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So! Have you seen it? Have you? Have you? And by "it," I mean, of course….
I didn't always feel this way about BBC Sherlock. I don't watch a lot of television, so I missed it entirely when it came out, and just accidentally happened onto it on PBS one night. It was the scene in "The Blind Banker" where Soo Lin is telling her story to Sherlock and John in the museum, and it just…didn't grab me at all. Several months later, I decided to try it again via iTunes, and I was hooked. Completely.
The Sherlockian world is so very broad, and there are so many stories and films and aspects of the hobby that it can, I think, accommodate just about every possible interest. Which is wonderful, but it means that, while you may find collecting pastiche is your passion, someone else may only be enraptured with the Canon and a collection of vintage pipes. The woman sitting next to you at a scion meeting may be the slash queen of Tumblr, while you may run a blog centered around John Barrymore. The man who irritates you with his continual chronology arguments may share your desire to collect every Holmes figurine ever, even the weird mugs with faces. What I'm trying to say is, appreciate others' Holmesian interests, even if (and perhaps especially if) you don't share them, and once in awhile, step out and try something new. You may find you like it.
Keep an open mind–like Anderson!
Question 1: List three Canon references which appeared in Sherlock:The Abominable Bride. Be sure to include the story in which each appeared!
Question 2: List 3 brides (or brides-to-be) whom Holmes helped in the Canon. Be sure to give the names of the stories in which they appear!
As always, send your answers via blog comment, Facebook PM or Twitter DM! Good luck!
Congratulations to Claire Danes, winner of the Otto Penzler volume of Sherlock Holmes short stories! The violin pieces submitted were: Hoffman's Barcarolle (several times), the Chopin piece Holmes sings in STUD, Mendelssohn's Lieder, "Gabrielle," from "The Private Life of Sherlock Holmes," Sarasate's program from REDH and, of course, Patrick Gower's opening from Granada Holmes.


text:
Q: query json in a string column bigquery I would like to know if it is possible to query a json string column that will unnest the products without having to specify the index.
Example
with mytable as (
select
'{"ecommerce":{"purchase":{"actionField":{"id":"T12345","affiliation":"Online Store","revenue":"35.43","tax":"4.90","shipping":"5.99","coupon":"SUMMER_SALE"},"products":[{"name":"Triblend Android T-Shirt","id":"12345","price":"15.25","brand":"Google","category":"Apparel","variant":"Gray","quantity":1,"coupon":""},{"name":"Donut Friday Scented T-Shirt","id":"67890","price":"33.75","brand":"Google","category":"Apparel","variant":"Black","quantity":1}]}}}' as eec
)
select
JSON_EXTRACT_SCALAR(eec, "$['ecommerce'].purchase.actionField.id") AS order_id,
JSON_EXTRACT_SCALAR(eec, "$['ecommerce'].purchase.products[0].name") AS product_1
from mytable
union all
select
JSON_EXTRACT_SCALAR(eec, "$['ecommerce'].purchase.actionField.id") AS order_id,
JSON_EXTRACT_SCALAR(eec, "$['ecommerce'].purchase.products[1].name") AS product_1
from mytable

Expected output
order_id    product_1
T12345  Triblend Android T-Shirt
T12345  Donut Friday Scented T-Shirt

enter image description here
But i would like to obtain this output without having to do a union of product[index] but have something that will repeat and unnest automatically as much element that there is inside product

A: #standardSQL
CREATE TEMP FUNCTION json2array(input STRING)
RETURNS ARRAY<STRING>
LANGUAGE js AS """
  return JSON.parse(input).map(x=>JSON.stringify(x));
""";
WITH mytable AS (
  SELECT
  '{"ecommerce":{"purchase":{"actionField":{"id":"T12345","affiliation":"Online Store","revenue":"35.43","tax":"4.90","shipping":"5.99","coupon":"SUMMER_SALE"},"products":[{"name":"Triblend Android T-Shirt","id":"12345","price":"15.25","brand":"Google","category":"Apparel","variant":"Gray","quantity":1,"coupon":""},{"name":"Donut Friday Scented T-Shirt","id":"67890","price":"33.75","brand":"Google","category":"Apparel","variant":"Black","quantity":1}]}}}' AS eec
)
SELECT
  JSON_EXTRACT_SCALAR(eec, "$['ecommerce'].purchase.actionField.id") AS order_id,
  JSON_EXTRACT_SCALAR(product, "$.name") AS product_1
FROM mytable,
UNNEST(json2array(JSON_EXTRACT(eec, "$['ecommerce'].purchase.products"))) product

with output
Row order_id    product_1
1   T12345      Triblend Android T-Shirt
2   T12345      Donut Friday Scented T-Shirt

redpajama_set_name: RedPajamaStackExchange


Between Tuesday the 1st and Wednesday the 3rd of July the adolesc ents of the parish took part in a live-in in a farmhouse in Ta' Kerċem. During this time, they took part in various games and on their last day they went to swim in Ħondoq ir-Rummien.
Children will be able to attend Mass at 9.00 a.m. from Monday to Friday (except Wednesday) as from Monday the 7th of July. All the parish children are encouraged to attend.
As from Tuesday the 8th of July at 12.30 p.m. parishioners are encouraged to come together to work on the 'għażżiel' in preparation for the feast of September.


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text:
Home > News > SLLWP closure undermines university promises
SLLWP closure undermines university promises
6th March 2010 News
The Students' Union has been left reeling from the latest departmental changes proposed by the University with the closure of the School of Lifelong Learning and Widening Participation (SLLWP) as an academic institution.
The department was established to support the local community and to widen participation in higher education for young people and adults of all ages. It played a key role in the PASSPORT to higher education scheme, providing a link between college and university life in the local area.
On 24 February, Senate accepted proposals to end the academic lifespan of the school, closing one section of the department, the Centre for Training and Development (CETAD). The Department of Continuing Education (DCE) was moved to the Division of Marketing and External Linkages, a subsection of Central Services, effectively ensuring that the department would no longer play a role at the university as an academic institution but rather as a subsection of university management.
This particular development is a direct contrast with the words of the Vice-Chancellor, Professor Paul Wellings. Speaking at the opening of the SLLWP, he said: "The new School of Lifelong Learning and Widening Participation should be seen as a portal for ideas and activities which will enable the University to reach out into the community and offer more flexible learning opportunities. The University has high expectations of where this venture will lead."
The move has been viewed by many as a betrayal of the original principles behind the establishment of the school and was opposed by the Students' Union in Senate.
"The laying down of the School for Lifelong Learning and Widening Participation and the Department of Continuing Education is lamentable. What's more, in spite of economic and political changes over recent years it seems an absolute about-face for the University," said LUSU President Michael Payne. "Whatever the future for widening participation work in the University – the message sent out by this move is not a praiseworthy one."
He continued to question the quality of consultation given to the students at Lancaster, adding: "There hasn't been a great deal of consultation but the Students' Union has nevertheless been very vocal, pro-active and supportive of the activities of DCE and SLLWP ever since the cut back of DCE activities over a year ago. That is something students can and should be proud of."
The Students' Union has stressed its continuing support for the department and its staff throughout its existence.
The move has also prompted some questions over the university's handling of the SLLWP, especially with regards to the movement of CETAD to the School of Health and Medicine from August 2009 which rendered many sections of the SLLWP ineffective as an individual academic institute.
LUSU Vice-President (Academic Affairs) Danny Ovens said of the move "I think CETAD moving to the School of Health and Medicine sealed the deal. I think it is important to note that the 'decline' is just in the departments connected to the school disappearing, not the importance of its activity."
He continued "I think it is essential that this 'move' to MEL [Marketing and External Linkages] doesn't hinder the activities of the PASSPORT scheme as it does some brilliant work on widening participation and shouldn't be lost in some broom closet in University House."
The PASSPORT scheme will be continued by the university where it is hoped its move to central services will ensure it remains a large component of the University's interaction with the local community.
Lancaster University, Lancaster University Management School, LUSU, Paul Wellings, senate
"A glorified school disco" – plans to move Grad Ball to the Great Hall fall flat among students
24th April 2018 James Gilmour
Ticket slump for Christmas balls despite colleges pulling out the stops
Grad Ball: What do you think?
Success for Sugarhouse as attendance records smashed
Interview: Full-Time Officers Team 2019/20
2nd April 2019 Domenica Giorgianni
Bailrigg FM complete 24 hour broadcast
Lancaster storms ahead in 'Votes for Students' campaign
LURPS to begin using cap guns

redpajama_set_name: RedPajamaCommonCrawl


text:
package org.iotivity.service.easysetup.core;

/**
 * This interface contains the connection details of the enrollee device
 */
public interface OnBoardingConnection {

    /**
     * This method returns the implementation of the Connection
     *
     * @return implementation
     */
    public Object getConnection();

    /**
     * This method returns description on the connectivity.
     *
     * @return description
     */
    public String getDesc();

    /**
     * This method is used to know if the device is connected or not
     *
     * @return True if connected or False
     */
    public boolean isConnected();

}

redpajama_set_name: RedPajamaGithub


text:
DC&MARVEL
Posted in DC&MARVEL
A Flip of a Coin – Symbolism Behind Harvey Dent's Coin Toss
Author: Eren Published Date: 10/14/2020
Comic book fanatics are well aware of the character of Two-Face, sometimes known as Harvey Dent. The former District Attorney of Gotham city is one of Batman's toughest foes and Bruce Wayne's former friend. For those who are new to this character, the name Two-Face comes from him having one normal side of his face while the other is horribly disfigured.
Comic book fans also know that the origin of him becoming Two-Face was during a trial when a convicted felon threw acid at his face. Deforming one of his sides and leaving it permanently scarred. Fans of the Batman Animated Series would say that half of Harvey's body was disfigured due to a chemical explosion. Burning the entire side of his body. It is the origin story that Christopher Nolan used in his Dark Knight movie but slightly modified.
Nevertheless, whichever origin story you go with the outcome is still the same. Harvey Dent has gone through a terrible ordeal by being disfigured and it has left his psyche scared as well as his body.
Pat Loika [CC BY (https://creativecommons.org/licenses/by/2.0)]
It leads him to form a dual personality, his inner demon and both of them struggle for dominance. The only wat they can decide how to act is with a toss of a coin. The scared side usually means that something bad is going to happen and Two-Face prevails. While the unsacred side of the coin means that Harvey is the one at the wheel and bad things are not going to happen.
But what is the significance of the coin toss? Is it a type of gamble? Is Harvey placing a bet every time he tosses it?
Harvey Dent was once viewed s Gotham's White Knight. He was constantly facing and determining the fate of bad guys. He was the one who wanted to make good on all the wrongdoings happening in the city. But all the crime and corruption can be overwhelming at times. Not being able to cope and fight against the situation can lead to some damaging problems.
This is what happened when Harvey's face got burnt. It unleashed a demon inside him that wanted to be the bad guy. But the good persona is still there to fight. The White Knight Harvey sill wants to make all the bad wrongs good.
The coin toss is the symbol of his duality. It is the fight between his inner demons and himself. The is both good and the bad, we all have that inside of us. With some people, one inside is more apparent than the other. While with Harvey, it is cut right down in the middle.
He is definitely struggling and he prefers to leave everything to fate than actually deciding by himself. But it is still a dangerous game to play. A string of bad luck, or tosses, will deepen the descent into madness.
So I would say that Harvey is playing a dangerous game and it is a gamble every time to coin leaves his hand and swings in the air.
Author: Eren
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Copyright © 2021 animerica-mag.com


text:
Eduard Grau (Barcelona, 1981) és un director de fotografia català. Dels dels 13 anys va tenir clar que es volia dedicar al cine, i per això va estudiar a l'ESCAC de Barcelona i a la NFTS de Beaconsfield (Anglaterra). En algunes entrevistes explica que gravar una pel·lícula amb Tom Ford li va canviar la vida. La seva fotografia sobre Buried de Rodrigo Cortés li va valdre una nominació als Premis Goya. També ha fet anuncis i videoclips com Born This Way de Lady Gaga.

Filmografia seleccionada
 2006 : Honor de cavalleria d'Albert Serra
 2009 : A Single Man de Tom Ford
 2010 : Buried de Rodrigo Cortés
 2010 : Finisterrae de Sergio Caballero
 2011 : La Maison des ombres de Nick Murphy
 2012 : No Place on Earth de Janet Tobias
 2014 : Suite française de Saul Dibb
 2015 : The Gift de Joel Edgerton
 2015 : Sufragistes de Sarah Gavron
 2017 : Quién Te Cantará de Carlos Vermut

Referències

Directors de fotografia catalans
Artistes barcelonins contemporanis

redpajama_set_name: RedPajamaWikipedia


text:
\section{Introduction}

In a complicated system such as our universe, which passes through various
stages of evolution and is presently undergoing an accelerated expansion,
driven supposedly by a dark energy (DE) component
\cite{CST-rev,FTH-rev,AT-book,wols-ed,MCGM-ed},
it is natural to ask: {\it will it ever get stabilized or atleast approach
to a quasi-stable stage asymptotically?} The best way to address this is to
resort to the dynamical analysis that enables one to assert the qualitative
description of evolving systems without the prior need of any initial
conditions. As is well-known, observations such as that of the Cosmic
Microwave Background (CMB)
\cite{wmap9-CP,wmap9-Fin,Planck15-CP,Planck15-DEMG,Planck18-CP},
large-scale structures (LSS)
\cite{rsdnev,rsdhou},
type-$1$a Supernovae
\cite{bet-SN,scol-SN},
and so on, generally concord to the cosmological constant ($\Lambda$) candidature
of the DE, or more specifically the $\Lambda$CDM model (where CDM stands for the
cold dark matter). However, because of the severe theoretical problems, viz.
{\em fine-tuning} and {\em coincidence} associated with $\Lambda$ (or the $\Lambda$CDM),
one's focus gets shifted to the dynamical DE, possibly from the scalar fields,
such as quintessence
\cite{CRS-quin,CLW-quin,tsuj-quin},
k-essence
\cite{AMS2000-kess,AMS2001-kess,MCLT-kess,schr-kess,SSSD-kess,SS-dquin},
etc., or from the so-called {\em modified gravity} (MG) scenarios
\cite{NO-mgDE,tsuj-mgDE,CFPS-mg,JBE-mgDE,papa-ed,NOO-mg,SSASB-MST},
that go beyond the realm of General Relativity (GR).

Apart from the extensive studies of the possible consequences of a dynamically
evolving DE, a considerable interest has recently been developed on the
interaction(s) or even the unification of the DE component with the matter
field(s), from various perspectives
\cite{WS-intDE,amend-intDE,CPR-intDE,FP-intDE,CW-intDE,CHOP-intDE,CHP2008-intDE,
CHP2009-intDE,amend-rev,ban-intDE,BBM-uDE,BBPP-uDE,GNP-uDE,FFKB-uDE,CDS-MMT,SDC-MMT}.
Most notable are the MG cosmological scenarios, or the scalar-tensor (ST)
equivalents thereof
\cite{FM-ST,frni-ST,BP-ST,BEPS-ST,TUMTY-ST,ENO-ST,CHL-ST,BGP-ST,ST-ST,ST-KT,ASBSS-MSTda,ASBSS-MSTpp},
which naturally give rise to the DE-matter (DEM) interactions under conformal
transformations. An interaction of such a sort can have a significant effect
on the cosmic expansion rate as well as on the formation of large-scales
structures. Hence, the stability analyses of the corresponding cosmological
solutions, on the physical ground (i.e. against the density perturbations)
as well as mathematically (i.e. in the phase space), are crucial.
While the former has been carried out comprehensively, for a typical (and well-motivated) ST equivalent MG scenario, in a preceding work
we focus on performing the latter (i.e. the dynamical analysis) in this
paper, by resorting to not just the background cosmological level, but to
that of the linear density perturbations as well.

The class of ST theories under consideration is the one with a specific
(quadratic) form of the non-minimal coupling between a scalar field $\phi$ and
the Ricci scalar curvature $R$, with the potential for $\phi$ being just in
the form of a mass $m$, in the original Jordan frame. The corresponding
Lagrangian has its equivalence with that of a wide range of MG scenarios,
starting from some variants of $f(R,\phi)$ to metric-scalar-torsion (MST)
formulations, and so on.
A conformal transformation to the Einstein frame, followed by a field
redefinition $\phi \to \varphi$, essentially leads to an exponential coupling of
$\varphi$ with matter (including the CDM), and hence an effective DEM interaction,
once $\varphi$ is considered to be the entity that induces the DE.

Now, the objective of the dynamical system analysis is to determine the critical
points (CPs), and examine their nature, or more specifically, the eigenvalues of
the linear perturbation matrix $\mathcal M$ corresponding to the autonomous set of
equations, which one suitably constructs out of the governing field equations.
The essence of the CPs, which are the equilibrium solutions of the autonomous
equations, lies in their predictability in asserting whether a system is stable
or not. Note however that more the number of autonomous variables (which
characterize the mathematical state of the system), more would be the number
of phase space trajectories representing the plausible ways in which the system
can evolve. In other words, an increased number of autonomous variables would
inevitably imply a less restrictive dynamical analysis and a better scope of
figuring out a stable configuration to which the system may transpire. In this
sense it is much desirable for one to be open to the possibility of the DEM
interactions, and also take the cosmological density perturbations in
consideration, so as to have additional autonomous variable(s) enhancing the
dimensionality of the phase space.
\cite{da-BLPS,da-land}.

As is well known, the LSS can more or less be described correctly just by the
matter density perturbations. The scalar field perturbations are generally too
small to make any significant contribution, at least at the sub-horizon scales
\cite{koiv-grow,PA-grow,PG-grow,GP-grow}.
In presence of DEM interaction, the field and matter density perturbations get
entangled with each other, such that the overall study of the formation of
LSS requires a full-fledged analysis of each perturbed component. Nevertheless,
being small in magnitude, one can at least incorporate the on-an-average
contribution of field perturbations on the matter density perturbations. Larger
is the coupling between the scalar field and matter, larger would be the field
contribution on the matter density perturbations. Hence, in view of the high
dominance of the latter at the sub-horizon scales, it become reasonable to
consider it (or more precisely, the matter density growth factor $f$) as a
autonomous variable for our dynamical analysis.
Suitably defining the other dimensionless autonomous variables, we set up
the full set of autonomous equations, considering for simplicity the overall
(visible + dark) matter content of the universe to be {\it dust}-like, albeit
with the interaction with the scalar field $\varphi$.

With the matter density perturbations taken into consideration, the system
is found to possess more CPs than in the case in which the analysis is
restricted to the background level cosmology. We qualitatively distinguish
the CPs by examining the eigenvalues of the linear perturbation matrix $\mathcal M$
for the autonomous set of equations, and solve the latter simultaneously to
obtain the phase space trajectories. In addition to having the matter
perturbation growth factor $f$ as a dynamical variable, we revert back to
the background level physical variables, viz. the matter density parameter
$\Omega^{(m)}$ and the DEM interaction parameter (or the $\varphi$-matter coupling
parameter) $n$. This then enables us to obtain the $\Omega^{(m)}\,$--$\,f$ phase
portraits, for $n = 0$ ($\Lambda$CDM), and for a certain fiducial setting $n =
0.1$.

A similar procedure can be followed while the growth factor is formulated in
terms of the growth index $\gamma$. However, for the interacting system one finds
an inconsistency with the parametrization $f=[\Omega^{(m)}]^\gamma$ commonly used in the
literature
\cite{WYF-grwpara,PAB-grwpara,BBS-grwpara,BP-grwindx,SBM-grwindx,bat-grwindx,
MBMDR-grwindx,PSG-grwindx,BA-grwindx,HKV-modsel,SS-MSTintDE}.
So, to cope with this, we use an alternative parametrization which we have
proposed in the earlier paper
\cite{MS-SS-IJMPD}.
From the corresponding phase portraits, we find that there can be only one
physically plausible trajectory which exhibits a very mild time-evolution
of $\gamma$ at high redshifts, which picks up gradually near the present epoch
and beyond. For this trajectory, the growth index is not much deviated from
the $\Lambda$CDM value $6/11$, throughout the expansion history of the universe,
which implies that for whatever interacting picture we resort to, the
overall (and observationally favoured) $\Lambda$CDM cosmology is not much
distorted.

\vspace{5pt}
\noindent
{\large \sl Conventions and Notations}: Throughout this paper, we use metric
signature $\, (-,+,+,+)$ and natural units, with the speed of light $c = 1$.
We denote and the gravitational coupling factor by $\, \kappa = \sqrt{8 \pi G_N}$,
where $G_N$ is the Newton's constant, the metric determinant by $g$, and the
values of parameters or functions at the present epoch by an affixed subscript
`$0$'.

\section{Interacting DE-matter framework}
\label{sec:intDEM}

Let us begin with a recourse to the DEM interaction(s) in the standard
Friedmann-Robertson-Walker (FRW) cosmological framework, limiting our
attention to the scenarios in which such interactions arise naturally.
Note that the very formulation of a gravitational theory, such as GR,
leaves the scope of such interactions. To be specific, the conservation
of the total energy-momentum tensor, does not insinuate to that for each
individual component in a multi-component system, such as the universe,
composed of baryons, CDM, DE, radiation etc.
Since the late-time evolution of the universe is effectively driven by
two main constituents, viz. the DE and the (visible + dark) matter, any
breach of their individual self-conservation (due to interactions) can
affect the dynamical stability of the universe. It is then natural to ask:
{\em what happens to such an interacting system if one slightly perturbs it
in the space of the autonomous variables about the critical points?}

Consider now a DE component, induced by a scalar field $\varphi$, is interacting
with matter. Let $T_{\mu\nu}^{(m)}$ and $T_{\mu\nu}^{(\varphi)}$ denote the respective
energy-momentum tensors. One has
\begin{equation} \label{cons-eqn}
\nabla\!^{\,\mu} \left[T_{\mu\nu}^{(m)} + T_{\mu\nu}^{(\varphi)}\right] = 0 \, .
\end{equation}
This in general implies
\begin{equation} \label{cons-eqn1}
\nabla\!^{\,\mu} \, T_{\mu\nu}^{(m)} =\, -\, \nabla\!^{\,\mu} \, T_{\mu\nu}^{(\varphi)}
=\, \mathcal Q_{\nu} \, ,
\end{equation}
where $\mathcal Q_{\nu}$ is an interaction four-vector.

Let us note that an interaction of type $\mathcal Q_{\nu}$ although seems to be
arbitrary in Eq. (\ref{cons-eqn}), arises naturally in the ST equivalent
formulations of MG theories. In other words, the non-minimally coupling
of a scalar field with the Ricci scalar $R$ in the Jordan frame induces
a coupling between scalar field and matter in the Einstein frame. In this
paper, we consider a class of such ST formulation for $\mathcal Q_{\nu}$ takes
the form
\cite{MS-SS-IJMPD}:
\begin{equation} \label{Q}
\mathcal Q_{\nu} = \kappa n \rho^{(m)} \varphi_{,\mu} \, , \quad \mbox{such that}
\quad \kappa \phi = e^{n\kappa\varphi} \, ,
\end{equation}
where $\kappa^2=8\pi G$, $n$ is a coupling parameter and $\rho^{(m)}$ is the matter
density.

For a background cosmological evolution described by the spatially flat
metric, and driven by a DEM interaction given by Eq. (\ref{Q}), the
corresponding Friedman and the Klein-Gordon equation for $\varphi$ are given as
\begin{eqnarray}
H^2 \equiv \left(\frac{\dot{a}}{a}\right)^2 = \frac{\kappa^2} 3 \left[ \rho^{(m)} +\rho^{(\varphi)} \right] \, , \label{Friedmann} \\
\ddot{\varphi} + 3H\dot{\varphi} + U_{,\varphi} = \frac{3n}{\kappa} H^2 \Omega^{(m)} \dot{\varphi} \, ,
\label{KG-eqn}
\end{eqnarray}
where an overdot represents derivative with respect to the cosmic time $t$,
$H$ is the Hubble parameter, $\rho^{(\varphi)}$ is the energy-density of field $\varphi$ and
$\Omega^{(m)}:= \kappa^2\rho^{(m)}/(3H^2)$ is the matter density parameter. Note that for our
analysis we consider an exponential potential of the form
$U(\varphi)=\Lambda e^{-2\kappa n\varphi}$ (where $\Lambda$ is the value of $U$ at present epoch
$t=t_0$), which lead to exact background solutions, as pointed out in ref.
\cite{MS-SS-IJMPD}.

Now, we know that a given matter density perturbation $\delta \rho^{(m)}$ plays the
all-important role in the LSS formation. However, in presence of an interaction
between matter and the DE induced by a field $\varphi$, a small but not necessarily
negligible effect of $\varphi$ on the LSS may result. This may be perceived from
the equations of motion of the matter density contrast $\delta^{(m)}:= \delta \rho^{(m)}/\rho^{(m)}$
given by
\cite{MS-SS-IJMPD},
\begin{equation} \label{ddmt}
\delta^{(m)}_{,NN}  +\left[2(1-2n^2)-\frac{3 \Omega^{(m)}}2 \right] \delta^{(m)}_{,N} = \frac{3(1+2n^2)}2 \Omega^{(m)} \delta^{(m)} \, ,
\end{equation}
where $N(t)= \ln a(t)$ is the number of e-foldings. Note that the rightmost
term, which is nothing but the source term for $\delta^{(m)}$ gets enhanced by a
factor of $2n^2$, regardless of its contribution in the drag-force due to
the background expansion (middle term). Let us emphasize that this effect is
not just crucial from the perspective of parametric estimations but also from
the dynamical behavior of the system even up to the perturbative level, as
more solutions are expected than in the case when the effects of DE perturbations
are ignorable.

\section{Critical points and their stability conditions}
\label{sec:CP}

In order to perform the dynamical system analysis, let us define following dimensionless quantities:
\begin{equation} \label{X,Y,U}
X:=\kappa \frac{\dot \varphi}{\sqrt{6}H} \, , \quad Y:= \kappa \frac{\sqrt{\Lambda} e^{-\kappa n \varphi }}{\sqrt{3}H} \, , \quad f:=  \frac{\delta^{(m)}_{,N}}\delta^{(m)} \, ,
\end{equation}
such that from Eq. (\ref{Friedmann}) it follows that
\begin{equation} \label{mdp}
\Omega^{(m)} = 1-X^2-Y^2 \, .
\end{equation}
The coupled set of first-order differential autonomous equations can be worked out as
\begin{eqnarray}
X_{,N}&=&-\frac{1} 2 \left(\sqrt{6} n-3 X\right) \left(X^2-Y^2-1\right) \, ,
\label{dxdn}
\\
Y_{,N}&=&-\frac{1}2 Y \left(2 \sqrt{6} n X-3 X^2+3 Y^2-3\right) \, ,
\label{dydn} \\
f_{,N}&=& -\frac{1} 2 \left[2f^2 -f\left(3(X^2-Y^2)-1+2\sqrt{6}nX \right) +3 (1+2n^2)(X^2+Y^2-1)  \right] \, .
\label{dudn}
\end{eqnarray}
where we have used Eqs. (\ref{Friedmann}), (\ref{KG-eqn}) and (\ref{ddmt}) and relation $dN=Hdt$. Taking the derivative of Eq. (\ref{mdp}) and using Eqs. (\ref{dxdn}) and (\ref{dydn}), we find that
\begin{equation} \label{dmpd}
\Omega^{(m)}_{,N} = \left(X^2+Y^2-1\right) \left(\sqrt{6} n X-3 X^2+3 Y^2\right)
\end{equation}

The solutions or the critical points (CP) of above set of Eqs. (\ref{dxdn}), (\ref{dydn}) and (\ref{dudn}) can be obtained by simultaneously putting
\begin{equation} \label{dynam-eqns}
\frac{dX}{dN}=\frac{dY}{dN}=\frac{df}{dN}=0 \, .
\end{equation}
The obtained CPs: ($X_c,Y_c$, $f_c$) are enlisted in table (\ref{tab:CP}). In that table, we have shown all the CPs together with their corresponding field energy density parameter $\Omega^{(\varphi)}$ and its equation of state parameter $w^{(\varphi)}:=p^{(\varphi)}/\rho^{(\varphi)}$. Also, due to the invariance in the form of autonomous Eqs. (\ref{dxdn}), (\ref{dydn}) and (\ref{dudn}) under the exchange of $Y \to -Y$, we represent both positive and negative $Y_c$ in a single CP. In other words, both signs will give identical set of eigenvalues and hence stability condition(s).

In order to obtain the stability condition(s) for CPs given in table (\ref{tab:CP}), one requires to add small perturbations ($|\zeta| \ll 1$) to each one of them and to check whether the system still remains intact or gets deviated from its original configuration.
Let us consider small perturbations around $X_c$, $Y_c$ and $f_c$:
\begin{equation} \label{pert-CP}
X=X_c + \delta X \, , \quad Y=Y_c+\delta Y \, \quad \mbox{and}\quad f=f_c+\delta f
\, ,
\end{equation}
such that $\zeta \equiv\{\delta X,\delta Y,\delta f\}$ satisfies the following relation:
\begin{equation}  \label{pert-CP-eqn}
\frac{d\zeta} {dN} = \mathcal{M}\, \zeta \, ,
\end{equation}
where $\mathcal{M}_{(3\times3)}$ is the Jacobian matrix evaluated at $\{X_c, Y_c, f_c\}$. Note that the sign of each eigenvalue of a matrix $\mathcal{M}$ for a given CP determines the stability of autonomous system of equations. In particular, if all the eigenvalues are negative (positive), then that CP is stable (unstable) or saddle otherwise, provided that each eigenvalue is real-valued and non-zero. In table (\ref{table:Eigenvalues}), we have shown all the obtained eigenvalues together with their stability condition(s). Let us now examine the stability criteria of all CPs one by one:

\begin{table*}
\centering
\renewcommand{\arraystretch}{1.6}
\begin{tabular}{|c|ccc|c|c|c|}
\hline
CP & $X_c$ & $Y_c$ & $f_c$ & $\Omega^{(\varphi)}$ & $w^{(\varphi)}$ & $\mathsf w$ \\
\hline\hline
(a) & $-1$ & $0$ & $0$ & $1$ & $1$ & $1$ \\
\hline
(b) & $1$ & $0$ & $0$ & $1$ & $1$  & $1$ \\
\hline
(c) & $-1$ & $0$ & $1-\sqrt{6n}$ & $1$ & $1$  & $1$  \\
\hline
(d) & $1$ & $0$ & $1+\sqrt{6n}$ & $1$ & $1$  & $1$ \\
\hline
(e) & $\sqrt{\rfra{3}{2n^2}}$ & $\pm \sqrt{\rfra{3}{2n^2}-1}$ & $2\!-\!\sqrt{6n^2-2-\rfra{9}{2n^2}}$ & $\rfra{3}{n^2}-1$ & $ \rfra{n^2}{3-n^2}$  &
$1$ \\
\hline
(f) & $\sqrt{\rfra{3}{2n^2}}$ & $\pm \sqrt{\rfra{3}{2n^2}-1}$ & $2\!+\!\sqrt{6n^2-2-\rfra{9}{2n^2}}$ & $\rfra{3}{n^2}-1$ &  $ \rfra{n^2}{3-n^2}$  & $1$ \\
\hline
(g) & $\sqrt{\rfra{2}3}\,n$ & $0$ & $n^2-\rfra{3}2$ & $\rfra{2n^2}3$ & $1$  & $\rfra{2n^2}3$ \\
\hline
(h) & $\sqrt{\rfra{2}3}\,n$ & $0$ & $1+2n^2$ & $\rfra{2n^2}3$ & $1$  & $\rfra{2n^2}3$ \\
\hline
(i) & $\sqrt{\rfra{2}3}\,n$ & $\pm\sqrt{1-\rfra{2n^2}3}$ & $0$ & $1$ & $\rfra{4n^2}3-1$  & $\rfra{4n^2}3-1$ \\
\hline
(j) & $\sqrt{\rfra{2}3}\,n$ & $\pm\sqrt{1-\rfra{2n^2}3}$ & $-2(1-2n^2)$ & $1$ & $\rfra{4n^2}3-1$  & $\rfra{4n^2}3-1$  \\
\hline
\hline
 \end{tabular}
 \caption{Critical points $X_c, Y_c$ and $f_c$ with their corresponding DE density and equation of state parameter of field $\varphi$ and total equation of state parameter of system.}
 \label{tab:CP}
\end{table*}

\begin{table*}[!htb]
\footnotesize
\centering
\renewcommand{\arraystretch}{2}
\begin{tabular}{|c|ccc|c|c|}
\hline
& \multicolumn{3}{c|}{} &  \\
CP  & & Eigenvalues  & & CP nature \\
\cline{2-4}
 & $\mu_1$ & $\mu_2$ & $\mu_3$ &   \\
\hline\hline
(a) & $1 - \sqrt{6}\,n$ & $3+\sqrt{6}\,n$ & $3+\sqrt{6}\,n$ &   Unstable or saddle \\
\hline
(b) & $1+\sqrt{6}\,n$ & $3-\sqrt{6}\,n$ & $3-\sqrt{6}\,n$ & Unstable or saddle \\
\hline
(c) & $-1+\sqrt{6}\,n$ & $3+\sqrt{6}\,n$ & $3+\sqrt{6}\,n$  & Stable or unstable or saddle \\
\hline
(d) & $-(1+\sqrt{6}\,n)$ & $3-\sqrt{6}\,n$ & $3-\sqrt{6}\,n$ & Stable or unstable or saddle \\
\hline
(e) & $\sqrt{24n^2\!-\!8\!-\!\rfra{18}{n^2}}$ & $\sqrt{\rfra{3}2}\left(
\rfra{3}n-2n\right)$ & $-\sqrt{\rfra{3}2}\left(\rfra{3}n-2n\right)$ & Saddle \\
\hline
(f) & $-\sqrt{24n^2\!-\!8\!-\!\rfra{18}{n^2}}$  & $\sqrt{\rfra{3}2}\left( \rfra{3}n-2n\right)$ & $-\sqrt{\rfra{3}2}\left( \rfra{3}n-2n\right)$ & Saddle \\
\hline
(g) &  $\rfra{5}2+n^2$ & $\rfra{3}2-n^2$ & $-\left(\rfra{3}2-n^2\right)$ & Saddle \\
\hline
(h) & $-\left(\rfra{5}2+n^2\right)$ & $\rfra{3}2-n^2$ & $-\left(\rfra{3}2-n^2\right)$ & Saddle \\
\hline
(i) & $-2(1-2n^2)$ & $-3+2n^2$ & $-3+2n^2$ &
Stable or unstable or saddle \\
\hline
(j) & $2(1-2n^2)$ & $-3+2n^2$ & $-3+2n^2$ & Stable or saddle \\
\hline
\end{tabular}
\caption{Eigenvalues of critical points of table (\ref{tab:CP}) and their corresponding nature. }
 \label{table:Eigenvalues}
\end{table*}

\begin{itemize}
{
\item CP(a): It is valid for all real values of $n$. It is
either unstable or saddle but its nature cannot be determined for
$n=-\sqrt{3/2}$ and $\sqrt{1/6}$. It corresponds to a scenario when the DE
density parameter $\Omega^{(\varphi)}=1$ and the growth of matter perturbations $f=0$.
But due to the steep-fluid like behavior of DE i.e. $w^{(\varphi)}=1$ the cosmic
acceleration is not possible. Hence, it is non-physical.

\item CP(b):  It is valid for all real values of parameter $n$.
This point is either unstable or saddle but its nature cannot be
determined for $n=-\sqrt{1/6}$ and $\sqrt{3/2}$. It corresponds to a state of the universe when $\Omega^{(\varphi)}=1$ and $f=0$. However, due to the steep fluid-like nature of DE and hence absence of cosmic acceleration, it is also not physical. Hence, it does not give physical description of the universe.

\item CP(c):  It is valid for all real values of parameter $n$.
This point could be stable, unstable or saddle but its nature cannot
be determined for $n=-\sqrt{1/6}$ and $\sqrt{3/2}$. Although, for small value of $n$ i.e. within the saddle parametric regime, this CP could be considered to lie in the deep-matter dominated era when $f\leq 1$. However, due to dominated DE i.e. $\Omega^{(\varphi)}=1$ and $\mathsf w=1$ it is not physically possible. Hence, it does not give physical description
of the universe.

\item CP(d): It is valid for all real values of parameter $n$.
This point could be stable, unstable or saddle but its nature cannot
be determined for $n=-\sqrt{1/6}$ and $\sqrt{3/2}$. Although, it is mathematically stable for $n>\sqrt{3/2}$ but due to the similar reasons mentioned for CP ($c$) it has no physical relevance.

\item CP(e): It is valid for all values of $n$ except when $n=0$.
This point is only saddle but its nature cannot be determined for $n= \pm \sqrt{3/2}$ and $\pm \sqrt{(1+2\sqrt{7})/6}$.
Since $\Omega^{(\varphi)}=-1+3/n^2$, therefore, one requires $\sqrt{3/2}<|n|<\sqrt{3}$
for a physically allowed region $0<\Omega^{(\varphi)} < 1$, which is too far from allowed observational constraints. Also, since it lacks the cosmic acceleration due to the stiff nature of DE it does not give
a physical description of the universe.

\item CP(f): It is valid for all values of $n$ excluding $n=0$.
This point is only saddle but its nature cannot be determined for $n= \pm \sqrt{3/2}$ and $\pm \sqrt{(1+2\sqrt{7})/6}$. Similar, to the reasons mentioned for CP ($e$), it is also not a physically relevant point.

\item CP(g): It is valid for all values of $n$. It is only a saddle point but its nature cannot be determined for $n= \pm \sqrt{3/2}$.
Due to its saddle behavior, it may correspond to the growth of matter perturbations, lets say, near the recombination epoch for $n>\sqrt{3/2}$.
Also, for $n\ll \sqrt{3/2}$, it corresponds to matter dominated epoch with $\mathsf w \to 0$, but then $f$ becomes negative, which results in decay of matter perturbations. Hence, this point does support growth of matter perturbations. Therefore, it is not a physically relevant point.

\item CP(h): It is valid for all values of $n$. It is only a saddle point but its nature cannot be determined for $n= \pm \sqrt{3/2}$.
Due to its saddle behavior, it may correspond to the growth of matter perturbations, lets say, near the recombination epoch while also providing a physical description of the background at that epoch i.e. $\mathsf w \ll 1$ for $n<\sqrt{3/2}$. Interestingly, this point indeed show that $f \leq 1$, which is one of the important feature of an interacting DEM scenario. Therefore, this point is of physical relevance in large redshifts.

\item CP(i): It is valid for all values of $n$. It could be
stable, unstable or saddle but its nature cannot be determined
for $n=\pm \sqrt{3/2}$ and $\pm \sqrt{1/2}$. For $n \in (-\sqrt{1/2},\sqrt{1/2})$ this point actually determines the state of the universe in which DE
becomes dominated and no evolution in the matter density perturbations can be observed. This point is a late-time attractor. For $n=0$, it corresponds to a perfect de-Sitter universe in the future. This point is of physical relevance in far future.

\item CP(j): It is valid for all values of $n$. It is either
stable or saddle but its nature cannot be determined for $n=\pm
\sqrt{3/2}$ and $\pm \sqrt{1/2}$. Although, at the background level, it correctly describe the DE dominated universe
but shows a negative growth of matter perturbations for  $n \in (-\sqrt{1/2},\sqrt{1/2})$. This results decay of the matter density contrast which is not possible. Hence, it is not physical.
}
\end{itemize}

\subsection{Numerical solutions of state variables }
\label{sub:numerical}

Now in order to depict the evolutionary behavior of dimensionless variables $X,Y$ and $f$, we numerically solve the coupled system of  Eqs. (\ref{dxdn}), (\ref{dydn}) and (\ref{dudn}). However, before proceeding ahead let us first note that the free parameter $n$ are very tightly constrained by both background as well as at the perturbative level observations. In fact, we have shown in
\cite{MS-SS-IJMPD}
that from the OHD and RSD-GOLD data, $n \leq 0.1$ upto $1\sigma$ confidence interval. Therefore, in order to be compatible with observational constraints, we take fiducial values: $n=0$ and $0.1$. Moreover, we take the initial conditions at $N=-7$ or $z=1095$:
\begin{equation}
X(-7)=0.01\, , \quad Y(-7)=4 \times 10^{-5} \, , \quad f(-7)=1.
\end{equation}
The obtained evolutionary profiles are shown in Fig. (\ref{fig:XYU}).
\begin{figure}[!t]
\centering
\begin{subfigure}{.49\textwidth}
  \centering
  \includegraphics[height=7cm, width=8.5cm]{fig_XYU_n-0-sol.pdf}
  \label{fig:XYU-a}
 \caption{$X(N)$, $Y(N)$ and $f(N)$ for $n=0$.}
\end{subfigure}
\begin{subfigure}{.48\textwidth}
  \centering
  \includegraphics[height=7cm, width=8.5cm]{fig_XYU_n-01-sol.pdf}
  \label{fig:XYU-b}
  \caption{$X(N)$, $Y(N)$ and $f(N)$ for $n=0.1$.}
\end{subfigure}
\caption{\footnotesize Functional profiles of $X$, $Y$ and $f$ for $n=0$ ($\Lambda$CDM) and $0.1$ in the range of $N\in[-7,5]$.}
\label{fig:XYU}
\end{figure}
In that figure one observes that $f$ tends to cross the value of unity
in large enough redshifts. One also finds that variable $X$ is quite small ($\ll 1$) and least evolving variable, whereas, $Y$ increases monotonically.

Here, let us emphasize that although in fig. (\ref{fig:XYU}) it seems
that $X$ is constant, but in general it is not true. In fact, we have explicitly shown in insets that for any
given initial conditions, $X$ always tends to settle to a value of the order of $n$. In other words, changing or perturbing its initial conditions
does not affect its evolutionary profile. Hence, no matter what initial condition for $X$ one will start with it always finds it near(or equal to) the magnitude of $n$ for any given epoch. It is also interesting to note that $X_c$ for this stable CP($i$) agrees with our previous obtained background solution of $\varphi(N)$
\cite{MS-SS-IJMPD}
i.e.,
\begin{equation} \label{vph_sol}
\varphi(N)=\frac{2 n N}\kappa \, , \quad \mbox{which implies,} \quad X = X_c=\sqrt{\frac{2}3}n \, .
\end{equation}
Hence, it confirms that this unique solution is both stable physically and dynamically of combined background and perturbative system.

\subsection{Phase space dynamics of matter perturbations}
\label{sub:phaseplane}

In order to depict the phase space dynamics of above system comprised of $X,Y$ and $f$ variables, it is suitable to replace $X,Y$ and their derivatives by $\Omega^{(m)}$ and $\Omega^{(m)}_{,N}$, respectively, by using Eq. (\ref{mdp}) and (\ref{dmpd}). In this way, not just our phase space dynamics will get reduced to two-dimensions but it will also be more reasonable to have a phase space dynamics between two physically observable quantities i.e., $\Omega^{(m)}$ and $f$.
Using above solution (\ref{vph_sol}) together with Eq. (\ref{dxdn}),
(\ref{dydn}) and (\ref{dudn}), one obtains a set of equations:
\begin{eqnarray} \label{dodn-sol}
\Omega^{(m)}_{,N} &=& -3 \left( 1 -\frac{2}3 n^2 - \Omega^{(m)} \right) \Omega^{(m)} \, , \\
f_{,N} &=& \frac{3}2 \Omega^{(m)} \left(1+2 n^2+f\right)-f \left(2-4 n^2+f\right)\, .
\label{dudn-sol}
\end{eqnarray}
\begin{figure}[!t]
\centering
\begin{subfigure}{.49\textwidth}
  \centering
  \includegraphics[height=7cm, width=8.5cm]{fig_Om-U-n_0.pdf}
  \label{fig:OmU-a}
 \caption{Phase portrait for $n=0$.}
\end{subfigure}
\begin{subfigure}{.48\textwidth}
  \centering
  \includegraphics[height=7cm, width=8.5cm]{fig_Om-U-n_01.pdf}
  \label{fig:OmU-b}
  \caption{Phase portrait for $n=0.1$.}
\end{subfigure}
\caption{\footnotesize Phase portrait in the space of the matter density
parameter and growth factor for $n=0$ ($\Lambda$CDM) and $0.1$.}
\label{fig:OmU}
\end{figure}
Solving them simultaneously by putting $\Omega^{(m)}_{,N}=f_{,N}=0$, we obtain the phase space dynamics between matter density parameter $\Omega^{(m)}$ and growth factor $f$.
In fig. (\ref{fig:OmU}), we show the phase portrait diagram for $\Lambda$CDM and $n=0.1$. In this figure, the nature of CPs are given as: ($i$) CP($P_1$) corresponds to the stable de-Sitter solution, ($ii$) : CP($P_2$) is a saddle point which may belongs to the deep matter-dominated era, ($iii$) : CP($P_3$) is a saddle, and ($iv$) : CP($P_4$) is an unstable point. Since, both $P_3$ and $P_4$ corresponds to the negative growth of matter perturbations and hence the matter density contrast will behave as: $\delta^{(m)}\ \propto
e^{-c N}$ (where $c$ is some positive integration constant), the LSS formation will not occur. Hence, they are physically irrelevant. Therefore, one only expect those trajectories that starts with $P_2$ and ends at $P_1$. In fact, these are the only trajectories which makes the universe to evolve in such a way that growth of matter perturbations declines as matter density parameter decreases with time.

After depicting the possible evolutionary profile of $f$ with $\Omega^{(m)}$, it is also necessary to figure out how does the evolutionary profile of growth index ($\gamma$) follows. Let us recall that, in literature, the growth index is related to the growth factor $f$ via the relation: $f =[\Omega^{(m)}] ^\gamma$. In general, $\gamma$ is a time-varied function and depends on the background parameters such as $\Omega^{(m)}$ and coupling parameters ($n$ in our case). As it can be noticed from fig. (\ref{fig:XYU}) that $f$ crosses the upper barrier of unity in presence of $n$, which is not possible in the above parametrization of $f$ as it is only limited to the minimally coupled scalar field scenarios, such as quintessence. In order to incorporate this feature, the above parametrization of $f$ requires modification. In fact, in
\cite{MS-SS-IJMPD},
we have shown that a suitable modification in the parametrization of $f$ can take the following form:
\begin{equation} \label{U-param}
f(N) = (1+2n^2)\left[\Omega^{(m)}\right]^{\gamma(N)} \, .
\end{equation}
Using this in the Eq. (\ref{dudn-sol}), we obtain a first-order differential equation for $\gamma(N)$:
\begin{equation} \label{dgammadn}
\gamma_{,N} = -\frac{4-3 (\Omega^{(m)})^{1-\gamma}+2 (\Omega^{(m)})^\gamma+6 \gamma  (\Omega^{(m)} -1)+4 n^2
   \left[(\Omega^{(m)}) ^\gamma +\gamma -2\right]-3 \Omega^{(m)}}{2 \log (\Omega^{(m)})} \, .
\end{equation}
\begin{figure}[!t]
\centering
\begin{subfigure}{.49\textwidth}
  \centering
  \includegraphics[height=7cm, width=8.5cm]{fig-om-gamma-n_0.pdf}
  \label{fig:Omgamma-a}
 \caption{Phase portrait for $n=0$.}
\end{subfigure}
\begin{subfigure}{.48\textwidth}
  \centering
  \includegraphics[height=7cm, width=8.5cm]{fig-om-gamma-n_01.pdf}
  \label{fig:Omgamma-b}
  \caption{Phase portrait for $n=0.1$.}
\end{subfigure}
\caption{\footnotesize Phase portrait in the space of the matter density
parameter and growth index for $n=0$ ($\Lambda$CDM) and $0.1$.}
\label{fig:Omgamma}
\end{figure}
By putting $\gamma_{,N}=\Omega^{(m)}_{,N}=0$, and then solving Eqs. (\ref{dodn-sol}) and
(\ref{dgammadn}) simultaneously we obtain the phase space diagram between
$\Omega^{(m)}$ and $\gamma$ (shown in fig. (\ref{fig:Omgamma})) for both $\Lambda$CDM and
$n=0.1$. In that
figure, the green vector-field (thick) is the only solution as it remains
finite over the entire range of $\Omega^{(m)} \in [0,1]$. All other solutions gets
diverge which either implies extensive amount of growth of matter
perturbations (when $\gamma\ll 0.555$) or no structure formation at all (when
$\gamma\gg 0.555$). This solution, thus, giving rise to consistent matter
perturbations growth at the present epoch is also consistent with the
theoretically calculated value $\gamma=6/11$ in the deep matter-dominated epoch.
It is interesting to note that this behavior of increasing $\gamma$ with $N$
agrees with our fitting function of $\gamma$ in previous paper
\cite{MS-SS-IJMPD},
which also depicts sharp increase in the value of $\gamma$ in future by remaining mildly dynamic atleast up to the present epoch.

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

We have performed a thorough dynamical analysis considering both the cosmological background as well as linear perturbative effects in order to find out the stability of the entire system. The system under consideration involves an interaction between DE and matter which has generated as a consequence of a non-minimal coupling (exponential in nature) between scalar field and matter sector in the Einstein frame.  Since this type of coupling can be obtained in various sort of modified gravity theories such as $f(\phi,R)$, metric-scalar-torsion theories, etc., which although have different theoretical motivations but due to the degeneracy at the solution level, they land up to an identical dynamical system. Therefore, it becomes important to perform a dynamical system not just considering the cosmological background but also the linear perturbative level which can incorporate a wide variety of modified gravity theories. Keeping that as an objective we have carried out our dynamical analysis in this paper.

In presence of an interacting DE, not just matter density ceases to evolve as a standard dust i.e. $a^{-3}(t)$ but its perturbations also experience  additional effects both from homogeneous (at background level) as well as the inhomogeneous (at perturbative level) DE. Similarly, the evolution of DE perturbations experience both homogeneous and inhomogeneous contribution from the matter sector. As a consequence, the evolution of cosmological perturbations becomes non-trivial. In fact, one observes following characteristic features in presence of an interacting DEM scenario : (i) enhancement of growth of matter perturbations, (ii) oscillations of DE perturbations about non-zero mean value, (iii) DE perturbations sources matter density perturbations, etc. These features
being absent in the standard minimally coupled scalar field scenarios, such as quintessence, demands to re-perform a dynamical analysis under given notable effects.

Our dynamical analysis consists of an interacting DEM FRW background and linear matter density perturbations over it. Since the observable large scale structure and its evolution lies in the deep sub-horizon regime, therefore, for our analysis we consider effects of cosmological perturbations in that regime only. It is also beneficial in a sense that matter density perturbations becomes scale-independent which makes dynamical analysis independent of the perturbation scale. Hence, we have three dimensionless variables in total : two for the background which is constructed out of the kinetic and potential term of the scalar field in the Friedmann equation, and one for perturbations which consists of scale-independent matter density perturbations with an on-an-average effect of field perturbations. By simultaneously solving equations for our dimensionless quantities, we obtain fourteen critical points, out of which four are identical which give rise to same mathematical and physical description of our universe. Hence, in total we have ten distinct critical points which is ofcourse more than one would obtain in case of minimally coupled scenario.

We find that out of these ten distinct critical points only four is mathematically stable under linear perturbations in the solution space. However, from physical point of view only one point is stable i.e. CP(i) of table (\ref{tab:CP}), which is supposed to lie in the far future when field energy density gets completely dominated and growth of structure formation reduces to zero. Also, we find that all points could be saddle for a certain parametric regime of coupling parameter $n$, however, physically only one such point is possible i.e. CP(h) which only gives rise to a matter dominated description of the universe with a large growth of matter perturbations at high redshifts. Moreover, by simultaneously solving the set of autonomous equations, we find that the late-time behavior of dimensionless variables agrees with the stable CP(i). In particular, it also verify our previous found solution of $\varphi$ in
\cite{MS-SS-IJMPD},
in which we have shown that this is the only real solution for a coupled set of background differential equations.

Moreover, by formulating background dimensionless variables in terms of matter density parameter we have constructed a phase space solution between two observable $\Omega^{(m)}$ and $f$. This we have done by fixing background dimensionless variables to that of the value corresponds to  stable CP(i). In that we find that only one physical trajectory is possible which leads to matter dominated to DE dominated era with decreasing $f$. However, a similar trajectory which also leads to the same configuration starts with negative $f$, which is not of physical importance. Further, in view of the necessity in the modifications in the analytical growth factor ansatz, we have also obtained a phase space dynamics of growth index $\gamma$ with $\Omega^{(m)}$. In that we find that there lies only one such solution which is weakly dynamical. While all other solutions gets diverge in a finite interval of time, it correctly describe the high redshift approximation of $\gamma$ i.e. $6/11$. Also, a speed up evolution of $\gamma$ is observed near the present epoch till the asymptotic future.

\section*{Acknowledgments}

The work of MKS was supported by the Council of Scientific and Industrial Research (CSIR), Government of India.

redpajama_set_name: RedPajamaArXiv


Begin each Skating Session with an All Skate in the Normal direction.
About 10 mins into the Session, stop the skating, give Welcome Greetings, and go over the Rink's Rules. Introduce Floor Guards, DJ and Manager. Give other announcements as useful, such as up coming events and specials.
Do the "Rink's Rules Review" again at the Session mid point. (captures the late comers) At this point, recognize all Birthday and other Anniversary folks.
Run songs in groups of 3's & 2's, 3 fast - 2 slow. This strategy encourages safety by helping to keep the overall skating speed under control.
Play only music that has a good skating rhythm, and no obscene language or negative messages. (like killing Cops, using drugs, hitting on women, etc.) Bad for Rink Image.
Music & Lighting must to be coordinated in speed, style & levels to the skating / special being done.
Do floor changes / Specials every 15 to 20 mins max. Variety is useful to break monotony, keep up interest, control/structure the crowd, and by default - the patrons get more options and activities to improve their skills and interest in skating.
During regular skating, the Floor Center Oval should be reserved for the Free-stylers (Spinners & Jumpers), Rexers, Jammers, Fancy Foot-workers, and Figures folks only. This is a safety issue, and a promotional issue, as it allows everyone to see and admire these types of skating, and allows folks to learn more skating skills. More interest, more fun, more patrons.
Change the order of programing for each session, so there are different activities for each one. Mixing it up will help keep the sessions fresh. You can have similar starts and endings for most sessions – like starting with an All Skate, and having a Moon Light Couples Skate to end each session.
If you have a floor that's large enough, you can do the Limbo, Spread Eagle, Center Dance, Cha Cha Slide, and a few other specials within the Floor Center Oval, while the other skaters continue around the outside of the Oval.
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For all competitive skate specials, give out prizes to the first three winning skaters.
At Session's end, Do the "Thank You for Skating With Us, and please come back to Skate with us", and reminders for upcoming events and sessions Announcements.
Adult Only Skate Sessions: Can be 18 & up, or 21 & up. Day, Evening, or Nite Sessions.
Use different music genres, like 50's/60's, 70's/80's, 90's/2000's; or by R&B, and Old School, Funk and Disco for Adult Only Themed Skate Sessions. Can also use these for Themed General Sessions.
Overnight Skate Sessions, perhaps in conjunction with Fund Raising events.
Live Band Skate Session Nites; Usually using local Bands.
Use E-Mail Subscriptions to the Rink's Web Site, with skate schedules, discounts, coupons, heads-up notices on upcoming special events, etc. You can send these out too.
Pay a flat fee, and you can skate all general sessions for a month (or a longer period).
"Adult Coffee Skate" for older retired folks in the mornings.
Mother's & Kids Skate Sessions; Usually in Mornings, Mid-day, or early Afternoons.
Mom's and Strollers Only Skate Sessions; for mom's with very young children.
Religious Groups Skate Sessions; Christian, Jewish, Muslim, Etc. May be done in conjunction with local Church Organizations.
Charity Skating Events for Local & National Charities, & other Fund Raising Events.
And On & On – for local Community needs.
Last edited by Skatervideoguy; December 20th, 2016 at 12:57 AM.
I thank rinks should have sessions just for shuffle skaters.
1beastnstl, thanks for your positive comment.
My experience suggests that that the greater range of opprotunities that skaters have to expand and hone their skills, the more fun they have and the longer they tend to skate. By watching other "better" skaters, that tends to encourage folks to want to try and do more, and better skating.
Simply going about in big Ovals, with few to Zero floor changes, and perhaps not seeing many skilled skaters - is a prescription for at best mediocrity, and at worst - eventual failure. In some Rinks, the prohibition on backwards skating - is pretty much a Death Sentence for any useful skating at all.
Holiday Guy, A specific Session for Shuffle Skaters only, might work if you you happen to have enough folks doing that type of skating. I've not yet been to a Rink that has enough Shuffle skaters to make a go of it. In some Rinks there may be those kind of numbers. That's why I have it as a Skate Special on my list of Skate Games & Specials in another Post.
Don't know if you've checked my other similar Posts, but you might give them a look.
Why do rinks have sessions anyway? Why can't they just open and close everyday and I can skate as long and when I want? Would people not show up? Not organized enough?
Guess it's too hard to set a price for that?
It's not like music can't be pre programmed now days?
Is this a hold over from DJ's and records? Organ music?
Why do rinks have sessions anyway? Why can't they just open and close everyday and I can skate as long and when I want?
Most Rinks I've attended simply did/do not have a large enough feeder population to support that many hours of Skating Sessions. Not to mention the Pricing, Staffing and Maintenance Costs Issues.
The City and area you're in dictates some of what you can do with that.
The Crystal Palace Rinks in Las Vegas have the most open Skating Session hours that I've encountered. http://skatevegas.com But then Las Vegas is a 24hr City.
Some places have a large screen on one side of the rink for showing videos with the music (i.e. skating to mixed music videos).
Live music from an organist is a plus. Like having a VJ to mix videos or DJ to mix music, the organist can react to the skating group and adjust the program to energize the crowd. I skate to organ music at least once a week.
Absolutely fantastic, you have certainly helped me!
There are other Rink recommendations scattered throughout my other Posts.
I try to put out recommendations that I have experienced, have done, and have researched. Effectiveness, efficiency, Safety, Skating Skills Attainment, and comfort of patrons are my goals - after more than 60yrs of Skating. Most all these items are either self explanatory, or easily demonstrated and taught. These are mostly measurable, by anyone who cares to investigate.
If you're interested in seeing the Rink that I usually attend, and some of folks I Skate with weekly - check out some of my Videos at https://www.youtube.com/channel/UCgv...ZFdQKxd4D0wcsg.
I will post and update this page. You gave me a chance to comment on decades of skate rink observations of what Goes Wrong and Could be Improved.
1st: Speaker Position for ALL to HEAR the Announcer.
Prior to your good ideas of 10 minutes into the session, the speakers must be positioned for all including the parents to hear. I have been in a few rinks where the speakers all face the floor and no one on the side lines can hear a darn word of the announcement. If the parents do not hear and EVEN sometimes if they do; the KIDS and crazy adults won't pay any attention.
2nd: Recorded Message like "Rink Rules"
I love the kids, yet they miss stuff when left to their own. Some over the years are great and some are ho ho ho hum. So the important stuff like no carrying children and other blanket stuff should be on a Taped/Computer File. And then do your idea of Putting Names to the Faces. I like IT.
BTW I have seen this message played at the END of the Session. I always laugh at how late they play it almost everyplace.
In the recorded announcement I would like it spelled out for some difficult adults that they MUST go to the management or the Skate Guard to rectify situations on the floor. No "Taking matters into your own hands"
On a side note the rink works better if this is done. NOW it does not work everyplace, yet it is a good idea.


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Hook Lighthouse, Wexford: The History Behind The World's Oldest Operational Lighthouse
One of my favourite things to do in Wexford is to take a spin out as far as Hook Head for a coffee and a ramble around Hook Lighthouse.
If you've never been to Hook Head, lash it onto your 'to-visit-sharpish' list. It's a gorgeous little corner of Ireland.
The historic Hook lighthouse is the OLDEST operational lighthouse in the world, which is pretty damn impressive!
However, even with all that history, many people visiting Wexford never venture out to Hook Head (it's a bit of a treck, but worth the spin), which is a shame.
In the guide below, you'll discover the history of Hook Lighthouse and you'll find a load of info on tours, opening hours and all that craic.
Photo by Hook Tourism via Failte Ireland
The story of Hook Head Lighthouse (the quick version)
Things to do at Hook
The story of Hook Lighthouse, Wexford (the longer version)
Recent years at the historic Wexford lighthouse
Hook Head webcam
I've popped in a short and a long version of the history of Hook Lighthouse, for those of you stuck for time. If you fancy reading the longer version (you should), scroll to the end:
A monastery was founded around 1.6 km north of Hook in the 5th century
The first Tower of Hook was built around 1172
Monks kept Hook Head Lighthouse running until civil war arrived in 1641
In 1667 the lighthouse was finally re-established
Over the years it fell into disrepair
It wasn't until 1791 that a contract to maintain it was granted
Various enhancements were introduced over the years (scroll down for more info)
In 1972, electricity took over
It was equipped with a radar beacon in 1974.
The station was converted to 'unwatched' in 1996.
Automatic Identification System functionality was introduced in 2009
I've visited Hook Lighthouse several times over the years. If you arrive on a fine day, there's plenty to do. If you arrive when it's p***ing down, you're a little more limited. Here are some suggestions:
1. Take a guided tour
If you want to go inside the lighthouse, you'll need to pay to get on a guided tour. I did this tour about 8 or 9 years ago, and it was brilliant (there's a picture from the visit hanging in front of me, randomly enough).
You can do a regular tour of the lighthouse which will take you on a journey through its history, or you can head off on one of the sunrise or sunset tours.
The Hook Lighthouse tours are led by local tour guides who'll fascinate you with the areas rich history. Visitors will pass through the lighthouse chambers and hear about those who inhabited the structure over the years.
The tour climaxes at the balcony, where there's a magnificent view of the surrounding countryside.
July & August: 09.30 a.m. – 7.00 p.m.
Rest of year: Daily 09.30 a.m. – 5.00 p.m.
Tours every half hour
Closed Christmas eve and Christmas Day
2. Head for a ramble along the coast
The last time that I visited Hook was during the summer of 2018, and the sun was blazing. We dodged the tour on this occasion, grabbed a cup of coffee and headed off on a ramble along the coast.
In the photo below (from the day mentioned), you can see how clear the waters surrounding Hook Lighthouse were. On the day that we were here, we perched ourselves on the rock and watched a group of divers swimming around.
3. Head for a coffee and a bite to eat
You'll find a little cafe in the visitor centre next to the lighthouse. If you visit on a wet day, forget all about rambling along the coast.
Head for the cafe, grab something warm, and gaze out of the windows at the waves crashing against the rocks.
Photo © The Irish Road Trip
The story of the Hook Head Lighthouse begins way back during the 5th century when a Welsh monk by the name of Dubhán founded a monastery around 1.6 km north of Hook Head.
It's said that Dubhán set up a beacon at Hook Point (this was possibly a pile of stones with burning timber on top) to act as a navigational light for ships entering the nearby Harbour.
The first lighthouse at Hook Head
Many years later, between 1170 and 1182, a Norman named Raymond LeGros landed at Baginbun (a headland in Wexford) where he found the remains of a Celtic fort.
After some time, LeGros constructed another fort. From this point, he battled and beat the Irish advance from Waterford. LeGros was later killed, but it's said by some that he left his mark on the area by constructing the first Tower of Hook, around 1172.
Another theory is that the first Hook Head Lighthouse was built by a chap named William Marshal, Earl of Pembroke, in 1245.
Monks and shipwrecks
The Hook Lighthouse (it was just a tower at this point) was placed in the hands of the monk and they kept it running until civil war arrived in 1641.
During this time nothing was done to maintain the beacon and shipwrecks at Hook Head became the norm. Mariners spent many years trying to convince the Governor of Duncannon Fort to help re-establish the beacon at Hook.
It wasn't until 1667 that Hook Lighthouse was finally re-established, and the waters nearby were made safe once again.
Falling into disrepair
During 1704, Hook Head Lighthouse was transferred to the Revenue Commissioners by Queen Anne. The lighthouse wasn't taken over for some time, as Henry Loftus, the owner of the supposedly haunted Loftus Hall, had a 21-year lease on it.
The lighthouse passed to Henry's son, who had a dispute with the Revenue Commissioners over the renewal of the lease. It's said that in 1726 he made a threat to extinguish the light unless rent was paid in advance by Revenue. The disagreement went on for two years.
Over the years Hook tower started to fall into disrepair. It wasn't until 1791 when Revenue granted a contract to maintain and staff Hook Point along with a number of other lighthouses in Ireland.
A new lease of life
A new lantern was installed along with 12 Argand oil lamps. This was enhanced later, in 1812 and again in 1864. A new lantern and fixed dioptric lens were installed and the lighthouse at Hook tipped on as strong as ever.
In 1871, the oil lamps were replaced by coal gas. This lasted for just under 30 years. In 1910, a revolving focal distance optic replaced the dioptric lens. Coal gas was given the boot, with vaporised paraffin taking its place. Many years later, in 1972, electricity took over.
Guns, bells and horns
Hook Head Lighthouse has used many different sounds to help with navigation over the years:
A fog signal bell (introduced in 1838)
A gun (introduced in 1872)
An explosive charge (introduced in 1905)
An electric horn (introduced in 1995)
Photo by Failte Ireland via Ireland's Content Pool
Hook Lighthouse was equipped with a radar beacon recently enough, in 1974. This signalled the coming-of-the-end to the 'old' way in which lighthouses were run in Ireland.
3 years later, in 1977, Lightkeepers families were removed from lighthouses across Ireland. 6 keepers were assigned to the lighthouse at Hook, with three on duty at any one time.
The end of the lighthouse keeper post at Hook arrived in 1996 when the last keepers were withdrawn. In January 2009, Automatic Identification System functionality was introduced to Hook Head Lighthouse.
If you want to see what this corner of Wexford looks like for yourself, without actually visiting, there's a Hook Head webcam that'll give you a peek.
There are several different webcams currently live on their official website.
Have you visiting Hook Lighthouse? What did you think? Let me know in the comments section below!
Tags Wexford
Kylemore Abbey, Ireland: The Tragic Tale Behind Ireland's Iconic Kylemore Castle
The 2020 Bucket List Road Trip: 4 Weeks Exploring Ireland (Full Route + Itinerary)


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Vienna's bar scene is really humming. New bars with all sorts of concepts are constantly being opened: in some, bartenders can be found slaving away over a hot stove making the ingredients for their cocktail creations themselves, while others focus on local spirits and bringing long-forgotten classics back to the table.
In a number of cocktail bars in the city, patrons will find all kinds of homemade essences bubbling away in the background. Dubbed "cuisine style", the latest trend to sweep the capital's bar scene has triggered a proliferation of homemade lemonades, infusions, syrups and liqueurs. DIY is back, and bartenders are doubling up as cooks. The Bruder is a good example, a bar and bistro in one. Hubert Peter, an adventurous bartender, works here. Large preserving jars filled with pickled fruit, roots and home-made liqueurs are everywhere to be seen and in no way just for decoration. The cocktails with beetroot and horseradish are particularly cool.
Herb garden in a cocktail glass.
At Heuer am Karlsplatz, giant preserving jars also decorate a shelving wall. The house specialty is fruit syrups based on vinegar, so-called 'shrubs'. This bar specializes in vinegar-based fruit syrups known as shrubs. Its homemade preserves, syrups and essences are also used in the restaurant's kitchens, while the herbs are picked fresh from the neighboring Karls Garten. At Botanical Garden the name says it all: the herbs and spices for the homemade infusions and juices are harvested from herb planters suspended directly above the cocktail bar.
Bar staff tending a hot stove are only one aspect of Vienna's new approach to mixology. Many bars are using local spirits to give their cocktails a special twist. By changing its recipe on multiple occasions over the years, Viennese vermouth producer Burschik (est. 1891) never grows old. As a result, Burschik's Vermouth has recently found its way back onto the capital's drinks menus in recent years – either unadulterated or as a mixer. Stockists include the Spelunke restaurant, which also has a fantastic cocktail bar.
Heuer am Karlsplatz and Kleinod also use local Burschik vermouth, as well as Wien Gin. Made by the three Kesselbrüder friends, the latter can be sampled at Erich's stylish, gold-paneled bar or the Salonplafond im MAK restaurant. With its Art Nouveau-inspired label this fantastic gin is also a popular souvenir.
It's not just products from Vienna that find their way into the capital's cocktail glasses – the rest of the country has a part to play too. Fruit brandies and sweet dessert wines are also suitable for cocktails, both of which are Tür 7's forte. Famous for preparing its own ingredients, this bar has a private living room atmosphere and prospective patrons should remember to make a reservation by phone. In a throwback to the days of prohibition, visitors have to ring the bell, speakeasy-style, before they can enter the premises.
A new generation of bartenders is revisiting the canon of classic cocktails. The recipes, some of which date back more than 100 years, are celebrating a comeback. Popular haunt Miranda Bar is among the places to pick up on this trend, serving long-forgotten cocktails – including numerous fizz mixes – in an inviting and contemporary atmosphere. Kleinod is another devotee of all things vintage, a theme that runs through its art deco interiors as well as its cocktails. It almost goes without saying that both of these bars make a lot of their ingredients themselves.
Various bars have struck the perfect mix, proving that excellent food and excellent drinks are a match made in heaven. Both aperitif aficionados, Roberto American Bar and the charming 1950s-style R&Bar, serve fantastic snacks with their cocktails.
Vintage cocktails are the order of the day in the contemporary Miranda Bar.
The gold-paneled bar at Erich provides a stylish focal point.


The topic of heart failure although extensively researched and reported, has several gray areas. This is especially seen in determining strategies for combating advanced heart failure. The global burden of ischemic heart disease is increasing at an alarming rate to almost pandemic proportions. Even in the presence of state-of the art technology for prompt and successful coronary reperfusion, the population progressing to ischemic cardiomyopathy is substantial. The treatment strategies for ischemic cardiomyopathy do not always ensure optimal clinical results, the reasons for which are varied. The proper patient selection is vital as is the tailoring of suitable therapy. The approach is thus multi-pronged, of which surgical ventricular restoration (SVR) has a definite role. Despite various randomized studies like the STICH-hypothesis 2, where there was a huge selection bias, with patients subjected to an erroneous technique of SVR, with inadequate ventricular remodelling assessment. The present need is to understand the unique anatomy of the left ventricle, its function and perturbations with the onset of heart failure, in order to plan therapeutic strategies to restore near normal anatomy and function. A study of the hemodynamics in ischemic cardiomyopathy is essential, as it differs in certain aspects from other causes of heart failure like idiopathic dilated cardiomyopathy and valvular heart diseases. Since information on these aspects are scattered in various scholarly articles, a concise source in the form of a book would be appropriate and useful for both cardiologists and cardiac surgeons.
This project represents a concise review of normal and abnormal cardiac anatomy and physiology, the evolution of SVR to EVLPP with validation of near normal ventricular restoration, plus act as a reference guide for cardiac surgeons with interest in SVR.


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Goals Soccer has plunged into a full-year loss for the first time in 12 years, the company announced on Monday (14 March).
The five-a-side centre specialist reported a £6.2m ($8.8m, €7.9m) loss in the year to 31 December 2015. The results sent shares plunging to 7.6% before stabilising to a total 2.5% fall.
Following the success Goals Soccer saw in 2014, largely due to enthusiasm for football boosted by the World Cup, the company saw costs creep up in 2015. Revenue also dropped, down 4.9% in 2015 to £33m. Like-for-like sales fell by 6.3%.
The company, which has opened 47 centres since it was founded in Scotland in 1987, said its results are dented by competition from new grant-aided full-size 3G pitches. Goals said league operators have an advantage because of the grants, as the lower costs allows the rivals to operate with lower prices.
"Twenty-fifteen was undoubtedly a disappointing year," chairman Nick Basing admitted, adding that he was confident about Goals' business model.
"I will be continuing to spearhead the ongoing review into every aspect of the business to develop a new strategy to improve performance and returns, partly based around a reinvestment programme to rejuvenate and grow the business."
Basing replaced Keith Edelman but Goals Soccer remains without a chief executive. At the moment, Keith Rogers, the company's manager director, is holding on to the reins at Goals.