I feel honored and humbled to join Dilla University leadership as the community celebrates the 20th Anniversary of DU. The anniversary presents an ideal opportunity to reflect on the achievements and challenges of our young university as well as to chart out a brighter future for DU. Since its founding in 1996, DU has registered significant growth in terms of size of student enrollment and staff size facilitating provision of instruction in ever increasing fields of study. The University's strategic plan envisages further growth in DU's size and stature with the ultimate goal of enhancing the University's overall performance in education and research as well as its national and regional standing.
With 34 universities up and running in various regions of the country and 11 more on the pipeline, higher education in Ethiopia has been undergoing a massive expansion process over the past two decades. While the expansion process has significantly enhanced access to education to the country's youthful population, Ethiopian universities have yet to effectively demonstrate their contribution to building a knowledge-based economy as well as the entrenchment of democratic values in Ethiopia. Indeed, Ethiopia's universities, which currently owe their existence to enormous public funding, require strategic vision and commitment to excellence to justify their social relevance and continued existence into the future.
Thus, the main challenge to DU's leadership and our community will be to ensure the relevance of DU and enhance its contributions to societal goals. To this end, the University needs to place a high premium on ensuring quality in the teaching-learning process as well as in research and community service. In the context of DU, ensuring quality requires upgrading the overall staff profile by enhancing the capacity of the University to attract, develop and retain qualified academic staff. In this connection, adequate attention will be paid to graduate expansion with a view to gradually transforming what is currently an essentially undergraduate institution into a research-oriented graduate University. Further, DU needs to continually embark on reform of curricula in order to provide relevant education as well as to foster the establishment of multidisciplinary centers of excellence in key areas central to the attainment of the mission of the University. The overall goal is to facilitate the fulfillment of DU's ambitious vision of becoming one of the top rated universities in eastern Africa. Our commitment to excellence is crucial to realize the ambition!
I wish you a happy 20th Anniversary!


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text:
Incunabula and the Keio University Library Collection
front pde
Vincentius Bellovacensis [Vincent de Beauvais], Speculum historiale ([Strassburg]: [The R-Printer (Adolf Rusch)], [1476-80]; also recorded as [c. 1473])
Speculum historiale
The 13th-century Dominican friar, Vincent de Beauvais, compiled a tripartite encyclopaedia entitled the Speculum maius at the request of Louis IX, the French king. It consists of the Speculum naturale, the Speculum doctrinale and the Speculum historiale (cf. IKUL 007, 009). Later, another compiler added the Speculum morale, drawing mainly on the Summa theologiae of Thomas Aquinas (cf. IKUL 008). Thus the Speculum maius came to comprise four parts by the advent of printing; the Speculum historiale was the most popular of all the four. The Keio copy is a voluminous folio edition weighing 16. 8 kg. It consists of four volumes, containing 155, 176, 175, and 191 folios respectively. All four volumes are bound together in 15th-century German calf on wooden boards. It includes the preface to the whole Speculum maius as well as the text of the Speculum historiale and a detailed list of its contents.
It has frequently been observed that, in incunabula, only the main text is printed with type, whereas the rubrication, such as the large initials, is added later by hand. In the Keio copy, the spaces for the initials have been left blank, and this shows the state of the book immediately after printing. It is presumed that it was left incomplete because of the great cost of inserting all the initials in this voluminous book by hand. An early owner did supply some by hand, in black ink, in imitation of calligraphic initials, but most of the spaces remain blank. There are occasional marginalia throughout the volume, particularly in the florilegium of passages from Ovid; in this section, one missing line is supplied, and a citation from the Remedia amoris is inscribed by hand in the margin. Moreover, an account of the battle in 1581 between Poland, under the rule of King Stefan Batoriy (reigned 1576-86), and the Grand Duchy of Moscow is added, along with a reference to Ovid in the margin of another page. There are two autographs of Germans on other pages, and one can therefore conjecture that the Keio copy was formerly owned in Germany or Eastern Europe.
(TM; trans. by YO)
The Speculum historiale is extant in various manuscripts, and provided material for numerous Latin and vernacular texts in late-medieval Europe. England was no exception. The Speculum historiale is primarily a treasury of exempla for preachers' sourcebooks. Various collections of anecdotes, such as the Gesta Romanorum (IKUL005) – a Latin collection written around the end of the 13th or the beginning of the 14th century and circulated around Europe – depended on the Speculum maius, and especially the Historiale. The Speculum historiale also influenced writers in late-medieval England: it is the source for 112 exempla in John Gower's Confessio amantis, for example (Welter, p. 208). Aiken claims that Geoffrey Chaucer owed his knowledge of medicine and alchemy, as demonstrated in the Canterbury Tales, to the Speculum maius. A number of anecdotes about historical or legendary figures in the Legend of Good Women and the Canterbury Tales were probably derived from the Historiale as well. The Historiale was also an important sourcebook for English and Scottish chronicles. Book IV of the Historiale is entirely devoted to Alexander the Great, and is the longest medieval chronicle written about him. The Polychronicon, by Ranulph Higden, and Mandeville's Travels both turn to the Historiale in recounting the episode of Alexander's visit to the speaking oracular trees named 'the trees of the sun and the moon'. As an anthology of medieval Alexandrian materials, Book IV of the Historiale also stands at the beginning of an important tradition. It is the first medieval text that condemns Alexander rather severely for the murder of the philosopher Callisthenes, who advised Alexander against adopting the oriental custom of monarch-worship. The Historiale compiles several isolated anecdotes into a fictitious historical narrative, and as a result the murder of Callisthenes could be interpreted as proof of the cruelty of Alexander, who grew vainglorious after visiting the oracles of Ammon, where he was told that he was the son of the god (Cary, pp. 114-15). The long account in the Historiale shaped the morally culpable figure of the medieval Alexander. However, in another chapter a seriously wounded Alexander is shown denying that he belongs to the divine race. This morally positive episode was entirely forgotten by medieval moralists until Vincent cited it (Cary, pp. 152-53). The influence of the Historiale has stretched not only to literature but also to the world of art: the wall paintings at Eton College and decorations in Winchester Cathedral contain motifs derived from it (Gill, pp. 4-5).
[Bibliography]
Aiken, Pauline, 'Vincent of Beauvais and Dame Pertelote's Knowledge of Medicine', Speculum, 10 (1935), 281-87
---, 'Vincent of Beauvais and Chaucer's Monk's Tale', Speculum, 17 (1942), 56-68
---, 'Vincent of Beauvais and Chaucer's Knowledge of Alchemy', Studies in Philology, 41 (1944), 371-89
Cary, George, The Medieval Alexander, ed. by D. J. A. Ross (1956; repr. Cambridge: Cambridge University Press, 1967)
Gill, Miriam, 'The Speculum historiale as a Source for Art: The Eton College Wall Paintings', Vincent of Beauvais Newsletter, 24 (1999), 4-8
Welter, J.-Th., L''exemplum' dans la littérature religieuse et didactique du Moyen Age, Bibliothèque de la société d'histoire ecclésiastique de la France (1927; repr. New York: AMS Press, 1973)
Wimsatt, W. K., Jr, 'Vincent of Beauvais and Chaucer's Cleopatra and Croesus', Speculum, 12 (1937), 375-81
(YO)
Vincentius Bellovacensis [Vincent de Beauvais]
[Strassburg]
[The R-Printer (Adolf Rusch)]
fº
[1476-80]; [c. 1473]
Contemporary German calf on wooden boards, rebacked.
Bibliographical Notes
Four parts bound in one; 155 leaves (of 156), lacking 1 blank at the beginning, 176 leaves, 175 leaves, 191 leaves (of 192), wanting 1 blank at the end; a stub remains between the 58th and 59th leaves (conjugate with the 54th leaf).
ISTC
iv00282000
Goff V282, HC 6245, IJL2 376, MB 15
143X@13@1
Acquisition Year
Its probable East European or German provenance can be seen in marginalia. On vol. I, fol.129, there is an early MS marginalia referring to the 1581 campaign of Stefan Batoriy, Polish King, against the Muscovites and connecting the event to passages from Ovid. On the lower margin of part 2, fol. 29v is written 'Joannes Leonardi Pfalizell' with the mention of the year 1599. On part 1, fol. 2r, a German ownership inscription dated 1821 can be seen, along with a signature by another owner erased by pen. On the front paste-down, there is a bookplate of Paul Helbronner (with the motto 'Perseverantia'), the famous Alpine cartographer at the beginning of the twentieth century. Maîtres Laurin – Guilloux – Buffetaud, Paris, 27 October 1997 sales catalogue. (Cf. Mostly British, p. 99).

redpajama_set_name: RedPajamaCommonCrawl


After reading Phil's thread on bots etc I have to say, how scary does it sound that Phil is saying that bots cannot be stopped and the tone was more of a trying to minimise the problem - it's like there is an acceptance of them and trying to minimise them rather than eradicating them is all that can be done.
This sort of honesty is refreshing and not something you ever hear Stars, Party etc admit to, but I think to the lay poker player and general casual player they won't take lightly to joining a site that admits it has bots but just try to minimise them.
After watching a recent pod cast by Joey Ingram (9mins 6 secs) he shows graphs were the bots on ACR has almost identical stats. They are winning at a scary rate. Their owner seemed to take a very flippant approach to the subject, not to mentioned the huge issue they faced on the MAC client.
Knowing they are rife on Party. ACR and Stars does not bode well.
I really think a great move forward to this is to have the RIO poker program interact on a regular basis with the user where the player has to click on things like "click the number 5 below" every x minutes - this is just another trivial click for the user and also lets RIO know the user is a real player. Who on earth would ever object to that? We can all give up a 1/20th of s sec for the sake of the integrity of the site.
The posting of the hands played after session is just a brilliant idea by they way. I think this will also show up any attempts where players are are colluding and teaming up in 3 and 4 bets in front of their targets.
I have to say I am really looking forward to the launch - I like many others play on Sky Poker in the UK as it is very secure, does not have bots to my knowledge, doesn't offer their data to 3rd party data mining site and so none of the PT4 and Holdem programs work on it - great deterrent.
The one thing they did brilliantly when they launches was they had their own live channel and used to review hands from players during cash or tourney play - the players would write in with hand ID's and the experts would review them. If you got your hand reviewed you felt great. They would also give updates live on the tourney running and show leaders boards etc. All great and made you feel part of it.
I have to say I am only joining RIO because Phil has his name to it - My human instincts tell me he is a stand up guy and is trying to take online poker to the next level - the ideas seen so far are very encouraging - I just hope he gets the support he deserves.


EDMONTON, Alberta (AP) Tinkering with their top line paid dividends for the Edmonton Oilers.
Reunited as linemates, Leon Draisaitl had a goal and two assists and Connor McDavid added three assists to lead Edmonton past the Montreal Canadiens 6-2 on Tuesday night.
Matt Benning, Drake Caggiula, Ryan Nugent-Hopkins, Darnell Nurse and Kris Russell also scored for the Oilers, who snapped a four-game skid that marked their longest of the season.
"Leon probably could have won the scoring race tonight alone had he found a way to finish on some of his opportunities," Edmonton coach Todd McLellan quipped. "Connor, Leon, Drake - a real good night there. They shared the puck well."
Mikko Koskinen made 27 saves as the Oilers inched back over .500 at 9-8-1.
"It's big," Russell said. "We have a few days between games and now the mood's going to be a lot better."
Caggiula thought McDavid in particular looked as though he had had more than enough of the losing streak.
"We've been on a bit of a skid here and he said, `No, I'm going to take the game into my hands today,'" Caggiula said. "There is a few times when you say, `Jeez, is he really doing that in an NHL game right now?'"
Max Domi and Andrew Shaw had the goals for the Canadiens (9-6-3), who have lost three of four.
"It was called a no-show on our part," coach Claude Julien said. "We are a fast-skating team so it wasn't about speed, but their team looked pretty fast when you give them the room that we did tonight. We didn't show up. It was a tough night, by far our worst game of the year."
Edmonton started the scoring on a fluke goal three minutes in as Benning's point shot was off the mark but came off the boards and hit Canadiens goalie Antti Niemi from behind and caromed in.
Montreal drew even a couple of minutes later just after killing off a penalty when Domi came in with a head of stream and picked the top corner to beat Koskinen for his 10th of the season.
Edmonton regained the advantage with 3 1/2 minutes left in the opening period as McDavid shrugged off a defender and fed the puck to Draisaitl, who scored his 11th goal.
The Canadiens tied it again just more than a minute into the second when Jonathan Drouin passed from behind the net to Shaw, who trickled a shot past Koskinen.
Caggiula fished a puck out of the air and scored for Edmonton two minutes later on a knuckleball of a shot.
Nugent-Hopkins scored on a deceptive wrister 5:36 into the third to make it 4-2.
Edmonton added to the lead on a long shot by Russell with eight minutes left and got its third goal from the blue line with two minutes to go when Nurse scored. That matched the number of goals the Oilers had from defensemen in the entire season coming into the game.
Niemi stopped 37 of Edmonton's 43 shots.
NOTES: It was the first of two meetings between the teams this season. Edmonton won both matchups last season. ... Draisaitl's goal was his seventh in the last 10 games. ... Canadiens captain Shea Weber (knee) skated for the first time Monday, but didn't take part in contact drills.
Canadiens: At Calgary on Thursday.
Oilers: At Calgary on Saturday.


text:
\section{Introduction}\label{1}
It has been found different classes of non-linear electrodynamic (NED) theories as candidates to the extension of Maxwell electrodynamics in the literature \cite{Plebanski,Boillat,Birula1,Birula2}. Also, one can find some recent developments in this area of research in Refs. \cite{Aschieri:2008ns,Dehghani:2021fwb,Sorokin:2021tge,Paixao:2205jsw}. Recently, it has been discovered a nonlinear extension of Maxwell's theory, which  not only is invariant under electromagnetic duality transformations, but also is a conformally invariant theory. This electromagnetic modification is known as the ModMax theory \cite{Bandos:2020jsw} and is described by the following Lagrangian density
\begin{eqnarray}\label{LMM}
{\cal L}_{MM}=\cosh(\gamma) \mathcal{S}+\sinh(\gamma) \sqrt{ \mathcal{S}^2+\mathcal{P}^2},
\end{eqnarray}
where $ \mathcal{S}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ and $\,\,\mathcal{P}=-\frac{1}{4}F_{\mu\nu}\tilde F^{\mu\nu}$ are two Lorentz invariant variables, which $\tilde{F}^{\mu \nu}= \frac{1}{2} \varepsilon^{\mu \nu\alpha \beta} F_{\alpha \beta}$ is the Hodge dual of the electromagnetic field strength $F_{\mu \nu}$. As is obvious from (1.1), the ModMax theory comprises a positive constant $\gamma$ so that for $\gamma=0$, it reduces to the Maxwell Lagrangian \cite{Kosyakov:2020wxv}. The generalization of the ModMax theory in the context of conformal invariant p-form gauge theories was studied in Ref. \cite{Bandos:2020hgy}.

It has been shown \cite{Bandos:2020jsw,Bandos:2020hgy} that the ModMax and the Born-Infeld theories can be combined into a duality-symmetric generalized Born-Infeld (GBI) electrodynamics as follows
\begin{equation}\label{GBI}
{\cal L}_{BI \gamma} = \frac{1}{\lambda} \Bigg[ 1 -  \sqrt{1 -  \lambda \Bigl( 2 {\cal L}_{MM}+\lambda \mathcal{P}^2 \Bigr)} \Bigg],
\end{equation}
where in the weak field limit $\lambda\to 0$, the Lagrangian \reef{GBI} reduces to the ModMax theory denoted by \reef{LMM}. More studies on the ModMax theory, such as the supersymmetric actions, the nonlinear generalizations, and solutions of field equations can be found in Refs.~\cite{Bandos:2021rqy,Kuzenko:2021cvx,Avetisyan:2021heg,Kruglov:2021bhs,BallonBordo:2020jtw,Nastase:2021uvc,Mkrtchyan:2205uvc,Barrientos:2022bzm}.  Also, an alternative form of the ModMax Lagrangian  in terms of axion-dilaton-like auxiliary scalar fields coupling has been proposed in Ref. \cite{Lechner:2022qhb}. ModMax theory has been more carefully studied and expanded in recent works \cite{Barrientos:2022bzm,Bokulic:2022cyk,Banerjee:202207bzm,Pantig20220bzm,Neves20220bzm}.

As well known, the Maxwell's field equations are invariant under the electric-magnetic $SO(2)$ duality transformations, however, the enhancement of this symmetry to the level of an action is another problematic task. This fact becomes even more complicated for theories featuring nonlinear interactions of the Maxwell fields. There are three approaches to construct theories of self-interacting Maxwell fields incorporating an electromagnetic $SO(2)$ duality invariance; the Gaillard and Zumino \cite{Gaillard:1981rj, Gaillard:1997rt} approach which has been further developed by Gibbons and Rasheed \cite{Gibbons:1995ap}, the non-covariant first order Hamiltonian approach based on the work by Henneaux and Teitelboim \cite{HT} and further followed by Deser, Gomberoff, Henneaux and Teitelboim \cite{Deser:1997mz,Deser:1997se}, and the third one is the PST method proposed by Pasti, Sorokin and Tonin \cite{Pasti:1995tn,Pasti:1995us,Pasti:1996vs,Pasti:1997gx}.

In the first systematic method to construct duality invariant theories, we have to define a doublet $\left(\tilde{F},G\right)$,  where $G_{\mu\nu}=-2\, \frac{\partial { \cal L} (\mathcal{S},\mathcal{P})}{\partial F^{\mu\nu}}$ is an antisymmetric tensor, to show that the equations of motion are invariant under $SO(2)$ rotations. This requirement imposes a non-linear differential equation; $G\tilde{G}-F\tilde{F}=0$, the so-called self-dual condition. In the second approach, $SO(2)$ duality is an exact and explicit symmetry of the Lagrangian of the theory. Such a NED theory has a doublet of electric and magnetic fields $(A_1,A_2)$ and the $SO(2)$ duality is an ordinary 2d rotation on these doublets. Finally in the PST approach, if one introduces an auxiliary scalar field and a compensating gauge symmetry, it is possible to formulate a manifestly covariant action with the correct degrees of freedom.

We will follow the Gaillard-Zumino approach and present the self-dual invariant actions for the ModMax and GBI (and axion-dilaton-GBI) theories which gives the correct equations of motion when one imposes the duality invariant condition as an extra constraint. As the main purpose of our study in this letter, we find a manifestly $SL(2,R)$ invariant structure for the energy-momentum tensors of the ModMax and also the GBI theories. In this respect, although the corresponding Lagrangians are not invariant under $SO(2)$ duality-symmetry, their derivatives with respect to the invariant parameters are. The invariant parameter could be a coupling constant or an external background field, such as the gravitational field, which does not change under duality rotations\cite{Gaillard:1981rj}.
In this approach if a theory satisfies the self-dual condition, then the physical objects of the theory such as equations of motion\cite{Green:1996qg}, energy-momentum tensor\cite{Gibbons:1995ap,BabaeiVelni:2016qea} and scattering amplitudes\cite{Babaei-Aghbolagh:2013hia,Garousi:2017fbe,BabaeiVelni:2019ptj} are (S-dual) invariant. We are also interested in finding corresponding self-dual invariant theory in the form that are manifestly $SL(2,R)$ invariant.
The self-dual invariant Lagrangian is given by the derivative of the original Lagrangian with respect to the constant parameters $\gamma$ and $\lambda$
 \cite{Aschieri:2013nda}.

The structure of this paper is organized as follows: in Sec. \ref{2}, we enhance $SO(2)$ duality in the ModMax  as well as the GBI theories to the non-compact $SL(2, R)$  group by coupling the electromagnetic field to the dilaton and axion fields. In Sec. \ref{secLinv}, we find the manifestly self-dual invariant actions for a general non-linear GBI theories of electrodynamics. We show that  the expansion of a self-dual invariant Lagrangian  in NED theories  is compatible with  the deformed Lagrangian constructed by both irrelevant and marginal  $T \bar{T}$ operators in NED theories. Finally, the Sec.~\ref{4} is devoted to giving a brief summary of results and identifies some directions for future researches.
\section{$SL(2,R)$ symmetry in axion-dilaton ModMax Theory}\label{2}

The equations of motion of free Maxwell theory
are $\partial_\nu F^{\mu \nu} =0$  and $\partial_\nu \tilde{F}^{\mu \nu} =0$.
These equations  obviously  are transformed into each other under the rotation of $ (F^{\mu \nu},\tilde{F}^{\mu \nu}) \to (\tilde{F}^{\mu \nu},-F^{\mu \nu})$. This duality, which is known as the electric-magnetic  duality, is a special case of the $SO(2)$ duality-symmetry. In this section we are going to study the $SO(2)$ symmetry  for the  energy-momentum tensor of the ModMax theory and its extension to $SL(2,R)$ symmetry  by coupling the electromagnetic field to some axion and dilaton fields. Consequently, we find the $SL(2,R)$ duality-symmetric structures for the energy-momentum tensor of the ModMax and  GBI theories.
\subsection{Electric-magnetic duality in the ModMax theory }
According to \cite{Gibbons:1995cv}, a general electric-magnetic duality rotation through an angle $\alpha$ is given by
\begin{equation}
\left\{
\begin{array}{rcl}
G_{\mu\nu} & \!\!\!\rightarrow \!\!\! & \cos\alpha ~G_{\mu\nu} + \sin\alpha ~\tilde{F} \\
& & \\
\tilde{F}_{\mu\nu} & \!\!\! \rightarrow \!\!\! & \cos\alpha ~\tilde{F}_{\mu\nu}-\sin\alpha ~
G_{\mu\nu}.
\end{array}
\right.
\label{Rot}
\end{equation}
This rotational symmetry is called the $SO(2)$-duality symmetry. Assume a finite $SO(2)$-duality rotation of the angle $\pi/2$, then we have the following relations for the fields and invariant parameters
\begin{equation}\label{GF}
(G^{\mu \nu},\tilde{F}^{\mu \nu}) \to (\tilde{F}^{\mu \nu},-G^{\mu \nu})\,,
\,\,\,\,\,\,\,\,\,\,\,(g_{\mu \nu},\gamma,\lambda) \to (g_{\mu \nu},\gamma,\lambda)\,.
\end{equation}
 One can find the $G$-tensor for the ModMax theory \reef{LMM} as follows
\begin{equation} \label{G}
 G_{\mu\nu}=-2\, \frac{\partial {\cal L}_{MM}}{\partial F^{\mu\nu}}=\cosh(\gamma) F_{\mu\nu} +\sinh(\gamma) \frac{ F_{\mu\nu}\mathcal{S}+ \tilde{F}_{\mu\nu} \mathcal{P}}{ \sqrt{  \mathcal{S}^2+\mathcal{P}^2 } }.
\end{equation}

It has been found in Ref. \cite{Gibbons:1995ap} that the energy-momentum tensor of a NED theory can be written as
\begin{equation}
\label{EMT1}
T_{\mu\nu}=g_{\mu\nu}\, {\cal L} (\mathcal{S},\mathcal{P})+{F_{\mu}}^ \rho G_{\nu \rho},
\end{equation}
therefore, by considering the above relation, the energy-momentum tensor of the ModMax theory \reef{LMM} can be found as
\begin{eqnarray}\label{Tmunu}
T_{\mu\nu}= T^{Max}_{\mu\nu} \bigg(\cosh(\gamma) + \frac{ \mathcal{S}}{ \sqrt{ \mathcal{S}^2+\mathcal{P}^2}} \sinh(\gamma)\bigg),
\end{eqnarray}
where $T^{Max}_{\mu\nu}=F_{\mu \rho}{F_{\nu}}^ \rho+ g_{\mu\nu} \mathcal{S}$ is the energy-momentum tensor of the Maxwell theory.
Now, in order to study the behavior of the energy-momentum tensor \reef{Tmunu} under the nonlinear duality transformations \reef{GF}, we define the following symmetric structure that is invariant under these transformations
\begin{eqnarray}\label{Nmunu}
{\cal N}_{\mu\nu}=G_{\mu}{}^{\alpha} G_{\alpha \nu}+\tilde{F}_{\mu}{}^{\alpha} \tilde{F}_{\alpha\nu}.
\end{eqnarray}

Also substituting the ModMax $G$-tensor \reef{G} in Eq.~\reef{Nmunu}, one can find the symmetric invariant structure corresponding to the ModMax theory
\begin{eqnarray}\label{NN}
{\cal N}_{\mu\nu} &=& -2 \cosh(\gamma) \Bigl(\cosh(\gamma) + \frac{\mathcal{S}}{ \sqrt{ \mathcal{S}^2+\mathcal{P}^2}} \sinh(\gamma)\Bigr) T^{Max}_{\mu\nu}\nonumber\\
&& + 2 \sinh(\gamma) \Bigl(\cosh(\gamma)  \sqrt{ \mathcal{S}^2+\mathcal{P}^2} +\sinh(\gamma) \mathcal{S}\Bigr) g_{\mu\nu}.
\end{eqnarray}
Considering the energy-momentum tensor \reef{Tmunu} and the invariant structure \reef{NN}, we obtain a manifestly invariant $T_{\mu\nu}$ of the form
\begin{equation}\label{manifest}
T_{\mu\nu}=- \frac{1}{2 \cosh( \gamma)} \Big[ {\cal N}_{\mu\nu} -\frac{1}{4}  {\cal N}_{\rho}{}^{\rho}  { g}_{\mu\nu} \Big],
\end{equation}
where $ {\cal N}_{\rho}{}^{\rho}=8\sinh(\gamma) \Bigl(\cosh(\gamma)  \sqrt{ \mathcal{S}^2+\mathcal{P}^2} + \sinh(\gamma) \mathcal{S}\Bigr)$ is the trace of symmetric structure ${\cal N}_{\mu\nu}$. Note also that ${\cal N}_{\rho}{}^{\rho}$ vanishes at $\gamma=0$ and we achieve the standard Maxwell energy-momentum tensor.
\subsection{ModMax theory  coupled to an axion-dilaton field }\label{MMAD}
Consider the following nonlinear $SL(2,R)$ transformation
\begin{eqnarray}
\label{sl2r}
\tau \rightarrow \frac{p\tau+q}{r\tau+s},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Lambda=\left(
\begin{array}{cc}
p&q  \\
r&s
\end{array}
\right)\in\, {SL(2,R)},
\end{eqnarray}
where  $ \tau = C_0 + i e ^ {- \phi_0}$ is a complex axion-dilaton field.
Now, for a Lagrangian in which the general NED theory is coupled with the axion field so that the equations of motion are $SL(2,R)$ invariant, one should  add  the term $\frac{1}{4} C_0  F_{\mu \nu} \tilde{F}^{\mu \nu}$ to the primary NED Lagrangian. On the other hand, the contribution of dilatonic field is given by the  field redefinition $\bar{F}_{\mu \nu} \to  e^{-\frac{\phi_0}{2}}F_{\mu \nu}$ \cite{Gaillard:1997rt}. Thus, the final theory appears an axion-dilatonic NED theory denoted by Lagrangian density $\hat{{\cal L}}(\tau , F )=\bar{{\cal L}}(e^{-\frac{\phi_0}{2}}F)+\frac{1}{4} C_0  F_{\mu \nu} \tilde{F}^{\mu \nu}$. In this regard, the $SO(2)$ duality group in Eq. \reef{GF} enhances  to the $SL(2,R)$ symmetry group.  Due to this consideration, one can find the ModMax theory coupled to axion-dilaton field as
\begin{eqnarray}\label{hatLMM}
\hat{{\cal L}}_{MM}(\tau , F )=\bar{{\cal L}}_{MM}(e^{-\frac{\phi_0}{2}}F )- C_0 \mathcal{P}=\cosh(\gamma) e^{-\phi_0} \mathcal{S} +\sinh (\gamma)\sqrt{e^{-2\phi_0}( \mathcal{S}^2+\mathcal{P}^2)}- C_0 \mathcal{P}.
\end{eqnarray}

It is clear that the above theory reduces to $\hat{{\cal L}}_{Max}(\tau , F )$ in the limit of $\gamma \to 0$. So, the energy-momentum tensor obtained from Lagrangian \reef{hatLMM}, only differs up to a dilaton field factor from the energy-momentum tensor of the pure ModMax theory . This  means that $\hat{T}_{\mu \nu}=  e^{-\phi_0} T_{\mu \nu}$  where the tensor $T_{\mu \nu}$  is given by Eq. \reef{Tmunu}. We can also find  the $G$-tensor corresponding to the axion-dilaton ModMax theory \reef{hatLMM} from $\hat{G}_{\mu\nu}=-2\, \frac{\partial \hat{{\cal L}}_{MM}(\tau , F )}{\partial F^{\mu\nu}}$. Thus, we have the  antisymmetric tensor  $\hat{G}_{\mu\nu}=\bar{G}_{\mu\nu}-C_0\tilde{F}_{\mu\nu}$ where  $\bar{G}_{\mu\nu}=-2\frac{\partial \bar{{\cal L}}_{MM}}{\partial F^{\mu\nu}}= e^{-\phi_0} G_{\mu \nu}$ and $G_{\mu \nu}$ is given in Eq. \reef{G}.

 According to \cite{Gibbons:1995ap}, for any NED theory one can extend the duality transformations \reef{GF} to the corresponding $SL(2,R)$ duality group. In the case of ModMax theory, the corresponding $SL(2,R)$ duality transformations is described by
\begin{equation}\label{SL1}
{\cal F}_{\mu\nu}\equiv\left(
\begin{array}{c}
\tilde{F}_{\mu\nu} \cr  \
\bar{G}_{\mu\nu}-C_0\tilde{F}_{\mu\nu}
\end{array}
\right) \rightarrow  (\Lambda^{-1})^T \left(
\begin{array}{c}
\tilde{F}_{\mu\nu}\cr  \
\bar{G}_{\mu\nu}-C_0\tilde{F}_{\mu\nu}
\end{array}
\right)\,,
\end{equation}
and\footnote{Note that the matrix ${\cal M}_0$ here is the inverse of the matrix ${\cal M}$ in \cite{Gibbons:1995ap}.}
\begin{equation}\label{SL2}
{\cal M}_0=e^{\phi_0}\left(\begin{array}{cc}
|\tau|^2 & C_0\\
C_0 & 1
\end{array}\right) ,\,\,\,\,\,\, {\cal M}_0\longrightarrow \Lambda {\cal M}_0\Lambda ^T\,.\end{equation}
Using the transformations \reef{SL1} and \reef{SL2}, we can find a symmetric structure in the form of $\hat{\cal N}_{\mu \nu}=({\cal F}^T)_\mu{}^\alpha {\cal M}_0 {\cal F}_{\alpha \nu }$ which is manifestly invariant under the $SL(2,R)$ duality transformations. That is,
\begin{eqnarray}\label{NAD}
\hat{\cal N}_{\mu \nu}&=&e^{-\phi_0}\,\tilde{F}_\mu{}^\alpha\,\tilde{F}_{\alpha \nu }+ e^{\phi_0}\, \bar{G}_{\mu}{}^\alpha \, \bar{G}_{\alpha \nu }\nonumber\\
&=& e^{-\phi_0} {\cal N}_{\mu\nu}\,,
\end{eqnarray}
where we have used $\bar{G}_{\mu\nu}= e^{-\phi_0} G_{\mu \nu}$ and Eq. \reef{Nmunu} to obtain the second line.
On the other hand, the energy-momentum tensor corresponding to Eq. \reef{hatLMM} is given as follows
\begin{eqnarray}\label{hatTmunui}
\hat{T}_{\mu \nu}=- \frac{1}{2 \cosh( \gamma)} \Big[ {\hat{{\cal N}}}_{\mu\nu} -\frac{1}{4}  {{\hat{{\cal N}}}_{\rho}}^{\rho}  { g}_{\mu\nu} \Big],
\end{eqnarray}
which is a manifestly  $SL(2,R)$ invariant structure in the axion-dilaton ModMax theory.
\subsection{$SL(2, R)$  symmetry in the GBI theory}

Following the discussion in the previous section, one can construct a GBI theory with $SL(2, R)$  symmetry group by adding an axion-dilaton field to $BI \gamma$ theory given by Eq. \reef{GBI}, i,e, as $\hat{{\cal L}}_{BI \gamma}(\tau , F )=\bar{{\cal L}}_{BI \gamma}(e^{-\frac{\phi_0}{2}}F )- C_0 \mathcal{P}$. Due to this identification we obtain
\begin{equation}\label{GBIAD}
\hat{{\cal L}}_{BI \gamma}(\tau, F)= \frac{1}{\lambda} \Bigg[ 1 -  \sqrt{1 -  \lambda \Bigl( 2 e^{-\phi_0} \Big(\cosh(\gamma)  \mathcal{S} +\sinh (\gamma)\sqrt{( \mathcal{S}^2+\mathcal{P}^2)}\Big)+\lambda e^{-2\phi_0} \mathcal{P}^2 \Bigr)} \Bigg]- C_0 \mathcal{P}.
\end{equation}

Similar to what we have a priori done for the ModMax theory, we are interested in the $SL(2,R)$ invariant structure for the energy-momentum of GBI theory in \reef{GBIAD}.
Therefore, using the standard definition of the energy-momentum tensor and after a straightforward calculation, we obtain
\begin{eqnarray}\label{Tbig}
\hat{T}_{\mu \nu}&=&e^{-\phi_0}\frac{F_{\mu \alpha} F_{\nu}{}^{\alpha} \bigl(\cosh(\gamma) \sqrt{ \mathcal{S}^2+\mathcal{P}^2} + \sinh(\gamma) \mathcal{S}\bigr)}{ x\sqrt{ \mathcal{S}^2+\mathcal{P}^2}}  \nonumber\\
&+& e^{-\phi_0} \frac{\bigl(2 \lambda \cosh(\gamma)\mathcal{S} \sqrt{ \mathcal{S}^2+\mathcal{P}^2}+\lambda \sinh(\gamma) (\mathcal{P}^2 + 2  \mathcal{S}^2)+ e^{\phi_0} ( x-1)\,\sqrt{ \mathcal{S}^2+\mathcal{P}^2} \bigr) \mathit{g}_{\mu \nu}}{\lambda  x \sqrt{ \mathcal{S}^2+\mathcal{P}^2}},
\end{eqnarray}
where, for simplicity, we use the shorthand $x=\sqrt{1 -  \lambda \Bigl( 2 e^{-\phi_0} \Big(\cosh(\gamma)  \mathcal{S} +\sinh (\gamma)\sqrt{( \mathcal{S}^2+\mathcal{P}^2)}\Big)+\lambda e^{-2\phi_0} \mathcal{P}^2 \Bigr)}$.
The $\lambda$-expansion of the above energy momentum tensor up to order $\lambda$ is as follows
\begin{eqnarray}\label{Tlanda}
\hat{T}_{\mu \nu}&=& e^{-\phi_0}\biggl(\cosh(\gamma) + \frac{ \sinh(\gamma) \mathcal{S}}{\sqrt{ \mathcal{S}^2+\mathcal{P}^2}}\biggr) T^{Max}_{\mu \nu} + e^{-2\phi_0} \lambda \biggl[ \Bigl(  \cosh(2 \gamma)\mathcal{S}+\frac{ \sinh(2 \gamma) (\mathcal{P}^2 + 2  \mathcal{S}^2)}{2 \sqrt{ \mathcal{S}^2+\mathcal{P}^2}}\Bigr) T^{Max}_{\mu \nu} \nonumber\\
&-&  \frac{1}{2} \Bigl( \cosh^2(\gamma) \mathcal{P}^2 + \mathcal{S} \bigl(\cosh(2 \gamma)\mathcal{S} - \sinh(2 \gamma) \sqrt{ \mathcal{S}^2+\mathcal{P}^2} \bigr)\Bigr) \mathit{g}_{\mu \nu}\biggr]+{\cal O} ( \lambda^2)+\ldots\, .
\end{eqnarray}
 The leading term in the expansion of \reef{Tlanda} is the energy-momentum tensor of the axion-dilaton ModMax theory that we found in terms of $SL(2,R)$ invariant structure in Eq. \reef{hatTmunui}.
In fact, if the Eq.~\reef{Tlanda} is to be invariant under the $SL(2,R)$ transformations, one should find it invariant at any order of $\lambda$ as well.

 In comparison to the axion-dilaton ModMax theory, the antisymmetric $G$-tensor of generalized theory in Eq. \reef{GBIAD} could be found from   $\hat{G}_{\mu\nu}=-2\, \frac{\partial \hat{{\cal L}}_{BI \gamma}(\tau, F)}{\partial F^{\mu\nu}}$. However, one can write the result as $\hat{G}_{\mu\nu}=\bar{G}_{\mu\nu}-C_0\tilde{F}_{\mu\nu}$, where
\begin{eqnarray}\label{GGne}\bar{G}_{\mu\nu}=\frac{e^{-\phi_0}\sinh(\gamma) (F_{\mu \nu}\mathcal{S} +\mathcal{P} \tilde{F}_{\mu \nu})}{\sqrt{ \mathcal{S}^2+\mathcal{P}^2} x} + \frac{ e^{-\phi_0} \cosh(\gamma) F_{\mu \nu} + e^{-2\phi_0} \lambda \mathcal{P} \tilde{F}_{\mu \nu}}{x}\, .
\end{eqnarray}
The $\lambda$-expansion of the above tensor up to order $\lambda$ is as follows
\begin{eqnarray}\label{Gbar}
\bar{G}_{\mu \nu}&=&e^{-\phi_0}\bigg[\cosh(\gamma) F_{\mu \nu} +  \frac{\sinh(\gamma) (F_{\mu \nu}\mathcal{S} + \tilde{F}_{\mu \nu} \mathcal{P})}{ \sqrt{ \mathcal{S}^2+\mathcal{P}^2}} \bigg]  +e^{-2\phi_0} \lambda \Bigl[ \cosh^2(\gamma) \tilde{F}_{\mu \nu} \mathcal{P}+ \frac{\cosh(\gamma)  \sinh(\gamma) \tilde{F}_{\mu \nu} \mathcal{P}\mathcal{S}}{ \sqrt{ \mathcal{S}^2+\mathcal{P}^2}}\nonumber  \\
&+&\frac{F_{\mu \nu} \bigl(2 \cosh(2 \gamma)\mathcal{S}  \sqrt{ \mathcal{S}^2+\mathcal{P}^2} +  \sinh(2 \gamma) (\mathcal{P}^2 + 2  \mathcal{S}^2)\bigr)}{2 \sqrt{ \mathcal{S}^2+\mathcal{P}^2}}\Bigr]+{\cal O} ( \lambda^2)+\ldots\,.
\end{eqnarray}

Substituting $\bar{G}_{\mu\nu}$ from \reef{Gbar} in Eq. \reef{NAD}, we find  the $SL(2,R)$ invariant structure corresponding to  \reef{GBIAD} up to order $\lambda$, i.e. ,
\begin{eqnarray}
\hat{\cal N}_{\mu\nu} &\!\!\!=\!\!\!&-2 e^{-\phi_0}\bigg[\cosh(\gamma) \Bigl(\cosh(\gamma) + \frac{\mathcal{S} \sinh(\gamma)}{ \sqrt{ \mathcal{S}^2+\mathcal{P}^2}}\Bigr) T^{Max}_{\mu\nu}- \sinh(\gamma) \Bigl(\cosh(\gamma)  \sqrt{ \mathcal{S}^2+\mathcal{P}^2} + \sinh(\gamma) \mathcal{S}\Bigr) g_{\mu\nu}\bigg]\nonumber \\
&\!\!\!+\!\!\!&-2  \lambda \, e^{-2\phi_0} \biggl[\Bigl(\cosh(3 \gamma)\mathcal{S}+ \frac{2 \cosh^2(\gamma) \sinh(\gamma) \mathcal{P}^2 }{ \sqrt{ \mathcal{S}^2+\mathcal{P}^2}} + \frac{ \sinh(3 \gamma)  \mathcal{S}^2}{ \sqrt{ \mathcal{S}^2+\mathcal{P}^2}}\Bigr) T^{Max}_{\mu\nu} \\
&\!\!\!-\!\!\!&\frac12\bigl(\cosh(\gamma) \mathcal{P}^2 + \cosh(3 \gamma) (\mathcal{P}^2 + 2  \mathcal{S}^2) + 2 \sinh(3 \gamma) \mathcal{S} \sqrt{ \mathcal{S}^2+\mathcal{P}^2}\bigr) g_{\mu\nu}\biggr]  +{\cal O} ( \lambda^2)+\ldots\, .\nonumber
\end{eqnarray}

Now, by considering this invariant structure and after performing some lengthy calculations we are able to recast the energy-momentum tensor of the axion-dilaton $BI\gamma$ theory in Eq. \reef{Tbig} in a manifestly $SL(2,R)$ invariant form. Therefore, we have
\begin{eqnarray}\labell{hatT}
\hat{T}_{\mu\nu}=- \frac{1}{2 \cosh( \gamma)} \Big( \hat{\cal N}_{\mu\nu} -\frac{1}{4}  \hat{\cal N}_{\rho}{}^{\rho}  { g}_{\mu\nu} \Big)
+\lambda \biggl[ a_0(\gamma)\, \hat{\cal N}_{\rho}{}^{\rho}\, \hat{\cal N}_{\mu \nu} +b_0(\gamma)\, \hat{\cal N}_{\mu}{}^{\rho}\, \hat{\cal N}_{\nu \rho} \biggr]+\ldots,
\end{eqnarray}
where the coefficients $a_0(\gamma)$ and $b_0(\gamma)$ are given by
\begin{eqnarray}a_0(\gamma)= -  \frac{1}{16 \cosh^2(\gamma)} +\frac{3}{16}, \,\,\,\,\,\,\,\,\,\,b_0(\gamma)= \frac{1}{4\cosh^2(\gamma)}- \frac{3}{8}\,. \nonumber
\end{eqnarray}
 Note that in Eq. \reef{hatT} we have also used the  identities $F_{\mu}{}^{\alpha} \tilde{F}_{\nu \alpha}=- g_{\mu \nu}  \mathcal{P}$ and $ F_{\mu}{}^{\alpha} F_{\alpha}{}^{\gamma} F_{ \gamma \beta} F^{\beta}{}_{\nu}= 2 F_{\mu}{}^{\alpha} F_{ \alpha \nu}\mathcal{S}+ \mathit{g}_{\mu \nu} \mathcal{P}^2$ that hold for any two-form $F$ . The explicit forms of the invariant structures that appear in Eq. \reef{hatT} have mentioned in the appendix.
\section{Manifestly self-dual invariant actions}\label{secLinv}
As mentioned before, the Lagrangian of the Maxwell and the ModMax theories are not invariant under the $SL(2,R)$ transformation, however, the derivatives of them with respect to an invariant parameter are invariant.
In this paper we deal with two coupling constants $\lambda$ and $\gamma$ and also the gravitational field. The energy-momentum tensor that obtains from the variation of the corresponding Lagrangian with respect to the gravitational field would be invariant under this duality\cite{Gaillard:1981rj,Gaillard:1997rt}. The non-zero contribution of the invariant action in electrodynamic theories comes from the variation of original theory with respect to the electromagnetic field strength \cite{Carrasco:2011jv}. It has been shown in \cite{Aschieri:2013nda} that the derivative of the original Lagrangian $\cal L$ with respect to the coupling constant $c$ in the form $c \frac{\partial{ \cal L}}{\partial c}$ produces the corresponding invariant theory $L^{c}_{inv}$. However, for a theory with a typical dimensionless coupling constant such as the ModMax theory, the corresponding derivative can produce an invariant theory.

Due to this fact and according to \reef{GBI}, consider the invariant Lagrangians $L^{\lambda}_{inv}$ and $L^{\gamma}_{inv}$ corresponding to the coupling constants $\lambda$ and $\gamma$, respectively. Obviously, the Lagrangians  of the Maxwell theory as well as the ModMax theory  are independent of $\lambda$, therefore there is no self-dual invariant Lagrangian of type $L^{\lambda}_{inv}$ for these theories. On the other hand, $L^{\gamma}_{inv}$ has non-vanishing contribution only for the ModMax theory, while the derivative of the Lagrangian density of Maxwell theory with respect to $\gamma$ vanishes. We ensue the following invariant action from the above discussion for the ModMax theory
\begin{eqnarray}\label{LinvMM}
\frac{\partial{\cal L}_{MM}}{\partial\gamma}={\cal L}^{\gamma}_{inv-MM} =\sinh(\gamma) \mathcal{S}+\cosh(\gamma) \sqrt{ \mathcal{S}^2+\mathcal{P}^2}.
\end{eqnarray}

Comparing the self-dual Lagrangian \reef{LinvMM} with Eq. \reef{NN}, we can find a  manifestly $SO(2)$ duality-symmetric form for the Lagrangian \reef{LinvMM} as follows
\begin{eqnarray}\label{NLinMM}
{\cal L}^{\gamma}_{inv-MM}= \frac{1}{8\sinh(\gamma)} \,\, {\cal N}_{\rho}{}^{\rho}.
\end{eqnarray}
It is noticed that there is a similar consideration for the GBI theory \reef{GBI} as well. In fact, variations of the GBI Lagrangian with respect to the  couplings $\lambda$ and $\gamma$, which are respectively denoted by ${\cal L}^{\lambda}_{inv-BI\gamma}$ and ${\cal L}^{\gamma}_{inv-BI\gamma}$, are invariant under $SO(2)$ duality group. In the rest of this section, we will derived the frameworks of these invariant theories in the context of $SL(2,R)$ structures.
\subsection{Self-dual invariant Lagrangian: ${\cal L}^{\lambda}_{inv-BI\gamma}$}
For the purpose of constructing a self-dual Lagrangian for GBI theory, let us briefly review the construction of this duality for an arbitrary electrodynamic theory ${\cal L}(\mathcal{S},\mathcal{P})$ which satisfies the duality invariant condition. Now, one can define an invariant Lagrangian ${\cal L}^{\lambda}_{inv}$ so that is self-dual as follows  \cite{Aschieri:2008ns,Aschieri:2013nda,Chemissany:2006qd,Carrasco:2011jv,Chemissany:2011yv}
\begin{eqnarray}\label{Linv}
 -\lambda\frac{\partial{\cal L}}{\partial\lambda}={\cal L}^{\lambda}_{inv}={\cal L}(\mathcal{S},\mathcal{P})+\frac{1}{4} F_{\mu\nu} G^{\mu\nu}.
\end{eqnarray}

As we mentioned above, due to the lack of dependence on $\lambda$, the invariant actions for both the Maxwell theory and the ModMax theory are trivial. For example, in the case of ModMax theory one can check this fact by inserting ${\cal L}_{MM}$ from \reef{LMM} and $ G^{\mu\nu}$ from \reef{G} in Eq. \reef{Linv}.
  Now, consider a family of  NED theories that led to the Maxwell theory (or the ModMax theory) at the first order of coupling constant, i.e. ${\cal O} ( F^2)$.
  It  seems that the  nonzero self-dual contribution of these NED theories appears from the order of ${\cal O} (F^4)$\cite{Carrasco:2011jv}, but we will show that this statement is not true for the ModMax theories.
By comparing \reef{EMT1} and \reef{Linv}, one can find that ${\cal L}^{\lambda}_{inv}= \frac{1}{4}  {T_{\mu}}^{\mu}$. This interesting result asserts that  the necessary condition for a NED theory to have nonzero contribution to self-dual Lagrangian \reef{Linv} is that ${T_{\mu}}^{\mu}\neq 0$.

The axion-dilaton $BI\gamma$ theory satisfies both duality invariant condition $G\tilde{G}-F\tilde{F}=0$ and the energy-momentum traceless condition ${T_{\mu}}^{\mu}\neq 0$. If we set $\phi_0 = 0$ and $C_0 =0$ in \reef{GBIAD} then we will find the corresponding duality invariant Lagrangian as follows
\begin{eqnarray}\label{GBinv}
{\cal L}^{\lambda}_{inv-BI\gamma}=  \frac{1}{\lambda} \Big[ 1 -  \frac{1}{\sqrt{1 -  \lambda \bigl( 2 {\cal L}_{MM}+\lambda \mathcal{P}^2 \bigr)}} +\lambda \frac{ {\cal L}_{MM}}{\sqrt{1 -  \lambda \bigl( 2 {\cal L}_{MM}+\lambda \mathcal{P}^2 \bigr)}}\Big].
\end{eqnarray}
In other words, a comparison between  Eq.~\reef{Tlanda} for $\phi_0 = 0$ and Eq.~\reef{GBinv} shows that the identity ${\cal L}^{\lambda}_{inv}= \frac{1}{4}  {T_{\mu}}^{\mu}$ works truly. Also, it is worth mentioning that  the $\lambda$-expansion of Lagrangian \reef{GBinv} starts from the order of $\lambda$, or equivalently  ${\cal O}( F^4)$.

Considering the duality invariant structure \reef{Nmunu} in which $G_{\mu\nu}$ is substituted from \reef{Gbar} with $\phi_0= 0$, we find the $\lambda$-expansion of action \reef{GBinv} in the form that is manifestly duality invariant
\begin{eqnarray}\label{Linvlambda}
{\cal L}^{\lambda}_{inv-BI\gamma}= - \frac{\lambda}{2^7 \sinh^2(\gamma)}  {\cal N}_{\mu}{}^{\mu} \, {\cal N}_{\nu}{}^{\nu}+ \frac{\lambda^2 \cosh(\gamma)}{ 2^9 \sinh^4(\gamma)}{\cal N}_{\mu}{}^{\mu} \, {\cal N}_{\nu}{}^{\nu}\, {\cal N}_{\alpha}{}^{\alpha}+\ldots ,
\end{eqnarray}
in which we have used the following identity
\begin{eqnarray}
{\cal N}_{\mu \nu} \, {\cal N}^{\mu \nu} =  \frac{\cosh(2 \gamma)}{4 \sinh^2(\gamma)}  {\cal N}_{\mu}{}^{\mu} \, {\cal N}_{\nu}{}^{\nu}.
\end{eqnarray}

As alluded before, the $SO(2)$-invariant action ${\cal L}^{\lambda}_{inv-BI\gamma}$ can be related to the action \reef{GBI} by a differential flow equation $ -\lambda\frac{\partial{\cal L}_{BI\gamma}}{\partial\lambda}={\cal L}^{\lambda}_{inv}$. Therefore, one can find the relationship of the duality invariant actions \reef{GBinv} to the  irrelevant $T \bar{T}$-like deformation of the GBI theory\cite{Babaei-Aghbolagh:2022uij,Ferko:2022iru,Conti:2022egv} as well as Born-Infeld theory \cite{Conti:2018jho,Ferko:2019oyv,Babaei-Aghbolagh:2020kjg} (the GBI theory at the limit of $\gamma \to 0$), that is ${\cal L}^{\lambda}_{inv}= - \frac18  \lambda O^{\lambda }_{T^2} $ where $O^{\lambda }_{T^2}=T_{\mu\nu}T^{\mu\nu}- \frac{1}{2} {T_{\mu}}^{\mu} {T_{\nu}}^{\nu}$ is an irrelevant $T\bar{T}$  operator.
\subsection{Self-dual invariant Lagrangian: ${\cal L}^{\gamma}_{inv-BI\gamma}$ }
In this subsection, we consider the second  self-dual invariant action coming from the derivative of GBI Lagrangian \reef{GBI} with respect to the parameter $\gamma$. One can find that
\begin{eqnarray}\label{Linvgamma}
{\cal L}^{\gamma}_{inv-BI\gamma}=\frac{\partial{\cal L}_{BI\gamma}}{\partial\gamma}=\frac{ \sinh(\gamma)\mathcal{S}+\cosh(\gamma) \sqrt{ \mathcal{S}^2+\mathcal{P}^2} }{\sqrt{1 -  \lambda \bigl( 2 {\cal L}_{MM}+\lambda \mathcal{P}^2 \bigr)}},
\end{eqnarray}
  therefore, expanding  the invariant action \reef{Linvgamma} up to order $\lambda$ yields that
\begin{eqnarray}\label{LinvGa}
{\cal L}^{\gamma}_{inv-BI\gamma} &\!\!\!=\!\!\!& \sinh(\gamma) \mathcal{S}+\cosh(\gamma) \sqrt{ \mathcal{S}^2+\mathcal{P}^2}  \\
&\!\!\!+\!\!\!& \lambda \biggl(\cosh(2 \gamma)\mathcal{S} \sqrt{ \mathcal{S}^2+\mathcal{P}^2} + \cosh(\gamma)  \sinh(\gamma)(\mathcal{P}^2 + 2  \mathcal{S}^2)\biggr) +{\cal O}( \lambda^2)+\ldots \,\,.\nonumber
\end{eqnarray}
As the same steps that we did in the previous subsection, by considering the invariant structure \reef{Nmunu} and $\phi_0= 0$, we find the above action in the form that is manifestly duality invariant as follows
\begin{eqnarray}\label{NLin}
{\cal L}^{\gamma}_{inv-BI\gamma}= \frac{1}{8\sinh(\gamma)} \,\, {\cal N}_{\mu}{}^{\mu} - \lambda \,\,\frac{1}{64} \frac{\cosh(\gamma)}{\sinh^3( \gamma)}\,\,  {\cal N}_{\mu}{}^{\mu} \, {\cal N}_{\nu}{}^{\nu} +{\cal O}( \lambda^2)+\ldots \,\,.
\end{eqnarray}

 Note that, the  self-dual invariant Lagrangian ${\cal L}^{\lambda}_{inv-BI\gamma} \reef{Linvlambda}$ starts from the order of $F^4$, but the self-dual invariant Lagrangian ${\cal L}^{\gamma}_{inv-BI\gamma}$ \reef{NLin} starts from  the order of $F^2$. Actually, the nonzero contribution of the invariant action ${\cal L}^{\gamma}_{inv-BI\gamma}$ at the Maxwell order can be interested because the other known NED theories have no contribution for their corresponding invariant actions at this order.

In general, by replacing ${\cal N}$ with $\hat{\cal N}$, the results that we have found in this section extend to the case of $\phi_0 \neq 0$ and $ C_0 \neq 0$. Doing so, we extend the $SO(2)$ duality invariant actions \reef{Linvlambda} and \reef{NLin}  to corresponding $SL(2,R)$-invariant cases.
It would be also of interest to notice that the self-dual invariant Lagrangian \reef{Linvgamma} can be related to a marginal  $T \bar{T}$-like deformation of the ModMax theory \cite{Babaei-Aghbolagh:2022uij}. One can show that ${\cal L}^{\gamma}_{inv-BI\gamma}= \frac12  O^{\gamma }_{T^2}$, where $O^{\gamma }_{T^2}=\sqrt{T_{\mu\nu}T^{\mu\nu}- \frac{1}{4} {T_{\mu}}^{\mu} {T_{\nu}}^{\nu}}$ is a marginal operator. It has been discussed in Refs. \cite{Conti:2022egv,Babaei-Aghbolagh:2022kfz,Ferko:2206jsw} that a dimensional reduction of the ModMax theory to two spacetime dimensions corresponds to the continuous marginal root-$T \bar{T}$ deformation of free bosons .

\section{Discussions}\label{4}
 In this paper, we studied the self-dual invariant structures of the conformal nonlinear modification of the Maxwell electrodynamics (ModMax theory) and its generalization. We found an $SL(2,R)$ invariant form for the energy-momentum tensor of the ModMax theory in terms of some $SL(2,R)$ duality-symmetric structures. We have also obtained the manifestly duality invariant structures for the energy-momentum  tensor of the GBI theory.

 In Sec.~\ref{secLinv}, we have proposed two actions which are invariant under $SL(2,R)$ symmetry, and consistent with two  irrelevant and marginal  $T\bar{T}$-like deformations. The first self-dual action \reef{GBinv} is related to the standard form of the expansion of the  GBI Lagrangian  with respect to $\lambda$, and the second one \reef{Linvgamma} concerns with the  expansion of the  GBI Lagrangian  with respect to $\gamma$, proposed in Ref.~\cite{Bandos:2020hgy}, which are manifestly $SL(2,R)$-duality invariant and include axion-dilaton couplings.

 As custom, the correlation functions are fundamental observables in QFTs. For instance,  it has been shown in Refs.~\cite{He:2020udl,He:2020cxp,He:2021bhj,Song He:2202oyv} that the deformed Lagrangian and stress tensor could obtained order by order from a $T \bar{T}$-product operator in two dimensional CFTs, which can be used to expand the partition function up to the second-order. It would be of interest to study the correlation functions in the context of the  ModMax theory and check out the consistency of  two self-dual actions and   $T \bar{T}$-like deformations.

 The extension of supersyemmetric theories to NED theories are also of great interest \cite{DP,Cecotti:1986gb}. For example, the supersymmetric extension of the Born-Infeld Lagrangian has been studied in Ref.~\cite{Bagger:1996wp} and in the case of ModMax theory this extension was discussed in Refs. \cite{Bandos:2021rqy,Kuzenko:2021cvx}.  Recently, I. Bandos et al. \cite{Bandos:2021rqy} (see also \cite{Kuzenko:2021cvx}) have shown that the Born-Infeld-like extension of  superModMax theory is described by the following superfield Lagrangian density
\begin{eqnarray}\label{susy-BI-MM}
\mathcal{L}_{{ susy}-{ BI \gamma}}
=
\frac{\cosh(\gamma)}{4}\left\{\int
d^2 \theta\, W^2 + \int d^2\bar{\theta} \,\bar{W}^2+\int d^2\theta d^2\bar{\theta}\, W^2 \bar{W}^2K(\mathbb{S},\mathbb{P})\right\}
~,~~~~~~
\end{eqnarray}
where $ K(\mathbb{S},\mathbb{P})$
is given by
\begin{eqnarray}
K(\mathbb{S},\mathbb{P}) =
 \frac{\frac{1}{\lambda} \big(1-\sqrt{1-2\lambda\left[\cosh(\gamma)\mathbb{S}+\sinh(\gamma)\sqrt{\mathbb{S}^2+\mathbb{P}^2}\right]- \lambda^2 \mathbb{P}^2}\big)-\cosh(\gamma)\mathbb{S}}{\cosh(\gamma)(\mathbb{S}^2+\mathbb{P}^2)}
~.~~~~~~
\end{eqnarray}
 Here, the superfields $\mathbb{S}$ and $\mathbb{P}$ are
\begin{eqnarray}
\mathbb{S}
=
-\frac{1}{16}(D^2 W^2+\bar{{D}}^2\bar{W}^2)~,
\quad \mathbb{P}
= \frac{i}{16}({D}^2 W^2-\bar{{D}}^2 \bar{W}^2)
~,
\label{superSandP}
\end{eqnarray}
where $W^2=W^{\alpha} W_{\alpha}$. $W_{\alpha}$ and its conjugate $\bar{W}_{\dot{\alpha}}=(W_\alpha)^*$, are anticommuting Weyl spinor chiral superfields satisfying in the conditions
\begin{gather}
\bar{{D}}_{\dot{\beta}}W_{\alpha} = 0, \quad {D}^{\alpha}W_{\alpha}=\bar{{D}}_{\dot{\alpha}}\bar{W}^{\dot{\alpha}} ~.
\end{gather}

In the limit  $\gamma=0$, the Lagrangian \eqref{susy-BI-MM} reduces to the Bagger-Galperin Lagrangian \cite{Bagger:1996wp}. The self-dual condition for electromagnetic duality invariance of generic nonlinear electrodynamic theories in Refs. \cite{Gaillard:1981rj,Gaillard:1997rt}, were generalized to superfield formulations of $N = 1$  supersymmetric theories in Refs.\cite{Kuzenko:2000tg,Kuzenko:2000uh}. For generic $N = 1$ theories described by a Lagrangian
$\mathcal{L}[W, \bar{W} ]$, the Kuzenko–Theisen duality-invariance condition is \cite{Kuzenko:2000tg,Kuzenko:2000uh}
\begin{eqnarray}\label{SDC}
Im\int d^4x\,d\theta^2\left( W^\alpha W_\alpha+M^\alpha M_\alpha \right)=0\, , \qquad
M_\alpha\equiv-2i\frac{\delta \mathcal{L}[W,\bar W]}{\delta W^\alpha} \, .
\end{eqnarray}
The  Born-Infeld-like extension of the supersymmetric ModMax theory in \reef{susy-BI-MM} satisfies the duality-invariance condition \reef{SDC}. Therefore, as shown in Sec. \ref{secLinv},  we expect to have two supersymmetric self-duality invariant actions for the $ N = 1 $ supersymmetric Born-Infeld-like theory.
These two actions are denoted by
\begin{eqnarray}
 \mathcal L_{{ susy}-inv- BI \gamma}^\lambda=- \lambda \frac{ \partial \mathcal L_{{ susy}- BI \gamma}}{\partial \lambda}
\,\,\,\,\,\,\,\,;\,\,\,\,\,\,\, \mathcal L_{{ susy}-inv- BI \gamma}^\gamma=\frac{ \partial \mathcal L_{{ susy}- BI \gamma}}{\partial \gamma}
~.
\label{flow-susy-BIMM}
\end{eqnarray}

 In \cite{Babaei-Aghbolagh:2020kjg} we have, a priori, studied the compatibility deformation of $N=2$ supersymmetric BI theory with $N=2$ supersymmetric self-dual invariant Lagrangian.
The $T\bar{T}$-deformation of the $N=1$ supersymmetric ModMax theory with operator $O_{T^2}^{\lambda}$, was initially considered  in Ref. ~\cite{Ferko:2022iru}.
 As final remark, extending the $N=1$ and $N=2$ supersymmetric self-dual invariant Lagrangian of the ModMax theory  can be investigated as a future work.

\section*{Acknowledgements}
We are very grateful to D. Sorokin for his kindly interest in this work and fruitful discussion. HBA is specially appreciate M.R.Garousi for his useful comments at the early stages of the work.
\section*{Appendix}\label{App}
In this appendix we write the explicit forms of invariant structures  $\hat{\cal N}_{\rho}{}^{\rho}$
 and $\hat{\cal N}_{\mu}{}^{\rho}\, \hat{\cal N}_{\nu \rho}$ that appear in Eq \reef{hatT}.
\begin{eqnarray}\label{Nhat}
\hat{\cal N}_{\rho}{}^{\rho}&=& 8\, e^{-\phi_0} \sinh(\gamma) \Bigl(\cosh(\gamma)  \sqrt{ \mathcal{S}^2+\mathcal{P}^2} + \sinh(\gamma)\mathcal{S}\Bigr)  \\
&+&  \lambda \, e^{-2\phi_0} \Bigl[4 \cosh(\gamma) \mathcal{P}^2 + 4 \cosh(3 \gamma) (\mathcal{P}^2 + 2  \mathcal{S}^2) + 8 \sinh(3 \gamma)\mathcal{S} \sqrt{ \mathcal{S}^2+\mathcal{P}^2}\Bigr]+{\cal O} ( \lambda^2)+\ldots\, ,\nonumber
\end{eqnarray}
and
\begin{eqnarray}
\hat{\cal N}_{\mu}{}^{\rho}\, \hat{\cal N}_{\nu \rho} &=&- e^{-2\phi_0}  \frac{2 \sinh(2 \gamma) \bigl(\mathcal{P}^2 + \cosh(2 \gamma) (\mathcal{P}^2 + 2  \mathcal{S}^2) + 2 \sinh(2 \gamma)\mathcal{S} \sqrt{ \mathcal{S}^2+\mathcal{P}^2}\bigr)}{\sqrt{ \mathcal{S}^2+\mathcal{P}^2}}  T^{Max}_{\mu \nu} \\
&+& 2 \, e^{-2\phi_0} \cosh(2 \gamma) \Bigl(\mathcal{P}^2 + \cosh(2 \gamma) (\mathcal{P}^2 + 2  \mathcal{S}^2) + 2 \sinh(2 \gamma)\mathcal{S} \sqrt{ \mathcal{S}^2+\mathcal{P}^2}\Bigr) g_{\mu \nu}  \nonumber \\
&+& \lambda \, e^{-3\phi_0}  \biggl[\Bigl(-2 \cosh(\gamma) \mathcal{P}^2 - 4 \cosh(3 \gamma) \mathcal{P}^2 - 2 \cosh(5 \gamma) (\mathcal{P}^2 + 4  \mathcal{S}^2) \nonumber \\
&+&  \frac{32 \cosh^2(\gamma)  \sinh(\gamma)\mathcal{P}^2\mathcal{S}}{\sqrt{ \mathcal{S}^2+\mathcal{P}^2}} -  \frac{24 \cosh^2(\gamma)  \sinh(3 \gamma)\mathcal{P}^2\mathcal{S}}{\sqrt{ \mathcal{S}^2+\mathcal{P}^2}} -  \frac{8  \sinh(5 \gamma)\mathcal{S}^3}{\sqrt{ \mathcal{S}^2+\mathcal{P}^2}}\Bigr) T^{Max}_{\mu \nu}  \nonumber \\
&+& \Bigl(-2 \cosh(\gamma) \mathcal{P}^2 \mathcal{S} + 4 \cosh(3 \gamma) \mathcal{P}^2\mathcal{S}+ \cosh(5 \gamma) (6 \mathcal{P}^2\mathcal{S}+ 8 \mathcal{S}^3) +  \sinh(\gamma)2 \mathcal{P}^2 \sqrt{ \mathcal{S}^2+\mathcal{P}^2}\nonumber \\
&+& \bigl(4 \mathcal{P}^2 \sinh(3 \gamma) + 2 \sinh(5 \gamma)(\mathcal{P}^2 + 4  \mathcal{S}^2)\bigr)\Bigr) \mathit{g}_{\mu \nu} \sqrt{ \mathcal{S}^2+\mathcal{P}^2}\biggr]+{\cal O} ( \lambda^2)+\ldots\, .\nonumber
\end{eqnarray}

\if{}
\bibliographystyle{abe}

redpajama_set_name: RedPajamaArXiv


It is time for Africans to practice some tough love on President Obama.
We did not need to hear these catch phrases that laced Obama's recent speech in Ghana. They are so benign that even Africa's leaders promised them with each stolen election.
We did not need to be told that "Africa's future is up to Africans." We have been taking responsibility for our future since the days of colonialism, efforts often frustrated by the US.
As Jeffrey Gettleman notes in his July 11, 2009 New York Times article, the US supported Congo's dictator, Mobuto Sese Seko for years, and in Somalia, "bungled a huge relief mission in the 1990s" leading eventually to an intervention weary US watching the Rwanda genocide unfold from the sidelines.
Instead, he met each of the most pertinent issues facing the continent with a generality. Take the issue of corruption. African leaders have stashed billions of dollars in European and American banks.
Patrick Bond, a South African political economist, estimates that the African elite had as of 2003 $80 billion sitting in Western banks.
Mobutu Sese Seko is said to have been so rich that he personally could have cleared Congo's debt. In the same way that the Obama administration has gone after Swiss Banks, might he not also have promised to help Africans retrieve this money?
This tidy sum has after all been siphoned out of the continent through the same corrupt channels Obama was speaking against.
But will this even be possible at a time when US citizens need more manufacturing jobs and Americans firms are barely competing with those from China and Japan?
And how will the Obama administration tackle US farmer subsidies that cost African farmers millions of dollars in lost revenue due to depressed markets?
It is well and good to give more aid to combat Aids. But surely, how about giving support so that we can manufacture cheap generic drugs, like India does?
This would mean breaking patent laws in cases of emergencies, something the WTO allows. We can use a powerful friend and partner here.
When Obama says that "we have seen countless examples of people taking control of their destiny, and making change from the bottom up" and gives the example of Kenya following the electoral violence last year, he does not mention that civil society organizations and social movements are generally under-funded, and unrecognized.
In Kenya human rights defenders continue to be killed and exiled even in these times of 'peace.' What "these brave Africans" need to hear is not what they know already – that destiny is in their hands – but rather what Obama can do to help them break away from the isolation that makes them so vulnerable.
But how is this different from Reagan's low intensity warfare and cold war proxy wars in Angola and Mozambique?
Instead of promising the obvious support of bringing war criminals to book, as an African aware of what past American logistical support meant, I would have liked to know how and why the US-African command Centre he defends and supports is going to be different.
That will certainly sound familiar to the American reader who listened to his campaign promises. Can the US export and import clean energy at the same time?
A question pertinent to Africans, yet not answered.
Africa has a population of 780 million people, and houses 52 countries. Africa can fit the US three times over and leave room for China.
Child mortality is on the rise, and half the population lives in dire poverty. Into the myriad of problems facing Africa, throw in the possibility of swine flu and Obama's speech is no longer one of tough love, but tough luck.
Africans did not need Africa sold to them; they needed a new and friendlier US foreign policy sold to them.
Mukoma wa Ngugi is a political columnist for the BBC Focus on Africa Magazine, the author of Hurling Words at Consciousness and the forthcoming, Nairobi Heat (Penguin Group South Africa, 2009).


Via Promobricks.de, LEGO has made the instructions of the set 40290 60 Years of the LEGO Brick available for download. This set will be given as a freebie with qualifying purchases from the 28th of January 2018 onwards and is anticipated to be quite a popular freebie among AFOL's.
We were previously alerted to this promotional set via the January 2018 Calendar for Lego Brand Retail Stores in the US and we can't wait to pick this set up.
Featuring mini build tributes to the classic sets below, this really feels like a premium and substantial freebie that's too good to miss.


text:
Home GROUP OF COMPANIES / Emami FMCG
Emami Ltd.
Established in 1974, Emami Limited is the flagship company of the Emami Group and one of the leading FMCG companies in India operating in beauty and healthcare spaces. The Company's enviable portfolio of over 300 products based on Ayurvedic formulations include many household brand names such as BoroPlus, Navratna, Fair and Handsome, Zandu balm, Mentho Plus balm, Fast Relief and Kesh King.
In 2008, Emami Limited made one of the most historic FMCG acquisitions of The Zandu Pharmaceutical Works Ltd, for Rs 730 cr. In 2009, the FMCG business of Zandu was merged with Emami Ltd. In June 2015, Emami Ltd made another big-ticket acquisition of Rs 1684 cr when it acquired the business of 'Kesh King' and stepped into the Ayurvedic hair & scalp care segment. In January 2019, the Company also acquired Creme 21, a German brand with strong roots & brand recall. In 2022, the Company went on to acquire 'Dermicool', one of the leading Prickly Heat and Cool Talc brands of India.
Emami has a strong distribution network of over 2800+ distributors through which its products are available over 4.9 million + retail outlets across India. The Company has footprints in more than 60 countries including SAARC, MENAP, SEA, Africa, Eastern Europe and the CIS countries.
One of the leading and fastest growing personal and healthcare businesses in India
Markets personal and health care products across 60 countries
Dedicated manufacturing units in India and abroad
Develops products suited to meet the diverse needs and preferences of consumers in different markets
BoroPlus antiseptic skin hygiene liquid


Make math matter to students in all grades using Math Tutor - Pre-Algebra Skills! This 80-page book provides step-by-step instructions of the most common math concepts and includes practice exercises reviews and vocabulary definitions. The book covers factoring positive and negative numbers order of operations variables exponents and formulas such as perimeter area and volume. It aligns with state national and Canadian provincial standards.


text:
Cobbs Creek Golf Course
West Philadelphia - Clear-cutting of over 100 acres of woodlands, riparian buffers and wetlands at the Cobbs Creek Golf Course brings a new threat of flooding to downstream residents and the environment in West Philadelphia. A developer, prior to receiving permission, clear-cut approximately 500 large (24" DBH and larger) trees. The same developer, the Cobbs Creek Foundation, is now seeking from the City Planning Commission (PCPC) to cut even more trees on steep slopes that could easily make flooding even worse. What is NOT surprising is that the City of Philadelphia is supportive of it!
The Darby and Cobbs Creek in Philadelphia and Delaware County is one of the most flood prone, flashy watersheds in Southeastern Pennsylvania. Floodwaters routinely rise-up so frequently that it's a common site to see sandbags stacked alongside homes ready to be deployed in front of garages and basement doors to prevent flood damage. Where is this flooding so prevalent? In the Overbrook, Kingsessing and Eastwick communities where predominantly people-of-color reside.
The flooding of the Darby and Cobbs is not new - it has been going on for decades. The City of Philadelphia knows it. The City just up-dated their All Hazard Mitigation Plan where flooding was noted to be one of the prominent threats to the City's well-being. The flooding of the Darby/Cobbs in Eastwick even got its own chapter (OEM Section 5.2.1.27)
There are two primary but separate issues with the development process of the Cobbs Creek Golf Course. The first is the destruction of the riparian buffer which now, obviously, needs to be restored and mitigated. In addition to the tree loss, 20+/- acres of wetlands were destroyed. The second is the developer's attempt to get further approval from the City of Philadelphia Planning Commission to clear even more trees off upland steep slopes.
DRN has partnered with several talented and focused grass-roots groups in West Philly and Eastwick. Towns in Delaware County are also concerned about this new source of floodwaters.
DRN recently submitted extensive comment to the U.S. Army Corps of Engineers to raise the alarm that this developer is now seeking to cut more trees down at this site and that the Corps may not be aware of. DRN's comment also addresses the the developer should not be allowed to profit off their destruction of the floodplain, riparian buffer and wetlands by selling-off mitigation credits to other developments elsewhere (i.e. mitigation credits). DRN and partners are also preparing to argue against the PCPC's approval of a re-zoning measure that would allow for a steep-slope overlay zone (PCPC Zoning Bill No. 220918) that would allow more tree clearing and more flooding.
ACE Mitigation Permit Extension Request (2022-12-19)


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Tag - Trouble in Shangri-La
About the Song
The catchy "My Heart" appears exclusively on the U.S. Barnes and Noble and international editions of Stevie Nicks' In Your Dreams (2011). Co-written by Michael Campbell, who provided the music track, Stevie...
Chris Lord-AlgeThe Pensado Awards
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On August 20, veteran music studio professional Chris Lord-Alge was honored at the Pensado Awards, an industry event acknowledging top music producers, engineers, and mixers. Stevie Nicks — along with Bruce Springsteen...
Trouble in Shangri-La (2001)
May 1, 2001 marks the fifth anniversary of the commercial release of Trouble in Shangri-La — Stevie Nicks' sixth solo album and her first full length release since 1994's Street Angel. Working with a...
Stevie Nicks in her own words
For years, superstar Stevie Nicks' life was fueled by cocaine. She talks to Cynthia McFadden about the successes and failures of her tumultuous life. Stevie Nicks has been in the public spotlight for 30 years as a...
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Q&A: Stevie Nicks
A fog is pouring over the Pacific Coast Highway toward Stevie Nicks' Southern California home, but the singer's mood could hardly be brighter. The Fleetwood Mac alumna's Trouble in Shangri-La has just entered the...
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Nicks' best solo work in years
By James Hunter Rolling Stone June 21, 2001 STEVIE NICKS: Trouble in Shangri-La (Reprise) * * * 1/2 (3 and a 1/2 stars out of 5) Stripped to the bare essentials, Stevie Nicks' music is just Nicks' articulate rasp and...
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Stevie Nicks unveils full summer itinerary
Stevie Nicks has confirmed more dates for a summer North American tour in support of her new Reprise album Trouble in Shangri-La. The trek will now begin July 6 in Burgettstown, Pa., and keep her on the road through...
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Retailers weigh in on Nicks, Wings campaigns
HMV in holding pattern for U.S., lays off 7; Retailers weigh in on Nicks, Wings campaigns Section: Merchants & Marketing Retail Track WHICH WAY: HMV continues to downsize its presence in the U.S. Two weeks ago, HMV...
Stevie's wonder
It's not much of a surprise when a current act such as Destiny's Child debuts at No. 1 on the albums chart. What might not have been as expected is the second-highest debut this issue, a posting at No. 5 for...
Sarah McLachlanTrouble in Shangri-La (2001)
Stevie Nicks takes care of herself
First she gave up sunbathing, then drugs Sitting two feet in front of Stevie Nicks, it is difficult to tell this is the same Fleetwood Mac siren who once lived the sex, drugs and rock 'n' roll lifestyle so severely that...
Face to Face: Stevie Nicks
Will you write this for me? He says, 'No, you write your songs yourself,'" sings Stevie Nicks in "That Made Me Stronger" on her soon-to-be-released Trouble In Shangri-La. The line references a conversation she had with...
Stevie Nicks: Queen of the stoned age
Fleetwood Mac's full-pelt excess and partner-swapping made for rock's most incredible soap opera. But there's one question everyone wants to ask Stevie Nicks. It concerns a large pile of cocaine, a...


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Using Laboratory DNA Tests to Confirm Ancestry
Nov 13, 2020 | Relationship
Once you've located unknown family members, wouldn't you like to know for sure if you're biologically related to them? Researching your family tree takes a lot of time and is associated with costs. Rest assured, DNA testing can confirm with certainty if ancestry information uncovered during research is true. It's a highly-accurate method for expanding your family tree.
Call us for a FREE confidential consultation at 800.929.0847. We're here to help.
DNA Tests for Ancestry
A DNA test provides vital information about biological relationships between potential living relatives, making the test an invaluable research tool for people interested in tracing their genealogy or ancestry. The results of DNA tests can link common ancestors and answer otherwise unanswered questions about a person's family tree. The results of a DNA test can help you find previously-unknown family members, and in some cases may reveal an unexpected truth about a paternal or maternal tie. Since the popularity of ancestry exploded, DDC has seen a marked increase in the number of people seeking to confirm possible biological relationships discovered during ancestry testing.
Accredited laboratories like DDC offer many options that help confirm questions about possible family relationships, including:
Maternity: While similar to a DNA paternity test, this analysis compares a child's DNA profile with their mother's
Grandparent: When the father isn't available for a paternity test, this test can determine the statistical probability of whether or not the alleged grandmother and/or grandfather is biologically related to tested children
Sibling: This test determines whether two individuals share one or both parents. Although an indirect way to prove or disprove paternity, it's useful when the alleged fathercan't be tested.
Aunt-Uncle: Called avuncular testing, this is a good option when the father isn't available for testing, because uncles and aunts share 25% of variable DNA with their nieces and nephews.
Family reconstruction: This type of testing involves gathering DNA samples from secondary sources in order to construct a DNA profile when direct samples aren't available. For example, the child's DNA profile is compared with DNA profiles of at least two of the potential father's close relatives (e.g. a full sibling or biological parent). If only two relatives are available, it's recommended that the mother participates to help strengthen results
Male Lineage (Y-STR): See below
Female Lineage (mtDNA): See below
Twins (Twin Zygosity): This DNA analysis definitively shows whether twins are identical or fraternal. This test can be helpful for establishing genetic compatibility for blood or tissue donation or determining if both twins inherited genetic defects.
DNA Testing Technology: Y-STR and mtDNA
Due to advances in DNA testing technology, tests can provide definitive answers about a person's ancestry. In addition to other forms of technology, accredited laboratories use a combination of two types of tests;the Y-STR Paternal Lineage Test and the mtDNA Maternal Lineage Test are useful for determining lines of ancestry to establish or rule out relationships between more distant relatives.
Using Y-STR Analysis
The Y-STR test is used to determine whether or not two or more males are related through their fathers via the paternal/male line (e.g. brothers, paternal cousins, grandfather/grandson, uncle/nephew, etc.). The results of this test are often used to provide additional evidence in paternity situations in which the possible father is not available for testing. The information carried on the Y chromosome remains relatively unchanged throughout generations. Because the Y chromosome follows the same father-to-son pattern as surnames in Western culture, this test is sometimes called the "Surname Test." A tangible timekeeper of history, the Y chromosome enables:
Tracing human evolution
Tracking migration patterns
Determining relatedness in groups of people
Answering paternity questions going back generations
Although Y-STR analysis cannot distinguish between males who belong to the same paternal line, it's useful for excluding males from a possible biological relationship. For example, if a male child and his possible paternal uncle are tested, their Y-STR profiles must match in order to determine a biological familial relationship. If they don't match, the potential uncle is excluded as a biological relative of the child.
NOTE: Because women do have a Y chromosome, individuals born as females cannot do this test. However they can ask their brother or father to test.
Using mtDNA Analysis
Mitochondrial DNA (mtDNA) is inherited through the maternal line throughout time, essentially unchanged. Unlike the Y chromosome that can only be passed down from father to son, females inherit mtDNA from their mother and pass on mtDNA to both their male and female children. Since mtDNA is maternally inherited, results from testing enable researchers to trace lineage far back in time. The mtDNA test shows the origin of maternal ancestors as well as individuals you're related to in the direct maternal line. Minor mutations that occur in mtDNA throughout generations make certain regions of mtDNA unique to a family line.
Again, while the Y-STR test can only be performed on males, the mtDNA test is applicable to both males and females. Because everyone from the same maternal line has similar mtDNA sequences, this test is an accurate method for establishing connections along the mother's familial line.
Not Sure Where to Start? Speak with DNA Diagnostics Center
Since every situation is unique, we highly recommend that you call our paternity experts at 1.800.929.0847 to ensure you order the test that's right for you. We're committed to helping you find the answers you need, and stand behind our promise of making the testing process convenient, confidential, accurate, and fast.
Call us for a FREE confidential consultation at 800-929-0847: We're here to help.
Do you have questions or comments about this topic? Share in the comments and we'll answer.
Vivian	on July 3, 2018 at 5:34 pm
I was contacted and told that family friends have reason to believe that my father is the father of 2 siblings. If i give them a DNA sample will that help them to find out the truth as to know if my father is also their father?
DDC	on July 5, 2018 at 10:32 am
Hi, Vivian. Yes, it can. Although with a half-sibling test you're more likely to get conclusive results if the mothers are able and willing to participate. Of course, the ultimate best scenario is for them to test with your father, if he's living and willing.
Katherine	on November 19, 2020 at 3:43 am
Hi do you have any offices in Durban South Africa
I am needing to prove that a British Father is the father of the South African child..
The child is in SA and the father is in the uk how do we go about thus and what is the procedure and cost
DDC	on November 19, 2020 at 8:50 am
Hi, Katherine. Yes, please contact our corporate partner in South Africa via this site: https://dnatest.co.za/
Barbara	on March 29, 2019 at 9:29 pm
I've wanted to see what kind of trible blood I have can DNA
Help see what type of bloodline I come from
DDC	on April 2, 2019 at 8:42 am
Hi, Barbara. There is no DNA test that can tell you what Native-American tribe you may or not be part of. Sorry!
Is There Such a Thing as an Infidelity Gene?
It's only natural for someone who has been cheated on to ask why their partner was unfaithful. A handful of studies have suggested that genes and hormones may predispose some men and women to infidelity. So that begs the question: Is cheating genetic? Here's a look at...
8 Mistakes You Don't Want to Make When Healing from Infidelity
Finding out that your spouse or partner has cheated can be devastating, but dealing with the betrayal irrationally could worsen the pain and delay healing. If you're wondering how to get over someone cheating on you, first ask yourself if you can forgive and trust...
10 Reasons Why Someone Might Cheat
Infidelity—or cheating on a partner—is alarmingly common, but knowing you're not alone doesn't make you feel any better. When infidelity hits close to home, it can be devastating. So why do people cheat in relationships and what does cheating say about a person? ...
How Much Does a DNA Test Cost?
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James Bain
James Bain spent more time in prison for a crime he didn't commit than nearly any other American exonerated through DNA evidence. He entered prison at age 19 and left a free man— at age 54.
In 1974, a nine-year-old boy who had been sleeping at home was dragged to a baseball field and raped. Despite Mr. Bain's alibi and conflicting serological evidence, he was convicted and sentenced to life in prison. The prosecution's case rested largely on the young victim's identification of Bain in a photo lineup.
In 2001, Mr. Bain took advantage of a new statute making it possible for certain cases to be reopened for DNA testing and filed motions with the court. He was denied five times. With the help of with Innocence Project Florida and Public Defender Bob Young, DNA testing by the DDC Forensics team took place in 2009. Analysis of the DNA on the boy's underwear excluded Mr. Bain, and he was ultimately exonerated.
Bain isn't angry about spending 35 years of his life in prison for a horrific crime he didn't commit. "How can I be?" he told the press. "You can't go back." Since his release, Bain has been a leading advocate for the innocence movement and shares his story with students across the country.
Largely based on being chosen from a photo lineup, Derrick Williams was convicted of kidnapping and raping an adult female in Palmetto, Florida in 1992. In 2011, he was exonerated with the help of DNA evidence.
On the night she was kidnapped and assaulted in an orange grove near her home, a woman managed to get away from her attacker while he was distracted and drove away in his car, leaving him behind. The car still contained his t-shirt, as well as a white cloth he'd been holding.
Inexplicably, most of the evidence collected at the time of the crime, including a rape kit, was destroyed by the Manatee County Sheriff's Office. Ultimately, Mr. Williams was convicted of the crimes, largely based on being chosen from a photo lineup. He served 18 years before his defense team with Innocence Project Florida worked to free him. Part of that process was contacting DDC to perform DNA testing.
The DDC Forensics team was able to exclude Mr. Williams through DNA analysis on one of the few remaining pieces of evidence that had not been destroyed: the t-shirt from the assailant's car, and after nearly two decades in prison, Mr. Williams' conviction was vacated.
Kerry Porter
Kerry Porter served 14 years for the Kentucky shooting death of his ex-girlfriend's husband: a crime he insisted he didn't commit. He was exonerated in 2011 after DNA testing by DDC excluded him.
Melanie Lowe, an attorney with the Kentucky Innocence Project worked on Mr. Porter's case for five years prior to contacting the DDC Forensics team to request forensic testing on previously-untested evidence: duct tape used to make a homemade gun silencer found at the crime scene.
DNA analysis of this key piece of evidence excluded Kerry Porter as a contributor to the genetic material identified on the duct tape. The Louisville Metro Police Department's cold-case unit worked closely with the Innocence Project and was also instrumental in the dismissal of charges and Mr. Porter's exoneration.
Mr. Porter was released from the Eastern Kentucky Correctional Complex around Christmastime in 2011 after wrongfully serving 14 years. His story is another example of how post-conviction DNA analysis, which often includes testing severely degraded or difficult samples, can free the innocent and change a life forever.
Cheydrick Britt
Cheydrick Britt served 9 years for a sexual-battery conviction in Florida. He was exonerated in 2013 after DNA testing by DDC showed that he did not contribute to the evidence on the victim's underwear.
"DNA testing is a powerful tool to help uncover the truth, and its value is especially evident in cases like this where it helps a wrongfully-convicted man be set free," said Dr. Julie Heinig, leader of the DDC Forensics team.
Mr. Britt rejoiced, "I would like to thank Dr. Heinig and her entire staff for the tireless efforts in pursuing justice. My family and I greatly appreciate their services and patience that they have showed in my case, and I am very proud of their work."
Mr. Britt's legal team, including attorneys with the Innocence Project Florida, meticulously reviewed and worked through the original testing files and requested testing. DNA testing conducted by DDC identified a DNA profile on the victim's underwear that excluded Mr. Britt. He was released from incarceration shortly before Christmas.
Raymond Towler
A routine traffic stop proved costly to Raymond Towler when he was identified as matching the suspect's description for the horrific assault and rape of two children in a Cleveland Park.
Despite a lack of evidence and a corroborated alibi, Mr. Towler was convicted of the crimes and sentenced to 12-40 years due to the young age of the victims. Despite three earlier attempts to prove his innocence using DNA testing, a breakthrough occurred when DDC Forensics was able to demonstrate that male DNA from one of the victims' underwear excluded Mr. Towler as the perpetrator.
Mistaken identification during a photo lineup put him in prison, but the power of DNA science freed him. An Ohio judge choked back tears as she ordered Mr. Towler's release on May 5, 2010. After nearly three decades behind bars for crimes he did not commit at the age of 24, he was exonerated and became a free man at the age of 52.
Ralph Daniel Wright, Jr.
The Supreme Court of Florida reversed a two-count murder conviction and death penalty sentence for the 2007 murder of Wright's ex-girlfriend Paula O'Connor and their son Alijah.
On July 6, 2007, Paula O'Conner and her 15-month-old son Alijah were found deceased in their Tampa, Florida home. The mother had been strangled and her baby had been suffocated. Ralph Wright Jr., then an Air Force sergeant stationed nearby, was O'Conner's boyfriend at the time, came under suspicion, and was charged with two counts of first-degree murder.
There had been disagreement between Wright and O'Conner over whether or not Alijah was Wright's biological child, and in addition to this motive, the state's case also focused on a single black glove that was at the crime scene. That glove was the same type issued for Wright's military unit, but DNA testing by DDC could not match the glove to Wright nor could it be absolutely determined whether or not that glove actually came from Wright's Air Force base.
Despite the lack of forensic evidence, Wright was found guilty on both murder counts and sentenced to death in 2014. The convictions were appealed to the Florida State Supreme Court. Based on many factors, including the results of the DNA testing performed by DDC, in May of 2017 the court unanimously vacated Wright's conviction for the murders, ruling that the evidence against him was "purely circumstantial."
Dewey Jones
Dewey Jones was convicted of robbing and killing 71-year old Neal Rankin, a family friend, at Rankin's Akron home in 1993. He always maintained his innocence in Rankin's murder.
Dewey Jones was convicted of robbing and killing 71-year old Neal Rankin, a family friend, at Rankin's Akron home in 1993. While Jones had prior convictions related to drug trafficking and passing bad checks, he always maintained his innocence in Rankin's murder.
In early 2012, the Forensics Team at DNA Diagnostics Center released its findings from the laboratory tests conducted on crime-scene DNA evidence. A partial male DNA profile was found on the piece of rope used to tie Rankin's wrists, the knife used to cut the rope, and pieces of Rankin's shirtsleeves. None of the DNA matched Jones' DNA.
In 2012, Summit County Judge Mary Margaret Rowland overturned the conviction and granted a new trial. After two more years of legal proceedings, Jones' charges for aggravated murder, kidnapping, and robbery were dismissed and he became a free man.
Robert McClendon
Robert McClendon served 18 years for the rape of a 10-year-old girl. He was exonerated in 2008 after DNA testing by DDC showed that he did not contribute to the evidence on the girl's underwear.
Robert McClendon always maintained he did not commit the crime of which he was accused. DNA testing performed by the forensics team at DDC enabled inmate McClendon to walk out of the Franklin County Courthouse a free man.
McClendon's incarceration was 1 of 30 cases in Ohio that were identified to have "legitimate claims of innocence" in a joint investigation conducted by The Columbus Dispatch and the Ohio Innocence Project. Of these, 14 were approved for DNA testing by the presiding judges and county prosecutor's offices.
In McClendon's case, DDC's forensic laboratory tested evidence submitted by the prosecutor. After traces of semen were found, a Y-STR DNA test was performed, which generates a DNA profile from the Y chromosome (found only in males). A comparison to the Y-STR profile from McClendon revealed that he could not have been the source of the DNA from the semen stain.
Lydell Grant
DNA test frees man serving life sentence for Houston murder, leads to new arrest.
New DNA testing in a 2010 Houston murder case has led to the exoneration of one man — after nine years behind bars — and the arrest of another.
Lydell Grant, 43, was found guilty of stabbing 28-year-old Aaron Scheerhoorn to death in the vicinity of a nightclub on the sworn testimony of multiple witnesses, according to reports. He was cleared last month — and released from prison — after DNA obtained from Sheerhorn's fingernails was tested, using new technology. He was serving a life sentence.
On Thursday, Jermarico Carter, 41, was charged with the murder after the same DNA linked him to the crime. Investigators got a match to his DNA using an FBI database containing the DNA of convicted criminals, Fox 26 Houston reported Sunday.
"On behalf of the Houston Police Department, I want to extend an apology to Mr. Grant and his family as they have waited for justice all these years," Houston Police Chief Art Acevedo said Friday on Twitter. Acevedo said Carter "has recently confessed to his role in Mr. Scheerhoorn's killing." Carter was in custody in Georgia on unrelated charges, the chief said.
Grant said Saturday that he wasn't mad at Grant for sitting in jail for a crime he committed, Fox 26 reported. "I'm not mad at him at all," he said, according to the station. "I forgive him because he know now what he did."
The station reported that as of Saturday Grant no longer had to wear an ankle monitor or abide by a curfew. Houston prosecutors said they will move for Grant's formal exoneration before the Texas Court of Criminal Appeals. Grant's lawyer Mike Ware of the Innocence Project of Texas was quoted by The Associated Press as saying that he believes erroneous witness identifications based on outdated and flawed techniques used by detectives helped to wrongly convict his client.
Mistaken witness identifications contributed to more than 70 percent of the more than 360 wrongful convictions in the U.S. that have been overturned by post-conviction DNA evidence, according to the Innocence Project.


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