text:
\section{Introduction}

General Relativity (GR) is an extensively accepted theory that was
proposed by Albert Einstein in 1915 in which he related matter and
spacetime with the help of Einstein field equations. In accordance
with his beliefs, there does not occur any contraction or expansion
in our universe since from the big bang. After a short time, the
red-shifted light has been received from some galaxies when Edwin
Hubble was trying to measure the velocities of those galaxies in
1929, and thus he concluded that our universe is in state of
expansion. Some fascinating discoveries have recently shown that our
cosmic dynamics needs a deeper understanding due to very high amount
(almost 95 percent) of dark matter and dark energy in our cosmos.
Many astrophysicists have expressed their interest in exploring
various ways of explaining the influence of our cosmic dark source
components. A number of modified gravitational theories have been
suggested recently which are significant for the study of dynamical
features of cosmic expansion. By substituting the generic function
of the Ricci scalar $R$ in place of $R$ in an action, the
straightforward generalization of GR has been obtained which is
known as $f(R)$ theory (for details on modified gravity please see
\cite{copeland2006dynamics,
nojiri2011unified,nojiri2017modified,capozziello2010beyond,capozziello2011extended,
de2010f,joyce2015beyond,cai2016f,bamba2015inflationary,yousaf2016influence,yousaf2016causes}).
To investigate the mysterious aspects of our cosmos, Nojiri and
Odintsov \cite{nojiri2007introduction} affirmed the consistency of
$f(R)$ theory by designing different models in this theory that are
compatible with certain solar system tests. In $f(R)$ theory, Bamba
\emph{et al.} \cite{bamba2012dark} discussed the features of several
cosmologies containing dark energy and $\Lambda$CDM-like universe by
considering different models. Different factors that could produce
irregularities in any self-gravitating object have been examined by
various researchers
\cite{konoplich1999formation,yousaf2019non,yousaf2019tilted,bhatti2017gravitational,sharif2019complexity,doi:10.1142/S0217732319503334,yadav2020existence},
and by taking some applications into consideration, they analyzed
rigorous solutions to the field equations in $f(R)$ theory. Yousaf
and his collaborators
\cite{yousaf2016electromagnetic,bhatti2017stabilitya,yousaf2020construction,bhatti2020stability,yousaf2020gravastars}
concluded that for some particular modified theory, the
gravitational models are quite appropriate for hosting compact stars
with smaller radii but having larger masses.

Harko \emph{et al.} \cite{harko2011f} first studied the effects of arbitrary coupling between matter and curvature by introducing $f(R,T)$ theory, where $T$ is the trace of stress-energy tensor. For this theory, they investigated the impact of additional force term on moving particles, and field equations are also evaluated with the help of variational principle. By considering some models in $f(R,T)$ theory, Baffou \emph{et al.} \cite{baffou2015cosmological} found their stability conditions, and established several cosmological solutions that suit observations. The more generalized form of $f(R,T)$ theory is known as $f(R,T,Q)$ theory, which was suggested by Haghani \emph{et al.} \cite{haghani2013further} and it incorporates the matter lagrangian consisting of the strong interaction between curvature and fluid. The equations of motion were also calculated by assuming Lagrange multiplier method in this theory. Odintsov and S\'{a}ez-G\'{o}mez \cite{odintsov2013f} noticed that the spacetime is strongly non-minimally coupled with geometry and found that the cosmic expansion may be the result of extra curvature terms of $f(R,T,Q)$ gravity. Ayuso \emph{et al.} \cite{ayuso2015consistency} inspected the stability conditions in $f(R,T,Q)$ gravity for certain specific scalar and vector fields, and also calculated higher order field equations. Baffou \emph{et al.} \cite{baffou2016exploring} analyzed the consistency criteria for $f(R,T,Q)$ theory via some particular solutions and claimed that one can better understand the early stages of our cosmic expansion through modified correction terms. In addition, by using computational techniques, they explored the stability of certain models in this theory. The gravitational collapse of a self-gravitating object was studied by Bhatti and his colleagues \cite{yousaf2016stability,Yousaf2017,yousaf2017stability,bhatti2019dissipative}, and some relations between physical variables containing in field equations and the Conformal tensor are also found.

A complexity can be defined as a combination of several components which can lead a stable self-gravitating system to be complicated. Here, the existing results in GR for zero complexity must be explained under the influence of extra curvature terms of $f(R,T,Q)$ theory. In various fields of science, different approaches to define a complexity can be observed. L\'{o}pez-Ruiz \emph{et al.} \cite{lopez1995statistical,calbet2001tendency,catalan2002features} defined this term by entropy and information amongst all those factors. Entropy of any system can be determined by measuring its disorderness, whereas the knowledge that one can gain by studying a system could be called as information. L\'{o}pez-Ruiz \emph{et al.} \cite{lopez1995statistical} then developed this definition by means of a term, i.e., disequilibrium.

The term complexity can be demonstrated in physics by assuming simplest structures, which by definition do not have any complications, such as isolated ideal gas and perfect crystal. By theirs very definition, such both structures are extreme in all aspects. The first system is entirely unstable in nature as it consists of randomly moving particles, and therefore gives maximum possible information because all particles have equal involvement in this system. In second system, the elements that construct this structure are organized in an orderly manner and therefore it provides minimum data set because to known its nature, it is sufficient to analyze its small portion. It can thus be seen in the former case that there is maximum disequilibrium, while in the later case it is found to be zero. The notion of complexity factor can also be used in astrophysics to examine the structural features of self-gravitating (cylindrical or spherical) systems. Usually, there can be various factors in any system such as pressure, heat dissipation and energy density which contribute in making the system more complex. The complexity of any system cannot be measured via its energy density only, unless the pressure component in the stress-energy tensor is taken into account.

Usually for cylindrical objects, the nature of several physical characteristics was analyzed at different scales. In particular, the phenomenon which encourage to consider the cylindrical symmetry are gravitational collapse and its radiations, rotating fluids and spinning massive structures. By definition of Birkhoff's theorem, outside a spherically self-gravitating object, there exists a vacuum and therefore its collapse does not cause to produce any gravitational waves. For this reason, one turns to a different fundamental symmetry, which is commonly known as cylindrical symmetry. By considering weak approximation fields, the solutions for gravitational waves produced by cylindrical stars have been constructed by Einstein and Rosen \cite{einstein1937gravitational}, and they also stated that such problems reduce to standard cylindrical waves in the case of Minkowski space. Many astrophysical facets have been investigated in relation to cylindrical symmetric propagation. Herrera and Santos \cite{herrera2005cylindrical} evaluated the smooth conditions to match the inner and outer geometries for gravitational collapse in cylindrical structure and sustained that the radial pressure at the boundary did not meet zero value. However, Herrera \emph{et al.} \cite{herrera2006matching} later noticed some errors in lengthy calculations, reporting that there would be no radial pressure at the boundary if such mistakes were corrected. Recently, Herrera \cite{herrera2017gibbs,herrera2020landauer} elaborated the physical interpretation of congruences as well as the fluid distribution in the study of astrophysical objects.

For static cylindrical self-gravitating star, Herrera \emph{et al.}
\cite{herrera2005static} established the matching criteria with the
Levi-Civita vacuum spacetime  and checked the regularity conditions
for them. They evaluated equations of motion and furthermore
demonstrated that an appropriate collection of conformally flat
solutions would represent an incompressible fluid. Sharif and Bhatti
\cite{sharif2012structure} found some structure scalars for charged
cylinder which contained anisotropic pressure and heat flux, and
examined its fundamental characteristics through those scalars. The
field equations and certain dynamical equations were also
constructed by considering the charge effects. By considering
cylindrical matter distribution by Sharif and Yousaf
\cite{sharif2012expansion}, vacuum cavity within the fluid
distribution was examined while they were studying the expansion
free condition. Their discussion focused on solutions which contain
a thin shell, and also on solutions that fulfil the Darmois
conditions on the boundary. The structural instability of an
isotropic cylindrical spacetime was studied by Sharif and Bhatti
\cite{sharif2014stability}, in which they observed the effects of
electromagnetic field. Their conclusion was that the energy density,
pressure and electric charge are the components which influence the
cylinder's stability. Yousaf \emph{et al.} \cite{yousaf2020study}
calculated the effective scalars by splitting the Riemann tensor
orthogonally in $f(R,T,Q)$ theory, found complexity factors for
cylindrical and charged spherical systems and also discussed
structural properties for these systems.

We have adopted a definition of complexity given in Herrera
\cite{herrera2018new} and want to generalize it $f(R,T,Q)$ theory of
gravity with static cylindrically symmetric spacetime. The scheme of
this paper is mentioned below. For $f(R,T,Q)$ gravity theory, in the
next section, we suggest some new definitions and the field
equations are also evaluated. Then in Sec. 3, we orthogonally split
the Riemann tensor to get some scalar factors, one of which is
claimed to be a complexity factor. After this, we evaluate the
condition to vanish the complexity factor in Sec. 4, and also
present some mathematical models in the context of $f(R,T,Q)$
theory. Eventually, all modified results and their effects have been
concluded in Sec. 5.

\section{Basic Definitions and Some Other Equations}

The fluid considered in this case is static self-gravitating
cylindrically symmetric which is coupled with electric charge and
pressure anisotropy. We study the stellar object in $f(R,T,Q)$
theory via some equations of motion, evaluate the C-energy as well
as Tolman mass functions and the required matching conditions on the
boundary surface $\Sigma$.

\subsection{Modified Field Equations}

For $f(R,T,Q)$ theory \cite{odintsov2013f,ayuso2015consistency,baffou2016exploring}, the Einstein-Hilbert action becomes after modification as
\begin{equation}\label{f1}
S=\frac{1}{2}\int \sqrt{-g}\left[f(R,T,Q)+L_{m}+L_{EM}\right]d^{4}x,
\end{equation}
where $L_{m}$ denotes the matter Lagrangian and the electromagnetic Lagrangian $L_{EM}$ contains the influence of electric charge.

Corresponding to \eqref{f1}, the field equations in this case are
\begin{equation}\label{f2}
G_{\rho\nu}=8\pi T_{\rho\nu}^{(eff)},
\end{equation}
where $G_{\rho\nu}$ is an Einstein tensor and $T_{\rho\nu}^{(eff)}$ could represent the stress energy tensor for $f(R,T,Q)$ theory,
\begin{align}
\nonumber T_{\rho\nu}^{(eff)}&=\frac{1}{f_{R}-L_{m}f_{Q}}\left[\left(f_{T}+\frac{1}{2}Rf_{Q}+1\right)X_{\rho\nu}+\left\{\frac{R}{2}(\frac{f}{R}-f_{R})-L_{m}f_{T}
\right.\right.\\\nonumber
&-\left.\frac{1}{2}\nabla_{\mu}\nabla_{\beta}(f_{Q}X^{\mu\beta})\right\}g_{\rho\nu}-\frac{1}{2}\Box(f_{Q}X_{\rho\nu})
-(g_{\rho\nu}\Box-\nabla_{\rho}\nabla_{\nu})f_{R}\\\label{f3}
&-2f_{Q}R_{\mu(\rho}X_{\nu)}^{\mu}+\nabla_{\mu}\nabla_{(\rho}[X_{\nu)}^{\mu}f_{Q}]+2(f_{Q}R^{\mu\beta}
+\left.f_{T}g^{\mu\beta})\frac{\partial^2 L_{m}}{\partial g^{\rho\nu}\partial g^{\mu\beta}}\right],
\end{align}
where $X_{\rho\nu}=T_{\rho\nu}^{(m)}+S_{\rho\nu}$. Also, $\nabla_{\rho}$ is the usual covariant derivative,~$R_{\rho\nu}$ is the Ricci tensor,~$g_{\rho\nu}$ is the metric tensor and $\Box\equiv g^{\gamma\alpha}\nabla_{\gamma}\nabla_{\alpha}$. Here, the subscripts $R,~T$ and $Q$ represent that their arguments are partially differentiated. The trace of usual stress energy tensor in GR provides a particular relation between $R$ and $T$. To study the charged fluid, the electromagnetic energy tensor $S_{\rho\nu}$ can be given as
\begin{equation}\label{f4}
  S_{\rho\nu}=\frac{1}{4\pi}\left(-F^{\gamma}_{\rho}F_{\nu\gamma}+\frac{1}{4}F^{\gamma\beta}F_{\gamma\beta}g_{\rho\nu}\right),
\end{equation}
where $F_{\rho\nu}=\phi_{\nu,\rho}-\phi_{\rho,\nu}$ is the Maxwell field tensor and in this case of static fluid, the four potential $\phi_{\rho}$ becomes as $\phi_{\rho}=\phi(r)\delta^{0}_{\rho}$. The electric charge becomes interior to radius $r$ as
\begin{equation}\label{f5}
  s(r)=4\pi \int_{0}^{r}\tau r^2Ydr,
\end{equation}
where $\tau$ denotes the charge density which depends on $r$ only. In this case, the matter Lagrangian shown in action \eqref{f1} is defined as $L_{m}=-\mu-\frac{1}{4}F^{\rho\nu}F_{\rho\nu}$, and after using the value of Maxwell tensor it becomes, $L_{m}=-\mu+\frac{s^2}{2r^4}$ in which $\mu$ is the energy density of the fluid.

Also, the usual energy-momentum tensor considered in \eqref{f3} is of the form
\begin{equation}\label{f6}
T_{\rho\nu}^{(m)}=\mu v_{\rho} v_{\nu}-Ph_{\rho\nu}+\Pi_{\rho\nu},
\end{equation}
where
\begin{equation}\label{f7}
\Pi_{\rho\nu}=\Pi\left(w_{\rho} w_{\nu}+\frac{1}{3}h_{\rho\nu}\right);\quad P=\frac{P_{r}+2P_{t}}{3},
\end{equation}
\begin{equation}\label{f8}
\Pi=P_{r}-P_{t};\quad h_{\rho\nu}=g_{\rho\nu}-v_\rho v_{\nu},
\end{equation}
and the 4-vector in radial direction $w^{\rho}$ is set as
\begin{equation}\label{f9}
w^\rho=(0,Y^{-1},0,0),
\end{equation}
with some properties $w^\mu v_{\mu}=0, w^\mu w_{\mu}=-1$.\\
Here, the term $\Pi$ represents anisotropic pressure which further contains the radial pressure $P_{r}$ and tangential pressure $P_{t}$. It should be noted that the pressure normally exists in three different directions for cylindrical anisotropic structure, but the pressure anisotropy considered in this case does not have the most general form.\\
Indeed, we found the following from Eq.\eqref{f3} as
\begin{align}\nonumber
&3\Box
f_R+\frac{1}{2}\Box(f_QX)-X(f_T+1)+\nabla_\pi\nabla_\rho(f_QX^{\pi\rho})+R(f_R-\frac{X}{2}f_Q)\\\nonumber
&+(Rf_Q+4f_T)\textit{L}_m-2f+2R_{\pi\rho}X^{\pi\rho}f_Q -2g^{\lambda\sigma}\frac{\partial^2\textit{L}_m}{\partial g^{\lambda\sigma}\partial g^{\pi\rho}}\left(f_Tg^{\pi\rho}+f_QR^{\pi\rho}\right)=0.
\end{align}
From above expression, the effects of $f(R,T)$ theory can be investigated with the assumption $Q=0$, while taking the vacuum case will help us to examine these complex results in $f(R)$ theory. Various authors \cite{odintsov2013f,ayuso2015consistency,baffou2016exploring} looked into the accurate analysis of their derivation and physical importance by examining our cosmic composition.

The 4-velocity vector in the comoving frame becomes
\begin{equation}\label{f10}
v^\rho=(X^{-1},0,0,0).
\end{equation}
The 4-acceleration $a^{\beta}=v^{\beta}_{;\pi}v^{\pi}$ gives only one non-disappearing component as follows
\begin{equation}\label{f11}
  a_{1}=-\frac{X'}{X}.
\end{equation}

In order to clarify the complexity, the appropriate inner space-time restricted by the hypersurface $\Sigma$ is indicated by a line element as
\begin{equation}\label{f12}
ds^2=-Y^2 dr^2-r^2(d\theta^2+\alpha^2 dz^2)+X^2 dt^2,
\end{equation}
where $X=X(r)$ and $Y=Y(r)$. Note that the above spacetime is not in
the most general form as it contains two independent functions, and
due to this assumption, the expressions for anisotropic tensor,
electric part of Weyl tensor and the scalar $Y_{TF}$ will be in
restricted forms. The most general form of the cylindrically
symmetric spacetime is given in \cite{Herrera2012}. For cylindrical
symmetry, we implement the following ranges on the above coordinates
as
\begin{equation}\nonumber
0\leq r, \quad 0\leq\theta\leq 2\pi, \quad -\infty< z< +\infty, \quad -\infty\leq t\leq +\infty.
\end{equation}

The cylindrical object \eqref{f12} delivers the field equations for $f(R,T,Q)$ theory as
\begin{align}\label{f13}
\frac{8\pi}{(f_{R}-L_{m}f_{Q})}\left[\mu^{(eff)}+Q_{0}^{(eff)}\right]&=-\frac{2Y'}{rY^3}+\frac{1}{r^2Y^2},\\\label{f14}
\frac{8\pi}{(f_{R}-L_{m}f_{Q})}\left[P_{r}^{(eff)}+Q_{1}^{(eff)}\right]&=-\frac{1}{r^2Y^2}-\frac{2X'}{rXY^2},\\\label{f15}
\frac{1}{(f_{R}-L_{m}f_{Q})}\left[P_{t}^{(eff)}+Q_{2}^{(eff)}\right]&=-\frac{X''}{XY^2}+\frac{Y'}{rY^3}+\frac{X'Y'}{XY^3}-\frac{X'}{rXY^2},
\end{align}
where $P_{r}^{(eff)}$,~$P_{t}^{(eff)}$ and $\mu^{(eff)}$ comprise the dark source terms of $f(R,T,Q)$ theory together with physical variables, while $Q_{1}^{(eff)}$,~$Q_{2}^{(eff)}$ and $Q_{0}^{(eff)}$ represent effective charge terms respectively. The values of all these quantities are set in Appendix A. Here, differentiation with respect to $r$ is shown by prime.

It is useful to say that the divergence of stress-energy tensor in $f(R,T,Q)$ theory is not conserved and therefore allowing all physical laws to be violated. Hence, the particles move in a non-geodesic path, owing to an extra force in its gravitational field. This can therefore be formalized as
\begin{align}\nonumber
\nabla^\lambda
X_{\lambda\epsilon}&=\frac{2}{Rf_Q+2f_T+1}\left[\nabla_\epsilon(\textit{L}_mf_T)
+\nabla_\lambda(f_QR^{\pi\lambda}X_{\pi\epsilon})-\frac{1}{2}(f_Tg_{\pi\rho}+f_QR_{\pi\rho})\right.\\\label{f16}
&\times\left.\nabla_\epsilon X^{\pi\rho}-G_{\lambda\epsilon}\nabla^\lambda(f_Q\textit{L}_m)\right].
\end{align}
From \eqref{f16}, the equation for the hydrostatic equilibrium can be read as
\begin{eqnarray}\nonumber
\left(\frac{P_{r}^{(eff)}+Q_{1}^{(eff)}}{H}\right)'&=&\frac{-X'}{HX}\left(\mu^{(eff)}+Q_{0}^{(eff)}+P^{(eff)}_{r}
+Q_{1}^{(eff)}\right)+\frac{2}{rH}\\\label{f17}
&\times&\left(P^{(eff)}_{t}+Q_{2}^{(eff)}-P^{(eff)}_{r}-Q_{1}^{(eff)}\right)+ZY^2,
\end{eqnarray}
where $H=f_{R}-L_{m}f_{Q}$. The term $Z$ represents extra curvature terms and refers to the constraints on the stress-energy tensor in $f(R,T,Q)$ theory whose value is given in Appendix A. It can be called as the generalized Tolman-Opphenheimer-Volkoff (TOV) equation which uses widely to study the compact stellar properties. In 1971, Bekenstein initially extended the TOV equation in GR for compact objects having the influence of electromagnetic field.

One can find the value of $\frac{X'}{X}$ from Eq.\eqref{f14} as
\begin{equation}\label{f18}
\frac{X'}{X}=\frac{4r^2}{r^2-8mr+4s^2}\left[-\frac{4\pi r}{H}\left(P_{r}^{(eff)}+Q_{1}^{(eff)}\right)+\frac{m}{r^2}-\frac{1}{8r}-\frac{s^2}{2r^3}\right].
\end{equation}
We will get the following by putting back Eq.\eqref{f18} in Eq.\eqref{f17},
\begin{align}\nonumber
\left(\frac{P_{r}^{(eff)}+Q_{1}^{(eff)}}{H}\right)'&=\frac{4r^2}{r^2-8mr+4s^2}\left[\frac{4\pi r}{H}\left(P_{r}^{(eff)}+Q_{1}^{(eff)}\right)-\frac{m}{r^2}\right.\\\nonumber
&+\left.\frac{1}{8r}+\frac{s^2}{2 r^3}\right]\left(\mu^{(eff)}+Q_{0}^{(eff)}+P^{(eff)}_{r}+Q_{1}^{(eff)}\right)\\\label{f19}
&+\frac{2}{rH}\left(P^{(eff)}_{t}+Q_{2}^{(eff)}-P^{(eff)}_{r}-Q_{1}^{(eff)}\right)+ZY^2,
\end{align}
in which the mass $m$ of the geometry is obtained as
\cite{PhysRev.136.B571}
\begin{equation}\label{f20}
m(r)\equiv
\tilde{E}=E\hat{l}=-\frac{r}{2Y^2}\left(1-\frac{Y^2}{4}\right)+\frac{s^2}{2r}.
\end{equation}
Equation \eqref{f20} can be written by combining it with
Eq.\eqref{f13} as
\begin{equation}\label{f21}
m(r)=\frac{r}{8}-4\pi\int_{0}^{r} \frac{\tilde{r}^2}{H}\left(\mu^{(eff)}+Q_{0}^{(eff)}\right) d\tilde{r}+\frac{s^2}{2r}.
\end{equation}

To study the cylindrical star, the line element which represents the exterior space-time is given as
\begin{equation}\label{f22}
ds^2=-\left(-\frac{2M(\nu)}{r}+\frac{S^2(\nu)}{r^2}\right)d\nu^2-2d\nu dr
+r^2(d\theta^2+\alpha^2 dz^2),
\end{equation}
where $M$ denotes its total mass. By succeeding the work done by Senovilla \cite{senovilla2013junction}, Yousaf \emph{et al.} \cite{Yousaf2017} introduced the corresponding criteria for smoothly matching of the space-times \eqref{f12} and \eqref{f22} in $f(R,T,Q)$ theory. Therefore, at the hypersurface $r=r_{\Sigma}=$constant, we impose some constraints on the effective radial pressure and its charge component separately as
\begin{equation}\label{f23}
\tilde{E}-M_=^{\Sigma}\frac{1}{8}; \quad [P_{r}]_=^{\Sigma}-F_{0}; \quad s^2{_=^{\Sigma}}\tilde{H}S^2-8\pi r^4F_{1}.
\end{equation}
The values of $F_{0}$ and $F_{1}$ are set out in Appendix A. Note that the conditions \eqref{f23} are necessarily required to match the inner and outer space-times on $\Sigma$.

\subsection{Some Curvature Tensors}

A well known tensor which is used to measure the space-time curvature named as the Riemann tensor can be written as a linear combination of the Weyl tensor $C_{\alpha\gamma\beta\rho}$, the Ricci tensor $R_{\gamma\rho}$ and the Ricci scalar $R$ as
\begin{eqnarray}\nonumber
R^\alpha_{\pi\beta\rho}&=&C^\alpha_{\pi\beta\rho}-\frac{1}{2}R_{\pi\beta}\delta^\alpha_{\rho}+\frac{1}{2}R_{\pi\rho}\delta^\alpha_{\beta}
-\frac{1}{2}R^\alpha_{\rho}g_{\pi\beta}+\frac{1}{2}R^\alpha_{\beta}g_{\pi\rho}\\\label{f24}
&-&\frac{1}{6}R\left(\delta^\alpha_{\beta}g_{\pi\rho}-g_{\pi\beta}\delta^\alpha_{\rho}\right).
\end{eqnarray}
One can define the electric and magnetic part of Weyl tensor separately as
\begin{equation}\label{f25}
E_{\gamma\beta}=C_{\gamma\delta\beta\pi}v^{\delta}v^{\pi}, \quad H_{\gamma\beta}=\frac{1}{2}\eta_{\gamma\delta\mu\pi}C^{\mu\pi}_{\quad\beta\sigma}v^{\delta}v^{\sigma}.
\end{equation}
The magnetic part of Weyl tensor described in Eq.\eqref{f25} vanishes, as we have considered static cylindrical structure \eqref{f12}. Thus another form of the Weyl tensor may be written as
\begin{equation}\label{f26}
C_{\pi\lambda\kappa\rho}=(g_{\pi\lambda\mu\nu}g_{\kappa\rho\gamma\beta}
-\eta_{\pi\lambda\mu\nu}\eta_{\kappa\rho\gamma\beta})v^\mu v^\gamma E^{\nu\beta},
\end{equation}
where $g_{\pi\lambda\alpha\nu}=g_{\pi\alpha}g_{\lambda\nu}-g_{\pi\nu}g_{\lambda\alpha}$, and $\eta_{\lambda\pi\alpha\nu}$ is the Levi-Civita tensor. The electric part $E_{\rho\nu}$ can further be written as
\begin{equation}\label{f27}
E_{\rho\nu}=E\left(w_{\rho}w_{\nu}+\frac{1}{3}h_{\rho\nu}\right),
\end{equation}
with its non-fading elements
\begin{equation}\label{f28}
E_{11}=\frac{2}{3}Y^2E, \quad E_{22}=-\frac{1}{3}r^2E,
\end{equation}
where
\begin{equation}\label{f29}
E=\frac{1}{2XY^2}\left[X''-\frac{X'Y'}{Y}+\frac{XY'}{rY}-\frac{X'}{r}+\frac{X}{r^2}\right],
\end{equation}
which must meet some conditions given below as
\begin{equation}\label{f30}
E^\rho_{\rho}=0,\quad E_{\rho\gamma}=E_{(\rho\gamma)},\quad E_{\rho\gamma}v^\gamma=0.
\end{equation}

\subsection{The Mass Function}

Here, we will evaluate few useful equations by following the definitions of C-energy \cite{thorne1965energy} and Tolman mass \cite{tolman1930use} in order to investigate some properties of cylindrical system. Such equations will further support us to make the relation between mass function and Conformal tensor. Equations \eqref{f2},~\eqref{f20},~\eqref{f24} and \eqref{f27} are used to write the following as
\begin{align}\nonumber
m&=\frac{r}{8}-\frac{4\pi r^3}{3H}\left(\mu^{(eff)}+Q^{(eff)}_{0}+P^{(eff)}_{t}+Q^{(eff)}_{2}
-P^{(eff)}_{r}-Q^{(eff)}_{1}\right)\\\label{f31}
&+\frac{s^2}{2r}-\frac{r^3E}{3}.
\end{align}
One may also write above expression for $E$ as
\begin{eqnarray}\nonumber
E&=&-\frac{4\pi}{r^3}\int_{0}^{r} \tilde{r}^3\left(\frac{\mu^{(eff)}+Q^{(eff)}_{0}}{H}\right)' d\tilde{r}+\frac{4\pi}{H} \left(P^{(eff)}_{r}+Q^{(eff)}_{1}\right.\\\label{f32}
&-&\left.P^{(eff)}_{t}-Q^{(eff)}_{2}\right).
\end{eqnarray}
Note that Eq.\eqref{f32} presents the relation of Weyl tensor with cylindrical structural properties, i.e., effective energy density inhomogeneity, effective local anisotropic pressure and electric charge of the fluid with modified corrections. One can insert Eq.\eqref{f32} in Eq.\eqref{f31} and get
\begin{equation}\label{f33}
m(r)=\frac{r}{8}+\frac{s^2}{2r}-\frac{4\pi r^3}{3H}\left(\mu^{(eff)}+Q^{(eff)}_{0}\right)+\frac{4\pi}{3}\int_{0}^{r} \tilde{r}^3 \left(\frac{\mu^{(eff)}+Q^{(eff)}_{0}}{H}\right)' d\tilde{r},
\end{equation}
which contributes to link the mass of our geometry with uniform energy density and charge distribution. The formula mentioned above, allows us to analyze the changes in mass function which are carried by effective inhomogeneous energy density and respective component of effective charge for self-gravitating system.

Another way in order to measure the total energy of static cylinder was suggested by Tolman \cite{tolman1930use} which is given as
\begin{equation}\label{f34}
m_{T}=4\pi \int_{0}^{r_{\Sigma}} r^2XY\left(T_{0}^{0(eff)}-T_{1}^{1(eff)}-2T_{2}^{2(eff)}\right)dr.
\end{equation}
With and without taking into consideration the effects of electric charge in self-gravitating structures, Bhatti \emph{et al.} \cite{zaeem2019energy,bhatti2019tolman} evaluated the Tolman mass function in $f(R)$ theory. Thus, within the cylindrical configuration of radius $r$, Eq.\eqref{f34} becomes
\begin{equation}\label{f35}
m_{T}=4\pi \int_{0}^{r} r^2XY\left(T_{0}^{0(eff)}-T_{1}^{1(eff)}-2T_{2}^{2(eff)}\right)dr.
\end{equation}
By using field equations in Eq.\eqref{f35}, we receive
\begin{equation}\label{f36}
m_{T}=-\frac{X'r^2}{Y}-4\pi \int_{0}^{r}\frac{XYr^2Q_{3}^{(eff)}}{H}dr,
\end{equation}
where $Q_{3}^{(eff)}=Q_{0}^{(eff)}+Q_{1}^{(eff)}+2Q_{2}^{(eff)}$, or it becomes by putting the value of $X'$ from Eq.\eqref{f18} as
\begin{equation}\label{f37}
m_{T}=XY\left[\frac{4\pi r^3}{H}\left(P_{r}^{(eff)}+Q_{1}^{(eff)}\right)-m+\frac{s^2}{2r}+\frac{r}{8}\right].
\end{equation}
This expression could be referred to as the effective gravitational mass. The gravitational acceleration $(a=-w^{\nu}a_{\nu})$ of a test molecule can be pursued instantaneously at rest in a static gravitational field as
\begin{equation}\label{f38}
a=\frac{X'}{XY}=-\frac{1}{Xr^2}\left[m_{T}+4\pi\int_{0}^{r}\frac{XYr^2Q_{3}^{(eff)}}{H}dr\right].
\end{equation}
Equation \eqref{f36} can be more appropriately expressed as
\begin{eqnarray}\nonumber
m_{T}&=&(m_{T})_{\Sigma}(\frac{r}{r_{\Sigma}})^3+r^3\int_{r}^{r_{\Sigma}}\frac{XY}{\tilde{r}}\left[\frac{4\pi}{H}\Pi^{(eff)}+E\right]d\tilde{r}\\\label{f39}
&+&4\pi r^3 \int_{r}^{r_{\Sigma}}\frac{XYQ_{4}^{(eff)}}{\tilde{r}H}d\tilde{r},
\end{eqnarray}
where $Q_{4}^{(eff)}=Q_{0}^{(eff)}+2Q_{1}^{(eff)}+Q_{2}^{(eff)}$. One can get from the above expression by using Eq.\eqref{f32} in it, as
\begin{eqnarray}\nonumber
m_{T}&=&(m_{T})_{\Sigma}(\frac{r}{r_{\Sigma}})^3+r^3\int_{r}^{r_{\Sigma}}\frac{XY}{\tilde{r}}\left[\frac{8\pi}{H}\Pi^{(eff)}
-\frac{4\pi}{\tilde{r}^3}\int_{0}^{r}\tilde{r^3}\left(\frac{\mu^{(eff)}}{H}\right)'d\tilde{r}\right]d\tilde{r}\\\label{f40}
&+&4\pi r^3 \int_{r}^{r_{\Sigma}}\frac{XYQ_{5}^{(eff)}}{\tilde{r}H}d\tilde{r},
\end{eqnarray}
where $Q_{5}^{(eff)}=Q_{0}^{(eff)}+3Q_{1}^{(eff)}$. Equation \eqref{f40} could be useful to interpret the relation between effective Tolman mass and some physical phenomenon like effective inhomogeneity of the energy density, effective local pressure anisotropy and effective form of electric charge in the static cylinder.

\section{The Orthogonal Splitting of The Riemann Tensor}

The orthogonal splitting of the Riemann tensor was first suggested by Bel \cite{bel1961inductions}
and Herrera \emph{et al.} \cite{herrera2004spherically} from which one can obtain the following three tensors as
\begin{eqnarray}\label{f41}
&&Y_{\delta\gamma}=R_{\delta\beta\gamma\alpha}u^\beta u^\alpha,\\\label{f42}
&&Z_{\delta\gamma}=*R_{\delta\beta\gamma\alpha}u^\beta u^\alpha=\frac{1}{2}\eta_{\delta\beta\epsilon\mu}R^{\epsilon\mu}_{\gamma\alpha}u^\beta u^\alpha,\\\label{f43}
&&X_{\delta\gamma}=*R^*_{\delta\beta\gamma\alpha}u^\beta u^\alpha=\frac{1}{2}\eta_{\delta\beta}^{\epsilon\mu}R^*_{\epsilon\mu\gamma\alpha}u^\beta u^\alpha,
\end{eqnarray}
where $*$ is the dual tensor and $\eta_{\delta\epsilon\mu\beta}$ is
known as Levi-Civita symbol which takes $0$ value for no
permutation,~$1$ for even permutation and $-1$ for odd permutaion.
Now we use field equations to rewrite the Riemann tensor \eqref{f24}
by means of above tensors \eqref{f41}-\eqref{f43} (see
\cite{gomez2007dynamical}) as
\begin{equation}\label{f44}
R^{\delta\beta}_{\quad\gamma\alpha}=C^{\delta\beta}_{\quad\gamma\alpha}+16\pi T^{(eff)[\delta}_{[\gamma} \delta_{\alpha]}^{\beta]}+8\pi T^{(eff)}\left(\frac{1}{3}\delta^{\delta}_{\quad[\gamma} \delta_{\alpha]}^{\beta}-\delta^{[\delta}_{\quad[\gamma} \delta_{\alpha]}^{\beta]}\right).
\end{equation}
The Riemann tensor \eqref{f44} can further be divided into three components after using Eq.\eqref{f3} in it as
\begin{equation}\label{f45}
R^{\delta\beta}_{\quad\gamma\alpha}=R^{\delta\beta}_{(I)\gamma\alpha}+R^{\delta\beta}_{(II)\gamma\alpha}+R^{\delta\beta}_{(III)\gamma\alpha},
\end{equation}
where
\begin{align}\nonumber
R^{\delta\beta}_{(I)\gamma\alpha}&=\frac{16\pi}{H}(f_{T}+\frac{1}{2}Rf_{Q}+1) \left[\left(\mu+\frac{s^2}{8\pi r^4}\right) v^{[\delta}v_{[\gamma} \delta_{\alpha]}^{\beta]}-\left(P+\frac{s^2}{24\pi r^4}\right)\right.\\\nonumber
&\times\left.h^{[\delta}_{\quad[\gamma} \delta_{\alpha]}^{\beta]}+\Pi^{[\delta}_{\quad[\gamma} \delta_{\alpha]}^{\beta]}-\frac{s^2}{4\pi r^4}\left(w^{[\delta}w_{[\gamma} \delta_{\alpha]}^{\beta]}+\frac{1}{3}h^{[\delta}_{\quad[\gamma} \delta_{\alpha]}^{\beta]}\right)\right]+\frac{8\pi}{H}\\\nonumber
&\times\left[(f_{T}+\frac{1}{2}Rf_{Q}+1)(\mu-3P)+4\left\{\frac{R}{2}\left(\frac{f}{R}
-f_{R}\right)+\left(\mu-\frac{s^2}{2r^4}\right)f_{T}\right.\right.\\\nonumber
&-\left.\frac{1}{2}\nabla_{\mu}\nabla_{\nu}(f_{Q}X^{\mu\nu})\right\}-3\Box f_{R}-\frac{1}{2}\Box\{f_{Q}(\mu-3P)\}+\nabla_{\mu}\nabla_{\pi}(f_{Q}X^{\mu\pi})\\\nonumber
&-\left.2f_{Q}R_{\mu\pi}X^{\mu\pi}+2g^{\pi\xi}(f_{Q}R^{\mu\nu}+f_{T}g^{\mu\nu})\frac{\partial^2L_{m}}{\partial g^{\pi\xi}\partial g^{\mu\nu}}\right]\\\label{f46}
&\times\left(\frac{1}{3}\delta^{\delta}_{\quad[\gamma} \delta_{\alpha]}^{\beta}-\delta^{[\delta}_{\quad[\gamma} \delta_{\alpha]}^{\beta]}\right),
\\\nonumber
R^{\delta\beta}_{(II)\gamma\alpha}&=\frac{4\pi}{H}\left[2\left\{\frac{R}{2}
\left(\frac{f}{R}-f_{R}\right)+\left(\mu-\frac{s^2}{2r^4}\right) f_{T}-\frac{1}{2}\nabla_{\mu}\nabla_{\rho}(f_{Q}X^{\mu\rho})\right\}\right.\\\nonumber
&\times\left(\delta^{\delta}_{\gamma} \delta_{\alpha}^{\beta}-\delta^{\delta}_{\alpha} \delta_{\gamma}^{\beta}\right)-\frac{1}{2}\Box\left\{f_{Q}\left(X^{\delta}_{\gamma} \delta_{\alpha}^{\beta}-X^{\delta}_{\alpha} \delta_{\gamma}^{\beta}-X^{\beta}_{\gamma} \delta_{\alpha}^{\delta}+X^{\beta}_{\alpha} \delta_{\gamma}^{\delta}\right)\right\}\\\nonumber
&-2\Box f_{R}\left(\delta^{\delta}_{\gamma} \delta_{\alpha}^{\beta}-\delta^{\delta}_{\alpha} \delta_{\gamma}^{\beta}\right)+\left( \delta_{\alpha}^{\beta}\nabla^{\delta}\nabla_{\gamma}- \delta_{\gamma}^{\beta}\nabla^{\delta}\nabla_{\alpha}- \delta_{\alpha}^{\delta}\nabla^{\beta}\nabla_{\gamma}\right.\\\nonumber
&+\left. \delta_{\gamma}^{\delta}\nabla^{\beta}\nabla_{\alpha}\right)f_{R}
-f_{Q}\left(R_{\mu\gamma}X^{\mu\delta} \delta_{\alpha}^{\beta}-R_{\mu\alpha}X^{\mu\delta} \delta_{\gamma}^{\beta}
-R_{\mu\gamma}X^{\mu\beta} \delta_{\alpha}^{\delta}\right.\\\nonumber
&+\left.R_{\mu\alpha}X^{\mu\beta} \delta_{\gamma}^{\delta}+R^{\delta}_{\mu}X^{\mu}_{\gamma} \delta_{\alpha}^{\beta}-R^{\delta}_{\mu}X^{\mu}_{\alpha} \delta_{\gamma}^{\beta}-R^{\beta}_{\mu}X^{\mu}_{\gamma} \delta_{\alpha}^{\delta}+R^{\beta}_{\mu}X^{\mu}_{\alpha} \delta_{\gamma}^{\delta}\right)\\\nonumber
&+\frac{1}{2}\nabla_{\mu}\nabla^{\delta}\left\{f_{Q}\left(X^{\mu}_{\gamma} \delta_{\alpha}^{\beta}-X^{\mu}_{\alpha} \delta_{\gamma}^{\beta}\right)\right\}+\frac{1}{2}\nabla_{\mu}\nabla_{\alpha}\left\{f_{Q}\left(X^{\beta\mu} \delta_{\gamma}^{\delta}-X^{\delta\mu} \delta_{\gamma}^{\beta}\right)\right\}\\\nonumber
&+\frac{1}{2}\nabla_{\mu}\nabla^{\beta}\left\{f_{Q}\left(X^{\mu}_{\alpha} \delta_{\gamma}^{\delta}-X^{\mu}_{\gamma} \delta_{\alpha}^{\delta}\right)\right\}+\frac{1}{2}\nabla_{\mu}\nabla_{\gamma}\left\{f_{Q}\left(X^{\delta\mu} \delta_{\alpha}^{\beta}-X^{\beta\mu} \delta_{\alpha}^{\delta}\right)\right\}\\\nonumber
&+2g^{\delta\beta}(f_{Q}R^{\mu\rho}+f_{T}g^{\mu\rho})\left\{\delta_{\alpha}^{\pi}\frac{\partial^2L_{m}}{\partial g^{\pi\gamma}\partial g^{\mu\rho}}-\delta_{\gamma}^{\pi}\frac{\partial^2L_{m}}{\partial g^{\pi\alpha}\partial g^{\mu\rho}}\right.\\\label{f47}
&-\delta_{\alpha}^{\pi}\frac{\partial^2L_{m}}{\partial g^{\pi\gamma}\partial g^{\mu\rho}}+\left.\left.\delta_{\gamma}^{\pi}\frac{\partial^2L_{m}}{\partial g^{\pi\alpha}\partial g^{\mu\rho}}\right\}\right],
\\\label{f48}
R^{\delta\beta}_{(III)\gamma\alpha}&=4v^{[\delta}v_{[\gamma} E_{\alpha]}^{\beta]}-\epsilon^{\delta\beta}_{\mu}\epsilon_{\gamma\alpha\rho}E^{\mu\rho},
\end{align}
with
\begin{equation}\label{f49}
\epsilon_{\mu\beta\rho}=v^\delta \eta_{\delta\mu\beta\rho},\quad \epsilon_{\mu\beta\gamma}v^\gamma=0,
\end{equation}
where the magnetic part of Weyl tensor has been disappeared for the considered static cylindrical symmetry.

It is possible to find the tensors $X_{\delta\gamma},~Y_{\delta\gamma}$ and $Z_{\delta\gamma}$ which encompass material variables, electric charge and extra curvature terms of $f(R,T,Q)$ gravity, as
\begin{align}\nonumber
Y_{\delta\gamma}&=E_{\delta\gamma}+\frac{1}{H}\left\{\frac{4\pi}{3}(\mu+3P)h_{\delta\gamma}+4\pi \Pi_{\delta\gamma}-\frac{s^2}{r^4}\left(w_{\delta}w_{\gamma}+\frac{1}{3}h_{\delta\gamma}\right)+\frac{s^2}{3r^4}h_{\delta\gamma}\right\}\\\nonumber
&\times(f_{T}+\frac{1}{2}Rf_{Q}+1)-\frac{8\pi}{3H}\left\{\frac{R}{2}\left(\frac{f}{R}-f_{R}\right)+\left(\mu-\frac{s^2}{2r^4}\right) f_{T}\right.\\\nonumber
&-\left.\frac{1}{2}\nabla_{\mu}\nabla_{\rho}(f_{Q}X^{\mu\rho})\right\}h_{\delta\gamma}+\frac{4\pi}{H}\left[-\frac{1}{2}\left\{\Box(f_{Q}X_{\delta\gamma})-v_{\delta}v_{\beta}\Box(f_{Q}X^{\beta}_{\gamma})\right.\right.\\\nonumber
&-\left.v_{\gamma}v^{\alpha}\Box(f_{Q}X_{\delta\alpha})
+g_{\delta\gamma}v_{\beta}v^{\alpha}\Box(f_{Q}X^{\beta}_{\alpha})\right\}
+(\nabla_{\delta}\nabla_{\gamma}f_{R}-v_{\delta}v_{\beta}\nabla^{\beta}\nabla_{\gamma}f_{R}\\\nonumber
&-v_{\gamma}v^{\alpha}\nabla_{\delta}\nabla_{\alpha}f_{R}
+g_{\delta\gamma}v_{\beta}v^{\alpha}\nabla^{\beta}\nabla_{\alpha}f_{R})
+2f_{Q}R_{\delta\mu}\left\{\left(P+\frac{s^2}{24\pi r^4}\right) h^{\mu}_{\gamma}\right.\\\nonumber
&-\left.\left(w^{\mu}w_{\gamma}+\frac{1}{3}h^{\mu}_{\gamma}\right)\left(\Pi-\frac{s^2}{4\pi r^4}\right)\right\}
-f_{Q}R^{\beta}_{\mu}\left\{\left(\mu+\frac{s^2}{8\pi r^4}\right) v^{\mu}v_{\beta}h_{\delta\gamma}\right.\\\nonumber
&\left.-v_{\delta}v_{\beta}\left(w^{\mu}w_{\gamma}+\frac{1}{3}h^{\mu}_{\gamma}\right)\left(\Pi-\frac{s^2}{4\pi r^4}\right)+v_{\delta}v_{\beta}\left(P+\frac{s^2}{24\pi r^4}\right) h^{\mu}_{\gamma}\right\}\\\nonumber
&-f_{Q}R_{\mu\alpha}\left\{\left(\mu+\frac{s^2}{8\pi r^4}\right) v^{\mu}v^{\alpha}h_{\delta\gamma}-v_{\gamma}v^{\alpha}\left(w^{\mu}w_{\delta}+\frac{1}{3}h^{\mu}_{\delta}\right)\left(\Pi-\frac{s^2}{4\pi r^4}\right)\right.\\\nonumber
&+\left.v_{\gamma}v^{\alpha}\left(P+\frac{s^2}{24\pi r^4}\right)h^{\mu}_{\delta}\right\}+\frac{1}{2}\{\nabla_{\mu}\nabla_{\delta}(f_{Q}X^{\mu}_{\gamma})
+g_{\delta\gamma}v_{\beta}v^{\alpha}\nabla_{\mu}\nabla^{\beta}(f_{Q}X^{\mu}_{\alpha})\\\nonumber
&+\nabla_{\mu}\nabla_{\gamma}(f_{Q}X^{\mu}_{\delta})
+g_{\delta\gamma}v_{\beta}v^{\alpha}\nabla_{\mu}\nabla_{\alpha}(f_{Q}X^{\mu\beta})-v_{\gamma}v^{\alpha}\nabla_{\mu}\nabla_{\delta}(f_{Q}X^{\mu}_{\alpha})\\\nonumber
&-v_{\beta}v_{\delta}\nabla_{\mu}\nabla_{\gamma}(f_{Q}X^{\beta\mu})-v_{\beta}v_{\delta}\nabla_{\mu}\nabla^{\beta}(f_{Q}X^{\mu}_{\gamma})
-v_{\gamma}v^{\alpha}\nabla_{\mu}\nabla_{\alpha}(f_{Q}X^{\mu}_{\delta})\}\\\nonumber
&+\left.2h^{\epsilon}_{\delta}(f_{Q}R^{\mu\rho}+f_{T}g^{\mu\rho})\frac{\partial^2L_{m}}{\partial g^{\epsilon\gamma}\partial g^{\mu\rho}}\right]+\frac{8\pi}{3H}\left[\frac{1}{2}\Box\{f_{Q}(\mu-3P)\}+2f_{Q}R_{\mu\epsilon}\right.\\\nonumber
&\times\left\{\left(\mu+\frac{s^2}{8\pi r^4}\right) v^{\mu}v^{\epsilon}
+\left(\Pi-\frac{s^2}{4\pi r^4}\right)\left(w^{\mu}w^{\epsilon}+\frac{1}{3}h^{\mu\epsilon}\right)-h^{\mu\epsilon}\right.\\\label{f50}
&\times\left.\left.\left(P+\frac{s^2}{24\pi r^4}\right)\right\}
-\nabla_{\mu}\nabla_{\epsilon}(f_{Q}X^{\mu\epsilon})-2g^{\epsilon\xi}(f_{Q}R^{\mu\rho}+f_{T}g^{\mu\rho})\frac{\partial^2L_{m}}{\partial g^{\epsilon\xi}\partial g^{\mu\rho}}\right]h_{\delta\gamma},
\\\nonumber
Z_{\delta\gamma}&=\frac{4\pi}{H}\left[\frac{1}{2}v^{\alpha}\Box(f_{Q}X^{\epsilon}_{\alpha})+f_{Q}\left(\mu+\frac{s^2}{8\pi r^4}\right) R_{\alpha}^{\epsilon}v^{\alpha}-v^{\alpha}\nabla^{\epsilon}\nabla_{\alpha}f_{R}\right.\\\nonumber
&+\frac{1}{3}f_{Q}\left(\Pi-\frac{s^2}{4\pi r^4}\right) R^{\epsilon}_{\alpha}v^{\alpha}-f_{Q}\left(P+\frac{s^2}{24\pi r^4}\right)R_{\alpha}^{\epsilon}v^{\alpha}-\frac{1}{2}v^{\alpha}\nabla_{\mu}\nabla_{\alpha}(f_{Q}X^{\mu\epsilon})\\\label{f51}
&-\left.\frac{1}{2}v^{\alpha}\nabla_{\mu}\nabla^{\epsilon}(f_{Q}X^{\mu}_{\alpha})\right]\epsilon_{\epsilon\gamma\delta},
\end{align}
and
\begin{align}\nonumber
X_{\delta\gamma}&=-E_{\delta\gamma}+\frac{1}{H}\left\{\frac{8\pi}{3}\mu h_{\delta\gamma}+4\pi \Pi_{\delta\gamma}-\frac{s^2}{r^4}\left(w_{\delta}w_{\gamma}+\frac{1}{3}h_{\delta\gamma}\right)+\frac{s^2}{3r^4}h_{\delta\gamma}\right\}\\\nonumber
&\times(f_{T}+\frac{1}{2}Rf_{Q}+1)+\frac{4\pi}{H}\left[\left\{\nabla^{\nu}\nabla_{\epsilon}f_{R}-\frac{1}{2}\Box(f_{Q}X^{\nu}_{\epsilon})
+\frac{1}{2}\nabla_{\mu}\nabla_{\epsilon}(f_{Q}X^{\mu\nu})\right.\right.\\\nonumber
&+\left.\left.\frac{1}{2}\nabla_{\mu}\nabla^{\nu}(f_{Q}X^{\mu}_{\epsilon})\right\}\epsilon^{\epsilon\alpha}_{\delta}\epsilon_{\nu\alpha\gamma}
+f_{Q}R_{\mu\epsilon}\left\{\left(P+\frac{s^2}{24\pi r^4}\right)-\frac{1}{3}\left(\Pi-\frac{s^2}{4\pi r^4}\right)\right\}\right.\\\nonumber
&\times\left.\epsilon^{\epsilon\alpha}_{\delta}\epsilon^{\mu}_{\alpha\gamma}+f_{Q}R^{\nu}_{\mu}\left\{\left(P+\frac{s^2}{24\pi r^4}\right)-\frac{1}{3}\left(\Pi-\frac{s^2}{4\pi r^4}\right)\right\}\epsilon^{\mu\alpha}_{\delta}\epsilon_{\nu\alpha\gamma}\right]\\\nonumber
&+\frac{8\pi}{3H}\left[\left\{\frac{R}{2}\left(\frac{f}{R}-f_{R}\right)+\left(\mu-\frac{s^2}{2r^4}\right) f_{T}-\frac{1}{2}\nabla_{\mu}\nabla_{\rho}(f_{Q}X^{\mu\rho})\right\}\right.\\\nonumber
&-\frac{1}{2}\Box
\{f_{Q}(\mu-3P)\}-2Rf_{Q}\left\{\frac{1}{3}\left(\Pi-\frac{s^2}{4\pi r^4}\right)-\left(P+\frac{s^2}{24\pi r^4}\right)\right\}\\\label{f52}
&+\left.\nabla_{\mu}\nabla_{\nu}(f_{Q}X^{\mu\nu})+2g^{\nu\epsilon}(f_{Q}R^{\mu\rho}+f_{T}g^{\mu\rho})\frac{\partial^2L_{m}}{\partial g^{\nu\epsilon}\partial g^{\mu\rho}}\right]h_{\delta\gamma}.
\end{align}
The three tensors \eqref{f50}-\eqref{f52} can further be expressed
with the help of following scalars $X_{T},~X_{TF},~Y_{T}$ and
$Y_{TF}$ whose values are given as
\begin{eqnarray}\label{f53}
X_{T}&=&\frac{1}{H}\left(8\pi\mu+\frac{s^2}{r^4}\right)\left(1+\frac{1}{2}Rf_{Q}+f_{T}\right)+\varphi_{1}^{(D)},
\\\label{f54}
X_{TF}&=&-E+\frac{1}{H}\left(4\pi\Pi-\frac{s^2}{r^4}\right)\left(1+\frac{1}{2}Rf_{Q}+f_{T}\right),
\\\label{f55}
Y_{T}&=&\frac{4\pi}{H}\left(\mu+3P_{r}-2\Pi+\frac{s^2}{4\pi r^4}\right)\left(1+\frac{1}{2}Rf_{Q}+f_{T}\right) +\varphi_{2}^{(D)},
\\\label{f56}
Y_{TF}&=&E+\frac{1}{H}\left(4\pi\Pi-\frac{s^2}{r^4}\right)\left(1+\frac{1}{2}Rf_{Q}+f_{T}\right)+\varphi_{3}^{(D)},
\end{eqnarray}
where
$\varphi_{3}^{(D)}=\frac{1}{w_{\delta}w_{\gamma}+\frac{1}{3}h_{\delta\gamma}}\varphi_{\delta\gamma}^{(D)}$.
These scalars were first determined and discussed in detail by
Herrera \emph{et al.} \cite{herrera2009structure}. Combining
Eqs.\eqref{f54} and \eqref{f56} with Eq.\eqref{f32} separately, we
have respectively
\begin{align}\nonumber
X_{TF}&=\frac{4\pi}{r^3}\int_{0}^{r}\tilde{r}^3\left(\frac{\mu^{(eff)}+Q_{0}^{(eff)}}{H}\right)'d\tilde{r}
-\frac{4\pi}{H}(\Pi^{(eff)}+Q_{1}^{(eff)}-Q_{2}^{(eff)})\\\label{f57}
&+\frac{1}{H}\left(4\pi\Pi-\frac{s^2}{r^4}\right)\left(1+\frac{1}{2}Rf_{Q}+f_{T}\right),
\\\nonumber
Y_{TF}&=-\frac{4\pi}{r^3}\int_{0}^{r}\tilde{r}^3\left(\frac{\mu^{(eff)}+Q_{0}^{(eff)}}{H}\right)'d\tilde{r}
+\frac{4\pi}{H}(\Pi^{(eff)}+Q_{1}^{(eff)}-Q_{2}^{(eff)})\\\label{f58}
&+\frac{1}{H}\left(4\pi\Pi-\frac{s^2}{r^4}\right)\left(1+\frac{1}{2}Rf_{Q}+f_{T}\right)+\varphi_{3}^{(D)},
\end{align}
whereas Appendix B contains the values of $\varphi_{1}^{(D)}$, $\varphi_{2}^{(D)}$ and $\varphi_{\delta\gamma}^{(D)}$.

By adding two above scalars $X_{TF}$ and $Y_{TF}$, we get local anisotropic pressure and electric charge along with corrections terms of $f(R,T,Q)$ theory, as
\begin{equation}\label{f59}
X_{TF}+Y_{TF}=\frac{2}{H}\left(4\pi\Pi-\frac{s^2}{r^4}\right)\left(1+\frac{1}{2}Rf_{Q}+f_{T}\right)+\varphi_{3}^{(D)}.
\end{equation}
The physical significance of the scalar function $Y_{TF}$ can be determined by utilizing Eq.\eqref{f56} in Eq.\eqref{f39} as
\begin{eqnarray}\nonumber
m_{T}&=&(m_{T})_{\Sigma}(\frac{r}{r_{\Sigma}})^3+r^3\int_{r}^{r_{\Sigma}}\frac{XY}{\tilde{r}}\left[Y_{TF}
+\frac{4\pi\Pi^{(eff)}}{H}-\frac{1}{H}\left(4\pi\Pi-\frac{s^2}{r^4}\right)\right.\\\label{f60}
&\times&\left.\left(1+\frac{1}{2}Rf_{Q}+f_{T}\right)-\varphi_{3}^{(D)}\right]d\tilde{r}+4\pi r^3\int_{r}^{r_{\Sigma}}\frac{XY}{\tilde{r}H}Q_{4}^{(eff)}d\tilde{r}.
\end{eqnarray}
One can observe by comparing Eq.\eqref{f39} with Eq.\eqref{f60} that the scalar $Y_{TF}$ contributes to determine the impact of anisotropic pressure, energy density inhomogeneity and electromagnetic field on the Tolman mass along with dark source terms of $f(R,T,Q)$ theory. Likewise, in terms of scalar factor $Y_{T}$, the Tolman mass \eqref{f35} takes the form as
\begin{eqnarray}\nonumber
m_{T}&=&\int_{0}^{r}XY\tilde{r}^2\left[Y_{T}-\frac{4\pi}{H}\left(\mu+3P_{r}-2\Pi+\frac{s^2}{4\pi r^4}\right)\left(\frac{1}{2}Rf_{Q}+f_{T}\right)\right.\\\label{f61}
&-&\left.\frac{4\pi s^2}{Hr^4}+\frac{4\pi}{H}\left(\mu^{(D)}+P_{r}^{(D)}+2P_{t}^{(D)}+Q_{3}^{(eff)}\right)-\varphi_{2}^{(D)}\right]d\tilde{r}.
\end{eqnarray}
Equation \eqref{f61} provides the relation among the scalar $Y_{T}$, matter variables and the Tolman mass which are influenced by modified correction terms. Herrera \emph{et al.} \cite{herrera2011meaning,Herrera2012} and Yousaf \emph{et al.} \cite{PhysRevD.95.024024,bhatti2017dynamical,bhatti2017evolution} observed the role of the scalar $Y_{T}$ on equation for evolution which is commonly called as Raychaudhuri equation, and thus, Eq.\eqref{f61} could be used to provide this equation through $m_{T}$ even for charged structure in $f(R,T,Q)$ theory.

\section{Matter Distribution With Disappearing Complexity Factor}

It should be obvious that various factors can contribute to make any static/non-static system more complex. For the considered structure, we indicate $Y_{TF}$ as a complexity factor in which certain components such as inhomogeneous energy density, anisotropic pressure and electric charge are participated under the presence of modified correction terms. In relevant field equations, five unknown functions $(\mu,P_{r},P_{t},X,Y)$ are to be found. Therefore, two more conditions are required for our further work and from them, one could be the disappering complexity factor condition (by putting $Y_{TF}=0$ in Eq.\eqref{f58}), as
\begin{align}\nonumber
\Pi&=\frac{H}{2}\left[\frac{1}{r^3}\int_{0}^{r}\tilde{r}^3\left(\frac{\mu^{(eff)}+Q_{0}^{(eff)}}{H}\right)'d\tilde{r}
-\frac{1}{H}(\Pi^{(D)}+Q_{1}^{(eff)}-Q_{2}^{(eff)})\right.\\\label{f62}
&-\left.\frac{1}{H}\left(f_{T}+\frac{1}{2}Rf_{Q}\right)\left(\Pi-\frac{s^2}{4\pi r^4}\right)+\frac{s^2}{4\pi Hr^4}-\frac{1}{4\pi}\varphi_{3}^{(D)}\right].
\end{align}
In modified gravity theory, Eq.\eqref{f62} could be regarded as a non-local equation of state. In next two subsections, different models of stellar objects are discussed.

\subsection{The Gokhroo and Mehra Ans\"{a}tz}

Gokhroo and Mehra \cite{gokhroo1994anisotropic} supposed a particular form of the energy density of a stellar object and analyzed the formation of its internal structure. So, we proceed their assumption in this case as
\begin{equation}\label{f63}
\frac{\mu^{(eff)}}{H}=\mu_{o}\left(1-\frac{Kr^2}{r_{\Sigma}^2}\right),
\end{equation}
where $K\in(0,1)$. One may write the following by using Eq.\eqref{f63} in Eq.\eqref{f21} as
\begin{equation}\label{f64}
m(r)=\frac{r}{8}+\frac{s^2}{2r}+\beta r^3\left(-\frac{1}{3}+\frac{Kr^2}{5r_{\Sigma}^2}\right)-4\pi \int_{0}^{r}\frac{\tilde{r}^2Q_{0}^{(eff)}}{H}d\tilde{r},
\end{equation}
and $\beta=4\pi\mu_{0}$. Also, by combining Eqs.\eqref{f20} and \eqref{f64}, we get
\begin{equation}\label{f65}
\frac{1}{Y}=\sqrt{\frac{2\beta r^3}{ r}\left(\frac{1}{3}-\frac{Kr^2}{5r_{\Sigma}^2}\right)
+\frac{8\pi}{r}\int_{0}^{r}\frac{\tilde{r}^2Q_{0}^{(eff)}}{H}d\tilde{r}}.
\end{equation}
Futher, from field equations \eqref{f14} and \eqref{f15} we have
\begin{eqnarray}\label{f66}
\frac{8\pi}{H}\left[\Pi^{(eff)}+Q_{1}^{(eff)}-Q_{2}^{(eff)}\right]=\frac{1}{Y^2}\left[-\frac{X'}{rX}
-\frac{X'Y'}{XY}-\frac{1}{r^2}+\frac{X''}{X}-\frac{Y'}{rY}\right].
\end{eqnarray}
We propose some new variables which can be useful as
\begin{equation}\label{f67}
X^2=e^{\int(2z(r)-2/r)dr},\quad Y^{-2}=y(r),
\end{equation}
and putting back these variables in Eq.\eqref{f66}, we get
\begin{eqnarray}\label{f68}
y'+y\left[\frac{2z'}{z}+2z+\frac{4}{r^2z}-\frac{6}{r}\right]&=&\frac{16\pi}{Hz}\left(\Pi^{(eff)}+Q_{1}^{(eff)}-Q_{2}^{(eff)}\right),
\end{eqnarray}
which looks to be in the proposed form by Ricatti. In $f(R,T,Q)$ theory, the line element \eqref{f12} appears in terms of new variables after the above expression has been integrated as
\begin{align}\nonumber
ds^2&=e^{\int(2z(r)-2/r)dr}dt^2\\\nonumber
&-\frac{z^2(r)e^{\int \left(\frac{4}{r^2z(r)}+2z(r)\right)dr}}{r^6\left[16\pi\int\left\{\frac{z\left(\Pi^{(eff)}+Q_{1}^{(eff)}-Q_{2}^{(eff)}\right)e^{\int \left(\frac{4}{r^2z(r)}+2z(r)\right)dr}}{Hr^6}\right\} dr+C\right]}dr^2\\\label{f69}
&-r^2(d\theta^2+\alpha^2 dz^2),
\end{align}
where $C$ presents as a constant of integration. In addition, the field equations \eqref{f13}-\eqref{f15} can also be expressed in terms of variables \eqref{f67} as
\begin{align}\label{f70}
\frac{4\pi}{H}\left[\mu^{(eff)}+Q_{0}^{(eff)}\right]&=\frac{1}{2r^2}-\frac{m'}{r^2}+\frac{s}{r^3}
\left(s'-\frac{s}{2r}\right),
\\\label{f71}
\frac{4\pi}{H}\left[P^{(eff)}_{r}+Q_{1}^{(eff)}\right]&=\left(\frac{1}{2}-\frac{4m}{r}\right)
\left(\frac{1}{4r^2}-\frac{z}{2r}\right)+\frac{s^2}{r^3}\left(\frac{1}{2r}-z\right),
\\\nonumber
\frac{4\pi}{H}\left[P^{(eff)}_{t}+Q_{0}^{(eff)}\right]&=\left(-\frac{1}{8}+\frac{m}{r}\right)\left(z'+z^2-\frac{3z}{2r}
+\frac{1}{r^2}\right)+\frac{m'z}{2r}\\\label{f72}
&-\frac{s^2}{2r^2}\left(z'+z^2-\frac{2z}{r}+\frac{1}{r^2}+\frac{s'z}{s}\right).
\end{align}
In cylindrical structure, Eqs.\eqref{f70}-\eqref{f72} may be helpful in understanding its mysterious and highly fascinating properties. Di Prisco \emph{et al.} \cite{di2011expansion} done the same work in the background of GR and calculated their corresponding solutions, whereas Sharif and Yousaf \cite{sharif2012shearfree,sharif2012expansion} extended this for spherical and cylindrical structures in modified framework. Also, Yousaf \cite{yousaf2017spherical} and Bhatti \cite{bhatti2016shear} found the more generalized results for Einstein-$\Lambda$ gravity. The conditions \eqref{f23} calculated at the boundary $\Sigma$ must satisfied by the obtained solutions to prevent the effective material variables from being of a singular nature.

\subsection{The Polytropic Equation With Zero Complexity Factor}

The polytropical equation of state is a key to analyze any self-gravitating system. To explain the effects of modified corrections in our model, we have to solve a system of equations for which it is required to combine this equation of state with the disappearing complexity factor condition. We will discuss here two individual cases of polytropes in charged cylindrical star, of which the first is
\begin{equation}\label{f73}
P^{(eff)}_{r}=\omega[\mu^{(eff)}]^{\rho}=\omega[\mu^{(eff)}]^{(1+1/n)}; \quad Y_{TF}=0,
\end{equation}
where $\omega$ and $\rho$ denote the polytropic constant and polytropic exponent. Also, $n$ is termed as polytropic index.

In order to solve the system of equations easily, we are introducing few variables to write these equations in dimensionless form as
\begin{equation}\label{f74}
\sigma=P^{(eff)}_{rc}/\mu^{(eff)}_{c},\quad r=\eta/A,\quad A^2=4\pi\mu^{(eff)}_{c}/\sigma(n+1),
\end{equation}
\begin{equation}\label{f75}
\phi^n=\mu^{(eff)}/\mu^{(eff)}_{c},\quad \nu(\eta)=m(r)A^3/(4\pi\mu^{(eff)}_{c}),
\end{equation}
where $\mu_{c}^{(eff)}$ and $P_{rc}^{(eff)}$ show that these quantities are calculated at the center. Also, at the boundary $r=r_{\Sigma}$, we have $\eta=\eta_{\Sigma}$ and thus $\phi(\eta_{\Sigma})=0$. After utilizing Eqs. \eqref{f74} and \eqref{f75} in TOV equation \eqref{f19}, it becomes
\begin{align}\nonumber
&\frac{\eta^2}{1+\sigma\phi}\left[1-\frac{8\nu\sigma(n+1)}{\eta}+\frac{4s^2A^2}{\eta^2}\right]\frac{d\phi}{d\eta}
+\frac{2\phi^{-n}\eta}{P^{(eff)}_{rc}(n+1)(1+\sigma\phi)}\left(\Pi^{(eff)}\right.\\\nonumber
&+\left.Q_{1}^{(eff)}-Q_{2}^{(eff)}\right)\left[1-\frac{8\nu\sigma(n+1)}{\eta}+\frac{4s^2A^2}{\eta^2}\right]
+4\left(1+\frac{Q_{5}^{(eff)}\phi^{-n}}{\mu_{c}^{(eff)}(1+\sigma\phi)}\right)\\\nonumber
&\times\left[\frac{\sigma\eta^3\phi^{n+1}}{H}+\frac{\eta^3Q_{1}^{(eff)}}{H\mu_{c}^{(eff)}}-\nu+\frac{A^2\eta}{32\pi\mu_{c}^{(eff)}}
+\frac{s^2A^4}{8\pi\eta A\mu_{c}^{(eff)}}\right]=\frac{\eta^2\phi^{-n}}{AP_{rc}^{(eff)}(n+1)}\\\label{f76}
&\times\frac{1}{(1+\sigma\phi)}\left[1-\frac{8\nu\sigma(n+1)}{\eta}+\frac{4s^2A^2}{\eta^2}\right]
\left\{Y^2ZH+\frac{A}{H}\frac{dH}{d\eta}\left(\phi^{n+1}P_{rc}^{(eff)}+Q_{1}^{(eff)}\right)\right\}.
\end{align}
By taking derivative of $\nu(\eta)$ and combining it with Eq.\eqref{f70}, we have
\begin{equation}\label{f77}
\frac{d\nu}{d\eta}=\frac{A^2}{32\pi\mu_{c}^{(eff)}}-\frac{\eta^2}{H}\left(\phi^n+\frac{Q_{0}^{(eff)}}
{\mu_{c}^{(eff)}}\right)+\frac{sA^4}{8\pi\eta\mu_{c}^{(eff)}}\left(2\frac{ds}{d\eta}-\frac{s}{\eta}\right).
\end{equation}
It should be mentioned that two ordinary differential equations (ODE's) \eqref{f76} and \eqref{f77} consist of three unknown functions $\phi,~\nu$ and $\Pi$. We require one more equation to have a unique solution of this system and thus the dimensionless form of vanishing complexity factor condition \eqref{f62} becomes
\begin{align}\nonumber
&\frac{6\Pi}{n\mu^{(eff)}_{c}}+\frac{2\eta}{n\mu^{(eff)}_{c}}\frac{d\Pi}{d\eta}=\phi^{n-1}\eta\frac{d\phi}{d\eta}
+\frac{2\Pi\eta}{Hn\mu^{(eff)}_{c}}\frac{dH}{d\eta}-\frac{3}{n\mu^{(eff)}_{c}}\left[\left(\Pi-\frac{s^2A^4}{4\pi\eta^4}\right)\right.\\\nonumber
&\times\left(f_{T}+\frac{1}{2}Rf_{Q}\right)-\left.\frac{s^2A^4}{4\pi\eta^4}+\Pi^{(D)}+Q_{1}^{(eff)}-Q_{2}^{(eff)}+\frac{H}{4\pi}\varphi_{3}^{(D)}\right]
+\frac{H\eta\phi^n}{n}\frac{d}{d\eta}\left(\frac{1}{H}\right)\\\nonumber
&+\frac{H\eta}{n\mu^{(eff)}_{c}}\left[\frac{d}{d\eta}\left(\frac{Q^{(eff)}_{0}}{H}\right)-\frac{d}{d\eta}\left\{\frac{1}{H}\left(\Pi^{(D)}+Q_{1}^{(eff)}
-Q_{2}^{(eff)}\right)\right\}-\frac{d}{d\eta}\left\{\frac{1}{H}\right.\right.\\\label{f78}
&\times\left.\left.\left(\Pi-\frac{s^2A^4}{4\pi\eta^4}\right)\left(f_{T}+\frac{1}{2}Rf_{Q}\right)\right\}+\frac{sA^4}{2\pi H\eta^4}\frac{ds}{d\eta}-\frac{s^2A^4}{4\pi H^2\eta^4}\frac{dH}{d\eta}-\frac{s^2A^4}{\pi H \eta^5}-\frac{1}{4\pi}\frac{d\varphi_{3}^{(D)}}{d\eta}\right].
\end{align}
We finally get a system of three ODE's \eqref{f76},~\eqref{f77} and \eqref{f78}. This system provides us a unique solution by putting arbitrary values of parameters $\sigma$ and $n$. One can study some properties of stellar object such as mass and density with the help of solution of the above system which may obtain for particular values of $\sigma$ and $n$.

We are now ready to consider the second case of the polytropic equation of state as $P^{(eff)}_{r}=\omega[\mu_{b}^{(eff)}]^{\rho}=\omega[\mu_{b}^{(eff)}]^{(1+1/n)}$, where $\mu_{b}$ indicate the baryonic (rest) mass density. Therefore, Eqs.\eqref{f76} and \eqref{f78} can be rewritten in this case as
\begin{align}\nonumber
&\frac{\eta^2}{1+\sigma\phi_{b}}\left[1-\frac{8\nu\sigma(n+1)}{\eta}+\frac{4s^2A^2}{\eta^2}\right]\frac{d\phi_{b}}{d\eta}
+\frac{2\phi_{b}^{-n}\eta}{P^{(eff)}_{rc}(n+1)(1+\sigma\phi_{b})}\left(\Pi^{(eff)}\right.\\\nonumber
&+\left.Q_{1}^{(eff)}-Q_{2}^{(eff)}\right)\left[1-\frac{8\nu\sigma(n+1)}{\eta}+\frac{4s^2A^2}{\eta^2}\right]
+4\left(1+\frac{Q_{5}^{(eff)}\phi_{b}^{-n}}{\mu_{bc}^{(eff)}(1+\sigma\phi_{b})}\right)\\\nonumber
&\times\left[\frac{\sigma\eta^3\phi_{b}^{n+1}}{H}+\frac{\eta^3Q_{1}^{(eff)}}{H\mu_{bc}^{(eff)}}-\nu+\frac{A^2\eta}{32\pi\mu_{bc}^{(eff)}}
+\frac{s^2A^4}{8\pi\eta A\mu_{bc}^{(eff)}}\right]=\frac{\eta^2\phi_{b}^{-n}}{AP_{rc}^{(eff)}(n+1)}\\\label{f79}
&\times\frac{1}{(1+\sigma\phi_{b})}\left[1-\frac{8\nu\sigma(n+1)}{\eta}+\frac{4s^2A^2}{\eta^2}\right]
\left\{Y^2ZH+\frac{A}{H}\frac{dH}{d\eta}\left(\phi_{b}^{n+1}P_{rc}^{(eff)}+Q_{1}^{(eff)}\right)\right\},
\end{align}
and
\begin{align}\nonumber
&\frac{6\Pi}{n\mu^{(eff)}_{bc}}+\frac{2\eta}{n\mu^{(eff)}_{bc}}\frac{d\Pi}{d\eta}=\phi_{b}^{n-1}\eta\left[1+K(n+1)
\left(\mu_{bc}^{(eff)}\right)^{1/n}\phi_{b}\right]\frac{d\phi_{b}}{d\eta}+\frac{2\Pi\eta}{Hn\mu^{(eff)}_{bc}}\\\nonumber
&\times\frac{dH}{d\eta}-\frac{3}{n\mu^{(eff)}_{bc}}\left[\left(\Pi-\frac{s^2A^4}{4\pi\eta^4}\right)\left(f_{T}
+\frac{1}{2}Rf_{Q}\right)-\frac{s^2A^4}{4\pi\eta^4}+\Pi^{(D)}+Q_{1}^{(eff)}\right.\\\nonumber
&-\left.Q_{2}^{(eff)}+\frac{H}{4\pi}\varphi_{3}^{(D)}\right]+\frac{H\eta\phi_{b}^n}{n}\frac{d}{d\eta}\left(\frac{1}{H}\right)+\frac{H\eta}{n\mu^{(eff)}_{bc}}
\left[\frac{d}{d\eta}\left(\frac{Q^{(eff)}_{0}}{H}\right)-\frac{d}{d\eta}\left\{\frac{1}{H}\left(\Pi^{(D)}\right.\right.\right.\\\nonumber
&+\left.\left.Q_{1}^{(eff)}-Q_{2}^{(eff)}\right)\right\}-\frac{d}{d\eta}\left\{\frac{1}{H}\left(\Pi-\frac{s^2A^4}{4\pi\eta^4}\right)
\left(f_{T}+\frac{1}{2}Rf_{Q}\right)\right\}+\frac{sA^4}{2\pi H\eta^4}\frac{ds}{d\eta}\\\label{f80}
&-\left.\frac{s^2A^4}{4\pi H^2\eta^4}\frac{dH}{d\eta}-\frac{s^2A^4}{\pi H \eta^5}-\frac{1}{4\pi}\frac{d\varphi_{3}^{(D)}}{d\eta}\right],
\end{align}
where $\phi_{b}^n=\mu_{b}^{(eff)}/\mu_{bc}^{(eff)}$.
\section{Conclusions}

The basic objective in doing this work is to analyze the self-gravitating system having cylindrical symmetry under $f(R,T,Q)$ corrections and for this reason, we considered anisotropic static charged geometry. The field equations containing $f(R,T,Q)$ corrections, C-energy and Tolman mass function are derived, and their relations with Conformal tensor are also developed. The tensors $X_{\rho\gamma},~Y_{\rho\gamma}$ and $Z_{\rho\gamma}$ are found from proper orthogonal splitting of the Riemann tensor and then received five modified scalar functions from them. After this, the effects of these scalars on the sustainability and evolution of cylindrical relativistic fluid were studied. Herrera \cite{herrera2018new} suggested the idea to find complexity of spherical star by preceding the assumption that the system with homogeneous energy density and isotropic pressure is simplest, i.e., there is no complexity. The corresponding modified scalar $Y_{TF}$ is therefore known as the complexity factor in this case. Thus, we are focused on some significant changes in $f(R,T,Q)$ theory.\\

(i) The effective scalar $Y_{TF}$ includes the effects of electric charge as we are studying charged cylindrical object.\\

(ii) For this charged case, the system becomes more complex (we can see from Eq.\eqref{f58}), because $Y_{TF}$ also contains the effective charge along with effective inhomogeneous energy density and effective pressure anisotropy.\\

(iii) The Tolman mass in $f(R,T,Q)$ theory could be measured via this scalar $Y_{TF}$ in terms of above effective physical quantities.\\

After calculating the Darmois junction conditions on the boundary $\Sigma$ in modified gravity, we briefly discussed the electric and magnetic parts of Weyl tensor. In addition, we have examined two different applications of stellar systems by adding modified corrections through vanishing of complexity factor \eqref{f58}. In first of them, we modified the form of energy density which was suggested by Gokhroo and Mehra to analyze the characteristics of stellar structures. In second example, we discussed the equation of state for polytropic substance and also added certain variables to make some equations dimensionless. For some realistic constraints, the obtained system of modified equations offer a solution that allows us to better understand the system. To get all our results in GR with charge free case, one can substitute $f(R,T,Q)=R$ and $L_{EM}=0$ in action function \eqref{f1}.

\section*{Appendix A}

The effective quantities appearing in field equations \eqref{f13}-\eqref{f15} are
\begin{align}\nonumber
\mu^{(eff)}&=\mu\left[1+2f_{T}+f_{Q}\left(\frac{3X'Y'}{2XY^3}-\frac{3X'}{rXY^2}+\frac{1}{2}R-\frac{3X''}{2XY^2}\right)-f'_{Q}\left(\frac{Y'}{2Y^3}\right.\right.\\\nonumber
&-\left.\left.\frac{1}{rY^2}\right)+\frac{f''_{Q}}{2Y^2}\right]
+\mu'\left[-f_{Q}\left(\frac{Y'}{2Y^3}-\frac{1}{rY^2}\right)+\frac{f'_{Q}}{Y^2}\right]
+\frac{\mu''f_{Q}}{2Y^2}+P_{r}\left[f_{Q}\right.\\\nonumber
&\times\left.\left(\frac{Y'}{rY^3}-\frac{1}{r^2Y^2}+\frac{X''}{2XY^2}-\frac{X'Y'}{2XY^3}\right)-f'_{Q}\left(\frac{2}{rY^2}
-\frac{Y'}{2Y^3}\right)-\frac{f''_{Q}}{2Y^2}\right]\\\nonumber
&-P'_{r}\left[f_{Q}\left(\frac{2}{rY^2}-\frac{Y'}{2Y^3}\right)
+\frac{f'_{Q}}{Y^2}\right]-\frac{P''_{r}f_{Q}}{2Y^2}+P_{t}\left[f_{Q}\left(\frac{1}{r^2Y^2}+\frac{X'}{rXY^2}\right.\right.\\\label{f81}
&-\left.\left.\frac{Y'}{rY^3}\right)+\frac{f'_{Q}}{rY^2}\right]+\frac{P'_{t}f_{Q}}{rY^2}+\frac{R}{2}\left(\frac{f}{R}-f_{R}\right)-f'_{R}\left(\frac{Y'}{Y^3}-\frac{2}{rY^2}\right)+\frac{f''_{R}}{Y^2},
\\\nonumber
P_{r}^{(eff)}&=\mu\left[-f_{T}+f_{Q}\left(\frac{X''}{2XY^2}+\frac{X'}{rXY^2}-\frac{X'Y'}{2XY^3}\right)
-\frac{f'_{Q}X'}{2XY^2}\right]-\frac{\mu'X'f_{Q}}{2XY^2}\\\nonumber
&+P_{r}\left[1+f_{T}+f_{Q}\left(\frac{2X'}{rXY^2}+\frac{1}{2}R+\frac{1}{r^2Y^2}+\frac{3X'Y'}{2XY^3}+\frac{3Y'}{rY^3}-\frac{3X''}{2XY^2}\right)\right.\\\nonumber
&+\left.f'_{Q}\left(\frac{1}{rY^2}+\frac{X'}{2XY^2}\right)\right]+P'_{r}\left[f_{Q}\left(\frac{1}{rY^2}+\frac{X'}{2XY^2}\right)\right]
+P_{t}\left[f_{Q}\left(\frac{Y'}{rY^3}\right.\right.\\\nonumber
&-\left.\left.\frac{1}{r^2Y^2}-\frac{X'}{rXY^2}\right)+\frac{3f'_{Q}}{2rY^2}\right]+\frac{P'_{t}f_{Q}}{rY^2}
+\frac{R}{2}\left(f_{R}-\frac{f}{R}\right)-f'_{R}\\\label{f82}
&\times\left(\frac{2}{rY^2}+\frac{X'}{XY^2}\right),
\\\nonumber
P_{t}^{(eff)}&=\mu\left[-f_{T}+f_{Q}\left(\frac{X''}{2XY^2}+\frac{X'}{rXY^2}-\frac{X'Y'}{2XY^3}\right)
+\frac{f'_{Q}X'}{2XY^2}\right]+\frac{\mu'X'f_{Q}}{2XY^2}\\\nonumber
&+P_{r}\left[f_{Q}\left(\frac{X''}{2XY^2}+\frac{X'}{rXY^2}-\frac{X'Y'}{2XY^3}\right)
+f'_{Q}\left(\frac{1}{rY^2}+\frac{X'}{XY^2}-\frac{Y'}{2Y^3}\right)\right.\\\nonumber
&+\left.\frac{f''_{Q}}{2Y^2}\right]+P'_{r}\left[f_{Q}\left(\frac{1}{rY^2}+\frac{X'}{XY^2}-\frac{Y'}{2Y^3}\right)+\frac{f'_{Q}}{Y^2}\right]
+\frac{P''_{r}f_{Q}}{2Y^2}+P_{t}\left[1+f_{T}\right.\\\nonumber
&+\left.\left.f_{Q}\left(\frac{2Y'}{rY^3}+\frac{1}{2}R-\frac{2}{r^2Y^2}-\frac{2X'}{rXY^2}\right)
-f'_{Q}\left(\frac{Y'}{2Y^3}-\frac{X'}{2XY^2}\right)+\frac{f''_{Q}}{2Y^2}\right]\right.\\\nonumber
&+P'_{t}\left[-f_{Q}\left(\frac{Y'}{2Y^3}-\frac{X'}{2XY^2}\right)
+\frac{f'_{Q}}{Y^2}\right]+\frac{P''_{t}f_{Q}}{2Y^2}+\frac{R}{2}\left(f_{R}-\frac{f}{R}\right)-f'_{R}\\\label{f83}
&\times\left(\frac{1}{rY^2}-\frac{Y'}{Y^3}+\frac{X'}{XY^2}\right)-\frac{f''_{R}}{Y^2},
\\\nonumber
Q_{0}^{(eff)}&=\left(1+\frac{1}{2}Rf_{Q}+f_{T}\right)\frac{s^2}{8\pi r^4}-\frac{s^2f_{T}}{2r^4}
+f''_{Q}\left(\frac{s^2}{8\pi r^4X^2}+\frac{s^2}{8\pi r^4Y^2}\right)\\\nonumber
&+f'_{Q}\left(\frac{s^2X'}{8\pi r^4XY^2}-\frac{s^2}{2\pi r^5Y^2}+\frac{ss'}{2\pi r^4Y^2}-\frac{s^2Y'}{8\pi r^4Y^3}-\frac{s^2Y'}{8\pi r^4X^2Y}\right.\\\nonumber
&+\left.\frac{ss'}{2\pi r^4X^2}-\frac{s^2}{2\pi r^5X^2}\right)+f_{Q}\left(\frac{ss'X'}{4\pi r^4XY^2}-\frac{ss'}{\pi r^5Y^2}+\frac{3s^2X'}{8\pi r^5XY^2}\right.\\\nonumber
&+\frac{s'^2}{4\pi r^4Y^2}+\frac{ss''}{4\pi r^4Y^2}-\frac{ss'Y'}{4\pi r^4Y^3}+\frac{3s^2}{4\pi r^6Y^2}+\frac{s^2Y'}{4\pi r^5Y^3}+\frac{s'^2}{4\pi r^4X^2}\\\label{f84}
&+\left.\frac{ss''}{4\pi r^4X^2}-\frac{ss'}{\pi r^5X^2}-\frac{ss'Y'}{4\pi r^4X^2Y}+\frac{s^2X''}{4\pi r^4X^3}
-\frac{s^2X'Y'}{4\pi r^4X^3Y}\right),
\\\nonumber
Q_{1}^{(eff)}&=-\left(1+\frac{1}{2}Rf_{Q}+f_{T}\right)\frac{s^2}{8\pi r^4}+\frac{s^2f_{T}}{2r^4}
-\frac{f'_{Q}s^2X'}{8\pi r^4XY^2}+f_{Q}\left(\frac{s^2X''}{4\pi r^4XY^2}\right.\\\nonumber
&-\frac{s^2X'Y'}{4\pi r^4XY^3}+\frac{ss'Y'}{4\pi r^4Y^3}-\frac{ss'X'}{4\pi r^4XY^2}+\frac{s^2X'}{8\pi r^5XY^2}-\frac{s^2}{4\pi r^6Y^2}\\\label{f85}
&-\left.\frac{3s^2Y'}{4\pi r^5Y^3}\right),
\\\nonumber
Q_{2}^{(eff)}&=\left(1+\frac{1}{2}Rf_{Q}+f_{T}\right)\frac{s^2}{8\pi r^4}+\frac{s^2f_{T}}{2r^4}-\frac{f'_{Q}s^2}{8\pi r^5Y^2}+f_{Q}\left(\frac{s^2Y'}{4\pi r^5Y^3}\right.\\\label{f86}
&-\left.\frac{ss'}{4\pi r^5Y^2}-\frac{3s^2X'}{8\pi r^5XY^3}+\frac{s^2}{4\pi r^6Y^2}\right).
\end{align}

The value of $Z$ which is arisen in Eq.\eqref{f17} is
\begin{align}\nonumber
Z &= \frac{2}{\left(2+Rf_{Q}+2f_{T}\right)}\left[\frac{f'_{Q}}{Y^2}\left(P_{r}-\frac{s^2}{8\pi r^4}\right)\left(\frac{Y'}{rY}+\frac{X'Y'}{2XY}+\frac{1}{2r^2}+\frac{X'}{rX}
\right)+\frac{1}{2Y^2}\right.\\\nonumber
&\times\left(P_{r}-\frac{s^2}{8\pi r^4}\right)\left\{f_{Q}\left(\frac{X''Y'}{XY}+\frac{2Y''}{rY}-\frac{6Y'^2}{rY^2}-\frac{9X'Y'^2}{XY^2}-\frac{X'^2Y'}{X^2Y}+\frac{X'Y''}{XY}\right.\right.\\\nonumber
&-\left.\left.\frac{4Y'}{r^2Y}
-\frac{2X'}{r^2X}+\frac{2X''}{rX}-\frac{4X'Y'}{rXY}-\frac{2X'^2}{rX^2}-\frac{2}{r^3}\right)+2f'_{T}Y^2\right\}+\frac{1}{2Y^2}\left(P'_{r}\right.\\\nonumber
&-\left.\frac{ss'}{4\pi r^4}+\frac{s^2}{2\pi r^5}\right)\left\{f_{Q}\left(\frac{X'Y'}{XY}
+\frac{2Y'}{rY}-\frac{X''}{X}\right)+f_{T}Y^2\right\}-\frac{3f_{T}}{2}\left(\mu'-\frac{ss'}{r^4}\right.\\\nonumber
&\left.+\frac{2s^2}{r^5}\right)- f'_{T}\left(\mu -\frac{s^2}{2r^4}\right)-\frac{f_{Q}}{XY^2}\left(\mu'+\frac{ss'}{4\pi r^4}-\frac{s^2}{2\pi r^5}\right)\left(\frac{Y'}{r}-\frac{X'Y'}{2Y}+\frac{X''}{2}\right)\\\nonumber
&-\frac{1}{r^2Y^2}\left(P'_{t}+\frac{ss'}{4\pi r^2}-\frac{s^2}{4\pi r^3}\right)\left\{f_{Q}\left(\frac{rY'}{Y}-1-\frac{rX'}{X}\right)-f_{T}r^2Y^2\right\}+\left(\frac{1}{r^2}\right.\\\label{f87}
&+\left.\left.\frac{2X'}{rX}\right)\left\{f_{Q}\left(\mu'-\frac{ss'}{r^4}+\frac{2s^2}{r^5}\right)+ f'_{Q}\left(\mu-\frac{s^2}{2r^4}\right)\right\}\right].
\end{align}

The modified correction terms $F_{0}$ and $F_{1}$ in Eq.\eqref{f23} are given as
\begin{eqnarray}\nonumber
F_{0}&=&\tilde{f}_{T}\left(P_{r}-\mu\right)+\tilde{f}_{Q}\left[\mu\left(\frac{X''}{2XY^2}+\frac{X'}{rXY^2}
-\frac{X'Y'}{2XY^3}\right)-\frac{\mu'X'}{2XY^2}\right.\\\nonumber
&+&P_{r}\left(\frac{2X'}{rXY^2}+\frac{1}{2}R-\frac{3X''}{2XY^2}
+\frac{1}{r^2Y^2}+\frac{3X'Y'}{2XY^3}+\frac{3Y'}{rY^3}\right)+P'_{r}\left(\frac{X'}{2XY^2}\right.\\\label{f88}
&+&\left.\left.\frac{1}{rY^2}\right)+P_{t}\left(\frac{Y'}{rY^3}-\frac{1}{r^2Y^2}-\frac{X'}{rXY^2}\right)
+\frac{P'_{t}}{rY^2}\right]+\frac{R}{2}\left(\tilde{f}_{R}-\frac{f}{R}\right),
\\\nonumber
F_{1}&=&-\left(\frac{1}{2}R\tilde{f}_{Q}+\tilde{f}_{T}\right)\frac{s^2}{8\pi r^4}+\frac{s^2\tilde{f}_{T}}{2r^4}
+\tilde{f}_{Q}\left(\frac{s^2X''}{4\pi r^4XY^2}-\frac{s^2X'Y'}{4\pi r^4XY^3}\right.\\\nonumber
&+&\frac{ss'Y'}{4\pi r^4Y^3}-\frac{ss'X'}{4\pi r^4XY^2}+\frac{s^2X'}{8\pi r^5XY^2}-\frac{s^2}{4\pi r^6Y^2}+\frac{s^2X'Y'}{r^4XY^3}-\frac{s^2X''}{r^4XY^2}\\\label{f89}
&-&\left.\frac{3s^2Y'}{4\pi r^5Y^3}\right).
\end{eqnarray}

\section*{Appendix B}

The effective terms $\varphi_{1}^{(D)},~\varphi_{2}^{(D)}$ and $\varphi_{\delta\gamma}^{(D)}$ appeared in structure scalars \eqref{f53}, \eqref{f55} and \eqref{f56} are given as
\begin{eqnarray}\nonumber
\varphi_{1}^{(D)}&=&\frac{4\pi}{H}\left[\left\{h^{\gamma}_{\nu}\Box(f_{Q}X^{\nu}_{\gamma})-2h^{\gamma}_{\nu}\nabla^{\nu}\nabla_{\gamma}f_{R}
-h^{\gamma}_{\nu}\nabla_{\pi}\nabla_{\gamma}(f_{Q}X^{\pi\nu})\right.\right.\\\nonumber
&-&\left.h^{\gamma}_{\nu}\nabla_{\pi}\nabla^{\nu}(f_{Q}X^{\pi}_{\gamma})\right\}
+2f_{Q}\left(R^{\nu}_{\pi}h^{\pi}_{\nu}+R_{\pi\gamma}h^{\gamma\pi}\right)\left\{\frac{1}{3}\left(\Pi-\frac{s^2}{4\pi r^4}\right)\right.\\\nonumber
&-&\left.\left.\left(P+\frac{s^2}{24\pi r^4}\right)\right\}\right]
+\frac{8\pi}{H}\left[\left\{\frac{R}{2}\left(\frac{f}{R}-f_{R}\right)+\left(\mu-\frac{s^2}{2r^4}\right) f_{T}\right.\right.\\\nonumber
&-&\left.\frac{1}{2}\nabla_{\pi}\nabla_{\rho}(f_{Q}X^{\pi\rho})\right\}+2Rf_{Q}\left\{\left(P+\frac{s^2}{24\pi r^4}\right)-\frac{1}{3}\left(\Pi-\frac{s^2}{4\pi r^4}\right)\right\}\\\nonumber
&+&\nabla_{\pi}\nabla_{\nu}(f_{Q}X^{\pi\nu})-\frac{1}{2}\Box\{f_{Q}(\mu-3P)\}+2g^{\nu\gamma}(f_{Q}R^{\pi\rho}+f_{T}g^{\pi\rho})\\\label{f90}
&\times&\left.\frac{\partial^2L_{m}}{\partial g^{\nu\gamma}\partial g^{\pi\rho}}\right],
\\\nonumber
\varphi_{2}^{(D)}&=&-\frac{8\pi}{H}\left\{\frac{R}{2}\left(\frac{f}{R}-f_{R}\right)+\left(\mu-\frac{s^2}{2r^4}\right) f_{T}-\frac{1}{2}\nabla_{\pi}\nabla_{\rho}(f_{Q}X^{\pi\rho})\right\}+\frac{4\pi}{H}\\\nonumber
&\times&\left[-\frac{1}{2}\left\{\Box(f_{Q}X)+3v^{\alpha}v_{\delta}\Box(f_{Q}X^{\delta}_{\alpha})-v^{\delta}v^{\alpha}\Box(f_{Q}X_{\delta\alpha})
\right\}+\left(\Box f_{R}+3v^{\alpha}v_{\delta}\right.\right.\\\nonumber
&\times&\left.\nabla^{\delta}\nabla_{\alpha}f_{R}-v^{\delta}v^{\alpha}\nabla_{\delta}\nabla_{\alpha}f_{R}\right)
-2f_{Q}R^{\alpha}_{\pi}\left\{\left(\Pi-\frac{s^2}{4\pi r^4}\right)\left(w^{\pi}w_{\alpha}
+\frac{1}{3}h^{\pi}_{\alpha}\right)\right.\\\nonumber
&-&\left.\left(P+\frac{s^2}{24\pi r^4}\right) h^{\pi}_{\alpha}\right\}-3f_{Q}\left(R_{\pi\alpha} v^{\pi}v^{\alpha}+R^{\delta}_{\pi}v^{\pi}v_{\delta}\right)\left(\mu+\frac{s^2}{8\pi r^4}\right)\\\nonumber
&+&\frac{1}{2}\{\nabla_{\pi}\nabla_{\delta}(f_{Q}X^{\pi\delta})+4v_{\delta}v^{\alpha}\nabla_{\pi}\nabla^{\delta}(f_{Q}X^{\pi}_{\alpha})
+\nabla_{\pi}\nabla_{\alpha}(f_{Q}X^{\pi\alpha})+4v_{\delta}v^{\alpha}\\\nonumber
&\times&\nabla_{\pi}\nabla_{\alpha}(f_{Q}X^{\pi\delta})-v_{\delta}v^{\alpha}\nabla_{\pi}\nabla_{\alpha}(f_{Q}X^{\delta\pi})
-v^{\delta}v^{\alpha}\nabla_{\pi}\nabla_{\delta}(f_{Q}X^{\pi}_{\alpha})-v_{\delta}v^{\alpha}\\\nonumber
&\times&\nabla_{\pi}\nabla^{\delta}(f_{Q}X^{\pi}_{\alpha})
-v^{\delta}v^{\alpha}\nabla_{\pi}\nabla_{\alpha}(f_{Q}X^{\pi}_{\delta})\}+2h^{\epsilon\alpha}(f_{Q}R^{\pi\nu}+f_{T}g^{\pi\nu})\\\nonumber
&\times&\left.\frac{\partial^2L_{m}}{\partial g^{\epsilon\alpha}\partial g^{\pi\nu}}\right]+\frac{8\pi}{H}\left[\frac{1}{2}\Box\{f_{Q}(\mu-3P)\}
-\nabla_{\pi}\nabla_{\rho}(f_{Q}X^{\pi\rho})+2f_{Q}R_{\pi\rho}X^{\pi\rho}\right.\\\label{f91}
&-&\left.2g^{\rho\xi}(f_{Q}R^{\pi\nu}+f_{T}g^{\pi\nu})\frac{\partial^2L_{m}}{\partial g^{\rho\xi}\partial g^{\pi\nu}}\right],
\\\nonumber
\varphi_{\delta\gamma}^{(D)}&=&-\frac{2\pi}{H}\left[h^{\lambda}_{\delta}h^{\mu}_{\gamma}\Box(f_{Q}X_{\lambda\mu})
-v_{\delta}v_{\gamma}v_{\beta}v^{\alpha}\Box(f_{Q}X^{\beta}_{\alpha})-\Box(f_{Q}X_{\delta\gamma})\right]+\frac{4\pi}{H}\\\nonumber
&\times&\left[(h^{\lambda}_{\delta}h^{\mu}_{\gamma}\nabla_{\mu}\nabla_{\lambda}f_{R}
-v_{\delta}v_{\gamma}v_{\beta}v^{\alpha}\nabla^{\beta}\nabla_{\alpha}f_{R}-\nabla_{\delta}\nabla_{\gamma}f_{R})-f_{Q}(R_{\lambda\pi}h^{\lambda}_{\delta}\right.\\\nonumber
&-&R_{\delta\pi})\left\{\left(\Pi-\frac{s^2}{4\pi r^4}\right)\left(w^{\pi}w_{\gamma}+\frac{1}{3}h^{\pi}_{\gamma}\right)-\left(P+\frac{s^2}{24\pi r^4}\right)h^{\pi}_{\gamma}\right\}+f_{Q}(R_{\pi\gamma}\\\nonumber
&-&R_{\pi\mu}h^{\mu}_{\gamma})\left\{\left(\Pi-\frac{s^2}{4\pi r^4}\right)\left(w^{\pi}w_{\delta}
+\frac{1}{3}h^{\pi}_{\delta}\right)-\left(P+\frac{s^2}{24\pi r^4}\right)h^{\pi}_{\delta}\right\}\\\nonumber
&+&\frac{1}{2}\{h^{\lambda}_{\delta}h^{\mu}_{\gamma}\nabla_{\pi}\nabla_{\lambda}(f_{Q}X^{\pi}_{\mu})
-\nabla_{\pi}\nabla_{\gamma}(f_{Q}X^{\pi}_{\delta})+h^{\lambda}_{\delta}h^{\mu}_{\gamma}\nabla_{\pi}\nabla_{\mu}(f_{Q}X^{\pi}_{\lambda})\\\nonumber
&-&\nabla_{\pi}\nabla_{\delta}(f_{Q}X^{\pi}_{\gamma})
-v_{\delta}v_{\gamma}v_{\beta}v^{\alpha}\nabla_{\pi}\nabla^{\beta}(f_{Q}X^{\pi}_{\alpha})
-v_{\delta}v_{\gamma}v_{\beta}v^{\alpha}\nabla_{\pi}\nabla_{\alpha}(f_{Q}X^{\pi\beta})\}\\\label{f92}
&+&\left.2(f_{Q}R^{\pi\nu}+f_{T}g^{\pi\nu})h^{\epsilon}_{\delta}\left\{h^{\mu}_{\gamma}\frac{\partial^2L_{m}}{\partial g^{\epsilon\mu}\partial g^{\pi\nu}}-\frac{\partial^2L_{m}}{\partial g^{\epsilon\gamma}\partial g^{\pi\nu}}\right\}\right].
\end{eqnarray}

\vspace{0.5cm}

{\bf Acknowledgments}

\vspace{0.25cm}

This work was supported by National Research Project for Universities (NRPU), Higher Education Commission, Pakistan
under the research project No. 8754/Punjab/NRPU/R\&D/HEC/2017.

\vspace{0.3cm}

redpajama_set_name: RedPajamaArXiv


LONDON (Reuters) - The European Central Bank may have to implement quantitative easing measures again in order to contain a worsening budget crisis in Italy, German asset management firm DWS said on Tuesday.
The ECB has been gradually winding down its stimulus program - launched to revive the euro zone economy and avoid deflation after the debt crisis - and its asset purchases will end this year.
"China, given its view toward data security, data usage and morality, is the place that they get ahead in terms of autonomous driving," said Schuh.
"Constructive" on the pound, he saw a boost from this relief potential as well as a chance of a win-win from Brexit for the UK in the longer term. "The very long term outcome could be a re-industrialization of Britain, a rethinking of the business model," said Schuh.
If fear in the market ramps up with no Brexit resolution by February or March and volatility is high, then Schuh said he would invest in both the pound and equities, adding that DWS is still a little bit under-invested toward a UK rebound.


In vertebrates, the genes encoding the class of transcription factors called homeobox genes are found in clusters named A, B, C, and D on four separate chromosomes. Expression of these proteins is spatially and temporally regulated during embryonic development. This gene is part of the A cluster on chromosome 7 and encodes a DNA-binding transcription factor which may regulate gene expression, morphogenesis, and differentiation. This gene is highly similar to the abdominal-B (Abd-B) gene of Drosophila. A specific translocation event which causes a fusion between this gene and the NUP98 gene has been associated with myeloid leukemogenesis. Read-through transcription exists between this gene and the upstream homeobox A10 (HOXA10) gene.


Synopsis The Bill repeals the Change of Names Registration Act 1923, the Legitimation Act 1909 and the Registration of Identity of Persons Act 1975 and makes consequential amendments to a number of other statutes as part of an important package of reforms proposed under the Births, Deaths and Marriages Registration Bill 1998 to achieve greater nationwide consistency in the registration of births, deaths and marriages.
2.	Comes into effect on the Proclamation of the Births, Deaths and Marriages Registration Bill (123-1).
3.	The Committee reported on May 27 1998.


The Certified Automation Cybersecurity Expert (CACE) and Certified Automation Cybersecurity Specialist (CACS) specialties were developed in response to the growing demand to secure industrial automation control systems (IACS) and SCADA systems by providing a means for people to develop and demonstrate competence in understanding and applying security standards and best practices.
WHY BECOME A CACE OR CACS?
Participation in the CACE / CACS certification specialties can provide many benefits to you and your company. CACE / CACS certification allows individuals to demonstrate their competency in a specialized field that is growing and in high demand. Attaining certification will make you a more valuable employee and can open the door to career advancement. CACE / CACS certified staff helps companies demonstrate their organization's expertise and their commitment to providing best-in-class services to their customers.
Control system cybersecurity is an emerging, highly specialized field of engineering. Its goal is to assure a systematic, lifecycle approach is used to assess, design, implement, and monitor methods to protect critical infrastructure from attack. The field of industrial automation and control system cybersecurity combines disciplines of basic control system engineering, risk assessment, safety system engineering, software (e.g. Microsoft) platforms, industrial networking, and IT network security.
While several programs exist for training and certification of each of these individual disciplines, there is no program today that focuses on the intersection of these disciplines as it applies to control system cybersecurity. The CACE / CACS focuses on the practical application of security practices and principles, not on knowledge of theory.
This certification is targeted for engineers and control system administrators who are involved in establishing, designing and implementing a comprehensive cybersecurity strategy for complex industrial control or SCADA systems. The specialty allows these individuals to demonstrate their competency in leading and managing complex ICS cybersecurity programs as well as evaluating and assessing the cybersecurity of existing ICS systems or system designs.
This certification is appropriate for engineers and technicians responsible for the design, installation & commissioning, and maintenance of industrial control or SCADA systems. The specialty allows these individuals to demonstrate their understanding and appreciation of IACS cybersecurity principles and best practices as they apply to operating and maintaining the security of their industrial automation and control systems.
Furthermore, the specialty evaluates the candidate's competency in understanding, applying and maintaining technical cybersecurity controls in their day-to-day activities.
Both specialties require the applicant to demonstrate that they have the requisite experience and knowledge to perform these job functions.


text:
Child Nutrition Programs(90)
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CEP: Community Eligibility Provision(1)
Demonstrations(7)
Direct Certification(13)
Farm Bill(1)
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HHFKA(2)
Nutrient Analysis(1)
Program Integrity(30)
WBSCM(1)
03/30/2018 Evaluation of Pilot Project for Procurement of Unprocessed Fruits and Vegetables
02/08/2018 Evaluation of Demonstration Projects to End Childhood Hunger (EDECH): Final Interim Evaluation Report
01/19/2017 Evaluation of the Elementary Schools Pilot Project for Canned, Frozen, or Dried Fruits and Vegetables in the FFVP
12/02/2016 Direct Certification in the National School Lunch Program Report to Congress: State Implementation Progress, School Year 2014-2015
11/29/2016 The Second Access, Participation, Eligibility, and Certification Study (APEC II): Estimating and Validating Statistical Models for Updating Estimates of Improper Payments in the NSLP and SBP
10/31/2016 Special Nutrition Program Operations Study, School Year 2013-14
10/31/2016 School Nutrition Program Operations Study, School Year 2012-13
10/24/2016 Examining the Potential to Expand Data Matching in the School Meal Program Eligibility and Verification Processes
08/29/2016 Approaches to Measuring Erroneous Payments in the National School Lunch Program and School Breakfast Program At The State Level
07/01/2016 Regional Office Review of Applications (RORA) for School Meals 2015
06/06/2016 Year 2 Demonstration Impacts of Using Medicaid Data to Directly Certify Students for Free School Meals
02/29/2016 Nutrient and Food Group Analysis of USDA Foods in Five of its Food and Nutrition Programs - 2014
01/28/2016 HHFKA Implementation Research Brief Series
05/05/2015 Diet Quality of American School Children by National School Lunch Program Participation Status: Data from the National Health and Nutrition Examination Survey, 2005-2010
05/04/2015 Program Error in the NSLP and SBP: Findings from the Second Access, Participation, Eligibility and Certification Study (APEC II)
01/27/2015 Evaluation of Demonstrations of National School Lunch Program and School Breakfast Program Direct Certification of Children Receiving Medicaid Benefits: Access Evaluation Report
01/27/2015 Evaluation of Demonstrations of National School Lunch Program and School Breakfast Program Direct Certification of Children Receiving Medicaid Benefits: Year 1 Report
01/27/2015 National School Lunch Program Direct Certification Study: State Practices and Performance Report
11/14/2014 Evaluation of the Impact of Wave 2 Incentives Demonstrations on Participation in the Summer Food Service Program (SFSP): FY 2012
08/29/2014 Regional Office Review of Applications ( RORA) for School Meals 2013
08/29/2014 National School Lunch Program Direct Certification Improvement Study
03/31/2014 School Foodservice Indirect Cost Study
03/20/2014 Special Nutrition Program Operations Study: State and School Food Authority Policies and Practices for School Meals Programs School Year 2011-12
02/25/2014 Community Eligibility Provision Evaluation
12/05/2013 Direct Certification in the National School Lunch Program: State Implementation Progress, School Year 2012–2013: Report to Congress
04/22/2013 School Associated Foodborne Outbreak Data
03/15/2013 Analysis of Verification Summary Data School Year 2008-2009
12/20/2012 Regional Office Review of Application (RORA) For School Meals 2011
12/19/2012 Using American Community Survey Data to Expand Access to School Meals Programs
12/19/2012 CACFP Assessment of Sponsor Tiering Determinations 2011
12/12/2012 School Nutrition Dietary Assessment Study IV
10/01/2012 Direct Certification in the National School Lunch Program: State Implementation Progress School Year 2011-2012: Report to Congress
07/01/2012 School Food Purchase Study-III: Nutritional Characteristics of School Food Acquisitions
04/19/2012 Modeling of High-Risk Indicators of Certification Error in the National School Lunch Program
04/10/2012 School Food Purchase Study III
01/02/2012 Addressing Child Hunger and Obesity in Indian Country: Report to Congress
01/01/2012 Nutrient and MyPyramid Analysis of USDA Foods in Five of Its Food and Nutrition Programs
11/02/2011 Direct Certification in the National School Lunch Program State Implementation Progress, School Year 2010-2011 Report to Congress
10/14/2011 Evaluation of the Fresh Fruit and Vegetable Program: Interim Report
11/01/2010 Direct Certification in the National School Lunch Program: State Progress in Implementation Report to Congress
10/26/2010 Feasibility of Wider Implementation of Direct Verification With Medicaid
09/17/2010 Developing and Evaluating Methods for Using American Community Survey Data to Support the School Meals Programs
06/25/2010 Selecting Policy Indicators And Developing Simulation Models for the National School Lunch and Breakfast Programs
03/30/2010 School Meals Building Blocks for Healthy Children

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Have We Forgotten The Kamala Mills Fire Accident?
Savitha Sampath
What Happened Till Date After The Gruesome Kamala Mills Fire Accident?
Fire hazards are common in India and there have been several cases that have resulted in casualties. The fire accident at Kamala Mills Compound in 1 Above Pub and Mojo's Bistro Pub located in the Lower Parel in Mumbai is one of the most infamous fire accident. The major fire accident claimed the lives of 14 people and more a dozen people were injured. The accident happened on 29th December 2017 at 12.30. Kamala Mills compound houses offices of many TV news channels and famous restaurants and pubs. We are approaching the one-year anniversary of the saddening fire accident. Let us have a look at the proceedings in the case so far and what actions have been taken by the government till date.
29th December 2017: FIR by Police
After the news of the fire broke out, The Mumbai police filed an FIR under section 304 of the Indian Penal Code (IPC) and several other sections have also been included in the FIR including common intention and endangering the life of others. Bombay Municipal Corporation mayor Vishvanath Mahadeshwar orders an in-depth inquiry to seek the culprits.[1]
Picture Source: DNA India
30th December 2017: Demolition Drive & Lookout Notice For Bar Owners
Just one day after the accident, BMC cracked down on the bars and pubs located in the Kamala Mills Compound with illegal extensions. 314 restaurants, bars, and pubs suffered at the arms of the bulldozer and 7 restaurants were sealed for violating the guidelines. Additionally, the police also issued lookout notices for the partners who owned 1Above restaurant that was the major cause of the fire. [2]
Series Of Arrests
Following the fire accident that took the lives of 14 people, the Mumbai Police went on an arrest spree and has arrested more than 8 people so far including Ramesh Govani, the owner of Kamala Mills Compound, Yug Tulli, one of the co-owners of The Mojo's Bistro, three owners and two managers of the 1Above Pub where most of the deaths took place. [3]
12th September 2018: High Court Recommends Strict Actions Against The Owners
A judicial committee appointed by the Mumbai High court has recommended strict actions against the owners of the Kamala Mills Compound and the two rooftop restaurants – Mojo Bistro and 1Above. According to the committee, "appropriate punitive actions" must be taken against the trio of the owners of the premises and pubs involved in the accident. [4]
17th May 2018: Bail Granted To co-owners of Kamala Mills Compound
On May 17, 2018, a vacation bench of the Bombay High court granted bail to Ramesh Govani and Ramesh Bhandari, the co-owners of the Kamala Mills Compound on the ground of the fact that both the co-owners were unaware of the illegalities that were incorporated in the two pubs involved in the accident. [5]
1st November 2018: Bombay High Court Rejects Bail Pleas of the Pub Owners
The Bombay High court rejected the bail pleas filed by the owners of Mojo Bistro and 1Above Pub. [6]
16th December 2018: Bail Granted To the Owners of 1 Above Pub
The Supreme Court granted bail to Kripesh Sanghavi and Jigar Sanghavi, the owners of the 1Above pub where the fire broke out last year. [7]
Inspections and other actions after the incident:
Post the gruesome incidents there were series of inspections took place, not only in Maharashtra but across major cities in India. In Mumbai, there have been 48434 fire calls received by the Mumbai Fire Brigade since 2008. [8] A whopping 1479 eateries, multiplexes, malls in Mumbai have been flouting fire safety norms. Following the inspection, 386 eateries have been demolished and 36 nightspots in Bandra have been shut down due to non-compliance of fire safety norms among the other violations. According to this report, in 2017 alone, there have been 4790 fire incidents across Mumbai.
Are the violators punished? Are they penalized for deviating the fire safety norms?
Authorities are yet to provide justice to the relatives of the deceased and the injured persons that gutted the Kamala Mill Compound in Mumbai.
This Incident is spoken for a while, then drained from people's memories as yet another fire accident. At least we should consider this as a lesson to revamp our age-old fire safety rules and have a proper implementation of fire safety norms. Only then can we prevent incidents like these in the future.
https://www.firstpost.com/india/kamala-mills-building-fire-mumbai-live-updates-major-fire-in-mumbais-lower-parel-leaves-at-least-14-dead-many-injured-4279843.html
https://www.hindustantimes.com/mumbai-news/mumbai-kamala-mills-fire-police-issue-look-out-notice-against-owners-of-1above-restaurant/story-DlnBDQQYt8pbu9oRshPucN.html
https://www.indiatoday.in/india/story/mumbai-kamala-mills-fire-managers-of-1-above-pub-arrested-1119780-2018-01-01
https://timesofindia.indiatimes.com/india/kamala-mills-fire-judicial-panel-recommends-action-against-owners/articleshow/65785850.cms
https://timesofindia.indiatimes.com/city/mumbai/bombay-hc-grants-kamala-mills-owners-bail-in-fire-case/articleshow/64197902.cms
https://www.ndtv.com/mumbai-news/kamala-mills-fire-bombay-high-court-rejects-bail-pleas-of-restaurant-owners-1941360
https://timesofindia.indiatimes.com/city/mumbai/kamala-mills-fire-accused-get-bail-after-nearly-one-year/articleshow/67110872.cms.
https://timesofindia.indiatimes.com/city/mumbai/fires-in-mumbai-claimed-614-lives-in-10-years/articleshow/65678859.cms
How does a fire spread? How can it get dangerous?
What are the important fire safety signs that you should be aware of in a public place?
Throwback on India's 5 worst fire accidents & why there is a prevailing laxity
Forest Fires In India- History & Stats On Rising Forest Fires In India
How to buy a fire extinguisher for your home
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This eBook will provide you the right data for your digital marketing strategy and help you implement the right tactics for your online success in Singapore.
It will help you evaluate your digital marketing budget through detailed overviews of advertising prices in Singapore for top keywords and key marketing segments on the leading digital marketing platforms: Google and Facebook.
It will allow you to appraise your own potential to sell your products and services directly via your website, compare your operations or select the right third-party platforms by presenting you a list of the top e-commerce websites in Singapore and estimated traffic.
It will help you get inspiration and define the most interesting activities, models or partners for your social media presence through convenient lists of the top Facebook pages and Twitter accounts of Singapore.
It will present you the most successful bloggers of Singapore to get inspiration and refine your own value-proposition, also allowing you to quickly select the ones to get in touch with, to follow and maybe collaborate for content creation and link-building.


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Home / Real-World Economics Review / The social environment as a cause in economics
The social environment as a cause in economics
Editor December 14, 2019 Real-World Economics Review
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From Blair Fix Have you noticed that economists are missing a word in their vocabulary? In microeconomics you'll see words like 'individual', 'utility' and 'maximize'. But you won't see the word 'environment' anywhere. It seems that in microeconomics, individuals maximize their utility in a void. [1] This lobotomy of the environment has led economists astray. By focusing only on differences between individuals, economists ignore a major part of human behavior. Individual differences are meaningful only in the context of an environment — especially a social environment. The environment as a cause To frame my argument, we need to leave economics behind for a moment. The problem is that economists treat the environment as something that is passive. It's a stock of resources or a trove
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from Blair Fix
Have you noticed that economists are missing a word in their vocabulary? In microeconomics you'll see words like 'individual', 'utility' and 'maximize'. But you won't see the word 'environment' anywhere. It seems that in microeconomics, individuals maximize their utility in a void. [1]
This lobotomy of the environment has led economists astray. By focusing only on differences between individuals, economists ignore a major part of human behavior. Individual differences are meaningful only in the context of an environment — especially a social environment.
The environment as a cause
To frame my argument, we need to leave economics behind for a moment. The problem is that economists treat the environment as something that is passive. It's a stock of resources or a trove of 'ecosystem services'. We can affect the environment, but it can't affect us.
In contrast, I want you to think of the environment as something that is active. To get into this headspace, I want you to think like an evolutionary biologist.
In Darwinian theory, the environment plays an active role in evolution. To have evolution by natural selection, we need (heritable) differences between individuals. But without an environment, these differences won't cause natural selection. Why? Because it's the organism's environment that determines if a trait is good or bad. The environment is an evolutionary cause.
Here's an example. In the 19th century, the peppered moth population in Britain changed drastically. Before the industrial revolution, most peppered moths were white. But as Britain industrialized, most peppered moths became black.
What caused this evolutionary change? Biologists think it was the moths' environment.
On natural-colored trees, white moths blend in with the bark. Black moths stick out. So in their 'natural' environment, white moths are better adapted. That's why they dominated the moth population before industrialization. But as Britain industrialized, soot turned trees black. This made black moths better camouflaged. In this altered environment, black moths flourished and white moths declined.
The peppered-moth story is a clear case where the environment caused a species to change. The story has two ingredients:
Individual variation
An environment that acts on this variation
To have natural selection, we need differences between individuals (here, moth color). But these differences gain meaning only in the context of the environment (natural or soot-colored trees). The lesson is that we can't separate individual variation from the environment in which it occurs.
Keeping individual variation, purging the environment
Having framed the 'environment' in an evolutionary context, I'll return to where economists go wrong. Economists purge the environment from their theory of human behavior (neoclassical microeconomics). Instead, they focus only on individual variation.
I'll use human capital theory to illustrate this environmental lobotomy. This theory explains income in terms of differences in human capital. Individuals with more human capital are more productive (the theory claims), and so earn more income.
Where is the environment in this theory? It's nowhere to be found! In the world of human capital theory, individual variation happens in a void.
The story of cow capital
To see the problems with lobotomizing the environment, I'm going to tell you a story about 'cow capital'. Think of this as a cautionary tale. It warns about the perils of purging the environment from a theory of animal behavior.
Our story begins with a biologist named Bob. Bob has set up camp on a farm and is studying the cows. These cows come in two colors: black and brown. Like a typical biologist, Bob is obsessed with fertility. So he asks: "Does cow fertility vary by color?"
To answer his question, Bob spends a year measuring cow reproduction. He finds that fertility does vary by color. Black cows have (on average) more offspring than brown cows.
Intrigued by this result, Bob tests for causation. He paints some black cows brown and some brown cows black. Then he measures the fertility of these painted cows.
His results are astonishing.
When brown cows are painted black, their fertility increases. And when black cows are painted brown, their fertility decreases. Color — whether natural or painted — affects cows' reproduction!
Bob knows a sexy result when he sees one. He writes up his finding and publishes it in Nature. Then he returns to the field.
Back in the ivory tower, an economist named Nancy reads Bob's paper. She thinks to herself: "Neoclassical theory can explain Bobs results!" So she writes a paper called "Cow Capital Theory".
Here's how the theory works. Color is a type of 'cow capital' — a skill that makes cows more productive. Nancy argues that black cows have more cow capital than brown cows. That's why they have more offspring. But not only is cow capital a skill, it's a skill that can be acquired. When brown cows get painted black, they acquire more cow capital. That's why they become more fertile.
Pleased with her theory, Nancy publishes it in the American Economic Review. Economists read Nancy's paper and celebrate the universality of human capital theory.
Meanwhile, Bob has discovered the real reason that black cows outbreed brown cows. The cows are owned by a farmer who breeds them for color! Every generation, the farmer chooses mostly black cows for breeding. The farmer is also nearly blind and can't tell the difference between natural-colored and painted cows.
Contrary to cow capital theory, black cows are not innately more fertile than brown cows. Instead, the cows live in an environment where being black is advantageous. The environment (the farmer) causes black cows to outbreed brown cows.
The moral of our story is this: remove the environment from a theory of animal behavior and you'll be led astray.
When all you see is individual variation, you attach undeserved significance to individual traits. Cow color becomes 'cow capital' — a magical source of fertility. The reality, however, is that the environment is what makes traits good or bad. Black cows don't have more cow capital. They're just favored by the farmer.
The social environment
In our story of cow capital, the most important part of the cows' environment was the farmer — another species. Among social animals like humans, though, much of the environment is created by the species itself. We'll call this the 'social environment'.
Like the physical environment, the social environment determines if a trait is good or bad. Again, it's worth contrasting this evolutionary view with neoclassical economics. In neoclassical theory, traits are significant in their own right.
Take education.
According to neoclassical theory, education makes people more productive, and this productivity makes them earn more income. In our evolutionary theory, however, education isn't innately connected to productivity. Instead, it's a trait that the social environment acts on.
Put a person with PhD in a hunter-gatherer society and they won't be rewarded for their higher education. Put the same person in an industrial society and they will be rewarded. It's the social environment, not the trait itself, that determines rewards.
Hierarchy: the farmer and the cows
Now that you've laughed at the story of cow capital, I'm going to show you that it's not that far-fetched. It's a good metaphor for how traits are rewarded in a hierarchy, and for how economists misinterpret these rewards.
In our story of cow capital, the farmer and the cows form a hierarchy. The farmer has power over the cows, who surrender their reproductive decisions to him. Because the farmer breeds mostly black cows, the black cows are rewarded.
The same logic applies in a human hierarchy. The farmer represents those with power. In a hierarchy, the powerful determine which traits get rewarded.
And the cows? They're a metaphor for the masses at the bottom of the hierarchy. After the powerful create the reward system, the masses are at the mercy of this environment.
And cow color? It's a metaphor for traits that are valued by the hierarchy. (Given humanity's history of racism, you could take the trait of color literally though.)
Having fleshed out the metaphor, let's look at hierarchy from a biologist's point of view. A hierarchy is a social environment characterized by concentrated power. In this environment, those with power get more resources. (I've proposed, that this is a universal feature of hierarchies). So you get rewarded by climbing the hierarchy.
What varies between hierarchies are the traits that get you power. Historically, most societies rewarded birthright. (Think of feudal caste systems in which rank was fixed at birth.) In contrast, modern societies award power by 'merit'. In these societies, you climb the hierarchy if you have credentials.
From the biologist's point of view, we have two factors at work in a hierarchy:
If this seems familiar, it's because it's the same two factors as in the peppered-moth story and the cow capital story.
Now let's look at hierarchy from the economist's point of view. To do this, we keep individual variation but we ignore the social environment. So let's forget that hierarchy exists. When people move up the hierarchy, we see only their traits and their reward.
In modern societies, we see that educated people are rewarded. So we propose that education makes you more productive.
We can apply the same thinking to hierarchies that reward birthright. We ignore the hierarchy and see only that family lineage determines rewards. We conclude that highborns are innately better than lowborns.
So neoclassical economists are actually continuing the tradition created by feudal apologists. Ignore the social environment and glorify individual traits. By doing so, you legitimize the social hierarchy.
[1] Yes, there is a sub-discipline of economics called 'environmental economics'. But this field is an afterthought, tacked onto microeconomics like Eeyore's tail. Environmental economics takes the principles of microeconomics and applies them to problems like pollution. What the field doesn't do is put the environment into economists' core theory of human behavior.
Uncategorized 2019-12-14
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