A Fragrance can be scientifically defined and then applied to suit a specific purpose. It is this ability to understand fragrance that has made us one of India's leader in the creation of Scent for use in products ranging from Fine Perfumes to Cosmetic & Toiletries.
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Sheer Sensuality, Crystal Transparency and Luminous Brightness are what we observe at Ultra.
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This fusion of quality products and innovation lends outstanding performance for us. we are committed for providing our employee with the knowledge, tool and skills needed to achieve excellence in customer satisfaction and impressive business performance. Our science and technology inputs have provided substantial value addition to the splendid world of fragrance and flavor.
Since our inception in 1987, our main objective is to nurture an environment where excellence is everyone's goal and delighting our customer is a Shared Vision.
Presents a whole new experience in myriad range of products that soothes/ stimulates your senses.


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Ewell Plaza Façade Design Finalized
JANUARY 18, 2022 | DEPARTMENT OF COMMUNITY PLANNING & ECONOMIC DEVELOPMENT We are excited to announce that we have reached the conclusion of the Ewell Plaza design process. The construction of the building at 151 North Queen Street is also nearing completion and in 2022 we will see the construction of the plaza landscape, the completion of a statue honoring Barney Ewell, and the installation of a colorful public art façade on the front of the garage and above the Lancaster Public Library. The Lancaster Parking Authority leadership and City Administration have received and accepted the final concept design for the public art façade, as shown in the updated renderings. The art façade has garnered a lot of attention and discussion over the months, so […]
Posted ByIN News
First Piece of Precast Concrete Erected at Christian Street Garage Project
Lancaster, PA – The first piece of precast concrete was erected at approximately 3:00 PM on April 5, 2021 at the Lancaster Parking Authority's Christian Street Garage project at 151 N. Queen Street. Over the next four months, over one thousand pieces of concrete will be erected to form the building which will include the Lancaster Public Library, Christian Street Garage, a retail space and an event terrace for the Holiday Inn.
Lancaster Online Article About Larry Cohen
See the recent article in Lancaster Online about Larry Cohen, Executive Director of the Lancaster Parking Authority, and his recent book about the parking industry, The Quirky World of Parking. Click Here to go to the Lancaster Online article
Lancaster Parking Authority's Executive Director Writes a Book about Parking
Lancaster, PA — Larry J. Cohen, Executive Director of the Lancaster Parking Authority, has written a book called The Quirky World of Parking about his experiences in the parking industry.
https://www.lancasterparkingauthority.com/wp-content/uploads/2021/03/WGAL-News-8-at-6-Larry-Cohen-Interview.mp4
Prince Street Garage Daily Parkers: Pay for Parking Before Returning to Vehicle
Pay machines installed in Prince Street Garage Lobbies
Ceremony marks demolition of annex, advent of Christian Street Garage, new library
At Tuesday's "demolition ceremony" at Ewell Plaza, Lancaster Parking Authority Executive Director Larry Cohen led the assembled officials, company representatives and community leaders in a brief chant of "Tear it down! Tear it down!" Read more.
Lancaster Parking Authority receives accreditation
The Lancaster Parking Authority is now an accredited parking organization. Read more.
AEDs now available within Lancaster parking garages
LANCASTER, Pa. — As part of a proactive safety initiative, you can now find Automated External Defibrillators (AEDs) within parking garages in the event of sudden cardiac arrest. Read more.
Lancaster Parking Authority installs AEDs in 6 parking garages
LANCASTER, Pa. (WHTM) – Lancaster Parking Authority has installed automated external defibrillators in its six parking garages across the city. Read more.
Here's what the Lancaster parking study says about downtown
On a typical weekday, a little under two-thirds of Lancaster's downtown off-street parking is occupied, according to Michael Connor of consulting firm Kimley-Horn. That, he said, is the "baseline condition" his team found when it counted cars in October as part of its parking study. But add in the demand from development projects that are coming online and from big events at the Lancaster County Convention Center, and the city's inventory of parking begins to look much tighter. Read more.

redpajama_set_name: RedPajamaCommonCrawl


Historic Langley Castle is preparing to stage a chilling theatrical production this autumn, as it helps celebrate the 200th anniversary of Mary Shelley's world-famous, Gothic horror novel, Frankenstein.
Langley Castle, which has been on the Northumbrian landscape since 1350, will be hosting a reimagining of Mary Shelley's classic storyline, performed by North East theatre company, Timeworks Theatre.
This theatre group is known for celebrating various anniversaries and historical events, by creating productions based on a wide variety of pieces, from comedy to drama and from children's stories to horror.
This theme will resonate and provoke responses, in a world in which cloning, genetic engineering and the unlocking of the secrets that DNA holds are never out of the news. Seymour's interpretation will portray both the terror of the novel's plot, as well as key events in the life of a writer who breathed life into such a terrifying creature.
Taking place spookily close to Halloween, on Friday October 26, this event can be attended at a cost of just 13.50. A fully-licensed bar will also be available throughout the evening and a pre-theatre menu is on offer to those who want to dine before the production.
A two-course meal will cost £15.95 and a three-course meal £19.95. Tickets and meals should all be booked in advance.
To entice more people who have always wanted to pop into Langley, but never dared, the castle has special rates on accommodation to coincide with the Frankenstein production. These cover three levels of comfort within the castle's 'CastleView' accommodation and range in price from £65 per person, to £85 per person, inclusive of full Northumbrian breakfast and a Langley Battlements Tour on the morning of Saturday October 27. The rate is based on two people sharing a double room. To book or for more information, visit www.langleycastle.co.uk or call 01434 688888.


Police are searching for two individuals they say broke into the Biggby in Fowlerville early Monday morning.
Police are searching for two suspects they say broke into a Biggby in Fowlerville early Monday morning.
According to owner David Roe, the pair did not steal anything from the store and were in and out in less than a minute.
He said the only damage the suspects caused was to the store's front window which he said will be repaired in the coming days.
Roe retired worked as an Ingham County Sheriff's Office deputy for 25 years before opening the coffeehouse three years ago.
"I was a cop for 25 years," he said. "They'll get caught."
This is the first time his store has been broken into and he hopes it will be the last.
"It kind of ticks me off to be honest," Roe said.
Roe said the shop is up and running despite the early morning disturbance.
"It's all part of owning a business," he said.
Livingston County Sheriff's Office Lieutenant Jim Lynch said deputies were dispatched to the coffeehouse at approximately 5:20 a.m. Monday morning.
"The officers were met by an employee who noticed broken glass when attempting ot open the store," he said.
Lynch said he is not aware of any other break-ins in the surrounding area.
April 1, 2019, 6:30 a.m.


Opening address by Dr Melanie Hillebrand, art historian and artist, former Director of the Nelson Mandela Metropolitan Art Museum.
An exhibition organized by ArtEC and the Nelson Mandela Metropolitan ART Museum to commemorate the centenary of the Eastern Province Society of Arts and Culture (now ARTEC). Artworks on display will be loaned from past members and will be selected from the Art Museum's permanent collection.


PURE HEALTH CLUBS offer a fresh unique opportunity for those embarking on quest for greater physical, mental or spiritual fulfillment.
At PURE, we believe physical activity is the baseline for ensuring a healthy body and mind, and that it is only with a healthy mind and body that one can achieve true happiness.
Whilst many feel exercise is a chore or' a must do activity PURE has changed the face of health clubs becoming the leading provider in innovative classes.
PURE's unique and extensive range of classes ensures a vibrancy to your exercise routine that couldn't exist before. Not only is your body having fun, and seeing astounding changes, but your mind is also on a high.
Bringing you the latest in exercise research and innovation, the best instructors, unlimited number and times of classes ensures your greatest chance achieving your desired results, the impossibility of failure and the ideal environment within which to develop the most important relationship you have in life – the one with yourself!
Very enjoyable. I am amazed that such small movements can have such an effect on the muscles. I always walk away feeling relaxed."
"I sit down in my job for most of the day and tended to get pain in my neck and back (bad posture). Since starting Pilates - no pain! My stomach is stronger and I am losing the flabby bits!"
Because of the nature of what we offer here at PURE Health Clubs (innovative, exciting classes in a myriad of disciplines), there are many different memberships available.
Every individual has different goals and as such, different needs from their health club. We are proud that we facilitate our memberships without pressure, and will individualise prices according to the specific needs and wants of our clients.
To find out what kind of structure will help maximise your health and wellbeing goals, we would love to meet you! We can introduce you to our friendly staff and let you get a feel for the atmosphere here at PURE. We guarantee the experience will be like no other gym, and that a visit to Pure, will be one you won't regret.


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'All My Children' and 'One Life to Live' Top the iTunes and Hulu Charts in Debut Week
The OnLine Network/TOLN
The OnLine Network released the following breakdown showing how popular "All My Children" and "One Life to Live" were during their debut week on Hulu and iTunes.
via Press Release
WEEKLY "ALL MY CHILDREN" AND "ONE LIFE TO LIVE" STATS AND FACTS
The OnLine Network's (TOLN) "All My Children" and "One Life To Live" topped all charts in its debut week and earned rave reviews across the board. The shows maintained an average of #2 across both iTunes and Hulu, along with 4.5/5 star audience review on all platforms!
Below is a summary of key statistics:
Day One (Launch Day):
· "AMC" #1 on both iTunes TV Seasons (dethroned "Mad Men" from #1 position) and #1 most popular episodes on Hulu
· "OLTL" #2 on iTunes TV Seasons and Most Popular episodes on Hulu
· "AMC" #1 on both iTunes TV Seasons and #1 most popular episodes on Hulu
· "AMC" #2 and "OLTL" #4 on iTunes (dethroned only by "Dexter")
· "AMC" #3 and "OLTL" #5 on Hulu
· "AMC" #2 and "OLTL" #4 on iTunes
Recap shows air today and numbers are strong heading into the weekend as viewers are likely to binge view.
CENTER STAGE: A View from the Top with Angelica McDaniel, Senior Vice President, CBS Daytime (Part 1 of 2)
RESOLVED: Who Did JR Shoot?; 'MORE All My Children' Debuts
Erin Chambers is Lucky to be on 'GH'
Lucky Spencer (Jonathan Jackson) may have found himself involved in more baby daddy drama courtesy of long-time love Elizabeth Webber (Rebecca Herbst), but ABC's "General Hospital" has cast actress Erin Chambers in the recurring role of Siobhan, who plays a part in the upcoming early fall Lucky storyline, according to an ABC representative.
Remembering 9/11: A Special SON Report
On September 11, 2001, the world witnessed the worst terrorist act in American history. It is a day none of us will ever forget. For the 10th Anniversary of 9/11, I asked those who were working within the soap opera industry if they would like to share their thoughts and memories about that horrific day. Here are the stories from former soap writers Tom Casiello, Tom King, and former soap writer/producer Lisa Connor.
'B&B's' Hope Snags A New Love Interest
Some of you are going to say it, others are going to shout it, but the thought that comes along with the news that Hope is getting a new love interest other than Scott Clifton's Liam Spencer is "Well, it's about freaking time!" Yep, Kim Matula's alter ego will finally be caught up in a romantic situation sans Bill's son. In fact, a rep confirms that another Emmy winner — Darin Brooks (ex-Max Brady, "Days of our Lives) — will be the one stealing Hope's heart!
OLTL First Look: LaChanze Comes to 'Life' in New Role
Facebook Twitter Flipboard Pinterest Reddit (SoapOperaNetwork.com) — When Broadway star and Tony Award winning actress LaChanze joins the…


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text:
package com.huawei.esdk.uc.jna.callback;

import com.huawei.esdk.uc.jna.struct.STVideoCallParam;
import com.sun.jna.win32.StdCallLibrary.StdCallCallback;

/**
 * 视频呼叫回调函数
 * @author lWX169831
 *
 */
public interface VideoCallEventCB extends StdCallCallback
{
    public void callBack(STVideoCallParam callParam);
}

redpajama_set_name: RedPajamaGithub


text:
Statement by Mary Ng
Mary Ng on IBM Canada
In the House of Commons on November 7th, 2017. See this statement in context.
IBM CanadaStatements By Members
November 7th, 2017 / 2 p.m.
Mary Ng Liberal Markham—Thornhill, ON
Mr. Speaker, I am pleased to rise today to highlight an outstanding company in my riding of Markham—Thornhill.
Celebrating 100 years in Canada this year, IBM is one of the world's most successful companies and a staple of the local economy in Markham since 1983. Over the last century, IBM has showcased what it means to be a truly innovative company and a leader in our community. Its innovation extends beyond the organization, with the IBM innovation space in the Markham Convergence Centre, home to the next generation of entrepreneurs and start-up companies working in the global marketplace.
I want to highlight Peyman Moeini, an entrepreneur in the innovation centre, who started as an engineering student in a campus-led accelerator and now an employer, whose company is scaling and growing. It is a really inspiring Canadian story.
I would like to take a moment to thank IBM for its remarkable work in Markham and across Canada, and for the work it will continue to do over the next 100 years.


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About Keegan Murray
Reporter for The Shetland Times. Interested in politics, literature and music. Born and bred Shetlander. Long suffering Newcastle United supporter.
Posts by Keegan Murray
0 comments, 20/01/2020, by Keegan Murray in News
The Scottish government has overturned a Shetland Islands Council decision to reject the extension of temporary planning permission for the Sella Ness accommodation camp. In May of last year the camp owner's Malthus Uniteam had its request to extend the 426-bedroom facility's planning permiss... Read more...
1 comment, 26/12/2019, by Keegan Murray in News, ST Online
The Coastguard helicopter was called out twice on Christmas Day to airlift people to the Shetland Mainland. The first call came just after 3am when the helicopter was called to an offshore rig 140 miles northeast of Sumburgh to collect a man who had been involved in an accident. The man was ta... Read more...
Sheriff demands answers after prisoners miss flight
Sheriff Ian Cruickshank has instructed a court clerk to write to the holders of Scotland's prisoner escort contract GEOAmey amid concerns that Shetland is receiving a "second rate" service. His instruction followed a day in Lerwick Sheriff Court in which proceedings were heavily delayed due... Read more...
Foodie festival returns in new home
The fifth annual Taste of Shetland festival got underway this morning (Saturday) with the celebration of local produce now taking up residence at a new venue. A somewhat scaled back incarnation of the foodie festival has found a new home at the Islesburgh Community Centre. But despite the smal... Read more...
Four candidate no shows at Town Hall hustings
11 comments, 22/11/2019, by Keegan Murray in News
The first of seven general elections hustings set to be held across the Orkney and Shetland constituency took place at Lerwick Town Hall this evening (Friday) – though only a third of the candidates actually attended. With less than three weeks remaining until the country goes to the polls for ... Read more...
Winners announced in two council by-elections
Stephen Flaws has emerged victorious in the Lerwick South by-election despite coming second place in first preference votes. Meanwhile, voters in Shetland Central have given their support to Moraig Lyall, who ran out a comfortable winner in the ward's by-election. Voters in the two wards went ... Read more...
First candidate declared for December election
0 comments, 30/10/2019, by Keegan Murray in News, ST Online
The first candidate to stand against Orkney and Shetland MP Alistair Carmichael in the December general election has been declared. Scottish Labour members in the Northern Isles have endorsed chairwoman of the Orkney Labour party Coilla Drake as their candidate for the forthcoming election. On... Read more...
Reduced vehicle capacity on Yell service
The Yell Sound ferry MV Dagalien will be out of service for up to two weeks from the evening Friday 25th October due to planned maintenance on her propulsion system. During this period the MV Fivla will operate alongside MV Daggri, running the Yell Sound service to its normal timetable but with a... Read more...
Time to 'do something about the Gilbert Bain', NHS board hears
A new hospital in Lerwick could be on the cards after health board members backed an assessment of the current Gilbert Bain Hospital in response to a report identifying a "need for change". The decision to approve the assessment process was taken at a meeting of the NHS Shetland board on Tuesday ... Read more...
'Funeral' procession mourns loss of landscape
Environmentalists "grieving the imminent loss of our landscape" took part in a "funeral" procession on Saturday afternoon. Around a dozen people participated in the 'Lament for the Lang Kames' event, walking over 5 miles from the Sandwater junction to Lower Voe - where a small crowd had gathe... Read more...


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Betting Guide to the World Series
The current Futures betting for the World Series is wide open which suggests no one MLB team has stood out over the early matches of the season. Each franchise plays 162 matches in the regular season and to date no team has completed 30 so the form has not settled down and in theory every team could qualify for the play-offs. The LA Dodgers are the favourites but 12 sides are priced between 7.00 and 21.00. Odds are updated after each day of matches and fluctuations in form are reflected in the odds on offer for all the divisions and leagues.
History of World Series
The World Series has been played since 1903 in MLB between the winner of the American League and the winner of the National League. Before 1969 the teams with the best records during the regular season from each league met in the final. Since then the post season involves a play-off series which culminates in the World Series. The finals have been played 110 times with the AL winning 63 and the NL 47. The winner is determined by results in a 7 match roster of matches.
Tactical bunt in baseball
The New York Yankees from the American League have won most World Series, 27 in total from 40 played. Their last appearance in the season decider was in 2009 when they were the overall champions of Major League Baseball. The next most successful franchise is the St Louis Cardinals but they have only won the World Series 11 times. No other team has a double digit number of championships in their record.
Bovada – Latest World Series odds
Twenty two different franchises, some under name changes, have won the World Series at least once. Eight more sides have reached the final and lost. The Texas Rangers and San Diego Padres have both lost the only two play-offs they have contested. San Francisco and Los Angeles have both lost in 12 World Series campaigns while the Yankees have the best strike rate of any team that has played in the World Series five times or more.
World Series Records
The Yankees have won at least two play-offs in every decade since the 1920s except the 1980's when they won none. The New York Giants played in each final from 1921 to 1924 which is a record and there are only three National League teams have won back-to-back end of season eliminators. The Chicago Cubs were the winners in 1908 and have not since added to their tally which is a record. The Washington Nationals and Seattle Mariners are the only two franchises to have not played in the World Series.
Bookmakers issue odds for the World Series and both leagues before the regular season begins and update these markets after each schedule of matches. In Europe these type of bets are called ante-post markets which are the same as the Futures offered by bookies in North America. It is also possible to place Futures bets on the winner of each division within the two leagues. It is possible to bet on the run line money line and totals for each individual match in MLB.
MLB Season to Date
The Yankees are still in the mix to win their 28th World Series after a gap of six years from their last appearance. They beat the Boston Red Sox 3-0 in a mini-series last weekend and have since been backed to win the finals and the American League pennant. They have a win-loss of record of 16-9 for the season and are joint third favourites with the Red Sox to win the league behind two American League Central teams who are just about tied in the division. The Detroit Tigers, Los Angeles Dodgers sand Washington Nationals are the first three in the latest betting for the World Series this season.
Latest MLB standings
The Dodgers are clear favourites to win the NL West and the shortest priced of any of the divisional favourites. They are on a four game winning streak, including a 3-0 beating of Arizona over last weekend. They are now the favourites to win the pennant though the St Louis Cardinals and Chicago Cubs will have their say in the final outcome. Chicago would have been shorter in the betting but for losing two of three matches at home to the Milwaukee Brewers.
Pitcher Statistics
Odds are quoted for all MLB matches during the regular season and the play-offs. The key to consistently backing winners is to compare the efficiency of each pitcher and the quality of their replacements in reserve. The pitcher is the most important player in any baseball team. His role is to throw the ball at the batters as quickly and accurately as possible so they have no chance of hitting it and getting a run. It's easy to keep track of readily available stats which can be used to find the best bets.
The Earned Runs Average provides the average number of runs a pitcher concedes through poor pitching during a game. It is an average for the current season to date. If you are assessing a game and team A's pitcher has an ERA of 2.5 and team B's an ERA of 5.6 you will see straight away that team A should have a significant advantage. To take the analysis further it is possible to compare the Total Runs Average to discover the total average number of runs a team concedes when a particular pitcher is playing for them.
There are so many games played in a baseball season that the best approach is to use a long-term strategy for the summer, using these statistics. Typical odds are 1.95 so if you bet 10 dollars each time and get 20% wrong over 100 matches the profit for the season could be over $500.
For a full range of baseball odds please visit 888Sport


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OMA in Hybrids series
We include some videos and pictures illustrating the OMA projects analized in the latest series of a+t magazine, devoted to mixed-use buildings.
De Rotterdam
A distance in time of eleven years separates the first design and the latest –shown in the pages of Hybrids III. Residential Mixed-use Buildings– of the De Rotterdam building. It involves a multipurpose complex located in Wilhelminapier, the wharf which aims to act as a counterpart, on the other side of the River Meuse, to downtown Rotterdam. The project maintains the bulk of the functions originally foreseen, even though public initiative has withdrawn from the project, and as a result, the cultural uses have disappeared from the programme.
The groundbreaking in late 2009 offers an optimistic vision of how to make the most of the downturn. According to the authors, the current decrease in building costs has allowed them to take on a project which eleven years ago was unachievable.
Wilhelminapier
Rotterdam. The Netherlands 1998-2011
Transformation of the Entrepôt Macdonald
Project published in Hybrids III. Residential Mixed-use Buildings
The northeast of Paris is made up of fragments of the city that jut out over a landscape of transport infrastructures. The urban plan for the area, designed by Dusapin Leclerc, looks to re-establish the connection between Paris and its outskirts by creating new centralities between cities. The Macdonald warehouses are to be converted into a huge hybrid boosted by tram and suburban train traffic. The existing structure will be used, enlarged to house various urban uses. The building, because of its large size, is comparable to the scale of Corviale or Karl Marx-Hof. Nevertheless, in face of the self-sufficient social condenser, which mixes use to satisfy only the needs of its residents, this project opens the city to achieve a mix of users.
OMA (Rem Koolhaas/Floris Alkemade) FAA+XDGA (Floris Alkemade/Xaveer de Geyter)
141-221, Boulevard Macdonald
Paris. France 2006-2013
Bryghusprojektet
Project published in Hybrids II. Low-rise Mixed-use buildings
The former Copenhagen harbour is going through a deep transformation to turn it into an axis of public facilities with a strong symbolic vocation (the Royal Theatre, the Opera or the future National Archive). On the last vacant lot on the water, the building reinforces pedestrian connections between the historic centre and the border of the canal, saving the existing motorway with a passage where accesses will be concentrated. The project raises the issue of disorderly combination of uses, breaking away from the typical hierarchical distribution of programmes, and thus driving true interaction. The resulting volume, from the stacking of smaller pieces, contrasts with the categorical scale of the large monolithic buildings that tend to close themselves to their own atria.
Christians Brygge
Copenhaguen. Denmark 2008-2013
Dubai Renaissance
Project published in Hybrids I. High-rise Mixed-use buildings
The Dubai Renaissance project raises a criticism of the profusion of icons in the contemporary city and could be located anywhere. It is a revision of the Modern skyscraper, although bigger and more daring: a monolithic envelope houses all programmes without them becoming apparent from the outside. On top of that, it rotates on a circular base.
Abstraction against form. In opposition to the Dubai model, the desire to make a city arises. Surrounded by skyscrapers that behave as vertical ghettos, Dubai Renaissance was designed to be a catalyst of city life: three large public areas are distributed along its section. A vertical street (an elevator core) connects them to the main floor without intermediate stops.
Ras Al Khor Wildlife Sanctuary
Dubai. United Arab Emirates 2006-
Jersey City looks to compete with its neighbouring Manhattan and is in the process of putting up another high-rise business district with much lower taxes and prices. 111 First Street consists of stacking up the programme in three different volumes on top of a socle. This base takes up 100% of the lot to avoid empty space between the building and the street. Each volume holds different uses and housing typologies: programmatic variations impose building formalisation. Volume rotation allows for the opening of exterior terraces. One of them, on the roof of the socle, can be directly accessed from the street so as to incorporate it in the circuit of public activities of the surrounding cultural district.
Jersey City. USA 2006-
Copenhagen, OMA, a+t 31. HYBRIDS I, Hybrid buildings, a+t 32. HYBRIDS II, Paris, New York, Patricia García, a+t 33-34. HYBRIDS III, Rotterdam, Dubai, OMA
HYBRIDS III. Index
a+t 32. HYBRIDS II. Low-rise Mixed-use Buildings. Index
a+t 31. HYBRIDS I. Index


The Rosses Walking Weekend takes place over the June Bank Holiday weekend and offers the participant a selection of guided walks of varying distance and terraine.
The walks willl take participants to some of the most amazing, beautiful, wild and tranquil spots of The Rosses, usually only reserved for adventerous locals and native wildlife. Check out our facebook page "Rosses Walking Weekend 2017" which is now live.


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text:
\section{Introduction}

For the nonlinear optical fiber channel, modeled by the nonlinear Schr\"odinger equation (NLSE), the nonlinear Fourier transform (NFT) maps a pulse from time domain into a so-called nonlinear Fourier domain where the signal transformation along the link can be described by simple equations~\cite{yousefi2014information}. This motivates the modulation of data in the nonlinear spectrum, which needs the computation of the inverse NFT and NFT at the transmitter and receiver, respectively. A special class of transmission pulses, considered here, are multi-solitons which are characterized in the nonlinear spectrum simply by some pairs of eigenvalue and corresponding spectral amplitude.
In the last few years, multi-soliton communication schemes have been studied and experimentally demonstrated in single polarization transmission, e.g. ~\cite{dong2015nonlinear,aref2015experimental,hari2016multieigenvalue,aref2016onoff,Geisler2016,buelow20167eigenvalues,Gui2017,Koch2019trans}, as well as in polarization-multiplexed transmission, e.g. \cite{Gaiarin2018,Gaiarin2018a,span2019efficient,chan2019experimental}.

 The Darboux transform (DT)~\cite{matveev1991darboux} is a common inverse NFT algorithm to generate multi-soliton signals iteratively.
Although this inverse NFT is rather simple with low complexity,
the forward NFT is a more challenging task in order to compute the eigenvalues and their spectral amplitudes.
Various numerical methods have been developed so far to compute the spectral amplitudes, see~\cite{yousefi2014information,aref2016control,Hari2016,Vasylchenkova2018,garcia2019statistics,Chimmalgi2018,chimmalgi2019fast,Medvedev20,vaibhav2019efficient,Mullyadzhanov2019}.
These algorithms are mainly based on numerically solving the Zakharov-Shabat system (ZSS) by a piecewise approximation method.
The approximation error is rectified usually by a rather large number of signal samples or by using higher-order approximation methods~\cite{Chimmalgi2018,chimmalgi2019fast,Medvedev20,vaibhav2019efficient,Mullyadzhanov2019}.
Higher-order approximation methods increase the computational complexity
but the error declines faster in terms of signal samples, allowing to find the nonlinear spectrum of a multi-soliton with tens of eigenvalues~\cite{Mullyadzhanov2019}.
Besides, the forward-backward (FB) method (called bi-directional method in \cite{Hari2016}) further reduces
the numerical errors of all one-step discretization NFT methods~\cite{aref2016control}.
The accuracy of this method is recently improved by some modification in \cite{Prins2019}.

Here, we propose a new approach for the forward NFT based on the Darboux transformation.
Although the DT is commonly known for adding eigenvalues, it is also able to remove eigenvalues from a given solitonic pulse\cite{Junmin1990}. Applying the DT carefully removes an eigenvalue from the pulse's nonlinear spectrum with a simple change on the remaining nonlinear spectrum of the pulse. As we explain in Sec.~\ref{sec:basic}, $a(\lambda)$
will be changed but the $b-$values remain the same.
While this property of the DT is known for a long time, it has not yet been used for the detection of the nonlinear spectrum.

We propose the Successive Eigenvalue Removal (SER) algorithm where the spectral coefficients are estimated one by one successively. Following a predefined order, an eigenvalue and its spectral coefficient are estimated and then removed from the nonlinear spectrum.
This process is repeated on the remaining pulse until all eigenvalues are removed.
This algorithm is beneficial as removing an eigenvalue can reduce the pulse duration as well as the bandwidth (smoother variations). Both reductions can decrease the number of numerically required pulse samples and thus decrease the total computational complexity of the NFT computation.

We discuss that the SER algorithm is quite robust against the error propagation drawback of successive algorithms in which usually an error in an early iteration influences the result of the following iterations and thus the error accumulates.
 We compare the SER algorithm to the conventional approach in terms of computational complexity and numerical accuracy by simulation, also in the presence of noise.
Note that the SER algorithm can be equipped with any aforementioned forward NFT algorithms and can improve the performance of those algorithms.
If the continuous spectrum contains a large fraction of the pulse energy (not the case for multi-solitons), the SER algorithm can still be helpful in computing the discrete spectrum by simplifying the pulse after an eigenvalue removal, but it may not lead to a reduction in pulse duration.

The paper is outlined as follows: In Sec.~\ref{sec:basic}, we review the basics of the NFT
and describe how the DT can remove eigenvalues from a nonlinear spectrum.
We explain the SER algorithm in Sec.~\ref{sec:SolDecom} and explain its robustness against coarse estimation of eigenvalues in Sec.~\ref{sec:analysis}. We compare its performance numerically to the conventional approaches in Sec.~\ref{sec:Sim_eval}. We draw some conclusions in Sec.~\ref{sec:concl}.

\section{Preliminaries of NFT and Darboux Transform}\label{sec:basic}
In a fiber link with ideal distributed amplification, the pulse propagation in a single polarization can be modeled by the nonlinear Schr\"odinger equation (NLSE),
\begin{equation}\label{eq:NLSE}
j\frac{\partial q(t,z)}{\partial z}=\frac{\partial^2 q(t,z)}{\partial t^2}+2\left\|q(t,z)\right\|^2 q(t,z).
\end{equation}
where $t$, $z$ and $q(t,z)$ are the normalized time, distance along the fiber and the normalized pulse envelope, respectively. On a single mode fiber with chromatic dispersion $\beta_2$ and Kerr nonlinearity $\gamma$, the physical pulse is obtained simply from
\begin{equation*}
p(\tau,\ell)=\sqrt{P_0}q\left(\frac{\tau}{T_0},\ell\frac{|\beta_2|}{2T_0^2}\right), \quad  \text{with } P_0T_0^2=\frac{|\beta_2|}{\gamma}
\end{equation*}
where $T_0$ is an arbitrary time-scale, $\tau$ and $\ell$ are the physical time and distance along the fiber.

\subsection{Basics of Nonlinear Fourier Transform}\label{sec:NFT}

A solution of \eqref{eq:NLSE} can be represented in a nonlinear spectrum via the Zakharov Shabat system (ZSS) \cite{shabat1972exact},
\begin{equation}\label{eq:ZS}
\frac{\partial}{\partial t}\left( \begin{matrix} \vartheta_1(\lambda;t,z) \\ \vartheta_2(\lambda;t,z) \end{matrix} \right)
= \left( \begin{matrix} -j\lambda & q(t,z) \\ -q^*(t,z) & j\lambda \end{matrix} \right)
\left( \begin{matrix} \vartheta_1(\lambda;t,z) \\ \vartheta_2(\lambda;t,z) \end{matrix} \right)
\end{equation}
with the boundary condition
\begin{equation}
\left( \begin{matrix} \vartheta_1(\lambda;t,z) \\ \vartheta_2(\lambda;t,z) \end{matrix} \right)\to \left( \begin{matrix} 1 \\ 0 \end{matrix} \right) \exp(-j\lambda t)
\qquad \mathrm{for} \quad t \to -\infty	\label{eq:boundary_condition_left}
\end{equation}
under the vanishing boundary assumption $q(t)\to 0$ as $t \to \pm \infty$.
The nonlinear (Jost) coefficients are defined as
\begin{align}
a(\lambda;z)&=\lim_{t\to +\infty} \vartheta_1(\lambda;t,z) \exp(j\lambda t)	\nonumber
\\
b(\lambda;z)&=\lim_{t\to +\infty} \vartheta_2(\lambda;t,z) \exp(-j\lambda t). \label{eq:spec_coeffs}
\end{align}
The crucial property of the nonlinear spectrum is
\begin{equation}\label{eq:evolution_z}
a(\lambda;z)=a(\lambda;0), \quad b(\lambda;z)=\exp(-4j\lambda^2 z)b(\lambda;0).
\end{equation}
This property indicates that the nonlinear spectrum for each $\lambda$
evolves independently in $z$ according to this simple equation. This suggests to modulate information on the nonlinear frequencies.
We drop the dependency on $z$, as it is simply multiplicative in \eqref{eq:evolution_z}.

The nonlinear spectrum of $q(t)$ consists of two parts:
\begin{itemize}
\item[(i)] the \emph{discrete spectrum} $\{(\lambda_k,b_k)\}$, where
the $N$ eigenvalues ${\lambda_k=\omega_k+j\sigma_k \in \mathbb{C}^+}$ ($\omega_k,\sigma_k \in \mathbb{R}$) are the roots of $a(\lambda)$ and spectral coefficients $b_k=b(\lambda_k)$.

\item[(ii)] the \emph{continuous spectrum} $Q_c(\lambda)=\frac{b(\lambda)}{a(\lambda)}$ for ${\lambda \in \mathbb{R}}$.
\end{itemize}

Note that the discrete spectrum is sometimes defined differently
like in \cite{yousefi2014information}. The above representation is chosen as $b_k$ are more preferable for data modulation~\cite{Aref2018,Gui2017,Buelow2018,yangzhang2019dual}.

There are various ways to compute the nonlinear spectrum by numerically solving \eqref{eq:ZS} \cite{yousefi2014information,wahls2015fast,Turitsyn2017,Chimmalgi2018,Medvedev20}. Practically, it is assumed that the signal $q(t)$ is truncated such that it is non-zero only inside the interval $t \in [T_-,T_+]$. Then the differential equation system \eqref{eq:ZS} can be solved by propagating $\boldsymbol{\vartheta}(\lambda;t)$ from the known solution \eqref{eq:boundary_condition_left} at the boundary $t=T_-$ to $t=T_+$. The time is discretized with step size $h$ as $t_m=T_- + m\cdot h$ where the sample index is $m=0,1,\dots,M-1$.

By applying a change of variables
\begin{equation}\label{eq:var_change}
\boldsymbol{\Psi}(\lambda;t)=\left( \begin{matrix} \vartheta_1(\lambda;t) \exp(j\lambda t) \\ \vartheta_2(\lambda;t) \exp(-j\lambda t)
\end{matrix} \right),
\end{equation}
and defining $\boldsymbol{w}^{(m)}=\boldsymbol{\Psi}(\lambda;t_m)$, one can iterate
\begin{equation}\label{eq:vec_update}
\boldsymbol{w}^{(m+1)} = \boldsymbol{S}_{m} \boldsymbol{w}^{(m)}
\end{equation}
from $\boldsymbol{w^{(0)}}=\boldsymbol{\Psi}(\lambda;T_-)=\left( 1,0 \right)^T$ (obtained from the boundary condition \eqref{eq:boundary_condition_left}) to $\boldsymbol{w^{(M)}}\approx \boldsymbol{\Psi}(\lambda;T_+)$. Then, according to \eqref{eq:spec_coeffs},\eqref{eq:var_change}, the spectral coefficients are obtained from
\begin{align}\label{eq:spec_coeffs_estimation}
\hat{a}(\lambda)=w^{(M)}_1,
\qquad
\hat{b}(\lambda)=w^{(M)}_2.
\end{align}
The scattering matrix $\boldsymbol{S}_{m}$ depends on the approximation method.
Using mid-point approximation, it becomes
\begin{equation}\label{eq:iteration_matrix}
\boldsymbol{S}_{m} = \left( \begin{matrix} \cos(|q_m h|) & \frac{q_m \sin(|q_m| h) \exp(2j\lambda t_m)}{|q_m|}  \\ -\frac{q_m^* \sin(|q_m| h) \exp(-2j\lambda t_m)}{|q_m|} & \cos(|q_m| h) \end{matrix} \right)
\end{equation}
where $q_m$ are the samples $q(t_m)$ at time $t_m$. Applying a higher-order approximation method, e.g. \cite{Chimmalgi2018,Medvedev20}, results in different $\boldsymbol{S}_{m}$ with smaller numerical errors.

The forward-backward method \cite{aref2016control} is a modification of the iteration \eqref{eq:vec_update}. In addition to propagating $\boldsymbol{w}^{(m)}$ from $t=T_-$ to $t=T_+$, it propagates another vector $\boldsymbol{u}^{(m)}$ from $t=T_+$ to $t=T_-$ with boundary condition
$\boldsymbol{u}^{(M)}=\left( 0,1 \right)^T$, i.e.
\begin{equation}\label{eq:vec_update_FB}
\boldsymbol{u}^{(m-1)} = \boldsymbol{S}_{m}^{-1} \boldsymbol{u}^{(m)}
\end{equation}
where $\boldsymbol{S}_{m}^{-1}$ is the inverse of $\boldsymbol{S}_{m}$, obtained by changing $h$ to $-h$ in $\boldsymbol{S}_{m}$.
Both iterations \eqref{eq:vec_update} and \eqref{eq:vec_update_FB} are repeated up to some index $0<p<M$. For $\lambda=\lambda_k$ being an eigenvalue,
the spectral coefficients are then obtained as
\begin{align}\label{eq:spec_coeffs_estimation_FB}
& \hat{a}(\lambda_k)=w_1^{(p)} u_2^{(p)} - u_1^{(p)} w_2^{(p)}\approx 0	\nonumber
\\
& \hat{b}(\lambda_k)=\frac{w_2^{(p)}}{u_2^{(p)}}
\end{align}

In the next section, we use the following approximation of $\boldsymbol{\vartheta}(\lambda_k;t)=\left(\vartheta_1(\lambda_k;t),\vartheta_2(\lambda_k;t) \right)^\mathrm{T}$. For $m<p$,
\begin{equation}
\hat{\boldsymbol{\vartheta}}(\lambda_k;t_m) \approx \left( \begin{matrix} w_1^{(m)} \exp(-j \lambda_k t_m) \\ w_2^{(m)} \exp(j \lambda_k t_m) \end{matrix} \right)
\end{equation}
while for $m>p$
\begin{equation}
\hat{\boldsymbol{\vartheta}}(\lambda_k;t_m) \approx \hat{b}(\lambda_k) \left( \begin{matrix} u_1^{(m)} \exp(-j\lambda_k t_m) \\ u_2^{(m)} \exp(j\lambda_k t_m) \end{matrix} \right)
\end{equation}
We have observed that $p=\frac{M}{2}$ gave rather precise approximations
for the pulses in the next sections. In general, one can choose $p$ according to the criterion in \cite{Prins2019}.

\subsection{Two Properties of the Darboux Transform}\label{sec:DT}

Multi-soliton pulses are specific solutions of the NLSE having only a discrete spectrum.
To numerically generate a multi-soliton pulse, an efficient algorithm is based on the DT.
It constructs a signal by adding eigenvalues one by one recursively and updating the pulse accordingly.
The DT is an elegant approach to modify a pulse by adding a new eigenvalue or also removing an existing one while the rest of the discrete spectrum as well as $b(\lambda)$, $\lambda \in \mathbb{R}$, are unchanged~\cite{Junmin1990}.
Let us briefly summarize the DT and its properties.
Assume a pulse $q^{(n)}(t)$ with $n$ eigenvalues $\Lambda^{(n)}=\{\lambda_1,\dots,\lambda_n\}$.
Consider a complex frequency $\mu\in\mathbb{C}^+$. Let $\boldsymbol{\vartheta}^{(n)}(\mu;t)$ denote a solution of \eqref{eq:ZS} for the pulse $q^{(n)}(t)$. By applying the following transformation,
\begin{equation}\label{eq:sig_update}
\tilde{q}(t) = q^{(n)}(t) +\frac{2j(\mu^*-\mu) \vartheta_{2}^{*(n)}(\mu;t)\vartheta_{1}^{(n)}(\mu;t)}{|\vartheta_1^{(n)}(\mu;t)|^2+|\vartheta_2^{(n)}(\mu;t)|^2}
\end{equation}
the spectrum of $\tilde{q}(t)$ is related to the one of $q^{(n)}(t)$ according to the following two cases:

\subsubsection{Adding Eigenvalues} If $\mu\not\in \Lambda^{(n)}$, then $\tilde{q}(t)$ has $n+1$ eigenvalues $\Lambda^{(n+1)}=\Lambda^{(n)}\cup\{\mu\}$. In this case, we write $\tilde{q}(t)=q^{(n+1)}(t)$.
As shown in \cite{aref2016control,Aref2018,Junmin1990}, the required condition to enforce
the desired $b_{n+1}$ is
\begin{equation}
    \lim_{t\to-\infty} e^{-j\mu t}\vartheta_2^{(n)}(\mu;t) = -b_{n+1}\lim_{t\to+\infty} e^{+j\mu t}\vartheta_1^{(n)}(\mu;t).
\end{equation}

Interestingly, $b(\lambda)$ for $\lambda\in\mathbb{R}$ and $b_k$ for $\lambda_k\in\Lambda^{(n)}$ of $q^{(n+1)}(t)$ stay the same as the ones of $q^{(n)}(t)$~\cite{Junmin1990}. This property is commonly used to generate a multi-soliton recursively~\cite{matveev1991darboux}, or to generate a pulse with a continuous and discrete spectrum~\cite{Aref2018}.

\subsubsection{Removing Eigenvalues} If $\mu\in \Lambda^{(n)}$, then $\tilde{q}(t)$ has $n-1$ eigenvalues $\Lambda^{(n-1)}=\Lambda^{(n)}\setminus\{\mu\}$. In this case, we write $\tilde{q}(t)=q^{(n-1)}(t)$.
Without loss of generality, assume that $\mu=\lambda_n$. Then the relation between the nonlinear spectrum of $q^{(n-1)}(t)$
and the one of $q^{(n)}(t)$ is given by,

\begin{align}
& a^{(n-1)}(\lambda)=\frac{\lambda-\lambda_n^*}{\lambda-\lambda_n}a^{(n)}(\lambda), \text{ for } \lambda\in\mathbb{C}\\
& \Lambda^{(n-1)}=\{\lambda_1,...,\lambda_{n-1}\} \\
& b^{(n-1)}(\lambda)=b^{(n)}(\lambda), \text{ for } \lambda\in\mathbb{R} \\
& b^{(n-1)}_k=b^{(n)}_k, \text{ for eigenvalue } \lambda_k,1\leq k\leq n-1.
\end{align}
Consequently, the remaining eigenvalues and corresponding spectral coefficients do not change if the signal update \eqref{eq:sig_update} is applied. In Fig.~\ref{fig:example_decomposition} we show an example of successively applying \eqref{eq:sig_update} on a multi-soliton pulse. The pulse has initially 3 eigenvalues which are removed successively in ascending order of their imaginary part.
After the last eigenvalue removal, no residual pulse is left.

\begin{figure}[tb]
\centering
\includegraphics{fig1.pdf}

\caption{Example of third order soliton decomposition with initial discrete spectrum $(\lambda_k,b_k)\in\{(1j,2.1),(1.5j,-0.09),(2j,0.13)\}$. The dashed pulse is resulted after removal of $\lambda_1=1j$ and the dotted pulse is resulted after removal of $\lambda_1=1j$ and $\lambda_2=1.5j$.
}
\label{fig:example_decomposition}
\end{figure}

\section{Successive Eigenvalue Removal Algorithm}\label{sec:SolDecom}

\subsection{Motivation}

Finding the discrete spectrum from \eqref{eq:ZS} is not an easy task. In practice, the tails of a pulse are truncated and then
\eqref{eq:ZS} is approximated from the pulse samples. These two steps cause some approximation errors in finding the eigenvalues and spectral amplitudes. A discretized NFT like \eqref{eq:vec_update} accumulates the approximation errors. This error is very sensitive to the sampling rate and the truncation. When a pulse has a large support, usually a large sampling rate is required to achieve small errors. A typical observation is that a larger number of eigenvalues in a pulse results in either faster variations of the pulse shape or a longer pulse duration or both.
Both effects increase the approximation errors.
In contrast, removing an eigenvalue may lead to less variations and a reduction in pulse duration.
Both effects decrease the number of required samples which leads to a lower computational complexity.

Let $q^{(n)}(t)$ denote a multi-soliton pulse with $n$ eigenvalues $\lambda_k=\omega_k+j\sigma_k$, $k=1,\dots,n$.
Assume that the eigenvalues are indexed in decreasing order of their imaginary part, i.e.
$\sigma_1>\sigma_{2}> \dots> \sigma_n$. It is shown in \cite{Span2019} that when $t\to\pm\infty$
\begin{align}
\label{eq:tail}
|q^{(n)}(t)| \sim & 4\sigma_n |b_n|^{\pm 1} \exp(\mp 2\sigma_n(t \mp t_s)),
\\
\mathrm{with} \qquad & t_s=\frac{1}{2\sigma_n}\ln \left( \left| \prod_{k=1}^{n-1} \frac{\lambda_n - \lambda_k^*}{\lambda_n - \lambda_k} \right| \right)
\label{eq:ts}.
\end{align}
When $|b_k|\sim 1$ for all eigenvalues (i.e. they are in the same order) and eigenvalues are not located
too close together (as in practice),
removing $\lambda_n$ (having the smallest imaginary part) decreases $t_s$ and also
increases the exponent of \eqref{eq:tail} resulting a faster decay in the tails.
Both changes can be seen in Fig.~\ref{fig:example_decomposition}.
Note that removing other eigenvalues reduces also the pulse duration slightly as $t_s$ decreases and the DT enables us to remove eigenvalues in any desired order.

\begin{figure}[tb]
\center
\includegraphics[scale=0.65]{block_diagram_decomposition_algorithm.pdf}
\caption{Block diagram of the Successive Eigenvalue Removal algorithm.}
\label{fig:algorithm}
\end{figure}

\subsection{Eigenvalue Removal and Truncation}

Let $q^{(n)}(t)$ denote a multi-soliton pulse
as defined in the previous section.
We define the effective pulse support as the smallest time interval $[T_-,T_+]$
such that $|q^{(n)}(t)|\leq 2\sigma_n\sqrt{\varepsilon}$ for $t \notin [T_-,T_+]$ and some small fixed $\varepsilon$.
The pulse tails outside $[T_-,T_+]$ are truncated and the pulse duration is then $T=T_+-T_-$. While other definitions for pulse duration can be taken as well, this definition has the advantage that
the truncation threshold scales the same as the pulse energy by normalization of the NLSE \eqref{eq:NLSE}.
For instance, the effective support of a first-order soliton with eigenvalue $j\sigma_n$ contains $\sqrt{1-\varepsilon}$ fraction of the pulse energy regardless of  $\sigma_n$ value.

As it is shown in Fig.~\ref{fig:algorithm}, the Successive Eigenvalue Removal (SER) algorithm computes the discrete spectrum of a multi-soliton $q^{(N)}(t)$ in few steps.
First, it requires an initial guess of eigenvalues
$\tilde{\lambda}_k$, $k=1,\dots ,N$ to determine the sequential order of removal
and the effective pulse support after each iteration.
We use the Fourier collocation (FC) method~\cite{yang2010nonlinear,garcia2019statistics} but one can use any other estimator, e.g. the one finding $\tilde{\lambda}_k$ from the continuous spectrum~\cite{aref2019efficient}.
This step can be skipped when an initial guess of eigenvalues is available.
Here, we remove the eigenvalues in the increasing order of their imaginary part as follows:

\begin{itemize}
    \item[(i)] [\emph{Refinement of eigenvalue}]
    Consider the eigenvalue with smallest imaginary part $\tilde{\lambda}_n$.
    If it is necessary, we improve the eigenvalue estimation further
    via a Newton zero search $a(\lambda)=0$ in the neighborhood of $\tilde{\lambda}_n$.
    For that, we estimate $a(\lambda)$ and $\frac{\partial a(\lambda)}{\partial \lambda}$ according to Sec.~\ref{sec:NFT} based on the current pulse $q^{(n)}(t)$. We end up with a more precise eigenvalue estimate $\hat{\lambda}_n$.

    \item[(ii)] [\emph{Spectral amplitude estimation}] We numerically compute the solution $\hat{\boldsymbol{\vartheta}}^{(n)}(\hat{\lambda}_n;t)$ for $t \in [T_-^{(n)},T_+^{(n)}]$ as described in Sec.\ref{sec:NFT}. Then, we obtain $\hat{b}(\hat{\lambda}_n)$ from \eqref{eq:spec_coeffs_estimation_FB}.

    \item[(iii)] [\emph{Eigenvalue removal}] We apply the Darboux update \eqref{eq:sig_update} to $q^{(n)}(t)$ using $\hat{\boldsymbol{\vartheta}}^{(n)}(\hat{\lambda}_n;t)$ for removing the eigenvalue $\lambda_n$ (the eigenvalue with smallest imaginary part) from the nonlinear spectrum. The resulting pulse is $q^{(n-1)}(t)$ and has $n-1$ eigenvalues.

    \item[(iv)] [\emph{Pulse truncation}] The pulse support is truncated to the interval $[T_-^{(n-1)},T_+^{(n-1)}]$ with the truncation threshold of $2\mathrm{Im}\{\hat{\lambda}_{n-1}\}\sqrt{\varepsilon}$.

\end{itemize}

We repeat these steps until all $N$ eigenvalues are removed.
Note that we assume a very small $\varepsilon$ to avoid a severe
pulse truncation causing changes in eigenvalues as well as $b-$values. We refer the interested readers to \cite{vaibhav2018nonlinear,aref2019nonlinear,gelash2019direct} for the spectral changes after truncation. In the case of severe truncation, the SER algorithm is still applicable and
will find the distorted spectrum (like any other NFT algorithm).

It is worth mentioning that the SER algorithm works in principle
for any removal order of eigenvalues. Here, we remove eigenvalues in
increasing order of their imaginary part. This is a suitable choice when $\frac{1}{\sigma_k}\ln(|b_k|)$
have almost the same magnitudes but it may not
be the best choice in other cases to improve complexity or accuracy.
For instance, when a pulse consists of two separate solitonic compenents in time-domain ($\frac{1}{\sigma_k}\ln(|b_k|)$ have large variance), then
it may be more efficient to remove first the eigenvalues of a solitonic component and then remove the ones of the other component.

\begin{figure}[tb]
\centering
\includegraphics{fig3a.pdf}
\includegraphics{fig3b.pdf}

\caption{Addition of $(\hat{\lambda}_n=0.5j+\delta,\hat{b}_n=\exp(5.88j))$
to a multi-soliton
(dashed line) with five eigenvalues $\Lambda^{(5)}=\{2.5j,2j,1.5j,1j,0.5j\}$, and $b_k=\exp(j\varphi_k)$ of randomly chosen $\varphi_k=0.24,4.90,0.58,3.98,0.09$, respectively. The modified pulses $\tilde{q}(t)$ (solid lines) have three parts: two first-order soliton with eigenvalue $\approx 0.5j$ and a multi-soliton with four eigenvalues
$\Lambda^{(4)}=\{2.5j,2j,1.5j,1j\}$ in the middle. The distance of two first-order solitons is $t^+_\delta+t^-_\delta=O(-\ln(|\delta|))$, given in \eqref{eq:t_delta}. Both figures show the same plots with different y-axis in linear (top) and logarithmic (bottom) scale.
}
\label{fig:6Sol_closeEV}
\end{figure}

\section{Analysis of Eigenvalue Estimation Error}\label{sec:analysis}

An existing eigenvalue $\lambda_n$ will be removed by DT from the discrete spectrum, if $\boldsymbol{\vartheta}^{(n)}(\mu;t)$ is computed exactly at $\mu=\lambda_n$. In this section,
we explain how the eigenvalue estimation error $\delta = \hat{\lambda}_n-\lambda_n$ affects the capability of the SER algorithm to remove the eigenvalue $\lambda_n$.
As before, we assume that $\lambda_n$ is the eigenvalue with the smallest imaginary part.

Since there is always an estimation error $\delta \neq 0$, the SER algorithm indeed adds a new eigenvalue $\hat{\lambda}_n$ to the spectrum.
The SER applies the DT signal update \eqref{eq:sig_update} at $\mu=\hat{\lambda}_n$ where $\hat{\mathbf{\vartheta}}^{(n)}(\mu;t)$ is numerically computed from the pulse $q^{(n)}(t)$.
Then, the resulting pulse $\tilde{q}(t)$ has an additional eigenvalue $\hat{\lambda}_n=\lambda_n+\delta$.
Note that there are two kinds of error: $(i)$ estimation error $\delta$ $(ii)$ numerical errors in $\hat{\mathbf{\vartheta}}^{(n)}(\hat{\lambda}_n;t)$. To study only the effect of $\delta$, we use the DT to generate a multi-soliton
with discrete spectrum $\{(\lambda_k,b_k)\}_{k=1}^n$ and additionally $\{(\hat{\lambda}_n,\hat{b}_n)\}$ for some arbitrary value $\hat{b}_n$.
When adding $\hat{\lambda}_n$ in the last iteration, one can generate
the exact $\mathbf{\vartheta}^{(n)}(\hat{\lambda}_n;t)$ as well as $\tilde{q}(t)$. The detailed algorithm is given in \cite{garcia2019statistics}
and we skip it here.

When $|\delta|$ is small enough, adding $\hat{\lambda}_n$ decomposes $\tilde{q}(t)$ into three separate parts in time domain: two first-order solitons with eigenvalues $\hat{\lambda}_n\approx \lambda_n$ and another pulse between them containing the rest of the discrete spectrum.
The separation effect is illustrated in Fig.~\ref{fig:6Sol_closeEV}.
A multi-soliton with
five eigenvalues $\Lambda^{(5)}=\{2.5j,2j,1.5j,1j,0.5j\}$ (dashed line)
is modified by an additional eigenvalue $\hat{\lambda}=0.5j+\delta$ for
two choices of $\delta$. One can observe that
the first-order solitons (sech-shape pulses in the tails) are pushed away from the middle part as $|\delta|$ decays.
We can analytically approximate the distance between two first-order solitons in terms of $\delta$
and spectral amplitude $\hat{b}_n$. Assume that $|\delta|\ll \sigma_n$ and denote $b_n=|b_n|\exp(j\varphi_n)$ and $\hat{b}_n=|\hat{b}_n|\exp(j\hat{\varphi}_n)$.
Similar to the analysis in \cite{Span2019}, we write
the first-order approximation of the DT update \eqref{eq:sig_update} to characterize
$\tilde{q}(t)$ when $t\to\pm\infty$,
\begin{align}
\tilde{q}(t) \sim &
-2\sigma_{n} e^{\mp j \varphi_{\delta}^{\pm} - 2\omega_n t}\mathrm{sech}\left( 2\sigma_{n}(t\mp t_{\delta}^{\pm})\right) \label{eq:qtilde_tails}
\\
\mathrm{with} \qquad & t_{\delta}^{\pm} \approx \frac{1}{2 \sigma_n} \ln \left( \left| \frac{2j\sigma_n}{\delta} \prod_{k=1}^{n-1} \frac{\lambda_n-\lambda_k^*}{\lambda_n-\lambda_k} \alpha^\pm \right| \right)	\label{eq:t_delta}
\\
& \varphi_{\delta}^{\pm} \approx \arg \left( \frac{2j\sigma_n}{\delta} \prod_{k=1}^{n-1} \frac{\lambda_n-\lambda_k^*}{\lambda_n-\lambda_k} \alpha^\pm \right)
\\
& \alpha^\pm= \left(|\hat{b}_n|^{\pm 1} e^{\pm j\hat{\varphi}_n} - |b_n|^{\pm 1} e^{\pm j\varphi_n}  \right)
\end{align}
We found in different simulations that the above $t_{\delta}^{+}$ and $t_{\delta}^{-}$ are very close to their numerical values. These values are shown in Fig.~\ref{fig:6Sol_closeEV} for $\delta=10^{-8}j$.

The distance between two first-order solitons is $t_{\delta}^{+} + t_{\delta}^{-}=O(\ln(\frac{1}{|\delta|}))$. Look again at Fig.~\ref{fig:6Sol_closeEV}. Let us find the separation points of the three parts of the signal,
denoted by $t_{\rm th}^{\pm}$. Assume that $|\delta|$ is very small such that
the two first-order solitons are far from the middle part. Then,
the middle part converges to a multi-soliton with $\{(\lambda_k,b_k)\}_{k=1}^{n-1}$.
The separation point $t_{\rm th}^{+}$ is approximately equal to the intersection point of the right tail of the middle multi-soliton, given in \eqref{eq:tail}, and the left tail of the first-order soliton on the right, given in \eqref{eq:qtilde_tails}. Similarly, we can approximate the separation point $t_{\rm th}^{-}$. Accordingly, we obtain
\begin{align}\label{eq:t_th}
& t_{\mathrm{th}}^{\pm} \approx \pm \frac{ 0.5 \ln \left(|b_{n-1}|^{\pm 1}\frac{\sigma_{n-1}}{\sigma_n} \right) + \sigma_n t_\delta^{\pm} + \sigma_{n-1} t_0 }{\sigma_n+\sigma_{n-1}}
\\
& \mathrm{where} \qquad
t_0=\frac{1}{2\sigma_{n-1}} \ln \left(\left| \prod_{k=1}^{n-2} \frac{\lambda_{n-1}-\lambda_k^*}{\lambda_{n-1}-\lambda_k} \right| \right)
\label{eq:t0}
\end{align}
and $\sigma_{n-1}=\text{Im}\{\lambda_{n-1}\}$.

Now we find a condition on $|\delta|$. After addition of $\hat{\lambda}_n$, the SER algorithm truncates $\tilde{q}(t)$ to the interval $[T_-^{(n-1)},T_+^{(n-1)}]$. A necessary condition for the truncated signal to contain only the middle part of the signal is $[T_-^{(n-1)},T_+^{(n-1)}]\subset[t_{\mathrm{th}}^{-},t_{\mathrm{th}}^{+}]$.
To satisfy this condition, $|\tilde{q}(t_{\mathrm{th}}^{\pm})|<2\sigma_{n-1}\sqrt{\varepsilon}$. From \eqref{eq:tail} and \eqref{eq:t_th}, we find,
\begin{align}
\left|\frac{\delta}{\sigma_n}\right| \leq\max\frac{|\alpha^\pm|\sqrt{\varepsilon}\sigma_{n-1}}{\sigma_n} \left(\frac{\sqrt{\varepsilon}}{2|b_{n-1}|^{\pm 1}}\right)^{\frac{\sigma_n}{\sigma_{n-1}}}e^{2\sigma_n(t_s-t_0)}
\end{align}
where $t_s$ and $t_0$ are given in \eqref{eq:ts} and \eqref{eq:t0}.

An interesting advantage of the SER algorithm is that it allows to \emph{detect} whether an eigenvalue has been estimated precisely enough. By removing the eigenvalue $\lambda_n$, the pulse energy must decrease by $4\text{Im}(\lambda_n)$. This can be verified by checking the pulse energy after the truncation. Such a capability is missing in other usual NFT algorithms~\cite{yousefi2014information,aref2016control,Hari2016,Vasylchenkova2018,garcia2019statistics,Chimmalgi2018,chimmalgi2019fast,Medvedev20,vaibhav2019efficient,Mullyadzhanov2019}.

\section{Simulation Results}\label{sec:Sim_eval}

We compare the SER algorithm to the common NFT methodology
that finds all eigenvalues and their
spectral coefficients separately without pulse modification.
We compare them in terms of computational complexity and estimation accuracy
for some exemplary multi-soliton pulses. Note that the absolute accuracy and/or complexity depends on the specific NFT method that is used. However, the SER algorithm can be applied with any underlying NFT method. Thus, although we evaluate all results based on the method in Sec.\ref{sec:NFT}, the advantage of the SER algorithm is quite general.

\begin{figure}[tb]
\includegraphics{fig4.pdf}

\caption{Pulse duration $T^{(n)}$ for $n$ remaining eigenvalues after each iteration of the SER algorithm. Three multi-soliton pulses are considered with
(a) $\Lambda_a^{(5)}=\{9j;7j;5j;3j;0.5j\}$, (b) $\Lambda_b^{(5)}=\{0.58j;0.56j;0.54j;0.52j;0.5j\}$ and (c) $\Lambda_c^{(5)}=\{2.5j;2j;1.5j;1j;0.5j\}$.}
\label{fig:5Soliton_Tmax_vs_N}
\end{figure}

\subsection{Pulse Duration and Complexity Reduction}

When the ZSS is solved numerically, the computational complexity scales linearly in terms of the pulse samples $M$.
The classical approaches solve the ZSS for all $N$ eigenvalues based on the same given pulse. Although all $N$ eigenvalues can be computed in parallel,
the total computational complexity is $C N M$ for some constant $C$ operation cost per sample\footnote{We assumed the eigenvalues are given.
To estimate the eigenvalues by a Newton search, the total complexity is multiplied by
the iteration number of the Newton method. We neglect this multiplication
factor for simplicity as it is almost the same for the SER algorithm and the other methods.}.

The SER algorithm truncates the pulse support after removing each eigenvalue. Recall that $T^{(n)}=T^{(n)}_+-T^{(n)}_-$ is the pulse duration when the pulse has still $n$ eigenvalues. Let $T^{(N)}$ denote the initial pulse duration of a pulse with $N$ eigenvalues. Then the computational complexity is $C M\frac{T^{(n)}}{T^{(N)}}$ if the same NFT algorithm for computing the spectral
coefficient of $\lambda_n$ is applied. The total complexity of the SER algorithm is then $\alpha_N C N M$ with $\alpha_N=\frac{\sum_{n=1}^NT^{(n)}}{N T^{(N)}}$.

The SER algorithm has a factor $\alpha_N$ smaller complexity than a classical approach. The complexity gain, however, depends on the nonlinear spectrum of the pulse.
Fig.~\ref{fig:5Soliton_Tmax_vs_N} shows how the pulse duration $T^{(n)}$ decreases after each SER iteration for three exemplary multi-solitons with $5$ eigenvalues.
Three sets of eigenvalues are chosen: $(a)$ $\Lambda_a^{(5)}=\{9j;7j;5j;3j;0.5j\}$, $(b)$  $\Lambda_b^{(5)}=\{0.58j;0.56j;0.54j;0.52j;0.5j\}$, and $(c)$ $\Lambda_c^{(5)}=\{2.5j;2j;1.5j;1j;0.5j\}$. For all three pulses, we let $b_k=(-1)^k$ and $\varepsilon=2\cdot 10^{-4}$.
For the chosen soliton examples, the complexity reduces by $\alpha_N$ $(a)$ $\approx 0.3$, $(b)$ $\approx 0.62$ and $(c)$ $\approx 0.46$, respectively. Note that from \eqref{eq:tail}, the pulse duration $T^{(n)}$ can be well approximated as
\begin{equation}
    T^{(n)}\approx 2t_s + \frac{1}{2\sigma_n}\ln\left(\frac{4}{\varepsilon}\right)
\end{equation}
for sufficiently small $\varepsilon$. We see that, up to the first-order approximation,
$T^{(n)}$ is independent of the spectral phases $\varphi_k$.

The FC method was not considered in the complexity analysis as its complexity is independent of the SER algorithm. Note that, before their refinement, only a coarse estimate of the eigenvalues is needed to determine their removal order. The FC can provide such a coarse estimate with only a few samples~\cite{garcia2019statistics}. Thus, although it has cubic complexity with respect to the number of samples, the SER's total complexity is not dominated by the FC method. If predefined eigenvalues are used for modulation, FC is not needed at all.

\subsection{Estimation Accuracy}

We compare the precision of the SER algorithm and the common NFT methodology in finding the spectral coefficients of a given pulse. To have a fair comparison, the SER algorithm should use the same NFT algorithm to find $\hat{\boldsymbol{\vartheta}}^{(n)}(\hat{\lambda}_n;t)$.
Here, we use the NFT algorithm of Sec.~\ref{sec:NFT}.
We consider the eigenvalue set $\Lambda_c^{(5)}$ with $b_k=\exp(j\varphi_k)$.
We generate $5000$ multi-soliton pulses $q^{(5)}(t)$ with randomly chosen $\varphi_k$ values.
Let us define the bandwidth $B$ of a pulse in (linear) Fourier domain
as the frequency support containing $99.99\%$ of the total energy.
All pulses are sampled with the same sampling rate of $f_s=4 \max_{\varphi_k}{B}$.
We further add a white Gaussian noise to the pulses according to a signal-to-noise ratio (SNR)
defined within the bandwidth $B_\mathrm{max}$. Then, we compute the spectral coefficients $\hat{b}(\hat{\lambda}_k)$ of the noisy pulses using both algorithms. Define $\hat{\varphi}_k=\arg \{\hat{b}(\hat{\lambda}_k)\}$.
Fig.~\ref{fig:5Soliton_variance_vs_snr} shows the variance of the phase error, $\mathrm{Var}\left(\hat{\varphi}_k - \varphi_k\right)$, in terms of SNR for both algorithms. We observe that both algorithms have quite the same precision while the SER algorithm has about
half the complexity.

\begin{figure}
\includegraphics{fig5.pdf}

\caption{Accuracy of the SER algorithm in comparison to the classical NFT algorithm of Sec.~\ref{sec:NFT} for the detection of the spectral amplitudes $b(\lambda_k)=\exp(j\phi_k)$.
The variance of the phase estimation error $\mathrm{Var}\left(\hat{\varphi}_k - \varphi_k\right)$
in terms of SNR for a multi-soliton pulse with $\Lambda_c^{(5)}=\{2.5j;2j;1.5j;1j;0.5j\}$ being perturbed by an additive white Gaussian noise is shown.
}
\label{fig:5Soliton_variance_vs_snr}
\end{figure}

The SER algorithm reduces the complexity by pulse truncation.
We now compare the performance of the SER and the classical algorithm when both algorithms have the same total number of samples (and thus the same complexity).
One interesting scenario is to investigate how the performance of the classical NFT algorithm is sensitive to the pulse truncation. The logic behind this scenario is to see
how the contribution of an eigenvalue is distributed over the support of an initial pulse.
We choose the pulse truncations according to the SER algorithm: $T^{(5)},T^{(4)},T^{(3)},T^{(2)},T^{(1)}$ as given in Fig.~\ref{fig:5Soliton_Tmax_vs_N}. The pulses are generated and sampled as explained before at SNR$=30\mathrm{dB}$.
We compute the spectral coefficients using the classical NFT methodology when applying the different truncations $T^{(n)}$. Fig.~\ref{fig:5Soliton_variance_vs_truncation} shows the respective $\mathrm{Var}\left(\hat{\varphi}_k - \varphi_k\right)$ for all eigenvalues with different truncation.
The dashed line indicates the performance of the SER algorithm.
The dotted line is the performance of the classical NFT algorithm when it applies the same truncation value $T^{(n)}$ as in the SER algorithm before finding each $\hat{b}(\hat{\lambda}_n)$. This way, it uses the same number of samples as the SER algorithm but its performance degrades significantly.
In fact, any pulse truncation causes a drastic growth in the estimation error without applying the SER pulse modification. On the other hand, by applying the pulse modification of SER algorithm, the nonlinear spectral contents will be condensed in a shorter time support.

\begin{figure}
\includegraphics{fig6.pdf}

\caption{Effect of a pulse truncation $T^{(n)}$ on the accuracy of a classical NFT algorithm to detect $b(\lambda_k)=\exp(j\phi_k)$. The variance of the phase estimation error $\mathrm{Var}\left(\hat{\varphi}_k - \varphi_k\right)$ is shown for a randomly phase modulated multi-soliton pulse with $\Lambda_c^{(5)}=\{2.5j;2j;1.5j;1j;0.5j\}$ with an additive white Gaussian noise perturbation at SNR$=30\mathrm{dB}$. The dashed line shows the result when using the SER algorithm. The dotted line is the accuracy of the classical NFT algorithm when applying the same truncation value $T^{(n)}$ as in the SER algorithm before finding the respective $\hat{b}(\hat{\lambda}_n)$.
}
\label{fig:5Soliton_variance_vs_truncation}
\end{figure}

\section{Conclusion}\label{sec:concl}

We proposed the successive eigenvalue removal algorithm to compute the discrete spectrum.
The algorithm computes the spectral coefficients successively and carefully
removes the previously computed eigenvalue and its spectral coefficient from the spectrum.
The algorithm peels off the eigenvalues of a pulse one by one to simplify the computation of the remaining discrete spectrum. As an application, we showed a considerable complexity gain of the SER algorithm compared to the conventional methodology without pulse modification in finding the spectral coefficients without sacrificing accuracy.

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


Houston, you had no problem in Alex Box Stadium when it counted.
BATON ROUGE – Houston, you had no problem in Alex Box Stadium when it counted.
The Houston Cougars, losers to LSU, 5-1, Saturday and trailers of a 10-0 LSU by 4-0 late Sunday at a raucous and full Box, returned the Tigers to earth with a bang Monday night by a 12-2 score. It was LSU's worst loss in an NCAA Regional since falling to Cal State Fullerton and then-coach Augie Garrido, 11-0, in 1992. And it was the Tigers' first elimination from an NCAA Regional after a 2-0 start in the history of a program long known as either the greatest or very close to that.
"The season coming to an end is the worst day of the year," a despondent LSU coach Paul Mainieri said Monday night. "It was awful that we had to have our worst game of the year on this date, but it is the way the game is. Just a very tough night."
The loss eliminated the Tigers from their own home Regional for the first time since 2005 and for only the fourth time since LSU started hosting such tournaments as its rite of summer in 1986. LSU won 13 of 15 NCAA Regionals hosted from 1986 through 2004 and 16 of 18 counting Super Regionals as well in 2000, 2003 and '04.
Now, the Tigers have been eliminated from the NCAA postseason twice in the last three years alone in the Box, known as the most difficult place to play in the country for visitors with LSU leading the nation in attendance for two decades. And both of those eliminations came with back-to-back losses – 5-4 in 11 innings Sunday and 12-2 Monday to Houston and 3-1 and 7-2 to tiny Stony Brook in the NCAA Super Regional in 2012.
"We were 2-0 in this tournament and seemingly in the drivers' seat," Mainieri lamented. "Unfortunately we just didn't get it done."
LSU, playing in its 28th home NCAA postseason tournament over the weekend, went into Sunday night's game against Houston after beating Southeastern Louisiana and Houston to improve to 99-29 overall in NCAA Regional and Super Regional play for a .773 winning percentage that has been No. 1 in the nation for decades. LSU entered the postseason as a top eight national seed for the third straight year and seventh time this century, which means it was expected to reach the College World Series in Omaha, Neb., later this month.
"I thought this team was one of the best teams in the country," LSU shortstop Alex Bregman said. But the Tigers, who finished third in the RPI-No. 1 Southeastern Conference and won the SEC Tournament on their way to a 10-game winning streak through Saturday, did not play on Sunday and Monday as they had previously.
"I do not think this game was a reflection of our season at all," Bregman said. "We know that we are a much better team than that. I think we were a better team than what we showed today."
Mainieri also felt his team (48-16) underachieved over the weekend compared to the regular season and SEC Tournament.
"This game today was not indicative of our season," he said. "It was as poortly a pitched game as we had all year. When I think of the 2014 season, I'm not going to think about this game."
Oh, to think of what might have been.
With one more win on either Sunday or Monday, the Tigers would have hosted Texas and legendary coach Augie Garrido – the Skip Bertman of the West – in a best-of-three Super Regional in the Box this weekend with LSU's 17th CWS appearance and Texas' 35th on the line. Texas, which won the NCAA Houston Regional over Texas A&M, is second to LSU in NCAA Regional and Super Regional winning percentage at 146-55 and .726. It would have been a College Baseball Brand Name Super "Bowl" Regional in Baton Rouge.
And it all came splashing down. The Cougars (48-16), who were on the cusp for the No. 8 national seed that LSU received, scored four in the eighth Sunday night to tie it and one in the 11th to win it 5-4. Then they put up a touchdown in the third inning Monday night off three relievers for a 9-2 lead and coasted through six pitchers in all to the 12-2 victory.
"I think we came out hot in the first inning," said Bregman, who singled and scored on Sean McMullen's two-RBI single as LSU jumped out to a 2-0 lead off Houston freshman starter Andrew Lantrip in the top of the first. "We scored two right off the bat."
But LSU went scoreless from the second through the ninth, quieting its stadium to echoes, and totaled eight hits to Houston's 12.
"We just did not really do anything after the first," Bregman said.
Hauntingly, LSU did a similar thing in the elimination game against Stony Brook two years ago. After tying that game 1-1 in the first on a Mason Katz home run before a rocking crowd, LSU went quietly and finished with three hits in a 7-2 loss.
Jared Robinson (5-1) relieved Lantrip in the third to four-hit the Tigers over six and one-third innings and struck out eight for the victory.
LSU freshman starting pitcher Alden Cartwright entered the game with five SEC starts to his credit, including the SEC title game against Florida just last week, and predicted an LSU victory Sunday night. But he supplied the worst start by a pitcher likely in Mainieri's eight years at LSU. He threw 11 pitches. Nine were balls. He walked two and hit a batter and was out of there.
Reliever Parker Bugg limited the damage to two runs, but he allowed four runs in Houston's seven-run third as he walked a batter, hit a batter to load the bases and hit another batter with the bases loaded.
Brady Domangue, who had not retired a batter since April 29 and had not pitched in a game since May 4, relieved and immediately looked as if he had not faced competition in several weeks. He allowed four hits and three runs with a walk for the 9-2 deficit.
"You can't help but be happy about a seven-run inning," said Robinson, who pitched from the fourth through the seventh without an LSU runner reaching second base.
Nate Fury, a much more used reliever than Domangue, finally got LSU out of the inning and pitched four and one-third innings of three-hit, shutout ball, but it was too late for LSU and for Fury, a senior. "I did not want to come out of the game. I was tired, but I wanted to pitch all night," he said.
"Never really gave LSU a chance to get back in the thing," said Houston coach Todd Whitting, whose team will play at Texas this weekend in the Super Regional.
"I talked about just hanging in there," Mainieri said of the growing deficit in the third after Cartwright's disastrous appearance. "It didn't work out as we planned. The third inning was a nightmare. I thought we were ready to play today."
As the innings went on in a three-hour, 37-minute game, LSU's dugout looked like it just wanted to go home.
"Just end it," a disgusted LSU fan yelled from the now spacious upper seating area at the Box as Houston tacked on three in the eighth for the 12-2 final.
"This is a program that I have dreamed of playing at for my whole life," said McMullen, a senior who played his last game. "And it is now at an end. There is really not much more I can say."
Just 24 hours prior, LSU led 4-0 and was six outs away from Fury and McMullen and others having more – maybe many more – games to play. From the eighth inning Sunday through the third inning Monday, Houston shut the Box with a thud, outscoring LSU, 14-2.
"To be able to steal momentum in this ballpark, which I think is the toughest place to play in college baseball, especially when you're facing an LSU that's the hottest team in baseball coming off an SEC championship," Whitting said and paused. "To grab that momentum Sunday and be able to play through it today, you've just got to tip your hat to our club. This place was rocking and rolling."
But now the Box and LSU are closed for summer.


UPS is a global leader in logistics, offering a broad range of solutions including transporting packages and freight; facilitating international trade, and deploying advanced technology to more efficiently manage the world of business. Headquartered in Atlanta, UPS serves more than 220 countries and territories worldwide. UPS is in the Netherlands since 1985 and has 13 facilities in the country. Within the Netherlands we have the two business units of UPS: Small Package and Supply Chain Solutions. Small Package is known for the brown busses which deliver the packages to the consumers, while Supply Chain Solutions offers all logistics solutions to the business.
The company can be found on the web at ups.com or pressroom.ups.com and its corporate blog can be found at longitudes.ups.com. To get UPS news direct, follow @UPS_News on Twitter.


text:
La stagione 2007-2008 della New Polibasket San Bonifacio è stata l'ottava consecutiva disputata in Serie A2 femminile.

Stagione
Sponsorizzata dalla Pakelo lubricants, la società veronese si è classificata all'undicesimo posto in Serie A2 e ha dovuto partecipare ai play-out. Nel primo turno ha eliminato la Tecno Allarmi Cervia e si è salvata.

Roster

Organigramma societario
La dirigenza è composta da:

 Presidente: Lorenzo Cortese
 Vicepresidente e addetto stampa: Tarcisio Caltran
 Segretario e resp. internet: Andrea Sordo
 Dirigenti: Pietro Baratello, Dino Dal Cer, Marino Piccino, Valentino Zorzetto
 Resp. settore giovanile: Marino Piccino

Note

Voci correlate
Serie A2 femminile FIP 2007-2008

Collegamenti esterni

2007
San Bonifacio

redpajama_set_name: RedPajamaWikipedia


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text:
Celebrating Korean American Writer, Min Jin Lee
Date: February 16, 2018Author: reneejohnsonwrites 6 Comments
The timing was perfect.
While the world's eyes were focused on PyeongChang, (which translates into land of prosperity), and the Winter Olympics, those of a small group from Women's National Book Association in Charlotte, North Carolina were dialed onto Min Jin Lee, and her latest novel, Pachinko.
Pachinko author, Min Jin Lee is seated in the center.
Blog author, Renee Johnson, is standing directly behind her.
Pachinko, a National Book Award Finalist, is a sweeping historical saga of a Korean family in Japan, much as the first of this trilogy–Free Food for Millionaires–focuses on Koreans in America. It was also selected as one of New York Times Notable Books of 2017 and Top Ten of the Year, a USA Today Top Ten of 2017, and numerous other accolades, along with being one of our book club choices.
Having missed the group discussion immediately after reading it, I was thrilled to be able to attend the luncheon with its author, Min Jin Lee.
Lee arrived looking fresh and beautiful in spite of her whirlwind, international book tour. Her friendliness and immediate ease with our group made us feel like we were having lunch with another gal pal, albeit an especially smart and talented one.
She seemed interested in us as well, quickly sketching a seating chart. With our names noted in the spots we occupied at the table, she then asked questions about our lives and backgrounds.
By the time we got around to discussing Pachinko, we were relaxed and eager to learn about her process and inspiration.
"History has failed us, but no matter," Lee began, in both the novel and our discussion of it. Our group took this one bold statement and explored the reasons why this is true for us all, not just Koreans exiled in Japan.
We didn't have to search far. Our association, WNBA, was founded in 1917 when women were excluded from joining the prestigious Bookseller's League. Not to be ignored, fifteen women met at Sherwood's Book Store in New York, forming a group of their own to give voice to female writers and readers.
We even found commonality in the inevitable presence of a "Hansu" character in most of our lives. (Read the book to discover reasons why Hansu was both captivating, and a character to be avoided.)
The thing I most admired about Pachinko and Lee's treatment of its historical significance, was her ability to deliver a fresh perspective on the Korean experience in the years from 1910 to 1989, while handling the events of that age with enough subtlety to make me forget I was traipsing through a history lesson.
If you'd like the opportunity to hear Min Jin Lee speak, you can catch her in one of the following cities:
Pittsburgh, PA at Alphabet City on February 20, 2018 at 8:00 p.m.
Washington, DC at Politics & Prose at The Wharf on February 26, 2018 at 7:00 p.m.
Cincinnati, OH at Mercantile Library on March 6, 2018 at 7:00 p.m.
Tucson, AZ during Tucson Festival of Books, March 11, 2018 at 11:30 a.m.
Hoboken, NJ at Little City Books on March 14, 2018 at 7:00 p.m.
Dartmouth, MA at Farm and Coast Market, March 15, 2018 at 7:00 p.m.
East Meadow, NY at East Meadow Public Library on March 20, 2018 at 8:00 p.m.
Nashville, TN at Parnassus Books (with Ann Patchett) on March 21, 2018 at 7:00 p.m.
And many more dates and cities. To find one near you, click here.
Renee Johnson is the author of Behind The Mask, Herald Angels, Acquisition, and The Haunting of William Gray. She is currently working on a historical novel, while editing a suspense novel which has international flair–an homage to her love of travel and foreign food. She lives on a farm in North Carolina with her husband, Tony Johnson, and two very spoiled German shepherds named Hansel and Gretel.
American LiteratureauthorsCharlottehistorical novelKoreaKorean-AmericanliteratureMin Jin LeeNorth CarolinaPachinkoWomen's National Book AssociationWriters and Readers
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6 thoughts on "Celebrating Korean American Writer, Min Jin Lee"
I will have to keep an eye out for Pachinko. It sounds like a fascinating read.
You won't be disappointed. Everyone in the book club loved it, and those of us who had the good fortune to have lunch with Min Jin Lee walked away adoring her as well. She followed her passion and instincts, and the result is a novel doused in history but straight from her heart.
—-Renee,
I love how involved you are….NO grass will ever grow under your feet, dear!
The book sounds great!
Sometimes it's a bit of a whirlwind, but it keeps me inspired. How are things with you? Busy too, I'd imagine.
Fascinating post, Renee. Thanks!
I'm glad you liked it, Linda. Fantastic novel from a lovely author!


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Home Football NFL Teams and Figures That Were Doomed to Failure
The NFL is the ultimate test of players, coaches and executives. For those that reach the summit of the sport, there is no better feeling than hoisting the Vince Lombardi Trophy.
However, only team per season gets the opportunity to taste the ultimate success in football. The other teams in the league have to wait and wonder whether they will ever win the Super Bowl. For a large portion of NFL players, coaches and executives, that moment never comes.
It can be a case of bad luck, poor talent, inefficient coaching, and ultimately not being good enough. The defining moment can come on one particular play or a game where the breaks don't quite fall your way. However, for some individuals it feels as though their hopes are over before they've even begun. Optics and opinion can play a vital role in the mood of a team and a fan base.
Certain coaching hires and player signings catch the eye for the wrong reasons, knowing that will not work out. Here are three examples of when it was clear from the off that the team and individual were doomed towards failure.
Brandon Weeden
Poor Brandon Weeden. Eyebrows were raised when the Cleveland Browns selected Weeden with the 22nd overall pick in the 2012 Draft.
The Browns endured years of misery picking quarterbacks since the franchise returned to the NFL in 1999. They had previously the 22nd selection in 2007 to acquire Brady Quinn, who had surprisingly fallen down the NFL Draft after being touted as a top-10 selection.
The Browns had hope he would fit the bill to turn them into perennial playoff contenders. Unfortunately for them, they were spectacularly wrong.
Therefore when Cleveland selected the 29-year-old Weeden with the 22nd selection five years later, Browns fans everywhere winced. Weeden was a solid starter for Oklahoma State in his college career, and although he was a mature rookie, hopes were high for his future.
However, his first moment on the field would epitomize his career as he got caught under the American flag during the pre-game ceremony of his NFL debut. It went downhill from there and he lasted only two seasons in Cleveland before being released. Weeden served as a backup around the league for Dallas, Houston and Tennessee before his career ended in 2018.
Adam Gase
Adam Gase had flamed out of his previous head-coaching position with the Miami Dolphins after three unsuccessful seasons. He built a reputation for being stubborn and forthright with his players, making him an unpopular figure in the locker room.
Gase was fired after the 2018 season with a 23-25 record, along with one post-season defeat to his name. The New York Jets were undeterred by his failure at their AFC East rivals and hired Gase in the hope of developing quarterback Sam Darnold.
Gase began life at the Jets with a bizarre press conference that would sum up his time in New York. His eyes caught the attention of social media for drifting all over the room, which prompted ridicule and Adam Gase memes aplenty after only a few hours on the job.
Gase was already facing an uphill battle reviving the fortunes of a struggling franchise, with his reputation already in doubt after his spell in Miami.
Matters did not improve in the locker room as he brought the same prickly attitude to the Jets, resulting in the departures of Le'Veon Bell and Jamal Adams.
Darnold failed to progress, which resulted in dire performances in 2020 for the Jets. They finished with a 2-14 record and Gase was consequently fired. He needs a break to salvage his coaching reputation in the future, while the Jets are a long way from being one of the leading contenders in the Super Bowl odds.
The Denver Broncos were outstanding during the 2013 on offense, breaking records aplenty. Peyton Manning threw 55 touchdown passes over the course of the campaign, eclipsing Tom Brady's record from the 2007 campaign. They defeated Brady and the Patriots to advance to Super Bowl XLVIII against the Seattle Seahawks in the hope of securing a second Lombardi Trophy for Manning and a first for the franchise since the 1998 season.
However, their hopes were dashed from the first play of the game when center Manny Ramirez snapped the ball over the head of Manning and the Broncos were forced to concede a safety.
From that point onwards the Seahawks dominated, easing to a 43-8 win. The Broncos did get their Super Bowl two years later in a victory over the Carolina Panthers, banishing the demons of their thrashing and sending Manning out of the NFL on a high note.
Sometimes destiny is written in the stars. Unfortunately for the Broncos and the two individuals – their fate was sealed before they even started.
Ray Lewis: Real "Crime" Is No Season
Devin Hester – The Bears MADE Me Play Wide Receiver


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