Scott with Saws on Skates Reveals Why a Paint Sprayer is Always the Better Option!
We DIYers have a love-hate relationship with polyurethane. We love it because it provides a clear, durable finish for our DIY projects, but we hate it because it's so tricky to apply with a paintbrush. Don't stir it too much because you'll create bubbles. Don't wipe your brush on the rim of the can because you'll introduce bubbles into the finish. Don't brush it too much because you'll create bumps. It feels like there are more don'ts than dos.
Did you know there's an easier way to apply polyurethane? The HomeRight Super Finish Max not only effortlessly sprays paint, but it also sprays polyurethane like a professional furniture shop! Spoiler alert... I will NEVER apply polyurethane with a brush EVER again! EVER!
For years I avoided polyurethane for my DIY projects because it was so difficult to apply. Earlier this year I got a HomeRight Super Finish Max to spray paint on an outdoor coffee table. It was so easy to use that I decided to paint some Adirondack chairs. Then I painted a dresser. I even painted the siding of my house.
There was one thing in the instructions that caught my eye. The HomeRight Super Finish Max could also be used to spray polyurethane. What?! The sprayer I just used to spray the siding of my house could also be used to spray a fine finish like poly? I had to give this a try!
The results are in and I'm totally convinced. I will NEVER apply poly with a brush EVER again! The finish on this wall-mounted wine bar is amazing. The poly was quick to apply and the piece looks like it came from a furniture factory!
A quality finish begins by properly prepping the surface of your furniture project. All of the polyurethane in the world won't cover the mistakes of a poorly prepped project. Start by thoroughly sanding the piece, fill any holes or gaps with wood filler and apply a coat of quality wood stain.
While it's drying, polyurethane attracts dust like a moth to a flame. Those moths, errrr I mean dust specks, can create bumps in the finish and will need to be sanded smooth. To reduce the chances of dust ruining your finish it's import to set up a dust-free environment and the best way to do that is with a HomeRight Spray Shelter.
The spray shelter not only protects surrounding objects from overspray but best of all it prevents dust from landing on your furniture project while it's drying. The spray shelter is also great for painting furniture projects.
The HomeRight Super Finish Max sprayer can go from spraying the siding of a house to spraying a fine finish like polyurethane because of the three tips included with the kit. The blue tip sprays stain and polyurethane in a 1″-6″ spray pattern, which is great for furniture projects. So that's the one we want to use. The tip is easy to install with the included wrench.
Spraying just the right amount of polyurethane is critical. The goal is to allow just enough poly to flow through the sprayer to evenly coat the piece and be able to overlap each stroke or pass without spraying too much or too little.
The volume control knob governs the amount of poly allowed to flow through the sprayer. Turning the knob towards the "+" allows more poly to flow and turning towards the "-" allows less poly to flow. Fine tuning to get the perfect amount is sooo easy thanks to the volume control knob on the HomeRight Super Finish Max sprayer.
Just like you would prime bare wood if you were going to paint, we need to "prime" or apply a base coat of polyurethane so we can later apply additional coats of poly. The base coat is simple to make. It's two parts polyurethane thinned or mixed with one part paint thinner. Start by adding the polyurethane to the container.
Then mix with the appropriate amount of thinner.
There are a couple of important things to keep in mind when spraying polyurethane. First, keep the sprayer 10″ to 12″ away from the surface you're spraying. Second, switch the trigger on as you approach the piece, follow through with a long, continuous movement and switch the trigger off when the sprayer is beyond the piece.
A long, continuous movement is important because if you start and stop while spraying the piece you will end up with more poly in some areas and less in others. Applying more poly in some areas could cause runs or drips in your finish. The goal is for an even coat that is uniformly thick over the entire piece.
Another way to ensure a uniformly thick coat is to keep a stiff wrist. Allowing your wrist to bend at the beginning and end of a stroke creates an arc. That arc applies more poly at the middle of the stroke and less at the beginning and end of the stroke. Keeping your wrist stiff will ensure that you spray an even coat that is uniformly thick over the entire piece.
The HomeRight Super Finish Max sprayer has three spray patterns which provides complete coverage for any surface you are spraying.
The horizontal spray pattern is the spray pattern I use the most. It's perfect for spraying surfaces like tabletops and the front and sides of dressers, etc.
The vertical spray pattern is perfect for spraying vertical surfaces like table legs.
The horizontal/vertical spray pattern can be used for both horizontal and vertical surfaces. It's great for spraying inside corners and edges of table tops.
Polyurethane has a strange characteristic. It doesn't like to stick to itself. We need to sand between coats in order to make the new coat stick to the previous coat. Sanding creates fine scratches and those fine scratches create a place for the next coat of poly to "bite" in to.
I use 220 grit sandpaper to lightly sand the poly. Remember, the keyword is lightly. We're just trying to scuff the poly. If you sand too aggressively you could sand right through the poly and even sand off the stain.
Thoroughly remove all of the sanding dust with a Shop-Vac.
How many coats of polyurethane should you spray? I would recommend to spray a base coat and then spray two to three regular coats to ensure a durable finish.
Thanks to the HomeRight Super Finish Max sprayer I will NEVER apply poly with a brush EVER again! Applying polyurethane with the sprayer is quick, easy and the result is nothing less than professional.
For more info about prepping projects, adjusting the volume control knob and more hints and tips for spraying polyurethane please visit Saws on Skates (https://sawsonskates.com/spray-polyurethane/) And please let me know if you have any questions about spraying polyurethane... I'm happy to help!


for complimentary Coffee & Dessert with your movie!
BIFF pre-screens some of the very best award winning films submitted for the film festival.
You can review and vote for inclusion in our upcoming film festival. We will also screen some of the most popular films from prior BIFF festivals along with some of the cinema classics.
Immediately following each BYR screening the filmmakers join us via live screencast for conversation.
You will have the uncommon opportunity to discuss the film with the filmmakers themselves. So join us for this exceptional opportunity to enjoy some of the very best in independent film being produced and meet the filmmakers themselves.
we transition to BIFF Outdoors.


text:
Nationwide Search Group - The source for medical sales and marketing professionals
The Search Process
Meet Our Group
Erick Hinz
Erick Hinz joined Nationwide Search Group in 2012 after two and a half years of medical sales with Lincare where he focused on selling respiratory equipment to hospital systems and family practice clinics in South Central Wisconsin. Erick worked with a variety of divisions in the medical industry including Pulmonology, Cardiology, Neurology, Internal Medicine, and Family Practice. Prior to Lincare, Erick worked for Aerotek for three years, which is one of the top recruiting firms in the country. Erick started off as a recruiter and moved his way to Account Manager where he learned how to seek out candidates with specific skill sets for his clients.
Erick's combination of recruiting and medical sales background gives him the ability to understand what skill sets his clients are looking for and how to properly screen potential candidates for their openings. Because of this and his relentless pursuit of qualified candidates, Erick grows strong, trustworthy relationships with his clients.
Erick received his Bachelor's Degree from The University of Wisconsin La Crosse where he studied Business and Marketing. Erick currently resides in Southeastern Wisconsin where he enjoys playing golf and basketball. He's an avid hunter with his family and friends in Northern Wisconsin and has a great passion for the Packers, Brewers and North Carolina basketball.
11801 W. Silver Spring Drive, Suite 107 | Milwaukee, WI 53225 | P: 414-258-2409
All Rights Reserved 2019, Nationwide Search Group - Admin Login
Web Site Development by Alt Media Studios

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: On macOS how do you make an smb volume mounted at boot I have a Mac running Monterey connected to a network that has a NAS (Synology, DSM 7.0.1-42218) that has smb shares.
I want my Mac to mount that smb share so that /Volumes/<smb share name> is always always always available as soon as possible during the macOS startup process, and never times out or unmounts for any reason, even in the case of disk failure or network failure.
I do not want this mount to lazily require a keychain UI interaction, I do not want this mount to lazily time out and is be unmounted after some time.
I want this smb share to be seen as the same thing as a local disk and cause errors in the same scenario that a local disk would.
Is there a way to do this?
There is no /etc/fstab in macOS, or similar config file that I can create.
I can write some apple script to "open" the smb share in Finder upon login, but this is after other stuff has run and is too late.
On macOS (Montery and later) how do I automount an smb volume that is on my network at boot time and keep it mounted forever?

redpajama_set_name: RedPajamaStackExchange


Nice and simple - before the first coffee of the day.
Stuck with a box and a block. Found the key but can't get it. No clues. Now what?
This is the weirdest escape-the-room game I've played as I can't even find a door and I can't find a clue. The clock might be one. I'll keep trying.
Well...I have no idea what to do. Any chance someone has a link to a walkthrough?
It's all about the clock on the wall. Click it, to get an idea what time it shows (on a 24hrs scale).
Take a closer look at the block, too.
Nice little escape game, Bart! Thanks for sharing it with us!
Yes! finally! wouldn't have gotten it if it weren't for you guys though.
Thx a lot for sharing, Bart! I really enjoyed playing it!
You need to put the look in the drawers to find a slip of paper and some number codes. Then go back to the clock.
Flipping the world around was certainly a neat touch in this escape. A good find thanks Bart!
Whew, I finally got one! Keep them coming Bart.
Games on no1game won't load if you're using an ad blocker.
Get another lever and put it back on the semicircular thing.
Oh, that was fun! Thanks, Bart!
I can't get the room to flip again after the second side. :( I'm stuck laying on my left!
I've got the key now what do i do with it?


I recently spoke with a young person from abroad who had been studying postgraduate phonetics at an English university. As we chatted, I couldn't stop myself trying to list the characteristics which together made up his strong foreign accent. Quite high on the list was the SQUARE vowel pronounced as eə. Despite his being a phonetician, and despite his having lived for years among native speakers who use the contemporary ɛː, he'd kept the old RP diphthong which I imagine had been taught to him by non-native schoolteachers, and which continues to be transcribed as such in textbooks and dictionaries.
I have to confess that my heart sank as he attributed monophthongal ɛː to "younger people". For many of Radio 4's traditional listeners, attributing a trait to "younger people" is liable to equate it with wearing your pants low enough to expose your rear. And some non-native listeners may have inferred that ɛː is a youthful fashion, not a recommendable standard pronunciation.
People do increasingly use a long monophthong for this vowel, rather than the schwa-tending diphthong implied by the standard symbol. What used to be a local-accent feature has become part of the mainstream. There are millions of English people, however, who still use a diphthong. To produce the distinction in pairs such as shed — shared EFL learners generally find it easier to make the square vowel diphthongal ([eə]) rather than to rely on length alone.
On the first point, my view is that the millions of eə-users in the Estuary area are maybe not the best BrE model for foreign learners. It'll be interesting to hear, over the coming years, what happens to higher-class ɛː and lower-class eə.
On the second point, there must be a lot of experience behind John's argument about EFL learners. But I'm not persuaded. What about aural competence? Do we teach students to produce eə while warning them that what they're more likely to hear from SSB natives is ɛː? Anyway, my students tend to be higher-level learners, and I can't bring myself to teach them a vowel that's too archaic for "younger people" like Jeremy Paxman, and makes foreigners sound foreigner still.
I think there are two aspects, maybe minor, that tend to be missed by those who flatly decla' the diphthong to be dead.
Firstly, the definition of the accent, in particular whether you count speakers of what John Wells called adoptive RP (GB, SSBE…).
Secondly, that – I think – the monophthong that is undoubtedly widespread today often enough consists of two single vowels in terms of intontation, an autodiphthong ©, as it were, which isn't the case with ɔ or o for older ʊə, even when these are long.
I wonder when eəin the mouth of an EFL learner will actually start to sound prole. Probably not that soon, especially if it doesn't come with other low-prestige features.
[eə] is rather like tapped [ɾ]. Depending on the accompanying features, [ɾ] either sounds like old Noel Coward RP, or like Scouse, or Scottish, or foreign.
Downton Abbey: I was also struck by the fairly modern pronunciations of some of the actresses/actors.
To [ɛɛ] or not to [ɛɛ]. I teach English phonetics to university students whose mother tongue is mostly German and who are in their twenties. I learnt to say /eə/ and /ʊə/, but am aware that I can't impose my 'OAP pronunciation' on them. So I convinced myself to say /ɔː/ in words like 'poor', but haven't managed yet to switch to [ɛɛ] in 'square'. So I feel like a split personality.
I think the centering diphthong pronunciations are very natural to German speakers because it is a common realization of /VV/ + /r/ in many varieties of German.
Your affirmation that "centring diphthongs are less natural and less stable" strikes me since I am speaking a Southern German variety that has featured stable centering diphthongs [iə], [yə] and [uə] for more than thousand years. But I guess you are right.
Thanks for reading and commenting. It's a good point, though I believe most Germans lost those Middle High German diphthongs. Who knows, maybe somewhere there's a forgotten enclave of the British Empire whose centring diphthongs will last a thousand years.
Prof. John Harris, who knows far more than I do about phonological variation and change, tells me he's skeptical about anything being very stable in Germanic long-vowel systems (including English). However, it is true that very natural systems like Spanish vowels are amazingly stable through the centuries and across the continents. And, in the grand scheme of Southern British English, it does seem that the centring diphthongs are/were just a "halfway house" in the loss of final r and l.
Mach's point about the realization of /r/ in German is a good one, though. The realization as a centering diphtong is the standard one even in middle Germany even on a modern theatre stage. For instance: http://www.schauspielfrankfurt.de/schwarzer_bereich/stuecke.php?SID=1000105 (bottom of the page). The older realization as [r] even after a vowel, though still recommended in one of the standard text books on stage speaking ("Der kleine Hey"), sounds very artificial today.
While I do recognize 'square' with [ɛɛ], I'd have to force myself consciously to not realize it as [eə]. I don't see that changing any time soon, since most of my quotidian exposure to English is in talking with other EFL speakers living in the Frankfurt area, whose German isn't good enough to hold a conversation at ease, although they have been living here (and been exposed to German phonology) for years. This is a rather common phenomenon here in Frankfurt, and I'm wondering whether there is such a thing as an "international English accent".
When I attempt to speak German I do my best to reproduce its centring diphthongs; I try not to impose my British monophthongs. I understand that Germans might be tempted to use their schwer vowel when they say English SQUARE, but it's wrong for two reasons in addition to the centring diphthong: 1. the German schwer vowel starts too high; 2. the quantity is wrong – even in old-fashioned RP, SQUARE is not a long vowel + schwa.
And that is in very clear, demonstrative speech. I have a hunch that syllables like those may sometimes be monophthongs in conversational German. What do you think?
Regarding the international pronunciation of English, there is a huge literature on English as an Intenational Language and English as Lingua Franca. Jennifer Jenkins is well known for proposing a "lingua franca core" of features – which would include rhoticity rather than centring diphthongs. I certainly don't believe that "Anglo-Saxon" native accents are superior; my posts are just to provide native-speaker data (and some analysis) for those who are interested.
I can't check my references right now. If I speak "Wort" in my "stage voice", I produce it as [vɔʁt] (or maybe as [vɔχt], I'm not quite sure). Curiously, though, "führt" very clearly becomes [fʏɐt]. Not sure, what I'd be doing in informal speech in either case. A monophthong in "Wort" would strike me as highly regional and non-standard. However, I doubt that the first (non-accented) syllable in "verstehen" would be anything but [fɐ], except in the slowest, most formal speech.
What do you think about Lipman's point about the vowel in "square" being an "autodiphthong"? Personally, I'd love to adopt a standard English accent, but I have a suspicion that my Korean, Finnish, French, Romanian, Italian etc. friends wouldn't understand me, if I produced it like the [ɛː] in German "Käse".
I'm rather more interested in what you hear people actually saying than in what the references say!
It isn't really "curious" if you have clear diphthongs in words like führt and schwer but more robust uvularity in words like Wort. The former have front vowels, and it's more of a stretch to get from a front vowel articulation to a uvular articulation; and syllable-final position is a weak position, so it's not surprising that the consonant lenites to a vowel. In a word like Wort, the vowel is back, much closer to the uvular position, so the consonant is under less pressure to weaken.
The regionality of the monophthongal pronunciations doesn't mean they don't exist and aren't interesting. It's often the case that non-standard variants point to the future of the standard dialect.
I can't comment on autodiphthongs until I hear some phonetic and/or phonological arguments in favour of them.
Regarding your last point, I presume that your international friends find old native speakers easier to understand than less old ones. Do you think London Underground should change its train announcements from Leicester Squ[ɛɛ], Edgw[ɛɛ] etc. to the old-fashioned pronunciations for the benefit of foreign tourists?
Regarding your last point: of course not! That's not the point. Rather, I'm trying to understand the phenomenon with a view on how I, as an EFL user, should try to cope with it. If I aim for [eə], this becomes in many circumstances involuntarily something like [ɛɐ]. But if I would aim for [ɛː], that would for me be the vowel in "Käse"; and if there should happen to be a subtle yet important difference in intonation, it would be more difficult to reproduce and would inevitably get lost in many speaking situation. Meaning, I might exchange something that is known, understood and more or less tolerated as a German accent with something yet unheard.
With regard to uvularization, shouldn't what you write also apply to /ɪr/ and /ʏr/? If I try "Gürtel" and "Wirt" — again, as if speaking on a stage –, I produce them both with an uvularization similar to "Wort". dwds.de agrees with me on the latter: http://www.dwds.de/?qu=Wirt (link to audio in the top left box). I guess, in informal speech, I'd expect all of them, "Wort", "Wirt" and "Gürtel", to have a diphthong.
I don't disagree with what you write on dialects.
I was only joking about the underground! What sort of subtle but important problem do you anticipate from using the Käse vowel in SQUARE? The only thing that occurs to me is that you might be less likely to use r-liaison (e.g. square inch) than if you have the Berg vowel as your model. As for something "yet unheard", why not be bold and give it a try! As for differences between führt and Wirt (if I understand right), I'm well out of my area and perhaps shouldn't venture any more suggestions!
My native language is Spanish and something similar happens to me. I pick /ɔː/ or /ʊə/ depending on the word (saying /pɔː/ and /pjʊə/ according to what modern dictionaries depict as preferable today) but I struggle to incorporate [ɛː] and erradicate [eə] because the same "modern" dictionaries are reluctant to keep up to date in this respect. Wells says that changes that can be inferred as a rule don't need to be shown in pronunciation dictionaries, but I disagree. It's difficult to be constantly reminding yourself that one symbol stands for another one. And in the second edition of his dictionary (LPD, copyright 2000, the one I've used since 2002) he didn't even include a note in the introduction mentioning [ɛː]. So it's no wonder I speak like an 80-year-old though I'm 28!
No, for the time being, the diphthongs are stable in German. The "zwerch-" above clearly has a diphthong, and the "wort" has a uvular fricative, which isn't the default pronunciation. Monophthongs are strongly associated with certain Hessian dialects.
1. Many thanks for confirming that truly monophthongal pronunciations exist – exactly what I expected. I'd be really grateful for a reference/link to some audio data for the progressive Hessian dialects you mention.
2. My point about Zwerch– is that the diphthong is slight, and would be a better model for SQUARE than a really clear diphthong as in schwer.
3. There is certainly no phonetic uvular fricative in that pronunciation of Wort. Maybe there is some uvularization which is hard to distinguish from the vowel; and the speaker seems to be using expressive breathiness. The reason I included the extract of the word's ending is because it sounds very like [ɔt], and certainly not like [ɔχt].
3. Here's a Hessian comedian. I only watched the first part, but there are examples everywhere. The striking feature is that there are no ex-r diphthongs nor compensatory lengthenings, even for those where the first part is further away from schwa etc., eg kurz [kʊts]. There are other dialects (some High German dialects on Low German ground) that have long monophthongs for some of the older diphthongs, but in non-regional colloquial German, they seem to be pretty stable.
Many thanks for the reference – sorry, but I think my spam blocker has deleted a link.
Not sure what you mean by "the second part". What I said about the syllables Zwerch– and Wort was that they "aren't very diphthongal", they "don't seem very rhotic", and they "don't seem to end in [ɐ]". All three points (depending on your interpretation of the word very!) are true, although the second point is weak because uvular approximation doesn't stand out much in the context of back vowels. The two syllables' endings are phonetically very different, but of course one can analyze them as phonologically ending in the same thing: I'm certainly not claiming (I couldn't, as a low-level amateur in German) that this Wort actually rhymes with Spott. I don't know whether trained phoneticians with zero knowledge of German would transcribe this utterance as [vɔʁt]. If I were to read [vɔʁt] as a non-linguistic "nonsense dictation" for students, I'd make the [ʁ] stronger and clearer.
Just for interest, here's a 0-6kHz spectrogram of the Zwerch– and Wort tokens. The diphthongal nature of the former is definite but small, mainly a downshift in F2 halfway through the vowel. This is followed by a big palatal fricative, heavily coloured by the preceding vowel's formants. In Wort, the F1 and F2 region is roughly constant in frequency up to the [t], hence the heavy auditory [ɔ] colour (not [ɐ]) right up to the [t]. The vowel seems noisy throughout, and becomes noisier as it progresses. Just before the [t] there's a brief increase in noise which I've highlighted with a dotted blue box, but it doesn't correspond to much that's audible (to me) and is nothing like the big [ç] in Zwerch-. Any r's here are weaker than the strong r's of rot and drohen and vowel-coloured by their "preceding" vowels. This is exactly the kind of thing which could lead (and apparently already has in some dialects) to true monophthongs.
Sorry for the confusion of the item numbers. By "second part" I meant the second part of the diphthongs in zwerch- and schwer in general, or in the pronunciation you'll most likely meet in a German EFL learner. The first part is different, but the second part is more important when it's about teaching the SQUARE vowel in English, isn't it?
What I hear in the sample is voiced (ʁ), that might be why the spectrogram is less clear.
Something puzzles me about Upton's views on this. In the chapter in Urban Voices that he wrote with Widdowson and Stoddart, only ɛə is given for Sheffield's SQUARE vowel. ɛ: is not mentioned. It strikes me as implausible that [ɛ:] had not become established in Sheffield by the '90s. [ɛ:] is originally the Midlands pronunciation. Petyt (based on fieldwork done in 1970-1) found that it was spreading to West Yorkshire. It must have gone through Sheffield in the process.
My best guess is that Widdowson and Stoddart disagreed with Upton on how to represent the SQUARE vowel of RP, and they won by 2-1 majority.
Your best guess could be true. This is a good example of how fortunate we are today to have multimedia. Back in the 90s we were more dependent on printed symbols, which are merely an imperfect tool in capturing the reality of sound systems. The result is that you can't be sure from that chapter whether the [ɛə] was real or the baggage of conservative R.P. But it shouldn't be too hard to find real recordings of 90s Sheffield. Or email Clive.
My comments were too lengthy to give here.
I would never cut out the end of a diphthong to falsify the data. The real audio data is by far the most important thing about this blog.
Here are a couple of adverts I caught on the telly. One is for the Belgravia Centre, which specializes in hair loss and scalp care, the other is for BUPA care homes for the elderly. My students can choose whichever pronunciation they find more suitable as a model.
Your blog has been extremely useful! I'm preparing (with /ɛː/) a Phonetics I final, and I've had to update some of my knowledge, especially these monopthongisations and the lowering of /æ/. You manage to make phonetics really interesting. Thanks!
Thanks for your kind words, Victoria – much appreciated – and good luck for your final.


text:
News : 11/23/2021 ~ The Abominable Snowmen to be Animated
Thread starter Kierra
Kierra
The BBC has announced that an animated version of the 1967 story The Abominable Snowmen will be released next year.
The story stars Patrick Troughton, Jack Watling, Frazer Hines and Deborah Watling. Originally shown between 30th September 1967 - Saturday 4th November 1967. it was the second story of Season 5 and features the Doctor's first encounter with the Yeti and the Great Intelligence. Most of the six-part story is missing from the BBC Archives, with only episode two still existing.
The release gives fans the opportunity to enjoy the six new fully animated presentations of this lost story, which will be released alongside the one surviving segment episode two.
https://www.doctorwho.tv/news/?article=missing-adventure-the-abominable-snowmen-animation


text:
Rhynchosia buettneri
Rhynchosia buettneri is a sprawling, twining undershrub with cylindrical, smooth branches around 25mm in diameter[
African Flowering Plants Database
http://www.ville-ge.ch/musinfo/bd/cjb/africa/recherche.php
Conservatoire et Jardin Botaniques.
Contains information on over 150,000 plant names (including synonyms) giving a description and habitat, plus a distribution map.
The Useful Plants of West Tropical Africa.
Burkil. H. M.
Brief descriptions and details of the uses of over 4,000 plants. A superb, if terse, resource, it is also available electronically on the Web - see http://www.aluka.org/
The plant is harvested from the wild for local use as a medicine. An ornamental species, it is worth cultivating for its long racemes of reddish or purplish flowers, which turn dark red-brown in drying[
Tropical Africa - Senegal to Nigeria, southern Central African Republic and northern DR Congo
Savanna; forest gallery with fine gravelly soil; colluvium in thalweg; karstic hollow[
Habit Climber
The leaves are used in the treatment of stomach troubles[
Agroforestry Uses:
We have seen no specific entry for this species, but all members of this genus have at least some merit for use as a ground cover and in local soil conservation projects[
The Leguminosae; A Source Book of Characteristics, Uses and Nodulation
Allen O.N.; Allen E.K.
An amazing and comprehensive work, giving a brief guide to the many genera of the family Fabaceae and also the principle uses of the genus.
Like many species within the family Fabaceae, once they have ripened and dried the seeds of this species may benefit from scarification before sowing in order to speed up and improve germination. This can usually be done by pouring a small amount of nearly boiling water on the seeds (being careful not to cook them!) and then soaking them for 12 - 24 hours in warm water. By this time they should have imbibed moisture and swollen - if they have not, then carefully make a nick in the seedcoat (being careful not to damage the embryo) and soak for a further 12 hours before sowing[
Cite as: Tropical Plants Database, Ken Fern. tropical.theferns.info. 2023-01-31. <tropical.theferns.info/viewtropical.php?id=Rhynchosia+buettneri>


This chartreuse silk chiffon scarf is just waiting for the perfect black outfit to compliment! It is made from silk chiffon and natural merino wool fibers using the process of wet felting. It was finished with lots of silk accents, beautiful hand-dyed wool locks and also includes the hand beaded flower accessory pictured.
Approximate size of garment is: 18" x 67"
Process photos are also available upon request.


As part of their centenary celebrations, the Botanical Society and the Kirstenbosch National Botanical Gardens held a silver tree planting ceremony on 24 July. The MCSA was honoured to have been invited and was represented by several Cape Town Section members. The MCSA extends congratulations to BotSoc and Kirstenbosch on their centenaries.
The Silver Tree Restoration Project aims to secure the future of the endangered Silver Tree (Leucadendron argenteum) as an icon of biodiversity conservation on Table Mountain. Thousands of saplings now line the newly dubbed 'Silver Tree Avenue' from Kirstenbosch National Botanical Garden's Rycroft Gate to the historic Cork Oak planting where Southern Cross Drive meets Rhodes Drive. These saplings will eventually create a beautifully shaded walkway and ultimately extend as far as Constantia Nek.
From the Natal Witness 29 July: Mount Amery has been officially identified as the summit terminus for the proposed Drakensberg cable way. A business plan will be created followed by an environmental impact assessment (EIA). The land is owned by the Ingonyama Trust and is mainly utilized by hikers based at the Amangwane Mnweni Hiking and Cultural Centre. According to Graham Muller of Graham Muller and Associates, the cable way will greatly increase tourism to the area. The cost of the construction is estimated at being between R420 - 500 million.
The Youth Leader Training Course scheduled for 25-31 August has been cancelled.
2012 Journal: Members can expect an early Journal, compliments of Journal Editor, Jenny Paterson.
2013 Journal: Any member who would be interested in being the Editor of the 2013 Journal, must please contact the MCSA Secretary (secretary@mcsa.org.za). The next Editor will be appointed in November.
Stellenbosch Section will be hosting the 2013 Joint Meet of the Western Cape Sections on Sunday 17 November. It will be a day walk in Jonkershoek: Up with Swartboskam to Klipspringervlakte and on to Pic-sans-Nom; down with Swartboskloof. Average fitness is required and there will be some scrambling. Come and enjoy the beauty of the place, the indigenous forest and the views over the Jonkershoek Valley, the Helderberg and the Cape Peninsula. Meet at 8:00 at the Jonkershoek gate. Remember to bring your Wild Card. Lea: Peter-Jan Randewijk (pjrandewijk@gmail.com, 021 883 8592 (h)).
The meet will take place on the day following the Annual Dinner also to be hosted by Stellenbosch Section at Clos Malverne Wine Estate in Stellenbosch: Saturday 16 November at 19h. Details on: http://goo.gl/OqHZR . Bookings: Paul Verhoeven (treasurer@ste.mcsa.org.za; 021 887 4113). Combine the two events and overnight in Stellenbosch! One option is Elsbeth's guesthouse (Ask Paul for more info).
Sondag 17 November - Daguitstappie in Jonkershoek: Op met Swartboskam na Klipspringervlakte na Pic-Sans-Nom en af met Swartboskloof.
Gemiddelde fiksheid stap en klouter met mooi natuurskoon. Inheemse woud en uitsigte oor Jonkershoekvallei, Helderberg en Kaapse skiereiland.
Ontmoet 08h00 by die Jonkershoekhek. Onthou wildcards. Stapleier Peter-Jan Randewijk (pjrandewijk@gmail.com, 021 883 8592 (h)).
MCSA members are invited to bring any national matters of concern to the attention of Petro Grobler (petrogrobler@telkomsa.net; president@mcsa.org.za). It would be good though first to discuss your concerns with your Section committee.
Any newsworthy items for inclusion in MCSA National News can be sent to the Editor, Bridget Carrick (bridget.carrick7@gmail.com).
For years mountaineers have been ridiculed for their beardy weirdy image, but now it seems that facial hair may actually enhance performance at high altitudes. Scientists from Wisconsin Institute of Physiological Performance Science conducted a high altitude field study which appears to show that growing a beard actually improves oxygen uptake at high altitudes.
Climbers on last year's Save The Yeti Everest expedition were kitted out with advanced, but lightweight respiratory monitoring equipment which they used, climbing without supplementary oxygen, at all altitudes from base-camp to the summit.
Some 50% of the climbers were clean shaven, with the others encouraged to grow a beard. The results were remarkable. At base-camp, ORI – Oxygen Respiratory Index – was comparable for both groups. By 7,500 metres, the bearded sample showed, on average, a 7.3% rise in ORI compared to the clean-shaven mountaineers, by 8,000 metres, the gap had grown to 10.7% and on the summit, bearded climbers were processing oxygen almost 15% more efficiently/. This could mean the difference between life and death in Everest's harsh climate. The longer the beard growth, the more effective it seemed to be.
9.5) The Korean climbing community and the UIAA Ice Climbing World Tour lost one of its most passionate voices with the sudden death of Kim Jaewook. He was a pioneer in ice climbing and the development of ice climbing competition in Korea, and a member of the UIAA Management Committee. The World Cup stop in Cheongsong, Korea launches the UIAA Ice Climbing World Tour, and which Jaewook was involved in organizing.
50 years ago a South African expedition of MCSA members successfully made 1st ascents of a number of peaks in the Andes. The climbers were: Don Good, Malcolm Griffin, Bob Reinecke, Harold Hill, Colas Coetzee, Robin Sandell.


Running rounding moonsault splash, at times from the second rope; Running or rebound clothesline, at times to an oncoming opponent; Bodyscissors, Enzuigiri, Fireman's carry slam, Handspring back elbow smash to a cornered opponent, Handstand corner foot choke, Handstand elbow drop, Kneeling jawbreaker, Low dropkick, Modified bow and arrow stretch.
Dana Brooke studied at Kent State University; graduating in 2011 with a major in fashion, merchandising & design, and a minor in business administration.
Brooke was a gymnast for about 18 years, however, several injuries—including ones in which she broke both her ankles—led her to transition from gymnastics to fitness competition & bodybuilding.
55(w)/136(l)/2(d) out of 193 matches.
"Look But Don't Touch" by Alex Jacke, Glynn Toliver, J Henry, Khaila Locke, Richard Stuckey and Sedrick Bolden, "Respectful" by CFO$ (NXT/WWE), "Chemical Mind" by Jason Davis (NXT; used while accompanying Emma), "Real Deal" by CFO$ (NXT/WWE; used while accompanying Emma), "Recognition" by CFO$ (WWE; used while teaming with and accompanying Charlotte), "Making Moves" by Jim Johnston (WWE; used as a member of Titus Worldwide).
In the years 2011 & 2012, Brooke won several championships with the NPC (National Physique Committee); earning a IFBB (International Federation of Bodybuilding & Fitness) pro card in '12.
Brooke partook in the European Arnold Classic in 2013, as well as the Arnold Sports Festival in 2015. Brooke landed the 5th spot in the 2017 Arnold Classic event's international category.
Ashley Sebera signed with the WWE in 2013, and was set to perform for the WWE's NXT brand as Dana Brooke.
Sebera didn't possess prior professional wrestling experience, however, brought her bodybuilding credentials and gymnastics background into the WWE.
Dana Brooke made her WWE television debut on the NXT brand at NXT TakeOver: Fatal 4-Way in 2014; appearing in a backstage segment with Tyler Breeze.
Brooke's first televised NXT Match came in 2015; which saw her defeat Blue Pants. She joined forces with Emma and continued to perform in alliance with the latter.
Brooke competed against Asuka at NXT TakeOver: Respect in October of 2015, in what was her last televised match. In March of 2016, Brooke managed Emma in the latter's match against Asuka—which in turn would be Brooke's last NXT appearance.
In May of 2016, Brooke attacked Becky Lynch, and realigned herself with Emma on SmackDown Live. An injury to Emma later that month, marked the end of the Brooke-Emma alliance.
On May 22nd, Brooke—dressed up as Ric Flair—came out to the ring during Charlotte Flair's match against Natalya. Brooke helped Charlotte retain her WWE Women's Championship. The segment marked Brooke's first WWE PPV appearance, as she aligned herself with Charlotte at the Extreme Rules PPV.
In March of 2017, Brooke turned face, and attacked Charlotte Flair on RAW; thereby ending their alliance.
In November of 2017, Brooke was defeated by Asuka in 3 seconds, whereby the former served as one half of the WWE record for the fastest submission finish in a Women's Match.
After a few months of teasing potentially becoming a part of Titus Worldwide—starting in July of 2017—Brooke finally joined the faction in a managerial role in November.
Brooke continues to perform as an in-ring competitor as well; having participated in the inaugural Women's Royal Rumble Match at the Royal Rumble PPV in January of 2018—eliminating Kairi Sane, before being tossed over the top rope and eliminated from the match by Torrie Wilson.


We may use a third party ad company to display advertisements on our site. These ads may contain cookies and are collected by the ad company, and we do not have access to this information.
We may also log IP addresses for the purposes of systems administration.
By using the website, you signify your acceptance of this Privacy Statement. If you are not comfortable with any policy described in this Privacy Statement, you may feel free to e-mail us with questions or concerns, but your only remedy is to discontinue use of the website. We reserve the right to modify this Privacy Statement at any time.


KATHLEEN SNOW - TWO GRIZZLY TITLES — Shakespeare & Co.
Wednesday, July 20th: Kathleen Snow reads from her popular nonfiction book Taken By Bear in Yellowstone: More Than a Century of Harrowing Encounters Between Grizzlies and Humans, and introduces her new mystery novel, Searching for Bear Eyes: A Yellowstone Park Mystery. We are lucky to have Kathleen sharing with us some Yellowstone artifacts! 7pm.
TAKEN BY BEAR IN YELLOWSTONE: More than a century of harrowing encounters between grizzlies and humans. Humans and grizzly bears have been coming into contact in Yellowstone National Park ever since it was founded in 1872. Most of these encounters have ended peacefully, but many have not. In order to most accurately tell the stories of those involved in the more deadly incidents, Kathleen Snow went directly to the source: the National Park Service archives. With help from personnel at park headquarters, Snow has collected more than 100 years' worth of hair-raising stories that read like crime scene investigations and provide hard-learned lessons in outdoor safety. A must-read for fans of Death in Yellowstone and anyone fascinated by human-animal interactions.
SEARCHING FOR BEAR EYES: A Yellowstone Park Mystery. Did a grizzly bear kill eElody Applegate? Or was it something else stalking the remote edges of Yellowstone Park? This mystery is narrated by the enigmatic X. Wenn, Yellowstone's Chief Criminal Investigator. A gifted tracker of bear and man, it's up to him to piece together the story of what may be a mauling by Wild Eye, one of the park's most fabled grizzlies, or the perfect murder. As X. inches toward a reality that he cannot accept, slivers of his own past come to the surface-- lost love, shadows of an abusive childhood, estrangement from his son. He assembles the evidence with meticulous precision, but can we trust him to face the truth?
Kathleen Snow is a member of the Montana Writers Guild and Outdoor Writers Association of America. Her nonfiction has appeared in Harper's Magazine, Women in Natural Resources, and other periodicals. Her fiction includes the new mystery, Searching for Bear Eyes, out from the University of Montana Press.


Our client is a creative agency who had been smashing various creative awards in the recent years. The agency and the Creative Team of the agency are ranked at top tier. They are now looking for a Senior English Copywriter to join them to keep the momentum and to further explore English-focus creative briefs and strategies.


text:
Buddhist master to be mummified
By Baldev Chauhan, BBC News, Jan 11, 2006
Tibetan Buddhists are against preserving bodies
Shimla, India -- Buddhist monks in the northern Indian state of Himachal Pradesh have decided to preserve the body of their master so he can be worshipped forever.
<< Buddhist monks in Dharamsala
Disciples of Yogi Amtrin, who died on 1 July last year aged 84, plan to mummify his body, breaking with tradition.
Tibetan Buddhists oppose preserving the dead, but his followers want to continue to be able to pay respects. Yogi Amtrin, who belonged to the minority Drupka Kargu sect, came to India from Tibet nearly 45 years ago.
Box of salt
He spent years teaching yoga and meditation at the Khampagar monastery at Tashijong in the Kangra valley - 260km (160 miles) away from the state capital, Shimla.
It will be placed in a glass chamber so that all his followers can pay their reverence to the great master as they did when he was alive
Monk Nawang Negi
Khampagar is considered to be the oldest Tibetan Buddhist monastery in India.
Since his death, the monk's body has been placed in a wooden box in a lotus sitting posture as if he is meditating.
"Yes we are preserving the body of Yogi Amtrin. Currently the body lies in a wooden box packed with salt which is changed every two months," said Nawang Negi, a senior monk at the monastery.
Salt removes the moisture from the body, preventing decay. Disciples say the mummification process will take a year.
"We cannot tell the exact time when the body will be ready. After it is completely dry it will be finally placed in a glass chamber so that all his followers can pay their reverence to the great master as they did when he was alive," Nawang Negi said.
Asked if the Indian authorities might object, Mr Negi said there were already three such preserved bodies in Himachal Pradesh, including one in Dharamsala.
Apart from Tibetans, Amtrin had many foreigners among his followers.
The monastery is about 50km from Dharamsala, the seat of the Tibetan government-in-exile led by spiritual leader the Dalai Lama.
In 1959, the first wave of Tibetan refugees fled to India from Tibet with the Dalai Lama after the Chinese occupation.


text:
\section{Introduction}	 \label{S1}
In this paper, we consider an importance sampling problem for a certain rare-event simulations involving the behavior of a diffusion process pertaining to a chain of distributed systems with random perturbations. We also assume that the distributed system formed by $n$-subsystems -- in which a small random perturbation enters only in the first subsystem and then subsequently transmitted to the other subsystems -- satisfies an appropriate H\"{o}rmander condition and, as a result of this, the corresponding diffusion process is assumed to have a transition probability density with strong Feller property (e.g., see \cite{ReyL00}, \cite{Soi00}, \cite{BefA15} and \cite{EckPR99} for related discussions from the physical point of view). Here we provide an efficient importance sampling estimator, with an exponential variance decay rate, for the asymptotics of the probabilities of the rare-event simulations involving the behavior of such a diffusion process which also ensures a minimum relative estimation error in the small noise limit case. The framework for such an analysis basically relies on the connection between the probability theory of large deviations (i.e., the asymptotics for exit probabilities from a given bounded open domain) and that of the values functions for a family of stochastic control problems associated with the underlying distributed system, where such a connection also provides a desirable computational paradigm -- based on an exponentially-tilted biasing distribution -- for constructing an efficient importance sampling estimator for rare-event simulations in the context of such a diffusion process. Moreover, as a by-product, the framework also allows us to derive a family of Hamilton-Jacobi-Bellman for which we also provide a solvability condition for the corresponding optimal control problem.

Here, it is worth mentioning that a number of interesting studies based on various approximations techniques from the theory of large deviations have provided a framework for constructing efficient importance sampling estimators for rare-event simulation problems involving the behavior of diffusion processes (e.g., \cite{BudD19}, \cite{DupE97}, \cite{DupW04}, \cite{RenV04} and \cite{VanW12} for additional discussions). The main approach followed in these studies is to construct exponentially-tilted biasing distributions, which was originally introduced for proving Cram\'{e}r's theorem and its extension, and later on it was found to be an efficient importance sampling distribution for certain problems with various approximations involving rare-events (e.g., see \cite{Sie76}, \cite{AsmG10}, \cite{Gar77} or \cite{DupE97} for detailed discussions). Note that the rationale behind our framework follows in some sense the settings of these papers. However, to our knowledge, the problem of rare-event simulations involving the behavior of a diffusion process pertaining to a chain of distributed systems has not been addressed in the context of stochastic control arguments along with asymptotic analysis in the small noise limit; and it is important because it provides a framework for constructing an efficient importance sampling estimator for rare-event simulations in the context of such a diffusion process.

This paper is organized as follows. In Section~\ref{S2}, we discuss our problem formulation, where we start by considering a diffusion process pertaining to a distributed systems with small random perturbations and we formalize our problem formulation. In this section, we also provide an asymptotic estimate on the exit probability using the Ventcel-Freidlin theorem \cite{VenFre70} (see also \cite[Chapter~4]{FreWe84}) and the stochastic control arguments from Fleming \cite{Fle78} (see also \cite{FlemRi75}). In particular, as we discuss in detail in Subsection~\ref{S2(2)}, such an asymptotic estimate relies on the interpretation of the exit probability function as a value function for a family of stochastic control problems that can be associated with the underlying distributed control system with small random perturbations. In Section~\ref{S3}, we provide our main result for an efficient importance sampling estimator of the rare-events involving the behavior of such a diffusion process. Finally, Section~\ref{S4} provides some concluding remarks.

\section{Problem setup and preliminary results} \label{S2}
In this section, we consider the following distributed system, which is formed by a chain of $n$ subsystems, with small random perturbations, i.e.,
\begin{eqnarray}
\left.\begin{array}{l}
d x_t^{\epsilon,1} = f_1(t, x_t^{\epsilon,1}, x_t^{\epsilon,2}, \ldots, x_t^{\epsilon,n}) dt + \sqrt{\epsilon} \, \sigma(t, x_t^{\epsilon,1}, x_t^{\epsilon,2}, \ldots, x_t^{\epsilon,n})dW_t \\
d x_t^{\epsilon,2} = f_2(t, x_t^{\epsilon,1}, x_t^{\epsilon,2}, \ldots, x_t^{\epsilon,n}) dt  \\
d x_t^{\epsilon,3} = f_3(t, x_t^{\epsilon,2}, x_t^{\epsilon,3}, \ldots, x_t^{\epsilon,n}) dt  \\
 \quad\quad\quad~~ \vdots  \\
d x_t^{\epsilon,n} = f_n(t, x_t^{\epsilon,n-1}, x_t^{\epsilon,n}) dt, ~\,\text{with} ~\, (x_s^{\epsilon,1}, x_s^{\epsilon,2}, \ldots, x_s^{\epsilon,n}), ~\, 0 \le s \le t \le T,
\end{array}\right. \label{Eq1}
\end{eqnarray}
where
\begin{itemize}
\item[-] $x_{\cdot}^{\epsilon,i}$ is an $\mathbb{R}^{d}$-valued diffusion process that corresponds to the $i$th-subsystem,
\item[-] the functions $f_i$ are uniformly Lipschitz, with bounded first derivatives,
\item[-] $\sigma \colon [0, \infty) \times \mathbb{R}^{nd} \rightarrow \mathbb{R}^{d \times d}$ is Lipschitz with the least eigenvalue of $\sigma\,\sigma^T$ uniformly bounded away from zero, i.e.,
\begin{align*}
 \sigma(t, x)\,\sigma^T(t, x) \ge \lambda I_{d \times d} , \quad \forall x \in \mathbb{R}^{nd}, \quad \forall t \ge 0,
\end{align*}
for some $\lambda > 0$,
\item[-] $W_{\cdot}$ (with $W_0 = 0$) is an $d$-dimensional standard Wiener process,
\item[-] $\epsilon$ is a small positive number that represents the level of random perturbation in the distributed system,
\end{itemize}
for $i = 1, 2, \ldots, n$.

Note that, from the physical point of view (e.g., see \cite{ReyL00}, \cite{Soi00}, \cite{BefA15} or \cite{EckPR99}), the functions $f_i$, for $i \ge 3$, cannot be dependent on the position variables $x_t^{1}, \ldots, x_t^{i-1}$ and, as a result of this, the random perturbation enters only in the first subsystem and then it subsequently passes through the second subsystem, the third subsystem, and so on to the $n$th-subsystem. Hence, such a distributed system is described by an $\mathbb{R}^{nd}$-valued diffusion process, which is degenerate in the sense that the backward operator associated with it is a degenerate parabolic equation.

In what follows, let us denote the solution of the stochastic differential equation (SDE) in \eqref{Eq1} by a bold face letter $\bigl(\mathbf{x}_t^{\epsilon}\bigr)_{t \ge 0} = \bigl(x_t^{\epsilon,1}, x_t^{\epsilon,2}, \ldots, x_t^{\epsilon,n}\bigr)_{t \ge 0}$ as an $\mathbb{R}^{nd}$-valued diffusion process and further rewrite the above SDE as follows
\begin{align}
d \mathbf{x}_t^{\epsilon} = \mathbf{f} (t, \mathbf{x}_t^{\epsilon}) + \sqrt{\epsilon}\,\mathbf{b}\,\sigma(t, \mathbf{x}_t^{\epsilon}) \, dW_t,
\end{align}
where $\mathbf{f} = (f_1, f_2, \ldots, f_n)$ is an $\mathbb{R}^{nd}$-valued function, $\mathbf{b}$ is an $nd \times d$ constant matrix that embeds $\mathbb{R}^{d}$ into $\mathbb{R}^{nd}$, i.e., $\mathbf{b} = (I_{d \times d}, 0, \ldots, 0)^T$ and, moreover, for $\mathbf{x} =(x_1,\, x_2, \, \ldots, \, x_n)$, we set $\mathbf{x}^{i,n} =(x_i,\, \ldots, \, x_n)$. Note that the corresponding backward operator for the diffusion process $\mathbf{x}_t^\epsilon$, when applied to a certain function $\upsilon^{\epsilon}(t, \mathbf{x})$, is given by
\begin{eqnarray}
 \partial_t \upsilon^{\epsilon} + \mathcal{L}^{\epsilon} \upsilon \triangleq \frac{\partial \upsilon^{\epsilon}}{\partial t} + \frac{\epsilon}{2} \operatorname{tr}\Bigl \{a\, \upsilon_{x^{1}x^{1}}^{\epsilon} \Bigr\} + \sum\nolimits_{i=1}^n \bigl\langle  f_{i},\, \upsilon_{x^{i}}^{\epsilon} \bigr\rangle, \label{Eq5}
\end{eqnarray}
where $a(t, \mathbf{x})=\sigma(t, \mathbf{x})\,\sigma^T(t, \mathbf{x})$.

Let $\Omega \in \mathbb{R}^{nd}$ be bounded open domains with smooth boundary (i.e., $\partial \Omega$ is a manifold of class $C^2$) and let $\Omega^{T}$ be an open set defined by
\begin{align*}
\Omega^{T} = (0, T) \times \Omega.
\end{align*}
Furthermore, let us denote by $C^{\infty}(\Omega^{T})$ the spaces of infinitely differentiable functions on $\Omega^{T}$ and by $C_0^{\infty}(\Omega^{T})$ the space of the functions $\phi \in C^{\infty}(\Omega^{T})$ with compact support in $\Omega^{T}$. A locally square integrable function $\upsilon^{\epsilon}(t, \mathbf{x})$ on $\Omega^{T}$ is said to be a distribution solution to the following equation
\begin{eqnarray}
 \partial_t \upsilon^{\epsilon} + \mathcal{L}^{\epsilon} \upsilon^{\epsilon} = 0,  \label{Eq6}
\end{eqnarray}
if, for any test function $\phi \in C_0^{\infty}(\Omega^{T})$, the following holds true
\begin{eqnarray}
 \int_{\Omega^{T}} \left(-\frac{\partial \phi} {\partial t} + {\mathcal{L}^{\epsilon}}^{\ast} \phi \right)\upsilon^{\epsilon} d \Omega^{T} = 0, \label{Eq7}
\end{eqnarray}
where $d \Omega^{T}$ denotes the Lebesgue measure on $\mathbb{R}^{nd + 1}$ and ${\mathcal{L}^{\epsilon}}^{\ast}$ is an adjoint operator corresponding to the infinitesimal generator $\mathcal{L}^{\epsilon}$ of the process $\mathbf{x}_t^\epsilon$, i.e.,
\begin{eqnarray}
 {\mathcal{L}^{\epsilon}}^{\ast} \phi = \frac{\epsilon}{2} \sum\nolimits_{j=1}^d \sum\nolimits_{m=1}^d \bigr(a_{j,m} \phi \bigr)_{x_j^{1} x_m^{1}} - \sum\nolimits_{i=1}^n \sum\nolimits_{m=1}^d \bigl(f_{i}\phi\bigr)_{x_m^{i}}.  \label{Eq8}
\end{eqnarray}

Moreover, we also assume that the following statements hold for the distributed system in \eqref{Eq1}.

\begin{assumption} \label{AS1} ~\\\vspace{-3mm}
\begin{enumerate} [(a)]
\item The function $\mathbf{f}$ is a bounded $C^{\infty}((0, \infty) \times \Omega)$-function, with bounded first derivatives. Moreover, $\sigma$ and $\sigma^{-1}$ are bounded $C^{\infty} \bigl((0, \infty) \times \mathbb{R}^{nd}\bigr)$-functions, with bounded first derivatives.
\item The backward operator in \eqref{Eq5} is hypoelliptic in $C^{\infty}((0, \infty) \times \Omega)$ (e.g., see \cite{Hor67} or \cite{Ell73}).
\item Let $n(\mathbf{x})$ be the outer normal vector to $\partial \Omega$ and, further, let $\Gamma^{+}$ and $\Gamma^{0}$ denote the sets of points $(t, \mathbf{x})$, with $\mathbf{x} \in \partial \Omega$, such that
\begin{align*}
\bigl\langle \mathbf{f}(t, \mathbf{x}),\, n(\mathbf{x}) \bigr\rangle
\end{align*}
is positive and zero, respectively.\footnote{Note that
\begin{align*}
&\mathbb{P}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon} \Bigl\{ \bigl( \tau^{\epsilon}, \mathbf{x}_{\tau^{\epsilon}}^{\epsilon}\bigr) \in \Gamma^{+} \bigcup \Gamma^{0},\,\, \tau^{\epsilon} < \infty \Bigr \} =1,  \quad \forall \bigl(s, \mathbf{x}_s^{\epsilon}\bigr) \in \Omega_{0}^{\infty}.
\end{align*}
where $\tau^{\epsilon} = \inf \bigl\{ t > s \, \bigl\vert \, \mathbf{x}_t^{\epsilon} \in \partial \Omega \bigr\}$. Moreover, if
\begin{align*}
&\mathbb{P}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon} \Bigl\{ \bigl(t, \mathbf{x}_t^{\epsilon}\Bigr) \in \Gamma^{0} ~ \text{for some} ~ t \in [s, T] \Bigr \} = 0, \quad \forall \bigl(s, \mathbf{x}_s^{\epsilon}\bigr) \in \Omega_{0}^{\infty},
\end{align*}
and $\tau^{\epsilon} \le T$, then we have $\bigl( \tau^{\epsilon},  \mathbf{x}_{\tau^{\epsilon}}^{\epsilon} \bigr) \in \Gamma^{+}$, almost surely (see \cite[Section~7]{StrVa72}).}
\end{enumerate}
\end{assumption}

\begin{remark}
Here we remark that, in general, the hypoellipticity assumption (i.e., the backward operator in \eqref{Eq5}) is related to a strong accessibility property of controllable nonlinear systems that are driven by white noise (e.g., see \cite{SusJu72} concerning the controllability of nonlinear systems, which is closely related to \cite{StrVa72} and \cite{IchKu74}; see also \cite[Section~3]{Ell73}). That is, the hypoellipticity assumption implies that the diffusion process $\mathbf{x}_t^{\epsilon}$ has a transition probability density with a strong Feller property.
\end{remark}

\subsection{Importance sampling} \label{S2(1)}
In this paper, we are mainly concerned with estimating the following quantity
\begin{eqnarray}
 \mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\Bigl[ \exp\Bigl( -\frac{1} {\epsilon} \Phi^{\epsilon}(\mathbf{x}^{\epsilon}) \Bigr) \Bigr], \label{Eqx.1}
\end{eqnarray}
where $\Phi^{\epsilon}$ is an appropriate functional on $C\bigl([0, T]; \mathbb{R}^{nd}\bigr)$ and $\mathbf{x}^{\epsilon}$ is a solution of the SDE in \eqref{Eq1} and our analysis is in the situation where the level of the random perturbation is small, i.e., $\epsilon \ll 1$, and the functional $\mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon} \bigl[\exp\bigl( -\frac{1} {\epsilon} \Phi^{\epsilon}(\mathbf{x}^{\epsilon}) \bigr)\bigr]$ is rapidly varying in $\mathbf{x}^{\epsilon}$. Note that the challenge presented by such an analysis of rare-event probabilities is well documented (see \cite{Buc04}, \cite{AsmG10} and \cite{BudD19} for additional discussions). In the following (and also in Subsection~\ref{S2(1)} and Section~\ref{S3}), we specifically consider the case when the functional $\Phi^{\epsilon}$ is bounded and nonnegative Lipschitz, with $\Phi^{\epsilon} = 0$, if $\mathbf{x}_t^{\epsilon} \in \Omega^T \subset C([0, T]: \mathbb{R}^{nd})$ and $\Phi^{\epsilon} = \infty$ otherwise; and we further consider analysis on the asymptotic estimates for exit probabilities from a given bounded open domain in the small noise limit case.

Consider the following simple estimator for the quantity of interest in \eqref{Eqx.1}
\begin{eqnarray}
 \rho(\epsilon) = \frac{1}{N} \sum\nolimits_{j=1}^N \exp\Bigl( -\frac{1} {\epsilon} \Phi^{\epsilon}( {\mathbf{x}^{\epsilon}}^{(j)}) \Bigr), \label{Eqx.2}
\end{eqnarray}
where $\Bigl\{ {\mathbf{x}^{\epsilon}}^{(j)} \Bigr\}_{j=1}^N$ are $N$-copies of independent samples of $\mathbf{x}^{\epsilon}$. Here we remark that such an estimator is unbiased in the sense that
\begin{align}
\mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\bigl[\rho(\epsilon)\bigr] = \mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon} \Bigl[ \exp\Bigl( -\frac{1} {\epsilon} \Phi^{\epsilon}(\mathbf{x}^{\epsilon}) \Bigr) \Bigr], \label{Eqx.3}
\end{align}
Moreover, its variance is given by
\begin{align}
\Var\left(\rho(\epsilon)\right) = \frac{1}{N} \left(  \mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon} \Bigl[ \exp\Bigl( -\frac{2} {\epsilon} \Phi^{\epsilon}(\mathbf{x}^{\epsilon}) \Bigr) \Bigr] -  \mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\Bigl[ \exp\Bigl( -\frac{1} {\epsilon} \Phi^{\epsilon}(\mathbf{x}^{\epsilon}) \Bigr) \Bigr]^2 \right ). \label{Eqx.4}
\end{align}
Then, we have the following for the relative estimation error
\begin{align}
\Rerr\left(\rho(\epsilon)\right) = \frac{\sqrt {\Var\left(\rho(\epsilon)\right)}}{\mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\bigl[\rho(\epsilon)\bigr]} \label{Eqx.5}
\end{align}
which can be further rewritten as follows
\begin{align}
\Rerr\left(\rho(\epsilon)\right) = \bigl(1/\sqrt{N}\bigr) \sqrt{\Delta\left(\rho(\epsilon)\right) - 1}, \label{Eqx.6}
\end{align}
where
\begin{align}
\Delta\left(\rho(\epsilon)\right) = \frac{\mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\Bigl[ \exp\Bigl( -\frac{2} {\epsilon} \Phi^{\epsilon}(\mathbf{x}^{\epsilon}) \Bigr) \Bigr]} {\mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\Bigl[ \exp\Bigl( -\frac{1} {\epsilon} \Phi^{\epsilon}(\mathbf{x}^{\epsilon}) \Bigr) \Bigr]^2 }.
\end{align}

Note that, as we might expect,  the relative estimation error may decrease with increasing the number of the sample size $N$. However, from Varahhan's lemma (e.g., see \cite{Var85}; see also \cite{Ven73} and \cite{VenFre70}), under suitable assumptions, we also have the following conditions
\begin{align}
  \limsup_{\epsilon \rightarrow 0} \epsilon\,\log  \mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\Bigl[ \exp\Bigl( -\frac{1} {\epsilon} \Phi^{\epsilon}(\mathbf{x}^{\epsilon}) \Bigr) \Bigr]  = -  \inf_{\substack {\varphi \in C_{sT}\bigl([s, T], \mathbb{R}^{nd}\bigr) \\ \varphi(s) = \mathbf{x}_s}} \Bigl \{I (\varphi)  + \Phi^{\epsilon}(\varphi) \Bigr\} \label{Eqx.7}
\end{align}
and
\begin{align}
  \limsup_{\epsilon \rightarrow 0} \epsilon\,\log  \mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\Bigl[ \exp\Bigl( -\frac{2} {\epsilon} \Phi^{\epsilon}(\mathbf{x}^{\epsilon}) \Bigr) \Bigr]  = -  \inf_{\substack {\varphi \in C_{sT}\bigl([s, T], \mathbb{R}^{nd}\bigr) \\ \varphi(s) = \mathbf{x}_s}} \Bigl \{I (\varphi)  + 2 \Phi^{\epsilon}(\varphi) \Bigr\} \label{Eqx.8}
\end{align}
where $C_{sT}\bigl([s, T], \mathbb{R}^{nd}\bigr)$ is the set of absolutely continuous functions from $[s, T]$ into $\mathbb{R}^{nd}$, with $0 \le s \le t \le T$, and $I(\varphi)$ is the rate functional for the diffusion process $\mathbf{x}_t^{\epsilon}$. From Jensen's inequality, the above equations in \eqref{Eqx.7} and \eqref{Eqx.8} also imply the following condition $\Delta\left(\rho(\epsilon)\right) \ge 1$.

In the following subsection, i.e., Subsection~\ref{S2(2)} (see also Section~\ref{S3}, the framework that we use to address rare-event simulation problem basically relies on the connection between the probability theory of large deviations (i.e., the asymptotics for exit probabilities) and the values functions for a family of stochastic control problems associated with the underlying distributed system. Note that such a connection provides a computational paradigm -- based on an exponentially-tilted biasing distribution -- for constructing an efficient importance sampling estimators for rare-event simulations that further improves the efficiency of Monte Carlo simulations.

\subsection{Exit probabilities and stochastic optimal control} \label{S2(2)}
Let $\mathbf{x}_t^{\epsilon}$, for $0 \le t \le T$, be the diffusion process associated with \eqref{Eq1} and consider the following boundary value problem
\begin{eqnarray}
\left.\begin{array}{c}
\partial_s \upsilon^{\epsilon} + \mathcal{L}^{\epsilon} \upsilon^{\epsilon} = 0 \quad \text{in} \quad \Omega^{T}  \\
 \upsilon^{\epsilon} \bigl(s, \mathbf{x} \bigr) = 1 \quad \text{on} \quad \Gamma_{T}^{+}  \\
 \upsilon^{\epsilon} \bigl(s, \mathbf{x} \bigr) = 0 \quad \text{on} \quad \{T\} \times \Omega
\end{array}\right\}   \label{Eq9}
\end{eqnarray}
where $\mathcal{L}^{\epsilon}$ is the backward operator in \eqref{Eq5} and
\begin{align*}
 \Gamma_{T}^{+} = \Bigl\{\bigl(s, \mathbf{x} \bigr) \in \Gamma^{+}\, \bigl \vert \, 0 < s \le T \Bigr\}.
\end{align*}
Further, let $\Omega^{0T}$ be the set consisting of $\Omega^{T} \bigcup \{T\} \times \Omega$, together with the boundary points $\bigl(s, \mathbf{x} \bigr) \in \Gamma^{+}$, with $0< s<T$. Then, the following proposition, whose proof is given in \cite{BefA15}, provides a solution to the exit probability $\mathbb{P}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon} \bigl\{ \tau^{\epsilon} \le T \bigr\}$ with which the diffusion process $\mathbf{x}_t^{\epsilon}$ exits from the domain $\Omega$.

\begin{proposition} \label{P1}
Suppose that the statements~(a)-(c) in the above assumption (i.e., Assumption~\ref{AS1}) hold true. Then, the exit probability $q^{\epsilon} (s, \mathbf{x}^{\epsilon}) = \mathbb{P}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\bigl\{ \tau^{\epsilon} \le T \bigr\}$ is a smooth solution to the boundary value problem in \eqref{Eq9} and, moreover, it is a continuous function on $\Omega^{0T}$.
\end{proposition}

Note that, from Proposition~\ref{P1}, the exit probability $q^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon}\bigr)$ is a smooth solution to the boundary value problem in \eqref{Eq9}. Further, if we introduce the following logarithmic transformation (e.g., see \cite{Fle78},\cite{EvaIsh85} or \cite{FlemRi75})
\begin{eqnarray}
 I^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon} \bigr) = -\epsilon \log q^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon} \bigr). \label{Eq17}
\end{eqnarray}
Then, using ideas from stochastic control theory (see \cite{Fle78} for similar arguments), we present results useful for proving the following asymptotic property
\begin{align}
 I^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon} \bigr) \rightarrow I^{0} \bigl(s, \mathbf{x}^{\epsilon}\bigr) \quad \text{as} \quad \epsilon \rightarrow 0. \label{Eq27}
\end{align}
The starting poin for such an analysis is to introduce a family of related stochastic control problems whose dynamic programming equation, for $\epsilon > 0$, is given below by \eqref{Eq31}. Then, this also allows us to reinterpret the exit probability function as a value function for a family of stochastic control problems that are associated with the underlying distributed system with small random perturbation.

In what follows, we consider the following boundary value problem
\begin{eqnarray}
\left.\begin{array}{c}
\partial_s g^{\epsilon} + \frac{\epsilon}{2} \operatorname{tr}\Bigl \{a\, g_{x^{1}x^{1}}^{\epsilon} \Bigr\} + \sum\nolimits_{j=1}^n \bigl\langle f_{j},\, g_{x^{j}}^{\epsilon}  \bigr\rangle = 0 \quad \text{in} \quad \Omega^{T} \\
g^{\epsilon} = \mathbb{E}_{s, \mathbf{x}}^{\epsilon} \Bigl \{ \exp\Bigl(-\frac{1} {\epsilon} \Phi^{\epsilon} \Bigr) \Bigr\} \quad \text{on} \quad \partial^{\ast} \Omega^{T}
\end{array}\right\}  \label{Eq28}
\end{eqnarray}
where $\Phi^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon} \bigr)$ is a bounded, nonnegative Lipschitz function such that
\begin{eqnarray}
\Phi^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon} \bigr) =0, \quad \forall \bigl(s, \mathbf{x}^{\epsilon} \bigr) \in \Gamma_{T}^{+}. \label{Eq29}
\end{eqnarray}
Observe that the function $g^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon} \bigr)$ is a smooth solution in $\Omega^{T}$ to the backward operator in \eqref{Eq5}; and it is also continuous on $\partial^{\ast} \Omega^{T}$. Moreover, if we introduce the following logarithm transformation (cf. Equation~\eqref{Eq17})
\begin{eqnarray}
  J^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon} \bigr) = - \epsilon \log g^{\epsilon}\bigl(s, \mathbf{x}^{\epsilon} \bigr). \label{Eq30}
\end{eqnarray}
Then,  $J^{\epsilon}\bigl(s, \mathbf{x}^{\epsilon}\bigr)$ satisfies the following dynamic programming equation
\begin{eqnarray}
\partial_s J^{\epsilon} + \frac{\epsilon}{2} \operatorname{tr}\Bigl \{a\, J_{x^{\epsilon,1} x^{\epsilon,1}}^{\epsilon} \Bigr\} + \sum\nolimits_{j=1}^n \bigl\langle f_{j},\,  J_{x^{\epsilon, j}}^{\epsilon} \bigr\rangle + H^{\epsilon} = 0, \quad  \text{in} \quad  \Omega^{T},  \label{Eq31}
\end{eqnarray}
where $H^{\epsilon} = H^{\epsilon}\bigl(s, \mathbf{x}^{\epsilon}, J_{x^{\epsilon,1}}^{\epsilon} \bigr)$ is given by
\begin{eqnarray}
H^{\epsilon}\bigl(s, \mathbf{x}^{\epsilon}, J_{x^{\epsilon,1}}^{\epsilon} \bigr) = \bigl\langle f_{1}(s, \mathbf{x}^{\epsilon}) ,\,  J_{x^{\epsilon,1}}^{\epsilon} \bigr\rangle - \frac{1}{2} \Bigl(J_{x^{\epsilon,1}}^{\epsilon}\Bigr)^T a(s, \mathbf{x}^{\epsilon}) J_{x^{\epsilon,1}}^{\epsilon}.  \label{Eq32}
\end{eqnarray}
Note that the duality relation between $H^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon},\,\cdot\bigr)$ and $L^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon},\,\cdot\bigr)$, i.e.,
\begin{eqnarray}
H^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon}, J_{x^{\epsilon,1}}^{\epsilon}\bigr) = \inf_{\hat{u}} \biggl \{L^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon}, \hat{u} \bigr) + \bigl\langle J_{x^{\epsilon,1}}^{\epsilon},\, \hat{u} \bigr\rangle \biggr\}, \label{Eq33}
\end{eqnarray}
with
\begin{align*}
L^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon}, \hat{u} \bigr) = \frac{1}{2} \Bigl \Vert f_{1}(s, \mathbf{x}^{\epsilon}) - \hat{u} \Bigr \Vert_{[a(s, \mathbf{x}^{\epsilon})]^{-1}}^2,
\end{align*}
where $\Vert\,\cdot \,\Vert_{[a(s, \mathbf{x}^{\epsilon})]^{-1}}^2$ denotes the Riemannian norm of a tangent vector.

Then, it is easy to see that $J^{\epsilon}\bigl(s, \mathbf{x}^{\epsilon} \bigr)$ is a solution in  $\Omega^{T}$, with $J^{\epsilon}=\Phi^{\epsilon}$ on $\partial^{\ast} \Omega^{T}$, to the dynamic programming in \eqref{Eq31}, where the latter is associated with the following stochastic control problem
\begin{align}
 J^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon}\bigr) = \inf_{\hat{u} \in \hat{U}(s, \mathbf{x}_s^{\epsilon})} \mathbb{E}_{s ,\mathbf{x}_s^{\epsilon}}\Biggl\{ \int_{s}^{\theta}  L^{\epsilon} \bigl(s,\mathbf{x}^{\epsilon}, \hat{u} \bigr)dt + \Phi^{\epsilon} \bigl(\theta, \mathbf{x}^{\epsilon} \bigr) \Biggr\}  \label{Eq34}
\end{align}
that corresponds to the following system of SDEs
\begin{eqnarray}
\left.\begin{array}{l}
d x_t^{\epsilon,1} = \hat{u}(t) dt + \sqrt{\epsilon} \,\sigma \bigl(t, \mathbf{x}^{\epsilon}(t)\bigr) dW_t \\
d x_t^{\epsilon,2} = f_2(t, \mathbf{x}_t^{\epsilon}) dt  \\
d x_t^{\epsilon,3} = f_3(t, \mathbf{x}_t^{\epsilon, 3,n}) dt  \\
 \quad\quad\quad~~ \vdots  \\
d x_t^{\epsilon,n} = f_n(t, \mathbf{x}_t^{\epsilon, n-1,n}) dt \\
~ \\
\quad \quad \text{with an initial condition} \quad \mathbf{x}_s^{\epsilon}
\end{array}\right. \label{Eq35}
\end{eqnarray}
 where $\mathbf{x}_t^{\epsilon, i,n} = (x^{\epsilon, i},\, \ldots, \, x^{\epsilon,n})$, with $i \ge 3$, and $\hat{U}(s, \mathbf{x}^{\epsilon})$ is a class of continuous functions for which $\theta \le T$ and  $(\theta, x_{\theta}^{\epsilon})\in \Gamma_{T}^{+}$.

In what follows, we provide bounds (i.e., the asymptotic lower and upper bounds) on the exit probability $q^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon}\bigr)$.

Define
\begin{eqnarray}
 I_{\Omega}^{\epsilon} \bigl(\bigl(s, \mathbf{x}^{\epsilon}\bigr); \, \partial \Omega \bigr) &=& -\lim_{\epsilon \rightarrow 0} \epsilon\,\log \mathbb{P}_{s, \mathbf{x}_s}^{\epsilon} \bigl \{ \mathbf{x}_{\theta}^{\epsilon} \in \partial \Omega \bigl\}, \notag \\
 & \triangleq& -\lim_{\epsilon \rightarrow 0} \epsilon\,\log q^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon} \bigr), \label{Eq36}
\end{eqnarray}
where $\theta$ (or $\theta=\tau^{\epsilon} \wedge T$) is the first exit-time of $\mathbf{x}_t^{\epsilon}$ from the domain $\Omega$. Furthermore, let us introduce the following supplementary minimization problem
\begin{eqnarray}
\tilde{I}_{\Omega}^{\epsilon} \bigl(s, \varphi, \theta\bigr) = \inf_{\varphi \in C_{sT}\bigl([s, T], \mathbb{R}^d\bigr), \theta \ge s} \int_{s}^{\theta} L^{\epsilon}\big(t, \varphi(t),\dot{\varphi}(t)\big)dt, \label{Eq37}
\end{eqnarray}
where the infimum is taken among all $\varphi(\cdot) \in C_{sT}\bigl([s, T], \mathbb{R}^d\bigr)$ (i.e., from the space of $\mathbb{R}^d$-valued (locally) absolutely continuous functions, with $\int_{s}^T \bigl\vert \dot{\varphi}(t) \bigr\vert^2 dt < \infty$ for each $T > s$) and $\theta \ge s > 0$ such that $\varphi(s)=x_s^{\epsilon,1}$, $\bigl(t, \varphi(t), \mathbf{x}_t^{\epsilon,2, n}\bigr) \in \Omega^{T}$, for all $t \in [s,\,\theta)$, and $\bigl(\theta, \varphi(\theta), \mathbf{x}_{\theta}^{\epsilon,2,n} \bigr) \in \Gamma_{T}^{+}$. Then, it is easy to see that
\begin{eqnarray}
 \tilde{I}_{\Omega}^{\epsilon} \bigl(s, \varphi, \theta\bigr) = I_{\Omega}^{\epsilon} \bigl(\bigl(s, \mathbf{x}^{\epsilon} \bigr); \,\partial \Omega \bigr). \label{Eq38}
\end{eqnarray}

Next, we state the following lemma that will be useful for proving Proposition~\ref{P2} (cf. \cite[Lemma~3.1]{Fle78}).
\begin{lemma} \label{L2}
If $\varphi \in C_{sT}\bigl([s, T], \mathbb{R}^d\bigr)$, for $s> 0$, and $\varphi(s)=x_s^{\epsilon,1}$, $\bigl(t, \varphi(t), \mathbf{x}_t^{\epsilon,2, n} \bigr) \in \Omega^{T}$, for all $t \in [s,T)$, then $\lim_{T \rightarrow \infty} \int_{s}^{T} L^{\epsilon,\ell}\big(t,\varphi(t),\dot{\varphi}(t)\big)dt = +\infty$.
\end{lemma}

Consider again the stochastic control problem in \eqref{Eq34} together with \eqref{Eq35}. Suppose that  $\Phi_M^{\epsilon}$ (with $\Phi_M^{\epsilon} \ge 0$) is class $C^2$ such that $\Phi_M^{\epsilon} \rightarrow +\infty$ as $M \rightarrow \infty$ uniformly on any compact subset of $\Omega^{T} \setminus \bar{\Gamma}_{T}^{+}$ and $\Phi_M^{\epsilon}$ on $\Gamma_{T}^{+}$. Further, if we let $J^{\epsilon} = J_{\Phi_M}^{\epsilon}$, when  $\Phi^{\epsilon} = \Phi_M^{\epsilon}$, then we have the following lemma.
\begin{lemma} \label{L3}
Suppose that Lemma~\ref{L2} holds, then we have
\begin{align}
 \liminf_{\substack{M \rightarrow \infty \\ \bigl(t, \mathbf{x}_t^{\epsilon} \bigr) \rightarrow \bigl(s, \mathbf{x}_s^{\epsilon} \bigr)}} J_{\Phi_M}^{\epsilon}(\bigl(s, \mathbf{x}^{\epsilon} \bigr))  \ge I^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon} \bigr). \label{Eq39}
\end{align}
\end{lemma}
Then, we have the following result.
\begin{proposition} \label{P2}
Suppose that Lemma~\ref{L2} holds, then we have
\begin{eqnarray}
  I^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon} \bigr) \rightarrow I^{0} \bigl(s, \mathbf{x}^{\epsilon} \bigr) \quad \text{as} \quad \epsilon \rightarrow 0, \label{Eq40}
\end{eqnarray}
uniformly for all $\bigl(s, x^{\epsilon,1}(s), x^{\epsilon,2}(s), \ldots, x^{\epsilon,\ell}(s) \bigr)$ in any compact subset $\bar{\Omega}_{\ell}^T$.
\end{proposition}

\begin{proof}
It is suffices to show the following conditions
\begin{eqnarray}
  \limsup_{\epsilon \rightarrow 0} \epsilon\,\log \mathbb{P}_{s, \mathbf{x}_s}^{\epsilon} \Bigl \{ \mathbf{x}_\theta^{\epsilon} \in \partial \Omega \Bigl\} \le -I_{\Omega}^{\epsilon} \bigl(\bigl(s, \mathbf{x}^{\epsilon} \bigr); \, \partial \Omega \bigr) \label{Eq41}
\end{eqnarray}
and
\begin{eqnarray}
 \liminf_{\epsilon \rightarrow 0} \epsilon\,\log \mathbb{P}_{s, x_{s}}^{\epsilon} \Bigl \{ \mathbf{x}_{\theta}^{\epsilon} \in \partial \Omega \Bigl\} \ge -I_{\Omega}^{\epsilon} \bigl(\bigl(s, \mathbf{x}^{\epsilon} \bigr); \, \partial \Omega \bigr), \label{Eq42}
\end{eqnarray}
uniformly for all $\bigl(s, \mathbf{x}_s^{\epsilon} \bigr)$ in any compact subset $\bar{\Omega}^T$.

Note that $I_{\Omega}^{\epsilon} \bigl(\bigl(s, \mathbf{x}^{\epsilon} \bigr); \, \partial \Omega \bigr) = I^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon} \bigr)$ (cf. Equation~\eqref{Eq38}), then the upper bound in \eqref{Eq41} can be verified using the Ventcel-Freidlin asymptotic estimates (see \cite[pp.\,332--334]{Frie06}, \cite{VenFre70} or \cite{Ven73}).

On the other hand, to prove the lower bound in \eqref{Eq42}, we introduce a penalty function $\Phi_M^{\epsilon}$ (with $\Phi_M^{\epsilon}\bigl(t, \mathbf{y}\bigr)=0$ for $\bigl(t, \mathbf{y}\bigr) \in \Gamma_{T}^{+}$); and write $g^{\epsilon}=g_{M}^{\epsilon}$ ($\equiv \mathbb{E}_{s, \mathbf{s}}^{\epsilon} \bigl \{\exp\bigl(-\frac{1} {\epsilon} \Phi_{M}^{\epsilon} \bigr) \bigr\}$) and $J^{\epsilon}=J_{\Phi_M}^{\epsilon}$, with $\Phi^{\epsilon} =\Phi_M^{\epsilon}$. From the boundary condition in \eqref{Eq28}, then, for each $M$, we have
\begin{eqnarray}
 g^{\epsilon}\bigl(s, \mathbf{x}^{\epsilon} \bigr) \le g_{M}^{\epsilon}\bigl(s, \mathbf{x}^{\epsilon} \bigr). \label{Eq43}
\end{eqnarray}
Using Lemma~\ref{L3} and noting further the following
\begin{eqnarray}
J_{\Phi_M}^{\epsilon} \bigl(s, \mathbf{x}^{\epsilon} \bigr) \ge I_{\Omega}^{\epsilon} \bigl(\bigl(s, \mathbf{x}^{\epsilon} \bigr); \, \partial \Omega \bigr).  \label{Eq44}
\end{eqnarray}
Then, the lower bound in \eqref{Eq42} holds uniformly for all $\bigl(s, \mathbf{x}_s^{\epsilon}\bigr)$ in any compact subset $\bar{\Omega}^T$. This completes the proof of Proposition~\ref{P2}.
\end{proof}

\section{Main results} \label{S3}
Let $\hat{\mathbf{x}}_t^{\epsilon}$ be the solution to the following SDE
\begin{align}
d \hat{\mathbf{x}}_t^{\epsilon} &= \mathbf{f} (t, \hat{\mathbf{x}}_t^{\epsilon} ) + \mathbf{b} \, \sigma(t, \hat{\mathbf{x}}_t^{\epsilon} ) \,v^{\epsilon}(t, \hat{\mathbf{x}}_t^{\epsilon}) + \sqrt{\epsilon}\,\mathbf{b} \, \sigma(t, \hat{\mathbf{x}}_t^{\epsilon} ) \, dW_t, \notag\\
&\quad \quad \text{with an initial condition} \quad \hat{\mathbf{x}}_s^{\epsilon} = \mathbf{x}_s^{\epsilon},  \label{Eq45}
\end{align}
where $v^{\epsilon}$ is an appropriate control function (which also depends on $\epsilon$) to be chosen so as to reduce the variance of the importance sampling estimator.

Let
\begin{align}
 z^{\epsilon} = \exp\Bigl( -\frac{1} {\sqrt{\epsilon}} \int_s^T  \bigl\langle v^{\epsilon}(t, \hat{\mathbf{x}}_t^{\epsilon}) ,\, dW_t \bigr\rangle  -  \frac{1} {2\epsilon} \int_s^T  \bigl \vert v^{\epsilon}(t, \hat{\mathbf{x}}_t^{\epsilon}) \bigr \vert^2 dt \Bigr).  \label{Eq46}
 \end{align}
The corresponding importance sampling estimator is then given by
\begin{eqnarray}
 \hat{\rho}(\epsilon) = \frac{1}{N} \sum\nolimits_{j=1}^N \exp\Bigl( -\frac{1} {\epsilon} \Phi^{\epsilon}(\hat{\mathbf{x}}^{\epsilon^{(j)}}) \Bigr) {z^{\epsilon}}^{(j)},  \label{Eq47}
\end{eqnarray}
where $\left\{\bigl(\hat{\mathbf{x}}^{\epsilon^{(j)}}, {z^{\epsilon}}^{(j)}\bigr)\right\}_{j=1}^N$ are $N$-copies of independent samples of $\bigl(\hat{\mathbf{x}}^{\epsilon}, z^{\epsilon}\bigr)$. Note that, for an appropriately chosen control function $v^{\epsilon}$, the above importance sampling estimator in \eqref{Eq47} is an unbiased estimator for \eqref{Eqx.2}, i.e.,
\begin{align}
\mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\bigl[\hat{\rho}(\epsilon)\bigr] &= \mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon} \Bigl[ \exp\Bigl( -\frac{1} {\epsilon} \Phi^{\epsilon}(\mathbf{x}^{\epsilon}) \Bigr) \Bigr] \notag\\
& \equiv \mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\bigl[\rho(\epsilon)\bigr].  \label{Eq48}
\end{align}
Moreover, the relative estimation error is given by
\begin{align}
\Rerr\left(\hat{\rho}(\epsilon)\right) &= \frac{\sqrt {\Var\left(\hat{\rho}(\epsilon)\right)}}{\mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\bigl[\hat{\rho}(\epsilon)\bigr]}  \label{Eq49}
\end{align}
which can be rewritten as follows
\begin{align}
\Rerr\left(\hat{\rho}(\epsilon)\right) &= \bigl(1/ \sqrt{N}\bigr) \sqrt{\Delta\left(\hat{\rho}(\epsilon)\right) - 1},  \label{Eq50}
\end{align}
where
\begin{align}
\Delta\left(\hat{\rho}(\epsilon)\right) = \frac{ \mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\Bigl[ \exp\Bigl( -\frac{2} {\epsilon} \Phi^{\epsilon}( \hat{\mathbf{x}}^{\epsilon}) \Bigr) \Bigr] \bigl(z^{\epsilon}\bigr)^2 } { \mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\Bigl[ \exp\Bigl( -\frac{1} {\epsilon} \Phi^{\epsilon}(\mathbf{x}^{\epsilon}) \Bigr) \Bigr]^2 }.  \label{Eq51}
\end{align}
Hence, in order to reduce the relative estimation error $\Rerr\left(\hat{\rho}(\epsilon)\right)$, we need to control the term $\Delta\left(\hat{\rho}(\epsilon)\right)$ in \eqref{Eq50}. Note that, from Jensen's inequality, we have the following condition
\begin{align}
  \limsup_{\epsilon \rightarrow 0} - \epsilon\,\log  \mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\Bigl[ \exp\Bigl( -\frac{2} {\epsilon} \Phi^{\epsilon}(\hat{\mathbf{x}}^{\epsilon}) \Bigr) \Bigr] \le  2 \lim_{\epsilon \rightarrow 0} - \epsilon\,\log  \mathbb{E}_{s, \mathbf{x}_s^{\epsilon}}^{\epsilon}\Bigl[ \exp\Bigl( -\frac{1} {\epsilon} \Phi^{\epsilon}(\hat{\mathbf{x}}^{\epsilon}) \Bigr) \Bigr]   \label{Eq52}
\end{align}
which also implies $\Delta\left(\hat{\rho}(\epsilon)\right) \ge 1$ with $\lim_{\epsilon \rightarrow 0} \Delta\left(\hat{\rho}(\epsilon)\right) = 1$. Moreover, the statement in \eqref{Eq49} further implies the following
\begin{align*}
\Rerr\left(\hat{\rho}(\epsilon)\right) = \frac{1}{\sqrt{N}} \exp\bigl(o(1)/\epsilon\bigr) \quad \text{as} \quad \epsilon \rightarrow 0,
\end{align*}
which is generally referred as asymptotic efficiency or optimality. In this paper, our main objective is to choose appropriately the control function $v^{\epsilon}$ in \eqref{Eq45}, so that the resulting importance sampling estimator achieves a minimum rate of error growth. For this reason, we introduce the following standard definition from simulation theory (e.g., see \cite{Buc04} or \cite{BudD19}) which is useful for interpreting our result.
\begin{definition}
An importance sampling estimator of the form \eqref{Eq47} is log-efficient (i.e., asymptotic efficiency or optimal) if
\begin{align}
 \lim_{\epsilon \rightarrow 0} - \epsilon \log \Delta\left(\hat{\rho}(\epsilon)\right) = 0.  \label{Eq53}
\end{align}
\end{definition}

Then, we state our main result as follows.
\begin{proposition} \label{P3}
Suppose that the importance sampling estimator $\hat{\rho}(\epsilon)$ in \eqref{Eq47}, with $v^{\epsilon} (t, \mathbf{x}) = -\sigma^T(t, \mathbf{x}) J_{x^{\epsilon,1}}^{\epsilon}(t, \mathbf{x})$, is uniformly log-efficient (i.e., asymptotic efficient), where $J^{\epsilon}(t, \mathbf{x})$ satisfies the corresponding dynamic programming equation in $\Omega^T$, w.r.t. the system in \eqref{Eq45}, with $J^{\epsilon} = \Phi^{\epsilon}$ on $\partial^{\ast} \Omega^T$. Then, there exits a set $\mathbb{A} \subset \mathbb{R}^{nd}$ such that the Hausedorf dimension of $\mathbb{A}^c$ is zero and
\begin{align}
 \lim_{\epsilon \rightarrow 0} \Rerr\left(\hat{\rho}(\epsilon)\right) = 0, \label{Eq54}
\end{align}
for all $x \in \mathbb{A}$.
 \end{proposition}

\section{Concluding remarks} \label{S4}
In this paper, we considered an importance sampling problem for a certain rare-event simulations involving the behavior of a diffusion process pertaining to a chain of distributed systems with random perturbations. In particular, we have provided an efficient importance sampling estimator, with an exponential variance decay rate, for the asymptotics of the probabilities of the rare-events involving the behavior of such a diffusion process on finite time intervals which also ensures a minimum relative estimation error in the small noise limit. The framework for such an analysis basically relies on the connection between the probability theory of large deviations and the values functions for a family of stochastic control problems associated with the underlying distributed system, where such a connection provides a computational paradigm -- based on an exponentially-tilted biasing distribution -- for constructing efficient importance sampling estimators for the rare-event simulation. Moreover, as a by-product, the framework also allows us to derive a family of Hamilton-Jacobi-Bellman for which we also provide a solvability condition for the corresponding optimal control problem.

redpajama_set_name: RedPajamaArXiv


text:
Samuel Cabrera Castañeda (* 15. August 1960 in Pacho; † 21. März 2022 ebenda) war ein kolumbianischer Radrennfahrer.

Sportliche Laufbahn
1980 siegte er in der Guatemala-Rundfahrt mit einem Etappensieg. 1981 fuhr er mit der Nationalmannschaft Kolumbiens die Tour de l'Avenir, die er auf dem 7. Platz des Gesamtklassements beendete. 1982 gewann er die Costa Rica-Rundfahrt. Dabei gewann er zwei Etappen. Im Clásico RCN wurde er beim Sieg von Luis Herrera Dritter. 1985 belegte er in der Tour de l'Avenir den 3. Platz hinter dem Sieger Martín Ramírez und gewann das Klassement der Bergwertung. 1988 kam der zweite Rang in der Vuelta Americas dazu, in der er eine Etappe gewann.

Von 1985 bis 1989 war er Berufsfahrer in den Radsportteams Café de Colombia und Reynolds.

Sechsmal bestritt er die Tour de France. 1983 wurde er 57., 1984 32., 1986 11., 1988 31. und 1989 49. der Gesamtwertung. 1987 schied er aus. Den Giro d'Italia beendete er 1989 als 47. Die Vuelta a España beendete er zweimal, 1986 als 35. und 1987 als 44. des Gesamtklassements. 1985 und 1988 schied er in dem Etappenrennen aus.

Weblinks

Einzelnachweise

Radsportler (Kolumbien)
Departamento de Cundinamarca
Kolumbianer
Geboren 1960
Gestorben 2022
Mann

redpajama_set_name: RedPajamaWikipedia


Editor's Note. The piece below is excerpted and adapted from a Circle of Blue article by Brett Walton.
If you live downstream from a dam, you hope that someone is maintaining it and monitoring its safety. This is not always the case. In Alabama, for example, all but 10% of the state's dams are privately owned and regulatory oversight is minimal.
The universe of American dams is expansive. There are tailings dams that hold back a slurry of mine wastes, stock ponds for irrigation or watering cattle, and artificial lakes for sailing and speedboats. There are dams to detain flood waters and dams to filter debris. Then there are the hydropower behemoths such as Grand Coulee and Hoover, symbols of 20th-century engineering might. Though iconic, these canyon-bridging concrete plugs are the minority. Most dams are small structures less than 25 feet tall made of packed dirt and rock and built more than 50 years ago.
Surprisingly little is known about why individual dams fail. Few states do autopsies to learn precisely what went wrong. That is why a Stanford University professor founded the National Performance of Dams Program in 1994. The program's goal is to learn from past failures so that managers can identify problems before they become tragedies. The program's researchers have found that the U.S. dam industry is far behind the nuclear power and oil and gas pipeline industries in the amount of data it collects.
Not every dam failure is judged by the same criteria. The United States has a three-tier rating system that classifies a dam based on the destruction resulting from failure. The rating system is used to set design standards; the greater the risk the stricter the codes. Low-hazard dams are expected to cause minimal property damage. It is considered acceptable if these dams, as long as they are accurately categorized, fail because the risks to life and property are low. Richland County, South Carolina notes that several of its dams will fail in a 50-year flood. Significant-hazard dams are a step up. They might destroy infrastructure or cause severe property damage if breached. The most worrisome category is high-hazard potential. A rupture at one of these dams could kill people.
A new risk on the industry's radar is climate change but engineers are still trying to figure out how droughts and severe storms will affect dam performance.
Eric Halpin, deputy dam safety officer for the U.S. Army Corps of Engineers, said that the key variable is how a dam's engineering responds to sharp shifts in weather.


Animals have a knack for saving other animals, but they've also been known to put themselves in the line of danger for humans , and we're not just talking about pet dogs that protect their homes from burglars.
From dolphins that rescued a surfer from sharks and a whale that helped a drowning athlete to an elephant that protected a young girl from a tsunami, these amazing, selfless animal heroes remind us once again of the unique species we co-habitat with that we need to protect.
Athlete Yang Yun was part of an underwater competition that required contestants to remain at to the bottom of a 20-foot arctic pool in China's Polar Land when the freezing temperatures caused her legs to cramp, leaving her unable to return to the surface.
That's when one of the tank's residents, a beluga whale named Mila grabbed Yun's leg and guided her up toward the air.
Flipper spent several seasons helping humans out of jams, but his rescues aren't entirely out of the ordinary for the dolphin population.
Stories of dolphins rescuing humans are everywhere, including the one about Todd Endris, a surfer who survived an attack by a great white shark that he said "came out of nowhere" off the coast of California.
After he was bitten, a pod of bottlenose dolphins circled Endris, holding off the shark until he could make it back to shore – just as they would for one of their own.
When a 3-year-old boy fell into the gorilla exhibit at the Brookfield Zoo in Illinois it wasn't just injuries from the 24-foot-drop that put him in danger: The exhibit had several adult gorillas on display that were also a threat.
But 7-year-old female gorilla Binti took over, picking up the boy and cradling him before leaving him gently by a door for zookeepers to rescue him. The toddler was transferred to the hospital and recovered.
Man's best friend isn't the kind to shy away from rescuing an owner, but it's still heartwarming to hear about pups that put themselves in harm's way for their masters.
In this case, it was 18-week-old puppy Pinky who grabbed the attention of a swarm of bees that was headed for her human companion, 9-year-old Richie Bragg: Pinky was stung more than 40 times but survived, and most likely saved Richie – who has a blood condition that prevents proper clotting – from needing serious medical attention.
When an injured elephant seal pup was rescued in 1994 at the animal center that would become the Aquarium of the Pacific, keepers were simply hoping she'd survive — she did, and became a favorite of visitors and locals with a reputation for being "a gentle giant," as volunteer Hugh Ryono writes on the Aquarium's blog.
But when Ryono fell while feeding seal pups a year later and found himself unable to get out of the way of the pups' aggression, it was Gimpy that stepped in, protecting Ryono from an attack (and earning a writeup in Reader's Digest).
While some sites dispute the truth of this story, The Guardian reports that, in 2005, a girl in Ethopia was kidnapped by seven men and beaten for a week — until three lions chased the men away and stood guard over the girl until police found her.
Along with mentioning Gimpy the seal in its "True Stories of Hero Pets" collection, Reader's Digest also brought our attention to Lurch, an African Watusi calf that intercepted his owner, Janice Wolf, while she was walking the paths of her Arkansas refuge.
Lurch refused to move out of the way and when Wolf grabbed his horns to guide him, he knocked her off balance – thereby preventing her from stepping on a coiled copperhead snake she was about to disturb.
It's not just shark attacks that inspire dolphins to swim into action and save humans: One rescuer, Filippo, was a popular resident of the Adriatic Sea off Italy's Manfredonia who became a spur-of-the-moment lifeguard when a 14-year-old boy fell into the ocean – without knowing how to swim.
Scotland's Daily Record reports that the dolphin pushed the boy to keep him above the water until his parents could pull him to safety.
Amber Mason, an eight-year-old girl from Bucks, England, was in Phuket, Thailand when the tsunami struck in 2005 – riding an elephant named Ningnong.
As the water rose, the elephant ran up the shore, taking the brunt of the impact and keeping Amber above the water.


Charles Callixte, Comte de Tusseau [d. 1864], Airvault, Deux-Sèvres, by 1861. Frédéric Spitzer [1815-1890], Paris, by 1881; (his estate sale, at his residence by Chevallier and Mannheim, Paris, 17 April-16 June 1893 [15 May], no. 664, sold for 10,200 francs); Goldsmith. (Lowengard, Paris); purchased 15 February 1906 by Peter A.B. Widener, Elkins Park, Pennsylvania; inheritance from Estate of Peter A.B. Widener by gift through power of appointment of Joseph E. Widener, Elkins Park; gift 1942 to NGA.
 Delange, Henri, and Carle Delange, Recueil de toutes les pièces connues jusqu'à ce jour de la faïence française dite de Henri II et Diane de Poitiers, Paris, 1861: 31.
 Edouard Garnier, "Collections de M. Spitzer: Céramique française," Gazette des beaux-arts, 2d ser., 24 (1881): 402 (one of five salts mentioned as in Spitzer's collection). The buyer's name is according to Edmond Bonnaffé, "Les faïences de Saint-Porchaire," GBA, 3rd ser., 13 (1895): 285.
 Widener collection records in NGA curatorial files.
Exposition retrospective de l'art français, Trocadéro, Paris, 1889, no. 1162.
The basic body of the salt is formed of slabs of white clay, stamped and incised with designs in clays that show through the glaze as orange brownish and flecked red; there is applied decoration in pale brown and orange-brownish clays. The salt is covered entirely in a somewhat crackled transparent glaze, and the design is heightened in green and purple red. There are repairs to the feet and some shrinkage, chipping, and cracking where the underside of the base meets the edge: there is, however, nothing to support the statement in Bonnaffé 1891 that "la base de cette saliére a été refait d'après une autre pièce de la série des faïences de Saint-Porchaire," repeated in the Widener collection records ("the base is remade from another example"); no evidence has been found that the salt is not essentially in its original condition. There are minor repairs to the protruding ridge.
 It seems tha Bonnaffé confused this salt with one now in Brussels, which is stated by Delange and Delange 1861, no. 16, to have lost its original foot. Anne-Marie Mariën Dugardin kindly made a special examination of the Brussels salt for me, which confirmed that its foot is a replacement.
Delange/Delange 1861, 31, no. 18.
Fillon, Benjamin. L'art de terre chez les Poitevins. Niort, 1864: 92, no. 26.
King, A.C. Henri Deux Ware, Photographs of twenty examples of this ware chiefly in English collections with an introductory notice. London (The Arundel Society), 1868: 8, nos. 50, 51, or 52 (mentioned).
Garnier, Edouard. "Collections de M. Spitzer: Céramique française." Gazette des Beaux-Arts 2d ser. 24 (1881): 402.
Bonnaffé, Edmond. "Les faïences de Saint-Porchaire." Gazette des Beaux-Arts 37 (1888): 327.
Exposition rétrospective de l'art français au Trocadéro. Lille, 1889: 186, no. 1162.
Exposition retrospective de l'art français. Exh. cat. Trocadéro, Paris, 1889: no. 1162.
Bonnaffé, Edmond. "Faïences de Saint-Porchaire dites de Henri II." In La collection Spitzer: Antiquité, moyen âge, renaissance. 6 vols. Paris, 1890-1892: 129, no. 5, repro.
Bonnaffé, Edmond. "Les faïences de Saint-Porchaire." Gazette des Beaux-Arts 3d ser. 13 (1895): 285.
McCall, George Henry. "Saint-Porchaire in America." Arts and Decoration 10 (April 1919): 316, repro.
Inventory of the Objects d'Art at Lynnewood Hall, Elkins Park, Pennsylvania, The Estate of the Late P.A.B. Widener. Philadelphia, 1935: 70, as c. 1560.
Works of Art from the Widener Collection. Foreword by David Finley and John Walker. National Gallery of Art, Washington, 1942: 14, as Henri II Ware, Salt cellar.
Wilson, Carolyn C. Renaissance Small Bronze Sculpture and Associated Decorative Arts at the National Gallery of Art. Washington, 1983: 204, no. 34, as c. 1555.
Schnitzer, Barbar K. "The Sixteenth-Century French Ceramic Ware Called Saint-Porchaire." Ph.D. dissertation, University of Michigan, Ann Arbor, 1987: 201, no. 21.
Distelberger, Rudolf, Alison Luchs, Philippe Verdier, and Timonthy H. Wilson. Western Decorative Arts, Part I: Medieval, Renaissance, and Historicizing Styles including Metalwork, Enamels, and Ceramics. The Collections of the National Gallery of Art Systematic Catalogue. Washington, D.C., 1993: 250-253, color repro. 251.
Sturman, Shelley and Barbour, Daphne. "'Saint-Porchaire' Ceramic Bodies." Studies in the History of Art 52 (1996): 80, 82, 87, repro. no. 1.