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A Colorado man is asking the state's attorney general to investigate officials in the town of Palisade, who he accuses in a lawsuit of creating an illegal monopoly by limiting the number of medical marijuana businesses to one.
If the lawsuit succeeds, it could become harder for Colorado municipalities to limit the number of dispensaries they allow. But an official with the Colorado Municipal League told the Grand Junction Sentinel that Colorado law allows municipalities to prohibit or limit marijuana and dispensaries.
Kevin King of Grand Junction filed the lawsuit this month in Mesa County District Court, charging that Palisade officials adopted an ordinance permitting just one dispensary in town in violation of the state constitution and local laws. The suit also alleges that the lone license was awarded without a bidding process, and that town officials colluded with the eventual licensee, the Colorado Alternative Healthcare Medical Marijuana Center.
In addition to requesting that the attorney general conduct a criminal investigation, the suit asks the judge to nullify the one dispensary limit, and suspend Colorado Alternative's license.


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Audited Accounts
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India's Modi shows way on cutting government
Prime minister shows the way to trimming fat from a federal leviathan
Publishers of law books occupy a sweet spot in the law. Without fail, the legal tomes expand every year. The last Congress added 284 statutes, and Congress has added 184 so far in this session. More than 20,000 bills have been signed into law, and they all had to go into the law books, requiring frequently updated volumes. The rarest law of all is one that repeals an obsolete statute.
Governments here could learn a thing or two from Narenda Modi, the new prime minister of India, who rode to office on the slogan, "minimum government, maximum governance." He's redeeming the promise by trimming the fat from the law code. The Centre for Civil Society, a freedom-oriented think tank in New Delhi, estimates that India's code is still cluttered with 3,000 useless directives, some of them from the colonial era. "Some of the laws on our books are laughable," says Ravi Shankar Prasad, the law minister. "Others have no place in a modern and democratic India."
One of the laws on the chopping block decrees that certain property in Calcutta may only be sold to the East India Company, which went out of business more than a century ago. Though they may seem harmless, obsolete statutes weigh down business and harm India's economic competitiveness. Campaigning this spring, Mr. Modi pledged that for every new law passed, he would see to it that government repeals 10 redundant laws. Mr. Modi wanted to toss 100 statutes in the bin in his first 100 days.
He didn't make it, but he's close. His government has revised 36 obsolete laws, and a bill is pending to eliminate 287 obsolete statutes this winter. His ministers have put 1,000 up for the shredder. Mr. Modi is clearly on to something. Here in America, a few states have sifted through their codes to identify pointless laws and regulations and marked them for elimination.
Gov. Sam Brownback of Kansas, a Republican, created the "Office of the Repealer" three years ago. The office, the first in the nation, eliminated scores of impractical regulations, bad laws and outdated executive orders. The Kansas law books are no longer cluttered by a state fee on pest-control operators, a process for appointing "jail matrons" and instructions for reinstating a sheriff after a lynching in his county.
Tennessee followed Kansas with a repealer last year. Rhode Island devised a weaker but useful method of ridding outmoded laws through a joint legislative committee. Even in Illinois, there's an interest in the idea.
Washington politicians can't resist seeing their names attached to new resolutions and acts of Congress, so there's never an incentive to do the right thing. Success in India and Kansas ought to be an inspiration to congressmen who say they're for leaner government.
Read the story on The Washington Times website.
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BuzzFeeds > Blog > Entertainment > Zendaya doesn't want her Disney fans to see 'Euphoria' on HBO
Zendaya doesn't want her Disney fans to see 'Euphoria' on HBO
Karol G seduces with a bikini with red hearts that overflows with her charms
Zendaya is one of the protagonists of 'Euphoria'.
Photo: Marcell Rev / HBO / Courtesy
If the 'Spider-man' franchise allowed him to make the leap to the big screen to officially become a movie star, the work of Zendaya in the series 'Euphoria' he was undoubtedly the one who managed to get rid of the youth idol label to claim his talent as an actress.
The story may focus on a 17-year-old teenager, but those who expected Zendaya to repeat a character similar to those she played at the time at the Disney factory were truly disappointed. The plot includes images of violence, sex and drug use that make it unsuitable for minors and now its protagonist has wanted to remind everyone of this in view of the premiere of the new episodes.
"I know I've said it before, but I want to repeat to everyone that 'Euphoria' is for an adult audience. This season, perhaps even more than the last, is very emotional and deals with issues that can be disturbing or difficult to see."She has assured on Instagram to recommend to anyone who feels affected by some of the things that her co-stars and she do on camera to seek professional help.
"Please watch the show only if you feel comfortable doing it. Take good care of yourselves and I want you to know that, in any case, you still deserve love and I continue to feel your support ".
• Zendaya is considering Tom Holland to appear in the series "Euphoria"
• "My Spider-man": The tender message that Zendaya dedicated to Tom Holland in networks
• Tom Holland and Zendaya broke the rule of not falling in love while working together on "Spider-Man: No Way Home"
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Ana Lorena Sánchez has not yet recovered from the scare that Mauricio Islas gave her between scenes of 'The Woman of my Life'


Statement of the Chairwoman and Ranking Member of the Committee on Ethics Regarding Representative John Conyers, Jr.
The Committee is aware of public allegations that Representative John Conyers, Jr. may have engaged in sexual harassment of members of his staff, discriminated against certain staff on the basis of age, and used official resources for impermissible personal purposes. The Committee, pursuant to Committee Rule 18(a), has begun an investigation and will gather additional information regarding these allegations.
The Committee notes that the mere fact that it is investigating these allegations, and publicly disclosing its review, does not itself indicate that any violation has occurred, or reflect any judgment on behalf of the Committee.


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Everyday Illuminations: Gemma Gorga's "Book of Minutes," Translated from Catalan by Sharon Dolin
September 3, 2019 · by smilkova · in Catalan · Leave a comment
By Mary Ann Newman
Unlike American poetry, Catalan poetry tends to shun the confessional or the directly personal. To revive a Lacanian phrase, Catalan poetry is always already political. A language and a literature that suffered continual interruptions owing to lost wars and various repressions—the 1714 War of Succession, the 1923 dictatorship of Primo de Rivera, the 1936-1939 Civil War and its aftermath, all of which entailed suppression of the language—are inevitably self-conscious. The very vehicle of poetry, of survival, was always under attack.
The great resurrectors of Catalan poetry—in the 19th and early 20th century—had to be nation-builders, coaxing the language phoenix from the ashes to resuscitate the epic, like Jacint Verdaguer, or to celebrate and sing themselves and others, like Joan Maragall, Catalonia's Whitman. Responsibility lay heavy on those first great restorers of the Catalan language, though they wore it with grace and joy. In the early 20th century, poets like J. V. Foix and Carles Riba were concerned with restoring classical influences where there had been no Enlightenment tradition. Those miraculous generations supplied new traditions to supplant the void between the early Renaissance lyricism of Ausiàs March and the 19th-century Renascence of the Catalan symbolic universe.
Despite the new hiatus of forty years of Spanish state repression of the Catalan language following the Civil War (1939-1975), generations of poets did, nonetheless, continue writing. They, too, were doomed to talk of their land: Pere Quart (the pen name of Joan Oliver) and Josep Carner wrote from exile and of exile; Salvador Espriu wrote of internal exile and imagined a land of freedom somewhere in the north. Joan Brossa, an extraordinary conceptual and visual poet, used his picture poems to denounce the dictator. They, too, were called upon to sustain the language and the land. Together, they guaranteed that the next generations would find a language to work with and a literature that envies none.
This summer in Brooklyn, Gemma Gorga and Ernest Farrés did a bilingual reading, orchestrated by Gorga's translator, Sharon Dolin, at the beloved Berl's Poetry Shop. Gorga and Dolin were reading from Book of Minutes, published this year in a bilingual edition by Oberlin Press; Farrés was reading from Lawrence Venuti's translation of his 2006 collection, Edward Hopper, published in 2009 by Graywolf Press. The association of the two poets is significant. Gorga and Farrés, born respectively in 1968 and 1967, experienced the tail end of the Franco regime, when the avant-garde was flourishing, resistance was the order of the day, and Franco would soon be a thing of the past—he died in 1975 (though his legacy lingered). As a democratic government emerged, their generation seemed to have left the syndrome of repression and reaction behind. It was not incumbent on them to save the language from oblivion, or to protect it from exile. So they were able to reinvent it from the perspective of the everyday, and not from the urgency of extinction. For a brief moment, Catalan poets seemed almost able to relax.
In his 2011 collection, Blitzkrieg, Ernest Farrés offers a useful emblem for this generational shift. The poem explodes the notion of a simple poetic "I":
Let's be real: the word
of the poet is
an objectivated word, the I
of the text need not
intersect with the real
I, poetry is a
bunch of lines
surrounded by white space. [1]
Gemma Gorga, too, eschews a simple "I," taking the oblique approach of finding a vehicle to allow distance while writing of feelings and perceptions refracted through an impassioned, but objectivated, I.
In Gorga's case, this vehicle is a Book of Minutes, the abbreviated structure of a "Book of Hours," an illuminated manuscript (and never has the notion of illumination been more apt) containing liturgical texts, usually intended for noblewomen's consumption. Gorga's background as a medievalist comes through here. Her Book of Minutes can indeed be seen as a book of devotions, daily meditations for the devout, which upends traditional devotion, substituting a passion for the details of intimate existence: the liturgies of household saints.
Sharon Dolin, a gifted, award-winning poet herself, has produced this translation as an act of love. She discovered a few poems from the Book of Minutes by chance in an anthology she encountered at an artists' residency in Barcelona (shout-out to the Institut Ramon Llull for supporting Anna Crowe's translation and Pere Ballart's edition of Six Catalan Poets). Dolin relates discovery, the intervention of chance, to her own translation and to Gorga's own method, quoting a conversation with the author in her translator's preface: "For me, poetry is a tool that opens things up and excavates, and a light that guides me into corners that I don't know." Gorga's Book of Minutes references these intense illuminations, reducing them to their essence, which she calls "glimpses." And yet, as Dolin observes, despite the medieval inspiration for its form, Book of Minutes "feels very 21st century in its range of diction: in one breath a poem talks about the soul, in the next, about diopters or benzodiazepine."
The form of Gorga's poems, her "minutes," is perfectly adapted to producing these glimpses. In many poems, one image gives way to another and another, in an exercise of poetic displacement, as in poem #8:
A simple day, a transparent day, a unicellular day, a day composed only of vowels, a day that fits in the palm of your hand, a day that will not sing when it dies. One day, simply without foreseeable consequences. (19)
Un dia senzill, un dia transparent, un dia unicel·lular, un dia fet només de vocals, un dia que hi cap al palmell de la mà, un dia que no cantarà quan mori. Un dia, simplement, sense conseqüències previsibles. (18)
The succession of images—simple, transparent, unicellular, composed of vowels, fitting in the palm, not singing, without a future—remits the reader to categories that do not overlap: luminescence, biological composition, language, size/space, music/joy, finally circling back to simplicity, now defined as an absence of futurity. Dolin has rendered the sequence beautifully.
Still, there is a nuance that has not been reflected, which has to do with the synonym pair senzill, which modifies "day" in the first phrase, and simple/simplement, which appears as an adverb inflecting "dia/day" in the last sentence. Senzill and simple can both be translated as "simple" in English, but, as usual, they are not exactly the same. If you wished to reflect the difference between the two words, you might translate the first phrase as "An ordinary day" or "An unassuming day." But if you decide to retain the coincidence in the target language, playing with the shift from the adjective "simple" to the adverb "simply"—a delightful and legitimate choice—it might have been more effective to place the adverb at the beginning of the phrase, modifying "day," and not after the initial phrase, modifying "consequences." It could work this way: "Simply a day, without foreseeable consequences." Or "One simple day, without foreseeable consequences," either of which would also pick up more effortlessly on the rhythm that Dolin has established previously.
This simple day is one of the oases in a collection also marked by great sensuality and passion, and much erotic anguish, still kept in check by the masterful use of images and, again, by the objectivated I. The images are startling, as in poem #27, which opens "Tears shed at dawn are sharp as diamonds cut with anger's chisel…" (57) ["Les llàgrimes que es vessen de matinada són cantelludes com diamants tallats amb el cisell de la ràbia…," 56]. In this poem, everything is off-kilter; the intuited anguish of the narrator is deflected onto external objects: "canaries chirp out of tune/els canaris refilen, fora de pentagrama", "coffee whistles off key/el cafè xiula, fora dels rails." I might have preferred a more literal "the coffee whistles off the rails" here, as I believe the coffee/rails image is intended to be jarring. But Dolin's elegant poetic sensibility and craft prevail over the quibbles of word choice, accomplishing a sweeping rendition of Gorga's immense range of references contained in a sequence of minutes.
To conclude, Book of Minutes ultimately channels both the I and the eye of a medievalist, training its observations on the experiences of 21st-century women. To some extent it vindicates the long bloodlines of women whose senses were circumscribed by religious text and image, establishing the subtle tension between the deeply religious aesthetics and the intrinsic erotics of Catholicism. This tension is then carried over by Gorga and Dolin into more everyday observations, always delivered through the five senses: touch, taste, smell, sight, and hearing.
As is only right in an illuminated manuscript, sight often dominates, so I will leave you with poem #23, in which the most intrinsic characteristic of these prose poems is made manifest, literally brought to light:
I raise the blinds so light may enter. I part the curtains so light may enter. I close my eyes so light may enter. (p. 49)
Aixeco la persiana perquè pugui entrar la llum. Enretiro la cortina perquè pugui entrar la llum. Tanco els ulls perquè pugui entrar la llum. (48)
In the Beginning was the Word, and the Word was Light. Captured by Gorga and Dolin.
Gorga, Gemma. Book of Minutes. Translated by Sharon Dolin, Oberlin College Press, 2019.
[1] Siguem francs: la paraula / del poeta és paraula / objectivada, el jo / del text no té per què / coincidir amb el jo / real, la poesia / és un manyoc de ratlles / envoltades de blanc. Taken from the web: http://www.editorialmeteora.com/libro/blitzkrieg/145. Translation mine.
Tags: Catalan, Catalan poetry, Gemma Gorga, literary reviews, Literary translation, Mary Ann Newman, Oberlin College Press, Poetry
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Al-Marwallah and Standards for Detention
31 Jul Al-Marwallah and Standards for Detention
31.07.08 | 1 Comment
Let me start by saying that I don't think I've substantially narrowed my detention criteria between the book and this discussion–though I am potentially amenable to doing so. The book is written for a general-interest audience and, consequently, at a higher-level of altitude than this discussion is taking place. Precisely to preserve the ability to have this discussion sort of more granular discussion, I wrote that the contours of the detainable class "will require careful legislative definition" (p. 163) and contented myself in the book with the most general principles for defining the class. I have, to be sure, been more specific in my discussion here as to some of the legislative criteria I might apply (emphasis on the word "might"), but I think they are well within the ambit of the principles I describe in the book.
Marty does not quote the sentence I actually intended as the centerpiece: "The core of the authority should more or less track the domestic power to lock up the insane: A person should be subject to administrative detention if he is both a member or associate of opposing forces and dangerous for that reason." In this discussion, I have added a third prong to this two-pronged test–the impracticability of criminal trial–and I have tried to give more texture (informed by Judge Wilkinson's opinion, Matt Waxman's paper, and numerous conversations with Marty) to the two pre-existing prongs. But I see this more in the realm of refinement and addition of greater detail than as a broad change. That said, if it makes Marty more comfortable, I'm delighted.
Incidentally, I might be amenable to what I think is a far more significant change–the addition of some sort of criminal predicate to the test, some kind of requirement that the government demonstrate by a preponderance of the evidence that the detainee's conduct violated some criminal law. Conversely, I might also be persuadable that my conduct-based second prong is too strict and that a showing a future dangerousness based solely on statements ("I will kill as many Americans as I can the moment I get the change," say) ought in combination with a showing of Al Qaeda affiliation to be adequate for detention.
But let me talk for a moment about Mr. Al-Marwallah, whose case I read under this regime very differently from the way Marty reads it. Al-Marwallah, were he detained today based on post-2004 conduct, could certainly be prosecuted for crimes (assuming the evidence were usable); taking training from Al Qaeda is now a crime, after all. Back in 2001, however, his conduct may not have been criminal. Yet he acknowledges taking sniper training at an Al Qaeda camp; he acknowledges being on the "back lines" of a fight in which we were on the other side, and he acknowledges being part of a military retreat with the enemy. Moreover, when asked whether he is a member of Al Qaeda, he says that he doesn't know but he does "know I am an Arab fighter." I would read such statements as an acknowledgement that he operated on behalf of Al Qaeda and the Taliban, in taking training from one organization and in being in some sense part of the other's military formation and retreat–all conduct dangerous to the security of the United States. The only hard question to my mind about such a person is the assessment of his future dangerousness–but that is fact-intensive inquiry about the individual. I cannot imagine that as a matter of constitutional law, the United States is disabled from detaining people who actively train with the enemy and participate in its military activities (its formations and its retreats, in this case).
In fact, I regard Al-Marwallah's detention under the laws of war as perfectly lawful based on nothing more than his own statements. Consider the hypothetical German in street clothes who acknowledged taking training from the Wermacht, being on the "back lines" in France, and retreating with the German Army before his capture and who cagily said–when asked whether he was a member of that force–"I don't know. I know I'm a fighter for the Aryan cause." We would have no doubt of his detainability under the laws of war. I can't imagine any American court would treat Al-Marwallah differently and I'm willing to bet Marty lunch at a restaurant of the winner's choice that however many detainees at Gitmo are ultimately deemed in habeas cases not to be enemy combatants, one of them will not be Bashir Nasir Al-Marwallah.
Why then is he evidence of the need for an adminstrative detention regime and not simply evidence that law of war detention is a reasonable approach? He's not. If all cases were this clear, law of war detention would be perfectly adequate for the war on terrorism. But some detainees aren't as honest as this guy. Some admit a whole lot less–either because they're lying or because they're telling the truth. Some of them are smarter. And some of them blur the lines of the laws of war a whole lot more. In other words, Al-Marwallah is a good example of the need for some non-criminal detention authority. His is not, however, the case that shows why that detention authority should not stem from the laws of war.
I understand but let us keep going down. Let us imagine that there is a guy who gives less information. For example, in WWII two guys in street clothes came across the battle lines where my dad was. One was black and one was white in northern Italy. They had some story. So what happens? My dad interrogates him (an GC III Article 5 setting now) and makes a judgment about whether they are a threat or not. If they are a threat they become security detainees. If not, they are sent on their way. This happened a large number of times in the First Gulf War. So next case, put these two guys on a street in Paris – you have nothing on them. You got nothing – you can ask for their papers and if they are in order you can have the Surete create a file to watch them. Now in between those spaces on battlefields or non-battlefields you have evidence levels that differ. What you are saying is that the evidence levels needed to detain somebody is analogizing the situations when people are insane. But the situation when someone is insane is that there is evidence… Read more »


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REVEALED: £200,000 available for improving Ottery play areas and other projects
Daniel Clerk
Councillor Geoff Pratt. Picture: Geoff Pratt
Nearly £90,000 earmarked for projects such as play areas and allotments is owed to Ottery Town Council (OTC) - with another £115,000 on the way.
The land behind Butts Road after a succesful appeal to put 130 homes on the site in 2013.
The authority was unaware of the money, despite the fact it had spent two years hounding East Devon District Council (EDDC) about the amount of available section 106 funds - cash paid by developers to mitigate the impact of new homes.
Councillor Geoff Pratt asked how much section 106 money EDDC held which was owed to OTC at a meeting on Wednesday, July 24.
It was revealed that £22,581 was available for play projects and £66,192.63 for generic open space projects - allotments, children's play or youth play etc.
A further £115,202 is about to be made available from the 130-home Butts Road development.
Cllr Pratt said: "The Ottery town clerk has been trying to get that information for the last two years without any success and it is only since I asked this question, a flurry of information has been provided and this total of £203,000 that we have has come to light.
"The lack of communication on behalf of the council is totally unacceptable and in future town and parishes should get proper replies when questions are asked.
"It seems that we have £200,000 to spend but the town clerk who was trying to get this has been ignored for years."


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Show Review: Whiskey Warehouse in Mansfield, Ohio
What an incredible and crazy year it's been so far! Where to start….
DOSE has been working with the Filthy Hands and Mushroomhead camp, a group of extremely talented people, tracking our first full length release. Going into the studio and not being "on the clock" was probably one of the biggest changes from past experiences. Having the time to free jam, try ideas, different tones and equipment was a blast. We've never spent so much time on things like initial tones for the making of an album.
At this point, the recording is 95% done. Just some minor adjustments, then mix / master and get it out to all of you who have been waiting to hear what we've got.
Compared to previous recording sessions (and while it's definitely cliche), damn… this is the best work we've ever done by leaps and bounds! Seriously, each of us went to places we didn't know existed in ourselves. I mean, how can you explain creative… I can't, but I know when something feels great. Can't wait for this thing to hit and it WILL HIT HARD and will also include a special surprise or two!
Outside of the studio, DOSE just wrapped up the music video for the single which is coming out very soon. Full production badassery! It was all shot in about 13 hours, long night but the crew was extremely helpful and supportive. And, they even fed us – nice! A cool thing I noticed was the camera guy asking "you have a few more left in you" after about the 23rd take and Jeff said "whatever it takes!" That pretty much says it all about the attitude for everything we are doing here.
The live show. We've been working on putting together lights and front-of-house backing tracks to enhance the experience. Syncing the lights with the music was another goal we've been working on. A full set is about an hour, so it takes some time. Like with any creative, you put it out there, then refine and tweak to really dial it in. The lights themselves bring another cool layer to the show and enhance the live experience. Pretty intense.
Gear was another big growing point. We're now playing with in-ear monitors and accumulating equipment which makes recording at our studio easier as well as at the show, and we are all psyched about the tones and sounds we are able to experiment with.
Probably one of the last big changes is the DOSE practice hall. We just left the hall we've been at for quite a few years for a much larger place. It's basically a full studio so we can just press record when we want to, put the riff on loop and play with ideas. We basically have a control room, live room and green room (it's actually green) and will have all the comforts of home away from home.
If you aren't connected to us on social media, you will want to do that now to stay tuned into the latest release dates and other adventures from DOSE. Oh, and if you're wondering what a caterpillar toy has to do with this, trust me, it's not what you think!
Rise Up (Official Music Video)
New Merchandise Now Available


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The Blog of Gene Luen Yang - Indonesian Smile
Thursday, May 6, 2010, 05:56 PM
I got copies of the Indonesian translation of The Eternal Smile a while ago, but I didn't flip through it until just now. They did a beautiful job on the production values-- it's kinda crazy seeing your own work in a language you don't understand. Ever wonder what a splash sounds like in Indonesia?
Speaking of The Eternal Smile, Amazon.com is offering the original English version for the amazing price of $6.61! Not bad for an Eisner nominee, right? (Er. Only a third of the book is nominated for an Eisner, but still. It's only a third of the original price!)
Sunday, May 2, 2010, 01:59 AM - Events
On Tuesday 5/4 at 4pm, I'll be at the Southgate Library in Sacramento talking about how I make comics. If you're in the area, come on by! Here's the address:
Southgate Library
6132 66th Ave.
What are you doing on Saturday 5/1/10?
Thursday, April 22, 2010, 05:10 PM - Events
Come to Mrs. Dalloway's, a cool indie bookstore in Berkeley! I'll be there on Saturday 5/1/10 at 7pm to talk about my two just-released books, Prime Baby and Animal Crackers. Here's the address:
Mrs. Dalloway's
The Rise of the Graphic Novel
Tuesday, April 20, 2010, 01:36 AM - Events
On the evening Monday 4/26, The Commonwealth Club will be hosting a panel discussion on Why Comic Books, Being So Freaking Awesome, Are Taking Over The Whole Freaking World (or as they put it, "The Rise of the Graphic Novel") with Lark Pien, Andy Hartzell, Dash Shaw, and me! Should be fun. Tickets are $7 if you're a student and $20 if you're not. You have to show school ID so you can't just say you're a student of life.
Want details? Go here.
Thursday, April 15, 2010, 01:24 AM
Two of my books were released this month!
First, First Second recently published Prime Baby, which collects a short story I did for the New York Times at the end of 2008.
Second, Animal Crackers from SLG Publishing collects my first two graphic novels Gordon Yamamoto and the King of the Geeks and Loyola Chin and the San Peligran Order. I've also added a short twelve-page comic at the end where I talk about my process.
Available RIGHT NOW! from your neighborhood comic shop, your local independent bookstore, or Amazon.com.
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Graphic Novel and Comic Writing & Illustrating Conference


50,000 investors are currently afraid of losing parts of their fortunes because of the insolvency of P&R (PR Insolvency), a Munich container investment company.
What happened in the PR Insolvency?
A lot of investors were shocked when P&R announced on their website on March 7 that no more investments were possible anymore. This started off a wave of speculations about what could have happened. It now seems to be clear that P&R's system of acquiring money has badly failed which led to the final bankruptcy.
How did P&R's system work?
An investor (e.g. Mr. Smith) concludes a purchase and rental contract with P&R Transport-Container GmbH.
Smith becomes the owner of the purchased containers.
At the same time 'P&R Transport-Container GmbH' rents the containers from Mr. Smith.
'P&R Transport-Container GmbH' does not have the purchased containers in stock but purchases them from the Swiss Sister-Company 'P&R Equipment & Finance Corp.' Between these two companies a general agreement is set up.
These two P&R Companies now conclude a sublease agreement.
So Mr. Smith is now the owner of the container. 'P&R Transport-Container GmbH' leases the container from Mr. Smith. 'P&R Equipment & Finance Corp.' subleases the container from 'P&R Transport-Container GmbH'. 'P&R Equipment & Finance Corp.' now rents the container out to other companies like Textainer, Dong Fang or Seacube and so gains further profit.
Additionally, P&R guaranteed that they would repurchase the sold containers to a price worth 65% of the selling price.
Most investors were certain that this system is absolutely safe and trustworthy because it is rather simple, has worked for more than 40 years and has gained profits of approx. € 3.5 Billion. But something must have gone wrong. Finance experts and analysists state that P&R was not able to gain the profits necessary to keep the system profitable without being dependent on the money of new investors.
In the past three years P&R lost nearly half a Billion Euros because they had to pay the investors' rents using company capital and not money they earned renting out their containers.
Furthermore, many five-year deals are expiring this year which means that P&R would have to pay up to € 650 Million while only having sold containers worth € 442 Million.
All this led to the insolvency of the company.
What can investors do and are there even more risks than not getting payed?
Of course, Mr. Smith will now be wondering what he will have to face in the upcoming time. The following might help to get a first impression on what could happen.
Should investors be afraid of facing further costs?
Not only could Mr. Smith could lose all the money he invested but is facing more risks: He would be liable for his containers to the point of "private insolvency." After P&R's failure, shipping lines or ports around the world could saddle him with storage or recycling costs.
So, it seems that immense costs might be waiting for investors after the P&R insolvency. What can they do?
Is it possible for Mr. Smith to get any damages from P&R itself?
Probably not. Due to the company's insolvency it is extremely hard if not impossibly to get any damages from the company anymore.
One possible way might be trying to receive damages from auditors or financial advisors who could have not fulfilled their duties of checking the company's financial status or advising possible investors correctly.
One more way of getting damages could be by asserting claims against 'BaFin', the so called "Bundesanstalt für Finanzdienstleistungsaufsicht" (Federal Financial Supervisory Authority). This agency is responsible for keeping upright a trustworthy financial system and for supervising the financial markets in Germany. They should have noticed that P&R was selling far more containers than they had in stock, which also makes the whole affair look like a snowball scheme only relying on the money of new investors.
We – SYLVENSTEIN-Lawyers – represent and advise many investors in the PR Insolvency and offer a broad range of actions. Next to claims arising from the liability of public authorities our specialist lawyer for financial law check claims for damages against certain financial advisors. We also keep contact to potential purchasers of claims who are willing to offer you a fair price for your claims.
type of investment/ offer number?
Do you have an legal expence insurance?
if your document exceeds 10 MB please send us an e-mail tobroecker [at] sylvenstein-law.de.
Sollte der Dateiupload nicht funktionieren, senden Sie uns die Unterlagen zu Ihrer Anfrage bitte einfach separat im Anhang einer E-Mail an broecker [at] sylvenstein-law.de. zu. Wir kümmern uns dann zeitnah um Ihr Anliegen.
this inquiry is free of charge.


Lamy Safari – F – A Hanukkah present from my in-laws. Love. Love. Love. Smooth as silk and I won't need to be institutionalized if I loose it. I use it for everything.
Pigma Micron – 08 – Began my love affair with these when my wife and I were looking for archival pens/inks with which to sign our Ketubah (Jewish wedding agreement: http://en.wikipedia.org/wiki/Ketubah). Well suited for brainstorming on legal pads.
White Label Retro 51 or Knock off? – While I have a visceral aversion to all things ballpoint, this pen just feels darn good in my hand. Planning on getting some R51 refills to see if I can swap it in to my regular rotation. I got it from a Nikon Rep when I ran the digital photography department at Adorama's rental shop.
Uni-ball Signo DX – .38 – Gift from my brother. These were the love I left for the majesty of the fountain pen. I still roll it around every once in a while, for old times sake.
Lanbitou 8012 – F – Peter, my friend and blogmate, lent me one of his and it was love at first write. It may be a cheap Chinese pen with very little hope of living a long life, but it writes like a dream and it's mine, all mine.
Pilot Vanishing Point – EF – Peter and his wife, Cat gave this to me as a wedding gift. If I lost it I might have to be institutionalized. My first fountain pen. It doesn't see a lot of action but I'm hoping my neurotic fear of damaging and or loosing it will soon wane enough for me to put it through its paces.
Baoer 519 – M – Very pretty and incredibly frustrating. I don't know if I got a defective nib but this thing skips like that's its job. All in all, it was six bucks. So I'd complain but who'd listen?
Kaweco Classic Sport – EF – My wife consulted with Peter and gave this to me for Christmas. I was surprised by how much smaller it is than I expected it to be. But still, it's a great pocket carry. And that Kaweco Sepia Brown ink is the bees knees.
Jared, thanks for the submission! I really appreciate all of your descriptions and stories behind each pen. It definitely adds a level of personality and meaning to your collection and it's awesome for everyone to be able to see. You have a pen for everything here, which is very important. All fountain pens are great, but if you need something archival, or have to sign a receipt, they may not always be up to the challenge. On top of that, it's a really cool picture!
You can find Jared blogging over at PenEnvy.net – make sure to check them out!


Lyon Fresco is a wall painting 20 m by 5 m. It depicts pioneers who lived in Albigny, such as the brothers Gabriel (1880-1973) and Charles (1882-1912) Voisin and Armand Zipfel (1883-1954) . It was realized in 2009 by the City of Creation.
A native of albigny, Armand Zipfel (1882-1954) was the first pilot to fly to Lyon on December 9, 1908 at the Doua military camp (the current INSA campus). Here at the controls of a biplane, made by him from the plans provided by the VOISIN brothers.
Gabriel and Charles VOISIN (1880-1973 and 1882-1912) industrialists and inventors of airplanes and cars, lived at quai de Villevert, 300 m from this place. Here you can see one of their cars : "VOISIN C11" of 14 hp.
The Castle of Albigny or "Tower of Gondebaud" was turn by turn: stronghold in the middle ages, watchtower, church, then school and town hall. it has belonged to private individuals since 1942.
The quarries of stones of the calcareous cliffs of Albigny and Couzon have furnished for decades stones of construction for the buildings and bridges of Lyon. The operation ceased around 1920.
Albigny is the only municipality in the Monts d'Or area that has chosen to be called "sur Saône" meaning "on the Saône" in English (other communities in this area have the "au Monts d'Or" suffix which means "of the mount of gold" in English). Apart from Saône river's charm and beauty, Its floods were dreadful in the past, the most famous one dates back to 1840 (level of flood visible by a sign in the corner of "Lefebvre-qai de Villevert", 200 m from here.
Wink to the memory of Remy Bordelet, (1931-2003) called "Remy", From Albigny who devoted his life to drawing and comics.


This page is part of my Home Automation project.
Modbus is a serial communication interface used in industrial automation. Modbus uses either EIA-485 (RS-485) or Ethernet as physical layer. In this document we consentrate in EIA-485.
An open source project libmodbus exists to provide modbus support under GPL license. The library supports Linux, Mac OSX, FreeBSD, QNX and Win32.
There exists several inexpensive RS232-RS485 and USB-RS485 adapters at the market. eBox2300 has physical serial port, thus I'm using a RS232-RS485 adapter sold at DealExtreme for 4€ ($5).
At the time of writing this document, libmodbus is not included in Debian packages, thus it need to be built from sources.
Download latest source package from libmodbus download site. Following instructions apply for version 3.0.1. Debian package build-essential is required for building.
Here is a simple test software that reads given number of register values from modbus slave and prints them to stdout. This application is made only to study libmodbus and test modbus interface.
Build the code with the command below. In Debian, at least following packages are needed: build-essential, pkg-config.


text:
Home All issues Volume 542 (June 2012) A&A, 542 (2012) L10 Full HTML
GREAT: early science results
Volume 542, June 2012
2. Observation and ...
Appendix A:
A&A 542, L10 (2012)
Dynamics and photodissociation region properties in IC 1396A
Y. Okada1, R. Güsten2, M. A. Requena-Torres2, M. Röllig1, P. Hartogh3, H.-W. Hübers4,5, T. Klein2, O. Ricken1, R. Simon1 and J. Stutzki1
1 I. Physikalisches Institut der Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany
e-mail: okada@ph1.uni-koeln.de
2 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany
3 Max-Planck-Institut für Sonnensystemforschung, Max-Planck-Straße 2, 37191 Katlenburg-Lindau, Germany
4 Deutsches Zentrum für Luft- und Raumfahrt, Institut für Planetenforschung, Rutherfordstraße 2, 12489 Berlin, Germany
5 Institut für Optik und Atomare Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
Accepted: 2 March 2012
Aims. We investigate the gas dynamics and the physical properties of photodissociation regions (PDRs) in IC 1396A, which is an illuminated bright-rimmed globule with internal structures created by young stellar objects.
Methods. Our mapping observations of the [C ii] emission in IC 1396A with GREAT onboard SOFIA revealed the detailed velocity structure of this region. We combined them with observations of the [C i] 3P1−3P0 and CO(4–3) emissions to study the dynamics of the different tracers and physical properties of the PDRs.
Results. The [C ii] emission generally matches the IRAC 8 μm, which traces the polycyclic aromatic hydrocarbon (PAH) emissions. The CO(4–3) emission peaks inside the globule, and the [C i] emission is strong in outer regions, following the 8 μm emission to some degree, but its peak is different from that of [C ii]. The [C ii] emitting gas shows a clear velocity gradient within the globule, which is not significant in the [C i] and CO(4–3) emission. Some clumps that are prominent in [C ii] emission appear to be blown away from the rim of the globule. The observed ratios of [C ii]/[C i] and [C ii]/CO(4–3) are compared to the KOSMA-τ PDR model, which indicates a density of 104–105 cm-3.
Key words: ISM: lines and bands / photon-dominated region (PDR) / ISM: individual objects: IC 1396
© ESO, 2012
IC 1396 is a large H ii region in the Cep OB2 association, excited by the O6.5V star HD 206267. Its border includes many bright-rimmed molecular clouds (Weikard et al. 1996), among which IC 1396A is the closest globule, 3.7 pc away from HD 206267 in the plane of the sky assuming a 750 pc distance to IC 1396 (Matthews 1979). IC 1396A has a more negative velocity (~−8 km s-1) compared to HD 206267 and several of the other bright-rimmed clouds, indicating that it is located in front of the star and moving toward us (Weikard et al. 1996). IC 1396A thus provides a simple example of photodissociation regions (PDRs) on a globule, illuminated by a single star from one side. Within the globule a cavity associated with the young stellar object (YSO) LkHα 349 is seen in IRAC 8 μm, 12CO and 13CO(1–0) maps (Nakano et al. 1989), and the extinction map obtained from the Two micron All-Sky Survey (2MASS) data (Reach et al. 2009). There are additional cavities associated with YSOs, and Reach et al. (2009) suggested that the globule is being reshaped from the inside, with each protostar residing in a "compartment" that it has blown out via its outflow. Therefore, high spatial resolution, as provided by the German REceiver for Astronomy at Terahertz frequencies (GREAT1; Heyminck et al. 2012) onboard the Stratospheric Observatory For Infrared Astronomy (SOFIA; Becklin & Gehrz 2009; Young et al. 2012) at THz-frequencies is essential for understanding the structure and evolution of the IC 1396A globule.
2. Observation and data reduction
2.1. [C ii] observations with SOFIA/GREAT
Fig. 1
Integrated ( − 5 to − 15 km s-1) intensity map of the [C ii] emission (color), overlayed with the contours of a) IRAC 8 μm, b) velocity-integrated [C i] 3P1−3P0 and c) CO(4–3). The contour spacing is (a) 10 MJy sr-1, b) 2 K km s-1, and c) 10 K km s-1. The (0, 0)-position is , 57°31′10′′ (J2000) at LkHα 349. Five asterisks in b) mark the positions of the spectra shown in Fig. 3. HD 206267, the exciting star, is located at (ΔRA, ΔDec) = (17.1′, − 1.8′) and illuminates the globule from the east.
Open with DEXTER
We mapped the [C ii] emission at 1900.5369 GHz (158 μm) with GREAT onboard SOFIA during the basic science flights on 22 and 28 July 2011. On the second flight we observed only the [C ii] emission with the L2 channel, whereas on the first flight the L1 channel was operated in parallel tuned to CO(12–11) at 1381.9951050 GHz. The observations were made in total-power on-the-fly (OTF) mode, with 1 s integration time at each dump, and 8′′ step size. The OFF position is located east and outside of the globule at , 57°31′10′′ (J2000).
Calibration was made by the standard pipeline (Guan et al. 2012). The forward efficiency is 0.95, and the beam efficiency is 0.54 and 0.51 for L1 and L2, respectively (Heyminck et al. 2012). Because the emission line is sufficiently narrow, only a linear baseline was fitted between −30 and −20 km s-1, and between 0 and 20 km s-1. After excluding data suffering from strong standing waves and discarding spectra with excess noise, the data were spectrally resampled to a 0.5 km s-1 resolution and spatially to 25′′ (0.09 pc at the distance of 750 pc) to obtain a better signal-to-noise ratio (S/N). In the following, we use the data from the XFFTS-backend, because the other backends show fully consistent and hence redundant data (see Heyminck et al. 2012,for details of the spectrometers).
Figure 1 shows the integrated line intensity, summed over the velocity range from −5 to −15 km s-1. The error in the line-integrated intensity, estimated from the rms noise of the baseline varies between 2–6 K km s-1 and depends on the number of times that the map point was observed. CO(12–11) is not detected, and the upper limit is 4–13 K km s-1 (see the Appendix for details of the derivation).
2.2. Complementary observations
Complementary observations of [C i] 3P1−3P0-line at 492.1606510 GHz and CO(4–3) at 461.0407682 GHz were performed between October 2000 and April 2001 using the CHAMP array receiver (Güsten et al. 1998) at the CSO. The array with 2 × 8 pixels was operated in frequency-switching mode (throw ± 25 MHz for CO and ± 10 MHz for [C i], rate 1 Hz). CO(4–3) data were recorded OTF, the weaker [C i] in raster mode. The auto-correlator back-end array in its high-resolution mode provided 2048 channels with 0.15 km s-1 spectral resolution. Main beams were 15′′ at 464 GHz and 14.5′′ at 492 GHz, with respective main beam efficiencies of 0.52 and 0.51 (see Philipp et al. 2006,for more details of the set-up). Both maps were convolved to 25′′ angular resolution. The integrated line intensities were obtained in the same manner as for the [C ii] emission (Figs. 1b, c).
3.1. Dynamics of the [C ii] emitting gas
Figure 1a shows an overlay of the [C ii] emission and the IRAC 8 μm emission, which traces polycyclic aromatic hydrocarbons (PAHs) in PDRs, showing a good spatial match between the two. The [C ii] emission also follows the local cavities originating from the YSO outflows mentioned in Sect. 1. One exception is the rim at the eastern edge of the globule, where the IRAC 8 μm has a strong peak while the [C ii] emission is not prominent. The spatial distributions of [C i] and CO(4–3) are different from [C ii]. CO(4–3) is widely distributed over the globule, having a shallow peak in the middle. The [C i] emission is distributed toward the southeast and farther out compared to CO(4–3), following the ridge-like structure from (ΔRA, ΔDec) ~ (1′, 0′) to (0′, − 2′) in the IRAC 8 μm emission. The [C ii] emission also follows this rim, but it is much stronger toward the northern part of the globule.
Wootten et al. (1983) showed the intensity distribution and the position-velocity diagram of CO(2–1) and 13CO(2–1) along the longitude cut that intersects LkHα 349 (located at offset (0, 0)). The CO(2–1) distribution is similar to that of CO(4–3), and 13CO(2–1) is similar to [C i] and [C ii] in the sense that the intensity drops around LkHα 349. On the other hand, Nakano et al. (1989) showed the cavity near LkHα 349 both in CO(1–0) and 13CO(1–0) although it appears as a much steeper hole in the latter. These differences are compatible with being caused by opacity: 12CO is more optically thick than 13CO, and 12CO(2–1) and (4–3) emissions have a higher optical depth than 12CO(1–0) in LTE gas with T ≳ 10 K, although detailed radiative transfer modeling is needed to quantify it.
The channel map of [C ii] (Fig. 2) shows that the gas near the rim of the globule is redshifted; going west and toward the center of the globule the velocity becomes more negative. With the globule in front of the exciting star and moving toward us as mentioned in Sect. 1, this velocity gradient implies that the western part, which is farther away from the illuminated rim, moves faster away from the exciting source. The upper right panel of Fig. 2 shows two [C ii] components with significantly higher negative velocity (see also the spectra at positions D and E in Fig. 3). In the southern component (position E) the local outflows from the YSOs may also affect the velocity field because there are several YSOs in the void-like structure from (ΔRA, ΔDec) ~ (1.5′, 0′) to (0.5′, − 2′) (Reach et al. 2009), and the IRAC 8 μm emission shows a cometary shape toward the southwest.
On the other hand, the [C i] and CO(4–3) emitting gas does not show a significant velocity gradient. Figure 3 shows the line profiles at the five positions marked in Fig. 1b. CO(4–3) has a slight blueshift at position E, which is also indicated by Wootten et al. (1983) in the CO(2–1) position-velocity diagram along the latitude cut. However, there is a clear difference between the line profiles of [C ii] and CO(4–3). Considering the spatial distribution discussed above, it seems that the gas emitting [C ii] is blown away from the rim. The numerical simulations of the evolution of cometary globules by Lefloch & Lazareff (1994) suggest that such a cometary regime lasts ~90% of the cloud's lifetime, and numerous small clumps that are ejected and accelerated along the globule appear as "fuzz" on the blue side of Lefloch & Lazareff's model position-velocity diagram. The observed structures match this cometary regime.
Channel maps of the [C ii] emission overlayed with the contours of IRAC 8 μm emission.
Spectra of [C ii], CO(4–3) and [C i] at the five positions marked in Fig. 1.
3.2. Comparison with PDR models
PDR model results.
We compared the observed integrated intensity ratios between [C ii], [C i] 3P1−3P0, and CO(4–3) with the KOSMA-τ PDR model (Röllig et al. 2006) at the five positions shown in Fig. 3. To exclude the contribution of the velocity components in [C ii] that do not match with the CO and [C i] emission, we fitted the [C ii] emission line with the CO(4–3) line profile at each position and adopted the integrated intensity of this fit result. At position E, the intensity obtained this way is 57% of the total integral of the [C ii]. We used a single-clump model, which has as free parameters the mean gas density (n), the far-ultraviolet (FUV; hν = 6–13.6 eV) flux (χ) in units of the Draine field (2.7 × 106 W m-2), and the mass of the single clump (m) in units of the solar mass (M⊙). We used two simplified model approaches because with three observed lines, there are only two independent ratios but three unknowns. In Model 1, we fixed χ at the value estimated from the luminosity of the exciting star HD 206267, 105.23 L⊙, as an O6.5V star, and the distance to this star from each observed position, assuming that 50% of the photons have an energy within 6 eV < hν < 13.6 eV. The calculated value of χ is listed in Table 1. Then we determined n and m by fitting the intensity ratios of [C ii]/[C i] and [C ii]/CO(4–3), as listed in Table 1. To be conservative, we estimated the error by assuming that each line intensity has a systematic uncertainty of 30%. At position B, no combination of n and m can explain the observed ratio. m is not well constrained because these ratios are insensitive to the mass; the layers that emit [C ii], [C i] , and CO(4–3) emissions are not the central core but outer layers. With χ = 102 and n = 105 cm-3, AV at the center is ~4 even for a clump with m = 10-3 M⊙. Increasing m corresponds to adding layers at deeper AV, which does not significantly change the ratio between the [C ii], [C i] , and CO(4–3) intensities. Therefore, we fixed log (m) in Model 2 to a value of − 1, i.e. 0.1 M⊙, and fitted n and χ. A clump with m = 0.1 M⊙ and n = 105 cm-3 has a radius of ~0.02 pc, which is ~5″ at the distance of 750 pc, small compared to the beam size. The obtained n from both models are consistent within the errors, giving 104–105 cm-3 at position B and ~105 cm-3 for the other positions. The fitted χ in Model 2 is consistent with the values estimated from the luminosity of the star for Model 1 except for position B. At position B, the very low [C ii]/[C i] ratio compared to the other positions is explained by a low UV field in Model 2. It implies that the UV radiation is shielded at the rim of the globule.
We presented mapping observations of the [C ii] emission at 1900.5369 GHz (158 μm) with GREAT onboard SOFIA in IC 1396A, which is an illuminated globule with internal structures created by embedded YSOs. The [C ii] emission closely follows the IRAC 8 μm emission, which traces the PAH emission. Together with complementary [C i] and CO(4–3) observations, we investigated the spatial distributions of the velocity-resolved emission. [C ii] shows significant velocity changes within the globule, while this is not the case for the [C i] and CO(4–3) emission. The spatial distribution is also different: CO(4–3) has a shallow peak at the center of the globule, the [C ii] and [C i] emission is strong at outer region of the globule, but their peaks show different locations. The spatial structure and velocity distribution is consistent with a scenario in which spatially unresolved clumps, which emit mainly [C ii], are blown away along the globule and are accelerated toward us. A PDR model analysis of the observed intensity ratios indicates densities of ~105 cm-3 and a UV flux consistent with excitation by the external star except for one position, where the UV is indicated to be low because of the shielding at the rim of the globule. IC 1396 is selected as one of the key regions to be observed by the GREAT consortium because it provides many bright-rimmed globules with simple exciting sources, and observations in wider regions are expected in future flights.
GREAT is a development by the MPI für Radioastronomie and the KOSMA/Universität zu Köln, in cooperation with the MPI für Sonnensystemforschung and the DLR Institut für Planetenforschung.
This work is based in part on observations made with the NASA/DLR Stratospheric Observatory for Infrared Astronomy. SOFIA Science
Mission Operations are conducted jointly by the Universities Space Research Association, Inc., under NASA contract NAS2-97001, and the Deutsches SOFIA Institut under DLR contract 50 OK 0901. We thank the SOFIA engineering and operations teams whose support has been essential for the GREAT accomplishments during basic science flights, and the DSI telescope engineering team.
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Appendix A: Estimate of uncertainties
We estimate the uncertainty or the upper limit of the integrated line intensity from the rms noise (σ) of the baseline as follows. σ = 1.1–2.9 K and 0.6–2.1 K in the 25″ resolution map for [C ii] and CO(12–11), respectively. At each map position, we create an artificial spectrum by generating normally distributed (Gaussian) random numbers with a standard deviation of σ. Then we integrate this spectrum over the velocity range from −5 to −15 km s-1, which is the same range to obtain the line integrated intensity, and obtain an intensity I (K km s-1). After repeating this trial 20 000 times, we obtain a clear Gaussian distribution of I, and adopt its standard deviation (σ2) as an uncertainty of the integrated line intensity at this map position. If 3σ2 is higher than the integrated line intensity, we consider it a non-detection, and adopt 3σ2 as an upper limit.
In the text
All Figures
IRAM and Gaia views of multi-episodic star formation in IC 1396A — The origin and dynamics of the Class 0 protostar at the edge of an HII region
Dynamics of different layers traced by [C ii] and CO in bright-rimmed clouds in IC 1396
Conditions and Impact of Star Formation R. Simon, R. Schaaf and J. Stutzki (eds) EAS Publications Series, 75–76 (2015) 163-166
The stellar population and complex structure of the bright-rimmed cloud IC 1396N
A&A 504, 97-107 (2009)
A Herschel view of IC 1396 A: Unveiling the different sequences of star formation
The Herschel/PACS view of the Cep OB2 region: Global protoplanetary disk evolution and clumpy star formation


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Alberta group uses its love of Ghostbusters franchise to help local charities
By Allison Bench	Global News
2:46 A look at the 2019 Edmonton Comic and Entertainment Expo
WATCH ABOVE: Phil Darlington speaks to the Alberta Ghostbusters group at the Edmonton Comic & Entertainment Expo. – Sep 20, 2019
Members of an Alberta group are taking their love of Ghostbusters and using it to raise money for a local cause close to their hearts.
The Alberta Ghostbusters consider themselves not only super fans of the iconic film series; but also a group that strives to make a difference in their local communities.
There are currently about 35 members spread across the province, and this weekend, some of them are at the Edmonton Expo collecting donations for the Stollery Children's Hospital Foundation.
"[We're] encouraging people to give donations and take pictures with us. We're also selling some promotional items like lanyards, stickers [and] mugs," said Alberta Ghostbusters member Mario Giguere.
The group raises thousands of dollars at its events and is registered as a non-profit. Members attend events all around the province throughout the year, including other major ones like K-Days and the Calgary Stampede.
Participants are committed to the cause but also to looking the part. They are all required to put together an authentic Ghostbusters uniform to join the group. Giguere says that their passion for the franchise makes it a fun project to do.
"The uniform wasn't too long, it's a U.S. Army flight suit," said Giguere.
"The pack was definitely the part that took the longest to build."
Tweet This Click to share quote on Twitter: "The pack was definitely the part that took the longest to build."
WATCH BELOW: Phil Darlington explores the sights and sounds of the Edmonton Comic & Entertainment Expo.
1:50 Superheroes and villains take over at the Edmonton Expo
Superheroes and villains take over at the Edmonton Expo – Sep 21, 2019
Not only are they hoping Edmonton Expo attendees will line up for photos; they've also collected items they're raffling off to raise even more for the Stollery.
"We even have a signed poster by Jason Reitman, who's currently filming the new movie," said Giguere.
Reitman is a Canadian director who is currently filming a sequel to the original two Ghostbusters films that were directed by his father, Ivan Reitman.
READ MORE: 'Ghostbusters' director Jason Reitman clarifies 'fans' remark
The Alberta Ghostbusters will be at the Edmonton Expo all weekend.
Stollery Children's Hospital tagcomics tagGhostbusters tagEdmonton Expo tagEdmonton Comic & Entertainment Expo tagalberta charity tagalberta ghostbusters tagGhostbuster tagedmonton costume tagghosbusters cosplay tagghostbusters costume tag
Canadian Coast Guard's oldest serving vessel decommissioning
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text:
\section{Introduction.}
Let $f:\R^n\rightarrow\R$, $n\geq 2$, be an analytic function. According to {\L}ojasiewicz \cite{lojasiewicz},
the limit set of a trajectory of the dynamical system $\dot{x}=\nabla f(x)$ is either empty or contains
a single critical point of $f$. So the family of integral curves which converge to a critical point is a natural object of study
in the theory of gradient dynamical systems.

Let $f:\R^n,0\rightarrow\R,0$ be an analytic function defined in a neighbourhood of the origin, having a critical point at $0$.
We shall write $T(f)$ for the set of integral curves which converge to the origin, and
$S(f)$ for the stable set of the origin, which is the union of all orbits of the solutions
that converge to the origin. By \cite{lojasiewicz}, the stable set in closed
near the origin.

In this paper we study the naturally occurring question whether the set $T(f)$ is infinite or
whether the interior of $S(f)$ is non-empty? (In the planar case these problems are equivalent.)
By Remark \ref{plik_2_1_1}, if $T(f)$ is infinite then it has the cardinality of the continuum.

In some cases the answer is rather obvious. If the hessian matrix of $f$ at the origin has at least
two negative eigenvalues then the dimension of the stable manifold at the origin is $\geq 2$,
and then $T(f)$ is infinite. If the origin is a local strict maximum then $\operatorname{int}S(f)\neq\emptyset$.
It is worth pointing out that according to Moussu \cite[Theorem 3]{moussu} the family $T(f)$ always contain
trajectories which are represented by real analytic curves converging to the origin. In some cases
the family of those analytic curves can be infinite.

Let $\omega:\R^n,0\rightarrow\R,0$ be the homogeneous initial form associated with $f$.
Put $\Omega=S^{n-1}\cap\{\omega<0\}$.
Applying the Moussu results \cite{moussu}  one may
show that $\dim\operatorname\,S(f)\geq 2$ if there exists at least one
non-degenerate critical point of $\omega|\Omega$ which is not a local minimum.
In that case the set $T(f)$ is infinite.
Moreover,  if  there exists at least one
non-degenerate critical point of $\omega|\Omega$ which is  a local maximum
then $\operatorname{int}S(f)\neq\emptyset$.

 Let $S_r=S^{n-1}_r\cap\{f<0\}$, where $S^{n-1}_r=\{x\in\R^n\ |\ |x|=r\}$, $0<r\ll 1$.
By \cite{nowelszafraniec1}, \cite{nowelszafraniec2}, if $T(f)$ is finite then each  cohomology group $H^i(S_r)$ is trivial for  $i\geq 1$.
Hence, if there exists $i\geq 1$ with $H^i(S_r)\neq 0$, then $T(f)$ is infinite.

However, there are examples  where $n=2$ and none of the above assumptions holds but $T(f)$ is infinite (see Example \ref{plik_8_3} ).

In the course of proving Thom's gradient conjecture Kurdyka {\it et al.} \cite{kurdykaetal} have applied advanced techniques
of analytic geometry so as to investigate geometric properties of trajectories converging to a the origin. In particular
they have proved that with each such a trajectory one can associate a pair $(\ell,a)\in L'(f)$, where $\ell$ is a characteristic
exponent of $f$, the number $a$ is an asymptotic critical value of $f/|x|^{\ell}$, and $L'(f)$
is a finite subset of $\Q^+\times \R_-$, where $\Q^+$ is the set of positive rationals and
$\R_-$ is the set of negative real numbers.

In \cite{dzedzejszafraniec} (see also Section \ref{rozdzial_22}) there is presented an intrinsic  filtration of $T(f)$ given  in terms
of characteristic exponents and asymptotic critical values of $f$. Unfortunately, these numbers are difficult
to compute. This is why in this paper we present methods which are more easy to apply.

The first main result of this paper shows that $T(f)$ is infinite if
$\rank\, H^0(S_r)< \rank\, H^0(\Omega)$ (see Theorem \ref{plik_8_1}),
i.e. if $S_r$ has less
connected components than $\Omega$.
As a corollary we shall show that the inequality $\chi(S_r)<\chi(\Omega)$ implies that $T(f)$ is infinite.
It is proper to add that there exist efficient methods of computing those Euler-Poincar\'e characteristics (see \cite{leckiszafraniec1}, \cite{szafraniec12}).
(These results have been earlier presented  in \cite{szafraniec13}.)

The second main result of this paper shows that $\operatorname{int}S(f)\neq\emptyset$  if
$\operatorname{rank} H^{n-2}(S_r)< \operatorname{rank} H^{n-2}(\Omega)$,
where $H^{n-2}(\cdot)$ is the $(n-2)$-th cohomology group with rational coefficients (see Theorem \ref{teor_1}).

Let $\Omega'=S^{n-1}\cap \{\omega\geq 0\}=S^{n-1}\setminus\Omega$, and $S_r'=S_r^{n-1}\cap \{f\geq 0\}=S_r^{n-1}\setminus S_r$, $0<r\ll 1$.
Sets $\Omega'$, $S_r'$ are compact and semianalytic, hence they are triagulable. By the Alexander duality theorem,
if $S_r'$ and $\Omega'$ are non-empty then
$\operatorname{rank}H_0(S_r')=1+\operatorname{rank}H^{n-2}(S_r)$ and
$\operatorname{rank}H_0(\Omega')=1+\operatorname{rank}H^{n-2}(\Omega)$.
Thus, if $S_r'$ has less connected components than $\Omega'$ then the interior of $S(f)$ is non-empty (see Theorem \ref{theor_2}).

Let $f$ be as above.
Assume that $g:\R^n,0\rightarrow\R,0$ an analytic function which is right-equivalent
to $f$.
We shall prove that $T(g)$ is infinite (resp. $\operatorname{int}S(g)\neq\emptyset$)
if $S_r$ has less connected components than $\Omega$ (resp. if $S_r'$ has less
connected components than $\Omega'$) (see Theorem \ref{cor_4_1}).

The paper is organized as follows. In Section \ref{rozdzial_01} we prove sufficient conditions which imply that a compact subset of the sphere has a non-empty interior. In Section \ref{rozdzial_1}
we study the homotopy type of some semi-analytic sets. In Section \ref{rozdzial_22}
we present properties of important geometric invariants
associated with trajectories of the gradient flow.
In Sections \ref{rozdzial_cardinality}, \ref{rozdzial_2}, we prove
the main results (Theorems \ref{plik_8_1}, \ref{teor_1}, \ref{theor_2}).
Section \ref{rozdzial_3} is devoted to functions right-equivalent to the ones that satisfy assumptions of those theorems.
References \cite{absilkurdyka, bohmetal,  goldstein,  kurdykaetal, kurdykaparusinski, lageman, nowelszafraniec2} present  significant related results and
applications.

\section{Sets with non-empty interior.}\label{rozdzial_01}

In this section we present some consequences of the Alexander duality theorem. The best reference here is \cite{spanier}.

\begin{lemma}\label{lem_1}
Suppose that $L\subset K$ are closed subsets of $S^{n-1}$, $n\geq 2$, and
$\operatorname{rank}\bar{H}^{n-2}(K)<\operatorname{rank}\bar{H}^{n-2}(L)<\infty$,
where $\bar{H}^{n-2}(\cdot)$ is the $(n-2)$-th \v{C}ech-Alexander-Spanier cohomology group with rational coefficients.
Then the interior of $K$ is non-empty.
\end{lemma}

\begin{proof} As $\bar{H}^{n-2}(L)\neq 0$ then sets $L$, $K$, $S^{n-1}\setminus L$ are not void.
If $K=S^{n-1}$ then the assertion holds. From now on we assume that $S^{n-1}\setminus K\neq\emptyset$ and $n\geq 3$.

By the Alexander duality theorem there are isomorphisms
$$\bar{H}^{n-2}(L)\simeq \tilde{H}_0(S^{n-1}\setminus L)\ , \ \bar{H}^{n-2}(K)\simeq \tilde{H}_0(S^{n-1}\setminus K),$$
where $\tilde{H}_0(\cdot)$ is the $0$-th reduced singular homology group with rational coefficients.

Then $S^{n-1}\setminus L$ is a disjoint union of open connected components $U_1,\ldots,U_\ell$,
where $\ell=1+\operatorname{rank}\tilde{H}_0(S^{n-1}\setminus L)=1+\operatorname{rank}\bar{H}^{n-2}(L)$, and
$S^{n-1}\setminus K$ is a disjoint union of open connected components $V_1,\cdots,V_k$,
where $k=1+\operatorname{rank}\tilde{H}_0(S^{n-1}\setminus K)=1+\operatorname{rank}\bar{H}^{n-2}(K)$.

Suppose that $U_i\setminus K\neq \emptyset$ for each $1\leq i\leq\ell$, so that
there are points $p_i\in U_i\setminus K$ and then $p_i\in V_{j(i)}$ for some $1\leq j(i)\leq k$.
As $V_{j(i)}$ is a connected subset of $U_1\cup\ldots\cup U_\ell$, then $V_{j(i)}\subset U_i$.

Because $U_i$ are pairwise disjoint, then  $V_{j(i)}$ are pairwise disjoint too. Hence $k\geq \ell$,
contrary to our claim.
Then at least one open connected component $U_i$ is a subset of $K$.

Similar arguments apply to the case where $n=2$.
\end{proof}

\begin{cor}\label{cor_1}
Suppose that $L\subset K\subset F$, where $L,K$ are compact, $n\geq 2$,
$\operatorname{rank}\bar{H}^{n-2}(K)<\operatorname{rank}\bar{H}^{n-2}(L)<\infty$,
and $F$ is an $(n-1)$-dimensional manifold homeomorphic to a subset of $S^{n-1}$.
Then the interior of $K$ is non-empty.
\end{cor}

\section{Homotopy type of semi-analytic sets.}\label{rozdzial_1}

Let $f:\R^n,0\rightarrow\R,0$ be an analytic function defined in an open neighbourhood
of the origin. Let $\Q^+$ denote the set of positive rationals.
For $\ell\in\Q^+$, $a<0$, $y<0$ and $r>0$ we shall write
$$B_r^n=\{x\in\R^n\ |\ |x|\leq r\}\ ,\ \  S_r^{n-1}=\{x\in\R^n\ |\ |x|=r\},$$
$$V^{\ell,a}=\{x\in\R^n\setminus\{0\}\ |\ f(x)\leq a|x|^\ell\} ,\ $$
$$S_r^{\ell,a}=S_r^{n-1}\cap V^{\ell,a}\ ,\ \ B_r^{\ell,a}=B_r^n\cap V^{\ell,a},\ \
F^{\ell,a}(y)=f^{-1}(y)\cap V^{\ell,a},$$
$$D^{\ell,a}(y)= f^{-1}([y,0))\cap V^{\ell,a}=\{ x\in V^{\ell,a}\ |\ y\leq f(x)<0\}.$$

\begin{lemma} \label{plik_6_1} Assume that $\ell\in\Q^+$ and $a<0$.
If $0<-y\ll r\ll 1$ then
the sets $S_r^{\ell,a}$ and $F^{\ell,a}(y)$ are homotopy
equivalent. In particular, the singular cohomology groups $H^*(S_r^{\ell,a})$ and
$H^*(F^{\ell,a}(y))$ are isomorphic.
\end{lemma}

\begin{proof} For $x\in  V^{\ell,a}\cup\{0\}$ lying sufficiently close to the origin we have $|x|^{1/2}\geq |f(x)|\geq |a|\cdot |x|^\ell$,
so that in particular functions $f(x)$, $|x|^2$  restricted to this set are proper.
According to the local triviality of proper analytic mappings between semi-analytic sets (see \cite{goreskymacpherson, hardt, hironaka}), they are locally trivial.
So there is $r_0>0$ such that $|x|:B_{r_0}^{\ell,a}\rightarrow (0,r_0]$ is a trivial fibration.
Hence the inclusion $S_r^{\ell,a}\subset B_r^{\ell,a}$ is a homotopy equivalence
for  each $0<r\leq  r_0$.

By similar arguments, there is $y_0<0$ such that $D^{\ell,a}(y_0)\subset B_{r_0}^{\ell,a}$ and
 $f:D^{\ell,a}(y_0)\rightarrow [y_0,0)$ is
a trivial fibration, so that the inclusion $F^{\ell,a}(y)\subset D^{\ell,a}(y)$
is a homotopy equivalence for each $y_0\leq y<0$.

 So, if $0<-y\ll r\ll 1$ then we may assume that $r\leq r_0$, $y_0\leq y$, and
$$D^{\ell,a}(y)\subset B_r^{\ell,a}\subset D^{\ell,a}(y_0)\subset B_{r_0}^{\ell,a}.$$
As inclusions $D^{\ell,a}(y)\subset D^{\ell,a}(y_0)$ and $B_r^{\ell,a}\subset B_{r_0}^{\ell,a}$
are homotopy equivalencies, then $D^{\ell,a}(y)\subset B_r^{\ell,a}$ is a homotopy
equivalence too. Then $F^{\ell,a}(y)$ is homotopy equivalent to $S_r^{\ell,a}$.
\end{proof}

 For $0<-y\ll r\ll 1$ we shall write
$$ F_r(y)=B_r^n\cap f^{-1}(y)\ ,\ \ S_r=\{x\in S_r^{n-1}\ |\ f(x)<0\}.$$
We call the set $F_r(y)$ the {\it real Milnor fibre}.
According to \cite{milnor}, it is either an $(n-1)$-dimensional compact manifold with boundary or an empty set.
Moreover, the sets $F_r(y)$ and $S_r$   are homotopy equivalent.

\begin{cor}\label{plik_6_2}
If $0<-y\ll r\ll 1$ then the cohomology groups $H^*(S_r)$ and $H^*(F_r(y))$ are isomorphic.
\end{cor}

\begin{lemma}\label{plik_6_2_1}
If $\ell\in\Q^+$, $a<0$ and $0<-y\ll r\ll 1$ then $F^{\ell,a}(y)=\{x\in F_r(y)\ |\ y\leq a|x|^\ell\}$.
In particular,  $F^{\ell,a}(y)\subset F_r(y)$.
\end{lemma}
\begin{proof}
If  $x\in F^{\ell,a}(y)$ then $y=f(x)\leq a|x|^\ell$, and then $|x|^\ell\leq y/a$.
If  $0<-y\ll r\ll 1$ then $|x|\leq r$, and then $x\in B_r^n\cap f^{-1}(y)=F_r(y)$.

If $x\in B_r^n\cap f^{-1}(y)$ and $y\leq a|x|^\ell$, then $x\in f^{-1}(y)\cap V^{\ell,a}=F^{\ell,a}(y)$.
Hence, if $0<-y\ll r\ll 1$ then $\{x\in F_r(y)\ |\ y\leq a|x|^\ell\}\subset F^{\ell,a}(y)$.
\end{proof}

Let $\omega$ be the initial form associated with $f$ and let $g=f-\omega$,
so that $f=\omega+g$. Denote by $d$ the degree of $\omega$. Hence $g=O(|x|^{d+1})$.

\begin{lemma}\label{plik_5_1} If $0<r\ll -a\ll 1$ then sets $S_r^{d,a}=S_r^{n-1}\cap\{f\leq a r^d\}$,
$S_r^{n-1}\cap\{\omega\leq a r^d\}$ and $\Omega=S^{n-1}\cap\{\omega<0\}$
have the same homotopy type.
\end{lemma}

\begin{proof}
For $r\in\R$ sufficiently close to zero and $x\in S^{n-1}$ we have
$$f(rx)=\omega(rx)+g(rx)=r^d\omega(x)+r^{d+1}G(x,r),$$
where $G(x,r)$ is an analytic function defined in an open neighbourhood of $S^{n-1}\times\{0\}$.
Put $H(x,r)=\omega(x)+r G(x,r)$, and $H_r=H(\cdot,r):S^{n-1}\rightarrow\R$.

By \cite[Corollary 2.8]{milnor}, there exists $a_0<0$ such that any $a_0<a<0$ is a regular value of $\omega|S^{n-1}$.
Hence there exists $r_0>0$ such that $a$ is a regular value of every $H_r$, where $-r_0<r<r_0$.
Then
$$\{(x,r)\in S^{n-1}\times (-r_0,r_0)\ |\ H(x,r)\leq a\}$$
is an $n$-dimensional manifold with boundary $S^{n-1}\times (-r_0,r_0)\cap H^{-1}(a)$.
By the implicit function theorem, the mapping $(x,r)\mapsto r$ restricted to both above manifolds
is a proper submersion. By Ehresmann's theorem, it is a locally trivial
fibration. Hence if $r$ is sufficiently close to zero then the manifolds
$ S^{n-1}\cap \{\omega\leq a\}=\{x\in S^{n-1}\ |\ H(x,0)\leq a\}$ and
$S^{n-1}\cap \{ H_r\leq a\}=\{x\in S^{n-1}\ |\ H(x,r)\leq a\}$ are homeomorphic.

 The set $S^{n-1}\cap\{\omega\leq a\}$ is a deformation retract
of $\Omega=S^{n-1}\cap\{\omega<0\}$,
so that these sets have the same homotopy type.

We have $f(rx)=r^d H_r(x)$. Hence $x\in S^{n-1}\cap\{H_r\leq a\}$ if and only if
$rx\in S_r^{n-1}\cap\{f\leq a r^d\}$,
and the proof is complete.
\end{proof}

\section{Geometric properties of trajectories.} \label{rozdzial_22}
In the beginning of this section we present some results obtained by Kurdyka {\it et al.} \cite{kurdykaetal}, \cite{kurdykaparusinski} in the course of proving Thom's gradient conjecture. In exposition and notation we follow closely these papers.

Let $f:\R^n,0\rightarrow\R,0$ be an analytic function defined in a neighbourhood of the origin,
having a critical point at $0$. The gradient $\nabla f(x)$ splits into its radial component
$\frac{\partial f}{\partial r}(x)\frac{x}{|x|}$ and the spherical one
$\nabla'f(x)=\nabla f(x)-\frac{\partial f}{\partial r}(x)\frac{x}{|x|}$.
We shall denote $\frac{\partial f}{\partial r}$ by $\partial_rf$.

For $\epsilon>0$ define
$W^{\epsilon}=\{x\ |\ f(x)\neq 0\ ,\ \epsilon |\nabla'f|\leq |\partial_r f|\}$.
There exists a finite subset of positive rationals $L(f)\subset\Q^+$ such that for any $\epsilon>0$ and any sequence
$W^{\epsilon}\ni x\rightarrow 0$ there is a subsequence $W^{\epsilon}\ni x'\rightarrow 0$
and $\ell\in L(f)$ such that
$$\frac{|x'|\ \partial_r f(x')}{f(x')}\ \rightarrow \ \ell\ .$$
Elements of $L(f)$ are called {\it characteristic exponents}.

Fix $\ell >0$, not necessarily in $L(f)$, and consider $F=f/|x|^{\ell}$ defined in the complement of the origin.
We say that $a\in\R$ is {\it  an asymptotic critical value} of $F$ at the origin if there exists a sequence $x\rightarrow 0$, $x\neq 0$,
such that
$$  |x|\cdot \left|\nabla   F(x)\right|\ \rightarrow 0\ \ ,\ \
F(x)=\frac{f(x)}{|x|^\ell}\ \rightarrow\ a\ . $$
The set of asymptotic critical values of $F$ is finite.

The real number $a\neq 0$ is an asymptotic critical value if and only if there exists a sequence
$x\rightarrow 0$, $x\neq 0$, such that
$$ \frac{|\nabla'f(x)|}{|\partial_r f(x)|}\ \rightarrow 0\ \ ,\ \
  \frac{f(x)}{|x|^\ell}\ \rightarrow\ a\ . $$

Hence the set
$$L'(f)\ =\ \{(\ell,a)\ |\ \ell\in L(f), a < 0\ \mbox{is an asymptotic critical value of}\ f/|x|^{\ell}\}$$
is a finite subset of $\Q^+\times \R_-$, where $\R_-$ is the set of negative real numbers.

We shall write $T(f)$ for the set of non-trivial trajectories of the gradient flow
$\dot{x}=\nabla f(x)$ converging to the origin. By Section 6 of \cite{kurdykaetal}, for every such a trajectory
$x(t)$, with $x(t)\rightarrow 0$,
there exists a unique pair $(\ell',a')\in L'(f)$ such that $f(x(t))/|x(t)|^{\ell'} \rightarrow a'$.
There is a natural partition of $T(f)$ associated with $L'(f)$. Namely for $(\ell',a')\in L'(f)$,
$$T^{\ell',a'}(f)=\{ x(t)\in T(f)\ |\ f(x(t))/|x(t)|^{\ell'}\rightarrow a'\ \mbox{ as } x(t)\rightarrow 0\}.$$
In the set $\Q^+\times\R_-$ we can introduce the lexicographic order
$$(\ell',a')\leq (\ell,a)\ \ \mbox{if}\ \ \ell'<\ell, \ \mbox{or}\ \ell'=\ell\ \mbox{and}\ a'\leq a.$$
Take $(\ell,a)\in \Q^+\times\R_-\setminus L'(f)$.
We  shall write
$$\tilde{T}^{\ell,a}(f)=\bigcup T^{\ell',a'}(f), \mbox{ where } (\ell',a')<(\ell,a) \mbox{ and }(\ell',a')\in L'(f).$$

According to \cite{nowelszafraniec1}, there are $0<-y\ll r\ll 1$ such that each trajectory  $x(t)\in T(f)$ intersects $F_r(y)$ transversally at exactly one point.
Let $\Gamma(f)\subset F_r(y)$ be the union of all those points. By \cite{lojasiewicz} the set $\Gamma(f)$ is closed subset of $F_r(y)$, hence it is compact.
So there is a natural one-to-one correspondence between trajectories in $T(f)$ and
points in $\Gamma(f)$. The same way one can define the set $\Gamma^{\ell',a'}(f)\subset F_r(y)$  (resp. $\tilde{\Gamma}^{\ell,a}(f)\subset F_r(y)$)
whose points are in one-to-one correspondence with trajectories from $T^{\ell',a'}(f)$ (resp. $\tilde{T}^{\ell,a}(f)$).
In particular, for $(\ell,a)\in \Q^+\times\R_-\setminus L'(f)$ the set
$$\tilde{\Gamma}^{\ell,a}(f)=\bigcup \Gamma^{\ell',a'}(f), \mbox{ where } (\ell',a')<(\ell,a) \mbox{ and }(\ell',a')\in L'(f),$$
is a subset of $\Gamma(f)$.

By \cite[Theorem 12]{nowelszafraniec1}, \cite[Theorem 6]{dzedzejszafraniec}  and Lemma \ref{plik_6_2_1} we have

\begin{theorem} \label{plik_2_1}
If $0<-y\ll r\ll 1$ then the inclusion $\Gamma(f)\subset F_r(y)$ induces an isomorphism
$$\bar{H}^*(\Gamma(f))\ \simeq\ H^*(F_r(y)),$$
where $\bar{H}^*(\cdot)$ is the \v{C}ech-Alexander-Spanier cohomology group
and $H^*(\cdot)$ is the singular  cohomology group..
In particular $\Gamma(f)$ has the same (finite) number of connected components as $F_r(y)$.

Moreover, for every $(\ell,a)\in\Q^+\times\R_-\setminus L'(f)$ the set
$\tilde\Gamma^{\ell,a}(f)$ is a compact subset of
$F^{\ell,a}(y)$. The inclusion
 induces an isomorphism
$$\bar{H}^*(\tilde\Gamma^{\ell,a}(f))\ \simeq\ H^*(F^{\ell,a}(y)).$$

\end{theorem}

\section{Cardinality of $T(f)$}\label{rozdzial_cardinality}

The cardinality of the set $T(f)$ is obviously the same as that of $\Gamma(f)$. In this section we shall present  simple topological
conditions which imply that $\Gamma(f)$ and  $T(f)$ are  infinite sets.

We shall write $S(f)$ for the stable set of the origin, which is the union of all orbits of the solutions
that converge to the origin.

\begin{rem}\label{plik_2_1_1}
If $\Gamma(f)$ is infinite then it contains at least one  compact and infinite connected component,
which is obviously not a zero-dimensional space.
If that is the case then the Menger-Urysohn dimension as well as the \v{C}ech-Lebesgue covering dimension of this component is at least one (see \cite{engelking}), sets $\Gamma(f)$
and $T(f)$ have the cardinality of the continuum, and the dimension of the stable set
$S(f)$ is at least two.
\end{rem}

By Lemma \ref{plik_6_1}, Corollary \ref{plik_6_2} and Theorem \ref{plik_2_1} we get

\begin{cor}\label{plik_2_3} There is an isomorphism
$\bar{H}^*(\Gamma(f))\ \simeq\ H^*(S_r)$.
In particular $\Gamma(f)$ has the same (finite) number of connected components as $S_r$.
If there exists $i\geq 1$ such that $H^i(S_r)\neq 0$ then $T(f)$ is infinite.
So, if $S_r\neq \emptyset$ and the Euler-Poincar\'e characteristic $\chi(S_r)\leq 0$,
then $T(f)$ is infinite.

Moreover, for every $(\ell,a)\in\Q^+\times\R_-\setminus L'(f)$, if $0< r\ll 1$ then
$$\bar{H}^*(\tilde\Gamma^{\ell,a}(f))\ \simeq\ H^*(S_r^{\ell,a}).$$

\end{cor}

\begin{ex}\label{plik_2_3_1} The polynomial $f(x,y,z)=x^3+x^2z-y^2$
is weighted homogeneous. Of course $S_r\neq\emptyset$.
By \cite [p.245]{szafraniec12}, the Euler-Poincar\'e characteristic $\chi(S_r^2\cap\{f\geq 0\})=2$.
By the Alexander duality theorem we have $\chi(S_r)=0$. Hence the set $T(f)$ is infinite.
\end{ex}

\begin{prop}\label{plik_7_1}  If $0<-a\ll 1$ then $\bar{H}^*(\tilde\Gamma^{d,a}(f))\simeq H^*(\Omega)$.
If $H^i(\Omega)\neq 0$ for some $i\geq 1$ then $T(f)$ is infinite.
\end{prop}
\begin{proof} As $L'(f)$ is finite, if $0<-a\ll  1$ then $(d,a)\not\in L'(f)$.
By  Corollary \ref{plik_2_3}  and Lemma \ref{plik_5_1}, if $0<r\ll -a$ then we have
$$\bar{H}^*(\tilde\Gamma^{d,a}(f))\ \simeq\ H^*(S_r^{d,a})\ \simeq\ H^*(\Omega) .$$
In particular, if $H^i(\Omega)\neq 0$ for some $i\geq 1$ then
 $\tilde\Gamma^{d,a}(f)$ is infinite.
Hence $\tilde{T}^{d,a}(f)$, as well as $T(f)$, is infinite.
\end{proof}

\begin{ex} Let $f(x,y,z)=z(x^2+y^2)+x^2y^2 z-z^4$. It is easy to see that $S_r=S_r^2\cap\{f<0\}$
is homeomorphic to a union of two disjoint 2-discs, so that $H^i(S_r)=0$ for $i\geq 1$.
As $\omega=z(x^2+y^2)$, then $\Omega$ is homeomorphic to $S^1\times(0,1)$,
and so $H^1(\Omega)\neq 0$. Hence $T(f)$ is infinite.
\end{ex}

\begin{cor}\label{plik_7_2}
If    $\Omega\neq \emptyset$  and  the  Euler-Poincar\'e characteristic  $\chi(\Omega)\leq 0$,  then $T(f)$ is infinite.
\end{cor}

\begin{rem}\label{plik_7_1_1} If $\omega$ is a quadratic form which may be reduced
to the diagonal form
$-x_1^2-\cdots-x_{i+1}^2+x_{i+2}^2+\cdots+x_j^2,$
where $i\geq 1$, then
the dimension of the stable manifold at the origin is at least two.
Hence $T(f)$ is infinite.

\end{rem}

Investigating the gradient flow in polar coordinates and applying arguments presented by Moussu in \cite[p.449]{moussu}
the reader may also prove the next proposition. (As its proof would require to introduce other techniques ,
so we omit it here.)

\begin{prop}\label{plik_7_3}
Suppose that  there exists a non-degenerate critical point of $\omega|\Omega$
which is not a local minimum. Then $T(f)$ is infinite.

In particular, if there exists a non-degenerate local maximum of $\omega|\Omega$
then the interior of the stable set of the origin is non-empty.
\end{prop}

\begin{ex}\label{ex_7_4} Let $f(x,y)=x^3+3xy^2+x^2 y^2$, so that $\omega=x^3+3xy^2$.
It is easy to see that $\omega|S^1$ has a non-degenerate  local maximum at $(-1,0)\in\Omega$.
Then the interior of the stable set of the origin is non-empty.
In particular $T(f)$ is infinite.
\end{ex}

The next theorem is the main result of this section.

\begin{theorem}\label{plik_8_1}
Suppose that $f:\R^n,0\rightarrow\R,0$ is an analytic function having
a critical point at the origin

If $\rank\,H^0(S_r)<\rank\, H^0(\Omega)$, i.e. the number of connected components of $S_r$
is smaller than the number of connected components of $\Omega$, then
the set of trajectories of the gradient flow $\dot{x}=\nabla f(x)$ converging to the origin
is infinite.
\end{theorem}
\begin{proof}
Suppoose, contrary to our claim, that $T(f)$ if finite. Then $\Gamma(f)$ is finite, and
 for any $(\ell,a)\in \Q^+\times\R_-\setminus L'(f)$ the set $\tilde\Gamma^{\ell,a}(f)$ is finite too. Hence
$\rank \bar{H}^0(\tilde\Gamma^{\ell,a}(f))$
equals the number of elements in $\tilde\Gamma^{\ell,a}(f)$.

By Lemma \ref{plik_5_1}, there exist $0<r\ll -a\ll 1$ such that $\Omega$ and  $S_r^{d,a}$ have the same homotopy type.
By Corollary \ref{plik_2_3}, the group $H^*(S_r)$ is isomorphic to $\bar{H}^*(\Gamma(f))$.
Hence $\rank H^0(S_r)$$=\rank \bar{H}^0(\Gamma(f))$
equals the number of elements in $\Gamma(f)$. Moreover,
$\rank H^0(\Omega)$$=\rank H^0(S_r^{d,a})$$=\rank \bar{H}^0(\tilde\Gamma^{d,a}(f))$
equals the number of elements in $\tilde\Gamma^{d,a}(f)$.

As $\tilde\Gamma^{d,a}(f)\subset \Gamma(f)$, then $\rank H^0(\Omega)\leq\rank H^0(S_r)$,
which contradicts the assumption.
\end{proof}

\begin{theorem}\label{plik_8_2}
If $\chi(S_r)<\chi(\Omega)$ then $T(f)$ is infinite.
\end{theorem}

\begin{proof} By Corollary \ref{plik_2_3} and Proposition \ref{plik_7_1}, it is enough to consider the case
where all cohomology groups $H^i(S_r)$, $H^i(\Omega)$, where $i\geq 1$, are trivial.

If that is the case then
$\rank H^0(S_r)=\chi(S_r)<\chi(\Omega)=\rank H^0(\Omega).$
By Theorem \ref{plik_8_1}, the set $T(f)$ is infinite.
\end{proof}

\begin{ex}\label{plik_8_3} Let $f(x,y)=x^3-y^2$, so that $\omega=-y^2$.
Then $\Omega=\{ (x,y)\in S^1\ |\ -y^2<0    \}= S^1\setminus \{(\pm 1, 0)\}$.
Obviously $\Omega$ has two connected components and $H^i(\Omega)=0$ for any $i\geq 1$.
The function $\omega |\Omega$ has exactly two critical (minimum) points at $(0,\pm 1)$,
so one cannot apply Proposition \ref{plik_7_3}.

As $S_r$ is homeomorphic to an interval, then by Theorem \ref{plik_8_1} the set $T(f)$ is infinite.
\end{ex}

\begin{ex}\label{plik_8_4} Let $f(x,y,z)=xyz-z^4$, so that $\omega=xyz$.
It is easy to see that $\Omega$ is homeomorphic to a disjoint union of four discs,
and $S_r$ is homeomorphic to a disjoint union of two discs. By Theorem \ref{plik_8_1} the set
$T(f)$ is infinite.
\end{ex}

\begin{ex}\label{plik_8_5} Let $f(x,y,z)=xyz+x^4y-2y^4z+3xz^4$,
so that $f$ has an isolated critical point at the origin and $\omega=xyz$.
Applying Andrzej {\L}\c ecki computer program (see \cite{leckiszafraniec1}) we have verified that
the local topological degree of the mapping
$$\R^3,0\,\ni (x,y,z)\ \mapsto\ -\nabla f(x,y,z)\, \in\R^3,0$$
equals zero. By \cite{khimshiashvili1}, \cite{khimshiashvili2}, the Euler-Poincar\'e characteristic $\chi(S_r^2\cap\{f\geq 0\})=1-0=1$.
By the Alexander duality theorem $\chi(S_r)=1$. By Theorem \ref{plik_8_2} the set $T(f)$ is infinite.
\end{ex}

\section{Interior of the stable set.}\label{rozdzial_2}

In this section we shall present simple topological conditions which imply that the interior
of the stable set $S(f)$ has a non-empty interior

The set $\Omega$ is semi-algebraic, hence $\operatorname{rank}H^{n-2}(\Omega)<\infty$.
By Theorem \ref{plik_2_1} and Proposition \ref{plik_7_1}, if $0<-a\ll 1$ then $\tilde{\Gamma}^{d,a}(f)$
is compact and \\$\operatorname{rank}\bar{H}^{n-2}(\tilde{\Gamma}^{d,a}(f))<\infty$.

\begin{rem}\label{plik_7_1_1} If $\omega$ is a quadratic form which can be reduced
to the diagonal form
$-x_1^2-\cdots-x_{i+1}^2+x_{i+2}^2+\cdots+x_j^2,$
where $i\geq 1$, then
$$\bar{H}^*(\tilde{\Gamma}^{d,a}(f))\ \simeq\ H^*(\Omega)\ \simeq H^*(S^i).$$

In that case
$\operatorname{rank}\bar{H}^{n-2}(\tilde{\Gamma}^{d,a}(f))=\operatorname{rank}H^{n-2}(S^i)>0$ if and only if
$\omega$ can be reduced to the diagonal form $-x_1^2-\cdots-x_{n-1}^2$.
\end{rem}

The next theorem is the main result of this section.

\begin{theorem}\label{teor_1}
Suppose that $f:{\mathbb R}^n,0\rightarrow{\mathbb R},0$, $n\geq 2$, is an analytic function defined
in an open neighbourhood of the origin. Suppose that
$\operatorname{rank}H^{n-2}(S_r)<\operatorname{rank}H^{n-2}(\Omega)$.
Then the stable set of the origin of the gradient flow $\dot{x}=\nabla f(x)$
has a non-empty interior.
\end{theorem}
\begin{proof}
By \cite[Lemma 5.10]{milnor}, if $0<-y\ll r\ll 1$ then the Milnor number $F_r(y)$ is homeomorphic to
an $(n-1)$-dimensional submanifold of $S_r^{n-1}$.

As $\tilde{\Gamma}^{d,a}(f)\subset\Gamma(f)$ are compact subsets of $F_r(y)$ with
$\operatorname{rank}\bar{H}^{n-2}(\Gamma(f))=\operatorname{rank}H^{n-2}(S_r)<
\operatorname{rank}H^{n-2}(\Omega)=\operatorname{rank}\bar{H}^{n-2}(\tilde{\Gamma}^{d,a}(f))<\infty,$
then by Corollary \ref{cor_1} the set $\Gamma(f)$ has a non-empty interior in $F_r(y)$.

Trajectories of the flow $\dot{x}=\nabla f(x)$ converging to the origin cut transversally $F_r(y)$ at point of $\Gamma(f)$.
Hence the stable set of the origin has a non-empty interior.
\end{proof}

Put $\Omega'=S^{n-1}\cap \{\omega\geq 0\}=S^{n-1}\setminus\Omega$, and $S_r'=S_r^{n-1}\cap \{f\geq 0\}=S_r^{n-1}\setminus S_r$, $0<r\ll 1$.
Sets $\Omega'$, $S_r'$ are compact and semianalytic, hence they are triagulable. By the Alexander duality theorem,
if $S_r'$ and $\Omega'$ are non-empty then
$\operatorname{rank}H_0(S_r')=1+\operatorname{rank}H^{n-2}(S_r)$ and
$\operatorname{rank}H_0(\Omega')=1+\operatorname{rank}H^{n-2}(\Omega)$.

\begin{theorem}\label{theor_2}
Suppose that the set $S_r'$ has less connected components than $\Omega'$.
Then the stable set of the origin of the gradient flow $\dot{x}=\nabla f(x)$
has a non-empty interior.
\end{theorem}
\begin{proof}
The set $\Omega'$ is obviously not empty. If $S_r'=\emptyset$ then the origin is a strict local maximum,
and then $\operatorname{int}S(f)\neq\emptyset$.

Suppose that $S_r'\neq\emptyset$.
Sets $S_r'$, $\Omega'$ are compact, semianalytic. So they are triangulable, and
the number of connected components of $S_r'$ (resp. $\Omega'$) equals the number
of its path-components which is $\operatorname{rank}H_0(S_r')$
(resp. $\operatorname{rank}H_0(\Omega')$).

By assumption,
$\operatorname{rank}H_0(S_r')<
\operatorname{rank}H_0(\Omega')$ and then
$\operatorname{rank}H^{n-2}(S_r)<\operatorname{rank}H^{n-2}(\Omega)$.
By Theorem \ref{teor_1}, the stable set $S(f)$ has a non-empty interior.
\end{proof}

\begin{ex}\label{plik_8_3_1} Let $f(x,y)=x^3-y^2$ be the same as in Example \ref{plik_8_3}.
Then $\Omega'=\{(-1,0),(1,0)\}$.
As $S_r'$ is homeomorphic to a closed  interval, then by Theorem \ref{theor_2} the interior
of $S(f)$ is non-empty.
\end{ex}

\begin{ex}
Let $f(x,y,z)=-x^2 y^2-z^4+x^5$. Then $\omega=-x^2 y^2-z^4$ and $\Omega'$
consists of four points. It is easy to see that $S_r'$ is homeomorphic to
a disjoint union of a closed disc and two points. By Theorem \ref{theor_2}
the interior of $S(f)$ is non-empty.
\end{ex}

\section{Right-equivalent functions}\label{rozdzial_3}
Let $g:\R^n,0\rightarrow\R,0$ be an analytic function which is right-equivalent to $f$, i.e. there exists
a $C^\infty$-diffeomorphism $\phi:{\mathbb R}^n,0\rightarrow {\mathbb R}^n,0$ defined
in an open neighbourhood of the origin such that $g=f\circ\phi$. Then in particular the derivative
$D\phi(0):{\mathbb R}^n\rightarrow{\mathbb R}^n$ is a linear isomorphism.

Let $\theta$ be the initial homogeneous form associated with $g$,
let $\Theta=S^{n-1}\cap\{\theta<0\}$,
and let $\Theta'=S^{n-1}\cap\{\theta\geq 0\}$.
It is easy to see that $\theta=\omega\circ D\phi(0)$. Hence sets
$\Omega$ and $\Theta$,  as well as $\Omega'$ and $\Theta'$, are homeomorphic.
Then $H^0(\Omega)\simeq H^0(\Theta)$ and
$H_0(\Omega')\simeq H_0(\Theta')$.

 Both $f$ and $g$ are analytic, hence there exists small $r_0>0$
such that for each $0<r\leq r_0$ the number of connected components
of $S_r'$ equals the number of connected components of $(B_r^n\setminus\{0\})\cap\{f\geq 0\}$,
and the the number of connected components of $S_r^{n-1}\cap\{g\geq 0\}$ equals
the number of connected components of $(B_r^n\setminus\{0\})\cap\{g\geq 0\}$.
As  $g=f\circ \phi$ then
$(B_{r}^n\setminus\{0\})\cap\{g\geq 0\}$ is homeomorphic to
$(\phi(B_{r}^n)\setminus\{0\})\cap\{f\geq 0\}$.

There exist $0<r_3<r_2<r_1<r_0$ such that
$\phi(B_{r_3}^n)\subset
B_{r_2}^n\subset \phi(B_{r_1}^n)
\subset B_{r_0}^n$.

The inclusion $(B_{r_3}^n\setminus\{0\})\cap\{g\geq 0\}\subset (B_{r_1}^n\setminus\{0\})\cap\{g\geq 0\}$
is a homotopy equivalence. Hence inclusions
$$(\phi(B_{r_3}^n)\setminus\{0\})\cap\{f\geq 0\}\subset
 (\phi(B_{r_1}^n)\setminus\{0\})\cap\{f\geq 0\},$$
$$(B_{r_2}^n\setminus\{0\})\cap\{f\geq 0\}
\subset ( B_{r_0}^n\setminus\{0\})\cap\{f\geq 0\}$$
are homotopy equivalencies, and then in particular sets $(B_{r_1}^n\setminus\{0\})\cap\{g\geq 0\}$,
$(\phi(B_{r_1}^n)\setminus\{0\})\cap\{f\geq 0\}$ and
$( B_{r_0}^n\setminus\{0\})\cap\{f\geq 0\}$ have the same number of connected components.

Hence sets $S_r'$ and $S_r^{n-1}\cap\{g\geq 0\}$  have the same number of connected components too. By similar arguments, the sets $S_r$ and $S_r^{n-1}\cap\{g<0\}$
have the same number of connected components too.
By Theorems \ref{plik_8_1}, \ref{theor_2} we get

\begin{theorem}\label{cor_4_1}
Suppose that an analytic function $g:\R^n,0\rightarrow \R,0$ is right-equivalent to $f$.
If $S_r$ has less connected components than $\Omega$ then $T(g)$ is infinite.
If $S_r'$ has less connected components than $\Omega'$ then $S(g)$ has a non-empty interior.
\end{theorem}

The next example demonstrates that the assumptions of Theorem \ref{cor_4_1} are significant.

\begin{ex}
Let $f(x,y)=x^3+3xy^2$, so that $S_r'$ and $\Omega'$ are homeomorphic. The same way as in Example \ref{ex_7_4} one can show that the interior
of $S(f)$ is non-empty. The function $g(x,y)=f(\sqrt{3}x,y)=3\sqrt{3}(x^3+xy^2)$ is right-equivalent
to $f$. Applying the polar coordinates one can show that $S(g)$ consists of a single trajectory,
so that its interior is empty.
\end{ex}

In the case where $g$ has an algebraically isolated critical point at the origin, one can compute its Milnor number
$\mu(g)=\dim_{\R}\R[[x_1,\ldots,x_n]]/\langle\partial g\rangle$, where $\langle \partial g\rangle$
is the ideal in $\R[[x_1,\ldots,x_n]]$ generated by $\partial g/\partial x_1,\ldots,\partial g/\partial x_n$ (see \cite{milnor}).

\begin{theorem}\label{ostatnietwierdzenie}
Let $g:\R^n,0\rightarrow\R,0$ be an analytic function having an algebraically isolated critical point
at the origin.
Suppose that $\mu(g)$ is even, and $\theta$ is a quadratic form which can be reduced to the diagonal form $-x_2^2-\cdots-x_n^2$.
Then the interior of $S(g)$ is non-empty.
\end{theorem}

\begin{proof} Applying standard methods of the singularities theory (see \cite{brockerlander})
one can show that $g$ is right-equivalent to $f=x_1^k -x_2^2-\ldots-x_n^2$, where $k=\mu(g)+1$. Then
$S_r'$ is homeomorphic to a closed $(n-2)$-dimensional closed ball and $\Omega'$ consists of two points.
By Theorem \ref{cor_4_1}, the set $S(g)$ has a non-empty interior.
\end{proof}

\begin{ex} Let $g(x,y,z,w)= x^5+z^5+2zw-x^2-y^2-z^2-w^2-2xyz-y^2z^2$. In this case $\mu(g)=4$, and
$\theta$ can be reduced to the diagonal form
$-y^2-z^2-w^2$.  By Theorem \ref{ostatnietwierdzenie}, the set $S(g)$ has a non-empty interior.
\end{ex}

redpajama_set_name: RedPajamaArXiv


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