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+{"text":"\\section{Introduction}\n\n\n\n\nNext generation cellular networks \\cite{QWu2017}\\cite{MAgiwal2016} are envisioned to have some unique beneficial characteristics, among which ultra-dense deployment \\cite{XGe2016} of network nodes is one of the most crucial paradigms. Ideally, ultra-dense deployment of nodes enhances the coverage and capacity of the system \\cite{LZhou2014}. However, there are still some problems with this concept if the deployment and system parameters are not properly optimized. Since the number of transmitters in such networks is very large, this causes huge interference. Consequently, energy-efficient transmission is one of the important research endeavors in this arena. In order to obtain fruitful outcome in this context, designing an effective and efficient radio access technology \\cite{TEdler2014} is one of the possible solutions. Non-orthogonal multiple access (NOMA) is considered as the most promising radio access technique for next generation wireless systems \\cite{ZDing2014}\\cite{KHIGUCHI2015}\\cite{VWong2017}. Through experimentation and theoretical analysis \\cite{Higuchi2013}\\cite{YEndo2012}\\cite{JUmehara2012}\\cite{NOtao2012}, it is proved that the NOMA technique is able to provide enhanced performance in different sectors of wireless communication systems comparing with other orthogonal multiple access (OMA) techniques. Conceptually, power-domain NOMA \\cite{SMRIslam2017} allows multiple users to occupy the same resource. This leads to additional interference for NOMA-equipped networks and their neighboring networks. Consequently, existing resource management techniques \\cite{DFeng2013} in conventional networks, especially the energy-efficient ones, need to be revisited due to the incorporation of additional interference this new technology brings. \n\nAlthough the NOMA technique allows multiple users to be superimposed on the same frequency channel, due to the usage of practical, finite-length and possibly non-Gaussian codes, it is not an optimal design to assign large number users to the same channel. Consequently, dedicated spectrum of a system needs to be subdivided into multiple subchannels in order to support increased number of users~\\cite{LDai2015}. At the same time, how to allocate these subchannels among the users in a multiplexed manner given the maximum allowable number of users that can utilize a subchannel simultaneously, is an important problem. While targeting on the maximization of the overall throughput, some research has been conducted on the downlink and uplink subchannel and power allocation in such NOMA systems. Based on some assumption of having constant power on the subchannels, existing works typically provide some heuristic solutions for the subchannel-user mapping problem. Once the subchannel-user mapping is done, in order to enhance the performance of the system further, different existing works have provided different schemes for the power allocation. For example, in~\\cite{YSaito2013, ABenjebbour2013}, the authors use the fractional transmit power allocation technique among users and apply the equal power allocation technique across subchannels.~\\cite{MRHojeij2015} uses the water filling-based approach for the power allocation, and in \\cite{PParida2014}, the authors use difference of convex (DC) programming-based approach~\\cite{NVucic2010} for the power allocation at both the user and subchannel levels. For uplink systems, there are some works as well, such as~\\cite{KKumaran2003, MAlImari2015, MMollanoori2014}. In~\\cite{MAlImari2015}, the authors used the iterative water filling idea~\\cite{WeiYu2004} for both the subchannel-user mapping problem and their granular power allocation. On the other hand, in~\\cite{MMollanoori2014}, the authors assumed that the resources are time slots instead of frequency channels. Although the work in~\\cite{MAlImari2015} can ensure that users can be assigned to multiple subchannels, each user is assigned to at most one resource block via the scheme in~\\cite{MMollanoori2014}. While the base station (BS) of all these works is assumed to be equipped with the half-duplex functionality, the authors in \\cite{YSun2017} have proposed a joint subchannel and power allocation scheme for such a system based on the assumption that the BS is equipped with the full-duplex functionality to serve uplink and downlink users simultaneously. \n\n\\begin{comment}\nAlthough NOMA technique allows multiple users to be superimposed on the same frequency channel, due to the error propagation in the successive interference cancellation (SIC) mechanism~\\cite{NIMiridakis2013}, it is not an optimal assign to assign large number users on the same channel. Consequently, dedicated spectrum of a system needs to be subdivided into multiple subchannels in order to support increased number of users. At the same time, how to allocate these subchannels among users in a multiplexed manner given the allowable maximum number of users that can use a subchannel simultaneously, is an important problem. While targeting on maximizing overall throughput of the system, some research has been conducted on the downlink and uplink subchannel and power allocation in such NOMA systems. Based on some assumption of having constant power on the subchannels, typically, existing works provide some heuristics for subchannel-user mapping task. Once the subchannel-user mapping is done, different existing works provide different schemes for power allocation. For example, for downlink systems, in \\cite{YSaito2013, ABenjebbour2013}, the authors use fractional transmit power allocation technique among users and equal power allocation across subchannels.~\\cite{MRHojeij2015} uses water filling based approach for power allocation, and in \\cite{PParida2014}, the authors use difference of convex (DC) programming-based approach for the power allocation in both user and subchannel levels. For uplink systems, there are some works as well, such as~\\cite{KKumaran2003, MMollanoori2014}. \n\nIn~\\cite{KKumaran2003}, the authors used iterative water filling idea~\\cite{WeiYu2004} for both subchannel and user mapping and their granular power allocation. On the other hand, in~\\cite{MMollanoori2014}, the authors assumed that the resource is time slot instead of frequency channel. Although the work in~\\cite{KKumaran2003} can ensure that users can obtain multiple subchannels, each user obtains at most one time slot via the scheme in~\\cite{MMollanoori2014}. \n\\end{comment}\n\nWith the increasing desire to have green communications in the recent years, reducing energy consumption has become the prime concern for researchers, and the fifth generation (5G) systems have also targeted energy efficiency as one of the major milestones to achieve~\\cite{AAbrol2016}. Although the minimization of used power reported in \\cite{LeiLei2016} could be one of the objectives of energy-efficient transmissions, the resultant solution of such a type of formulation is not spectrum-efficient. In order to achieve spectrum efficiency with the minimal power, the ideal objective of an energy-efficient transmission is to maximize the achievable bits of a channel under the unit Joule of power consumption~\\cite{OJumira2012}. For a NOMA-equipped single cell, which has two users, an energy-efficient power allocation scheme is studied in~\\cite{SHan2014}. In this work, the authors found the relationship between energy efficiency and spectrum efficiency under the total system power constraint. At the same time, energy efficiency is also studied for NOMA-equipped MIMO systems in~\\cite{QSun2015}. The closest to ours and the most recent energy-efficient resource allocation scheme appeared in~\\cite{FFang2016}. The system model that this paper considered is similar to ours, and our target is to enhance the performance of the proposed scheme in this paper further. This work has proposed an energy-efficient downlink subchannel and power allocation scheme for a cellular network under total power constraint at the BS with a number of drawbacks. For example, through their scheme, each user cannot be assigned to multiple subchannels, the concept of which fails to exploit the multi-user diversity of wireless systems~\\cite{DavidTse2005}. Moreover, each subchannel can be assigned to at most two users, which also conflicts with the NOMA concept. Furthermore, they have used the DC programming-based approach in order to allocate power across all the subchannels and users of the system. Actually, through this power allocation, the entire power at the BS is used up, and hence the system fails to reach the optimal energy-efficient state. On the other hand, they also have utilized the DC programming-based approach in order to calculate the optimal instantaneous rate of a subchannel, which is an essential intermediate step of their subchannel-user mapping algorithm. However, the DC programming-based approach is appropriate only in this case because of provisioning at most two users to a subchannel. If the allowable number of users per subchannel is increased, the DC programming-based approach is not appropriate to solve this problem any more as the problem can no longer be expressed as a difference of convex functions under this realistic consideration.\n\n\n\n\n\nThe contribution of this paper is the design of an energy-efficient downlink subchannel and power allocation scheme in a cellular network with enhanced performance compared to the work in~\\cite{FFang2016}. Since this problem considers subchannels assignment which are associated with discrete variables in the formulated problem, the problem is NP-hard in general. Moreover, even if the subchannel assignment information is known, the power allocation problem is non-convex~\\cite{DPBertsekas1999}. As a result, joint subchannel assignment and power allocation of this problem can be considered as an mixed integer non-linear programming (MINLP) problem. Considering the importance of the overall energy efficiency for such a system from the perspective of green communications, since the existing solution of~\\cite{FFang2016} is approximate, the investigation of the better ones is required. Consequently, we find that decomposing the problem into a subchannel allocation problem followed by a power loading problem and then solving each problem individually while considering inter-dependency, is an elegant approach to solve this problem. In the first step, under the assumption that total power of the BS is subdivided equally among all the subchannels, we solve the subchannel-user mapping problem via a many-to-many matching model~\\cite{KHamidouche2014, ARoth1984}. However, unlike the scheme in~\\cite{FFang2016}, our proposed subchannel-user mapping scheme can utilize the multi-user diversity efficiently. On the other hand, one of the assumptions of the subchannel-users mapping algorithm in~\\cite{FFang2016} is, at most $2$ users can be assigned to a subchannel. Because of this assumption, the power allocation problem of a subchannel is amenable to the DC programming-based approach. If a subchannel is assigned to more than $2$ users, the power allocation problem is no longer amenable to the DC programming-based approach. In contrast, our proposed scheme is general, and the number of users assigned to a subchannel is arbitrary. Our discovery to solve this problem via the geometric programming (GP)-based~\\cite{Boyd2007, MChiang2005} approach is one of the specific contributions of this paper. Then, in the second step, once the subchannel-user mapping information are known, we adopt the GP-based approach in order to allocate power across all subchannel-user slots upon approximating the energy efficiency of system by a ratio of two posynomials. The DC programming-based approach in~\\cite{FFang2016} allocates power across the subchannels while ignoring the detailed granularity of the user power level. Our proposed power loading scheme using the GP-based approach allocates power to each user's data stream on each of its allocated subchannels. Compared to the most relevant existing work~\\cite{FFang2016}, the major contributions of the paper are summarized as follows. \n\n\n\n\n\\begin{itemize}\n\n\\item For the subchannel-user mapping task, unlike the one in~\\cite{FFang2016}, we have adopted a many-to-many matching model that exploits the multi-user diversity of wireless systems efficiently. Through this scheme, not only each subchannel can be used to serve multiple users, but also each user can utilize multiple subchannels. However, assigning more and more users to a subchannel cannot necessarily enhance the energy efficiency of the system further. Our proposed algorithm can smartly adapt user assignment to each subchannel given the information of some maximal allowable number of users to each subchannel.\n\n\\item While solving the subchannel-user mapping problem, it is required to determine the optimal instantaneous sum-rate of a subchannel in the intermediate decision making steps. When a subchannel can serve at most $2$ users, the approach in~\\cite{FFang2016} is appropriate to determine the instantaneous sum-rate of a subchannel. However, in practice, having more than $2$ users on a subchannel can enhance the system performance. On the other hand, when a subchannel is assigned to more than $2$ users, the DC programming-based approach is no longer a suitable method to determine the optimal instantaneous sum-rate of that subchannel. This is because the objective function cannot be expressed as a difference of convex functions in this case. However, this problem is amenable to GP via the single condensation heuristic method after approximating the objective function of the problem by a ratio of two posynomials. Consequently, upon the approximation, we have adopted the single condensation method to determine the optimal instantaneous sum-rate of each subchannel while making the decision about each subchannel-user assignment.\n\n\\item Once the subchannel-user mapping information are known from the first step, the DC programming-based approach in~\\cite{FFang2016} can allocate power optimally across the subchannels. However, this power allocation scheme ignores the detailed granularity of each subchannel-user slot, and the entire power of the BS is used up. While targeting on the optimality and considering the detailed granularity of each subchannel-user slot, we approximate the original problem via replacing the log function by the first term of its approximated series~\\cite{RNave} to facilitate the problem solving via the GP-based approach. Upon approximating the objective function of the problem via a ratio of two posynomials, we adopt the GP-based single condensation heuristic to solve the approximated problem. The proposed scheme allocates power to each user's data stream on each of its allocated subchannels. As a result, the total required power, to reach the optimal overall energy-efficient state, is much less than the total power of the BS. From the perspective of green communications, this phenomenon of our power loading scheme is a big advantage over the DC programming-based approach. Although the proposed GP-based approach provides fine-grained power allocation across each subchannel and user with higher energy efficiency, this approach is computationally intensive with the growing number of users and subchannels due to the implementation limitation of the off-the-shelf GP solvers~\\cite{cvx}. Therefore, based on the insights of the optimal solution, we also have proposed a computationally-efficient suboptimal solution for the fine-grained energy-efficient power loading problem, which has a polynomial time complexity.\n\n\\item Extensive simulation has been conducted in order to verify the effectiveness of our proposed resource allocation scheme. The results demonstrate that our scheme always outperforms the scheme in~\\cite{FFang2016} under various realistic scenarios.\n\n\\end{itemize}   \n\n\n\nThe rest of the paper is organized as follows. Along with the background information and the description of the system, in Section~\\ref{sec:sysmodel}, we formulate our energy-efficient resource allocation problem. The detailed solution approach is provided in Section~\\ref{sec:sol}. Followed by the simulation methodology, we evaluate the performance of our proposed resource allocation schemes in Section~\\ref{sec:eval}. Finally, Section~\\ref{sec:concl} concludes the paper with some implication.    \n\n\n\\section{System Model and Problem Formulation}\n\\label{sec:sysmodel}\n\nIn this paper, we consider a downlink scenario of a cellular network, which has one BS. Time is divided into frames, and the entire pre-assigned spectrum for the system is divided into $N$ subchannels in each frame. The resultant subchannels are the elements of a set, denoted by \\textbf{N}. There are $M$ users in the system, and the corresponding set holding these users is denoted by \\textbf{M}. Using the subchannels in set \\textbf{N} as the transmission media, the BS transmits data to the users in set \\textbf{M}. Both the BS and the users in the system are equipped with NOMA technologies. The BS transmits its data to a set of users using the superposition coding (SC) technique over a set of subchannels. Whereas, the receivers (i.e., the users) apply the SIC technique on each subchannel to decode the superimposed signals for extracting their own individual signal. However, before the downlink transmission operation, it is required to schedule subchannels and power across the users optimally so that the overall energy efficiency of the system is maximized. We assume that the scheduling scheme in the system is centralized, and the BS is appointed to conduct the entire scheduling operation. To develop this scheduling scheme, since the entire channel state information (CSI) of the system is required, the BS is aware of all these information. At the beginning of each time frame, all users send their CSI to the BS via some reliable control channels. The CSI of the subchannels in set \\textbf{N} follows the block fading model~\\cite{DavidTse2005}, where the CSI of each subchannel is constant for a time slot and varies in an i.i.d. manner from time slot to time slot. The BS has the maximal power constraint, denoted by $p^{\\mbox{max}}$. \n\n\n\nWe assume that the BS assigns $M_n$ users to the $n$th subchannel, and the corresponding set holding these users is denoted by $\\textbf{M}_n$. Given this, let denote the symbol transmitted by the BS on subchannel $n$ as $x_n$, and it is given by $\\sum_{m \\in \\textbf{M}_n}\\sqrt{p_m^n}{s_m}$. Here, $s_m$ is the modulated symbol of the $m$th user on subchannel $n$ and $p_m^n$ is the power level assigned to user $m$ on subchannel $n$. Consequently, the received signal of user $m$ on subchannel $n$ can be represented as\n\n\n\n\\[\ny_m^n = \\sqrt{p_m^n}h_m^ns_m + \\displaystyle\\sum_{i=1, i \\neq m}\\sqrt{p_i^n}h_m^ns_i + z_n,\n\\]\n\n\n\n\\noindent \nwhere $h_m^n$ is the channel gain of user $m$ on the $n$th subchannel. $z_n$ is the noise power over subchannel $n$, which follows Additive White Gaussian Noise (AWGN)~\\cite{KMcClaning2000} distribution with mean zero and variance ${\\sigma}_n^2$, i.e., $z_n \\approx {\\cal{CN}}(0, {\\sigma}_n^2)$. The noise power of subchannel $n$ is statistically same for all users. In NOMA systems, each subchannel is shared by multiple users. Consequently, each user on subchannel $n$ receives its signal as well as the the interference signals from the other users that share the same subchannel. Let us denote the set holding the users that cause interference to user $m$ on subchannel $n$ as $\\textbf{M}_n^m$. Given the information of set $\\textbf{M}_n^m$, the received signal-to-interference-ratio (SINR) of the $m$th user on subchannel $n$ is given by\n\n\n\\begin{equation}\n\\mbox{SINR}_m^n = \\frac{p_m^n|h_m^n|^2}{{\\sigma}_n^2 + \\displaystyle\\sum_{i \\in \\textbf{M}_n^m}p_i^n|h_m^n|^2} = \\frac{p_m^n{g_m^n}}{{\\sigma}_n^2 + \\displaystyle\\sum_{i \\in \\textbf{M}_n^m}p_i^ng_m^n},\n\\end{equation}\n\n\\noindent\nwhere ${\\sigma}_n^2 = E[|z_n|^2]$ is the noise power on subchannel $n$, and $g_m^n = |h_m^n|^2$ represents the power gain of the $m$th user on subchannel $n$. \n\n\nSince the NOMA users are equipped with the SIC technique, the way a NOMA user retrieves information depends on the order of the allocated power level among all the NOMA users in the corresponding subchannel. In other words, a particular NOMA user can decode the information of other users that are assigned to larger power level compared to itself. On the other hand, this user can decode its own information considering the power level of other remaining users (that are assigned to lower power level compared to itself) as noise. Therefore, if the order of the power level assigned to the NOMA users are known, the decoding order is straightforward for a particular NOMA user and vice versa. It is clear that the optimal decoding order of the NOMA users and their power assignment are coupled to each other and is not a straightforward problem. However, in order to reduced the complexity of the proposed solution scheme in this paper, we intend to fix the order of the power level among the NOMA users while solving the entire energy efficiency problem. Ideally, in order to maximize the energy efficiency of the system, the optimal power allocation among the NOMA users of a subchannel should follow the water filling-based approach. On the other hand, when the fairness of the system is considered, the order of the ideal power allocation should be in-between the water filling-based approach and the opposite of the water filling one. If we would fix the order of the power level among the NOMA users according to the water filling norm, it is very likely that an ideal power loading scheme\\footnote{The proposed power loading scheme in this paper is particularly optimal for the applications that work in the low SNR regime.} assigns very low power to the users with worse channel condition. In this case, those users (with worse channel condition) may not be able to decode their information or this is not a fair attitude to those users. Therefore, while considering the fairness of the system, we have adopted the opposite norm of the water filling technique so that the users with worse channel have fair provision. In other words, in order to reduce the overall computational complexity of the solution schemes and considering the fairness of the system, similar to~\\cite{FFang2016}, the users with larger gain are assigned to lower power level compared to the users with lower gain. As a result, following the norm of the SIC technique, the set of users that cause interference to user $m$ on subchannel $n$ is given by $\\textbf{M}_n^m = \\{m'|_{g_{m'}^n > g_m^n}, m' \\in \\textbf{M}_n\\}$. Consequently, the sum-rate of subchannel $n$, denoted by $R_n(\\textbf{M}_n, \\{p_m^n\\})$, can be obtained by the Shannon's capacity formula~\\cite{HMichiel2001}.\n\n\\[\nR_n(\\textbf{M}_n, \\{p_m^n\\}) = \\displaystyle\\sum_{m \\in \\textbf{M}_n}\\mbox{log}_2\\left(1 + \\mbox{SINR}_m^n\\right).\n\\]\n\n\n\nIf the circuit power consumption on subchannel $n$ is denoted by $p_c$, the energy efficiency of subchannel $n$ is given by\n\n\\[\nE_n = \\frac{R_n(\\textbf{M}_n, \\{p_m^n\\})}{p_c + \\sum_{m \\in \\textbf{M}_n}p_m^n}.\n\\]     \n\n\n\n\nIn this work, given the power constraint of the BS, our objective is to allocate the subchannels in set \\textbf{N} across all the users in set \\textbf{M} so that the aggregate energy efficiency of all subchannels, $\\sum_{n \\in \\textbf{N}}E_n$, is maximized. Clearly, this is an optimization problem. To formulate this problem, we define binary variables ${\\beta}_m^n, m \\in \\textbf{M}_n, n \\in \\textbf{N}$. ${\\beta}_m^n = 1$ implies that subchannel $n$ is allocated to user $m$, and ${\\beta}_m^n = 0$ means the other case. Ideally, more and more users are assigned to a subchannel, the better the system performance. However, due to the increasing level of interference with more and more assigned users and the decoding complication of the SIC technique, not necessarily more users assigned to a subchannel will enhance the system energy efficiency. While giving weight to this observation and insight, we assume that maximum $K$ users can be assigned to a subchannel\\footnote{Setting $K$ to $M$, the multi-user diversity of a subchannel can be fully exploited. Due to the limited power level and the superposition-coded interference level caused by other users, an ideal resource allocation scheme should smartly select the users that are beneficial for that subchannel in terms of energy efficiency}. Therefore, if the achieved instantaneous rate of user $m$ is $R_m$ and its minimum required rate is $R_m^{\\mbox{min}}$ (to satisfy its quality-of-service requirement), the energy-efficient downlink subchannel and power allocation problem in this context can be formulated as (\\ref{eq:opt-prob1})-(\\ref{eq:opt-prob5}). Note that the aggregate power level of subchannel $n$ is denoted by $p_n$, and $\\sum_{m \\in \\textbf{M}_n}p_m^n = p_n$.\n\n\n\n\\begin{eqnarray}\n\\label{eq:opt-prob1} &\\displaystyle\\max_{\\{p_m^n, {\\beta}_m^n\\}}\\displaystyle\\sum_{n \\in \\textbf{N}}\\frac{1}{p_c + \\displaystyle\\sum_{m \\in \\textbf{M}_n}p_m^n}\\displaystyle\\sum_{m \\in \\textbf{M}_n}{\\beta}_m^n\\mbox{log}\\left(1 + \\mbox{SINR}_m^n\\right), \\\\\n& \\nonumber \\mbox{subject to} \\\\\n\\label{eq:opt-prob2} & \\displaystyle\\sum_{m \\in \\textbf{M}}{\\beta}_m^n \\le K,~ \\forall{n \\in \\textbf{N}},~~~\\displaystyle\\sum_{n \\in \\textbf{N}}{\\beta}_m^n \\le N,~ \\forall{m \\in \\textbf{M}}, \\\\\n\\label{eq:opt-prob3} & {\\beta}_m^n \\in \\{0, 1\\},~ \\forall{m \\in \\textbf{M}},~ \\forall{n \\in \\textbf{N}}, \\\\\n\\label{eq:opt-prob4} & \\displaystyle\\sum_{n \\in \\textbf{N}}\\displaystyle\\sum_{m \\in \\textbf{M}_n}{\\beta}_m^np_m^n \\le p^{\\mbox{max}},~ p_m^n \\ge 0, \\\\\n\\label{eq:opt-prob5}& R_m  \\ge R_m^{\\mbox{min}}, \\forall{m \\in \\textbf{M}}.\n\\end{eqnarray}\n\n\nIn the above formulation, there are two types of variables, i.e., $\\{{\\beta}_m^n\\}$ and $\\{p_m^n\\}$. $\\{{\\beta}_m^n\\}$ are the set of discrete variables, and the problem is NP-hard because of these variables. The NP-hardness property of this problem can be proved by mapping it to the classical relaxed bin-packing problem~\\cite{SMartello1990}. Let us assume that the subchannels in set \\textbf{N} are the possible bins, the users in set \\textbf{M} are items and the capacity of each bin is $K$. Now, if we assume that an item can reside into multiple bins and if we transform the objective of the problem from minimizing the number of used bins to maximizing the system energy efficiency, reduction of our problem to the relaxed bin packing problem is completed. Thus, the NP-hardness property of our problem is proved. On the other hand, even if the information of set $\\{{\\beta}_m^n\\}$ are known, it is straightforward to prove that the problem is non-convex with respect to (w.r.t.) $\\{p_m^n\\}$. Moreover, in order to reduce the complexity of the solution method proposed in the following section, the ordering of the power level and gain of the users in set $\\textbf{M}_n$ are considered to be fixed and given as follows. For any $2$ users $m$ and $m'$ in set $\\textbf{M}_n$, if they follow $g_m^n > g_{m'}^n$, they must hold $p_m^n < p_{m'}^n$.  \n\n\n\n\n\\section{Solution Approach}\n\\label{sec:sol}\n\n\nIn this section, we explore the solution approach of the energy-efficient downlink resource allocation problem of a NOMA system described in (\\ref{eq:opt-prob1})-(\\ref{eq:opt-prob5}). Apparently, due to the discrete nature of subchannel assignment (i.e., variables $\\{{\\beta}_m^n\\}$) and the continuous nature of power assignment (i.e., variables $\\{p_m^n\\}$), this is an MINLP problem. Furthermore, the assignment issue of users on the same subchannel brings further complication in the solution strategy. Therefore, we realize that decomposing the problem into a subchannel allocation subproblem followed by a power loading subproblem and then solving each subproblem based on the insights of the optimal solution, is a good strategy to solve the joint problem. Consequently, based on the assumption that the maximal power of the BS ($p^{\\mbox{max}}$) is equally subdivided among all the subchannels, we solve the subchannel-user mapping subproblem using a many-to-many matching model. Compared to the one-to-many matching model adopted in~\\cite{FFang2016}, we believe that a many-to-many one can capture the structure of the problem well. This is because one user can be assigned to multiple subchannels in order to exploit the multi-user diversity of wireless systems, and one subchannel can be assigned to multiple users to take the advantages of the NOMA technology. While solving the subchannel-user mapping subproblem, given the power constraint of each subchannel, it is required to assign power optimally among the allocated users so that the overall rate of that subchannel is maximized. For this purpose, we have noticed that if the value of $K$ is larger than $2$, the resultant optimization problem cannot be written as the difference of convex functions, and hence the DC programming-based approach is not appropriate to solve this problem. However, we found that GP is a well-fit technique to solve this problem after approximating the objective function of the original problem by a ratio of two posynomials. For the solution of the second part of the problem, we assume that the subchannel-user mapping information are available. Given this information,~\\cite{FFang2016} has utilized the DC programming-based approach to allocate power across the subchannels while ignoring the detailed granularity of the user power level. On the attempt of finding an elegant solution for the power loading subproblem, we find that the GP-based approach is a good one as well upon approximating the problem by a ratio of two posynomials. \n\n\\subsection{Subchannel and User Mapping Scheme}\nIntuitively, assignment of many subchannels to a user and allocating multiple users to a subchannel (to follow the guidelines of the NOMA technique) is expected to enhance the overall energy efficiency of the system. However, due to the usage of practical, finite-length and possibly non-Gaussian codes, we assume that maximum $K$ users can be assigned to a subchannel. Given the total power constraint of the BS, this problem is NP-hard. The nature of the problem implies that a many-to-many matching model~\\cite{KHamidouche2014, ARoth1984} is appropriate to capture the aforementioned behavior. Given that maximum $K$ users can be multiplexed to a subchannel, $M$ users in set \\textbf{M} and $N$ subchannels in set \\textbf{N} are two sets of players of this many-to-many matching model. Note that each user $m$ can have infinite ($N$ in practice) subchannels if possible. In this case, since the BS has the maximal power constraint $p^{max}$, this should be subdivided equally among all the subchannels. \n\n\\noindent\n\\textbf{Definition 1:} A many-to-many matching model $\\mu$ is a mapping from set \\textbf{M} to set \\textbf{N} such that every $m \\in \\textbf{M}$ and $n \\in \\textbf{N}$ satisfy the following properties:\n\n\\begin{itemize}\n\n\\item ${\\mu}(m) \\subseteq \\textbf{N}$ and ${\\mu}(n) \\subseteq \\textbf{M}$\n\n\\item $|{\\mu}(m)| \\le \\infty, \\forall{m \\in \\textbf{M}}$\n\n\\item $|{\\mu}(n)| \\le K, \\forall{n \\in \\textbf{N}}$\n\n\\item $n \\in {\\mu}(m)$ if and only if $m \\in {\\mu}(n)$\n\n\\end{itemize}\n\n\\noindent\nwhere ${\\mu}(m)$ is the set of partners for user $m$ and ${\\mu}(n)$ is the set of partners for subchannel $n$ under the matching model $\\mu$. The definition states that each user in set \\textbf{M} is matched to a subset of subchannels in set \\textbf{N}, and vice versa. However, before accomplishing these assignment operations, each user needs to have a preference list based on some criteria. The criterion of constructing the preference list for the users is based on their gain from the subchannels, which is similar to that in~\\cite{FFang2016}. The preference of each subchannel $n$ is based on the overall benefit (i.e., energy efficiency) of the system. For example, if user $m$ chooses subchannel $n$, this subchannel only accepts this user if and only the energy efficiency of the system is enhanced by this allocation. \n\n\n\nTo solve our subchannel-user mapping problem, we are interested to look at a stable solution, in which there are no players that are not matched to one another but they all prefer to be partners. Since the subchannel players give preference to the overall energy efficiency of the system while choosing partners from set \\textbf{M}, a stable solution is considered to be an elegant solution of this problem\\footnote{At this point, none of the players can enhance their performance further by choosing alternative partners, and hence this point is considered as a stable point.}. In the many-to-many matching models~\\cite{ARoth1984}, many stability concepts can be considered based on the number of players that can improve their utility by forming new partners among one another. However, due to the large number of players ($\\textbf{M} \\cup \\textbf{N}$) in our problem, identifying optimal subset of partners for a player is computationally intractable. Consequently, we choose to solve the matching problem by identifying the partners one by one from the opposite set. This way of choosing the partners in the matching relation brings pair-wise stability. In \\textit{Definition 1} and \\textit{Definition 2}, we highlight some properties of a pair-wise stable matching relation. For the sake of these definitions, we define some notations. For example, $\\textbf{C}_m(\\textbf{N})$ denotes the choice set of user $m$, which basically follows $\\textbf{C}_m(\\textbf{N}) \\subseteq \\textbf{N}$. Moreover, ${\\textbf{M}'}_n {\\succ}_m {\\textbf{M}''}_n$ implies that subchannel $n$ prefers the users in set  ${\\textbf{M}'}_n$ over that in set ${\\textbf{M}''}_n$.\n\n\\noindent\n\\textbf{Definition 1:} Consider that pair $(m,n)$ is not the element of matching model $\\mu$, that is $m \\not\\in {\\mu}(n)$ and $n \\not\\in {\\mu}(m)$. Now consider another pair $(\\phi, \\varphi)$, that satisfies $\\phi \\in \\textbf{C}_m({\\mu}(n) \\cup \\{n\\})$ and $\\varphi \\in \\textbf{C}_n({\\mu}(n) \\cup \\{m\\})$. For the matching relation $\\mu$ to be pairwise stable, it is not possible that both $\\phi {\\succ}_m {\\mu}(m)$ and $\\varphi {\\succ}_n {\\mu}(n)$ are satisfied.\n\n\n\\noindent\n\\textbf{Definition 2:} Let ${\\textbf{M}}_n$ (${\\textbf{M}}_n \\in \\textbf{M}$) be the set of matched users of subchannel $n$ under the matching relation $\\mu$, and $|{\\textbf{M}}_n| = K$. Consider $m$ is one of the matched users in set ${\\textbf{M}}_n$. Now, another user $m' \\in \\textbf{M}$ has come to be matched with subchannel $n$. Note that $m' \\not\\in {\\textbf{M}}_n$. However, if user $m$ is replaced by user $m'$, the performance of subchannel $n$ is enhanced. Therefore, in order to have stable matching, ${\\textbf{M}}_n = ({\\textbf{M}}_n \/\\ m) \\cup m'$ should hold. This phenomenon of stable matching is called substitutability. \n\n\n\n\nWhile satisfying the properties of a stable many-to-many matching relation (e.g, substitutability~\\cite{ARoth1984}), we have proposed an algorithm in \\textit{Algorithm~\\ref{alg:sc-usr-map}}. In order to bring the stability in this matching relation or enhance the overall energy efficiency of the system, we have adopted a few heuristics or strategies. The description of the algorithm is as follows. In order to realize the outcome of each step of the algorithm, we introduce one more type of set variable: ${\\bm{\\Omega}}_m$ that holds the allocated subchannels for user $m$. First, ${\\bm{\\Omega}}_m, m \\in \\textbf{M}$ and $\\textbf{M}_n, n \\in \\textbf{N}$ are initialized with $\\emptyset$. These sets are gradually filled up as we go through the iterations in between step $3$ and step $26$. At the initialization phase, each user $m \\in \\textbf{M}$ also constructs its subchannel preference list based on the descending order of their gain. Then, inside the outer-most loop (between step $3$ and step $26$), if no assignment is possible, the algorithm terminates\\footnote{At this point, it is assumed that the system has reached a stable situation or the improvement of energy efficiency is no longer possible.}. Inside the inner loop (between step $4$ and step $25$), each user $m$ chooses its most preferred unallocated (and not rejected already) subchannel $n$. At this point, two conditions are possible. The first condition is that the number of allocated users to subchannel $n$ can be less than $K$ (maximum allowable number of users per subchannel), and the second condition is the other case. If the first condition is true, we can apply the addition strategy for this subchannel-user assignment: user $m$ can be added to this subchannel and this strategy is added to set \\textbf{S} (which was initialized before sorting out the possible strategies for use $m$). Whereas, for the second condition, only substitution strategy (already allocated user $m'$ can be replaced by new user $m$) is possible. After filling the strategy set \\textbf{S}, no matter the number of allocated users to subchannel $n$ is less than or equal to $K$, the elements of \\textbf{S} are filtered based on the previous sum-rate of subchannel $n$ before this new possible assignment. The filtered strategy set is \\textbf{CS} in \\textit{Algorithm~\\ref{alg:sc-usr-map}}. Finally, $s^{\\mbox{Best}}$ strategy is chosen based on the sum-rate (${\\Gamma}_{s}^n$ in \\textit{Algorithm~\\ref{alg:sc-usr-map}}) each strategy achieves. If $s^{\\mbox{Best}}$ is empty, the innermost loop continues, and the next user is chosen from set \\textbf{M} for building its possible strategy set. For the other case, the corresponding strategy is executed. As a result, set ${\\bm{\\Omega}}_m$, set ${\\bm{\\Omega}}_{m'}$ (only for the substitution strategy) and set $\\textbf{M}_n$ are updated. By analyzing the algorithm, we conclude \\textit{Proposition 1}, \\textit{Proposition 2} and \\textit{Theorem 1}. \n\n\n\n\\begin{algorithm}\n\\caption{The energy-efficient subchannel-user mapping algorithm via a many-to-many matching model.}\n\\label{alg:sc-usr-map}\n\\begin{algorithmic}[1]\n\\STATE ${\\bm{\\Omega}}_m \\leftarrow \\emptyset, \\forall{m \\in \\textbf{M}}$; $\\textbf{M}_n \\leftarrow \\emptyset, \\forall{n \\in \\textbf{N}}$.\n\\STATE Each user $m \\in \\textbf{M}$ produces its preference list in the descending order of its gain on subchannel $n \\in \\textbf{N}$\n\n\\REPEAT\n\\FOR{$m \\in \\textbf{M}$}\n\n\\STATE $n$ $\\leftarrow$ The best unallocated (and not rejected previously) subchannel of user $m$. \n\\STATE ${\\Gamma}_{\\mbox{prev}}^n \\leftarrow$ The aggregate rate of subchannel $n$.\n\\STATE $\\textbf{S} \\leftarrow \\emptyset$, $\\textbf{CS} \\leftarrow \\emptyset$, and $\\textbf{TRate} \\leftarrow \\emptyset$. \n\\IF{The number of assigned users on subchannel $n$ is less than $K$}\n\\STATE Construct strategy $s$ by adding user $m$ to set $\\textbf{M}_n$, and add $s$ to set \\textbf{S}.\n\\ELSIF{The number of assigned users on subchannel $n$ is equal to $K$}\n\\STATE Construct each strategy $s$ by replacing each user $m' \\in \\textbf{M}_n$ by user $m$, and add $s$ to set \\textbf{S}.\n\\ENDIF\n\\FOR{$s \\in \\textbf{S}$}\n\\STATE Due to strategy $s$, determine the power level of user $m$ in $\\textbf{M}_n$ using (18)~\\cite{FFang2016} or the approach in Section~\\ref{sssec:GP-indiv}, and ${\\Gamma}_{s}^n \\leftarrow$ The aggregate rate of subchannel $n$ due to strategy $s$.\n\\STATE Add ${\\Gamma}_{s}^n$ to set \\textbf{TRate} and add strategy $s$ to \\textbf{CS} if and only if ${\\Gamma}_{s}^n - {\\Gamma}_{\\mbox{prev}}^n > 0$.\n\\ENDFOR\n\n\\STATE $s^{\\mbox{Best}} \\leftarrow \\argmax_{s \\in \\textbf{CS}}\\textbf{TRate}(s)$.\n\\IF{$s^{\\mbox{Best}}$ is the substitution strategy}\n\\STATE ${\\bm{\\Omega}}_m \\leftarrow {\\bm{\\Omega}}_m \\cup \\{n\\}$, ${\\bm{\\Omega}}_{m'} \\leftarrow {\\bm{\\Omega}}_{m'} \/\\ \\{n\\}$, and $\\textbf{M}_n \\leftarrow (\\textbf{M}_n \/\\ \\{m'\\}) \\cup \\{m\\}$. \\COMMENT{$m'$ is the to-be-replaced user and $m$ is the new user on subchannel $n$}\n\\STATE Update the preference list of user $m$ and user $m'$.\n\\ELSIF{$s^{\\mbox{Best}}$ is the addition strategy}\n\\STATE ${\\bm{\\Omega}}_m \\leftarrow {\\bm{\\Omega}}_m \\cup \\{n\\}, \\textbf{M}_n \\leftarrow \\textbf{M}_n \\cup \\{m\\}$.\n\\STATE Update the preference list of user $m$.\n\\ENDIF\n\\ENDFOR\n\\UNTIL{The overall performance of the system cannot be enhanced}\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\n\n\n\n\n\n\n\n\\noindent\n\\textbf{Proposition 1:} Given a certain value of $K$, allocating more and more users to a subchannel cannot enhance the energy efficiency further.\n\n\n\\noindent\n\\textbf{Proof:} See \\textit{Appendix A}.\n\n\n\n\\noindent\n\\textbf{Proposition 2:} Once a user is rejected while to be assigned with a subchannel through the addition or the substitution strategy, that rejection is final. More precisely, even if that user wants to be assigned to that particular subchannel later, this assignment no longer enhances the system energy efficiency.\n\n\n\n\\noindent\n\\textbf{Proof:} See \\textit{Appendix B}.\n\n\n\n\\noindent\n\\textbf{Theorem 1:} The subchannel-user mapping algorithm (i.e., \\textit{Algorithm 1}) is guaranteed to converge to a pair-wise stable matching relation.\n\n\\noindent\n\\textbf{Proof:} See \\textit{Appendix C}.\n\n\\noindent\n\\textbf{\\textit{Computational Complexity of \\textit{Algorithm~\\ref{alg:sc-usr-map}}:}} As discussed and proved in \\textit{Proposition 2}, if user $m$ is rejected by subchannel $n$ once, that user does not propose subchannel $n$ to be matched with anymore. This is because the energy efficiency of this subchannel $n$ is not enhanced in any way if previously rejected user is given preference in the following iterations. The enhancement of the energy efficiency for each subchannel implies the enhancement of the overall energy efficiency in the system. The termination of the outer-most loop (between step $3$ and step $26$) depends on the improvement of the overall energy efficiency. To be precise, if user $m$ in the inner loop proposes a subchannel $n$ and the corresponding $s^{\\mbox{Best}}$ results in empty value, the next user in set \\textbf{M} builds its possible strategy set. If none of the users in set \\textbf{M} can be matched with any of the subchannel in set \\textbf{N}, the outer-most loop terminates. Therefore, intuitively, each user $m \\in \\textbf{M}$ approaches at most $N$ subchannels to be matched with. Consequently, the outer-most loop runs at most $MN$ times. On the other hand, the complexity of finding the best strategy for each user inside the inner loop is $\\mbox{O}(K)$, and the corresponding reason is as follows. If the number of users assigned to subchannel $n$ is less than $K$, only one strategy (the addition strategy) is possible. If the number of users assigned to subchannel $n$ is exactly $K$, then $K$ strategies are possible due to the $K$ possible replacement policies. Since the value of $K$ is usually not large in practice, we consider the complexity of finding the best strategy as constant. The remaining operations inside the inner-most loop happen in constant time, and so we can ignore the complexity of these operations as well. Consequently, the overall complexity of the algorithm is $\\mbox{O}(MN)$.\n\n\n \n\\subsection{Power Allocation Scheme}\n\\label{ssec:GP}\n\nFrom \\textit{Algorithm~\\ref{alg:sc-usr-map}}, we know the subchannel-user mapping information, i.e., $\\textbf{M}_n, n \\in \\textbf{N}$ and ${\\bm{\\Omega}}_m, m \\in \\textbf{M}$. This information is derived based on the assumption that the maximal power level $p^{\\mbox{max}}$ of the BS is equally subdivided among all subchannels, i.e., $p_n = p^{\\mbox{max}}\/N, n \\in \\textbf{N}$. However, in (\\ref{eq:opt-prob1}), we see that the instantaneous rate of user $m$ assigned to subchannel $n$ is the increasing function of $p_m^n$ and the decreasing function of the interference power level caused by other users, i.e., $p_i^n, i \\in \\textbf{M}_n^m$. Consequently, even if the information about $\\textbf{M}_n, n \\in \\textbf{N}$ and ${\\bm{\\Omega}}_m, m \\in \\textbf{M}$ are known, the power allocation across all subchannel-user slots, i.e., $p_m^n, m \\in \\textbf{M}, n \\in \\textbf{N}$, is a non-convex optimization problem. In the literature, the transformation of such a type of problem to convex form is not reported yet, and hence the dual-based method~\\cite{CWang2016} or the bisection search~\\cite{QSun22015} cannot be adopted to solve this problem. Therefore, our next objective is to design a method that allocates power across all the subchannel-user slots in such a manner that near-optimality is achieved. We have adopted the GP-based optimization technique to solve this power allocation problem, the description of which is provided in the following discussions.\n\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{log_expansion2.eps}\n    \\caption{A sample verification of natural log series expansion~\\cite{RNave}.}\n    \\label{fig:log-expansion}\n  \\end{center}\n\\end{figure}\n\n\nA GP~\\cite{Boyd2007, MChiang2005} is a type of mathematical optimization problem with the characteristics that the objective and constraint functions must be monomial(s) or posynomial(s). Two problems that we have discussed in the following have a certain form, such as $\\frac{G_n(q_1,\\cdots,q_T)}{G_d(q_1,\\cdots,q_T)}$. Here, both functions $G_n(q_1,\\cdots,q_T)$ and $G_d(q_1,\\cdots,q_T)$ are posynomials; $q_1,\\cdots,q_T$ are the variables in the problem; and $T$ is the number of variables. This type of function is still not amenable to GP. To make this problem amenable to GP, there are some heuristics, such as the single condensation method and the double condensation method~\\cite{Boyd2007, MChiang2005}. We plan to apply the single condensation method in this context, which requires the denominator (i.e., $G_d(q_1,\\cdots,q_T)$) to be approximated via some monomial ${\\lambda}\\prod_{i=1}^T{q_i}^{a_i}$. Given the values of $q_1,\\cdots,q_T$, the unknown variables of this monomial can be approximated by $a_i = \\frac{q_i}{G_d(.)}\\frac{{\\partial}G_d(.)}{{\\partial}q_i}$ and $\\lambda = \\frac{G_d(.)}{\\prod_{i=1}^T{q_i}^{a_i}}$. As a result, the approximated objective function becomes the ratio of a posynomial and a monomial, which is a posynomial. Consequently, we require an iterative process in order to solve this problem step by step. The steps of this iterative process are provided in Section III-A of~\\cite{RRuby20152}.\n\n\n\n\n\\begin{comment}\n\n\\begin{enumerate}\n\\item Set the initial values for variables $q_1,\\cdots,q_T$.\n\\item Determine the values of $a_1,\\cdots,a_T$, and $\\lambda$.\n\\item Solve the problem associated with resultant objective function, i.e., the ratio of the posynomial and the monomial, via GP.\n\\item If the absolute difference between the current and previous values of the variables is less than or equal to some small number, i.e., $\\epsilon$, go to step $2$, and if otherwise, stop the iterative process. \n\\end{enumerate}\n\nThe final values of the variables obtained in the last iteration of the iterative process is the solution of our defined optimization problem.\n\\end{comment}\n\n\n\n\n\\subsubsection{Joint Power Allocation among all Subchannels}\n\\label{sssec:GP-joint}\n\nOnce the subchannel-user assignment information, i.e., $\\textbf{M}_n, n \\in \\textbf{M}_n$, in order to obtain the energy-efficient power level of each  subchannel-user tuple given the power constraint at the BS, we need to solve an optimization problem, which is as follows.\n\n\n\n\n\n\\begin{eqnarray}\n\\label{eq:GP-joint}\n\\nonumber & \\displaystyle\\max_{\\{p_m^n\\}}\\displaystyle\\sum_{n \\in \\textbf{N}}\\frac{1}{p_c+\\displaystyle\\sum_{m \\in \\textbf{M}_n}p_m^n}\\displaystyle\\sum_{m \\in \\textbf{M}_n}\\mbox{log}_2\\left(1 + \\frac{g_m^np_m^n}{{\\sigma}_n^2 + \\displaystyle\\sum_{i\\in\\textbf{M}_n^m}p_i^ng_m^n}\\right),  \\\\\n\\nonumber & \\mbox{s.t.}~\\displaystyle\\sum_{n \\in \\textbf{N}}\\displaystyle\\sum_{m \\in \\textbf{M}_n}p_m^n \\le p^{max}, \\\\\n& \\displaystyle\\sum_{m\\in \\textbf{M}_n}p_m^n \\ge p_n^{\\mbox{min}},~\\forall{n \\in \\textbf{N}}.\n\\end{eqnarray}\n\n\n\\begin{equation}\n\\mbox{log}_2\\left(1 + \\frac{g_m^np_m^n}{{\\sigma}_n^2 + \\displaystyle\\sum_{i\\in\\textbf{M}_n^m}p_i^ng_m^n}\\right) = \\frac{\\frac{g_m^np_m^n}{{\\sigma}_n^2 + \\displaystyle\\sum_{i\\in\\textbf{M}_n^m}p_i^ng_m^n}}{2 + \\frac{g_m^np_m^n}{{\\sigma}_n^2 + \\displaystyle\\sum_{i\\in\\textbf{M}_n^m}p_i^ng_m^n}}\n\\end{equation}\n\n\n\n\n\nThe relations in~(\\ref{eq:opt-prob5}) basically are associated with $M$ equations. The number of variables (i.e., $\\{p_m^n\\}_{m \\in \\textbf{M}_n, n \\in \\textbf{N}}$) in all these equations is $\\sum_{n \\in \\textbf{N}}|\\textbf{M}_n|$. However, using (18)~\\cite{FFang2016}, the number of variables in all these equations can be reduced to $N$. Therefore, for $M \\ge N$, intuitively, by solving all these equations jointly, the relations in the last constraint of~(\\ref{eq:GP-joint}) are achievable. For the other case (i.e., $M < N$), the last constraint of~(\\ref{eq:GP-joint}) can be approximated as well based on some simple assumption. Moreover, on subchannel $n$, the set of users, $\\textbf{M}_n^m$, that cause interference to user $m$ is determined based on the decoding order described in Section~\\ref{sec:sysmodel}. Clearly, the objective function here is not amenable to GP (not even any GP-based heuristic method) because of the energy efficiency factor $\\frac{1}{p_c+\\sum_{m \\in \\textbf{M}_n}p_m^n}$. Therefore, we have decided to expand the log series. Since the SINR of user $m$ on subchannel $n$ is larger than zero, we can expand the log function according to $\\mbox{ln}(1+x) = 2\\left[\\frac{x}{2+x} + \\frac{1}{3}(\\frac{x}{2+x})^3+\\cdots\\right]$~\\cite{RNave}. The accuracy of the log expansion is verified in Fig.~\\ref{fig:log-expansion}. For the sake of simplicity, using the first term of the expanded log function, we transform the original objective function to a form, which becomes the ratio of two posynomials. As shown in Fig.~\\ref{fig:log-expansion}, both the original log function and the first term of the expanded series are increasing with the increasing value of $x$. Moreover, the difference between the original log function and the first term of the log series increases with the increasing value of $x$. However, this difference is negligible when the value of $x$ is $\\le 3$. If we look at the objective function in ~(\\ref{eq:GP-joint}), $x$ is mapped to the SINR term of each user on a particular subchannel. The SINR of a user on a subchannel is a function of its and other users' gain and assigned power level. If the values of gain and maximal power level of the system are such that the SINR of a user on a subchannel does not exceed $5$ dB (the absolute value of which is around 3), the resultant approximated problem using the first term of the log series does not deviate much from the original problem. On the other hand, the search range of the SINR for a user over a subchannel may exceed $5$ dB at higher gains and a larger power level of the system. However, the first term of the log series dominates the entire log function. Moreover, since the nature of the log function as well as the first term of the log series are increasing, the sum of the log function as well as the sum of the first term (of the log series) are increasing. Therefore, when the search range of a SINR exceeds $5$ dB, although the approximated function is deviated from the original function, the resultant solution of the approximated problem should not be very bad. The accuracy of the resultant solution can be improved by approximating the problem using the first two terms of the log series instead of only the first one. Since the problem is a maximization problem, we apply the inverse function\\footnote{The inverse function of $\\frac{A}{B}$ is $\\frac{B}{A}$.} to transform it to the minimization one. Consequently, the problem becomes solvable via GP, as described in Section~\\ref{ssec:GP}. The outcome of the proposed fine-grained power allocation scheme strengthens the well-established truth~\\cite{SHan2014} that the two objectives of maximizing the energy efficiency and maximizing the spectrum efficiency cannot be achieved simultaneously at the same operating point.\n\n\n\n\n\n\n\n\\begin{comment}\n\n\\noindent\n\\textbf{Proposition 3:} If the overall energy efficiency achieved by a certain power allocation scheme is better than that of some benchmark scheme, it does not necessarily implies that the overall throughput achieved by that scheme will be better compared to the benchmark scheme.\n\n\\noindent\n\\textbf{Proof:} See \\textit{Appendix D}.\n\\end{comment}\n\n\n\n\\subsubsection{Power Allocation of a Subchannel}\n\\label{sssec:GP-indiv}\n\n\nFor subchannel $n$, once $\\textbf{M}_n$ and $p_n$ are known, we would like to determine $\\{p_m^n\\}_{m \\in {\\textbf{M}}_n}$. If $K=2$,~\\cite{FFang2016} has proved that this problem can be solved using the DC programming-based approach. However, if $K > 2$, this problem can no longer be solvable via this approach. To prove this statement, let us assume $K=3$, and $g_1^n > g_2^n > g_3^n$ holds for subchannel $n$. Under this circumstances, the instantaneous rate of the $3$rd user on subchannel $n$ can be given by $\\mbox{log}_2(1 + g_3^np_3^n\/(1+g_3^np_1^n+g_3^np_2^n))$. The interference term in the instantaneous rate of the $3$rd user is a barrier to express the problem in terms of a difference of convex functions. If the value of $K$ is even larger, the nonconforming nature of the problem to the DC programming-based approach becomes even more severe. In general, the energy-efficient power allocation problem of subchannel $n$ can be written as follows.\n\n\n\n\\begin{eqnarray}\n\\label{eq:GP-indiv1}&  \\displaystyle\\max_{\\{p_m^n\\}}\\frac{1}{p_c+\\displaystyle\\sum_{m \\in \\textbf{M}_n}p_m^n}\\displaystyle\\sum_{m \\in \\textbf{M}_n}\\mbox{log}_2\\left(1 + \\frac{g_m^np_m^n}{{\\sigma}_n^2 + \\sum_{i\\in\\textbf{M}_n^m}p_i^ng_m^n}\\right), \\\\\n\\label{eq:GP-indiv2}&\\mbox{s.t.}~\\displaystyle\\sum_{m \\in \\textbf{M}_n}p_m^n \\le p_n.\n\\end{eqnarray}\n\nExpanding the log function and then taking the first term of the series, the objective function in (\\ref{eq:GP-indiv1})-(\\ref{eq:GP-indiv2}) becomes a ratio of two posynomials. Upon applying the inverse function, the objective function is transformed to a form (i.e., $\\frac{G_n(.)}{G_d(.)}$) as described in Section~\\ref{ssec:GP}. \n\n\n\\subsubsection{Joint Computationally-Efficient Suboptimal Power Allocation among all Subchannels}\n\\label{sssec:sopt-joint}\n\n\n\n\n\nThe solution of the fine-grained power allocation problem in Section~\\ref{sssec:GP-joint} is computationally intensive due to the cumbersome iterative process of the single condensation method as well as the implementation complication of the off-the-shelf GP solvers~\\cite{cvx}. The complexity of this iterative process is proportional to the number of degrees of freedom in the system. If the number of users and subchannels in the system is moderately large, the solution of the problem cannot be obtained in a reasonable amount of time. Consequently, in this subsection, we have proposed another suboptimal solution for the fine-grained power allocation across the subchannels and users with much less computational overhead even with the off-the-shelf GP solvers~\\cite{cvx}. The mechanism of this suboptimal solution is developed based on the insights of the optimal solution, the description of which is given as follows. The idea of this suboptimal solution is to obtain the energy-efficient solution of the subchannels one by one. For this, using (18)~\\cite{FFang2016}, we plan to approximate the problem in (\\ref{eq:GP-indiv1})-(\\ref{eq:GP-indiv2}) to a one-variable optimization problem no matter the value of $|\\textbf{M}_n|$ is. If the energy-efficient power level of subchannel $n$ is $p_n$, using (18)~\\cite{FFang2016}, we have $p_m^n = {\\gamma}_m^np_n,~m \\in \\textbf{M}_n$, where $\\sum_{m\\in\\textbf{M}_n}{\\gamma}_m^n = 1$. Then, we can transform the problem in (\\ref{eq:GP-indiv1})-(\\ref{eq:GP-indiv2}) to either the problem in (\\ref{eq:sopt-DC1})-(\\ref{eq:sopt-DC2}) or that in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}). Since we do not know the allocated power level of subchannel $n$ in advance, we assume that the maximal power level at the BS, $p^{\\mbox{max}}$, is the upper limit to be allocated at the beginning. It is proved in~\\cite{FFang2016} that $\\mbox{log}_2\\left(1 + p_nA_m(\\{{\\gamma}_m^n\\},\\{g_m^n\\},{\\sigma}^2) \\right)\/(p_c+\\sum_{m \\in \\textbf{M}_n}{\\gamma}_m^np_n)$ is a concave function, and hence the DC-programming-based approach can be used to solve the problem in (\\ref{eq:sopt-DC1})-(\\ref{eq:sopt-DC2}). On the other hand, in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}), both the $C(\\{{\\gamma}_m^n\\},\\{g_m^n\\},p_n,{\\sigma}^2)$ and $D(\\{{\\gamma}_m^n\\},\\{g_m^n\\},p_n,{\\sigma}^2)$ are the polynomial functions of variable $p_n$. Using the first term of the log series, this problem can be transformed to a form which is the ratio of two posynomials as well. Consequently, we can adopt the GP-based single condensation method to obtain an elegant energy efficiency of subchannel $n$. From our experiments, the latter solution approach provides better solution for this problem, and so we adopt it as a part of our proposed suboptimal energy-efficient power allocation scheme. The number of iterations required for the single condensation method is constant as the number of optimization variable is one in this case no matter the value of $|\\textbf{M}_n|$ for subchannel $n$ is. \n\n\\begin{eqnarray}\n\\label{eq:sopt-DC1}&  \\displaystyle\\max_{p_n}\\frac{1}{p_c+\\displaystyle\\sum_{m \\in \\textbf{M}_n}{\\gamma}_m^np_n}\\displaystyle\\sum_{m\\in\\textbf{M}_n}\\frac{\\mbox{log}_2\\left(1 + p_nA_m(\\{{\\gamma}_m^n\\},\\{g_m^n\\},{\\sigma}^2) \\right)}{\\mbox{log}_2\\left(1 + p_nB_m(\\{{\\gamma}_m^n\\},\\{g_m^n\\},{\\sigma}^2)\\right)}, \\\\\n\\label{eq:sopt-DC2}&\\mbox{s.t.}~\\displaystyle\\sum_{m \\in \\textbf{M}_n}{\\gamma}_m^np_n \\le p^{\\mbox{max}}.\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\label{eq:sopt-GP1} &  \\displaystyle\\max_{p_n}\\frac{1}{p_c+\\displaystyle\\sum_{m \\in \\textbf{M}_n}{\\gamma}_m^np_n}\\mbox{log}_2\\left(1 + \\frac{C(\\{{\\gamma}_m^n\\},\\{g_m^n\\},p_n,{\\sigma}^2)}{D(\\{{\\gamma}_m^n\\},\\{g_m^n\\},p_n,{\\sigma}^2)}\\right), \\\\\n\\label{eq:sopt-GP2}&\\mbox{s.t.}~\\displaystyle\\sum_{m \\in \\textbf{M}_n}{\\gamma}_m^np_n \\le p^{\\mbox{max}}.\n\\end{eqnarray}\n\n\n\n\n\\begin{comment}\n\nconvex nature of the energy efficiency problem of a subchannel. Given the power constraint of $p^{\\mbox{max}}$, the energy-efficient power allocation of subchannel $n$ can be obtained by solving the optimization problem in~(\\ref{eq:sopt-indiv}). \n\\begin{eqnarray}\n\\label{eq:sopt-indiv}\n\\nonumber &  \\displaystyle\\max_{\\{p_m^n\\}}\\displaystyle\\sum_{m \\in \\textbf{M}_n}\\mbox{log}_2\\left(1 + \\frac{g_m^np_m^n}{{\\sigma}_n^2 + \\sum_{i\\in\\textbf{M}_n^m}p_i^ng_m^n}\\right), \\\\\n&\\displaystyle\\sum_{m \\in \\textbf{M}_n}p_m^n \\le p^{\\mbox{max}}.\n\\end{eqnarray}\nIt is proved in~\\cite{FFang2016} that the problem\nin~(\\ref{eq:sopt-indiv}) is convex, and hence it is straightforward to find the solution of this problem.\n\n\\end{comment}\n\nAfter solving the problem in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) for all the subchannels, the resultant total power $\\sum_{n \\in \\textbf{N}}\\sum_{m \\in {\\textbf{M}}_n}{\\gamma}_m^np_n$ can be $>$ or $\\le$ $p^{\\mbox{max}}$. For the latter case, the resultant total power, $\\{{\\gamma}_m^np_n\\}_{n \\in {\\textbf{M}}_n, n \\in \\textbf{N}}$, can be considered as an elegant solution. On the other hand, for the former case, we need to revisit the solution of the optimization problem further. In order to develop a good solution for this case, we need to know the energy efficiency rate per unit of power level for all subchannels. Once these information are known, we can develop a suboptimal scheme, that assigns unit power to the subchannels one by one until the total allocated power to all the subchannels results in $p^{\\mbox{max}}$. In order to know which subchannel has higher rate of energy efficiency, we construct an energy efficiency rate matrix \\textbf{EEM}, the dimension of which is $(\\lfloor{p^{\\mbox{max}}\/{\\Delta}}\\rfloor+1) \\times N$. The $1$st row of this \\textbf{EEM} matrix contains the energy efficiency of the subchannels given that each subchannel is assigned with $\\Delta$ level of power. Each element of the $2$nd row contains the subtracted energy\nefficiency when the corresponding subchannel is allocated with\n$2\\Delta$ and $\\Delta$ level of power. Finally, each element of the last row contains the subtracted energy efficiency of the\ncorresponding subchannel when it is assigned to $\\lfloor{p^{\\mbox{max}}\/{\\Delta}}\\rfloor \\Delta$ and $p^{\\mbox{max}}$ level of power. The energy efficiency of each subchannel is computed using the formulation in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) given the maximal power constraint designated for each element in matrix \\textbf{EEM}. Based on the information of this matrix, power is assigned to each subchannel one by one in an iterative manner. For example, in one iteration, $(a, n),~a \\in [1,\n\\lfloor{p^{\\mbox{max}}\/{\\Delta}}\\rfloor+1]$ tuple in \\textbf{EEM} matrix has been selected since it has the highest value. In this case, subchannel $n$ is assigned to $a\\Delta$ level of power. This process continues until the allocated power level to all subchannels, $\\sum_{n \\in \\textbf{N}}p_n$, is equal to $p^{\\mbox{max}}$. The detailed  steps of the entire iterative process are provided in \\textit{Algorithm \\ref{alg:sopt-pwr-alloc}}.\n\n\\begin{algorithm}[h!]\n\\caption{The computationally-efficient fine-grained energy-efficient power allocation scheme.}\n\\label{alg:sopt-pwr-alloc}\n\\begin{algorithmic}[1]\n\\FOR{$n \\in \\textbf{N}$}\n\\STATE Solve the problem in~(\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) for subchannel $n$.\n\\ENDFOR\n\n\\IF{$\\sum_{n \\in \\textbf{N}}\\sum_{m \\in \\textbf{M}_n}{\\beta}_m^np_n > p^{\\mbox{max}}$}\n\\STATE Construct matrix \\textbf{EEM}.\n\\REPEAT\n\\STATE Select an un-marked tuple $(a, n)$ from matrix \\textbf{EEM}, which\nhas the highest non-zero value.\n\\IF{$a \\in [1, \\lfloor{p^{\\mbox{max}}\/{\\Delta}}\\rfloor]$}\n\\STATE Assign $a\\Delta$ level of power to subchannel $n$.\n\\ELSE\n\\STATE Assign $p^{\\mbox{max}}$ level of power to subchannel $n$.\n\\ENDIF\n\\STATE Mark tuple $(a,n)$ in matrix \\textbf{EEM}.\n\\UNTIL{$\\sum_{n \\in \\textbf{N}}\\sum_{m \\in {\\textbf{M}}_n}{\\beta}_m^np_n  = p^{\\mbox{max}}$}\n\\ENDIF\n\\end{algorithmic}\n\\end{algorithm}\n\n\\noindent\n\\textbf{Computational complexity of \\textit{Algorithm\n\\ref{alg:sopt-pwr-alloc}}:} Ideally, the GP-based single condensation method is supposed to adopt a conventional optimization solver, such as the interior-point method to solve the transformed convex problem in each iteration. Since the number of iterations is constant for the single condensation method to solve the problem in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}), its computational complexity is equivalent to the complexity of the adopted convex optimization solver. In~\\cite{YNesterov1994}, it is shown that the interior-point method can obtain the $\\epsilon$-optimal solution in polynomial time, and hence we can say that the complexity of the single condensation method to solve the problem in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) has a polynomial time complexity. Moreover, although the off-the-shelf GP solvers do not solve a GP problem in a native manner, since the number of variable(s) of the problem in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) is one, the computational complexity of the resultant solution is constant and quick unlike that in Section~\\ref{sssec:GP-joint}. The steps in between $1$ and  $3$, the problem in~(\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) needs to be solved $N$ times. If the resultant total power does not satisfy the power constraint, we need to construct matrix \\textbf{EEM}. Except the $1$st row, for each tuple of matrix\n\\textbf{EEM}, the problem in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) needs to be solved twice while replacing $p^{\\mbox{max}}$ by different values. Therefore, the process of constructing matrix \\textbf{EEM} takes a polynomial time multiplied with $(\\lfloor{p^{\\mbox{max}}\/{\\Delta}}\\rfloor+1) \\times N$. Since each tuple is marked once it is selected for the assignment of power, the steps in between $6$ and $14$ run at most $(\\lfloor{p^{\\mbox{max}}\/{\\Delta}}\\rfloor+1) \\times N$ times. As a result, the complexity of constructing matrix \\textbf{EEM} is dominated by the rest of the other steps in \\textit{Algorithm\n\\ref{alg:sopt-pwr-alloc}}.\n\n\n\n\n\n\n\n\n \\section{Performance Evaluation}\n \\label{sec:eval}\n\nIn this section, via simulation, we evaluate the performance of our proposed energy-efficient downlink resource allocation schemes. Followed by the methodology, we exhibit the detailed outcome of the simulation in order to verify the effectiveness of the proposed schemes.\n \n \\subsection{Simulation Setup}\n \nThe cellular network, that we consider, is isolated from the neighboring networks. It has a circular-like shape and suitable for an office environment. We place the BS at the center of the network, and the users are uniformly spread surrounding the BS within $500$ m distance. Typically, the minimum relative distance between two emploees in an office environment could be $40$ m, and hence we set the minimum distance between two users as $40$ m. Although we set a minimum relative distance between two users, the performance of each NOMA user is independent of this distance~\\cite{MBasit2018}. We also set the minimum distance from each user to the BS as $50$ m. As mentioned previously, continuous time is divided into frames. Each time frame is equivalent to $1$ s. During each time frame, the designated spectrum is equally subdivided among $20$ subchannels, and these subchannels are available to be allocated among $M$ users in the system. Each subchannel is assumed to have $200$ KHz bandwidth. According to~\\cite{XQiu1999}, the theoretical limit of the channel capacity is given by $\\frac{-1.5}{\\mbox{ln}(5P_b)}$, where $P_b$ denotes the Bit Error Rate (BER). The BER for each subchannel is configured as $10^{-6}$. We consider that the shadow and Rayleigh are two main fading components of a wireless channel between the BS and a user in the system. The shadow fading follows the log-normal distribution with variance $3.76$.\n\nIn order to calculate the quantity of the shadow fading over a subchannel, we assume the reference distance as $1$ km and the SNR for this reference distance is $28$ dB. The Rayleigh fading for all users over all subchannels follows the Rayleigh distribution with $0$ mean and $4.3$ variance. Using all these parameters, the gain of each subchannel for a user towards the BS is computed following ($22$) in~\\cite{RRuby2015}. We employ the SIC technique in~\\cite{SVanka2012, NIMiridakis2013} for a user in order to decode the SC-coded signal of each subchannel transmitted by the BS. \n\n\n \n\nIn addition to implementing our proposed resource allocation schemes, we have implemented the scheme in~\\cite{FFang2016}. In the following, this scheme is referred as ``Scheme in~\\cite{FFang2016}''. In order to show that the many-to-many matching model applied in the subchannel-user mapping algorithm outperforms the one-to-many matching model, we have one version (i.e., Proposed Scheme-1) which uses \\textit{Algorithm~\\ref{alg:sc-usr-map}} to solve the subchannel-user mapping problem and the DC programming-based approach for the final power allocation across the subchannels and users. Note that in this version, for step $14$ of \\textit{Algorithm~\\ref{alg:sc-usr-map}}, ($18$)~\\cite{FFang2016} is used. On the other hand, in order to show the superiority of our fine-grained power allocation scheme via the GP technique over the DC programming-based power allocation approach, we have another version (i.e., Proposed Scheme-$2$). This version adopts the subchannel-user mapping algorithm in~\\cite{FFang2016} for the first step, and adopts the scheme in Section~\\ref{sssec:GP-joint} in order to have power allocation across each allocated subchannel-user slot. We have three more versions of our proposed schemes, namely Scheme-$3$, Scheme-$4$ and Scheme-$5$, in order to exhibit a tradeoff between optimality and computational complexity. Both Scheme-$3$ and Scheme-$4$ adopt the GP technique in Section~\\ref{sssec:GP-joint} for the final power allocation. However, Scheme-$3$ uses ($18$)~\\cite{FFang2016} for step $14$ in \\textit{Algorithm~\\ref{alg:sc-usr-map}}, and Scheme-$4$ uses the GP technique in Section~\\ref{sssec:GP-indiv}. Consequently, step $14$ in \\textit{Algorithm~\\ref{alg:sc-usr-map}} happens in constant time for Scheme-$3$ version. On the other hand, this operation of Scheme-$4$ has polynomial time complexity w.r.t. $\\textbf{M}_n$. Since the maximum value of $\\textbf{M}_n$ is $K$ (e.g., $4$ in our simulation), this complexity of Scheme-$4$ can be considered negligible, but provides better solution compared to Scheme-$3$. Moreover, our Scheme-$5$ version uses the technique in Section~\\ref{sssec:GP-indiv} for step $14$ of the subchannel-user mapping algorithm, and uses \\textit{Algorithm 2} for the final fine-grained power allocation across the subchannels and users. The complexity of the suboptimal power allocation algorithm (i.e., \\textit{Algorithm 2}) is elaborately explained in Section~\\ref{sssec:sopt-joint}. Apparently, both \\textit{Algorithm 2} and the fine-grained power allocation scheme in Section~\\ref{sssec:GP-joint} have polynomial time complexity. However, since the complexity of the GP technique is computationally intensive with the large number of optimization variables (e.g., $\\sum_{n \\in \\textbf{N}}|\\textbf{M}_n|$), the proposed Scheme-$3$ or Scheme-$4$ may run very slowly with the growing number of users and subchannels in the system. On the other hand, although step $2$ of \\textit{Algorithm 2} is associated with solving the problem in~(\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) via the GP technique, the corresponding problem has only one optimization variable which results in constant time complexity for this step. Other steps of \\textit{Algorithm 2} have constant time complexity, and hence the complexity of Scheme-$5$ can be considered negligible compared to Scheme-$3$ or Scheme-$4$. However, because of the intense search of the fine-grained power allocation scheme in Section~\\ref{sssec:GP-joint}, Scheme-$3$ and Scheme-$4$ (especially Scheme-$4$) achieve much better performance compared to Scheme-$5$, which has been verified in the subsequent discussions. Finally, in order to demonstrate the global optimal performance, we apply the brute-force search on both the subchannel-user mapping and power allocation problems for a system with $10$ and $20$ users. In the following subsection, for each data point, we conduct the simulation over $10000$ times.\n\n \n \\subsection{Simulation Results}\n \n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{syseff_usrs.eps}\n    \\caption{Comparison of overall energy efficiency with the increasing number of users.}\n    \\label{fig:syseff-usrs}\n  \\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\n\nFor $K = 4$, $p^{\\mbox{max}} = 23~\\mbox{dBW}$ and $p_c = 1.75~\\mbox{dBW}$, Fig.~\\ref{fig:syseff-usrs} presents the increasing energy efficiency with the increasing number of users. Given the constant power level at the BS, the more the users, the higher the sum-rate, and hence the larger the efficiency is. This trend is similar for all the schemes no matter it is ours or the scheme in~\\cite{FFang2016}. Via the many-to-many matching model, if a user has better gain over many subchannels compared to other users, that user can be assigned to as many subchannels as possible if such assignments increase the system energy efficiency. On the other hand, via the one-to-many matching model, it is possible that a user with worse gain is forced to be assigned to a subchannel, despite such assignment is not necessarily beneficial for the system energy efficiency. This is because each user can be assigned to at most one subchannel via this matching model. As a result, both intuitively and empirically, we see the evidence of enhanced performance for our Scheme-$1$ compared to that in~\\cite{FFang2016}, even when we use (18)~\\cite{FFang2016} for step $14$ in \\textit{Algorithm~\\ref{alg:sc-usr-map}} and the DC programming-based approach for the final power allocation. To provide further evidence, for $M = 40$, we plot Fig.~\\ref{fig:usrs-vs-sc} and Fig.~\\ref{fig:scs-vs-usr}, which are the outcome of the subchannel-user mapping algorithms. In Fig.~\\ref{fig:scs-vs-usr}, we compare the number of allocated subchannels to each individual user (the users are sorted in the ascending order of their distance from the BS) between the one-to-many and our model. From our observation, we see that via the many-to-many matching model, a user can be assigned to multiple subchannels, and this number depends on the subchannels over which that particular user can achieve enhanced energy efficiency compared to other users. Our channel model is such that the quantity of the shadow fading is dominated by that of the Rayleigh fading. Therefore, a user that is closer to the BS is likely to have better gain over more subchannels compared to a farther user. Therefore, in Fig.~\\ref{fig:scs-vs-usr}, we see that a user that is the closest to the BS is assigned to the largest number of subchannels. On the other hand, based on the dynamics of users' gain, in order to enhance the energy efficiency further, some subchannels accommodate more users compared to other subchannels.\n\n\n\n\n\\begin{comment}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{usrs_vs_sc.eps}\n    \\caption{Comparison of allocated users to each individual subchannel when $M=40$.}\n    \\label{fig:usrs-vs-sc}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{scs_vs_usr.eps}\n    \\caption{Comparison of allocated subchannels to each individual user when $M=40$.}\n    \\label{fig:scs-vs-usr}\n  \\end{center}\n\\end{figure}\n\\end{comment}\n\n\n\n\\begin{figure}[h!]%\n    \\centering\n    \\subfloat[Comparison of allocated users to each individual subchannel.\\label{fig:usrs-vs-sc}]{{\\includegraphics[scale=0.6]{usrs_vs_sc.eps} }}%\n   ~\n    \\subfloat[Comparison of allocated subchannels to each individual user.\\label{fig:scs-vs-usr}]{{\\includegraphics[scale=0.6]{scs_vs_usr.eps} }}%\n    \\caption{Detailed subchannel-user assignment study when $M=40$.}%\n   \n\\end{figure}\n\n\n\n\n\n\nIn Fig.~\\ref{fig:usrs-vs-sc}, we compare the number of assigned users to each individual subchannel between two subchanel-user mapping algorithms. In this figure, although $K=4$, not necessarily all subchannels have $4$ allocated users. In general, it is seen that if the difference of gain between any two assigned users over any subchannel is larger compared to other subchannels, that subchannel achieves better sum-rate as well as better energy efficiency. The numerical degree of users' gain over a subchannel also plays a crucial role in the decision whether additional user will be allocated to that subchannel or not. Because of these user dynamics, some subchannels cannot be assigned to many users as the allocation of more and more users may decrease the sum-rate as well as its energy efficiency, which is proved in \\textit{Proposition 1}. Due to the structure of the formulation, unlike the DC programming-based approach, the GP technique can provide fine-grained energy-efficient power allocation across all subchannel-user tuples. Hence, the energy efficiency is much better for this case (i.e., Scheme-$3$) even if we use (18)~\\cite{FFang2016} for step $14$ of \\textit{Algorithm~\\ref{alg:sc-usr-map}}. If we use the GP technique instead of (18)~\\cite{FFang2016} for step $14$ in \\textit{Algorithm~\\ref{alg:sc-usr-map}} (i.e., Scheme-$4$), we obtain better organization in the subchannel-user map due to the better power allocation among the users of each subchannel. Therefore, this scheme has the best performance compared to all the others. The elegance of the fine-grained GP technique is further evident from the enhanced performance of Scheme-$2$ over the scheme in \\cite{FFang2016}. In Scheme-$2$, we take the same subchannel-user mapping algorithm (via the one-to-many matching model) as that in \\cite{FFang2016}. On the other hand, we argued previously that the fine-grained power allocation using the GP technique is computationally quite intensive. Consequently, we developed \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}} in order to have fine-grained energy-efficient power allocation across all the allocated subchannel-user tuples in a low complexity manner. Because of adopting the insights of the optimal solution, the resource allocation scheme via this algorithm (i.e., Scheme-$5$) has very close performance to that of the Scheme-$3$ and Scheme-$4$ versions, and obviously outperforms the scheme with the DC programming-based approach.\n\n\n\n\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{syspwr_usrs.eps}\n    \\caption{Comparison of total power consumption with the increasing number of users.}\n    \\label{fig:syspwr-usrs}\n  \\end{center}\n\\end{figure}\n\n\n\n\n\nGiven $p^{\\mbox{max}} = 23~\\mbox{dBW}$ and $p_c = 1.75~\\mbox{dBW}$, Fig.~\\ref{fig:syspwr-usrs} presents total power consumption with the increasing number of users. This is natural that the more the users in the system, the diversity of the users that spread among all the allocated subchannels (i.e., multi-user diversity) increases. Therefore, with the increasing number of users, it is more likely that each subchannel is assigned to at least one user with good channel. On the other hand, from our observation, it is seen that if a subchannel has at least one user with good channel, it requires less power for that subchannel to reach the optimal energy-efficient state. Consequently, a lower power level should be required to reach the optimal energy-efficient state for a system with more users compared to that with less users. This observation and insights hold for both the GP technique and \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}}. However, since the mechanism of \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}} is somewhat suboptimal (although developed based on the insights of the optimal solution) compared to that in Section~\\ref{sssec:GP-joint}, total consumed power in this case is slightly larger compared to the other one. The findings of this figure are quite interesting in a sense that the schemes with the fine-grained power allocation via the GP technique and \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}} use much less power compared to that with the DC programming-based approach. If we compare Fig.~\\ref{fig:syseff-usrs} with this figure, it becomes even more interesting as our schemes achieve much better energy efficiency using much less power compared to that with the DC programming-based approach. On the other hand, the schemes with the DC programming-based approach use full power of the BS, but incur much less energy efficiency. From the perspective of green communications, the results presented in this figure verify our original motivation towards pursuing this work.\n\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{systhr_usrs.eps}\n    \\caption{Comparison of energy-efficient total throughput with the increasing number of users.}\n    \\label{fig:systhr-usrs}\n  \\end{center}\n\\end{figure}\n\n\nSimilar to our work, the conventional definition of energy efficiency is achievable blocks of bits from a channel under the usage of unit power level. The Shannon's information capacity theorem has already established that two objectives, i.e., minimizing the consumed energy and maximizing the spectral efficiency are not achievable simultaneously at the same operating point. Consequently, under the consideration of fixed circuit power, there always exist two separate optimal points in the energy efficiency versus spectrum efficiency curve.  In Fig.~\\ref{fig:systhr-usrs}, we compare the energy-efficient total throughput acheievd by all aforementioned schemes. The more the users, the better the utilization of limited resources because of the enhanced multi-user diversity. Hence, we see the increasing trend in the energy-efficient total throughput, achieved by all the schemes, with the increasing number of users. However, due to the fact in the Shannon's information capacity theorem, since the scheme in~\\cite{FFang2016} does not achieve the optimal energy-efficient state, it is possible that the total energy-efficient throughput achieved by this scheme is larger than our schemes. This is what observed in this figure.\n\n\n\n\n\n\n\n\\begin{comment}\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{syseff_pwr.eps}\n    \\caption{Comparison of overall energy efficiency with the increasing power at the BS.}\n    \\label{fig:syseff-pwr}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{syspwr_pwr.eps}\n    \\caption{Comparison of total power consumption with the increasing power at the BS.}\n    \\label{fig:syspwr-pwr}\n  \\end{center}\n\\end{figure}\n\\end{comment}\n\n\\begin{figure}[h!]%\n    \\centering\n    \\subfloat[Overall energy efficiency.\\label{fig:syseff-pwr}]{{\\includegraphics[scale=0.6]{syseff_pwr.eps} }}%\n   ~\n    \\subfloat[Total power consumption.\\label{fig:syspwr-pwr}]{{\\includegraphics[scale=0.6]{syspwr_pwr.eps} }}%\n    \\caption{Comparison of overall energy efficiency and total power consumption with the increasing power at the BS.}%\n   \n\\end{figure}\n\n\nIn Fig.~\\ref{fig:syseff-pwr}, given $M = 40$ and $p_c = 1.75~\\mbox{dBW}$, we show the energy efficiency with the increasing power at the BS. As we saw in the previous results that the scheme in~\\cite{FFang2016} fails to exploit the multi-user diversity of wireless systems as well as uses the DC programming-based approach to obtain the coarse-grained power allocation, this scheme has the lowest energy efficiency no matter the power constraint of the BS is. For this case, we see that with the increasing power level, the trend of energy efficiency is decreasing. This is because there is a tradeoff between the transmission capacity and the energy-efficient power consumption. Whereas, for our case, since the GP technique provides a fine-grained elegant power allocation, the overall energy efficiency is much better compared to the benchmark scheme. Moreover, since the GP technique provides the unique solution while consuming much less power, no matter we increase the power level of the BS, the energy efficiency remains same at the unique point. Similar trend is observed in the case of our suboptimal \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}} although the energy efficiency achieved by this scheme is slightly lower compared to the fine-grained GP-based power allocation scheme. In order to show the power consumption for this case, we plot Fig.~\\ref{fig:syspwr-pwr}. Since the optimal energy-efficient state of the system is unique and our proposed GP technique is able to search this state to some extent, we see the constant level of used power no matter how much power the BS has. In the similar manner, given the subchannel-user mapping matrix, via our suboptimal \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}}, we obtain an elegant energy-efficient point for each individual subchannel. Consequently, although this suboptimal scheme does not achieve as good solution as by that in Section~\\ref{sssec:GP-joint}, the suboptimal unique energy efficiency is still achieved at the unique power level. Whereas, the schemes, which use the DC programming-based approach, consume full power of the BS. Therefore, in this case, we see the increasing power consumption with the increasing total power at the BS.\n\n\n\\begin{comment}\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{figures\/syseff_pc.eps}\n    \\caption{Comparison of overall energy efficiency with the increasing $p_c$.}\n    \\label{fig:syseff-pc}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{figures\/syspwr_pc.eps}\n    \\caption{Comparison of total power consumption with the increasing $p_c$.}\n    \\label{fig:syspwr-pc}\n  \\end{center}\n\\end{figure}\n\\end{comment}\n\n\\begin{figure}[h!]%\n    \\centering\n    \\subfloat[Overall energy efficiency.\\label{fig:syseff-pc}]{{\\includegraphics[scale=0.6]{syseff_pc.eps} }}%\n   ~\n    \\subfloat[Total power consumption.\\label{fig:syspwr-pc}]{{\\includegraphics[scale=0.6]{syspwr_pc.eps} }}%\n    \\caption{Comparison of overall energy efficiency and total power consumption with the increasing $p_c$.}%\n   \n\\end{figure}\n\n\n\nIn Fig.~\\ref{fig:syseff-pc}, given $p^{\\mbox{max}} = 23~\\mbox{dBW}$ and $M = 40$, we show the decreasing overall energy efficiency with the increasing circuit power consumption. This is a natural trend achieved by all the schemes. Since circuit power is not used in enhancing system throughput but is added as consumed power, the overall energy efficiency is decreasing with the increasing circuit power. However, our subchannel-user mapping scheme can better exploit the multi-user diversity of wireless systems and we use the GP technique for the final fine-grained energy-efficient power allocation, our schemes, even via our suboptimal \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}}, always outperform the scheme in~\\cite{FFang2016}. In order to show the total power consumption in this case, we plot Fig.~\\ref{fig:syspwr-pc} with the increasing $p_c$. Increasing $p_c$ means, a subchannel requires higher power to reach its energy-efficient state. Consequently, the optimal energy efficiency is achieved at larger power level with the increasing $p_c$. For the similar reason, via our suboptimal \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}}, the same trend is observed with the increasing $p_c$. On the other hand, since the DC programming-based approach uses full power of the BS no matter the value of $p_c$ is, the total power consumption is constant (i.e., $p^{\\mbox{max}}$) via the scheme in~\\cite{FFang2016} and our Scheme-$1$ version.    \n\n\\section{Conclusion}\n\\label{sec:concl}\n\nIn this paper, we proposed an energy-efficient downlink subchannel and power allocation scheme for NOMA systems with enhanced performance compared to the most relevant existing work in the literature. Due to the discrete nature of subchannel assignment and the characteristics of the NOMA technique, this is an MINLP problem. Therefore, similar to an existing work, we solved the problem via decomposing it into a subchannel allocation subproblem followed by a power loading subproblem. However, unlike the existing work, via a many-to-many matching model, we better exploited the multi-user diversity of wireless systems in the solution of the subchannel-user mapping subproblem. In the second step, unlike the DC programming-based approach, via the GP technique, we were able to allocate the power level across the allocated subchannel-user slots in a fine-grained manner such that better energy efficiency is achieved compared to the benchmark scheme. Since the fine-grained power allocation via the GP technique is computationally intensive using the off-the-shelf GP solvers, we also proposed a suboptimal fine-grained power allocation algorithm with much lower computational complexity. Under various realistic scenarios, extensive simulation had been conducted to verify that our scheme (even via our computationally-efficient suboptimal power allocation algorithm) can outperform the existing scheme while consuming much less power in the system. \n\n\nBesides achieving an elegant energy-efficient state via the better resource allocation schemes, the implication of this work is extended to a certain extent. Via our schemes (even the suboptimal one), since better energy efficiency is achieved at a lower power level, the interference effect to the neighboring networks is expected to be mitigated. At the same time, unused power in the system can be used for other purposes.\n\n\\begin{appendices}\n\n\\section{Proof of Proposition 1}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{prop1_proof.eps}\n    \\caption{A sample example that $[R_n(\\{1,2,3\\})-R_n(\\{1,2\\})]$ can be both positive and negative.}\n    \\label{fig:prop1-proof}\n  \\end{center}\n\\end{figure}\n\nConsider a subchannel $n$, and it has two allocated users, indexed by $1$ and $2$. Moreover, we set $K = 4$, and there is one assumption, i.e., $R_n(\\{1, 2\\}) > R_n(\\{1\\})$. It implies that the sum-rate of user $1$ and user $2$ is larger than that of only user $1$. Now, user $3$ has come to be assigned with subchannel $n$. There are two possible conditions for this assignment, which are $R_n(\\{1,2,3\\}) > R_n(\\{1,2\\})$ and $R_n(\\{1,2,3\\}) < R_n(\\{1,2\\})$. According to step $15$ of \\textit{Algorithm~\\ref{alg:sc-usr-map}}, we only consider this user to construct an addition strategy if and only if the former case is true. In practice, both the conditions for any subchannel $n$ can happen. This statement can be proved from the result of Fig.~\\ref{fig:prop1-proof}. In this figure, we plot $R_n(\\{1,2,3\\}) - R_n(\\{1,2\\})$ w.r.t. $g_3^n$. It is obvious that $R_n(\\{1,2,3\\}) - R_n(\\{1,2\\})$ can be both positive and negative. The energy efficiency of subchannel $n$ is enhanced if and only if the aforementioned value is positive, and user $3$ is not added to subchannel $n$ for the negative case. This completes the proof. Note that $g_1^n = 0.4141,~g_2^n = 6.2512$ and $p_n = 50$ in Fig.~\\ref{fig:prop1-proof}. \n\n\\section{Proof of Proposition 2}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{prop2_proof.eps}\n    \\caption{A sample example that $R_n(\\{1,2\\})$ is increasing w.r.t. $g_2^n$.}\n    \\label{fig:prop2-proof}\n  \\end{center}\n\\end{figure}\n\nWithout loss of generality, let us assume $K=2$. Consider that subchannel $n$ has already $2$ users, indexed by $1$ and $2$, and $g_1^n > g_2^n$ holds. At this point, user $3$ has come to be assigned with subchannel $n$ with $g_3^n > g_2^n~\\mbox{and}~g_1^n > g_3^n$. Since the nature of function $R_n(\\{1,2\\})$ is increasing w.r.t. $g_2^n$ (as shown in Fig.~\\ref{fig:prop2-proof}), we can conclude $R_n(\\{1,3\\}) > R_n(\\{1,2\\})$. Moreover, using (18)~\\cite{FFang2016}\nand then simplifying, $R_n(\\{1,3\\})$ and $R_n(\\{3,2\\})$ are given by\n\n\\begin{eqnarray}\n\\label{prop2:f}  & R_n(\\{1,3\\}) = \\\\\n& \\nonumber \\mbox{log}_2\\left((1+\\frac{p_n(g_1^n)^{1+\\alpha}}{(g_1^n)^{\\alpha}+(g_3^n)^{\\alpha}})(1\n+ \\frac{p_n(g_1^n)^{1+\\alpha}}{(g_1^n)^{\\alpha}+(g_3^n)^{\\alpha}+p_ng_3^n(g_1^n)^{\\alpha}})\\right),\n\\\\\n\\label{prop2:l} & R_n(\\{3,2\\}) = \\\\\n& \\nonumber \\mbox{log}_2\\left((1+\\frac{p_n(g_3^n)^{1+\\alpha}}{(g_2^n)^{\\alpha}+(g_3^n)^{\\alpha}})(1\n+ \\frac{p_n(g_2^n)^{1+\\alpha}}{(g_2^n)^{\\alpha}+(g_3^n)^{\\alpha}+p_ng_2^n(g_3^n)^{\\alpha}})\\right),~\\mbox{respectively}.\n\\end{eqnarray}\n\nSince $g_1^n > g_3^n > g_2^n$, from (\\ref{prop2:f}) and\n(\\ref{prop2:l}), for any $\\alpha < 0$, it is straightforward to say $R_n(\\{1,3\\}) > R_n(\\{3,2\\})$. Now, if another user $4$ comes, two possible cases are possible. The first case is, user $4$ will replace further user $3$ if both $g_4^n > g_3^n~\\mbox{and}~g_1^n > g_4^n$ hold. In order to prove this proposition, we need to show that rejected user $2$ cannot match with subchannel $n$ further as the assignment of user $2$ to subchannel $n$ cannot enhance the energy efficiency any more. For this case, since $g_3^n > g_2^n$, via substituting user $3$ by user $4$ in (\\ref{prop2:f}) and (\\ref{prop2:l}), $R_n(\\{1,4\\}) > R_n(\\{2,4\\})$ can be\nproved. For the sake of soundness, the second possible case is, user $4$ approaches subchannel $n$ with $g_4^n < g_3^n$. In this case, user $3$ is not substituted by user $4$ as $R_n(\\{1,4\\})$ is an increasing function of $g_4^n$ according to Fig.~\\ref{fig:prop2-proof}. Now, our objective is to prove $R_n(\\{1,3\\}) > R_n(\\{3,4\\}) > R_n(\\{2,4\\})$. While substituting user $2$ by user $4$ in (\\ref{prop2:f}) and (\\ref{prop2:l}),\n$R_n(\\{1,3\\}) > R_n(\\{3,4\\})$ is straightforward as $g_1^n > g_3^n > g_4^n$ is true.  In the similar manner,  $R_n(\\{3,4\\}) > R_n(\\{2,4\\})$ can be proved as well due to the $g_3^n > g_4^n$ and $g_3^n > g_2^n$ relations. Thus, the proof of this proposition is completed.\n\n\n\n\n\\section{Proof of Theorem 1}\n\nLet us prove this theorem by contradiction. Consider a matching relation $\\mu$ and a user-subchannel pair $(m, n)$, which satisfy $m \\not\\in {\\mu}(n)$ and $n \\not\\in {\\mu}(m)$. Although this pair is not matched, both user $m$ and subchannel $n$ prefer each other over the remaining other subchannels and users, respectively. Now, according to the steps of \\textit{Algorithm \\ref{alg:sc-usr-map}}, two possible cases can happen. The first case is, $|\\bm{\\Omega}_m| < K$, and user $m$ proposes subchannel $n$ to be matched with (step $5$ in \\textit{Algorithm \\ref{alg:sc-usr-map}}). Over receiving the proposal, according to \\textit{Proposition 1}, if the energy efficiency of subchannel $n$ is enhanced by adding user $m$, subchannel $n$ is paired with user $m$, otherwise not. From the latter case, it can be concluded that although user $m$ prefers subchannel $n$ over the remaining other subchannels, subchannel $n$ does not prefer user $m$ over the remaining other users. The second case is, subchhanel $n$ is already matched with user $m$ through the addition strategy. However, later, in some iteration, user $m'$ proposes subchannel $n$ to be matched with, and subchannel $n$ prefers user $m'$ over user $m$ because of the enhanced energy efficiency. Consequently, user $m$ will be replaced by user $m'$ (step $19$ - step $20$) in ${\\mu}(n)$. From this discussion, it can be concluded that if both the user $m$ and subchannel $n$ prefer each other over the remaining other players, there is no way that a matching relation will not be established between them. Consequently, the initial statement of this proof is shown to be false. This concludes the proof of this theorem. \n\n\\begin{comment}\n\n\\section{Proof of Proposition 3}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=1.0\\columnwidth]{figures\/eeth_figure.eps}\n    \\caption{Comparison of total energy efficiency (EE) and total throughput (TH) for the same increasing level of $p_1$ and $p_2$.}\n    \\label{fig:EE-vs-TH}\n  \\end{center}\n\\end{figure}\n\n\n\nThis statement can be proved by contradiction. We\nassume that the two objectives of maximizing energy efficiency and maximizing spectrum efficiency can be achieved simultaneously. Without loss of generality, let assume $K = 2$, $\\textbf{N} = \\{1,2\\}$, $\\textbf{M} = \\{1,2,3,4\\}$, $\\textbf{M}_1 = \\{1,2\\}$,\n$\\textbf{M}_2 = \\{3,4\\}$. For the given value of $p_n$, each user in subchannel $n$ obtains power level based on (18)~\\cite{FFang2016}. Fig.~\\ref{fig:EE-vs-TH} plots the overall throughput and energy efficiency for different values of $p_1$ and $p_2$. From the figure, it is obvious that there are certain values $p_1$ and $p_2$ for which the energy efficiency is better than the overall throughput, and there are other values of those parameters for which the overall throughput is better. More specifically, when $p_1 = p_2 = 4.9$, the overall throughput is better than the energy efficiency (right subfigure), and when $p_1 = 3.5, p_2 = 4.1$, the energy efficiency is\nbetter than the overall throughput (left subfigure). Therefore, the optimal\nenergy-efficient power allocation is associated with the latter case, and the throughput-optimal power allocation belongs to the former case. This completes the proof.\n\n\\end{comment}\n\n\\end{appendices}\n\n\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe puzzle of dark matter remains an outstanding problem of particle physics. One of the more attractive approaches to this problem exploits the fact that weakly interacting massive particles (WIMPs) in thermal equilibrium produce the relic dark matter abundance in the right ballpark. Particles of this type appear in many extensions of the Standard Model. In this work, we explore the possibility that the Standard Model is connected to the dark sector through the Higgs portal~\\cite{Silveira:1985rk},\n\\begin{equation}\nV_{\\rm portal}= \\lambda_{h \\phi} H^\\dagger H \\phi^\\dagger \\phi \\;,\n\\label{eq1}\n\\end{equation}\nwhere $H$ is the Higgs field and $\\phi$ is the ``hidden Higgs'', that is, the field responsible for breaking the gauge symmetry of the hidden sector. We assume the dark sector to be endowed with U(1) or SU(N) gauge symmetry. In that case, the massive gauge fields can play the role of WIMP--type dark matter quite naturally.\nIndeed, they are weakly coupled to the Standard Model and, owing to inherent discrete symmetries, can be stable.\n\nThe U(1) case was considered in Ref.~\\cite{Lebedev:2011iq} where the stabilising $Z_2$ symmetry was found to be related to charge conjugation.\nThe SU(2) example was worked out in Ref.~\\cite{Hambye:2008bq} and the DM stability was attributed to a custodial SO(3). In this work, we extend these ideas to larger SU(N) Lie groups and uncover the nature of the underlying discrete symmetries. We focus on weakly coupled theories, although\nconfining hidden sectors represent a viable alternative~\\cite{Hambye:2009fg,Boddy:2014yra}.\nFinally, we perform a comprehensive study of DM phenomenology in all of these cases, which includes direct and indirect DM detection as well as an analysis of the DM relic abundance. Our main conclusion is that massive hidden gauge fields serve as viable and attractive DM candidates.\n\n\n\\section{Massive U(1) and SU(2) gauge fields as vector DM}\nIn this section, we review the cases of massive U(1) and SU(2) gauge fields as vector DM and identify the underlying symmetries leading to stability of vector dark matter.\nIn what follows we assume that the hidden sector consists only of gauge fields and of scalar multiplets which are necessary to make these gauge fields massive.\n\n\\subsection{Hidden U(1)}\nAn Abelian gauge sector provides the simplest example of the vector DM model endowed\nwith a natural $Z_2$ symmetry~\\cite{Lebedev:2011iq}. In this case, the $Z_2$ corresponds to the charge conjugation symmetry.\n \n Consider a U(1) gauge theory with a single charged scalar $\\phi$,\n \\begin{equation}\n {\\cal L_{\\rm hidden}}= -{1\\over 4} F_{\\mu\\nu} F^{ \\mu\\nu} + (D_\\mu \\phi)^\\dagger D^\\mu \\phi -V(\\phi) \\;,\n \\end{equation}\n where $F_{\\mu\\nu}$ is the field strength tensor of the gauge field $A_\\mu$ and \n $V(\\phi)$ is the scalar potential. For easier comparison to the non--Abelian case, we take the charge of $\\phi $ to be +1\/2. Suppose at the minimum of the scalar potential\n $\\phi$ develops a VEV, $\\langle \\phi \\rangle= 1\/\\sqrt{2}~ \\tilde v$ . The imaginary part of $\\phi$ gets eaten by the gauge field which\n now acquires the mass $m_A= \\tilde g \\tilde v \/2$, where $\\tilde g$ is the gauge coupling.\n The real part of $\\phi$ remains as a degree of freedom. Denoting it by $\\rho$ and normalising it canonically, $\\phi=1\/\\sqrt{2}~ (\\rho +\\tilde v)$ , we get the following gauge--scalar interactions:\n \\begin{eqnarray}\n&& \\Delta {\\cal L}_{\\rm s-g}= {\\tilde g^2\\over 4} \\tilde v \\rho \\; A_\\mu A^{ \\mu} +\n {\\tilde g^2\\over 8} \\rho^2 \\; A_\\mu A^{\\mu} \\;. \n \\end{eqnarray}\nThe system possesses the $Z_2$ symmetry \n\\begin{equation}\nA_\\mu \\rightarrow - A_\\mu ~,\n\\end{equation} \nwhich is the usual charge conjugation symmetry. In terms of the original scalar field, this symmetry acts as $\\phi \\rightarrow \\phi^*$ and $A_\\mu \\rightarrow - A_\\mu$, which is preserved by both the Lagrangian and the vacuum. The $Z_2$ makes the massive gauge field stable.\nNote that this symmetry applies to a sequestered U(1) which has no tree\nlevel mixing with the hypercharge, in which case no mixing is generated radiatively either.\n\nInteractions with the visible sector proceed through the Higgs portal coupling\n\\begin{equation}\n {\\cal L_{\\rm portal}}= - \\lambda_{h \\phi} \\vert H \\vert^2 \\vert \\phi \\vert^2 \\;,\n \\end{equation}\nwhich also leads to the Higgs mixing with $\\rho$. \nIn the unitary gauge, the Higgs field is given by $H^T= (0, v+h)\/\\sqrt{2}$.\nThe fields $\\rho$ and $h$ are then to be expressed in terms of the mass eigenstates $h_{1,2}$ as follows: \n\\begin{eqnarray}\n&& \\rho= - h_1 \\; \\sin\\theta + h_2 \\; \\cos\\theta \\;, \\nonumber\\\\\n&& h = h_1 \\; \\cos\\theta + h_2 \\; \\sin\\theta \\;,\n \\end{eqnarray}\nwhere the mixing angle $\\theta$ is constrained by various experiments, most notably, by LEP and LHC. \nThe upper bound on $\\sin \\theta$ depends on the mass of the heavier state $h_2$ and is around 0.3 for $m_{h_2}$ of the order of a TeV, see e.g.~figure 3 of Ref.~\\cite{Falkowski:2015iwa} for details.\nThe lighter state $h_1$ is identified with the 125 GeV Higgs, while the mass of the second state can vary in a wide range.\n\nAs the Higgs portal necessarily preserves the $Z_2$ symmetry, $A_\\mu$ \nis a viable DM candidate.\nAll the relevant scattering processes proceed through an exchange of $h_1$ and $h_2$, which include DM annihilation into the SM particles as well as DM scattering on nucleons.\n \n \n\\subsection{Hidden SU(2)}\nThe U(1) considerations can easily be extended to SU(2), albeit with a modification of the stabilising symmetry.\nConsider an SU(2) gauge theory with one doublet $\\phi$,\n \\begin{equation}\n {\\cal L_{\\rm hidden}}= -{1\\over 4} F_{\\mu\\nu}^a F^{a \\; \\mu\\nu} + (D_\\mu \\phi)^\\dagger D^\\mu \\phi -V(\\phi) \\;,\n \\end{equation}\nwhere $a=1,2,3$.\nThe potential is assumed to have a minimum at a nonzero VEV of $\\phi$. In the unitary gauge, $\\phi$ takes the form \n\\begin{equation}\n\\phi = {1\\over \\sqrt{2}}\\; \\left( \n\\begin{matrix}\n 0 \\\\\n \\rho +\\tilde v\n\\end{matrix}\n\\right)~,\n\\end{equation}\nwith $\\rho$ being a real field and $\\tilde v$ being the VEV. Denoting the gauge coupling by $\\tilde g$,\nthis leads to the gauge boson mass $m_A= \\tilde g \\tilde v\/2$. The scalar--gauge and gauge--gauge field interactions are given by\n\\begin{eqnarray}\n&& \\Delta {\\cal L}_{\\rm s-g}= {\\tilde g^2\\over 4} \\tilde v \\rho \\; A_\\mu^a A^{ a\\; \\mu} +\n {\\tilde g^2\\over 8} \\rho^2 \\; A_\\mu^a A^{ a\\; \\mu} \\;, \\nonumber \\\\\n&& \\Delta {\\cal L}_{\\rm g-g} = - \\tilde g \\epsilon^{abc} (\\partial_\\mu A_\\nu^a) A^{\\mu \\;b}\nA^{\\nu \\;c} -{\\tilde g^2\\over 4} \\left( (A_\\mu^a A^{\\mu \\;a})^2 -\n A_\\mu^a A_\\nu^a \\; A^{\\mu \\;b} A^{\\nu \\;b} \\right) ~. \n\\end{eqnarray} \nAlthough the triple gauge vertex breaks the parity of the previous section,\nit follows that the system possesses a $Z_2 \\times Z_2'$ symmetry,\n\\begin{eqnarray}\n&& Z_2: ~~ A^1_\\mu \\rightarrow - A^1_\\mu ~~,~~ A^2_\\mu \\rightarrow - A^2_\\mu \\;, \\nonumber \\\\\n&& Z_2':~~ A^1_\\mu \\rightarrow - A^1_\\mu ~~,~~ A^3_\\mu \\rightarrow - A^3_\\mu \\;.\n\\end{eqnarray}\nAs a result, all three $A^a_\\mu$ fields are stable and can play the role of dark matter.\nWhile the above symmetry is \nsufficient to ensure stability of DM, it actually generalises in this simple case to a custodial \nSO(3)~\\cite{Hambye:2008bq}. As we will see, for larger SU(N) groups, it is the discrete\nsymmetry that plays a crucial role. The first $Z_2$ is associated with a gauge transformation, while the $Z_2'$ generalises the charge conjugation symmetry, i.e.~it\ncorresponds to complex conjugation of the group elements. \n\n\nAs before, the dark sector couples to the visible one through the Higgs portal,\n$\\lambda_{h \\phi} \\vert H \\vert^2 \\vert \\phi \\vert^2 $. The discussion of the Higgs mixing with $\\rho$ of the previous section applies here as well.\nClearly, the $Z_2 \\times Z_2'$ symmetry is preserved by the Higgs portal and the hidden gauge fields couple to the visible sector only in pairs. \n\n \n\\section{Extension to SU(3)}\nWhile the SU(2) case is straightforward, larger SU(N) groups exhibit more complicated breaking patterns. In a phenomenologically viable set--up, the symmetry must be broken completely to avoid the existence of massless fields (barring confinement). One possibility \nwould be to break SU(3) by two scalar multiplets in the fundamental representation, i.e.~triplets. One may also explore other options involving SU(3) tensors. As we detail below,\nour conclusion is that a single irreducible representation with two indices cannot break SU(3) completely, which leaves the two triplet option as the minimal one. \n\n\n\\subsection{Breaking SU(3) by tensor fields}\nThe lowest order SU(3) tensor whose generic VEV can break SU(3) completely is the symmetric tensor $\\phi_{ij}$, that is {\\bf 6} of SU(3). \nGauge transformations act on $\\phi_{ij}$ as\n\\begin{equation}\n\\phi \\rightarrow U \\phi \\, U^T \\;.\n\\end{equation}\nBy virtue of Takagi's matrix decomposition, this allows one to bring $\\phi_{ij}$\nto the diagonal form, \n\\begin{equation}\n\\phi = \\left( \n\\begin{matrix}\n \\phi_1 & 0 & 0 \\\\\n 0 & \\phi_2 & 0 \\\\\n 0 & 0 & \\phi_3\n\\end{matrix}\n\\right)~,\n\\end{equation}\nwhere $\\phi_1, \\phi_2, \\phi_3$ are real up to an overall complex phase.\nIf the VEVs of $\\phi_1, \\phi_2 $ and $\\phi_3$ are all different, SU(3) is broken completely. However,\nwhen some of them coincide, the residual gauge group is at least SO(2).\n\nIn order to determine what VEVs are possible, let us write down the most general gauge invariant potential for $\\phi_{ij}$.\nIt is easy to convince oneself that the potential has the form\n\\begin{equation}\nV= m^2 \\; {\\rm Tr} \\phi^\\dagger \\phi + \\lambda_1 {\\rm Tr} (\\phi^\\dagger \\phi)^2\n+ \\lambda_2 \\left( {\\rm Tr} \\phi^\\dagger \\phi \\right)^2 +\n \\left( \\mu \\; {\\rm Det} \\phi + {\\rm h.c.} \\right) \\;,\n\\end{equation}\nwhere $m^2$ can be negative.\nThe minima of this potential determine the $\\phi_i$ VEVs.\nActing upon $V$ with the operator $\\phi_i \\partial\/\\partial \\phi_i$ (no summation over $i$), one finds that all nonzero $ \\langle \\phi_i \\rangle $ satisfy the same equation.\nThis implies that $ \\langle \\phi_i \\rangle $ are either degenerate or zero. \nFor the case $\\mu=0$,\nsuch behaviour has been noticed in~\\cite{Li:1973mq,Jetzer:1983ij}.\nWe find (analytically and numerically) that it persists in the case of nonzero $\\mu$ as well.\nAs a result, the residual gauge symmetry is at least SO(2) and the corresponding gauge bosons remain massless.\\footnote{ Ref.~\\cite{D'Eramo:2012rr} has considered $\\langle \\phi \\rangle$ proportional to the unit matrix, which entails unbroken SO(3). The corresponding gauge bosons may be confined in glueballs.}\nThus, the model with a single symmetric tensor is unrealistic. \n\nOne may also consider the possibility of breaking SU(3) by an antisymmetric tensor.\nBy virtue of Youla's decomposition, it can be gauge--transformed to the block--diagonal form \n\\begin{equation}\n\\phi = \\left( \n\\begin{matrix}\n 0 & \\phi_1 & 0 \\\\\n -\\phi_1 & 0 & 0 \\\\\n 0 & 0 & 0\n\\end{matrix}\n\\right)~.\n\\end{equation} \nTherefore, the residual gauge symmetry is at least Sp(1)=SU(2), which can also be understood via the equivalence between the antisymmetric tensor and the (anti)funda\\-mental representation. Again we find that the model is unrealistic.\n\nSimilarly, complete SU(3) breaking cannot be achieved by a VEV of an adjoint scalar.\nIn this case, the group rank is preserved and thus there are massless gauge fields.\n\nThe next simplest option is to combine the symmetric and antisymmetric tensors. This system would have 9 complex degrees of freedom, while two SU(3) triplets only have 6.\nThus SU(3) breaking with two triplets is minimal and sufficient for our purposes. \n\n\n\\subsection{Breaking SU(3) by triplets and $Z_2 \\times Z_2'$ }\nMisaligned VEVs of two triplets break SU(3) completely. This breaking pattern can be understood in stages: the first triplet VEV reduces the symmetry to SU(2), while the second breaks the remaining SU(2). VEVs misaligned in SU(3) space represent a generic situation, that is, they result from the minimisation of a general scalar potential consistent with SU(3) symmetry. Therefore, such a breaking pattern is phenomenologically viable. \n\nBefore going into details, let us identify the Lie group discrete symmetries which eventually lead to DM stability. One way to find them is to analyse the SU(3) structure constants (using the usual Gell-Mann basis),\n\\begin{eqnarray}\n&& f^{123}=1 \\;, \\nonumber\\\\\n&& f^{147}=-f^{156}= f^{246}=f^{257}=f^{345}=-f^{367}={1\\over 2} \\;, \\nonumber\\\\\n&& f^{458}=f^{678}= {\\sqrt{3}\\over 2} \\;.\n\\end{eqnarray}\nIdentifying the transformation properties of the generators with those of the gauge fields, we define the ``parity'' transformation as\n\\begin{equation}\nA^a_\\mu \\rightarrow \\eta (a) A^a_\\mu \\;.\n\\end{equation}\nIt is easy to see that the structure constants are invariant if the parities are\n\\begin{eqnarray}\n Z_2: ~~&&\\eta (a) = -1 ~~{\\rm for }~~a=1,2,4,5 ~, \\nonumber \\\\\n &&\\eta (a) = +1 ~~{\\rm for }~~a=3,6,7,8 ~,\n\\end{eqnarray}\nand also\n\\begin{eqnarray}\n Z_2': ~~&&\\eta (a) = -1 ~~{\\rm for }~~a=1,3,4,6,8 ~, \\nonumber \\\\\n &&\\eta (a) = +1 ~~{\\rm for }~~a=2,5,7 ~.\n\\end{eqnarray}\nOne may notice that the first $Z_2$ acts on the off--diagonal generators with nonzero entries in the first row, while the second reflects the real SU(3) generators.\nIn Section \\ref{SUN}, we will show that these symmetries generalise to arbitrary SU(N) and \nthat the first $Z_2$ is a gauge transformation, whereas the second corresponds to an outer automorphism of the group, i.e.~complex conjugation of the group elements.\n\nThese symmetries are inherited by the Yang--Mills Lagrangian. If {\\it CP} is conserved, they are also preserved in the matter sector leading to stable dark matter. Below we study the relevant interactions in detail.\n \n\n\\subsection{Explicit example}\nConsider an SM extension by two complex fields $\\phi_1$ and $\\phi_2$ transforming as triplets of hidden gauge SU(3).\nThe Lagrangian of the model is\n\\begin{equation}\n{\\cal L}_{\\rm SM} + {\\cal L}_{\\rm portal} + {\\cal L}_{\\rm hidden} \\;,\n\\end{equation}\nwhere \n\\begin{subequations}\n\\bal\n-{\\cal L}_{\\rm SM} &\\supset V_{\\rm SM} =\\frac{\\lambda_{H}}{2} |H|^4+m_{H}^2 |H|^2 \\;,\n \\\\\n-{\\cal L}_{\\rm portal} &= V_{\\rm portal}= \\lambda_{H11} \\, |H|^2 |\\phi_1|^2 + \\lambda_{H22} \\, |H|^2 | \\phi_2|^2 - ( \\lambda_{H12} \\, |H|^2 \\phi_1^\\dagger \\phi_2 +{\\textrm{h.c.}})\\;,\n \\\\\n{\\cal L}_{\\rm hidden} &= - \\frac12 \\textrm{tr} \\{G_{\\mu \\nu} G^{\\mu \\nu}\\} + |D_\\mu \\phi_1|^2 + |D_\\mu \\phi_2|^2 -V_{\\rm hidden} \\,.\n\\eal\n\\end{subequations}\nHere, $G_{\\mu \\nu}=\\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu + i \\tilde g [A_\\mu,A_\\nu]$ is the field strength tensor of the SU(3) gauge fields $A_\\mu^a$, $D_\\mu \\phi_i = \\partial_\\mu \\phi_i + i \\tilde g A_{\\mu} \\phi_i$ is the covariant derivative of $\\phi_i$, $H$ is the Higgs doublet, and the most general renormalisable hidden sector scalar potential is given by \n\\bal\nV_{\\rm hidden}(\\phi_1,\\phi_2) &=\nm_{11}^2 |\\phi_1|^2\n+ m_{22}^2 |\\phi_2|^2\n- ( m_{12}^2 \\phi_1^\\dagger \\phi_2 + {\\textrm{h.c.}} )\n\\nonumber \\\\ \n& \n+ \\frac{\\lambda_1}{2} |\\phi_1|^4\n+ \\frac{\\lambda_2}{2} |\\phi_2|^4\n+ \\lambda_3 |\\phi_1|^2 |\\phi_2|^2\n+ \\lambda_4 | \\phi_1^\\dagger\\phi_2 |^2\n\\nonumber \\\\ \n& \n+ \\left[\n\\frac{ \\lambda_5}{2} ( \\phi_1^\\dagger\\phi_2 )^2\n+ \\lambda_6 |\\phi_1|^2\n( \\phi_1^\\dagger\\phi_2)\n+ \\lambda_7 |\\phi_2|^2\n( \\phi_1^\\dagger\\phi_2 )\n+ {\\textrm{h.c.}} \\right] \\,.\n\\eal\nUsing SU(3) gauge freedom, 5 real degrees of freedom of $\\phi_1$ and 3 real degrees of freedom of $\\phi_2$ can be removed. Therefore, \nin the unitary gauge $\\phi_1$, $\\phi_2$ read\n\\begin{equation} \\label{unitarygauge}\n\\phi_1={1\\over \\sqrt{2}} \\,\n\\left( \\begin{array}{c}\n0\\\\0\\\\v_1+\\varphi_1\n\\end{array} \\right) \\,,\n\\quad \n\\phi_2= {1 \\over \\sqrt{2}}\\,\n\\left( \\begin{array}{c}\n0\\\\v_2+\\varphi_2\\\\(v_3+\\varphi_3) + i (v_4+\\varphi_4)\n\\end{array} \\right) ~,\n\\end{equation}\nwhere the $v_i$ are real VEVs and $\\varphi_{1-4} $ are real scalar fields.\nAnalogously, we express the Higgs field in the unitary gauge as $H^T= (0,v+h)\/\\sqrt{2}$.\n\n\nIn what follows, we make two assumptions which are crucial for stability of vector dark matter:\n\\begin{itemize}\n\\item\nthe scalar potential is {\\it CP} invariant \n\\item\nthe VEVs of $\\phi_1$, $\\phi_2$ are real.\n\\end{itemize}\nThe first condition implies that the scalar couplings are real, while the second assumes\nthat no spontaneous {\\it CP} violation occurs ($v_4=0$). As a result, {\\it CP}--even and {\\it CP}--odd fields do not mix.\n\nIn this case, the $Z_2 \\times Z_2'$ symmetry extends to the Higgs sector as well. \nUnder the first $Z_2$ all $\\varphi_i$ are even, while the second $Z_2'$ \n reflects $\\varphi_4$ and leaves the other fields intact. As we detail in Section \\ref{SUN},\nthis assignment follows from the explicit form of the first $Z_2$ as a gauge transformation and the fact that the second $Z_2'$ acts as complex conjugation. In any case, these are explicit symmetries of the Lagrangian and the vacuum. \nThe full list of the parities is presented in Table~\\ref{parities}.\nClearly, the lightest states with non--trivial parities cannot decay to the Standard Model particles.\n \\begin{table}[h]\n \\begin{center}\n \\begin{tabular}{|c|c|}\n\\hline\n fields & $Z_2 \\times Z_2'$\n \\\\ \\hline \\hline \n$h, \\varphi^1, \\varphi^2, \\varphi^3, A_\\mu^7$ & $(+,+)$\n\\\\ \\hline \n$A_\\mu^2,A_\\mu^5$& $(-,+)$\n\\\\ \\hline \n$A_\\mu^1,A_\\mu^4$& $(-,-)$\n\\\\ \\hline \n$\\varphi^4,A_\\mu^3,A_\\mu^6, A_\\mu^8$& $(+,-)$\n\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{parities} $Z_2 \\times Z_2'$ parities of the scalars and dark gauge bosons.\n}\n \\end{table}\n \n \nWe now discuss the Lagrangian in more detail, starting with the covariant derivates of $\\phi_1$ and $\\phi_2$,\n\\begin{equation} \\label{dphi}\n|D_\\mu \\phi_i|^2 = |\\partial_\\mu \\phi_i|^2 + i \\tilde g A_\\mu^a \\left((\\partial^\\mu \\phi_i)^\\dagger T^a \\phi_i - {\\textrm{h.c.}} \\right)\n+\\tilde g^2 A_\\mu^a A^{\\mu b} \\, \\phi_i^\\dagger T^a T^b \\phi_i \\,.\n\\end{equation}\nInserting the parametrization~(\\ref{unitarygauge}) with $v_4=0$,\nwe get the kinetic terms for the scalars\n\\begin{equation} \n|\\partial_\\mu \\phi_1|^2+ |\\partial_\\mu \\phi_2|^2 \n=\n\\frac12\n\\sum_{i=1}^4\n(\\partial_\\mu \\varphi_i)^2 \\,,\n\\end{equation}\nthe mass terms for the gauge fields, the mixing terms \nas well as the gauge--scalar interactions.\nLet us first discuss the terms quadratic in the fields.\nThe third term on the r.h.s. of Eq.~(\\ref{dphi}) contains the mass terms for the gauge bosons, \n\\begin{equation}\n{\\cal L} \\supset\n\\frac12 \\left( \\vec{A}_{(1,4)}^T {\\cal M}_{(1,4)} \\vec{A}_{(1,4)}\n+ \\vec{A}_{(2,5)}^T {\\cal M}_{(2,5)} \\vec{A}_{(2,5)}\n+ \\vec{A}_{(3,6,8)}^T {\\cal M}_{(3,6,8)} \\vec{A}_{(3,6,8)}\n+ {\\cal M}_{(7)} A_\\mu^7 A^{\\mu 7} \\right),\n\\end{equation}\nwhere we have used the shorthand notation \n$\\vec{A}_{(1,4)}^T \\equiv (A_\\mu^1,A_\\mu^4)$ and similarly for the other gauge bosons.\nThe mass--squared matrices are\n\\bal \\label{Amasses}\n{\\cal M}_{(1,4)}&={\\cal M}_{(2,5)}\n= \\frac{\\tilde g^2}{4}\n\\left(\n\\begin{array}{cc}\nv_2^2&v_2 v_3\n\\\\\nv_2 v_3&v_1^2+v_3^2\n\\end{array}\n\\right)~,\n\\nonumber \\\\\n{\\cal M}_{(3,6,8)}\n&= \\frac{\\tilde g^2}{4}\n\\left(\n\\begin{array}{ccc}\nv_2^2&-v_2 v_3&-v_2^2\/\\sqrt{3}\n\\\\\n-v_2 v_3&v_1^2+v_2^2+v_3^2&-v_2 v_3\/\\sqrt{3}\n\\\\\n-v_2^2\/\\sqrt{3}&-v_2 v_3\/\\sqrt{3}&(4 v_1^2+v_2^2+4 v_3^2)\/3\n\\end{array}\n\\right)~,\n\\nonumber \\\\\n{\\cal M}_{(7)}\n&= \\frac{\\tilde g^2}{4} ( v_1^2+v_2^2+v_3^2 ) \\,.\n\\eal\nThe second term on the r.h.s. of Eq.~(\\ref{dphi}) contains the $A_\\mu \\partial^\\mu \\varphi$ mixing terms\n\\begin{equation}\n{\\cal L} \\supset \\frac{\\tilde g v_2}{2} A^{\\mu 6} \\partial_\\mu \\varphi_4\n-\\frac{\\tilde g v_3}{\\sqrt{3}} A^{\\mu 8} \\partial_\\mu \\varphi_4\n+ \\frac{\\tilde g v_3}{2} A^{\\mu 7} \\partial_\\mu \\varphi_2\n- \\frac{\\tilde g v_2}{2} A^{\\mu 7} \\partial_\\mu \\varphi_3 \\,.\n\\end{equation}\nIn general, terms of the type $\\kappa_{ai} A_\\mu^a \\partial^\\mu \\varphi^i$ can be removed by the field redefinition \n\\begin{equation} \\label{removemix}\n\\tilde A_\\mu^a = A_\\mu^a + \\partial_\\mu Y^a \\;,\n\\quad \\textrm{where} \\quad\nY^a\\equiv ({\\cal M}^{-1})_{ab}\\, \\kappa_{bi}\\,\\varphi^i\\,.\n\\end{equation}\nWe then have\n\\bal\n&- \\frac12 \\textrm{tr} \\{G_{\\mu \\nu} G^{\\mu \\nu}\\} + \\frac12 (\\partial_\\mu \\varphi^i)^2 + \\frac12 {\\cal M}_{ab} A_\\mu^a A^{\\mu b} + \\kappa_{ai} A_\\mu^a \\partial^\\mu \\varphi^i =\n\\nonumber\n\\\\\n& - \\frac12 \\textrm{tr} \\{\\tilde G_{\\mu \\nu} \\tilde G^{\\mu \\nu}\\} + \\frac12 \\gamma_{ij} (\\partial_\\mu \\varphi^i) (\\partial^\\mu \\varphi^j) + \\frac12 {\\cal M}_{ab} \\tilde A_\\mu^a \\tilde A^{\\mu b} \\,,\n\\eal\nwhere\n\\begin{subequations}\n\\bal\n\\tilde G_{\\mu \\nu} &= \\partial_\\mu \\tilde A_\\nu - \\partial_\\nu \\tilde A_\\mu + i \\tilde g [\\tilde A_\\mu - \\partial_\\mu Y,\\tilde A_\\nu - \\partial_\\nu Y] \\, ,\n\\\\\n\\gamma_{ij} &=\\delta_{ij} - \\kappa^T_{ia}{\\cal M}^{-1}_{ab}\\kappa_{bj} \\, ,\n\\eal\n\\end{subequations}\nand we have defined $Y=Y^a T^a$.\nThe kinetic terms for the vector fields are still canonically normalised, unlike those for the scalar fields.\nThe latter can be normalised canonically by a further field redefinition\n\\begin{equation} \\label{normalizescalars}\n\\tilde \\varphi^i = \\omega_{ik} \\varphi^k \\;, \\quad \\textrm{where} \\quad (\\omega^T \\omega)_{ij}= \\gamma_{ij} \\,.\n\\end{equation}\nNote that the term $- \\frac12 \\textrm{tr} \\{\\tilde G_{\\mu \\nu} \\tilde G^{\\mu \\nu}\\} $ includes additional couplings of the gauge fields to the scalars.\nThe resulting vertices are obtained by \n replacing $A_\\mu^a$ in the triple and quartic gauge boson terms by $-\\partial_\\mu Y^a$\n such that interactions of the type \n $(\\tilde A)^3 \\partial Y, (\\tilde A)^2 (\\partial Y)^2,$ etc. arise.\n The analysis of the general case is very cumbersome, so further we will focus on the simple case of $v_3 = 0$ which retains all the relevant physics.\n In that case, the couplings involving $\\partial Y$ play no role in the DM phenomenology.\n \n\n\\subsection{Detailed study for $v_{3,4} = 0$}\nAs long as $v_1$ and $v_2$ are nonzero, SU(3) is broken completely. Hence it suffices\nto consider the case $v_3 = 0$, which simplifies the analysis.\nThen the only mixing term among the gauge bosons is $ A^3_\\mu A^{\\mu 8}$ and the only\ngauge--scalar mixing terms are \n$ A^6_\\mu \\partial^\\mu \\varphi_4$, $ A^7_\\mu \\partial^\\mu \\varphi_3$.\n\nIn what follows, we restrict ourselves to the case $v_3 = 0$.\n\n\n\\subsubsection{Masses}\n\n\n\\paragraph{Gauge boson masses}\n\nThe gauge boson mass eigenstates are\n\\bal\n\\left( \\begin{array}{c}\nA'^3_\\mu\\\\\nA'^8_\\mu\n\\end{array} \\right)\n=\n \\left( \\begin{array}{c}\n\\cos \\alpha \\, A^3_\\mu + \\sin \\alpha \\, A^8_\\mu\\\\\n\\cos \\alpha \\, A^8_\\mu - \\sin \\alpha \\, A^3_\\mu\n\\end{array} \\right)\\,,\n\\quad\n\\textrm{where}\n\\quad\n\\tan2 \\alpha = \\frac{\\sqrt{3} v_2^2}{2 v_1^2-v_2^2} \\;,\n\\eal\nand the masses are\\footnote{Note that\n$\\frac{\\tan \\alpha}{\\sqrt{3}}=\\frac{v_2^2}{4 v_1^2} + \\ensuremath{\\mathcal O}\\left(\\frac{v_2^6}{v_1^6}\\right)$,\n$\\cos \\alpha = 1- \\frac{3 v_2^4}{32 v_1^4} + \\ensuremath{\\mathcal O}\\left(\\frac{v_2^8}{v_1^8}\\right)$\nand\n$\\sin \\alpha = \\frac{\\sqrt{3} v_2^2}{4 v_1^2} + \\ensuremath{\\mathcal O}\\left(\\frac{v_2^6}{v_1^6}\\right)$.\n}\n\\begin{equation}\nm^2_{A'^3}=\\frac{\\tilde g^2 v_2^2}{4} \\Big(1-\\frac{\\tan \\alpha}{\\sqrt{3}}\\Big) \\,,\n\\qquad\nm^2_{A'^8}=\\frac{\\tilde g^2 v_1^2}{3} \\frac{1}{1-\\frac{\\tan \\alpha}{\\sqrt{3}}} \\,,\n\\label{A3prime-mass}\n\\end{equation}\nwhile the other gauge boson masses can be read off directly from Eq.~(\\ref{Amasses}) for $v_3=0$:\n\\begin{equation}\nm^2_{A^1}=m^2_{A^2}=\\frac{\\tilde g^2 }{4}v_2^2\\;,\n\\quad\nm^2_{A^4}=m^2_{A^5}=\\frac{\\tilde g^2 }{4}v_1^2\\;,\n\\quad\nm^2_{A^6}=m^2_{A^7}=\\frac{\\tilde g^2 }{4}(v_1^2+v_2^2) \\;.\n\\end{equation}\nFor $v_2< v_1$, the light fields are $A_\\mu^{1,2}$ and $A_\\mu^{\\prime 3}$, with the latter being the lightest.\nIt is instructive\nto consider the case $v_2^2 \\ll v_1^2$, so that $\\tan \\alpha $ is small and positive.\nThen $A'^3_\\mu$ is slightly lighter than $A^1_\\mu$ and $A^2_\\mu$ by a factor of $(1-\\frac{\\tan \\alpha}{\\sqrt{3}})^{1\/2}$, while the other five dark gauge bosons are all much heavier, by a factor of order $v_1\/v_2$. The mass degeneracy between $A^1_\\mu$ and $A^2_\\mu$ persists at loop level by symmetry arguments (see Section~\\ref{SUN}).\n\nOne can easily verify that the heavier states $A^{4-7}_\\mu$ and $A^{\\prime 8}_\\mu$ \nall decay into the light states and the SM particles. The decay proceeds via emission\n(off--shell or on--shell) of the {\\it CP} even scalars which couple to the SM Higgs\n and thus to all other SM fields.\n\nThe 3 lightest states all have different $Z_2 \\times Z_2^\\prime$ parities such that they cannot decay into each other by emitting SM particles. The only scalar with negative parity is $\\varphi_4$, however it is generally heavy (see below) and does not contribute to the above decay. Hence, $A_\\mu^{1,2}$ and $A_\\mu^{\\prime 3}$ are stable.\\footnote{The parities allow for a decay of one DM component into two others, however this is forbidden kinematically. }\n\n\n\\paragraph{Gauge boson - scalar mixing}\n\nAccording to Eq.~(\\ref{removemix}), the $A_\\mu \\partial^\\mu \\varphi$ mixing terms are removed by the redefinition (which does not affect the gauge boson masses)\n\\bal\n\\tilde A^6_\\mu&=A^6_\\mu+ \\partial_\\mu Y^6 \\;, \\quad \\textrm{where}\\quad Y^6=\\frac2{\\tilde g} \\frac{v_2}{v_1^2+v_2^2} \\varphi^4\\;,\n\\nonumber\n\\\\\n\\tilde A^7_\\mu&=A^7_\\mu+ \\partial_\\mu Y^7 \\;, \\quad \\textrm{where}\\quad Y^7=-\\frac2{\\tilde g} \\frac{v_2}{v_1^2+v_2^2} \\varphi^3\\,,\n\\eal\nwhile, according to Eq.~(\\ref{normalizescalars}), the canonically normalised scalars are\n\\begin{equation}\n\\tilde \\varphi^3 = \\frac{v_1}{\\sqrt{v_1^2+v_2^2}} \\varphi^3 \\,,\n\\quad\n\\tilde \\varphi^4 = \\frac{v_1}{\\sqrt{v_1^2+v_2^2}} \\varphi^4 \\,.\n\\end{equation}\n\n\n\\paragraph{Scalar masses}\nThe general scalar potential has many parameters. To make our discussion more transparent, let us assume the symmetry \n $\\phi_2 \\to - \\phi_2$ which does not affect the essence of our considerations and \n requires $m_{12}^2=\\lambda_{H12}=\\lambda_6=\\lambda_7=0$.\nIn this case, it turns out that the potential has no local minima with all of $v,v_1,v_2, v_3$ being nonzero such that setting $v_3=0$ is actually required.\nIt should be noted that in practice we are considering the limit in which \nthe above quantities are very small but nonzero such that the decay channels for heavy particles, e.g. $\\tilde\\varphi_3$, into the SM fields are open. \n\n\nThe VEVs $v,v_1,v_2$ can be expressed in a compact form using the matrix\n\\bal\n\\mathbf \\Lambda \\equiv\n\\left( \\begin{array}{ccc} \n\\lambda_H & \\lambda_{H11} & \\lambda_{H22} \\\\\n\\lambda_{H11} & \\lambda_1 & \\lambda_3 \\\\\n\\lambda_{H22} & \\lambda_3 & \\lambda_2 \n\\end{array} \\right) \n\\eal\nas well as the matrices $\\mathbf \\Lambda_{ij}$ defined as $(-1)^{i+j}$ times the matrix obtained by deleting the $i$-th row and $j$-th column of $\\mathbf \\Lambda$, i.e.~$\\det \\mathbf \\Lambda_{ij}$ is the $(i,j)$-cofactor of $\\mathbf \\Lambda$.\nOne finds\n\\bal\nv^2&= - 2 ( m_H^2 \\det{\\mathbf \\Lambda_{11}} +m_{11}^2 \\det{\\mathbf \\Lambda_{21}}+m_{22}^2 \\det{\\mathbf \\Lambda_{31}})\/\\det{\\mathbf \\Lambda} \\,, \\nonumber\n\\\\\nv_1^2&= -2 ( m_H^2 \\det{\\mathbf \\Lambda_{12}} +m_{11}^2 \\det{\\mathbf \\Lambda_{22}}+m_{22}^2 \\det{\\mathbf \\Lambda_{32}})\/\\det{\\mathbf \\Lambda} \\,, \\nonumber\n\\\\\nv_2^2&= -2 ( m_H^2 \\det{\\mathbf \\Lambda_{13}} +m_{11}^2 \\det{\\mathbf \\Lambda_{23}}+m_{22}^2 \\det{\\mathbf \\Lambda_{33}})\/\\det{\\mathbf \\Lambda} \\,.\n\\eal\nThe mass terms for the scalars are\n\\begin{equation}\n- {\\cal L} \\supset \\frac12 \\Phi^T \\mathbf{m}^2 \\Phi + \\frac14 (\\lambda_4+\\lambda_5) (v_1^2+v_2^2) \\, \\tilde \\varphi_3^2 + \\frac14 (\\lambda_4-\\lambda_5) (v_1^2+v_2^2) \\, \\tilde \\varphi_4^2 \\;,\n\\label{phi4-mass}\n\\end{equation}\nwhere $\\Phi^T\\equiv (h,\\varphi_1,\\varphi_2)$ and\n\\bal\n\\mathbf{m}^2 =\n\\left( \\begin{array}{ccc} \n\\lambda_H v^2 & \\lambda_{H11} v v_1 & \\lambda_{H22} v v_2 \\\\\n\\lambda_{H11} v v_1 & \\lambda_1 v_1^2& \\lambda_3 v_1 v_2\\\\\n\\lambda_{H22} v v_2& \\lambda_3 v_1 v_2 & \\lambda_2 v_2^2\\\\\n\\end{array} \\right) \\,.\n\\eal\nNote that $\\tilde \\varphi_3$ and $\\tilde \\varphi_4$ are generally heavier than $A^1,A^2,A'^3$ since their masses involve $v_1$\n(unless $\\lambda_{4,5}$ are very small). \n\nThe matrix $\\mathbf{m}^2$ is positive definite if and only if $\\det \\mathbf \\Lambda>0$, $\\det \\mathbf \\Lambda_{33}\\equiv \\lambda_H \\lambda_1- \\lambda_{H11}^2>0$ and $\\lambda_H>0$.\nIt can be diagonalised by an orthogonal transformation\n$O^T \\mathbf{m}^2 O = \\textrm{diag}(m_{H_1}^2,m_{H_2}^2,m_{H_3}^2) ,$\nwhere\n\\bal\nO&= \\left( \\begin{array}{ccc}\nc_{12} c_{13}& s_{12} & c_{12}s_{13}\\\\\n-c_{13} c_{23} s_{12} - s_{13} s_{23} & c_{12} c_{23} &-c_{23} s_{12} s_{13}+ c_{13} s_{23} \\\\\n-c_{23} s_{13} + c_{13} s_{12} s_{23} &-c_{12} s_{23} &c_{13} c_{23} + s_{12} s_{13} s_{23}\n\\end{array} \\right) \\;\n\\eal\nand we have used the abbreviation $s_{ij} \\equiv \\sin \\theta_{ij}, c_{ij} \\equiv \\cos \\theta_{ij}$. Instead of providing the most general formulae, let us focus on a simplified case.\n Suppose that the $(12)$ and $(23)$ entries of $\\mathbf{m}^2 $ are much smaller than the other matrix elements, in other words, that $\\lambda_{H11}$\nand $\\lambda_3$ are very small. In this case, the Higgs mixing with\n$\\varphi_2$ is the dominant one. The reason behind this choice is that \nthe DM constituents $A^1,A^2$ and $A'^3$ all have a significant coupling to $\\varphi_2$, see Table~\\ref{AAscalar}.\nGiven that the mixing between the Higgs and $\\varphi_2$ is substantial,\nthis facilitates DM annihilation into the SM fields. \nClearly, similar considerations also apply to the general case with all the angles $\\theta_{ij}$ being significant.\n\n\nAssuming small $\\mathbf{m}^2_{12},\\mathbf{m}^2_{23}$, one finds\n\\begin{subequations}\n\\bal\n& \\theta_{12} \\approx p_{32}\\, \\mathbf{m}^2_{12} + q \\, \\mathbf{m}^2_{23} \\;,\n\\\\\n& \\theta_{23} \\approx p_{21}\\, \\mathbf{m}^2_{23} + q \\, \\mathbf{m}^2_{12} \\;,\n\\\\\n& \\tan 2 \\theta_{13} \\approx \\frac{2 \\lambda_{H22} v v_2}{\\lambda_2 v_2^2-\\lambda_H v^2} \\,,\n\\eal\n\\end{subequations}\nwhere $p_{ij}=(\\mathbf{m}^2_{ii}-\\mathbf{m}^2_{jj})\/s$, $q=\\mathbf{m}^2_{13}\/s$ and $s=(\\mathbf{m}^2_{13})^2+(\\mathbf{m}^2_{11}-\\mathbf{m}^2_{22})(\\mathbf{m}^2_{22}-\\mathbf{m}^2_{33})$.\nThe mass eigenstates are\n\\bal\n\\left( \\begin{array}{c}\nh_1\\\\\n{\\cal H}\\\\\nh_2\n\\end{array} \\right)\n\\equiv O^T \\Phi\n\\approx\n \\left( \\begin{array}{c}\nc_{13} h - s_{13} \\varphi_2 \\\\\n\\varphi_1 \\\\\nc_{13} \\varphi_2 + s_{13} h\n\\end{array} \\right)\n- \\theta_{12}\n \\left( \\begin{array}{c}\n c_{13} \\varphi_1 \\\\\n-h \\\\\ns_{13} \\varphi_1\n\\end{array} \\right)\n- \\theta_{23}\n \\left( \\begin{array}{c}\n s_{13} \\varphi_1 \\\\\n \\varphi_2\\\\\n-c_{13} \\varphi_1\n\\end{array} \\right)\n\\,. \\label{rotation-matrix}\n\\eal\nand the mass--squared eigenvalues are \n\\begin{subequations}\n\\bal\n& m^2_{h_1,h_2} \\approx \\frac12 ( \\lambda_2 v_2^2+\\lambda_H v^2) \\mp \\frac{ \\lambda_2 v_2^2-\\lambda_H v^2}{2 \\cos 2 \\theta_{13}}\n\\\\\n& m^2_{\\cal H} \\approx \\lambda_1 v_1^2 \\,.\n\\eal\n\\end{subequations}\nThe eigenstates $h_1,h_2$ are typically the lighter ones, while ${\\cal H}$\nis heavy (unless $\\lambda_1$ is very small). In our analysis of DM phenomenology, we retain only the former states.\nIn summary, the relevant light fields are the DM components $A^1,A^2$ and $A'^3$ as well as the mediators $h_1,h_2$ which link the dark sector to the SM fermions and gauge bosons.\n \n\n\\subsubsection{Couplings}\n\nThe full list of the couplings is not necessary for our DM studies.\nThe important couplings are those with two gauge bosons and one or two scalars at the vertex. In terms of the variables $\\tilde A_\\mu^a $\nand $\\tilde \\varphi^i$, most of these are listed in Table~\\ref{AAscalar} \nand Table~\\ref{AAscalarscalar}. For our applications, the couplings of $h_1$ and $h_2$ are obtained from these tables using the relation\n (\\ref{rotation-matrix}), in which one may neglect $\\theta_{12}$\n and $\\theta_{23}$.\n\nWe focus on the case $v_1 \\gg v_2$ so that\n DM consists of $A^1,A^2$ and $A'^3$. Other fields with non--trivial parities\n decay into these states and the SM particles. For instance, \n the processes $\\tilde \\varphi_4 \\rightarrow A^1 A^2 \\, +$~SM and \n $\\tilde \\varphi_4 \\rightarrow A^{3\\prime}\\, +$~SM are allowed. \n (When $v_1$ and $v_2$ are close, the composition of DM depends on the mass splittings.)\n DM annihilation and scattering proceeds through an exchange of $h_1$\n and $h_2$. Therefore, only the vertices involving these fields \n play a significant role.\n\n\n\\section{Generalisation to arbitrary SU(N)} \\label{SUN}\nSU(N) is broken completely by generic VEVs of $N-1$ fields $\\phi_i$ in the fundamental representation.\nThe $\\phi_i$'s can be gauge--transformed to the form\n\\begin{equation}\n\\phi_1= \\left( \\begin{matrix} \n   0\\\\ \n   0 \\\\\n   ...\\\\\n   0 \\\\\n   \\rho_1 \n  \\end{matrix}\n\\right) ~~,~~\n\\phi_2= \\left( \\begin{matrix} \n   0\\\\ \n   0 \\\\\n   ...\\\\\n   \\rho_2^{(1)} \\\\\n   \\rho_2^{(2)} e^{i \\xi_2}\n   \\end{matrix} \n\\right) ~~,~~...~~,~~\n\\phi_{N-1}= \\left( \\begin{matrix}\n    0\\\\ \n   \\rho_{N-1}^{(1)} \\\\\n   ...\\\\\n   \\rho_{N-1}^{(N-2)} e^{i \\xi_{N-1}^{(N-3)}}\\\\ \n   \\rho_{N-1}^{(N-1)} e^{i \\xi_{N-1}^{(N-2)}}\n   \\end{matrix} \n\\right) ~~.\n\\end{equation}\nHere the radial fields $\\rho_i^{(j)}$ and the phases $\\xi_i^{(j)}$ are real. We label the scalars such that \nthe lightest gauge fields are associated with the SU(2) subgroup which gets broken\nat the last stage by a VEV of $\\rho_{N-1}^{(1)}$. We assume that the VEVs as well as the couplings in the scalar potential are all real so that {\\it CP} is preserved in the hidden sector.\n\nThe generalisation of the $Z_2 \\times Z_2'$ parity to SU(N) is as follows. \nThe transformation properties of the gauge fields are identified with those of the corresponding SU(N) generators. \nThe basis of the $N(N-1)$ off--diagonal generators $T^{ab}, \\tilde T^{ab}$ can be chosen as \n\\begin{eqnarray}\n&& ( T^{ab} )_{ij}= \\delta_{ia} \\delta_{jb} + \\delta_{ib} \\delta_{ja} \\;, \\nonumber\\\\\n&& ( \\tilde T^{ab})_{ij}= -i \\delta_{ia} \\delta_{jb} + i \\delta_{ib} \\delta_{ja} \\;,\n\\end{eqnarray}\nwhere $a=1,..,N-1$ and $b=2,..,N$.\nWith the Cartan generators denoted by $H^\\alpha$, the $Z_2$ associated with \ncomplex conjugation of the group elements acts as\n\\begin{eqnarray}\n&& T^{ab} \\rightarrow -T^{ab} ~~,~~ \\tilde T^{ab} \\rightarrow \\tilde T^{ab}\n~~,~~ H^\\alpha \\rightarrow - H^\\alpha \\;. \\label{Z21N}\n\\end{eqnarray}\nThis is a well known outer automorphism of SU(N) which entails the corresponding symmetry of the Yang--Mills Lagrangian.\n\nAnother $Z_2$ can be defined by reflecting the off--diagonal generators containing nonzero elements in the first row:\n\\begin{eqnarray}\n&& T^{1a} \\rightarrow -T^{1a} ~~,~~ \\tilde T^{1a} \\rightarrow - \\tilde T^{1a}\n~~, \\nonumber\\\\\n&& T^{bc} \\rightarrow T^{bc} ~~~~~,~~ \\tilde T^{bc} \\rightarrow \\tilde T^{bc}\n~~ ~~~(b,c \\geq 2), \\nonumber\\\\\n&& H^\\alpha \\rightarrow H^\\alpha \\;. \\label{Z22N}\n\\end{eqnarray}\nIt is easy to show that this $Z_2$ is an inner automorphism. Indeed, it corresponds to the group transformation with\n\\begin{equation}\nU= e^{ { i \\pi \\over N}} \\; {\\rm diag}(-1,1,...1) \\;. \\label{U}\n\\end{equation}\nThe gauge fields $A^{1-3}_\\mu$ associated with the \nupper left SU(2) block \n\\begin{equation}\nT^{12},~~\\tilde T^{12},~~ H^1= {\\rm diag}(1,-1,0,...,0)\n\\end{equation}\ntransform under these parities the same way as did the SU(2) gauge fields\nunder $Z_2 \\times Z_2'$ of the previous section. The above gauge transformation of course leaves the Yang--Mills Lagrangian invariant.\n\nThese symmetries are preserved by gauge interactions with scalars in our set--up. The $Z_2$\nassociated with complex conjugation acts on scalars by reflecting the complex phases,\nwhich therefore correspond to odd fields under the transformation~(\\ref{Z21N}). This symmetry is guaranteed by {\\it CP} invariance of the Lagrangian and preserved by the vacuum (assuming no spontaneous {\\it CP} violation).\nThe second $Z_2$ is a gauge transformation. On vectors of the form \n$(0,a_1,...,a_{N-1})$, it acts as multiplication by an overall \nconstant phase which cancels in all the Lagrangian terms. It is therefore a valid symmetry in the broken phase as well. \n\nAs long as the $\\phi_i$ have a zero first component,\nthe interaction vertices contain an even number of $T^{1a}$ and $\\tilde T^{1a}$.\n The gauge fields associated with $a>2$ are heavier than those corresponding to $a=2$. By virtue of \n the vertices involving $T^{12} T^{1k}$ $(k>2)$ and the matter fields, they decay \n to the lighter fields such that only the final SU(2) block remains stable. \n (Similar considerations also apply to other heavy gauge fields.)\nThen DM is composed mostly of the aforementioned $A^{1-3}_\\mu$ whose stability is enforced by $Z_2 \\times Z_2'$.\\footnote{As before, we take the ``phase'' fields to be heavy since they get\ntheir masses from large VEVs $\\langle \\phi_i \\rangle$, unlike $A^{1-3}_\\mu$. In the presence of more than $N-1$ fundamentals, this logic no longer applies and the light phase fields can constitute DM.} \n \nAs in the SU(3) case, DM consists of three components, two of which are degenerate in mass,\n\\begin{equation}\nm_{ A^{1\\prime }} =m_{ A^{2\\prime }} \\not= m_{ A^{3\\prime }} \\;,\n\\end{equation}\nwhere $ A^{1\\prime -3 \\prime}_\\mu$ are the mass eigenstates consisting mostly of $ A^{1-3 }_\\mu$\nwith some admixture of other gauge fields (see the SU(3) example). The degeneracy persists at loop level. This can be seen as follows. The SU(N) Lie algebra possesses a discrete symmetry which interchanges the real and imaginary generators with nonzero entries in the first row,\n\\begin{equation}\nT^{1a} \\rightarrow \\tilde T^{1a} ~~,~~ \\tilde T^{1a} \\rightarrow - T^{1a}~~,\n\\label{A1-A2}\n\\end{equation}\nwhile all the other generators remain intact.\nThis is achieved by the group transformation\n\\begin{equation}\nU'= e^{ { -i \\pi \\over 2 N}} \\; {\\rm diag}(i,1,...1) \\;,\n\\end{equation}\nwhich can be recognised as the square root of $U$ in (\\ref{U}).\nThis gauge transformation acts on $\\phi_i$ with a zero first entry as an overall\nconstant phase multiplication. Since such a phase cancels in all of the Lagrangian terms,\n(\\ref{A1-A2}) remains a valid symmetry even in the broken phase.\n\n Consider now the mass matrix for the gauge fields associated with $T^{1a}$ and $\\tilde T^{1a}$. By virtue of (\\ref{Z22N}), only fields corresponding to $T^{1a}$ and $\\tilde T^{1a}$\n can mix, while (\\ref{Z21N}) forbids a mixing between the tilded and untilded fields.\n The resulting mass matrix for $T^{1a}$ is then identical to that of $\\tilde T^{1a}$\n according to (\\ref{A1-A2}). Analogous considerations apply to the kinetic terms.\n Hence the lightest eigenstates have the same mass. \n \n The resulting DM phenomenology is analogous to that for the SU(3) case. \n \n\n\\section{Dark matter phenomenology }\n\n In what follows, we consider direct and indirect detection as well as relic abundance constraints on vector DM. \n In the U(1) and SU(2) cases, all the scattering processes are mediated by $h_1$ and $h_2$. For SU(3) and larger groups, further states can contribute.\nHowever, we make the simplifying assumption that the Higgs mixing with those states is small and\/or such states are heavy. In that case, it suffices to consider an exchange of $h_1$ and $h_2$ only. We note that earlier phenomenological analyses of vector dark matter \nhave appeared in various contexts~\\cite{Kanemura:2010sh}-\\cite{Duch:2014xda}.\nThe current collider searches for dark matter, e.g. in the form of a monojet plus missing energy, do not set further useful constraints on the model, see e.g. Ref.~\\cite{Kim:2015hda}.\n\n\n\\subsection{U(1) dark matter}\nLet us start with the U(1) case. The relevant terms in the Lagrangian are\n\\bal\n{\\cal L} &\\supset {1\\over 2} m_A^2 A_\\mu A^\\mu + {\\tilde g \\, m_A \\over 2} \n\\left(-h_1 \\sin\\theta + h_2 \\cos\\theta \\right) A_\\mu A^\\mu \\nonumber \\\\\n&+\n{\\tilde g^2 \\over 8 } \\left( h_1^2 \\sin^2 \\theta -2 h_1 h_2 \\sin\\theta \\;\\cos\\theta +\nh_2^2 \\cos^2 \\theta \\right) A_\\mu A^\\mu \\;.\n\\label{DMinteractions}\n\\eal\nThe couplings of $h_1$ and $h_2$ to the SM fields are those of the SM Higgs up\nto the suppression factors of $\\cos\\theta$ and $\\sin\\theta$, respectively.\nA phenomenological analysis of this model in the decoupling limit $m_{h_2} \\gg m_{h_1}$,\n$\\sin\\theta \\rightarrow 0$ can be found in~\\cite{Kanemura:2010sh} and \n\\cite{Lebedev:2011iq,Djouadi:2011aa}. \nRelated studies have also appeared in~\\cite{Farzan:2012hh,Baek:2012se}. \n\n\nThe DM scattering on nucleons proceeds through the $t$--channel exchange of $h_{1,2}$\n and leads to the following spin--independent cross\nsection (see e.g.~\\cite{Lebedev:2011iq}),\n\\begin{equation}\n\\sigma^{\\rm SI}_{A-N}= {g^2 \\tilde g^2 \\over 16 \\pi} \\; {m_N^4 f_N^2 \\over m_W^2} \\;\n{ (m_{h_2}^2 - m_{h_1}^2 )^2 \\sin^2 \\theta \\; \\cos^2 \\theta \\over m_{h_1}^4 m_{h_2}^4} \\;,\n\\end{equation}\nwhere $m_N$ is the nucleon mass and $f_N \\simeq 0.3$ parametrizes the Higgs--nucleon coupling. One should keep in mind that there is significant uncertainty in $f_N$ and here\nwe use the value somewhat smaller than the one assumed in~\\cite{Lebedev:2011iq}. \nAs expected, $h_1$ and $h_2$ contribute with opposite signs. Since \n$m_{h_2}^2 \\gg m_{h_1}^2$ in realistic cases, the cancellation is not very significant.\n %\n\\begin{figure}[h] \n\\centering{\n\\includegraphics[scale=0.283]{DMDMtoH.pdf}\n\\includegraphics[scale=0.283]{DMDMtoHtoHH.pdf}\n\\includegraphics[scale=0.283]{DMDMtoHH_Tchannel.pdf}\n\\includegraphics[scale=0.283]{DMDMtoHH.pdf}\n }\n\\caption{ \\label{diagrams1}\nLeading diagrams for vector DM annihilation.\n}\n\\end{figure}\n\n\nThe calculation of the DM annihilation cross section is more involved due to a few contributing diagrams (Fig.~\\ref{diagrams1}). To compute the DM relic abundance, we use the software package micrOMEGAs 4.1.8~\\cite{Belanger:2014vza}. It is important to note that the leading $s$--channel annihilation amplitude\ndue to the $h_{1,2}$--exchange is proportional to the factor\n \\begin{equation}\n {\\cal A}^{\\rm annih} \\propto \\sin\\theta \\;\\cos\\theta \\; \\left( {1\\over s-m_{h_1}^2 } -\n {1\\over s-m_{h_2}^2 } \\right) \\;,\n\\end{equation}\nwhere the $s$--parameter can be approximated by $s \\simeq 4 m_A^2$. Therefore,\nthe amplitude is highly suppressed at $m_A \\gg m_{h_2}$. \nAlthough other annihilation channels remain available, this makes DM annihilation\ninefficient for heavy masses and the corresponding parameter space is challenged by the direct detection constraint. \n\nClearly, the $s$--channel annihilation becomes very efficient around the resonances,\n$m_A \\simeq m_{h_1}\/2$ and $m_A \\simeq m_{h_2}\/2$. In this case, a very small gauge coupling is sufficient to obtain the right relic abundance. \nThe first resonance is quite narrow due to the small width of the SM Higgs, whereas \nthe second resonance is broad since many decay channels are available to $h_2$. For \n$m_{h_2} > 2 m_{h_1}$, the decay $h_2 \\rightarrow h_1 h_1$ becomes important. Its significance depends on $\\lambda_{h \\phi}$ of Eq.~(\\ref{eq1}) with BR${_{h_2 \\rightarrow h_1 h_1}}$ increasing for larger $\\lambda_{h \\phi}$ (see the explicit formulae in~\\cite{Falkowski:2015iwa}). The resonance is widened further by the thermal averaging over the DM momentum.\n\nThe most important features of DM annihilation are associated with the $s$--channel.\nOther channels play a less significant role. Similar considerations apply to the indirect detection constraint due to the gamma ray emission in the process of DM annihilation (FERMI). \n\nThe last constraint we impose is of theoretical nature. We require that the theory be perturbative at the TeV scale. One way to enforce it is to demand perturbative unitarity at tree level, for instance, in the process $h_i h_i \\rightarrow h_j h_j$. The resulting constraint was estimated in~\\cite{Chen:2014ask} to be \n\\begin{equation}\n\\lambda_i < {\\cal O} (4\\pi\/3) \\;,\n\\label{unitarity}\n\\end{equation} \nwhere $\\lambda_i$ are the scalar quartic couplings.\nWe define the quartic couplings involving $\\phi$ by \n$\\Delta V_{\\rm quart} = \\lambda_{h \\phi} \\vert H \\vert^2 \\vert \\phi \\vert^2 +\n{1\\over 2} \\lambda_{\\phi} \\vert \\phi \\vert^4 $. They can be expressed in terms of \nthe masses, the gauge coupling and the mixing angle as (see e.g.~\\cite{Falkowski:2015iwa}) \n\\begin{eqnarray}\n&& \\lambda_{h \\phi}= \\tilde g ~\\sin 2 \\theta~ { m_{h_2}^2-m_{h_1}^2 \\over 4 v m_A } \\;,\n\\nonumber \\\\ \n&& \\lambda_{\\phi}= \\frac{ 4 \\, \\lambda_{h \\phi}^2 }{\\sin^2 2 \\theta } \\frac{v^2}{m_{h_2}^2-m_{h_1}^2} \\left(\\frac{m_{h_2}^2}{m_{h_2}^2-m_{h_1}^2} -\\sin^2 \\theta \\right) \\;,\n\\end{eqnarray} \n where we have used $\\tilde v =2 m_A\/ \\tilde g $. \nThis implies that both $\\lambda_{h \\phi}$ and $\\lambda_\\phi$ become large for heavy $h_2$ \nor for light dark matter. As a result, Eq.~(\\ref{unitarity}) imposes an important constraint \non our model. We further require the standard perturbativity constraint \n\\begin{equation}\n{\\tilde g^2 \\over 4 \\pi} < 1 ~,\n\\end{equation} \nwhich we find less significant for our purposes. \n\\begin{figure}[h] \n\\centering{\n\\includegraphics[scale=0.43]{U1ST03MHp280DM1TeV.pdf}\n\\includegraphics[scale=0.43]{U1ST03MHp1000DM2TeV.pdf}\n\\includegraphics[scale=0.43]{U1ST02MHp3000DM3TeV.pdf}\n\\includegraphics[scale=0.43]{SU2ST03MHp280DM1TeV.pdf}\n\\includegraphics[scale=0.43]{SU2ST03MHp1000DM2TeV.pdf}\n\\includegraphics[scale=0.43]{SU2ST02MHp3000DM3TeV.pdf}\n}\n\\caption{ \\label{plots}\nConstraints on the gauge coupling $\\tilde g$ vs DM mass $m_A$ for U(1) and SU(2).\nThe area between the red lines is favoured by the DM relic abundance, while the regions above \nthe dashed blue, dotted black and green lines are ruled out by direct detection, indirect detection and perturbativity of $\\lambda_i$, respectively.\n{\\it Upper row:} U(1) dark matter with $\\sin\\theta=0.3, m_{h_2}= 280$ GeV (left),\n $\\sin\\theta=0.3, m_{h_2}= 1$ TeV (center), $\\sin\\theta=0.2, m_{h_2}= 3$ TeV (right).\n {\\it Lower row:} SU(2) dark matter with $\\sin\\theta=0.3, m_{h_2}= 280$ GeV (left),\n $\\sin\\theta=0.3, m_{h_2}= 1$ TeV (center), $\\sin\\theta=0.2, m_{h_2}= 3$ TeV (right). \n}\n\\end{figure}\n\nOur results for U(1) DM are presented in Fig.~\\ref{plots}, upper row. \nWe include the constraints from PLANCK~\\cite{Ade:2015xua} (relic abundance), LUX~\\cite{Akerib:2013tjd} (direct detection),\nFERMI~\\cite{Ackermann:2015zua} (indirect detection) and perturbativity of $\\lambda_i$.\nThe area between the red lines is consistent with the thermal relic DM abundance measured by PLANCK. The LUX data provide the strongest constraint on the allowed parameter space. \nThe FERMI bound is typically relevant for light DM, while the perturbativity bound becomes\nimportant for heavy $h_2$. \nIn each panel, the mixing angle is chosen such that, for a given $m_{h_2}$,\nit is consistent with the LHC and EW precision data~\\cite{Falkowski:2015iwa}. Heavier\n$h_2$ imply smaller $\\sin\\theta$, so we take $\\sin\\theta=0.3$ for the left and center panels,\nand $\\sin\\theta=0.2$ for the right panel.\n\n\nThe two dips are associated with the resonant annihilation through $h_1$ and $h_2$. The second\nresonance gets broader with increasing $h_2$ due to the increase in $\\lambda_{h \\phi}$ and\navailability of the decay $h_2 \\rightarrow h_1 h_1$. The area around this resonance constitutes the largest parameter space consistent with all of the constraints. For $m_{h_2}= 280$ GeV,\nthe allowed DM mass range is about 100 GeV; for $m_{h_2}= 1$ TeV, it widens to 1 TeV, and\nfor $m_{h_2}= 3$ TeV, it reaches more than 3 TeV. The resonance is broadened by the thermal averaging\nover the DM momentum, so even though it appears very broad for $m_{h_2}= 3$ TeV,\nit is still consistent with perturbativity. \n\nThe dip associated with the resonant annihilation through $h_1$ is quite narrow and does not open up further significant areas of parameter space. \nOther features of the PLANCK curve are local peaks corresponding to the kinematic opening of additional annihilation channels. For instance, the peak at $m_A \\sim 80$ GeV is associated with the $W^+ W^-$ final state. There are further visible peaks at $(m_{h_2} + m_{h_1})\/2$ and $m_{h_2}$. \n\n\nAway from the resonances, there appears to be a further allowed region at $m_A > m_{h_2}$.\nSince the $h_2$ production is not suppressed by $\\sin\\theta$, \nin this case the $t$--channel annihilation $AA \\rightarrow h_2 h_2$\nand the quartic interactions dominate.\nThe Planck--allowed strip is dangerously close to the LUX bound, so the conclusion\ndepends strongly on the nucleon--Higgs coupling $f_N \\simeq 0.3$, which suffers from substantial uncertainties.\n\n\n\\subsection{SU(2) dark matter}\nAside from the gauge self--interactions, the Lagrangian (\\ref{DMinteractions}) applies\nto the SU(2) case as well, up to the summation over the 3 species, $A_\\mu A^\\mu \n\\rightarrow A_\\mu^a A^{a \\mu }$. The main change compared to the U(1) case is that \nthe annihilation cross section decreases since only the species with the same group index \ncan annihilate through the Higgs--like states. In order to keep the same relic abundance, one needs to increase the gauge coupling. Since the $s$--channel annihilation through $h_{1,2}$\noften dominates, this amounts approximately to\n\\begin{equation}\n\\tilde g \\rightarrow \\sqrt{3} \\tilde g \\;,\n\\end{equation}\nwhereas if \n the $t$--channel and the quartic interactions dominate, the rescaling factor is closer\n to $3^{1\/4}$. The direct detection constraint remains the same as in the U(1) case since particles with different group indices scatter the same way on nucleons.\nThis decreases somewhat the allowed parameter space compared to the Abelian case (Fig.~\\ref{plots}).\n\nNon--Abelian DM features a semi--annihilation channel $AA \\rightarrow A h_{1,2}$\n(Fig.~\\ref{diagrams2}).\nIn some regimes, for example at large $\\tilde g$ and small $\\sin\\theta$, it can even dominate~\\cite{D'Eramo:2012rr} (see also~\\cite{Arina:2009uq,Khoze:2014xha}). \nUsing the analytical results of~\\cite{D'Eramo:2012rr},\nwe find that DM semi--annihilation is insignificant in the relevant parameter regions.\nFor example, at $m_A > m_{h_2}$ the $\\sin\\theta$--unsuppressed and potentially important channel $AA \\rightarrow A h_2$ opens up, yet it is dominated by $AA \\rightarrow h_2 h_2$.\nAlso, around the resonances the gauge coupling is rather small which diminishes the relative importance of semi--annihilation. \n\n\\begin{figure}[h] \n\\centering{\n\\includegraphics[scale=0.32]{DMDMtoDMH_Schannel.pdf} \\hspace{0.8cm}\n\\includegraphics[scale=0.32]{DMDMtoDMH_Tchannel.pdf}\n }\n\\caption{ \\label{diagrams2}\nSemiannihilation of vector DM.\n}\n\\end{figure}\n\nNo firm conclusion can be reached as to whether the region $m_A > m_{h_2}$ is allowed.\nAs stated above, the uncertainties in $f_N$ play a critical role due to the proximity of the Planck band and the LUX bound.\n\n\n\\subsection{SU(3) dark matter}\nIn the SU(3) case, DM is composed again of 3 species with two of them being degenerate \n($A^1_\\mu, A^2_\\mu$) in mass and the third one being lighter ($A^{3\\prime}_\\mu$). This is a result of the mixing between the gauge bosons corresponding to the Cartan generators of SU(3). Therefore, while the couplings of $A^1_\\mu, A^2_\\mu$ to the Higgs like scalars remain the same\nas in the SU(2) case, the coupling of $A^{3\\prime}_\\mu$ changes. In terms of\n$m_A \\equiv m_{A^{1,2}}$, the relevant Lagrangian reads\n\\bal\n{\\cal L} & \\supset \\frac12 m_{A^{}}^2 \\Big(\\sum_{a=1,2} A^a_\\mu A^{a\\mu}+\\Big(1-{\\tan\\alpha \\over \\sqrt{3}}\\Big) A'^3_\\mu A'^{3\\mu}\\Big)\n\\\\\n&+ \\frac{ \\tilde g \\, m_{A^{}}}2 (- h_1 \\sin\\theta + h_2 \\cos\\theta) \\Big(\\sum_{a=1,2} A^a_\\mu A^{a\\mu}+\\Big(\\cos\\alpha-{\\sin\\alpha\\over \\sqrt{3}}\\Big)^2 A'^3_\\mu A'^{3\\mu}\\Big)\n\\nonumber\n\\\\\n&+ \\frac{ \\tilde g^2}8 (h_1^2 \\sin^2\\theta -2 h_1 h_2 \\sin\\theta \\cos\\theta + h_2^2 \\cos^2 \\theta) \\Big( \\! \\sum_{a=1,2} \\!\\! A^a_\\mu A^{a\\mu} \\! + \\! \\Big(\\cos \\alpha-{\\sin \\alpha \\over\\sqrt{3}}\\Big)^2 A'^3_\\mu A'^{3\\mu}\\Big) .\n\\nonumber\n\\eal\nThe mass of the lighter DM component is reduced by the factor \n$( 1- {1\\over \\sqrt{3}} \\tan\\alpha )^{1\/2} $ compared to that of \n$A^1_\\mu$ and $A^2_\\mu$, while the gauge--scalar couplings decrease by a factor\n$( \\cos\\alpha- {1\\over \\sqrt{3}} \\sin\\alpha )^{ 2}$.\nTherefore, the lighter state has a smaller annihilation cross section. \nNote that \n$\\sin \\alpha \\simeq \\frac{\\sqrt{3} v_2^2}{4 v_1^2}$ and even a factor \nof two difference in the triplet VEVs leads to a rather small $\\alpha \\sim 10^{-1}$.\nIn that case, there is no tangible difference between the SU(2) and the SU(3) analyses.\n %\n\\begin{figure}[t] \n\\centering{\n\\includegraphics[scale=0.6]{SU3.pdf}\n}\n\\caption{ \\label{plot2}\n Illustration of the main features of SU(3) vs SU(2) DM in the $extreme$ case. Here $v_1\/v_2 = 1.1$ and $\\sin\\theta=0.3, m_{h_2}= 280$ GeV. In the SU(3) case, $M_{\\rm DM}$ stands for the mass of the (dominant) lighter component $A_\\mu^{3\\prime}$.\n See Fig.~\\ref{plots} for further details. \n \\label{su3plot}\n}\n\\end{figure}\n\nTo understand where differences can appear, it is instructive to consider the limit $v_1 \\simeq v_2$. Although in this case there are further relatively light states\nthat can mediate DM annihilation (e.g. $A_\\mu^{8\\prime}$), let us consider the simplified example in which only the same states are allowed to contribute in the SU(2) and SU(3) \nset--ups. The main features (Fig.~\\ref{su3plot}) are that the gauge coupling must be larger in the SU(3) case\nin order to allow for efficient annihilation of $A^{3\\prime}_\\mu$ and that the resonant\ndips are slightly shifted due to a different freeze--out temperature (see e.g.~\\cite{Griest:1990kh}). The DM density today is dominated by the lighter component. Since it couples to nucleons weaker than $A^{1,2}_\\mu$ do, the direct detection constraint relaxes. Understanding further features of the model\nwould require precise knowledge of the spectrum and the couplings, which we\nrelegate to future work~\\cite{nextpaper}.\n\n\nFinally, one should keep in mind that \n there exists the coupling $A^{3\\prime}_\\mu \\; \\tilde\\varphi_3 \\;\\tilde\\varphi_4$,\nwhere $A^{3\\prime}$ and $\\tilde\\varphi_4$ have the same parities.\nIf $\\tilde\\varphi_4$\nwere light, that is, $\\lambda_4-\\lambda_5 \\ll 1$ and\/or $v_1 \\sim v_2$ (see Eq.~(\\ref{phi4-mass})), \nthe decay $A^{3\\prime}_\\mu \\rightarrow \\tilde\\varphi_4 \\, +$~SM would occur. In that case, DM would consist of both the vector and scalar components.\nWe defer a detailed study of this scenario to future work~\\cite{nextpaper}.\n\n\n\\section{Summary and conclusions}\nIn this paper, we have considered the possibility that the hidden sector enjoys SU(N) gauge symmetry and couples to the Standard Model through the Higgs portal. We find that when endowed with a ``minimal'' matter content, such hidden sectors lead naturally to stable vector dark matter. The underlying Lie group symmetries which stabilise DM \nare associated with complex conjugation of the group elements and discrete gauge transformations.\n\nWe require complete breaking of hidden SU(N) by scalar multiplets to avoid massless states (barring confinement in some cases). That can be done in a minimal fashion by introducing $N-1$ scalar multiplets in the fundamental representation, which develop generic VEVs.\nIf the scalar sector preserves {\\it CP}, the above--mentioned discrete symmetries of the Lie group \ngeneralise to full--fledged symmetries of the model and lead to stable gauge fields.\nWhen sufficiently light, they constitute all of dark matter. \nIn this case, DM consists of 3 components \nassociated with an SU(2) subgroup which hosts the lightest gauge fields $A_\\mu^{1\\prime}$, $A_\\mu^{2\\prime}$ and $A_\\mu^{3\\prime}$. Two of them ($A_\\mu^{1\\prime}, A_\\mu^{2\\prime}$) are\nalways degenerate in mass, while for $N=2$ all 3 components have the same mass.\n\nWe have performed phenomenological analyses of U(1), SU(2) and SU(3) gauge field dark matter. We find that there are vast regions of parameter space where all of the relevant constraints are satisfied. In many of these regions, DM annihilation is facilitated by \nthe broad resonances associated with the Higgs--like scalars.\nWe also find that the SU(3) case appears very similar to that of SU(2), \nunless the scalar VEVs breaking SU(3) are close in magnitude.\n\n\n\\vspace{10pt}\n{\\bf Acknowledgements.} \n The authors are indebted to Sasha Pukhov and Genevieve Belanger for their\nhelp with the new version of micrOMEGAs 4.1.8. C.G. and O.L. acknowledge support\nfrom the Academy of Finland, project ``The Higgs boson and the Cosmos''.\n\nThis work was also supported by the Spanish MICINN's Consolider-Ingenio 2010\nProgramme under grant Multi-Dark {\\bf CSD2009-00064}, the contract\n{\\bf FPA2010-17747} and the France-US PICS no. 06482. \nY.M. acknowledges\npartial support from the European Union FP7 ITN INVISIBLES (Marie Curie\nActions, PITN- GA-2011- 289442) and the ERC advanced grants\n Higgs@LHC and MassTeV. This\nresearch was also supported in part by the Research Executive Agency\n(REA) of the European Union under the Grant Agreement PITN-GA2012-316704\n(``HiggsTools\"). \n\nThe authors would like to thank the Instituto de Fisica Teorica (IFT\nUAM-CSIC) in Madrid for its support via the Centro de Excelencia Severo\nOchoa Program under Grant SEV-2012-0249, during the Program\n``Identification of Dark Matter with a Cross-Disciplinary Approach'' where\nsome of the ideas presented in this paper were developed.\n\n\n\\begin{appendix}\n\\section*{Appendix}\n\n\n\\section{Hidden SU(3) vector--scalar couplings for $v_3=0$}\nThe tables below provide a list of most important gauge--scalar couplings. These\nare relevant to DM phenomenology as well as to understanding the decay channels of the heavier gauge fields.\n\n %\n \\begin{table}[h]\n \\begin{center}\n \\begin{tabular}{|c|c||c|c||c|}\n\\hline\na&b&i&j&coeff. of $\\tilde A_\\mu^a \\tilde A^{\\mu b} \\tilde \\varphi^i \\tilde \\varphi^j$\n\\\\\n\\hline \\hline\n$4$&$4$&$1$&$1$& \\multirow{4}{*}{$\\tilde g^2\/8$}\n\\\\\n$5$&$5$&$1$&$1$&\n\\\\\n$6$&$6$&$1$&$1$&\n\\\\\n$7$&$7$&$1$&$1$&\n\\\\\n\\hline\n$8$&$8$&$1$&$1$& $\\tilde g^2\/6$\n\\\\\n\\hline \\hline\n$1$&$1$&$2$&$2$&\\multirow{5}{*}{$\\tilde g^2\/8$}\n\\\\\n$2$&$2$&$2$&$2$&\n\\\\\n$3$&$3$&$2$&$2$& \n\\\\\n$6$&$6$&$2$&$2$& \n\\\\\n$7$&$7$&$2$&$2$& \n\\\\\n\\hline\n$8$&$8$&$2$&$2$&$\\tilde g^2\/24$\n\\\\\n\\hline\n$3$&$8$&$2$&$2$&$- \\frac{1}{4 \\sqrt{3}}\\tilde g^2$\n\\\\\n\\hline \n\\end{tabular}\n\\quad\n \\begin{tabular}{|c|c||c|c||c|}\n\\hline\na&b&i&j&coeff. of $\\tilde A_\\mu^a \\tilde A^{\\mu b} \\tilde \\varphi^i \\tilde \\varphi^j$\n\\\\\n\\hline \\hline\n$4$&$4$&$3$&$3$& \\multirow{4}{*}{$\\frac 18 \\tilde g^2 \\frac{v_1^2+v_2^2}{v_1^2}$}\n\\\\\n$5$&$5$&$3$&$3$&\n\\\\\n$6$&$6$&$3$&$3$&\n\\\\\n$7$&$7$&$3$&$3$&\n\\\\\n\\hline\n$8$&$8$&$3$&$3$& $\\frac16 \\tilde g^2 \\frac{v_1^2+v_2^2}{v_1^2}$\n\\\\\n\\hline \\hline\n$4$&$4$&$4$&$4$& \\multirow{4}{*}{$\\frac18 \\tilde g^2 \\frac{v_1^2+v_2^2}{v_1^2}$}\n\\\\\n$5$&$5$&$4$&$4$&\n\\\\\n$6$&$6$&$4$&$4$&\n\\\\\n$7$&$7$&$4$&$4$&\n\\\\\n\\hline\n$8$&$8$&$4$&$4$& $\\frac16 \\tilde g^2 \\frac{v_1^2+v_2^2}{v_1^2}$\n\\\\\n\\hline \\hline\n$1$&$4$&$2$&$3$& \\multirow{2}{*}{$\\frac14 \\tilde g^2 \\frac{\\sqrt{v_1^2+v_2^2}}{v_1}$}\n\\\\\n$2$&$5$&$2$&$3$&\n\\\\\n\\hline\n$3$&$6$&$2$&$3$&$-\\frac14 \\tilde g^2 \\frac{\\sqrt{v_1^2+v_2^2}}{v_1}$\n\\\\\n\\hline\n$6$&$8$&$2$&$3$& $- \\frac{1}{4 \\sqrt{3}}\\tilde g^2 \\frac{\\sqrt{v_1^2+v_2^2}}{v_1}$\n\\\\\n\\hline \\hline\n$1$&$5$&$2$&$4$& $\\frac14 \\tilde g^2 \\frac{\\sqrt{v_1^2+v_2^2}}{v_1}$\n\\\\\n\\hline\n$2$&$4$&$2$&$4$&\\multirow{2}{*}{$-\\frac14 \\tilde g^2 \\frac{\\sqrt{v_1^2+v_2^2}}{v_1}$}\n\\\\\n$3$&$7$&$2$&$4$&\n\\\\\n\\hline\n$7$&$8$&$2$&$4$& $- \\frac{1}{4 \\sqrt{3}}\\tilde g^2 \\frac{\\sqrt{v_1^2+v_2^2}}{v_1}$\n\\\\\n\\hline \n\\end{tabular}\n\\end{center}\n\\caption{\\label{AAscalarscalar} Non--derivative couplings $\\tilde A_\\mu^a \\tilde A^{\\mu b} \\tilde \\varphi^i \\tilde \\varphi^j$.\n}\n \\end{table}\n\n \\begin{table}[h]\n \\begin{center}\n \\begin{tabular}{|c|c||c||c|}\n\\hline\na&b&i&coeff. of $\\tilde A_\\mu^a \\tilde A^{\\mu b} \\tilde \\varphi^i$\n\\\\\n\\hline \\hline\n$4$&$4$&$1$& \\multirow{4}{*}{$\\frac{1}{4}\\tilde g^2 v_1$}\n\\\\\n$5$&$5$&$1$& \n\\\\\n$6$&$6$&$1$& \n\\\\\n$7$&$7$&$1$& \n\\\\\n\\hline\n$8$&$8$&$1$& $\\frac{1}{3}\\tilde g^2 v_1$\n\\\\\n\\hline \\hline\n$1$&$1$&$2$& \\multirow{5}{*}{$\\frac{1}{4}\\tilde g^2 v_2 $}\n\\\\\n$2$&$2$&$2$& \n\\\\\n$3$&$3$&$2$& \n\\\\\n$6$&$6$&$2$& \n\\\\\n$7$&$7$&$2$& \n\\\\\n\\hline\n$8$&$8$&$2$& $\\frac{1}{12}\\tilde g^2 v_2 $\n\\\\\n\\hline\n$3$&$8$&$2$& $- \\frac{1}{2 \\sqrt{3}}\\tilde g^2 v_2 $\n\\\\\n\\hline \n\\end{tabular}\n\\quad\n \\begin{tabular}{|c|c||c||c|}\n\\hline\na&b&i&coeff. of $\\tilde A_\\mu^a \\tilde A^{\\mu b} \\tilde \\varphi^i$\n\\\\\n\\hline \\hline\n$1$&$4$&$3$& \\multirow{2}{*}{$\\frac{1}{4}\\tilde g^2 \\frac{v_2 \\sqrt{v_1^2+v_2^2} }{v_1} $}\n\\\\\n$2$&$5$&$3$& \n\\\\\n\\hline\n$3$&$6$&$3$& $-\\frac{1}{4}\\tilde g^2 \\frac{v_2 \\sqrt{v_1^2+v_2^2} }{v_1} $\n\\\\\n\\hline\n$6$&$8$&$3$& $- \\frac{1}{4 \\sqrt{3}}\\tilde g^2 \\frac{v_2 \\sqrt{v_1^2+v_2^2} }{v_1} $\n\\\\\n\\hline \n\\hline\n$1$&$5$&$4$& $\\frac{1}{4}\\tilde g^2 \\frac{v_2 \\sqrt{v_1^2+v_2^2} }{v_1} $\n\\\\\n\\hline\n$2$&$4$&$4$& \\multirow{2}{*}{$-\\frac{1}{4}\\tilde g^2 \\frac{v_2 \\sqrt{v_1^2+v_2^2} }{v_1} $}\n\\\\\n$3$&$7$&$4$& \n\\\\\n\\hline\n$7$&$8$&$4$& $- \\frac{1}{4 \\sqrt{3}}\\tilde g^2 \\frac{v_2 \\sqrt{v_1^2+v_2^2} }{v_1} $\n\\\\\n\\hline \n\\end{tabular}\n\\end{center}\n\\caption{\\label{AAscalar} Non--derivative couplings $\\tilde A_\\mu^a \\tilde A^{\\mu b} \\tilde \\varphi^i$. \n}\n \\end{table}\n %\n\n\n\n\\end{appendix} \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $\\mathbb{R}$  be the real field. A tensor can be regarded as a high-order generalization of a matrix, which takes a form\n\\begin{equation*}\n\\mathcal{A}=(a_{i_1i_2\\dots i_m}),\\quad a_{i_1i_2\\dots i_m}\\in\\mathbb{R},\\quad 1\\leq i_1,i_2,\\ldots,i_m\\leq n.\n\\end{equation*}\nSuch a multi-array $\\mathcal{A}$ is said to be an $m$-order $n$-dimensional real tensor. We denote the set of all $m$-order $n$-dimensional real tensor\nby $\\mathbb{R}^{[m,n]}$.\nLet $x\\in\\mathbb{R}^{n}$, the $n$ dimensional vector $\\mathcal{A}x^{m-1}$ is defined as \\cite{Qi1}:\n\\begin{equation}\\label{eq2}\n (\\mathcal{A}x^{m-1})_{i}=\\sum\\limits_{i_2,\\ldots,i_m}^{n}a_{ii_2,\\ldots,i_m}x_{i_2}\\ldots x_{i_m},\\; i=1,2,\\ldots,n,\n\\end{equation}\nwhere $x_i$ denotes the $i$th component of $x$.\n\nFor any $q\\in\\mathbb{R}^{n} $, we consider the tensor complementarity problem, a special class of nonlinear complementarity problems, denoted by TCP$(\\mathcal{A},q)$: finding $x\\in\\mathbb{R}^{n}$ such that\n \\[x\\geq 0,\\quad\\mathcal{A}x^{m-1}+q\\geq0,\\quad x^{\\top}(\\mathcal{A}x^{m-1}+q)=0. \\]\nThis is a generalization of the linear complementarity problem. So far many\nresearchers have paid attention to this topic \\cite{Bai2016Global,Che2016Positive,Ding2015P,Du2018,Gowda2016Z,Huang2017Formulating,Luo2017The,Song2015Properties,Song2016Tensor,Xie2017An} because of its applications such\nas DNA micro-arrays, communication and $n$-person non-cooperative game \\cite{Huang2017Formulating, Luo2017The}. In \\cite{Song2015Properties}, Song and Qi showed that TCP$(\\mathcal{A}, q)$  has a solution if and only if $\\mathcal{A}$ is nonnegative with its diagonal entries being positive. Song and Qi \\cite{Song2016Tensor} discussed the solution of TCP$(\\mathcal{A}, q)$, when $\\mathcal{A}$ is strictly semi-positive.\n Che, Qi and Wei \\cite{Che2016Positive} discussed the existence and uniqueness of solution\nof TCP$(\\mathcal{A}, q)$ with some special tensors. Luo, Qi and Xiu \\cite{Luo2017The} obtained the sparsest solutions to TCP$(\\mathcal{A}, q)$ with a $\\mathcal{Z}$-tensor. Song and Yu \\cite{Song2016Properties} obtained global\nupper bounds of the solution of the TCP$(\\mathcal{A}, q)$ with a strictly semi-positive\ntensor. Gowda, Luo, Qi and Xiu \\cite{Gowda2016Z} studied the various equivalent conditions for the existence of solution to TCP$(\\mathcal{A}, q)$ with a $\\mathcal{Z}$-tensor.\nDing, Luo and Qi \\cite{Ding2015P} showed the properties of TCP$(\\mathcal{A}, q)$ with a $P$-tensor.\nBai, Huang and Wang \\cite{Bai2016Global} considered the global uniqueness and solvability\nfor TCP$(\\mathcal{A}, q)$ with a strong $P$-tensor. Wang, Huang and Bai \\cite{Wang2016Exceptionally} gave the solvability of TCP$(\\mathcal{A}, q)$ with exceptionally regular tensors.\n\nNumerical algorithms for solving tensor complementarity problems have been\nproposed recently. Xie, Li and Xu \\cite{Xie2017An} presented numerical methods for finding the least solution\nto the TCP$(\\mathcal{A}, q)$ with a $\\mathcal{Z}$-tensor. Liu, Li and Vong \\cite{Liu2017Tensor} proposed the modulus equation for TCP$(\\mathcal{A}, q)$ and based on this equation, they developed the corresponding nonsmooth Newton's method for solving TCP$(\\mathcal{A}, q)$.  Huang and Qi \\cite{Huang2017Formulating} proposed a smoothing type algorithm. Han \\cite{Han2018} introduced a Kojima-Megiddo-Mizuno type continuation method for solving TCP$(\\mathcal{A}, q)$.\nDu et al. \\cite{Du2018} showed that the tensor absolute equation is equivalent\nto a generalized tensor complementarity problem and proposed an inexact Levenberg-Marquardt method for solving the tensor absolute equation. Also, Du and Zhang \\cite{Du2018a} gave a mixed integer programming model to solve the TCP$(\\mathcal{A}, q)$.\n\n\nThe introduction of dynamic models in optimization started in 1980s\n\\cite{chua1984nonlinear,hopfield1985neural}. Since then, significant research results have been achieved for various optimization problems,\nsuch as linear programming \\cite{zak1995solving}, quadratic programming \\cite{bouzerdoum1993neural}, linear complementarity problems \\cite{LIZHI19999}, and nonlinear programming \\cite{rodriguezvazquez1990nonlinear}. The essence of dynamic approach for optimization is\nto establish an energy function (nonnegative).\n The dynamic system is normally in the form of first-order ordinary differential\nequations. It is expected that for an initial state, the dynamic system will approach its static\nstate (or equilibrium point) which corresponds the solution of the underlying optimization problem.\nAn important requirement is that the energy function decreases monotonically as the dynamic system\napproaches an equilibrium point.\n\n\nThe gradient dynamical system (GDS) has now been regarded as a powerful alternative for online computation \\cite{Zhang2006A},\nlinear complementarity problems \\cite{LIZHI19999} and nonlinear complementarity problems \\cite{LIAO2001},  in view of its high speed processing nature and its convenience of hardware implementation in practical applications \\cite{Feng2006Gradient,Ramezani2013Nonlinear}. To effectively solve the linear complementarity problem and nonlinear complementarity problems, linear gradient dynamical system (LGDS) is thus obtained \\cite{LIZHI19999,LIAO2001}. Note\nthat the LGDS with application to online linear (nonlinear) complementarity problems solving have been investigated by the previous work \\cite{LIZHI19999,LIAO2001}.\nThe existence and the convergence of the trajectory of the dynamical system are addressed in detail in \\cite{LIZHI19999,LIAO2001}. In addition, Liao, Qi and Qi \\cite{LIAO2001} also explore the stability properties, such as the stability in the sense of Lyapunov, the asymptotic stability and the exponential stability, for the dynamical system model.\nHowever, to the best of our knowledge, there exists few research results on solving the TCP$(\\mathcal{A}, q)$ via the nonlinear gradient dynamical system (NGDS). Motivated by this\nreason, we thus design, propose and investigate different NGDS for solving TCP$(\\mathcal{A}, q)$ by defining error-monitoring functions. It is theoretically proved that defined  NGDS converge to the theoretical solution.\nThrough illustrative computer-simulation examples, the efficacy and the superiority of the proposed dynamical system model for online computation of the TCP$(\\mathcal{A}, q)$ with $\\mathcal{A}x^{m-1}+q$ is a $P$-function is well-verified.\n\nThe main contributions of the paper are listed as follows.\n\n(1) One type of NGDS for solving the TCP$(\\mathcal{A}, q)$ with $\\mathcal{A}x^{m-1}+q$ for any $q\\in \\mathbb{R}^{n}$ is a $P$-function are presented;\n\n(2) Theoretical analysis shows the convergence of the presented gradient neural networks to the theoretical solution of the TCP$(\\mathcal{A}, q)$ with $\\mathcal{A}x^{m-1}+q$ for any $q\\in \\mathbb{R}^{n}$ is a $P$-function;\n\n(3) Computer simulation results via illustrative examples are presented, compared and\ndiscussed. Generated numerical results comparatively substantiate that the NGDS with nonlinear activation function are much more efficient in solving the TCP$(\\mathcal{A}, q)$ with $\\mathcal{A}x^{m-1}+q$ for any $q\\in \\mathbb{R}^{n}$ is a $P$-function, as compared to the NGDS with linear activation function proposed in this paper.\n\nThis paper is organized as follows. In Section \\ref{sec2}, we recall some preliminary definitions and\nresults. Dynamical system models with different nonlinear activation functions for online solution of the TCP$(\\mathcal{A}, q)$ with $\\mathcal{A}x^{m-1}+q$  for any $q\\in \\mathbb{R}^{n}$ is a $P$-function are presented in Section \\ref{sec3}. Convergence properties of the presented dynamical system models will be discussed in Section \\ref{sec4}.\nIllustrative numerical examples are presented in Section \\ref{Examples}.\n\n\n\\section{Preliminaries}\\label{sec2}\nHan \\cite{HAN201749} gave a method to partially symmetrize tensor $\\mathcal{A}=(a_{i_1i_2\\ldots i_m})\\in \\mathbb{R}^{[m,n]}$ with respect to the indices $i_2\\ldots i_m$, which will be used in sequel. In detail, the partially symmetrized tensor $\\mathcal{\\widehat{A}}=(\\widehat{a}_{i_1i_2\\ldots i_m})$ as follows\n\\[\\widehat{a}_{i_1i_2\\ldots i_m}=\\frac{1}{(m-1)!}\\sum\\limits_{\\pi}a_{i_1\\pi(i_2\\ldots i_m)},\\]\nwhere the sum is over all the permutations $\\pi(i_2\\ldots i_m)$. For any $\\mathcal{A}\\in \\mathbb{R}^{[m,n]}$, we can get a partially symmetrized tensor $\\mathcal{\\widehat{A}}\\in \\mathbb{R}^{[m,n]}$ such that $\\mathcal{A}x^{m-1}=\\mathcal{\\widehat{A}}x^{m-1}$, by an averaging procedure.\n\n\n\\subsection{Function tensors and matrices}\nWe first recall the definitions of $P$-matrix, $P$-tensor and $P$-function as follows.\n\\begin{lemma}{\\bf (\\cite{Berman1994Plemmons})}\\label{pmat}\nLet $A\\in \\mathbb{R}^{n\\times n}$, then\n $A$ is called a $P$-matrix if all the principal minors of $A$ are positive.\n\n\\end{lemma}\n\n\n\nThroughout the paper, we assume that $F:\\mathbb{R}^n\\rightarrow \\mathbb{R}^n$ is a continuously differentiable function.\n\\begin{definition}{\\bf (\\cite{Facchinei2003Finite})}\\label{pfunction}\nA function $F:K\\subseteq \\mathbb{R}^n\\rightarrow \\mathbb{R}^n$ is called a\n $P$-function on $K$ if for all $x,y\\in \\mathbb{R}^n$ with $x\\neq y$, it holds that\n\\begin{equation}\\label{peq1}\n \\max_{i}(x_i-y_i)[F_{i}(x)-F_{i}(y)]>0.\n\\end{equation}\n\\end{definition}\n\\begin{lemma}{\\bf (\\cite{Facchinei2003Finite})}\\label{plema}\n$F$:\\;$\\Omega\\supset K\\subseteq \\mathbb{R}^n\\rightarrow \\mathbb{R}^n$ be continuously differentiable on the\nopen set $\\Omega$ containing the set $K$. $F$ is a $P$-function on $K$ if and only if Jacobian matrix $F'(x)$ is a $P$-matrix for all $x\\in K$.\n\n\n\\end{lemma}\n\\begin{definition}{\\bf (\\cite{Bai2016Global,Liu2017Tensor})}\\label{ptensor}\nLet $\\mathcal{A}\\in\\mathbb{R}^{[m,n]}$, then $\\mathcal{A}$ is called\\\\\n{\\rm (i)} a $P$-tensor, if for each nonzero $x\\in \\mathbb{R}^n$ there exists index $i$ such that\n\\begin{equation}\\label{peq2}\n x_i(\\mathcal{A}x^{m-1})_i> 0;\n\\end{equation}\n{\\rm (ii)} a strong $P$-tensor, if and only if $\\mathcal{A}x^{m-1}$ is a $P$-function;\\\\\n{\\rm (iii)} a strong strictly semi-positive tensor if $\\mathcal{A}x^{m-1}+q$ is a $P$-function in $\\mathbb{R}_{+}^{n}$ for any $q\\in \\mathbb{R}^n$, where $\\mathbb{R}_{+}^{n}$ denotes the set of all the nonnegative vectors.\n\\end{definition}\nFrom the definitions of the $P$-tensor, the strong $P$-tensor and the strong strictly semi-positive tensor, it is easy to see that every strong $P$-tensor must be a $P$-tensor, a strong strictly semi-positive tensor is a $P$-tensor. In additional, we can obtain if for any $q\\in \\mathbb{R}^n$, $\\mathcal{A}x^{m-1}+q$ is a $P$-function, then tensor $\\mathcal{A}$ is a $P$-tensor, furthermore, we have Jacobian matrix $(\\mathcal{A}x^{m-1})'$ is a $P$-matrix.\n\nBai, Huang, and Wang \\cite{Bai2016Global} proved that a TCP possesses the\nglobal uniqueness and solvability property if the tensor is a strong $P$-tensor.  Liu {\\it et al.} \\cite{Liu2017Tensor} showed that a TCP possesses the\nglobal uniqueness and solvability property if the tensor is strong strictly semi-positive tensor.  We summarize their results in the following theorem.\n\\begin{lemma}{\\bf (\\cite{Bai2016Global,Liu2017Tensor})}\\label{pexist} Let $\\mathcal{A}\\in \\mathbb{R}^{[m,n]}$,\\\\\n{\\rm (i)} if $\\mathcal{A}$ is a $P$-tensor, then, for any $q\\in \\mathbb{R}^{n}$, the solution set of $TCP(\\mathcal{A},q)$ is nonempty and compact;\\\\\n{\\rm (ii)} if $\\mathcal{A}$ is a strong $P$-tensor, then $TCP(\\mathcal{A},q)$ has the global uniqueness and solvability property;\\\\\n{\\rm (iii)} if $\\mathcal{A}$ is a strong strictly semi-positive tensor, then $TCP(\\mathcal{A},q)$ has the global uniqueness and solvability property.\n\\end{lemma}\n\\subsection{The Fischer-Burmeister NCP function}\nBelow, we introduce the classical nonlinear complementarity problem (NCP). It will be\nshown that the tensor complementarity problem TCP$(\\mathcal{A},q)$ is a\nspecial kind of nonlinear complementarity problem.\n\\begin{definition}{\\bf (\\cite{Facchinei2003Finite})}\\label{ncp}\nGiven a mapping $F$ : $\\mathbb{R}^n\\rightarrow \\mathbb{R}^n$, the nonlinear complementarity problem,\ndenoted by NCP(F), is to find a vector $x\\in \\mathbb{R}^n $ satisfying\n\\begin{equation}\\label{eqncp}\n  x\\geq 0,\\quad F(x)\\geq 0,\\quad x^{\\top}F(x)=0.\n\\end{equation}\n\\end{definition}\nNote that if $F(x)=\\mathcal{A}x^{m-1}+q$, then NCP reduces to TCP$(\\mathcal{A},q)$.\n\nMany solution methods developed for NCP or related problems are based on\nreformulating them as a system of equations using so-called NCP-functions. Here,\na function $\\phi:\\;\\mathbb{R}^2\\rightarrow \\mathbb{R}$ is called an NCP-function if\n\\begin{equation}\\label{ncpfun}\n \\phi(a,b)=0\\Leftrightarrow ab=0,\\quad a\\geq 0, \\quad b\\geq 0.\n\\end{equation}\nHere, we use the following Fischer-Burmeister NCP-function \\cite{ Fischer1992A},\n\\[\\phi(a,b)=\\sqrt{a^2+b^2}-a-b,\\]\nwhich are widely used in nonlinear complementarity problems.\n\n\\begin{lemma}\\label{lefb}\nThe square of $\\phi(a,b)$ is continuously differentiable; $\\phi(a,b)$ is twice continuously differentiable everywhere except at the origin; but it is strongly semismooth at the origin.\n\\end{lemma}\nBy a simple calculation procedure, we have\n\\[\\partial \\phi(a,b)=\\left\\{\\begin{array}{lc}\n                             \\left(\\frac{a}{\\sqrt{a^2+b^2}}-1,\\frac{b}{\\sqrt{a^2+b^2}}-1\\right), & a^2+b^2\\neq 0, \\\\\n                             (\\alpha-1,\\beta-1)\\;\\text{with}\\; \\alpha^2+\\beta^2\\leq 1, & a^2+b^2=0.\n                           \\end{array}\\right.\n\\]\nFrom Lemma \\ref{lefb}, we have $\\partial \\phi(a,b)$ is a singleton except at the region.\n\\subsection{Activation function}\\label{subsec2.3}\n\nThe matrix-valued activation function $\\mathcal{F}(E)$, $E=(e_{ij})$, is defined as $(f(e_{ij}))$, $i,j=1,2,\\ldots,n$, where $f(\\cdot)$ is a scalar-valued monotonically-increasing odd function. The following real-valued linear and nonlinear odd and monotonically increasing functions $f(\\cdot)$ are widely used.\\\\\n\nLinear function\n  $$f_{\\rm{lin}}(x)=x;$$\n\nBipolar-sigmoid function\n$$\n f_{\\rm{bs}} (x,q)=\\frac{1+\\exp(-q)}{1-\\exp(-q)}\\frac{1-\\exp(-qx)}{1+\\exp(-qx)},\\;q>2;\n$$\n\nPower-sigmoid function\n$$\n f_{\\rm{ps}}(x,p,q)=\\left\\{\\begin{array}{cl}\n                         x^p, & \\mathrm{if }\\ |x|\\geq1 \\\\\n                         \\frac{1+\\exp(-q)}{1-\\exp(-q)}\\frac{1-\\exp(-qx)}{1+\\exp(-qx)},    &  \\mathrm{otherwise}\n                       \\end{array},\\right.\\;q\\geq2,\\;p\\geq3;\n$$\n\nSmooth power-sigmoid function\n$$\nf_{\\rm{sps}}(x,p,q)=\\frac{1}{2}x^{p}+\\frac{1+\\exp(-q)}{1-\\exp(-q)}\\frac{1-\\exp(-qx)}{1+\\exp(-qx)},\\;p\\geq3,\\;q>2.\n$$\n\nIn general, any monotonically increasing odd activation function $f(\\cdot)$  can be used for\nthe construction of the dynamical system.\nAs it was shown in \\cite{Li2013Accelerating,Zhang2009Global}, the\nconvergence rate can be thoroughly improved by an appropriate activation function. So far,\nthe influence of various nonlinear activation functions was investigated for different dynamical system models. We\ninvestigate this scenario on several dynamical system models which are introduced in this paper.\n\n\n\\section{Nonlinear dynamical system methods}\\label{sec3}\nIn this section, the error monitoring function is designed for deriving a gradient dynamical system (GDS). Specifically, by defining different EFs, different GDSs can be obtained for online solution of the  TCP$(\\mathcal{A},q)$. We construct nonlinear gradient dynamical system models, called NGDS, and consider their convergence.\n\nSince  TCP$(\\mathcal{A},q)$ can be equivalently reformulated as finding a solution of the following equation:\n\\begin{equation}\\label{ERRf}\n \\Phi(x)=\\left(\n           \\begin{array}{c}\n             \\phi(x_1,(\\mathcal{A}x^{m-1}+q)_1) \\\\\n                 \\phi(x_2,(\\mathcal{A}x^{m-1}+q)_2) \\\\\n             \\vdots \\\\\n            \\phi(x_n,(\\mathcal{A}x^{m-1}+q)_n) \\\\\n           \\end{array}\n         \\right)=0.\n\\end{equation}\nWe note that $\\Phi(x)$ is locally Lipschitz continuous everywhere, so that Clarke's \\cite{Clarke1983Optimization} generalized Jacobian\n$\\partial\\Phi(x)$ is well defined at any point.\n\nThus, we can define the error monitoring function as\n\\[\\varepsilon(t)=\\varepsilon(x(t))=\\frac{1}{2}\\|\\Phi(x(t))\\|_{2}^2.\\]\nThe function $\\varepsilon(x(t))$ is continuously differentiable \\cite{facchinei1997a}, which follows from the semi-smoothness of $\\Phi(x)$.\n\n\nIn order to force $\\varepsilon(t)$ to converge to zero, the\nnegative of the gradient (i.e., -$\\partial(\\varepsilon(t))\/\\partial x$) is used as the\ndescent direction, which leads to the so-called GDS design formula in the form of a first-order differential\nequation:\n\\begin{equation}\\label{GNNF}\n\\frac{\\mathrm{d}x}{\\mathrm{d}t}=-\\gamma\\frac{\\partial(\\varepsilon(t))}{\\partial x}=-\\gamma V^{\\top}\\Phi(x),\n\\end{equation}\nwhere  $V\\in\\partial\\Phi(x)$ and $\\gamma$ is a positive scaling constant. Note that $\\gamma$ corresponds to the reciprocal of a capacitance parameter, of\nwhich the value should be set as large as the hardware\nwould permit, or appropriately large for modeling and experimental purposes. The dynamic equation (\\ref{GNNF}) will be simply termed the LGDS model.\n\n\n\nFollowing the principle of nonlinear activation in the LGDS model defined in Section \\ref{subsec2.3}, the conventional LGDS model (\\ref{GNNF}) can be improved\ninto the following nonlinear GDS model by exploiting a nonlinear activation function array\n$\\mathcal{F}(\\cdot)$:\n\\begin{equation}\\label{GNN-I2}\n\\frac{\\mathrm{d}x}{\\mathrm{d}t}=-\\gamma V^{\\top}\\mathcal{F}(\\Phi(x)),\n\\end{equation}\nwhere $\\mathcal{F}(\\cdot)$ denotes a matrix-valued activation function. The dynamic equation (\\ref{GNN-I2}) will be simply termed the NGDS model.\n\nWe give the precise definition of $V$ which is necessary for the implementation of our model.\n\\begin{lemma}{\\bf(\\cite{deluca1996a})}\nAny $V$ with the following structure is an element of $\\partial\\Phi(x)$\n\\[V=D_{a}(x)+(m-1)D_{b}(x)\\mathcal{\\widehat{A}}x^{m-2},\\]\nwhere $D_{a}(x)=$diag$(a_1(x),\\ldots,a_n(x))$, $D_{b}(x)=$diag$(b_1(x),\\ldots,b_n(x))$ are diagonal matrices whose $i$th diagonal elements are given by\n\\[a_i(x)=\\frac{x_i}{\\sqrt{x_i^2+(\\mathcal{A}x^{m-1}+q)_i^2}}-1,\\quad b_i(x)=\\frac{(\\mathcal{A}x^{m-1}+q)_i}{\\sqrt{x_i^2+(\\mathcal{A}x^{m-1}+q)_i^2}}-1\\]\nif $(x_i,(\\mathcal{A}x^{m-1}+q)_i)\\neq (0,0)$, and by\n\\[a_i(x)=\\alpha_i-1,\\quad b_i(x)=\\beta_i-1\\]\nfor every $(\\alpha_i,\\beta_i)\\in \\mathbb{R}^2$ such that $\\alpha_i^2+\\beta_i^2\\leq 1$ if $(x_i,(\\mathcal{A}x^{m-1}+q)_i)= (0,0)$.\n\\end{lemma}\n\n\n\\section{Stability analysis}\\label{sec4}\nIn this section, we address the stability issues on the dynamical system (\\ref{GNN-I2}) to the solution of the  TCP$(\\mathcal{A},q)$.\n\nFor given $x_{*}\\in \\mathbb{R}^{n}$ with $x_{*}$ is a solution of the  TCP$(\\mathcal{A},q)$, define $\\delta=\\min\\|x_{*}-u\\|_2$ for any $u\\in \\mathbb{R}^{n}$. Thus, for any $0<\\widehat{\\delta}\\leq\\delta$, we define a neighbourhood of $x_{*}$ as\n\\begin{equation}\\label{BQOP:equation16}\n\\mathbb{B}(x_{*};\\widehat{\\delta}):=\\{x\\mid\\|x-x_{*}\\|_2\\leq\\widehat{\\delta}\\},\n\\end{equation}\nwhere $x\\in\\mathbb{R}^n$.\n\nNow we recall some stability results from \\cite{Zabczyk2015Mathematical} on the following differential equation:\n\\begin{equation}\\label{eqnnew}\n \\frac{\\mathrm{d}x(t)}{\\mathrm{d}t}=f(x(t)), \\quad x(t_0)\\in\\mathbb{R}^n.\n\\end{equation}\n\nThe following classical results on the existence and uniqueness of the solution to (\\ref{eqnnew}) hold.\n\\begin{definition}{\\bf(\\cite{Zabczyk2015Mathematical})} \\label{def4.1}\nLet $x(t)$ be a solution of (\\ref{eqnnew}). An isolated equilibrium point $x^*$ is Lyapunov stable if for any $x(t_0)$ and any scalar $\\epsilon>0$ there exists a $\\hat{\\delta}>0$\nso that if $x(t_0)\\in \\mathbb{B}(x_{*},\\widehat{\\delta})$ then $\\|x(t)-x_*\\|_2<\\epsilon$ for $t\\geq t_0$.\n\\end{definition}\n\\begin{definition}{\\bf(\\cite{Zabczyk2015Mathematical})} \\label{def4.2}\n An isolated equilibrium point $x_*$ is said to be asymptotic stable\nif in addition to being a Lyapunov stable it has the property that\n$x(t)\\rightarrow x_*$ as $t\\rightarrow +\\infty $, if $x(t_0)\\in \\mathbb{B}(x_{*},\\hat{\\delta})$.\n\\end{definition}\n\nThen we focus on a particular case where the equilibrium point is isolated. Let $S$ denote the solution set of the  TCP$(\\mathcal{A},q)$ and $x\\in S$  implies $\\Phi(x)=0$. Togather with   $\\mathcal{F}(\\cdot)$ is odd and monotonically increasing function, we have $\\mathcal{F}(\\Phi(x))=0$, consequently, $\\frac{\\mathrm{d}x}{\\mathrm{d}t}=0$. Hence, we have the following result.\n\\begin{theorem}\\label{theq}\nEvery solution to the TCP$(\\mathcal{A},q)$ is an equilibrium point of the dynamical system (\\ref{GNN-I2}).\nConversely; if $x\\in \\mathbb{R}^n$ is an equilibrium of (\\ref{GNN-I2}) and for any $q\\in \\mathbb{R}^n$, $\\mathcal{A}x^{m-1}+q$ is a $P$-function, then $x\\in S$.\n\\end{theorem}\n\\begin{proof}\nWe only need to address the second part of the Theorem. Since  $\\mathcal{A}x^{m-1}+q$ is a continuously differentiable $P $-function, it's Jacobian matrix is a $P $-matrix for all $x\\in \\mathbb{R}^{n}$. Analogy to the proof of  \\cite[Corollary  4.4]{deluca1996a}, we have  a stationary point of (\\ref{GNN-I2}) is a solution to the TCP$(\\mathcal{A},q)$.\nWe complete our proof.\n\\end{proof}\n\nNote that if $\\mathcal{A}$ is a strong $P$-tensor (strong strictly semi-positive tensor), then $TCP(\\mathcal{A},q)$ has the global uniqueness and solvability property. Together with a strong $P$-tensor (strong strictly semi-positive tensor) is a $P$-tensor, we obtain the following results.\n\\begin{theorem}\\label{strongp}\nLet $\\mathcal{A}$ be a strong $P$-tensor (strong strictly semi-positive tensor) and $x$ be a solution of the TCP$(\\mathcal{A},q)$, then $x$ is an unique equilibrium point of the dynamical system (\\ref{GNN-I2}).\n\\end{theorem}\n\\begin{proof}\nSince $\\mathcal{A}$ is a strong $P$-tensor (strong strictly semi-positive tensor), then $TCP(\\mathcal{A},q)$ has the global uniqueness and solvability property. That is if $x$ is a solution of the TCP$(\\mathcal{A},q)$, then $x$ is a unique solution of the TCP$(\\mathcal{A},q)$. From the Theorem \\ref{theq}, we get $x$ is an unique equilibrium point of the neural network (\\ref{GNN-I2}). The proof is thus completed.\n\\end{proof}\nWe have the following result of the convergence of the NGDS model.\n\\begin{theorem} \\label{theorem4.1}\nGiven $\\widehat{\\delta}>0$ and $x_{*}$ be a solution of the TCP$(\\mathcal{A},q)$, if nonzero vector $x(0)\\in \\mathbb{B}(x_{*};\\widehat{\\delta})$ and  $\\mathcal{A}x(t)^{m-1}+q$ is a $P$-function for any $q\\in \\mathbb{R}^n$, then the state $x(t)$ of the {\\em NGDS} model (\\ref{GNN-I2}), starting from the initial state $x(0)\\in \\mathbb{B}(x_{*};\\widehat{\\delta})$, converges to the solution $x_*\\in \\mathbb{R}^{n}$ of the  TCP$(\\mathcal{A},q)$.\n\\end{theorem}\n\\begin{proof} We construct the following Lyapunov function:\n\\begin{equation}\\label{lyap}\n  L(t)=\\varepsilon(x(t))=\\frac{1}{2}\\|\\Phi(x(t))\\|_{2}^2\\geq 0,\n\\end{equation}\nwith its time derivative being\n\\begin{equation}\\label{eq4.2}\n\\begin{array}{ccl}\n \\frac{\\mathrm{d}L(t)}{\\mathrm{d}t}&= &\\frac{1}{2}\\frac{\\mathrm{d}}{dt} \\mathrm{Tr}\\left(\\Phi(x(t))^{\\top}\\Phi(x(t))\\right)\\\\\n &= &\\left(\\Phi(x(t))^{\\top}\\frac{\\mathrm{d}\\Phi(x(t)))}{\\mathrm{d}t}\\right)\\\\\n  &=&\\left(\\Phi(x(t))^{\\top}V\\frac{\\mathrm{d}x}{\\mathrm{d}t}\\right)\\\\\n    &=&-\\gamma\\left(\\Phi(x(t))^{\\top}VV^{\\top}\\mathcal{F}(\\Phi(x(t)))\\right)\\\\\n &=&-\\gamma\\mathrm{Tr}\\left(VV^{\\top}\\Phi(x(t))^{\\top}\\mathcal{F}(\\Phi(x(t)))\\right).\\\\\n \\end{array}\n \\end{equation}\nSince $V$ is a $P$-matrix, which follows from $\\mathcal{A}x^{m-1}+q$ is a $P$-function for any $q\\in \\mathbb{R}^n$, then, $VV^{\\top}$ is a symmetric positive definite matrix, and then, we have\n\n\\[\\begin{array}{cll}\n     \\lambda_{\\min}\\mathrm{Tr}\\left(\\Phi(x(t))^{\\top}\\mathcal{F}(\\Phi(x(t)))\\right) & \\leq& \\mathrm{Tr}\\left(VV^{\\top}\\Phi(x(t))^{\\top}\\mathcal{F}(\\Phi(x(t)))\\right) \\\\\n     & \\leq & \\lambda_{\\max}\\mathrm{Tr}\\left(\\Phi(x(t))^{\\top}\\mathcal{F}(\\Phi(x(t)))\\right),\n  \\end{array}\n\\]\nwhere $ \\lambda_{\\min}$ and $ \\lambda_{\\max}$ denote the smallest eigenvalue and the largest eigenvalue of matrix $VV^{\\top}$, respectively.  $\\mathrm{Tr}(A)$ is the trace of the matrix $A$.\n\nSince the scalar-valued function $f(\\cdot)$ is an odd and monotonically increasing function, it immediately follows $f(-x)=-f(x)$ and\n\\[f(x)\\left\\{\\begin{array}{cl}\n               >0 ,& \\mathrm{if}\\; x>0, \\\\\n               =0, & \\mathrm{if} \\;x=0, \\\\\n               <0 ,& \\mathrm{if} \\;x<0,\n             \\end{array}\n\\right. \\]\nwhich implies\n\\[xf(x)\\left\\{\\begin{array}{cl}\n               >0, & \\mathrm{if}\\; x\\neq 0, \\\\\n               =0,& \\mathrm{if} \\;x=0.\n             \\end{array}\n\\right. \\]\nIt follows that\n\\[\\mathrm{Tr}\\left(\\Phi(x(t))^{\\top}\\mathcal{F}(\\Phi(x(t)))\\right)\\left\\{\\begin{array}{cl}\n               >0, & \\mathrm{if}\\; \\Phi(x(t))\\neq 0, \\\\\n               =0, & \\mathrm{if} \\;\\Phi(x(t))=0.\n             \\end{array}\n\\right.\\]\nDue to the fact that the design parameter satisfies $\\alpha>0$, in view of (\\ref{eq4.2}), it follows that\n\\[\\frac{\\mathrm{d}L_1(t)}{\\mathrm{d}t}\\left\\{\\begin{array}{cl}\n              <0, & \\mathrm{if}\\; \\Phi(x(t))\\neq 0, \\\\\n               =0, & \\mathrm{if} \\;\\Phi(x(t))=0.\n              \n             \\end{array}\n\\right.\\]\n\nBy the Lyapunov theory,  $\\Phi(x(t))$ can converge to zero; or, equivalently speaking, state $x(t)$ of NGDS model (\\ref{GNN-I2}) is asymptotic stable at one of solutions $x_{*}$ with $\\Phi(x_{*})=0$ starting from an initial state $x(0)\\in \\mathbb{B}(x_{*};\\widehat{\\delta})$. The proof is thus complete.\n\\end{proof}\n\nWe summarize the convergence result of the NGDS  method (\\ref{GNN-I2}) when $\\mathcal{A}$\nis strong $P$-tensor (strong strictly semi-positive tensor) in the following theorem.\n\\begin{theorem} \\label{theorem4.2}\nSuppose that $\\mathcal{A}\\in \\mathbb{R}^{[m,n]}$ is a strong $P$-tensor (strong strictly semi-positive tensor) and $q\\in \\mathbb{R}^n$. The state $x(t)$\nof the {\\em NGDS} model (\\ref{GNN-I2}), starting from an arbitrary initial state $x(0)\\in \\mathbb{R}^{n}$, converges to the unique solution $x_*\\in \\mathbb{R}^{n}$ of the TCP$(\\mathcal{A},q)$.\n\\end{theorem}\n\\begin{proof}\nSince $\\mathcal{A}$ is a strong $P$-tensor (strong strictly semi-positive tensor), from Theorem \\ref{strongp}, we know $TCP(\\mathcal{A},q)$, for given $q\\in \\mathbb{R}^{n}$, has the global uniqueness and solvability property. Together with the proof of Theorem \\ref{theorem4.1}, we can show that the state $x(t)$\nof the { NGDS} model (\\ref{GNN-I2}), starting from an arbitrary initial state $x(0)\\in \\mathbb{R}^{n}$, converges to the unique solution $x_*\\in \\mathbb{R}^{n}$ of the TCP$(\\mathcal{A},q)$. We complete our proof.\n\\end{proof}\n\\section{Numerical examples}\\label{Examples}\nIn this section, some computer-simulation examples are\ndemonstrated to verify the efficacy and the superiority of\nthe proposed neural network models. We apply the NGDS to the TCP$(\\mathcal{A},q)$.\n\n\nAll computations are carried out in Matlab Version 2014a, which has a unit roundoff $ 2^{-53}\\approx 1.1\\times 10^{-16}$, on a laptop with Intel Core(TM) i5-4200M CPU (2.50GHz) and 7.89GB RAM.\n\n\n\\begin{exmple} \\label{eg1}\nConsider the tensor $\\mathcal{A}\\in \\mathbb{R}^{[4,2]}$, from \\cite{Bai2016Global,Liu2017Tensor}, defined by:\n\\[a_{1111}=1,\\;a_{1222}=-1,\\;a_{1122}=1,\\;a_{2222}=1,\\;a_{2111}=-1,\\;a_{2211}=1,\\]\nand $a_{i_1i_2i_3i_4}=0$ otherwise. This tensor is a $P$-tensor, but not a strong $P$-tensor \\cite{Bai2016Global}. However, it is a strong strictly semi-positive tensor \\cite{Liu2017Tensor}.\n\\end{exmple}\nIn our tests, we take different vectors $q\\in \\mathbb{R}^2$ and initial vector as $x_0={\\rm{rand}}(2,1)$. Trajectories of state variables corresponding to the LGDS with $\\gamma=10$ are shown in Figure \\ref{figure 1} (a), (c), (e) and Figure \\ref{figure 2} (a) and (c), respectively.\nResidual errors\n\\begin{equation}\\label{eq6.2}\n \\mathrm{Res}=\\|\\Phi(x)\\|_2,\n\\end{equation}\n\\begin{figure}[H]\n\\centering\n\\subfigure[$\\gamma=10$, $q=(-5,-3)^{\\top}$.]{\\includegraphics[width=3in, height=1.8in]{ex1q1state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(-5,-3)^{\\top}$.]{\\includegraphics[width=3in, height=1.8in]{ex1q1error.jpg}}\\\\\n\\subfigure[$\\gamma=10$, $q=(-5,3)^{\\top}$.]{\\includegraphics[width=3in, height=1.8in]{ex1q2state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(-5,3)^{\\top}$.]{\\includegraphics[width=3in, height=1.8in]{ex1q2error.jpg}}\\\\\n\\subfigure[$\\gamma=10$, $q=(5,3)^{\\top}$.]{\\includegraphics[width=3in, height=1.8in]{ex1q3state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(5,3)^{\\top}$.]{\\includegraphics[width=3in, height=1.8in]{ex1q3error.jpg}}\n\\caption{Trajectories of state variables and residual errors of the NGDSs for Example \\ref{eg1}.}\\label{figure 1}\n\\end{figure}\n\n\nderived by employing the NGDS with $\\gamma=1\\times 10^6$  shown in Figures \\ref{figure 1} (b), (d), (f) and Figure \\ref{figure 2} (b) and (d), respectively, where  blue stars indicate that $f(\\cdot)$ is the linear function, red stars  correspond to the bipolar-sigmoid function $f_{bs}(x,5)$,\npink stars denote that power-sigmoid function $f_{ps}(x,3,5)$  and green stars are generated using $f_{sps}(x,3,7)$ as the smooth power-sigmoid function.\n\\begin{figure}[!h]\n\\centering\n\\subfigure[$\\gamma=10$, $q=(2,-3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex1q4state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(2,-3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex1q4error.jpg}}\\\\\n\\subfigure[$\\gamma=10$, $q=(0,-5)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex1q5state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(0,-5)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex1q5error.jpg}}\\\\\n\\caption{Trajectories of state variables and residual errors of the NGDSs for Example \\ref{eg1}.}\\label{figure 2}\n\\end{figure}\n\\begin{exmple} \\label{eg2}\nLet $\\mathcal{A}\\in \\mathbb{R}^{[5,3]}$ be defined by $a_{kkkkk}=k$ for $k=1,2,3$ and\n $a_{i_1i_2i_3i_4i_5}=0$ otherwise. This tensor is both a strong $P$-tensor \\cite{Bai2016Global} and a strong strictly semi-positive \\cite{Han2018}.\n\\end{exmple}\nFor different vectors $q\\in \\mathbb{R}^3$, we take  initial vector as $x_0={\\rm{rand}}(3,1)$. Trajectories of state variables corresponding to the LGDS with different $\\gamma$ are shown in Figure \\ref{figure 3} (a), (c), (e) and Figure \\ref{figure 4} (a), (c) and (e), respectively.\nResidual errors (\\ref{eq6.2})\nderived by employing the NGDS with $\\gamma=1\\times 10^6$  shown in Figures \\ref{figure 3} (b), (d), (f) and Figure \\ref{figure 4} (b), (d) and (f), respectively.\n\n\\begin{figure}[H]\n\\centering\n\\subfigure[$\\gamma=1000$, $q=(1,2,3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q1state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(1,2,3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q1error.jpg}}\\\\\n\\subfigure[$\\gamma=1000$, $q=(1,-2,3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q2state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(1,-2,3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q2error.jpg}}\\\\\n\\subfigure[$\\gamma=100$, $q=(-3,-2,-3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q3state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(-3,-2,-3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q3error.jpg}}\n\\caption{Trajectories of state variables and residual errors of the NGDSs for Example \\ref{eg2}.}\\label{figure 3}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\subfigure[$\\gamma=100$, $q=(3,3,3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q4state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(3,3,3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q4error.jpg}}\\\\\n\\subfigure[$\\gamma=100$, $q=(-3,-1,-2)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q5state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(-3,-1,-2)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q5error.jpg}}\\\\\n\\subfigure[$\\gamma=100$, $q=(-1,-1,-2)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q6state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(-1,-1,-2)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q6error.jpg}}\n\\caption{Trajectories of state variables and residual errors of the NGDSs for Example \\ref{eg2}.}\\label{figure 4}\n\\end{figure}\n\nWhere  blue stars indicate that $f(\\cdot)$ is the linear function, red stars  correspond to the bipolar-sigmoid function $f_{bs}(x,7)$,\npink stars denote that power-sigmoid function $f_{ps}(x,5,7)$  and green stars are generated using $f_{sps}(x,5,9)$ as the smooth power-sigmoid function, respectively.\n\n\n\n\n\\begin{exmple} \\label{eg3}\nLet $\\mathcal{A}\\in \\mathbb{R}^{[4,2]}$ be defined by\n\\[a_{1111}=1,\\;a_{1112}=-2,\\;a_{1122}=1,\\;a_{2222}=1,\\]\nand all other $a_{i_1i_2i_3i_4}=0$. This tensor is  a $P$-tensor \\cite{Bai2016Global}.\n\\end{exmple}\nTaking $q=(0,-1)^{\\top}$, it is easy to see that $x_{*}=(0,1)^{\\top}$ and $x_{*}=(1,1)^{\\top}$ are the solutions to the TCP$(\\mathcal{A},q)$.\n\nWe choose  initial vector as $x_0=(0.1,0.5)^{\\top}$ for $x_{*}=(0,1)^{\\top}$ and $x_0=(1.5,1.1)^{\\top}$ for $x_{*}=(1,1)^{\\top}$. Trajectories of state variables corresponding to the LGDS with  $\\gamma=100$ and  $\\gamma=10000$ are shown in Figure \\ref{figure 5} (a) and Figure \\ref{figure 6} (a), respectively.\n\n\n\nResidual errors (\\ref{eq6.2}) derived by employing the NGDS with $\\gamma=1\\times 10^6$  shown in Figures \\ref{figure 5} (b) and Figures \\ref{figure 6} (b), where  blue stars indicate that $f(\\cdot)$ is the linear function, red stars  correspond to the bipolar-sigmoid function $f_{bs}(x,7)$,\npink stars denote that power-sigmoid function $f_{ps}(x,5,9)$  and green stars are generated using $f_{sps}(x,7,11)$ as the smooth power-sigmoid function, respectively.\n\n\\begin{figure}[H]\n\\centering\n\\subfigure[$\\gamma=100$, $x_{*}=(0,1)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex3q2state1.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $x_{*}=(0,1)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex3q1error.jpg}}\n\\caption{Trajectories of state variables and residual errors of the NGDSs for Example \\ref{eg3}.}\\label{figure 5}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\subfigure[$\\gamma=10000$, $x_{*}=(1,1)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex3q2state2.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $x_{*}=(1,1)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex3q2error2.jpg}}\n\\caption{Trajectories of state variables and residual errors of the NGDSs for Example \\ref{eg3}.}\\label{figure 6}\n\\end{figure}\n\nFrom the computer simulation results derived in the illustrative examples, the following conclusions can be highlighted:\n\n1. The NGDS models with different types of activation functions  presented in section 3, exactly and efficiently solve the TCP$({\\mathcal A},q)$ with $P$-tensors, strong $P$-tensor and strong strictly semi-positive tensor.\n\n2. NGDS models could achieve different performances if different activation function arrays are used. In general, the convergence\nperformance of nonlinear activation functions is superior to that of the\nlinear activation function.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\\label{intro}\n\t\t\nThe Moran construction is a typical way to generate self-similar fractals, and has been studied extensively in the literature (e.g. \\cite{Moran},  \\cite{Beardon}, \\cite{hutch}, \\cite{Mandelbrot},\\cite{Li}, \\cite{Su}, \\cite{inhom}, and references therein).  In this paper, we extend ideas from iterated function systems (IFS) and Moran constructions by describing a new process that allows for the functions to be updated at every iteration while still maintaining the computational simplicity of an IFS.  This process provides more variance in the limit sets (such as non-self-similarity) using an analogous approach to an IFS procedure.  We also give estimates of the Hausdorff dimension of the limit sets created from such a process, and provide concrete examples.\n\t\t\nThe classic construction of Moran sets was introduced in \\cite{Moran}.  We reproduce the definition here with a more current interpretation to introduce notations.\n\t\n\tLet $\\{n_k\\}_{k\\geq1}$ be a sequence of positive integers for $k \\geq 1$.  For any $k \\in {\\mathbb{N}}$, define \n\t\\begin{equation}\n\t\\label{Dk}\n\tD_{k} = \\{ (i_1, i_2, \\cdots, i_k) : 1 \\leq i_j \\leq n_k, 1 \\leq j \\leq k\\} \\text{ and } D = \\bigcup_{k \\geq 0} D_{k}. \n\t\\end{equation} \n\tWe define $D_0 = \\emptyset$.  Let $\\sigma = (\\sigma_1, \\cdots, \\sigma_k) \\in D_k$ and $\\tau = (\\tau_1, \\cdots, \\tau_m) \\in D_m$, then denote $\\sigma * \\tau = (\\sigma_1, \\cdots, \\sigma_k, \\tau_1, \\cdots, \\tau_m)$.\n\tUsing this notation, we may express\n\t\\begin{equation}\n\t\\label{dksigma}\n\tD_{k} =\\{ \\sigma * j | \\sigma \\in D_{k-1}, 1 \\leq j \\leq n_k\\}\n\t\\end{equation} to emphasize the process of moving between generations.\n\t\n\tSuppose $\\mathcal J := \\{J_{\\sigma} : \\sigma \\in D\\}$ is a collection of subsets of ${\\mathbb{R}}^N$.  \n\tSet \n\t\\begin{equation}\n\t\\label{eqn: Limit_set}\n\tE_{k} = \\bigcup_{\\sigma \\in D_{k}} J_{\\sigma}, \\text{ and } F = \\bigcap_{k \\geq 0} E_{k}.\n\t\\end{equation}\n\tWe call $F$ the limit set associated with the collection $\\mathcal{J}$.\n\t\n\t\n\t\\begin{definition}[\\cite{Su}]\n\t\tSuppose that $J \\subset {\\mathbb{R}}^N$ is a compact set with nonempty interior.  Let $\\{n_k\\}_{k \\geq 1}$ be a sequence of positive integers, and $\\{ \\Phi_k\\}_{k \\geq 1}$ be a sequence of positive real vectors with \n\t\t\\begin{equation}\n\t\t\\Phi_k = (c_{k,1}, c_{k,2}, \\dots, c_{k,n_k}), \\sum_{1 \\leq j \\leq n_k} c_{k,j} \\leq 1, k \\in {\\mathbb{N}}.\n\t\t\\end{equation} \n\t\tSuppose that ${\\mathcal F} := \\{J_\\sigma : \\sigma \\in D\\}$ is a collection of subsets of ${\\mathbb{R}}^N,$ where $D$ is given in (\\ref{Dk}).  We say that the collection ${\\mathcal F}$ fulfills the Moran Structure provided it satisfies the following Moran Structure Conditions (MSC):\n\t\t\\begin{enumerate}\n\t\t\t\\item[MSC(1)] $J_{\\emptyset} = J.$\n\t\t\t\\item[MSC(2)] For any $\\sigma \\in D$, $J_\\sigma$ is geometrically similar to $J$.  That is, there exists a similarity $S_\\sigma: {\\mathbb{R}}^N \\to {\\mathbb{R}}^N$ such that $J_\\sigma = S_\\sigma(J)$. \n\t\t\t\\item[MSC(3)] For any $k \\geq 0$ and $\\sigma \\in D_k$, $J_{\\sigma \\star 1}, \\dots , J_{\\sigma \\star n_k}$ are subsets of $J_\\sigma$, and $int(J_{\\sigma \\star i}) \\cap int(J_{\\sigma \\star j}) = \\emptyset$ for $i \\neq j$.\n\t\t\t\\item[MSC(4)] For any $k \\geq 1$ and $\\sigma \\in D_{k-1}, 1 \\leq j \\leq n_k,$ \n\t\t\t\\begin{equation}\n\t\t\t\\label{ckj}\n\t\t\t\\dfrac{diam(J_{\\sigma \\star j})}{diam(J_{\\sigma})} = c_{k,j}.\t\n\t\t\t\\end{equation}\n\t\t\t\n\t\t\\end{enumerate}\n\t\t\n\t\tFor the collection ${\\mathcal F}$ fulfilling the MSC, the limit set $F$ given in (\\ref{eqn: Limit_set}) is a nonempty compact set. This limit set $F$ is called the Moran set associate with the collection ${\\mathcal F}$.  This Moran set is self-similar, and has been studied extensively by many authors with various approaches (e.g. \\cite{Moran}, \\cite{hutch}, \\cite{Falconer}, \\cite{Beardon}, \\cite{peres}).\n\t\\end{definition}\n\t\n   \tSeveral approaches have been used to relax MSC in order to create more general limit sets.  The dimension (e.g. Hausdorff, Box, Packing, \\dots) of these sets has been a fruitful area of study.  For example, in \\cite{mcmullen}, MSC(2) has been expanded to affine maps.  In this setting, however, calculations of the dimension of some limit sets can become particularly difficult.  One could also study the limit sets generated by infinitely many similarities, as in \\cite{mauldin}. In \\cite{Li}, the authors  removed MSC(2), but required $\\overbar{int(J_\\sigma)} = J_\\sigma$ in their construction, and studied the dimension of the resulting fractals.  In \\cite{inhom}, Holland and Zhang studied a construction that replaced similarity maps in MSC(2) with a more general class of functions that are not necessarily contractions.  In \\cite{PW}, Pesin and Weiss removed the requirement for similarities from MSC(2), but also relaxed MSC(3) from non-intersecting basic sets to non-intersecting balls contained in the basic sets.  In particular they pursued sufficient conditions for which the Box-counting and Hausdorff dimensions coincide.  For more examples of modifications to the Moran set definition, see \\cite{Su} and the references therein.\n\t\n\tA special case of Moran sets can be constructed from an iterated function system (IFS).  An iterated function system $\\{S_1, S_2, \\cdots, S_m\\}$ is a finite family of similarities for a fixed natural number $m\\geq2$ (see \\cite{Falconer} for more details and applications).  In MSC(2), define $n_k=m$ and set $S_{\\sigma} = S_{i_k} \\circ S_{i_{k-1}} \\circ \\cdots \\circ S_{i_1}$ for $\\sigma = (i_1, i_2, \\cdots, i_k) \\in D$.  Then the resulting Moran set is self-similar and agrees with the attractor of the IFS $\\{S_1, S_2, \\cdots, S_m\\}$.  The dimension of the limit set can be quickly calculated from the Moran-Hutchinson formula in \\cite{hutch}.  Using iterated function systems is a popular way to construct fractals, and has been used to great effect (e.g. \\cite{barnsley}, \\cite{Falconer}, \\cite{hutch}, \\cite{BV} ).   \n\t\n\tA natural question arises: Can we construct more general fractals (e.g. non-self-similar Moran type sets) using an analogous approach while preserving the computational simplicity of the IFS?  In this paper, we present a method to do so.\n\t\n\tWe first make the following observations.  Note that in the above construction,\n\t\\begin{equation}\n\t\\label{Jsigmai}\n\tJ_{\\sigma * i} = S_i (J_\\sigma), \\text{ for all } i=1, ... m, \\text{ and } \\sigma \\in D.\n\t\\end{equation}\n\t\n\tSuppose that there is a tuning parameter in the expression of the function $S_i$ (e.g. the coefficients $a_i, b_i$ in a linear function $S_i(x) = a_i x+ b_i$).  One can tune the values of the parameter to get a comparable function.  When $J_{\\sigma}$ is given, applying the comparable function to $J_{\\sigma}$, as in equation (\\ref{Jsigmai}), will not significantly change the computational complexity of constructing $J_{\\sigma * i}$. The advantage of doing this at each iteration is that we introduce some variance into the limit set.  Another observation is about which space the functions are defined. In classical IFS constructions, the functions are usually defined on all of the ambient space ${\\mathbb{R}}^N$ (as in \\cite{inhom}, the functions are $C^{1+\\alpha}$ diffeomorphisms on ${\\mathbb{R}}^N$).  For our construction, we wish to relax the condition MSC(2) as well.  Instead of restricting our attention to functions of higher regularity defined on the whole ambient space ${\\mathbb{R}}^N$, we use maps from a collection of subsets to itself.  \n\n\t  This article is organized as follows.  In section 2 we find bounds for the Hausdorff dimension of the limit sets in a general metric space setting of a collection of bounded sets, not necessarily satisfying the MSC conditions. Then in section 3 we formulate the general setup for the construction of Moran-type limit sets using the ideas from a modified IFS procedure, as discussed in the previous paragraph.  In our construction we relax MSC(2) so that the limit set is not necessarily self-similar.  More importantly, we drop MSC(4) from the construction process so that there are no limitations on the ratios of the diameters of the sets.  Specifically, the ratio $\\displaystyle \\dfrac{diam(J_{\\sigma * j})}{diam(J_\\sigma)}$ in (\\ref{ckj}) is not limited to depend on just $k$ and $j$, but varies with $\\sigma$.  This change allows us to produce a mosaic of possible fractals.  An important observation is that the computational complexity of generating these fractals is the same as using an analogous, standard IFS.  In section 4 we give estimates of the Hausdorff dimension of the limit sets created from the general construction.  Finally, in section 5 we apply the results to specific examples, including modifications of the Cantor set, the Sierpinski triangle, and the Menger sponge.\n\t  \n\t\\section{Hausdorff Dimension of the Limit Sets}\n\tIn this section we investigate the Hausdorff dimension $\\dim_H(F)$ of the fractals $F$ defined in (\\ref{eqn: Limit_set}), which is not necessarily satisfying MSC conditions.  To start, we determine an upper bound for the dimension of the limit set $F$ by considering the step-wise relative ratios between the diameters of sets.\n\t\n\t\\begin{proposition}\\label{thm: upper}\n\t\tSuppose $\\mathcal J := \\{J_{\\sigma} : \\sigma \\in D\\}$ is a collection of bounded subsets of a metric space $(X,d)$, and $s>0$. \n\t\tLet\t$E_{k} = \\bigcup_{\\sigma \\in D_{k}} J_{\\sigma}, \\text{ and } F = \\bigcap_{k \\geq 0} E_{k}$ be defined as in (\\ref{eqn: Limit_set}).  If there exists a sequence of positive numbers $\\{c_k\\}_{k=1}^{\\infty}$ such that\n\t\t\\[\\liminf_{k\\rightarrow \\infty} \\prod_{i=1}^k c_i =0\\]\n\t\tand\n\t\t\\begin{equation}\n\t\t\t\\label{eqn: c}\n\t\t\t\\sum_{j=1}^{n_k} \\left (\\textsl{diam}(J_{\\sigma*j})\\right)^s \\leq c_k \\left( \\textsl{diam}(J_\\sigma)\\right)^s,\n\t\t\\end{equation}\n\t\tfor all $\\sigma\\in D_{k-1}$ and all $k=1,2,\\cdots$, then $dim_H(F) \\le s$.\n\t\\end{proposition}\n\t\n\t\\begin{proof}\n\t\tWe prove by using mathematical induction that for $k=1,2,\\cdots,$\n\t\t\\begin{equation}\n\t\t\t\\label{general_form}\n\t\t\t\\sum_{\\sigma \\in D_k}(\\textsl{diam}(J_{\\sigma}))^s \\le \\left(\\prod_{i=1}^k c_i \\right) (diam(J_\\emptyset))^s.\n\t\t\\end{equation}\n\t\tWhen $k=1$, (\\ref{general_form}) follows from (\\ref{eqn: c}). Now assume (\\ref{general_form}) is true for some $k\\ge 1$.  Then by (\\ref{dksigma}), (\\ref{eqn: c}), and (\\ref{general_form}),\n\t\t\n\t\t\\begin{eqnarray*}\n\t\t\t\\sum_{\\sigma\\in D_{k+1}} (\\textsl{diam}(J_{\\sigma}))^s &=&\\sum_{\\sigma\\in D_k} \\left(\\sum_{j=1}^{n_{k+1}} (\\textsl{diam}(J_{\\sigma*j}))^s\\right)\\\\\n\t\t\t &\\le& c_{k+1} \\sum_{\\sigma\\in D_k}(\\textsl{diam}(J_{\\sigma}))^s \\le \\left(\\prod_{i=1}^{k+1} c_i \\right)(\\textsl{diam}(J_\\emptyset))^s\n\t\t\\end{eqnarray*}\n\t\t\n\t\tas desired. By the induction principle, (\\ref{general_form}) holds for all $k=1,2,\\cdots.$\n\t\t\n\t\tFor each $k$, set\n\t\t\\[\\delta_k=\\max\\{\\textsl{diam}(J_\\sigma): \\sigma\\in D_k\\}>0.\\]\n\t\tThen, by (\\ref{general_form}),  $\\delta_k\\le  \\left(\\prod_{i=1}^k c_i \\right)^{1\/s}\\textsl{diam}(J_\\emptyset)$.  Moreover, by (\\ref{general_form})\n\t\t\\[\\mathcal{H}_{\\delta_k}^{s}(F) \\le \\mathcal{H}_{\\delta_k}^{s}(E_k)\\le \\sum_{\\sigma\\in D_k} \\alpha(s)\\left(\\frac{\\textsl{diam}(J_\\sigma)}{2}\\right)^s \\le \\left(\\prod_{i=1}^k c_i \\right)\\alpha(s)\\left(\\frac{\\textsl{diam}(J_\\emptyset)}{2}\\right)^s.\\]\n\t\tSince $\\liminf_{k\\rightarrow \\infty} \\prod_{i=1}^k c_i =0$, there exists a sequence $\\{k_t\\}_{t=1}^\\infty$ such that\\\\\n\t\t$\\lim_{t\\rightarrow \\infty} \\prod_{i=1}^{k_t} c_i =0$. Thus, $\\delta_{k_t}\\rightarrow 0$ as $t\\rightarrow \\infty$, and\n\t\t$\\mathcal{H}^{s}(F) =\\lim_{t\\rightarrow \\infty}\\mathcal{H}_{\\delta_{k_t}}^{s}(F)=0 $, and hence $\\dim_H(F) \\le s$.\n\t\\end{proof}\n\t\n\tConversely, a lower bound on the Hausdorff dimension of the limit set $F$ can also be obtained as follows. \n\t\n\t\\begin{proposition}\\label{thm:lower}\n\t\tSuppose $\\mathcal J := \\{J_{\\sigma} : \\sigma \\in D\\}$ is a collection of compact subsets of Euclidean space ${\\mathbb{R}}^N$, and let $F$ be the limit set of $\\mathcal{J}$ as given in (\\ref{eqn: Limit_set}).  If for some $s>0$,\n\t\t\\begin{equation}\n\t\t\t\\label{lower_eqn}\\sum_{j=1}^{n_k} \\textsl{diam}(J_{\\sigma*j})^s\n\t\t\t\\geq  \\textsl{diam}(J_\\sigma)^s\n\t\t\\end{equation}\n\t\tfor all $\\sigma\\in D_{k-1}$ and all $k=1,2,\\cdots$, then $dim_H(F)\\ge s$.\n\t\\end{proposition}\n\t\n\t\\begin{proof}\n\t\t\n\t\tWe first show that under condition (\\ref{lower_eqn}), there exists a probability measure $\\mu$ on ${\\mathbb{R}}^N$ concentrated on $F$ such that for each Borel subset $B$ of ${\\mathbb{R}}^N$, \n\t\t\\begin{equation}\\label{mu_c} \\mu(B)\\le \\left(\\frac{\\textsl{diam}(B)}{\\textsl{diam}(J_\\emptyset)}\\right)^s.\\end{equation}\n\t\t\n\t\tLet $\\mu(J_\\emptyset)=1$, and for each $\\sigma \\in D_k$ for $k>0$ and $i=1, \\cdots, n_k$, we inductively set\n\t\t\\[\\mu(J_{\\sigma *i}) =\\frac{\\textsl{diam}(J_{\\sigma*i})^s}{\\sum_{j=1}^{n_k} \\textsl{diam}(J_{\\sigma *j})^s} \\mu(J_{\\sigma}).\\]\n\t\tThen by Proposition 1.7 in \\cite{Falconer}, $\\mu$ can be uniquely extended to a probability measure on ${\\mathbb{R}}^N$, concentrated on $F$.\n\t\tFor any Borel set $B$, the value\n\t\t\\[ \\mu(B) = \\inf \\left \\{ \\sum_{i=1}^{\\infty} \\mu(J_{\\sigma_i}) : B \\cap F \\subset \\bigcup_{i=1}^{\\infty} J_{\\sigma_i} \\text{ and } J_{\\sigma_i} \\in \\mathcal J \\right \\}. \\]\n\t\tThus, to prove (\\ref{mu_c}) for each Borel set $B$, it is sufficient to prove (\\ref{mu_c}) for $J_{\\sigma}$, $ \\forall \\sigma \\in D$.  We proceed by using induction on $k$ when $\\sigma \\in D_k$. It is clear for $k=0$.  Now assume that (\\ref{mu_c}) holds for each $\\sigma \\in D_k$ for some $k$.  Then by induction assumption and (\\ref{lower_eqn}), for each $i=1,\\cdots, n_{k+1}$,\n\t\t\\begin{align*}\n\t\t\t\\mu(J_{\\sigma*i}) &=\\frac{\\textsl{diam}(J_{\\sigma*i})^s}{\\sum_{j=1}^{n_k} \\textsl{diam}(J_{\\sigma*j})^s} \\mu(J_{\\sigma}) \\\\\n\t\t\t&\\leq \\frac{\\textsl{diam}(J_{\\sigma*i})^s}{\\sum_{j=1}^{n_k} \\textsl{diam}(J_{\\sigma*j})^s} \\left(\\frac{\\textsl{diam}(J_{\\sigma})}{\\textsl{diam}(J_{\\emptyset})}\\right)^s\\\\\n\t\t\t&\\leq  \\left(\\frac{\\textsl{diam}(J_{\\sigma*i})}{\\textsl{diam}(J_\\emptyset)}\\right)^s.\n\t\t\\end{align*}\n\t\tThis proves inequality (\\ref{mu_c}).\n\t\t\n\t\t\n\t\tNow, for any $\\delta>0$, let $\\{B_i\\}$ be any collection of closed balls with $\\textsl{diam}(B_i)\\le \\delta$ and $F\\subseteq \\cup_i B_i$. Then, by (\\ref{mu_c}),\n\t\t\\[\\sum_i  \\alpha(s)\\left(\\frac{\\textsl{diam}(B_i)}{2}\\right)^s \n\t\t\\ge \\alpha(s)\\left(\\frac{\\textsl{diam}(J_\\emptyset)}{2}\\right)^s \\sum_i  \\mu(B_i)\\ge c \\mu \\left(\\bigcup_i B_i \\right)\\ge c\\mu(F) = c,\\]\n\t\twhere $c=\\alpha(s)\\left(\\frac{\\textsl{diam}(J_\\emptyset)}{2}\\right)^s$.\n\t\tThus, $\\mathcal{H}^{s}(F) =\\lim_{\\delta \\rightarrow 0}\\mathcal{H}_{\\delta}^{s}(F)\\ge c>0 $, and hence $\\dim_H(F) \\ge s$.\n\t\\end{proof}\n\t\n\t\\section{General Setup of ${\\mathcal F}$-Limit sets}\n\t\\label{general setup}\n\t\n\tWe now formalize the ideas from section 1 to give a description of the construction of such fractals.  We concentrate on the maps in order to take advantage of the computational nature of an IFS, but allow for the maps to be updated and changed at each iteration.\n\t\n\tIn this section let $\\mathcal{X}$ be a collection of nonempty compact subsets of a metric space.\n\t\n\t\\begin{definition}\n\t  A mapping $f: \\mathcal{X} \\to \\mathcal{X}$ is called a \\textit{compression} on $\\mathcal{X}$ if $f(E) \\subseteq E$ for each $E \\in \\mathcal{X}$. \n\t\\end{definition}\n\nFor each natural number $m$, let $$\\mathcal{C}_m(\\mathcal{X}) = \\{ (f^{(1)}, f^{(2)}, \\dots, f^{(m)}): f_i \\text{ is a compression on } \\mathcal{X}, i=1, \\dots, m\\}.$$\n\n\t\\begin{definition}\n\t\tLet $\\mathcal{M}$ be a nonempty set.  A mapping \t\n\t\t\\begin{eqnarray}\n\t\t&\\mathcal{F}: &\\mathcal{M} \\to \\mathcal{C}_m(\\mathcal{X})  \\\\\n\t\t&&k \\to f_k = (f_k^{(1)}, f_k^{(2)}, \\cdots , f_k^{(m)}). \\label{f_k}\n\t\t\\end{eqnarray} is called a marking of $\\mathcal{C}_m(\\mathcal{X})$ by $\\mathcal{M}$.\n\tEach element $ k \\in \\mathcal{M}$ is called the marker of $f_{k}$.\n\t\t\n\t\\end{definition}\n\nGiven a marking ${\\mathcal F}$ and an initial set $E_0 \\in \\mathcal{X}$, we will construct a generalized Moran set from any sequence of markers in $\\mathcal{M}$.  Note that any sequence $\\{k_{\\ell}\\}_{\\ell = 0}^{\\infty}$ in $\\mathcal{M}$ can be represented as a mapping from the ordered set $D$ to $\\mathcal{M}$.\n\n\n\\begin{definition}\n\t\\label{3.0.3}\n\tLet $\\mathcal{F}$ be a marking of $\\mathcal{C}_m(\\mathcal{X})$ by $\\mathcal{M}$, let $E_0$ be any element in $\\mathcal{X}$, and $D$ be as in (\\ref{Dk}).  Suppose $\\vec{k} : D \\to \\mathcal{M}$ is a map sending $\\sigma$ to $k_{\\sigma}$. \n\tFor each $\\sigma \\in D$ and $1 \\leq j \\leq m$, we recursively define $J_{\\emptyset} = E_0$ and\n\t\\begin{equation}  \n\t\\label{Jsigj}\n\tJ_{\\sigma * j} = f_{\\k_{ \\sigma}}^{(j)} (J_{\\sigma}), \n\t\\end{equation}\n\t\n\twhere $f_{k_{\\sigma}}$ is given by $\\mathcal{F}$ as in (\\ref{f_k}).\n\t\n\tThe limit set $\\displaystyle F = \\bigcap_{k\\geq 1} \\bigcup_{\\sigma \\in D_k} J_{\\sigma}$ associated with $\\mathcal{J}(\\vec{k})=\\{J_\\sigma: \\sigma \\in D\\}$ is called the $\\mathcal{F}$-limit set generated by $\\vec{k}$ with the initial set $ E_0$.  \n\t\n\t\n\\end{definition}\n\nWe now make two observations relating the concepts of an ${\\mathcal F}$-limit set with the attractor of an IFS.\t\n\n\t\tWe first observe that the attractor of an IFS $\\{ S_1, S_2, \\dots, S_m\\}$ on a closed subset $\\Delta$ of ${\\mathbb{R}}^N$ can be viewed as an ${\\mathcal F}$-limit set as follows.\n\t\t\n\t\tLet $\\mathcal{X} = \\{ E : E \\text{ is a non-empty compact subset of } \\Delta,  S_i(E) \\subseteq E, \\text{ for all } i\\}$.  Since each $S_i$ is a contraction on $\\Delta$, the set $E_r := \\Delta \\cap \\overbar{B(0,r)}$ is a non-empty compact subset of $\\Delta$, and $S_i(E_r) \\subseteq E_r$ for each $i$ when $r$ is sufficiently large.  In other words, $E_r \\in \\mathcal{X}$ for sufficiently large $r$.  Also, each contraction map $S_i$ acting on $\\Delta$ naturally determines a map $f^{(i)} :\\mathcal{X} \\to \\mathcal{X}$ given by \n\t\t\\begin{equation}f^{(i)}(E) = S_i(E) := \\left \\{ S_i(x) | x \\in E \\subseteq \\Delta \\right \\} \\label{fie}\\end{equation} \n\t\t\\noindent for each $E \\in \\mathcal{X}$.  Since $f^{(i)}(E) = S_i(E) \\subseteq E$, $f^{(i)}$ is a compression for each $i$.  Set $$ f=(f^{(1)}, f^{(2)}, \\dots , f^{(m)} ).$$  For any non-empty set $\\mathcal{M}$, define the marking ${\\mathcal F}$ of $\\mathcal{C}_{m}(\\mathcal{X})$ to be the constant function ${\\mathcal F}(k)=f$ for all $k \\in \\mathcal{M}$.  \n\t\t Thus, for each $\\sigma \\in D_k$ and $i=1, \\dots ,m$, we have that $J_{\\sigma * i} = S_i(J_{\\sigma})$ from (\\ref{Jsigj}). As a result, for any map $\\vec{k}: D \\to \\mathcal{M}$, the collection $\\mathcal{J}(\\vec{k}) = \\{ J_{\\sigma} : \\sigma \\in D\\}$ is independent of the choice of $\\vec{k}$.  Thus, the associated $\\mathcal{F}$-limit set $\\displaystyle F = \\bigcap_{k\\geq 1} \\bigcup_{\\sigma \\in D_k} J_{\\sigma}$  agrees with the attractor of the given IFS $\\{ S_1, S_2, \\dots, S_m\\}$.\n\n\t\n\tConversely, let $\\mathcal{F}$ be a marking of $\\mathcal{C}_m(\\mathcal{X})$ by $\\mathcal{M}$ where $\\mathcal{X}$ is a collection of non-empty compact subsets of $\\Delta$.  Suppose there is a mapping $\\vec{k} : D \\to \\mathcal{M}$ such that the sequence $\\{ f_{k_{\\sigma}}\\}_{\\sigma \\in D}$ is constant in $\\mathcal{C}_{m}(\\mathcal{X})$ (i.e. there exists an $f \\in \\mathcal{C}_{m}(\\mathcal{X})$ such that $f_{k_{\\sigma}} =f$ for all $\\sigma \\in D$)  and for each $i=1,2,\\dots,m$, there exists a contraction $S_i$ on $\\Delta$ such that equation (\\ref{fie}) holds for each $E \\in \\mathcal{X}$. Then the $\\mathcal{F}$-limit set $F$ generated by $\\vec{k}$ is the attractor of the IFS $\\{S_1, S_2, \\dots, S_m \\}$.\n\t\t\nTherefore, choosing $\\vec{k}: D \\to \\mathcal{M}$ to be a constant map will result in a limit set $F$ that is the attractor of an IFS.\n\tIn the above sense, our approach is a generalization of the standard IFS construction.\n\t\n\t\tAn important observation is that replacing $\\{k_{\\sigma}\\}_{\\sigma \\in D}$ by another sequence  $\\{\\tilde{k}_{\\sigma}\\}_{\\sigma \\in D}$ in (\\ref{Jsigj}) will not change the computational complexity of the construction of $\\mathcal{J}(\\vec{k})$.  Thus, generating the limit set $F$ will have a similar computational complexity as generating the attractor of a comparable IFS.\n\t\n\t In the following section we will compute the Hausdorff dimension of the constructed ${\\mathcal F}$-limit sets.  In section 5 we will provide examples along with their dimensions.\n\t\n\t\\section{Hausdorff dimensions of ${\\mathcal F}$-Limit sets}\n\t\\label{dimensions}\n\t\n\tAs indicated in Propositions \\ref{thm: upper} and \\ref{thm:lower} in section 2, the relative ratio between the diameters of the sets plays an important role in the calculation of the dimension of the limit set.  Therefore, we introduce the following definition.\n\t\n\t\\begin{definition}\n\t\tFor any compression $g: \\mathcal{X}\\to \\mathcal{X}$, define\n\t\t\\begin{equation}\n\t\t\tU(g)=\\sup_{E\\in \\mathcal{X}} \\frac{\\textsl{diam}(g(E))}{\\textsl{diam}(E)}, \\text{ and } L(g)=\\inf_{E\\in \\mathcal{X}} \\frac{\\textsl{diam}(g(E))}{\\textsl{diam}(E)}.\n\t\t\\end{equation}\n\t\\end{definition}\n\tNote that, for each $E\\in \\mathcal{X}$,\n\t\\begin{equation}\n\t\\label{LgUg}\n\tL(g)\\cdot \\textsl{diam}(E)\\le \\textsl{diam}(g(E)) \\le U(g) \\cdot \\textsl{diam}(E).\n\t\\end{equation}\n\t\n\tFor any $\\mathbf{k}\\in \\mathcal{M}$ and $f_{\\mathbf{k}} = (f_{\\k}^{(1)}, f_{\\k}^{(2)}, \\cdots , f_{\\k}^{(m)}) \\in \\mathcal{C}_m(\\mathcal{X})$, define\n\t\\[\\mathbf{U_k}=\\left( U(f_{\\mathbf{k}}^{(1)}), \\cdots,  U(f_{\\mathbf{k}}^{(m)})\\right) \\in \\mathbb{R}^m,\\]\n\tand \n\t\\[ \\mathbf{L_k}=\\left( L(f_{\\mathbf{k}}^{(1)}), \\cdots,  L(f_{\\mathbf{k}}^{(m)})\\right) \\in \\mathbb{R}^m.\\]\n\t\n\tAlso, for each $x=(x_1, \\cdots, x_m)\\in \\mathbb{R}^m$ and $s>0$, denote\n\t\\[||x||_s=\\left(\\sum_{i=1}^m |x_i|^s\\right)^{\\frac{1}{s}}.\\]\n\t\n\tThese notations, Proposition \\ref{thm: upper} and Proposition \\ref{thm:lower} motivate our main theorem.\n\t\\begin{theorem} \\label{thm:ratio_bounds}\n\t\tLet $F$ be the ${\\mathcal F}$-limit set generated by a sequence $\\{ \\k_{\\sigma} \\}_{\\sigma \\in D}$ with initial set $J_{\\emptyset}$, and $s> 0$. \n\t\t\\begin{enumerate}\n\t\t\t\\item[(a)] If \n\t\t\t\\[\\inf_{\\sigma \\in D} \\{||\\mathbf{L}_{\\k_\\sigma}||_{s}\\}\\ge 1,\\]\n\t\t\tthen $\\dim_H(F) \\ge s$.\n\t\t\t\\item[(b)] If\n\t\t\t\\[\\sup_{\\sigma \\in D}\\{||\\mathbf{U}_{\\k_\\sigma}||_{s}\\}<1,\\]\n\t\t\tthen $\\dim_H(F)\\le s$.\n\t\t\\end{enumerate}\n\t\\end{theorem}\n\t\n\t\\begin{proof}\n\t\t(a)  By \\ref{Jsigj} and \\ref{LgUg}, for all $\\sigma \\in D$, \n\t\t\\begin{eqnarray*}\n\t\t\t& &\\sum_{j=1}^{m} \\textsl{diam}(J_{\\sigma * j})^s =\\sum_{j=1}^{m} \\textsl{diam}\\left(f_{\\k_{\\sigma}}^{(j)}(J_{\\sigma})\\right)^s \n\t\t\t\\ge  \\sum_{j=1}^{m}\\left(L(f_{\\k_\\sigma}^{(j)})\\right)^s \\textsl{diam}(J_{\\sigma})^s \\ge \\textsl{diam}(J_{\\sigma})^s.\n\t\t\\end{eqnarray*}\n\t\tThus, by Proposition \\ref{thm:lower}, $\\dim_H(F)\\ge s$.\n\t\t\n\t\t(b) Similarly, for all $\\sigma \\in D$, \n\t\t\\[\\sum_{j=1}^{m} \\textsl{diam}(J_{\\sigma*j})^s \\le \\sum_{j=1}^{m}\\left(U(f_{\\k_\\sigma}^{(j)})\\right)^s \\textsl{diam}(J_{\\sigma})^s \\le c \\cdot \\textsl{diam}(J_{\\sigma})^s,\\]\n\t\n\t\twhere\n\t\t\\[c:=\\sup_\\sigma\\{(||\\mathbf{U}_{\\k_\\sigma}||_{s})^s\\}<1.\\]\n\t\tBy Proposition \\ref{thm: upper}, $\\dim_H(F)\\le s$.\n\t\\end{proof}\n\t\n\t\n\t\n\t\n\t\\begin{remark}\n\t\t\n\t\tFor practical reasons, we find that it is more convenient to represent the mapping $\\vec{k} : D \\to \\mathcal{M}$ by a sequence $\\{k_{\\ell}\\}_{\\ell=0}^{\\infty} \\subseteq \\mathcal{M}$.  For each $\\sigma=(i_1, i_2, \\dots, i_k) \\in D_k$, let \n\t\t\\begin{equation}\\ell(\\sigma) = \\sum_{p=0}^{k-1} m^p i_{k-p}\\end{equation}\n\t\tbe the ordering of $\\sigma$ in the ordered set $D$.  Using this notation, we can rewrite Definition \\ref{3.0.3} as follows. \n\t\t\n\t\t\\noindent \\textbf{Definition \\ref{3.0.3}'.}\n\t\tLet ${\\mathcal F}$ be a marking of $\\mathcal{C}_m(\\mathcal{X})$ by $\\mathcal{M}$, let $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty} $ be a sequence in $\\mathcal{M}$, and $E_0 \\in \\mathcal{X}$ be a starting set.  For each $\\ell =0,1, 2,\\cdots$ and $j=1,2,\\cdots, m$, we iteratively denote the set\n\t\t\\[E_{m\\ell+j}=f_{\\k_\\ell}^{(j)}(E_\\ell) \\in \\mathcal{X},\\]\n\t\twhere $f_{\\k_{\\ell}}$ is given by $\\mathcal{F}$ as in (\\ref{f_k}).\n\t\t\n\t\tLet $\\mathcal{G}_m(0)=0$ and for $n \\ge 1$,\n\t\t\\begin{equation} \\label{eqn:p_m}\n\t\t\\mathcal{G}_m(n)=m+m^2+\\cdots+m^n=\\frac{m^{n+1}-m}{m-1}\n\t\t\\end{equation}  \n\t\tdenote the number of sets in the $n^{th}$ generation, i.e. the cardinality of $D_n$.\n\t\t\n\t\tThe limit set \\begin{equation} \n\t\t F=\\bigcap_{n=1}^{\\infty} \\bigcup_{\\ell=\\mathcal{G}_m(n-1)+1}^{\\mathcal{G}_m(n)} E_{\\ell} \\label{F_limit_set} \\end{equation}\n\t\tis called the ${\\mathcal F}$-limit set generated by the triple $({\\mathcal F}, \\{\\k_{\\ell} \\}_{\\ell =0}^{\\infty} , E_0)$. \n\t\t\n\t\tIn the following, we will use the notation from Definition 3.0.3' to describe the construction of the ${\\mathcal F}$-limit sets.  Clearly, using this notation, Theorem \\ref{thm:ratio_bounds} simply says that if\t$\\displaystyle \\inf_{\\ell} \\{||\\mathbf{L}_{\\k_{\\ell}}||_{s}\\}\\ge 1,$ then $\\dim_H(F) \\ge s$, and if $ \\displaystyle \\sup_{\\ell}\\{||\\mathbf{U}_{\\k_\\ell}||_{s}\\}<1,$\tthen $\\dim_H(F)\\le s$.\n\t\n\t\\end{remark}\n\n\n\tWhen both $\\{||\\mathbf{L}_{\\k_\\ell} ||_s\\}_{\\ell=0}^{\\infty}$ and $\\{||\\mathbf{U}_{\\k_\\ell} ||_s\\}_{\\ell =0}^{\\infty}$ are convergent sequences, the following corollary enables us to quickly estimate the dimension of $F$.\n\n\n\n\t\\begin{corollary}\\label{cor:limit_sup_inf}\n\t\tLet $F$ be the limit set generated by the triple $({\\mathcal F}, \\{ \\k_{\\ell} \\}_{\\ell =0}^{\\infty} , E_0)$. Then,\n\t\t\n\t\t\\begin{equation}\\label{eqn: s_star}\n\t\t\ts_* \\le \\dim_H(F)\\le s^*,\n\t\t\\end{equation}\n\t\twhere \n\t\t\\[s_*=\\sup \\left\\{s: \\liminf_{\\ell \\rightarrow \\infty}\\{||\\mathbf{L}_{\\k_\\ell} ||_s\\}>1 \\right\\}\\text{, and } \n\t\ts^*=\\inf \\left \\{s: \\limsup_{\\ell \\rightarrow \\infty}\\{||\\mathbf{U}_{\\k_\\ell}||_s\\}<1 \\right\\} .\\]\n\t\\end{corollary}\n\t\\begin{proof}\n\t\tFor any $0< s< s_*$, by the definition of $s_*$,\n\t\t\\[\\liminf_{\\ell \\rightarrow \\infty}\\{||\\mathbf{L}_{\\k_\\ell} ||_s\\}>1.\\]\n\t\tThus, when $\\ell_*\\in \\mathbb{N}$ is large enough,\n\t\t\\[\\inf_{\\ell\\ge \\ell_*}\\{||\\mathbf{L}_{\\k_\\ell}||_s \\} \\ge 1, \\qquad \\text{i.e. } \\inf_{\\ell\\ge 0}\\{||\\mathbf{L}_{\\k_{\\ell_*+\\ell}}||_s\\} \\ge 1.\\]\n\t\tSince $F\\cap E_{\\ell_*}$ is the set generated by\n\t\tthe triple $({\\mathcal F}, \\{\\k_{\\ell_*+\\ell} \\}_{\\ell =0}^{\\infty} , E_{\\ell_*})$,  by Theorem \\ref{thm:ratio_bounds}, it follows that $\\dim_H(F\\cap E_{\\ell_*}) \\ge s$ for any $\\ell_*$ large enough. This implies that $\\dim_H(F) \\ge s$ for any $s<s_*$ and hence $\\dim_H(F) \\ge s_*$.\n\t\t\n\t\tSimilarly, we also have $\\dim_H(F)\\le s^*$.\n\t\\end{proof}\n\t\n\tIn the following corollaries, we will see that bounds of the dimension of $F$ can also be obtained from corresponding bounds on $\\mathbf{L}_{\\k_\\ell}$ and $\\mathbf{U}_{\\k_\\ell}$. \n\t\n\t\\textbf{Notation.} For any two points $x=(x_1,\\cdots, x_m)$ and $y=(y_1,\\cdots, y_m)$ in $\\mathbb{R}^m$, we say $x \\le y$ if $x_i \\le y_i$ for each $i=1,\\cdots, m$.\n\t\n\t\\begin{corollary} \\label{cor: bound_on_L_U}\n\t\tLet $\\mathbf{t}=(t_1,\\cdots, t_m) $  and $\\mathbf{r}=(r_1,\\cdots, r_m) $ be two points in $(0,1)^m \\subset \\mathbb{R}^m$. Let $s_*$ and $s^*$ be the solutions to $||\\mathbf{t}||_{s_*}=1$, and $||\\mathbf{r}||_{s^*}=1$ respectively,  i.e. \n\t\t\\[t_1^{s_*}+t_2^{s_*}+\\cdots+t_m^{s_*}=1,  \\text{ and } r_1^{s^*}+r_2^{s^*}+\\cdots+r_m^{s^*}=1.\\]\n\t\t\\begin{enumerate} \n\t\t\t\\item[(a)] If $\\mathbf{L}_{\\k_\\ell}\\ge \\mathbf{t}$ for all $\\ell$, then $\\dim_H(F)\\ge s_*$.\n\t\t\t\n\t\t\t\\item[(b)]If $\\mathbf{U}_{\\k_\\ell} \\le \\mathbf{r}$ for all $\\ell$, then\n\t\t\t$\\dim_H(F) \\le s^*$. \n\t\t\t\n\t\t\t\\item[(c)] If $ \\mathbf{L}_{\\k_\\ell}=\\mathbf{r} =\\mathbf{U}_{\\k_\\ell}$ for all $\\ell$, then $\\dim_H(F)=s^*$.\n\t\t\\end{enumerate}\n\t\\end{corollary}\n\t\\begin{proof}\n\t\t(a) Let $0< s<s_*$. Then,\n\t\t\\[\\inf_\\ell\\{||\\mathbf{L}_{\\k_\\ell}||_{s}\\}\\ge ||\\mathbf{t}||_s \\ge ||\\mathbf{t}||_{s_*}= 1.\\]\n\t\tThus, by Theorem \\ref{thm:ratio_bounds}, $\\dim_H(F)\\ge s$ for any $s<s_*$, and hence $\\dim_H(F)\\ge s_*$.\n\t\t\n\t\t(b) Similarly,  let $0<s^*<s$. Then,\n\t\t\\[\\sup_\\ell\\{||\\mathbf{U}_{\\k_\\ell}||_{s}\\}\\le ||\\mathbf{r}||_s < ||\\mathbf{r}||_{s^*}= 1.\\]\n\t\tThus, by Theorem \\ref{thm:ratio_bounds},  $\\dim_H(F)\\le s$ for any $s>s^*$, and hence $\\dim_H(F)\\le s^*$.\n\t\t\n\t\t(c) follows from (a) and (b).\n\t\\end{proof}\n\t\n\tA special case of Corollary \\ref{cor: bound_on_L_U} gives the following explicit formulas for the bounds on the dimension of $F$.\n\t\\begin{corollary}\\label{cor:uniform_bounds}\n\t\tLet $F$ be the limit set generated by the triple $({\\mathcal F}, \\{ \\k_{\\ell} \\}_{\\ell =0}^{\\infty} , E_0)$.  Let\n\t\t\\[\\mathbf{t}=(t, \\cdots, t) \\text{ and } \\mathbf{r}=(r, \\cdots, r),\\]\n\t\tfor some $0<t, r<1$. \n\t\t\\begin{enumerate}\n\t\t\t\\item[(a)] If $\\mathbf{L}_{\\k_\\ell}\\ge \\mathbf{t}$ for all $\\ell$, then $\\dim_H(F)\\ge \\frac{\\log{m}}{-\\log t}$.\n\t\t\t\n\t\t\t\\item[(b)]If $\\mathbf{U}_{\\k_\\ell} \\le \\mathbf{r}$ for all $\\ell$, then\n\t\t\t$\\dim_H(F) \\le \\frac{\\log{m}}{-\\log r}$. \n\t\t\t\n\t\t\t\\item[(c)] If $ \\mathbf{L}_{\\k_\\ell}=\\mathbf{r} =\\mathbf{U}_{\\k_\\ell}$ for all $\\ell$, then $\\dim_H(F)=\\frac{\\log{m}}{-\\log r}$.\n\t\t\\end{enumerate}\n\t\\end{corollary}\n\t\n\tOther types of bounds on $\\mathbf{L}_{\\k_\\ell}$ and $\\mathbf{U}_{\\k_\\ell}$ can also be used to provide bounds on $\\dim_H(F)$, as indicated by the following result.\n\t\\begin{corollary}\\label{cor: mean}\n\t\tLet $F$ be the limit set generated by the triple $({\\mathcal F}, \\{\\k_{\\ell} \\}_{\\ell =0}^{\\infty} , E_0)$. \n\t\t\\begin{enumerate}\n\t\t\t\\item[(a)] If \n\t\t\t\\[w:=\\inf_{\\ell}\\{||\\mathbf{L}_{\\k_\\ell}||_1\\} \\ge 1,\\]\n\t\t\tthen $\\dim_H(F) \\ge \\frac{\\log(m)}{\\log(m)-\\log(w)}$.\n\t\t\t\\item[(b)]\n\t\t\tIf \n\t\t\t\\[u:=\\sup_{\\ell}\\{||\\mathbf{U}_{\\k_\\ell}||_1\\} < 1,\\]\n\t\t\tthen $\\dim_H(F) \\le \\frac{\\log(m)}{\\log(m)-\\log(u)}$.\n\t\t\\end{enumerate}\n\t\\end{corollary}\n\t\\begin{proof}\n\t\t(a). In this case, for $s = \\frac{\\log(m)}{\\log(m)-\\log(w)} \\ge 1$, we have\n\t\t\\[\\frac{\\sum_{j=1}^{m} \\left(L\\left(f_{\\k_\\ell}^{(j)}\\right) \\right)^{s}}{m}\\ge \\left(\\frac{\\sum_{j=1}^{m}L\\left(f_{\\k_\\ell}^{(j)}\\right)}{m}\\right)^s \\ge \\left(\\frac{w}{m}\\right)^s\\]\n\t\tfor each $\\ell$. Thus,\n\t\t\\[\\inf_\\ell\\{||\\mathbf{L}_{\\k_\\ell}||_s\\}\\ge m^{\\frac{1}{s}}\\frac{w}{m} = 1,\\]\n\t\tthen by Theorem \\ref{thm:ratio_bounds}, $\\dim_H(F) \\ge s$.\n\t\t\n\t\t(b).  In this case, for any $1 \\ge s > \\frac{\\log(m)}{\\log(m)-\\log(u)}$, we have\n\t\t\\[\\frac{\\sum_{j=1}^{m}\\left(U\\left(f_{\\k_\\ell}^{(j)}\\right)\\right)^{s}}{m}\\le \\left(\\frac{\\sum_{j=1}^{m}U\\left(f_{\\k_\\ell}^{(j)}\\right)}{m}\\right)^s \\le \\left(\\frac{u}{m}\\right)^s\\]\n\t\tfor each $\\ell$. Thus,\n\t\t\\[\\sup_\\ell\\{||\\mathbf{U}_{\\k_\\ell}||_s\\}\\le m^{\\frac{1}{s}}\\frac{u}{m} <1.\\]\n\t\tBy Theorem \\ref{thm:ratio_bounds}, $\\dim_H(F) \\le s$. Hence, $\\dim_H(F) \\le \\frac{\\log(m)}{\\log(m)-\\log(u)}$.\n\t\\end{proof}\n\t\n\tNote that this corollary generally provides better bounds on $\\dim_H(F)$ than those obtained from directly applying Theorem \\ref{thm:ratio_bounds}.\n\t\n\t\n\t\n\t\\section{Examples of ${\\mathcal F}$-Limit sets}\n\t\n\tIn this section we describe the construction of both classical fractals and generalized Moran sets in the language of Section \\ref{general setup}, and calculate the dimension using the results from Section \\ref{dimensions}.  \n\t\n\t\\begin{subsection}{Cantor-Like Sets}\n\t\n\tWe first consider Cantor-like sets.  Let $$\\mathcal{X} = \\{ [a,b] : a,b \\in {\\mathbb{R}}\\}$$ be the collection of closed intervals, $m=2$, and let $\\mathcal{M} = [0,1]^2 \\subseteq {\\mathbb{R}}$. For each $\\mathbf{k}=(k^{(1)},k^{(2)}) \\in \\mathcal{M}$, we consider the following two maps, \n\t\\begin{eqnarray*}\n\t\tf^{(1)}_\\mathbf{k} : & \\mathcal{X} &\\to \\mathcal{X} \\\\\n\t\t&[a,b]   &\\mapsto  [a, k^{(1)}(b-a)+a]\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\t\tf^{(2)}_\\mathbf{k} : &\\mathcal{X} &\\to \\mathcal{X} \\\\\n\t\t& [a,b]   &\\mapsto [k^{(2)}(a-b)+b, b].\n\t\\end{eqnarray*}\n\n\n\n\tNote that both $f^{(1)}_\\k$ and $f^{(2)}_{\\k}$ are compression maps for any $\\k \\in \\mathcal{M}$.  Thus, this defines a marking\n\t\t\\begin{eqnarray*}\n\t\t\t{\\mathcal F} : &\\mathcal{M} &\\to \\mathcal{C}_{2}(\\mathcal{X}) \\\\\n\t\t&\\k   &\\mapsto  f_{\\k}= (f^{(1)}_{\\k}, f^{(2)}_{\\k}).\n\t\\end{eqnarray*}\n\t\n\t\n\tHere, for each $\\mathbf{k}=(k^{(1)}, k^{(2)}) \\in \\mathcal{M}$, one can clearly see that \\[\\textsl{diam} \\left(f^{(i)}_\\mathbf{k}([a,b])\\right)=k^{(i)} \\cdot \\textsl{diam}([a,b]).\\]  Thus, $L \\left(f^{(i)}_\\mathbf{k} \\right)=k^{(i)}=U\\left(f^{(i)}_\\mathbf{k}\\right)$, and hence \\begin{equation}\\label{cantor_Lk} \\mathbf{L_k=k=U_k}.\\end{equation}\n\t\n\n  \n  Let $E_0 = [0,1] \\in \\mathcal{X}$ be fixed. For any sequence $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty} \\in \\mathcal{M}$, we define the following:\n\t\n\t\\begin{eqnarray*}\n\t\t&E^{(0)} &= E_0\\\\\n\t\t&E^{(1)} &= f^{(1)}_{\\mathbf{k}_0}  \\left( E_0 \\right) \\cup  f^{(2)}_{\\mathbf{k}_0}  \\left( E_0 \\right) =: E_{1} \\cup E_{2}\\\\\n\t\t&E^{(2)} &= f^{(1)}_{\\mathbf{k}_1}  \\left( E_{1} \\right) \\cup  f^{(2)}_{\\mathbf{k}_1}  \\left( E_{1} \\right) \\cup  f^{(1)}_{\\mathbf{k}_2}  \\left( E_{2} \\right) \\cup  f^{(2)}_{\\mathbf{k}_2}  \\left( E_{2} \\right) \\\\\n\t\t& &:=\\qquad E_{3} \\qquad  \\cup \\quad  E_{4} \\quad \\cup \\quad  E_{5} \\quad  \\cup \\quad  E_{6}\\\\\n\t\t&  \\vdots \\\\\n\t\t&E^{(n)}&= \\bigcup_{i=2^{n-1}-1}^{2^n-2} \\left(f_{\\k_i}^{(1)}(E_i) \\cup f_{\\k_i}^{(2)}(E_i) \\right):=\\bigcup_{i=2^{n-1}-1}^{2^n-2} \\left(E_{2i+1} \\cup E_{2i+2} \\right)=\\bigcup_{ \\ell= 2^n-1}^{2(2^n-1)} E_{\\ell}.\n\t\\end{eqnarray*}\n\n\tNote that when $\\mathbf{k}_{\\ell}=(\\frac{1}{3}, \\frac{1}{3})$ for all $\\ell$, $E^{(n)} $ is the $n^{th}$-generation of the Cantor set ${\\mathcal C}$ and $\\displaystyle F =\\lim_{n \\to \\infty} E^{(n)} = \\bigcap_{n} E^{(n)} ={\\mathcal C} $. \n\t\n\t\n\t\t Observe that the process of constructing the sequence  $\\{E^{(n)}\\}_{n=0}^{\\infty}$ is independent of the values of $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}$. To allow for more general outcomes, we can update the linear functions $f^{(1)}_\\k$ and $f^{(2)}_\\k$ simply by changing the value of $\\k$ at each stage of the construction, which does not change the computational complexity of the process.  Using this idea, we now construct some examples of Cantor-like sets by choosing suitable sequences $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}$.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=4in]{compare3.png} \n\t\t\\caption{Comparison of classical Cantor set (blue) and new Cantor-like set (red) }\n\t\t\\label{comparecant}\n\t\\end{figure}\n\t\n\t\\begin{example}\\label{example: 1}\n\t\tLet $\\mathbf{k}_{\\ell} = \\left(\\frac{\\ell+1}{4\\ell+6}, \\frac{2\\ell +5}{8\\ell+16}\\right)$ for $\\ell \\geq 0$, and let $F$ be the ${\\mathcal F}$-limit set generated by the triple $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}, E_0)$.  In Figure \\ref{comparecant} we plot the usual Cantor set ${\\mathcal C}$ (in blue) below the set $F$ (in red) to illustrate the comparison. We can see that the set $F$ has the same basic shape as the Cantor set ${\\mathcal C}$, but is no longer strictly self-similar.\n\t\t\n\t\tIn order to compute the Hausdorff dimension of the new Cantor-like set $F$, we apply Corollary \\ref{cor:limit_sup_inf}. Note that by \\ref{cantor_Lk},\n\t\t\\[\\lim_{\\ell \\rightarrow \\infty}||\\mathbf{L}_{\\mathbf{k}_\\ell}||_s=\\lim_{\\ell \\rightarrow \\infty}||\\mathbf{k}_\\ell ||_s=\\frac{2^{\\frac{1}{s}}}{4}.\\]\n\t\tSo,  \n\t\t\\[\n\t\ts_* =\\sup_s\\{\\liminf_{\\ell \\rightarrow \\infty}||\\mathbf{L}_{\\mathbf{k}_\\ell}||_s>1\\} = \\sup_s \\left\\{\\frac{2^{\\frac{1}{s}}}{4}>1\\right\\}=\\frac{1}{2}.\n\t\t\\]\n\t\tSimilarly, we also have $s^*=\\frac{1}{2}$. By (\\ref{eqn: s_star}), $\\dim_H(F)=\\frac{1}{2}$.\n\t\t\n\t\\end{example}\n\t\n\tIn the next example, we will construct a random Cantor-like set as follows.\n\t\n\t\\begin{example} \\label{example: 2}\n\t For each $\\ell\\ge 0$, we take $\\mathbf{k}_{\\ell}=\\left(q_\\ell, \\frac{1}{2}-q_\\ell \\right)$ where $q_\\ell$ is a random number between $\\frac{1}{8}$ and $\\frac{3}{8}$. Let $F$ be the corresponding ${\\mathcal F}$-limit set generated by the triple $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}, E_0)$.  We plot $F$ in Figure \\ref{cantorrand}.  In this example, the total length of the $n^{th}$ generation $E^{(n)}$ is chosen to be $(\\frac{1}{2})^n$, while the scaling factors of the left subintervals at each stage are randomly chosen.\n\t\t\n\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=3.1in]{example2_new.png} \n\t\t\t\\caption{A randomly generated Cantor-like set}\n\t\t\t\\label{cantorrand}\n\t\t\\end{figure}\n\t\t\n\t\tWe now estimate the dimension of $F$. By (\\ref{cantor_Lk}),\n\t\t\\[\\left(\\frac{1}{8},\\frac{1}{8}\\right)\\le \\mathbf{L}_{\\mathbf{k}_\\ell} =\\mathbf{k}_\\ell=\\mathbf{U}_{\\mathbf{k}_\\ell} \\le \\left(\\frac{3}{8},\\frac{3}{8}\\right).\\]\n\t\tBy Corollary \\ref{cor:uniform_bounds}, \n\t\t\\[\\frac{\\log(2)}{-\\log(1\/8)}\\le \\dim_H(F) \\le \\frac{\\log(2)}{-\\log(3\/8)}.\\]\n\t\tThat is,\n\t\t\\[\\frac{1}{3}\\le \\dim_H(F) \\le \\frac{\\log(2)}{\\log(8\/3)}\\approx 0.7067.\\]\n\t\t\n\t\t\n\t\\end{example}\n\t\n\t\n\t\\begin{example} \\label{example: 3}\n\tIn this example, we create a sequence $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}$ that results in a limit set with a given measure, e.g. 1\/3.   Of course, the classic example of such a limiting set is the fat Cantor set.  For a different approach, let $\\sum_{n=0}^{\\infty} a_n$ be any convergent series of positive terms with limit $L$.  We consider a sequence $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}$ defined in the following way.  \n\t\t\n\t\tLet $n\\geq1$ be the generation of the construction and for each $\\ell$ with $2^{n-1}-1 \\leq \\ell \\leq 2^{n}-2$, define $\\mathbf{k}_\\ell=(b_n, b_n)$ where $$\\displaystyle b_1:= \\dfrac{\\frac{3}{2}L-a_0}{2\\left(\\frac{3}{2}L\\right)} \\quad \\text{ and } \\quad b_n := \\dfrac{\\frac{3}{2}L-\\sum_{i=0}^{n-1} a_i}{2\\left(\\frac{3}{2}L-\\sum_{i=0}^{n-2}a_i\\right)} \\text{ for } n\\ge 2.$$ \n\t\tWith this sequence $\\{\\k_{\\ell}\\}_{\\ell =0}^{\\infty}$, one can find that the length of each interval in the $n^{th}$ generation is $$b_1 b_2 \\cdots b_n=\\dfrac{\\frac{3}{2}L-\\sum_{i=0}^{n-1} a_i}{2^n \\cdot \\frac{3}{2}L}.$$  Thus, the total length of the $n^{th}$ generation is \\[ \\dfrac{\\frac{3}{2}L-\\sum_{i=0}^{n-1} a_i}{\\frac{3}{2}L}=1-\\frac{2}{3L}\\sum_{i=0}^{n-1} a_i \\] which converges to 1\/3 as desired.  \n\t\t\n\t\tAs an example, we take the convergent series $\\displaystyle \\sum_{n=0}^{\\infty} \\dfrac{1}{n!} = e$ and use it to create the ${\\mathcal F}$-limit set $F$ with measure 1\/3.  The first few generations are shown in Figure \\ref{example3cant}.\n\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=3.1in]{example3_1.png} \n\t\t\t\\caption{Fractal of measure $\\frac{1}{3}$ created by using $\\sum_{n=0}^{\\infty} \\frac{1}{n!} = e$  }\n\t\t\t\\label{example3cant}\n\t\t\\end{figure}\n\t\\end{example}\n\t\\end{subsection}\n\t\n\t\\begin{subsection}{Sierpinski Triangle}\n\t\tThe Sierpinski triangle is another well known fractal.\n\t\t\n\t\tFollowing the general setup in Section \\ref{general setup}, we take \\begin{equation} \\mathcal{X} = \\{ (A,B,C) | A,B,C \\in {\\mathbb{R}}^2\\} \\label{xdefinesierp} \\end{equation} representing the collection of all triangles $\\Delta ABC$ in ${\\mathbb{R}}^2$, $m=3$, and $\\mathcal{M} = [0,1]^6 \\subseteq {\\mathbb{R}}^6$.  For each $\\mathbf{k}=\\left(k^{(1)}, k^{(2)}, k^{(3)}, k^{(4)}, k^{(5)}, k^{(6)}\\right) \\in \\mathcal{M}$ and $i=1,2,3$ we can define affine transformations $f_{\\k}^{(i)} : \\mathcal{X} \\to \\mathcal{X}$ as \n\t\t\\begin{eqnarray*}\n\t\t\tf^{(1)}_{\\k} (A,B,C) &= &(A,A+k^{(1)}(B-A), A+k^{(2)}(C-A))\\\\\n\t\t\tf^{(2)}_{\\k} (A,B,C) &= &(B+k^{(4)}(A-B),B, B+k^{(3)}(C-B))\\\\\n\t\t\tf^{(3)}_{\\k} (A,B,C) &= &(C+k^{(5)}(A-C),C+k^{(6)}(B-C), C)\n\t\t\\end{eqnarray*}\n\t\tfor every $(A,B,C) \\in \\mathcal{X}$.\n\t\t\n\n\t\tNote that each $f^{(i)}_\\k$ is a compression map for $i=1,2,3$ and  any $\\k \\in \\mathcal{M}$.  Thus, this defines a marking\n\t\t\\begin{eqnarray*}\n\t\t\t{\\mathcal F} : &\\mathcal{M} &\\to \\mathcal{C}_{3}(\\mathcal{X}) \\\\\n\t\t\t&\\k   &\\mapsto  f_{\\k}= (f^{(1)}_{\\k}, f^{(2)}_{\\k}, f^{(3)}_{\\k}).\n\t\t\\end{eqnarray*}\n\t\n\t \n\t\n\t\n\tOf course, to prevent overlaps we can require that $k^{(1)} + k^{(4)} \\leq 1, k^{(2)} +k^{(5)} \\leq 1, k^{(3)}+k^{(6)} \\leq 1$. When each of the inequalities are strict, the images of $f^{(i)}_{\\k}$ are three disconnected triangles, as illustrated in Figure (\\ref{disconnect}). When all equalities hold, the images are connected, as illustrated in Figure (\\ref{connect}). \n\t\n\t\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\t\t\\includegraphics[width=2in]{sierpinski_mod.png}\n\t\t\t\t\\caption{}\n\t\t\t\t\\label{disconnect}\n\t\t\t\\end{subfigure}\n\t\t\t~\n\t\t\n\t\t\t\\qquad \\qquad\n\t\t\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\t\t\\includegraphics[width=2in]{sierpinski_mod_2.png}\n\t\t\t\t\\caption{}\n\t\t\t\t\\label{connect}\n\t\t\t\\end{subfigure}\n\t\t\t\\caption{First generation of disconnected and connected triangles}\\label{gen1}\n\t\t\\end{figure}\n\t\t\n\tIn the case of the connected sets, the values of $\\mathbf{k}=\\left(k^{(1)}, k^{(2)}, k^{(3)}, k^{(4)}, k^{(5)}, k^{(6)}\\right) $ are determined by $k^{(1)}, k^{(2)}, k^{(3)}$ since $k^{(4)} = 1-k^{(1)}$, $k^{(5)}=1-k^{(2)}$, $k^{(6)}=1-k^{(3)}$.  In this sense, we may also view $\\k=\\left(k^{(1)}, k^{(2)}, k^{(3)}\\right)$ as a vector in $[0,1]^3 \\subseteq {\\mathbb{R}}^3$.  \n\t\n\t\n\t\n\t\tTo create the normal Sierpinski triangle, we choose\n\t\\begin{equation}\n\tE_0 = \\begin{bmatrix}\n\t-1\/2 & 1\/2 & 0\\\\\n\t0&0&\\sqrt{3}\/2\n\t\\end{bmatrix},\n\t\\label{e0sierp}\n\t\\end{equation}\n\tthe equilateral triangle of unit side length, and $\\k_{\\ell} \\in \\mathcal{M}$ to be the constant sequence $\\k_{\\ell} =\\k=(1\/2, 1\/2, 1\/2, 1\/2, 1\/2, 1\/2)$ so that each iteration maps a triangle to three triangles of half the side length with the desired translation.  In this case the ${\\mathcal F}$-limit set generated by $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}, E_0)$ corresponds to the standard Sierpinski Triangle.  \n\t\n\tTo generate Sierpinski-like fractals, we now adjust the values of the marking parameters $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}$.  For each $\\mathbf{k}=(k^{(1)}, k^{(2)},\\cdots, k^{(6)})\\in \\mathcal{M}$ and $1\\le i \\le 3$, \n\t\\begin{eqnarray*}\n\t\tU\\left(f_{\\mathbf{k}}^{(i)}\\right)=\\sup_{(A,B,C)\\in \\mathcal{X}}\\frac{diam \\left(f_{\\mathbf{k}}^{(i)}(A,B,C)\\right)}{diam\\left((A,B,C)\\right)} =\\max\\left \\{k^{(2i-1)}, k^{(2i)} \\right\\},\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\t\tL\\left(f_{\\mathbf{k}}^{(i)} \\right)=\\inf_{(A,B,C)\\in \\mathcal{X}}\\frac{diam \\left(f_{\\mathbf{k}}^{(i)}(A,B,C)\\right)}{diam\\left((A,B,C)\\right)} =\\min \\left \\{k^{(2i-1)}, k^{(2i)}\\right \\}.\n\t\\end{eqnarray*}\n\t\nWhen $\\k$ is bounded, i.e. if $\\lambda \\le k^{(j)} \\le \\Lambda<1$ for all $j=1,\\cdots ,6$, then\n\\[\\mathbf{U_k} \\le \\mathbf{r}:=(r,\\cdots,r) \\text{ and } \\mathbf{L_k}\\ge \\mathbf{s}:=(s,\\cdots,s),\\]\nwhere $r=\\max\\{1-\\lambda, \\Lambda\\}$ and $s=\\min\\{1-\\lambda, \\Lambda\\}$.\n\t\t\n\t\tFollowing our general process, we construct some random Sierpinski-like sets by introducing randomness into the choice of the sequence $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}$.\n\t\t\\begin{example}\\label{example: 4}\n\t\t\tLet $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty} =\\left \\{\\left (k_\\ell^{(1)}, k_\\ell^{(2)}, k_\\ell^{(3)} \\right )\\right \\}_{\\ell=0}^{\\infty}$ be a sequence in $[0,1]^3$ with each $k_{\\ell}^{(i)}$ a random number between given numbers $\\lambda$ and $\\Lambda$ for each $i=1,2,3$.  Let $F$ be the ${\\mathcal F}$-limit set generated by $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}, E_0)$.  Then the $6^{th}$ generation of the construction results in images like Figure \\ref{rand1}. Here, in Figure \\ref{rand1a}, $\\lambda=\\frac{1}{4}$ and $\\Lambda=\\frac{3}{4}$;  while in Figure \\ref{rand1b}, $\\lambda=0.45$ and $\\Lambda=0.55$. Note that the sets are no longer self-similar.\n\t\t\\end{example}\n\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\t\\begin{subfigure}[b]{0.42\\textwidth}\n\t\t\t\t\\includegraphics[width=2.5in]{sierp_rand_4.png}\n\t\t\t\t\\caption{Each $ k_{\\ell}^{(i)}$ is random in $[\\frac{1}{4}, \\frac{3}{4}]$.}\n\t\t\t\t\\label{rand1a}\n\t\t\t\\end{subfigure}\n\t\t\t~\n\t\t\t\n\t\t\n\t\t\t  \\qquad \\qquad\n\t\t\t\\begin{subfigure}[b]{0.42\\textwidth}\n\t\t\t\t\\includegraphics[width=2.5in]{2dim_example1.png}\n\t\t\t\t\\caption{Each $ k_{\\ell}^{(i)}$ is random in $[0.45, 0.55]$.}\n\t\t\t\t\\label{rand1b}\n\t\t\t\\end{subfigure}\n\t\t\t\\caption{Generation 6 of Random Sierpinski triangle}\\label{rand1}\n\t\t\\end{figure}\n\n\t\t\n\t\tIn Figure \\ref{rand1b}, we pick $\\lambda=0.45$ and $\\Lambda=0.55$. By Corollary \\ref{cor:uniform_bounds},\n\t\t\\[\\frac{\\log(m)}{-\\log(s)}\\le \\dim_H(F) \\le \\frac{\\log(m)}{-\\log(r)},\\]\n\t\twhere $m=3$, $r=0.55$ and $s=0.45$. That is,\n\t\t\\[1.3758 \\le \\dim_H(F)\\le 1.8377.\\]\n\t\t\n\t\t\n\t\t\\begin{example}\\label{example: 5}\n\t\t\t\n\t\t\tAs in Example \\ref{example: 4}, but replacing $E_0$ with  $\\tilde{E_0} = \\begin{bmatrix} 0 & 1 & 0\\\\0 & 0&1\\end{bmatrix}$, the $7^{th}$ generation of the construction results in an image like Figure \\ref{rand2}, when $\\lambda=\\frac{1}{4}$ and $\\Lambda=\\frac{3}{4}$.\n\t\t\n\t\t\\end{example}\n\t\t\t\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\n\t\t\t\\includegraphics[width=3in]{sierp_right_3.png} \n\t\t\t\\caption{Generation 7 of a Random Sierpinski triangle}\n\t\t\t\\label{rand2}\n\t\t\\end{figure}\t\n\n\t\t\\begin{example}\\label{example: 5new}  \n\t\t\tFor each $\\ell=0,1, \\cdots$, let $\\mathbf{k}_\\ell=\\left(k_{\\ell}^{(1)}, k_{\\ell}^{(2)}, \\cdots, k_{\\ell}^{(6)}\\right)$ where\n\t\t\t\\begin{eqnarray*}\n\t\t\t\tk_{\\ell}^{(1)} &=&\\frac{1}{2}+\\frac{a_\\ell}{\\sqrt{\\ell+1}},\\quad  k_{\\ell}^{(2)}=1-k_{\\ell}^{(1)},\\\\\n\t\t\t\tk_{\\ell}^{(3)} &=& \\frac{1}{2}+\\frac{b_\\ell}{\\sqrt{\\ell+1}}, \\quad k_{\\ell}^{(4)}=1-k_{\\ell}^{(3)},\\\\\n\t\t\t\tk_{\\ell}^{(5)} &=& \\frac{1}{2}+\\frac{c_\\ell}{\\ell+1}, \\quad k_{\\ell}^{(6)}=1-k_{\\ell}^{(5)}.\n\t\t\t\\end{eqnarray*}\n\t\t\tfor random numbers $a_\\ell, b_\\ell, c_\\ell \\in [-\\frac{1}{3}, \\frac{1}{3}]$.  Let $F$ be the ${\\mathcal F}$-limit set $F$ generated by $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}, E_0)$.\n\t\t\tThen the seventh generation of the construction of $F$ results in an image like Figure \\ref{rand3}. \n\t\t\t\n\t\t\t\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\n\t\t\t\\includegraphics[width=3in]{sierp_controlled.png} \n\t\t\t\\caption{Generation 6 of a Sierpinski-type triangle with controlled dimension}\n\t\t\t\\label{rand3}\n\t\t\\end{figure}\n\t\t\n\t\t\tIn this case, we can calculate the exact value of the Hausdorff dimension of $F$.  Indeed, by Corollary \\ref{cor:limit_sup_inf},\n\t\t\t\\[\\lim_{\\ell \\rightarrow \\infty} (||\\mathbf{U}_{\\mathbf{k}_\\ell}||_s)^s=\\frac{3}{2^s}=\\lim_{\\ell \\rightarrow \\infty} (||\\mathbf{L}_{\\mathbf{k}_\\ell}||_s)^s.\\]\n\t\t\tThus, $\\dim_H(F)=\\frac{\\log(3)}{\\log(2)}$.\n\t\t\\end{example}\n\n\t\\end{subsection}\n\t\n\t\\begin{subsection}{Menger Sponge}\n\t\t\n\tLet \\begin{equation} \\mathcal{X} = \\left \\{ (O,A,B,C) | O,A,B,C \\in {\\mathbb{R}}^3 \\right \\} \\label{xdefinemeng} \\end{equation} representing the collection of all rectangular prisms $(OABC)$ in ${\\mathbb{R}}^3$, $m=20,$ and\n\t\t$$\\mathcal{M} = \\left \\{ \\left(k^{(1)}, k^{(2)}, k^{(3)}, k^{(4)}, k^{(5)}, k^{(6)}\\right) \\in [0,1]^6 :  k^{(1)} \\leq k^{(2)},  k^{(3)} \\leq k^{(4)}, k^{(5)} \\leq k^{(6)} \\right \\}.$$\n\t  For each $\\k \\in \\mathcal{M}$ and $i =1, 2, \\dots , 20$, we can define affine transformations $f^{(i)}_{\\k} : \\mathcal{X} \\to \\mathcal{X}$ as follows.  \n\t\t\n\t\tFor any $\\k=(k^{(1)}, k^{(2)}, k^{(3)}, k^{(4)}, k^{(5)}, k^{(6)}) \\in \\mathcal{M}$, define  \n\t\t$$ T = \\begin{bmatrix} 0& k^{(1)}& k^{(2)}& 1 \\end{bmatrix}, \\quad R= \\begin{bmatrix} 0& k^{(3)}& k^{(4)}& 1 \\end{bmatrix}, \\quad S= \\begin{bmatrix} 0& k^{(5)}& k^{(6)}& 1 \\end{bmatrix}.$$\n\t\tLet\n\t\t\\begin{eqnarray*}\n\t\t\tI&=& \\{ (a,b,c) | 1 \\leq a,b,c\\leq3 \\text{ with } a,b,c \\in {\\mathbb{Z}}, \\text{and no two of $a,b,c$ equal to 2} \\}.\n\t\t\\end{eqnarray*}\n\t\tFor each $(a,b,c) \\in I$ and $\\k \\in \\mathcal{M}$, define \n\t\t\n\t\t\\begin{equation*}\n\t\t\tM_{\\k}(a,b,c)=\\begin{bmatrix} \n\t\t\t\t1-( T(a) + R(b) +S(c)) & T(a) & R(b) & S(c)\\\\\n\t\t\t\t1-( T(a+1) + R(b) +S(c)) & T(a+1) & R(b) & S(c)\\\\\n\t\t\t\t1-( T(a) + R(b+1) +S(c)) & T(a) & R(b+1) & S(c)\\\\\n\t\t\t\t1-( T(a) + R(b) +S(c+1)) & T(a) & R(b) & S(c+1)\\\\\n\t\t\t\\end{bmatrix}.\n\t\t\\end{equation*}\n\t\t\n\t\tNote that the set $I$ contains 20 elements, so we can express it as $$I = \\{ (a_i, b_i, c_i) | 1 \\leq i \\leq 20\\}.$$\n\t\t\n\t\tFor each $\\k \\in \\mathcal{M}$ and $1 \\leq i \\leq 20$, we consider the affine transformation $f^{(i)}_{\\k} : \\mathcal{X} \\to \\mathcal{X}$ given by\n\t\t\\begin{equation} \n\t\t\tf^{(i)}_{\\k} (O,A,B,C)  = M_{\\k} (a_i,b_i,c_i)  \\begin{bmatrix} O\\\\A\\\\B\\{\\mathcal C}\\end{bmatrix} \n\t\t\t\\label{fdefinemeng_f} \n\t\t\\end{equation}\n\t\t\n\t\tfor every $(O, A, B, C) \\in \\mathcal{X}$.  \n\t\t\n\t\tNote that for $i=1, \\dots, 20$ and $\\k \\in \\mathcal{M}$, $f^{(i)}_{\\k}$ is a compression.   Thus, we can define a marking ${\\mathcal F}$  by\n\t\t\t\\begin{eqnarray*}\n\t\t\t{\\mathcal F} : &\\mathcal{M} &\\to \\mathcal{C}_{20}(\\mathcal{X}) \\\\\n\t\t\t&\\k   &\\mapsto  f_{\\k}= (f^{(1)}_{\\k}, \\dots , f^{(20)}_{\\k}).\n\t\t\\end{eqnarray*}\n\t\n\t\tUsing this, for any starting rectangular prism $E_0 = (O,A,B,C) \\in \\mathcal{X}$, we can generate a sequence of sets that follows a similar construction to the Menger Sponge.\n\t\t\n\t\t\\begin{example}\\label{example: 7}\n\t\t\t\\label{standardmeng}\n\t\t  Let\n\t\t\t\n\t\t\t\\begin{equation}\n\t\t\t\tE_0 = \\begin{bmatrix}\n\t\t\t\t\t0 & 1 & 0 & 0\\\\\n\t\t\t\t\t0&0&1&0\\\\\n\t\t\t\t\t0&0&0&1\n\t\t\t\t\\end{bmatrix}\n\t\t\t\t\\label{e0meng}\n\t\t\t\\end{equation}\n\t\t\tbe the cube of unit side length and choose $\\k_{\\ell} \\in \\mathcal{M}$ to be the constant sequence $\\k_{\\ell} =\\k=(1\/3, 2\/3, 1\/3, 2\/3, 1\/3, 2\/3)$.  Then the ${\\mathcal F}$-limit set $F$ generated by the triple $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell =0}^{\\infty}, E_0 )$ is the classical Menger sponge.  \\end{example}\n\t\t\n\t\tNow we consider variations of Menger Sponge. For each $\\mathbf{k}=(k^{(1)}, k^{(2)}, \\cdots, k^{(6)})\\in \\mathcal{M}$ and $1\\le i \\le 20$, \n\t\t\\begin{eqnarray*}\n\t\t\tU\\left(f_{\\mathbf{k}}^{(i)}\\right)&=&\\sup_{(O,A,B,C)\\in \\mathcal{X}}\\frac{diam \\left(f_{\\mathbf{k}}^{(i)}(O,A,B,C)\\right)}{diam\\left((O,A,B,C)\\right)} \\\\\n\t\t\t&=&\\sup_{(O,A,B,C)\\in \\mathcal{X}}\\frac{diam \\left(M_{\\mathbf{k}}(a_i, b_i, c_i)[O,A,B,C]'\\right)}{diam\\left((O,A,B,C)\\right)} \\\\\n\t\t\t&=&\\max\\{T(a_{i+1})-T(a_i), R(b_{i+1})-R(b_i), S(c_{i+1})-S(c_i)\\}.\n\t\t\\end{eqnarray*}\n\t\tSimilarly,\n\t\t\\begin{eqnarray*}\n\t\t\tL\\left(f_{\\mathbf{k}}^{(i)}\\right)=\\min\\{T(a_{i+1})-T(a_i), R(b_{i+1})-R(b_i), S(c_{i+1})-S(c_i)\\}.\n\t\t\\end{eqnarray*}\n\t\tWhen $k^{(2j)}=1-k^{(2j-1)}$ for each $j=1, 2, 3$, it is easy to check that\n\t\t\\begin{eqnarray*}\n\t\t\t\\sum_{i=1}^{20}U(f_{\\mathbf{k}}^{(i)})^s&=&\\sum_{i=1}^{20}\\max\\{T(a_{i+1})-T(a_i), R(b_{i+1})-R(b_i), S(c_{i+1})-S(c_i)\\}^s\\\\\n\t\t\t&=&8\\max\\{k^{(1)}, k^{(3)}, k^{(5)}\\}^s+4\\max\\{1-2k^{(1)}, k^{(3)}, k^{(5)}\\}^s  \\\\\n\t\t\t& & +4\\max\\{k^{(1)}, 1-2k^{(3)}, k^{(5)}\\}^s +4\\max\\{k^{(1)}, k^{(3)}, 1-2k^{(5)}\\}^s.\n\t\t\\end{eqnarray*}\n\t\t\n\t\t\n\t\t\\begin{example}\\label{example: 8}\n\t\t Let $$\\tilde{E_0} = \\begin{bmatrix} 0& 3 & 0 & 0\\\\ 0 &0&1&0\\\\0&0&0&2\\end{bmatrix}.$$ Let $\\left(k^{(1)}, k^{(2)}, k^{(3)}, k^{(4)}, k^{(5)}, k^{(6)}\\right) \\in \\mathcal{M}$ where each $k^{(i)}$ is a random number in $[0,1]$, but still satisfying the condition $k^{(1)} \\leq k^{(2)}, k^{(3)} \\leq k^{(4)}, k^{(5)} \\leq k^{(6)}$.  Then the first generation $E^{(1)}$ of the construction results in a set like Figure \\ref{meng1}.  \n\t\t\t\\begin{figure}[h]\n\t\t\t\t\\centering\n\t\t\t\t\\includegraphics[width=2.5in]{gen_1_6.png}\n\t\t\t\t\\caption{First generation of a randomly generated Menger sponge}\n\t\t\t\t\\label{meng1}\n\t\t\t\\end{figure}\n\t\t\\end{example}\n\t\t\n\t\t\\begin{example}\\label{example: 9}\n\t\t\t  Let $\\k_{\\ell} =\\left(k_\\ell^{(1)}, k_\\ell^{(2)}, k_\\ell^{(3)}, k_\\ell^{(4)}, k_\\ell^{(5)}, k_\\ell^{(6)}\\right) \\in \\mathcal{M}$ with each $k_\\ell^{(2j-1)}$ a random number between given parameters $\\lambda$ and $\\Lambda$ and $k_\\ell^{(2j)}=1-k_\\ell^{(2j-1)}$ for each $j=1,2,3$.  Let $F$ be the ${\\mathcal F}$-limit set generated by $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell =0}^{\\infty}, E_0)$. Then the third iteration of the construction of $F$ results in images like Figure \\ref{meng3}. Here, in Figure \\ref{meng3a} the parameters $\\lambda=0$ and $\\Lambda=\\frac{1}{2}$, while in Figure \\ref{meng3b} the parameters $\\lambda=0.32$ and $\\Lambda=0.35$.\n\t\t\t\n\t\t\t\n\t\t\t\\begin{figure}[h]\n\t\t\t\t\\centering\n\t\t\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\t\t\\includegraphics[width=3.6in]{random_3rd_generation.png}\n\t\t\t\t\t\\caption{$\\lambda=0, \\quad \\Lambda= \\frac{1}{2}$}\n\t\t\t\t\t\\label{meng3a}\n\t\t\t\t\\end{subfigure}\n\t\t\t\n\t\t\t\t~\n\t\t\t\n\t\t\t\t  \\qquad \\qquad\n\t\t\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\t\t\\includegraphics[width=2.5in]{3dim_example4.png}\n\t\t\t\t\t\\caption{$\\lambda=0.32, \\Lambda= 0.35$}\n\t\t\t\t\t\\label{meng3b}\n\t\t\t\t\\end{subfigure}\n\t\t\t\t\\caption{Generation 3 of random Menger sponge}\\label{meng3}\n\t\t\t\\end{figure}\n\t\t\\end{example}\n\t\t\n\t\tWe now calculate the dimension of the limit fractal $F$ illustrated by Figure \\ref{meng3b} in Example \\ref{example: 9}.\n\t\tNote that in general, when $\\lambda \\le k^{(2j-1)} \\le \\Lambda$ for each $j=1, 2, 3$, it follows that\n\t\t\\[\n\t\t(||\\mathbf{U_k}||_s)^s=\\sum_{i=1}^{20}U\\left(f_{\\mathbf{k}}^{(i)}\\right)^s \\le 8\\Lambda^s +12\\max\\{1-2\\lambda, \\Lambda\\}^s.\n\t\t\\]\n\t\tSimilarly,\n\t\t\\begin{eqnarray*}\n\t\t\t(||\\mathbf{L_k}||_s)^s \\ge 8\\lambda^s +12\\min\\{1-2\\Lambda, \\lambda\\}^s. \n\t\t\\end{eqnarray*}\n\t\tIn particular, when $\\lambda=0.32$ and $\\Lambda=0.35$,  for any $s>2.901$,\n\t\t\\begin{eqnarray*}\n\t\t\t(||\\mathbf{U_k}||_s)^s &\\le & 8\\Lambda^s+12\\max\\{1-2\\lambda, \\Lambda\\}^s \\le 8*0.35^s+12*0.36^s \\\\\n\t\t\t&<& 8*0.35^{2.901}+12*0.36^{2.901}\\approx 1.000.\n\t\t\\end{eqnarray*}\n\t\tBy Theorem \\ref{thm:ratio_bounds}, $\\dim_H(F)\\le 2.901$. Similarly, for any $s \\le 2.546$,\n\t\t\\begin{eqnarray*}\n\t\t\t(||\\mathbf{L_k}||_s)^s\n\t\t\t&\\ge & 8\\lambda^s+12\\min\\{1-2\\Lambda, \\lambda\\}^s \\\\\n\t\t\t&\\ge & 8*0.32^s+12*0.3^s \\ge 8*0.32^{2.546}+12*0.3^{2.546}\\approx 1.000.\n\t\t\\end{eqnarray*}\n\t\tBy Theorem \\ref{thm:ratio_bounds} again, $\\dim_H(F)\\ge 2.546$. As a result,\n\t\t\\[2.546 \\le \\dim_H(F)\\le 2.901.\\]\n\t\t\n\t\t\n\t\t\\begin{example}\n\t\t\tFor each $\\ell \\ge 0$, let $\\mathbf{k}_\\ell=\\left(k_{\\ell}^{(1)}, k_{\\ell}^{(2)}, \\cdots, k_{\\ell}^{(6)}\\right)$ where\n\t\t\t\\begin{eqnarray*}\n\t\t\t\tk_{\\ell}^{(1)} &=&\\frac{1}{3}+\\frac{(-1)^\\ell}{12(\\ell+1)},\\quad  k_{\\ell}^{(2)}=1-k_{\\ell}^{(1)},\\\\\n\t\t\t\tk_{\\ell}^{(3)} &=& \\frac{1}{3}-\\frac{(-1)^\\ell}{6(\\ell+1)}, \\quad k_{\\ell}^{(4)}=1-k_{\\ell}^{(3)},\\\\\n\t\t\t\tk_{\\ell}^{(5)} &=& \\frac{1}{3}+\\frac{(-1)^\\ell}{18(\\ell+1)}, \\quad k_{\\ell}^{(6)}=1-k_{\\ell}^{(5)}.\n\t\t\t\\end{eqnarray*}\n\t\tLet $F$ be the ${\\mathcal F}$-limit set generated by $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell =0}^{\\infty}, E_0)$.  Then the third generation of the construction of $F$ leads to an image like Figure \\ref{meng4}.\n\t\t\t\n\t\t\t\\begin{figure}[h] \n\t\t\t\t\\centering\n\t\t\t\t\\includegraphics[width=3.5in]{3dim_example3.png} \n\t\t\t\t\\caption{Generation 3 of random Menger sponge with controlled dimension}\n\t\t\t\t\\label{meng4}\n\t\t\t\\end{figure}\n\t\t\\end{example}\n\t\t\n\t\tIn this case, we can still calculate the exact Hasudorff dimension of $F$. By direct computation, \n\t\t\\[\\lim_{\\ell \\rightarrow \\infty} (||\\mathbf{U}_{\\mathbf{k}_\\ell}||_s)^s=\\frac{20}{3^s}=\\lim_{\\ell \\rightarrow \\infty} (||\\mathbf{L}_{\\mathbf{k}_\\ell}||_s)^s.\\]\n\t\tThus, by Corollary \\ref{cor:limit_sup_inf}, $\\dim_H(F)=\\frac{\\log(20)}{\\log(3)} \\approx 2.7268$.\n\t\\end{subsection}\n\t\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nReaction-diffusion equations of the form\n\\begin{equation}\\label{eqn-classical}\nu_{t}=u_{xx}+f(t,x,u),\\quad (t,x)\\in\\mathbb{R}\\times\\mathbb{R}\n\\end{equation}\nare widely used to model diffusive systems in applied sciences.\nThe nonlinearity $f$ arising from many diffusive systems in biology or physics possesses two zeros representing two states, say $0$ and $1$, that is, $f(t,x,0)=f(t,x,1)=0$ for all $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$.\nSince the pioneering works of Fish (see \\cite{Fish37}) and Kolmogorov, Petrowsky and Piscunov (see \\cite{KPP37}) on the traveling waves of \\eqref{eqn-classical} connecting the two constant states, i.e., $u\\equiv 0$ and $u\\equiv 1$, in the case $f(t,x,u)=u(1-u)$, a vast amount of literature has been carried out to the understanding of front-like solutions connecting $u\\equiv 1$ and $u\\equiv 0$ in such an equation and its generalized forms. We refer to \\cite{ArWe75,ArWe78,FiMc77,FiMc80,Kam76,Uch78,Xi00} and references therein for works in homogeneous media, i.e., $f(t,x,u)=f(u)$. Recently, there is a lot of progress concerning \\eqref{eqn-classical} in heterogeneous media. We refer to \\cite{BeHa02,DiHaZh14,MeRoSi10,MNRR09,Na14,NoRy09,NRRZ12,TaZhZl14,We02,Zl12} and references therein for works in space heterogeneous media, i.e., $f(t,x,u)=f(x,u)$, and to \\cite{AlBaCh99,NaRo12,Sh99-1,Sh99-2,Sh06,Sh11,ShSh14,ShSh14-1} and references therein for works in time heterogeneous media, i.e., $f(t,x,u)=f(t,x)$. There are also some works in space-time heterogeneous media (see e.g. \\cite{KoSh14,LiZh1,LiZh2,Na09,Na10, Sh04,Sh11-1,We02}), but it remains widely open.\n\n\nWhen using equation \\eqref{eqn-classical} to model a diffusive system in applied sciences, it is implicitly assumed that the internal\ninteraction range of organisms in the system is infinitesimal  and that the internal dispersal can be described by random walk.\n However, in practice,  a diffusive system may exhibit long range internal interaction. Equation \\eqref{eqn-classical} is then no long suitable to model such a system. More precisely, the random dispersal operator $\\partial_{xx}$ is no long suitable. As a substitute, the nonlocal dispersal operator is introduced (see e.g. \\cite{Fi03,GHHMV05} for some background) and we are now concerned with the following integral equation\n\\begin{equation}\\label{main-eqn}\nu_{t}=J\\ast u-u+f(t,x,u),\\quad (t,x)\\in\\mathbb{R}\\times\\mathbb{R},\n\\end{equation}\nwhere $J$ is a convolution kernel and $[J\\ast u](x)=\\int_{\\mathbb{R}}J(x-y)u(y)dy=\\int_{\\mathbb{R}}J(y)u(x-y)dy$.\nThere is also a great amount of research toward  the understanding of front-like solutions of \\eqref{main-eqn} connecting $u\\equiv 0$ and $u\\equiv 1$. See \\cite{BaCh06, BaCh02, BaFiReWa97,CaCh04,Ch97,Cov-thesis,CoDu05,CoDu07,Sc80} and references therein for the study in the  homogeneous case $f(t,x,u)\\equiv f(u)$.  See \\cite{BaCh99, FCh} for the study in the case that $f(t,x,u)\\equiv f(t,u)$ is periodic or almost periodic in $t$. In \\cite{CDM13,ShZh10,ShZh12-1,ShZh12-2}, the authors investigated \\eqref{main-eqn} in space periodic monostable media, i.e., $f(t,x,u)=f(x,u)$ is of monostable type and periodic in $x$, and proved the existence of spreading speeds and periodic traveling waves. In \\cite{RaShZh}, the authors studied the existence of spreading speeds and traveling waves of \\eqref{main-eqn} in space-time periodic monostable media. Very recently, both Berestycki, Coville and Vo (see \\cite{BCV14}), and Lim and Zlato\\v{s} (see \\cite{LiZl14}) investigated \\eqref{main-eqn} in space heterogeneous monostable media. While Berestycki, Coville and Vo studied principal eigenvalue, positive solution and long-time behavior of solutions, Lim and Zlato\\v{s} proved the existence of transition fronts in the sense of Berestycki-Hamel (see \\cite{BeHa07,BeHa12}). In \\cite{ShSh14-2,ShSh14-3}, the authors studied \\eqref{main-eqn} in the time heterogeneous media of ignition type, and prove the existence, regularity and stability of transition fronts.\n\n\nHowever, comparing to the classical random dispersal case, i.e., \\eqref{eqn-classical}, results concerning front propagation for \\eqref{main-eqn} are still very limited. One of the difficulties arising in the study of front propagation dynamics of \\eqref{main-eqn} is that solutions of \\eqref{main-eqn} do not become smoother as time $t$ elapses due to the fact that the semigroup generated by the nonlocal dispersal operator $u\\mapsto J\\ast u-u$ has no regularizing effect. The objective of the present paper is to investigate the space regularity of transition fronts of \\eqref{main-eqn} with various nonlinearities, including the monostable nonlinearity, the ignition nonlinearity, and the bistable nonlinearity. The results to be developed in this paper have important applications in the study of stability, uniqueness, asymptotic, etc. of transition fronts of \\eqref{main-eqn}.\n\nTo state the main results of this paper, we first introduce two standard hypotheses.\n\n\n\\begin{itemize}\n\\item[\\bf{(H1)}] $J:\\mathbb{R}\\to\\mathbb{R}$ satisfies $J\\not\\equiv0$, $J\\in C^{1}$, $J(x)=J(-x)\\geq0$ for $x\\in\\mathbb{R}$, $\\int_{\\mathbb{R}}J(x)dx=1$ and\n\\begin{equation*}\n\\int_{\\mathbb{R}}J(x)e^{\\gamma x}dx<\\infty,\\quad \\int_{\\mathbb{R}}|J'(x)|e^{\\gamma x}dx<\\infty,\\quad\\forall\\gamma\\in\\mathbb{R}.\n\\end{equation*}\n\\end{itemize}\n\n\\begin{itemize}\n\\item[\\bf{(H2)}] There exist $C^{2}$ functions $f_{B}:[0,1]\\to\\mathbb{R}$ and  $f_{M}:[0,1]\\to\\mathbb{R}$ such that\n\\begin{equation*}\nf_{B}(u)\\leq f(t,x,u)\\leq f_{M}(u),\\quad (t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1];\n\\end{equation*}\nmoreover, the following conditions hold:\n\\begin{itemize}\n\\item $f:\\mathbb{R}\\times\\mathbb{R}\\times[0,1]\\to\\mathbb{R}$ is continuous and continuously differentiable in $x$ and $u$, and satisfies\n\\begin{equation*}\n\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{x}(t,x,u)|<\\infty\\quad\\text{and}\\quad\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{u}(t,x,u)|<\\infty;\n\\end{equation*}\n\n\\item there exist $\\theta_{1}\\in(0,1)$ such that $f_{u}(t,x,u)\\leq0$ for all $(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[\\theta_{1},1]$;\n\n\\item $f_{B}$ is of standard bistable type, that is, $f_{B}(0)=f_{B}(\\theta)=f_{B}(1)=0$ for some $\\theta\\in(0,1)$, $f_{B}(u)<0$ for $u\\in(0,\\theta)$, $f_{B}(u)>0$ for $u\\in(\\theta,1)$ and $\\int_{0}^{1}f_{B}(u)du>0$; moreover, $u_{t}=J\\ast u-u+f_{B}(u)$ admits a traveling wave $\\phi_{B}(x-c_{B}t)$ with $\\phi_{B}(-\\infty)=1$, $\\phi_{B}(\\infty)=0$ and $c_{B}\\neq0$;\n\n\\item  $f_{M}$ is of standard monostable type, that is, $f_{M}(0)=f_{M}(1)=0$ and $f_{M}(u)>0$ for $u\\in(0,1)$.\n\\end{itemize}\n\\end{itemize}\n\n\nWe remark that $c_{B}$ must be positive. Observe that nonlinear functions satisfying (H2) include the monostable nonlinearity, the ignition nonlinearity, and the bistable nonlinearity satisfying the conditions below.\n\n\\begin{itemize}\n\\item[\\rm(i)] Monostable or Fisher-KPP nonlinearity.\n\n\\item[] Standard monostable nonlinearity $f(\\cdot)$: $f(0)=f(1)=0$, $f(u)>0$ for $u\\in(0,1)$, and $\\frac{f(u)}{u}$ is decreasing in $u$.\n\n\\item[] General monostable nonlinearity $f(\\cdot,\\cdot,\\cdot)$: $f_{\\min}(u)\\le f(t,x,u)\\le f_{\\max}(u)$ for $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and $u\\in[0,1]$, where $f_{\\min}(\\cdot)$ and $f_{\\max}(u)$ are two standard monostable nonlinearities, and $\\frac{f(t,x,u)}{u}$ decreasing in $u$ for any $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$.\n\n\\item[{\\rm (ii)}] Ignition nonlinearity.\n\n\\item[] Standard ignition nonlinearity $f(\\cdot)$: $f(u)=0$ for $u\\in [0,\\theta]\\cup\\{1\\}$,\n$f(u)>0$ for $u\\in(\\theta,1)$, and $f_u(1)<0$, where $\\theta\\in(0,1)$ is referred to as the {\\it ignition temperature}.\n\n\\item[] General ignition nonlinearity $f(\\cdot,\\cdot,\\cdot)$: $f_{\\min}(u)\\le f(t,x,u)\\le f_{\\max}(u)$ for $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and $u\\in[0,1]$,\nwhere $f_{\\min}(\\cdot)$ and $f_{\\max}(\\cdot)$ are two standard ignition nonlinearities.\n\n\\item[{\\rm (iii)}] Bistable nonlinearity.\n\n\\item[] Standard bistable nonlinearity $f(\\cdot)$: $f(0)=f(\\theta)=f(1)=0$, $f(u)<0$ for $u\\in(0,\\theta)$, $f(u)>0$ for $u\\in(\\theta,1)$, and $f_u(0)<0$, $f_u(1)<0$, $f_u(\\theta)>0$,\nwhere $\\theta\\in(0,1)$.\n\n\\item[] General bistable nonlinearity $f(\\cdot,\\cdot,\\cdot)$: $f_{\\min}(u)\\le f(t,x,u)\\le f_{\\max}(u)$ for $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and $u\\in [0,1]$, where $f_{\\min}(\\cdot)$ and $f_{\\max}(\\cdot)$ are two standard bistable nonlinearities with $\\int_{0}^{1}f_{\\min}(u)du>0$ and $u_{t}=J\\ast u-u+f_{\\min}(u)$ having traveling waves with nonzero speed.\n \\end{itemize}\n\n\nWe remark that (H2) can also be applied to a general bistable nonlinearity $f(t,x,u)$ with $\\int_0^1 f_{\\max}(u)du<0$ and $u_{t}=J\\ast u-u+f_{\\max}(u)$ having traveling waves with nonzero speed. In fact, let $v(t,x)=1-u(t,x)$. Then, $v(t,x)$ satisfies\n$$\nv_t=J\\ast v-v+\\tilde f(t,x,v),\\quad (t,x)\\in\\mathbb{R}\\times\\mathbb{R},\n$$\nwhere $\\tilde f(t,x,v)=-f(t,x,1-v)$. Hence\n$$\n\\tilde f_{\\min}(v)\\le \\tilde f(t,x,v)\\le \\tilde f_{\\max}(v),\\quad (t,x,v)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]\n$$\nwhere $\\tilde f_{\\min}(v)=-f_{\\max}(1-v)$ and $\\tilde f_{\\max}(v)=-f_{\\min}(1-v)$.\nClearly, $\\tilde f_{\\min}(\\cdot)$ and $\\tilde f_{\\max}(\\cdot)$ are two standard bistable nonlinearities and\n$\\int_0^1 \\tilde f_{\\min}(v)dv>0$ and $u_{t}=J\\ast u-u+\\tilde{f}_{\\min}(u)$ admits traveling waves with nonzero speed.\n\nWe also remark that, besides monostable, ignition, and bistable nonlinearities, nonlinear functions satisfying (H2) include those having more than one zeros between $0$ and $1$.\n\n\nNext, we recall the definition of transition fronts of \\eqref{main-eqn} connecting $u\\equiv 0$ and $u\\equiv 1$.\n\n\\begin{defn}\\label{transition-front-defn}\nSuppose that $f(t,x,0)=f(t,x,1)=0$ for all $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$. A global-in-time solution $u(t,x)$ of \\eqref{main-eqn} is called a (right-moving) {\\rm transition front} (connecting $0$ and $1$) in the sense of Berestycki-Hamel (see \\cite{BeHa07,BeHa12}, also see \\cite{Sh99-1,Sh99-2}) if $u(t,x)\\in(0,1)$ for all $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and there exists a function $X:\\mathbb{R}\\to\\mathbb{R}$, called {\\rm interface location function}, such that\n\\begin{equation*}\n\\lim_{x\\to-\\infty}u(t,x+X(t))=1\\,\\,\\text{and}\\,\\,\\lim_{x\\to\\infty}u(t,x+X(t))=0\\,\\,\\text{uniformly in}\\,\\,t\\in\\mathbb{R}.\n\\end{equation*}\n\\end{defn}\n\nThe notion of a transition front is a proper generalization of a traveling wave in homogeneous media and a periodic (or pulsating) traveling wave in periodic media. The interface location function $X(t)$ tells the position of the transition front $u(t,x)$ as time $t$ elapses, while the uniform-in-$t$ limits (the essential property in the definition) shows the \\textit{bounded interface width}, that is,\n\\begin{equation*}\n\\forall\\,\\,0<\\epsilon_{1}\\leq\\epsilon_{2}<1,\\quad\\sup_{t\\in\\mathbb{R}}{\\rm diam}\\{x\\in\\mathbb{R}|\\epsilon_{1}\\leq u(t,x)\\leq\\epsilon_{2}\\}<\\infty.\n\\end{equation*}\nNotice, if $\\xi(t)$ is a bounded function, then $X(t)+\\xi(t)$ is also an interface location function. Thus, interface location function is not unique. But, it is easy to check that if $Y(t)$ is another interface location function, then $X(t)-Y(t)$ is a bounded function. Hence, interface location functions are unique up to addition by bounded functions.\n\nWe see that neither the definition nor the equation \\eqref{main-eqn} guarantees any space regularity of transition fronts. In fact, there is even no guarantee that a transition front $u(t,x)$ of \\eqref{main-eqn} is continuous in $x$ (we refer to \\cite{BaFiReWa97} for the existence of discontinuous traveling waves in the bistable case). This lack of space regularity indeed causes a lot of troubles in studying transition fronts because $\\rm(i)$ space regularity of approximating solutions is required to ensure the convergence to transition fronts; $\\rm(ii)$ space regularity of transition fronts lays the foundation for applying various techniques for reaction-diffusion equations to nonlocal equations, and hence, for further studying various qualitative properties such as stability and uniqueness. Hence, it is very important to study the space regularity of special solutions.\n\nIn the present paper, we intend to establish some general criteria concerning the space regularity of transition fronts of \\eqref{main-eqn}. More precisely, we want to know whether a transition front $u(t,x)$ of \\eqref{main-eqn} is continuously differentiable in $x$. To state our first result, we further introduce the following hypothesis.\n\n\\begin{itemize}\n\\item[\\bf{(H3)}] There exist $\\theta_{0}\\in(0,\\theta_{1})$ and $\\kappa_{0}>0$ such that\n\\begin{equation*}\nf_{u}(t,x,u)\\leq1-\\kappa_{0},\\quad (t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,\\theta_{0}].\n\\end{equation*}\n\\end{itemize}\n\nWe prove\n\n\\begin{thm}\\label{thm-regularity-of-tf}\nSuppose (H1)-(H3). Let $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn}. Then, $u(t,x)$ is regular in space in the following sense:  for any $t\\in\\mathbb{R}$, $u(t,x)$ is continuously differentiable in $x$ and satisfies $\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}|u_{x}(t,x)|<\\infty$.\n\\end{thm}\n\nWe remark that Theorem \\ref{thm-regularity-of-tf} can be proven essentially due to the assumptions $\\int_{0}^{1}f_{B}(u)du>0$ and the existence of traveling waves of \n\\begin{equation}\\label{bistable-tw-eqn-000000}\nu_{t}=J\\ast u-u+f_{B}(u)\n\\end{equation}\nwith nonzero speed in (H2), which corresponds to the unbalanced case in the bistable case. If we drop these assumptions, then, in the bistable case, there is no hope that Theorem \\ref{thm-regularity-of-tf} is true without additional assumptions on $f(t,x,u)$, since discontinuous traveling waves of \\eqref{bistable-tw-eqn-000000} with zero speed were constructed in \\cite{BaFiReWa97}, where necessary conditions for the existence of discontinuous traveling waves are also given. Thus, our results are kind of sharp and are compatible with the results for traveling waves in the monostable case, the ignition case and the unbalanced bistable case. It is worthwhile to point out that in the case \\eqref{bistable-tw-eqn-000000} admits discontinuous traveling waves, it may also admits non-monotone waves (see \\cite[Theorem 5.4]{Ch97}).\n\nAlthough the assumptions on $f_{B}$ are automatically true in the monostable case and the ignition case, it does cause some restrictions in the bistable case, and hence, there remains an interesting problem, that is, whether transition fronts in the bistable case are regular in space when \\eqref{bistable-tw-eqn-000000} admits only smooth stationary waves, i.e., smooth traveling waves with speed zero. \n\nThe proof of Theorem \\ref{thm-regularity-of-tf} is based on the rightward propagation estimate of transition fronts and the analysis of the  growth and the  decay of $\\frac{u(t,x+\\eta)-u(t,x)}{\\eta}$. The rightward propagation estimate reads as\n\\begin{equation}\\label{right-propagation-intro}\nX(t)-X(t_{0})\\geq c_{1}(t-t_{0}-T_{1}),\\quad t\\geq t_{0}\n\\end{equation}\nfor some $c_{1}>0$ and $T_{1}>0$, which is established in Theorem \\ref{lem-propagation-estimate} (to show \\eqref{right-propagation-intro},  we need $\\int_{0}^{1}f_{B}(u)du>0$ and the existence of traveling waves of \\eqref{bistable-tw-eqn-000000} with nonzero speed). To control the behavior of $\\frac{u(t,x+\\eta)-u(t,x)}{\\eta}$ when $x$ is close to $\\infty$, we need (H3). A key ingredient in proving Theorem \\ref{thm-regularity-of-tf}, i.e., controlling the term $\\frac{u(t,x+\\eta)-u(t,x)}{\\eta}$, is the observation that for fixed $x$, the term $\\frac{u(t,x+\\eta)-u(t,x)}{\\eta}$ can only grow for a period of time that is independent of $x$. Moreover, since we directly study transition fronts, which may not come from approximating solutions, we are lack of a priori information, which immediately causes the possible blow-up behavior of $\\frac{u(t_{0},x+\\eta)-u(t_{0},x)}{\\eta}$ as $\\eta\\to0$ at the initial time $t_{0}$. To overcome this technical difficulty, we utilize the fact that transition fronts are global-in-time, which means we can take $t_{0}$ to approach $-\\infty$ along subsequences.\n\nClearly, (H3) rules out many monostable nonlinearities, and it does not cover all Fisher-KPP nonlinearities. Our next result is trying to cover the monostable nonlinearities at the cost of putting some restrictions on transition fronts. We prove\n\n\\begin{thm}\\label{thm-regularity-of-tf-ii}\nSuppose (H1) and (H2). Suppose, in addition, that\n\\begin{equation*}\n\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{xu}(t,x,u)|<\\infty\\quad\\text{and}\\quad\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{uu}(t,x,u)|<\\infty.\n\\end{equation*}\nLet $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} satisfying\n\\begin{equation}\\label{harnack-type-intro}\nu(t,x)\\leq Ce^{r|x-y|}u(t,y),\\quad (t,x,y)\\in\\mathbb{R}\\times\\mathbb{R}\\times\\mathbb{R}\n\\end{equation}\nfor some $C>0$ and $r>0$. Then, $u(t,x)$ is regular in space in the following sense:  for any $t\\in\\mathbb{R}$, $u(t,x)$ is continuously differentiable in $x$ and satisfies $\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\frac{|u_{x}(t,x)|}{u(t,x)}<\\infty$.\n\\end{thm}\n\nThe key assumption in Theorem \\ref{thm-regularity-of-tf-ii} is the Harnack-type inequality \\eqref{harnack-type-intro}, which is the case for some transition fronts in the Fisher-KPP case (see \\cite{LiZl14,ShSh14-kpp}). The importance of \\eqref{harnack-type-intro} lies in the fact that it allows the comparison of $J\\ast u$ and $J'\\ast u$ with $u$. More precisely, by \\eqref{harnack-type-intro}, we find $\\frac{1}{C}\\int_{\\mathbb{R}}J(x)e^{-r|x|}dx\\leq\\frac{J\\ast u}{u}\\leq C\\int_{\\mathbb{R}}J(x)e^{r|x|}dx$ and similarly for $J'\\ast u$, which plays crucial roles in controlling $\\frac{u(t,x+\\eta)-u(t,x)}{\\eta u(t,x)}$.\n\nThe next result gives space regularity of transition fronts under the exact decay assumption.\n\n\\begin{thm}\\label{thm-regularity-of-tf-iii}\nSuppose (H1) and (H2). Suppose, in addition, that\n\\begin{equation*}\n\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{xu}(t,x,u)|<\\infty\\quad\\text{and}\\quad\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{uu}(t,x,u)|<\\infty.\n\\end{equation*}\nLet $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$. There exists $r_{0}>0$ such that if \n\\begin{equation}\\label{condition-exact-decay-intro}\n\\lim_{x\\to\\infty}\\frac{u(t,x+X(t))}{e^{-rx}}=1\\quad\\text{uniformly in}\\quad t\\in\\mathbb{R}\n\\end{equation}\nfor some  $r\\in(0,r_{0}]$,\nthen, $u(t,x)$ is regular in space in the following sense:  for any $t\\in\\mathbb{R}$, $u(t,x)$ is continuously differentiable in $x$ and satisfies $\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\frac{|u_{x}(t,x)|}{u(t,x)}<\\infty$.\n\\end{thm}\n\nWe remark that the exact decay assumption \\eqref{condition-exact-decay-intro} is the case for some transition fronts in the Fisher-KPP case (see \\cite{LiZl14,ShSh14-kpp}). The importance of \\eqref{condition-exact-decay-intro} lies in the fact that it allows the comparison of $J\\ast u$ and $J'\\ast u$ with $u$ near $x=\\infty$. Note that if the limit in \\eqref{condition-exact-decay-intro} is some other positive number instead of $1$, then we only need to shift the exponential function on the bottom correspondingly. The number $r_{0}$ corresponds to the possible decay rate of Fisher-KPP transition fronts, and thus, it would be interesting to determine the optimal $r_{0}$.\n\n\nThe study of space regularity of transition fronts of \\eqref{main-eqn} was initiated in \\cite{ShSh14-3}, where the space regularity of well-constructed transition fronts in time heterogeneous media of ignition type is obtained. Those well-constructed transition fronts are uniformly Lipschitz continuous in space and their interface location functions can be chosen to be continuously differentiable with uniformly positive first order derivatives. These properties, which may not be true for an arbitrary transition front, play important roles in the study of space regularity of well-constructed transition fronts in \\cite{ShSh14-3}. Our results, Theorem \\ref{thm-regularity-of-tf}, Theorem \\ref{thm-regularity-of-tf-ii} and Theorem \\ref{thm-regularity-of-tf-iii}, generalize the work in \\cite{ShSh14-3} to arbitrary transition fronts (with some additional assumptions in Theorem \\ref{thm-regularity-of-tf-ii} and Theorem \\ref{thm-regularity-of-tf-iii}) of \\eqref{main-eqn} with a large class of nonlinearities. As already shown in \\cite{ShSh14-3}, space regularity of transition fronts is of great significance in the study of stability, which together with uniqueness and asymptotic speeds, will be studied elsewhere.\n\n\nFinally, we study the existence of continuously differentiable and increasing interface location functions. As mentioned before, if $X(t)$ is an interface location function of a transition front $u(t,x)$ of \\eqref{main-eqn}, then for any bounded function $\\xi(t)$, $X(t)+\\xi(t)$ is also an interface location function of $u(t,x)$. Hence, interface location functions of a transition front are not unique and may not be continuous. But, near each interface location function, we are able to find a continuously differentiable and increasing function, as the new interface location function. This is given by the following\n\n\\begin{thm}\\label{thm-modified-interface-location}\nSuppose (H1) and (H2). Let $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$ satisfying\n\\begin{equation}\\label{upper-avg-speed}\nX(t)-X(t_{0})\\leq c_{2}(t-t_{0}+T_{2}),\\quad t\\geq t_{0}\n\\end{equation}\nfor some $c_{2}>0$ and $T_{2}>0$. Then, there are constants $0<\\tilde{c}_{\\min}\\leq\\tilde{c}_{\\max}<\\infty$ and a continuously differentiable function $\\tilde{X}(t)$ satisfying\n\\begin{equation*}\n\\tilde{c}_{\\min}\\leq\\dot{\\tilde{X}}(t)\\leq \\tilde{c}_{\\max},\\quad t\\in\\mathbb{R}\n\\end{equation*}\nsuch that\n\\begin{equation*}\n\\sup_{t\\in\\mathbb{R}}|\\tilde{X}(t)-X(t)|<\\infty.\n\\end{equation*}\nIn particular, $\\tilde{X}(t)$ is also an interface location function of $u(t,x)$.\n\\end{thm}\n\nWe see that (H1) and (H2) do not ensure the space regularity of  transition fronts. It will be clear later that the proof of Theorem \\ref{thm-modified-interface-location} does not need the space regularity of transition fronts.\n\nWe refer to $\\tilde{X}(t)$ in Theorem \\ref{thm-modified-interface-location} as the modified interface location function, which gives a better characterization of the propagation of transition fronts and has been verified to be of great technical importance in studying the stability of transition fronts in time heterogeneous media (see e.g. \\cite{ShSh14-1,ShSh14-2,ShSh14-3}). Moreover, for reaction-diffusion equations in space heterogeneous media, the rightmost interface location at some constant value continuously moves to the right (see e.g. \\cite{MeRoSi10,MNRR09,NoRy09,Zl13}). But for nonlocal equations in space heterogeneous media, the rightmost interface location at some constant value jumps in general due to the nonlocality, which makes the modified interface location function more important.\n\nThe condition \\eqref{upper-avg-speed} is a technical assumption saying a transition front moves to the right at most at linear speed in the average sense, which together with \\eqref{right-propagation-intro} allow us to find the modified interface location function. Arguing as in the proof of Theorem \\ref{lem-propagation-estimate}, we readily check that the condition \\eqref{upper-avg-speed} is always true in the bistable case. This condition can also be verified in other cases including the monostable case and the ignition case as in the following two corollaries.\n\n\\begin{cor}\\label{cor-modified-interface}\nSuppose (H1) and (H2). Let $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn}. If there exists $\\tilde{\\theta}\\in(0,\\theta)$ such that\n\\begin{equation*}\nf(t,x,u)\\leq0,\\quad(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,\\tilde{\\theta}],\n\\end{equation*}\nthen \\eqref{upper-avg-speed} is true. In particular, the conclusions of Theorem \\ref{thm-modified-interface-location} hold.\n\\end{cor}\n\nClearly, Corollary \\ref{cor-modified-interface} covers, in particular, all bistable and ignition nonlinearities, but rules out all monostable nonlinearities, which are covered by the next result.\n\n\\begin{cor}\\label{cor-modified-interface-ii}\nSuppose (H1) and (H2). Suppose, in addition, that $J$ is compactly supported. Let $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$. If there exist $r>0$ and $h>0$ such that\n\\begin{equation*}\nu(t,x+X(t))\\leq e^{-r(x-h)},\\quad (t,x)\\in\\mathbb{R}\\times\\mathbb{R},\n\\end{equation*}\nthen \\eqref{upper-avg-speed} is true. In particular, the conclusions of Theorem \\ref{thm-modified-interface-location} hold.\n\\end{cor}\n\nWe remark that in the monostable case, uniform exponential decay as $x\\to\\infty$ may be necessary for transition fronts to travel at linear speeds, since slower decay near $x=\\infty$ may cause super-linear propagation (see \\cite{HaRo10}). It is worthwhile to point out that in the bistable case, a discontinuous traveling wave may not converge to zero exponentially as $x\\to\\infty$ (see \\cite[Theorem 5.1]{BaChm99}). Here, we need $J$ to be compactly supported, since we will use results obtained in \\cite{CaCh04} and \\cite{ShZh12-2} in the proof, which were proven when $J$ is compactly supported. It should be pointed out that the arguments and results in\n\\cite{ShZh12-2} can be extended to dispersal kernels $J$ which are not compactly supported, but satisfy (H1) (see \\cite{AZh}).\n\n\nAs a direct application of the results, in particular, Theorem \\ref{thm-regularity-of-tf} and Corollary \\ref{cor-modified-interface}, obtained in the present paper, we study in \\cite{ShSh14-bistable} the equation \\eqref{main-eqn} in time heterogeneous media of general bistable type, that is, $f(t,x,u)=f(t,u)$ satisfies\n\\begin{equation*}\nf_{\\min}(u)\\leq f(t,u)\\leq f_{\\max}(u),\\quad (t,u)\\in\\mathbb{R}\\times[0,1],\n\\end{equation*}\nwhere the $C^{2}$ functions $f_{\\min}$ and $f_{\\max}$ are standard bistable nonlinearities on $[0,1]$ with the unbalanced condition $\\int_{0}^{1}f_{\\min}(u)du>0$. Provided that there is a space nonincreasing transition front, we show by means of Theorem \\ref{thm-regularity-of-tf} and Corollary \\ref{cor-modified-interface} that (i) all transition fronts are asymptotically stable and enjoy decaying estimates; (ii) transition fronts are unique up to space shifts; (iii) all transition fronts become periodic traveling waves in the periodic media and have asymptotic speeds in the uniquely ergodic media. The assumption on the existence of a space nonincreasing transition front can be verified if, for example, $f(t,u)$ is of standard bistable type in $u$ for each $t$ and their middle zeros are the same.\n\n\nThe rest of the paper is organized as follows. In Section \\ref{sec-prop-estimate}, we establish the rightward propagation estimate of transition fronts. We will see, in particular, that any transition fronts moves front left infinity to right infinity as time goes from $-\\infty$ to $\\infty$. In Section \\ref{subsec-unconditional-regularity}, we prove Theorem \\ref{thm-regularity-of-tf}. In Section \\ref{sec-proof-2}, we prove Theorem \\ref{thm-regularity-of-tf-ii}. In Section \\ref{sec-proof-iafjiaf}, we prove Theorem \\ref{thm-regularity-of-tf-iii}. In Section \\ref{sec-proof-thm-cor}, we prove Theorem \\ref{thm-modified-interface-location}, Corollary \\ref{cor-modified-interface} and Corollary \\ref{cor-modified-interface-ii}.  We end up the present paper with an appendix, Appendix \\ref{app-ig-tw}, on ignition traveling waves.\n\n\n\n\n\n\\section{Rightward propagation estimates}\\label{sec-prop-estimate}\n\nIn this section, we study the rightward propagation estimates of transition fronts. Throughout this section, we assume (H1) and (H2).\n\nIn what follows in this section, $u(t,x)$ will be an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$. For $\\lambda\\in(0,1)$, we define $X_{\\lambda}^{-}(t)$ and $X_{\\lambda}^{+}(t)$ by setting\n\\begin{equation}\\label{defn-interface-locations}\n\\begin{split}\nX_{\\lambda}^{-}(t)&=\\sup\\{x\\in\\mathbb{R}|u(t,y)>\\lambda,\\,\\,\\forall y\\leq x\\},\\\\\n X_{\\lambda}^{+}(t)&=\\inf\\{x\\in\\mathbb{R}|u(t,y)<\\lambda,\\,\\,\\forall y\\geq x\\}.\n\\end{split}\n\\end{equation}\nNote that if $u(t,x)$ is continuous in $x$, then $X_{\\lambda}^{-}(t)$ and $X_{\\lambda}^{+}(t)$ are nothing but the leftmost and rightmost interface locations at $\\lambda$. Trivially, $X_{\\lambda}^{-}(t)\\leq X_{\\lambda}^{+}(t)$ and $X_{\\lambda}^{\\pm}(t)$ are decreasing in $\\lambda$. Due to the possible discontinuity of $u(t,x)$ in $x$, it may happen that $u(t,X_{\\lambda}^{-}(t))<\\lambda$ or $u(t,X_{\\lambda}^{+}(t))>\\lambda$.\n\nFrom the definition of transition fronts, we have the following simple lemma.\n\n\\begin{lem}\\label{lem-bounded-interface-width}\nThe following statements hold:\n\\begin{itemize}\n\\item[\\rm(i)] for any $0<\\lambda_{1}\\leq\\lambda_{2}<1$, there holds $\\sup_{t\\in\\mathbb{R}}[X^{+}_{\\lambda_{1}}(t)-X^{-}_{\\lambda_{2}}(t)]<\\infty$;\n\n\\item[\\rm(ii)] for any $\\lambda\\in(0,1)$, there hold $\\sup_{t\\in\\mathbb{R}}|X(t)-X^{\\pm}_{\\lambda}(t)|<\\infty$.\n\\end{itemize}\n\\end{lem}\n\\begin{proof}\n$\\rm(i)$ By the uniform-in-$t$ limits $\\lim_{x\\to-\\infty}u(t,x+X(t))=1$ and $\\lim_{x\\to\\infty}u(t,x+X(t))=0$, there exist $x_{1}$ and $x_{2}$ such that $u(t,x+X(t))>\\lambda_{2}$ for all $x\\leq x_{2}$ and $t\\in\\mathbb{R}$, and $u(t,x+X(t))<\\lambda_{1}$ for all $x\\geq x_{1}$ and $t\\in\\mathbb{R}$. It then follows from the definition of $X_{\\lambda_{2}}^{-}(t)$ and $X_{\\lambda_{1}}^{+}(t)$ that $x_{2}+X(t)\\leq X_{\\lambda_{2}}^{-}(t)$ and $x_{1}+X(t)\\geq X_{\\lambda_{1}}^{+}(t)$ for all $t\\in\\mathbb{R}$. The result then follows.\n\n$\\rm(ii)$ Let $\\lambda_{1}=\\lambda=\\lambda_{2}$ in the proof of $\\rm(i)$, we have  $x_{2}+X(t)\\leq X_{\\lambda}^{-}(t)$ and $x_{1}+X(t)\\geq X_{\\lambda}^{+}(t)$ for all $t\\in\\mathbb{R}$. In particular,\n\\begin{equation*}\nx_{2}+X(t)\\leq X_{\\lambda}^{-}(t)\\leq X_{\\lambda}^{+}(t)\\leq x_{1}+X(t),\\quad t\\in\\mathbb{R}.\n\\end{equation*}\nThis completes the proof.\n\\end{proof}\n\nThe next result gives the rightward propagation estimate of $u(t,x)$ in terms of $X(t)$.\n\n\\begin{thm}\\label{lem-propagation-estimate}\nThere exist $c_{1}>0$ and $T_{1}>0$ such that\n\\begin{equation*}\nX(t)-X(t_{0})\\geq c_{1}(t-t_{0}-T_{1}),\\quad t\\geq t_{0}.\n\\end{equation*}\n\\end{thm}\n\\begin{proof}\nFix some $\\lambda\\in(\\theta,1)$. We  write $X^{-}(t)=X^{-}_{\\lambda}(t)$. Since $\\sup_{t\\in\\mathbb{R}}|X(t)-X^{-}(t)|<\\infty$ by Lemma \\ref{lem-bounded-interface-width}, it suffices to show\n\\begin{equation}\\label{propagation-estimate-1}\nX^{-}(t)-X^{-}(t_{0})\\geq c(t-t_{0}-T),\\quad t\\geq t_{0}\n\\end{equation}\nfor some $c>0$ and $T>0$.\n\nRecall $f_{B}$ is as in (H2). Let $(c_{B},\\phi_{B})$ with $c_{B}>0$ be the unique solution of\n\\begin{equation*}\n\\begin{cases}\nJ\\ast\\phi-\\phi+c\\phi_{x}+f_{B}(\\phi)=0,\\\\\n\\phi_{x}<0,\\,\\, \\phi(0)=\\theta,\\,\\, \\phi(-\\infty)=1\\,\\,\\text{and}\\quad\\phi(\\infty)=0\n\\end{cases}\n\\end{equation*}\n(see \\cite{BaFiReWa97} for the existence and uniqueness of $(c_B,\\phi_B)$). That is, $c_{B}$ is the unique speed and $\\phi_{B}$ is the normalized profile of traveling waves of\n\\begin{equation}\\label{eqn-bistable-homo-B}\nu_{t}=J\\ast u-u+f_{B}(u).\n\\end{equation}\n\nLet $u_{0}:\\mathbb{R}\\to[0,1]$ be a uniformly continuous and nonincreasing function satisfying\n\\begin{equation}\\label{some-special-initial-data-132184}\nu_{0}(x)=\\begin{cases}\n\\lambda,\\quad& x\\leq x_{0},\\\\\n0,\\quad & x\\geq0,\n\\end{cases}\n\\end{equation}\nwhere $x_{0}<0$ is fixed. By the definition of $X^{-}(t)$, we see that for any $t_{0}\\in\\mathbb{R}$, there holds $u(t_{0},x+X^{-}(t_{0}))\\geq u_{0}(x)$ for all $x\\in\\mathbb{R}$, and then, by $f(t,x,u)\\geq f_{B}(u)$ and the comparison principle, we find\n\\begin{equation}\\label{result-after-cp-42492}\nu(t,x+X^{-}(t_{0}))\\geq u_{B}(t-t_{0},x;u_{0}),\\quad x\\in\\mathbb{R},\\,\\,t\\geq t_{0},\n\\end{equation}\nwhere $u_{B}(t,x;u_{0})$ is the unique solution to \\eqref{eqn-bistable-homo-B} with $u_{B}(0,\\cdot;u_{0})=u_{0}$. By the choice of $u_{0}$ and the stability of bistable traveling waves (see e.g. \\cite{BaFiReWa97}), there are constants $x_{B}=x_{B}(\\lambda)\\in\\mathbb{R}$, $q_{B}=q_{B}(\\lambda)>0$ and $\\omega_{B}>0$ such that\n\\begin{equation*}\nu_{B}(t-t_{0},x;u_{0})\\geq\\phi_{B}(x-x_{B}-c_{B}(t-t_{0}))-q_{B}e^{-\\omega_{B}(t-t_{0})}, \\quad x\\in\\mathbb{R},\\,\\,t\\geq t_{0}.\n\\end{equation*}\nHence,\n\\begin{equation*}\nu(t,x+X^{-}(t_{0}))\\geq \\phi_{B}(x-x_{B}-c_{B}(t-t_{0}))-q_{B}e^{-\\omega_{B}(t-t_{0})}, \\quad x\\in\\mathbb{R},\\,\\,t\\geq t_{0}.\n\\end{equation*}\nLet $T_{0}=T_{0}(\\lambda)>0$ be such that $q_{B}e^{-\\omega_{B}T_{0}}=\\frac{1-\\lambda}{2}$ (making $q_{B}$ larger so that $q_{B}>\\frac{1-\\lambda}{2}$ if necessary) and denote by $\\xi_{B}(\\frac{1+\\lambda}{2})$ the unique point such that $\\phi_{B}(\\xi_{B}(\\frac{1+\\lambda}{2}))=\\frac{1+\\lambda}{2}$. Setting $x_{*}=x_{B}+c_{B}(t-t_{0})+\\xi_{B}(\\frac{1+\\lambda}{2})$, the monotonicity of $\\phi_{B}$ implies that for all $t\\geq t_{0}+T_{0}$ and $x\\leq x^{*}-1$\n\\begin{equation*}\n\\begin{split}\nu(t,x+X^{-}(t_{0}))&\\geq\\phi_{B}(x_{*}-1-x_{B}-c_{B}(t-t_{0}))-q_{B}e^{-\\omega_{B}T_{0}}\\\\\n&>\\phi_{B}(x_{*}-x_{B}-c_{B}(t-t_{0}))-q_{B}e^{-\\omega_{B}T_{0}}\\\\\n&=\\phi_{B}(\\xi_{B}(\\frac{1+\\lambda}{2}))-q_{B}e^{-\\omega_{B}T_{0}}=\\lambda.\n\\end{split}\n\\end{equation*}\nThis says that $x_{*}-1+X^{-}(t_{0})\\leq X^{-}(t)$ for all $t\\geq t_{0}+T_{0}$, that is,\n\\begin{equation}\\label{estimate-long-time}\nX^{-}(t)-X^{-}(t_{0})\\geq x_{B}-1+c_{B}(t-t_{0})+\\xi_{B}(\\frac{1+\\lambda}{2}),\\quad t\\geq t_{0}+T_{0}.\n\\end{equation}\n\nWe now estimate $X^{-}(t)-X^{-}(t_{0})$ for $t\\in[t_{0},t_{0}+T_{0}]$. We claim that there exists $z=z(T_{0})<0$ such that\n\\begin{equation}\\label{estimate-finite-time}\nX^{-}(t)-X^{-}(t_{0})\\geq z,\\quad t\\in[t_{0},t_{0}+T_{0}].\n\\end{equation}\nLet $u_{B}(t,x;u_{0})$ and $u_{B}(t;\\lambda):=u_{B}(t,x;\\lambda)$ be solutions of \\eqref{eqn-bistable-homo-B} with $u_{B}(0,x;u_{0})=u_{0}(x)$ and $u_{B}(0;\\lambda)=u_{B}(0,x;\\lambda)\\equiv\\lambda$, respectively. By the comparison principle, we have $u_{B}(t,x;u_{0})<u_{B}(t;\\lambda)$ for all $x\\in\\mathbb{R}$ and $t>0$, and $u_{B}(t,x;u_{0})$ is strictly decreasing in $x$ for $t>0$.\n\nWe see that for any $t>0$, $u_{B}(t,-\\infty;u_{0})=u_{B}(t;\\lambda)$. This is because that $\\frac{d}{dt}u_{B}(t,-\\infty;u_{0})=f_{B}(u_{B}(t,-\\infty;u_{0}))$ for $t>0$ and $u_{B}(0,-\\infty;u_{0})=\\lambda$. Since $\\lambda\\in(\\theta,1)$, as a solution of the ODE $u_{t}=f_{B}(u)$, $u_{B}(t;\\lambda)$ is strictly increasing in $t$, which implies that $u_{B}(t,-\\infty;u_{0})=u_{B}(t;\\lambda)>\\lambda$ for $t>0$. As a result, for any $t>0$ there exists a unique $\\xi_{B}(t)\\in\\mathbb{R}$ such that $u_{B}(t,\\xi_{B}(t);u_{0})=\\lambda$. Moreover, $\\xi_{B}(t)$ is continuous in $t$.\n\nSetting $x_{**}=\\xi_{B}(t-t_{0})$, we find $u(t,x+X^{-}(t_{0}))>\\lambda$ for all $x\\leq x_{**}$ by the monotonicity of $u_{B}(t,x;u_{0})$ in $x$, which  together with \\eqref{result-after-cp-42492}\nimplies that\n\\begin{equation*}\nX^{-}(t)\\geq x_{**}+X^{-}(t_{0})=\\xi_{B}(t-t_{0})+X^{-}(t_{0}),\\quad t>t_{0}.\n\\end{equation*}\nThus, \\eqref{estimate-finite-time} follows if $\\inf_{t\\in(t_{0},t_{0}+T_{0}]}\\xi_{B}(t-t_{0})>-\\infty$, that is,\n\\begin{equation}\\label{not-to-negative-finity}\n\\inf_{t\\in(0,T_{0}]}\\xi_{B}(t)>-\\infty.\n\\end{equation}\n\nWe now show \\eqref{not-to-negative-finity}. Since $u_{0}(x)=\\lambda$ for $x\\leq x_{0}$, continuity with respect to the initial data\n (in sup norm) implies that for any $\\epsilon>0$ there exists $\\delta>0$ such that\n\\begin{equation*}\nu_{B}(t;\\lambda)-\\lambda\\leq\\epsilon\\quad\\text{and}\\quad\\sup_{x\\leq x_{0}}[u_{B}(t;\\lambda)-u_{B}(t,x;u_{0})]=u_{B}(t;\\lambda)-u_{B}(t,x_{0};u_{0})\\leq\\epsilon\n\\end{equation*}\nfor all $t\\in[0,\\delta]$, where the equality is due to monotonicity. By (H1), $J$ concentrates near $0$ and decays very fast as $x\\to\\pm\\infty$. Thus, we can choose $x_{1}=x_{1}(\\epsilon)<<x_{0}$ such that\n\\begin{equation*}\n\\int_{-\\infty}^{x_{0}}J(x-y)dy\\geq1-\\epsilon,\\quad x\\le x_1.\n\\end{equation*}\nNow, for any $x\\leq x_{1}$ and $t\\in(0,\\delta]$, we have\n\\begin{equation*}\n\\begin{split}\n\\frac{d}{dt}u_{B}(t,x;u_{0})&=\\int_{\\mathbb{R}}J(x-y)u_{B}(t,y;u_{0})dy-u_{B}(t,x;u_{0})+f_{B}(u_{B}(t,x;u_{0}))\\\\\n&\\geq\\int_{-\\infty}^{x_{0}}J(x-y)u_{B}(t,y;u_{0})dy-u_{B}(t,x;u_{0})+f_{B}(u_{B}(t,x;u_{0}))\\\\\n&\\geq(1-\\epsilon)\\inf_{x\\leq x_{0}}u_{B}(t,x;u_{0})-u_{B}(t;\\lambda)+f_{B}(u_{B}(t,x;u_{0}))\\\\\n&=-(1-\\epsilon)\\sup_{x\\leq x_{0}}[u_{B}(t;\\lambda)-u_{B}(t,x;u_{0})]-\\epsilon u_{B}(t;\\lambda)+f_{B}(u_{B}(t,x;u_{0}))\\\\\n&\\geq-\\epsilon(1-\\epsilon)-\\epsilon(\\lambda+\\epsilon)+f_{B}(u_{B}(t,x;u_{0}))>0\n\\end{split}\n\\end{equation*}\nif we choose $\\epsilon>0$ sufficiently small, since then $f_{B}(u_{B}(t,x;u_{0}))$ is close to $f_{B}(\\lambda)$, which is positive. This simply means that $u_{B}(t,x;u_{0})>\\lambda$ for all $x\\leq x_{1}$ and $t\\in(0,\\delta]$, which implies that $\\xi_{B}(t)>x_{1}$ for $t\\in(0,\\delta]$. The continuity of $\\xi_{B}$ then leads to \\eqref{not-to-negative-finity}. This proves \\eqref{estimate-finite-time}. \\eqref{propagation-estimate-1} then follows from \\eqref{estimate-long-time} and \\eqref{estimate-finite-time}. This completes the proof.\n\\end{proof}\n\nAs a simple consequence of Theorem \\ref{lem-propagation-estimate}, we have\n\n\\begin{cor}\\label{cor-propagation-estimate}\nThere holds $X(t)\\to\\pm\\infty$ as $t\\to\\pm\\infty$. In particular,  $u(t,x)\\to1$ as $t\\to\\infty$ and $u(t,x)\\to0$ as $t\\to-\\infty$ locally uniformly in $x$.\n\\end{cor}\n\\begin{proof}\nWe have from Lemma \\ref{lem-propagation-estimate} that\n\\begin{equation*}\nX(t)-X(t_{0})\\geq c_{1}(t-t_{0}-T_{1}) ,\\quad t\\geq t_{0}.\n\\end{equation*}\nSetting $t\\to\\infty$ in the above estimate, we find $X(t)\\to\\infty$ as $t\\to\\infty$. Setting $t_{0}\\to-\\infty$, we find $X(t_{0})\\to-\\infty$ as $t_{0}\\to-\\infty$.\n\\end{proof}\n\nThis corollary shows that any transition front travels from the left infinity to the right infinity. Thus, steady-state-like transition fronts, blocking the propagations of solutions, do not exist.\n\n\n\\section{Proof of Theorem \\ref{thm-regularity-of-tf}}\\label{subsec-unconditional-regularity}\n\nWe prove Theorem \\ref{thm-regularity-of-tf} in this section. Throughout this section, we assume that (H1)-(H3) hold and $u(t,x)$ is an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$.\n\nTo prove Theorem \\ref{thm-regularity-of-tf}, we first do some preparations and prove several lemmas. Fix some $0<\\delta_{0}\\ll1$. For $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and $\\eta\\in\\mathbb{R}$ with $0<|\\eta|\\leq\\delta_{0}$, we set\n\\begin{equation*}\nv^{\\eta}(t,x)=\\frac{u(t,x+\\eta)-u(t,x)}{\\eta}.\n\\end{equation*}\nIt is easy to see that $v^{\\eta}(t,x)$ satisfies\n\\begin{equation}\\label{lipschitz-eqn-987654}\nv^{\\eta}_{t}(t,x)=b^{\\eta}(t,x)-v^{\\eta}(t,x)+a^{\\eta}(t,x)v^{\\eta}(t,x)+\\tilde{a}^{\\eta}(t,x),\n\\end{equation}\nwhere\n\\begin{equation*}\n\\begin{split}\na^{\\eta}(t,x)&=\\frac{f(t,x,u(t,x+\\eta))-f(t,x,u(t,x))}{u(t,x+\\eta)-u(t,x)},\\\\\n\\tilde{a}^{\\eta}(t,x)&=\\frac{f(t,x+\\eta,u(t,x+\\eta))-f(t,x,u(t,x+\\eta))}{\\eta},\\\\\nb^{\\eta}(t,x)&=\\int_{\\mathbb{R}}J(x-y)v^{\\eta}(t,y)dy=\\int_{\\mathbb{R}}\\frac{J(x-y+\\eta)-J(x-y)}{\\eta}u(t,y)dy.\n\\end{split}\n\\end{equation*}\nHence, for any fixed $x$, treating \\eqref{lipschitz-eqn-987654} as an ODE in the variable $t$, we find from the variation of constants formula that for any $t\\geq t_{0}$\n\\begin{equation}\\label{diff-integral-sol}\n\\begin{split}\nv^{\\eta}(t,x)&=v^{\\eta}(t_{0},x)e^{-\\int_{t_{0}}^{t}(1-a^{\\eta}(s,x))ds}+\\int_{t_{0}}^{t}b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\\\\n&\\quad\\quad+\\int_{t_{0}}^{t}\\tilde{a}^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau.\n\\end{split}\n\\end{equation}\n\n\nMoreover, we set\n\\begin{equation*}\nL_{0}=1+\\delta_{0}+\\sup_{t\\in\\mathbb{R}}|X(t)-X_{\\theta_{0}}^{+}(t)|\\quad\\text{and}\\quad L_{1}=1+\\delta_{0}+\\sup_{t\\in\\mathbb{R}}|X(t)-X_{\\theta_{1}}^{-}(t)|,\n\\end{equation*}\nwhere $\\theta_{1}$ and $\\theta_{0}$ are as in (H2) and (H3), respectively. By Lemma \\ref{lem-bounded-interface-width}, $L_{0}<\\infty$ and $L_{1}<\\infty$. We also set\n\\begin{equation*}\n\\begin{split}\nI_{l}(t)&=(-\\infty,X(t)-L_{1}),\\\\\nI_{m}(t)&=[X(t)-L_{1},X(t)+L_{0}],\\\\\n I_{r}(t)&=(X(t)+L_{0},\\infty)\n\\end{split}\n\\end{equation*}\nfor $t\\in\\mathbb{R}$. Clearly, $I_{l}(t)$, $I_{m}(t)$ and $I_{r}(t)$ are disjoint and $I_{l}(t)\\cup I_{m}(t)\\cup I_{r}(t)=\\mathbb{R}$. Since $X(t)$ may jump, so do $I_{l}(t)$, $I_{m}(t)$ and $I_{r}(t)$.\n\nSince $X(t)\\to\\pm\\infty$ as $t\\to\\pm\\infty$ by Corollary \\ref{cor-propagation-estimate}, for any fixed $x\\in\\mathbb{R}$, there hold $x\\in I_{r}(t)$ for all $t\\ll-1$ and $x\\in I_{l}(t)$ for all $t\\gg1$. Thus, for any fixed $x\\in\\mathbb{R}$,\n\\begin{equation*}\n\\begin{split}\nt_{\\rm first}(x)&=\\sup\\big\\{\\tilde{t}\\in\\mathbb{R}\\big|x\\in I_{r}(t)\\,\\,\\text{for all}\\,\\,t\\leq \\tilde{t}\\big\\},\\\\\nt_{\\rm last}(x)&=\\inf\\big\\{\\tilde{t}\\in\\mathbb{R}\\big|x\\in I_{l}(t)\\,\\,\\text{for all}\\,\\,t\\geq \\tilde{t}\\big\\}\n\\end{split}\n\\end{equation*}\nare well-defined. We see that if $X(t)$ is continuous, then $t_{\\rm first}(x)$ and $t_{\\rm last}(x)$ are the first time and the last time that $x$ is in $I_{m}(t)$. Clearly, $-\\infty<t_{\\rm first}(x)\\leq t_{\\rm last}(x)<\\infty$ (notice, $t_{\\rm first}(x)=t_{\\rm last}(x)$ may happen since $X(t)$ may jump). Although the functions $x\\mapsto t_{\\rm first}(x)$ and $x\\mapsto t_{\\rm last}(x)$ are unbounded, their difference is a bounded function as given by\n\n\\begin{lem}\nThere holds\n\\begin{equation}\\label{lipschitz-growth-time}\nT:=\\sup_{x\\in\\mathbb{R}}[t_{\\rm last}(x)-t_{\\rm first}(x)]<\\infty.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nTo see this, we suppose $t_{\\rm first}(x)<t_{\\rm last}(x)$. Due to possible jumps, we consider two cases.\n\nIf $x\\notin I_{r}(t_{\\rm first}(x))$, then $x\\in I_{l}(t_{\\rm first}(x))\\cup I_{m}(t_{\\rm first}(x))$, that is, $x\\leq X(t_{\\rm first}(x))+L_{0}$. Thus, for all $t\\geq t_{\\rm first}(x)+T_{1}+\\frac{L_{0}+L_{1}+1}{c_{1}}$, we see from Theorem \\ref{lem-propagation-estimate} that\n\\begin{equation*}\nx\\leq X(t_{\\rm first}(x))+L_{0}\\leq X(t)-c_{1}(t-t_{\\rm first}(x)-T_{1})+L_{0}\\leq X(t)-L_{1}-1.\n\\end{equation*}\nThis, implies that $x\\in I_{l}(t)$ for all $t\\geq t_{\\rm first}(x)+T_{1}+\\frac{L_{0}+L_{1}+1}{c_{1}}$, and hence, by definition\n\\begin{equation}\\label{growth-time-1}\nt_{\\rm last}(x)\\leq t_{\\rm first}(x)+T_{1}+\\frac{L_{0}+L_{1}+1}{c_{1}}.\n\\end{equation}\n\nIf $x\\in I_{r}(t_{\\rm first}(x))$, then we can find a sequence $\\{t_{n}\\}$ satisfying $t_{n}>t_{\\rm first}(x)$, $t_{n}\\to t_{\\rm first}(x)$ as $n\\to\\infty$ and $x\\notin I_{r}(t_{n}(x))$. Then, similar arguments as in the case $x\\notin I_{r}(t_{\\rm first}(x))$ lead to $t_{\\rm last}(x)\\leq t_{n}+T_{1}+\\frac{L_{0}+L_{1}+1}{c_{1}}$. Passing to the limit $n\\to\\infty$, we find \\eqref{growth-time-1} again. Hence, we have shown \\eqref{lipschitz-growth-time}.\n\\end{proof}\n\nTo control $v^{\\eta}(t,x)$, it is crucial to control $1-a^{\\eta}(t,x)$, which is achieved by means of $t_{\\rm first}(x)$ and $t_{\\rm last}(x)$ and is given in the following\n\n\\begin{lem}\nFor any $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and $0<|\\eta|\\leq\\delta_{0}$, there holds\n\\begin{equation}\\label{lipschitz-coefficient-1}\na^{\\eta}(t,x)\\leq\\begin{cases}\n1-\\kappa_{0},&\\quad t< t_{\\rm first}(x),\\\\\n0,&\\quad t> t_{\\rm last}(x).\n\\end{cases}\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nBy the definition of $t_{\\rm first}(x)$ and $t_{\\rm last}(x)$, we see\n\\begin{equation}\\label{result-from-defn}\nx\\in\\begin{cases}\nI_{r}(t),&\\quad t< t_{\\rm first}(x),\\\\\nI_{l}(t),&\\quad t> t_{\\rm last}(x).\n\\end{cases}\n\\end{equation}\nBy (H2), (H3) and the choices of $L_{0}$ and $L_{1}$, we find that for any $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and $0<|\\eta|\\leq\\delta_{0}$\n\\begin{equation*}\na^{\\eta}(t,x)\\leq\\begin{cases}\n1-\\kappa_{0},&\\quad x\\in I_{r}(t),\\\\\n0,&\\quad x\\in I_{l}(t).\n\\end{cases}\n\\end{equation*}\nThe lemma then follows from \\eqref{result-from-defn}.\n\\end{proof}\n\nThe above lemma says that for any fixed $x$, the solution $v^{\\eta}(t,x)$ of the ODE \\eqref{lipschitz-eqn-987654} can only grow for a period of time that is not longer than $T$.\n\nIn the next lemma, we show that $u(t,x)$ is continuous in space.\n\n\\begin{lem}\\label{lem-tf-continuity}\nFor any $t\\in\\mathbb{R}$, $u(t,x)$ is continuous in $x$. Moreover, there holds\n\\begin{equation*}\n\\sup_{t\\in\\mathbb{R}}\\sup_{x\\neq y}\\bigg|\\frac{u(t,x)-u(t,y)}{x-y}\\bigg|<\\infty.\n\\end{equation*}\n\\end{lem}\n\\begin{proof}\nSince $0<u(t,x)<1$, we trivially have\n\\begin{equation*}\n\\sup_{t\\in\\mathbb{R}}\\sup_{|x-y|\\geq\\delta_{0}}\\bigg|\\frac{u(t,x)-u(t,y)}{x-y}\\bigg|<\\infty.\n\\end{equation*}\nThus, we only need to show\n\\begin{equation}\\label{local-lipschitz-uniform-190}\n\\sup_{t\\in\\mathbb{R}}\\sup_{0<|x-y|\\leq\\delta_{0}}\\bigg|\\frac{u(t,x)-u(t,y)}{x-y}\\bigg|<\\infty.\n\\end{equation}\nTo do so, we consider \\eqref{diff-integral-sol} for some fixed $x\\in\\mathbb{R}$. We are going to take $t_{0}\\to-\\infty$ along some subsequence, and so $t_{0}\\ll t_{\\rm first}(x)$. For $t$, there are  three cases: $t< t_{\\rm first}(x)$, $t\\in[t_{\\rm first}(x),t_{\\rm last}(x)]$ and $t> t_{\\rm last}(x)$. Here, we only consider the case $t> t_{\\rm last}(x)$; other two cases can be treated similarly and are simpler.\n\nWe first note\n\\begin{equation*}\nM_{0}:=\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0<|\\eta|\\leq\\delta_{0}}\\Big[|b^{\\eta}(t,x)|+|\\tilde{a}^{\\eta}(t,x)|\\Big]<\\infty\n\\end{equation*}\nand the following uniform-in-$\\eta$ estimates hold:\n\\begin{equation}\\label{lipschitz-estimates}\n\\begin{split}\ne^{-\\int_{r}^{t_{\\rm first}(x)}(1-a^{\\eta}(s,x))ds}&\\leq e^{-\\kappa_{0}(t_{\\rm first}(x)-r)},\\quad r\\leq t_{\\rm first}(x),\\\\\ne^{-\\int_{r}^{t_{\\rm last}(x)}(1-a^{\\eta}(s,x))ds}&\\leq e^{C_{a}T},\\quad r\\in[t_{\\rm first}(x),t_{\\rm last}(x)],\\\\\ne^{-\\int_{r}^{t}(1-a^{\\eta}(s,x))ds}&\\leq e^{-(t-r)},\\quad r\\in[t_{\\rm last}(x),t],\n\\end{split}\n\\end{equation}\nwhere $C_{a}:=\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0<|\\eta|\\leq\\delta_{0}}|1-a^{\\eta}(t,x)|<\\infty$ by (H2). They are simple consequences of \\eqref{lipschitz-growth-time} and \\eqref{lipschitz-coefficient-1}.\n\nFor the second and third terms on the right-hand side of \\eqref{diff-integral-sol}, we have\n\\begin{equation*}\n\\begin{split}\n&\\frac{1}{M_{0}}\\bigg|\\int_{t_{0}}^{t}\\big[b^{\\eta}(\\tau,x)+\\tilde{a}^{\\eta}(\\tau,x)\\big]e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\bigg|\\leq \\int_{t_{0}}^{t}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\\\\n&\\quad\\quad=\\int_{t_{0}}^{t_{\\rm first}(x)}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau+\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau+\\int_{t_{\\rm last}(x)}^{t}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau.\n\\end{split}\n\\end{equation*}\nFor the three terms on the right-hand side of the above estimate, we use \\eqref{lipschitz-estimates} to deduce\n\\begin{equation*}\n\\begin{split}\n&\\int_{t_{0}}^{t_{\\rm first}(x)}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\\\\n&\\quad\\quad=\\int_{t_{0}}^{t_{\\rm first}(x)}e^{-\\int_{\\tau}^{t_{\\rm first}(x)}(1-a^{\\eta}(s,x))ds}e^{-\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}(1-a^{\\eta}(s,x))ds}e^{-\\int_{t_{\\rm last}(x)}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\\\\n&\\quad\\quad\\leq\\int_{t_{0}}^{t_{\\rm first}(x)}e^{-\\kappa_{0}(t_{\\rm first}(x)-\\tau)}e^{C_{a}T}e^{-(t-t_{\\rm last}(x))}d\\tau\\leq\\frac{e^{C_{a}T}}{\\kappa_{0}},\n\\end{split}\n\\end{equation*}\n\n\\begin{equation*}\n\\begin{split}\n\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau&=\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}e^{-\\int_{\\tau}^{t_{\\rm last}(x)}(1-a^{\\eta}(s,x))ds}e^{-\\int_{t_{\\rm last}(x)}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\\\\n&\\leq\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}e^{C_{a}T}e^{-(t-t_{\\rm last}(x))}d\\tau\\leq Te^{C_{a}T}\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n\\int_{t_{\\rm last}(x)}^{t}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\leq\\int_{t_{\\rm last}(x)}^{t}e^{-(t-\\tau)}d\\tau\\leq1.\n\\end{split}\n\\end{equation*}\nHence, we have shown\n\\begin{equation}\\label{first-estimate-190}\n\\bigg|\\int_{t_{0}}^{t}\\big[b^{\\eta}(\\tau,x)+\\tilde{a}^{\\eta}(\\tau,x)\\big]e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\bigg|\\leq M_{0}\\bigg(\\frac{e^{C_{a}T}}{\\kappa_{0}}+Te^{C_{a}T}+1\\bigg).\n\\end{equation}\nNotice the above estimate is universal.\n\nFor the first term on the right hand side of \\eqref{diff-integral-sol}, we choose $t_{0}$ such that $t_{\\rm first}(x)-t_{0}=\\frac{1}{|\\eta|}$ and claim that\n\\begin{equation}\\label{diff-claim-2}\nv^{\\eta}(t_{0},x)e^{-\\int_{t_{0}}^{t}(1-a^{\\eta}(s,x))ds}\\to0\\quad\\text{as}\\quad\\eta\\to0.\n\\end{equation}\nIn fact, from $|v^{\\eta}(t_{0},x)|\\leq\\frac{1}{|\\eta|}$ and \\eqref{lipschitz-estimates}, we see\n\\begin{equation*}\n\\begin{split}\n\\bigg|v^{\\eta}(t_{0},x)e^{-\\int_{t_{0}}^{t}(1-a^{\\eta}(s,x))ds}\\bigg|&\\leq\\frac{1}{|\\eta|}e^{-[\\int_{t_{0}}^{t_{\\rm first}(x)}+\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}+\\int_{t_{\\rm last}(x)}^{t}](1-a^{\\eta}(s,x))ds}\\\\\n&\\leq\\frac{1}{|\\eta|}e^{-\\kappa_{0}(t_{\\rm first}(x)-t_{0})}e^{C_{a}T}e^{-(t-t_{\\rm last}(x))}\\\\\n&\\leq\\frac{1}{|\\eta|}e^{-\\frac{\\kappa_{0}}{|\\eta|}}e^{C_{a}T}\\to0\\quad\\text{as}\\quad\\eta\\to0.\n\\end{split}\n\\end{equation*}\n\nConsequently, choosing $t_{0}$ such that $t_{\\rm first}(x)-t_{0}=\\frac{1}{|\\eta|}$, we conclude from \\eqref{diff-integral-sol}, \\eqref{first-estimate-190} and \\eqref{diff-claim-2} that\n\\begin{equation*}\n\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0<|\\eta|\\leq\\delta_{0}}|v^{\\eta}(t,x)|\\leq\\tilde{C}+M_{0}\\bigg(\\frac{e^{C_{a}T}}{\\kappa_{0}}+Te^{C_{a}T}+1\\bigg),\n\\end{equation*}\nwhere $\\tilde{C}=\\tilde{C}(\\kappa_{0},\\delta_{0},T)>0$. This proves \\eqref{local-lipschitz-uniform-190}, and hence, the lemma follows.\n\\end{proof}\n\n\nWe see that for any $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$, as $\\eta\\to0$, we have\n\\begin{equation}\\label{diff-limits-1}\n\\begin{split}\n&a^{\\eta}(t,x)\\to f_{u}(t,x,u(t,x)),\\\\\n&\\tilde{a}^{\\eta}(t,x)\\to f_{x}(t,x,u(t,x)),\\\\\n&b^{\\eta}(t,x)\\to\\int_{\\mathbb{R}}J'(x-y)u(t,y)dy.\n\\end{split}\n\\end{equation}\nNotice the first two convergence in \\eqref{diff-limits-1} need the continuity of $u(t,x)$ in $x$. By \\eqref{diff-limits-1}, \\eqref{lipschitz-coefficient-1} holds with $a^{\\eta}(t,x)$ replaced by $f_{u}(t,x,u(t,x))$, that is,\n\\begin{equation}\\label{lipschitz-coefficient-2}\nf_{u}(t,x,u(t,x))\\leq\\begin{cases}\n1-\\kappa_{0},&\\quad t< t_{\\rm first}(x),\\\\\n0,&\\quad t> t_{\\rm last}(x).\n\\end{cases}\n\\end{equation}\n\n\nNow, we prove Theorem \\ref{thm-regularity-of-tf}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm-regularity-of-tf}]\n\nLet us consider \\eqref{diff-integral-sol}. We are going to prove the existence of the limit $\\lim_{\\eta\\to0}v^{\\eta}(t,x)$. As in the proof of Lemma \\ref{lem-tf-continuity}, we assume $t_{0}\\ll t_{\\rm first}(x)$ and $t>t_{\\rm last}(x)$ in the rest of the proof. We treat three terms on the right-hand side of \\eqref{diff-integral-sol} separately.\n\n\nFor the second term on the right-hand side of \\eqref{diff-integral-sol}, we claim\n\\begin{equation}\\label{diff-claim-1}\n\\begin{split}\n&\\int_{t_{0}}^{t}b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\to\\int_{t_{0}}^{t}\\bigg(\\int_{\\mathbb{R}}J'(x-y)u(\\tau,y)dy\\bigg)e^{-\\int_{\\tau}^{t}(1-f_{u}(s,x,u(s,x)))ds}d\\tau\\\\\n&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad \\text{as}\\,\\, \\eta\\to0\\,\\,\\text{ uniformly in}\\,\\, t_{0}\\ll t_{\\rm first}(x).\n\\end{split}\n\\end{equation}\nTo see this, we notice\n\\begin{equation*}\n\\begin{split}\n&\\bigg|\\int_{t_{0}}^{t}b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau-\\int_{t_{0}}^{t}\\bigg(\\int_{\\mathbb{R}}J'(x-y)u(\\tau,y)dy\\bigg)e^{-\\int_{\\tau}^{t}(1-f_{u}(s,x,u(s,x)))ds}d\\tau\\bigg|\\\\\n&\\quad\\quad\\leq\\int_{-\\infty}^{t}\\bigg|b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}-\\bigg(\\int_{\\mathbb{R}}J'(x-y)u(\\tau,y)dy\\bigg)e^{-\\int_{\\tau}^{t}(1-f_{u}(s,x,u(s,x)))ds}\\bigg|d\\tau.\n\\end{split}\n\\end{equation*}\nBy \\eqref{diff-limits-1}, the integrand converges to $0$ as $\\eta\\to0$ pointwise. Thus, by dominated convergence theorem, we only need to make sure that the integrand is controlled by some integrable function that is independent of $\\eta$. Writing\n\\begin{equation*}\nb^{0}(\\tau,x)=\\int_{\\mathbb{R}}J'(x-y)u(\\tau,y)dy\\quad\\text{and} \\quad a^{0}(s,x)=f_{u}(s,x,u(s,x)),\n\\end{equation*}\nwe only need to make sure that the function\n\\begin{equation*}\n\\tau\\mapsto\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}\\bigg|b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}\\bigg|\n\\end{equation*}\nis integrable over $(-\\infty,t]$.\n\nSetting $M:=\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}|b^{\\eta}(t,x)|<\\infty$, we have\n\\begin{equation*}\n\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}\\bigg|b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}\\bigg|\\leq M\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}.\n\\end{equation*}\nTo bound the last integral uniformly in $0\\leq|\\eta|\\leq\\delta_{0}$, according to \\eqref{lipschitz-estimates} and \\eqref{lipschitz-coefficient-2}, we consider three cases:\n\n\\paragraph{\\textbf{Case i. $\\tau< t_{\\rm first}(x)$}} In this case,\n\\begin{equation*}\n\\begin{split}\n\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}&=\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}e^{-[\\int_{\\tau}^{t_{\\rm first}(x)}+\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}+\\int_{t_{\\rm last}(x)}^{t}](1-a^{\\eta}(s,x))ds}\\\\\n&\\leq e^{-\\kappa_{0}(t_{\\rm first}(x)-\\tau)}e^{C_{a}T}e^{-(t-t_{\\rm last}(x))};\n\\end{split}\n\\end{equation*}\n\n\\paragraph{\\textbf{Case ii. $\\tau\\in[t_{\\rm first}(x),t_{\\rm last}(x)]$}} In this case,\n\\begin{equation*}\n\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}=\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}e^{-[\\int_{\\tau}^{t_{\\rm last}(x)}+\\int_{t_{\\rm last}(x)}^{t}](1-a^{\\eta}(s,x))ds}\\leq e^{C_{a}T}e^{-(t-t_{\\rm last}(x))};\n\\end{equation*}\n\n\\paragraph{\\textbf{Case iii. $\\tau\\in(t_{\\rm last}(x),t]$}} In this case, $\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}\\leq e^{-(t-\\tau)}$.\n\nThus, setting\n\\begin{equation*}\nh(\\tau)=\\begin{cases}\ne^{-\\kappa_{0}(t_{\\rm first}(x)-\\tau)}e^{C_{a}T}e^{-(t-t_{\\rm last}(x))},&\\quad\\tau< t_{\\rm first}(x)\\\\\ne^{C_{a}T}e^{-(t-t_{\\rm last}(x))},&\\quad\\tau\\in[t_{\\rm first}(x),t_{\\rm last}(x)]\\\\\ne^{-(t-\\tau)},&\\quad\\tau\\in(t_{\\rm last}(x),t],\n\\end{cases}\n\\end{equation*}\nwe find for any $\\tau\\in(-\\infty,t]$\n\\begin{equation*}\n\\begin{split}\n&\\sup_{0<|\\eta|\\leq\\delta_{0}}\\bigg|b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}-\\bigg(\\int_{\\mathbb{R}}J'(x-y)u(\\tau,y)dy\\bigg)e^{-\\int_{\\tau}^{t}(1-f_{u}(s,x,u(s,x)))ds}\\bigg|\\\\\n&\\quad\\quad\\leq2\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}\\bigg|b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}\\bigg|\\leq 2h(\\tau).\n\\end{split}\n\\end{equation*}\nTo show \\eqref{diff-claim-1}, it remains to show $\\int_{-\\infty}^{t}h(\\tau)d\\tau<\\infty$. But, we readily compute\n\\begin{equation}\\label{diff-uniform-bound}\n\\begin{split}\n\\int_{-\\infty}^{t}h(\\tau)d\\tau&=\\int_{-\\infty}^{t_{\\rm first}(x)}h(\\tau)d\\tau+\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}h(\\tau)d\\tau+\\int_{t_{\\rm last}(x)}^{t}h(\\tau)d\\tau\\\\\n&\\leq\\int_{-\\infty}^{t_{\\rm first}(x)}e^{-\\kappa_{0}(t_{\\rm first}(x)-\\tau)}e^{C_{a}T}e^{-(t-t_{\\rm last}(x))}d\\tau\\\\\n&\\quad\\quad+\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}e^{C_{a}T}e^{-(t-t_{\\rm last}(x))}d\\tau+\\int_{t_{\\rm last}(x)}^{t}e^{-(t-\\tau)}d\\tau\\\\\n&\\leq\\frac{e^{C_{a}T}}{\\kappa_{0}}+Te^{C_{a}T}+1.\n\\end{split}\n\\end{equation}\nThus, we have shown \\eqref{diff-claim-1}. Note that the last bound is uniform in $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$.\n\nFor the third term on the right hand side of \\eqref{diff-integral-sol}, we claim\n\\begin{equation}\\label{diff-claim-3}\n\\begin{split}\n&\\int_{t_{0}}^{t}\\tilde{a}^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\to\\int_{t_{0}}^{t}f_{x}(t,x,u(t,x))e^{-\\int_{\\tau}^{t}(1-f_{u}(s,x,u(s,x)))ds}d\\tau\\\\\n&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad \\text{as}\\,\\, \\eta\\to0\\,\\,\\text{ uniformly in}\\,\\, t_{0}\\ll t_{\\rm first}(x).\n\\end{split}\n\\end{equation}\nThe proof of \\eqref{diff-claim-3} is similar to that of \\eqref{diff-claim-1}. So, we omit it. Notice\n\\begin{equation}\\label{diff-uniform-bound-1}\n\\begin{split}\n&\\int_{-\\infty}^{t}\\bigg|f_{x}(t,x,u(t,x))e^{-\\int_{\\tau}^{t}(1-f_{u}(s,x,u(s,x)))ds}\\bigg|d\\tau\\\\\n&\\quad\\quad\\leq\\bigg[\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{x}(t,x,u)|\\bigg]\\int_{-\\infty}^{t}h(\\tau)d\\tau\\\\\n&\\quad\\quad\\leq\\bigg[\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{x}(t,x,u)|\\bigg]\\bigg(\\frac{e^{C_{a}T}}{\\kappa_{0}}+Te^{C_{a}T}+1\\bigg).\n\\end{split}\n\\end{equation}\n\nFor the first term on the right hand side of \\eqref{diff-integral-sol}, we have \\eqref{diff-claim-2}, that is,\n\\begin{equation}\\label{diff-claim-4}\n\\text{with}\\quad t_{\\rm first}(x)-t_{0}=\\frac{1}{|\\eta|},\\quad v^{\\eta}(t_{0},x)e^{-\\int_{t_{0}}^{t}(1-a^{\\eta}(s,x))ds}\\to0\\quad\\text{as}\\quad\\eta\\to0.\n\\end{equation}\n\n\nHence, choosing $t_{0}$ such that $t_{\\rm first}(x)-t_{0}=\\frac{1}{|\\eta|}$ and passing to the limit $\\eta\\to0$ in \\eqref{diff-integral-sol}, we conclude from \\eqref{diff-claim-1}, \\eqref{diff-claim-3} and \\eqref{diff-claim-4} that\n\\begin{equation}\\label{a-formula-for-ux}\n\\begin{split}\nu_{x}(t,x)&=\\lim_{\\eta\\to0}v^{\\eta}(t,x)\\\\\n&=\\int_{-\\infty}^{t}\\bigg[\\int_{\\mathbb{R}}J'(x-y)u(\\tau,y)dy+f_{x}(\\tau,x,u(\\tau,x))\\bigg]e^{-\\int_{\\tau}^{t}(1-f_{u}(s,x,u(s,x)))ds}d\\tau.\n\\end{split}\n\\end{equation}\nFrom which, we see that $u_{x}(t,x)$ is continuous in $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$. Moreover, by \\eqref{diff-uniform-bound}, \\eqref{diff-uniform-bound-1} and \\eqref{a-formula-for-ux}, we have $\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}|u_{x}(t,x)|<\\infty$. This completes the proof.\n\\end{proof}\n\n\n\\begin{rem}\\label{rem-a-formula-for-ux}\nFrom \\eqref{a-formula-for-ux}, \\eqref{diff-uniform-bound} and \\eqref{diff-uniform-bound-1}, we see\n\\begin{equation*}\n\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}|u_{x}(t,x)|\\leq\\bigg[\\|J'\\|_{L^{1}(\\mathbb{R})}+\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{x}(t,x,u)|\\bigg]\\bigg(\\frac{e^{C_{a}T}}{\\kappa_{0}}+Te^{C_{a}T}+1\\bigg),\n\\end{equation*}\nwhere $C_{a}$ depends only on $\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{u}(t,x,u)|$ and $T$ is controlled by \\eqref{growth-time-1}, and hence, $T$ depends only on $f_{B}$ and the shape of $u(t,x)$.\n\\end{rem}\n\n\n\\section{Proof of Theorem \\ref{thm-regularity-of-tf-ii}}\\label{sec-proof-2}\n\nThis whole section is devoted to the proof of Theorem \\ref{thm-regularity-of-tf-ii}. Let $u(t,x)$ be a transition front as in the statement of Theorem \\ref{thm-regularity-of-tf-ii}. Hence, there exist $C>0$ and $r>0$ such that\n\\begin{equation}\\label{harnack-type-inequality}\nu(t,x)\\leq Ce^{r|x-y|}u(t,y),\\quad (t,x,y)\\in\\mathbb{R}\\times\\mathbb{R}\\times\\mathbb{R}.\n\\end{equation}\n\nLet $X(t)$ and $X^{\\pm}_{\\lambda}(t)$ be interface location functions of $u(t,x)$, where $X^{\\pm}_{\\lambda}(t)$ are given in \\eqref{defn-interface-locations}. As in \\cite{LiZl14}, for $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and $\\eta\\in\\mathbb{R}$ with $0<|\\eta|\\leq\\delta_{0}\\ll1$, we consider\n\\begin{equation*}\nv^{\\eta}(t,x)=\\frac{u(t,x+\\eta)-u(t,x)}{\\eta}\\quad\\text{and}\\quad w^{\\eta}(t,x)=\\frac{v^{\\eta}(t,x)}{u(t,x)}.\n\\end{equation*}\nWe readily check $w^{\\eta}(t,x)$ satisfies\n\\begin{equation}\\label{eqn-9182}\nw^{\\eta}_{t}=a_{1}^{\\eta}+a_{2}^{\\eta}w^{\\eta},\n\\end{equation}\nwhere $a_{1}^{\\eta}=\\frac{J\\ast v^{\\eta}}{u}+\\frac{\\tilde{a}^{\\eta}}{u}$ and $a_{2}^{\\eta}=-\\frac{J\\ast u}{u}+a^{\\eta}-\\frac{f}{u}$ with\n\\begin{equation}\\label{1-term}\n\\tilde{a}^{\\eta}=\\frac{f(t,x+\\eta,u(t,x+\\eta))-f(t,x,u(t,x+\\eta))}{\\eta}\n\\end{equation}\nand\n\\begin{equation}\\label{2-term}\na^{\\eta}=\\frac{f(t,x,u(t,x+\\eta))-f(t,x,u(t,x))}{u(t,x+\\eta)-u(t,x)}.\n\\end{equation}\n\nTo bound the solution of \\eqref{eqn-9182}, we first analyze $a_{1}^{\\eta}$ and $a_{2}^{\\eta}$. For $a^{\\eta}_{1}$, we first see\n\\begin{equation*}\n\\bigg|\\frac{J\\ast v^{\\eta}}{u}\\bigg|=\\frac{1}{u(t,x)}\\bigg|\\int_{\\mathbb{R}}\\frac{J(x-y+\\eta)-J(x-y)}{\\eta}u(t,y)dy\\bigg|\\leq C\\int_{\\mathbb{R}}\\bigg|\\frac{J(x+\\eta)-J(x)}{\\eta}\\bigg|e^{r|x|}dx,\n\\end{equation*}\nwhere we used \\eqref{harnack-type-inequality}. Next, setting $C_{1}:=\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{xu}(t,x,u)|$, we have\n\\begin{equation*}\n\\begin{split}\n\\bigg|\\frac{\\tilde{a}^{\\eta}}{u}\\bigg|\\leq\\frac{|f_{x}(t,x+\\eta_{*},u(t,x+\\eta))|}{u(t,x)}\\leq C_{1}\\frac{u(t,x+\\eta)}{u(t,x)}\\leq C_{1}Ce^{r|\\eta|},\n\\end{split}\n\\end{equation*}\nwhere we used Taylor expansion, the fact $f_{x}(t,x,0)=0$ and \\eqref{harnack-type-inequality}. Hence,\n\\begin{equation}\\label{a-1}\nC_{2}:=\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0<|\\eta|\\leq\\delta_{0}}|a_{1}^{\\eta}|<\\infty.\n\\end{equation}\n\nFor $a_{2}^{\\eta}$, we first see from \\eqref{harnack-type-inequality} that\n\\begin{equation*}\n\\frac{1}{C}\\int_{\\mathbb{R}}J(x)e^{-r|x|}dx\\leq\\frac{J\\ast u}{u}\\leq C\\int_{\\mathbb{R}}J(x)e^{r|x|}dx,\n\\end{equation*}\nand thus, setting $C_{3}:=\\frac{1}{C}\\int_{\\mathbb{R}}J(x)e^{-r|x|}dx$ and $C_{4}:=C\\int_{\\mathbb{R}}J(x)e^{r|x|}dx$, we find\n\\begin{equation*}\n-C_{4}\\leq-\\frac{J\\ast u}{u}\\leq-C_{3}.\n\\end{equation*}\nTo control the term $a^{\\eta}-\\frac{f}{u}$ in $a_{2}^{\\eta}$, we set\n\\begin{equation*}\n\\tilde{\\theta}_{0}:=\\min\\bigg\\{\\frac{\\theta_{1}}{2},\\frac{C_{3}}{2}\\Big[\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f(t,x,u)|\\Big]^{-1}\\bigg\\},\n\\end{equation*}\nand define\n\\begin{equation*}\nL_{0}=1+\\delta_{0}+\\sup_{t\\in\\mathbb{R}}|X(t)-X^{+}_{\\tilde{\\theta}_{0}}(t)|\\quad\\text{ and}\\quad L_{1}=1+\\delta_{0}+\\sup_{t\\in\\mathbb{R}}|X(t)-X^{-}_{\\theta_{1}}(t)|.\n\\end{equation*}\nAs in the proof of Theorem \\ref{thm-regularity-of-tf}, we set\n\\begin{equation*}\n\\begin{split}\nI_{l}(t)&=(-\\infty,X(t)-L_{1}),\\\\\nI_{m}(t)&=[X(t)-L_{1},X(t)+L_{0}],\\\\\n I_{r}(t)&=(X(t)+L_{0},\\infty)\n\\end{split}\n\\end{equation*}\nfor $t\\in\\mathbb{R}$, and for any fixed $x\\in\\mathbb{R}$, define\n\\begin{equation*}\n\\begin{split}\nt_{\\rm first}(x)&=\\sup\\{\\tilde{t}\\in\\mathbb{R}|x\\in I_{r}(t)\\,\\,\\text{for all}\\,\\,t\\leq \\tilde{t}\\},\\\\\nt_{\\rm last}(x)&=\\inf\\{\\tilde{t}\\in\\mathbb{R}|x\\in I_{l}(t)\\,\\,\\text{for all}\\,\\,t\\geq \\tilde{t}\\}.\n\\end{split}\n\\end{equation*}\nThen, there hold $T:=\\sup_{x\\in\\mathbb{R}}[t_{\\rm last}(x)-t_{\\rm first}(x)]<\\infty$ and for all $x\\in\\mathbb{R}$\n\\begin{equation*}\\label{result-from-defn-2}\nx\\in\\begin{cases}\nI_{r}(t),&\\quad t< t_{\\rm first}(x),\\\\\nI_{l}(t),&\\quad t> t_{\\rm last}(x).\n\\end{cases}\n\\end{equation*}\n\nNow, for $0<|\\eta|\\leq\\delta_{0}$, we have\n\\begin{itemize}\n\\item if $t< t_{\\rm first}(x)$, then $x\\in I_{r}(t)$, in particular, $x\\geq X^{+}_{\\tilde{\\theta}_{0}}(t)+\\delta_{0}$, and hence, $u(t,x)\\leq\\tilde{\\theta}_{0}$ and $u(t,x+\\eta)\\leq\\tilde{\\theta}_{0}$; it then follows from Taylor expansion that\n\\begin{equation*}\n\\bigg|a^{\\eta}-\\frac{f}{u}\\bigg|=|f_{u}(t,x,u_{*})-f_{u}(t,x,u_{**})|\\leq|u_{*}-u_{**}|\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{uu}(t,x,u)|\\leq\\frac{C_{3}}{2},\n\\end{equation*}\nwhere $u_{*}$ is between $u(t,x)$ and $u(t,x+\\eta)$, and $u_{**}$ is between $0$ and $u(t,x)$, and hence, both $u_{*}$ and $u_{**}$ are between $0$ and $\\tilde{\\theta}_{0}$, so $|u_{*}-u_{**}|\\leq\\tilde{\\theta}_{0}$;\n\n\\item if $t> t_{\\rm last}(x)$, then $x\\in I_{l}(x)$, in particular, $x\\leq X^{-}_{\\theta_{1}}(t)-\\delta$, and hence, $u(t,x)\\geq\\theta_{1}$ and $u(t,x+\\eta)\\geq\\theta_{1}$; it then follows from (H2) that $a^{\\eta}\\leq0$, which leads to $a^{\\eta}-\\frac{f}{u}\\leq0$;\n\n\\item if $t\\in[t_{\\rm first}(x),t_{\\rm last}(x)]$, then $|a^{\\eta}-\\frac{f}{u}|\\leq2\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{u}(t,x,u)|$.\n\\end{itemize}\nTherefore, we have the following for $a_{2}^{\\eta}$: for $0<|\\eta|\\leq\\delta_{0}$ and $x\\in\\mathbb{R}$\n\\begin{equation}\\label{a-2}\na_{2}^{\\eta}\\leq\\begin{cases}\n-\\frac{C_{3}}{2},\\quad& t\\leq t_{\\rm first}(x),\\\\\nC_{5},\\quad & t_{\\rm first}(x)\\leq t\\leq t_{\\rm last}(x),\\\\\n-C_{3},\\quad & t\\geq t_{\\rm last}(x),\n\\end{cases}\n\\end{equation}\nwhere $C_{5}:=C_{4}+2\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{u}(t,x,u)|$.\n\nWith the help of \\eqref{a-1} and \\eqref{a-2}, we are now able to bound the solution of \\eqref{eqn-9182}. Notice, the solution of \\eqref{eqn-9182} can be written as\n\\begin{equation}\\label{eqn-integral-9812}\nw^{\\eta}(t,x)=e^{\\int_{t_{0}}^{t}a_{2}^{\\eta}(s,x)ds}w^{\\eta}(t_{0},x)+\\int_{t_{0}}^{t}e^{\\int_{\\tau}^{t}a_{2}^{\\eta}(s,x)ds}a_{1}^{\\eta}(\\tau,x)d\\tau,\\quad x\\in\\mathbb{R},\\quad t\\geq t_{0}.\n\\end{equation}\n\nUsing \\eqref{a-1}, \\eqref{a-2} and \\eqref{eqn-integral-9812}, we first argue as in the proof of Lemma \\ref{lem-tf-continuity} to conclude that\n\\begin{equation*}\n\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0<|\\eta|\\leq\\delta_{0}}\\bigg|\\frac{u(t,x+\\eta)-u(t,x)}{\\eta u(t,x)}\\bigg|=\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0<|\\eta|\\leq\\delta_{0}}|w^{\\eta}(t,x)|<\\infty,\n\\end{equation*}\nwhich in particular implies that $u(t,x)$ is continuous in $x$, since $u(t,x)\\in(0,1)$.\n\n\nNow, we set $a^{0}_{1}=\\frac{J'\\ast u}{u}+f_{x}$ and $a_{2}^{0}=-\\frac{J\\ast u}{u}+f_{u}-\\frac{f}{u}$. Using the continuity of $u(t,x)$ in $x$, we have the pointwise limits $a^{\\eta}_{1}\\to a^{0}_{1}$\nand $a^{\\eta}_{2}\\to a^{0}_{2}$ as $\\eta\\to0$. Then, using \\eqref{a-1} and \\eqref{a-2}, we can show that as \\eqref{diff-claim-1},\n\\begin{equation*}\n\\int_{t_{0}}^{t}e^{\\int_{\\tau}^{t}a_{2}^{\\eta}(s,x)ds}a_{1}^{\\eta}(\\tau,x)d\\tau\\to\\int_{t_{0}}^{t}e^{\\int_{\\tau}^{t}a_{2}^{0}(s,x)ds}a_{1}^{0}(\\tau,x)d\\tau\n\\end{equation*}\nuniformly in $t_{0}\\ll t_{\\rm first}(x)$ as $\\eta\\to0$, and as \\eqref{diff-claim-2},\n\\begin{equation*}\n\\text{with}\\,\\,t_{\\rm first}(x)-t_{0}=\\frac{1}{|\\eta|},\\quad e^{\\int_{t_{0}}^{t}a_{2}^{\\eta}(s,x)ds}w^{\\eta}(t_{0},x)\\to0\\quad\\text{as}\\quad\\eta\\to0.\n\\end{equation*}\nHence, setting $t_{\\rm first}(x)-t_{0}=\\frac{1}{|\\eta|}$ in \\eqref{eqn-integral-9812} and passing to the limit $\\eta\\to0$, we find\n\\begin{equation*}\n\\frac{u_{x}(t,x)}{u(t,x)}=\\lim_{\\eta\\to0}w^{\\eta}(t,x)=\\int_{-\\infty}^{t}e^{\\int_{\\tau}^{t}a_{2}^{0}(s,x)ds}a_{1}^{0}(\\tau,x)d\\tau,\\quad (t,x)\\in\\mathbb{R}\\times\\mathbb{R}.\n\\end{equation*}\nThis completes the proof.\n\n\n\n\\section{Proof of Theorem \\ref{thm-regularity-of-tf-iii}}\\label{sec-proof-iafjiaf}\n\nWe prove Theorem \\ref{thm-regularity-of-tf-iii} in this section. Throughout this section, we assume (H1) and (H2). To prove Theorem \\ref{thm-regularity-of-tf-iii}, we need the following three lemmas. For $r>0$, let\n\\begin{equation*}\n\\Gamma_{r}(x)=\\min\\{1,e^{-rx}\\}=\\begin{cases}\n1,\\quad&x\\leq0,\\\\\ne^{-rx},\\quad&x\\geq0.\n\\end{cases}\n\\end{equation*}\n\n\\begin{lem}\\label{lem-090-1}\nThere exist two continuous functions $M:(0,\\infty)\\to(0,\\infty)$ and $\\gamma:(0,\\infty)\\to(0,\\infty)$ with $\\gamma(r)\\to0$ such that\n\\begin{equation*}\n\\bigg|\\frac{[J\\ast\\Gamma_{r}](x)}{\\Gamma_{r}(x)}-1\\bigg|\\leq\\gamma(r),\\quad x\\geq M(r)\n\\end{equation*}\nfor all $r>0$.\n\\end{lem}\n\\begin{proof}\nFix $r>0$ and write $\\Gamma=\\Gamma_{r}$. We see\n\\begin{equation*}\n\\begin{split}\n\\frac{[J\\ast\\Gamma](x)}{\\Gamma(x)}-1&=\\int_{-\\infty}^{0}J(x-y)\\frac{\\Gamma(y)}{\\Gamma(x)}dy+\\int_{0}^{\\infty}J(x-y)\\frac{\\Gamma(y)}{\\Gamma(x)}dy-1\\\\\n&=\\int_{-\\infty}^{0}J(x-y)\\frac{\\Gamma(y)}{\\Gamma(x)}dy+\\int_{0}^{\\infty}J(x-y)e^{r(x-y)}dy-1\\\\\n&=\\int_{-\\infty}^{0}J(x-y)\\frac{\\Gamma(y)}{\\Gamma(x)}dy+\\bigg[\\int_{\\mathbb{R}}J(y)e^{ry}dy-1\\bigg]-\\int_{x}^{\\infty}J(y)e^{ry}dy.\n\\end{split}\n\\end{equation*}\nDue to the decay of $J$ at $\\pm\\infty$, it is not hard to see that $\\lim_{x\\to\\infty}\\int_{-\\infty}^{0}J(x-y)\\frac{\\Gamma(y)}{\\Gamma(x)}dy=0$. In fact, for all large $x$,\n\\begin{equation*}\n\\int_{-\\infty}^{0}J(x-y)\\frac{\\Gamma(y)}{\\Gamma(x)}dy=\\int_{-\\infty}^{0}J(x-y)e^{rx}dy\\leq\\int_{-\\infty}^{0}J(x-y)e^{r(x-y)}dy\\to0\\,\\,\\text{as}\\,\\,x\\to\\infty.\n\\end{equation*}\nSince $\\int_{\\mathbb{R}}J(y)dy=1$, we find $\\lim_{r\\to0}\\int_{\\mathbb{R}}J(y)e^{ry}dy=1$ by dominated convergence theorem. Clearly, $\\lim_{x\\to\\infty}\\int_{x}^{\\infty}J(y)e^{ry}dy=0$. The lemma then follows.\n\\end{proof}\n\n\\begin{lem}\\label{lem-090-2}\nThere exists $r_{0}>0$ such that if $r\\in(0,r_{0}]$ is such that\n\\begin{equation}\\label{exact-decay-condition}\n\\lim_{x\\to\\infty}\\frac{u(t,x+X(t))}{e^{-rx}}=1\\quad\\text{uniformly in}\\quad t\\in\\mathbb{R},\n\\end{equation}\nthen, there exists $M(r)>0$ such that\n\\begin{equation}\\label{result-1}\n\\frac{1}{2}\\leq\\frac{[J\\ast u(t,\\cdot+X(t))](x)}{u(t,x+X(t))}\\leq\\frac{3}{2},\\quad x\\geq M(r)\\quad \\text{and}\\quad t\\in\\mathbb{R},\n\\end{equation}\nand\n\\begin{equation}\\label{result-2}\n\\frac{|[J'\\ast u(t,\\cdot+X(t))](x)|}{u(t,x+X(t))}\\leq2,\\quad x\\geq M(r)\\quad \\text{and}\\quad t\\in\\mathbb{R}.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nLet $u=u(t,x+X(t))$. For $r\\in(0,r_{0}]$, where $r_{0}$ is to be chosen, we let $\\Gamma=\\Gamma_{r}$. Write\n\\begin{equation*}\n\\begin{split}\n\\frac{J\\ast u}{u}-1&=\\frac{J\\ast\\Gamma}{\\Gamma}\\bigg[\\frac{J\\ast(u-\\Gamma)}{J\\ast\\Gamma}+1\\bigg]\\frac{\\Gamma}{u}-1\\\\\n&=\\bigg[\\frac{J\\ast\\Gamma}{\\Gamma}-1\\bigg]\\bigg[\\frac{J\\ast(u-\\Gamma)}{J\\ast\\Gamma}+1\\bigg]\\frac{\\Gamma}{u}+\\frac{J\\ast(u-\\Gamma)}{J\\ast\\Gamma}\\frac{\\Gamma}{u}+\\bigg[\\frac{\\Gamma}{u}-1\\bigg].\n\\end{split}\n\\end{equation*}\nBy Lemma \\ref{lem-090-1} and \\eqref{exact-decay-condition}, we only need to treat the term $\\frac{J\\ast(u-\\Gamma)}{J\\ast\\Gamma}$ for large $x$.\n\nBy \\eqref{exact-decay-condition}, for any $\\epsilon>0$, there exists $M(\\epsilon,r)>0$ such that\n\\begin{equation*}\n|u(t,x+X(t))-e^{-rx}|\\leq\\epsilon e^{-rx},\\quad x\\geq M(\\epsilon,r)\\quad \\text{and}\\quad t\\in\\mathbb{R}.\n\\end{equation*}\nThen, for $\\epsilon>0$, we have\n\\begin{equation*}\n\\begin{split}\n\\bigg|\\frac{J\\ast(u-\\Gamma)}{J\\ast\\Gamma}\\bigg|&=\\frac{\\Gamma}{J\\ast\\Gamma}\\bigg|\\frac{J\\ast(u-\\Gamma)}{\\Gamma}\\bigg|\\\\\n&\\leq\\frac{\\Gamma}{J\\ast\\Gamma}\\bigg[\\int_{-\\infty}^{M(\\epsilon,r)}J(x-y)\\frac{|u(t,y+X(t))-\\Gamma(y)|}{\\Gamma(x)}dy\\\\\n&\\quad\\quad\\quad\\quad\\quad+\\int_{M(\\epsilon,r)}^{\\infty}J(x-y)\\frac{|u(t,y+X(t))-e^{-ry}|}{e^{-rx}}dy\\bigg]\\\\\n&\\leq\\frac{\\Gamma}{J\\ast\\Gamma}\\bigg[\\int_{-\\infty}^{M(\\epsilon,r)}J(x-y)\\frac{|u(t,y+X(t))-\\Gamma(y)|}{\\Gamma(x)}dy+\\epsilon\\int_{M(\\epsilon,r)}^{\\infty}J(x-y)e^{r(x-y)}dy\\bigg]\\\\\n&\\leq\\frac{\\Gamma}{J\\ast\\Gamma}\\bigg[\\int_{-\\infty}^{M(\\epsilon,r)}J(x-y)\\frac{|u(t,y+X(t))-\\Gamma(y)|}{\\Gamma(x)}dy+\\epsilon\\int_{\\mathbb{R}}J(y)e^{ry}dy\\bigg]\n\\end{split}\n\\end{equation*}\nDue to the decay of $J$ at $\\pm\\infty$, we have\n\\begin{equation*}\n\\lim_{x\\to\\infty}\\int_{-\\infty}^{M(\\epsilon,r)}J(x-y)\\frac{|u(t,y+X(t))-\\Gamma(y)|}{\\Gamma(x)}dy=0\\quad\\text{uniformly in}\\quad t\\in\\mathbb{R}.\n\\end{equation*}\nIt then follows from Lemma \\ref{lem-090-1} that for any $\\epsilon>0$, there exists $\\tilde{M}(\\epsilon,r)>0$ such that\n\\begin{equation*}\n\\bigg|\\frac{J\\ast(u-\\Gamma)}{J\\ast\\Gamma}\\bigg|\\leq\\tilde{\\gamma}(r)\\bigg[\\frac{\\epsilon}{2}+\\epsilon\\int_{\\mathbb{R}}J(y)e^{ry}dy\\bigg],\\quad x\\geq\\tilde{M}(\\epsilon,r)\\quad\\text{and}\\quad t\\in\\mathbb{R},\n\\end{equation*}\nwhere $\\tilde{\\gamma}(r)\\to0$ as $r\\to0$. The result \\eqref{result-1} follows by first fixing $\\epsilon$, say $\\epsilon=\\frac{1}{100}$, and then choosing $r_{0}$ small so that if $r\\in(0,r_{0}]$, then $|\\frac{J\\ast u}{u}-1|\\leq\\frac{1}{2}$ for all large $x$ depending on $r$. Note that the term $\\frac{J\\ast\\Gamma}{\\Gamma}-1$ and Lemma \\ref{lem-090-1} restrict $r_{0}$. Similar arguments lead to \\eqref{result-2}, since\n\\begin{equation*}\n\\bigg|\\frac{J'\\ast u}{u}\\bigg|\\leq\\frac{|J'|\\ast\\Gamma}{\\Gamma}\\bigg[\\frac{|J'|\\ast(u-\\Gamma)}{|J'|\\ast\\Gamma}+1\\bigg]\\frac{\\Gamma}{u}.\n\\end{equation*}\nThis completes the proof.\n\\end{proof}\n\n\\begin{lem}\\label{lem-away-from-0}\nLet $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$. Then, there holds\n\\begin{equation*}\n\\forall L>0,\\quad \\inf_{t\\in\\mathbb{R}}\\inf_{x\\leq L+X(t)}u(t,x)>0.\n\\end{equation*}\n\\end{lem}\n\\begin{proof}\nFix $L>0$ and $x_{*}>0$. Let $\\lambda_{1}\\in(\\frac{1}{2},1)$ and $\\lambda_{2}\\in(0,\\frac{1}{10})$. We define $u_{0}^{B}:\\mathbb{R}\\to[0,1]$ and $u_{0}^{M}:\\mathbb{R}\\to[0,1]$ by setting\n\\begin{equation*}\nu_{0}^{B}(x)=\\begin{cases}\n\\lambda_{1},\\quad &x\\leq-x_{*},\\\\\n-\\frac{\\lambda_{1}}{x_{*}}x,\\quad& x\\in[-x_{*},0],\\\\\n0,\\quad &x\\geq0\n\\end{cases}\n\\quad\\text{and}\\quad\nu_{0}^{M}(x)=\\begin{cases}\n1,\\quad &x\\leq0,\\\\\n\\frac{\\lambda_{2}-1}{x_{*}}x+1,\\quad&x\\in[0,x_{*}],\\\\\n\\lambda_{2},\\quad &x\\geq x_{*}.\n\\end{cases}\n\\end{equation*}\nClearly, $u_{0}^{B}(\\cdot-X^{-}_{\\lambda_{1}}(t))\\leq u(t,\\cdot)\\leq u_{0}^{M}(\\cdot-X^{+}_{\\lambda_{2}}(t))$ for all $t\\in\\mathbb{R}$. Now, denote by $u_{B}(t,x;u_{0}^{B})$ and $u_{M}(t,x;u_{0}^{M})$ the solutions of $u_{t}=J\\ast u-u+f_{B}(u)$ and $u_{t}=J\\ast u-u+f_{M}(u)$, respectively, with initial data $u_{B}(0,\\cdot;u_{0}^{B})=u_{0}^{B}$ and $u_{M}(0,\\cdot;u_{0}^{M})=u_{0}^{M}$. It then follows from comparison principle and homogeneity that\n\\begin{equation}\\label{some-result-8472828373}\nu_{B}(T,x-X^{-}_{\\lambda_{1}}(t-T);u_{0}^{B})\\leq u(t,x)\\leq u_{M}(T,x-X^{+}_{\\lambda_{2}}(t-T);u_{0}^{M}),\\quad (t,x)\\in\\mathbb{R}\\times\\mathbb{R}\n\\end{equation}\nfor all $T\\geq0$. Now, we consider a small $T$ and let $\\lambda>0$ be small. Let $\\xi_{\\lambda}^{B}(T)$ be such that $u_{B}(T,\\xi_{\\lambda}^{B}(T);u_{0}^{B})=\\lambda$. We then see from the first inequality in \\eqref{some-result-8472828373} that if $x\\leq\\xi^{B}_{\\lambda}(T)+X^{-}_{\\lambda_{1}}(t-T)-1$, then the monotonicity of $u_{B}(t,x;u_{0}^{B})$ in $x$ yields\n\\begin{equation*}\nu(t,x)\\geq u_{B}(T,\\xi_{\\lambda}^{B}(T)-1;u_{0}^{B})>u_{B}(T,\\xi_{\\lambda}^{B}(T);u_{0}^{B})=\\lambda,\n\\end{equation*}\nwhich then leads to\n\\begin{equation*}\nX^{-}_{\\lambda}(t)\\geq\\xi^{B}_{\\lambda}(T)+X^{-}_{\\lambda_{1}}(t-T)-1,\\quad t\\in\\mathbb{R}.\n\\end{equation*}\nNote that if we can find some $C>0$ such that\n\\begin{equation}\\label{a-key-step-9828318317}\nX^{-}_{\\lambda_{1}}(t-T)\\geq X(t)-C,\\quad t\\in\\mathbb{R},\n\\end{equation}\nthen we can make $\\lambda$ closer to $0$ such that $\\xi^{B}_{\\lambda}(T)$ is so large that $X_{\\lambda}^{-}(t)\\geq L+X(t)+1$ for all $t\\in\\mathbb{R}$, which then leads to\n\\begin{equation*}\n\\inf_{t\\in\\mathbb{R}}\\inf_{x\\leq L+X(t)}u(t,x)\\geq \\inf_{t\\in\\mathbb{R}}\\inf_{x\\leq X_{\\lambda}^{-}(t)-1}u(t,x)>\\lambda>0.\n\\end{equation*}\nHence, to finish the proof, we only need to show \\eqref{a-key-step-9828318317}.\n\nWe now show \\eqref{a-key-step-9828318317}. Let us look at the interface locations for $u(t,x)$ and $u_{M}(t,x;u_{0}^{M})$ at $\\frac{1}{2}$. From the second inequality in \\eqref{some-result-8472828373}, we see\n\\begin{equation}\\label{some-inequalities-198314717419741}\n X^{+}_{\\frac{1}{2}}(t)\\leq \\xi^{M}_{\\frac{1}{2}}(T)+X^{+}_{\\lambda_{2}}(t-T)+1,\\quad t\\in\\mathbb{R},\n\\end{equation}\nwhere $\\xi^{M}_{\\frac{1}{2}}(T)$ is such that $u(T,\\xi^{M}_{\\frac{1}{2}}(T);u_{0}^{M})=\\frac{1}{2}$. Notice choosing $T$ or $\\lambda_{2}$ smaller, we can guarantee that $\\xi^{M}_{\\frac{1}{2}}(T)$ is well-defined. We then deduce from \\eqref{some-inequalities-198314717419741} that\n\\begin{equation*}\n\\begin{split}\nX_{\\lambda_{1}}^{-}(t-T)&\\geq X_{\\lambda_{2}}^{+}(t-T)-C_{1}\\\\\n&\\geq X^{+}_{\\frac{1}{2}}(t)-\\xi_{\\frac{1}{2}}^{M}(T)-1-C_{1}\\\\\n&\\geq X(t)-C_{2}-\\xi_{\\frac{1}{2}}^{M}(T)-1-C_{1}\n\\end{split}\n\\end{equation*}\nfor all $t\\in\\mathbb{R}$, where $C_{1}=\\sup_{t\\in\\mathbb{R}}|X_{\\lambda_{1}}^{-}(t)-X_{\\lambda_{2}}^{+}(t)|$ and $C_{2}=\\sup_{t\\in\\mathbb{R}}|X_{\\frac{1}{2}}^{+}(t)-X(t)|$. Setting $C=C_{2}+\\xi_{\\frac{1}{2}}^{M}(T)+1+C_{1}$, we find \\eqref{a-key-step-9828318317}, and hence, the lemma follows.\n\\end{proof}\n\n\nWe are ready to prove Theorem \\ref{thm-regularity-of-tf-iii}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm-regularity-of-tf-iii}]\nLet $r_{0}>0$ be as in Lemma \\ref{lem-090-2} and fix $r\\in(0,r_{0}]$. Let $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} satisfying\n\\begin{equation}\\label{exact-decay-condition-134141}\n\\lim_{x\\to\\infty}\\frac{u(t,x+X(t))}{e^{-rx}}=1\\quad\\text{uniformly in}\\quad t\\in\\mathbb{R}.\n\\end{equation}\n\nTo prove the theorem, we proceed as in the proof of Theorem \\ref{thm-regularity-of-tf-ii}. Thus, we only need to bound\n$a_{1}^{\\eta}=\\frac{J\\ast v^{\\eta}}{u}+\\frac{\\tilde{a}^{\\eta}}{u}$\nas in \\eqref{a-1} and estimate $a_{2}^{\\eta}=-\\frac{J\\ast u}{u}+a^{\\eta}-\\frac{f}{u}$ as in \\eqref{a-2}, where $\\tilde{a}^{\\eta}$ and $a^{\\eta}$ are given in \\eqref{1-term} and \\eqref{2-term}, respectively.\n\nFor $a_{1}^{\\eta}$, we have\n\\begin{equation*}\n|a_{1}^{\\eta}|\\leq\\bigg|\\frac{J\\ast v^{\\eta}}{u}\\bigg|+\\bigg|\\frac{\\tilde{a}^{\\eta}}{u}\\bigg|\\leq\\frac{1}{u(t,x)}\\int_{\\mathbb{R}}\\frac{|J(x-y+\\eta)-J(x-y)|}{\\eta}u(t,y)dy+C_{1}\\frac{u(t,x+\\eta)}{u(t,x)},\n\\end{equation*}\nwhere $C_{1}=\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{xu}(t,x,u)|$. For a sufficiently large $M_{1}>0$, we see from \\eqref{result-2} and \\eqref{exact-decay-condition-134141} that\n\\begin{equation*}\n\\sup_{t\\in\\mathbb{R}}\\sup_{x\\geq M_{1}+X(t)}\\sup_{0<|\\eta|\\leq\\delta_{0}}|a_{1}^{\\eta}|<\\infty.\n\\end{equation*}\nSince $\\inf_{t\\in\\mathbb{R}}\\inf_{x\\leq M_{1}+X(t)}u(t,x)>0$ by Lemma \\ref{lem-away-from-0}, we have\n\\begin{equation*}\n\\sup_{t\\in\\mathbb{R}}\\sup_{x\\leq M_{1}+X(t)}\\sup_{0<|\\eta|\\leq\\delta_{0}}|a_{1}^{\\eta}|<\\infty.\n\\end{equation*}\nHence, we obtain\n\\begin{equation*}\n\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0<|\\eta|\\leq\\delta_{0}}|a_{1}^{\\eta}|<\\infty.\n\\end{equation*}\n\nFor $a^{\\eta}_{2}$, we first see from \\eqref{result-1} that we can find a sufficiently large $M_{2}>0$ such that\n\\begin{equation*}\n\\frac{1}{2}\\leq\\frac{J\\ast u}{u}\\leq\\frac{3}{2},\\quad x\\geq M_{2}+X(t)\\quad\\text{and}\\quad t\\in\\mathbb{R}.\n\\end{equation*}\nSince $\\inf_{t\\in\\mathbb{R}}\\inf_{x\\leq M_{2}+X(t)}u(t,x)>0$ by Lemma \\ref{lem-away-from-0}, there exist $C_{1}>0$ and $C_{2}>0$ such that\n\\begin{equation*}\nC_{1}\\leq\\frac{J\\ast u}{u}\\leq C_{2},\\quad x\\leq M_{2}+X(t)\\quad\\text{and}\\quad t\\in\\mathbb{R}.\n\\end{equation*}\nThen, setting $C_{3}=\\min\\{\\frac{1}{2},C_{1}\\}$ and $C_{4}=\\max\\{\\frac{3}{2},C_{2}\\}$, we have\n\\begin{equation*}\n-C_{4}\\leq-\\frac{J\\ast u}{u}\\leq-C_{3},\\quad(t,x)\\in\\mathbb{R}\\times\\mathbb{R}.\n\\end{equation*}\nWe then follow the arguments as in the proof of Theorem \\ref{thm-regularity-of-tf-ii} to conclude an estimate for $a_{2}^{\\eta}$ as in \\eqref{a-2}.\n\nThe rest of the proof can be done along the same line as in the proof of Theorem \\ref{thm-regularity-of-tf-ii} and then we complete the proof.\n\\end{proof}\n\n\n\\section{Proof of Theorem \\ref{thm-modified-interface-location} and Corollaries \\ref{cor-modified-interface}-\\ref{cor-modified-interface-ii}}\\label{sec-proof-thm-cor}\n\nIn this section, we prove Theorem \\ref{thm-modified-interface-location},  Corollary \\ref{cor-modified-interface} and Corollary \\ref{cor-modified-interface-ii}. We first prove Theorem \\ref{thm-modified-interface-location}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm-modified-interface-location}]\nLet $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$ as in the statement of Theorem \\ref{thm-modified-interface-location}. Then, by Theorem \\ref{lem-propagation-estimate} and the assumption, we have\n\\begin{equation}\\label{lower-upper-estimate}\nc_{1}(t-t_{0}-T_{1})\\leq X(t)-X(t_{0})\\leq c_{2}(t-t_{0}+T_{2}),\\quad t\\geq t_{0}.\n\\end{equation}\nWe modify $X(t)$ within two steps by means of \\eqref{lower-upper-estimate}. The first step gives a continuous modification. The second step gives the continuously differentiable modification as in the statement of the theorem. We remark that two inequalities in \\eqref{lower-upper-estimate} play different roles in the following arguments. While the first inequality in \\eqref{lower-upper-estimate} pushes $X(t)$ move to the right, the second inequality in \\eqref{lower-upper-estimate} controls the possible jumps of $X(t)$.\n\n\\paragraph{\\bf{Step 1.}} We show there is a continuous function $\\tilde{X}:\\mathbb{R}\\to\\mathbb{R}$ such that $\\sup_{t\\in\\mathbb{R}}|\\tilde{X}(t)-X(t)|<\\infty$. Fix some $T>0$. At $t=0$, let\n\\begin{equation*}\nZ^{+}(t;0)=X(0)+c_{2}(T+T_{2})+\\frac{c_{1}}{2}t,\\quad t\\geq0\n\\end{equation*}\nBy the second inequality in \\eqref{lower-upper-estimate}, $X(t)<Z^{+}(t;0)$ for all $[0,T]$. By the first inequality in \\eqref{lower-upper-estimate}, we have $X(t)>Z^{+}(t;0)$ for all large $t$. Define $T_{1}^{+}=\\inf\\{t\\geq0|X(t)\\geq Z^{+}(t;0)\\}$. By \\eqref{lower-upper-estimate}, it is easy to see that $T_{1}^{+}\\in[T,\\frac{c_{2}(T+T_{2})+c_{1}T_{1}}{c_{1}\/2}]$. At the moment $T_{1}^{+}$, $X(t)$ may jump, but, due to the second inequality in \\eqref{lower-upper-estimate}, the jump is at most $c_{2}T_{2}$. Thus, we obtain\n\\begin{equation*}\n\\begin{split}\nX(t)&<Z^{+}(t;0)\\,\\,\\text{for}\\,\\,t\\in[0,T_{1}^{+}),\\\\\nX(T_{1}^{+})&\\in[Z^{+}(T_{1}^{+};0)-c_{2}T_{2},Z^{+}(T_{1}^{+};0)+c_{2}T_{2}).\n\\end{split}\n\\end{equation*}\n\n\nNext, at $t=T_{1}^{+}$, let\n\\begin{equation*}\nZ^{+}(t;T_{1}^{+})=X(T_{1}^{+})+c_{2}(T+T_{2})+\\frac{c_{1}}{2}(t-T_{1}^{+}),\\quad t\\geq T_{1}^{+}.\n\\end{equation*}\nThen, $T_{2}^{+}=\\inf\\{t\\geq T_{1}^{+}|X(t)\\geq Z^{+}(t;T_{1}^{+})\\}$ is well-defined, and $T_{2}^{+}-T_{1}^{+}\\in[T,\\frac{c_{2}(T+T_{2})+c_{1}T_{1}}{c_{1}\/2}]$. Moreover, there hold\n\\begin{equation*}\n\\begin{split}\nX(t)&<Z^{+}(t;T_{1}^{+})\\,\\,\\text{for}\\,\\,t\\in[T_{1}^{+},T_{2}^{+}),\\\\ X(T_{2}^{+})&\\in[Z^{+}(T_{2}^{+};T_{1}^{+})-c_{2}T_{2},Z^{+}(T_{2}^{+};T_{1}^{+})+c_{2}T_{2}).\n\\end{split}\n\\end{equation*}\n\nRepeating the above arguments, we obtain the following, there is a sequence of times $\\{T_{n-1}^{+}\\}_{n\\geq1}$ satisfying $T_{0}^{+}=0$, $T_{n}^{+}-T_{n-1}^{+}\\in[T,\\frac{c_{2}(T+T_{2})+c_{1}T_{1}}{c_{1}\/2}]$  and\n\\begin{equation}\\label{an-estimate-0198932r592}\n\\begin{split}\nX(t)&<Z^{+}(t;T_{n-1}^{+})\\,\\,\\text{for}\\,\\,t\\in[T_{n-1}^{+},T_{n}^{+}),\\\\\nX(T_{n}^{+})&\\in[Z^{+}(T_{n}^{+};T_{n-1}^{+}),Z^{+}(T_{n}^{+};T_{n-1}^{+})+c_{2}T_{2}),\n\\end{split}\n\\end{equation}\nfor all $n\\geq1$, where $Z^{+}(t;T_{n-1}^{+})=X(T_{n-1}^{+})+c_{2}(T+T_{2})+\\frac{c_{1}}{2}(t-T_{n-1}^{+})$.\n\nWe define $Z^{+}:[0,\\infty)\\to\\mathbb{R}$ by setting\n\\begin{equation*}\nZ^{+}(t)=Z^{+}(t;T_{n-1}^{+}),\\quad t\\in[T^{+}_{n-1},T^{+}_{n}),\\quad n\\geq1\n\\end{equation*}\nSince $\\sup_{n\\geq1}[T^{+}_{n-1},T^{+}_{n})=[0,\\infty)$, $Z^{+}(t)$ is well-defined. It follows from \\eqref{an-estimate-0198932r592} that $X(t)<Z^{+}(t)$ for all $t\\geq0$. Moreover, for $t\\in[T^{+}_{n-1},T^{+}_{n})$,\n\\begin{equation*}\n\\begin{split}\nZ^{+}(t)-X(t)&\\leq X(T_{n-1}^{+})+c_{2}(T+T_{2})+\\frac{c_{1}}{2}(t-T_{n-1}^{+})-[X(T_{n-1}^{+})+c_{1}(t-T_{n-1}^{+}-T_{1})]\\\\\n&\\leq c_{2}(T+T_{2})-\\frac{c_{1}}{2}(t-T_{n-1}^{+})+c_{1}T_{1}\\leq c_{2}(T+T_{2})+c_{1}T_{1}.\n\\end{split}\n\\end{equation*}\nHence, $0<Z^{+}(t)-X(t)\\leq c_{2}(T+T_{2})+c_{1}T_{1}$ for all $t\\in[0,\\infty)$. Modifying $Z^{+}(t)$ near $T_{n-1}^{+}$ for $n\\geq1$, we find a continuous function $\\tilde{Z}^{+}:[0,\\infty)\\to\\mathbb{R}$ such that $\\sup_{t\\in[0,\\infty)}|\\tilde{Z}^{+}(t)-X(t)|<\\infty$.\n\nClearly, we can mimic the above arguments to find a continuous function $\\tilde{Z}^{-}:(-\\infty,0]\\to\\mathbb{R}$ such that $\\sup_{t\\in(-\\infty,0]}|\\tilde{Z}^{-}(t)-X(t)|<\\infty$. Combining $\\tilde{Z}^{\\pm}(t)$ and modifying near $0$, we find a continuous function $\\tilde{X}:\\mathbb{R}\\to\\mathbb{R}$ such that $\\sup_{t\\in\\mathbb{R}}|\\tilde{X}(t)-X(t)|<\\infty$.\n\n\\paragraph{\\bf{Step 2.}} By Step 1, we assume, without loss of generality, that $X(t)$ is continuous. We proceed as in Step 1.\n\nFix any $t_{0}\\in\\mathbb{R}$ and consider it as an initial moment. At the initial moment $t_{0}$, we define $Z(t;t_{0})=X(t_{0})+C_{0}+\\frac{c_{1}}{2}(t-t_{0})$ for $ t\\geq t_{0}$, where $C_{0}>0$ is so large that $C_{0}>c_{2}T_{2}$. Clearly, $X(t_{0})<Z(t_{0};t_{0})$. By the first inequality in \\eqref{lower-upper-estimate} and continuity, $X(t)$ will hit $Z(t;t_{0})$ sometime after $t_{0}$. Let $T_{1}(t_{0})$ be the first time that $X(t)$ hits $Z(t;t_{0})$, that is, $T_{1}(t_{0})=\\min\\big\\{t\\geq t_{0}\\big|X(t)=Z(t;t_{0})\\big\\}$. It follows that $X(t)<Z(t;t_{0})$ for $t\\in[t_{0},T_{1}(t_{0}))$ and $X(T_1(t_{0}))=Z(T_{1}(t_{0});t_{0})$. Moreover, $T_{1}(t_{0})-t_{0}\\in\\Big[\\frac{C_{0}-c_{2}T_{2}}{c_{2}-c_{1}\/2},\\frac{C_{0}+c_{1}T_{1}}{c_{1}\/2}\\Big]$, which is a simple result of \\eqref{lower-upper-estimate}.\n\nNow, at the moment $T_{1}(t_{0})$, we define $Z(t;T_{1}(t_{0}))=X(T_{1}(t_{0}))+C_{0}+\\frac{c_{1}}{2}(t-T_{1}(t_{0}))$ for $t\\geq T_{1}(t_{0})$. Similarly, $X(T_{1}(t_{0}))<Z(T_{1}(t_{0});T_{1}(t_{0}))$ and $X(t)$ will hit $Z(t;T_{1}(t_{0}))$ sometime after $T_{1}(t_{0})$. Denote by $T_{2}(t_{0})$ the first time that $X(t)$ hits $Z(t;T_{1}(t_{0}))$. Then, $X(t)<Z(t;T_{1}(t_{0}))\\,\\,\\text{for}\\,\\,t\\in[T_{1}(t_{0}),T_{2}(t_{0}))$ and $X(T_{2}(t_{0}))=Z(T_{2}(t_{0});T_{1}(t_{0}))$, and $T_{2}(t_{0})-T_{1}(t_{0})\\in\\Big[\\frac{C_{0}-c_{2}T_{2}}{c_{2}-c_{1}\/2},\\frac{C_{0}+c_{1}T_{1}}{c_{1}\/2}\\Big]$.\n\n\nRepeating the above arguments, we obtain the following: there is a sequence of times $\\{T_{n-1}(t_{0})\\}_{n\\in\\mathbb{N}}$ satisfying $T_{0}(t_{0})=t_{0}$ and\n\\begin{equation}\\label{uniform-time-interval}\nT_{n}(t_{0})-T_{n-1}(t_{0})\\in\\bigg[\\frac{C_{0}-c_{2}T_{2}}{c_{2}-c_{1}\/2},\\frac{C_{0}+c_{1}T_{1}}{c_{1}\/2}\\bigg],\\quad\\forall\\,\\,n\\in\\mathbb{N},\n\\end{equation}\nand  for any $n\\in\\mathbb{N}$\n\\begin{equation*\n\\begin{split}\nX(t)<Z(t;T_{n-1}(t_{0}))\\,\\,\\text{for}\\,\\,t\\in[T_{n-1}(t_{0}),T_{n}(t_{0}))\\quad\\text{and}\\quad X(T_{n}(t_{0}))=Z(T_{n}(t_{0});T_{n-1}(t_{0})),\n\\end{split}\n\\end{equation*}\nwhere $Z(t;T_{n-1}(t_{0}))=X(T_{n-1}(t_{0}))+C_{0}+\\frac{c_{1}}{2}(t-T_{n-1}(t_{0}))$. Moreover, for any $n\\in\\mathbb{N}$ and $t\\in[T_{n-1}(t_{0}),T_{n}(t_{0}))$, we conclude from \\eqref{lower-upper-estimate} that\n\\begin{equation*}\n\\begin{split}\n&Z(t;T_{n-1}(t_{0}))-X(t)\\\\\n&\\quad\\quad\\leq X(T_{n-1}(t_{0}))+C_{0}+\\frac{c_{1}}{2}(t-T_{n-1}(t_{0}))-\\big[X(T_{n-1}(t_{0}))+c_{1}(t-T_{n-1}(t_{0})-T_{1})\\big]\\\\\n&\\quad\\quad=C_{0}+c_{1}T_{1}-\\frac{c_{1}}{2}(t-T_{n-1}(t_{0}))\\leq C_{0}+c_{1}T_{1}.\n\\end{split}\n\\end{equation*}\n\nNext, define $\\tilde{Z}(\\cdot;t_{0}):[t_{0},\\infty)\\rightarrow\\mathbb{R}$ by setting\n\\begin{equation}\\label{definition-new-fun}\n\\tilde{Z}(t;t_{0})=Z(t;T_{n-1}(t_{0}))\\quad\\text{for}\\,\\, t\\in[T_{n-1}(t_{0}),T_{n}(t_{0})),\\,\\,n\\in\\mathbb{N}.\n\\end{equation}\nSince $[t_{0},\\infty)=\\cup_{n\\in\\mathbb{N}}[T_{n-1}(t_{0}),T_{n}(t_{0}))$ by \\eqref{uniform-time-interval}, $\\tilde{Z}(t;t_{0})$ is well-defined for all $t\\geq t_{0}$. Notice $\\tilde{Z}(t;t_{0})$ is strictly increasing and is linear on $[T_{n-1}(t_{0}),T_{n}(t_{0}))$ with slope $\\frac{c_{1}}{2}$ for each $n\\in\\mathbb{N}$, and satisfies\n\\begin{equation*}\n0\\leq\\tilde{Z}(t;t_{0})-X(t)\\leq C_{0}+c_{1}T_{1},\\quad t\\geq t_{0}.\n\\end{equation*}\n\nDue to \\eqref{uniform-time-interval}, we can modify $\\tilde{Z}(t;t_{0})$ near each $T_{n}(t_{0})$ for $n\\in\\mathbb{N}$ as follows. Fix some $\\delta_{*}\\in\\Big(0,\\frac{1}{2}\\frac{C_{0}-c_{2}T_{2}}{c_{2}-c_{1}\/2}\\Big)$. We modify $\\tilde{Z}(t;s)$ by redefining it on the intervals $(T_{n}(t_{0})-\\delta_{*},T_{n}(t_{0}))$, $n\\in\\mathbb{N}$ as follows: define\n\\begin{equation*}\nX(t;t_{0})=\\begin{cases}\n\\tilde{Z}(t;t_{0}),\\quad t\\in[t_{0},\\infty)\\backslash\\cup_{n\\in\\mathbb{N}}(T_{n}(t_{0})-\\delta_{*},T_{n}(t_{0})),\\\\\nX(T_{n}(t_{0}))+\\delta(t-T_{n}(t_{0})),\\quad t\\in(T_{n}(t_{0})-\\delta_{*},T_{n}(t_{0})),\\,\\,n\\in\\mathbb{N},\n\\end{cases}\n\\end{equation*}\nwhere $\\delta:[-\\delta_{*},0]\\rightarrow[-\\frac{1}{2}c_{1}\\delta_{*},C_{0}]$ is twice continuously differentiable and satisfies\n\\begin{equation*}\n\\begin{split}\n&\\delta(-\\delta_{*})=-\\frac{c_{1}}{2}\\delta_{*},\\quad\\delta(0)=C_{0},\\\\\n&\\dot{\\delta}(-\\delta_{*})=\\frac{c_{1}}{2}=\\dot{\\delta}(0),\\quad\\dot{\\delta}(t)\\geq\\frac{c_{1}}{2}\\,\\,\\text{for}\\,\\,t\\in(-\\delta_{*},0)\\quad\\text{and}\\\\\n&\\ddot{\\delta}(-\\delta_{*})=0=\\ddot{\\delta}(0).\n\\end{split}\n\\end{equation*}\nThe existence of such a function $\\delta(t)$ is clear. Moreover, there exist $c_{\\max}=c_{\\max}(\\delta_{*})>0$ and $\\tilde{c}_{\\max}=\\tilde{c}_{\\max}(\\delta_{*})>0$ such that $\\dot{\\delta}(t)\\leq c_{\\max}$ and $|\\ddot{\\delta}(t)|\\leq\\tilde{c}_{\\max}$ for $t\\in(-\\delta_{*},0)$. Notice the above modification is independent of $n\\in\\mathbb{N}$ and $t_{0}$. Hence, $X(t;t_{0})$ satisfies the following uniform in $t_{0}$ properties:\n\\begin{itemize}\n\\item $0\\leq X(t;t_{0})-X(t)\\leq d_{\\max}$ for some $d_{\\max}>0$,\n\\item $\\frac{c_{1}}{2}\\leq\\dot{X}(t;t_{0})\\leq c_{\\max}$,\n\\item $|\\ddot{X}(t;t_{0})|\\leq\\tilde{c}_{\\max}$.\n\\end{itemize}\nSince $X(t)$ is continuous, so locally bounded, we may apply Arzel\\`{a}-Ascoli theorem to conclude the existence of some continuously differentiable function $\\tilde{X}:\\mathbb{R}\\to\\mathbb{R}$ such that $X(t;t_{0})\\to \\tilde{X}(t)$ and $\\dot{X}(t;t_{0})\\to\\dot{\\tilde{X}}(t)$ locally uniformly in $t$ as $t_{0}\\to-\\infty$ along some subsequence. It's easy to see that $\\tilde{X}(t)$ satisfies all the properties as in the statement of the theorem.\n\\end{proof}\n\nNext, we prove Corollary \\ref{cor-modified-interface}. Recall that for a given transition front $u(t,x)$ of \\eqref{main-eqn}, $X^{\\pm}_{\\lambda}(t)$ are defined in \\eqref{defn-interface-locations}.\n\n\\begin{proof}[Proof of Corollary \\ref{cor-modified-interface}]\nWe modify the proof of Theorem \\ref{lem-propagation-estimate}. Let $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$. Since $f(t,x,u)\\leq0$ for $(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,\\tilde{\\theta}]$, we can find a function $f_{I}(u)$ such that $f(t,x,u)\\leq f_{I}(u)$ for $(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]$, where $f_{I}:[0,1]\\to\\mathbb{R}$ is $C^{2}$ and is of standard ignition type, that is, there exists $\\theta_{I}\\in(0,1)$ such that\n\\begin{equation*}\nf_{I}(u)=0,\\,\\,u\\in[0,\\theta_{I}]\\cup\\{1\\},\\quad f_{I}(u)>0,\\,\\,u\\in(\\theta_{I},1)\\quad\\text{and}\\quad f_{I}'(1)<0.\n\\end{equation*}\n\n\n\n\nFix some $\\lambda\\in(\\theta,1)$. We  write $X^{+}(t)=X^{+}_{\\lambda}(t)$. Since $\\sup_{t\\in\\mathbb{R}}|X(t)-X^{+}(t)|<\\infty$ by Lemma \\ref{lem-bounded-interface-width}, it suffices to show\n\\begin{equation}\\label{propagation-estimate-1-ig}\nX^{+}(t)-X^{+}(t_{0})\\leq c(t-t_{0}+T),\\quad t\\geq t_{0}\n\\end{equation}\nfor some $c>0$ and $T>0$.\n\nTo do so, we fix some $\\tilde{\\theta}_{I}\\in(0,\\theta_{I})$. Let $(c_{I},\\phi_{I})$ with $c_{I}>0$ be the unique solution of\n\\begin{equation*}\n\\begin{cases}\nJ\\ast\\phi-\\phi+c\\phi_{x}+f_{I}(\\phi)=0,\\\\\n\\phi_{x}<0,\\,\\,\\phi(0)=\\theta_{I},\\,\\,\\phi(-\\infty)=1\\,\\,\\text{and}\\,\\,\\phi(\\infty)=\\tilde{\\theta}_{I}.\n\\end{cases}\n\\end{equation*}\nNote that $\\phi_{I}$ connects $\\tilde{\\theta}_{I}$ and $1$ instead of $0$ and $1$ (see Appendix \\ref{app-ig-tw} for more properties about $\\phi_{I}$; in Appendix \\ref{app-ig-tw}, we consider traveling waves connecting $0$ and $1$, but by simple shift, all results there apply here).\n\nLet $u_{0}:\\mathbb{R}\\to[0,1]$ be a uniformly continuous and nonincreasing function satisfying $u_{0}(x)=1$ for $x\\leq 0$ and $u_{0}(x)=\\tilde{\\theta}_{I}$ for $x\\geq x_{0}$, where $x_{0}>0$ is fixed. Clearly, $u(t_{0},\\cdot+X^{+}_{\\tilde{\\theta}_{I}}(t_{0}))\\leq u_{0}$. Applying comparison principle and Lemma \\ref{lem-ignition-property}, we find\n\\begin{equation*}\nu(t,x+X^{+}_{\\tilde{\\theta}_{I}}(t_{0}))\\leq u_{I}(t-t_{0},x;u_{0})\\leq\\phi_{I}(x-c_{I}(t-t_{0})-\\xi_{I})+\\epsilon_{I}e^{-\\omega_{I}(t-t_{0})},\\quad x\\in\\mathbb{R},\\quad t\\geq t_{0}.\n\\end{equation*}\nLet $\\xi_{I}(\\frac{\\lambda}{2})$ be the unique point such that $\\phi_{I}(\\xi_{I}(\\frac{\\lambda}{2}))=\\frac{\\lambda}{2}$ and $T>0$ be such that $\\epsilon_{I}e^{-\\omega_{I}T}=\\frac{\\lambda}{2}$ (we may make $\\epsilon_{I}>\\frac{\\lambda}{2}$ if necessary). Setting $x_{*}=c_{I}(t-t_{0})+\\xi_{I}+\\xi_{I}(\\frac{\\lambda}{2})$, we conclude from the monotonicity of $\\phi_{I}$ that for $x\\geq x_{*}+1$ and $t\\geq t_{0}+T$,\n\\begin{equation*}\n\\begin{split}\nu(t,x+X^{+}_{\\tilde{\\theta}_{I}}(t_{0}))&\\leq\\phi_{I}(x_{*}+1-c_{I}(t-t_{0})-\\xi_{I})+\\epsilon_{I}e^{-\\omega_{I}T}\\\\\n&<\\phi_{I}(x_{*}-c_{I}(t-t_{0})-\\xi_{I})+\\epsilon_{I}e^{-\\omega_{I}T}=\\lambda.\n\\end{split}\n\\end{equation*}\nIt then follows from the definition of $X^{+}(t)$ that\n\\begin{equation*}\nX^{+}(t)\\leq x_{*}+1+X^{+}_{\\tilde{\\theta}_{I}}(t_{0})=c_{I}(t-t_{0})+\\xi_{I}+\\xi_{I}(\\frac{\\lambda}{2})+1+X^{+}_{\\tilde{\\theta}_{I}}(t_{0}),\\quad t\\geq t_{0}+T.\n\\end{equation*}\nSetting $C_{*}:=\\sup_{t_{0}\\in\\mathbb{R}}|X^{+}(t_{0})-X^{+}_{\\tilde{\\theta}_{I}}(t_{0})|<\\infty$ due to Lemma \\ref{lem-bounded-interface-width}, we conclude\n\\begin{equation*}\nX^{+}(t)-X^{+}(t_{0})\\leq c_{I}(t-t_{0})+\\xi_{I}+\\xi_{I}(\\frac{\\lambda}{2})+1+C_{*},\\quad t\\geq t_{0}+T.\n\\end{equation*}\n\nIt remains to show that\n\\begin{equation}\\label{what-to-show}\nX^{+}(t)-X^{+}(t_{0})\\leq\\xi_{*},\\quad t\\in[t_{0},t_{0}+T]\n\\end{equation}\nfor some $\\xi_{*}>0$ independent of $t_{0}$. To do so, let $\\tilde{u}_{0}$ be the $u_{0}$ in the proof of Theorem \\ref{lem-propagation-estimate}. Then, we have $\\tilde{u}_{0}(\\cdot-X^{-}(t_{0}))\\leq u(t_{0},\\cdot)\\leq u_{0}(\\cdot-X^{+}_{\\tilde{\\theta}_{I}}(t_{0}))$, where $X^{-}(t)=X^{-}_{\\lambda}(t)$. Since $f_{B}\\leq f\\leq f_{I}$, we apply comparison principle to conclude that\n\\begin{equation*}\nu_{B}(t-t_{0},x-X^{-}(t_{0});\\tilde{u}_{0})\\leq u(t,x)\\leq u_{I}(t-t_{0},x-X^{+}_{\\tilde{\\theta}_{I}}(t_{0});u_{0}),\\quad x\\in\\mathbb{R},\\quad t\\geq t_{0}.\n\\end{equation*}\nWe then conclude \\eqref{what-to-show} from the continuity of $u_{B}(t-t_{0},x-X^{-}(t_{0})$ and $u_{I}(t-t_{0},x-X^{+}_{\\tilde{\\theta}_{I}}(t_{0});u_{0})$, and the fact $\\sup_{t_{0}\\in\\mathbb{R}}|X^{-}(t_{0})-X^{+}_{\\tilde{\\theta}_{I}}(t_{0})|<\\infty$ due to Lemma \\ref{lem-bounded-interface-width}. This completes the proof.\n\\end{proof}\n\nFinally, we prove Corollary \\ref{cor-modified-interface-ii}.\n\n\\begin{proof}[Proof of Corollary \\ref{cor-modified-interface-ii}]\nNote first we can find a $C^{2}$ Fisher-KPP nonlinearity $f_{\\rm KPP}:[0,1]\\to\\mathbb{R}$ such that $f(t,x,u)\\leq f_{\\rm KPP}(u)$ for all $(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]$. Let $u(t,x)$ be the transition front as in the statement of Corollary \\ref{cor-modified-interface-ii}, that is, there exist $r>0$ and $h>0$ such that\n\\begin{equation*}\nu(t_{0},x+X(t_{0}))\\leq e^{-r(x-h)},\\quad (t_{0},x)\\in\\mathbb{R}\\times\\mathbb{R},\n\\end{equation*}\n\nFixed $\\lambda\\in(0,1)$. Setting $h_{0}:=h+\\sup_{t_{0}\\in\\mathbb{R}}|X(t_{0})-X^{+}_{\\lambda}(t_{0})|<\\infty$, we find\n\\begin{equation*}\nu(t_{0},x+X^{+}_{\\lambda}(t_{0})+h_{0})\\leq e^{-rx},\\quad (t_{0},x)\\in\\mathbb{R}\\times\\mathbb{R}.\n\\end{equation*}\nThen, we can find some uniformly continuous function $u_{0}:\\mathbb{R}\\to[0,1]$ satisfying\n\\begin{equation*}\n\\lim_{x\\to-\\infty}u_{0}(x)=1\\quad\\text{and}\\quad\\lim_{x\\to\\infty}\\frac{u_{0}(x)}{e^{-rx}}=1\n\\end{equation*}\nsuch that\n\\begin{equation*}\nu(t_{0},x+X^{+}_{\\lambda}(t_{0})+h_{0})\\leq u_{0}(x),\\quad (t_{0},x)\\in\\mathbb{R}\\times\\mathbb{R}.\n\\end{equation*}\nNote that we may assume, without loss of generality, that $r$ is so small that it is the decay rate of some traveling wave of $\\phi_{r}(x-c_{r}t)$ (satisfying $\\phi_{r}(-\\infty)=1$ and $\\phi_{r}(\\infty)=0$) with speed $c_{r}=\\frac{\\int_{\\mathbb{R}}J(y)e^{ry}dy-1+f_{\\rm KPP}'(0)}{r}>0$ of\n\\begin{equation}\\label{eqn-kpp-homo}\nu_{t}=J\\ast u-u+f_{\\rm KPP}(u),\n\\end{equation}\nthat is, $\\lim_{x\\to\\infty}\\frac{\\phi_{r}(x)}{e^{-rx}}=1$ (see \\cite{CaCh04} and \\cite{ShZh12-2}). In particular, we have\n\\begin{equation}\\label{exact-decay-rate-2015201}\n\\lim_{x\\to\\infty}\\frac{u_{0}(x)}{\\phi_{r}(x)}=1.\n\\end{equation}\nMoreover, there holds\n\\begin{equation}\\label{exact-decay-rate-20152015}\n\\lim_{x\\to\\infty}\\frac{\\phi_{r}'(x)}{\\phi_{r}(x)}=-r.\n\\end{equation}\nTo see this, we notice $\\frac{J\\ast\\phi_{r}}{\\phi_{r}}-1+c_{r}\\frac{\\phi_{r}'}{\\phi_{r}}+\\frac{f_{\\rm KPP}(\\phi_{r})}{\\phi_{r}}=0$. Clearly, $\\lim_{x\\to\\infty}\\frac{f_{\\rm KPP}(\\phi_{r}(x))}{\\phi_{r}(x)}=f_{\\rm KPP}'(0)$. For $\\frac{J\\ast\\phi_{r}}{\\phi_{r}}$, we have\n\\begin{equation*}\n\\frac{[J\\ast\\phi_{r}](x)}{\\phi_{r}(x)}=\\frac{e^{-rx}}{\\phi_{r}(x)}\\int_{\\mathbb{R}}J(y)e^{ry}\\frac{\\phi_{r}(x-y)}{e^{-r(x-y)}}dy\\to J(y)e^{ry}dy\\quad\\text{as}\\quad x\\to\\infty\n\\end{equation*}\nby \\eqref{exact-decay-rate-2015201} and dominated convergence theorem. From which, we conclude \\eqref{exact-decay-rate-20152015}.\n\nThen, arguing as in the proof of Corollary \\ref{cor-modified-interface}, we conclude the result from the stability of $\\phi_{r}(x-c_{r}t)$, that is,\n\\begin{equation}\\label{stability-kpp}\n\\lim_{t\\to\\infty}\\bigg|\\frac{u_{\\rm KPP}(t,x;u_{0})}{\\phi_{r}(x-c_{r}t)}-1\\bigg|=0,\n\\end{equation}\nwhere $u_{\\rm KPP}(t,x;u_{0})$ is the solution of \\eqref{eqn-kpp-homo} with initial data $u_{\\rm KPP}(0,\\cdot;u_{0})=u_{0}$. We remark that \\eqref{stability-kpp} follows from \\cite[Theorem 2.6]{ShZh12-2}. Also, by means of \\eqref{exact-decay-rate-2015201} and \\eqref{exact-decay-rate-20152015}, it can be proven as that of \\cite[Theorem 1.3]{ShSh14-kpp}.\n\\end{proof}\n\n\n\n\\section*{Acknowledgements} \n\nThe authors would like to thank the referee for carefully reading the manuscript, pointing out some problems that we were not aware of, and drawing our attention to the reference \\cite{BaChm99}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}