text:
Q: Unable to set current page on NextJS navigation I have the following Navigation component in my NextJS project;
const navigation = [
    { name: 'Dashboard', href: '/dashboard', icon: Squares2X2Icon, current: true },
    { name: 'Orders', href: '/orders', icon: UsersIcon, current: false },
    { name: 'Products', href: '/products', icon: FolderIcon, current: false },
]

function classNames(...classes) {
    return classes.filter(Boolean).join(' ')
}

const Navigation = () => {
    return (
        <>
            {navigation.map((item) => (
                <a
                    key={item.name}
                    href={item.href}
                    className={classNames(
                        item.current
                            ? 'bg-gray-100 text-gray-900'
                            : 'text-gray-600 hover:bg-gray-50 hover:text-gray-900',
                        'group flex items-center px-2 py-2 text-sm font-medium rounded-md'
                    )}
                >
                    <item.icon
                        className={classNames(
                            item.current
                                ? 'text-gray-500'
                                : 'text-gray-400 group-hover:text-gray-500',
                            'mr-4 flex-shrink-0 h-6 w-6'
                        )}
                        aria-hidden="true"
                    />
                    {item.name}
                </a>
            ))}
        </>
    )
}

export default Navigation

This is copied from the Tailwind CSS site ... When I am on the Dashboard page, the navigation link is current, but when i go to Orders or Products the 'current' navigation stays on the Dashboard.
Is there any way to dynamically change this based on the page i am currently on?

A:

const navigation = [
    { name: 'Dashboard', href: '/dashboard', icon: Squares2X2Icon, current: true },
    { name: 'Orders', href: '/orders', icon: UsersIcon, current: false },
    { name: 'Products', href: '/products', icon: FolderIcon, current: false },
]

function classNames(...classes) {
    return classes.filter(Boolean).join(' ')
}

const Navigation = () => {
    const router = useRouter();
    return (
        <>
            {navigation.map((item) => {
               const isCurrent = router.pathname === item.href
                return <a
                    key={item.name}
                    href={item.href}
                    className={classNames(
                        isCurrent
                            ? 'bg-gray-100 text-gray-900'
                            : 'text-gray-600 hover:bg-gray-50 hover:text-gray-900',
                        'group flex items-center px-2 py-2 text-sm font-medium rounded-md'
                    )}
                >
                    <item.icon
                        className={classNames(
                            isCurrent
                                ? 'text-gray-500'
                                : 'text-gray-400 group-hover:text-gray-500',
                            'mr-4 flex-shrink-0 h-6 w-6'
                        )}
                        aria-hidden="true"
                    />
                    {item.name}
                </a>
            })}
        </>
    )
}

export default Navigation

Better way to get current page path through next/router
const router = useRouter();
with that you can make a condition like below and detect what is current page
const isCurrent = router.pathname === item.href

redpajama_set_name: RedPajamaStackExchange


This is my first jewelry of the new year. I want to say, my choice is correct. I ..
Better than I expected.Happy new year ..
My order on December 23,i received it today,it is very nice. As a late Christma ..
This is a timely parcel indeed! I think it will be a wonderful Christmas! ..


With Warren Buffett at its helm, the country's fifth-biggest dealership group is about to get a lot bigger.
When Buffett signed a deal to buy Van Tuyl Group last week through his Berkshire Hathaway Inc. investment company, he clearly signaled that the top four dealership groups are in his sights. When a CNBC host mentioned AutoNation Inc. and said it was the country's biggest auto retailer, Buffett quipped: "Temporarily."
"This is just the beginning for Berkshire Hathaway Automotive," Buffett said of the new venture, which is being called the biggest acquisition by far in automotive retail history. Terms weren't disclosed, but outsiders estimated the all-cash price tag at over $4 billion.
The Van Tuyl operation can be "scaled up a lot from where it is," said Buffett, who predicted that he and Van Tuyl's leaders will hear from "hundreds of dealerships" interested in selling in the next year.
Larry Van Tuyl, who will become chairman of the new group, told Automotive News that he started getting those calls from potential sellers right after the news broke Thursday, Oct. 2.


Happy New Year! As 2019 begins, I'm thinking about time.
I ushered in the New Year in Mainz, Germany. Believe it or not, it was the first time I have ever celebrated New Year's in Germany. As we watched the fireworks over the Rhine from the window of our hotel room, I could not help but to be impressed. The Silvester revelers certainly put even Los Angeles to shame in terms of the length and duration of the firework show. Rather than just a central-show, everyone brought and launched their own. It was so much fun.
Little Augustine slept through the first 20 minutes of the fireworks. It was only when we opened the window (my wife and I both like the sulfur-laced smell of the air during a fireworks show) that the increased noise woke him up. You should have seen his face. He was amazed and delighted. It marked such a happy family memory.
And it reminded me, as I looked down on him, that time is running fast. You see that when you compare the pictures of him when we flew SWISS First Class exactly one year ago to him today. The idea that 01/01 wipes the slate clean is simply not true. Sure, it marks one arbitrary day to restart, but life groans on…it never stops.
Isn't that the sobering truth? Time is an ever-rolling stream and it will eventually claim all of us. Thus, it matters all the more what we do with our time, our talents, and our resources.As I look ahead, I don't want to waste time in the pursuit of money and vanity when it comes at so great an expense. Hard work is a virtue, and so is the reward it brings…yet they are not ends in themselves, merely means to an end.
My analysis of the black man ejected from a hotel in Portland was quite eye-opening. Not so much in the negative reaction of many commenters, for I expected that. Instead, it was the level of passion and detail in which many responded. Read through the comments and consider all the time it took to compose such stinging rebuttals to my own analysis. There are so many issues in which agreement will never be found, including this topic. I'm not out to earn everyone's respect. That seems a pretty bad way to go about life. But when I write about any topic, I do want people to respect the point of view, even if it they strongly disagree with it. In order for that to occur, I need to be more empathetic. I need to be more understanding. That takes time. And time is an ever-rolling stream.
A few days ago I was in Colmar, France to visit a Christmas Market. Unlike in Germany, in France Christmas markets continue the week after Christmas. It was a very cold day, but the hot roasted chestnuts warmed all of us. I know I got taken to the cleaners, but there's something fun about shopping at these little wooden booths and we took care of much of next year's Christmas shopping.
And while 01/01 may not really represent a new start, every day–in one sense–marks a new start. Resolutions that are broken need not be left until 2020. Rather, they can restart the moment they are broken. Life is not a linear continuum to perfection. Quite the opposite, it is marked by peaks and valleys, a time for everything under the sun.
As time flies by, my resolution this year is to continue to learn, continue to empathize, and continue to do my best to show dignity to others.
We'll discuss blog-specific goals tomorrow. For now, Happy New Year and let's all try to be a little nicer to one another this year, even when it is not deserved.
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That's unfortunate. You are missing out on so many things if you listen to the worst news stories and only go where you think is perfectly safe.
I'm also "missing out" on terrorist attacks. I (and my family) prefer the latter. As long as various western european governments do not care about keeping people safe, I will not spend my money in these countries.
I approve this comment. Don't be a fool and subject your family to unnecessary risk! I wouldn't fly if someone paid me ten million bucks.
Hey Matt. I just wanted to comment and let you know it's posts like this that remind me even though you might have a very different perspective than mine at times, we're all human and we could all do a little better at working to be more compassionate and understanding with one another. Despite how I feel some of your articles are framed/written overall I think your showing that your doing what you can to try and learn from others perspectives and it's appreciated. It's because of this and continual quality content as well as posts about your family reminding me of our similarities that keep me coming back to your blog even though I may disagree with you on topics. Your one of the view places where I know I will likely disagree with your perspective but still want to read it to try and understand better. I only hope you continue to do the same in return. Best wishes to you and your family and safe travels to you and everyone else in 2019!
Thanks William. I appreciate your comment and thank you for continuing to read my blog. I hope to continue to gain insight from you in 2019.
I read all your posts. Reading this one it occurs to me you might find useful something I found useful this past year. I suggest that you consider reading more by travelers of color, women of color, LGBTQ travelers, and others who face situations that are unlike your own. Empathy is hard to build when you are defending yourself. As you understandably were in the comments to your other post. But if you regularly read about the experiences of those not like you empathy grows. At least I found mine did. I also made time last year for a regular volunteer commitment with a community I was unfamiliar with and to start learning another new language, two things I did think I had time for because of travel. But I found I did. I'm not the most emphatic person. It's hard for me. I have to work at it. But being more present to listening to others and hearing that they might experience something I don't was helpful. When I only read news pieces about bad things happening to people like the guest at the doubletree I didn't see their perspective as clearly. I still have tons of work to do. I hope you take these suggestions in the spirit they are intended, from one fellow traveler trying to improve to another. Every day is indeed a new opportunity for empathy and growth.
Thanks Emily. Well-noted. I agree with you sentiment.
Thanks for this — the style of the post reminded me a bit of Peggy Noonan's WSJ column. I hope that comes across as a complement.
Would love to see a picture of your son.
Noonan is one of my favorite columnists!


Home Studios has recently been recognized for their outstanding interior design at a new LA Bar, Bibo Ergo Sum. We were lucky enough to have designed custom bar tap handles and our Volupta small vessals as candle holders. Check out the article feature on Sight Unseen's website here.


Edgar Pieterse is an urbanist. He endeavours to remain grounded in the tough and messy realities of cities—invariably always on the move—working with materialist and aesthetic optics. His work is rooted in two South African cities, Johannesburg and Cape Town but he also tracks African cities as part of larger discourses on sustainable urban transitions and southern urbanism.


text:
National Maritime Museum Manuscripts : Fifth voyage of Bombay Castle, Bombay and China, (Captain Archibald Hamilton, Managing owner Joseph Dorin): accounts 1802...
Hamilton, Archibald
National Maritime Museum Manuscripts

One of three documents from the papers of the East India Company Captain Archibald Hamilton, who from 1802 commanded two voyages in the East India Company Ship Bombay Castle that sailed to Bombay and China along the well-established trade networks as part of the East India Company's lucrative business in the Indian Ocean and the Pacific. The journey Bombay Castle has received most attention for is its involvement in the 1804 Battle of Pulo Aura - a confrontation initiated by French attempts to raid the cargo-laden British vessels at the Straits of Malacca.

These records, however, do not contain information about military skirmishes, rather, they contain the instructions for, and attempts at, calculating longitude by lunar and stellar observation and by timekeeper. What is perhaps most interesting is the list of instructions contained in  (HMN/31) that outline in quite basic terms the method of calculating longitude by lunar or stellar observation. It should be remembered that these techniques were not necessarily 'new' but they were difficult, and by the early nineteenth century no standard procedure had been established of who should perform them or when they should be done: on British voyages of discovery, for instance, it was often Board of Longitude appointed astronomers who were tasked to carry out these actions. These techniques for finding longitude also ran up against a real reluctance by officers both from the Royal Navy and from the East India Company to attend naval schools and academies to learn the necessary mathematics.

By comparing the instructions  (HMN/31) with the rough calculations (in HMN/29 and  (HMN/44)), we can also see that the ongoing trialling of timekeepers was totally dependant upon astronomical technique in order to produce rates of the going of these timekeepers and to judge how well they were performing (and therefore how much they were to be trusted) - something which was still occurring in the early nineteenth century. This information tended to be used in three ways: by the clock and watchmakers themselves to promote their instruments both to the private market of officers on Royal Navy and East India Company ships; by the Board of Longitude in their attempt to discern between the quality of different timekeepers and the usefulness of timekeepers as a practical solution for finding longitude (even at the beginning of the nineteenth century); and by officers themselves who, by the early nineteenth century were demonstrating their capacity as conspicuous consumers of timekeepers by writing about the various advantages and/or deficiencies of timekeepers in magazines and journals such as the New Monthly Magazine.

Eóin Phillips
History and Philosophy of Science
University of Cambridge

redpajama_set_name: RedPajamaCommonCrawl


text:
The federal government has scrapped all planned parliament sitting days between March and August. Image by Lukas Coch/AAP PHOTOS
'Better things to do' than parliament
By Katina Curtis, AAP Senior Political Writer
The Morrison government has declared it has better things to do than sitting in parliament as it deals with the response to the coronavirus.
Parliament has been recalled for a single day on Wednesday to pass legislation giving the legal underpinning to the $130 billion wage subsidy for businesses hit by the virus crisis.
But the government continues to insist its scrapping of all planned sitting days between March and August is the most sensible thing.
Australia is alone among similar democracies in having dumped so much of its parliamentary time.
Leader of the House Christian Porter says parliament is "a very flexible organ of government" and that is how it's being used.
"But why would we set down a regular sitting schedule over the coming weeks and months, in the most irregular time Australia has ever known? What is the point of that?" he told ABC Radio National on Monday.
"If people want to sit out there during the greatest economic crisis Australia's experienced and read practice and procedure of the House of Representatives, good luck to them. But we've got better things to do."
His opposition counterpart Tony Burke said Mr Porter was effectively ridiculing his own job as the government's manager of parliamentary business.
"At the moment you've got parliament behaving at its best and the government's response is they'd rather not have parliament," he told reporters.
"I know Scott Morrison doesn't like the parliament much, but at the moment we have a public show of national unity every time it meets."
Labor wants to have more sitting days scheduled in a few weeks' time to update legislation as needed, saying it was inevitable with such vast sums of money being spent in such a short time that there would be things not done right.
It is also negotiating with crossbenchers and the government to set up a special Senate committee to oversee the response to the coronavirus crisis, but Mr Burke said that would be no substitute for proper parliamentary oversight.
Centre Alliance senator Rex Patrick has proposed the upper house should switch to a default sitting calendar where a decision can be made before each session whether it needs to be abandoned on health grounds.
"That way the default is to sit unless there is a good reason not to," he told AAP.
He said parliament must continue its vital scrutiny responsibilities, through committees which can meet using video-conferencing.
The government doesn't hold the majority in the Senate, so if Senator Patrick's suggestion wins the backing of Labor and the Greens it has a good chance of happening.
Constitutional expert George Williams says the Australian approach is a poor contrast to what is happening elsewhere.
He also points out parliament continued to sit during World War I and II.
"The current crisis demands that we place high levels of trust in our leaders. However, it would be naive and foolish to rely on trust alone," Prof Williams wrote in The Australian.
"Taking parliament out of the equation is unwise when governments are exercising authoritarian powers. There is a lot that can go wrong."


Mr P. Hamilton (Headteacher) is our Child Protection Lead Officer, and Mrs J. Lewis (Deputy Headteacher and Y5/6 Teacher) is our Deputy Child Protection Officer.
Keeping children safe in Ysgol Maes-y-Llan is the responsibility of all staff.
We all have a personal responsibility to notice when a child or young person may be at risk of harm and pass our concerns to someone who can act to protect them.
At Ysgol Maes-y-Llan we take the welfare of children in our care very seriously.
We get asked by many parents about how they can best support their child with their Math's homework, as procedures for performing different calculations have changed since they were at school. Below is a copy of the calculation policies we follow at Maes-y-Llan showing, step-by-step, how skills are introduced and then developed.
The policy aims to identify the progression in each of the four operations that children will typically follow. Each stage builds upon previous experience. It recognises that children will develop their mathematical skills at different rates and have their own individual learning styles.
They will develop calculation skills through a combination of practical, oral and mental activities. Although the focus of this policy is on pencil and paper procedures, it is important to recognise that in every written method there is an element of mental processing. Written calculation strategies will therefore be taught alongside mental calculation strategies and should be seen as complementary to and not as separate from them. Informal written recording will take place regularly and is an important part of learning and understanding. More formal written methods follow only when the child is able to use a wide range of mental calculation strategies.
We, with help with some of your children, will be going through the calculations policies on the 25th of February 2016 at 6pm. All parents / carers will be welcome. We will circulate more details closer to the time.
Trouble remembering how many days in each month?


First and foremost the Netzarim are NOT Christian, It's NOT a play on words and NOT Noahides or Karites, But the Restored original (1st century pre-Christian) B'nai Netzarim Ha'Derech of historical Ribi Eashoa before the Roman Hellenist Apostasy of 135CE.
(1) Isaiah 11:1 Isaiah 11:1 And there shall come forth a rod out of the stem of Jesse, and a Netzer (Branch) shall grow out of his roots.
Our Savior, Eashoa haMoshiach, is the Netzer in our understanding.
(2) Matthew (Mattyah) 2:23. "He will be called a Netzari (Nazoraios).
We here learn that the Savior was a Netzari.
(4) Acts (Ma'asei) 11:26. It was at Antioch that the disciples were first called "Christians". Netzari ≠ Christian. Antioch was a pagan city. It was populated by Greeks. "Christian" means "Smeared One." This title was probably not complimentary, but meant as an insult, and is only mentioned as an aside ∗. The Savior and his disciples were Israelites and spoke the language of Israel. But "Christian" is a Greek word; one that the disciples would never use, though pagan Greeks would. "Christian" is found only three times in the Greek Scriptures. It is not found at all in the Scriptures of the Israelite language.
Nazorean, Nazarene, or following the sect of the Nazarenes mentioned in Mattyah (Matthew) 2:23 and. Netzer means "branch" Yeshayahu (Isaiah 11:1). It also means guardian or keeper, as in "keeper of the Torah," the commandments of Father Yahweh.
Netzarim Believers do not emphasize theology or doctrine as the standard of "Faith", but righteous character and the action of true Faith. Torah observance by Faith is action demonstrating the anointing of the Ruach HaKodesh (Set-Apart Spirit). If one says they have Faith, but has NO love for Torah, their Faith is vanity.
Ya'aqob 1:22-25 And become doers of the word and not hearers only, deceiving yourselves.
Ya'aqob 2:18 Yes, a man will say, "You have faith, and I have worked. "Show me your faith without works, and I by my works will show you my faith.
∗ aside (not comparable). To or on one side so as to be out of the way. [quotations▽]. a remark that is not directly related to the main topic of discussion.


text:
Q: Bash: passing different lists of arguments to a function I use this function to feed some filenames to another command.
function fun(){
  find "${@}" -print0 | xargs -r0 other_command
}

When invoked, all arguments a passed to find in order to filter the filenames ( -type, -name, etc.)
Is there some way to pass some of those arguments to other_command ?
Variable amount of arguments if possible.
Something like this
fun [list of aguments for find] [list of aguments for other_command]   # pseudo-phantasy syntax

Is it possible?

A: Pass a couple of arrays by "nameref" to the function.
fun() {
  local -n first_args="$1"
  local -n second_args="$2"
  local -i idx
  for idx in "${!first_args[@]}"; do
    printf 'first arg %d: %s\n' "$idx" "${first_args[idx]}"
  done
  for idx in "${!second_args[@]}"; do
    printf 'second arg %d: %s\n' "$idx" "${second_args[idx]}"
  done
  echo 'All first args:' "${first_args[@]}"
  echo 'All second args:' "${second_args[@]}"
}

one_arg_pack=(--{a..c}{0..2})
another_arg_pack=('blah blah' /some/path --whatever 'a b c')

fun one_arg_pack another_arg_pack


text:
Q: Why are convolutions so widely used in functional analysis Functional Analysis has never been particularly intuitive to me, but in recent years I'm starting to see how it aids Fourier Analysis, Operator Theory and PDEs, but for the life of me I don't get the fixation on the convolution operation. I know it has some applications in engineering but it dates back to the time of Laplace. How is it used in mathematics?
One example I can think of is if you have a family of good kernels/mollifiers $D_n$, $f\in L^p$ then $D_n * f$ are smooth and converge to $f$.

A: First of all, there's no "fixation" whatsoever. No more than there is a "fixation" with addition. Or with numbers. Or with vector spaces. Actually way way less, as convolution does not feature prominently.
In any case, convolution appears very naturally in lots of contexts. Off the top of my head, and with the caveat that I don't know/use/see convolution much:

*

*in Probability, the distribution of the sum of two independent random variables is the convolution of the distributions;

*when you multiply two polynomials, the coefficients of the products are obtained as the convolution of the coefficients;

*in Group Theory, $\ell^1(G)$ is made naturally an algebra with convolution as the product. "Natural" because the canonical basis satisfies $e_{gh}=e_g*e_h$;

*as mentioned by copper.hat, the Fourier transform takes convolution to pointwise product;

*as mentioned in the question, convolution allows one to show that $C^\infty$ functions  with compact support are dense in $L^p(\mathbb R^n)$.


text:
1098 км — топоним в России.

По происхождению — название по расстоянию от узловых станций, от которых начинается километраж на линиях, где расположены платформы, разъезды, обгонные пункты и пр.

 1098 км — остановочный пункт Восточно-Сибирской железной дороги в Республике Бурятия. Код 903346.
 1098 км, Блок-пост 1098 км — путевой пост Куйбышевской железной дороги в Самарской области.

Примечания

redpajama_set_name: RedPajamaWikipedia


text:
Home World S. in Syria. Ahead of Shoaib's Arrival: A Deadly Attack in Damascus
S. in Syria. Ahead of Shoaib's Arrival: A Deadly Attack in Damascus
Will Smith Feb 17, 2022 0 Comments
In Damascus, a Syrian soldier was killed and 11 others were injured in an explosion near Customs Square. "At around 7:25 this morning, an IED exploded on a military bus near Customs Square in Damascus. As a result, one soldier was killed and 11 others were wounded, "said Sanaa, the official Syrian agency's military source.
It is noteworthy that last October, two IEDs exploded on a bus at the Jisr al-Royce point in Damascus, killing 14 people.
The latest attack comes at a time when Russian Defense Minister Sergei Shoigu is expected to arrive in the Syrian capital. The Russian military said in a statement on Tuesday that it had met with Syrian President Bashar al-Assad to inspect Russian air bases in the country.
According to pictures released by the Russian military, S. at the beginning of the meeting in Damascus. Soikovum b. Assad both shook hands lovingly and everyone smiled. The Russian defense minister said in a statement that he had "informed the Syrian president of the maneuvers carried out by the Russian navy in the eastern part of the Mediterranean." They also noted the military and technical cooperation between the two allies in the fight against international terrorism and "Russia's humanitarian assistance to the Syrian people affected by US and other Western sanctions." Proof. S. Choigou, a strong man in Putin's organization, also inspected the Russian air base at Hmeimim in western Syria.
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Curfew lifted in two Ebola affected districts
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Tunisians elect a marginalized parliament in the shadow of President Syed
About the Author: Will Smith
"Social mediaholic. Tv fanatic. Gamer. Professional explorer. Amateur music junkie."


text:
Wexford 3rd July 2015
165D Stena Line Horizon S'chase
The Stewards requested a report from C.Byrnes, trainer as to the apparent improvement in form of Arnaud, placed first, as compared with its recent form. The trainer reported that his charge disappointed the last day in a competitive race at Aintree, when on his head throughout. He added that the drop into a less competitive race with fewer runners, helped his charge return to form. He concluded by saying that his horse's seem in better form at the moment. Having considered the evidence the Stewards noted the explanation offered.
165E Stena Line Europe Beginners S'chase
M.P. Walsh, rider of Tiger Bay Lady, trained by Peter Fahey, reported to the Stewards' Secretaries that his mount refused to race after a fair start had been effected, may have resented the first time blinkers.
The Turf Club Veterinary Officer examined Is Love Alive (GB), trained by Paul Nolan, at the request of the Stewards and reported the animal to be slightly lame post race.
165F Carrickbyrne H'cap S'chase
The Stewards fined D.M. Christie, trainer, €350 for lodging a passport in respect of Bullock Harbour, in which the vaccinations did not conform with the provisions of Rule 91. The Stewards allowed the horse to run on receipt of the trainer's written assurance that the vaccinations were in order.
The Stewards received a report from Turf Club Starter,Mr.P.F.Graffin, that L.P. Dempsey, rider of Illtakeitfromhere contributed to the false start. Evidence was heard from the starter and rider concerned. Having viewed the recording of the incident and considered the evidence, the Stewards were of the opinion that the rider was in breach of Rule 208, in that he anticipated the start and disobeyed the starters instructions and having considered his record, suspended him for three race days.
A.E. Lynch, rider of Carpet Elegance trained by Thomas J.Farrell, reported to the Clerk of Scales that his mount was badly hampered in running.
165G Campile INH Flat Race
The Stewards enquired into an incident in the straight involving Don'tdropmein, ridden by Mr R.J. Kiely, placed second and Hows Patsy, ridden by Mr D. Skehan, placed first. Evidence was heard from the riders concerned. Having viewed the recording of the race and considered the evidence, the Stewards made no alteration to the judges placings and took no further action.
Andrew Slattery, trainer of Don'tdropmein, reported to the Stewards' that his charge lost his right fore shoe in running.


text:
Q: Code first Type per hierarchy I use EF 5 with the code first approach. Now I try to define a "code" table in which I want to have several different codes (like address code, medium code, etc.). In this table I just have the following properties: ID (Guid), Name (String), Description (String) and a discriminator (in this case something like the type of the code: address code, medium code, etc.).
So I defined the following base class:
public abstract class Code : EntityBase
{
    public string Name { get; set; }
    public string Beschreibung { get; set; }
}

Then I derived two classes from code
public class AddressCode : Code {}
public class MediumCode : Code {}

The class EntityBase is abstract and just defines the Id property, we use it for every POCO class...
The goal is that I can use AddressCode as a property on my address POCO class:
public class Adresse : EntityBase
{
    #region Properties
    public string Name1 { get; set; }

    public virtual AddressCode AddressCode { get; set; }
    #endregion
}

The question now is, how can I explain EF how to do that? Anyone can help?
Thanks
Marco

Thanks for your answer!
I tried to do it like you said. Unfortunately I get an error because of my EntityBase class:
public abstract class EntityBase
{
    #region Properties

    public virtual Guid Id { get; set; }

    public virtual bool IsValid
    {
        get
        {
            {
                return Validate();
            }
        }
    }

    [NotMappedAttribute]
    public virtual IList<ValidationFailure> ValidationFailures { get; set; }

    #endregion

    #region Methods

    private bool Validate()
    {
        var validatorFactory = new AttributedValidatorFactory();
        IValidator validator = validatorFactory.GetValidator(GetType());

        if (validator == null)
        {
            return true;
        }

        ValidationResult validationResult = validator.Validate(this);

        ValidationFailures = validationResult.Errors;

        return validationResult.IsValid;
    }

    #endregion
}

The error message is:
You cannot use Ignore method on the property 'ValidationFailures' on type 'Entities.AdresseCode' because this type inherits from the type 'Entities.EntityBase' where this property is mapped. To exclude this property from your model, use NotMappedAttribute or Ignore method on the base type.
As you can see I already defined the property ValidationFailures as NotMapped but still I get this error.. Do you have an idea?
Thanks
Marco

A: Just create a context (derived from DbContext)
public class AddressesDb : DbContext
{
    public DbSet<Code> Codes { get; set; }
    public DbSet<Adresse> Adressen { get; set; }
}

And (when used in code) EF will create a database with default table and column names. It will create a discriminator column of type text (nvarchar) which will contain the names of the classes that derive from Code.
If you want different names and/or types you should either use data annotations or fluent API to configure these.

A: Finally I got it work!
In the DBContext be aware to define DbSets for the code derived classes before all the other POCO's and then it works!


restlessrisa: Seed Paper in ACTION!
"And he shall plant in the hearts of the children the promises made to the fathers, and the hearts of the children shall turn to their fathers."
So, I thought... how perfect to have the favor be a heart that you can plant!!!
These are a few shots of the table before we put the dessert on.
I know... I have an obsession with gingham, especially since it is $1.50 a yard at Wal-mart!
See the seed banner? The seed packets were $.20 each, so I thought - why not?
Here is the framed scripture.
The next few shots are of ways sisters in the ward display their family history.
We started the night with a great speaker. Christy, you were AWESOME!
Then Becca's brother taught us how to index. He was great!
After the indexing, we moved into the primary room for refreshments, and to see the display.
Yes, I am including a picture of the forks. Well, not the forks only, but the hankies under the forks. DO YOU LOVE THEM???
I got them at DI one day, and felt SO LUCKY! I am glad I got to use them finally!
It was a nice night. Attendance was poor, but it was fun to chat with those that were there, and to learn something new.
You did such a wonderful job. It was lovely. Thanks for everything!!!
What a great idea for a "weeknight RS activity"! I love the hankies and seed packet banner.


text:
def _read_readme():

    with open('docs/readme.rst', 'r') as readme:
        return readme.read()

def _substitute(readme):

    readme = (readme.replace('_static', 'docs/_static')
              .replace('.. testcode::', '.. code-block:: python')
              .replace('.. testoutput::\n   :hide:', ''))

    return readme

def _write(readme):

    with open('readme.rst', 'w') as out:
        out.write(readme)

if __name__ == '__main__':

    readme = _read_readme()
    readme = _substitute(readme)
    _write(readme)

redpajama_set_name: RedPajamaGithub


text:
Lana Del Rey – "Ride"
News September 25, 2012 11:07 AM By Tom Breihan
In the first months of the year, the biggest story in music might've been Born To Die, Lana Del Rey's spectacular mess of a debut album, which mangled the mysterious drugged-out grace of her excellent early singles and completed her transformation into self-styled stereotype punchline. But the album does have its fans, and they'll be happy to learn that LDR will release a massive "Paradise Edition" of Born To Die this fall.
The new edition will be available in a bunch of different versions, including a massive box set, with tons of bonus tracks, many of which have shown up online in recent months. And one key brand-new track is "Ride," an absolutely gorgeous string-drenched ballad that Rick Rubin produced, one that moves LDR back to the power of "Video Games" and "Blue Jeans" while simultaneously pushing her into a grand Adele crossover-soul-pop zone. It's really nice. There's hope for this lady yet! Hear it and watch a teaser for the Paradise Edition, featuring LDR making LDR faces, below. Also, the eight new songs from the Paradise Edition will be available on their own, and we've got the tracklist for that below.
01 "Ride"
02 "American"
03 "Cola"
04 "Body Electric"
05 "Blue Velvet"
06 "Gods And Monsters"
07 "Yayo"
08 "Bel Air"
The Paradise Edition of Born To Die is out 11/13 on Interscope. For more info on the various different available versions of it, check Spin.


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text:
Home » Amidst a dismal San Jose Sharks season, a new leader emerges
Amidst a dismal San Jose Sharks season, a new leader emerges
International, News
SAN JOSE – Just 15 seconds after he scored his ninth goal of the year, matching his total from last season, Nico Sturm squared off with Brett Ritchie to the left of the San Jose Sharks' net.
It didn't matter to Sturm that he had just one fighting major in the NHL prior to Tuesday's game with the Calgary Flames, or that he was giving up about 10 pounds to Ritchie, a much more experienced pugilist.
Sturm had just seen Ritchie flatten Marc-Edouard Vlasic against the boards and perhaps without much regard for his own well-being, immediately challenged him, throwing his gloves down on the ice.
Did Sturm's goal and fight help spark a Sharks turnaround after they allowed two goals in the first 30 seconds of the opening period? Maybe.
The Sharks came back to tie the game early in the third period on a Timo Meier goal, only to see Nazem Kadri score with 9:56 left in regulation time, helping lift the Flames to a 7-3 win at SAP Center.
Milan Lucic, Tyler Toffoli, and Kadri all added insurance goals in the final 3:17 of regulation time.
But there's no question that Sturm's actions provided some leadership after a disastrous start.
Sturm's goal and fight were among the few highlights for the Sharks on Tuesday, as wingers Alexander Barabanov and Time Meier also scored and goalie James Reimer finished with 32 saves as the Sharks lost their third straight game and fell to 3-10-5 on home ice.
The Sharks host the Minnesota Wild, Sturm's first NHL team, on Thursday in their last game before the Christmas break.
Toffoli scored just 15 seconds into the first period after some shoddy defensive zone coverage, and Dillon Dube scored at the 30-second mark after a giveaway by Erik Karlsson.
Sturm's goal came at te 14:58 mark, as he hustled to the front of the Flames' net before he took a pass from Meier and chipped the puck past goalie Jacob Markstrom.
Sturm's scrap was his second in the NHL, as he previously fought Zach Sanford in March 2021. His only other fight on record came in 2016 when he was with the Tri-City Storm of the USHL.
The Sharks' 10-18-6 record represents their worst start to a season after 34 games since 1995-96, when they were 7-23-4.
The Sharks announced Tuesday morning that forward Luke Kunin had undergone surgery earlier in the day to repair a torn anterior cruciate ligament in his right knee.
Kunin, 25, suffered the injury in last week's game against the Arizona Coyotes and will miss the rest of the season, although it appears he'll be ready for the start of next season's training camp in September. Kunin, in the first year of a two-year, $5.5 million contract, had 13 points in 31 games, playing mostly a middle-six forward role.
Noah Gregor, for now, has been been given that role as he started Tuesday's game on the Sharks' third line with Nico Sturm and Evgeny Svechnikov.
"It's an opportunity for whoever raises their hand and grabs that (ice time)," Quinn said. "Obviously (Gregor's) done a good job since he's been in and I've liked some of his game, so it's a great opportunity for him."
Monday night, the NHL handed Sharks No. 1 center Tomas Hertl a two-game suspension for his high stick on Flames center Elias Lindholm. Hertl, the Sharks' second leading scorer with 32 points in 33 games, will be eligible to play again Dec. 27 when the Sharks are in Vancouver to play the Canucks.
Tags:General News


Need Sweetest Day gift ideas? We have a wide variety of "sweet" flowers and plants to choose from. From roses to plants , we have various bouquets to make someone smile. Earthly Enchantments delivers locally to Munster, IN or nationwide.


This is a gorgeous season. Nature is a blaze of color and everything seems to come into balance. Night and day are again equal. There is a bountiful harvest to be thankful for, yet we must plan for the sparse times ahead. This is a time of generosity and conservation. So, how do you share these values with your children? You can plan Mabon activities for the whole family to enjoy.
Mabon (also Harvest Home, Alban Elfed or Winter Finding) is celebrated at the Autumnal Equinox. This is the second harvest festival of the year, that of fruits and vegetables. Mabon is the Welsh God of all things wild and free. He is also associated with the Sun God whose power dies on this day.
*Have a potluck feast with a group of friends and loved ones to celebrate the abundance of the season. Feel the warmth of being part of a community.
*Adopt someone in a nursing home. As a family, take your special person baked goodies and colored pictures. Read them books or tell them stories.
*Walk around your neighborhood picking up garbage. Do what you can to improve your home and prepare for winter.
*Pick a subject that interests the whole family. Go to the library or find other resources and study that subject. Together, share what you've learned.
*Look at old family photo albums or scrapbooks. Try to tell stories about each person in the pictures.
*Leave an apple on the grave of an ancestor. Cut an apple in half to show your children the star inside. This is a reminder that all life is renewed in some way.
*Bake cored apples filled with butter and cinnamon as a special treat.
*Create decorations for your front door out of colored leaves, pinecones, nuts, acorns and Indian Corn bundles.
*Take a walk in a wild place. Gather seedpods and dried plants. Sing songs and talk about all the things you've done over the summer. Make plans for the winter.
*Honor the birds and small animals in the wilderness or by your home by making a birdfeeder or mandala filled with seeds and grain.
*Make rattles out of empty gourds and sunflower seeds or seeds collected from nature walks. Use the rattles to make music or scare away bad dreams.
*Look at your family habits and figure out what you can do to improve your conservation habits. Can you use less water or recycle more of your garbage?
*Make a Vine God (stick-type male figure with a hollow body) filled with foil-wrapped cornbread and sacrifice him on the campfire (or barbecue!). Give thanks to the god for his sacrifice and enjoy the cornbread!
Heather Osterman is the Family Services Coordinator for the Aquarian Tabernacle Church. For more information on pagan oriented activities and events for children and families, please contact her at ATCchild@A... or ATC at (360) 793-1945 between 9 a. m. and 9 p. m.


Educate - Motivate - Perform - Fly Educate - I will teach you fundamental principles regarding strength, hypertrophy and muscular endurance training and a variety of cardio routines. Motivate - I will help you set goals, I will hold you accountable and I will push you to your highest potential. Perform - Through consistent action with progression and variation you will get results. Effort is at the foundation of my philosophy. Fly - Through developing positive and consistent habits you will become fitness independent and you will no longer need my services. You will become your motivation and the motivation for others.
Jeff works with all people and listens to their goals and ideas to determine the appropriate approach and plan. Everyone is different and so is every body. He keeps an open mind and looks at every situation as being unique due to the unique qualities of each individual.


I got the shingles vaccine, Zostavax, today. It was expensive but anyone who has lived with someone with shingles or has known anyone with shingles would probably think it was worth it to prevent getting it. Of course, I could still get shingles, but the vaccine would make it a milder version.
With RLS we don't need any problems that could be prevented. I only found out about this vaccine recently and called to request it right away.
I agree with this completely. I got the vaccine last year.
It's on my list when I'm old enough .
I got the Shingles vaccine at age 60, my wife at age 56. Six years after getting the vaccine I got shingles, but thankfully it was a mild case. I understand that there is a newer Zostavax that has been developed and is even more effective.
Here in UK we get the vaccine free from age 70. A quick google shows that a private vaccine costs around £170.
TimG, the vaccine does not guarantee protection but luckily you got a mild case. The vaccine helps make sure it is milder than it would be. From what I understand, it also doesn't last forever and immunity is less as a person gets older.
Lucky UK people, the vax costs NZ$190 in New Zealand but that's better than getting a full blown case of shingles.
My understanding is that here in UK it is offered free to over 70s, but is not available once 80 years is reached.
Estimates that I've seen state that about 30% fewer would get shingles and the remainder would have milder symptoms.


text:
M1 Postcards
Postcards from the United Kingdom
Roman Pharos, Dover
25 October 2021 by admin
Dover Castle and Pharos
Publisher: Valentine.
Google Street View.
Dover Castle.
…within the walls of the medieval castle stands a much older building, dating from a time when Britain was an outpost of the Roman Empire. Around 2,000 years ago, in the early 2nd century AD, the Romans built a pharos, or lighthouse, here. This would have guided the ships of a Roman fleet into the harbour below. Not only is the Dover pharos the most complete standing Roman building in England, it's also one of only three lighthouses to survive from the whole of the former Roman empire.
Seventy years after the Roman invasion in AD 43, construction of a fort began at the mouth of the river Dour. This was Dubris, a fort for the classis Britannica, a Roman fleet that patrolled the eastern Channel. Though building stopped suddenly, it began again around AD 130 and the fort was completed. The Romans built an octagonal tower-like lighthouse on Castle Hill around the same time [as the fort], with another on the opposite hill, the Western Heights. These lighthouses supported fire beacons to act as navigation lights for ships approaching the narrow river mouth, enabling them to find a quayside outside the fort. The fort at Dubris was demolished around AD 215 and a new one constructed around AD 270, which may have continued in use, along with the lighthouses, into the 5th century. The pharos was later reused for the church of St Mary in Castro as a chapel and bell tower, and can still be seen.
Roman pharos on the western heights of Dover (GB), inside view, 1893 (from Wikimedia Commons).
The Roman pharos or lighthouse at Dover was probably built in the first century A.D. A similar lighthouse was built on the Western Heights and at night guided Roman ships into the port of Dubris. The tower was octagonal outside and rectangular inside rising to a height of perhaps 80 feet (24m). It had eight storeys each set back 1 foot (0.3m) from the one below, which gave the whole structure the appearance of an extended telescope. Only the first four Roman storeys remain, the present topmost storey being a fifteenth century reconstruction. The present splayed shape of the pharos is a result of the severe weathering it suffers in exposed position and mediaeval refacing.
AD 46. Built under the Emperor Claudius. This guided the Roman fleet round to the port of Richborough. In mediaeval times it was used as a belfry to the Church of St Mary Sub-Castro. 4 storeys, 3 being Roman and the top storey and remains of battlements mediaeval. An octagonal tower with originally vertical stepped walls rising in tiers set back each within the last, now almost smoothed. Rubble with a facing of green sandstone and tufa and levelled at an interval of 7 courses with a double course of brick set in hard pink mortar. Round-headed windows with a small recessed spy-hole inside them.
Categories Kent, Lighthouses, Roman, Ruins & Abandoned buildings Leave a comment
Lighthouse, Ailsa Craig
31 August 2021 25 August 2020 by admin
Lighthouse, Ailsa Craig, Girvan
Posted 1921-22
Publisher: The Philco Publishing Company, London
Google Street View (approximate)
Ailsa Craig Lighthouse is located on Ailsa Craig, an island in the Firth of Clyde, just offshore from Girvan, South Ayrshire, Scotland. In 1881, petitions were received by the Commissioners of Northern Lighthouses from Lloyds and the Scottish Shipmasters Association requesting the erection of two fog signals and a lighthouse on Ailsa Craig. The Board of Trade and Trinity House both agreed to the proposal and work commenced the following year. The light was first exhibited on the night of 15 June 1886, an oil burning light which remained in use until 24 January 1911, when it was converted to incandescent.
Until wireless telephone communications were established on Ailsa Craig in 1935, the lightkeepers and employees of Ailsa Craig Granites Ltd used to depend on pigeons for the conveyance of messages. A pigeon house was established at Girvan Green, where the town council established a parking place for cars and buses in 1935. The pigeons were provided by the Lighthouse Boatman at that time, who received an annual payment of £4.00. When a doctor or supplies were required urgently in stormy weather when it was impossible to have messages taken by carrier pigeon, a system of signals by fire was used. One fire on the castle path showing the Lighthouse to the North indicated "bring doctor for Lighthouse"; two fires on the castle path (one at the same place as the Lighthouse fire, and the other 20-30 yards above it), meant "bring doctor for Quarry Company"; one fire at the north end of the Castle Flat showing the Lighthouse to the South indicated the provisions were required.
Northern Lighthouse Board
Categories Lighthouses, South Ayrshire, Victorian Leave a comment
Smeaton Tower, Plymouth
5 August 2018 by admin
6609 Smeaton Tower – Plymouth – Judges Ltd.
1922 Produced by Judges Ltd
Categories Devon, Lighthouses Tags no words Leave a comment
Master Pages
English Castles
This is just British postcards. The rest of the world (including Crown Dependencies & Overseas Territories) is at: 432 Postcards
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Monte Estoril, Portugal
S.S. Paris
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Boating, sailing & rowing
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text:
\section{Introduction}

\noindent
The Yang--Baxter equation is a cornerstone in the theory of integrable quantum systems. It facilitates the construction of commuting transfer matrices which generate sets of conserved operators. A limiting procedure may be performed to obtain the {\it classical} Yang--Baxter equation \cite{bd82}. Despite the name, this equation also has a long history in the study of integrable {\it quantum} systems \cite{eks93,j89,j89a}. In some cases these models have been formulated through the notion of a {\it Gaudin algebra} \cite{bt07,g76,osdr05}, which turns out to be an essentially equivalent approach up to generalisation of the classical Yang-Baxter equation \cite{bdp05}. Below, the use of a non-standard form of the classical Yang--Baxter equation will be employed, which affords solutions that do not have the property of being {\it skew-symmetric}. A particular case will be identified within a class of solutions of this equation, and analysed in detail.

The motivation for this study arises from a recent construction of an integrable generalisation, and Bethe Ansatz solution, of the $p+ip$-pairing Hamiltonian \cite{lil16}.
With the benefit of some hindsight, it will be shown below that this result can be re-derived in a more direct manner. Not only does this provide a clearer understanding of the integrability of the model derived in \cite{lil16}, it also opens up new avenues of applications. As an example, an integrable, multi-species, boson tunneling model will be derived.
Following the experimental realisation of a two-species Bose-Einstein condensate in 2008 \cite{tbdcmi08}, there has been considerable effort to study such systems. Tunneling models for two-species systems were quickly formulated \cite{cno11,qtf10,sbnhec09,sp09}, and the general area of research continues to attract significant activity
\cite{cnpv15, cvct15, lcd14,wpc16,ws15}. Having an integrable model for multi-species tunneling offers an opportunity for a different perspective on tunneling phenomena.

The integrable model which will be derived below is described in terms of $2L$ mutually commuting sets of canonical boson operators, representing $2L$ degrees of freedom for the system.  These operators are labeled by two indices, $j=1,...,L$ denoting the boson species, and $X=A,B$ denoting the two potential wells between which tunneling occurs. The operators satisfy the relations
\begin{align*}
[b^{\phantom{\dagger}}_{j,X}, \,b^{\phantom{\dagger}}_{k,Y}]&=0, \\
[b^{{\dagger}}_{j,X}, \,b^\dagger_{k,Y}]&=0, \\
[b^{\phantom{\dagger}}_{j,X}, \,b^\dagger_{k,Y}]&=\delta_{jk}\delta_{XY}I.
\end{align*}
Number operators for each species in each well are given by
\begin{align*}
N_{j,X}=b^\dagger_{j,X}b^{\phantom{\dagger}}_{j,X}.
\end{align*}
Set
\begin{align*}
N_{X}&=\sum_{j=1}^L N_{j,X}, \\
N_j&=N_{j,A}+N_{j,B}.
\end{align*}
The Hamiltonian is given by
\begin{align}
H=U(N_A-N_B)^2+\mu(N_A-N_B)+\sum_{j=1}^L {\mathcal E}_j \left(b^\dagger_{j,A}b^{\phantom{\dagger}}_{j,B}
+b^\dagger_{j,B}b^{\phantom{\dagger}}_{j,A}\right).
\label{ham}
\end{align}
The coupling parameter $U$ characterising the inter-boson interactions is positive for repulsive interactions and negative for attractive interactions. The well bias, such as that due to the presence of an applied external field, is parametrised by $\mu$. The parameters ${\mathcal E}_j$ are the amplitudes associated with each species for tunneling between the wells. The signs of ${\mathcal E}_j$ are not important, as they can be changed by a unitary transformation $b_{j,X} \mapsto -b_{j,X}$.
There are no constraints imposed on any of the coupling parameters, they are all free, real variables.

When $L=1$ (\ref{ham}) reduces to the two-site Bose--Hubbard model, which endures as the subject of widespread investigation, e.g. \cite{b16,cdp16,gks14,safr15,css14,wzhql16}. Thus (\ref{ham}) represents an $L$-species generalisation. This Hamiltonian commutes with each of the operators $N_j$, $j=1,..,L$, indicating that the number of particles for each species is conserved. The main objective below is to construct an additional $L$ conserved operators, establishing that the model is integrable.

In Sect. 2 a general Lie-algebraic approach will be taken to construct a family of commuting transfer matrices. It will be seen that the classical Yang-Baxter equation emerges in a natural way, in a form that does not assume skew-symmetry. This form has been promoted by Skrypnyk in a series of works \cite{s06,s07,s07a,s09,s10,s14,s15}. Relationship to the usual form of the Yang--Baxter equation will be discussed. A study of some provisional elements of the associated representation theory will lead to the identification of a distinguished solution of the classical Yang--Baxter equation with an exotic symmetry, in a sense which will be defined.
In Sect 3. attention turns to the construction of the model, and its solution. Due to the lack of an apparent reference state, application of the algebraic Bethe Ansatz to solve the model in general is prohibitive. Instead the Bethe Ansatz solution will first be derived in the limiting, but non-trivial, case when $N_j=1$ for all $j=1,...,L$.  Based on this result the general solution will be obtained through a tensor product construction.
Concluding remarks are offered in Sect. 4.

\section{Commuting transfer matrices from the classical Yang-Baxter equation} \label{}
\noindent
Let $\{e^j_k:j,k=1,...,n\}$ denote the standard basis elements for ${\rm End}({\mathbb C}^n)$. Introduce an abstract, infinite-dimensional complex Lie algebra ${\mathcal G}$ with generating functions
$$\{T^j_k(u)=\sum_{l=-\infty}^{\infty}t^j_k[l]u^l:u\in{\mathbb C}, j,k=1,...,n\}$$ and set
\begin{align*}
\mathcal T(u)&=\sum_{j,k=1}^n T^j_k(u)e_j^k.
\end{align*}
For $u,\,v\in{\mathbb C}$, let $r(u,v)\in\,{\rm End}({\mathbb C}^n\otimes {\mathbb C}^n)$,  which is expressible as
\begin{align*}
r(u,v)&=\sum_{j,k,l,p=1}^n r^{jl}_{kp}(u,v)e^k_j\otimes e^p_l.
\end{align*}
The symmetry properties
\begin{align}
[r_{12}(u,v),\,r_{21}(v,u)]&=0, \label{comm1}\\
[r_{12}(u,v)^{{\rm t}_2},\,r_{21}(v,u)^{{\rm t}_2}]&=0,  \label{comm2}
\end{align}
where ${\rm t}_2$ denotes the partial transpose taken in the second component of the tensor product, are assumed.
These assumptions provide for a different approach than those followed in \cite{s06,s07,s07a,s09,s10,s14,s15}.
The commutation relations imposed on ${\mathcal G}$ are compactly given by
\begin{align}
\left[ \mathcal T_1(u),\,\mathcal T_2(v)\right]=\left[ \mathcal T_2(v),\,r_{12}(u,v)   \right]-\left[\mathcal T_1(u),\,r_{21}(v,u)  \right]
\label{comm}
\end{align}
where the subscripts denote the actions on tensor components of ${\mathbb C}^n \otimes {\mathbb C}^n$.
Assuming that the Jacobi identity holds, such that an associative enveloping algebra can be constructed in the usual way,
it follows from the commutation relations (\ref{comm}) that the transfer matrix
\begin{align}
t(u)={\rm tr}\left( (\mathcal T(u))^2 \right)=\sum_{j,k=1}^nT^j_k(u)T^k_j(u)
\label{tm}
\end{align}
satisfies
$$\left[t(u),\,t(v)\right]=0 \qquad \forall\,u,\,v\in\,{\mathbb C}.$$
This result does not depend on any properties of $r(u,v)$ other than (\ref{comm1},\ref{comm2}), as well as (\ref{comm}) being compatible with the Jacobi identity.
It can be shown that a {\it sufficient} condition for the Jacobi identity to hold is the imposition that the non-standard classical Yang-Baxter equation (CYBE)
\begin{align}
\left[r_{12}(u,v),\,r_{23}(v,w)\right]-\left[r_{21}(v,u),\,r_{13}(u,w)\right]+\left[r_{13}(u,w),\,r_{23}(v,w)\right] =0
\label{cybe}
\end{align}
holds. Recall that for the complex vector space ${\mathbb C}^m$,  a Lie algebra representation $\pi:{\mathcal G}\rightarrow {\rm End}({\mathbb C}^m)$ is required to preserve the commutator in the sense
\begin{align}
\pi\left(\left[T^j_k(u),\,T^p_l(v)\right]\right)&=\left[\pi(T^j_k(u)),\,\pi(T^p_l(v))\right].
\label{rep}
\end{align}
The imposition of the CYBE also provides the notion of an {\it adjoint} representation $\pi_{\rm ad}:{\mathcal G}\rightarrow {\rm End}({\mathbb C}^n)$. The identification
\begin{align}
\pi_{\rm ad}(T^{j}_{k}(u))=\sum_{l,p=1}^n r^{jl}_{kp}(u,v)\otimes e^p_l
\label{adjoint}
\end{align}
satisfies (\ref{rep}) as a consequence of (\ref{comm}) and (\ref{cybe}).

The above construction is non-standard in the sense that there is no assumption that the solution of (\ref{cybe}) possesses the {\it skew-symmetry} property
\begin{align*}
r(u,v)=-r(v,u).
\end{align*}
Consequently such solutions lie outside the Belavin-Drinfel'd classification of solutions \cite{bd82}. The skew-symmetry property arises naturally in cases where a solution of (\ref{cybe}) is obtained from a solution $R(u,v)\in{\rm End}({\mathbb C}^n\otimes {\mathbb C}^n)$ of the Yang-Baxter equation
\begin{align*}
R_{12}(u,v)R_{13}(u,w)R_{23}(v,w)=R_{23}(v,w)R_{13}(u,w)R_{12}(u,v)
\end{align*}
with the unitarity condition
\begin{align}
R_{12}(u,v)R_{21}(v,u)=I\otimes I,
\label{unitarity}
\end{align}
where $I$ is the identity operator. If $R(u,v)$ depends on some parameter $\eta$, and admits an expansion $R(u,v)\sim I\otimes I+\eta \,{\mathfrak r}(u,v)+ O(\eta^2)$ as $\eta\rightarrow 0$,
 then ${\mathfrak r}(u,v)$ satisfies
\begin{align}
\left[{\mathfrak r}_{12}(u,v),\,{\mathfrak r}_{23}(v,w)\right]+\left[{\mathfrak r}_{12}(u,v),\,{\mathfrak r}_{13}(u,w)\right]
+\left[{\mathfrak r}_{13}(u,w),\,{\mathfrak r}_{23}(v,w)\right] =0
\label{sscybe}
\end{align}
while it follows from (\ref{unitarity}) that
\begin{align}
{\mathfrak r}_{12}(u,v)+{\mathfrak r}_{21}(v,u)=0.
\label{cunitarity}
\end{align}
It is then seen that (\ref{sscybe}) and (\ref{cunitarity}) combined are equivalent to (\ref{cybe}).

As an example, let $P: {\mathbb C}^n \otimes {\mathbb C}^n \rightarrow {\mathbb C}^n \otimes {\mathbb C}^n $ denote the permutation operator with the action
\begin{align*}
P(x \otimes y)=(y \otimes x) \qquad \forall \, x,\,y\in\, {\mathbb C}^n.
\end{align*}
A simple solution of (\ref{cybe}) is
\begin{align}
r(u,v)=\frac{1}{u-v}P.
\label{rat}
\end{align}
This solution has the property (\ref{cunitarity}), so it also provides a solution of (\ref{sscybe}).

Hereafter set $n=2$. One family of solutions of (\ref{cybe}) which does not generally possess the skew-symmetry property
(\ref{cunitarity}),  but does satisfy (\ref{comm1},\ref{comm2}), was presented by Skrypnyk \cite{s09}. This solution can be represented in matrix form as
\begin{align}
r(u,v)
&= \left(\begin{array}{ccccc} \displaystyle
\frac{u^2+v^2}{u^2-v^2}+c &0&|&0&0 \\
0&0 &|&\displaystyle\frac{2uv}{u^2-v^2}&0 \\
-&-&~&-&- \\
0&\displaystyle\frac{2uv}{u^2-v^2}&|&0 &0 \\
0&0&|&0&\displaystyle\frac{u^2+v^2}{u^2-v^2}+c  \end{array} \right)
\label{nonss}
\end{align}
where $c\in\,{\mathbb C}$ is arbitrary. The above solution is skew-symmetric only for the particular choice $c=0$. In this instance the algebra $\mathcal G$ coincides with the trigonometric/hyperbolic $sl(2)$ Gaudin algebra \cite{osdr05,vdv14}. The primary interest here is in the case $c=1$ which, as will be explained below, gives rise to an exotic symmetry. For this choice
\begin{align}
r(u,v)&= \left(\begin{array}{ccccc}
\displaystyle\frac{2u^2}{u^2-v^2} &0&|&0&0 \\
0&0 &|&\displaystyle\frac{2uv}{u^2-v^2}&0 \\
-&-&~&-&- \\
0&\displaystyle\frac{2uv}{u^2-v^2}&|&0 &0 \\
0&0&|&0&\displaystyle\frac{2u^2}{u^2-v^2}  \end{array} \right).
\label{exo}
\end{align}
This solution may also be expressed as
\begin{align*}
r(u,v)=\left(\frac{u}{u-v}+\frac{u}{u+v}\sigma_1 \sigma_2\right)P
\end{align*}
where $\sigma={\rm diag}(1,-1)$. This form is convenient for calculations that follow in Subsect. 2.1.

\subsection{Symmetries}
\noindent
Symmetries of solutions of (\ref{cybe}) may be characterised in terms of the one-dimensional representations $\pi_0$ of their associated Lie algebra ${\mathcal G}$, i.e. $\pi_0:{\mathcal G}\rightarrow {\mathbb C}$. From (\ref{rep}) it is seen that for any one-dimensional representation
\begin{align*}
\pi_0\left(\left[T^j_k(u),\,T^p_l(v)\right]\right)&=0.
\end{align*}
Equivalently, a one-dimensional representation may be expressed as a mapping $\rho(\mathcal T(u))=\mathcal J(u)\in {\rm End}({\mathbb C}^2)$
satisfying
\begin{align}
0=\left[\mathcal J_2(v),\,r_{12}(u,v)\right]-\left[\mathcal J_1(u),\,r_{21}(v,u)   \right].
\label{1d}
\end{align}
In \cite{s14} such a solution is termed a {\it generalised shift element}.

For the case of the solution (\ref{rat}), it is seen that all constant elements of ${\rm End}({\mathbb C}^2)$ satisfy (\ref{1d}). It is therefore natural to identify the symmetry of (\ref{rat}) as $gl(2)$. On the other hand for generic values of $c$ in (\ref{nonss}), solutions of (\ref{1d}) are given by constant diagonal elements of ${\rm End}({\mathbb C}^2)$. In this instance the symmetry of (\ref{nonss}) is identified as $u(1)$. For the specific choice $c=1$ in (\ref{nonss}) leading to (\ref{exo}), an exotic symmetry emerges. Setting $\mathcal J(u)=\mathcal D+u\mathcal O$ where $\mathcal D$ is diagonal and $\mathcal O$ is off-diagonal it is found that (\ref{1d}) is satisfied by $\mathcal J(u)$. First
\begin{align*}
\left[\mathcal J_2(v),\,r_{12}(u,v)\right]&-\left[\mathcal J_1(u),\,r_{21}(v,u)  \right] \\
&=u(\mathcal D_2+v\mathcal O_2)\left(\frac{1}{u-v}+\frac{1}{u+v}\sigma_1 \sigma_2\right)P
\\
&\qquad
-u\left(\frac{1}{u-v}+\frac{1}{u+v}\sigma_1 \sigma_2\right)P(\mathcal D_2+v\mathcal O_2) \\
&\qquad-v(\mathcal D_1+u\mathcal O_1)\left(\frac{1}{v-u}+\frac{1}{u+v}\sigma_1 \sigma_2\right)P \\
& \qquad +v\left(\frac{1}{v-u}+\frac{1}{u+v}\sigma_1 \sigma_2\right)P(\mathcal D_1+u\mathcal O_1) \\
&=(\mathcal D_2-\mathcal D_1)P+\sigma_1 \sigma_2(\mathcal D_2-\mathcal D_1)P
\end{align*}
where simplification to the final expression has relied on the fact that $\sigma\mathcal O=-\mathcal O\sigma$.
Now every diagonal operator on ${\mathbb C}^2$ may be expressed as $\mathcal D=aI +b\sigma$ for some $a,b\in\,{\mathbb C}$. It then follows that
\begin{align*}
\left[\mathcal J_2(v),\,r_{12}(u,v)\right]-\left[\mathcal J_1(u),\,r_{21}(v,u) \right]
&=b(\sigma_2-\sigma_1)P+b\sigma_1\sigma_2(\sigma_2-\sigma_1)P \\
&=b(\sigma_2-\sigma_1)P+b(\sigma_1-\sigma_2)P \\
&=0.
\end{align*}
Because this solution for $\mathcal J(u)$ has an explicit dependence on the parameter $u$, it is considered to be exotic compared to the $gl(2)$ symmetry of (\ref{rat}) and the $u(1)$ symmetry of (\ref{nonss}). For later use the notation $\overline{\mathcal{G}}$ will be adopted for the  Lie algebra associated with the case of the solution (\ref{exo}).

\subsection{General representations}
\noindent
General representations of $\overline{{\mathcal G}}$ are first obtained by an evaluation homomorphism $\rho_z$ to the enveloping algebra $U(sl(2))$ of $sl(2)$. Adopting the convention for the $sl(2)$ generators $\{S^z,\,S^+,\,S^-\}$ to satisfy the commutation relations
\begin{align*}
[S^z,\,S^{\pm}]=\pm S^{\pm}, \qquad [S^+,\, S^-]=2S^z,
\end{align*}
$\rho_z$ has the action
\begin{align*}
\rho_z(T^1_1(u))&=\frac{u^2}{u^2-z^2}(I+2S^z),   \\
\rho_z(T^1_2(u))&=\frac{uz}{u^2-z^2}S^+,    \\
\rho_z(T^2_1(u))&=\frac{uz}{u^2-z^2}S^-,  \\
\rho_z(T^2_2(u))&=\frac{2u^2}{u^2-z^2}(I-2S^z),
\end{align*}
preserving the commutation relations (\ref{comm}) for the solution (\ref{exo}). Every representation of $sl(2)$ then affords a
a representation of $\overline{\mathcal{G}}$. For the choice of the spin-1/2 representation of $sl(2)$ this coincides with the adjoint representation (\ref{adjoint}).

A second evaluation homomorphism to the enveloping algebra $U(sl(2))$, denoted $\rho_{(0)}$, is deduced from the one-dimensional representation. Specifically,
\begin{align*}
\rho_{(0)}(T^1_1(u))&=\alpha I,   \\
\rho_{(0)}(T^1_2(u))&=u\beta I,    \\
\rho_{(0)}(T^2_1(u))&=u\gamma I,  \\
\rho_{(0)}(T^2_2(u))&=\delta I,
\end{align*}
where $\alpha,\,\beta,\,\gamma,\,\delta\in\,{\mathbb C}$ are arbitrary, which again preserves the commutation relations (\ref{comm}) for the solution (\ref{exo}). Taking the tensor product of these evaluation homomorphisms provides evaluation homomorphisms acting on tensor product of copies of $U(sl(2))$. The general map
\begin{align*}
\rho^{\otimes L}=\rho_{(0)}\otimes \rho_{z_1}\otimes \rho_{z_2}\otimes ...\otimes \rho_{z_L}
\end{align*}
has the form
\begin{align*}
\rho^{\otimes L}(T^1_1(u))&=\alpha I+\sum_{j=1}^L\frac{u^2}{u^2-z_j^2}(I+2S_j^z),   \\
\rho^{\otimes L}(T^1_2(u))&=u\beta I+\sum_{j=1}^L\frac{2uz_j}{u^2-z_j^2}S_j^+,    \\
\rho^{\otimes L}(T^2_1(u))&=u\gamma I+\sum_{j=1}^L\frac{2uz_j}{u^2-z_j^2}S_j^-,  \\
\rho^{\otimes L}(T^2_2(u))&=\delta I+\sum_{j=1}^L\frac{u^2}{u^2-z_j^2}(I-2S_j^z).
\end{align*}
It then follows that $\tau(u)=\rho^{\otimes L}(t(u))\in\,U(sl(2))^{\otimes L}$ satisfying
\begin{align*}
[\tau(u),\,\tau(v)]=0, \qquad \forall\,u,\,v\in\,{\mathbb C}
\end{align*}
is a transfer matrix expressed in terms of spin operators.

Hereafter set $\delta=-\alpha$, and $\gamma=\beta$ with $\alpha,\,\beta\in\,{\mathbb R}$, giving
\begin{align*}
\tau(u)&=\left(\alpha I+\sum_{j=1}^L\frac{u^2}{u^2-z_j^2}(I+2S_j^z)\right)^2+\left(-\alpha I+\sum_{j=1}^L\frac{u^2}{u^2-z_j^2}(I-2S_j^z)\right)^2   \\
&\qquad+\left(u\beta I+\sum_{j=1}^L\frac{2uz_j}{u^2-z_j^2}S_j^+\right)
\left(u\beta I+\sum_{k=1}^L\frac{2uz_k}{u^2-z_k^2}S_k^-\right) \\
&\qquad +\left(u\beta I+\sum_{j=1}^L\frac{2uz_j}{u^2-z_j^2}S_j^-\right)\left(u\beta I+\sum_{k=1}^L\frac{2uz_k}{u^2-z_k^2}S_k^+\right) \\
&= 2(\alpha^2+\beta^2u^2)I  + 2u^4\left(\sum_{j=1}^L\frac{1}{u^2-z_j^2}\right)^2I
\\
&\qquad\qquad\qquad\qquad
+4u^2\sum_{j=1}^L\frac{z_j^2}{(u^2-z_j^2)^2}C_j+4\sum_{j=1}^L\frac{u^2}{u^2-z_j^2}\mathcal T_j.
\end{align*}
where $C=2(S^z)^2+S^+S^-+S^-S^+$ is the $sl(2)$ Casimir element and
\begin{align}
\mathcal T_j=2(S_j^z)^2+&2\alpha S_j^z+\beta z_j (S_j^++S_j^-) \nonumber \\
+&\sum_{k\neq j}^L\left(\frac{4z^2_j}{z_j^2-z_k^2}S_j^zS_k^z+\frac{2z_jz_k}{z_j^2-z_k^2}(S_j^+S_k^-+S_j^-S_k^+)\right).
\label{ttees}
\end{align}
With the restriction $z_j\in\,{\mathbb R}, \,j=1,...,L$, which is imposed from now on, the $\{\mathcal T_j:j=1,...,L\}$ form a set of mutually commuting, self-adjoint operators.  Consequently they are simultaneously diagonalisable.
It is seen that the union of this set, the identity operator, and the set of Casimir elements $\{C_j:j=1,...,L\}$ is the full set of functionally independent operators that can be extracted from the transfer matrix.

Matrix forms of transfer matrix are obtained by choosing a representation of $sl(2)$ for each copy in the tensor product
$U(sl(2))^{\otimes L}$. It should be noted that these representations need not be isomorphic, and throughout the parameters
$\alpha,\,\beta, z_1,\,z_2,\,...,z_L$ are arbitrary real parameters. This provides a range of potential applications for constructing integrable models.

\section{Integrable, multi-species, boson tunneling model}

\noindent
Having identified the exotic symmetry of the solution (\ref{exo}), and outlined the construction of representations of $\overline{\mathcal{G}}$, attention turns towards an application through the construction of an integrable, multi-species, boson tunneling model. The Hamiltonian is defined as
\begin{align}
H&=2U\sum_{j=1}^L \mathcal T_j \nonumber \\
&= 4U\left(\sum_{j=1}^L S_j^z\right)^2+4U\alpha\sum_{j=1}^L S_j^z +2{U\beta}\sum_{j=1}^L z_j(S_j^++S_j^-).
\label{sum}
\end{align}
Next the spin operators are represented in terms of boson operators through the Jordan-Schwinger map
\begin{align*}
S^z_j&=\frac{1}{2}(N_{j,A}-N_{j,B}), \\
S^+_j&=b^\dagger_{j,A}b_{j,B},  \\
S^-_j&=b^\dagger_{j,B}b_{j,A}
\end{align*}
where the spin $s_j$ of each representation is determined by $2s_j=N_j$.
This maps (\ref{sum}) to (\ref{ham}) with the identification $\mu=2U\alpha$ and ${\mathcal E}_j=2U\beta z_j$.
For simplicity it will be assumed hereafter that the $z_j$, and consequently the ${\mathcal E}_j$, are distinct parameters.

Since the operators (\ref{ttees}) commute with (\ref{sum}), application of the Jordan-Schwinger map to the $\mathcal T_j$ yields a set of $L$ operators which necessarily commute with (\ref{ham}). Along with the $N_j$, which also commute with (\ref{ham}), the construction for the model is one which establishes that it is integrable since the number of independent conserved operators is the same as the number of degrees of freedom. Closely related to the property of integrability is the notion of an exact solution. The next step is to derive an explicit form for such a solution.

\subsection{Bethe Ansatz solution}
\noindent
Below it will be shown that the eigenvalues $\Lambda_j$ of the conserved operators (\ref{ttees})  have the form
\begin{align}
\Lambda_j&=2s_j^2+2\alpha s_j+\sum_{k\neq j}^L\frac{4s_js_k z_j^2}{z_j^2-z_k^2} -2\sum_{n=1}^N\frac{s_j z_j^2}{z_j^2-v_n}
\label{eigs}
\end{align}
such that the parameters $\{v_n:l=1,...,N\}$, known as the Bethe roots, satisfy the Bethe Ansatz equations
\begin{align}
\alpha-1+\sum_{l=1}^L\frac{2v_n s_l}{v_n-z_l^2}&=\sum_{m\neq n}^N\frac{2v_n}{v_n-v_m}
+\frac{\displaystyle \beta^2\prod_{l=1}^L(v_n-z_l^2)^{2s_l}} {\displaystyle 4\prod_{m\neq n}^N(v_n-v_m)}.
\label{bae}
\end{align}
Consequently, the energy eigenvalues $E$ for the Hamiltonian (\ref{ham}) have the form
\begin{align*}
E&=2U\sum_{j=1}^L \Lambda_j \\
&=UN^2+\mu N-4U\sum_{j=1}^L\sum_{n=1}^N \frac{ s_j z_j^2}{z_j^2-v_n}.
\end{align*}

Let the representation space associated with highest weight $s$ be denoted $V(s)$, such that for a system with $N_j$ bosons of each species $j$ the Hilbert space is
\begin{align*}
V(N_1/2)\otimes V(N_2/2)\otimes ... \otimes V(N_L/2)
\end{align*}
The tensor product state consisting of all the highest weight states is not an eigenstate of the transfer matrix,
except in the case when $\beta=0$. Consequently, the algebraic Bethe Ansatz method of solution for calculating eigenvalue expressions of the operators
(\ref{tees}) cannot be easily applied due to the absence of an apparent reference state.
Here a different approach is taken, based on operator identities in the first instance for a specific case. Using the method of proof by induction, the result will be extended to the general case by examining tensor product decompositions.

\subsection{$N$-species case}
\noindent
First the case where $N_j=1$ for all $j=1,...,L$, so $N=L$, will be considered. The quasi-classical limit of the {\it Off-diagonal} Bethe Ansatz approach \cite{wycs15} was utilised in \cite{lil16} to obtain the solution in this instance. While the physical interpretation of the $p+ip$ pairing Hamiltonian adopted in \cite{lil16} is a completely different context to the one considered here, at the mathematical level it is entirely equivalent and can be seen as the case where all copies of $sl(2)$ algebras are represented by spin 1/2.  It was found that the eigenvalues $\lambda_j$ are given by (\ref{eigs}) subject to the Bethe Ansatz equations (\ref{bae}) with the specialisation $s_j=1/2$ for all $j=1,...,L$. In \cite{cdv16} a method which generalises the algebraic Bethe Ansatz was used to confirm the result.

The approaches of \cite{cdv16,lil16} are quite technical. Below a different approach will be taken, following \cite{l16}, which reproduces the equations in a streamlined manner. To simplify calculations, first define the operators
\begin{align}
T_j=2\alpha S_j^z&+\beta z_j (S_j^++S_j^-) \nonumber \\
&+\sum_{k\neq j}^L\left(\frac{z^2_j}{z_j^2-z_k^2}(4S_j^zS_k^z-I)+\frac{2z_jz_k}{z_j^2-z_k^2}(S_j^+S_k^-+S_j^-S_k^+)\right)
\label{tees}
\end{align}
which, in the spin-1/2 case, only differ from (\ref{ttees}) by constant terms.
The starting point uses a result given in \cite{cdv16} that the operators $T_j$ satisfy the
quadratic identities
\begin{align*}
T_j^2=\alpha^2+\beta^2z_j^2-2z_j^2\sum_{k\neq j}^L\frac{1}{z_j^2-z_k^2}(T_j-T_k).
\end{align*}
Since these are operator identities it follows that the eigenvalues $\lambda_j$ of the operators $T_j$ satisfy analogous relations:
\begin{align}
\lambda_j^2=\alpha^2+\beta^2z_j^2-2z_j^2\sum_{k\neq j}^L\frac{\lambda_j-\lambda_k}{z_j^2-z_k^2}.
\label{quad}
\end{align}
Define $Q(u)$ to be a polynomial of order $L$ satisfying
\begin{align*}
2z_j^2Q'(z_j^2)+(\lambda_j-\alpha)Q(z_j^2)=0, \qquad j=1,...,L.
\end{align*}
Since this is a homogeneous linear system of $L$ equations for the $L+1$ coefficients
of the polynomial $Q(u)$, there exists a non-trivial solution. Set
\begin{align*}
Q(u)=\prod_{k=1}^L(u-v_k),
\end{align*}
which assumes that the roots of $Q(u)$ do not have multiplicities. The validity of this assumption will be reviewed later.
Now, provided $Q(z_j^2)\neq 0$ for all $j=1,...,L$, then
\begin{align}
\lambda_j^2
&= \alpha^2-4\alpha z_j^2\frac{Q'(z_j^2)}{Q(z_j^2)}+4z_j^4\left( \frac{Q'(z^2_j)}{Q(z^2_j)}\right)^2 \nonumber \\
&= \alpha^2-4\alpha z_j^2\frac{Q'(z^2_j)}{Q(z^2_j)}+4z_j^4\sum_{k=1}^L\frac{1}{(z_j^2-v_k)^2} +4z_j^4\frac{Q''(z^2_j)}{Q(z^2_j)}.
\label{sq2}
\end{align}

On the other hand from (\ref{quad})
\begin{align}\lambda_j^2&=\alpha^2+\beta^2z_j^2
-2z_j^2\sum_{k\neq j}^L\frac{\lambda_j-\lambda_k}{z^2_j-z^2_k}\nonumber\\
&=\alpha^2+\beta^2z_j^2+4z_j^2\sum_{l=1}^L\sum_{k\neq j}^L\frac{v_l}{(z^2_k-v_l)(v_l-z^2_j)} \nonumber\\
&=\alpha^2+\beta^2 z_j^2+4z_j^2\sum_{l=1}^L\frac{v_l}{z^2_j-v_l}\frac{P'(v_l)}{P(v_l)} \nonumber\\
&\qquad\quad +4z_j^2\sum_{l=1}^L\frac{z_j^2}{(v_l-z^2_j)^2}
+4z_j^2\sum_{l=1}^L\frac{1}{v_l-z_j^2}\nonumber\\
&=\alpha^2+\beta^2 z_j^2+\sum_{l=1}^L\frac{4z_j^2 v_l}{z^2_j-v_l}\frac{P'(v_l)}{P(v_l)} +\sum_{l=1}^L\frac{4z_j^4}{(v_l-z^2_j)^2}
-4z_j^2\frac{Q'(z_j^2)}{Q(z_j^2)}\label{sq1}
\end{align}
where
\begin{align*}
P(u)=\prod_{j=1}^L(u-z_j^2).
\end{align*}
For equality of the expressions (\ref{sq2}) and (\ref{sq1}) it is required that
\begin{align}
\frac{\beta^2}{4} +\sum_{l=1}^L\frac{v_l}{z^2_j-v_l}\frac{P'(v_l)}{P(v_l)}
=(1-\alpha) \frac{Q'(z^2_j)}{Q(z^2_j)}+z_j^2\frac{Q''(z^2_j)}{Q(z^2_j)}.
\label{zeros}
\end{align}
Set
\begin{align}
S(u)=uQ''(u)+(1-\alpha) Q'(u)-\left(\frac{\beta^2}{4}+\sum_{j=1}^L\frac{v_j}{u-v_j}\frac{P'(v_l)}{P(v_l)}\right)Q(u)
\label{ess}
\end{align}
which is a polynomial of order $L$. It follows from (\ref{zeros}) that
\begin{align*}
S(z_j^2)=0,\qquad j=1,..., L
\end{align*}
which, along with the consideration of the asymptotic behaviour of (\ref{ess}) as $u\rightarrow\infty$, establishes that
\begin{align}
S(u)&=-\frac{\beta^2}{4}P(u).
\label{essp}
\end{align}
Evaluating $S(v_n)$ through (\ref{ess}) and (\ref{essp}) and equating these expressions then yields the Bethe Ansatz equations
\begin{align}
\alpha-1+v_n\frac{P'(v_n)}{P(v_n)}=v_n\frac{Q''(v_n)}{Q'(v_n)}+\frac{\beta^2}{4}\frac{P(v_n)}{Q'(v_n)},
\qquad n=1,...,L
\label{poly}
\end{align}
which are equivalent to (\ref{bae}). It can be verified that (\ref{eigs}) holds.

It needs addressing at this point that the assumptions  that the roots of $Q(u)$ do not have multiplicities, and that $Q(z_j^2)\neq 0$, are not always valid. In the $\beta=0$ limit numerical solutions of the  Bethe Ansatz equations confirm that both of these situations occur \cite{dilsz10,ilsz09,lmm15,rdo10}. Two Bethe roots may coalesce at a point $z_j^2$ through which they transition between being real-valued and a complex-conjugate pair. Moreover in this limit some of the $L$ Bethe roots must be seen as being infinite-valued, and there are well-identified points at which several additional roots collapse to be zero-valued or infinite-valued. All of these instances may be interpreted as singular solutions of (\ref{bae}), but can be simply regularised by expressing (\ref{poly}) in a polynomial form
\begin{align*}
(\alpha-1)P(v_n)Q'(v_n)+v_n{P'(v_n)}{Q'(v_n)}=v_n{Q''(v_n)}{P(v_n)}+\frac{\beta^2}{4}{P(v_n)^2}.
\end{align*}
So, while the above {\it derivation} of the Bethe Ansatz equations breaks down when the roots of $Q(u)$ have multiplicities, or $Q(z_j^2)=0$, indicators from available numerical studies \cite{dilsz10,ilsz09,lmm15,rdo10} suggest that the equations are nonetheless valid.

\subsection{The general case}
\noindent
The validity of the formulae (\ref{eigs},\ref{bae}) in general follows by an inductive argument on each of the species number labels $N_j$, starting with the case $N_j=1$ for all $j=1,...,N$ which has been established above.
Every finite-dimensional irreducible $sl(2)$ module is contained in some tensor product of spin-1/2 modules, and in particular
\begin{align*}
V(N_1/2)\otimes ...V(N_L/2) \subseteq V(1/2)^{\otimes N}
\end{align*}
where $N=N_1+...+N_L$ as before.
Now consider a system of $M\geq L$ species, for which it is determined that
for $j\neq M-1,\,M$
\begin{align}
\lim_{z_M\rightarrow z_{M-1}}\mathcal T_j=&2(S_j^z)^2+2\alpha S_j^z+\beta z_j (S_j^++S_j^-) \nonumber \\
&\quad+\sum_{k\neq j}^{M-2}\left(\frac{4z^2_j}{z_j^2-z_k^2}S_j^zS_k^z+\frac{2z_jz_k}{z_j^2-z_k^2}(S_j^+S_k^-+S_j^-S_k^+)\right)
\nonumber \\
&\qquad +\frac{4z^2_j}{z_j^2-z_{M-1}^2}S_j^z(S_{M-1}^z+S_M^z) \nonumber \\
&\qquad+\frac{2z_jz_{M-1}}{z_j^2-z_{M-1}^2}(S_j^+(S_{M-1}^-+S_M^-)
+S_j^-(S_{M-1}^++S_M^+))
\label{mod1}
\end{align}
while
\begin{align}
\lim_{z_M\rightarrow z_{M-1}}(\mathcal T_{M-1}+\mathcal T_M)&=2(S_{M-1}^z+S_M^z)^2+2\alpha (S_{M-1}^z+S_M^z) \nonumber \\
&\qquad+\beta z_j (S_{M-1}^++S_M^++S_{M-1}^-+S_M^-) \nonumber \\
&\qquad+\sum_{k=1}^{M-2}\frac{2z_{M-1}z_k}{z_{M-1}^2-z_k^2}(S_{M-1}^++S_M^+)S_k^-
\nonumber  \\
&\qquad+\sum_{k=1}^{M-2}\frac{2z_{M-1}z_k}{z_{M-1}^2-z_k^2}(S_{M-1}^-+S_M^-)S_k^+
\nonumber  \\
&\qquad+\sum_{k=1}^{M-2}\frac{4z^2_{M-1}}{z_{M-1}^2-z_k^2}(S_{M-1}^z+S_M^z)S_k^z .
\label{mod2}
\end{align}
Noting the result
\begin{align*}
V(s_{M-1}+s_M)\subseteq V(s_{M-1})\otimes V(s_M),
\end{align*}
let $\mathcal P_M$ denote the projection onto the component $V(s_{M-1}+s_M)$.
The above expressions (\ref{mod1},\ref{mod2}) show that the general form of the conserved operators (\ref{ttees}) is preserved via
\begin{align*}
\tilde{\mathcal T}_j&=\lim_{z_M\rightarrow z_{M-1}}\mathcal P_M\mathcal T_j \mathcal P_M, \qquad j=1,...,M-2, \\
\tilde{\mathcal T}_{M-1}&= \lim_{z_M\rightarrow z_{M-1}}\mathcal P_M(\mathcal T_{M-1}+\mathcal T_M)\mathcal P_M
\end{align*}
whereby the tilded notation refers to a new system of $N$ particles of $M-1$ species, such that $\tilde{N}_{M-1}=N_{M-1}+N_M$ and $\tilde{N}_j=N_j$ otherwise.
It should be noted that a projection onto any other irreducible component of the tensor product $V(s_{M-1})\otimes V(s_M)$ gives rise to a system where the number of particles is strictly less that $N$. This is not an appropriate step for establishing the desired result, since it requires a change in the number of Bethe roots.

Suppose that (\ref{eigs},\ref{bae}) hold for the $M$-species system. Letting $\tilde{\Lambda}_j$ denote the eigenvalues for $\tilde{\mathcal T}_j$, it is required to show that (\ref{eigs},\ref{bae}) hold for the new system of $M-1$ species with the tilded notation. For (\ref{bae}) the result follows simply from the relations
\begin{align*}
\lim_{z_M\rightarrow z_{M-1}}\sum_{l=1}^{M}\frac{2s_l}{v_n-z_l^2}
&=
\frac{2(s_{M-1}+s_{M})}{v_n-z_{M-1}^2}+\sum_{l=1}^{M-2}\frac{2s_l}{v_n-z_l^2}, \\
\lim_{z_M\rightarrow z_{M-1}}\prod_{l=1}^M(v_n-z_l^2)^{2s_l}
&=(v_n-z_{M-1})^{2s_{M-1}+2s_M}{\prod_{l=1}^{M-2}(v_n-z_l^2)^{2s_l}}.
\end{align*}
In a similar manner it follows that ({\ref{eigs}) holds with the observations
\begin{align*}
\tilde{\Lambda}_j&= \lim_{z_M\rightarrow z_{M-1}}\Lambda_j, \\
\tilde{\Lambda}_{M-1}&= \lim_{z_M\rightarrow z_{M-1}}(\Lambda_{M-1}+\Lambda_M).
\end{align*}
Finally note that at each step the labels $N_j$ can be reordered without loss of generality, such that limits (\ref{mod1},\ref{mod2}) can be taken. It therefore follows by induction on each $N_j=2s_j$ that (\ref{eigs},\ref{bae}) hold in general.

\section{Conclusion}

\noindent
This work first identified a particular solution of a non-standard form of the classical Yang-Baxter equation. This solution is unusual in that it admits a symmetry which is more general than $u(1)$-symmetry, but less general than $gl(2)$-symmetry. The solution was used for the construction of an integrable, multi-species, boson tunneling model, and the Bethe Ansatz equations characterising the spectrum were derived.

Having obtained the exact solution, there are several avenues for ongoing work to take place. One promising course is to adapt integral equation techniques \cite{dilsz10,rdo10} to investigate ground-state properties in the limit of large particle numbers. This will be the subject of a future study.

\section*{Acknowledgments}

\noindent
This work was supported by the Australian Research Council through Discovery Project DP150101294. I thank Taras Skrypnyk for helpful comments.

redpajama_set_name: RedPajamaArXiv


Trichophytic closure, or Tricho for short, is a technique used in closing wounds. Its major advantage is that the wound in the donor area becomes practically invisible. The technique is used in FUT for closing the wound resulting from the removal of the strip.
Wound openings are closed in such a way that hairs can later on grow out of the scar, thus making it practically invisible. This is the reason why Tricho is the preferred technique in corrective or reconstructive treatment or when the patient likes to wear his hair very short.


text:
Did the coronavirus just end the apartment boom? This investor thinks so.
Worried about overpaying in a declining market, Chicago's Origin Investments has "indefinitely postponed" $241 million in apartment deals. "What we're trying to do is keep our eyes on the horizon and protect our investors."
It's not clear yet how badly the coronavirus will infect the commercial real estate market, but it looks bad enough already to Origin Investments, which has put more than $241 million in apartment deals on hold.
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Worried about overpaying in a declining market, the Chicago-based real estate investor has "indefinitely postponed" the acquisitions eyed for three of its funds, Origin wrote in email to its investors this week. The firm pulled out of pending deals in Houston, Denver, Charlotte, N.C., Orlando, Fla., and Austin, Texas, said Origin Principal Michael Episcope.
"The market has changed, and our pricing needs to reflect the new normal, because closing at these prices today would undoubtedly lead to lower performance than our initial projections," Episcope and Origin Principal David Scherer wrote in the email.
Translation: We don't want to buy in an apartment market that's going to get worse, not better.
Investors everywhere are trying to understand the impact of what could be the biggest Black Swan event of their lifetimes as the coronavirus, now officially a pandemic, sweeps through the U.S. economy.
Though Origin may be an outlier, its pullback underscores the much higher risk investors face as they try to navigate a real estate market that's looking more fragile by the day after being so good for so long. Demand for apartments, office, retail and other commercial space would decline if the economy slows down or dips into a recession, depressing rents and occupancies.
"What we're trying to do is keep our eyes on the horizon and protect our investors," Episcope said in an interview. "I hope we are premature, but I don't think we are."
Yet the cup is half full to others. With investors piling into the bond market, bond yields and interest rates have dropped, creating an opportunity to borrow cheaply for real estate investors that want to buy or refinance. Pete Evans, a Chicago apartment broker, hasn't seen a slowdown at all.
"The lending environment is amazing," said Evans, president of the eastern region and managing partner of the Chicago office at Moran & Co.
Today, apartment investors can get a 10-year mortgage at a 2.9 percent interest rate with five years of interest-only payments, he said.
"That's pretty close to unprecedented," he said. "It just jacks your return."
The other good news is that lenders in recent years haven't gotten carried away ladling out debt to low-quality borrowers, a big problem before the last real estate bust. So landlords may be in better shape to weather a slump.
"Sure, investors in CRE will feel pain if the real side of the economy falters," Jim Costello, senior vice president of Real Capital Analytics, a New York-based research firm, writes in a recent report. "Some tenants will be late with rent checks, some may even go out of business. Property income is going to be challenged. However, there have been fewer investors in CRE in this economic expansion using such risky debt."
Many investors will wait for the smoke to clear before they make their next move. Big out-of-town institutional investors are likely to hold off on acquisitions here for two reasons: They don't known what the future holds, and, with the coronavirus spreading, they won't get on a plane to Chicago to do a property tour, said Ron DeVries, senior managing director in the Chicago office of Integra Realty Resources, an appraisal and consulting firm.
"That market is just going to be in pause mode," he said. Investors are asking "Do we get through this in the next 60 days and start to see the other side? Is there a clear path, or are there still a lot of question marks 60 days from now?"
Origin isn't waiting to find out.
"We believe this is a time to take risk off the table and we are fortunate to be in the latter stages of selling four properties in our closed funds," Episcope and Scherer wrote in their email.
Origin hasn't bailed out completely. On March 9, the firm closed on a $65.8 million acquisition of Monroe Aberdeen Place, a 120-unit apartment complex in the West Loop. Because of its location and position at the high end of the market, the property "is pretty well protected," from a downturn, Episcope said.
Origin also is past the point of no return with Pilsen Gateway, a 202-unit apartment project in the University Village/Little Italy neighborhood that it recently launched in a joint venture with Chicago developer Cedar Street.
It is hard to overstate how good the apartment market has been to investors over the past decade. An index of U.S. apartment values compiled by research firm Green Street Advisors has risen more than 60 percent since 2014, and 8 percent in the past year. Episcope believes the market got a little frothy in some places, with prices 5 to 10 percent too high.
Now, Origin is waiting for prices to fall and preparing to pounce after they do.
"We will continue to raise as much capital as possible in the months ahead so we can play offense when the time is right," the firm's principals wrote in their email. "Our goal is to position our firm to take advantage of a new pricing environment that is about to emerge, and our team will continue to look for deals that have outsized return potential."