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CBN: At 57.0 PMI, Manufacturing Sector Expanded in June
Business | 5 years ago
• Says business confidence was stable in May
Kunle Aderinokun
The Central Bank of Nigeria has said the manufacturing sector expanded in the month of June.
The apex bank, which disclosed this in its Purchasing Managers' Index (PMI) Survey Report for June 2018, obtained by THISDAY, pointed out that Manufacturing PMI in the review month stood at 57.0 index points.
According to the CBN, the index point of 57.0 for the manufacturing in June indicated that the sector expanded for the 15th consecutive month.
The Manufacturing and Non-Manufacturing PMI Report on businesses is based on survey responses, indicating the changes in the level of business activities in the current month compared with the preceding month.
The June 2018 PMI survey was conducted by the Statistics Department of the Central Bank of Nigeria during the period June 8 – 19, 2018. The respondents were purchasing and supply executives of manufacturing and non-manufacturing organisations in 31 locations in Nigeria
A composite PMI above 50 points indicates that the manufacturing/non-manufacturing economy is generally expanding, 50 points indicates no change and below 50 points indicates that it is generally contracting. The subsectors reporting growth are listed in the order of highest to lowest growth, while those reporting contraction are listed in the order of the highest to the lowest contraction.
The CBN explained in the report that, the Manufacturing PMI grew at a faster rate when compared to the index in the previous month. "Of the 14 subsectors surveyed, 10 reported growth in the review month in the following order: paper products; furniture & related products; printing & related support activities; food, beverage & tobacco products; plastics & rubber products; electrical equipment; textile, apparel, leather & footwear; chemical & pharmaceutical products; petroleum & coal products and nonmetallic mineral products. The transportation equipment; fabricated metal products; primary metal; and cement subsectors declined in the review month.â€
The CBN report further revealed that at 59.2 points, the production level index for the manufacturing sector grew for the sixteenth consecutive month in June 2018. This also indicated a faster growth in the current month, when compared to its level in the preceding month. According to the apex bank, 10 of the 14 manufacturing subsectors recorded increase in production level, 1 remained unchanged, while the remaining 3 recorded declines in production level during the review month.
Assessing the employment, the CBN stated that the employment level index in June 2018 stood at 55.4 points, indicating growth in employment level for the 14th consecutive month. The PMI report noted: "Of the 14 subsectors, 7 reported increased employment level, 4 remained unchanged while 3 reported reduced employment level in the review month.â€
For the non-manufacturing PMI, the CBN noted that, the composite PMI for the nonmanufacturing sector stood at 57.5 points in June 2018, indicating expansion in the non-manufacturing PMI for the fourteenth consecutive month.
Pointing out that, "The index grew at a faster rate when compared to that in May 2018,†the apex bank said, " Fourteen of the 17 subsectors recorded growth in the following order: repair, maintenance/ washing of motor vehicles; agriculture; information & communication; professional, scientific, & technical services; finance & insurance; utilities; water supply, sewage & waste management; health care & social assistance; real estate rental & leasing; electricity, gas, steam & air conditioning supply; wholesale/retail trade; construction; management of companies; and transportation & warehousing. The arts, entertainment & recreation subsector remained unchanged, while the accommodation & food services; and educational services subsectors recorded contraction in the review period.â€
In terms of business activity, the CBN reported that at 59.1 points, the business activity index grew for the fifteenth consecutive month, indicating expansion in nonmanufacturing business activity in June 2018. "The index grew at a faster rate when compared to its level in the previous month. Twelve subsectors recorded growth in business activity, 3 remained unchanged while 2 declined in the review month.â€
Another CBN report called the Monthly Business Expectations Survey (BES) Report for May 2018 put the business confidence index at 28.9 points, indicating respondents' overall optimism on the economy and unchanged from April.
The May 2018 BES, which was carried out from May 11-22, 2018, had a sample size of 650 businesses covering services, industry, wholesale/retail trade and construction. A response rate of 91.8per cent was achieved.
According to the report, the business climate is directly correlated to a country's macroeconomic environment; this latest survey indicates that there is now a more stable macroeconomic outlook. Respondents saw the main business constraints as inadequate power supplies, the unfavourable economic climate and high interest rates.
The CBN explained that the optimism in May was driven by the opinion of respondents from the services (17.6 points) and industrial (9.2 points) sectors while those from the construction and wholesale/trade sectors represented just 1.2 and 0.9 points respectively of the total confidence index.
Analysts at FBNQuest Capital noted that "the trade sector is probably the best barometer for economic activity across the country. Based on the national accounts for Q1 2018, it performed better than the previous quarter in that it contracted at a slower rate.â€
"Despite the obvious easing of inflationary pressures, consumer demand remains soft, which helps to explain slow activity within the sector and by extension its low contribution to overall business confidence in May,†they pointed out. Â
Continuing, the FBNQuest analysts said, "Although access to credit remains a challenge for businesses, respondents were more optimistic with regards to financial conditions. Given that banks do not have the luxury of FX gains and 20 per cent+ treasury bill yields to boost their earnings this year, the need to extend credit is likely to grow in the near to medium term. However, we doubt lending rates will decline from current levels (i.e. above 20per cent).â€
The survey pointed towards a positive outlook for employment. The services, industrial and wholesale/retail sectors have the best prospects for job creation at 30.0, 23.6 and 18.2 points respectively. Additionally, the services and construction sectors indicate the greater disposition towards expansion.
FBNQuest Capital noted further, "As for inflation, the general expectation is for a slowdown in the headline rate for June. This is similar to our thinking. We expect the rate to slow to 10.8 per cent y/y, from 11.6 per cent in May.â€
The survey also captured the outlook for business confidence in June. Based on data from respondents, the business confidence index is projected at 65.5 points.
According to the analysts, "business confidence is likely to remain positive. However, consumers remain cautious with their spending, resulting in soft demand. If consumption trends remain unfavourable, this could have an adverse effect on overall business confidence.â€
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AlphaBay Market was shut down by the law enforcement, alleged mastermind committed suicide in jail
July 14, 2017 By Pierluigi Paganini
The AlphaBay Market was shut down by the law enforcement, exactly like its predecessor Silk Road, and the alleged mastermind committed suicide in jail.
AlphaBay Market was considered the largest marketplace on the Dark Web for drugs, guns, and other illegal goods competing against the likes of Abraxas, Dream, and Hansa.
The AlphaBay Market went down last week on Tuesday, July 4th, without any explanation, many users that have purchased products on the marketplace fear the Exit-Scam.
A few days after the disconcerting disappearance, the Wall Street Journal reported that the black marketplace shut down after authorities in the United States, Canada, and Thailand conducted several raids and arrested Alexandre Cazes, who is suspected to be an operators of the AlphaBay Market.
"An online marketplace that sold illegal goods on the so-called Dark Web was shut last week following action by international authorities, according to people familiar with the matter." states the WSJ.
Cazes was resident in Canada, but he was arrested by law enforcement in Thailand. Police also seized "four Lamborghini cars and three houses worth about 400 million baht ($11.7 million) in total."
Local authorities have taken him into custody in Bangkok on July 5th, on the same day the police executed two raids on residences in Quebec, Canada.
"Investigators with the RCMP executed search warrants in Montreal and Trois-Rivières Wednesday morning, reportedly in connection with the sale of merchandise on the "Dark Web" of the internet." reported the Montreal Gazette.
Now there is the dramatic part of the story, Cazes was awaiting extradition to the United States when a guard found him dead in the jail cell on Wednesday.
According to the Chiang Rai Times, Cazes has hanged himself using a towel. AlphaBay was considered the "the new Silk Road," it emerged in 2014 and became of the most prominent dark web marketplaces.
"Narcotics Suppression Bureau (NSB) Pol Maj Gen Sunthon Chalermkiat told Thai media yesterday that an initial examination of the body of Canadian Alexander Cazes, points to suicide." reports the Chiangraitimes.
"A duty officer noticed a towel hanging from the toilet door in his cell about 7am, but could not see him, police said. The officer unlocked and entered the cell and found Cazes dead in the toilet."
The case anyway has not been resolved, law enforcement has to discover if there were other operators behind the AlphaBay Market and which were their roles.
Don't forget also that there is a lot of money still stored inside its wallets and probably someone could access them.
"It is also unclear as to what Cazes may have told the authorities in regards to AlphaBay, its users, or other people working on the platform." continues the Chiangrai Times.
"On Reddit, there is some wild speculation as to whether or not these claims are true. It is hard to determine if Cazes is the real AlphaBay admin, but all of the information seems to hint at that outcome. His arrest coincides with the platform going down, and it would also explain why AlphaBay has not resurfaced. On the platform, the admin is known as "deSnakes", which very well could indeed by Cazes."
We can consider the case closed, the AlphaBay Market was shut down by the law enforcement, exactly like its predecessor Silk Road.
(Security Affairs – AlphaBay Market, Dark Web)
AlphaBay MarketBlack marketplaceCybercrimeDark Webdrugshut down
Pierluigi Paganini is member of the ENISA (European Union Agency for Network and Information Security) Threat Landscape Stakeholder Group and Cyber G7 Group, he is also a Security Evangelist, Security Analyst and Freelance Writer. Editor-in-Chief at "Cyber Defense Magazine", Pierluigi is a cyber security expert with over 20 years experience in the field, he is Certified Ethical Hacker at EC Council in London. The passion for writing and a strong belief that security is founded on sharing and awareness led Pierluigi to find the security blog "Security Affairs" recently named a Top National Security Resource for US. Pierluigi is a member of the "The Hacker News" team and he is a writer for some major publications in the field such as Cyber War Zone, ICTTF, Infosec Island, Infosec Institute, The Hacker News Magazine and for many other Security magazines. Author of the Books "The Deep Dark Web" and "Digital Virtual Currency and Bitcoin".
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A 20 years-old vulnerability in Kerberos, dubbed Orpheus' Lyre, was parched this week for both Microsoft and Linux distros. A...

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Witcher 2 HD Screenshots revealed along with System Requirements and Box Art.
By Hassan Mujtaba
Apr 6, 2011 12:51 EDT
New HD Screenshots of Witcher 2: Assassins of Kings have been revealed along with Box Art and Recommended/Minimum System Requirements. Witcher 2: Assassins of Kings is the sequel to the highly successful "The Witcher" and is a Role Playing Game being developed by CD Projekt Red. The game is scheduled for release on May 17 2011 exclusively on the PC however there were some reports that the game will be released later on consoles after the PC release but this hasn't been confirmed yet.
Following are the PC requirements of the game, Although DX11 support isn't mentioned but from the visuals it looks like the game features tessellation and DOF but the max listed in hardware is DX10 support. An interview done with CD Projeckt Key revealed that they'll be adding DX11 enhancements which you can see here. The game also features Nvidia 3D vision Support as seen on the box art.
Mininum System Requirements:
• OS: Windows XP SP2 / Windows Vista SP2 / Windows 7 (32/64-bit)
• Processor: Intel Core 2 Duo 2.2 Ghz or AMD Athlon 64 X2 5000+
• Memory: 1 GB Windows XP / 2 GB Windows Vista and Windows 7
• Video Card: 512 MB RAM, supporting Pixel Shader 3.0 (Nvidia GeForce 8800 or ATI Radeon HD3850)
Recommended System Requirements:
• Processor: Intel Core 2 Quad or AMD Phenom X4
• Video Card: 1 GB RAM, supporting Pixel Shader 3.0 (Nvidia GeForce GTX260 or ATI Radeon HD4850)
Below are screenshots of the map. This game offers vast room to explorer than the first Witcher game, Just look at the size difference of the map from both games.
Intel Steps Up The Marketing Game With Leaked 9th Generation Core i9 Packaging
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« Color-Coded MP3 Album
Ambient Haze MP3s »
Free Amon Tobin MP3
[ December 6, 2004 / Department: downstream / Leave a comment ]
Free online downloads from major labels and long-running independent labels have decreased in regularity as pay-per-song services have gone mainstream, notably with Apple's iTunes Music Store. Was a time, you could regularly surf through the sites of maverick electronic labels like Ninja Tune and Warp and find full-length songs, for free, as promotional items. Warp launched its own download service earlier this year, Bleep.com, and since then a number of labels have signed on to use Bleep as the main commercial conduit for their MP3s, including not only Ninja Tune but One Little Indian, Schematic, Mego, Def Jux and others. And while many of these labels still offer the occasional free downloads as well, the site of Ninja Tune, for one example, rarely if ever these days posts MP3s on its Downloads page, and when it does, they hang around for a long time. Right now, the Ninja Downloads page has tons of wallpapers and screensavers, and just one little MP3, but it's a good one: the first track ("Intro") off Amon Tobin's first full-length release as a DJ: Solid Steel Presents Amon Tobin: Recorded Live, which came out this past July. It's one of a handful of tracks he composed for the compilation, a set from an Australian gig that also includes material by AFX, Dizzee Rascal and the Velvet Underground ("Venus in Furs," in case you're wondering). Tobin's track is a hard, dense chunk of techno, less rhythmically resplendent than his best work, but still worthwhile for in its sheer thoroughness. Check out the Ninja Tune Downloads page at ninjatune.net/downloads, and download the Tobin track directly here.
[ Tags: free, live-performance / Leave a comment ]


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Revenge of the QR code
by Russ Michaels | Feb 21, 2022 | Jibber Jabber, Tech Stuff | 0 comments
Remember those hazy "before times" of late 2019 and early 2020, when the QR Code was just this strange techy barcode image you saw here and there but didn't actually know what it was and barely anyone used them. A QR Code seemed passé—almost like putting "e" or "I" at the beginning of a new device's name and considering it techy and hip, but nobody really took it seriously.
Then along came a deadly global pandemic, when no one wanted to touch a restaurant menu and "contactless ordering" became a thing, and the QR Code had its revenge—even starring in one of the hottest Super Bowl ads of 2022.
But the QR Code wasn't originally designed for ordering pizza and burgers to your table. The first QR Codes were actually developed in Japan in the early 1990s by Denso Wave, a Toyota subsidiary, to expedite automobile manufacturing. In 1992, engineer Masahiro Hara was tasked with inventing a new bar code that could hold more data. the current barcode that could hold only 20 characters wasn't adequate for use in managing Toyota's massive parts inventory. Workers were found themselves scanning up to 1,000 codes a day as they made do by placing multiple bar codes on a single part.
Hara needed a compact code that could hold more data and be scanned quickly. Inspiration struck during a lunchtime game of the strategy game Go. Looking at the game board, he realized how much data its grid pattern could contain—and that by adding a position detection pattern, such a code could always be scanned accurately.
How QR codes work
Three corners of the QR Code contain the finder pattern, a nested series of black and white squares that, when detected by an optical scanner and interpreted by software, allow the scanning device to determine the orientation of the QR Code. Two other patterns are also present. The alignment pattern, smaller squares containing yet smaller squares, is used in all but the smallest codes to determine if the QR Code has been distorted, and the timing pattern, a row and column of alternating black and white squares connecting the large squares of the finder pattern, serves as the coordinate system of the QR Code.
The largest possible code, Version 40, allowed under the QR Code standard is a matrix of 177 × 177 pixels, and the smallest, Version 1, is 21 × 21 pixels. A Version 40 QR Code can contain 7,089 numeric characters or 4,296 alphanumeric characters. (For comparison, this article contains 2,006 alphanumeric characters.) Denso Wave also created a smaller version of the QR Code for use in tracking small objects. Called Micro QR Code, it comes in four versions ranging in size from 11 × 11 to 17 × 17 pixels.
As the COVID-19 pandemic evolves, the future for QR Code use is anybody's guess—but because we've all discovered its conveniences with ordering food and smartphones are so ubiquitous, it's hard to imagine they will fade away. Plus, QR Codes are now part of the cybercriminal's arsenal, with the FBI issuing a warning about malicious code use in January.
If you want to know more about QR codes and how they are used, checkout this page on wikipedia.
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\section{Introduction}
Defects are internal boundaries separating different regions of a bulk theory. When crossing the defect, the fields of the system undergo a discontinuous jump but preserve a particular set of properties. Of particular interest are defects preserving properties that enable us to keep computational control over the total system. In the context of integrable models, that property is integrability.  Adding defects to integrable models is especially interesting considering that they can help us move towards less symmetric or `ideal' configurations and consequently allows one to probe new features in these systems.  Here, we will address the question whether it is possible at all to introduce defects in a class of integrable classical  field theories that are deformed and thus already feature a reduced set of (global) symmetries.

In $d=1+1$ integrable classical field theories, an integrable defect is introduced as an internal discontinuity or boundary condition at a fixed point $x_0$ in the spatial direction, see figure \ref{fig:defect_conv}. Although this definition of a defect might seem to compromise integrability from the onset, obviously breaking translation invariance and thus ruining already the conservation of momentum, it is possible to introduce defects whilst preserving integrability.
Indeed, the study of integrable defects in integrable field theories were initiated in the Lagrangian formalism in \cite{Bowcock:2003dr,Bowcock:2004my}, as an extra term in the total Lagrangian. The defect contribution to the total Lagrangian is fixed by demanding it modifies the momentum as to be again conserved. At first sight this approach seems to be only engineered to salvage the conservation of the first charge and nothing in principle guarantees that the defect will preserve sufficiently many conserved charges to warrant for the total system to remain integrable\footnote{In this context, one can distinguish two types of defects. Type I defects do not carry any additional degree freedom and realise defects that only require fields of the original theory. It was realised however, that the definition of type I was not sufficient to realise defects in certain integrable systems. The solution is to allow for the defects to interact through their own degree of freedom (localised at the defect location) \cite{Corrigan:2009vm}. These are called type II or dynamical defects.}. In support of this approach to add defects to integrable field theories was the crucial observation that the defect conditions at the defect location coincide with ``frozen'' B\"acklund transformations\cite{Corrigan:2005iq}. Using this insight, it was shown in  \cite{Caudrelier:2008zz} that the integrable defect indeed preserved integrability by constructing the generating function for the whole tower of modified conserved charges. In addition, this observation gave a new momentum to the study of integrable defects as one can study the B\"acklund transformations for the integrable system to introduce integrable systems defects, see e.g \cite{Gomes:2015hra,Aguirre:2016zrj,Caudrelier:2014gsa}. Although this approach guarantees the conservation of an infinite set of conserved charges (i.e. \textit{weak} integrability) the question of the involution of the charges ( or \textit{strong}/Liouville integrability) is difficult to address due to the divergence in the Poisson brackets at the localised defect, see \cite{Caudrelier:2014oia} for a discussion.

\begin{figure}
\centering
\tikzset{every picture/.style={line width=0.75pt}}

\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]

\draw    (225.24,129.3) -- (498.24,130.28) ;
\draw [shift={(500.24,130.29)}, rotate = 180.21] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-4.9) .. controls (6.95,-2.3) and (3.31,-0.67) .. (0,0) .. controls (3.31,0.67) and (6.95,2.3) .. (10.93,4.9)   ;
\draw [shift={(223.24,129.29)}, rotate = 0.21] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-4.9) .. controls (6.95,-2.3) and (3.31,-0.67) .. (0,0) .. controls (3.31,0.67) and (6.95,2.3) .. (10.93,4.9)   ;
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (357.55,129.79) .. controls (357.55,127.47) and (359.43,125.59) .. (361.74,125.59) .. controls (364.06,125.59) and (365.94,127.47) .. (365.94,129.79) .. controls (365.94,132.11) and (364.06,133.99) .. (361.74,133.99) .. controls (359.43,133.99) and (357.55,132.11) .. (357.55,129.79) -- cycle ;

\draw (268,102.93) node [anchor=north west][inner sep=0.75pt]   [align=left] {system$\displaystyle _{1}$};
\draw (401,102.93) node [anchor=north west][inner sep=0.75pt]   [align=left] {system$\displaystyle _{2}$};
\draw (355,136.24) node [anchor=north west][inner sep=0.75pt]  [font=\small]  {$x_{0}$};
\end{tikzpicture}
\caption{The system on the left (right) of the defect located at $x=x_0$ will be denoted by the superscript $1$ ($2$).}\label{fig:defect_conv}
\end{figure}

A second, tangent approach, first initiated by \cite{Habibullin:2007ww} and systematically studied in several integrable system in \cite{Avan:2011sr,Doikou:2012fc,Doikou:2016oej}, imposes Liouville integrability from the outset. The idea is to construct a modified monodromy matrix for the total system by gluing the two systems through a defect contribution such that the Sklyanin quadratic algebra remains satisfied. One has in addition to impose consistency relation, dubbed ``sewing/jumping conditions'', between the fields in the bulk theories and those of the defect matrix.  Although this approach warrants for the involution and conservation of the charges, it can clearly only be applied to integrable systems that are ultralocal, or alternatively, admit an ultralocal representation of the Poisson algebra of their Lax connection.  This method was first concretely realised for the non-linear Schr\"odinger model \cite{Avan:2011sr} and later in many other systems, amongst which the Faddeev-Reshetikhin model or ultralocal form of the $SU(2)$ principal chiral model \cite{Doikou:2012fc}.

Let us finally remark that although these two approaches seem diametrically different, both can be in principle used to implement integrable defects in a given integrable system and one should expect them to be somehow related. Indeed, the B\"acklund defects and the Liouville defects were indeed reconciled and clarified their relation in \cite{Caudrelier:2014oia} by using a multisymplectic approach.

In this letter we will make the first steps towards understanding how integrable defects can be introduced in a class of (1+1)-dimensional integrable field theories that feature target spaces with deformed symmetries, known as Yang-Baxter models. In particular, we show how the two methods above have to be modified to describe integrable defects Yang-Baxter models. In section \ref{sec:YBmodels}, we review some basic properties of Yang-Baxter models, setting the stage for the addition of defects in Yang-Baxter models. In section \ref{sec:monodromy_def}, using the modified monodromy matrix approach, we construct a continuous family of Liouville integrable defects for the $S^2$ non-split Yang-Baxter model. In section \ref{sec:baecklund}, we show how B\"acklund transformation can be constructed for the split and non-split Yang-Baxter deformation. Using these we specialise to the defect location and give the form of the defect matrix. We illustrate the construction and provide explicit expressions for the non-split $SU(2)$ Yang-Baxter model.

\section{Yang-Baxter models 101}\label{sec:YBmodels}
 In this section, we will concisely review the Yang-Baxter models and the concepts we need to introduce integrable defects for these models. Extensive discussions on Yang-Baxter models including their relation to generalised dualities can be found in the recent reviews \cite{Thompson:2019ipl,Hoare:2021dix} or the original papers \cite{Klimcik:2002zj,Klimcik:2008eq}.

 \subsubsection*{The principal chiral model}
For a semisimple Lie group $G$, the principal chiral model (PCM) is described by a map $g:\Sigma \rightarrow G$ with the action
\begin{align}
	S_\mathrm{PCM}= \int \mathrm d^2 \sigma \, \langle g^{-1}\partial_- g,g^{-1}\partial_+ g \rangle\,,
\end{align}
where $\langle -,-\rangle=\mathrm{tr}(-,-)$ is the Killing form on the Lie algebra $\frak g$ of $G$. The global invariance of the action under left and right translation of the principal chiral field $g$ under a constant group element implies the existence of two conserved currents $J_\pm^R =-\partial_\pm gg^{-1}$ and $J_\pm^L =g^{-1}\partial_\pm g$. The equations of motion of the principal chiral model take the form of a flatness and conservation equation for the current
\begin{align}
	\partial_+ J_-^{L/R}+\partial_- J_+^{L/R}=0\,, \quad \partial_+ J_-^{L/R}-\partial_- J_+^{L/R} +[J_+^{L/R},J_-^{L/R}] =0\,.
\end{align}
In what follows we will work exclusively with the left current $J_\pm\equiv J^{L}_\pm$.
Any flat and conserved current capturing the dynamics of a system can be used to construct the Mikhailov and Zakharov Lax connection
\begin{align}\label{eq:Lax_PCM}
	\mathscr L_\pm(x,t;\xi)=-\frac{J_\pm}{1\pm\xi} \,,
\end{align}
whose flatness is equivalent to the compatibility condition for an auxiliary linear system by the equation
\begin{align}\label{eq:aux_system_PCM}
	\partial_\pm \Psi(x,t;\xi)= - \mathscr L_\pm(x,t;\xi)\Psi(x,t;\xi)\,,
\end{align}
with solutions taking valued in the loop algebra $\mathpzc L \frak g\equiv C^\infty(S^1,\frak g)$.
 \subsubsection*{The Yang-Baxter models}

 Yang-Baxter models are constructed starting from a Lie group $G$ and a linear map $R:\frak g\rightarrow \frak g$ verifying the modified classical Yang-Baxter equation (mCYBE)
\begin{align}\label{eq:mCYBE}
	[RX,RY]-R([RX,Y]+[X,RY])=-c^2[X,Y]\,.
\end{align}
Trivial rescaling of the linear map $R$ leads to three possible values of $c\in \{0,1,i\}$. Solutions corresponding to these three values are called homogeneous, split and non-split, respectively.
The action of the Yang-Baxter model, as an integrable deformation of the principal chiral model (PCM), is described by a map $g:\Sigma \rightarrow M$, for $M$ a Lie group manifold or coset space. When $g$ is valued in a Lie group\footnote{For coset spaces $M$ the action is structurally similar but includes suitable projectors.} that action takes the form \cite{Klimcik:2002zj}
\begin{align}\label{eq:actYB}
	S_\mathrm{YB}= \int \mathrm d^2 \sigma \, \langle g^{-1}\partial_- g, \left(1-	\eta R_g\right)^{-1}g^{-1}\partial_+ g \rangle\,,
\end{align}
where $\Sigma$ is the two-dimensional worldsheet and $R_g=\mathrm{Ad}_{g^{-1}}R\mathrm{Ad}_{g}$.  The existence of a Lax connection in both cases relies heavily on $R$ solving the mCYBE. Historically the non-split Yang-Baxter models are also sometimes called $\eta$-models.

 \subsubsection*{Conserved currents of the Yang=Baxter models}
 The deformation operator $(1-\eta R)$ in the Yang-Baxter model in eq. \eqref{eq:actYB} breaks the initial global symmetry group $G_L\times G_R$ to $G_L\times U(1)^{\mathrm{rk}(G)}$, hence breaking the right-invariant symmetry. The left current remains unbroken and takes on the form
 \begin{align}\label{eq:conservation_YB_current}
 	\partial_+ J_-^\eta+\partial_- J_+^\eta=0\,, \quad \text{where}\quad J^\eta_\pm=  \frac{1\pm c\eta}{1\pm \eta R_g}g^{-1}_\eta\partial_\pm g_\eta\,.
 \end{align}
 Using the modified Yang-Baxter equation and the equations of motion of the YB model one can show that in fact this current is not only conserved but also flat
 \begin{align}\label{eq:flatness_YB_current}
 \partial_+ J_-^\eta-\partial_- J_+^\eta +[J_+^\eta,J_-^\eta] =0\,.
\end{align}
We will see in the last section a different, more straightforward argument to the conservation and flatness of the current $J^\eta$.

 \subsubsection*{Yang-Baxter Lax connection}
Using this conserved current \eqref{eq:conservation_YB_current}, one can construct the associated Mikhailov-Zakharov Lax connection which is given by
\begin{align}
	\mathscr L^\eta_\pm = \frac{1}{1\mp \xi}\frac{1\pm c\eta}{1\pm \eta R_g}g^{-1}_\eta\partial_\pm g_\eta=\frac{1}{1\mp \xi} J^\eta_\pm\,.
\end{align}
Flatness of this connection
\begin{align}
 \partial_+ \mathscr L^\eta_-(\xi)-\partial_- \mathscr L^\eta_+(\xi) +[\mathscr L^\eta_+(\xi),\mathscr L^\eta_-(\xi)] =0\,,
\end{align}
encodes the equations \eqref{eq:conservation_YB_current} and \eqref{eq:flatness_YB_current} summarising the dynamics of the Yang-Baxter model and ensuring the existence of an infinite number of conserved charges.

 \subsubsection*{Some properties of the $R$-matrix}
Given a solution $R$ of the mCYBE \eqref{eq:mCYBE}, one can show that $\frak g$ admits a second Lie bracket $[-,-]_{ R}=([ R-,-]+[-, R-])$. We will denote this second Lie algebra $\frak g_{ R}=(\frak g,[-,-]_{ R})$, where here $\frak g$ is understood to be the vector space. The couple $(\frak g,\frak g_R)$ is called a bialgebra and they can be embedded together into the so-called Drinfel'd double algebra $\frak d=\frak g\oplus \frak g_R$. It turns out that the solution of the mCYBE $R$ can be used to define an injective homomorphism of $\frak g_{R}$ into $\frak d$ when $R$ is a split or non-split solution. In particular
	\begin{align}\label{eq:isom_gR}
	\frak g_{R} \hookrightarrow \mathfrak d: 	 X \mapsto \begin{cases}
 	&\gamma ({R}^+X,{R}^-X)\quad \text{(split)}\\
 	&\gamma {R}^\pm X \qquad \qquad \text{(non-split)}
 \end{cases}\,,
	\end{align}
		where $\gamma$ is an arbitrary constant and $R^\pm=R\pm c1$, see e.g. \cite{semenov2008integrable,Vicedo:2015pna} for a detailed discussion.
We will need these maps to extract the Lax connections of the Yang-Baxter-models from the extended solutions of the principal chiral model.

Let us finally briefly summarise the form of the double Lie algebra $\frak d$ in the three types of solutions for the mCYBE \eqref{eq:mCYBE}.
\begin{itemize}
\item Split case ($c=i$)	: The solution is unique and the double $\frak d$ is isomorphic to the complexified algebra $\frak g^\mathbb{C}$. The  algebra $\frak g$ and its dual algebra $\frak g_R$ can be retrieved by identifying $\frak g$ as the real form of $\frak g^\mathbb{C}$ and performing the Iwasawa decomposition $\frak g^\mathbb{C}=\frak g\oplus \frak{an}=\frak g\oplus \frak g_R$.
\item Non-split case ($c=1$)	: The solution is also unique and the double $\frak d$ is isomorphic to $\frak g\oplus \frak g$. The original algebra $\frak g$ is embedded diagonally in the double $\frak g\cong \frak g^\delta=\{(x,x)\mid x\in \frak g\}\subset \frak d$. To extract the dual algebra we use the isomorphisms \eqref{eq:isom_gR}, $\frak g_R=\{R^+X,R^-X\mid X\in \frak g\}\subset \frak g_\delta$.
\item Homogeneous case ($c=0$): For the homogeneous case, the solution is in general not unique. When restricting to skew-symmetric solutions, one can show that the solutions are in bijective correspondence with so-called quasi-Frobenius Lie subalgebras of $\frak g$ \cite{stolin1991rational,stolin1999rational}.
\end{itemize}
In this letter, we will consider  split and non-split solutions exclusively.

At the level of the group, the decomposition at the level the algebra discussed above might not be true globally. Here however we will assume that the double is simple\footnote{or we can also restrict ourselves to the so called main cell of a double coset decomposition explained in \cite{alekseev1994symplectic}. We also take the $\frak g$ algebras for which the mCYBE is solved to be semi-simple.} such that the we can perform the decomposition $D=GG_R$ at the level of the group unhindered.

 \subsubsection*{Ultralocality and Hamiltonian formulation}
Both the principal chiral model and its integrable deformations suffer from non-ultralocality. That is, the equal-time Poisson-bracket of their Lax connection does not take the Sklyanin form but instead features an extra term containing the derivative of the Dirac-delta distribution
\begin{align}
	\{\mathscr L(x;\xi)\overset{\otimes}{,}\mathscr L(y;\mu)\}=[r(\xi,\mu),\mathscr L(x;\xi)\otimes \mathscr L(y;\mu)]\delta_{xy}+A(x,y;\xi,\mu)\partial_x\delta_{xy}\,,
\end{align}
for some function $A(x,y;\xi,\mu)$, $\delta_{xy}=\delta(x-y)$ and where the $r$-matrix has to be antisymmetric.
However, as was first proposed by Maillet in \cite{Maillet:1985fn,Maillet:1985ek}, for some non-ultralocal systems, amongst which the principal chiral model and the Yang-Baxter models, the Poisson-Bracket can be written in a form generalising the Sklyanin identity and ensures the involution of the conserved charges. The generalisation is achieved by allowing for the $r$-matrix to be non-skew-symmetric. The failure of this new $r$-matrix to be skew-symmetric is measured by the so-called twist function $\varphi(\xi)$ that depends on the spectral parameter $\xi$.
More explicitly, we have that the $r$-matrix entering the Maillet bracket deforms the canonical $r$-matrix $r^0(\xi,\mu)$ on the loop algebra of $\frak g$:
\begin{align}
	r_{12}(\xi,\mu)=r_{12}^0(\xi,\mu)\varphi^{-1}(\xi)\,.
	\end{align}
In this letter we will not need the explicit form of the Maillet bracket but merely need the expressions for the twist functions.

Sigma-model are generically non-ultralocal and a prime example is the principal chiral model. As an ultralocal model, the Poisson algebra of its Lax connection is thus characterised by its twist function which is given by
  \begin{align}\label{eq:twistfct_PCM}
  	\varphi_\mathrm{PCM}(\xi)= \frac{1-\xi^2}{\xi^2}\,.
  \end{align}
Note already that the twist function features a double pole at the origin. This fact will be important when introducing the dressing method in section \ref{sec:PCM_dressing}. In \cite{Delduc:2013fga}, an Hamiltonian perspective on the deformation of the principal chiral model leading to the Yang-Baxter models was proposed to prove the strong or Hamiltonian integrability of these models. In this picture, and in contrast to the Lagrangian perspective discussed above, the deformation is achieved by keeping the Lax of the principal chiral model unchanged but deforming the twist function. Reality conditions allows for the double pole of the principal chiral model to split symmetrically, either settling on the complex axis of the spectral plane at $\xi=\pm i\eta$ or on the real axis at $\xi=\pm \eta$. In particular the twist function for the split and non-split Yang-Baxter models (up to an overall constant) is given by
   \begin{align}\label{eq:twistfct_YB}
  	\varphi_\mathrm{YB}(\xi)= \frac{1-\xi^2}{\xi^2-c^2\eta^2}\,.
  \end{align}
The pair of poles on the so-called complex branch realise the non-split ($c=i$), while the poles on the real branch lead to the split ($c=1$) Yang-Baxter model.
For the homogeneous Yang-Baxter models, the effect of the deformation in fact does not change the pole structure of the principal chiral model twist function, but rather reshuffles the expression of the Lax in terms of its defining fields.

\section{Liouville integrable defects}\label{sec:monodromy_def}

In this section we first review how integrable defects can be introduced by means of a modified monodromy matrix together with a set of regularisation conditions. We then briefly introduce the sausage model, which is equivalent to the non-split Yang-Baxter model on the two-sphere, and its ultralocal formulation. Having set the ground, we show how Liouville integrable defects can be introduced in the Yang-Baxter model on $S^2$. Notably, the Yang-Baxter model on $S^2$ admits a continuous family of integrable defects.  This feature is not shared by Liouville integrable defects of the undeformed principal chiral. Indeed the presence of an additional parameter can be traced back to the trigonometric r-matrix underlying the Sklyanin identity of the deformed model.

\subsection{Modified monodromy approach to integrable defects}

This method introduces defects by inserting a defect operator in the monodromy, leading to a ``modified monodromy''  \cite{Avan:2011sr}. The challenge however is as we will see that this method assumes ultralocality, ruling out its naive applicability to integrable sigma-models. Denote by $\mathscr L(\xi)$ the Lax connection of a fixed (ultralocal) integrable system whose Sklyanin identity is verified for an $r$-matrix $r(\xi)$. We introduce a spectral-parameter dependent defect operator $\mathcal{D}(x_0,\xi)$ localised at some space point $x_0$ incorporating non-trivial sewing conditions between the bulk theories on either side. This is done by invoking monodromy matrices of systems on both sides of the defect at $x_0$ and glue them together with an Ansatz for the defect matrix $\mathcal D(x_0,\xi)$. Following the notation set in Figure \ref{fig:defect_conv}, the total monodromy matrix reads \cite{Avan:2010xd}
\begin{align}
	\mathcal T(L,-L,\xi)&=T^{(2)}(L,x_0^+,\xi)\mathcal D(x_0,\xi)T^{(1)}(x_0^-,-L,\xi)\notag\\
	&=\mathcal P\exp\left(\int^L_{x_0^+} \mathscr L^{(2)}_x(x)\right)\mathcal D(x_0,\xi)\mathcal P\exp\left(\int^{x_0^-}_{-L}\mathscr L^{(1)}_x(x)\right)\,,
\end{align}
where $x_0^\pm$ are infinitesimally close to the left/right of the defect location $x=x_0$.
Imposing Liouville integrability on the total system by demanding the modified monodromy to verify the quadratic Poisson algebra or Sklyanin identity
\begin{align*}
	\lbrace \mathcal T(L,-L,\xi)\stackrel{\otimes}{,} \mathcal T(L,-L,\mu )\rbrace =\left[ r(\xi-\mu), \mathcal T(L,-L,\xi)\otimes\mathcal T(L,-L,\mu )\right]\,,
\end{align*}
requires for the defect operator to verify the same quadratic algebra
\begin{align}\label{eq:sklyanin_defect}
	\lbrace \mathcal D(\xi)\stackrel{\otimes}{,} \mathcal D(\mu) \rbrace =\left[ r(\xi-\mu), \mathcal D(\xi)\otimes\mathcal D(\mu)\right]\,,
\end{align}
 where $r(\xi)$ is the same r-matrix as for the bulk theories. In addition, one has to impose sewing conditions across the defect to avoid singular contributions from the zero curvature condition of the Lax connection \cite{Avan:2008xr,Avan:2010xd}. In particular, at the location of the defect the zero curvature condition translates into the condition
 \begin{align}\label{eq:jumping_condition}
 	\frac{\mathrm d}{\mathrm d t}\mathcal D(x_0)= \tilde{\mathscr{L}}_t^{(2)}(x_0)\mathcal D(x_0)-\mathcal D(x_0)  \tilde{ \mathscr{L}}_t^{(1)}(x_0)\,,
 \end{align}
 where the matrices $\tilde{ \mathscr{L}}_t^{(i)}(x_0)$ are the time components of the Lax pair evaluated at the defect location and are derived by demanding  analyticity at the defect, i.e. $ { \mathscr{L}}_t^{(i)}(x_0^\pm)\rightarrow  \tilde{ \mathscr{L}}_t^{(i)}(x_0)$.  That is the condition \eqref{eq:jumping_condition} effectively describes the jump/sewing over the defect.

\subsection{Liouville integrable defects in the $S^2$  non-split Yang-Baxter model}

The so-called $2d$ ``sausage model'' was first written down in \cite{Fateev:1992tk} as an integrable deformation of the $O(3)$ non-linear sigma-model and shortly after shown to admit a Lax form \cite{Lukyanov:2012zt}. Later, in \cite{Delduc:2013fga,Hoare:2014pna}, it was shown to fit in the Yang-Baxter model family as the non-split Yang-Baxter deformation on the two-sphere $SU(2)/U(1)\cong S^2$. The sausage model is an integrable deformation of the $O(3)$ non-linear sigma-model. Although in this formulation this integrable deformation of the (coset) principal chiral model is non-ultralocal, it is possible to reformulate its dynamics in an ultralocal form. Inspired by the existence of an ultralocal  form of the $O(3)$ model obtained by a ``gauge'' transformation \cite{Bytsko:1994ae}, the authors in \cite{Bazhanov:2017nzh} constructed an ultralocal version of the sausage model.

We will use the coordinates in terms of the sausage model given in  \cite{Bazhanov:2017nzh}, which we review in this section. That is the $S^2$ Yang-Baxter model or sausage model is given by the Lagrangian
\begin{align}
	\mathcal L=\frac{1}{2}\frac{\partial_\mu \mathbf n \partial^\mu \mathbf n}{\kappa^{-1}-\kappa n^2_3}\,,
\end{align}
where $\kappa \in (0,1)$ is the deformation parameter and $\mathbf n=(n_1,n_2,n_3)$ is a three-dimensional vector such that $\mathbf n \cdot \mathbf n =1$. Although naively non-ultralocal, a different flat Lax connection can be shown to satisfy an ultralocal Poisson algebra, see \cite{Bazhanov:2017nzh}. The Sklynanin identity is then satisfied for this new Lax connection with the original trigonometric r-matrix
\begin{align}\label{eq:trig_rmatrix}
	r(\mu)=\frac{1}{\sinh \mu}(\sigma_1\otimes \sigma_1 +\sigma_2\otimes \sigma_2 + \cosh \mu \,\sigma_3\otimes \sigma_3)\,.
\end{align}
In what follows we will refer to this system as the sausage model but with the understanding each time that it is equivalent to the $S^2$ non-split Yang-Baxter model.

Given the ultralocal representation of the 2d sausage model, we propose the following Ansatz for the defect of the $\eta$-deformed $S^2$-model\footnote{This particular form of a matrix verifying versions of the Sklyanin identity and in different contexts has already appeared in \cite{Avan:2010xd,Bazhanov:2017nzh}, what is new here is its interpretation as a defect matrix for the $\eta$-deformed two-sphere.}
\begin{align}\label{eq:Ansatz_defect_sausage}
	\mathcal D(\xi)=\xi \mathds 1 +
\frac{1}{2}\left(
\begin{array}{cc}
 \mathcal{F} (\xi )  \zeta^z&  \mathcal{G}(\xi ) \zeta^-  \\
 \mathcal{G}(\xi ) \zeta^+  & -\mathcal{F}(\xi ) \zeta^z \\
\end{array}
\right)\,,
\end{align}
where the variables $\zeta^i$, upon requiring this defect Ansatz to satisfy the Sklyanin identity, have to generate an $\frak{sl}(2)$ algebra
\begin{align}\label{eq:sl2alg}
	\{\zeta^z,\zeta^\pm\}= \pm 2 \zeta^\pm\,,\quad \{\zeta^+,\zeta^-\}=\zeta^z\,.
\end{align}
and the functions $\mathcal{F}(\mu )$ and $\mathcal{G}(\mu) $ are given by
\begin{gather}
\begin{aligned}
\mathcal{F}(\mu ) &= \frac{\mu(1+\delta )\mu^{-1}+(1-\delta ) \mu }{(1-\delta ) \mu -(1+\delta )\mu^{-1}}\\
\mathcal{G}(\mu)&= \frac{2 \mu\sqrt{1-\delta ^2}}{(1-\delta ) \mu -(1+\delta )\mu^{-1}}\,,
\end{aligned}
\end{gather}
where $\delta$ is an arbitrary constant.
It can be readily checked to verify the Sklyanin identity \eqref{eq:sklyanin_defect} for the trigonometric $r$-matrix given in \eqref{eq:trig_rmatrix}. Let us note that in comparison to the defect Ansatz proposed in \cite{Doikou:2012fc} for the (undeformed) $SU(2)$ principal chiral, the sausage model admits a continuous family of defects parametrised by the parameter $\delta$. This freedom is a direct consequence of the sausage model being described by a trigonometric rather than rational r-matrix. Conversely the defect Ansatz \eqref{eq:Ansatz_defect_sausage} does not solve the Sklyanin identity for the rational r-matrix considered in \cite{Doikou:2012fc} for the undeformed $SU(2)$ principal chiral model. Finally let us remark that the parameter $\delta$ that enters the defect Ansatz cannot be identified with the deformation parameter $\kappa$, as one can check that this identification is incompatible with the sewing conditions, that we now discuss.

Satisfying only the Sklyanin identity  \eqref{eq:sklyanin_defect} for the defect is not sufficient. Integrability is only preserved provided the additional sewing or jumping conditions can be solved for an adequate form of the time component Lax at the location of the defect given in \eqref{eq:jumping_condition}. In order to do so, first note that in general $\mathscr{L}_t$ is given in terms of the monodromy matrix by \cite{Avan:2011sr,Faddeev:2007}
\begin{equation}\label{eq:V from monodromy}
\mathscr{L}_t(x,\xi,\mu)=t^{-1}(\xi) \mathrm{tr}_a \left(\mathcal{T}_a(L,x,\xi) r_{ab}(\xi,\mu)  \mathcal{T}_a(x,-L,\xi) \right)\,.
\end{equation}
The subscript $a$ is a tensor product space index, such that $\mathcal{T}_a, \mathcal{D}_a$ are given as $\mathcal{T}\otimes \mathds{1}$ and $\mathcal{D} \otimes \mathds{1}$ and $t(\lambda)=\mathrm{tr}(\mathcal{T}(L,-L,\lambda) )=\mathrm{tr}(T^{(2)}(L,x_0,\lambda)\mathcal{D}(x_0,\lambda)T^{(1)}(x_0,-L,\lambda))$.
Depending on whether the above expression is evaluated on the bulk or at the defect location $x_0$ will lead to a generically different results. Looking at the right bulk theory and its limit towards the defect location, we have that
\begin{gather}
\begin{aligned}
\mathscr{L}_t^{(2)}(x,\xi,\mu)& = \frac{t^{-1}(\xi)}{\xi/\mu-(\xi/\mu)^{-1}}\left(M_o + \frac{\xi/\mu-(\xi/\mu)^{-1}}{4} (M_d - \det(M_d)M_d^{-1} \right)\,,\\
\tilde{\mathscr{L}_t}^{(2)}(x,\xi,\mu)& =  \frac{t^{-1}(\xi)}{\xi/\mu-(\xi/\mu)^{-1}}\left(N_o + \frac{\xi/\mu-(\xi/\mu)^{-1}}{4} (N_d - \det(N_d)N_d^{-1} \right)\,.
\end{aligned}
\end{gather}
where  $M_d$ and $M_o$ denote, respectively, the diagonal and off-diagonal parts of $M$ the matrices $M$ and $N$ given by
\begin{align*}
M&=\ T_a^{(2)}(x,x_0,\xi) \mathcal{D}_a( x_0,\xi)T_a^{(1)}(x_0,-L,\xi)T_a^{(2)}(L,x,\xi)\,,\\
N&=\mathcal{D}_a( x_0,\xi) \ T_a^{(1)}(x_0,-L,\xi)T_a^{(2)}(L,x_0,\xi)\,.
\end{align*}
The sewing conditions are obtained by imposing that the bulk fields glue continuously along the defect, i.e
\begin{equation}\label{eq:sewing condition}
\lim_{x \to x_0} \mathscr{L}_t^{(2)}(x) = \tilde{\mathscr{L}_t}^{(2)}(x_0)\,.
\end{equation}
This will lead to the sewing condition which will involve the bulk fields as well as the $\frak{sl}(2)$-variables $\zeta^i$ given in \eqref{eq:sl2alg} that are associated to the defect itself. One has similar relations for time-component of the left bulk and defect Lax connections  $\mathscr{L}_t^{(1)},\tilde{\mathscr{L}_t}^{(1)}$.

Calculating the sewing conditions from the gluing condition in eq. (\ref{eq:sewing condition}) is carried out by expanding the monodromy matrix order by order in powers of the spectral parameter.
A generic expansion of the monodromy is given by  \cite{Faddeev:2007}
\begin{equation}\label{eq:monodromy_expansion}
T^{(1/2)}(x,y,\xi) = (\mathds 1+W^{(1/2)}(x))e^{Z^{(1/2)}(x,y)}(\mathds 1+W^{(1/2)}(y))^{-1}\,,
\end{equation}
where $W$ and $Z$ are, respectively, anti-diagonal and diagonal matrices that are analytic in the spectral parameter $\xi$ around the poles of the Lax connection
\begin{align*}
W(\xi)&=\sum_{n=0}^{\infty} W_n (\xi -\xi_p)^n\,, \qquad W_n=\left(
\begin{array}{cc}
0 & u_n\\
v_n  & 0 \\
\end{array}
\right)\,,\\
Z(\xi)&=\sum_{n=-1}^{\infty}Z_n(\xi-\xi_p)^n\,, \qquad Z_n=\left(
\begin{array}{cc}
 q_n &0\\
0& r_n  \\
\end{array}
\right)\,,
\end{align*}
where $\xi_p$ is a fixed pole of the Lax connection and the matrix elements $q_n,r_n,u_n,v_n$ are yet to be determined\footnote{In fact as we will see, to zeroth order, the matrix elements from the $Z$ matrix will not play a role and consequently won't enter the sewing condition.}. Using this form of the matrices $W(\xi)$ and $Z(\xi)$, we get sewing conditions relating their various matrix elements and the fields $\zeta^i$ featured in the defect Ansatz. The Lax connection of the ultralocal sausage model has a pole at $\xi_p=\frac{\sqrt{1+\kappa}}{\sqrt{1-\kappa }}$, expanding the monodromy matrix as in eq. \eqref{eq:monodromy_expansion} around $\xi_p$ to zeroth order:
\begin{align}
[\mathscr{L}_t^{(2)}]_0&=\frac{1}{\mathfrak{J_-}(\xi)}
\left(
\begin{array}{cc}
 \frac{1}{4}\mathfrak{J}_+(\xi) &
   - \xi\,  u^{(2)}_0\\
  \xi \,v^{(2)}_0 &
  -  \frac{1}{4}\mathfrak{J}_+(\xi)\\
\end{array}
\right)\,,
\end{align}
for the expression on the bulk. The subscript 0 denotes the zeroth order contribution and the overall denominator is given by
\begin{align}
	\mathfrak{J}_\pm(\xi)= \kappa_\pm(\xi)\left(1\pm u^{(2)}_0 v^{(2)}_0\right)\,,\;\; \text{with}\;\; \kappa_\pm(\xi)=\xi_p(1\pm (1-\kappa)\xi^2)\,.
\end{align}
At the defect location we get
\begin{align}\label{eq:t_Lax_2_def_loc}
[\tilde{\mathscr{L}_t}^{(2)}]_0=\frac{1}{\tilde{\mathfrak{J}}_-(\xi)}
\left(
\begin{array}{cc}
 \frac{1}{4}\tilde{\mathfrak{J}}{}_+(\xi) & -
 \left(\Omega_--Z^- v^{(1)}_0\right) \xi u^{(2)}_0 \\
\left( \Omega_+v^{(1)}_0-Z ^+ \right)\xi& - \frac{1}{4}\tilde{\mathfrak{J}}{}_+(\xi) \\
\end{array}
\right)
\end{align}
where $\Omega_\pm=2(\delta- \kappa) \pm (1-\delta  \kappa)  \zeta ^z$ and $Z^\pm =\sqrt{1-\delta ^2}  \sqrt{1-\kappa ^2} \zeta ^\pm$. The overall scaling factor in \eqref{eq:t_Lax_2_def_loc} is given by
\begin{align}
\tilde{\mathfrak{J}}_\pm(\xi) &= \kappa_\pm(\xi) \left( \Omega_--Z^-v_0^{(1)}-(\Omega_+v_0^{(1)}-Z^+)u_0^{(2)}\right)
\end{align}
Imposing the gluing or jumping constraint in eq. (\ref{eq:sewing condition}) for the time-component of the Lax connection right of the defect $\mathscr L_t^{(2)}$ and at the defect $\tilde{\mathscr{L}}_t^{(2)}$ we get the following sewing condition
\begin{align}\label{eq:sewing bulk 2}
v^{(1)}_0 v^{(2)}_0 Z ^-&=Z ^++ \Omega_+v^{(1)}_0 - \Omega_-v^{(2)}_0\,.
\end{align}
 Performing the same computation for the fields on the system left of the defect, we get
\begin{align}\label{eq:sewing bulk 1}
u^{(1)}_0 u^{(2)}_0 Z^+ &=Z ^-- \Omega_+u^{(1)}_0 + \Omega_-u^{(2)}_0\,.
\end{align}
We see that the two conditions are related by exchanging
$Z^+ \leftrightarrow Z^-$ (or equivalently $\zeta^+ \leftrightarrow \zeta^-$) and
$v^{(i)}_0 \leftrightarrow -u^{(i)}_0$. A similar symmetry was already noticed in \cite{Doikou:2012fc} for the ultralocal $SU(2)$ principal chiral model.

The defect \eqref{eq:Ansatz_defect_sausage} for the $S^2$ non-split Yang-Baxter model features two continuous parameters: the deformation parameter $\kappa$ of the sausage model and the arbitrary constant $\delta$ carried by the defect itself. The latter is a novel feature of the deformed model in comparison with the defect for the undeformed $SU(2)$ principal chiral mdoel presented in \cite{Doikou:2012fc}. Finally, using the expansion of the monodromy matrix given in eq. \eqref{eq:monodromy_expansion} together with the defect matrix, one can extract the defect contribution to the lowest modified conserved charge
\begin{align}
\mathfrak D_0 &= \frac{\xi_p \left( \Omega_--Z^-v_0^{(1)}-(\Omega_+v_0^{(1)}-Z^+)u_0^{(2)}\right)}{2(\kappa-\delta)(1-u_0^{(2)}v_0^{(2)})} \,.
\end{align}
Using the sewing condition at lowest order in eq. \eqref{eq:sewing bulk 2} simplifies the corresponding defect contribution to
\begin{align}
\mathfrak D_0 &= \frac{\xi_p\left( Z^+v_0^{(2)} -\Omega_+ \right)}{2(\kappa-\delta)}  \,.
\end{align}
The derivation of the higher order charged can then performed in a similar way, each time computing at the same time the sewing condition of the corresponding order.

\section{B\"acklund transformations for inhomogeneous Yang-Baxter models}\label{sec:baecklund}
 A B\"acklund transformation\footnote{One can differentiate between auto-B\"acklund and B\"acklund transformations, where the suffix ``auto'' refers to the case where the differential system relates two solutions of the \textit{same} non-linear differential equation. Since we will only consider auto-B\"acklund transformations, relating solutions of the equations of motion of the Yang-Baxter model, we will just write ``B\"acklund'' throughout.} is a system of differential equations relating two different solutions of a given non-linear differential equation describing e.g. the dynamics of a physical system or the defining relations of certain surfaces. Strikingly, B\"acklund transformations appear to be closely related to integrable systems as the differential equations that admit such transformations often coincide with that of an integrable system, see \cite{rogers2002backlund} for an extensive review. The main application of B\"acklund transformations is as a solution generating technique. Indeed, given a solution, often taken to be the vacuum or trivial solution, the B\"acklund transformation yields a differential equation that is often linear or of a lower order than the equations of motion.  In the context of integrable defects, the defect condition coincide with a B\"acklund transformation ``frozen'' at the defect location, thus relating the two solution separated by a defect \cite{Corrigan:2005iq}. The fields\footnote{Note that, to avoid a concatenation of super- and subscripts, we will in what follows change the notation slightly compared to the previous section. The fields in the system on the left/right of the defect will be denote without/with a tilde instead of a subscript (1)/(2).} $g$ and $\tilde g$ on the left and the right of the system are then related at the location of the defect $x=x_0$ by a defect matrix $K(x_0,t)$, that is
 \begin{align}\label{eq:defect_matrix}
 	\tilde g(x_0,t)=K(x_0,t) g(x_0,t)\,,
 \end{align}
 where $K(x_0,t)$ is the defect matrix that is related to the particular B\"acklund transformation.

 In this section we will first review the dressing method and the related B\"acklund transformations in the principal chiral model, setting the ground for deriving defects in the Yang-Baxter model. To construct the latter, we need to adopt the Hamiltonian approach to integrable deformations: the deformation is seen as changing the pole structure of the twist function and thus the Poisson brackets of the field in the cotangent space. The Yang-Baxter fields $g,\tilde g:\Sigma\rightarrow G$ on both side of the defect, as well as the defect matrix itself, are then constructed by first evaluating a dressed solution of the principal chiral model at the poles of the twist function and a subsequent group decomposition.

 Let us also mention that although defects have never been studied in Yang-Baxter models before, in \cite{Driezen:2018glg}  integrable boundary configuration on the worldsheet, and the corresponding D-branes on the target space, were derived for the related $\lambda$-deformation and then mapped via Poisson-Lie T-duality to the non-split Yang-Baxter model for $G=SU(2)$.
\subsection{Dressing method and B\"acklund transformations in the PCM}\label{sec:PCM_dressing}
It was first observed in \cite{zakharov1978relativistically} and \cite{shabat1972exact} that, using the auxiliary problem one can obtain new solution of the auxiliary problem in eq. \eqref{eq:auxproblem} by ``dressing'' a known solution.  The starting point is the auxiliary problem
\begin{align}\label{eq:auxproblem}
	\partial_\pm\Psi(x,t;\xi)=-\mathscr L_\pm(x,t;\xi) \Psi(x,t;\xi)\,.
\end{align}
that encodes the flatness condition of the Lax connection as its compatibility condition. Considering the auxiliary problem for the principal chiral model, that is for the Lax connection given in \eqref{eq:Lax_PCM}, note that the principal chiral field can be extracted from the extended solution by evaluating at $\xi=0$, indeed $\Psi(x,t;0)=g$. Denoting a known solution by $\Psi$ and the new putative solution by $\tilde \Psi$, the dressing Ansatz takes the form
\begin{align}\label{eq:dressing_PCM}
	\tilde \Psi(x,t;\xi)=\chi(x,t;\xi)\Psi(x,t;\xi)\,,
\end{align}
where $\chi(x,t;\xi)$ is a spectral parameter dependent function called the dressing factor which particular form is derived by demanding $\tilde \Psi(x,t;\xi)$ to remain a solution of the auxiliary problem and depends also on the specific target space. The chiral field of the dressed solution can then be extracted from the dressed solution by evaluating the relation \eqref{eq:dressing_PCM} at $\xi=0$, i.e. $\tilde g=\chi(0)g$. The dressing factor $\chi(\xi)$ and its inverse are required to be meromorphic functions in the spectral parameter \cite{Ogielski:1979hv,harnad1984backlund}
\begin{align}
	\chi(\xi)=\mathds 1+\sum_{i=1}^M\frac{Q_i}{\xi-\xi_i}\,,\qquad  \chi(\xi)^{-1}=\mathds 1+\sum_{i=1}^M\frac{R_i}{\xi-\xi_i}\,,
\end{align}
featuring a finite set of simple poles $\{\xi_i\}_{i=1}^K$ and $\{\xi_i\}_{i=1}^K$ where $K$ is an integer.

The matrices $R_i$ and $Q_i$ are functions only of the worldsheet coordinates and not the spectral parameter. The dressing function is normalised by requiring $\chi(\infty)=\chi^{-1}(\infty)=I$.
Additional analyticity constraint on the residues $Q_i$ and $R_i$ leads to a derivation of their general form, see \cite{harnad1984backlund}. More details on the minimum number of poles and their reality conditions  depend on the target space and can be found in that reference.

Plugging the dressed wave function in eq. \eqref{eq:dressing_PCM} back into the auxiliary system \eqref{eq:aux_system_PCM}, demanding the system to remain invariant and for the currents \eqref{eq:Lax_PCM} featured in the Lax connection to remain independent of the spectral parameter, the currents have to transform as
\begin{align}\label{eq:pre-Baecklund_PCM}
	\tilde J_\pm= \chi^{-1}(\xi)J_\pm\chi(\xi)+(1\mp\xi)\partial_\pm\chi^{-1}(\xi)\chi(\xi)\,.
\end{align}
Subsequently evaluating equation \eqref{eq:pre-Baecklund_PCM} at the value $\xi=0$ of the spectral parameter, yields an equation in the algebra $\frak g$ relating the currents of the seed solution $g$ and the dressed solution $\tilde g$
\begin{align}
	\tilde J_\pm= \chi^{-1}(0)J_\pm\chi(0)+\partial_\pm\chi^{-1}(0)\chi(0)\,.
\end{align}
This relation is known as the B\"acklund transformation for the principal chiral model, transforming the current of the principal chiral field of the seed solution $g:\Sigma\rightarrow G$ and that of the dressed solution $\tilde g:\Sigma\rightarrow G$. In the following section we show how, by lifting the construction to the Drinfel'd double and considering an ``auxiliary principal chiral model'' there, one can construct B\"acklund transformations for the split and non-split Yang-Baxter models.

\subsection{B\"acklund transformations for Yang-Baxter models}

Our strategy to identify B\"acklund transformations and thus defect matrices for the Yang-Baxter model will be to use the Hamiltonian approach summarised at the end of section \ref{sec:YBmodels} rather than the Lagrangian approach. In the Hamiltonian setting the starting point are the undeformed principal chiral model Lax connection and its twist function that encodes the non-ultralocal form of its Poisson algebra. The deformed model in the split and non-split case is then obtained by moving the double poles of the Lax matrix whilst preserving certain reality conditions.

As was first observed in \cite{Klimcik:2008eq,Klimcik:2014bta} for the non-split case, the Yang-Baxter field $g:\Sigma\rightarrow G$ can by obtained by evaluating the extended solution of the auxiliary problem of the \textit{principal chiral model} at $\xi=\pm c\eta$, rather than $\xi=0$, and subsequently performing an Iwasawa decomposition $G^\mathbb{C}=AN\cdot G$. The Yang-Baxter field $g_\eta:\Sigma \rightarrow G$ is then identified with the contribution from the compact real subgroup of $G^\mathbb{C}$, i.e.
\begin{align}
G^\mathbb{C} \ni \Psi (\pm c\eta)=b_\eta g_{\eta}\in  AN\cdot G\,.
\end{align}
Later this very observation was explained and generalised to solutions $c=1$ of the mCYBE by using the Hamiltonian formulation of the Yang-Baxter models. The value $\xi=-i\eta$ used in \cite{Klimcik:2014bta}, or more generally $\xi=\pm c\eta$, was identified as the pole of the twist function controlling the deformation \cite{Delduc:2013fga}, see also \cite{Vicedo:2015pna}. In the split case, the wave function takes values in the real double and we have the corresponding decomposition $\Psi(\pm \eta)=g^\delta  b\in G^\delta\cdot G_R=G\times G$. Note that this procedure is consistent with the observation made in the previous section: evaluating the extended solution at the location of a pole of the twist function yields the principal chiral field $g_0$ or Yang-Baxter field $g_\eta$, except that in the latter one has to perform an additional group decomposition.  We will briefly review this argument, as it will be critical to define B\"acklund transformations for Yang-Baxter models.

Starting from the auxiliary problem
 \begin{align}\label{eq:aux_problem_PCM_LCG}
	\mathscr L_\pm^\mathrm{PCM}  (\pm c\eta)\equiv - \frac{g_0^{-1}\partial_\pm g_0}{1 \pm c\eta}&= -\Psi ^{-1}(\pm c\eta)\partial_\pm \Psi (\pm c\eta)\,,
\end{align}
we can consider this equation now for $\xi$ taking values in $\mathbb C$. In particular, we can now evaluate the wave function $\Psi (\xi)$ at both the real or complex values $\xi=\pm c\eta$ corresponding to the poles of the twist functions \eqref{eq:twistfct_YB} of the split and non-split Yang-Baxter models. Without loss of generality and to avoid a profusion of signs, in what follows  we will fix the sign to $\xi=+c\eta$.  Remarkably, after performing the decomposition, the conserved current of the principal chiral model $J^0_\pm$ turns out to be proportional to the unbroken current of the Yang-Baxter model
 \begin{align}\label{eq:idfct_Jeta_PCMcurrent}
 		\frac{g_0^{-1}\partial_\pm g_0}{1\mp c\eta}= \frac{1\pm c\eta}{1\pm \eta R_g}g^{-1}_\eta\partial_\pm g_\eta= J_\pm^\eta\,,
\end{align}
where the derivation uses that {\color{black} $R\pm c:\frak g_R\hookrightarrow \frak d$} is an injective homomorphism \eqref{eq:isom_gR}, and thus that in particular for any $b\in AN$ one can write $b^{-1}\partial_\pm b=\eta (R- c )X_\pm$ for some $X_\pm\in \frak g$ and the factor $\eta$ is there for convenience. Note that the identity \eqref{eq:idfct_Jeta_PCMcurrent} provides direct proof that the current $J^\eta$ is both conserved and flat, since $J^0_\pm=g_0^{-1}\partial_\pm g_0\propto J^\eta_\pm$. This was exactly the observation mentioned earlier that was used in \cite{Klimcik:2014bta} to re-derive the Lax connection for the (non-split) Yang-Baxter model and construct that of the bi-Yang-Baxter model.

This suggests that we have to consider dressing solutions for the auxiliary system \eqref{eq:aux_problem_PCM_LCG} for the PCM with fields taking values in the double of the split or non split solution. By evaluating at the poles of the Yang-Baxter twist function we will then obtain B\"acklund transformation for the deformed systems. For a seed solution $\Psi $ of the PCM consider the dressed solution for a particular allowed dressing factor $\chi(\xi)$:
\begin{align}\label{eq:dress_Psi}
	\widetilde \Psi (\xi)=\chi(\xi)\Psi (\xi)\,.
\end{align}
In the last section we haver reviewed how the respective currents are related through eq. \eqref{eq:pre-Baecklund_PCM} valued in loop group $\mathpzc L\frak g^\mathbb{C}$. However, now we can instead evaluate this equation at the location of the poles of the deformed twist function $\xi=c\eta$ and thus find a relation between the currents of the Yang-Baxter model
\begin{align}
	\widetilde J_\pm^\eta=\chi(c\eta)^{-1} J_\pm^\eta\chi(c\eta)+(1\mp c\eta)\chi(c\eta)^{-1}\partial_\pm \chi(c\eta) \,.
\end{align}
We can now both extract the Yang-Baxter fields $g_\eta, \tilde g_\eta:\Sigma\rightarrow G$ and interpret the transition between the two fields as crossing a defect as in \eqref{eq:defect_matrix}.
As explained above, the group element $\tilde g_\eta$ can be obtained by performing an Iwasawa decomposition of the right hand side of
\begin{align}
	\tilde\Psi(c\eta)=\tilde b_\eta \tilde g_\eta= \chi(c\eta)\Psi(c\eta)\,.
\end{align}
In addition, after some reshuffling, we can identify the defect matrix $K$ as a `dressed' dressing factor evaluated at the poles of the twist function
\begin{align}\label{eq:defect_YB}
	\tilde g_\eta = \Big(\tilde b_\eta^{-1}\chi(c\eta)b_\eta\Big) g_\eta\equiv Kg_\eta\,.
\end{align}

We will close this section with a couple of comments. Let us first point out that the B\"acklund transformations are interesting in se and not only as an instrument for constructing defects. B\"acklund transformations offer an alternative way to the monodromy matrix expansion to generate the tower of conserved charges underlying the integrable property of the model. Given a periodic solution of the B\"acklund equations, one can construct an infinite number of local conserved currents of the seed solution  \cite{Pohlmeyer:1975nb}. Systematic methods to extract the charges using this method for the PCM were first developed perturbatively in \cite{Ogielski:1979hv} and later a non-perturbative method was presented in \cite{harnad1984backlund}. The relation between the charges generated from the monodromy matrix and the B\"acklund transformations was studied in \cite{Arutyunov:2005nk} in the particular context of strings propagating in AdS$_5\times S^5$.

Although we have shown here how B\"acklund transformations can be realised for Yang-Baxter models, the dressing method itself for Yang-Baxter models does not seem to straightforwardly follow from the results presented here. This generalisation of dressing method would require one to solve the auxiliary problem with the Yang-Baxter Lax connection together with a way to extract the Yang-Baxter field directly. Here we obtained the B\"acklund transformations for the Yang-Baxter-models by starting from the extended problem for the principal chiral model and subsequently reduced its dressed solution to a solution of the Yang-Baxter-model. Note however that the dressing method has already been considered in the context of the $\lambda$-deformation in \cite{Appadu:2017xku}.

\subsection{Defect matrix for the $SU(2)$ non-split Yang-Baxter model}
In this section we provide an example of two Yang-Baxter fields valued in $SU(2)$ that can be separated by a defect and give the corresponding defect matrix $K$.  We thus need to consider the auxiliary problem of the PCM for $SU(2)^\mathbb{C}=SL(2,\mathbb{C})$. The procedure starts from a vacuum solution, i.e. a principal chiral field $g:\Sigma\rightarrow SL(2,\mathbb{C})$ such that
\begin{align}\label{eq:vacuum_eqs}
	\partial_-J_{0,+}=\partial_+J_{0,-}=[J_{0,+},J_{0,-}]=0\,.
\end{align}
The auxiliary problem for a vacuum solution is easily integred, yielding the solution $\Psi_0(\xi)$ to the auxiliary problem.
To dress this solution, we consider the simplest possible dressing factor, which for $SL(2,\mathbb{C})$ takes the form \cite{harnad1984quadratic}
\begin{align}\label{eq:dressing_factor}
	\chi(\xi)=\left(\mathds 1+\frac{\mu_0-\xi_0}{\xi-\xi_0}P \right)\,,
\end{align}
where $\mu_0$ and $\xi_0$ are constants taken to be valued  on the unit circle and $P$ is a projector. The latter is computed by choosing two arbitrary 2-dimensional vectors $m$ and $n$ such that $n^\dagger m$ is non-singular, in terms of which it is computed via
\begin{align}
	P=\frac{\Psi_0(\mu_0)m\,n^\dagger\Psi_0(\xi_0)^{-1}}{n^\dagger \Psi_0(\xi_0)^{-1}\Psi(\mu_0)m}\,.
\end{align}
One must be careful, as the dressing factor is not necessarily on element of $SL(2,\mathbb{C})$ and thus neither \`a priori the dressed solution $\Psi(\xi)=\chi(\xi)\Psi_0(\xi)$. To mend this, one adds a normalisation factor to \eqref{eq:dressing_factor} ensuring the dressing factor has unit determinant\footnote{ As was remarked in \cite{harnad1984quadratic}, the dressing factor for the PCM auxiliary problem in $SL(2,\mathbb C)$ (or more generally $GL(2,\mathbb C)$) might feature singularities producing a singular solution from a non-singular seed solution. Although nothing points to these solution being physical not sound, we will abstain from considering any singular dressing factors.}.

\noindent
Consider the following $SL(2,\mathbb C)$ ``vacuum solution''
\begin{align}
g_0=\left(
\begin{array}{cc}
 1 & x_- +x_+ \\
 0  & 1 \\
\end{array}
\right)\,, \quad \text{with} \;\,J_{0,\pm}= \left(
\begin{array}{cc}
 0 & 1 \\
 0 & 0 \\
\end{array}
\right)\,,
\end{align}
where we have denoted the light-cone coordinates by $x_\pm=(t\pm ix)/2$. The extended solution can be readily integrated, yielding
\begin{align}\label{eq:example_psi_vac}
\Psi_0(\xi)=\left(
\begin{array}{cc}
 1 &\frac{x_+}{1-\xi}+ \frac{x_- }{1+\xi} \\
 0 & 1 \\
\end{array}
\right)\,.
\end{align}
To build the dressing factor \eqref{eq:dressing_factor}, we chose two equal vectors $m=n=(1,0)^t$ for which the projector takes the form
\begin{align}
	P= \left(
\begin{array}{cc}
 0 & \frac{x_+ }{1-\mu_0}+\frac{x_- }{1+\mu_0} \\
 0 & 1 \\
\end{array}
\right)\,.
\end{align}
We will take for simplicity $\mu_0=\bar{\lambda}_0=\exp(i\gamma)$ for $\gamma$ an (real) angle. Then the normalisation factor $\chi(\xi)\rightarrow \alpha^{-1}\chi(\xi)$ ensuring that the dressed solution still has unit determinant is
\begin{align}
	\alpha= \sqrt{1+\frac{1-e^{2i\gamma}}{1-ie^{i\gamma}\eta}}\,.
\end{align}
Computing with this information the dressed extended solution \eqref{eq:dressing_PCM} with the dressing function \eqref{eq:dressing_factor} evaluated at the pole of the twist function $\xi=i\eta$ yields
\begin{align}\label{eq:example_psi_dress}
\Psi(\xi)=\left(
\begin{array}{cc}
\alpha^{-1} &\frac{  \left(t-\eta x)(t+i e^{i \gamma } x\right) }{\left(1+\eta ^2\right) \left(1- i\eta e^{i\gamma}\right)}\alpha^{-1} \\
 0 & \alpha \\
\end{array}
\right) = \left(
\begin{array}{cc}
\alpha^{-1} &\alpha^{-1}\beta  \\
 0 & \alpha \\
\end{array}
\right)  \,.
\end{align}
We can now identify the corresponding Yang-Baxter fields $g_\eta, \tilde g_\eta$ and interpret them as two systems separated by a defect. Decomposing the vacuum solution \eqref{eq:example_psi_vac} at $\xi=i\eta$, we get for the Yang-Baxter map of the system left from the defect
\begin{align}
	g_\eta= \rho^{-1} \left(
\begin{array}{cc}
 1+\eta ^2 & (t-\eta  x) \\
 - (t-\eta  x) & 1+\eta ^2 \\
\end{array}
\right)\,,\quad \rho^2=\left(1+\eta ^2\right)^2+ (t-\eta  x)^2\,.
\end{align}
and from \eqref{eq:example_psi_dress}, the Iwasawa decomposition yields the map characterising the system on the right
\begin{align}
	\tilde g_\eta=\frac{1}{\sqrt{1+|\beta|^2}} \left(
\begin{array}{cc}
\left(\alpha/\bar\alpha\right)^{1/2} & \bar \beta \left(\alpha/\bar\alpha\right)^{1/2} \\
 - \beta \left(\alpha/\bar\alpha\right)^{-1/2} &\left(\alpha/\bar\alpha\right)^{-1/2}  \\
\end{array}
\right) \,,
\end{align}
where $d$ has been defined in \eqref{eq:example_psi_dress}.
Finally we can interpret these two Yang-Baxter fields as being divided by a defect captured by  the defect matrix $K=\tilde b_\eta^{-1} \chi(i\eta)b_\eta$  given by
\begin{align}
	K=  \frac{1}{\sqrt{1+|\beta|^2} \rho}\left(
\begin{array}{cc}
\frac{(1+|\beta|^2)(\rho^2+(1+\eta^2)^2\beta)}{(1+\eta^2)\alpha |\alpha|}+\frac{\alpha^4(t-\eta x)\bar \beta}{ \alpha |\alpha|}&\frac{\beta(1+|\beta|^2)}{\alpha |\alpha|}\frac{(1+\eta^2)^2}{t-x\eta}+\frac{\bar \beta \alpha^2}{|\alpha|} (1+\eta^2) \\
(t-x\eta)|\alpha|& (1+\eta^2)|\alpha|\\
\end{array}
\right)\,.
\end{align}

\section{Conclusions and outlook}

In this short letter, we made the first steps towards implementing integrable defects in a class of  integrable deformations known as Yang-Baxter models. We first considered Liouville integrable defects by modifying the monodromy matrix to include the contribution of a defect matrix. We applied this method to the non-ultra local representation of the $S^2$ non-split Yang-Baxter model. A major hurdle for the implementation of Liouville integrable defects is the non-ultra locality of the Yang-Baxter models. This method has as starting assumptions that the monodromy matrices of the systems on both side of the defect satisfy the Sklyanin identity. To apply this method one has to only consider non-ultralocal models or instead rely on the existence of an ultralocal representation, if it even exists\footnote{One could wonder if the non-ultra locality of the family of principal chiral models is a fundamental property with a few ``happy'' coincides for which we can write down an ultralocal representation or rather that it is the product of a bad choice of ``coordinates'' and that all integrable sigma-models are in fact not fundamentally non-ultralocal.}. As such, the possible application of this method are, due to the absence of a general ultralocal formulation of Yang-Baxter models, quite constrained and we have presented the defect matrix for the only Yang-Baxter model that to our knowledge is known to admit an explicit ultralocal representation. Recent developments, seem to suggest that the ultralocality of non-linear sigma-models might be more of technical, rather than a fundamental, obstacle. For symmetric space sigma models for example, the original regularisation procedure of Faddeev and Reshetikhin can be generalised and one can alleviate the non-ultralocality of these models \cite{Delduc:2012qb}. One could also start from the 4d Chern-Simons approach \cite{Costello:2019tri,Delduc:2019whp} to integrable field theories that encodes both ultralocal and non-ultralocal indiscriminately.

Using a different approach to integrable defect, we showed how B\"acklund transformations for the split and non-split Yang-Baxter models can be obtained by using the Hamiltonian perspective on integrable deformation and using as starting point the auxiliary problem of the principal chiral model.  Following the interpretation of a defect as a ``frozen'' B\"acklund transformation of the defect we give the expression for the defect matrix relating two Yang-Baxter fields. As an example we constructed two Yang-Baxter fields for a deformed $SU(2)$ target space that can be separated by an integrable defect and write down the associated  defect matrix.

The B\"acklund approach can be applied to a much wider range of Yang-Baxter models compared to the modified monodromy method  as it does not rely on the systems being ultralocal. Here the challenge would be to generalise the derived B\"acklund transformation to the Yang-Baxter fields on deformed coset spaces, extending their applicability to holographic backgrounds as well as enabling us to construct B\"acklund defects in the $S^2$ non-split Yang-Baxter model. An obvious strategy in that direction would be to try to generalised the dressing method to Yang-Baxter model directly rather than rely on the dressing method of the principal chiral model and a subsequent reduction as discussed in section \ref{sec:baecklund}. The relevance of such a generalisation of the dressing method to Yang-Baxter models is manifold. It would e.g. give us a direct window towards exploring giant magnons \cite{Spradlin:2006wk,Kalousios:2006xy,Appadu:2017xku} in Yang-Baxter deformed target spaces and thus offer a new direction towards understanding the dual gauge theories via the spectrum of the corresponding integrable spin chain. In addition, a dressing method for Yang-Baxter models could provide a way to derive, from first principles, the uniton solutions for the $SU(2)$ non-split Yang-Baxter model found in \cite{Demulder:2016mja}. We plan to report on this particular direction in the future.

\acknowledgments
SD would like to thank the organisers of the conference and school ``Duality, integrability and deformations 2021'' in Santiago de Compostela for providing a stimulating environment where part of this work was carried out. TR thanks the Max-Planck-Institut f\"ur Physik for hospitality during the finalisation of this letter. In addition we would like to thank Ana L. Retore, Giacomo Piccinini and Sibylle Driezen for discussions. We also express our gratitude towards Alessandra Gnecchi, Daniel C. Thompson and  Dieter L\"ust for comments on the draft of this letter.

\bibliographystyle{JHEP}

redpajama_set_name: RedPajamaArXiv


text:
Q: get characters (not separated with spaces) in array using scanf() I need to store characters from user's input in  array, BUT not one by one. User will input them as one line like this;
....

I need to save each dot in array, but I can't do this:
scanf("%s%s%s%s", &s[0], &s[1], &s[2], &s[3])

because user can enter N number of dots. So it must be dynamic, I guess.

A: scanf() is a C runtime function.  In C++, you should be using std::cin instead.  For instance, with std::getline().  You can treat the returned std::string like an array of characters.

A:
User will input them as one line like this; .... I need to save each dot in array,

C solution:
Define upper sane bound like 1000 and use a scanset "%[]".
// Read up to 1000 `.`
char dot[1000 + 1];
if (scanf(" %1000[.]") == 1) {
  //  Success
  puts(dot);
}

Additional code needed if other non-., non -white-space characters need to be handled.

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text:
Отретур () насеље је и општина у северној Француској у региону Горња Нормандија, у департману Приморска Сена која припада префектури Руан.

По подацима из 2011. године у општини је живело 697 становника, а густина насељености је износила 183,42 становника/-{km²}-. Општина се простире на површини од 3,8 -{km²}-. Налази се на средњој надморској висини од 149 метара (максималној 152 -{m}-, а минималној 126 -{m}-).

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Ravindranath, G. G., Department of Surgery, Viswabharthi Medical College, Kurnool, Andhra Pradesh (India)
Ravindranath, G. G., Department of General Surgery, Vishwabharathi Medical College, RT Nagar, Kurnool, Andhra Pradesh, India
Ravindranath, G. G., Department of General Surgery, Vishwabharathi Medical College,R.T. Nagar,Kurnool, India
Ravishankar, Jeyaraj, Department of Transfusion Medicine, Government Villupuram Medical College and Hospital, Villupuram, Tamil Nadu, India (India)
Ravishankar, K. S., Department of Surgery, Sree Balaji Medical College and Hospital, Chennai, Tamil Nadu, India (India)
Ravishankar, N., Department of Surgery, JSS Medical College, Mysuru, Karnataka, India
Ravishankar, Nagaraj, Department of Surgery, JSS Medical College, Mysuru, Karnataka, India
Ravishankar, Nagaraj, Department of General Surgery, JSS Academy of Higher Education and Research, Mysuru, Karnataka, India
Ravishankar, Nagaraj, Department of General Surgery, JSS Hospital, Mysuru, Karnataka, India
Rawal, Gireesha, Department of Pathology, Vardhman Mahavir Medical College and Safdarjung Hospital, New Delhi, India
Rawal, Gireesha, Department of Pathology, Vardhman Mahavir Medical College and Safdarjung Hospital, New Delhi, India (India)
Rawal, Krutik, Department of Urology, Ruby Hall Clinic, Pune, Maharashtra, India
Rawat, Anshu, Department of Surgery, Dr. D.Y. Patil Vidyapeeth University, Pimpri, Pune, Maharashtra, India (India)
Rawat, Anshu, Department of General Surgery, Dr D.Y. Patil Medical College, Hospital and Research Centre, Dr. D. Y. Patil Vidyapeeth, Pimpri, Pune, Maharashtra, India (India)
Rawat, J. D., Department of Paediatrics Surgery, Kings George's Medical University (KGMU), Lucknow, Uttar Pradesh, India
Rawat, J. D., Department of Pediatric Surgery, KGMU, Lucknow, Uttar Pradesh, India
Rawat, Jiledar, Department of Pediatric Surgery, King George's Medical University, Lucknow-226003, India (India)
Rawat, Jiledar, Department of Pediatric Surgery, King George's Medical University, Lucknow, Uttar Pradesh, India
Rawat, Kinnari A. V., Department of Plastic Surgery, Himalayan Institute of Medical Sciences, SRHU, Dehradun, Uttarakhand, India
Rawat, Kinnari A. V., Department of Department of Plastic Surgery, Himalayan Institute of Medical Sciences, SRHU, Dehradun, Uttarakhand, India
Ray, Aniket, Department of General Surgery, DY Patil School of Medicine, Navi Mumbai, Maharashtra, India
Ray, Jitendra Prasad, Department of Surgery, Himalayan Institute of Medical Sciences, Dehradun, Uttarakhand, India (India)
Ray, Jitendra Prasad, Department of Surgery, SRH University, Jollygrant, Dehradun, India
Ray, Jitendra P., Department of General Surgery, Himalayan Institute of Medical Sciences, SRHU, Dehradun, Uttarakhand, India
Ray, M. S., Department of Surgery, Military Hospital, Jalandhar Cantt, Jalandhar, Punjab 144005, India


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Home » POLTERGEIST Review
POLTERGEIST Review
vetswebHorror, Thriller May 22, 2015
Director: Gil Kenan
Writer: David Lindsay-Abaire (screenplay), Steven Spielberg (story)
Cast: Sam Rockwell, Rosemarie DeWitt, Saxon Sharbino, Kyle Catlett, Kennedi Clements, Jared Harris, Jane Adams, Susan Heyward, Nicholas Braun, Karen Ivany, Patrick Garrow, Doug MacLeod, Eve Crawford, L.A. Lopes, Soma Bhatia, John Stoneham Jr. Kathryn Greenwood, Molly Kidder
MPAA Rating: PG-13
Production Company: Metro-Goldwyn-Mayer (MGM), Ghost House Pictures, Vertigo Entertainment, TSG Entertainment, Fox 2000 Pictures, 20th Century Fox
Budget: N/A
While many loathed the idea of a Poltergeist remake, some greeted the news with cheers. The ones cheering most likely are from the younger demographic, and after watching the film, I believe that's definitely the film's target audience. Not having seen the original, I can't compare the two, but I am now very curious about how this title received its classic horror movie status back in 1982. This status raises so many questions as to how the original caused so many nightmares, and yet the remake only brought forth laughs.
The Poltergeist remake is entertaining to say the least, but it does not remotely approach being scary. It doesn't even succeed with cheap jump scares, and fails to bring forth any sense of danger. It's a light-hearted comedy that wastes its few shocking moments on the trailers, then dishes you a childish rendition of ghouls and ghosts in the actual film.
Eric Bowmen (Sam Rockwell) takes his wonderful and loving family to a new home in the suburbs, and this is most likely the biggest mistake he's ever made. When we meet his family, we sense the love and warmth between them, yet the emotions feel a little misplaced. Although Eric is the father, he sometimes comes across as a fun uncle or an older brother rather than the man of the house. Still, you feel the passion between him and his lovely wife Amy (Rosemarie DeWitt), and their children Griffin (Kyle Catlett) and Kendra (Saxon Sharbino) are hilarious, soaking up everything in their surroundings like a sponge.
Griffin is my favorite character of the bunch. He does exactly what you would want him to in this type of film, and you root for him throughout his adventure. He's curious and smart and also contains the latest drone technology the market has to offer. Kendra, his younger sister, is the most adorable thing around. She's very different too, seeing that when the closet speaks to her she takes it as normal instead of being frightened. It's hard not to love the family, even though the relationships between them seem peculiar at times; they do the best with the cards they're dealt, and that's all you can ask for.
The film, I repeat, is quite the opposite of frightening. I found myself and the rest of the auditorium laughing the whole time. I'm not sure if this is the goal of the movie, but if it is, it succeeds with a high rank. Every encounter the family faced and reacted to brought tears of joy.
No tension is developed, and nothing is at stake to add weight to the story. There's a sense that everything is going to be ok, and that's not a good feeling to produce in your audience when you're trying to convey that the family's in grave danger. With expectations already being low, my mood wasn't crushed due the tone of the film not living up to what the trailers promised. It was even quite refreshing to receive something I hadn't expected.
On a positive note, the film doesn't waste any time getting to the magic and mayhem. When the special effects hit, they are cranked up all the way and don't let up until the credits roll. This is also where all the fun is. From what I've heard about the original film, the remake hits all the key points of its predecessor, but it feels like a check list: with no stakes involved, many events just seem to be happening because they happen in the original. It's still fun to witness, but of course you can't take the film seriously.
There's always that dumb character in scary movies who deserves to die. Why this is so is beyond my understanding. I like watching characters that actually move the story along, but here they slow it down. Just as you can't expect the movie to be scary, you can't expect to take the characters seriously either.
Carrigan Burke (Jared Harris) is one such character and it sucks, because I appreciate his work as an actor. He's supposedly been a real professional in dealing with the Poltergeist, knowing all the ins and outs. It's as though such things happen to him every weekend, which we know isn't the point. He conveniently has a free schedule and decides to visit the home when he's across the nation. In addition to his stupidity, his grand plan to rescue Kendra from captivity involves Griffin's drone bot flying off and saving the day. This alone knocks the film down a few notches, seeing that if this device didn't exist, Carrigan's plan would've fallen through. So there you have it, folks. If you have a Poltergeist in your home, make sure you have a flying drone to help you save the day.
So if you're a horror junkie and want the next scary movie to add to your list, the 2015 remake of Poltergeist is definitely not it. In all honesty, this doesn't mean the film is horrible. It just matters what you're looking for when going in. It's definitely very entertaining, and you'll laugh from the beginning to the end. Take that information going in and you might have a good time. If this movie was marketed as a spoof or a comedy of some sort, it would nearly be a masterpiece. You won't need any overnight guest sleeping at your place after watching the movie, and getting through the night with the lights off will be easy as it can be.
My Rating : 4 / 10
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Table No. 2 Restaurant to open on the Avenue of Fashion Easter Sunday
Melody Baetens
The Avenue of Fashion is getting a new upscale restaurant this month with the debut of Table No. 2.
It's the latest venture from chef Omar Mitchell and his twin brother Shimar Mitchell. They will host the first service, a buffet brunch, on Easter Sunday, April 21. The all-you-can-eat meal will be served noon-4 p.m.
Table No. 2 will serve a brunch-style buffet every Sunday going forward for $32.95.
Once open for regular service, the menu will feature seafood like Lake Michigan walleye, crab legs, lobster, salmon, seafood pasta Alfredo, grilled oysters Rockefeller and sesame-crusted Ahi tuna. The fine dining spot will also serve a shareable seafood tower and jumbo crab cakes.
Beyond seafood, guests will find a tableside Caesar salad, steaks, short ribs, rack of lamb and southern-fried chicken.
Chef Mitchell, who was on an episode of Food Network's popular "Chopped" show in 2017, also ran the Great Lakes Burger Bar in Detroit and I Luv Cheese in Clinton Township.
Table No. 2 Restaurant is at 18925 Livernois in Detroit. Call (313) 340-9550 or visit tablenumber2.com to make reservations for April 21 or later.
mbaetens@detroitnews.com
Twitter: @melodybaetens


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Denmark / Philippines
Philippine Village Thankful for Danish Water
April 29, 2007 April 29, 2007 - by Marie Varming - Leave a Comment
To many Filipinos Denmark is a distant country, but to the villagers in Cadiz, the water running from their tabs is Danish. Entering the village, the first thing you see is a big Danish flag painted on the main water tank of the village.
In 1996, a Danida project was initiated in five Philippine villages.
"The aim was to improve the health of the populations by introducing sanitation and potable water to every household", says engineer and customer service assistant at Cadiz Water Department, Winston Makilan. The project costs were approximately 65 million Danish kroner and the main part; 61 million kroner were sponsored by Danida. The rest of the means came from local authorities. The consumers only pays for their actual consumption.
"The project is especially for the poor", Winston Makilan explains.
"They have not paid for any of the construction of this non-profit project", he says.
Consumer participation
The project was unusual in one particular way: It was demand driven.
"That means that all the consumers in the villages had to participate in the foundation of the cisterns, latrines and faucets that were built", Winston Makilan explains.
According to en evaluation report on the project that meant that the projects was somewhat delayed due to the Philippine 'wait and
see-attitude'.
In one village, La Castellana, the project eventually had to be stopped because of 'lack og support from the local government', as it says in the evaluation report. All the other four villages – all located in the island of Negros in the central Philippines – the project was a success for the 130,000 people who were affected by the project.
We love Denmark
In the office of the local water authorities Danish band Michael Learns to Rock is flowing from the speakers.
"We love Denmark", Winston Makilan laughs with reference both to the music and the local water works.
Danida is no longer active in The Philippines and the project has not seen a Dane since it was finished in year 2000, but the villagers still remember Denmark as the country that gave them clean water and sanitation.
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I can not believe it's been over a year since I last wrote a post! So much has happened and so much has changed in my life in the past year. I have definitely been pretty busy. My boys are now 4 and 2.5, I can not believe how fast time has flown!
I promise I have had good reasons for my absence and that, starting now, I will be posting and updating this blog as often as I am able to. In my next few posts, I will be filling you all in on what's going on in my life right now. I also plan to start doing reviews and giveaways again as well as writing more informational posts. Stay tuned for what's to come on Diary of a Natural Mom and, if you haven't yet liked my Facebook page, please do so by clicking here. I look forward to interacting with all of you again and getting back to what this blog was originally intended to be!