tuberculosis ( tb ) is the second leading cause of death from an infectious disease worldwide .active tb refers to disease that occurs in someone infected with _mycobacterium tuberculosis_. it is characterized by signs or symptoms of active disease , or both , and is distinct from latent tuberculosis infection , which occurs without signs or symptoms of active disease .only individuals with active tb can transmit the infection .many people with active tb do not experience typical tb symptoms in the early stages of the disease .these individuals are unlikely to seek care early , and may not be properly diagnosed when seeking care .delays to diagnosis of active tb present a major obstacle to the control of a tb epidemic , it may worsen the disease , increase the risk of death and enhance tuberculosis transmission to the community . both patient andthe health system may be responsible for the diagnosis delay .efforts should be done in patient knowledge / awareness about tb , and health care systems should improve case finding strategies to reduce the delay in diagnosis of active tb .mathematical models are an important tool in analyzing the spread and control of infectious diseases .there are several mathematical dynamic models for tb , see , e.g. , . in this paperwe consider the mathematical model for tb proposed in .we introduce a discrete time delay which represents the delay on the diagnosis of individuals with active tb and commencement of treatment .the stability of the disease free and endemic equilibriums is analyzed for any time delay .optimal control theory has been successfully applied to tb mathematical models ( see , e.g. , and references cited therein ) .we propose and analyze an optimal control problem where the control system is the mathematical model from , but with a time delay in the state variable that represents individuals with active tb , and introduce two control functions .the control functions represent the fraction of early and persistent latent individuals that are treated for tb .treatment of latent tb infection greatly reduces the risk that tb infection will progress to active tb disease .certain groups are at very high risk of developing active tb disease once infected .every effort should be made to begin appropriate treatment and to ensure completion of the entire course of treatment for latent tb infection .treatment of latent tb infection should be initiated after the possibility of tb disease has been excluded .it can take 2 to 8 weeks after tb infection for the body s immune system to react to tuberculin and for the infected to be detected , which justifies the introduction of a time delay on the control associated to treatment of early latent individuals .on the other hand , delays in the treatment of latent tb may also occur due to clinical and demographic patient and health care services characteristics . for these reasons ,we consider discrete time delays in both control functions . to our knowledge , this work is the first to apply optimal control theory to a tb model with time delay in state and control variables .the paper is organized as follows . in section [ sec:2 ]we formulate the tb model with state delay .the stability of the disease free equilibrium is analyzed in section [ sec:3 ] while stability of the endemic equilibrium is investigated in section [ sec:4 ] .optimal control of tb with state and control delays is carried out in section [ sec:5 ] and some numerical results given in section [ sec:6 ] .we end with section [ sec:7 ] of conclusions .in this section we consider a tb mathematical model proposed in , where reinfection and post - exposure interventions for tuberculosis are considered .the model divides the total population into five categories : susceptible ( ) ; early latent ( ) , , individuals recently infected ( less than two years ) but not infectious ; infected ( ) , , individuals who have active tb and are infectious ; persistent latent ( ) , , individuals who were infected and remain latent ; and recovered ( ) , , individuals who were previously infected and have been treated . as in , we assume that the total population , , with , is constant in time . in other words ,we assume that the birth and death rates are equal and there are no tb - related deaths .we introduce a discrete time - delay in the state variable , denoted by , that represents the delay to diagnosis and commencement of treatment of active tb infection , the initial conditions for system ( [ tbmodel : delay : state ] ) are  , where with the banach space , { \mathbb{r}}^5 \right) ] into . from biological meaning , we further assume that for .throughout this paper , we focus on the dynamics of the solutions of ( [ tbmodel : delay : state ] ) in the restricted region in this region , the usual local existence , uniqueness and continuation results apply .hence , a unique solution of ( [ tbmodel : delay : state ] ) with initial condition ( [ eq : init : cond : moddelay ] ) exists for all time .consider the solutions of ( [ tbmodel : delay : state ] ) with , , for all  ] onehas and since and , we conclude that there exists a unique in the interval -\infty , \bar{\lambda}_1[ ] that _ minimizes _ either the cost functional in or in subject to the dynamic constraints , initial conditions and control constraints .necessary optimality conditions for optimal control problems with multiple time delays in control and state variables may be found , e.g. , in . here, we discuss the _ maximum principle _ in order to display the controls and the switching functions in a convenient way . to define the hamiltonian for the delayed control problem, we introduce the delayed state variable and the delayed control variables , . using the adjoint variable ,the hamiltonian for the objective and the control system is given by \quad + \lambda_{l_1 } ( \frac{\beta}{n } ( s + \sigma l_2 + \sigma_r r ) - ( \delta + \tau_1 + \epsilon_1 v_1 + \mu ) l_1)\\[1 mm ] \quad + \lambda_i ( \phi \delta l_1 + \omega l_2 + \omega_r r - \tau_0 y_3 - \mu i ) \\[1 mm ] \quad + \lambda_{l_2 } ( ( 1 - \phi ) \deltal_1- \sigma \frac{\beta}{n } i l_2 - ( \omega + \epsilon_2 v_2 + \tau_2 + \mu)l_2 ) . \end{array}\ ] ] we obtain the adjoint equations  ] ,  ] , where subscripts denote partial derivatives and } ] .note that only the equation for contains the _ advanced _ time . since the terminal state is free , the transversality conditions are to characterize the optimal controls and , we introduce the following _ switching functions _ for : + \chi_{\,[0,t - d_{u_k}]}(t+d_{u_k})\ ,  h_{v_k}[t+d_{u_k}]\\[1 mm ] & = \ ; \left\ { \begin{array}{lcl } -w_k - \epsilon_k \lambda_{l_k}(t+d_{u_k } )  l_k(t+d_{u_k } ) & \mbox{for } & 0 \leq t \leq t - d_{u_k},\\[1 mm ] -w_k & \mbox{for } &  t - d_{u_k } \leq t \leq t. \end{array } \right .\end{array}\ ] ] then the maximum condition for the optimal controls , is equivalent to the maximization , , which gives the control law 0 & & \mbox{if } \quad \phi_k(t ) < 0 ,  \\[1 mm ] { \rm singular } & &  \mbox{if } \quad \phi_k(t ) = 0 \;\ ; \mbox{on } \ ; i_s \subset [ 0,t ] , \end{array } \right .\quad k=1,2.\ ] ] we do not discuss singular controls further , since both in the non - delayed and the delayed control problem we did not find singular arcs . in view of the transversality conditions , the terminal value of the switching function is for .hence , we may conclude from the control law that holds on a terminal interval .we choose the numerical approach `` first discretize then optimize '' to solve both the non - delayed and delayed optimal control problem .the discretization of the control problem on a fine grid leads to a large - scale nonlinear programming problem ( nlp ) that can be conveniently formulated with the help of the applied modeling programming language ampl .ampl can be linked to several powerful optimization solvers .we use the interior - point optimization solver ipopt developed by wchter and biegler .details of discretization methods for delayed control problems may be found in .the subsequent computations for the terminal time have been performed with to grid points using the trapezoidal rule as integration method .choosing the error tolerance in ipopt , we can expect that the state variables are correct up to or decimal digits .since lagrange multipliers are computed a posteriori in ipopt , we can not expect more than 6 correct decimal digits in the adjoint variables .also , the control package nudocccs developed by bskens ( cf . also ) provides a highly efficient method for solving the discretized control problem , because it allows to implement higher order integration methods .however , so far nudocccs can only be implemented for non - delayed control problems . for the non - delayed tb control problem, we obtained only bang - bang controls .an important feature of nudocccs is the fact that it provides an efficient method for optimizing the switching times of bang - bang controls using the arc - parametrization method .this approach is called the induced optimization problem ( iop ) for bang - bang controls .nudocccs then allows for a check of second - order sufficient conditions of the iop , whereby the second - order sufficient conditions for bang - bang controls can be verified with high accuracy ; cf . .first , we consider the optimal control of non - delayed tb model , where formally we put .the numerical solutions serve as reference solutions , which later will be compared with the solutions to the delayed control problem .we choose the weights in the objective ( [ j1-objective ] ) and the parameter ( ) in table [ table : parameters ] .the discretization approach shows that controls are bang - bang and with only one switch at , : to obtain a refined solution , we solve the iop with respect to the switching times and using the arc - parametrization method and the code nudocccs .we get the following numerical results : the initial value of the adjoint variable is computed as the control and state trajectories are displayed in figure [ label : of : fig:2 ] .objective and weights ._ top row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) susceptible individuals and recovered individuals , ( c ) infectious individuals ._ bottom row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) early latent , ( c ) persistent latent .,title="fig:",scaledwidth=31.0% ]  objective and weights ._ top row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) susceptible individuals and recovered individuals , ( c ) infectious individuals . _ bottom row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) early latent , ( c ) persistent latent .,title="fig:",scaledwidth=31.0% ]  objective and weights . _top row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) susceptible individuals and recovered individuals , ( c ) infectious individuals . _ bottom row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) early latent , ( c ) persistent latent .,title="fig:",scaledwidth=31.0% ] +  objective and weights . _ top row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) susceptible individuals and recovered individuals , ( c ) infectious individuals . _ bottom row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) early latent , ( c ) persistent latent .,title="fig:",scaledwidth=31.0% ]  objective and weights ._ top row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) susceptible individuals and recovered individuals , ( c ) infectious individuals . _ bottom row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) early latent , ( c ) persistent latent .,title="fig:",scaledwidth=31.0% ]  objective and weights . _top row _ : ( a ) control and ( scaled ) switching function satisfying the control law for ,( b ) susceptible individuals and recovered individuals , ( c ) infectious individuals . _ bottom row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) early latent , ( c ) persistent latent .,title="fig:",scaledwidth=31.0% ]  the hessian of the lagrangian for the iop is positive definite : .it can be seen from figure [ label : of : fig:2 ] , top row ( a ) and bottom row ( a ) , that the switching functions satisfy the strict bang - bang property ( cf . ) corresponding to the maximum principle : for , , for .hence , the bang - bang controls characterized by the data satisfies the second - order sufficient conditions in ( * ? ? ?* chap . 7 ) and thus provides a strict strong minimum .figure [ label : of : fig:3 ] displays the comparison of the optimal controls for the functionals and .the state variables are nearly identical , since the control variables differ only a terminal interval .also the objective values are very close : , .note that the controls for are _ continuous _ , since the strict legendre clebsch condition holds and the hamiltonian has a unique minimum .the proof , that second - order sufficient conditions ( ssc ) are satisfied for the controls corresponding to , is quite elaborated since it requires to test whether an associated matrix riccati equation has a bounded solution ; cf . .   and for the -type objective and -type objective with weights .,title="fig:",scaledwidth=40.0% ]  and for the -type objective and -type objective with weights .,title="fig:",scaledwidth=40.0% ]  when we increase the weights and , the control stays to be bang - bang with only one switching time which gets smaller , whereas the control may have an additional zero arc at the beginning .e.g. , for we obtain the controls the objective value and the switching times are computed as , , , and .the optimal controls are shown in figure [ label : of : fig:4 ] .   and for the -type objective with weights .,title="fig:",scaledwidth=40.0% ]  and for the -type objective with weights .,title="fig:",scaledwidth=40.0% ]   to see more distinctively the difference between delayed and non - delayed solutions , we consider state and control delays with values at their upper bounds in , that is , , , and .again , we choose the weights in the objective ( [ j1-objective ] ) and the parameter in table [ table : parameters ] , for which .the discretization approach with grid points and the trapezoidal rule as integration method yields the following bang - bang controls with only one switch as in the non - delayed case : we obtain the numerical results , , , , , , , and .however , in contrast to the non - delayed case , we are not able to optimize the switching times directly because the time - transformation in the arc - parametrization method can not be applied to the delayed problem . the initial value of the adjoint variable is computed as . when comparing the results for the delayed problem with those in for the non - delayed problem , we notice the surprising fact that the terminal values , , and the switching times , are significantly smaller in the delayed problem on the expense that the terminal value is significantly higher .-objective , and delays ._ top row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) susceptible individuals and recovered individuals , ( c ) infectious individuals . _ bottom row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) early latent , ( c ) persistent latent .,title="fig:",scaledwidth=32.0% ] -objective , and delays . _ top row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) susceptible individuals and recovered individuals , ( c ) infectious individuals . _ bottom row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) early latent , ( c ) persistent latent .,title="fig:",scaledwidth=29.0% ] -objective , and delays . _ top row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) susceptible individuals and recovered individuals , ( c ) infectious individuals . _ bottom row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) early latent , ( c ) persistent latent .,title="fig:",scaledwidth=29.0% ] + -objective , and delays . _ top row _ : ( a )control and ( scaled ) switching function satisfying the control law for , ( b ) susceptible individuals and recovered individuals , ( c ) infectious individuals . _ bottom row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) early latent , ( c ) persistent latent .,title="fig:",scaledwidth=32.0% ] -objective , and delays ._ top row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) susceptible individuals and recovered individuals , ( c ) infectious individuals . _ bottom row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) early latent , ( c ) persistent latent .,title="fig:",scaledwidth=30.0% ] -objective , and delays . _ top row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) susceptible individuals and recovered individuals , ( c ) infectious individuals . _ bottom row _ : ( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) early latent , ( c ) persistent latent .,title="fig:",scaledwidth=30.0% ]  as in the non - delayed problem , the switching functions satisfy the strict bang - bang property related to the maximum principle : however , we are not aware in the literature of any type of _ sufficient conditions _ which could be applied to the extremal solution shown in figure [ label : of : fig:5 ] .we also compared the extremal solutions for the -type objective and the -type objective .since the controls are very similar to those in figure [ label : of : fig:3 ] , they are not displayed here .figure [ label : of : fig:6 ] shows the extremal controls for the -objective for the increased weights .objective , and delays , .( a ) control and ( scaled ) switching function satisfying the control law for , ( b ) control and ( scaled ) switching function satisfying the control law for .,title="fig:",width=196,height=143 ]  objective , and delays , .( a ) control and ( scaled ) switching function satisfying the control law for ,( b ) control and ( scaled ) switching function satisfying the control law for .,title="fig:",width=196,height=143 ]  the most significant influence on the optimal controls is exerted by the transmission coefficient .   and in the delayed tb model with objective , weights and delays .,title="fig:",width=196,height=151 ]  and in the delayed tb model with objective , weights and delays .,title="fig:",width=196,height=151 ]  it can be clearly seen that the increase of the transmission coefficient triggers a substantial increase in the switching times of the bang - bang controls for ( cf .figure [ label : of : fig:7 ] ) as could be expected from the equation for .let us perform in this case a more detailed sensitivity analysis of the trajectories with respect to the parameter  ] by a homotopy method and plot the objective and the terminal state as a function of .the numerical results are shown in figure [ label : of : fig:8 ] .objective and weights for parameters  ] .displayed are the objective value and the terminal states , , , , .,title="fig:",scaledwidth=31.0% ]  objective and weights for parameters  ] .displayed are the objective value and the terminal states , , , , .,title="fig:",scaledwidth=31.0% ]  objective and weights for parameters  ] .displayed are the objective value and the terminal states , , , , .,title="fig:",scaledwidth=31.0% ]we introduced a discrete time delay in a mathematical model for tuberculosis ( tb ) , which represents the delay on the diagnosis of active tb infection and commencement of treatment .the delay on the diagnosis of active tb has important negative consequences on tb control and eradication .the later the treatment of active tb starts , the more people can be infected and may die from tb .the introduction of a time delay on the state variable of active tb infected individuals is therefore justified from the epidemiological point of view .when a time delay is introduced into a mathematical model , the stability of its disease free and endemic equilibriums may change .we proved that the disease free equilibrium ( dfe ) of the tb model with delay in the state variable is unstable for any time delay , whenever the basic reproduction number is greater than one .we derived conditions under which the model is locally asymptotically stable for and proved that there exists at least one positive time delay such that the dfe is unstable for . despite of this, we also proved that for the concrete time delay , the set of parameters of table [ table : parameters ] , and a transmission coefficient for which , the dfe is locally asymptotically stable .the value ( 36.5 days ) fits the data reported in the literature for the delay in the diagnosis of active tb . for the endemic equilibrium ( ee ), we considered that the transmission coefficient is such that and proved the local stability of the specific ee associated to our set of parameters and discrete time delay .we proposed an optimal control problem where the control system is the mathematical model for tb with time delay in the state variable and where the control functions and represent , respectively , the effort on early detection and treatment of recently infected individuals , and the application of prophylactic treatment to persistent latent individuals .the objective is to minimize the number of individuals with active and persistent latent tb as well as the cost associated to the implementation of the control measures .human immune system can take from two to eight weeks to react to tb infection and detection . moreover , prophylactic treatment of early and persistent latent individuals may face resistance from patients and health care services . based on these facts , we introduced discrete time delays in the controls and . to our knowledge , this is the first time an optimal control problem for tb with delays in the state and control variables is investigated .firstly , we considered the non - delayed case ( ) and compared the solutions for and objectives .our results show that the optimal state variables are nearly identical for both objectives , the control variables differing only on a terminal interval .the optimal values of the objective functionals are also very close .we observed that for the objective the optimal control may have an additional zero arc at the beginning if the weight constants associated to the implementation of the control policies and are big enough .for the delayed optimal control problem , our focus was on the objective since it seems to be the more realistic and , comparing the extremal solutions of and objectives , the differences are similar to the non - delayed case . in the delayed case , the switching times and are significantly smaller and the costs are also smaller , when compared to the non - delayed case . from an epidemiological point of view , when we introduce delays in the tb model , the optimal solutions for the reduction of the number of individuals with active and persistent latent tb infection are associated to control policies that are less costly and can be implemented in a shorter period of time .associated to these positive facts , we observed that if a delay is introduced in the state variable and in the controls , the number of individuals with active tb infection at the end of five years is reduced in approximately 45.2 per cent , when compared to the non - delayed case .similarly , the number of persistent latent individuals at the end of five years is also reduced in approximately 60.1 per cent .moreover , the terminal number of susceptible and recovered individuals is bigger in the delayed case . through a sensitivity analysis, we observed that the transmission coefficient has a significant influence on the optimal controls and the cost functional , and that the number of active infected individuals and the number of early and persistent latent individuals also increase with .the number of recovered individuals increases for  ] , which means that for the control measures are no longer effective and should be rethought .this work was partially supported by fct within project toccata , reference ptdc / eei - aut/2933/2014 .silva and torres were also supported by cidma and project uid / mat/04106/2013 ; silva by the post - doc grant sfrh / bpd/72061/2010 .the authors are very grateful to anonymous reviewers for their careful reading and helpful comments .( mr1781710 ) [ 10.1016/s0377 - 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Abstract: we introduce delays in a tuberculosis ( tb ) model , representing the time delay on the diagnosis and commencement of treatment of individuals with active tb infection . the stability of the disease free and endemic equilibriums is investigated for any time delay . corresponding optimal control problems , with time delays in both state and control variables , are formulated and studied . although it is well - known that there is a delay between two to eight weeks between tb infection and reaction of body s immune system to tuberculin , delays for the active infected to be detected and treated , and delays on the treatment of persistent latent individuals due to clinical and patient reasons , which clearly justifies the introduction of time delays on state and control measures , our work seems to be the first to consider such time - delays for tb and apply time - delay optimal control to carry out the optimality analysis .   cristiana j. silva  helmut maurer  delfim f. m. torres


the open nature of the wireless channel facilitates a multiuser transmission , but also incurs the security problem . recently , as a complement of traditional upper - layer encryption techniques , physical layer security ( phy - security ) has been proposed to realize secure communications by making use of the characteristics of wireless channels , i.e. , noise , fading and interference .  from an information - theoretic viewpoint ,the performance of phy - security is determined by the rate difference between the legitimate channel and the eavesdropper channel  .therefore , to enhance wireless security , it makes sense to simultaneously increase the legitimate channel rate and decrease the eavesdropper channel rate .inspired by this , various physical layer techniques have been introduced to improve the secrecy performance .wherein , mimo relaying technique gains considerable attention due to the following two reasons .first , the relay can provide a diversity gain and shorten the accessing distance , and thus improve the secrecy performance .second , mimo techniques , such as spatial beamforming , can reduce the information leakage to the eavesdropper .the beamforming schemes at the mimo relay based on amplify - and - forward ( af ) and decode - and - forward ( df ) relaying protocols were presented in and , respectively .it is worth pointing out that the optimal beam design at the relay requires global channel state information ( csi ) .yet , the csi , especially eavesdropper csi is difficult to obtain , since the eavesdropper is usually passive and keeps silent .therefore , it is impossible to realize absolutely secure communications over fading channels . in thiscontext , statistical secrecy performance metrics , e.g. , secrecy outage capacity and interception probability , are adopted to evaluate wireless security .recently , an advanced mimo relaying technology , namely massive mimo ( m - mimo ) relaying , is introduced into secure communications to further improve the secrecy performance . even without eavesdropper csi ,m - mimo relaying can generate a very high - resolution spatial beam , and thus the information leakage to the eavesdropper is quite small .more importantly , the secrecy performance can be enhanced by simply adding more antennas . hence , the challenging issue of short - distance interception in secure communications can be well solved .note that in two - hop secure communications , the transmit power at the relay has a great impact on the secrecy performance , since it affects the signal quality at both the destination and the eavesdropper .an optimal power allocation scheme for a multi - carrier two - hop single - antenna relaying network was given in by maximizing the sum secrecy rate .however , the power allocation for a multi - antenna relay , especially an m - mimo relay , is still an open issue . in this paper, we focus on power allocation for df m - mimo secure relaying systems under very practical assumptions , i.e. , no eavesdropper csi and imperfect legitimate csi .the contributions of this paper are three - fold :  1 .  we reveal the relation between the secrecy outage capacity and a defined relative distance - dependent path loss , and then derive the condition for a nonnegative secrecy outage capacity and the constraint on the minimum number of antennas .we derive closed - form expressions for the optimal power at the relay in the sense of maximizing the secrecy outage capacity and minimizing the interception probability , respectively .we present clear insights into the secrecy performance through asymptotic analysis .we show that the maximum secrecy outage capacity is an increasing function of the source power while the minimum interception probability keeps constant .the rest of this paper is organized as follows .we first give an overview of the df m - mimo secure relaying system in section ii , and then derive two optimal power allocation schemes for the relay in section iii . in section iv, we present simulation results to validate the effectiveness of the proposed schemes .finally , we conclude the paper in section v.we consider a time division duplex ( tdd ) two - hop massive mimo ( m - mimo ) relaying system , where a single antenna source transmits message to a single antenna destination with the aid of a relay with antennas , while a passive single antenna eavesdropper intends to intercept the information .note that the number of antennas at the relay in such an m - mimo relaying system is quite large , i.e. , or even bigger .it is assumed that there is no direct transmission between the source and the destination due to a long propagation distance .the relay system works in a half - duplex mode , which means any successful transmission requires two time slots .specifically , the source sends the signal to the relay in the first time slot , and then the relay forwards the post - processing signal to the destination in the second time slot . in this paper , we assume the eavesdropper is far from the source and close to the relay , since it was concluded that the signal comes from the relay directly .therefore , it is reasonable to assume that the eavesdropper only monitors the transmission from the relay to the destination .we use , and to represent the channels from the source to the relay , the relay to the destination , and the relay to the eavesdropper , respectively , where , and are the distance - dependent path losses and , , and denote the channel fading coefficient vectors with independent and identically distributed ( i.i.d . )zero mean and unit variance complex gaussian entries .it is assumed that the channels remain constant during a time slot and fade independently over slots .thus , the received signal at the relay in the first time slot can be expressed as where is the normalized gaussian distributed transmit symbol , is the transmit power at the source , and stands for the additive white gaussian noise with unit variance at the relay .we assume that the relay has perfect csi about by channel estimation .then , maximum ratio combination ( mrc ) decoding is performed to recover the information .specifically , the received signal is multiplied by a vector .  during the second time slot, the relay forwards the decoded signal through maximum ratio transmission ( mrt ) .we assume that the relay has partial csi about due to channel reciprocity impairment in tdd systems .the relation between the estimated csi and the real csi is given by where is the error vector with i.i.d .zero mean and unit variance complex gaussian entries , and is independent of . , scaling from to , is the correlation coefficient between and .thus , the received signals at the destination and the eavesdropper are given by and respectively , where is the transmit power of the relay , is the forwarded signal with being a mrt beamforming vector . and are additive white gaussian noise ( awgn ) samples with unit variance at the destination and the eavesdropper , respectively .in this section , we aim to optimize the secrecy performance through allocating the relay power , since it affects the signal quality at both the destination and the eavesdropper .in what follows , we analyze and design the power allocation schemes in the sense of maximizing the secrecy outage capacity and minimizing the interception probability , respectively .   based on the received signal in ( [ eqn1 ] ) , when performing mrc decoding at the relay , the channel capacity from the source to the relay can be expressed as where is the spectral bandwidth and is the signal - to - noise ratio ( snr ) .similarly , according to ( [ eqn3 ] ) and ( [ eqn4 ] ) , channel capacities from the relay to the destination and from the relay to the eavesdropper are given by and respectively , where and .then , for the secrecy outage capacity in a df m - mimo secrecy relaying system , we have the following theorem :  _ theorem 1 _ : given an outage probability bound by , the secrecy outage capacity is approximated as , when the number of relay antennas is large.  please refer to appendix i.  note that the secrecy outage capacity may be negative from a pure mathematical view .hence , it makes sense to find the condition that the secrecy outage capacity is nonnegative .  for notional simplicity ,we let , where is defined as the relative distance - dependent path loss .then , the secrecy outage capacity can be rewritten as    observing the above secrecy outage capacity , we get the following theorem :  _ theorem 2 _ : if and only if , the secrecy outage capacity in such a df m - mimo secure relaying system in presence of imperfect csi is nonnegative.  please refer to appendix ii .notice that is the precondition for power allocation in such a secure relaying system . given channel conditions and outage probability requirements , in order to fulfill ,the number of antennas must be bigger than . for an m - mimo relaying system, it is always possible to meet the condition of by adding more antennas , which is one of its main advantages . in what follows, we only consider the case of .based on theorem 1 , the secrecy outage capacity maximization power allocation is equivalent to the following optimization problem : where is the transmit power constraint at the relay .  for the optimal solution of the above optimization problem , we have the following theorem :  _ theorem 3 _ : the optimal power at the relay is , and the corresponding maximum secrecy outage capacity is .please refer to appendix iii ._ remark _ : it is found that when eavesdropper csi is unavailable , it is optimal for the df secure relaying system to let the two hops have the same channel capacity , resulting in .   in this subsection, we analyze the optimal power allocation from the perspective of minimizing the interception probability . in general ,interception probability is defined as the probability of information leakage , namely the probability of .then , interception probability is equivalent to the secrecy outage probability when in ( [ equ5 ] ) .thus , it can be computed as where is the legitimate channel capacity .then , interception probability minimization power allocation can be described as the following optimization problem : since is a monotonously increasing function of and is equivalent to , can be transformed to the following problem :  by solving the above optimization problem , we have the following theorem :  _ theorem 4 _ : from the perspective of minimizing interception probability , the optimal transmit power at the relay belongs to a region  ] .second , when , the optimization problem is reduced as  the optimal solution of is , since the objective function is a decreasing function of .        c. jeong , i - m .kim , and d. kim , joint secure beamforming design at the source and the relay for an amplify - and - forward mimo untrusted relay system , " _ ieee trans . signal process .1 , pp . 310 - 325 , jan .2012 .b. m. hochwald , t. l. marzetta , and v. tarokh , multiple - antenna channel hardening and its implications for rate - feedback and scheduling , " _ ieee trans .inf . theory _1893 - 1909 , sept . 2004

Abstract: in this paper , we address the problem of optimal power allocation at the relay in two - hop secure communications under practical conditions . to guarantee secure communication during the long - distance transmission , the massive mimo ( m - mimo ) relaying techniques are explored to significantly enhance wireless security . the focus of this paper is on the analysis and design of optimal power assignment for a decode - and - forward ( df ) m - mimo relay , so as to maximize the secrecy outage capacity and minimize the interception probability , respectively . our study reveals the condition for a nonnegative the secrecy outage capacity , obtains closed - form expressions for optimal power , and presents the asymptotic characteristics of secrecy performance . finally , simulation results validate the effectiveness of the proposed schemes .


computational social choice studies computational aspects of voting . for example , how does a coalition of agents compute a manipulation ? can we compile these votes into a more compact form ? how do we decide if we have elicited enough votes from the agents to be able to declare the result ?whilst there has been a very active research community studying these sort of questions for well known voting rules like plurality and borda , there are other less well known rules that might deserve attention .in particular , we put forward two historical voting rules due to nanson and baldwin which are related to borda voting .there are several reasons to consider these two rules .firstly , they have features that might appeal to the two opposing camps that support borda and condorcet .in particular , both rules are condorcet consistent as they elect the candidate who beats all others in pairwise elections .secondly , both rules are elimination style procedures where candidates are successively removed .other elimination procedures like stv and plurality with runoff are computationally hard to manipulate ( in the case of stv , with or without weights on the votes , whilst in the case of plurality with runoff , only in the case of weighted votes ) .we might therefore expect nanson s and baldwin s rules to be computationally hard to manipulate .thirdly , statistical analysis suggest that , whilst the borda rule is vulnerable to manipulation , nanson s rule is particularly resistant . we might expect baldwin to be similarly resistant .finally , the two rules have been used in real elections in the universitiy of melbourne ( between 1926 and 1982 ) , the university of adelaide ( since 1968 ) , and the state of michigan ( in the 1920s ) .it is perhaps therefore somewhat surprising that neither rule has received much attention till now in the computational social choice literature .let be the set of _ candidates _ ( or _ alternatives _ ) .a linear order on is a transitive , antisymmetric , and total relation on .the set of all linear orders on is denoted by .an -voter profile on consists of linear orders on .that is , , where for every , .the set of all -profiles is denoted by .we let denote the number of candidates .a ( deterministic ) _ voting rule _ is a function that maps any profile on to a unique winning candidate , that is , . in this paper , if not mentioned otherwise , ties are broken in the fixed order .  _( positional ) scoring rules _ are commonly used voting rules .each positional scoring rule is identified by a _ scoring vector _ of integers , for any vote and any candidate , let , where is the rank of in . for any profile ,let .the rule selects such that the total score is maximized .we assume scores are integers and decreasing ._ borda _ is the positional scoring rule that corresponds to the scoring vector .we write for the borda score given to candidate from the profile of votes , and where is obvious from the context .when voters are weighted ( that is , each voter is associated with a positive real number as the weight ) , a positional scoring rule selects the candidate that maximizes the weighted total score .the _ unweighted ( coalitional ) manipulation _ problem is defined as follows . an instance is a tuple , where is a voting rule , is the non - manipulators profile , is the candidate preferred by the manipulators , and is the set of manipulators .we are asked whether there exists a profile for the manipulators such that .the _ weighted ( coalitional ) manipulation _ is defined similarly , where the weights of the voters ( both non - manipulators and manipulators ) are also given as inputs .as is common in the literature , we break ties in favour of the coalition of the manipulators where appropriate .the borda rule has several good properties .for instance , it is monotonic as increasing the score for a candidate only helps them win .also it never elects the condorcet loser ( a candidate that loses to all others in a majority of head to head elections ) .however , it may not elect the condorcet winner ( a candidate that beats all others in a majority of head to head elections ) .nanson s and baldwin s rules , by comparison , always elect the condorcet winner when it exists .nanson s and baldwin s rules are derived from the borda rule .nanson s rule eliminates all those candidates with less than the average borda score  .the rule is then repeated with the reduced set of candidates until there is a single candidate left .a closely related voting rule proposed by baldwin successively eliminates the candidate with the lowest borda score until one candidate remains .the two rules are closely related , and indeed are sometimes confused .one of the most appealing properties of nanson s and baldwin s rules is that they are condorcet consistent , i.e. they elect the condorcet winner .this follows from the fact that the borda score of the condorcet winner is never below the average borda score .both rules possess several other desirable properties including the majority criterion and the condorcet loser criterion .there are also properties which distinguish them apart .for instance , nanson s rule satisfies reversal symmetry ( i.e. if there is a unique winner and voters reverse their vote then the winner changes ) but baldwin s rule does not .we start by considering the computational complexity of manipulating both these rules with unweighted votes .we prove that the coalitional manipulation problem is np - complete for both rules even with a single manipulator .computational intractability with a single manipulator is known only for a small number of other voting rules including the second order copeland rule  , stv  and ranked pairs  .in contrast , when there are two or more manipulators , unweighted coalitional manipulation is hard for some other common voting rules  .our results therefore significantly increase the size of the set of voting rules used in practice that are known to be np - hard to manipulate with a single manipulator .this also contrasts to borda where computing a manipulation with a single manipulator is polynomial  . adding elimination rounds to borda to get nanson s or baldwin s rules increases the computational complexity of computing a manipulation with one manipulator from polynomial to np - hard .our results are proved by reductions from the exact 3-cover ( x3c ) problem .an x3c instance contains two sets : and , where and for all , and .we are asked whether there exists a subset of such that each element in is in exactly one of the 3-sets in .[ thm : baldwin ] with unweighted votes , the coalitional manipulation problem under baldwin s rule is np - complete even when there is only one manipulator .  *proof : * we sketch a reduction from x3c . given an x3c instance ,we let the set of candidates be , where is the candidate that the manipulator wants to make the winner , , and and are additional candidates .members of correspond to the 3-sets in .let .  the profile contains two parts : , which is used to control the changes in the score differences between candidates , after a set of candidates are removed , and , which is used to balance the score differences between the candidates .we define the votes where is a total order in which the candidates in are in a pre - defined lexicographic order , and is the reverse .we make the following observations on . for any set of candidates and any pair of candidates , here is the pair of votes obtained from by removing all candidates in . in words ,the formula states that after is removed , the score difference between and is increased by if and only if and is removed ; it is decreased by if and only if and is removed ; for any other cases , the score difference does not change .moreover , for any , and .we next show how to use to construct the first part of the profile .let , that is , . is composed of the following votes : ( 1 ) for each and each , there are copies of ; ( 2 ) for each , there are copies of ; ( 3 ) there are copies of .it is not hard to verify that , and for any , . is composed of the following votes : ( 1 ) for each , there are copies of ; ( 2 ) for each , there are copies of ; ( 3 ) there are copies of .let .we make the following observations on the borda scores of the candidates in . for any , ;   for any , ;   .suppose the x3c instance has a solution , denoted by ( after reordering the sets in ) .then , we let the manipulator vote for :    in the first rounds , all candidates in and drop out .then drops out . in the following rounds the candidates in drop out .finally , loses to in their pairwise election , which means that is the winner .suppose the manipulator can cast a vote to make the winner .we first note that must be eliminated in the final round since its score is higher than in all previous rounds . in the round when is eliminated, the score of should be no more than the score of .we note that and the score difference can only be reduced by the manipulator ranking below , and by eliminating before .however , by ranking below , the score difference is reduced by no more than .therefore , before drops out , all candidates in must have already dropped out .we note that for any , .therefore , for each , there exists with who is removed before . for any such , none of the candidates in can drop out before ( otherwise the score of can not be less than before drops out ) , and in the next three rounds the candidates in drop out .it follows that the set of candidates in that drop out before any candidate in corresponds to an exact cover of .  with unweighted votes ,the coalitional manipulation problem under nanson s rule is np - complete even when there is only one manipulator .the proof uses the same gadget that is used in the proof of theorem [ thm : baldwin ] . due to the space constraints, the proof can be found in an online technical report .if the number of candidates is bounded , then manipulation is np - hard to compute when votes are weighted .baldwin s rule appears more computationally difficult than nanson s rule . prove that baldwin s requires only 3 candidates to be np - hard , whilst we prove here that nanson s rule is polynomial to manipulate with 3 candidates and requires at least 4 candidates to be np - hard .it follows that computing a manipulation is np - hard for both rules when votes are unweighted , the number of candidates is small and there is uncertainty about how agents have voted in the form of a probability distribution .note that the coalition manipulation problem for borda with weighted votes is np - hard for 3 or more candidates .thus , somewhat surprisingly , adding an elimination round to borda , which gives us nanson s rule , decreases the computational complexity of computing a manipulation with 3 manipulators from np - hard to polynomial .[ t : baldwin_constructive ] with nanson s rule and weighted votes , the coalition manipulation problem is np - complete for just 4 candidates .* proof : * the proof is by a reduction from partition , where we are given a group of integers with sum , and we are asked whether there is way to partition the group into two groups , the elements in each of which sum to . for any partition instance , we construct a coalition manipulation problem with 4 candidates ( , , and ) where is again the candidate that the manipulators wish to win . we suppose the non - manipulators have voted as follows : for each of , , and , for and , and 1 each for , , and .the total scores from non - manipulators are as follows : , and . for each integer , we have a member of the manipulating coalition with weight .  now, suppose there is a solution to the partition instance .let the manipulators corresponding to the integers in one half of the partition vote , and let the others vote .all scores are now ( which is also the average ) . by the tie - breaking rule, wins in the first round .thus the manipulators can make win if a perfect partition exists .conversely , suppose there is a successful manipulation .clearly , can not be eliminated in the first round . to ensure this, all manipulators must put in first place .next , we show that if is not a joint winner of the first round , can not win overall .we consider all possible sets of candidates that could be eliminated in the first round .there are 6 cases . in the first case ,only is eliminated in the first round .the scores from non - manipulators in the second round are as follows : , and .the average score is .even with the maximum possible score from the manipulators , is eliminated .this contradicts the assumption that wins . in the second case ,only is eliminated in the first round .as and are not eliminated in the first round , the manipulators have to cast votes that put in first place and in second place . with such manipulating votes , the scores in the second round are : , and .the average score is .hence , is eliminated . in the next round, is eliminated as , and the average score is .this contradicts the assumption that wins . in the third case ,only is eliminated in the first round .this case is symmetric to the second case . in the fourth case , and eliminated in the first round .the case when and are eliminated is symmetric . in the second round ,the scores from non - manipulators are and .the score from the manipulators can not prevent being eliminated .this contradicts the assumption that wins . in the fifth case , and eliminated in the first round . however ,in the first round , the score and receive from the non - manipulators is .one of them will get at least points from manipulators . this will give them greater than the average score of .hence , at least one of them is not eliminated . in the sixth and final case , , and all eliminated in the first round .this case is again impossible by the same argument as the last case .the only way for to win is to have a tie with all candidates in the first round . as we observed above, the manipulators have to put in first place , and in second place . in turn, both and have to get exactly points from the manipulators .hence , there exists a solution to the partition instance .  clearly , it is polynomial to compute a manipulation of baldwin s rule with 2 candidates ( since this case degenerates to majority voting ) . with nanson s rule , on the other hand , it is polynomial with up to 3 candidates .  with nanson s rule and weighted votes , the coalition manipulation problem is polynomial for up to 3 candidates .* proof : * consider an election with 3 candidates ( and ) in which the manipulators want to win .we prove that the optimal strategy is for the manipulators either all to vote or all to vote .if does not win using one of these two votes , then can not win .therefore we simply try out the two votes and compute if wins in either case .suppose the manipulators can make win .we first note that there is no loss for them to raise to the first position , while keeping the other parts of their preferences the same . by doing so, the score of goes up and the scores of and go down .the only possible change in the elimination process is that now both and drop out in the first round , so that still wins .  now , suppose that all manipulators rank in their top positions .let denote the manipulators profile that makes win .because nanson s rule never selects the condorcet loser , can not be beaten by both and in pairwise elections .without loss of generality , suppose beats .we argue that if all manipulators vote , then still wins . for the sake of contradiction , suppose all manipulators vote but does not win . as the manipulators still rank in their top positions , the score of in the first round is the same as in .therefore , must enter ( and lose ) the second round .hence , only is eliminated in the first round , and in the second round beats . however , having the manipulators vote only lowers s score in the first round , compared to the case where they vote .hence , when the manipulators vote , also enters the second round and then beats , which is a contradiction .therefore , if the manipulators can make win , then they can make win by all voting , or all voting .one way to deal with computational intractability is to treat computing a manipulation as an approximation problem where we try to minimize the number of manipulators .we therefore considered five approximation methods .these are either derived from methods used with borda or are specifically designed for the elimination style of nanson s and baldwin s rules .: : :  the desired candidate is put first , and the other candidates are  reverse ordered by their current borda score .we repeat this  construction until the desired candidate wins .was used to manipulate  the borda rule in .: : :  this method was proposed for the borda rule . unlike which  constructs votes one by one , we construct votes in any order using a  bin packing heuristic which puts the next largest borda score into the  `` best '' available vote .we start with a target number of  manipulators .simple counting arguments will lower bound this number ,  and we can increae it until we have a successful manipulation .we  construct votes for the manipulators in which the desired candidate is  in first place .we take the other borda scores of the manipulators in  decreasing order , and assign them to the candidate with the lowest  current borda score who has been assigned less than the required  number of scores .a perfect matching algorithm then converts the sets  of borda scores for the candidates into a set of manipulating votes .: : :  this method was also proposed for the borda rule .we again  have a target number of manipulators , and construct votes for the  manipulators in which the desired candidate is in first place .we take  the other borda scores of the manipulators in decreasing order , and  assign them to the candidate with the current lowest _ average _ borda  score who has less than the required number of scores .the intuition  is that if every score was of average size , we would have a perfect  fit .if more than one candidate has the same lowest average borda  score and can accommodate the next score , we tie - break on the  candidate with the fewest scores .examples of and can be found in   .: : :  we repeatedly construct votes in which the desired candidate is put in  first place , and the other candidates in the reverse of the current  elimination order . for instance, the first candidate eliminated is put  in last place . for nansons rule , we order candidates eliminated in  the same round by their borda score in that round . : : :  we repeatedly construct votes in which the desired candidate is put in  first place , and the other candidates in the current elimination  order . for instance, the first candidate eliminated is put in second  place . for nansons rule , we order candidates eliminated in the same  round by the inverse of their borda score in that round .the intuition behind is to move the desired candidate up the elimination order whilst keeping the rest of the order unchanged . with, the intuition is to move the desired candidate up the elimination order , and to assign the largest borda scores to the least dangerous candidates .it is easy to show that all methods will eventually compute a manipulation of nanson s or baldwin s rule in which the desired candidate wins .  with borda voting ,good bounds are known on the quality of approximation that is achievable .in particular , proved that never requires more than one extra manipulator than optimal .baldwin s and nanson s rules appear more difficult to approximate within such bounds .we can give examples where all five methods compute a manipulation that use several more manipulators than is optimal .indeed , even with a fixed number of candidates , can require an unbounded number of extra manipulators .[ t : path_exm_baldwin ] with baldwin s rule , there exists an election with 7 candidates and votes where computes a manipulation with at least more votes than is optimal .* proof : * ( sketch ) consider an election over , , , , , and where is the candidate that the manipulators wish to win .we define as the pair of votes : , where is some fixed ordering of the other candidates and is its reverse .the non - manipulators cast the following votes : copies of , , , and .in addition , there are copies of the votes : and . if manipulators vote identically then wins .this provides an upper bound on the size of the optimal manipulation .after the non - manipulators have voted , , and .will put in first place .we suppose is a multiple of 2 , but more complex arguments can be given in other cases .after manipulating votes have been constructed , the scores of candidates to are level at and is leveled at . from then on ,the manipulators put in first place and alternate the order of the other candidates .at least votes are therefore required for to move out of last place .asymptotically this result is as bad as we could expect .any election can be manipulated with votes by simply reversing all previous votes , and this proof demonstrates that may use more votes than is optimal ..[table : urn_opt ] percentage of urn elections with 5 candidates where the heuristic finds the optimal manipulation . [ cols="^,^,^,^,^,^",options="header " , ]   our second set of experiments used larger problems .this amplifies the differences between the different approximation methods ( but means we are unable to compute the optimal manipulation using brute force search ) .problems have between and candidates , and the same number of votes as candidates .we tested 6000 instances , 1000 at each problem size .tables [ table : uni_av_baldwin][table : urn_av_nanson ] show the results for the average number of manipulators .the results show that overall works slightly better than and , which themselves outperform the other two methods especially for problems with large number of candidates . we observe a similar picture with nanson s rule .this contrasts with the borda rule where and do much better than . in most casesis less effective than except urn elections with nanson s rule .these experimental results suggest that baldwin s and nanson s rules are harder to manipulate in practice than borda .approximation methods that work well on the borda rule are significantly less effective on these rules .overall , , and appear to offer the best performance , though no heuristic dominates . prove that a class of voting rules which use repeated ballots and eliminate one candidate in each round are condorcet consistent .they illustrate this class with the _ weakest link _ rule in which the candidate with the fewest ballots in each round is eliminated . has proposed a variant of single transferable vote where first place votes , candidates are successively eliminated based on their _ original _ borda score . unlike nanson s and baldwin s rules , this method does not recalculate the borda score based on the new reduced set of candidates .for any condorcet consistent rule ( and thus for nanson s and baldwin s rule ) , showed that many types of control and manipulation are polynomial to compute when votes are single peaked .with unweighted votes , we have proven that nanson s and baldwin s rules are np - hard to manipulate with one manipulator .this increases by two thirds the number of rules known to be np - hard to manipulate with just a single manipulator . with weighted votes , on the other hand ,we have proven that nanson s rule is np - hard to manipulate with just a small number of candidates and a coalition of manipulators .we have also proposed a number of approximation methods for manipulating nanson s and baldwin s rules .our experiments suggest that both rules are difficult to manipulate in practice .there are many other interesting open questions coming from these results .for example , are there other elimination style voting rules which are computationally difficult to manipulate ? as a second example , with nanson s and baldwin s rule what is the computational complexity of other types of control like the addition / deletion of candidates , and the addition / deletion of voters ? as a third example , we could add elimination rounds to other scoring rules .do such rules have interesting computational properties ?nina narodytska is supported by the asian office of aerospace research and development through grant aoard-104123 .toby walsh is funded by the australian department of broadband , communications and the digital economy and the arc .lirong xia acknowledges a james b. duke fellowship and vincent conitzer s nsf career 0953756 and iis-0812113 , and an alfred p. sloan fellowship for support .we thank all aaai-11 reviewers for their helpful comments and suggestions .betzler , n. ; niedermeier , r. ; and woeginger , g. 2011 .unweighted coalitional manipulation under the borda rule is np - hard . in _ijcai-11_. brandt , f. ; brill , m. ; hemaspaandra , e. ; and hemaspaandra , l. 2010. bypassing combinatorial protections : polynomial - time algorithms for single - peaked electorates . in _ aaai-10 _, 715722 .chamberlin , j. 1985 .an investigation into the relative manipulability of four voting systems .30:195203 .davies , j. ; katsirelos , g. ; narodytska , n. ; and walsh , t. 2011 .complexity of and algorithms for borda manipulation . in _faliszewski , p. ; hemaspaandra , e. ; and schnoor , h. 2008 .copeland voting : ties matter . in _ aamas-08 _ , 983990 .

Abstract: nanson s and baldwin s voting rules select a winner by successively eliminating candidates with low borda scores . we show that these rules have a number of desirable computational properties . in particular , with unweighted votes , it is np - hard to manipulate either rule with one manipulator , whilst with weighted votes , it is np - hard to manipulate either rule with a small number of candidates and a coalition of manipulators . as only a couple of other voting rules are known to be np - hard to manipulate with a single manipulator , nanson s and baldwin s rules appear to be particularly resistant to manipulation from a theoretical perspective . we also propose a number of approximation methods for manipulating these two rules . experiments demonstrate that both rules are often difficult to manipulate in practice . these results suggest that elimination style voting rules deserve further study .


dynamical systems are mathematical models describing the evolution of systems in terms of equation of motion with sensitive initial values. the application areas of dynamical systems can be found e.g. in many disciplines in physics , population dynamics in biology , and chemical kinetics in chemistry .dynamic systems theory has also found its way into subjects outside the fields of mathematics and natural sciences , exemplarily we would like to mention mathematical economy and finance .poincar made a kick start to the subject of dynamical systems in his pioneering work in 1890 .later , in the 1920 s , and introduced the dynamics of complex analytic maps .on the one hand , studies like , , cartwright and littlewood find their origin in physics , like e.g. the three body problem in astronomy , whereas on the other hand stephan smale provided a purely mathematics motivated approach .e. n. lorenz observed that very simple differential equations become chaotic under certain circumstances .the system proposed by lorenz shows a very complex dynamical behaviour and displays the well - known two - scroll butterfly - shape .the dynamic equations of the lorenz system are given as where the parameters , , and are assumed to be positive .lorenz used exemplarily the values , and to demonstrate the systems chaotic behaviour .the study on `` the equation for continuous chaos '' by rssler can be seen as another landmark in the discussion of 3d dynamic systems .the rssler system and the burke shaw system have the property of two unstable saddle foci in common .other chaotic systems exhibiting a similar simple structure as the lorenz system , without being topologically equivalent , are proposed by chen and l and chen . discusses the transition between lorenz and chen attractors .furthermore , yang and chen introduced another chaotic system with three fixed points : one saddle and two stable fixed points .yang et al . and pehlivan et al . introduced and analysed chaotic systems similar to the lorenz , chen , and yang - chen systems , with two different fixed points , i.e. two stable node - foci .chaotic systems have found their ways also into many applications in engineering , such as electronic circuits - .modern pacemakers actually rely nowadays on these chaotic circuits .dynamical systems have also been discussed in the framework of various non - newtonian calcului , like fractional calculus , geometric multiplicative calculus , and bigeometric multiplicative calculus .jun guo lu transformed the l system into fractional calculus , investigating chaotic behaviour of fractional - order of the l system numerically .aniszewska applied multiplicative calculus to the rssler system and showed chaotic behaviour of multiplicative rssler .as an application of the bigeometric multiplicative runge - kutta method , the bigeometric multiplicative rssler system was solved in .this study introduces a new chaotic attractor , found by modification of the lorenz system by a quadratic term .detailed numerical and theoretical analysis reveals that the proposed system shows chaotic behaviour and the property of a two - scroll attractor like the lorenz attractor .section [ sec:1 ] introduces the modified quadratic lorenz attractor , discusses and analyses its properties by determining the equilibria points , and lyapunov exponents theoretically as well as numerically . in section 3, the modified quadratic lorenz attractor is translated into geometric and bigeometric calculus , and the solutions of the the system are obtained using the corresponding multiplicative runge - kutta methods and . finally , this paper closes with the summary of the obtained results .this paper presents a new chaotic system derived from the lorenz system .the system is generated by the following simple three - dimensional system :    where , , and are variables and , , and are real parameters . in the new proposed chaotic system ,all the equations have some differences compared to the original lorenz system . in order to see the differences between the two systems we can compare the equations - and - one by one .evidently , in the lorenz system equation is linear , whereas equation is non - linear .furthermore , equations and are both non - linear , where the -dependence of equation is cancelled .the most significant difference can be observed comparing equations and . in the term squared compared to equation .the initial values and the parameters of the system are chosen as and , with varying .then it has been observed that the behavior of the new chaotic system changes depending on the different values of as shown in figure [ fig : three graphs ] .  0.4 , and various parameter values , title="fig : " ]  0.4 , and various parameter values , title="fig : " ]  0.4 , and various parameter values , title="fig : " ]  0.4 , and various parameter values , title="fig : " ]  0.4 , and various parameter values , title="fig : " ]  as a result of the calculations , as it can be seen in figure [ fig : three graphs ], it has been observed that the new system is chaotic for the parameters the analysis of the new chaotic system will be done according to those parameters .the first step to analyze a chaotic system is to find the equilibrium points . in order to determine the equilibrium points of the proposed system -, we need to solve the system    thus , the solution of the system with respect to , , give the equilibria points as :  {\beta\,\rho^3},\sqrt[4]{\frac{\beta}{\rho}},\rho\right ) \\\it e^-&=&\left(-\sqrt[4]{\beta\,\rho^3},-\sqrt[4]{\frac{\beta}{\rho}},\rho\right)\end{aligned}\ ] ]  for the parameters chosen in , the numerical values of the equilibria points are    in order to decide on the stability of the new proposed system , the eigenvalues of the jacobian matrix must be analyzed .the jacobian matrix for this system - can be easily obtained as   the expressions for the eigenvalues of the jacobian matrix are very long and complicated .as we are only interested in the numerical values of the eigenvalues at the equilibria points - for the given parameters , the eigenvalues are stated in the table below :  .eigenvalues of the jacobian at the equilibrium points [ cols="<,^,^,^ " , ]   as the eigenvalues and for the equilibrium point are both negative , the system is unstable at this equilibrium point .the eigenvalues corresponding to the equilibrium point will be the same with the eigenvalues of , because of the quadratic nature of the system . since is a negative real number and and are two complex conjugate eigenvalues with positive real parts , equilibrium points and are unstable according to .the system - has a natural symmetry and is invariant under the coordinate transformation which persists for all values of the system parameters .so , system - has rotation symmetry about the .let , , and in the system - .then we get for the vector field    consequently the divergence of the vector field yields to :    note that is a negative value , so the system is a dissipative system and an exponential rate is :    from , it can be seen that a volume element is contracted by the flow into a volume element at the time .the lyapunov exponents generally refer to the average exponential rates of divergence or convergence of nearby trajectories in the phase space .the important part is that if there is at least one positive lyapunov exponent , the system can be defined to be chaotic .according to the detailed numerical and theoretical analysis , the lyapunov exponents are found to be , , and .therefore , the lyapunov dimension of this system is :    this result is consistent with the findings in , i.e. that the lyapunov dimension is in the range 2 - 3 .equation means that the system - is a dissipative system , and the lyapunov dimensions of the system are fractional . having a strange attractor and positive lyapunov exponent , it is obvious that the system is a 3d chaotic system .[ h ]     time series analysis of the system - according to axes are listed in the figure [ fig : wf ] . for the solutions of the systemthe runge - kutta method was employed .  0.4 , , respectively , title="fig : " ]  0.4 , , respectively , title="fig : " ]  0.4 , , respectively , title="fig : " ]obviously the time series show that the functions , , and are not periodic , indicating that the system is chaotic .the projections of the system - , on the various axes is given in the figure [ fig : pp ] .  0.4   0.4   0.4the chaotic attractors are displayed in the figure [ fig : pp ] .it appears that the new attractor exhibits an interesting complex chaotic dynamics behavior .as an application of the dynamical systems , the newly formed chaotic system can also be written in the sense of geometric and the bigeometric calculus .   the geometric chaotic system corresponding to system - , can be written as    on the other hand , the bigeometric chaotic system corresponding to system - will be    in order to solve the system , one can use the geometric runge - kutta method defined in , while the system can be solved by using the bigeometric runge - kutta method which was defined in . by choosing the same values for the parameters , such as , with varying and the initial values as , we can see that the solutions of the geometric chaotic system and the bigeometric chaotic system will be similar to the ones that we get from the solutions of the chaotic system - .the following figure shows the chaotic behaviour of the geometric and the bigeometric dynamic systems when the parameters are chosen as , and ., width=302 ]the proposed modified quadratic lorenz attractor was analysed theoretically and numerically showing that this system is chaotic .following the standardized analysis method for chaotic systems , we determined first the equilibria points and the eigenvalues of the jacobian matrix at these equilibria points to get a first indication about the stability of the proposed system .as the eigenvalues are either non - positive real numbers , or complex numbers with positive real parts , we can conclude that this system is not stable at the equilibria points .moreover , we could identify that the proposed system has a rotational symmetry about the -axis , and shows dissipative behaviour contracting the volume element to .the lyapunov exponents yield to , and , showing the chaotic nature of the system .the fractional dimension of the system has also been given .overall , the analysis shows that this is a new chaotic system with two scrolls .32 [ 1]#1 [ 1]`#1 ` urlstyle [ 1]doi : # 1     ml cartwright and je littlewood . on non - linear differential equations of the second order : ii .the equation y+kf(y,y+g(y , k)=p(t)=p_1(t)+kp_2(t);k>0,f(y)>1 . _annals of mathematics _ , pages 472494 , 1947 .a. a. n. kolmogorov . .in g. casati and j. ford , editors , _ stochastic behavior in classical and quantum hamiltonian systems _ , volume 93 of _ lecture notes in physics , berlin springer verlag _ , pages 5156 , 1979 .doi : 10.1007/bfb0021737 .ihsan pehlivan and yilmaz uyarolu . a new chaotic attractor from general lorenz system family and its electronic experimental implementation ._ turkish journal of electrical engineering & computer sciences _ , 180 ( 2):0 171184 , 2010 .simin yu , jinhu l , wallace ks tang , and guanrong chen . a general multiscroll lorenz system family and its realization via digital signal processors ._ chaos : an interdisciplinary journal of nonlinear science _ , 160 ( 3):0 033126 , 2006 .

Abstract: this study introduces a modified quadratic lorenz attractor . the properties of this new chaotic system are analysed and discussed in detail , by determining the equilibria points , the eigenvalues of the jacobian , and the lyapunov exponents . the numerical simulations , the time series analysis , and the projections to the -plane , -plane , and -plane are conducted to highlight the chaotic behaviour . the multiplicative form of the new system is also presented and the simulations are conducted using multiplicative runge - kutta methods .


space - time modulation techniques to achieve transmit diversity have been proposed in the literature . in most cases , it is assumed that the channel state information ( csi ) is perfectly known at the receiver , . this is a reasonable assumption whenthe channel changes slowly and can be estimated by transmitting known training symbols . however, this is not always possible , and there is a tradeoff between frame length and accuracy of the channel estimation .therefore , the effects of channel estimation error make it desirable to use schemes that avoid such an estimation .prior work has proposed many differential space - time coding schemes in which neither the transmitter nor the receiver knows the csi .the first differential coding schemes based on orthogonal designs for multiple transmit antennas were proposed in and with about 3-db loss in performance compared to the corresponding coherent detection .other examples of differential modulation schemes using space - time block codes ( stbcs ) and linear decoding complexity were proposed in - . a rate - one differential modulation scheme based on the quasi - orthogonal space - time block codes( qostbcs ) can be found in .multi - user detection schemes with simple coherent detection structures for multiple access channels ( macs ) have garnered significant attention - .the main goal is to design a low complexity interference cancelation method for a mac with users using only receive antennas .this is done for transmit antennas in and for users in using the properties of orthogonal space - time block codes ( ostbcs ) . to solve the problem for any number of users , any constellation , and any number of transmit antennas, presents a method utilizing qostbcs with a moderate increase in decoding complexity .space - time / frequency code design criteria for fading mimo macs and a code construction for two users have been derived in .differential modulation schemes for two - user mac systems have been proposed in .these schemes have a high decoding complexity . in , we proposed low complexity differential modulation schemes for two - user mimo systems that achieve full transmit diversity .moreover , we presented additional differential decoding schemes that provide full diversity , outperform the existing differential schemes , and work for any square ostbc ( ostbc ) .all the existing multi - user differential schemes assume the transmission of the data by the users to be perfectly synchronized in time . to the best of our knowledge ,a differential modulation scheme for asynchronous multi - user systems does not exist in the literature .in this paper , we design differential detection schemes for asynchronous multi - user mimo systems where neither the transmitters nor the receiver knows the channel .our main results are as follows :  1 .  with a slow rayleigh fading channel model for an asynchronous multi - user system, we present a differential encoder and derive novel low complexity differential decoders by performing interference cancelation in time and employing different decoding methods .the decoding complexity of these schemes grows linearly with the number of users .we also present additional differential decoding schemes that perform significantly better than our low complexity decoders and outperform the existing synchronous differential schemes , but need higher decoding complexity compared to our low complexity decoders .all the proposed decoders work for any square ostbc , any constant amplitude constellation , any number of users , and any number of receive antennas .we analyze the diversity of our schemes and derive conditions under which the proposed schemes provide full diversity . for the cases of two and four transmit antennas , we provide examples of psk constellations to achieve full diversity .simulation results show that the proposed differential detection schemes provide good performance .the rest of the paper is organized as follows . in section [ systemmodel ] , we introduce the system model . in section [ encoding ] , we present the differential encoding for our asynchronous differential modulation schemes .the differential decoding schemes are put forward in section [ decoding ] .we analyze the diversity of our schemes in section [ diversitygainanalysis ] .simulation results are provided in section [ simulations ] , and section [ conclusions ] concludes the paper .  * notation : * we use boldface capital letters to denote matrices , boldface small letters to denote vectors , and super - scripts and to denote conjugate and conjugate transpose , respectively . indicates the frobenius norm , and  ] where snr is the ratio of the average transmit power to the noise power .let then , the received samples can be written in a matrix form as where ,  , are , and matrices , respectively , and is a matrix given by for the sake of simplicity , in this paper we consider the case where and the pulse - shaping filter is a rectangular pulse then , it can be easily seen from ( [ eq0006 ] ) that therefore , in this case , using ( [ eq0007 ] ) , ( [ eq0009 ] ) and ( [ eq0011 ] ) , becomes         \smash{\overbrace{0 \cdots 0}^{j-1 \textrm{times } } } & \smash{\overbrace{1 \cdots 1}^{j\ \textrm{times } } } & 0 \cdots 0 & \cdots & 0 \cdots 0 & 0 \cdots 0 \\         0 \cdots 0 & 0 \cdots 0 & \smash{\overbrace{1 \cdots 1}^{j\ \textrm{times } } } & \cdots & 0 \cdots 0 & 0 \cdots 0 \\         \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\         0 \cdots 0 & 0 \cdots 0 & 0 \cdots 0 & \cdots & \smash{\overbrace{1 \cdots 1}^{j\ \textrm{times } } } & \smash{\overbrace{0 \cdots 0}^{j - j+1 \textrm{times } } } \\       \end{pmatrix } , } \label{eq0012}\ ] ] and , , become independent zero - mean complex gaussian random vectors with covariance matrices = \frac{(\textrm{snr})^{-1 } t_s}{\tau_{i+1,i } } \cdot \bm{i}_m\smash{\underbrace{\phantom {   \begin{matrix}\ 1 \cdots 1\end{matrix}}}_{\text{}}}\smash{\underbrace{\phantom {   \begin{matrix}0 \cdots 0 & 0 \cdots 0 & \\ \cdots \ \ & 0 \cdots 0\end{matrix}}}_{\text{}}} ] is invertible .also , since its inverse must be a full rank matrix , multiplying its inverse by must result in a matrix with the same rank as .therefore , using lemma c.1 and the definition of in ( [ eq0314 ] ) , by multiplying by \bm{i}_n \ \ \bm{p}_1 \\ \bm{i}_n \ \\bm{p}_3 \end{pmatrix}^{-1} ] , by lemma c.1 , \bm{i}_n \ \ \bm{p}_1\bm{p}_2 \\ \bm{i}_n \ \\bm{p}_1\bm{p}_4 \end{pmatrix} ] from the left we obtain \bm{i}_n \ \\bm{p}_1\bm{p}_2 \\ \bm{i}_n \ \ \bm{p}_1\bm{p}_4 \end{pmatrix}^{-1 } \cdot \bm{g}(\bm{p}_1 , \bm{p}_2 , \bm{p}_3 , \bm{p}_4 )\\ = \begin{pmatrix}[0.75 ]         \bm{i}_n + \bm{p}_1\bm{p}_2\left(\frac{n[\bm{p}_1(\bm{p}_4 - \bm{p}_2)]^\dagger}{\| \bm{p}_1(\bm{p}_4 - \bm{p}_2 ) \|_f^2}\right ) \ \ - \bm{p}_1\bm{p}_2\left(\frac{n[\bm{p}_1(\bm{p}_4 - \bm{p}_2)]^\dagger}{\| \bm{p}_1(\bm{p}_4 - \bm{p}_2 ) \|_f^2}\right ) \\-\frac{n[\bm{p}_1(\bm{p}_4 - \bm{p}_2)]^\dagger}{\| \bm{p}_1(\bm{p}_4 - \bm{p}_2 ) \|_f^2 } \qquad \qquad \frac{n[\bm{p}_1(\bm{p}_4 - \bm{p}_2)]^\dagger}{\| \bm{p}_1(\bm{p}_4 - \bm{p}_2 ) \|_f^2 }       \end{pmatrix } \\\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \\begin{pmatrix}[0.75 ]         \bm{i}_n & \bm{p}_1 & \bm{p}_1 \bm{p}_2 \\\bm{i}_n & \bm{p}_1 & \bm{p}_1 \bm{p}_4       \end{pmatrix } \cdot \bm{\bar{a } } \\= \begin{pmatrix}[0.75 ]         \bm{i}_n & \bm{p}_1 & \bm{0}_n \\\bm{0}_n & \bm{0}_n & \bm{i}_n       \end{pmatrix } \cdot \bm{\bar{a } } , \end{array } } \label{eq0409}\ ] ] which must be of the same rank as . once again , since is a full rank matrix , multiplying ( [ eq0409 ] ) by from the right yields a matrix with the same rank as ( [ eq0409 ] ) , and thus , given by \bm{i}_n \ \ \bm{p}_1\bm{p}_2 \\ \bm{i}_n \ \\bm{p}_1\bm{p}_4 \end{pmatrix}^{-1 } \cdot \bm{g}(\bm{p}_1 , \bm{p}_2 , \bm{p}_3 , \bm{p}_4 ) \cdot \bm{b}_1^{-1 } = \begin{pmatrix}[0.75 ]         \bm{i}_n & \bm{p}_1 & \bm{0}_n \\\bm{0}_n & \bm{0}_n & \bm{i}_n       \end{pmatrix } \cdot \bm{b}_2 . }\label{eq0410}\ ] ] then proceeding similarly to the procedure described in ( [ eq0405])-([eq0408 ] ) for case 1 , we find that has full row rank if and only if .note that this condition is a special case of ( [ eq0340 ] ) when .therefore , ( [ eq0340 ] ) is a sufficient condition for to have full row rank in case 2 .also , we showed that ( [ eq0340 ] ) is a necessary and sufficient condition for to have full row rank in case 1 .thus , ( [ eq0340 ] ) is a necessary and sufficient condition in the general case for to have full row rank for any possible data matrices with .a. f. naguib , n. seshadri , and a. r. calderbank , `` applications of space - time block codes and interference suppression for high capacity and high data rate wireless systems , '' in _ proc .32nd asilomar conf .signals , syst ._ , nov . 1998 , vol .1803 - 1810 .a. stamoulis , n. al - dhahir , and a. r. calderbank , `` further results on interference cancellation and space - time block codes , '' in _ proc .35th asilomar conf .signals , syst .2001 , vol .257 - 261 .m. brehler and m. k. varanasi , `` asymptotic error probability analysis of quadratic receivers in rayleigh - fading channels with applications to a unified analysis of coherent and noncoherent space - time receivers , '' _ ieee trans .inf . theory _2383 - 2399 , sep . 2001 .sina poorkasmaei received the b.s .degree in electrical and computer engineering from the american university in dubai , uae , in 2007 , and the m.s .degree in electrical engineering and computer science from the university of california , irvine , ca , usa , in 2008 , where he is currently working toward the ph.d . degree .his research interests include wireless communications and signal processing .hamid jafarkhani ( f06 ) is a chancellor s professor with the department of electrical engineering and computer science , university of california , where he is also the director of the center for pervasive communications and computing and the conexant - broadcom endowed chair .he is the author of the book _ space - time coding : theory and practice_.  dr .jafarkhani is a fellow of aaas . in 1984, he ranked first in the nationwide entrance examination of iranian universities .he was the corecipient of the american division award of the 1995 texas instruments dsp solutions challenge .he was the recipient of an nsf career award in 2003 .he was the recipient of the uci distinguished mid - career faculty award for research in 2006 and the school of engineering fariborz maseeh best faculty research award in 2007 .in addition , he was the corecipient of the 2002 best paper award of iswc , the 2006 ieee marconi best paper award in wireless communications , the 2009 best paper award of the journal of communications and networks , the 2012 ieee globecom best paper award ( communication theory symposium ) , the 2013 ieee eric e. sumner award , and the 2014 ieee communications society award for advances in communication .he is listed as a highly cited researcher in http://www.isihighlycited.com . during 1997 - 2007 , according to the thomson scientific , he was one of the top 10 most cited researchers in the field of `` computer science . 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Abstract: we propose several differential decoding schemes for asynchronous multi - user mimo systems based on orthogonal space - time block codes ( ostbcs ) where neither the transmitters nor the receiver has knowledge of the channel . first , we derive novel low complexity differential decoders by performing interference cancelation in time and employing different decoding methods . the decoding complexity of these schemes grows linearly with the number of users . we then present additional differential decoding schemes that perform significantly better than our low complexity decoders and outperform the existing synchronous differential schemes but require higher decoding complexity compared to our low complexity decoders . the proposed schemes work for any square ostbc , any constant amplitude constellation , any number of users , and any number of receive antennas . furthermore , we analyze the diversity of the proposed schemes and derive conditions under which our schemes provide full diversity . for the cases of two and four transmit antennas , we provide examples of psk constellations to achieve full diversity . simulation results show that our differential schemes provide good performance . to the best of our knowledge , the proposed differential detection schemes are the first differential schemes for asynchronous multi - user systems .   poorkasmaei and jafarkhani : asynchronous orthogonal differential decoding for multiple access channels  differential detection , multi - user detection , interference suppression , synchronization , space - time block coding .