diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzndxt" "b/data_all_eng_slimpj/shuffled/split2/finalzzndxt" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzndxt" @@ -0,0 +1,5 @@ +{"text":"\\section*{Abstract}\nThe symport of lactose and $\\rm H^+$ is an important physiological process in $E.\\ coli$, for it is closely related to cellular energy supply. In this paper, the symport of $\\rm H^+$ and lactose by $E.\\ coli$ LacY protein is computationally simulated using a newly proposed cotransport model that takes the ``leakage\" phenomenon (uncoupled sugar translocation) into account and also satisfies the static head equilibrium condition. Then, we study the equilibrium properties of this cotransport process including equilibrium solution and the time required to reach equilibrium state, when varying the parameters of the initial state of cotransport system. It can be found that in the presence of leakage, $\\rm H^+$ and lactose will reach their equilibrium state separately, but the intensity of ``leakage\" has almost no effect on the equilibrium solution, while the stronger leakage is, the shorter the time required for $\\rm H^+$ and lactose to reach equilibrium. For $E.\\ coli$ cells with different periplasm and cytoplasm volumes, when cotransport is performed at constant initial particle concentration, the time for cytoplasm pH to be stabilized increases monotonically with the periplasm to cytoplasm volume ratio. For a certain $E.\\ coli$ cell, as it continues to lose water and contract, the time for cytoplasm pH to be stabilized by cotransport also increases monotonically when the cell survives. The above phenomena and other findings in this paper may help us to not only further validate or improve the model, but also deepen our understanding of the cotransport process of $E.\\ coli$ LacY protein.\n\n\n\n\n\n\\section*{Introduction}\n\tLacY protein (lactose permease) of $Escherichia\\ coli$ is a kind of transport protein involved in the secondary active transport of hydrogen ions and lactose molecules. The protein uses the energy stored in the $\\rm H^+$ electrochemical potential to cotransport hydrogen ions into the cytoplasm along with $\\rm\\beta-galactosides$, such as lactose\\cite{guan2006lessons}. Since large amounts of lactose are required to sustain life activities in $E.\\ coli$ cells, the intracellular lactose concentration is usually higher than the environment, and the above cotransport process is usually inverse to the lactose concentration gradient\\cite{abramson2003structure}. Possible mechanisms and mathematical models for the cotransport process have been extensively studied, starting roughly from Cohen and Rickenberg's report in 1955\\cite{kramer2014membrane}. Among the numerous research results so far, one of the most significant achievement, and also a mechanism now generally accepted, is the ``six-state\" mechanism described by Kaback $et\\ al.$ in their 2001 article\\cite{kaback2001kamikaze}, which is baesd on a more universal cotransport mechanism proposed by Jardetzky in 1966\\cite{jardetzky1966simple}. The so-called ``six-state\" mechanism refers to the fact that the cotransporter lactose permease have six different functional conformations (states) in the cotransport process. The LacY protein begins the reaction cycle at a outward-facing state 1, quickly binds a hydrogen ion, turning to state 2, then continues to trap a lactose molecule and turns to state 3, during which the cotransporter remains in the outward-facing state. After the combination of particles, the cotransporter makes a rapid conformational change to inward-facing, followed by detachment of the lactose molecule to state 5, then to state 6 by shedding the hydrogen ion, and finally returns to state 1 by another rapid conformational change. All above processes are naturally reversible. However, the former conventional mechanism is challenged by some subsequent experimental results\\cite{andrini2008leak}, where uncoupled transport is observed and determined, $i.e.$ , the two kinds of particles involved in cotransport do not follow the previous stoichiometric, and one kind has a uniport-like``leakage\" phenomenon. It also poses a problem for how to modify previous existing mathematical models that simulate this ``six-state\" mechanism. For the symport of sodium ions with glucose by SGLT1 protein, which is similar to the cotransport process of $E.\\ coli$ LacY protein, several possible mechanisms leading to the leakage phenomenon have been proposed in the literature\\cite{centelles1991energetic}, among which the simpler one that has also been used many times to modify mathematical models, is to allow the transition between cotransporter states 2 and 5. Nevertheless, some literatures such as\\cite{naftalin2010reassessment} propose that any determinate modified cotransport model by allowing the transition between states 2 and 5 cannot satisfy a certain thermodynamic condition (static head equilibrium condition). Then Barreto $et\\ al.$ propose a statistical mechanical model based on the above way of modifying conventional mechanism on the work of paper\\cite{barreto2019transport}, which is a random-walk model and satisfies the static head equilibrium condition in the case of leakage\\cite{barreto2020random}.\n\n\tBased on such a model, we are finally able to computationally simulate the transport process of $E.\\ coli$ LacY protein in the presence of leakage, and find some distinct equilibrium state properties in the presence and absence of leakage, such as the equilibrium solution and the convergence rate to the equilibrium solution. In this paper, we first briefly restate and analyze the model in article\\cite{barreto2020random} and predict the results of the computational simulations in the following. Next, we vary the intensity of the leakage, the volume of $E.\\ coli$ periplasm and cytoplasm, the initial concentration of the particles and the number (density) of cotransporter to study the variation of the corresponding equilibrium solution and the time required to reach equilibrium. Finally we find some very similar or different phenomena in the presence and absence of leakage.\n\n\\section*{A brief statement and analysis of the model}\n\tThe model in article\\cite{barreto2020random} considers a closed system of periplasm and cytoplasm, the volumes of which are $V_p$ and $V_c$ respectively. Periplasm and cytoplasm are separated by a cell membrane with cotransporters embedded, across which there is a membrane potential $\\Delta\\Psi$ that we assume a constant (cytoplasm lower). The total number of cotransporters on the cell membrane is fixed to $n$, and they are all assumed to be independent of each other. Let $n_k,k=1,2\\cdots6,$ represent the number of cotransporters in the current state $k$, respectively, and the sum of these six terms is clearly the fixed value $n$. Here I use an illustration Fig~\\ref{1} to graphically show the six-state mechanism described in the introduction section for the readers to understand it more visually.\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.7]{Fig_1_new.jpg}\n\\caption{{\\bf The modified six-state mechanism of cotransport.}\nAn alternating access model of $E.\\ coli$ $\\rm lactose-H^+$ cotransport by Jardetzky modified with the transition between cotransporter states 2 and 5. A simple schematic of the ``six-state\" mechanism of cotransporter LacY protein of $E.\\ coli$ during cotransport is shown in the figure above. Numbers 1 to 6 indicate state 1 to state 6, which are the six effective conformations of LacY protein. The blue double-head arrows refer to the six states of cotransporter in relation to each other, and the yellow dashed arrow refers to the way to modify the present model\u2014allowing the transition between cotransporter states 2 and 5.}\n\\label{1}\n\\end{figure}\n\n\t$N_c^{\\rm H^+}(t)$ and $N_c^{\\rm L}(t)$ represent the current number of hydrogen ions and lactose molecules in cytoplasm, and $N_p^{\\rm H^+}(t)$ and $N_p^{\\rm L}(t)$ similarly for periplasm. When it is not necessary to specify one kind of particle or one of periplasm and cytoplasm, we will later use $S$ to refer to either of the two kinds of particles involved in cotransport and $l$ refer to periplasm or cytoplasm. Since the system is closed, the total number of $S$ particles in all states and positions is always equal to the number we put in at the beginning, as a constant value, denoted $N_S$. $\\xi$ represents the intensity of leakage (similar to the ratio of leakage current to cotransport current excluding leakage) and takes the value in $[0,1]$. $\\xi=0$ means no leakage occurs, $\\xi=1$ means the cotransporter in state 2 has an equal probability to take a conformation change to state 1,3 or 5, and the larger the value of $\\xi$, the more obvious the leakage phenomenon is. The master equations of the model consists of ten differential equations related to each other, the exact form of which is not repeated here; please move to the original \\cite{barreto2020random}. Here we perform a brief analysis of the model.\n\n\tThe original \\cite{barreto2020random} first makes the left-hand side quotient of the master equation tend to 0 by making $t\\rightarrow\\infty$, which causes all variables to take equilibrium solutions (or asymptotic solutions), and the solutions when $\\xi=0$ and $\\xi\\neq0$ have the following form,\n\\begin{displaymath}\nN_{l,\\xi=0}^S=C_0V_l\\left(\\frac{\\alpha_{K+1K}}{\\alpha_{KK+1}}\\frac{n_{K+1}}{n_K}\\right)^{h\/\\nu_S} ,\n\\end{displaymath}\t\n\\begin{displaymath}\nN_{l,\\xi\\neq0}^S=\\left(1-\\frac{\\xi}{3}\\right)^{\\tilde{h}\/\\nu_S}\\!\\!\\!\\!\\!\\!\\!\\!\\!C_0V_l\\left(\\frac{\\alpha_{K+1K}}{\\alpha_{KK+1}}\\frac{n_{K+1}}{n_K}\\right)^{h\/\\nu_S} .\n\\end{displaymath}\n\t$\\nu_S$ is the number of particle $S$ transported in one cycle of cotransporter in figure\\ref{1}, here equals 1 for $\\rm H^+$ and lactose. $h, K, \\tilde{h}, \\alpha_{pq}$ are all constants decided by $S$ and $l$, and the details are shown in article\\cite{barreto2020random}. After obtaining the above results, the paper \\cite{barreto2020random} states that $N_{l,\\xi\\neq0}^S=\\left(1-\\frac{\\xi}{3}\\right)^{\\tilde{h}\/\\nu_S}N_{l,\\xi=0}^S$, which leads to the equilibrium solution for all values of $\\xi$. However, a careful observation shows that the above two equations are essentially the relationship between the equilibrium solution of the particle numbers $N_{l,\\xi}^S$ and the cotransporter numbers $n_K$. The equilibrium solutions of the cotransporters are not necessarily the same for different values of $\\xi$ (in practice, it is clear from the calculations below that they are indeed not the same), so the conclusion of the original equilibrium solution in \\cite{barreto2020random} does not hold.\n\n\tIn fact, we can get new information about the equilibrium solution by the Gibbs free energy change as well as the electrochemical gradient. The original\\cite{barreto2020random} proves that the equilibrium solution satisfies the following static head equilibrium condition when $\\xi=0$, \n\\begin{displaymath}\n\\frac{N_c^A\/V_c}{N_p^A\/V_p}=\\left(\\frac{N_p^B\/V_p}{N_c^B\/V_c}\\right)^{(2f-1)\\nu_B\/\\nu_A}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\times\\exp\\left(-\\frac{1}{k_BT}\\left[Q_A+(2f-1)\\frac{\\nu_A}{\\nu_B}Q_B\\right]\\Delta\\Psi\\right).\n\\end{displaymath}\n\tHere $f=1$ for lactose permease and $A,B$ are particles transported by cotransporter. And the following scaling relationships between equilibrium solutions are obtained in the same way when $\\xi\\neq0$ (also satisfying the static head equilibrium),\n\\begin{displaymath}\n\\frac{N_c^S\/V_c}{N_p^S\/V_p}=\\exp\\left(-\\frac{1}{k_BT}Q_S\\Delta\\Psi\\right) .\n\\end{displaymath}\n\t$Q_S$ is the charge of a particle $S$, $k_B$ is the Boltzmann constant, and $T$ is the is the ambient temperature of $E.\\ coli$. The above two relations are important constraints on the equilibrium solution, since in reality the number of cotransporters is much smaller than the number of particles involved in cotransport, and the total number of particles is constant during the transport, $i.e.$ the following two equations hold at all times during the transport, \n\\begin{displaymath}\nN_A-N_c^A-N_p^A=\\nu_A[n_3+n_4+(n_2+n_5)f],\n\\end{displaymath}\n\\begin{displaymath}\nN_B-N_c^B-N_p^B=\\nu_B[(n_3+n_4)f+(1-f)(n_1+n_6)].\n\\end{displaymath}\n\tThen we can see that under the general condition that the total number of cotransporters is very small, the equilibrium solution changes quite little with the parameter $\\xi$ when $\\xi\\neq0$. Only if the total number of cotransporters is not small relative to the total number of particles and also the distribution of cotransporters in each state varies considerably when $\\xi$ changes does the equilibrium solution change significantly. The computational verification of the above theoretical descriptions will be carried out below, while the above conclusions give directional hints for our later calculations.\n\\section*{Computational simulations and equilibrium property studies using the above model}\n\tIn the following of this paper, computational simulations of the $E.\\ coli$ LacY protein 1:1 symport processes of $\\rm H^+$ and lactose are performed with the aforementioned model. The values used in the calculation are from the article \\cite{barreto2020random}\\cite{barreto2019transport} or selected according to the reality, as detailed in the calculation section later. We use the finite difference method to solve the differential master equations of the model numerically\\cite{bathe2006finite}, that is, we use the relation ${dN(t)}\/{dt}\\approx{(N(t+\\Delta t)-N(t))}\/{\\Delta t}$ to transform the differential equation into a difference equation for calculation, and in subsequent calculations in this paper we take the time step $\\Delta t=0.1$\n\t\\subsection*{Effect of parameter $\\xi$ (leakage intensity) on equilibrium properties}\n\tThe $\\xi=0$ and $\\xi\\neq0$ cases are too different to be studied together, so we focus on the $\\xi\\neq0$ case. First, the correlation between the equilibrium solution and $\\xi$ will be verified. The following Fig~\\ref{2} shows the correlation between $\\xi$ and the equilibrium solutions as well as that between $\\xi$ and the time to reach equilibrium states for $\\rm H^+$ and lactose at a fixed initial condition, respectively.\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.8]{Fig_2.jpg}\n\\caption{{\\bf Equilibrium properties of lactose permease cotransport in $E.\\ coli$ as $\\xi$ changes.}\nOnly the $\\xi\\neq0$ cases are shown with the following initial conditions: $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+}=10^5, N_p^{\\rm L}=2.5\\times10^4, N_c^{\\rm L}=7.5\\times10^4, N_c^{\\rm H^+}=10^3$. And also, we set $V_c=V_p=10^6\/C_0$, $C_0$ is a constant with the unit of $\\rm nm^{-3}$. The meaning of $C_0$ is the same as which in \\cite{barreto2020random}, similarly hereinafter. (a) The equilibium solution of $\\rm H^+$ in periplasm and cytoplasm. (b) The equilibium solution of lactose in periplasm and cytoplasm. (c) Time used to reach equilbrium state for $\\rm H^+$ and lactose with the unit of $\\Delta t$.}\n\\label{2}\n\\end{figure}\n\n\tThe reason why the initial conditions being so is that, according to the article \\cite{wilks2007ph}, $E.\\ coli$ cells are generally in an environment that causes the pH of periplasm about 1.8 to 2.1 higher than that of cytoplasm. From the above Fig~\\ref{2}(a) and Fig~\\ref{2}(b), it can be found that in the case that $\\xi$ is not very small ($\\geq0.1$), the above general conclusions about the equilibrium solution (insignificant variation of the equilibrium solution with $\\xi$ and general proportionality between the equilibrium solutions) are correct within error permissibility. In fact, when the total number of cotransporters is increased (in this example we just need to increase $n_1$), the error will be significantly reduced and the proportional relationship between the equilibrium solutions will become more obvious and precise.\n\n\tAnother important feature about cotransport is the time required to reach the equilibrium solution. Although the equilibrium solution is asymptotic and theoretically not achievable in finite time, we can still consider that equilibrium has been reached when the change rate or the first-order backward difference quotient of the solution is small enough in our simulation. When there is no leakage ($\\xi=0$), it is easy to find that $\\rm H^+$ and lactose reach equilibrium at the same time by calculation, which is also easy to see from the above theoretical analysis. However, when leakage does exist ($\\xi\\neq0$), the time for the two kinds of particles to reach equilibrium is seperated, which can also be inferred by splitting the full reaction cycle into two independent transport process. The above Fig~\\ref{2}(c) shows the relationship between the time of two kinds of particles to reach equilibrium and $\\xi$ when $\\xi\\in(0,1]$.\n\n\tFig~\\ref{2}(c) is calculated by considering that when $[N_c^S(t)-N_c^S(t-\\Delta t)]\/N_c^S(t)\\leq10^{-9}$ holds for almost every discrete time moments from some t (for over $99.9999\\%$ of time moments), the particle $S$ reach the equilibrium. Subsequent calculations of the time to reach the equilibrium are similar, but accuracy may be adjusted as needed. The reasons for not using the first-order backward difference quotient and instead using change rate are the poor estimation of the range of the difference quotient and the fact that the difference quotient in this model is not guaranteed to be consistently smaller than the required accuracy after a certain time. As can be seen in the figure, the time for $\\rm H^+$ to reach equilibrium is always shorter than the time for lactose when $\\xi\\neq0$. Because we can formally split the overall transort process in the presence of leakage, the full reaction process can be viewed as a combination of a leakage-free cotransport process and a separate uniport process of $\\rm H^+$, with little interference between them. Then $\\rm H^+$ has two transport processes transporting it, while lactose has only one. So $\\rm H^+$ can reach equilibrium with the net transport of the two processes canceling each other, while lactose has not reached equilibrium at that point. It can also be observed that the time for the two kinds of particles to reach equilibrium do not have a consistent trend with the change of $\\xi$. Both of them are large when $\\xi$ is small, after that it decreases rapidly with the increase of $\\xi$, and the change is not obvious after $\\xi>0.2$. But the equilibrium time of lactose has a small rebound after $\\xi>0.5$, while $\\rm H^+$ continues to remain monotonically decreasing. The reason for this phenomenon is not clear, and it is speculated that there may be some calculation errors. Another point worth mentioning is that particle numbers converge much faster when $\\xi=0$ than $\\xi\\neq0$. The addition of the transition between cotransporter states 2 and 5 ($i.e.$ , making $\\xi\\neq0$) makes the convergence of the system much slower.\n\\subsection*{Effect of periplasm and cytoplasm volumes $V_p,V_c$ on equilibrium properties}\n\tThe previous arithmetic examples have been calculated assuming $V_p=V_c$, and as seen before, the proportionality satisfied by the equilibrium solution when $\\xi\\neq0$ is actually a proportionality between the particle concentrations in the two reaction chambers (periplasm and cytoplasm), so we consider changing the periplasm and cytoplasm volume ratios to verify the conclusion and observe if there are new phenomena. Also due to the previous conclusions on the equilibrium solution and the convergence rate in the case of $\\xi\\neq0$, we can only study the $\\xi=0$ and $\\xi=1$ cases. The equilibrium solutions of the $\\xi=0$ case are shown in the following Fig~\\ref{3}. \n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.7]{Fig_3.jpg}\n\\caption{{\\bf The equilibium solutions for $\\xi=0$ when $V_c$ or $V_p$ is fixed and the other one varies.}\n(a) The equilibium solutions of $\\rm H^+$ and lactose when $V_c$ is fixed to $10^6\/C_0$ and $V_p\/V_c>1$, with initial conditions $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+} =10^5$, $N_p^{\\rm L}=2.5\\times10^4$, $N_c^{\\rm L}=7.5\\times10^4, N_c^{\\rm H^+}=10^3$. (b) The equilibium solutions of $\\rm H^+$ and lactose when $V_c$ is fixed to $10^6\/C_0$ and $V_p\/V_c<1$, with the same initial conditions before.}\n\\label{3}\n\\end{figure}\n\n\tFrom Fig~\\ref{3}, the following phenomenon can be found that under the condition of fixing the number of particles involved in cotransport ($i.e.$ fixing $N_c^S,N_p^S$ in initial conditions), the change of $V_c$ seems to have less effect on the equilibrium solution than $V_p$, and the system equilibrium solution is more insensitive to the change of $V_c$. When changing other initial conditions and making $V_p$ a constant value to change $V_c\/V_p$, it is found that this phnomenon is actually more accurately formulated as that changing $V_c$ or $V_p$ only under the condition that $V_p\/V_c>1$ has a more pronounced effect on the equilibrium solution when $\\xi=0$, thus showing an insensitivity to the increase in the denominator $V_c$. After adjusting the parameters and performing a lot of calculations, it is found that this property is somewhat related to the membrane potential. In this case the direction of potential decrease is from periplasm to cytoplasm, and the membrane potential is $-100mV$. However, when decreasing this potential difference, this phenomenon becomes less and less obvious, especially when reversing the membrane potential, $i.e.$ making the potential of cytoplasm higher than that of periplasm, the significances of $V_c$ and $V_p$ in the above properties are switched, and $V_p$ becomes the one that has less influence on the equilibrium solution of the system. \n\n\tThe above phenomena can be explained from the theoretical analysis, for which we consider the Gibbs free energy variation at the starting moment $\\Delta G$. For this cotransport process of the $E.\\ coli$ LacY protein, we can rewrite the free energy variation in terms of the sum of the chemical potential multiplying the stoichiometry of the two particles \\cite{nelson2008lehninger}, bringing the data and slightly transforming then we have, \n\\begin{displaymath}\n\\Delta G=k_BT\\ln\\left(\\frac{N_c^{\\rm H^+}N_c^{\\rm L}}{N_p^{\\rm H^+}N_p^{\\rm L}}\\right)+Q_{\\rm H^+}\\Delta\\Psi+2k_BT\\ln(V_p\/V_c).\n\\end{displaymath}\n\tFor a fixed initial condition, the first of the three terms on the right-hand side of the above equation is constant, and based on the data we use in calculation and the data corresponding to the environment in which $E.\\ coli$ is usually found, this term is usually in the $(-4k_BT,k_BT)$ interval, and the second term is usually around $-3.9k_BT$ \\cite{lo2007nonequivalence}. When changing $V_p$ or $V_c$, the absolute value of $\\Delta G$ can be reduced only if $V_p\/V_c>1$, and making $\\left|\\Delta G\\right|$ decrease produces a larger rate of change than making it increase when the amount of change in $V_p\/V_c$ is the same. The equilibrium solution is reached when $\\Delta G$ is 0, and it is easy to find that $\\Delta G$ varies monotonically during the cotransport process. So $\\Delta G$ can actually represent the \"gap\" between the initial state and the equilibrium state, and the gap between the equilibrium solutions is larger when the difference between $\\Delta G$ is larger. And since the negative membrane potential in this process contributes positively (also not small proportion) to $\\Delta G$, it produces a very different property when the membrane potential decreases or even reverses.\n\n\tNext, according to common biological sense, we will observe the relationship between $V_p\/V_c$ or $V_c\/V_p$ and the time required for the system to reach equilibrium when $V_p\/V_c<1$. In the following, the case of $\\xi=0$ comes first. Inspired by the above phenomena, we fix $V_c$ and $V_p$ separately and change the other one to calculate. For practical reasons, particle numbers should not be fixed as in the case of equilibrium solution just probed before, but the concentration of particles in periplasm and cytoplasm in initial conditions should be fixed, so that the results are more realistic. The following Fig~\\ref{4}(a) and Fig~\\ref{4}(b) are plotted with $V_p\/V_c$ as the horizontal coordinate (with $V_c$ fixed) and $V_c\/V_p$ as the horizontal coordinate (with $V_p$ fixed), respectively, and the time to reach the equilibrium solution as the vertical coordinate.\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.9]{Fig_4_new.jpg}\n\\caption{{\\bf Time to reach equilibrium for $\\xi=0$ when $V_c$ or $V_p$ is fixed and the other one varies.}\nTime used to reach equilbrium state with the unit of $\\Delta t$ when the initial concentrations of $\\rm H^+$ and lactose are fixed in periplasm and cytoplasm, which has the initial conditions $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+}\/V_p=C_0\/10, N_c^{\\rm H^+}\/V_c=C_0\/1000, N_p^{\\rm L}\/V_p=C_0\/40, N_c^{\\rm L}\/V_c=3C_0\/40$, $\\xi=0$. (a) $V_c$ fixed to $10^6\/C_0$. (b) $V_p$ fixed to $10^6\/C_0$.}\n\\label{4}\n\\end{figure}\n\n\tWhen the change rate of $N_c^S$ is always less than $10^{-9}$, the particle $S$ is considered to reach equilibrium in the above Fig~\\ref{4}. It is still clear to find different effects of $V_p$ and $V_c$ on the time required to reach equilibrium, and the effect of $V_c$ is still much smaller than that of $V_p$. Increasing $V_p$ when $V_c$ is fixed can monotonically increase the time for the system to reach equilibrium. However, changing $V_c$ when $V_p$ is fixed has a less significant effect on the convergence rate. This phenomenon can still be explained using the free energy variation $\\Delta G$, because although $\\Delta G$ is constant as particle concentration is constant, the net change of particles transported through the cotransporter is different as the number of particles involved in cotransport expands with $V_p$ or $V_c$ increasing. Based on the initial conditions we give and the general conditions in practice, the $\\rm H^+$ concentration is much higher in periplasm than in cytoplasm (possibly by an order of magnitude or more), and lactose concentration is generally higher in cytoplasm than in periplasm. In the case of $\\xi=0$, the net transport is a 1:1 cotransport for both kinds of particles from periplasm to cytoplasm, so when periplasm volume $V_p$ fixed and cytoplasm volume $V_c$ varied, the net transport does not change much and therefore the time to reach equilibrium does not change significantly. But in contrast the net transport increases almost linearly when $V_c$ fixed and $V_p$ varied, so that the time required to reach equilibrium also increases monotonically with the volume ratio $V_p\/V_c$. \n\n\tThe above is the case of $\\xi=0$. And for the case of $\\xi=1$, it is verified that the conclusion about the ratio of equilibrium solutions, $i.e.$ , ${(N_c^S\/V_c)}\/{(N_p^S\/V_p)}=\\rm const.$ , holds for both $\\rm H^+$ and lactose particles within error permissibility when fixing the number of each particle in the initial condition. Then the next concern is the time required to reach the equilibrium solution when $\\xi=1$, and we still require a fixed concentration of the particles in periplasm and cyptoplasm in the initial conditions. The following figures Fig~\\ref{5}(a) and Fig~\\ref{5}(b) show the relations between $V_p\/V_c$ ($V_c$ fixed) or $V_c\/V_p$ ($V_p$ fixed), and the time to reach the equilibrium solution, respectively, and we consider that the particle $S$ reaches equilibrium when the rate of change of $N_c^S$ is always less than $10^{-8}$. It is easy to see that in the case of $\\xi=1$ there are some phenomena that do not match our expectations. For example, with $V_c$ fixed, the time to reach equilibrium for both kinds of particles increase monotonically with the increase of $V_p$; however, with $V_p$ fixed, the equilibrium time for $\\rm H^+$ decreases monotonically with $V_c$ increasing, but the equilibrium time for lactose shows an increase followed by a decrease and takes a maximum around $V_c\/V_p=4$. \n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.9]{Fig_5_new.jpg}\n\\caption{{\\bf Time to reach equilibrium for $\\xi=1$ when $V_c$ or $V_p$ is fixed and the other one varies.}\n Different time for $\\rm H^+$ and lactose to reach equilbrium state with the unit of $\\Delta t$ when the initial concentrations of $\\rm H^+$ and lactose are fixed in periplasm and cytoplasm, which has the initial conditions $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+}\/V_p=C_0\/10, N_c^{\\rm H^+}\/V_c=C_0\/1000, N_p^{\\rm L}\/V_p=C_0\/40, N_c^{\\rm L}\/V_c=3C_0\/40$, $\\xi=1$. (a) $V_c$ fixed to $10^6\/C_0$. (b) $V_p$ fixed to $10^6\/C_0$.}\n\\label{5}\n\\end{figure}\n\n\tThe above two phenomena are quite anomalous, but they can still be broadly explained as follows. Since the concentration of $\\rm H^+$ and lactose on both sides of the cell membrane and the number of cotransporters are all constant, the transport rate of the cotransporter changes little when varying volume ratio, so the main difference in the equilibrium time comes from the number of particles by net cotransport. The $\\rm H^+$ concentration is considerably higher in periplasm than in cytoplasm, with lactose concentration the opposite. Therefore, according to the proportional relationship that needs to be satisfied by the equilibrium solution, the net transport of $\\rm H^+$ is from periplasm to cytoplasm, which eventually makes the $\\rm H^+$ concentration in cytoplasm much higher than in periplasm. Then increasing $V_p$ causes the net transport of $\\rm H^+$ to increase significantly, so the equilibrium time also increases significantly and monotonically with $V_p\/V_c$ up. Meanwhile, the net transport of lactose is from cytoplasm to periplasm, so the effect of changing $V_p$ on the net transport of lactose is positive, and therefore the equilibrium time increases with $V_p\/V_c$ up, but asymptotic phenomena occur as $V_p\/V_c$ continues to increase. Thus, the increasing rate of the time for lactose to reach equilibrium slows down rapidly as $V_p\/V_c$ increases, but is still greater than the $\\rm H^+$ equilibrium time.\n\t\n\tWhen varying $V_c$, the case of $\\rm H^+$ is similar to the earlier discussion and easily explained as monotonically decreasing with $V_c\/V_p$, but the case of lactose is more complicated. Due to the proportionality of the equilibrium solution, when $V_c$ is very large or small, the concentration of lactose in cytoplasm at equilibrium will be close to the initial concentration in cytoplasm or periplasm, respectively, so that the net transport is similar and both are small, but $V_c\/V_p$ of moderate size may produce a large net transport. This may lead to a phenomenon that equilibrium time of lactose increases and then decreases with $V_c\/V_p$ up as shown in Fig~\\ref{5}(b), but the exact situation still needs to be supported by experimental data. From the above discussion we can find that in the case of $\\xi\\neq0$, the correlation between the equilibrium of the two particles is very weak, not only the equilibrium solution itself is irrelevant, but also the correlation of the time required to reach the equilibrium solution is very low, which is exactly an important property of this model. This leads a slightly questioning of the model when $\\xi\\neq0$.\n\n\tAbove we set the volume ratio of periplasm and cytoplasm as the horizontal coordinate to calculate, but for a fixed $E.\\ coli$ cell (or cells from the same population), periplasm is the portion between the cell wall (cytoderm) and the cell membrane, and cyptoplasm is the portion within the cell membrane other than the nuclear region, and the sum of the volumes $V_c+V_p$ varies very little due to the rigidity of the cytoderm. Therefore, we continue to investigate its effect on the convergence rate by taking the percentage of periplasm to the sum of periplasm and cyptoplasm volumes as the horizontal coordinate while fixing the sum of the two volumes. In this case, we choose to fix the initial concentration and initial number of particles in periplasm and cyptoplasm, respectively, and calculate the time required for different particles to reach equilibrium. The following Fig~\\ref{6} is the image of the time required to reach equilibrium as the periplasm fraction changes for the case $\\xi=0$, where it is still considered that the particle $S$ reaches the equilibrium solution when the change rate of $N_c^S$ is always less than $10^{-9}$. As seen from the experimental data in article \\cite{stock1977periplasmic}\\cite{graham1991periplasmic}, the range of horizontal ordinates in Fig~\\ref{6} already includes the volume fraction of periplasm (8\\%-40\\%) in which $E.\\ coli$ can survive.\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.9]{Fig_6.jpg}\n\\caption{{\\bf Time to reach equilibrium with the variation of periplasm fraction $\\frac{V_p}{V_p+V_c}$, $\\xi=0$.}\nThe sum of volumes of periplasm and cytoplasm is fixed, $V_c+V_p=2\\times10^6\/C_0$. (a) Initial particle concentration fixed, and the initial conditions are like $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+}\/V_p=C_0\/10, N_c^{\\rm H^+}\/V_c=C_0\/1000, N_p^{\\rm L}\/V_p=C_0\/40, N_c^{\\rm L}\/V_c=3C_0\/40$. (b) Initial particle population fixed, and the initial conditions are like $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+} =10^5, N_p^{\\rm L}=2.5\\times10^4, N_c^{\\rm L}=7.5\\times10^4, N_c^{\\rm H^+}=10^3$.}\n\\label{6}\n\\end{figure}\n\n\tIt can be found that the two graphs have some similarity, both appearing to be approximately linearly increasing followed by monotonically decreasing. Although it may seem strange, the above case of $\\xi=0$ can actually be roughly explained by the standard theoretical methods. With a fixed initial concentration, the $\\rm H^+$ concentration in cytoplasm increases continuously as the volume fraction of periplasm increases due to the higher concentration of $\\rm H^+$ in periplasm than in cyplasm in the initial conditions .However, the increase does not exceed the initial $\\rm H^+$ concentration in periplasm, meanwhile the volume of cyptoplasm decreases accordingly. If the volume difference between periplasm and cytoplasm is too large, and the $\\rm H^+$ concentrations in the smaller part of the volume at equilibrium will be close to the initial values of the larger part, then the net transport will be neither large, so the net transport of hydrogen ions is likely to increase and then decrease as the volume fraction of periplasm increases. Therefore, the time to reach equilibrium is likely to increase and then decrease at constant initial concentration and constant cotransport rate.\n\n\tWith a fixed initial particle number, $\\left|\\Delta G\\right|$ decreases as the fraction of periplasm increases, resulting in a decrease in the net amount of particles transported, but the change in volume also decreases the transport rate of cotransporters from periplasm to cytoplasm but increases the rate of reverse direction, with a decrease in the (initial) net transport rate. A qualitative analysis of the form reveals a more pronounced change in the net transport rate (than in the net amount of particles transported), but the change rate of the net transport rate gradually decreases as the fraction of periplasm increases, thus possibly leading to an increase and then a decrease in the time required to reach equilibrium. The above analyses and explanations are qualitative and still need to be verified or denied by experimental data. The above is the case of $\\xi=0$, while the following Fig~\\ref{7}(a) and Fig~\\ref{7}(b) are the case of $\\xi=1$, which are surprisingly not similar, unlike the similarity of the previous Fig~\\ref{6}(a) and Fig~\\ref{6}(b).\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.9]{Fig_7.jpg}\n\\caption{{\\bf Time to reach equilibrium with the variation of periplasm fraction $\\frac{V_p}{V_p+V_c}$, $\\xi=1$.}\n Different time for $\\rm H^+$ and lactose to reach equilbrium state with the unit of $\\Delta t$. The sum of volumes of periplasm and cytoplasm is fixed, $V_c+V_p=2\\times10^6\/C_0$. (a) Initial particle concentration fixed, and the initial conditions are like $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+}\/V_p=C_0\/10, N_c^{\\rm H^+}\/V_c=C_0\/1000, N_p^{\\rm L}\/V_p=C_0\/40, N_c^{\\rm L}\/V_c=3C_0\/40$. (b) Initial particle population fixed, and the initial conditions are like $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+} =10^5, N_p^{\\rm L}=2.5\\times10^4, N_c^{\\rm L}=7.5\\times10^4, N_c^{\\rm H^+}=10^3$.}\n\\label{7}\n\\end{figure}\n\n\tA closer observation reveals that although the equilibrium time images of lactose are disparate, the equilibrium time images of $\\rm H^+$ are still similar in shape. This is primarily due to the fact that in the presence of leakage, as mentioned earlier, the equilibrium of the two kinds of particles has little influence on each other. And no matter initial concentration or initial particle numbers is fixed, in the range of periplasm volume fraction in our calculation (${V_p}\/{(V_p+V_c)}\\in[0.1,0.9]$), the net transport direction of hydrogen ions is from periplasm to cytoplasm according to $\\Delta G$. Increasing the periplasm volume proportion in the presence of leakage, when the initial concentration is fixed, the (initial) net transport rate remains constant and the net transport amount increases, while when the initial particle number is fixed, the (initial) net transport rate decreases and the net transport volume does not change much from the aforementioned proportionality of the equilibrium solution, so the time required to reach equilibrium in both cases shows a monotonically increasing state. \n\n\tThe case of lactose is more complicated, especially in the condition of fixed initial particle number, where there are three inflection points and also the order for $\\rm H^+$ and lactose to reach equilibrium changes with different volume fraction of periplasm. We can give the following explanation for the minimal value point generated near ${V_p}\/{(V_p+V_c)}=0.25$ under the condition of fixed initial particle numbers. Since the initial value of $N_c^{\\rm L}\/N_p^{\\rm L}={1}\/{3}$, when $V_c\/V_p$ is around 3, the initial condition of lactose is actually very close to the equilibrium state because the total number of cotransporters is small compared to the number of particles. $N_c^{\\rm L}$ and $N_p^{\\rm L}$ change very little during the cotransport process and can be considered as reaching equilibrium very early. Changing the initial value of lactose and computationally verifying, it is found that this phenomenon is practically universal. When the volume ratio is adjusted so that the initial value of lactose is close to the equilibrium solution, there is always a significant reduction in the time for lactose to reach equilibrium, causing lactose's reaching equilibrium before $\\rm H^+$. More generally, a similar phenomenon occurs for both kinds of particles. Nevertheless, it should still be noted that for the general realistic initial values, as in the previous analysis, $\\rm H^+$ usually reaches equilibrium before lactose because of the additional leakage current regulation, which also complements and explains the above inference. Regarding the remaining phenomena embodied in the two figures, it is difficult for the author to give a reasonable explanation here, and I can only leave it to the experimental data to verify or deny.\n\t\\subsection*{Effect of changing the initial concentration and initial distribution of particles on equilibrium properties}\n\tNow we will explore the effect of the initial state on the equilibrium properties. Inspired by the previous section, we can observe the effect of changing the initial state on $\\Delta G$ to determine the effect on the equilibrium solution and the time required to reach equilibrium. We still study only the $\\xi=0,1$ cases in the following, as in the previous section. If the number of particles in the initial state is expanded linearly, and $\\Delta G$ does not change in this case, then it can be presumed that the equilibrium solution and the time to reach equilibrium should also increase roughly linearly, which is verified to be correct when $\\xi=0,1$ after calculation, and we will not elaborate it here. In the following we will consider the effect of changing the initial distribution of particles in periplasm and cytoplasm. Since both volumes $V_c,V_p$ are kept constant, we do not need to care about the difference between the two conditions of initial concentrations and initial particle numbers. The following Fig~\\ref{8} shows the change of $\\rm H^+$ and lactose equilibrium fractions in periplasm with the variation of $\\rm H^+$ initial fraction in periplasm. \n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=1.0]{Fig_8.jpg}\n\\caption{{\\bf Equilibrium fractions of particles in periplasm as initial fraction of $\\rm H^+$ in periplasm changes.}\nThe initial conditions are $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+}+N_c^{\\rm H^+}=10^5, N_p^{\\rm L}=2.5\\times10^4, N_c^{\\rm L}=7.5\\times10^4$. And also, we set $V_c=V_p=10^6\/C_0$, $\\xi=0,1$. There is always enough time for particles to stabilize so the final fractions equals the equilibrium fractions. (a) Final $\\rm H^+$ fraction. (b) Final lactose fraction.}\n\\label{8}\n\\end{figure}\n\n\tIt can be seen that the case of $\\xi=1$ conforms well to our expectation, and the equilibrium solution scale relation hardly changes with the initial conditions. In the case of $\\xi=0$, the equilibrium proportion of $\\rm H^+$ in periplasm increases monotonically with the initial proportion of $\\rm H^+$ in periplasm, and it can be observed that equilibrium value is always lower than the initial value. Meanwhile, the proportion of lactose in periplasm in the equilibrium solution decreases monotonically with the proportion of $\\rm H^+$ in periplasm in the initial condition. Both phenomena are consistent with common biological sense. When the $\\rm H^+$ concentration in periplasm is higher than in cytoplasm, $\\rm H^+$ provides a sufficient electrochemical potential for the transportation of lactose to cytoplasm, which allows $E.\\ coli$ to take up lactose against the concentration gradient of it. But doing so depletes $\\rm H^+$ in periplasm, making the $\\rm H^+$ concentration gap between periplasm and cytoplasm smaller when equilibrium is reached. The larger the initial $\\rm H^+$ concentration difference between periplasm and cyptoplasm, the more $\\rm H^+$ can be used and the more lactose is transported to cyptoplasm. Also, because of the need to maintain a high concentration of lactose in the cytoplasm, the periplasm must maintain a sufficiently high $\\rm H^+$ concentration at equilibrium to counteract the chemical potential of lactose. Due to the fact that some of the $\\rm H^+ $ are transported into cytoplasm with lactose, the $\\rm H^+$ concentration difference at equilibrium is lower than in the initial condition. The above two phenomena are thus explained. In the following, we consider the change in the initial proportion of lactose in periplasm, corresponding to the change in the proportion of two particles in periplasm at equilibrium, and the figure \\ref{11} is shown below. The following Fig~\\ref{9} show the variations in $\\rm H^+$ and lactose equilibrium fraction as the initial fraction of lactose in periplasm changes.\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=1.0]{Fig_9.jpg}\n\\caption{{\\bf Equilibrium fractions of particles in periplasm as initial fraction of lactose in periplasm changes.}\nThe initial conditions are $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+}=9\\times10^4, N_c^{\\rm H^+}=10^4, N_p^{\\rm L}+N_c^{\\rm L}=10^5$. And also, we set $V_c=V_p=10^6\/C_0$, $\\xi=0,1$. There is always enough time for particles to stabilize so the final fractions equals the equilibrium fractions. (a) Final $\\rm H^+$ fraction. (b) Final lactose fraction.}\n\\label{9}\n\\end{figure}\n\n\tThe above two plots obtained by varying the initial distribution of lactose are also fully interpretable in a similar way as above. When $\\xi=0$, as the initial lactose molecules in periplasm keep increasing, the absolute value of the free energy change of cotransport in the initial state $\\left|\\Delta G\\right|$ increases, and more lactose molecules are transported into cytoplasm. As the two particles are transported into cytoplasm in strictly equal amounts when $\\xi=0$, there will be a monotonic decrease in the amount of hydrogen ions and a slow monotonic increase in the amount of remaining lactose in periplasm at equilibrium with initial fraction of lactose in periplasm increasing. The $\\xi=1$ cases still meet the expectation and we do not elaborate on it.\n\t\\subsection*{Effect of changing the total number of cotransporters on equilibrium properties}\n\tFrom the master equations of the system, we know that the first order derivatives of both particle numbers and cotransporter numbers in different states are linearly related to the current values of some states of the cotransporters. So we guess that the convergence rate should be linearly related to the total number of cotransporters with constant initial values, and then the time required to reach the equilibrium state should be inversely proportional to cotransporter numbers. After the calculation we have Fig~\\ref{10} for $\\xi=0,1$ cases.\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=1.0]{Fig_10_new.jpg}\n\\caption{{\\bf Time to reach equilibium with the variation of the total number of cotransporters.}\nThe initial conditions are $N_p^{\\rm H^+}=10^5, N_c^{\\rm H^+}=10^3, N_p^{\\rm L}=2.5\\times10^4, N_c^{\\rm L}=7.5\\times10^4, V_c=V_p=10^6\/C_0$. All cotransporters are supposed to be in state 1 initially. (a) The $\\xi=0$ case. (b) The $\\xi=1$ case.}\n\\label{10}\n\\end{figure}\n\n\tAlthough the negative correlation is evident in both plots, a closer observation does not show the inverse proportion relationship as we would expect. The reason for this is unclear and needs to be verified or disproved by experimental data.\n\\section*{Conclusions and further discussions of the above results}\n\tFor the cotransport process of $E.\\ coli$ LacY protein symporting $\\rm H^+$ and lactose, we apply the random-walk model proposed in article \\cite{barreto2020random} to computationally simulate and find, validate or predict the following phenomena. \\begin{itemize}\n\t\\item In the absence of leakage, the concentrations of the two particles $\\rm H^+$ and lactose reach stability simultaneously; in the presence of leakage, the time required for the two particles to reach the equilibrium state is separated. In the realistic situation where the number of cotranporters is small relative to the number of transported particles, the intensity of leakage, if leakage exists, hardly affects the equilibrium state of cotransport, but affects the time for both particles to reach the equilibrium state, and generally the more pronounced leakage is, the shorter the time to reach equilibrium. \n\t\\item In the absence of leakage, fixing the initial number or concentration of $\\rm H^+$ and lactose in periplasm and cytoplasm, for homogenous $E.\\ coli$ cells with different periplasm and cytoplasm volumes, the periplasm volume but not the cytoplasm volume has a greater effect on the equilibrium state and the time required to reach equilibrium. In other words, when periplasm volumes are similar and cytoplasm volumes are different, cells have similar equilibrium states and need similar time to reach equilibrium by this cotransport process, but not vice versa. \n\t\\item In the presence of leakage, fixing the initial concentration of $\\rm H^+$ and lactose in periplasm and cytoplasm, for homogenous $E.\\ coli$ cells with different periplasm and cytoplasm volumes, the time required for the pH of cytoplasm to stabilize increases monotonically with the periplasm to cytoplasm volume ratio increasing. Meanwhile, the time for the lactose concentration to reach equilibrium has a more complex relationship with the volume ratio of periplasm and cytoplasm as shown in Fig~\\ref{5}.\n\t\\item For a certain $E.\\ coli$ cell (or cells with similar size from the same population), fixing the initial number or concentration of $\\rm H^+$ and lactose in periplasm and cytoplasm, the time for the pH of cytoplasm to stabilize increases monotonically as the cell loses water regardless of the presence or absence of leakage (under the condition of cell survival), whereas the time for the concentration of lactose to stabilize varies more complexly, as seen in Fig~\\ref{6} and Fig~\\ref{7}.\n\t\\item For $E.\\ coli$ cells from the same population, the concentration ratio of particles in periplasm and cyptoplasm in equilibrium state does not vary with the initial state concentration ratio if leakage exists.\n\t\\item For different subpecies of $E.\\ coli$, the time for the cotransport process to reach equilibrium is negatively but not inversely correlated with the amount of the cotransporter LacY with initial concentrations of $\\rm H^+$ and lactose in periplasm and cytoplasm fixed.\n\\end{itemize}\n\tMany of the above phenomena can be qualitatively explained, but some of them still can not be well explained and need to be verified or negated by experimental data.\n\n\tAfter summarizing the results above, it is easy to notice that in the case of the parameter $\\xi=0$, $i.e.$ , no leakage, the results are mostly consistent with biological or chemical intuition, but the appearance of some properties for $\\xi\\neq0$ case is quite anomalous. The biggest problem is the equilibrium solution. When $\\xi\\neq0$, the equilibrium solution almost strictly satisfies the concentration proportionality under the general condition that the number of cotransporters is much smaller than the number of particles to be transported. For the cotransport of $E.\\ coli$ LacY protein, the concentration of lactose in periplasm and cytoplasm at equilibrium is almost equal because the lactose molecule is neutral, and this relationship does not change with other conditions such as the initial value. This phenomenon may be somewhat different from the reality. Actually for this model in $\\xi\\neq0$ cases, any neutral particles involved in cotransport has equal concentration in periplasm and cytoplasm, which is no different from ordinary diffusion and far from the energy-consuming active transport. Next, from a kinetic point of view, there are still some parts of the model that need discussions when the parameter $\\xi\\neq0$. Although the essence of the model is cotransport, we find that the correlation of the time required for two particles to reach equilibrium is not strong. If the initial state of a certain particle happens to satisfy the proportionality relation that the equilibrium state should satisfy, it will soon equilibrate and is hardly controlled by the other particle. From this point of view, the two kinds of particles are almost independent of each other. At this point, the effect of on lactose transport is more like a facilitated transport, which is also used to explain the uncoupled cotransport in recent years, such as literature \\cite{kaback2015chemiosmotic} and \\cite{\nkaback2019takes}. In contrast, there is no case where some particle reaches equilibrium by itself when $\\xi=0$. Therefore, although changing the parameter $\\xi$ gives the model good kinetic properties, such as adjusting $\\xi$ to control the speed of convergence, the model still has some defects. The parameter $\\xi$ is introduced to solve the problem of leakage current, but this solution is not perfect, and we may still need to think about other solutions. And furthermore, actually there is no experimental evidence showing this uncoupled sugar translocation (``leakage\") could happen before the concentrations of lactose across the cytoplasmic membrane reach equilibrium. In other words, all the above calculations and explanations are based on a shared assumption, and the specific mechanism of cotransporter LacY still lacks knowledge.\n\\section*{Acknowledgement}\n\tI would like to give my sincere gratitude to Prof. Yunxin Zhang from the School of Mathematical Sciences, Fudan University. From my freshman year in Fudan, I have been in contact with Professor Zhang, who is although not my tutor but still gives me much selfless guidance and help in my academic learning and undergraduate studies. It was with his advice and help that I was able to complete this paper as an undergraduate. He taught me almost all the basics about academic writing, read my drafts and made many valuable suggestions. Once again, my sincere thanks to all the people who helped me during the completion of this paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\bigskip We are interested in the regularity of weak solutions to the\nviscous incompressible magnetohydrodynamics (MHD) equations in $\\mathbb{R}%\n^{3}$\n\\begin{equation}\n\\left\\{\n\\begin{array}{c}\n\\partial _{t}u+(u\\cdot \\nabla )u-\\left( b\\cdot \\nabla \\right) b-\\Delta\nu+\\nabla \\pi =0, \\\\\n\\partial _{t}b+(u\\cdot \\nabla )b-(b\\cdot \\nabla )u-\\Delta b=0, \\\\\n\\nabla \\cdot u=\\nabla \\cdot b=0, \\\\\nu(x,0)=u_{0}(x),\\text{ \\ }b(x,0)=b_{0}(x),%\n\\end{array}%\n\\right. \\label{eq1.1}\n\\end{equation}%\nwhere $u=(u_{1},u_{2},u_{3})$ is the velocity field, $b=(b_{1},b_{2},b_{3})$\nis the magnetic field, and $\\pi $ is the scalar pressure, while $u_{0}$ and $%\nb_{0}$ are the corresponding initial data satisfying $\\nabla \\cdot\nu_{0}=\\nabla \\cdot b_{0}=0$ in the sense of distribution.\n\nLocal existence and uniqueness theories of solutions to the MHD equations\nhave been studied by many mathematicians and physicists (see, e.g., \\cite%\n{CW, DL, ST}). But due to the presence of Navier-Stokes equations in the\nsystem (\\ref{eq1.1}) whether this unique local solution can exist globally\nis an outstanding challenge problem. For this reason, there are many\nregularity criteria of weak solutions for the MHD equations has been\ninvestigated by many authors over past years (see e.g., \\cite{DJZ, D, FJNZ,\nG1, GR1, GR2, GRZ, LD, NGZ, Z1, Z2} and references therein). Note that the\nliteratures listed here are far from being complete, we refer the readers to\nsee for example \\cite{GR20, JZ1, JZ2, JZ3, JZ4} for expositions and more\nreferences.\n\nMore recently, Beir\\~{a}o and Yang \\cite{BY} proved the following regularity\ncriterion for the mixed pressure-velocity in Lorentz spaces for Leray-Hopf\nweak solutions to 3D Navier-Stokes equations\n\\begin{equation}\n{\\frac{\\pi }{\\left( e^{-\\left\\vert x\\right\\vert ^{2}}+\\left\\vert\nu\\right\\vert \\right) ^{\\theta }}\\in L}^{p}(0,T;L^{q,\\infty }(\\mathbb{R}%\n^{3})),\\text{ \\ where \\ }0\\leq \\theta \\leq 1\\text{ and }\\frac{2}{p}+\\frac{3}{%\nq}=2-\\theta , \\label{eq7}\n\\end{equation}%\nwhere $L^{q,\\infty }(\\mathbb{R}^{3})$ denotes the Lorentz space (c.f. \\cite%\n{Tri}).\n\nMotivated by the recent work of \\cite{BY}, the purpose of this note is to\nestablish the regularity for the MHD equations (\\ref{eq1.1}) with the mixed\npressure-velocity-magnetic in Lorentz spaces. Our main result can be stated\nas follows:\n\n\\begin{thm}\n\\label{th1}Suppose that $(u_{0},b_{0})\\in L^{2}(\\mathbb{R}^{3})\\cap L^{4}(%\n\\mathbb{R}^{3})$ with $\\nabla \\cdot u_{0}=\\nabla \\cdot b_{0}=0$ in the sense\nof distribution.\\ Let $\\left( u,b\\right) $ be a weak solution to the MHD\nequations on some interval $\\left[ 0,T\\right] $ with $00,\\text{ }p>0,\\text{ }q>0,\n\\end{equation*}\nwe have%\n\\begin{eqnarray*}\nK &=&\\int_{\\mathbb{R}^{3}}\\left\\vert \\pi \\right\\vert ^{\\lambda }V^{-\\lambda\n\\theta }\\left\\vert \\pi \\right\\vert ^{2-\\lambda }V^{\\lambda \\theta\n}(\\left\\vert u\\right\\vert +\\left\\vert b\\right\\vert )^{2}dx \\\\\n&\\leq &\\int_{\\mathbb{R}^{3}}\\left\\vert \\widetilde{\\pi }\\right\\vert ^{\\lambda\n}\\left\\vert \\pi \\right\\vert ^{2-\\lambda }V^{2+\\lambda \\theta }dx \\\\\n&\\leq &\\left\\Vert \\left\\vert \\widetilde{\\pi }\\right\\vert ^{\\lambda\n}\\right\\Vert _{L^{\\frac{q}{\\lambda },\\infty }}\\left\\Vert \\left\\vert \\pi\n\\right\\vert ^{2-\\lambda }\\right\\Vert _{L^{s,\\frac{2}{2-\\lambda }}}\\left\\Vert\nV^{2\\lambda }\\right\\Vert _{L^{r,\\frac{2}{\\lambda }}} \\\\\n&=&\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\lambda\n}\\left\\Vert \\pi \\right\\Vert _{L^{s(2-\\lambda ),2}}^{2-\\lambda }\\left\\Vert\nV^{2}\\right\\Vert _{L^{\\lambda r,2}}^{\\lambda },\n\\end{eqnarray*}%\nwhere\n\\begin{equation*}\n\\frac{\\lambda }{q}+\\frac{1}{s}+\\frac{1}{r}=1\\text{ \\ and \\ }\\lambda =\\frac{2%\n}{2-\\theta }.\n\\end{equation*}%\nBy (\\ref{eq120}), we have%\n\\begin{eqnarray*}\nK &\\leq &\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\lambda\n}\\left( \\left\\Vert \\left\\vert u\\right\\vert ^{2}\\right\\Vert _{L^{s(2-\\lambda\n),2}}+\\left\\Vert \\left\\vert b\\right\\vert ^{2}\\right\\Vert _{L^{s(2-\\lambda\n),2}}\\right) ^{2-\\lambda }\\left\\Vert V^{2}\\right\\Vert _{L^{\\lambda\nr,2}}^{\\lambda } \\\\\n&\\leq &C\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\lambda\n}\\left\\Vert V^{2}\\right\\Vert _{L^{s(2-\\lambda ),2}}^{2-\\lambda }\\left\\Vert\nV^{2}\\right\\Vert _{L^{\\lambda r,2}}^{\\lambda }.\n\\end{eqnarray*}%\nBy the interpolation and Sobolev inequalities in Lorentz spaces, it follows\nthat%\n\\begin{equation}\n\\left\\{\n\\begin{array}{c}\n\\left\\Vert V^{2}\\right\\Vert _{L^{s(2-\\lambda ),2}}\\leq C\\left\\Vert\nV^{2}\\right\\Vert _{L^{2,2}}^{1-\\delta _{1}}\\left\\Vert V^{2}\\right\\Vert\n_{L^{6,2}}^{\\delta _{1}}\\leq C\\left\\Vert V^{2}\\right\\Vert _{L^{2}}^{1-\\delta\n_{1}}\\left\\Vert \\nabla V^{2}\\right\\Vert _{L^{2}}^{\\delta _{1}}, \\\\\n\\left\\Vert V^{2}\\right\\Vert _{L^{\\lambda r,2}}\\leq C\\left\\Vert\nV^{2}\\right\\Vert _{L^{2,2}}^{1-\\delta _{2}}\\left\\Vert V^{2}\\right\\Vert\n_{L^{6,2}}^{\\delta _{2}}\\leq C\\left\\Vert V^{2}\\right\\Vert _{L^{2}}^{1-\\delta\n_{2}}\\left\\Vert \\nabla V^{2}\\right\\Vert _{L^{2}}^{\\delta _{2}},%\n\\end{array}%\n\\right. \\label{eq6.6}\n\\end{equation}%\nwhere $0<\\delta _{1},\\delta _{2}<1$ and\n\\begin{equation*}\n\\frac{1}{s(2-\\lambda )}=\\frac{1-\\delta _{1}}{2}+\\frac{\\delta _{1}}{6},\\text{\n\\ }\\frac{1}{\\lambda r}=\\frac{1-\\delta _{2}}{2}+\\frac{\\delta _{2}}{6}.\n\\end{equation*}%\nHence from (\\ref{eq6.6}) and Young inequality, it follows that%\n\\begin{eqnarray*}\nK &\\leq &C\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\lambda\n}\\left\\Vert V^{2}\\right\\Vert _{L^{2}}^{(2-\\lambda )(1-\\delta _{1})+\\lambda\n(1-\\delta _{2})}\\left\\Vert \\nabla V^{2}\\right\\Vert _{L^{2}}^{(2-\\lambda\n)\\delta _{1}+\\lambda \\delta _{2}} \\\\\n&\\leq &C\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\frac{%\n2\\lambda }{2-(2-\\lambda )\\delta _{1}-\\lambda \\delta _{2}}}\\left\\Vert\nV^{2}\\right\\Vert _{L^{2}}^{2}+\\frac{1}{2}\\left\\Vert \\nabla V^{2}\\right\\Vert\n_{L^{2}}^{2}.\n\\end{eqnarray*}%\nDue to the definition of $V$, we see that\n\\begin{equation*}\n\\left\\Vert V^{2}\\right\\Vert _{L^{2}}^{2}\\leq C(1+\\left\\Vert \\left\\vert\nu\\right\\vert +\\left\\vert b\\right\\vert \\right\\Vert _{L^{2}}^{2}+\\left\\Vert\n\\left\\vert u\\right\\vert ^{2}+\\left\\vert b\\right\\vert ^{2}\\right\\Vert\n_{L^{2}}^{2}),\n\\end{equation*}%\nand%\n\\begin{equation*}\n\\left\\Vert \\nabla V^{2}\\right\\Vert _{L^{2}}^{2}\\leq C(1+\\left\\Vert\n\\left\\vert u\\right\\vert +\\left\\vert b\\right\\vert \\right\\Vert\n_{L^{2}}^{2}+\\left\\Vert \\nabla (\\left\\vert u\\right\\vert +\\left\\vert\nb\\right\\vert )\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla (\\left\\vert\nu\\right\\vert ^{2}+\\left\\vert b\\right\\vert ^{2})\\right\\Vert _{L^{2}}^{2}).\n\\end{equation*}%\nConsequently, we get%\n\\begin{eqnarray*}\nK &\\leq &C\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\frac{%\n2\\lambda }{2-(2-\\lambda )\\delta _{1}-\\lambda \\delta _{2}}}(1+\\left\\Vert\n\\left\\vert u\\right\\vert +\\left\\vert b\\right\\vert \\right\\Vert\n_{L^{2}}^{2}+\\left\\Vert \\left\\vert u\\right\\vert ^{2}+\\left\\vert b\\right\\vert\n^{2}\\right\\Vert _{L^{2}}^{2}) \\\\\n&&+C(1+\\left\\Vert \\left\\vert u\\right\\vert +\\left\\vert b\\right\\vert\n\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla (\\left\\vert u\\right\\vert\n+\\left\\vert b\\right\\vert )\\right\\Vert _{L^{2}}^{2})+\\frac{1}{2}\\left\\Vert\n\\nabla (\\left\\vert u\\right\\vert ^{2}+\\left\\vert b\\right\\vert\n^{2})\\right\\Vert _{L^{2}}^{2} \\\\\n&\\leq &C\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\frac{%\n2\\lambda }{2-(2-\\lambda )\\delta _{1}-\\lambda \\delta _{2}}}(1+\\left\\Vert\nu\\right\\Vert _{L^{2}}^{2}+\\left\\Vert b\\right\\Vert _{L^{2}}^{2}+\\left\\Vert\nu\\right\\Vert _{L^{4}}^{4}+\\left\\Vert b\\right\\Vert _{L^{4}}^{4}) \\\\\n&&+C(1+\\left\\Vert u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert b\\right\\Vert\n_{L^{2}}^{2}+\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla\nb\\right\\Vert _{L^{2}}^{2})+\\frac{1}{2}\\left\\Vert \\nabla \\left\\vert\nu\\right\\vert ^{2}\\right\\Vert _{L^{2}}^{2}+\\frac{1}{2}\\left\\Vert \\nabla\n\\left\\vert b\\right\\vert ^{2}\\right\\Vert _{L^{2}}^{2}.\n\\end{eqnarray*}%\nSince $(u,b)$ is a weak solution to (\\ref{eq1.1}), then $(u,b)$ satisfies%\n\\begin{equation*}\n(u,b)\\in L^{\\infty }(0,T;L^{2}(\\mathbb{R}^{3}))\\cap L^{2}(0,T;H^{1}(\\mathbb{R%\n}^{3})).\n\\end{equation*}%\nInserting the above estimates into (\\ref{eq21}), we obtain\n\\begin{eqnarray*}\n&&\\frac{d}{dt}(\\left\\Vert u\\right\\Vert _{L^{4}}^{4}+\\left\\Vert b\\right\\Vert\n_{L^{4}}^{4})+\\left\\Vert \\nabla \\left\\vert u\\right\\vert ^{2}\\right\\Vert\n_{L^{2}}^{2}+\\left\\Vert \\nabla \\left\\vert b\\right\\vert ^{2}\\right\\Vert\n_{L^{2}}^{2} \\\\\n&&+2\\left\\Vert \\left\\vert u\\right\\vert \\left\\vert \\nabla u\\right\\vert\n\\right\\Vert _{L^{2}}^{2}+2\\left\\Vert \\left\\vert b\\right\\vert \\left\\vert\n\\nabla b\\right\\vert \\right\\Vert _{L^{2}}^{2}+2\\left\\Vert \\left\\vert\nu\\right\\vert \\left\\vert \\nabla b\\right\\vert \\right\\Vert\n_{L^{2}}^{2}+2\\left\\Vert \\left\\vert b\\right\\vert \\left\\vert \\nabla\nu\\right\\vert \\right\\Vert _{L^{2}}^{2} \\\\\n&\\leq &C\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\frac{%\n2\\lambda }{2-(2-\\lambda )\\delta _{1}-\\lambda \\delta _{2}}}(1+\\left\\Vert\nu\\right\\Vert _{L^{2}}^{2}+\\left\\Vert b\\right\\Vert _{L^{2}}^{2}+\\left\\Vert\nu\\right\\Vert _{L^{4}}^{4}+\\left\\Vert b\\right\\Vert _{L^{4}}^{4}) \\\\\n&&+C(1+\\left\\Vert u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert b\\right\\Vert\n_{L^{2}}^{2}+\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla\nb\\right\\Vert _{L^{2}}^{2}) \\\\\n&\\leq &C\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\frac{%\n2\\lambda }{2-(2-\\lambda )\\delta _{1}-\\lambda \\delta _{2}}}(1+\\left\\Vert\nu\\right\\Vert _{L^{4}}^{4}+\\left\\Vert b\\right\\Vert\n_{L^{4}}^{4})+C(1+\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert\n\\nabla b\\right\\Vert _{L^{2}}^{2}),\n\\end{eqnarray*}%\nUsing Gronwall's inequality with the assumption (\\ref{eq15}), we deduce that%\n\\begin{equation*}\n(u,b)\\in L^{\\infty }(0,T;L^{4}(\\mathbb{R}^{3}))\\subset L^{8}(0,T;L^{4}(%\n\\mathbb{R}^{3})).\n\\end{equation*}%\nWe complete the proof of Theorem \\ref{th1}.\n\\end{pf}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec1}\n Recently, CDF-II collaboration published a new result in W boson mass with increased precision $M^W_{CDF} =80.4335 \\pm 0.0094 ~\\rm{GeV}$ which deviates from SM prediction by $7\\sigma$\\cite{CDF:2022hxs}. The SM prediction for the W boson mass is $ M^W_{SM}=80.357 \\pm 0.006 ~\\rm {GeV}$ \\cite{ParticleDataGroup:2020ssz}.\nNeedless to say a better understanding of SM calculations, and also more accurate measurements are needed. Nevertheless, a lot of new suggestions have been proposed to explain this anomaly \\cite{Ghorbani:2022vtv,Cheng:2022aau,Borah:2022zim,Arcadi:2022dmt,Nagao:2022oin,Kawamura:2022uft,Fan:2022dck,Lu:2022bgw,Athron:2022qpo,Yuan:2022cpw,Strumia:2022qkt,Yang:2022gvz,deBlas:2022hdk,Du:2022pbp,Tang:2022pxh,Cacciapaglia:2022xih,Blennow:2022yfm,Sakurai:2022hwh,Fan:2022yly,Liu:2022jdq,Lee:2022nqz,Bagnaschi:2022whn,Paul:2022dds,Bahl:2022xzi,Asadi:2022xiy,DiLuzio:2022xns,Athron:2022isz,Gu:2022htv,Babu:2022pdn,Heo:2022dey,Du:2022brr,Cheung:2022zsb,Crivellin:2022fdf,Endo:2022kiw,Biekotter:2022abc,Balkin:2022glu,Han:2022juu,Ahn:2022xeq,Zheng:2022irz,Ghoshal:2022vzo,FileviezPerez:2022lxp,Mondal:2022xdy,Borah:2022obi,Chowdhury:2022moc}. In\\cite{Zhang:2022nnh}, the extra $U(1)$ gauge field mix with the $U(1$) hypercharge via gauge kinetic term, and this kinetic mixing can generate an enhancement of the W boson mass. Another approach for the $U(1)$ extension of SM is to consider an additional scalar field in which the new field can enhance W-boson mass via the loop corrections.\n\nIt is known that the SM is an incomplete theory in part due to its inability to explain phenomena such as DM, neutrino mass, and the baryon asymmetry of the Universe. In this paper, we consider a $U(1)$ extension of SM in which the vector field does not mix with the SM gauge field and consider the scalar field that plays the role of mediator between the dark sector and SM. This field shift W-boson mass in loop corrections. We study this simple extension of the SM to explain the W boson mass enhancement and also offer a viable DM candidate with mass ranging from $1~\\rm GeV$ to $2~\\rm TeV$. In the following, we apply various phenomenological constraints such as invisible Higgs decay mode and direct detection experiment in our analysis.\n\n\nIn the context of SM, the vacuum stability and perturbativity have resulted in theoretical bounds on the Higgs mass. The Higgs mass value, together with other relevant parameters such as the top quark mass, affects on the behaviour of Higgs potential at very high energy scales, in particular for the sake of electroweak vacuum stability \\cite{Bezrukov:2012sa},\\cite{Buttazzo:2013uya}.\nThis is because, for Higgs and top mass values, the Higgs quartic coupling can be very small or even negative. Since the top Yukawa coupling dependency is strong and very subtle, there are different views on an explanation of this issue in literature, some of them favouring\\cite{Bezrukov:2012sa} and some others disfavouring \\cite{Degrassi:2012ry}. In the following, we also ask how the model behaves at a high energy scale and is it computationally reliable? Standard treatment is the study of the running of the coupling constants in terms of the mass scale $\\Lambda$ via the RGE. The three standard problems to consider are the positivity, perturbativity of the coupling constants and the vacuum stability of the model. These issues have been studied in the literature in the presence of a scalar extension of SM \\cite{Ghorbani:2017qwf,Gonderinger:2012rd} and it was shown that the vacuum stability requirement can affect the DM relic density. Here, we discuss the requirement of vacuum stability and RGE of the model.\n\nThis work is organized as follows: After the introduction, we introduce the model.\nIn section~3, we study conditions of vacuum stability and calculate the RGEs of the model. In section~4, we study the contribution of the model to W-boson mass and probe the consistent parameter space of the model with CDF-II measurement. Invisible Higgs decay constraint on the model study in section.~5. In section.~6, we find the allowed regions in the parameter space which will give rise to the correct DM relic density. We describe combined results consistent with constraints provided and examine the RGEs numerically in section~7 . Section 8 contains our conclusions.\n\n\n\n\n\n\\section{The Model} \\label{sec2}\nIn our model, beyond the SM, we employ two new fields to furnish the model: a scalar field $S$ which has a unit charge under a dark $U(1)$ gauge symmetry with a dark photon vector field $ V_{\\mu} $. The model has a $Z_2$ discrete flavor symmetry, under which $V_{\\mu}$ is odd and all the other fields are even. $Z_2$ symmetry forbids the kinetic mixing between the vector field $ V_{\\mu} $ and SM $ U_{Y}(1) $ gauge boson $ B_{\\mu} $, i.e., $ V_{\\mu \\nu} B_{\\mu \\nu} $. Therefore, the vector field $ V_{\\mu} $ is stable and can be considered a DM candidate. The Lagrangian one can write with assumption is:\n\\begin{equation}\n {\\cal L} ={\\cal L}_{SM} + (D'_{\\mu} S)^{*} (D'^{\\mu} S) - V(H,S) - \\frac{1}{4} V_{\\mu \\nu} V^{\\mu \\nu} , \\label{2-2}\n\\end{equation}\nwhere $ {\\cal L} _{SM} $ is the SM Lagrangian without the Higgs potential term and\n\\begin{align}\n& D'_{\\mu} S= (\\partial_{\\mu} + i g_v V_{\\mu}) S,\\nonumber \\\\\n& V_{\\mu \\nu}= \\partial_{\\mu} V_{\\nu} - \\partial_{\\nu} V_{\\mu},\\nonumber \\end{align}\nand the potential which is renormalizable and invariant\nunder gauge and $ Z_{2} $ symmetry is:\n\\begin{equation}\nV(H,S) = -\\mu_{H}^2 H^{\\dagger}H-\\mu_{S}^2 S^*S+\\lambda_{H} (H^{\\dagger}H)^{2} + \\lambda_{S} (S^*S)^{2} + \\lambda_{S H} (S^*S) (H^{\\dagger}H). \\label{2-3}\n\\end{equation}\nNote that the quartic portal interaction, $ \\lambda_{SH} (S^*S) (H^{\\dagger}H) $, is the only connection between the dark sector and the SM.\n\nSM Higgs field $ H $, as well as dark scalar $S$, can receive VEVs breaking respectively the electroweak and $ U'_{D}(1) $ symmetries.\nIn the unitary gauge, the imaginary component of $S$ can be absorbed as the longitudinal component of $ V_{\\mu} $.\nIn this gauge, we can write\n\\begin{equation}\nH = \\frac{1}{\\sqrt{2}} \\begin{pmatrix}\n0 \\\\ h_{1} \\end{pmatrix} \\, \\, \\, {\\rm and} \\, \\, \\, S = \\frac{1}{\\sqrt{2}} h_{2} , \\label{2-4}\n\\end{equation}\nwhere $ h_{1} $ and $ h_{2} $ are real scalar fields which can get VEVs.\nThe tree level potential in unitary gauge is:\n\\begin{equation}\nV_{\\text{tree}}(h_{1},h_{2})=-\\frac{1}{2} \\mu _H^2 h_1^2-\\frac{1}{2} \\mu _S^2 h_2^2 +\\frac{1}{4} \\lambda _H h_1^4 +\\frac{1}{4} \\lambda _S h_2^4+\\frac{1}{4} \\lambda _{SH} h_1^2 h_2^2.\n\\end{equation}\nGiven differentiable $ V_{\\text{tree}} $, one can obtain\nthe Hessian matrix, $ {\\cal{H}}_{ij}(h_{1},h_{2})=\\frac{\\partial^{2}V_{\\text{tree}}}{\\partial h_{i} \\partial h_{j} }$.\nIn order to get the mass spectrum of the model, it is necessary to consider the sufficient conditions for a local minimum:\n\\begin{align}\n& \\nabla V_{\\text{tree}} = 0 \\label{minimum1} \\\\\n& \\det {\\cal{H}} > 0 \\label{minimum2} \\\\\n& {\\cal{H}}_{11} > 0 \\label{minimum3}\n\\end{align}\nto occur at a point $ (\\nu_{1},\\nu_{2}) $. Note that Eq. (\\ref{minimum2}) and Eq. (\\ref{minimum3}) also imply that $ {\\cal{H}}_{22} > 0 $.\nEq. (\\ref{minimum1}) leads to\n\\begin{align}\n& \\mu _H^2= \\lambda _H \\nu _1^2+ \\frac{1}{2} \\lambda _{SH} \\nu _2^2 , \\nonumber\\\\\n& \\mu _S^2=\\lambda _S \\nu _2^2+ \\frac{1}{2} \\lambda _{SH} \\nu _1^2 \\label{mini}\n\\end{align}\nEq. (\\ref{mini}) leads to the non-diagonal mass matrix $\\cal{H}$ as follows:\n\\begin{equation}\n{\\cal{H}}(\\nu_{1},\\nu_{2})= \\left(\n\\begin{array}{cc}\n 2 \\lambda _H \\nu _1^2 & \\lambda _{SH} \\nu _1 \\nu _2 \\\\\n \\lambda _{SH} \\nu _1 \\nu _2 & 2 \\lambda _S \\nu _2^2 \\\\\n\\end{array}\n\\right) \\label{hess}\n\\end{equation}\nTherefore, according to the conditions $ {\\cal{H}}_{11} > 0 $ and $ {\\cal{H}}_{22} > 0 $ and Eq. (\\ref{minimum2}) we should have\n\\begin{equation}\n\\lambda_{H} > 0 \\, \\, \\, , \\, \\, \\, \\lambda_{S} > 0 \\, \\, \\, , \\, \\, \\, \\lambda_{SH}^{2} < 4 \\lambda_{H} \\lambda_{S}\n\\end{equation}\nNow by substituting $ h_1 \\rightarrow \\nu_1 + h_1 $ and $ h_2 \\rightarrow \\nu_2 + h_2 $, the fields $ h_1 $ and $ h_1 $ mix with each other and they can be rewritten by the mass eigenstates $ H_1 $ and $ H_1 $ as\n\\begin{equation}\n\\begin{pmatrix}\nh_{1}\\\\h_{2}\\end{pmatrix}\n =\\begin{pmatrix} cos \\alpha~~~ sin \\alpha \\\\-sin \\alpha ~~~~~cos \\alpha\n \\end{pmatrix}\\begin{pmatrix}\nH_1 \\\\ H_{2}\n\\end{pmatrix}, \\label{matri}\n\\end{equation}\nwhere $ \\alpha $ is the mixing angle. After symmetry breaking, we have\n\\begin{align}\n& \\nu _2=\\frac{M_V}{g_v} \\nonumber,~~~~~~~~~~ \\sin\\alpha=\\frac{\\nu_1}{\\sqrt{\\nu _1^2+\\nu_2^2}} \\\\\n& \\lambda _H=\\frac{\\cos ^2\\alpha M_{H_1}^2+\\sin ^2\\alpha M_{H_2}^2}{2 \\nu _1^2} \\nonumber \\\\\n& \\lambda _S=\\frac{\\sin ^2\\alpha M_{H_1}^2+\\cos ^2\\alpha M_{H_2}^2}{2 \\nu _2^2} \\nonumber \\\\\n& \\lambda _{SH}=\\frac{ \\left(M_{H_2}^2-M_{H_1}^2\\right) \\sin \\alpha \\cos \\alpha}{\\nu _1 \\nu _2} \\label{cons}\n\\end{align}\nThe mass eigenstates of scalar fields can be written as following:\n\\begin{align}\nM^2_{H_{2},H_{1}}=\\lambda_H \\nu_1^2+\\lambda_S \\nu_2^2 \\pm \\sqrt{(\\lambda_H \\nu_1^2-\\lambda_S \\nu_2^2)^2+\\lambda_{SH}^2\\nu_1^2\\nu_2^2},\n\\label{mas}\n\\end{align}\nwhere we take $ M_{H_1} = 125 $ GeV and $ \\nu _1 = 246 $ GeV. Note that, beside of SM parameters, the model has only three free parameters $ g_v $, $ M_{H_2} $ and $ M_V $.\n\n\\section{Vacuum stability and RGE}\n\nA prominent feature of the study of high energy physics is the evolution of the coupling constants with energy. This has become an incentive to further strengthen theories such as the GUT and supersymmetry by merging couplings at high energies\\cite{Wulzer:2019max}. Renormalization Group Equation(RGE) describes the behavior of quantities with energy. After the discovery of the Higgs particle by ATLAS and CMS experiments at the LHC in 2012, the vacuum stability study has been done more clearly\\cite{Ghorbani:2017qwf,Gonderinger:2012rd,Abada:2013pca,Baek:2012uj,Ghorbani:2021rgs,Duch:2015jta}. In the SM, the Higgs quartic coupling becomes negative at the scale $10^{10}$ GeV, and the Higgs non-zero VEV is no longer a minimum of the theory. The reason for this is that the top quark has a large negative contribution to the RGE for $\\lambda_{H}$. We will show the running of $\\lambda_{H}$ in SM in the next sections.\n\n\n\nThere are three types of theoretical constraints on the couplings in the model. The first one is related to the perturbativity condition for couplings, which is satisfied when $\\lambda_i< 4\\pi$. The second is vacuum stability of the model dictates some other constraints on the couplings such that for self-coupling constants. In this regard, we should have $\\lambda_i$ and $\\lambda_H>0$. On the other hand, by adding a new scalar mediator the vacuum structure of the model will be modified. The third condition is positivity where potential must be well-defined and positive at all scales. The requirement of positivity for the potential implies the following relations :\n\\begin{equation}\n\\lambda_H>0 , \\lambda_S>0 , \\lambda_{SH}>-2 \\sqrt{\\lambda_H \\lambda_S }\n\\end{equation}\nAlso, the common investigation for vacuum stability analyses in the literature begins with the RGE improved potential and choice of the renormalization scale to minimize the one-loop potential. In this light, we consider the running of the coupling constants with energy. The Model is implemented in SARAH \\cite{Staub:2015kfa} to compute $\\beta$ functions and their runnings. We calculate the one-loop RGE and one-loop $\\beta$ functions for scalar couplings and dark coupling including the following relationships:\n\n\\begin{align}\n& (16\\pi^2)\\beta_{\\lambda_{S}} = -20\\lambda_{S}^2 -2\\lambda_{SH}^2 -6g_v^4 -12g_v^2 \\lambda_{S} ,\\nonumber \\\\\n& (16\\pi^2)\\beta_{\\lambda_{SH}}= -\\frac{3}{2}g_{1}^2 \\lambda_{SH} -\\frac{9}{2}g_{2}^2 \\lambda_{SH} -12\\lambda_{SH}\\lambda_{H} -8\\lambda_{SH}\\lambda_{S} -4\\lambda_{SH}^2 + 6\\lambda_{SH}\\lambda_{t}^2 -6g_v^2 \\lambda_{SH} \\nonumber ,\\\\\n& (16\\pi^2)\\beta_{\\lambda_{H}}= -\\frac{3}{8}g_{1}^4 -\\frac{3}{4}g_{1}^2 g_{2}^2 -\\frac{9}{8}g_{2}^4 -3g_{1}^2 \\lambda_{H} -9g_{2}^2 \\lambda_{H} -24\\lambda_{H}^2 -\\lambda_{SH}^2 +12\\lambda_{H}\\lambda_{t}^2 +6\\lambda_{t}^4, \\nonumber \\\\\n& (16\\pi^2)\\beta_{g_v}= \\frac{1}{3}g_v^3.\n\\label{RGE}\n\\end{align}\nwhere $\\beta_a\\equiv \\mu \\frac{da}{d\\mu}$ that $\\mu$ is the renormalization scale with initial value $\\mu_0=100~\\rm GeV$. The $\\beta$ functions of couplings, $g_1$, $g_2$ and $g_3$ are given to one-loop order by:\n\\begin{align}\n& (16\\pi^2)\\beta_{g_{1}}= \\frac{41}{6} g_{1}^3 , \\nonumber \\\\\n& (16\\pi^2)\\beta_{g_{2}}= -\\frac{19}{6} g_{2}^3 , \\nonumber \\\\\n& (16\\pi^2)\\beta_{g_{3}}= -7 g_{3}^3 .\n\\end{align}\nAmong the Yukawa couplings of SM, the top quark has the largest contribution compared with other fermions in the SM. Therefore, we set all the SM Yukawa couplings equal to zero and consider only the top quark coupling. The RGE of top quark Yukawa coupling is given to one-loop order by\n\\begin{align}\n(16\\pi^2)\\beta_{\\lambda_t}= -\\frac{17}{12}g_{1}^2 \\lambda_t -\\frac{9}{4}g_2^2 \\lambda_{t} -8g_3^2 \\lambda_t +\\frac{9}{2}\\lambda_t^3 .\n\\end{align}\n\nIn the following, we first study experimental constraints on the model, and then in the final section, we investigate conditions of the Higgs stability and RGE of coupling parameters of the model.\n\n\n\n\n\n\\section{$\\rm W$-Mass Anomaly}\nTo explain the CDF-II anomaly, we study W-mass correction in the context of the model. The corrections of new physics to the W-boson mass can be written in terms of the Peskin-Takeuchi oblique parameters $S, T,$ and $U$. The Peskin\u2013Takeuchi parameters are only sensitive to new physics that contribute to the oblique corrections, i.e., the vacuum polarization corrections to four-fermion scattering processes. In general, the SM contribution to an oblique parameter is subtracted from the new physics contribution to define the oblique parameter. The effects of $S, T,$ and $U$ on $W$ boson mass can be expressed as follows\n \\cite{Peskin:1991sw}:\n\\begin{equation}\n\\Delta M_W^2 = \\frac{ M_{SM}^2}{c_W^2 - s_W^2} \\left(\n-\\frac{ \\alpha S}{2} + c_W^2 \\alpha T +\\frac{c_W^2-s_W^2}{4s_W^2} \\alpha U \\right)\\,.\n\\label{WMco}\n\\end{equation}\nwhere $M_{SM}$ is SM of W-boson mass, $c_W$ and $s_W$ are cosine and sine of Weinberg angle. Oblique parameters $S, T,$ and $U$ for our model are as follows \\cite{Grimus:2008nb} :\n\\begin{equation}\n\\alpha S=\\frac{g^2 sin ^2\\alpha}{96 {\\pi}^2 }[(ln {M_{H_2}}^2 + G( {M_{H_2}}^2 ,{M_{Z}}^2 ))-(ln {M_{H_1}}^2 + G({M_{H_1}}^2 ,{M_{Z}}^2 )) ] ,\n\\end{equation}\n\\begin{equation}\n\\alpha T=\\frac{3g^2 sin ^2\\alpha}{64\\pi^2 s_W^2 M_W^2 }[(F(M_Z^2 ,M^2_{H_2}) - F(M_W^2 ,M_{H_2}^2)) - (F(M_Z^2 ,M_{H1}^2)-F(M_W^2 ,M_{H_1}^2))] ,\n\\end{equation}\n\\begin{equation}\n\\alpha U=\\frac{g^2 sin ^2\\alpha}{96\\pi^2 }[(G( M_{H_2}^2 ,M_W^2 )-G( M_{H_2}^2 ,M_{Z}^2 ))-(G(M_{H_1}^2 ,M_{W}^2 )-G(M_{H_1}^2 ,M_{Z}^2))] ,\n\\end{equation}\nwhere\n\\begin{equation}\ng^2= 4\\pi \\alpha_{QED}\n\\end{equation}\n\\begin{eqnarray}\nF(x,y)=\\bigg\\lbrace \\begin{array}{cc}\n{\\frac{x+y}{2}-\\frac{xy}{x-y}ln\\frac{x}{y}}&{for~x\\neq y ,}\\\\{0}&{for~ x=y ,}\n\\end{array} ,\n\\end{eqnarray}\n\\begin{equation}\nG(x,y)=-\\frac{79}{3}+ 9\\frac{x}{y} -2\\frac{x^2}{y^2} +(-10+18\\frac{x}{y}-6\\frac{x^2}{y^2}+\\frac{x^3}{y^3}-9\\frac{x+y}{x-y}) ln\\frac{x}{y} +(12-4\\frac{x}{y}+\\frac{x^2}{y^2}) \\frac{f(x,x^2 -4xy)}{y} ,\n\\end{equation}\n\\begin{eqnarray}\nf(a,b)=\\bigg\\lbrace \\begin{array}{cc}\n{\\sqrt b ln\\vert \\frac{a-\\sqrt b}{a+\\sqrt b}\\vert}&{for~ b>0}\\\\{0}&{for~ b=0}\\\\{2\\sqrt{-b}~ Arctan \\frac{\\sqrt{-b}}{a}}&{for~ b<0} .\n\\end{array} ,\n\\end{eqnarray}\n\n\n\n\\begin{figure\n\t\\begin{center}\n\t\t\\centerline{\\hspace{0cm}\\epsfig{figure=ScaterWmass.eps,width=12cm}}\n\t\t\\centerline{\\vspace{-0.2cm}}\n\t\t\\caption{Scatter points depict allowed range of parameters space of the model consistent with W-boson mass measurement.} \\label{Wmass}\n\t\\end{center}\n\\end{figure}\n\n To study the model parameter space, other than theoretical constraints (such as perturbativity condition and vacuum stability), it is necessary to consider the experimental upper limit on mixing angle $\\alpha$. For low masses, $M_{H_2} < 5~\\rm GeV$, the strongest limit comes from decay $B\\rightarrow K\\ell\\ell$\\cite{LHCb:2012juf,Belle:2009zue}. It was shown that for this range of parameters, $\\sin\\alpha$ should be smaller than $10^{-3}$. Between $5-12~\\rm GeV$, the constraint on $sin \\alpha< 0.5$ is imposed by the decay of a low-mass Higgs boson in radiative decay of the Y and the DELPHI searches for a light Higgs in Z-decay\\cite{DELPHI:1990vtb,BaBar:2012wey}. For above this mass range to about $65~\\rm GeV$, an overall result of the Higgs signal strength measured by ATLAS and CMS \\cite{ATLAS:2016neq} severely constrains the mixing angle to values smaller than $sin\\alpha<0.12$. For the larger values of $M_{H_2}$, the lower limit is set by the LHC constraints on the mixing angle in which $sin \\alpha\\leq0.44$\\cite{Farzinnia:2013pga,Farzinnia:2014xia}. In the following, for low mass $M_{H_2}$ in which decay of SM Higgs like to $H_2$ is kinematically possible, we consider severe upper limit on mixing angle and for large $M_{H_2}$ mass, we relax this bound. As was discussed in the previous section, the present model has three free parameters, $ g_v $, $ M_{H_2} $ and $ M_V$. In addition to the mixing angle constraints, we also make the following choices for the mass parameters:\n \\begin{itemize}\n \t\\item The DM mass $M_V$ is between $1-2000~\\rm GeV$;\n \t\\item The mediator scalar mass ($M_{H_2}$) is between $1-500~\\rm GeV$;\n \\end{itemize}\nWe scan over the three-dimensional parameters $ g_v $, $ M_{H_2} $, and $ M_V $ to probe a consistent range of parameters space with observables.\n In figure.~\\ref{Wmass}, we depict the allowed range of parameters of the model which is consistent with CDF measurement for W-boson mass. Note that, for a large value of $M_{H_2}$, we choose $sin\\alpha\\leq0.44$ on the mixing angle, and for $M_{H_2}<65~\\rm GeV$, we suppose $\\alpha<6.9^{~\\circ}$ to satisfy ATLAS and CMS upper limit on the mixing angle. As is seen in the figure, the $\\rm W$-Mass measurement, for $M_{H_2}\\lesssim 4.5~\\rm GeV$ and $M_{H_2}\\gtrsim 124~\\rm GeV$, excludes the parameters space of the model. However, for $M_{H_2}$ between $4.5-124~\\rm GeV$ and $M_V$ between $1-2000~\\rm GeV$, the model is consistent with the W-mass anomaly.\n\n\\section{Invisible Higgs decay}\nIn the model, SM Higgs-like can decay invisibly into a pair of DM if kinematically allowed. Also, it can decay to another Higgs boson for $M_{H_2}<1\/2M_{H_1}$. Therefore, $H_1$ can contribute to the invisible decay mode with a branching ratio:\n\\begin{eqnarray}\nBr(H_1\\rightarrow \\rm Invisible)& =\\frac{\\Gamma(H_1\\rightarrow 2VDM)+\\Gamma(H_1\\rightarrow 2H_2)}{\\Gamma(h)_{SM}+\\Gamma(H_1\\rightarrow 2VDM)+\\Gamma(H_1\\rightarrow 2H_2)},\n\\label{decayinv1}\n\\end{eqnarray}\n\n\nwhere $\\Gamma(h)_{SM}=4.15 ~ \\rm [MeV]$ is total width of Higgs boson \\cite{LHCHiggsCrossSectionWorkingGroup:2011wcg}. The partial width for processes $H_1\\rightarrow 2VDM$ and $H_1\\rightarrow 2H_2$ are given by:\n\\begin{eqnarray}\n\\Gamma(H_1\\rightarrow 2VDM)& =\\frac{g_v^4v^2_2 sin^2{\\alpha}}{8\\pi M_{H_1}}\\sqrt{1-\\frac{4M^2_{V}}{M^2_{H_1}}}.\n\\label{decayinv1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\Gamma(H_1\\rightarrow 2H_2)& =\\frac{a^2}{8\\pi M_{H_1}}\\sqrt{1-\\frac{4M^2_{H_2}}{M^2_{H_1}}}.\n\\label{decayinv1}\n\\end{eqnarray}\n\nwhere $a=(1\/2cos^3\\alpha -sin^2\\alpha cos\\alpha)v_1$.\nThe SM prediction for the branching ratio of the Higgs boson decaying to invisible particles which coming from process $h\\rightarrow ZZ^*\\rightarrow 4\\nu$ \\cite{Denner:2011mq},\\cite{Dittmaier:2012vm},\\cite{Brein:2003wg},\\cite{LHCHiggsCrossSectionWorkingGroup:2013rie} is, $1.2\\times10^{-3}.$\nCMS Collaboration has reported the observed (expected) upper limit\non the invisible branching fraction of the Higgs boson to be $0.18 (0.10)$ at the $95\\%$ confidence level, by assuming the SM production cross section \\cite{CMS:2022qva}. A Similar analysis was performed by ATLAS collaboration in which an observed upper limit of $0.145$ is placed on the branching fraction of its decay into invisible particles at a $95\\%$ confidence level\\cite{ATLAS:2022yvh}.\n\n\\begin{figure\n\t\\begin{center}\n\t\t\\centerline{\\hspace{0cm}\\epsfig{figure=Invisible.eps,width=12cm}}\n\t\t\\centerline{\\vspace{-0.2cm}}\n\t\t\\caption{ The cross points depict allowed region which is consistent with invisible Higgs decay at \\cite{CMS:2022qva}.\n\t\t} \\label{Invisible}\n\t\\end{center}\n\\end{figure}\n\nFigure.~\\ref{Invisible}, shows the allowed range of parameters by considering CMS\\cite{CMS:2022qva} upper limit for invisible Higgs mode. In this figure, we consider LHC bound on the mixing angle $\\sin\\alpha<0.12$. For $M_{H_2}<1\/2M_{SM}$ CMS upper limit excludes the parameters space. Note that this upper limit practically can not constraint the model for low mass DM and also part of the parameters space in which invisible Higgs decay is forbidden.\n\n\n\\section{Relic Density}\nThe evolution of the number density of DM particles with time is governed by the Boltzmann equation. In this regard, we calculate the relic density\nnumerically for the VDM particle by implementing the model into micrOMEGAs \\cite{Belanger:2014vza}. We investigate viable parameter space which satisfies constraints from observed DM relic density ( according to the data of Planck collaboration \\cite{Planck:2014egr}):\n\\begin{equation} \\label{44}\n\\Omega_{DM} h^{2} = 0.1199 \\pm 0.0027.\n\\end{equation}\n\n\\begin{figure\n\t\\begin{center}\n\t\t\\centerline{\\hspace{0cm}\\epsfig{figure=Relic.eps,width=12cm}}\n\t\t\\centerline{\\vspace{-0.2cm}}\n\t\t\\caption{The allowed range of parameter space consistent with DM relic density.} \\label{Relic}\n\t\\end{center}\n\\end{figure}\n\nThe allowed range of parameter space corresponding to this constraint is depicted in figure \\ref{Relic}. As seen in the figure, for small values of VDM mass ($ M_V\\lesssim 22~ \\rm GeV$), relic density measurement excludes the model.\n\n\n\n\n\\section{Final Results}\nBefore, we present a combined analysis of all constraints, let us turn our attention to the direct detection of VDM in the model. In the model, at the tree level, a VDM particle can interact elastically with a nucleon either through $H_1$ or via $H_2$ exchange\\cite{YaserAyazi:2019caf,YaserAyazi:2018lrv}. Presently, the XENON1T experiment \\cite{XENON:2018voc} excludes new parameter space for the WIMP-nucleon spin-independent elastic scatter cross-section above 6 GeV with a minimum of $ 4.1\\times10^{-47} cm^{2}$ at 30 GeV. We restrict the model with these results. The direct detection restrictions and constraints discussed in the previous sections are summarized in figure.~\\ref{final}.\nThe cross points show allowed region consistent with relic density, W-mass anomaly, direct detection as well as the invisible decay rate.\n\nThe outcome of imposing these experimental constraints on the model is for a large portion of VDM mass values, narrow region of scalar mediator $H_2$ ($100~\\rm GeV \\lesssim M_{H_2}\\lesssim 124~\\rm GeV$), and $0124~\\rm GeV$ the model is respectively excluded by invisible Higgs upper limit and W-mass CDF-II measurement.\n\n\\begin{figure\n\t\\begin{center}\n\t\t\\centerline{\\hspace{0cm}\\epsfig{figure=final.eps,width=12cm}}\n\t\t\\centerline{\\vspace{-0.2cm}}\n\t\t\\caption{Final results for allowed ranges of parameters of the model. The cross points depict allowed region which is consistent with relic density, direct detection, W mass anomaly and invisible decay rate.} \\label{final}\n\t\\end{center}\n\\end{figure}\n\nIn section.~3, we analyse RGEs of coupling of the model. We show that adding a new VDM and scalar mediator, the RGEs will be modified. It is interesting to see behaviour of RGEs of couplings in Planck scale. We solve the RG equations numerically and determine the RG evolution of the couplings of the models. For input parameters, we pick benchmark points for parameters of the model that are consistent with all constraints considered previously in the paper. Similar analyses with different input parameters have been performed for $U(1)$ extension of SM in Ref\\cite{Duch:2015jta}.\tThe running of couplings up to the Planck scale have been shown in Figures.(\\ref{RGE}~.a-c). In Figure.~(\\ref{RGE}~.d), we compared running of $\\lambda_H$ in the model with SM Higgs coupling. As it is known, SM Higgs coupling will be negative for $\\mu> 10^{10}~\\rm GeV$. This means, it is not possible to establish all three conditions (perturbativity, vacuum stability and positivity) simultaneously in any scale. It is remarkable that the SM stability problem (positivity of $\\lambda_H$) is solved in the model. This issue arises that $\\lambda_{SH}$ changes very little in our model. This leads small changes for $\\lambda_H$ and as a result, $\\lambda_H$ remains positive until the Planck scale.\n\n\n\\begin{figure\n\t\\begin{center}\n\t\t\\centerline{\\hspace{0cm}\\epsfig{figure=RGE-Plot.eps,width=8cm}\\hspace{0.5cm}\\hspace{0cm}\\epsfig{figure=RGE-Plot1.eps,width=8cm}}\n\t\t\\centerline{\\vspace{0.2cm}\\hspace{1.5cm}(a)\\hspace{8cm}(b)}\n\t\t\\centerline{\\hspace{0cm}\\epsfig{figure=RGE-Plot2.eps,width=8cm}\\hspace{0.5cm}\\hspace{0cm}\\epsfig{figure=plpt.eps,width=8cm}}\t\t\n\t\t\\centerline{\\vspace{0.2cm}\\hspace{1.5cm}(c)\\hspace{8cm}(d)}\n\t\t\\centerline{\\vspace{-0.2cm}}\n\t\t\\caption{Running of couplings of the model up to Planck scale. We select sample point in which all the experimental constraints considered in the paper are satisfied.} \\label{RGE}\n\t\\end{center}\n\\end{figure}\n\n\\section{Conclusions} \\label{sec7}\n\nWe proposed a model to explain the W boson mass anomaly reported by the CDF-II collaboration. We studied an $U(1)$ extension of the SM including a VDM candidate and a scalar mediator. In the model, there is no kinetic mixing between the VDM field and SM Z-boson, but scalar field exchange between SM and dark side. To explain the W mass anomaly one needs extra degrees of freedom that affects on W-boson mass. In the model, the one-loop corrections induced by the new\nscalar can shift the W boson mass. We have also imposed constraints on the Higgs mixing angle and other parameters of the model by investigating of relic density of DM, invisible Higgs decay mode at LHC and direct detection of DM. We have shown that the model for a large part of VDM mass values, scalar mediator mass range $100~\\rm GeV \\lesssim M_{H_2}\\lesssim 124~\\rm GeV$ and $00,\n\\end{align*}\nwhere ${\\mathcal{E}}_{p,q}(t)=e^{-\\frac{p}{t}-\\frac{q}{1-t}}$.\nFor $p=q$ we denote $B_{p,q}$ by $B_{p}$ and for $p=q=0$, we get classical Beta function defined by\n\\begin{eqnarray}\\label{beta}\nB(x,y)=\\int\\limits_{0}^{1}t^{x-1}(1-t)^{y-1}dt, (\\Re(x)>0, \\Re(y)>0).\n\\end{eqnarray}\n\nFor the case $m-1<\\Re(\\mu)0$ and $\\Re(q)>0$. It is clear that when $\\lambda=\\rho$, then (\\ref{FEfrac}) reduce to (\\ref{Efrac}).\\\\\n\nThe Gauss hypergeometric function which is defined (see \\cite{Rainville1960}) as\n\\begin{eqnarray}\\label{Hyper}\n_2F_1(\\sigma_1,\\sigma_2;\\sigma_3;z)=\\sum\\limits_{n=0}^{\\infty}\\frac{(\\sigma_1)_n(\\sigma_2)_n}{(\\sigma_3)_n}\\frac{z^n}{n!}, (|z|<1),\n\\end{eqnarray}\n $\\Big(\\sigma_1, \\sigma_2, \\sigma_3\\in\\mathbb{C}$ and $\\sigma_3\\neq0,-1,-2,-3,\\cdots\\Big)$.\n The integral representation of hypergeometric hypergeometric function is defined by\n\\begin{eqnarray}\\label{Ihyper}\n_2F_1(\\sigma_1,\\sigma_2;\\sigma_3;z)=\\frac{\\Gamma(\\sigma_3)}{\\Gamma(\\sigma_2)\\Gamma(\\sigma_3-\\sigma_2)}\n\\int_0^1t^{\\sigma_2-1}(1-t)^{\\sigma_3-\\sigma_2-1}(1-zt)^{-\\sigma_1}dt,\n\\end{eqnarray}\n$\\Big(\\Re(\\sigma_3)>\\Re(\\sigma_2)>0, |\\arg(1-z)|<\\pi\\Big)$.\\\\\nThe Appell series or bivariate hypergeometric series and its integral representation is respectively defined by\n\\begin{eqnarray}\\label{CAppell}\nF_{1}(\\sigma_1,\\sigma_2,\\sigma_3;\\sigma_4;x, y)=\\sum\\limits_{m,n=0}^{\\infty}\\frac{(\\sigma_1)_{m+n}(\\sigma_2)_{m}(\\sigma_3)_{n}x^{m}y^{n}}{(\\sigma_4)_{m+n}m!n!};\n\\end{eqnarray}\n for all $ \\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4\\in \\mathbb{C}, \\sigma_4\\neq 0,-1,-2,-3,\\cdots, \\quad |x|<1, |y|<1$.\\\\\n\\begin{align}\\label{FIAppell}\nF_{1}\\Big(\\sigma_1,\\sigma_2, \\sigma_3,\\sigma_4;x,y\\Big)&=\\frac{\\Gamma(\\sigma_4)}{\\Gamma(\\sigma_1)\\Gamma(\\sigma_4-\\sigma_1)}\\notag\\\\\n&\\times\\int\\limits_{0}^{1}t^{\\sigma_1-1}(1-t)^{\\sigma_4-\\sigma_1-1}(1-xt)^{-\\sigma_2}(1-yt)^{-\\sigma_3}dt\n\\end{align}\n$\\Re(\\sigma_4)>\\Re(\\sigma_1)>0$, $|\\arg(1-x)|<\\pi$ and $|\\arg(1-y)|<\\pi$.\\\\\n\nChaudhry et al. \\cite{Chaudhry1997} introduced the extended Beta function is defined by\n\\begin{eqnarray}\\label{Ebeta}\nB(\\sigma_1,\\sigma_2;p)=B_p(\\sigma_1,\\sigma_2)=\\int\\limits_{0}^{1}t^{\\sigma_1-1}(1-t)^{\\sigma_2-1}\ne^{-\\frac{p}{t(1-t)}}dt\n\\end{eqnarray}\n(where $\\Re(p)>0, \\Re(\\sigma_1)>0, \\Re(\\sigma_2)>0$) respectively. When $p=0$, then $B(\\sigma_1,\\sigma_2;0)=B(\\sigma_1,\\sigma_2)$.\\\\\nThe extended hypergeometric function introduced in \\cite{Chaudhrya2004} by using the definition of extended Beta function $B_p(\\delta_1,\\delta_2)$ as follows:\n\\begin{eqnarray}\\label{Ehyper}\nF_p(\\sigma_1,\\sigma_2;\\sigma_3;z)=\\sum\\limits_{n=0}^{\\infty}\\frac{B_p(\\sigma_2+n, \\sigma_3-\\sigma_2)}{B(\\sigma_2, \\sigma_3-\\sigma_2)}(\\sigma_1)_n\\frac{z^n}{n!},\n\\end{eqnarray}\nwhere $p\\geq0$ and $\\Re(\\sigma_3)>\\Re(\\sigma_2)>0$, $|z|<1$.\\\\\nIn the same paper, they defined the following integral representations of extended hypergeometric and confluent hypergeometric functions as\n\\begin{align}\\label{IEhyper}\nF_p(\\sigma_1,\\sigma_2;\\sigma_3;z)&=\\frac{1}{B(\\sigma_2, \\sigma_3-\\sigma_2)}\\notag\\\\\n&\\times\\int_0^1t^{\\sigma_2-1}(1-t)^{\\sigma_3-\\sigma_2-1}(1-zt)^{-\\sigma_1}\\exp\\Big(\\frac{-p}{t(1-t)}\\Big)dt,\n\\end{align}\n$$\\Big(p\\geq0, \\Re(\\sigma_3)>\\Re(\\sigma_2)>0, |\\arg(1-z)|<\\pi\\Big).$$\nThe extended Appell's function is defined by (see \\cite{Ozerslan})\n\\begin{eqnarray}\\label{EAppell}\nF_1(\\sigma_1,\\sigma_2,\\sigma_3;\\sigma_4;x,y;p)=\\sum\\limits_{m,n=0}^{\\infty}\\frac{B_p(\\sigma_1+m+n, \\sigma_4-\\sigma_1)}{B(\\sigma_1, \\sigma_4-\\sigma_1)}(\\sigma_2)_m(\\sigma_3)_n\\frac{x^my^n}{m!n!}\n\\end{eqnarray}\nwhere $p\\geq0$ and $\\Re(\\sigma_4)>\\Re(\\sigma_1)>0$ and $|x|,|y|<1$.\\\\\n \\\"{O}zerslan and \\\"{O}zergin \\cite{Ozerslan} defined its integral representation by\n\\begin{align}\\label{IEAppell}\nF_1(\\sigma_1,\\sigma_2,\\sigma_3;\\sigma_4;z;p)&=\\frac{1}{B(\\sigma_1, \\sigma_4-\\sigma_1)}\\notag\\\\\n&\\times\\int_0^1t^{\\sigma_1-1}(1-t)^{\\sigma_4-\\sigma_1-1}(1-xt)^{-\\sigma_2}(1-yt)^{-\\sigma_3}\n\\exp\\Big(\\frac{-p}{t(1-t)}\\Big)dt,\\\\\n&\\Big(\\Re(p)>0, \\Re(\\sigma_4)>\\Re(\\sigma_1)>0, |\\arg(1-x)|<\\pi, |\\arg(1-y)|<\\pi\\Big)\\notag.\n\\end{align}\n\nIt is clear that when $p=0$, then the equations (\\ref{Ehyper})-(\\ref{IEAppell}) reduce to the well known hypergeometric, confluent hypergeometric and Appell's series and their integral representation respectively.\\\\\nVery recently Shadab et al. \\cite{Choi2017} introduced a new and modified extension of Beta function as:\n\\begin{eqnarray}\\label{Cbeta}\nB^{\\alpha}_{p}(\\sigma_1,\\sigma_2)=\\int_0^1t^{\\sigma_1-1}(1-t)^{\\sigma_2-1}E_{\\alpha}\\Big(-\\frac{p}{t(1-t)}\\Big)dt,\n\\end{eqnarray}\nwhere $\\Re(\\sigma_1)>0$, $\\Re(\\sigma_2)>0$ and $E_{\\alpha}\\Big(.\\Big)$ is the Mittag-Leffler function defined by\n\\begin{eqnarray}\nE_{\\alpha}\\Big(z\\Big)=\\sum_{n=0}^\\infty\\frac{z^n}{\\Gamma(\\alpha n+1)}.\n\\end{eqnarray}\nObviously, when $\\alpha=1$ then $B^{1}_{p}(x,y)=B_p(x,y)$ is the extended Beta function (see\\cite{Chaudhry1997}). Similarly, when when $\\alpha=1$ and $p=0$, then $B^{1}_{0}(x,y)=B_0(x,y)$ is the classical Beta function.\\\\\nShadab et al. \\cite{Choi2017} also defined extended hypergeometric function and its integral representation\n\\begin{align}\\label{pqhyper}\nF_{p}^{\\alpha}(\\sigma_1,\\sigma_2;\\sigma_3;z)={}_2F_{1}\\Big(\\sigma_1,\\sigma_2;\\sigma_3;z;p,\\alpha\\Big)\\notag&=\n\\sum_{n=0}^\\infty(\\sigma_1)_n\\frac{B_{p}^{\\alpha}(\\sigma_2+n,\\sigma_3-\\sigma_2)}{B(\\sigma_2,\\sigma_3-\\sigma_2)}\n\\frac{z^n}{n!}\\notag\\\\\n=&\\sum_{n=0}^\\infty(\\sigma_1)_n\\frac{B(\\sigma_2+n,\\sigma_3-\\sigma_2; p,\\alpha)}{B(\\sigma_2,\\sigma_3-\\sigma_2)}\n\\frac{z^n}{n!}\n\\end{align}\nwhere $p,\\alpha\\geq0$, $\\sigma_1,\\sigma_2,\\sigma_3\\in\\mathbb{C}$ and $|z|<1$.\n \\begin{align}\\label{pqIhyper}\nF_{p}^{\\alpha}(\\sigma_1,\\sigma_2;\\sigma_3;z)&=\\frac{1}{\\beta(\\sigma_2;\\sigma_3-\\sigma_2)}\\notag\\\\\n&\\times\\int_0^1t^{\\sigma_2-1}\n(1-t)^{\\sigma_3-\\sigma_2-1}(1-tz)^{-\\sigma_1}E_{\\alpha}\\Big(-\\frac{p}{t(1-t)}\\Big)dt,\n\\end{align}\nwhere $\\Re(p)>0$, $\\Re(\\alpha)>0$, $\\Re(\\sigma_3)>\\Re(\\sigma_2)>0$. Obviously when $\\alpha=1$, then the hypergeometric function (\\ref{pqhyper}) will reduce to the extended hypergeometric function (\\ref{Ehyper}) and similarly when $\\alpha=1$ and $p=0$ then the hypergeometric function (\\ref{pqhyper}) will reduce to the hypergeometric function (\\ref{Hyper}).\\\\\n\nFor various extensions and generalizations of Beta function and hypergeometric functions the interested readers may refer to the recent work of researchers (see e. g., \\cite{Choi2014,Mubeen2017,Ozerslan,Ozergin}).\n\\section{ extension of Appell's functions and its integral representations}\n\nWe start the section by deriving the relation of \\eqref{Cbeta} with multi-index Mittag-Leffler function \\cite{Kir1} as follows:\n\n\n\\begin{proposition}\nFor $\\Re(p), \\Re(\\sigma_1), \\Re(\\sigma_2), \\Re(\\alpha) > 0$ the following relation holds true:\n\\begin{align}\nB^{\\alpha}_{p}(\\sigma_1,\\sigma_2)=\\frac{\\pi sin\\pi(\\sigma_1+\\sigma_2)}{(sin\\pi\\sigma_1)(sin\\pi\\sigma_2)}E_{(\\alpha,1), (1, 1-\\sigma_1), (1,1-\\sigma_2)}^{(2,1-\\sigma_1-\\sigma_2)}(-p)\n\\end{align}\nwhere $E_{(\\alpha,1), (1, 1-\\sigma_1), (1,1-\\sigma_2)}^{(2,1-\\sigma_1-\\sigma_2)}(-p)$ is the multi-index Mittag-Leffler function \\cite{Kir1}.\n\\end{proposition}\n\n{\\bf Proof}\nUsing the definition of Beta function and reduction theorem of Gamma function, we get\n\\begin{align*}\nB^{\\alpha}_{p}(\\sigma_1,\\sigma_2)&=\\sum_{n=0}^{\\infty}\\frac{(-p)^n}{\\Gamma(\\alpha n+1)}\\int_{0}^{1}t^{\\sigma_1-1}(1-t)^{\\sigma_2-1}\\frac{1}{t^n(1-t)^n}dt\\\\\n&=\\sum_{n=0}^{\\infty}\\frac{(-p)^n}{\\Gamma(\\alpha n+1)}\\int_{0}^{1}t^{\\sigma_1-n-1}(1-t)^{\\sigma_2-n-1}dt\\\\\n&=\\sum_{n=0}^{\\infty}\\frac{(-p)^n}{\\Gamma(\\alpha n+1)}B(\\sigma_1-n, \\sigma_2-n)\\\\\n&=\\sum_{n=0}^{\\infty}\\frac{(-p)^n}{\\Gamma(\\alpha n+1)}\\frac{\\Gamma(\\sigma_1-n)\\Gamma(\\sigma_2-n)}{\\Gamma(\\sigma_1+\\sigma_2-2n)}\\\\\n&=\\frac{\\Gamma(-\\sigma_1)\\Gamma(1+\\sigma_1)\\Gamma(-\\sigma_2)\\Gamma(1+\\sigma_2)}{\\Gamma(-\\sigma_1-\\sigma_2)\\Gamma(1+\\sigma_1+\\sigma_2)}\\\\\n&\\times \\sum_{n=0}^{\\infty} \\frac{\\Gamma(2n+1-\\sigma_1-\\sigma_2)(-p)^n}{\\Gamma(\\alpha n+1)\\Gamma(n+1-\\sigma_1)\\Gamma(n+1-\\sigma_2)}\n\\end{align*}\nNow, using the Euler's reflection formula on Gamma function,\n\\begin{align}\\label{rel1}\n{\\Gamma(r)\\Gamma(1-r)}=\\frac{\\pi}{\\sin(\\pi r)},\n\\end{align}\nwe get the desired result.\n\nNext, we used the definition (\\ref{Cbeta}) and consider the following modified extension Appell's functions.\n\\begin{definition}\\label{def2}\nThe modified extended Appell's function $F_1$ is defined by\n\\begin{align}\\label{Appell}\nF_{1,p}^{\\alpha}(\\sigma_1,\\sigma_2,\\sigma_3;\\sigma_4;x,y)&=\nF_{1}(\\sigma_1,\\sigma_2,\\sigma_3;\\sigma_4;x,y;p,\\alpha)\\notag\\\\&=\\sum_{m,n=0}^\\infty(\\sigma_2)_m(\\sigma_3)_n\n\\frac{B_{p}^{\\alpha}(\\sigma_1+m+n,\\sigma_4-\\sigma_1)}{B(\\sigma_1,\\sigma_4-\\sigma_1)}\n\\frac{x^m}{m!}\\frac{y^n}{n!}\\notag\\\\\n=&\\sum_{m,n=0}^\\infty(\\sigma_2)_m(\\sigma_3)_n\\frac{B(\\sigma_1+m+n,\\sigma_4-\\sigma_1; p,\\alpha)}{B(\\sigma_1,\\sigma_4-\\sigma_1)}\n\\frac{x^m}{m!}\\frac{y^n}{n!}\n\\end{align}\nwhere $p,\\alpha\\geq0$, $\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4\\in\\mathbb{C}$ and $|x|<1$, $|y|<1$.\n\\end{definition}\n\\begin{remark}\nSetting $\\alpha=1$ in (\\ref{Appell}), then we get the extended Appell's functions (see \\cite{Ozerslan}).\n\\end{remark}\nNow, we derive the following proposition\n\\begin{proposition} For $p,q >0$, $00$, $\\Re(\\alpha)>0$, $\\Re(\\sigma_4)>\\Re(\\sigma_1)>0$.\n\\end{theorem}\n\\begin{proof}\nUsing the definition (\\ref{Cbeta}) in (\\ref{Appell}), we have\n\\begin{align}\\label{PS2}\nF_1\\Big(\\sigma_1,\\sigma_2,\\sigma_3;\\sigma_4;x,y;p,\\alpha\\Big)&=\\frac{1}{B(\\sigma_1;\\sigma_4-\\sigma_1)}\n\\int_0^1t^{\\sigma_1-1}(1-t)^{\\sigma_4-\\sigma_1-1}\\notag\\\\\n\\times&\\Big(\\sum_{m,n=0}^\\infty\\frac{(\\sigma_2)_m(\\sigma_3)_n(tx)^m(ty)^n}{m!n!}\\Big) dt.\n\\end{align}\n Since\n\\begin{align}\\label{PS}\n\\sum_{m,n=0}^\\infty\\frac{(\\sigma_2)_m(\\sigma_3)_n(x)^m(y)^n}{m!n!}=(1-tx)^{-\\sigma_2}(1-ty)^{-\\sigma_3}.\n\\end{align}\n\nThus by using (\\ref{PS}) in (\\ref{PS2}), we get the desired result.\n\\end{proof}\n\\section{Extension of fractional derivative operator}\nIn this section, we define a new and modified extension of Riemann-Liouville fractional derivative and obtain its related results.\n\\begin{definition}\\label{frac}\n\\begin{eqnarray}\\label{eq7}\n\\mathfrak{D}_{z;p}^{\\mu;\\alpha}\\{f(z)\\}=\\frac{1}{\\Gamma(-\\mu)}\\int_0^zf(t)(z-t)^{-\\mu-1}E_\\alpha\\Big(-\\frac{pz^2}{t(z-t)}\\Big)dt, \\Re(\\mu)<0.\n\\end{eqnarray}\nFor the case $m-1<\\Re(\\mu)0$ and assume that the function $f(z)$ is analytic at the origin with its Maclaurin expansion given by\n$f(z)=\\sum_{n=0}^{\\infty} a_n z^n$ where $|z|<\\delta$ for some $\\delta\\in \\mathbb{R^+}$. Then\n\\begin{align}\n\\mathfrak{D}_{z;p}^{\\mu;\\alpha}\\{f(z)\\}=\\mathfrak{D}_{z}^{\\mu}\\{f(z);p,\\alpha\\}&=\\sum_{n=0}^\\infty a_n\\mathfrak{D}_{z}^{\\mu}\\{z^n;p,\\alpha\\}\\notag\\\\\n=&\\frac{1}{\\Gamma(-\\mu)}\\sum_{n=0}^\\infty a_nB_p^\\alpha(n+1,-\\mu)z^{n-\\mu}.\n\\end{align}\n\\end{theorem}\n\n\\begin{proof}\nUsing the series expansion of the function $f(z)$ in (\\ref{eq7}) gives\n\\begin{eqnarray*}\n\\mathfrak{D}_{z}^{\\mu}\\{f(z);p,\\alpha\\}=\\frac{1}{\\Gamma(-\\mu)}\\int_0^z\\sum_{n=0}^\\infty a_nt^n(z-t)^{-\\mu-1}\\,\nE_\\alpha\\Big(-\\frac{pz^2}{t(z-t)}\\Big) dt.\n\\end{eqnarray*}\nThe series is uniformly convergent on any closed disk centered at the origin with its radius smaller than $\\delta$, so does on the line segment from $0$ to a fixed $z$ for $|z|<\\delta$. Thus it guarantee terms by terms integration as follows\n\\begin{eqnarray*}\n\\mathfrak{D}_{z}^{\\mu}\\{f(z);p,\\alpha\\}&=&\\sum_{n=0}^\\infty a_n\\Big\\{\\frac{1}{\\Gamma(-\\mu)}\\int_0^z t^n(z-t)^{-\\mu-1}\\,\nE_\\alpha\\Big(-\\frac{pz^2}{uz(z-uz)}\\Big) dt\\\\\n&=&\\sum_{n=0}^\\infty a_n\\mathfrak{D}_{z}^{\\mu}\\{z^n;p,\\alpha\\}.\n\\end{eqnarray*}\nNow, applying Theorem \\ref{th1}, we get\n\\begin{eqnarray*}\n\\mathfrak{D}_{z}^{\\mu}\\{f(z);p,\\alpha\\}\n&=&\\frac{1}{\\Gamma(-\\mu)}\\sum_{n=0}^\\infty a_nB_p^\\alpha(n+1,-\\mu)z^{n-\\mu}, \\Re(\\mu)<0.\n\\end{eqnarray*}\nwhich is the required proof.\n\\end{proof}\n\\begin{example}\nThe following result holds true:\n\\begin{align}\n\\mathfrak{D}_{z}^{\\mu}\\{e^z;p,\\alpha\\}=\\frac{z^{-\\mu}}{\\Gamma(-\\mu)}\\sum_{n=0}^\\infty B_p^\\alpha(n+1,-\\mu)\\frac{z^n}{n!}.\n\\end{align}\nUsing the power series of $\\exp(z)$ and applying Theorem \\ref{th2a}, we have\n\\begin{align}\n\\mathfrak{D}_{z}^{\\mu}\\{e^z;p,\\alpha\\}=\\sum_{n=0}^\\infty \\frac{1}{n!}\\mathfrak{D}_{z}^{\\mu}\\{z^{n};p,\\alpha\\}.\n\\end{align}\nNow, applying Theorem \\ref{th1}, we get the desired result.\n\\end{example}\n\\begin{theorem}\\label{th3}\nThe following formula holds true:\n\\begin{eqnarray}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta-1}(1-z)^{-\\beta};p,\\alpha\\}=\n\\frac{\\Gamma(\\eta)}{\\Gamma(\\mu)}z^{\\mu-1}\\,_2F_{1;p}^{\\alpha}\n\\Big(\\beta, \\eta;\\mu;z\\Big),\n\\end{eqnarray}\nwhere $\\Re(\\mu)> \\Re(\\eta)>0$ and $|z|<1$.\n\\end{theorem}\n\\begin{proof}\nBy direct calculation, we have\n\\begin{align*}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta-1}(1-z)^{-\\beta};p,\\alpha\\}\n&=\\frac{1}{\\Gamma(\\mu-\\eta)}\\int_0^zt^{\\eta-1}(1-t)^{-\\beta}(z-t)^{\\mu-\\eta-1}E_\\alpha\\Big(-\\frac{pz^2}{t(z-t)}\\Big) dt\\\\\n&=\\frac{z^{\\mu-\\eta-1}}{\\Gamma(\\mu-\\eta)}\\int_0^zt^{\\eta-1}(1-t)^{-\\beta}(1-\\frac{t}{z})^{\\mu-\\eta-1}\nE_\\alpha\\Big(-\\frac{pz^2}{t(z-t)}\\Big) dt.\n\\end{align*}\nSubstituting $t=zu$ in the above equation, we get\n \\begin{align*}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta-1}(1-z)^{-\\beta};p,\\alpha\\}\n&=\\frac{z^{\\mu-1}}{\\Gamma(\\mu-\\eta)}\\int_0^1u^{\\eta-1}(1-uz)^{-\\beta}(1-u)^{\\mu-\\eta-1}E_\\alpha\\Big(-\\frac{pz^2}{uz(z-uz)}\\Big) du\\\\\n=&\\frac{z^{\\mu-1}}{\\Gamma(\\mu-\\eta)}\\int_0^1u^{\\eta-1}(1-uz)^{-\\beta}(1-u)^{\\mu-\\eta-1}E_\\alpha\\Big(-\\frac{p}{u(1-u)}\\Big) du\n\\end{align*}\nUsing (\\ref{pqIhyper}) and after simplification we get the required proof.\n\\end{proof}\n\\begin{theorem}\\label{th4}\nThe following formula holds true:\n\\begin{align}\\label{fd3}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta-1}(1-az)^{-\\alpha}(1-bz)^{-\\beta};p,\\alpha\\}\n=\\frac{\\Gamma(\\eta)}{\\Gamma(\\mu)}z^{\\mu-1}F_{1}\\Big(\\eta,\\alpha,\\beta;\\mu;az,bz;p,\\alpha\\Big),\n\\end{align}\nwhere $\\Re(\\mu)> \\Re(\\eta)>0$, $\\Re(\\alpha)>0$, $\\Re(\\beta)>0$, $|az|<1$ and $|bz|<1$.\n\\end{theorem}\n\\begin{proof}\nConsider the following power series expansion\n\\begin{eqnarray*}\n(1-az)^{-\\alpha}(1-bz)^{-\\beta}=\\sum_{m=0}^\\infty\\sum_{n=0}^\\infty(\\alpha)_m(\\beta)_n\\frac{(az)^m}{m!}\\frac{(bz)^n}{n!}.\n\\end{eqnarray*}\nNow, applying Theorem \\ref{th3}, we obtain\n\\begin{align*}\n&\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta-1}(1-az)^{-\\alpha}(1-bz)^{-\\beta};p,\\alpha\\}\\\\\n&=\\sum_{m=0}^\\infty\\sum_{n=0}^\\infty(\\alpha)_m(\\beta)_n\\frac{(a)^m}{m!}\\frac{(b)^n}{n!}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta+m+n-1};p,\\alpha\\}.\n\\end{align*}\nUsing Theorem \\ref{th1}, we have\n\\begin{align*}\n&\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta-1}(1-az)^{-\\alpha}(1-bz)^{-\\beta};p,\\alpha\\}\\\\\n&=\\sum_{m=0}^\\infty\\sum_{n=0}^\\infty(\\alpha)_m(\\beta)_n\\frac{(a)^m}{m!}\\frac{(b)^n}{n!}\n\\frac{B_{p}^{\\alpha}(\\eta+m+n,\\mu-\\eta)}\n{\\Gamma(\\mu-\\eta)}z^{\\mu+m+n-1}.\n\\end{align*}\nNow, applying (\\ref{Appell}), we get\n\\begin{align*}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta-1}(1-az)^{-\\alpha}(1-bz)^{-\\beta};p,\\alpha\\}\n=\\frac{\\Gamma(\\eta)}{\\Gamma(\\mu)}z^{\\mu-1}F_{1}\\Big(\\eta,\\alpha,\\beta;\\mu;az,bz;p,\\alpha\\Big).\n\\end{align*}\n\\end{proof}\n\\begin{theorem}\\label{th5}\nThe following Mellin transform formula holds true:\n\\begin{eqnarray}\\label{Mellin}\nM\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}(z^{\\eta});p\\rightarrow r,\\Big\\}=\\frac{\\pi}{\\sin(\\pi r)}\\frac{z^{\\eta-\\mu}}{\\Gamma(-\\mu)\\Gamma(1-r\\alpha)}B(\\eta+r+1,-\\mu+r),\n\\end{eqnarray}\nwhere $\\Re(\\eta)>-1$, $\\Re(\\mu)<0$, $\\Re(r)>0$.\n\\end{theorem}\n\\begin{proof}\nApplying the Mellin transform on definition (\\ref{eq7}), we have\n\\begin{align*}\n&M\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}(z^{\\eta});p\\rightarrow r\\Big\\}=\\int_0^\\infty p^{r-1}\\mathfrak{D}_{z;p}^{\\mu;\\alpha}(z^{\\eta})dp\\\\\n&=\\frac{1}{\\Gamma(-\\mu)}\\int_0^\\infty p^{r-1}\\Big\\{\\int_0^z t^\\eta(z-t)^{-\\mu-1}\\,\nE_\\alpha\\Big(-\\frac{pz^2}{t(z-t)}\\Big)dt\\Big\\}dp \\\\\n&=\\frac{z^{-\\mu-1}}{\\Gamma(-\\mu)}\\int_0^\\infty p^{r-1}\\Big\\{\\int_0^z t^\\eta(1-\\frac{t}{z})^{-\\mu-1}\\,\nE_\\alpha\\Big(-\\frac{pz^2}{t(z-t)}\\Big)dt\\Big\\}dp \\\\\n&=\\frac{z^{\\eta-\\mu}}{\\Gamma(-\\mu)}\\int_0^\\infty p^{r-1}\\Big\\{\\int_0^1 u^\\eta(1-u)^{-\\mu-1}\\,\nE_\\alpha\\Big(-\\frac{p}{u(1-u)}\\Big)du\\Big\\}dp\n\\end{align*}\nFrom the uniform convergence of the integral, the order of integration can be interchanged. Thus, we have\n\\begin{align}\\label{Th-eqn-Mellin}\nM\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}(z^{\\eta});p\\rightarrow r\\Big\\}&=\\frac{z^{\\eta-\\mu}}{\\Gamma(-\\mu)}\\notag\\\\\n&\\times\\int_0^1 u^\\eta(1-u)^{-\\mu-1}\\Big(\\int_0^\\infty p^{r-1}\\,E_\\alpha\\Big(-\\frac{p}{u(1-u)}\\Big)dp\\Big)du.\n\\end{align}\nLetting $v=\\frac{p}{u(1-u)}$, \\eqref{Th-eqn-Mellin} reduces to\n\\begin{align*}\n&M\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}(z^{\\eta});p\\rightarrow r\\Big\\}=\\frac{z^{\\eta-\\mu}}{\\Gamma(-\\mu)}\\int_0^1 u^{\\eta+r}(1-u)^{-\\mu+r-1}\\Big(\\int_0^\\infty v^{r-1}\\,E_\\alpha\\Big(-v\\Big)dv\\Big)du.\n\\end{align*}\nBy using the following formula,\n\\begin{align}\\label{gam}\n\\int_{0}^{\\infty}v^{r-1}E_{\\alpha,\\gamma}^{\\delta}(-wv)dv=\\frac{\\Gamma(r)\\Gamma(\\delta-r)}{\\Gamma(\\delta)w^{r}\\Gamma(\\gamma-r\\alpha)},\n\\end{align}\n for $\\gamma=\\delta=1$ and $w=1$, we have\n\\begin{align*}\nM\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}(z^{\\eta});p\\rightarrow r\\Big\\}&=\\frac{z^{\\eta-\\mu}\\Gamma(r)\\Gamma(1-r)}{\\Gamma(-\\mu)\\Gamma(1-r\\alpha)}\\int_0^1 u^{\\eta+r}(1-u)^{-\\mu+r-1}du\\\\\n&=\\frac{z^{\\eta-\\mu}\\Gamma(r)\\Gamma(1-r)}{\\Gamma(-\\mu)\\Gamma(1-r\\alpha)}B(\\eta+r+1,-\\mu+r),\n\\end{align*}\nNow, using \\eqref{rel1} we get the desired result.\n\\end{proof}\n\\begin{theorem}\\label{th6}\nThe following Mellin transform formula holds true:\n\\begin{align}\\label{Mellin1}\nM\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}((1-z)^{\\alpha});p\\rightarrow r\\Big\\}&=z^{-\\mu}\\frac{\\pi}{\\sin(\\pi r)}\\frac{B(1+r,-\\mu+r)}{\\Gamma(-\\mu)\\Gamma(1-r\\alpha)}\\notag\\\\\n&\\times_2F_1\\Big(\\lambda,r+1;1-\\mu+2r;z\\Big),\n\\end{align}\nwhere $\\Re(p)>0$, $\\Re(\\mu)<0$, $\\Re(r)>0$.\n\\end{theorem}\n\\begin{proof}\nApplying Theorem \\ref{th5} with $\\eta=n$, we can write\n\\begin{align*}\nM\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}((1-z)^{\\lambda});p\\rightarrow r\\Big\\}&=\n\\sum_{n=0}^\\infty\\frac{(\\lambda)_n}{n!}\nM\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}(z^{n});p\\rightarrow r\\Big\\}\\\\\n&=\\frac{\\Gamma(r)\\Gamma(1-r)}{\\Gamma(-\\mu)\\Gamma(1-r\\alpha)}\\sum_{n=0}^\\infty\\frac{(\\lambda)_n}{n!}\nB(n+r+1,-\\mu+r)z^{n-\\mu}\\\\\n&=z^{-\\mu}\\frac{\\Gamma(r)\\Gamma(1-r)}{\\Gamma(-\\mu)\\Gamma(1-r\\alpha)}\\sum_{n=0}^\\infty\nB(n+r+1,-\\mu+r)\\frac{(\\lambda)_nz^{n}}{n!}.\n\\end{align*}\nIn view of \\eqref{rel1}, we obtain the required result.\n\\end{proof}\n\\section{Generating relations}\nIn this section, we derive generating relations of linear and bilinear type for the extended hypergeometric functions.\n\\begin{theorem}\nThe following generating relation holds true:\n\\begin{align}\\label{fd4}\n\\sum_{n=0}^\\infty\\frac{(\\lambda)_n}{n!}\\,_2F_{1;p}^{\\alpha}\\Big(\\lambda+n,\\beta;\\gamma;z\\Big)t^n\n=(1-t)^{-\\lambda}\\,_2F_{1;p}^{\\alpha}\\Big(\\lambda,\\beta;\\gamma;\\frac{z}{1-t}\\Big),\n\\end{align}\nwhere $|z|<\\min(1, |1-t|)$, $\\Re(\\lambda)>0$, $\\Re(\\alpha)>0$, $\\Re(\\gamma)>\\Re(\\beta)>0$.\n\n\\end{theorem}\n\\begin{proof}\nConsider the following series identity\n\\begin{eqnarray*}\n[(1-z)-t]^{-\\lambda}=(1-t)^{-\\lambda}[1-\\frac{z}{1-t}]^{-\\lambda}.\n\\end{eqnarray*}\nThus, the power series expansion yields\n\\begin{eqnarray}\\label{fd5}\n\\sum_{n=0}^\\infty\\frac{(\\lambda)_n}{n!}(1-z)^{-\\lambda}\\Big(\\frac{t}{1-z}\\Big)^n=(1-t)^{-\\lambda}[1-\\frac{z}{1-t}]^{-\\lambda}.\n\\end{eqnarray}\nMultiplying both sides of (\\ref{fd5}) by $z^{\\beta-1}$ and then applying the operator $\\mathfrak{D}_{z;p}^{\\beta-\\gamma;\\alpha}$ on both sides, we have\n\\begin{align*}\n\\mathfrak{D}_{z;p}^{\\beta-\\gamma;\\alpha}\\Big[\\sum_{n=0}^\\infty\\frac{(\\lambda)_n}{n!}\n(1-z)^{-\\lambda}\\Big(\\frac{t}{1-z}\\Big)^nz^{\\beta-1}\\Big]=(1-t)^{-\\lambda}\\mathfrak{D}_{z;p}^{\\beta-\\gamma;\\alpha}\n\\Big[z^{\\beta-1}\\Big(1-\\frac{z}{1-t}\\Big)^{-\\lambda}\\Big].\n\\end{align*}\nInterchanging the order of summation and the operator $\\mathfrak{D}_{z;p}^{\\beta-\\gamma;\\alpha}$, we have\n\\begin{align*}\n\\sum_{n=0}^\\infty\\frac{(\\lambda)_n}{n!}\\mathfrak{D}_{z;p}^{\\beta-\\gamma;\\alpha}\\Big[\nz^{\\beta-1}(1-z)^{-\\lambda-n}\\Big]t^n=(1-t)^{-\\lambda}\\mathfrak{D}_{z;p}^{\\beta-\\gamma;\\alpha}\n\\Big[z^{\\beta-1}\\Big(1-\\frac{z}{1-t}\\Big)^{-\\lambda}\\Big].\n\\end{align*}\nThus by applying Theorem \\ref{th3}, we obtain the required result.\n\\end{proof}\n\\begin{theorem}\nThe following generating relation holds true:\n\\begin{align}\\label{fd6}\n\\sum_{n=0}^\\infty\\frac{(\\beta)_n}{n!}\\,_2F_{1;p}^{\\alpha}\\Big(\\delta-n,\\beta;\\gamma;z\\Big)t^n\n=(1-t)^{-\\beta}F_{1}\\Big(\\lambda,\\delta,\\beta;\\gamma;-\\frac{zt}{1-t};p,\\alpha\\Big),\n\\end{align}\nwhere $|z|<\\frac{1}{1+|t|}$, $\\Re(\\delta)>0$, $\\Re(\\beta)>0$, $\\Re(\\gamma)>\\Re(\\lambda)>0$.\n\n\\end{theorem}\n\\begin{proof}\nConsider the series identity\n\\begin{eqnarray*}\n[1-(1-z)t]^{-\\beta}=(1-t)^{-\\beta}\\Big[1+\\frac{zt}{1-t}\\Big]^{-\\beta}.\n\\end{eqnarray*}\nUsing the power series expansion to the left sides, we have\n\\begin{eqnarray}\\label{fd7}\n\\sum_{n=0}^\\infty\\frac{(\\beta)_n}{n!}(1-z)^nt^n=(1-t)^{-\\beta}\\Big[1-\\frac{-zt}{1-t}\\Big]^{-\\beta}.\n\\end{eqnarray}\nMultiplying both sides of (\\ref{fd7}) by $z^{\\alpha-1}(1-z)^{-\\delta}$ and applying the operator $\\mathfrak{D}_{z;p}^{\\lambda-\\gamma;\\alpha}$ on both sides, we have\n\\begin{eqnarray*}\n\\mathfrak{D}_{z;p}^{\\lambda-\\gamma;\\alpha}\\Big[\\sum_{n=0}^\\infty\\frac{(\\beta)_n}{n!}z^{\\alpha-1}(1-z)^{-\\delta+n}t^n\\Big]\n=(1-t)^{-\\beta}\\mathfrak{D}_{z;p}^{\\lambda-\\gamma;\\alpha}\\Big[z^{\\lambda-1}(1-z)^{-\\delta}\\Big(1-\\frac{-zt}{1-t}\\Big)^{-\\beta}\n\\Big],\n\\end{eqnarray*}\nwhere $\\Re(\\lambda)>0$ and $|zt|<|1-t|$,\nthus by Theorem \\ref{th2}, we have\n\\begin{eqnarray*}\n\\sum_{n=0}^\\infty\\frac{(\\beta)_n}{n!}\\mathfrak{D}_{z;p}^{\\lambda-\\gamma;\\alpha}\\Big[z^{\\lambda-1}(1-z)^{-\\delta+n}\\Big]t^n\n=(1-t)^{-\\beta}\\mathfrak{D}_{z;p}^{\\lambda-\\gamma;\\alpha}\\Big[z^{\\lambda-1}(1-z)^{-\\delta}\\Big(1-\\frac{-zt}{1-t}\\Big)^{-\\beta}\n\\Big].\n\\end{eqnarray*}\nApplying Theorem \\ref{th4} on both sides, we get the required result.\n\\end{proof}\n\n\\begin{theorem}\\label{th7}\nThe following result holds true:\n\\begin{eqnarray}\\label{fd8}\n\\mathfrak{D}_{z;p}^{\\eta-\\mu;\\alpha}\\Big[z^{\\eta-1}E_{\\gamma,\\delta}^{\\mu}(z)\\Big]=\\frac{z^{\\mu-1}}{\\Gamma(\\mu-\\eta)}\n\\sum_{n=0}^\\infty\\frac{(\\mu)_n}{\\Gamma(\\gamma n+\\delta)}B_{p}^{\\alpha}(\\eta+n,\\mu-\\eta)\\frac{z^n}{n!},\n\\end{eqnarray}\nwhere $\\gamma,\\delta,\\mu, \\alpha\\in\\mathbb{C}$, $\\Re(p)>0$, $\\Re(\\mu)>\\Re(\\eta)>0$ and $E_{\\gamma,\\delta}^{\\mu}(z)$ is Mittag-Leffler function (see \\cite{Prabhakar}) defined as:\n\\begin{eqnarray}\\label{fd9}\nE_{\\gamma,\\delta}^{\\mu}(z)=\\sum_{n=0}^\\infty\\frac{(\\mu)_n}{\\Gamma(\\gamma n+\\delta)}\\frac{z^n}{n!}, \\Re(\\gamma)>0.\n\\end{eqnarray}\n\\begin{proof}\nUsing (\\ref{fd9}) in (\\ref{fd8}), we have\n\\begin{eqnarray*}\n\\mathfrak{D}_{z;p}^{\\eta-\\mu;\\alpha}\\Big[z^{\\eta-1}E_{\\gamma,\\delta}^{\\mu}(z)\\Big]=\n\\mathfrak{D}_{z;p}^{\\eta-\\mu;\\alpha}\\Big[z^{\\eta-1}\\Big\\{\\sum_{n=0}^\\infty\\frac{(\\mu)_n}{\\Gamma(\\gamma n+\\delta)}\\frac{z^n}{n!}\\Big\\}\\Big].\n\\end{eqnarray*}\nBy Theorem \\ref{th2}, we have\n\\begin{eqnarray*}\n\\mathfrak{D}_{z;p}^{\\eta-\\mu;\\alpha}\\Big[z^{\\eta-1}E_{\\gamma,\\delta}^{\\mu}(z)\\Big]=\n\\sum_{n=0}^\\infty\\frac{(\\mu)_n}{\\Gamma(\\gamma n+\\delta)}\\Big\\{\\mathfrak{D}_{z;p}^{\\eta-\\mu;\\alpha}\\Big[z^{\\eta+n-1}\\Big]\\Big\\}.\n\\end{eqnarray*}\nApplying Theorem \\ref{th1}, we get the required proof.\n\\end{proof}\n\\end{theorem}\n\\begin{remark}\nIn \\cite{Mehrez} Mehrez and Tomovski introduced a new $p$-Mittag-Leffler function defined by\n\\begin{align}\\label{MT}\nE_{\\lambda,\\gamma,\\delta;p}^{(\\mu,\\eta,\\omega)}(z)=\\sum_{k=0}^\\infty\\frac{(\\mu)_k}{[\\Gamma(\\gamma k+\\delta)]^\\lambda}\n\\frac{B_p(\\eta+k,\\omega-\\eta)}{B(\\eta,\\omega-\\eta)}\\frac{z^k}{k!}, (z \\in \\mathbb{C})\n\\end{align}\n$$(\\mu,\\,\\eta,\\,\\omega,\\,\\gamma,\\,\\delta,\\,\\lambda>0,\\, \\Re(p)>0).$$\n\n\\end{remark}\nHence we get the following corollary.\n\\begin{corollary} The following formula holds true for $E_{\\gamma,\\delta}^{\\mu}(z)$:\n\\begin{align}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\Big\\{z^{\\eta-1}E_{\\gamma,\\delta}^{\\mu}(z);p\\Big\\}=\nz^{\\mu-1}\\frac{\\Gamma(\\eta)}{\\Gamma(\\mu)}E_{1,\\gamma,\\delta;p}^{(\\mu,\\eta,\\mu)}(z),\n\\end{align}\n$$(\\mu,\\eta,\\,\\gamma,\\,\\delta>0,\\, \\Re(p)>0).$$\n\\end{corollary}\n\n\\begin{theorem}\nThe following result holds true:\n\\begin{align}\\label{fd10}\n\\mathfrak{D}_{z;p}^{\\eta-\\mu;\\alpha}\\Big\\{z^{\\eta-1}\\,_m\\Psi_n\\left[\n \\begin{array}{cc}\n (\\alpha_i,A_i)_{1,m}; & \\\\\n & |z \\\\\n (\\beta_j,B_j)_{1,n}; & \\\\\n \\end{array}\n \\right]\n\\Big\\}&=\\frac{z^{\\mu-1}}{\\Gamma(\\mu-\\eta)}\\notag\\\\\n&\\times\\sum_{k=0}^\\infty\\frac{\\prod_{i=1}^{m}\\Gamma(\\alpha_i+A_ik)}{\\prod_{j=1}^{n}\\Gamma(\\beta_j+B_jk)}\nB_{p}^{\\alpha}(\\eta+k,\\mu-\\eta)\\frac{z^k}{k!},\n\\end{align}\nwhere $\\Re(p)>0$, $\\Re(\\alpha)>0$, $\\Re(\\mu)>\\Re(\\eta)>0$ and $_m\\Psi_n(z)$ represents the Fox-Wright function (see \\cite{Kilbas}, pp. 56-58)\n\\begin{eqnarray}\\label{fd11}\n_m\\Psi_n(z)=\\,_m\\Psi_n\\left[\n \\begin{array}{cc}\n (\\alpha_i,A_i)_{1,m}; & \\\\\n & |z \\\\\n (\\beta_j,B_j)_{1,n}; & \\\\\n\\end{array}\n\\right]=\\sum_{k=0}^\\infty\\frac{\\prod_{i=1}^{m}\\Gamma(\\alpha_i+A_ik)}{\\prod_{j=1}^{n}\\Gamma(\\beta_j+B_jk)}\\frac{z^k}{k!}.\n\\end{eqnarray}\n\\end{theorem}\n\\begin{proof}\nApplying Theorem \\ref{th1} and followed the same procedure used in Theorem \\ref{th7}, we get the desired result.\n\\end{proof}\n\\begin{remark}\nIn \\cite{Sharma} Sharma and Devi introduced extended Wright generalized\nhypergeometric function defined by\n\\begin{align}\\label{fd12}\n_{m+1}\\Psi_{n+1}(z;p)&=\\,_{m+1}\\Psi_{n+1}\\left[\n \\begin{array}{cc}\n (\\alpha_i,A_i)_{1,m},(\\gamma,1); & \\\\\n & |(z;p) \\\\\n (\\beta_j,B_j)_{1,n}; (c,1) & \\\\\n\\end{array}\n\\right]\\notag\\\\=&\\frac{1}{\\Gamma(c-\\gamma)}\\sum_{k=0}^\\infty\\frac{\\prod_{i=1}^{m}\\Gamma(\\alpha_i+A_ik)}{\\prod_{j=1}^{n}\n\\Gamma(\\beta_j+B_jk)}\\frac{B_p(\\gamma+k,c-\\gamma)z^k}{k!},\n\\end{align}\n$$(\\Re(p)>0,\\, \\Re(c)>\\Re(\\gamma)>0).$$\n\\end{remark}\nHence we get the following corollary.\n\\begin{corollary} The following results holds true for the Fox-Wright\nhypergeometric function:\n\\begin{align}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\Big\\{z^{\\eta-1}\\,_{m}\\Psi_{n}\\left[\n \\begin{array}{cc}\n (\\alpha_i,A_i)_{1,m}; & \\\\\n & |z \\\\\n (\\beta_j,B_j)_{1,n}; & \\\\\n \\end{array}\n \\right]\n;p\\Big\\}&=z^{\\mu-1}{}_{m+1}\\Psi_{n+1}\\left[\n \\begin{array}{cc}\n (\\alpha_i,A_i)_{1,m},(\\eta,1); & \\\\\n & |(z;p) \\\\\n (\\beta_j,B_j)_{1,n}; (\\mu,1) & \\\\\n\\end{array}\n\\right],\n\\end{align}\n$$(\\Re(p)>0,\\, \\Re(\\mu)>\\Re(\\eta)>0).$$\n\\end{corollary}\n\\section{Concluding remarks}\nIn this paper, we established a modified extension of Riemnn-Liouville fractional derivative operator. We conclude that when $\\alpha=1$, then all the results established in this paper will reduce to the results associated with classical Riemnn-Liouville derivative operator (see \\cite{Ozerslan}).\nSimilarly, if we letting $\\alpha=1$ and $p=0$ then all the results established in this paper will reduce to the results associated with classical Riemnn-Liouville fractional derivative operator (see \\cite{Kilbas}).\n\n\\begin{remark}\nThe preprint of this paper is available at 'https:\/\/arxiv.org\/abs\/1801.05001'.\n\\end{remark}\n\n\\textbf{Acknowledgments}\\\\\nThe authors would like to express profound gratitude to Prof. Virginia Kiryakova for valuable comments and remarks which improved the final version of this paper\\\\\n\n\n\n\\vskip 20pt\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nCataclysmic Variables are semi-detached binaries built of a white dwarf (WD) which is accreting mass from a Roche-lobe filling main sequence (MS) star. Mass transfer is driven by the loss of angular momentum in absence of strong magnetic fields, and the transferred material forms an accretion disc surrounding the central WD (see e.g., \\citealt{warner95-1}, \\citealt{hellier_book} and \\citealt{kniggeetal11-1} for comprehensive reviews). Since the structure of both components is relatively simple, CVs are one of the best sources to test our understanding of many astrophysical phenomena involving evolution of compact, interacting binaries and accretion phenomena. Their study helps to resolve standing discrepancies between current population models and observations in many present and complex topics including black hole binaries, short gamma-ray bursts, X-ray transients, milli-second pulsars and Supernovae Ia. \n\nThe orbital period distribution is the main tool to study the evolution of CVs, as it presents features in key points that allow us to understand their behaviour. As a consequence of the angular momentum loss and the mechanisms driving it, CVs move from long orbital periods and high mass transfer rates to short orbital periods and low mass transfer rates \\citealt{paczynski+sienkiewicz83-1}; \\citealt{Townsley09}; \\citealt{GoliaschNelson15}; \\citealt{palaetal7}). The evolution proceeds in this way until the system reaches the ``period minimum'' at $\\sim$ 76-80 minutes (\\citealt{knigge06-1}; \\citealt{gaensickeetal09}) in which the donor turns into a brown dwarf. Consequently, the orbital separation and period now increases as the mass transfer continues, becoming in the so-called period bouncers, faint systems with short orbital periods. On their way to the period minimum, observations show an abrupt drop in the number of systems with periods between 2 and 3h, referred to as the period gap. Below this range (Porb $<$ 2\\,h) systems have low mass-transfer rates governed by gravitational radiation (GR) \\citep{patterson84-1} while the higher mass-transfer rates above the gap (Porb $>$ 3\\,h) are a consequence of the stronger magnetic braking (MB) (\\citealt{rappaportetal83-1}; \\citealt{spruit+ritter83-1}; \\citealt{hameuryetal88-2}; \\citealt{davis08}). The standard explanation suggests that MB switches off when a CV has evolved down to 3h, the secondary contracts to its thermal equilibrium and detaches from its Roche lobe. Such systems crossing the gap are known as detached Cataclysmic Variables (dCVs). The continuing angular momentum loss by GR shrinks the orbit until at a period of about 2h, the Roche lobe makes contact with the stellar surface again and mass transfer is re-established albeit at a lower level. \n\nDuring this evolution, the CV will change appearance: The relative contribution of the WD, the secondary star and the accretion disc or stream makes for unique colours \\citep[e.g.][]{szkodyetal02-2}. The systems thus occupy distinct locations in colour-colour diagrams with respect to single stars. Due to the relatively small sample of CVs and the inherent difficulty of any source in obtaining its distance, it has not been possible so far to perform an analysis of the CV absolute magnitude distribution. \nWith the arrival of {\\it Gaia},\nthis has changed. Already \\cite{Pala19} show the advances that {\\it Gaia} parallaxes bring to the understanding of CVs, and we now have the data to study CVs in the HR-diagram.\n\nThe paper is organised as such: In section \\ref{sec:gaia}, we explain how we use {\\it Gaia} to define our CV sample. In section \\ref{sec:HRdiagram} the results are presented for all CVs and the trends with the orbital periods and the subtypes are discussed. Finally, we present our summary in \\ref{sec:Summary}.\n\n\n\\section{The Catalogue of {\\it Gaia} DR2 and the cross-match with CV catalogue}\\label{sec:gaia}\n\nThe goal of the {\\it Gaia} \nspace mission is to make the largest, most precise three-dimensional map of the Milky Way to-date by detecting and measuring the motion and parallax of each star in its orbit around the centre of the Galaxy. To this means, the three filters $G$, $G_{BP}$ and $G_{RP}$ are observed at several epochs over a period of about 670 days of mission operations. For details, see \\cite{GaiaDR2}.\n\nThe second data release (GDR2 hereafter) is based on 22 months of observations and provides positions, parallaxes and proper motions for 1.3 billion sources up to G $\\sim$ 20 magnitudes. This kind of data allows the derivation of distances and absolute magnitudes to study the position of all objects in the global HR-diagram.\n\n\\subsection{Deriving absolute magnitudes}\\label{subsubsec:Abs_mag}\n\nOne of the aims of this paper is finding the CV locus in the HR-diagram. We make use of GDR2 data to compute their absolute magnitudes. \nThe absolute magnitude $M$ of an object is given by:\n\\begin{equation}\nM = m + 5 - 5\\log(d) + A,\n\\label{eq:M}\n\\end{equation}\n\nwhere $m$ is the apparent magnitude, $A$ is the interstellar extinction and $d$, the distance to the source which can be obtained by the GDR2 data. \nGDR2 provides weighted mean fluxes\\footnote{Weighted means are used because flux errors on different epochs may vary depending on the configuration of each observation, see \\cite{CarrascoGDR1} and \\cite{RielloGDR2} for detailed information.} and, as CVs are variable stars, this procedure has an effect on their $G\n, $G_{BP}$ and $G_{RP}$ values. The degree of impact might be determined by comparing the 670-days length of the {\\it Gaia}-mission and the cycle of variation length for every CV subtype. The highest impact is on Dwarf Novae systems as they can have outbursts even on a weekly base (\\citealt{SterkenBook} and references therein). The low recurrence in Novae, Nova-like and Magnetic CVs should have no significant impact on the overall sample.\n\nInferring the distance from the {\\it Gaia} DR2 parallax is not a trivial issue. Distance can be derived as the inverse of the parallax, only if the parallax error is lower than 20\\% and by doing so we would discard 80\\% of the sources (\\citealt{Luri18}). We used instead the distances inferred by \\citet{Bailer-Jones18} who compute distances and their uncertainties through a probabilistic analysis based on the Bayes theorem and adopting an exponentially decreasing space density\nprior\\footnote{For a detailed explanation of this approach and an analysis of applying this technique refer to \\citet{Bailer-Jones18}. For a discussion of the use of different priors, see\n\\citet{Bailer-Jones18},\\citet{Luri18},\\citet{igoshevetal2016},\\citet{astraatmadjaetal16}.\n} \nfor the 1.33 billion sources from GDR2. These are available using ADQL\\footnote{http:\/\/gaia.ari.uni-heidelberg.de\/tap.html} and we here use them to derive the absolute magnitudes through Equation~\\ref{eq:M}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The CV sample}\\label{sec:CVsample}\n\n\nThe Catalog and Atlas of Cataclysmic Variables \\citep{DownesCat} includes all objects which have been classified as a CV at some point in time. Although it was frozen on February 1st, 2006, it is one of the main references among the community, providing coordinates, proper motion, type, chart, spectral and period references for all 1830 sources when available. In order to obtain the purest sample, we discarded from this catalogue the objects designated as ``NON-CV'', which are stars that have been previously identified as CVs but later confuted, and those with the extensions ``:'' and ``::'' because their classification is not conclusive.\n\nThe Catalog of Cataclysmic Binaries, Low-Mass X-Ray Binaries and Related Objects \\citep{RKCat} which only contains objects with a measured period, is updated up to December 31st, 2015 and it provides coordinates, apparent magnitudes, orbital parameters, stellar parameters of the components and other characteristic properties for 1429 CVs. In this case uncertain values are followed by only one ``:'' and have been discarded as well. \n\nBoth catalogues have been merged into a final sample of 1920 CVs, out of which 1187 are contained in the GDR2 footprint. The density studies of CVs in the HR-diagram\nwere done using this full sample. For 839 of these systems, the orbital period is known, and for 1130 systems, the subtype is unambiguously known (see Tab.\\ref{tab:subtypes_sample}).\n\n\n\\begin{table}\n\\scriptsize\n\t\\centering\n\t\\caption{\\label{tab:subtypes_sample} Distribution of the CV sample utilised by subtype.}\n\t\\begin{tabular}{lcccc} \n\t\t\\hline\nCV subtype & Periods & Main & \\multicolumn{2}{c}{Centroid position in HRD}\\\\\n& sample & sample & $G_{BP}-G_{RP}$ & $G_{abs}$\\\\\n\t\t\\hline\n\t\tNovalike & 76 & 119 & 0.37 & 5.63 \\\\\n\t\tDwarf Novae & 484 & 688 & 0.64 & 9.49 \\\\\n\t\tOld Novae & 77 & 119 & 0.79 & 5.58 \\\\\n\t\tPolar & 75 & 135 & 0.83 & 9.67 \\\\\n\t\tIntermediate Polar & 51 & 69 & 0.59 & 5.61 \\\\\n\t\t\\hline\n\t\tTotal Sample & 839 & 1130 & &\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\n\n\n\n\\section{CVs in the HR-diagram}\\label{sec:HRdiagram}\n\n\n\n\n\\subsection{The impact of the orbital period}\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}\n \\centering\n\t \\begin{tabular}{c c}\n\t \\includegraphics[width=0.45\\textwidth]{H-Rdiag_periods_rast.pdf}&\t\\includegraphics[width=0.42\\textwidth]{H-R_density_rast.pdf}\t\n\t \\end{tabular}\n \\caption{In grey, all stars from {\\it Gaia}'s 2nd data release are plotted in the HR-diagrams. On the left side, the CVs period distribution, the CVs from our sample with known orbital periods (see Section \\ref{sec:CVsample}) are plotted in larger dots. The colour of each dot refers to the orbital period as given in the bar at the right of the panel. CVs with larger periods lie close to the main sequence path getting shorter while approaching the white dwarfs area. On the right, the density distribution of our whole sample of CVs (brown dots), surfaces with different tones of blue represent areas of equal density. On the x- and y-axis the marginal distributions are shown. CVs lie on average between the MS path and the WDs with a high density area peaking at $G_{BP}-G_{RP} \\sim 0.56$ and $G_{abs} \\sim 10.15$. Such area corresponds to the overpopulation below the period gap as reflected in the left panel by black dots, CVs with orbital period below 2h\n \\label{fig:HRper_densityMap}}\n\\end{figure*}\n\nLeft panel of Fig. \\ref{fig:HRper_densityMap} displays the CV locus in the HR-diagram of all CVs for which an orbital period is known (839 systems). The orbital period of each system is represented by the colour of the symbol as defined in the auxiliary axis. The CVs lie on average between the main sequence stars and the WDs.\nA clear trend is seen on their position with the orbital period: CVs with longer periods fall close to the main sequence path, while, as the orbital period decreases, they approach the WDs region.\nThis behaviour can be understood from the contribution of the secondary star. On average, a Roche-lobe filling secondary star is larger and brighter for longer orbital periods, while the WD does not change much during the secular CV evolution. Hence, the contribution of the secondary should be more dominant for longer orbital periods. Systems below the period gap, are instead dominated by their WD, as the secondary becomes only visible in the near infrared and does not contribute to the {\\it Gaia} colour. The contribution of the accretion disc should change colour and magnitude depending on the sub-type and will be discussed in the next subsection.\n\nThe right panel of Fig. \\ref{fig:HRper_densityMap} shows the locus of all CVs of our sample defined in Section \\ref{sec:CVsample} within {\\it Gaia's} HR-diagram. On the x- and y-axis, the respective projected density is plotted. A high density area is well distinguishable at $G_{BP}-G_{RP} \\sim 0.56$ and $G_{abs} \\sim 10.15$ (values obtained from the mode of the marginal distributions) which corresponds to the population below the period gap.\n\n\n\\subsection{The locus depending on the subtype}\nFigure \\ref{fig:HR_per_types} exhibits the distribution of every CV subtype on the HR-diagram. Bivariate Gaussian distributions are computed for 1 and 3 $\\sigma$ given by\n\n\\begin{equation}\np(x, y | \\mu_x, \\mu_y, \\sigma_x, \\sigma_y, \\sigma_{xy}) = \\frac{1}{2 \\pi \\sigma_x \\sigma_y \\sqrt{1-\\rho^2}}exp \\left(\\frac{-z^2}{2(1-\\rho^2)}\\right),\n\\label{eq:bivariate1}\n\\end{equation}\n\nwhere\n\n\\begin{equation}\nz^2 = \\frac{(x-\\mu_x)^2}{\\sigma_x^2}+\\frac{(y-\\mu_y)^2}{\\sigma_y^2}-2\\rho \\frac{(x-\\mu_x)(y-\\mu_y)}{\\sigma_x \\sigma_y}, \n\\label{eq:bivariate2}\n\\end{equation}\n\n\\begin{equation}\n\\rho = \\frac{\\sigma_{xy}}{\\sigma_x \\sigma_y},\n\\label{eq:bivariate3}\n\\end{equation}\n\nusing the median instead of the mean and the interquartile range to estimate variances in order to avoid the impact of outliers. The results are given in Table \\ref{tab:subtypes_sample}.\n\n\\begin{figure*\n\\center\n\t\\begin{tabular}{c c c}\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_types_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Nova_like_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Dwarf_Novae_rast.pdf}\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Novae_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Polars_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Intermediate_Polars_rast.pdf}\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{NL_N_hist_short.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{AM_IP_hist_short.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{DN_WZSge_hist_short.pdf}\t\n\t\\end{tabular}\n\\caption{The distribution of CV subtypes in the HR-diagram. Top-left panel shows all subtypes together, after that every subtype separately. The dashed ellipses represent 1 and 3 $\\sigma$ of each subtype bivariate Gaussian distribution. The sample utilized here is composed by all CVs in the Ritter\\&Kolb and Downes catalogues (see Section \\ref{sec:CVsample}) whose subtype is unambiguously known and are included in the {\\it Gaia} footprint; 119 Nova-likes, 688 Dwarf Novae, 119 Novae, 135 Polars and 69 Intermediate Polars. On the bottom, the period histograms for each subtype. \\label{fig:HR_per_types}}\n\\end{figure*}\n\nNova-likes are dominated by a high mass-transfer accretion disc, that usually \novershines\nthe WD and the secondary star at optical and even infrared wavelengths. Their colour and final absolute magnitude mainly depends on the inclination with respect to the line of sight. In the HR-diagram, they concentrate around $G_{abs}=5.63$ and $G_{BP}-G_{RP}=0.37$, i.e. on the blue and bright corner of all CVs. A similar locus but with a much higher scatter is occupied by the old novae and by intermediate polars. This can be explained by the eclectic composition of these two sub-groups which also contain a large fraction of novalike stars. \n\nIn contrast, polars which do not accrete mass through a disc, are much fainter and their colour and magnitude will depend on the nature of the secondary. In the HR-diagram they scatter around $G_{abs}=9.67$ and $G_{BP}-G_{RP}=0.83$\nrepresenting the reddest and faintest of all the CV subgroups.\n\nDwarf novae (DNe) occupy the whole region between MS stars and WDs with the centroid being at $G_{abs}=9.49$ and $G_{BP}-G_{RP}=0.64$. Since the secondary star in these systems can be anything from an early K-type star down to a brown dwarf, the range in colours and magnitude is not surprising. In addition, these sources are characterised by undergoing \nregular outbursts increasing their brightness and blueness. \nAs discussed in Section \\ref{sec:gaia}, the given magnitude is a weighted mean of several epochs and thus also increases the spread of this distribution. A detailed study of the DNe locus depending on their subtype and outburst state can be done following the next {\\it Gaia} release when individual measurements and epochs become available\n\n\nWZ Sge-type objects deserve a separate mention, a class of DNe characterised by great outburst amplitudes, slow declines and long intervals between outbursts compared with ordinary DNe. These kind of systems have been considered to be period bouncer candidates \\citep{Patterson11}, some of them extensively investigated in this regard (QZ Lib, \\citealt{Pala18}; J122221 \\citealt{vitaly17} and \\citealt{kato13b}; J184228 \\citealt{kato13b}; J075418 and J230425 \\citealt{Nakata14}). We have plotted a sample of 71 of such systems in the upper-right panel of Fig. \\ref{fig:HR_per_types}, along with the rest of DNe, and it can be seen that they concentrate near the WDs area. This is consistent with them being period bouncers or similar systems, as these are the CVs with the lowest mass transfer and faintest secondary stars. The disc is only visible in some emission lines, the secondary does not contribute to the optical range at all.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Detached CVs}\n\n\\begin{figure\n\t\\includegraphics[width=0.425\\textwidth]{HR_WDM4-6.pdf}\n\\caption{\\label{fig:dCVs} Systems comprised by a WD and an M4-6. The figure contains the 39 objects from the sample \nby \\citep{Zorotovic16} present in {\\it Gaia} (see text). In green are those within the period gap and in brown the rest. All of them follow a trend but the objects within the gap are on average fainter.\n}\n\\end{figure}\n\nAnother question we can address is finding the locus occupied by the so-called detached cataclysmic variables (dCVs) crossing the orbital period gap. A first approach could be made by finding the area with boundaries in 2 and 3h in the left panel of Fig. \\ref{fig:HRper_densityMap} using\nregression techniques. However, since dCVs no longer \ncontain an accretion disc, they should appear fainter than regular CVs of the same period. \n\nDue to the continuous mass loss, the donor is being driven out of equilibrium and secondaries in CVs just above the period gap are bloated up to 30\\% with respect to regular MS stars \\citep{kniggeetal11-1}. When the mass transfer stops, the secondary shrinks towards its thermal equilibrium radius to nearly its equivalent for MS stars \\citep{stevehowell-01} and hence we expect secondaries in dCVs to be comparable to single MS stars of the same type.\n\nSince the mass transfer ceases, the mass and spectral type of the donor star stays constant during the interval in which the binary is detached. In regular CVs this happens at $M_{sec} = 0.2 \\pm 0.02$ $M\\odot$ \\citep{knigge06-1} and spectral type $\\sim M6$ \\citep{rebassaetal2007}, though variations occur depending on the moment in which the CV started the mass transfer and the time passed as CV until the secondary becomes fully convective. In the extreme case of a CV starting the mass transfer within the period gap range, the donor star type will be that of a fully convective isolated M star, which, according to \\cite{Chabrier-97}, occurs at $M_{sec} \\sim0.35$ $M\\odot$ and spectral type M4. We thus assume that the secondary of dCVs is in the range M4-M6.\n\nSo far the only observational evidence for the existence of dCVs come from \\cite{Zorotovic16}, who show that the orbital period distribution of detached close binaries consisting of a WD and an M4-M6 secondary star cannot be produced by Post Common Envelope Binaries alone, but a contribution of dCVs is needed to explain the peak between 2 and 3h. They also show that the systems inside this peak have a higher average mass than would be expected for normal WDMS systems. Still, with only 52 such systems known in total (WDMS systems with secondary spectral types in the range M4-6 and orbital periods below 10h) and 12 between 2 and 3h, the significance is not very high.\n\n\nWe distinguish two groups, the sources with orbital periods corresponding to those of the period gap (2-3h) and therefore, more likely to be dCVs, and the rest with periods outside this range. In Fig. \\ref{fig:dCVs} they are plotted in the HR-diagram, the former appear\nfainter compared to the latter. This can be explained by the higher WD masses in CVs, and consequently in dCVs, than in PCEBs, making them smaller in size and surface and contributing in a lesser extent on the brightness\nof the whole system.\n\n\n\\section{Summary}\\label{sec:Summary}\nWe have analyzed the evolutionary cycle of CVs from a statistical perspective using {\\it Gaia} DR2 data in conjunction with the HR-diagram tool. We have reported the discovery of a trend of the period and mass accretion with colour and absolute magnitude. We have also investigated their density distribution as a whole population, peaking at $G_{BP}-G_{RP} \\sim 0.56$ and $G_{abs} \\sim 10.15$, and the contribution of the main CV subtypes to this regard, highlighting the location of WZ Sge systems, which are period bouncer candidates. Finally, we have identified the location and a trend among systems comprised of a WD and secondary in the range M4-M6, which correspond with dCVs, CVs going through the orbital period gap.\n\n\\section*{Acknowledgements}\nThis research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France (DOI: 10.26093\/cds\/vizier). The original description of the VizieR service was published in 2000, A\\&AS 143, 23.\n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Introduction}\n\nThis is a simple template for authors to write new MNRAS papers.\nSee \\texttt{mnras\\_sample.tex} for a more complex example, and \\texttt{mnras\\_guide.tex}\nfor a full user guide.\n\nAll papers should start with an Introduction section, which sets the work\nin context, cites relevant earlier studies in the field by \\citet{Others2013},\nand describes the problem the authors aim to solve \\citep[e.g.][]{Author2012}.\n\n\\section{Methods, Observations, Simulations etc.}\n\nNormally the next section describes the techniques the authors used.\nIt is frequently split into subsections, such as Section~\\ref{sec:maths} below.\n\n\\subsection{Maths}\n\\label{sec:maths}\n\nSimple mathematics can be inserted into the flow of the text e.g. $2\\times3=6$\nor $v=220$\\,km\\,s$^{-1}$, but more complicated expressions should be entered\nas a numbered equation:\n\n\\begin{equation}\n x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}.\n\t\\label{eq:quadratic}\n\\end{equation}\n\nRefer back to them as e.g. equation~(\\ref{eq:quadratic}).\n\n\\subsection{Figures and tables}\n\nFigures and tables should be placed at logical positions in the text. 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Captions appear above each table.\n\tRemember to define the quantities, symbols and units used.}\n\t\\label{tab:example_table}\n\t\\begin{tabular}{lccr}\n\t\t\\hline\n\t\tA & B & C & D\\\\\n\t\t\\hline\n\t\t1 & 2 & 3 & 4\\\\\n\t\t2 & 4 & 6 & 8\\\\\n\t\t3 & 5 & 7 & 9\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\n\\section{Conclusions}\n\nThe last numbered section should briefly summarise what has been done, and describe\nthe final conclusions which the authors draw from their work.\n\n\\section*{Acknowledgements}\n\nThe Acknowledgements section is not numbered. 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Other,$^{2}$\nand Third Author$^{2,3}$\n\\\\\n$^{1}$Affiliation 1\\\\\n$^{2}$Affiliation 2\\\\\n$^{3}$Affiliation 3}\n\\end{verbatim}\nAffiliations should be in the format `Department, Institution, Street Address, City and Postal Code, Country'.\n\nEmail addresses can be inserted with the \\verb'\\thanks{}' command which adds a title page footnote.\nIf you want to list more than one email, put them all in the same \\verb'\\thanks' and use \\verb'\\footnotemark[]' to refer to the same footnote multiple times.\nPresent addresses (if different to those where the work was performed) can also be added with a \\verb'\\thanks' command.\n\n\\subsection{Abstract and keywords}\n\nThe abstract is entered in an \\verb'abstract' environment:\n\\begin{verbatim}\n\\begin{abstract}\nThe abstract of the paper.\n\\end{abstract}\n\\end{verbatim}\n\\noindent Note that there is a word limit on the length of abstracts.\nFor the current word limit, see the journal instructions to authors$^{\\ref{foot:itas}}$.\n\nImmediately following the abstract, a set of keywords is entered in a \\verb'keywords' environment:\n\\begin{verbatim}\n\\begin{keywords}\nkeyword 1 -- keyword 2 -- keyword 3\n\\end{keywords}\n\\end{verbatim}\n\\noindent There is a list of permitted keywords, which is agreed between all the major astronomy journals and revised every few years.\nDo \\emph{not} make up new keywords!\nFor the current list of allowed keywords, see the journal's instructions to authors$^{\\ref{foot:itas}}$.\n\n\\section{Sections and lists}\n\nSections and lists are generally the same as in the standard \\LaTeX\\ classes.\n\n\\subsection{Sections}\n\\label{sec:sections}\nSections are entered in the usual way, using \\verb'\\section{}' and its variants. 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If a paper has a large number of equations, it may be better to number them by section (2.1, 2.2 etc.). To do this, add the command \\verb'\\numberwithin{equation}{section}' to the preamble.\n\nIt is also possible to produce un-numbered equations by using the \\LaTeX\\ built-in \\verb'\\['\\textellipsis\\verb'\\]' and \\verb'$$'\\textellipsis\\verb'$$' commands; however MNRAS requires that all equations are numbered, so these commands should be avoided.\n\n\\subsection{Special symbols}\n\n\n\\begin{table}\n \\caption{Additional commands for special symbols commonly used in astronomy. These can be used anywhere.}\n \\label{tab:anysymbols}\n \\begin{tabular}{lll}\n \\hline\n Command & Output & Meaning\\\\\n \\hline\n \\verb'\\sun' & \\sun & Sun, solar\\\\[2pt]\n \\verb'\\earth' & \\earth & Earth, terrestrial\\\\[2pt]\n \\verb'\\micron' & \\micron & microns\\\\[2pt]\n \\verb'\\degr' & \\degr & degrees\\\\[2pt]\n \\verb'\\arcmin' & \\arcmin & arcminutes\\\\[2pt]\n \\verb'\\arcsec' & \\arcsec & arcseconds\\\\[2pt]\n \\verb'\\fdg' & \\fdg & fraction of a degree\\\\[2pt]\n \\verb'\\farcm' & \\farcm & fraction of an arcminute\\\\[2pt]\n \\verb'\\farcs' & \\farcs & fraction of an arcsecond\\\\[2pt]\n \\verb'\\fd' & \\fd & fraction of a day\\\\[2pt]\n \\verb'\\fh' & \\fh & fraction of an hour\\\\[2pt]\n \\verb'\\fm' & \\fm & fraction of a minute\\\\[2pt]\n \\verb'\\fs' & \\fs & fraction of a second\\\\[2pt]\n \\verb'\\fp' & \\fp & fraction of a period\\\\[2pt]\n \\verb'\\diameter' & \\diameter & diameter\\\\[2pt]\n \\verb'\\sq' & \\sq & square, Q.E.D.\\\\[2pt]\n \\hline\n \\end{tabular}\n\\end{table}\n\n\\begin{table}\n \\caption{Additional commands for mathematical symbols. These can only be used in maths mode.}\n \\label{tab:mathssymbols}\n \\begin{tabular}{lll}\n \\hline\n Command & Output & Meaning\\\\\n \\hline\n \\verb'\\upi' & $\\upi$ & upright pi\\\\[2pt]\n \\verb'\\umu' & $\\umu$ & upright mu\\\\[2pt]\n \\verb'\\upartial' & $\\upartial$ & upright partial derivative\\\\[2pt]\n \\verb'\\lid' & $\\lid$ & less than or equal to\\\\[2pt]\n \\verb'\\gid' & $\\gid$ & greater than or equal to\\\\[2pt]\n \\verb'\\la' & $\\la$ & less than of order\\\\[2pt]\n \\verb'\\ga' & $\\ga$ & greater than of order\\\\[2pt]\n \\verb'\\loa' & $\\loa$ & less than approximately\\\\[2pt]\n \\verb'\\goa' & $\\goa$ & greater than approximately\\\\[2pt]\n \\verb'\\cor' & $\\cor$ & corresponds to\\\\[2pt]\n \\verb'\\sol' & $\\sol$ & similar to or less than\\\\[2pt]\n \\verb'\\sog' & $\\sog$ & similar to or greater than\\\\[2pt]\n \\verb'\\lse' & $\\lse$ & less than or homotopic to \\\\[2pt]\n \\verb'\\gse' & $\\gse$ & greater than or homotopic to\\\\[2pt]\n \\verb'\\getsto' & $\\getsto$ & from over to\\\\[2pt]\n \\verb'\\grole' & $\\grole$ & greater over less\\\\[2pt]\n \\verb'\\leogr' & $\\leogr$ & less over greater\\\\\n \\hline\n \\end{tabular}\n\\end{table}\n\nSome additional symbols of common use in astronomy have been added in the MNRAS class. These are shown in tables~\\ref{tab:anysymbols}--\\ref{tab:mathssymbols}. The command names are -- as far as possible -- the same as those used in other major astronomy journals.\n\nMany other mathematical symbols are also available, either built into \\LaTeX\\ or via additional packages. If you want to insert a specific symbol but don't know the \\LaTeX\\ command, we recommend using the Detexify website\\footnote{\\url{http:\/\/detexify.kirelabs.org}}.\n\nSometimes font or coding limitations mean a symbol may not get smaller when used in sub- or superscripts, and will therefore be displayed at the wrong size. There is no need to worry about this as it will be corrected by the typesetter during production.\n\nTo produce bold symbols in mathematics, use \\verb'\\bmath' for simple variables, and the \\verb'bm' package for more complex symbols (see section~\\ref{sec:packages}). Vectors are set in bold italic, using \\verb'\\mathbfit{}'.\n\nFor matrices, use \\verb'\\mathbfss{}' to produce a bold sans-serif font e.g. \\mathbfss{H}; this works even outside maths mode, but not all symbols are available (e.g. Greek). For $\\nabla$ (del, used in gradients, divergence etc.) use \\verb'$\\nabla$'.\n\n\\subsection{Ions}\n\nA new \\verb'\\ion{}{}' command has been added to the class file, for the correct typesetting of ionisation states.\nFor example, to typeset singly ionised calcium use \\verb'\\ion{Ca}{ii}', which produces \\ion{Ca}{ii}.\n\n\\section{Figures and tables}\n\\label{sec:fig_table}\nFigures and tables (collectively called `floats') are mostly the same as built into \\LaTeX.\n\n\\subsection{Basic examples}\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{example}\n \\caption{An example figure.}\n \\label{fig:example}\n\\end{figure}\nFigures are inserted in the usual way using a \\verb'figure' environment and \\verb'\\includegraphics'. The example Figure~\\ref{fig:example} was generated using the code:\n\\begin{verbatim}\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{example}\n \\caption{An example figure.}\n \\label{fig:example}\n\\end{figure}\n\\end{verbatim}\n\n\\begin{table}\n \\caption{An example table.}\n \\label{tab:example}\n \\begin{tabular}{lcc}\n \\hline\n Star & Mass & Luminosity\\\\\n & $M_{\\sun}$ & $L_{\\sun}$\\\\\n \\hline\n Sun & 1.00 & 1.00\\\\\n $\\alpha$~Cen~A & 1.10 & 1.52\\\\\n $\\epsilon$~Eri & 0.82 & 0.34\\\\\n \\hline\n \\end{tabular}\n\\end{table}\nThe example Table~\\ref{tab:example} was generated using the code:\n\\begin{verbatim}\n\\begin{table}\n \\caption{An example table.}\n \\label{tab:example}\n \\begin{tabular}{lcc}\n \\hline\n Star & Mass & Luminosity\\\\\n & $M_{\\sun}$ & $L_{\\sun}$\\\\\n \\hline\n Sun & 1.00 & 1.00\\\\\n $\\alpha$~Cen~A & 1.10 & 1.52\\\\\n $\\epsilon$~Eri & 0.82 & 0.34\\\\\n \\hline\n \\end{tabular}\n\\end{table}\n\\end{verbatim}\n\n\\subsection{Captions and placement}\nCaptions go \\emph{above} tables but \\emph{below} figures, as in the examples above.\n\nThe \\LaTeX\\ float placement commands \\verb'[htbp]' are intentionally disabled.\nLayout of figures and tables will be adjusted by the publisher during the production process, so authors should not concern themselves with placement to avoid disappointment and wasted effort.\nSimply place the \\LaTeX\\ code close to where the figure or table is first mentioned in the text and leave exact placement to the publishers.\n\nBy default a figure or table will occupy one column of the page.\nTo produce a wider version which covers both columns, use the \\verb'figure*' or \\verb'table*' environment.\n\nIf a figure or table is too long to fit on a single page it can be split it into several parts.\nCreate an additional figure or table which uses \\verb'\\contcaption{}' instead of \\verb'\\caption{}'.\nThis will automatically correct the numbering and add `\\emph{continued}' at the start of the caption.\n\\begin{table}\n \\contcaption{A table continued from the previous one.}\n \\label{tab:continued}\n \\begin{tabular}{lcc}\n \\hline\n Star & Mass & Luminosity\\\\\n & $M_{\\sun}$ & $L_{\\sun}$\\\\\n \\hline\n $\\tau$~Cet & 0.78 & 0.52\\\\\n $\\delta$~Pav & 0.99 & 1.22\\\\\n $\\sigma$~Dra & 0.87 & 0.43\\\\\n \\hline\n \\end{tabular}\n\\end{table}\nTable~\\ref{tab:continued} was generated using the code:\n\n\\begin{verbatim}\n\\begin{table}\n \\contcaption{A table continued from the previous one.}\n \\label{tab:continued}\n \\begin{tabular}{lcc}\n \\hline\n Star & Mass & Luminosity\\\\\n & $M_{\\sun}$ & $L_{\\sun}$\\\\\n \\hline\n $\\tau$~Cet & 0.78 & 0.52\\\\\n $\\delta$~Pav & 0.99 & 1.22\\\\\n $\\sigma$~Dra & 0.87 & 0.43\\\\\n \\hline\n \\end{tabular}\n\\end{table}\n\\end{verbatim}\n\nTo produce a landscape figure or table, use the \\verb'pdflscape' package and the \\verb'landscape' environment.\nThe landscape Table~\\ref{tab:landscape} was produced using the code:\n\\begin{verbatim}\n\\begin{landscape}\n \\begin{table}\n \\caption{An example landscape table.}\n \\label{tab:landscape}\n \\begin{tabular}{cccccccccc}\n \\hline\n Header & Header & ...\\\\\n Unit & Unit & ...\\\\\n \\hline\n Data & Data & ...\\\\\n Data & Data & ...\\\\\n ...\\\\\n \\hline\n \\end{tabular}\n \\end{table}\n\\end{landscape}\n\\end{verbatim}\nUnfortunately this method will force a page break before the table appears.\nMore complicated solutions are possible, but authors shouldn't worry about this.\n\n\\begin{landscape}\n \\begin{table}\n \\caption{An example landscape table.}\n \\label{tab:landscape}\n \\begin{tabular}{cccccccccc}\n \\hline\n Header & Header & Header & Header & Header & Header & Header & Header & Header & Header\\\\\n Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit \\\\\n \\hline\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n \\hline\n \\end{tabular}\n \\end{table}\n\\end{landscape}\n\n\\section{References and citations}\n\n\\subsection{Cross-referencing}\n\nThe usual \\LaTeX\\ commands \\verb'\\label{}' and \\verb'\\ref{}' can be used for cross-referencing within the same paper.\nWe recommend that you use these whenever relevant, rather than writing out the section or figure numbers explicitly.\nThis ensures that cross-references are updated whenever the numbering changes (e.g. during revision) and provides clickable links (if available in your compiler).\n\nIt is best to give each section, figure and table a logical label.\nFor example, Table~\\ref{tab:mathssymbols} has the label \\verb'tab:mathssymbols', whilst section~\\ref{sec:packages} has the label \\verb'sec:packages'.\nAdd the label \\emph{after} the section or caption command, as in the examples in sections~\\ref{sec:sections} and \\ref{sec:fig_table}.\nEnter the cross-reference with a non-breaking space between the type of object and the number, like this: \\verb'see Figure~\\ref{fig:example}'.\n\nThe \\verb'\\autoref{}' command can be used to automatically fill out the type of object, saving on typing.\nIt also causes the link to cover the whole phrase rather than just the number, but for that reason is only suitable for single cross-references rather than ranges.\nFor example, \\verb'\\autoref{tab:journal_abbr}' produces \\autoref{tab:journal_abbr}.\n\n\\subsection{Citations}\n\\label{sec:cite}\n\nMNRAS uses the Harvard -- author (year) -- citation style, e.g. \\citet{author2013}.\nThis is implemented in \\LaTeX\\ via the \\verb'natbib' package, which in turn is included via the \\verb'usenatbib' package option (see section~\\ref{sec:options}), which should be used in all papers.\n\nEach entry in the reference list has a `key' (see section~\\ref{sec:ref_list}) which is used to generate citations.\nThere are two basic \\verb'natbib' commands:\n\\begin{description}\n \\item \\verb'\\citet{key}' produces an in-text citation: \\citet{author2013}\n \\item \\verb'\\citep{key}' produces a bracketed (parenthetical) citation: \\citep{author2013}\n\\end{description}\nCitations will include clickable links to the relevant entry in the reference list, if supported by your \\LaTeX\\ compiler.\n\n\\defcitealias{smith2014}{Paper~I}\n\\begin{table*}\n \\caption{Common citation commands, provided by the \\texttt{natbib} package.}\n \\label{tab:natbib}\n \\begin{tabular}{lll}\n \\hline\n Command & Ouput & Note\\\\\n \\hline\n \\verb'\\citet{key}' & \\citet{smith2014} & \\\\\n \\verb'\\citep{key}' & \\citep{smith2014} & \\\\\n \\verb'\\citep{key,key2}' & \\citep{smith2014,jones2015} & Multiple papers\\\\\n \\verb'\\citet[table 4]{key}' & \\citet[table 4]{smith2014} & \\\\\n \\verb'\\citep[see][figure 7]{key}' & \\citep[see][figure 7]{smith2014} & \\\\\n \\verb'\\citealt{key}' & \\citealt{smith2014} & For use with manual brackets\\\\\n \\verb'\\citeauthor{key}' & \\citeauthor{smith2014} & If already cited in close proximity\\\\\n \\verb'\\defcitealias{key}{Paper~I}' & & Define an alias (doesn't work in floats)\\\\\n \\verb'\\citetalias{key}' & \\citetalias{smith2014} & \\\\\n \\verb'\\citepalias{key}' & \\citepalias{smith2014} & \\\\\n \\hline\n \\end{tabular}\n\\end{table*}\n\nThere are a number of other \\verb'natbib' commands which can be used for more complicated citations.\nThe most commonly used ones are listed in Table~\\ref{tab:natbib}.\nFor full guidance on their use, consult the \\verb'natbib' documentation\\footnote{\\url{http:\/\/www.ctan.org\/pkg\/natbib}}.\n\nIf a reference has several authors, \\verb'natbib' will automatically use `et al.' if there are more than two authors. However, if a paper has exactly three authors, MNRAS style is to list all three on the first citation and use `et al.' thereafter. If you are using \\bibtex\\ (see section~\\ref{sec:ref_list}) then this is handled automatically. If not, the \\verb'\\citet*{}' and \\verb'\\citep*{}' commands can be used at the first citation to include all of the authors.\n\n\\subsection{The list of references}\n\\label{sec:ref_list}\n\nIt is possible to enter references manually using the usual \\LaTeX\\ commands, but we strongly encourage authors to use \\bibtex\\ instead.\n\\bibtex\\ ensures that the reference list is updated automatically as references are added or removed from the paper, puts them in the correct format, saves on typing, and the same reference file can be used for many different papers -- saving time hunting down reference details.\nAn MNRAS \\bibtex\\ style file, \\verb'mnras.bst', is distributed as part of this package.\nThe rest of this section will assume you are using \\bibtex.\n\nReferences are entered into a separate \\verb'.bib' file in standard \\bibtex\\ formatting.\nThis can be done manually, or there are several software packages which make editing the \\verb'.bib' file much easier.\nWe particularly recommend \\textsc{JabRef}\\footnote{\\url{http:\/\/jabref.sourceforge.net\/}}, which works on all major operating systems.\n\\bibtex\\ entries can be obtained from the NASA Astrophysics Data System\\footnote{\\label{foot:ads}\\url{http:\/\/adsabs.harvard.edu}} (ADS) by clicking on `Bibtex entry for this abstract' on any entry.\nSimply copy this into your \\verb'.bib' file or into the `BibTeX source' tab in \\textsc{JabRef}.\n\nEach entry in the \\verb'.bib' file must specify a unique `key' to identify the paper, the format of which is up to the author.\nSimply cite it in the usual way, as described in section~\\ref{sec:cite}, using the specified key.\nCompile the paper as usual, but add an extra step to run the \\texttt{bibtex} command.\nConsult the documentation for your compiler or latex distribution.\n\nCorrect formatting of the reference list will be handled by \\bibtex\\ in almost all cases, provided that the correct information was entered into the \\verb'.bib' file.\nNote that ADS entries are not always correct, particularly for older papers and conference proceedings, so may need to be edited.\nIf in doubt, or if you are producing the reference list manually, see the MNRAS instructions to authors$^{\\ref{foot:itas}}$ for the current guidelines on how to format the list of references.\n\n\\section{Appendices and online material}\n\nTo start an appendix, simply place the \\verb'\n\\section{Introduction}\n\nCataclysmic Variables are semi-detached binaries built of a white dwarf (WD) which is accreting mass from a Roche-lobe filling main sequence (MS) star. Mass transfer is driven by the loss of angular momentum in absence of strong magnetic fields, and the transferred material forms an accretion disc surrounding the central WD (see e.g., \\citealt{warner95-1}, \\citealt{hellier_book} and \\citealt{kniggeetal11-1} for comprehensive reviews). Since the structure of both components is relatively simple, CVs are one of the best sources to test our understanding of many astrophysical phenomena involving evolution of compact, interacting binaries and accretion phenomena. Their study helps to resolve standing discrepancies between current population models and observations in many present and complex topics including black hole binaries, short gamma-ray bursts, X-ray transients, milli-second pulsars and Supernovae Ia. \n\nThe orbital period distribution is the main tool to study the evolution of CVs, as it presents features in key points that allow us to understand their behaviour. As a consequence of the angular momentum loss and the mechanisms driving it, CVs move from long orbital periods and high mass transfer rates to short orbital periods and low mass transfer rates \\citealt{paczynski+sienkiewicz83-1}; \\citealt{Townsley09}; \\citealt{GoliaschNelson15}; \\citealt{palaetal7}). The evolution proceeds in this way until the system reaches the ``period minimum'' at $\\sim$ 76-80 minutes (\\citealt{knigge06-1}; \\citealt{gaensickeetal09}) in which the donor turns into a brown dwarf. Consequently, the orbital separation and period now increases as the mass transfer continues, becoming in the so-called period bouncers, faint systems with short orbital periods. On their way to the period minimum, observations show an abrupt drop in the number of systems with periods between 2 and 3h, referred to as the period gap. Below this range (Porb $<$ 2\\,h) systems have low mass-transfer rates governed by gravitational radiation (GR) \\citep{patterson84-1} while the higher mass-transfer rates above the gap (Porb $>$ 3\\,h) are a consequence of the stronger magnetic braking (MB) (\\citealt{rappaportetal83-1}; \\citealt{spruit+ritter83-1}; \\citealt{hameuryetal88-2}; \\citealt{davis08}). The standard explanation suggests that MB switches off when a CV has evolved down to 3h, the secondary contracts to its thermal equilibrium and detaches from its Roche lobe. Such systems crossing the gap are known as detached Cataclysmic Variables (dCVs). The continuing angular momentum loss by GR shrinks the orbit until at a period of about 2h, the Roche lobe makes contact with the stellar surface again and mass transfer is re-established albeit at a lower level. \n\nDuring this evolution, the CV will change appearance: The relative contribution of the WD, the secondary star and the accretion disc or stream makes for unique colours \\citep[e.g.][]{szkodyetal02-2}. The systems thus occupy distinct locations in colour-colour diagrams with respect to single stars. Due to the relatively small sample of CVs and the inherent difficulty of any source in obtaining its distance, it has not been possible so far to perform an analysis of the CV absolute magnitude distribution. \nWith the arrival of {\\it Gaia},\nthis has changed. Already \\cite{Pala19} show the advances that {\\it Gaia} parallaxes bring to the understanding of CVs, and we now have the data to study CVs in the HR-diagram.\n\nThe paper is organised as such: In section \\ref{sec:gaia}, we explain how we use {\\it Gaia} to define our CV sample. In section \\ref{sec:HRdiagram} the results are presented for all CVs and the trends with the orbital periods and the subtypes are discussed. Finally, we present our summary in \\ref{sec:Summary}.\n\n\n\\section{The Catalogue of {\\it Gaia} DR2 and the cross-match with CV catalogue}\\label{sec:gaia}\n\nThe goal of the {\\it Gaia} \nspace mission is to make the largest, most precise three-dimensional map of the Milky Way to-date by detecting and measuring the motion and parallax of each star in its orbit around the centre of the Galaxy. To this means, the three filters $G$, $G_{BP}$ and $G_{RP}$ are observed at several epochs over a period of about 670 days of mission operations. For details, see \\cite{GaiaDR2}.\n\nThe second data release (GDR2 hereafter) is based on 22 months of observations and provides positions, parallaxes and proper motions for 1.3 billion sources up to G $\\sim$ 20 magnitudes. This kind of data allows the derivation of distances and absolute magnitudes to study the position of all objects in the global HR-diagram.\n\n\\subsection{Deriving absolute magnitudes}\\label{subsubsec:Abs_mag}\n\nOne of the aims of this paper is finding the CV locus in the HR-diagram. We make use of GDR2 data to compute their absolute magnitudes. \nThe absolute magnitude $M$ of an object is given by:\n\\begin{equation}\nM = m + 5 - 5\\log(d) + A,\n\\label{eq:M}\n\\end{equation}\n\nwhere $m$ is the apparent magnitude, $A$ is the interstellar extinction and $d$, the distance to the source which can be obtained by the GDR2 data. \nGDR2 provides weighted mean fluxes\\footnote{Weighted means are used because flux errors on different epochs may vary depending on the configuration of each observation, see \\cite{CarrascoGDR1} and \\cite{RielloGDR2} for detailed information.} and, as CVs are variable stars, this procedure has an effect on their $G\n, $G_{BP}$ and $G_{RP}$ values. The degree of impact might be determined by comparing the 670-days length of the {\\it Gaia}-mission and the cycle of variation length for every CV subtype. The highest impact is on Dwarf Novae systems as they can have outbursts even on a weekly base (\\citealt{SterkenBook} and references therein). The low recurrence in Novae, Nova-like and Magnetic CVs should have no significant impact on the overall sample.\n\nInferring the distance from the {\\it Gaia} DR2 parallax is not a trivial issue. Distance can be derived as the inverse of the parallax, only if the parallax error is lower than 20\\% and by doing so we would discard 80\\% of the sources (\\citealt{Luri18}). We used instead the distances inferred by \\citet{Bailer-Jones18} who compute distances and their uncertainties through a probabilistic analysis based on the Bayes theorem and adopting an exponentially decreasing space density\nprior\\footnote{For a detailed explanation of this approach and an analysis of applying this technique refer to \\citet{Bailer-Jones18}. For a discussion of the use of different priors, see\n\\citet{Bailer-Jones18},\\citet{Luri18},\\citet{igoshevetal2016},\\citet{astraatmadjaetal16}.\n} \nfor the 1.33 billion sources from GDR2. These are available using ADQL\\footnote{http:\/\/gaia.ari.uni-heidelberg.de\/tap.html} and we here use them to derive the absolute magnitudes through Equation~\\ref{eq:M}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The CV sample}\\label{sec:CVsample}\n\n\nThe Catalog and Atlas of Cataclysmic Variables \\citep{DownesCat} includes all objects which have been classified as a CV at some point in time. Although it was frozen on February 1st, 2006, it is one of the main references among the community, providing coordinates, proper motion, type, chart, spectral and period references for all 1830 sources when available. In order to obtain the purest sample, we discarded from this catalogue the objects designated as ``NON-CV'', which are stars that have been previously identified as CVs but later confuted, and those with the extensions ``:'' and ``::'' because their classification is not conclusive.\n\nThe Catalog of Cataclysmic Binaries, Low-Mass X-Ray Binaries and Related Objects \\citep{RKCat} which only contains objects with a measured period, is updated up to December 31st, 2015 and it provides coordinates, apparent magnitudes, orbital parameters, stellar parameters of the components and other characteristic properties for 1429 CVs. In this case uncertain values are followed by only one ``:'' and have been discarded as well. \n\nBoth catalogues have been merged into a final sample of 1920 CVs, out of which 1187 are contained in the GDR2 footprint. The density studies of CVs in the HR-diagram\nwere done using this full sample. For 839 of these systems, the orbital period is known, and for 1130 systems, the subtype is unambiguously known (see Tab.\\ref{tab:subtypes_sample}).\n\n\n\\begin{table}\n\\scriptsize\n\t\\centering\n\t\\caption{\\label{tab:subtypes_sample} Distribution of the CV sample utilised by subtype.}\n\t\\begin{tabular}{lcccc} \n\t\t\\hline\nCV subtype & Periods & Main & \\multicolumn{2}{c}{Centroid position in HRD}\\\\\n& sample & sample & $G_{BP}-G_{RP}$ & $G_{abs}$\\\\\n\t\t\\hline\n\t\tNovalike & 76 & 119 & 0.37 & 5.63 \\\\\n\t\tDwarf Novae & 484 & 688 & 0.64 & 9.49 \\\\\n\t\tOld Novae & 77 & 119 & 0.79 & 5.58 \\\\\n\t\tPolar & 75 & 135 & 0.83 & 9.67 \\\\\n\t\tIntermediate Polar & 51 & 69 & 0.59 & 5.61 \\\\\n\t\t\\hline\n\t\tTotal Sample & 839 & 1130 & &\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\n\n\n\n\\section{CVs in the HR-diagram}\\label{sec:HRdiagram}\n\n\n\n\n\\subsection{The impact of the orbital period}\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}\n \\centering\n\t \\begin{tabular}{c c}\n\t \\includegraphics[width=0.45\\textwidth]{H-Rdiag_periods_rast.pdf}&\t\\includegraphics[width=0.42\\textwidth]{H-R_density_rast.pdf}\t\n\t \\end{tabular}\n \\caption{In grey, all stars from {\\it Gaia}'s 2nd data release are plotted in the HR-diagrams. On the left side, the CVs period distribution, the CVs from our sample with known orbital periods (see Section \\ref{sec:CVsample}) are plotted in larger dots. The colour of each dot refers to the orbital period as given in the bar at the right of the panel. CVs with larger periods lie close to the main sequence path getting shorter while approaching the white dwarfs area. On the right, the density distribution of our whole sample of CVs (brown dots), surfaces with different tones of blue represent areas of equal density. On the x- and y-axis the marginal distributions are shown. CVs lie on average between the MS path and the WDs with a high density area peaking at $G_{BP}-G_{RP} \\sim 0.56$ and $G_{abs} \\sim 10.15$. Such area corresponds to the overpopulation below the period gap as reflected in the left panel by black dots, CVs with orbital period below 2h\n \\label{fig:HRper_densityMap}}\n\\end{figure*}\n\nLeft panel of Fig. \\ref{fig:HRper_densityMap} displays the CV locus in the HR-diagram of all CVs for which an orbital period is known (839 systems). The orbital period of each system is represented by the colour of the symbol as defined in the auxiliary axis. The CVs lie on average between the main sequence stars and the WDs.\nA clear trend is seen on their position with the orbital period: CVs with longer periods fall close to the main sequence path, while, as the orbital period decreases, they approach the WDs region.\nThis behaviour can be understood from the contribution of the secondary star. On average, a Roche-lobe filling secondary star is larger and brighter for longer orbital periods, while the WD does not change much during the secular CV evolution. Hence, the contribution of the secondary should be more dominant for longer orbital periods. Systems below the period gap, are instead dominated by their WD, as the secondary becomes only visible in the near infrared and does not contribute to the {\\it Gaia} colour. The contribution of the accretion disc should change colour and magnitude depending on the sub-type and will be discussed in the next subsection.\n\nThe right panel of Fig. \\ref{fig:HRper_densityMap} shows the locus of all CVs of our sample defined in Section \\ref{sec:CVsample} within {\\it Gaia's} HR-diagram. On the x- and y-axis, the respective projected density is plotted. A high density area is well distinguishable at $G_{BP}-G_{RP} \\sim 0.56$ and $G_{abs} \\sim 10.15$ (values obtained from the mode of the marginal distributions) which corresponds to the population below the period gap.\n\n\n\\subsection{The locus depending on the subtype}\nFigure \\ref{fig:HR_per_types} exhibits the distribution of every CV subtype on the HR-diagram. Bivariate Gaussian distributions are computed for 1 and 3 $\\sigma$ given by\n\n\\begin{equation}\np(x, y | \\mu_x, \\mu_y, \\sigma_x, \\sigma_y, \\sigma_{xy}) = \\frac{1}{2 \\pi \\sigma_x \\sigma_y \\sqrt{1-\\rho^2}}exp \\left(\\frac{-z^2}{2(1-\\rho^2)}\\right),\n\\label{eq:bivariate1}\n\\end{equation}\n\nwhere\n\n\\begin{equation}\nz^2 = \\frac{(x-\\mu_x)^2}{\\sigma_x^2}+\\frac{(y-\\mu_y)^2}{\\sigma_y^2}-2\\rho \\frac{(x-\\mu_x)(y-\\mu_y)}{\\sigma_x \\sigma_y}, \n\\label{eq:bivariate2}\n\\end{equation}\n\n\\begin{equation}\n\\rho = \\frac{\\sigma_{xy}}{\\sigma_x \\sigma_y},\n\\label{eq:bivariate3}\n\\end{equation}\n\nusing the median instead of the mean and the interquartile range to estimate variances in order to avoid the impact of outliers. The results are given in Table \\ref{tab:subtypes_sample}.\n\n\\begin{figure*\n\\center\n\t\\begin{tabular}{c c c}\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_types_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Nova_like_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Dwarf_Novae_rast.pdf}\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Novae_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Polars_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Intermediate_Polars_rast.pdf}\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{NL_N_hist_short.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{AM_IP_hist_short.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{DN_WZSge_hist_short.pdf}\t\n\t\\end{tabular}\n\\caption{The distribution of CV subtypes in the HR-diagram. Top-left panel shows all subtypes together, after that every subtype separately. The dashed ellipses represent 1 and 3 $\\sigma$ of each subtype bivariate Gaussian distribution. The sample utilized here is composed by all CVs in the Ritter\\&Kolb and Downes catalogues (see Section \\ref{sec:CVsample}) whose subtype is unambiguously known and are included in the {\\it Gaia} footprint; 119 Nova-likes, 688 Dwarf Novae, 119 Novae, 135 Polars and 69 Intermediate Polars. On the bottom, the period histograms for each subtype. \\label{fig:HR_per_types}}\n\\end{figure*}\n\nNova-likes are dominated by a high mass-transfer accretion disc, that usually \novershines\nthe WD and the secondary star at optical and even infrared wavelengths. Their colour and final absolute magnitude mainly depends on the inclination with respect to the line of sight. In the HR-diagram, they concentrate around $G_{abs}=5.63$ and $G_{BP}-G_{RP}=0.37$, i.e. on the blue and bright corner of all CVs. A similar locus but with a much higher scatter is occupied by the old novae and by intermediate polars. This can be explained by the eclectic composition of these two sub-groups which also contain a large fraction of novalike stars. \n\nIn contrast, polars which do not accrete mass through a disc, are much fainter and their colour and magnitude will depend on the nature of the secondary. In the HR-diagram they scatter around $G_{abs}=9.67$ and $G_{BP}-G_{RP}=0.83$\nrepresenting the reddest and faintest of all the CV subgroups.\n\nDwarf novae (DNe) occupy the whole region between MS stars and WDs with the centroid being at $G_{abs}=9.49$ and $G_{BP}-G_{RP}=0.64$. Since the secondary star in these systems can be anything from an early K-type star down to a brown dwarf, the range in colours and magnitude is not surprising. In addition, these sources are characterised by undergoing \nregular outbursts increasing their brightness and blueness. \nAs discussed in Section \\ref{sec:gaia}, the given magnitude is a weighted mean of several epochs and thus also increases the spread of this distribution. A detailed study of the DNe locus depending on their subtype and outburst state can be done following the next {\\it Gaia} release when individual measurements and epochs become available\n\n\nWZ Sge-type objects deserve a separate mention, a class of DNe characterised by great outburst amplitudes, slow declines and long intervals between outbursts compared with ordinary DNe. These kind of systems have been considered to be period bouncer candidates \\citep{Patterson11}, some of them extensively investigated in this regard (QZ Lib, \\citealt{Pala18}; J122221 \\citealt{vitaly17} and \\citealt{kato13b}; J184228 \\citealt{kato13b}; J075418 and J230425 \\citealt{Nakata14}). We have plotted a sample of 71 of such systems in the upper-right panel of Fig. \\ref{fig:HR_per_types}, along with the rest of DNe, and it can be seen that they concentrate near the WDs area. This is consistent with them being period bouncers or similar systems, as these are the CVs with the lowest mass transfer and faintest secondary stars. The disc is only visible in some emission lines, the secondary does not contribute to the optical range at all.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Detached CVs}\n\n\\begin{figure\n\t\\includegraphics[width=0.425\\textwidth]{HR_WDM4-6.pdf}\n\\caption{\\label{fig:dCVs} Systems comprised by a WD and an M4-6. The figure contains the 39 objects from the sample \nby \\citep{Zorotovic16} present in {\\it Gaia} (see text). In green are those within the period gap and in brown the rest. All of them follow a trend but the objects within the gap are on average fainter.\n}\n\\end{figure}\n\nAnother question we can address is finding the locus occupied by the so-called detached cataclysmic variables (dCVs) crossing the orbital period gap. A first approach could be made by finding the area with boundaries in 2 and 3h in the left panel of Fig. \\ref{fig:HRper_densityMap} using\nregression techniques. However, since dCVs no longer \ncontain an accretion disc, they should appear fainter than regular CVs of the same period. \n\nDue to the continuous mass loss, the donor is being driven out of equilibrium and secondaries in CVs just above the period gap are bloated up to 30\\% with respect to regular MS stars \\citep{kniggeetal11-1}. When the mass transfer stops, the secondary shrinks towards its thermal equilibrium radius to nearly its equivalent for MS stars \\citep{stevehowell-01} and hence we expect secondaries in dCVs to be comparable to single MS stars of the same type.\n\nSince the mass transfer ceases, the mass and spectral type of the donor star stays constant during the interval in which the binary is detached. In regular CVs this happens at $M_{sec} = 0.2 \\pm 0.02$ $M\\odot$ \\citep{knigge06-1} and spectral type $\\sim M6$ \\citep{rebassaetal2007}, though variations occur depending on the moment in which the CV started the mass transfer and the time passed as CV until the secondary becomes fully convective. In the extreme case of a CV starting the mass transfer within the period gap range, the donor star type will be that of a fully convective isolated M star, which, according to \\cite{Chabrier-97}, occurs at $M_{sec} \\sim0.35$ $M\\odot$ and spectral type M4. We thus assume that the secondary of dCVs is in the range M4-M6.\n\nSo far the only observational evidence for the existence of dCVs come from \\cite{Zorotovic16}, who show that the orbital period distribution of detached close binaries consisting of a WD and an M4-M6 secondary star cannot be produced by Post Common Envelope Binaries alone, but a contribution of dCVs is needed to explain the peak between 2 and 3h. They also show that the systems inside this peak have a higher average mass than would be expected for normal WDMS systems. Still, with only 52 such systems known in total (WDMS systems with secondary spectral types in the range M4-6 and orbital periods below 10h) and 12 between 2 and 3h, the significance is not very high.\n\n\nWe distinguish two groups, the sources with orbital periods corresponding to those of the period gap (2-3h) and therefore, more likely to be dCVs, and the rest with periods outside this range. In Fig. \\ref{fig:dCVs} they are plotted in the HR-diagram, the former appear\nfainter compared to the latter. This can be explained by the higher WD masses in CVs, and consequently in dCVs, than in PCEBs, making them smaller in size and surface and contributing in a lesser extent on the brightness\nof the whole system.\n\n\n\\section{Summary}\\label{sec:Summary}\nWe have analyzed the evolutionary cycle of CVs from a statistical perspective using {\\it Gaia} DR2 data in conjunction with the HR-diagram tool. We have reported the discovery of a trend of the period and mass accretion with colour and absolute magnitude. We have also investigated their density distribution as a whole population, peaking at $G_{BP}-G_{RP} \\sim 0.56$ and $G_{abs} \\sim 10.15$, and the contribution of the main CV subtypes to this regard, highlighting the location of WZ Sge systems, which are period bouncer candidates. Finally, we have identified the location and a trend among systems comprised of a WD and secondary in the range M4-M6, which correspond with dCVs, CVs going through the orbital period gap.\n\n\\section*{Acknowledgements}\nThis research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France (DOI: 10.26093\/cds\/vizier). The original description of the VizieR service was published in 2000, A\\&AS 143, 23.\n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Introduction}\n\nThe journal \\textit{Monthly Notices of the Royal Astronomical Society} (MNRAS) encourages authors to prepare their papers using \\LaTeX.\nThe style file \\verb'mnras.cls' can be used to approximate the final appearance of the journal, and provides numerous features to simplify the preparation of papers.\nThis document, \\verb'mnras_guide.tex', provides guidance on using that style file and the features it enables.\n\nThis is not a general guide on how to use \\LaTeX, of which many excellent examples already exist.\nWe particularly recommend \\textit{Wikibooks \\LaTeX}\\footnote{\\url{https:\/\/en.wikibooks.org\/wiki\/LaTeX}}, a collaborative online textbook which is of use to both beginners and experts.\nAlternatively there are several other online resources, and most academic libraries also hold suitable beginner's guides.\n\nFor guidance on the contents of papers, journal style, and how to submit a paper, see the MNRAS Instructions to Authors\\footnote{\\label{foot:itas}\\url{http:\/\/www.oxfordjournals.org\/our_journals\/mnras\/for_authors\/}}.\nOnly technical issues with the \\LaTeX\\ class are considered here.\n\n\n\\section{Obtaining and installing the MNRAS package}\nSome \\LaTeX\\ distributions come with the MNRAS package by default.\nIf yours does not, you can either install it using your distribution's package manager, or download it from the Comprehensive \\TeX\\ Archive Network\\footnote{\\url{http:\/\/www.ctan.org\/tex-archive\/macros\/latex\/contrib\/mnras}} (CTAN).\n\nThe files can either be installed permanently by placing them in the appropriate directory (consult the documentation for your \\LaTeX\\ distribution), or used temporarily by placing them in the working directory for your paper.\n\nTo use the MNRAS package, simply specify \\verb'mnras' as the document class at the start of a \\verb'.tex' file:\n\n\\begin{verbatim}\n\\documentclass{mnras}\n\\end{verbatim}\nThen compile \\LaTeX\\ (and if necessary \\bibtex) in the usual way.\n\n\\section{Preparing and submitting a paper}\nWe recommend that you start with a copy of the \\texttt{mnras\\_template.tex} file.\nRename the file, update the information on the title page, and then work on the text of your paper.\nGuidelines for content, style etc. are given in the instructions to authors on the journal's website$^{\\ref{foot:itas}}$.\nNote that this document does not follow all the aspects of MNRAS journal style (e.g. it has a table of contents).\n\nIf a paper is accepted, it is professionally typeset and copyedited by the publishers.\nIt is therefore likely that minor changes to presentation will occur.\nFor this reason, we ask authors to ignore minor details such as slightly long lines, extra blank spaces, or misplaced figures, because these details will be dealt with during the production process.\n\nPapers must be submitted electronically via the online submission system; paper submissions are not permitted.\nFor full guidance on how to submit a paper, see the instructions to authors.\n\n\\section{Class options}\n\\label{sec:options}\nThere are several options which can be added to the document class line like this:\n\n\\begin{verbatim}\n\\documentclass[option1,option2]{mnras}\n\\end{verbatim}\nThe available options are:\n\\begin{itemize}\n\\item \\verb'letters' -- used for papers in the journal's Letters section.\n\\item \\verb'onecolumn' -- single column, instead of the default two columns. This should be used {\\it only} if necessary for the display of numerous very long equations.\n\\item \\verb'doublespacing' -- text has double line spacing. Please don't submit papers in this format.\n\\item \\verb'referee' -- \\textit{(deprecated)} single column, double spaced, larger text, bigger margins. Please don't submit papers in this format.\n\\item \\verb'galley' -- \\textit{(deprecated)} no running headers, no attempt to align the bottom of columns.\n\\item \\verb'landscape' -- \\textit{(deprecated)} sets the whole document on landscape paper.\n\\item \\verb\"usenatbib\" -- \\textit{(all papers should use this)} this uses Patrick Daly's \\verb\"natbib.sty\" package for citations.\n\\item \\verb\"usegraphicx\" -- \\textit{(most papers will need this)} includes the \\verb'graphicx' package, for inclusion of figures and images.\n\\item \\verb'useAMS' -- adds support for upright Greek characters \\verb'\\upi', \\verb'\\umu' and \\verb'\\upartial' ($\\upi$, $\\umu$ and $\\upartial$). Only these three are included, if you require other symbols you will need to include the \\verb'amsmath' or \\verb'amsymb' packages (see section~\\ref{sec:packages}).\n\\item \\verb\"usedcolumn\" -- includes the package \\verb\"dcolumn\", which includes two new types of column alignment for use in tables.\n\\end{itemize}\n\nSome of these options are deprecated and retained for backwards compatibility only.\nOthers are used in almost all papers, but again are retained as options to ensure that papers written decades ago will continue to compile without problems.\nIf you want to include any other packages, see section~\\ref{sec:packages}.\n\n\\section{Title page}\n\nIf you are using \\texttt{mnras\\_template.tex} the necessary code for generating the title page, headers and footers is already present.\nSimply edit the title, author list, institutions, abstract and keywords as described below.\n\n\\subsection{Title}\nThere are two forms of the title: the full version used on the first page, and a short version which is used in the header of other odd-numbered pages (the `running head').\nEnter them with \\verb'\\title[]{}' like this:\n\\begin{verbatim}\n\\title[Running head]{Full title of the paper}\n\\end{verbatim}\nThe full title can be multiple lines (use \\verb'\\\\' to start a new line) and may be as long as necessary, although we encourage authors to use concise titles. The running head must be $\\le~45$ characters on a single line.\n\nSee appendix~\\ref{sec:advanced} for more complicated examples.\n\n\\subsection{Authors and institutions}\n\nLike the title, there are two forms of author list: the full version which appears on the title page, and a short form which appears in the header of the even-numbered pages. Enter them using the \\verb'\\author[]{}' command.\n\nIf the author list is more than one line long, start a new line using \\verb'\\newauthor'. Use \\verb'\\\\' to start the institution list. Affiliations for each author should be indicated with a superscript number, and correspond to the list of institutions below the author list.\n\nFor example, if I were to write a paper with two coauthors at another institution, one of whom also works at a third location:\n\\begin{verbatim}\n\\author[K. T. Smith et al.]{\nKeith T. Smith,$^{1}$\nA. N. Other,$^{2}$\nand Third Author$^{2,3}$\n\\\\\n$^{1}$Affiliation 1\\\\\n$^{2}$Affiliation 2\\\\\n$^{3}$Affiliation 3}\n\\end{verbatim}\nAffiliations should be in the format `Department, Institution, Street Address, City and Postal Code, Country'.\n\nEmail addresses can be inserted with the \\verb'\\thanks{}' command which adds a title page footnote.\nIf you want to list more than one email, put them all in the same \\verb'\\thanks' and use \\verb'\\footnotemark[]' to refer to the same footnote multiple times.\nPresent addresses (if different to those where the work was performed) can also be added with a \\verb'\\thanks' command.\n\n\\subsection{Abstract and keywords}\n\nThe abstract is entered in an \\verb'abstract' environment:\n\\begin{verbatim}\n\\begin{abstract}\nThe abstract of the paper.\n\\end{abstract}\n\\end{verbatim}\n\\noindent Note that there is a word limit on the length of abstracts.\nFor the current word limit, see the journal instructions to authors$^{\\ref{foot:itas}}$.\n\nImmediately following the abstract, a set of keywords is entered in a \\verb'keywords' environment:\n\\begin{verbatim}\n\\begin{keywords}\nkeyword 1 -- keyword 2 -- keyword 3\n\\end{keywords}\n\\end{verbatim}\n\\noindent There is a list of permitted keywords, which is agreed between all the major astronomy journals and revised every few years.\nDo \\emph{not} make up new keywords!\nFor the current list of allowed keywords, see the journal's instructions to authors$^{\\ref{foot:itas}}$.\n\n\\section{Sections and lists}\n\nSections and lists are generally the same as in the standard \\LaTeX\\ classes.\n\n\\subsection{Sections}\n\\label{sec:sections}\nSections are entered in the usual way, using \\verb'\\section{}' and its variants. It is possible to nest up to four section levels:\n\\begin{verbatim}\n\\section{Main section}\n \\subsection{Subsection}\n \\subsubsection{Subsubsection}\n \\paragraph{Lowest level section}\n\\end{verbatim}\n\\noindent The other \\LaTeX\\ sectioning commands \\verb'\\part', \\verb'\\chapter' and \\verb'\\subparagraph{}' are deprecated and should not be used.\n\nSome sections are not numbered as part of journal style (e.g. the Acknowledgements).\nTo insert an unnumbered section use the `starred' version of the command: \\verb'\\section*{}'.\n\nSee appendix~\\ref{sec:advanced} for more complicated examples.\n\n\\subsection{Lists}\n\nTwo forms of lists can be used in MNRAS -- numbered and unnumbered.\n\nFor a numbered list, use the \\verb'enumerate' environment:\n\\begin{verbatim}\n\\begin{enumerate}\n \\item First item\n \\item Second item\n \\item etc.\n\\end{enumerate}\n\\end{verbatim}\n\\noindent which produces\n\\begin{enumerate}\n \\item First item\n \\item Second item\n \\item etc.\n\\end{enumerate}\nNote that the list uses lowercase Roman numerals, rather than the \\LaTeX\\ default Arabic numerals.\n\nFor an unnumbered list, use the \\verb'description' environment without the optional argument:\n\\begin{verbatim}\n\\begin{description}\n \\item First item\n \\item Second item\n \\item etc.\n\\end{description}\n\\end{verbatim}\n\\noindent which produces\n\\begin{description}\n \\item First item\n \\item Second item\n \\item etc.\n\\end{description}\n\nBulleted lists using the \\verb'itemize' environment should not be used in MNRAS; it is retained for backwards compatibility only.\n\n\\section{Mathematics and symbols}\n\nThe MNRAS class mostly adopts standard \\LaTeX\\ handling of mathematics, which is briefly summarised here.\nSee also section~\\ref{sec:packages} for packages that support more advanced mathematics.\n\nMathematics can be inserted into the running text using the syntax \\verb'$1+1=2$', which produces $1+1=2$.\nUse this only for short expressions or when referring to mathematical quantities; equations should be entered as described below.\n\n\\subsection{Equations}\nEquations should be entered using the \\verb'equation' environment, which automatically numbers them:\n\n\\begin{verbatim}\n\\begin{equation}\n a^2=b^2+c^2\n\\end{equation}\n\\end{verbatim}\n\\noindent which produces\n\\begin{equation}\n a^2=b^2+c^2\n\\end{equation}\n\nBy default, the equations are numbered sequentially throughout the whole paper. If a paper has a large number of equations, it may be better to number them by section (2.1, 2.2 etc.). To do this, add the command \\verb'\\numberwithin{equation}{section}' to the preamble.\n\nIt is also possible to produce un-numbered equations by using the \\LaTeX\\ built-in \\verb'\\['\\textellipsis\\verb'\\]' and \\verb'$$'\\textellipsis\\verb'$$' commands; however MNRAS requires that all equations are numbered, so these commands should be avoided.\n\n\\subsection{Special symbols}\n\n\n\\begin{table}\n \\caption{Additional commands for special symbols commonly used in astronomy. These can be used anywhere.}\n \\label{tab:anysymbols}\n \\begin{tabular}{lll}\n \\hline\n Command & Output & Meaning\\\\\n \\hline\n \\verb'\\sun' & \\sun & Sun, solar\\\\[2pt]\n \\verb'\\earth' & \\earth & Earth, terrestrial\\\\[2pt]\n \\verb'\\micron' & \\micron & microns\\\\[2pt]\n \\verb'\\degr' & \\degr & degrees\\\\[2pt]\n \\verb'\\arcmin' & \\arcmin & arcminutes\\\\[2pt]\n \\verb'\\arcsec' & \\arcsec & arcseconds\\\\[2pt]\n \\verb'\\fdg' & \\fdg & fraction of a degree\\\\[2pt]\n \\verb'\\farcm' & \\farcm & fraction of an arcminute\\\\[2pt]\n \\verb'\\farcs' & \\farcs & fraction of an arcsecond\\\\[2pt]\n \\verb'\\fd' & \\fd & fraction of a day\\\\[2pt]\n \\verb'\\fh' & \\fh & fraction of an hour\\\\[2pt]\n \\verb'\\fm' & \\fm & fraction of a minute\\\\[2pt]\n \\verb'\\fs' & \\fs & fraction of a second\\\\[2pt]\n \\verb'\\fp' & \\fp & fraction of a period\\\\[2pt]\n \\verb'\\diameter' & \\diameter & diameter\\\\[2pt]\n \\verb'\\sq' & \\sq & square, Q.E.D.\\\\[2pt]\n \\hline\n \\end{tabular}\n\\end{table}\n\n\\begin{table}\n \\caption{Additional commands for mathematical symbols. These can only be used in maths mode.}\n \\label{tab:mathssymbols}\n \\begin{tabular}{lll}\n \\hline\n Command & Output & Meaning\\\\\n \\hline\n \\verb'\\upi' & $\\upi$ & upright pi\\\\[2pt]\n \\verb'\\umu' & $\\umu$ & upright mu\\\\[2pt]\n \\verb'\\upartial' & $\\upartial$ & upright partial derivative\\\\[2pt]\n \\verb'\\lid' & $\\lid$ & less than or equal to\\\\[2pt]\n \\verb'\\gid' & $\\gid$ & greater than or equal to\\\\[2pt]\n \\verb'\\la' & $\\la$ & less than of order\\\\[2pt]\n \\verb'\\ga' & $\\ga$ & greater than of order\\\\[2pt]\n \\verb'\\loa' & $\\loa$ & less than approximately\\\\[2pt]\n \\verb'\\goa' & $\\goa$ & greater than approximately\\\\[2pt]\n \\verb'\\cor' & $\\cor$ & corresponds to\\\\[2pt]\n \\verb'\\sol' & $\\sol$ & similar to or less than\\\\[2pt]\n \\verb'\\sog' & $\\sog$ & similar to or greater than\\\\[2pt]\n \\verb'\\lse' & $\\lse$ & less than or homotopic to \\\\[2pt]\n \\verb'\\gse' & $\\gse$ & greater than or homotopic to\\\\[2pt]\n \\verb'\\getsto' & $\\getsto$ & from over to\\\\[2pt]\n \\verb'\\grole' & $\\grole$ & greater over less\\\\[2pt]\n \\verb'\\leogr' & $\\leogr$ & less over greater\\\\\n \\hline\n \\end{tabular}\n\\end{table}\n\nSome additional symbols of common use in astronomy have been added in the MNRAS class. These are shown in tables~\\ref{tab:anysymbols}--\\ref{tab:mathssymbols}. The command names are -- as far as possible -- the same as those used in other major astronomy journals.\n\nMany other mathematical symbols are also available, either built into \\LaTeX\\ or via additional packages. If you want to insert a specific symbol but don't know the \\LaTeX\\ command, we recommend using the Detexify website\\footnote{\\url{http:\/\/detexify.kirelabs.org}}.\n\nSometimes font or coding limitations mean a symbol may not get smaller when used in sub- or superscripts, and will therefore be displayed at the wrong size. There is no need to worry about this as it will be corrected by the typesetter during production.\n\nTo produce bold symbols in mathematics, use \\verb'\\bmath' for simple variables, and the \\verb'bm' package for more complex symbols (see section~\\ref{sec:packages}). Vectors are set in bold italic, using \\verb'\\mathbfit{}'.\n\nFor matrices, use \\verb'\\mathbfss{}' to produce a bold sans-serif font e.g. \\mathbfss{H}; this works even outside maths mode, but not all symbols are available (e.g. Greek). For $\\nabla$ (del, used in gradients, divergence etc.) use \\verb'$\\nabla$'.\n\n\\subsection{Ions}\n\nA new \\verb'\\ion{}{}' command has been added to the class file, for the correct typesetting of ionisation states.\nFor example, to typeset singly ionised calcium use \\verb'\\ion{Ca}{ii}', which produces \\ion{Ca}{ii}.\n\n\\section{Figures and tables}\n\\label{sec:fig_table}\nFigures and tables (collectively called `floats') are mostly the same as built into \\LaTeX.\n\n\\subsection{Basic examples}\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{example}\n \\caption{An example figure.}\n \\label{fig:example}\n\\end{figure}\nFigures are inserted in the usual way using a \\verb'figure' environment and \\verb'\\includegraphics'. The example Figure~\\ref{fig:example} was generated using the code:\n\\begin{verbatim}\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{example}\n \\caption{An example figure.}\n \\label{fig:example}\n\\end{figure}\n\\end{verbatim}\n\n\\begin{table}\n \\caption{An example table.}\n \\label{tab:example}\n \\begin{tabular}{lcc}\n \\hline\n Star & Mass & Luminosity\\\\\n & $M_{\\sun}$ & $L_{\\sun}$\\\\\n \\hline\n Sun & 1.00 & 1.00\\\\\n $\\alpha$~Cen~A & 1.10 & 1.52\\\\\n $\\epsilon$~Eri & 0.82 & 0.34\\\\\n \\hline\n \\end{tabular}\n\\end{table}\nThe example Table~\\ref{tab:example} was generated using the code:\n\\begin{verbatim}\n\\begin{table}\n \\caption{An example table.}\n \\label{tab:example}\n \\begin{tabular}{lcc}\n \\hline\n Star & Mass & Luminosity\\\\\n & $M_{\\sun}$ & $L_{\\sun}$\\\\\n \\hline\n Sun & 1.00 & 1.00\\\\\n $\\alpha$~Cen~A & 1.10 & 1.52\\\\\n $\\epsilon$~Eri & 0.82 & 0.34\\\\\n \\hline\n \\end{tabular}\n\\end{table}\n\\end{verbatim}\n\n\\subsection{Captions and placement}\nCaptions go \\emph{above} tables but \\emph{below} figures, as in the examples above.\n\nThe \\LaTeX\\ float placement commands \\verb'[htbp]' are intentionally disabled.\nLayout of figures and tables will be adjusted by the publisher during the production process, so authors should not concern themselves with placement to avoid disappointment and wasted effort.\nSimply place the \\LaTeX\\ code close to where the figure or table is first mentioned in the text and leave exact placement to the publishers.\n\nBy default a figure or table will occupy one column of the page.\nTo produce a wider version which covers both columns, use the \\verb'figure*' or \\verb'table*' environment.\n\nIf a figure or table is too long to fit on a single page it can be split it into several parts.\nCreate an additional figure or table which uses \\verb'\\contcaption{}' instead of \\verb'\\caption{}'.\nThis will automatically correct the numbering and add `\\emph{continued}' at the start of the caption.\n\\begin{table}\n \\contcaption{A table continued from the previous one.}\n \\label{tab:continued}\n \\begin{tabular}{lcc}\n \\hline\n Star & Mass & Luminosity\\\\\n & $M_{\\sun}$ & $L_{\\sun}$\\\\\n \\hline\n $\\tau$~Cet & 0.78 & 0.52\\\\\n $\\delta$~Pav & 0.99 & 1.22\\\\\n $\\sigma$~Dra & 0.87 & 0.43\\\\\n \\hline\n \\end{tabular}\n\\end{table}\nTable~\\ref{tab:continued} was generated using the code:\n\n\\begin{verbatim}\n\\begin{table}\n \\contcaption{A table continued from the previous one.}\n \\label{tab:continued}\n \\begin{tabular}{lcc}\n \\hline\n Star & Mass & Luminosity\\\\\n & $M_{\\sun}$ & $L_{\\sun}$\\\\\n \\hline\n $\\tau$~Cet & 0.78 & 0.52\\\\\n $\\delta$~Pav & 0.99 & 1.22\\\\\n $\\sigma$~Dra & 0.87 & 0.43\\\\\n \\hline\n \\end{tabular}\n\\end{table}\n\\end{verbatim}\n\nTo produce a landscape figure or table, use the \\verb'pdflscape' package and the \\verb'landscape' environment.\nThe landscape Table~\\ref{tab:landscape} was produced using the code:\n\\begin{verbatim}\n\\begin{landscape}\n \\begin{table}\n \\caption{An example landscape table.}\n \\label{tab:landscape}\n \\begin{tabular}{cccccccccc}\n \\hline\n Header & Header & ...\\\\\n Unit & Unit & ...\\\\\n \\hline\n Data & Data & ...\\\\\n Data & Data & ...\\\\\n ...\\\\\n \\hline\n \\end{tabular}\n \\end{table}\n\\end{landscape}\n\\end{verbatim}\nUnfortunately this method will force a page break before the table appears.\nMore complicated solutions are possible, but authors shouldn't worry about this.\n\n\\begin{landscape}\n \\begin{table}\n \\caption{An example landscape table.}\n \\label{tab:landscape}\n \\begin{tabular}{cccccccccc}\n \\hline\n Header & Header & Header & Header & Header & Header & Header & Header & Header & Header\\\\\n Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit \\\\\n \\hline\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n \\hline\n \\end{tabular}\n \\end{table}\n\\end{landscape}\n\n\\section{References and citations}\n\n\\subsection{Cross-referencing}\n\nThe usual \\LaTeX\\ commands \\verb'\\label{}' and \\verb'\\ref{}' can be used for cross-referencing within the same paper.\nWe recommend that you use these whenever relevant, rather than writing out the section or figure numbers explicitly.\nThis ensures that cross-references are updated whenever the numbering changes (e.g. during revision) and provides clickable links (if available in your compiler).\n\nIt is best to give each section, figure and table a logical label.\nFor example, Table~\\ref{tab:mathssymbols} has the label \\verb'tab:mathssymbols', whilst section~\\ref{sec:packages} has the label \\verb'sec:packages'.\nAdd the label \\emph{after} the section or caption command, as in the examples in sections~\\ref{sec:sections} and \\ref{sec:fig_table}.\nEnter the cross-reference with a non-breaking space between the type of object and the number, like this: \\verb'see Figure~\\ref{fig:example}'.\n\nThe \\verb'\\autoref{}' command can be used to automatically fill out the type of object, saving on typing.\nIt also causes the link to cover the whole phrase rather than just the number, but for that reason is only suitable for single cross-references rather than ranges.\nFor example, \\verb'\\autoref{tab:journal_abbr}' produces \\autoref{tab:journal_abbr}.\n\n\\subsection{Citations}\n\\label{sec:cite}\n\nMNRAS uses the Harvard -- author (year) -- citation style, e.g. \\citet{author2013}.\nThis is implemented in \\LaTeX\\ via the \\verb'natbib' package, which in turn is included via the \\verb'usenatbib' package option (see section~\\ref{sec:options}), which should be used in all papers.\n\nEach entry in the reference list has a `key' (see section~\\ref{sec:ref_list}) which is used to generate citations.\nThere are two basic \\verb'natbib' commands:\n\\begin{description}\n \\item \\verb'\\citet{key}' produces an in-text citation: \\citet{author2013}\n \\item \\verb'\\citep{key}' produces a bracketed (parenthetical) citation: \\citep{author2013}\n\\end{description}\nCitations will include clickable links to the relevant entry in the reference list, if supported by your \\LaTeX\\ compiler.\n\n\\defcitealias{smith2014}{Paper~I}\n\\begin{table*}\n \\caption{Common citation commands, provided by the \\texttt{natbib} package.}\n \\label{tab:natbib}\n \\begin{tabular}{lll}\n \\hline\n Command & Ouput & Note\\\\\n \\hline\n \\verb'\\citet{key}' & \\citet{smith2014} & \\\\\n \\verb'\\citep{key}' & \\citep{smith2014} & \\\\\n \\verb'\\citep{key,key2}' & \\citep{smith2014,jones2015} & Multiple papers\\\\\n \\verb'\\citet[table 4]{key}' & \\citet[table 4]{smith2014} & \\\\\n \\verb'\\citep[see][figure 7]{key}' & \\citep[see][figure 7]{smith2014} & \\\\\n \\verb'\\citealt{key}' & \\citealt{smith2014} & For use with manual brackets\\\\\n \\verb'\\citeauthor{key}' & \\citeauthor{smith2014} & If already cited in close proximity\\\\\n \\verb'\\defcitealias{key}{Paper~I}' & & Define an alias (doesn't work in floats)\\\\\n \\verb'\\citetalias{key}' & \\citetalias{smith2014} & \\\\\n \\verb'\\citepalias{key}' & \\citepalias{smith2014} & \\\\\n \\hline\n \\end{tabular}\n\\end{table*}\n\nThere are a number of other \\verb'natbib' commands which can be used for more complicated citations.\nThe most commonly used ones are listed in Table~\\ref{tab:natbib}.\nFor full guidance on their use, consult the \\verb'natbib' documentation\\footnote{\\url{http:\/\/www.ctan.org\/pkg\/natbib}}.\n\nIf a reference has several authors, \\verb'natbib' will automatically use `et al.' if there are more than two authors. However, if a paper has exactly three authors, MNRAS style is to list all three on the first citation and use `et al.' thereafter. If you are using \\bibtex\\ (see section~\\ref{sec:ref_list}) then this is handled automatically. If not, the \\verb'\\citet*{}' and \\verb'\\citep*{}' commands can be used at the first citation to include all of the authors.\n\n\\subsection{The list of references}\n\\label{sec:ref_list}\n\nIt is possible to enter references manually using the usual \\LaTeX\\ commands, but we strongly encourage authors to use \\bibtex\\ instead.\n\\bibtex\\ ensures that the reference list is updated automatically as references are added or removed from the paper, puts them in the correct format, saves on typing, and the same reference file can be used for many different papers -- saving time hunting down reference details.\nAn MNRAS \\bibtex\\ style file, \\verb'mnras.bst', is distributed as part of this package.\nThe rest of this section will assume you are using \\bibtex.\n\nReferences are entered into a separate \\verb'.bib' file in standard \\bibtex\\ formatting.\nThis can be done manually, or there are several software packages which make editing the \\verb'.bib' file much easier.\nWe particularly recommend \\textsc{JabRef}\\footnote{\\url{http:\/\/jabref.sourceforge.net\/}}, which works on all major operating systems.\n\\bibtex\\ entries can be obtained from the NASA Astrophysics Data System\\footnote{\\label{foot:ads}\\url{http:\/\/adsabs.harvard.edu}} (ADS) by clicking on `Bibtex entry for this abstract' on any entry.\nSimply copy this into your \\verb'.bib' file or into the `BibTeX source' tab in \\textsc{JabRef}.\n\nEach entry in the \\verb'.bib' file must specify a unique `key' to identify the paper, the format of which is up to the author.\nSimply cite it in the usual way, as described in section~\\ref{sec:cite}, using the specified key.\nCompile the paper as usual, but add an extra step to run the \\texttt{bibtex} command.\nConsult the documentation for your compiler or latex distribution.\n\nCorrect formatting of the reference list will be handled by \\bibtex\\ in almost all cases, provided that the correct information was entered into the \\verb'.bib' file.\nNote that ADS entries are not always correct, particularly for older papers and conference proceedings, so may need to be edited.\nIf in doubt, or if you are producing the reference list manually, see the MNRAS instructions to authors$^{\\ref{foot:itas}}$ for the current guidelines on how to format the list of references.\n\n\\section{Appendices and online material}\n\nTo start an appendix, simply place the \\verb'\n\\section{Introduction}\n\nThis is a simple template for authors to write new MNRAS papers.\nSee \\texttt{mnras\\_sample.tex} for a more complex example, and \\texttt{mnras\\_guide.tex}\nfor a full user guide.\n\nAll papers should start with an Introduction section, which sets the work\nin context, cites relevant earlier studies in the field by \\citet{Others2013},\nand describes the problem the authors aim to solve \\citep[e.g.][]{Author2012}.\n\n\\section{Methods, Observations, Simulations etc.}\n\nNormally the next section describes the techniques the authors used.\nIt is frequently split into subsections, such as Section~\\ref{sec:maths} below.\n\n\\subsection{Maths}\n\\label{sec:maths}\n\nSimple mathematics can be inserted into the flow of the text e.g. $2\\times3=6$\nor $v=220$\\,km\\,s$^{-1}$, but more complicated expressions should be entered\nas a numbered equation:\n\n\\begin{equation}\n x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}.\n\t\\label{eq:quadratic}\n\\end{equation}\n\nRefer back to them as e.g. equation~(\\ref{eq:quadratic}).\n\n\\subsection{Figures and tables}\n\nFigures and tables should be placed at logical positions in the text. Don't\nworry about the exact layout, which will be handled by the publishers.\n\nFigures are referred to as e.g. Fig.~\\ref{fig:example_figure}, and tables as\ne.g. Table~\\ref{tab:example_table}.\n\n\\begin{figure}\n\n\n\n\t\\includegraphics[width=\\columnwidth]{example}\n \\caption{This is an example figure. Captions appear below each figure.\n\tGive enough detail for the reader to understand what they're looking at,\n\tbut leave detailed discussion to the main body of the text.}\n \\label{fig:example_figure}\n\\end{figure}\n\n\\begin{table}\n\t\\centering\n\t\\caption{This is an example table. Captions appear above each table.\n\tRemember to define the quantities, symbols and units used.}\n\t\\label{tab:example_table}\n\t\\begin{tabular}{lccr}\n\t\t\\hline\n\t\tA & B & C & D\\\\\n\t\t\\hline\n\t\t1 & 2 & 3 & 4\\\\\n\t\t2 & 4 & 6 & 8\\\\\n\t\t3 & 5 & 7 & 9\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\n\\section{Conclusions}\n\nThe last numbered section should briefly summarise what has been done, and describe\nthe final conclusions which the authors draw from their work.\n\n\\section*{Acknowledgements}\n\nThe Acknowledgements section is not numbered. Here you can thank helpful\ncolleagues, acknowledge funding agencies, telescopes and facilities used etc.\nTry to keep it short.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}