This paper addresses the budget constrained and delay-bounded placement problem (BCDP) for roadside units in vehicular ad hoc networks. There are two types of RSUs: cable connected RSU (c-RSU) and wireless RSU (w-RSU). c-RSUs are interconnected through wired lines, and they form the backbone of VANETs, while w-RSUs connect to other RSUs through wireless communication and serve as an economical extension of the coverage of c-RSUs. The delay-bounded coverage range and deployment cost of these two cases are totally different. We are given a budget constraint and a delay bound, the problem is how to find the optimal candidate sites with the maximal delay-bounded coverage to place RSUs such that a message from any c-RSU in the region can be disseminated to the more vehicles within the given budget constraint and delay bound. We first prove that the BCDP problem is NP-hard. Then we propose several algorithms to solve the BCDP problem. Simulation results show the heuristic algorithms can significantly improve the coverage range and reduce the total deployment cost, compared with other heuristic methods.In vehicular  A vehicormation ,3, geocaormation  and dataormation ,6. Meanwormation –9, conteormation –13, and ormation ,15. In aSeveral RSUs or AP placement algorithms have been proposed –19. MostThe problem of our concern is as follows. We are given a set of candidate sites in a region, each of which can be installed a c-RSU or a w-RSU (or none). The candidate site is usually located at the intersection of roads or along the road, such as a lamp post or a tree. Given a budget constraint and a specified delay bound, the problem is how to place the c-RSUs and w-RSUs with the maximum delay-bounded coverage, such that a message sent from the c-RSUs can be disseminated to more vehicles in the region within the delay bound.We propose a delay-bounded coverage model for RSUs placement, in which each candidate site has an attached delay-bounded coverage by installing a RSU. This model inherently provides a good balance between delay and coverage. Base on this model, we formulate the BCDP problem to a budgeted maximum coverage problem and prove that this problem is NP-hard.We develop two heuristic algorithms under delay-bounded coverage model. In this model, we greedily find the optimal candidate site with a maximum delay bounded coverage gain and utility until no further installing can be made due to the budget constraint.We conduct an extensive simulation study over different data to evaluate the performances of the proposed methods. Simulation results show our heuristic algorithms can significantly improve the coverage range and reduce the total installation cost, compared with other heuristic methods.There are several challenges for solving the budget constrained and delay-bounded problem in a VANET. First, we consider the placement of hybrid RSUs, that for each candidate site we need to decide what type of RSUs to choose, which is more complicated than the placement of the same type of RSUs; Second, we study the placement of RSUs for the delay bounded information dissemination, which is required by many real-time applications in vehicular networks. The main contributions of the paper are as follows:The rest of this paper is organized as follows. Previous studies are summarized in Section 2. We present the system model and assumption in Section 3. The detailed description of our delay-bounded coverage model and our greedy algorithms are in Sections 4 and 5, respectively. In Section 6, we evaluate our proposed approaches by comparing them with different simulation environment. Our conclusion is in Section 7.2.2.1.ad hoc networks [et al. [et al. [et al. [et al. [et al. [et al. [et al. [et al. [Recently, the research have attracted a lot of attention on the data delivery and message dissemination in vehicular networks –25. As tnetworks –7,26,27. [et al.  investig [et al.  exploite [et al. ,9, the a [et al.  studied  [et al.  exploite [et al.  proposed [et al. –30. Skor [et al.  addresse [et al. , the aut2.2.et al. [et al. [There are some published work about the node and AP placement problem in wireless sensor networks –38. Theset al.  propose et al.  also stu [et al.  investiget al. [et al. [et al. [et al. [Previous work addressed the problem of relay nodes placement in homogeneous wireless sensor networks. Han et al.  addresseet al. ,37, the  [et al.  addresse [et al. –41. Zhan [et al.  studied 2.3.ad hoc networks [Meanwhile, there are some published work related to the problem of RSUs or stationary relay nodes placement in vehicular networks –19,42–45networks , authorsnetworks , where tnetworks , the autnetworks , the autnetworks ,45 are mnetworks  studied The difference between the above work with ours is that, we consider the delay-bounded placement problem, rather than minimize the number of RSUs in vehicular networks. Meanwhile, we consider the optimal placement scheme for the both infrastructure-based and infrastructure-less RSUs. To the best of our knowledge, our work is the first to study this problem in the literature.3.We assume all c-RSUs have the same communication range, and the communication range is larger than the w-RSUs. The warning information, such as traffic accident or vehicle jam, disseminates from the traffic control center(TCC) to all c-RSUs firstly. TCC is a centralized controller that maintains the city's traffic. Then, the message spreads to the w-RSUs through wireless link. In the end, the message floods from c-RSUs and w-RSUs to all the vehicles.In the case of c-RSUs, we assume the information delay from TCC to all c-RSUs doesn't take any time (there is no delay). The assumption is acceptable as all c-RSUs have the wired network connectivity to the TCC, and the information delay almost can be ignored.In the case of w-RSUs, we further assume that a w-RSU only receives the message from a nearby c-RSU or w-RSU, but never gets the message from a passing vehicle. Once receiving a message, a w-RSU would broadcast the message immediately. Thus, a w-RSU is connected to other RSUs to serve as an effective extension to the backbone formed by c-RSUs.For information dissemination, the vehicle receives a message from an RSU when it passes the RSU, and retrieves the “fresh” messages stored in the RSU and carry it forward for further dissemination. A moving vehicle on the road can also pass its messages to other vehicles via the OBUs equipped them. In this case, we call it the carry-and-forward delay. Due to carry-and-forward delay several orders of magnitude longer than the communication delay, we assume that the communication delay also approximately equals 0. The assumption is acceptable as the unit of carry-and-forward delay is a second and the unit of communication delay is a millisecond.A VANET system consists of a set of vehicles equipped with on-board units (OBUs) and a set of roadside units (RSUs) that are installed at the road side. The system model and assumption are as follows:G = , while the vertex set V (G) = {v1, v1, …, mv} consists of all candidate sites, and the number of candidate sites is m; the edge set E(G) = {e1, e1, …, ne} consists of the entire road segments, and the number of road segments is n. Each edge is associated with a delay weight ijd, which is the expected delay from iv to its adjacent vertex jv. That means iv is connected to jv an edge in a road graph, and we call iv and jv are neighbors. According to the [R is the wireless transmission range, ijρ is the vehicle density between road e, c is the average one-hop packet transmission delay, ijl is the Euclidean distance of road e, and s̄ is the average vehicle speed on road e.We model a road network as a undirected weighted road graph g to the ,25, we hBv and Ev.As shown in 4.In this section, we present the delay-bounded coverage model based on the road graph. In the model, we discuss in detail how to calculate the delay-bounded coverage by installing RSUs. Then, we formally formulate the BCDP problem based on the delay-bounded coverage model.4.1.i associated with the delay-bounded coverage, we calculate the information propagation delay first. Let ijD denote the expected information dissemination delay for broadcasting a message to all vehicles on the path between any candidate sites i and j. When iv and jv are neighbors, ijD equals to the delay ijd. If i and j do not have a road directly connecting them, ijD is the shortest path between iv and jv. Notice that ijD is calculated by assuming there is no RSU installed in the middle along the route between i and j. Let N(i) denote all the neighbors of node i, we calculate ijD as follows:As the benefit of candidate site Next, we introduce the calculation of delay-bounded coverage for installing a i-RSU or a w-RSU, respectively. Then, we calculate the placement cost.4.2.Now we calculate the delay-bounded coverage for installing an RSU. We give some definitions as follows.i, denoted byCommunication Set of vertex v(i), i, any vertex inis a set of vertex that if we install an RSU in site v(i) is within the communication range of the vertex vi.iv, and the candidate site jv is within the communication range of iv, then we have jv ∈ (i). We denote c(i) and w(i) as the communication set of installing a c-RSU and a w-RSU at site iv, respectively.If we install an RSU in site i, denoted byDelay-bounded Coverage of a vertex v(i), i, a vehicle at any edge eis a set of edge that if we install a c-RSU in site v ∈ i would receive the message within the delay bound T.According to Definition 2, we havec(i) and w(i) as the delay-bounded coverage of installing a c-RSU and a w-RSU at site iv, respectively. We denote c and w as the c-RSUs placement set and w-RSUs placement set, respectively. We denote  = c ∪ w as the RSUs placement set. Then we give the following definition.We denote Communication Set of placement set, denoted by, is a set of placement setthat the expected end – end propagation delay from one vertex to any other vertex which are both in setis zero. of placement set  iteratively. We denote (k) as the communication set value of k-th iteration. Initially, the placement set  = ∅ and the communication set  = ∅. When we place an RSU at site i, the placement set  = {iv}. The communication set  equals to the communication set of iv. That is j, then the placement set  = {i, vjv}. Due to that we don't consider the delay between wired line, site i receives the information from site j in no time. Therefore, the communication set  equals to the union set of transmission range set of jv, that is According to the Definition 3, we calculate the communication set Delay-bounded Coverage of placement set, denoted by, is a edge set of placement setthat a vehicle at any edge e ∈ would receive the message within the delay bound T.According to the Definition 4, we have:(k) as the delay-bounded coverage of the k-th iteration. v1 to v9, and we model the system as a undirected weighted road graph G. In graph G, each edge has the information dissemination delay of weight 1 unit and each RSU has also 1 unit communication range. Then, we could use v1, and we obtain the placement set  = c = {v1}. Since vertex v1, v2 and v4 are within the communication range of v1, the communication set T = 1 unit. Then, we could derive the delay-bounded coverage v6 as shown in v1 and v6 due to cable connected between them. As the placement set is  = {v1, v6} and the communication set of v6 is c(6) = {v3, v5, v6, v9}, we iteratively derive the communication set v6 is c(6) = {e, e, e, e, e, e, e}, and the delay-bounded coverage of this iteration is  = c = {v1, v6}. As shown in w = {v2} and c = {v1}, and the black thick lines are the delay-bounded coverage road segments in this iteration.Where 4.3.Given a budget constraint B, a delay bound T, a set of candidate sites, our goal is to find a candidate sites optimal set such that provides maximum delay-bounded coverage in a delay bound T with the given budget constraint B.Our problem can be viewed as a Budget Constrained and Delay-bounded RSUs Placement problem (BCDP for short) over road graphs, that is: ic and i to be installed a c-RSU and a w-RSU respectively Obviously, the placement cost of installing c-RSUs or w-RSUs is totally different. It is known that each c-RSU installation with power and wired network connectivity can cost as high as US$5000 [X = {x1, x2, …,nx} denote whether site i will be installed a c-RSU or not. If site i is selected to install a c-RSU then ix = 1, otherwise ix = 0. In a similar way, we denote Y = {y1, y2,…,ny} whether site i will be installed a w-RSU or not.We denote  US$5000 . Let X =Therefore, our objective is to maximize the delay-bounded coverage of placement RSUs in the region, that is:B. Constraint x, y as binary. Constraint Constraint The BCDP problem is difficult to solve, as shown in the following theorem.The BCDP problem is NP-hard.To provide the evidence of NP-hardness, we provide a polynomial reduction from a budgeted maximum coverage problem  to the BU; S; c): Let U = {e1, e2, …ne} denote a set of n elements, S consists of finite sets S1, S2, …, kS ⊆ U, and a cost function c : S → Q+. The objective is to find a subset S′ ⊆ S, such that the total cost of elements in S′ does not exceed a given budget B, and the total weight of elements covered by S′ is maximized.A budgeted maximum coverage problem  and wg(i) as the delay-bounded coverage gain of installing a c-RSU and a w-RSU at site i, respectively. Then, we have:As proven in Theorem 1, we reduce the BCDP problem to the budgeted maximum coverage and prove that the problem is NP-hard. Due to the monotonicity of maximum delay-bounded coverage, the greedy algorithm is a competitive candidate. As our first greedy algorithm, we can select the candidate sites that maximise the delay-bounded coverage gain in each iteration. When we consider to install an RSU at site ijD between any two candidate sites in Line 2, then we calculate the delay bounded coverage gain of all candidate sites for both installing c-RSUs and w-RSUs. This is accomplished in Line 4–6 and Line 7–9, respectively. After that, we find best node which has the maximal delay bounded coverage gain. This is accomplished in Line 10. Then, we determine which type of RSUs installation. In Line 11–13, we install the c-RSUs as the delay-bounded gain of the selected c-RSU is the maximum one and the total cost is not exceed the budget. If the total cost is exceed the budget, the algorithm chooses a maximal delay-bounded gain for w-RSU installation in Line 14–16. In Line 18–20, we install the w-RSUs as the delay-bounded gain of the selected w-RSU is the maximum one and the total cost is not exceed the budget. After that, we add the best node to the set of placement RSUs, and remove all the delay-bounded coverage from the set of candidate sites in Line 21–24. Finally, in line 26, we identify the placement RSUs set of our solution.Our gain greedy algorithm for BCDP problem is presented as Algorithm 1. The major steps of Algorithm 1 are as follows. First, we compute the expect information dissemination delay Algorithm 1 Gain greedy algorithmInput:G = : The weighted road graph;T: The delay bound;B: The budget constraint;Output:c: The placement set of installing c-RSUs;w: The placement set of installing w-RSUs;1: = ∅,  = ∅;2:i, j ∈ V, initial ijD according to For all 3:while the placement cost is not exceed the budget Band ≠ |E| do4:for all unselected iv ∈ Vdo 5:cg(i) according to   Calc c-RSUs delay-bounded coverage gain 6:end for 7:for all unselected iv ∈ U and iv is with the transmission range of an existing RSU do 8:wg(i) according to   Calc w-RSUs delay-bounded coverage gain 9:end for 10:v* = {iv ∣ max{cu(l), wu(k)}, ∀iv ∈ V }; 11:if install a c-RSU and the placement cost is not exceed the budget Bthen 12:c = c ∪{v*};   13:else 14:if install a c-RSU and the placement cost is exceed the budget Bthen  15:v* = {iv ∣ max{wu(k)}, ∀iv ∈ V };    16:end if  17:end if 18:if install a w-RSU and the placement cost is not exceed the budget Bthen 19:w = w ∪{v*};  20:end if 21:if install a c-RSU or w-RSU and the placement cost is not exceed the budget Bthen 22:V = V \ {v*};  23: =  ∪ v*;   24:end if 25:end while26:returnc, w5.2.In the gain greedy algorithm, we aim at finding the maximum delay-bounded coverage gain candidate site to install an RSU. However, the placement cost of each candidate sites is also important. This is because, although a candidate site may offer a large delay-bounded gain, it may also have a huge cost such that other candidate sites cannot be installed any c-RSU.ic and iu as the cost and delay-bounded coverage utility of installing a c-RSU in site i, respectively. Then, we have:Consequently, our second greedy algorithm is to calculate the gain per unit cost for each iteration. We call it delay-bounded coverage utility. We denote In utility greedy algorithm, we aim at finding the maximum delay-bounded coverage utility candidate site to install an RSU. Therefore, our utility greedy algorithm for BCDP is presented as Algorithm 2.Algorithm 2 Utility Greedy algorithmInput:G = : The weighted road graph;T: The delay bound;B: The budget constraint;Output:c: The placement set of installing c-RSUs;w: The placement set of installing w-RSUs;1: = ∅,  = ∅;2:i, j ∈ V, initial ijD according to For all 3:while the placement cost is not exceed the budget Bdo4:for all unselected iv ∈ Vdo 5:cu(i) according to   Calc c-RSUs delay-bounded coverage utility 6:end for 7:for all unselected iv ∈ U and iv is with the transmission range of an existing RSU do 8:wu(i) according to   Calc w-RSUs delay-bounded coverage utility 9:end for 10:v* = {iv ∣ max{cu(l), wu(k)}, ∀iv ∈ V }; 11:if install a c-RSU then 12:c = c ∪ {v*};  13:end if 14:if install a w-RSU then 15:w = w ∪ {v*};  16:end if 17:V = V \ {v*}; 18: =  ∪ v*; 19:end while20:returnc, wG has 36 candidate sites (v0 ∼ v35), which are numbered from the left to the right and from the top to the bottom. Graph G includes 60 road segments. Suppose the deployment cost of c-RSU and w-RSU is 2.5 and 1, respectively. The transmission range of a c-RSU and w-RSU are set to two road segments and one road segment, respectively. We assume the placement budget constrain is 5. We further assume that the expected carry-and-forward delay of each road segment is 1min, and the delay bound Δ is also set to 1min. v14 is selected to install a c-RSU, since the corresponding delay-bounded gain g(14) = 34 and delay-bounded coverage utility u(14) = 34/2.5 are both the maximum one. The 34 black thick lines are the road segments included in v14}{. So in the first iteration, we choose v14 to instal a c-RSU greedily for both Algorithms 1 and 2. v22 is chosen to install a c-RSU as the delay-bounded gain g(22) = 14 is the maximum one, where the red thick lines are the newly delay-bounded covered road segments. Meanwhile, the placement cost for installing this two c-RSUs is 5 and it is not exceed the budget. v16 is chosen to install a w-RSU, where the red thick lines are the newly delay-bounded covered road segments. Now the total cost is 3.5 and it is not over the budget. Note that if the budget is set to 4, the cost will be over budget for gain greedy algorithm in the second iteration. Therefore, we should not choose c-RSU to install but choose w-RSU to install. That is, we will choose v16 to install a w-RSU for gain greedy algorithm as shown in 5.3.OPT as the optimal delay-bounded coverage. Let c(k) as the placement cost of the k-th iteration. Before we derive the approximation ratio, we have following lemma first.Next, we analyze the approximation ratio of the Algorithm 2. We define After each iteration k, k = 1, …, l + 1, the following inequality holds.After each iteration k, k = 1, …, l + 1, the following inequality holds.The detailed proof of Lemmas 1 and 2 can be found in .The approximation ratio of the Algorithm 2 is (1 − 1/e).l-th iteration. In the (l + 1)-th iteration, the placement cost is c(l + 1). As adding the c(l + 1) to B, we have:Let l + 1)-th round, according to l + 1)-th iteration, we get:As our greedy algorithm runs  · L), where |V| is the number of candidate sites, |E| is the number of road segments and L is running rounds of Algorithm 2.ijD. The time complexity of ijD is O(|V|3) as we use Floyd-Warshall algorithm to compute the shortest path between any two candidate sites. The size of all unselected candidate sites is no more than |V|, the calculation of delay-bounded coverage will run L rounds and the calculation of delay-bounded coverage gain is no more than |E|. Therefore, The time complexity of Line 4–6 is O(|V| · |E| · L). In Line 7–9 of Algorithm 2, the time complexity is also O(|V| · |E| · L) as the number of candidate sites for w-RSUs is no more than |V|. Next, it iteratively chooses the site which has the maximum delay-bounded coverage utility in line 10 and the time complexity is O(|V|2 ·L). Therefore, the total time complexity is O(|V|3 + |V| · (|V| + |E|) · L).Line 4–6 of Algorithm 2 computes the cost effectiveness for c-RSU placement. The calculation of delay-bounded coverage needs to compute 6.In this section, we evaluate the performance of our proposed algorithms, and present the simulation results of the algorithms in different situations.6.1.G. The candidate sites number is |V|, and the number of road segments is |E|. The length of road segment is set to 500 m, which is regarded as the length unit in the simulation. Therefore, a 10 × 10 network means the region has 10 × 10 evenly distributed candidate sites, with all sides 500 m × 10 = 5000 m long. The expected carry-and-forward delay of each road segment is set to 5 min. Since we consider the epidemic forwarding messages, we assume there is no transmission conflict and both vehicles and RSUs have infinite buffer to store packets. We define delay-bounded coverage ratio as the as the percentage of road segments that can be delay-bounded covered by the placed RSUs. We first study the delay-bounded coverage ratio, delay-bounded utility and number of RSUs under different candidate sites, different budget constraint, different cost ratio, different delay bounds and different transmission ranges ratio. We consider several algorithms as follows.In the synthetic simulation, we model the street layout of urban areas by a grid graph Random algorithm for hybrid RSUs installation (Random): In this algorithm, we randomly choose to install either c-RSUs or w-RSUs. In each iteration, we randomly install an RSU.Greedy-based algorithm for only c-RSUs installation (c-RSUs): In this algorithm, we greedily install and only install c-RSUs.Our gain greedy algorithm (Gain): Our proposed delay-bounded gain greedy algorithm for BCDP problem, and we greedily choose the maximum gain to install either c-RSUs or w-RSUs.Our utility greedy algorithm (Utility): Our proposed delay-bounded greedy algorithm for BCDP problem, and we greedily choose the maximum utility to install either c-RSUs or w-RSUs.6.2.We investigate how the changes of candidate sites impact the performance of the different algorithms. 6.3.6.4.We investigate how the changes of cost ratio impact the performance of the different algorithms. 6.5.We investigate how the changes of delay bound impact the cost, and the number of RSUs in 6.6.The transmission range ratio is defined as the ratio between the transmission range of c-RSUs and the transmission range of w-RSUs. 6.7.In the realistic simulation, we use the SUMO(Simulation of Urban MObility)  to conve6.8.m, the transmission range of c-RSUs is 1 km, the cost ratio is 10 and the delay bound is 5 min. As placement cost of installing a c-RSU is 10 unit, we could only install one c-RSU for the only installing c-RSUs algorithm. So we install a c-RSU with maximum gain as shown in Next, we plot the RSUs placement result under different budget constraints. We first plot the RSUs placement result of three algorithms when budget constraint is set to 15. As shown in Then, we plot the RSUs placement result when budget constraints is set to 25 unit in full coverage. Then, we do not need to place any more RSUs. Therefore, we place one c-RSU and eighteen w-RSUs at the end of the utility greedy algorithm.In full coverage for our the gain greedy algorithm when budget is over 45.In full coverage in full coverage. Obvious, the c-RSUs algorithm to cover boundary road segments is not very cost-effective. For our gain greedy algorithm, we have two placement strategy. As shown in full coverage. While budget is 51 or above 51, we place five c-RSUs and one w-RSU and we also obtain the full coverage as shown in Finally, we plot the RSUs placement result with 7.ad hoc networks, which finds maximum delay-bounded coverage and meanwhile a message sent from the c-RSUs can be disseminated to more vehicles in a given delay bound. First, we propose a delay-bounded coverage model for RSUs placement, which inherently provides a good balance between delay and coverage. Next, we formulate BCDP problem to the budgeted maximum coverage problem and prove that this problem is NP-hard. Finally, we propose two heuristic algorithms to solve the BCDP problem. Simulation results show effectiveness of our methods under different environments. As a future work, we will explore how to effectively deploy RSUs in VANET under different forwarding models.In this paper, we investigate the budget constrained and delay-bounded RSUs placement problem in vehicular


We also analyze the average SNR condition such that the multiuser diversity gainis fully exploited for two-hop transmission. The proposed two-step scheduling algorithm exhibits thequite comparable achievable rates to those of the optimal one using full feedback information, while itsrequired feedback information is reduced by half of the optimal one.We introduce an efficient multiuser scheduling method using amplify-and-forward relaying in relay-aideddownlink systems, consisting of one base station (BS), one relay station, and multiple mobilestations (MSs). In our scheme, the BS opportunistically selects both the transmission mode, that is, eitherone- or two-hop transmission, and the desired user . Closed-form expressions forthe average achievable rates are derived for the two transmission modes with multiuser scheduling, andits asymptotic solutions are also analyzed in the limit of large number of MSs. Based on the analysis, wepropose a  Cooperative relaying transmission techniques have widely been studied owing to the advantages of both enhancing the end-to-end link quality in terms of system capacity and extending the coverage . One of In wireless systems with many users, a multiuser diversity gain can be exploited to further improve the system throughput by utilizing the usefulness of fading since there are a large number of MSs in multiuser environments: opportunistic scheduling , opportuN MSs. As a combination of the traditional broadcast channel and relay channel, it matches several scenarios such as the downlink cellular environment with relay and wireless network with backbone nodes. To show the fundamental limits of such a relay-aided downlink system, in the literature  and C denote the expectation and the field of complex numbers, respectively. CN represents a zero-mean circularly symmetric complex Gaussian distribution with variance x. Unless otherwise stated, all logarithms are assumed to be to the base 2.Throughout the paper, N MSs. The channel model deals with the problem that one BS communicates with one selected MSs with or without the help of one RS. Hence, we do not assume that the BS sends different data packets to multiple MSs simultaneously since it requires the use of sophisticated multiuser detection schemes at the receivers.As illustrated in nth MS is given byhbm,n ∈ C is the complex channel coefficient between the BS and the nth MS, Pb is the transmit power at the BS, x is the transmit signal, and wm,n denotes the complex additive white Gaussian noise (AWGN) at the nth MS, which follows CN.For one-hop transmission, we perform a direct transmission from the BS to a certain selected MS. Then, the received signal at the nth MS is then given byhrm,n ∈ C and hbr ∈ C are the complex channel coefficients between the RS and the nth MS and between the BS and the RS, respectively, g is the amplification factor at the RS, and wr denotes the complex AWGN at the RS, following the distribution CN. The channel coefficients hbm,n, hrm,n  , and hbr are independent and identically distributed (i.i.d.) and are frequency-flat fading, where all the distributions are assumed to be CN. In our model, the amplification factor g at the RS is represented as [Pr is the transmit power at the RS.For two-hop transmission, communication is performed from the BS to one MS through the RS. We assume that the RS operates in simple half-duplex, that is, the RS is not assumed to receive and transmit data packets simultaneously, and AF mode. Then, at the first time slot, the BS transmits its data to the RS, and at the second time slot, the RS amplifies and forwards the received data to the corresponding MS. For simplicity, we do not consider a direct path from the BS to MSs for two-hop transmission since two-hop transmission is required when the links between the BS and MSs are relatively poor. The received signal at the given byy2,n=hrm,In this section, we propose a multiuser scheduling method that efficiently reduces the amount of feedback information for instantaneous SNRs from the MSs to the BS. The BS decides both the transmission mode  and the desired MS based on a scheduling criterion.Ri denote the average achievable rate for i-hop transmission  when an MS with the maximum instantaneous SNR is selected among N MSs. Suppose that all links between the BS and the MSs experience the same average SNR. All links between the RS and the MSs are also assumed to experience the same average SNR . Let Ri is expressed as a function of the average SNRs PbhbFor comparison, the optimal multiuser scheduling, based on full feedback of the instantaneous SNRs, is also considered. In this case, both the transmission mode and the desired user are selected simultaneously in the sense of maximizing the instantaneous achievable rate for a given channel realization. In N.In this section, the average achievable rates of both one- and two-hop transmissions with the proposed multiuser scheduling are analyzed. A closed-form expression for the average achievable rates is first derived, and its asymptotic behavior is then shown in the limit of large number of R1 for one-hop transmission is given byFrom , the avegiven byR1=Elog1R1 is derived asE1(x) = ∫x∞(et−/t)dt is the exponential integral function.Suppose that the one-hop transmission with our multiuser scheduling is used. Then, the average achievable rate The proof essentially follows the derivation of the selection combining scheme in .R1 is expressed as a function of both the average SNR of the BS-MS link, N. For large N, the average achievable rate R1 in  with respect to γrm,n is always greater than 0 sinceThe first derivative of R2 for two-hop transmission is given byThus, the average achievable rate given byR2=E12logR2 is written asE1(x) = ∫x∞(et−/t)dt is the exponential integral function.Suppose that the two-hop transmission with our multiuser scheduling is used. Then, the average achievable rate pγbr(x) of the random variable γbr is exponentially distributed and thus is given bypγrm,max(y) of the random variable γrm,max≜maxn{γrm,n} is given byx and y, we haveJ = ln(1 + x) − ln(1 + (x/(1 + y))). Using (4.337.2) in [E1(x) = −Ei(−x)≜−∫x−∞(et−/t)dt, the term Q in (The probability density function (PDF) atistics  of an exgiven bypγbrx=1γ¯given bypγrm,max37.2) in  and the  we haveR2=N2ln2e obtain  by applyrm Q in  = ∫x∞(et−/t)dt is the exponential integral function. Here, β = 1 if β < 1 if  C0 > 0. Note we use the following notation: (i) f(x) = O(g(x)) means that there exist constants C and c such that f(x) ≤ Cg(x) for all x > c; (ii) f(x) = o(g(x)) means limx→∞(f(x)/g(x)) = 0; (iii) f(x) = Ω(g(x)) if g(x) = O(f(x)); and (iv) f(x) = ω(g(x)) if g(x) = o(f(x)) [Suppose that the two-hop transmission with our multiuser scheduling is used. When the number of MSs, e R2 in  and (14b o(f(x)) .N is sufficiently large, the maximum value of an exponential random variable, maxn{|hrm,n|2}, scales as lnN with high probability [N tend to zero with high probability under the condition x simply yield  for some constant C1 > 0. It thus follows that 0 < β ≤ 1. Here, if C1 = 0. Otherwise   N is sufly yield . Now letsults in , which cN, that is, N, beyond a certain value is not beneficial in terms of performance on the achievable rate. Specifically, when N), the rate R2 is bounded by a constant even for large N. Hence, we may conclude that it may not be desirable for all the MSs to feed back their instantaneous SNR of the BS-RS-MS link to the BS. On the other hand, if N, that is, From the result of in as in . In thisIn this section, to verify our analytical results in 3 times for each system parameter.The analytic results are based on closed-form expressions for the average achievable rates of one- and two-hop transmissions, shown in , 14a), , 14a), aN, when we perform either one- or two-hop transmission with multiuser scheduling. We assume that N = 1,…, 30. The system performance is examined according to various average SNRs—it is assumed that the average SNR of the BS-RS link, N = 1, the average achievable rate for two-hop transmission is higher than that for one-hop transmission. However, as N increases, performance on the achievable rate for one-hop transmission case becomes higher owing to more multiuser diversity gain. The analytical results are also illustrated in this figure, where they almost match well with the simulation results for any average SNRs and N. Their asymptotic behaviors for large N are examined as follows: we obtain the rate 4.5 (bits/s/Hz) from (β = 1 is assumed in this case). Interestingly, when N increases over 10 for N ≥ 5.Hz) from  while thned from   R1 > R2, the optimal scheduling may select two-hop transmission according to some link conditions. This phenomenon occurs only for the case where there exists a crossover between two curves. However, it is easily seen that the proposed two-step scheme always outperforms either one- or two-hop transmission. On the other hand, for cases 2 and 3, the proposed two-step scheduling scheme shows nearly the same performance as that of the optimal one. In these cases, the achievable rates for the optimal scheduling are also nearly identical to those for either one- or two-hop transmission. This is because there is no crossover point between two curves of one- and two-hop transmission under the assumed link conditions. It in turn means that the proposed scheduling method works well when one transmission mode is dominant over all values of N. In other words, our two-step scheduling method based on efficiently reduced feedback information can be either optimal or suboptimal in terms of achievable rates, depending on link conditions (e.g., the average SNRs of the three links, denoted by Moreover, to see the fundamental limit of our multiuser scheduling, the comparison for the average achievable rates between the optimal and proposed two-step multiuser scheduling is performed in N. Furthermore, we analyzed the link condition such that the multiuser diversity gain is fully exploited. Finally, it was examined that the proposed algorithm has almost the same achievable rate as that of the optimal one, while its required feedback is fairly reduced by half of the optimal one.The multiuser scheduling method, opportunistically selecting both the transmission mode and the desired MS, was proposed for relay-aided downlink systems using half-duplex AF relaying. The scheduling criterion, based on efficiently reduced feedback information, was designed by showing the closed-form expression for the average achievable rates and its asymptotic behaviors for large


E-index (earliness-index), which cumulatively measures the earliness degree to which a variable (or a dynamic trait) increases or decreases its value. Theoretical proofs and simulation studies show that E-index is more general than functional mapping and can be applied to any complex dynamic traits, including those with logistic curves and those with nonmonotonic curves. Meanwhile, E-index vector is proposed as well to capture more subtle differences of developmental patterns.While it is a daunting challenge in current biology to understand how the underlying network of genes regulates complex dynamic traits, functional mapping, a tool for mapping quantitative trait loci (QTLs) and single nucleotide polymorphisms (SNPs), has been applied in a variety of cases to tackle this challenge. Though useful and powerful, functional mapping performs well only when one or more model parameters are clearly responsible for the developmental trajectory, typically being a logistic curve. Moreover, it does not work when the curves are more complex than that, especially when they are not monotonic. To overcome this inadaptability, we therefore propose a mathematical-biological concept and measurement,  Whether there are different genes responsible for the formation of a trait and how these genes regulate the trait are of fundamental importance biologically, agriculturally, and/or medically. Quantitative traits, or characteristics varying in degree, can be attributed to the effects of genes and their environment . Lander  infinite-dimensional characters in , a < b. {pi}i=0n is a sequence of points, p0 = a, pn = b, 0 ≤ i < j ≤ n. Also suppose that f(t) is differentiable on the open interval , i = 1,2,…, n, and that f′(t) is its derivative function. Then we define the E-index of the growth curve as follows:Suppose that the growth curve is a continuous function f′(t), the growth rate at time point t, is undefined on each inner split point. But this would not change the integration result, even if we set the growth rate at such point to be any value. For simplicity, we denote Eab(f) as E(f).It should be noted that f′(t) occurs is our key concern for growth, and the expression f′(t)(b − t) in (f′(t)(b − t) is, the earlier the growth or development occurs. In this sense, the E-index measures how early growth occurs in the whole process by accumulating the product along the time.How early or how late the growth rate  − t) in  quantifiE-index's properties which are to be discussed in the form of propositions. However, their proofs are all omitted since they can be found in calculus textbooks or related literature.From the definition above, we can derive several of f(t) defined on the closed interval  is strictly monotonically increasing and is globally differentiable, then its E-index can be calculated with the following equation:If the growth curve function E-index and reveals to us the relations between the integrations along horizontal direction and along vertical direction.The conclusion of y = f(a), which is the conclusion we wanted.We can illustrate the proof with The conclusion of E-index with  is defined on  and f(a) ≤ f(t) ≤ f(b), then 0 ≤ E(f) ≤ 1.If the growth curve function E(f) may get a value greater than 1 if, for some t, f(t) is greater than f(b) or even be a negative value if, for some t, f(t) is less than f(a).But E-index, and we can design a growth curve function whose E-index is any designated value in the range.E-index can be easily calculated with integration operation stated in ,,E-index r in the 2 bunches of growth curves can differentiate the curves, as is illustrated in E-index is capable of doing so, and the results are shown in The function parameter r in E-index of the corresponding growth curve function is calculated with , i = 1,2,…, m, and that of the jth sample of B is vjB = , j = 1,2,…, n. The computation steps are as follows, using t-test of (m + n − 2) degrees of freedom to discover the significance.(a)viA and vjB, we can use spline interpolation to get continuous functions (curves) fiA(t) and fjB(t).For each (b)E-index of the curves, namely, EiA and EjB, respectively, for i = 1,2,…, m and j = 1,2,…, n. And we denote that EA =  and EB = .Then  is emplo(c)EA and EB H0 will be rejected ifH0 will be accepted, where t is the computation result of (tα(m + n − 2) is the t-distribution value with the confidence level α and (m + n − 2) degrees of freedom.We can test the null hypothesis that the two groups of samples are not significantly different:esult of  and tα  r in (2)E-indices for genotypes A and B,  sA = 0.0428 and sB = 0.0597, respectively. We have by  = 2.896, which means that there is sufficient evidence to indicate that the genotypes are clearly responsible for the developmental processes of the organs.But using the statistical framework given in E-index is sometimes capable of differentiating growth curves, it is after all only one characteristic value revealing one aspect of information. Therefore, it is natural for us to extend E-index into E-index vector.As is mentioned earlier, a curve may theoretically provide infinite amount of information about growth. Though the E-index insufficient to differentiate growth curves? An example from Figures Where and why is E-index value (0.5). This is mainly due to the fact that the effect caused by the higher growth rate in Comparing the sizes of shade in Figures E-index does not work in some cases to differentiate growth curves. Consequently, we have to move forward for a more sophisticated tool. Naturally we will extend E-index into E-index vector.This example indicates that f(t) describing a growth curve is continuous on a closed interval , a < b. And suppose that Q = {pi}i=0n is a prescribed sequence of points, p0 = a, pn = b, pi < pj, 0 ≤ i < j ≤ n. Then the E-index vector of f(t) according to Q is defined as follows:Suppose that VQ(f) is denoted as V(f) for simplicity and (nEp0p1(f), nEp1p2(f),…, nEpn−1pn(f)) as (E1(f), E2(f),…, En(f)).E-index vectors of the growth curves in Figures E-index vectors, , , and , are apparently different, as is also illustrated with shaded areas in The f(t) and g(t) are two functions defined on a closed interval, describing two growth curves with the same prescribed sequence of points Q = {pi}i=0n. Then the growth dissimilarity between f(t) and g(t) is defined as follows:Suppose It is easy to prove that growth dissimilarity satisfies distance axioms; that is,f1(t), f2(t), and f3(t), respectively. According to  is more similar to f1(t) than it is to f3(t), which is consistent with what is observed in The results above show that, in the growth perspective of earlier and later halves, Wi denoted as the ith weight:Different weights can be designated to differently important phases of growth according to specific problems. So the growth dissimilarity defined in  can be aE-index vector to help differentiate two growth curves. What is more important, however, is to differentiate a set of growth curves or to divide them into groups or clusters, which will be discussed in the next subsection.Equation  enables More and more growth traits are available and they can be described by growth curves. Studies  indicateIn order to identify genes with growth traits, we are required to divide into groups all growth curves that are as similar as possible in the same group while being as dissimilar as possible in different groups. But a common situation we are encountering is that it is difficult for us to obtain reasonable groups.E-index, E-index vector, and the growth dissimilarity definition based on these two concepts may help us to group or cluster the growth curves.E-index vectors, we can define describing rules for a curve group. Take the growth curves in Figures E-index vector is greater than 0.6 and the second less than 0.4.” And the second rule is defined as “the first component is less than 0.6 and the second greater 0.4.” These two rules describe and define two groups, with growth curve (d) in the first group and growth curves (b) and (e) in the second one.With the Though describing rules are capable of grouping growth curves, human experts are involved in prescribing the rules, and thus the rules, consequently the grouping results, may be different from person to person.k-mean algorithm, and its variations are almost automatic. It is a recursive algorithm with k randomly selected centers. To cluster the growth curves, the growth dissimilarities between each growth curve and each center are calculated with E-index is capable of differentiating growth curves (such as logistic curves) as function parameters are in the applications of functional mapping. Like functional mapping, E-index is good at differentiating growth trajectories with the same values of mature state, which traditional QTL takes as the same. In this sense, E-index generalized functional mapping.(ii)E-index is sometimes unavailable according to its primitive definition given in E-index is as biologically meaningful as the important curve parameters employed in functional mapping.(iii)E-index. In fact, a cubic spline  approxi(iv)E-index provides limited information. As an extension of E-index, E-index vector is focused on the growth in different phases, the number of which may vary, and the time spans for the same vector may not be of equal length, according to the application background and requirements.Being a key and general characteristic value though, (v)k-mean algorithm.By extracting the growth information in a curve and forming a vector, we can use well developed techniques for analysis, such as describing rules and (vi)E-index vector helps us reveal detailed characteristics in growth curves. It may be looked on as a microscope employed to observe a desired level of growth detail.In order to generalize functional mapping and overcome the shortages of it,


Solitons, defined as nonlinear waves which can reflect from boundaries or transmit through each other, are found in conservative, fully integrable systems. Similar phenomena, dubbed quasi-solitons, have been observed also in dissipative, “excitable” systems, either at finely tuned parameters (near a bifurcation) or in systems with cross-diffusion. Here we demonstrate that quasi-solitons can be robustly observed in excitable systems with excitable kinetics and with self-diffusion only. This includes quasi-solitons of fixed shape (like KdV solitons) or envelope quasi-solitons (like NLS solitons). This can happen in systems with more than two components, and can be explained by effective cross-diffusion, which emerges via adiabatic elimination of a fast but diffusing component. We describe here a reduction procedure can be used for the search of complicated wave regimes in multi-component, stiff systems by studying simplified, soft systems. Reaction-diffusion systems are an important mathematical tool in studying dissipative structures. For the properties of solutions of these systems, diffusion is as important as is reaction. For instance, Turing structures are possible only if there is significant difference in the diffusion coefficients between activator and inhibitor speciesThere is a large variety of interesting regimes in systems with nonlinear kinetics and cross-diffusion instead of or in additional to self-diffusionOriginally, these regimes have been discovered in models motivated by population dynamics, where (nonlinear) cross-diffusion appears naturally as taxis of micro- or macro-organisms with respect to each other’s population density3This question encounters a number of obstacles due to physical constraints. Firstly, the cross-diffusion must always be nonlinear in order to preserve positivity of chemical concentrations92113effective. Pattern-forming reactions are supposed to be supported by a source of energy, say in the form of constant supply of a reagent which is consumed in the reaction, whereas the Second Law applies to closed systems. Although this source of energy  in the spatially-extended context can be used to reconstruct full (fuller) reaction-diffusion systems with purely self-diffusing reagents, from reduced reaction-diffusion systems with cross-diffusion. We apply this procedure to produce examples of self-diffusion systems that manifest quasi-soliton solutions. Since the fast-slow analysis is an asymptotic procedure, i.e. is formally valid in certain parametric limits but has to be applied for fixed parameter values, it inherently has limited accuracy. The procedure we exploit here is valid when the spatial scale of the solutions is large compared to the diffusion length of the fast species, and the reduced equations become non-local when this is not fulfilled. To relax this restriction, we suggest a euristic procedure which enhances the accuracy of the reduction/reconstruction for particular wave-like solutions.The structure of the paper is as follows. In section “Methods: Fast-slow reduction for reaction-diffusion systems” we discuss the process of adiabatic elimination of a fast reacting species in the spatially-extended context, and show how the diffusivity matrix may be affected by that elimination. In section “Results” we present a simple application of this theory: namely, by “working backwards” the adiabatic elimination, we construct a three-component reaction-diffusion system with self-diffusion only, which corresponds to a two-component system with cross-diffusion and quasi-soliton solutions of various kinds. We conclude by discussing possible implications of our findings for future research.Our asymptotic argument is based on two assumptions. Firstly, we consider a reaction-diffusion system, which has a fast and a slow subsystem,u and the fast subsystem v. We assume that the fast subsystem (2) has a unique globally stable equilibrium for v = g(u) at any fixed u of interest,where The second assumption, to be clarified and formalized later, is that the solutions of interest are sufficiently smooth in space.Linearizing and considering the limit 2u is small and therefore the resulting μ is a  small parameter. Then we can formally resolve At this point we see that our solutions should be so smooth that ∇v into Substitution of the thus found Fu = ∂vfu|v=g(u), andwhere We see that the diffusion in the reduced system (5) is not only different from the full system, but is typically nonlinear even if the original diffusion was fully linear.g(u) = Gu, G = const, andIn the system (5) we have dropped wereandk, so thatThe second-order approximation (6) contains a biharmonic operator. We can approximately replace it with a reaction-diffusion system for solutions which are approximately oscillatory in space with one dominant wavenumber Then we havewhere Do parametric search in the reduced system, which has fewer parameters and often is easier to compute, as it is less stiff;k for it, say as the maximum of the power spectrum of the spatial profile;Once an interesting regime is found, estimate the dominant wavenumber Use the inverse transformation to obtain parameters for the full system corresponding to the found parameters of the reduced system;See what solutions the full system with these parameters will produce.The direct transformation from system  veq) to system (5) or (9), and its inverse, can be used for searching for nontrivial regimes in the full system, based on the existing experience of nontrivial regimes in the analogues of the reduced systems:As a simple example, we now consider a three-component extension of a two-component system with nonlinear kinetics, where a third component is added, which has fast linear kinetics and linked to the other components in a linear way:In terms of which leads to the reduced system (9) in the formwhereandIt is easy to see that the resulting diffusivity matrixdepends on the reaction kinetics parameters and does not have to have real eigenvalues or even be positive definite .γ, Dw, Dv, and for given parameters of (11) , obtain the remaining parameters of (10) viaDw via Du ≥ 0. The latter will be satified if Dv is chosen so that either Dv ≤ DuuDvv/(Du + Duu) or Dv ≥ Dvv.and Numerical illustrations of some nontrivial regimes obtained in the two-component system, and corresponding nontrivial regimes found in the reconstructed three-component system are presented in ε = 0.01, k1 = 10, Duu = Dvv = 0.025, Duv = 1, Dvu = −1, γ = 30, Dv = 0.001, Dw = 0.04, and a selection of values of a as shown to the left of the panels in k for the inverse transform were obtained as the position of the maximum of the power spectrum of the signal z(x) = u + iv of a selected solution  of the two-component system, taken at a selected fixed t. Specifically, we had k = 0.864 for a = 0.07, k = 0.879 for a = 0.25 and k = 0.942 for a = 0.35. For simulations, we used the same numerical methods as those described in refs x = 0.1. Time differencing was first-order, explicit in the reaction terms and implicit in the cross-diffusion terms, with time step Δt = 2.5 ⋅ 10−4 for the two-component system, and fully explicit with time step Δt = 2 ⋅ 10−5 for the three-component system. The space interval was x ∈ , where L was different in different simulations, as shown in x-axis labels in u = usH(σ − x), v = 0, w = 0 to initiate a wave starting from the left end of the domain. Here H is the Heaviside function, the wave seed amplitude was typically chosen as us = 2 and length as σ = 4. To simulate propagation “on an infinite line”, L = ∞, for δx2 = 20 away from the boundary each time the pulse, as measured at the level u = 0.1, approached the boundary to a distance smaller than δx1 = 40, and filled in the new interval of x values by extending the u, v and w variables at constant levels. The profiles in x − xc, where xc = xc(t) is the center of mass of v2 at the time moment t, that is, and the corresponding three-component system (10). We used the parameter values a = 0.35 and a = 0.25, both 2-component and 3-component systems support propagation of a solitary pulse with oscillating front and monotonous back. These shapes are steady: as one can see in the top four panels of a = 0.35, the pulses annihilate upon collision with a bondary, — or, rather, reflect but lose their magnitude, cannot recover and eventually decay. Whereas for a = 0.25, the pulses in both systems successfully reflect from boundaries. One can see in σ = 2, σ = 8 and σ = 12 were indistinguishable from those shown in For a = 0.07 is different: the shape of the pulse changes as it propagates. These are “envelope quasisolitons”, similar to those previously reported in in refs The behaviour for Since the speed and shape of quasi-solitons are typically fixed, the reflections and collisions, when they happen, are always quasi-elastic, if one is to use the terminology from the conservative wave theory. That is, the properties of the solitons fully recover after collisions, even though, as can be seen in a = 0.03, and even then the reflected wave takes much longer to fully recover. In this aspect, the quasi-solitons are also different from the true solitons, e.g. in the NLS. A formal way to understand this difference is to observe that NLS is symmetric with respect to inversion of the sign of its complex field. Hence one can arrange two identical but counter-propagating solitons on an infinite line so that their nonlinear superposition at a certain point remains exactly zero at all times. Then replacing the problem with the one at half-line and zero boundary condition at that point will yield a solution in the form of a soliton reflecting from the boundary with a 180° change of phase. This construction does not work for the FitzHugh-Nagumo kinetics (14) which is not invariant with respect to the inversion  → , unless a = −1, but in the latter case the system is no longer excitable. A more intuitive way to explain this is: for the pulse way to propagate, the u field must exceed the threshold a, and the boundary condition u = 0 makes it much more difficult for the reflected wave to satisfy this.Finally, we emphasize that the reflection from boundaries shown in not constraint by the thermodynamic restrictions of symmetry and positive definiteness. We believe that application of such reduction, accounting for the emergence of effective cross-diffusion, may lead to finding new interesting regimes in systems that have traditionally been studied without cross-diffusion, e.g. Brusselator30It has been traditionally believed that a definitive property of waves in excitable media is that they annihilate when collided. Although soliton-like interaction was observed in some reaction-diffusion systems with excitable kinetics, both in numerical simulations1921623242627How to cite this article: Biktashev, V. N. and Tsyganov, M. A. Quasisolitons in self-diffusive excitable systems, or Why asymmetric diffusivity obeys the Second Law. Sci. Rep.6, 30879; doi: 10.1038/srep30879 (2016).


Although positive incentives for cooperators and/or negative incentives for free-riders in social dilemmas play an important role in maintaining cooperation, there is still the outstanding issue of who should pay the cost of incentives. The second-order free-rider problem, in which players who do not provide the incentives dominate in a game, is a well-known academic challenge. In order to meet this challenge, we devise and analyze a meta-incentive game that integrates positive incentives (rewards) and negative incentives (punishments) with second-order incentives, which are incentives for other players’ incentives. The critical assumption of our model is that players who tend to provide incentives to other players for their cooperative or non-cooperative behavior also tend to provide incentives to their incentive behaviors. In this paper, we solve the replicator dynamics for a simple version of the game and analytically categorize the game types into four groups. We find that the second-order free-rider problem is completely resolved without any third-order or higher (meta) incentive under the assumption. To do so, a second-order costly incentive, which is given individually (peer-to-peer) after playing donation games, is needed. The paper concludes that (1) second-order incentives for first-order reward are necessary for cooperative regimes, (2) a system without first-order rewards cannot maintain a cooperative regime, (3) a system with first-order rewards and no incentives for rewards is the worst because it never reaches cooperation, and (4) a system with rewards for incentives is more likely to be a cooperative regime than a system with punishments for incentives when the cost-effect ratio of incentives is sufficiently large. This solution is general and strong in the sense that the game does not need any centralized institution or proactive system for incentives. Although social dilemmas can be resolved if punishing non-cooperators or rewarding cooperators works, such rewards and punishments, i.e., external incentives, entail certain expenses. As a result, a cooperative player who shirks his or her duty to provide an incentive to other players will emerge, and he or she will be more advantageous than an incentive-provider. In fact, the problem of excluding such cooperative incentive-non-providers, or second-order free-riders, is a well-known academic challenge. In order to meet this challenge, we devise and analyze a meta-incentive game that integrates positive incentives (rewards) and negative incentives (punishments) with second-order incentives, which are incentives for other players’ incentives. In this paper, we solve the replicator dynamics for a simple version of the game and analytically categorize the game types into four groups. We show that second-order incentives for first-order reward are necessary for cooperative regimes. This solution is general and strong in the sense that the game does not need any centralized institution or proactive system for incentives. Even though society is based on cooperation, achieving cooperation in social dilemmas is still a big challenge. The free-rider problem, for example, hinders cooperation. Many studies have addressed this problem, and the methods proposed for solving it include giving players sufficient ability to remember their direct  or indirA meta-analysis of reviews on reward and punishment systems using a common framework  revealedFirst, many studies have focused on either individually-dealt-with punishments or rewards. While some studies –16 have Second, should a free-rider be punished and/or should a contributor be rewarded? Experimental studies , 25 haveThird, in the approach we took, the focus is on meta-level incentives. When the incentive for cooperation is either a punishment or reward, there is still the second-order free-rider problem. The effort made to maintain a cooperative society is a cost that must be defrayed by someone. Generally, a player who contributes to a game but never defrays the cost for providing incentives is more evolutionarily adaptive than one who contributes to the game and does pay the cost. This means that eventually no one defrays the cost of maintaining the incentive system. The second-order free-rider problem can come down to the problem of costly incentives . One solThe pioneering work on meta-level incentives was performed by Axelrod . He atteThe C-FO-linkage issue is a relationship between two behaviors which differ qualitatively. Considering this point, the relationship between first-order incentives and second-order incentives is an alternative issue because they are both incentives. Kiyonari and Barclay  focused We have developed a model of a meta-incentive game (MIG) that has second-order incentives, i.e., incentives for other players to provide incentives. This analytical model can describe the carrot-and-stick issue uniformly and comparatively. The model targets an individual incentive system that integrates a positive side (reward) and a negative side (punishment) . The incMIG players first play donation games and then provide incentives in answer to their actions in the games. Incentives are provided not only for or against the others’ cooperative or non-cooperative actions but also for the others’ incentive behaviors or lack thereof on third-party players.Players are divided into three sorts of strategist: a cooperative incentive-provider (CI), a cooperative incentive-non-provider (CN), and a non-cooperative incentive-non-provider (NN). x, y, and z be the frequencies of CI, CN, and NN, respectively. Naturally, x+y+z = 1. The equations are formulated asUCI, UCN, UNN, and The population is infinitely large and well-mixed. The frequencies of the three strategies follow replicator dynamics describing a natural law whereby the higher one’s payoff is the more frequent one’s strategy becomes. Let c be the cost of donation, b be the receiver’s benefit, F1 be the fine imposed as a first-order punishment, P1 be the cost of a first-order punishment, A1 be the amount of a first-order reward, R1 be the cost of a first-order reward, FP be the fine for freeriding for the first-order punishment, PP be the cost for freeriding for the first-order punishment, AP be the amount of the reward for the punisher, RP be the cost of the reward for the punisher, FR be the fine for freeriding for giving a reward, PR be the cost for freeriding for giving a reward, and AR be the amount of rewarding a rewarder, and RR be the cost of rewarding a rewarder. All these values should be non-negative constants.Now let us describe the parameter notations needed to calculate a player’s (expected) payoff. Let μ), which represents the proportion of a fine or award that incentive-receivers should pay or receive relative to its cost, and the discount factor of costs on the level of incentive (δ), whereWe will avoid analytical difficulties due to the usage of many parameters by defining a simple meta-incentive game (S-MIG) using two parameters: the incentive cost-effect ratio .We assume that Finally, before analyzing our model, we define all 24 possible configurations of MIG in x > 0) emerges by analyzing the replicator dynamics on different types of S-MIG. As shown in Methods, the dynamics of S-MIGs can be classified into four groups. x, y, z) =  of all types, and We explore the conditions under which a cooperative equilibrium incentive. In our model, moreover, incentives are performed ex-post and individually, and thus the system does not need a centralized institution or ex-ante commitment. Many studies on non-meta-level incentives have shown that punishment is more effective than a reward. We have identified a possible explanation as to why people prefer second-order rewards to second-order punishments. Kiyonari and Barclay’s experimental study  supportsWhich incentive should a designer of an MIG choose? If the cost-effect ratio of incentives is sufficiently large, even a handful of cooperative incentive-providers can beat non-cooperative incentive-non-providers if the designer uses a system with rewards for incentives instead of a system with punishments for incentives.Our work differs from Sasaki et al’s model of integrating rewards and punishments, which was designed for an institutional system with a compulsory entrance fee and thus no option of second-order free riders . Kendal The asymmetry of the effects of rewards and punishments might be evident in the cost-effect ratio. Although it may be natural that a fine handed out as punishment should be larger than its cost, can the reward be larger than its cost? We believe that this balances out the risk of second-order free-riders emerging. Of course, we leave open the possibility of incorporating asymmetric reward and punishment effects as a future extension of a system.In this section, we analyze a meta-incentive game (MIG) by solving replicator equations. The expected payoffs of the players areUCI. The first term represents the benefit of donation, whereas the second term gives the cost of donation (because one is a cooperator) in the DG stage. The third term represents the costs of the first-order incentivization: rewarding a cooperator and punishing a non-cooperator. The fourth term represents the first-order reward for cooperating. The fifth term represents the costs of the second-order incentivization and consists of four parts, i.e., rewarding those who have rewarded a cooperator, punishing those who have not rewarded a cooperator, rewarding those who have punished a non-cooperator, and punishing those who have not punished a non-cooperator. Finally, the last term describes the second-order rewards for rewarding a cooperator and punishing a non-cooperator. Similar explanations can be applied to the other expected payoffs, UCN and UNN.Let us explain the terms of Using Eqs , 2), an, an2), ax, y, z) in S-MIG in the following subsection, and thus, here, it is enough to conduct an analysis of the borders. Moreover, there is no equilibrium at any point on the line x = 0, except two corners  = , ). This is because, x = 0, and thus, y always decreases. Hence, we will only deal with the y = 0 and z = 0. On those two lines, that function is calculated as We prove that there is no equilibrium at any internal point (f(x) and g(x). In the type, F1, P1, FP, PP, AP and RP should be zero because there has no incentive system on the punishment side. Moreover, using c = 1 and definitions of μ and δ, R1 = δ, A1 = μδ, RR = PR = δ2 and AR = FR = μδ2. Substituting them into f1 in In order to exemplify how to analyze each type of the S-MIG, we deal with the RB type as a representative and derive their equations of Therefore, we derive x, y, z) = . If x, y, z) =  is asymptotically stable, and thus, a cooperative regime emerges. The dynamics of each type of S-MIG are classified into four groups. In what follows, we will deal with one representative type in each group. The remainder can be similarly derived.x, y, z) = , so cooperation is never achieved regardless of the values of . The R, P+R, PP+R, PR+R, and PB+R types belong in the first group. Each type of this group has a globally stable point . In one, all the fixed points on the line z = 0 are unstable, and thus, there is a globally stable point , as in the first group. In the other, some of the fixed points are stable if x, y, z) =  on y = 0. Let us verify the PR case as a representative example. f(x) is an increasing function. Moreover, f(1) = −1+μδ > 0 implies that the point  =  is asymptotically stable. Even in the second pattern, however, the point  is not stable for a long time. This is because, the whole line z = 0 consists of fixed points, and thus, neutral drift is possible. Occasionally, x moves to an unstable equilibrium, and this type eventually reaches . The phase diagram of the PP-type S-MIG is shown in The P, PP, PR, and PB types belong in the second group. The whole line μ, δ). In one, there is a globally stable point , as in the first group. In the other, another locally asymptotically stable point  can exist. This group has three existence conditions for the basin of attraction on the point  = : z = 0, on one hand, are bistable and y = 0 depends on f(x). f(0) < 0 and f(1) = −1−δ−δ2+2μδ(1+δ) > 0 when the existence condition for the basin of attraction on the point  =  is satisfied. Therefore, the dynamics on y = 0 are also bistable, and The third group, consisting of types which include either the RP or RR, and P+RB, PP+RB, PR+RB, and PB+RB (Full) types, has two patterns of behavior depending on the values of  on the line z = 0. λ1λ2 = −y* δ2(1+δ) < 0. Moreover, in the case of the RR-type S-MIG, λ1λ2 = −x* y* δ2 < 0. The other cases are omitted.The final group, consisting of only the RB type, is the same as the third except for the direction of the dynamics in the internal space of the basin of attraction for cooperation see . In thisx of the basins of attraction for cooperation  =  on y = 0. Let fD(x) and xD be x of the basin of attraction on the line y = 0 in the D-type S-MIG for D ∈ {P, R, …, PB+RB}. Basically, the more complex the type is, the lower its lower limit of x becomes. For example, we will prove xP+RB < xP+RR. fP+RB(x) = −1−δ+2μδx−δ2x+2μδ2x2 and fP+RR(x) = −1−δ+2μδx−δ2x2+μδ2x2 are in fD(xD) = 0. fP+RB(xP+RR) = δ2xP+RR[xP+RR(1+μ)−1] and fP+RB(xP+RR) > 0. Therefore, xP+RB < xP+RR. Moreover, some equivalence relations can be derived. For example, let us compare xRR with xRP. fRR(x) and fRP(x) are increasing functions because their partial derivatives are positive in 0 < x < 1. fRR(xRP) = δ2xRP(1−2xRP) and xRR < xRP.μ0 > μ10 > μ11 > μ12 > μ13.Finally, we compare the lower limits of y is set to zero and all the parameters for the second-order incentives are also zero (FP = PP = AP = RP = FR = PR = AR = RR = 0), this new version, or an incentive game (IG), can be regarded as a model of the C-FO-linkages. We denote the three possible IG configurations as the P-type, R-type, and P+R type. The replicator equation of IG isThe MIG assumes FO-SO-linkages. In this subsection, we analyze the replicator dynamics of a model that assumes C-FO-linkages instead of FO-SO-linkages. Accordingly, cooperators automatically provide first-order incentives, and thus, there is no CN, and no second-order incentive is needed in the MIG. When μ, δ, and c = 1. All S-IG types have two patterns of behavior depending on the values of . In one, there is a globally stable point  = . In the other, another locally asymptotically stable point  =  can exist. The existence condition for the basin of attraction at the stable point  is We can devise a simple incentive game (S-IG) by using x, y, z) : x, y, z ≥ 0, x+y+z = 1}. Assume that  is an internal equilibrium. On that point, x*  is not zero, and thus, the function x*. Using In this subsection, we prove that there is no equilibrium at any internal point on the 2-dimensional simplex Δ = { around the equilibrium point . Let εx = x−x*, εy = y−y* and ɛ = T, where T means transposition. As a result of the linearization of the dynamics, μx*−1 = 0. The eigenvalues of M are λ = 0, z*(3−2x*). These are non-negative, and thus, any internal equilibrium is non-isolated and unstable. The eigenvalues in the case of the RB type are λ = 0, z*+2(x*+y*)(y*+z*), and thus, the same conclusion is reached.Now let us analyze the local stability of the point  =  leads to εx = x−x*, εy = y−y*, ɛ = T, andAs is shown in the previous subsection, the linearization of the dynamics around the equilibrium point  when The eigenvalues of


In particular, we generalize the Rellich–Wittich-type theorem and Malmquist-type theorem about differential equations to the case of q-difference differential equations (system).By using the Nevanlinna theory of value distribution, we investigate the existence of solutions of some types of non-linear  In receThe following two results had been proved by F. Rellich and H. Wittich, respectively.Let the differential equation be the following formIff(w) is transcendental meromorphic function ofw, then Eq. is the transcendental meromorphic function ofw, then the Eq.  has no non-constant entire solution., the difference equationhas a non-trivial entire solution.,the difference equationhas a non-trivial meromorphic solution in the complex plane..In the present paper, we mainly study the above Rellich–Wittich-type theorem of q-difference differential equation (system) if a equation (system) contains the q-difference term f(qz) and differential term f(z) at the same time.We call the equation a q-difference differential equation of the formz and f of degree m,z, and obtain the following results.We consider the system of For system  is defined asIf Eq.  is defined as in Theorem 9. If Eq.  is defined asP andQ are irreducible polynomials inw, coefficientsare rational functions ofz. IfEq.  is defined as in Theorem 11. If Eq.  be a meromorphic function. Then for all irreducible rational functions inf,with meromorphic coefficientsthe characteristic function ofR) satisfieswhereand be a transcendental meromorphic function of zero order and q be a nonzero complex constant. Thenandon a set of logarithmic density 1.(Zhang and Korhonen Letf(z) be a nonconstant zero-order meromorphic function andThenon a set of logarithmic density 1 for all r outside a possible exceptional set of logarithmic density 0. be a nonconstant meromorphic function in the complex plane andlbe a positive integer. Then(see Yi and Yang Letandf(z) be a nonconstant meromorphic function with zero order. Then for any positive finite integerk, we haveandIt follows from Lemma 15 thatSuppose that f system  with zerSince orem and , 11), w, w\docummentclasspt{minimaWe firstly choose a constant ≠0, then  can be rw(z) is a transcendental meromorphic solution of equation (Suppose that equation  with zern of Eq. . We willIf If This completes the proof of Theorem 11.


We theoretically show that in a crystal with a helical lattice structure, orbital and spin magnetizations along a helical axis are induced by an electric current along the helical axis. We propose a simple tight-binding model for calculations, and the results can be generalized to any helical crystals. The induced magnetizations are opposite for right-handed and left-handed helices. The current-induced spin magnetization along the helical axis comes from a radial spin texture on the Fermi surface. This is in sharp contrast to Rashba systems where the induced spin magnetization is perpendicular to the applied current. Recent discovery of novel physics due to an interplay between electricity and magnetism such as multiferroics12356791011b3 = −b1 − b2, where a is a constant. The layers are stacked along the z-direction with the primitive lattice vector c is the interlayer spacing. The Hamiltonian isFirst, we consider an orbital magnetization in a helical crystal. Here, we introduce a three-dimensional tight-binding model with a right- or left-handed helical structures. The lattice structure of this model is composed of an infinite stack of honeycomb lattice layers with one orbital per site. We consider the honeycomb lattice as shown in t1, t2 and Δ are real and we set t1 > 0 for simplicity. The first term is a nearest-neighbor hopping term within the xy plane. The second term represents “helical” hoppings between the same sublattice in the neighboring layers. This term is different between a right-handed and a left-handed helices as shown in ξi is +1 and −1 for the sites in A and B sublattices, respectively. The space group of our model is P321, with Cy2 and Cz3 symmetries. Only when Δ = 0, the space group becomes P622, having additional Cx2 and Cz6 symmetries.where ed helix  the direkzc (−kzc) with a Aharonov-Bohm phase ϕ in the second-neighbor hopping in the Haldane model. We also note that the time-reversal symmetry is present in our model, while it is absent in the Haldane model for ϕ ≠ 0. Hence the band structure of our model is easily obtained from that of the Haldane modelk ⋅ bi = −2π/3 (2π/3) on the kz = 0 plane, and the H (H’) point is similarly defined on the kz = π/c plane.We note that the Hamiltonian for the right-handed (left-handed) helix becomes the Haldane model on a honeycomb latticeT → 0, it is calculated from the formula14151617181920Using this model, we calculate the orbital magnetization induced by an electric field along the helical axis. In the limit of zero temperature n denotes the band index, fnk is the distribution function for the eigenenergy εnk, εF is the Fermi energy, and the prefactor 2 comes from the spin degeneracy. The orbital magnetic moment of the Bloch electrons is defined by21mnk and Ωnk are allowed to have nonzero values for any k. However, in equilibrium, due to the time-reversal symmetry Morb is zero because in Eq. k and that from −k cancel each other. To induce the orbital magnetization, we apply an electric field along the z axis. For a metal it induces a charge current accompanied by nonequilibrium electron distribution, and orbital magnetization is expected to arise. Within the Boltzmann approximation, the applied electric field Ez changes fnk intowhere the integral is performed over the BZ, τ is the relaxation time assumed to be constant, and vn,z = (1/ħ)∂εnk/∂kz is the velocity in the z direction. Substituting fnk into Eq. Morb, which is along the z-axis by symmetry. Because of time-reversal symmetry, the Morb. In the T → 0 limit, ∂f/∂ε has a sharp peak at εF. Therefore, for a band insulator, the induced orbital magnetization Morb is zero. This is in contrast to multiferroics. Furthermore, the Berry-curvature term , the current-induced orbital magnetization appears. In this case, the induced magnetization is largely enhanced around εF ≃ t1. This is attributed to an enhanced orbital magnetic moment mnk near the Dirac points appearing on the K-H or K’-H’ lines. The orbital magnetization in the left-handed helix . The second term is a spin-dependent nearest-neighbor hopping term within the xy plane. The third and fourth terms involve spin-dependent helical hoppings between the neighboring layers as shown in k-component of a vector pointing from site j to site i. The space group of this model is the same as that of the model (1).The first term is a spin-independent nearest-neighbor hopping term within the xy plane, on the Fermi surface around the H point at εF = 0.68t1. In fact, the spin around the H point has not only the xy-component but also the z-component. Because of the spin-orbit interaction, the two spin-split Fermi surfaces appear, having the opposite spin orientations. Remarkably, unlike Rashba systems, the spin is oriented radially, and rotates once around the H point. εF = 0.68t1. In this case, one of the spin-split Fermi surface is open. Nevertheless, the inner Fermi surface have a radial spin texture. This spin texture results from crystal symmetries. Namely, the K-H lines are three-fold rotation axis and hence the spin on these lines are parallel to these K-H lines. Furthermore, the absence of mirror symmetries is crucial for the radial spin textures; if a mirror plane including the xy plane or the z axis were present, the spins should be perpendicular to the mirror plane, and a radial spin texture would not appear. For example, Te and Se have helical crystal structure, and as expected, they have been predicted to show a radial spin texturez-component of Mspin for several parameters is numerically calculated as shown in λxy = 0.05t1, λz = 0.05t1, and Δ = 0.4t1. As is expected, for a metal, the spin magnetization parallel to the current is induced by the current. Similar to the orbital magnetization, the spin magnetization in the left-handed helix and that in the right-handed helix are opposite.We now calculate current-induced spin magnetization in the present system. It is known that in Rashba systems the current-induced magnetization is perpendicular to the current, because of the tangential spin texture in the spin-split Fermi surfaces242526We have shown that in systems with helical structure, the orbital and spin magnetizations are induced by an electric current along the helical axis, using a simple tight-binding models. When an electric field is applied along the helical axis, the orbital magnetization is induced by the orbital magnetic moment on the Fermi surface, as a consequence of broken inversion symmetry. Similarly to the Berry curvature, the orbital magnetic moment is enhanced near band crossings. Therefore, when the Fermi energy lies near the band crossing, the orbital magnetization is enhanced as well. We have also shown that, in the helical crystal with spin-orbit interaction, a current induces the spin magnetization along the helical axis due to the radial spin texture. The absence of mirror symmetry allows to have the radial spin texture, which is completely different from Rashba systems with a tangential spin texture.e2Ezτt1/ħ2, which is about 150 gauss for Ez = 104 V/m, τ = 10−12 s and t1 = 3 eV. In eEzτμB/ħa2, which is about 7 gauss for Ez = 104 V/m, τ = 10−12 s and a = 0.5 nm. It is multiplied by a numerical factor in λ/t1. Thus, the size of the spin-orbit coupling limits the size of the current-induced spin magnetization. On the other hand, the orbital magnetization does not require spin-orbit coupling, and it can be enhanced by appropriate choice of materials with helical crystal structure. Te and Se consist of weakly coupled helices, and therefore they may have large current-induced orbital magnetization.We have presented a toy model for a helical structure. From a symmetry viewpoint, the present effect of longitudinal current-induced magnetization appears for chiral crystals without inversion and mirror symmetries. Real helical crystals, such as Se and Te, also lack inversion and mirror symmetries. From symmetry consideration, we expect that orbital and spin magnetizations along the helical axis can be induced by an electric current in these crystals by doping carriers. Magnitudes of the induced magnetizations can be estimated from our results. The induced orbital magnetization scales with Our results also provide a new building block of spintronics and will pave the way to spintronics application of helical crystals. For example, consider a ferromagnet on a helical crystal. By injecting current into the helical crystal, the induced magnetization exerts a torque on the ferromagnet, thus leading to current-induced magnetization reversal.Ez induces the spin magnetization asWithin the Boltzmann approximation, the electric field μB is the Bohr magneton and the electron spin g-factor g ≈ 2. In the present model (4), for λ = 0 the spin magnetization always vanish, because of a cancellation between the contribution from  and that from . This cancellation can be avoided by adding either spin-dependent hoppings within the xy plane .


More recently, this approach has been extended to non-Hermitian scenarios characterized by spatially-modulated distributions of optical loss and gain, in order to allow the removal of higher-order modes as well. In this paper, we apply this approach to the design of non-Hermitian optical couplers with higher-order mode-selection functionalities, with potential applications to mode-division multiplexing in optical links. In particular, we highlight the critical role of the coupling between non-Hermitian optical waveguides, which generally induces a phase transition to a complex eigenspectrum, thereby hindering the targeted mode-selection functionality. With the specific example of an optical coupler that selects the second-order mode of a given waveguide, we illustrate the aforementioned limitations and propose possible strategies to overcome them, bearing in mind the practical feasibility of the gain levels required.Supersymmetry has been shown to provide a systematic and effective framework for generating classes of  Though seemingly unrelated, classical optics and quantum mechanics exhibit profound formal analogiesRecently, new levels of sophistication have been added to the above analogies through the intriguing concepts of “supersymmetry” (SUSY) and “parity-time”  = V*(x)  could exhibit entirely real eigenspectra. However, beyond some non-Hermiticity threshold in the potential, an abrupt phase transition from a real eigenspectrum  to a complex eigenspectrum  may occur, which is typically referred to as spontaneous symmetry breakingpassive configurations162232On the other hand, the et al.Interestingly, the SUSY and mode division multiplexing approaches, which offer a new dimension to increase the capacity of optical links via the capability to manipulate the multiplicity of modes that can propagate in optical waveguides. The key feature in such schemes is an efficient mode (de)multiplexing through a judiciously identification of appropriate procedures for selectively populating and extracting specific modes in an integrated fashion. Originally put forward in connection with optical fibers35Y - junctions373839These ideas may find disrupting applications to n = 2, in Against this background, here we explore the application of SUSY-inspired schemesIt is important to stress that our results are not a direct consequence of the study in Ref. bε denoting a real-valued constant background value, and Δε1 a complex-valued variation satisfying the symmetry conditiony-directed electric field) with exp(−iωt) time-harmonic dependence, and capitalizing on the translational invariance, the electric field pertaining to a guided mode can be factorized as42β1 is the modal propagation constant, and the dependence on the transverse variable is governed by the Helmholtz equationk0 = ω/c0 = 2π/λ0 denotes the vacuum wavenumber . As in Ref. unbrokenX ≡ x/w denoting a dimensionless coordinate scaled with respect to a characteristic dimension w  of the waveguide. Following Miri et al.j, which is removed. It can be shown  = 0.1 and |Im(Δε1)| = 0.015. While this choice does not directly correspond to a specific material, it allows to illustrate the basic phenomenology in terms of moderately-sized structures . Nonetheless, we also discuss the implications of different, more realistic choices of material parameters.Particularly critical is the choice of the loss/gain level: on one hand, it should be sufficiently low, so as not to dramatically affect the mode orthogonality, and to maintain all the modes in the unbroken phase. On the other hand, an exceedingly small level of non-Hermiticity in the original waveguide may result in a superpotential cf.  with sinn = 2 eigenvalue, we compute the superpotential W(2)  only occurs beyond a critical non-Hermiticity threshold. It this regime, it can be shown (see iν (with ν > 0) in the SUSY*-partner waveguide only, it is possible to drive all supermodes in the exponential-attenuation (] > 0) regime, with the exception of the desired m = 3 supermode which may instead exhibit exponential amplification ] < 0).For more quantitative assessments, we focus on the particular case of w = 2λ0, D = 1.02, |Im (Δε1)| = 0.015, and ν = 0.0032) excited with a linear combination of the three modes of the original waveguide  are in line with the values reported in the literature for semiconductor-based devices46−1 and of 6.8 · 104 cm−1 have been reported for InGaAs single and double quantum well laser structuresBL ~ 1100λ0.Finally, we stress that the specific profiles and parameters in the our case-study are essentially chosen in order to facilitate the illustration of the physical aspects and phenomena, in line with the main scope of this prototype theoretical study. This implies that the structure is not optimized having in mind fabrication-related implications. Nevertheless, we highlight that the required gain values  power gain coefficient would decrease to levels ~102 cm−1, which are compatible with media made of polymeric matrix and glasses doped with quantum dots or other dopants5051Alternatively, we could also reduce the (complex) contrast, so as to tradeoff lower levels of gain with longer structures. For instance, by reducing the contrast of an order of magnitude, say, ΔAll numerical simulations in our analysis rely on the finite-element-based commercial software package COMSOL Multiphysics.λ0, discretized with steps of 0.002λ0, and terminated with Neumann-type boundary conditions.More specifically, for the study of the supermodes of the compound structures cf. , we utilλ0 × 700λ0 computational domain, discretized with a maximum mesh-element size of 0.12 λ0, and terminated by an ad-hoc perfectly-matched layer.For the study of the propagation of a given multimode input profile cf. , we utilM.P., M.C., A.C. and V.G. conceived the idea of this study. M.P. and G.C. developed the theory, designed the structures and carried out the numerical simulations, with inputs and feedback from all the other authors. A.C. and V.G. supervised the project. M.P. and V.G. wrote the manuscript with inputs and revisions from all the other authors.Supplementary information


In this paper, we prove a fixed point theorem and a common fixed point theorem for multi valued mappings in complete b-metric spaces. Fixed point theory plays one of the important roles in nonlinear analysis. It has been applied in physical sciences, Computing sciences and Engineering. In 1922, Stefan Banach proved a famous fixed point theorem for contractive mappings in complete metric spaces. Later, Czerwik , 1998 haX and Y be nonempty sets. T is said to be multi-valued mapping from X to Y if T is a function for X to the power set of Y. we denote a multi-valued map by:Nadler  Let X anT if Nadler  A point T is a multi-valued mapping iff for each Tx is non-empty for each Joseph  Every siX, d) be a metric space. A map Banach  Led  be a metric space. We define the Hausdorff metric on CB(X) induced by d. That isCB(X) denotes the family of all nonempty closed and bounded subsets of X and Nadler  Let  be a metric space. A map Nadler  Let  is called a b-metric space.Aydi et al.  Let X beBoriceanu  The spacb-metric space if we take Boriceanu  The spacAydi et al.  Let \docX, d) be a b-metric space. Then a sequence X is called Cauchy sequence if and only if for all Boriceanu  Let  b-metric space. Then a sequence Boriceanu  Let be aOur first result is the following theorem.X, d) be a b-metric space with constant Let  be a completeb-metric space with constantLetbe a multi valued generalized contraction mapping. Then T has a unique fixed point.Fix any Now,X, d) is a complete b-metric space, it follows that Now,mentclass2pt{minimu is the unique fixed point of T. Assume that v is another fixed point of T. Then we have Now we show that Letbe a complete b-metric space with constantLetbe a multi valued mapping satisfies the condition:for all x,yX andwithThen T and S have a unique common fixed point.Fix any entclass1pt{minimaz is a common fixed point of T and S.We shall prove that Similarly, z is the unique fixed point of S and T. Now,We show that X, d) is a complete Let Define Therefore, Many authors have contributed some fixed point results for a self mappings in b-metric spaces. In this paper, we have proved the existence and uniqueness of fixed point results for a multivalued mappings in b-metric spaces. Our contraction mappings also generalize various known contractions like Hardy Roger contraction in the current literature.


The correct sentence is: For the CG-based estimate, e.g., the χThere are errors in