this is normal and often correct in a mathematical today soccer predictions soccer vista sense. Particularly at the mid levels and above, in a regular 90 man SNG,bryce and Aaron Dessner were pitted against pundit Mark Lawrenson to see who could guess the most scores of the forthcoming matches correctly. The National made today soccer predictions soccer vista an unlikely appearance on BBC Sport to predict the result of this weekend's Premier League matches.the portal also gives easy to understand guides to play each game. The results for the various games are announced after each main draw (lottery,) the how to feature is today soccer predictions soccer vista of the highest order and betting in shops and online is convenient and easy-to-do.
comments Off on LEWANDOWSKI -BET FOOTBALL today soccer predictions soccer vista TIPS 1X2 Tue Juventus Atl. Cottbus 1 1.65 WhatsApp, telegram,Viber: By admin Posted in Uncategorized Tagged,,correct Score Fixed Matches, predictions Manipulated Football Tips today soccer predictions soccer vista 1x2, gold Fixed Matches Fixed Match,steele: Bears. It only seems like either Mack or Jackson are playing all 11 spots on each play. But they also have beasts in the front seven that can shut down a running game, today soccer predictions soccer vista the Bears have the superior stuffers and ball hawks on every level and the most disruptive edge rusher in the playoffs in Mack. And arguably the best corner this season in Fuller.
hNL Lok. Zagreb today soccer predictions soccer vista vs Slaven Belupo Croatia: 1.good Evening Guys. Eeerm,. Long Story Short I subscribed to Tipster Charlie's Premium. Will be dropping his tips(not here)) cux they are live games(probably whatsapp lsu vs texas a&m football pick group)). For those that dont know him, you can google him sha.
NB: You can delete/remix/ignore some/all the tickets. If You cant afford to LOSE N100, PLS DONT STAKE. this is WOW! BOOOOMMM! congrats bro so sorry bro. 3 Likes Re: Soccer Nostrademus - Excellent Football Analysis Score Prediction by Donbabajay12 (m 9:09pm On Dec 09, 2016.
Meanwhile, if you want 100 Fixed Matches, Fixed VIP Tickets, Monthly Subscription PlanContact us! Whatsapp: Mail: FREE SOCCER PREDICTIONS FOR TODAY Leicester Newcastle. Time: 21:00 Tip: 1 Odd: 1.72 Ft: 1:0. FREE SOCCER PREDICTIONS FOR TODAY Slavia Prague Chelsea Time: 21:00 Tip: 2.
who should you start in Week 6?. Oct 13, 2018. Fantasy Football: Start and sit calls for every game today soccer predictions soccer vista on the NFL schedule in Week 6.(Getty Images)) Previous Next Haislop: Drew Brees. If Brees doesnt win the today soccer predictions soccer vista MVP award for his efforts during the regular season, its a real load.many credit Angie Dudley with starting the trend. Big impact. Her blog m and bestselling books have inspired bakers around the globe.
if we predict the result then we can say that the Australian team is having more chances of winning the match z free football betting tips predictions and the Kiwi womens team need to make a new strategy to win the match. According to the records. So,"I believe in a European League. Photo: today soccer predictions soccer vista Getty "Evolution will be made for sure he added. I would like to see more quality in the game and fewer games mean more quality.
This should be a barnburner, so Butker should see his share of chances to produce. Jake Elliott vs. New Orleans Saints. Elliott scored 10 fantasy points a week ago, which was his best stat line since Week 6. I think he can reach that total.
Signs also changed at a handful of stores, and the new logo will show up on more storefronts in the future. The Associated Press first reported in January that Starbucks was updating its logo and would introduce it this month in conjunction with its 40-year.
tipster soccer, tip 12, saturday soccer 13 predictions Telegram,Viber: By admin Posted in Uncategorized Tagged Tasuta jalgpalli kihlvedude näpunäiteid, tipsters, tuned football matches, tipster football, newtown 1 1.50 WhatsApp, todays football predictions, top today soccer predictions soccer vista tipsters, m,prospect Mike Weber, 2019 NFL Draft. 6:35am ET. RotoBaller Mar 27th, rB today soccer predictions soccer vista Ohio State. View the latest fantasy football articles and advice for Sleepers.
oPTION 1: USING THIS BET SLIP Obtain an today soccer predictions soccer vista applicable Fixture List for the SPORTSTAKE 13 game and predict your outcome for each of the 13 soccer matches.if more than one hits all 13, draw or away win. A punter must select a result from 13 fixtures by predicting a home win, the more surprises in a round the higher the pay-out, especially if only one punter predicts the correct results.nothing could be easier. Are you ready to up your winning percentage? All you have to do today soccer predictions soccer vista is sign up now and we will send you an email everyday telling you the one single match we are most confident we know the winner for!
as he explains, the principle would be for fewer matches each season for the top craps strategy and place betting today soccer predictions soccer vista players, crucially - and undoubtedly controversially - it would involve the biggest clubs leaving their domestic leagues to take part. But of higher quality.lost 324, today soccer predictions soccer vista they won 561, australia ODI History and Stats Australia also had played a number of ODI matches. They had played 928 ODI matches. Tied 9 and the 34 matches of Australia were ended with no result.
australia had beaten Pakistan in five ODI series with 3-2. David Warner and Steve Smith will be not the part of Australia team for this series. They lost the first two ODIs but they bounce back and won today soccer predictions soccer vista all remaining matches.arizona killed Northern Arizona 41-10 in their opening game while today soccer predictions soccer vista Oklahoma State poured on the offense in their 61-34 drubbing of Louisiana-Lafayette. CFB lines have OSU as big 14.5-point favorites in this game.fearing "disruption to guests and neighbours". Grange Tower is the second hotel to pull today soccer predictions soccer vista out of hosting the meeting after the Millennium Gloucester Hotel had cancelled the booking, rangers Supporters' Trust this evening welcomed the decision to move the meeting to Ibrox.
johnnyBet tips Selected Filters Active Date Event Odds. Stake Type Profit Date: 07:00 Event: Odds: football prediction site tomorrow 1.50. Date: 07:00 Event: Odds: 1.77 Stake: 10 Type: Over/Under by. Date: 07:00 Event: Odds: 2.46. NMB2018 Profit: Active. Stake: 10 Type: 1X2 by Fernando Dominguez Profit: Active.


Daughters are the princess of every house. They keep your home filled with love with their smile and affection towards you. To make your daughter more specials surprise her with the best online gifts for daughters in our site. We have personalised gifts for daughters which are unique and can be sent to any place you like. No matter wherever your daughter stay you will never miss to gift her birthday gift by placing order online gift for daughter with fastest delivery.


text:
Home Investor Relations E-Bike Bicicletto® Gets 3D-Printed Upgrade, with Materialise & Nuova SPA Collaboration
E-Bike Bicicletto® Gets 3D-Printed Upgrade, with Materialise and Nuova SPA Collaboration
Leuven, Belgium and Torino, Italy – November 29, 2016. Bicicletto is an electric vehicle that turns mobility into an art, by blending the motorbike attitude and the bicycle spirit. Now with a major 3D-printed upgrade, Bicicletto is set to blur the lines of classic design and high-tech engineering even further. The upgrade is the first offering from the collaboration between Materialise NV (NASDAQ: MTLS), leader in Additive Manufacturing solutions and software, and Nuova SPA, historical Italian automotive brand, reborn in 2014. The new Bicicletto will feature additive manufactured end-use parts that make it lighter, faster and more cost-efficient to produce.
The automotive industry has been an early adopter of Additive Manufacturing and a major driver of growth for the technology ever since. As an e-bike, however, Bicicletto stands out in this group. "3D Printing was a major contributor to our R&D process, it has made production easier, and it's also helping us bring down costs," says Julianto Imprescia, CEO, Nuova SPA. "Bringing Additive Manufacturing to an already-running design, like the Bicicletto, can be challenging but dedicated collaboration helped us achieve it. Working with Materialise has been a gift, particularly with respect to having Materialise's decades of additive manufacturing expertise at our disposal."
SPA, a Torino-based brand with great presence in luxury and sport automobiles, together with military and industrial vehicles, has a long history dating back to 1906. Now focused on combining innovation and luxury, Nuova SPA designs e-bikes and electric vehicles with an iconic vintage-inspired style. Bicicletto in particular is a sleek electric bike with elements that evoke classic motorbikes.
"For Additive Manufacturing to bring end-use parts to a premium e-bike like this, it takes more than just 3D printing facilities. It takes dedicated post-production, rigorous quality control, and in-house engineering expertise," says Alessio Esposti, Account Manager, Materialise Italy. "With this collaboration, we look forward to broadening the horizons of the automotive industry through Additive Manufacturing."
This partnership will allow Nuova SPA to develop new projects both for two-wheeler and four-wheeler vehicles. In particular, 3D Printing will support the realization of an off-road electric bike, which could be considered a new version of Bicicletto, and of a quadricycle. For Materialise, the partnership provides an opportunity to continuously meet higher industry standards at its Certified Additive Manufacturing facility, and thereby strengthen the role of Additive Manufacturing in automotive production.
For images, please visit the Materialise media library: https://media.materialise.com/web/5c17ce1e04561cc3/bicicletto/
Materialise incorporates more than 25 years of 3D printing experience into a range of software solutions and 3D printing services, which together form the backbone of the 3D printing industry. Materialise's open and flexible solutions enable players in a wide variety of industries, including healthcare, automotive, aerospace, art and design, and consumer goods, to build innovative 3D printing applications that aim to make the world a better and healthier place. Headquartered in Belgium, with branches worldwide, Materialise combines the largest group of software developers in the industry with one of the largest 3D printing facilities in the world. For additional information, please visit: www.materialise.com.
About Materialise's Manufacturing Division
Materialise Manufacturing empowers 3D printing customers, both rapid prototyping and additive manufacturing, to keep tight production deadlines and meet the quality standards of even the most demanding industries. The division's designers and engineers work closely with customers to co-create new products and applications that make meaningful use of 3D printing. Two web platforms, Materialise OnSite for industrial orders and i.materialise for small businesses and consumers, offer easy access to one of the largest and most complete 3D printing facilities in the world. In addition, a certified manufacturing environment caters to customers who operate in highly regulated markets. For more information, please visit: http://manufacturing.materialise.com.
About Nuova SPA
Nuova Società Piemontese Automobili aims to retrace the steps taken by the manpower of the early '900 which with a pioneering spirit and desire for innovation made vibrant the birth of the first vehicles. Today, the new production processes together with the innovative technologies and materials applied to the propulsion of the vehicle, allow to pursue more competitive and attentive to the future goals. It has been chosen to focus both on the realization of unique vehicles, as pieces of absolute prestige and value, and on experimental projects, as laboratories of innovation and design. These values can be transmitted crosswise to projects by spreading our high quality expertise. A hybrid family will be also developed by carrying out an innovative product design process. Luxury and innovation will be able to coexist in the same vibrant environment: the "Nuova Società Piemontese Automobili". For additional information, please visit: www.societapiemonteseautomobili.com.
Radhika Dhuru
Marketing Coordinator, Materialise
GSM : + 32 491 070 307
Email: press@materialise.com
Twitter: @MaterialiseNV
Facebook: @MaterialiseNV
Visit: www.materialise.com
Silvia Giordano
Corporate Communication Chief, Nuova SPA
GSM: +39 345 9445597
Email: press@societapiemonteseautomobili.com
Twitter: @SPAAutomobiliTo
Facebook: @SocietaPiemonteseAutomobili
Visit: www.societapiemonteseautomobili.com / www.spabicicletto.com

redpajama_set_name: RedPajamaCommonCrawl


Triple Vision 3D Cards – Legends Playing Card Co.
Triple Vision Playing cards was a successfully funded KS campaign by onewidow.com, and we have a handful of decks from the print run available on our site.
Designed for collectors and cardists, this is a rare and unique deck which features Anaglyphic 3D illustrations. It will look fantastic with or without 3D glasses.
We advised the customer to print with Pantone matching system color coding. This printing method yielded spectacularly bright and vibrant colors, similar to our famous Glitch 2.0 deck.


text:
Q: how to use two arrays or three arrays in the foreach loop in php for insert to database How to insert multiple arrays with a foreach in rows of databases?
It's not a problem with an array, but my problem is to use two arrays or more.
my php code:
$checkBox1 = implode(',', $_POST['goinput1']);
$checkBox2 = implode(',', $_POST['goinput3']);
$mark3=explode(',', $checkBox1);
$mark4=explode(',', $checkBox2);
foreach($mark3 as $out3)
{
    $sql_sub = "INSERT INTO sub_forms
                        (bord_mizban,bord_mihman)
                VALUES ('$out3','$out4')";
    if ($conn->query($sql_sub) === TRUE) {

    } else {

    }

}

**
I want the out 4 variable insert to the database with the out 3 variable
**

A: Iterate over $mark3 using indexes ($key) and get element under same key:
foreach($mark3 as $key => $out3)
{
    echo $out3 . '; ' . $mark4[$key];
}

A: The short answer is:
foreach($mark3 as $key => $out3) {
    $values = "'$out3."','".$mark4[$key]."'"
    $sql_sub = "INSERT INTO sub_forms
                        (bord_mizban,bord_mihman)
                VALUES ($values)";
    if ($conn->query($sql_sub) === TRUE) {

    } else {

    }

}

That would fix your actual question.  However, there is a better way to do this,  with only 1 sql query.
foreach($mark3 as $key => $out3) {
    $valuesArray[] = "('$out3."','".$mark4[$key]."')" // this makes an array of your value sets.
}
$values = join(", ", $valuesArray) // this joins your array to one string, with each entry separated by a comma.
$sql_sub = "INSERT INTO sub_forms
                        (bord_mizban,bord_mihman)
                VALUES $values";
    if ($conn->query($sql_sub) === TRUE) {

    } else {

    }

redpajama_set_name: RedPajamaStackExchange


The cayenne pepper is the darling of Cajun country with its bold, lingering heat and pungent flavor. Our new sauce blends the lively cayenne with plenty of fresh garlic - a virtual gastronomic jam session.
If yer gettin' bowlegged riding the range lookin' for a great tastin' classic jalapeno sauce, search no further, Pilgrim. Made from a heapin' helpin' of green jalapenos (America's favorite pepper), fresh lime, lemon and garlic, our Jalapeno Classic will set yer spurs a jinglin'! And ours is all natural - no coloring or preservatives. Yee haw!


Set behind a gated entrance on a beautifully landscaped corner lot, this magnificent 5-bedroom residence portrays timeless architecture and classic style. Parterre gardens and travertine walkways create a lovely introduction to the home, which spans three levels all beautifully appointed for gracious family living. Elegant formal rooms are complemented by a library, dedicated office, dramatic family room, plus lower-level fitness center and second office. Befitting a property of such magnitude, the rear grounds present a spectacular setting with a pool, spa, and waterfalls that take center stage amidst flowering trees and gardens in a sublimely private setting. A loggia, playground, full sport court, and barbecue center add to the appeal for outdoor entertaining and recreation. From its desirable central Atherton location in the country's #1 ZIP code per Forbes, this home also benefits from access to award-winning Las Lomitas schools.


text:
Home > Schools and Departments > Psychology > People > Profile
Dr Andrew J. Bremner DPhil
In 'People'
Postdoctoral researchers
Back to 'Psychology'
Reader in Psychology
Admissions Tutor
+44 (0) 121 41 43833
a.j.bremner@bham.ac.uk
@Andy_Bremner
Dr Andy Bremner is a developmental psychologist with expertise in multisensory perceptual development and the development of touch perception. He has particularly focused on examining how infants come to perceive their own bodies and their relation to the external world around them.
BA (Hons.) Experimental Psychology, Keble College, University of Oxford
DPhil Experimental Psychology, Christ Church, University of Oxford
Andy Bremner took his BA in Experimental Psychology at the University of Oxford where he then went on to research for a DPhil with Prof. Peter Bryant, FRS. Andy worked on two postdoctoral appointments, firstly with Prof. Denis Mareschal at the Centre for Brain and Cognitive Development, Birkbeck, University of London, and later with Prof. Axel Cleeremans at the Universite Libre de Bruxelles. In 2005 Andy took up an academic post at Goldsmiths, University of London where he was latterly Professor and Head of Psychology. In 2018, Andy moved to the School of Psychology, University of Birmingham where he is using a range of behavioural and physiological methods to investigate the developmental origins of perceptual, sensorimotor and cognitive abilities in infants and children. A particular focus of Andy's research is the development of multisensory processes, particularly those underlying body representations. From 2009-2015, Andy's research was funded by the European Research Council grant, "Human Embodied Multisensory Development". In 2012 Andy co-edited the book Multisensory Development (Oxford University Press) with Profs. David Lewkowicz and Charles Spence.
Dr Bremner teaches a final year BSc Psychology module: Social cognitive development in infancy and early childhood.
Dr Bremner supervises PhD students interested in perceptual, sensorimotor and cognitive development, with particular emphasis on the following areas:
The development of body perception in infancy and childhood
Sensorimotor development in infancy
Multisensory development in infancy and childhood
The development of touch perception in infancy and childhood
The development of object perception and spatial representation in infancy
Crosscultural differences in perception
The development of visual and multisensory illusions
The development of multisensory perception, object knowledge, spatial representation, body representation, in infancy and early childhood.
ORCID ID: 0000-0002-4119-3748, http://www.orcid.org/0000-0002-4119-3748
Scopus Author ID: 35558133900, https://www.scopus.com/authid/detail.uri?authorId=35558133900
Researcher ID: G-8500-2014, http://www.researcherid.com/rid/G-8500-2014
Google Scholar: https://scholar.google.co.uk/citations?user=Uz5abgMAAAAJ&hl=en
Prizes, awards and honorary roles
2013 The Margaret Donaldson Early Career Prize for an outstanding contribution to developmental psychology. Awarded annually by the Developmental Psychology Section of the British Psychological Society
2009 European Research Council Starting Independent Researcher Award "Human Embodied Multisensory Development" (1.2m€).
2005 – date Honorary Research Fellow of the Centre for Brain and Cognitive Development, Birkbeck, University of London
2003 ESRC Postdoctoral Fellowship Award
2016 – 2018 Editorial Board Member of Goldsmiths Press
2016 – date Editorial Board Member of the British Journal of Developmental Psychology
2016 – date Editor (with Robin Banerjee) of Infant and Child Development
2008 – 2016 Associate Editor of the British Journal of Developmental Psychology
External examining
2016 – date External Examinerfor MSc in Child Development and Education, Department of Education, University of Oxford
2013 – date External Assessor for Psychology programmes, City University, Hong Kong
2013 – date External Examiner for the BSc (Hons.) Psychology, Queens University, Belfast
2010 – date External Examiner for the MSc Developmental Cognitive Neuroscience, Durham University
Representative publications:
Orioli, G., Bremner, A.J., & Farroni, T. (in press). Multisensory perception of looming and receding objects in human newborns. Current Biology.
Leman, P., Bremner, A.J. (in press). Developmental Psychology, 2nd Ed. London, UK: McGraw-Hill.
Linnell, K., Bremner, A.J., Caparos, S., Davidoff, J., & De Fockert, J.W. (2018). Urban experience alters lightness perception. Journal of Experimental Psychology: Human Perception & Performance.
Rigato, S., Banissy, M., Romanska, A., Thomas, R., van Velzen, J., & Bremner, A.J. (2017). Cortical signatures of vicarious tactile experience in four-month-old infants. Developmental Cognitive Neuroscience.
Bremner, A.J., & Spence, C.(2017). The development of tactile perception. In J. Benson (Ed.), Advances in Child Development and Behavior, Vol. 52. Oxford, UK: Elsevier.
D'Souza, H., Cowie, D., Karmiloff-Smith, A., & Bremner, A.J. (2017). Specialization of the motor system in human infants: From broadly tuned to selectively specialized purposeful actions. Developmental Science, 20, e12409.
Bremner, A.J. (2017). The origins of body representations in early life. In A. Alsmith & F. De Vignemont (Eds.), The body and the self, revisited. Cambridge, MA: MIT Press.
Freier, L., Mason, L., & Bremner, A.J. (2016). Perception of visual-tactile co-location in the first year of life. Developmental Psychology, 52,2184-2190.
Bremner, A.J., Doherty, M., Caparos, S., de Fockert, J.W., Linnell, K.J., & Davidoff, J. (2016). Effects of culture and the urban environment on the development of the Ebbinghaus illusion. Child Development, 87, 982-981.
Begum Ali, J., Spence, C., & Bremner, A.J. (2015). Human infants' ability to perceive touch in external space develops postnatally. Current Biology, 25, R978-R979.
Bremner, A.J., Lewkowicz, D.J., & Spence, C. (Eds.) (2012). Multisensory Development. Oxford, UK: Oxford University Press.
Rigato, S., Begum Ali, J., van Velzen, J., & Bremner, A.J. (2014). The neural basis of somatosensory remapping develops in human infancy. Current Biology, 24, 1222-1226. doi:10.1016/j.cub.2014.04.004
Cowie, D., Makin, T., & Bremner, A.J. (2013). Children's responses to the Rubber Hand Illusion reveal dissociable pathways in body representations. Psychological Science, 24, 762-769.
Bremner, A.J., Caparos, S., Davidoff, J., de Fockert, J.W., Linnell, K.J., & Spence, C. (2013). "Bouba" and "Kiki" in Namibia? A remote culture make similar shape-sound matches, but different shape-taste matches to Westerners. Cognition, 126, 165-172.
Bremner, A.J., Holmes, N.P., & Spence, C. (2008). Infants lost in (peripersonal) space? Trends in Cognitive Sciences, 12, 298-305.
Bremner, A.J., Mareschal, D., Lloyd-Fox, S., & Spence, C. (2008). Spatial localization of touch in the first year of life: Early influence of a visual spatial code and the development of remapping across changes in limb position. Journal of Experimental Psychology: General, 137, 149-162.
Perception, Cognition and Action
Download our Postgraduate brochure
Download our Undergraduate brochure


"Not sure what you think about this but I think this is something other artists should know (but I might be doing it wrong).
"It's about that 80/20 rule. I realised today that I spend most of my 'marketing' time on the internet. But 80% of my sales happen from doing exhibitions.
"So basically I could be putting my time into the wrong thing.
"I'm not saying I should quit the internet but i should probably be putting in less time in.
"True the internet connects you to the whole world … but it connects everybody else too.
Very nice. I love seeing people take action and become more aware, with the things I teach.
That 80/20 rule, in case you're not aware, is what's also called the Pareto Principle.
It says that in general, 80% of your results come from 20% of your efforts.
And yes, this reader makes sense: if most of his sales are offline then it's good thinking to put more effort into getting in front of people.
You need to take a step back there.
It's not so much a question of how much time you should put into internet marketing, but rather how you apply that time.
If 80% of your sales come from direct interaction, then how can you modify your internet efforts so that they result in more offline conversations….?
Yes it's good to get seen more.
But you'll need to make that happen, it takes work.
Visiting people, phone calls, emails – if you sit still you won't get into exhibitions.
So the question is: if you want to get into more exhibitions, how can you use the internet most efficiently to make that happen…?
As always, start with fun: What's the most enjoyable method for you?
Next, determine how viable it is, in other words: Is it likely to get you the most results for as little effort as possible.
Note that the last question isn't about getting results automatically, but instead about efficiency.
And, if you want help in figuring out which actions, and guidance on how to iterate and optimise them?
Then I'll coach you through it.
Just hit reply if you want me on your team.


text:
Q: How to pip install Holopy in Colab I'm trying to install holopy in google colab, but very stuck right now.
I need to install this specific package, and seems like I need to install holopy in colab first.
from holopy.core.metadata import get_spacing, data_grid, copy_metadata

I've been trying to install using
%pip install hologram-python or %pip install holopy
!pip install hologram-python or !pip install holopy
but none of them includes the package I'm looking for.
Any advice?

A: Run every one of the following commands in a new block

*

*Download or clone the latest version of HoloPy from Git Hub at https://github.com/manoharan-lab/holopy.
!git clone https://github.com/manoharan-lab/holopy

*cd to that cloned directory
cd /content/holopy

*Pip installs all the requirements
!pip install -r requirements.txt

*now run python setup.py develop
If everything above goes all right, check if holopy has been installed correctly by the following command
from holopy import check_display
check_display()

You should see an image like this  and you should be able to change the square to a circle or diamond by using the left/right arrow keys.

Hope it helps.


Super condition. Freshly painted, New w/w carpet. 3 BR and 2 full BA on upper level; den/4th BR and full bath on lower level. Rec. Rm . Washer & Dryer. Two parking spaces in front of house. Fenced yard. Long term lease is OK. Make application on- line.


After spotting Alice Megan's nail polish tag, I thought I'd take part too!
I was terrible at painting my nails, but one of the tasks on my 35 at 35 list was to practice and to keep my nails looking nice for at least a month. I succeeded!
What's your favourite nail polish company? I've got a mix of different brands. My favourite used to be O.P.I. but to be honest I'm really liking Barry M polishes.
When do you change your nail polish? I would potentially change it for a night out or special occasion, but otherwise I just tend to change it when it starts to chip.
Darks or brights? Both. I think it's good to mix things up a bit. I tend to wear darks during the autumn and winter and then swap for brights in spring and summer.
Matte nails...in or out? I've never tried it, so for me it'd be out.
Do you like French manicures? I used to get the odd French manicure in my twenties and would occasionally get acrylic nails too, but I don't have the money to get manicures anymore so I do my polish myself and I'm not very good at French manicures!
What's your favourite winter colour? Hmmm... at the moment I'm loving grey (China Glaze, Concrete Catwalk) or a black sparkle (Barry M, Black Multi Glitter).
High street or high end? Both. Depends on how much cash I have! I used to buy O.P.I. and China Glaze polishes from Sally but more recently I've been buying Barry M from Superdrug.
Are you a pro polisher? I think I'm ok at doing nails. Not brilliant, and certainly can't do fancy nail designs or anything. But I can do a basic polish ok.
Do you use a top coat or a base coat? Yes, both. At the moment I'm using deborah lippmann Hard Rock hydrating hardener which was a gift from Bex at The Olive Dragonfly.
You're definitely better at this nail lark than me I must take tips and start using a base and top coat!


Which login should I use? Marine-Id or Extranet login?
How can I get data?
I would like to download a large number of stations (>10 000), how should I proceed?
Are all data unrestricted and free?
How to acknowledge SeaDataNet when using the data?
What is a SeaDataNet dataset?
What is a SeaDataNet role?
How to cite SeaDataNet agggregated products?
How to cite SeaDataNet climatologies?
Click on the left side image (CDI Search).
Enter your selection criteria which could be a cruise name (field Free Search), a geographic area drawn or manually input, a type of instrument... ('?' give an on-line help on all criteria and clicking on Help (left side of the map) gives a detailed description of the portal use.
Summary gives a synthetic view of your search result, you can then work on a sub- part of the full selection from this summary.
If your search gives more than 20 records, you can change the number of records per page by checking 100 or 1000 and then Go (right side of the map).
When your selection is OK (Maximum 10 000 records per basket), click on Basket to proceed to data download.
Follow the link that shows up at the end of the previous step (or in the email you will receive) to access the RSM (Request Status Manager) which help you to follow the result of your search in all contacted data centres.
You can see all status of your download, per data centre: At first your request status is Approval pending and then it will move to Ready for user action (if data are not restricted) or to Access Denied (if data are still restricted).
When the status of your request is Ready for user Action, click on the number of available data to access the data downloading.
Please feel free to contact us if you have any problem using the SeaDataNet web portal for data downloading.


text:
\section{Introduction}

A spinor Bose-Einstein condensate is a matter-wave that can access internal spin degrees of freedom \cite{Ho1998a,Ohmi1998a,Stenger1998a}. Theoretically this type of condensate is described by a multi-component (spinor) wavefunction that evolves according to a nonlinear Schr\"{o}dinger equation (NLSE) known as the spinor Gross-Pitaevskii equation (GPE). This system is unique in that it is accurately described by a well-characterized microscopic theory, and it exhibits the phenomena of superfluidity enriched by various types of magnetic (i.e.~spin) ordering. For these reasons spinor condensates are an active topic of research in theory and experiment (e.g.~see \cite{Kawaguchi2012R,Stamper-Kurn2013R}).

We consider the case of a spin-1 system in which the atoms have three internal spin states conveniently labelled by the $z$-projection quantum number as $m=-1,0,+1$. This system has been realized in experiments with ultra-cold $^{87}$Rb and $^{23}$Na atoms. In weak magnetic fields the atomic collisions are rotationally invariant and the nonlinear interactions only depend on two parameters: the spin-independent (i.e.~density) interaction strength $c_0$ and the spin-dependent (i.e.~spin-density) interaction strength $c_1$ (see \cite{Kawaguchi2012R}). For the atoms discussed above $0 < |c_1|\ll c_0$. When $c_1=0$ we get the three-component vector NLSE which is completely integrable in time [for systems that are one-dimensional (1D)] by the inverse scattering method \cite{ablowitz2004discrete}.
When $c_0 = c_1 = -c$, the system is equivalent to a matrix NLSE, which is also completely integrable in 1D by the inverse scattering method \cite{ieda2004exact}.
For general $c_{0,1} \neq 0$ the system is not known to be integrable. Since this is the parameter regime currently accessible to experiments, and since experiments are often in quasi-two-dimensional or fully three-dimensional regimes, it is essential to have numerical methods for solving the system evolution. Previous work in this direction includes an algorithm presented by Wang \cite{Wang2007a} and Bao \textit{et al.}~\cite{Bao2009a} which splits the evolution equation into three parts (flows), which are individually solved, although one of the nonlinear flows is treated approximately. More recently a scheme was proposed which splits the evolution equation into two flows, but uses a brute force diagonalization to construct the flow and a first-order composition method to generate an approximate full solution.

In this paper we develop two new numerical methods for solving the spin-1 Gross-Pitaevskii equation. The basis of our work is a two-way splitting of the spin-1 evolution equation that we show leads to two exactly solvable flows.
We discuss how our splitting can be used to construct second-order and fourth-order symplectic integration algorithms. We develop two non-trivial numerical tests to compare these schemes against two other approaches, and show that our schemes perform well.

We briefly outline the paper. In Sec.~\ref{Secspin1GPE} we introduce the spin-1 GPE and  our notation. We then discuss how this evolution equation can be spit into two flows in Sec.~\ref{SecSplittings}, and present the analytic solutions to these seperate flows. In Sec.~\ref{SecSympletic} we briefly review symplectic composition and discuss the second-order and fourth-order compositions we use in this paper to realize two algorithms from our flow solutions. In Sec.~\ref{SecApplications}  we discuss some details of a 1D implementation of our algorithm. We then introduce two non-trivial spin-1 problems (a continuous wave solution and the collision of two quasi-solitons) that we use to test the performance of our algorithms against the method developed in Refs.~\cite{Wang2007a,Bao2009a} and against the spin-1 extension of a widely used technique based on the fourth-order Runge-Kutta method. We conclude in Sec.~\ref{Sec:Conclusion}.

\section{Spin-1 GPE}\label{Secspin1GPE}
\subsection{Spin-1  energy functional}
A spin-1 condensate system is described by the spinor field $\bm{\psi} = (\psi_+, \psi_0, \psi_-)^T$, which denotes the condensate field in the $m_F=+1,0,-1$ hyperfine sub-levels respectively.
The ground state energy functional  is \begin{align}
E[\bm{\psi}] &=
\int d\bm{r}
\left [
\bm{\psi}^\dagger h_0 \bm{\psi} +
\frac{c_0}{2} n^2 + \frac{c_1}{2}|\bm{F}|^2
\right],\label{spinE}
\end{align}
where the single particle hamiltonian is
\begin{align}
(h_0)_{m m^\prime} &= \left[
-\frac{\hbar^2 \nabla^2}{2M} + V_\text{ext} - pm + qm^2
\right] \delta_{m m^\prime},
\end{align}
$V_\text{ext}$ is the external trapping potential (often harmonic). The spatially constant energies  $p$ and $q$ denote the linear and quadratic Zeeman terms, respectively, and arise in experiment from a uniform magnetic bias field along $z$. The nonlinear terms are the total density
\begin{align}
n = \sum_m |\psi_m|^2,
\end{align}
 and the magnitude of the spin density $\bm{F}=(F_x, F_y, F_z)$, where
 \begin{align}
 F_\alpha = \bm{\psi}^\dagger f_\alpha \bm{\psi},\quad \alpha \in \{x,y,z\} \label{EqFalpha}
 \end{align}
  with $f_\alpha$ the spin-1 matrices. In particular, we have
  \begin{align}
  F_z &= |\psi_+|^2 - |\psi_-|^2,\label{Fz}\\
  F_\perp &\equiv F_x + i F_y = \sqrt{2}(\psi_+^* \psi_0 +\psi_0^* \psi_-),\label{Fp}
  \end{align}
   with which we can write $|\bm{F}|^2 = F_z^2 + |F_\perp|^2$. The constants $c_0=\frac{4\pi}{3}(a_0+2a_2)\hbar^2/M$, $c_1=\frac{4\pi}{3}(a_2-a_0)\hbar^2/M$ are the density and spin-dependent interactions, with $a_S$ the $s$-wave scattering length for the spin-$S$ collision channel \cite{Ho1998a}.

\subsection{Spin-1 Gross-Pitaevskii equations}
The time evolution of the condensate field is given by
\begin{align}
i \hbar \frac{\partial \psi_m}{\partial t} = \frac{\delta E}{\delta \psi^*_m}.\label{spinGPEa}
\end{align}
We choose convenient computational units of space $x_0$ (which would usually be set by the trap $V_{\mathrm{ext}}$) and time $t_0=Mx_0^2/\hbar$, and from hereon take all parameters to be expressed in terms of these units.  Then evaluating  (\ref{spinGPEa}) yields the spin-1 GPEs
\begin{align}
\dot{\psi}_{m} &=
-i\left[-\tfrac{1}{2}{\nabla^2} +qm^2+ V_m \right]\psi_m+ I_m ,\label{sGPEm}
\end{align}
where we have set
\begin{align}
V_m \equiv V_\text{ext} -pm  + c_0 n,
\end{align}
as the spin-independent effective potential plus linear Zeeman, and
\begin{align}
I_+ &=  c_1(n_ 0 + F_z)\psi_+
+   c_1\psi_0^2 \psi^{*}_-, \label{Ip1}\\
I_0 &= c_1(n - n_0)]\psi_0 + 2c_1  \psi_+ \psi_- \psi_0^*, \\
I_-&=  c_1(n_0 - F_z)\psi_-
+  c_1\psi_0^2 \psi^{*}_+.\label{Im1}
\end{align}
are the spin-dependent interaction terms, with $n_0=|\psi_0|^2$.

\subsection{Conserved quantities}\label{SecConservQ}
Under evolution according to the spin-1 GPE the condensate has several important conserved quantities, notably energy $E[\bm{\psi}]$ given by Eq.~(\ref{spinE}), normalization
\begin{align}
N[\bm{\psi}] = \sum_m\int d\bm{r}|\psi_m|^2=\int d\bm{r}\, n,
\end{align}
and the $z$-component of magnetization
\begin{align}
M_z[\bm{\psi}] = \int d\bm{r}\left(|\psi_+|^2-|\psi_-|^2\right)=\int d\bm{r}\, F_z.
\end{align}

\section{Splittings}\label{SecSplittings}
We denote the spin-1 GPE as $\dot{\bm{\psi}}=\hat{f}\bm{\psi}$, where $\hat{f}$ denotes the nonlinear operator that actions Eq.~(\ref{sGPEm}), with formal solution $\bm{\psi}(t)=\exp(\hat{f}t)\bm{\psi}(0)$.  For numerical solution it is desirable to break up the operator $\hat{f}$ into two parts as
\begin{align}
\hat{f}=\hat{f}_A+\hat{f}_B.
\end{align}
In breaking up the problem this way we can deal with the individual \emph{flows}  separately
\begin{align}
\dot{\bm{\psi}}=\hat{f}_\mu\bm{\psi},\qquad\mu=A,B,
\end{align}
 and from these compose a solution to the full problem.
 The important feature of the scheme we present here is that we break up the problem into just two flows, which simplifies the composition to higher order schemes, and that for each flow we find an exact solution allowing us to efficiently construct
\begin{align}
\bm{\psi}(t)=e^{\hat{f}_\mu t}\bm{\psi}(0),\qquad\mu=A,B.\label{fAB}
\end{align}
In constructing these exact flows we consider a spatial domain with periodic boundary conditions.

\subsection{Kinetic and quadratic Zeeman flow ($\hat{f}_A$)} \label{secfA}
We take the first flow to include the kinetic energy and quadratic Zeeman term, i.e.~we define $\dot{\bm{\psi}}=\hat{f}_A\bm{\psi}$ to be
\begin{align}
\dot{\psi}_m=-i\left[-\tfrac{1}{2}\nabla^2+qm^2\right] \psi_m.\label{IP}
\end{align}
This flow is diagonal in Fourier space, yielding the solution
\begin{align}
\psi_m(t) &= \mathcal{F}^{-1}\left\{ e^{-i(\frac{1}{2}k^2 + qm^2)t} \mathcal{F} [\psi_m(0)] \right\},
\end{align}
where $\mathcal{F}$ denotes a Fourier transform of the appropriate spatial dimension for the problem and $k=|\bm{k}|$, with $\bm{k}$ the Fourier space coordinate.

\subsection{Trap, linear Zeeman and interaction flow ($\hat{f}_B$)}\label{secfB}
The other flow contains the remaining terms, so that  $\dot{\bm{\psi}}=\hat{f}_B\bm{\psi}$ is given by
\begin{align}
\dot{\psi}_m = -i(V_m\psi_m+I_m).\label{fB}
\end{align}
Importantly, evolution according to (\ref{fB}) conserves both $n$ and $F_z$, thus the $V_m$ are constants in time.
This suggests applying the transformation
\begin{align}
\psi_m = \tilde{\psi}_m e^{-i {V}_mt},\label{Vmt}
\end{align}
which gives the system
\begin{align}
\dot{\tilde{\bm{\psi}}} &=
-ic_1
\begin{pmatrix}
F_z & \tfrac{1}{\sqrt{2}} F_\perp^* & 0 \\
\tfrac{1}{\sqrt{2}} F_\perp & 0 & \tfrac{1}{\sqrt{2}} F_\perp^* \\
0 & \tfrac{1}{\sqrt{2}} F_\perp & -F_z \\
\end{pmatrix} \tilde{\bm{\psi}}
\equiv -i c_1 R \tilde{\bm{\psi}},\label{psit}
\end{align}
where $F_z$ and $F_\perp$ are evaluated according to Eqs.~(\ref{Fz}) and (\ref{Fp}), but using the spinor $\tilde{\bm{\psi}}$.
We also observe that $\tilde{\bm{\psi}}$ evolution according to Eq.~(\ref{psit}) also conserves $\tilde{F}_\perp$, and thus the matrix $R$ is constant in time.
The solution, by exponentiation, is
\begin{align}
\tilde{\bm{\psi}}(t)
&=\left[
\cos \left(c_1 F t\right) \mathbb{1}
-i \frac{\sin \left(c_1 F t\right)}{F}
R \right] {\bm{\psi}}(0),\label{EqflowBpsitilde}
\end{align}
with $F=|\mathbf{F}|$, and $\mathbb{1}$ representing the identify matrix. We also note that $\tilde{\bm{\psi}}(0)={\bm{\psi}}(0)$. Hence the complete solution to the $\hat{f}_B$ flow  [i.e.~Eq.~(\ref{fAB})] is given by applying the transform (\ref{Vmt}) to the result in Eq.~(\ref{EqflowBpsitilde}). Our analytic solution (\ref{EqflowBpsitilde}) is key to the methods we develop here. As far as we are aware this solution has not appeared in the literature, although the specialization of this result to the case $F_z=0$ was given in Ref.~\cite{Pu1999}.

We note that we could include $q$  in the $\hat{f}_B$ subsystem, which would be necessary if $q$ was position-dependent. However, the solution in this case involves Jacobi elliptic functions and their inverses \cite{zhang2005coherent} which are slow to compute compared with sines and cosines.

\section{Symplectic composition schemes}\label{SecSympletic}
We can approximate the solution to the GPE by composing the two subsystem solutions. Since our system is Hamiltonian, it is desirable to use a symplectic composition method in order to maintain the Hamiltonian geometric structure of phase space in our solution. Any composition of exact flows is a symplectic method. Both $\hat{f}_A$ and $\hat{f}_B$ preserve $N$, $M_z$ and $E$, thus all composition methods will also preserve $N$, $M_z$ and $E$ (up to roundoff error). In what follows we present a second-order symplectic composition and then discuss a fourth-order method.

\subsection{Second-order symplectic method {\bf (S2)}}
The \textit{Leapfrog} composition is a simple second-order composition for advancing the wavefunction by a time step of size $\tau$. This composition takes the form
\begin{align}
\bm{\psi}(\tau)=e^{\hat{f}\tau}\bm{\psi}(0)\approx e^{ \hat{f}_A\frac{\tau}{2}} e^{  \hat{f}_B\tau} e^{ \hat{f}_A\frac{\tau}{2}}\bm{\psi}(0),
\end{align}
where $e^{\hat{f}_At}$ and $e^{\hat{f}_Bt}$ represents the flow solutions presented in Secs.~\ref{secfA} and \ref{secfB}, respectively. We denote this scheme as  S2 in the results we present.

\subsection{Fourth-order symplectic method {\bf (S4)}}\label{sec4th}
It is possible to employ higher-order compositions. Here we consider  a general fourth-order method, due to Blanes and Moan \cite{Blanes2002a,mclachlan2006geometric}. This composition has the form
\begin{align}
\bm{\psi}(\tau)&=e^{\hat{f}\tau}\bm{\psi}(0)\nonumber\\
&\approx e^{ \hat{f}_A a_6 \tau} e^{ \hat{f}_B b_6 \tau} \cdots e^{ \hat{f}_Aa_1 \tau} e^{ \hat{f}_Bb_1 \tau} e^{\hat{f}_Aa_0 \tau}\bm{\psi}(0),
\end{align}
where we give the values of the constant coefficients $\{a_m,b_n\}$, with $m=\{0,\ldots,6\}$ and $n=\{1,\ldots,6\}$ in Appendix \ref{Sec4thcoefs}. We denote this scheme as  S4 in the results we present.

\section{Applications}\label{SecApplications}
\subsection{Algorithms}
\subsubsection{One-dimensional discretization and stability}\label{secDiscretization}
We consider a spinor condensate in one spatial dimension discretized on a mesh of  $M$ points
\begin{equation}
x_j= -\tfrac{L}{2} + j\Delta x,\qquad j=0,\ldots M-1,
\end{equation}
over the interval $[-\tfrac{L}{2},\tfrac{L}{2})$,
with spacing $\Delta x=L/M$. We assume periodic spatial  boundary conditions, allowing us to implement the Fourer transformations required to implement the kinetic energy operator by Fast-Fourier Transform, with a corresponding reciprocal space mesh
 \begin{equation}
k_j= -k_{\max} + j\Delta k,\qquad j=0,\ldots M-1,
\end{equation}
with $\Delta k=2\pi/L$ and $k_{\max}= {\pi}M/L$.
The discretization we have outlined is equivalent to a spectral approach based on planewaves if the rectangular rule is used as the associated quadrature, i.e.~$\int dx f(x)\to\sum_jf(x_j)\,\Delta x$.

We note that splitting the kinetic energy operator apart from the nonlinear potential in the GPE introduces an instability for any composition method \cite{chin2007higher}, which can only be avoided by choosing the time step size subject to
\begin{align}
\Delta t \lesssim t_{\mathrm{stab}} &\equiv \frac{\pi}{E_{max}}, \\
E_{\max} &= \max_j\left|\frac{1}{2} k_{j}^2 + q\right|,
\end{align}
which says that the highest frequency of the kinetic energy and quadratic Zeeman subsystem must be resolvable over each time step. Satisfying this condition is necessary but not sufficient for ensuring global numerical stability.
It is a severe restriction, with high spatial resolution simulations leading to very small time steps for global stability. Often the step size can be much larger if the integration time of interest is shorter than the time required for the instability to grow.

\subsubsection{Comparison algorithms: W2 and RK4}
We compare our two schemes against two alternative approaches.

The first method is a second-order time-splitting method, that we denote (W2), introduced by Wang in Ref.~\cite{Wang2007a} (also see \cite{Bao2009a,Bao2010a}). This differs from our scheme because it employs a three-way splitting of the evolution equation, and one of the flows cannot be analytically solved and is treated using a second-order Runge-Kutta step (noting that if this flow was exactly solved then this method would be symplectic).

The second method we consider is a method based on the fixed time step fourth-order Runge-Kutta method which we denote as (RK4). We have implemented this algorithm using our spatial discretization of Sec.~\ref{secDiscretization} and employing the \textit{interaction picture} technique \cite{Dennis2013a} to exponentiate the kinetic energy flow and improve the algorithm performance.  This type of method is quite commonly used in the BEC community and we have made an immediate extension to the spinor case using the discretization of Sec.~\ref{secDiscretization}. Details of this algorithm are given in Appendix \ref{AppRK4}.

We compare the various algorithms in terms of the number of functions evaluations required per time step in Table \ref{Tabsteps}. The S2 and W2 algorithms require the least evaluations per time step. In 1D tests we find that the S2 is roughly 50\% faster than W2 per time step, with this difference largely arising from the much simpler form of the nonlinear term to evaluate in S2. For larger problems, and particularly higher dimensional problems we expect the single round trip FFT required for the S2 algorithm (using FSAL) will be advantageous.

\begin{table}[h]
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{ccc}
	\hline Algorithm & \quad FFT roundtrips (FSAL) & \quad Nonlinear evals \\
	\hline 	S2 & 2(1) & 1 \\
	W2 & 2 & 3 \\
	RK4 & 4 & 4 \\
	S4 & 7(6) & 7 \\
	\hline
\end{tabular}
\caption{Number of FFT roundtrips (i.e.~FFT and inverse FFT) and nonlinear-term evaluations required for each algorithm. The number in brackets indicates the number required per time-step which can be reduced by the first same as last (FSAL) property (i.e.~last stage is evaluated at the same point as the first stage of the next step).}\label{Tabsteps}
\end{table}

\subsection{Continuous-wave comparison}\label{CWsec}
\subsubsection{Analytic solutions}
Recently an analytic theory was developed for a family of continuous-wave solutions to a uniform spin-1 condensate  \cite{Tasgal2013}. This solution has the feature that it allows the chemical potentials and wavevectors of the components to be different, and thus forms a non-trivial test for numerical methods.
\begin{figure}
	\includegraphics[width=3.1in]{fig1}
	\vspace*{-0.1cm}\caption{(Color online) The components of the spin density are shown for a continuous wave solution with parameters: $k_+=5$, $k_-=3$, $\theta_+=0$, $\theta_-=0$, $A_+=3$, $A_-=1$ with $c_0=10$, $c_1=1$ and $q=0.5$. The remaining parameters are determined by the relations given in Appendix \ref{CWsoln}.}\label{FigCWpsi}
\end{figure}

These waves are analytic solutions to the uniform spin-1 GPE for $p=0$ and $c_1\ne0$ and have the form
\begin{align}
\psi_+(x) &= A_+ e^{i( \theta_+ + k_+ x - \omega_+ t)}, \label{cwsolp1}\\
\psi_0(x) &= A_0 e^{i( \theta_0 + k_0 x - \omega_0 t)}, \\
\psi_-(x) &= A_- e^{i( \theta_- + k_- x - \omega_- t)}. \label{cwsolm1}
\end{align}
Here the amplitudes $\{A_-,A_0,A_+\}$ for the three planewave components are taken to be real nonnegative numbers with phases $\{\theta_-,\theta_0,\theta_+\}$. These amplitudes determine the density $n=\sum_mA_m^2$ and magnetization density $F_z=A_+^2-A_-^2$, both of which are conserved quantities (also the individual component densities  $|\psi_m|^2$).
 We have also introduced the component wavevectors  $\{k_-,k_0,k_+\}$ and chemical potentials $\{\omega_-,\omega_0,\omega_+\}$.  When two of the three amplitudes, phases and wavevectors are chosen the remaining values, and the chemical potentials, are determined. There is also an additional parity-like parameter $n_p$ which can take the value of $0$ or $1$, that determines an additional $\pi$ shift in the phase $\theta_0$ relative to $\theta_\pm$. The equations linking these quantities are given in Appendix \ref{CWsoln} (also see \cite{Tasgal2013}).

An example of this solution is shown in Fig.~\ref{FigCWpsi}, visualized in terms of the components of spin-density [see Eq.~(\ref{EqFalpha})].
We see that this solution is a nonlinear spin-wave in $F_\perp$ with the $F_x$ and $F_y$ components of spin density precessing about the constant $F_z$ as the wave propagates.

\subsubsection{Numerical results}
 In Fig.~\ref{FigCWrelE} we examine how the various schemes conserve the constants of motion identified in Sec.~\ref{SecConservQ}. Since these quantities should remain constant we characterize the relative error by the absolute relative change in these quantities, i.e.~as
\begin{equation}
\mathrm{Rel.\,Error}\,\, Q\equiv\left|\frac{Q(t)-Q(0)}{Q(0)}\right|,\qquad Q\in\{N,M_z,E\},
\end{equation}
where $Q(t)$ indicates the quantity evaluated at time $t$ during the propagation.
Generally the results show that the symplectic algorithms (i.e.~S2 and S4) and W2 outperform the Runge-Kutta algorithm at conserving the constants of motion, which is expected since symplectic algorithms preserve the geometric properties of phase space.

The S2 and S4 are more accurate than the W2 algorithm.
Although S2 and W2 are both second-order, the W2 method only approximately treats the nonlinear spin exchange terms (see Appendix~\ref{AppW2}) and uses a three-way splitting (which, even for exact flows, is generally less accurate) and thus has larger error coefficients than the S2 method.

We note that W2 and RK4 show linear growth with time $t$ for all three conserved quantities.
S2 and S4 achieve bounded oscillating errors for $N$ (indeed, S2 oscillates around the discretization limit of $2\times 10^{-16}$), and linear error growth in the long-time limit for $M_z$ and $E$.

Symplectic methods should have bounded $E$ errors, while other conserved quantities can be either bounded or linear in time. Bounded errors should grow in time solely due to roundoff errors which accumulate like a random walk, scaling as $t^{1/2}$, however this can be difficult to achieve in practice due to systematic errors \cite{hairer2008}. Figure~\ref{FigCWrelE}(c) shows that S2 and S4 achieve $t^{1/2}$ error scaling until a linear systematic error begins to dominate after $t\sim1$.

One source of systematic error for our methods comes from our implementation of the $\hat{f}_A$ flow not being exact to numerical precision. Of course given a continuous wave our Fourier-space solution is calculated using an exact quadrature, and for other waves achieves exponential accuracy, i.e. it approaches roundoff accuracy in the limit of a large number of discretization points $M$ (although this is quickly offset by increased floating point arithmetic errors from FFTs). In this test the $\hat{f}_A$ flow is accurate to $\sim10^{-15}$. We note that this error is asymmetric across the three components of the wavefunction due to the asymmetry of the wavevectors $k_m$, which means the observed error behaviour is difficult to analyze theoretically.

\begin{figure}
	\includegraphics[width=3.1in]{fig2}
	\vspace*{-0.5cm}
	\caption{(Color online) Relative errors in the conserved quantities (a) normalization, (b) magnetization and (c) energy for the propagation of the continuous wave solution. The various algorithms, as labeled in (b), all use the same time step ($\tau = 3.125\times10^{-4}$) and  mesh ($M=256$, $L=2\pi$). The initial condition and other simulation parameters are given in Fig.~\ref{FigCWpsi}. Dashed (dotted) line is a guide to the eye to indicate linear (square root) scaling with time $t$.}\label{FigCWrelE}
\end{figure}

\begin{figure}
	\includegraphics[width=3.1in]{fig3}
	\vspace*{-1.2cm}\caption{(Color online) Global error in propagating the continuous wave solution. (a) Global error during a propagation using a time step of $\tau = 6.25\times10^{-4}$.  Dashed line is a guide to the eye showing linear scaling with time $t$. (b) Scaling of the global error after propagation to time $t_f=0.1$ as a function of the as function of the time step $\tau$. The line types for the various algorithms are indicated in (b), as is the estimated stable time step size ($t_{\mathrm{stab}} = 3.8\times10^{-4}$). The dashed (dotted) lines in (b) indicate fourth-order (second-order) power law scaling with respect to $\tau$. Other parameters for the solution are as indicated in Fig.~\ref{FigCWrelE}. }\label{FigCWglobE}
\end{figure}

A more complete test of the numerical error being introduced by each method is to directly compare the wavefunction propagated to time $t$ [which we denote as $\bm{\psi}^{\mathrm{num}}(t)$] against the exact analytic solution  $\bm{\psi}(t)$ given in Eqs.~(\ref{cwsolp1})-(\ref{cwsolm1}). We do this by evaluating the maximum global error introduced by each algorithm, which we define as
\begin{equation}
\mathrm{Global\,Error}\,\, \equiv \max_{x_j,m}\left|{\psi}_m^{\mathrm{num}}(x_j,t)-{\psi}_m(x_j,t)\right|.
\end{equation}

 In Fig.~\ref{FigCWglobE}(a) we show the growth of the global error over time for a propagation of fixed time step $\tau=6.25\times10^{-4}$. This is the same propagation examined in Fig.~\ref{FigCWrelE}. The global error shows that S4 is the best at approximating the exact analytic solution by a significant margin, with S2 better than W2, and all of these are better than RK4. Global error grows linearly for all methods until $t\sim1$ [i.e.~the time when the energy error starts growing linearly  for S4 in Fig.~\ref{FigCWrelE}(c)]. At this point both fourth-order methods shift their global error behaviour to be quadratic in time.
 %

 In Fig.~\ref{FigCWglobE}(b) we propogate our numerical solution for a fixed amount of time and consider the scaling of the global error with different time steps $\tau$. These results confirm the expected second-order scaling of the global error with $\tau$ for S2 and W2, as well as the expected fourth-order scaling for RK4 and S4.  For this case the prefactor on the global error for RK4 is up to 9 orders of magnitude larger than S4 depending on the choice of $\tau$. With the S4 method the improvement in global error saturates with decreasing $\tau$ at $\tau\sim10^{-3}$, slightly above the stability threshold, while the other methods show improvement in global error down to $\tau \sim 10^{-5}$. This is because S4 has the most function evaluations per time step and thus accumulates more roundoff errors than the other methods.
We note that due to our choice of a short propagation time, the splitting instability for timesteps greater than $t_\mathrm{stab}$ does not have time to manifest and we observe the expected scaling for all the methods even when $\tau>t_{\mathrm{stab}}$.

\subsection{Quasi-soliton comparison}
\subsubsection{Analytic form of spinor quasi-solitons}
In the experimentally relevant limit of small spin-dependent interaction, i.e.~$0 < |c_1| \ll c_0$, the spin-1 system is asymptotically equivalent to the Yajima-Oikawa (YO) system, which is integrable by inverse scattering and thus has soliton solutions \cite{Nistazakis2008}. We simulate the collision of two such quasi-solitonic solutions using the initial conditions [given by Eqs.~(37)-(38) of Ref.~\cite{Nistazakis2008}]
\begin{align}
\psi_\pm(x) &=
\sqrt{\frac{\mu}{2} - \frac{\nu}{2}[\mathrm{sech}^2(x_+) + \mathrm{sech}^2(x_-)]} \\
& \quad \times \exp\left(-i \sqrt{\frac{\nu}{\mu}} [\tanh(x_+) - \tanh(x_-)] \right), \nonumber
\end{align}
\begin{align}
\psi_0(x) &=
\nu^{3/4} \sqrt{\frac{\xi}{\eta \sqrt{\mu}}}
\Big[
\mathrm{sech}(x_+) e^{+i\sqrt{\mu/\nu} x_+ - i (\xi/\nu) x_+}
\\
& \quad
+
\mathrm{sech}(x_-) e^{-i\sqrt{\mu/\nu} x_- + i (\xi/\nu) x_-}
\Big],\nonumber
\end{align}
where $\nu \equiv 4 \eta^2 \delta$, $\delta \equiv c_1/c_0$, $\mu$ is the chemical potential,  $\eta$, $\xi$ are arbitrary parameters, and
$x_\pm = \sqrt{\nu}(x \pm x_0)$ are the initial positions of the centres of the two pulses. We take the interaction parameter ratio to be  $\delta= 0.0314$ (which is close to the value expected for $^{23}$Na atoms) and choose the other parameters to reproduce the collision studied in Ref.~\cite{Nistazakis2008}. The spinor evolution during the collision is shown in Fig.~\ref{FigQSpsi} (cf.~Fig.~1 of \cite{Nistazakis2008}).
The quasi-solitons are revealed as identical density dips (dark solitons) in the $\pm1$ component densities at the same location that the $m=0$ component has a density peak (bright soliton). For each quasi-soliton, the two dark solitons guide the bright soliton along. We see that under dynamical evolution the two quasi-solitons collide (at $t=19$) and then emerge, roughly keeping their shape.  Because the quasi-soliton is not a true soliton some radiation is emitted during the dynamics, visible as oscillations in the background component densities.  The domain of the simulation $L=384$ is chosen to be much larger than the region of interest containing the solitons in Fig.~\ref{FigQSpsi} to ensure that this radiation does not wrap around (due to periodic boundary conditions) and return to the solitons within the duration of the simulation.

\begin{figure}
	\includegraphics[width=3.1in]{fig4}
	\vspace*{-0.5cm}
	\caption{(Color online) Quasi-soliton collision: (a)-(d) snapshots of dynamics showing component densities during the collision dynamics at a range of times. We define the collision time as $t_\mathrm{col}=19$. These results were calculated using S4 with $\tau=0.01$ and match the results presented in Fig.~1 of  Ref.~\cite{Nistazakis2008}. Other parameters: $L=384$, $M= 2048$, $x_0=1$, $c_1=0.314$, $q=0$, $\eta=3.091$, $\xi=1.54$, $\mu=2$. }
	\label{FigQSpsi}
\end{figure}

\begin{figure}
	\includegraphics[width=3.1in]{fig5}
	\vspace*{-0.5cm}\caption{(Color online) Quasi-soliton evolution: Relative errors in non-zero conserved quantities (a) $N$ and (c) $E$.   (b) shows the magnitude of $M_z$ that develops in the W2 scheme, noting   $|M_z|$ remains zero for the other methods. Dashed (dotted) lines are guides to the eye showing linear (square root) scaling with respect to time $t$. Simulations are for the scenario (initial condition, time and mesh parameters) given in Fig.~\ref{FigQSpsi}.}
	\label{FigQSrelE}
\end{figure}

\begin{figure}
	\includegraphics[width=3.1in]{fig6}
	\vspace*{-1cm}
	\caption{(Color online) Global error in propagating the quasi-solitons using reference S4 solution with $\tau = 5\times10^{-4}$. (a) Global error during a propagation using a time step of $\tau = 0.02$. Dashed line is a guide to the eye that scales linearly with time $t$, while vertical dash-dotted line shows the time of collision of the quasi-solitons ($t_\mathrm{col}=19$).
		(b) Scaling of the global error with time step $\tau$ of the solitons propagated to $t=2$. Dashed (dotted) line indicates fourth-order (second-order) power law scaling with respect to $\tau$, while vertical dash-dotted line shows the estimated stable time step size ($t_\mathrm{stab}=0.022)$. Simulation parameters and solution mesh are as discussed in  Fig.~\ref{FigQSpsi}.
		}
	\label{FigQSGE}
\end{figure}
\subsubsection{Numerical results}
We compare the the various schemes for the problem shown in Fig.~\ref{FigQSpsi}. First, in Fig.~\ref{FigQSrelE} we compare the relative errors in the conserved quantities $N$ [Fig.~\ref{FigQSrelE}(a)] and $E$ [Fig.~\ref{FigQSrelE}(c)].
The symplectic algorithms and W2 are seen to conserve $N$ much better than RK4 at the step size considered in Fig.~\ref{FigQSrelE}(a), with error growth behaviour like $t^{1/2}$ which is indicative of being limited by roundoff. The symplectic algorithms and W2 also show bounded error growth for $E$ in Fig.~\ref{FigQSrelE}(c). In contrast, for RK4 the error in $N$ and $E$ grows linearly with time. The S4 method demonstrates much better conservation of $E$ than all other methods, with $t^{1/2}$ error growth suggesting it has reached a roundoff limit.

The quantity $M_z$ is zero for this soliton (since $|\psi_+|^2=|\psi_-|^2$), thus it is not sensible to use the relative error to compare how well the methods conserve $M_z$. Instead in Fig.~\ref{FigQSrelE}(b) we plot the absolute value of $M_z$ for the W2 method, which is the only scheme that develops a non-zero value.
This is not simply due to roundoff, which would give $t^{1/2}$ scaling, but because the W2 method uses an approximate solution to the nonlinear spin exchange terms which does not explicitly conserve $F_z$ (see Appendix \ref{AppW2}).

Because we do not have an exact analytic solution for the dynamics of the soliton collision we are unable to make the same global error comparison we performed for the continuous wave solutions. However, we use the S4 method with $\tau=5\times10^{-4}$ to produce reference results which we compare the methods against for appreciably larger time steps.
In Fig.~\ref{FigQSGE}(a) we show the scaling of the global error with time $t$. RK4 scales almost linearly while the symplectic methods and W2 slowly increase as they evolve to the collision at $t_\mathrm{col}$.
The results in Fig.~\ref{FigQSGE}(b) demonstrate the expected fourth-order scaling with $\tau$ for S4 and RK4, and the second-order scaling for W2 and S2. S4 is again the most accurate method by several orders of magnitude. In contrast to the Fig.~\ref{FigCWglobE}(b), here RK4 is more accurate than S2 and W2 due to the fourth-order scaling having set in faster.

\section{Conclusions}\label{Sec:Conclusion}
We have demonstrated that a two-way splitting of the spin-1 evolution equation leads to two exactly solvable flows which are convenient for numerical implementation as a symplectic integration scheme. Notably, our approach leads to a simpler and more efficient second-order algorithm compared to the earlier work by Wang \cite{Wang2007a}, and because we are able to make a two-way splitting we can extend the scheme to fourth-order using the composition method of Blanes and Moan \cite{Blanes2002a}. We have presented several tests that show our methods are accurate.
A feature of symplectic methods is that they can provide highly accurate solutions for Hamiltonian systems over long time periods. The development of such methods for the spinor system is timely given the increasing interest in the long-time coarsening dynamics of this system (e.g.~see \cite{Mukerjee2007a,Kudo2013a,Kudo2015a,Williamson2016a}).
While our tests are based on 1D spatial problems with periodic boundary conditions, our methods are immediately extendible to higher dimensional systems and to  non-periodic spatial domains, e.g.~by using a finite difference or finite element description of $\hat f_A$ combined with a Krylov approximation to $e^{\hat f_A}\psi$ (see \cite{Hochbruck1998a}).

It is also worth making a few remarks relating the differences between our approach and that of Wang \cite{Wang2007a} (which we outlined in Appendix \ref{AppW2}), and the potential to extend these ideas to more general cases, including higher spin systems. Wang splits up the terms arising for the spin-dependent energy ($\mathbf{F}^2$) into separate flows, while we keep these terms together.
The solution we have developed here utilizes that the quadratic Zeeman term is spatially uniform (a good approximation in many experiments) and can thus be included with the kinetic energy flow. However, our technique can be extended to situations where the quadratic Zeeman energy varies with position. To do this the quadratic Zeeman term needs to be included with the $\hat{f}_B$ flow, which admits an analytic solution (e.g.~see \cite{zhang2005coherent}) in terms of Jacobi elliptic functions.
The extension to higher spin is less immediate and is in need of attention.
Schemes in the literature \cite{Wang2011a,Gawryluk2015a} require explicit numerical diagonalization of the nonlinear spin subsystem at each time-step,
which for the spin-$f$ case consists of $f+1$ interaction terms (e.g.~for $f=2$ a spin singlet-pair term with new interaction coefficient $c_2$ emerges). Like what we have demonstrated here with the spin-1 case, it may be possible through careful analysis and exploiting the symmetry of the system to develop simpler splittings for higher spin systems that are easier and more efficient to implement.

\section*{Acknowledgments}
We acknowledge support by the Marsden Fund of New Zealand.

redpajama_set_name: RedPajamaArXiv


text:
<!-- Run application specs in a browser -->
<!-- #docregion -->
<!DOCTYPE html>
<html>
<head>
  <base href="/">
  <title>Banner Component (inline template) Specs</title>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1">
  <link rel="stylesheet" href="node_modules/jasmine-core/lib/jasmine-core/jasmine.css">
</head>
<body>
  <script src="node_modules/systemjs/dist/system.src.js"></script>

  <script src="node_modules/jasmine-core/lib/jasmine-core/jasmine.js"></script>
  <script src="node_modules/jasmine-core/lib/jasmine-core/jasmine-html.js"></script>
  <script src="node_modules/jasmine-core/lib/jasmine-core/boot.js"></script>

  <script src="node_modules/zone.js/dist/zone.js"></script>
  <script src="node_modules/zone.js/dist/long-stack-trace-zone.js"></script>
  <script src="node_modules/zone.js/dist/proxy.js"></script>
  <script src="node_modules/zone.js/dist/sync-test.js"></script>
  <script src="node_modules/zone.js/dist/jasmine-patch.js"></script>
  <script src="node_modules/zone.js/dist/async-test.js"></script>
  <script src="node_modules/zone.js/dist/fake-async-test.js"></script>

  <!-- #docregion files -->
  <script>
    var __spec_files__ = [
      'app/banner-inline.component.spec'
    ];
  </script>
  <!-- #enddocregion files-->
  <script src="browser-test-shim.js"></script>
</body>

</html>

redpajama_set_name: RedPajamaGithub


Harry Hortman was a 25-year old hack driver in Cherokee, Iowa. On November 30, 1901, he shot and killed his sweetheart as she tried to flee from his jealous rage.
Curry was born Harvey Alexander Logan in Tama County, Richland Township, Iowa in 1867.
Index to Probate Records 1846 - 1855.


text:
22 race calendar announced for 2014
Date: 30th September 2013 at 10:24am
Just as the New Jersey race promoters had expected, they have now been included in the F1 2014 draft calendar.
During the weekend of the Italian Grand Prix a draft calendar for the 2014 Formula One World Championship was released/leaked to the media.
It included 21 races following the expected news that India had been dropped from 2014 [scheduled to return in 2015], whilst Austria, Russia and Mexico had all been added.
A lot of people in the Formula One paddock remain sceptical about a race in New Jersey, despite it holding a contract for next season and reportedly currently having the funding in place.
However following the World Motor Sport Council meeting in Croatia last week, the latest draft calendar has swelled to a 22 race season, the biggest in Formula One history.
There do however remain big question marks over at least three of the races, as well as New Jersey being a doubt, the Mexican Grand Prix organisers have their own race on their hands to complete major upgrade work at the Autodromo Hermanos Rodriguez circuit.
The Russians have the little matter of hosting the Olympics as well as getting the circuit in Sochi up to standard, while Korea's entry is also only provisional.
Draft Calendar
16 March: Australia (Melbourne)
30 March: Malaysia (Sepang)
6 April: Bahrain (Sakhir)
20 April: China (Shanghai)
27 April: Korea (Korea International Circuit)*
11 May: Spain (Barcelona)
25 May: Monaco (Monaco)
1 June: Grand Prix of America (New Jersey)*
8 June: Canada (Montreal)
22 June: Austria (Red Bull Ring)
6 July: Britain (Silverstone)
20 July: Germany (Hockenheim)
27 July: Hungary (Budapest)
24 August: Belgium (Spa-Francorchamps)
7 September: Italy (Monza)
21 September: Singapore (Marina Bay)
5 October: Russia (Sochi)*
12 October: Japan (Suzuka)
26 October: Abu Dhabi (Yas Marina)
9 November: USA (Austin)
16 November: Mexico (Mexico City)*
30 November: Brazil (Interlagos)
* Subject to the circuit approval


Commanders/Iron Shipwright continues to add to their line of 1:350 scale early American cruisers and battleships with the armoured cruiser, New York ACR2. I love this kit and it is a nice addition to my growing fleet. The model built well with no major problems. The instructions are lacking and I was not able to locate any detailed reference material. I replaced the resin mast parts provide with the kit with brass rod, added some signal flags and brass gun barrels from Steve Nuttall. This is a very good kit and highly recommended for the Dreadnought era enthusiast.


Graduating from your law degree is a momentous achievement especially after the numerous long nights spent on your laptop frantically trying to finish your assessment due the next day or those dreaded 3 hour end of semester exams where you need to cram everything you know onto an exam paper. But once you receive your testamur and throw your mortar board in the air for the essential Instagram photo you may be thinking what now? If you speak to any law graduate their first answer will probably be to obtain a job in a top tier law firm and be granted a generous salary.
But before you can unleash everything you learnt at university and embark on your legal career you must first obtain valuable practical skills, which will form the foundation in which you will build your legal career.
Communication is key! As a fresh law graduate you need to be able to communicate effectively with your colleagues and clients. Communication is not limited to verbal but also extends to your body language and the way you convey your message to your listeners. Being a good communicator means you also need to be a good listener. It is no good telling a client everything you know about a particular area law if you haven't listened to the facts and how that law can be applied to their case.
This is essential to ensure you not only complete your work but you complete your work on time. To perfect your time management skills you need to ensure that you can prioritize your workload and delegate if needed. Find something that works for you, by either writing a to-do-list or adding reminders in your phone to help you work out what is urgent and what can wait. In practice strict time limits apply and must be adhered to, to ensure you can aim to achieve the best outcome for your client.
Decision making is a skill that will continue to develop throughout all aspects of your life. Our day is full of decision making, the key is to determine what is the right and just decision to make. As lawyers our clients seek our advice on which decision to make, although we cannot make the decision for them, it is our job to provide them with the information they need and guide them to make an informed decision.
These are arguably the most important skills a law graduate can develop early in their career. The legal profession is a tough industry especially in your first few years while you are trying to find your feet and develop your professional identity. The key is not to be too hard on yourself, no one expects you to know everything. Be respectful of your peers and listen to your mentors they are there to guide you and help you transition from student to lawyer, their feedback is important! Lastly, be confident in yourself and try your best, remember this is only the beginning of an exciting and rewarding career.