigma$ (Theorem \[BLMS stability\]). As [@Huy], we consider the group ${\mathrm{Aut}}(\D_X,\sigma)$ (resp. ${\mathrm{Aut}}_{\mathrm{s}}(\D_X,\sigma)$) of autoequivalences (resp. symplectic autoequivalences) of Fourier-Mukai type on $\D_X$ (Definition \[FM-type\], Definition \[stabilizer cubic\]). The first result in our paper is as follow.
\[intromain1\] The group homomorphism $${\mathrm{Aut}}(X) \to {\mathrm{Aut}}(\D_X), f \mapsto f_*$$ induces the isomorphisms of groups $${\mathrm{Aut}}(X) {\xrightarrow{\sim}}{\mathrm{Aut}}(\D_X,\sigma),$$ $${\mathrm{Aut}}_{\mathrm{s}}(X) {\xrightarrow{\sim}}{\mathrm{Aut}}_{\mathrm{s}}(\D_X, \sigma),$$ where ${\mathrm{Aut}}_{\mathrm{s}}(X)$ is the symplectic automorphism group of $X$.
By Theorem \[intromain1\], we can regard the (symplectic) automorphism groups of cubic fourfolds as symmetries of two dimensional Calabi-Yau categories.
The second result of our paper is the comparison of automorphisms of cubic fourfolds with automorphisms of K3 surfaces. Relations between cubic fourfolds and K3 surfaces are studied via Hodge theory [@Has00], derived categories [@Kuz] and geometry of irreducible holomorphic symplectic manifolds [@BD], [@Add]. From point of view of Hodge theory, a labeled cubic fourfold is often related to a polarized K3 surface [@Has00]. For a positive integer $d$, a labeled cubic fourfold $(X,K)$ of discriminant $d$ is a pair of a cubic fourfold $X$ and a rank two primitive sublattice $K \subset H^{2,2}(X,\ZZ)$ such that the lattice $K$ contains the square $H^2$ of the hyperplane class $H$ and the discriminant of $K$ is equal to $d$. Hassett [@Has00] introduced the two arithmetic conditions on an integer $d$ as follow.
- $d>6$ and $d \equiv 0$ or $2$ (mod $6$)
- $d$ is not divisible by $4$, $9$, or any odd prime $p \equiv 2$ (mod $3$)
Hassett [@Has00] proved that for a labeled cubic fourfold $(X,K)$ of discriminant $d$, the integer $d$ satisfies $(*)$ and $(**)$ if and only if there is a polarized K3 surface $(S,h)$ of degree $d$ such that a Hodge isometry $K^\perp(-1) \simeq H^2_{\mathrm{prim}}(S,\ZZ)$ exists, where $K^\perp$ is the orthogonal complement of the sublattice $K$ in the middle cohomology group $H^4(X,\ZZ)$ of $X$ and the primitive cohomology $H^2_{\mathrm{prim}}(S,\ZZ)$ of $S$ is the orthogonal complement of the ample class $h$ in the second cohomology $H^2(S,\ZZ)$ of $S$. On the other hand, Kuznetsov [@Kuz] proposed the following conjecture for derived categories of cubic fourfolds and K3 surfaces.
\[rationality\] Let $X$ be a cubic fourfold. Then $X$ is rational if and only if there is a K3 surface $S$ such that $\D_X \simeq D^b(S)$.
The rationality problem of cubic fourfolds is one of long standing problems in algebraic geometry. Conjecturally, very general cubic fourfolds are irrational. However, there are no known irrational cubic fourfolds so far. Conjecture \[rationality\] implies that very general cubic fourfolds are irrational. Addington and Thomas [@AT] proved that Hodge theoretical relations [@Has00] and categorical relations [@Kuz] are equivalent in some sense (See also [@BLMNPS]). In the second result of our paper, we will treat Hodge theoretical relations [@Has00] and categorical relations [@Kuz] simultaneously. For a labeled cubic fourfold $(X,K)$ and an associated K3 surface $(S,h)$ in the sense of [@Has00], we will construct the isomorphism between the labeled automorphism group ${\mathrm{Aut}}(X,K)$ of the cubic fourfold $X$ and the polarized automorphism group ${\mathrm{Aut}}(S,h)$ of the K3 surface $S$ via derived categories of $X$ and $S$. The labeled automorphism group ${\mathrm{Aut}}(X,K)$ of a labeled cubic fourfold $(X,K)$ consists of automorphisms of the cubic fourfold $X$ acting on the sublattice $K$ identically. Similarly, the polarized automorphism group ${\mathrm{Aut}}(S,h)$ of a polarized K3 surface $(S,h)$ consists of automorphisms of the K3 surface $S$ fixing on the ample class $h$. We introduce certain lattices associated to labeled cubic fourfolds and polarized K3 surfaces following [@AT]. For a labeled cubic fourfold $(X,K)$, there is the rank three primitive sublattice $L_K$ of the Mukai lattice $H^*(\D_X,\ZZ)$ of the Kuznetsov component $\D_X$ such that the orthogonal complement $L^\perp_K$ in $H^*(\D_X,\ZZ)$ is Hodge isometric to $K^\perp(-1)$ (Remark \[label Mukai\]). For a polarized K3 surface $(S,h)$, there is the rank three primitive sublattice $L_h$ of the Mukai lattice $H^*(S,\ZZ)$ of $S$ such that the orthogonal complement $L^\perp_h$ in $H^*(S,\ZZ)$ is Hodge isometric to the primitive cohomology $H^2_{\mathrm{prim}}(S,\ZZ)$ of $S$ (Section 7). The following is the second result of our paper.
\[intromain2\] Let $d$ be an integer satisfying $(*)$ and $(**)$. For a labeled cubic fourfold $(X,K)$ of discriminant $d$, there is a polarized K3 surface $(S,h)$ of degree $d$ such that the following hold.
- There exists the object $\E \in D^b(S \times X)$ such that the Fourier-Mukai functor $\Phi_\E: D^b(S) \to \D_X$ associated to the Fourier-Mukai kernel $\E$ is an equivalence. The cohomological Fourier-Mukai transform $\Phi^H_\E: H^*(S,\ZZ) {\xrightarrow{\sim}}H^*(\D_X,\ZZ)$ induces the isometry $$\Phi^H_\E|_{L_h}: L_h {\xrightarrow{\sim}}L_K$$ and the Hodge isometry $$\Phi^H_\E|_{L^\perp_h}: L^\perp_h {\xrightarrow{\sim}}L^\perp_K.$$
- There is a stability condition $\sigma_X \in {\mathrm{Stab}}^*(S)$ such that the group homomorphism $$(-)_\E: {\mathrm{Aut}}(X) \to {\mathrm{Aut}}(D^b(S),\sigma_X), f \mapsto f_\E:=\Phi^{-1}_\E \circ f_* \circ \Phi_\E$$ is an isomorphism of groups. Moreover, the restriction of $(-)_\E$ induces the isomorphisms $$(-)_\E : {\mathrm{Aut}}(X,K) {\xrightarrow{\sim}}{\mathrm{Aut}}(S, h),$$ $$(-)_\E : {\mathrm{Aut}}_{\mathrm{s}}(X,K) {\xrightarrow{\sim}}{\mathrm{Aut}}_{\mathrm{s}}(S, h)$$ of groups. In particular, for any automorphism $f \in {\mathrm{Aut}}(X,K)$, there is an unique isomorphism $f_\E \in {\mathrm{Aut}}(S,h)$ such that the following diagram commutes. $$\xymatrix{D^b(S) \ar[d]_{f_\E} \ar[r]^{\Phi_\E} & \D_X \ar[d]^{f_*} \\
D^b(S) \ar[r]^{\Phi_\E} & \D_X }$$
The key of the proof of Theorem \[intromain2\] is the construction of a polarized K3 surface $(S,h)$ in terms of moduli spaces of stable objects in the Kuznetsov component $\D_X$ as [@BLMNPS]. The stability condition $\sigma_X$ in Theorem \[intromain2\] is induced by the stability condition on $\D_X$ constructed in [@BLMS]. The first group isomorphism in Theorem \[intromain2\] (2) is deduced from Theorem \[intromain1\]. Labeled automorphisms of a labeled cubic fourfold $(X,K)$ induce autoequivalences of the Kuznetsov component $\D_X$. Such autoequivalences induce automorphisms of the moduli space $S$ of stable objects in $\D_X$ and fix the polarization.
The third result in our paper is about the relation between the existence of non-trivial symplectic automorphisms of cubic fourfolds and the existence of associated K3 surfaces (cf. [@Laza], [@LZ]).
\[intromain3\] Let $X$ be a cubic fourfold. If the symplectic automorphism group ${\mathrm{Aut}}_{\mathrm{s}}(X)$ of $X$ is not isomorphic to the trivial group $1$ or the cyclic group $\ZZ/2\ZZ$ of order $2$, there is a K3 surface $S$ such that $\D_X \simeq D^b(S)$.
The proof relies on the classification of symplectic automorphism groups of cubic fourfolds in [@LZ] and the lattice theoretic technique in [@Nik] and [@Mor]. Using Theorem \[intromain2\], we can find examples of finite symplectic autoequivalences of K3 surfaces, which are not conjugate to symplectic automorphisms of K3 surfaces. For example, there is the K3 surface and the symplectic autoequivalence of order $9$ from the symplectic automorphism of order $9$ on the Fermat cubic fourfold (Example \[Fermat\]). From the Klein cubic fourfold, we can construct the K3 surface and the symplectic autoequivalence of order $11$ (Example \[Klein\]). Typical examples of autoequivalences of derived categories of K3 surfaces are shifts, tensoring line bundles, automorphisms and spherical twists. Since shifts, tensoring line bundles and spherical twists have infinite order, it is not easy to give non-trivial examples of finite symplectic autoequivalences in terms of K3 surfaces. We can construct other examples using Theorem 1.2 and Theorem 1.8 in [@LZ].
Notation
--------
We work over the complex number field $\mathbb{C}$. For a smooth projective variety $X$, we denote the bounded derived category of coherent sheaves on $X$ by $D^b(X)$. For a smooth projective variety $Z$ and an object $E \in D^b(Z)$, we define the Mukai vector $v_Z(E)$ of $E$ by $$v_Z(E):=\mathrm{ch}(E) \cdot \sqrt{\mathrm{td}(Z)}.$$ For objects $E, F \in D^b(X)$, we denote $E \otimes F:=E \otimes^\mathbf{L}F$ for simplicity. For smooth projective varieties $X$ and $Y$ and an object $\E \in D^b(X \times Y)$, the Fourier-Mukai functor $\Phi_\E: D^b(X) \to D^b(Y)$ is defined by $$\Phi_\E(E):=\mathbf{R}p_*(q^*E \otimes \E),$$ where $p:X \times Y \to Y$ and $q:X \times Y \to X$ are projections and $E \in D^b(X)$. Then the cohomological Fourier-Mukai transform $\Phi^H_\E: H^*(X,\QQ) \to H^*(Y,\QQ)$ is the linear map defined by $$\Phi^H_\E(\alpha):=p_*(q^*\alpha \cdot v_{X \times Y}(\E)).$$ If $X$ and $Y$ are K3 surfaces, the the cohomological Fourier-Mukai transform induces $\Phi^H_\E: H^*(X,\ZZ) {\xrightarrow{\sim}}H^*(Y,\ZZ)$. For a triangulated category $\D$ and an exceptional object $E \in \D$, the left mutation functor $\mathbf{L}_E: \D \to E^\perp$ and the $\mathbf{R}_E: \D \to {}^\perp E$ with respect to $E$ fit into the exact triangles $$\mathbf{R}\mathrm{Hom}(E, F) \otimes E \to F \to \mathbf{L}_E(F), \mathbf{R}_E(F) \to F \to \mathbf{R}\mathrm{Hom}(F,E)^*\otimes E$$ for $F \in \D$. We assume that cubic fourfolds are smooth. We assume that K3 surfaces are projective. For a cubic fourfold $X$, denote the hyperplane class of $X$ by $H$. The middle primitive cohomology $H^4_{\mathrm{prim}}(X,\ZZ)$ of $X$ is the orthogonal complement $\langle H^2 \rangle^\perp$ of $H^2$ in $H^4(X,\ZZ)$. The hyperbolic plane lattice $U$ is a lattice determined by the Gramian matrix $$\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix}.$$ The root lattice $E_8$ is positive definite.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author would like to thanks Lie Fu, Wahei Hara, Seung-Jo Jung and Naoki Koseki for valuable conversation. The part of this work was done while the author was staying at Max Planck Institute for Mathematics in Bonn. The author is grateful to Max Planck Institute for Mathematics for its hospitality and financial support. This work is also supported by Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS) in RIKEN and JSPS KAKENHI Grant number 19K14520.
Cubic fourfolds and K3 surfaces
===============================
In this section, we review relations between cubic fourfolds and K3 surfaces via derived categories and Hodge theory.
Mukai lattice of K3 surfaces
----------------------------
In this subsection, we recall Mukai lattices of K3 surfaces. Mukai lattices are important in the study of derived categories of K3 surfaces.
Let $S$ be a K3 surface. The cohomology group $H^*(S,\ZZ)$ of $S$ has a structure of a lattice given by $$((r_1, c_1, m_1), (r_2, c_2, m_2)):=c_1c_2-r_1m_2-r_2m_1.$$ Here, note that $H^*(S,\ZZ)=H^0(S,\ZZ) \oplus H^2(S,\ZZ) \oplus H^4(S,\ZZ)$. The lattice $(H^*(S,\ZZ),(-,-))$ is called the Mukai lattice of $S$ and $(-,-)$ is called the Mukai pairing on $S$. The Mukai lattice $(H^*(S,\ZZ),(-,-))$ is isometric to the even unimodular lattice $U^{\oplus4}\oplus E_8(-1)^{\oplus2}$ of signature $(4,20)$. Moreover, the Mukai lattice $H^*(S,\ZZ)$ of $S$ has a weight two Hodge structure $\widetilde{H}(S)$ given by
$$\widetilde{H}^{2,0}(S):=H^{2,0}(S),$$ $$\widetilde{H}^{1,1}(S):=\bigoplus_{p=0}^{2}H^{p,p}(S),$$ $$\widetilde{H}^{0,2}(S):=H^{0,2}(S).$$ The integral part $\widetilde{H}^{1,1}(S,\ZZ):=\widetilde{H}^{1,1}(S) \cap H^*(S,\ZZ)$ is called the algebraic Mukai lattice of $S$. Note that $\widetilde{H}^{1,1}(S,\ZZ)=H^0(S,\ZZ) \oplus \mathrm{NS}(S) \oplus H^4(S,\ZZ)$. So we have $\mathrm{rk}\widetilde{H}^{1,1}(S,\ZZ)=\rho(S)+2$ and $T_S=\widetilde{H}^{1,1}(S,\ZZ)^\perp$, where $T_S$ is the transcendental lattice of $S$. For an object $E \in D^b(S)$, the Mukai vector $v(E)$ of $E$ is defined by $v(E):=\mathrm{ch}(E)\sqrt{\mathrm{td}(S)} \in \widetilde{H}^{1,1}(S,\ZZ)$. Then the homomorphism $v:K_0(S) \to \widetilde{H}^{1,1}(S,\ZZ)$ is surjective.
Mukai lattices for cubic fourfolds
----------------------------------
In this subsection, we introduce Mukai lattices for cubic fourfolds following Addington and Thomas [@AT].
Let $X$ be a cubic fourfold. The derived category $D^b(X)$ of $X$ has a semi-orthogonal decomposition $$D^b(X)=\langle \D_X, \OO_X, \OO_X(1), \OO_X(2) \rangle.$$ The admissible subcategory $\D_X$ is called the Kuznetsov component of $X$. Kuznetsov [@Kuz] proved that $\D_X$ is a $2$-dimensional Calabi-Yau category, that is the Serre functor of $\D_X$ is isomorphic to the double shift functor $[2]$. We denote the topological K-group of $X$ by $K_{\mathrm{top}}(X)$. For an element $\alpha \in K_{\mathrm{top}}(X)$, we define the Mukai vector $v_X(\alpha)$ of $\alpha$ as $v_X(\alpha):=\mathrm{ch}(\alpha)\sqrt{\mathrm{td}(X)} \in H^*(X,\QQ)$. For elements $\alpha, \beta \in K_{\mathrm{top}}(X)$, we have the topological Euler characteristic $\chi_{\mathrm{top}}(\alpha, \beta) \in \ZZ$. Addington and Thomas introduced the Mukai lattice of $\D_X$.
\[Mukai lattice\] We define the cohomology group of $\D_X$ as $$H^*(\D_X, \ZZ):=\{\alpha \in K_{\mathrm{top}}(X) \mid \chi_{\mathrm{top}}([\OO_X(k)], \alpha)=0, \text{for} \ k=0,1, 2\}.$$ Let $(-,-)$ be the restriction of $-\chi_{\mathrm{top}}(-,-)$ to $H^*(\D_X,\ZZ)$. The pair $(H^*(\D_X,\ZZ), (-,-))$ is a lattice isometric to the even unimodular lattice $U^{\oplus4}\oplus E_8(-1)^{\oplus2}$ of signature $(4,20)$. The lattice $(H^*(\D_X,\ZZ), (-,-))$ is called the Mukai lattice of $\D_X$. Moreover, $H^*(\D_X,\ZZ)$ has a weight two Hodge structure $\widetilde{H}(\D_X)$ given by $$\widetilde{H}^{2,0}(\D_X):=v^{-1}_X\bigl( H^{3,1}(X)\bigr),$$ $$\widetilde{H}^{1,1}(\D_X):=v^{-1}_X\Bigl(\bigoplus_{p=0}^{4}H^{p,p}(X)\Bigr),$$ $$\widetilde{H}^{0,2}(\D_X):=v^{-1}_X\bigl( H^{1,3}(X)\bigr).$$ The integral part $\widetilde{H}^{1,1}(\D_X,\ZZ):=\widetilde{H}^{1,1}(\D_X) \cap H^*(\D_X,\ZZ)$ is called the algebraic Mukai lattice of $\D_X$. The transcendental lattice $T_{\D_X}$ of $\D_X$ is defined by the orthogonal complement $T_{\D_X}:=\widetilde{H}^{1,1}(\D_X,\ZZ)^\perp$ in $H^*(\D_X,\ZZ)$.
We consider Mukai vectors for objects in the Kuznetsov component.
\[v\]
We denote the natural map $K_0(\D_X) \to H^*(\D_X,\ZZ)$ by $v$. By Proposition 2.4 in [@AT], the image of $v$ is equal to the algebraic Mukai lattice $\widetilde{H}^{1,1}(\D_X,\ZZ)$. For an object $E \in \D_X$, we define the Mukai vector $v(E)$ of $E$ as $v(E):=v([E])$. The map $v: K_0(\D_X) \to \widetilde{H}^{1,1}(\D_X,\ZZ)$ is different from the map $v_X: K_{\mathrm{top}}(X) \to H^*(X,\QQ)$.
The algebraic Mukai lattice $\widetilde{H}^{1,1}(\D_X,\ZZ)$ of $\D_X$ always contains the certain primitive sublattice of rank two. We see the definition of this sublattice. Let $i: \D_X \to D^b(X)$ be the inclusion functor and $i^*:D^b(X) \to \D_X$ the left adjoint functor of $i$. For an integer $k \in \ZZ$, we define the element by $$\lambda_k:=[i^*\OO_{\mathrm{line}}(k)] \in \widetilde{H}^{1,1}(\D_X,\ZZ).$$ The Gramian matrix of the sublattice $\langle\lambda_1, \lambda_2 \rangle$ of $\widetilde{H}^{1,1}(\D_X,\ZZ)$ is $$\left( \begin{array}{cc}
2 & -1 \\
-1 & 2
\end{array} \right),$$ Denote the sublattice $\langle\lambda_1, \lambda_2 \rangle$ of $\widetilde{H}^{1,1}(\D_X,\ZZ)$ by $A_2$. Since the algebraic Mukai lattice $\widetilde{H}^{1,1}(\D_X,\ZZ)$ of $\D_X$ contains $A_2$, we have $\mathrm{rk}\widetilde{H}^{1,1}(\D_X,\ZZ)\geq 2$. The sublattice $A_2$ is related to the primitive cohomology lattice $H^4_{\mathrm{prim}}(X,\ZZ)$ of $X$.
\[prim\] Let $A_2^{\perp}$ be the orthogonal complement of $A_2$ in $H^*(\D_X,\ZZ)$. Then we have the Hodge isometry $v_X: A_2^{\perp} {\xrightarrow{\sim}}H^4_{\mathrm{prim}}(X,\ZZ)(-1)$.
Picard numbers of Kuznetsov components and K3 surfaces
------------------------------------------------------
In this subsection, we introduce Picard numbers of Kuznetsov components.
Let $X$ be a cubic fourfold. The following is the definition of the Picard number of the Kuznetsov component $\D_X$.
The Picard number $\rho(\D_X)$ of the Kuznetsov component $\D_X$ of $X$ is $$\rho(\D_X):=\mathrm{rk}\widetilde{H}^{1,1}(\D_X,\ZZ)-2.$$
Since $\mathrm{rk}\widetilde{H}^{1,1}(\D_X,\ZZ)\geq 2$, we have $\rho(\D_X)\geq 0$. We can describe the Picard number of $\D_X$ in terms of Hodge structure on $H^4(X,\ZZ)$.
\[2,2\] The equality $\rho(\D_X)=\mathrm{rk}H^{2,2}(X,\ZZ)-1$ holds.
It is deduced from Definition \[Mukai lattice\] and Proposition \[prim\].
Picard numbers of Kuznetsov components are compatible with Picard numbers of K3 surfaces.
\[Picard number\] If there is a K3 surface $S$ such that $\D_X \simeq D^b(S)$, we have a Hodge isometry ${H}^*(\D_X,\ZZ) \simeq {H}^*(S,\ZZ)$. So we obtain $\rho(\D_X)=\rho(S)\geq 1$ and $T_{\D_X} \simeq T_S$.
For a cubic fourfold $X$, there is a necessary and sufficient condition of the existence of a K3 surface $S$ such that $\D_X \simeq D^b(S)$ in terms of lattice theory.
\[derivedTorelli\] There is an embedding of the hyperbolic lattice $U$ into the algebraic Mukai lattice $\widetilde{H}^{1,1}(\D_X,\ZZ)$ of $\D_X$ if and only if there is a K3 surface such that $\D_X \simeq D^b(S)$.
In Theorem \[derivedTorelli\], the hyperbolic lattice $U$ is corresponding to the sublattice $H^0(S,\ZZ)\oplus H^4(S,\ZZ)$ of the algebraic Mukai lattice $\widetilde{H}^{1,1}(S,\ZZ)$ of a K3 surface $S$.
Special cubic fourfolds
-----------------------
In this subsection, we recall the notion of special cubic fourfolds following Hassett [@Has00].
Let $X$ be a cubic fourfold. If there is a K3 surface $S$ such that $\D_X \simeq D^b(S)$, then we have $\rho(\D_X)\geq1$ by Remark \[Picard number\]. A cubic fourfold $X$ is special if we have $\rho(\D_X)\geq1$, equivalently $\mathrm{rk}H^{2,2}(X,\ZZ)\geq2$ by Remark \[2,2\]. The following is the analogue of the notion of polarized K3 surfaces.
\[labeled cubic fourfold\] For a positive integer $d$, a labeled cubic fourfold $(X,K)$ of discriminant $d$ is a pair of a special cubic fourfold $X$ and a rank two primitive sublattice $K \subset H^{2,2}(X,\ZZ)$ such that $K$ contains $H^2$ and $\mathrm{disc}(K)=d$. A cubic fourfold $X$ is a special cubic fourfold of discriminant $d$ if there is a rank two primitive sublattice $K \subset H^{2,2}(X,\ZZ)$ such that $(X,K)$ is a labeled cubic fourfold of discriminant $d$.
We interpret labeled cubic fourfolds of discriminant $d$ via Mukai lattices.
\[label Mukai\] Let $(X,K)$ be a labeled cubic fourfold of discriminant $d$. We define the rank three primitive sublattice $L_K$ of the algebraic Mukai lattice $\widetilde{H}^{1,1}(\D_X,\ZZ)$ such that $\mathrm{disc}L_K=d$ as follow. There is a class $T \in H^{2,2}_{\mathrm{prim}}(X,\ZZ)$ such that $K \cap H^{2,2}_{\mathrm{prim}}(X,\ZZ)=\ZZ \cdot T$. Using Proposition \[prim\], we define the class $\kappa_T \in A^\perp_2$ by $\kappa_T:=v^{-1}_X(T)$. Let $L_K \subset \widetilde{H}^{1,1}(\D_X,\ZZ)$ be the saturation of the sublattice generated by $A_2$ and $\kappa_T$. By Proposition \[prim\] and the definition of $L_K$, we have the Hodge isometry $v_X: L_K^\perp {\xrightarrow{\sim}}K^\perp(-1)$, where we take orthogonal complements in $H^*(\D_X,\ZZ)$ and $H^4(X,\ZZ)(-1)$ respectively. Since $K$ and $L_K$ are primitive sublattices of unimodular lattices $H^4(X,\ZZ)$ and $H^*(\D_X,\ZZ)$ respectively, we have $$d=\mathrm{disc}(K)=\mathrm{disc}\bigr(K(-1)\bigl)=\mathrm{disc}\bigl(K^\perp(-1)\bigr)=\mathrm{disc}(L_K).$$
The moduli space $\C$ of cubic fourfolds is a twenty dimensional quasi-projective variety. For a positive integer $d$, denote the subset of special cubic fourfolds of discriminant $d$ by $\C_d$. Hassett [@Has00] introduced the two arithmetic conditions on an integer $d$ as follow.
- $d>6$ and $d \equiv 0$ or $2$ (mod $6$)
- $d$ is not divisible by $4$, $9$, or any odd prime $p \equiv 2$ (mod $3$)
Hassett [@Has00] proved that the condition ($*$) is equivalent to the non-emptyness of $\C_d$.
If an integer $d$ satisfies $(*)$, then the subset $\C_d$ is a subvariety of codimension one in $\C$.
The condition $(**)$ is related to K3 surfaces. For labeled cubic fourfolds and polarized K3 surfaces, Hassett [@Has00] proved the following theorem.
\[Hassett K3\] Assume that an integer satisfies $(*)$. Let $(X,K)$ be a labeled cubic fourfold of discriminant $d$. The integer $d$ satisfies $(**)$ if and only if there is a polarized K3 surface $(S,h)$ of degree $d$ such that we have a Hodge isometry $K^\perp(-1) \simeq h^\perp$. Here, we take orthogonal complements of $K$ and $h$ in $H^4(X,\ZZ)$ and $H^2(S,\ZZ)$ respectively.
In the context of derived categories, the following is known.
Let $X$ be a cubic fourfold. There is a K3 surface $S$ such that $\D_X \simeq D^b(S)$ if and only if there is an integer $d$ satisfying $(*)$ and $(**)$ such that $X \in \C_d$.
Stability conditions
====================
In this section, we review facts about stability conditions on Kuznetsov components of cubic fourfolds and derived categories of K3 surfaces.
Weak stability conditions
-------------------------
In this subsection, we introduce the notion of (weak) stability conditions following [@Bri07] and [@BLMS].
Let $\D$ be a triangulated category over $\CC$. The definition of weak stability conditions and stability conditions on $\D$ is the following.
Fix a finitely generated free abelian group $\Lambda$ and a surjective group homomorphism $\mathrm{cl}:K_0(\D) \to \Lambda$. A weak stability condition on $\D$ [(]{}*with respect to $\Lambda$[)]{} is a pair $\sigma=(Z,\A)$ of a group homomorphism $Z: \Lambda \to \CC$ [(]{}*called a central charge[)]{} *and the heart of a bounded t-structure $\A$ in $\D$ such that the following three properties hold.***
- For any object $E \in \A$, we have $\mathrm{Im}Z(\mathrm{cl}(E))\geq 0$ and if $\mathrm{Im}Z(\mathrm{cl}(E))=0$, then $Z(\mathrm{cl}(E)) \in \RR_{\leq0}$ holds. For simplicity, we will denote $Z(\mathrm{cl}(E))$ by $Z(E)$ for an object $E \in \D$.
We prepare terminologies to state (ii) *and (iii)*. For $E \in \A$ with $\mathrm{Im}Z(E)>0$, we define the slope $\mu_\sigma(E)$ of $E$ with respect to $\sigma$ as $$\mu_\sigma(E):=-\frac{\mathrm{Re}Z(E)}{\mathrm{Im}Z(E)}.$$ For $E \in \A$ with $\mathrm{Im}Z(E)=0$, we put $\mu_\sigma(E):=\infty$. For a nonzero object $E \in \A$, $E$ is $\sigma$-semistable if for any subobject $F$ of $E$ in $\A$, we have $\mu_\sigma(F)\leq \mu_\sigma(E)$.**
- For any $E \in \A$, there exists a filtration $$0=E_0 \subset E_1 \subset \cdot \cdot \cdot \subset E_n=E$$ in $\A$ such that for any $1 \leq k \leq n$, the quotient $F_k:=E_k/E_{k-1}$ is $\sigma$-semistable with $$\mu_\sigma(F_1)>\mu_\sigma(F_2)>\cdot \cdot \cdot >\mu_\sigma(F_n) .$$ This filtration is called a Harder-Narasimhan filtration of $E$ with respect to $\sigma$.
- There exists a quadratic form $Q$ on $\Lambda \otimes \RR$ such that $Q|_{\mathrm{Ker}Z}$ is negative definite and $Q(\mathrm{cl}(E))\geq0$ for any $\sigma$-semistable object $E \in \A$. This property is called the support property.
A weak stability condition $\sigma=(Z,\A)$ on $\D$ is a stability condition on $\D$ if $Z(E)\neq 0$ holds for any nonzero object $E \in \A$.
Let $\sigma=(Z,\A)$ be a weak stability condition on $\D$ with respect to $\Lambda$. If $\mathrm{rk}\Lambda=2$ and $Z:\Lambda \to \CC$ is injective, any positive semi-definite quadratic form $Q$ on $\Lambda \otimes \RR$ satisfies the support property.
Bridgeland [@Bri07] proved that the set ${\mathrm{Stab}}(\D)$ of stability conditions on $\D$ with respect to $\Lambda$ has a structure of a complex manifold.
Let $\sigma=(Z,\A)$ be a stability condition on $\D$ with respect to $\Lambda$. For an object $E \in \A \setminus\{0\}$, we define the phase $\phi_\sigma(E)$ of $E$ with respect to $\sigma$ by $$\phi_{\sigma}(E):=\frac{1}{\pi}\mathrm{arg}Z(E) \in (0, 1].$$ For a real number $\phi \in (0,1]$, we define the full subcategory $\P_\sigma(\phi)$ in $\A$ by $$\P_\sigma(\phi):=\{E \in \A \mid \text{$E$ is $\sigma$-semistable with $\phi_\sigma(E)=\phi$}\} \cup \{0\}.$$ For any real number $\phi \in (0,1]$, the full subcategory $\P_\sigma(\phi)$ is an abelian subcategory of $\A$.
We use weak stability conditions to construct new hearts of bounded t-structures. By the existence of Harder-Narasimhan filtrations, we have the following torsion pairs and they produce new hearts of bounded t-structures.
\[tilt\] Let $\sigma=(Z,\A)$ be a weak stability condition on $\D$. For $\mu \in \RR$, we define a torsion pair $(\T^\mu_\sigma, \F^\mu_\sigma)$ on $\A$ as $$\T^\mu_\sigma:=\langle E \in \A \mid \text{$E$ is $\sigma$-semistable with $\mu_\sigma(E)>\mu \rangle$} ,$$ $$\F^\mu_\sigma:=\langle E \in \A \mid \text{$E$ is $\sigma$-semistable with $\mu_\sigma(E) \leq \mu$} \rangle,$$ where $\langle - \rangle$ is the extension closure. We define the heart $\A^{\mu}_\sigma$ of a bounded t-structure as $$\A^\mu_\sigma:=\langle \F^\mu_\sigma[1], \T^\mu_\sigma \rangle.$$ We say that $\A^\mu_\sigma$ is obtained by the tilting of $\A$ with respect to the torsion pair $(\T^\mu_\sigma, \F^\mu_\sigma)$.
Stability conditions on K3 surfaces
-----------------------------------
In this subsection, we recall examples of stability conditions on derived categories of K3 surfaces.
Let $S$ be a K3 surface. Using the group homomorphism $v: K_0(S) \to \widetilde{H}^{1,1}(S,\ZZ)$, we consider only stability conditions on $D^b(S)$ with respect to $\widetilde{H}^{1,1}(S,\ZZ)$. Take $\RR$-divisors $\beta, \omega \in \mathrm{NS}(S)_{\RR}$ such that $\omega$ is an ample class. The weak stability condition $\sigma_\omega=(Z_\omega, \mathrm{Coh}(S))$ is given by $$Z_\omega(E):=i \mathrm{ch}_0(E)-\mathrm{ch}_1(E)\cdot \omega$$ for an object $E \in D^b(S)$. It is nothing but the slope stability on $\mathrm{Coh}(S)$. Using Definition \[tilt\], we define the heart $\A_{\beta,\omega}$ of a bounded t-structure on $D^b(S)$ by $$\A_{\beta,\omega}:=\mathrm{Coh}(S)^{\beta \cdot \omega}_{\sigma_{\omega}}.$$ Let $Z_{\beta,\omega}: \widetilde{H}^{1,1}(S,\ZZ) \to \CC$ be the group homomorphism defined by $$Z_{\beta,\omega}(w):=(e^{\beta+i\omega},w),$$ where $w \in \widetilde{H}^{1,1}(S,\ZZ)$. Bridgeland [@Bri08] proved the following theorem.
\[stability K3\] If $Z_{\beta, \omega}(E) \notin \RR_{<0}$ holds for any spherical sheaf $E$ on $S$, then the pair $\sigma_{\beta,\omega}:=(Z_{\beta,\omega}, \A_{\beta,\omega})$ is a stability condition on $D^b(S)$. If $\omega^2>2$, we have $Z_{\beta, \omega}(E) \notin \RR_{<0}$ for any spherical sheaf $E$ on $S$.
Spaces of stability conditions on K3 surfaces
---------------------------------------------
In this subsection, we recall structures of spaces of stability conditions on K3 surfaces.
Let $S$ be a K3 surface. Let ${\mathrm{Stab}}(S)$ be the space of stability conditions on $D^b(S)$ with respect to $\widetilde{H}^{1,1}(S, \ZZ)$. We define the subset $\P(S)$ of $\widetilde{H}^{1,1}(S, \ZZ)\otimes \CC$ as $$\P(S):=\{\Omega \in \widetilde{H}^{1,1}(S, \ZZ)\otimes \CC \mid \text{$\langle \mathrm{Re}(\Omega), \mathrm{Im}(\Omega) \rangle_{\mathbb{R}}$ is a positive definite plane} \}.$$ Let $\P^+(S)$ be the connected component of $\P(S)$ containing $e^{i\omega}$, where $\omega$ is an ample divisor on $S$. Let $\Delta_S$ be the set of $(-2)$-classes in $\widetilde{H}^{1,1}(S, \ZZ)$. We define the subset $\P^+_0(S)$ of $\P^+(S)$ as $$\P^+_0(S):=\P^+(S)\setminus \bigcup_{\delta \in \Delta_S} \delta^\perp.$$ We consider the action of the autoequivalence group ${\mathrm{Aut}}(D^b(S))$ on ${\mathrm{Stab}}(S)$.
For an autoequivalence $\Phi \in {\mathrm{Aut}}(D^b(S))$ and a stability condition\
$\sigma=(Z,\A) \in {\mathrm{Stab}}(S)$, we define the stability condition $\Phi\sigma$ by $$\Phi\sigma:=(Z \circ (\Phi^H)^{-1}, \Phi(\A)) \in {\mathrm{Stab}}(S).$$
Since the Mukai pairing $(-,-)$ on $\widetilde{H}^{1,1}(S, \ZZ)$ is non-degenerate, for a stability condition $\sigma=(Z,\A) \in \mathrm{Stab}(S)$, there is the unique element $\Omega_Z \in \widetilde{H}^{1,1}(S, \ZZ)\otimes \CC$ such that $Z(-)=(\Omega_Z,-)$. Let ${\mathrm{Stab}}^*(S) \subset {\mathrm{Stab}}(S)$ be the connected component containing stability conditions in Theorem \[stability K3\]. Bridgeland [@Bri08] proved the following theorem.
\[BriK3\] For a stability condition $\sigma=(Z,\A) \in {\mathrm{Stab}}^*(S)$, put $\pi_S(\sigma):=\Omega_Z$. Then $\pi_S$ induces the covering map $\pi_S: {\mathrm{Stab}}^*(S) \to \P^+_0(S)$. We define $${\mathrm{Aut}}^0(D^b(S)):=\{\Phi \in {\mathrm{Aut}}(D^b(S)) \mid \Phi^H=\mathrm{id}, \Phi({\mathrm{Stab}}^*(S)) \subset {\mathrm{Stab}}^*(S)\}.$$ Then the natural homomorphism ${\mathrm{Aut}}^0(D^b(S)) \to \mathrm{Deck}(\pi_S)$ is an isomorphism, where $\mathrm{Deck}(\pi_S)$ is the group of deck transformations of the covering map $\pi_S$.
Clifford algebra associated to a line on a cubic fourfold
---------------------------------------------------------
In this subsection, we recall the Clifford algebra on the projective space $\mathbb{P}^3$ associated to a line on a cubic fourfold in [@Kuz08] and Section 7 in [@BLMS].
Let $X$ be a cubic fourfold. Take a line $l \subset X$. Consider the blowing up $p_l: \mathrm{Bl}_lX \to X$ of $X$ along the line $l$. We have the embedding $j_l: \mathrm{Bl}_lX \to \mathrm{Bl}_l\mathbb{P}^5$, where $\mathrm{Bl}_l\mathbb{P}^5 \to \mathbb{P}^5$ is the blowing up of $\mathbb{P}^5$ along the line $l$. The linear projection from $l$ induces the following commutative diagram. $$\xymatrix{ &\mathrm{Bl}_lX \ar@{^{(}-{>}} [r]^{j_l} \ar[d]_{p_l}& \mathrm{Bl}_l{\mathbb{P}^5} \ar[d] \ar[rd]^{q_l} && \\ l \ar@{^{(}-{>}} [r] & X \ar@{^{(}-{>}} [r] & \mathbb{P}^5 \ar@{.{>}} [r]& \mathbb{P}^3 & }$$ Denote $h:=\mathrm{c}_1(\OO_{\mathbb{P}^3}(1))$. Note that $q_l:\mathrm{Bl}_l\mathbb{P}^5 \to \mathbb{P}^3$ is a $\mathbb{P}^2$-bundle via the isomorphism $\mathrm{Bl}_l\mathbb{P}^5 \simeq \mathbb{P}(\OO_{\mathbb{P}^3}^{\oplus2} \oplus \OO_{\mathbb{P}^3}(-h))$. The composition $\pi_l:=q_l \circ j_l: \mathrm{Bl}_lX \to \mathbb{P}^3$ is a conic fibration. Let $\mathcal{B}^l_0$ (resp. $\mathcal{B}^l_1$) be the even part (resp. the odd part) of the sheaf of Clifford algebras on $\mathbb{P}^3$ associated to $\pi_L$. For $k \in \ZZ$, we define the $\mathcal{B}^l_0$-bimodule $\B^l_k$ by $\B^l_k=\mathcal{B}^l_{k-2} \otimes \OO_{\mathbb{P}^3}(h)$. Let $\mathrm{Coh}(\mathbb{P}^3, \mathcal{B}^l_0)$ be the category of coherent right $\mathcal{B}^l_0$-modules and define $D^b(\mathbb{P}^3, \mathcal{B}^l_0):=D^b(\mathrm{Coh}(\mathbb{P}^3, \mathcal{B}^l_0))$. Note that $\mathbf{L}p_l^*: D^b(X) \to D^b(\mathrm{Bl}_lX)$ is a fully faithful functor. There exists a coherent sheaf $\mathcal{E}_l$ of right $\pi^*_l\mathcal{B}^l_0$-modules on $\mathrm{Bl}_lX$ such that $\Psi_l(-):=\mathbf{R}{\pi_l}_*(-\otimes \OO_{\mathrm{Bl}_lX}(h) \otimes \mathcal{E}_l)[1]$ is a fully faithful functor from $\mathbf{L}p_l^*\D_X$ to $D^b(\PP^3,\mathcal{B}^l_0)$ and there is a semi-orthogonal decomposition $D^b(\PP^3,\mathcal{B}^l_0)=\langle \Psi_l(\mathbf{L}p_l^*\D_X), \B^l_1, \B^l_2, \B^l_3 \rangle$. (See [@BLMS], Proposition 7.7.)
\[exact\] By Section 4 in [@Kuz08], the coherent right $\B^L_0$-module $\E_L$ fits into the exact sequence $$0 \to q^*_l\B^l_{-1}(-2H) \to q^*_l\B^l_0(-H) \to j_{l*}\E_l \to 0.$$
Stability conditions on Kuznetsov components
--------------------------------------------
In this subsection, we recall the examples of stability conditions on Kuznetsov components of cubic fourfolds in [@BLMS].
Let $X$ be a cubic fourfold. Using the group homomorphism $v: K_0(\D_X) \to \widetilde{H}^{1,1}(\D_X, \ZZ)$ in Remark \[v\], we consider only stability conditions on $\D_X$ with respect to $\widetilde{H}^{1,1}(\D_X, \ZZ)$. Fix a line $l$ on $X$. We define the Chern character map ${\mathrm{ch}}_{\mathcal{B}^l_0} : K_0(D^b(\mathbb{P}^3, \mathcal{B}^l_0)) \to H^*(\mathbb{P}^3, \QQ)$ as $${\mathrm{ch}}_{\mathcal{B}^l_0}(E):={\mathrm{ch}}(\mathrm{forg}(E))\Bigl(1-\frac{11}{32}h^2\Bigr),$$ where $E\in D^b(\mathbb{P}^3,\mathcal{B}^l_0)$ and $\mathrm{forg}:D^b(\mathbb{P}^3, \mathcal{B}^l_0) \to D^b(\mathbb{P}^3)$ is the forgetful functor. For $\beta \in \RR$, we define the twisted Chern character map ${\mathrm{ch}}^{\beta}_{\mathcal{B}^l_0}: K_0(D^b(\mathbb{P}^3, \mathcal{B}^l_0)) \to H^*(\mathbb{P}^3, \RR)$ as ${\mathrm{ch}^{\beta}_{\mathcal{B}^l_0}}:=e^{-\beta h}{\mathrm{ch}}_{\mathcal{B}^l_0}$. The chern character map ${\mathrm{ch}}_{\mathcal{B}^l_0}$ is same as ${\mathrm{ch}}^0_{\B^l_0}$. Using the isomorphism $$\RR^4 {\xrightarrow{\sim}}H^*(\PP^3,\RR), (x_1,x_2,x_3,x_4) \mapsto (x_1, x_2h, x_3h^2, x_4h^3),$$ we regard ${\mathrm{ch}^{\beta}_{\mathcal{B}^l_0}}(E)=({\mathrm{ch}}^{\beta}_{\mathcal{B}^l_0,0}(E),{\mathrm{ch}^{\beta}_{{\mathcal{B}^l_0},1}}(E),{\mathrm{ch}^{\beta}_{{\mathcal{B}^l_0},2}}(E),{\mathrm{ch}^{\beta}_{{\mathcal{B}^l_0},3}}(E)) \in \RR^4$ for $E\in D^b(\PP^3,\mathcal{B}^l_0)$. For $j=1,2$, we define the finitely generated free abelian group $\Lambda^j_{\mathcal{B}^l_0}$ as $$\Lambda^j_{\mathcal{B}^l_0}:=\langle {\mathrm{ch}}_{\mathcal{B}^l_0,k}(E)\mid E \in D^b(\PP^3,\mathcal{B}^l_0),0 \leq k \leq j \rangle_\ZZ \subset \RR^4.$$ Note that $\mathrm{rk}\Lambda^j_{\mathcal{B}^l_0}=1+j$. We define a weak stability condition $\sigma_{\mathrm{slope}}=(Z_{\mathrm{slope}},\mathrm{Coh}(\PP^3,\mathcal{B}^l_0))$ on $D^b(\PP^3,\mathcal{B}^l_0)$ with respect to $\Lambda^j_{\mathcal{B}^l_0}$ as $$Z_{\mathrm{slope}}(E):=i {\mathrm{ch}}_{\mathcal{B}^l_0,0}(E)-{\mathrm{ch}}_{\mathcal{B}^l_0,1}(E)$$ for $E \in D^b(\PP^3,\B^l_0)$. It is nothing but the slope stability. As Definition \[tilt\], for $\beta \in \RR$, consider the heart $\mathrm{Coh}^\beta(\PP^3,\B^l_0)$ of a bounded t-structure in $D^b(\PP^3,\B^l_0)$ defined by $$\mathrm{Coh}^\beta(\PP^3,\B^l_0) :=\mathrm{Coh}(\PP^3,\B^l_0)^{\beta}_{\sigma_{\mathrm{slope}}}.$$ For $\alpha>0$ and $\beta \in \RR$, we can define a weak stability condition $\sigma_{\alpha, \beta}=(Z_{\alpha,\beta}, \mathrm{Coh}^\beta(\PP^3,\B^l_0))$ with respect to $\Lambda^2_{\mathcal{B}^l_0}$ by $$Z_{\alpha, \beta}(E):=i{\mathrm{ch}^{\beta}_{{\mathcal{B}^l_0},1}}(E)+\frac{1}{2}\alpha^2{\mathrm{ch}^{\beta}_{\mathcal{B}^l_0,0}}(E)-{\mathrm{ch}^{\beta}_{{\mathcal{B}^l_0},2}}(E)$$ for $E \in D^b(\PP^3,\B^l_0)$. (See [@BLMS], Proposition 9.3.) Consider the heart $\mathrm{Coh}^0_{\alpha,\beta}(\PP^3,\B^l_0)$ of a bounded t-structure on $D^b(\PP^3,\B^l_0)$, that is defined by $$\mathrm{Coh}^0_{\alpha,\beta}(\PP^3,\B^l_0):=\mathrm{Coh}^\beta(\PP^3,\B^l_0))^0_{\sigma_{\alpha,\beta}}.$$ Let $\widetilde{\Psi_l}: \D_X {\xrightarrow{\sim}}\Psi(\mathbf{L}p_l^*\D_X)$ be the equivalence induced by the fully faithful functor $\Psi_l \circ \mathbf{L}p_l^* :\D_X \to D^b(\PP^3, \B^l_0)$.
\[BLMS stability\] For $0<\alpha<1/4$, we define $$\A^l_\alpha:=\widetilde{\Psi_{l}}^{-1}(\mathrm{Coh}^0_{\alpha,-1}(\PP^3,\B^l_0) \cap \widetilde{\Psi_l}(\D_X)),$$ $$Z^l_\alpha(E):=-iZ_{\alpha,-1}(\widetilde{\Psi_l}(E)), E \in \D_X.$$ Then $\sigma^l_\alpha=(Z^l_\alpha, \A^l_\alpha)$ is a stability condition on $\D_X$ with respect to $\widetilde{H}^{1,1}(\D_X, \ZZ)$. For any lines $l$ and $l'$ on $X$, we have $\sigma^l_\alpha=\sigma^{l'}_\alpha$. Denote the stability condition $\sigma^l_\alpha$ on $\D_X$ by $\sigma_\alpha=(Z_\alpha, \A_\alpha)$.
Spaces of stability conditions on Kuznetsov components
------------------------------------------------------
In this subsection, we recall structures of spaces of stability conditions on Kuznetsov components.
Let $X$ be a cubic fourfold. Let ${\mathrm{Stab}}(\D_X)$ be the space of stability conditions on $\D_X$ with respect to $\widetilde{H}^{1,1}(\D_X, \ZZ)$. We define the subset $\P(\D_X)$ of $\widetilde{H}^{1,1}(\D_X, \ZZ)\otimes \CC$ as $$\P(\D_X):=\{\Omega \in \widetilde{H}^{1,1}(\D_X, \ZZ)\otimes \CC \mid \text{$\langle \mathrm{Re}(\Omega), \mathrm{Im}(\Omega) \rangle_{\mathbb{R}}$ is a positive definite plane} \}.$$ Since the Mukai pairing $(-,-)$ on $H^*(\D_X,\ZZ)$ is non-degenerate, for a stability condition $\sigma=(Z,\A) \in \mathrm{Stab}(\D_X)$, there is the unique element $\Omega_Z \in \widetilde{H}^{1,1}(\D_X, \ZZ)\otimes \CC$ such that $Z(-)=(\Omega_Z,-)$. Let $\P^+(\D_X)$ be the connected component of $\P(\D_X)$ containing $\Omega_{Z_\alpha}$ for $0<\alpha <1/4$. Here, $\sigma_\alpha$ is the stability condition in Theorem \[BLMS stability\]. Let $\Delta_X$ be the set of $(-2)$-classes in $\widetilde{H}^{1,1}(\D_X, \ZZ)$. We define the subset $\P^+_0(\D_X)$ of $\P^+(\D_X)$ as $$\P^+_0(\D_X):=\P^+(\D_X)\setminus \bigcup_{\delta \in \Delta_X} \delta^\perp.$$ As the case of K3 surfaces, we consider the action of the autoequivalence group of $\D_X$ on ${\mathrm{Stab}}(\D_X)$. To consider the action, we need the following.
\[FM-type\] An autoequivalence $\Phi:\D_X \to \D_X$ is called Fourier-Mukai type if there exists $\mathcal{E} \in D^b(X \times X)$ such that the following diagram commutes. $$\xymatrix{D^b(X) \ar[d]_{i^*} \ar[r]^{\Phi_{\mathcal{E}}} & D^b(X) \\
\D_X \ar[r]^{\Phi} & \D_X \ar[u]_{i}}$$ Here, $\Phi_{\mathcal{E}}$ is the Fourier-Mukai functor with the Fourier-Mukai kernel $\mathcal{E}$. We denote the group of autoequivalences of Fourier-Mukai type by ${\mathrm{Aut}}^{{\mathrm{FM}}}(\D_X)$. For $\Phi \in {\mathrm{Aut}}^{{\mathrm{FM}}}(\D_X)$, $\Phi^H: H^*(\D_X, \ZZ) \to H^*(\D_X,\ZZ)$ is the Hodge isometry induced by $\Phi$.
We will consider only autoequivalences of Fourier-Mukai type.
For an autoequivalence $\Phi \in {\mathrm{Aut}}^{\mathrm{FM}}(\D_X)$ and a stability condition $\sigma=(Z,\A) \in {\mathrm{Stab}}(\D_X)$, we define the stability condition $\Phi\sigma:=(Z \circ (\Phi^H)^{-1}, \Phi(\A)) \in {\mathrm{Stab}}(\D_X)$.
Let $\mathrm{Stab}^*(\D_X) \subset {\mathrm{Stab}}(\D_X)$ be the connected component containing stability conditions in Theorem \[BLMS stability\]. Then the following holds.
For a stability condition $\sigma=(Z,\A) \in {\mathrm{Stab}}^*(\D_X)$, put $\pi_X(\sigma):=\Omega_Z$. Then $\pi_X$ induces the covering map $\pi_X: {\mathrm{Stab}}^*(\D_X) \to \P^+_0(\D_X)$.
Autoequivalences of K3 surfaces and stability conditions
========================================================
In this section, we introduce certain subgroups of autoequivalence groups of K3 surfaces related to stability conditions following [@Huy]. We will study polarized automorphisms of K3 surfaces and their relation with stability conditions.
Subgroups of autoequivalence groups of K3 surfaces
--------------------------------------------------
In this subsection, we see the definition of groups that we are interested in.
Let $S$ be a K3 surface. First, we recall the notion of symplectic automorphism group ${\mathrm{Aut}}_{\mathrm{s}}(S)$ of $S$ and the symplectic autoequivalence group ${\mathrm{Aut}}_{\mathrm{s}}(D^b(S))$.
An automorphism $f \in {\mathrm{Aut}}(S)$ of $S$ is a symplectic automorphism of $S$ if the pullback $f^*$ of $f$ acts on $H^{2,0}(S)$ trivially. The group ${\mathrm{Aut}}_{\mathrm{s}}(S)$ of symplectic automorphisms of $S$ is called the symplectic automorphism group of $S$. An autoequivalence $\Phi \in {\mathrm{Aut}}(D^b(S))$ is a symplectic autoequivalence of $D^b(S)$ if the cohomological Fourier-Mukai transform $\Phi^H$ acts on $\widetilde{H}^{2,0}(S)$ trivially. The group ${\mathrm{Aut}}_{\mathrm{s}}(D^b(S))$ of symplectic autoequivalences of $D^b(S)$ is called the symplectic autoequivalence group of $D^b(S)$.
Huybrechts [@Huy] studied the following subgroups of $D^b(S)$.
\[stabilizer K3\] For a stability condition $\sigma \in {\mathrm{Stab}}^*(S)$, we define the group ${\mathrm{Aut}}(D^b(S),\sigma)$ of autoequivalences fixing $\sigma$ by $${\mathrm{Aut}}(D^b(S),\sigma):=\{\Phi \in {\mathrm{Aut}}(D^b(S)) \mid \Phi\sigma=\sigma\}.$$ Denote the intersection of ${\mathrm{Aut}}(D^b(S),\sigma)$ and ${\mathrm{Aut}}_{\mathrm{s}}(D^b(S))$ by ${\mathrm{Aut}}_{\mathrm{s}}(D^b(S),\sigma)$.
The groups in Definition \[stabilizer K3\] can be described in terms of Mukai latices.
\[stabilizer Hodge K3\] For a linear subspace $W \subset H^*(S,\RR)$, we define $$\mathrm{O}_{\mathrm{Hodge}}(H^*(S,\ZZ),W):=\{\varphi \in \mathrm{O}_{\mathrm{Hodge}}(H^*(S,\ZZ)) \mid \varphi |_W=\mathrm{id}_W\}.$$ The weight two Hodge structure on $H^*(S,\ZZ)$ defines the positive definite plane $P_S$ given by $$P_S:=\bigl(\widetilde{H}^{2,0}(S)\oplus\widetilde{H}^{0,2}(S)\bigr)\cap H^*(S,\RR).$$ For a stability condition $\sigma=(Z,\A) \in {\mathrm{Stab}}^*(S)$, we have the positive definite plane $P_\sigma$ defined by $$P_\sigma:=\langle \mathrm{Re}\bigl(\pi_S(\sigma)\bigl), \mathrm{Im}\bigl(\pi_S(\sigma)\bigr) \rangle_{\mathbb{R}}.$$ For a stability condition $\sigma \in {\mathrm{Stab}}^*(S)$, we put $$\Pi_\sigma:=P_S \oplus P_\sigma.$$
Huybrechts proved the following proposition.
\[symmetry lattice\] Let $\sigma \in {\mathrm{Stab}}^*(S)$ be a stability condition on $D^b(S)$. Then we have the isomorphism $$(-)^H:{\mathrm{Aut}}(D^b(S),\sigma) {\xrightarrow{\sim}}\mathrm{O}_{\mathrm{Hodge}}(H^*(S,\ZZ),P_\sigma), \Phi \mapsto \Phi^H$$ of groups ${\mathrm{Aut}}(D^b(S),\sigma)$ and $\mathrm{O}_{\mathrm{Hodge}}(H^*(S,\ZZ),P_\sigma)$. The restriction of this isomorphism to ${\mathrm{Aut}}_{\mathrm{s}}(D^b(S),\sigma)$ induces the isomorphism $$(-)^H: {\mathrm{Aut}}_{\mathrm{s}}(D^b(S),\sigma) {\xrightarrow{\sim}}\mathrm{O}_{\mathrm{Hodge}}(H^*(S,\ZZ),\Pi_\sigma)$$ of groups ${\mathrm{Aut}}_{\mathrm{s}}(D^b(S),\sigma)$ and $\mathrm{O}_{\mathrm{Hodge}}(H^*(S,\ZZ),\Pi_\sigma)$.
Polarized automorphisms of K3 surfaces
--------------------------------------
In this subsection, we study stability conditions on K3 surfaces fixed by polarized automorphisms. First, we define polarized automorphisms of K3 surfaces.
For a polarized K3 surface $(S,h)$, we define the group ${\mathrm{Aut}}(S,h)$ of automorphisms of $(S,h)$ by $${\mathrm{Aut}}(S,h):=\{f \in {\mathrm{Aut}}(S) \mid f^*h=h \}.$$ An automorphism $f \in {\mathrm{Aut}}(S)$ of a K3 surface $S$ is called a polarized automorphism of $S$ if there is a primitive ample divisor $h \in \mathrm{NS}(S)$ such that $f \in {\mathrm{Aut}}(S,h)$. Denote the intersection of ${\mathrm{Aut}}(S,h)$ and ${\mathrm{Aut}}_{\mathrm{s}}(S)$ by ${\mathrm{Aut}}_{\mathrm{s}}(S,h)$.
Let $(S,h)$ be a polarized K3 surface. Using the polarization $h$, we consider the following stability conditions on $D^b(S)$.
\[polarized stability\] Take real numbers $\alpha,\beta \in \RR$ such that $\alpha>0$ and $e^{\beta h+i\alpha h} \in \P^+_0(S)$. We define the stability condition $\sigma_{\alpha,\beta}=(Z_{\alpha.\beta}, \A_{\alpha,\beta})$ by $\sigma_{\alpha, \beta}:=\sigma_{\beta h, \alpha h}$ in Theorem \[stability K3\].
The stability conditions in Definition \[polarized stability\] is fixed by polarized automorphisms in ${\mathrm{Aut}}(S,h)$.
\[fix K3\] Take real numbers $\alpha,\beta \in \RR$ such that $\alpha>0$ and $e^{\beta h+i\alpha h} \in \P^+_0(S)$. For any automorphism $f \in {\mathrm{Aut}}(S,h)$, we have $f_*\sigma_{\alpha,\beta}=\sigma_{\alpha,\beta}$. In particular, we have the natural inclusions $${\mathrm{Aut}}(S,h) \hookrightarrow {\mathrm{Aut}}(D^b(S),\sigma_{\alpha,\beta}),$$ and $${\mathrm{Aut}}_{\mathrm{s}}(S,h) \hookrightarrow {\mathrm{Aut}}_{\mathrm{s}}(D^b(S),\sigma_{\alpha,\beta}).$$
Let $f \in {\mathrm{Aut}}(S,h)$ be a polarized automorphism of $S$. Note that $f_*h=h$. Recall that $Z_{\alpha,\beta}(v)=(e^{\beta h+i\alpha h},v)$ for an object $v \in \widetilde{H}^{1,1}(S,\ZZ)$. We have $$\begin{aligned}
Z_{\alpha, \beta}(f^*(v))
&=& (e^{\alpha h + i\beta h}, f^*(v))\\
&=& (f_*(e^{\alpha h+i \beta h}),v)\\
&=& (e^{\alpha h+i \beta h}, v)\\
&=& Z_{\alpha,\beta}(v). \end{aligned}$$ Moreover, $f_*$ preserves the torsion pair on the abelian category $\mathrm{Coh}(S)$ in Definition \[tilt\] and Theorem \[stability K3\]. So $f_*$ preserves the heart $\A_{\alpha, \beta}$ of the bounded t-structure on $D^b(S)$.
Automorphisms of cubic fourfolds and stability conditons
========================================================
Section 5 is almost parallel to Section 4. In this section, we introduce certain subgroups of autoequivalence groups of Kuznetsov components of cubic fourfolds related to stability conditions. We will study automorphisms of cubic fourfolds and their relation with stability conditions.
Subgroups of autoequivalence groups of Kuznetsov components
-----------------------------------------------------------
In this subsection, we introduce subgroups of autoequivalence groups of Kuznetsov components as Subsection 4.1.
Let $X$ be a cubic fourfold. First, we compare the automorphism group ${\mathrm{Aut}}(X)$ with the autoequivalence group ${\mathrm{Aut}}(\D_X)$ of $\D_X$. For an automorphism $f \in {\mathrm{Aut}}(X)$, we have the autoequivalence $f_* \in {\mathrm{Aut}}^{\mathrm{FM}}(\D_X)$ since $f_*\OO_X(k)$ for any integer $k \in \ZZ$. So we obtain the group homomorphism $\rho_1: {\mathrm{Aut}}(X) \to {\mathrm{Aut}}^{\mathrm{FM}}(\D_X)$. Note that the admissible subcategory $\D_X$ fits into the semi-orthogonal decomposition $$D^b(X)=\langle \OO_X(-1),\D_X,\OO_X,\OO_X(1) \rangle$$ by the Serre duality on $X$. We define the projection functor $\mathrm{pr}:D^b(X) \to \D_X$ by $\mathrm{pr}:=\mathbf{R}_{\OO_X(-1)}\mathbf{L}_{\OO_X}\mathbf{L}_{\OO_X(1)}$. Recall the following property of the projection functor $\mathrm{pr}:D^b(X) \to \D_X$.
\[pr\] If $x \neq y \in X$, the object $\mathrm{pr}(\OO_x)$ is not isomorphic to $\mathrm{pr}(\OO_y)$.
Lemma \[pr\] implies the injectivity of $\rho_1$.
\[cubic injective\] The homomorphism $\rho_1$ is injective.
Since $f_*\OO_X(k) \simeq \OO_X(k)$ for $k \in \ZZ$, for an automorphism $f \in {\mathrm{Aut}}(X)$, we have $f_* \circ \mathrm{pr} \simeq \mathrm{pr} \circ f_* $. So we obtain $f_* \circ \mathrm{pr}(\OO_x) \simeq \mathrm{pr}(\OO_{f(x)})$ for any point $x \in X$. By Lemma \[pr\], the homomorphism $\rho_1$ is injective.
We define the symplectic automorphism group of $X$ and the symplectic autoequivalence group of $\D_X$.
An automorphism $f \in {\mathrm{Aut}}(X)$ is a symplectic automorphism of $X$ if $f^*$ acts on $H^{3,1}(X)$ trivially. The group ${\mathrm{Aut}}_{\mathrm{s}}(X)$ of symplectic automorphisms of $X$ is called the symplectic automorphism group of $X$. An autoequivalence $\Phi \in {\mathrm{Aut}}^{\mathrm{FM}}(\D_X)$ is a symplectic autoequivalence if $\Phi^H$ acts on $\widetilde{H}^{2,0}(\D_X)$ trivially. The group ${\mathrm{Aut}}^{\mathrm{FM}}_{\mathrm{s}}(\D_X)$ of symplectic autoequivalences of $\D_X$ is called the symplectic autoequivalence group of $\D_X$.
Symplectic automorphisms of cubic fourfolds are compatible with symplectic autoequivalences of Kuznetsov components.
Let $f \in {\mathrm{Aut}}(X)$ be an automorphism of $X$. By Definition \[Mukai lattice\], $f \in {\mathrm{Aut}}_{\mathrm{s}}(X)$ if and only if $f_* \in {\mathrm{Aut}}^{\mathrm{FM}}(\D_X)$.
As Definition \[stabilizer K3\], we introduce the following groups.
\[stabilizer cubic\] For a stability condition $\sigma \in {\mathrm{Stab}}^*(\D_X)$, we define the group ${\mathrm{Aut}}(\D_X,\sigma)$ of autoequivalences fixing $\sigma$ by $${\mathrm{Aut}}(\D_X,\sigma):=\{\Phi \in {\mathrm{Aut}}^{\mathrm{FM}}(\D_X) \mid \Phi\sigma=\sigma\}.$$ Denote the intersection of ${\mathrm{Aut}}(\D_X,\sigma)$ and ${\mathrm{Aut}}^{\mathrm{FM}}_{\mathrm{s}}(\D_X)$ by ${\mathrm{Aut}}_{\mathrm{s}}(\D_X,\sigma)$.
As in Definition \[stabilizer Hodge K3\], we define the following groups in terms of Hodge theory.
\[stabilizer Hodge cubic\] For a linear subspace $W \subset H^*(\D_X,\RR)$, we define $$\mathrm{O}_{\mathrm{Hodge}}(H^*(\D_X,\ZZ),W):=\{\varphi \in \mathrm{O}_{\mathrm{Hodge}}(H^*(\D_X,\ZZ)) \mid \varphi |_W=\mathrm{id}_W\}.$$ The weight two Hodge structure on $H^*(\D_X,\ZZ)$ defines the positive definite plane $P_X$ given by $$P_X:=\bigl(\widetilde{H}^{2,0}(\D_X)\oplus\widetilde{H}^{0,2}(\D_X)\bigr)\cap H^*(\D_X,\RR).$$ For a stability condition $\sigma=(Z,\A) \in {\mathrm{Stab}}^*(\D_X)$, we have the positive definite plane $P_\sigma$ defined by $$P_\sigma:=\langle \mathrm{Re}\bigl(\pi_X(\sigma)\bigl), \mathrm{Im}\bigl(\pi_X(\sigma)\bigr) \rangle_{\mathbb{R}}.$$ For a stability condition $\sigma \in {\mathrm{Stab}}^*(\D_X)$, we put $$\Pi_\sigma:=P_X \oplus P_\sigma.$$
For stability conditions in Theorem \[BLMS stability\], the corresponding positive definite planes come from $A_2$.
\[A2\] For a real number $0<\alpha<1/4$, we have $P_{\sigma_\alpha}=A_2 \otimes \RR$.
Automorphisms of cubic fourfolds
--------------------------------
In this subsection, we prove that automorphisms of cubic fourfolds fix stability conditions in Theorem \[BLMS stability\]. The goal of this subsection is the following proposition.
\[fix cubic\] Fix a real number $0<\alpha<1/4$. For an automorphism $f \in {\mathrm{Aut}}(X)$ of $X$, we have $f_*\sigma_\alpha=\sigma_\alpha$. In particular, we have the natural inclusions $${\mathrm{Aut}}(X) \hookrightarrow {\mathrm{Aut}}(\D_X,\sigma_\alpha),$$ and $${\mathrm{Aut}}_{\mathrm{s}}(X) \hookrightarrow {\mathrm{Aut}}_{\mathrm{s}}(\D_X,\sigma_\alpha).$$
In Section 7, we will see the above inclusions are isomorphisms. To prove Proposition \[fix cubic\], we study relations between automorphisms of cubic fourfolds and sheaves of Clifford algebras. Fix a line $l$ on $X$. An automorphism $f \in {\mathrm{Aut}}(X)$ of $X$ induces $f_l: \mathrm{Bl}_lX {\xrightarrow{\sim}}\mathrm{Bl}_{f(l)}X$ and $\tilde{f}_l \in {\mathrm{Aut}}(\mathbb{P}^3)$ such that we have the following commutative diagrams.
$$\xymatrix{
&\mathrm{Bl}_lX \ar[r]^{f_l} \ar[d]_{p_l} \ar@{}[dr]^{} & \mathrm{Bl}_{f(l)}X \ar[d]^{p_{f(l)}}&&&\mathrm{Bl}_lX \ar[r]^{f_l} \ar[d]_{\pi_l} & \mathrm{Bl}_{f(l)}X \ar[d]^{\pi_{f(l)}}\\
&X \ar[r]^{f} & X && &\mathbb{P}^3 \ar[r]^{\tilde{f}_l} & \mathbb{P}^3
}$$
The following is the relation between automorphisms of $X$ and the sheaves of Clifford algebras on $\mathbb{P}^3$.
\[Clifford\] Let $k \in \ZZ$ be an integer. For an automorphism $f \in {\mathrm{Aut}}(X)$ of $X$, we have $\tilde{f}_{l*}\B^l_k \simeq \B^{f(l)}_k$.
In the proof of Lemma 3.2 in [@Kuz08], this is observed in the more general situation. (Cf. Lemma 7.2 in [@BLMS]).
The fully faithful functor in Subsection 3.4 is compatible with automorphisms of $X$.
\[automorphisms\] Let $f \in {\mathrm{Aut}}(X)$ be an automorphism of $X$. Then we have the following commutative diagram. $$\xymatrix{\D_X \ar[d]_{\Psi_l \circ \mathbf{L}p^*_l} \ar[r]^{f_*} & \D_X \ar[d]^{\Psi_{f(l)} \circ \mathbf{L}p^*_{f(l)}} \\
D^b(\mathbb{P}^3, \B^l_0) \ar[r]^{\tilde{f}_{l*}} & D^b(\mathbb{P}^3,\B^{f(l)}_0)}$$
By Lemma \[Clifford\], we have the equivalence $\tilde{f}_{l*}: D^b(\mathbb{P}^3, \B^l_0) {\xrightarrow{\sim}}D^b(\mathbb{P}^3, \B^{f(l)}_0)$. By the definition of $f_l$ and $\tilde{f}_l$, Lemma \[Clifford\] and the exact sequence (A), we have $f_{l*}\E_l \simeq \E_{f(l)}$. So we obtain the desired commutative diagram.
We prove Proposition \[fix cubic\].
Let $f \in {\mathrm{Aut}}(X)$ be an automorphism of $X$. Note that $f^*$ acts on $A_2$ trivially. By Proposition \[A2\], we have $Z_\alpha \circ f^*=Z_\alpha$. By Lemma \[Clifford\] and Proposition \[automorphisms\], the equivalence $\tilde{f}_{l*}: D^b(\PP^3,\B^l_0) {\xrightarrow{\sim}}D^b(\PP^3,\B^{f(l)}_0)$ induces the equivalence $$\tilde{f}_{l*}: \Psi_l \circ \mathbf{L}p^*_l(\D_X) {\xrightarrow{\sim}}\Psi_{f(l)} \circ \mathbf{L}^*_{f(l)}(\D_X).$$ Since $\tilde{f}_{l*}$ is compatible with tilting (cf. Definition \[tilt\]) in the construction of $\A^l_\alpha$, we have $\tilde{f}_{l*}\A^l_\alpha=\A^{f(l)}_\alpha$. So we obtain $f_*\sigma^l_\alpha=\sigma^{f(l)}_\alpha$. By Theorem \[BLMS stability\], we have $f_*\sigma_\alpha=\sigma_\alpha$.
Automorphism groups of cubic fourfolds and Kuznetsov components
===============================================================
In this section, we characterize automorphism groups of cubic fourfolds as subgroups of autoequivalence groups of Kuznetsov components.
Automorphisms of cubic fourfolds as autoequivalences
----------------------------------------------------
In this subsection, we give the statement of the theorem which is main in this section. Let $X$ be a cubic fourfold. Fix a real number $0<\alpha<1/4$. Put $\sigma:=\sigma_\alpha \in {\mathrm{Stab}}^*(\D_X)$. By Proposition \[cubic injective\] and Proposition \[fix cubic\], the group homomorphism $\rho_1: {\mathrm{Aut}}(X) \to {\mathrm{Aut}}(\D_X, \sigma)$ is injective. We define the group homomorphism $\rho_2: {\mathrm{Aut}}(\D_X,\sigma) \to \mathrm{O}_{\mathrm{Hodge}}(H^*(\D_X,\ZZ), P_\sigma)$ by $\rho_2(\Phi):=\Phi^H$. The goal of this section is the following theorem.
\[K3 sigma model\] The group homomorphisms $$\rho_1: {\mathrm{Aut}}(X) \to {\mathrm{Aut}}(\D_X, \sigma),$$ $$\rho_2: {\mathrm{Aut}}(\D_X,\sigma) \to \mathrm{O}_{\mathrm{Hodge}}(H^*(\D_X,\ZZ), P_\sigma)$$ are isomorphisms. In particular, we have isomorphisms $$\rho_1: {\mathrm{Aut}}_{\mathrm{s}}(X) \to {\mathrm{Aut}}_{\mathrm{s}}(\D_X, \sigma),$$ $$\rho_2: {\mathrm{Aut}}_{\mathrm{s}}(\D_X,\sigma) \to \mathrm{O}_{\mathrm{Hodge}}(H^*(\D_X,\ZZ), \Pi_\sigma).$$
By Proposition \[cubic injective\], it is enough to show that $\rho_2 \circ \rho_1$ is surjective and $\rho_2$ is injective.
First, we prove the surjectivity of $\rho_2 \circ \rho_1$.
\[surj\] The composition $\rho_2 \circ \rho_1: {\mathrm{Aut}}(X) \to \mathrm{O}_{\mathrm{Hodge}}(H^*(\D_X,\ZZ), P_\sigma)$ of $\rho_1$ and $\rho_2$ is surjective.
By Proposition \[A2\], we have $\mathrm{O}_{\mathrm{Hodge}}(H^*(\D_X,\ZZ), P_\sigma)=\mathrm{O}_{\mathrm{Hodge}}(H^*(\D_X,\ZZ), A_2 \otimes \RR)$. By Proposition \[prim\], there is the isomorphism $$v: \mathrm{O}_{\mathrm{Hodge}}(H^*(\D_X,\ZZ), A_2 \otimes \RR) \xrightarrow{\sim} \mathrm{O}_{\mathrm{Hodge}}(H^*_{\mathrm{prim}}(X,\ZZ)).$$ Under this isomorphism, the natural homomorphism ${\mathrm{Aut}}(X) \to \mathrm{O}_{\mathrm{Hodge}}(H^4_{\mathrm{prim}}(X,\ZZ))$ is compatible with $\rho_2 \circ \rho_1$. By the Torelli theorem for cubic fourfolds, we obtain the statement.
We will prove the following proposition.
\[inj2\] The homomorphism $\rho_2 : {\mathrm{Aut}}(\D_X, \sigma) \to \mathrm{O}_{\mathrm{Hodge}}(H^*(\D_X,\ZZ), P_\sigma)$ is injective. Equivalently, if $\Phi \in {\mathrm{Aut}}^{{\mathrm{FM}}}(\D_X)$ satisfies $\Phi\sigma=\sigma$ and $\Phi^H=\mathrm{id}$, we have $\Phi=\mathrm{id}_{\D_X}$ in ${\mathrm{Aut}}(\D_X,\sigma)$.
To prove Proposition \[inj2\], we study the Fano scheme $F(X)$ of lines on $X$.
Fano schemes of lines
---------------------
In this subsection, we recall the description of Fano schemes of lines on cubic fourfolds in terms of moduli spaces of stable objects in Kuznetsov components.
Let $X$ be a cubic fourfold. The Fano scheme $F(X)$ of lines on $X$ can be described by the moduli theory on the Kuznetsov component $\D_X$. For a line $l$ on $X$, there is the exact sequence $$0 \to F_l \to \OO_X^{\oplus4} \to I_l(1) \to 0,$$ where $I_l$ is the ideal sheaf of $l$ in $X$. Then we have $i^*(\O_l(1)) \simeq F_l[2]$. (Lemma 5.1 in [@KM]) We consider the autoequivalence $\Xi:= \mathbf{R}_{\mathcal{O}_X(-1)}(-\otimes \O_X(-1)) \in {\mathrm{Aut}}^{\mathrm{FM}}(\D_X)$ on $\D_X$. For a line $l \in F(X)$, we define the object $E_l \in \D_X$ as $E_l:=\Xi(F_l)$. For a line $l \in F(X)$, the object $E_l$ fits into the exact triangle $\OO_X(-1)[1] \to E_l \to I_l$ ([@MS], Section 2.3). Li, pertusi and Zhao proved the following theorem.
\[Fano\] Take a real number $0<\alpha<1/4$. Then we have an isomorphism $u: F(X) {\xrightarrow{\sim}}M_{\sigma_\alpha}(v)$ given by $u([l]):=E_l$, where $v:=\lambda_1+\lambda_2$.
Denote the Fano scheme of lines on $X$ by $F(X)$. For the universal line $$\F(X):=\{ (x, l) \in X \times F(X) \mid x \in l\}$$ on $X$, consider the natural projections $p: \F(X) \to F(X)$ and $q: \F(X) \to X$. We define an exact functor $\Sigma: \D_X \to D^b(F(X))$ as $\Sigma:=\mathbf{R}p_* \circ \mathbf{L}q^* \circ i$. Denote the right adjoint functor of $\Sigma$ by $\Sigma_R: D^b(F(X)) \to \D_X$. Addington proved the following proposition.
\[for universal family\] For $[l] \in F(X)$, we have $\Sigma_R(\mathcal{O}_l(1)) \simeq F_l[1]$
We modify the adjoint functors $\Sigma$ and $\Sigma_R$ for this situation.
Take a real number $0<\alpha<1/4$ and put $\sigma:=\sigma_\alpha$. We define the exact functors $P^*: \D_X \to D^b(M_\sigma(v))$ and $P_*: D^b(M_\sigma(v)) \to \D_X$ by $$P^*:=u_* \circ \Sigma \circ \Xi^{-1} \circ [1]: \D_X \to D^b(M_\sigma(v)),$$ $$P_*:=[-1] \circ \Xi \circ \Sigma_R \circ u^*: D^b(M_\sigma(v)) \to \D_X.$$ Then $P_*$ is the right adjoint functor of $P^*$.
The above functors are related to an universal family of the moduli space $M_\sigma(v)$.
\[univP\] Since $\Xi \in {\mathrm{Aut}}^{\mathrm{FM}}(\D_X)$, the composition $i \circ P_*: D^b(M_\sigma(v)) \to D^b(X)$ is a Fourier-Mukai functor, that is, there is an object $\U \in D^b(M_\sigma(v) \times X)$ such that $i \circ P_* \simeq \Phi_\U$. By Proposition \[for universal family\], the object $\U$ is an universal family of the moduli space $M_\sigma(v)$ over $X$.
We will need the following result about kernels of actions of automorphism groups on cohomology groups for irreducible holomorphic symplectic manifolds.
\[ker\] Take a positive integer $n>0$. Let $M$ be an irreducible holomorphic symplectic manifold deformation equivalent to Hilbert scheme of $n$-points on K3 surfaces. Consider the group homomorphism $$\rho: {\mathrm{Aut}}(M) \to \mathrm{O}(H^2(M,\ZZ)), f \mapsto f^*.$$ Then we have $\mathrm{Ker}(\rho)=1$.
Let $\Sigma^H: H^*(\D_X, \QQ) \to H^*(F(X),\QQ)$ be the linear map induced by $\Sigma: \D_X \to D^b(F(X))$.
\[AJ\] The restriction $\Sigma^H|_{A_2^\perp} A_2^\perp {\xrightarrow{\sim}}H^2_{\mathrm{prim}}(F(X),\ZZ)$ is a Hodge isometry, where $H^2_{\mathrm{prim}}(F(X),\ZZ)$ is the primitive cohomology group of $F(X)$ with respect to the Plücker polarization of $F(X)$.
The Abel-Jacobi map $p_*q^*: H^4_{\mathrm{prim}}(X,\ZZ)(-1) {\xrightarrow{\sim}}H^2_{\mathrm{prim}}(F(X),\ZZ)$ is the Hodge isometry. By Proposition \[prim\] and the definition of $\Sigma$, we have $\Sigma^H_{{A_2}^\perp}=p_*q^*v$.
For an autoequivalence $\Phi \in \mathrm{Ker}(\rho_2)$, we have isomorphisms $\Phi \circ P_* \simeq P_*$ and $\P^* \circ \Phi \simeq P^*$.
Let $\Phi \in {\mathrm{Aut}}^{{\mathrm{FM}}}(\D_X)$ be an autoequivalence satisfying $\Phi\sigma=\sigma$ and $\Phi^H=\mathrm{id}$. Then $\Phi$ induces an automorphism $\Phi_{\sigma,v}: M_\sigma(v) {\xrightarrow{\sim}}M_\sigma(v), [E] \mapsto [\Phi(E)]$ such that the following diagram commutes. $$\xymatrix{v^{\perp} \ar[d]_{\theta_{\sigma,v}} \ar[r]^{\Phi^H} & v^\perp \ar[d]_{\theta_{\sigma, v}} \\
H^2(M_\sigma(v),\mathbb{Z}) \ar[r]^{{\Phi_{\sigma,v}}_*} & H^2(M_{\sigma}(v),\mathbb{Z})}$$ Since $\Phi^H=\mathrm{id}$, we have ${\Phi_{\sigma,v}}_*=\mathrm{id}$. By Theorem \[ker\], we obtain $\Phi_{\sigma,v}=\mathrm{id}_{M_\sigma(v)}$. Since $\Phi$ is of Fourier-Mukai type, there is an object $\E \in D^b(X \times X)$ such that $\Phi \simeq i \circ \Phi_\E \circ i^*$. By Remark \[univP\], we have $\Phi_\U \simeq i \circ P_*$, where $\U \in D^b(M_\sigma(v) \times X)$ is an universal family of $M_\sigma(v)$. For $[E] \in M_\sigma(v)$, we have $$\begin{aligned}
\Phi_\E \circ \Phi_\U(\O_{[E]})
&\simeq&i^* \circ \Phi \circ i(E)\\
&\simeq&E. \end{aligned}$$ The convolution product $\E \circ \U$, that is a Fourier-Mukai kernel of the composition $\Phi_\E \circ \Phi_\U$, is also an universal family of $M_\sigma(v)$. So there is a line bundle $L \in \mathrm{Pic}(M_\sigma(v))$ such that $\E \circ \U \simeq \U \otimes p_M^*L$, where $p_M: M_\sigma(v) \times X \to M_\sigma(v)$ is the projection. Since $i^* \circ i \simeq \mathrm{id}_{\D_X}$ and $\Phi_\E \circ \Phi_\U \simeq \Phi_\U \circ (- \otimes L)$, we obtain $\Phi \circ P_* \simeq P_* \circ (-\otimes L)$. By the uniqueness of left adjoint functors, we have $P^* \circ \Phi^{-1} \simeq (- \otimes L^{-1}) \circ P^*$ and it induces $(-\otimes L) \circ P^* \simeq P^* \circ \Phi$. Let $(P^*)^H: H^*(\D_X,\mathbb{Q}) \to H^*(M_\sigma(v), \mathbb{Q})$ be the linear map induced by $P^*:\D_X \to D^b(M_\sigma(v))$. Then we have $e^L \circ (P^*)^H=(P^*)^H$. Take a non-zero divisor $D \in \mathrm{NS}(M_\sigma(v))$. By Proposition \[AJ\], there is an element $w \in \widetilde{H}^{1,1}(\D_X,\ZZ)$ such that $D=(P^*)^H(w)$. Then we have $$\begin{aligned}
D
&=& (P^*)^H(w)\\
&=&e^L \cdot (P^*)^H(w)\\
&=& D+L\cdot D+\frac{1}{2}L^2 \cdot D+\frac{1}{6}L^3 \cdot D . \end{aligned}$$ By the Verbitsky’s result ([@Ver], Theorem 1.3), the natural map $\mathrm{Sym}^2(H^2(M_\sigma(v),\CC)) \to H^4(M_\sigma(v),\CC)$ is injective. So we have $L=0$. Therefore, we have $\Phi \circ P_* \simeq P_*$ and $P^* \circ \Phi \simeq P^*$.
Comonads
--------
In this subsection, we collect definitions and basic properties of comodules over comonads following [@elagin1]. Let $\mathcal{C}$ be a category.
A [*comonad*]{} $\mathbb{T}=(T,\varepsilon,\delta)$ on $\mathcal{C}$ consists of an endo-functor $T:\mathcal{C}\rightarrow\mathcal{C}$ and morphisms $\varepsilon:T\rightarrow {\rm id}_{\mathcal{C}}$ and $\delta:T\rightarrow T^2$ of functors such that the following diagrams commute.
$$\xymatrix{
&T \ar[r]^{\delta} \ar[d]_{\delta} \ar@{=}[dr]^{{\rm id}_T} & T^2 \ar[d]^{T\varepsilon}&&&T \ar[r]^{\delta} \ar[d]_{\delta} & T^2 \ar[d]^{T\delta}\\
&T^2 \ar[r]^{\varepsilon T} & T && &T^2 \ar[r]^{\delta T} & T^3
}$$
From adjoint functors, we can construct comonads.
\[adj\] Let $P=(P^*\dashv P_*)$ be adjoint functors $P^*:\mathcal{C}'\rightarrow\mathcal{C}$ and $P_*:\mathcal{C}\rightarrow\mathcal{C}'$. Denote the unit and the counit by $\eta_{P} :{\rm id}_{\mathcal{C}'}\rightarrow P_*\circ P^*$ and $\varepsilon_{P}:P^* \circ P_*\rightarrow {\rm id}_{\mathcal{C}}$ respectively. We have an endo-functor $T_P:=P^*\circ P_*$ and morphisms $\delta_P:=P^*\eta_P P_*$ of functors. Then the triple $\mathbb{T}(P):=(T_P,\varepsilon_{P},\delta_P)$ is a comonad on $\mathcal{C}$.
For a comonad, we have the notion of comodules over the comonad.
\[comodule\] Let $\mathbb{T}=(T,\varepsilon,\delta)$ be a comonad on $\mathcal{C}$. A [*comodule*]{} over $\mathbb{T}$ is a pair $(C,\theta_C)$ of an object $C\in\mathcal{C}$ and a morphism $\theta_{C}:C\rightarrow T(C)$ such that
- $\varepsilon(C)\circ\theta_C={\rm id}_C$, and
- the following diagram commutes.
$$\begin{CD}
C@>{\theta_C}>>T(C)\\
@V{\theta_C}VV @VVT({\theta_C})V\\
T(C)@>{\delta(C)}>>T^2(C).
\end{CD}$$
Let $\mathbb{T}=(T,\varepsilon,\delta)$ be a comonad on $\mathcal{C}$. We define the category $\mathcal{C}_{\mathbb{T}}$ of comodules over $\mathbb{T}$ as follow.
- The set $\mathrm{Ob}(\mathcal{C}_{\mathbb{T}})$ of objects in $\mathcal{C}_{\mathbb{T}}$ consists of comodules over $\mathbb{T}$.
- For comodules $(C_1, \theta_{C_1}),(C_2, \theta_{C_2}) \in \mathrm{Ob}(\mathcal{C}_{\mathbb{T}})$, $${\rm Hom}_{\C_{\mathbb{T}}}((C_1,\theta_{C_1}),(C_2,\theta_{C_2})):=\{f\in{{\rm Hom}}_{\C}(C_1,C_2) \mid T(f)\circ\theta_{C_1}=\theta_{C_2}\circ f \}.$$
We have the following natural functors.
Let $\mathbb{T}=(T,\varepsilon, \delta)$ be a comonad on $\C$. We define a functor $Q_*: \C \to \C_{\mathbb{T}}$ as follow.
- For an object $C \in \C$, set $Q_*(C):=(T(C), \delta(C))$.
- For a morphism $f \in {{\rm Hom}}_{\C}(C_1,C_2)$, set $Q_*(f):=T(f)$.
We define a functor $Q^*:\C_{\mathbb{T}} \to \C$ as the forgetful functor. Then we have adjoint functors $Q=(Q^*\dashv Q_*)$.
\[comparison functor\] Let $P=(P^*\dashv P_*)$ be adjoint functors $P^*:\mathcal{C}\rightarrow\mathcal{C}'$ and $P_*:\mathcal{C}'\rightarrow\mathcal{C}$. Then there exists a functor $\Gamma_P: \C' \to \C_{\mathbb{T}(P)}$ unique up to an isomorphism such that $\Gamma_P \circ P_* \simeq Q_*$ and $Q^* \circ \Gamma_P \simeq P^*$. The functor $\Gamma_P: \C' \to \C_{\mathbb{T}(P)}$ is called the comparison functor.
The following proposition gives sufficient conditions for a comparison functor to be an equivalence.
\[comparison theorem\] Let $P=(P^*\dashv P_*)$ be adjoint functors $P^*:\mathcal{C}\rightarrow\mathcal{C}'$ and $P_*:\mathcal{C}'\rightarrow\mathcal{C}$. If $\mathcal{C}$ is idempotent complete and the functor morphism $\eta_P :{\rm id}_{\mathcal{C}}\rightarrow P_* \circ P^*$ is a split mono, i.e. there exists a functor morphism $\zeta:P_* \circ P^*\rightarrow {\rm id}_{\mathcal{C}}$ such that $\zeta\circ\eta={\rm id}$, then $\Gamma_{P}:\mathcal{C}\rightarrow\mathcal{D}_{\mathbb{T}(P)}$ is an equivalence.
Application of comonads
-----------------------
In this subsection, we prove that $\rho_2$ is injective as an application of comonads. Let $X$ be a cubic fourfold. Fix a real number $0< \alpha < 1/4$ and put $v:=[E_l]$, where $l$ is a line on $X$. Addington [@Add] proved the following theorem.
\[RF\] The unit $\eta: \mathrm{id}_{\D_X} \to \Sigma_R \circ \Sigma$ is a split mono and $\Sigma_R \circ \Sigma \simeq \mathrm{id}_{\D_X} \oplus [-2]$.
As a consequence of Theorem \[RF\], we have the following statement.
\[split mono\] The unit $\eta: \mathrm{id}_{\D_X} \to P_* \circ P^*$ is a split mono and $P_* \circ P^* \simeq \mathrm{id}_{\D_X} \oplus [-2]$.
We prove the Proposition \[inj2\].
As in Example \[adj\], let $\mathbb{T}(P)$ be the comonad on $D^b(M_\sigma(v))$ determined by the adjoint pair $P=(P^* \dashv P_*)$. By Theorem \[comparison functor\], there is a comparison functor $\Gamma_P: \D_X \to D^b(M_\sigma(v))_{\mathbb{T}(P)}$ unique up to an isomorphism such that $\Gamma_P \circ P_* \simeq Q_*$ and $Q^* \circ \Gamma_P \simeq P^*$. Since $\Phi \circ P_* \simeq P_*$ and $P^* \circ \Phi \simeq P^*$, we have $\Gamma_P \circ \Phi \circ P_* \simeq Q_*$ and $Q^* \circ \Gamma_P \circ \Phi \simeq P^*$ So the composition $\Gamma_P \circ \Phi$ is also a comparison functor. Due to the uniqueness of comparison functors, we have $\Gamma_P \simeq \Gamma_P \circ \Phi$. By Corollary \[split mono\] and Theorem \[comparison theorem\], the comparison functor $\Gamma_P$ is an equivalence. Therefore, we obtain $\Phi \simeq \mathrm{id}_{\D_X}$.
As a corollary of Proposition \[inj2\], we have
We define $${\mathrm{Aut}}^0(\D_X):=\{\Phi \in {\mathrm{Aut}}^{{\mathrm{FM}}}(\D_X) \mid \Phi^H=\mathrm{id}, \Phi({\mathrm{Stab}}^*(\D_X)) \subset {\mathrm{Stab}}^*(\D_X)\}.$$ Then the natural homomorphism ${\mathrm{Aut}}^0(\D_X) \to \mathrm{Deck}(\pi)$ is injective, where $\mathrm{Deck}(\pi)$ is the group of deck transformations of the covering $\pi$.
Take $\Phi \in \mathrm{Ker}\bigl({\mathrm{Aut}}^0(\D_X) \to \mathrm{Deck}(\pi)\bigr)$. Then $\Phi \sigma=\sigma$ holds. By Proposition \[inj2\], we have $\Phi=\mathrm{id}_{\D_X}$.
Automorphisms of cubic fourfolds and K3 surfaces
================================================
In this section, we compare automorphisms of cubic fourfolds and autoequivalences of derived categories of K3 surfaces. For labeled automorphisms of cubic fourfolds, they induce polarized automorphisms of K3 surfaces. We introduce the notion of labeled automorphisms of cubic fourfolds.
For a labeled cubic fourfold $(X,K)$, we define the labeled automorphism group ${\mathrm{Aut}}(X,K)$ of $(X,K)$ by $${\mathrm{Aut}}(X,K):=\{f \in {\mathrm{Aut}}(X) \mid f^*|_K=\mathrm{id}_K\}.$$ An automorphism $f \in {\mathrm{Aut}}(X)$ is called a labeled automorphism of $X$ if there is a rank two primitive sublattice $K \subset H^{2,2}(X,\ZZ)$ such that $(X, K)$ is a labeled cubic fourfold and $f \in {\mathrm{Aut}}(X,K)$.
We study polarized K3 surfaces associated to labeled cubic fourfolds from point of view of moduli spaces of stable objects in Kuznetsov components.
Take an integer $d$ that satisfies conditions $(*)$ and $(**)$. Let $(X,K)$ be a labeled cubic fourfold of discriminant $d$. By Remark \[label Mukai\] and Theorem \[Hassett K3\], there is a polarized K3 surface $(S,h)$ of degree $d$ such that there is a Hodge isometry $\varphi: L^\perp_K {\xrightarrow{\sim}}L^\perp_h$, where $L_h$ is the sublatitce $H^0(S,\ZZ)\oplus \ZZ \cdot h \oplus H^4(S,\ZZ)$ of the Mukai lattice $H^*(S,\ZZ)$ of $S$. By Theorem 1.14.4 in [@Nik], there is a Hodge isometry $\tilde{\varphi}: H^*(\D_X,\ZZ) {\xrightarrow{\sim}}H^*(S,\ZZ)$ such that $\tilde{\varphi}|_{L^\perp_K}=\mathrm{id}_{L^\perp_K}$. Then $\tilde{\varphi}$ induces the isometry $\tilde{\varphi}|_{L_K}: L_K {\xrightarrow{\sim}}L_h$. We will reconstruct the polarized K3 surface $(S,h)$ in terms of moduli spaces of stable objects in $\D_X$. Then we will obtain the nice equivalence $\D^b(S) {\xrightarrow{\sim}}\D_X$ in the context of both Hodge theory and moduli theory. First, we choose a Mukai vector and a stability condition on $\D_X$ to specify the moduli space. We put $v:=\tilde{\varphi}^{-1}(0,0,1) \in \widetilde{H}^{1,1}(\D_X,\ZZ)$. Note that the Mukai vector $v$ is a primitive isotropic. Replacing $\tilde{\varphi}$ by $\bigl(\mathrm{id}_{H^0} \oplus -\mathrm{id}_{H^2} \oplus \mathrm{id}_{H^4}\bigr) \circ \tilde{\varphi}$ if necessary, we may assume that $\Omega_1:=\tilde{\varphi}^{-1}(e^{ih})$ is contained in $\P^+_0(\D_X)$. Here, we use $d>6$ and Theorem \[stability K3\] and Theorem \[BriK3\]. We define $\Omega_n:=\tilde{\varphi}^{-1}(e^{inh}) \in \P^+_0(\D_X)$ for a positive integer $n$.
For a positive integer $n$, we define the group homomorphism $$Z_n: \widetilde{H}^{1,1}(\D_X,\ZZ) \to \CC, w \mapsto (\Omega_n,w).$$
The following lemma will be used in the proof of Lemma \[generic\].
\[Bri finite\] Fix $C>0$. For $\Omega \in \P^+_0(\D_X)$, there are finitely many elements $w \in \widetilde{H}^{1,1}(\D_X,\ZZ)$ such that $w^2 \geq -2$ and $|(\Omega,w)|\leq C$.
Recall that a stability condition $\sigma \in {\mathrm{Stab}}^*(\D_X)$ is $v$-generic if and only if a $\sigma$-semistable object $E \in \D_X$ with $v(E)=v$ is $\sigma$-stable by the primitivity of $v$.
\[generic\] For a positive integer $n$, let $\sigma_n=(Z_n, \A_n) \in {\mathrm{Stab}}^*(S)$ be a stability condition on $S$. Then there is a positive integer $N$ such that $\sigma_n$ is $v$-generic for any integer $n \geq N$.
By Lemma \[Bri finite\], the set $$\Gamma:=\{w \in \widetilde{H}^{1,1}(\D_X,\ZZ) \mid w^2 \geq -2, |(\Omega_1,w)|\leq 1 \}$$ is a finite set. For $w \in \widetilde{H}^{1,1}(\D_X,\ZZ)$, let $w_0$ be the $H^0$-part of $\tilde{\varphi}(w)\in \widetilde{H}^{1,1}(S,\ZZ)$. We define $N:=\mathrm{max}\{|w_0|\mid w \in \Gamma\}+1$. Take an integer $n \geq N$. Let $E$ be a $\sigma_n$-stable object with $v(E)=v$ and $\phi_{\sigma_n}(E)=1$. By the definitions of $\Omega_n$ and $v$, we have $Z_n(v(E))=-1$. Assume that there is a subobject $A$ of $E$ in $\A_n$ such that $A$ is $\sigma_n$-stable and $\phi_{\sigma_n}(A)=1$. Denote $(r,c,m):=\tilde{\varphi}(v(A))$. Due to $\phi_{\sigma_n}(A)=1$, we have $h \cdot c=0$. Since $h$ is ample and $\mathrm{sign}(\mathrm{NS}(S))=(1,\rho(S))$, we have $c^2<-2$. If $r=0$, then $v(A)^2=c^2<-2$ holds and this is contradiction. So the integer $r$ is not zero. Since $Z_n(A)$ and $Z_n(E/A)$ are negative real numbers and $Z_n(A)+Z_n(E/A)=-1$, we have $|Z_n(v(A))|\leq1$. Denote $(r,c,m):=\tilde{\varphi}(v(A))$. Then we obtain $$\begin{aligned}
Z_{n}(v(A))
&=& (\Omega_n, v(A))\\
&=& (e^{inh},(r,c,m))\\
&=& -m+\frac{1}{2}ndr\\
&=& (e^{ih},(rn,c,m))\\
&=& (\Omega_1,\tilde{\varphi}(rn,c,m)). \end{aligned}$$ So $\tilde{\varphi}(rn,c,m) \in \Gamma$ holds. Note that $|rn| \geq N$ holds. By the definition of $N$, this is contradiction. Therefore, $E$ is a $\sigma_n$-stable object.
We choose a $v$-generic stability condition $\sigma_n$ such that $\sigma_n$ is fixed by the action of ${\mathrm{Aut}}(X,K)$.
\[good heart\] Let $n$ be a positive integer. There is a stability condition $\sigma_n=(Z_n,\A_n) \in {\mathrm{Stab}}^*(\D_X)$ such that $f_*\sigma_n=\sigma_n$ holds for any automorphism $f \in {\mathrm{Aut}}(X,K)$.
Recall that we have the isometry $\tilde{\varphi}|_{L_K}: L_K {\xrightarrow{\sim}}L_h$. Since $e^{inh} \in L_h\otimes \CC$, the class $\Omega_n$ is contained in $L_K \otimes \CC$. Since $A_2 \subset L_K$, there are no $(-2)$ classes in $L^\perp_K$ by Theorem \[BLMS stability\] and Proposition \[A2\]. So the intersection $\P^+_0(\D_X) \cap (L_K \otimes \CC)$ is path-connected. Fix $0<\alpha<1/4$. Take a path $\gamma$ from $\pi_X(\sigma_\alpha)$ to $\Omega_n$. By Theorem \[BriK3\], the path $\gamma$ has the unique lift $\tilde{\gamma}: [0,1] \to {\mathrm{Stab}}^*(S)$ such that $\tilde{\gamma}(0)=\sigma_\alpha$. By the definition of $L_K$ and ${\mathrm{Aut}}(X,K)$, $f^*$ acts on the lattice $L_K$ trivially. By Proposition \[fix cubic\], $f_* \circ \tilde{\gamma}$ is also a lift of $\gamma$ starting from $\sigma_\alpha$. We define the stability condition $\sigma_n=(Z_n,\A_n) \in {\mathrm{Stab}}^*(\D_X)$ by $\sigma_n:=\tilde{\gamma}(1)$. By the uniqueness of lifts, we have $f_* \sigma_n=\sigma_n$ for any automorphism $f \in {\mathrm{Aut}}(X,K)$.
Fix a sufficiently large integer $n>0$ as in Lemma \[generic\]. Take a stability condition $\sigma=(Z,\A):=\sigma_n$ in Proposition \[good heart\]. Since $v$ is a primitive isotropic and $\sigma$ is $v$-generic, the moduli space $M_\sigma(v)$ of $\sigma$-stable objects with Mukai vector $v$ is a K3 surface. There is an isotropic $v' \in \widetilde{H}^{1,1}(\D_X,\ZZ)$ such that $(v,v')=-1$. Hence, $M_\sigma(v)$ is the fine moduli space. Take an universal family $\U \in D^b(M_\sigma(v) \times X)$ of $M_\sigma(v)$ over $X$. Then we have the equivalence $\Phi_\U: D^b(M_\sigma(v)) {\xrightarrow{\sim}}\D_X$. By Theorem 1.3 in [@BM1], we have the ample divisor $l_\sigma$ on $M_\sigma(v)$ such that $l_\sigma \cdot C=\mathrm{Im}Z(\Phi_\U(\mathcal{O}_C))$ holds for any curve $C$ on $M_\sigma(v)$. We denote the ample divisor $l_\sigma/n$ by $\omega$.
\[isomorphism\] There are an isomorphism $t: M_\sigma(v) {\xrightarrow{\sim}}S$ and the universal family $\U \in D^b(M_\sigma(v)\times X)$ of $M_\sigma(v)$ over $X$ such that $t^*h=\omega$ and $t_*=\tilde{\varphi}\circ \Phi^H_\U$.
By the definition of the Mukai vector $v$, we have $\tilde{\varphi} \circ \Phi^H_\U(0,0,1)=(0,0,1)$. Since $(\tilde{\varphi} \circ \Phi^H_\U)^{-1}(1,0,0)$ is an isotropic and $((\tilde{\varphi} \circ \Phi^H_\U)^{-1}(1,0,0), (0,0,1))=-1$, there is a line bundle $L$ on $M_\sigma(v)$ such that $(\tilde{\varphi} \circ \Phi^H_\U)^{-1}(1,0,0)=(1,L,L^2/2)$. Equivalently, we have $\tilde{\varphi} \circ \Phi^H_\U(1,L,L^2/2)=(1,0,0)$. Let $p_M: M_\sigma(v) \times X \to M_\sigma(v)$ be the projection. Replacing $\U$ by $\U \otimes p^*_M L^{-1}$, we may assume that $\tilde{\varphi} \circ \Phi^H_\U(1,0,0)=(1,0,0)$ holds. The restriction of $\tilde{\varphi} \circ \Phi^H_\U$ to the second cohomology group $H^2(M_\sigma(v),\ZZ)$ of $M_\sigma(v)$ is the Hodge isometry $\tilde{\varphi} \circ \Phi^H_\U|_{H^2(M_\sigma(v),\ZZ)}: H^2(M_\sigma(v),\ZZ) {\xrightarrow{\sim}}H^2(S,\ZZ)$. Take a curve $C$ on $M_\sigma(v)$. We have $$\begin{aligned}
l_\sigma \cdot C
&=& Z(\Phi_\U(\mathcal{O}_C))\\
&=& (\mathrm{Im}(\Omega_n), v(\Phi_\U(\mathcal{O}_C)))\\
&=& (\tilde{\varphi}^{-1}(0,nh,0),\Phi^H_\U(v(\mathcal{O}_C)))\\
&=& ((\tilde{\varphi} \circ \Phi^H_\U)^{-1}(0,nh,0), (0,C,\chi(\mathcal{O}_C))\\
&=& (\tilde{\varphi} \circ \Phi^H_\U)^{-1}(0,nh,0) \cdot C. \end{aligned}$$ So we have $(\tilde{\varphi} \circ \Phi^H_\U)^{-1}(0,h,0)=(0,\omega,0)$, equivalently, $\tilde{\varphi} \circ \Phi^H_\U(0,\omega,0)=(0,h,0)$. By the Torelli theorem for K3 surfaces, there is an isomorphism $t: M_\sigma(v) {\xrightarrow{\sim}}S$ such that $t_*=\tilde{\varphi} \circ \Phi^H_\U$. In particular, we have $t^*h=\omega$.
From now on, take the universal family $\U$ as in Proposition \[isomorphism\]. By Proposition \[isomorphism\], we obtain the following.
\[sublattice L\] Let $L_\omega$ be the sublattice $H^0(M_\sigma(v),\ZZ) \oplus \ZZ \cdot \omega \oplus H^4(M_\sigma(v),\ZZ)$ of $H^*(\M_\sigma(v),\ZZ)$. The Hodge isometry $\Phi^H_\U: H^*(M_\sigma(v),\ZZ) {\xrightarrow{\sim}}H^*(\D_X,\ZZ)$ induces the isometry $$\Phi^H_\U|_{L_\omega}:L_\omega {\xrightarrow{\sim}}L_K$$ and the Hodge isometry $$\Phi^H_\U|_{L^\perp_\omega}: L^\perp_\omega {\xrightarrow{\sim}}L^\perp_K.$$
The equivalence $\Phi_\U: D^b(M_\sigma(v)) {\xrightarrow{\sim}}\D_X$ induces the isomorphism between the distinguished connected components of the spaces of stability conditions.
\[dist connected\] For a stability condition $\tau:=(W,\B) \in {\mathrm{Stab}}(\D_X)$, we define the stability condition $\Phi^*_\U\tau \in {\mathrm{Stab}}(M_\sigma(v))$ by $$\Phi^*_\U\tau:=(W \circ \Phi^H_\U, \Phi^{-1}_\U(\B)).$$ Then we have the isomorphism $$\Phi^*_\U: {\mathrm{Stab}}^*(\D_X) {\xrightarrow{\sim}}{\mathrm{Stab}}^*(M_\sigma(v)).$$
Let $\sigma$ be the stability condition as in Lemma \[generic\]. By the construction of $\sigma$ and Proposition \[isomorphism\], we have $Z\circ \Phi^H_\U(-)=(e^{in\omega},-)$. By Lemma \[generic\], for any point $[E] \in M_\sigma(v)$, we have the $\sigma$-stable object $\Phi_\U(\mathcal{O}_{[E]}) \simeq E$. By Proposition 10.3 in [@Bri08], we have $\Phi^*_\U\sigma=\sigma_{0, n\omega}$ as in \[stability K3\]. So we obtain the isomorphism $\Phi^*_\U: {\mathrm{Stab}}^*(\D_X) {\xrightarrow{\sim}}{\mathrm{Stab}}^*(M_\sigma(v))$.
Using results in the previous section and this subsection, we obtain the following theorem.
\[group iso cubic K3\] There is a stability condition $\sigma_X \in {\mathrm{Stab}}^*(M_\sigma(v))$ such that we have the isomorphism $$(-)_\U: {\mathrm{Aut}}(X) {\xrightarrow{\sim}}{\mathrm{Aut}}(D^b(M_\sigma(v)),\sigma_X), f \mapsto f_\U:=\Phi^{-1}_\U \circ f_* \circ \Phi_\U$$ of groups. Moreover, the restriction of $(-)_\U$ to the labeled automorphism group ${\mathrm{Aut}}(X,K)$ of $(X,K)$ induces the isomorphism $$(-)_\U : {\mathrm{Aut}}(X,K) {\xrightarrow{\sim}}{\mathrm{Aut}}(M_\sigma(v), \omega)$$ of groups.
Fix a real number $0<\alpha<1/4$. Put $\sigma_X:=\Phi^*_\U \sigma_\alpha \in {\mathrm{Stab}}(\D_X)$. By Proposition \[dist connected\], the stability condition $\sigma_X$ is in the distinguished connected component ${\mathrm{Stab}}^*(M_\sigma(v))$. By Proposition \[symmetry lattice\] and Theorem \[K3 sigma model\], we have the isomorphism $$(-)_\E: {\mathrm{Aut}}(X) {\xrightarrow{\sim}}{\mathrm{Aut}}(D^b(M_\sigma(v)),\sigma_X), f \mapsto f_\U:=\Phi^{-1}_\U \circ f_* \circ \Phi_\U.$$ Let $f \in {\mathrm{Aut}}(X,K)$ be an automorphism. Due to Proposition \[good heart\] and $v \in L_K$, for an automorphism $f \in {\mathrm{Aut}}(X,K)$, we obtain the automorphism $$f_{\sigma,v}: M_\sigma(v) {\xrightarrow{\sim}}M_\sigma(v), [E] \mapsto [f_*E].$$ Take a point $[E] \in M_\sigma(v)$. By Proposition \[good heart\] and $v \in L_K$, we have $$\Phi^{-1}_\U \circ f_* \circ \Phi_\U(\mathcal{O}_{[E]}) \simeq \mathcal{O}_{[f_*E]}.$$ There is a line bundle $L$ on $M_\sigma(v)$ such that $\Phi_\U \circ f_{\sigma,v*} \circ (-\otimes L) \simeq f_* \circ \Phi_\U $. Since $\Phi^H_\U(1,0,0)= \tilde{\varphi}(1,0,0)$ in $L_K$, we have $\Phi^H_\U(1,L,L^2/2)=\Phi^H_\U(1,0,0)$. So $L=0$ holds. Due to $\Phi^H_\U(0,\omega,0)=\tilde{\varphi}(0,h,0)$ in $L_K$, we obtain $f^*_{\sigma,v}\omega=\omega$. Hence, we have $f_{\sigma,v*}=f_\U$. We shall prove that $(-)_\U: {\mathrm{Aut}}(X,K) \hookrightarrow {\mathrm{Aut}}(M_\sigma(v),\omega)$ is surjective. Take an automorphism $g \in {\mathrm{Aut}}(M_\sigma(v), \omega)$. By Proposition \[prim\] and Remark \[sublattice L\], $\Phi^H_\U \circ g_* \circ (\Phi^{-1}_\U)^H$ induces the Hodge isometry $\Phi^H_\U \circ g_* \circ (\Phi^{-1}_\U)^H|_{A^\perp_2}: H^4_{\mathrm{prim}}(X,\ZZ) {\xrightarrow{\sim}}H^4_{\mathrm{prim}}(X,\ZZ)$. By the Torelli theorem for cubic fourfolds [@Voi], there is an automorphism $f \in {\mathrm{Aut}}(X)$ such that $f_*=\Phi^H_\U \circ g_* \circ (\Phi^{-1}_\U)^H|_{A^\perp_2}$. By Remark \[sublattice L\], we have $f \in {\mathrm{Aut}}(X,K)$. Then the equality $f_{\sigma,v*}=g_*$ holds. By the Torelli theorem for K3 surfaces, we obtain $f_{\sigma,v}=g$. Therefore, the homomorphism $(-)_{\U}: {\mathrm{Aut}}(X,K) \hookrightarrow {\mathrm{Aut}}(M_\sigma(v),\omega)$ is surjective.
Therefore, we have obtained Theorem \[intromain2\] from Proposition \[isomorphism\], Remark \[sublattice L\] and Theorem \[group iso cubic K3\].
Symplectic automorphisms of cubic fourfolds and associated K3 surfaces
======================================================================
In this section, we study relations between symplectic automorphisms of cubic fourfolds and autoequivalences of derived categories of K3 surfaces.
Let $X$ be a cubic fourfold and put $G:={\mathrm{Aut}}_{\mathrm{s}}(X)$. We define the coinvariant sublattice $S_G(X)$ of $H^4(X,\ZZ)$ by $S_G(X):=(H^4(X,\ZZ)^G)^\perp$. Let $T_X$ be the orthogonal complement of $H^{2,2}(X,\ZZ)$ in $H^4(X,\ZZ)$. By Proposition \[prim\], we have $T_{\D_X} \simeq T_X(-1)$.
For a symplectic automorphism $f \in G$ of $X$, we have $f^*|_{T_X}=\mathrm{id}_{T_X}$.
The proof is the same as the case of K3 surfaces. See [@HuyK3] (Remark 3.3 in Section 3 and Remark 1.2 in Section 15).
\[pic vs coinv\] The inequality $\rho(\D_X) \geq \mathrm{rk}(S_G(X))$ holds.
Since $H^2 \in H^4(X,\ZZ)^G$ and $T_X \subset H^4(X,\ZZ)^G$, we have $S_G(X) \subset H^{2,2}_{\mathrm{prim}}(X,\ZZ)$.
Existence of non-trivial symplectic automorphisms of $X$ is the strong constraint for the Picard number of the Kuznetsov component.
If $G$ is not isomorphic to the trivial group $1$ or the cyclic group $\ZZ/2\ZZ$ of order two, then we have $\rho(\D_X) \geq 12$.
Assume that $G$ is not isomorphic to the trivial group $1$ or the cyclic group $\ZZ/2\ZZ$ of order two. By Theorem 1.2 in [@LZ], we have $ \mathrm{rk}(S_G(X)) \geq 12$. By Lemma \[pic vs coinv\], we obtain the inequality $\rho(\D_X) \geq 12$.
We note the following.
\[symplectic exist\] If $G$ is not isomorphic to the trivial group $1$ or $\ZZ/2\ZZ$, then there exists the unique K3 surface $S$ such that $\D_X \simeq D^b(S)$.
By Corollary 2.10 in [@Mor], there is the unique K3 surface $S$ such that $T_{\D_X}$ has the unique primitive embedding $T_{\D_X} \hookrightarrow H^2(S,\ZZ)$ up to isometries of $H^2(S,\ZZ)$ such that $T_{\D_X}=T_S$. Now, we have two primitive embeddings $$T_{\D_X} \hookrightarrow H^2(S,\ZZ) \hookrightarrow H^*(S,\ZZ),$$ $$T_{\D_X} \hookrightarrow H^*(\D_X,\ZZ).$$ For the primitive embedding $T_{\D_X} \hookrightarrow H^2(S,\ZZ) \hookrightarrow H^*(S,\ZZ)$, we have $U \simeq H^0(S,\ZZ) \oplus H^4(S,\ZZ) \subset T^\perp_{\D_X}$. By Remark 1.13 in [@HuyK3], there is a Hodge isometry $H^*(S,\ZZ) \simeq H^*(\D_X,\ZZ)$. So we have $U \subset \widetilde{H}^{1,1}(\D_X,\ZZ)$. By Theorem \[derivedTorelli\] and the uniqueness of $S$, we obtain $\D_X \simeq D^b(S)$.
The following is the list of orders of finite symplectic automorphisms of K3 surfaces.
\[order K3\] Let $S$ be a K3 surface. If $f$ is a symplectic automorphism of $S$ with the finite order, then we have $1 \leq \mathrm{ord}(f) \leq 8$.
We can find examples of finite symplectic autoequivalences of K3 surfaces via symplectic automorphisms of cubic fourfolds. In some cases, they are not conjugate to symplectic automorphisms of K3 surfaces.
\[Fermat\] Let $X$ be the Fermat cubic fourfold, that is defined by the equation $$x^3_1+x^3_2+x^3_3+x^3_4+x^3_5+x^3_6=0.$$ By Theorem 1.8 (1) in [@LZ], the Fermat cubic fourfold $X$ is the unique cubic fourfold with a symplectic automorphism $f_9$ of order $9$. By Theorem \[symplectic exist\], there is the unique K3 surface $S$ such that $\D_X \simeq D^b(S)$. Then $f_9$ induces the symplectic autoequivalence $\Phi_9 \in {\mathrm{Aut}}_{\mathrm{s}}(D^b(S))$ of order $9$. By Remark \[order K3\], $\Phi_9$ is not conjugate to symplectic automorphisms of $S$.
\[Klein\] Let $X$ be the Klein cubic fourfold, that is defined by the equation $$x^3_1+x^2_2x_3+x^2_3x_4+x^2_4x_5+x^2_5x_6+x^2_6x_2=0.$$ This is the triple cover of $\mathbb{P}^4$ branched along the Klein cubic threefold in [@Adl]. By Theorem 1.8 (5) in [@LZ], the Klein cubic fourfold $X$ is the unique cubic fourfold with a symplectic automorphism $f_{11}$ of order $11$ and its symplectic automorphism group is the finite simple group $L_2(11)$ (cf. [@Adl]). By Theorem \[symplectic exist\], there is the unique K3 surface $S$ such that $\D_X \simeq D^b(S)$. Then $f_{11}$ induces the symplectic autoequivalence $\Phi_{11} \in {\mathrm{Aut}}_{\mathrm{s}}(D^b(S))$ of order $11$. By Remark \[order K3\], $\Phi_{11}$ is not conjugate to symplectic automorphisms of $S$.
See Theorem 1.2 and 1.8 in [@LZ] for other interesting cubic fourfolds.
[99]{}
N. Addington, *[On two rationality conjectures for cubic fourfolds]{}, [Mathematical Research Letters 23 (2016), pp. 1-13, arXiv:1405.4902v3]{}*
N. Addington, *New derived symmetries of some hyperKähler varieties, Alg. Geom. 3(2): 223-260, (2016).*
A. Adler, *On the automorphism group of certain cubic threefold, Amer. J. Math. 199 (1978), no. 6, 1275-1280.*
N. Addington, B. Hassett, Y. Tschinkel, A. Barilly-Alvarado, *[Cubic fourfolds fibered in sextic delPezzo surfaces]{}, [arXiv: 1606.05321]{}*
N. Addington and R. Thomas, *[Hodge theory and derived categories of cubic fourfolds]{}, [Duke Math. J. 163(10):1885-1927. 2014]{}, [arXiv:1211.3758]{}.*
M. Barr, C. Wells, *Toposes, triples and theories, Reprints in Theory and Applications of Categories, No. 12, 2005.*
A. Beauville, *Some remarks on Kähler manifolds with $c_1=0$, Progr. Math. vol. 39 (1983), 1-26.*
A. Bayer, M. Lahoz, E. Macrì and P. Stellari, *Stability conditions on Kuznetsov components, Appendix by A. Bayer, E. Macrì, P. Stellari and X. Zhao, arXiv:1703.10839.*
A. Bayer, M. Lahoz, E. Macrì, H. Nuer, A. Perry and P. Stellari, *Stability conditions in families, arXiv:1902.08184.*
A. Bayer and E. Macrì, *Projectivity and birational geometry of Bridgeland moduli spaces, J. Amer. Math. Soc. 27 (2014), no. 3, 707-752.*
A. Beauville and R. Donagi, *[La variété des droites d’une hypersurface cubique de dimension 4]{}, [C. R. Acad. Sci. Paris. Sér. I Math.]{} **[301]{} [(1985)]{}, 703-706.***
T. Bridgeland, *[Stability conditions on triangulated categories,]{} Ann. of Math 166 (2007), 317-345.*
T. Bridgeland, *Stability conditions on K3 surfaces, Duke Math. J. 141 (2008), 241-191.*
M. C. N. Cheng, F. Ferrari, S. M. Harrison, N. M. Paquette, *Landau-Ginzburg orbifolds and symmetries of K3 CFTs, J. High Energy Phys. 1701 (2017) 046.*
A. D. Elagin, *Cohomological descent theory for a morphism of stacks and for equivariant derived categories, Sb. Math. 202 (2011), no. 3-4, 495-526.*
T. Eguchi, H. Ooguri and Y. Tachikawa, *Note on the K3 surfaces and the Mathieu group $M_{24}$, Exper. Math. 20 (2011) 91-96.*
M. R. Gaberdiel, S. Hohenegger and R. Volpato, *Symmetries of K3 *$\sigma$-models, Commun. Num. Theor. Phys. 6 (2012) 1.**
B. Hassett, *[Special cubic fourfolds]{}, [Compositio Math.]{} **[120]{} [(2000), 1-23]{}.***
B. Hassett and Y. Tschinkel, *Hodge theory and lagrangian planes on generalized Kummer fourfolds, Mosc. Math. J. vol. 13(2013), no. 1 33-56.*
D. Huybrechts, *On derived categories of K3 surfaces, symplectic automorphism groups and the Conway group, Adv. Stud. Pure Math. Development of Moduli Theory, Tokyo: Mathematical society of Japan (2016), 387-405.*
D. Huybrechts, *Lectures on K3 surfaces, (Cambridge Studies in Advanced Mathematics). Cambridge: Cambridge University Press.(2016) doi:10.1017/CBO9781316594193*
A. Kuznetsov, *Derived categories of quadric fibrations and intersections of quadrics, Adv. Math, 218(5): 1340-1369 (2008). A. Kuznetsov, *[Derived categories of cubic fourfolds]{}, [in: Cohomological and geometric approaches to rationality problems, Progr.Math.]{} **[282]{} [(2010), 219-243.]{}****
A. Kuznetsov and D. Markushevich, *Symplectic structures on moduli spaces of sheaves via Atiyah class, Journal of geometry and physics, Volume 59, Issue 7 (2009), 843-860.*
R. Laza, *Maximally algebraic potentially irrational cubic fourfolds, arXiv:1805.04063.*
C. Li, L. Pertusi and X. Zhao, *Twisted cubics on cubic fourfolds and stability conditions, arXiv:1802.01134.*
R. Laza, Z. Zheng, *Automorphisms and periods of cubic fourfolds, arXiv:1905.11547.*
E. Macrì, P. Stellari, *Fano varieties of cubic fourfolds containing a plane, Math Ann, 354, (2012), 1147-1176.*
D. R. Morrison, *On K3 surfaces with large Picard number, Invent. Math. 75 (1984), no. 1, 105-121.*
S. Mukai, *Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math. 94 (1988), no. 1, 183-221.*
V. Nikulin, *Integer symmetric bilinear forms and their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 111-177, 238.*
V. Nikulin, *Finite groups of automorphisms of Kählerian surfaces of type K3, (English translation). Moscow Math. Soc. 38, 71-137 (1980).*
D. Orlov, *Derived categories of coherent sheaves and triangulated categories of singularities , Algebra, arithmetic and geometry: in honor of Yu. I. Manin. Vol. II, 503-531, Progr. Math., 270, Birkhäuger Boston, Inc., Boston, MA, 2009.*
G. Ouchi, *[Lagrangian embeddings of cubic fourfolds containing a plane,]{} [Compositio Math.]{} Volume 153, Issue 5 (2017), pp. 947-972.*
M. Verbitsky, *Cohomology of compact hyperKähler manifolds and its applications, Geom. Funct. Anal. 6 (1996) 601-611.*
C. Voisin, *Théorème de Torelli pour les cubiques de $\mathbb{P}^5$, Invent. Math., 86(3) (1986), 577-601.*
| 2023-12-15T01:27:17.760571 | https://example.com/article/5425 |
FILED
United States Court of Appeals
PUBLISH Tenth Circuit
UNITED STATES COURT OF APPEALS December 30, 2014
Elisabeth A. Shumaker
TENTH CIRCUIT Clerk of Court
UNITED STATES OF AMERICA,
Plaintiff - Appellee,
v. No. 13-5045
JAMES HOWARD BURNS,
Defendant - Appellant.
Appeal from the United States District Court
for the Northern District of Oklahoma
(D.C. No. 4:12-CR-00053-CVE-1)
Barry L. Derryberry, Office of the Federal Public Defender, Tulsa,
Oklahoma (Julia L. O’Connell, Federal Public Defender, and Stephen J.
Greubel, Assistant Federal Public Defender, Tulsa, Oklahoma, on the
briefs) for Appellant-Defendant.
Leena Alam, Assistant United States Attorney, Tulsa, Oklahoma (Danny C.
Williams, Sr., United States Attorney, and Matthew P. Cyran, Assistant
United States Attorney, Tulsa, Oklahoma, on the brief) for Appellee-
Plaintiff.
Before GORSUCH, SEYMOUR, and BACHARACH, Circuit Judges.
BACHARACH, Circuit Judge.
Mr. James Burns was convicted of possession and attempted
possession of child pornography. See 18 U.S.C. § 2252(a)(4)(B), (b)(2).
He was sentenced to 63 months in prison, followed by 5 years of
supervised release. This appeal involves one of the conditions of his
supervised release. That condition requires approval of the probation
department before Mr. Burns can have any contact with minors, including
his youngest daughter (S.B.).
This restriction intrudes on Mr. Burns’s constitutional right to
familial association. Because of this intrusion on a constitutional right,
the district court should have made particularized findings before
restricting Mr. Burns’s contact with his daughter. Because the district
court failed to make these findings, we reverse on plain error. 1
I. Plain Error
Because Mr. Burns did not object to the condition in district court,
we review only for plain error. United States v. Mike, 632 F.3d 686, 691
(10th Cir. 2011). To establish plain error, Mr. Burns must show an error
that is plain, that affects substantial rights, and that seriously affects the
fairness, integrity, or public reputation of judicial proceedings. United
States v. Gonzalez-Huerta, 403 F.3d 727, 732 (10th Cir. 2005).
1
The district court’s error was understandable because of defense
counsel’s failure to object.
2
Mr. Burns has established plain error. The district court restricted
his contact with a family member without making the constitutionally
required findings. If the court had addressed these findings, there is a
reasonable probability that the court wouldn’t have restricted Mr. Burns’s
contact with S.B. Thus, we vacate the restriction that requires probation
office approval for Mr. Burns to contact his daughter and remand for
reconsideration of this restriction.
A. Error
Mr. Burns has shown that the district court made an error. 2
A district court has broad discretion to impose conditions on
supervised release. United States v. Mike, 632 F.3d 686, 692 (10th Cir.
2011). But when a court imposes a special condition that invades a
fundamental right or liberty interest, the court must justify the condition
with compelling circumstances. Id. at 1284.
Mr. Burns has a fundamental liberty interest that is invaded by the
special condition, for “a father has a fundamental liberty interest in
maintaining his familial relationship with his [child].” United States v.
Edgin, 92 F.3d 1044, 1049 (10th Cir. 1996). Thus, the circumstances had
2
In oral argument, the government argued for the first time that the
claim is precluded under the invited error doctrine. Oral Arg. 17:28-44.
But, the government had not raised this argument in its brief. Thus, the
government waived reliance on the invited error doctrine. See United
States v. Rivera-Nevarez, 418 F.3d 1104, 1112 n.2 (10th Cir. 2005)
(“[I]ssues raised for the first time at oral argument are waived.”).
3
to be compelling before the district court could restrict Mr. Burns’s contact
with S.B. See United States v. Lonjose, 663 F.3d 1292, 1303 (10th Cir.
2011) (stating that a similar condition interfered with the right of familial
association); Edgin, 92 F.3d at 1049 (remanding for reconsideration of a
sentence when the court prohibited contact without justification).
The district court failed to make the required findings, as the
government conceded in oral argument. Oral Arg. 23:38-47; 31:00-22.
Thus, the district court erred by failing to justify the invasion on Mr.
Burns’s fundamental right of familial association.
B. Plain
Mr. Burns must also show that the error was plain. “An error is plain
if it is ‘clear and obvious under current law.’” United States v. Brown, 316
F.3d 1151, 1158 (10th Cir. 2003) (quoting United States v. Fabiano, 169
F.3d 1299, 1302-03 (10th Cir. 1999)).
Our precedents unambiguously require supporting findings when
courts impose special conditions of supervised release. Edgin, 92 F.3d at
1049; United States v. Smith, 606 F.3d 1270, 1283 (10th Cir. 2010); United
States v. Hahn, 551 F.3d 977, 982-83 (10th Cir. 2008). This precedent was
“clear and obvious” when Mr. Burns was sentenced in April 2013. Thus,
the error was plain under current law.
4
C. Affects Substantial Rights
Mr. Burns has satisfied his burden to show that the court’s error
affects his substantial rights. But for the district court’s error, a
reasonable probability exists that the court would not have restricted Mr.
Burns’s contact with S.B.
An error affects substantial rights if there is a reasonable probability
that the error affected the outcome of the proceedings. United States v.
Marcus, 560 U.S. 258, 262 (2010). In the sentencing context, we ask: Is
there a reasonable probability that but for the court’s error, Mr. Burns
would have received a lesser sentence? See United States v. Trujillo-
Terrazas, 405 F.3d 814, 819 (10th Cir. 2005) (reviewing an application of
the sentencing guidelines for plain error).
To resolve this question, we analyze the requirements for imposing a
special condition of supervised release. Courts may impose a special
condition if two requirements are met:
1. The condition is reasonably related to the nature and
circumstances of the offense and the history and characteristics
of the defendant.
2. The condition involves no greater deprivation of liberty than is
reasonably necessary.
United States v. Hahn, 551 F.3d 977, 983 (10th Cir. 2008). To be valid,
the condition must satisfy both requirements. Hahn, 551 F.3d at 983-84.
5
The condition likely fails under the second requirement. At
sentencing, neither the pretrial services officer nor government counsel
spoke of a need to restrict Mr. Burns’s right to visit his daughter. 3 As a
result, if the district court had addressed the issue, it probably would not
have restricted contact with S.B.
Because of the burden on Mr. Burns’s constitutional right of familial
association, the restriction is valid only if Mr. Burns presents a danger to
S.B. United States v. Lonjose, 663 F.3d 1292, 1303 (10th Cir. 2011). But,
the record is not sufficient for us to make this determination in the first
instance. There is no evidence that Mr. Burns has abused or sexually
molested children, and the record indicates that Mr. Burns has a positive
relationship with four of his five children. II App. at 8-9. Thus, there was
little to support a restriction on Mr. Burns’s contact with S.B.
The government argues that the condition was appropriate because it
provides an alternative for Mr. Burns “to pursue contact with his daughter,
while still providing needed protection for her and other children.”
Aplee’s Br. at 16. But the district court probably would have rejected this
3
The presentence report referred to the possibility of special
conditions, but did not identify them. Instead, the presentence report
referred to the “Special Sex Offender Conditions.” These conditions were
not listed in the presentence report. Instead, the document cited the
court’s website for a list of the conditions. The website contains a list of
potential special conditions listed in a general order issued by the district
court in 2008. See In re Special Condition of Probation & Supervised
Release, Gen. Order No. 08-3 (N.D. Okla. Mar. 28, 2008).
6
alternative in light of the absence of any evidence or argument on a need to
protect S.B. from Mr. Burns. See Lonjose, 663 F.3d at 1303 (stating that a
restriction on contact with the defendant’s young relatives violated the
Constitution notwithstanding the allowance of contact upon permission by
a probation officer); 4 see also United States v. Doyle, 711 F.3d 729, 736
(6th Cir. 2013) (holding that the imposition of special conditions of
supervised release for a sex offender, without the required findings,
affected the defendant’s substantial rights because a reasonable probability
existed that the court may not have imposed the special conditions if it had
explained the basis for the conditions or made sure that the record
supported the conditions); United States v. Perazza-Mercado, 553 F.3d 65,
78 (1st Cir. 2009) (holding that the erroneous imposition of a condition of
supervised release, which banned possession of all pornography by a sex
offender, affected the defendant’s substantial rights because the lack of
explanation for the ban created a reasonable probability that the court
might not have imposed the condition if it had explained the basis for the
condition or made sure that “the record illuminated the basis for the
condition”).
4
The author of the Lonjose opinion was the judge who imposed Mr.
Burns’s sentence.
7
D. Seriously Affects Judicial Proceeding
Finally, Mr. Burns has shown that the error “‘seriously affects the
fairness, integrity, or public reputation of judicial proceedings.’” United
States v. Rosales-Miranda, 755 F.3d 1253, 1262 (10th Cir. 2014) (quoting
United States v. Figueroa-Labrada, 720 F.3d 1258, 1266 (10th Cir. 2013)).
As discussed above, the district court likely would have softened the
contact restrictions if the issue had been raised. As a result, the error
seriously affected the fairness and integrity of the proceedings. See United
States v. Doyle, 711 F.3d 729, 736 (6th Cir. 2013) (holding that the
imposition of special conditions of supervised release for a sex offender,
without the required findings, affected the “fairness, integrity, or public
reputation of the proceedings because these conditions were likely more
severe than the ones the district court would have imposed had it fulfilled
its obligation to explain its reasoning for imposing any special
conditions”).
Because the error undermines the fairness, integrity or public
reputation of judicial proceedings, we conclude that Mr. Burns has
satisfied the final prong of plain-error review.
II. Conclusion
The district court committed plain error by restricting Mr. Burns’s
contact with his daughter without the required findings. Thus, we remand
8
for reconsideration of the supervised-release condition requiring Mr. Burns
to obtain permission from the probation office before he can contact S.B.
9
| 2024-03-08T01:27:17.760571 | https://example.com/article/3789 |
This morning there was a hole in Miles’ poop bag. I’m just gonna let that sink in for a minute, let the ramifications of such shake out like a snow globe into perfect clarity.
What I’m trying to say is that whilst picking up after my dog, grabbing aggressively so as not to leave any trace of his butt leavings on our neighbor’s sidewalk, there was an unforeseen tear and my right forefinger came in contact with fecal matter the consistency of peanut butter.
Yes, I screamed.
Totally unfazed, my 19-pound Pug pranced off to the next bush all the more perfect to pee on while I was left gagging and dry-heaving. Call it Karma (but don’t call it bad Karma because that’s redundant).
Anyway, with three more blocks till home, I was disgusted with life in general. Miles, of course, was peachy. I try not to take things out on him even when they’re sorta kinda his fault, a practice of patience that makes it less stupid when people suggest that “before you get a kid, get a dog.”
Like when he’s left little dog penis stains on a shirt I was planning to wear that he just had to lay on because it was on the floor, or when he bit a huge rip in a trash bag loaded with moldy spaghetti because obviously the smell was intoxicating. Instead of yelling, I find my center and usually a mop.
But not this time. Not with butt barf all on the finger I use to do stuff with. So when we got home, I stomped to the bathroom and slammed the door, vigorously rubbing three types of soap (dishwashing, fancy, and laundry) into my hands.
It wasn’t long before I heard the familiar whimper at the door. Miles HATES it when I’m in the bathroom alone. If I don’t let him watch me pee (because that’s all I do in there) then he thinks I’ve abandoned him to life alone out in the living room for a gut-wrenching commercial break. He can’t stand it.
So, of course, I let him in.
I don’t know what it is about dog owner/dog time in the toilet but in our house it trumps the dog park, and it’s somehow the opposite of weird. He watches and I wee. Or I wash my face, put on my makeup, blow my nose, whatever.
The bathroom is the only room he can’t stand to not be a part of. I’m typing this from our office/storage/guestroom while Miles is chilling on his mound of blankets next to the TV in the living room totally sated. But if I make a move for the bathroom, I’ll hear the tap tap tap of his nails against the hardwood before I can get my pants down (OK, fine I don’t have pants on, but you get the picture).
At first, I thought I was engendering this behavior, having grown up in a totally naked house and being therefore quite boundaryless.
“Close the friggin’ door!” is a familiar exclamation I’ve heard from roomies to lovers.
Other pet owners, though, have told me that their dogs practically live in the poop shack. They’re mini voyeurs.
One guy told me, “Mister Tibbs would hop into the shower with me if I let him.” All you need is one “yes” to make something a trend, so I’m rolling with it. Now if only I could teach Miles to pass the toilet paper, we’d be good to go. Or is that too much?
Editor’s note: Dogster readers, does your dog come into the bathroom with you?
Got a Doghouse Confessional to share?
We’re looking for intensely personal stories from our readers about life with their dogs. E-mail confess@dogster.com, and you might become a published Dogster Magazine author! | 2024-01-08T01:27:17.760571 | https://example.com/article/4445 |
"We demand that this candidate exercise control to preserve the serenity that has been prevailing till now," Solitoki Esso, the secretary general of the ruling party, said.
Call for calm
Election officials sorted through the ballots on Friday, after calling on citizens to remain calm while results from Thursday's poll are tallied.
In a statement broadcast on state television, the independent electoral commission urged candidates and voters to "exercise patience and serenity while the commission makes every effort to centralize the results from various polling stations", the Associated Press news agency reported.
It said the commission will release provisional results by Saturday.
The race pitted six opposition candidates against the incumbent, Faure Gnassingbe, 43, who came to power in 2005 after the death of his father, General Gnassingbe Eyadema, whose dictatorial rule lasted 38 years.
Opposition candidates
Voters in the capital Lome had characterised the poll as an attempt by "the palm tree" to "uproot the maize".
The maize is the emblem of the RPT, and the palm tree is the symbol of the UFC, the main party of the divided opposition.
Gnassingbe, a former mines minister and financial adviser under his father, is seeking a second-term mandate but the opposition dismisses him as a candidate of "a system" that froze development over the past 43 years.
Gnassingbe vowed that this poll will raise Togo to new heights, on the basis of "a state of law".
With the main opposition party [the UFC] expressing concern over possible electoral fraud, forty international observers were deployed by the African Union, 130 by the European Union and 150 civilians and 146 soldiers by the Economic Community of West African States to oversee the elections.
The commission had vowed to stage a free and fair poll, devoid of violence, and there were no reports of of any incidents during the polling, police said.
Election officials were trying to prevent a situation similar to Togo's presidential election in 2005 when hundreds of people died in post-election violence.
The violence that followed the disputed vote in 2005 left up to 800 dead according to various sources, but the UN put the toll at 400 to 500 deaths.
Yet parliamentary elections two years later were peaceful, raising hopes of an end to Togo's long history of political violence and leading to the restoration of foreign aid.
Related
After 13 years, opposition parties participate in polls seen as free and fair.
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global license for no consideration to use, reuse, delete or publish comments, in
accordance with Community Rules & Guidelines and Terms and Conditions. | 2023-11-24T01:27:17.760571 | https://example.com/article/3291 |
BREAKING: At least 8 people dead as minibus plunges into ravine in HORROR crash
AT LEAST eight people have been killed and 15 seriously injured after a minibus veered off the road and plunged into a ravine at a mountainous region in western Guatemala, it has emerged.Daily Express :: News Feed | 2024-01-07T01:27:17.760571 | https://example.com/article/8750 |
Q:
Not getting data in return with AJAX
This is the data I'm trying to get
public function dohvatiZadatak($id){
$this->id = $id;
$conn = $this->connect()->prepare("SELECT * FROM zadatci WHERE id=:id");
$conn->bindParam('id', $this->id);
$conn->execute();
$zadatak = $conn->fetch(PDO::FETCH_ASSOC);
return json_encode(array('zadatak' => $zadatak));
}
Ajax request
function urediZadatak(id){
$("#uredi-zadatak-modal").modal('show');
$.ajax({
type: "GET",
url: "../zadatci/uredizadatak.php?id="+id,
data: id,
async: false,
success: function(response){
console.log(response.zadatak);
}
});
}
I'm getting nothing in return. How do I get that data I'm returning with PHP?
A:
In order for the response to be sent back from PHP you need to use echo. On your code you should add the following to the place calling dohvatiZadatak($id):
echo dohvatiZadatak($id);
And than on your front-end you can use:
function urediZadatak(id){
$("#uredi-zadatak-modal").modal('show');
$.ajax({
type: "GET",
url: "../zadatci/uredizadatak.php?id="+id,
data: id,
async: false,
success: function(response){
var parsedResponse = JSON.parse(response);
// use the response as parsedResponse.key
}
});
}
| 2024-02-21T01:27:17.760571 | https://example.com/article/3122 |
Q:
How do you get rid of page numbers on \part pages?
I am writing a book of puzzles in LaTeX, using the book class. The \thispagestyle{empty} command has successfully removed page numbers on all the \chapter pages, but so far no combination of \newpage and \thispagestyle{empty} before and/or after a \part command has been able to remove the page number at the center of the bottom of the \part page.
\newpage
\tableofcontents
\thispagestyle{empty}
\newpage
\thispagestyle{empty}
\newpage
\thispagestyle{empty}
\part{}
\thispagestyle{empty}
\chapter{}
\thispagestyle{empty}
The code above produces a table of contents page, then a blank page, then a \part page, then a blank page, then a \chapter page. There are no page numbers except on the \part page, and it appears to be incredibly reluctant to disappear, as you can see from the number of empty page commands surrounding the \part command!
Any help is appreciated.
Thank you.
A:
The first page of a part or chapter explicitly uses the page style plain. If you want to change this behaviour, you can make plain to mean the same as empty by adding the following lines to your preamble:
\makeatletter
\let\ps@plain\ps@empty
\makeatother
After this, page style plain is no longer available. A more flexible solution is to define
\makeatletter
\let\origps@plain\ps@plain
\newcommand\MakePlainPagestyleEmpty{\let\ps@plain\ps@empty}
\newcommand\MakePlainPagestylePlain{\let\ps@plain\origps@plain}
\makeatother
Now you can equate plain with empty using \MakePlainPagestyleEmpty and revert it again by \MakePlainPagestylePlain.
Note that you can also switch the default page style using
\pagestyle{empty}
and later switch back to
\pagestyle{headings}
Try the following code:
\documentclass{book}
\makeatletter
\let\origps@plain\ps@plain
\newcommand\MakePlainPagestyleEmpty{\let\ps@plain\ps@empty}
\newcommand\MakePlainPagestylePlain{\let\ps@plain\origps@plain}
\makeatother
\usepackage{blindtext}
\begin{document}
\pagestyle{empty}
\MakePlainPagestyleEmpty
\tableofcontents
\part{part}
\chapter{chapter}
\Blindtext
% Revert to standard behaviour of the book class
\chapter{chapter}
\MakePlainPagestylePlain
\pagestyle{headings}
\Blindtext
\end{document}
A:
Since you're using the book document class, all you need to do is issue the following two instructions in the preamble:
\usepackage{etoolbox}
\patchcmd{\part}{\thispagestyle{plain}}{\thispagestyle{empty}}{}{}
A full MWE:
\documentclass{book}
\usepackage{etoolbox}
\patchcmd{\part}{\thispagestyle{plain}}{\thispagestyle{empty}}{}{}
\begin{document}
\tableofcontents
\part{Uno}
\chapter{A}
\part{Due}
\chapter{B}
\chapter{C}
\end{document}
| 2023-10-12T01:27:17.760571 | https://example.com/article/3881 |
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com.badlogic.gdx.scenes.scene2d.ui.SelectBox$SelectBoxStyle: {
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com.badlogic.gdx.scenes.scene2d.ui.SplitPane$SplitPaneStyle: {
default-vertical: { handle: splitpane_vertical },
default-horizontal: { handle: splitpane_horizontal }
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com.badlogic.gdx.scenes.scene2d.ui.Window$WindowStyle: {
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dialog: { titleFont: default-font, background: window, titleFontColor: text-light, stageBackground: dialogDim },
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dialog-thin: { titleFont: thin-font, background: window, titleFontColor: text-light, stageBackground: dialogDim }
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com.badlogic.gdx.scenes.scene2d.ui.Slider$SliderStyle: {
default-horizontal: { background: scrubber_track, knob: scrubber_control_normal},
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com.badlogic.gdx.scenes.scene2d.ui.Label$LabelStyle: {
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com.badlogic.gdx.scenes.scene2d.ui.TextField$TextFieldStyle: {
default: { selection: textfield_selection, background: textfield_default, focusedBackground: textfield_focused, disabledBackground: textfield_disabled, font: default-font, fontColor: text-light, focusedFontColor: text-light, disabledFontColor: text-light, cursor: textfield_cursor, messageFont: default-font, messageFontColor: text-light },
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com.badlogic.gdx.scenes.scene2d.ui.CheckBox$CheckBoxStyle: {
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com.badlogic.gdx.scenes.scene2d.ui.List$ListStyle: {
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com.badlogic.gdx.scenes.scene2d.ui.Touchpad$TouchpadStyle: {
default: { background: text, knob: scrubber_control_normal }
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com.badlogic.gdx.scenes.scene2d.ui.Tree$TreeStyle: {
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| 2024-05-26T01:27:17.760571 | https://example.com/article/8086 |
s picked without replacement from {c: 6, h: 2, x: 10}.
1/612
Two letters picked without replacement from {z: 6, b: 10, s: 4}. What is prob of sequence bz?
3/19
Three letters picked without replacement from {j: 3, h: 1, s: 3, x: 7, p: 1, l: 3}. What is prob of sequence jjp?
1/816
Two letters picked without replacement from {c: 1, b: 1, a: 1, j: 2, w: 2}. Give prob of sequence ca.
1/42
Three letters picked without replacement from {b: 1, w: 1, m: 2, y: 1, q: 1, i: 2}. What is prob of sequence bym?
1/168
What is prob of sequence eeg when three letters picked without replacement from ggkeeekgggekeek?
5/91
Three letters picked without replacement from jsjsjssjss. Give prob of sequence sjs.
1/6
Two letters picked without replacement from bkcckkcbcbcnbc. Give prob of sequence kc.
9/91
What is prob of sequence aa when two letters picked without replacement from aasaasssaa?
1/3
What is prob of sequence ikg when three letters picked without replacement from {e: 1, f: 4, i: 1, r: 3, g: 4, k: 1}?
1/546
Calculate prob of sequence hgg when three letters picked without replacement from {h: 3, g: 8}.
28/165
Calculate prob of sequence kkf when three letters picked without replacement from kkpkfappfkkaa.
10/429
Two letters picked without replacement from {m: 2, z: 13, j: 1, i: 1}. What is prob of sequence zm?
13/136
What is prob of sequence ffl when three letters picked without replacement from {f: 3, i: 1, g: 4, l: 2}?
1/60
Calculate prob of sequence yj when two letters picked without replacement from {y: 2, m: 1, w: 1, j: 1}.
1/10
What is prob of sequence eeep when four letters picked without replacement from pepeeepe?
3/28
Three letters picked without replacement from rrwwrpptprdrc. Give prob of sequence tdw.
1/858
Two letters picked without replacement from miimmikkiks. Give prob of sequence im.
6/55
What is prob of sequence ybt when three letters picked without replacement from ylybllyyttlyct?
5/728
Calculate prob of sequence raq when three letters picked without replacement from ttrpqac.
1/210
Two letters picked without replacement from {e: 2, g: 1}. What is prob of sequence gg?
0
What is prob of sequence kvjj when four letters picked without replacement from jkkvvvkj?
3/280
Calculate prob of sequence aat when three letters picked without replacement from {t: 5, a: 3}.
5/56
Calculate prob of sequence pyzy when four letters picked without replacement from {g: 2, i: 1, z: 1, y: 2, p: 1}.
1/420
Four letters picked without replacement from {n: 8, m: 6, s: 3, d: 2}. What is prob of sequence ddms?
1/2584
What is prob of sequence aa when two letters picked without replacement from {f: 3, b: 2, d: 1, p: 3, z: 2, a: 4}?
2/35
Four letters picked without replacement from {i: 1, y: 2, q: 1, u: 1}. Give prob of sequence qiyy.
1/60
What is prob of sequence hyy when three letters picked without replacement from {h: 2, y: 2}?
1/6
Two letters picked without replacement from goolmowrwoogg. Give prob of sequence wr.
1/78
Calculate prob of sequence yyc when three letters picked without replacement from {x: 1, c: 11, m: 1, y: 6}.
55/969
Calculate prob of sequence sik when three letters picked without replacement from {k: 7, i: 1, f: 1, o: 1, s: 3}.
7/572
Calculate prob of sequence giit when four letters picked without replacement from ttittigiii.
1/63
What is prob of sequence yee when three letters picked without replacement from {h: 1, y: 1, o: 1, e: 4, x: 2, j: 3}?
1/110
Four letters picked without replacement from xbxvzxvvxzvzzxxvzvx. What is prob of sequence bzzz?
5/7752
Calculate prob of sequence ydxu when four letters picked without replacement from {u: 6, k: 1, y: 4, x: 1, d: 4}.
1/455
Three letters picked without replacement from {a: 3, b: 6, h: 2, v: 8}. Give prob of sequence ahb.
2/323
What is prob of sequence du when two letters picked without replacement from edddoeuoeddaoe?
5/182
Four letters picked without replacement from {n: 1, j: 1, l: 7, w: 4}. Give prob of sequence wlwn.
7/1430
Calculate prob of sequence oooo when four letters picked without replacement from ooododoodooooo.
30/91
Calculate prob of sequence nu when two letters picked without replacement from gznzu.
1/20
Calculate prob of sequence eqz when three letters picked without replacement from bxexaazqexbx.
1/660
Two letters picked without replacement from nnnlnnnnnnnlnnnnnnnn. What is prob of sequence ln?
9/95
What is prob of sequence lx when two letters picked without replacement from lllxxlllllxllxxl?
11/48
Three letters picked without replacement from {r: 1, z: 2, q: 2, n: 2, h: 1}. Give prob of sequence nrn.
1/168
Three letters picked without replacement from duuuuuuuuuuruuudruud. What is prob of sequence rur?
1/228
Calculate prob of sequence mmzz when four letters picked without replacement from {z: 2, m: 9}.
1/55
Calculate prob of sequence yfrr when four letters picked without replacement from vvvjvyfjfrvrvvvvjf.
1/12240
What is prob of sequence vww when three letters picked without replacement from vvvvvwvwvwwwwvwv?
9/80
What is prob of sequence vsv when three letters picked without replacement from {v: 5, s: 3, x: 1}?
5/42
Four letters picked without replacement from {n: 9, a: 2}. Give prob of sequence nnnn.
21/55
Four letters picked without replacement from ttttttt. What is prob of sequence tttt?
1
Two letters picked without replacement from aaannaaqaqna. What is prob of sequence qn?
1/22
Four letters picked without replacement from dddddd. What is prob of sequence dddd?
1
Three letters picked without replacement from pknxnnpndngdxnxnpngn. What is prob of sequence xdg?
1/570
Three letters picked without replacement from {l: 4, j: 1, r: 1, u: 5, n: 3, g: 1}. What is prob of sequence ngl?
2/455
What is prob of sequence xqyx when four letters picked without replacement from {x: 3, m: 1, q: 3, y: 3}?
3/280
Three letters picked without replacement from {g: 5, h: 13}. What is prob of sequence ggh?
65/1224
Four letters picked without replacement from {i: 2, n: 1, j: 2, b: 2, p: 12}. Give prob of sequence pinp.
11/3876
What is prob of sequence rrrr when four letters picked without replacement from {r: 5}?
1
Two letters picked without replacement from {d: 9, z: 1, p: 1}. What is prob of sequence zd?
9/110
Three letters picked without replacement from {r: 2, o: 4, h: 5}. Give prob of sequence hor.
4/99
Three letters picked without replacement from {g: 3, n: 8, l: 1, d: 6, m: 1}. Give prob of sequence ndl.
8/969
What is prob of sequence ti when two letters picked without replacement from {v: 1, t: 1, i: 2, r: 7, l: 6}?
1/136
Calculate prob of sequence kq when two letters picked without replacement from eekeeqeeeeeeeeeeee.
1/306
What is prob of sequence nzn when three letters picked without replacement from nnnnnnnnnznn?
1/12
Two letters picked without replacement from wungmga. Give prob of sequence wm.
1/42
Calculate prob of sequence hh when two letters picked without replacement from heeeeeeueheheehuehee.
1/19
What is prob of sequence rrrj when four letters picked without replacement from rjrrrrrjrrj?
7/55
What is prob of sequence lf when two letters picked without replacement from {w: 1, f: 10, c: 1, l: 5}?
25/136
Two letters picked without replacement from {z: 9, d: 2, l: 5}. Give prob of sequence ll.
1/12
What is prob of sequence ddll when four letters picked without replacement from {l: 9, d: 11}?
22/323
What is prob of sequence cf when two letters picked without replacement from mhpfmhpcfcif?
1/22
Four letters picked without replacement from hkykkyyhhhhhhyhkyhyh. What is prob of sequence yyyk?
4/969
Calculate prob of sequence fd when two letters picked without replacement from {z: 4, d: 4, f: 2}.
4/45
What is prob of sequence oomm when four letters picked without replacement from {v: 2, u: 2, l: 2, m: 2, q: 1, o: 5}?
5/3003
Calculate prob of sequence rlh when three letters picked without replacement from hjfrfhfhltjhfflh.
1/336
What is prob of sequence by when two letters picked without replacement from {y: 2, l: 4, p: 3, b: 1}?
1/45
What is prob of sequence qtq when three letters picked without replacement from btgtgtgqgqgggghgaggg?
1/1140
Three letters picked without replacement from qqqmqfmmmmmvqqoqfmq. What is prob of sequence qff?
8/2907
What is prob of sequence tnvt when four letters p | 2024-07-01T01:27:17.760571 | https://example.com/article/8777 |
1. Field of the Invention
The present invention relates to a digitized data processing apparatus which edits and converts digitized data obtained by scanning a model shape to be used in the preparation of an NC (numerical control) working program.
2. Description of the Prior Art
Conventionally, for formation of a mold having a freely curved surface, a model having the same shape as the mold is previously formed and profiling of the model is performed with a profile working machine. To form male and female mold parts, however, inverted models are required which are formed of plaster and which are difficult to form with accuracy. Moreover, the profiling speed using the profile working machine is considerably low.
With the advent of NC machine tools, the advantages of NC machining in terms of speed and reproducing performance have become known and some attempts have therefore been made to apply NC techniques to the formation of molds. For example, a digitizer apparatus has been developed which scans the surface of a model shape with a stylus (i.e., a probe) to obtain digitized data denoting stylus coordinate values, and which edits and converts the digitized data to prepare an NC working program. However, the NC working program thus formed only enables a process of working with a tool having the same shape as the stylus. That is, the conventional profile working process is merely separated into two processes associated through the NC working program.
An improved digitized data processing apparatus has also been developed which is capable of converting digitized data to enable working with a tool having a shape which is different from that of the stylus, which is capable of extracting a part of the digitized data corresponding to a particular work portion, and which is capable of preparing an NC working program for an inverted shape from the digitized data obtained by measuring the model without utilizing the digitized data on the inverted model. The above digitized data processing apparatus can prepare, by computation, suitable digitized data which is equivalent to corresponding data obtained with a digitizer apparatus based on equalizing the shape of the stylus to that of the desired tool and restricting the scanning range.
FIG. 1 is a block diagram of an example of the prior art digitized data processing apparatus. Digitized data DD (hereinafter referred to as "digitized measurement data") is stored in a digitized data memory means 2 via a digitized data input means 1. The digitized measurement data DD is read out of the digitized data memory means 2 and input to a digitized data editing means 3 where it is edited in accordance with an editing instruction DS input from a digitized data editing instruction input means 4 to form digitized data DO representing a tool path for working a work piece (which data is hereinafter referred to as "digitized tool path data"). The digitized tool path data DO is stored in the digitized data memory means 2. The digitized tool path data DO is read out of the digitized data memory means 2 and input to a digitized data NC program conversion means 5. An NC working program is thereby prepared and output.
The above digitized data processing apparatus enables an inverting processing to be performed as one of the steps for editing the digitized measurement data. That is, it can obtain, from the digitized measurement data of one model, digitized tool path data for the model shape (hereinafter referred to as "normal shape") and digitized tool path data for a shape formed by inverting the model shape (hereinafter referred to as "inverted shape"). Examination of the size of a normal shape work piece formed on the basis of a normal shape NC working program prepared by converting the digitized tool path data for the normal shape is performed by comparison measurement using a three-dimensional measuring apparatus. However, examination of the size of an inverted shape work piece formed on the basis of an inverted shape NC working program prepared by converting digitized tool path data for the inverted shape cannot be performed since there is no inverted shape model. | 2024-01-05T01:27:17.760571 | https://example.com/article/4657 |
Recently there has been much weeping, wailing and gnashing of teeth on TV, social media and in general conversations concerning what many refer to as “disrespect” surrounding players in the National Football League (NFL) who have “taken a knee” during the playing of our national anthem or shown some other form of behaviour which a large group of people seem to find offensive. While this seems to bother a large segment of people, I find that I really don’t care.
Here again, we have a prime example of how the masses are manipulated into emotional rather than cognitive responses by a tyrannical government desperate to appear relevant amidst a sea of tyranny. Continue reading →
In my recent article about the destruction of Southern history (and eventually all real US history) I noted that part of that destruction was the promotion of the “reality show” mentality and the promotion of continual sports extravaganzas. People can now watch continuous wall-to-wall sports programs literally every weekend as well as about three nights per week–giving them no time to think about anything of any greater depth than the batting averages of their favorite baseball players or the running yardage of their favorite football players. And then we have basketball, track, hockey, tiddly-winks and a host of others also.
And while there is nothing wrong with a little recreational viewing of various sports events (I like to watch a good rodeo once in awhile myself) it has gotten to the point in this country now where the sports events are almost a form of idolatry. Continue reading →
I am hearing (as is the case when CUTS are talked about) that any loss of tax receipts will add more to the national debt.
This is easily solved and we can be left with money left over if we follow a simple formula as do those Americans who live on limited income and do so without going into debt beyond having a mortgage on a house and possibly a loan for a vehicle.Continue reading →
Never before since I can remember have so many been affected in North America in such a short period of time. First came Hurricane Harvey. That was less than 3 weeks ago. That left Houston and the Gulf Areas East of Houston pretty much flooded or wind damaged/destroyed. Then came Hurricane Irma which has affected a lot of folks in Florida. Even those who did not have property damage lost electricity or found themselves on the road for days trying to avoid Hurricane Irma. Now Mexico has been hit with two rather large earthquakes. The last one has greatly affected Mexico City. I figure from looking at pictures from above the city and the resulting dust clouds that occurred when buildings collapsed that at least 10% of the buildings there collapsed. Those who lived further North were not as affected, but they were affected to some degree nonetheless. I am down about $50 so far from paying higher gasoline prices. That affect is nothing compared to those further south of me in Houston and elsewhere. But it is an affect nonetheless. Continue reading →
Double standards for the protection of the leftist elites.
“Once Upon A Time, In a Galaxy Far, Far Away” the left wing of the political spectrum were the vanguard of opposing the globalist warfare matrix. The anti-war movement changed the world in the 1960’s and stopped the bloody carnage in Viet Nam. Judging by today’s standards, that world is forgotten and the replacement culture has an attitude that preserving the empire is necessary no matter how trumped up the rationale is to maintain perpetual conflict. Choose the newest enemy; Jihad Islam, Russian nationalism, Chinese supremacy or North Korean belligerence and any dedicated internationalist can spin an argument why fighting the next war is necessary.
Now we all know just how half-crazed the foreign policy of mentally deranged “Bomber” McCain and his NeoCon warmonger comrades has been over the decades, but what is the excuse from the Looney Left for abandoning their moral argument that underpinned opposition to the establishment’s persistent drive to maintain a permanent war environment?
Remember the Democratic left is the party that purged the righteous and enlightened foreign policy opposition from the likes of Dennis Kucinich, Cynthia McKinney and Eric Massa; forcing each from office. Throwing them under the bus is much different from having them sitting in the back of the coach. Continue reading →
Just a few thoughts from an America loving simple man on the whole situation that we find ourselves in…
To all of the anti-Trump hateful leftists that will undoubtedly troll this post… Our eyes are closed no longer. We see you for what you are. We see your destructive and divisive agenda/ideology, AND WE REJECT IT…. Continue reading →
The Grand Old Party is about to commit suicide.
All this talk about Trump this, and Trump that, masks a far bigger political controversy. The Republican Party leadership in Washington, D.C., has fundamentally betrayed its constituents and they are about to learn that they’ve been double-crossed — for years.
Every Republican candidate’s stock speech sounds the same, the thunderous roar about a government out of control, federal spending out of control (insert charts and graphs and why, if you stack hundred dollar bills, they will reach the edge of the universe), federal taxes out of control (insert comparisons to socialist countries), the federal bureaucracy out of control (insert metaphors about chains, yokes, and the like), the family shattered with federal funding of abortion a crime against humanity (watch for it — there! The heart-wrenching sob), and our military is emasculated. Continue reading →
“I, Jeffrey Bennett, do solemnly swear (or affirm) that I will support and defend the Constitution of the United States against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; and that I will obey the orders of the President of the United States and the orders of the officers appointed over me, according to regulations and the Uniform Code of Military Justice. So help me God.” ~ November 7, 1965
What occurred in Charlottesville North Carolina was a sad, tragic event especially with the death of a young woman Heather Heyer.
We can all agree that extremist fringe groups on both sides of the ideological spectrum; White Nationalists, Antifa, Neo-Nazi’s, White Supremacists, Black Lives Matter (BLM), anarchists and who knows what else prowls the streets of unrest, should be denounced and are anathema to the American ethos. Continue reading →
“Every record has been destroyed or falsified, every book rewritten, every picture has been repainted, every statue and street building has been renamed, every date has been altered. And the process is continuing day by day and minute by minute. History has stopped. Nothing exists except an endless present in which the Party is always right.” ― George Orwell, 1984
We are headed in Orwell’s direction if we don’t get smart about censoring the past. One of the things the internet has done is make more information available to people.
Someone says this about that person, and another person says that based on some additional research. Each side gets a voice.Continue reading →
Originally posted on Federal Observer on April 20, 2017 as a reply toIs America In The Throes Of A Nasty Divorce?, by Ron Ewart. In a rare move – we felt that the author’s commentary warranted a posting of it’s own. ~ J.B.
I don’t think that there’s any doubt, that the American Union will eventually be dissolved. How that happens is what is in question. Will it be a violent dissolution, or will it be peaceful. Will each state become an independent nation, or will some states join in new, but much smaller Unions? Continue reading →
“These Juramentado attacks were materially reduced in number by a practice that the Mohamedans held in abhorrence. The bodies were publicly buried in the same grave with a dead pig. It was not pleasant to have to take such measures, but the prospect of going to hell instead of heaven sometimes deterred the would-be assassins.”
These were the words of General Pershing in his autobiography. According to the New York Times (despite having reported on it at the time) it’s a myth. The Washington Post agrees. They both cite fact checks by Politifact and Snopes. The problem with all the fact checks is… the facts. Continue reading →
I don’t live in an echo chamber, partly because there aren’t enough people out there who think like me, but also because I constantly and intentionally attempt to challenge my worldview by reading stuff from all over the political map. I ingest as much as I can from a wide variety of intelligent sources, picking and choosing what makes sense to me, and then synthesizing it the best I can.
Though I’m certainly grounded in certain key principles, my perspective on specific issues remains malleable as I take in additional information and perspectives. I try to accept and acknowledge my own ignorance and view life as a journey of constant mental, emotionally and spiritual growth. If I’m not growing my capacity in all of those realms until the day I die, I’m doing it wrong. Life should be seen as a battle against one’s own ignorance, as opposed to an obsession with the ignorance of others. You can’t legislate morality, nor can you legislate wisdom. The only way the world will improve on a long-term sustainable basis is if more of us get wise. That’s a personal journey and it’s our individual duty to accept it. Continue reading →
Charlottesville was a huge victory for the hard-Left Southern Poverty Law Center (SPLC). And that’s not good for anyone who loves freedom.
The driver of the car who plowed into a crowd of Leftist demonstrators in Charlottesville Saturday was a neo-Nazi, and on Monday President Trump denounced the Ku Klux Klan, “neo-Nazis, white supremacists, and other hate groups,” which he rightly said were “repugnant to everything we hold dear as Americans.” Continue reading →
Life without oil, natural gas and coal would most likely be nasty, brutish and short.
Al Gore’s new movie, a New York Times article on the final Obama Era “manmade climate disaster” report, and a piece saying wrathful people twelve years from now will hang hundreds of “climate deniers” are a tiny sample of Climate Hysteria and Anti-Trump Resistance rising to a crescendo. If we don’t end our evil fossil-fuel-burning lifestyles and go 100% renewable Right Now, we are doomed, they rail.
Maybe it’s our educational system, our cargo cult’s easy access to food and technology far from farms, mines and factories, or the end-of-days propaganda constantly pounded into our heads. Whatever the reason, far too many people have a pitiful grasp of reality: natural climate fluctuations throughout Earth history; the intricate, often fragile sources of things we take for granted; and what life would really be like in the utopian fossil-fuel-free future they dream of. Let’s take a short journey into that idyllic realm. Continue reading →
How many of these points can you share? So much of this sounds so familiar…..
When I was 8 during the summer back in the summer of 1968, we’d swim in the bayou, fish all day, live in the woods near my home playing guns (liberals of today would have kittens), football (no helmets or pads), basketball and baseball. We caught crawfish and our dads would boil them in a huge picnic with the corn and potatoes. We’d wave to the shrimp boats and the party boats headed to the Gulf and they’d blow their horns to us on the bank.
I never really watched a lot of TV and never had a reason to do so. I did chores, such as clipping the beautiful hedges that surrounded our parcel like a living fence and mowing the grass under our giant live oak tree with an old push mower. I started mowing the grass when I was 8 and I got an allowance.
My parents had a big, screened in porch that overlooked the bayou and we’d have sleep outs on it. We’d sneak outside and look at the massive amount stars overhead. Continue reading →
You may well be already aware of what I am going to say here, but I felt the need to say it anyway. Lest it come across to you in the wrong manner, let me state that my wife and I both voted for you, and we pray for you, that what you are trying to do will be successful. There are people in your own administration that cannot honestly say that much.
Regarding this entire “Russian collusion” narrative you have asked several good questions which I need to comment briefly on. It should be obvious to you that Mr. Mueller, in some vain attempt to tie you to something illegal or treasonous, will investigate you, your family, your friends, including all the family pets (to see if you ever owned a Russian wolfhound) back for at least three generations, and no matter how tenuous whatever they manage to dig up is, it will be parroted by the mainstream media as a world crime of gigantic proportions. I think we both realize that. Continue reading →
~ Forward ~Tesla sales have plummeted to near zero in Hong Kong and Denmark, as generous subsidies evaporated. Its $7,500-per-car federal rebate (taxpayer subsidy) is about to start its death spiral. So California is halfway toward enacting legislation that would provide $3 billion in state point-of-sale rebates: as much as $30,000 or even $40,000 per car. The legislature apparently wants to continue ensuring that all families help perpetuate programs that thus far have transferred 90% of all electric car subsidies to the wealthiest 20% of families.
Meanwhile, ardent renewable energy aficionados insist that the key to a wind and solar future is battery backup systems … which are just around the corner. Not so fast, says technology guru Mark Mills. Storing 12 hours worth of household and business electricity demand per day, plus charging up 1.4 billion currently gasoline-powered vehicles, would require 1,250 years of production from every existing lithium battery factory worldwide. It’s not going to happen, he says. And that’s just the beginning of the subsidy and sustainability fantasies we must deal with. ~Paul DriessenContinue reading →
Following President Trump’s February executive order, the Department of Education has been reviewing regulations and guidance to make sure that they “adequately protect students.” Men on college and university campuses are students too, and they greatly need protection from the regulations and guidance of the Obama-era Office for Civil Rights, which remain in force at this writing. Campus drinking is a problem, but it is not the most serious problem (“Trump education official: Parties usually ‘both drunk’ in campus sexual assault cases,” July 12, 2017). Here is the most serious problem.
The situation regarding anything even remotely sexual on college and university campuses today is entirely controlled by ideologues steeped in former President Barack Obama identity and sexual politics. Those who caused the current situation to exist are “true believers” who must be replaced by centrists open to evidence and reason. Continue reading →
Rational thinking and moral conduct are rare in a society that has no respect for other individuals or succumb to the dictates of an all powerful government. Without free will, the dignity of the person is removed from the social relationship. Tyranny is not simply a state of mind, but is an imposition of compulsion designed to control the behavior of subjects of whatever form of government claiming legitimate rule.
Mary Ann McGrail contends that Shakespeare viewed “tyranny is a perpetual political and human problem“.
“She locates Shakespeare’s expansive definition of tyranny between the definitions accepted by classical and modern political philosophy. Is tyranny always the worst of all possible political regimes, as Aristotle argues in his Politics? Or is disguised tyranny, as Machiavelli proposes, potentially the best regime possible?
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~ QUOTABLE ~
The phrase “the Fall of America” suggests some cataclysmic event ended the American Empire which had stretched from Maine to California and Florida to Washington. But at the end, there was no straining at the gates, no barbarian horde that dispatched the Empire in one fell swoop. Rather, the Empire fell slowly, as a result of challenges from within and without, and changing over the course of hundreds of years until its form was unrecognizable.
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~ NOTICE OF COPYRIGHT ~
We make every attempt to respect the rights of others. If you feel that something here has infringed your work please let us know and we will correct it immediately. It is not always easy to determine the status of material posted to the Internet with regard to fair use and public domain. | 2023-09-08T01:27:17.760571 | https://example.com/article/5011 |
Kate Middleton isn’t the alone aristocratic who has a affection for recycling clothes and cossack she’s beat abounding times before.
For Princess Eugenie’s bells yesterday, Prince Harry appeared to action the aforementioned brace of shoes he wore for a friend’s bells in August.
The Duke of Sus, 34, donned fleet socks and archetypal atramentous covering dress shoes for his cousin’s big day. Upon aing examination, the prince’s brogues could use a repair. The larboard shoe, in particular, featured a ample aperture and a hardly beat heel.
Prince Harry wore the aforementioned shoes aback in August back he stepped out for the bells of his adolescence friend Charlie van Straubenzee in Surrey, England. In fact, he chose a actual agnate head-to-toe attending for both occasions.
Want more?
Happy 34th Birthday, Prince Harry: How His Style Has Changed Over the Years | 2023-10-29T01:27:17.760571 | https://example.com/article/7100 |
Macrophage brain infiltration in experimental autoimmune encephalomyelitis is not completely compromised by suppressed T-cell invasion: in vivo magnetic resonance imaging illustration in effective anti-VLA-4 antibody treatment.
Large inflammatory infiltrates of T cells, macrophages and B cells in the central nervous system (CNS) contribute to the pathogenesis of multiple sclerosis (MS). The passage of T cells through the blood-brain barrier can be suppressed with antibodies directed against alpha-4 integrins (VLA-4) that mediate T-cell adherence. This treatment, in phase III of clinical trial evaluation, reduces lesion development in MS patients. In the ongoing inflammatory disease process the consequences of T-cell inhibitory anti-VLA-4 antibodies on inflammatory compounds are still poorly investigated. We show that anti-VLA-4 antibody treatment during the late preclinical phase of the acute experimental autoimmune encephalomyelitis (EAE) MS rat model interrupts T-cell egress out of the vascular compartment and suppresses clinical disease and histological alterations but macrophage recruitment in the CNS is not fully compromised. Among the treated EAE animals not developing disease, none presented foci of T-cell infiltration in CNS. However, in 75% of the treated EAE rats monocyte ingress in CNS was observed in vivo by magnetic resonance imaging with the ultrasmall superparamagnetic iron oxide contrast agent. Our data shed new light on the role of remaining macrophage brain infiltration in an induced but interrupted T-cell-mediated EAE disease process. | 2024-06-02T01:27:17.760571 | https://example.com/article/8577 |
Time-dose effects in the matching of adjacent radiotherapy fields.
When adjacent radiotherapy fields are treated in different times, the dose fraction at the junction of the fields is different from that within the portals. When the width of the gab between the fields is adjusted to produce equality of dose at the junction, it will not give equality with respect to radiobiological effect. The authors have used Ellis's NSD equation to take into account time-dose relationships in the matching of adjacent fields. Distribution of physical dose and biologically effective dose in the region near the junction of opposed pairs of parallel 60Co fields are given. | 2024-01-22T01:27:17.760571 | https://example.com/article/5126 |
Q:
Deleting an Object in c++ implies deletion of its members?
I am quite new to c++ and I have a question.
If I have a class containing a (pointer to a) vector:
class myClass {
public:
int* direction;
myClass(int d){direction=new int[d];}
}
When I create an object and delete it in the main:
int main(){
int d;
myClass* myvec;
myvec = new myClass(d);
delete myvec;
}
Also the destructor for myvec->direction has been coherently called and the memory freed? Or do I have to write an apposite method for that?
I hope the question is clear...
A:
If you've allocated memory with new you need to delete it too, like this:
class myClass {
int* direction;
public:
myClass(int d) : direction(new int[d]) {}
~myClass() {
delete[] direction;
}
}
But you'd also need to write a copy constructor and copy assignment operator, and in C++11 and later also a move constructor and move assignment operator, for this to be working good. Otherwise, you'd risk the default versions of those copying the raw pointer when you use instances of this class.
Take a look at the rule of three/five/zero.
You'd be much better off using a std::vector<int> instead of a raw pointer.
| 2024-06-01T01:27:17.760571 | https://example.com/article/4837 |
"============================================================================
"File: slim.vim
"Description: Syntax checking plugin for syntastic.vim
"Maintainer: Martin Grenfell <martin.grenfell at gmail dot com>
"License: This program is free software. It comes without any warranty,
" to the extent permitted by applicable law. You can redistribute
" it and/or modify it under the terms of the Do What The Fuck You
" Want To Public License, Version 2, as published by Sam Hocevar.
" See http://sam.zoy.org/wtfpl/COPYING for more details.
"
"============================================================================
if exists("g:loaded_syntastic_slim_slimrb_checker")
finish
endif
let g:loaded_syntastic_slim_slimrb_checker=1
function! SyntaxCheckers_slim_slimrb_IsAvailable()
return executable("slimrb")
endfunction
function! s:SlimrbVersion()
if !exists('s:slimrb_version')
let output = system("slimrb --version 2>/dev/null")
let output = substitute(output, '\n$', '', '')
let output = substitute(output, '^slim ', '', 'i')
let s:slimrb_version = split(output, '\.')
end
return s:slimrb_version
endfunction
function! SyntaxCheckers_slim_slimrb_GetLocList()
let makeprg = syntastic#makeprg#build({
\ 'exe': 'slimrb',
\ 'args': '-c',
\ 'subchecker': 'slimrb' })
if syntastic#util#versionIsAtLeast(s:SlimrbVersion(), [1,3,1])
let errorformat =
\ '%C\ %#%f\, Line %l\, Column %c,'.
\ '%-G\ %.%#,'.
\ '%ESlim::Parser::SyntaxError: %m,'.
\ '%+C%.%#'
else
let errorformat =
\ '%C\ %#%f\, Line %l,'.
\ '%-G\ %.%#,'.
\ '%ESlim::Parser::SyntaxError: %m,'.
\ '%+C%.%#'
endif
return SyntasticMake({ 'makeprg': makeprg, 'errorformat': errorformat })
endfunction
call g:SyntasticRegistry.CreateAndRegisterChecker({
\ 'filetype': 'slim',
\ 'name': 'slimrb'})
| 2024-06-12T01:27:17.760571 | https://example.com/article/5864 |
Abstract
Politicians regularly make statements to the effect that there are two, three or even four generations, within the same family, that have never worked. Politicians are also on record as stating that there is a something for nothing culture. These concepts of generations and cultures of worklessness regularly appear in the traditional print media, internationally and in academia. Where the reasons for holding a belief are adequate and the belief formed from the available evidence is adequate, the belief may be understood as properly held. Where this is not the case the belief is improperly held. This paper uses the British Household Panel Survey to link information on individuals and between generations to assess whether belief in generations and cultures of worklessness are ‘properly held’. The level of intergenerational worklessness is quantified and the circiumstances of those out of work assessed. The descriptive analysis confirms previous research which finds no evidence for the generations and cultures theses. In the absence of corroborating evidence it is argued that the concepts of generations and cultures of worklessness should be treated as improperly held beliefs and it is time they are done away with. | 2024-05-12T01:27:17.760571 | https://example.com/article/1062 |
37 Wn. App. 251 (1984)
680 P.2d 63
THE STATE OF WASHINGTON, Respondent,
v.
ROBERT B. GONZALES, Appellant.
No. 5540-0-III.
The Court of Appeals of Washington, Division Three.
April 5, 1984.
*252 Katherine M. Steele, for appellant (appointed counsel for appeal).
C.J. Rabideau, Prosecuting Attorney, and Philip A. Meyers and Pamela Cameron, Deputies, for respondent.
GREEN, J.
Defendant was a resident of the Department of Corrections Tri-Cities Work Release Facility in Pasco. On the morning of April 3, 1982, he checked out of the facility for work, noting the name of the employer and the time he would return. Upon attempting to verify his notations, correction officials learned the listed place of employment was closed and he was not there. Defendant returned to the facility approximately 2 1/2 hours late. Upon being told his work release status was suspended, he ran from the facility. Defendant was later apprehended, charged and convicted of escape in the first degree under RCW 9A.76.110(1). He appeals.
First, defendant claims the conviction should be set aside on the theory of double jeopardy. This contention is based upon the court dismissing but later reinstating the escape charge. He argues the dismissal was tantamount to an acquittal and the subsequent conviction placed him twice in jeopardy. We disagree.
[1] The dismissal was based upon the court's pretrial determination that defendant could only be charged under RCW 72.65.070 for willful failure to return to the facility under State v. Danforth, 97 Wn.2d 255, 643 P.2d 882 (1982). On reconsideration, the court determined its ruling was incorrect because the defendant returned to the facility, *253 but thereafter left without permission. In light of these circumstances, the court determined the dismissal should not have been granted and therefore the escape charge was reinstated. This determination was made before the jury had been impaneled or the court had begun receiving the evidence; hence, jeopardy did not attach. United States v. Martin Linen Supply Co., 430 U.S. 564, 51 L.Ed.2d 642, 97 S.Ct. 1349, 1353 (1977).[1]
[2] Likewise, the contention defendant should have been charged for willful failure to return to the facility, rather than escape, is not meritorious. Defendant did return; his running from the facility without permission gives rise to the escape charge. See State v. Thompson, 35 Wn. App. 766, 769, 669 P.2d 1270 (1983).
Finally, defendant contends the State failed to prove the elements of first degree escape, RCW 9A.76.110. That statute provides:
(1) A person is guilty of escape in the first degree if, being detained pursuant to a conviction of a felony or an equivalent juvenile offense, he escapes from custody or a detention facility.
He claims he was not in lawful custody because the convictions which led to his incarceration were based on constitutionally infirm guilty pleas. These contentions must be rejected.
[3] We agree with the observation in Judge Scholfield's dissent in State v. Thompson, supra at 774:
The gravamen of the crime of escape is a defendant's defiance of lawful custodial authority and deliberate violation of a judgment directing his or her confinement. The orderly and rational administration of the criminal justice system requires that the judgment be treated as a valid conviction until a court with jurisdiction rules otherwise.
The escape statute, RCW 9A.76.110, requires only that the escape occur while defendant is being detained "pursuant *254 to a conviction of a felony". Thompson was being detained "pursuant to a conviction of a felony" whether or not the conviction is subject to attack. The language of the statute contains no suggestion of a legislative intent that the conviction could not be used in a prosecution for first degree escape if it was vulnerable to collateral attack.
This approach appears to be supported by the weight of authority[2] and, in our view, is the better rule. As the court in Commonwealth v. Stanley, 265 Pa. Super. 194, 203, 401 A.2d 1166, 1171 (1979), observed:
The reason for this rule is obvious the difficulties of prison administration would be intolerable if each prisoner was permitted to "go over the wall" as a means of testing the legality of his incarceration, rather than utilizing the customary means of administrative and judicial redress.
(Footnote omitted.) Defendant was detained pursuant to convictions for two felonies at the time he left the facility without permission. So long as the convictions had not been set aside, they were valid and he was lawfully confined. All the elements of the statute were proved. For the foregoing reasons, we decline to follow the reasoning of the majority in State v. Thompson, supra.
Nevertheless, we have reviewed the record relating to the two prior convictions and find no error.
*255 Affirmed.
MUNSON, C.J., and THOMPSON, J., concur.
Review granted by Supreme Court June 8, 1984.
NOTES
[1] State v. Dowling, 98 Wn.2d 542, 656 P.2d 497 (1983), relied on by defendant, is distinguishable. That case involved a motion to dismiss at the close of the State's case.
[2] See, e.g., United States v. Pereira, 574 F.2d 103, 106 n. 6 (2d Cir.1978) and cases cited therein; Louis v. Nelson, 560 F. Supp. 899, 901 (S.D. Fla. 1983); In re Estrada, 63 Cal.2d 740, 408 P.2d 948, 48 Cal. Rptr. 172, 178 (1966); State v. Handran, 92 Idaho 579, 448 P.2d 193 (1968); Carpenter v. State, 178 Ind. App. 446, 382 N.E.2d 1026, 1028 (1978); Henderson v. State, 198 Kan. 655, 426 P.2d 92, 94 (1967); State v. Perry, 364 So.2d 900, 901 (La. 1978); Eaton v. State, 302 A.2d 588, 594 (Me. 1973); In re Lynch, 379 Mass. 757, 760 n. 2, 400 N.E.2d 854, 857 (1980) and cases cited therein; People v. Holt, 54 Mich. App. 60, 220 N.W.2d 205, 206 (1974); State v. Lopez, 79 N.M. 235, 441 P.2d 764 (1968); People ex rel. Haines v. Hunt, 229 A.D. 419, 242 N.Y.S. 105 (1930); State v. Speaks, 17 Ohio App.2d 129, 244 N.E.2d 799, 801 (1969); Phillips v. State, 622 P.2d 719, 720 (Okla. Crim. App. 1980); Commonwealth v. Stanley, 265 Pa. Super. 194, 401 A.2d 1166, 1171 (1979); Annot., 70 A.L.R.2d 1430 (1960 & Supp. 1978).
| 2023-08-14T01:27:17.760571 | https://example.com/article/7114 |
---
abstract: 'We give a direct derivation of the distribution of the maximum and the location of the maximum of one-sided and two-sided Brownian motion with a negative parabolic drift. The argument uses a relation between integrals of special functions, in particular involving integrals with respect to functions which can be called “incomplete Scorer functions". The relation is proved by showing that both integrals, as a function of two parameters, satisfy the same extended heat equation, and the maximum principle is used to show that these solution must therefore have the stated relation. Once this relation is established, a direct derivation of the distribution of the maximum and location of the maximum of Brownian motion minus a parabola is possible, leading to a considerable shortening of the original proofs.'
address:
- |
Delft University\
,
- |
University of Chicago\
,
- |
CWI\
,
author:
-
-
-
bibliography:
- 'cupbook.bib'
title: 'Chernoff’s distribution and differential equations of parabolic and Airy type'
---
Introduction {#section:intro}
============
Let $\{W(t):\,t\in\R\}$ be standard two-sided Brownian motion, originating from zero. The determination of the distribution of the (almost surely unique) location of the maximum of $\{W(t)-t^2:t\in\R\}$ has a long history, which probably started with Chernoff’s paper [@chernoff:64] in a study of the limit distribution of an estimator of the mode of a distribution. In the latter paper the density of the location of the maximum of $\{W(t)-t^2:t\in\R\}$, which we will denote by $$\label{def_Z}
Z=\argmax_t\{W(t)-t^2,\,t\in\R\},$$ is characterized in the following way. Let $u(t,x)$ be the solution of the heat equation $$\frac{\partial}{\partial t}u(t,x)=-\tfrac12\frac{\partial^2}{\partial x^2}u(t,x),$$ for $x\le t^2$, under the boundary conditions $$u(t,x)\ge0,\qquad u(t,t^2)\stackrel{\mbox{def}}=\lim_{x\uparrow t^2}u(t,x)=1,\qquad (t,x)\in\R^2,\qquad \lim_{x\downarrow -\infty}u(t,x)=0,\qquad t\in\R.$$ Furthermore, let the function $u_2$ be defined by $$u_2(t)=\lim_{x\uparrow t^2}\frac{\partial}{\partial x}u(t,x).$$ Then the density of (\[def\_Z\]) is given by $$\label{f_Z}
f_Z(t)=\tfrac12u_2(t)u_2(-t),\,t\in\R.$$
The original attempts to compute the density $f_Z$ were based on numerically solving the heat equation above, but it soon became clear that this method did not produce a very accurate solution, mainly because of the rather awkward boundary conditions. However, around 1984 the connection with Airy functions was discovered and this connection was exploited to give analytic solutions in the papers [@daniels:85], [@nico:85] and [@gro:89], which were all written in 1984, although the last paper appeared much later.
There seems to be a recent revival of interest in this area of research, see, e.g., [@svante:10], [@piet:10b], [@piet:11d], [@nico_piet:11], [@leandro:12] and [@svante:13]. Also, the main theorem (Theorem 2.3) in [@dembo:08]) uses Theorem 3.1 of [@gro:89] in an essential way. These recent papers (except [@leandro:12]) rely a lot on the results in [@daniels:85] and [@gro:89], but it seems fair to say that the derivation of these results in [@daniels:85] and [@gro:89] is not a simple matter. The most natural approach still seems to use the Cameron-Martin-Girsanov formula for making the transition from Brownian motion with drift to Brownian motion without drift, and next to use the Feynman-Kac formula for determining the distribution of the Radon-Nikodym derivative of the Brownian motion with parabolic drift with respect to the Brownian motion without drift from the corresponding second order differential equation. This is the approach followed in [@gro:89]. However, the completion of these arguments used a lot of machinery which one would prefer to avoid. For this reason we give an alternative approach in the present paper.
The starting point of our approach is Theorem 2.1 in [@gro:89], which is given below for convenience. Theorem 2.1 in [@gro:89] in fact deals with the process $\{W(t)-ct^2:t\in\R\}$ for an arbitrary positive constant $c>0$, but since we can always deduce the results for general $c$ from the case $c=1$, using Brownian scaling, see, e.g., [@svante:13], we take for convenience $c=1$ in the theorem below. Another simplification is that we consider first hitting times of $0$ for processes starting at $x<0$ instead of first hitting times of $a$ of processes starting at $x<a$ for an arbitrary $a\in\R$, using space homogeneity. We made slight changes of notation, in particular the function $h_x$, $x>0$, of [@gro:89] is again denoted by $h_{x}$, but now with a negative argument, so $h_x$ in our paper corresponds to $h_{-x}$ in [@gro:89].
\[th:stopping\_time\] Let, for $s\in\R$ and $x<0$, $Q^{(s,x)}$ be the probability measure on the Borel $\s$-field of $C([s,\infty):\R)$, corresponding to the process $\{X(t):t\ge s\}$, where $X(t)=W(t)-t^2$, starting at position $x$ at time $s$, and where $\{W(t):t\ge s\}$ is Brownian motion, starting at $x+s^2$ at time $s$. Let the first passage time $\t_0$ of the process $X$ be defined by $$\t_0=\inf\{t\ge s:X(t)=0\},$$ where, as usual, we define $\t_0=\infty$, if $\{t\ge s:X(t)=0\}=\emptyset$. Then
1. $$\begin{aligned}
Q^{(s,x)}\left\{\t_0\in dt\right\}=e^{-\tfrac23\bigl(t^3-s^3\bigr)+2sx}\psi_{x}(t-s)
E^0\left\{e^{-2\int_0^{t-s}B(u)\,du}\Bigm|B(t-s)=-x\right\}\,dt,\end{aligned}$$
where $B$ is a process, starting at zero at time $0$, with corresponding expectation $E^0$, and where $\psi_z(u)=\bigl(2\pi u^3\bigr)^{-1/2}z\exp\bigl(-z^2/(2u)\bigr),\,u,z>0,$ is the value at $u$ of the density of the first passage time through zero of Brownian motion, starting at $z$ at time $0$.
2. $$\begin{aligned}
Q^{(s,x)}\left\{\t_0\in dt\right\}=e^{-\tfrac23\bigl(t^3-s^3\bigr)+2sx}h_{x}(t-s)\,dt,\end{aligned}$$
where the function $h_{x}:\R_+\to\R_+$ has Laplace transform $$\begin{aligned}
\hat h_{x}(\l)=\int_0^{\infty}e^{-\l u}h_{x}(u)\,du
=\ai\bigl(\xi-4^{1/3}x\bigr)/\ai(\xi),\,\quad \xi=2^{-1/3}\l>0,\end{aligned}$$ and $\ai$ denotes the Airy function $\ai$.
[Note that the function $h_x$ in the definition of the density of the stopping time $\t_0$ has by part (ii) of Theorem \[th:stopping\_time\] the representation $$\label{def_h_x}
h_{x}(t)=\frac1{2\pi}\int_{v=-\infty}^{\infty}e^{itv}\frac{\ai(i2^{-1/3}v-4^{1/3}x)}{\ai(i2^{-1/3}v)}\,dv,\qquad\,t>0.$$ This representation is obtained by inverting the Laplace transform and will be used in Section \[section:max+locmax\] and the proof of Lemma \[lemma:psi\_phi\]. ]{}
[ Theorem \[th:stopping\_time\] occurs in different forms in the literature, see, e.g., Theorem 2.1 in [@paavo:88]. For convenience of the reader, we give a short self-contained proof of Theorem \[th:stopping\_time\] in Appendix A. The interpretation in terms of Bessel process is not really necessary, but this naturally leads to an interpretation in terms of Brownian excursions, further explored in Section 4 of [@gro:89]. ]{}
Theorem \[th:stopping\_time\] should in principle be sufficient to derive the density $f_Z$ of (\[f\_Z\]), since, defining $$q(s)=\lim_{x\uparrow0}\frac{\partial}{\partial x}Q^{(s,x)}\left\{X_t<0,\,\forall t\ge s\right\}
=\lim_{x\uparrow0}\frac{\partial}{\partial x}Q^{(s,x)}\left\{\t_0=\infty\right\},$$ we find: $$f_Z(s)=\tfrac12q(s)q(-s),$$ following a line of reasoning similar to the derivation of (\[f\_Z\]) in [@chernoff:64] (in Chernoff’s argument, which is based on a random local perturbation of the starting point $(s,x)$ and the ensuing convolution equation, the factor $1/2$ can be interpreted as the expectation of the squared maximum of the standard Brownian bridge). Moreover, by (ii) of Theorem \[th:stopping\_time\] we have, using inversion of the Laplace transform along the imaginary axis: $$\begin{aligned}
\label{fundamental_relation}
&Q^{(s,x)}\left\{\t_0=\infty\right\}=1-Q^{(s,x)}\left\{\t_0<\infty\right\}
=1-\int_{t=s}^{\infty}Q^{(s,x)}\left\{\t_0\in dt\right\}\nonumber\\
&=1-\int_{t=s}^{\infty}e^{-\tfrac23\bigl(t^3-s^3\bigr)+2sx}
h_{x}(t-s)\,dt\nonumber\\
&=1-\frac{e^{2sx+\tfrac23s^3}}{2\pi}\
\int_{v=-\infty}^{\infty}\frac{\ai\bigl(2^{-1/3}iv-4^{1/3}x\bigr)}{\ai\bigl(2^{-1/3}iv\bigr)}\int_{t=0}^{\infty}e^{itv-\tfrac23(s+t)^3}\,dt\,dv.\end{aligned}$$ So we would be done if we can deal with the properties of the integral in the last line.
However, the latter integral has some unpleasant properties. Taking the special case $s=0$, the integral reduces to: $$\begin{aligned}
&\frac1{2\pi}\int_{v=-\infty}^{\infty}\frac{\ai\bigl(2^{-1/3}iv-4^{1/3}x\bigr)}{\ai\bigl(2^{-1/3}iv\bigr)}\int_{t=0}^{\infty}e^{ivt-\tfrac23t^3}\,dt\,dv
=\tfrac12\int_{v=-\infty}^{\infty}\frac{\ai\bigl(iv-4^{1/3}x\bigr)}{\ai\bigl(iv\bigr)}\,\hi(iv)\,dv,\end{aligned}$$ where $\hi$ denotes Scorer’s function $\hi$ (the transition of the coefficient $2/3$ of $t^3$ in the left-hand side to the coefficient $1/3$ of $t^3$ in the definition of Scorer’s function was made by changes of variables in $t$ and $v$). But to treat the behavior of this integral (and its derivative with respect to $x$) as $x\uparrow0$, we can not take limits inside the integral sign, since we then end up with divergent integrals. For the function $\hi$ has the asymptotic expansion: $$\hi(z)\sim -\frac{1}{\pi z}\sum _{{k=0}}^{{\infty}}\frac{(3k)!}{k!(3z^{3})^{k}},\,\qquad\,|\text{ph}(-z)|<\tfrac23\pi-\d$$ for $\d>0$ arbitrarily small, where $\text{ph}(-z)$ denotes the phase of $-z$, and if we put $x$ equal to zero inside the integral we are stuck with a non-integrable integrand, whereas in fact: $$\lim_{x\uparrow0}\tfrac12\int_{v=-\infty}^{\infty}\frac{\ai\bigl(iv-4^{1/3}x\bigr)}{\ai\bigl(iv\bigr)}\,\hi(iv)\,dv=\lim_{x\uparrow0}Q^{(0,x)}\left\{\t_0<\infty\right\}=1.$$
For this reason part (ii) of Theorem \[th:stopping\_time\] was not directly used in the derivation of density $f_Z$ in [@gro:89], but instead the limit $$Q^{(s,x)}\left\{\t_0=\infty\right\}=\lim_{t\to\infty}Q^{(s,x)}\left\{\t_0>t\right\}$$ was computed by first determining the transition density $$Q^{(s,x)}\left\{X_t^{\partial}\in dy\right\},\,t>s,\,x,y<0,$$ of the process $X_t^{\partial}$, which is the process $X_t$, killed when reaching $0$. The details of this computation were given in the appendix of [@gro:89], giving the result: $$\label{result89}
Q^{(s,x)}\left\{\t_0=\infty\right\}=\frac{e^{\tfrac23s^3+2sx}}{4^{1/3}}\int_{v=-\infty}^{\infty}e^{-isv}\frac{\ai(i\xi)\bi\bigl(i\xi-4^{1/3}x\bigr)-\ai\bigl(i\xi-4^{1/3}x\bigr)\bi\bigl(i\xi\bigr)}{\ai\bigl(i\xi\bigr)}\,dv,$$ where $\xi=2^{-1/3}v$, see Theorem 3.1 of [@gro:89]. So by (\[fundamental\_relation\]) we must have the analytic relation $$\begin{aligned}
\label{analytic_relation}
&\frac{e^{\tfrac23s^3+2sx}}{2\pi}\int_{v=-\infty}^{\infty}\frac{\ai\bigl(i\xi-4^{1/3}x\bigr)}{\ai\bigl(i\xi\bigr)}\int_{t=0}^{\infty}e^{itv-\tfrac23(s+t)^3}\,dt\,dv\nonumber\\
&=1-\frac{e^{\tfrac23s^3+2sx}}{4^{1/3}}\int_{v=-\infty}^{\infty}e^{-isv}\frac{\ai\bigl(i\xi\bigr)\bi\bigl(i\xi-4^{1/3}x\bigr)-\ai\bigl(i\xi-4^{1/3}x\bigr)\bi\bigl(i\xi\bigr)}{\ai\bigl(i\xi\bigr)}\,dv,\qquad\xi=2^{-1/3}v.\end{aligned}$$
Conversely, if we can prove the analytic relation (\[analytic\_relation\]), we have an easy road to Theorem 3.1 of [@gro:89] and the derivation of the density $f_Z$. We call the function $$z\mapsto\frac1{\pi}\int_{t=s}^{\infty}e^{tz-\tfrac13t^3}\,dt$$ an [*incomplete Scorer function*]{}, corresponding to the (complete) Scorer function $$z\mapsto\hi(z)=\frac1{\pi}\int_{t=0}^{\infty}e^{tz-\tfrac13t^3}\,dt.$$
In the present paper we first prove in Section \[section:analytic\_relation\] relation (\[analytic\_relation\]) by showing that both integrals, as a function of the parameters $s$ and $x$, satisfy the same extended heat equation. Section \[section:max+locmax\] discusses the derivation of the distribution of the maximum and location of maximum of one-sided or two-sided Brownian motion with a negative parabolic drift from these results. The appendices contain further details on the results.
A parabolic partial differential equation and the analytic relation (\[analytic\_relation\]) {#section:analytic_relation}
============================================================================================
We start with the following lemma.
\[lemma:function1\] Let the function $f:\R\times(-\infty,0)\to \R$ be defined by $$\label{an_representation1}
f(s,x)=\frac1{2\pi}\int_{v=-\infty}^{\infty}\frac{\ai\bigl(i\xi-4^{1/3}x\bigr)}{\ai\bigl(i\xi\bigr)}\int_{t=0}^{\infty}e^{itv-\tfrac23(s+t)^3}\,dt\,dv,\qquad \xi=2^{-1/3}v.$$ Then $f$ satisfies the partial differential equation $$\label{pde1}
\frac{\partial}{\partial s}f(s,x)
=-\tfrac12\frac{\partial^2}{\partial x^2}f(s,x)-2xf(s,x).$$ Moreover $0\le f(s,x)\le e^{-2sx-\tfrac23s^3}$ and $$\label{boundary_condition1}
\lim_{x\uparrow0}f(s,x)=e^{-\tfrac23s^3},\qquad \lim_{s\to\infty}f(s,x)=0,\quad\,x<0,\qquad\lim_{x\to-\infty}e^{2sx}f(s,x)=0,\quad\, s\in\R.$$
The proof follows from the following observations.\
**First Observation:** Let $\tau_{A}=\inf\{t:X_{t}=A\}$, and define $u (s,x)=u(s,x;A)= Q^{(s,x)}\{\tau_{A}<\infty \}$. The process $\{X_{t}:t\ge s\}$ under $Q^{(s,x)}$ is a diffusion process with a time-dependent generator, obtained by subtracting $2t(d/dx)$ from the generator of standard Brownian motion. Consequently, standard arguments from Markov process theory yield: $$\label{eq:generatorEqn}
\frac{\partial u}{\partial s} = -\frac{1}{2}
\frac{\partial^{2}u}{\partial x^{2}} +2s \frac{\partial
u}{\partial x}$$ in the region $x<A$.
**Second Observation:** Let $u (s,x)$ and $f (s,x)$ be functions satisfying the relation $$\label{eq:expTilt}
f (s,x)=e^{-2sx-\tfrac23s^3}u (s,x).$$ Then $u$ satisfies the PDE if and only if $f$ satisfies the PDE (\[pde1\]), as can be seen by routine calculus.\
**Third Observation:** Relation (\[fundamental\_relation\]) (which follows from Theorem \[th:stopping\_time\] by the Laplace inversion (\[def\_h\_x\])) shows that the function $u
(s,x)=Q^{(s,x)}\{\tau_{0}<\infty \}$ is related to the function $f
(s,x)$ of the lemma by the transformation above. Since $u$ satisfies , it now follows immediately that $f$ satisfies (\[pde1\]).\
The boundary conditions follow immediately from the probabilistic interpretation of the function $u$.
It turns out that the right-hand side of (\[analytic\_relation\]) has a more convenient representation, which generalizes relation (2.3) of Lemma 2.2 in [@nico_piet:11] (see also Remark 2.1 in [@nico_piet:11] on the equivalent relation (5.10) in [@svante:10]).
\[lemma:function2\_rep\] Let the function $g:\R\times(-\infty,0]\to \R$ be defined by $$\label{an_representation2}
g(s,x)=\frac1{4^{1/3}}\int_{v=-\infty}^{\infty}e^{-isv}\frac{\ai(i\xi)\bi\bigl(i\xi-4^{1/3}x\bigr)-\ai\bigl(i\xi-4^{1/3}x\bigr)\bi(i\xi)}{\ai(i\xi)}\,dv,\qquad \xi=2^{-1/3}v.$$ Then $g$ has the alternative representation $$\label{function2_rep}
g(s,x)=\frac{e^{-2sx}}{2\pi}\int_{u=-\infty}^{\infty}\frac{\int_{y=0}^{-4^{1/3}x}e^{-2^{1/3}s(iu+y)}\ai(iu+y)\,dy}{\ai(iu)^2}\,du.$$
By the definition of the function $g$, we have: $$\begin{aligned}
g(s,x)=2^{-1/3}\int_{-\infty}^{\infty}e^{-2^{1/3}isu}\frac{\ai(iu)\bi\bigl(iu-4^{1/3}x\bigr)-\bi(iu)\ai\bigl(iu-4^{1/3}x\bigr)}{\ai(iu)}\,du.\end{aligned}$$ For simplicity of notation, we consider instead: $$\begin{aligned}
\tilde g(s,x)&=\tfrac12\int_{-\infty}^{\infty}e^{-isu}\frac{\ai(iu)\bi(iu+x)-\bi(iu)\ai(iu+x)}{\ai(iu)}\,du
,\quad s\in\R,\,x>0.\end{aligned}$$ It is shown in Section \[section:appendixB\] that the function $x\mapsto \tilde g(s,x)$ satisfies the first order differential equation: $$\begin{aligned}
\label{DE_g}
\frac{\partial}{\partial x}\tilde g(s,x)=s\tilde g(s,x)+\frac1{2\pi}\int_{u=-\infty}^{\infty}e^{-isu}\frac{\ai(iu+x)}{\ai(iu)^2}\,du.\end{aligned}$$ So if $\tilde g(s,0)=0$, the solution is given by: $$\tilde g(s,x)=\frac{e^{sx}}{2\pi}\int_{u=-\infty}^{\infty}\frac{\int_{y=0}^x e^{-s(iu+y)}\ai(iu+y)\,dy}{\ai(iu)^2}\,du.$$ Transferring this result to the function $g$ and using $g(s,0)=0$, we get that the corresponding linear differential equation for $x\mapsto g(s,x)$ has the solution given by (\[function2\_rep\]).
\[lemma:function2\] Let the function $g:\R\times(-\infty,0]\to \R$ be defined as in Lemma \[lemma:function2\_rep\]. Then $g$ satisfies the partial differential equation $$\label{pde2}
\frac{\partial}{\partial s}g(s,x)
=-\tfrac12\frac{\partial^2}{\partial x^2}g(s,x)-2xg(s,x).$$ Moreover: $$\label{boundary_condition2}
\lim_{x\uparrow0}g(s,x)=0,\qquad\lim_{x\to-\infty}e^{2sx}g(s,x)=e^{-\tfrac23s^3},\qquad s>0.$$
We have: $$\begin{aligned}
&\frac{\partial^2}{\partial x^2}\frac{\ai(i\xi)\bi\bigl(i\xi-4^{1/3}x\bigr)-\ai\bigl(i\xi-4^{1/3}x\bigr)\bi(i\xi)}{\ai(i\xi)}\\
&=2(iv-2x)\frac{\ai(i\xi)\bi\bigl(i\xi-4^{1/3}x\bigr)-\ai\bigl(i\xi-4^{1/3}x\bigr)\bi(i\xi)}{\ai(i\xi)}\,,\qquad \xi=2^{-1/3}v.\end{aligned}$$ We also have: $$\frac{\partial}{\partial s}g(s,x)=-\frac1{4^{1/3}}\int_{v=-\infty}^{\infty}ive^{-isv}\frac{\ai(i\xi)\bi\bigl(i\xi-4^{1/3}x\bigr)-\ai\bigl(i\xi-4^{1/3}x\bigr)\bi(i\xi)}{\ai(i\xi)}\,dv\,.$$ This yields (\[pde2\]). It is clear from the definition (\[an\_representation2\]) that $\lim_{x\uparrow0}g(s,x)=0$ for all $s\in\R$. A stronger version of the second part of (\[boundary\_condition2\]) is proved in Section \[section:appendixC\].
The preceding two lemmas give the desired result (\[analytic\_relation\]).
\[th:solution\_relation\]
1. Let the functions $f$ and $g$ be defined as in Lemmas \[lemma:function1\] to \[lemma:function2\]. Then we have: $$f(s,x)=e^{-2sx-\tfrac23s^3}-g(s,x),\quad s\in\R,\quad x\le0,$$ where $f(s,0)$ is defined by taking the limit of $f(s,x)$, as $x\uparrow0$.
2. Let, for $s\in\R$ and $x\le0$, $\{X_t:t\in\R\}=\{W(t)-t^2:t\in\R\}$ be Brownian motion with a negative parabolic drift, starting at $x$ at time $s$, with corresponding probability measure $Q^{(s,x)}$. Then $$\label{rep1}
Q^{(s,x)}\{\t_0<\infty\}=e^{2sx+\tfrac23s^3}f(s,x),$$ and $$\label{rep2}
Q^{(s,x)}\{\t_0=\infty\}=e^{2sx+\tfrac23s^3}g(s,x)=\frac{e^{\tfrac23s^3}}{2\pi}\int_{u=-\infty}^{\infty}\frac{\int_{y=0}^{-4^{1/3}x}e^{-2^{1/3}s(iu+y)}\ai(iu+y)\,dy}{\ai(iu)^2}\,du.$$
(i). The function $$(s,x)\mapsto e^{-2sx-\tfrac23s^3},\quad\,(s,x)\in\R^2,$$ satisfies the same partial differential equation as the functions $f$ and $g $ of Lemmas \[lemma:function1\] and \[lemma:function2\]. We have to show: $$\label{maximum_princ}
h(s,x)\stackrel{\text{\small def}}=f(s,x)+g(s,x)-e^{-2sx-\tfrac23s^3}=0,\qquad s\in\R,\qquad x\le0,$$ defining $f(s,0)$ and $g(s,0)$ by the limits of $f(s,x)$ and $g(s,x)$ as $x\uparrow0$, respectively. To show that (\[maximum\_princ\]) holds, we use the maximum principle.
First of all, (\[maximum\_princ\]) holds for all $s$ if $x=0$ by Lemmas \[lemma:function1\] and \[lemma:function2\]. It is shown in Section \[section:appendixC\] that also $$\label{max_bound1}
\lim_{x\to-\infty}h(s,x)=0,\qquad\forall s\in\R,$$ and $$\label{max_bound2}
\lim_{s\to\infty}h(s,x)=0,\qquad\forall x<0.$$ We now consider an infinite rectangle $R_c=\{(s,x) : s\ge c, x \le 0\}$, for some $c\in\R$. Suppose that $h$ attains a strictly positive maximum over $R_c$ at an interior point $(s_0,x_0)\in R_c^0$. Then $\partial_1h(s_0,x_0)=0$, denoting the derivative w.r.t. the $i$th argument by $\partial_i$. Hence, since $h$ satisfies the same partial differential equation as $f$ and $g$, we get: $$0=\partial_1h(s_0,x_0)=-\tfrac12\partial_2^2h(s_0,x_0)-2x_0h(s_0,x_0),$$ implying $$\partial_2^2h(s_0,x_0)=-4x_0h(s_0,x_0)>0,$$ since $x_0<0$. But this contradicts the assumption that $h$ attains its maximum at $(s_0,x_0)$. Similarly, if $h$ attains a strictly negative minimum at an interior point $(s_0,x_0)\in R_c^0$, we would get $\partial_2^2h(s_0,x_0)<0$, again giving a contradiction. So a strictly positive maximum or strictly negative minimum over $R_c$ can only be attained on the line $s=c$. Suppose that a strictly positive maximum is attained at the point $(c,x_0)$, where $x_0<0$. Then we must have: $\partial_1 h(c,x_0)\le0$, implying by the partial differential equation for $h$: $$\partial_2^2 h(c,x_0)\ge -4x_0h(c,x_0)>0,$$ contradicting the assumption that $h$ attains its maximum on the line $s=c$ at the point $(c,x_0)$.
In a similar way we get a contradiction if we assume that $h$ attains a strictly negative minimum on the line $s=c$. So the conclusion is that $h$ is identically zero on $R_c$. Since the argument holds for all $c\in\R$, we get that the function $h$ is identically zero on $\R\times(-\infty,0]$.\
(ii) This follows from (\[fundamental\_relation\]), Lemmas \[lemma:function1\] to \[lemma:function2\], and (i).
The distribution of the maximum and location of maximum of one-sided and two-sided Brownian motion with parabolic drift. {#section:max+locmax}
========================================================================================================================
Let $M$ denote the maximum of the process $\{X_t=W(t)-t^2:t\ge s\}$, starting at $x$ at time $s$, with corresponding probability measure $Q^{(s,x)}$. Moreover, let, with a slight abuse of notation, $\t_M$ denote the location of the maximum $M$ of this process. The following theorem gives the joint distribution of $\t_M$ and $M$ under $Q^{(s,x)}$.
\[th:max+locmax\] Let the function $k$ be defined by $$\label{def_k}
k(s,x)=\frac{\partial}{\partial x}Q^{(s,x)}\left\{\t_0<\infty\right\},$$ where $Q^{(s,x)}$ is the probability measure, corresponding to the process $\{X_t=W(t)-t^2:t\ge s\}$, starting at $x$ at time $s$. Moreover, let $k(t,0)=\lim_{x\uparrow0}k(t,x)$ for all $t\in\R$. Then
1. $$\label{stopping_time_dist}
Q^{(s,x)}\left\{\t_0<\infty\right\}
=e^{\tfrac23s^3+2sx}f(s,x)=1-e^{\tfrac23s^3+2sx}g(s,x)$$
where the functions $f$ and $g$ are defined as in Lemma \[lemma:function1\] and Lemma \[lemma:function2\], respectively, and $$\label{k(s,0)}
k(s,0)=\lim_{x\downarrow0}\frac{\partial}{\partial x}Q^{(s,x)}\left\{\t_0<\infty\right\}=\frac{e^{\tfrac23s^3}}{4^{1/3}\pi}\int_{v=-\infty}^{\infty}\frac{e^{-isv}}{\ai(i2^{-1/3}v)}\,dv.$$
2. The function $a\mapsto k(s,x-a)$ is the density of the maximum $M$ at $a>x$ under the probability measure $Q^{(s,x)}$.
3. The joint density of $\t_M$ and $M$ is given by: $$\begin{aligned}
\label{joint1}
f_{(\t_M,M)}(t,a)
&=e^{-\tfrac23\bigl(t^3-s^3\bigr)+2s(x-a)}h_{x-a}(t-s)k(t,0)\nonumber\\
&=\frac{e^{-\tfrac23s^3+2s(x-a)}h_{x-a}(t-s)}{\pi}\int_{v=-\infty}^{\infty}\frac{e^{-itv}}{\ai(i\xi)}\,dv,\quad a>x,\quad t>s,\end{aligned}$$ where $h_{x-a}$ is defined as in part (ii) of Theorem \[th:stopping\_time\], that is: $$h_{x-a}(u)=\frac1{2\pi}\int_{v=-\infty}^{\infty}e^{iuv}
\frac{\ai\bigl(i2^{-1/3}v-4^{1/3}(x-a)\bigr)}{\ai\bigl(i2^{-1/3}v\bigr)}\,dv.$$
\(i) (\[stopping\_time\_dist\]) is relation (\[analytic\_relation\]), which follows from Theorem \[th:solution\_relation\] in Section \[section:analytic\_relation\], and (\[k(s,0)\]) follows from the representation in the right-hand side of (\[analytic\_relation\]) by taking the derivative w.r.t. $x$, letting $x\downarrow0$ and using that the Wronskian of the two solutions $\ai$ and $\bi$ of the Airy differential equation equals $1/\pi$.\
(ii) By a space homogeneity argument, the density of the maximum $M$ under $Q^{(s,x)}$ is given by $$\begin{aligned}
&-\frac{\partial}{\partial a}Q^{(s,x)}\{M>a\}=-\frac{\partial}{\partial a}Q^{(s,x)}\{\t_a<\infty\}
=-\frac{\partial}{\partial a}Q^{(s,x-a)}\{\t_0<\infty\}\\
&=\frac{\partial}{\partial x}Q^{(s,x-a)}\{\t_0<\infty\}=k(s,x-a).\end{aligned}$$ (iii) Since, if the process starts at $(s,x)$, with $s<t$ and $x<a$, we only can have $\t_M<t$ and $M<a$ if $M=b\in(x,a)$ and $\t_M=u\in(s,t)$, we have: $$\begin{aligned}
Q^{(s,x)}\left\{\t_M<t,\,M<a\right\}=\int_{b=x}^a\int_{u=s}^t Q^{(s,x)}\left\{\t_b\in du\right\}k(u,0)\,db,\end{aligned}$$ where $k(u,0)$ corresponds to the event that the maximum of the path, started at $b$ at time $u$, stays below $b$. Hence differentiation gives: $$\begin{aligned}
f^{(s,x)}_{\t_M,M}(t,a)\,dt&= Q^{(s,x)}\left\{\t_a\in dt\right\}k(t,0)
=Q^{(s,x-a)}\left\{\t_0\in dt\right\}k(t,0)\\
&=e^{-\tfrac23\bigl(t^3-s^3\bigr)+2s(x-a)}h_{x-a}(t-s)k(t,0)\,dt,\end{aligned}$$ where we use part (ii) of Theorem \[th:stopping\_time\] in the last equality.
As a corollary we get the corresponding result for two-sided Brownian motion.
\[cor:loc+locmax\_BM\] Let $X_t=W(t)-t^2$, where $\{W(t):t\in\R\}$ is two-sided Brownian motion, originating from zero. Furthermore, let $M$ and $\t_M$ be the maximum and the location of the maximum of the process $\{X_t:t\in\R\}$, respectively. Then the joint density of $(\t_M,M)$ is given by $$\label{joint-2sided}
f_{(\t_M,M)}(t,a)=h_{-a}(|t|)g(0,-a)\f(|t|),$$ where $h_{-a}$ is defined as in part (ii) of Theorem \[th:stopping\_time\], $g(t,-a)$ by (\[an\_representation2\]) of Lemma \[lemma:function2\], and $\phi$ by: $$\label{phi}
\f(t)=\frac{1}{4^{1/3}\pi}\int_{v=-\infty}^{\infty}\frac{e^{-itv}}{\ai(i2^{-1/3}v)}\,dv,\,t\in\R.$$
Let $t>0$ and let $\M_+$ and $\t_{M+}$ be the maximum and the location of the maximum for the one-side process to the right of zero. By part (iii) of Theorem \[th:max+locmax\], the density of $(\t_{M+},M_+)$ is given by (\[joint1\]), which, since $s=x=0$, boils down to $$\frac{h_{-a}(t)}{4^{1/3}\pi}\int_{v=-\infty}^{\infty}\frac{e^{-itv}}{\ai(i\xi)}\,dv.$$ If we want to turn this into the density of the global maximum and location of maximum on $\R$, we have to multiply this density with the probability that the maximum left of zero is less than $M+$, which means, using a symmetry argument, that the density becomes: $$f_{(\t_M,M)}(t,a)=\frac{h_{-a}(t)Q^{(0,0)}\{\t_a=\infty\}}{4^{1/3}\pi}\int_{v=-\infty}^{\infty}\frac{e^{-itv}}{\ai(i\xi)}\,dv
=\frac{h_{-a}(t)Q^{(0,-a)}\{\t_0=\infty\}}{4^{1/3}\pi}\int_{v=-\infty}^{\infty}\frac{e^{-itv}}{\ai(i\xi)}\,dv,$$ where $\xi=2^{-1/3}v$. By (\[stopping\_time\_dist\]) of Theorem \[th:max+locmax\] we now get: $$f_{(\t_M,M)}(t,a)
=\frac{h_{-a}(t)g(0,-a)}{4^{1/3}\pi}\int_{v=-\infty}^{\infty}\frac{e^{-itv}}{\ai(i\xi)}\,dv
=h_{-a}(t)g(0,-a)\f(t).$$ The case where the maximum is reached to the left of zero is treated in a similar way.
[Note that the function $\f$, defined by (\[phi\]), has the following probabilistic interpretation: $$\label{phi_interpretation}
\f(t)=-e^{-\tfrac23t^3}\frac{\partial}{\partial x}Q^{(t,x)}\{\t_0=\infty\}\biggr|_{x=0}.$$ This interpretation can perhaps easiest be seen from the representation (\[rep2\]) in Theorem \[th:max+locmax\]. The function defines the density of the location of the maximum, as is seen in the following Corollary \[cor:Chernoff\]. ]{}
\[cor:Chernoff\] Let $X_t=W(t)-t^2$, where $\{W(t):t\in\R\}$ is two-sided Brownian motion, originating from zero. Then the density of the location of the maximum $\t_M$ is given by: $$f_{\t_M}(t)=\tfrac12\f(t)\f(-t),$$ where $\f$ is defined by (\[phi\]).
We have by Corollary \[cor:loc+locmax\_BM\] and Theorem \[th:max+locmax\] for $t\in\R$: $$\begin{aligned}
f_{\t_M}(t)&=\f(t)\int_{x=0}^{\infty}h_{-x}(t)g(0,-x)\,dx.\end{aligned}$$ Now let $\psi$ be defined by $$\label{def_psi}
\psi(t)=\int_{x=0}^{\infty}h_{-x}(t)g(0,-x)\,dx.$$ Then we have to show: $$\label{key_Chernoff}
\psi(t)=\tfrac12\f(-t).$$ Since, using a time reversal argument, the density obviously has to be symmetric, we only have to prove (\[key\_Chernoff\]) for all $t\ge0$. The equality is derived in the proof of Lemma \[lemma:psi\_phi\] in Section \[section:appendixD\] by an (asymptotic) analytic argument.
[Note that the integrand on the right-hand side of (\[def\_psi\]) is the product of the density of the first hitting time $\t_{x}$ under $Q^{(0,0)}$ and the probability that the drifting process stays below $x$ under $Q^{(0,0)}$. The latter probability can also be interpreted as the probability that the process $\{X_t=W(t)-t^2,\,t\le0\}$, starting at zero and running to the left, stays below $x$. So, intuitively, the product $h_{-x}(t)g(0,-x)\f(t)$ corresponds to paths of two-sided Brownian motion minus a parabola, having their first hitting time of $x$ at time $t>0$ (the factor $h_{-x}(t)g(0,-x)$), and staying below $x$ on the interval $[t,\infty)$ (the factor $\f(t)$). The factor $\exp\{-\tfrac23t^3\}$ in (\[phi\_interpretation\]) disappears, since by part (ii) of Theorem \[th:stopping\_time\], $$h_{-x}(t)\,dt=e^{\tfrac23t^3}Q^{(0,x)}\left\{\t_0\in dt\right\}.$$ The factor $\tfrac12$ in front of the product is introduced by going from a derivative in the space variable $x$ in (\[phi\_interpretation\]) to a derivative in the time variable $t$. This seems a bit different from the way the factor $\tfrac12$ entered in Chernoff’s argument as the expectation of the squared maximum of the Brownian bridge.]{}
Concluding remarks {#section:conclusion}
==================
We gave a direct approach to Chernoff’s theorem and other results of this type, using the Feynman-Kac formula with a stopping time and the analytic relation (\[analytic\_relation\]). Relation (\[analytic\_relation\]) is proved by showing that the integrals in this relation satisfy the parabolic partial differential equation (\[pde1\]) as a function of the parameters $s$ (time) and $x$ (space) and by an application of the maximum principle. As shown in [@piet:10b] and [@svante:10], these results also give the distribution of the maximum of Brownian motion minus a parabola itself, both for the one-sided and two-sided case. An asymptotic development of the tail of the distribution of this maximum is given in [@nico_piet:11]. We hope that the direct approach of the present paper will make these results more accessible. The original proofs in [@gro:89] were rather long and technical, and lacked this property.
It is proved in [@piet:11d] and again in [@svante:13] that the maximum $M$ and the location of the maximum $\t_M$ of two-sided Brownian motion minus the parabola $t^2$ satisfy the relation $$E\t_M^2=\tfrac13EM.$$ This result is generalized in [@leandro:12], where the relation is proved not using Airy functions, and where a completely general result of this type is given for drifting Brownian motion. More results for moments and combinatorics for Airy integrals are given in [@svante:13]. The latter paper ends with a series of conjectures and open problems in this area.
For computational purposes, the representation (\[function2\_rep\]) in Lemma \[lemma:function2\_rep\] seems the best choice, since we lose in this way the inconvenient difference of products of the Airy functions $\ai$ and $\bi$ on the right side of (\[analytic\_relation\]), which are not integrable along the imaginary axis by themselves, and we also do not have the trouble near zero that the function on the left of relation (\[analytic\_relation\]), further analyzed in Lemma \[lemma:function1\], is exhibiting.
Appendix A {#section:FK}
==========
In this section we prove Theorem \[th:stopping\_time\]. We start with the Feynman-Kac part.
Let, for $\l>0$, $u_{\l}$ be the unique non-negative solution of the boundary problem $$\label{Airy-equation}
\tfrac12u''(x)-\bigl(\l-2x\bigr)u(x)=0,\,x<0,\qquad\lim_{x\uparrow0} u(x)=1,\qquad u(x)\le1,\, x\le0.$$ The unique solution of (\[Airy-equation\]) is given by $$\label{u_lambda}
u_{\l}(x)=\frac{\ai\bigl(2^{-1/3}\l-4^{1/3}x\bigr)}{\ai\bigl(2^{-1/3}\l\bigr)},\,x\le0.$$ We now consider the process $$Y_t=e^{-\int_{v=s}^t \bigl(\l-2X_v\bigr)\,dv}u_{\l}(X_t),$$ where $X_t$ is standard Brownian motion, starting at $x<0$ at time $0$. By Itô’s formula and (\[Airy-equation\]) we have: $$du(X_t)=u_{\l}'(X_t)dX_t+\tfrac12u_{\l}''(X_t)\,dt=u_{\l}'(X_t)dX_t+\bigl(\l-2X_t\bigr)u_{\l}(X_t)\,dt.$$ So we get: $$\begin{aligned}
dY_t&=-u_{\l}(X_t)e^{-\int_{v=0}^t \bigl(\l-2X_v\bigr)\,dv}\bigl(\l-2X_t\bigr)\,dt\\
&\qquad\qquad+e^{-\int_{v=0}^t \bigl(\l-2X_v\bigr)\,dv}\left\{u_{\l}'(X_t)dX_t+\bigl(\l-2X_t\bigr)u_{\l}(X_t)\,dt\right\}\\
&=e^{-\int_{v=0}^t \bigl(\l-2X_v\bigr)\,dv}u_{\l}'(X_t)\,dX_t,\end{aligned}$$ implying that $Y_t$ is a local martingale and that $$Y_{\t_0}-x=\int_{t=0}^{\t_0}e^{-\int_{v=0}^t \bigl(\l-2X_v\bigr)\,dv}u_{\l}'(X_t)\,dX_t.$$ Moreover, $t\mapsto Y_{t\wedge \tau_0}$ is a bounded martingale, and hence: $$E^{x}e^{-\int_{v=0}^{\t_0}\bigl(\l-2X_v\bigr)\,dv}
=E^{x}e^{-\int_{v=0}^{\t_0}\bigl(\l-2X_v\bigr)\,dv}u_{\l}(X_{\t_0})=E^{x}Y_{\t_0}=Y_s=u_{\l}(x),$$ with the conclusion that $$\int_{t=0}^{\infty}e^{-\l t}E^x\left\{e^{\int_{v=0}^{t} 2X_v\,dv}\Bigm|\t_0=t\right\}P^{x}\left\{\t_0\in dt\right\}=u_{\l}(x)=\frac{\ai\bigl(2^{-1/3}\l-4^{1/3}x\bigr)}{\ai\bigl(2^{-1/3}\l\bigr)},\quad x<0,$$ where $P^x$ denotes the probability measure of Brownian motion in standard scale, starting at $x$ at time $0$.
We now turn to the Cameron-Martin-Girsanov part of the proof. Suppose that $P^{(s,x)}$ and $Q^{(s,x)}$ are the probability measures of continuous paths from $(s,x)$ such that:
1. under $P^{(s,x)}$, $\{X_t:t\ge s\}$ is standard Brownian motion with $X_s=x$,
2. under $Q^{(s,x)}$, $\{X_t:t\ge s\}$ is standard Brownian motion $-t^2$, with $X_s=x$.
Then, by the Cameron-Martin-Girsanov formula, $Q^{(s,x)}<<P^{(s,x)}$ on $\{{\cal F}_t:t\ge s\}$, where ${\cal F}_t=\s\{X_u:u\in[s,t]\}$ and $$\frac{dQ^{(s,x)}}{dP^{(s,x)}}\Bigr|_{{\cal F}_t}=Z_t,$$ where $$Z_t=\exp\left\{-2\int_s^t u\,dX_u-\tfrac23\bigl(t^3-s^3\bigr)\right\}=
\exp\left\{2\int_s^t X_u\,du-2\bigl(tX_t-sX_s\bigr)-\tfrac23\bigl(t^3-s^3\bigr)\right\},\quad t\ge s.$$ This implies that we have: $$\begin{aligned}
&Q^{(s,x)}\left\{\t_0\in dt\right\}\\
&=\exp\left\{2sx-\tfrac23\bigl(t^3-s^3\bigr)\right\}E^{P^{(s,x)}}\left\{\exp\left\{2\int_s^t X_u\,du\right\}\bigm|\t_0=t\right\}
P^{(s,x)}\left\{\t_0\in dt\right\}.\end{aligned}$$ This is part (i) of Theorem \[th:stopping\_time\]. The interpretation in terms of a Bessel process, given in (i) of Theorem \[th:stopping\_time\] is standard (and further detailed in [@gro:89]).
Moreover, by time homogeneity and the Feynman-Kac argument, given above, we have $$\begin{aligned}
&E^{P^{(s,x)}}\left\{\exp\left\{2\int_s^t X_u\,du\right\}\bigm|\t_0=t\right\}
P^{(s,x)}\left\{\t_0\in dt\right\}\\
&=E^{P^{(0,x)}}\left\{\exp\left\{2\int_0^{t-s} X_u\,du\right\}\bigm|\t_0=t-s\right\}
P^{(0,x))}\left\{s+\t_0\in dt\right\}=h_x(t-s)\,dt\end{aligned}$$ where the function $h_{x}:\R_+\to\R_+$ has Laplace transform $$\begin{aligned}
\hat h_{x}(\l)=\int_0^{\infty}e^{-\l u}h_{x}(u)\,du
=\ai\bigl(\xi-4^{1/3}x\bigr)/\ai(\xi),\,\quad \xi=2^{-1/3}\l>0.\end{aligned}$$ This gives part (ii) of Theorem \[th:stopping\_time\].
Appendix B {#section:appendixB}
==========
\[lemma:DE\_g\] Let, for $s\in\R$ and $x\ge0$, the function $\tilde g$ be defined by $$\label{def_tilde_g}
\tilde g(s,x)=\tfrac12\int_{-\infty}^{\infty}e^{-isu}\frac{\ai(iu)\bi(iu+x)-\bi(iu)\ai(iu+x)}{\ai(iu)}\,du.$$ Then, for each $s\in\R$, the function $x\mapsto\tilde g(s,x)$ satisfies the differential equation $$\label{DE_g2}
\frac{\partial}{\partial x}\tilde g(s,x)=s\tilde g(s,x)+\frac1{2\pi}\int_{u=-\infty}^{\infty}e^{-isu}\frac{\ai(iu+x)}{\ai(iu)^2}\,du.$$
We have: $$\begin{aligned}
\tilde g(s,x)&=\tfrac12\int_{-\infty}^{\infty}e^{-isu}\frac{\ai(iu)\bi(iu+x)-\bi(iu)\ai(iu+x)}{\ai(iu)}\,du\\
&=e^{-i\pi/6}\int_{0}^{\infty}e^{-isu}\left\{\ai\bigl(e^{-i\pi/6}u+e^{-2i\pi/3}x\bigr)-\frac{\ai\bigl(e^{-i\pi/6}u\bigr)\ai(iu+x)}{\ai(iu)}\right\}\,du\\
&\qquad\qquad+e^{i\pi/6}\int_{0}^{\infty}e^{isu}\left\{\ai\bigl(e^{i\pi/6}u+e^{2i\pi/3}x\bigr)-\frac{\ai\bigl(e^{i\pi/6}u\bigr)\ai(-iu+x)}{\ai(-iu)}\right\}\,du
,\quad s\in\R,\quad x>0.\end{aligned}$$ We now get: $$\begin{aligned}
&\frac{\partial}{\partial x}e^{-i\pi/6}\int_{0}^{\infty}e^{-isu}\ai\bigl(e^{-i\pi/6}u+e^{-2i\pi/3}x\bigr)\,du
=e^{-5i\pi/6}\int_{0}^{\infty}e^{-isu}\ai'\bigl(e^{-i\pi/6}u+e^{-2i\pi/3}x\bigr)\,du\\
&=\left[e^{-2i\pi/3}e^{-isu}\ai\bigl(e^{-i\pi/6}u+e^{-2i\pi/3}x\bigr)\right]_{u=0}^{\infty}
+ise^{-2i\pi/3}\int_{0}^{\infty}e^{-isu}\ai\bigl(e^{-i\pi/6}u+e^{-2i\pi/3}x\bigr)\,du\\
&=-e^{-2i\pi/3}\ai\bigl(e^{-2i\pi/3}x\bigr)
+ise^{-2i\pi/3}\int_{0}^{\infty}e^{-isu}\ai\bigl(e^{-i\pi/6}u+e^{-2i\pi/3}x\bigr)\,du.\end{aligned}$$ Similarly, $$\begin{aligned}
&\frac{\partial}{\partial x}e^{i\pi/6}\int_{-\infty}^{0}e^{-isu}\ai\bigl(e^{i\pi/6}u+e^{2i\pi/3}x\bigr)\,du\\
&=-e^{2i\pi/3}\ai\bigl(e^{2i\pi/3}x\bigr)
-ise^{2i\pi/3}\int_{0}^{\infty}e^{isu}\ai\bigl(e^{i\pi/6}u+e^{2i\pi/3}x\bigr)\,du.\end{aligned}$$ This implies: $$\begin{aligned}
&\frac{\partial}{\partial x}
\left\{e^{-i\pi/6}\int_{0}^{\infty}e^{-isu}\ai\bigl(e^{-i\pi/6}u+e^{-2i\pi/3}x\bigr)\,du
+e^{i\pi/6}\int_{0}^{\infty}e^{isu}\ai\bigl(e^{i\pi/6}u+e^{2i\pi/3}x\bigr)\,du\right\}\\
&=\ai(x)+se^{-i\pi/6}\int_{0}^{\infty}e^{-isu}\ai\bigl(e^{-i\pi/6}u+e^{-2i\pi/3}x\bigr)\,du
+se^{i\pi/6}\int_{0}^{\infty}e^{isu}\ai\bigl(e^{i\pi/6}u+e^{2i\pi/3}x\bigr)\,du,\end{aligned}$$ using $$\ai(x)=-e^{-2i\pi/3}\ai\bigl(e^{-2i\pi/3}x\bigr)-e^{2i\pi/3}\ai\bigl(e^{2i\pi/3}x\bigr).$$ Furthermore, $$\begin{aligned}
&e^{-i\pi/6}\int_{0}^{\infty}e^{-isu}\frac{\ai\bigl(e^{-i\pi/6}u\bigr)\ai'(iu+x)}{\ai(iu)}\,du\\
&=\left[e^{-i\pi/6}e^{-isu}\frac{\ai\bigl(e^{-i\pi/6}u\bigr)\ai(iu+x)}{i\ai(iu)}\right]_{u=0}^{\infty}
-e^{-i\pi/6}\int_{0}^{\infty}\ai(iu+x)\frac{d}{du}\left(e^{-isu}\frac{\ai\bigl(e^{-i\pi/6}u\bigr)}{i\ai(iu)}\right)\,du\\
&=e^{i\pi/3}\ai(x)-\int_{0}^{\infty}\ai(iu+x)\frac{d}{du}\left(\frac{e^{-i\pi/6}\ai\bigl(e^{-i\pi/6}u\bigr)}{i\ai(iu)}\right)\,du\\
&\qquad\qquad\qquad\qquad\qquad\qquad+se^{-i\pi/6}\int_{0}^{\infty}e^{-isu}\frac{\ai\bigl(e^{-i\pi/6}u\bigr)\ai(iu+x)}{i\ai(iu)}\,du.\end{aligned}$$ Since $$\begin{aligned}
\label{h(u)}
h(u)\stackrel{\text{\small def}}=\frac{d}{du}\left(\frac{e^{-i\pi/6}\ai\bigl(e^{-i\pi/6}u\bigr)}{i\ai(iu)}\right)=\frac1{2\pi\ai(iu)^2}\,,\end{aligned}$$ (see the proof of Lemma 2.2 in [@nico_piet:11]), the conclusion now follows.
Appendix C {#section:appendixC}
==========
We first note that, for $s=0$, the limit of $g(s,x)$, as $x\to-\infty$, has been studied in [@nico_piet:11], since it describes the behavior of the maximum of one-sided Brownian motion minus a parabola, starting at time $s=0$. However, this analysis started from the probabilistic interpretation of the right-hand side of relation (\[analytic\_relation\]), which is something we cannot do at this point, since we want to derive this interpretation without assuming (\[analytic\_relation\]). But we can use similar techniques for the asymptotic analysis.
Using the representation (\[function2\_rep\]) in Lemma \[lemma:function2\_rep\], we can write: $$e^{2sx}g(s,x)=\frac{1}{2\pi}\int_{u=-\infty}^{\infty}\frac{\int_{y=0}^{-4^{1/3}x}e^{-2^{1/3}s(iu+y)}\ai(iu+y)\,dy}{\ai(iu)^2}\,du\,.$$ So we get: $$\label{Airy_limit}
\lim_{x\to-\infty}e^{2sx}g(s,x)
=\frac{1}{2\pi}\int_{u=-\infty}^{\infty}\frac{\int_{y=0}^{\infty}e^{-2^{1/3}s(iu+y)}\ai(iu+y)\,dy}{\ai(iu)^2}\,du.$$
We now have the following lemma.
\[lemma:Laplace\_transform\] $$\label{Laplace_transforms}
\frac{1}{2\pi}\int_{u=-\infty}^{\infty}\frac{\int_{y=0}^{\infty}e^{-2^{1/3}s(iu+y)}\ai(iu+y)\,dy}{\ai(iu)^2}\,du
=e^{-\tfrac23s^3},\,\forall s\in\R.$$
Note that $$\begin{aligned}
&\frac{\partial}{\partial s}\int_{y=0}^{\infty}e^{-2^{1/3}s(iu+y)}\ai(iu+y)\,dy=
-2^{1/3}\int_{y=0}^{\infty}e^{-2^{1/3}s(iu+y)}(iu+y)\ai(iu+y)\,dy\\
&=-2^{1/3}\int_{y=0}^{\infty}e^{-2^{1/3}s(iu+y)}\ai''(iu+y)\,dy\\
&=2^{1/3}\ai'(iu)e^{-2^{1/3}isu}+2^{2/3}s\ai(iu)e^{-2^{1/3}isu}-2s^2\int_{y=0}^{\infty}e^{-2^{1/3}s(iu+y)}\ai(iu+y)\,dy.\end{aligned}$$ Hence, defining the functions $p$ and $q$ by $$p(s)=\frac{1}{2\pi}\int_{u=-\infty}^{\infty}\frac{\int_{y=0}^{\infty}e^{-2^{1/3}s(iu+y)}\ai(iu+y)\,dy}{\ai(iu)^2}\,du,\,s\in\R,$$ and $$q(s)=\frac{1}{2\pi}\int_{u=-\infty}^{\infty}\frac{2^{1/3}\ai'(iu)e^{-2^{1/3}isu}+2^{2/3}s\ai(iu)e^{-2^{1/3}isu}}{\ai(iu)^2}\,du,\,s\in\R,$$ we get: $$\begin{aligned}
p'(s)=-2s^2p(s)+q(s).\end{aligned}$$ But we have, by integration by parts, $$\begin{aligned}
&\int_{u=-\infty}^{\infty}\frac{2^{2/3}s\ai(iu)e^{-2^{1/3}isu}}{\ai(iu)^2}\,du
=\int_{u=-\infty}^{\infty}\frac{2^{2/3}se^{-2^{1/3}isu}}{\ai(iu)}\,du
=-2^{1/3}\int_{u=-\infty}^{\infty}\frac{e^{-2^{1/3}isu}\ai'(iu)}{\ai(iu)^2}\,du,\end{aligned}$$ implying $q(s)\equiv0$, and hence: $$p(s)=ce^{-\tfrac23s^3},$$ for some constant $c$. For $s=0$, we get: $$p(0)=\frac{1}{2\pi}\int_{u=-\infty}^{\infty}\frac{\int_{y=0}^{\infty}\ai(iu+y)\,dy}{\ai(iu)^2}\,du.$$
By Cauchy’s theorem, we now have, for $u\in\R$, $$\begin{aligned}
\int_{y=0}^{\infty}\ai(iu+y)\,dy=\int_{y=0}^{\infty}\ai(y)\,dy-i\int_{y=0}^{u}\ai(iy)\,dy
=\frac13-i\int_{y=0}^{u}\ai(iy)\,dy,\end{aligned}$$ where the integrals are interpreted as (directed) Riemann integrals if $u<0$. Hence, since $$\frac{1}{2\pi}\int_{u=-\infty}^{\infty}\frac1{\ai(iu)^2}\,du=1,$$ (see Appendix D), we get: $$p(0)=1/3-\frac{1}{2\pi}\int_{u=-\infty}^{\infty}\frac{i\int_{y=0}^{u}\ai(iv)\,dv}{\ai(iu)^2}\,du=\frac13+\frac23=1,$$ so $c=1$. The conclusion of the lemma now follows.
Relation (\[Airy\_limit\]) and Lemma \[lemma:Laplace\_transform\] yield the second part of (\[boundary\_condition2\]). We therefore have: $$\begin{aligned}
e^{2sx}g(s,x)=e^{-\tfrac23s^3}-\frac{1}{2\pi}\int_{u=-\infty}^{\infty}\frac{\int_{y=-4^{1/3}x}^{\infty}e^{-2^{1/3}s(iu+y)}\ai(iu+y)\,dy}{\ai(iu)^2}\,du,\end{aligned}$$ implying $$\begin{aligned}
\frac{\partial}{\partial x}\left\{e^{2sx}g(s,x)\right\}=-\frac{1}{2^{1/3}\pi}\int_{u=-\infty}^{\infty}\frac{e^{-2^{1/3}s(iu-4^{1/3}x)}\ai(iu-4^{1/3}x)\,dy}{\ai(iu)^2}\,du=O\left(e^{-\tfrac23|x|^{3/2}}\right).\end{aligned}$$ This, in turn, implies that also $$\begin{aligned}
&e^{-2sx-\tfrac23s^3}-g(s,x)=e^{-2sx}\left\{e^{-\tfrac23s^3}-e^{2sx}g(s,x)\right\}\\
&=\frac{e^{-2sx}}{2\pi}\int_{u=-\infty}^{\infty}\frac{\int_{y=-4^{1/3}x}^{\infty}e^{-2^{1/3}s(iu+y)}\ai(iu+y)\,dy}{\ai(iu)^2}\,du
=O\left(e^{-\tfrac23|x|^{3/2}-2sx}\right)=O\left(e^{-\tfrac23|x|^{3/2}(1+o(1))}\right),\end{aligned}$$ and hence, for all $s\in\R$, $$\begin{aligned}
g(s,x)=e^{-\tfrac23s^3-2sx}+o(1),\,x\to-\infty.\end{aligned}$$
The analysis of $f(s,x)$ is easier and shows, by standard methods, that also, for all $s\in\R$, $$e^{2sx}f(s,x)=O\left(e^{-\tfrac23|x|^{3/2}(1+o(1))}\right),\,x\to-\infty,$$ implying $$\lim_{x\to-\infty}f(s,x)=0,\,\forall s\in\R.$$
Finally, again using the representation (\[function2\_rep\]) in Lemma \[lemma:function2\_rep\], we get, for each $x<0$: $$\begin{aligned}
g(s,x)=\frac{e^{-2sx}}{2\pi}\int_{y=0}^{-4^{1/3}x}\left\{\int_{u=-\infty}^{\infty}\frac{e^{-2^{1/3}s(iu+y)}\ai(iu+y)\,du}{\ai(iu)^2}\,du\right\}\,dy.\end{aligned}$$ We write: $$\frac1{2\pi}\int_{u=-\infty}^{\infty}\frac{e^{-2^{1/3}s(iu+y)}\ai(iu+y)\,du}{\ai(iu)^2}\,du
=\frac1{2\pi i}\int_{u=-i\infty}^{i\infty}\frac{e^{-2^{1/3}s(u+y)}\ai(u+y)\,du}{\ai(u)^2}\,du.$$ Shifting the integration path of the latter integral to the right, letting it pass through the approximate saddle point $2^{-2/3}s^2$ on the real axis, we get for fixed $y>0$ by the asymptotic expansion of the Airy function: $$\begin{aligned}
\frac1{2\pi i}\int_{u=-i\infty}^{i\infty}\frac{e^{-2^{1/3}s(u+y)}\ai(u+y)}{\ai(u)^2}\,du
= O\left(s^{3/4}e^{-\tfrac23s^3}\right),\end{aligned}$$ yielding $$\lim_{s\to\infty}g(s,x)=0,\,\forall x<0.$$ The relation $$\lim_{s\to\infty}f(s,x)=0,\,\forall x<0,$$ follows in a similar way.
Appendix D {#section:appendixD}
==========
We have not been able to find a direct reference for the nice relation $$\label{AiryInt}
\frac{1}{2\pi}\int_{u=-\infty}^{\infty}\frac1{\ai(iu)^2}\,du=1,$$ so we provide a proof here. The indefinite integral of $1/\ai(iu)^2$ appears in [@prudnikov:90], p. 30, but further work is needed to obtain (\[AiryInt\]). Let $h$ be given by (\[h(u)\]). We use $$\begin{aligned}
\label{defhu}
h(u)=\frac1{2\pi\ai(iu)^2}=\frac{d g(u)}{du},\quad g(u)=\frac{e^{-i\pi/6}\ai\bigl(e^{-i\pi/6}u\bigr)}{i\ai(iu)}\,.\end{aligned}$$ For the proof we use integration by parts and we need the asymptotic behavior of $g(u)$ as $u\to\pm\infty$.
We have as $z\to\infty$ $$\label{asyai}
\ai(z)\sim \frac{e^{-\zeta}}{2\sqrt{\pi}z^{\frac14}},\quad |\text{ph}(z)|<\pi, \quad \zeta=\frac23z^{\frac32}.$$ This gives, as $u\to+\infty$, $$\begin{aligned}
\label{asyuplus}
g(u)\sim -ie^{-\frac43u^{\frac32}e^{\frac14\pi i}},\end{aligned}$$ which is exponentially small and does not give a contribution when integrating by parts.
For $u\to-\infty$ we write $$\label{gminu}
g(-u)=\frac{e^{-i\pi/6}\ai\bigl(e^{5i\pi/6}u\bigr)}{i\ai(-iu)}\,,$$ and we have, using (\[asyai\]), $$g(-u)\sim -1,\quad u\to+\infty.$$ It follows that we need more details of the asymptotic behavior of the Airy function by using the complete expansion $$\ai(z)\sim \frac{e^{-\zeta}}{2\sqrt{\pi}z^{\frac14}}\sum_{k=0}^\infty (-1)^k\frac{u_k}{\zeta^k},\quad |\text{ph}(z)| <\pi,$$ with $\zeta$ given in (\[asyai\]), and the coefficients $u_k$ do not depend on $z$. The first few are $u_0=1$ and $u_1=\frac{5}{72}$.
For $z=e^{5i\pi/6}u$ and for $z=-iu$ we have the same $\zeta=\frac23u^{3/2} e^{5i\pi/4}=\frac23u^{3/2} e^{-3i\pi/4}$, and we see that the asymptotic series of the Airy functions in (\[gminu\]) are the same as well. We conclude $$g(-u)=-1+O\left(u^{-k}\right),\quad \forall k, \quad u\to+\infty.$$ This proves the relation in (\[AiryInt\]).
Appendix E {#section:appendixE}
==========
\[lemma:psi\_phi\] Let the functions $\f$ and $\psi$ be defined by (\[phi\]) and (\[def\_psi\]), respectively. Then, for all $t\ge0$, $$\psi(t)=\tfrac12\f(-t).$$
By (\[def\_h\_x\]) we have for $t\ge0$ and $x>0$: $$h_{-x}(t)=\frac1{2\pi}\int_{v=-\infty}^{\infty}e^{itv}\frac{\ai(i2^{-1/3}v+4^{1/3}x)}{\ai(i2^{-1/3}v)}\,dv,$$ and by Lemma \[lemma:function2\_rep\] we have: $$g(0,-x)=\frac1{2\pi}\int_{u=-\infty}^{\infty}\frac{\int_{y=0}^{4^{1/3}x}\ai(iu+y)\,dy}{\ai(iu)^2}\,du.$$ Hence we get: $$\begin{aligned}
\label{def_psi2}
\psi(t)&=\frac1{4\pi^2}\int_{x=0}^{\infty}\left\{\int_{v=-\infty}^{\infty}e^{itv}\frac{\ai(i2^{-1/3}v+4^{1/3}x)}{\ai(i2^{-1/3}v)}\,dv\int_{u=-\infty}^{\infty}\frac{\int_{y=0}^{4^{1/3}x}\ai(iu+y)\,dy}{\ai(iu)^2}\,du\right\}\,dx\nonumber\\
&=\frac1{2^{1/3}\cdot4\pi^2}\int_{x=0}^{\infty}\left\{\int_{v=-\infty}^{\infty}e^{i2^{1/3}tv}\frac{\int_{y=x}^{\infty}\ai(iv+y)\,dy}{\ai(iv)}\,dv\int_{u=-\infty}^{\infty}\frac{\ai(iu+x)}{\ai(iu)^2}\,du\right\}\,dx\nonumber\\
&=\frac1{2^{1/3}\cdot4\pi^2}\int_{x=0}^{\infty}\left\{\int_{v=-\infty}^{\infty}e^{i2^{1/3}t(v-u)}\frac{\int_{y=x}^{\infty}\ai(iv+y)\,dy}{\ai(iv)}\,dv\int_{u=-\infty}^{\infty}\frac{e^{i2^{1/3}tu}\ai(iu+x)}{\ai(iu)^2}\,du\right\}\,dx.\end{aligned}$$
Let $t>0$. For convenience of notation, we consider $\psi(2^{-1/3}t)$ instead of $\psi(t)$ (this makes the first factor in the inner integral in the expression on the right-hand side $\exp\{itv\}$ instead of $\exp\{i2^{1/3}tv\}$). Introducing an extra parameter $\e>0$, we first study the behavior of: $$\begin{aligned}
&\int_{y=x\e}^{\infty}\int_{v=-\infty}^{\infty}e^{it(v-u)}\frac{\ai(iv+y)}{\ai(iv)}\,dv\,dy
=\frac1{i}\int_{y=x\e}^{\infty}\int_{v=-i\infty}^{i\infty}e^{tv}\frac{\ai(v+u+y)}{\ai(v+u)}\,dv\,dy.\end{aligned}$$ We move the integration path of the inner integral to a path, parallel to the imaginary axis, and passing through the (approximate saddle) point $\tfrac14(y/t)^2$ on the real axis. This yields, for $y>0$ and $|v|\to\infty$, $$\begin{aligned}
&\frac1i\int_{v=-i\infty}^{i\infty}e^{tv}\frac{\ai(v+u+y)}{\ai(v+u)}\,dv\sim
\exp\left\{-\frac{y^2}{4t}\right\}\frac{y\sqrt{\pi}}{t^{3/2}}\,.\end{aligned}$$ Integration of the expression on the right-hand side on the interval $[x\e,\infty)$ gives: $$\begin{aligned}
\int_{y=x\e}^{\infty}\exp\left\{-\frac{y^2}{4t\e^{2}}\right\}\frac{y\sqrt{\pi}}{t^{3/2}}\,dy=\frac{2e^{-\tfrac14\e^2x^2/t}\sqrt{\pi}}{\sqrt{t}}\,.\end{aligned}$$ So we find: $$\begin{aligned}
\psi(2^{-1/3}t)
&=\lim_{\e\downarrow0}\frac{\e}{2^{1/3}\cdot4\pi^2}\int_{x=0}^{\infty}\int_{y=\e x}^{\infty}\left\{\int_{v=-\infty}^{\infty}e^{it(v-u)}\frac{\ai(iv+y)}{\ai(iv)}\,dv\,dy\int_{u=-\infty}^{\infty}\frac{e^{i2^{1/3}tu}\ai(iu+\e x)}{\ai(iu)^2}\,du\right\}\,dx\\
&=\lim_{\e\downarrow0}\frac{\e}{2^{1/3}\cdot4\pi^2}\int_{x=0}^{\infty}\left\{\sqrt{\pi}\int_0^{\infty}\frac{e^{-\tfrac14\e^2x^2/t}}{\sqrt{t}}\int_{u=-\infty}^{\infty}\frac{e^{itu}\ai(iu+x\e)}{\ai(iu)^2}\,du\right\}\,dx\\
&=\frac{1}{2^{4/3}\pi}\int_{u=-\infty}^{\infty}\frac{e^{itu}}{\ai(iu)}\,du
=\frac12\frac{1}{2^{1/3}\pi}\int_{u=-\infty}^{\infty}\frac{e^{itu}}{\ai(iu)}\,du,\end{aligned}$$ noting that the contribution to the integrals on the right-hand side of the first equality over the region $\{v:|v|\le M\}$ tends to zero for each $M>0$, which implies that we can shift to the asymptotic expansion of the Airy function $\ai$ in the first inner integral. Hence: $$\psi(t)=\frac12\frac{1}{2^{1/3}\pi}\int_{u=-\infty}^{\infty}\frac{e^{i2^{1/3}tu}}{\ai(iu)}\,du=\frac12\frac{1}{4^{1/3}\pi}\int_{u=-\infty}^{\infty}\frac{e^{itu}}{\ai(i2^{-1/3}u)}\,du=\tfrac12\f(-t).$$ At $t=0$ we get $$\psi(0)=\lim_{t\downarrow0}\psi(t)
=\tfrac12\lim_{t\downarrow0}\f(-t)=\tfrac12\f(0),$$ which shows that the relation also holds at $t=0$.
| 2024-05-26T01:27:17.760571 | https://example.com/article/2441 |
# Copyright (C) 2016 Antoine Carme <Antoine.Carme@Laposte.net>
# All rights reserved.
# This file is part of the Python Automatic Forecasting (PyAF) library and is made available under
# the terms of the 3 Clause BSD license
import pandas as pd
import numpy as np
from io import BytesIO
import base64
SIGNAL_COLOR='green'
FORECAST_COLOR='blue'
RESIDUE_COLOR='red'
COMPONENT_COLOR='navy'
SHADED_COLOR='turquoise'
UPPER_COLOR='grey'
LOWER_COLOR='black'
def add_patched_legend(ax , names):
# matplotlib does not like labels starting with '_'
patched_names = []
for name in names:
# remove leading '_' => here, this is almost OK: no signal transformation
patched_name = name[2:] if(name.startswith('__')) else name
patched_name = patched_name[1:] if(patched_name.startswith('_')) else patched_name
patched_names = patched_names + [ patched_name ]
ax.legend(patched_names)
def decomp_plot(df, time, signal, estimator, residue, name = None, format='png', max_length = 1000) :
assert(df.shape[0] > 0)
assert(df.shape[1] > 0)
assert(time in df.columns)
assert(signal in df.columns)
assert(estimator in df.columns)
assert(residue in df.columns)
import matplotlib
# print("MATPLOTLIB_BACKEND", matplotlib.get_backend())
# matplotlib.use('Agg')
import matplotlib.pyplot as plt
df1 = df.tail(max_length);
if(name is not None):
plt.switch_backend('Agg')
fig, axs = plt.subplots(ncols=2, figsize=(32, 16))
lColor = COMPONENT_COLOR;
if(name is not None and name.endswith("Forecast")):
lColor = FORECAST_COLOR;
df1.plot.line(time, [signal, estimator, residue],
color=[SIGNAL_COLOR, lColor, RESIDUE_COLOR],
ax=axs[0] , grid = True, legend=False)
add_patched_legend(axs[0] , [signal, estimator, residue])
residues = df1[residue].values
import scipy.stats as scistats
resid = residues[~np.isnan(residues)]
scistats.probplot(resid, dist="norm", plot=axs[1])
if(name is not None):
plt.switch_backend('Agg')
fig.savefig(name + '_decomp_output.' + format)
plt.close(fig)
def decomp_plot_as_png_base64(df, time, signal, estimator, residue, name = None, max_length = 1000) :
assert(df.shape[0] > 0)
assert(df.shape[1] > 0)
assert(time in df.columns)
assert(signal in df.columns)
assert(estimator in df.columns)
assert(residue in df.columns)
import matplotlib
# matplotlib.use('Agg')
import matplotlib.pyplot as plt
plt.switch_backend('Agg')
df1 = df.tail(max_length);
fig, axs = plt.subplots(ncols=2, figsize=(16, 8))
lColor = COMPONENT_COLOR;
if(name is not None and name.endswith("Forecast")):
lColor = FORECAST_COLOR;
df1.plot.line(time, [signal, estimator, residue],
color=[SIGNAL_COLOR, lColor, RESIDUE_COLOR],
ax=axs[0] , grid = True, legend = False)
add_patched_legend(axs[0] , [signal, estimator, residue])
residues = df1[residue].values
import scipy.stats as scistats
resid = residues[~np.isnan(residues)]
scistats.probplot(resid, dist="norm", plot=axs[1])
figfile = BytesIO()
fig.savefig(figfile, format='png')
figfile.seek(0) # rewind to beginning of file
figdata_png = base64.b64encode(figfile.getvalue())
plt.close(fig)
return figdata_png.decode('utf8')
def prediction_interval_plot(df, time, signal, estimator, lower, upper, name = None, format='png', max_length = 1000) :
assert(df.shape[0] > 0)
assert(df.shape[1] > 0)
assert(time in df.columns)
assert(signal in df.columns)
assert(estimator in df.columns)
assert(lower in df.columns)
assert(upper in df.columns)
df1 = df.tail(max_length).copy();
lMin = np.mean(df1[signal]) - np.std(df1[signal]) * 3;
lMax = np.mean(df1[signal]) + np.std(df1[signal]) * 3;
df1[lower] = df1[lower].apply(lambda x : x if (np.isnan(x) or x >= lMin) else np.nan);
df1[upper] = df1[upper].apply(lambda x : x if (np.isnan(x) or x <= lMax) else np.nan);
# last value of the signal
lLastSignalPos = df1[signal].dropna().tail(1).index[0];
lEstimtorValue = df1[estimator][lLastSignalPos];
df1.loc[lLastSignalPos , lower] = lEstimtorValue;
df1.loc[lLastSignalPos , upper] = lEstimtorValue;
import matplotlib
# matplotlib.use('Agg')
import matplotlib.pyplot as plt
if(name is not None):
plt.switch_backend('Agg')
fig, axs = plt.subplots(ncols=1, figsize=(16, 8))
df1.plot.line(time, [signal, estimator, lower, upper],
color=[SIGNAL_COLOR, FORECAST_COLOR, LOWER_COLOR, UPPER_COLOR],
ax=axs, grid = True, legend=False)
add_patched_legend(axs , [signal, estimator, lower, upper])
x = df1[time];
type1 = np.dtype(x)
if(type1.kind == 'M'):
x = x.apply(lambda t : t.date());
axs.fill_between(x.values, df1[lower], df1[upper], color=SHADED_COLOR, alpha=.2)
if(name is not None):
plt.switch_backend('Agg')
fig.savefig(name + '_prediction_intervals_output.' + format)
plt.close(fig)
def prediction_interval_plot_as_png_base64(df, time, signal, estimator, lower, upper, name = None, max_length = 1000) :
assert(df.shape[0] > 0)
assert(df.shape[1] > 0)
assert(time in df.columns)
assert(signal in df.columns)
assert(estimator in df.columns)
assert(lower in df.columns)
assert(upper in df.columns)
df1 = df.tail(max_length).copy();
lMin = np.mean(df1[signal]) - np.std(df1[signal]) * 3;
lMax = np.mean(df1[signal]) + np.std(df1[signal]) * 3;
df1[lower] = df1[lower].apply(lambda x : x if (np.isnan(x) or x >= lMin) else np.nan);
df1[upper] = df1[upper].apply(lambda x : x if (np.isnan(x) or x <= lMax) else np.nan);
# last value of the signal
lLastSignalPos = df1[signal].dropna().tail(1).index;
lEstimtorValue = df1[estimator][lLastSignalPos];
df1.loc[lLastSignalPos , lower] = lEstimtorValue;
df1.loc[lLastSignalPos , upper] = lEstimtorValue;
import matplotlib
# matplotlib.use('Agg')
import matplotlib.pyplot as plt
plt.switch_backend('Agg')
fig, axs = plt.subplots(ncols=1, figsize=(16, 8))
df1.plot.line(time, [signal, estimator, lower, upper],
color=[SIGNAL_COLOR, FORECAST_COLOR, FORECAST_COLOR, FORECAST_COLOR],
ax=axs, grid = True, legend=False)
add_patched_legend(axs , [signal, estimator, lower, upper])
x = df1[time];
type1 = np.dtype(x)
if(type1.kind == 'M'):
x = x.apply(lambda t : t.date());
axs.fill_between(x.values, df1[lower], df1[upper], color=SHADED_COLOR, alpha=.5)
figfile = BytesIO()
fig.savefig(figfile, format='png')
plt.close(fig)
figfile.seek(0) # rewind to beginning of file
figdata_png = base64.b64encode(figfile.getvalue())
return figdata_png.decode('utf8')
def qqplot_residues(df , residue):
pass
def build_record_label(labels_list):
out = "<f0>" + str(labels_list[0]);
i = 1;
for l in labels_list[1:]:
out = out + " | <f" + str(i) + "> " + str(l) ;
i = i + 1;
return out + "";
def plot_hierarchy(structure , iAnnotations, name):
import pydot
graph = pydot.Dot(graph_type='graph', rankdir='LR', fontsize="12.0");
graph.set_node_defaults(shape='record')
lLevelsReversed = sorted(structure.keys(), reverse=True);
for level in lLevelsReversed:
color = '#%02x%02x%02x' % (255, 255, 127 + int(128 * (1.0 - (level + 1.0) / len(lLevelsReversed))));
for col in structure[level].keys():
lLabel = col if iAnnotations is None else str(iAnnotations[col]);
if iAnnotations is not None:
lLabel = build_record_label(iAnnotations[col]);
node_col = pydot.Node(col, label=lLabel, style="filled", fillcolor=color, fontsize="12.0")
graph.add_node(node_col);
for col1 in structure[level][col]:
lLabel1 = col1
if iAnnotations is not None:
lLabel1 = build_record_label(iAnnotations[col1]);
color1 = '#%02x%02x%02x' % (255, 255, 128 + int(128 * (1.0 - (level + 2.0) / len(lLevelsReversed))));
node_col1 = pydot.Node(col1, label=lLabel1, style="filled",
fillcolor=color1, fontsize="12.0")
graph.add_node(node_col1);
lEdgeLabel = "";
if iAnnotations is not None:
lEdgeLabel = iAnnotations[col + "_" + col1];
lEdge = pydot.Edge(node_col, node_col1, color="red", label=lEdgeLabel, fontsize="12.0")
graph.add_edge(lEdge)
# print(graph.obj_dict)
if(name is not None):
graph.write_png(name);
else:
from IPython.display import Image, display
plot1 = Image(graph.create_png())
display(plot1)
| 2024-06-06T01:27:17.760571 | https://example.com/article/9182 |
"Call the police." "I need change for a dollar." "No change without purchase." "What's the cheapest thing you've got?" "A two-ounce bag of chips." "$5.99." "$5.99?" "!" "What a rip-off!" "Someone should shoot you!" "I was hoping we'd miss the 3:00 holdup." "I don't know what to do for dinner now." "Maybe we could shop somewhere else." "There's a farm stand next door." "Interesting." "It's like a corn dog without the dog." "Put whatever you want in this burlap sack." "You may not." "That's my baby carrier." "They so traumatized, I'm takin' them for a walk in they stroller." "Well, I think this family should try more vegetarian meals." "We'd feel better and live longer." "Vegetables, you've been promoted from side dish to entree." "I don't miss meat at all." "This portobello mushroom eats like a steak." "A rubbery, fungus-like steak." "Marge, I'm giving birth to a food baby." "I feel like something crawled inside me and took a crap." "Bart, don't use that word..." "Crap." "I brought you some weak tea and dry toast." "No fair." "Lisa didn't get sick." "It's 'cause your bodies are so used to processed food, it's a shock when you eat vegetables full of vitamins, minerals and trace amounts of bug feces." "I think it's staying down." "It's staying down." "It's not staying down." "Wait." "Yup, it's staying down." "Oh, no, it's not." "Yes, it is." "No, it's not." "Yup." "No." "Yup." "No." "Yup." "No." "Yup, no, yup, no, yup, no." "Stop it." "You'll all feel better if you get a little sleep." "Now, close your eyes." "Where's my bucket?" "Hush, little baby, don't say a word" "Mama's gonna buy you a mockingbird." "And if that mockingbird don't sing" "and if that diamond ring turns brass Mama's gonna buy you a looking glass..." "Lisa sings so sweetly, and Bart is my pillow." "Everyone's good for something." "Looks like you guys are on the mend." "Yeah, 'cause we're through with vegetables." "From now on, I'm only eating food that I know had a soul." "Hey, hey, kids." "Do you have what it takes to be a singing sensation?" "A dynamite voice?" "Ruthless, pushy parents?" "A void in your self-esteem that can only be filled by applause?" "Oh, God, I know that void." "Then you were born to enter Krusty's L'il Starmaker Singing Competition." "Not affiliated with American Idol." "We've never even heard of American Idol." "The winner will be animated into an Itchy and Scratchy cartoon." "So enter today, especially if you're a funny-looking kid who doesn't know he sucks." "I'm-a comin', Krusty." "Lisa, you should enter that contest." "You've got a great voice." "It made us all feel better." "Thanks, Bart." "That's really sweet." "I think I puked up all of my meanness." "Praise the Lord!" "I'm on the road to recovery." "I hope we didn't camp out here for nothing." "It's like every kid in Springfield showed up." "Don't worry about the competition, sweetie." "Whenever I enter a contest, I tell myself I'm going to win because sometimes, if you believe hard enough, God cuts you a break." "After all, He works for us." "Our prayers pay His salary." "And if I win, I'll be on Itchy and Scratchy." "Ah-ha, en garde." "Parry, thrust." "Nicely done, Wind-dancer." "But seriously, cruelty to animals is no laughing matter." "You can make a difference." "Spay and neuter your pets." "And remember, save the violence for cartoons." "For details, log onto PETA.org." "All right, kids, let's do this quick." "After this, I got to record 27 seasons of DVD commentaries, and I remember nothing." "Today, we'll pick the lucky children who will get to compete on the air." "I was going to be your celebrity judge, but then I realized I'd have to pay attention." "So instead, we'll use the Applause-O-Meter." "Now, let's get this over with!" "A-B-C-D-E-F-G" "How I wonder what you are." "Thank you." "I can't believe I used to date him." "Lis, you're a shoo-in." "Hush, little baby, don't say a word Mama's gonna buy you a mockingbird" "That's gonna sing, yes, it is, y'all" "And if that mockingbird don't sing" "Mama's gonna buy you a diamond ring" "That's gonna shine." "Shine" "Oh, it shines so bright." "Lisa, that sounds like a fancier version of the song you're going to sing." "And she's giving us the emotional experience of a lifetime." "Oh, thank you for this angel, Lord." "This contest is moot." "A Li'l Star has been made." "When she sings a "C," it's a "C."" "Unlike a certain glee club I've wasted my life on." "I may not know much, but I do know talent when my producers point it out to me." "We have our first contestant:" "Clarissa Wellington!" "There's no way I'm going to beat her." "Why?" "Just because she sings like Whitney Houston brought to life?" "Don't lose hope, sweetie." "I'm going to write you a song so great, you can't lose." "And that's not just the beer talking." "Your father's in here, too." "When a man loves a woman..." "Which one are you, the man or the woman?" "Questioning the kid's sexuality-- well done." "Here you go, honey." "Sing this." "How did you write a song so quickly?" "Much of the tune is plagiarized." "Now go, go, go." "I've been to Paris and London and Tokyo town" "But one crazy burg has 'em all beat hands down..." "Jacksonville?" "I'm talkin' Springfield." "You can buy chimichangas" "Talkin' Springfield" "The chicks have big gazongas?" "There's tires on fire" "A guy named Apu" "And Skinner, and Grandpa" "And ol' Disco Stu." "Did I forget to mention you?" "You?" "!" "That's me!" "I'm talkin' Springfield where nobody sucks." "Except for Flanders." "I get it, I get it." "She's good." "Looks like they love you, kid, and the audience is always right." "I wish I could say the same about my stockbroker." "Ah, what do you jerks know?" "That's a quality joke." "You did it, sweetie." "No, we did it, Dad." "She's right, it was all me." "I really do like working with you, Dad." "Will you keep helping me?" "Of course, sweetie." "You and I are going to write and sing our way out of this godforsaken hellhole." "But your song said you liked Springfield." "I wrote it about Shelbyville, then changed the names." "Good job, sweetie." "Every time you hit that note, I'll shoot a glass with my BB gun." "I really appreciate your taking such an interest in me, Dad." "Honey, our time together is precious to me." "I didn't want to say anything, but I don't think I'm going to be around much longer." "Really?" "What's wrong?" "Well, Moe's getting a big screen TV in February, just in time for the Pro Bowl." "Maybe this time my beloved blue shirts will win." "Is it possible to get a green spotlight?" "Sorry, sweetie, we don't have green." "Let Daddy go talk to the man." "May I have a word with you?" "Is that green enough for you, sweetie?" "Cause I can make it greener." "No, it's fine." "Alas, my love." "You do me wrong" "To cast me off discourteously...." "That's Cameron." "Girls go crazy over him." "He's cute, unthreatening, and his smile brought a puppy back to life." "Don't worry, honey." "The song I wrote you is so schmaltzy, it'll make "Moon River" sound like a farting orangutan." "My kitty died on Christmas Eve" "Daddy told me to be brave" "But instead of singing carols I was digging" "Snowball's grave..." "God, I miss Lisa's cat so much." "All right, you know how it works." "Every week, we eliminate one contestant based on the votes cast by you,the audience." "Disclaimer:" "All ballots were lost and vote totals made up." "The first contestant eliminated is..." "Katie Anderson." "Katie, you were the first victim of the Buzz Cola trapdoor." "Fall into the flavor hole!" "And the next contestant to be eliminated is..." "Clarissa." "Yeah, it's you, honey." "Please don't cry." "Keep crying." "It's dynamite." "Sweetie, tonight, you were perfect." "Marge, Lisa and I were just discussing how to improve tonight's performance." "Now you're confusing her by telling her she was perfect." "Mr. Simpson, I brought those Twizzlers you asked for." "My little girl likes Red Vines!" "You little...!" "Mr. Simpson, please stop!" "You think I can't kill you?" "There's a hundred little punks graduating from Syracuse this year that would beg to kiss my ass." "Let go." "Get out of here." "He's crazy." "Hey, these are Red Vines." "Here you go, sweetie." "Daddy got you your candy." "Homie, I'm worried you're turning into some kind of super-cuckoo stage mother." "Yeah." "For once, couldn't you just turn into a good father?" "Would that be so hard?" "Hey, give me a break." "Living through Lisa is the healthiest thing that ever happened to me." "But, Dad, you don't need to help me by humiliating people." "You love sausage, but you hate to see it getting made." "I don't love sausage." "Then would you like to see it getting made?" "Dad... you're fired." "Fired?" "Me?" "Everything I did, I did for you." "Dad, I'm sorry you're hurt, but you left me no choice." "You were obnoxious at a level not even permitted in show business." "Do you know the hours I worked?" "The people I had to yell at?" "The tires I had to slash?" "No one asked you to yell and slash!" "It's called schmoozing!" "Bart, you want to go to the video arcade?" "Hell, I'll even go shoe shopping." "Hey, come back!" "I'm calming down!" "We'll have family fun!" "Family fun!" "Hey, everybody, I brought a guest." "Someone who appreciates my in-your-face management style." "Smart move." "Soon the whole world will be lining up to suck my nose." "After I make my new client a star." "Hey, Simpson family." "How can I touch your hearts tonight?" "You're working with Cameron now?" "His name isn't Cameron anymore." "He's now "Johnny Rainbow."" "Mr. Simpson, about that..." "I'm not Mr. Simpson." "That ship has sailed." "I'm Colonel Cool." "And I'm the captain on this rocket to the stars." "So, are you a colonel or a captain?" "Neither." "I'm both." "All right," "I wanna teach you a little something about jazz hands." "Now don't stop till it's as natural as breathing." "Dad looks so happy." "Actually, I think your father's still upset about the way things went down." "Marge." "Bart." "Maggie." "Santa's Little Helper." "Lisa." "Dad, I don't want things to be awful between us." "I made you some cookies." "I don't think cookies are going to make me feel better." "Oh." "Oh, God." "They're delicious." "Oh, I'm so happy." "Oh, God, they're..." "They're gone." "Hi." "This lighting's a little bright." "And , could I get a red wash in the background?" "Sorry, hon, this is all we got." "Who's next?" "We are." "Okay Morty, gimme a rose-colored spot, tight to mid-chest and an aqua background with an astronaut on a surfboard." "Sure thing, Mr. S." "Lisa, at the end of your song, remember to look into the spotlight." "It makes your eyes twinkle." "Thanks, Dad." "You're welcome." "This is our final night." "30 contestants have already fallen through our Buzz Cola trapdoor." "Let us out." "Hey, you're supposed to clean this out every night." "What do you mean, talk to your shop steward?" "Now let's welcome the first of our two finalists," "Lisa Simpson." "Ladies and gentlemen, I have a confession to make." "I didn't have my usual songwriter tonight." "So I took a stab at writing my own song." "Something that expresses what's really in my heart." "Bernie, if you please." "I'm in the final two I should be happy." "But all I want to do is spend more time with my pappy" "Now that you're gone, Dad" "I miss you so much and your threats against teamsters" "And techies and such your management style is like Attila the Hun." "You were vicious, malicious." "But you got the job done." "I'm sorry I hurt you." "But please don't be sad." "You're no longer my coach." "But you're always my dad." "Even the Applause-O-Meter is crying." "Oh, God, that's battery acid." "Dad, you know the best part about this whole thing has been the time we've spent together." "I don't care if I win, just as long as you're in my corner again." "I always was, honey." "Look." "Cameron's about to sing the song I wrote for him." "He's about to learn the most important lesson in the music business:" "Don't trust people in the music business." "I'm a privileged boy." "It's great, I gotta tell ya." "A privileged boy." "My dad can buy and sell ya." "It really doesn't matter." "That you're on the list in front of me." "I'm gonna get your table." "Cause I always tip the maitre d'" "And then I'll go to Yale." "Because I am a legacy." "I'm better than you." "You suck, Johnny Rainbow." "I believed in you." "You sabotaged Cameron for me." "Sure did." "I'd do anything for you, honey." "Especially if it's easy." "Well, I think you're the sweetest dad in the whole world." "I'll always be there." "Not even death will stop me." "Now you're getting creepy." "How 'bout I love you, honey." "Good." | 2024-05-12T01:27:17.760571 | https://example.com/article/7525 |
Type A aortic dissection with arch entry tear: Surgical experience in 104 patients over a 12-year period.
To evaluate the efficacy of the frozen elephant trunk (FET) and total arch replacement (TAR) technique (FET + TAR) in the management of type A aortic dissection (TAAD) with arch entry tear. Clinical outcomes were analyzed for 104 TAAD patients with arch entry tear undergoing FET + TAR at 4.7 ± 3.5 days from symptom onset and compared with 728 TAAD patients with entry tears in elsewhere. The mean patient age was 49.3 ± 9.3 years, compared with 45.6 ± 10.8 years in other TAAD patients (P < .001). There were 84 men (80.8%). Hypertension was seen in 84.6% (88/104). Operative mortality was 8.6% (9/104). Spinal cord injury occurred in 3 cases (2.9%), stroke in 2 (1.9%), renal failure in 4 (3.8%) and limb ischemia in 2 (1.9%). Follow-up was 100% (95/95) at mean 5.6 ± 2.6 years (range 1.3-11.6). Late death occurred in 2 cases (1.9%). Survival and freedom from late adverse events were 89.2% (95% confidence interval [CI], 81.3%-93.9%) and 85.0% (95% CI, 76.3%-90.8%) at 8 years, respectively. Both the stented and unstented distal aortic segments showed significant trends of false lumen shrinkage and true lumen expansion over time (P < .001). Of the 65 CT scans at mean 4.6 ± 2.9 years postoperatively, the false lumen was completely obliterated in 63. Risk factors for arch entry tear were hypertension (odds ratio [OR], 2.091; 95% CI, 1.186-3.687; P = .011) and age (OR, 1.025; 95% CI, 1.002-1.048; P = .032). TAAD with arch entry tear was treated safely and durably by FET + TAR. Although patients with arch entry tear were somewhat older than other patients, operative mortality was not substantially higher despite their older age and arch location of entry tear. These results argue favorably for the use of the FET + TAR technique in the management of TAAD patients with arch entry tears. | 2024-02-25T01:27:17.760571 | https://example.com/article/9207 |
Q:
Defining equation with variable in the exponent
I am trying to define and plot the following equation.
f(x,y) = Sin(x^4) – Cos(12x^y) + 0.7x
is this the correct way to do it?
f[x_, y_] := Sin[x^4] - Cos[12 x^y] + 0.7 x
Would this be a correct representation of the equation?
I'm trying to find out if any value of y will make f(x,y) positive, even when x is negative.
A:
Yes that is correct. You might want to do a 3D plot:
f[x_, y_] := Sin[x^4] - Cos[12 x^y] + 0.7 x
Plot3D[f[x,y],{x,-2,2},{y,-2,2}]
This will plot f for -2<x<2 and -2<y<2. Good luck!
| 2024-02-13T01:27:17.760571 | https://example.com/article/2410 |
1956–57 Atlantic Coast Conference men's basketball season
The following contains a list of games within the 1956-57 season.
Final standings
ACC Tournament
See 1957 ACC Men's Basketball Tournament
NCAA tournament
Round of 23
North Carolina 90, Yale 74
Regional semi-finals
North Carolina 87, Canisius 75
Regional final
North Carolina 67, Syracuse 58
National semi-finals
North Carolina 74, Michigan St 70 (3ot)
National championship
North Carolina 54, Kansas 53 (3ot)
ACC's NCAA record
6–0
NIT
League rules prevented ACC teams from playing in the NIT, 1954–1966 | 2023-08-01T01:27:17.760571 | https://example.com/article/9767 |
Significance Statement
======================
This study shows that the neurotransmitter dopamine plays important roles in attributes of social groups such as social hierarchy through one of the dopamine receptors, D1 receptor. In particular, we found that low D1 receptor function, although it has been known to cause detrimental effects such as cognitive dysfunction, could still yield beneficial changes such as facilitation of social dominance in animals living in social groups. Moreover, such effects were different between rodents and primates. Although the effects were more substantial in rodents, they were less clear in primates, which appeared to be due to contributions of other social factors, such as social affiliation, in determining social hierarchy. These results raise a possibility that low D1 receptor function may be under the balance between benefits and disadvantages in social group environments.
Introduction {#s1}
============
Humans and many animals organize into social groups and follow a social hierarchy ([@CIT0005]). In a hierarchical social group, social class determines the behavioral fitness, such as health and reproduction, of subjects in the group ([@CIT0045]; [@CIT0028]). Accumulating evidence suggests that neural transmission of monoamines, such as dopamine (DA), is involved in social hierarchy, although how monoamine signaling plays roles in social hierarchy construction and maintenance has largely remained elusive. For instance, in nonhuman primates, striatal DA D2 receptor availability is correlated with social class ([@CIT0034]). Consistent with this, human functional imaging studies have reported that striatal D2 availability is correlated with social dominance ([@CIT0004]) and social status ([@CIT0030]). In addition, D2 function also mediates social dominance in rodents ([@CIT0019]). The roles of D1 function in social hierarchy remain less clear. However, a recent human positron emission tomography study has also reported that subjects with low D1 receptor availability in the limbic-striatal regions exhibit higher aggression and socially dominant personalities ([@CIT0038]), suggesting that D1 and D2 receptors may have opposite functions for determining social hierarchy.
Several macaque species ([@CIT0008]) and rodents ([@CIT0026]) construct a linear hierarchy in their societies. Behavioral factors, such as aggression and impulsivity, which are behavioral traits associated with psychiatric disorders ([@CIT0042]), play roles in determining the social class of an animal. Since social hierarchy is an emerging property consequence of competition among subjects in a group, more highly aggressive and impulsive nonhuman primates and mice may gain higher social status ([@CIT0027]; [@CIT0032]; [@CIT0010]; [@CIT0021]). DA D1 receptor signaling is important for cognitive and affective functions ([@CIT0047]). In contrast, a mouse strain with low D1 expression exhibits higher aggression than another strain with high D1 expression ([@CIT0006]). Moreover, blockade of the D1 receptor also induces impulsivity ([@CIT0050]). Thus, although low D1 receptor signaling causes cognitive dysfunctions, low D1 signaling is also expected to augment impulsivity and aggression, which may assist in attaining higher social dominance, and consequently higher social class, in a social hierarchy.
In this study, we examined the effects of the pharmacological manipulation to inhibit D1 receptor function in socially housed mice and macaques, two different species, both of which exhibit rigorous linear social hierarchies in their social groups. We have hypothesized that administration of the D1 receptor antagonist augmented social dominance of the drug-treated animals in both rodent and primate social groups.
Methods {#s2}
=======
Subjects {#s3}
--------
Mice and nonhuman primates were used in this study. All experiments were conducted in accordance with the Guidelines for Proper Conduct of Animal Experiments by the Science Council of Japan and were approved by the Kyoto University Primate Research Institute animal ethics committee. A total of 116 adult male CD1 mice were used for the experiments. Four mice per cage were grouped and housed together in a cage. To extend the rodent findings to nonhuman primates, which are thought to have more complex social relationships with hierarchy and that are closely related to humans, we also conducted the experiments using a social group consisting of 6 Japanese macaques (*Macaca fuscata*). These macaques were 3 years old each, and the group consisted of 4 males and 2 females (supplementary Table 1). Further detail on the animals is provided in the supplementary Materials.
Social Rank Test in Mice {#s4}
------------------------
To determine social hierarchy of mice housed in a group, the tube rank test, which is similar to that used in other studies ([@CIT0051]), was conducted. In this test, an apparatus consisting of a transparent plexiglass tube connected to transparent boxes at each end was used. A pair of mice from the same group were taken and simultaneously placed with one mouse into each of the boxes on either side of the tube. When mice from each side reached the middle of the tube, the partition wall was removed. The mouse that caused the other to retreat was designated as the "winner," and the mouse that was retreated out of the tube was designated as the "loser"; they were scored with +1 or 0, respectively. Tournaments of all possible combinations of matches by pairs of mice in each group (e.g., in a group of 4 mice denoted as "A-D", combinations are 6 pairs of A-B, A-C, A-D, B-C, B-D, C-D) were conducted once per day, 5 times (5 trials). Social dominance was then quantified by David's score (DS) ([@CIT0007]). DS is defined as
> *The proportion of wins by individual *i* in his interactions with another individual *j*, (P~ij~) is the number of times that i defeats j, (α~ij~) divided by the total number of interactions between i and j (n~ij~), i.e. P~ij~=α~ij~/n~ij~. The proportion of losses by i in interactions with j, P~ji~=1-P~ij~\...DS for each member, i, of a group is calculated with the formula: DS=w+w~2~-l-l~2~, where w represents the sum of i's P~ij~ values, w~2~ represents the summed w values (weighted by the appropriate P~ij~ values) of those individuals with which i interacted, l represents the sum of i's P~ji~ values and l~2~ represents the summed l values (weighted by the appropriate P~ji~ values) of those individuals with which i interacted\...* ([@CIT0012]).
An example of DS calculations is shown in supplementary Table 2.
Administration of the DA D1 receptor antagonist SCH23390 (SCH) at the dose of 0.1 mg/kg was given to mice i.p. As a control, an equivalent volume of saline (SAL) was administered. Twenty-five groups of 4 mice each (100 mice total) were tested. The 20 groups were divided into 4 sets of drug administration conditions. In one set of 5 groups, the drug was administered to first-rank mice in each group; in another set of 5 groups, the drug was administered to second-rank mice in each group, and so on. In these 20 groups, the drug-administered mice were returned to the groups, and housed together immediately after drug administration. In an additional 5 groups in which the second-rank mice received drug administration, the drug-administered mice were isolated from the groups for 6 hours after drug administration, by which the drug effects were expected to be largely decreased, and then returned to the groups. After training, the tube rank test for the baseline (BASE) condition was conducted to determine the social rank of each mouse in each group (supplemental Figure 1a). Then, SAL was given once per day for 5 days in mice at each rank in their home cages and was followed by the tube rank test for 5 days. After confirming no change in social rank with SAL administration, SCH was given once per day for 5 days in the mouse that had received SAL beforehand. After 5 days of repeated SCH administration, the tube rank test was conducted again for another 5 days. No SAL or SCH was administered during the tube rank test. Further detail on the method is provided in the supplementary Materials.
Social Behavior Test in Mice {#s5}
----------------------------
Although social affiliations have been suggested to underlie determination of social class in a hierarchy in primates ([@CIT0040]; [@CIT0016]), it has remained elusive in rodents. Thus, alterations of social hierarchy by drug administration may be consequences of behavioral changes, such as enhancement of impulsivity and aggression, but also alterations of social affiliative bonds. To investigate this issue, we further examined the effects of SCH administration on motivation to socially interact with mates in mice with the 3-chamber social preference test modified from that used in other studies ([@CIT0033]; [@CIT0037]). In this test, subject mice were placed in the middle chamber, which was connected to 2 other chambers on each side. The mice were allowed to freely enter into these chambers through the openings on the walls. Either (1) normal adult mice of the same gender with the test-subject mice and that had no previous contact with the test-subject mice, or (2) cage mates that were housed together with the test-subject mice were placed in a metal mesh cage that was positioned in the center of one of the sides of the chamber. On the other side of the chamber, an identical mesh cage without mice was placed. The amount of time that the test-subject mice spent on each side of the chamber was measured for 10 minutes. Animals that spent more time in the side of the chamber with the trapped mice were considered to be more strongly motivated to interact with mates.
A total of 56 mice were used in this test, of which 16 mice were paired with unfamiliar trapped mice (having no previous interaction), and 20 pairs of one higher and one lower ranked mice in the same home cages (which were also used for the tube rank test) were subjected to the test with SAL and SCH administration. Further detail on the method is provided in the supplementary Materials.
Other Behavioral Tests in Mice {#s6}
------------------------------
D1 receptor signaling plays critical roles in cognitive functions, such as working memory ([@CIT0052]; [@CIT0046]), behavioral inhibition ([@CIT0043]; [@CIT0050]), and behavioral flexibility ([@CIT0039]). We investigated the effects of SCH administration on the associations between social rank and these nonsocial cognitive factors to address how alterations in them may contribute to social hierarchical changes using food foraging and elevated plus maze tests.
The random foraging task, which is similar to that used in other studies ([@CIT0003]; [@CIT0011]; [@CIT0018]), was conducted to examine working memory and behavioral flexibility. In this task, the radial 8-arm maze was used. Mice were placed in the central arena of a maze that was connected to 8 arms. At the beginning of the test, a piece of a cereal was baited at the end of each arm. Mice were allowed to freely explore the maze for 5 minutes or until consuming all baited cereals, whichever happened first. The number of times that mice reentered arms that they had already visited was recorded. Repeated entries into the same arms were divided into 2 categories. One category was re-entries into the same arm at least 2 nonconsecutive times (NC entrance). This pattern of entries into the same arms was measured as a reflection of working memory deficit. The other category was reentries into the same arm consecutively (C entrance). This pattern of entries into the same arm was measured as a reflection of behavioral flexibility deficit. SCH and SAL were given to animals approximately 10 minutes before starting tests.
The elevated plus maze task was conducted to examine inhibition of inappropriate behaviors. Mice were placed in the central crossing area of the arms and were allowed to freely enter into the arms for 5 minutes. Studies, including ours ([@CIT0014]; [@CIT0049]; [@CIT0023]), have shown that the elevated plus maze can be used to examine behavioral inhibition, because entering into the opened arms is considered to be an inappropriate behavior and thereby should be inhibited. Accordingly, we measured the number of times that mice entered into the opened and enclosed arms. SCH and SAL were given to animals approximately 10 minutes before starting tests. Further detail on the method is provided in the supplementary Materials.
Behavioral Observations and Recordings in Nonhuman Primates {#s7}
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Behavioral observations were conducted with the focal animal sampling method. Observations and recordings were conducted for 15 min/d per monkey at the frequency of 3 or 4 d/week for 4 weeks (14 days of sampling) for the BASE observations, followed by another 4 weeks of observations after chronic SAL administration and then 4 additional weeks of observations after chronic SCH administration (supplementary Figure 1b). For later analysis of video recordings, 15-minute recordings were segmented into 10 seconds, and the presence or absence of specific behaviors in each segment was counted. Locomotion was separately quantified by subtracting the amount of time that animals were motionless from the whole 15-minute recording period. We measured "individual behavior" (goal-directed action, stereotypy, agonistic display, scanning, locomotion, unusual) of the drug-treated subject, and "social behavior" (affiliation, aggression, social inhibition, escape, mounting) of each animal in the group. Items of behavioral assessments and their detailed criteria are described in [Table 1](#T1){ref-type="table"}. Further detail on the method is provided in the supplementary Materials.
######
Items of behavioral assessment used in primates.
Individual behavior Definition
---------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Goal-directed action Any behavioral action that an observer can predict a target that a subject is approaching a target at least 1 sec before reaching the target. These actions include approaching to (1) a drinking fountain for drinking water, (2) other subjects or objects in the cage, and (3) a specific area within the cage (e.g., the area where a subject can see outside of the cage through the window).
Stereotypy \(1\) Repetitive circular running at the same orbit in the cage for more than 2 rounds or (2) Repeated licking of an iron bar at constant speed for more than 2 rounds
Agonistic display Banging (hard striking with making loud noises) any object in the cage (e.g. an iron bar), floor, or wall of the cage more than twice repeatedly.
Scanning Sustained gazing at others for longer than 1 sec
Unusual Behavior that is not observed in a normal condition, e.g., falling asleep or ataxia.
Locomotion The amount of time a subject is moving, which is calculated by subtraction of stillness time from the whole recording time
Social behavior Definition
Affiliation \(1\) Proximate sitting (sitting with the body touching another animal, or less than a 30-cm distance between subjects for longer than 1 sec), (2) Grooming (for longer than 1 sec), or (3) Playing (biting, hitting, grabbing without facial expressions associated with aggression). Although biting, hitting, and grabbing are also actions associated with aggression, these actions exhibited in the context of playing behavior are clearly distinct, i.e., with a subject making these actions much slower and softer than when the actions are made in the context of aggression.
Aggression Either (1) biting, (2) hitting, (3) grabbing, or (4) threatening with an open-mouth facial expression by an attacker and a simultaneous bared-teeth display or scream by an attack-recipient
Social inhibition Stopping on-going (planed) actions due to interference by others
Escape Changing locomotive orbit to avoid others
Mounting Mounting on the back of others
Social Rank Determination in Nonhuman Primates {#s8}
----------------------------------------------
A typically employed index that determines social rank is the direction of aggressive attacks, which are usually made from higher to lower social rank subjects ([@CIT0002]). Thus, social rank was estimated by counting the number of aggressive attacks from one monkey to other cage mates. DS was also calculated with the number of aggressive attacks from an initiator to a receiver.
Another index for social rank is food incentive priority ([@CIT0041]). Thus, we also conducted a food priority test to further determine social rank. In this test, food presentation (a portion of a sweet potato) was given at roughly equal distance from all subjects, and the order of obtaining a food was recorded. After one subject accessed and consumed a food, the next sweet potato portion was presented immediately. This process was repeated until the last subject obtained a piece of a sweet potato. However, to emphasize the priority for accessing a food, the portioned size of the food was consecutively reduced to be approximately two-thirds smaller than the formerly presented food portion. Ten trials at the frequency of one trial per day were conducted in each of the control and drug administration conditions. Quantification of social rank was attempted by scoring (food priority score or FPS) for the orders of food acquiring from 6 to 1 points with 6 points for the first access. Further detail on the method is provided in the supplementary Materials.
Drug Administration in Nonhuman Primates {#s9}
----------------------------------------
SCH was administered in the subject that was at second rank in the social group, based on the observation that the increase of social rank with SCH administration was the largest in second-rank subjects in mice. The drug was delivered by subcutaneous implantation of an ALZET osmotic pump, with which drug delivery was expected to persist for up to 4 weeks ([@CIT0017]). The concentration of SCH was adjusted to be approximately 0.1 mg/kg/d. As a control treatment, an equivalent volume of 0.9% saline in an osmotic pump was also implanted for 1 month before the drug was administered. Further detail on the method is provided in the supplementary Materials.
Data Analysis {#s10}
-------------
Drug effects on social hierarchy and affiliations were analyzed with ANOVA with repeated measures. In addition, drug effects on the linearity and steepness of the hierarchy were analyzed with linear regression analysis, with the linearity expressed as Pearson product-moment correlation coefficients, and the steepness as slopes of linear regressions. Concurrence of social affiliations was calculated by absolute difference of normalized affiliative contacts for each observation between specific bonds (denoted as "similarity index"). Lower index values indicated higher similarity between the contacts, with an index score=0 the similarity identical. Further detail on the data analysis is provided in the supplementary Materials.
Results {#s11}
=======
Social Hierarchy of Mice {#s12}
------------------------
A stable, linear social hierarchy was established in all groups under the BASE condition ([Figure 1a](#F1){ref-type="fig"}-[b](#F1){ref-type="fig"}). Three-way repeated-measures ANOVA with the condition (i.e., drug treatment in different rank of mice; n=20 groups in total, n=5 groups for SAL/SCH administration at each rank) and rank (i.e., original rank of each mouse in a group) as independent variables and the treatment (BASE vs SAL vs SCH) as repeated measures revealed the significant effects of SCH administration on DS in the interactions of treatment x rank and treatment x rank x condition (F~3,64~=0.00, *P*=1.00 in condition; F~3,64~=869.6, *P*\<.001 in rank; F~9,64~=3.73, *P*\<.001 in condition x rank; F~2,128~=0.00, *P*=1.00 in treatment; F~6,128~=0.00, *P*=1.00 in treatment x condition; F~6,128~=2.48, *P*=.026 in treatment x rank; F~18,128~=11.24, *P*\<.001 in treatment x condition x rank). Posthoc Tukey analysis has revealed that in a set of 5 groups in which SCH was administered to the first-rank mice, DS was not significantly altered at any rank ([Figure 1c](#F1){ref-type="fig"}). In another set of the 5 groups in which SCH was administered to the second-rank mice, DS was significantly increased in the drug-treated mice (*P*\<.001 in BASE vs SCH, *P*\<.001 in SAL vs SCH) ([Figure 1c](#F1){ref-type="fig"}). SCH administration of the second-rank mice also decreased DS of the first-rank mice (*P*\<.001 in BASE vs SCH, *P*\<.001 in SAL vs SCH; [Figure 1c](#F1){ref-type="fig"}) and increased DS of the fourth-rank mice (*P*=.001 in SAL vs SCH; BASE vs SCH was marginally significant with *P*=.075; [Figure 1c](#F1){ref-type="fig"}) within the same groups. In the groups in which the third-rank mice received SCH administration, DS was significantly increased only in the drug-treated mice (*P*=.033 in BASE vs SCH, *P*=.033 in SAL vs SCH) ([Figure 1c](#F1){ref-type="fig"}). In a set of the groups in which SCH was administered to the fourth-rank mice, DS was not altered in the drug-treated mice, but a significant decrease in DS was observed in the third-rank mice (*P*\<.001 in BASE vs SCH, *P*=.033 in SAL vs SCH).
{#F1}
Further analysis revealed that SCH administration in mice at the second rank but not other ranks significantly decreased the linearity (F~3,16~=8.61, *P*=.001 in rank; F~2,32~=33.8, *P*\<.001 in treatment; F~6,32~=13.4, *P*\<.001 in interaction; *P*\<.001 in BASE vs SCH at second rank, *P*\<.001 in SAL vs SCH at second rank; [Figure 1d](#F1){ref-type="fig"}-[e](#F1){ref-type="fig"}) and steepness (F~3,16~=1.05, *P*=.398 in rank; F~2,32~=3.41, *P*=.045 in treatment; F~6,32~=3.29, *P*=.012 in interaction; *P*=.003 in SAL vs SCH at second rank; [Figure 1d](#F1){ref-type="fig"}-[f](#F1){ref-type="fig"}) of the hierarchy, which may be associated with the SCH effects on DS most pronounced in the second-rank mice.
To further elucidate an insight on how social hierarchy was changed with D1 antagonist administration, the effects of drug administration was examined in the second-rank mice, but the drug-administered subjects were isolated, although this manipulation also became the confounding factor of repeated acute social isolation, until the drug effects were waned before returning to the groups. DS of the drug-administered mice in this condition was unchanged ([Figure 1g](#F1){ref-type="fig"}).
These results suggest that pharmacological manipulation to inhibit D1 receptor signaling may facilitate social dominance in the drug-administered mice at the middle, but not in the highest or lowest social rank in their social groups. Such alterations in the drug-administered mice may simultaneously alter the social status of other mates within the same social group. Moreover, such social dominance alterations may also involve altered social interactions between the drug-administered mice and other mates in the groups immediately after drug administration.
Behavioral Factors Associated with Social Hierarchical Changes in Mice {#s13}
----------------------------------------------------------------------
### Social Behavior {#s14}
First, we investigated the interactions between test-subject mice and unfamiliar trapped mice that were not from the home cage of the test subject mice (n=16). Test subject mice that received SAL but not with SCH administration stayed longer in the area where there were trapped mice than in the area where mates were absent (F~1,30~=7.12, *P*=.012 in area; F~1,30~=1.00, *P*=.325 in treatment; F~1,30~=5.14, *P*=.031 in interaction; *P*=.004 in social vs nonsocial area under SAL condition) ([Figure 2a](#F2){ref-type="fig"}). Accordingly, social preference, which was expressed as the ratio of time in the social to the non-social area was significantly decreased with SCH administration (paired t-test, t~15~=2.16, *P*=.047; [Figure 2a](#F2){ref-type="fig"}).
{#F2}
We then examined the interactions between test-subject mice and familiar trapped mice selected from the home cages of the test subject mice (n=20). In this condition, test-subject mice that received both SAL and SCH administration stayed longer in the area with other mice than in the area with no mates (F~1,38~=19.3, *P*\<.001 in area; F~1,38~=0.111, *P*=.741 in treatment; F~1,38~=0.803, *P*=.376 in interaction; *P*=.006 and *P*=.048 in the social vs nonsocial area with SAL and SCH administration, respectively; [Figure 2b](#F2){ref-type="fig"}), and consequently, social preference was not different between the SAL and SCH conditions (t~19~=0.383, *P*=.406; [Figure 2b](#F2){ref-type="fig"}). Further analysis was conducted in which social dominance of the test-subject and trapped mice was considered, that is, a pair of high and low dominance ranked test-subject and trapped mice, respectively (n=10) and vice versa (n=10). SCH administration did not alter longer time spent in the social area compared with the nonsocial area in either low→high (t~9~=0.189, *P*=.854; [Figure 2c](#F2){ref-type="fig"}) or high→low pairs (t~9~=0.331, *P*=.748; [Figure 2c](#F2){ref-type="fig"}). Social preference was discovered to be higher when social dominance between the test-subject mice and trapped mice was closer in the SAL condition (r=−0.724, *P*=.018 in high→low rank; r=−0.743, *P*=.014 in low→high rank; [Figure 2d](#F2){ref-type="fig"}). In contrast, a correlation was not observed in the SCH condition ([Figure 2d](#F2){ref-type="fig"}). However, statistical analyses for comparison of the correlation coefficients between the SAL and SCH conditions did not reveal any significant difference (F~2,16~=1.75, *P*=.206 in high→low; F~2,16~=2.12, *P*=.153 in low→high).
These results suggest that the roles of D1 receptor signaling on social behaviors in rodents may depend on the social contexts, such as familiarity and social status of interacting mates. Since D1 receptor antagonist administration did not affect social preference with mates that were housed together in the same groups, the SCH effects to alter social dominance in mice may not involve social affiliative changes.
### Impulsivity and Cognitive Functions {#s15}
The random foraging test using the radial 8-arm maze was conducted to examine the effects of SCH administration on working memory (NC entrance) and behavioral flexibility (C entrance) and their relations to social dominance. SCH administration increased the number of both NC (t~15~=3.06, *P*=.008; [Figure 3a](#F3){ref-type="fig"}) and C (t~15~=2.44, *P*=.028; [Figure 3a](#F3){ref-type="fig"}) entries compared with those with SAL administration. Significant positive correlations were found between DS and the number of both NC (r=0.622, *P*=.010; [Figure 3b](#F3){ref-type="fig"}) and C (r=0.656, *P*=.008; [Figure 3b](#F3){ref-type="fig"}) entries in the SAL condition, which indicates that higher-ranked mice tended to perform worse in the test. In contrast, no such correlation was observed with SCH administration ([Figure 3b](#F3){ref-type="fig"}). Statistical analyses for comparison of the correlation coefficients between the SAL and SCH conditions revealed significant (F~2,28~=3.79, *P*=.035) and marginally significant (F~2,28~=2.79, *P*=.079) differences in the number of C and NC entries, respectively.
{#F3}
The elevated plus maze test was also conducted to examine behavioral inhibition as measured by the number of inappropriate actions, that is, entering into the opened-arms of the maze, during the test. The number of opened-arm entries was significantly higher with SCH than with SAL administration (t~15~=3.38, *P*=.004) ([Figure 3c](#F3){ref-type="fig"}). Moreover, statistically significant correlations were observed between the number of opened-arm entries and DS in the SAL (r=0.591, *P*=.016; [Figure 3d](#F3){ref-type="fig"}) and SCH (r=0.573, *P*=.020; [Figure 3d](#F3){ref-type="fig"}) conditions, which indicates that higher social rank mice exhibit more impulsive behavior.
These results suggest that impairments of behavioral inhibition resulting in higher impulsivity may be involved in promotion of social rank with SCH administration, whereas deficits in other cognitive functions may not have significant impacts on determining social hierarchy in rodents.
Social Hierarchy of Nonhuman Primates {#s16}
-------------------------------------
Social rank was determined by aggression between subjects and the food priority test. The highest-ranked subject (denoted as "A" in [Figure 4a](#F4){ref-type="fig"}) exhibited aggressions over all other members in the group. The second-ranked subject (denoted as "B" in [Figure 4a](#F4){ref-type="fig"}) exhibited aggressions to others except the highest ranked subject. Aggressions were observed only from higher to lower ranked subjects ([Figure 4a](#F4){ref-type="fig"}). The social rank, which was determined by aggressions, was further confirmed with the food priority test in which the order of food access in each subject was recorded and quantified by the FPS ([Figure 4e](#F4){ref-type="fig"}). Social rank was consistent independent of whether it was determined by aggressions or the food priority test ([Figure 4a](#F4){ref-type="fig"},[e](#F1){ref-type="fig"}).
{#F4}
Aggressions from the first to the second (i.e., the drug-treated) and third-rank subjects as well as from the second to the third rank subjects were observed in the BASE and SAL conditions, but these were diminished when SCH administration was given to the second-rank subject ([Figure 4a](#F4){ref-type="fig"}-[b](#F1){ref-type="fig"}). The DS that was calculated with aggressions illustrated a linear hierarchical relationship in the group under the BASE and SAL conditions ([Figure 4c](#F4){ref-type="fig"}-[d](#F1){ref-type="fig"}). SCH administration resulted in the reduction of the linearity and steepness of the hierarchy ([Figure 4c](#F4){ref-type="fig"}-[d](#F1){ref-type="fig"}). In the food priority test, the second- and third-rank subjects were able to obtain foods before the first-rank subject 4 times out of 10 trials for each of them in the SCH condition, which had not been observed in the SAL condition (Fisher's exact test, χ^2^=5, df=1, *P*=.025 in SAL vs SCH; [Figure 4e](#F4){ref-type="fig"}). The FPS of the first-rank subject was significantly decreased (Wilcoxon matched pairs test, Z=2.023, *P*=.043 in SAL vs SCH; [Figure 4f](#F4){ref-type="fig"}) but was unaffected in others, including the drug-administered (second rank) subject.
These results suggest that SCH administration in the second-rank subject may not affect social dominance of the drug-administered subject but attenuates dominance of the highest-ranked subject over others in the group, and this may thereby promote stronger competitions among higher-ranked subjects.
Individual Behavior of the Nonhuman Primate with Drug Administration {#s17}
--------------------------------------------------------------------
The effects of SCH administration were confirmed with several behavioral alterations in the drug-administered subject. SCH administration caused a significant decrease of goal-directed actions (F~2,\ 12~ =24.76, *P\<*.001; posthoc Tukey test, *P\<*.001 in BASE vs SCH, *P*=.002 in SAL vs SCH; [Figure 5a](#F5){ref-type="fig"}) and an increase in the number of agonistic displays (F~2,\ 12~ =9.795, *P*=.003; *P*=.001 in BASE vs SCH, *P*=.005 in SAL vs SCH; [Figure 5c](#F5){ref-type="fig"}), which indicates heightened impulsive and aggressive behaviors, respectively, in the drug-treated subject. These alterations persisted for approximately 1 month of the observation period ([Figure 5b](#F5){ref-type="fig"},[d](#F5){ref-type="fig"}). None of the stereotypy, scanning, or locomotion was altered by SCH administration ([Figure 5e](#F5){ref-type="fig"}-[g](#F5){ref-type="fig"}).
{#F5}
Social Behaviors of Nonhuman Primates {#s18}
-------------------------------------
Social affiliations (i.e., proximate sitting, grooming, and playing; see [Table 1](#T1){ref-type="table"} for the details of the criteria on these behaviors) between subjects were also altered by SCH administration into the second-rank subject. Because affiliative contacts occurred at relatively variable frequencies even between the BASE and SAL conditions ([Figure 6a](#F6){ref-type="fig"}), the drug effects were determined when affiliations in the SCH condition were significantly different compared with both SAL and BASE conditions. The alterations matching with this criterion were the affiliative contacts from the first- to second-rank subjects (F~2,12~ =10.16, *P*=.003; *P*=.005 in BASE vs SCH, *P*=.001 in SAL vs SCH) and from the third- to fifth-rank subjects (F~2,12~ =8.771, *P*=.004; *P*=.02 in BASE vs SCH, *P*=.042 in SAL vs SCH) ([Figure 6a](#F6){ref-type="fig"}). Although these affiliative bonds between first- and second- and between third- and fifth-rank subjects were independent in the BASE and SAL conditions, concurrences between them had emerged in the SCH condition ([Figure 6b](#F6){ref-type="fig"}), which was evidenced by a significantly smaller similarity index score in the SCH condition than in the BASE and SAL conditions (1-way ANOVA, F~2,39~ =8.088, *P*=.001; *P*=.001 in BASE vs SCH, *P*=.014 in SAL vs SCH) ([Figure 6c](#F6){ref-type="fig"}). SCH administration in the second-rank subject did not alter other social behaviors (supplementary Figure 2a-c).
{#F6}
These results suggest that SCH administration in one member of a group may cause complex affiliative social relationship alterations not only between the drug-treated subject and other mates, but also between subjects that are not directly related to the drug-treated subject (supplementary Figure 3a-c).
Discussion {#s19}
==========
In this study, by comparing mice and macaque social groups, we examined whether the mechanisms underlying social hierarchy were mutual across the 2 species. We found that D1 antagonist administration facilitated social dominance in the drug-treated mice. In contrast, in nonhuman primates, although D1 antagonist administration into the second-rank macaque in the social group did not alter social dominance of the drug-treated subject, this manipulation resulted in greater competitiveness between the drug-treated subject and other higher ranked (i.e., dominant) subjects, along with the alterations of social affiliations that consequently worked beneficially for the drug-treated subject. Thus, the findings suggest that D1 antagonist administration induces beneficial effects in both mice and macaques living in social groups, although the underlying processes may differ between species. Our study found that there was a loss of social dominance in the first-rank subject and a gain of social dominance in the third-rank subject after drug administration in the second-rank subject, which could be associated with altered social affiliation relationships. This is consistent with the studies suggesting that affiliative abilities are important in the attainment of dominance for primate species ([@CIT0040]; [@CIT0016]). Such social relationship alterations with SCH administration into the second-rank macaque within the group did not cause a major devastating effect in each individual macaque in the group, as no significant changes in the amount of chronic stress hormones in the drug-treated subject and other mates were observed (supplementary Methods; supplementary Results; supplementary Figure 4a-b, for stress hormone assays in macaques). In contrast, the role of affiliative ability in social dominance was less clear in the mouse social groups. We also found that social preference between subjects housed in the same group was not altered by D1 receptor antagonist, suggesting that social affiliations may be less important in determination of a social hierarchy in rodents than in primates. These findings suggest that rodents and primates may have different levels of social complexity (social tactics including various aspects of interactions, such as affiliation and dominance, among subjects in a group).
Social hierarchy of animals and humans has been shown to be related to various factors such as kin relationship, age, and body size/weight ([@CIT0021]). However, these factors had little effect in our study, because the animals used in this study had no kin relationships and were age matched. Moreover, social group dynamics in nonhuman primates were also independent of the gender of subjects, as we did not observe gender-specific social status and affiliative relationships in the group consisting of male and female macaques.
In mice, the SCH effects were not uniform in all drug-treated animals but were dependent on social rank in the groups. Facilitations of social dominance were observed in the second- and third-rank mice, but not in the first- and fourth-rank mice. One of possible explanations for no increase of social dominance in the first-rank mice may be the ceiling effect. In contrast, absence of increase in social dominance by SCH administration in the fourth-rank mice may involve other factors whose effects to place the mice into the lowest rank are substantially larger than the effects yielded by SCH administration.
Several studies have investigated the impacts of pharmacological manipulation of DA transmission on social behaviors of nonhuman primates housed in social groups. Amphetamine administration in such group-housed nonhuman primates decreases social interactions with other mates in the drug-treated subjects, which is reversed by the D1 antagonist ([@CIT0009]; [@CIT0031]). Consistent with these previous studies, we found no apparent decrease in social interactions of the drug-treated Japanese macaque with other group members following D1 receptor antagonist administration. Nonetheless, these findings contradict the study showing that D1 receptor stimulation facilitates social interactions in rodents ([@CIT0015]). Importantly, we found that the effects of D1 antagonist administration on social preference in mice was different when the drug-treated subjects interacted with familiar vs unfamiliar mates. Moreover, the difference of social dominance between the drug-treated subjects and other mates also affected social preference. Thus, social contexts, such as familiarity and social rank, play important roles on DA-dependent regulation of social behaviors. In this regard, also note that administration of a low dose of phencyclidine, which is used as an animal model of schizophrenia, causes deteriorative effects, such as "decreased" social interactions in individually housed rodents and nonhuman primates ([@CIT0044]; [@CIT0029]). In contrast, the same drug treatment has been found to actually "increase" social affiliations with other mates without altering aggressions in socially housed nonhuman primates ([@CIT0024], [@CIT0025]).
D1 receptors are distributed in several brain areas, including the prefrontal cortex (PFC), limbic structures, and the basal ganglia nuclei ([@CIT0013]). This study did not unveil which brain regions and neural mechanisms were involved in the alterations of social hierarchy. In rodents, glutamatergic transmission in the PFC has been shown to be associated with social hierarchy ([@CIT0051]), suggesting that the PFC is one of the promising targets. However, in nonhuman primates, the anterior and dorsal parts of the PFC gray matters have been demonstrated to correlate to not only social class of each subject in a group but also social group size ([@CIT0035]). In contrast, the volumes of the subcortical structures where D1 receptors are expressed, such as the striatum, amygdala, and hypothalamus, are associated sorely with social status of each subject in a group ([@CIT0035]). Collectively, DA-dependent information processing in the PFC and its striatal and limbic pathways may be the most promising neural substrates that determine social group dynamics.
In addition to DA, serotonin (5-HT) is another neurotransmitter involved in social dominance and social affiliations, and thereby, 5-HT and DA would work together to orchestrate social group dynamics. Administration of selective 5-HT reuptake inhibitor has been shown to attenuate dominant and submissive behaviors in the dominant and subordinate rank, respectively, of primates ([@CIT0048]) and other animals ([@CIT0022]). In contrast, selective 5-HT reuptake inhibitor administration has also been demonstrated to facilitate social affiliations in humans ([@CIT0020]). Collectively, increasing 5-HT transmission may result in social groups of less clear hierarchy and stronger social affiliative bonds.
In the social contexts of nonhuman primates and rodents and possibly other animals that organize into social hierarchy, behavioral changes caused by D1 receptor inhibition, which causes cognitive dysfunction, cannot simply be interpreted as "deficits." Thus, a counter-balance could be established in D1 receptor signaling for cognitive deficits and facilitation of social dominance. Lower social ranked animals may perform better at foraging to facilitate reproductive fitness in a social group. This may also be associated with the facet of cognitive functions playing crucial roles in food foraging, and this may be relaxed for animals in social groups where resources can be shared compared with those in solitary environments. Finally, the evolutionary origins of psychiatric disorders that may involve D1 receptor alterations, such as schizophrenia ([@CIT0036]; [@CIT0001]), may be explained by such balancing selection mechanisms.
In conclusion, our study suggests that D1 receptor functions play significant roles in social group dynamics, such as social hierarchy. In rodents, low D1 functions facilitate social dominance in subjects housed in social groups. In contrast, D1 facilitation of social dominance is less clear in nonhuman primate social groups, which may be because social hierarchy determination in nonhuman primates may involve more complex social interaction dynamics than those in rodents.
Statement of Interest
=====================
None.
Supplementary Material
======================
######
Click here for additional data file.
We thank Drs. Mary Ann Raghanty and Michael A. Huffman for comments and editing of the manuscript, and Ms. Tsukasa Obora and the staff of the Center for Human Evolution Modeling Research at Kyoto University Primate Research Institute for technical assistance.
This work was supported by Kyoto University Young Investigator Step-up and Core-stage Research Grants, Chubei Ito Foundation Research Grant, Sumitomo Foundation Research Grant, Inamori Foundation Research Grant, Institute of Seizon & Life Sciences Research Grant, JSPS Grant-in-Aid for challenging Exploratory Research \#26640044 (Y.G.), JSPS Research Fellowship for Young Scientists \#15J01210 (Y.Y.), and National Research Foundation of Korea \#NRF-2014R1A1A3052796 (Y.A.L.).
[^1]: Correspondence: Yukiori Goto, PhD, Primate Research Institute, Kyoto University, 41--2 Kanrin, Inuyama, Aichi 484--8506 Japan (<goto.yukiori.5c@kyoto-u.ac.jp>).
| 2023-11-06T01:27:17.760571 | https://example.com/article/5891 |
Put on your bras, shave your armpits and quit your bitching.
W hy do so many women insist on carrying out this war against men? Absolute gender equality isn't going to happen; we have gender equality now. Even if in a million years, men and women somehow made this unrealistic ideal happen, there will always be a sexual distinction between men and women causing some sort of inequality (if only on the level of basic physical needs). Or should I say women and men, as not to imply a male superiority? Why the hell should people go out of their way to be politically correct and use this "he/she" nonsense so a few chicks with language complexes won't be offended? Oops, I said "chicks". Damn.
W hy the hell do women get offended when they're called chicks? I don't see how that word can be remotely offensive in anyway. But, some women think it's derogatory and belittling. Some women are so petty, that they resent any male implication in the english language. Who cares? What if guys suddenly felt like bitching and wanted to eradicate all the derogatory male phrases from the language? Buster, Pal, Buddy, Stud, Hunk. Oooh, don't call me a buster, I'll be offended.
W ho's to say what's offensive anyway? Just because a few feminist extremists think that something's offensive, does the whole society have to change their way of doing things? I don't want mother nature being called mother nature anymore, but rather father nature. I don't want ships to be referred to as female anymore, but rather male. The phrase "she's a good ship" offends me. I don't want liberty to be a lady. Why does it have to be lady liberty? Why do people say "she's beautiful" when referring to cars? Why not he? Who cares? It's just the way things have always been. It's not meant to be offensive, so why doesn't the offended party pull their head out of their ass, and stop bitching about it.
W hy there will never be absolute gender equality is because of the inherent contradictions between equality and liberty. For example, if a guy wanted to say a joke about women at work, or hung pornography in the work place, it would make the women work in an environment that may seem hostile to them. If someone at work frequently referred to women as chicks, and a woman was offended by it, the woman would then be working in a hostile environment. If a woman wants to be treated equally in the work place, then she shouldn't be offended at what guys usually talk about. I think it's unreasonable for all women to be content with whatever people do at work, and for guys to go out of their way to change their lifestyles to conform to what's politically correct. It's impossible to live your life without being offended at something. Violence, foul language, pornography, sex, religion, whatever it is, you're bound to be offended by it sooner or later. So rather than bitching about it, just deal with it and move on. Men aren't out to get women; we're not the bad guys.
W hy would anyone do anything that's degrading to themselves? Would you strip down in front of a crowd of people? Probably not. But would you do it if they were paying you $250 per hour? Or if they paid you $1000 per hour? Maybe then. It doesn't seem so degrading when there's a huge incentive for you to do it. So why do some feminist extremists think pornography is degrading to women? Obviously the women in pornography don't think it's degrading, otherwise they wouldn't do it. Their dignity has a price, and they were willing to sell it. They don't represent all women in general, but only the few who chose to go into that business, just as women who choose not to go into pornography don't represent the women who do. So the phrase "pornography is degrading to women" doesn't make sense.
W hat's the point of not shaving your armpits and not wearing your bras? If you like yourself better with hairy armpits, then by all means don't shave. But if you're not shaving to make some stupid point, you'r a fool. Nobody cares. You won't be a social outcast if you don't shave your armpits. Same thing goes for not wearing a bra. If you'd rather not wear a bra, then don't. If you want to walk around in public without your shirt on, then by all means do it. I'm sure most guys wouldn't mind. If you want that kind of attention, then walk around naked. It won't help your image with men that already think of women as little more than sex objects. What straight man wouldn't like to see a nude woman? I think women, nude or not, are attractive, as do most guys. I'd consider someone looking at me in admiration more of a compliment than anything else. Some feminists go ballistic when a guy looks at her breasts. Why? Not wearing a bra just draws more attention to a woman's breasts. While women shouldn't have to worry about some pervert always staring at them, it's bound to happen. Wait.. I've got an idea. If you don't like guys staring at your breasts, WEAR A BRA. It's hard not to notice a girl when she has a pair of daggers poking out of her chest. I referred to nipples as daggers.. Oops. There I go again, offending you crazy feminists. While most guys don't practice self-dicipline and don't respect women enough not to stare at their breasts, a woman shouldn't go ape shit when she catches a guy looking at her. Most women try to accentuate parts of their body that men find attractive anyway. Is this so bad? Who doesn't like to feel attractive? While it can be overdone, it's really been blown out of proportion by feminist extremists.
F eminism is in a lot of ways like fascism. Your average Fascist will disregard any scientific argument unless the conclusion supports his existing belief. The ideology comes first and the Fascist looks for anything to back it up, no matter how trivial, unreliable or discredited. Much like today's feminists and their ideology. Fascists attempt to rationalize their beliefs and portray them as truth by twisting the facts. A fascist might, for example, cast blame for unemployment and work discontent on immigrants "stealing" their jobs. Feminists similarly cast blame for women's lower average pay onto another party (men). Both feminists and fascists are quick to cast blame on someone else for anything that goes wrong in their lives. Most feminists seem to conform to feminist stereotypes. I can usually pick out a feminist in a croud of women. She'll usually have short hair, regular pants, a regular shirt, and an unbathed look; she'll look very much like a stereotypical guy. I think why a feminist might appear like this is to make a statement that "if men can do it and be accepted, then women should be able to". How bold, to go around and look like a stereotypical guy as opposed to a stereotypical girl. Who cares? Either way, you're an ass for thinking anybody cares about the statement you're making. If you're trying to prove a point to the average guy that's only concerned with women as sex objects, you're wasting your time. A guy that's concerned with women as sex objects is going to be concerned with women as sex objects regardless of how you look. Not every woman will share feminist ideals, so a possible argument that "if all women did it, then guys would have to respect us" isn't very realistic. If you really want to make a point, surgically remove your breasts. Or is that going too far? Feminism serves as nothing more than a wedge to further seperate the sexes, segregating men and women into cultures that wouldn't otherwise exist.
I 'm pretty damn sick of hearing feminists bitch about men being paid more then women. If 100 male chemical engineers that worked for fortune 500 companies were compared to 100 female chemical engineers that worked for fortune 500 companies, their pay would probably be the same, if not very close. If it were not, then sexual discrimination would probably be a good candidate as to why it's not. The reason why women get paid less than men on average is because women and men simply prefer different occupations, and different lifestyles. If a woman decides to have a child, chances are that she would end up taking more time off from work to take care of her child than would a guy (if only for child labor alone). It's unfair, but usually the case. Women are usually more family oriented, and nurturing, while men are usually less sensitive and work oriented. All this is of course changing, but it's to be expected that women get paid less than men. If for example, pay for men in the armed forces was compared to pay for women in the armed forces, the men would probably swamp the women in pay comparison simply because of the militariy's past male domination & segregation policies. That does not necessarily imply the same segregation today.
I work at a telemarketing company. I'm a network operator. I've been with the company for almost 4 years (3 yrs, 9 months), and I get paid less than almost every woman in my facility, except for two. I get paid less than people with only a fraction of the experience I have. I'm living proof that not all women get paid less than men. Bottom line: feminazis, stop your bitching. It's nobody's fault but yours if you can't do as well as a man. Take responsibility for your own life.
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© 1997-2017 by Maddox | 2024-07-26T01:27:17.760571 | https://example.com/article/7553 |
User Reviews
Strengths: The bearings really stand out. They are up there with the best, and the seals are now on their second season of wet weather abuse.
The cups are well made, and the design even stands out a bit, which is rather rare for a headset.
It is quite light for the strength.
The graphics looks rather nice.
They are very cheap, compared to the high quality.
Weaknesses: None
Bottom Line:
This headset is the best value I have seen so far.
The quality matches all Raceface any day, at less than half the cost.
Maybe it's not the ideal choice for a CC bike, as you can find some that are lighter, and still strong enough for that, but for AM-FR-DH and DJ, it is ideal.
Mine has taken a beating for a couple of years now, and it's still like new.
Fireeye is somewhat unknown, at least around here, but the more of their stuff I get, the more I like it.
Their customer service is top notch.
I can't tell how good they are at dealing with warranty claims, cause none of my Fireeye stuff has failed so far. | 2024-07-16T01:27:17.760571 | https://example.com/article/7249 |
Q:
Lighting a LED slowly with GPIO
What is a good way to make an LED turn on and off slowly using a Raspberry Pi? Im trying to hook up an led to become a Tardis lamp, but since the Pi doesnt have ADC/DAC, Im kind of stuck. What are some generic ways to do it? I don't have a wide selection of components lying around, and would prefer not to buy anything, if possible.
The best I have found is this, which seems kind of excessive for a single LED.
A:
You can do this with the PWM pin on the Pi.
Here's a link to a blog post dealing with this that uses the C WiringPi library to do it:
http://lifebydesignuk.wordpress.com/2013/02/20/how-to-breathing-led-in-python-for-raspberrypi/
I would paste the code in here but the editor isn't co-operating.
A:
You could also achieve this effect with a simple 555-timer chip a capacitor and some resistors.
I myself plan on using the following circuit for my tardis project:
http://www.youtube.com/watch?feature=player_detailpage&v=qLAi7hkDuYw&t=307
A:
To achieve slow dimming or brightening of your LED you will require (as others have posted before) PWM.
Introduction to PWM
There are primarily two types of modulation software and hardware. There exist several libraries which can achieve a software enabled modulation whose performance lies in between software and hardware. You could read up more about the different types of modulation on this answer here.
Solution to your query (without having to buy or install anything)
The finer your application the more advanced the PWM you need to use, for your purpose (LEDs) the precision required is not too great and thus it can be achieved by simple GPIO modulation.
import RPi.GPIO as GPIO
import time
GPIO.setmode(GPIO.BCM) # choose BCM or BOARD numbering schemes. I use BCM
GPIO.setup(26, GPIO.OUT)# set GPIO 26 as output for led
led= GPIO.PWM(26,100) # create object led for PWM on port 25 at 100 Hertz
led.start(0) # start led on 0 percent duty cycle (off)
pause_time = 0.02 # you can change this to slow down/speed up
try:
while True:
for i in range(0,101): # 101 because it stops when it finishes 100
led.ChangeDutyCycle(i)
sleep(pause_time)
for i in range(100,-1,-1): # from 100 to zero in steps of -1
led.ChangeDutyCycle(i)
sleep(pause_time)
except KeyboardInterrupt:
led.stop() # stop the led PWM output
GPIO.cleanup() # clean up GPIO on CTRL+C exit
This code will increase and then decrease the brightness. You can, of course, edit the code to your needs.
| 2024-04-15T01:27:17.760571 | https://example.com/article/2105 |
Two mechanisms in the stimulus suffix effect.
The stimulus suffix is a redundant item presented immediately after a stimulus list. Its effect is the selective impairment of recall of the final items in a serially recalled, auditorily presented list of unconnected items. Two experiments indicate that there was no difference between the effects of suffixes .5 and 1.0 sec after the end of a digit list presented at a rate of one digit/sec. This suggests that the effect of the suffix, in this case the vowel sound "ah," is not a simple function of its time of arrival after the final digit, as has been thought. The possibility of more complex factors was supported in a further experiment which showed a slight reduction in the size of the suffix effect by repeating the suffix three times. | 2024-06-18T01:27:17.760571 | https://example.com/article/9362 |
Volume 14, Issue 8
April 2008 Theme
AUDIENCE PARTICIPATIONS
Give all the Cubs a slip of paper – each the same size and shape as the others.
Then ask everyone to write a short sentence of four or five words. The words should be written plainly and should not be shown to any other person.
Then instruct them to fold their papers and bring them to someone previously selected to act as the ‘guardian’.
No one, not even the guardian, should attempt to read the papers, still folded.
As you gravely close your eyes, place the folded paper against your forehead and remain a moment in deep thought. Then call out any sentence that has occurred to you and as who wrote it.
One of the Cubs, who is the accomplice, and who did not write a sentence nor submit a slip of paper, admits authorship of the sentence.
Then unfold the paper, apparently to verify his announcement (and read the sentence to yourself).
Next place the paper in your left hand and ask the guardian for another.
Repeat the same preliminaries and then call out the words written on the previous paper, which you have had the opportunity to read. This will be a bona fide answer and one of the Cubs will have to admit to writing the sentence.
Keep the performance up in this manner until all the player’s slips of paper have been read.
In order for the trick to be successful, the accomplice must be careful to conceal from the audience the fact that he did not include a sentence in the collection given to the guardian.
JARED AND HIS MAGIC SHOWSam Houston Area Council
Directions: Divide the group in two parts. Then tell them, every time you hear the word Magic, I want the Left side to say Abracadabra and then the Right side to say Alacazam!
Jared was a Cub Scout. He liked MAGIC shows. He decided that he would put on his own MAGIC show.
He practiced and practiced all his MAGIC tricks. Finally the big day came. He was going to put on his MAGIC show for is family and friends.
His brother liked MAGIC tricks too! He thought it would be fun to pull a trick on Jared. He talked to his sister Shari who was going to help Jared with his MAGIC tricks, and asked her to help him with his plan.
The time came and Jared started his show. He showed some MAGIC tricks with cards. Then he showed a MAGIC trick using a handkerchief and some flowers.
The next MAGIC trick he did was to pull a rabbit out of a hat. Then Jared showed the MAGIC number trick.
Now it was time for his last MAGIC trick. He was going to make Shari disappear.
Jared had Shari get in a MAGIC box. Jared shut the door and said the MAGIC words. He opened the door and she was gone. He closed the door and told everyone he would make her come back. Again, Jared said the MAGIC words and opened the door and guess what? She was still gone. Jared was very surprised. His brother was laughing. Jared’s brother came out to help him. They both said the MAGIC words and opened the door and there she was with a big smile on her face.
Jared’s brother said, “The MAGIC trick was on you. You thought she was really gone and she wasn’t.”
The Reluctant RabbitCapital Area Council
Divide audience into four groups. Assign each group a word and a response. Practice as you make assignments.
Joe was very busy. It was only a few days until the Pack Meeting and he was still trying to perfect his disappearing Rabbit trick. All the other Cub Scouts already had their tricks ready. But Joe was having a little trouble. He had his black top Hat and a very nice little Rabbit and Joe even knew exactly how to do the Rabbit in the Hat trick, but the Rabbit would not cooperate.
Just then, Joe’sDog came bounding into the room. “Woof,” said the Dog. He was a very large Dog and the minute the Rabbit saw him, he bolted under the bed, toppling the Hat and just about knocking JOE off the bed. “Dog,” shouted Joe, “Get down! You keep scaring my Rabbit. And I already have enough trouble with him.” The Dog hung his head in shame. He was really a very good Dog and did not mean to scare the Rabbit. Joe reached under the bed and pulled out the Rabbit. Joe petted the Rabbit ad soon he was calm. “Now, let’s practice,” Joe said. He took the Hat and after waving the magic wand a few times he put the Rabbit into the Hat. But the Rabbit would not stay in the Hat. Joe’sDog sat quietly and watched. Nothing Joe did seemed to help.
Joe was getting very frustrated. He put the Hat on his own head and sat down on the bed. The Hat fell down around Joe’s eyes so he could not see. “Boy it’s dark in here,” he said. That was when it hit him. The Rabbit had claustrophobia and was afraid of the dark! Every time Joe put him in the Hat, the Rabbit got very nervous and tried to escape. Joe patted his Dog on the head and thought and thought. He didn’t have time to train another Rabbit. What could he do?
Then Joe had a bright idea. He opened the top drawer of his bureau and began searching through all his stuff. His Dog came over to help. Soon Joe found what he was looking for—his penlight. He put it in the bottom of the Hat and turned on the light.
Then he took the Rabbit waved the magic wand and stuffed the Rabbit into the Hat. This time, with the light to keep him calm, the Rabbit stayed.
“Hurray!” shouted Joe. “Now I can do my trick!” And sure enough, the Rabbit had disappeared; the Hat was empty! Even the Dog was surprised. | 2024-01-07T01:27:17.760571 | https://example.com/article/9378 |
Attention, Filmmakers: Here's 12 Tips for Directing Your First Feature Film
Six year ago, I decided to make a comedy about the paradox of family – how it can simultaneously be our source of strength and the root of all our problems, and for better or worse, it’s forever. This week my debut feature film, “The Discoverers,” rolls out on VOD and DVD capping off an award-winning festival run and theatrical release.
Making my first feature was one of the most difficult and rewarding experiences of my life. It was honor and privilege to work with an all-star cast and crew of 200+ creative collaborators who helped me learn these lessons that I hope will inspire other first-timers to realize their vision.
Here are 12 things i learned along the way:
1. Feel it.
Ultimately, as the director, your job is to become the emotional truth barometer on set. So when the camera starts rolling, forget about everything else and just try to feel the performance on each take. I liked to stand as close to the lens as our AC would let me to feel what the camera is recording. From editing to finding music to color correct and the mix, feel the movie you want the audience to experience.
2. Find your key frames.
Prepping “The Discoverers,” I shotlisted and storyboarded, but I knew we wouldn’t have time to do everything I imagined on paper. I developed a key frame system, which became extremely helpful on set. The idea is to try to distill each scene down to a single image that represents what the scene must achieve dramatically. It may be an extreme wide shot that acts like a punch in the gut, a lyrical tracking shot, or the close up you’re saving for emotional impact. When you have to start combining shots on set, you’ll know the essential image you’ll need for each scene to serve its dramatic function.
3. Your film is only as good as your collaborators.
I think it was Elia Kazan who first said 90% of directing is casting. While it’s certainly true with actors, I think the same principal applies to your creative crew. Have a clear aesthetic vision and create a space where people can bring their talent and experience to do their best work. Find collaborators who will challenge your ideas and push you creatively. We had ambitious goals and limited resources, but everyone stretched themselves to make something much grander than any one of us could have imagined.
4. Make a lookbook.
I started collecting images when I was first writing “The Discoverers,” which I eventually made into a printed book. It’s an opportunity to show people the film you want to make: how light, color, and form will function, character sketches, location ideas, tone exploration, how you’ll shoot different locations, etc. It was a helpful tool to show investors, became a great conversation starter when interviewing creative keys and became the keystone for refining the film’s aesthetic design. I suggest getting a cheap portable USB scanner for image collection. I made pilgrimages to the Mid-Manhattan Picture Library to leaf through their vast image files and scoured the Columbia and NYPL arts libraries to scan photographs, paintings, and other visual inspiration. I also took film screen-grabs and combed Flickr, Google and Getty Images for additional material.
5. Know your film’s spine.
Your film’s spine will drive every aesthetic decision you make, so develop a clear and focused throughline. In prep and on set, you will be asked thousands of questions: what should the protagonist’s house look like; which jacket should they wear; what color should we paint the walls; what kind of camera language should we use; what about lighting? With a cohesive spine, you can vet each choice through that prism. Since “The Discoverers” was a return to the pastoral story where each of the characters stripped from their urban selves saw themselves and each other in a new light, we developed an aesthetic arc with four movements (city, suburbs, forest, Pacific) where light, color, camera movement, and form could evolve over the course of their journey.
READ MORE: Griffin Dunne Kickstarts a Documentary About (His Aunt) Joan Didion
6. Help your actors build characters before coming to set.
Hopefully you’ll get some rehearsal time, but even before your actors get to set, you can help them build their characters. Take this time to learn their language – are they method or Meisner, what’s their process? I shared my visual research, music, books, and movies, we talked backstories, and I gave them homework. I asked Griffin Dunne, who plays an academic, to immerse himself in Lewis and Clark literature and sent him on a research expedition to a historical archive. For Cara Buono, I asked her to read about Alice Waters, speak with Etsy crafters, and chat with Hollywood Farmers Market vendors. I had Devon Graye read Rilke’s Letters To A Young Poet, hang out with skaters, and send me drawings. Madeleine Martin went vegan and became a Mother Jones, Howard Zinn and Barbara Ehrenreich expert.
7. Always carry a pocketful of toys.
I started collecting toys to experiment with blocking and camera movement when I was making short films. In the beginning, spaceships and animals were actor stand-ins until I upgraded to wooden artist models you can find at any art supply store. During prep I blocked out scenes with my DP and we’d photograph reference shots. On set I’d always carry a pocketful of toys and matchbox cars to illustrate a tracking shot or where the camera would go. As a parting gift, my Production Design team gave me a new set of toys, so I’m ready for the next film.
8. Over-prepare, then leave it all behind.
The one thing you have before the camera starts rolling is time. Time to breakdown your script, storyboard and pre-visualize. Once you get on set and are in real locations with actors breathing life into your characters, there never seems to be enough hours in the day. Not to worry, you can throw out all your prep work and respond and adapt to the magic that collaboration brings, confident that you know what you need.
9. Understand everyone’s job.
In film school we worked on each other’s crews to cycle through every role on set and learn from each other. Try to master everything you can, from budgeting to scheduling, camera to sound, grip and lighting, how to transcode and edit. It will empower you to know what is possible and enable you to make creative decisions when faced with resource constraints.
10. Be open to ideas, and protect your initial inspiration.
When you’re writing, get as much feedback as you can. Same goes for test screenings to refine the cut. During prep, go through the script with your actors, and make script changes if something revelatory emerges. Be open to new ideas wherever they come from, but know the film you want to make and protect your initial inspiration that brought everyone together to make the movie.
11. You set the tone.
Embrace the experience of making a movie and bring your energy, inspiration, and excitement to set. Try to create an environment where your cast and crew feel free to take risks and make great work. At wrap time, I also tried to personally thank each and every cast and crew member for their work each day. Filmmaking is a collaborative medium and everybody should be treated with respect and have their efforts acknowledged.
12. Don’t compromise (until the last minute).
Your creative keys don’t want you to compromise. Neither do your actors. They want you to be bold and ambitious and take chances. Most likely, they aren’t doing your project for the money, they were moved by the script or creative opportunities the film presented, so don’t compromise – at least until the last minute. Then, be realistic and embrace the idea that often the best creative solutions come from limitations.
Beginning today, Schwarz’s directorial debut “The Discoverers” is available on iTunes, DVD and VOD. Find out more at www.discoverersmovie.com.
READ MORE: Griffin Dunne Talks Acting in “The Discoverers”
Sign Up: Stay on top of the latest breaking film and TV news! Sign up for our Email Newsletters here. | 2023-08-11T01:27:17.760571 | https://example.com/article/3052 |
SEOUL (Reuters) - Dozens of South Korean conscientious objectors were released from jail on parole on Friday after a landmark court decision that conscience is a valid reason to refuse mandatory military service.
Slideshow ( 6 images )
The Supreme Court, South Korea’s top court, overturned earlier in November about five decades of precedents that punished conscientious objectors, a change that is in step with shifting attitudes as relations with North Korea thaw.
An official from the Ministry of Justice said on Friday the ministry decided to release on parole 58 conscientious objectors who had been in jail for six months or more.
That would reduce the number of conscientious objectors in jail from 71 to 13, South Korean online media Money Today reported, citing an unidentified law enforcement source.
The parolees will be doing community service, the justice ministry official said.
Among those released, eight clean-shaven young men, some dressed in suits, stood in front of Daegu Detention Centre to pose for media photographs. The parolees were released from several detention centers around the country.
“I fully understand critical views on conscientious objection, given that in South Korea we have a duty of national defense,” one of the men, a 26-year-old surnamed Kim, told Yonhap news agency.
“Whether it’s alternative service or community service, we hope to diligently do the work given to us and change views from negative to positive,” he said.
The 58 released on Friday had been serving jail terms of up to three years under the Supreme Court’s previous interpretation of the Military Service Act that said religious beliefs or conscience were not valid reasons to refuse mandatory service.
Conscientious objectors were soon expected to have the option of alternative service, although the period and type of service have yet to be decided.
South Korea’s defense ministry is holding its second public hearing on alternative service on Friday, a ministry statement said.
The hearing is discussing whether requiring conscientious objectors to serve 36 months of alternative service is punitive or is needed to deter potential abuse of conscientious objection to shirk mandatory military service.
The 36 months of alternative service compares with 18 months of mandatory army service by 2022. | 2023-11-29T01:27:17.760571 | https://example.com/article/5779 |
Pages
Thursday, March 12, 2009
Halladay still perfect
Sounds like your everyday Roy Halladay start, doesn’t it? With his sparking 0.00 ERA throughout 3 starts and 9 innings of work in Spring Training, Halladay has yet to give up a single run. Between mowing down and embarrassing opposing hitters, Halladay talked to the Toronto Sun and sounded pretty optimistic about this upcoming season:
"I think we aren't expected to do as much as we've done in the past. But I think we have a chance to be a lot more surprising than we have in the past. If some of these young guys we have pan out and become what we think they can be, I think we have a chance of being a lot better than mediocre.”
Hell yes, that’s what you want to hear from your number one player on the team. None of this “fourth place” bullshit. Halladay is going all the way, and so are the Blue Jays. I guess things aren’t looking so grim anymore, and J.P. Ricciardi saved a boatload of cash by not offering A.J. Burnett ridiculous amounts of money to stay in Toronto. So far, I whole-heartedly approve of that decision.
And as if Halladay’s pitch repertoire wasn’t scary enough, it sounds like he’s working on perfecting a changeup too, as he threw 9 in yesterday’s 2-2 draw with the Pittsburgh Pirates. Add a killer changeup to his 2-seam fastball, cutter, and curveball, and he has a deadly pitch repertoire that’s nearly un-hittable.
To the Baseball Writers Association of America, you might as well go ahead write Roy Halladay’s name on your ballot for the AL Cy Young Award, because he’s not leaving town unless he packs up another Cy.
EBL, if I lived in Toronto and I knew Doc was pitching that night, I'd run my ass to the Rogers Centre. I've actually only ever seen him pitch at the Rogers Centre once. It was Sept 5th last year vs. the Rays, and he was phenomenal. Totally worth the price of admission.
Matthias, it should be interesting to see Halladay's pitch placement/repetoire this year. I agree that he is very underappreciated, and should be viewed as a Greek God of baseball.
How to Contact the Blue Jay Hunter
About the Blue Jay Hunter
The Blue Jay Hunter is a blog about the Toronto Blue Jays, which takes a look at the team under the microscope. Mixed in with just a hint of humour, and a shred of dignity. I also have an affinity for baseball moustaches. | 2023-10-26T01:27:17.760571 | https://example.com/article/5801 |
Yasin Sancak
Yasin Sancak (born September 9, 1978 in İstanbul) is a Turkish volleyball player. He is 199 cm and plays as middle blocker. He played 63 times for the national team and also played for Netaş, Erdemir, Arkas Spor, Galatasaray, Ziraat Bankası, Fenerbahçe. He is signed for Galatasaray
Honours and awards
4 times Turkish Men's Volleyball League Champion
1 time Turkish Cup Champion
2005 Summer Universiade Champion
2009 Balkan Cup Champion
External links
Galatasaray Official Website
References
Category:1979 births
Category:Living people
Category:People from Istanbul
Category:Turkish men's volleyball players
Category:Galatasaray S.K. (men's volleyball) players
Category:Fenerbahçe volleyballers
Category:Arkas Spor volleyball players
Category:Ziraat Bankası volleyball players
Category:Universiade medalists in volleyball
Category:Universiade gold medalists for Turkey | 2024-05-01T01:27:17.760571 | https://example.com/article/3349 |
Q:
Foursquare missing file upload / InvalidPhotoFormat error while uploading photo through api
I'm trying to add a photo to foursquare page using api: https://api.foursquare.com/v2/photos/add and following node.js code:
var accessToken = "myAccessToken";
var platformProfileId = "4squarePageId";
var b64content = "somebase64stringrepresentationofimage";
var url = "https://api.foursquare.com/v2/photos/add";
var formObj = {'oauth_token': accessToken, v: '20151009', 'pageId': platformProfileId, 'photo': b64content};
request({
url: url, //URL to hit
form: formObj, //form data
method: 'POST',
headers: { 'Content-Type': 'image/jpeg' }
}, function(error, response, body){
if(error) {
console.log(error);
return cb(error);
} else {
if(typeof body != 'object') {
body = JSON.parse(body);
}
console.log(body);
if(('meta' in body) && ('code' in body['meta']) && (body['meta']['code'] != 200)) {
return callback_inner("error");
}
var mediaIdStr = body['response']['id'];
return callback_inner(null, mediaIdStr);
}
});
I'm getting following response:
{ meta:
{ code: 400,
errorType: 'other',
errorDetail: 'Missing file upload',
requestId: '561fe6c1498e097824456e38' },
notifications: [ { type: 'notificationTray', item: [Object] } ],
response: {} }
Can anyone please tell me where am I doing wrong ?
Update:
var queryObj = {'oauth_token': accessToken, v: '20151009', 'pageId': platformProfileId};
request({
url: url, //URL to hit
qs: queryObj, //query obj
method: 'POST',
headers: { 'Content-Type': 'image/jpeg' },
body: b64content
}, function(error, response, body){
if(error) {
console.log(error);
return cb(error);
} else {
if(typeof body != 'object') {
body = JSON.parse(body);
}
console.log(body);
if(('meta' in body) && ('code' in body['meta']) && (body['meta']['code'] != 200)) {
return callback_inner("error");
}
var mediaIdStr = body['response']['id'];
return callback_inner(null, mediaIdStr);
}
});
Tried sending image as post message body but even then it's not working.
Update 2:
var b64mediaFilesArr = results.C;
async.map(b64mediaFilesArr, function(b64content, callback_inner){
var imagename = new Date() + '.jpg';
var url = "https://api.foursquare.com/v2/photos/add";
var formObj = {
'oauth_token': accessToken,
'v': '20151009',
'pageId': platformProfileId,
'photo': {
value: b64content,
options: {
filename: imagename,
contentType: 'image/jpeg'
}
}
};
request({
url: url, //URL to hit
formData: formObj, //form data
method: 'POST',
headers: { 'Content-Type': 'image/jpeg' }
}, function(error, response, body){
if(error) {
console.log(error);
return cb(error);
} else {
if(typeof body != 'object') {
body = JSON.parse(body);
}
console.log(body);
if(('meta' in body) && ('code' in body['meta']) && (body['meta']['code'] != 200)) {
return callback_inner("error");
}
var mediaIdStr = body['response']['id'];
return callback_inner(null, mediaIdStr);
}
});
If I use above code, then there is change in the response:
{ meta:
{ code: 400,
errorType: 'param_error',
errorDetail: 'InvalidPhotoFormat: Unable to determine photo type',
requestId: '56207798498ee45703ab6059' },
notifications: [ { type: 'notificationTray', item: [Object] } ],
response: {} }
I'm going crazy after this. Can anyone please help me out ?
Solution
In addition to below accepted answer, I solved base64 encoded problem. For those of you using base64 encoded image data in your web app, you need to send original binary rep of image to Foursquare. This SO answer helped me to do that. Convert Binary.toString('encode64') back to Binary
A:
These are the request options that worked for me:
var options = {
'url': 'https://api.foursquare.com/v2/photos/add',
'qs': {
'v': '20161001',
'oauth_token': ACCESS_TOKEN,
'venueId': VENUE_ID
},
'formData': {
'file': {
'value': RAW_IMAGE_BUFFER,
'options': {
'filename': 'topsecret.jpg',
'contentType': 'image/jpg'
}
}
},
'json': true
};
Then just call:
request.post(options, function(error, response, body){})
A:
photo parameter does not exist. photo is response field.
The image data is sent as the POST message body on the HTTP request.
EDIT
You use request? Refer to https://github.com/request/request#multipartform-data-multipart-form-uploads
You don't need encode into base64.
| 2024-01-15T01:27:17.760571 | https://example.com/article/1742 |
Order-4 octahedral honeycomb
In the geometry of hyperbolic 3-space, the order-4 octahedral honeycomb is a regular paracompact honeycomb. It is called paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {3,4,4}, it has four octahedra, {3,4} around each edge, and infinite octahedra around each vertex in a square tiling {4,4} vertex arrangement.
Symmetry
A half symmetry construction, [3,4,4,1+], exists as {3,41,1}, with alternating two types (colors) of octahedral cells. ↔ . A second half symmetry, [3,4,1+,4]: ↔ . A higher index subsymmetry, [3,4,4*], index 8, exists with a pyramidal fundamental domain, [((3,∞,3)),((3,∞,3))]: .
This honeycomb contains , that tile 2-hypercycle surfaces, similar to the paracompact tiling or
Related polytopes and honeycombs
It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.
There are fifteen uniform honeycombs in the [4,4,3] Coxeter group family, including this regular form.
It is a part of a sequence of honeycombs with a square tiling vertex figure:
It a part of a sequence of regular polychora and honeycombs with octahedral cells.
Rectified order-4 octahedral honeycomb
The rectified order-4 octahedral honeycomb, t1{3,4,4}, has cuboctahedron and square tiling facets, with a square prism vertex figure.
Truncated order-4 octahedral honeycomb
The truncated order-4 octahedral honeycomb, t0,1{3,4,4}, has truncated octahedron and square tiling facets, with a square pyramid vertex figure.
Cantellated order-4 octahedral honeycomb
The cantellated order-4 octahedral honeycomb, t0,2{3,4,4}, has rhombicuboctahedron and square tiling facets, with a triangular prism vertex figure.
Cantitruncated order-4 octahedral honeycomb
The cantitruncated order-4 octahedral honeycomb, t0,1,2{3,4,4}, has truncated cuboctahedron and square tiling facets, with a tetrahedron vertex figure.
Runcitruncated order-4 octahedral honeycomb
The runcitruncated order-4 octahedral honeycomb, t0,1,3{3,4,4}, has truncated octahedron and square tiling facets, with a square pyramid vertex figure.
Snub order-4 octahedral honeycomb
The snub order-4 octahedral honeycomb, s{3,4,4}, has Coxeter diagram . It is a scaliform honeycomb, with square pyramid, square tilings, and icosahedra.
See also
Convex uniform honeycombs in hyperbolic space
List of regular polytopes
References
Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, , (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
Jeffrey R. Weeks The Shape of Space, 2nd edition (Chapter 16-17: Geometries on Three-manifolds I,II)
Norman Johnson Uniform Polytopes, Manuscript
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
N.W. Johnson: Geometries and Transformations, (2015) Chapter 13: Hyperbolic Coxeter groups
Norman W. Johnson and Asia Ivic Weiss Quadratic Integers and Coxeter Groups PDF Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336
Category:Honeycombs (geometry) | 2024-07-24T01:27:17.760571 | https://example.com/article/8481 |
At least 25 people were shot, five of them fatally, in Chicago between Friday and Sunday, police said.
The weekend violence began just one day after 13 people were injured in a shooting at a Chicago park.
In a Sunday evening incident, a chicken delivery driver says he was shot in the hip after a gunman came out of a gangway at started firing at about 10:30 p.m. near East 99th Street and South Paxton Avenue.
Police said they do not believe the gunman was trying to rob the delivery man.
Earlier in the evening, near West 101st and South State streets, a Chicago police sergeant shot and wounded a man who refused to drop a weapon, officials said. The man was shot in the foot and transported in good condition to Roseland Community Hospital.
A police officer was also involved in an auto accident in the same location, fire media said. The officer, whose age and gender was not immediately available, was taken in good condition to Little Company of Mary Hospital.
Earlier in the day, two men were shot near 49th Street and Langley Avenue, according to fire media.
A 19-year-old man was taken in serious-to-critical condition to John H. Stroger Jr., Hospital of Cook County after that shooting, which happened at about 2:15 p.m., and another man, whose age was not immediately known, was taken to Jackson Park Hospital, fire media said.
In another shooting, a 23-year-old man was shot in the head while on the 7200 block of South Wentworth Avenue just before 1:15 p.m., police said. The man was taken in critical condition to Stroger Hospital, according to Chicago Police News Affairs Officer Veejay Zala.
At around 3 a.m., two people were shot in the 6400 block of South Bell Avenue. Police said a 26-year-old man was sitting on the front porch of a residence with a group of others when a white van drove by and an unknown offender fired shots.
The man was struck twice, once in each leg, and was taken to Holy Cross Hospital in good condition, police said. A second victim, a 24-year-old woman, suffered a graze wound to the left leg and was also treated at Holy Cross Hsopital.
Less than an hour earlier, a 28-year-old man was shot while in a car in the 3100 block of West Lyndale Street. Police said the man was traveling westbound on Lyndale when someone emerged from an alley and fired at him, striking him in the ankle.
He was taken to Advocate Illinois Masonic Hospital in stable condition.
Just before 2 a.m., a 17-year-old man was walking in the 0-100 block of North Mason Avenue was heard shots and felt pain. The teen was taken by a family member to Oak Park Hospital with a wound to his left foot, police said.
Around midnight Sunday, a 40-year-old man was shot in the left leg after a gunman fired two shots at him in the 1200 block of North Spaulding Avenue, police said.
The man was taken in stable condition to Stroger Hospital. No further details were immediately available on the shooting.
On Saturday, authorities said a 16-year-old girl was shot around 10 p.m. while standing with a group of friends in the playlot of a park in the 1200 block of West 92nd Street.
Two men walked out of an alley and opened fire, police said.
The teen was taken to Advocate Christ Medical Center with a gunshot wound to her right arm. She was listed in stable condition, according to Chicago Police News Affairs Officer Michael Sullivan.
Just after 5 p.m. Saturday, three people were shot in two separate attacks. A 19-year-old man was shot in the hand and hip around 5:20 p.m. in the 1200 block of West 69th Street, police said.
His condition was not immediately known and details surrounding the shooting were not available, Sullivan said.
At the same time, two people were shot in the Englewood neighborhood. Police said the shooting took place near 69th Street and South Racine Avenue, where a 19-year-old man and a 22-year-old man were shot.
Chicago Police News Affairs said their conditions were not known and no further details were available on the shooting.
Around 5:35 a.m. Saturday, in the 11800 block of South Lowe Avenue, a 51-year-old man was shot in the stomach. The man claimed he heard gunshots and felt pain, according to Chicago Police News Affairs Officer Jose Estrada.
He was transported in serious condition to Advocate Christ Medical Center in Oak Lawn, according to authorities.
Around 4 a.m. Saturday, a man was shot over a parking spot in Chicago's Gold Coast neighborhood. The shooting occurred in the 0-100 block of East Chestnut Street when a 41-year-old man was shot in the abdomen, police said.
The shooter fled in a white car and the man was taken in serious condition to Northwestern Memorial Hospital, Sullivan said.
An hour earlier, two men were shot in the 1200 block of West Washburn Avenue, police said. A 22-year-old man was standing on the sidewalk when he heard shots and felt pain, Sullivan said. The man, later identified as Jamal Coleman, of the 5400 block of West Monroe Street, was taken to John H. Stroger Hospital and later died.
A second victim, a 23-year-old man, drove himself to the hospital with a wound to the leg. and said he'd been hurt in the same Washburn Avenue attack.
Around 1:20 a.m., police said a 30-year-old man was exiting the driver’s side of a vehicle when another car drove by and fired several shots. The man was transported to Mount Sinai Hospital in serious condition.
On Friday, a 37-year-old man was walking in an alley around 11:10 p.m. in the 6000 block of South Halsted Street when an offender fired shots from behind, police said.
Robert Hart, 37, of the 5800 block of South Shields Avenue was taken to Mount Sinai Hospital and died short after midnight.
Earlier Friday, a 21-year-old man was fatally shot in the head in the 6600 block of South Honore Street. Police said an offender wearing a black hoodie and blue jeans fired shots at the man around 6:30 p.m. and fled the scene.
Brenden Temple, 21, of the 4000 block of South Indiana Avenue, was pronounced dead at Stroger Hospital.
Fifteen minutes earlier, an 18-year-old man was shot in the 2200 block of East 71st Street. Jaleel Pearson, of the 6800 block of South Artesian Avenue, was shot in the chest and arm and died at Stroger Hostpital.
Around 5:30 p.m., a man was shot during a police-involved shooting on the city’s Northwest Side. Police said an officer observed a car matching the description of a vehicle wanted in connection with a burglary in the 6100 block of West Belmont Avenue.
The officer pursued the vehicle and another responding sergeant attempted to curb it in the 5600 block of West Roscoe Street.
The sergeant then exited his vehicle and the offender reversed and intentionally rammed the squad car. The sergeant approached the vehicle, but the offender accelerated forward toward the officer and the officer fired, according to a statement from police.
The man was taken into custody and transported to an area hospital for his injuries. Angel Torres, 44, was charged with felony burglary, aggravated assault and armed violence, police said Saturday.
Earlier, a shooting in the Portage Park neighborhood left a man in his 20s with a gunshot wound to the foot, according to Chicago Police News Affairs Officer Daniel O'Brien.
The man was shot around 1:52 p.m. in the 3700 block of North Mobile Avenue and was taken to Our Lady of Resurrection Medical Center in good condition, O'Brien said.
A teen was fatally shot minutes earlier around 1:35 p.m. in the 6900 block of South Vernon Avenue.
The teen was shot in the back and pronounced dead at the scene, O'Brien said.
More Local Stories: | 2023-12-03T01:27:17.760571 | https://example.com/article/8966 |
Tag Archive
Designers: Richard Garfield Publisher: IELLO Year: 2017 MSRP: $49.99 Players: 2-4 Play Time: 45 Min Ages: 12+ Rules Available Online: Yes BGG: Bunny Kingdom Prelude The Bunny King has sent out his Rabbit Lords to take control of territories in the New World and turn them into successful Fiefs. The Lords must build cities, gather resources, and complete tasks for the king to be build up the …
Designers: Maxime Rambourg Publisher: IELLO Year: 2017 MSRP: $39.99 Players: 2-6 Play Time: 30-40min Ages: 8+ Rules Available Online: Yes BGG: Arena for the Gods Prelude Champions from 6 different mythos enter the arena to fight for the entertainment of the Gods, handpicked to fight for their skill in combat. Before the battle, each must bid for the best equipment using their life force, …
IELLO is known for publishing quality games, and Arena for the Gods is no exception. Not only is the boxed backed full of cardboard, dice, counters, and cards, but it also comes with a removeable bowl tucked into the insert that can be used to store tokens while playing. There’s also a King of Tokyo card tossed into the card … | 2023-12-11T01:27:17.760571 | https://example.com/article/8607 |
Q:
Funcionalidade de download com requisição via Ajax
Estou fazendo uma função de download com requisição via ajax em minha aplicação, porém não está funcionando.
Se eu acessar o endereço utilizando o método GET com parâmetro no endereço, o download funciona perfeitamente.
download.php?opcao=download
Mas se eu acessar o arquivo download.php via ajax com o script abaixo o download não é realizado.
index.php
<head>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
</head>
<div class="container">
<form method ="" id="upload_form" >
<input id="upload_button" type='button' onclick="download()" name="upload_button" class="btn btn-info" value="Download" />
<br><br>
</form>
</div>
<br>
<script language="javascript">
function download(){
$.ajax({
type: "GET",
url: "download.php",
data: {
'opcao': 'download'
}
});
}
</script>
download.php
<?php
$reqmethod = $_SERVER['REQUEST_METHOD'];
$opcao = verificarMetodo($reqmethod);
if($opcao == 'download'){
$enderecoEntrega = './';
$nomeArquivo = "teste.pdf";
// Tempo máximo de execução
// Parâmetro 0 para as conexões lentas
set_time_limit(0);
baixarArquivo($enderecoEntrega, $nomeArquivo);
}
function verificarMetodo($reqmethod){
if($reqmethod == 'POST'){
$opcao = filter_input(INPUT_POST, "opcao");
}else{
$opcao = filter_input(INPUT_GET, "opcao");
}
return $opcao;
}
function baixarArquivo($caminho, $nome){
$validacao = false;
$nomeArquivo = "{$nome}";
$arquivo = "{$caminho}/{$nome}";
if(!empty($nomeArquivo) && file_exists($arquivo)){
header("Content-Type: application/pdf");
header("Content-Disposition: attachment; filename='{$nomeArquivo}'");
$bool = readfile($arquivo);
}
return $bool;
}
?>
Estou fazendo essa funcionalidade via ajaxpara não ter atualização na tela durando o download, e apesar de eu utilizar apenas um parâmetro na variável data do JavaScript, penso em utilizar os parâmetros com nome do arquivo e endereço para então ser realizado um $enderecoEntrega = filter_input(INPUT_GET, "enderecoEntrega"); e $nomeArquivo= filter_input(INPUT_GET, "nomeArquivo"); e ficar dinâmico o endereço e arquivo do meu download.
Se tiverem uma forma alternativa para solucionar meu problema, podem ficar a vontade para postar como resposta.
A:
O Ajax não funciona desta forma. As funções executadas no PHP só surtem efeito diretos para o usuário se a página PHP for aberta diretamente no navegador. Como o Ajax faz uma requisição para a página em segundo plano, por assim dizer, tudo que acontece lá no PHP fica lá. O que o Ajax faz é esperar (ou não) alguma resposta em forma de dados (uma string ou um JSON, por exemplo).
O que você teria que fazer é ativar o download do arquivo pelo próprio JavaScript assim que o Ajax receber como resposta de que a requisição foi bem sucedida. No caso do jQuery se dá através do callback success:
success: function(data){
// faz alguma coisa
}
Onde o valor de data é algo retornado do PHP requisitado. Só que no seu caso o data seria vazio porque o PHP não está retornando nada (seja por um echo ou códigos HTML). Para exemplificar, vamos supor que o arquivo download.php tenha apenas o código abaixo:
<?
echo "olá";
?>
Neste caso, o valor de data seria olá. No seu caso, o que você poderia fazer é retornar um echo com o caminho do arquivo a ser baixado, por exemplo:
if(!empty($nomeArquivo) && file_exists($arquivo)){
echo $arquivo;
}
Neste caso o valor de data seria o caminho do arquivo e você pode forçar o download com o código JavaScript abaixo, que cria um elemento de link <a> com o atributo download e fazer um click automático, forçando o download do arquivo, isso tudo dentro do success::
function download(){
$.ajax({
type: "GET",
url: "download.php",
data: {
'opcao': 'download'
},
success: function(data){
var a = document.createElement("a");
a.download = data;
a.href = data;
a.click();
}
});
}
Porém vale ressaltar que o atributo download não tem suporte total nos navegadores da Microsoft (IE e Edge). No caso desses navegadores, em vez de abrir o diálogo para baixar o arquivo, será perguntado se você deseja abrir ou salvar o arquivo.
Firefox
No Firefox (versão 64.0.2) o click() só funcionou se o elemento for adicionado ao body. Adicione o elemento e após o click, remova-o:
function download(){
$.ajax({
type: "GET",
url: "download.php",
data: {
'opcao': 'download'
},
success: function(data){
var a = document.createElement("a");
a.download = data;
a.href = data;
document.body.appendChild(a); // adiciona ao body
a.click();
a.outerHTML = ''; // remove o elemento
}
});
}
| 2024-03-03T01:27:17.760571 | https://example.com/article/7950 |
Inland Empire 66ers
The Inland Empire 66ers of San Bernardino are a minor league baseball team in San Bernardino, California. They are the Class A Advanced affiliate of the Los Angeles Angels and play in the California League. The 66ers play home games at San Manuel Stadium.
Franchise history
The 66ers began when the Salinas Spurs moved to San Bernardino and bought the name of the San Bernardino Spirit from the Rancho Cucamonga Quakes who had just left town. The team was bought by Elmore Sports Group soon afterwards. The team originally played at Fiscalini Field until 1996 when the team moved to San Bernardino Stadium, nicknamed "The Ranch" in concert with the team's new name, the San Bernardino Stampede.
Later, the stadium was re-christened San Manuel Stadium sponsored by the San Manuel Band of Mission Indians with their casino.
The team name was later changed to the team's current name, the Inland Empire 66ers, taken from the historic U. S. Route 66 that ran through San Bernardino.
In September 2006, the 66ers announced they were renewing its affiliation with the Los Angeles Dodgers beginning in the 2007 season. That affiliation would last through the end of the 2010 season.
In September 2010, the 66ers agreed to a two-year affiliation with the Los Angeles Angels to be the MLB club's minor league "High A" ball affiliate.
The team was once unique for having a National Public Radio affiliate, KVCR-FM, as the station that broadcast its games on radio. However, in 2008 the team changed its station to commercial talk station KCAA.
Year-by-year record
San Bernardino Spirit (1993–1995)
San Bernardino Stampede (1996–2002)
Inland Empire 66ers (2003–present)
Roster
Notable alumni
Hall of Fame alumni
Ken Griffey, Jr. (1988) Inducted, 2016
Notable alumni
Adrián Beltré (1996) 4 x MLB All-Star
Zach Borenstein (MVP)
Dave Burba (1988)
Asdrúbal Cabrera (2005) 2 x MLB All-Star
Tom Candiotti (1996)
Shin-Soo Choo (2002-2003)
Jeff Cirillo (2003) 2 x MLB All-Star
Del Crandall (1996-1997, MGR) 11 x MLB All-Star
Rich Dauer (1988, MGR) MLB All-Star
Ivan DeJesus (1992, MGR)
Rafael Furcal (2007, 2010) 3 x MLB All-Star; 2000 NL Rookie of the Year
Mark Grudzielanek (1999) MLB All-Star
Kenley Jansen (2009-2010) 2 x MLB All-Star
Mike Hampton (1992) 2 x MLB All-Star
Mark Harmon
Matt Herges (1995)
Félix Hernández
Orel Hershiser (2000) 1988 World Series Most Valuable Player; 1988 NL Cy Young Award
Todd Hollandsworth (1995, 1997, 1999) 1996 NL Rookie of the Year
Eric Karros (1998) N1992 L Rookie of the Year
Paul Konerko (1995) 6 x MLB All-Star
Ted Lilly (1997) 2 x MLB All-Star
Brandon Morrow (2006)
Gregg Olson MLB All Star; 1989 NL Rookie of the Year
Scott Radinsky (1995)
Rene Rivera
David Ross (2000)
Carlos Santana (2008)
Bill Swift (1989) 1992 NL ERA Leader
Luis Valbuena (2006)
Ismael Valdez (1998)
Omar Vizquel (1990) 11 x Gold Glove; 3 x MLB All-Star
Tim Wallach (1995)(1998, MGR) 5 x MLB All-Star
Devon White (2000) 7 x Gold Glove; 3 x MLB All-Star
Terry Whitfield (1987)
C.J. Wilson (1986) 2 x MLB All-Star
Steve Yeager, coach
External links
Inland Empire 66ers
Category:Professional baseball teams in California
01
Category:Sports in San Bernardino, California
Category:Sports in the Inland Empire
Category:California League teams
Category:Baseball teams established in 1987
Category:Los Angeles Angels minor league affiliates
Category:Los Angeles Angels of Anaheim minor league affiliates
Category:Los Angeles Dodgers minor league affiliates
Category:San Francisco Giants minor league affiliates
Category:Seattle Mariners minor league affiliates
Category:1987 establishments in California | 2023-10-22T01:27:17.760571 | https://example.com/article/6732 |
Choose the United States Four Major League's most popular teams' apparel online, including NFL/NBA/NHL/MLB jerseys and other kind of cheap sports jerseys. Enjoy the low prices and quality service. Looking forward to your visit!
Draft He Holds The NHL Record Evgeny Kuznetsov Jersey For Most Points In A Season By A German Player
If I were a team in the Flyers’ position, of not really being a playoff team and not really having Martin Jones jersey a super-bright future with the current core of guys who’ve been the big contributors, I’d definitely consider trading Giroux while his value will still be pretty high. Will teams want to give up a lot for a 30-year-old with five years left on a deal paying him $8.275 AAV? Doubt it, but
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Despite these important issues — and, hey, every team’s got ’em, right? — the Rangers will be a fun, exciting, fast-paced team to watch in 2017-18, whether they are a Stanley Cup favorite TJ Oshie jersey Discount or not.
One thing to nhl jersey size 52 Alex Ovechkin jersey know:?The Oilers took Draisaitl with the third overall pick in the 2014 draft. He holds the NHL record for most points in a season by a German player: 77 in 2016-17.
Despite this list not being jerseys from china nfl hats in any order, there is no arguing that the list should start the man dubbed as cheap nhl jerseys Mr. Hockey. Howe’s career spanned Washington Capitals road jersey across decades — yes, decades — and different leagues, becoming the face of not cheap hockey jerseys only the NHL, but hockey everywhere.
The guys Brent Burns jersey men know that they’re chasing points right now. Everything we talk about is to try and improve our position for the playoffs and home ice, Nashville coach Peter Laviolette said.
First of all thanks for sending a separate college hockey question. Second, Ohio State is better than anyone had any right to expect (they entered the season with literally no goaltending experience) and Penn State would be good if they had any goaltending at all. | 2023-12-17T01:27:17.760571 | https://example.com/article/3203 |
➲ 2012 SEPTEMBER NO PURCHASE CLUB - Page 9
A great theme for avoiding the temptation to buy and to maintain what exists. We often do not wear what we have as it is lost in the wardrobe. More creative maintenance could also mean changing buttons (ordinary to mother of pearl etc), or even stitching the button in a new way. Perhaps women are good at circulating the wardrobe. Thanks for the idea. Lastly, re-engineering clothes to enhance life need not be ruled out.
I did need 2 tshirts from BR because I gained a bit of weight and my shirts did not fit and were attacked by a rodent while being pilled up from not wearing them. I think that does count as an emergency as my undershirts are wife beaters and i cant wear those out in public unless i live in Miami.
It will be cheating, though I would be really interested in how you will now deem this an emergency.
No you may not spent it.
What? WHAT? Oh my. Photos please, because 6 months from now I want you to come back to this thread and ask yourself if it was worth it.
You guys need to toughen up and buckle down here. Seriously, if we were all going to climb Mt Everest together your bodies would clutter up Base camp. Please be strong. We are not even half way yet. No more spending!
I accept the judge's decision!
Still spend free here and can't think of anything I desperately need I the next two weeks.
I placed 3 Bontoni orders during trunk show but didnt pay a dime.
I did need 2 tshirts from BR because I gained a bit of weight and my shirts did not fit and were attacked by a rodent while being pilled up from not wearing them. I think that does count as an emergency as my undershirts are wife beaters and i cant wear those out in public unless i live in Miami.
NPC'ers let me know if i am still eligible or if I fail.
I think you might be out for placing the Bontoni order. I believe the rule is that you can pick up/pay for orders already placed, but that you cannot place any new orders.
I did need 2 tshirts from BR because I gained a bit of weight and my shirts did not fit and were attacked by a rodent while being pilled up from not wearing them. I think that does count as an emergency as my undershirts are wife beaters and i cant wear those out in public unless i live in Miami.
NPC'ers let me know if i am still eligible or if I fail.
I think you might be out for placing the Bontoni order. I believe the rule is that you can pick up/pay for orders already placed, but that you cannot place any new orders.
Reporting in, I'm still going strong, but I almost lost. I walked into C&J at the Burlington Arcade, intending to take a look at the Skye 2 boot in cordovan. Unfortunately the shipment was delayed from the factory, but I tried on the original Skye and the pleasant manager almost managed to convince me to put down payment for the Skye 2 boot. I stayed strong and gave them my phone number instead of my credit card. They expected delivery in the beginning of October, so I just might get them once this no purchase month is over.
I'm going strong, although after viewing my Charles Tyrwhitt midnight blue tie in a mirror, I almost wanted to order a midnight blue woven fine rib reppe tie and a green untipped woven wool solid - both from Drakes.
No! I've just noted the beautiful lilac shirts on the Drakes website. Must. Hold. Fast. My credit card has taken a whack this month, from taking out my Chinese language partner to dinner and drinks for weeks and weeks, and then taking out a Taiwanese girl for drinks and dinner the other night.
In truth, I'll be fine until October. Then I'll need to purchase those ties, think about finding a dark olive Sports Coat, and a lilac and a lavender shirt. | 2023-12-30T01:27:17.760571 | https://example.com/article/7170 |
In The
Court of Appeals
Ninth District of Texas at Beaumont
____________________
NO. 09-07-00487-CR
NO. 09-07-00488-CR
____________________
HERMAN NEAL KIRKMAN, Appellant
V.
THE STATE OF TEXAS, Appellee
On Appeal from the 252nd District Court
Jefferson County, Texas
Trial Cause Nos. 99592, 98724
MEMORANDUM OPINION
Appellant Herman Neal Kirkman (1) appeals from the trial court's orders revoking his
community supervision in two cases. Kirkman raises four issues for our consideration. We
affirm.
Background
Pursuant to plea bargain agreements, appellant Herman Neal Kirkman pled guilty to
deadly conduct and evading arrest or detention using a vehicle. In each case, the trial court
found the evidence was sufficient to find Kirkman guilty, but deferred further proceedings,
placed Kirkman on community supervision for two years, and assessed a fine of $500.
Condition number ten of each deferred adjudication order provided as follows: "Do not use
or possess any drug, except under the order of your doctor."
The State subsequently filed a motion to revoke Kirkman's unadjudicated community
supervision in each case. In both cases, the State's motion to revoke alleged that Kirkman
violated condition 10 of the deferred adjudication order because he was arrested for
possession of a controlled substance in Jefferson County on June 8, 2007. At the hearing on
the motions to revoke, Kirkman pled "true" to being arrested for unlawful possession of a
controlled substance. However, he denied being in unlawful possession of a controlled
substance. The State introduced evidence from Officer Rector of the Beaumont Police
Department. Officer Rector testified that on June 7, 2007, he observed a traffic violation by
the driver of a vehicle in which Kirkman was a passenger, and he stopped the vehicle. As
Officer Rector was stopping the vehicle, Kirkman opened the passenger door and quickly got
out of the car, and Officer Rector instructed Kirkman to get back into the vehicle. Officer
Rector testified, "it was very suspicious to me that he wanted to get out of the car rather
quickly, and the driver did not do that." Kirkman got back into the vehicle, and as Officer
Rector approached the vehicle on the passenger side, Kirkman opened the glove
compartment, pulled out a sunglass case "as if it was . . . burning hot, [and] he tossed it over
towards the driver and it landed in the driver's lap." According to Officer Rector, the driver
immediately tossed the case back to Kirkman, and Kirkman "just immediately threw it into
the glove compartment and closed the glove compartment."
Officer Rector testified that he was fearful because the sunglass case was large enough
to contain a small revolver, so he attempted to recover the case to determine if it in fact
contained a weapon. When the suspects handed Officer Rector the case, he opened it and
saw that it contained "multiple small off-white . . . rocks." According to Officer Rector,
Kirkman was extremely nervous, especially when he and the driver were handling the box,
and Kirkman did not want to make eye contact with Officer Rector. Officer Rector believed
the material in the sunglass case was a controlled substance, so he arrested the two suspects
and took the sunglass case and its contents to the police station. The State also introduced
the testimony of Emily Esquivel, a forensic analyst at the Jefferson County Crime Lab.
Esquivel testified that she analyzed the substance and determined that it was 3.99 grams of
cocaine.
Kirkman testified that on June 7, 2007, he was a passenger in a car driven by
Desmond Jones. According to Kirkman, after Officer Rector signaled the vehicle to stop,
Jones told Kirkman "that he had drugs in the case[,]" and Kirkman put the case into the glove
compartment. Kirkman explained that prior to the stop, the case was sitting on the seat
between him and the driver, and he put the case in the glove box because the driver was
"trying to hide it." Kirkman also testified that he did not know the box contained drugs until
the driver told him so, and up to that point, he did not have possession, control, or custody
of the drugs. Kirkman testified that Officer Rector asked him and Jones whose drugs were
in the case, and both Kirkman and Jones denied possessing the drugs.
At the conclusion of the hearing on the motions to revoke, the trial court found the
evidence sufficient to find count one true, revoked Kirkman's unadjudicated probation in
both cases, and found Kirkman guilty of deadly conduct and evading arrest or detention by
using a vehicle. In the evading arrest case, the trial court assessed punishment at two years
of confinement in a state jail facility. In the deadly conduct case, the trial court assessed
punishment at ten years of confinement in the Institutional Division, and the trial court
ordered that the sentences would run consecutively. Kirkman then filed these appeals, in
which he raises the same issues in each case for our consideration.
Issues One and Two
In issues one and two, Kirkman contends the trial court abused its discretion by
revoking his community supervision on the basis of allegations that were not included in the
State's motion to revoke, thereby violating his rights under the Fifth and Fourteenth
Amendments of the U.S. Constitution, as well as article I, section 19 of the Texas
Constitution and Article 1.04 of the Texas Code of Criminal Procedure. Specifically,
Kirkman argues that the motion to revoke did not provide him fair notice because, although
the motion to revoke only alleged that he violated condition ten, the trial court found that he
violated condition one, which mandated that he not break the law of Texas or any other state.
While the nunc pro tunc adjudication of guilt in the record contains a clerical error as it
references condition one, it is clear from the record that the State sought to revoke Kirkman's
probation only for violation of condition ten and the trial court found that he had violated
condition ten. We address these issues together.
Kirkman did not present the arguments he raises in issues one and two to the trial
court. Instead, Kirkman's objected "it's not a violation of probation to be arrested because
there's no presumption of guilt or there's no evidence of guilt. Merely being arrested -
anybody can be arrested at any time[.]" From this record, we find Kirkman has failed to
preserve these issues for our review. See Tex. R. App. P. 33.1(a); Gordon v. State, 575
S.W.2d 529, 531 (Tex. Crim. App. 1978) (When appellant raises lack of adequate notice in
a motion to revoke, he must file a motion to quash to preserve error.); see also Wilson v.
State, 71 S.W.3d 346, 349 (Tex. Crim. App. 2002) (An issue on appeal must comport with
a proper trial objection.).
Even if Kirkman had properly preserved these issues, he has not demonstrated that the
motions to revoke failed to provide him with fair notice. In each case, the motions to revoke
alleged that he had violated condition ten, which prohibited him from possessing any drug
except under his doctor's orders, because he had been arrested for possession of a controlled
substance. The allegations in a motion to revoke community supervision do not require the
same particularity as those in an indictment or information; however, "in all fairness the
allegations as to violation of probation should be fully and clearly set forth in the revocation
motion, so that the defendant and his counsel might be informed as to that upon which he
will be called to defend." Garner v. State, 545 S.W.2d 178, 179 (Tex. Crim. App. 1977).
We hold that the written motion to revoke gave Kirkman fair notice that the State intended
to prove he had possessed a controlled substance, which is a violation of Texas law. See id.;
see also generally Tex. Health & Safety Code Ann. § 481.115 (Vernon 2003). We
overrule issues one and two.
Issues Three and Four
In his third and fourth issues, Kirkman contends the trial court violated his due
process and due course of law rights under the U.S. and Texas constitutions by finding that
the evidence was sufficient to revoke his community supervision. We address these issues
together.
In a revocation proceeding, the State must establish by a preponderance of the
evidence that the defendant violated the conditions of the supervision order. Cardona v.
State, 665 S.W.2d 492, 493 (Tex. Crim. App. 1984). If the State meets its burden of proof,
it is within the trial court's discretion to revoke community supervision. Id. We review the
trial court's decision to revoke community supervision for an abuse of discretion. Jackson
v. State, 645 S.W.2d 303, 305 (Tex. Crim. App. 1983).
"To prove unlawful possession of a controlled substance, the State must first prove
appellant exercised actual care, control and management over the contraband and second,
that appellant had knowledge the substance in his possession was contraband." Nixon v.
State, 928 S.W.2d 212, 215 (Tex. App.--Beaumont 1996, no pet.) (citing King v. State, 895
S.W.2d 701, 703 (Tex. Crim. App. 1995)). The State need not prove exclusive possession
of the contraband, since control over contraband may be jointly exercised by more than one
person. McGoldrick v. State, 682 S.W.2d 573, 578 (Tex. Crim. App. 1985); State v. Derrow,
981 S.W.2d 776, 779 (Tex. App.--Houston [1st Dist.] 1998, pet. ref'd). However, "mere
presence at a place where contraband is being used or possessed by others does not justify
finding that a person is in joint possession or is a party to an offense." Roberson v. State, 80
S.W.3d 730, 735 (Tex. App.--Houston [1st Dist.] 2002, pet. ref'd).
When an accused is not in exclusive possession of the location where contraband is
found, additional independent facts and circumstances may affirmatively link him to the
contraband. Nixon, 928 S.W.2d at 215. An affirmative link may be established through
either direct or circumstantial evidence, and it must show that the accused's connection to
the contraband was more than fortuitous. Poindexter v. State, 153 S.W.3d 402, 405-06 (Tex.
Crim. App. 2005). Factors which tend to establish affirmative links include:
(1) the contraband was in plain view;
(2) the accused was the owner of the premises in which the contraband was
found;
(3) the contraband was conveniently accessible to the accused;
(4) the contraband was found in close proximity to the accused;
(5) a strong residual odor of the contraband was present;
(6) paraphernalia to use the contraband was in view or found near the accused;
(7) the physical condition of the accused indicated recent consumption of
the contraband in question;
(8) conduct by the accused indicated a consciousness of guilt;
(9) the accused had a special connection to the contraband;
(10) the place where the contraband was found was enclosed;
(11) the occupants of the premises gave conflicting statements about
relevant matters; and
(12) affirmative statements connect the accused to the contraband.
Nixon, 928 S.W.2d at 215. "It is . . . not the number of links that is dispositive, but rather the
logical force of all the evidence, direct and circumstantial." Evans v. State, 202 S.W.3d 158,
162 (Tex. Crim. App. 2006).
In this case, Officer Rector testified that Kirkman initially fled the vehicle, which
Officer Rector viewed as suspicious. In addition, Kirkman pulled out the sunglass case from
the glove compartment and quickly tossed it toward the driver. Kirkman was on the
passenger side of the vehicle, and the sunglass case was in the glovebox, which would have
been easily accessible to Kirkman. In addition, Officer Rector testified that Kirkman was
nervous and did not want to make eye contact. We conclude that sufficient affirmative links
exist in this case. See Nixon, 928 S.W.2d at 215. The evidence is sufficient to support the
trial court's decision to revoke Kirkman's community supervision. Accordingly, we overrule
issues three and four. Having overruled all of Kirkman's issues on appeal, we affirm the trial
court's judgments.
AFFIRMED.
__________________________________
CHARLES KREGER
Justice
Submitted on July 29, 2008
Opinion Delivered February 4, 2009
Do not publish
Before McKeithen, C.J., Kreger and Horton, JJ.
1. On the judgment in each case, appellant's surname is spelled "Kirkmon."
| 2023-09-20T01:27:17.760571 | https://example.com/article/7510 |
Surgical management of gallstone pancreatitis in children.
Because of the low incidence of gallstone pancreatitis in children, we sought to examine effects of varied practice patterns on outcomes. A retrospective review was performed on all patients undergoing cholecystectomy for a diagnosis of gallstone pancreatitis from January 2000 to June 2011. Demographics, diagnostic strategies, operative approaches, length of stay, and morbidity were compared between Group 1, who had cholecystectomy performed during the admission of diagnosis, and Group 2, who underwent cholecystectomy subsequently. Cholecystectomy was performed for gallstone pancreatitis in 41 patients, of whom 29 (70.7%) patients were female. Ultrasound was performed in all cases, revealing cholelithiasis in 37 (90.2%). There were 22 patients in Group 1 and 19 in Group 2. Mean age and body mass index did not vary between groups. Endoscopic retrograde cholangiopancreatography was performed in 14 patients (8 in Group 1 and 6 in Group 2), of these procedures 11 were prior to cholecystectomy, 2 were after cholecystectomy, and 1 was both. Total number of hospital days attributed to the diagnosis of gallstone pancreatitis was 8.9 ± 6.5 in Group 1 compared with 14.0 ± 14.4 in Group 2 (P = .15). There were 7 patients (36.8%) in Group 2 who required readmission for recurrent pancreatitis prior to their operation. This represents the largest reported series of cholecystectomy for gallstone pancreatitis in children. Our results support the use of laparoscopic cholecystectomy during the initial hospitalization as is recommended in the adult literature, and this approach may decrease the total hospital stay. | 2023-10-30T01:27:17.760571 | https://example.com/article/5106 |
European Historic Houses Association
European Historic Houses Association is an international organization based in Brussels, Belgium, and registered in the Netherlands. Through consultancy with the European Commission and the European Council, it has oversight over twenty-four national historic houses associations. Its scope includes conservation of historic houses and cultural heritage, including cultural goods.
Other affiliated organizations through the Council of Europe include Europa Nostra, European Confederation of Conservator-Restorers' Organisations (ECCO), and European Network for Conservation-Restoration Education (ENCoRE). While previous presidents of the association include Antoine, 13th Prince of Ligne and Jean, Grand Duke of Luxembourg, in 2017, the Executive President was Count Rodolphe de Looz-Corswarem.
References
Bibliography
External links
Official website
European Historic Houses Association at Union of International Associations
Category:Historic houses
Category:History organisations based in Belgium
Category:Cultural organisations based in Belgium
Category:Organisations based in Brussels | 2024-05-31T01:27:17.760571 | https://example.com/article/2388 |
/** A Java stacktrace text grammar for ANTLR v3
* see http://java.sun.com/javase/6/docs/api/java/lang/Throwable.html
*
* Version 1.0 -- initial release March 9, 2008 (requires 3.0 or higher)
*
* Primary authors: Luca Dall'Olio, Christian Cassano, Gabriele Contini
*/
grammar StackTraceText;
options{
output=AST;
}
tokens {
DOT='.';
AT='at';
CAUSED_BY='Caused by:';
MORE='more';
ELLIPSIS='...';
COLON=':';
NATIVE_METHOD='Native Method';
UNKNOWN_SOURCE='Unknown Source';
INIT='<init>';
}
@rulecatch { } /** Message will catch anything */
startRule
: stackTrace EOF;
stackTrace
: messageLine+ stackTraceLine* causedByLine?
;
stackTraceLine
: (atLine|ellipsisLine)
;
atLine
: AT qualifiedMethod '(' classFile (COLON Number)? ')'
;
causedByLine
: CAUSED_BY stackTrace
;
ellipsisLine : ELLIPSIS Number MORE
;
messageLine
: (qualifiedClass message?)
;
qualifiedClass: packagePath? className innerClassName*;
innerClassName
: ('$' className)
;
classFile
: (identifier '.java' | NATIVE_METHOD | UNKNOWN_SOURCE)
;
/** method name may be missing, I think in ctors */
qualifiedMethod
: qualifiedClass DOT (methodName|constructor)?;
constructor
: INIT
;
methodName
: identifier
;
packagePath : (identifier DOT)+;
className : JavaWord;
identifier : JavaWord;
message : COLON (options {greedy=false;}: .)*;
Number : Digit+
;
JavaWord : (JavaCharacter)+;
fragment
JavaCharacter
: (CapitalLetter
| NonCapitalLetter | Symbol
| Digit)
;
fragment NonCapitalLetter : 'a'..'z';
fragment CapitalLetter
: 'A'..'Z'
;
fragment Symbol : '_'
;
fragment Digit : '0'..'9';
WS : (' '|'\r'|'\t'|'\u000C'|'\n') {$channel=HIDDEN;}
;
| 2024-06-03T01:27:17.760571 | https://example.com/article/9574 |
/*
Copyright (C) 2011 Mark Chandler (Desura Net Pty Ltd)
Copyright (C) 2014 Bad Juju Games, Inc.
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 3 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., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
Contact us at legal@badjuju.com.
*/
#include "Common.h"
#include "WebCore.h"
#include "XMLMacros.h"
#ifdef WIN32
#include <Wininet.h>
#endif
#include "mcfcore/MCFI.h"
#include "mcfcore/MCFMain.h"
#include "sqlite3x.hpp"
#include "sql/WebCoreSql.h"
static gcString g_szRootDomain = "desura.com";
namespace WebCore
{
extern gcString genUserAgent();
WebCoreClass::WebCoreClass()
: m_ImageCache()
, m_RefCount()
, m_bDebuggingOut(false)
, m_bUserAuth(false)
, m_bValidateCert(true)
, m_mSessLock()
, m_szAppDataPath("")
, m_szIdCookie("")
, m_szUserAgent(genUserAgent())
, m_uiUserId(0)
, onCookieUpdateEvent()
, onLoggedOutEvent()
{
#ifdef DEBUG
m_bDebuggingOut = true;
#endif
}
WebCoreClass::~WebCoreClass()
{
m_ImageCache.saveToDb();
}
void WebCoreClass::enableDebugging(bool state)
{
m_bDebuggingOut = state;
}
void WebCoreClass::init(const char* appDataPath)
{
init(appDataPath, nullptr);
}
void WebCoreClass::init(const char* appDataPath, const char* szProviderUrl)
{
gcTrace("App data path {0}, Provider {1}", appDataPath, szProviderUrl);
gcString strProvUrl(szProviderUrl);
if (strProvUrl.size() == 0)
setUrlDomain("desura.com");
else
setUrlDomain(strProvUrl.c_str());
m_szAppDataPath = appDataPath;
createWebCoreDbTables(appDataPath);
m_ImageCache.init(appDataPath);
m_ImageCache.loadFromDb();
}
void WebCoreClass::setUrlDomain(const char* domain)
{
g_szRootDomain = domain;
m_bValidateCert = g_szRootDomain == "desura.com";
}
gcString WebCoreClass::getMCFDownloadUrl()
{
return gcString("http://api.") + g_szRootDomain + "/2/itemdownloadurl";
}
gcString WebCoreClass::getPassWordReminderUrl()
{
return gcString("http://api.") + g_szRootDomain + "/1/memberpasswordreminder";
}
gcString WebCoreClass::getLoginUrl()
{
return gcString("https://secure.") + g_szRootDomain + "/3/memberlogin";
}
gcString WebCoreClass::getRefreshUrl()
{
return gcString("http://api.") + g_szRootDomain + "/1/refresh";
}
gcString WebCoreClass::getUpdatePollUrl()
{
return gcString("http://api.") + g_szRootDomain + "/2/updatepoll";
}
gcString WebCoreClass::getNameLookUpUrl()
{
return gcString("http://api.") + g_szRootDomain + "/1/itemnamelookup";
}
gcString WebCoreClass::getMcfUploadUrl()
{
return gcString("http://api.") + g_szRootDomain + "/1/itemupload";
}
gcString WebCoreClass::getItemInfoUrl()
{
return gcString("http://api.") + g_szRootDomain + "/2/item";
}
gcString WebCoreClass::getInstalledWizardUrl()
{
return gcString("http://api.") + g_szRootDomain + "/2/itemwizard";
}
gcString WebCoreClass::getUpdateAccountUrl()
{
return gcString("http://api.") + g_szRootDomain + "/1/iteminstall";
}
gcString WebCoreClass::getCDKeyUrl()
{
return gcString("http://api.") + g_szRootDomain + "/1/cdkey";
}
gcString WebCoreClass::getMemberDataUrl()
{
return gcString("http://api.") + g_szRootDomain + "/1/memberdata";
}
gcString WebCoreClass::getAppUpdateUrl()
{
return gcString("http://api.") + g_szRootDomain + "/1/appupdate";
}
void WebCoreClass::setCookie(const char* sess)
{
if (!sess)
return;
gcString strSess(sess);
{
std::lock_guard<std::mutex> l(m_mSessLock);
if (strSess == gcString(m_szSessCookie))
return;
Safe::strncpy(const_cast<char*>(m_szSessCookie.data()), m_szSessCookie.size(), strSess.c_str(), strSess.size());
}
onCookieUpdateEvent();
}
void WebCoreClass::setCookies(gcRefPtr<CookieCallbackI> pCallback)
{
gcAssert(pCallback);
if (!pCallback)
return;
auto strRoot = getUrl(WebCore::Root);
(*pCallback.get())(strRoot.c_str(), "freeman", m_szIdCookie.c_str());
(*pCallback.get())(strRoot.c_str(), "masterchief", gcString(m_szSessCookie).c_str());
(*pCallback.get())(strRoot.c_str(), "AWSELB", gcString(m_AWSELBCookie).c_str());
}
void WebCoreClass::setWCCookies(HttpHandle& hh)
{
std::lock_guard<std::mutex> l(m_mSessLock);
hh->addCookie("freeman", m_szIdCookie.c_str());
hh->addCookie("masterchief", gcString(m_szSessCookie).c_str());
hh->addCookie("AWSELB", gcString(m_AWSELBCookie).c_str());
hh->setUserAgent(getUserAgent());
}
void WebCoreClass::clearNameCache()
{
try
{
sqlite3x::sqlite3_connection db(getWebCoreDb(m_szAppDataPath.c_str()).c_str());
db.executenonquery("DELETE FROM namecache");
}
catch (std::exception &ex)
{
Warning("Failed to clear namecache table: {0}\n", ex.what());
}
}
gcString WebCoreClass::getUrl(WebCoreUrl id)
{
gcString url("http://www.");
url += g_szRootDomain;
switch (id)
{
case Welcome:
url += "?firsttime=t";
break;
case Cart:
url = "https://secure." + g_szRootDomain + "/cart";
break;
case Purchase:
url += "/cart/history";
break;
case Gifts:
url += "/gifts";
break;
case ActivateGame:
url += "/gifts";
break;
case Games:
url += "/games";
break;
case Mods:
url += "/mods";
break;
case Community:
url += "/community";
break;
case Development:
url += "/development";
break;
case Support:
url += "/support";
break;
case Help:
url += "/forum";
break;
case McfUpload:
url = gcString("http://api.{0}/1/itemupload", g_szRootDomain);
break;
case ThreadWatch:
url += "/forum/board/thread-watch";
break;
case Inbox:
url += "/messages/inbox";
break;
case Updates:
url += "/messages/updates";
break;
case LinuxToolHelp:
url += "/tutorials/linux-tool-help";
break;
case ListKeys:
url += "/collection";
break;
case AppChangeLog:
url += "/app/changelog";
break;
case ComplexModTutorial:
url += "/groups/desura/tutorials/complex-mod-installing";
break;
case PlayJavaScript:
url += "/app/playjs";
break;
case Root:
break;
};
return url;
}
}
| 2024-04-08T01:27:17.760571 | https://example.com/article/3463 |
The present invention relates to apparatus for cutting tubes. It is particularly adapted to cutting paper tubes. Paper tubes have a number of uses in various industries. They are of particular utility in the electrical and electronics industries where they are used for forms for coils, transformers, inductors and the like. The size of the paper tube varies with the application, but paper tubes have been made ranging in size from less than a quarter of an inch O.D. to as great as seven inches O.D. Further, depending upon the application, the thickness of the wall of the tube may vary from a very thin tube to one having a wall thickness of 175 mils. Different techniques have been used for cutting paper tubes in the industry. For example, for small, thin-walled tubes which are flexible and somewhat difficult to saw, a snap cutter of the type disclosed in my copending application for "Snap-Cutter for Paper Tubes", Ser. No. 806,106, filed June 13, 1977, are successful. Where the tubes are made on an automatic winding machine, such a machine may be equipped with conventional saws for cutting the tubes into the longer lengths normally found in the initial manufacturing stages. That is, a plurality of tapes are wound onto an arbor in a continuous fashion and laminated together adhesively. In this initial stage, the tubes are cut to relatively long lengths, and taken to a separate location for cutting into the shorter, more precise lengths required by the customer.
Another technique used in cutting smaller thin-walled tubes is to hold the tubes in such a fashion that they can be surface-driven at a relatively high speed and then moved into cutting position. Although this technique may be used for smaller tubes, it, as well as other known techniques, have proven difficult in the cutting of larger tubes (because of the design involved, up to seven inches O.D., for example) as well as in the cutting of thick tubes (having a wall thickness of approximately 170 mils. and an O.D. of approximately 31/2 inches).
According to the present invention, apparatus includes a cradle for holding a tube to be cut. The cradle is mounted for rotation about its axis. A plurality of saws are mounted on a shaft having its axis parallel to the axis of the cradle, and the saws are driven at high speed during operation. The cradle is rotated between a loading position in which the tubes are placed in the cradle, and a cutting position adjacent the saws. Surface drive means are provided adjacent the cutting station for spinning the tube held in the cradle. Thus, the cradle can be rotated at a fairly high speed from the loading position through the cutting position and back to the loading position. In other words, one complete revolution of the cradle defines a cutting cycle.
In the case of thick-walled paper tubes (for example, tubes having a thickness in the range of 0.100-0.200 inches), the cradle is brought to a stop for a limited period of time of the order of two seconds just as the tube enters the cutting station to enable saws to make partial peripheral cuts in the tube. This momentary interruption in the rotation of the cradle enables the saws to cut thick-walled tubes without causing a bend in the tube, and while permitting the cradle to rotate through its cycle at a high speed, except for the momentary stop at the initiation of cutting. The high-speed rotation of the cradle is important when it is considered that paper tubes of this type are cut at the rate of 150-200 tubes per hour. Such savings in time in the operating cycle of the machine greatly increase production.
The present invention thus provides for tube-cutting apparatus which is adaptable for a wide range of outer diameters and wall thicknesses of paper tubes. Other features and advantages of the present invention will be apparent to persons skilled in the art from the following detailed description of a preferred embodiment accompanied by the attached drawing wherein identical reference numerals will refer to like parts in the various views. | 2024-01-09T01:27:17.760571 | https://example.com/article/9992 |
Q:
Cauchy sequence of functions and the limit inferior
I'm trying to understand a step in a proof. I don't get a special trick that is used several times in the book I am reading, so this does not get out of my head. I try to explain the prerequisites and what I don't understand:
Let $Y$ be a Banach space and let $S$ be a set. For a mapping $f:S\to Y$ let
$\|f\| := \sup_{x\in S}|f(x)|$. Let $B(S;Y)$ be the set of all mappings $g:S\to Y$ with $\|g\| < \infty$. I don't understand a step in the proof of the fact that this is a complete metric space.
Let $(f_k)_{k\in\mathbb{N}}$ be a Cauchy sequence in $B(S;Y)$. Let $x\in S$. Since $|f_k(x)-f_l(x)| \rightarrow 0$ for $k,l\rightarrow \infty$, we can define the pointwise limit $f$ of the sequence by $f(x) := \lim_{k\to\infty}f_k(x)$. Now here is the step that I don't understand:
The author of the book (and the proof) states that
$\lim_{l\to\infty}|f_l(x)-f_k(x)| \le \lim\inf_{l\to\infty}\|f_l-f_k\|<\infty$.
Why is that? What has the $\lim\inf$ to do here. And why is this finite? I think that it has something to do with subsequences of Cauchy sequences, but I don't understand it.
This $\lim\inf$-trick is used several times in the book so that it seems a bit important to me.
Thank you very much in advance. I'm glad for every help. This does not get out of my head.
A:
Assume we are give two sequences $\{a_r:r\in\mathbb{N}\}$ and $\{b_r:r\in\mathbb{N}\}$ such that for all $r\in\mathbb{N}$ we have $a_r\leq b_r$. Fix some $l\in\mathbb{N}$ and consider the last inequality for all $r\geq l$. After taking infimum on the left side over all $r\geq l$ we get
$$
\inf\limits_{r\geq l}a_r \leq b_r
$$
Then we take infimum on the right side over all $r\geq l$, and we get
$$
\inf\limits_{r\geq l}a_r \leq \inf\limits_{r\geq l}b_r
$$
Note that sequences $\{\inf\limits_{r\geq l}a_r:l\in\mathbb{N}\}$ and $\{\inf\limits_{r\geq l}b_r:l\in\mathbb{N}\}$ are non decreasing hence they have limits (finite or infinite). Lets take this limits, then we get
$$
\lim\limits_{l\to\infty}\inf\limits_{r\geq l}a_r \leq \lim\limits_{l\to\infty}\inf\limits_{r\geq l}b_r
$$
It is known that $\liminf\limits_{l\to\infty}x_l=\lim\limits_{l\to\infty}\inf\limits\limits_{r\geq l}x_r$ (sometimes this equality is taken as definition of $\liminf$), so
$$
\liminf\limits_{l\to\infty}a_l \leq \liminf\limits_{l\to\infty}b_l\tag{1}
$$
Another interesting fact: if a sequnces $\{x_l:l\in\mathbb{N}\}$ have a limit then it is equal to limit inferioir and limit superioir
$$
\lim\limits_{l\to\infty}x_l=\liminf\limits_{l\to\infty}x_l=\limsup\limits_{l\to\infty}x_l\tag{2}
$$
Now we proceed to your question. Fix $k\in\mathbb{N}$ and consider sequences
$$
a_l=|f_l(x)-f_k(x)|\qquad b_l=\Vert f_l-f_k\Vert\qquad\text{ where }\qquad l\in\mathbb{N}
$$
Since $a_l\leq b_l$ for all $l\in\mathbb{N}$ from (1) we get
$$
\liminf\limits_{l\to\infty}|f_l(x)-f_k(x)| \leq \liminf\limits_{l\to\infty}\Vert f_l-f_k\Vert
$$
By construction sequence $\{a_n:n\in\mathbb{N}\}$ is convergent. Indeed
$$
\lim\limits_{l\to\infty} a_l=
\lim\limits_{l\to\infty} |f_l(x)-f_k(x)|=
|\lim\limits_{l\to\infty}f_l(x)-f_k(x)|=
|f(x)-f_k(x)|
$$
hence from $(2)$ we get
$$
\lim\limits_{l\to\infty}|f_l(x)-f_k(x)|=\liminf\limits_{l\to\infty}|f_l(x)-f_k(x)| \leq \liminf\limits_{l\to\infty}\Vert f_l-f_k\Vert
$$
| 2024-04-10T01:27:17.760571 | https://example.com/article/6207 |
Q:
Tastypie Accessing Query Before Data Gets Replaced by Request
I am using a PATCH request in tastypie to update a field in my model but when this PATCH goes through I am trying to append the data that is already in the database rather than replacing the value entirely.
Example:
In a blog post, a user flags a post as inappropriate. You want to get this user's UserID and append it to a field, flaggedUserID. flaggedUserID will contain this user's ID along with anyone else who has flagged the post previously.
Where I'm struggling is in finding a hook in tastypie through which I can access the query BEFORE the value is copied from the request into the data from the query. I tried the "alter_deserialized_detail_data(self, request, data)" hook but the value has already been replaced in the data parameter that is passed to this function.
Anyone have any suggestions?
A:
After reading through the source code of tastypie this is the solution I came to:
The copying of data from the request to the query seems to be happening in the "update_in_place" function. By overriding this function in the following way I was able to accomplish the results I was looking for:
def update_in_place(self, request, original_bundle, new_data):
"""
Update the object in original_bundle in-place using new_data.
"""
"""Right here I am checking if the parameter flaggedUserID is in the
request (PATCH saves request data to request.body). If this is false then
copy the new data into the query, otherwise jump to
alter_deserialized_detail_data where I am handling the appending of the data
from the request. """
if 'flaggedUserID' not in request.body:
original_bundle.data.update(**dict_strip_unicode_keys(new_data))
# Now we've got a bundle with the new data sitting in it and we're
# we're basically in the same spot as a PUT request. SO the rest of this
# function is cribbed from put_detail.
self.alter_deserialized_detail_data(request, original_bundle.data)
kwargs = {
self._meta.detail_uri_name: self.get_bundle_detail_data(original_bundle),
'request': request,
}
return self.obj_update(bundle=original_bundle, **kwargs)
Depending on the objective I would probably recommend cleaning this up a little bit more. I just wanted to show this code in its most basic form without having to post multiple functions and start confusing people.
| 2024-06-14T01:27:17.760571 | https://example.com/article/6883 |
Love Is All You Need? toured twenty cities in tribute to true events depicted in the film, with all proceeds going to the non-profits: including The Matthew Shepard Foundation, Tyler Clementi Foundation, Equality House, Planting Peace, Columbia City Ballet, Center on Halsted, and others across the country.
"I had a lot of offers to distribute my movie through the usual channels, but this film is about more than making money. It's about making a difference," said Director/Co-Writer K Rocco Shields.
Shields chose to self-distribute the film with a new indie platform Distribber and donate what would have been the distribution fees to her nonprofit partners. Once the film reaches one million paid downloads, the fund will be activated and money disbursed to the nonprofit recipients. Additionally, the film will pay in perpetuity a percentage of its revenues to various anti-hate and equality measures with non-profit partners.
"It's time to stop asking for change, and start making change ourselves. The charities we partnered with need this money to help the people in their communities stand up to bullying, prejudice and discrimination," K. Rocco Shields.
The film releases nationwide, Friday, Dec 9th on Itunes, Google Play, and Amazon
Co-written by David Tillman, this film is a love story taking place in a parallel world where homosexualliy is mainstream and heterosexuals are bullied and ostracized for who they love. "I could have used anything to illustrate my point: race, religion, social status but I chose the most universal concept I could think of: love," K. Rocco Shields.
The story unfolds in a small Indiana town that is rocked to its core when Jude, (Briana Evigan) the star quarterback of the local university, strikes up a forbidden love affair with Ryan, (Tyler Blackburn) a sports journalism major. When the couple is outed by Jude's girlfriend (Emily Osment), the community's powerful religious leader (Elisabeth Röhm) begins a vitriolic crusade from the pulpit against all heterosexuals setting off a series of catastrophic events in lives of an 11 year old girl (Kyla Kennedy), experiencing her first crush on a boy, and a teacher (Jeremy Sisto) who tries to teach his students about tolerance.
The star-studded cast includes Briana Evigan from the wildly successful Step Up Series, Tyler Blackburn from Free Form's Pretty Little Liars, Emily Osment from the hugely popular Young & Hungry on Free Form, Jeremy Sisto from ABC's Suburgatory, SAG award winning actress Elisabeth Röhm from American Hustle, Katherine LaNasa who will be starring in the upcoming Bravo series Imposters, Ana Ortiz ALMA best actress award winner for Ugly Betty, and Kyla Kenedy first seen in AMC's The Walking Dead and now playing Minnie Driver's daughter on ABC'S new hit comedy Speechless.
For more information about Love Is All You Need? and to view the trailer, visit www.loveisallyouneedthemovie.com, Love Is All You Need? on Facebook and @liayn_themovie on Instagram and Twitter.
View the original short film 'Love is All You Need?' on YouTube.
Press Contact:
David Sperber
Publicist / GENIUS PICTURES
[email protected]
818-441-1025
Video - https://www.youtube.com/watch?v=qMOONBCqzG8
SOURCE Love Is All You Need?
Related Links
http://www.loveisallyouneedthemovie.com/
| 2023-12-05T01:27:17.760571 | https://example.com/article/5520 |
The study of various parameters affecting the ion exchange of Cu2+, Zn2+, Ni2+, Cd2+, and Pb2+ from aqueous solution on Dowex 50W synthetic resin.
A gel resin containing sulfonate groups (Dowex 50W) was investigated for its sorption properties towards copper, zinc, nickel, cadmium and lead metal ions. The use of selective ion exchange to recover metals from aqueous solution has been studied. The ion exchange behavior of five metals on Dowex 50W, depending on pH, temperature, and contact time and adsorbate amount was studied. Experimental measurements have been made on the batch sorption of toxic metals from aqueous solutions using cation exchanger Dowex 50W. The maximum recoveries (about 97%) Cu(2+), Zn(2+), Ni(2+), Cd(2+) and (about 80%) Pb(2+) were found at pH ranges 8-9. The amount of sorbed metal ion was calculated as 4.1, 4.6, 4.7, 4.8, and 4.7mequiv./gram dry resin for Pb(2+), Cu(2+), Zn(2+), Cd(2+), and Ni(2+), respectively. The precision of the method was examined at under optimum conditions. Selectivity increased in the series: Pb>Cd>Cu>Zn>Ni. It has been observed that, selectivity of the -SO(3)H group of the resin increases with atomic number, valance, degree of ionization of the exchanged metals. The equilibrium ion exchange capacity of resin for metal ions was measured and explored by using Freundlich and Langmuir isotherms. Langmuir type sorption isotherm was suitable for equilibrium studies. | 2024-06-20T01:27:17.760571 | https://example.com/article/2216 |
1. Field of the Invention
This invention relates to seagoing vehicles, and more particularly to means for controlling their attitude in the water.
2. Description of the Prior art
Previously, attitude control, especially with respect to submersible vehicles, was accomplished by systems involving the pumping of liquid mercury from one container to another; ballast tanks taking on sea water as ballast and injecting compressed air into the tanks to force out and displace the appropriate amount of such sea water; and the shifting of solid weights by complex mechanical means.
The prior techniques has disadvantages in that mercury is a lethal material and its use on the open seas poses a serious ecological hazard. Water ballast systems are complex, are not precise, and take up a considerable amount of inboard space. Shifting weight systems tend to be complex. When installed aboard the vehicle, such systems take up a significant amount of the limited available space, whereas externally mounted systems present maintenance problems and are susceptible to damage.
The present invention overcomes these disadvantages by providing an ecologically safe, compact, and structurally simple attitude control system for a seagoing vehicle. | 2024-03-22T01:27:17.760571 | https://example.com/article/3277 |
Sekimizu Wants Polar Code, Ballast Water Action
By The Maritime Executive 10-14-2014 08:47:00
Koji Sekimizu, IMO secretary-general, has highlighted the importance of the Polar Code and the Ballast Water Management Convention in his opening remarks at MEPC67 this week.
The Maritime Safety Committee approved the draft Polar Code in May with a view to adopting it and associated SOLAS changes at MSC 94 in November. “The principal remaining task for this Committee is now to finalize Part II of the Code addressing pollution prevention measures and the associated MARPOL amendments,” says Sekimizu. He concedes that there are still several challenges to be overcome, but hopes that entry into force will remain on target for January 2017.
“This will be timely to demonstrate to the outside world that the organization is once again taking the lead as a global standard-setter – by pioneering the first-ever internationally binding regime covering the full range of vessel design, construction, equipment, operational and training matters, environmental protection and search and rescue in the remote and inhospitable waters surrounding the two poles.”
For the ballast water management convention, Sekimizu noted a number of positive developments since MEPC66. Four additional States (Japan, Jordan, Tonga and the Republic of Congo) have become contracting parties. The total number of ratifications or accessions now stands at 42, representing 32.02 percent of the gross tonnage of the world’s merchant fleet.
“The new sub-committee on Implementation of IMO Instruments, which held its first session in July, finalized its work on the intensively discussed draft Guidelines for port State control under the 2004 Ballast Water Management Convention. I therefore urge the committee to resolve the outstanding issue of sampling and indicative analysis of ballast water and to adopt these long-awaited guidelines at this session.
“Also at this session the committee will be addressing industry concerns pertaining to the effective implementation of the convention, including legal clarity over the type approval of ballast water management systems. In this context, the committee will be invited to consider a study on the implementation of the ballast water performance standard described in regulation D-2 of the convention. Conducting this study, for which the secretariat has developed a draft plan and terms of reference, should contribute to further facilitating implementation and encouraging ratification of the convention.”
IMO has put a lot of effort into the convention since 2004, says Sekimizu, including the multi-million dollar, GEF/UNDP/IMO GloBallast project, assisting developing countries to prepare for the implementation of the convention.
“I trust that throughout this week’s discussions the committee will stay firmly focused on the ultimate objective of the Convention – to prevent the global spread of invasive aquatic species through ships’ ballast waters and sediments, which is inherently linked to the expansion of international shipping as world trade continues to grow and we do it under the IMO system, that is the implementation of agreed global standards and IMO measures under the IMO convention.”
Also on the agenda is the Hong Kong Convention for the safe and environmentally sound recycling of ships. Norway, France and the Republic of the Congo are the only states to have ratified the convention to date “Once again I urge Member States that have not yet done so to ratify or accede to the Hong Kong Convention and thus promote its entry into force in the near future for the benefits of the environment, the shipping industry and the ship recycling countries.”
Regarding air emissions, the committee is expected to respond to concerns raised at MSC93 last May about the safety risks associated with poor fuel oil quality. The key issue for this meeting is to reconcile the diverging views on what quality control measures such as existing provisions in MARPOL Annex VI or the need for an amendment to the Annex. An alternative option, not necessarily favoured by Sekimizu is the development of a non-mandatory guidance based on good practice taking account of existing procedures in some of the world’s leading bunkering ports. “I would like to urge the committee to consider how such non-mandatory guidance could effectively be implemented and enforced without legally binding requirements. Fuel oil quality is of paramount importance for safe shipping and to avoid pollution and IMO should regulate this.”
Additionally, the committee is expected to consider a definition of Black Carbon, which is important in the context of work to address the adverse impact of ships’ emissions on the Arctic. The committee is also expected to review and approve the Third IMO Greenhouse Gas Study (2014). “It is encouraging to see that contribution from international shipping to the Global GHG emission was reduced from 2.7% to 2.2% in 2012 and that the positive impact of IMO agreed emission reduction measures is highlighted in the study. Shipping has a great potential for growth to meet the demand of the world economy but shipping has also, a great potential to significantly reduce GHG emissions, while achieving further growth of maritime transport.
“All this and other IMO on-going work on greenhouse gas emissions, such as the accelerated implementation of the Energy Efficiency Design Index to extend its application to a wide range of ship types, is ample evidence of the organization’s contribution to address climate change concerns. Most recently, these concerns were the focus of a one-day Climate Summit convened by UN Secretary-General Ban Ki-moon in New York, on 23 September, under the banner “There is no Plan B for climate change action as there is no Planet B”. We, at IMO, can certainly be confident as well as proud to publicize our ground-breaking contribution to climate change abatement with regard to international shipping.” | 2023-10-01T01:27:17.760571 | https://example.com/article/5503 |
related
related
Bringing literal street style to your pad, these easily-applied vinyl sheets're large-format reproductions of actual photos of cool-looking walls found around Norway (in your choice of smooth or stucco finish), and run from clean cinderblock jobs, to graffiti-tagged industrial facades, to weathered brick, 'cause you don't have to be a Commodore to appreciate a brick house. | 2023-12-01T01:27:17.760571 | https://example.com/article/3697 |
Introduction: Why Have an Alternative Care Site?
================================================
During the current COVID-19 pandemic, the limited surge capacity of the healthcare system is being quickly overwhelmed. Similar scenarios play out when an institution's systems fail---or when local or regional disasters occur. In these situations, it becomes necessary to use one or more alternative care sites (ACS). Situated in a variety of non-healthcare structures, ACS may be used for ambulatory, acute, subacute, or chronic care. Developing alternative care facilities is the disaster-planning step that moves communities from talking to doing. This commitment pays real dividends when a disaster of any magnitude strikes.
Any available site may initially be used to provide healthcare after a disaster. After a massive earthquake in Turkey in 1999, for example, makeshift medical centers were set up on street corners and in ruined buildings.[@b1-wjem-21-484] Five months after Hurricane Katrina, when the media spotlight had dimmed, the New Orleans Convention Center still housed a makeshift medical center. The "emergency rooms" consisted of six military-surplus tents in which about 5000 patients, many of them uninsured poor, were treated each month.[@b2-wjem-21-484]
After 9/11, Canadian emergency medicine resident physicians used a New York City high school as a medical facility. They later wrote:
> *When we arrived at Stuyvesant High School, we found that there was limited electrical power because many local power grids were knocked out . . . Thus, many of the more intricate procedures, such as suturing, needed to be performed under the illumination of hand-held flashlights. The layout of the triage was very simple. In the main foyer of the school were approximately 8 stretchers clustered together in a makeshift patient care area. Surrounding the cots were dozens of large boxes of unorganized medical equipment, many unopened. The injured arrived either by ambulance or were ambulatory, and promptly triaged. Each stretcher, when possible, was staffed with two nurses, an attending physician, a resident and a medical student. As there were few survivors of the direct disaster, all patients we assessed were emergency response personnel and volunteer rescue workers.*[@b3-wjem-21-484]
As the current pandemic enveloped the world, China rapidly built new hospitals, and military units and civilians deployed tent facilities. In the US, states are considering using hotels, ice rinks, stadiums, nursing homes, ships, and recently closed hospitals as alternative care facilities, while the federal government promises that large temporary hospitals will soon be available in the most-affected areas.[@b4-wjem-21-484] Having to do this during a crisis demonstrates our dismal preparation for what is a rare, but predictable, event.
Issues in Establishing an ACS
=============================
Several inter-related issues must be resolved when establishing a viable ACS. These include the following:
- Who will decide to open the facility? Under whose authority will the site be established and run? This should normally be decided at a regional level.
- Who will direct the facility's operations?
- What types of patients will the facility house and treat? Will the ACS be used to decompress hospitals or nursing homes, or to provide primary care? What patient acuity will it accept? Will it accept oxygen-dependent patients?
- Which available facility will be used for this type of medical care? What will be the selection criteria? How will approval for use be gained? What will be the infrastructure dependencies? Or will the site be a previously designated portable/temporary shelter?
- Who will staff the ACS, including medical support and volunteer staff?
- What durable and disposable supplies and equipment will be available? In what quantities?
- What operational support (meals, sanitation needs, and infrastructure) will be required?
- What policies and what patient documentation will be used?
- Who will decide to close the ACS? What criteria will be used?
Implementation Difficulties
===========================
Planning for ACS is difficult, and takes personnel time, financial resources, and political capital. For these reasons, most regions have abandoned any real ACS planning. Rather, they use a conceptual model, such as "we can use the stadium," without any further thought. Only when a disaster strikes do they find that the "planning" was woefully inadequate. As a demonstration of how poorly prepared we are, the American Medical Association sent a letter to Congressional leadership on March 19, 2020, demanding funding for frontline treatment: "Providing the funding for the capacity to care for mildly or moderately sick Covid-19 patients in an alternative care site when they cannot appropriately care for themselves at home, such as outpatient facilities or large structures in the community that are in close proximity to a hospital, will provide additional capacity for sicker Covid-19 patients that need more intensive care. This is a crucial step in ensuring we have as much inpatient capacity as possible to respond to the sickest Covid-19 patients."[@b5-wjem-21-484]
In addition to a lack of finances and leadership, another major stumbling block is that, to be effective, multiple groups that traditionally have not done so, must work together.
Who Controls, Opens, and Closes the Site?
=========================================
The single most important issue for the successful establishment of an ACS is determining its ownership, command, and control. These are political issues that should be decided at a local or regional (as opposed to institutional) level. Decisions must be made about the individual(s) who can decide whether, when, and where an ACS should be opened, as well as about who has the authority to operate the site.[@b6-wjem-21-484],[@b7-wjem-21-484]
Deciding to open an ACS is bound up with bureaucratic and financial implications. While an individual hospital can make the decision alone, the decision will usually be regional, with support from many sources, especially the government
If a hospital decides to open an ACS as part of its emergency operations plan, they assume an enormous burden that few except the largest institutions can manage. This includes the need to find an acceptable facility; to staff and equip it; to establish policies and manage it; to coordinate operations with emergency medical services, the Red Cross, and other community emergency assets; and, lastly, to finance it. While there may be many fewer bureaucratic obstacles to "going it alone," the sheer magnitude of such an operation presents formidable barriers.
In some cases, where the need is obvious and no leadership has either prepared for or is willing to support establishing an ACS, individual clinicians and support staff may need to open an ACS on their own. That was the case in St. Bernard Parish (county) after Hurricane Katrina, when three family practitioners opened the only medical facility in two parishes in what had been the lobby of the ExxonMobil refinery, located on the highest point in the parish. Using equipment and supplies salvaged from the flood water (with the help of the St. Bernard Paris Sheriff's Office), they provided care for a week after the storm. They recruited nurses who were willing to stay and help, as well as two family practice residents from a nearby state (one was a relative). Eventually, a disaster medical assistance team (DMAT; AZ-1, from Arizona) assumed responsibility for that ACS.
An exit strategy and exit criteria should be built into the initial plan. The decision to close an ACS is much easier if pre-set and widely understood guidelines control the process. However, ACS closure can be overtaken by events, such as when multiple ACS facilities (opened after Katrina) had to be abandoned when Hurricane Rita threatened the area.
How Will the ACS be Used?
=========================
Most of the decisions about an ACS (staffing, equipment, supplies, and type of structure) flow from the way it will be used. Therefore, that is the first major decision. Possible uses include the following: [@b6-wjem-21-484], [@b7-wjem-21-484]
- Facility in which to quarantine or isolate patients
- Facility to house low-acuity patients from hospitals and nursing homes
- Ambulatory care/vaccination clinic
- Primary triage point to decide where and how patients can best be treated
- Acute care inpatient facility
- Place to provide palliative care for hospitalized patients
- Facility to house patients discharged from hospitals so that they can be released earlier than usual
- Combination of many or all the above functions
Note that many alternative care facilities have multiple functions. Some develop as the situation progresses, although these should not exceed the facility's structural, staffing, or logistical capabilities.
Which Type of Facility to Use?
==============================
Selecting facilities to serve as an ACS is an imprecise science and varies with the event. Most commonly used are facilities of opportunity, or "buildings of convenience," which are non-medical structures that can be adapted into an ACS.[@b6-wjem-21-484], [@b7-wjem-21-484] The building selection process works best if there is first a clear idea of the role planned for the ACS. Buildings typically used as an ACS are listed in [Table 1](#t1-wjem-21-484){ref-type="table"}. An often-forgotten fact is that if an ACS can accommodate patients from nursing home/long-term care facilities, those beds can be converted for acute care use (often with an existing oxygen supply).
If there are options when selecting an ACS, especially if it selected in advance of a disaster, some basic questions must be answered to get the best possible facility. Of course, even the best facility will still need much improvisation to make it functional. ([Table 2](#t2-wjem-21-484){ref-type="table"}).
Structural Issues: Selection Criteria
=====================================
Rate possible alternative care facilities using pre-set criteria ([Table 3](#t3-wjem-21-484){ref-type="table"}). Give each criterion a rating (1 to 5) based on how close it comes to the same criterion in a hospital. Sites with the highest total ranking should be the first choice for an ACS. The lack or inadequacy of some criteria may eliminate the structure from consideration, such as the inability to secure the structure or to fulfill most of the criteria listed under "Utilities."
An example of a rating procedure for each criterion is as follows: 5 = equal to or same as hospital; 4 = similar to that of a hospital, but with some limitations (ie, quantity/condition); 3 = similar to that of a hospital, but with major limitations (ie, quantity/condition); 2 = not similar to that of a hospital and would require minimal modifications to make the facility useable; 1 = not similar to that of a hospital and would require major modifications to make the facility useable; 0 = does not exist in this facility or is not applicable to this event.
Before use begins, facilities' internal space should be laid out in an organized fashion. A grid system allows clinicians to make "rounds" and know exactly where to find a patient (eg, bed A4). Public health issues are critical (eg, safe food and water supply, sanitation, latrine resources). [@b6-wjem-21-484],[@b7-wjem-21-484]
Staffing and Security
=====================
Once a suitable facility is identified, staffing and emergency privileging of healthcare professionals become issues. Staff may be volunteers, off-duty providers from the primary facility, retired clinicians, military personnel, or designated members of disaster response teams (eg, DMAT). [Table 4](#t4-wjem-21-484){ref-type="table"} lists the ideal staffing for each 12-hour shift in a 50-bed inpatient ACS.
Staff members face several issues, including that of arranging for provisions to house and feed the patients. If volunteers are used (as they should be), they should have their own coordinator who understands that some volunteers may not want to do certain tasks (eg, colostomy care, diaper changes). Establish who is going to do what. Note that placing an ACS near a college or university enlarges the potential workforce (eg, sports teams, fraternities, religious groups) to help carry patients, set up equipment, and so forth.[@b6-wjem-21-484],[@b7-wjem-21-484]
In chaotic situations, security becomes an extremely important concern, especially since local law enforcement will be stretched thin. Establish a system to identify staff members, patients, and their families. "Planners must develop robust security plans. It is helpful if security personnel have previous experience in dealing with patients, especially those with behavior disorders. The best potential source of security staff would be off-duty hospital security personnel, but these individuals may not be available. Other potential sources would include on- or off-duty police officers, activated members of the National Guard, or volunteers."[@b6-wjem-21-484] "Security makes patients and staff members feel safe and keeps out troublemakers. Having uniformed people on site (even Reserve Officer Training Corps \[ROTC\] cadets) makes a real difference."[@b6-wjem-21-484]
Many of these positions are interchangeable. For the ACS to function optimally, everyone must be willing to do any job for which they are qualified.
Supplies and Equipment
======================
Supplies and equipment for an ACS will vary with its mission and range from extremely primitive to similar to those found in the basic hospital (not including the operating rooms or radiology). Conversely, the lack of specific items, such as oxygen, may limit an ACS's role. Oxygen and pharmaceuticals are often difficult to obtain during a disaster.
Oxygen
======
While generally taken for granted in modern (developed-world) hospitals, oxygen is an expensive, difficult-to-acquire commodity. [Table 5](#t5-wjem-21-484){ref-type="table"} lists the costs, power requirements, and flow rates for some typical portable oxygen delivery systems. In the least-developed countries, the solution is often to use oxygen concentrators, although facility-size units are not commonly available unless purchased in advance. The most common solution is to not accept oxygen-dependent patients in an ACS. Obtaining oxygen from industrial sources may also solve the problem in resource-poor situations. Because industrial-grade oxygen is purer than medical grade, biomedical engineers may be needed to help adapt the connections to medical equipment.[@b10-wjem-21-484] Reducing oxygen use in patients who may not really need it may also be necessary.
Pharmaceuticals
===============
The nature of the ACS responsibilities (acute and chronic care) and the patient population will determine what pharmaceuticals are needed. In St. Bernard Parish after Hurricane Katrina, for example, the ACS pharmacy (containing items that Sheriff's officers scrounged from the non-flooded shelves in local pharmacies) was used primarily to supply chronic medications to members of the military and rescue teams. Many medications, despite pharmaceutical company claims and clinician habits, have adequate substitutes that are often readily available. When such needs arise, pharmacists should automatically suggest them for use.[@b12-wjem-21-484] Most ACS will also need to stock medications to provide acute respiratory therapy, acute hemodynamic support, pain control, anxiolytics, antibiotic coverage, and behavioral health maintenance. [@b6-wjem-21-484],[@b7-wjem-21-484]
Alternative Care Site Tools
===========================
In March 2020, the US Department of Health and Human Services Office of the Assistant Secretary for Preparedness and Response issued the *Federal Healthcare Resilience Task Force Alternate Care Site (ACS) Toolkit* 2020. It contains a detailed description of the staffing and, most importantly, a checklist to assure that the site is appropriate as an ACS.[@b13-wjem-21-484] Their Healthcare Emergency Preparedness Information Gateway has collected a wide variety of free monographs and teaching tools about ACS (Topic Collection: Alternate Care Sites (including shelter medical care).[@b14-wjem-21-484] The best of these is the California Department of Public Health Government-Authorized Alternate Care Site Training Guide.[@b7-wjem-21-484] In 2009, the Agency for Healthcare Research and Quqlity published three free downloadable resources to help identify and run alternative care sites. One is a monograph, *Disaster Alternate Care Facilities: Selection and Operation*, and two are interactive tools, *Disaster Alternate Care Facility Selection Tool*, and the *Alternate Care Facility Patient Selection Tool*[@b6-wjem-21-484]).
*Section Editor:* Mark I. Langdorf, MD, MHPE
Full text available through open access at <http://escholarship.org/uc/uciem_westjem>
Disclaimer: Due to the rapidly evolving nature of this outbreak, and in the interests of rapid dissemination of reliable, actionable information, this paper went through expedited peer review. Additionally, information should be considered current only at the time of publication and may evolve as the science develops.
*Conflicts of Interest*: By the *West*JEM article submission agreement, all authors are required to disclose all affiliations, funding sources and financial or management relationships that could be perceived as potential sources of bias. No author has professional or financial relationships with any companies that are relevant to this study. There are no conflicts of interest or sources of funding to declare.
######
Buildings/structures typically used as alternative care sites during disasters.[@b6-wjem-21-484],[@b7-wjem-21-484]
---------------------------------- ---------------------------- ---------------------------------
Adult detention facility Aircraft hanger Church
Community/recreation center Convalescent care facility Fairground
Government building Hotel/motel Meeting hall
Military facility National Guard armory Same-day surgical center/clinic
School Shuttered hospital Sports facility/stadium
Trailer/tent (military or other)
---------------------------------- ---------------------------- ---------------------------------
######
Questions to ask when selecting an alternative healthcare site.[@b8-wjem-21-484]
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Will the structure accommodate the expected number of patients and staff, and the planned activities?Is the structure located in a relatively safe area (culturally and geographically)? Is it structurally sound?Is it easily accessible by ambulance, foot, and automobile/public transportation?Is there adequate electrical power (plus back-up power or the capacity to tie in to large portable generators)?Is there adequate potable water, ventilation, refrigeration, and lighting? Are the ventilation and lighting systems on the back-up generator? Are there also other back-up electrical outlets for critical equipment, such as ventilators?Are there kitchen facilities adequate for the number of people expected (patients, staff, visitors)?Is the entire patient care area wheelchair/stretcher accessible? If elevators are needed, are they on the back-up power system?Will there be separate space for other necessary functions, such as staff sleeping/rest areas, communications center, command center, waiting area, security office, pharmacy, equipment supply and storage areas, chapel/family counseling area, and a morgue?Can the building be secured? Can you control patient and staff traffic?Are there phone and computer access lines? Will cellular phones and radios (two-way, ambulance, public sector, and ham) work within the building without interference?Can lights be dimmed in sleeping and patient care areas?Are the doors \> 33 inches wide to permit ambulance stretchers to move through them?Are there areas to load and unload patients and supplies? Ideally, these will accommodate forklifts.Is there parking for patients, staff, and visitors?Are toilet and shower facilities adequate for the anticipated number of patients, staff, and visitors? [(\*)](#tfn1-wjem-21-484){ref-type="table-fn"}Does the facility have oxygen or will it be readily available? [(\*)](#tfn1-wjem-21-484){ref-type="table-fn"}Is the facility easy to clean for patient use?
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Important and often overlooked in planning alternative care sites.
######
Criteria to consider in alternative care site selection.
Infrastructure
------------------------------------------------------------------------ ---------------------------------------------------------
Door sizes and stairways adequate for gurneys Floors
Parking for staff and visitors Roof
Ventilation Walls
Toilet facilities/showers (number of) Loading dock
Total space and layout
Auxiliary spaces (pharmacy, counselors, chapel) Staff areas
Lab specimen handling area Equipment/supply storage area
Mortuary holding area Family area
Patient decontamination areas Food supply and prep area
Pharmacy area Patient care/ward areas
Utilities
Electrical power (Backup present? Adequate for anticipated equipment?) Air conditioning (Sufficient for the number of people?)
Lighting Heating
Refrigeration Water (Hot?)
Communications
Communication (number of phones, local/long distance, intercom) Wired for information technology and Internet access
Two-way radio capability to main facility Other services
Other factors
Ability to lock down facility Laundry
Oxygen delivery capability Biohazard and other waste disposal
Ownership/other uses during disaster Proximity to hospitals
Accessibility/proximity to public transportation Liability insurance coverage
######
Idealized staffing for a 50-bed alternative care site per 12-hour shift.[@b6-wjem-21-484],[@b7-wjem-21-484]
------------------------ ---------------------------------------------------------------- -------------------------------------------------
Physician, 1 Physician Extender (Physician Assistant/Nurse Practitioner), 1 Registered Nurse or Licensed Practical Nurse, 6
Health Technician, 4 Unit Secretary, 2 Respiratory Therapist, 1
Case Manager, 1 Social Worker, 1 Medical Assistant/Phlebotomist, 1
Food Service, 2 Chaplain/Pastoral, 1 Day Care/Pet Care, 1
Volunteer, 4 Engineering/Maintenance, 0.25 Biomedical Engineer, 0.25
Security, 2 Housekeeper, 2 Lab, 1
Patient Transporter, 2
------------------------ ---------------------------------------------------------------- -------------------------------------------------
######
Oxygen equipment typically available.[@b11-wjem-21-484]
Oxygen generation systems Oxygen flow rate (L/min) Power required (kW) Cost of unit \$1000 Oxygen purity (%)
------------------------------------------------------ -------------------------- --------------------- --------------------- -------------------
Expeditionary deployable oxygen concentration system 120 8 131 93 ± 3
Portable therapeutic oxygen concentration system 45 7 40 93+
Portable oxygen generation system 33 12 35 93--95
Patient ventilation oxygen concentration system 20 4.3 35 93 ± 3
Home oxygen compressor 3 0.2 2.5 93 ± 3
| 2024-06-19T01:27:17.760571 | https://example.com/article/2626 |
Top News: Simkiss starts slow and gets a bit quicker at the Midlands XC Champs.
Following complaints of high electricity bills from fans trying to read MCKEP athlete Rich Simkiss’s race reports in their entirety, Simkiss attempts to write a brief few words on today’s exploits at Newbold Commyn in the 2014 Midlands XC Championships:
I started at a reasonable, speed, possibly a bit slower than I should have, possibly not.
I instantly regretted not sourcing 15mm spikes and using the 12’s I had sat in my kit bag.
I fell on my arse coming down beacon hill on lap 1, sliding a good 10-15m… Sadly not entirely in the right direction and losing me ground to Phillip Gould.
Wasn’t sure if I’d catch Gould or Siggers, but Siggers faded and I caught him on the last lap.
Weather was insane, cannot remember a thunderstorm like that in years.
Had a nice battle with Seb Palmer of RNAC over the last lap, he passed me on the approach to the finish but I got my gurn on and thunder thigh’ed my way past him in a perfectly executed sprint finish.
Satisfied with performance, getting a good few scalps, but should really be higher up the field.
Non competing McCarthy took on the role of photographer for the day, though such changeable conditions were not conducive to producing his usual art gallery worthy masterpieces, and he had to rely on the ‘auto’ setting.
Somewhat scared and confused by the flashes of lightning paired with the immediate deafening thunder, McCarthy found solace in a large hi-vis coat; because Hi-vis makes you safe. Always. | 2024-01-12T01:27:17.760571 | https://example.com/article/6088 |
With a collapsed roof that has sunk into the building, covering about five different recycling machines floating inside a pool of water that has flooded it due to years of rainfall and negligence, the current state of Osogbo plastic waste recycling plant reveals how government’s investments go to waste in various parts of the country.
Twenty- six plastic waste recycling plants located in 26 cities across Nigeria, whose contracts the Federal Government awarded in 2009 to eradicate the problem of plastic waste, are at different stages of deterioration, but the government is oblivious of this fact despite the huge investment on the project, investigations by the ICIR have revealed.
Nigeria is not catching up with the movement to eliminate plastic waste, but the government would like the world to believe otherwise.
At the event marking the 2018 World Environment Day, which focused on eradication of plastic waste, the Minister of State for Environment, Ibrahim Jibril, announced with pride that Nigeria was moving towards eradicating plastic waste with the establishment of recycling plants – perhaps in keeping with Rwanda and South Africa examples: the two countries have banned the use of plastic for packaging and as bags.
“At present, a total of eight plants have already been completed and handed over to the states while 18 others are at various stages of completion,” the minister disclosed at a press conference.
But this statement is grossly inaccurate and the minister seems oblivious of the fact.
Minister of State for Environment, Ibrahim Jibril: He is unaware of the state of the plastic recycling plants, yet he told the world they were completed
Jibril indeed was unaware that the plastic waste recycling plants he said were completed or ongoing, at different locations were wasting away despite the government’s huge investment in the project.
In 2009, the administration of former President Goodluck Jonathan awarded the contract for the procurement and installation of 26 multipurpose plastic recycling plants in 26 cities including the Federal Capital Territory, Abuja.
Before the end of that administration− the projects, in 2013 were recorded completed and waiting for commissioning in a document prepared by the Ecological Fund Office (EFO) and submitted to the Presidency.
The contract was funded through the EFO whose mandate among others is to reduce ecological problems nationwide to the barest minimum and ensure judicious and equitable utilization of the Fund.
The government awarded the contract at the sum of N15million each to one contractor, Abdul Essentials Services Limited, under the supervision of the Federal Ministry of Environment.
The 26 plants which were envisioned to take care of plastic waste in those cities, cost N392 million in total.
When the ICIR reporter visited states like Osun, Ekiti, Lagos and Kaduna, which were among those the government claimed were completed and handed over to state governments, findings reveal typical examples of a culture of waste by both the federal and state government as well as lack of synergy between the tiers of government in project execution and management.
With a collapsed roof that has sunk into the building, covering about five different recycling machines floating inside a pool of water that has flooded it due to years of rainfall and negligence, the current state of Osogbo plastic waste recycling plant reveals how government’s investments go to waste in various parts of the country.
Sitting in the midst of a heap of refuse at Osogbo landfill along Iwo road, in Osun State capital, the N15million recycling plant installed in 2013 already has been overtaken by weeds.
To a visitor, there is nothing to suggest that a N15 million worth of a project lie fallow in the bush.
Facilities in the plant included a toilet and bathroom, a changing room for operators and three ceiling fans.
The state government was expected to provide access road, water and electricity to the plant. They were not provided.
Some components of the plants were also vandalized by some unknown persons due to the porous nature of the dumpsite, the ICIR learned from workers at the site.
Like the Minister of Environment, Ganiyu Oyeladun, General Manager of Osun State Waste Management Agency, whose office is in charge of waste management in Osun State, told the ICIR that he was unaware of the plant, and did not know of its sorry state.
All he knew about the only recycling plant in the state was that it “was a project by the Federal Ministry of Environment that was badly executed.”
He told the ICIR how officials of the Federal Ministry of Environment came looking for a suitable place to site the project back in 2009.
“The agency gave out the dumpsite where the machine was installed. That was the level of involvement of Osun State Waste Management Agency,” he said.
For him, the project was poorly executed by the contractor because key components such as cooling system, dryer and water for washing plastic waste were not part of what was installed.
And the state government never operated the plant− to the detriment of the people of the state.
But Yaqub Abdul, a representative of the contractor refuted the claim that the contract was badly done. He told the ICIR that both the Federal Government and Osun State Government abandoned the project.
He insisted that the plant was installed and test run before it was handed over to the state government.
“It is unfortunate the kind of government we have in Nigeria, after handing over they just abandoned the plant. The state did not take over; they did not make use of the machine,” Abdul said.
When asked about the key components that were alleged not to be included in what was installed, he said those “parts were not part of the bill of quantity the Federal Government gave to the contractor.”
Another staff of the contracted company who did not want his name to be mentioned said the government was told of those components by the contractor but was ignored.
“You know that project was done during the time of Jonathan, the kind of project they gave us, they did not complete, they were supposed to include all those things you mentioned; even there was supposed to be dryer,” he says.
“They didn’t put all that in our bill of quantity. We told them all that that time, I think they just use that project to siphon money. We did what we could do, we did our best and we test run the machine. We handed over the project. We completed that of Osun State.”
Some officials of the agency revealed to the ICIR that the plant was indeed test run by the contractor after the installation.
“Yes, the contractor rented a generating set and we supplied him with some plastic bottles and polythene to test the plant,” an official of the agency who did not want his name mentioned said.
It was never used after that day. Nobody bothered to operate it or ask after it, because as the official said, “Since our ogas don’t have interest in it, it has been like that.”
In Ekiti, recycling plant is moribund
The moribund Ekiti State recycling plant
YEKEEN AKINWALE
Like its Osun State counterpart, the Ekiti State government does not attach much importance to the plastic waste recycling plant located inside the state’s main dumpsite at Ilokun village.
It is under the supervision of Ekiti State Waste Management Agency, but the plant is not on the priority list of the state government.
Since September 2012 that the plant was handed over to the state government for operation, there has been no budgetary allocation to run it efficiently.
Rather, the plant survives on a pittance from the annual budget.
The Director of Operations at the agency, Osalusi Ayoola who oversees the operation of the plant told the ICIR that the state government has no special budget for the plant. Hence, the plant runs only whenever there is a little fund to buy diesel to power the generator set.
The machine has the capacity to produce three tons (3000kg) of plastic pellets per week which translates to about 12000kg in a month.
The National grid was recently extended to Ilokun area, but the plant is not connected. Presently, only the crusher, a machine that crushes plastic waste, works at the plant because the main recycling machine has broken down over a year ago, the ICIR gathered. The pumping machine that supplies water for washing of plastic waste broke down even earlier.
“There are months that we don’t work,” said Olu Ajayi, the operator of the plant.
Pet bottles take over everywhere in Ado-Ekiti
YEKEEN AKINWALE
At the current price per kilogram of pellets which is N150 − the recycling plant can generate as much as N1.8million in a month and N21.6million in a year if well managed according to the operator. Ajayi confirmed that 50 per cent of solid waste generated in the state is plastic waste; he, however, revealed that the plant works only when there is diesel to power the generator.
Scavengers who pick plastic waste from the dumpsite were paid N15,000 by the state government in the beginning. Suberu Umar who had worked there as an operator since 2012 confirmed this. The pay, he lamented, has since been slashed to N10,000.
Ekiti State government under the former governor Ayodele Fayose was already considering selling the plant to a private investor for better utilisation, according to Ayoola.
Where is the Lagos N15m plastic waste recycling plant?
Indiscriminate disposal of plastic waste is a comm practice in Lagos
YEKEEN AKINWALE
For hours, men of Mushin Local Government sweated in the sun to evacuate the blocked drainage in front of Chief Bode’s house on Coker road at Ilupeju Estate. For each load of waste they evacuated, there were more plastics than any other waste.
By the time, they finished packing the waste, the roadside was lined with dirty empty plastic waste.
Blocked waterways and canals, and flooding are permanent features of Lagos, Nigeria’s commercial nerve centre. This is a reflection of the waste disposal and collection system in the state.
At Ilupeju estate, Lagos, a mixture of the residential and industrial estate, residents contend with blocked drainage and flood. “It’s a major source of worry for us here,” Bolu, an Earth Scientist and a resident in the area lamented.
Despite the half-hearted job done by the municipal men, some residents expressed relief that the drainage was free and water could flow freely because each day it rains, the whole road is flooded.
At Coker roundabout, Ikeja, plastic waste are not only environmental threats, they are also health risks
YEKEEN AKINWALE
But a resident was indifferent or not so relieved.
“I’m not bothered by those plastics blocking the drainage,” says Chief Bode, “those plastics have always been there and once it rains, yes the road is flooded but the water will go after some time and the plastic will be there.”
He had lived on the street for well over 20 years, and the drainage around there have always been sources of worry. They are always taken over by waste, particularly plastics, thereby causing floods whenever rain falls.
But that’s not Chief Bode’s real grievance. His angst was the failure of the municipal officials to remove the waste from the street after the evacuation.
“Soon, those plastics would still find their way back into the drainage,” he said, “and that’s my anger.” “The plastics have been there and they came to remove them and left them out there. I won’t pack it, let them pack it.”
He admitted that Coker roundabout which is just opposite his house and the drainage is river-like whenever there is a deluge of water from the sky, but “if they had left the waste in the drainage, the water would have been flowing and the plastics would have been there.”
“Municipal do the evacuation and one of the greatest disservices they do is not to pack what they brought out,” Bolu also voiced his discontent about the act.
From Ojodu, Ogba, Surulere, Ikorodu, Egbeda to the eyebrow areas such as Lekki, Ajah, Victory Island and Parkview, Lagos city is plagued by blocked waterways, drainages, canals and other areas that are clogged by plastic waste.
As Nigeria’s commercial nerve centre, Lagos State hosts over 2000 bottled and sachet water manufacturing and distribution companies.
According to the Head of Lagos State Department of Public Affairs, Mukaila Sanusi, Lagos state houses about 2,000 Industrial complexes, 15,000 commercial ventures, and a burgeoning middle class with high purchasing power.
He discloses that waste generated in Lagos is more than 13,000 tons per day; and 12 per cent of the waste consists of plastic materials in the form of soft drink plastic, water and other consumer goods packaging.
“According to the World Bank, the generation of solid waste is tied to population, income and urbanization. If the report by this body which puts per capita waste generation rate at 1.2 kg per person per day is anything to go by, waste generated in Lagos far outweighs the official figure of 13,000 tons per day,” he says.
Plastic waste management should ordinarily not be a source of worry for Lagosian like Chief Bode and other residents.
A plastic waste recycling plant that was installed in the state some five years ago should have taken care of that worry.
The plant was among those awarded in 2009 which were ready for commissioning in 2013 according to EFO. But it was strange that it could not be traced in Lagos State.
Authorities at the state’s Ministry of Environment and Waste Management agency were not aware of such project in Lagos metropolis or any other places around it.
The General Manager, Lagos State Waste Management Agency (LAWMA), Segun Adeniji, says he was not informed of any plastic waste recycling plant executed by the Federal Government in the state.
“Ha, I’m not aware o, honestly, I’m not aware,” he says. “What I’m aware of is a brisket machine installed by the Federal Government and that one has to do with the wood waste.
“That’s what I’m aware of and I’m not aware of that one, except I will ask from my colleagues. I was not in LAWMA in 2013, I got here in late 2015. If there is anything like that I would have known, having spent more than two years here.”
Asked if the state government has a plastic recycling plant, Adeniji replied, “the recycling plant is a private facility. It is a Public Private Partnership (PPP), West Africa Energy came on board, partners with the state government.
“They bring machinery to do recycling; they sort plastics, bottles and resell to those that require them. The plant is at Igando. They call it materials recovery facility.”
The Contractor confirms to the ICIR that the plant was indeed installed in Lagos state, but he could not state the specific location.
“We installed the plant in Lagos State. But I can tell you that the machine has been removed,” he says.
INFOGRAPHIC: N392 million plastic waste recycling plant
Already, there is a Food and Beverage Recycling Alliance (FBRA) in Lagos that is committed to cleaning up of post-consumer plastic waste in the state. The Alliance consists of Nigerian Bottling Company Limited, the Coca-Cola Company Nigeria, Nigerian Breweries Plc, Seven-Up Bottling Company Limited and Nestle Nigeria Plc.
In July 2018, FBRA signed a Memorandum of Understanding (MoU) with the Lagos State Government, through the Ministry of Transportation, to rid the state’s waterways of plastic and packaging waste.
The MoU is a three-year partnership between Lagos State and the FBRA to clean-up and prevents waste pollution from plastics and other food and beverage packaging, on Lagos State’s inland waterways.
The programme focuses on evacuation for recycling of packaging waste collected from the four inland waterways: Five-cowrie Creek to Lekki; Marina through Elegbata and Osborne to Oworonshoki, waterways from Apapa through Kirikiri, Mile 2, Festac to Oke-Afa, and the Ikorodu Axis, which covers Ipakodo, Ibeshe, Baiyeku, Ijede and Badore.
Kaduna plant never worked since installation
Abandoned: Kaduna plant has never been operated since it was first test run by the contractor
YEKEEN AKINWALE
Like Osun State, Kaduna State plant located inside the Gonigora Kaduna’s main dump site has never worked since 2012 when it was test-ran.
“The plant is dead,” says an official of Kaduna Environmental Protection Authority (KEPA) who is in charge of the dumpsite situated along Abuja Kaduna express road.
“Nothing is there, it has never worked since the day it was tested, adds the official who did not want his name to be mentioned because he did not have the authority to speak to the press.
With some samples of plastic crushed during the test-running still lying at the mouth of one of the machines, probably− the crusher, other machines were covered in dust.
Wasting away: The N15million plant that the state government keeps budgeting money for without operating it
YEKEEN AKINWALE
The General Manager of KEPA, Yusuf Rigasa who was appointed on February 9, 2106, by Governor Nasir El-Rufai said the state government has earmarked N19million in its 2017 and 2018 budget to resuscitate the plant.
“The project was handed over to the then Commissioner of Environment,” Rigasa says. He told The ICIR that the Federal Government started the project but did not complete it.
FG, State governments not interested in the project – Contractor
Failure: Osun State government failed to provide complementary facilities to operate the plant
YEKEEN AKINWALE
After installing nine recycling plants in nine states, Abdul Ishaq, the Chairman of Abdul Essential Services Limited, told the ICIR that his firm was frustrated by both the agents of Federal and State governments that he could not continue with the project.
He maintained that the failure of the Federal Government to pay according to work done and penchant for demanding kickbacks by officials of Ecological Fund Office, Federal Ministry of Environment, and state governments’ officials were responsible for why the rest of the project was abandoned.
“The frustration on the job was from government officials; they never paid us according to the value of the job done, before they pay us, you have to balance the equation.
“You understand what I’m saying. When you finish balancing the equation and the scatter level they scattered the job to, you are left with nothing.”
They paid over N320million and that is the money for us and money for the consultant, money for the Ecological Fund agency and money for the Ministry of Environment, he told the ICIR.
Untapped revenue: Despite its huge economic potentials, Ekiti State has no budget line for the recycling plant
YEKEEN AKINWALE
“I have what is called the Financial Outlay that shows how much they pay us every time. But you know what? These guys are wicked, in the bill of quantities, what they paid me was not even up to the cost of the machine. The machine alone was over 80 per cent of the job,” Abdul alleges.
He specifically mentioned Rivers, Cross River, Delta and Oyo states where, according to him, state officials were demanding for money before they could allocate land where the plant would be sited.
Abdul revealed that many of the states have the machines installed already. “We have in Borno, Katsina, Oyo, Delta, Rivers and other states.”
But the Federal Ministry of Environment that supervised the project has not responded to all the allegations.
A Freedom of Information (FOI) request dated August 29, 2018, and addressed to the Minister of Environment requesting for the contract details of the project was not responded to even at the expiration of the seven working days allowed by law.
However, after a period of two months and a week, the Ministry responded to the FOI request in a four-page document and two Video CDs documenting when Lagos and Katsina states recycling plants were test-run. The document was dated November 5, 2018.
The Federal Ministry of Environment is reputed for its notoriety for not responding to FOI requests. The Ministry came last in 2016 in the ranking of government ministries, agencies and commissions’ response to FOI request conducted by the Public-Private Development Centre (PPDC). It was second to the last in 2017 and 2018 respectively.
When the ICIR reporter contacted the Ministry’s Director of Public Communications, Mohammed Sagir, for a response on the state of the plants and allegations by the contractor, he simply directed him to write a formal letter.
But when he was told an FOI request was pending in the Ministry, he claimed that such has not been brought to his desk as he is also new in the Ministry.
The following day that this reporter went back to the Ministry with the acknowledged copy of the FOI request, he was told to go and follow it up at the office of the Minister.
“We will treat it whenever it is brought here, but you can go and follow it up at the Minister’s office a female staff at the Public Communications office told this reporter.
Findings by the ICIR revealed that almost all the plants installed by the contractor in states like Lagos, Borno, Ekiti, Katsina, Osogbo, Niger, Kwara, Kaduna, Oyo have either been vandalized, stolen or abandoned to rot away.
In the response to the FOI request which further confirmed that the Ministry was unaware of the state of the project contrary to claims by the Minister, the document stated that all the 26 machines have been delivered to the beneficiary states, noting that 21 equipment buildings were completed.
The copy of the response of the Federal Ministry of Environment to the Freedom of Information Request
YEKEEN AKINWALE
It added that the project was completed in nine states − Katsina, Kaduna, Ilorin, FCT, Bauchi, Osogbo, Ado-Ekiti, Ibadan and Lagos, while the training of personnel was also carried out.
Despite claiming that all the 26 machines have been delivered, the Ministry in its response said, three states, Cross River, Delta and Ebonyi were yet to provide land as promised at the beginning of the project.
Twelve states where the equipment buildings have been completed are yet to provide water, electricity for the installation and test running of the machines, the Ministry said in the document but failed to name those states. And it is not clear if the nine states where the government claimed the project was completed were among them.
It confirmed that there is an outstanding of N69.77million debt to be paid to the contractor as it was paid N322.53million, which the document said represents 82.21 per cent of the contract sum.
The revenue potentials that Nigerian government failed to harness
Waste to wealth: Oluwalana used to sort plastic bottles at the dumpsite, now she supplies plastic bottles to recycling plants and makes money
YEKEEN AKINWALE
Rachael Oluwalana was once a worker at Saje dumpsite in Abeokuta, Ogun State. She was always sorting out plastic waste for merchants who take it to recycling plants in Lagos.
“I used to work here for like two years, but from my daily contributions I started my own business about three months ago,” she said
Selling at N50 per kilogram, she told the ICIR that her income in a month revolves around N200,000 and N250,000.
“We used to sell a kilogram at N34 but it has been increased to N50. The buyer used to deduct like 9 per cent or 5 per cent for the dirtiness of the plastic and we pay commission to the owner of the plant, West Africa Energy. We pay N3000 on 2500kg.
In the plastic waste value chain, Oluwalana is just one aside from the scavengers who pick the paste− the loaders, transporters and recycling plants operators and pellets buyers− it has parts of the solution to the unemployment problem in Nigeria which stands at 18.8 per cent as at third quarter of 2017.
Plastic waste recycling is definitely a huge revenue generation potential that government is not harnessing.
In Kaduna, plastic recycling is a booming business
YEKEEN AKINWALE
According to Ishaq, the type of recycling plant that his firm installed has the capacity to produce the 10,000 kg of pellets per day and an average of 70,000kilogram in a week noting that the energy requirement of the machine is 200kilowatts and can run on 100kva generator.
At N150 per kilogram of pellets, Nigeria can generate as much as N42million from the sales of pellets in a month and as much as N504billion in a year from just a recycling plant if well managed.
The government is not taking this advantage, whereas other countries of the world are already making revenue out of plastic waste recycling.
In Haiti, the Philippines, and Brazil for example; the Plastic Bank pays people to collect plastic garbage.
They can trade it for groceries, fuel, and even school fees, or receive payment in the blockchain. The goal is to help people out of poverty and combat plastic pollution in the ocean, Forbes reported.
At a private recycling plant in Kaduna, bagged plastic pellets waiting to be transported to final destinations
YEKEEN AKINWALE
The bank pays them more than other recycling schemes because it sells the plastic on to corporations at a premium as an ethically sound raw material called social plastic. Since 2014, it has processed 3.2 million kilograms (32,000 tons) of plastic.
Some schools in Haiti, Forbes reports, also accept plastic to be used toward tuition.
“If a family making $1,000 a year gets $60 from plastic, it’s a profound change in their lives,” says David Katz, founder of the Vancouver, BC-based company. (The company’s term for the currency is Social Plastic).
According to Daily Mail, 8 million tons of plastic enter the world’s oceans every year. Eighty per cent of it comes from developing countries like Nigeria.
The Plastic Bank wants to address the root cause of plastic pollution, rather than clean up once it’s already in the ocean.
“We should realize the value of plastic while it’s still on land then transfer that value to the world’s most disadvantaged,” Chikere says.
“Nigerian government needs to be in the mainstream of nations seeking to build a globally functioning recycling economy so that fewer new plastics are created and less are disposed of in an uncontrolled manner,” he concluded
Banning the use of plastic in Rwanda has worked well. And the country is a now used as UNEP case study for other countries to emulate
In 2004, the Rwandan Ministry of Environment, concerned by the improper disposal of plastic bags, as they were often burned or clogged drainage systems, commissioned a baseline study which revealed that plastic bag litter was threatening agricultural production, contaminating water sources, killing fish and creating visual pollution.
In 2008 the Rwandan government banned the manufacturing, use, sale and importation of all plastic bags. Paper bags replaced plastic ones, and citizens also started using reusable bags made of cotton.
Along with the new ban, tax incentives were provided to companies willing to invest in plastic recycling equipment or in the manufacturing of environmentally friendly bags.
Not only can eliminating plastic trash save lives, but it also boosts economies, reports the Global Citizen which chronicles the success recorded by Rwanda after banning plastic.
Rwanda, according to the report, has seen an increase in tourism and compared to Uganda, a much cleaner environment.
Rwanda relies heavily on tourism and by eliminating plastics bags can promote eco-tourism and a cleaner greener environment than neighbouring countries. 1.2 million tourists visited Rwanda in 2014–an increase of 4 per cent in the last year.
“Eight per cent (177,000 jobs) in Rwanda are in the tourism sector. And tourism brought in 305 million (USD) last year.
“In addition, Rwanda’s efforts to eliminate plastic bags and overall plastic waste by banning it saved funds that would have been needed to pay government employees to clean up plastic waste.
“So the ban was a pretty smart move economically.”
If Nigeria must get it right the management of plastic waste, Isiaka Amoo, a professor of Analytical and Environmental Chemistry at the Federal University of Technology, Akure, says the Federal and State governments must work together.
Huge business: Already, private individuals are making a living out of plastic waste business in Osogbo, Osun State
YEKEEN AKINWALE
“There is no national policy on plastic waste management in Nigeria as of now. I don’t think there is one,” he says.
“At the Federal level, we have the Senate and the House of Representatives, at our local level, we have House of Assembly here. We can have the law at the top and bye-law at the state level so they can work hand-in-hand in order to have effective ways of managing these plastic wastes.”
He says that plastic waste, besides recycling can also be used to generate energy as used in China.
“We can recycle plastic and it can be used again. Apart from this, you can produce energy from it. When you burn plastic, you see the heat that comes out of it. It shows the amount of heat energy in it. I think in China or India, they are using plastic to generate power. And the by-product that comes from it, they use it to tar their road.”
Another problem is that we don’t have a continuous government; if there is continuity, the new government ought to take over abandoned projects and continue with them.
As it is done in Rwanda, Amoo says the Federal Government can also introduce Public Private Partnership (PPP) to the management of plastic waste in the country.
“We need Public Private Partnership (PPP) at all level, it is a menace, Federal Government and state governments cannot do it alone. If the attitude of government workers is nothing to write home about, it is better to have PPP,” he says.
“The policy can be made by the Federal Government and the private people should continue with the policy.”
On his part, Chikere avers that political engagement is a powerful lever for setting the right incentives to change. “Developing a circular economy is just a matter of political will,” he quips.
The plastic waste menace Nigeria is not dealing with
Plastic waste everywhere: Once it rains, plastic waste takes over everywhere and the government is not dealing with it
YEKEEN AKINWALE
“It was as if all the plastic in Ife were directed here,” says Adewale Mathew, who for two hours, with the help of his friends and clients, spent two hours picking plastic waste that took over his fitness centre situated along Ede road very close to Obafemi Awolowo University, Osun State.
It was after a heavy rainfall. Water flooded everywhere – from Ile-Ife down to Ede road. Roads and houses were all taken over by water.
That day, Monday, July 30, 2018, residents counted their losses but one thing littered everywhere – it was a plastic waste. Pet bottles, take-away foils, pure water sachets and polythene covered the walkways, road and all available space.
“We have to wait for the flood to come down,” he narrated. “Immediately the flood subsided, we have to pick the plastics.”
“I have clients and friends who joined me in picking the plastic, it took us like two hours, we just have to pick and pick, gather them and burn them all,” Mathew told the ICIR, pointing to flooded parts of his fitness centre.
The amount of plastic that came with the flood was unimaginable for him. “I can’t even count, more than 500 plastics covering everywhere because they could not flow under the bridge, so they were washed here and at the edge of the bridge.”
His prayer every time it’s about to rain is ‘for there not to be flood’ because as he said, “Any time there is a flood, we must see plastics.”
Mathew’s experience after the flood underscores the menace of plastic waste which he admits constitutes a serious environmental challenge.
Nigeria has no national policy on plastic waste management− the Minister of Environment of State disclosed that the Federal Government was working on a national policy on plastic waste management to regulate use and disposal of plastic waste in the country.
This is unlike fellow African countries − Kenya, Rwanda, Mauritania and Morocco, that now have a total ban of plastic bags and packaging.
In February 2017, ten countries signed up with the United Nations Environment Assembly (UNEA) to ban the use of plastic bags or containers.
Each year, more than 8 million tonnes of plastic end up in the oceans, wreaking havoc on marine wildlife, fisheries and tourism, and costing at least $8 billion in damage to marine ecosystems. Up to 80 per cent of all litter in our oceans is made of plastic, the UN agency said.
“According to some estimates, at the rate we are dumping items such as plastic bottles, bags and cups after a single use, by 2050 oceans will carry more plastic than fish and an estimated 99 per cent of seabirds will have ingested plastic,” says UN Environment Program in an article titled “UN declares war on ocean plastic.
The threat by plastic waste, environmentalists say is very real.
“If individual countries fail to clean up their plastic waste, our seas will be choked and this will almost certainly affect the world’s ability to generate oxygen, sustain the food chain and indeed all life on the planet,” they said at the UNEA’s #CleanSeas campaign in 2017.
The environmental impacts of plastic waste alongside the looming threat to human survival are many, says Chigozie Chikere, a Training Facilitator at the Institute of Maritime Studies, University of Lagos.
Scientists across the globe are increasingly finding wildlife that has been killed after ingesting or becoming entangled with plastic. Ninety per cent of seabirds, for example, have been found to have plastic in their bellies, The New Republic reports.
It says the estimated 19 billion pounds of plastic that end up in the ocean every year is expected to double by 2025.
“These plastics will not only kill more animals; they’ll decimate coral reefs, and damage human health as microplastics enter the food chain.
“They’ll create more and bigger dead zones where nothing can live, harm biodiversity, and change ecosystems. There will likely be additional, unknown impacts; researchers have only been studying ocean plastics for less than two decades.”
Chikere explained that only one per cent of ocean plastic is actually found floating on the surface.
“In fact, the plastic concentration on the ocean floor is 1,000 times greater than on the surface.”
This investigation was supported by John D. and Catherine T. MacArthur Foundation and the International Centre for Investigative Reporting, ICIR.
SaharaReporters.com is an outstanding, groundbreaking news website that encourages citizen journalists to report ongoing corruption and government malfeasance in Africa. Using photos, text, and video dynamically, the site informs and prompts concerned African citizens and activists globally to act, denouncing officially-sanctioned corruption, the material impoverishment of its citizenry, defilement of the environment, and the callous disregard of the democratic principles enshrined in the constitution. | 2023-09-22T01:27:17.760571 | https://example.com/article/5442 |
/*
* All or portions of this file Copyright (c) Amazon.com, Inc. or its affiliates or
* its licensors.
*
* For complete copyright EntityRef license terms please see the LICENSE at the root of this
* distribution (the "License"). All use of this software is governed by the License,
* or, if provided, by the license below or the license accompanying this file. Do not
* remove or modify any license notices. This file is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
*
*/
#pragma once
#include <AzTest/AzTest.h>
namespace ScriptCanvasTests
{
class EntityRefTestEvents : public AZ::EBusTraits
{
public:
static const AZ::EBusAddressPolicy AddressPolicy = AZ::EBusAddressPolicy::ById;
using BusIdType = AZ::EntityId;
virtual void TestEvent(AZ::EntityId, bool) = 0;
};
using EntityRefTestEventBus = AZ::EBus<EntityRefTestEvents>;
class TestComponent : public AZ::Component
, EntityRefTestEventBus::Handler
{
public:
AZ_COMPONENT(TestComponent, "{527680AE-BF46-4BC8-A923-A39B458A3B53}", AZ::Component);
void Init() override {}
void Activate() override
{
EntityRefTestEventBus::Handler::BusConnect(GetEntityId());
}
void Deactivate() override
{
EntityRefTestEventBus::Handler::BusDisconnect();
}
void TestEvent(AZ::EntityId referencedEntity, bool shouldBeSelf) override
{
// validate entity
if (shouldBeSelf)
{
EXPECT_TRUE(GetEntity()->GetId() == referencedEntity);
}
AZ_TracePrintf("Script Canvas", "TestEvent handled by: %s\n", GetEntity()->GetName().c_str());
}
static void Reflect(AZ::ReflectContext* reflection)
{
AZ::SerializeContext* serializeContext = azrtti_cast<AZ::SerializeContext*>(reflection);
if (serializeContext)
{
serializeContext->Class<TestComponent, AZ::Component>()
->Version(0)
;
}
AZ::BehaviorContext* behaviorContext = azrtti_cast<AZ::BehaviorContext*>(reflection);
if (behaviorContext)
{
behaviorContext->EBus<EntityRefTestEventBus>("EntityRefTestEventBus")
->Event("TestEvent", &EntityRefTestEvents::TestEvent)
;
}
}
};
}
| 2024-05-16T01:27:17.760571 | https://example.com/article/6834 |
Impacting factors for recovery of erectile function within 1 year following robotic-assisted laparoscopic radical prostatectomy.
Neurovascular bundle preservation generally results in good postoperative sexual function after radical prostatectomy. However, erectile function (EF) after radical prostatectomy is still a significant concern. The same surgical technique often results in different EF outcomes. We evaluated factors that correlate with recovery of EF within 1 year after robotic-assisted laparoscopic radical prostatectomy (RALP). From January 2008 to May 2009, 145 consecutive patients underwent RALP by one surgeon. Patients were followed postoperatively at 3-month intervals and assessed for EF recovery, defined as an erection sufficient for penetrative intercourse with satisfaction. Baseline demographics, medical comorbidities, degree of nerve sparing, and perioperative and postoperative variables were recorded. Univariate and multivariate analyses were used to determine factors associated with EF recovery. Postoperative sexual outcomes were attained prospectively via our erectile state questionnaire. Complete follow-up EF data were available on 89 men. Within 1-year follow-up, 56 men (62.9%) recovered EF and 33 men (37.1%) did not. In univariate logistic regression analysis, race (black), diabetes mellitus, hyperlipidemia, and clinical T2 carcinoma of the prostate were associated with diminished EF. Higher-preoperative Sexual Health Inventory for Men score and incremental nerve sparing (enhanced lateral prostatic fascia sparing) were associated with higher odds of recovering potency. In multivariate analysis, hyperlipidemia was primary comorbidity associated with diminished EF, and bilateral nerve sparing with a minimum unilateral-enhanced status was the impacting factor for EF recovery within 1 year after surgery. Bilateral nerve preservation with a minimum unilateral-enhanced status is associated with improved recovery of EF, and hyperlipidemia is a significant negative predictive factor of postoperative EF recovery within 1 year following RALP. Therefore, it is important to control hyperlipidemia as well as to use the proper surgical technique in maximizing EF recovery within 1 year after radical prostatectomy. | 2024-03-01T01:27:17.760571 | https://example.com/article/1323 |
Education information
Nowadays, Malta is one of the most popular destination for spending summer holidays. Foreign students often choose Malta as a perfect place for rest and effective studies. There is a great deal of language summer camp in Malta. SMAPSE offers you 6 best summer camps for international students during summer vacations. You can find many interesting studying programs and courses. A trip to the picturesque Mediterranean island will be remembered for every young traveler and will be an excellent choice for foreign students to improve command of English.
Advantages of studying at 6 best summer language camps in Malta
Modern summer languahe camps in Malta offer many exciting activities. Foreign students will like adventure games, classical excursions and water sports. In all this diversity of entertainment, communication with teachers, coaches and peers is held in English. Therefore, it's a great practice for international students to catch contemporary language, slang and develop active vocabulary.
It's worth noting that studying at language camps in Malta is really affordable and inexpensive. Moreover, studying programs are high-quality and modern. You can choose courses by different intensity and duration. Malta is a perfect place for studying during whole year, there is mild climate, comfort weather and a lot of opportunities for a bright, unforgettable, interesting and useful holiday.
How to choose a language camp in Malta?
To choose one of 6 best language camps in Malta could be challenging. It's important to consider student's interest, hobbies and wishes. You can allow your chils choose several language camps and, then, make decesion together.
According to SMAPSE experts, best studying programs offer IELS/LAL Malta, Sprachcaffe, Inlingua, EC Malta. Schools are located in tourist areas of the capital region. Students can choose to visit camps in a picturesque bay with stunning scenery or in the entertainment center of St. Julian's. All language camps have convenient location, offer high-quality services, effective and balanced education, extracurricular programme.
Do you doubt? At any time you can contact SMAPSE experts. They are always ready to answer all your questions. Also, they can select a language camp and effective vacation programme according to interests of the child, his needs and ambitions.
Tuition fees for studying at TOP-6 summer language camps in Malta
Cost in most cases includes a full range of services, including curriculum, school materials, accommodation, meals, regular excursions and activities, sports games, transfers and insurance. Depending on the prestige, location and popularity of the camp, the option of accommodation, the price for everything varies on average from 450 to 750 euros per week. | 2023-09-15T01:27:17.760571 | https://example.com/article/5931 |
/*
Copyright The Kubeform Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
*/
// Code generated by Kubeform. DO NOT EDIT.
package v1alpha1
import (
base "kubeform.dev/kubeform/apis/base/v1alpha1"
core "k8s.io/api/core/v1"
metav1 "k8s.io/apimachinery/pkg/apis/meta/v1"
)
// +genclient
// +k8s:openapi-gen=true
// +k8s:deepcopy-gen:interfaces=k8s.io/apimachinery/pkg/runtime.Object
// +kubebuilder:object:root=true
// +kubebuilder:subresource:status
// +kubebuilder:printcolumn:name="Phase",type=string,JSONPath=`.status.phase`
type AppEngineStandardAppVersion struct {
metav1.TypeMeta `json:",inline,omitempty"`
metav1.ObjectMeta `json:"metadata,omitempty"`
Spec AppEngineStandardAppVersionSpec `json:"spec,omitempty"`
Status AppEngineStandardAppVersionStatus `json:"status,omitempty"`
}
type AppEngineStandardAppVersionSpecDeploymentFiles struct {
Name string `json:"name" tf:"name"`
// +optional
Sha1Sum string `json:"sha1Sum,omitempty" tf:"sha1_sum,omitempty"`
// +optional
SourceURL string `json:"sourceURL,omitempty" tf:"source_url,omitempty"`
}
type AppEngineStandardAppVersionSpecDeploymentZip struct {
// +optional
FilesCount int64 `json:"filesCount,omitempty" tf:"files_count,omitempty"`
// +optional
SourceURL string `json:"sourceURL,omitempty" tf:"source_url,omitempty"`
}
type AppEngineStandardAppVersionSpecDeployment struct {
// +optional
Files []AppEngineStandardAppVersionSpecDeploymentFiles `json:"files,omitempty" tf:"files,omitempty"`
// +optional
// +kubebuilder:validation:MaxItems=1
Zip []AppEngineStandardAppVersionSpecDeploymentZip `json:"zip,omitempty" tf:"zip,omitempty"`
}
type AppEngineStandardAppVersionSpecEntrypoint struct {
// +optional
Shell string `json:"shell,omitempty" tf:"shell,omitempty"`
}
type AppEngineStandardAppVersionSpecHandlersScript struct {
// +optional
ScriptPath string `json:"scriptPath,omitempty" tf:"script_path,omitempty"`
}
type AppEngineStandardAppVersionSpecHandlersStaticFiles struct {
// +optional
ApplicationReadable bool `json:"applicationReadable,omitempty" tf:"application_readable,omitempty"`
// +optional
Expiration string `json:"expiration,omitempty" tf:"expiration,omitempty"`
// +optional
HttpHeaders map[string]string `json:"httpHeaders,omitempty" tf:"http_headers,omitempty"`
// +optional
MimeType string `json:"mimeType,omitempty" tf:"mime_type,omitempty"`
// +optional
Path string `json:"path,omitempty" tf:"path,omitempty"`
// +optional
RequireMatchingFile bool `json:"requireMatchingFile,omitempty" tf:"require_matching_file,omitempty"`
// +optional
UploadPathRegex string `json:"uploadPathRegex,omitempty" tf:"upload_path_regex,omitempty"`
}
type AppEngineStandardAppVersionSpecHandlers struct {
// +optional
AuthFailAction string `json:"authFailAction,omitempty" tf:"auth_fail_action,omitempty"`
// +optional
Login string `json:"login,omitempty" tf:"login,omitempty"`
// +optional
RedirectHTTPResponseCode string `json:"redirectHTTPResponseCode,omitempty" tf:"redirect_http_response_code,omitempty"`
// +optional
// +kubebuilder:validation:MaxItems=1
Script []AppEngineStandardAppVersionSpecHandlersScript `json:"script,omitempty" tf:"script,omitempty"`
// +optional
SecurityLevel string `json:"securityLevel,omitempty" tf:"security_level,omitempty"`
// +optional
// +kubebuilder:validation:MaxItems=1
StaticFiles []AppEngineStandardAppVersionSpecHandlersStaticFiles `json:"staticFiles,omitempty" tf:"static_files,omitempty"`
// +optional
UrlRegex string `json:"urlRegex,omitempty" tf:"url_regex,omitempty"`
}
type AppEngineStandardAppVersionSpecLibraries struct {
// +optional
Name string `json:"name,omitempty" tf:"name,omitempty"`
// +optional
Version string `json:"version,omitempty" tf:"version,omitempty"`
}
type AppEngineStandardAppVersionSpec struct {
ProviderRef core.LocalObjectReference `json:"providerRef" tf:"-"`
ID string `json:"id,omitempty" tf:"id,omitempty"`
// +optional
// +kubebuilder:validation:MaxItems=1
Deployment []AppEngineStandardAppVersionSpecDeployment `json:"deployment,omitempty" tf:"deployment,omitempty"`
// +optional
// +kubebuilder:validation:MaxItems=1
Entrypoint []AppEngineStandardAppVersionSpecEntrypoint `json:"entrypoint,omitempty" tf:"entrypoint,omitempty"`
// +optional
EnvVariables map[string]string `json:"envVariables,omitempty" tf:"env_variables,omitempty"`
// +optional
Handlers []AppEngineStandardAppVersionSpecHandlers `json:"handlers,omitempty" tf:"handlers,omitempty"`
// +optional
Libraries []AppEngineStandardAppVersionSpecLibraries `json:"libraries,omitempty" tf:"libraries,omitempty"`
// +optional
Name string `json:"name,omitempty" tf:"name,omitempty"`
// +optional
NoopOnDestroy bool `json:"noopOnDestroy,omitempty" tf:"noop_on_destroy,omitempty"`
// +optional
Project string `json:"project,omitempty" tf:"project,omitempty"`
Runtime string `json:"runtime" tf:"runtime"`
// +optional
RuntimeAPIVersion string `json:"runtimeAPIVersion,omitempty" tf:"runtime_api_version,omitempty"`
// +optional
Service string `json:"service,omitempty" tf:"service,omitempty"`
// +optional
Threadsafe bool `json:"threadsafe,omitempty" tf:"threadsafe,omitempty"`
// +optional
VersionID string `json:"versionID,omitempty" tf:"version_id,omitempty"`
}
type AppEngineStandardAppVersionStatus struct {
// Resource generation, which is updated on mutation by the API Server.
// +optional
ObservedGeneration int64 `json:"observedGeneration,omitempty"`
// +optional
Output *AppEngineStandardAppVersionSpec `json:"output,omitempty"`
// +optional
State *base.State `json:"state,omitempty"`
// +optional
Phase base.Phase `json:"phase,omitempty"`
}
// +k8s:deepcopy-gen:interfaces=k8s.io/apimachinery/pkg/runtime.Object
// +kubebuilder:object:root=true
// AppEngineStandardAppVersionList is a list of AppEngineStandardAppVersions
type AppEngineStandardAppVersionList struct {
metav1.TypeMeta `json:",inline"`
metav1.ListMeta `json:"metadata,omitempty"`
// Items is a list of AppEngineStandardAppVersion CRD objects
Items []AppEngineStandardAppVersion `json:"items,omitempty"`
}
| 2023-09-10T01:27:17.760571 | https://example.com/article/8386 |
On a cold February night in Stamford, Connecticut, the chill in the air is doing nothing to dampen patrons' thirst for Half Full Brewery's craft beers. The brewery's tasting room is serving a steady flow of customers looking to sip a glass of its Bright Ale, Pursuit IPA or Onward Pale Ale. The mood may be mellow, but it belies Half Full's rapid growth. Opened in August 2012, Half Full saw its revenue top $350,000 by the end of 2013. Last year it hit $750,000, and this year owner Conor Horrigan expects it to top $1 million. As demand has increased for Half Full's beers, the brewery is looking to hire more workers.
"I would say in the next 60 days we're looking to almost double our staff," said Horrigan, who currently counts seven employees on his payroll.
Half Full is not the only craft brewer looking for workers. The Brewers Association trade group said here in the U.S. one or two craft breweries open every single day, and growth in this industry shows little sign of slowing down.
A worker at the Half Full Brewery in Stamford, Conn. Karina Frayter | CNBC
Defined as small, independent and traditional, these brewers will make less than 6 million barrels a year, but small is increasingly mighty in the world of beer. Bart Watson, chief economist at the Brewers Association, said craft beers accounted for about 12 to 13 percent of all beer sales by volume last year, a number he forecasts could reach 20 percent by 2020. As volume increases, so too will the need for workers to fill a whole host of positions at these businesses, creating what Watson estimates will be tens of thousands of new jobs.
"The types of positions that brewers are looking to fill are really going to vary by the size of our brewer," said Watson. "So a small brewery is going to need a jack-of-all-trades. They're going to need somebody who is both a chemical engineer and a forklift driver, whereas, as you move up to a regional craft brewer, they're going to have much more specific positions — lab technicians, that chemical engineer, or a head brewer, somebody that understands production processes and can help in the canning line."
I would say in the next 60 days we're looking to almost double our staff Conor Horrigan, owner, Half Full Brewery
While humans have been making beer for what historians believe is well more than 5,000 years, there is a shortage of workers experienced in this ancient craft. It takes years to become a master brewer and while a lot of people are familiar with the product, they lack the know-how to produce it.
"I would say that the toughest thing in this industry, as it grows, is finding people with an advanced level of knowledge," said Horrigan, the 34-year old himself a refugee from Wall Street, where he once was a trader for Bear Stearns.
The need to fill that knowledge gap is the primary reason San Diego State University created a program where students can receive a professional certificate in the business of craft beer. With San Diego home to more than 100 craft brewers, local businesses wanted to hire qualified people, and find ways to ensure the quality of the product and the process among new entrants. The brewers' concerns and input helped SDSU develop the 13-course program.
A worker at the Half Full Brewery in Stamford, Conn. Karina Frayter | CNBC
"So every student needs to start with (the course) Exploring Craft Beer. That gives everybody the same level of education coming in, and then they can choose their path," said Program Director Giana Rodriguez. "Those that are really interested in the brewery start-up piece can take our Brewery Startup One and Brewery Startup Two."
Rodriguez said these courses teach students how to write a business and marketing plan, how to get the financing to start the brewery and how to handle front-of-the house management which includes tasting room management and draft systems. Additionally, students who are more focused on the hospitality side of the business can take courses on various beer styles, and beer and food tasting.
"We get the emails saying we want to hire directly from your group because we know that they can talk about beer," said Rodriguez.
More than 600 students have taken at least of one the program's courses over the last three years, and 144 have received their certificates, said Rodriguez.
"We have a mix of students who are in a career looking to change careers who are really trying to get their foot in the door in the industry," she said. "We have some entrepreneurs that come in who are actually looking to invest into the industry and are starting to make connections and learn about the craft."
Twenty-seven-year-old Ashley Benson is currently enrolled in the program. A home brewer for five years, she never considered a job in the industry until she started seeing ads for openings at craft breweries in and around San Diego. She has completed one course of the SDSU program and is enrolled in another. A social media specialist for pet supplies retailer Petco, Benson said she plans to pursue the certificate, with her sight set on landing a marketing position.
"I'm hoping to switch as soon as possible, but I think in the next year would be a good goal," she said. "And then in five years I'm just hoping to kind of have enough knowledge to either be doing marketing for a good craft brewery or be looking to open my own craft brewery."
Ashley Benson in Beer Styles class at San Diego State University. Source: Issac Cadriel
The positions craft brewers are looking to fill these days are on the business and service side, include salespeople to work with distributors, tasting room managers and tasting room staff, said Rodriguez. As for what those jobs and others in the brewing operations pay, it depends on the worker's experience and skill set.
"Service industry jobs are going to be comparable to other service industry jobs, so a job in a brewpub is going to look like a job in a restaurant," said the Brewers Association's Watson. "Highly skilled technical brewers can be making you know, as much as six figures if they're working in a large regional brewery."
The pay also may vary by region. Horrigan said he pays his workers more because the the cost of living on Connecticut is higher than the national average.
Beer Styles class at San Diego State University. Source: Issac Cadriel | 2024-02-22T01:27:17.760571 | https://example.com/article/7006 |
Amino and carboxy functionalized modified nucleosides: a potential class of inhibitors for angiogenin.
The 3'-amino and carboxy functionalize thymidines execute their ribonucleolytic inhibition activity for angiogenin. These modified nucleosidic molecules inhibit the ribonucleolytic activity of angiogenin in a competitive manner like the other conventional nucleotidic inhibitors, which have been confirmed from kinetic experiments. The improved inhibition constant (Ki) values 427 ± 7, 775 ± 6 μM clearly indicate modified nucleosides are an obvious option for the designing of inhibitors of angiogenesis process. The chorioallantoic membrane (CAM) assay qualitatively suggests that amino functionalized nucleosides have an effective potency to inhibited angiogenin-induced angiogenesis. Docking studies further demonstrate the interaction of their polar amino group with the P1 site residues of angiogenin, i.e., His-13 and His-114 residues. | 2024-05-05T01:27:17.760571 | https://example.com/article/4010 |
Q:
How to create many to many relationship between two tables using Flask-SQLAlchemy
I am having users and forums tables. User has contributed in many forums and Forum has many users
My database tables
users
+----+------------+
| id | name |
+----+------------+
| 1 | user_one |
| 2 | user_two |
| 3 | user_three |
+----+------------+
forums
+----+--------+
| id | name |
+----+--------+
| 1 | forum1 |
| 2 | forum2 |
| 3 | forum3 |
+----+--------+
users_forums
+---------+----------+
| user_id | forum_id |
+---------+----------+
| 1 | 1 |
| 1 | 3 |
| 3 | 2 |
| 2 | 1 |
+---------+----------+
A:
Ref - https://docs.sqlalchemy.org/en/13/orm/basic_relationships.html#many-to-many.
You can create the UserForum model or users_forums table to pass secondary option in relationship.
from flask_sqlalchemy import SQLAlchemy
from sqlalchemy import Column, Integer, String, ForeignKey
from sqlalchemy.orm import relationship
db = SQLAlchemy()
class User(db.Model):
__tablename__ = 'users'
id = Column(Integer, primary_key=True)
name = Column(String(24))
forums = relationship("Forum", secondary='users_forums')
class Forum(db.Model):
__tablename__ = 'forums'
id = Column(Integer, primary_key=True)
name = Column(String(24))
users = relationship("User", secondary='users_forums')
class UserForum(db.Model):
__tablename__ = 'users_forums'
user_id = Column(Integer, ForeignKey('users.id'))
forum_id = Column(Integer, ForeignKey('forums.id'))
using as follow
user = User.query.get(1)
print(user)
<User 1>
forums = user.forums
print(forums)
[<Forum 1>, <Forum 2>]
| 2023-10-27T01:27:17.760571 | https://example.com/article/2122 |
Biological and biochemical characteristics of Fusobacterium necrophorum leukocidin.
The production of a leukocidal exotoxin by 18-hour dialysis cultures of Fusobacterium necrophorum was detected in vitro by a cytotoxicity assay, using bovine leukocytes isolated from peripheral blood. Biological characteristics were determined while maximizing the in vitro assay. Biochemical characteristics of heat stability and stability to degradative enzymes were investigated and indicated that the leukocidin is protein in nature. The characteristics indicate that this leukocidin is different from previous descriptions in the literature of the characteristics of a leukotoxin produced by F necrophorum. | 2023-10-31T01:27:17.760571 | https://example.com/article/3473 |
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"Date": [
"Mon, 21 May 2018 23:15:43 GMT"
],
"Pragma": [
"no-cache"
],
"Server": [
"Microsoft-IIS/10.0"
],
"Set-Cookie": [
"ARRAffinity=69b320be9c048dc3cc8f55882e47807e3fab384c4889f0246ba753cc7ed02ff5;Path=/;HttpOnly;Domain=api.botframework.com"
],
"Request-Context": [
"appId=cid-v1:234f01b1-3f0f-49c0-b49c-c528b5cef7a8"
],
"X-Powered-By": [
"ASP.NET"
],
"X-Content-Type-Options": [
"nosniff"
]
},
"StatusCode": 200
}
],
"Names": {},
"Variables": {}
} | 2024-01-30T01:27:17.760571 | https://example.com/article/6863 |
Latin Angels Season 3
About
Follow top bikini models, movie and television actresses, celebrities and famous Latinas as they set sail on an array of adventures to the most alluring locations. Every episode will showcase glamorous swimsuit models on various photo shoots. From the beautiful white sand beaches of the Bahamas, to the exotic islands of Puerto Rico, Dominican Republic and Jamaica, Latín Angels destinations are captivating. | 2024-05-09T01:27:17.760571 | https://example.com/article/5934 |
The evolution of a programmer's job Watch Now
I cut my programming teeth on IBM 360 Assembler. This shouldn't be anyone's first language. In computing's early years, the only languages were machine and assembler. In those days, computing science really was "science." Clearly, there needed to be an easier language for programming those hulking early mainframes. That language, named in September 1959, became Common Business-Oriented Language (COBOL).
The credit for coming up with the basic idea goes not to Grace Hopper, although she contributed to the language and promoted it, but to Mary Hawes. She was a Burroughs Corporation programmer who saw a need for a computer language. In March 1959, Hawes proposed that a new computer language be created. It would have an English-like vocabulary that could be used across different computers to perform basic business tasks.
(Image: IEEE History Center)
Hawes talked Hopper and others into creating a vendor-neutral interoperable computer language. Hopper suggested they approach the Department of Defense (DoD) for funding and as a potential customer for the unnamed language.
Business IT experts agreed, and in May 1959, 41 computer users and manufacturers met at the Pentagon. There, they formed the Short Range Committee of the Conference on Data Systems Languages (CODASYL).
Drawing on earlier business computer languages such as Remington Rand UNIVAC's FLOW-MATIC, which was largely the work of Grace Hopper, and IBM's Commercial Translator, the committee established that COBOL-written programs should resemble ordinary English.
But, even with the support of the DoD, IBM, and UNIVAC, COBOL's path forward wasn't clear. Honeywell proposed its own language, FACT, as the business programming language of the future. For a brief time, it appeared the earlier business developers would be FACT rather than COBOL programmers, but the hardware of the day couldn't support FACT. So, COBOL once more took the lead.
By that September, COBOL's basic syntax was nailed down, and COBOL programs were running by the summer of 1960. In December 1960, COBOL programs proved to be truly interoperable by running on computers from two different vendors. COBOL was on its way to becoming the first truly commercial programming language.
It would still be the business language of choice until well into the 1980s. And it's not done yet.
In 2016, the Government Accountability Office (GAO) reported the Department of Homeland Security, Department of Veterans Affairs, and the Social Security Administration, to name just three, were still using COBOL. According to a COBOL consulting company, which goes by the delightful name, COBOL Cowboys, 200 billion lines of COBOL code are still in use today and that 90% of Fortune 500 companies still having COBOL code keeping the lights on. And, if you've received cash out of an ATM recently, it's almost certain COBOL was running behind the scenes.
Today, COBOL is kept up and running by Micro Focus. In an e-mail interview, Derek Britton, Micro Focus's global director of product marketing, application modernization, and connectivity, said:
"While market sizing is difficult to specify with any accuracy, we do know the number of organizations running COBOL systems today is in the tens of thousands. It is impossible to estimate the tens of millions of end users who interface with COBOL-based applications on a daily basis, but the language's reliance is clearly seen with its use in 70 percent of global transaction processing systems."
What does that mean? Britton explained:
"Any time you phone a call center, any time you transfer money, or check your account, or pay a mortgage, or renew or get an insurance quote, or when contacting a government department, or shipping a parcel, or ordering some flowers, or buying something online at a whole range of retailers, or booking a vacation, or a flight, or trading stocks, or even checking your favorite baseball team's seasonal statistics, you are interacting with COBOL."
That's because the largest number of businesses using COBOL are financial institutions. This includes "banking, insurance and wealth management/equities trading. Second is government services (federal, provincial, local)."
So, while COBOL is nearing retirement age, the language itself is still far from being put out to pasture. In fact, some people are moving their COBOL applications to the cloud. At this rate, COBOL programs will outlive us all.
Related Stories: | 2024-03-18T01:27:17.760571 | https://example.com/article/5249 |
Samuel John Welsh (September 17, 1879 – October 25, 1950) was a civil engineer, canal manager, and politician from Vinton, located west of the Calcasieu Parish seat of government in Lake Charles in far southwestern Louisiana.
A native of New Castle in Lawrence County in far western Pennsylvania, Welsh studied civil engineering and relocated to Vinton in 1902 after his marriage to the former Margaret Friggle (1881-1940)[1] of Oil City in Venango County, also in western Pennsylvania. The couple had five sons: John Edgar (1903-1980) of Lake Charles, James Herbert (1906-1977) and Robert Bruce (1910-2000), both of Baton Rouge, Samuel John, Jr. (1915-2000), of Houston, Texas, and William Max (1920-2004) of Sulphur, also in Calcasieu Parish. Welsh was a railroad surveyor and the manager of the Sabine Canal Company. An active Democrat, Welsh served on the Calcasieu Parish Police Jury from 1916 to 1920 and was a member of the Calcasieu Parish School Board from 1936 to 1950 and was the board president at the time of his death. He was also a member and past president of the Vinton Drainage Board. He was a member of the Methodist Church, Rotary International, and the Masonic lodge. For a short time prior to his death, he was a director of the West Calcasieu-Cameron Hospital.[2]
Welsh went missing on October 25, 1950; his body was found four days later in Dynamite Slough in Vinton.[3] He is interred beside his first wife at Big Woods Cemetery in rural Edgerly near Vinton.[2] His second wife, Fannie B. Campbell Welsh (1908-1964), who was twenty-nine years his junior, is interred at Forest Park Cemetery in Houston, Texas.[4] The S. J. Welsh Middle School in Lake Charles bears his name.[2] | 2023-08-22T01:27:17.760571 | https://example.com/article/1058 |
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